LMFDB - Embedded newform 1850.2.b.i.149.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1850,2,Mod(149,1850)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1850, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1850.149");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1850 = 2 \cdot 5^{2} \cdot 37 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1850.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:	$14.7723243739$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{13})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 7x^{2} + 9 $ x^4 + 7*x^2 + 9
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 74)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			149.2
Root			$1.30278i$ of defining polynomial
Character	$\chi$	$=$	1850.149
Dual form			1850.2.b.i.149.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{2} +0.302776i q^{3} -1.00000 q^{4} +0.302776 q^{6} +4.60555i q^{7} +1.00000i q^{8} +2.90833 q^{9} +O(q^{10})$	\(q-1.00000i q^{2} +0.302776i q^{3} -1.00000 q^{4} +0.302776 q^{6} +4.60555i q^{7} +1.00000i q^{8} +2.90833 q^{9} +1.30278 q^{11} -0.302776i q^{12} +2.30278i q^{13} +4.60555 q^{14} +1.00000 q^{16} -6.00000i q^{17} -2.90833i q^{18} -2.00000 q^{19} -1.39445 q^{21} -1.30278i q^{22} +6.90833i q^{23} -0.302776 q^{24} +2.30278 q^{26} +1.78890i q^{27} -4.60555i q^{28} -6.90833 q^{29} +3.30278 q^{31} -1.00000i q^{32} +0.394449i q^{33} -6.00000 q^{34} -2.90833 q^{36} +1.00000i q^{37} +2.00000i q^{38} -0.697224 q^{39} -0.908327 q^{41} +1.39445i q^{42} +6.60555i q^{43} -1.30278 q^{44} +6.90833 q^{46} -2.60555i q^{47} +0.302776i q^{48} -14.2111 q^{49} +1.81665 q^{51} -2.30278i q^{52} +6.00000i q^{53} +1.78890 q^{54} -4.60555 q^{56} -0.605551i q^{57} +6.90833i q^{58} -3.39445 q^{59} -10.5139 q^{61} -3.30278i q^{62} +13.3944i q^{63} -1.00000 q^{64} +0.394449 q^{66} +14.5139i q^{67} +6.00000i q^{68} -2.09167 q^{69} +6.00000 q^{71} +2.90833i q^{72} +8.69722i q^{73} +1.00000 q^{74} +2.00000 q^{76} +6.00000i q^{77} +0.697224i q^{78} +16.1194 q^{79} +8.18335 q^{81} +0.908327i q^{82} -17.2111i q^{83} +1.39445 q^{84} +6.60555 q^{86} -2.09167i q^{87} +1.30278i q^{88} -5.21110 q^{89} -10.6056 q^{91} -6.90833i q^{92} +1.00000i q^{93} -2.60555 q^{94} +0.302776 q^{96} +12.4222i q^{97} +14.2111i q^{98} +3.78890 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{4} - 6 q^{6} - 10 q^{9}+O(q^{10}) $ 4 * q - 4 * q^4 - 6 * q^6 - 10 * q^9	$ 4 q - 4 q^{4} - 6 q^{6} - 10 q^{9} - 2 q^{11} + 4 q^{14} + 4 q^{16} - 8 q^{19} - 20 q^{21} + 6 q^{24} + 2 q^{26} - 6 q^{29} + 6 q^{31} - 24 q^{34} + 10 q^{36} - 10 q^{39} + 18 q^{41} + 2 q^{44} + 6 q^{46} - 28 q^{49} - 36 q^{51} + 36 q^{54} - 4 q^{56} - 28 q^{59} - 6 q^{61} - 4 q^{64} + 16 q^{66} - 30 q^{69} + 24 q^{71} + 4 q^{74} + 8 q^{76} + 14 q^{79} + 76 q^{81} + 20 q^{84} + 12 q^{86} + 8 q^{89} - 28 q^{91} + 4 q^{94} - 6 q^{96} + 44 q^{99}+O(q^{100}) $ 4 * q - 4 * q^4 - 6 * q^6 - 10 * q^9 - 2 * q^11 + 4 * q^14 + 4 * q^16 - 8 * q^19 - 20 * q^21 + 6 * q^24 + 2 * q^26 - 6 * q^29 + 6 * q^31 - 24 * q^34 + 10 * q^36 - 10 * q^39 + 18 * q^41 + 2 * q^44 + 6 * q^46 - 28 * q^49 - 36 * q^51 + 36 * q^54 - 4 * q^56 - 28 * q^59 - 6 * q^61 - 4 * q^64 + 16 * q^66 - 30 * q^69 + 24 * q^71 + 4 * q^74 + 8 * q^76 + 14 * q^79 + 76 * q^81 + 20 * q^84 + 12 * q^86 + 8 * q^89 - 28 * q^91 + 4 * q^94 - 6 * q^96 + 44 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1850\mathbb{Z}\right)^\times$.

$n$	$1001$	$1777$
$\chi(n)$	$1$	$-1$

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 $

$2$	− 1.00000i	− 0.707107i
$2$	− 1.00000i	− 0.707107i
$3$	0.302776i	0.174808i	0.996173	+	0.0874038i	$0.0278570\pi$
$3$	0.302776i	0.174808i	−0.996173	+	0.0874038i	$0.972143\pi$
$4$	−1.00000	−0.500000
$5$	0	0
$5$	0	0
$6$	0.302776	0.123608
$7$	4.60555i	1.74073i	0.492403	+	0.870367i	$0.336119\pi$
$7$	4.60555i	1.74073i	−0.492403	+	0.870367i	$0.663881\pi$
$8$	1.00000i	0.353553i
$9$	2.90833	0.969442
$10$	0	0
$11$	1.30278	0.392802	0.196401	−	0.980524i	$-0.437075\pi$
$11$	1.30278	0.392802	0.196401	+	0.980524i	$0.437075\pi$
$12$	− 0.302776i	− 0.0874038i
$13$	2.30278i	0.638675i	0.947641	+	0.319338i	$0.103460\pi$
$13$	2.30278i	0.638675i	−0.947641	+	0.319338i	$0.896540\pi$
$14$	4.60555	1.23089
$15$	0	0
$16$	1.00000	0.250000
$17$	− 6.00000i	− 1.45521i	−0.685994	−	0.727607i	$-0.740633\pi$
$17$	− 6.00000i	− 1.45521i	0.685994	−	0.727607i	$-0.259367\pi$
$18$	− 2.90833i	− 0.685499i
$19$	−2.00000	−0.458831	−0.229416	−	0.973329i	$-0.573682\pi$
$19$	−2.00000	−0.458831	−0.229416	+	0.973329i	$0.573682\pi$
$20$	0	0
$21$	−1.39445	−0.304294
$22$	− 1.30278i	− 0.277753i
$23$	6.90833i	1.44049i	0.693722	+	0.720243i	$0.255970\pi$
$23$	6.90833i	1.44049i	−0.693722	+	0.720243i	$0.744030\pi$
$24$	−0.302776	−0.0618038
$25$	0	0
$26$	2.30278	0.451611
$27$	1.78890i	0.344273i
$28$	− 4.60555i	− 0.870367i
$29$	−6.90833	−1.28284	−0.641422	−	0.767188i	$-0.721655\pi$
$29$	−6.90833	−1.28284	−0.641422	+	0.767188i	$0.721655\pi$
$30$	0	0
$31$	3.30278	0.593196	0.296598	−	0.955002i	$-0.404148\pi$
$31$	3.30278	0.593196	0.296598	+	0.955002i	$0.404148\pi$
$32$	− 1.00000i	− 0.176777i
$33$	0.394449i	0.0686647i
$34$	−6.00000	−1.02899
$35$	0	0
$36$	−2.90833	−0.484721
$37$	1.00000i	0.164399i
$37$	1.00000i	0.164399i
$38$	2.00000i	0.324443i
$39$	−0.697224	−0.111645
$40$	0	0
$41$	−0.908327	−0.141857	−0.0709284	−	0.997481i	$-0.522596\pi$
$41$	−0.908327	−0.141857	−0.0709284	+	0.997481i	$0.522596\pi$
$42$	1.39445i	0.215168i
$43$	6.60555i	1.00734i	0.863897	+	0.503669i	$0.168017\pi$
$43$	6.60555i	1.00734i	−0.863897	+	0.503669i	$0.831983\pi$
$44$	−1.30278	−0.196401
$45$	0	0
$46$	6.90833	1.01858
$47$	− 2.60555i	− 0.380059i	−0.981778	−	0.190029i	$-0.939142\pi$
$47$	− 2.60555i	− 0.380059i	0.981778	−	0.190029i	$-0.0608583\pi$
$48$	0.302776i	0.0437019i
$49$	−14.2111	−2.03016
$50$	0	0
$51$	1.81665	0.254382
$52$	− 2.30278i	− 0.319338i
$53$	6.00000i	0.824163i	0.911147	+	0.412082i	$0.135198\pi$
$53$	6.00000i	0.824163i	−0.911147	+	0.412082i	$0.864802\pi$
$54$	1.78890	0.243438
$55$	0	0
$56$	−4.60555	−0.615443
$57$	− 0.605551i	− 0.0802072i
$58$	6.90833i	0.907108i
$59$	−3.39445	−0.441920	−0.220960	−	0.975283i	$-0.570919\pi$
$59$	−3.39445	−0.441920	−0.220960	+	0.975283i	$0.570919\pi$
$60$	0	0
$61$	−10.5139	−1.34616	−0.673082	−	0.739568i	$-0.735030\pi$
$61$	−10.5139	−1.34616	−0.673082	+	0.739568i	$0.735030\pi$
$62$	− 3.30278i	− 0.419453i
$63$	13.3944i	1.68754i
$64$	−1.00000	−0.125000
$65$	0	0
$66$	0.394449	0.0485533
$67$	14.5139i	1.77315i	0.462583	+	0.886576i	$0.346923\pi$
$67$	14.5139i	1.77315i	−0.462583	+	0.886576i	$0.653077\pi$
$68$	6.00000i	0.727607i
$69$	−2.09167	−0.251808
$70$	0	0
$71$	6.00000	0.712069	0.356034	−	0.934473i	$-0.384129\pi$
$71$	6.00000	0.712069	0.356034	+	0.934473i	$0.384129\pi$
$72$	2.90833i	0.342750i
$73$	8.69722i	1.01793i	0.860786	+	0.508967i	$0.169972\pi$
$73$	8.69722i	1.01793i	−0.860786	+	0.508967i	$0.830028\pi$
$74$	1.00000	0.116248
$75$	0	0
$76$	2.00000	0.229416
$77$	6.00000i	0.683763i
$78$	0.697224i	0.0789451i
$79$	16.1194	1.81358	0.906789	−	0.421585i	$-0.138526\pi$
$79$	16.1194	1.81358	0.906789	+	0.421585i	$0.138526\pi$
$80$	0	0
$81$	8.18335	0.909261
$82$	0.908327i	0.100308i
$83$	− 17.2111i	− 1.88916i	−0.328276	−	0.944582i	$-0.606467\pi$
$83$	− 17.2111i	− 1.88916i	0.328276	−	0.944582i	$-0.393533\pi$
$84$	1.39445	0.152147
$85$	0	0
$86$	6.60555	0.712295
$87$	− 2.09167i	− 0.224251i
$88$	1.30278i	0.138876i
$89$	−5.21110	−0.552376	−0.276188	−	0.961104i	$-0.589071\pi$
$89$	−5.21110	−0.552376	−0.276188	+	0.961104i	$0.589071\pi$
$90$	0	0
$91$	−10.6056	−1.11176
$92$	− 6.90833i	− 0.720243i
$93$	1.00000i	0.103695i
$94$	−2.60555	−0.268742
$95$	0	0
$96$	0.302776	0.0309019
$97$	12.4222i	1.26128i	0.776074	+	0.630642i	$0.217208\pi$
$97$	12.4222i	1.26128i	−0.776074	+	0.630642i	$0.782792\pi$
$98$	14.2111i	1.43554i
$99$	3.78890	0.380799
$100$	0	0
$101$	16.4222	1.63407	0.817035	−	0.576588i	$-0.195616\pi$
$101$	16.4222	1.63407	0.817035	+	0.576588i	$0.195616\pi$
$102$	− 1.81665i	− 0.179876i
$103$	− 3.30278i	− 0.325432i	−0.986673	−	0.162716i	$-0.947975\pi$
$103$	− 3.30278i	− 0.325432i	0.986673	−	0.162716i	$-0.0520255\pi$
$104$	−2.30278	−0.225806
$105$	0	0
$106$	6.00000	0.582772
$107$	4.30278i	0.415965i	0.978133	+	0.207983i	$0.0666897\pi$
$107$	4.30278i	0.415965i	−0.978133	+	0.207983i	$0.933310\pi$
$108$	− 1.78890i	− 0.172137i
$109$	−2.00000	−0.191565	−0.0957826	−	0.995402i	$-0.530535\pi$
$109$	−2.00000	−0.191565	−0.0957826	+	0.995402i	$0.530535\pi$
$110$	0	0
$111$	−0.302776	−0.0287382
$112$	4.60555i	0.435184i
$113$	− 11.2111i	− 1.05465i	−0.849663	−	0.527326i	$-0.823195\pi$
$113$	− 11.2111i	− 1.05465i	0.849663	−	0.527326i	$-0.176805\pi$
$114$	−0.605551	−0.0567151
$115$	0	0
$116$	6.90833	0.641422
$117$	6.69722i	0.619159i
$118$	3.39445i	0.312484i
$119$	27.6333	2.53314
$120$	0	0
$121$	−9.30278	−0.845707
$122$	10.5139i	0.951882i
$123$	− 0.275019i	− 0.0247977i
$124$	−3.30278	−0.296598
$125$	0	0
$126$	13.3944	1.19327
$127$	− 4.78890i	− 0.424946i	−0.977167	−	0.212473i	$-0.931848\pi$
$127$	− 4.78890i	− 0.424946i	0.977167	−	0.212473i	$-0.0681517\pi$
$128$	1.00000i	0.0883883i
$129$	−2.00000	−0.176090
$130$	0	0
$131$	3.39445	0.296574	0.148287	−	0.988944i	$-0.452624\pi$
$131$	3.39445	0.296574	0.148287	+	0.988944i	$0.452624\pi$
$132$	− 0.394449i	− 0.0343324i
$133$	− 9.21110i	− 0.798704i
$134$	14.5139	1.25381
$135$	0	0
$136$	6.00000	0.514496
$137$	− 9.90833i	− 0.846525i	−0.906007	−	0.423263i	$-0.860885\pi$
$137$	− 9.90833i	− 0.846525i	0.906007	−	0.423263i	$-0.139115\pi$
$138$	2.09167i	0.178055i
$139$	−8.90833	−0.755594	−0.377797	−	0.925888i	$-0.623318\pi$
$139$	−8.90833	−0.755594	−0.377797	+	0.925888i	$0.623318\pi$
$140$	0	0
$141$	0.788897	0.0664372
$142$	− 6.00000i	− 0.503509i
$143$	3.00000i	0.250873i
$144$	2.90833	0.242361
$145$	0	0
$146$	8.69722	0.719787
$147$	− 4.30278i	− 0.354887i
$148$	− 1.00000i	− 0.0821995i
$149$	1.81665	0.148826	0.0744130	−	0.997228i	$-0.476292\pi$
$149$	1.81665	0.148826	0.0744130	+	0.997228i	$0.476292\pi$
$150$	0	0
$151$	−13.3944	−1.09002	−0.545012	−	0.838428i	$-0.683475\pi$
$151$	−13.3944	−1.09002	−0.545012	+	0.838428i	$0.683475\pi$
$152$	− 2.00000i	− 0.162221i
$153$	− 17.4500i	− 1.41075i
$154$	6.00000	0.483494
$155$	0	0
$156$	0.697224	0.0558226
$157$	7.21110i	0.575509i	0.957704	+	0.287754i	$0.0929087\pi$
$157$	7.21110i	0.575509i	−0.957704	+	0.287754i	$0.907091\pi$
$158$	− 16.1194i	− 1.28239i
$159$	−1.81665	−0.144070
$160$	0	0
$161$	−31.8167	−2.50750
$162$	− 8.18335i	− 0.642944i
$163$	20.4222i	1.59959i	0.600273	+	0.799795i	$0.295059\pi$
$163$	20.4222i	1.59959i	−0.600273	+	0.799795i	$0.704941\pi$
$164$	0.908327	0.0709284
$165$	0	0
$166$	−17.2111	−1.33584
$167$	12.5139i	0.968353i	0.874970	+	0.484176i	$0.160881\pi$
$167$	12.5139i	0.968353i	−0.874970	+	0.484176i	$0.839119\pi$
$168$	− 1.39445i	− 0.107584i
$169$	7.69722	0.592094
$170$	0	0
$171$	−5.81665	−0.444811
$172$	− 6.60555i	− 0.503669i
$173$	23.2111i	1.76471i	0.470587	+	0.882354i	$0.344042\pi$
$173$	23.2111i	1.76471i	−0.470587	+	0.882354i	$0.655958\pi$
$174$	−2.09167	−0.158569
$175$	0	0
$176$	1.30278	0.0982004
$177$	− 1.02776i	− 0.0772509i
$178$	5.21110i	0.390589i
$179$	−7.81665	−0.584244	−0.292122	−	0.956381i	$-0.594361\pi$
$179$	−7.81665	−0.584244	−0.292122	+	0.956381i	$0.594361\pi$
$180$	0	0
$181$	20.0000	1.48659	0.743294	−	0.668965i	$-0.233262\pi$
$181$	20.0000	1.48659	0.743294	+	0.668965i	$0.233262\pi$
$182$	10.6056i	0.786136i
$183$	− 3.18335i	− 0.235320i
$184$	−6.90833	−0.509289
$185$	0	0
$186$	1.00000	0.0733236
$187$	− 7.81665i	− 0.571610i
$188$	2.60555i	0.190029i
$189$	−8.23886	−0.599289
$190$	0	0
$191$	12.5139	0.905472	0.452736	−	0.891644i	$-0.350448\pi$
$191$	12.5139	0.905472	0.452736	+	0.891644i	$0.350448\pi$
$192$	− 0.302776i	− 0.0218509i
$193$	4.00000i	0.287926i	0.989583	+	0.143963i	$0.0459847\pi$
$193$	4.00000i	0.287926i	−0.989583	+	0.143963i	$0.954015\pi$
$194$	12.4222	0.891862
$195$	0	0
$196$	14.2111	1.01508
$197$	− 6.00000i	− 0.427482i	−0.976890	−	0.213741i	$-0.931435\pi$
$197$	− 6.00000i	− 0.427482i	0.976890	−	0.213741i	$-0.0685649\pi$
$198$	− 3.78890i	− 0.269265i
$199$	2.42221	0.171706	0.0858528	−	0.996308i	$-0.472639\pi$
$199$	2.42221	0.171706	0.0858528	+	0.996308i	$0.472639\pi$
$200$	0	0
$201$	−4.39445	−0.309961
$202$	− 16.4222i	− 1.15546i
$203$	− 31.8167i	− 2.23309i
$204$	−1.81665	−0.127191
$205$	0	0
$206$	−3.30278	−0.230115
$207$	20.0917i	1.39647i
$208$	2.30278i	0.159669i
$209$	−2.60555	−0.180230
$210$	0	0
$211$	6.69722	0.461056	0.230528	−	0.973066i	$-0.425955\pi$
$211$	6.69722	0.461056	0.230528	+	0.973066i	$0.425955\pi$
$212$	− 6.00000i	− 0.412082i
$213$	1.81665i	0.124475i
$214$	4.30278	0.294132
$215$	0	0
$216$	−1.78890	−0.121719
$217$	15.2111i	1.03260i
$218$	2.00000i	0.135457i
$219$	−2.63331	−0.177942
$220$	0	0
$221$	13.8167	0.929409
$222$	0.302776i	0.0203210i
$223$	− 15.8167i	− 1.05916i	−0.848260	−	0.529581i	$-0.822349\pi$
$223$	− 15.8167i	− 1.05916i	0.848260	−	0.529581i	$-0.177651\pi$
$224$	4.60555	0.307721
$225$	0	0
$226$	−11.2111	−0.745751
$227$	− 7.81665i	− 0.518810i	−0.965769	−	0.259405i	$-0.916474\pi$
$227$	− 7.81665i	− 0.518810i	0.965769	−	0.259405i	$-0.0835264\pi$
$228$	0.605551i	0.0401036i
$229$	−17.3944	−1.14946	−0.574729	−	0.818344i	$-0.694892\pi$
$229$	−17.3944	−1.14946	−0.574729	+	0.818344i	$0.694892\pi$
$230$	0	0
$231$	−1.81665	−0.119527
$232$	− 6.90833i	− 0.453554i
$233$	9.51388i	0.623275i	0.950201	+	0.311637i	$0.100877\pi$
$233$	9.51388i	0.623275i	−0.950201	+	0.311637i	$0.899123\pi$
$234$	6.69722	0.437811
$235$	0	0
$236$	3.39445	0.220960
$237$	4.88057i	0.317027i
$238$	− 27.6333i	− 1.79120i
$239$	−0.513878	−0.0332400	−0.0166200	−	0.999862i	$-0.505291\pi$
$239$	−0.513878	−0.0332400	−0.0166200	+	0.999862i	$0.505291\pi$
$240$	0	0
$241$	8.00000	0.515325	0.257663	−	0.966235i	$-0.417048\pi$
$241$	8.00000	0.515325	0.257663	+	0.966235i	$0.417048\pi$
$242$	9.30278i	0.598005i
$243$	7.84441i	0.503219i
$244$	10.5139	0.673082
$245$	0	0
$246$	−0.275019	−0.0175346
$247$	− 4.60555i	− 0.293044i
$248$	3.30278i	0.209726i
$249$	5.21110	0.330240
$250$	0	0
$251$	−6.78890	−0.428511	−0.214256	−	0.976778i	$-0.568733\pi$
$251$	−6.78890	−0.428511	−0.214256	+	0.976778i	$0.568733\pi$
$252$	− 13.3944i	− 0.843771i
$253$	9.00000i	0.565825i
$254$	−4.78890	−0.300482
$255$	0	0
$256$	1.00000	0.0625000
$257$	− 11.2111i	− 0.699329i	−0.936875	−	0.349665i	$-0.886296\pi$
$257$	− 11.2111i	− 0.699329i	0.936875	−	0.349665i	$-0.113704\pi$
$258$	2.00000i	0.124515i
$259$	−4.60555	−0.286175
$260$	0	0
$261$	−20.0917	−1.24364
$262$	− 3.39445i	− 0.209710i
$263$	7.81665i	0.481996i	0.970526	+	0.240998i	$0.0774746\pi$
$263$	7.81665i	0.481996i	−0.970526	+	0.240998i	$0.922525\pi$
$264$	−0.394449	−0.0242766
$265$	0	0
$266$	−9.21110	−0.564769
$267$	− 1.57779i	− 0.0965595i
$268$	− 14.5139i	− 0.886576i
$269$	6.78890	0.413926	0.206963	−	0.978349i	$-0.433642\pi$
$269$	6.78890	0.413926	0.206963	+	0.978349i	$0.433642\pi$
$270$	0	0
$271$	6.42221	0.390121	0.195061	−	0.980791i	$-0.437510\pi$
$271$	6.42221	0.390121	0.195061	+	0.980791i	$0.437510\pi$
$272$	− 6.00000i	− 0.363803i
$273$	− 3.21110i	− 0.194345i
$274$	−9.90833	−0.598584
$275$	0	0
$276$	2.09167	0.125904
$277$	− 25.1194i	− 1.50928i	−0.656139	−	0.754640i	$-0.727811\pi$
$277$	− 25.1194i	− 1.50928i	0.656139	−	0.754640i	$-0.272189\pi$
$278$	8.90833i	0.534286i
$279$	9.60555	0.575069
$280$	0	0
$281$	−12.0000	−0.715860	−0.357930	−	0.933748i	$-0.616517\pi$
$281$	−12.0000	−0.715860	−0.357930	+	0.933748i	$0.616517\pi$
$282$	− 0.788897i	− 0.0469782i
$283$	− 17.3944i	− 1.03399i	−0.855988	−	0.516996i	$-0.827050\pi$
$283$	− 17.3944i	− 1.03399i	0.855988	−	0.516996i	$-0.172950\pi$
$284$	−6.00000	−0.356034
$285$	0	0
$286$	3.00000	0.177394
$287$	− 4.18335i	− 0.246935i
$288$	− 2.90833i	− 0.171375i
$289$	−19.0000	−1.11765
$290$	0	0
$291$	−3.76114	−0.220482
$292$	− 8.69722i	− 0.508967i
$293$	25.0278i	1.46214i	0.682304	+	0.731069i	$0.260978\pi$
$293$	25.0278i	1.46214i	−0.682304	+	0.731069i	$0.739022\pi$
$294$	−4.30278	−0.250943
$295$	0	0
$296$	−1.00000	−0.0581238
$297$	2.33053i	0.135231i
$298$	− 1.81665i	− 0.105236i
$299$	−15.9083	−0.920002
$300$	0	0
$301$	−30.4222	−1.75351
$302$	13.3944i	0.770764i
$303$	4.97224i	0.285648i
$304$	−2.00000	−0.114708
$305$	0	0
$306$	−17.4500	−0.997548
$307$	7.09167i	0.404743i	0.979309	+	0.202372i	$0.0648649\pi$
$307$	7.09167i	0.404743i	−0.979309	+	0.202372i	$0.935135\pi$
$308$	− 6.00000i	− 0.341882i
$309$	1.00000	0.0568880
$310$	0	0
$311$	5.09167	0.288722	0.144361	−	0.989525i	$-0.453887\pi$
$311$	5.09167	0.288722	0.144361	+	0.989525i	$0.453887\pi$
$312$	− 0.697224i	− 0.0394726i
$313$	− 27.0278i	− 1.52770i	−0.645394	−	0.763850i	$-0.723307\pi$
$313$	− 27.0278i	− 1.52770i	0.645394	−	0.763850i	$-0.276693\pi$
$314$	7.21110	0.406946
$315$	0	0
$316$	−16.1194	−0.906789
$317$	− 5.21110i	− 0.292685i	−0.989234	−	0.146342i	$-0.953250\pi$
$317$	− 5.21110i	− 0.292685i	0.989234	−	0.146342i	$-0.0467501\pi$
$318$	1.81665i	0.101873i
$319$	−9.00000	−0.503903
$320$	0	0
$321$	−1.30278	−0.0727138
$322$	31.8167i	1.77307i
$323$	12.0000i	0.667698i
$324$	−8.18335	−0.454630
$325$	0	0
$326$	20.4222	1.13108
$327$	− 0.605551i	− 0.0334871i
$328$	− 0.908327i	− 0.0501540i
$329$	12.0000	0.661581
$330$	0	0
$331$	1.21110	0.0665682	0.0332841	−	0.999446i	$-0.489403\pi$
$331$	1.21110	0.0665682	0.0332841	+	0.999446i	$0.489403\pi$
$332$	17.2111i	0.944582i
$333$	2.90833i	0.159375i
$334$	12.5139	0.684729
$335$	0	0
$336$	−1.39445	−0.0760734
$337$	− 19.1194i	− 1.04150i	−0.853709	−	0.520751i	$-0.825652\pi$
$337$	− 19.1194i	− 1.04150i	0.853709	−	0.520751i	$-0.174348\pi$
$338$	− 7.69722i	− 0.418674i
$339$	3.39445	0.184361
$340$	0	0
$341$	4.30278	0.233008
$342$	5.81665i	0.314529i
$343$	− 33.2111i	− 1.79323i
$344$	−6.60555	−0.356147
$345$	0	0
$346$	23.2111	1.24784
$347$	31.8167i	1.70801i	0.520267	+	0.854004i	$0.325832\pi$
$347$	31.8167i	1.70801i	−0.520267	+	0.854004i	$0.674168\pi$
$348$	2.09167i	0.112125i
$349$	22.2389	1.19042	0.595209	−	0.803571i	$-0.297069\pi$
$349$	22.2389	1.19042	0.595209	+	0.803571i	$0.297069\pi$
$350$	0	0
$351$	−4.11943	−0.219879
$352$	− 1.30278i	− 0.0694382i
$353$	− 31.8167i	− 1.69343i	−0.532047	−	0.846715i	$-0.678577\pi$
$353$	− 31.8167i	− 1.69343i	0.532047	−	0.846715i	$-0.321423\pi$
$354$	−1.02776	−0.0546246
$355$	0	0
$356$	5.21110	0.276188
$357$	8.36669i	0.442812i
$358$	7.81665i	0.413123i
$359$	11.2111	0.591699	0.295850	−	0.955235i	$-0.404397\pi$
$359$	11.2111	0.591699	0.295850	+	0.955235i	$0.404397\pi$
$360$	0	0
$361$	−15.0000	−0.789474
$362$	− 20.0000i	− 1.05118i
$363$	− 2.81665i	− 0.147836i
$364$	10.6056	0.555882
$365$	0	0
$366$	−3.18335	−0.166396
$367$	− 17.8167i	− 0.930022i	−0.885305	−	0.465011i	$-0.846050\pi$
$367$	− 17.8167i	− 0.930022i	0.885305	−	0.465011i	$-0.153950\pi$
$368$	6.90833i	0.360121i
$369$	−2.64171	−0.137522
$370$	0	0
$371$	−27.6333	−1.43465
$372$	− 1.00000i	− 0.0518476i
$373$	− 3.81665i	− 0.197619i	−0.995106	−	0.0988094i	$-0.968497\pi$
$373$	− 3.81665i	− 0.197619i	0.995106	−	0.0988094i	$-0.0315034\pi$
$374$	−7.81665	−0.404190
$375$	0	0
$376$	2.60555	0.134371
$377$	− 15.9083i	− 0.819321i
$378$	8.23886i	0.423761i
$379$	15.3305	0.787477	0.393738	−	0.919223i	$-0.371182\pi$
$379$	15.3305	0.787477	0.393738	+	0.919223i	$0.371182\pi$
$380$	0	0
$381$	1.44996	0.0742838
$382$	− 12.5139i	− 0.640266i
$383$	− 20.8444i	− 1.06510i	−0.846399	−	0.532550i	$-0.821234\pi$
$383$	− 20.8444i	− 1.06510i	0.846399	−	0.532550i	$-0.178766\pi$
$384$	−0.302776	−0.0154510
$385$	0	0
$386$	4.00000	0.203595
$387$	19.2111i	0.976555i
$388$	− 12.4222i	− 0.630642i
$389$	11.8806	0.602369	0.301184	−	0.953566i	$-0.402618\pi$
$389$	11.8806	0.602369	0.301184	+	0.953566i	$0.402618\pi$
$390$	0	0
$391$	41.4500	2.09621
$392$	− 14.2111i	− 0.717769i
$393$	1.02776i	0.0518435i
$394$	−6.00000	−0.302276
$395$	0	0
$396$	−3.78890	−0.190399
$397$	27.8167i	1.39608i	0.716060	+	0.698039i	$0.245944\pi$
$397$	27.8167i	1.39608i	−0.716060	+	0.698039i	$0.754056\pi$
$398$	− 2.42221i	− 0.121414i
$399$	2.78890	0.139620
$400$	0	0
$401$	13.8167	0.689971	0.344985	−	0.938608i	$-0.387884\pi$
$401$	13.8167	0.689971	0.344985	+	0.938608i	$0.387884\pi$
$402$	4.39445i	0.219175i
$403$	7.60555i	0.378859i
$404$	−16.4222	−0.817035
$405$	0	0
$406$	−31.8167	−1.57903
$407$	1.30278i	0.0645762i
$408$	1.81665i	0.0899378i
$409$	5.02776	0.248607	0.124303	−	0.992244i	$-0.460330\pi$
$409$	5.02776	0.248607	0.124303	+	0.992244i	$0.460330\pi$
$410$	0	0
$411$	3.00000	0.147979
$412$	3.30278i	0.162716i
$413$	− 15.6333i	− 0.769265i
$414$	20.0917	0.987452
$415$	0	0
$416$	2.30278	0.112903
$417$	− 2.69722i	− 0.132084i
$418$	2.60555i	0.127442i
$419$	25.1472	1.22852	0.614260	−	0.789104i	$-0.289455\pi$
$419$	25.1472	1.22852	0.614260	+	0.789104i	$0.289455\pi$
$420$	0	0
$421$	28.7250	1.39997	0.699985	−	0.714158i	$-0.253190\pi$
$421$	28.7250	1.39997	0.699985	+	0.714158i	$0.253190\pi$
$422$	− 6.69722i	− 0.326016i
$423$	− 7.57779i	− 0.368445i
$424$	−6.00000	−0.291386
$425$	0	0
$426$	1.81665	0.0880172
$427$	− 48.4222i	− 2.34331i
$428$	− 4.30278i	− 0.207983i
$429$	−0.908327	−0.0438544
$430$	0	0
$431$	−5.21110	−0.251010	−0.125505	−	0.992093i	$-0.540055\pi$
$431$	−5.21110	−0.251010	−0.125505	+	0.992093i	$0.540055\pi$
$432$	1.78890i	0.0860684i
$433$	11.9361i	0.573612i	0.957989	+	0.286806i	$0.0925934\pi$
$433$	11.9361i	0.573612i	−0.957989	+	0.286806i	$0.907407\pi$
$434$	15.2111	0.730156
$435$	0	0
$436$	2.00000	0.0957826
$437$	− 13.8167i	− 0.660940i
$438$	2.63331i	0.125824i
$439$	9.33053	0.445322	0.222661	−	0.974896i	$-0.428526\pi$
$439$	9.33053	0.445322	0.222661	+	0.974896i	$0.428526\pi$
$440$	0	0
$441$	−41.3305	−1.96812
$442$	− 13.8167i	− 0.657191i
$443$	− 0.275019i	− 0.0130666i	−0.999979	−	0.00653328i	$-0.997920\pi$
$443$	− 0.275019i	− 0.0130666i	0.999979	−	0.00653328i	$-0.00207962\pi$
$444$	0.302776	0.0143691
$445$	0	0
$446$	−15.8167	−0.748940
$447$	0.550039i	0.0260159i
$448$	− 4.60555i	− 0.217592i
$449$	0.788897	0.0372304	0.0186152	−	0.999827i	$-0.494074\pi$
$449$	0.788897	0.0372304	0.0186152	+	0.999827i	$0.494074\pi$
$450$	0	0
$451$	−1.18335	−0.0557216
$452$	11.2111i	0.527326i
$453$	− 4.05551i	− 0.190545i
$454$	−7.81665	−0.366854
$455$	0	0
$456$	0.605551	0.0283575
$457$	4.60555i	0.215439i	0.994181	+	0.107719i	$0.0343548\pi$
$457$	4.60555i	0.215439i	−0.994181	+	0.107719i	$0.965645\pi$
$458$	17.3944i	0.812789i
$459$	10.7334	0.500991
$460$	0	0
$461$	−16.4222	−0.764858	−0.382429	−	0.923985i	$-0.624912\pi$
$461$	−16.4222	−0.764858	−0.382429	+	0.923985i	$0.624912\pi$
$462$	1.81665i	0.0845184i
$463$	− 30.3028i	− 1.40829i	−0.710056	−	0.704145i	$-0.751331\pi$
$463$	− 30.3028i	− 1.40829i	0.710056	−	0.704145i	$-0.248669\pi$
$464$	−6.90833	−0.320711
$465$	0	0
$466$	9.51388	0.440722
$467$	0	0	1.00000			$0$
$467$	0	0	−1.00000			$\pi$
$468$	− 6.69722i	− 0.309579i
$469$	−66.8444	−3.08659
$470$	0	0
$471$	−2.18335	−0.100603
$472$	− 3.39445i	− 0.156242i
$473$	8.60555i	0.395684i
$474$	4.88057	0.224172
$475$	0	0
$476$	−27.6333	−1.26657
$477$	17.4500i	0.798979i
$478$	0.513878i	0.0235042i
$479$	−12.1194	−0.553751	−0.276875	−	0.960906i	$-0.589299\pi$
$479$	−12.1194	−0.553751	−0.276875	+	0.960906i	$0.589299\pi$
$480$	0	0
$481$	−2.30278	−0.104998
$482$	− 8.00000i	− 0.364390i
$483$	− 9.63331i	− 0.438331i
$484$	9.30278	0.422853
$485$	0	0
$486$	7.84441	0.355830
$487$	− 22.7889i	− 1.03266i	−0.856389	−	0.516332i	$-0.827297\pi$
$487$	− 22.7889i	− 1.03266i	0.856389	−	0.516332i	$-0.172703\pi$
$488$	− 10.5139i	− 0.475941i
$489$	−6.18335	−0.279621
$490$	0	0
$491$	−14.7250	−0.664529	−0.332265	−	0.943186i	$-0.607813\pi$
$491$	−14.7250	−0.664529	−0.332265	+	0.943186i	$0.607813\pi$
$492$	0.275019i	0.0123988i
$493$	41.4500i	1.86681i
$494$	−4.60555	−0.207214
$495$	0	0
$496$	3.30278	0.148299
$497$	27.6333i	1.23952i
$498$	− 5.21110i	− 0.233515i
$499$	−8.23886	−0.368822	−0.184411	−	0.982849i	$-0.559038\pi$
$499$	−8.23886	−0.368822	−0.184411	+	0.982849i	$0.559038\pi$
$500$	0	0
$501$	−3.78890	−0.169275
$502$	6.78890i	0.303003i
$503$	24.5139i	1.09302i	0.837453	+	0.546510i	$0.184044\pi$
$503$	24.5139i	1.09302i	−0.837453	+	0.546510i	$0.815956\pi$
$504$	−13.3944	−0.596636
$505$	0	0
$506$	9.00000	0.400099
$507$	2.33053i	0.103503i
$508$	4.78890i	0.212473i
$509$	25.8167	1.14430	0.572152	−	0.820148i	$-0.306109\pi$
$509$	25.8167	1.14430	0.572152	+	0.820148i	$0.306109\pi$
$510$	0	0
$511$	−40.0555	−1.77195
$512$	− 1.00000i	− 0.0441942i
$513$	− 3.57779i	− 0.157964i
$514$	−11.2111	−0.494501
$515$	0	0
$516$	2.00000	0.0880451
$517$	− 3.39445i	− 0.149288i
$518$	4.60555i	0.202356i
$519$	−7.02776	−0.308484
$520$	0	0
$521$	−9.63331	−0.422043	−0.211021	−	0.977481i	$-0.567679\pi$
$521$	−9.63331	−0.422043	−0.211021	+	0.977481i	$0.567679\pi$
$522$	20.0917i	0.879389i
$523$	− 32.2389i	− 1.40971i	−0.709353	−	0.704853i	$-0.751013\pi$
$523$	− 32.2389i	− 1.40971i	0.709353	−	0.704853i	$-0.248987\pi$
$524$	−3.39445	−0.148287
$525$	0	0
$526$	7.81665	0.340822
$527$	− 19.8167i	− 0.863227i
$528$	0.394449i	0.0171662i
$529$	−24.7250	−1.07500
$530$	0	0
$531$	−9.87217	−0.428416
$532$	9.21110i	0.399352i
$533$	− 2.09167i	− 0.0906004i
$534$	−1.57779	−0.0682779
$535$	0	0
$536$	−14.5139	−0.626904
$537$	− 2.36669i	− 0.102130i
$538$	− 6.78890i	− 0.292690i
$539$	−18.5139	−0.797449
$540$	0	0
$541$	−20.9361	−0.900113	−0.450056	−	0.893000i	$-0.648596\pi$
$541$	−20.9361	−0.900113	−0.450056	+	0.893000i	$0.648596\pi$
$542$	− 6.42221i	− 0.275857i
$543$	6.05551i	0.259867i
$544$	−6.00000	−0.257248
$545$	0	0
$546$	−3.21110	−0.137423
$547$	− 13.3944i	− 0.572705i	−0.958124	−	0.286353i	$-0.907557\pi$
$547$	− 13.3944i	− 0.572705i	0.958124	−	0.286353i	$-0.0924429\pi$
$548$	9.90833i	0.423263i
$549$	−30.5778	−1.30503
$550$	0	0
$551$	13.8167	0.588609
$552$	− 2.09167i	− 0.0890275i
$553$	74.2389i	3.15696i
$554$	−25.1194	−1.06722
$555$	0	0
$556$	8.90833	0.377797
$557$	− 6.51388i	− 0.276002i	−0.990432	−	0.138001i	$-0.955932\pi$
$557$	− 6.51388i	− 0.276002i	0.990432	−	0.138001i	$-0.0440677\pi$
$558$	− 9.60555i	− 0.406635i
$559$	−15.2111	−0.643361
$560$	0	0
$561$	2.36669	0.0999218
$562$	12.0000i	0.506189i
$563$	− 44.0555i	− 1.85672i	−0.371684	−	0.928359i	$-0.621220\pi$
$563$	− 44.0555i	− 1.85672i	0.371684	−	0.928359i	$-0.378780\pi$
$564$	−0.788897	−0.0332186
$565$	0	0
$566$	−17.3944	−0.731143
$567$	37.6888i	1.58278i
$568$	6.00000i	0.251754i
$569$	10.4222	0.436922	0.218461	−	0.975846i	$-0.429896\pi$
$569$	10.4222	0.436922	0.218461	+	0.975846i	$0.429896\pi$
$570$	0	0
$571$	−20.3028	−0.849645	−0.424822	−	0.905277i	$-0.639663\pi$
$571$	−20.3028	−0.849645	−0.424822	+	0.905277i	$0.639663\pi$
$572$	− 3.00000i	− 0.125436i
$573$	3.78890i	0.158283i
$574$	−4.18335	−0.174609
$575$	0	0
$576$	−2.90833	−0.121180
$577$	− 28.2389i	− 1.17560i	−0.809007	−	0.587800i	$-0.799994\pi$
$577$	− 28.2389i	− 1.17560i	0.809007	−	0.587800i	$-0.200006\pi$
$578$	19.0000i	0.790296i
$579$	−1.21110	−0.0503317
$580$	0	0
$581$	79.2666	3.28853
$582$	3.76114i	0.155904i
$583$	7.81665i	0.323733i
$584$	−8.69722	−0.359894
$585$	0	0
$586$	25.0278	1.03389
$587$	− 2.36669i	− 0.0976838i	−0.998807	−	0.0488419i	$-0.984447\pi$
$587$	− 2.36669i	− 0.0976838i	0.998807	−	0.0488419i	$-0.0155531\pi$
$588$	4.30278i	0.177443i
$589$	−6.60555	−0.272177
$590$	0	0
$591$	1.81665	0.0747272
$592$	1.00000i	0.0410997i
$593$	− 36.5139i	− 1.49945i	−0.661752	−	0.749723i	$-0.730187\pi$
$593$	− 36.5139i	− 1.49945i	0.661752	−	0.749723i	$-0.269813\pi$
$594$	2.33053	0.0956229
$595$	0	0
$596$	−1.81665	−0.0744130
$597$	0.733385i	0.0300154i
$598$	15.9083i	0.650540i
$599$	35.2111	1.43869	0.719343	−	0.694655i	$-0.244443\pi$
$599$	35.2111	1.43869	0.719343	+	0.694655i	$0.244443\pi$
$600$	0	0
$601$	−20.6972	−0.844257	−0.422129	−	0.906536i	$-0.638717\pi$
$601$	−20.6972	−0.844257	−0.422129	+	0.906536i	$0.638717\pi$
$602$	30.4222i	1.23992i
$603$	42.2111i	1.71897i
$604$	13.3944	0.545012
$605$	0	0
$606$	4.97224	0.201984
$607$	− 31.5139i	− 1.27911i	−0.768746	−	0.639554i	$-0.779119\pi$
$607$	− 31.5139i	− 1.27911i	0.768746	−	0.639554i	$-0.220881\pi$
$608$	2.00000i	0.0811107i
$609$	9.63331	0.390361
$610$	0	0
$611$	6.00000	0.242734
$612$	17.4500i	0.705373i
$613$	8.18335i	0.330522i	0.986250	+	0.165261i	$0.0528467\pi$
$613$	8.18335i	0.330522i	−0.986250	+	0.165261i	$0.947153\pi$
$614$	7.09167	0.286197
$615$	0	0
$616$	−6.00000	−0.241747
$617$	47.5694i	1.91507i	0.288314	+	0.957536i	$0.406905\pi$
$617$	47.5694i	1.91507i	−0.288314	+	0.957536i	$0.593095\pi$
$618$	− 1.00000i	− 0.0402259i
$619$	2.69722	0.108411	0.0542053	−	0.998530i	$-0.482737\pi$
$619$	2.69722	0.108411	0.0542053	+	0.998530i	$0.482737\pi$
$620$	0	0
$621$	−12.3583	−0.495921
$622$	− 5.09167i	− 0.204157i
$623$	− 24.0000i	− 0.961540i
$624$	−0.697224	−0.0279113
$625$	0	0
$626$	−27.0278	−1.08025
$627$	− 0.788897i	− 0.0315055i
$628$	− 7.21110i	− 0.287754i
$629$	6.00000	0.239236
$630$	0	0
$631$	18.3028	0.728622	0.364311	−	0.931277i	$-0.381305\pi$
$631$	18.3028	0.728622	0.364311	+	0.931277i	$0.381305\pi$
$632$	16.1194i	0.641196i
$633$	2.02776i	0.0805961i
$634$	−5.21110	−0.206959
$635$	0	0
$636$	1.81665	0.0720350
$637$	− 32.7250i	− 1.29661i
$638$	9.00000i	0.356313i
$639$	17.4500	0.690310
$640$	0	0
$641$	−2.48612	−0.0981959	−0.0490980	−	0.998794i	$-0.515635\pi$
$641$	−2.48612	−0.0981959	−0.0490980	+	0.998794i	$0.515635\pi$
$642$	1.30278i	0.0514165i
$643$	29.8167i	1.17585i	0.808914	+	0.587927i	$0.200056\pi$
$643$	29.8167i	1.17585i	−0.808914	+	0.587927i	$0.799944\pi$
$644$	31.8167	1.25375
$645$	0	0
$646$	12.0000	0.472134
$647$	− 25.9361i	− 1.01965i	−0.860277	−	0.509826i	$-0.829710\pi$
$647$	− 25.9361i	− 1.01965i	0.860277	−	0.509826i	$-0.170290\pi$
$648$	8.18335i	0.321472i
$649$	−4.42221	−0.173587
$650$	0	0
$651$	−4.60555	−0.180506
$652$	− 20.4222i	− 0.799795i
$653$	− 6.90833i	− 0.270344i	−0.990822	−	0.135172i	$-0.956841\pi$
$653$	− 6.90833i	− 0.270344i	0.990822	−	0.135172i	$-0.0431587\pi$
$654$	−0.605551	−0.0236789
$655$	0	0
$656$	−0.908327	−0.0354642
$657$	25.2944i	0.986827i
$658$	− 12.0000i	− 0.467809i
$659$	42.1194	1.64074	0.820370	−	0.571833i	$-0.193767\pi$
$659$	42.1194	1.64074	0.820370	+	0.571833i	$0.193767\pi$
$660$	0	0
$661$	−12.4861	−0.485654	−0.242827	−	0.970070i	$-0.578075\pi$
$661$	−12.4861	−0.485654	−0.242827	+	0.970070i	$0.578075\pi$
$662$	− 1.21110i	− 0.0470708i
$663$	4.18335i	0.162468i
$664$	17.2111	0.667920
$665$	0	0
$666$	2.90833	0.112695
$667$	− 47.7250i	− 1.84792i
$668$	− 12.5139i	− 0.484176i
$669$	4.78890	0.185149
$670$	0	0
$671$	−13.6972	−0.528775
$672$	1.39445i	0.0537920i
$673$	− 24.3028i	− 0.936803i	−0.883516	−	0.468402i	$-0.844830\pi$
$673$	− 24.3028i	− 0.936803i	0.883516	−	0.468402i	$-0.155170\pi$
$674$	−19.1194	−0.736453
$675$	0	0
$676$	−7.69722	−0.296047
$677$	− 36.2389i	− 1.39277i	−0.717667	−	0.696386i	$-0.754790\pi$
$677$	− 36.2389i	− 1.39277i	0.717667	−	0.696386i	$-0.245210\pi$
$678$	− 3.39445i	− 0.130363i
$679$	−57.2111	−2.19556
$680$	0	0
$681$	2.36669	0.0906918
$682$	− 4.30278i	− 0.164762i
$683$	− 12.0000i	− 0.459167i	−0.973289	−	0.229584i	$-0.926264\pi$
$683$	− 12.0000i	− 0.459167i	0.973289	−	0.229584i	$-0.0737364\pi$
$684$	5.81665	0.222405
$685$	0	0
$686$	−33.2111	−1.26801
$687$	− 5.26662i	− 0.200934i
$688$	6.60555i	0.251834i
$689$	−13.8167	−0.526373
$690$	0	0
$691$	8.00000	0.304334	0.152167	−	0.988355i	$-0.451375\pi$
$691$	8.00000	0.304334	0.152167	+	0.988355i	$0.451375\pi$
$692$	− 23.2111i	− 0.882354i
$693$	17.4500i	0.662869i
$694$	31.8167	1.20774
$695$	0	0
$696$	2.09167	0.0792847
$697$	5.44996i	0.206432i
$698$	− 22.2389i	− 0.841753i
$699$	−2.88057	−0.108953
$700$	0	0
$701$	14.8806	0.562031	0.281016	−	0.959703i	$-0.409329\pi$
$701$	14.8806	0.562031	0.281016	+	0.959703i	$0.409329\pi$
$702$	4.11943i	0.155478i
$703$	− 2.00000i	− 0.0754314i
$704$	−1.30278	−0.0491002
$705$	0	0
$706$	−31.8167	−1.19744
$707$	75.6333i	2.84448i
$708$	1.02776i	0.0386254i
$709$	1.66947	0.0626982	0.0313491	−	0.999508i	$-0.490020\pi$
$709$	1.66947	0.0626982	0.0313491	+	0.999508i	$0.490020\pi$
$710$	0	0
$711$	46.8806	1.75816
$712$	− 5.21110i	− 0.195294i
$713$	22.8167i	0.854490i
$714$	8.36669	0.313116
$715$	0	0
$716$	7.81665	0.292122
$717$	− 0.155590i	− 0.00581061i
$718$	− 11.2111i	− 0.418395i
$719$	8.36669	0.312025	0.156012	−	0.987755i	$-0.450136\pi$
$719$	8.36669	0.312025	0.156012	+	0.987755i	$0.450136\pi$
$720$	0	0
$721$	15.2111	0.566491
$722$	15.0000i	0.558242i
$723$	2.42221i	0.0900828i
$724$	−20.0000	−0.743294
$725$	0	0
$726$	−2.81665	−0.104536
$727$	29.9083i	1.10924i	0.832104	+	0.554619i	$0.187136\pi$
$727$	29.9083i	1.10924i	−0.832104	+	0.554619i	$0.812864\pi$
$728$	− 10.6056i	− 0.393068i
$729$	22.1749	0.821294
$730$	0	0
$731$	39.6333	1.46589
$732$	3.18335i	0.117660i
$733$	− 29.6333i	− 1.09453i	−0.836959	−	0.547266i	$-0.815669\pi$
$733$	− 29.6333i	− 1.09453i	0.836959	−	0.547266i	$-0.184331\pi$
$734$	−17.8167	−0.657625
$735$	0	0
$736$	6.90833	0.254644
$737$	18.9083i	0.696497i
$738$	2.64171i	0.0972427i
$739$	42.3305	1.55715	0.778577	−	0.627549i	$-0.215942\pi$
$739$	42.3305	1.55715	0.778577	+	0.627549i	$0.215942\pi$
$740$	0	0
$741$	1.39445	0.0512264
$742$	27.6333i	1.01445i
$743$	− 35.4500i	− 1.30053i	−0.759706	−	0.650266i	$-0.774657\pi$
$743$	− 35.4500i	− 1.30053i	0.759706	−	0.650266i	$-0.225343\pi$
$744$	−1.00000	−0.0366618
$745$	0	0
$746$	−3.81665	−0.139738
$747$	− 50.0555i	− 1.83144i
$748$	7.81665i	0.285805i
$749$	−19.8167	−0.724085
$750$	0	0
$751$	14.0000	0.510867	0.255434	−	0.966827i	$-0.417782\pi$
$751$	14.0000	0.510867	0.255434	+	0.966827i	$0.417782\pi$
$752$	− 2.60555i	− 0.0950147i
$753$	− 2.05551i	− 0.0749070i
$754$	−15.9083	−0.579347
$755$	0	0
$756$	8.23886	0.299644
$757$	9.30278i	0.338115i	0.985606	+	0.169058i	$0.0540724\pi$
$757$	9.30278i	0.338115i	−0.985606	+	0.169058i	$0.945928\pi$
$758$	− 15.3305i	− 0.556830i
$759$	−2.72498	−0.0989105
$760$	0	0
$761$	42.1194	1.52683	0.763414	−	0.645909i	$-0.223522\pi$
$761$	42.1194	1.52683	0.763414	+	0.645909i	$0.223522\pi$
$762$	− 1.44996i	− 0.0525266i
$763$	− 9.21110i	− 0.333464i
$764$	−12.5139	−0.452736
$765$	0	0
$766$	−20.8444	−0.753139
$767$	− 7.81665i	− 0.282243i
$768$	0.302776i	0.0109255i
$769$	22.0000	0.793340	0.396670	−	0.917961i	$-0.370166\pi$
$769$	22.0000	0.793340	0.396670	+	0.917961i	$0.370166\pi$
$770$	0	0
$771$	3.39445	0.122248
$772$	− 4.00000i	− 0.143963i
$773$	50.0555i	1.80037i	0.435506	+	0.900186i	$0.356569\pi$
$773$	50.0555i	1.80037i	−0.435506	+	0.900186i	$0.643431\pi$
$774$	19.2111	0.690529
$775$	0	0
$776$	−12.4222	−0.445931
$777$	− 1.39445i	− 0.0500256i
$778$	− 11.8806i	− 0.425939i
$779$	1.81665	0.0650884
$780$	0	0
$781$	7.81665	0.279702
$782$	− 41.4500i	− 1.48225i
$783$	− 12.3583i	− 0.441649i
$784$	−14.2111	−0.507539
$785$	0	0
$786$	1.02776	0.0366589
$787$	25.2111i	0.898679i	0.893361	+	0.449339i	$0.148341\pi$
$787$	25.2111i	0.898679i	−0.893361	+	0.449339i	$0.851659\pi$
$788$	6.00000i	0.213741i
$789$	−2.36669	−0.0842565
$790$	0	0
$791$	51.6333	1.83587
$792$	3.78890i	0.134633i
$793$	− 24.2111i	− 0.859761i
$794$	27.8167	0.987176
$795$	0	0
$796$	−2.42221	−0.0858528
$797$	− 17.3305i	− 0.613879i	−0.951729	−	0.306939i	$-0.900695\pi$
$797$	− 17.3305i	− 0.613879i	0.951729	−	0.306939i	$-0.0993049\pi$
$798$	− 2.78890i	− 0.0987259i
$799$	−15.6333	−0.553067
$800$	0	0
$801$	−15.1556	−0.535496
$802$	− 13.8167i	− 0.487883i
$803$	11.3305i	0.399846i
$804$	4.39445	0.154980
$805$	0	0
$806$	7.60555	0.267894
$807$	2.05551i	0.0723575i
$808$	16.4222i	0.577731i
$809$	−29.4500	−1.03541	−0.517703	−	0.855561i	$-0.673213\pi$
$809$	−29.4500	−1.03541	−0.517703	+	0.855561i	$0.673213\pi$
$810$	0	0
$811$	54.1472	1.90136	0.950682	−	0.310166i	$-0.100385\pi$
$811$	54.1472	1.90136	0.950682	+	0.310166i	$0.100385\pi$
$812$	31.8167i	1.11655i
$813$	1.94449i	0.0681961i
$814$	1.30278	0.0456623
$815$	0	0
$816$	1.81665	0.0635956
$817$	− 13.2111i	− 0.462198i
$818$	− 5.02776i	− 0.175791i
$819$	−30.8444	−1.07779
$820$	0	0
$821$	11.2111	0.391270	0.195635	−	0.980677i	$-0.437323\pi$
$821$	11.2111	0.391270	0.195635	+	0.980677i	$0.437323\pi$
$822$	− 3.00000i	− 0.104637i
$823$	12.8444i	0.447728i	0.974620	+	0.223864i	$0.0718671\pi$
$823$	12.8444i	0.447728i	−0.974620	+	0.223864i	$0.928133\pi$
$824$	3.30278	0.115058
$825$	0	0
$826$	−15.6333	−0.543952
$827$	27.3944i	0.952598i	0.879283	+	0.476299i	$0.158022\pi$
$827$	27.3944i	0.952598i	−0.879283	+	0.476299i	$0.841978\pi$
$828$	− 20.0917i	− 0.698234i
$829$	−4.72498	−0.164105	−0.0820527	−	0.996628i	$-0.526148\pi$
$829$	−4.72498	−0.164105	−0.0820527	+	0.996628i	$0.526148\pi$
$830$	0	0
$831$	7.60555	0.263834
$832$	− 2.30278i	− 0.0798344i
$833$	85.2666i	2.95431i
$834$	−2.69722	−0.0933972
$835$	0	0
$836$	2.60555	0.0901149
$837$	5.90833i	0.204222i
$838$	− 25.1472i	− 0.868695i
$839$	49.0278	1.69263	0.846313	−	0.532686i	$-0.178817\pi$
$839$	49.0278	1.69263	0.846313	+	0.532686i	$0.178817\pi$
$840$	0	0
$841$	18.7250	0.645689
$842$	− 28.7250i	− 0.989928i
$843$	− 3.63331i	− 0.125138i
$844$	−6.69722	−0.230528
$845$	0	0
$846$	−7.57779	−0.260530
$847$	− 42.8444i	− 1.47215i
$848$	6.00000i	0.206041i
$849$	5.26662	0.180750
$850$	0	0
$851$	−6.90833	−0.236814
$852$	− 1.81665i	− 0.0622375i
$853$	11.5416i	0.395178i	0.980285	+	0.197589i	$0.0633111\pi$
$853$	11.5416i	0.395178i	−0.980285	+	0.197589i	$0.936689\pi$
$854$	−48.4222	−1.65697
$855$	0	0
$856$	−4.30278	−0.147066
$857$	− 14.8444i	− 0.507075i	−0.967326	−	0.253538i	$-0.918406\pi$
$857$	− 14.8444i	− 0.507075i	0.967326	−	0.253538i	$-0.0815942\pi$
$858$	0.908327i	0.0310098i
$859$	24.0555	0.820764	0.410382	−	0.911914i	$-0.365395\pi$
$859$	24.0555	0.820764	0.410382	+	0.911914i	$0.365395\pi$
$860$	0	0
$861$	1.26662	0.0431661
$862$	5.21110i	0.177491i
$863$	− 12.0000i	− 0.408485i	−0.978920	−	0.204242i	$-0.934527\pi$
$863$	− 12.0000i	− 0.408485i	0.978920	−	0.204242i	$-0.0654731\pi$
$864$	1.78890	0.0608595
$865$	0	0
$866$	11.9361	0.405605
$867$	− 5.75274i	− 0.195373i
$868$	− 15.2111i	− 0.516298i
$869$	21.0000	0.712376
$870$	0	0
$871$	−33.4222	−1.13247
$872$	− 2.00000i	− 0.0677285i
$873$	36.1278i	1.22274i
$874$	−13.8167	−0.467355
$875$	0	0
$876$	2.63331	0.0889712
$877$	7.21110i	0.243502i	0.992561	+	0.121751i	$0.0388509\pi$
$877$	7.21110i	0.243502i	−0.992561	+	0.121751i	$0.961149\pi$
$878$	− 9.33053i	− 0.314890i
$879$	−7.57779	−0.255593
$880$	0	0
$881$	25.5416	0.860520	0.430260	−	0.902705i	$-0.358422\pi$
$881$	25.5416	0.860520	0.430260	+	0.902705i	$0.358422\pi$
$882$	41.3305i	1.39167i
$883$	2.42221i	0.0815137i	0.999169	+	0.0407568i	$0.0129769\pi$
$883$	2.42221i	0.0815137i	−0.999169	+	0.0407568i	$0.987023\pi$
$884$	−13.8167	−0.464704
$885$	0	0
$886$	−0.275019	−0.00923945
$887$	28.4222i	0.954324i	0.878815	+	0.477162i	$0.158335\pi$
$887$	28.4222i	0.954324i	−0.878815	+	0.477162i	$0.841665\pi$
$888$	− 0.302776i	− 0.0101605i
$889$	22.0555	0.739718
$890$	0	0
$891$	10.6611	0.357159
$892$	15.8167i	0.529581i
$893$	5.21110i	0.174383i
$894$	0.550039	0.0183960
$895$	0	0
$896$	−4.60555	−0.153861
$897$	− 4.81665i	− 0.160823i
$898$	− 0.788897i	− 0.0263258i
$899$	−22.8167	−0.760978
$900$	0	0
$901$	36.0000	1.19933
$902$	1.18335i	0.0394011i
$903$	− 9.21110i	− 0.306526i
$904$	11.2111	0.372876
$905$	0	0
$906$	−4.05551	−0.134735
$907$	26.0000i	0.863316i	0.902037	+	0.431658i	$0.142071\pi$
$907$	26.0000i	0.863316i	−0.902037	+	0.431658i	$0.857929\pi$
$908$	7.81665i	0.259405i
$909$	47.7611	1.58414
$910$	0	0
$911$	−46.4222	−1.53804	−0.769018	−	0.639227i	$-0.779254\pi$
$911$	−46.4222	−1.53804	−0.769018	+	0.639227i	$0.779254\pi$
$912$	− 0.605551i	− 0.0200518i
$913$	− 22.4222i	− 0.742067i
$914$	4.60555	0.152338
$915$	0	0
$916$	17.3944	0.574729
$917$	15.6333i	0.516257i
$918$	− 10.7334i	− 0.354254i
$919$	38.4222	1.26743	0.633716	−	0.773566i	$-0.281529\pi$
$919$	38.4222	1.26743	0.633716	+	0.773566i	$0.281529\pi$
$920$	0	0
$921$	−2.14719	−0.0707522
$922$	16.4222i	0.540837i
$923$	13.8167i	0.454781i
$924$	1.81665	0.0597635
$925$	0	0
$926$	−30.3028	−0.995811
$927$	− 9.60555i	− 0.315488i
$928$	6.90833i	0.226777i
$929$	36.5139	1.19798	0.598991	−	0.800756i	$-0.295569\pi$
$929$	36.5139	1.19798	0.598991	+	0.800756i	$0.295569\pi$
$930$	0	0
$931$	28.4222	0.931500
$932$	− 9.51388i	− 0.311637i
$933$	1.54163i	0.0504708i
$934$	0	0
$935$	0	0
$936$	−6.69722	−0.218906
$937$	− 28.9083i	− 0.944394i	−0.881493	−	0.472197i	$-0.843461\pi$
$937$	− 28.9083i	− 0.944394i	0.881493	−	0.472197i	$-0.156539\pi$
$938$	66.8444i	2.18255i
$939$	8.18335	0.267053
$940$	0	0
$941$	−7.81665	−0.254816	−0.127408	−	0.991850i	$-0.540666\pi$
$941$	−7.81665	−0.254816	−0.127408	+	0.991850i	$0.540666\pi$
$942$	2.18335i	0.0711373i
$943$	− 6.27502i	− 0.204343i
$944$	−3.39445	−0.110480
$945$	0	0
$946$	8.60555	0.279791
$947$	39.6333i	1.28791i	0.765064	+	0.643955i	$0.222707\pi$
$947$	39.6333i	1.28791i	−0.765064	+	0.643955i	$0.777293\pi$
$948$	− 4.88057i	− 0.158514i
$949$	−20.0278	−0.650128
$950$	0	0
$951$	1.57779	0.0511635
$952$	27.6333i	0.895601i
$953$	18.7527i	0.607461i	0.952758	+	0.303730i	$0.0982322\pi$
$953$	18.7527i	0.607461i	−0.952758	+	0.303730i	$0.901768\pi$
$954$	17.4500	0.564963
$955$	0	0
$956$	0.513878	0.0166200
$957$	− 2.72498i	− 0.0880861i
$958$	12.1194i	0.391561i
$959$	45.6333	1.47358
$960$	0	0
$961$	−20.0917	−0.648118
$962$	2.30278i	0.0742445i
$963$	12.5139i	0.403254i
$964$	−8.00000	−0.257663
$965$	0	0
$966$	−9.63331	−0.309947
$967$	25.7250i	0.827260i	0.910445	+	0.413630i	$0.135739\pi$
$967$	25.7250i	0.827260i	−0.910445	+	0.413630i	$0.864261\pi$
$968$	− 9.30278i	− 0.299003i
$969$	−3.63331	−0.116719
$970$	0	0
$971$	31.5416	1.01222	0.506110	−	0.862469i	$-0.331083\pi$
$971$	31.5416	1.01222	0.506110	+	0.862469i	$0.331083\pi$
$972$	− 7.84441i	− 0.251610i
$973$	− 41.0278i	− 1.31529i
$974$	−22.7889	−0.730203
$975$	0	0
$976$	−10.5139	−0.336541
$977$	18.0000i	0.575871i	0.957650	+	0.287936i	$0.0929689\pi$
$977$	18.0000i	0.575871i	−0.957650	+	0.287936i	$0.907031\pi$
$978$	6.18335i	0.197722i
$979$	−6.78890	−0.216974
$980$	0	0
$981$	−5.81665	−0.185711
$982$	14.7250i	0.469893i
$983$	12.0000i	0.382741i	0.981518	+	0.191370i	$0.0612931\pi$
$983$	12.0000i	0.382741i	−0.981518	+	0.191370i	$0.938707\pi$
$984$	0.275019	0.00876729
$985$	0	0
$986$	41.4500	1.32004
$987$	3.63331i	0.115649i
$988$	4.60555i	0.146522i
$989$	−45.6333	−1.45105
$990$	0	0
$991$	54.3028	1.72498	0.862492	−	0.506070i	$-0.168902\pi$
$991$	54.3028	1.72498	0.862492	+	0.506070i	$0.168902\pi$
$992$	− 3.30278i	− 0.104863i
$993$	0.366692i	0.0116366i
$994$	27.6333	0.876475
$995$	0	0
$996$	−5.21110	−0.165120
$997$	− 23.5778i	− 0.746716i	−0.927687	−	0.373358i	$-0.878206\pi$
$997$	− 23.5778i	− 0.746716i	0.927687	−	0.373358i	$-0.121794\pi$
$998$	8.23886i	0.260797i
$999$	−1.78890	−0.0565982

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1850.2.b.i.149.2		4
5.2	odd	4		1850.2.a.u.1.2		2
5.3	odd	4		74.2.a.a.1.1	✓	2
5.4	even	2	inner	1850.2.b.i.149.3		4
15.8	even	4		666.2.a.j.1.1		2
20.3	even	4		592.2.a.f.1.2		2
35.13	even	4		3626.2.a.a.1.2		2
40.3	even	4		2368.2.a.ba.1.1		2
40.13	odd	4		2368.2.a.s.1.2		2
55.43	even	4		8954.2.a.p.1.1		2
60.23	odd	4		5328.2.a.bf.1.1		2
185.73	odd	4		2738.2.a.l.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
74.2.a.a.1.1	✓	2	5.3	odd	4
592.2.a.f.1.2		2	20.3	even	4
666.2.a.j.1.1		2	15.8	even	4
1850.2.a.u.1.2		2	5.2	odd	4
1850.2.b.i.149.2		4	1.1	even	1	trivial
1850.2.b.i.149.3		4	5.4	even	2	inner
2368.2.a.s.1.2		2	40.13	odd	4
2368.2.a.ba.1.1		2	40.3	even	4
2738.2.a.l.1.1		2	185.73	odd	4
3626.2.a.a.1.2		2	35.13	even	4
5328.2.a.bf.1.1		2	60.23	odd	4
8954.2.a.p.1.1		2	55.43	even	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:	\( N \)	\(=\)	\( 1850 = 2 \cdot 5^{2} \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1850.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} +0.302776i q^{3} -1.00000 q^{4} +0.302776 q^{6} +4.60555i q^{7} +1.00000i q^{8} +2.90833 q^{9} +O(q^{10})\)	\(q-1.00000i q^{2} +0.302776i q^{3} -1.00000 q^{4} +0.302776 q^{6} +4.60555i q^{7} +1.00000i q^{8} +2.90833 q^{9} +1.30278 q^{11} -0.302776i q^{12} +2.30278i q^{13} +4.60555 q^{14} +1.00000 q^{16} -6.00000i q^{17} -2.90833i q^{18} -2.00000 q^{19} -1.39445 q^{21} -1.30278i q^{22} +6.90833i q^{23} -0.302776 q^{24} +2.30278 q^{26} +1.78890i q^{27} -4.60555i q^{28} -6.90833 q^{29} +3.30278 q^{31} -1.00000i q^{32} +0.394449i q^{33} -6.00000 q^{34} -2.90833 q^{36} +1.00000i q^{37} +2.00000i q^{38} -0.697224 q^{39} -0.908327 q^{41} +1.39445i q^{42} +6.60555i q^{43} -1.30278 q^{44} +6.90833 q^{46} -2.60555i q^{47} +0.302776i q^{48} -14.2111 q^{49} +1.81665 q^{51} -2.30278i q^{52} +6.00000i q^{53} +1.78890 q^{54} -4.60555 q^{56} -0.605551i q^{57} +6.90833i q^{58} -3.39445 q^{59} -10.5139 q^{61} -3.30278i q^{62} +13.3944i q^{63} -1.00000 q^{64} +0.394449 q^{66} +14.5139i q^{67} +6.00000i q^{68} -2.09167 q^{69} +6.00000 q^{71} +2.90833i q^{72} +8.69722i q^{73} +1.00000 q^{74} +2.00000 q^{76} +6.00000i q^{77} +0.697224i q^{78} +16.1194 q^{79} +8.18335 q^{81} +0.908327i q^{82} -17.2111i q^{83} +1.39445 q^{84} +6.60555 q^{86} -2.09167i q^{87} +1.30278i q^{88} -5.21110 q^{89} -10.6056 q^{91} -6.90833i q^{92} +1.00000i q^{93} -2.60555 q^{94} +0.302776 q^{96} +12.4222i q^{97} +14.2111i q^{98} +3.78890 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{4} - 6 q^{6} - 10 q^{9}+O(q^{10}) \) 4 * q - 4 * q^4 - 6 * q^6 - 10 * q^9	\( 4 q - 4 q^{4} - 6 q^{6} - 10 q^{9} - 2 q^{11} + 4 q^{14} + 4 q^{16} - 8 q^{19} - 20 q^{21} + 6 q^{24} + 2 q^{26} - 6 q^{29} + 6 q^{31} - 24 q^{34} + 10 q^{36} - 10 q^{39} + 18 q^{41} + 2 q^{44} + 6 q^{46} - 28 q^{49} - 36 q^{51} + 36 q^{54} - 4 q^{56} - 28 q^{59} - 6 q^{61} - 4 q^{64} + 16 q^{66} - 30 q^{69} + 24 q^{71} + 4 q^{74} + 8 q^{76} + 14 q^{79} + 76 q^{81} + 20 q^{84} + 12 q^{86} + 8 q^{89} - 28 q^{91} + 4 q^{94} - 6 q^{96} + 44 q^{99}+O(q^{100}) \) 4 * q - 4 * q^4 - 6 * q^6 - 10 * q^9 - 2 * q^11 + 4 * q^14 + 4 * q^16 - 8 * q^19 - 20 * q^21 + 6 * q^24 + 2 * q^26 - 6 * q^29 + 6 * q^31 - 24 * q^34 + 10 * q^36 - 10 * q^39 + 18 * q^41 + 2 * q^44 + 6 * q^46 - 28 * q^49 - 36 * q^51 + 36 * q^54 - 4 * q^56 - 28 * q^59 - 6 * q^61 - 4 * q^64 + 16 * q^66 - 30 * q^69 + 24 * q^71 + 4 * q^74 + 8 * q^76 + 14 * q^79 + 76 * q^81 + 20 * q^84 + 12 * q^86 + 8 * q^89 - 28 * q^91 + 4 * q^94 - 6 * q^96 + 44 * q^99

Introduction

L-functions

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