LMFDB - Embedded newform 1850.2.b.i.149.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1850,2,Mod(149,1850)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://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);

sage: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")

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:traces = [4,0,0,-4,0,-6,0,0,-10,0,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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			149.4
Root			$2.30278i$ of defining polynomial
Character	$\chi$	$=$	1850.149
Dual form			1850.2.b.i.149.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.00000i q^{2} +3.30278i q^{3} -1.00000 q^{4} -3.30278 q^{6} +2.60555i q^{7} -1.00000i q^{8} -7.90833 q^{9} -2.30278 q^{11} -3.30278i q^{12} +1.30278i q^{13} -2.60555 q^{14} +1.00000 q^{16} +6.00000i q^{17} -7.90833i q^{18} -2.00000 q^{19} -8.60555 q^{21} -2.30278i q^{22} +3.90833i q^{23} +3.30278 q^{24} -1.30278 q^{26} -16.2111i q^{27} -2.60555i q^{28} +3.90833 q^{29} -0.302776 q^{31} +1.00000i q^{32} -7.60555i q^{33} -6.00000 q^{34} +7.90833 q^{36} -1.00000i q^{37} -2.00000i q^{38} -4.30278 q^{39} +9.90833 q^{41} -8.60555i q^{42} +0.605551i q^{43} +2.30278 q^{44} -3.90833 q^{46} -4.60555i q^{47} +3.30278i q^{48} +0.211103 q^{49} -19.8167 q^{51} -1.30278i q^{52} -6.00000i q^{53} +16.2111 q^{54} +2.60555 q^{56} -6.60555i q^{57} +3.90833i q^{58} -10.6056 q^{59} +7.51388 q^{61} -0.302776i q^{62} -20.6056i q^{63} -1.00000 q^{64} +7.60555 q^{66} +3.51388i q^{67} -6.00000i q^{68} -12.9083 q^{69} +6.00000 q^{71} +7.90833i q^{72} -12.3028i q^{73} +1.00000 q^{74} +2.00000 q^{76} -6.00000i q^{77} -4.30278i q^{78} -9.11943 q^{79} +29.8167 q^{81} +9.90833i q^{82} +2.78890i q^{83} +8.60555 q^{84} -0.605551 q^{86} +12.9083i q^{87} +2.30278i q^{88} +9.21110 q^{89} -3.39445 q^{91} -3.90833i q^{92} -1.00000i q^{93} +4.60555 q^{94} -3.30278 q^{96} +16.4222i q^{97} +0.211103i q^{98} +18.2111 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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}+ \cdots + 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

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
$3$	3.30278i	1.90686i	0.301617	+	0.953429i	$0.402474\pi$
$3$	3.30278i	1.90686i	−0.301617	+	0.953429i	$0.597526\pi$
$4$	−1.00000	−0.500000
$5$	0	0
$5$	0	0
$6$	−3.30278	−1.34835
$7$	2.60555i	0.984806i	0.870367	+	0.492403i	$0.163881\pi$
$7$	2.60555i	0.984806i	−0.870367	+	0.492403i	$0.836119\pi$
$8$	− 1.00000i	− 0.353553i
$9$	−7.90833	−2.63611
$10$	0	0
$11$	−2.30278	−0.694313	−0.347156	−	0.937807i	$-0.612853\pi$
$11$	−2.30278	−0.694313	−0.347156	+	0.937807i	$0.612853\pi$
$12$	− 3.30278i	− 0.953429i
$13$	1.30278i	0.361325i	0.983545	+	0.180662i	$0.0578242\pi$
$13$	1.30278i	0.361325i	−0.983545	+	0.180662i	$0.942176\pi$
$14$	−2.60555	−0.696363
$15$	0	0
$16$	1.00000	0.250000
$17$	6.00000i	1.45521i	0.685994	+	0.727607i	$0.259367\pi$
$17$	6.00000i	1.45521i	−0.685994	+	0.727607i	$0.740633\pi$
$18$	− 7.90833i	− 1.86401i
$19$	−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
$21$	−8.60555	−1.87789
$22$	− 2.30278i	− 0.490953i
$23$	3.90833i	0.814942i	0.913218	+	0.407471i	$0.133589\pi$
$23$	3.90833i	0.814942i	−0.913218	+	0.407471i	$0.866411\pi$
$24$	3.30278	0.674176
$25$	0	0
$26$	−1.30278	−0.255495
$27$	− 16.2111i	− 3.11983i
$28$	− 2.60555i	− 0.492403i
$29$	3.90833	0.725758	0.362879	−	0.931836i	$-0.381794\pi$
$29$	3.90833	0.725758	0.362879	+	0.931836i	$0.381794\pi$
$30$	0	0
$31$	−0.302776	−0.0543801	−0.0271901	−	0.999630i	$-0.508656\pi$
$31$	−0.302776	−0.0543801	−0.0271901	+	0.999630i	$0.508656\pi$
$32$	1.00000i	0.176777i
$33$	− 7.60555i	− 1.32396i
$34$	−6.00000	−1.02899
$35$	0	0
$36$	7.90833	1.31805
$37$	− 1.00000i	− 0.164399i
$37$	− 1.00000i	− 0.164399i
$38$	− 2.00000i	− 0.324443i
$39$	−4.30278	−0.688996
$40$	0	0
$41$	9.90833	1.54742	0.773710	−	0.633540i	$-0.218399\pi$
$41$	9.90833	1.54742	0.773710	+	0.633540i	$0.218399\pi$
$42$	− 8.60555i	− 1.32787i
$43$	0.605551i	0.0923457i	0.998933	+	0.0461729i	$0.0147025\pi$
$43$	0.605551i	0.0923457i	−0.998933	+	0.0461729i	$0.985297\pi$
$44$	2.30278	0.347156
$45$	0	0
$46$	−3.90833	−0.576251
$47$	− 4.60555i	− 0.671789i	−0.941900	−	0.335894i	$-0.890961\pi$
$47$	− 4.60555i	− 0.671789i	0.941900	−	0.335894i	$-0.109039\pi$
$48$	3.30278i	0.476715i
$49$	0.211103	0.0301575
$50$	0	0
$51$	−19.8167	−2.77489
$52$	− 1.30278i	− 0.180662i
$53$	− 6.00000i	− 0.824163i	−0.911147	−	0.412082i	$-0.864802\pi$
$53$	− 6.00000i	− 0.824163i	0.911147	−	0.412082i	$-0.135198\pi$
$54$	16.2111	2.20605
$55$	0	0
$56$	2.60555	0.348181
$57$	− 6.60555i	− 0.874927i
$58$	3.90833i	0.513188i
$59$	−10.6056	−1.38073	−0.690363	−	0.723464i	$-0.742549\pi$
$59$	−10.6056	−1.38073	−0.690363	+	0.723464i	$0.742549\pi$
$60$	0	0
$61$	7.51388	0.962054	0.481027	−	0.876706i	$-0.340264\pi$
$61$	7.51388	0.962054	0.481027	+	0.876706i	$0.340264\pi$
$62$	− 0.302776i	− 0.0384525i
$63$	− 20.6056i	− 2.59606i
$64$	−1.00000	−0.125000
$65$	0	0
$66$	7.60555	0.936179
$67$	3.51388i	0.429289i	0.976692	+	0.214644i	$0.0688592\pi$
$67$	3.51388i	0.429289i	−0.976692	+	0.214644i	$0.931141\pi$
$68$	− 6.00000i	− 0.727607i
$69$	−12.9083	−1.55398
$70$	0	0
$71$	6.00000	0.712069	0.356034	−	0.934473i	$-0.384129\pi$
$71$	6.00000	0.712069	0.356034	+	0.934473i	$0.384129\pi$
$72$	7.90833i	0.932005i
$73$	− 12.3028i	− 1.43993i	−0.694010	−	0.719965i	$-0.744158\pi$
$73$	− 12.3028i	− 1.43993i	0.694010	−	0.719965i	$-0.255842\pi$
$74$	1.00000	0.116248
$75$	0	0
$76$	2.00000	0.229416
$77$	− 6.00000i	− 0.683763i
$78$	− 4.30278i	− 0.487193i
$79$	−9.11943	−1.02602	−0.513008	−	0.858384i	$-0.671469\pi$
$79$	−9.11943	−1.02602	−0.513008	+	0.858384i	$0.671469\pi$
$80$	0	0
$81$	29.8167	3.31296
$82$	9.90833i	1.09419i
$83$	2.78890i	0.306121i	0.988217	+	0.153061i	$0.0489130\pi$
$83$	2.78890i	0.306121i	−0.988217	+	0.153061i	$0.951087\pi$
$84$	8.60555	0.938943
$85$	0	0
$86$	−0.605551	−0.0652983
$87$	12.9083i	1.38392i
$88$	2.30278i	0.245477i
$89$	9.21110	0.976375	0.488187	−	0.872739i	$-0.337658\pi$
$89$	9.21110	0.976375	0.488187	+	0.872739i	$0.337658\pi$
$90$	0	0
$91$	−3.39445	−0.355835
$92$	− 3.90833i	− 0.407471i
$93$	− 1.00000i	− 0.103695i
$94$	4.60555	0.475026
$95$	0	0
$96$	−3.30278	−0.337088
$97$	16.4222i	1.66742i	0.552201	+	0.833711i	$0.313788\pi$
$97$	16.4222i	1.66742i	−0.552201	+	0.833711i	$0.686212\pi$
$98$	0.211103i	0.0213246i
$99$	18.2111	1.83028
$100$	0	0
$101$	−12.4222	−1.23606	−0.618028	−	0.786156i	$-0.712068\pi$
$101$	−12.4222	−1.23606	−0.618028	+	0.786156i	$0.712068\pi$
$102$	− 19.8167i	− 1.96214i
$103$	− 0.302776i	− 0.0298334i	−0.999889	−	0.0149167i	$-0.995252\pi$
$103$	− 0.302776i	− 0.0298334i	0.999889	−	0.0149167i	$-0.00474830\pi$
$104$	1.30278	0.127748
$105$	0	0
$106$	6.00000	0.582772
$107$	− 0.697224i	− 0.0674032i	−0.999432	−	0.0337016i	$-0.989270\pi$
$107$	− 0.697224i	− 0.0674032i	0.999432	−	0.0337016i	$-0.0107296\pi$
$108$	16.2111i	1.55991i
$109$	−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
$111$	3.30278	0.313486
$112$	2.60555i	0.246201i
$113$	− 3.21110i	− 0.302075i	−0.988528	−	0.151038i	$-0.951739\pi$
$113$	− 3.21110i	− 0.302075i	0.988528	−	0.151038i	$-0.0482614\pi$
$114$	6.60555	0.618667
$115$	0	0
$116$	−3.90833	−0.362879
$117$	− 10.3028i	− 0.952492i
$118$	− 10.6056i	− 0.976320i
$119$	−15.6333	−1.43310
$120$	0	0
$121$	−5.69722	−0.517929
$122$	7.51388i	0.680275i
$123$	32.7250i	2.95071i
$124$	0.302776	0.0271901
$125$	0	0
$126$	20.6056	1.83569
$127$	19.2111i	1.70471i	0.522964	+	0.852355i	$0.324826\pi$
$127$	19.2111i	1.70471i	−0.522964	+	0.852355i	$0.675174\pi$
$128$	− 1.00000i	− 0.0883883i
$129$	−2.00000	−0.176090
$130$	0	0
$131$	10.6056	0.926611	0.463306	−	0.886199i	$-0.346663\pi$
$131$	10.6056	0.926611	0.463306	+	0.886199i	$0.346663\pi$
$132$	7.60555i	0.661978i
$133$	− 5.21110i	− 0.451860i
$134$	−3.51388	−0.303553
$135$	0	0
$136$	6.00000	0.514496
$137$	− 0.908327i	− 0.0776036i	−0.999247	−	0.0388018i	$-0.987646\pi$
$137$	− 0.908327i	− 0.0776036i	0.999247	−	0.0388018i	$-0.0123541\pi$
$138$	− 12.9083i	− 1.09883i
$139$	1.90833	0.161862	0.0809311	−	0.996720i	$-0.474211\pi$
$139$	1.90833	0.161862	0.0809311	+	0.996720i	$0.474211\pi$
$140$	0	0
$141$	15.2111	1.28101
$142$	6.00000i	0.503509i
$143$	− 3.00000i	− 0.250873i
$144$	−7.90833	−0.659027
$145$	0	0
$146$	12.3028	1.01818
$147$	0.697224i	0.0575061i
$148$	1.00000i	0.0821995i
$149$	−19.8167	−1.62344	−0.811722	−	0.584044i	$-0.801469\pi$
$149$	−19.8167	−1.62344	−0.811722	+	0.584044i	$0.801469\pi$
$150$	0	0
$151$	−20.6056	−1.67686	−0.838428	−	0.545012i	$-0.816525\pi$
$151$	−20.6056	−1.67686	−0.838428	+	0.545012i	$0.816525\pi$
$152$	2.00000i	0.162221i
$153$	− 47.4500i	− 3.83610i
$154$	6.00000	0.483494
$155$	0	0
$156$	4.30278	0.344498
$157$	7.21110i	0.575509i	0.957704	+	0.287754i	$0.0929087\pi$
$157$	7.21110i	0.575509i	−0.957704	+	0.287754i	$0.907091\pi$
$158$	− 9.11943i	− 0.725503i
$159$	19.8167	1.57156
$160$	0	0
$161$	−10.1833	−0.802560
$162$	29.8167i	2.34262i
$163$	8.42221i	0.659678i	0.944037	+	0.329839i	$0.106994\pi$
$163$	8.42221i	0.659678i	−0.944037	+	0.329839i	$0.893006\pi$
$164$	−9.90833	−0.773710
$165$	0	0
$166$	−2.78890	−0.216460
$167$	5.51388i	0.426677i	0.976978	+	0.213338i	$0.0684337\pi$
$167$	5.51388i	0.426677i	−0.976978	+	0.213338i	$0.931566\pi$
$168$	8.60555i	0.663933i
$169$	11.3028	0.869444
$170$	0	0
$171$	15.8167	1.20953
$172$	− 0.605551i	− 0.0461729i
$173$	− 8.78890i	− 0.668207i	−0.942536	−	0.334104i	$-0.891566\pi$
$173$	− 8.78890i	− 0.668207i	0.942536	−	0.334104i	$-0.108434\pi$
$174$	−12.9083	−0.978578
$175$	0	0
$176$	−2.30278	−0.173578
$177$	− 35.0278i	− 2.63285i
$178$	9.21110i	0.690401i
$179$	13.8167	1.03271	0.516353	−	0.856376i	$-0.327289\pi$
$179$	13.8167	1.03271	0.516353	+	0.856376i	$0.327289\pi$
$180$	0	0
$181$	20.0000	1.48659	0.743294	−	0.668965i	$-0.233262\pi$
$181$	20.0000	1.48659	0.743294	+	0.668965i	$0.233262\pi$
$182$	− 3.39445i	− 0.251613i
$183$	24.8167i	1.83450i
$184$	3.90833	0.288126
$185$	0	0
$186$	1.00000	0.0733236
$187$	− 13.8167i	− 1.01037i
$188$	4.60555i	0.335894i
$189$	42.2389	3.07242
$190$	0	0
$191$	−5.51388	−0.398970	−0.199485	−	0.979901i	$-0.563927\pi$
$191$	−5.51388	−0.398970	−0.199485	+	0.979901i	$0.563927\pi$
$192$	− 3.30278i	− 0.238357i
$193$	− 4.00000i	− 0.287926i	−0.989583	−	0.143963i	$-0.954015\pi$
$193$	− 4.00000i	− 0.287926i	0.989583	−	0.143963i	$-0.0459847\pi$
$194$	−16.4222	−1.17905
$195$	0	0
$196$	−0.211103	−0.0150788
$197$	6.00000i	0.427482i	0.976890	+	0.213741i	$0.0685649\pi$
$197$	6.00000i	0.427482i	−0.976890	+	0.213741i	$0.931435\pi$
$198$	18.2111i	1.29421i
$199$	−26.4222	−1.87302	−0.936510	−	0.350640i	$-0.885964\pi$
$199$	−26.4222	−1.87302	−0.936510	+	0.350640i	$0.885964\pi$
$200$	0	0
$201$	−11.6056	−0.818592
$202$	− 12.4222i	− 0.874023i
$203$	10.1833i	0.714731i
$204$	19.8167	1.38744
$205$	0	0
$206$	0.302776	0.0210954
$207$	− 30.9083i	− 2.14828i
$208$	1.30278i	0.0903312i
$209$	4.60555	0.318573
$210$	0	0
$211$	10.3028	0.709272	0.354636	−	0.935004i	$-0.384605\pi$
$211$	10.3028	0.709272	0.354636	+	0.935004i	$0.384605\pi$
$212$	6.00000i	0.412082i
$213$	19.8167i	1.35781i
$214$	0.697224	0.0476613
$215$	0	0
$216$	−16.2111	−1.10303
$217$	− 0.788897i	− 0.0535538i
$218$	− 2.00000i	− 0.135457i
$219$	40.6333	2.74574
$220$	0	0
$221$	−7.81665	−0.525805
$222$	3.30278i	0.221668i
$223$	− 5.81665i	− 0.389512i	−0.980852	−	0.194756i	$-0.937609\pi$
$223$	− 5.81665i	− 0.389512i	0.980852	−	0.194756i	$-0.0623915\pi$
$224$	−2.60555	−0.174091
$225$	0	0
$226$	3.21110	0.213599
$227$	− 13.8167i	− 0.917044i	−0.888683	−	0.458522i	$-0.848379\pi$
$227$	− 13.8167i	− 0.917044i	0.888683	−	0.458522i	$-0.151621\pi$
$228$	6.60555i	0.437463i
$229$	−24.6056	−1.62598	−0.812990	−	0.582277i	$-0.802162\pi$
$229$	−24.6056	−1.62598	−0.812990	+	0.582277i	$0.802162\pi$
$230$	0	0
$231$	19.8167	1.30384
$232$	− 3.90833i	− 0.256594i
$233$	8.51388i	0.557763i	0.960326	+	0.278881i	$0.0899636\pi$
$233$	8.51388i	0.557763i	−0.960326	+	0.278881i	$0.910036\pi$
$234$	10.3028	0.673514
$235$	0	0
$236$	10.6056	0.690363
$237$	− 30.1194i	− 1.95647i
$238$	− 15.6333i	− 1.01336i
$239$	17.5139	1.13288	0.566439	−	0.824103i	$-0.308321\pi$
$239$	17.5139	1.13288	0.566439	+	0.824103i	$0.308321\pi$
$240$	0	0
$241$	8.00000	0.515325	0.257663	−	0.966235i	$-0.417048\pi$
$241$	8.00000	0.515325	0.257663	+	0.966235i	$0.417048\pi$
$242$	− 5.69722i	− 0.366231i
$243$	49.8444i	3.19752i
$244$	−7.51388	−0.481027
$245$	0	0
$246$	−32.7250	−2.08647
$247$	− 2.60555i	− 0.165787i
$248$	0.302776i	0.0192263i
$249$	−9.21110	−0.583730
$250$	0	0
$251$	−21.2111	−1.33883	−0.669416	−	0.742887i	$-0.733456\pi$
$251$	−21.2111	−1.33883	−0.669416	+	0.742887i	$0.733456\pi$
$252$	20.6056i	1.29803i
$253$	− 9.00000i	− 0.565825i
$254$	−19.2111	−1.20541
$255$	0	0
$256$	1.00000	0.0625000
$257$	− 3.21110i	− 0.200303i	−0.994972	−	0.100152i	$-0.968067\pi$
$257$	− 3.21110i	− 0.200303i	0.994972	−	0.100152i	$-0.0319328\pi$
$258$	− 2.00000i	− 0.124515i
$259$	2.60555	0.161901
$260$	0	0
$261$	−30.9083	−1.91318
$262$	10.6056i	0.655213i
$263$	13.8167i	0.851971i	0.904730	+	0.425986i	$0.140073\pi$
$263$	13.8167i	0.851971i	−0.904730	+	0.425986i	$0.859927\pi$
$264$	−7.60555	−0.468089
$265$	0	0
$266$	5.21110	0.319513
$267$	30.4222i	1.86181i
$268$	− 3.51388i	− 0.214644i
$269$	21.2111	1.29326	0.646632	−	0.762802i	$-0.276177\pi$
$269$	21.2111	1.29326	0.646632	+	0.762802i	$0.276177\pi$
$270$	0	0
$271$	−22.4222	−1.36205	−0.681026	−	0.732259i	$-0.738466\pi$
$271$	−22.4222	−1.36205	−0.681026	+	0.732259i	$0.738466\pi$
$272$	6.00000i	0.363803i
$273$	− 11.2111i	− 0.678527i
$274$	0.908327	0.0548740
$275$	0	0
$276$	12.9083	0.776990
$277$	− 0.119429i	− 0.00717582i	−0.999994	−	0.00358791i	$-0.998858\pi$
$277$	− 0.119429i	− 0.00717582i	0.999994	−	0.00358791i	$-0.00114207\pi$
$278$	1.90833i	0.114454i
$279$	2.39445	0.143352
$280$	0	0
$281$	−12.0000	−0.715860	−0.357930	−	0.933748i	$-0.616517\pi$
$281$	−12.0000	−0.715860	−0.357930	+	0.933748i	$0.616517\pi$
$282$	15.2111i	0.905808i
$283$	24.6056i	1.46265i	0.682030	+	0.731324i	$0.261097\pi$
$283$	24.6056i	1.46265i	−0.682030	+	0.731324i	$0.738903\pi$
$284$	−6.00000	−0.356034
$285$	0	0
$286$	3.00000	0.177394
$287$	25.8167i	1.52391i
$288$	− 7.90833i	− 0.466003i
$289$	−19.0000	−1.11765
$290$	0	0
$291$	−54.2389	−3.17954
$292$	12.3028i	0.719965i
$293$	11.0278i	0.644248i	0.946697	+	0.322124i	$0.104397\pi$
$293$	11.0278i	0.644248i	−0.946697	+	0.322124i	$0.895603\pi$
$294$	−0.697224	−0.0406630
$295$	0	0
$296$	−1.00000	−0.0581238
$297$	37.3305i	2.16614i
$298$	− 19.8167i	− 1.14795i
$299$	−5.09167	−0.294459
$300$	0	0
$301$	−1.57779	−0.0909426
$302$	− 20.6056i	− 1.18572i
$303$	− 41.0278i	− 2.35698i
$304$	−2.00000	−0.114708
$305$	0	0
$306$	47.4500	2.71253
$307$	− 17.9083i	− 1.02208i	−0.859556	−	0.511041i	$-0.829260\pi$
$307$	− 17.9083i	− 1.02208i	0.859556	−	0.511041i	$-0.170740\pi$
$308$	6.00000i	0.341882i
$309$	1.00000	0.0568880
$310$	0	0
$311$	15.9083	0.902078	0.451039	−	0.892504i	$-0.351053\pi$
$311$	15.9083	0.902078	0.451039	+	0.892504i	$0.351053\pi$
$312$	4.30278i	0.243597i
$313$	− 9.02776i	− 0.510279i	−0.966904	−	0.255139i	$-0.917879\pi$
$313$	− 9.02776i	− 0.510279i	0.966904	−	0.255139i	$-0.0821214\pi$
$314$	−7.21110	−0.406946
$315$	0	0
$316$	9.11943	0.513008
$317$	− 9.21110i	− 0.517347i	−0.965965	−	0.258674i	$-0.916715\pi$
$317$	− 9.21110i	− 0.517347i	0.965965	−	0.258674i	$-0.0832854\pi$
$318$	19.8167i	1.11126i
$319$	−9.00000	−0.503903
$320$	0	0
$321$	2.30278	0.128528
$322$	− 10.1833i	− 0.567496i
$323$	− 12.0000i	− 0.667698i
$324$	−29.8167	−1.65648
$325$	0	0
$326$	−8.42221	−0.466463
$327$	− 6.60555i	− 0.365288i
$328$	− 9.90833i	− 0.547096i
$329$	12.0000	0.661581
$330$	0	0
$331$	−13.2111	−0.726148	−0.363074	−	0.931760i	$-0.618273\pi$
$331$	−13.2111	−0.726148	−0.363074	+	0.931760i	$0.618273\pi$
$332$	− 2.78890i	− 0.153061i
$333$	7.90833i	0.433374i
$334$	−5.51388	−0.301706
$335$	0	0
$336$	−8.60555	−0.469471
$337$	− 6.11943i	− 0.333347i	−0.986012	−	0.166673i	$-0.946697\pi$
$337$	− 6.11943i	− 0.333347i	0.986012	−	0.166673i	$-0.0533025\pi$
$338$	11.3028i	0.614790i
$339$	10.6056	0.576014
$340$	0	0
$341$	0.697224	0.0377568
$342$	15.8167i	0.855267i
$343$	18.7889i	1.01451i
$344$	0.605551	0.0326491
$345$	0	0
$346$	8.78890	0.472494
$347$	− 10.1833i	− 0.546671i	−0.961919	−	0.273335i	$-0.911873\pi$
$347$	− 10.1833i	− 0.546671i	0.961919	−	0.273335i	$-0.0881269\pi$
$348$	− 12.9083i	− 0.691959i
$349$	−28.2389	−1.51159	−0.755796	−	0.654807i	$-0.772750\pi$
$349$	−28.2389	−1.51159	−0.755796	+	0.654807i	$0.772750\pi$
$350$	0	0
$351$	21.1194	1.12727
$352$	− 2.30278i	− 0.122738i
$353$	10.1833i	0.542005i	0.962579	+	0.271002i	$0.0873551\pi$
$353$	10.1833i	0.542005i	−0.962579	+	0.271002i	$0.912645\pi$
$354$	35.0278	1.86170
$355$	0	0
$356$	−9.21110	−0.488187
$357$	− 51.6333i	− 2.73272i
$358$	13.8167i	0.730233i
$359$	−3.21110	−0.169476	−0.0847378	−	0.996403i	$-0.527005\pi$
$359$	−3.21110	−0.169476	−0.0847378	+	0.996403i	$0.527005\pi$
$360$	0	0
$361$	−15.0000	−0.789474
$362$	20.0000i	1.05118i
$363$	− 18.8167i	− 0.987618i
$364$	3.39445	0.177917
$365$	0	0
$366$	−24.8167	−1.29719
$367$	− 3.81665i	− 0.199228i	−0.995026	−	0.0996139i	$-0.968239\pi$
$367$	− 3.81665i	− 0.199228i	0.995026	−	0.0996139i	$-0.0317607\pi$
$368$	3.90833i	0.203736i
$369$	−78.3583	−4.07917
$370$	0	0
$371$	15.6333	0.811641
$372$	1.00000i	0.0518476i
$373$	− 17.8167i	− 0.922511i	−0.887267	−	0.461256i	$-0.847399\pi$
$373$	− 17.8167i	− 0.922511i	0.887267	−	0.461256i	$-0.152601\pi$
$374$	13.8167	0.714442
$375$	0	0
$376$	−4.60555	−0.237513
$377$	5.09167i	0.262235i
$378$	42.2389i	2.17253i
$379$	−24.3305	−1.24978	−0.624888	−	0.780715i	$-0.714855\pi$
$379$	−24.3305	−1.24978	−0.624888	+	0.780715i	$0.714855\pi$
$380$	0	0
$381$	−63.4500	−3.25064
$382$	− 5.51388i	− 0.282115i
$383$	− 36.8444i	− 1.88266i	−0.337486	−	0.941331i	$-0.609576\pi$
$383$	− 36.8444i	− 1.88266i	0.337486	−	0.941331i	$-0.390424\pi$
$384$	3.30278	0.168544
$385$	0	0
$386$	4.00000	0.203595
$387$	− 4.78890i	− 0.243433i
$388$	− 16.4222i	− 0.833711i
$389$	37.1194	1.88203	0.941015	−	0.338365i	$-0.109874\pi$
$389$	37.1194	1.88203	0.941015	+	0.338365i	$0.109874\pi$
$390$	0	0
$391$	−23.4500	−1.18592
$392$	− 0.211103i	− 0.0106623i
$393$	35.0278i	1.76692i
$394$	−6.00000	−0.302276
$395$	0	0
$396$	−18.2111	−0.915142
$397$	− 6.18335i	− 0.310333i	−0.987888	−	0.155167i	$-0.950409\pi$
$397$	− 6.18335i	− 0.310333i	0.987888	−	0.155167i	$-0.0495914\pi$
$398$	− 26.4222i	− 1.32443i
$399$	17.2111	0.861633
$400$	0	0
$401$	−7.81665	−0.390345	−0.195173	−	0.980769i	$-0.562527\pi$
$401$	−7.81665	−0.390345	−0.195173	+	0.980769i	$0.562527\pi$
$402$	− 11.6056i	− 0.578832i
$403$	− 0.394449i	− 0.0196489i
$404$	12.4222	0.618028
$405$	0	0
$406$	−10.1833	−0.505391
$407$	2.30278i	0.114144i
$408$	19.8167i	0.981071i
$409$	−31.0278	−1.53422	−0.767112	−	0.641513i	$-0.778307\pi$
$409$	−31.0278	−1.53422	−0.767112	+	0.641513i	$0.778307\pi$
$410$	0	0
$411$	3.00000	0.147979
$412$	0.302776i	0.0149167i
$413$	− 27.6333i	− 1.35975i
$414$	30.9083	1.51906
$415$	0	0
$416$	−1.30278	−0.0638738
$417$	6.30278i	0.308648i
$418$	4.60555i	0.225265i
$419$	−36.1472	−1.76591	−0.882953	−	0.469462i	$-0.844448\pi$
$419$	−36.1472	−1.76591	−0.882953	+	0.469462i	$0.844448\pi$
$420$	0	0
$421$	−3.72498	−0.181544	−0.0907722	−	0.995872i	$-0.528934\pi$
$421$	−3.72498	−0.181544	−0.0907722	+	0.995872i	$0.528934\pi$
$422$	10.3028i	0.501531i
$423$	36.4222i	1.77091i
$424$	−6.00000	−0.291386
$425$	0	0
$426$	−19.8167	−0.960120
$427$	19.5778i	0.947436i
$428$	0.697224i	0.0337016i
$429$	9.90833	0.478379
$430$	0	0
$431$	9.21110	0.443683	0.221842	−	0.975083i	$-0.428793\pi$
$431$	9.21110	0.443683	0.221842	+	0.975083i	$0.428793\pi$
$432$	− 16.2111i	− 0.779957i
$433$	34.9361i	1.67892i	0.543421	+	0.839461i	$0.317129\pi$
$433$	34.9361i	1.67892i	−0.543421	+	0.839461i	$0.682871\pi$
$434$	0.788897	0.0378683
$435$	0	0
$436$	2.00000	0.0957826
$437$	− 7.81665i	− 0.373921i
$438$	40.6333i	1.94153i
$439$	−30.3305	−1.44760	−0.723799	−	0.690011i	$-0.757606\pi$
$439$	−30.3305	−1.44760	−0.723799	+	0.690011i	$0.757606\pi$
$440$	0	0
$441$	−1.66947	−0.0794985
$442$	− 7.81665i	− 0.371800i
$443$	32.7250i	1.55481i	0.629000	+	0.777405i	$0.283465\pi$
$443$	32.7250i	1.55481i	−0.629000	+	0.777405i	$0.716535\pi$
$444$	−3.30278	−0.156743
$445$	0	0
$446$	5.81665	0.275427
$447$	− 65.4500i	− 3.09568i
$448$	− 2.60555i	− 0.123101i
$449$	15.2111	0.717856	0.358928	−	0.933365i	$-0.383142\pi$
$449$	15.2111	0.717856	0.358928	+	0.933365i	$0.383142\pi$
$450$	0	0
$451$	−22.8167	−1.07439
$452$	3.21110i	0.151038i
$453$	− 68.0555i	− 3.19753i
$454$	13.8167	0.648448
$455$	0	0
$456$	−6.60555	−0.309333
$457$	2.60555i	0.121883i	0.998141	+	0.0609413i	$0.0194102\pi$
$457$	2.60555i	0.121883i	−0.998141	+	0.0609413i	$0.980590\pi$
$458$	− 24.6056i	− 1.14974i
$459$	97.2666	4.54002
$460$	0	0
$461$	12.4222	0.578560	0.289280	−	0.957245i	$-0.406584\pi$
$461$	12.4222	0.578560	0.289280	+	0.957245i	$0.406584\pi$
$462$	19.8167i	0.921954i
$463$	26.6972i	1.24073i	0.784315	+	0.620363i	$0.213015\pi$
$463$	26.6972i	1.24073i	−0.784315	+	0.620363i	$0.786985\pi$
$464$	3.90833	0.181440
$465$	0	0
$466$	−8.51388	−0.394398
$467$	0	0	1.00000			$0$
$467$	0	0	−1.00000			$\pi$
$468$	10.3028i	0.476246i
$469$	−9.15559	−0.422766
$470$	0	0
$471$	−23.8167	−1.09741
$472$	10.6056i	0.488160i
$473$	− 1.39445i	− 0.0641168i
$474$	30.1194	1.38343
$475$	0	0
$476$	15.6333	0.716551
$477$	47.4500i	2.17258i
$478$	17.5139i	0.801066i
$479$	13.1194	0.599442	0.299721	−	0.954027i	$-0.403106\pi$
$479$	13.1194	0.599442	0.299721	+	0.954027i	$0.403106\pi$
$480$	0	0
$481$	1.30278	0.0594015
$482$	8.00000i	0.364390i
$483$	− 33.6333i	− 1.53037i
$484$	5.69722	0.258965
$485$	0	0
$486$	−49.8444	−2.26099
$487$	37.2111i	1.68620i	0.537760	+	0.843098i	$0.319271\pi$
$487$	37.2111i	1.68620i	−0.537760	+	0.843098i	$0.680729\pi$
$488$	− 7.51388i	− 0.340137i
$489$	−27.8167	−1.25791
$490$	0	0
$491$	17.7250	0.799917	0.399959	−	0.916533i	$-0.369025\pi$
$491$	17.7250	0.799917	0.399959	+	0.916533i	$0.369025\pi$
$492$	− 32.7250i	− 1.47536i
$493$	23.4500i	1.05613i
$494$	2.60555	0.117229
$495$	0	0
$496$	−0.302776	−0.0135950
$497$	15.6333i	0.701250i
$498$	− 9.21110i	− 0.412759i
$499$	42.2389	1.89087	0.945436	−	0.325809i	$-0.105637\pi$
$499$	42.2389	1.89087	0.945436	+	0.325809i	$0.105637\pi$
$500$	0	0
$501$	−18.2111	−0.813612
$502$	− 21.2111i	− 0.946698i
$503$	− 6.48612i	− 0.289202i	−0.989490	−	0.144601i	$-0.953810\pi$
$503$	− 6.48612i	− 0.289202i	0.989490	−	0.144601i	$-0.0461898\pi$
$504$	−20.6056	−0.917844
$505$	0	0
$506$	9.00000	0.400099
$507$	37.3305i	1.65791i
$508$	− 19.2111i	− 0.852355i
$509$	4.18335	0.185424	0.0927118	−	0.995693i	$-0.470446\pi$
$509$	4.18335	0.185424	0.0927118	+	0.995693i	$0.470446\pi$
$510$	0	0
$511$	32.0555	1.41805
$512$	1.00000i	0.0441942i
$513$	32.4222i	1.43148i
$514$	3.21110	0.141636
$515$	0	0
$516$	2.00000	0.0880451
$517$	10.6056i	0.466432i
$518$	2.60555i	0.114481i
$519$	29.0278	1.27418
$520$	0	0
$521$	33.6333	1.47350	0.736751	−	0.676164i	$-0.236359\pi$
$521$	33.6333	1.47350	0.736751	+	0.676164i	$0.236359\pi$
$522$	− 30.9083i	− 1.35282i
$523$	− 18.2389i	− 0.797530i	−0.917053	−	0.398765i	$-0.869439\pi$
$523$	− 18.2389i	− 0.797530i	0.917053	−	0.398765i	$-0.130561\pi$
$524$	−10.6056	−0.463306
$525$	0	0
$526$	−13.8167	−0.602435
$527$	− 1.81665i	− 0.0791347i
$528$	− 7.60555i	− 0.330989i
$529$	7.72498	0.335869
$530$	0	0
$531$	83.8722	3.63974
$532$	5.21110i	0.225930i
$533$	12.9083i	0.559122i
$534$	−30.4222	−1.31650
$535$	0	0
$536$	3.51388	0.151776
$537$	45.6333i	1.96922i
$538$	21.2111i	0.914476i
$539$	−0.486122	−0.0209387
$540$	0	0
$541$	25.9361	1.11508	0.557540	−	0.830150i	$-0.311745\pi$
$541$	25.9361	1.11508	0.557540	+	0.830150i	$0.311745\pi$
$542$	− 22.4222i	− 0.963116i
$543$	66.0555i	2.83471i
$544$	−6.00000	−0.257248
$545$	0	0
$546$	11.2111	0.479791
$547$	20.6056i	0.881030i	0.897745	+	0.440515i	$0.145204\pi$
$547$	20.6056i	0.881030i	−0.897745	+	0.440515i	$0.854796\pi$
$548$	0.908327i	0.0388018i
$549$	−59.4222	−2.53608
$550$	0	0
$551$	−7.81665	−0.333001
$552$	12.9083i	0.549415i
$553$	− 23.7611i	− 1.01043i
$554$	0.119429	0.00507407
$555$	0	0
$556$	−1.90833	−0.0809311
$557$	− 11.5139i	− 0.487859i	−0.969793	−	0.243929i	$-0.921564\pi$
$557$	− 11.5139i	− 0.487859i	0.969793	−	0.243929i	$-0.0784365\pi$
$558$	2.39445i	0.101365i
$559$	−0.788897	−0.0333668
$560$	0	0
$561$	45.6333	1.92664
$562$	− 12.0000i	− 0.506189i
$563$	− 28.0555i	− 1.18240i	−0.806525	−	0.591199i	$-0.798655\pi$
$563$	− 28.0555i	− 1.18240i	0.806525	−	0.591199i	$-0.201345\pi$
$564$	−15.2111	−0.640503
$565$	0	0
$566$	−24.6056	−1.03425
$567$	77.6888i	3.26262i
$568$	− 6.00000i	− 0.251754i
$569$	−18.4222	−0.772299	−0.386150	−	0.922436i	$-0.626195\pi$
$569$	−18.4222	−0.772299	−0.386150	+	0.922436i	$0.626195\pi$
$570$	0	0
$571$	−16.6972	−0.698757	−0.349379	−	0.936982i	$-0.613607\pi$
$571$	−16.6972	−0.698757	−0.349379	+	0.936982i	$0.613607\pi$
$572$	3.00000i	0.125436i
$573$	− 18.2111i	− 0.760780i
$574$	−25.8167	−1.07757
$575$	0	0
$576$	7.90833	0.329514
$577$	− 22.2389i	− 0.925816i	−0.886406	−	0.462908i	$-0.846806\pi$
$577$	− 22.2389i	− 0.925816i	0.886406	−	0.462908i	$-0.153194\pi$
$578$	− 19.0000i	− 0.790296i
$579$	13.2111	0.549035
$580$	0	0
$581$	−7.26662	−0.301470
$582$	− 54.2389i	− 2.24827i
$583$	13.8167i	0.572227i
$584$	−12.3028	−0.509092
$585$	0	0
$586$	−11.0278	−0.455552
$587$	45.6333i	1.88349i	0.336330	+	0.941744i	$0.390814\pi$
$587$	45.6333i	1.88349i	−0.336330	+	0.941744i	$0.609186\pi$
$588$	− 0.697224i	− 0.0287530i
$589$	0.605551	0.0249513
$590$	0	0
$591$	−19.8167	−0.815148
$592$	− 1.00000i	− 0.0410997i
$593$	18.4861i	0.759134i	0.925164	+	0.379567i	$0.123927\pi$
$593$	18.4861i	0.759134i	−0.925164	+	0.379567i	$0.876073\pi$
$594$	−37.3305	−1.53169
$595$	0	0
$596$	19.8167	0.811722
$597$	− 87.2666i	− 3.57158i
$598$	− 5.09167i	− 0.208214i
$599$	20.7889	0.849411	0.424706	−	0.905331i	$-0.360378\pi$
$599$	20.7889	0.849411	0.424706	+	0.905331i	$0.360378\pi$
$600$	0	0
$601$	−24.3028	−0.991331	−0.495665	−	0.868514i	$-0.665076\pi$
$601$	−24.3028	−0.991331	−0.495665	+	0.868514i	$0.665076\pi$
$602$	− 1.57779i	− 0.0643061i
$603$	− 27.7889i	− 1.13165i
$604$	20.6056	0.838428
$605$	0	0
$606$	41.0278	1.66664
$607$	13.4861i	0.547385i	0.961817	+	0.273692i	$0.0882450\pi$
$607$	13.4861i	0.547385i	−0.961817	+	0.273692i	$0.911755\pi$
$608$	− 2.00000i	− 0.0811107i
$609$	−33.6333	−1.36289
$610$	0	0
$611$	6.00000	0.242734
$612$	47.4500i	1.91805i
$613$	− 29.8167i	− 1.20428i	−0.798389	−	0.602142i	$-0.794314\pi$
$613$	− 29.8167i	− 1.20428i	0.798389	−	0.602142i	$-0.205686\pi$
$614$	17.9083	0.722721
$615$	0	0
$616$	−6.00000	−0.241747
$617$	42.5694i	1.71378i	0.515500	+	0.856890i	$0.327606\pi$
$617$	42.5694i	1.71378i	−0.515500	+	0.856890i	$0.672394\pi$
$618$	1.00000i	0.0402259i
$619$	6.30278	0.253330	0.126665	−	0.991946i	$-0.459573\pi$
$619$	6.30278	0.253330	0.126665	+	0.991946i	$0.459573\pi$
$620$	0	0
$621$	63.3583	2.54248
$622$	15.9083i	0.637866i
$623$	24.0000i	0.961540i
$624$	−4.30278	−0.172249
$625$	0	0
$626$	9.02776	0.360822
$627$	15.2111i	0.607473i
$628$	− 7.21110i	− 0.287754i
$629$	6.00000	0.239236
$630$	0	0
$631$	14.6972	0.585087	0.292544	−	0.956252i	$-0.405498\pi$
$631$	14.6972	0.585087	0.292544	+	0.956252i	$0.405498\pi$
$632$	9.11943i	0.362751i
$633$	34.0278i	1.35248i
$634$	9.21110	0.365820
$635$	0	0
$636$	−19.8167	−0.785781
$637$	0.275019i	0.0108967i
$638$	− 9.00000i	− 0.356313i
$639$	−47.4500	−1.87709
$640$	0	0
$641$	−20.5139	−0.810249	−0.405125	−	0.914261i	$-0.632772\pi$
$641$	−20.5139	−0.810249	−0.405125	+	0.914261i	$0.632772\pi$
$642$	2.30278i	0.0908833i
$643$	− 8.18335i	− 0.322720i	−0.986896	−	0.161360i	$-0.948412\pi$
$643$	− 8.18335i	− 0.322720i	0.986896	−	0.161360i	$-0.0515880\pi$
$644$	10.1833	0.401280
$645$	0	0
$646$	12.0000	0.472134
$647$	− 20.9361i	− 0.823082i	−0.911391	−	0.411541i	$-0.864991\pi$
$647$	− 20.9361i	− 0.823082i	0.911391	−	0.411541i	$-0.135009\pi$
$648$	− 29.8167i	− 1.17131i
$649$	24.4222	0.958655
$650$	0	0
$651$	2.60555	0.102120
$652$	− 8.42221i	− 0.329839i
$653$	− 3.90833i	− 0.152945i	−0.997072	−	0.0764723i	$-0.975634\pi$
$653$	− 3.90833i	− 0.152945i	0.997072	−	0.0764723i	$-0.0243657\pi$
$654$	6.60555	0.258297
$655$	0	0
$656$	9.90833	0.386855
$657$	97.2944i	3.79581i
$658$	12.0000i	0.467809i
$659$	16.8806	0.657574	0.328787	−	0.944404i	$-0.393360\pi$
$659$	16.8806	0.657574	0.328787	+	0.944404i	$0.393360\pi$
$660$	0	0
$661$	−30.5139	−1.18685	−0.593426	−	0.804888i	$-0.702225\pi$
$661$	−30.5139	−1.18685	−0.593426	+	0.804888i	$0.702225\pi$
$662$	− 13.2111i	− 0.513464i
$663$	− 25.8167i	− 1.00264i
$664$	2.78890	0.108230
$665$	0	0
$666$	−7.90833	−0.306441
$667$	15.2750i	0.591451i
$668$	− 5.51388i	− 0.213338i
$669$	19.2111	0.742744
$670$	0	0
$671$	−17.3028	−0.667966
$672$	− 8.60555i	− 0.331966i
$673$	20.6972i	0.797819i	0.916990	+	0.398910i	$0.130611\pi$
$673$	20.6972i	0.797819i	−0.916990	+	0.398910i	$0.869389\pi$
$674$	6.11943	0.235712
$675$	0	0
$676$	−11.3028	−0.434722
$677$	− 14.2389i	− 0.547244i	−0.961837	−	0.273622i	$-0.911778\pi$
$677$	− 14.2389i	− 0.547244i	0.961837	−	0.273622i	$-0.0882217\pi$
$678$	10.6056i	0.407304i
$679$	−42.7889	−1.64209
$680$	0	0
$681$	45.6333	1.74867
$682$	0.697224i	0.0266981i
$683$	12.0000i	0.459167i	0.973289	+	0.229584i	$0.0737364\pi$
$683$	12.0000i	0.459167i	−0.973289	+	0.229584i	$0.926264\pi$
$684$	−15.8167	−0.604765
$685$	0	0
$686$	−18.7889	−0.717363
$687$	− 81.2666i	− 3.10051i
$688$	0.605551i	0.0230864i
$689$	7.81665	0.297791
$690$	0	0
$691$	8.00000	0.304334	0.152167	−	0.988355i	$-0.451375\pi$
$691$	8.00000	0.304334	0.152167	+	0.988355i	$0.451375\pi$
$692$	8.78890i	0.334104i
$693$	47.4500i	1.80247i
$694$	10.1833	0.386555
$695$	0	0
$696$	12.9083	0.489289
$697$	59.4500i	2.25183i
$698$	− 28.2389i	− 1.06886i
$699$	−28.1194	−1.06357
$700$	0	0
$701$	40.1194	1.51529	0.757645	−	0.652667i	$-0.226350\pi$
$701$	40.1194	1.51529	0.757645	+	0.652667i	$0.226350\pi$
$702$	21.1194i	0.797101i
$703$	2.00000i	0.0754314i
$704$	2.30278	0.0867891
$705$	0	0
$706$	−10.1833	−0.383255
$707$	− 32.3667i	− 1.21727i
$708$	35.0278i	1.31642i
$709$	41.3305	1.55220	0.776100	−	0.630609i	$-0.217195\pi$
$709$	41.3305	1.55220	0.776100	+	0.630609i	$0.217195\pi$
$710$	0	0
$711$	72.1194	2.70469
$712$	− 9.21110i	− 0.345201i
$713$	− 1.18335i	− 0.0443167i
$714$	51.6333	1.93233
$715$	0	0
$716$	−13.8167	−0.516353
$717$	57.8444i	2.16024i
$718$	− 3.21110i	− 0.119837i
$719$	51.6333	1.92560	0.962799	−	0.270220i	$-0.0870963\pi$
$719$	51.6333	1.92560	0.962799	+	0.270220i	$0.0870963\pi$
$720$	0	0
$721$	0.788897	0.0293801
$722$	− 15.0000i	− 0.558242i
$723$	26.4222i	0.982652i
$724$	−20.0000	−0.743294
$725$	0	0
$726$	18.8167	0.698352
$727$	− 19.0917i	− 0.708071i	−0.935232	−	0.354035i	$-0.884809\pi$
$727$	− 19.0917i	− 0.708071i	0.935232	−	0.354035i	$-0.115191\pi$
$728$	3.39445i	0.125807i
$729$	−75.1749	−2.78426
$730$	0	0
$731$	−3.63331	−0.134383
$732$	− 24.8167i	− 0.917250i
$733$	− 13.6333i	− 0.503558i	−0.967785	−	0.251779i	$-0.918984\pi$
$733$	− 13.6333i	− 0.503558i	0.967785	−	0.251779i	$-0.0810156\pi$
$734$	3.81665	0.140875
$735$	0	0
$736$	−3.90833	−0.144063
$737$	− 8.09167i	− 0.298061i
$738$	− 78.3583i	− 2.88441i
$739$	2.66947	0.0981980	0.0490990	−	0.998794i	$-0.484365\pi$
$739$	2.66947	0.0981980	0.0490990	+	0.998794i	$0.484365\pi$
$740$	0	0
$741$	8.60555	0.316133
$742$	15.6333i	0.573917i
$743$	− 29.4500i	− 1.08041i	−0.841532	−	0.540207i	$-0.818346\pi$
$743$	− 29.4500i	− 1.08041i	0.841532	−	0.540207i	$-0.181654\pi$
$744$	−1.00000	−0.0366618
$745$	0	0
$746$	17.8167	0.652314
$747$	− 22.0555i	− 0.806969i
$748$	13.8167i	0.505187i
$749$	1.81665	0.0663791
$750$	0	0
$751$	14.0000	0.510867	0.255434	−	0.966827i	$-0.417782\pi$
$751$	14.0000	0.510867	0.255434	+	0.966827i	$0.417782\pi$
$752$	− 4.60555i	− 0.167947i
$753$	− 70.0555i	− 2.55296i
$754$	−5.09167	−0.185428
$755$	0	0
$756$	−42.2389	−1.53621
$757$	− 5.69722i	− 0.207069i	−0.994626	−	0.103535i	$-0.966985\pi$
$757$	− 5.69722i	− 0.207069i	0.994626	−	0.103535i	$-0.0330152\pi$
$758$	− 24.3305i	− 0.883725i
$759$	29.7250	1.07895
$760$	0	0
$761$	16.8806	0.611920	0.305960	−	0.952044i	$-0.401023\pi$
$761$	16.8806	0.611920	0.305960	+	0.952044i	$0.401023\pi$
$762$	− 63.4500i	− 2.29855i
$763$	− 5.21110i	− 0.188655i
$764$	5.51388	0.199485
$765$	0	0
$766$	36.8444	1.33124
$767$	− 13.8167i	− 0.498890i
$768$	3.30278i	0.119179i
$769$	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
$771$	10.6056	0.381950
$772$	4.00000i	0.143963i
$773$	22.0555i	0.793282i	0.917974	+	0.396641i	$0.129824\pi$
$773$	22.0555i	0.793282i	−0.917974	+	0.396641i	$0.870176\pi$
$774$	4.78890	0.172133
$775$	0	0
$776$	16.4222	0.589523
$777$	8.60555i	0.308722i
$778$	37.1194i	1.33080i
$779$	−19.8167	−0.710005
$780$	0	0
$781$	−13.8167	−0.494399
$782$	− 23.4500i	− 0.838569i
$783$	− 63.3583i	− 2.26424i
$784$	0.211103	0.00753938
$785$	0	0
$786$	−35.0278	−1.24940
$787$	− 10.7889i	− 0.384583i	−0.981338	−	0.192291i	$-0.938408\pi$
$787$	− 10.7889i	− 0.384583i	0.981338	−	0.192291i	$-0.0615919\pi$
$788$	− 6.00000i	− 0.213741i
$789$	−45.6333	−1.62459
$790$	0	0
$791$	8.36669	0.297485
$792$	− 18.2111i	− 0.647103i
$793$	9.78890i	0.347614i
$794$	6.18335	0.219439
$795$	0	0
$796$	26.4222	0.936510
$797$	− 22.3305i	− 0.790988i	−0.918469	−	0.395494i	$-0.870573\pi$
$797$	− 22.3305i	− 0.790988i	0.918469	−	0.395494i	$-0.129427\pi$
$798$	17.2111i	0.609266i
$799$	27.6333	0.977596
$800$	0	0
$801$	−72.8444	−2.57383
$802$	− 7.81665i	− 0.276016i
$803$	28.3305i	0.999763i
$804$	11.6056	0.409296
$805$	0	0
$806$	0.394449	0.0138939
$807$	70.0555i	2.46607i
$808$	12.4222i	0.437012i
$809$	35.4500	1.24635	0.623177	−	0.782081i	$-0.285842\pi$
$809$	35.4500	1.24635	0.623177	+	0.782081i	$0.285842\pi$
$810$	0	0
$811$	−7.14719	−0.250972	−0.125486	−	0.992095i	$-0.540049\pi$
$811$	−7.14719	−0.250972	−0.125486	+	0.992095i	$0.540049\pi$
$812$	− 10.1833i	− 0.357365i
$813$	− 74.0555i	− 2.59724i
$814$	−2.30278	−0.0807122
$815$	0	0
$816$	−19.8167	−0.693722
$817$	− 1.21110i	− 0.0423711i
$818$	− 31.0278i	− 1.08486i
$819$	26.8444	0.938020
$820$	0	0
$821$	−3.21110	−0.112068	−0.0560341	−	0.998429i	$-0.517846\pi$
$821$	−3.21110	−0.112068	−0.0560341	+	0.998429i	$0.517846\pi$
$822$	3.00000i	0.104637i
$823$	44.8444i	1.56318i	0.623794	+	0.781589i	$0.285590\pi$
$823$	44.8444i	1.56318i	−0.623794	+	0.781589i	$0.714410\pi$
$824$	−0.302776	−0.0105477
$825$	0	0
$826$	27.6333	0.961486
$827$	− 34.6056i	− 1.20335i	−0.798740	−	0.601676i	$-0.794500\pi$
$827$	− 34.6056i	− 1.20335i	0.798740	−	0.601676i	$-0.205500\pi$
$828$	30.9083i	1.07414i
$829$	27.7250	0.962928	0.481464	−	0.876466i	$-0.340105\pi$
$829$	27.7250	0.962928	0.481464	+	0.876466i	$0.340105\pi$
$830$	0	0
$831$	0.394449	0.0136833
$832$	− 1.30278i	− 0.0451656i
$833$	1.26662i	0.0438856i
$834$	−6.30278	−0.218247
$835$	0	0
$836$	−4.60555	−0.159286
$837$	4.90833i	0.169657i
$838$	− 36.1472i	− 1.24868i
$839$	12.9722	0.447852	0.223926	−	0.974606i	$-0.428113\pi$
$839$	12.9722	0.447852	0.223926	+	0.974606i	$0.428113\pi$
$840$	0	0
$841$	−13.7250	−0.473275
$842$	− 3.72498i	− 0.128371i
$843$	− 39.6333i	− 1.36504i
$844$	−10.3028	−0.354636
$845$	0	0
$846$	−36.4222	−1.25222
$847$	− 14.8444i	− 0.510060i
$848$	− 6.00000i	− 0.206041i
$849$	−81.2666	−2.78906
$850$	0	0
$851$	3.90833	0.133976
$852$	− 19.8167i	− 0.678907i
$853$	42.5416i	1.45660i	0.685260	+	0.728299i	$0.259689\pi$
$853$	42.5416i	1.45660i	−0.685260	+	0.728299i	$0.740311\pi$
$854$	−19.5778	−0.669938
$855$	0	0
$856$	−0.697224	−0.0238306
$857$	− 42.8444i	− 1.46354i	−0.681553	−	0.731769i	$-0.738695\pi$
$857$	− 42.8444i	− 1.46354i	0.681553	−	0.731769i	$-0.261305\pi$
$858$	9.90833i	0.338265i
$859$	−48.0555	−1.63963	−0.819816	−	0.572626i	$-0.805925\pi$
$859$	−48.0555	−1.63963	−0.819816	+	0.572626i	$0.805925\pi$
$860$	0	0
$861$	−85.2666	−2.90588
$862$	9.21110i	0.313731i
$863$	12.0000i	0.408485i	0.978920	+	0.204242i	$0.0654731\pi$
$863$	12.0000i	0.408485i	−0.978920	+	0.204242i	$0.934527\pi$
$864$	16.2111	0.551513
$865$	0	0
$866$	−34.9361	−1.18718
$867$	− 62.7527i	− 2.13119i
$868$	0.788897i	0.0267769i
$869$	21.0000	0.712376
$870$	0	0
$871$	−4.57779	−0.155113
$872$	2.00000i	0.0677285i
$873$	− 129.872i	− 4.39551i
$874$	7.81665	0.264402
$875$	0	0
$876$	−40.6333	−1.37287
$877$	7.21110i	0.243502i	0.992561	+	0.121751i	$0.0388509\pi$
$877$	7.21110i	0.243502i	−0.992561	+	0.121751i	$0.961149\pi$
$878$	− 30.3305i	− 1.02361i
$879$	−36.4222	−1.22849
$880$	0	0
$881$	−28.5416	−0.961592	−0.480796	−	0.876832i	$-0.659652\pi$
$881$	−28.5416	−0.961592	−0.480796	+	0.876832i	$0.659652\pi$
$882$	− 1.66947i	− 0.0562139i
$883$	26.4222i	0.889178i	0.895735	+	0.444589i	$0.146650\pi$
$883$	26.4222i	0.889178i	−0.895735	+	0.444589i	$0.853350\pi$
$884$	7.81665	0.262903
$885$	0	0
$886$	−32.7250	−1.09942
$887$	0.422205i	0.0141763i	0.999975	+	0.00708813i	$0.00225624\pi$
$887$	0.422205i	0.0141763i	−0.999975	+	0.00708813i	$0.997744\pi$
$888$	− 3.30278i	− 0.110834i
$889$	−50.0555	−1.67881
$890$	0	0
$891$	−68.6611	−2.30023
$892$	5.81665i	0.194756i
$893$	9.21110i	0.308238i
$894$	65.4500	2.18897
$895$	0	0
$896$	2.60555	0.0870454
$897$	− 16.8167i	− 0.561492i
$898$	15.2111i	0.507601i
$899$	−1.18335	−0.0394668
$900$	0	0
$901$	36.0000	1.19933
$902$	− 22.8167i	− 0.759711i
$903$	− 5.21110i	− 0.173415i
$904$	−3.21110	−0.106800
$905$	0	0
$906$	68.0555	2.26099
$907$	− 26.0000i	− 0.863316i	−0.902037	−	0.431658i	$-0.857929\pi$
$907$	− 26.0000i	− 0.863316i	0.902037	−	0.431658i	$-0.142071\pi$
$908$	13.8167i	0.458522i
$909$	98.2389	3.25838
$910$	0	0
$911$	−17.5778	−0.582378	−0.291189	−	0.956665i	$-0.594051\pi$
$911$	−17.5778	−0.582378	−0.291189	+	0.956665i	$0.594051\pi$
$912$	− 6.60555i	− 0.218732i
$913$	− 6.42221i	− 0.212544i
$914$	−2.60555	−0.0861840
$915$	0	0
$916$	24.6056	0.812990
$917$	27.6333i	0.912532i
$918$	97.2666i	3.21028i
$919$	9.57779	0.315942	0.157971	−	0.987444i	$-0.449505\pi$
$919$	9.57779	0.315942	0.157971	+	0.987444i	$0.449505\pi$
$920$	0	0
$921$	59.1472	1.94897
$922$	12.4222i	0.409104i
$923$	7.81665i	0.257288i
$924$	−19.8167	−0.651920
$925$	0	0
$926$	−26.6972	−0.877325
$927$	2.39445i	0.0786440i
$928$	3.90833i	0.128297i
$929$	18.4861	0.606510	0.303255	−	0.952909i	$-0.401927\pi$
$929$	18.4861	0.606510	0.303255	+	0.952909i	$0.401927\pi$
$930$	0	0
$931$	−0.422205	−0.0138372
$932$	− 8.51388i	− 0.278881i
$933$	52.5416i	1.72014i
$934$	0	0
$935$	0	0
$936$	−10.3028	−0.336757
$937$	18.0917i	0.591029i	0.955338	+	0.295515i	$0.0954911\pi$
$937$	18.0917i	0.591029i	−0.955338	+	0.295515i	$0.904509\pi$
$938$	− 9.15559i	− 0.298941i
$939$	29.8167	0.973030
$940$	0	0
$941$	13.8167	0.450410	0.225205	−	0.974311i	$-0.427695\pi$
$941$	13.8167	0.450410	0.225205	+	0.974311i	$0.427695\pi$
$942$	− 23.8167i	− 0.775989i
$943$	38.7250i	1.26106i
$944$	−10.6056	−0.345181
$945$	0	0
$946$	1.39445	0.0453374
$947$	3.63331i	0.118067i	0.998256	+	0.0590333i	$0.0188018\pi$
$947$	3.63331i	0.118067i	−0.998256	+	0.0590333i	$0.981198\pi$
$948$	30.1194i	0.978234i
$949$	16.0278	0.520283
$950$	0	0
$951$	30.4222	0.986508
$952$	15.6333i	0.506678i
$953$	49.7527i	1.61165i	0.592154	+	0.805825i	$0.298278\pi$
$953$	49.7527i	1.61165i	−0.592154	+	0.805825i	$0.701722\pi$
$954$	−47.4500	−1.53625
$955$	0	0
$956$	−17.5139	−0.566439
$957$	− 29.7250i	− 0.960872i
$958$	13.1194i	0.423870i
$959$	2.36669	0.0764245
$960$	0	0
$961$	−30.9083	−0.997043
$962$	1.30278i	0.0420032i
$963$	5.51388i	0.177682i
$964$	−8.00000	−0.257663
$965$	0	0
$966$	33.6333	1.08213
$967$	6.72498i	0.216261i	0.994137	+	0.108130i	$0.0344864\pi$
$967$	6.72498i	0.216261i	−0.994137	+	0.108130i	$0.965514\pi$
$968$	5.69722i	0.183116i
$969$	39.6333	1.27321
$970$	0	0
$971$	−22.5416	−0.723395	−0.361698	−	0.932295i	$-0.617803\pi$
$971$	−22.5416	−0.723395	−0.361698	+	0.932295i	$0.617803\pi$
$972$	− 49.8444i	− 1.59876i
$973$	4.97224i	0.159403i
$974$	−37.2111	−1.19232
$975$	0	0
$976$	7.51388	0.240513
$977$	− 18.0000i	− 0.575871i	−0.957650	−	0.287936i	$-0.907031\pi$
$977$	− 18.0000i	− 0.575871i	0.957650	−	0.287936i	$-0.0929689\pi$
$978$	− 27.8167i	− 0.889479i
$979$	−21.2111	−0.677910
$980$	0	0
$981$	15.8167	0.504987
$982$	17.7250i	0.565627i
$983$	− 12.0000i	− 0.382741i	−0.981518	−	0.191370i	$-0.938707\pi$
$983$	− 12.0000i	− 0.382741i	0.981518	−	0.191370i	$-0.0612931\pi$
$984$	32.7250	1.04323
$985$	0	0
$986$	−23.4500	−0.746799
$987$	39.6333i	1.26154i
$988$	2.60555i	0.0828936i
$989$	−2.36669	−0.0752564
$990$	0	0
$991$	50.6972	1.61045	0.805225	−	0.592969i	$-0.202044\pi$
$991$	50.6972	1.61045	0.805225	+	0.592969i	$0.202044\pi$
$992$	− 0.302776i	− 0.00961314i
$993$	− 43.6333i	− 1.38466i
$994$	−15.6333	−0.495858
$995$	0	0
$996$	9.21110	0.291865
$997$	52.4222i	1.66023i	0.557594	+	0.830114i	$0.311725\pi$
$997$	52.4222i	1.66023i	−0.557594	+	0.830114i	$0.688275\pi$
$998$	42.2389i	1.33705i
$999$	−16.2111	−0.512897

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1850.2.b.i.149.4		4
5.2	odd	4		74.2.a.a.1.2	✓	2
5.3	odd	4		1850.2.a.u.1.1		2
5.4	even	2	inner	1850.2.b.i.149.1		4
15.2	even	4		666.2.a.j.1.2		2
20.7	even	4		592.2.a.f.1.1		2
35.27	even	4		3626.2.a.a.1.1		2
40.27	even	4		2368.2.a.ba.1.2		2
40.37	odd	4		2368.2.a.s.1.1		2
55.32	even	4		8954.2.a.p.1.2		2
60.47	odd	4		5328.2.a.bf.1.2		2
185.147	odd	4		2738.2.a.l.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
74.2.a.a.1.2	✓	2	5.2	odd	4
592.2.a.f.1.1		2	20.7	even	4
666.2.a.j.1.2		2	15.2	even	4
1850.2.a.u.1.1		2	5.3	odd	4
1850.2.b.i.149.1		4	5.4	even	2	inner
1850.2.b.i.149.4		4	1.1	even	1	trivial
2368.2.a.s.1.1		2	40.37	odd	4
2368.2.a.ba.1.2		2	40.27	even	4
2738.2.a.l.1.2		2	185.147	odd	4
3626.2.a.a.1.1		2	35.27	even	4
5328.2.a.bf.1.2		2	60.47	odd	4
8954.2.a.p.1.2		2	55.32	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1850 = 2 \cdot 5^{2} \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1850.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} +3.30278i q^{3} -1.00000 q^{4} -3.30278 q^{6} +2.60555i q^{7} -1.00000i q^{8} -7.90833 q^{9} -2.30278 q^{11} -3.30278i q^{12} +1.30278i q^{13} -2.60555 q^{14} +1.00000 q^{16} +6.00000i q^{17} -7.90833i q^{18} -2.00000 q^{19} -8.60555 q^{21} -2.30278i q^{22} +3.90833i q^{23} +3.30278 q^{24} -1.30278 q^{26} -16.2111i q^{27} -2.60555i q^{28} +3.90833 q^{29} -0.302776 q^{31} +1.00000i q^{32} -7.60555i q^{33} -6.00000 q^{34} +7.90833 q^{36} -1.00000i q^{37} -2.00000i q^{38} -4.30278 q^{39} +9.90833 q^{41} -8.60555i q^{42} +0.605551i q^{43} +2.30278 q^{44} -3.90833 q^{46} -4.60555i q^{47} +3.30278i q^{48} +0.211103 q^{49} -19.8167 q^{51} -1.30278i q^{52} -6.00000i q^{53} +16.2111 q^{54} +2.60555 q^{56} -6.60555i q^{57} +3.90833i q^{58} -10.6056 q^{59} +7.51388 q^{61} -0.302776i q^{62} -20.6056i q^{63} -1.00000 q^{64} +7.60555 q^{66} +3.51388i q^{67} -6.00000i q^{68} -12.9083 q^{69} +6.00000 q^{71} +7.90833i q^{72} -12.3028i q^{73} +1.00000 q^{74} +2.00000 q^{76} -6.00000i q^{77} -4.30278i q^{78} -9.11943 q^{79} +29.8167 q^{81} +9.90833i q^{82} +2.78890i q^{83} +8.60555 q^{84} -0.605551 q^{86} +12.9083i q^{87} +2.30278i q^{88} +9.21110 q^{89} -3.39445 q^{91} -3.90833i q^{92} -1.00000i q^{93} +4.60555 q^{94} -3.30278 q^{96} +16.4222i q^{97} +0.211103i q^{98} +18.2111 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{4} - 6 q^{6} - 10 q^{9} - 2 q^{11} + 4 q^{14} + 4 q^{16} - 8 q^{19} - 20 q^{21} + 6 q^{24} + 2 q^{26} - 6 q^{29} + 6 q^{31} - 24 q^{34} + 10 q^{36} - 10 q^{39} + 18 q^{41} + 2 q^{44} + 6 q^{46}+ \cdots + 44 q^{99}+O(q^{100}) \) 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

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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