LMFDB - Embedded newform 115.4.a.e.1.2

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			1.2
Root			$-2.49214$ 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)

$f(q)$	$=$	$q-1.49214 q^{2} +9.02447 q^{3} -5.77352 q^{4} -5.00000 q^{5} -13.4658 q^{6} +4.33445 q^{7} +20.5520 q^{8} +54.4411 q^{9} +O(q^{10})$	\(q-1.49214 q^{2} +9.02447 q^{3} -5.77352 q^{4} -5.00000 q^{5} -13.4658 q^{6} +4.33445 q^{7} +20.5520 q^{8} +54.4411 q^{9} +7.46070 q^{10} +48.1836 q^{11} -52.1029 q^{12} +53.0174 q^{13} -6.46761 q^{14} -45.1224 q^{15} +15.5216 q^{16} -61.0120 q^{17} -81.2338 q^{18} +12.0115 q^{19} +28.8676 q^{20} +39.1161 q^{21} -71.8968 q^{22} -23.0000 q^{23} +185.471 q^{24} +25.0000 q^{25} -79.1094 q^{26} +247.642 q^{27} -25.0250 q^{28} +198.213 q^{29} +67.3289 q^{30} -11.9661 q^{31} -187.577 q^{32} +434.832 q^{33} +91.0386 q^{34} -21.6723 q^{35} -314.317 q^{36} -230.018 q^{37} -17.9229 q^{38} +478.454 q^{39} -102.760 q^{40} -393.486 q^{41} -58.3668 q^{42} -167.496 q^{43} -278.189 q^{44} -272.206 q^{45} +34.3192 q^{46} +531.002 q^{47} +140.074 q^{48} -324.213 q^{49} -37.3035 q^{50} -550.602 q^{51} -306.097 q^{52} +373.570 q^{53} -369.516 q^{54} -240.918 q^{55} +89.0818 q^{56} +108.398 q^{57} -295.762 q^{58} -825.880 q^{59} +260.515 q^{60} -632.806 q^{61} +17.8551 q^{62} +235.972 q^{63} +155.718 q^{64} -265.087 q^{65} -648.830 q^{66} -38.7503 q^{67} +352.254 q^{68} -207.563 q^{69} +32.3381 q^{70} +1109.37 q^{71} +1118.88 q^{72} -446.171 q^{73} +343.219 q^{74} +225.612 q^{75} -69.3488 q^{76} +208.850 q^{77} -713.921 q^{78} +642.608 q^{79} -77.6081 q^{80} +764.926 q^{81} +587.137 q^{82} -292.575 q^{83} -225.838 q^{84} +305.060 q^{85} +249.927 q^{86} +1788.77 q^{87} +990.271 q^{88} -273.272 q^{89} +406.169 q^{90} +229.801 q^{91} +132.791 q^{92} -107.988 q^{93} -792.329 q^{94} -60.0577 q^{95} -1692.78 q^{96} -959.139 q^{97} +483.771 q^{98} +2623.17 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−1.49214	−0.527551	−0.263776	−	0.964584i	$-0.584968\pi$
$2$	−1.49214	−0.527551	−0.263776	+	0.964584i	$0.584968\pi$
$3$	9.02447	1.73676	0.868380	−	0.495899i	$-0.165161\pi$
$3$	9.02447	1.73676	0.868380	+	0.495899i	$0.165161\pi$
$4$	−5.77352	−0.721689
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	−13.4658	−0.916231
$7$	4.33445	0.234038	0.117019	−	0.993130i	$-0.462666\pi$
$7$	4.33445	0.234038	0.117019	+	0.993130i	$0.462666\pi$
$8$	20.5520	0.908280
$9$	54.4411	2.01634
$10$	7.46070	0.235928
$11$	48.1836	1.32072	0.660360	−	0.750949i	$-0.270404\pi$
$11$	48.1836	1.32072	0.660360	+	0.750949i	$0.270404\pi$
$12$	−52.1029	−1.25340
$13$	53.0174	1.13111	0.565553	−	0.824712i	$-0.308663\pi$
$13$	53.0174	1.13111	0.565553	+	0.824712i	$0.308663\pi$
$14$	−6.46761	−0.123467
$15$	−45.1224	−0.776703
$16$	15.5216	0.242525
$17$	−61.0120	−0.870447	−0.435223	−	0.900323i	$-0.643331\pi$
$17$	−61.0120	−0.870447	−0.435223	+	0.900323i	$0.643331\pi$
$18$	−81.2338	−1.06372
$19$	12.0115	0.145034	0.0725168	−	0.997367i	$-0.476897\pi$
$19$	12.0115	0.145034	0.0725168	+	0.997367i	$0.476897\pi$
$20$	28.8676	0.322749
$21$	39.1161	0.406469
$22$	−71.8968	−0.696747
$23$	−23.0000	−0.208514
$23$	−23.0000	−0.208514
$24$	185.471	1.57746
$25$	25.0000	0.200000
$26$	−79.1094	−0.596716
$27$	247.642	1.76514
$28$	−25.0250	−0.168903
$29$	198.213	1.26922	0.634609	−	0.772834i	$-0.281161\pi$
$29$	198.213	1.26922	0.634609	+	0.772834i	$0.281161\pi$
$30$	67.3289	0.409751
$31$	−11.9661	−0.0693284	−0.0346642	−	0.999399i	$-0.511036\pi$
$31$	−11.9661	−0.0693284	−0.0346642	+	0.999399i	$0.511036\pi$
$32$	−187.577	−1.03622
$33$	434.832	2.29377
$34$	91.0386	0.459205
$35$	−21.6723	−0.104665
$36$	−314.317	−1.45517
$37$	−230.018	−1.02202	−0.511010	−	0.859575i	$-0.670728\pi$
$37$	−230.018	−1.02202	−0.511010	+	0.859575i	$0.670728\pi$
$38$	−17.9229	−0.0765126
$39$	478.454	1.96446
$40$	−102.760	−0.406195
$41$	−393.486	−1.49883	−0.749417	−	0.662098i	$-0.769666\pi$
$41$	−393.486	−1.49883	−0.749417	+	0.662098i	$0.769666\pi$
$42$	−58.3668	−0.214433
$43$	−167.496	−0.594019	−0.297010	−	0.954874i	$-0.595989\pi$
$43$	−167.496	−0.594019	−0.297010	+	0.954874i	$0.595989\pi$
$44$	−278.189	−0.953149
$45$	−272.206	−0.901734
$46$	34.3192	0.110002
$47$	531.002	1.64797	0.823984	−	0.566612i	$-0.191746\pi$
$47$	531.002	1.64797	0.823984	+	0.566612i	$0.191746\pi$
$48$	140.074	0.421208
$49$	−324.213	−0.945226
$50$	−37.3035	−0.105510
$51$	−550.602	−1.51176
$52$	−306.097	−0.816307
$53$	373.570	0.968186	0.484093	−	0.875017i	$-0.339150\pi$
$53$	373.570	0.968186	0.484093	+	0.875017i	$0.339150\pi$
$54$	−369.516	−0.931200
$55$	−240.918	−0.590644
$56$	89.0818	0.212572
$57$	108.398	0.251889
$58$	−295.762	−0.669578
$59$	−825.880	−1.82238	−0.911190	−	0.411987i	$-0.864835\pi$
$59$	−825.880	−1.82238	−0.911190	+	0.411987i	$0.864835\pi$
$60$	260.515	0.560538
$61$	−632.806	−1.32824	−0.664119	−	0.747627i	$-0.731193\pi$
$61$	−632.806	−1.32824	−0.664119	+	0.747627i	$0.731193\pi$
$62$	17.8551	0.0365743
$63$	235.972	0.471900
$64$	155.718	0.304136
$65$	−265.087	−0.505846
$66$	−648.830	−1.21008
$67$	−38.7503	−0.0706583	−0.0353292	−	0.999376i	$-0.511248\pi$
$67$	−38.7503	−0.0706583	−0.0353292	+	0.999376i	$0.511248\pi$
$68$	352.254	0.628192
$69$	−207.563	−0.362140
$70$	32.3381	0.0552163
$71$	1109.37	1.85434	0.927168	−	0.374645i	$-0.122236\pi$
$71$	1109.37	1.85434	0.927168	+	0.374645i	$0.122236\pi$
$72$	1118.88	1.83140
$73$	−446.171	−0.715348	−0.357674	−	0.933847i	$-0.616430\pi$
$73$	−446.171	−0.715348	−0.357674	+	0.933847i	$0.616430\pi$
$74$	343.219	0.539168
$75$	225.612	0.347352
$76$	−69.3488	−0.104669
$77$	208.850	0.309099
$78$	−713.921	−1.03635
$79$	642.608	0.915179	0.457589	−	0.889164i	$-0.348713\pi$
$79$	642.608	0.915179	0.457589	+	0.889164i	$0.348713\pi$
$80$	−77.6081	−0.108461
$81$	764.926	1.04928
$82$	587.137	0.790712
$83$	−292.575	−0.386919	−0.193460	−	0.981108i	$-0.561971\pi$
$83$	−292.575	−0.386919	−0.193460	+	0.981108i	$0.561971\pi$
$84$	−225.838	−0.293344
$85$	305.060	0.389276
$86$	249.927	0.313376
$87$	1788.77	2.20433
$88$	990.271	1.19958
$89$	−273.272	−0.325469	−0.162735	−	0.986670i	$-0.552031\pi$
$89$	−273.272	−0.325469	−0.162735	+	0.986670i	$0.552031\pi$
$90$	406.169	0.475711
$91$	229.801	0.264722
$92$	132.791	0.150483
$93$	−107.988	−0.120407
$94$	−792.329	−0.869388
$95$	−60.0577	−0.0648610
$96$	−1692.78	−1.79967
$97$	−959.139	−1.00398	−0.501989	−	0.864874i	$-0.667398\pi$
$97$	−959.139	−1.00398	−0.501989	+	0.864874i	$0.667398\pi$
$98$	483.771	0.498655
$99$	2623.17	2.66302
$100$	−144.338	−0.144338
$101$	−339.881	−0.334845	−0.167423	−	0.985885i	$-0.553544\pi$
$101$	−339.881	−0.334845	−0.167423	+	0.985885i	$0.553544\pi$
$102$	821.575	0.797530
$103$	−316.048	−0.302341	−0.151170	−	0.988508i	$-0.548304\pi$
$103$	−316.048	−0.302341	−0.151170	+	0.988508i	$0.548304\pi$
$104$	1089.61	1.02736
$105$	−195.581	−0.181778
$106$	−557.420	−0.510768
$107$	105.833	0.0956196	0.0478098	−	0.998856i	$-0.484776\pi$
$107$	105.833	0.0956196	0.0478098	+	0.998856i	$0.484776\pi$
$108$	−1429.76	−1.27388
$109$	−1903.06	−1.67229	−0.836146	−	0.548508i	$-0.815196\pi$
$109$	−1903.06	−1.67229	−0.836146	+	0.548508i	$0.815196\pi$
$110$	359.484	0.311595
$111$	−2075.79	−1.77500
$112$	67.2777	0.0567602
$113$	−94.0121	−0.0782647	−0.0391324	−	0.999234i	$-0.512459\pi$
$113$	−94.0121	−0.0782647	−0.0391324	+	0.999234i	$0.512459\pi$
$114$	−161.745	−0.132884
$115$	115.000	0.0932505
$116$	−1144.39	−0.915981
$117$	2886.33	2.28069
$118$	1232.33	0.961399
$119$	−264.454	−0.203718
$120$	−927.356	−0.705464
$121$	990.663	0.744300
$122$	944.236	0.700714
$123$	−3551.00	−2.60312
$124$	69.0866	0.0500336
$125$	−125.000	−0.0894427
$126$	−352.104	−0.248952
$127$	286.165	0.199945	0.0999727	−	0.994990i	$-0.468124\pi$
$127$	286.165	0.199945	0.0999727	+	0.994990i	$0.468124\pi$
$128$	1268.26	0.875777
$129$	−1511.56	−1.03167
$130$	395.547	0.266860
$131$	1226.25	0.817844	0.408922	−	0.912569i	$-0.365905\pi$
$131$	1226.25	0.817844	0.408922	+	0.912569i	$0.365905\pi$
$132$	−2510.51	−1.65539
$133$	52.0635	0.0339434
$134$	57.8210	0.0372759
$135$	−1238.21	−0.789393
$136$	−1253.92	−0.790609
$137$	3159.26	1.97017	0.985086	−	0.172061i	$-0.0550426\pi$
$137$	3159.26	1.97017	0.985086	+	0.172061i	$0.0550426\pi$
$138$	309.713	0.191047
$139$	1095.59	0.668540	0.334270	−	0.942477i	$-0.391510\pi$
$139$	1095.59	0.668540	0.334270	+	0.942477i	$0.391510\pi$
$140$	125.125	0.0755357
$141$	4792.01	2.86213
$142$	−1655.33	−0.978258
$143$	2554.57	1.49387
$144$	845.014	0.489013
$145$	−991.067	−0.567611
$146$	665.750	0.377383
$147$	−2925.85	−1.64163
$148$	1328.01	0.737581
$149$	2227.60	1.22478	0.612390	−	0.790556i	$-0.290208\pi$
$149$	2227.60	1.22478	0.612390	+	0.790556i	$0.290208\pi$
$150$	−336.645	−0.183246
$151$	−972.366	−0.524040	−0.262020	−	0.965062i	$-0.584389\pi$
$151$	−972.366	−0.524040	−0.262020	+	0.965062i	$0.584389\pi$
$152$	246.862	0.131731
$153$	−3321.56	−1.75511
$154$	−311.633	−0.163066
$155$	59.8306	0.0310046
$156$	−2762.36	−1.41773
$157$	−2872.71	−1.46030	−0.730151	−	0.683285i	$-0.760551\pi$
$157$	−2872.71	−1.46030	−0.730151	+	0.683285i	$0.760551\pi$
$158$	−958.862	−0.482804
$159$	3371.28	1.68151
$160$	937.883	0.463414
$161$	−99.6924	−0.0488004
$162$	−1141.38	−0.553550
$163$	1458.04	0.700631	0.350315	−	0.936632i	$-0.386074\pi$
$163$	1458.04	0.700631	0.350315	+	0.936632i	$0.386074\pi$
$164$	2271.80	1.08169
$165$	−2174.16	−1.02581
$166$	436.563	0.204120
$167$	−533.984	−0.247431	−0.123715	−	0.992318i	$-0.539481\pi$
$167$	−533.984	−0.247431	−0.123715	+	0.992318i	$0.539481\pi$
$168$	803.916	0.369187
$169$	613.842	0.279400
$170$	−455.193	−0.205363
$171$	653.922	0.292437
$172$	967.038	0.428697
$173$	2127.49	0.934973	0.467487	−	0.884000i	$-0.345160\pi$
$173$	2127.49	0.934973	0.467487	+	0.884000i	$0.345160\pi$
$174$	−2669.10	−1.16290
$175$	108.361	0.0468077
$176$	747.888	0.320308
$177$	−7453.13	−3.16504
$178$	407.760	0.171702
$179$	−2827.30	−1.18057	−0.590286	−	0.807194i	$-0.700985\pi$
$179$	−2827.30	−1.18057	−0.590286	+	0.807194i	$0.700985\pi$
$180$	1571.58	0.650772
$181$	284.441	0.116809	0.0584043	−	0.998293i	$-0.481399\pi$
$181$	284.441	0.116809	0.0584043	+	0.998293i	$0.481399\pi$
$182$	−342.896	−0.139655
$183$	−5710.75	−2.30683
$184$	−472.697	−0.189389
$185$	1150.09	0.457061
$186$	161.133	0.0635208
$187$	−2939.78	−1.14962
$188$	−3065.75	−1.18932
$189$	1073.39	0.413110
$190$	89.6146	0.0342175
$191$	−2683.22	−1.01650	−0.508248	−	0.861211i	$-0.669707\pi$
$191$	−2683.22	−1.01650	−0.508248	+	0.861211i	$0.669707\pi$
$192$	1405.27	0.528212
$193$	2173.31	0.810563	0.405281	−	0.914192i	$-0.367174\pi$
$193$	2173.31	0.810563	0.405281	+	0.914192i	$0.367174\pi$
$194$	1431.17	0.529650
$195$	−2392.27	−0.878533
$196$	1871.85	0.682160
$197$	4564.42	1.65077	0.825385	−	0.564570i	$-0.190958\pi$
$197$	4564.42	1.65077	0.825385	+	0.564570i	$0.190958\pi$
$198$	−3914.14	−1.40488
$199$	−3842.37	−1.36873	−0.684367	−	0.729138i	$-0.739921\pi$
$199$	−3842.37	−1.36873	−0.684367	+	0.729138i	$0.739921\pi$
$200$	513.801	0.181656
$201$	−349.701	−0.122717
$202$	507.150	0.176648
$203$	859.147	0.297046
$204$	3178.91	1.09102
$205$	1967.43	0.670299
$206$	471.588	0.159500
$207$	−1252.15	−0.420436
$208$	822.915	0.274322
$209$	578.760	0.191549
$210$	291.834	0.0958974
$211$	679.119	0.221576	0.110788	−	0.993844i	$-0.464663\pi$
$211$	679.119	0.221576	0.110788	+	0.993844i	$0.464663\pi$
$212$	−2156.81	−0.698729
$213$	10011.5	3.22054
$214$	−157.918	−0.0504442
$215$	837.478	0.265654
$216$	5089.54	1.60324
$217$	−51.8666	−0.0162255
$218$	2839.63	0.882220
$219$	−4026.46	−1.24239
$220$	1390.94	0.426261
$221$	−3234.70	−0.984567
$222$	3097.37	0.936405
$223$	2187.32	0.656832	0.328416	−	0.944533i	$-0.393485\pi$
$223$	2187.32	0.656832	0.328416	+	0.944533i	$0.393485\pi$
$224$	−813.042	−0.242516
$225$	1361.03	0.403268
$226$	140.279	0.0412887
$227$	−5797.66	−1.69517	−0.847586	−	0.530658i	$-0.821945\pi$
$227$	−5797.66	−1.69517	−0.847586	+	0.530658i	$0.821945\pi$
$228$	−625.837	−0.181785
$229$	−6874.42	−1.98373	−0.991866	−	0.127286i	$-0.959374\pi$
$229$	−6874.42	−1.98373	−0.991866	+	0.127286i	$0.959374\pi$
$230$	−171.596	−0.0491944
$231$	1884.76	0.536831
$232$	4073.69	1.15280
$233$	−4274.38	−1.20182	−0.600910	−	0.799317i	$-0.705195\pi$
$233$	−4274.38	−1.20182	−0.600910	+	0.799317i	$0.705195\pi$
$234$	−4306.80	−1.20318
$235$	−2655.01	−0.736994
$236$	4768.23	1.31519
$237$	5799.20	1.58945
$238$	394.602	0.107472
$239$	2416.98	0.654148	0.327074	−	0.944999i	$-0.393937\pi$
$239$	2416.98	0.654148	0.327074	+	0.944999i	$0.393937\pi$
$240$	−700.372	−0.188370
$241$	4246.06	1.13491	0.567453	−	0.823406i	$-0.307929\pi$
$241$	4246.06	1.13491	0.567453	+	0.823406i	$0.307929\pi$
$242$	−1478.21	−0.392656
$243$	216.725	0.0572136
$244$	3653.52	0.958576
$245$	1621.06	0.422718
$246$	5298.60	1.37328
$247$	636.821	0.164048
$248$	−245.928	−0.0629696
$249$	−2640.34	−0.671986
$250$	186.518	0.0471856
$251$	−4815.45	−1.21095	−0.605476	−	0.795864i	$-0.707017\pi$
$251$	−4815.45	−1.21095	−0.605476	+	0.795864i	$0.707017\pi$
$252$	−1362.39	−0.340566
$253$	−1108.22	−0.275389
$254$	−426.999	−0.105481
$255$	2753.01	0.676079
$256$	−3138.17	−0.766154
$257$	5309.07	1.28860	0.644301	−	0.764772i	$-0.277149\pi$
$257$	5309.07	1.28860	0.644301	+	0.764772i	$0.277149\pi$
$258$	2255.46	0.544259
$259$	−997.002	−0.239192
$260$	1530.48	0.365064
$261$	10791.0	2.55917
$262$	−1829.73	−0.431455
$263$	−1045.05	−0.245022	−0.122511	−	0.992467i	$-0.539095\pi$
$263$	−1045.05	−0.245022	−0.122511	+	0.992467i	$0.539095\pi$
$264$	8936.68	2.08339
$265$	−1867.85	−0.432986
$266$	−77.6860	−0.0179069
$267$	−2466.14	−0.565263
$268$	223.726	0.0509934
$269$	−3226.10	−0.731221	−0.365611	−	0.930768i	$-0.619140\pi$
$269$	−3226.10	−0.731221	−0.365611	+	0.930768i	$0.619140\pi$
$270$	1847.58	0.416445
$271$	2535.37	0.568312	0.284156	−	0.958778i	$-0.408287\pi$
$271$	2535.37	0.568312	0.284156	+	0.958778i	$0.408287\pi$
$272$	−947.005	−0.211105
$273$	2073.84	0.459759
$274$	−4714.06	−1.03937
$275$	1204.59	0.264144
$276$	1198.37	0.261352
$277$	−6453.28	−1.39978	−0.699891	−	0.714249i	$-0.746768\pi$
$277$	−6453.28	−1.39978	−0.699891	+	0.714249i	$0.746768\pi$
$278$	−1634.78	−0.352689
$279$	−651.449	−0.139789
$280$	−445.409	−0.0950652
$281$	−7534.53	−1.59955	−0.799773	−	0.600302i	$-0.795047\pi$
$281$	−7534.53	−1.59955	−0.799773	+	0.600302i	$0.795047\pi$
$282$	−7150.35	−1.50992
$283$	4584.47	0.962964	0.481482	−	0.876456i	$-0.340099\pi$
$283$	4584.47	0.962964	0.481482	+	0.876456i	$0.340099\pi$
$284$	−6404.96	−1.33826
$285$	−541.989	−0.112648
$286$	−3811.78	−0.788095
$287$	−1705.55	−0.350785
$288$	−10211.9	−2.08938
$289$	−1190.53	−0.242323
$290$	1478.81	0.299444
$291$	−8655.72	−1.74367
$292$	2575.98	0.516259
$293$	6890.03	1.37379	0.686894	−	0.726758i	$-0.258974\pi$
$293$	6890.03	1.37379	0.686894	+	0.726758i	$0.258974\pi$
$294$	4365.78	0.866045
$295$	4129.40	0.814993
$296$	−4727.33	−0.928279
$297$	11932.3	2.33125
$298$	−3323.89	−0.646134
$299$	−1219.40	−0.235852
$300$	−1302.57	−0.250680
$301$	−726.001	−0.139023
$302$	1450.91	0.276458
$303$	−3067.24	−0.581546
$304$	186.438	0.0351743
$305$	3164.03	0.594006
$306$	4956.24	0.925913
$307$	−1461.21	−0.271646	−0.135823	−	0.990733i	$-0.543368\pi$
$307$	−1461.21	−0.271646	−0.135823	+	0.990733i	$0.543368\pi$
$308$	−1205.80	−0.223074
$309$	−2852.16	−0.525094
$310$	−89.2757	−0.0163565
$311$	−750.293	−0.136801	−0.0684007	−	0.997658i	$-0.521790\pi$
$311$	−750.293	−0.136801	−0.0684007	+	0.997658i	$0.521790\pi$
$312$	9833.20	1.78428
$313$	8615.99	1.55592	0.777962	−	0.628311i	$-0.216254\pi$
$313$	8615.99	1.55592	0.777962	+	0.628311i	$0.216254\pi$
$314$	4286.49	0.770385
$315$	−1179.86	−0.211040
$316$	−3710.11	−0.660475
$317$	356.372	0.0631415	0.0315707	−	0.999502i	$-0.489949\pi$
$317$	356.372	0.0631415	0.0315707	+	0.999502i	$0.489949\pi$
$318$	−5030.42	−0.887081
$319$	9550.64	1.67628
$320$	−778.589	−0.136014
$321$	955.090	0.166068
$322$	148.755	0.0257447
$323$	−732.849	−0.126244
$324$	−4416.31	−0.757255
$325$	1325.43	0.226221
$326$	−2175.61	−0.369619
$327$	−17174.1	−2.90437
$328$	−8086.93	−1.36136
$329$	2301.60	0.385688
$330$	3244.15	0.541166
$331$	7389.98	1.22716	0.613580	−	0.789633i	$-0.289729\pi$
$331$	7389.98	1.22716	0.613580	+	0.789633i	$0.289729\pi$
$332$	1689.19	0.279236
$333$	−12522.4	−2.06074
$334$	796.779	0.130532
$335$	193.752	0.0315994
$336$	607.146	0.0985789
$337$	10614.8	1.71580	0.857898	−	0.513820i	$-0.171770\pi$
$337$	10614.8	1.71580	0.857898	+	0.513820i	$0.171770\pi$
$338$	−915.939	−0.147398
$339$	−848.410	−0.135927
$340$	−1761.27	−0.280936
$341$	−576.572	−0.0915633
$342$	−975.744	−0.154275
$343$	−2892.00	−0.455258
$344$	−3442.37	−0.539536
$345$	1037.81	0.161954
$346$	−3174.52	−0.493246
$347$	−3472.76	−0.537256	−0.268628	−	0.963244i	$-0.586570\pi$
$347$	−3472.76	−0.537256	−0.268628	+	0.963244i	$0.586570\pi$
$348$	−10327.5	−1.59084
$349$	4820.48	0.739353	0.369677	−	0.929161i	$-0.379469\pi$
$349$	4820.48	0.739353	0.369677	+	0.929161i	$0.379469\pi$
$350$	−161.690	−0.0246935
$351$	13129.3	1.99656
$352$	−9038.12	−1.36856
$353$	3056.54	0.460859	0.230430	−	0.973089i	$-0.425987\pi$
$353$	3056.54	0.460859	0.230430	+	0.973089i	$0.425987\pi$
$354$	11121.1	1.66972
$355$	−5546.85	−0.829284
$356$	1577.74	0.234888
$357$	−2386.56	−0.353809
$358$	4218.73	0.622812
$359$	−10754.6	−1.58108	−0.790541	−	0.612409i	$-0.790201\pi$
$359$	−10754.6	−1.58108	−0.790541	+	0.612409i	$0.790201\pi$
$360$	−5594.38	−0.819026
$361$	−6714.72	−0.978965
$362$	−424.426	−0.0616225
$363$	8940.21	1.29267
$364$	−1326.76	−0.191047
$365$	2230.86	0.319913
$366$	8521.24	1.21697
$367$	−10901.7	−1.55058	−0.775292	−	0.631602i	$-0.782398\pi$
$367$	−10901.7	−1.55058	−0.775292	+	0.631602i	$0.782398\pi$
$368$	−356.997	−0.0505700
$369$	−21421.8	−3.02216
$370$	−1716.10	−0.241123
$371$	1619.22	0.226593
$372$	623.470	0.0868963
$373$	98.0242	0.0136072	0.00680362	−	0.999977i	$-0.497834\pi$
$373$	98.0242	0.0136072	0.00680362	+	0.999977i	$0.497834\pi$
$374$	4386.57	0.606481
$375$	−1128.06	−0.155341
$376$	10913.2	1.49682
$377$	10508.8	1.43562
$378$	−1601.65	−0.217937
$379$	−8819.20	−1.19528	−0.597641	−	0.801764i	$-0.703895\pi$
$379$	−8819.20	−1.19528	−0.597641	+	0.801764i	$0.703895\pi$
$380$	346.744	0.0468095
$381$	2582.49	0.347257
$382$	4003.74	0.536254
$383$	7622.61	1.01696	0.508482	−	0.861073i	$-0.330207\pi$
$383$	7622.61	1.01696	0.508482	+	0.861073i	$0.330207\pi$
$384$	11445.4	1.52101
$385$	−1044.25	−0.138233
$386$	−3242.89	−0.427614
$387$	−9118.64	−1.19774
$388$	5537.60	0.724560
$389$	−495.174	−0.0645406	−0.0322703	−	0.999479i	$-0.510274\pi$
$389$	−495.174	−0.0645406	−0.0322703	+	0.999479i	$0.510274\pi$
$390$	3569.60	0.463471
$391$	1403.28	0.181501
$392$	−6663.22	−0.858530
$393$	11066.2	1.42040
$394$	−6810.76	−0.870866
$395$	−3213.04	−0.409280
$396$	−15144.9	−1.92187
$397$	−8510.26	−1.07586	−0.537931	−	0.842989i	$-0.680794\pi$
$397$	−8510.26	−1.07586	−0.537931	+	0.842989i	$0.680794\pi$
$398$	5733.35	0.722078
$399$	469.845	0.0589516
$400$	388.040	0.0485050
$401$	−8022.24	−0.999031	−0.499516	−	0.866305i	$-0.666489\pi$
$401$	−8022.24	−0.999031	−0.499516	+	0.866305i	$0.666489\pi$
$402$	521.804	0.0647393
$403$	−634.413	−0.0784177
$404$	1962.31	0.241654
$405$	−3824.63	−0.469253
$406$	−1281.97	−0.156707
$407$	−11083.1	−1.34980
$408$	−11316.0	−1.37310
$409$	−2089.24	−0.252583	−0.126291	−	0.991993i	$-0.540307\pi$
$409$	−2089.24	−0.252583	−0.126291	+	0.991993i	$0.540307\pi$
$410$	−2935.68	−0.353617
$411$	28510.6	3.42172
$412$	1824.71	0.218196
$413$	−3579.74	−0.426507
$414$	1868.38	0.221801
$415$	1462.88	0.173036
$416$	−9944.82	−1.17208
$417$	9887.15	1.16109
$418$	−863.591	−0.101052
$419$	8022.73	0.935408	0.467704	−	0.883885i	$-0.345081\pi$
$419$	8022.73	0.935408	0.467704	+	0.883885i	$0.345081\pi$
$420$	1129.19	0.131188
$421$	5382.92	0.623153	0.311577	−	0.950221i	$-0.399143\pi$
$421$	5382.92	0.623153	0.311577	+	0.950221i	$0.399143\pi$
$422$	−1013.34	−0.116893
$423$	28908.3	3.32286
$424$	7677.63	0.879383
$425$	−1525.30	−0.174089
$426$	−14938.5	−1.69900
$427$	−2742.87	−0.310859
$428$	−611.030	−0.0690076
$429$	23053.6	2.59450
$430$	−1249.63	−0.140146
$431$	12908.1	1.44261	0.721303	−	0.692620i	$-0.243544\pi$
$431$	12908.1	1.44261	0.721303	+	0.692620i	$0.243544\pi$
$432$	3843.80	0.428090
$433$	4964.66	0.551008	0.275504	−	0.961300i	$-0.411155\pi$
$433$	4964.66	0.551008	0.275504	+	0.961300i	$0.411155\pi$
$434$	77.3923	0.00855979
$435$	−8943.86	−0.985805
$436$	10987.3	1.20687
$437$	−276.265	−0.0302416
$438$	6008.05	0.655424
$439$	6214.52	0.675633	0.337816	−	0.941212i	$-0.390312\pi$
$439$	6214.52	0.675633	0.337816	+	0.941212i	$0.390312\pi$
$440$	−4951.36	−0.536470
$441$	−17650.5	−1.90590
$442$	4826.63	0.519410
$443$	−11783.3	−1.26375	−0.631875	−	0.775071i	$-0.717714\pi$
$443$	−11783.3	−1.26375	−0.631875	+	0.775071i	$0.717714\pi$
$444$	11984.6	1.28100
$445$	1366.36	0.145554
$446$	−3263.78	−0.346513
$447$	20102.9	2.12715
$448$	674.952	0.0711796
$449$	13387.6	1.40713	0.703563	−	0.710633i	$-0.251591\pi$
$449$	13387.6	1.40713	0.703563	+	0.710633i	$0.251591\pi$
$450$	−2030.85	−0.212744
$451$	−18959.6	−1.97954
$452$	542.780	0.0564828
$453$	−8775.09	−0.910132
$454$	8650.92	0.894291
$455$	−1149.01	−0.118387
$456$	2227.80	0.228785
$457$	15194.2	1.55526	0.777632	−	0.628720i	$-0.216421\pi$
$457$	15194.2	1.55526	0.777632	+	0.628720i	$0.216421\pi$
$458$	10257.6	1.04652
$459$	−15109.1	−1.53646
$460$	−663.954	−0.0672979
$461$	6087.94	0.615063	0.307531	−	0.951538i	$-0.400497\pi$
$461$	6087.94	0.615063	0.307531	+	0.951538i	$0.400497\pi$
$462$	−2812.32	−0.283206
$463$	14645.2	1.47002	0.735012	−	0.678055i	$-0.237177\pi$
$463$	14645.2	1.47002	0.735012	+	0.678055i	$0.237177\pi$
$464$	3076.59	0.307817
$465$	539.940	0.0538476
$466$	6377.98	0.634022
$467$	2233.21	0.221286	0.110643	−	0.993860i	$-0.464709\pi$
$467$	2233.21	0.221286	0.110643	+	0.993860i	$0.464709\pi$
$468$	−16664.2	−1.64595
$469$	−167.961	−0.0165368
$470$	3961.65	0.388802
$471$	−25924.7	−2.53620
$472$	−16973.5	−1.65523
$473$	−8070.54	−0.784533
$474$	−8653.23	−0.838515
$475$	300.289	0.0290067
$476$	1526.83	0.147021
$477$	20337.6	1.95219
$478$	−3606.47	−0.345097
$479$	−152.988	−0.0145934	−0.00729668	−	0.999973i	$-0.502323\pi$
$479$	−152.988	−0.0145934	−0.00729668	+	0.999973i	$0.502323\pi$
$480$	8463.90	0.804839
$481$	−12194.9	−1.15601
$482$	−6335.71	−0.598722
$483$	−899.671	−0.0847546
$484$	−5719.61	−0.537153
$485$	4795.69	0.448992
$486$	−323.384	−0.0301831
$487$	8018.35	0.746090	0.373045	−	0.927813i	$-0.378314\pi$
$487$	8018.35	0.746090	0.373045	+	0.927813i	$0.378314\pi$
$488$	−13005.5	−1.20641
$489$	13158.1	1.21683
$490$	−2418.85	−0.223005
$491$	16353.2	1.50307	0.751536	−	0.659692i	$-0.229313\pi$
$491$	16353.2	1.50307	0.751536	+	0.659692i	$0.229313\pi$
$492$	20501.8	1.87864
$493$	−12093.4	−1.10479
$494$	−950.226	−0.0865439
$495$	−13115.9	−1.19094
$496$	−185.734	−0.0168139
$497$	4808.51	0.433986
$498$	3939.76	0.354507
$499$	−14615.1	−1.31115	−0.655573	−	0.755132i	$-0.727573\pi$
$499$	−14615.1	−1.31115	−0.655573	+	0.755132i	$0.727573\pi$
$500$	721.689	0.0645499
$501$	−4818.92	−0.429728
$502$	7185.34	0.638839
$503$	−5505.02	−0.487985	−0.243993	−	0.969777i	$-0.578457\pi$
$503$	−5505.02	−0.487985	−0.243993	+	0.969777i	$0.578457\pi$
$504$	4849.71	0.428618
$505$	1699.40	0.149747
$506$	1653.63	0.145282
$507$	5539.60	0.485251
$508$	−1652.18	−0.144299
$509$	−7765.56	−0.676233	−0.338116	−	0.941104i	$-0.609790\pi$
$509$	−7765.56	−0.676233	−0.338116	+	0.941104i	$0.609790\pi$
$510$	−4107.88	−0.356666
$511$	−1933.91	−0.167419
$512$	−5463.50	−0.471591
$513$	2974.56	0.256004
$514$	−7921.88	−0.679804
$515$	1580.24	0.135211
$516$	8727.01	0.744545
$517$	25585.6	2.17650
$518$	1487.67	0.126186
$519$	19199.5	1.62382
$520$	−5448.07	−0.459450
$521$	−9808.85	−0.824824	−0.412412	−	0.910997i	$-0.635314\pi$
$521$	−9808.85	−0.824824	−0.412412	+	0.910997i	$0.635314\pi$
$522$	−16101.6	−1.35009
$523$	5173.83	0.432573	0.216287	−	0.976330i	$-0.430605\pi$
$523$	5173.83	0.432573	0.216287	+	0.976330i	$0.430605\pi$
$524$	−7079.75	−0.590229
$525$	977.904	0.0812937
$526$	1559.37	0.129262
$527$	730.078	0.0603467
$528$	6749.29	0.556298
$529$	529.000	0.0434783
$530$	2787.10	0.228422
$531$	−44961.8	−3.67453
$532$	−300.589	−0.0244966
$533$	−20861.6	−1.69534
$534$	3679.82	0.298205
$535$	−529.167	−0.0427624
$536$	−796.398	−0.0641775
$537$	−25514.9	−2.05037
$538$	4813.79	0.385757
$539$	−15621.7	−1.24838
$540$	7148.82	0.569696
$541$	14529.6	1.15467	0.577336	−	0.816507i	$-0.304092\pi$
$541$	14529.6	1.15467	0.577336	+	0.816507i	$0.304092\pi$
$542$	−3783.12	−0.299814
$543$	2566.93	0.202868
$544$	11444.4	0.901978
$545$	9515.28	0.747871
$546$	−3094.45	−0.242547
$547$	−8519.93	−0.665971	−0.332985	−	0.942932i	$-0.608056\pi$
$547$	−8519.93	−0.665971	−0.332985	+	0.942932i	$0.608056\pi$
$548$	−18240.0	−1.42185
$549$	−34450.7	−2.67818
$550$	−1797.42	−0.139349
$551$	2380.85	0.184079
$552$	−4265.84	−0.328924
$553$	2785.36	0.214187
$554$	9629.20	0.738457
$555$	10379.0	0.793805
$556$	−6325.43	−0.482478
$557$	14706.3	1.11872	0.559360	−	0.828925i	$-0.311047\pi$
$557$	14706.3	1.11872	0.559360	+	0.828925i	$0.311047\pi$
$558$	972.054	0.0737461
$559$	−8880.17	−0.671899
$560$	−336.388	−0.0253839
$561$	−26530.0	−1.99661
$562$	11242.6	0.843843
$563$	3719.27	0.278416	0.139208	−	0.990263i	$-0.455544\pi$
$563$	3719.27	0.278416	0.139208	+	0.990263i	$0.455544\pi$
$564$	−27666.7	−2.06557
$565$	470.060	0.0350010
$566$	−6840.68	−0.508013
$567$	3315.53	0.245572
$568$	22799.8	1.68426
$569$	−16399.1	−1.20823	−0.604116	−	0.796896i	$-0.706474\pi$
$569$	−16399.1	−1.20823	−0.604116	+	0.796896i	$0.706474\pi$
$570$	808.724	0.0594276
$571$	2180.66	0.159821	0.0799105	−	0.996802i	$-0.474537\pi$
$571$	2180.66	0.159821	0.0799105	+	0.996802i	$0.474537\pi$
$572$	−14748.9	−1.07811
$573$	−24214.6	−1.76541
$574$	2544.91	0.185057
$575$	−575.000	−0.0417029
$576$	8477.46	0.613242
$577$	10558.4	0.761790	0.380895	−	0.924618i	$-0.375616\pi$
$577$	10558.4	0.761790	0.380895	+	0.924618i	$0.375616\pi$
$578$	1776.44	0.127838
$579$	19613.0	1.40775
$580$	5721.94	0.409639
$581$	−1268.15	−0.0905540
$582$	12915.6	0.919875
$583$	18000.0	1.27870
$584$	−9169.72	−0.649736
$585$	−14431.6	−1.01996
$586$	−10280.9	−0.724743
$587$	21251.7	1.49430	0.747149	−	0.664656i	$-0.231422\pi$
$587$	21251.7	1.49430	0.747149	+	0.664656i	$0.231422\pi$
$588$	16892.4	1.18475
$589$	−143.732	−0.0100549
$590$	−6161.65	−0.429951
$591$	41191.5	2.86699
$592$	−3570.25	−0.247865
$593$	−9358.61	−0.648081	−0.324041	−	0.946043i	$-0.605041\pi$
$593$	−9358.61	−0.648081	−0.324041	+	0.946043i	$0.605041\pi$
$594$	−17804.6	−1.22985
$595$	1322.27	0.0911054
$596$	−12861.1	−0.883910
$597$	−34675.3	−2.37716
$598$	1819.52	0.124424
$599$	2392.62	0.163205	0.0816025	−	0.996665i	$-0.473996\pi$
$599$	2392.62	0.163205	0.0816025	+	0.996665i	$0.473996\pi$
$600$	4636.78	0.315493
$601$	−13565.8	−0.920730	−0.460365	−	0.887730i	$-0.652281\pi$
$601$	−13565.8	−0.920730	−0.460365	+	0.887730i	$0.652281\pi$
$602$	1083.30	0.0733420
$603$	−2109.61	−0.142471
$604$	5613.97	0.378194
$605$	−4953.31	−0.332861
$606$	4576.76	0.306796
$607$	10454.9	0.699097	0.349548	−	0.936918i	$-0.386335\pi$
$607$	10454.9	0.699097	0.349548	+	0.936918i	$0.386335\pi$
$608$	−2253.08	−0.150287
$609$	7753.35	0.515897
$610$	−4721.18	−0.313369
$611$	28152.3	1.86403
$612$	19177.1	1.26665
$613$	17858.8	1.17669	0.588344	−	0.808611i	$-0.299780\pi$
$613$	17858.8	1.17669	0.588344	+	0.808611i	$0.299780\pi$
$614$	2180.33	0.143307
$615$	17755.0	1.16415
$616$	4292.28	0.280748
$617$	13120.3	0.856082	0.428041	−	0.903759i	$-0.359204\pi$
$617$	13120.3	0.856082	0.428041	+	0.903759i	$0.359204\pi$
$618$	4255.83	0.277014
$619$	−7081.28	−0.459807	−0.229904	−	0.973213i	$-0.573841\pi$
$619$	−7081.28	−0.459807	−0.229904	+	0.973213i	$0.573841\pi$
$620$	−345.433	−0.0223757
$621$	−5695.76	−0.368056
$622$	1119.54	0.0721697
$623$	−1184.48	−0.0761723
$624$	7426.38	0.476431
$625$	625.000	0.0400000
$626$	−12856.3	−0.820830
$627$	5223.00	0.332674
$628$	16585.7	1.05389
$629$	14033.9	0.889613
$630$	1760.52	0.111335
$631$	−11411.8	−0.719961	−0.359981	−	0.932960i	$-0.617217\pi$
$631$	−11411.8	−0.719961	−0.359981	+	0.932960i	$0.617217\pi$
$632$	13206.9	0.831238
$633$	6128.69	0.384824
$634$	−531.757	−0.0333104
$635$	−1430.83	−0.0894183
$636$	−19464.1	−1.21353
$637$	−17188.9	−1.06915
$638$	−14250.9	−0.884324
$639$	60395.3	3.73897
$640$	−6341.30	−0.391659
$641$	2755.34	0.169781	0.0848903	−	0.996390i	$-0.472946\pi$
$641$	2755.34	0.169781	0.0848903	+	0.996390i	$0.472946\pi$
$642$	−1425.13	−0.0876096
$643$	−23204.2	−1.42315	−0.711574	−	0.702611i	$-0.752017\pi$
$643$	−23204.2	−1.42315	−0.711574	+	0.702611i	$0.752017\pi$
$644$	575.576	0.0352187
$645$	7557.80	0.461377
$646$	1093.51	0.0666002
$647$	30680.4	1.86425	0.932127	−	0.362133i	$-0.117951\pi$
$647$	30680.4	1.86425	0.932127	+	0.362133i	$0.117951\pi$
$648$	15720.8	0.953040
$649$	−39793.9	−2.40685
$650$	−1977.73	−0.119343
$651$	−468.069	−0.0281798
$652$	−8418.04	−0.505638
$653$	−11329.3	−0.678942	−0.339471	−	0.940617i	$-0.610248\pi$
$653$	−11329.3	−0.678942	−0.339471	+	0.940617i	$0.610248\pi$
$654$	25626.1	1.53220
$655$	−6131.23	−0.365751
$656$	−6107.54	−0.363505
$657$	−24290.1	−1.44238
$658$	−3434.31	−0.203470
$659$	13794.3	0.815400	0.407700	−	0.913116i	$-0.366331\pi$
$659$	13794.3	0.815400	0.407700	+	0.913116i	$0.366331\pi$
$660$	12552.5	0.740314
$661$	28141.0	1.65591	0.827955	−	0.560795i	$-0.189504\pi$
$661$	28141.0	1.65591	0.827955	+	0.560795i	$0.189504\pi$
$662$	−11026.9	−0.647390
$663$	−29191.5	−1.70996
$664$	−6013.01	−0.351431
$665$	−260.317	−0.0151800
$666$	18685.2	1.08714
$667$	−4558.91	−0.264650
$668$	3082.96	0.178568
$669$	19739.4	1.14076
$670$	−289.105	−0.0166703
$671$	−30490.9	−1.75423
$672$	−7337.27	−0.421193
$673$	933.716	0.0534801	0.0267400	−	0.999642i	$-0.491487\pi$
$673$	933.716	0.0534801	0.0267400	+	0.999642i	$0.491487\pi$
$674$	−15838.7	−0.905171
$675$	6191.04	0.353027
$676$	−3544.03	−0.201640
$677$	−10766.7	−0.611220	−0.305610	−	0.952157i	$-0.598860\pi$
$677$	−10766.7	−0.611220	−0.305610	+	0.952157i	$0.598860\pi$
$678$	1265.95	0.0717085
$679$	−4157.34	−0.234969
$680$	6269.61	0.353571
$681$	−52320.8	−2.94411
$682$	860.326	0.0483044
$683$	−6057.52	−0.339363	−0.169681	−	0.985499i	$-0.554274\pi$
$683$	−6057.52	−0.339363	−0.169681	+	0.985499i	$0.554274\pi$
$684$	−3775.43	−0.211048
$685$	−15796.3	−0.881088
$686$	4315.27	0.240172
$687$	−62038.1	−3.44527
$688$	−2599.80	−0.144065
$689$	19805.7	1.09512
$690$	−1548.57	−0.0854389
$691$	17012.4	0.936586	0.468293	−	0.883573i	$-0.344869\pi$
$691$	17012.4	0.936586	0.468293	+	0.883573i	$0.344869\pi$
$692$	−12283.1	−0.674760
$693$	11370.0	0.623248
$694$	5181.85	0.283430
$695$	−5477.97	−0.298980
$696$	36762.9	2.00215
$697$	24007.4	1.30466
$698$	−7192.83	−0.390047
$699$	−38574.0	−2.08727
$700$	−625.626	−0.0337806
$701$	11722.7	0.631613	0.315806	−	0.948824i	$-0.397725\pi$
$701$	11722.7	0.631613	0.315806	+	0.948824i	$0.397725\pi$
$702$	−19590.8	−1.05329
$703$	−2762.87	−0.148227
$704$	7503.05	0.401679
$705$	−23960.0	−1.27998
$706$	−4560.79	−0.243127
$707$	−1473.20	−0.0783667
$708$	43030.8	2.28417
$709$	−6613.52	−0.350319	−0.175159	−	0.984540i	$-0.556044\pi$
$709$	−6613.52	−0.350319	−0.175159	+	0.984540i	$0.556044\pi$
$710$	8276.67	0.437490
$711$	34984.3	1.84531
$712$	−5616.29	−0.295617
$713$	275.221	0.0144560
$714$	3561.08	0.186653
$715$	−12772.8	−0.668080
$716$	16323.5	0.852006
$717$	21812.0	1.13610
$718$	16047.4	0.834102
$719$	4633.47	0.240333	0.120166	−	0.992754i	$-0.461657\pi$
$719$	4633.47	0.240333	0.120166	+	0.992754i	$0.461657\pi$
$720$	−4225.07	−0.218693
$721$	−1369.89	−0.0707594
$722$	10019.3	0.516455
$723$	38318.4	1.97106
$724$	−1642.23	−0.0842995
$725$	4955.34	0.253844
$726$	−13340.1	−0.681950
$727$	−29659.7	−1.51309	−0.756547	−	0.653939i	$-0.773115\pi$
$727$	−29659.7	−1.51309	−0.756547	+	0.653939i	$0.773115\pi$
$728$	4722.88	0.240442
$729$	−18697.2	−0.949914
$730$	−3328.75	−0.168771
$731$	10219.2	0.517062
$732$	32971.1	1.66482
$733$	7552.02	0.380546	0.190273	−	0.981731i	$-0.439063\pi$
$733$	7552.02	0.380546	0.190273	+	0.981731i	$0.439063\pi$
$734$	16266.9	0.818013
$735$	14629.2	0.734160
$736$	4314.26	0.216068
$737$	−1867.13	−0.0933198
$738$	31964.4	1.59434
$739$	27931.7	1.39037	0.695187	−	0.718829i	$-0.255322\pi$
$739$	27931.7	1.39037	0.695187	+	0.718829i	$0.255322\pi$
$740$	−6640.06	−0.329856
$741$	5746.97	0.284913
$742$	−2416.11	−0.119539
$743$	−32808.3	−1.61995	−0.809974	−	0.586466i	$-0.800519\pi$
$743$	−32808.3	−1.61995	−0.809974	+	0.586466i	$0.800519\pi$
$744$	−2219.37	−0.109363
$745$	−11138.0	−0.547738
$746$	−146.266	−0.00717852
$747$	−15928.1	−0.780160
$748$	16972.9	0.829666
$749$	458.730	0.0223787
$750$	1683.22	0.0819502
$751$	1086.34	0.0527842	0.0263921	−	0.999652i	$-0.491598\pi$
$751$	1086.34	0.0527842	0.0263921	+	0.999652i	$0.491598\pi$
$752$	8242.00	0.399674
$753$	−43456.9	−2.10313
$754$	−15680.5	−0.757363
$755$	4861.83	0.234358
$756$	−6197.24	−0.298137
$757$	−2008.83	−0.0964492	−0.0482246	−	0.998837i	$-0.515356\pi$
$757$	−2008.83	−0.0964492	−0.0482246	+	0.998837i	$0.515356\pi$
$758$	13159.5	0.630573
$759$	−10001.1	−0.478285
$760$	−1234.31	−0.0589119
$761$	37676.6	1.79471	0.897356	−	0.441308i	$-0.145485\pi$
$761$	37676.6	1.79471	0.897356	+	0.441308i	$0.145485\pi$
$762$	−3853.44	−0.183196
$763$	−8248.71	−0.391380
$764$	15491.6	0.733594
$765$	16607.8	0.784911
$766$	−11374.0	−0.536501
$767$	−43786.0	−2.06130
$768$	−28320.3	−1.33063
$769$	−20834.3	−0.976988	−0.488494	−	0.872567i	$-0.662454\pi$
$769$	−20834.3	−0.976988	−0.488494	+	0.872567i	$0.662454\pi$
$770$	1558.17	0.0729252
$771$	47911.5	2.23799
$772$	−12547.7	−0.584975
$773$	25149.2	1.17019	0.585094	−	0.810965i	$-0.301058\pi$
$773$	25149.2	1.17019	0.585094	+	0.810965i	$0.301058\pi$
$774$	13606.3	0.631871
$775$	−299.153	−0.0138657
$776$	−19712.2	−0.911892
$777$	−8997.41	−0.415419
$778$	738.869	0.0340485
$779$	−4726.37	−0.217381
$780$	13811.8	0.634028
$781$	53453.4	2.44906
$782$	−2093.89	−0.0957509
$783$	49085.9	2.24034
$784$	−5032.30	−0.229241
$785$	14363.6	0.653067
$786$	−16512.4	−0.749333
$787$	30570.6	1.38465	0.692327	−	0.721584i	$-0.256585\pi$
$787$	30570.6	1.38465	0.692327	+	0.721584i	$0.256585\pi$
$788$	−26352.8	−1.19134
$789$	−9431.06	−0.425544
$790$	4794.31	0.215916
$791$	−407.491	−0.0183169
$792$	53911.5	2.41876
$793$	−33549.7	−1.50238
$794$	12698.5	0.567573
$795$	−16856.4	−0.751993
$796$	22184.0	0.987801
$797$	1413.52	0.0628225	0.0314112	−	0.999507i	$-0.490000\pi$
$797$	1413.52	0.0628225	0.0314112	+	0.999507i	$0.490000\pi$
$798$	−701.075	−0.0311000
$799$	−32397.5	−1.43447
$800$	−4689.42	−0.207245
$801$	−14877.2	−0.656256
$802$	11970.3	0.527040
$803$	−21498.2	−0.944774
$804$	2019.01	0.0885633
$805$	498.462	0.0218242
$806$	946.633	0.0413694
$807$	−29113.8	−1.26996
$808$	−6985.24	−0.304133
$809$	−13022.6	−0.565946	−0.282973	−	0.959128i	$-0.591321\pi$
$809$	−13022.6	−0.565946	−0.282973	+	0.959128i	$0.591321\pi$
$810$	5706.88	0.247555
$811$	6693.81	0.289829	0.144915	−	0.989444i	$-0.453709\pi$
$811$	6693.81	0.289829	0.144915	+	0.989444i	$0.453709\pi$
$812$	−4960.30	−0.214375
$813$	22880.3	0.987021
$814$	16537.5	0.712089
$815$	−7290.22	−0.313332
$816$	−8546.22	−0.366639
$817$	−2011.88	−0.0861527
$818$	3117.44	0.133250
$819$	12510.6	0.533769
$820$	−11359.0	−0.483748
$821$	33631.6	1.42966	0.714830	−	0.699299i	$-0.246504\pi$
$821$	33631.6	1.42966	0.714830	+	0.699299i	$0.246504\pi$
$822$	−42541.9	−1.80513
$823$	6220.84	0.263481	0.131740	−	0.991284i	$-0.457943\pi$
$823$	6220.84	0.263481	0.131740	+	0.991284i	$0.457943\pi$
$824$	−6495.42	−0.274610
$825$	10870.8	0.458755
$826$	5341.47	0.225004
$827$	−794.732	−0.0334166	−0.0167083	−	0.999860i	$-0.505319\pi$
$827$	−794.732	−0.0334166	−0.0167083	+	0.999860i	$0.505319\pi$
$828$	7229.28	0.303424
$829$	−25334.6	−1.06141	−0.530703	−	0.847558i	$-0.678072\pi$
$829$	−25334.6	−1.06141	−0.530703	+	0.847558i	$0.678072\pi$
$830$	−2182.82	−0.0912852
$831$	−58237.4	−2.43109
$832$	8255.75	0.344010
$833$	19780.9	0.822769
$834$	−14753.0	−0.612537
$835$	2669.92	0.110654
$836$	−3341.48	−0.138239
$837$	−2963.31	−0.122374
$838$	−11971.0	−0.493476
$839$	5311.49	0.218561	0.109281	−	0.994011i	$-0.465145\pi$
$839$	5311.49	0.218561	0.109281	+	0.994011i	$0.465145\pi$
$840$	−4019.58	−0.165106
$841$	14899.6	0.610913
$842$	−8032.07	−0.328745
$843$	−67995.2	−2.77803
$844$	−3920.90	−0.159909
$845$	−3069.21	−0.124952
$846$	−43135.3	−1.75298
$847$	4293.98	0.174195
$848$	5798.41	0.234809
$849$	41372.5	1.67244
$850$	2275.96	0.0918411
$851$	5290.41	0.213106
$852$	−57801.4	−2.32423
$853$	12483.5	0.501086	0.250543	−	0.968105i	$-0.419391\pi$
$853$	12483.5	0.501086	0.250543	+	0.968105i	$0.419391\pi$
$854$	4092.75	0.163994
$855$	−3269.61	−0.130782
$856$	2175.09	0.0868493
$857$	−27914.8	−1.11266	−0.556331	−	0.830961i	$-0.687791\pi$
$857$	−27914.8	−1.11266	−0.556331	+	0.830961i	$0.687791\pi$
$858$	−34399.3	−1.36873
$859$	−25257.2	−1.00322	−0.501610	−	0.865094i	$-0.667259\pi$
$859$	−25257.2	−1.00322	−0.501610	+	0.865094i	$0.667259\pi$
$860$	−4835.19	−0.191719
$861$	−15391.7	−0.609229
$862$	−19260.8	−0.761049
$863$	−23774.9	−0.937784	−0.468892	−	0.883255i	$-0.655347\pi$
$863$	−23774.9	−0.937784	−0.468892	+	0.883255i	$0.655347\pi$
$864$	−46451.8	−1.82908
$865$	−10637.5	−0.418133
$866$	−7407.97	−0.290685
$867$	−10743.9	−0.420856
$868$	299.453	0.0117098
$869$	30963.2	1.20869
$870$	13345.5	0.520063
$871$	−2054.44	−0.0799220
$872$	−39111.7	−1.51891
$873$	−52216.6	−2.02436
$874$	412.227	0.0159540
$875$	−541.806	−0.0209330
$876$	23246.8	0.896619
$877$	11398.5	0.438883	0.219441	−	0.975626i	$-0.429577\pi$
$877$	11398.5	0.438883	0.219441	+	0.975626i	$0.429577\pi$
$878$	−9272.94	−0.356431
$879$	62178.9	2.38594
$880$	−3739.44	−0.143246
$881$	28842.1	1.10297	0.551484	−	0.834186i	$-0.314062\pi$
$881$	28842.1	1.10297	0.551484	+	0.834186i	$0.314062\pi$
$882$	26337.0	1.00546
$883$	−26840.4	−1.02293	−0.511467	−	0.859303i	$-0.670898\pi$
$883$	−26840.4	−1.02293	−0.511467	+	0.859303i	$0.670898\pi$
$884$	18675.6	0.710552
$885$	37265.7	1.41545
$886$	17582.3	0.666693
$887$	9384.94	0.355260	0.177630	−	0.984097i	$-0.443157\pi$
$887$	9384.94	0.355260	0.177630	+	0.984097i	$0.443157\pi$
$888$	−42661.7	−1.61220
$889$	1240.37	0.0467949
$890$	−2038.80	−0.0767874
$891$	36856.9	1.38581
$892$	−12628.5	−0.474029
$893$	6378.15	0.239011
$894$	−29996.4	−1.12218
$895$	14136.5	0.527968
$896$	5497.21	0.204965
$897$	−11004.4	−0.409618
$898$	−19976.2	−0.742332
$899$	−2371.85	−0.0879928
$900$	−7857.92	−0.291034
$901$	−22792.3	−0.842754
$902$	28290.4	1.04431
$903$	−6551.78	−0.241450
$904$	−1932.14	−0.0710863
$905$	−1422.21	−0.0522384
$906$	13093.7	0.480141
$907$	36100.4	1.32160	0.660802	−	0.750560i	$-0.270216\pi$
$907$	36100.4	1.32160	0.660802	+	0.750560i	$0.270216\pi$
$908$	33472.9	1.22339
$909$	−18503.5	−0.675162
$910$	1714.48	0.0624554
$911$	−21644.8	−0.787183	−0.393591	−	0.919285i	$-0.628768\pi$
$911$	−21644.8	−0.787183	−0.393591	+	0.919285i	$0.628768\pi$
$912$	1682.51	0.0610893
$913$	−14097.3	−0.511012
$914$	−22671.9	−0.820482
$915$	28553.7	1.03165
$916$	39689.6	1.43164
$917$	5315.10	0.191407
$918$	22544.9	0.810560
$919$	−15016.8	−0.539018	−0.269509	−	0.962998i	$-0.586861\pi$
$919$	−15016.8	−0.539018	−0.269509	+	0.962998i	$0.586861\pi$
$920$	2363.48	0.0846975
$921$	−13186.6	−0.471785
$922$	−9084.07	−0.324477
$923$	58815.8	2.09745
$924$	−10881.7	−0.387425
$925$	−5750.45	−0.204404
$926$	−21852.7	−0.775513
$927$	−17206.0	−0.609621
$928$	−37180.2	−1.31519
$929$	7097.82	0.250670	0.125335	−	0.992115i	$-0.460000\pi$
$929$	7097.82	0.250670	0.125335	+	0.992115i	$0.460000\pi$
$930$	−805.667	−0.0284074
$931$	−3894.29	−0.137089
$932$	24678.2	0.867341
$933$	−6771.00	−0.237591
$934$	−3332.26	−0.116740
$935$	14698.9	0.514124
$936$	59319.8	2.07151
$937$	8007.69	0.279189	0.139594	−	0.990209i	$-0.455420\pi$
$937$	8007.69	0.279189	0.139594	+	0.990209i	$0.455420\pi$
$938$	250.622	0.00872399
$939$	77754.7	2.70227
$940$	15328.7	0.531881
$941$	12382.9	0.428980	0.214490	−	0.976726i	$-0.431191\pi$
$941$	12382.9	0.428980	0.214490	+	0.976726i	$0.431191\pi$
$942$	38683.4	1.33797
$943$	9050.18	0.312528
$944$	−12819.0	−0.441973
$945$	−5366.95	−0.184748
$946$	12042.4	0.413881
$947$	−14760.8	−0.506506	−0.253253	−	0.967400i	$-0.581501\pi$
$947$	−14760.8	−0.506506	−0.253253	+	0.967400i	$0.581501\pi$
$948$	−33481.8	−1.14709
$949$	−23654.8	−0.809134
$950$	−448.073	−0.0153025
$951$	3216.07	0.109662
$952$	−5435.06	−0.185033
$953$	37251.9	1.26622	0.633110	−	0.774062i	$-0.281778\pi$
$953$	37251.9	1.26622	0.633110	+	0.774062i	$0.281778\pi$
$954$	−30346.6	−1.02988
$955$	13416.1	0.454591
$956$	−13954.5	−0.472092
$957$	86189.5	2.91130
$958$	228.280	0.00769875
$959$	13693.7	0.461096
$960$	−7026.36	−0.236224
$961$	−29647.8	−0.995194
$962$	18196.6	0.609856
$963$	5761.69	0.192801
$964$	−24514.7	−0.819050
$965$	−10866.6	−0.362495
$966$	1342.44	0.0447124
$967$	13383.4	0.445069	0.222534	−	0.974925i	$-0.428567\pi$
$967$	13383.4	0.445069	0.222534	+	0.974925i	$0.428567\pi$
$968$	20360.1	0.676032
$969$	−6613.57	−0.219256
$970$	−7155.85	−0.236867
$971$	−19630.5	−0.648788	−0.324394	−	0.945922i	$-0.605160\pi$
$971$	−19630.5	−0.648788	−0.324394	+	0.945922i	$0.605160\pi$
$972$	−1251.26	−0.0412905
$973$	4748.80	0.156464
$974$	−11964.5	−0.393601
$975$	11961.3	0.392892
$976$	−9822.18	−0.322131
$977$	34825.7	1.14040	0.570200	−	0.821506i	$-0.306866\pi$
$977$	34825.7	1.14040	0.570200	+	0.821506i	$0.306866\pi$
$978$	−19633.7	−0.641939
$979$	−13167.2	−0.429854
$980$	−9359.23	−0.305071
$981$	−103605.	−3.37190
$982$	−24401.2	−0.792948
$983$	39998.8	1.29783	0.648913	−	0.760862i	$-0.275224\pi$
$983$	39998.8	1.29783	0.648913	+	0.760862i	$0.275224\pi$
$984$	−72980.3	−2.36436
$985$	−22822.1	−0.738247
$986$	18045.1	0.582832
$987$	20770.7	0.669848
$988$	−3676.69	−0.118392
$989$	3852.40	0.123862
$990$	19570.7	0.628281
$991$	−22462.6	−0.720029	−0.360015	−	0.932947i	$-0.617228\pi$
$991$	−22462.6	−0.720029	−0.360015	+	0.932947i	$0.617228\pi$
$992$	2244.57	0.0718398
$993$	66690.7	2.13128
$994$	−7174.97	−0.228950
$995$	19211.8	0.612116
$996$	15244.0	0.484965
$997$	7603.55	0.241532	0.120766	−	0.992681i	$-0.461465\pi$
$997$	7603.55	0.241532	0.120766	+	0.992681i	$0.461465\pi$
$998$	21807.8	0.691697
$999$	−56962.0	−1.80400

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	115.4.a.e.1.2	✓	5
3.2	odd	2		1035.4.a.k.1.4		5
4.3	odd	2		1840.4.a.n.1.1		5
5.2	odd	4		575.4.b.i.24.4		10
5.3	odd	4		575.4.b.i.24.7		10
5.4	even	2		575.4.a.j.1.4		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
115.4.a.e.1.2	✓	5	1.1	even	1	trivial
575.4.a.j.1.4		5	5.4	even	2
575.4.b.i.24.4		10	5.2	odd	4
575.4.b.i.24.7		10	5.3	odd	4
1035.4.a.k.1.4		5	3.2	odd	2
1840.4.a.n.1.1		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)

\(f(q)\)	\(=\)	\(q-1.49214 q^{2} +9.02447 q^{3} -5.77352 q^{4} -5.00000 q^{5} -13.4658 q^{6} +4.33445 q^{7} +20.5520 q^{8} +54.4411 q^{9} +O(q^{10})\)	\(q-1.49214 q^{2} +9.02447 q^{3} -5.77352 q^{4} -5.00000 q^{5} -13.4658 q^{6} +4.33445 q^{7} +20.5520 q^{8} +54.4411 q^{9} +7.46070 q^{10} +48.1836 q^{11} -52.1029 q^{12} +53.0174 q^{13} -6.46761 q^{14} -45.1224 q^{15} +15.5216 q^{16} -61.0120 q^{17} -81.2338 q^{18} +12.0115 q^{19} +28.8676 q^{20} +39.1161 q^{21} -71.8968 q^{22} -23.0000 q^{23} +185.471 q^{24} +25.0000 q^{25} -79.1094 q^{26} +247.642 q^{27} -25.0250 q^{28} +198.213 q^{29} +67.3289 q^{30} -11.9661 q^{31} -187.577 q^{32} +434.832 q^{33} +91.0386 q^{34} -21.6723 q^{35} -314.317 q^{36} -230.018 q^{37} -17.9229 q^{38} +478.454 q^{39} -102.760 q^{40} -393.486 q^{41} -58.3668 q^{42} -167.496 q^{43} -278.189 q^{44} -272.206 q^{45} +34.3192 q^{46} +531.002 q^{47} +140.074 q^{48} -324.213 q^{49} -37.3035 q^{50} -550.602 q^{51} -306.097 q^{52} +373.570 q^{53} -369.516 q^{54} -240.918 q^{55} +89.0818 q^{56} +108.398 q^{57} -295.762 q^{58} -825.880 q^{59} +260.515 q^{60} -632.806 q^{61} +17.8551 q^{62} +235.972 q^{63} +155.718 q^{64} -265.087 q^{65} -648.830 q^{66} -38.7503 q^{67} +352.254 q^{68} -207.563 q^{69} +32.3381 q^{70} +1109.37 q^{71} +1118.88 q^{72} -446.171 q^{73} +343.219 q^{74} +225.612 q^{75} -69.3488 q^{76} +208.850 q^{77} -713.921 q^{78} +642.608 q^{79} -77.6081 q^{80} +764.926 q^{81} +587.137 q^{82} -292.575 q^{83} -225.838 q^{84} +305.060 q^{85} +249.927 q^{86} +1788.77 q^{87} +990.271 q^{88} -273.272 q^{89} +406.169 q^{90} +229.801 q^{91} +132.791 q^{92} -107.988 q^{93} -792.329 q^{94} -60.0577 q^{95} -1692.78 q^{96} -959.139 q^{97} +483.771 q^{98} +2623.17 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 6 q^{2} + 4 q^{3} + 22 q^{4} - 25 q^{5} + 19 q^{6} - 3 q^{7} + 138 q^{8} + 77 q^{9}+O(q^{10}) \) 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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