LMFDB - Embedded newform 403.2.a.e.1.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [403,2,Mod(1,403)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(403, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("403.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 403 = 13 \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	403.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:	$3.21797120146$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{7} - 11x^{6} + 10x^{5} + 37x^{4} - 33x^{3} - 36x^{2} + 33x - 4 $ x^8 - x^7 - 11x^6 + 10x^5 + 37x^4 - 33x^3 - 36x^2 + 33x - 4
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Root			$-2.14079$ of defining polynomial
Character	$\chi$	$=$	403.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.14079 q^{2} +0.217190 q^{3} +2.58298 q^{4} -0.434938 q^{5} +0.464957 q^{6} +3.26287 q^{7} +1.24803 q^{8} -2.95283 q^{9} +O(q^{10})$	\(q+2.14079 q^{2} +0.217190 q^{3} +2.58298 q^{4} -0.434938 q^{5} +0.464957 q^{6} +3.26287 q^{7} +1.24803 q^{8} -2.95283 q^{9} -0.931109 q^{10} +4.50684 q^{11} +0.560996 q^{12} +1.00000 q^{13} +6.98512 q^{14} -0.0944640 q^{15} -2.49419 q^{16} -3.22707 q^{17} -6.32138 q^{18} +0.614865 q^{19} -1.12343 q^{20} +0.708663 q^{21} +9.64818 q^{22} +3.65374 q^{23} +0.271059 q^{24} -4.81083 q^{25} +2.14079 q^{26} -1.29289 q^{27} +8.42792 q^{28} -6.70994 q^{29} -0.202227 q^{30} -1.00000 q^{31} -7.83558 q^{32} +0.978839 q^{33} -6.90848 q^{34} -1.41915 q^{35} -7.62709 q^{36} -3.95309 q^{37} +1.31630 q^{38} +0.217190 q^{39} -0.542814 q^{40} +5.58573 q^{41} +1.51710 q^{42} -5.96799 q^{43} +11.6410 q^{44} +1.28430 q^{45} +7.82188 q^{46} +1.93997 q^{47} -0.541712 q^{48} +3.64634 q^{49} -10.2990 q^{50} -0.700887 q^{51} +2.58298 q^{52} +1.45251 q^{53} -2.76781 q^{54} -1.96019 q^{55} +4.07216 q^{56} +0.133542 q^{57} -14.3646 q^{58} +6.76571 q^{59} -0.243998 q^{60} -11.2702 q^{61} -2.14079 q^{62} -9.63471 q^{63} -11.7860 q^{64} -0.434938 q^{65} +2.09549 q^{66} +13.2135 q^{67} -8.33545 q^{68} +0.793554 q^{69} -3.03809 q^{70} +4.46071 q^{71} -3.68521 q^{72} -9.84245 q^{73} -8.46273 q^{74} -1.04486 q^{75} +1.58818 q^{76} +14.7052 q^{77} +0.464957 q^{78} -12.3413 q^{79} +1.08482 q^{80} +8.57768 q^{81} +11.9579 q^{82} +0.728296 q^{83} +1.83046 q^{84} +1.40357 q^{85} -12.7762 q^{86} -1.45733 q^{87} +5.62466 q^{88} +7.64021 q^{89} +2.74941 q^{90} +3.26287 q^{91} +9.43751 q^{92} -0.217190 q^{93} +4.15306 q^{94} -0.267428 q^{95} -1.70181 q^{96} +16.1772 q^{97} +7.80605 q^{98} -13.3079 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - q^{2} + 7 q^{3} + 7 q^{4} + 11 q^{5} - 2 q^{7} + 9 q^{9}+O(q^{10}) $ 8 * q - q^2 + 7 * q^3 + 7 * q^4 + 11 * q^5 - 2 * q^7 + 9 * q^9	\( 8 q - q^{2} + 7 q^{3} + 7 q^{4} + 11 q^{5} - 2 q^{7} + 9 q^{9} - 2 q^{10} - 2 q^{11} + 19 q^{12} + 8 q^{13} + 3 q^{14} + 6 q^{15} - 7 q^{16} + 7 q^{17} - 12 q^{18} - 5 q^{19} + 6 q^{20} + 8 q^{21} - 4 q^{22} + 14 q^{23} - 17 q^{24} + 17 q^{25} - q^{26} + 7 q^{27} - 9 q^{28} + 12 q^{29} - 9 q^{30} - 8 q^{31} - 21 q^{32} + 10 q^{33} - 12 q^{34} - q^{35} + 11 q^{36} + 2 q^{37} + 24 q^{38} + 7 q^{39} - 19 q^{40} + 13 q^{41} - 27 q^{42} - 5 q^{43} + 22 q^{44} + 19 q^{45} + 17 q^{46} + 23 q^{47} + 3 q^{48} + 26 q^{49} - 26 q^{50} + 18 q^{51} + 7 q^{52} + 25 q^{53} - 36 q^{54} - 17 q^{55} + 8 q^{56} - 35 q^{57} - 29 q^{58} - 5 q^{59} + 71 q^{60} - 9 q^{61} + q^{62} - 37 q^{63} - 14 q^{64} + 11 q^{65} - 41 q^{66} + 22 q^{67} - 6 q^{68} - 7 q^{69} - 29 q^{70} - 17 q^{71} - 34 q^{72} - 27 q^{73} + 14 q^{74} - 33 q^{75} - 36 q^{76} + 31 q^{77} - 23 q^{79} + 9 q^{80} - 12 q^{81} + 18 q^{82} - 25 q^{83} - 62 q^{84} + 13 q^{85} + 11 q^{86} + 26 q^{87} + 5 q^{88} + 2 q^{89} - 14 q^{90} - 2 q^{91} - 20 q^{92} - 7 q^{93} - 38 q^{94} + 3 q^{95} - 52 q^{96} - 15 q^{97} + 39 q^{98} - 8 q^{99}+O(q^{100}) \) 8 * q - q^2 + 7 * q^3 + 7 * q^4 + 11 * q^5 - 2 * q^7 + 9 * q^9 - 2 * q^10 - 2 * q^11 + 19 * q^12 + 8 * q^13 + 3 * q^14 + 6 * q^15 - 7 * q^16 + 7 * q^17 - 12 * q^18 - 5 * q^19 + 6 * q^20 + 8 * q^21 - 4 * q^22 + 14 * q^23 - 17 * q^24 + 17 * q^25 - q^26 + 7 * q^27 - 9 * q^28 + 12 * q^29 - 9 * q^30 - 8 * q^31 - 21 * q^32 + 10 * q^33 - 12 * q^34 - q^35 + 11 * q^36 + 2 * q^37 + 24 * q^38 + 7 * q^39 - 19 * q^40 + 13 * q^41 - 27 * q^42 - 5 * q^43 + 22 * q^44 + 19 * q^45 + 17 * q^46 + 23 * q^47 + 3 * q^48 + 26 * q^49 - 26 * q^50 + 18 * q^51 + 7 * q^52 + 25 * q^53 - 36 * q^54 - 17 * q^55 + 8 * q^56 - 35 * q^57 - 29 * q^58 - 5 * q^59 + 71 * q^60 - 9 * q^61 + q^62 - 37 * q^63 - 14 * q^64 + 11 * q^65 - 41 * q^66 + 22 * q^67 - 6 * q^68 - 7 * q^69 - 29 * q^70 - 17 * q^71 - 34 * q^72 - 27 * q^73 + 14 * q^74 - 33 * q^75 - 36 * q^76 + 31 * q^77 - 23 * q^79 + 9 * q^80 - 12 * q^81 + 18 * q^82 - 25 * q^83 - 62 * q^84 + 13 * q^85 + 11 * q^86 + 26 * q^87 + 5 * q^88 + 2 * q^89 - 14 * q^90 - 2 * q^91 - 20 * q^92 - 7 * q^93 - 38 * q^94 + 3 * q^95 - 52 * q^96 - 15 * q^97 + 39 * q^98 - 8 * 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 $

$2$	2.14079	1.51377	0.756883	−	0.653550i	$-0.226721\pi$
$2$	2.14079	1.51377	0.756883	+	0.653550i	$0.226721\pi$
$3$	0.217190	0.125395	0.0626973	−	0.998033i	$-0.480030\pi$
$3$	0.217190	0.125395	0.0626973	+	0.998033i	$0.480030\pi$
$4$	2.58298	1.29149
$5$	−0.434938	−0.194510	−0.0972550	−	0.995259i	$-0.531006\pi$
$5$	−0.434938	−0.194510	−0.0972550	+	0.995259i	$0.531006\pi$
$6$	0.464957	0.189818
$7$	3.26287	1.23325	0.616625	−	0.787257i	$-0.288499\pi$
$7$	3.26287	1.23325	0.616625	+	0.787257i	$0.288499\pi$
$8$	1.24803	0.441245
$9$	−2.95283	−0.984276
$10$	−0.931109	−0.294443
$11$	4.50684	1.35886	0.679431	−	0.733739i	$-0.262227\pi$
$11$	4.50684	1.35886	0.679431	+	0.733739i	$0.262227\pi$
$12$	0.560996	0.161946
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	6.98512	1.86685
$15$	−0.0944640	−0.0243905
$16$	−2.49419	−0.623547
$17$	−3.22707	−0.782680	−0.391340	−	0.920246i	$-0.627988\pi$
$17$	−3.22707	−0.782680	−0.391340	+	0.920246i	$0.627988\pi$
$18$	−6.32138	−1.48996
$19$	0.614865	0.141060	0.0705299	−	0.997510i	$-0.477531\pi$
$19$	0.614865	0.141060	0.0705299	+	0.997510i	$0.477531\pi$
$20$	−1.12343	−0.251207
$21$	0.708663	0.154643
$22$	9.64818	2.05700
$23$	3.65374	0.761857	0.380928	−	0.924605i	$-0.375604\pi$
$23$	3.65374	0.761857	0.380928	+	0.924605i	$0.375604\pi$
$24$	0.271059	0.0553297
$25$	−4.81083	−0.962166
$26$	2.14079	0.419843
$27$	−1.29289	−0.248817
$28$	8.42792	1.59273
$29$	−6.70994	−1.24600	−0.623002	−	0.782220i	$-0.714087\pi$
$29$	−6.70994	−1.24600	−0.623002	+	0.782220i	$0.714087\pi$
$30$	−0.202227	−0.0369215
$31$	−1.00000	−0.179605
$31$	−1.00000	−0.179605
$32$	−7.83558	−1.38515
$33$	0.978839	0.170394
$34$	−6.90848	−1.18479
$35$	−1.41915	−0.239879
$36$	−7.62709	−1.27118
$37$	−3.95309	−0.649884	−0.324942	−	0.945734i	$-0.605345\pi$
$37$	−3.95309	−0.649884	−0.324942	+	0.945734i	$0.605345\pi$
$38$	1.31630	0.213531
$39$	0.217190	0.0347782
$40$	−0.542814	−0.0858265
$41$	5.58573	0.872345	0.436172	−	0.899863i	$-0.356334\pi$
$41$	5.58573	0.872345	0.436172	+	0.899863i	$0.356334\pi$
$42$	1.51710	0.234093
$43$	−5.96799	−0.910109	−0.455055	−	0.890463i	$-0.650380\pi$
$43$	−5.96799	−0.910109	−0.455055	+	0.890463i	$0.650380\pi$
$44$	11.6410	1.75495
$45$	1.28430	0.191452
$46$	7.82188	1.15327
$47$	1.93997	0.282973	0.141487	−	0.989940i	$-0.454812\pi$
$47$	1.93997	0.282973	0.141487	+	0.989940i	$0.454812\pi$
$48$	−0.541712	−0.0781894
$49$	3.64634	0.520906
$50$	−10.2990	−1.45649
$51$	−0.700887	−0.0981438
$52$	2.58298	0.358194
$53$	1.45251	0.199517	0.0997585	−	0.995012i	$-0.468193\pi$
$53$	1.45251	0.199517	0.0997585	+	0.995012i	$0.468193\pi$
$54$	−2.76781	−0.376651
$55$	−1.96019	−0.264312
$56$	4.07216	0.544165
$57$	0.133542	0.0176881
$58$	−14.3646	−1.88616
$59$	6.76571	0.880820	0.440410	−	0.897797i	$-0.354833\pi$
$59$	6.76571	0.880820	0.440410	+	0.897797i	$0.354833\pi$
$60$	−0.243998	−0.0315000
$61$	−11.2702	−1.44301	−0.721503	−	0.692411i	$-0.756549\pi$
$61$	−11.2702	−1.44301	−0.721503	+	0.692411i	$0.756549\pi$
$62$	−2.14079	−0.271880
$63$	−9.63471	−1.21386
$64$	−11.7860	−1.47324
$65$	−0.434938	−0.0539474
$66$	2.09549	0.257937
$67$	13.2135	1.61429	0.807143	−	0.590356i	$-0.201013\pi$
$67$	13.2135	1.61429	0.807143	+	0.590356i	$0.201013\pi$
$68$	−8.33545	−1.01082
$69$	0.793554	0.0955327
$70$	−3.03809	−0.363121
$71$	4.46071	0.529389	0.264695	−	0.964332i	$-0.414729\pi$
$71$	4.46071	0.529389	0.264695	+	0.964332i	$0.414729\pi$
$72$	−3.68521	−0.434307
$73$	−9.84245	−1.15197	−0.575986	−	0.817460i	$-0.695382\pi$
$73$	−9.84245	−1.15197	−0.575986	+	0.817460i	$0.695382\pi$
$74$	−8.46273	−0.983772
$75$	−1.04486	−0.120650
$76$	1.58818	0.182177
$77$	14.7052	1.67582
$78$	0.464957	0.0526461
$79$	−12.3413	−1.38850	−0.694251	−	0.719733i	$-0.744264\pi$
$79$	−12.3413	−1.38850	−0.694251	+	0.719733i	$0.744264\pi$
$80$	1.08482	0.121286
$81$	8.57768	0.953076
$82$	11.9579	1.32053
$83$	0.728296	0.0799409	0.0399704	−	0.999201i	$-0.487274\pi$
$83$	0.728296	0.0799409	0.0399704	+	0.999201i	$0.487274\pi$
$84$	1.83046	0.199719
$85$	1.40357	0.152239
$86$	−12.7762	−1.37769
$87$	−1.45733	−0.156242
$88$	5.62466	0.599591
$89$	7.64021	0.809861	0.404930	−	0.914348i	$-0.367296\pi$
$89$	7.64021	0.809861	0.404930	+	0.914348i	$0.367296\pi$
$90$	2.74941	0.289813
$91$	3.26287	0.342042
$92$	9.43751	0.983929
$93$	−0.217190	−0.0225215
$94$	4.15306	0.428356
$95$	−0.267428	−0.0274375
$96$	−1.70181	−0.173690
$97$	16.1772	1.64255	0.821273	−	0.570535i	$-0.193264\pi$
$97$	16.1772	1.64255	0.821273	+	0.570535i	$0.193264\pi$
$98$	7.80605	0.788530
$99$	−13.3079	−1.33750
$100$	−12.4263	−1.24263
$101$	13.7761	1.37077	0.685387	−	0.728179i	$-0.259633\pi$
$101$	13.7761	1.37077	0.685387	+	0.728179i	$0.259633\pi$
$102$	−1.50045	−0.148567
$103$	2.76343	0.272288	0.136144	−	0.990689i	$-0.456529\pi$
$103$	2.76343	0.272288	0.136144	+	0.990689i	$0.456529\pi$
$104$	1.24803	0.122379
$105$	−0.308224	−0.0300796
$106$	3.10951	0.302022
$107$	−16.8710	−1.63098	−0.815492	−	0.578768i	$-0.803534\pi$
$107$	−16.8710	−1.63098	−0.815492	+	0.578768i	$0.803534\pi$
$108$	−3.33951	−0.321345
$109$	−6.05007	−0.579491	−0.289746	−	0.957104i	$-0.593571\pi$
$109$	−6.05007	−0.579491	−0.289746	+	0.957104i	$0.593571\pi$
$110$	−4.19636	−0.400107
$111$	−0.858571	−0.0814919
$112$	−8.13822	−0.768989
$113$	−0.0170631	−0.00160516	−0.000802579	−	1.00000i	$-0.500255\pi$
$113$	−0.0170631	−0.00160516	−0.000802579		1.00000i	$0.500255\pi$
$114$	0.285886	0.0267757
$115$	−1.58915	−0.148189
$116$	−17.3316	−1.60920
$117$	−2.95283	−0.272989
$118$	14.4839	1.33336
$119$	−10.5295	−0.965240
$120$	−0.117894	−0.0107622
$121$	9.31157	0.846506
$122$	−24.1272	−2.18438
$123$	1.21316	0.109387
$124$	−2.58298	−0.231958
$125$	4.26710	0.381661
$126$	−20.6259	−1.83750
$127$	18.5029	1.64187	0.820934	−	0.571023i	$-0.193453\pi$
$127$	18.5029	1.64187	0.820934	+	0.571023i	$0.193453\pi$
$128$	−9.56007	−0.844999
$129$	−1.29619	−0.114123
$130$	−0.931109	−0.0816637
$131$	21.7214	1.89781	0.948904	−	0.315563i	$-0.102193\pi$
$131$	21.7214	1.89781	0.948904	+	0.315563i	$0.102193\pi$
$132$	2.52832	0.220062
$133$	2.00623	0.173962
$134$	28.2873	2.44365
$135$	0.562328	0.0483975
$136$	−4.02748	−0.345353
$137$	−1.15271	−0.0984827	−0.0492414	−	0.998787i	$-0.515680\pi$
$137$	−1.15271	−0.0984827	−0.0492414	+	0.998787i	$0.515680\pi$
$138$	1.69883	0.144614
$139$	5.72543	0.485625	0.242813	−	0.970073i	$-0.421930\pi$
$139$	5.72543	0.485625	0.242813	+	0.970073i	$0.421930\pi$
$140$	−3.66562	−0.309801
$141$	0.421341	0.0354833
$142$	9.54944	0.801371
$143$	4.50684	0.376881
$144$	7.36491	0.613742
$145$	2.91840	0.242360
$146$	−21.0706	−1.74382
$147$	0.791949	0.0653188
$148$	−10.2107	−0.839317
$149$	−2.58197	−0.211524	−0.105762	−	0.994391i	$-0.533728\pi$
$149$	−2.58197	−0.211524	−0.105762	+	0.994391i	$0.533728\pi$
$150$	−2.23683	−0.182636
$151$	13.6608	1.11170	0.555852	−	0.831281i	$-0.312392\pi$
$151$	13.6608	1.11170	0.555852	+	0.831281i	$0.312392\pi$
$152$	0.767369	0.0622419
$153$	9.52899	0.770373
$154$	31.4808	2.53680
$155$	0.434938	0.0349350
$156$	0.560996	0.0449156
$157$	−17.4577	−1.39327	−0.696637	−	0.717423i	$-0.745321\pi$
$157$	−17.4577	−1.39327	−0.696637	+	0.717423i	$0.745321\pi$
$158$	−26.4200	−2.10187
$159$	0.315469	0.0250183
$160$	3.40799	0.269425
$161$	11.9217	0.939560
$162$	18.3630	1.44273
$163$	−1.46408	−0.114675	−0.0573377	−	0.998355i	$-0.518261\pi$
$163$	−1.46408	−0.114675	−0.0573377	+	0.998355i	$0.518261\pi$
$164$	14.4278	1.12662
$165$	−0.425734	−0.0331433
$166$	1.55913	0.121012
$167$	−18.8122	−1.45573	−0.727866	−	0.685720i	$-0.759488\pi$
$167$	−18.8122	−1.45573	−0.727866	+	0.685720i	$0.759488\pi$
$168$	0.884431	0.0682354
$169$	1.00000	0.0769231
$170$	3.00476	0.230454
$171$	−1.81559	−0.138842
$172$	−15.4152	−1.17540
$173$	12.1197	0.921443	0.460722	−	0.887545i	$-0.347591\pi$
$173$	12.1197	0.921443	0.460722	+	0.887545i	$0.347591\pi$
$174$	−3.11984	−0.236514
$175$	−15.6971	−1.18659
$176$	−11.2409	−0.847314
$177$	1.46944	0.110450
$178$	16.3561	1.22594
$179$	−24.7196	−1.84763	−0.923814	−	0.382842i	$-0.874945\pi$
$179$	−24.7196	−1.84763	−0.923814	+	0.382842i	$0.874945\pi$
$180$	3.31731	0.247257
$181$	17.4372	1.29610	0.648050	−	0.761598i	$-0.275585\pi$
$181$	17.4372	1.29610	0.648050	+	0.761598i	$0.275585\pi$
$182$	6.98512	0.517772
$183$	−2.44778	−0.180945
$184$	4.55997	0.336165
$185$	1.71935	0.126409
$186$	−0.464957	−0.0340923
$187$	−14.5439	−1.06355
$188$	5.01089	0.365457
$189$	−4.21855	−0.306854
$190$	−0.572507	−0.0415340
$191$	8.83485	0.639268	0.319634	−	0.947541i	$-0.396440\pi$
$191$	8.83485	0.639268	0.319634	+	0.947541i	$0.396440\pi$
$192$	−2.55979	−0.184737
$193$	−6.67055	−0.480157	−0.240078	−	0.970753i	$-0.577173\pi$
$193$	−6.67055	−0.480157	−0.240078	+	0.970753i	$0.577173\pi$
$194$	34.6320	2.48643
$195$	−0.0944640	−0.00676471
$196$	9.41842	0.672744
$197$	15.8952	1.13249	0.566244	−	0.824238i	$-0.308396\pi$
$197$	15.8952	1.13249	0.566244	+	0.824238i	$0.308396\pi$
$198$	−28.4894	−2.02466
$199$	9.15082	0.648685	0.324342	−	0.945940i	$-0.394857\pi$
$199$	9.15082	0.648685	0.324342	+	0.945940i	$0.394857\pi$
$200$	−6.00405	−0.424551
$201$	2.86984	0.202423
$202$	29.4917	2.07503
$203$	−21.8937	−1.53664
$204$	−1.81037	−0.126752
$205$	−2.42944	−0.169680
$206$	5.91591	0.412181
$207$	−10.7889	−0.749877
$208$	−2.49419	−0.172941
$209$	2.77110	0.191681
$210$	−0.659843	−0.0455335
$211$	−25.3399	−1.74447	−0.872237	−	0.489083i	$-0.837331\pi$
$211$	−25.3399	−1.74447	−0.872237	+	0.489083i	$0.837331\pi$
$212$	3.75179	0.257674
$213$	0.968821	0.0663825
$214$	−36.1173	−2.46893
$215$	2.59570	0.177025
$216$	−1.61357	−0.109789
$217$	−3.26287	−0.221498
$218$	−12.9519	−0.877214
$219$	−2.13768	−0.144451
$220$	−5.06313	−0.341356
$221$	−3.22707	−0.217076
$222$	−1.83802	−0.123360
$223$	−17.4240	−1.16680	−0.583398	−	0.812186i	$-0.698277\pi$
$223$	−17.4240	−1.16680	−0.583398	+	0.812186i	$0.698277\pi$
$224$	−25.5665	−1.70824
$225$	14.2056	0.947037
$226$	−0.0365284	−0.00242983
$227$	25.8749	1.71738	0.858689	−	0.512497i	$-0.171280\pi$
$227$	25.8749	1.71738	0.858689	+	0.512497i	$0.171280\pi$
$228$	0.344937	0.0228440
$229$	15.3224	1.01254	0.506268	−	0.862376i	$-0.331025\pi$
$229$	15.3224	1.01254	0.506268	+	0.862376i	$0.331025\pi$
$230$	−3.40203	−0.224323
$231$	3.19383	0.210138
$232$	−8.37419	−0.549793
$233$	8.27485	0.542103	0.271052	−	0.962565i	$-0.412629\pi$
$233$	8.27485	0.542103	0.271052	+	0.962565i	$0.412629\pi$
$234$	−6.32138	−0.413242
$235$	−0.843765	−0.0550411
$236$	17.4757	1.13757
$237$	−2.68040	−0.174111
$238$	−22.5415	−1.46115
$239$	−1.03651	−0.0670464	−0.0335232	−	0.999438i	$-0.510673\pi$
$239$	−1.03651	−0.0670464	−0.0335232	+	0.999438i	$0.510673\pi$
$240$	0.235611	0.0152086
$241$	−8.17026	−0.526293	−0.263146	−	0.964756i	$-0.584760\pi$
$241$	−8.17026	−0.526293	−0.263146	+	0.964756i	$0.584760\pi$
$242$	19.9341	1.28141
$243$	5.74167	0.368328
$244$	−29.1108	−1.86363
$245$	−1.58593	−0.101321
$246$	2.59713	0.165587
$247$	0.614865	0.0391229
$248$	−1.24803	−0.0792499
$249$	0.158179	0.0100242
$250$	9.13495	0.577745
$251$	7.28745	0.459980	0.229990	−	0.973193i	$-0.426131\pi$
$251$	7.28745	0.459980	0.229990	+	0.973193i	$0.426131\pi$
$252$	−24.8862	−1.56768
$253$	16.4668	1.03526
$254$	39.6108	2.48540
$255$	0.304842	0.0190899
$256$	3.10582	0.194114
$257$	−12.6816	−0.791054	−0.395527	−	0.918454i	$-0.629438\pi$
$257$	−12.6816	−0.791054	−0.395527	+	0.918454i	$0.629438\pi$
$258$	−2.77486	−0.172755
$259$	−12.8984	−0.801469
$260$	−1.12343	−0.0696724
$261$	19.8133	1.22641
$262$	46.5010	2.87284
$263$	−1.11913	−0.0690086	−0.0345043	−	0.999405i	$-0.510985\pi$
$263$	−1.11913	−0.0690086	−0.0345043	+	0.999405i	$0.510985\pi$
$264$	1.22162	0.0751854
$265$	−0.631749	−0.0388080
$266$	4.29491	0.263338
$267$	1.65938	0.101552
$268$	34.1301	2.08483
$269$	3.18517	0.194203	0.0971015	−	0.995274i	$-0.469043\pi$
$269$	3.18517	0.194203	0.0971015	+	0.995274i	$0.469043\pi$
$270$	1.20383	0.0732625
$271$	−3.72477	−0.226264	−0.113132	−	0.993580i	$-0.536088\pi$
$271$	−3.72477	−0.226264	−0.113132	+	0.993580i	$0.536088\pi$
$272$	8.04892	0.488037
$273$	0.708663	0.0428902
$274$	−2.46771	−0.149080
$275$	−21.6816	−1.30745
$276$	2.04973	0.123379
$277$	22.5857	1.35704	0.678522	−	0.734580i	$-0.262621\pi$
$277$	22.5857	1.35704	0.678522	+	0.734580i	$0.262621\pi$
$278$	12.2569	0.735123
$279$	2.95283	0.176781
$280$	−1.77113	−0.105846
$281$	−13.2604	−0.791050	−0.395525	−	0.918455i	$-0.629437\pi$
$281$	−13.2604	−0.791050	−0.395525	+	0.918455i	$0.629437\pi$
$282$	0.902002	0.0537135
$283$	9.68068	0.575456	0.287728	−	0.957712i	$-0.407100\pi$
$283$	9.68068	0.575456	0.287728	+	0.957712i	$0.407100\pi$
$284$	11.5219	0.683700
$285$	−0.0580826	−0.00344052
$286$	9.64818	0.570509
$287$	18.2255	1.07582
$288$	23.1371	1.36337
$289$	−6.58601	−0.387413
$290$	6.24769	0.366877
$291$	3.51352	0.205966
$292$	−25.4228	−1.48776
$293$	−3.57209	−0.208684	−0.104342	−	0.994541i	$-0.533274\pi$
$293$	−3.57209	−0.208684	−0.104342	+	0.994541i	$0.533274\pi$
$294$	1.69539	0.0988774
$295$	−2.94266	−0.171328
$296$	−4.93357	−0.286758
$297$	−5.82686	−0.338109
$298$	−5.52746	−0.320197
$299$	3.65374	0.211301
$300$	−2.69886	−0.155819
$301$	−19.4728	−1.12239
$302$	29.2450	1.68286
$303$	2.99203	0.171888
$304$	−1.53359	−0.0879574
$305$	4.90185	0.280679
$306$	20.3995	1.16616
$307$	16.8970	0.964362	0.482181	−	0.876072i	$-0.339845\pi$
$307$	16.8970	0.964362	0.482181	+	0.876072i	$0.339845\pi$
$308$	37.9833	2.16430
$309$	0.600188	0.0341435
$310$	0.931109	0.0528835
$311$	−4.16430	−0.236136	−0.118068	−	0.993006i	$-0.537670\pi$
$311$	−4.16430	−0.236136	−0.118068	+	0.993006i	$0.537670\pi$
$312$	0.271059	0.0153457
$313$	−28.8840	−1.63262	−0.816311	−	0.577613i	$-0.803984\pi$
$313$	−28.8840	−1.63262	−0.816311	+	0.577613i	$0.803984\pi$
$314$	−37.3732	−2.10909
$315$	4.19050	0.236108
$316$	−31.8772	−1.79323
$317$	18.3632	1.03138	0.515689	−	0.856776i	$-0.327536\pi$
$317$	18.3632	1.03138	0.515689	+	0.856776i	$0.327536\pi$
$318$	0.675353	0.0378719
$319$	−30.2406	−1.69315
$320$	5.12615	0.286561
$321$	−3.66422	−0.204517
$322$	25.5218	1.42227
$323$	−1.98421	−0.110405
$324$	22.1559	1.23089
$325$	−4.81083	−0.266857
$326$	−3.13428	−0.173592
$327$	−1.31401	−0.0726651
$328$	6.97115	0.384917
$329$	6.32987	0.348977
$330$	−0.911406	−0.0501712
$331$	17.3104	0.951467	0.475734	−	0.879589i	$-0.342183\pi$
$331$	17.3104	0.951467	0.475734	+	0.879589i	$0.342183\pi$
$332$	1.88117	0.103243
$333$	11.6728	0.639665
$334$	−40.2729	−2.20364
$335$	−5.74705	−0.313995
$336$	−1.76754	−0.0964271
$337$	−32.6040	−1.77605	−0.888026	−	0.459794i	$-0.847923\pi$
$337$	−32.6040	−1.77605	−0.888026	+	0.459794i	$0.847923\pi$
$338$	2.14079	0.116444
$339$	−0.00370592	−0.000201278 0
$340$	3.62540	0.196615
$341$	−4.50684	−0.244059
$342$	−3.88680	−0.210174
$343$	−10.9426	−0.590842
$344$	−7.44822	−0.401581
$345$	−0.345146	−0.0185821
$346$	25.9457	1.39485
$347$	9.92326	0.532709	0.266354	−	0.963875i	$-0.414181\pi$
$347$	9.92326	0.532709	0.266354	+	0.963875i	$0.414181\pi$
$348$	−3.76425	−0.201785
$349$	−0.114265	−0.00611649	−0.00305824	−	0.999995i	$-0.500973\pi$
$349$	−0.114265	−0.00611649	−0.00305824	+	0.999995i	$0.500973\pi$
$350$	−33.6042	−1.79622
$351$	−1.29289	−0.0690096
$352$	−35.3137	−1.88223
$353$	−2.01931	−0.107477	−0.0537385	−	0.998555i	$-0.517114\pi$
$353$	−2.01931	−0.107477	−0.0537385	+	0.998555i	$0.517114\pi$
$354$	3.14577	0.167196
$355$	−1.94013	−0.102971
$356$	19.7345	1.04593
$357$	−2.28691	−0.121036
$358$	−52.9194	−2.79688
$359$	−14.2036	−0.749639	−0.374820	−	0.927098i	$-0.622295\pi$
$359$	−14.2036	−0.749639	−0.374820	+	0.927098i	$0.622295\pi$
$360$	1.60284	0.0844770
$361$	−18.6219	−0.980102
$362$	37.3294	1.96199
$363$	2.02238	0.106147
$364$	8.42792	0.441743
$365$	4.28085	0.224070
$366$	−5.24018	−0.273909
$367$	−7.45365	−0.389077	−0.194539	−	0.980895i	$-0.562321\pi$
$367$	−7.45365	−0.389077	−0.194539	+	0.980895i	$0.562321\pi$
$368$	−9.11310	−0.475053
$369$	−16.4937	−0.858628
$370$	3.68076	0.191353
$371$	4.73934	0.246054
$372$	−0.560996	−0.0290863
$373$	−10.2164	−0.528985	−0.264492	−	0.964388i	$-0.585204\pi$
$373$	−10.2164	−0.528985	−0.264492	+	0.964388i	$0.585204\pi$
$374$	−31.1354	−1.60997
$375$	0.926770	0.0478582
$376$	2.42113	0.124860
$377$	−6.70994	−0.345579
$378$	−9.03102	−0.464506
$379$	−26.6346	−1.36813	−0.684064	−	0.729422i	$-0.739789\pi$
$379$	−26.6346	−1.36813	−0.684064	+	0.729422i	$0.739789\pi$
$380$	−0.690760	−0.0354352
$381$	4.01864	0.205881
$382$	18.9136	0.967702
$383$	−12.6423	−0.645994	−0.322997	−	0.946400i	$-0.604690\pi$
$383$	−12.6423	−0.645994	−0.322997	+	0.946400i	$0.604690\pi$
$384$	−2.07635	−0.105958
$385$	−6.39586	−0.325963
$386$	−14.2802	−0.726845
$387$	17.6224	0.895799
$388$	41.7853	2.12133
$389$	19.7259	1.00014	0.500070	−	0.865985i	$-0.333308\pi$
$389$	19.7259	1.00014	0.500070	+	0.865985i	$0.333308\pi$
$390$	−0.202227	−0.0102402
$391$	−11.7909	−0.596290
$392$	4.55074	0.229847
$393$	4.71767	0.237975
$394$	34.0283	1.71432
$395$	5.36768	0.270077
$396$	−34.3740	−1.72736
$397$	−1.25251	−0.0628615	−0.0314308	−	0.999506i	$-0.510006\pi$
$397$	−1.25251	−0.0628615	−0.0314308	+	0.999506i	$0.510006\pi$
$398$	19.5900	0.981957
$399$	0.435732	0.0218139
$400$	11.9991	0.599955
$401$	−16.3846	−0.818207	−0.409104	−	0.912488i	$-0.634159\pi$
$401$	−16.3846	−0.818207	−0.409104	+	0.912488i	$0.634159\pi$
$402$	6.14371	0.306421
$403$	−1.00000	−0.0498135
$404$	35.5833	1.77034
$405$	−3.73076	−0.185383
$406$	−46.8697	−2.32611
$407$	−17.8159	−0.883102
$408$	−0.874727	−0.0433054
$409$	8.71781	0.431068	0.215534	−	0.976496i	$-0.430851\pi$
$409$	8.71781	0.431068	0.215534	+	0.976496i	$0.430851\pi$
$410$	−5.20093	−0.256855
$411$	−0.250357	−0.0123492
$412$	7.13786	0.351657
$413$	22.0756	1.08627
$414$	−23.0967	−1.13514
$415$	−0.316763	−0.0155493
$416$	−7.83558	−0.384171
$417$	1.24351	0.0608947
$418$	5.93233	0.290160
$419$	33.9289	1.65753	0.828767	−	0.559595i	$-0.189043\pi$
$419$	33.9289	1.65753	0.828767	+	0.559595i	$0.189043\pi$
$420$	−0.796135	−0.0388474
$421$	−24.1220	−1.17564	−0.587818	−	0.808993i	$-0.700013\pi$
$421$	−24.1220	−1.17564	−0.587818	+	0.808993i	$0.700013\pi$
$422$	−54.2475	−2.64073
$423$	−5.72839	−0.278524
$424$	1.81277	0.0880358
$425$	15.5249	0.753068
$426$	2.07404	0.100488
$427$	−36.7734	−1.77959
$428$	−43.5775	−2.10640
$429$	0.978839	0.0472588
$430$	5.55685	0.267975
$431$	−9.44242	−0.454825	−0.227413	−	0.973798i	$-0.573027\pi$
$431$	−9.44242	−0.454825	−0.227413	+	0.973798i	$0.573027\pi$
$432$	3.22472	0.155149
$433$	−5.96402	−0.286612	−0.143306	−	0.989678i	$-0.545773\pi$
$433$	−5.96402	−0.286612	−0.143306	+	0.989678i	$0.545773\pi$
$434$	−6.98512	−0.335297
$435$	0.633847	0.0303907
$436$	−15.6272	−0.748406
$437$	2.24656	0.107467
$438$	−4.57632	−0.218665
$439$	−29.3104	−1.39891	−0.699454	−	0.714677i	$-0.746574\pi$
$439$	−29.3104	−1.39891	−0.699454	+	0.714677i	$0.746574\pi$
$440$	−2.44637	−0.116626
$441$	−10.7670	−0.512716
$442$	−6.90848	−0.328603
$443$	−9.68841	−0.460310	−0.230155	−	0.973154i	$-0.573923\pi$
$443$	−9.68841	−0.460310	−0.230155	+	0.973154i	$0.573923\pi$
$444$	−2.21767	−0.105246
$445$	−3.32301	−0.157526
$446$	−37.3011	−1.76626
$447$	−0.560778	−0.0265239
$448$	−38.4561	−1.81688
$449$	−16.5406	−0.780598	−0.390299	−	0.920688i	$-0.627628\pi$
$449$	−16.5406	−0.780598	−0.390299	+	0.920688i	$0.627628\pi$
$450$	30.4111	1.43359
$451$	25.1740	1.18540
$452$	−0.0440735	−0.00207304
$453$	2.96700	0.139402
$454$	55.3927	2.59971
$455$	−1.41915	−0.0665306
$456$	0.166665	0.00780479
$457$	−12.5124	−0.585304	−0.292652	−	0.956219i	$-0.594538\pi$
$457$	−12.5124	−0.585304	−0.292652	+	0.956219i	$0.594538\pi$
$458$	32.8021	1.53274
$459$	4.17226	0.194744
$460$	−4.10473	−0.191384
$461$	−35.1691	−1.63799	−0.818993	−	0.573803i	$-0.805467\pi$
$461$	−35.1691	−1.63799	−0.818993	+	0.573803i	$0.805467\pi$
$462$	6.83731	0.318100
$463$	1.00356	0.0466396	0.0233198	−	0.999728i	$-0.492576\pi$
$463$	1.00356	0.0466396	0.0233198	+	0.999728i	$0.492576\pi$
$464$	16.7358	0.776942
$465$	0.0944640	0.00438066
$466$	17.7147	0.820618
$467$	−26.3026	−1.21714	−0.608570	−	0.793501i	$-0.708256\pi$
$467$	−26.3026	−1.21714	−0.608570	+	0.793501i	$0.708256\pi$
$468$	−7.62709	−0.352562
$469$	43.1140	1.99082
$470$	−1.80632	−0.0833194
$471$	−3.79163	−0.174709
$472$	8.44379	0.388657
$473$	−26.8967	−1.23671
$474$	−5.73816	−0.263563
$475$	−2.95801	−0.135723
$476$	−27.1975	−1.24660
$477$	−4.28900	−0.196380
$478$	−2.21895	−0.101493
$479$	20.3614	0.930337	0.465168	−	0.885222i	$-0.345994\pi$
$479$	20.3614	0.930337	0.465168	+	0.885222i	$0.345994\pi$
$480$	0.740180	0.0337845
$481$	−3.95309	−0.180245
$482$	−17.4908	−0.796684
$483$	2.58927	0.117816
$484$	24.0515	1.09325
$485$	−7.03607	−0.319492
$486$	12.2917	0.557563
$487$	30.3756	1.37645	0.688226	−	0.725497i	$-0.258390\pi$
$487$	30.3756	1.37645	0.688226	+	0.725497i	$0.258390\pi$
$488$	−14.0656	−0.636719
$489$	−0.317982	−0.0143797
$490$	−3.39515	−0.153377
$491$	1.21101	0.0546523	0.0273261	−	0.999627i	$-0.491301\pi$
$491$	1.21101	0.0546523	0.0273261	+	0.999627i	$0.491301\pi$
$492$	3.13357	0.141272
$493$	21.6534	0.975222
$494$	1.31630	0.0592230
$495$	5.78811	0.260156
$496$	2.49419	0.111992
$497$	14.5547	0.652869
$498$	0.338627	0.0151742
$499$	2.45735	0.110006	0.0550030	−	0.998486i	$-0.482483\pi$
$499$	2.45735	0.110006	0.0550030	+	0.998486i	$0.482483\pi$
$500$	11.0218	0.492910
$501$	−4.08582	−0.182541
$502$	15.6009	0.696302
$503$	32.8106	1.46295	0.731475	−	0.681868i	$-0.238832\pi$
$503$	32.8106	1.46295	0.731475	+	0.681868i	$0.238832\pi$
$504$	−12.0244	−0.535609
$505$	−5.99174	−0.266629
$506$	35.2519	1.56714
$507$	0.217190	0.00964574
$508$	47.7926	2.12045
$509$	−13.1701	−0.583754	−0.291877	−	0.956456i	$-0.594280\pi$
$509$	−13.1701	−0.583754	−0.291877	+	0.956456i	$0.594280\pi$
$510$	0.652602	0.0288977
$511$	−32.1147	−1.42067
$512$	25.7690	1.13884
$513$	−0.794955	−0.0350981
$514$	−27.1486	−1.19747
$515$	−1.20192	−0.0529628
$516$	−3.34802	−0.147388
$517$	8.74312	0.384522
$518$	−27.6128	−1.21324
$519$	2.63227	0.115544
$520$	−0.542814	−0.0238040
$521$	34.9324	1.53042	0.765208	−	0.643783i	$-0.222636\pi$
$521$	34.9324	1.53042	0.765208	+	0.643783i	$0.222636\pi$
$522$	42.4161	1.85650
$523$	−38.0898	−1.66555	−0.832775	−	0.553611i	$-0.813249\pi$
$523$	−38.0898	−1.66555	−0.832775	+	0.553611i	$0.813249\pi$
$524$	56.1059	2.45100
$525$	−3.40926	−0.148792
$526$	−2.39582	−0.104463
$527$	3.22707	0.140573
$528$	−2.44141	−0.106249
$529$	−9.65021	−0.419574
$530$	−1.35244	−0.0587463
$531$	−19.9780	−0.866970
$532$	5.18204	0.224670
$533$	5.58573	0.241945
$534$	3.55237	0.153726
$535$	7.33784	0.317243
$536$	16.4908	0.712295
$537$	−5.36884	−0.231683
$538$	6.81877	0.293978
$539$	16.4335	0.707840
$540$	1.45248	0.0625048
$541$	−1.90581	−0.0819373	−0.0409686	−	0.999160i	$-0.513044\pi$
$541$	−1.90581	−0.0819373	−0.0409686	+	0.999160i	$0.513044\pi$
$542$	−7.97395	−0.342510
$543$	3.78719	0.162524
$544$	25.2860	1.08413
$545$	2.63140	0.112717
$546$	1.51710	0.0649258
$547$	−36.8714	−1.57651	−0.788253	−	0.615352i	$-0.789014\pi$
$547$	−36.8714	−1.57651	−0.788253	+	0.615352i	$0.789014\pi$
$548$	−2.97742	−0.127189
$549$	33.2791	1.42032
$550$	−46.4158	−1.97917
$551$	−4.12571	−0.175761
$552$	0.990378	0.0421533
$553$	−40.2680	−1.71237
$554$	48.3513	2.05425
$555$	0.373425	0.0158510
$556$	14.7887	0.627179
$557$	−25.1882	−1.06726	−0.533629	−	0.845719i	$-0.679172\pi$
$557$	−25.1882	−1.06726	−0.533629	+	0.845719i	$0.679172\pi$
$558$	6.32138	0.267605
$559$	−5.96799	−0.252419
$560$	3.53962	0.149576
$561$	−3.15878	−0.133364
$562$	−28.3878	−1.19747
$563$	22.2420	0.937387	0.468693	−	0.883361i	$-0.344725\pi$
$563$	22.2420	0.937387	0.468693	+	0.883361i	$0.344725\pi$
$564$	1.08831	0.0458263
$565$	0.00742137	0.000312219 0
$566$	20.7243	0.871106
$567$	27.9879	1.17538
$568$	5.56709	0.233590
$569$	43.9251	1.84143	0.920717	−	0.390230i	$-0.127605\pi$
$569$	43.9251	1.84143	0.920717	+	0.390230i	$0.127605\pi$
$570$	−0.124343	−0.00520814
$571$	−26.4232	−1.10578	−0.552888	−	0.833256i	$-0.686474\pi$
$571$	−26.4232	−1.10578	−0.552888	+	0.833256i	$0.686474\pi$
$572$	11.6410	0.486737
$573$	1.91884	0.0801607
$574$	39.0170	1.62854
$575$	−17.5775	−0.733032
$576$	34.8019	1.45008
$577$	−7.94674	−0.330827	−0.165413	−	0.986224i	$-0.552896\pi$
$577$	−7.94674	−0.330827	−0.165413	+	0.986224i	$0.552896\pi$
$578$	−14.0993	−0.586452
$579$	−1.44878	−0.0602091
$580$	7.53817	0.313005
$581$	2.37634	0.0985871
$582$	7.52171	0.311785
$583$	6.54620	0.271116
$584$	−12.2837	−0.508302
$585$	1.28430	0.0530991
$586$	−7.64709	−0.315898
$587$	3.37263	0.139204	0.0696018	−	0.997575i	$-0.477827\pi$
$587$	3.37263	0.139204	0.0696018	+	0.997575i	$0.477827\pi$
$588$	2.04558	0.0843585
$589$	−0.614865	−0.0253351
$590$	−6.29961	−0.259351
$591$	3.45228	0.142008
$592$	9.85974	0.405233
$593$	0.257218	0.0105627	0.00528135	−	0.999986i	$-0.498319\pi$
$593$	0.257218	0.0105627	0.00528135	+	0.999986i	$0.498319\pi$
$594$	−12.4741	−0.511817
$595$	4.57969	0.187749
$596$	−6.66918	−0.273180
$597$	1.98747	0.0813415
$598$	7.82188	0.319860
$599$	−26.5991	−1.08681	−0.543405	−	0.839471i	$-0.682865\pi$
$599$	−26.5991	−1.08681	−0.543405	+	0.839471i	$0.682865\pi$
$600$	−1.30402	−0.0532363
$601$	−19.6836	−0.802912	−0.401456	−	0.915878i	$-0.631496\pi$
$601$	−19.6836	−0.802912	−0.401456	+	0.915878i	$0.631496\pi$
$602$	−41.6871	−1.69904
$603$	−39.0172	−1.58890
$604$	35.2856	1.43575
$605$	−4.04995	−0.164654
$606$	6.40530	0.260198
$607$	44.9999	1.82649	0.913245	−	0.407410i	$-0.133568\pi$
$607$	44.9999	1.82649	0.913245	+	0.407410i	$0.133568\pi$
$608$	−4.81783	−0.195389
$609$	−4.75508	−0.192686
$610$	10.4938	0.424883
$611$	1.93997	0.0784827
$612$	24.6131	0.994927
$613$	15.0147	0.606436	0.303218	−	0.952921i	$-0.401939\pi$
$613$	15.0147	0.606436	0.303218	+	0.952921i	$0.401939\pi$
$614$	36.1729	1.45982
$615$	−0.527650	−0.0212769
$616$	18.3525	0.739445
$617$	16.0922	0.647846	0.323923	−	0.946083i	$-0.394998\pi$
$617$	16.0922	0.647846	0.323923	+	0.946083i	$0.394998\pi$
$618$	1.28488	0.0516853
$619$	24.0427	0.966357	0.483179	−	0.875522i	$-0.339482\pi$
$619$	24.0427	0.966357	0.483179	+	0.875522i	$0.339482\pi$
$620$	1.12343	0.0451182
$621$	−4.72389	−0.189563
$622$	−8.91489	−0.357455
$623$	24.9290	0.998761
$624$	−0.541712	−0.0216858
$625$	22.1982	0.887929
$626$	−61.8346	−2.47141
$627$	0.601854	0.0240357
$628$	−45.0928	−1.79940
$629$	12.7569	0.508651
$630$	8.97096	0.357412
$631$	−8.21496	−0.327033	−0.163516	−	0.986541i	$-0.552284\pi$
$631$	−8.21496	−0.327033	−0.163516	+	0.986541i	$0.552284\pi$
$632$	−15.4023	−0.612669
$633$	−5.50358	−0.218748
$634$	39.3117	1.56127
$635$	−8.04761	−0.319360
$636$	0.814850	0.0323109
$637$	3.64634	0.144473
$638$	−64.7387	−2.56303
$639$	−13.1717	−0.521065
$640$	4.15803	0.164361
$641$	−7.51297	−0.296745	−0.148372	−	0.988932i	$-0.547403\pi$
$641$	−7.51297	−0.296745	−0.148372	+	0.988932i	$0.547403\pi$
$642$	−7.84431	−0.309590
$643$	5.91691	0.233340	0.116670	−	0.993171i	$-0.462778\pi$
$643$	5.91691	0.233340	0.116670	+	0.993171i	$0.462778\pi$
$644$	30.7934	1.21343
$645$	0.563760	0.0221980
$646$	−4.24778	−0.167127
$647$	18.8047	0.739289	0.369645	−	0.929173i	$-0.379479\pi$
$647$	18.8047	0.739289	0.369645	+	0.929173i	$0.379479\pi$
$648$	10.7052	0.420540
$649$	30.4919	1.19691
$650$	−10.2990	−0.403959
$651$	−0.708663	−0.0277747
$652$	−3.78167	−0.148102
$653$	36.0448	1.41054	0.705271	−	0.708938i	$-0.250826\pi$
$653$	36.0448	1.41054	0.705271	+	0.708938i	$0.250826\pi$
$654$	−2.81302	−0.109998
$655$	−9.44746	−0.369143
$656$	−13.9319	−0.543948
$657$	29.0631	1.13386
$658$	13.5509	0.528270
$659$	2.28678	0.0890802	0.0445401	−	0.999008i	$-0.485818\pi$
$659$	2.28678	0.0890802	0.0445401	+	0.999008i	$0.485818\pi$
$660$	−1.09966	−0.0428042
$661$	47.1402	1.83354	0.916770	−	0.399416i	$-0.130787\pi$
$661$	47.1402	1.83354	0.916770	+	0.399416i	$0.130787\pi$
$662$	37.0580	1.44030
$663$	−0.700887	−0.0272202
$664$	0.908934	0.0352735
$665$	−0.872584	−0.0338373
$666$	24.9890	0.968304
$667$	−24.5163	−0.949277
$668$	−48.5915	−1.88006
$669$	−3.78431	−0.146310
$670$	−12.3032	−0.475314
$671$	−50.7931	−1.96085
$672$	−5.55279	−0.214203
$673$	24.7563	0.954287	0.477143	−	0.878825i	$-0.341672\pi$
$673$	24.7563	0.954287	0.477143	+	0.878825i	$0.341672\pi$
$674$	−69.7982	−2.68853
$675$	6.21989	0.239404
$676$	2.58298	0.0993452
$677$	50.1565	1.92767	0.963835	−	0.266500i	$-0.0858674\pi$
$677$	50.1565	1.92767	0.963835	+	0.266500i	$0.0858674\pi$
$678$	−0.00793360	−0.000304688 0
$679$	52.7842	2.02567
$680$	1.75170	0.0671746
$681$	5.61977	0.215350
$682$	−9.64818	−0.369448
$683$	43.6197	1.66906	0.834530	−	0.550963i	$-0.185739\pi$
$683$	43.6197	1.66906	0.834530	+	0.550963i	$0.185739\pi$
$684$	−4.68963	−0.179312
$685$	0.501357	0.0191559
$686$	−23.4257	−0.894397
$687$	3.32788	0.126966
$688$	14.8853	0.567496
$689$	1.45251	0.0553360
$690$	−0.738886	−0.0281289
$691$	43.8806	1.66930	0.834648	−	0.550784i	$-0.185671\pi$
$691$	43.8806	1.66930	0.834648	+	0.550784i	$0.185671\pi$
$692$	31.3049	1.19003
$693$	−43.4220	−1.64947
$694$	21.2436	0.806396
$695$	−2.49021	−0.0944589
$696$	−1.81879	−0.0689410
$697$	−18.0256	−0.682766
$698$	−0.244618	−0.00925893
$699$	1.79721	0.0679768
$700$	−40.5453	−1.53247
$701$	9.78593	0.369609	0.184805	−	0.982775i	$-0.440835\pi$
$701$	9.78593	0.369609	0.184805	+	0.982775i	$0.440835\pi$
$702$	−2.76781	−0.104464
$703$	−2.43062	−0.0916725
$704$	−53.1174	−2.00194
$705$	−0.183257	−0.00690186
$706$	−4.32291	−0.162695
$707$	44.9497	1.69051
$708$	3.79553	0.142645
$709$	−45.0119	−1.69046	−0.845228	−	0.534406i	$-0.820535\pi$
$709$	−45.0119	−1.69046	−0.845228	+	0.534406i	$0.820535\pi$
$710$	−4.15341	−0.155875
$711$	36.4416	1.36667
$712$	9.53520	0.357347
$713$	−3.65374	−0.136833
$714$	−4.89578	−0.183220
$715$	−1.96019	−0.0733070
$716$	−63.8501	−2.38619
$717$	−0.225120	−0.00840725
$718$	−30.4070	−1.13478
$719$	−18.0945	−0.674809	−0.337405	−	0.941360i	$-0.609549\pi$
$719$	−18.0945	−0.674809	−0.337405	+	0.941360i	$0.609549\pi$
$720$	−3.20327	−0.119379
$721$	9.01671	0.335800
$722$	−39.8656	−1.48365
$723$	−1.77450	−0.0659943
$724$	45.0400	1.67390
$725$	32.2804	1.19886
$726$	4.32948	0.160682
$727$	1.20004	0.0445068	0.0222534	−	0.999752i	$-0.492916\pi$
$727$	1.20004	0.0445068	0.0222534	+	0.999752i	$0.492916\pi$
$728$	4.07216	0.150924
$729$	−24.4860	−0.906889
$730$	9.16440	0.339190
$731$	19.2591	0.712324
$732$	−6.32256	−0.233689
$733$	18.0235	0.665712	0.332856	−	0.942978i	$-0.391988\pi$
$733$	18.0235	0.665712	0.332856	+	0.942978i	$0.391988\pi$
$734$	−15.9567	−0.588972
$735$	−0.344448	−0.0127052
$736$	−28.6292	−1.05528
$737$	59.5511	2.19359
$738$	−35.3095	−1.29976
$739$	22.7338	0.836278	0.418139	−	0.908383i	$-0.362683\pi$
$739$	22.7338	0.836278	0.418139	+	0.908383i	$0.362683\pi$
$740$	4.44103	0.163256
$741$	0.133542	0.00490580
$742$	10.1459	0.372469
$743$	−26.0795	−0.956764	−0.478382	−	0.878152i	$-0.658777\pi$
$743$	−26.0795	−0.956764	−0.478382	+	0.878152i	$0.658777\pi$
$744$	−0.271059	−0.00993750
$745$	1.12300	0.0411434
$746$	−21.8711	−0.800759
$747$	−2.15053	−0.0786839
$748$	−37.5665	−1.37357
$749$	−55.0480	−2.01141
$750$	1.98402	0.0724461
$751$	24.8379	0.906349	0.453174	−	0.891422i	$-0.350291\pi$
$751$	24.8379	0.906349	0.453174	+	0.891422i	$0.350291\pi$
$752$	−4.83864	−0.176447
$753$	1.58276	0.0576790
$754$	−14.3646	−0.523126
$755$	−5.94161	−0.216237
$756$	−10.8964	−0.396299
$757$	18.4042	0.668913	0.334457	−	0.942411i	$-0.391447\pi$
$757$	18.4042	0.668913	0.334457	+	0.942411i	$0.391447\pi$
$758$	−57.0191	−2.07103
$759$	3.57642	0.129816
$760$	−0.333758	−0.0121067
$761$	−44.2883	−1.60545	−0.802725	−	0.596350i	$-0.796617\pi$
$761$	−44.2883	−1.60545	−0.802725	+	0.596350i	$0.796617\pi$
$762$	8.60307	0.311656
$763$	−19.7406	−0.714658
$764$	22.8202	0.825606
$765$	−4.14451	−0.149845
$766$	−27.0646	−0.977883
$767$	6.76571	0.244296
$768$	0.674553	0.0243408
$769$	−8.14338	−0.293658	−0.146829	−	0.989162i	$-0.546907\pi$
$769$	−8.14338	−0.293658	−0.146829	+	0.989162i	$0.546907\pi$
$770$	−13.6922	−0.493432
$771$	−2.75431	−0.0991939
$772$	−17.2299	−0.620117
$773$	32.8412	1.18122	0.590609	−	0.806958i	$-0.298888\pi$
$773$	32.8412	1.18122	0.590609	+	0.806958i	$0.298888\pi$
$774$	37.7259	1.35603
$775$	4.81083	0.172810
$776$	20.1896	0.724765
$777$	−2.80141	−0.100500
$778$	42.2289	1.51398
$779$	3.43447	0.123053
$780$	−0.243998	−0.00873654
$781$	20.1037	0.719367
$782$	−25.2417	−0.902643
$783$	8.67524	0.310028
$784$	−9.09467	−0.324809
$785$	7.59300	0.271006
$786$	10.0995	0.360238
$787$	12.4701	0.444511	0.222255	−	0.974988i	$-0.428658\pi$
$787$	12.4701	0.444511	0.222255	+	0.974988i	$0.428658\pi$
$788$	41.0570	1.46259
$789$	−0.243064	−0.00865331
$790$	11.4911	0.408834
$791$	−0.0556746	−0.00197956
$792$	−16.6087	−0.590163
$793$	−11.2702	−0.400218
$794$	−2.68135	−0.0951577
$795$	−0.137209	−0.00486632
$796$	23.6364	0.837768
$797$	5.85388	0.207355	0.103678	−	0.994611i	$-0.466939\pi$
$797$	5.85388	0.207355	0.103678	+	0.994611i	$0.466939\pi$
$798$	0.932810	0.0330211
$799$	−6.26041	−0.221478
$800$	37.6957	1.33274
$801$	−22.5602	−0.797126
$802$	−35.0759	−1.23857
$803$	−44.3583	−1.56537
$804$	7.41272	0.261426
$805$	−5.18519	−0.182754
$806$	−2.14079	−0.0754061
$807$	0.691786	0.0243520
$808$	17.1930	0.604846
$809$	−26.8217	−0.943002	−0.471501	−	0.881866i	$-0.656288\pi$
$809$	−26.8217	−0.943002	−0.471501	+	0.881866i	$0.656288\pi$
$810$	−7.98676	−0.280626
$811$	−43.6595	−1.53309	−0.766545	−	0.642190i	$-0.778026\pi$
$811$	−43.6595	−1.53309	−0.766545	+	0.642190i	$0.778026\pi$
$812$	−56.5508	−1.98455
$813$	−0.808982	−0.0283723
$814$	−38.1401	−1.33681
$815$	0.636782	0.0223055
$816$	1.74814	0.0611972
$817$	−3.66951	−0.128380
$818$	18.6630	0.652536
$819$	−9.63471	−0.336664
$820$	−6.27520	−0.219139
$821$	23.7241	0.827976	0.413988	−	0.910282i	$-0.364135\pi$
$821$	23.7241	0.827976	0.413988	+	0.910282i	$0.364135\pi$
$822$	−0.535961	−0.0186938
$823$	−7.26516	−0.253247	−0.126624	−	0.991951i	$-0.540414\pi$
$823$	−7.26516	−0.253247	−0.126624	+	0.991951i	$0.540414\pi$
$824$	3.44883	0.120146
$825$	−4.70903	−0.163947
$826$	47.2593	1.64436
$827$	−21.7786	−0.757315	−0.378658	−	0.925537i	$-0.623614\pi$
$827$	−21.7786	−0.757315	−0.378658	+	0.925537i	$0.623614\pi$
$828$	−27.8674	−0.968458
$829$	27.7111	0.962446	0.481223	−	0.876598i	$-0.340193\pi$
$829$	27.7111	0.962446	0.481223	+	0.876598i	$0.340193\pi$
$830$	−0.678123	−0.0235380
$831$	4.90539	0.170166
$832$	−11.7860	−0.408604
$833$	−11.7670	−0.407703
$834$	2.66208	0.0921804
$835$	8.18213	0.283154
$836$	7.15767	0.247553
$837$	1.29289	0.0446889
$838$	72.6345	2.50912
$839$	−25.8270	−0.891646	−0.445823	−	0.895121i	$-0.647089\pi$
$839$	−25.8270	−0.891646	−0.445823	+	0.895121i	$0.647089\pi$
$840$	−0.384672	−0.0132725
$841$	16.0233	0.552527
$842$	−51.6402	−1.77964
$843$	−2.88003	−0.0991934
$844$	−65.4525	−2.25297
$845$	−0.434938	−0.0149623
$846$	−12.2633	−0.421620
$847$	30.3825	1.04395
$848$	−3.62282	−0.124408
$849$	2.10254	0.0721591
$850$	33.2355	1.13997
$851$	−14.4435	−0.495118
$852$	2.50244	0.0857322
$853$	14.7461	0.504895	0.252448	−	0.967611i	$-0.418764\pi$
$853$	14.7461	0.504895	0.252448	+	0.967611i	$0.418764\pi$
$854$	−78.7240	−2.69388
$855$	0.789669	0.0270061
$856$	−21.0555	−0.719663
$857$	−12.4661	−0.425833	−0.212917	−	0.977070i	$-0.568296\pi$
$857$	−12.4661	−0.425833	−0.212917	+	0.977070i	$0.568296\pi$
$858$	2.09549	0.0715387
$859$	−2.99735	−0.102268	−0.0511341	−	0.998692i	$-0.516284\pi$
$859$	−2.99735	−0.102268	−0.0511341	+	0.998692i	$0.516284\pi$
$860$	6.70463	0.228626
$861$	3.95840	0.134902
$862$	−20.2142	−0.688499
$863$	27.8543	0.948170	0.474085	−	0.880479i	$-0.342779\pi$
$863$	27.8543	0.948170	0.474085	+	0.880479i	$0.342779\pi$
$864$	10.1306	0.344649
$865$	−5.27131	−0.179230
$866$	−12.7677	−0.433864
$867$	−1.43041	−0.0485794
$868$	−8.42792	−0.286062
$869$	−55.6201	−1.88678
$870$	1.35693	0.0460044
$871$	13.2135	0.447722
$872$	−7.55065	−0.255697
$873$	−47.7685	−1.61672
$874$	4.80940	0.162680
$875$	13.9230	0.470683
$876$	−5.52158	−0.186557
$877$	19.6476	0.663452	0.331726	−	0.943376i	$-0.392369\pi$
$877$	19.6476	0.663452	0.331726	+	0.943376i	$0.392369\pi$
$878$	−62.7474	−2.11762
$879$	−0.775822	−0.0261678
$880$	4.88909	0.164811
$881$	−54.1200	−1.82335	−0.911675	−	0.410913i	$-0.865210\pi$
$881$	−54.1200	−1.82335	−0.911675	+	0.410913i	$0.865210\pi$
$882$	−23.0499	−0.776132
$883$	−14.7926	−0.497810	−0.248905	−	0.968528i	$-0.580071\pi$
$883$	−14.7926	−0.497810	−0.248905	+	0.968528i	$0.580071\pi$
$884$	−8.33545	−0.280351
$885$	−0.639116	−0.0214836
$886$	−20.7408	−0.696802
$887$	−44.7176	−1.50147	−0.750736	−	0.660603i	$-0.770301\pi$
$887$	−44.7176	−1.50147	−0.750736	+	0.660603i	$0.770301\pi$
$888$	−1.07152	−0.0359579
$889$	60.3727	2.02483
$890$	−7.11387	−0.238457
$891$	38.6582	1.29510
$892$	−45.0057	−1.50690
$893$	1.19282	0.0399162
$894$	−1.20051	−0.0401510
$895$	10.7515	0.359382
$896$	−31.1933	−1.04210
$897$	0.793554	0.0264960
$898$	−35.4099	−1.18164
$899$	6.70994	0.223789
$900$	36.6926	1.22309
$901$	−4.68734	−0.156158
$902$	53.8921	1.79441
$903$	−4.22929	−0.140742
$904$	−0.0212952	−0.000708267 0
$905$	−7.58411	−0.252104
$906$	6.35171	0.211021
$907$	−5.54945	−0.184267	−0.0921333	−	0.995747i	$-0.529369\pi$
$907$	−5.54945	−0.184267	−0.0921333	+	0.995747i	$0.529369\pi$
$908$	66.8343	2.21797
$909$	−40.6785	−1.34922
$910$	−3.03809	−0.100712
$911$	−30.5521	−1.01224	−0.506119	−	0.862464i	$-0.668920\pi$
$911$	−30.5521	−1.01224	−0.506119	+	0.862464i	$0.668920\pi$
$912$	−0.333080	−0.0110294
$913$	3.28231	0.108629
$914$	−26.7864	−0.886014
$915$	1.06463	0.0351957
$916$	39.5775	1.30768
$917$	70.8742	2.34047
$918$	8.93192	0.294797
$919$	31.8807	1.05165	0.525823	−	0.850594i	$-0.323757\pi$
$919$	31.8807	1.05165	0.525823	+	0.850594i	$0.323757\pi$
$920$	−1.98330	−0.0653875
$921$	3.66985	0.120926
$922$	−75.2895	−2.47953
$923$	4.46071	0.146826
$924$	8.24958	0.271391
$925$	19.0176	0.625296
$926$	2.14842	0.0706014
$927$	−8.15992	−0.268007
$928$	52.5763	1.72590
$929$	−19.2324	−0.630993	−0.315497	−	0.948927i	$-0.602171\pi$
$929$	−19.2324	−0.630993	−0.315497	+	0.948927i	$0.602171\pi$
$930$	0.202227	0.00663130
$931$	2.24201	0.0734789
$932$	21.3737	0.700120
$933$	−0.904444	−0.0296102
$934$	−56.3083	−1.84246
$935$	6.32568	0.206872
$936$	−3.68521	−0.120455
$937$	−11.7430	−0.383628	−0.191814	−	0.981431i	$-0.561437\pi$
$937$	−11.7430	−0.383628	−0.191814	+	0.981431i	$0.561437\pi$
$938$	92.2979	3.01363
$939$	−6.27332	−0.204722
$940$	−2.17942	−0.0710850
$941$	−44.3583	−1.44604	−0.723019	−	0.690828i	$-0.757246\pi$
$941$	−44.3583	−1.44604	−0.723019	+	0.690828i	$0.757246\pi$
$942$	−8.11708	−0.264469
$943$	20.4088	0.664602
$944$	−16.8749	−0.549233
$945$	1.83480	0.0596862
$946$	−57.5802	−1.87209
$947$	21.7921	0.708148	0.354074	−	0.935217i	$-0.384796\pi$
$947$	21.7921	0.708148	0.354074	+	0.935217i	$0.384796\pi$
$948$	−6.92340	−0.224862
$949$	−9.84245	−0.319500
$950$	−6.33248	−0.205453
$951$	3.98829	0.129329
$952$	−13.1411	−0.425907
$953$	50.8773	1.64808	0.824039	−	0.566533i	$-0.191715\pi$
$953$	50.8773	1.64808	0.824039	+	0.566533i	$0.191715\pi$
$954$	−9.18184	−0.297273
$955$	−3.84261	−0.124344
$956$	−2.67729	−0.0865896
$957$	−6.56795	−0.212312
$958$	43.5895	1.40831
$959$	−3.76115	−0.121454
$960$	1.11335	0.0359332
$961$	1.00000	0.0322581
$962$	−8.46273	−0.272849
$963$	49.8173	1.60534
$964$	−21.1036	−0.679701
$965$	2.90127	0.0933953
$966$	5.54307	0.178345
$967$	30.4121	0.977986	0.488993	−	0.872288i	$-0.337364\pi$
$967$	30.4121	0.977986	0.488993	+	0.872288i	$0.337364\pi$
$968$	11.6211	0.373516
$969$	−0.430951	−0.0138441
$970$	−15.0627	−0.483636
$971$	−25.7328	−0.825805	−0.412902	−	0.910775i	$-0.635485\pi$
$971$	−25.7328	−0.825805	−0.412902	+	0.910775i	$0.635485\pi$
$972$	14.8306	0.475691
$973$	18.6814	0.598897
$974$	65.0278	2.08363
$975$	−1.04486	−0.0334624
$976$	28.1101	0.899782
$977$	−49.7489	−1.59161	−0.795804	−	0.605554i	$-0.792952\pi$
$977$	−49.7489	−1.59161	−0.795804	+	0.605554i	$0.792952\pi$
$978$	−0.680733	−0.0217674
$979$	34.4332	1.10049
$980$	−4.09642	−0.130855
$981$	17.8648	0.570380
$982$	2.59252	0.0827307
$983$	33.1292	1.05666	0.528329	−	0.849040i	$-0.322819\pi$
$983$	33.1292	1.05666	0.528329	+	0.849040i	$0.322819\pi$
$984$	1.51406	0.0482666
$985$	−6.91343	−0.220280
$986$	46.3555	1.47626
$987$	1.37478	0.0437598
$988$	1.58818	0.0505268
$989$	−21.8054	−0.693373
$990$	12.3911	0.393816
$991$	−35.2579	−1.12000	−0.560002	−	0.828492i	$-0.689199\pi$
$991$	−35.2579	−1.12000	−0.560002	+	0.828492i	$0.689199\pi$
$992$	7.83558	0.248780
$993$	3.75965	0.119309
$994$	31.1586	0.988291
$995$	−3.98004	−0.126176
$996$	0.408571	0.0129461
$997$	−45.4250	−1.43862	−0.719311	−	0.694688i	$-0.755543\pi$
$997$	−45.4250	−1.43862	−0.719311	+	0.694688i	$0.755543\pi$
$998$	5.26067	0.166523
$999$	5.11092	0.161702

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	403.2.a.e.1.8	✓	8
3.2	odd	2		3627.2.a.p.1.1		8
4.3	odd	2		6448.2.a.bd.1.6		8
13.12	even	2		5239.2.a.i.1.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
403.2.a.e.1.8	✓	8	1.1	even	1	trivial
3627.2.a.p.1.1		8	3.2	odd	2
5239.2.a.i.1.1		8	13.12	even	2
6448.2.a.bd.1.6		8	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:	\( N \)	\(=\)	\( 403 = 13 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	403.a (trivial)

\(f(q)\)	\(=\)	\(q+2.14079 q^{2} +0.217190 q^{3} +2.58298 q^{4} -0.434938 q^{5} +0.464957 q^{6} +3.26287 q^{7} +1.24803 q^{8} -2.95283 q^{9} +O(q^{10})\)	\(q+2.14079 q^{2} +0.217190 q^{3} +2.58298 q^{4} -0.434938 q^{5} +0.464957 q^{6} +3.26287 q^{7} +1.24803 q^{8} -2.95283 q^{9} -0.931109 q^{10} +4.50684 q^{11} +0.560996 q^{12} +1.00000 q^{13} +6.98512 q^{14} -0.0944640 q^{15} -2.49419 q^{16} -3.22707 q^{17} -6.32138 q^{18} +0.614865 q^{19} -1.12343 q^{20} +0.708663 q^{21} +9.64818 q^{22} +3.65374 q^{23} +0.271059 q^{24} -4.81083 q^{25} +2.14079 q^{26} -1.29289 q^{27} +8.42792 q^{28} -6.70994 q^{29} -0.202227 q^{30} -1.00000 q^{31} -7.83558 q^{32} +0.978839 q^{33} -6.90848 q^{34} -1.41915 q^{35} -7.62709 q^{36} -3.95309 q^{37} +1.31630 q^{38} +0.217190 q^{39} -0.542814 q^{40} +5.58573 q^{41} +1.51710 q^{42} -5.96799 q^{43} +11.6410 q^{44} +1.28430 q^{45} +7.82188 q^{46} +1.93997 q^{47} -0.541712 q^{48} +3.64634 q^{49} -10.2990 q^{50} -0.700887 q^{51} +2.58298 q^{52} +1.45251 q^{53} -2.76781 q^{54} -1.96019 q^{55} +4.07216 q^{56} +0.133542 q^{57} -14.3646 q^{58} +6.76571 q^{59} -0.243998 q^{60} -11.2702 q^{61} -2.14079 q^{62} -9.63471 q^{63} -11.7860 q^{64} -0.434938 q^{65} +2.09549 q^{66} +13.2135 q^{67} -8.33545 q^{68} +0.793554 q^{69} -3.03809 q^{70} +4.46071 q^{71} -3.68521 q^{72} -9.84245 q^{73} -8.46273 q^{74} -1.04486 q^{75} +1.58818 q^{76} +14.7052 q^{77} +0.464957 q^{78} -12.3413 q^{79} +1.08482 q^{80} +8.57768 q^{81} +11.9579 q^{82} +0.728296 q^{83} +1.83046 q^{84} +1.40357 q^{85} -12.7762 q^{86} -1.45733 q^{87} +5.62466 q^{88} +7.64021 q^{89} +2.74941 q^{90} +3.26287 q^{91} +9.43751 q^{92} -0.217190 q^{93} +4.15306 q^{94} -0.267428 q^{95} -1.70181 q^{96} +16.1772 q^{97} +7.80605 q^{98} -13.3079 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - q^{2} + 7 q^{3} + 7 q^{4} + 11 q^{5} - 2 q^{7} + 9 q^{9}+O(q^{10}) \) 8 * q - q^2 + 7 * q^3 + 7 * q^4 + 11 * q^5 - 2 * q^7 + 9 * q^9	\( 8 q - q^{2} + 7 q^{3} + 7 q^{4} + 11 q^{5} - 2 q^{7} + 9 q^{9} - 2 q^{10} - 2 q^{11} + 19 q^{12} + 8 q^{13} + 3 q^{14} + 6 q^{15} - 7 q^{16} + 7 q^{17} - 12 q^{18} - 5 q^{19} + 6 q^{20} + 8 q^{21} - 4 q^{22} + 14 q^{23} - 17 q^{24} + 17 q^{25} - q^{26} + 7 q^{27} - 9 q^{28} + 12 q^{29} - 9 q^{30} - 8 q^{31} - 21 q^{32} + 10 q^{33} - 12 q^{34} - q^{35} + 11 q^{36} + 2 q^{37} + 24 q^{38} + 7 q^{39} - 19 q^{40} + 13 q^{41} - 27 q^{42} - 5 q^{43} + 22 q^{44} + 19 q^{45} + 17 q^{46} + 23 q^{47} + 3 q^{48} + 26 q^{49} - 26 q^{50} + 18 q^{51} + 7 q^{52} + 25 q^{53} - 36 q^{54} - 17 q^{55} + 8 q^{56} - 35 q^{57} - 29 q^{58} - 5 q^{59} + 71 q^{60} - 9 q^{61} + q^{62} - 37 q^{63} - 14 q^{64} + 11 q^{65} - 41 q^{66} + 22 q^{67} - 6 q^{68} - 7 q^{69} - 29 q^{70} - 17 q^{71} - 34 q^{72} - 27 q^{73} + 14 q^{74} - 33 q^{75} - 36 q^{76} + 31 q^{77} - 23 q^{79} + 9 q^{80} - 12 q^{81} + 18 q^{82} - 25 q^{83} - 62 q^{84} + 13 q^{85} + 11 q^{86} + 26 q^{87} + 5 q^{88} + 2 q^{89} - 14 q^{90} - 2 q^{91} - 20 q^{92} - 7 q^{93} - 38 q^{94} + 3 q^{95} - 52 q^{96} - 15 q^{97} + 39 q^{98} - 8 q^{99}+O(q^{100}) \) 8 * q - q^2 + 7 * q^3 + 7 * q^4 + 11 * q^5 - 2 * q^7 + 9 * q^9 - 2 * q^10 - 2 * q^11 + 19 * q^12 + 8 * q^13 + 3 * q^14 + 6 * q^15 - 7 * q^16 + 7 * q^17 - 12 * q^18 - 5 * q^19 + 6 * q^20 + 8 * q^21 - 4 * q^22 + 14 * q^23 - 17 * q^24 + 17 * q^25 - q^26 + 7 * q^27 - 9 * q^28 + 12 * q^29 - 9 * q^30 - 8 * q^31 - 21 * q^32 + 10 * q^33 - 12 * q^34 - q^35 + 11 * q^36 + 2 * q^37 + 24 * q^38 + 7 * q^39 - 19 * q^40 + 13 * q^41 - 27 * q^42 - 5 * q^43 + 22 * q^44 + 19 * q^45 + 17 * q^46 + 23 * q^47 + 3 * q^48 + 26 * q^49 - 26 * q^50 + 18 * q^51 + 7 * q^52 + 25 * q^53 - 36 * q^54 - 17 * q^55 + 8 * q^56 - 35 * q^57 - 29 * q^58 - 5 * q^59 + 71 * q^60 - 9 * q^61 + q^62 - 37 * q^63 - 14 * q^64 + 11 * q^65 - 41 * q^66 + 22 * q^67 - 6 * q^68 - 7 * q^69 - 29 * q^70 - 17 * q^71 - 34 * q^72 - 27 * q^73 + 14 * q^74 - 33 * q^75 - 36 * q^76 + 31 * q^77 - 23 * q^79 + 9 * q^80 - 12 * q^81 + 18 * q^82 - 25 * q^83 - 62 * q^84 + 13 * q^85 + 11 * q^86 + 26 * q^87 + 5 * q^88 + 2 * q^89 - 14 * q^90 - 2 * q^91 - 20 * q^92 - 7 * q^93 - 38 * q^94 + 3 * q^95 - 52 * q^96 - 15 * q^97 + 39 * q^98 - 8 * q^99

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