LMFDB - Embedded newform 1850.2.a.u.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1850,2,Mod(1,1850)]

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([0, 0]))

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

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

chi := DirichletCharacter("1850.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1850 = 2 \cdot 5^{2} \cdot 37 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1850.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$14.7723243739$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{13}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 3 $ x^2 - x - 3
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 74)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.30278$ of defining polynomial
Character	$\chi$	$=$	1850.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.00000 q^{2} +0.302776 q^{3} +1.00000 q^{4} +0.302776 q^{6} -4.60555 q^{7} +1.00000 q^{8} -2.90833 q^{9} +1.30278 q^{11} +0.302776 q^{12} +2.30278 q^{13} -4.60555 q^{14} +1.00000 q^{16} +6.00000 q^{17} -2.90833 q^{18} +2.00000 q^{19} -1.39445 q^{21} +1.30278 q^{22} +6.90833 q^{23} +0.302776 q^{24} +2.30278 q^{26} -1.78890 q^{27} -4.60555 q^{28} +6.90833 q^{29} +3.30278 q^{31} +1.00000 q^{32} +0.394449 q^{33} +6.00000 q^{34} -2.90833 q^{36} -1.00000 q^{37} +2.00000 q^{38} +0.697224 q^{39} -0.908327 q^{41} -1.39445 q^{42} +6.60555 q^{43} +1.30278 q^{44} +6.90833 q^{46} +2.60555 q^{47} +0.302776 q^{48} +14.2111 q^{49} +1.81665 q^{51} +2.30278 q^{52} +6.00000 q^{53} -1.78890 q^{54} -4.60555 q^{56} +0.605551 q^{57} +6.90833 q^{58} +3.39445 q^{59} -10.5139 q^{61} +3.30278 q^{62} +13.3944 q^{63} +1.00000 q^{64} +0.394449 q^{66} -14.5139 q^{67} +6.00000 q^{68} +2.09167 q^{69} +6.00000 q^{71} -2.90833 q^{72} +8.69722 q^{73} -1.00000 q^{74} +2.00000 q^{76} -6.00000 q^{77} +0.697224 q^{78} -16.1194 q^{79} +8.18335 q^{81} -0.908327 q^{82} -17.2111 q^{83} -1.39445 q^{84} +6.60555 q^{86} +2.09167 q^{87} +1.30278 q^{88} +5.21110 q^{89} -10.6056 q^{91} +6.90833 q^{92} +1.00000 q^{93} +2.60555 q^{94} +0.302776 q^{96} -12.4222 q^{97} +14.2111 q^{98} -3.78890 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} - 3 q^{3} + 2 q^{4} - 3 q^{6} - 2 q^{7} + 2 q^{8} + 5 q^{9} - q^{11} - 3 q^{12} + q^{13} - 2 q^{14} + 2 q^{16} + 12 q^{17} + 5 q^{18} + 4 q^{19} - 10 q^{21} - q^{22} + 3 q^{23} - 3 q^{24}+ \cdots - 22 q^{99}+O(q^{100}) $ 2 * q + 2 * q^2 - 3 * q^3 + 2 * q^4 - 3 * q^6 - 2 * q^7 + 2 * q^8 + 5 * q^9 - q^11 - 3 * q^12 + q^13 - 2 * q^14 + 2 * q^16 + 12 * q^17 + 5 * q^18 + 4 * q^19 - 10 * q^21 - q^22 + 3 * q^23 - 3 * q^24 + q^26 - 18 * q^27 - 2 * q^28 + 3 * q^29 + 3 * q^31 + 2 * q^32 + 8 * q^33 + 12 * q^34 + 5 * q^36 - 2 * q^37 + 4 * q^38 + 5 * q^39 + 9 * q^41 - 10 * q^42 + 6 * q^43 - q^44 + 3 * q^46 - 2 * q^47 - 3 * q^48 + 14 * q^49 - 18 * q^51 + q^52 + 12 * q^53 - 18 * q^54 - 2 * q^56 - 6 * q^57 + 3 * q^58 + 14 * q^59 - 3 * q^61 + 3 * q^62 + 34 * q^63 + 2 * q^64 + 8 * q^66 - 11 * q^67 + 12 * q^68 + 15 * q^69 + 12 * q^71 + 5 * q^72 + 21 * q^73 - 2 * q^74 + 4 * q^76 - 12 * q^77 + 5 * q^78 - 7 * q^79 + 38 * q^81 + 9 * q^82 - 20 * q^83 - 10 * q^84 + 6 * q^86 + 15 * q^87 - q^88 - 4 * q^89 - 14 * q^91 + 3 * q^92 + 2 * q^93 - 2 * q^94 - 3 * q^96 + 4 * q^97 + 14 * q^98 - 22 * q^99

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 $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	0.302776	0.174808	0.0874038	−	0.996173i	$-0.472143\pi$
$3$	0.302776	0.174808	0.0874038	+	0.996173i	$0.472143\pi$
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	0.302776	0.123608
$7$	−4.60555	−1.74073	−0.870367	−	0.492403i	$-0.836119\pi$
$7$	−4.60555	−1.74073	−0.870367	+	0.492403i	$0.836119\pi$
$8$	1.00000	0.353553
$9$	−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	$0.437075\pi$
$12$	0.302776	0.0874038
$13$	2.30278	0.638675	0.319338	−	0.947641i	$-0.396540\pi$
$13$	2.30278	0.638675	0.319338	+	0.947641i	$0.396540\pi$
$14$	−4.60555	−1.23089
$15$	0	0
$16$	1.00000	0.250000
$17$	6.00000	1.45521	0.727607	−	0.685994i	$-0.240633\pi$
$17$	6.00000	1.45521	0.727607	+	0.685994i	$0.240633\pi$
$18$	−2.90833	−0.685499
$19$	2.00000	0.458831	0.229416	−	0.973329i	$-0.426318\pi$
$19$	2.00000	0.458831	0.229416	+	0.973329i	$0.426318\pi$
$20$	0	0
$21$	−1.39445	−0.304294
$22$	1.30278	0.277753
$23$	6.90833	1.44049	0.720243	−	0.693722i	$-0.244030\pi$
$23$	6.90833	1.44049	0.720243	+	0.693722i	$0.244030\pi$
$24$	0.302776	0.0618038
$25$	0	0
$26$	2.30278	0.451611
$27$	−1.78890	−0.344273
$28$	−4.60555	−0.870367
$29$	6.90833	1.28284	0.641422	−	0.767188i	$-0.278345\pi$
$29$	6.90833	1.28284	0.641422	+	0.767188i	$0.278345\pi$
$30$	0	0
$31$	3.30278	0.593196	0.296598	−	0.955002i	$-0.404148\pi$
$31$	3.30278	0.593196	0.296598	+	0.955002i	$0.404148\pi$
$32$	1.00000	0.176777
$33$	0.394449	0.0686647
$34$	6.00000	1.02899
$35$	0	0
$36$	−2.90833	−0.484721
$37$	−1.00000	−0.164399
$37$	−1.00000	−0.164399
$38$	2.00000	0.324443
$39$	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	$0.522596\pi$
$42$	−1.39445	−0.215168
$43$	6.60555	1.00734	0.503669	−	0.863897i	$-0.331983\pi$
$43$	6.60555	1.00734	0.503669	+	0.863897i	$0.331983\pi$
$44$	1.30278	0.196401
$45$	0	0
$46$	6.90833	1.01858
$47$	2.60555	0.380059	0.190029	−	0.981778i	$-0.439142\pi$
$47$	2.60555	0.380059	0.190029	+	0.981778i	$0.439142\pi$
$48$	0.302776	0.0437019
$49$	14.2111	2.03016
$50$	0	0
$51$	1.81665	0.254382
$52$	2.30278	0.319338
$53$	6.00000	0.824163	0.412082	−	0.911147i	$-0.364802\pi$
$53$	6.00000	0.824163	0.412082	+	0.911147i	$0.364802\pi$
$54$	−1.78890	−0.243438
$55$	0	0
$56$	−4.60555	−0.615443
$57$	0.605551	0.0802072
$58$	6.90833	0.907108
$59$	3.39445	0.441920	0.220960	−	0.975283i	$-0.429081\pi$
$59$	3.39445	0.441920	0.220960	+	0.975283i	$0.429081\pi$
$60$	0	0
$61$	−10.5139	−1.34616	−0.673082	−	0.739568i	$-0.735030\pi$
$61$	−10.5139	−1.34616	−0.673082	+	0.739568i	$0.735030\pi$
$62$	3.30278	0.419453
$63$	13.3944	1.68754
$64$	1.00000	0.125000
$65$	0	0
$66$	0.394449	0.0485533
$67$	−14.5139	−1.77315	−0.886576	−	0.462583i	$-0.846923\pi$
$67$	−14.5139	−1.77315	−0.886576	+	0.462583i	$0.846923\pi$
$68$	6.00000	0.727607
$69$	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	$0.384129\pi$
$72$	−2.90833	−0.342750
$73$	8.69722	1.01793	0.508967	−	0.860786i	$-0.330028\pi$
$73$	8.69722	1.01793	0.508967	+	0.860786i	$0.330028\pi$
$74$	−1.00000	−0.116248
$75$	0	0
$76$	2.00000	0.229416
$77$	−6.00000	−0.683763
$78$	0.697224	0.0789451
$79$	−16.1194	−1.81358	−0.906789	−	0.421585i	$-0.861474\pi$
$79$	−16.1194	−1.81358	−0.906789	+	0.421585i	$0.861474\pi$
$80$	0	0
$81$	8.18335	0.909261
$82$	−0.908327	−0.100308
$83$	−17.2111	−1.88916	−0.944582	−	0.328276i	$-0.893533\pi$
$83$	−17.2111	−1.88916	−0.944582	+	0.328276i	$0.893533\pi$
$84$	−1.39445	−0.152147
$85$	0	0
$86$	6.60555	0.712295
$87$	2.09167	0.224251
$88$	1.30278	0.138876
$89$	5.21110	0.552376	0.276188	−	0.961104i	$-0.410929\pi$
$89$	5.21110	0.552376	0.276188	+	0.961104i	$0.410929\pi$
$90$	0	0
$91$	−10.6056	−1.11176
$92$	6.90833	0.720243
$93$	1.00000	0.103695
$94$	2.60555	0.268742
$95$	0	0
$96$	0.302776	0.0309019
$97$	−12.4222	−1.26128	−0.630642	−	0.776074i	$-0.717208\pi$
$97$	−12.4222	−1.26128	−0.630642	+	0.776074i	$0.717208\pi$
$98$	14.2111	1.43554
$99$	−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	$0.195616\pi$
$102$	1.81665	0.179876
$103$	−3.30278	−0.325432	−0.162716	−	0.986673i	$-0.552025\pi$
$103$	−3.30278	−0.325432	−0.162716	+	0.986673i	$0.552025\pi$
$104$	2.30278	0.225806
$105$	0	0
$106$	6.00000	0.582772
$107$	−4.30278	−0.415965	−0.207983	−	0.978133i	$-0.566690\pi$
$107$	−4.30278	−0.415965	−0.207983	+	0.978133i	$0.566690\pi$
$108$	−1.78890	−0.172137
$109$	2.00000	0.191565	0.0957826	−	0.995402i	$-0.469465\pi$
$109$	2.00000	0.191565	0.0957826	+	0.995402i	$0.469465\pi$
$110$	0	0
$111$	−0.302776	−0.0287382
$112$	−4.60555	−0.435184
$113$	−11.2111	−1.05465	−0.527326	−	0.849663i	$-0.676805\pi$
$113$	−11.2111	−1.05465	−0.527326	+	0.849663i	$0.676805\pi$
$114$	0.605551	0.0567151
$115$	0	0
$116$	6.90833	0.641422
$117$	−6.69722	−0.619159
$118$	3.39445	0.312484
$119$	−27.6333	−2.53314
$120$	0	0
$121$	−9.30278	−0.845707
$122$	−10.5139	−0.951882
$123$	−0.275019	−0.0247977
$124$	3.30278	0.296598
$125$	0	0
$126$	13.3944	1.19327
$127$	4.78890	0.424946	0.212473	−	0.977167i	$-0.431848\pi$
$127$	4.78890	0.424946	0.212473	+	0.977167i	$0.431848\pi$
$128$	1.00000	0.0883883
$129$	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	$0.452624\pi$
$132$	0.394449	0.0343324
$133$	−9.21110	−0.798704
$134$	−14.5139	−1.25381
$135$	0	0
$136$	6.00000	0.514496
$137$	9.90833	0.846525	0.423263	−	0.906007i	$-0.360885\pi$
$137$	9.90833	0.846525	0.423263	+	0.906007i	$0.360885\pi$
$138$	2.09167	0.178055
$139$	8.90833	0.755594	0.377797	−	0.925888i	$-0.376682\pi$
$139$	8.90833	0.755594	0.377797	+	0.925888i	$0.376682\pi$
$140$	0	0
$141$	0.788897	0.0664372
$142$	6.00000	0.503509
$143$	3.00000	0.250873
$144$	−2.90833	−0.242361
$145$	0	0
$146$	8.69722	0.719787
$147$	4.30278	0.354887
$148$	−1.00000	−0.0821995
$149$	−1.81665	−0.148826	−0.0744130	−	0.997228i	$-0.523708\pi$
$149$	−1.81665	−0.148826	−0.0744130	+	0.997228i	$0.523708\pi$
$150$	0	0
$151$	−13.3944	−1.09002	−0.545012	−	0.838428i	$-0.683475\pi$
$151$	−13.3944	−1.09002	−0.545012	+	0.838428i	$0.683475\pi$
$152$	2.00000	0.162221
$153$	−17.4500	−1.41075
$154$	−6.00000	−0.483494
$155$	0	0
$156$	0.697224	0.0558226
$157$	−7.21110	−0.575509	−0.287754	−	0.957704i	$-0.592909\pi$
$157$	−7.21110	−0.575509	−0.287754	+	0.957704i	$0.592909\pi$
$158$	−16.1194	−1.28239
$159$	1.81665	0.144070
$160$	0	0
$161$	−31.8167	−2.50750
$162$	8.18335	0.642944
$163$	20.4222	1.59959	0.799795	−	0.600273i	$-0.204941\pi$
$163$	20.4222	1.59959	0.799795	+	0.600273i	$0.204941\pi$
$164$	−0.908327	−0.0709284
$165$	0	0
$166$	−17.2111	−1.33584
$167$	−12.5139	−0.968353	−0.484176	−	0.874970i	$-0.660881\pi$
$167$	−12.5139	−0.968353	−0.484176	+	0.874970i	$0.660881\pi$
$168$	−1.39445	−0.107584
$169$	−7.69722	−0.592094
$170$	0	0
$171$	−5.81665	−0.444811
$172$	6.60555	0.503669
$173$	23.2111	1.76471	0.882354	−	0.470587i	$-0.155958\pi$
$173$	23.2111	1.76471	0.882354	+	0.470587i	$0.155958\pi$
$174$	2.09167	0.158569
$175$	0	0
$176$	1.30278	0.0982004
$177$	1.02776	0.0772509
$178$	5.21110	0.390589
$179$	7.81665	0.584244	0.292122	−	0.956381i	$-0.405639\pi$
$179$	7.81665	0.584244	0.292122	+	0.956381i	$0.405639\pi$
$180$	0	0
$181$	20.0000	1.48659	0.743294	−	0.668965i	$-0.233262\pi$
$181$	20.0000	1.48659	0.743294	+	0.668965i	$0.233262\pi$
$182$	−10.6056	−0.786136
$183$	−3.18335	−0.235320
$184$	6.90833	0.509289
$185$	0	0
$186$	1.00000	0.0733236
$187$	7.81665	0.571610
$188$	2.60555	0.190029
$189$	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	$0.350448\pi$
$192$	0.302776	0.0218509
$193$	4.00000	0.287926	0.143963	−	0.989583i	$-0.454015\pi$
$193$	4.00000	0.287926	0.143963	+	0.989583i	$0.454015\pi$
$194$	−12.4222	−0.891862
$195$	0	0
$196$	14.2111	1.01508
$197$	6.00000	0.427482	0.213741	−	0.976890i	$-0.431435\pi$
$197$	6.00000	0.427482	0.213741	+	0.976890i	$0.431435\pi$
$198$	−3.78890	−0.269265
$199$	−2.42221	−0.171706	−0.0858528	−	0.996308i	$-0.527361\pi$
$199$	−2.42221	−0.171706	−0.0858528	+	0.996308i	$0.527361\pi$
$200$	0	0
$201$	−4.39445	−0.309961
$202$	16.4222	1.15546
$203$	−31.8167	−2.23309
$204$	1.81665	0.127191
$205$	0	0
$206$	−3.30278	−0.230115
$207$	−20.0917	−1.39647
$208$	2.30278	0.159669
$209$	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	$0.425955\pi$
$212$	6.00000	0.412082
$213$	1.81665	0.124475
$214$	−4.30278	−0.294132
$215$	0	0
$216$	−1.78890	−0.121719
$217$	−15.2111	−1.03260
$218$	2.00000	0.135457
$219$	2.63331	0.177942
$220$	0	0
$221$	13.8167	0.929409
$222$	−0.302776	−0.0203210
$223$	−15.8167	−1.05916	−0.529581	−	0.848260i	$-0.677651\pi$
$223$	−15.8167	−1.05916	−0.529581	+	0.848260i	$0.677651\pi$
$224$	−4.60555	−0.307721
$225$	0	0
$226$	−11.2111	−0.745751
$227$	7.81665	0.518810	0.259405	−	0.965769i	$-0.416474\pi$
$227$	7.81665	0.518810	0.259405	+	0.965769i	$0.416474\pi$
$228$	0.605551	0.0401036
$229$	17.3944	1.14946	0.574729	−	0.818344i	$-0.305108\pi$
$229$	17.3944	1.14946	0.574729	+	0.818344i	$0.305108\pi$
$230$	0	0
$231$	−1.81665	−0.119527
$232$	6.90833	0.453554
$233$	9.51388	0.623275	0.311637	−	0.950201i	$-0.399123\pi$
$233$	9.51388	0.623275	0.311637	+	0.950201i	$0.399123\pi$
$234$	−6.69722	−0.437811
$235$	0	0
$236$	3.39445	0.220960
$237$	−4.88057	−0.317027
$238$	−27.6333	−1.79120
$239$	0.513878	0.0332400	0.0166200	−	0.999862i	$-0.494709\pi$
$239$	0.513878	0.0332400	0.0166200	+	0.999862i	$0.494709\pi$
$240$	0	0
$241$	8.00000	0.515325	0.257663	−	0.966235i	$-0.417048\pi$
$241$	8.00000	0.515325	0.257663	+	0.966235i	$0.417048\pi$
$242$	−9.30278	−0.598005
$243$	7.84441	0.503219
$244$	−10.5139	−0.673082
$245$	0	0
$246$	−0.275019	−0.0175346
$247$	4.60555	0.293044
$248$	3.30278	0.209726
$249$	−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	$0.568733\pi$
$252$	13.3944	0.843771
$253$	9.00000	0.565825
$254$	4.78890	0.300482
$255$	0	0
$256$	1.00000	0.0625000
$257$	11.2111	0.699329	0.349665	−	0.936875i	$-0.386296\pi$
$257$	11.2111	0.699329	0.349665	+	0.936875i	$0.386296\pi$
$258$	2.00000	0.124515
$259$	4.60555	0.286175
$260$	0	0
$261$	−20.0917	−1.24364
$262$	3.39445	0.209710
$263$	7.81665	0.481996	0.240998	−	0.970526i	$-0.422525\pi$
$263$	7.81665	0.481996	0.240998	+	0.970526i	$0.422525\pi$
$264$	0.394449	0.0242766
$265$	0	0
$266$	−9.21110	−0.564769
$267$	1.57779	0.0965595
$268$	−14.5139	−0.886576
$269$	−6.78890	−0.413926	−0.206963	−	0.978349i	$-0.566358\pi$
$269$	−6.78890	−0.413926	−0.206963	+	0.978349i	$0.566358\pi$
$270$	0	0
$271$	6.42221	0.390121	0.195061	−	0.980791i	$-0.437510\pi$
$271$	6.42221	0.390121	0.195061	+	0.980791i	$0.437510\pi$
$272$	6.00000	0.363803
$273$	−3.21110	−0.194345
$274$	9.90833	0.598584
$275$	0	0
$276$	2.09167	0.125904
$277$	25.1194	1.50928	0.754640	−	0.656139i	$-0.227811\pi$
$277$	25.1194	1.50928	0.754640	+	0.656139i	$0.227811\pi$
$278$	8.90833	0.534286
$279$	−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	$0.616517\pi$
$282$	0.788897	0.0469782
$283$	−17.3944	−1.03399	−0.516996	−	0.855988i	$-0.672950\pi$
$283$	−17.3944	−1.03399	−0.516996	+	0.855988i	$0.672950\pi$
$284$	6.00000	0.356034
$285$	0	0
$286$	3.00000	0.177394
$287$	4.18335	0.246935
$288$	−2.90833	−0.171375
$289$	19.0000	1.11765
$290$	0	0
$291$	−3.76114	−0.220482
$292$	8.69722	0.508967
$293$	25.0278	1.46214	0.731069	−	0.682304i	$-0.239022\pi$
$293$	25.0278	1.46214	0.731069	+	0.682304i	$0.239022\pi$
$294$	4.30278	0.250943
$295$	0	0
$296$	−1.00000	−0.0581238
$297$	−2.33053	−0.135231
$298$	−1.81665	−0.105236
$299$	15.9083	0.920002
$300$	0	0
$301$	−30.4222	−1.75351
$302$	−13.3944	−0.770764
$303$	4.97224	0.285648
$304$	2.00000	0.114708
$305$	0	0
$306$	−17.4500	−0.997548
$307$	−7.09167	−0.404743	−0.202372	−	0.979309i	$-0.564865\pi$
$307$	−7.09167	−0.404743	−0.202372	+	0.979309i	$0.564865\pi$
$308$	−6.00000	−0.341882
$309$	−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	$0.453887\pi$
$312$	0.697224	0.0394726
$313$	−27.0278	−1.52770	−0.763850	−	0.645394i	$-0.776693\pi$
$313$	−27.0278	−1.52770	−0.763850	+	0.645394i	$0.776693\pi$
$314$	−7.21110	−0.406946
$315$	0	0
$316$	−16.1194	−0.906789
$317$	5.21110	0.292685	0.146342	−	0.989234i	$-0.453250\pi$
$317$	5.21110	0.292685	0.146342	+	0.989234i	$0.453250\pi$
$318$	1.81665	0.101873
$319$	9.00000	0.503903
$320$	0	0
$321$	−1.30278	−0.0727138
$322$	−31.8167	−1.77307
$323$	12.0000	0.667698
$324$	8.18335	0.454630
$325$	0	0
$326$	20.4222	1.13108
$327$	0.605551	0.0334871
$328$	−0.908327	−0.0501540
$329$	−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	$0.489403\pi$
$332$	−17.2111	−0.944582
$333$	2.90833	0.159375
$334$	−12.5139	−0.684729
$335$	0	0
$336$	−1.39445	−0.0760734
$337$	19.1194	1.04150	0.520751	−	0.853709i	$-0.325652\pi$
$337$	19.1194	1.04150	0.520751	+	0.853709i	$0.325652\pi$
$338$	−7.69722	−0.418674
$339$	−3.39445	−0.184361
$340$	0	0
$341$	4.30278	0.233008
$342$	−5.81665	−0.314529
$343$	−33.2111	−1.79323
$344$	6.60555	0.356147
$345$	0	0
$346$	23.2111	1.24784
$347$	−31.8167	−1.70801	−0.854004	−	0.520267i	$-0.825832\pi$
$347$	−31.8167	−1.70801	−0.854004	+	0.520267i	$0.825832\pi$
$348$	2.09167	0.112125
$349$	−22.2389	−1.19042	−0.595209	−	0.803571i	$-0.702931\pi$
$349$	−22.2389	−1.19042	−0.595209	+	0.803571i	$0.702931\pi$
$350$	0	0
$351$	−4.11943	−0.219879
$352$	1.30278	0.0694382
$353$	−31.8167	−1.69343	−0.846715	−	0.532047i	$-0.821423\pi$
$353$	−31.8167	−1.69343	−0.846715	+	0.532047i	$0.821423\pi$
$354$	1.02776	0.0546246
$355$	0	0
$356$	5.21110	0.276188
$357$	−8.36669	−0.442812
$358$	7.81665	0.413123
$359$	−11.2111	−0.591699	−0.295850	−	0.955235i	$-0.595603\pi$
$359$	−11.2111	−0.591699	−0.295850	+	0.955235i	$0.595603\pi$
$360$	0	0
$361$	−15.0000	−0.789474
$362$	20.0000	1.05118
$363$	−2.81665	−0.147836
$364$	−10.6056	−0.555882
$365$	0	0
$366$	−3.18335	−0.166396
$367$	17.8167	0.930022	0.465011	−	0.885305i	$-0.346050\pi$
$367$	17.8167	0.930022	0.465011	+	0.885305i	$0.346050\pi$
$368$	6.90833	0.360121
$369$	2.64171	0.137522
$370$	0	0
$371$	−27.6333	−1.43465
$372$	1.00000	0.0518476
$373$	−3.81665	−0.197619	−0.0988094	−	0.995106i	$-0.531503\pi$
$373$	−3.81665	−0.197619	−0.0988094	+	0.995106i	$0.531503\pi$
$374$	7.81665	0.404190
$375$	0	0
$376$	2.60555	0.134371
$377$	15.9083	0.819321
$378$	8.23886	0.423761
$379$	−15.3305	−0.787477	−0.393738	−	0.919223i	$-0.628818\pi$
$379$	−15.3305	−0.787477	−0.393738	+	0.919223i	$0.628818\pi$
$380$	0	0
$381$	1.44996	0.0742838
$382$	12.5139	0.640266
$383$	−20.8444	−1.06510	−0.532550	−	0.846399i	$-0.678766\pi$
$383$	−20.8444	−1.06510	−0.532550	+	0.846399i	$0.678766\pi$
$384$	0.302776	0.0154510
$385$	0	0
$386$	4.00000	0.203595
$387$	−19.2111	−0.976555
$388$	−12.4222	−0.630642
$389$	−11.8806	−0.602369	−0.301184	−	0.953566i	$-0.597382\pi$
$389$	−11.8806	−0.602369	−0.301184	+	0.953566i	$0.597382\pi$
$390$	0	0
$391$	41.4500	2.09621
$392$	14.2111	0.717769
$393$	1.02776	0.0518435
$394$	6.00000	0.302276
$395$	0	0
$396$	−3.78890	−0.190399
$397$	−27.8167	−1.39608	−0.698039	−	0.716060i	$-0.745944\pi$
$397$	−27.8167	−1.39608	−0.698039	+	0.716060i	$0.745944\pi$
$398$	−2.42221	−0.121414
$399$	−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	$0.387884\pi$
$402$	−4.39445	−0.219175
$403$	7.60555	0.378859
$404$	16.4222	0.817035
$405$	0	0
$406$	−31.8167	−1.57903
$407$	−1.30278	−0.0645762
$408$	1.81665	0.0899378
$409$	−5.02776	−0.248607	−0.124303	−	0.992244i	$-0.539670\pi$
$409$	−5.02776	−0.248607	−0.124303	+	0.992244i	$0.539670\pi$
$410$	0	0
$411$	3.00000	0.147979
$412$	−3.30278	−0.162716
$413$	−15.6333	−0.769265
$414$	−20.0917	−0.987452
$415$	0	0
$416$	2.30278	0.112903
$417$	2.69722	0.132084
$418$	2.60555	0.127442
$419$	−25.1472	−1.22852	−0.614260	−	0.789104i	$-0.710545\pi$
$419$	−25.1472	−1.22852	−0.614260	+	0.789104i	$0.710545\pi$
$420$	0	0
$421$	28.7250	1.39997	0.699985	−	0.714158i	$-0.253190\pi$
$421$	28.7250	1.39997	0.699985	+	0.714158i	$0.253190\pi$
$422$	6.69722	0.326016
$423$	−7.57779	−0.368445
$424$	6.00000	0.291386
$425$	0	0
$426$	1.81665	0.0880172
$427$	48.4222	2.34331
$428$	−4.30278	−0.207983
$429$	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	$0.540055\pi$
$432$	−1.78890	−0.0860684
$433$	11.9361	0.573612	0.286806	−	0.957989i	$-0.407407\pi$
$433$	11.9361	0.573612	0.286806	+	0.957989i	$0.407407\pi$
$434$	−15.2111	−0.730156
$435$	0	0
$436$	2.00000	0.0957826
$437$	13.8167	0.660940
$438$	2.63331	0.125824
$439$	−9.33053	−0.445322	−0.222661	−	0.974896i	$-0.571474\pi$
$439$	−9.33053	−0.445322	−0.222661	+	0.974896i	$0.571474\pi$
$440$	0	0
$441$	−41.3305	−1.96812
$442$	13.8167	0.657191
$443$	−0.275019	−0.0130666	−0.00653328	−	0.999979i	$-0.502080\pi$
$443$	−0.275019	−0.0130666	−0.00653328	+	0.999979i	$0.502080\pi$
$444$	−0.302776	−0.0143691
$445$	0	0
$446$	−15.8167	−0.748940
$447$	−0.550039	−0.0260159
$448$	−4.60555	−0.217592
$449$	−0.788897	−0.0372304	−0.0186152	−	0.999827i	$-0.505926\pi$
$449$	−0.788897	−0.0372304	−0.0186152	+	0.999827i	$0.505926\pi$
$450$	0	0
$451$	−1.18335	−0.0557216
$452$	−11.2111	−0.527326
$453$	−4.05551	−0.190545
$454$	7.81665	0.366854
$455$	0	0
$456$	0.605551	0.0283575
$457$	−4.60555	−0.215439	−0.107719	−	0.994181i	$-0.534355\pi$
$457$	−4.60555	−0.215439	−0.107719	+	0.994181i	$0.534355\pi$
$458$	17.3944	0.812789
$459$	−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	$0.624912\pi$
$462$	−1.81665	−0.0845184
$463$	−30.3028	−1.40829	−0.704145	−	0.710056i	$-0.748669\pi$
$463$	−30.3028	−1.40829	−0.704145	+	0.710056i	$0.748669\pi$
$464$	6.90833	0.320711
$465$	0	0
$466$	9.51388	0.440722
$467$	0	0		−	1.00000i	$-0.5\pi$
$467$	0	0			1.00000i	$0.5\pi$
$468$	−6.69722	−0.309579
$469$	66.8444	3.08659
$470$	0	0
$471$	−2.18335	−0.100603
$472$	3.39445	0.156242
$473$	8.60555	0.395684
$474$	−4.88057	−0.224172
$475$	0	0
$476$	−27.6333	−1.26657
$477$	−17.4500	−0.798979
$478$	0.513878	0.0235042
$479$	12.1194	0.553751	0.276875	−	0.960906i	$-0.410701\pi$
$479$	12.1194	0.553751	0.276875	+	0.960906i	$0.410701\pi$
$480$	0	0
$481$	−2.30278	−0.104998
$482$	8.00000	0.364390
$483$	−9.63331	−0.438331
$484$	−9.30278	−0.422853
$485$	0	0
$486$	7.84441	0.355830
$487$	22.7889	1.03266	0.516332	−	0.856389i	$-0.327297\pi$
$487$	22.7889	1.03266	0.516332	+	0.856389i	$0.327297\pi$
$488$	−10.5139	−0.475941
$489$	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	$0.607813\pi$
$492$	−0.275019	−0.0123988
$493$	41.4500	1.86681
$494$	4.60555	0.207214
$495$	0	0
$496$	3.30278	0.148299
$497$	−27.6333	−1.23952
$498$	−5.21110	−0.233515
$499$	8.23886	0.368822	0.184411	−	0.982849i	$-0.440962\pi$
$499$	8.23886	0.368822	0.184411	+	0.982849i	$0.440962\pi$
$500$	0	0
$501$	−3.78890	−0.169275
$502$	−6.78890	−0.303003
$503$	24.5139	1.09302	0.546510	−	0.837453i	$-0.315956\pi$
$503$	24.5139	1.09302	0.546510	+	0.837453i	$0.315956\pi$
$504$	13.3944	0.596636
$505$	0	0
$506$	9.00000	0.400099
$507$	−2.33053	−0.103503
$508$	4.78890	0.212473
$509$	−25.8167	−1.14430	−0.572152	−	0.820148i	$-0.693891\pi$
$509$	−25.8167	−1.14430	−0.572152	+	0.820148i	$0.693891\pi$
$510$	0	0
$511$	−40.0555	−1.77195
$512$	1.00000	0.0441942
$513$	−3.57779	−0.157964
$514$	11.2111	0.494501
$515$	0	0
$516$	2.00000	0.0880451
$517$	3.39445	0.149288
$518$	4.60555	0.202356
$519$	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	$0.567679\pi$
$522$	−20.0917	−0.879389
$523$	−32.2389	−1.40971	−0.704853	−	0.709353i	$-0.748987\pi$
$523$	−32.2389	−1.40971	−0.704853	+	0.709353i	$0.748987\pi$
$524$	3.39445	0.148287
$525$	0	0
$526$	7.81665	0.340822
$527$	19.8167	0.863227
$528$	0.394449	0.0171662
$529$	24.7250	1.07500
$530$	0	0
$531$	−9.87217	−0.428416
$532$	−9.21110	−0.399352
$533$	−2.09167	−0.0906004
$534$	1.57779	0.0682779
$535$	0	0
$536$	−14.5139	−0.626904
$537$	2.36669	0.102130
$538$	−6.78890	−0.292690
$539$	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	$0.648596\pi$
$542$	6.42221	0.275857
$543$	6.05551	0.259867
$544$	6.00000	0.257248
$545$	0	0
$546$	−3.21110	−0.137423
$547$	13.3944	0.572705	0.286353	−	0.958124i	$-0.407557\pi$
$547$	13.3944	0.572705	0.286353	+	0.958124i	$0.407557\pi$
$548$	9.90833	0.423263
$549$	30.5778	1.30503
$550$	0	0
$551$	13.8167	0.588609
$552$	2.09167	0.0890275
$553$	74.2389	3.15696
$554$	25.1194	1.06722
$555$	0	0
$556$	8.90833	0.377797
$557$	6.51388	0.276002	0.138001	−	0.990432i	$-0.455932\pi$
$557$	6.51388	0.276002	0.138001	+	0.990432i	$0.455932\pi$
$558$	−9.60555	−0.406635
$559$	15.2111	0.643361
$560$	0	0
$561$	2.36669	0.0999218
$562$	−12.0000	−0.506189
$563$	−44.0555	−1.85672	−0.928359	−	0.371684i	$-0.878780\pi$
$563$	−44.0555	−1.85672	−0.928359	+	0.371684i	$0.878780\pi$
$564$	0.788897	0.0332186
$565$	0	0
$566$	−17.3944	−0.731143
$567$	−37.6888	−1.58278
$568$	6.00000	0.251754
$569$	−10.4222	−0.436922	−0.218461	−	0.975846i	$-0.570104\pi$
$569$	−10.4222	−0.436922	−0.218461	+	0.975846i	$0.570104\pi$
$570$	0	0
$571$	−20.3028	−0.849645	−0.424822	−	0.905277i	$-0.639663\pi$
$571$	−20.3028	−0.849645	−0.424822	+	0.905277i	$0.639663\pi$
$572$	3.00000	0.125436
$573$	3.78890	0.158283
$574$	4.18335	0.174609
$575$	0	0
$576$	−2.90833	−0.121180
$577$	28.2389	1.17560	0.587800	−	0.809007i	$-0.299994\pi$
$577$	28.2389	1.17560	0.587800	+	0.809007i	$0.299994\pi$
$578$	19.0000	0.790296
$579$	1.21110	0.0503317
$580$	0	0
$581$	79.2666	3.28853
$582$	−3.76114	−0.155904
$583$	7.81665	0.323733
$584$	8.69722	0.359894
$585$	0	0
$586$	25.0278	1.03389
$587$	2.36669	0.0976838	0.0488419	−	0.998807i	$-0.484447\pi$
$587$	2.36669	0.0976838	0.0488419	+	0.998807i	$0.484447\pi$
$588$	4.30278	0.177443
$589$	6.60555	0.272177
$590$	0	0
$591$	1.81665	0.0747272
$592$	−1.00000	−0.0410997
$593$	−36.5139	−1.49945	−0.749723	−	0.661752i	$-0.769813\pi$
$593$	−36.5139	−1.49945	−0.749723	+	0.661752i	$0.769813\pi$
$594$	−2.33053	−0.0956229
$595$	0	0
$596$	−1.81665	−0.0744130
$597$	−0.733385	−0.0300154
$598$	15.9083	0.650540
$599$	−35.2111	−1.43869	−0.719343	−	0.694655i	$-0.755557\pi$
$599$	−35.2111	−1.43869	−0.719343	+	0.694655i	$0.755557\pi$
$600$	0	0
$601$	−20.6972	−0.844257	−0.422129	−	0.906536i	$-0.638717\pi$
$601$	−20.6972	−0.844257	−0.422129	+	0.906536i	$0.638717\pi$
$602$	−30.4222	−1.23992
$603$	42.2111	1.71897
$604$	−13.3944	−0.545012
$605$	0	0
$606$	4.97224	0.201984
$607$	31.5139	1.27911	0.639554	−	0.768746i	$-0.279119\pi$
$607$	31.5139	1.27911	0.639554	+	0.768746i	$0.279119\pi$
$608$	2.00000	0.0811107
$609$	−9.63331	−0.390361
$610$	0	0
$611$	6.00000	0.242734
$612$	−17.4500	−0.705373
$613$	8.18335	0.330522	0.165261	−	0.986250i	$-0.447153\pi$
$613$	8.18335	0.330522	0.165261	+	0.986250i	$0.447153\pi$
$614$	−7.09167	−0.286197
$615$	0	0
$616$	−6.00000	−0.241747
$617$	−47.5694	−1.91507	−0.957536	−	0.288314i	$-0.906905\pi$
$617$	−47.5694	−1.91507	−0.957536	+	0.288314i	$0.906905\pi$
$618$	−1.00000	−0.0402259
$619$	−2.69722	−0.108411	−0.0542053	−	0.998530i	$-0.517263\pi$
$619$	−2.69722	−0.108411	−0.0542053	+	0.998530i	$0.517263\pi$
$620$	0	0
$621$	−12.3583	−0.495921
$622$	5.09167	0.204157
$623$	−24.0000	−0.961540
$624$	0.697224	0.0279113
$625$	0	0
$626$	−27.0278	−1.08025
$627$	0.788897	0.0315055
$628$	−7.21110	−0.287754
$629$	−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	$0.381305\pi$
$632$	−16.1194	−0.641196
$633$	2.02776	0.0805961
$634$	5.21110	0.206959
$635$	0	0
$636$	1.81665	0.0720350
$637$	32.7250	1.29661
$638$	9.00000	0.356313
$639$	−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	$0.515635\pi$
$642$	−1.30278	−0.0514165
$643$	29.8167	1.17585	0.587927	−	0.808914i	$-0.299944\pi$
$643$	29.8167	1.17585	0.587927	+	0.808914i	$0.299944\pi$
$644$	−31.8167	−1.25375
$645$	0	0
$646$	12.0000	0.472134
$647$	25.9361	1.01965	0.509826	−	0.860277i	$-0.329710\pi$
$647$	25.9361	1.01965	0.509826	+	0.860277i	$0.329710\pi$
$648$	8.18335	0.321472
$649$	4.42221	0.173587
$650$	0	0
$651$	−4.60555	−0.180506
$652$	20.4222	0.799795
$653$	−6.90833	−0.270344	−0.135172	−	0.990822i	$-0.543159\pi$
$653$	−6.90833	−0.270344	−0.135172	+	0.990822i	$0.543159\pi$
$654$	0.605551	0.0236789
$655$	0	0
$656$	−0.908327	−0.0354642
$657$	−25.2944	−0.986827
$658$	−12.0000	−0.467809
$659$	−42.1194	−1.64074	−0.820370	−	0.571833i	$-0.806233\pi$
$659$	−42.1194	−1.64074	−0.820370	+	0.571833i	$0.806233\pi$
$660$	0	0
$661$	−12.4861	−0.485654	−0.242827	−	0.970070i	$-0.578075\pi$
$661$	−12.4861	−0.485654	−0.242827	+	0.970070i	$0.578075\pi$
$662$	1.21110	0.0470708
$663$	4.18335	0.162468
$664$	−17.2111	−0.667920
$665$	0	0
$666$	2.90833	0.112695
$667$	47.7250	1.84792
$668$	−12.5139	−0.484176
$669$	−4.78890	−0.185149
$670$	0	0
$671$	−13.6972	−0.528775
$672$	−1.39445	−0.0537920
$673$	−24.3028	−0.936803	−0.468402	−	0.883516i	$-0.655170\pi$
$673$	−24.3028	−0.936803	−0.468402	+	0.883516i	$0.655170\pi$
$674$	19.1194	0.736453
$675$	0	0
$676$	−7.69722	−0.296047
$677$	36.2389	1.39277	0.696386	−	0.717667i	$-0.254790\pi$
$677$	36.2389	1.39277	0.696386	+	0.717667i	$0.254790\pi$
$678$	−3.39445	−0.130363
$679$	57.2111	2.19556
$680$	0	0
$681$	2.36669	0.0906918
$682$	4.30278	0.164762
$683$	−12.0000	−0.459167	−0.229584	−	0.973289i	$-0.573736\pi$
$683$	−12.0000	−0.459167	−0.229584	+	0.973289i	$0.573736\pi$
$684$	−5.81665	−0.222405
$685$	0	0
$686$	−33.2111	−1.26801
$687$	5.26662	0.200934
$688$	6.60555	0.251834
$689$	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	$0.451375\pi$
$692$	23.2111	0.882354
$693$	17.4500	0.662869
$694$	−31.8167	−1.20774
$695$	0	0
$696$	2.09167	0.0792847
$697$	−5.44996	−0.206432
$698$	−22.2389	−0.841753
$699$	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	$0.409329\pi$
$702$	−4.11943	−0.155478
$703$	−2.00000	−0.0754314
$704$	1.30278	0.0491002
$705$	0	0
$706$	−31.8167	−1.19744
$707$	−75.6333	−2.84448
$708$	1.02776	0.0386254
$709$	−1.66947	−0.0626982	−0.0313491	−	0.999508i	$-0.509980\pi$
$709$	−1.66947	−0.0626982	−0.0313491	+	0.999508i	$0.509980\pi$
$710$	0	0
$711$	46.8806	1.75816
$712$	5.21110	0.195294
$713$	22.8167	0.854490
$714$	−8.36669	−0.313116
$715$	0	0
$716$	7.81665	0.292122
$717$	0.155590	0.00581061
$718$	−11.2111	−0.418395
$719$	−8.36669	−0.312025	−0.156012	−	0.987755i	$-0.549864\pi$
$719$	−8.36669	−0.312025	−0.156012	+	0.987755i	$0.549864\pi$
$720$	0	0
$721$	15.2111	0.566491
$722$	−15.0000	−0.558242
$723$	2.42221	0.0900828
$724$	20.0000	0.743294
$725$	0	0
$726$	−2.81665	−0.104536
$727$	−29.9083	−1.10924	−0.554619	−	0.832104i	$-0.687136\pi$
$727$	−29.9083	−1.10924	−0.554619	+	0.832104i	$0.687136\pi$
$728$	−10.6056	−0.393068
$729$	−22.1749	−0.821294
$730$	0	0
$731$	39.6333	1.46589
$732$	−3.18335	−0.117660
$733$	−29.6333	−1.09453	−0.547266	−	0.836959i	$-0.684331\pi$
$733$	−29.6333	−1.09453	−0.547266	+	0.836959i	$0.684331\pi$
$734$	17.8167	0.657625
$735$	0	0
$736$	6.90833	0.254644
$737$	−18.9083	−0.696497
$738$	2.64171	0.0972427
$739$	−42.3305	−1.55715	−0.778577	−	0.627549i	$-0.784058\pi$
$739$	−42.3305	−1.55715	−0.778577	+	0.627549i	$0.784058\pi$
$740$	0	0
$741$	1.39445	0.0512264
$742$	−27.6333	−1.01445
$743$	−35.4500	−1.30053	−0.650266	−	0.759706i	$-0.725343\pi$
$743$	−35.4500	−1.30053	−0.650266	+	0.759706i	$0.725343\pi$
$744$	1.00000	0.0366618
$745$	0	0
$746$	−3.81665	−0.139738
$747$	50.0555	1.83144
$748$	7.81665	0.285805
$749$	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	$0.417782\pi$
$752$	2.60555	0.0950147
$753$	−2.05551	−0.0749070
$754$	15.9083	0.579347
$755$	0	0
$756$	8.23886	0.299644
$757$	−9.30278	−0.338115	−0.169058	−	0.985606i	$-0.554072\pi$
$757$	−9.30278	−0.338115	−0.169058	+	0.985606i	$0.554072\pi$
$758$	−15.3305	−0.556830
$759$	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	$0.223522\pi$
$762$	1.44996	0.0525266
$763$	−9.21110	−0.333464
$764$	12.5139	0.452736
$765$	0	0
$766$	−20.8444	−0.753139
$767$	7.81665	0.282243
$768$	0.302776	0.0109255
$769$	−22.0000	−0.793340	−0.396670	−	0.917961i	$-0.629834\pi$
$769$	−22.0000	−0.793340	−0.396670	+	0.917961i	$0.629834\pi$
$770$	0	0
$771$	3.39445	0.122248
$772$	4.00000	0.143963
$773$	50.0555	1.80037	0.900186	−	0.435506i	$-0.143431\pi$
$773$	50.0555	1.80037	0.900186	+	0.435506i	$0.143431\pi$
$774$	−19.2111	−0.690529
$775$	0	0
$776$	−12.4222	−0.445931
$777$	1.39445	0.0500256
$778$	−11.8806	−0.425939
$779$	−1.81665	−0.0650884
$780$	0	0
$781$	7.81665	0.279702
$782$	41.4500	1.48225
$783$	−12.3583	−0.441649
$784$	14.2111	0.507539
$785$	0	0
$786$	1.02776	0.0366589
$787$	−25.2111	−0.898679	−0.449339	−	0.893361i	$-0.648341\pi$
$787$	−25.2111	−0.898679	−0.449339	+	0.893361i	$0.648341\pi$
$788$	6.00000	0.213741
$789$	2.36669	0.0842565
$790$	0	0
$791$	51.6333	1.83587
$792$	−3.78890	−0.134633
$793$	−24.2111	−0.859761
$794$	−27.8167	−0.987176
$795$	0	0
$796$	−2.42221	−0.0858528
$797$	17.3305	0.613879	0.306939	−	0.951729i	$-0.400695\pi$
$797$	17.3305	0.613879	0.306939	+	0.951729i	$0.400695\pi$
$798$	−2.78890	−0.0987259
$799$	15.6333	0.553067
$800$	0	0
$801$	−15.1556	−0.535496
$802$	13.8167	0.487883
$803$	11.3305	0.399846
$804$	−4.39445	−0.154980
$805$	0	0
$806$	7.60555	0.267894
$807$	−2.05551	−0.0723575
$808$	16.4222	0.577731
$809$	29.4500	1.03541	0.517703	−	0.855561i	$-0.326787\pi$
$809$	29.4500	1.03541	0.517703	+	0.855561i	$0.326787\pi$
$810$	0	0
$811$	54.1472	1.90136	0.950682	−	0.310166i	$-0.100385\pi$
$811$	54.1472	1.90136	0.950682	+	0.310166i	$0.100385\pi$
$812$	−31.8167	−1.11655
$813$	1.94449	0.0681961
$814$	−1.30278	−0.0456623
$815$	0	0
$816$	1.81665	0.0635956
$817$	13.2111	0.462198
$818$	−5.02776	−0.175791
$819$	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	$0.437323\pi$
$822$	3.00000	0.104637
$823$	12.8444	0.447728	0.223864	−	0.974620i	$-0.428133\pi$
$823$	12.8444	0.447728	0.223864	+	0.974620i	$0.428133\pi$
$824$	−3.30278	−0.115058
$825$	0	0
$826$	−15.6333	−0.543952
$827$	−27.3944	−0.952598	−0.476299	−	0.879283i	$-0.658022\pi$
$827$	−27.3944	−0.952598	−0.476299	+	0.879283i	$0.658022\pi$
$828$	−20.0917	−0.698234
$829$	4.72498	0.164105	0.0820527	−	0.996628i	$-0.473852\pi$
$829$	4.72498	0.164105	0.0820527	+	0.996628i	$0.473852\pi$
$830$	0	0
$831$	7.60555	0.263834
$832$	2.30278	0.0798344
$833$	85.2666	2.95431
$834$	2.69722	0.0933972
$835$	0	0
$836$	2.60555	0.0901149
$837$	−5.90833	−0.204222
$838$	−25.1472	−0.868695
$839$	−49.0278	−1.69263	−0.846313	−	0.532686i	$-0.821183\pi$
$839$	−49.0278	−1.69263	−0.846313	+	0.532686i	$0.821183\pi$
$840$	0	0
$841$	18.7250	0.645689
$842$	28.7250	0.989928
$843$	−3.63331	−0.125138
$844$	6.69722	0.230528
$845$	0	0
$846$	−7.57779	−0.260530
$847$	42.8444	1.47215
$848$	6.00000	0.206041
$849$	−5.26662	−0.180750
$850$	0	0
$851$	−6.90833	−0.236814
$852$	1.81665	0.0622375
$853$	11.5416	0.395178	0.197589	−	0.980285i	$-0.436689\pi$
$853$	11.5416	0.395178	0.197589	+	0.980285i	$0.436689\pi$
$854$	48.4222	1.65697
$855$	0	0
$856$	−4.30278	−0.147066
$857$	14.8444	0.507075	0.253538	−	0.967326i	$-0.418406\pi$
$857$	14.8444	0.507075	0.253538	+	0.967326i	$0.418406\pi$
$858$	0.908327	0.0310098
$859$	−24.0555	−0.820764	−0.410382	−	0.911914i	$-0.634605\pi$
$859$	−24.0555	−0.820764	−0.410382	+	0.911914i	$0.634605\pi$
$860$	0	0
$861$	1.26662	0.0431661
$862$	−5.21110	−0.177491
$863$	−12.0000	−0.408485	−0.204242	−	0.978920i	$-0.565473\pi$
$863$	−12.0000	−0.408485	−0.204242	+	0.978920i	$0.565473\pi$
$864$	−1.78890	−0.0608595
$865$	0	0
$866$	11.9361	0.405605
$867$	5.75274	0.195373
$868$	−15.2111	−0.516298
$869$	−21.0000	−0.712376
$870$	0	0
$871$	−33.4222	−1.13247
$872$	2.00000	0.0677285
$873$	36.1278	1.22274
$874$	13.8167	0.467355
$875$	0	0
$876$	2.63331	0.0889712
$877$	−7.21110	−0.243502	−0.121751	−	0.992561i	$-0.538851\pi$
$877$	−7.21110	−0.243502	−0.121751	+	0.992561i	$0.538851\pi$
$878$	−9.33053	−0.314890
$879$	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	$0.358422\pi$
$882$	−41.3305	−1.39167
$883$	2.42221	0.0815137	0.0407568	−	0.999169i	$-0.487023\pi$
$883$	2.42221	0.0815137	0.0407568	+	0.999169i	$0.487023\pi$
$884$	13.8167	0.464704
$885$	0	0
$886$	−0.275019	−0.00923945
$887$	−28.4222	−0.954324	−0.477162	−	0.878815i	$-0.658335\pi$
$887$	−28.4222	−0.954324	−0.477162	+	0.878815i	$0.658335\pi$
$888$	−0.302776	−0.0101605
$889$	−22.0555	−0.739718
$890$	0	0
$891$	10.6611	0.357159
$892$	−15.8167	−0.529581
$893$	5.21110	0.174383
$894$	−0.550039	−0.0183960
$895$	0	0
$896$	−4.60555	−0.153861
$897$	4.81665	0.160823
$898$	−0.788897	−0.0263258
$899$	22.8167	0.760978
$900$	0	0
$901$	36.0000	1.19933
$902$	−1.18335	−0.0394011
$903$	−9.21110	−0.306526
$904$	−11.2111	−0.372876
$905$	0	0
$906$	−4.05551	−0.134735
$907$	−26.0000	−0.863316	−0.431658	−	0.902037i	$-0.642071\pi$
$907$	−26.0000	−0.863316	−0.431658	+	0.902037i	$0.642071\pi$
$908$	7.81665	0.259405
$909$	−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	$0.779254\pi$
$912$	0.605551	0.0200518
$913$	−22.4222	−0.742067
$914$	−4.60555	−0.152338
$915$	0	0
$916$	17.3944	0.574729
$917$	−15.6333	−0.516257
$918$	−10.7334	−0.354254
$919$	−38.4222	−1.26743	−0.633716	−	0.773566i	$-0.718471\pi$
$919$	−38.4222	−1.26743	−0.633716	+	0.773566i	$0.718471\pi$
$920$	0	0
$921$	−2.14719	−0.0707522
$922$	−16.4222	−0.540837
$923$	13.8167	0.454781
$924$	−1.81665	−0.0597635
$925$	0	0
$926$	−30.3028	−0.995811
$927$	9.60555	0.315488
$928$	6.90833	0.226777
$929$	−36.5139	−1.19798	−0.598991	−	0.800756i	$-0.704431\pi$
$929$	−36.5139	−1.19798	−0.598991	+	0.800756i	$0.704431\pi$
$930$	0	0
$931$	28.4222	0.931500
$932$	9.51388	0.311637
$933$	1.54163	0.0504708
$934$	0	0
$935$	0	0
$936$	−6.69722	−0.218906
$937$	28.9083	0.944394	0.472197	−	0.881493i	$-0.343461\pi$
$937$	28.9083	0.944394	0.472197	+	0.881493i	$0.343461\pi$
$938$	66.8444	2.18255
$939$	−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	$0.540666\pi$
$942$	−2.18335	−0.0711373
$943$	−6.27502	−0.204343
$944$	3.39445	0.110480
$945$	0	0
$946$	8.60555	0.279791
$947$	−39.6333	−1.28791	−0.643955	−	0.765064i	$-0.722707\pi$
$947$	−39.6333	−1.28791	−0.643955	+	0.765064i	$0.722707\pi$
$948$	−4.88057	−0.158514
$949$	20.0278	0.650128
$950$	0	0
$951$	1.57779	0.0511635
$952$	−27.6333	−0.895601
$953$	18.7527	0.607461	0.303730	−	0.952758i	$-0.401768\pi$
$953$	18.7527	0.607461	0.303730	+	0.952758i	$0.401768\pi$
$954$	−17.4500	−0.564963
$955$	0	0
$956$	0.513878	0.0166200
$957$	2.72498	0.0880861
$958$	12.1194	0.391561
$959$	−45.6333	−1.47358
$960$	0	0
$961$	−20.0917	−0.648118
$962$	−2.30278	−0.0742445
$963$	12.5139	0.403254
$964$	8.00000	0.257663
$965$	0	0
$966$	−9.63331	−0.309947
$967$	−25.7250	−0.827260	−0.413630	−	0.910445i	$-0.635739\pi$
$967$	−25.7250	−0.827260	−0.413630	+	0.910445i	$0.635739\pi$
$968$	−9.30278	−0.299003
$969$	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	$0.331083\pi$
$972$	7.84441	0.251610
$973$	−41.0278	−1.31529
$974$	22.7889	0.730203
$975$	0	0
$976$	−10.5139	−0.336541
$977$	−18.0000	−0.575871	−0.287936	−	0.957650i	$-0.592969\pi$
$977$	−18.0000	−0.575871	−0.287936	+	0.957650i	$0.592969\pi$
$978$	6.18335	0.197722
$979$	6.78890	0.216974
$980$	0	0
$981$	−5.81665	−0.185711
$982$	−14.7250	−0.469893
$983$	12.0000	0.382741	0.191370	−	0.981518i	$-0.438707\pi$
$983$	12.0000	0.382741	0.191370	+	0.981518i	$0.438707\pi$
$984$	−0.275019	−0.00876729
$985$	0	0
$986$	41.4500	1.32004
$987$	−3.63331	−0.115649
$988$	4.60555	0.146522
$989$	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	$0.168902\pi$
$992$	3.30278	0.104863
$993$	0.366692	0.0116366
$994$	−27.6333	−0.876475
$995$	0	0
$996$	−5.21110	−0.165120
$997$	23.5778	0.746716	0.373358	−	0.927687i	$-0.378206\pi$
$997$	23.5778	0.746716	0.373358	+	0.927687i	$0.378206\pi$
$998$	8.23886	0.260797
$999$	1.78890	0.0565982

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1850.2.a.u.1.2		2
5.2	odd	4		1850.2.b.i.149.3		4
5.3	odd	4		1850.2.b.i.149.2		4
5.4	even	2		74.2.a.a.1.1	✓	2
15.14	odd	2		666.2.a.j.1.1		2
20.19	odd	2		592.2.a.f.1.2		2
35.34	odd	2		3626.2.a.a.1.2		2
40.19	odd	2		2368.2.a.ba.1.1		2
40.29	even	2		2368.2.a.s.1.2		2
55.54	odd	2		8954.2.a.p.1.1		2
60.59	even	2		5328.2.a.bf.1.1		2
185.184	even	2		2738.2.a.l.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
74.2.a.a.1.1	✓	2	5.4	even	2
592.2.a.f.1.2		2	20.19	odd	2
666.2.a.j.1.1		2	15.14	odd	2
1850.2.a.u.1.2		2	1.1	even	1	trivial
1850.2.b.i.149.2		4	5.3	odd	4
1850.2.b.i.149.3		4	5.2	odd	4
2368.2.a.s.1.2		2	40.29	even	2
2368.2.a.ba.1.1		2	40.19	odd	2
2738.2.a.l.1.1		2	185.184	even	2
3626.2.a.a.1.2		2	35.34	odd	2
5328.2.a.bf.1.1		2	60.59	even	2
8954.2.a.p.1.1		2	55.54	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1850 = 2 \cdot 5^{2} \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1850.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +0.302776 q^{3} +1.00000 q^{4} +0.302776 q^{6} -4.60555 q^{7} +1.00000 q^{8} -2.90833 q^{9} +1.30278 q^{11} +0.302776 q^{12} +2.30278 q^{13} -4.60555 q^{14} +1.00000 q^{16} +6.00000 q^{17} -2.90833 q^{18} +2.00000 q^{19} -1.39445 q^{21} +1.30278 q^{22} +6.90833 q^{23} +0.302776 q^{24} +2.30278 q^{26} -1.78890 q^{27} -4.60555 q^{28} +6.90833 q^{29} +3.30278 q^{31} +1.00000 q^{32} +0.394449 q^{33} +6.00000 q^{34} -2.90833 q^{36} -1.00000 q^{37} +2.00000 q^{38} +0.697224 q^{39} -0.908327 q^{41} -1.39445 q^{42} +6.60555 q^{43} +1.30278 q^{44} +6.90833 q^{46} +2.60555 q^{47} +0.302776 q^{48} +14.2111 q^{49} +1.81665 q^{51} +2.30278 q^{52} +6.00000 q^{53} -1.78890 q^{54} -4.60555 q^{56} +0.605551 q^{57} +6.90833 q^{58} +3.39445 q^{59} -10.5139 q^{61} +3.30278 q^{62} +13.3944 q^{63} +1.00000 q^{64} +0.394449 q^{66} -14.5139 q^{67} +6.00000 q^{68} +2.09167 q^{69} +6.00000 q^{71} -2.90833 q^{72} +8.69722 q^{73} -1.00000 q^{74} +2.00000 q^{76} -6.00000 q^{77} +0.697224 q^{78} -16.1194 q^{79} +8.18335 q^{81} -0.908327 q^{82} -17.2111 q^{83} -1.39445 q^{84} +6.60555 q^{86} +2.09167 q^{87} +1.30278 q^{88} +5.21110 q^{89} -10.6056 q^{91} +6.90833 q^{92} +1.00000 q^{93} +2.60555 q^{94} +0.302776 q^{96} -12.4222 q^{97} +14.2111 q^{98} -3.78890 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} - 3 q^{3} + 2 q^{4} - 3 q^{6} - 2 q^{7} + 2 q^{8} + 5 q^{9} - q^{11} - 3 q^{12} + q^{13} - 2 q^{14} + 2 q^{16} + 12 q^{17} + 5 q^{18} + 4 q^{19} - 10 q^{21} - q^{22} + 3 q^{23} - 3 q^{24}+ \cdots - 22 q^{99}+O(q^{100}) \) 2 * q + 2 * q^2 - 3 * q^3 + 2 * q^4 - 3 * q^6 - 2 * q^7 + 2 * q^8 + 5 * q^9 - q^11 - 3 * q^12 + q^13 - 2 * q^14 + 2 * q^16 + 12 * q^17 + 5 * q^18 + 4 * q^19 - 10 * q^21 - q^22 + 3 * q^23 - 3 * q^24 + q^26 - 18 * q^27 - 2 * q^28 + 3 * q^29 + 3 * q^31 + 2 * q^32 + 8 * q^33 + 12 * q^34 + 5 * q^36 - 2 * q^37 + 4 * q^38 + 5 * q^39 + 9 * q^41 - 10 * q^42 + 6 * q^43 - q^44 + 3 * q^46 - 2 * q^47 - 3 * q^48 + 14 * q^49 - 18 * q^51 + q^52 + 12 * q^53 - 18 * q^54 - 2 * q^56 - 6 * q^57 + 3 * q^58 + 14 * q^59 - 3 * q^61 + 3 * q^62 + 34 * q^63 + 2 * q^64 + 8 * q^66 - 11 * q^67 + 12 * q^68 + 15 * q^69 + 12 * q^71 + 5 * q^72 + 21 * q^73 - 2 * q^74 + 4 * q^76 - 12 * q^77 + 5 * q^78 - 7 * q^79 + 38 * q^81 + 9 * q^82 - 20 * q^83 - 10 * q^84 + 6 * q^86 + 15 * q^87 - q^88 - 4 * q^89 - 14 * q^91 + 3 * q^92 + 2 * q^93 - 2 * q^94 - 3 * q^96 + 4 * q^97 + 14 * q^98 - 22 * q^99

Introduction

L-functions

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