LMFDB - Embedded newform 115.4.a.e.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [115,4,Mod(1,115)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(115, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("115.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 115 = 5 \cdot 23 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	115.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$6.78521965066$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	$\mathbb{Q}[x]/(x^{5} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 27x^{3} + 7x^{2} + 168x + 92 $ x^5 - x^4 - 27x^3 + 7x^2 + 168*x + 92
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-0.595043$ of defining polynomial
Character	$\chi$	$=$	115.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.404957 q^{2} -7.11323 q^{3} -7.83601 q^{4} -5.00000 q^{5} -2.88055 q^{6} +13.7888 q^{7} -6.41290 q^{8} +23.5981 q^{9} +O(q^{10})$	\(q+0.404957 q^{2} -7.11323 q^{3} -7.83601 q^{4} -5.00000 q^{5} -2.88055 q^{6} +13.7888 q^{7} -6.41290 q^{8} +23.5981 q^{9} -2.02479 q^{10} +24.2317 q^{11} +55.7394 q^{12} +3.05016 q^{13} +5.58389 q^{14} +35.5662 q^{15} +60.0911 q^{16} +63.1126 q^{17} +9.55621 q^{18} -2.07770 q^{19} +39.1800 q^{20} -98.0832 q^{21} +9.81282 q^{22} -23.0000 q^{23} +45.6165 q^{24} +25.0000 q^{25} +1.23518 q^{26} +24.1987 q^{27} -108.049 q^{28} -8.16397 q^{29} +14.4028 q^{30} -156.989 q^{31} +75.6376 q^{32} -172.366 q^{33} +25.5579 q^{34} -68.9442 q^{35} -184.915 q^{36} +302.801 q^{37} -0.841380 q^{38} -21.6965 q^{39} +32.0645 q^{40} -42.7514 q^{41} -39.7195 q^{42} +215.265 q^{43} -189.880 q^{44} -117.990 q^{45} -9.31401 q^{46} +247.096 q^{47} -427.442 q^{48} -152.868 q^{49} +10.1239 q^{50} -448.935 q^{51} -23.9010 q^{52} +600.400 q^{53} +9.79943 q^{54} -121.159 q^{55} -88.4265 q^{56} +14.7792 q^{57} -3.30606 q^{58} +92.2014 q^{59} -278.697 q^{60} +532.635 q^{61} -63.5740 q^{62} +325.390 q^{63} -450.099 q^{64} -15.2508 q^{65} -69.8009 q^{66} +30.3010 q^{67} -494.551 q^{68} +163.604 q^{69} -27.9194 q^{70} -736.349 q^{71} -151.332 q^{72} +349.936 q^{73} +122.622 q^{74} -177.831 q^{75} +16.2809 q^{76} +334.127 q^{77} -8.78614 q^{78} +301.545 q^{79} -300.456 q^{80} -809.279 q^{81} -17.3125 q^{82} +139.488 q^{83} +768.581 q^{84} -315.563 q^{85} +87.1732 q^{86} +58.0722 q^{87} -155.396 q^{88} +859.551 q^{89} -47.7810 q^{90} +42.0581 q^{91} +180.228 q^{92} +1116.70 q^{93} +100.063 q^{94} +10.3885 q^{95} -538.028 q^{96} -927.475 q^{97} -61.9051 q^{98} +571.823 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 6 q^{2} + 4 q^{3} + 22 q^{4} - 25 q^{5} + 19 q^{6} - 3 q^{7} + 138 q^{8} + 77 q^{9}+O(q^{10}) $ 5 * q + 6 * q^2 + 4 * q^3 + 22 * q^4 - 25 * q^5 + 19 * q^6 - 3 * q^7 + 138 * q^8 + 77 * q^9	\( 5 q + 6 q^{2} + 4 q^{3} + 22 q^{4} - 25 q^{5} + 19 q^{6} - 3 q^{7} + 138 q^{8} + 77 q^{9} - 30 q^{10} + 23 q^{11} + 47 q^{12} + 132 q^{13} + 93 q^{14} - 20 q^{15} + 282 q^{16} + 23 q^{17} - 15 q^{18} - 161 q^{19} - 110 q^{20} - 60 q^{21} + 193 q^{22} - 115 q^{23} + 105 q^{24} + 125 q^{25} - 257 q^{26} + 577 q^{27} + 17 q^{28} + 401 q^{29} - 95 q^{30} + 32 q^{31} + 670 q^{32} + 189 q^{33} - 663 q^{34} + 15 q^{35} - 659 q^{36} - 38 q^{37} - 875 q^{38} + 335 q^{39} - 690 q^{40} - 12 q^{41} - 798 q^{42} - 566 q^{43} + 47 q^{44} - 385 q^{45} - 138 q^{46} + 919 q^{47} - 773 q^{48} - 738 q^{49} + 150 q^{50} - 993 q^{51} - 305 q^{52} + 1156 q^{53} - 8 q^{54} - 115 q^{55} + 343 q^{56} + 114 q^{57} - 1042 q^{58} + 1324 q^{59} - 235 q^{60} - 1673 q^{61} + 565 q^{62} + 270 q^{63} + 2466 q^{64} - 660 q^{65} - 2781 q^{66} + 558 q^{67} - 2267 q^{68} - 92 q^{69} - 465 q^{70} - 108 q^{71} - 789 q^{72} + 1173 q^{73} + 1458 q^{74} + 100 q^{75} - 3477 q^{76} + 2608 q^{77} + 704 q^{78} + 656 q^{79} - 1410 q^{80} - 319 q^{81} + 3505 q^{82} - 82 q^{83} - 718 q^{84} - 115 q^{85} + 112 q^{86} + 2389 q^{87} + 2397 q^{88} + 570 q^{89} + 75 q^{90} - 1589 q^{91} - 506 q^{92} + 911 q^{93} - 948 q^{94} + 805 q^{95} - 5991 q^{96} + 633 q^{97} - 2555 q^{98} + 2021 q^{99}+O(q^{100}) \) 5 * q + 6 * q^2 + 4 * q^3 + 22 * q^4 - 25 * q^5 + 19 * q^6 - 3 * q^7 + 138 * q^8 + 77 * q^9 - 30 * q^10 + 23 * q^11 + 47 * q^12 + 132 * q^13 + 93 * q^14 - 20 * q^15 + 282 * q^16 + 23 * q^17 - 15 * q^18 - 161 * q^19 - 110 * q^20 - 60 * q^21 + 193 * q^22 - 115 * q^23 + 105 * q^24 + 125 * q^25 - 257 * q^26 + 577 * q^27 + 17 * q^28 + 401 * q^29 - 95 * q^30 + 32 * q^31 + 670 * q^32 + 189 * q^33 - 663 * q^34 + 15 * q^35 - 659 * q^36 - 38 * q^37 - 875 * q^38 + 335 * q^39 - 690 * q^40 - 12 * q^41 - 798 * q^42 - 566 * q^43 + 47 * q^44 - 385 * q^45 - 138 * q^46 + 919 * q^47 - 773 * q^48 - 738 * q^49 + 150 * q^50 - 993 * q^51 - 305 * q^52 + 1156 * q^53 - 8 * q^54 - 115 * q^55 + 343 * q^56 + 114 * q^57 - 1042 * q^58 + 1324 * q^59 - 235 * q^60 - 1673 * q^61 + 565 * q^62 + 270 * q^63 + 2466 * q^64 - 660 * q^65 - 2781 * q^66 + 558 * q^67 - 2267 * q^68 - 92 * q^69 - 465 * q^70 - 108 * q^71 - 789 * q^72 + 1173 * q^73 + 1458 * q^74 + 100 * q^75 - 3477 * q^76 + 2608 * q^77 + 704 * q^78 + 656 * q^79 - 1410 * q^80 - 319 * q^81 + 3505 * q^82 - 82 * q^83 - 718 * q^84 - 115 * q^85 + 112 * q^86 + 2389 * q^87 + 2397 * q^88 + 570 * q^89 + 75 * q^90 - 1589 * q^91 - 506 * q^92 + 911 * q^93 - 948 * q^94 + 805 * q^95 - 5991 * q^96 + 633 * q^97 - 2555 * q^98 + 2021 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0.404957	0.143174	0.0715870	−	0.997434i	$-0.477194\pi$
$2$	0.404957	0.143174	0.0715870	+	0.997434i	$0.477194\pi$
$3$	−7.11323	−1.36894	−0.684471	−	0.729040i	$-0.739967\pi$
$3$	−7.11323	−1.36894	−0.684471	+	0.729040i	$0.739967\pi$
$4$	−7.83601	−0.979501
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	−2.88055	−0.195997
$7$	13.7888	0.744527	0.372263	−	0.928127i	$-0.378582\pi$
$7$	13.7888	0.744527	0.372263	+	0.928127i	$0.378582\pi$
$8$	−6.41290	−0.283413
$9$	23.5981	0.874003
$10$	−2.02479	−0.0640293
$11$	24.2317	0.664195	0.332098	−	0.943245i	$-0.392244\pi$
$11$	24.2317	0.664195	0.332098	+	0.943245i	$0.392244\pi$
$12$	55.7394	1.34088
$13$	3.05016	0.0650739	0.0325370	−	0.999471i	$-0.489641\pi$
$13$	3.05016	0.0650739	0.0325370	+	0.999471i	$0.489641\pi$
$14$	5.58389	0.106597
$15$	35.5662	0.612210
$16$	60.0911	0.938924
$17$	63.1126	0.900415	0.450208	−	0.892924i	$-0.351350\pi$
$17$	63.1126	0.900415	0.450208	+	0.892924i	$0.351350\pi$
$18$	9.55621	0.125134
$19$	−2.07770	−0.0250872	−0.0125436	−	0.999921i	$-0.503993\pi$
$19$	−2.07770	−0.0250872	−0.0125436	+	0.999921i	$0.503993\pi$
$20$	39.1800	0.438046
$21$	−98.0832	−1.01921
$22$	9.81282	0.0950955
$23$	−23.0000	−0.208514
$23$	−23.0000	−0.208514
$24$	45.6165	0.387976
$25$	25.0000	0.200000
$26$	1.23518	0.00931689
$27$	24.1987	0.172483
$28$	−108.049	−0.729265
$29$	−8.16397	−0.0522762	−0.0261381	−	0.999658i	$-0.508321\pi$
$29$	−8.16397	−0.0522762	−0.0261381	+	0.999658i	$0.508321\pi$
$30$	14.4028	0.0876525
$31$	−156.989	−0.909553	−0.454776	−	0.890606i	$-0.650281\pi$
$31$	−156.989	−0.909553	−0.454776	+	0.890606i	$0.650281\pi$
$32$	75.6376	0.417842
$33$	−172.366	−0.909245
$34$	25.5579	0.128916
$35$	−68.9442	−0.332963
$36$	−184.915	−0.856087
$37$	302.801	1.34541	0.672706	−	0.739910i	$-0.265132\pi$
$37$	302.801	1.34541	0.672706	+	0.739910i	$0.265132\pi$
$38$	−0.841380	−0.00359184
$39$	−21.6965	−0.0890824
$40$	32.0645	0.126746
$41$	−42.7514	−0.162845	−0.0814225	−	0.996680i	$-0.525946\pi$
$41$	−42.7514	−0.162845	−0.0814225	+	0.996680i	$0.525946\pi$
$42$	−39.7195	−0.145925
$43$	215.265	0.763434	0.381717	−	0.924279i	$-0.375333\pi$
$43$	215.265	0.763434	0.381717	+	0.924279i	$0.375333\pi$
$44$	−189.880	−0.650580
$45$	−117.990	−0.390866
$46$	−9.31401	−0.0298538
$47$	247.096	0.766866	0.383433	−	0.923569i	$-0.374742\pi$
$47$	247.096	0.766866	0.383433	+	0.923569i	$0.374742\pi$
$48$	−427.442	−1.28533
$49$	−152.868	−0.445680
$50$	10.1239	0.0286348
$51$	−448.935	−1.23262
$52$	−23.9010	−0.0637400
$53$	600.400	1.55606	0.778031	−	0.628225i	$-0.216218\pi$
$53$	600.400	1.55606	0.778031	+	0.628225i	$0.216218\pi$
$54$	9.79943	0.0246951
$55$	−121.159	−0.297037
$56$	−88.4265	−0.211009
$57$	14.7792	0.0343430
$58$	−3.30606	−0.00748460
$59$	92.2014	0.203451	0.101725	−	0.994813i	$-0.467564\pi$
$59$	92.2014	0.203451	0.101725	+	0.994813i	$0.467564\pi$
$60$	−278.697	−0.599660
$61$	532.635	1.11798	0.558991	−	0.829174i	$-0.311189\pi$
$61$	532.635	1.11798	0.558991	+	0.829174i	$0.311189\pi$
$62$	−63.5740	−0.130224
$63$	325.390	0.650719
$64$	−450.099	−0.879100
$65$	−15.2508	−0.0291019
$66$	−69.8009	−0.130180
$67$	30.3010	0.0552515	0.0276258	−	0.999618i	$-0.491205\pi$
$67$	30.3010	0.0552515	0.0276258	+	0.999618i	$0.491205\pi$
$68$	−494.551	−0.881958
$69$	163.604	0.285444
$70$	−27.9194	−0.0476716
$71$	−736.349	−1.23083	−0.615413	−	0.788205i	$-0.711011\pi$
$71$	−736.349	−1.23083	−0.615413	+	0.788205i	$0.711011\pi$
$72$	−151.332	−0.247704
$73$	349.936	0.561053	0.280527	−	0.959846i	$-0.409491\pi$
$73$	349.936	0.561053	0.280527	+	0.959846i	$0.409491\pi$
$74$	122.622	0.192628
$75$	−177.831	−0.273788
$76$	16.2809	0.0245730
$77$	334.127	0.494511
$78$	−8.78614	−0.0127543
$79$	301.545	0.429449	0.214725	−	0.976675i	$-0.431115\pi$
$79$	301.545	0.429449	0.214725	+	0.976675i	$0.431115\pi$
$80$	−300.456	−0.419900
$81$	−809.279	−1.11012
$82$	−17.3125	−0.0233152
$83$	139.488	0.184468	0.0922340	−	0.995737i	$-0.470599\pi$
$83$	139.488	0.184468	0.0922340	+	0.995737i	$0.470599\pi$
$84$	768.581	0.998322
$85$	−315.563	−0.402678
$86$	87.1732	0.109304
$87$	58.0722	0.0715631
$88$	−155.396	−0.188242
$89$	859.551	1.02373	0.511866	−	0.859065i	$-0.328954\pi$
$89$	859.551	1.02373	0.511866	+	0.859065i	$0.328954\pi$
$90$	−47.7810	−0.0559618
$91$	42.0581	0.0484493
$92$	180.228	0.204240
$93$	1116.70	1.24513
$94$	100.063	0.109795
$95$	10.3885	0.0112193
$96$	−538.028	−0.572002
$97$	−927.475	−0.970833	−0.485417	−	0.874283i	$-0.661332\pi$
$97$	−927.475	−0.970833	−0.485417	+	0.874283i	$0.661332\pi$
$98$	−61.9051	−0.0638097
$99$	571.823	0.580508
$100$	−195.900	−0.195900
$101$	1713.34	1.68796	0.843979	−	0.536376i	$-0.180207\pi$
$101$	1713.34	1.68796	0.843979	+	0.536376i	$0.180207\pi$
$102$	−181.799	−0.176479
$103$	−930.516	−0.890160	−0.445080	−	0.895491i	$-0.646825\pi$
$103$	−930.516	−0.890160	−0.445080	+	0.895491i	$0.646825\pi$
$104$	−19.5604	−0.0184428
$105$	490.416	0.455806
$106$	243.136	0.222788
$107$	1514.91	1.36871	0.684355	−	0.729149i	$-0.260084\pi$
$107$	1514.91	1.36871	0.684355	+	0.729149i	$0.260084\pi$
$108$	−189.621	−0.168947
$109$	1748.76	1.53670	0.768352	−	0.640027i	$-0.221077\pi$
$109$	1748.76	1.53670	0.768352	+	0.640027i	$0.221077\pi$
$110$	−49.0641	−0.0425280
$111$	−2153.90	−1.84179
$112$	828.586	0.699054
$113$	2026.23	1.68683	0.843416	−	0.537261i	$-0.180541\pi$
$113$	2026.23	1.68683	0.843416	+	0.537261i	$0.180541\pi$
$114$	5.98493	0.00491702
$115$	115.000	0.0932505
$116$	63.9729	0.0512046
$117$	71.9778	0.0568748
$118$	37.3376	0.0291289
$119$	870.249	0.670383
$120$	−228.082	−0.173508
$121$	−743.822	−0.558845
$122$	215.694	0.160066
$123$	304.100	0.222925
$124$	1230.17	0.890908
$125$	−125.000	−0.0894427
$126$	131.769	0.0931660
$127$	−2126.49	−1.48579	−0.742897	−	0.669406i	$-0.766549\pi$
$127$	−2126.49	−1.48579	−0.742897	+	0.669406i	$0.766549\pi$
$128$	−787.371	−0.543707
$129$	−1531.23	−1.04510
$130$	−6.17591	−0.00416664
$131$	−1494.23	−0.996576	−0.498288	−	0.867012i	$-0.666038\pi$
$131$	−1494.23	−0.996576	−0.498288	+	0.867012i	$0.666038\pi$
$132$	1350.66	0.890606
$133$	−28.6491	−0.0186781
$134$	12.2706	0.00791058
$135$	−120.993	−0.0771367
$136$	−404.735	−0.255189
$137$	2265.31	1.41269	0.706346	−	0.707867i	$-0.250342\pi$
$137$	2265.31	1.41269	0.706346	+	0.707867i	$0.250342\pi$
$138$	66.2527	0.0408682
$139$	−2918.66	−1.78099	−0.890496	−	0.454991i	$-0.849642\pi$
$139$	−2918.66	−1.78099	−0.890496	+	0.454991i	$0.849642\pi$
$140$	540.247	0.326137
$141$	−1757.65	−1.04980
$142$	−298.190	−0.176222
$143$	73.9106	0.0432218
$144$	1418.03	0.820622
$145$	40.8198	0.0233786
$146$	141.709	0.0803282
$147$	1087.39	0.610110
$148$	−2372.75	−1.31783
$149$	−549.400	−0.302071	−0.151036	−	0.988528i	$-0.548261\pi$
$149$	−549.400	−0.302071	−0.151036	+	0.988528i	$0.548261\pi$
$150$	−72.0139	−0.0391994
$151$	335.721	0.180931	0.0904654	−	0.995900i	$-0.471165\pi$
$151$	335.721	0.180931	0.0904654	+	0.995900i	$0.471165\pi$
$152$	13.3241	0.00711005
$153$	1489.34	0.786965
$154$	135.307	0.0708011
$155$	784.947	0.406764
$156$	170.014	0.0872563
$157$	−1593.69	−0.810130	−0.405065	−	0.914288i	$-0.632751\pi$
$157$	−1593.69	−0.810130	−0.405065	+	0.914288i	$0.632751\pi$
$158$	122.113	0.0614860
$159$	−4270.79	−2.13016
$160$	−378.188	−0.186865
$161$	−317.143	−0.155245
$162$	−327.723	−0.158941
$163$	−2767.63	−1.32992	−0.664962	−	0.746877i	$-0.731552\pi$
$163$	−2767.63	−1.32992	−0.664962	+	0.746877i	$0.731552\pi$
$164$	335.000	0.159507
$165$	861.830	0.406627
$166$	56.4868	0.0264110
$167$	282.867	0.131071	0.0655357	−	0.997850i	$-0.479124\pi$
$167$	282.867	0.131071	0.0655357	+	0.997850i	$0.479124\pi$
$168$	628.998	0.288859
$169$	−2187.70	−0.995765
$170$	−127.790	−0.0576530
$171$	−49.0297	−0.0219263
$172$	−1686.82	−0.747784
$173$	−2331.63	−1.02468	−0.512342	−	0.858782i	$-0.671222\pi$
$173$	−2331.63	−1.02468	−0.512342	+	0.858782i	$0.671222\pi$
$174$	23.5168	0.0102460
$175$	344.721	0.148905
$176$	1456.11	0.623629
$177$	−655.850	−0.278512
$178$	348.081	0.146572
$179$	109.140	0.0455726	0.0227863	−	0.999740i	$-0.492746\pi$
$179$	109.140	0.0455726	0.0227863	+	0.999740i	$0.492746\pi$
$180$	924.574	0.382854
$181$	1476.85	0.606483	0.303242	−	0.952914i	$-0.401931\pi$
$181$	1476.85	0.606483	0.303242	+	0.952914i	$0.401931\pi$
$182$	17.0317	0.00693667
$183$	−3788.76	−1.53045
$184$	147.497	0.0590957
$185$	−1514.01	−0.601686
$186$	452.217	0.178270
$187$	1529.33	0.598051
$188$	−1936.25	−0.751147
$189$	333.672	0.128418
$190$	4.20690	0.00160632
$191$	2032.16	0.769853	0.384926	−	0.922947i	$-0.374227\pi$
$191$	2032.16	0.769853	0.384926	+	0.922947i	$0.374227\pi$
$192$	3201.66	1.20344
$193$	3883.64	1.44845	0.724224	−	0.689565i	$-0.242198\pi$
$193$	3883.64	1.44845	0.724224	+	0.689565i	$0.242198\pi$
$194$	−375.588	−0.138998
$195$	108.482	0.0398389
$196$	1197.88	0.436544
$197$	3580.64	1.29498	0.647488	−	0.762076i	$-0.275820\pi$
$197$	3580.64	1.29498	0.647488	+	0.762076i	$0.275820\pi$
$198$	231.564	0.0831137
$199$	−2831.17	−1.00853	−0.504263	−	0.863550i	$-0.668236\pi$
$199$	−2831.17	−1.00853	−0.504263	+	0.863550i	$0.668236\pi$
$200$	−160.323	−0.0566826
$201$	−215.538	−0.0756362
$202$	693.830	0.241672
$203$	−112.572	−0.0389211
$204$	3517.86	1.20735
$205$	213.757	0.0728265
$206$	−376.819	−0.127448
$207$	−542.756	−0.182242
$208$	183.287	0.0610995
$209$	−50.3463	−0.0166628
$210$	198.597	0.0652596
$211$	2903.80	0.947421	0.473711	−	0.880681i	$-0.342914\pi$
$211$	2903.80	0.947421	0.473711	+	0.880681i	$0.342914\pi$
$212$	−4704.74	−1.52417
$213$	5237.82	1.68493
$214$	613.474	0.195964
$215$	−1076.33	−0.341418
$216$	−155.184	−0.0488839
$217$	−2164.70	−0.677186
$218$	708.173	0.220016
$219$	−2489.17	−0.768049
$220$	949.401	0.290948
$221$	192.503	0.0585935
$222$	−872.236	−0.263697
$223$	454.760	0.136560	0.0682802	−	0.997666i	$-0.478249\pi$
$223$	454.760	0.136560	0.0682802	+	0.997666i	$0.478249\pi$
$224$	1042.95	0.311095
$225$	589.952	0.174801
$226$	820.538	0.241510
$227$	−2103.24	−0.614966	−0.307483	−	0.951554i	$-0.599487\pi$
$227$	−2103.24	−0.614966	−0.307483	+	0.951554i	$0.599487\pi$
$228$	−115.810	−0.0336390
$229$	−4647.97	−1.34125	−0.670625	−	0.741796i	$-0.733974\pi$
$229$	−4647.97	−1.34125	−0.670625	+	0.741796i	$0.733974\pi$
$230$	46.5701	0.0133510
$231$	−2376.73	−0.676957
$232$	52.3548	0.0148158
$233$	131.118	0.0368661	0.0184331	−	0.999830i	$-0.494132\pi$
$233$	131.118	0.0368661	0.0184331	+	0.999830i	$0.494132\pi$
$234$	29.1479	0.00814299
$235$	−1235.48	−0.342953
$236$	−722.491	−0.199280
$237$	−2144.96	−0.587891
$238$	352.414	0.0959814
$239$	3467.77	0.938541	0.469271	−	0.883054i	$-0.344517\pi$
$239$	3467.77	0.938541	0.469271	+	0.883054i	$0.344517\pi$
$240$	2137.21	0.574818
$241$	5818.91	1.55531	0.777653	−	0.628694i	$-0.216410\pi$
$241$	5818.91	1.55531	0.777653	+	0.628694i	$0.216410\pi$
$242$	−301.216	−0.0800120
$243$	5103.22	1.34721
$244$	−4173.73	−1.09506
$245$	764.341	0.199314
$246$	123.148	0.0319171
$247$	−6.33731	−0.00163252
$248$	1006.76	0.257779
$249$	−992.213	−0.252526
$250$	−50.6196	−0.0128059
$251$	4633.30	1.16515	0.582573	−	0.812779i	$-0.302046\pi$
$251$	4633.30	1.16515	0.582573	+	0.812779i	$0.302046\pi$
$252$	−2549.76	−0.637380
$253$	−557.330	−0.138494
$254$	−861.139	−0.212727
$255$	2244.67	0.551243
$256$	3281.94	0.801255
$257$	−5262.78	−1.27737	−0.638683	−	0.769470i	$-0.720520\pi$
$257$	−5262.78	−1.27737	−0.638683	+	0.769470i	$0.720520\pi$
$258$	−620.083	−0.149631
$259$	4175.28	1.00170
$260$	119.505	0.0285054
$261$	−192.654	−0.0456896
$262$	−605.099	−0.142684
$263$	1890.26	0.443189	0.221594	−	0.975139i	$-0.428874\pi$
$263$	1890.26	0.443189	0.221594	+	0.975139i	$0.428874\pi$
$264$	1105.37	0.257692
$265$	−3002.00	−0.695892
$266$	−11.6016	−0.00267422
$267$	−6114.18	−1.40143
$268$	−237.439	−0.0541189
$269$	7472.17	1.69363	0.846815	−	0.531888i	$-0.178517\pi$
$269$	7472.17	1.69363	0.846815	+	0.531888i	$0.178517\pi$
$270$	−48.9971	−0.0110440
$271$	1634.38	0.366352	0.183176	−	0.983080i	$-0.441362\pi$
$271$	1634.38	0.366352	0.183176	+	0.983080i	$0.441362\pi$
$272$	3792.51	0.845421
$273$	−299.169	−0.0663243
$274$	917.355	0.202261
$275$	605.794	0.132839
$276$	−1282.01	−0.279593
$277$	−590.262	−0.128034	−0.0640170	−	0.997949i	$-0.520391\pi$
$277$	−590.262	−0.128034	−0.0640170	+	0.997949i	$0.520391\pi$
$278$	−1181.93	−0.254992
$279$	−3704.65	−0.794952
$280$	442.132	0.0943659
$281$	2501.14	0.530980	0.265490	−	0.964114i	$-0.414466\pi$
$281$	2501.14	0.530980	0.265490	+	0.964114i	$0.414466\pi$
$282$	−711.775	−0.150303
$283$	−803.901	−0.168859	−0.0844293	−	0.996429i	$-0.526907\pi$
$283$	−803.901	−0.168859	−0.0844293	+	0.996429i	$0.526907\pi$
$284$	5770.04	1.20559
$285$	−73.8959	−0.0153586
$286$	29.9306	0.00618823
$287$	−589.491	−0.121242
$288$	1784.90	0.365195
$289$	−929.797	−0.189252
$290$	16.5303	0.00334721
$291$	6597.35	1.32901
$292$	−2742.10	−0.549552
$293$	−6332.54	−1.26263	−0.631316	−	0.775526i	$-0.717485\pi$
$293$	−6332.54	−1.26263	−0.631316	+	0.775526i	$0.717485\pi$
$294$	440.345	0.0873518
$295$	−461.007	−0.0909860
$296$	−1941.84	−0.381307
$297$	586.376	0.114562
$298$	−222.483	−0.0432487
$299$	−70.1536	−0.0135688
$300$	1393.48	0.268176
$301$	2968.26	0.568397
$302$	135.952	0.0259046
$303$	−12187.4	−2.31072
$304$	−124.851	−0.0235550
$305$	−2663.17	−0.499977
$306$	603.117	0.112673
$307$	7317.73	1.36041	0.680203	−	0.733024i	$-0.261892\pi$
$307$	7317.73	1.36041	0.680203	+	0.733024i	$0.261892\pi$
$308$	−2618.23	−0.484374
$309$	6618.97	1.21858
$310$	317.870	0.0582381
$311$	2838.86	0.517611	0.258805	−	0.965930i	$-0.416671\pi$
$311$	2838.86	0.517611	0.258805	+	0.965930i	$0.416671\pi$
$312$	139.137	0.0252471
$313$	160.132	0.0289175	0.0144588	−	0.999895i	$-0.495397\pi$
$313$	160.132	0.0289175	0.0144588	+	0.999895i	$0.495397\pi$
$314$	−645.376	−0.115989
$315$	−1626.95	−0.291010
$316$	−2362.91	−0.420646
$317$	−3330.78	−0.590142	−0.295071	−	0.955475i	$-0.595343\pi$
$317$	−3330.78	−0.590142	−0.295071	+	0.955475i	$0.595343\pi$
$318$	−1729.49	−0.304983
$319$	−197.827	−0.0347216
$320$	2250.50	0.393145
$321$	−10775.9	−1.87369
$322$	−128.429	−0.0222270
$323$	−131.129	−0.0225889
$324$	6341.52	1.08737
$325$	76.2539	0.0130148
$326$	−1120.77	−0.190410
$327$	−12439.3	−2.10366
$328$	274.161	0.0461524
$329$	3407.17	0.570953
$330$	349.004	0.0582184
$331$	7337.39	1.21843	0.609214	−	0.793006i	$-0.291485\pi$
$331$	7337.39	1.21843	0.609214	+	0.793006i	$0.291485\pi$
$332$	−1093.03	−0.180687
$333$	7145.53	1.17589
$334$	114.549	0.0187660
$335$	−151.505	−0.0247092
$336$	−5893.93	−0.956965
$337$	−7160.54	−1.15745	−0.578723	−	0.815524i	$-0.696449\pi$
$337$	−7160.54	−1.15745	−0.578723	+	0.815524i	$0.696449\pi$
$338$	−885.923	−0.142568
$339$	−14413.1	−2.30918
$340$	2472.76	0.394424
$341$	−3804.13	−0.604121
$342$	−19.8549	−0.00313928
$343$	−6837.44	−1.07635
$344$	−1380.48	−0.216367
$345$	−818.022	−0.127655
$346$	−944.209	−0.146708
$347$	6740.51	1.04279	0.521397	−	0.853314i	$-0.325411\pi$
$347$	6740.51	1.04279	0.521397	+	0.853314i	$0.325411\pi$
$348$	−455.054	−0.0700962
$349$	−10173.9	−1.56045	−0.780224	−	0.625501i	$-0.784895\pi$
$349$	−10173.9	−1.56045	−0.780224	+	0.625501i	$0.784895\pi$
$350$	139.597	0.0213194
$351$	73.8097	0.0112241
$352$	1832.83	0.277529
$353$	−2552.87	−0.384917	−0.192458	−	0.981305i	$-0.561646\pi$
$353$	−2552.87	−0.384917	−0.192458	+	0.981305i	$0.561646\pi$
$354$	−265.591	−0.0398757
$355$	3681.75	0.550442
$356$	−6735.45	−1.00275
$357$	−6190.28	−0.917716
$358$	44.1970	0.00652481
$359$	−6677.47	−0.981681	−0.490841	−	0.871249i	$-0.663310\pi$
$359$	−6677.47	−0.981681	−0.490841	+	0.871249i	$0.663310\pi$
$360$	756.661	0.110776
$361$	−6854.68	−0.999371
$362$	598.062	0.0868326
$363$	5290.98	0.765026
$364$	−329.567	−0.0474561
$365$	−1749.68	−0.250911
$366$	−1534.28	−0.219121
$367$	11722.8	1.66738	0.833688	−	0.552236i	$-0.186225\pi$
$367$	11722.8	1.66738	0.833688	+	0.552236i	$0.186225\pi$
$368$	−1382.10	−0.195779
$369$	−1008.85	−0.142327
$370$	−613.108	−0.0861458
$371$	8278.82	1.15853
$372$	−8750.49	−1.21960
$373$	8070.71	1.12034	0.560168	−	0.828379i	$-0.310736\pi$
$373$	8070.71	1.12034	0.560168	+	0.828379i	$0.310736\pi$
$374$	619.313	0.0856254
$375$	889.154	0.122442
$376$	−1584.61	−0.217340
$377$	−24.9014	−0.00340182
$378$	135.123	0.0183861
$379$	−6693.10	−0.907128	−0.453564	−	0.891224i	$-0.649848\pi$
$379$	−6693.10	−0.907128	−0.453564	+	0.891224i	$0.649848\pi$
$380$	−81.4044	−0.0109894
$381$	15126.2	2.03397
$382$	822.937	0.110223
$383$	−2502.49	−0.333867	−0.166934	−	0.985968i	$-0.553387\pi$
$383$	−2502.49	−0.333867	−0.166934	+	0.985968i	$0.553387\pi$
$384$	5600.76	0.744303
$385$	−1670.64	−0.221152
$386$	1572.71	0.207380
$387$	5079.85	0.667243
$388$	7267.71	0.950933
$389$	5487.69	0.715262	0.357631	−	0.933863i	$-0.383585\pi$
$389$	5487.69	0.715262	0.357631	+	0.933863i	$0.383585\pi$
$390$	43.9307	0.00570389
$391$	−1451.59	−0.187750
$392$	980.329	0.126311
$393$	10628.8	1.36426
$394$	1450.01	0.185407
$395$	−1507.73	−0.192056
$396$	−4480.81	−0.568609
$397$	7760.91	0.981130	0.490565	−	0.871405i	$-0.336790\pi$
$397$	7760.91	0.981130	0.490565	+	0.871405i	$0.336790\pi$
$398$	−1146.50	−0.144395
$399$	203.787	0.0255693
$400$	1502.28	0.187785
$401$	14485.8	1.80395	0.901977	−	0.431785i	$-0.142116\pi$
$401$	14485.8	1.80395	0.901977	+	0.431785i	$0.142116\pi$
$402$	−87.2836	−0.0108291
$403$	−478.842	−0.0591882
$404$	−13425.8	−1.65336
$405$	4046.39	0.496462
$406$	−45.5867	−0.00557248
$407$	7337.41	0.893616
$408$	2878.98	0.349340
$409$	6664.83	0.805758	0.402879	−	0.915253i	$-0.368010\pi$
$409$	6664.83	0.805758	0.402879	+	0.915253i	$0.368010\pi$
$410$	86.5624	0.0104269
$411$	−16113.7	−1.93389
$412$	7291.53	0.871912
$413$	1271.35	0.151475
$414$	−219.793	−0.0260923
$415$	−697.442	−0.0824966
$416$	230.706	0.0271906
$417$	20761.1	2.43807
$418$	−20.3881	−0.00238568
$419$	−8437.43	−0.983760	−0.491880	−	0.870663i	$-0.663690\pi$
$419$	−8437.43	−0.983760	−0.491880	+	0.870663i	$0.663690\pi$
$420$	−3842.90	−0.446463
$421$	13893.2	1.60834	0.804171	−	0.594397i	$-0.202609\pi$
$421$	13893.2	1.60834	0.804171	+	0.594397i	$0.202609\pi$
$422$	1175.91	0.135646
$423$	5831.00	0.670243
$424$	−3850.31	−0.441008
$425$	1577.82	0.180083
$426$	2121.09	0.241238
$427$	7344.41	0.832368
$428$	−11870.9	−1.34065
$429$	−525.743	−0.0591681
$430$	−435.866	−0.0488822
$431$	−10611.5	−1.18594	−0.592969	−	0.805225i	$-0.702044\pi$
$431$	−10611.5	−1.18594	−0.592969	+	0.805225i	$0.702044\pi$
$432$	1454.13	0.161948
$433$	−7569.77	−0.840139	−0.420069	−	0.907492i	$-0.637994\pi$
$433$	−7569.77	−0.840139	−0.420069	+	0.907492i	$0.637994\pi$
$434$	−876.611	−0.0969555
$435$	−290.361	−0.0320040
$436$	−13703.3	−1.50520
$437$	47.7871	0.00523105
$438$	−1008.01	−0.109965
$439$	−11794.9	−1.28232	−0.641162	−	0.767406i	$-0.721547\pi$
$439$	−11794.9	−1.28232	−0.641162	+	0.767406i	$0.721547\pi$
$440$	776.980	0.0841842
$441$	−3607.39	−0.389525
$442$	77.9556	0.00838907
$443$	11424.0	1.22521	0.612606	−	0.790388i	$-0.290121\pi$
$443$	11424.0	1.22521	0.612606	+	0.790388i	$0.290121\pi$
$444$	16878.0	1.80404
$445$	−4297.75	−0.457827
$446$	184.158	0.0195519
$447$	3908.01	0.413518
$448$	−6206.34	−0.654513
$449$	8862.74	0.931533	0.465767	−	0.884908i	$-0.345779\pi$
$449$	8862.74	0.931533	0.465767	+	0.884908i	$0.345779\pi$
$450$	238.905	0.0250269
$451$	−1035.94	−0.108161
$452$	−15877.6	−1.65225
$453$	−2388.06	−0.247684
$454$	−851.724	−0.0880471
$455$	−210.290	−0.0216672
$456$	−94.7774	−0.00973324
$457$	5187.22	0.530958	0.265479	−	0.964117i	$-0.414470\pi$
$457$	5187.22	0.530958	0.265479	+	0.964117i	$0.414470\pi$
$458$	−1882.23	−0.192032
$459$	1527.24	0.155306
$460$	−901.141	−0.0913390
$461$	−16816.7	−1.69898	−0.849491	−	0.527603i	$-0.823091\pi$
$461$	−16816.7	−1.69898	−0.849491	+	0.527603i	$0.823091\pi$
$462$	−962.472	−0.0969226
$463$	6351.26	0.637512	0.318756	−	0.947837i	$-0.396735\pi$
$463$	6351.26	0.637512	0.318756	+	0.947837i	$0.396735\pi$
$464$	−490.582	−0.0490834
$465$	−5583.51	−0.556837
$466$	53.0970	0.00527827
$467$	1057.02	0.104739	0.0523693	−	0.998628i	$-0.483323\pi$
$467$	1057.02	0.104739	0.0523693	+	0.998628i	$0.483323\pi$
$468$	−564.019	−0.0557089
$469$	417.815	0.0411363
$470$	−500.317	−0.0491020
$471$	11336.3	1.10902
$472$	−591.279	−0.0576606
$473$	5216.25	0.507069
$474$	−868.618	−0.0841707
$475$	−51.9425	−0.00501745
$476$	−6819.28	−0.656641
$477$	14168.3	1.36000
$478$	1404.30	0.134375
$479$	−10138.2	−0.967073	−0.483537	−	0.875324i	$-0.660648\pi$
$479$	−10138.2	−0.967073	−0.483537	+	0.875324i	$0.660648\pi$
$480$	2690.14	0.255807
$481$	923.591	0.0875512
$482$	2356.41	0.222679
$483$	2255.91	0.212521
$484$	5828.60	0.547389
$485$	4637.38	0.434170
$486$	2066.59	0.192885
$487$	−8874.74	−0.825776	−0.412888	−	0.910782i	$-0.635480\pi$
$487$	−8874.74	−0.825776	−0.412888	+	0.910782i	$0.635480\pi$
$488$	−3415.74	−0.316851
$489$	19686.8	1.82059
$490$	309.525	0.0285366
$491$	−21452.2	−1.97174	−0.985869	−	0.167521i	$-0.946424\pi$
$491$	−21452.2	−1.97174	−0.985869	+	0.167521i	$0.946424\pi$
$492$	−2382.93	−0.218356
$493$	−515.249	−0.0470703
$494$	−2.56634	−0.000233735 0
$495$	−2859.11	−0.259611
$496$	−9433.67	−0.854001
$497$	−10153.4	−0.916382
$498$	−401.804	−0.0361551
$499$	9696.91	0.869926	0.434963	−	0.900448i	$-0.356761\pi$
$499$	9696.91	0.869926	0.434963	+	0.900448i	$0.356761\pi$
$500$	979.501	0.0876093
$501$	−2012.10	−0.179429
$502$	1876.29	0.166818
$503$	−8892.32	−0.788249	−0.394124	−	0.919057i	$-0.628952\pi$
$503$	−8892.32	−0.788249	−0.394124	+	0.919057i	$0.628952\pi$
$504$	−2086.69	−0.184422
$505$	−8566.71	−0.754878
$506$	−225.695	−0.0198288
$507$	15561.6	1.36315
$508$	16663.2	1.45534
$509$	−1084.94	−0.0944778	−0.0472389	−	0.998884i	$-0.515042\pi$
$509$	−1084.94	−0.0944778	−0.0472389	+	0.998884i	$0.515042\pi$
$510$	908.997	0.0789236
$511$	4825.20	0.417719
$512$	7628.02	0.658426
$513$	−50.2776	−0.00432712
$514$	−2131.20	−0.182885
$515$	4652.58	0.398091
$516$	11998.7	1.02367
$517$	5987.58	0.509349
$518$	1690.81	0.143417
$519$	16585.4	1.40273
$520$	97.8018	0.00824787
$521$	4578.35	0.384993	0.192496	−	0.981298i	$-0.438342\pi$
$521$	4578.35	0.384993	0.192496	+	0.981298i	$0.438342\pi$
$522$	−78.0166	−0.00654156
$523$	7298.28	0.610194	0.305097	−	0.952321i	$-0.401311\pi$
$523$	7298.28	0.610194	0.305097	+	0.952321i	$0.401311\pi$
$524$	11708.8	0.976148
$525$	−2452.08	−0.203843
$526$	765.476	0.0634531
$527$	−9908.02	−0.818975
$528$	−10357.7	−0.853712
$529$	529.000	0.0434783
$530$	−1215.68	−0.0996337
$531$	2175.77	0.177817
$532$	224.494	0.0182952
$533$	−130.398	−0.0105970
$534$	−2475.98	−0.200648
$535$	−7574.56	−0.612106
$536$	−194.317	−0.0156590
$537$	−776.338	−0.0623863
$538$	3025.91	0.242484
$539$	−3704.26	−0.296018
$540$	948.106	0.0755555
$541$	−14971.5	−1.18979	−0.594895	−	0.803803i	$-0.702806\pi$
$541$	−14971.5	−1.18979	−0.594895	+	0.803803i	$0.702806\pi$
$542$	661.854	0.0524521
$543$	−10505.2	−0.830241
$544$	4773.69	0.376232
$545$	−8743.80	−0.687235
$546$	−121.151	−0.00949591
$547$	−19510.9	−1.52509	−0.762547	−	0.646932i	$-0.776052\pi$
$547$	−19510.9	−1.52509	−0.762547	+	0.646932i	$0.776052\pi$
$548$	−17751.0	−1.38373
$549$	12569.2	0.977119
$550$	245.320	0.0190191
$551$	16.9623	0.00131147
$552$	−1049.18	−0.0808986
$553$	4157.96	0.319737
$554$	−239.031	−0.0183311
$555$	10769.5	0.823674
$556$	22870.7	1.74448
$557$	−13098.1	−0.996378	−0.498189	−	0.867068i	$-0.666001\pi$
$557$	−13098.1	−0.996378	−0.498189	+	0.867068i	$0.666001\pi$
$558$	−1500.22	−0.113816
$559$	656.592	0.0496796
$560$	−4142.93	−0.312626
$561$	−10878.5	−0.818698
$562$	1012.85	0.0760226
$563$	−4086.19	−0.305883	−0.152942	−	0.988235i	$-0.548875\pi$
$563$	−4086.19	−0.305883	−0.152942	+	0.988235i	$0.548875\pi$
$564$	13773.0	1.02828
$565$	−10131.2	−0.754374
$566$	−325.546	−0.0241762
$567$	−11159.0	−0.826516
$568$	4722.14	0.348832
$569$	−5021.20	−0.369946	−0.184973	−	0.982744i	$-0.559220\pi$
$569$	−5021.20	−0.369946	−0.184973	+	0.982744i	$0.559220\pi$
$570$	−29.9247	−0.00219896
$571$	−8277.56	−0.606664	−0.303332	−	0.952885i	$-0.598099\pi$
$571$	−8277.56	−0.606664	−0.303332	+	0.952885i	$0.598099\pi$
$572$	−579.164	−0.0423358
$573$	−14455.2	−1.05388
$574$	−238.719	−0.0173588
$575$	−575.000	−0.0417029
$576$	−10621.5	−0.768336
$577$	−19548.4	−1.41042	−0.705210	−	0.708998i	$-0.749147\pi$
$577$	−19548.4	−1.41042	−0.705210	+	0.708998i	$0.749147\pi$
$578$	−376.528	−0.0270960
$579$	−27625.2	−1.98284
$580$	−319.865	−0.0228994
$581$	1923.38	0.137341
$582$	2671.64	0.190280
$583$	14548.7	1.03353
$584$	−2244.10	−0.159010
$585$	−359.889	−0.0254352
$586$	−2564.41	−0.180776
$587$	−17479.7	−1.22907	−0.614536	−	0.788889i	$-0.710657\pi$
$587$	−17479.7	−1.22907	−0.614536	+	0.788889i	$0.710657\pi$
$588$	−8520.77	−0.597603
$589$	326.177	0.0228182
$590$	−186.688	−0.0130268
$591$	−25469.9	−1.77275
$592$	18195.7	1.26324
$593$	513.405	0.0355531	0.0177766	−	0.999842i	$-0.494341\pi$
$593$	513.405	0.0355531	0.0177766	+	0.999842i	$0.494341\pi$
$594$	237.457	0.0164023
$595$	−4351.25	−0.299805
$596$	4305.10	0.295879
$597$	20138.8	1.38061
$598$	−28.4092	−0.00194271
$599$	13706.8	0.934964	0.467482	−	0.884003i	$-0.345161\pi$
$599$	13706.8	0.934964	0.467482	+	0.884003i	$0.345161\pi$
$600$	1140.41	0.0775952
$601$	−24403.7	−1.65632	−0.828159	−	0.560493i	$-0.810612\pi$
$601$	−24403.7	−1.65632	−0.828159	+	0.560493i	$0.810612\pi$
$602$	1202.02	0.0813796
$603$	715.045	0.0482900
$604$	−2630.71	−0.177222
$605$	3719.11	0.249923
$606$	−4935.37	−0.330835
$607$	16304.7	1.09026	0.545129	−	0.838352i	$-0.316481\pi$
$607$	16304.7	1.09026	0.545129	+	0.838352i	$0.316481\pi$
$608$	−157.152	−0.0104825
$609$	800.748	0.0532807
$610$	−1078.47	−0.0715837
$611$	753.683	0.0499030
$612$	−11670.5	−0.770834
$613$	7754.74	0.510948	0.255474	−	0.966816i	$-0.417769\pi$
$613$	7754.74	0.510948	0.255474	+	0.966816i	$0.417769\pi$
$614$	2963.36	0.194775
$615$	−1520.50	−0.0996952
$616$	−2142.73	−0.140151
$617$	−1574.90	−0.102760	−0.0513801	−	0.998679i	$-0.516362\pi$
$617$	−1574.90	−0.102760	−0.0513801	+	0.998679i	$0.516362\pi$
$618$	2680.40	0.174468
$619$	−9160.25	−0.594800	−0.297400	−	0.954753i	$-0.596120\pi$
$619$	−9160.25	−0.594800	−0.297400	+	0.954753i	$0.596120\pi$
$620$	−6150.85	−0.398426
$621$	−556.570	−0.0359652
$622$	1149.62	0.0741083
$623$	11852.2	0.762196
$624$	−1303.76	−0.0836416
$625$	625.000	0.0400000
$626$	64.8465	0.00414024
$627$	358.125	0.0228104
$628$	12488.2	0.793523
$629$	19110.6	1.21143
$630$	−658.845	−0.0416651
$631$	11663.9	0.735871	0.367935	−	0.929851i	$-0.380065\pi$
$631$	11663.9	0.735871	0.367935	+	0.929851i	$0.380065\pi$
$632$	−1933.78	−0.121712
$633$	−20655.4	−1.29697
$634$	−1348.82	−0.0844930
$635$	10632.5	0.664467
$636$	33465.9	2.08649
$637$	−466.272	−0.0290021
$638$	−80.1115	−0.00497123
$639$	−17376.4	−1.07574
$640$	3936.86	0.243153
$641$	−27074.5	−1.66830	−0.834148	−	0.551541i	$-0.814040\pi$
$641$	−27074.5	−1.66830	−0.834148	+	0.551541i	$0.814040\pi$
$642$	−4363.78	−0.268263
$643$	−4463.82	−0.273773	−0.136886	−	0.990587i	$-0.543710\pi$
$643$	−4463.82	−0.273773	−0.136886	+	0.990587i	$0.543710\pi$
$644$	2485.14	0.152062
$645$	7656.16	0.467381
$646$	−53.1017	−0.00323415
$647$	−11755.0	−0.714277	−0.357139	−	0.934051i	$-0.616248\pi$
$647$	−11755.0	−0.714277	−0.357139	+	0.934051i	$0.616248\pi$
$648$	5189.83	0.314623
$649$	2234.20	0.135131
$650$	30.8796	0.00186338
$651$	15398.0	0.927029
$652$	21687.2	1.30266
$653$	−1236.64	−0.0741094	−0.0370547	−	0.999313i	$-0.511798\pi$
$653$	−1236.64	−0.0741094	−0.0370547	+	0.999313i	$0.511798\pi$
$654$	−5037.40	−0.301189
$655$	7471.15	0.445682
$656$	−2568.98	−0.152899
$657$	8257.81	0.490362
$658$	1379.76	0.0817456
$659$	23646.9	1.39780	0.698901	−	0.715218i	$-0.253673\pi$
$659$	23646.9	1.39780	0.698901	+	0.715218i	$0.253673\pi$
$660$	−6753.31	−0.398291
$661$	10150.5	0.597290	0.298645	−	0.954364i	$-0.403465\pi$
$661$	10150.5	0.597290	0.298645	+	0.954364i	$0.403465\pi$
$662$	2971.33	0.174447
$663$	−1369.32	−0.0802112
$664$	−894.526	−0.0522806
$665$	143.245	0.00835311
$666$	2893.63	0.168357
$667$	187.771	0.0109003
$668$	−2216.55	−0.128385
$669$	−3234.81	−0.186943
$670$	−61.3530	−0.00353772
$671$	12906.7	0.742558
$672$	−7418.77	−0.425871
$673$	13941.8	0.798540	0.399270	−	0.916833i	$-0.369264\pi$
$673$	13941.8	0.798540	0.399270	+	0.916833i	$0.369264\pi$
$674$	−2899.71	−0.165716
$675$	604.967	0.0344966
$676$	17142.8	0.975353
$677$	−13370.4	−0.759035	−0.379518	−	0.925185i	$-0.623910\pi$
$677$	−13370.4	−0.759035	−0.379518	+	0.925185i	$0.623910\pi$
$678$	−5836.68	−0.330614
$679$	−12788.8	−0.722812
$680$	2023.68	0.114124
$681$	14960.9	0.841852
$682$	−1540.51	−0.0864943
$683$	15659.3	0.877288	0.438644	−	0.898661i	$-0.355459\pi$
$683$	15659.3	0.877288	0.438644	+	0.898661i	$0.355459\pi$
$684$	384.198	0.0214768
$685$	−11326.6	−0.631775
$686$	−2768.87	−0.154105
$687$	33062.1	1.83609
$688$	12935.5	0.716806
$689$	1831.31	0.101259
$690$	−331.264	−0.0182768
$691$	−9631.82	−0.530263	−0.265131	−	0.964212i	$-0.585415\pi$
$691$	−9631.82	−0.530263	−0.265131	+	0.964212i	$0.585415\pi$
$692$	18270.7	1.00368
$693$	7884.77	0.432204
$694$	2729.62	0.149301
$695$	14593.3	0.796484
$696$	−372.412	−0.0202819
$697$	−2698.15	−0.146628
$698$	−4119.99	−0.223415
$699$	−932.671	−0.0504676
$700$	−2701.24	−0.145853
$701$	21140.9	1.13906	0.569530	−	0.821971i	$-0.307125\pi$
$701$	21140.9	1.13906	0.569530	+	0.821971i	$0.307125\pi$
$702$	29.8898	0.00160700
$703$	−629.131	−0.0337527
$704$	−10906.7	−0.583894
$705$	8788.27	0.469483
$706$	−1033.80	−0.0551101
$707$	23625.0	1.25673
$708$	5139.25	0.272803
$709$	−15917.4	−0.843144	−0.421572	−	0.906795i	$-0.638522\pi$
$709$	−15917.4	−0.843144	−0.421572	+	0.906795i	$0.638522\pi$
$710$	1490.95	0.0788089
$711$	7115.89	0.375340
$712$	−5512.22	−0.290139
$713$	3610.76	0.189655
$714$	−2506.80	−0.131393
$715$	−369.553	−0.0193294
$716$	−855.221	−0.0446384
$717$	−24667.1	−1.28481
$718$	−2704.09	−0.140551
$719$	3323.34	0.172378	0.0861889	−	0.996279i	$-0.472531\pi$
$719$	3323.34	0.172378	0.0861889	+	0.996279i	$0.472531\pi$
$720$	−7090.17	−0.366993
$721$	−12830.7	−0.662748
$722$	−2775.85	−0.143084
$723$	−41391.2	−2.12912
$724$	−11572.6	−0.594051
$725$	−204.099	−0.0104552
$726$	2142.62	0.109532
$727$	−9877.52	−0.503902	−0.251951	−	0.967740i	$-0.581072\pi$
$727$	−9877.52	−0.503902	−0.251951	+	0.967740i	$0.581072\pi$
$728$	−269.714	−0.0137312
$729$	−14449.9	−0.734130
$730$	−708.545	−0.0359239
$731$	13586.0	0.687407
$732$	29688.7	1.49908
$733$	28951.5	1.45886	0.729432	−	0.684053i	$-0.239784\pi$
$733$	28951.5	1.45886	0.729432	+	0.684053i	$0.239784\pi$
$734$	4747.24	0.238725
$735$	−5436.93	−0.272849
$736$	−1739.66	−0.0871262
$737$	734.246	0.0366978
$738$	−408.541	−0.0203775
$739$	−31009.5	−1.54358	−0.771788	−	0.635880i	$-0.780637\pi$
$739$	−31009.5	−1.54358	−0.771788	+	0.635880i	$0.780637\pi$
$740$	11863.8	0.589353
$741$	45.0788	0.00223483
$742$	3352.57	0.165871
$743$	−13761.0	−0.679465	−0.339733	−	0.940522i	$-0.610337\pi$
$743$	−13761.0	−0.679465	−0.339733	+	0.940522i	$0.610337\pi$
$744$	−7161.31	−0.352885
$745$	2747.00	0.135090
$746$	3268.29	0.160403
$747$	3291.66	0.161226
$748$	−11983.8	−0.585792
$749$	20888.9	1.01904
$750$	360.069	0.0175305
$751$	32197.4	1.56445	0.782223	−	0.622998i	$-0.214086\pi$
$751$	32197.4	1.56445	0.782223	+	0.622998i	$0.214086\pi$
$752$	14848.3	0.720029
$753$	−32957.8	−1.59502
$754$	−10.0840	−0.000487052 0
$755$	−1678.60	−0.0809148
$756$	−2614.65	−0.125786
$757$	26139.9	1.25505	0.627524	−	0.778597i	$-0.284068\pi$
$757$	26139.9	1.25505	0.627524	+	0.778597i	$0.284068\pi$
$758$	−2710.42	−0.129877
$759$	3964.42	0.189591
$760$	−66.6205	−0.00317971
$761$	−24004.0	−1.14342	−0.571712	−	0.820454i	$-0.693721\pi$
$761$	−24004.0	−1.14342	−0.571712	+	0.820454i	$0.693721\pi$
$762$	6125.48	0.291211
$763$	24113.3	1.14412
$764$	−15924.0	−0.754072
$765$	−7446.68	−0.351942
$766$	−1013.40	−0.0478011
$767$	281.229	0.0132393
$768$	−23345.2	−1.09687
$769$	−33733.4	−1.58187	−0.790934	−	0.611902i	$-0.790405\pi$
$769$	−33733.4	−1.58187	−0.790934	+	0.611902i	$0.790405\pi$
$770$	−676.537	−0.0316632
$771$	37435.3	1.74864
$772$	−30432.2	−1.41876
$773$	40247.1	1.87269	0.936344	−	0.351083i	$-0.114186\pi$
$773$	40247.1	1.87269	0.936344	+	0.351083i	$0.114186\pi$
$774$	2057.12	0.0955318
$775$	−3924.74	−0.181911
$776$	5947.81	0.275147
$777$	−29699.7	−1.37126
$778$	2222.28	0.102407
$779$	88.8246	0.00408533
$780$	−850.069	−0.0390222
$781$	−17843.0	−0.817508
$782$	−587.832	−0.0268808
$783$	−197.557	−0.00901676
$784$	−9186.02	−0.418459
$785$	7968.45	0.362301
$786$	4304.21	0.195326
$787$	−16327.3	−0.739522	−0.369761	−	0.929127i	$-0.620560\pi$
$787$	−16327.3	−0.739522	−0.369761	+	0.929127i	$0.620560\pi$
$788$	−28057.9	−1.26843
$789$	−13445.9	−0.606700
$790$	−610.565	−0.0274974
$791$	27939.4	1.25589
$792$	−3667.04	−0.164524
$793$	1624.62	0.0727515
$794$	3142.83	0.140472
$795$	21353.9	0.952636
$796$	22185.1	0.987852
$797$	−29358.8	−1.30482	−0.652410	−	0.757866i	$-0.726242\pi$
$797$	−29358.8	−1.30482	−0.652410	+	0.757866i	$0.726242\pi$
$798$	82.5252	0.00366085
$799$	15594.9	0.690498
$800$	1890.94	0.0835685
$801$	20283.7	0.894745
$802$	5866.12	0.258279
$803$	8479.56	0.372649
$804$	1688.96	0.0740857
$805$	1585.72	0.0694275
$806$	−193.911	−0.00847420
$807$	−53151.3	−2.31848
$808$	−10987.5	−0.478389
$809$	−2426.36	−0.105447	−0.0527234	−	0.998609i	$-0.516790\pi$
$809$	−2426.36	−0.105447	−0.0527234	+	0.998609i	$0.516790\pi$
$810$	1638.62	0.0710804
$811$	−31170.9	−1.34964	−0.674819	−	0.737983i	$-0.735778\pi$
$811$	−31170.9	−1.34964	−0.674819	+	0.737983i	$0.735778\pi$
$812$	882.112	0.0381232
$813$	−11625.7	−0.501515
$814$	2971.34	0.127943
$815$	13838.2	0.594760
$816$	−26977.0	−1.15733
$817$	−447.257	−0.0191524
$818$	2698.97	0.115363
$819$	992.490	0.0423448
$820$	−1675.00	−0.0713336
$821$	−6679.93	−0.283960	−0.141980	−	0.989870i	$-0.545347\pi$
$821$	−6679.93	−0.283960	−0.141980	+	0.989870i	$0.545347\pi$
$822$	−6525.36	−0.276883
$823$	29299.3	1.24096	0.620480	−	0.784222i	$-0.286938\pi$
$823$	29299.3	1.24096	0.620480	+	0.784222i	$0.286938\pi$
$824$	5967.31	0.252283
$825$	−4309.15	−0.181849
$826$	514.842	0.0216872
$827$	39806.7	1.67378	0.836889	−	0.547372i	$-0.184372\pi$
$827$	39806.7	1.67378	0.836889	+	0.547372i	$0.184372\pi$
$828$	4253.04	0.178506
$829$	−17854.1	−0.748007	−0.374004	−	0.927427i	$-0.622015\pi$
$829$	−17854.1	−0.748007	−0.374004	+	0.927427i	$0.622015\pi$
$830$	−282.434	−0.0118114
$831$	4198.67	0.175271
$832$	−1372.87	−0.0572065
$833$	−9647.91	−0.401297
$834$	8407.37	0.349069
$835$	−1414.34	−0.0586169
$836$	394.514	0.0163212
$837$	−3798.94	−0.156882
$838$	−3416.80	−0.140849
$839$	−6656.92	−0.273924	−0.136962	−	0.990576i	$-0.543734\pi$
$839$	−6656.92	−0.273924	−0.136962	+	0.990576i	$0.543734\pi$
$840$	−3144.99	−0.129181
$841$	−24322.3	−0.997267
$842$	5626.14	0.230273
$843$	−17791.2	−0.726881
$844$	−22754.2	−0.928000
$845$	10938.5	0.445320
$846$	2361.31	0.0959614
$847$	−10256.4	−0.416075
$848$	36078.7	1.46102
$849$	5718.34	0.231158
$850$	638.948	0.0257832
$851$	−6964.43	−0.280538
$852$	−41043.6	−1.65039
$853$	−30008.7	−1.20455	−0.602273	−	0.798290i	$-0.705738\pi$
$853$	−30008.7	−1.20455	−0.602273	+	0.798290i	$0.705738\pi$
$854$	2974.17	0.119173
$855$	245.149	0.00980574
$856$	−9714.98	−0.387910
$857$	24281.5	0.967843	0.483922	−	0.875111i	$-0.339212\pi$
$857$	24281.5	0.967843	0.483922	+	0.875111i	$0.339212\pi$
$858$	−212.903	−0.00847133
$859$	30635.5	1.21685	0.608423	−	0.793613i	$-0.291802\pi$
$859$	30635.5	1.21685	0.608423	+	0.793613i	$0.291802\pi$
$860$	8434.10	0.334419
$861$	4193.19	0.165974
$862$	−4297.22	−0.169796
$863$	−26572.1	−1.04812	−0.524058	−	0.851683i	$-0.675582\pi$
$863$	−26572.1	−1.04812	−0.524058	+	0.851683i	$0.675582\pi$
$864$	1830.33	0.0720707
$865$	11658.1	0.458253
$866$	−3065.43	−0.120286
$867$	6613.87	0.259076
$868$	16962.6	0.663305
$869$	7306.97	0.285238
$870$	−117.584	−0.00458214
$871$	92.4227	0.00359543
$872$	−11214.6	−0.435522
$873$	−21886.6	−0.848511
$874$	19.3517	0.000748950 0
$875$	−1723.60	−0.0665925
$876$	19505.2	0.752305
$877$	−37158.4	−1.43073	−0.715366	−	0.698750i	$-0.753740\pi$
$877$	−37158.4	−1.43073	−0.715366	+	0.698750i	$0.753740\pi$
$878$	−4776.43	−0.183595
$879$	45044.9	1.72847
$880$	−7280.57	−0.278895
$881$	−3353.12	−0.128229	−0.0641143	−	0.997943i	$-0.520422\pi$
$881$	−3353.12	−0.128229	−0.0641143	+	0.997943i	$0.520422\pi$
$882$	−1460.84	−0.0557699
$883$	16292.5	0.620936	0.310468	−	0.950584i	$-0.399514\pi$
$883$	16292.5	0.620936	0.310468	+	0.950584i	$0.399514\pi$
$884$	−1508.46	−0.0573924
$885$	3279.25	0.124555
$886$	4626.22	0.175418
$887$	−5949.82	−0.225226	−0.112613	−	0.993639i	$-0.535922\pi$
$887$	−5949.82	−0.225226	−0.112613	+	0.993639i	$0.535922\pi$
$888$	13812.7	0.521988
$889$	−29321.9	−1.10621
$890$	−1740.41	−0.0655489
$891$	−19610.2	−0.737338
$892$	−3563.50	−0.133761
$893$	−513.393	−0.0192386
$894$	1582.58	0.0592050
$895$	−545.700	−0.0203807
$896$	−10856.9	−0.404804
$897$	499.019	0.0185750
$898$	3589.03	0.133371
$899$	1281.66	0.0475480
$900$	−4622.87	−0.171217
$901$	37892.8	1.40110
$902$	−419.512	−0.0154858
$903$	−21113.9	−0.778102
$904$	−12994.0	−0.478070
$905$	−7384.26	−0.271228
$906$	−967.062	−0.0354619
$907$	−44105.2	−1.61465	−0.807326	−	0.590106i	$-0.799086\pi$
$907$	−44105.2	−1.61465	−0.807326	+	0.590106i	$0.799086\pi$
$908$	16481.0	0.602360
$909$	40431.6	1.47528
$910$	−85.1586	−0.00310218
$911$	4385.18	0.159481	0.0797407	−	0.996816i	$-0.474591\pi$
$911$	4385.18	0.159481	0.0797407	+	0.996816i	$0.474591\pi$
$912$	888.097	0.0322454
$913$	3380.05	0.122523
$914$	2100.60	0.0760194
$915$	18943.8	0.684439
$916$	36421.5	1.31376
$917$	−20603.7	−0.741978
$918$	618.468	0.0222358
$919$	30027.0	1.07780	0.538901	−	0.842369i	$-0.318840\pi$
$919$	30027.0	1.07780	0.538901	+	0.842369i	$0.318840\pi$
$920$	−737.484	−0.0264284
$921$	−52052.7	−1.86232
$922$	−6810.03	−0.243250
$923$	−2245.98	−0.0800946
$924$	18624.1	0.663080
$925$	7570.03	0.269082
$926$	2571.99	0.0912751
$927$	−21958.4	−0.778002
$928$	−617.503	−0.0218432
$929$	−5457.52	−0.192740	−0.0963700	−	0.995346i	$-0.530723\pi$
$929$	−5457.52	−0.192740	−0.0963700	+	0.995346i	$0.530723\pi$
$930$	−2261.08	−0.0797246
$931$	317.614	0.0111809
$932$	−1027.44	−0.0361104
$933$	−20193.5	−0.708579
$934$	428.046	0.0149958
$935$	−7646.65	−0.267457
$936$	−461.587	−0.0161191
$937$	3039.15	0.105960	0.0529802	−	0.998596i	$-0.483128\pi$
$937$	3039.15	0.105960	0.0529802	+	0.998596i	$0.483128\pi$
$938$	169.197	0.00588964
$939$	−1139.05	−0.0395864
$940$	9681.25	0.335923
$941$	25791.0	0.893479	0.446740	−	0.894664i	$-0.352585\pi$
$941$	25791.0	0.893479	0.446740	+	0.894664i	$0.352585\pi$
$942$	4590.71	0.158783
$943$	983.282	0.0339555
$944$	5540.48	0.191025
$945$	−1668.36	−0.0574304
$946$	2112.36	0.0725991
$947$	18339.3	0.629299	0.314650	−	0.949208i	$-0.398113\pi$
$947$	18339.3	0.629299	0.314650	+	0.949208i	$0.398113\pi$
$948$	16807.9	0.575840
$949$	1067.36	0.0365099
$950$	−21.0345	−0.000718368 0
$951$	23692.6	0.807871
$952$	−5580.83	−0.189995
$953$	12658.7	0.430278	0.215139	−	0.976583i	$-0.430979\pi$
$953$	12658.7	0.430278	0.215139	+	0.976583i	$0.430979\pi$
$954$	5737.55	0.194717
$955$	−10160.8	−0.344289
$956$	−27173.5	−0.919302
$957$	1407.19	0.0475319
$958$	−4105.55	−0.138460
$959$	31236.0	1.05179
$960$	−16008.3	−0.538193
$961$	−5145.31	−0.172714
$962$	374.015	0.0125351
$963$	35749.0	1.19626
$964$	−45597.0	−1.52342
$965$	−19418.2	−0.647765
$966$	913.548	0.0304275
$967$	50470.3	1.67840	0.839201	−	0.543822i	$-0.183023\pi$
$967$	50470.3	1.67840	0.839201	+	0.543822i	$0.183023\pi$
$968$	4770.06	0.158384
$969$	932.752	0.0309229
$970$	1877.94	0.0621618
$971$	30778.2	1.01722	0.508610	−	0.860997i	$-0.330160\pi$
$971$	30778.2	1.01722	0.508610	+	0.860997i	$0.330160\pi$
$972$	−39988.9	−1.31959
$973$	−40245.0	−1.32600
$974$	−3593.89	−0.118230
$975$	−542.412	−0.0178165
$976$	32006.6	1.04970
$977$	−23575.5	−0.772004	−0.386002	−	0.922498i	$-0.626144\pi$
$977$	−23575.5	−0.772004	−0.386002	+	0.922498i	$0.626144\pi$
$978$	7972.31	0.260661
$979$	20828.4	0.679958
$980$	−5989.38	−0.195228
$981$	41267.4	1.34308
$982$	−8687.21	−0.282301
$983$	−44798.3	−1.45355	−0.726777	−	0.686874i	$-0.758982\pi$
$983$	−44798.3	−1.45355	−0.726777	+	0.686874i	$0.758982\pi$
$984$	−1950.17	−0.0631799
$985$	−17903.2	−0.579131
$986$	−208.654	−0.00673924
$987$	−24236.0	−0.781601
$988$	49.6592	0.00159906
$989$	−4951.10	−0.159187
$990$	−1157.82	−0.0371696
$991$	−12153.5	−0.389574	−0.194787	−	0.980846i	$-0.562402\pi$
$991$	−12153.5	−0.389574	−0.194787	+	0.980846i	$0.562402\pi$
$992$	−11874.3	−0.380050
$993$	−52192.6	−1.66796
$994$	−4111.69	−0.131202
$995$	14155.9	0.451026
$996$	7774.99	0.247350
$997$	36538.1	1.16065	0.580327	−	0.814383i	$-0.302925\pi$
$997$	36538.1	1.16065	0.580327	+	0.814383i	$0.302925\pi$
$998$	3926.83	0.124551
$999$	7327.40	0.232061

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	115.4.a.e.1.3	✓	5
3.2	odd	2		1035.4.a.k.1.3		5
4.3	odd	2		1840.4.a.n.1.5		5
5.2	odd	4		575.4.b.i.24.6		10
5.3	odd	4		575.4.b.i.24.5		10
5.4	even	2		575.4.a.j.1.3		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
115.4.a.e.1.3	✓	5	1.1	even	1	trivial
575.4.a.j.1.3		5	5.4	even	2
575.4.b.i.24.5		10	5.3	odd	4
575.4.b.i.24.6		10	5.2	odd	4
1035.4.a.k.1.3		5	3.2	odd	2
1840.4.a.n.1.5		5	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 115 = 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	115.a (trivial)

\(f(q)\)	\(=\)	\(q+0.404957 q^{2} -7.11323 q^{3} -7.83601 q^{4} -5.00000 q^{5} -2.88055 q^{6} +13.7888 q^{7} -6.41290 q^{8} +23.5981 q^{9} +O(q^{10})\)	\(q+0.404957 q^{2} -7.11323 q^{3} -7.83601 q^{4} -5.00000 q^{5} -2.88055 q^{6} +13.7888 q^{7} -6.41290 q^{8} +23.5981 q^{9} -2.02479 q^{10} +24.2317 q^{11} +55.7394 q^{12} +3.05016 q^{13} +5.58389 q^{14} +35.5662 q^{15} +60.0911 q^{16} +63.1126 q^{17} +9.55621 q^{18} -2.07770 q^{19} +39.1800 q^{20} -98.0832 q^{21} +9.81282 q^{22} -23.0000 q^{23} +45.6165 q^{24} +25.0000 q^{25} +1.23518 q^{26} +24.1987 q^{27} -108.049 q^{28} -8.16397 q^{29} +14.4028 q^{30} -156.989 q^{31} +75.6376 q^{32} -172.366 q^{33} +25.5579 q^{34} -68.9442 q^{35} -184.915 q^{36} +302.801 q^{37} -0.841380 q^{38} -21.6965 q^{39} +32.0645 q^{40} -42.7514 q^{41} -39.7195 q^{42} +215.265 q^{43} -189.880 q^{44} -117.990 q^{45} -9.31401 q^{46} +247.096 q^{47} -427.442 q^{48} -152.868 q^{49} +10.1239 q^{50} -448.935 q^{51} -23.9010 q^{52} +600.400 q^{53} +9.79943 q^{54} -121.159 q^{55} -88.4265 q^{56} +14.7792 q^{57} -3.30606 q^{58} +92.2014 q^{59} -278.697 q^{60} +532.635 q^{61} -63.5740 q^{62} +325.390 q^{63} -450.099 q^{64} -15.2508 q^{65} -69.8009 q^{66} +30.3010 q^{67} -494.551 q^{68} +163.604 q^{69} -27.9194 q^{70} -736.349 q^{71} -151.332 q^{72} +349.936 q^{73} +122.622 q^{74} -177.831 q^{75} +16.2809 q^{76} +334.127 q^{77} -8.78614 q^{78} +301.545 q^{79} -300.456 q^{80} -809.279 q^{81} -17.3125 q^{82} +139.488 q^{83} +768.581 q^{84} -315.563 q^{85} +87.1732 q^{86} +58.0722 q^{87} -155.396 q^{88} +859.551 q^{89} -47.7810 q^{90} +42.0581 q^{91} +180.228 q^{92} +1116.70 q^{93} +100.063 q^{94} +10.3885 q^{95} -538.028 q^{96} -927.475 q^{97} -61.9051 q^{98} +571.823 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 6 q^{2} + 4 q^{3} + 22 q^{4} - 25 q^{5} + 19 q^{6} - 3 q^{7} + 138 q^{8} + 77 q^{9}+O(q^{10}) \) 5 * q + 6 * q^2 + 4 * q^3 + 22 * q^4 - 25 * q^5 + 19 * q^6 - 3 * q^7 + 138 * q^8 + 77 * q^9	\( 5 q + 6 q^{2} + 4 q^{3} + 22 q^{4} - 25 q^{5} + 19 q^{6} - 3 q^{7} + 138 q^{8} + 77 q^{9} - 30 q^{10} + 23 q^{11} + 47 q^{12} + 132 q^{13} + 93 q^{14} - 20 q^{15} + 282 q^{16} + 23 q^{17} - 15 q^{18} - 161 q^{19} - 110 q^{20} - 60 q^{21} + 193 q^{22} - 115 q^{23} + 105 q^{24} + 125 q^{25} - 257 q^{26} + 577 q^{27} + 17 q^{28} + 401 q^{29} - 95 q^{30} + 32 q^{31} + 670 q^{32} + 189 q^{33} - 663 q^{34} + 15 q^{35} - 659 q^{36} - 38 q^{37} - 875 q^{38} + 335 q^{39} - 690 q^{40} - 12 q^{41} - 798 q^{42} - 566 q^{43} + 47 q^{44} - 385 q^{45} - 138 q^{46} + 919 q^{47} - 773 q^{48} - 738 q^{49} + 150 q^{50} - 993 q^{51} - 305 q^{52} + 1156 q^{53} - 8 q^{54} - 115 q^{55} + 343 q^{56} + 114 q^{57} - 1042 q^{58} + 1324 q^{59} - 235 q^{60} - 1673 q^{61} + 565 q^{62} + 270 q^{63} + 2466 q^{64} - 660 q^{65} - 2781 q^{66} + 558 q^{67} - 2267 q^{68} - 92 q^{69} - 465 q^{70} - 108 q^{71} - 789 q^{72} + 1173 q^{73} + 1458 q^{74} + 100 q^{75} - 3477 q^{76} + 2608 q^{77} + 704 q^{78} + 656 q^{79} - 1410 q^{80} - 319 q^{81} + 3505 q^{82} - 82 q^{83} - 718 q^{84} - 115 q^{85} + 112 q^{86} + 2389 q^{87} + 2397 q^{88} + 570 q^{89} + 75 q^{90} - 1589 q^{91} - 506 q^{92} + 911 q^{93} - 948 q^{94} + 805 q^{95} - 5991 q^{96} + 633 q^{97} - 2555 q^{98} + 2021 q^{99}+O(q^{100}) \) 5 * q + 6 * q^2 + 4 * q^3 + 22 * q^4 - 25 * q^5 + 19 * q^6 - 3 * q^7 + 138 * q^8 + 77 * q^9 - 30 * q^10 + 23 * q^11 + 47 * q^12 + 132 * q^13 + 93 * q^14 - 20 * q^15 + 282 * q^16 + 23 * q^17 - 15 * q^18 - 161 * q^19 - 110 * q^20 - 60 * q^21 + 193 * q^22 - 115 * q^23 + 105 * q^24 + 125 * q^25 - 257 * q^26 + 577 * q^27 + 17 * q^28 + 401 * q^29 - 95 * q^30 + 32 * q^31 + 670 * q^32 + 189 * q^33 - 663 * q^34 + 15 * q^35 - 659 * q^36 - 38 * q^37 - 875 * q^38 + 335 * q^39 - 690 * q^40 - 12 * q^41 - 798 * q^42 - 566 * q^43 + 47 * q^44 - 385 * q^45 - 138 * q^46 + 919 * q^47 - 773 * q^48 - 738 * q^49 + 150 * q^50 - 993 * q^51 - 305 * q^52 + 1156 * q^53 - 8 * q^54 - 115 * q^55 + 343 * q^56 + 114 * q^57 - 1042 * q^58 + 1324 * q^59 - 235 * q^60 - 1673 * q^61 + 565 * q^62 + 270 * q^63 + 2466 * q^64 - 660 * q^65 - 2781 * q^66 + 558 * q^67 - 2267 * q^68 - 92 * q^69 - 465 * q^70 - 108 * q^71 - 789 * q^72 + 1173 * q^73 + 1458 * q^74 + 100 * q^75 - 3477 * q^76 + 2608 * q^77 + 704 * q^78 + 656 * q^79 - 1410 * q^80 - 319 * q^81 + 3505 * q^82 - 82 * q^83 - 718 * q^84 - 115 * q^85 + 112 * q^86 + 2389 * q^87 + 2397 * q^88 + 570 * q^89 + 75 * q^90 - 1589 * q^91 - 506 * q^92 + 911 * q^93 - 948 * q^94 + 805 * q^95 - 5991 * q^96 + 633 * q^97 - 2555 * q^98 + 2021 * q^99

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