LMFDB - Embedded newform 1045.4.a.b.1.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1045,4,Mod(1,1045)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1045, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("1045.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1045 = 5 \cdot 11 \cdot 19 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1045.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:	$61.6569959560$
Analytic rank:	$1$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - 8 x^{19} - 82 x^{18} + 700 x^{17} + 2826 x^{16} - 25467 x^{15} - 53768 x^{14} + 498499 x^{13} + \cdots + 17756160 $ x^20 - 8x^19 - 82x^18 + 700x^17 + 2826x^16 - 25467x^15 - 53768x^14 + 498499x^13 + 623613x^12 - 5673747x^11 - 4539454x^10 + 37893109x^9 + 19879768x^8 - 143049638x^7 - 45064360x^6 + 280461480x^5 + 43097920x^4 - 240447168x^3 - 36068096x^2 + 74703872*x + 17756160
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{3}\cdot 7 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.9
Root			$-0.731640$ of defining polynomial
Character	$\chi$	$=$	1045.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.73164 q^{2} -4.39818 q^{3} -5.00142 q^{4} +5.00000 q^{5} +7.61606 q^{6} +22.5500 q^{7} +22.5138 q^{8} -7.65602 q^{9} +O(q^{10})$	\(q-1.73164 q^{2} -4.39818 q^{3} -5.00142 q^{4} +5.00000 q^{5} +7.61606 q^{6} +22.5500 q^{7} +22.5138 q^{8} -7.65602 q^{9} -8.65820 q^{10} +11.0000 q^{11} +21.9972 q^{12} -6.13662 q^{13} -39.0485 q^{14} -21.9909 q^{15} +1.02564 q^{16} -101.241 q^{17} +13.2575 q^{18} +19.0000 q^{19} -25.0071 q^{20} -99.1791 q^{21} -19.0480 q^{22} -53.5896 q^{23} -99.0196 q^{24} +25.0000 q^{25} +10.6264 q^{26} +152.423 q^{27} -112.782 q^{28} -259.880 q^{29} +38.0803 q^{30} +180.380 q^{31} -181.886 q^{32} -48.3800 q^{33} +175.312 q^{34} +112.750 q^{35} +38.2910 q^{36} +351.353 q^{37} -32.9012 q^{38} +26.9899 q^{39} +112.569 q^{40} -425.891 q^{41} +171.742 q^{42} +264.640 q^{43} -55.0157 q^{44} -38.2801 q^{45} +92.7979 q^{46} +294.823 q^{47} -4.51096 q^{48} +165.504 q^{49} -43.2910 q^{50} +445.275 q^{51} +30.6918 q^{52} -187.648 q^{53} -263.942 q^{54} +55.0000 q^{55} +507.686 q^{56} -83.5654 q^{57} +450.019 q^{58} +428.689 q^{59} +109.986 q^{60} -197.962 q^{61} -312.353 q^{62} -172.643 q^{63} +306.756 q^{64} -30.6831 q^{65} +83.7767 q^{66} +675.426 q^{67} +506.348 q^{68} +235.697 q^{69} -195.243 q^{70} -163.454 q^{71} -172.366 q^{72} -595.017 q^{73} -608.417 q^{74} -109.954 q^{75} -95.0271 q^{76} +248.050 q^{77} -46.7369 q^{78} +339.195 q^{79} +5.12821 q^{80} -463.673 q^{81} +737.490 q^{82} +1008.70 q^{83} +496.037 q^{84} -506.204 q^{85} -458.261 q^{86} +1143.00 q^{87} +247.652 q^{88} -641.390 q^{89} +66.2873 q^{90} -138.381 q^{91} +268.024 q^{92} -793.344 q^{93} -510.526 q^{94} +95.0000 q^{95} +799.968 q^{96} -640.865 q^{97} -286.593 q^{98} -84.2162 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q - 12 q^{2} - 21 q^{3} + 72 q^{4} + 100 q^{5} - 45 q^{6} - 131 q^{7} - 108 q^{8} + 159 q^{9}+O(q^{10}) $ 20 * q - 12 * q^2 - 21 * q^3 + 72 * q^4 + 100 * q^5 - 45 * q^6 - 131 * q^7 - 108 * q^8 + 159 * q^9	\( 20 q - 12 q^{2} - 21 q^{3} + 72 q^{4} + 100 q^{5} - 45 q^{6} - 131 q^{7} - 108 q^{8} + 159 q^{9} - 60 q^{10} + 220 q^{11} - 196 q^{12} - 223 q^{13} - 11 q^{14} - 105 q^{15} + 380 q^{16} - 471 q^{17} + 113 q^{18} + 380 q^{19} + 360 q^{20} - 57 q^{21} - 132 q^{22} - 653 q^{23} - 486 q^{24} + 500 q^{25} - 145 q^{26} - 177 q^{27} - 747 q^{28} + 51 q^{29} - 225 q^{30} - 90 q^{31} - 381 q^{32} - 231 q^{33} + 517 q^{34} - 655 q^{35} + 875 q^{36} - 96 q^{37} - 228 q^{38} + 97 q^{39} - 540 q^{40} - 1284 q^{41} - 638 q^{42} - 1592 q^{43} + 792 q^{44} + 795 q^{45} - 35 q^{46} - 2030 q^{47} - 1471 q^{48} + 765 q^{49} - 300 q^{50} - 185 q^{51} - 3242 q^{52} - 943 q^{53} - 730 q^{54} + 1100 q^{55} + 887 q^{56} - 399 q^{57} - 492 q^{58} - 515 q^{59} - 980 q^{60} + 446 q^{61} - 770 q^{62} - 3980 q^{63} + 2526 q^{64} - 1115 q^{65} - 495 q^{66} - 1719 q^{67} - 1808 q^{68} + 317 q^{69} - 55 q^{70} - 90 q^{71} + 1058 q^{72} - 3763 q^{73} - 1313 q^{74} - 525 q^{75} + 1368 q^{76} - 1441 q^{77} + 4286 q^{78} + 2 q^{79} + 1900 q^{80} + 308 q^{81} - 3830 q^{82} - 3436 q^{83} + 5734 q^{84} - 2355 q^{85} + 1922 q^{86} - 2719 q^{87} - 1188 q^{88} + 1700 q^{89} + 565 q^{90} + 2007 q^{91} - 3980 q^{92} - 2276 q^{93} + 2700 q^{94} + 1900 q^{95} - 6252 q^{96} - 1956 q^{97} - 2356 q^{98} + 1749 q^{99}+O(q^{100}) \) 20 * q - 12 * q^2 - 21 * q^3 + 72 * q^4 + 100 * q^5 - 45 * q^6 - 131 * q^7 - 108 * q^8 + 159 * q^9 - 60 * q^10 + 220 * q^11 - 196 * q^12 - 223 * q^13 - 11 * q^14 - 105 * q^15 + 380 * q^16 - 471 * q^17 + 113 * q^18 + 380 * q^19 + 360 * q^20 - 57 * q^21 - 132 * q^22 - 653 * q^23 - 486 * q^24 + 500 * q^25 - 145 * q^26 - 177 * q^27 - 747 * q^28 + 51 * q^29 - 225 * q^30 - 90 * q^31 - 381 * q^32 - 231 * q^33 + 517 * q^34 - 655 * q^35 + 875 * q^36 - 96 * q^37 - 228 * q^38 + 97 * q^39 - 540 * q^40 - 1284 * q^41 - 638 * q^42 - 1592 * q^43 + 792 * q^44 + 795 * q^45 - 35 * q^46 - 2030 * q^47 - 1471 * q^48 + 765 * q^49 - 300 * q^50 - 185 * q^51 - 3242 * q^52 - 943 * q^53 - 730 * q^54 + 1100 * q^55 + 887 * q^56 - 399 * q^57 - 492 * q^58 - 515 * q^59 - 980 * q^60 + 446 * q^61 - 770 * q^62 - 3980 * q^63 + 2526 * q^64 - 1115 * q^65 - 495 * q^66 - 1719 * q^67 - 1808 * q^68 + 317 * q^69 - 55 * q^70 - 90 * q^71 + 1058 * q^72 - 3763 * q^73 - 1313 * q^74 - 525 * q^75 + 1368 * q^76 - 1441 * q^77 + 4286 * q^78 + 2 * q^79 + 1900 * q^80 + 308 * q^81 - 3830 * q^82 - 3436 * q^83 + 5734 * q^84 - 2355 * q^85 + 1922 * q^86 - 2719 * q^87 - 1188 * q^88 + 1700 * q^89 + 565 * q^90 + 2007 * q^91 - 3980 * q^92 - 2276 * q^93 + 2700 * q^94 + 1900 * q^95 - 6252 * q^96 - 1956 * q^97 - 2356 * q^98 + 1749 * 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 $

$2$	−1.73164	−0.612227	−0.306114	−	0.951995i	$-0.599029\pi$
$2$	−1.73164	−0.612227	−0.306114	+	0.951995i	$0.599029\pi$
$3$	−4.39818	−0.846430	−0.423215	−	0.906029i	$-0.639098\pi$
$3$	−4.39818	−0.846430	−0.423215	+	0.906029i	$0.639098\pi$
$4$	−5.00142	−0.625178
$5$	5.00000	0.447214
$5$	5.00000	0.447214
$6$	7.61606	0.518207
$7$	22.5500	1.21759	0.608794	−	0.793329i	$-0.291654\pi$
$7$	22.5500	1.21759	0.608794	+	0.793329i	$0.291654\pi$
$8$	22.5138	0.994978
$9$	−7.65602	−0.283556
$10$	−8.65820	−0.273796
$11$	11.0000	0.301511
$11$	11.0000	0.301511
$12$	21.9972	0.529169
$13$	−6.13662	−0.130922	−0.0654612	−	0.997855i	$-0.520852\pi$
$13$	−6.13662	−0.130922	−0.0654612	+	0.997855i	$0.520852\pi$
$14$	−39.0485	−0.745440
$15$	−21.9909	−0.378535
$16$	1.02564	0.0160257
$17$	−101.241	−1.44438	−0.722191	−	0.691694i	$-0.756865\pi$
$17$	−101.241	−1.44438	−0.722191	+	0.691694i	$0.756865\pi$
$18$	13.2575	0.173601
$19$	19.0000	0.229416
$19$	19.0000	0.229416
$20$	−25.0071	−0.279588
$21$	−99.1791	−1.03060
$22$	−19.0480	−0.184593
$23$	−53.5896	−0.485835	−0.242917	−	0.970047i	$-0.578104\pi$
$23$	−53.5896	−0.485835	−0.242917	+	0.970047i	$0.578104\pi$
$24$	−99.0196	−0.842179
$25$	25.0000	0.200000
$26$	10.6264	0.0801542
$27$	152.423	1.08644
$28$	−112.782	−0.761209
$29$	−259.880	−1.66409	−0.832044	−	0.554709i	$-0.812829\pi$
$29$	−259.880	−1.66409	−0.832044	+	0.554709i	$0.812829\pi$
$30$	38.0803	0.231749
$31$	180.380	1.04507	0.522536	−	0.852617i	$-0.324986\pi$
$31$	180.380	1.04507	0.522536	+	0.852617i	$0.324986\pi$
$32$	−181.886	−1.00479
$33$	−48.3800	−0.255208
$34$	175.312	0.884289
$35$	112.750	0.544521
$36$	38.2910	0.177273
$37$	351.353	1.56114	0.780568	−	0.625070i	$-0.214930\pi$
$37$	351.353	1.56114	0.780568	+	0.625070i	$0.214930\pi$
$38$	−32.9012	−0.140455
$39$	26.9899	0.110817
$40$	112.569	0.444968
$41$	−425.891	−1.62227	−0.811134	−	0.584860i	$-0.801149\pi$
$41$	−425.891	−1.62227	−0.811134	+	0.584860i	$0.801149\pi$
$42$	171.742	0.630962
$43$	264.640	0.938540	0.469270	−	0.883055i	$-0.344517\pi$
$43$	264.640	0.938540	0.469270	+	0.883055i	$0.344517\pi$
$44$	−55.0157	−0.188498
$45$	−38.2801	−0.126810
$46$	92.7979	0.297441
$47$	294.823	0.914985	0.457492	−	0.889214i	$-0.348748\pi$
$47$	294.823	0.914985	0.457492	+	0.889214i	$0.348748\pi$
$48$	−4.51096	−0.0135646
$49$	165.504	0.482518
$50$	−43.2910	−0.122445
$51$	445.275	1.22257
$52$	30.6918	0.0818498
$53$	−187.648	−0.486330	−0.243165	−	0.969985i	$-0.578186\pi$
$53$	−187.648	−0.486330	−0.243165	+	0.969985i	$0.578186\pi$
$54$	−263.942	−0.665148
$55$	55.0000	0.134840
$56$	507.686	1.21147
$57$	−83.5654	−0.194184
$58$	450.019	1.01880
$59$	428.689	0.945942	0.472971	−	0.881078i	$-0.343181\pi$
$59$	428.689	0.945942	0.472971	+	0.881078i	$0.343181\pi$
$60$	109.986	0.236652
$61$	−197.962	−0.415515	−0.207757	−	0.978180i	$-0.566616\pi$
$61$	−197.962	−0.415515	−0.207757	+	0.978180i	$0.566616\pi$
$62$	−312.353	−0.639821
$63$	−172.643	−0.345254
$64$	306.756	0.599133
$65$	−30.6831	−0.0585503
$66$	83.7767	0.156245
$67$	675.426	1.23159	0.615794	−	0.787907i	$-0.288835\pi$
$67$	675.426	1.23159	0.615794	+	0.787907i	$0.288835\pi$
$68$	506.348	0.902995
$69$	235.697	0.411225
$70$	−195.243	−0.333371
$71$	−163.454	−0.273217	−0.136608	−	0.990625i	$-0.543620\pi$
$71$	−163.454	−0.273217	−0.136608	+	0.990625i	$0.543620\pi$
$72$	−172.366	−0.282132
$73$	−595.017	−0.953993	−0.476997	−	0.878905i	$-0.658275\pi$
$73$	−595.017	−0.953993	−0.476997	+	0.878905i	$0.658275\pi$
$74$	−608.417	−0.955770
$75$	−109.954	−0.169286
$76$	−95.0271	−0.143426
$77$	248.050	0.367116
$78$	−46.7369	−0.0678450
$79$	339.195	0.483069	0.241534	−	0.970392i	$-0.422349\pi$
$79$	339.195	0.483069	0.241534	+	0.970392i	$0.422349\pi$
$80$	5.12821	0.00716690
$81$	−463.673	−0.636040
$82$	737.490	0.993196
$83$	1008.70	1.33397	0.666983	−	0.745073i	$-0.267585\pi$
$83$	1008.70	1.33397	0.666983	+	0.745073i	$0.267585\pi$
$84$	496.037	0.644310
$85$	−506.204	−0.645947
$86$	−458.261	−0.574599
$87$	1143.00	1.40853
$88$	247.652	0.299997
$89$	−641.390	−0.763902	−0.381951	−	0.924183i	$-0.624748\pi$
$89$	−641.390	−0.763902	−0.381951	+	0.924183i	$0.624748\pi$
$90$	66.2873	0.0776367
$91$	−138.381	−0.159409
$92$	268.024	0.303733
$93$	−793.344	−0.884580
$94$	−510.526	−0.560178
$95$	95.0000	0.102598
$96$	799.968	0.850484
$97$	−640.865	−0.670824	−0.335412	−	0.942071i	$-0.608876\pi$
$97$	−640.865	−0.670824	−0.335412	+	0.942071i	$0.608876\pi$
$98$	−286.593	−0.295411
$99$	−84.2162	−0.0854954
$100$	−125.036	−0.125036
$101$	1126.39	1.10970	0.554852	−	0.831949i	$-0.312775\pi$
$101$	1126.39	1.10970	0.554852	+	0.831949i	$0.312775\pi$
$102$	−771.055	−0.748489
$103$	−1821.65	−1.74265	−0.871325	−	0.490707i	$-0.836738\pi$
$103$	−1821.65	−1.74265	−0.871325	+	0.490707i	$0.836738\pi$
$104$	−138.158	−0.130265
$105$	−495.895	−0.460899
$106$	324.939	0.297744
$107$	−1049.07	−0.947822	−0.473911	−	0.880573i	$-0.657158\pi$
$107$	−1049.07	−0.947822	−0.473911	+	0.880573i	$0.657158\pi$
$108$	−762.334	−0.679219
$109$	−308.854	−0.271402	−0.135701	−	0.990750i	$-0.543329\pi$
$109$	−308.854	−0.271402	−0.135701	+	0.990750i	$0.543329\pi$
$110$	−95.2402	−0.0825527
$111$	−1545.31	−1.32139
$112$	23.1283	0.0195126
$113$	408.000	0.339658	0.169829	−	0.985474i	$-0.445678\pi$
$113$	408.000	0.339658	0.169829	+	0.985474i	$0.445678\pi$
$114$	144.705	0.118885
$115$	−267.948	−0.217272
$116$	1299.77	1.04035
$117$	46.9821	0.0371239
$118$	−742.335	−0.579132
$119$	−2282.98	−1.75866
$120$	−495.098	−0.376634
$121$	121.000	0.0909091
$122$	342.798	0.254389
$123$	1873.14	1.37314
$124$	−902.157	−0.653356
$125$	125.000	0.0894427
$126$	298.956	0.211374
$127$	−574.808	−0.401622	−0.200811	−	0.979630i	$-0.564358\pi$
$127$	−574.808	−0.401622	−0.200811	+	0.979630i	$0.564358\pi$
$128$	923.899	0.637984
$129$	−1163.93	−0.794408
$130$	53.1321	0.0358461
$131$	−610.718	−0.407318	−0.203659	−	0.979042i	$-0.565283\pi$
$131$	−610.718	−0.407318	−0.203659	+	0.979042i	$0.565283\pi$
$132$	241.969	0.159551
$133$	428.451	0.279334
$134$	−1169.59	−0.754012
$135$	762.117	0.485871
$136$	−2279.31	−1.43713
$137$	−1057.49	−0.659471	−0.329736	−	0.944073i	$-0.606960\pi$
$137$	−1057.49	−0.659471	−0.329736	+	0.944073i	$0.606960\pi$
$138$	−408.142	−0.251763
$139$	−1676.41	−1.02296	−0.511480	−	0.859295i	$-0.670903\pi$
$139$	−1676.41	−1.02296	−0.511480	+	0.859295i	$0.670903\pi$
$140$	−563.911	−0.340423
$141$	−1296.68	−0.774470
$142$	283.043	0.167271
$143$	−67.5028	−0.0394746
$144$	−7.85234	−0.00454418
$145$	−1299.40	−0.744203
$146$	1030.36	0.584061
$147$	−727.915	−0.408418
$148$	−1757.27	−0.975989
$149$	−2221.35	−1.22134	−0.610672	−	0.791884i	$-0.709101\pi$
$149$	−2221.35	−1.22134	−0.610672	+	0.791884i	$0.709101\pi$
$150$	190.402	0.103641
$151$	2406.09	1.29672	0.648359	−	0.761335i	$-0.275456\pi$
$151$	2406.09	1.29672	0.648359	+	0.761335i	$0.275456\pi$
$152$	427.762	0.228264
$153$	775.101	0.409563
$154$	−429.534	−0.224759
$155$	901.900	0.467370
$156$	−134.988	−0.0692801
$157$	1802.23	0.916138	0.458069	−	0.888917i	$-0.348541\pi$
$157$	1802.23	0.916138	0.458069	+	0.888917i	$0.348541\pi$
$158$	−587.364	−0.295748
$159$	825.311	0.411644
$160$	−909.431	−0.449355
$161$	−1208.45	−0.591546
$162$	802.914	0.389401
$163$	−2607.31	−1.25288	−0.626442	−	0.779468i	$-0.715490\pi$
$163$	−2607.31	−1.25288	−0.626442	+	0.779468i	$0.715490\pi$
$164$	2130.06	1.01421
$165$	−241.900	−0.114133
$166$	−1746.70	−0.816690
$167$	−2010.02	−0.931378	−0.465689	−	0.884948i	$-0.654194\pi$
$167$	−2010.02	−0.931378	−0.465689	+	0.884948i	$0.654194\pi$
$168$	−2232.90	−1.02543
$169$	−2159.34	−0.982859
$170$	876.562	0.395466
$171$	−145.464	−0.0650523
$172$	−1323.58	−0.586754
$173$	−3483.98	−1.53111	−0.765556	−	0.643370i	$-0.777536\pi$
$173$	−3483.98	−1.53111	−0.765556	+	0.643370i	$0.777536\pi$
$174$	−1979.26	−0.862343
$175$	563.751	0.243517
$176$	11.2821	0.00483192
$177$	−1885.45	−0.800674
$178$	1110.66	0.467681
$179$	888.885	0.371164	0.185582	−	0.982629i	$-0.440583\pi$
$179$	888.885	0.371164	0.185582	+	0.982629i	$0.440583\pi$
$180$	191.455	0.0792790
$181$	4377.00	1.79746	0.898730	−	0.438503i	$-0.144491\pi$
$181$	4377.00	1.79746	0.898730	+	0.438503i	$0.144491\pi$
$182$	239.626	0.0975948
$183$	870.671	0.351704
$184$	−1206.50	−0.483395
$185$	1756.76	0.698162
$186$	1373.79	0.541564
$187$	−1113.65	−0.435497
$188$	−1474.53	−0.572028
$189$	3437.15	1.32284
$190$	−164.506	−0.0628132
$191$	−778.415	−0.294891	−0.147445	−	0.989070i	$-0.547105\pi$
$191$	−778.415	−0.294891	−0.147445	+	0.989070i	$0.547105\pi$
$192$	−1349.17	−0.507125
$193$	−3720.00	−1.38742	−0.693709	−	0.720256i	$-0.744024\pi$
$193$	−3720.00	−1.38742	−0.693709	+	0.720256i	$0.744024\pi$
$194$	1109.75	0.410697
$195$	134.950	0.0495587
$196$	−827.754	−0.301660
$197$	−549.462	−0.198718	−0.0993592	−	0.995052i	$-0.531679\pi$
$197$	−549.462	−0.198718	−0.0993592	+	0.995052i	$0.531679\pi$
$198$	145.832	0.0523426
$199$	−860.225	−0.306431	−0.153215	−	0.988193i	$-0.548963\pi$
$199$	−860.225	−0.306431	−0.153215	+	0.988193i	$0.548963\pi$
$200$	562.845	0.198996
$201$	−2970.65	−1.04245
$202$	−1950.50	−0.679391
$203$	−5860.31	−2.02617
$204$	−2227.01	−0.764322
$205$	−2129.45	−0.725500
$206$	3154.45	1.06690
$207$	410.283	0.137762
$208$	−6.29398	−0.00209812
$209$	209.000	0.0691714
$210$	858.712	0.282175
$211$	3707.62	1.20968	0.604842	−	0.796346i	$-0.293236\pi$
$211$	3707.62	1.20968	0.604842	+	0.796346i	$0.293236\pi$
$212$	938.509	0.304043
$213$	718.898	0.231259
$214$	1816.60	0.580282
$215$	1323.20	0.419728
$216$	3431.63	1.08098
$217$	4067.58	1.27247
$218$	534.823	0.166160
$219$	2616.99	0.807489
$220$	−275.078	−0.0842990
$221$	621.276	0.189102
$222$	2675.93	0.808993
$223$	2027.88	0.608955	0.304478	−	0.952519i	$-0.401518\pi$
$223$	2027.88	0.608955	0.304478	+	0.952519i	$0.401518\pi$
$224$	−4101.54	−1.22342
$225$	−191.401	−0.0567113
$226$	−706.508	−0.207948
$227$	732.454	0.214162	0.107081	−	0.994250i	$-0.465850\pi$
$227$	732.454	0.214162	0.107081	+	0.994250i	$0.465850\pi$
$228$	417.946	0.121400
$229$	2425.85	0.700020	0.350010	−	0.936746i	$-0.386178\pi$
$229$	2425.85	0.700020	0.350010	+	0.936746i	$0.386178\pi$
$230$	463.989	0.133020
$231$	−1090.97	−0.310738
$232$	−5850.89	−1.65573
$233$	−1089.07	−0.306212	−0.153106	−	0.988210i	$-0.548928\pi$
$233$	−1089.07	−0.306212	−0.153106	+	0.988210i	$0.548928\pi$
$234$	−81.3560	−0.0227282
$235$	1474.11	0.409194
$236$	−2144.06	−0.591382
$237$	−1491.84	−0.408884
$238$	3953.30	1.07670
$239$	−305.603	−0.0827105	−0.0413552	−	0.999145i	$-0.513168\pi$
$239$	−305.603	−0.0827105	−0.0413552	+	0.999145i	$0.513168\pi$
$240$	−22.5548	−0.00606628
$241$	−1848.34	−0.494032	−0.247016	−	0.969011i	$-0.579450\pi$
$241$	−1848.34	−0.494032	−0.247016	+	0.969011i	$0.579450\pi$
$242$	−209.528	−0.0556570
$243$	−2076.12	−0.548078
$244$	990.090	0.259771
$245$	827.519	0.215789
$246$	−3243.61	−0.840671
$247$	−116.596	−0.0300357
$248$	4061.04	1.03982
$249$	−4436.44	−1.12911
$250$	−216.455	−0.0547593
$251$	−2753.01	−0.692304	−0.346152	−	0.938178i	$-0.612512\pi$
$251$	−2753.01	−0.692304	−0.346152	+	0.938178i	$0.612512\pi$
$252$	863.463	0.215846
$253$	−589.486	−0.146485
$254$	995.361	0.245884
$255$	2226.37	0.546749
$256$	−4053.91	−0.989724
$257$	−1724.88	−0.418659	−0.209329	−	0.977845i	$-0.567128\pi$
$257$	−1724.88	−0.418659	−0.209329	+	0.977845i	$0.567128\pi$
$258$	2015.51	0.486358
$259$	7923.02	1.90082
$260$	153.459	0.0366044
$261$	1989.65	0.471863
$262$	1057.54	0.249371
$263$	−1589.11	−0.372581	−0.186290	−	0.982495i	$-0.559647\pi$
$263$	−1589.11	−0.372581	−0.186290	+	0.982495i	$0.559647\pi$
$264$	−1089.22	−0.253927
$265$	−938.241	−0.217493
$266$	−741.922	−0.171016
$267$	2820.95	0.646589
$268$	−3378.09	−0.769962
$269$	1132.15	0.256611	0.128306	−	0.991735i	$-0.459046\pi$
$269$	1132.15	0.256611	0.128306	+	0.991735i	$0.459046\pi$
$270$	−1319.71	−0.297463
$271$	−1453.65	−0.325840	−0.162920	−	0.986639i	$-0.552091\pi$
$271$	−1453.65	−0.325840	−0.162920	+	0.986639i	$0.552091\pi$
$272$	−103.837	−0.0231472
$273$	608.624	0.134929
$274$	1831.19	0.403746
$275$	275.000	0.0603023
$276$	−1178.82	−0.257089
$277$	−5267.04	−1.14248	−0.571238	−	0.820785i	$-0.693537\pi$
$277$	−5267.04	−1.14248	−0.571238	+	0.820785i	$0.693537\pi$
$278$	2902.94	0.626284
$279$	−1380.99	−0.296337
$280$	2538.43	0.541787
$281$	6437.07	1.36656	0.683280	−	0.730157i	$-0.260553\pi$
$281$	6437.07	1.36656	0.683280	+	0.730157i	$0.260553\pi$
$282$	2245.39	0.474152
$283$	−2420.72	−0.508470	−0.254235	−	0.967143i	$-0.581824\pi$
$283$	−2420.72	−0.508470	−0.254235	+	0.967143i	$0.581824\pi$
$284$	817.501	0.170809
$285$	−417.827	−0.0868419
$286$	116.891	0.0241674
$287$	−9603.85	−1.97525
$288$	1392.53	0.284914
$289$	5336.68	1.08624
$290$	2250.10	0.455621
$291$	2818.64	0.567806
$292$	2975.93	0.596416
$293$	−4884.84	−0.973978	−0.486989	−	0.873408i	$-0.661905\pi$
$293$	−4884.84	−0.973978	−0.486989	+	0.873408i	$0.661905\pi$
$294$	1260.49	0.250044
$295$	2143.45	0.423038
$296$	7910.28	1.55330
$297$	1676.66	0.327574
$298$	3846.58	0.747740
$299$	328.859	0.0636067
$300$	549.929	0.105834
$301$	5967.64	1.14275
$302$	−4166.47	−0.793886
$303$	−4954.07	−0.939287
$304$	19.4872	0.00367654
$305$	−989.808	−0.185824
$306$	−1342.20	−0.250746
$307$	−6602.20	−1.22739	−0.613693	−	0.789545i	$-0.710317\pi$
$307$	−6602.20	−1.22739	−0.613693	+	0.789545i	$0.710317\pi$
$308$	−1240.60	−0.229513
$309$	8011.96	1.47503
$310$	−1561.77	−0.286137
$311$	−3255.64	−0.593603	−0.296802	−	0.954939i	$-0.595920\pi$
$311$	−3255.64	−0.593603	−0.296802	+	0.954939i	$0.595920\pi$
$312$	607.646	0.110260
$313$	−6981.52	−1.26076	−0.630382	−	0.776285i	$-0.717102\pi$
$313$	−6981.52	−1.26076	−0.630382	+	0.776285i	$0.717102\pi$
$314$	−3120.82	−0.560885
$315$	−863.217	−0.154403
$316$	−1696.46	−0.302004
$317$	−1837.46	−0.325558	−0.162779	−	0.986663i	$-0.552046\pi$
$317$	−1837.46	−0.325558	−0.162779	+	0.986663i	$0.552046\pi$
$318$	−1429.14	−0.252020
$319$	−2858.68	−0.501742
$320$	1533.78	0.267941
$321$	4613.98	0.802265
$322$	2092.59	0.362161
$323$	−1923.57	−0.331364
$324$	2319.02	0.397638
$325$	−153.415	−0.0261845
$326$	4514.92	0.767050
$327$	1358.39	0.229723
$328$	−9588.41	−1.61412
$329$	6648.26	1.11407
$330$	418.883	0.0698751
$331$	−9887.26	−1.64185	−0.820926	−	0.571035i	$-0.806542\pi$
$331$	−9887.26	−1.64185	−0.820926	+	0.571035i	$0.806542\pi$
$332$	−5044.94	−0.833967
$333$	−2689.97	−0.442670
$334$	3480.63	0.570215
$335$	3377.13	0.550783
$336$	−101.722	−0.0165161
$337$	−932.334	−0.150705	−0.0753523	−	0.997157i	$-0.524008\pi$
$337$	−932.334	−0.150705	−0.0753523	+	0.997157i	$0.524008\pi$
$338$	3739.20	0.601733
$339$	−1794.46	−0.287497
$340$	2531.74	0.403832
$341$	1984.18	0.315101
$342$	251.892	0.0398268
$343$	−4002.55	−0.630079
$344$	5958.05	0.933826
$345$	1178.48	0.183906
$346$	6033.00	0.937388
$347$	3406.90	0.527067	0.263533	−	0.964650i	$-0.415112\pi$
$347$	3406.90	0.527067	0.263533	+	0.964650i	$0.415112\pi$
$348$	−5716.63	−0.880585
$349$	−10036.0	−1.53930	−0.769651	−	0.638465i	$-0.779570\pi$
$349$	−10036.0	−1.53930	−0.769651	+	0.638465i	$0.779570\pi$
$350$	−976.213	−0.149088
$351$	−935.364	−0.142239
$352$	−2000.75	−0.302955
$353$	−7174.48	−1.08175	−0.540877	−	0.841102i	$-0.681907\pi$
$353$	−7174.48	−1.08175	−0.540877	+	0.841102i	$0.681907\pi$
$354$	3264.92	0.490194
$355$	−817.268	−0.122186
$356$	3207.87	0.477575
$357$	10041.0	1.48858
$358$	−1539.23	−0.227237
$359$	−7886.72	−1.15946	−0.579729	−	0.814810i	$-0.696841\pi$
$359$	−7886.72	−1.15946	−0.579729	+	0.814810i	$0.696841\pi$
$360$	−861.830	−0.126173
$361$	361.000	0.0526316
$362$	−7579.39	−1.10045
$363$	−532.180	−0.0769482
$364$	692.102	0.0996593
$365$	−2975.09	−0.426639
$366$	−1507.69	−0.215323
$367$	11174.2	1.58934	0.794668	−	0.607044i	$-0.207645\pi$
$367$	11174.2	1.58934	0.794668	+	0.607044i	$0.207645\pi$
$368$	−54.9638	−0.00778583
$369$	3260.63	0.460004
$370$	−3042.08	−0.427433
$371$	−4231.47	−0.592149
$372$	3967.85	0.553020
$373$	−11987.5	−1.66404	−0.832022	−	0.554742i	$-0.812817\pi$
$373$	−11987.5	−1.66404	−0.832022	+	0.554742i	$0.812817\pi$
$374$	1928.44	0.266623
$375$	−549.772	−0.0757070
$376$	6637.57	0.910390
$377$	1594.79	0.217866
$378$	−5951.91	−0.809876
$379$	5228.39	0.708613	0.354306	−	0.935129i	$-0.384717\pi$
$379$	5228.39	0.708613	0.354306	+	0.935129i	$0.384717\pi$
$380$	−475.135	−0.0641419
$381$	2528.11	0.339945
$382$	1347.93	0.180540
$383$	7202.50	0.960916	0.480458	−	0.877018i	$-0.340470\pi$
$383$	7202.50	0.960916	0.480458	+	0.877018i	$0.340470\pi$
$384$	−4063.47	−0.540008
$385$	1240.25	0.164179
$386$	6441.70	0.849414
$387$	−2026.09	−0.266129
$388$	3205.24	0.419385
$389$	−9463.35	−1.23345	−0.616723	−	0.787180i	$-0.711540\pi$
$389$	−9463.35	−1.23345	−0.616723	+	0.787180i	$0.711540\pi$
$390$	−233.684	−0.0303412
$391$	5425.45	0.701731
$392$	3726.11	0.480095
$393$	2686.05	0.344766
$394$	951.470	0.121661
$395$	1695.98	0.216035
$396$	421.201	0.0534499
$397$	−2451.61	−0.309931	−0.154965	−	0.987920i	$-0.549527\pi$
$397$	−2451.61	−0.309931	−0.154965	+	0.987920i	$0.549527\pi$
$398$	1489.60	0.187605
$399$	−1884.40	−0.236436
$400$	25.6411	0.00320513
$401$	8709.44	1.08461	0.542305	−	0.840182i	$-0.317552\pi$
$401$	8709.44	1.08461	0.542305	+	0.840182i	$0.317552\pi$
$402$	5144.09	0.638218
$403$	−1106.92	−0.136823
$404$	−5633.56	−0.693763
$405$	−2318.36	−0.284446
$406$	10147.9	1.24048
$407$	3864.88	0.470700
$408$	10024.8	1.21643
$409$	9984.38	1.20708	0.603540	−	0.797332i	$-0.293756\pi$
$409$	9984.38	1.20708	0.603540	+	0.797332i	$0.293756\pi$
$410$	3687.45	0.444171
$411$	4651.03	0.558196
$412$	9110.86	1.08947
$413$	9666.96	1.15177
$414$	−710.462	−0.0843414
$415$	5043.50	0.596568
$416$	1116.17	0.131549
$417$	7373.17	0.865865
$418$	−361.913	−0.0423486
$419$	13313.8	1.55232	0.776161	−	0.630535i	$-0.217164\pi$
$419$	13313.8	1.55232	0.776161	+	0.630535i	$0.217164\pi$
$420$	2480.18	0.288144
$421$	8153.40	0.943877	0.471939	−	0.881631i	$-0.343554\pi$
$421$	8153.40	0.943877	0.471939	+	0.881631i	$0.343554\pi$
$422$	−6420.26	−0.740601
$423$	−2257.17	−0.259450
$424$	−4224.67	−0.483887
$425$	−2531.02	−0.288876
$426$	−1244.87	−0.141583
$427$	−4464.04	−0.505925
$428$	5246.82	0.592557
$429$	296.889	0.0334125
$430$	−2291.30	−0.256969
$431$	5213.55	0.582663	0.291331	−	0.956622i	$-0.405902\pi$
$431$	5213.55	0.582663	0.291331	+	0.956622i	$0.405902\pi$
$432$	156.332	0.0174109
$433$	−17494.6	−1.94165	−0.970825	−	0.239788i	$-0.922922\pi$
$433$	−17494.6	−1.94165	−0.970825	+	0.239788i	$0.922922\pi$
$434$	−7043.57	−0.779038
$435$	5715.00	0.629916
$436$	1544.71	0.169675
$437$	−1018.20	−0.111458
$438$	−4531.69	−0.494366
$439$	12803.0	1.39192	0.695959	−	0.718081i	$-0.254979\pi$
$439$	12803.0	1.39192	0.695959	+	0.718081i	$0.254979\pi$
$440$	1238.26	0.134163
$441$	−1267.10	−0.136821
$442$	−1075.83	−0.115773
$443$	8566.84	0.918788	0.459394	−	0.888233i	$-0.348067\pi$
$443$	8566.84	0.918788	0.459394	+	0.888233i	$0.348067\pi$
$444$	7728.77	0.826106
$445$	−3206.95	−0.341627
$446$	−3511.56	−0.372819
$447$	9769.90	1.03378
$448$	6917.36	0.729497
$449$	−7801.46	−0.819986	−0.409993	−	0.912089i	$-0.634469\pi$
$449$	−7801.46	−0.819986	−0.409993	+	0.912089i	$0.634469\pi$
$450$	331.437	0.0347202
$451$	−4684.80	−0.489132
$452$	−2040.58	−0.212347
$453$	−10582.4	−1.09758
$454$	−1268.35	−0.131116
$455$	−691.904	−0.0712901
$456$	−1881.37	−0.193209
$457$	−4451.62	−0.455663	−0.227832	−	0.973701i	$-0.573164\pi$
$457$	−4451.62	−0.455663	−0.227832	+	0.973701i	$0.573164\pi$
$458$	−4200.69	−0.428571
$459$	−15431.5	−1.56923
$460$	1340.12	0.135834
$461$	−7113.29	−0.718653	−0.359326	−	0.933212i	$-0.616994\pi$
$461$	−7113.29	−0.718653	−0.359326	+	0.933212i	$0.616994\pi$
$462$	1889.17	0.190242
$463$	2120.35	0.212832	0.106416	−	0.994322i	$-0.466063\pi$
$463$	2120.35	0.212832	0.106416	+	0.994322i	$0.466063\pi$
$464$	−266.544	−0.0266681
$465$	−3966.72	−0.395596
$466$	1885.88	0.187471
$467$	−7417.92	−0.735033	−0.367517	−	0.930017i	$-0.619792\pi$
$467$	−7417.92	−0.735033	−0.367517	+	0.930017i	$0.619792\pi$
$468$	−234.977	−0.0232090
$469$	15230.9	1.49957
$470$	−2552.63	−0.250519
$471$	−7926.54	−0.775447
$472$	9651.42	0.941192
$473$	2911.04	0.282980
$474$	2583.33	0.250330
$475$	475.000	0.0458831
$476$	11418.2	1.09948
$477$	1436.64	0.137902
$478$	529.194	0.0506376
$479$	−13326.6	−1.27121	−0.635604	−	0.772015i	$-0.719249\pi$
$479$	−13326.6	−1.27121	−0.635604	+	0.772015i	$0.719249\pi$
$480$	3999.84	0.380348
$481$	−2156.12	−0.204388
$482$	3200.65	0.302460
$483$	5314.97	0.500703
$484$	−605.172	−0.0568344
$485$	−3204.32	−0.300002
$486$	3595.08	0.335548
$487$	−2275.70	−0.211749	−0.105875	−	0.994379i	$-0.533764\pi$
$487$	−2275.70	−0.211749	−0.105875	+	0.994379i	$0.533764\pi$
$488$	−4456.86	−0.413428
$489$	11467.4	1.06048
$490$	−1432.96	−0.132112
$491$	−14493.8	−1.33217	−0.666084	−	0.745877i	$-0.732031\pi$
$491$	−14493.8	−1.33217	−0.666084	+	0.745877i	$0.732031\pi$
$492$	−9368.39	−0.858455
$493$	26310.5	2.40358
$494$	201.902	0.0183886
$495$	−421.081	−0.0382347
$496$	185.006	0.0167480
$497$	−3685.88	−0.332665
$498$	7682.32	0.691271
$499$	−4311.08	−0.386754	−0.193377	−	0.981124i	$-0.561944\pi$
$499$	−4311.08	−0.386754	−0.193377	+	0.981124i	$0.561944\pi$
$500$	−625.178	−0.0559176
$501$	8840.44	0.788347
$502$	4767.22	0.423847
$503$	4703.73	0.416957	0.208478	−	0.978027i	$-0.433149\pi$
$503$	4703.73	0.416957	0.208478	+	0.978027i	$0.433149\pi$
$504$	−3886.86	−0.343521
$505$	5631.96	0.496275
$506$	1020.78	0.0896819
$507$	9497.17	0.831922
$508$	2874.86	0.251085
$509$	5497.22	0.478704	0.239352	−	0.970933i	$-0.423065\pi$
$509$	5497.22	0.478704	0.239352	+	0.970933i	$0.423065\pi$
$510$	−3855.28	−0.334734
$511$	−13417.7	−1.16157
$512$	−371.279	−0.0320476
$513$	2896.04	0.249247
$514$	2986.88	0.256314
$515$	−9108.27	−0.779336
$516$	5821.33	0.496647
$517$	3243.05	0.275878
$518$	−13719.8	−1.16373
$519$	15323.2	1.29598
$520$	−690.792	−0.0582562
$521$	−18017.3	−1.51507	−0.757536	−	0.652793i	$-0.773597\pi$
$521$	−18017.3	−1.51507	−0.757536	+	0.652793i	$0.773597\pi$
$522$	−3445.36	−0.288887
$523$	16610.3	1.38875	0.694377	−	0.719611i	$-0.255680\pi$
$523$	16610.3	1.38875	0.694377	+	0.719611i	$0.255680\pi$
$524$	3054.46	0.254646
$525$	−2479.48	−0.206120
$526$	2751.77	0.228104
$527$	−18261.8	−1.50948
$528$	−49.6206	−0.00408988
$529$	−9295.16	−0.763964
$530$	1624.70	0.133155
$531$	−3282.05	−0.268228
$532$	−2142.86	−0.174633
$533$	2613.53	0.212391
$534$	−4884.87	−0.395859
$535$	−5245.33	−0.423879
$536$	15206.4	1.22540
$537$	−3909.48	−0.314164
$538$	−1960.48	−0.157104
$539$	1820.54	0.145485
$540$	−3811.67	−0.303756
$541$	−23414.5	−1.86076	−0.930379	−	0.366600i	$-0.880522\pi$
$541$	−23414.5	−1.86076	−0.930379	+	0.366600i	$0.880522\pi$
$542$	2517.19	0.199488
$543$	−19250.8	−1.52142
$544$	18414.3	1.45130
$545$	−1544.27	−0.121375
$546$	−1053.92	−0.0826071
$547$	22791.0	1.78148	0.890742	−	0.454509i	$-0.150185\pi$
$547$	22791.0	1.78148	0.890742	+	0.454509i	$0.150185\pi$
$548$	5288.96	0.412287
$549$	1515.60	0.117822
$550$	−476.201	−0.0369187
$551$	−4937.73	−0.381768
$552$	5306.42	0.409160
$553$	7648.86	0.588178
$554$	9120.62	0.699455
$555$	−7726.57	−0.590945
$556$	8384.45	0.639533
$557$	8235.75	0.626499	0.313250	−	0.949671i	$-0.398582\pi$
$557$	8235.75	0.626499	0.313250	+	0.949671i	$0.398582\pi$
$558$	2391.38	0.181425
$559$	−1623.99	−0.122876
$560$	115.641	0.00872632
$561$	4898.02	0.368618
$562$	−11146.7	−0.836645
$563$	−18336.5	−1.37263	−0.686316	−	0.727303i	$-0.740773\pi$
$563$	−18336.5	−1.37263	−0.686316	+	0.727303i	$0.740773\pi$
$564$	6485.26	0.484182
$565$	2040.00	0.151900
$566$	4191.82	0.311299
$567$	−10455.8	−0.774433
$568$	−3679.96	−0.271844
$569$	15232.1	1.12225	0.561127	−	0.827730i	$-0.310368\pi$
$569$	15232.1	1.12225	0.561127	+	0.827730i	$0.310368\pi$
$570$	723.526	0.0531669
$571$	−25222.4	−1.84855	−0.924276	−	0.381725i	$-0.875330\pi$
$571$	−25222.4	−1.84855	−0.924276	+	0.381725i	$0.875330\pi$
$572$	337.610	0.0246787
$573$	3423.61	0.249604
$574$	16630.4	1.20930
$575$	−1339.74	−0.0971670
$576$	−2348.53	−0.169888
$577$	−3609.62	−0.260434	−0.130217	−	0.991486i	$-0.541567\pi$
$577$	−3609.62	−0.260434	−0.130217	+	0.991486i	$0.541567\pi$
$578$	−9241.21	−0.665024
$579$	16361.2	1.17435
$580$	6498.86	0.465259
$581$	22746.2	1.62422
$582$	−4880.87	−0.347626
$583$	−2064.13	−0.146634
$584$	−13396.1	−0.949202
$585$	234.910	0.0166023
$586$	8458.78	0.596295
$587$	−642.453	−0.0451736	−0.0225868	−	0.999745i	$-0.507190\pi$
$587$	−642.453	−0.0451736	−0.0225868	+	0.999745i	$0.507190\pi$
$588$	3640.61	0.255334
$589$	3427.22	0.239756
$590$	−3711.68	−0.258995
$591$	2416.63	0.168201
$592$	360.363	0.0250183
$593$	−10566.0	−0.731690	−0.365845	−	0.930676i	$-0.619220\pi$
$593$	−10566.0	−0.731690	−0.365845	+	0.930676i	$0.619220\pi$
$594$	−2903.37	−0.200550
$595$	−11414.9	−0.786497
$596$	11109.9	0.763557
$597$	3783.42	0.259372
$598$	−569.465	−0.0389417
$599$	5715.85	0.389888	0.194944	−	0.980814i	$-0.437547\pi$
$599$	5715.85	0.389888	0.194944	+	0.980814i	$0.437547\pi$
$600$	−2475.49	−0.168436
$601$	4006.59	0.271934	0.135967	−	0.990713i	$-0.456586\pi$
$601$	4006.59	0.271934	0.135967	+	0.990713i	$0.456586\pi$
$602$	−10333.8	−0.699625
$603$	−5171.08	−0.349225
$604$	−12033.9	−0.810680
$605$	605.000	0.0406558
$606$	8578.67	0.575057
$607$	24695.4	1.65133	0.825663	−	0.564164i	$-0.190801\pi$
$607$	24695.4	1.65133	0.825663	+	0.564164i	$0.190801\pi$
$608$	−3455.84	−0.230514
$609$	25774.7	1.71501
$610$	1713.99	0.113766
$611$	−1809.21	−0.119792
$612$	−3876.61	−0.256050
$613$	4004.09	0.263824	0.131912	−	0.991261i	$-0.457888\pi$
$613$	4004.09	0.263824	0.131912	+	0.991261i	$0.457888\pi$
$614$	11432.6	0.751439
$615$	9365.72	0.614085
$616$	5584.55	0.365273
$617$	6903.38	0.450437	0.225219	−	0.974308i	$-0.427690\pi$
$617$	6903.38	0.450437	0.225219	+	0.974308i	$0.427690\pi$
$618$	−13873.8	−0.903053
$619$	20569.2	1.33562	0.667809	−	0.744333i	$-0.267232\pi$
$619$	20569.2	1.33562	0.667809	+	0.744333i	$0.267232\pi$
$620$	−4510.79	−0.292190
$621$	−8168.31	−0.527831
$622$	5637.60	0.363420
$623$	−14463.4	−0.930117
$624$	27.6820	0.00177591
$625$	625.000	0.0400000
$626$	12089.5	0.771874
$627$	−919.219	−0.0585488
$628$	−9013.73	−0.572750
$629$	−35571.2	−2.25488
$630$	1494.78	0.0945294
$631$	13414.1	0.846285	0.423143	−	0.906063i	$-0.360927\pi$
$631$	13414.1	0.846285	0.423143	+	0.906063i	$0.360927\pi$
$632$	7636.57	0.480643
$633$	−16306.8	−1.02391
$634$	3181.82	0.199316
$635$	−2874.04	−0.179611
$636$	−4127.73	−0.257351
$637$	−1015.63	−0.0631724
$638$	4950.21	0.307180
$639$	1251.40	0.0774723
$640$	4619.49	0.285315
$641$	−1080.84	−0.0665998	−0.0332999	−	0.999445i	$-0.510602\pi$
$641$	−1080.84	−0.0665998	−0.0332999	+	0.999445i	$0.510602\pi$
$642$	−7989.74	−0.491168
$643$	−26168.8	−1.60497	−0.802486	−	0.596671i	$-0.796490\pi$
$643$	−26168.8	−1.60497	−0.802486	+	0.596671i	$0.796490\pi$
$644$	6043.96	0.369822
$645$	−5819.67	−0.355270
$646$	3330.94	0.202870
$647$	−21680.3	−1.31737	−0.658686	−	0.752418i	$-0.728888\pi$
$647$	−21680.3	−1.31737	−0.658686	+	0.752418i	$0.728888\pi$
$648$	−10439.0	−0.632845
$649$	4715.58	0.285212
$650$	265.660	0.0160308
$651$	−17889.9	−1.07705
$652$	13040.3	0.783276
$653$	12736.6	0.763282	0.381641	−	0.924311i	$-0.375359\pi$
$653$	12736.6	0.763282	0.381641	+	0.924311i	$0.375359\pi$
$654$	−2352.25	−0.140642
$655$	−3053.59	−0.182158
$656$	−436.812	−0.0259979
$657$	4555.47	0.270511
$658$	−11512.4	−0.682066
$659$	3263.45	0.192907	0.0964537	−	0.995337i	$-0.469250\pi$
$659$	3263.45	0.192907	0.0964537	+	0.995337i	$0.469250\pi$
$660$	1209.84	0.0713532
$661$	2133.61	0.125549	0.0627744	−	0.998028i	$-0.480005\pi$
$661$	2133.61	0.125549	0.0627744	+	0.998028i	$0.480005\pi$
$662$	17121.2	1.00519
$663$	−2732.48	−0.160062
$664$	22709.7	1.32727
$665$	2142.25	0.124922
$666$	4658.05	0.271015
$667$	13926.9	0.808472
$668$	10053.0	0.582277
$669$	−8918.99	−0.515438
$670$	−5847.97	−0.337204
$671$	−2177.58	−0.125282
$672$	18039.3	1.03554
$673$	−9648.34	−0.552624	−0.276312	−	0.961068i	$-0.589112\pi$
$673$	−9648.34	−0.552624	−0.276312	+	0.961068i	$0.589112\pi$
$674$	1614.47	0.0922655
$675$	3810.58	0.217288
$676$	10799.8	0.614462
$677$	−17809.5	−1.01104	−0.505519	−	0.862815i	$-0.668699\pi$
$677$	−17809.5	−1.01104	−0.505519	+	0.862815i	$0.668699\pi$
$678$	3107.35	0.176013
$679$	−14451.5	−0.816787
$680$	−11396.6	−0.642703
$681$	−3221.47	−0.181273
$682$	−3435.89	−0.192913
$683$	−13375.4	−0.749333	−0.374667	−	0.927160i	$-0.622243\pi$
$683$	−13375.4	−0.749333	−0.374667	+	0.927160i	$0.622243\pi$
$684$	727.529	0.0406693
$685$	−5287.45	−0.294924
$686$	6930.97	0.385752
$687$	−10669.3	−0.592518
$688$	271.426	0.0150407
$689$	1151.53	0.0636715
$690$	−2040.71	−0.112592
$691$	−15026.8	−0.827273	−0.413636	−	0.910442i	$-0.635742\pi$
$691$	−15026.8	−0.827273	−0.413636	+	0.910442i	$0.635742\pi$
$692$	17424.9	0.957217
$693$	−1899.08	−0.104098
$694$	−5899.53	−0.322685
$695$	−8382.07	−0.457482
$696$	25733.3	1.40146
$697$	43117.5	2.34317
$698$	17378.8	0.942402
$699$	4789.93	0.259187
$700$	−2819.56	−0.152242
$701$	27649.5	1.48974	0.744869	−	0.667211i	$-0.232512\pi$
$701$	27649.5	1.48974	0.744869	+	0.667211i	$0.232512\pi$
$702$	1619.71	0.0870828
$703$	6675.71	0.358149
$704$	3374.32	0.180646
$705$	−6483.41	−0.346354
$706$	12423.6	0.662279
$707$	25400.2	1.35116
$708$	9429.95	0.500564
$709$	−12515.7	−0.662960	−0.331480	−	0.943462i	$-0.607548\pi$
$709$	−12515.7	−0.662960	−0.331480	+	0.943462i	$0.607548\pi$
$710$	1415.21	0.0748057
$711$	−2596.89	−0.136977
$712$	−14440.1	−0.760065
$713$	−9666.50	−0.507732
$714$	−17387.3	−0.911350
$715$	−337.514	−0.0176536
$716$	−4445.69	−0.232044
$717$	1344.10	0.0700086
$718$	13657.0	0.709851
$719$	16534.5	0.857625	0.428812	−	0.903394i	$-0.358932\pi$
$719$	16534.5	0.857625	0.428812	+	0.903394i	$0.358932\pi$
$720$	−39.2617	−0.00203222
$721$	−41078.3	−2.12183
$722$	−625.122	−0.0322225
$723$	8129.31	0.418164
$724$	−21891.3	−1.12373
$725$	−6497.01	−0.332818
$726$	921.543	0.0471098
$727$	24510.6	1.25041	0.625205	−	0.780461i	$-0.285015\pi$
$727$	24510.6	1.25041	0.625205	+	0.780461i	$0.285015\pi$
$728$	−3115.48	−0.158609
$729$	21650.3	1.09995
$730$	5151.78	0.261200
$731$	−26792.3	−1.35561
$732$	−4354.59	−0.219878
$733$	−11162.2	−0.562461	−0.281231	−	0.959640i	$-0.590743\pi$
$733$	−11162.2	−0.562461	−0.281231	+	0.959640i	$0.590743\pi$
$734$	−19349.6	−0.973035
$735$	−3639.58	−0.182650
$736$	9747.21	0.488162
$737$	7429.69	0.371338
$738$	−5646.24	−0.281627
$739$	13354.6	0.664761	0.332380	−	0.943145i	$-0.392148\pi$
$739$	13354.6	0.664761	0.332380	+	0.943145i	$0.392148\pi$
$740$	−8786.33	−0.436475
$741$	512.809	0.0254231
$742$	7327.39	0.362529
$743$	−25033.2	−1.23604	−0.618020	−	0.786163i	$-0.712065\pi$
$743$	−25033.2	−1.23604	−0.618020	+	0.786163i	$0.712065\pi$
$744$	−17861.2	−0.880137
$745$	−11106.8	−0.546202
$746$	20758.0	1.01877
$747$	−7722.63	−0.378255
$748$	5569.83	0.272263
$749$	−23656.4	−1.15406
$750$	952.008	0.0463499
$751$	−5466.79	−0.265627	−0.132814	−	0.991141i	$-0.542401\pi$
$751$	−5466.79	−0.265627	−0.132814	+	0.991141i	$0.542401\pi$
$752$	302.383	0.0146632
$753$	12108.2	0.585987
$754$	−2761.59	−0.133384
$755$	12030.4	0.579910
$756$	−17190.7	−0.827008
$757$	−36094.0	−1.73297	−0.866486	−	0.499202i	$-0.833627\pi$
$757$	−36094.0	−1.73297	−0.866486	+	0.499202i	$0.833627\pi$
$758$	−9053.68	−0.433832
$759$	2592.66	0.123989
$760$	2138.81	0.102083
$761$	−17091.7	−0.814155	−0.407078	−	0.913394i	$-0.633452\pi$
$761$	−17091.7	−0.814155	−0.407078	+	0.913394i	$0.633452\pi$
$762$	−4377.78	−0.208124
$763$	−6964.66	−0.330456
$764$	3893.18	0.184359
$765$	3875.51	0.183162
$766$	−12472.1	−0.588299
$767$	−2630.70	−0.123845
$768$	17829.8	0.837732
$769$	−20631.2	−0.967466	−0.483733	−	0.875216i	$-0.660719\pi$
$769$	−20631.2	−0.967466	−0.483733	+	0.875216i	$0.660719\pi$
$770$	−2147.67	−0.100515
$771$	7586.35	0.354365
$772$	18605.3	0.867383
$773$	28426.9	1.32270	0.661348	−	0.750079i	$-0.269985\pi$
$773$	28426.9	1.32270	0.661348	+	0.750079i	$0.269985\pi$
$774$	3508.46	0.162931
$775$	4509.50	0.209014
$776$	−14428.3	−0.667456
$777$	−34846.9	−1.60891
$778$	16387.1	0.755149
$779$	−8091.93	−0.372174
$780$	−674.941	−0.0309830
$781$	−1797.99	−0.0823779
$782$	−9394.92	−0.429619
$783$	−39611.8	−1.80793
$784$	169.748	0.00773268
$785$	9011.16	0.409710
$786$	−4651.26	−0.211075
$787$	18363.6	0.831758	0.415879	−	0.909420i	$-0.363474\pi$
$787$	18363.6	0.831758	0.415879	+	0.909420i	$0.363474\pi$
$788$	2748.09	0.124234
$789$	6989.19	0.315363
$790$	−2936.82	−0.132262
$791$	9200.40	0.413563
$792$	−1896.03	−0.0850661
$793$	1214.81	0.0544002
$794$	4245.30	0.189748
$795$	4126.55	0.184093
$796$	4302.35	0.191574
$797$	−20538.0	−0.912788	−0.456394	−	0.889778i	$-0.650859\pi$
$797$	−20538.0	−0.912788	−0.456394	+	0.889778i	$0.650859\pi$
$798$	3263.11	0.144753
$799$	−29848.0	−1.32159
$800$	−4547.16	−0.200958
$801$	4910.50	0.216609
$802$	−15081.6	−0.664028
$803$	−6545.19	−0.287640
$804$	14857.5	0.651719
$805$	−6042.23	−0.264548
$806$	1916.79	0.0837669
$807$	−4979.40	−0.217204
$808$	25359.3	1.10413
$809$	25023.3	1.08748	0.543742	−	0.839253i	$-0.317007\pi$
$809$	25023.3	1.08748	0.543742	+	0.839253i	$0.317007\pi$
$810$	4014.57	0.174145
$811$	13208.1	0.571887	0.285944	−	0.958246i	$-0.407693\pi$
$811$	13208.1	0.571887	0.285944	+	0.958246i	$0.407693\pi$
$812$	29309.9	1.26672
$813$	6393.40	0.275801
$814$	−6692.58	−0.288176
$815$	−13036.5	−0.560307
$816$	456.693	0.0195925
$817$	5028.16	0.215316
$818$	−17289.4	−0.739007
$819$	1059.45	0.0452016
$820$	10650.3	0.453567
$821$	−11868.2	−0.504510	−0.252255	−	0.967661i	$-0.581172\pi$
$821$	−11868.2	−0.504510	−0.252255	+	0.967661i	$0.581172\pi$
$822$	−8053.91	−0.341743
$823$	−30329.6	−1.28460	−0.642299	−	0.766454i	$-0.722019\pi$
$823$	−30329.6	−1.28460	−0.642299	+	0.766454i	$0.722019\pi$
$824$	−41012.3	−1.73390
$825$	−1209.50	−0.0510416
$826$	−16739.7	−0.705143
$827$	−14119.5	−0.593693	−0.296847	−	0.954925i	$-0.595935\pi$
$827$	−14119.5	−0.593693	−0.296847	+	0.954925i	$0.595935\pi$
$828$	−2052.00	−0.0861255
$829$	−32637.9	−1.36738	−0.683692	−	0.729771i	$-0.739627\pi$
$829$	−32637.9	−1.36738	−0.683692	+	0.729771i	$0.739627\pi$
$830$	−8733.52	−0.365235
$831$	23165.4	0.967026
$832$	−1882.45	−0.0784400
$833$	−16755.7	−0.696940
$834$	−12767.7	−0.530106
$835$	−10050.1	−0.416525
$836$	−1045.30	−0.0432445
$837$	27494.1	1.13541
$838$	−23054.7	−0.950373
$839$	−16323.8	−0.671705	−0.335853	−	0.941915i	$-0.609024\pi$
$839$	−16323.8	−0.671705	−0.335853	+	0.941915i	$0.609024\pi$
$840$	−11164.5	−0.458585
$841$	43148.8	1.76919
$842$	−14118.8	−0.577867
$843$	−28311.4	−1.15670
$844$	−18543.4	−0.756268
$845$	−10796.7	−0.439548
$846$	3908.60	0.158842
$847$	2728.55	0.110690
$848$	−192.460	−0.00779376
$849$	10646.8	0.430384
$850$	4382.81	0.176858
$851$	−18828.9	−0.758455
$852$	−3595.52	−0.144578
$853$	692.340	0.0277905	0.0138952	−	0.999903i	$-0.495577\pi$
$853$	692.340	0.0277905	0.0138952	+	0.999903i	$0.495577\pi$
$854$	7730.11	0.309741
$855$	−727.322	−0.0290923
$856$	−23618.4	−0.943062
$857$	41950.7	1.67212	0.836062	−	0.548636i	$-0.184853\pi$
$857$	41950.7	1.67212	0.836062	+	0.548636i	$0.184853\pi$
$858$	−514.105	−0.0204560
$859$	−7681.37	−0.305105	−0.152552	−	0.988295i	$-0.548749\pi$
$859$	−7681.37	−0.305105	−0.152552	+	0.988295i	$0.548749\pi$
$860$	−6617.88	−0.262405
$861$	42239.5	1.67191
$862$	−9027.98	−0.356722
$863$	−40100.1	−1.58172	−0.790860	−	0.611997i	$-0.790366\pi$
$863$	−40100.1	−1.58172	−0.790860	+	0.611997i	$0.790366\pi$
$864$	−27723.7	−1.09164
$865$	−17419.9	−0.684734
$866$	30294.3	1.18873
$867$	−23471.7	−0.919424
$868$	−20343.7	−0.795518
$869$	3731.15	0.145651
$870$	−9896.32	−0.385651
$871$	−4144.83	−0.161243
$872$	−6953.46	−0.270039
$873$	4906.48	0.190217
$874$	1763.16	0.0682377
$875$	2818.75	0.108904
$876$	−13088.7	−0.504824
$877$	36891.8	1.42046	0.710232	−	0.703968i	$-0.248590\pi$
$877$	36891.8	1.42046	0.710232	+	0.703968i	$0.248590\pi$
$878$	−22170.1	−0.852170
$879$	21484.4	0.824404
$880$	56.4104	0.00216090
$881$	21457.6	0.820574	0.410287	−	0.911956i	$-0.365428\pi$
$881$	21457.6	0.820574	0.410287	+	0.911956i	$0.365428\pi$
$882$	2194.16	0.0837656
$883$	11942.6	0.455154	0.227577	−	0.973760i	$-0.426920\pi$
$883$	11942.6	0.455154	0.227577	+	0.973760i	$0.426920\pi$
$884$	−3107.26	−0.118222
$885$	−9427.26	−0.358072
$886$	−14834.7	−0.562507
$887$	−23466.3	−0.888297	−0.444149	−	0.895953i	$-0.646494\pi$
$887$	−23466.3	−0.888297	−0.444149	+	0.895953i	$0.646494\pi$
$888$	−34790.8	−1.31476
$889$	−12961.9	−0.489010
$890$	5553.29	0.209153
$891$	−5100.40	−0.191773
$892$	−10142.3	−0.380706
$893$	5601.63	0.209912
$894$	−16918.0	−0.632909
$895$	4444.43	0.165990
$896$	20833.9	0.776801
$897$	−1446.38	−0.0538386
$898$	13509.3	0.502017
$899$	−46877.2	−1.73909
$900$	957.275	0.0354546
$901$	18997.6	0.702446
$902$	8112.39	0.299460
$903$	−26246.7	−0.967261
$904$	9185.61	0.337952
$905$	21885.0	0.803848
$906$	18324.9	0.671969
$907$	9422.58	0.344952	0.172476	−	0.985014i	$-0.444823\pi$
$907$	9422.58	0.344952	0.172476	+	0.985014i	$0.444823\pi$
$908$	−3663.32	−0.133889
$909$	−8623.68	−0.314664
$910$	1198.13	0.0436457
$911$	−2734.39	−0.0994451	−0.0497225	−	0.998763i	$-0.515834\pi$
$911$	−2734.39	−0.0994451	−0.0497225	+	0.998763i	$0.515834\pi$
$912$	−85.7083	−0.00311193
$913$	11095.7	0.402206
$914$	7708.61	0.278969
$915$	4353.35	0.157287
$916$	−12132.7	−0.437637
$917$	−13771.7	−0.495945
$918$	26721.7	0.960728
$919$	35688.4	1.28101	0.640507	−	0.767952i	$-0.278724\pi$
$919$	35688.4	1.28101	0.640507	+	0.767952i	$0.278724\pi$
$920$	−6032.52	−0.216181
$921$	29037.7	1.03890
$922$	12317.7	0.439979
$923$	1003.05	0.0357702
$924$	5456.40	0.194267
$925$	8783.82	0.312227
$926$	−3671.68	−0.130301
$927$	13946.6	0.494139
$928$	47268.7	1.67206
$929$	−7800.69	−0.275492	−0.137746	−	0.990468i	$-0.543986\pi$
$929$	−7800.69	−0.275492	−0.137746	+	0.990468i	$0.543986\pi$
$930$	6868.93	0.242195
$931$	3144.57	0.110697
$932$	5446.91	0.191437
$933$	14318.9	0.502444
$934$	12845.2	0.450007
$935$	−5568.24	−0.194760
$936$	1057.74	0.0369374
$937$	−30849.2	−1.07556	−0.537780	−	0.843085i	$-0.680737\pi$
$937$	−30849.2	−1.07556	−0.537780	+	0.843085i	$0.680737\pi$
$938$	−26374.4	−0.918075
$939$	30706.0	1.06715
$940$	−7372.66	−0.255819
$941$	−11852.5	−0.410606	−0.205303	−	0.978699i	$-0.565818\pi$
$941$	−11852.5	−0.410606	−0.205303	+	0.978699i	$0.565818\pi$
$942$	13725.9	0.474750
$943$	22823.3	0.788154
$944$	439.682	0.0151594
$945$	17185.8	0.591590
$946$	−5040.87	−0.173248
$947$	−10772.2	−0.369641	−0.184821	−	0.982772i	$-0.559170\pi$
$947$	−10772.2	−0.369641	−0.184821	+	0.982772i	$0.559170\pi$
$948$	7461.33	0.255625
$949$	3651.39	0.124899
$950$	−822.529	−0.0280909
$951$	8081.47	0.275562
$952$	−51398.5	−1.74983
$953$	31045.8	1.05527	0.527636	−	0.849471i	$-0.323079\pi$
$953$	31045.8	1.05527	0.527636	+	0.849471i	$0.323079\pi$
$954$	−2487.74	−0.0844273
$955$	−3892.07	−0.131879
$956$	1528.45	0.0517088
$957$	12573.0	0.424689
$958$	23076.9	0.778268
$959$	−23846.4	−0.802963
$960$	−6745.85	−0.226793
$961$	2745.97	0.0921745
$962$	3733.62	0.125132
$963$	8031.66	0.268761
$964$	9244.31	0.308858
$965$	−18600.0	−0.620472
$966$	−9203.61	−0.306544
$967$	−34803.0	−1.15738	−0.578692	−	0.815546i	$-0.696437\pi$
$967$	−34803.0	−1.15738	−0.578692	+	0.815546i	$0.696437\pi$
$968$	2724.17	0.0904525
$969$	8460.22	0.280476
$970$	5548.74	0.183669
$971$	−9688.44	−0.320203	−0.160101	−	0.987101i	$-0.551182\pi$
$971$	−9688.44	−0.320203	−0.160101	+	0.987101i	$0.551182\pi$
$972$	10383.5	0.342646
$973$	−37803.2	−1.24554
$974$	3940.69	0.129639
$975$	674.749	0.0221633
$976$	−203.038	−0.00665890
$977$	−2160.80	−0.0707577	−0.0353788	−	0.999374i	$-0.511264\pi$
$977$	−2160.80	−0.0707577	−0.0353788	+	0.999374i	$0.511264\pi$
$978$	−19857.4	−0.649254
$979$	−7055.29	−0.230325
$980$	−4138.77	−0.134906
$981$	2364.59	0.0769577
$982$	25098.0	0.815589
$983$	−49252.4	−1.59807	−0.799037	−	0.601282i	$-0.794657\pi$
$983$	−49252.4	−1.59807	−0.799037	+	0.601282i	$0.794657\pi$
$984$	42171.6	1.36624
$985$	−2747.31	−0.0888696
$986$	−45560.3	−1.47154
$987$	−29240.2	−0.942985
$988$	583.145	0.0187776
$989$	−14181.9	−0.455975
$990$	729.161	0.0234083
$991$	−4685.21	−0.150182	−0.0750912	−	0.997177i	$-0.523925\pi$
$991$	−4685.21	−0.150182	−0.0750912	+	0.997177i	$0.523925\pi$
$992$	−32808.7	−1.05008
$993$	43485.9	1.38971
$994$	6382.62	0.203666
$995$	−4301.12	−0.137040
$996$	22188.5	0.705894
$997$	12293.3	0.390503	0.195251	−	0.980753i	$-0.437448\pi$
$997$	12293.3	0.390503	0.195251	+	0.980753i	$0.437448\pi$
$998$	7465.24	0.236782
$999$	53554.4	1.69608

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1045.4.a.b.1.9	✓	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1045.4.a.b.1.9	✓	20	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 1045 = 5 \cdot 11 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1045.a (trivial)

\(f(q)\)	\(=\)	\(q-1.73164 q^{2} -4.39818 q^{3} -5.00142 q^{4} +5.00000 q^{5} +7.61606 q^{6} +22.5500 q^{7} +22.5138 q^{8} -7.65602 q^{9} +O(q^{10})\)	\(q-1.73164 q^{2} -4.39818 q^{3} -5.00142 q^{4} +5.00000 q^{5} +7.61606 q^{6} +22.5500 q^{7} +22.5138 q^{8} -7.65602 q^{9} -8.65820 q^{10} +11.0000 q^{11} +21.9972 q^{12} -6.13662 q^{13} -39.0485 q^{14} -21.9909 q^{15} +1.02564 q^{16} -101.241 q^{17} +13.2575 q^{18} +19.0000 q^{19} -25.0071 q^{20} -99.1791 q^{21} -19.0480 q^{22} -53.5896 q^{23} -99.0196 q^{24} +25.0000 q^{25} +10.6264 q^{26} +152.423 q^{27} -112.782 q^{28} -259.880 q^{29} +38.0803 q^{30} +180.380 q^{31} -181.886 q^{32} -48.3800 q^{33} +175.312 q^{34} +112.750 q^{35} +38.2910 q^{36} +351.353 q^{37} -32.9012 q^{38} +26.9899 q^{39} +112.569 q^{40} -425.891 q^{41} +171.742 q^{42} +264.640 q^{43} -55.0157 q^{44} -38.2801 q^{45} +92.7979 q^{46} +294.823 q^{47} -4.51096 q^{48} +165.504 q^{49} -43.2910 q^{50} +445.275 q^{51} +30.6918 q^{52} -187.648 q^{53} -263.942 q^{54} +55.0000 q^{55} +507.686 q^{56} -83.5654 q^{57} +450.019 q^{58} +428.689 q^{59} +109.986 q^{60} -197.962 q^{61} -312.353 q^{62} -172.643 q^{63} +306.756 q^{64} -30.6831 q^{65} +83.7767 q^{66} +675.426 q^{67} +506.348 q^{68} +235.697 q^{69} -195.243 q^{70} -163.454 q^{71} -172.366 q^{72} -595.017 q^{73} -608.417 q^{74} -109.954 q^{75} -95.0271 q^{76} +248.050 q^{77} -46.7369 q^{78} +339.195 q^{79} +5.12821 q^{80} -463.673 q^{81} +737.490 q^{82} +1008.70 q^{83} +496.037 q^{84} -506.204 q^{85} -458.261 q^{86} +1143.00 q^{87} +247.652 q^{88} -641.390 q^{89} +66.2873 q^{90} -138.381 q^{91} +268.024 q^{92} -793.344 q^{93} -510.526 q^{94} +95.0000 q^{95} +799.968 q^{96} -640.865 q^{97} -286.593 q^{98} -84.2162 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 12 q^{2} - 21 q^{3} + 72 q^{4} + 100 q^{5} - 45 q^{6} - 131 q^{7} - 108 q^{8} + 159 q^{9}+O(q^{10}) \) 20 * q - 12 * q^2 - 21 * q^3 + 72 * q^4 + 100 * q^5 - 45 * q^6 - 131 * q^7 - 108 * q^8 + 159 * q^9	\( 20 q - 12 q^{2} - 21 q^{3} + 72 q^{4} + 100 q^{5} - 45 q^{6} - 131 q^{7} - 108 q^{8} + 159 q^{9} - 60 q^{10} + 220 q^{11} - 196 q^{12} - 223 q^{13} - 11 q^{14} - 105 q^{15} + 380 q^{16} - 471 q^{17} + 113 q^{18} + 380 q^{19} + 360 q^{20} - 57 q^{21} - 132 q^{22} - 653 q^{23} - 486 q^{24} + 500 q^{25} - 145 q^{26} - 177 q^{27} - 747 q^{28} + 51 q^{29} - 225 q^{30} - 90 q^{31} - 381 q^{32} - 231 q^{33} + 517 q^{34} - 655 q^{35} + 875 q^{36} - 96 q^{37} - 228 q^{38} + 97 q^{39} - 540 q^{40} - 1284 q^{41} - 638 q^{42} - 1592 q^{43} + 792 q^{44} + 795 q^{45} - 35 q^{46} - 2030 q^{47} - 1471 q^{48} + 765 q^{49} - 300 q^{50} - 185 q^{51} - 3242 q^{52} - 943 q^{53} - 730 q^{54} + 1100 q^{55} + 887 q^{56} - 399 q^{57} - 492 q^{58} - 515 q^{59} - 980 q^{60} + 446 q^{61} - 770 q^{62} - 3980 q^{63} + 2526 q^{64} - 1115 q^{65} - 495 q^{66} - 1719 q^{67} - 1808 q^{68} + 317 q^{69} - 55 q^{70} - 90 q^{71} + 1058 q^{72} - 3763 q^{73} - 1313 q^{74} - 525 q^{75} + 1368 q^{76} - 1441 q^{77} + 4286 q^{78} + 2 q^{79} + 1900 q^{80} + 308 q^{81} - 3830 q^{82} - 3436 q^{83} + 5734 q^{84} - 2355 q^{85} + 1922 q^{86} - 2719 q^{87} - 1188 q^{88} + 1700 q^{89} + 565 q^{90} + 2007 q^{91} - 3980 q^{92} - 2276 q^{93} + 2700 q^{94} + 1900 q^{95} - 6252 q^{96} - 1956 q^{97} - 2356 q^{98} + 1749 q^{99}+O(q^{100}) \) 20 * q - 12 * q^2 - 21 * q^3 + 72 * q^4 + 100 * q^5 - 45 * q^6 - 131 * q^7 - 108 * q^8 + 159 * q^9 - 60 * q^10 + 220 * q^11 - 196 * q^12 - 223 * q^13 - 11 * q^14 - 105 * q^15 + 380 * q^16 - 471 * q^17 + 113 * q^18 + 380 * q^19 + 360 * q^20 - 57 * q^21 - 132 * q^22 - 653 * q^23 - 486 * q^24 + 500 * q^25 - 145 * q^26 - 177 * q^27 - 747 * q^28 + 51 * q^29 - 225 * q^30 - 90 * q^31 - 381 * q^32 - 231 * q^33 + 517 * q^34 - 655 * q^35 + 875 * q^36 - 96 * q^37 - 228 * q^38 + 97 * q^39 - 540 * q^40 - 1284 * q^41 - 638 * q^42 - 1592 * q^43 + 792 * q^44 + 795 * q^45 - 35 * q^46 - 2030 * q^47 - 1471 * q^48 + 765 * q^49 - 300 * q^50 - 185 * q^51 - 3242 * q^52 - 943 * q^53 - 730 * q^54 + 1100 * q^55 + 887 * q^56 - 399 * q^57 - 492 * q^58 - 515 * q^59 - 980 * q^60 + 446 * q^61 - 770 * q^62 - 3980 * q^63 + 2526 * q^64 - 1115 * q^65 - 495 * q^66 - 1719 * q^67 - 1808 * q^68 + 317 * q^69 - 55 * q^70 - 90 * q^71 + 1058 * q^72 - 3763 * q^73 - 1313 * q^74 - 525 * q^75 + 1368 * q^76 - 1441 * q^77 + 4286 * q^78 + 2 * q^79 + 1900 * q^80 + 308 * q^81 - 3830 * q^82 - 3436 * q^83 + 5734 * q^84 - 2355 * q^85 + 1922 * q^86 - 2719 * q^87 - 1188 * q^88 + 1700 * q^89 + 565 * q^90 + 2007 * q^91 - 3980 * q^92 - 2276 * q^93 + 2700 * q^94 + 1900 * q^95 - 6252 * q^96 - 1956 * q^97 - 2356 * q^98 + 1749 * q^99

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