LMFDB - Embedded newform 1900.2.c.d.1749.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1900,2,Mod(1749,1900)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1900, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("1900.1749");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1900 = 2^{2} \cdot 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1900.c (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:	$15.1715763840$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{8})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 1 $ x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 380)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1749.1
Root			$-0.707107 + 0.707107i$ of defining polynomial
Character	$\chi$	$=$	1900.1749
Dual form			1900.2.c.d.1749.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.41421i q^{3} -0.828427i q^{7} -8.65685 q^{9} +O(q^{10})$	$q-3.41421i q^{3} -0.828427i q^{7} -8.65685 q^{9} -2.00000 q^{11} -6.24264i q^{13} -0.828427i q^{17} +1.00000 q^{19} -2.82843 q^{21} -6.00000i q^{23} +19.3137i q^{27} +6.48528 q^{29} -6.82843 q^{31} +6.82843i q^{33} +1.75736i q^{37} -21.3137 q^{39} +3.65685 q^{41} +4.82843i q^{43} +4.82843i q^{47} +6.31371 q^{49} -2.82843 q^{51} +9.07107i q^{53} -3.41421i q^{57} -13.6569 q^{59} -13.6569 q^{61} +7.17157i q^{63} +3.41421i q^{67} -20.4853 q^{69} +5.17157 q^{71} -2.48528i q^{73} +1.65685i q^{77} -1.65685 q^{79} +39.9706 q^{81} -13.3137i q^{83} -22.1421i q^{87} +6.48528 q^{89} -5.17157 q^{91} +23.3137i q^{93} +10.2426i q^{97} +17.3137 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 12 q^{9}+O(q^{10}) $ 4 * q - 12 * q^9	$ 4 q - 12 q^{9} - 8 q^{11} + 4 q^{19} - 8 q^{29} - 16 q^{31} - 40 q^{39} - 8 q^{41} - 20 q^{49} - 32 q^{59} - 32 q^{61} - 48 q^{69} + 32 q^{71} + 16 q^{79} + 92 q^{81} - 8 q^{89} - 32 q^{91} + 24 q^{99}+O(q^{100}) $ 4 * q - 12 * q^9 - 8 * q^11 + 4 * q^19 - 8 * q^29 - 16 * q^31 - 40 * q^39 - 8 * q^41 - 20 * q^49 - 32 * q^59 - 32 * q^61 - 48 * q^69 + 32 * q^71 + 16 * q^79 + 92 * q^81 - 8 * q^89 - 32 * q^91 + 24 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1900\mathbb{Z}\right)^\times$.

$n$	$77$	$401$	$951$
$\chi(n)$	$-1$	$1$	$1$

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 $

$2$	0	0
$2$	0	0
$3$	− 3.41421i	− 1.97120i	−0.169102	−	0.985599i	$-0.554087\pi$
$3$	− 3.41421i	− 1.97120i	0.169102	−	0.985599i	$-0.445913\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 0.828427i	− 0.313116i	−0.987669	−	0.156558i	$-0.949960\pi$
$7$	− 0.828427i	− 0.313116i	0.987669	−	0.156558i	$-0.0500398\pi$
$8$	0	0
$9$	−8.65685	−2.88562
$10$	0	0
$11$	−2.00000	−0.603023	−0.301511	−	0.953463i	$-0.597491\pi$
$11$	−2.00000	−0.603023	−0.301511	+	0.953463i	$0.597491\pi$
$12$	0	0
$13$	− 6.24264i	− 1.73140i	−0.500566	−	0.865699i	$-0.666875\pi$
$13$	− 6.24264i	− 1.73140i	0.500566	−	0.865699i	$-0.333125\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 0.828427i	− 0.200923i	−0.994941	−	0.100462i	$-0.967968\pi$
$17$	− 0.828427i	− 0.200923i	0.994941	−	0.100462i	$-0.0320319\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	−2.82843	−0.617213
$22$	0	0
$23$	− 6.00000i	− 1.25109i	−0.780189	−	0.625543i	$-0.784877\pi$
$23$	− 6.00000i	− 1.25109i	0.780189	−	0.625543i	$-0.215123\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	19.3137i	3.71692i
$28$	0	0
$29$	6.48528	1.20429	0.602143	−	0.798388i	$-0.294314\pi$
$29$	6.48528	1.20429	0.602143	+	0.798388i	$0.294314\pi$
$30$	0	0
$31$	−6.82843	−1.22642	−0.613211	−	0.789919i	$-0.710122\pi$
$31$	−6.82843	−1.22642	−0.613211	+	0.789919i	$0.710122\pi$
$32$	0	0
$33$	6.82843i	1.18868i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	1.75736i	0.288908i	0.989512	+	0.144454i	$0.0461426\pi$
$37$	1.75736i	0.288908i	−0.989512	+	0.144454i	$0.953857\pi$
$38$	0	0
$39$	−21.3137	−3.41292
$40$	0	0
$41$	3.65685	0.571105	0.285552	−	0.958363i	$-0.407823\pi$
$41$	3.65685	0.571105	0.285552	+	0.958363i	$0.407823\pi$
$42$	0	0
$43$	4.82843i	0.736328i	0.929761	+	0.368164i	$0.120014\pi$
$43$	4.82843i	0.736328i	−0.929761	+	0.368164i	$0.879986\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	4.82843i	0.704298i	0.935944	+	0.352149i	$0.114549\pi$
$47$	4.82843i	0.704298i	−0.935944	+	0.352149i	$0.885451\pi$
$48$	0	0
$49$	6.31371	0.901958
$50$	0	0
$51$	−2.82843	−0.396059
$52$	0	0
$53$	9.07107i	1.24601i	0.782219	+	0.623003i	$0.214088\pi$
$53$	9.07107i	1.24601i	−0.782219	+	0.623003i	$0.785912\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	− 3.41421i	− 0.452224i
$58$	0	0
$59$	−13.6569	−1.77797	−0.888985	−	0.457935i	$-0.848589\pi$
$59$	−13.6569	−1.77797	−0.888985	+	0.457935i	$0.848589\pi$
$60$	0	0
$61$	−13.6569	−1.74858	−0.874291	−	0.485403i	$-0.838673\pi$
$61$	−13.6569	−1.74858	−0.874291	+	0.485403i	$0.838673\pi$
$62$	0	0
$63$	7.17157i	0.903533i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	3.41421i	0.417113i	0.978010	+	0.208556i	$0.0668764\pi$
$67$	3.41421i	0.417113i	−0.978010	+	0.208556i	$0.933124\pi$
$68$	0	0
$69$	−20.4853	−2.46614
$70$	0	0
$71$	5.17157	0.613753	0.306876	−	0.951749i	$-0.400716\pi$
$71$	5.17157	0.613753	0.306876	+	0.951749i	$0.400716\pi$
$72$	0	0
$73$	− 2.48528i	− 0.290880i	−0.989367	−	0.145440i	$-0.953540\pi$
$73$	− 2.48528i	− 0.290880i	0.989367	−	0.145440i	$-0.0464598\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	1.65685i	0.188816i
$78$	0	0
$79$	−1.65685	−0.186411	−0.0932053	−	0.995647i	$-0.529711\pi$
$79$	−1.65685	−0.186411	−0.0932053	+	0.995647i	$0.529711\pi$
$80$	0	0
$81$	39.9706	4.44117
$82$	0	0
$83$	− 13.3137i	− 1.46137i	−0.682715	−	0.730685i	$-0.739201\pi$
$83$	− 13.3137i	− 1.46137i	0.682715	−	0.730685i	$-0.260799\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	− 22.1421i	− 2.37389i
$88$	0	0
$89$	6.48528	0.687438	0.343719	−	0.939072i	$-0.388313\pi$
$89$	6.48528	0.687438	0.343719	+	0.939072i	$0.388313\pi$
$90$	0	0
$91$	−5.17157	−0.542128
$92$	0	0
$93$	23.3137i	2.41752i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	10.2426i	1.03998i	0.854172	+	0.519991i	$0.174065\pi$
$97$	10.2426i	1.03998i	−0.854172	+	0.519991i	$0.825935\pi$
$98$	0	0
$99$	17.3137	1.74009
$100$	0	0
$101$	4.00000	0.398015	0.199007	−	0.979998i	$-0.436228\pi$
$101$	4.00000	0.398015	0.199007	+	0.979998i	$0.436228\pi$
$102$	0	0
$103$	3.89949i	0.384229i	0.981373	+	0.192114i	$0.0615344\pi$
$103$	3.89949i	0.384229i	−0.981373	+	0.192114i	$0.938466\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 7.41421i	− 0.716759i	−0.933576	−	0.358380i	$-0.883329\pi$
$107$	− 7.41421i	− 0.716759i	0.933576	−	0.358380i	$-0.116671\pi$
$108$	0	0
$109$	3.17157	0.303782	0.151891	−	0.988397i	$-0.451464\pi$
$109$	3.17157	0.303782	0.151891	+	0.988397i	$0.451464\pi$
$110$	0	0
$111$	6.00000	0.569495
$112$	0	0
$113$	− 9.07107i	− 0.853334i	−0.904409	−	0.426667i	$-0.859688\pi$
$113$	− 9.07107i	− 0.853334i	0.904409	−	0.426667i	$-0.140312\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	54.0416i	4.99615i
$118$	0	0
$119$	−0.686292	−0.0629122
$120$	0	0
$121$	−7.00000	−0.636364
$122$	0	0
$123$	− 12.4853i	− 1.12576i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 9.55635i	− 0.847989i	−0.905665	−	0.423994i	$-0.860628\pi$
$127$	− 9.55635i	− 0.847989i	0.905665	−	0.423994i	$-0.139372\pi$
$128$	0	0
$129$	16.4853	1.45145
$130$	0	0
$131$	5.65685	0.494242	0.247121	−	0.968985i	$-0.420516\pi$
$131$	5.65685	0.494242	0.247121	+	0.968985i	$0.420516\pi$
$132$	0	0
$133$	− 0.828427i	− 0.0718337i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 8.14214i	− 0.695630i	−0.937563	−	0.347815i	$-0.886924\pi$
$137$	− 8.14214i	− 0.695630i	0.937563	−	0.347815i	$-0.113076\pi$
$138$	0	0
$139$	−5.31371	−0.450703	−0.225351	−	0.974278i	$-0.572353\pi$
$139$	−5.31371	−0.450703	−0.225351	+	0.974278i	$0.572353\pi$
$140$	0	0
$141$	16.4853	1.38831
$142$	0	0
$143$	12.4853i	1.04407i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 21.5563i	− 1.77794i
$148$	0	0
$149$	8.00000	0.655386	0.327693	−	0.944784i	$-0.393729\pi$
$149$	8.00000	0.655386	0.327693	+	0.944784i	$0.393729\pi$
$150$	0	0
$151$	−10.1421	−0.825355	−0.412678	−	0.910877i	$-0.635406\pi$
$151$	−10.1421	−0.825355	−0.412678	+	0.910877i	$0.635406\pi$
$152$	0	0
$153$	7.17157i	0.579787i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 8.82843i	− 0.704585i	−0.935890	−	0.352293i	$-0.885402\pi$
$157$	− 8.82843i	− 0.704585i	0.935890	−	0.352293i	$-0.114598\pi$
$158$	0	0
$159$	30.9706	2.45613
$160$	0	0
$161$	−4.97056	−0.391735
$162$	0	0
$163$	14.4853i	1.13457i	0.823520	+	0.567287i	$0.192007\pi$
$163$	14.4853i	1.13457i	−0.823520	+	0.567287i	$0.807993\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 14.7279i	− 1.13968i	−0.821755	−	0.569840i	$-0.807005\pi$
$167$	− 14.7279i	− 1.13968i	0.821755	−	0.569840i	$-0.192995\pi$
$168$	0	0
$169$	−25.9706	−1.99774
$170$	0	0
$171$	−8.65685	−0.662006
$172$	0	0
$173$	15.8995i	1.20882i	0.796675	+	0.604408i	$0.206590\pi$
$173$	15.8995i	1.20882i	−0.796675	+	0.604408i	$0.793410\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	46.6274i	3.50473i
$178$	0	0
$179$	−10.3431	−0.773083	−0.386542	−	0.922272i	$-0.626330\pi$
$179$	−10.3431	−0.773083	−0.386542	+	0.922272i	$0.626330\pi$
$180$	0	0
$181$	−18.0000	−1.33793	−0.668965	−	0.743294i	$-0.733262\pi$
$181$	−18.0000	−1.33793	−0.668965	+	0.743294i	$0.733262\pi$
$182$	0	0
$183$	46.6274i	3.44680i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	1.65685i	0.121161i
$188$	0	0
$189$	16.0000	1.16383
$190$	0	0
$191$	4.00000	0.289430	0.144715	−	0.989473i	$-0.453773\pi$
$191$	4.00000	0.289430	0.144715	+	0.989473i	$0.453773\pi$
$192$	0	0
$193$	3.41421i	0.245760i	0.992422	+	0.122880i	$0.0392131\pi$
$193$	3.41421i	0.245760i	−0.992422	+	0.122880i	$0.960787\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 17.3137i	− 1.23355i	−0.787139	−	0.616775i	$-0.788439\pi$
$197$	− 17.3137i	− 1.23355i	0.787139	−	0.616775i	$-0.211561\pi$
$198$	0	0
$199$	21.6569	1.53521	0.767607	−	0.640921i	$-0.221447\pi$
$199$	21.6569	1.53521	0.767607	+	0.640921i	$0.221447\pi$
$200$	0	0
$201$	11.6569	0.822211
$202$	0	0
$203$	− 5.37258i	− 0.377081i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	51.9411i	3.61016i
$208$	0	0
$209$	−2.00000	−0.138343
$210$	0	0
$211$	−28.4853	−1.96101	−0.980504	−	0.196500i	$-0.937042\pi$
$211$	−28.4853	−1.96101	−0.980504	+	0.196500i	$0.937042\pi$
$212$	0	0
$213$	− 17.6569i	− 1.20983i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	5.65685i	0.384012i
$218$	0	0
$219$	−8.48528	−0.573382
$220$	0	0
$221$	−5.17157	−0.347878
$222$	0	0
$223$	− 23.2132i	− 1.55447i	−0.629210	−	0.777236i	$-0.716621\pi$
$223$	− 23.2132i	− 1.55447i	0.629210	−	0.777236i	$-0.283379\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 23.4142i	− 1.55406i	−0.629466	−	0.777028i	$-0.716726\pi$
$227$	− 23.4142i	− 1.55406i	0.629466	−	0.777028i	$-0.283274\pi$
$228$	0	0
$229$	10.3431	0.683494	0.341747	−	0.939792i	$-0.388981\pi$
$229$	10.3431	0.683494	0.341747	+	0.939792i	$0.388981\pi$
$230$	0	0
$231$	5.65685	0.372194
$232$	0	0
$233$	10.0000i	0.655122i	0.944830	+	0.327561i	$0.106227\pi$
$233$	10.0000i	0.655122i	−0.944830	+	0.327561i	$0.893773\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	5.65685i	0.367452i
$238$	0	0
$239$	−21.6569	−1.40087	−0.700433	−	0.713718i	$-0.747010\pi$
$239$	−21.6569	−1.40087	−0.700433	+	0.713718i	$0.747010\pi$
$240$	0	0
$241$	26.2843	1.69312	0.846559	−	0.532294i	$-0.178670\pi$
$241$	26.2843	1.69312	0.846559	+	0.532294i	$0.178670\pi$
$242$	0	0
$243$	− 78.5269i	− 5.03750i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 6.24264i	− 0.397210i
$248$	0	0
$249$	−45.4558	−2.88065
$250$	0	0
$251$	−2.34315	−0.147898	−0.0739490	−	0.997262i	$-0.523560\pi$
$251$	−2.34315	−0.147898	−0.0739490	+	0.997262i	$0.523560\pi$
$252$	0	0
$253$	12.0000i	0.754434i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 6.24264i	− 0.389405i	−0.980862	−	0.194703i	$-0.937626\pi$
$257$	− 6.24264i	− 0.389405i	0.980862	−	0.194703i	$-0.0623742\pi$
$258$	0	0
$259$	1.45584	0.0904618
$260$	0	0
$261$	−56.1421	−3.47511
$262$	0	0
$263$	− 7.65685i	− 0.472142i	−0.971736	−	0.236071i	$-0.924140\pi$
$263$	− 7.65685i	− 0.472142i	0.971736	−	0.236071i	$-0.0758598\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	− 22.1421i	− 1.35508i
$268$	0	0
$269$	5.51472	0.336238	0.168119	−	0.985767i	$-0.446231\pi$
$269$	5.51472	0.336238	0.168119	+	0.985767i	$0.446231\pi$
$270$	0	0
$271$	12.3431	0.749793	0.374896	−	0.927067i	$-0.377678\pi$
$271$	12.3431	0.749793	0.374896	+	0.927067i	$0.377678\pi$
$272$	0	0
$273$	17.6569i	1.06864i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 11.1716i	− 0.671235i	−0.941998	−	0.335617i	$-0.891055\pi$
$277$	− 11.1716i	− 0.671235i	0.941998	−	0.335617i	$-0.108945\pi$
$278$	0	0
$279$	59.1127	3.53898
$280$	0	0
$281$	−14.9706	−0.893069	−0.446534	−	0.894766i	$-0.647342\pi$
$281$	−14.9706	−0.893069	−0.446534	+	0.894766i	$0.647342\pi$
$282$	0	0
$283$	− 25.3137i	− 1.50474i	−0.658739	−	0.752372i	$-0.728910\pi$
$283$	− 25.3137i	− 1.50474i	0.658739	−	0.752372i	$-0.271090\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 3.02944i	− 0.178822i
$288$	0	0
$289$	16.3137	0.959630
$290$	0	0
$291$	34.9706	2.05001
$292$	0	0
$293$	− 20.5858i	− 1.20263i	−0.799010	−	0.601317i	$-0.794643\pi$
$293$	− 20.5858i	− 1.20263i	0.799010	−	0.601317i	$-0.205357\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	− 38.6274i	− 2.24139i
$298$	0	0
$299$	−37.4558	−2.16613
$300$	0	0
$301$	4.00000	0.230556
$302$	0	0
$303$	− 13.6569i	− 0.784566i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	11.8995i	0.679140i	0.940581	+	0.339570i	$0.110282\pi$
$307$	11.8995i	0.679140i	−0.940581	+	0.339570i	$0.889718\pi$
$308$	0	0
$309$	13.3137	0.757390
$310$	0	0
$311$	−14.0000	−0.793867	−0.396934	−	0.917847i	$-0.629926\pi$
$311$	−14.0000	−0.793867	−0.396934	+	0.917847i	$0.629926\pi$
$312$	0	0
$313$	18.0000i	1.01742i	0.860938	+	0.508710i	$0.169877\pi$
$313$	18.0000i	1.01742i	−0.860938	+	0.508710i	$0.830123\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	14.2426i	0.799946i	0.916527	+	0.399973i	$0.130981\pi$
$317$	14.2426i	0.799946i	−0.916527	+	0.399973i	$0.869019\pi$
$318$	0	0
$319$	−12.9706	−0.726212
$320$	0	0
$321$	−25.3137	−1.41287
$322$	0	0
$323$	− 0.828427i	− 0.0460949i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 10.8284i	− 0.598813i
$328$	0	0
$329$	4.00000	0.220527
$330$	0	0
$331$	6.82843	0.375324	0.187662	−	0.982234i	$-0.439909\pi$
$331$	6.82843	0.375324	0.187662	+	0.982234i	$0.439909\pi$
$332$	0	0
$333$	− 15.2132i	− 0.833678i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 26.2426i	− 1.42953i	−0.699366	−	0.714764i	$-0.746534\pi$
$337$	− 26.2426i	− 1.42953i	0.699366	−	0.714764i	$-0.253466\pi$
$338$	0	0
$339$	−30.9706	−1.68209
$340$	0	0
$341$	13.6569	0.739560
$342$	0	0
$343$	− 11.0294i	− 0.595534i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	10.9706i	0.588931i	0.955662	+	0.294465i	$0.0951416\pi$
$347$	10.9706i	0.588931i	−0.955662	+	0.294465i	$0.904858\pi$
$348$	0	0
$349$	11.6569	0.623977	0.311989	−	0.950086i	$-0.399005\pi$
$349$	11.6569	0.623977	0.311989	+	0.950086i	$0.399005\pi$
$350$	0	0
$351$	120.569	6.43547
$352$	0	0
$353$	− 32.1421i	− 1.71075i	−0.518007	−	0.855377i	$-0.673326\pi$
$353$	− 32.1421i	− 1.71075i	0.518007	−	0.855377i	$-0.326674\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	2.34315i	0.124012i
$358$	0	0
$359$	−1.31371	−0.0693349	−0.0346674	−	0.999399i	$-0.511037\pi$
$359$	−1.31371	−0.0693349	−0.0346674	+	0.999399i	$0.511037\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	23.8995i	1.25440i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 0.142136i	− 0.00741942i	−0.999993	−	0.00370971i	$-0.998819\pi$
$367$	− 0.142136i	− 0.00741942i	0.999993	−	0.00370971i	$-0.00118084\pi$
$368$	0	0
$369$	−31.6569	−1.64799
$370$	0	0
$371$	7.51472	0.390145
$372$	0	0
$373$	− 21.5563i	− 1.11615i	−0.829792	−	0.558073i	$-0.811541\pi$
$373$	− 21.5563i	− 1.11615i	0.829792	−	0.558073i	$-0.188459\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 40.4853i	− 2.08510i
$378$	0	0
$379$	−33.6569	−1.72884	−0.864418	−	0.502773i	$-0.832313\pi$
$379$	−33.6569	−1.72884	−0.864418	+	0.502773i	$0.832313\pi$
$380$	0	0
$381$	−32.6274	−1.67155
$382$	0	0
$383$	17.0711i	0.872291i	0.899876	+	0.436145i	$0.143657\pi$
$383$	17.0711i	0.872291i	−0.899876	+	0.436145i	$0.856343\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 41.7990i	− 2.12476i
$388$	0	0
$389$	−16.6274	−0.843044	−0.421522	−	0.906818i	$-0.638504\pi$
$389$	−16.6274	−0.843044	−0.421522	+	0.906818i	$0.638504\pi$
$390$	0	0
$391$	−4.97056	−0.251372
$392$	0	0
$393$	− 19.3137i	− 0.974248i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	3.85786i	0.193621i	0.995303	+	0.0968103i	$0.0308640\pi$
$397$	3.85786i	0.193621i	−0.995303	+	0.0968103i	$0.969136\pi$
$398$	0	0
$399$	−2.82843	−0.141598
$400$	0	0
$401$	−29.3137	−1.46386	−0.731928	−	0.681382i	$-0.761379\pi$
$401$	−29.3137	−1.46386	−0.731928	+	0.681382i	$0.761379\pi$
$402$	0	0
$403$	42.6274i	2.12342i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 3.51472i	− 0.174218i
$408$	0	0
$409$	−26.4853	−1.30961	−0.654806	−	0.755797i	$-0.727250\pi$
$409$	−26.4853	−1.30961	−0.654806	+	0.755797i	$0.727250\pi$
$410$	0	0
$411$	−27.7990	−1.37122
$412$	0	0
$413$	11.3137i	0.556711i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	18.1421i	0.888424i
$418$	0	0
$419$	1.65685	0.0809426	0.0404713	−	0.999181i	$-0.487114\pi$
$419$	1.65685	0.0809426	0.0404713	+	0.999181i	$0.487114\pi$
$420$	0	0
$421$	−19.6569	−0.958016	−0.479008	−	0.877810i	$-0.659004\pi$
$421$	−19.6569	−0.958016	−0.479008	+	0.877810i	$0.659004\pi$
$422$	0	0
$423$	− 41.7990i	− 2.03234i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	11.3137i	0.547509i
$428$	0	0
$429$	42.6274	2.05807
$430$	0	0
$431$	−17.4558	−0.840818	−0.420409	−	0.907335i	$-0.638113\pi$
$431$	−17.4558	−0.840818	−0.420409	+	0.907335i	$0.638113\pi$
$432$	0	0
$433$	27.2132i	1.30778i	0.756588	+	0.653892i	$0.226865\pi$
$433$	27.2132i	1.30778i	−0.756588	+	0.653892i	$0.773135\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 6.00000i	− 0.287019i
$438$	0	0
$439$	18.3431	0.875471	0.437735	−	0.899104i	$-0.355781\pi$
$439$	18.3431	0.875471	0.437735	+	0.899104i	$0.355781\pi$
$440$	0	0
$441$	−54.6569	−2.60271
$442$	0	0
$443$	− 22.2843i	− 1.05876i	−0.848386	−	0.529379i	$-0.822425\pi$
$443$	− 22.2843i	− 1.05876i	0.848386	−	0.529379i	$-0.177575\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	− 27.3137i	− 1.29189i
$448$	0	0
$449$	30.4853	1.43869	0.719345	−	0.694653i	$-0.244442\pi$
$449$	30.4853	1.43869	0.719345	+	0.694653i	$0.244442\pi$
$450$	0	0
$451$	−7.31371	−0.344389
$452$	0	0
$453$	34.6274i	1.62694i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	14.9706i	0.700293i	0.936695	+	0.350147i	$0.113868\pi$
$457$	14.9706i	0.700293i	−0.936695	+	0.350147i	$0.886132\pi$
$458$	0	0
$459$	16.0000	0.746816
$460$	0	0
$461$	17.3137	0.806380	0.403190	−	0.915116i	$-0.367901\pi$
$461$	17.3137	0.806380	0.403190	+	0.915116i	$0.367901\pi$
$462$	0	0
$463$	13.3137i	0.618741i	0.950942	+	0.309370i	$0.100118\pi$
$463$	13.3137i	0.618741i	−0.950942	+	0.309370i	$0.899882\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	13.3137i	0.616085i	0.951373	+	0.308042i	$0.0996739\pi$
$467$	13.3137i	0.616085i	−0.951373	+	0.308042i	$0.900326\pi$
$468$	0	0
$469$	2.82843	0.130605
$470$	0	0
$471$	−30.1421	−1.38888
$472$	0	0
$473$	− 9.65685i	− 0.444023i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	− 78.5269i	− 3.59550i
$478$	0	0
$479$	10.0000	0.456912	0.228456	−	0.973554i	$-0.426632\pi$
$479$	10.0000	0.456912	0.228456	+	0.973554i	$0.426632\pi$
$480$	0	0
$481$	10.9706	0.500215
$482$	0	0
$483$	16.9706i	0.772187i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	12.3848i	0.561208i	0.959824	+	0.280604i	$0.0905347\pi$
$487$	12.3848i	0.561208i	−0.959824	+	0.280604i	$0.909465\pi$
$488$	0	0
$489$	49.4558	2.23647
$490$	0	0
$491$	12.6863	0.572524	0.286262	−	0.958151i	$-0.407587\pi$
$491$	12.6863	0.572524	0.286262	+	0.958151i	$0.407587\pi$
$492$	0	0
$493$	− 5.37258i	− 0.241969i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 4.28427i	− 0.192176i
$498$	0	0
$499$	−6.97056	−0.312045	−0.156023	−	0.987753i	$-0.549867\pi$
$499$	−6.97056	−0.312045	−0.156023	+	0.987753i	$0.549867\pi$
$500$	0	0
$501$	−50.2843	−2.24654
$502$	0	0
$503$	− 30.9706i	− 1.38091i	−0.723376	−	0.690455i	$-0.757411\pi$
$503$	− 30.9706i	− 1.38091i	0.723376	−	0.690455i	$-0.242589\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	88.6690i	3.93793i
$508$	0	0
$509$	−5.79899	−0.257036	−0.128518	−	0.991707i	$-0.541022\pi$
$509$	−5.79899	−0.257036	−0.128518	+	0.991707i	$0.541022\pi$
$510$	0	0
$511$	−2.05887	−0.0910792
$512$	0	0
$513$	19.3137i	0.852721i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 9.65685i	− 0.424708i
$518$	0	0
$519$	54.2843	2.38282
$520$	0	0
$521$	39.6569	1.73740	0.868699	−	0.495340i	$-0.164957\pi$
$521$	39.6569	1.73740	0.868699	+	0.495340i	$0.164957\pi$
$522$	0	0
$523$	18.7279i	0.818915i	0.912329	+	0.409457i	$0.134282\pi$
$523$	18.7279i	0.818915i	−0.912329	+	0.409457i	$0.865718\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	5.65685i	0.246416i
$528$	0	0
$529$	−13.0000	−0.565217
$530$	0	0
$531$	118.225	5.13055
$532$	0	0
$533$	− 22.8284i	− 0.988809i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	35.3137i	1.52390i
$538$	0	0
$539$	−12.6274	−0.543901
$540$	0	0
$541$	−11.3137	−0.486414	−0.243207	−	0.969974i	$-0.578199\pi$
$541$	−11.3137	−0.486414	−0.243207	+	0.969974i	$0.578199\pi$
$542$	0	0
$543$	61.4558i	2.63732i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	0.384776i	0.0164518i	0.999966	+	0.00822592i	$0.00261842\pi$
$547$	0.384776i	0.0164518i	−0.999966	+	0.00822592i	$0.997382\pi$
$548$	0	0
$549$	118.225	5.04574
$550$	0	0
$551$	6.48528	0.276282
$552$	0	0
$553$	1.37258i	0.0583682i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 2.20101i	− 0.0932598i	−0.998912	−	0.0466299i	$-0.985152\pi$
$557$	− 2.20101i	− 0.0932598i	0.998912	−	0.0466299i	$-0.0148481\pi$
$558$	0	0
$559$	30.1421	1.27488
$560$	0	0
$561$	5.65685	0.238833
$562$	0	0
$563$	− 8.58579i	− 0.361848i	−0.983497	−	0.180924i	$-0.942091\pi$
$563$	− 8.58579i	− 0.361848i	0.983497	−	0.180924i	$-0.0579088\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	− 33.1127i	− 1.39060i
$568$	0	0
$569$	4.82843	0.202418	0.101209	−	0.994865i	$-0.467729\pi$
$569$	4.82843	0.202418	0.101209	+	0.994865i	$0.467729\pi$
$570$	0	0
$571$	−37.3137	−1.56153	−0.780765	−	0.624825i	$-0.785170\pi$
$571$	−37.3137	−1.56153	−0.780765	+	0.624825i	$0.785170\pi$
$572$	0	0
$573$	− 13.6569i	− 0.570523i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	6.00000i	0.249783i	0.992170	+	0.124892i	$0.0398583\pi$
$577$	6.00000i	0.249783i	−0.992170	+	0.124892i	$0.960142\pi$
$578$	0	0
$579$	11.6569	0.484442
$580$	0	0
$581$	−11.0294	−0.457578
$582$	0	0
$583$	− 18.1421i	− 0.751370i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 18.9706i	− 0.782999i	−0.920178	−	0.391499i	$-0.871956\pi$
$587$	− 18.9706i	− 0.782999i	0.920178	−	0.391499i	$-0.128044\pi$
$588$	0	0
$589$	−6.82843	−0.281360
$590$	0	0
$591$	−59.1127	−2.43157
$592$	0	0
$593$	− 36.6274i	− 1.50411i	−0.659102	−	0.752054i	$-0.729063\pi$
$593$	− 36.6274i	− 1.50411i	0.659102	−	0.752054i	$-0.270937\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 73.9411i	− 3.02621i
$598$	0	0
$599$	27.3137	1.11601	0.558004	−	0.829838i	$-0.311567\pi$
$599$	27.3137	1.11601	0.558004	+	0.829838i	$0.311567\pi$
$600$	0	0
$601$	−18.0000	−0.734235	−0.367118	−	0.930175i	$-0.619655\pi$
$601$	−18.0000	−0.734235	−0.367118	+	0.930175i	$0.619655\pi$
$602$	0	0
$603$	− 29.5563i	− 1.20363i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	9.07107i	0.368183i	0.982909	+	0.184092i	$0.0589343\pi$
$607$	9.07107i	0.368183i	−0.982909	+	0.184092i	$0.941066\pi$
$608$	0	0
$609$	−18.3431	−0.743302
$610$	0	0
$611$	30.1421	1.21942
$612$	0	0
$613$	34.4853i	1.39285i	0.717631	+	0.696424i	$0.245227\pi$
$613$	34.4853i	1.39285i	−0.717631	+	0.696424i	$0.754773\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	45.7990i	1.84380i	0.387430	+	0.921899i	$0.373363\pi$
$617$	45.7990i	1.84380i	−0.387430	+	0.921899i	$0.626637\pi$
$618$	0	0
$619$	−18.0000	−0.723481	−0.361741	−	0.932279i	$-0.617817\pi$
$619$	−18.0000	−0.723481	−0.361741	+	0.932279i	$0.617817\pi$
$620$	0	0
$621$	115.882	4.65019
$622$	0	0
$623$	− 5.37258i	− 0.215248i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	6.82843i	0.272701i
$628$	0	0
$629$	1.45584	0.0580483
$630$	0	0
$631$	7.65685	0.304815	0.152407	−	0.988318i	$-0.451297\pi$
$631$	7.65685	0.304815	0.152407	+	0.988318i	$0.451297\pi$
$632$	0	0
$633$	97.2548i	3.86553i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 39.4142i	− 1.56165i
$638$	0	0
$639$	−44.7696	−1.77106
$640$	0	0
$641$	1.31371	0.0518884	0.0259442	−	0.999663i	$-0.491741\pi$
$641$	1.31371	0.0518884	0.0259442	+	0.999663i	$0.491741\pi$
$642$	0	0
$643$	− 20.6274i	− 0.813466i	−0.913547	−	0.406733i	$-0.866668\pi$
$643$	− 20.6274i	− 0.813466i	0.913547	−	0.406733i	$-0.133332\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 29.3137i	− 1.15244i	−0.817294	−	0.576220i	$-0.804527\pi$
$647$	− 29.3137i	− 1.15244i	0.817294	−	0.576220i	$-0.195473\pi$
$648$	0	0
$649$	27.3137	1.07216
$650$	0	0
$651$	19.3137	0.756964
$652$	0	0
$653$	− 25.7990i	− 1.00959i	−0.863239	−	0.504796i	$-0.831568\pi$
$653$	− 25.7990i	− 1.00959i	0.863239	−	0.504796i	$-0.168432\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	21.5147i	0.839369i
$658$	0	0
$659$	−10.6274	−0.413985	−0.206993	−	0.978342i	$-0.566368\pi$
$659$	−10.6274	−0.413985	−0.206993	+	0.978342i	$0.566368\pi$
$660$	0	0
$661$	32.6274	1.26906	0.634530	−	0.772898i	$-0.281194\pi$
$661$	32.6274	1.26906	0.634530	+	0.772898i	$0.281194\pi$
$662$	0	0
$663$	17.6569i	0.685735i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 38.9117i	− 1.50667i
$668$	0	0
$669$	−79.2548	−3.06417
$670$	0	0
$671$	27.3137	1.05443
$672$	0	0
$673$	1.75736i	0.0677412i	0.999426	+	0.0338706i	$0.0107834\pi$
$673$	1.75736i	0.0677412i	−0.999426	+	0.0338706i	$0.989217\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	5.55635i	0.213548i	0.994283	+	0.106774i	$0.0340521\pi$
$677$	5.55635i	0.213548i	−0.994283	+	0.106774i	$0.965948\pi$
$678$	0	0
$679$	8.48528	0.325635
$680$	0	0
$681$	−79.9411	−3.06335
$682$	0	0
$683$	− 13.2721i	− 0.507842i	−0.967225	−	0.253921i	$-0.918280\pi$
$683$	− 13.2721i	− 0.507842i	0.967225	−	0.253921i	$-0.0817203\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	− 35.3137i	− 1.34730i
$688$	0	0
$689$	56.6274	2.15733
$690$	0	0
$691$	−4.34315	−0.165221	−0.0826105	−	0.996582i	$-0.526326\pi$
$691$	−4.34315	−0.165221	−0.0826105	+	0.996582i	$0.526326\pi$
$692$	0	0
$693$	− 14.3431i	− 0.544851i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 3.02944i	− 0.114748i
$698$	0	0
$699$	34.1421	1.29137
$700$	0	0
$701$	38.3431	1.44820	0.724100	−	0.689695i	$-0.242255\pi$
$701$	38.3431	1.44820	0.724100	+	0.689695i	$0.242255\pi$
$702$	0	0
$703$	1.75736i	0.0662801i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 3.31371i	− 0.124625i
$708$	0	0
$709$	14.9706	0.562231	0.281116	−	0.959674i	$-0.409296\pi$
$709$	14.9706	0.562231	0.281116	+	0.959674i	$0.409296\pi$
$710$	0	0
$711$	14.3431	0.537910
$712$	0	0
$713$	40.9706i	1.53436i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	73.9411i	2.76138i
$718$	0	0
$719$	18.2843	0.681888	0.340944	−	0.940084i	$-0.389253\pi$
$719$	18.2843	0.681888	0.340944	+	0.940084i	$0.389253\pi$
$720$	0	0
$721$	3.23045	0.120308
$722$	0	0
$723$	− 89.7401i	− 3.33747i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	46.4853i	1.72404i	0.506871	+	0.862022i	$0.330802\pi$
$727$	46.4853i	1.72404i	−0.506871	+	0.862022i	$0.669198\pi$
$728$	0	0
$729$	−148.196	−5.48874
$730$	0	0
$731$	4.00000	0.147945
$732$	0	0
$733$	28.3431i	1.04688i	0.852063	+	0.523439i	$0.175351\pi$
$733$	28.3431i	1.04688i	−0.852063	+	0.523439i	$0.824649\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 6.82843i	− 0.251528i
$738$	0	0
$739$	10.6274	0.390936	0.195468	−	0.980710i	$-0.437377\pi$
$739$	10.6274	0.390936	0.195468	+	0.980710i	$0.437377\pi$
$740$	0	0
$741$	−21.3137	−0.782979
$742$	0	0
$743$	− 34.0416i	− 1.24887i	−0.781078	−	0.624433i	$-0.785330\pi$
$743$	− 34.0416i	− 1.24887i	0.781078	−	0.624433i	$-0.214670\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	115.255i	4.21695i
$748$	0	0
$749$	−6.14214	−0.224429
$750$	0	0
$751$	−14.1421	−0.516054	−0.258027	−	0.966138i	$-0.583072\pi$
$751$	−14.1421	−0.516054	−0.258027	+	0.966138i	$0.583072\pi$
$752$	0	0
$753$	8.00000i	0.291536i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 8.34315i	− 0.303237i	−0.988439	−	0.151618i	$-0.951552\pi$
$757$	− 8.34315i	− 0.303237i	0.988439	−	0.151618i	$-0.0484485\pi$
$758$	0	0
$759$	40.9706	1.48714
$760$	0	0
$761$	−14.6274	−0.530243	−0.265122	−	0.964215i	$-0.585412\pi$
$761$	−14.6274	−0.530243	−0.265122	+	0.964215i	$0.585412\pi$
$762$	0	0
$763$	− 2.62742i	− 0.0951189i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	85.2548i	3.07837i
$768$	0	0
$769$	6.62742	0.238991	0.119495	−	0.992835i	$-0.461872\pi$
$769$	6.62742	0.238991	0.119495	+	0.992835i	$0.461872\pi$
$770$	0	0
$771$	−21.3137	−0.767594
$772$	0	0
$773$	− 19.8995i	− 0.715735i	−0.933772	−	0.357868i	$-0.883504\pi$
$773$	− 19.8995i	− 0.715735i	0.933772	−	0.357868i	$-0.116496\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	− 4.97056i	− 0.178318i
$778$	0	0
$779$	3.65685	0.131020
$780$	0	0
$781$	−10.3431	−0.370107
$782$	0	0
$783$	125.255i	4.47624i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 45.8406i	− 1.63404i	−0.576608	−	0.817021i	$-0.695624\pi$
$787$	− 45.8406i	− 1.63404i	0.576608	−	0.817021i	$-0.304376\pi$
$788$	0	0
$789$	−26.1421	−0.930685
$790$	0	0
$791$	−7.51472	−0.267193
$792$	0	0
$793$	85.2548i	3.02749i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	7.89949i	0.279814i	0.990165	+	0.139907i	$0.0446804\pi$
$797$	7.89949i	0.279814i	−0.990165	+	0.139907i	$0.955320\pi$
$798$	0	0
$799$	4.00000	0.141510
$800$	0	0
$801$	−56.1421	−1.98368
$802$	0	0
$803$	4.97056i	0.175407i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 18.8284i	− 0.662792i
$808$	0	0
$809$	−51.2548	−1.80202	−0.901012	−	0.433794i	$-0.857175\pi$
$809$	−51.2548	−1.80202	−0.901012	+	0.433794i	$0.857175\pi$
$810$	0	0
$811$	−19.7990	−0.695237	−0.347618	−	0.937636i	$-0.613009\pi$
$811$	−19.7990	−0.695237	−0.347618	+	0.937636i	$0.613009\pi$
$812$	0	0
$813$	− 42.1421i	− 1.47799i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	4.82843i	0.168925i
$818$	0	0
$819$	44.7696	1.56437
$820$	0	0
$821$	10.0000	0.349002	0.174501	−	0.984657i	$-0.444169\pi$
$821$	10.0000	0.349002	0.174501	+	0.984657i	$0.444169\pi$
$822$	0	0
$823$	− 20.8284i	− 0.726033i	−0.931783	−	0.363017i	$-0.881747\pi$
$823$	− 20.8284i	− 0.726033i	0.931783	−	0.363017i	$-0.118253\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	14.9289i	0.519130i	0.965726	+	0.259565i	$0.0835792\pi$
$827$	14.9289i	0.519130i	−0.965726	+	0.259565i	$0.916421\pi$
$828$	0	0
$829$	−44.4264	−1.54299	−0.771496	−	0.636234i	$-0.780491\pi$
$829$	−44.4264	−1.54299	−0.771496	+	0.636234i	$0.780491\pi$
$830$	0	0
$831$	−38.1421	−1.32314
$832$	0	0
$833$	− 5.23045i	− 0.181224i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	− 131.882i	− 4.55852i
$838$	0	0
$839$	35.5980	1.22898	0.614489	−	0.788925i	$-0.289362\pi$
$839$	35.5980	1.22898	0.614489	+	0.788925i	$0.289362\pi$
$840$	0	0
$841$	13.0589	0.450306
$842$	0	0
$843$	51.1127i	1.76041i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	5.79899i	0.199256i
$848$	0	0
$849$	−86.4264	−2.96615
$850$	0	0
$851$	10.5442	0.361449
$852$	0	0
$853$	42.0000i	1.43805i	0.694983	+	0.719026i	$0.255412\pi$
$853$	42.0000i	1.43805i	−0.694983	+	0.719026i	$0.744588\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 15.2132i	− 0.519673i	−0.965653	−	0.259837i	$-0.916331\pi$
$857$	− 15.2132i	− 0.519673i	0.965653	−	0.259837i	$-0.0836687\pi$
$858$	0	0
$859$	−32.2843	−1.10153	−0.550763	−	0.834662i	$-0.685663\pi$
$859$	−32.2843	−1.10153	−0.550763	+	0.834662i	$0.685663\pi$
$860$	0	0
$861$	−10.3431	−0.352493
$862$	0	0
$863$	15.8995i	0.541225i	0.962688	+	0.270613i	$0.0872262\pi$
$863$	15.8995i	0.541225i	−0.962688	+	0.270613i	$0.912774\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 55.6985i	− 1.89162i
$868$	0	0
$869$	3.31371	0.112410
$870$	0	0
$871$	21.3137	0.722187
$872$	0	0
$873$	− 88.6690i	− 3.00099i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	22.9289i	0.774255i	0.922026	+	0.387128i	$0.126533\pi$
$877$	22.9289i	0.774255i	−0.922026	+	0.387128i	$0.873467\pi$
$878$	0	0
$879$	−70.2843	−2.37063
$880$	0	0
$881$	−12.2843	−0.413868	−0.206934	−	0.978355i	$-0.566348\pi$
$881$	−12.2843	−0.413868	−0.206934	+	0.978355i	$0.566348\pi$
$882$	0	0
$883$	− 44.8284i	− 1.50860i	−0.656532	−	0.754298i	$-0.727977\pi$
$883$	− 44.8284i	− 1.50860i	0.656532	−	0.754298i	$-0.272023\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	34.9289i	1.17280i	0.810022	+	0.586399i	$0.199455\pi$
$887$	34.9289i	1.17280i	−0.810022	+	0.586399i	$0.800545\pi$
$888$	0	0
$889$	−7.91674	−0.265519
$890$	0	0
$891$	−79.9411	−2.67813
$892$	0	0
$893$	4.82843i	0.161577i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	127.882i	4.26986i
$898$	0	0
$899$	−44.2843	−1.47696
$900$	0	0
$901$	7.51472	0.250352
$902$	0	0
$903$	− 13.6569i	− 0.454472i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 16.3848i	− 0.544048i	−0.962291	−	0.272024i	$-0.912307\pi$
$907$	− 16.3848i	− 0.544048i	0.962291	−	0.272024i	$-0.0876929\pi$
$908$	0	0
$909$	−34.6274	−1.14852
$910$	0	0
$911$	43.7990	1.45113	0.725563	−	0.688156i	$-0.241580\pi$
$911$	43.7990	1.45113	0.725563	+	0.688156i	$0.241580\pi$
$912$	0	0
$913$	26.6274i	0.881239i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 4.68629i	− 0.154755i
$918$	0	0
$919$	−12.6863	−0.418482	−0.209241	−	0.977864i	$-0.567099\pi$
$919$	−12.6863	−0.418482	−0.209241	+	0.977864i	$0.567099\pi$
$920$	0	0
$921$	40.6274	1.33872
$922$	0	0
$923$	− 32.2843i	− 1.06265i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 33.7574i	− 1.10874i
$928$	0	0
$929$	−29.3137	−0.961752	−0.480876	−	0.876789i	$-0.659681\pi$
$929$	−29.3137	−0.961752	−0.480876	+	0.876789i	$0.659681\pi$
$930$	0	0
$931$	6.31371	0.206923
$932$	0	0
$933$	47.7990i	1.56487i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	11.8579i	0.387380i	0.981063	+	0.193690i	$0.0620455\pi$
$937$	11.8579i	0.387380i	−0.981063	+	0.193690i	$0.937954\pi$
$938$	0	0
$939$	61.4558	2.00554
$940$	0	0
$941$	26.2843	0.856843	0.428421	−	0.903579i	$-0.359070\pi$
$941$	26.2843	0.856843	0.428421	+	0.903579i	$0.359070\pi$
$942$	0	0
$943$	− 21.9411i	− 0.714501i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	19.6569i	0.638762i	0.947626	+	0.319381i	$0.103475\pi$
$947$	19.6569i	0.638762i	−0.947626	+	0.319381i	$0.896525\pi$
$948$	0	0
$949$	−15.5147	−0.503629
$950$	0	0
$951$	48.6274	1.57685
$952$	0	0
$953$	14.2426i	0.461364i	0.973029	+	0.230682i	$0.0740958\pi$
$953$	14.2426i	0.461364i	−0.973029	+	0.230682i	$0.925904\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	44.2843i	1.43151i
$958$	0	0
$959$	−6.74517	−0.217813
$960$	0	0
$961$	15.6274	0.504110
$962$	0	0
$963$	64.1838i	2.06829i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	24.6274i	0.791964i	0.918258	+	0.395982i	$0.129596\pi$
$967$	24.6274i	0.791964i	−0.918258	+	0.395982i	$0.870404\pi$
$968$	0	0
$969$	−2.82843	−0.0908622
$970$	0	0
$971$	27.1127	0.870088	0.435044	−	0.900409i	$-0.356733\pi$
$971$	27.1127	0.870088	0.435044	+	0.900409i	$0.356733\pi$
$972$	0	0
$973$	4.40202i	0.141122i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 34.7279i	− 1.11104i	−0.831502	−	0.555522i	$-0.812518\pi$
$977$	− 34.7279i	− 1.11104i	0.831502	−	0.555522i	$-0.187482\pi$
$978$	0	0
$979$	−12.9706	−0.414541
$980$	0	0
$981$	−27.4558	−0.876598
$982$	0	0
$983$	− 20.5858i	− 0.656585i	−0.944576	−	0.328292i	$-0.893527\pi$
$983$	− 20.5858i	− 0.656585i	0.944576	−	0.328292i	$-0.106473\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	− 13.6569i	− 0.434702i
$988$	0	0
$989$	28.9706	0.921210
$990$	0	0
$991$	11.1127	0.353006	0.176503	−	0.984300i	$-0.443521\pi$
$991$	11.1127	0.353006	0.176503	+	0.984300i	$0.443521\pi$
$992$	0	0
$993$	− 23.3137i	− 0.739838i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	30.4853i	0.965479i	0.875764	+	0.482739i	$0.160358\pi$
$997$	30.4853i	0.965479i	−0.875764	+	0.482739i	$0.839642\pi$
$998$	0	0
$999$	−33.9411	−1.07385

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1900.2.c.d.1749.1		4
5.2	odd	4		380.2.a.c.1.1	✓	2
5.3	odd	4		1900.2.a.e.1.2		2
5.4	even	2	inner	1900.2.c.d.1749.4		4
15.2	even	4		3420.2.a.g.1.2		2
20.3	even	4		7600.2.a.u.1.1		2
20.7	even	4		1520.2.a.o.1.2		2
40.27	even	4		6080.2.a.y.1.1		2
40.37	odd	4		6080.2.a.bl.1.2		2
95.37	even	4		7220.2.a.m.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
380.2.a.c.1.1	✓	2	5.2	odd	4
1520.2.a.o.1.2		2	20.7	even	4
1900.2.a.e.1.2		2	5.3	odd	4
1900.2.c.d.1749.1		4	1.1	even	1	trivial
1900.2.c.d.1749.4		4	5.4	even	2	inner
3420.2.a.g.1.2		2	15.2	even	4
6080.2.a.y.1.1		2	40.27	even	4
6080.2.a.bl.1.2		2	40.37	odd	4
7220.2.a.m.1.2		2	95.37	even	4
7600.2.a.u.1.1		2	20.3	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:	\( N \)	\(=\)	\( 1900 = 2^{2} \cdot 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1900.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-3.41421i q^{3} -0.828427i q^{7} -8.65685 q^{9} +O(q^{10})\)	\(q-3.41421i q^{3} -0.828427i q^{7} -8.65685 q^{9} -2.00000 q^{11} -6.24264i q^{13} -0.828427i q^{17} +1.00000 q^{19} -2.82843 q^{21} -6.00000i q^{23} +19.3137i q^{27} +6.48528 q^{29} -6.82843 q^{31} +6.82843i q^{33} +1.75736i q^{37} -21.3137 q^{39} +3.65685 q^{41} +4.82843i q^{43} +4.82843i q^{47} +6.31371 q^{49} -2.82843 q^{51} +9.07107i q^{53} -3.41421i q^{57} -13.6569 q^{59} -13.6569 q^{61} +7.17157i q^{63} +3.41421i q^{67} -20.4853 q^{69} +5.17157 q^{71} -2.48528i q^{73} +1.65685i q^{77} -1.65685 q^{79} +39.9706 q^{81} -13.3137i q^{83} -22.1421i q^{87} +6.48528 q^{89} -5.17157 q^{91} +23.3137i q^{93} +10.2426i q^{97} +17.3137 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 12 q^{9}+O(q^{10}) \) 4 * q - 12 * q^9	\( 4 q - 12 q^{9} - 8 q^{11} + 4 q^{19} - 8 q^{29} - 16 q^{31} - 40 q^{39} - 8 q^{41} - 20 q^{49} - 32 q^{59} - 32 q^{61} - 48 q^{69} + 32 q^{71} + 16 q^{79} + 92 q^{81} - 8 q^{89} - 32 q^{91} + 24 q^{99}+O(q^{100}) \) 4 * q - 12 * q^9 - 8 * q^11 + 4 * q^19 - 8 * q^29 - 16 * q^31 - 40 * q^39 - 8 * q^41 - 20 * q^49 - 32 * q^59 - 32 * q^61 - 48 * q^69 + 32 * q^71 + 16 * q^79 + 92 * q^81 - 8 * q^89 - 32 * q^91 + 24 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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