LMFDB - Embedded newform 4900.2.e.t.2549.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4900,2,Mod(2549,4900)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4900, 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("4900.2549");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4900 = 2^{2} \cdot 5^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4900.e (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:	$39.1266969904$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.4227136.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} + 9x^{4} + 22x^{2} + 9 $ x^6 + 9x^4 + 22x^2 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 3^{2} $
Twist minimal:	no (minimal twist has level 700)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2549.2
Root			$-1.91223i$ of defining polynomial
Character	$\chi$	$=$	4900.2549
Dual form			4900.2.e.t.2549.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.56885i q^{3} -3.59899 q^{9} +O(q^{10})$	$q-2.56885i q^{3} -3.59899 q^{9} -1.56885 q^{11} +5.56885i q^{13} -7.16784i q^{17} -3.16784 q^{19} +5.73669i q^{23} +1.53871i q^{27} -1.96986 q^{29} +0.969861 q^{31} +4.03014i q^{33} +6.70655i q^{37} +14.3055 q^{39} -8.87439 q^{41} -4.59899i q^{43} -0.401012i q^{47} -18.4131 q^{51} +9.53871i q^{53} +8.13770i q^{57} +4.56885 q^{59} +15.3055 q^{61} -0.862301i q^{67} +14.7367 q^{69} -12.1377 q^{71} +4.00000i q^{73} +12.8744 q^{79} -6.84425 q^{81} +17.1076i q^{83} +5.06028i q^{87} +5.59899 q^{89} -2.49143i q^{93} -0.233174i q^{97} +5.64627 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 16 q^{9}+O(q^{10}) $ 6 * q - 16 * q^9	$ 6 q - 16 q^{9} + 8 q^{11} + 4 q^{19} - 6 q^{31} + 28 q^{39} + 22 q^{41} - 6 q^{51} + 10 q^{59} + 34 q^{61} + 48 q^{69} - 38 q^{71} + 2 q^{79} + 46 q^{81} + 28 q^{89} - 42 q^{99}+O(q^{100}) $ 6 * q - 16 * q^9 + 8 * q^11 + 4 * q^19 - 6 * q^31 + 28 * q^39 + 22 * q^41 - 6 * q^51 + 10 * q^59 + 34 * q^61 + 48 * q^69 - 38 * q^71 + 2 * q^79 + 46 * q^81 + 28 * q^89 - 42 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4900\mathbb{Z}\right)^\times$.

$n$	$101$	$1177$	$2451$
$\chi(n)$	$1$	$-1$	$1$

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
$3$	− 2.56885i	− 1.48313i	−0.670883	−	0.741563i	$-0.734085\pi$
$3$	− 2.56885i	− 1.48313i	0.670883	−	0.741563i	$-0.265915\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	−3.59899	−1.19966
$10$	0	0
$11$	−1.56885	−0.473026	−0.236513	−	0.971628i	$-0.576005\pi$
$11$	−1.56885	−0.473026	−0.236513	+	0.971628i	$0.576005\pi$
$12$	0	0
$13$	5.56885i	1.54452i	0.635306	+	0.772260i	$0.280874\pi$
$13$	5.56885i	1.54452i	−0.635306	+	0.772260i	$0.719126\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 7.16784i	− 1.73846i	−0.494411	−	0.869228i	$-0.664616\pi$
$17$	− 7.16784i	− 1.73846i	0.494411	−	0.869228i	$-0.335384\pi$
$18$	0	0
$19$	−3.16784	−0.726752	−0.363376	−	0.931643i	$-0.618376\pi$
$19$	−3.16784	−0.726752	−0.363376	+	0.931643i	$0.618376\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	5.73669i	1.19618i	0.801428	+	0.598091i	$0.204074\pi$
$23$	5.73669i	1.19618i	−0.801428	+	0.598091i	$0.795926\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	1.53871i	0.296125i
$28$	0	0
$29$	−1.96986	−0.365794	−0.182897	−	0.983132i	$-0.558547\pi$
$29$	−1.96986	−0.365794	−0.182897	+	0.983132i	$0.558547\pi$
$30$	0	0
$31$	0.969861	0.174192	0.0870961	−	0.996200i	$-0.472241\pi$
$31$	0.969861	0.174192	0.0870961	+	0.996200i	$0.472241\pi$
$32$	0	0
$33$	4.03014i	0.701557i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	6.70655i	1.10255i	0.834324	+	0.551275i	$0.185858\pi$
$37$	6.70655i	1.10255i	−0.834324	+	0.551275i	$0.814142\pi$
$38$	0	0
$39$	14.3055	2.29072
$40$	0	0
$41$	−8.87439	−1.38595	−0.692973	−	0.720963i	$-0.743700\pi$
$41$	−8.87439	−1.38595	−0.692973	+	0.720963i	$0.743700\pi$
$42$	0	0
$43$	− 4.59899i	− 0.701339i	−0.936499	−	0.350670i	$-0.885954\pi$
$43$	− 4.59899i	− 0.701339i	0.936499	−	0.350670i	$-0.114046\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 0.401012i	− 0.0584936i	−0.999572	−	0.0292468i	$-0.990689\pi$
$47$	− 0.401012i	− 0.0584936i	0.999572	−	0.0292468i	$-0.00931087\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	−18.4131	−2.57835
$52$	0	0
$53$	9.53871i	1.31024i	0.755524	+	0.655121i	$0.227383\pi$
$53$	9.53871i	1.31024i	−0.755524	+	0.655121i	$0.772617\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	8.13770i	1.07786i
$58$	0	0
$59$	4.56885	0.594814	0.297407	−	0.954751i	$-0.403878\pi$
$59$	4.56885	0.594814	0.297407	+	0.954751i	$0.403878\pi$
$60$	0	0
$61$	15.3055	1.95967	0.979837	−	0.199800i	$-0.0640294\pi$
$61$	15.3055	1.95967	0.979837	+	0.199800i	$0.0640294\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 0.862301i	− 0.105347i	−0.998612	−	0.0526734i	$-0.983226\pi$
$67$	− 0.862301i	− 0.105347i	0.998612	−	0.0526734i	$-0.0167742\pi$
$68$	0	0
$69$	14.7367	1.77409
$70$	0	0
$71$	−12.1377	−1.44048	−0.720240	−	0.693725i	$-0.755968\pi$
$71$	−12.1377	−1.44048	−0.720240	+	0.693725i	$0.755968\pi$
$72$	0	0
$73$	4.00000i	0.468165i	0.972217	+	0.234082i	$0.0752085\pi$
$73$	4.00000i	0.468165i	−0.972217	+	0.234082i	$0.924791\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	12.8744	1.44848	0.724241	−	0.689547i	$-0.242190\pi$
$79$	12.8744	1.44848	0.724241	+	0.689547i	$0.242190\pi$
$80$	0	0
$81$	−6.84425	−0.760472
$82$	0	0
$83$	17.1076i	1.87780i	0.344192	+	0.938899i	$0.388153\pi$
$83$	17.1076i	1.87780i	−0.344192	+	0.938899i	$0.611847\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	5.06028i	0.542519i
$88$	0	0
$89$	5.59899	0.593492	0.296746	−	0.954957i	$-0.404099\pi$
$89$	5.59899	0.593492	0.296746	+	0.954957i	$0.404099\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	− 2.49143i	− 0.258349i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 0.233174i	− 0.0236752i	−0.999930	−	0.0118376i	$-0.996232\pi$
$97$	− 0.233174i	− 0.0236752i	0.999930	−	0.0118376i	$-0.00376811\pi$
$98$	0	0
$99$	5.64627	0.567472
$100$	0	0
$101$	12.5035	1.24415	0.622073	−	0.782959i	$-0.286291\pi$
$101$	12.5035	1.24415	0.622073	+	0.782959i	$0.286291\pi$
$102$	0	0
$103$	3.37087i	0.332142i	0.986114	+	0.166071i	$0.0531081\pi$
$103$	3.37087i	0.332142i	−0.986114	+	0.166071i	$0.946892\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	17.7367i	1.71467i	0.514759	+	0.857335i	$0.327882\pi$
$107$	17.7367i	1.71467i	−0.514759	+	0.857335i	$0.672118\pi$
$108$	0	0
$109$	−5.90453	−0.565551	−0.282775	−	0.959186i	$-0.591255\pi$
$109$	−5.90453	−0.565551	−0.282775	+	0.959186i	$0.591255\pi$
$110$	0	0
$111$	17.2281	1.63522
$112$	0	0
$113$	15.6412i	1.47140i	0.677307	+	0.735701i	$0.263147\pi$
$113$	15.6412i	1.47140i	−0.677307	+	0.735701i	$0.736853\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	− 20.0422i	− 1.85290i
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−8.53871	−0.776246
$122$	0	0
$123$	22.7970i	2.05553i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 12.8744i	− 1.14242i	−0.820805	−	0.571209i	$-0.806475\pi$
$127$	− 12.8744i	− 1.14242i	0.820805	−	0.571209i	$-0.193525\pi$
$128$	0	0
$129$	−11.8141	−1.04017
$130$	0	0
$131$	−0.228115	−0.0199305	−0.00996526	−	0.999950i	$-0.503172\pi$
$131$	−0.228115	−0.0199305	−0.00996526	+	0.999950i	$0.503172\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	0	0
$137$	9.67641i	0.826712i	0.910570	+	0.413356i	$0.135643\pi$
$137$	9.67641i	0.826712i	−0.910570	+	0.413356i	$0.864357\pi$
$138$	0	0
$139$	12.3709	1.04928	0.524642	−	0.851323i	$-0.324199\pi$
$139$	12.3709	1.04928	0.524642	+	0.851323i	$0.324199\pi$
$140$	0	0
$141$	−1.03014	−0.0867533
$142$	0	0
$143$	− 8.73669i	− 0.730599i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−11.9096	−0.975671	−0.487836	−	0.872936i	$-0.662213\pi$
$149$	−11.9096	−0.975671	−0.487836	+	0.872936i	$0.662213\pi$
$150$	0	0
$151$	9.96986	0.811336	0.405668	−	0.914021i	$-0.367039\pi$
$151$	9.96986	0.811336	0.405668	+	0.914021i	$0.367039\pi$
$152$	0	0
$153$	25.7970i	2.08556i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	12.0422i	0.961074i	0.876974	+	0.480537i	$0.159558\pi$
$157$	12.0422i	0.961074i	−0.876974	+	0.480537i	$0.840442\pi$
$158$	0	0
$159$	24.5035	1.94326
$160$	0	0
$161$	0	0
$162$	0	0
$163$	− 16.7367i	− 1.31092i	−0.755231	−	0.655459i	$-0.772475\pi$
$163$	− 16.7367i	− 1.31092i	0.755231	−	0.655459i	$-0.227525\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	6.76683i	0.523633i	0.965118	+	0.261816i	$0.0843215\pi$
$167$	6.76683i	0.523633i	−0.965118	+	0.261816i	$0.915679\pi$
$168$	0	0
$169$	−18.0121	−1.38555
$170$	0	0
$171$	11.4010	0.871857
$172$	0	0
$173$	4.69446i	0.356913i	0.983948	+	0.178457i	$0.0571104\pi$
$173$	4.69446i	0.356913i	−0.983948	+	0.178457i	$0.942890\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	− 11.7367i	− 0.882183i
$178$	0	0
$179$	−11.4734	−0.857560	−0.428780	−	0.903409i	$-0.641057\pi$
$179$	−11.4734	−0.857560	−0.428780	+	0.903409i	$0.641057\pi$
$180$	0	0
$181$	0.832162	0.0618541	0.0309271	−	0.999522i	$-0.490154\pi$
$181$	0.832162	0.0618541	0.0309271	+	0.999522i	$0.490154\pi$
$182$	0	0
$183$	− 39.3176i	− 2.90644i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	11.2453i	0.822335i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	6.93972	0.502141	0.251070	−	0.967969i	$-0.419217\pi$
$191$	6.93972	0.502141	0.251070	+	0.967969i	$0.419217\pi$
$192$	0	0
$193$	19.6764i	1.41634i	0.706042	+	0.708169i	$0.250479\pi$
$193$	19.6764i	1.41634i	−0.706042	+	0.708169i	$0.749521\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	16.0422i	1.14296i	0.820616	+	0.571481i	$0.193631\pi$
$197$	16.0422i	1.14296i	−0.820616	+	0.571481i	$0.806369\pi$
$198$	0	0
$199$	10.6764	0.756831	0.378415	−	0.925636i	$-0.376469\pi$
$199$	10.6764	0.756831	0.378415	+	0.925636i	$0.376469\pi$
$200$	0	0
$201$	−2.21512	−0.156243
$202$	0	0
$203$	0	0
$204$	0	0
$205$	0	0
$206$	0	0
$207$	− 20.6463i	− 1.43502i
$208$	0	0
$209$	4.96986	0.343772
$210$	0	0
$211$	11.3658	0.782455	0.391227	−	0.920294i	$-0.372051\pi$
$211$	11.3658	0.782455	0.391227	+	0.920294i	$0.372051\pi$
$212$	0	0
$213$	31.1799i	2.13641i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	0	0
$218$	0	0
$219$	10.2754	0.694347
$220$	0	0
$221$	39.9166	2.68508
$222$	0	0
$223$	− 6.04222i	− 0.404617i	−0.979322	−	0.202309i	$-0.935156\pi$
$223$	− 6.04222i	− 0.404617i	0.979322	−	0.202309i	$-0.0648444\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 21.6412i	− 1.43638i	−0.695848	−	0.718189i	$-0.744971\pi$
$227$	− 21.6412i	− 1.43638i	0.695848	−	0.718189i	$-0.255029\pi$
$228$	0	0
$229$	5.43115	0.358901	0.179450	−	0.983767i	$-0.442568\pi$
$229$	5.43115	0.358901	0.179450	+	0.983767i	$0.442568\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 6.93972i	− 0.454636i	−0.973821	−	0.227318i	$-0.927004\pi$
$233$	− 6.93972i	− 0.454636i	0.973821	−	0.227318i	$-0.0729957\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	− 33.0724i	− 2.14828i
$238$	0	0
$239$	−1.69446	−0.109606	−0.0548028	−	0.998497i	$-0.517453\pi$
$239$	−1.69446	−0.109606	−0.0548028	+	0.998497i	$0.517453\pi$
$240$	0	0
$241$	−14.9347	−0.962026	−0.481013	−	0.876713i	$-0.659731\pi$
$241$	−14.9347	−0.962026	−0.481013	+	0.876713i	$0.659731\pi$
$242$	0	0
$243$	22.1980i	1.42400i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 17.6412i	− 1.12248i
$248$	0	0
$249$	43.9468	2.78501
$250$	0	0
$251$	8.90958	0.562368	0.281184	−	0.959654i	$-0.409273\pi$
$251$	8.90958	0.562368	0.281184	+	0.959654i	$0.409273\pi$
$252$	0	0
$253$	− 9.00000i	− 0.565825i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 9.66432i	− 0.602844i	−0.953491	−	0.301422i	$-0.902539\pi$
$257$	− 9.66432i	− 0.602844i	0.953491	−	0.301422i	$-0.0974613\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	7.08951	0.438830
$262$	0	0
$263$	− 6.10250i	− 0.376296i	−0.982141	−	0.188148i	$-0.939751\pi$
$263$	− 6.10250i	− 0.376296i	0.982141	−	0.188148i	$-0.0602485\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	− 14.3830i	− 0.880223i
$268$	0	0
$269$	−22.5337	−1.37390	−0.686951	−	0.726704i	$-0.741051\pi$
$269$	−22.5337	−1.37390	−0.686951	+	0.726704i	$0.741051\pi$
$270$	0	0
$271$	4.46129	0.271004	0.135502	−	0.990777i	$-0.456735\pi$
$271$	4.46129	0.271004	0.135502	+	0.990777i	$0.456735\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 6.50857i	− 0.391062i	−0.980698	−	0.195531i	$-0.937357\pi$
$277$	− 6.50857i	− 0.391062i	0.980698	−	0.195531i	$-0.0626431\pi$
$278$	0	0
$279$	−3.49052	−0.208972
$280$	0	0
$281$	−11.1980	−0.668015	−0.334008	−	0.942570i	$-0.608401\pi$
$281$	−11.1980	−0.668015	−0.334008	+	0.942570i	$0.608401\pi$
$282$	0	0
$283$	− 7.64627i	− 0.454524i	−0.973834	−	0.227262i	$-0.927023\pi$
$283$	− 7.64627i	− 0.454524i	0.973834	−	0.227262i	$-0.0729773\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−34.3779	−2.02223
$290$	0	0
$291$	−0.598988	−0.0351133
$292$	0	0
$293$	6.12561i	0.357862i	0.983862	+	0.178931i	$0.0572639\pi$
$293$	6.12561i	0.357862i	−0.983862	+	0.178931i	$0.942736\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	− 2.41401i	− 0.140075i
$298$	0	0
$299$	−31.9468	−1.84753
$300$	0	0
$301$	0	0
$302$	0	0
$303$	− 32.1196i	− 1.84523i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 26.4080i	− 1.50719i	−0.657341	−	0.753593i	$-0.728319\pi$
$307$	− 26.4080i	− 1.50719i	0.657341	−	0.753593i	$-0.271681\pi$
$308$	0	0
$309$	8.65927	0.492608
$310$	0	0
$311$	−19.3528	−1.09740	−0.548699	−	0.836020i	$-0.684877\pi$
$311$	−19.3528	−1.09740	−0.548699	+	0.836020i	$0.684877\pi$
$312$	0	0
$313$	8.70655i	0.492123i	0.969254	+	0.246062i	$0.0791366\pi$
$313$	8.70655i	0.492123i	−0.969254	+	0.246062i	$0.920863\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 15.2754i	− 0.857952i	−0.903316	−	0.428976i	$-0.858875\pi$
$317$	− 15.2754i	− 0.857952i	0.903316	−	0.428976i	$-0.141125\pi$
$318$	0	0
$319$	3.09042	0.173030
$320$	0	0
$321$	45.5629	2.54307
$322$	0	0
$323$	22.7065i	1.26343i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	15.1678i	0.838783i
$328$	0	0
$329$	0	0
$330$	0	0
$331$	16.7488	0.920596	0.460298	−	0.887765i	$-0.347743\pi$
$331$	16.7488	0.920596	0.460298	+	0.887765i	$0.347743\pi$
$332$	0	0
$333$	− 24.1368i	− 1.32269i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	31.2453i	1.70204i	0.525135	+	0.851019i	$0.324015\pi$
$337$	31.2453i	1.70204i	−0.525135	+	0.851019i	$0.675985\pi$
$338$	0	0
$339$	40.1799	2.18227
$340$	0	0
$341$	−1.52157	−0.0823974
$342$	0	0
$343$	0	0
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 12.1025i	− 0.649696i	−0.945766	−	0.324848i	$-0.894687\pi$
$347$	− 12.1025i	− 0.649696i	0.945766	−	0.324848i	$-0.105313\pi$
$348$	0	0
$349$	3.87439	0.207391	0.103696	−	0.994609i	$-0.466933\pi$
$349$	3.87439	0.207391	0.103696	+	0.994609i	$0.466933\pi$
$350$	0	0
$351$	−8.56885	−0.457371
$352$	0	0
$353$	− 2.56379i	− 0.136457i	−0.997670	−	0.0682284i	$-0.978265\pi$
$353$	− 2.56379i	− 0.136457i	0.997670	−	0.0682284i	$-0.0217347\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−20.8392	−1.09985	−0.549925	−	0.835214i	$-0.685344\pi$
$359$	−20.8392	−1.09985	−0.549925	+	0.835214i	$0.685344\pi$
$360$	0	0
$361$	−8.96480	−0.471832
$362$	0	0
$363$	21.9347i	1.15127i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	15.8322i	0.826432i	0.910633	+	0.413216i	$0.135595\pi$
$367$	15.8322i	0.826432i	−0.910633	+	0.413216i	$0.864405\pi$
$368$	0	0
$369$	31.9388	1.66267
$370$	0	0
$371$	0	0
$372$	0	0
$373$	22.5035i	1.16519i	0.812763	+	0.582594i	$0.197962\pi$
$373$	22.5035i	1.16519i	−0.812763	+	0.582594i	$0.802038\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 10.9699i	− 0.564977i
$378$	0	0
$379$	7.36581	0.378356	0.189178	−	0.981943i	$-0.439418\pi$
$379$	7.36581	0.378356	0.189178	+	0.981943i	$0.439418\pi$
$380$	0	0
$381$	−33.0724	−1.69435
$382$	0	0
$383$	33.6533i	1.71960i	0.510628	+	0.859802i	$0.329413\pi$
$383$	33.6533i	1.71960i	−0.510628	+	0.859802i	$0.670587\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	16.5517i	0.841370i
$388$	0	0
$389$	−17.4855	−0.886548	−0.443274	−	0.896386i	$-0.646183\pi$
$389$	−17.4855	−0.886548	−0.443274	+	0.896386i	$0.646183\pi$
$390$	0	0
$391$	41.1196	2.07951
$392$	0	0
$393$	0.585994i	0.0295595i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	7.59899i	0.381382i	0.981650	+	0.190691i	$0.0610729\pi$
$397$	7.59899i	0.381382i	−0.981650	+	0.190691i	$0.938927\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	10.8442	0.541536	0.270768	−	0.962645i	$-0.412722\pi$
$401$	10.8442	0.541536	0.270768	+	0.962645i	$0.412722\pi$
$402$	0	0
$403$	5.40101i	0.269044i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 10.5216i	− 0.521535i
$408$	0	0
$409$	16.8693	0.834135	0.417067	−	0.908876i	$-0.363058\pi$
$409$	16.8693	0.834135	0.417067	+	0.908876i	$0.363058\pi$
$410$	0	0
$411$	24.8572	1.22612
$412$	0	0
$413$	0	0
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 31.7789i	− 1.55622i
$418$	0	0
$419$	4.03014	0.196885	0.0984426	−	0.995143i	$-0.468614\pi$
$419$	4.03014	0.196885	0.0984426	+	0.995143i	$0.468614\pi$
$420$	0	0
$421$	25.1196	1.22426	0.612128	−	0.790758i	$-0.290314\pi$
$421$	25.1196	1.22426	0.612128	+	0.790758i	$0.290314\pi$
$422$	0	0
$423$	1.44324i	0.0701726i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−22.4432	−1.08357
$430$	0	0
$431$	30.2402	1.45662	0.728310	−	0.685248i	$-0.240306\pi$
$431$	30.2402	1.45662	0.728310	+	0.685248i	$0.240306\pi$
$432$	0	0
$433$	− 22.9769i	− 1.10420i	−0.833778	−	0.552099i	$-0.813827\pi$
$433$	− 22.9769i	− 1.10420i	0.833778	−	0.552099i	$-0.186173\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 18.1729i	− 0.869328i
$438$	0	0
$439$	−11.9518	−0.570429	−0.285214	−	0.958464i	$-0.592065\pi$
$439$	−11.9518	−0.570429	−0.285214	+	0.958464i	$0.592065\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 14.1076i	− 0.670270i	−0.942170	−	0.335135i	$-0.891218\pi$
$443$	− 14.1076i	− 0.670270i	0.942170	−	0.335135i	$-0.108782\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	30.5939i	1.44704i
$448$	0	0
$449$	−37.9468	−1.79082	−0.895409	−	0.445245i	$-0.853117\pi$
$449$	−37.9468	−1.79082	−0.895409	+	0.445245i	$0.853117\pi$
$450$	0	0
$451$	13.9226	0.655589
$452$	0	0
$453$	− 25.6111i	− 1.20331i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	18.5457i	0.867533i	0.901025	+	0.433767i	$0.142816\pi$
$457$	18.5457i	0.867533i	−0.901025	+	0.433767i	$0.857184\pi$
$458$	0	0
$459$	11.0292	0.514800
$460$	0	0
$461$	−23.2453	−1.08264	−0.541320	−	0.840817i	$-0.682075\pi$
$461$	−23.2453	−1.08264	−0.541320	+	0.840817i	$0.682075\pi$
$462$	0	0
$463$	7.59393i	0.352920i	0.984308	+	0.176460i	$0.0564646\pi$
$463$	7.59393i	0.352920i	−0.984308	+	0.176460i	$0.943535\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 28.1678i	− 1.30345i	−0.758454	−	0.651726i	$-0.774045\pi$
$467$	− 28.1678i	− 1.30345i	0.758454	−	0.651726i	$-0.225955\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	30.9347	1.42539
$472$	0	0
$473$	7.21512i	0.331752i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	− 34.3297i	− 1.57185i
$478$	0	0
$479$	34.1799	1.56172	0.780860	−	0.624706i	$-0.214781\pi$
$479$	34.1799	1.56172	0.780860	+	0.624706i	$0.214781\pi$
$480$	0	0
$481$	−37.3478	−1.70291
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	11.5035i	0.521274i	0.965437	+	0.260637i	$0.0839326\pi$
$487$	11.5035i	0.521274i	−0.965437	+	0.260637i	$0.916067\pi$
$488$	0	0
$489$	−42.9940	−1.94426
$490$	0	0
$491$	−25.5337	−1.15232	−0.576159	−	0.817338i	$-0.695449\pi$
$491$	−25.5337	−1.15232	−0.576159	+	0.817338i	$0.695449\pi$
$492$	0	0
$493$	14.1196i	0.635917i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	11.3960	0.510153	0.255076	−	0.966921i	$-0.417899\pi$
$499$	11.3960	0.510153	0.255076	+	0.966921i	$0.417899\pi$
$500$	0	0
$501$	17.3830	0.776613
$502$	0	0
$503$	− 2.49649i	− 0.111313i	−0.998450	−	0.0556564i	$-0.982275\pi$
$503$	− 2.49649i	− 0.111313i	0.998450	−	0.0556564i	$-0.0177251\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	46.2703i	2.05494i
$508$	0	0
$509$	−11.5438	−0.511669	−0.255834	−	0.966721i	$-0.582350\pi$
$509$	−11.5438	−0.511669	−0.255834	+	0.966721i	$0.582350\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	− 4.87439i	− 0.215209i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	0.629127i	0.0276690i
$518$	0	0
$519$	12.0594	0.529348
$520$	0	0
$521$	23.7488	1.04045	0.520226	−	0.854028i	$-0.325848\pi$
$521$	23.7488	1.04045	0.520226	+	0.854028i	$0.325848\pi$
$522$	0	0
$523$	16.1377i	0.705652i	0.935689	+	0.352826i	$0.114779\pi$
$523$	16.1377i	0.705652i	−0.935689	+	0.352826i	$0.885221\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 6.95181i	− 0.302826i
$528$	0	0
$529$	−9.90958	−0.430851
$530$	0	0
$531$	−16.4432	−0.713576
$532$	0	0
$533$	− 49.4201i	− 2.14062i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	29.4734i	1.27187i
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−1.53871	−0.0661543	−0.0330772	−	0.999453i	$-0.510531\pi$
$541$	−1.53871	−0.0661543	−0.0330772	+	0.999453i	$0.510531\pi$
$542$	0	0
$543$	− 2.13770i	− 0.0917375i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	14.3779i	0.614755i	0.951588	+	0.307377i	$0.0994514\pi$
$547$	14.3779i	0.614755i	−0.951588	+	0.307377i	$0.900549\pi$
$548$	0	0
$549$	−55.0844	−2.35095
$550$	0	0
$551$	6.24020	0.265842
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0	0
$557$	16.9347i	0.717545i	0.933425	+	0.358772i	$0.116805\pi$
$557$	16.9347i	0.717545i	−0.933425	+	0.358772i	$0.883195\pi$
$558$	0	0
$559$	25.6111	1.08323
$560$	0	0
$561$	28.8874	1.21963
$562$	0	0
$563$	0.664324i	0.0279979i	0.999902	+	0.0139990i	$0.00445615\pi$
$563$	0.664324i	0.0279979i	−0.999902	+	0.0139990i	$0.995544\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	0	0
$569$	41.0372	1.72037	0.860184	−	0.509984i	$-0.170349\pi$
$569$	41.0372	1.72037	0.860184	+	0.509984i	$0.170349\pi$
$570$	0	0
$571$	−28.6291	−1.19809	−0.599046	−	0.800715i	$-0.704453\pi$
$571$	−28.6291	−1.19809	−0.599046	+	0.800715i	$0.704453\pi$
$572$	0	0
$573$	− 17.8271i	− 0.744738i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 31.4604i	− 1.30971i	−0.755753	−	0.654856i	$-0.772729\pi$
$577$	− 31.4604i	− 1.30971i	0.755753	−	0.654856i	$-0.227271\pi$
$578$	0	0
$579$	50.5457	2.10061
$580$	0	0
$581$	0	0
$582$	0	0
$583$	− 14.9648i	− 0.619779i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	9.53871i	0.393705i	0.980433	+	0.196852i	$0.0630720\pi$
$587$	9.53871i	0.393705i	−0.980433	+	0.196852i	$0.936928\pi$
$588$	0	0
$589$	−3.07236	−0.126595
$590$	0	0
$591$	41.2101	1.69516
$592$	0	0
$593$	9.93972i	0.408175i	0.978953	+	0.204088i	$0.0654228\pi$
$593$	9.93972i	0.408175i	−0.978953	+	0.204088i	$0.934577\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 27.4261i	− 1.12248i
$598$	0	0
$599$	−9.41310	−0.384609	−0.192304	−	0.981335i	$-0.561596\pi$
$599$	−9.41310	−0.384609	−0.192304	+	0.981335i	$0.561596\pi$
$600$	0	0
$601$	−3.00506	−0.122579	−0.0612894	−	0.998120i	$-0.519521\pi$
$601$	−3.00506	−0.122579	−0.0612894	+	0.998120i	$0.519521\pi$
$602$	0	0
$603$	3.10341i	0.126381i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	18.8322i	0.764374i	0.924085	+	0.382187i	$0.124829\pi$
$607$	18.8322i	0.764374i	−0.924085	+	0.382187i	$0.875171\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	2.23317	0.0903445
$612$	0	0
$613$	15.8392i	0.639739i	0.947462	+	0.319869i	$0.103639\pi$
$613$	15.8392i	0.639739i	−0.947462	+	0.319869i	$0.896361\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	0.778912i	0.0313578i	0.999877	+	0.0156789i	$0.00499096\pi$
$617$	0.778912i	0.0313578i	−0.999877	+	0.0156789i	$0.995009\pi$
$618$	0	0
$619$	−9.94675	−0.399794	−0.199897	−	0.979817i	$-0.564061\pi$
$619$	−9.94675	−0.399794	−0.199897	+	0.979817i	$0.564061\pi$
$620$	0	0
$621$	−8.82710	−0.354219
$622$	0	0
$623$	0	0
$624$	0	0
$625$	0	0
$626$	0	0
$627$	− 12.7668i	− 0.509858i
$628$	0	0
$629$	48.0715	1.91673
$630$	0	0
$631$	−12.5638	−0.500157	−0.250078	−	0.968226i	$-0.580456\pi$
$631$	−12.5638	−0.500157	−0.250078	+	0.968226i	$0.580456\pi$
$632$	0	0
$633$	− 29.1971i	− 1.16048i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	0	0
$638$	0	0
$639$	43.6834	1.72809
$640$	0	0
$641$	−11.5207	−0.455039	−0.227519	−	0.973774i	$-0.573061\pi$
$641$	−11.5207	−0.455039	−0.227519	+	0.973774i	$0.573061\pi$
$642$	0	0
$643$	− 9.74175i	− 0.384177i	−0.981378	−	0.192088i	$-0.938474\pi$
$643$	− 9.74175i	− 0.384177i	0.981378	−	0.192088i	$-0.0615261\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	25.7538i	1.01249i	0.862390	+	0.506244i	$0.168966\pi$
$647$	25.7538i	1.01249i	−0.862390	+	0.506244i	$0.831034\pi$
$648$	0	0
$649$	−7.16784	−0.281362
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 15.1929i	− 0.594545i	−0.954793	−	0.297272i	$-0.903923\pi$
$653$	− 15.1929i	− 0.594545i	0.954793	−	0.297272i	$-0.0960770\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	− 14.3960i	− 0.561640i
$658$	0	0
$659$	14.6894	0.572218	0.286109	−	0.958197i	$-0.407638\pi$
$659$	14.6894	0.572218	0.286109	+	0.958197i	$0.407638\pi$
$660$	0	0
$661$	4.33568	0.168638	0.0843191	−	0.996439i	$-0.473128\pi$
$661$	4.33568	0.168638	0.0843191	+	0.996439i	$0.473128\pi$
$662$	0	0
$663$	− 102.540i	− 3.98231i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 11.3005i	− 0.437556i
$668$	0	0
$669$	−15.5216	−0.600098
$670$	0	0
$671$	−24.0121	−0.926976
$672$	0	0
$673$	34.1971i	1.31820i	0.752055	+	0.659100i	$0.229063\pi$
$673$	34.1971i	1.31820i	−0.752055	+	0.659100i	$0.770937\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 18.0904i	− 0.695271i	−0.937630	−	0.347636i	$-0.886985\pi$
$677$	− 18.0904i	− 0.695271i	0.937630	−	0.347636i	$-0.113015\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−55.5930	−2.13033
$682$	0	0
$683$	− 24.1256i	− 0.923141i	−0.887104	−	0.461570i	$-0.847286\pi$
$683$	− 24.1256i	− 0.923141i	0.887104	−	0.461570i	$-0.152714\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	− 13.9518i	− 0.532295i
$688$	0	0
$689$	−53.1196	−2.02370
$690$	0	0
$691$	17.4483	0.663764	0.331882	−	0.943321i	$-0.392316\pi$
$691$	17.4483	0.663764	0.331882	+	0.943321i	$0.392316\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	63.6102i	2.40941i
$698$	0	0
$699$	−17.8271	−0.674283
$700$	0	0
$701$	20.7015	0.781885	0.390942	−	0.920415i	$-0.372149\pi$
$701$	20.7015	0.781885	0.390942	+	0.920415i	$0.372149\pi$
$702$	0	0
$703$	− 21.2453i	− 0.801280i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−7.09547	−0.266476	−0.133238	−	0.991084i	$-0.542537\pi$
$709$	−7.09547	−0.266476	−0.133238	+	0.991084i	$0.542537\pi$
$710$	0	0
$711$	−46.3348	−1.73769
$712$	0	0
$713$	5.56379i	0.208366i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	4.35282i	0.162559i
$718$	0	0
$719$	−52.4432	−1.95580	−0.977901	−	0.209067i	$-0.932957\pi$
$719$	−52.4432	−1.95580	−0.977901	+	0.209067i	$0.932957\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	38.3649i	1.42681i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	29.9045i	1.10910i	0.832151	+	0.554549i	$0.187109\pi$
$727$	29.9045i	1.10910i	−0.832151	+	0.554549i	$0.812891\pi$
$728$	0	0
$729$	36.4905	1.35150
$730$	0	0
$731$	−32.9648	−1.21925
$732$	0	0
$733$	5.58094i	0.206137i	0.994674	+	0.103068i	$0.0328660\pi$
$733$	5.58094i	0.206137i	−0.994674	+	0.103068i	$0.967134\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	1.35282i	0.0498318i
$738$	0	0
$739$	−40.7488	−1.49897	−0.749484	−	0.662023i	$-0.769698\pi$
$739$	−40.7488	−1.49897	−0.749484	+	0.662023i	$0.769698\pi$
$740$	0	0
$741$	−45.3176	−1.66478
$742$	0	0
$743$	50.5327i	1.85387i	0.375226	+	0.926933i	$0.377565\pi$
$743$	50.5327i	1.85387i	−0.375226	+	0.926933i	$0.622435\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	− 61.5699i	− 2.25273i
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−30.7136	−1.12075	−0.560377	−	0.828238i	$-0.689344\pi$
$751$	−30.7136	−1.12075	−0.560377	+	0.828238i	$0.689344\pi$
$752$	0	0
$753$	− 22.8874i	− 0.834063i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	20.7116i	0.752776i	0.926462	+	0.376388i	$0.122834\pi$
$757$	20.7116i	0.752776i	−0.926462	+	0.376388i	$0.877166\pi$
$758$	0	0
$759$	−23.1196	−0.839190
$760$	0	0
$761$	−34.3528	−1.24529	−0.622644	−	0.782505i	$-0.713942\pi$
$761$	−34.3528	−1.24529	−0.622644	+	0.782505i	$0.713942\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	25.4432i	0.918702i
$768$	0	0
$769$	−46.8392	−1.68906	−0.844532	−	0.535505i	$-0.820121\pi$
$769$	−46.8392	−1.68906	−0.844532	+	0.535505i	$0.820121\pi$
$770$	0	0
$771$	−24.8262	−0.894094
$772$	0	0
$773$	33.3306i	1.19882i	0.800443	+	0.599409i	$0.204598\pi$
$773$	33.3306i	1.19882i	−0.800443	+	0.599409i	$0.795402\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	28.1126	1.00724
$780$	0	0
$781$	19.0422	0.681384
$782$	0	0
$783$	− 3.03105i	− 0.108321i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 8.40804i	− 0.299714i	−0.988708	−	0.149857i	$-0.952119\pi$
$787$	− 8.40804i	− 0.299714i	0.988708	−	0.149857i	$-0.0478814\pi$
$788$	0	0
$789$	−15.6764	−0.558095
$790$	0	0
$791$	0	0
$792$	0	0
$793$	85.2342i	3.02676i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 16.2935i	− 0.577144i	−0.957458	−	0.288572i	$-0.906820\pi$
$797$	− 16.2935i	− 0.577144i	0.957458	−	0.288572i	$-0.0931804\pi$
$798$	0	0
$799$	−2.87439	−0.101688
$800$	0	0
$801$	−20.1507	−0.711990
$802$	0	0
$803$	− 6.27540i	− 0.221454i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	57.8856i	2.03767i
$808$	0	0
$809$	15.5035	0.545075	0.272537	−	0.962145i	$-0.412137\pi$
$809$	15.5035	0.545075	0.272537	+	0.962145i	$0.412137\pi$
$810$	0	0
$811$	−36.6232	−1.28601	−0.643007	−	0.765861i	$-0.722313\pi$
$811$	−36.6232	−1.28601	−0.643007	+	0.765861i	$0.722313\pi$
$812$	0	0
$813$	− 11.4604i	− 0.401933i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	14.5688i	0.509700i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−43.3297	−1.51222	−0.756109	−	0.654446i	$-0.772902\pi$
$821$	−43.3297	−1.51222	−0.756109	+	0.654446i	$0.772902\pi$
$822$	0	0
$823$	41.0543i	1.43106i	0.698580	+	0.715532i	$0.253815\pi$
$823$	41.0543i	1.43106i	−0.698580	+	0.715532i	$0.746185\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 30.0473i	− 1.04485i	−0.852686	−	0.522423i	$-0.825028\pi$
$827$	− 30.0473i	− 1.04485i	0.852686	−	0.522423i	$-0.174972\pi$
$828$	0	0
$829$	42.0292	1.45974	0.729868	−	0.683588i	$-0.239582\pi$
$829$	42.0292	1.45974	0.729868	+	0.683588i	$0.239582\pi$
$830$	0	0
$831$	−16.7195	−0.579995
$832$	0	0
$833$	0	0
$834$	0	0
$835$	0	0
$836$	0	0
$837$	1.49234i	0.0515827i
$838$	0	0
$839$	−3.38893	−0.116999	−0.0584994	−	0.998287i	$-0.518632\pi$
$839$	−3.38893	−0.116999	−0.0584994	+	0.998287i	$0.518632\pi$
$840$	0	0
$841$	−25.1196	−0.866195
$842$	0	0
$843$	28.7659i	0.990751i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−19.6421	−0.674116
$850$	0	0
$851$	−38.4734	−1.31885
$852$	0	0
$853$	25.1267i	0.860321i	0.902752	+	0.430160i	$0.141543\pi$
$853$	25.1267i	0.860321i	−0.902752	+	0.430160i	$0.858457\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 47.7367i	− 1.63065i	−0.579001	−	0.815327i	$-0.696557\pi$
$857$	− 47.7367i	− 1.63065i	0.579001	−	0.815327i	$-0.303443\pi$
$858$	0	0
$859$	41.0422	1.40034	0.700171	−	0.713975i	$-0.253107\pi$
$859$	41.0422	1.40034	0.700171	+	0.713975i	$0.253107\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	31.1326i	1.05977i	0.848070	+	0.529884i	$0.177764\pi$
$863$	31.1326i	1.05977i	−0.848070	+	0.529884i	$0.822236\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	88.3117i	2.99922i
$868$	0	0
$869$	−20.1980	−0.685169
$870$	0	0
$871$	4.80202	0.162710
$872$	0	0
$873$	0.839190i	0.0284023i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 7.93972i	− 0.268105i	−0.990974	−	0.134053i	$-0.957201\pi$
$877$	− 7.93972i	− 0.268105i	0.990974	−	0.134053i	$-0.0427991\pi$
$878$	0	0
$879$	15.7358	0.530755
$880$	0	0
$881$	−29.3709	−0.989530	−0.494765	−	0.869027i	$-0.664746\pi$
$881$	−29.3709	−0.989530	−0.494765	+	0.869027i	$0.664746\pi$
$882$	0	0
$883$	− 26.9718i	− 0.907674i	−0.891085	−	0.453837i	$-0.850055\pi$
$883$	− 26.9718i	− 0.907674i	0.891085	−	0.453837i	$-0.149945\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	23.4855i	0.788565i	0.918989	+	0.394282i	$0.129007\pi$
$887$	23.4855i	0.788565i	−0.918989	+	0.394282i	$0.870993\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	10.7376	0.359723
$892$	0	0
$893$	1.27034i	0.0425103i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	82.0664i	2.74012i
$898$	0	0
$899$	−1.91049	−0.0637185
$900$	0	0
$901$	68.3719	2.27780
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 9.58690i	− 0.318328i	−0.987252	−	0.159164i	$-0.949120\pi$
$907$	− 9.58690i	− 0.318328i	0.987252	−	0.159164i	$-0.0508798\pi$
$908$	0	0
$909$	−45.0000	−1.49256
$910$	0	0
$911$	−3.78994	−0.125566	−0.0627831	−	0.998027i	$-0.519998\pi$
$911$	−3.78994	−0.125566	−0.0627831	+	0.998027i	$0.519998\pi$
$912$	0	0
$913$	− 26.8392i	− 0.888248i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−23.3900	−0.771564	−0.385782	−	0.922590i	$-0.626068\pi$
$919$	−23.3900	−0.771564	−0.385782	+	0.922590i	$0.626068\pi$
$920$	0	0
$921$	−67.8383	−2.23535
$922$	0	0
$923$	− 67.5930i	− 2.22485i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 12.1317i	− 0.398458i
$928$	0	0
$929$	−50.2573	−1.64889	−0.824445	−	0.565942i	$-0.808513\pi$
$929$	−50.2573	−1.64889	−0.824445	+	0.565942i	$0.808513\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	49.7145i	1.62758i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	2.43621i	0.0795875i	0.999208	+	0.0397937i	$0.0126701\pi$
$937$	2.43621i	0.0795875i	−0.999208	+	0.0397937i	$0.987330\pi$
$938$	0	0
$939$	22.3658	0.729881
$940$	0	0
$941$	−23.9648	−0.781230	−0.390615	−	0.920554i	$-0.627738\pi$
$941$	−23.9648	−0.781230	−0.390615	+	0.920554i	$0.627738\pi$
$942$	0	0
$943$	− 50.9096i	− 1.65784i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	51.1619i	1.66254i	0.555871	+	0.831269i	$0.312385\pi$
$947$	51.1619i	1.66254i	−0.555871	+	0.831269i	$0.687615\pi$
$948$	0	0
$949$	−22.2754	−0.723090
$950$	0	0
$951$	−39.2402	−1.27245
$952$	0	0
$953$	33.3306i	1.07968i	0.841766	+	0.539842i	$0.181516\pi$
$953$	33.3306i	1.07968i	−0.841766	+	0.539842i	$0.818484\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	− 7.93881i	− 0.256625i
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−30.0594	−0.969657
$962$	0	0
$963$	− 63.8341i	− 2.05703i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	31.8392i	1.02388i	0.859021	+	0.511940i	$0.171073\pi$
$967$	31.8392i	1.02388i	−0.859021	+	0.511940i	$0.828927\pi$
$968$	0	0
$969$	58.3297	1.87382
$970$	0	0
$971$	1.94675	0.0624742	0.0312371	−	0.999512i	$-0.490055\pi$
$971$	1.94675	0.0624742	0.0312371	+	0.999512i	$0.490055\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	50.1196i	1.60347i	0.597680	+	0.801735i	$0.296089\pi$
$977$	50.1196i	1.60347i	−0.597680	+	0.801735i	$0.703911\pi$
$978$	0	0
$979$	−8.78397	−0.280737
$980$	0	0
$981$	21.2503	0.678470
$982$	0	0
$983$	− 47.9217i	− 1.52846i	−0.644941	−	0.764232i	$-0.723118\pi$
$983$	− 47.9217i	− 1.52846i	0.644941	−	0.764232i	$-0.276882\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	26.3830	0.838929
$990$	0	0
$991$	8.86933	0.281743	0.140872	−	0.990028i	$-0.455009\pi$
$991$	8.86933	0.281743	0.140872	+	0.990028i	$0.455009\pi$
$992$	0	0
$993$	− 43.0251i	− 1.36536i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 12.2101i	− 0.386697i	−0.981130	−	0.193348i	$-0.938065\pi$
$997$	− 12.2101i	− 0.386697i	0.981130	−	0.193348i	$-0.0619347\pi$
$998$	0	0
$999$	−10.3194	−0.326493

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4900.2.e.t.2549.2		6
5.2	odd	4		4900.2.a.bc.1.1		3
5.3	odd	4		4900.2.a.ba.1.3		3
5.4	even	2	inner	4900.2.e.t.2549.5		6
7.2	even	3		700.2.r.d.249.5		12
7.4	even	3		700.2.r.d.149.2		12
7.6	odd	2		4900.2.e.s.2549.5		6
35.2	odd	12		700.2.i.d.501.3	yes	6
35.4	even	6		700.2.r.d.149.5		12
35.9	even	6		700.2.r.d.249.2		12
35.13	even	4		4900.2.a.bd.1.1		3
35.18	odd	12		700.2.i.e.401.1	yes	6
35.23	odd	12		700.2.i.e.501.1	yes	6
35.27	even	4		4900.2.a.bb.1.3		3
35.32	odd	12		700.2.i.d.401.3	✓	6
35.34	odd	2		4900.2.e.s.2549.2		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
700.2.i.d.401.3	✓	6	35.32	odd	12
700.2.i.d.501.3	yes	6	35.2	odd	12
700.2.i.e.401.1	yes	6	35.18	odd	12
700.2.i.e.501.1	yes	6	35.23	odd	12
700.2.r.d.149.2		12	7.4	even	3
700.2.r.d.149.5		12	35.4	even	6
700.2.r.d.249.2		12	35.9	even	6
700.2.r.d.249.5		12	7.2	even	3
4900.2.a.ba.1.3		3	5.3	odd	4
4900.2.a.bb.1.3		3	35.27	even	4
4900.2.a.bc.1.1		3	5.2	odd	4
4900.2.a.bd.1.1		3	35.13	even	4
4900.2.e.s.2549.2		6	35.34	odd	2
4900.2.e.s.2549.5		6	7.6	odd	2
4900.2.e.t.2549.2		6	1.1	even	1	trivial
4900.2.e.t.2549.5		6	5.4	even	2	inner

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:	\( N \)	\(=\)	\( 4900 = 2^{2} \cdot 5^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4900.e (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-2.56885i q^{3} -3.59899 q^{9} +O(q^{10})\)	\(q-2.56885i q^{3} -3.59899 q^{9} -1.56885 q^{11} +5.56885i q^{13} -7.16784i q^{17} -3.16784 q^{19} +5.73669i q^{23} +1.53871i q^{27} -1.96986 q^{29} +0.969861 q^{31} +4.03014i q^{33} +6.70655i q^{37} +14.3055 q^{39} -8.87439 q^{41} -4.59899i q^{43} -0.401012i q^{47} -18.4131 q^{51} +9.53871i q^{53} +8.13770i q^{57} +4.56885 q^{59} +15.3055 q^{61} -0.862301i q^{67} +14.7367 q^{69} -12.1377 q^{71} +4.00000i q^{73} +12.8744 q^{79} -6.84425 q^{81} +17.1076i q^{83} +5.06028i q^{87} +5.59899 q^{89} -2.49143i q^{93} -0.233174i q^{97} +5.64627 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 16 q^{9}+O(q^{10}) \) 6 * q - 16 * q^9	\( 6 q - 16 q^{9} + 8 q^{11} + 4 q^{19} - 6 q^{31} + 28 q^{39} + 22 q^{41} - 6 q^{51} + 10 q^{59} + 34 q^{61} + 48 q^{69} - 38 q^{71} + 2 q^{79} + 46 q^{81} + 28 q^{89} - 42 q^{99}+O(q^{100}) \) 6 * q - 16 * q^9 + 8 * q^11 + 4 * q^19 - 6 * q^31 + 28 * q^39 + 22 * q^41 - 6 * q^51 + 10 * q^59 + 34 * q^61 + 48 * q^69 - 38 * q^71 + 2 * q^79 + 46 * q^81 + 28 * q^89 - 42 * q^99

Introduction

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