LMFDB - Embedded newform 425.4.b.f.324.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [425,4,Mod(324,425)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(425, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0]))

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

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

chi := DirichletCharacter("425.324");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 425 = 5^{2} \cdot 17 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	425.b (of order $2$, degree $1$, not minimal)

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:	no
Analytic conductor:	$25.0758117524$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.27793984.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{3} + 49x^{2} - 14x + 2 $ x^6 - 2x^3 + 49x^2 - 14*x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 17)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			324.6
Root			$1.79483 - 1.79483i$ of defining polynomial
Character	$\chi$	$=$	425.324
Dual form			425.4.b.f.324.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+5.03251i q^{2} +8.47535i q^{3} -17.3261 q^{4} -42.6523 q^{6} -3.81828i q^{7} -46.9339i q^{8} -44.8316 q^{9} +O(q^{10})$	\(q+5.03251i q^{2} +8.47535i q^{3} -17.3261 q^{4} -42.6523 q^{6} -3.81828i q^{7} -46.9339i q^{8} -44.8316 q^{9} -52.3720 q^{11} -146.845i q^{12} -8.06025i q^{13} +19.2156 q^{14} +97.5862 q^{16} +17.0000i q^{17} -225.616i q^{18} +66.5154 q^{19} +32.3613 q^{21} -263.563i q^{22} +180.226i q^{23} +397.782 q^{24} +40.5633 q^{26} -151.129i q^{27} +66.1562i q^{28} +41.2800 q^{29} -34.9114 q^{31} +115.632i q^{32} -443.871i q^{33} -85.5527 q^{34} +776.759 q^{36} -130.368i q^{37} +334.739i q^{38} +68.3134 q^{39} -17.9081 q^{41} +162.859i q^{42} +277.620i q^{43} +907.405 q^{44} -906.987 q^{46} -463.789i q^{47} +827.078i q^{48} +328.421 q^{49} -144.081 q^{51} +139.653i q^{52} -329.944i q^{53} +760.560 q^{54} -179.207 q^{56} +563.741i q^{57} +207.742i q^{58} -678.656 q^{59} +340.280 q^{61} -175.692i q^{62} +171.180i q^{63} +198.770 q^{64} +2233.79 q^{66} -15.3925i q^{67} -294.545i q^{68} -1527.48 q^{69} -670.203 q^{71} +2104.12i q^{72} +193.480i q^{73} +656.080 q^{74} -1152.46 q^{76} +199.971i q^{77} +343.788i q^{78} -1080.15 q^{79} +70.4207 q^{81} -90.1229i q^{82} -865.668i q^{83} -560.697 q^{84} -1397.13 q^{86} +349.863i q^{87} +2458.02i q^{88} -1129.46 q^{89} -30.7763 q^{91} -3122.61i q^{92} -295.886i q^{93} +2334.02 q^{94} -980.023 q^{96} +379.412i q^{97} +1652.78i q^{98} +2347.92 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 50 q^{4} - 148 q^{6} - 118 q^{9}+O(q^{10}) $ 6 * q - 50 * q^4 - 148 * q^6 - 118 * q^9	$ 6 q - 50 q^{4} - 148 q^{6} - 118 q^{9} - 56 q^{11} - 184 q^{14} + 274 q^{16} - 160 q^{19} - 384 q^{21} + 1332 q^{24} + 52 q^{26} + 912 q^{29} + 460 q^{31} + 34 q^{34} + 2626 q^{36} - 536 q^{39} - 588 q^{41} + 2244 q^{44} - 1408 q^{46} + 538 q^{49} - 136 q^{51} + 2200 q^{54} + 1368 q^{56} - 1272 q^{59} - 168 q^{61} + 1838 q^{64} + 4936 q^{66} - 1152 q^{69} - 804 q^{71} - 1672 q^{74} - 1816 q^{76} + 1188 q^{79} - 1010 q^{81} + 4080 q^{84} - 2528 q^{86} + 340 q^{89} - 2032 q^{91} + 4032 q^{94} + 1356 q^{96} + 5840 q^{99}+O(q^{100}) $ 6 * q - 50 * q^4 - 148 * q^6 - 118 * q^9 - 56 * q^11 - 184 * q^14 + 274 * q^16 - 160 * q^19 - 384 * q^21 + 1332 * q^24 + 52 * q^26 + 912 * q^29 + 460 * q^31 + 34 * q^34 + 2626 * q^36 - 536 * q^39 - 588 * q^41 + 2244 * q^44 - 1408 * q^46 + 538 * q^49 - 136 * q^51 + 2200 * q^54 + 1368 * q^56 - 1272 * q^59 - 168 * q^61 + 1838 * q^64 + 4936 * q^66 - 1152 * q^69 - 804 * q^71 - 1672 * q^74 - 1816 * q^76 + 1188 * q^79 - 1010 * q^81 + 4080 * q^84 - 2528 * q^86 + 340 * q^89 - 2032 * q^91 + 4032 * q^94 + 1356 * q^96 + 5840 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/425\mathbb{Z}\right)^\times$.

$n$	$52$	$326$
$\chi(n)$	$-1$	$1$

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 $

$2$	5.03251i	1.77926i	0.456681	+	0.889630i	$0.349038\pi$
$2$	5.03251i	1.77926i	−0.456681	+	0.889630i	$0.650962\pi$
$3$	8.47535i	1.63108i	0.578699	+	0.815541i	$0.303561\pi$
$3$	8.47535i	1.63108i	−0.578699	+	0.815541i	$0.696439\pi$
$4$	−17.3261	−2.16577
$5$	0	0
$5$	0	0
$6$	−42.6523	−2.90212
$7$	− 3.81828i	− 0.206168i	−0.994673	−	0.103084i	$-0.967129\pi$
$7$	− 3.81828i	− 0.206168i	0.994673	−	0.103084i	$-0.0328711\pi$
$8$	− 46.9339i	− 2.07421i
$9$	−44.8316	−1.66043
$10$	0	0
$11$	−52.3720	−1.43552	−0.717761	−	0.696289i	$-0.754833\pi$
$11$	−52.3720	−1.43552	−0.717761	+	0.696289i	$0.754833\pi$
$12$	− 146.845i	− 3.53255i
$13$	− 8.06025i	− 0.171962i	−0.996297	−	0.0859811i	$-0.972598\pi$
$13$	− 8.06025i	− 0.171962i	0.996297	−	0.0859811i	$-0.0274025\pi$
$14$	19.2156	0.366827
$15$	0	0
$16$	97.5862	1.52478
$17$	17.0000i	0.242536i
$17$	17.0000i	0.242536i
$18$	− 225.616i	− 2.95434i
$19$	66.5154	0.803141	0.401570	−	0.915828i	$-0.368465\pi$
$19$	66.5154	0.803141	0.401570	+	0.915828i	$0.368465\pi$
$20$	0	0
$21$	32.3613	0.336277
$22$	− 263.563i	− 2.55417i
$23$	180.226i	1.63390i	0.576711	+	0.816948i	$0.304336\pi$
$23$	180.226i	1.63390i	−0.576711	+	0.816948i	$0.695664\pi$
$24$	397.782	3.38320
$25$	0	0
$26$	40.5633	0.305966
$27$	− 151.129i	− 1.07722i
$28$	66.1562i	0.446512i
$29$	41.2800	0.264328	0.132164	−	0.991228i	$-0.457807\pi$
$29$	41.2800	0.264328	0.132164	+	0.991228i	$0.457807\pi$
$30$	0	0
$31$	−34.9114	−0.202267	−0.101133	−	0.994873i	$-0.532247\pi$
$31$	−34.9114	−0.202267	−0.101133	+	0.994873i	$0.532247\pi$
$32$	115.632i	0.638783i
$33$	− 443.871i	− 2.34146i
$34$	−85.5527	−0.431534
$35$	0	0
$36$	776.759	3.59611
$37$	− 130.368i	− 0.579255i	−0.957139	−	0.289627i	$-0.906469\pi$
$37$	− 130.368i	− 0.579255i	0.957139	−	0.289627i	$-0.0935314\pi$
$38$	334.739i	1.42900i
$39$	68.3134	0.280485
$40$	0	0
$41$	−17.9081	−0.0682142	−0.0341071	−	0.999418i	$-0.510859\pi$
$41$	−17.9081	−0.0682142	−0.0341071	+	0.999418i	$0.510859\pi$
$42$	162.859i	0.598325i
$43$	277.620i	0.984573i	0.870433	+	0.492287i	$0.163839\pi$
$43$	277.620i	0.984573i	−0.870433	+	0.492287i	$0.836161\pi$
$44$	907.405	3.10901
$45$	0	0
$46$	−906.987	−2.90713
$47$	− 463.789i	− 1.43937i	−0.694299	−	0.719687i	$-0.744285\pi$
$47$	− 463.789i	− 1.43937i	0.694299	−	0.719687i	$-0.255715\pi$
$48$	827.078i	2.48705i
$49$	328.421	0.957495
$50$	0	0
$51$	−144.081	−0.395596
$52$	139.653i	0.372431i
$53$	− 329.944i	− 0.855118i	−0.903987	−	0.427559i	$-0.859374\pi$
$53$	− 329.944i	− 0.855118i	0.903987	−	0.427559i	$-0.140626\pi$
$54$	760.560	1.91665
$55$	0	0
$56$	−179.207	−0.427635
$57$	563.741i	1.30999i
$58$	207.742i	0.470308i
$59$	−678.656	−1.49752	−0.748759	−	0.662843i	$-0.769350\pi$
$59$	−678.656	−1.49752	−0.748759	+	0.662843i	$0.769350\pi$
$60$	0	0
$61$	340.280	0.714237	0.357118	−	0.934059i	$-0.383759\pi$
$61$	340.280	0.714237	0.357118	+	0.934059i	$0.383759\pi$
$62$	− 175.692i	− 0.359885i
$63$	171.180i	0.342328i
$64$	198.770	0.388223
$65$	0	0
$66$	2233.79	4.16606
$67$	− 15.3925i	− 0.0280671i	−0.999902	−	0.0140336i	$-0.995533\pi$
$67$	− 15.3925i	− 0.0280671i	0.999902	−	0.0140336i	$-0.00446717\pi$
$68$	− 294.545i	− 0.525276i
$69$	−1527.48	−2.66502
$70$	0	0
$71$	−670.203	−1.12026	−0.560130	−	0.828405i	$-0.689249\pi$
$71$	−670.203	−1.12026	−0.560130	+	0.828405i	$0.689249\pi$
$72$	2104.12i	3.44408i
$73$	193.480i	0.310207i	0.987898	+	0.155103i	$0.0495711\pi$
$73$	193.480i	0.310207i	−0.987898	+	0.155103i	$0.950429\pi$
$74$	656.080	1.03065
$75$	0	0
$76$	−1152.46	−1.73942
$77$	199.971i	0.295959i
$78$	343.788i	0.499055i
$79$	−1080.15	−1.53831	−0.769156	−	0.639061i	$-0.779323\pi$
$79$	−1080.15	−1.53831	−0.769156	+	0.639061i	$0.779323\pi$
$80$	0	0
$81$	70.4207	0.0965990
$82$	− 90.1229i	− 0.121371i
$83$	− 865.668i	− 1.14481i	−0.819970	−	0.572406i	$-0.806010\pi$
$83$	− 865.668i	− 1.14481i	0.819970	−	0.572406i	$-0.193990\pi$
$84$	−560.697	−0.728298
$85$	0	0
$86$	−1397.13	−1.75181
$87$	349.863i	0.431141i
$88$	2458.02i	2.97757i
$89$	−1129.46	−1.34520	−0.672599	−	0.740008i	$-0.734822\pi$
$89$	−1129.46	−1.34520	−0.672599	+	0.740008i	$0.734822\pi$
$90$	0	0
$91$	−30.7763	−0.0354531
$92$	− 3122.61i	− 3.53864i
$93$	− 295.886i	− 0.329914i
$94$	2334.02	2.56102
$95$	0	0
$96$	−980.023	−1.04191
$97$	379.412i	0.397149i	0.980086	+	0.198574i	$0.0636311\pi$
$97$	379.412i	0.397149i	−0.980086	+	0.198574i	$0.936369\pi$
$98$	1652.78i	1.70363i
$99$	2347.92	2.38359
$100$	0	0
$101$	131.732	0.129780	0.0648902	−	0.997892i	$-0.479330\pi$
$101$	131.732	0.129780	0.0648902	+	0.997892i	$0.479330\pi$
$102$	− 725.089i	− 0.703868i
$103$	195.988i	0.187488i	0.995596	+	0.0937442i	$0.0298836\pi$
$103$	195.988i	0.187488i	−0.995596	+	0.0937442i	$0.970116\pi$
$104$	−378.299	−0.356685
$105$	0	0
$106$	1660.45	1.52148
$107$	485.147i	0.438326i	0.975688	+	0.219163i	$0.0703327\pi$
$107$	485.147i	0.438326i	−0.975688	+	0.219163i	$0.929667\pi$
$108$	2618.49i	2.33300i
$109$	1255.12	1.10292	0.551460	−	0.834201i	$-0.314071\pi$
$109$	1255.12	1.10292	0.551460	+	0.834201i	$0.314071\pi$
$110$	0	0
$111$	1104.92	0.944812
$112$	− 372.612i	− 0.314362i
$113$	− 1013.35i	− 0.843612i	−0.906686	−	0.421806i	$-0.861396\pi$
$113$	− 1013.35i	− 0.843612i	0.906686	−	0.421806i	$-0.138604\pi$
$114$	−2837.03	−2.33081
$115$	0	0
$116$	−715.224	−0.572473
$117$	361.354i	0.285531i
$118$	− 3415.34i	− 2.66447i
$119$	64.9108	0.0500031
$120$	0	0
$121$	1411.83	1.06073
$122$	1712.46i	1.27081i
$123$	− 151.778i	− 0.111263i
$124$	604.880	0.438063
$125$	0	0
$126$	−861.464	−0.609090
$127$	− 1927.72i	− 1.34691i	−0.739227	−	0.673456i	$-0.764809\pi$
$127$	− 1927.72i	− 1.34691i	0.739227	−	0.673456i	$-0.235191\pi$
$128$	1925.37i	1.32953i
$129$	−2352.93	−1.60592
$130$	0	0
$131$	−406.738	−0.271274	−0.135637	−	0.990759i	$-0.543308\pi$
$131$	−406.738	−0.271274	−0.135637	+	0.990759i	$0.543308\pi$
$132$	7690.58i	5.07105i
$133$	− 253.975i	− 0.165582i
$134$	77.4631	0.0499387
$135$	0	0
$136$	797.877	0.503069
$137$	130.552i	0.0814149i	0.999171	+	0.0407074i	$0.0129612\pi$
$137$	130.552i	0.0814149i	−0.999171	+	0.0407074i	$0.987039\pi$
$138$	− 7687.03i	− 4.74177i
$139$	−2073.54	−1.26529	−0.632644	−	0.774443i	$-0.718030\pi$
$139$	−2073.54	−1.26529	−0.632644	+	0.774443i	$0.718030\pi$
$140$	0	0
$141$	3930.78	2.34774
$142$	− 3372.80i	− 1.99323i
$143$	422.131i	0.246856i
$144$	−4374.95	−2.53180
$145$	0	0
$146$	−973.689	−0.551939
$147$	2783.48i	1.56175i
$148$	2258.78i	1.25453i
$149$	1852.73	1.01867	0.509334	−	0.860569i	$-0.329892\pi$
$149$	1852.73	1.01867	0.509334	+	0.860569i	$0.329892\pi$
$150$	0	0
$151$	2050.86	1.10527	0.552637	−	0.833422i	$-0.313622\pi$
$151$	2050.86	1.10527	0.552637	+	0.833422i	$0.313622\pi$
$152$	− 3121.83i	− 1.66588i
$153$	− 762.138i	− 0.402714i
$154$	−1006.36	−0.526588
$155$	0	0
$156$	−1183.61	−0.607465
$157$	262.991i	0.133688i	0.997763	+	0.0668438i	$0.0212929\pi$
$157$	262.991i	0.133688i	−0.997763	+	0.0668438i	$0.978707\pi$
$158$	− 5435.88i	− 2.73706i
$159$	2796.39	1.39477
$160$	0	0
$161$	688.152	0.336857
$162$	354.393i	0.171875i
$163$	− 1444.98i	− 0.694354i	−0.937800	−	0.347177i	$-0.887140\pi$
$163$	− 1444.98i	− 0.694354i	0.937800	−	0.347177i	$-0.112860\pi$
$164$	310.279	0.147736
$165$	0	0
$166$	4356.48	2.03692
$167$	501.565i	0.232409i	0.993225	+	0.116204i	$0.0370728\pi$
$167$	501.565i	0.232409i	−0.993225	+	0.116204i	$0.962927\pi$
$168$	− 1518.84i	− 0.697508i
$169$	2132.03	0.970429
$170$	0	0
$171$	−2981.99	−1.33356
$172$	− 4810.08i	− 2.13236i
$173$	− 2590.14i	− 1.13829i	−0.822237	−	0.569146i	$-0.807274\pi$
$173$	− 2590.14i	− 1.13829i	0.822237	−	0.569146i	$-0.192726\pi$
$174$	−1760.69	−0.767112
$175$	0	0
$176$	−5110.79	−2.18886
$177$	− 5751.85i	− 2.44257i
$178$	− 5684.02i	− 2.39346i
$179$	−2165.65	−0.904294	−0.452147	−	0.891943i	$-0.649342\pi$
$179$	−2165.65	−0.904294	−0.452147	+	0.891943i	$0.649342\pi$
$180$	0	0
$181$	−1925.56	−0.790750	−0.395375	−	0.918520i	$-0.629385\pi$
$181$	−1925.56	−0.790750	−0.395375	+	0.918520i	$0.629385\pi$
$182$	− 154.882i	− 0.0630803i
$183$	2884.00i	1.16498i
$184$	8458.69	3.38904
$185$	0	0
$186$	1489.05	0.587003
$187$	− 890.324i	− 0.348165i
$188$	8035.68i	3.11735i
$189$	−577.055	−0.222088
$190$	0	0
$191$	−2783.52	−1.05449	−0.527247	−	0.849712i	$-0.676776\pi$
$191$	−2783.52	−1.05449	−0.527247	+	0.849712i	$0.676776\pi$
$192$	1684.65i	0.633223i
$193$	2258.27i	0.842246i	0.907004	+	0.421123i	$0.138364\pi$
$193$	2258.27i	0.842246i	−0.907004	+	0.421123i	$0.861636\pi$
$194$	−1909.39	−0.706631
$195$	0	0
$196$	−5690.27	−2.07371
$197$	1270.70i	0.459560i	0.973243	+	0.229780i	$0.0738007\pi$
$197$	1270.70i	0.459560i	−0.973243	+	0.229780i	$0.926199\pi$
$198$	11815.9i	4.24102i
$199$	4794.36	1.70786	0.853928	−	0.520392i	$-0.174214\pi$
$199$	4794.36	1.70786	0.853928	+	0.520392i	$0.174214\pi$
$200$	0	0
$201$	130.457	0.0457798
$202$	662.942i	0.230913i
$203$	− 157.619i	− 0.0544960i
$204$	2496.37	0.856769
$205$	0	0
$206$	−986.313	−0.333591
$207$	− 8079.80i	− 2.71297i
$208$	− 786.569i	− 0.262205i
$209$	−3483.54	−1.15293
$210$	0	0
$211$	−2807.00	−0.915837	−0.457918	−	0.888994i	$-0.651405\pi$
$211$	−2807.00	−0.915837	−0.457918	+	0.888994i	$0.651405\pi$
$212$	5716.66i	1.85199i
$213$	− 5680.21i	− 1.82724i
$214$	−2441.50	−0.779896
$215$	0	0
$216$	−7093.09	−2.23437
$217$	133.302i	0.0417009i
$218$	6316.38i	1.96238i
$219$	−1639.81	−0.505973
$220$	0	0
$221$	137.024	0.0417070
$222$	5560.51i	1.68107i
$223$	4684.30i	1.40665i	0.710866	+	0.703327i	$0.248303\pi$
$223$	4684.30i	1.40665i	−0.710866	+	0.703327i	$0.751697\pi$
$224$	441.516	0.131697
$225$	0	0
$226$	5099.70	1.50101
$227$	1395.72i	0.408095i	0.978961	+	0.204047i	$0.0654096\pi$
$227$	1395.72i	0.408095i	−0.978961	+	0.204047i	$0.934590\pi$
$228$	− 9767.47i	− 2.83713i
$229$	−894.638	−0.258163	−0.129082	−	0.991634i	$-0.541203\pi$
$229$	−894.638	−0.258163	−0.129082	+	0.991634i	$0.541203\pi$
$230$	0	0
$231$	−1694.83	−0.482733
$232$	− 1937.43i	− 0.548270i
$233$	1196.13i	0.336313i	0.985760	+	0.168156i	$0.0537814\pi$
$233$	1196.13i	0.336313i	−0.985760	+	0.168156i	$0.946219\pi$
$234$	−1818.52	−0.508035
$235$	0	0
$236$	11758.5	3.24328
$237$	− 9154.67i	− 2.50911i
$238$	326.664i	0.0889685i
$239$	−4948.82	−1.33938	−0.669691	−	0.742639i	$-0.733574\pi$
$239$	−4948.82	−1.33938	−0.669691	+	0.742639i	$0.733574\pi$
$240$	0	0
$241$	−6702.73	−1.79154	−0.895770	−	0.444518i	$-0.853375\pi$
$241$	−6702.73	−1.79154	−0.895770	+	0.444518i	$0.853375\pi$
$242$	7105.03i	1.88731i
$243$	− 3483.65i	− 0.919656i
$244$	−5895.75	−1.54687
$245$	0	0
$246$	763.824	0.197966
$247$	− 536.130i	− 0.138110i
$248$	1638.53i	0.419543i
$249$	7336.85	1.86728
$250$	0	0
$251$	−4756.08	−1.19602	−0.598010	−	0.801489i	$-0.704042\pi$
$251$	−4756.08	−1.19602	−0.598010	+	0.801489i	$0.704042\pi$
$252$	− 2965.89i	− 0.741402i
$253$	− 9438.77i	− 2.34550i
$254$	9701.29	2.39651
$255$	0	0
$256$	−8099.28	−1.97736
$257$	− 2892.84i	− 0.702143i	−0.936349	−	0.351071i	$-0.885817\pi$
$257$	− 2892.84i	− 0.702143i	0.936349	−	0.351071i	$-0.114183\pi$
$258$	− 11841.1i	− 2.85735i
$259$	−497.784	−0.119424
$260$	0	0
$261$	−1850.65	−0.438898
$262$	− 2046.92i	− 0.482667i
$263$	5415.48i	1.26971i	0.772633	+	0.634853i	$0.218939\pi$
$263$	5415.48i	1.26971i	−0.772633	+	0.634853i	$0.781061\pi$
$264$	−20832.6	−4.85666
$265$	0	0
$266$	1278.13	0.294613
$267$	− 9572.58i	− 2.19413i
$268$	266.693i	0.0607869i
$269$	−5787.00	−1.31167	−0.655835	−	0.754904i	$-0.727683\pi$
$269$	−5787.00	−1.31167	−0.655835	+	0.754904i	$0.727683\pi$
$270$	0	0
$271$	5465.13	1.22503	0.612515	−	0.790459i	$-0.290158\pi$
$271$	5465.13	1.22503	0.612515	+	0.790459i	$0.290158\pi$
$272$	1658.97i	0.369815i
$273$	− 260.840i	− 0.0578270i
$274$	−657.006	−0.144858
$275$	0	0
$276$	26465.3	5.77182
$277$	1207.65i	0.261952i	0.991386	+	0.130976i	$0.0418111\pi$
$277$	1207.65i	0.261952i	−0.991386	+	0.130976i	$0.958189\pi$
$278$	− 10435.1i	− 2.25128i
$279$	1565.13	0.335850
$280$	0	0
$281$	−1197.18	−0.254155	−0.127077	−	0.991893i	$-0.540560\pi$
$281$	−1197.18	−0.254155	−0.127077	+	0.991893i	$0.540560\pi$
$282$	19781.7i	4.17724i
$283$	3164.73i	0.664748i	0.943148	+	0.332374i	$0.107850\pi$
$283$	3164.73i	0.664748i	−0.943148	+	0.332374i	$0.892150\pi$
$284$	11612.0	2.42622
$285$	0	0
$286$	−2124.38	−0.439221
$287$	68.3784i	0.0140636i
$288$	− 5183.98i	− 1.06066i
$289$	−289.000	−0.0588235
$290$	0	0
$291$	−3215.65	−0.647782
$292$	− 3352.26i	− 0.671836i
$293$	7456.21i	1.48668i	0.668915	+	0.743339i	$0.266759\pi$
$293$	7456.21i	1.48668i	−0.668915	+	0.743339i	$0.733241\pi$
$294$	−14007.9	−2.77877
$295$	0	0
$296$	−6118.70	−1.20149
$297$	7914.94i	1.54637i
$298$	9323.89i	1.81248i
$299$	1452.66	0.280969
$300$	0	0
$301$	1060.03	0.202988
$302$	10321.0i	1.96657i
$303$	1116.47i	0.211683i
$304$	6490.98	1.22462
$305$	0	0
$306$	3835.46	0.716532
$307$	6535.48i	1.21498i	0.794327	+	0.607491i	$0.207824\pi$
$307$	6535.48i	1.21498i	−0.794327	+	0.607491i	$0.792176\pi$
$308$	− 3464.73i	− 0.640978i
$309$	−1661.07	−0.305809
$310$	0	0
$311$	−8935.89	−1.62928	−0.814642	−	0.579963i	$-0.803067\pi$
$311$	−8935.89	−1.62928	−0.814642	+	0.579963i	$0.803067\pi$
$312$	− 3206.22i	− 0.581783i
$313$	− 2628.71i	− 0.474707i	−0.971423	−	0.237353i	$-0.923720\pi$
$313$	− 2628.71i	− 0.474707i	0.971423	−	0.237353i	$-0.0762799\pi$
$314$	−1323.50	−0.237865
$315$	0	0
$316$	18714.9	3.33163
$317$	− 4268.54i	− 0.756293i	−0.925746	−	0.378147i	$-0.876562\pi$
$317$	− 4268.54i	− 0.756293i	0.925746	−	0.378147i	$-0.123438\pi$
$318$	14072.9i	2.48166i
$319$	−2161.92	−0.379449
$320$	0	0
$321$	−4111.79	−0.714946
$322$	3463.13i	0.599357i
$323$	1130.76i	0.194790i
$324$	−1220.12	−0.209211
$325$	0	0
$326$	7271.89	1.23544
$327$	10637.5i	1.79895i
$328$	840.500i	0.141490i
$329$	−1770.88	−0.296753
$330$	0	0
$331$	992.298	0.164778	0.0823892	−	0.996600i	$-0.473745\pi$
$331$	992.298	0.164778	0.0823892	+	0.996600i	$0.473745\pi$
$332$	14998.7i	2.47940i
$333$	5844.63i	0.961812i
$334$	−2524.13	−0.413516
$335$	0	0
$336$	3158.02	0.512750
$337$	− 8042.26i	− 1.29997i	−0.759947	−	0.649985i	$-0.774775\pi$
$337$	− 8042.26i	− 1.29997i	0.759947	−	0.649985i	$-0.225225\pi$
$338$	10729.5i	1.72665i
$339$	8588.52	1.37600
$340$	0	0
$341$	1828.38	0.290359
$342$	− 15006.9i	− 2.37275i
$343$	− 2563.68i	− 0.403573i
$344$	13029.8	2.04221
$345$	0	0
$346$	13034.9	2.02532
$347$	7414.16i	1.14701i	0.819202	+	0.573506i	$0.194417\pi$
$347$	7414.16i	1.14701i	−0.819202	+	0.573506i	$0.805583\pi$
$348$	− 6061.78i	− 0.933751i
$349$	859.194	0.131781	0.0658905	−	0.997827i	$-0.479011\pi$
$349$	859.194	0.131781	0.0658905	+	0.997827i	$0.479011\pi$
$350$	0	0
$351$	−1218.14	−0.185241
$352$	− 6055.89i	− 0.916988i
$353$	569.084i	0.0858053i	0.999079	+	0.0429027i	$0.0136605\pi$
$353$	569.084i	0.0858053i	−0.999079	+	0.0429027i	$0.986339\pi$
$354$	28946.3	4.34598
$355$	0	0
$356$	19569.2	2.91339
$357$	550.142i	0.0815592i
$358$	− 10898.7i	− 1.60897i
$359$	5005.21	0.735835	0.367918	−	0.929858i	$-0.380071\pi$
$359$	5005.21	0.735835	0.367918	+	0.929858i	$0.380071\pi$
$360$	0	0
$361$	−2434.71	−0.354965
$362$	− 9690.40i	− 1.40695i
$363$	11965.7i	1.73013i
$364$	533.235	0.0767833
$365$	0	0
$366$	−14513.7	−2.07280
$367$	10975.3i	1.56105i	0.625127	+	0.780523i	$0.285047\pi$
$367$	10975.3i	1.56105i	−0.625127	+	0.780523i	$0.714953\pi$
$368$	17587.5i	2.49134i
$369$	802.851	0.113265
$370$	0	0
$371$	−1259.82	−0.176298
$372$	5126.57i	0.714517i
$373$	− 3211.72i	− 0.445835i	−0.974837	−	0.222918i	$-0.928442\pi$
$373$	− 3211.72i	− 0.445835i	0.974837	−	0.222918i	$-0.0715581\pi$
$374$	4480.56	0.619477
$375$	0	0
$376$	−21767.4	−2.98556
$377$	− 332.727i	− 0.0454544i
$378$	− 2904.03i	− 0.395152i
$379$	−8051.48	−1.09123	−0.545616	−	0.838035i	$-0.683704\pi$
$379$	−8051.48	−1.09123	−0.545616	+	0.838035i	$0.683704\pi$
$380$	0	0
$381$	16338.1	2.19692
$382$	− 14008.1i	− 1.87622i
$383$	− 2584.16i	− 0.344763i	−0.985030	−	0.172382i	$-0.944854\pi$
$383$	− 2584.16i	− 0.344763i	0.985030	−	0.172382i	$-0.0551462\pi$
$384$	−16318.2	−2.16858
$385$	0	0
$386$	−11364.7	−1.49858
$387$	− 12446.2i	− 1.63482i
$388$	− 6573.74i	− 0.860132i
$389$	5174.31	0.674417	0.337208	−	0.941430i	$-0.390517\pi$
$389$	5174.31	0.674417	0.337208	+	0.941430i	$0.390517\pi$
$390$	0	0
$391$	−3063.83	−0.396278
$392$	− 15414.1i	− 1.98604i
$393$	− 3447.25i	− 0.442470i
$394$	−6394.79	−0.817677
$395$	0	0
$396$	−40680.4	−5.16230
$397$	5149.36i	0.650980i	0.945545	+	0.325490i	$0.105529\pi$
$397$	5149.36i	0.650980i	−0.945545	+	0.325490i	$0.894471\pi$
$398$	24127.7i	3.03872i
$399$	2152.53	0.270078
$400$	0	0
$401$	8700.49	1.08350	0.541748	−	0.840541i	$-0.317763\pi$
$401$	8700.49	1.08350	0.541748	+	0.840541i	$0.317763\pi$
$402$	656.527i	0.0814542i
$403$	281.394i	0.0347823i
$404$	−2282.41	−0.281074
$405$	0	0
$406$	793.219	0.0969625
$407$	6827.65i	0.831533i
$408$	6762.29i	0.820547i
$409$	−12346.0	−1.49260	−0.746299	−	0.665611i	$-0.768171\pi$
$409$	−12346.0	−1.49260	−0.746299	+	0.665611i	$0.768171\pi$
$410$	0	0
$411$	−1106.48	−0.132794
$412$	− 3395.72i	− 0.406056i
$413$	2591.30i	0.308740i
$414$	40661.7	4.82708
$415$	0	0
$416$	932.023	0.109847
$417$	− 17574.0i	− 2.06379i
$418$	− 17531.0i	− 2.05136i
$419$	5763.33	0.671974	0.335987	−	0.941867i	$-0.390930\pi$
$419$	5763.33	0.671974	0.335987	+	0.941867i	$0.390930\pi$
$420$	0	0
$421$	−1876.12	−0.217188	−0.108594	−	0.994086i	$-0.534635\pi$
$421$	−1876.12	−0.217188	−0.108594	+	0.994086i	$0.534635\pi$
$422$	− 14126.2i	− 1.62951i
$423$	20792.4i	2.38998i
$424$	−15485.6	−1.77369
$425$	0	0
$426$	28585.7	3.25113
$427$	− 1299.29i	− 0.147253i
$428$	− 8405.72i	− 0.949313i
$429$	−3577.71	−0.402642
$430$	0	0
$431$	83.9299	0.00937996	0.00468998	−	0.999989i	$-0.498507\pi$
$431$	83.9299	0.00937996	0.00468998	+	0.999989i	$0.498507\pi$
$432$	− 14748.1i	− 1.64252i
$433$	− 15345.0i	− 1.70308i	−0.524291	−	0.851539i	$-0.675669\pi$
$433$	− 15345.0i	− 1.70308i	0.524291	−	0.851539i	$-0.324331\pi$
$434$	−670.842	−0.0741968
$435$	0	0
$436$	−21746.3	−2.38867
$437$	11987.8i	1.31225i
$438$	− 8252.36i	− 0.900258i
$439$	−3064.74	−0.333194	−0.166597	−	0.986025i	$-0.553278\pi$
$439$	−3064.74	−0.333194	−0.166597	+	0.986025i	$0.553278\pi$
$440$	0	0
$441$	−14723.6	−1.58985
$442$	689.575i	0.0742076i
$443$	− 1792.97i	− 0.192295i	−0.995367	−	0.0961474i	$-0.969348\pi$
$443$	− 1792.97i	− 0.192295i	0.995367	−	0.0961474i	$-0.0306520\pi$
$444$	−19144.0	−2.04624
$445$	0	0
$446$	−23573.8	−2.50281
$447$	15702.6i	1.66153i
$448$	− 758.960i	− 0.0800391i
$449$	−2499.19	−0.262681	−0.131341	−	0.991337i	$-0.541928\pi$
$449$	−2499.19	−0.262681	−0.131341	+	0.991337i	$0.541928\pi$
$450$	0	0
$451$	937.885	0.0979231
$452$	17557.5i	1.82707i
$453$	17381.7i	1.80279i
$454$	−7024.00	−0.726107
$455$	0	0
$456$	26458.6	2.71719
$457$	− 14784.4i	− 1.51331i	−0.653813	−	0.756656i	$-0.726832\pi$
$457$	− 14784.4i	− 1.51331i	0.653813	−	0.756656i	$-0.273168\pi$
$458$	− 4502.28i	− 0.459340i
$459$	2569.20	0.261263
$460$	0	0
$461$	−17746.9	−1.79297	−0.896483	−	0.443078i	$-0.853887\pi$
$461$	−17746.9	−1.79297	−0.896483	+	0.443078i	$0.853887\pi$
$462$	− 8529.23i	− 0.858909i
$463$	18486.4i	1.85559i	0.373096	+	0.927793i	$0.378296\pi$
$463$	18486.4i	1.85559i	−0.373096	+	0.927793i	$0.621704\pi$
$464$	4028.36	0.403043
$465$	0	0
$466$	−6019.52	−0.598388
$467$	− 7406.57i	− 0.733908i	−0.930239	−	0.366954i	$-0.880401\pi$
$467$	− 7406.57i	− 0.733908i	0.930239	−	0.366954i	$-0.119599\pi$
$468$	− 6260.87i	− 0.618395i
$469$	−58.7731	−0.00578655
$470$	0	0
$471$	−2228.94	−0.218055
$472$	31852.0i	3.10616i
$473$	− 14539.5i	− 1.41338i
$474$	46071.0	4.46437
$475$	0	0
$476$	−1124.65	−0.108295
$477$	14791.9i	1.41986i
$478$	− 24905.0i	− 2.38311i
$479$	18550.9	1.76955	0.884775	−	0.466019i	$-0.154312\pi$
$479$	18550.9	1.76955	0.884775	+	0.466019i	$0.154312\pi$
$480$	0	0
$481$	−1050.80	−0.0996100
$482$	− 33731.6i	− 3.18762i
$483$	5832.34i	0.549442i
$484$	−24461.5	−2.29729
$485$	0	0
$486$	17531.5	1.63631
$487$	− 10203.4i	− 0.949406i	−0.880146	−	0.474703i	$-0.842556\pi$
$487$	− 10203.4i	− 0.949406i	0.880146	−	0.474703i	$-0.157444\pi$
$488$	− 15970.7i	− 1.48147i
$489$	12246.7	1.13255
$490$	0	0
$491$	−1247.46	−0.114658	−0.0573290	−	0.998355i	$-0.518258\pi$
$491$	−1247.46	−0.114658	−0.0573290	+	0.998355i	$0.518258\pi$
$492$	2629.73i	0.240970i
$493$	701.760i	0.0641089i
$494$	2698.08	0.245734
$495$	0	0
$496$	−3406.87	−0.308413
$497$	2559.03i	0.230962i
$498$	36922.7i	3.32238i
$499$	−70.0303	−0.00628254	−0.00314127	−	0.999995i	$-0.501000\pi$
$499$	−70.0303	−0.00628254	−0.00314127	+	0.999995i	$0.501000\pi$
$500$	0	0
$501$	−4250.94	−0.379078
$502$	− 23935.0i	− 2.12803i
$503$	1444.29i	0.128028i	0.997949	+	0.0640138i	$0.0203902\pi$
$503$	1444.29i	0.128028i	−0.997949	+	0.0640138i	$0.979610\pi$
$504$	8034.15	0.710058
$505$	0	0
$506$	47500.7	4.17325
$507$	18069.7i	1.58285i
$508$	33400.0i	2.91710i
$509$	−14272.8	−1.24289	−0.621445	−	0.783458i	$-0.713454\pi$
$509$	−14272.8	−1.24289	−0.621445	+	0.783458i	$0.713454\pi$
$510$	0	0
$511$	738.761	0.0639547
$512$	− 25356.7i	− 2.18871i
$513$	− 10052.4i	− 0.865157i
$514$	14558.3	1.24929
$515$	0	0
$516$	40767.2	3.47805
$517$	24289.6i	2.06625i
$518$	− 2505.10i	− 0.212486i
$519$	21952.3	1.85665
$520$	0	0
$521$	14874.0	1.25075	0.625376	−	0.780324i	$-0.284946\pi$
$521$	14874.0	1.25075	0.625376	+	0.780324i	$0.284946\pi$
$522$	− 9313.42i	− 0.780914i
$523$	− 8142.90i	− 0.680811i	−0.940279	−	0.340406i	$-0.889436\pi$
$523$	− 8142.90i	− 0.680811i	0.940279	−	0.340406i	$-0.110564\pi$
$524$	7047.21	0.587517
$525$	0	0
$526$	−27253.4	−2.25914
$527$	− 593.494i	− 0.0490569i
$528$	− 43315.7i	− 3.57022i
$529$	−20314.2	−1.66962
$530$	0	0
$531$	30425.3	2.48652
$532$	4400.40i	0.358612i
$533$	144.344i	0.0117303i
$534$	48174.1	3.90393
$535$	0	0
$536$	−722.432	−0.0582170
$537$	− 18354.7i	− 1.47498i
$538$	− 29123.1i	− 2.33380i
$539$	−17200.0	−1.37451
$540$	0	0
$541$	3179.67	0.252689	0.126344	−	0.991986i	$-0.459676\pi$
$541$	3179.67	0.252689	0.126344	+	0.991986i	$0.459676\pi$
$542$	27503.3i	2.17965i
$543$	− 16319.8i	− 1.28978i
$544$	−1965.75	−0.154928
$545$	0	0
$546$	1312.68	0.102889
$547$	− 2107.07i	− 0.164702i	−0.996603	−	0.0823509i	$-0.973757\pi$
$547$	− 2107.07i	− 0.164702i	0.996603	−	0.0823509i	$-0.0262428\pi$
$548$	− 2261.97i	− 0.176326i
$549$	−15255.3	−1.18594
$550$	0	0
$551$	2745.76	0.212292
$552$	71690.4i	5.52780i
$553$	4124.33i	0.317151i
$554$	−6077.51	−0.466081
$555$	0	0
$556$	35926.4	2.74032
$557$	− 467.382i	− 0.0355540i	−0.999842	−	0.0177770i	$-0.994341\pi$
$557$	− 467.382i	− 0.0355540i	0.999842	−	0.0177770i	$-0.00565890\pi$
$558$	7876.55i	0.597565i
$559$	2237.69	0.169309
$560$	0	0
$561$	7545.81	0.567887
$562$	− 6024.80i	− 0.452208i
$563$	14612.6i	1.09387i	0.837175	+	0.546935i	$0.184206\pi$
$563$	14612.6i	1.09387i	−0.837175	+	0.546935i	$0.815794\pi$
$564$	−68105.2	−5.08466
$565$	0	0
$566$	−15926.5	−1.18276
$567$	− 268.886i	− 0.0199156i
$568$	31455.3i	2.32365i
$569$	−11602.3	−0.854821	−0.427410	−	0.904058i	$-0.640574\pi$
$569$	−11602.3	−0.854821	−0.427410	+	0.904058i	$0.640574\pi$
$570$	0	0
$571$	−10534.9	−0.772104	−0.386052	−	0.922477i	$-0.626161\pi$
$571$	−10534.9	−0.772104	−0.386052	+	0.922477i	$0.626161\pi$
$572$	− 7313.91i	− 0.534633i
$573$	− 23591.3i	− 1.71997i
$574$	−344.115	−0.0250228
$575$	0	0
$576$	−8911.18	−0.644617
$577$	− 14404.7i	− 1.03930i	−0.854379	−	0.519650i	$-0.826062\pi$
$577$	− 14404.7i	− 1.03930i	0.854379	−	0.519650i	$-0.173938\pi$
$578$	− 1454.40i	− 0.104662i
$579$	−19139.6	−1.37377
$580$	0	0
$581$	−3305.37	−0.236024
$582$	− 16182.8i	− 1.15257i
$583$	17279.8i	1.22754i
$584$	9080.77	0.643433
$585$	0	0
$586$	−37523.5	−2.64519
$587$	11004.9i	0.773799i	0.922122	+	0.386900i	$0.126454\pi$
$587$	11004.9i	0.773799i	−0.922122	+	0.386900i	$0.873546\pi$
$588$	− 48227.0i	− 3.38240i
$589$	−2322.14	−0.162449
$590$	0	0
$591$	−10769.6	−0.749581
$592$	− 12722.2i	− 0.883239i
$593$	1853.59i	0.128361i	0.997938	+	0.0641804i	$0.0204433\pi$
$593$	1853.59i	0.128361i	−0.997938	+	0.0641804i	$0.979557\pi$
$594$	−39832.0	−2.75139
$595$	0	0
$596$	−32100.7	−2.20620
$597$	40633.9i	2.78565i
$598$	7310.53i	0.499916i
$599$	−19074.7	−1.30112	−0.650559	−	0.759456i	$-0.725465\pi$
$599$	−19074.7	−1.30112	−0.650559	+	0.759456i	$0.725465\pi$
$600$	0	0
$601$	−27776.0	−1.88520	−0.942600	−	0.333923i	$-0.891627\pi$
$601$	−27776.0	−1.88520	−0.942600	+	0.333923i	$0.891627\pi$
$602$	5334.62i	0.361168i
$603$	690.073i	0.0466035i
$604$	−35533.4	−2.39377
$605$	0	0
$606$	−5618.67	−0.376638
$607$	− 18728.3i	− 1.25232i	−0.779695	−	0.626159i	$-0.784626\pi$
$607$	− 18728.3i	− 1.25232i	0.779695	−	0.626159i	$-0.215374\pi$
$608$	7691.32i	0.513033i
$609$	1335.88	0.0888874
$610$	0	0
$611$	−3738.25	−0.247518
$612$	13204.9i	0.872184i
$613$	− 24405.3i	− 1.60802i	−0.594613	−	0.804012i	$-0.702695\pi$
$613$	− 24405.3i	− 1.60802i	0.594613	−	0.804012i	$-0.297305\pi$
$614$	−32889.8	−2.16177
$615$	0	0
$616$	9385.43	0.613880
$617$	22516.4i	1.46917i	0.678518	+	0.734584i	$0.262623\pi$
$617$	22516.4i	1.46917i	−0.678518	+	0.734584i	$0.737377\pi$
$618$	− 8359.35i	− 0.544114i
$619$	5146.53	0.334179	0.167089	−	0.985942i	$-0.446563\pi$
$619$	5146.53	0.334179	0.167089	+	0.985942i	$0.446563\pi$
$620$	0	0
$621$	27237.4	1.76006
$622$	− 44969.9i	− 2.89892i
$623$	4312.60i	0.277337i
$624$	6666.45	0.427679
$625$	0	0
$626$	13229.0	0.844627
$627$	− 29524.3i	− 1.88052i
$628$	− 4556.62i	− 0.289536i
$629$	2216.26	0.140490
$630$	0	0
$631$	−3858.77	−0.243447	−0.121724	−	0.992564i	$-0.538842\pi$
$631$	−3858.77	−0.243447	−0.121724	+	0.992564i	$0.538842\pi$
$632$	50695.8i	3.19078i
$633$	− 23790.3i	− 1.49381i
$634$	21481.5	1.34564
$635$	0	0
$636$	−48450.7	−3.02075
$637$	− 2647.15i	− 0.164653i
$638$	− 10879.9i	− 0.675138i
$639$	30046.3	1.86011
$640$	0	0
$641$	18689.3	1.15161	0.575805	−	0.817587i	$-0.304689\pi$
$641$	18689.3	1.15161	0.575805	+	0.817587i	$0.304689\pi$
$642$	− 20692.6i	− 1.27208i
$643$	26473.5i	1.62366i	0.583893	+	0.811831i	$0.301529\pi$
$643$	26473.5i	1.62366i	−0.583893	+	0.811831i	$0.698471\pi$
$644$	−11923.0	−0.729555
$645$	0	0
$646$	−5690.57	−0.346583
$647$	− 14397.7i	− 0.874855i	−0.899254	−	0.437427i	$-0.855890\pi$
$647$	− 14397.7i	− 0.874855i	0.899254	−	0.437427i	$-0.144110\pi$
$648$	− 3305.12i	− 0.200366i
$649$	35542.6	2.14972
$650$	0	0
$651$	−1129.78	−0.0680177
$652$	25036.0i	1.50381i
$653$	20939.5i	1.25486i	0.778672	+	0.627431i	$0.215893\pi$
$653$	20939.5i	1.25486i	−0.778672	+	0.627431i	$0.784107\pi$
$654$	−53533.6	−3.20081
$655$	0	0
$656$	−1747.59	−0.104012
$657$	− 8674.02i	− 0.515077i
$658$	− 8911.96i	− 0.528001i
$659$	−4031.76	−0.238323	−0.119162	−	0.992875i	$-0.538021\pi$
$659$	−4031.76	−0.238323	−0.119162	+	0.992875i	$0.538021\pi$
$660$	0	0
$661$	6691.52	0.393752	0.196876	−	0.980428i	$-0.436920\pi$
$661$	6691.52	0.393752	0.196876	+	0.980428i	$0.436920\pi$
$662$	4993.75i	0.293184i
$663$	1161.33i	0.0680275i
$664$	−40629.2	−2.37458
$665$	0	0
$666$	−29413.1	−1.71131
$667$	7439.71i	0.431884i
$668$	− 8690.19i	− 0.503344i
$669$	−39701.1	−2.29437
$670$	0	0
$671$	−17821.2	−1.02530
$672$	3742.01i	0.214808i
$673$	10319.2i	0.591048i	0.955335	+	0.295524i	$0.0954942\pi$
$673$	10319.2i	0.591048i	−0.955335	+	0.295524i	$0.904506\pi$
$674$	40472.7	2.31298
$675$	0	0
$676$	−36939.9	−2.10172
$677$	19813.3i	1.12480i	0.826866	+	0.562398i	$0.190121\pi$
$677$	19813.3i	1.12480i	−0.826866	+	0.562398i	$0.809879\pi$
$678$	43221.8i	2.44826i
$679$	1448.70	0.0818793
$680$	0	0
$681$	−11829.3	−0.665636
$682$	9201.33i	0.516624i
$683$	5924.61i	0.331916i	0.986133	+	0.165958i	$0.0530717\pi$
$683$	5924.61i	0.331916i	−0.986133	+	0.165958i	$0.946928\pi$
$684$	51666.4	2.88818
$685$	0	0
$686$	12901.7	0.718061
$687$	− 7582.38i	− 0.421085i
$688$	27091.9i	1.50126i
$689$	−2659.43	−0.147048
$690$	0	0
$691$	1973.16	0.108629	0.0543143	−	0.998524i	$-0.482703\pi$
$691$	1973.16	0.108629	0.0543143	+	0.998524i	$0.482703\pi$
$692$	44877.1i	2.46528i
$693$	− 8965.03i	− 0.491419i
$694$	−37311.8	−2.04083
$695$	0	0
$696$	16420.4	0.894274
$697$	− 304.439i	− 0.0165444i
$698$	4323.90i	0.234473i
$699$	−10137.6	−0.548554
$700$	0	0
$701$	−12840.1	−0.691815	−0.345907	−	0.938269i	$-0.612429\pi$
$701$	−12840.1	−0.691815	−0.345907	+	0.938269i	$0.612429\pi$
$702$	− 6130.30i	− 0.329591i
$703$	− 8671.50i	− 0.465223i
$704$	−10410.0	−0.557302
$705$	0	0
$706$	−2863.92	−0.152670
$707$	− 502.990i	− 0.0267566i
$708$	99657.5i	5.29005i
$709$	27749.7	1.46990	0.734952	−	0.678119i	$-0.237204\pi$
$709$	27749.7	1.46990	0.734952	+	0.678119i	$0.237204\pi$
$710$	0	0
$711$	48425.0	2.55426
$712$	53010.0i	2.79022i
$713$	− 6291.92i	− 0.330483i
$714$	−2768.60	−0.145115
$715$	0	0
$716$	37522.4	1.95849
$717$	− 41943.0i	− 2.18464i
$718$	25188.8i	1.30924i
$719$	−16888.3	−0.875979	−0.437989	−	0.898980i	$-0.644309\pi$
$719$	−16888.3	−0.875979	−0.437989	+	0.898980i	$0.644309\pi$
$720$	0	0
$721$	748.339	0.0386541
$722$	− 12252.7i	− 0.631575i
$723$	− 56808.0i	− 2.92215i
$724$	33362.6	1.71258
$725$	0	0
$726$	−60217.6	−3.07835
$727$	− 2135.25i	− 0.108930i	−0.998516	−	0.0544649i	$-0.982655\pi$
$727$	− 2135.25i	− 0.108930i	0.998516	−	0.0544649i	$-0.0173453\pi$
$728$	1444.45i	0.0735371i
$729$	31426.5	1.59663
$730$	0	0
$731$	−4719.54	−0.238794
$732$	− 49968.6i	− 2.52308i
$733$	4795.27i	0.241633i	0.992675	+	0.120817i	$0.0385513\pi$
$733$	4795.27i	0.241633i	−0.992675	+	0.120817i	$0.961449\pi$
$734$	−55233.1	−2.77751
$735$	0	0
$736$	−20839.9	−1.04371
$737$	806.138i	0.0402910i
$738$	4040.36i	0.201528i
$739$	32747.6	1.63010	0.815048	−	0.579393i	$-0.196710\pi$
$739$	32747.6	1.63010	0.815048	+	0.579393i	$0.196710\pi$
$740$	0	0
$741$	4543.89	0.225269
$742$	− 6340.05i	− 0.313680i
$743$	12299.4i	0.607298i	0.952784	+	0.303649i	$0.0982050\pi$
$743$	12299.4i	0.607298i	−0.952784	+	0.303649i	$0.901795\pi$
$744$	−13887.1	−0.684309
$745$	0	0
$746$	16163.0	0.793257
$747$	38809.3i	1.90088i
$748$	15425.9i	0.754046i
$749$	1852.43	0.0903688
$750$	0	0
$751$	30102.6	1.46266	0.731332	−	0.682021i	$-0.238899\pi$
$751$	30102.6	1.46266	0.731332	+	0.682021i	$0.238899\pi$
$752$	− 45259.4i	− 2.19474i
$753$	− 40309.4i	− 1.95081i
$754$	1674.45	0.0808753
$755$	0	0
$756$	9998.14	0.480990
$757$	− 38826.3i	− 1.86416i	−0.362257	−	0.932078i	$-0.617994\pi$
$757$	− 38826.3i	− 1.86416i	0.362257	−	0.932078i	$-0.382006\pi$
$758$	− 40519.2i	− 1.94159i
$759$	79996.9	3.82570
$760$	0	0
$761$	19981.6	0.951815	0.475907	−	0.879495i	$-0.342120\pi$
$761$	19981.6	0.951815	0.475907	+	0.879495i	$0.342120\pi$
$762$	82221.8i	3.90890i
$763$	− 4792.39i	− 0.227387i
$764$	48227.7	2.28379
$765$	0	0
$766$	13004.8	0.613424
$767$	5470.14i	0.257517i
$768$	− 68644.2i	− 3.22524i
$769$	22407.7	1.05077	0.525384	−	0.850865i	$-0.323922\pi$
$769$	22407.7	1.05077	0.525384	+	0.850865i	$0.323922\pi$
$770$	0	0
$771$	24517.9	1.14525
$772$	− 39127.0i	− 1.82411i
$773$	− 6902.77i	− 0.321184i	−0.987021	−	0.160592i	$-0.948660\pi$
$773$	− 6902.77i	− 0.321184i	0.987021	−	0.160592i	$-0.0513403\pi$
$774$	62635.4	2.90876
$775$	0	0
$776$	17807.3	0.823768
$777$	− 4218.89i	− 0.194790i
$778$	26039.8i	1.19996i
$779$	−1191.17	−0.0547856
$780$	0	0
$781$	35099.9	1.60816
$782$	− 15418.8i	− 0.705082i
$783$	− 6238.62i	− 0.284738i
$784$	32049.3	1.45997
$785$	0	0
$786$	17348.3	0.787270
$787$	22185.9i	1.00488i	0.864611	+	0.502442i	$0.167565\pi$
$787$	22185.9i	1.00488i	−0.864611	+	0.502442i	$0.832435\pi$
$788$	− 22016.3i	− 0.995301i
$789$	−45898.1	−2.07099
$790$	0	0
$791$	−3869.27	−0.173926
$792$	− 110197.i	− 4.94405i
$793$	− 2742.74i	− 0.122822i
$794$	−25914.2	−1.15826
$795$	0	0
$796$	−83067.8	−3.69882
$797$	16291.1i	0.724040i	0.932170	+	0.362020i	$0.117913\pi$
$797$	16291.1i	0.724040i	−0.932170	+	0.362020i	$0.882087\pi$
$798$	10832.6i	0.480539i
$799$	7884.41	0.349100
$800$	0	0
$801$	50635.5	2.23361
$802$	43785.3i	1.92782i
$803$	− 10132.9i	− 0.445309i
$804$	−2260.32	−0.0991485
$805$	0	0
$806$	−1416.12	−0.0618867
$807$	− 49046.8i	− 2.13944i
$808$	− 6182.70i	− 0.269191i
$809$	−17696.8	−0.769082	−0.384541	−	0.923108i	$-0.625640\pi$
$809$	−17696.8	−0.769082	−0.384541	+	0.923108i	$0.625640\pi$
$810$	0	0
$811$	−3095.34	−0.134022	−0.0670111	−	0.997752i	$-0.521346\pi$
$811$	−3095.34	−0.134022	−0.0670111	+	0.997752i	$0.521346\pi$
$812$	2730.93i	0.118026i
$813$	46318.9i	1.99812i
$814$	−34360.2	−1.47951
$815$	0	0
$816$	−14060.3	−0.603198
$817$	18466.0i	0.790751i
$818$	− 62131.5i	− 2.65572i
$819$	1379.75	0.0588674
$820$	0	0
$821$	12323.5	0.523864	0.261932	−	0.965086i	$-0.415640\pi$
$821$	12323.5	0.523864	0.261932	+	0.965086i	$0.415640\pi$
$822$	− 5568.36i	− 0.236276i
$823$	− 34436.5i	− 1.45854i	−0.684225	−	0.729271i	$-0.739860\pi$
$823$	− 34436.5i	− 1.45854i	0.684225	−	0.729271i	$-0.260140\pi$
$824$	9198.50	0.388889
$825$	0	0
$826$	−13040.8	−0.549329
$827$	− 18761.6i	− 0.788880i	−0.918922	−	0.394440i	$-0.870939\pi$
$827$	− 18761.6i	− 0.788880i	0.918922	−	0.394440i	$-0.129061\pi$
$828$	139992.i	5.87567i
$829$	−22423.8	−0.939457	−0.469728	−	0.882811i	$-0.655648\pi$
$829$	−22423.8	−0.939457	−0.469728	+	0.882811i	$0.655648\pi$
$830$	0	0
$831$	−10235.3	−0.427265
$832$	− 1602.13i	− 0.0667596i
$833$	5583.15i	0.232227i
$834$	88441.1	3.67202
$835$	0	0
$836$	60356.4	2.49697
$837$	5276.13i	0.217885i
$838$	29004.0i	1.19562i
$839$	−9128.63	−0.375632	−0.187816	−	0.982204i	$-0.560141\pi$
$839$	−9128.63	−0.375632	−0.187816	+	0.982204i	$0.560141\pi$
$840$	0	0
$841$	−22685.0	−0.930131
$842$	− 9441.58i	− 0.386435i
$843$	− 10146.5i	− 0.414547i
$844$	48634.4	1.98349
$845$	0	0
$846$	−104638.	−4.25240
$847$	− 5390.75i	− 0.218688i
$848$	− 32198.0i	− 1.30387i
$849$	−26822.2	−1.08426
$850$	0	0
$851$	23495.7	0.946442
$852$	98416.1i	3.95737i
$853$	27204.8i	1.09200i	0.837786	+	0.545999i	$0.183850\pi$
$853$	27204.8i	1.09200i	−0.837786	+	0.545999i	$0.816150\pi$
$854$	6538.68	0.262001
$855$	0	0
$856$	22769.8	0.909179
$857$	38060.0i	1.51704i	0.651649	+	0.758520i	$0.274077\pi$
$857$	38060.0i	1.51704i	−0.651649	+	0.758520i	$0.725923\pi$
$858$	− 18004.9i	− 0.716405i
$859$	33326.2	1.32372	0.661860	−	0.749627i	$-0.269767\pi$
$859$	33326.2	1.32372	0.661860	+	0.749627i	$0.269767\pi$
$860$	0	0
$861$	−579.531	−0.0229389
$862$	422.378i	0.0166894i
$863$	− 41724.2i	− 1.64578i	−0.568201	−	0.822890i	$-0.692360\pi$
$863$	− 41724.2i	− 1.64578i	0.568201	−	0.822890i	$-0.307640\pi$
$864$	17475.4	0.688108
$865$	0	0
$866$	77223.8	3.03022
$867$	− 2449.38i	− 0.0959460i
$868$	− 2309.60i	− 0.0903146i
$869$	56569.7	2.20828
$870$	0	0
$871$	−124.068	−0.00482649
$872$	− 58907.5i	− 2.28768i
$873$	− 17009.6i	− 0.659438i
$874$	−60328.6	−2.33483
$875$	0	0
$876$	28411.6	1.09582
$877$	49337.3i	1.89966i	0.312767	+	0.949830i	$0.398744\pi$
$877$	49337.3i	1.89966i	−0.312767	+	0.949830i	$0.601256\pi$
$878$	− 15423.3i	− 0.592838i
$879$	−63194.0	−2.42489
$880$	0	0
$881$	8845.46	0.338265	0.169132	−	0.985593i	$-0.445903\pi$
$881$	8845.46	0.338265	0.169132	+	0.985593i	$0.445903\pi$
$882$	− 74096.8i	− 2.82876i
$883$	14724.2i	0.561165i	0.959830	+	0.280582i	$0.0905276\pi$
$883$	14724.2i	0.561165i	−0.959830	+	0.280582i	$0.909472\pi$
$884$	−2374.10	−0.0903277
$885$	0	0
$886$	9023.14	0.342143
$887$	− 3864.38i	− 0.146283i	−0.997322	−	0.0731415i	$-0.976698\pi$
$887$	− 3864.38i	− 0.146283i	0.997322	−	0.0731415i	$-0.0233025\pi$
$888$	− 51858.1i	− 1.95974i
$889$	−7360.60	−0.277690
$890$	0	0
$891$	−3688.07	−0.138670
$892$	− 81160.9i	− 3.04649i
$893$	− 30849.1i	− 1.15602i
$894$	−79023.2	−2.95630
$895$	0	0
$896$	7351.61	0.274107
$897$	12311.8i	0.458283i
$898$	− 12577.2i	− 0.467379i
$899$	−1441.14	−0.0534648
$900$	0	0
$901$	5609.04	0.207397
$902$	4719.92i	0.174231i
$903$	8984.15i	0.331089i
$904$	−47560.6	−1.74982
$905$	0	0
$906$	−87473.7	−3.20764
$907$	743.409i	0.0272155i	0.999907	+	0.0136078i	$0.00433162\pi$
$907$	743.409i	0.0272155i	−0.999907	+	0.0136078i	$0.995668\pi$
$908$	− 24182.5i	− 0.883839i
$909$	−5905.76	−0.215491
$910$	0	0
$911$	16291.0	0.592475	0.296238	−	0.955114i	$-0.404268\pi$
$911$	16291.0	0.592475	0.296238	+	0.955114i	$0.404268\pi$
$912$	55013.4i	1.99745i
$913$	45336.8i	1.64340i
$914$	74402.5	2.69258
$915$	0	0
$916$	15500.6	0.559122
$917$	1553.04i	0.0559280i
$918$	12929.5i	0.464856i
$919$	6188.99	0.222150	0.111075	−	0.993812i	$-0.464571\pi$
$919$	6188.99	0.222150	0.111075	+	0.993812i	$0.464571\pi$
$920$	0	0
$921$	−55390.5	−1.98174
$922$	− 89311.6i	− 3.19015i
$923$	5402.00i	0.192643i
$924$	29364.8	1.04549
$925$	0	0
$926$	−93033.0	−3.30157
$927$	− 8786.47i	− 0.311311i
$928$	4773.30i	0.168848i
$929$	31661.7	1.11818	0.559089	−	0.829108i	$-0.311151\pi$
$929$	31661.7	1.11818	0.559089	+	0.829108i	$0.311151\pi$
$930$	0	0
$931$	21845.0	0.769003
$932$	− 20724.3i	− 0.728376i
$933$	− 75734.8i	− 2.65750i
$934$	37273.6	1.30581
$935$	0	0
$936$	16959.8	0.592251
$937$	− 35010.5i	− 1.22064i	−0.792153	−	0.610322i	$-0.791040\pi$
$937$	− 35010.5i	− 1.22064i	0.792153	−	0.610322i	$-0.208960\pi$
$938$	− 295.776i	− 0.0102958i
$939$	22279.2	0.774286
$940$	0	0
$941$	−45625.8	−1.58061	−0.790307	−	0.612711i	$-0.790079\pi$
$941$	−45625.8	−1.58061	−0.790307	+	0.612711i	$0.790079\pi$
$942$	− 11217.2i	− 0.387977i
$943$	− 3227.51i	− 0.111455i
$944$	−66227.5	−2.28339
$945$	0	0
$946$	73170.2	2.51477
$947$	21508.4i	0.738044i	0.929421	+	0.369022i	$0.120307\pi$
$947$	21508.4i	0.738044i	−0.929421	+	0.369022i	$0.879693\pi$
$948$	158615.i	5.43416i
$949$	1559.50	0.0533439
$950$	0	0
$951$	36177.4	1.23358
$952$	− 3046.52i	− 0.103717i
$953$	35686.7i	1.21302i	0.795076	+	0.606509i	$0.207431\pi$
$953$	35686.7i	1.21302i	−0.795076	+	0.606509i	$0.792569\pi$
$954$	−74440.5	−2.52631
$955$	0	0
$956$	85744.0	2.90079
$957$	− 18323.0i	− 0.618912i
$958$	93357.8i	3.14849i
$959$	498.486	0.0167851
$960$	0	0
$961$	−28572.2	−0.959088
$962$	− 5288.17i	− 0.177232i
$963$	− 21749.9i	− 0.727810i
$964$	116133.	3.88006
$965$	0	0
$966$	−29351.3	−0.977600
$967$	3731.33i	0.124086i	0.998073	+	0.0620432i	$0.0197617\pi$
$967$	3731.33i	0.124086i	−0.998073	+	0.0620432i	$0.980238\pi$
$968$	− 66262.5i	− 2.20016i
$969$	−9583.60	−0.317719
$970$	0	0
$971$	17645.1	0.583171	0.291585	−	0.956545i	$-0.405817\pi$
$971$	17645.1	0.583171	0.291585	+	0.956545i	$0.405817\pi$
$972$	60358.3i	1.99176i
$973$	7917.35i	0.260862i
$974$	51348.7	1.68924
$975$	0	0
$976$	33206.7	1.08906
$977$	− 24941.2i	− 0.816723i	−0.912820	−	0.408362i	$-0.866100\pi$
$977$	− 24941.2i	− 0.816723i	0.912820	−	0.408362i	$-0.133900\pi$
$978$	61631.8i	2.01510i
$979$	59152.1	1.93106
$980$	0	0
$981$	−56268.9	−1.83132
$982$	− 6277.85i	− 0.204006i
$983$	− 22506.2i	− 0.730252i	−0.930958	−	0.365126i	$-0.881026\pi$
$983$	− 22506.2i	− 0.730252i	0.930958	−	0.365126i	$-0.118974\pi$
$984$	−7123.53	−0.230782
$985$	0	0
$986$	−3531.62	−0.114066
$987$	− 15008.8i	− 0.484029i
$988$	9289.07i	0.299114i
$989$	−50034.2	−1.60869
$990$	0	0
$991$	32694.1	1.04799	0.523997	−	0.851720i	$-0.324440\pi$
$991$	32694.1	1.04799	0.523997	+	0.851720i	$0.324440\pi$
$992$	− 4036.88i	− 0.129205i
$993$	8410.08i	0.268767i
$994$	−12878.3	−0.410941
$995$	0	0
$996$	−127119.	−4.04410
$997$	− 18248.8i	− 0.579686i	−0.957074	−	0.289843i	$-0.906397\pi$
$997$	− 18248.8i	− 0.579686i	0.957074	−	0.289843i	$-0.0936030\pi$
$998$	− 352.428i	− 0.0111783i
$999$	−19702.5	−0.623983

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	425.4.b.f.324.6		6
5.2	odd	4		17.4.a.b.1.1	✓	3
5.3	odd	4		425.4.a.g.1.3		3
5.4	even	2	inner	425.4.b.f.324.1		6
15.2	even	4		153.4.a.g.1.3		3
20.7	even	4		272.4.a.h.1.1		3
35.27	even	4		833.4.a.d.1.1		3
40.27	even	4		1088.4.a.x.1.3		3
40.37	odd	4		1088.4.a.v.1.1		3
55.32	even	4		2057.4.a.e.1.3		3
60.47	odd	4		2448.4.a.bi.1.2		3
85.47	odd	4		289.4.b.b.288.6		6
85.67	odd	4		289.4.a.b.1.1		3
85.72	odd	4		289.4.b.b.288.5		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
17.4.a.b.1.1	✓	3	5.2	odd	4
153.4.a.g.1.3		3	15.2	even	4
272.4.a.h.1.1		3	20.7	even	4
289.4.a.b.1.1		3	85.67	odd	4
289.4.b.b.288.5		6	85.72	odd	4
289.4.b.b.288.6		6	85.47	odd	4
425.4.a.g.1.3		3	5.3	odd	4
425.4.b.f.324.1		6	5.4	even	2	inner
425.4.b.f.324.6		6	1.1	even	1	trivial
833.4.a.d.1.1		3	35.27	even	4
1088.4.a.v.1.1		3	40.37	odd	4
1088.4.a.x.1.3		3	40.27	even	4
2057.4.a.e.1.3		3	55.32	even	4
2448.4.a.bi.1.2		3	60.47	odd	4

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:	\( N \)	\(=\)	\( 425 = 5^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	425.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+5.03251i q^{2} +8.47535i q^{3} -17.3261 q^{4} -42.6523 q^{6} -3.81828i q^{7} -46.9339i q^{8} -44.8316 q^{9} +O(q^{10})\)	\(q+5.03251i q^{2} +8.47535i q^{3} -17.3261 q^{4} -42.6523 q^{6} -3.81828i q^{7} -46.9339i q^{8} -44.8316 q^{9} -52.3720 q^{11} -146.845i q^{12} -8.06025i q^{13} +19.2156 q^{14} +97.5862 q^{16} +17.0000i q^{17} -225.616i q^{18} +66.5154 q^{19} +32.3613 q^{21} -263.563i q^{22} +180.226i q^{23} +397.782 q^{24} +40.5633 q^{26} -151.129i q^{27} +66.1562i q^{28} +41.2800 q^{29} -34.9114 q^{31} +115.632i q^{32} -443.871i q^{33} -85.5527 q^{34} +776.759 q^{36} -130.368i q^{37} +334.739i q^{38} +68.3134 q^{39} -17.9081 q^{41} +162.859i q^{42} +277.620i q^{43} +907.405 q^{44} -906.987 q^{46} -463.789i q^{47} +827.078i q^{48} +328.421 q^{49} -144.081 q^{51} +139.653i q^{52} -329.944i q^{53} +760.560 q^{54} -179.207 q^{56} +563.741i q^{57} +207.742i q^{58} -678.656 q^{59} +340.280 q^{61} -175.692i q^{62} +171.180i q^{63} +198.770 q^{64} +2233.79 q^{66} -15.3925i q^{67} -294.545i q^{68} -1527.48 q^{69} -670.203 q^{71} +2104.12i q^{72} +193.480i q^{73} +656.080 q^{74} -1152.46 q^{76} +199.971i q^{77} +343.788i q^{78} -1080.15 q^{79} +70.4207 q^{81} -90.1229i q^{82} -865.668i q^{83} -560.697 q^{84} -1397.13 q^{86} +349.863i q^{87} +2458.02i q^{88} -1129.46 q^{89} -30.7763 q^{91} -3122.61i q^{92} -295.886i q^{93} +2334.02 q^{94} -980.023 q^{96} +379.412i q^{97} +1652.78i q^{98} +2347.92 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 50 q^{4} - 148 q^{6} - 118 q^{9}+O(q^{10}) \) 6 * q - 50 * q^4 - 148 * q^6 - 118 * q^9	\( 6 q - 50 q^{4} - 148 q^{6} - 118 q^{9} - 56 q^{11} - 184 q^{14} + 274 q^{16} - 160 q^{19} - 384 q^{21} + 1332 q^{24} + 52 q^{26} + 912 q^{29} + 460 q^{31} + 34 q^{34} + 2626 q^{36} - 536 q^{39} - 588 q^{41} + 2244 q^{44} - 1408 q^{46} + 538 q^{49} - 136 q^{51} + 2200 q^{54} + 1368 q^{56} - 1272 q^{59} - 168 q^{61} + 1838 q^{64} + 4936 q^{66} - 1152 q^{69} - 804 q^{71} - 1672 q^{74} - 1816 q^{76} + 1188 q^{79} - 1010 q^{81} + 4080 q^{84} - 2528 q^{86} + 340 q^{89} - 2032 q^{91} + 4032 q^{94} + 1356 q^{96} + 5840 q^{99}+O(q^{100}) \) 6 * q - 50 * q^4 - 148 * q^6 - 118 * q^9 - 56 * q^11 - 184 * q^14 + 274 * q^16 - 160 * q^19 - 384 * q^21 + 1332 * q^24 + 52 * q^26 + 912 * q^29 + 460 * q^31 + 34 * q^34 + 2626 * q^36 - 536 * q^39 - 588 * q^41 + 2244 * q^44 - 1408 * q^46 + 538 * q^49 - 136 * q^51 + 2200 * q^54 + 1368 * q^56 - 1272 * q^59 - 168 * q^61 + 1838 * q^64 + 4936 * q^66 - 1152 * q^69 - 804 * q^71 - 1672 * q^74 - 1816 * q^76 + 1188 * q^79 - 1010 * q^81 + 4080 * q^84 - 2528 * q^86 + 340 * q^89 - 2032 * q^91 + 4032 * q^94 + 1356 * q^96 + 5840 * q^99

Introduction

L-functions

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