LMFDB - Embedded newform 3900.2.bm.a.2257.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3900,2,Mod(2257,3900)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3900, base_ring=CyclotomicField(4))

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

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

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

chi := DirichletCharacter("3900.2257");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3900 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3900.bm (of order $4$, degree $2$, 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:	$31.1416567883$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$8$ over $\Q(i)$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} + 127x^{12} + 1449x^{8} + 632x^{4} + 16 $ x^16 + 127x^12 + 1449x^8 + 632*x^4 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{8} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			2257.2
Root			$0.571301 + 0.571301i$ of defining polynomial
Character	$\chi$	$=$	3900.2257
Dual form			3900.2.bm.a.2293.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.707107 + 0.707107i) q^{3} -0.607786 q^{7} -1.00000i q^{9} +O(q^{10})$	$q+(-0.707107 + 0.707107i) q^{3} -0.607786 q^{7} -1.00000i q^{9} +(-1.42977 + 1.42977i) q^{11} +(-0.652944 - 3.54594i) q^{13} +(1.08844 - 1.08844i) q^{17} +(-3.47542 + 3.47542i) q^{19} +(0.429770 - 0.429770i) q^{21} +(6.00343 + 6.00343i) q^{23} +(0.707107 + 0.707107i) q^{27} -6.79765i q^{29} +(3.01471 + 3.01471i) q^{31} -2.02200i q^{33} +8.46233 q^{37} +(2.96906 + 2.04565i) q^{39} +(-4.50834 - 4.50834i) q^{41} +(2.02200 + 2.02200i) q^{43} -5.00531 q^{47} -6.63060 q^{49} +1.53929i q^{51} +(-4.48089 + 4.48089i) q^{53} -4.91499i q^{57} +(-8.02942 - 8.02942i) q^{59} -14.4282 q^{61} +0.607786i q^{63} +8.85067i q^{67} -8.49014 q^{69} +(6.66154 + 6.66154i) q^{71} +9.92030i q^{73} +(0.868995 - 0.868995i) q^{77} -11.4774i q^{79} -1.00000 q^{81} -10.2649 q^{83} +(4.80667 + 4.80667i) q^{87} +(-3.64880 - 3.64880i) q^{89} +(0.396850 + 2.15517i) q^{91} -4.26345 q^{93} -1.28014i q^{97} +(1.42977 + 1.42977i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q+O(q^{10}) $ 16 * q	$ 16 q - 16 q^{11} - 8 q^{19} - 32 q^{31} + 8 q^{39} + 8 q^{49} + 32 q^{59} - 24 q^{61} - 8 q^{69} + 32 q^{71} - 16 q^{81} + 24 q^{91} + 16 q^{99}+O(q^{100}) $ 16 * q - 16 * q^11 - 8 * q^19 - 32 * q^31 + 8 * q^39 + 8 * q^49 + 32 * q^59 - 24 * q^61 - 8 * q^69 + 32 * q^71 - 16 * q^81 + 24 * q^91 + 16 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3900\mathbb{Z}\right)^\times$.

$n$	$301$	$1301$	$1951$	$3277$
$\chi(n)$	$e\left(\frac{1}{4}\right)$	$1$	$1$	$e\left(\frac{1}{4}\right)$

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
$3$	−0.707107	+	0.707107i	−0.408248	+	0.408248i
$3$	−0.707107	+	0.707107i	−0.408248	+	0.408248i
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	−0.607786			−0.229722			−0.114861	−	0.993382i	$-0.536642\pi$
$7$	−0.607786			−0.229722			−0.114861	+	0.993382i	$0.536642\pi$
$8$		0			0
$9$		−	1.00000i		−	0.333333i
$10$		0			0
$11$	−1.42977	+	1.42977i	−0.431092	+	0.431092i	−0.889000	−	0.457908i	$-0.848599\pi$
$11$	−1.42977	+	1.42977i	−0.431092	+	0.431092i	0.457908	+	0.889000i	$0.348599\pi$
$12$		0			0
$13$	−0.652944	−	3.54594i	−0.181094	−	0.983466i
$13$	−0.652944	−	3.54594i	−0.181094	−	0.983466i
$14$		0			0
$15$		0			0
$16$		0			0
$17$	1.08844	−	1.08844i	0.263986	−	0.263986i	−0.562686	−	0.826671i	$-0.690232\pi$
$17$	1.08844	−	1.08844i	0.263986	−	0.263986i	0.826671	+	0.562686i	$0.190232\pi$
$18$		0			0
$19$	−3.47542	+	3.47542i	−0.797317	+	0.797317i	−0.982672	−	0.185355i	$-0.940657\pi$
$19$	−3.47542	+	3.47542i	−0.797317	+	0.797317i	0.185355	+	0.982672i	$0.440657\pi$
$20$		0			0
$21$	0.429770	−	0.429770i	0.0937835	−	0.0937835i
$22$		0			0
$23$	6.00343	+	6.00343i	1.25180	+	1.25180i	0.954911	+	0.296891i	$0.0959497\pi$
$23$	6.00343	+	6.00343i	1.25180	+	1.25180i	0.296891	+	0.954911i	$0.404050\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$	0.707107	+	0.707107i	0.136083	+	0.136083i
$28$		0			0
$29$		−	6.79765i		−	1.26229i	−0.775664	−	0.631146i	$-0.782585\pi$
$29$		−	6.79765i		−	1.26229i	0.775664	−	0.631146i	$-0.217415\pi$
$30$		0			0
$31$	3.01471	+	3.01471i	0.541458	+	0.541458i	0.923956	−	0.382498i	$-0.124936\pi$
$31$	3.01471	+	3.01471i	0.541458	+	0.541458i	−0.382498	+	0.923956i	$0.624936\pi$
$32$		0			0
$33$		−	2.02200i		−	0.351985i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	8.46233			1.39120			0.695599	−	0.718430i	$-0.255139\pi$
$37$	8.46233			1.39120			0.695599	+	0.718430i	$0.255139\pi$
$38$		0			0
$39$	2.96906	+	2.04565i	0.475430	+	0.327567i
$40$		0			0
$41$	−4.50834	−	4.50834i	−0.704085	−	0.704085i	0.261200	−	0.965285i	$-0.415882\pi$
$41$	−4.50834	−	4.50834i	−0.704085	−	0.704085i	−0.965285	+	0.261200i	$0.915882\pi$
$42$		0			0
$43$	2.02200	+	2.02200i	0.308352	+	0.308352i	0.844270	−	0.535918i	$-0.180034\pi$
$43$	2.02200	+	2.02200i	0.308352	+	0.308352i	−0.535918	+	0.844270i	$0.680034\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−5.00531			−0.730099			−0.365050	−	0.930988i	$-0.618948\pi$
$47$	−5.00531			−0.730099			−0.365050	+	0.930988i	$0.618948\pi$
$48$		0			0
$49$	−6.63060			−0.947228
$50$		0			0
$51$			1.53929i			0.215543i
$52$		0			0
$53$	−4.48089	+	4.48089i	−0.615498	+	0.615498i	−0.944373	−	0.328875i	$-0.893330\pi$
$53$	−4.48089	+	4.48089i	−0.615498	+	0.615498i	0.328875	+	0.944373i	$0.393330\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		−	4.91499i		−	0.651007i
$58$		0			0
$59$	−8.02942	−	8.02942i	−1.04534	−	1.04534i	−0.998922	−	0.0464197i	$-0.985219\pi$
$59$	−8.02942	−	8.02942i	−1.04534	−	1.04534i	−0.0464197	−	0.998922i	$-0.514781\pi$
$60$		0			0
$61$	−14.4282			−1.84735			−0.923674	−	0.383179i	$-0.874829\pi$
$61$	−14.4282			−1.84735			−0.923674	+	0.383179i	$0.874829\pi$
$62$		0			0
$63$			0.607786i			0.0765739i
$64$		0			0
$65$		0			0
$66$		0			0
$67$			8.85067i			1.08128i	0.841254	+	0.540640i	$0.181818\pi$
$67$			8.85067i			1.08128i	−0.841254	+	0.540640i	$0.818182\pi$
$68$		0			0
$69$	−8.49014			−1.02209
$70$		0			0
$71$	6.66154	+	6.66154i	0.790579	+	0.790579i	0.981588	−	0.191009i	$-0.0611760\pi$
$71$	6.66154	+	6.66154i	0.790579	+	0.790579i	−0.191009	+	0.981588i	$0.561176\pi$
$72$		0			0
$73$			9.92030i			1.16108i	0.814231	+	0.580542i	$0.197159\pi$
$73$			9.92030i			1.16108i	−0.814231	+	0.580542i	$0.802841\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	0.868995	−	0.868995i	0.0990311	−	0.0990311i
$78$		0			0
$79$		−	11.4774i		−	1.29131i	−0.763630	−	0.645654i	$-0.776585\pi$
$79$		−	11.4774i		−	1.29131i	0.763630	−	0.645654i	$-0.223415\pi$
$80$		0			0
$81$	−1.00000			−0.111111
$82$		0			0
$83$	−10.2649			−1.12672			−0.563358	−	0.826213i	$-0.690491\pi$
$83$	−10.2649			−1.12672			−0.563358	+	0.826213i	$0.690491\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	4.80667	+	4.80667i	0.515329	+	0.515329i
$88$		0			0
$89$	−3.64880	−	3.64880i	−0.386772	−	0.386772i	0.486762	−	0.873535i	$-0.338178\pi$
$89$	−3.64880	−	3.64880i	−0.386772	−	0.386772i	−0.873535	+	0.486762i	$0.838178\pi$
$90$		0			0
$91$	0.396850	+	2.15517i	0.0416012	+	0.225923i
$92$		0			0
$93$	−4.26345			−0.442099
$94$		0			0
$95$		0			0
$96$		0			0
$97$		−	1.28014i		−	0.129978i	−0.997886	−	0.0649891i	$-0.979299\pi$
$97$		−	1.28014i		−	0.129978i	0.997886	−	0.0649891i	$-0.0207013\pi$
$98$		0			0
$99$	1.42977	+	1.42977i	0.143697	+	0.143697i
$100$		0			0
$101$			13.8398i			1.37711i	0.725183	+	0.688556i	$0.241755\pi$
$101$			13.8398i			1.37711i	−0.725183	+	0.688556i	$0.758245\pi$
$102$		0			0
$103$	−0.651547	−	0.651547i	−0.0641988	−	0.0641988i	0.674278	−	0.738477i	$-0.264455\pi$
$103$	−0.651547	−	0.651547i	−0.0641988	−	0.0641988i	−0.738477	+	0.674278i	$0.764455\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	2.61098	+	2.61098i	0.252413	+	0.252413i	0.821959	−	0.569546i	$-0.192881\pi$
$107$	2.61098	+	2.61098i	0.252413	+	0.252413i	−0.569546	+	0.821959i	$0.692881\pi$
$108$		0			0
$109$	3.70634	−	3.70634i	0.355003	−	0.355003i	−0.506964	−	0.861967i	$-0.669232\pi$
$109$	3.70634	−	3.70634i	0.355003	−	0.355003i	0.861967	+	0.506964i	$0.169232\pi$
$110$		0			0
$111$	−5.98377	+	5.98377i	−0.567954	+	0.567954i
$112$		0			0
$113$	3.30214	−	3.30214i	0.310639	−	0.310639i	−0.534518	−	0.845157i	$-0.679507\pi$
$113$	3.30214	−	3.30214i	0.310639	−	0.310639i	0.845157	+	0.534518i	$0.179507\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	−3.54594	+	0.652944i	−0.327822	+	0.0603647i
$118$		0			0
$119$	−0.661539	+	0.661539i	−0.0606432	+	0.0606432i
$120$		0			0
$121$			6.91152i			0.628320i
$122$		0			0
$123$	6.37576			0.574883
$124$		0			0
$125$		0			0
$126$		0			0
$127$	14.0726	−	14.0726i	1.24874	−	1.24874i	0.292468	−	0.956275i	$-0.405523\pi$
$127$	14.0726	−	14.0726i	1.24874	−	1.24874i	0.956275	−	0.292468i	$-0.0944766\pi$
$128$		0			0
$129$	−2.85954			−0.251768
$130$		0			0
$131$	−15.6572			−1.36798			−0.683988	−	0.729494i	$-0.739756\pi$
$131$	−15.6572			−1.36798			−0.683988	+	0.729494i	$0.739756\pi$
$132$		0			0
$133$	2.11232	−	2.11232i	0.183161	−	0.183161i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−18.2277			−1.55730			−0.778651	−	0.627458i	$-0.784096\pi$
$137$	−18.2277			−1.55730			−0.778651	+	0.627458i	$0.784096\pi$
$138$		0			0
$139$		−	19.2259i		−	1.63072i	−0.578954	−	0.815360i	$-0.696539\pi$
$139$		−	19.2259i		−	1.63072i	0.578954	−	0.815360i	$-0.303461\pi$
$140$		0			0
$141$	3.53929	−	3.53929i	0.298062	−	0.298062i
$142$		0			0
$143$	6.00343	+	4.13631i	0.502032	+	0.345896i
$144$		0			0
$145$		0			0
$146$		0			0
$147$	4.68854	−	4.68854i	0.386704	−	0.386704i
$148$		0			0
$149$	−12.4719	+	12.4719i	−1.02174	+	1.02174i	−0.0219817	+	0.999758i	$0.506998\pi$
$149$	−12.4719	+	12.4719i	−1.02174	+	1.02174i	−0.999758	+	0.0219817i	$0.993002\pi$
$150$		0			0
$151$	−6.18177	+	6.18177i	−0.503065	+	0.503065i	−0.912389	−	0.409324i	$-0.865765\pi$
$151$	−6.18177	+	6.18177i	−0.503065	+	0.503065i	0.409324	+	0.912389i	$0.365765\pi$
$152$		0			0
$153$	−1.08844	−	1.08844i	−0.0879952	−	0.0879952i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−7.80798	−	7.80798i	−0.623145	−	0.623145i	0.323189	−	0.946334i	$-0.395245\pi$
$157$	−7.80798	−	7.80798i	−0.623145	−	0.623145i	−0.946334	+	0.323189i	$0.895245\pi$
$158$		0			0
$159$		−	6.33694i		−	0.502552i
$160$		0			0
$161$	−3.64880	−	3.64880i	−0.287566	−	0.287566i
$162$		0			0
$163$		−	14.7915i		−	1.15856i	−0.815128	−	0.579281i	$-0.803333\pi$
$163$		−	14.7915i		−	1.15856i	0.815128	−	0.579281i	$-0.196667\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−25.4099			−1.96628			−0.983140	−	0.182855i	$-0.941466\pi$
$167$	−25.4099			−1.96628			−0.983140	+	0.182855i	$0.941466\pi$
$168$		0			0
$169$	−12.1473	+	4.63060i	−0.934410	+	0.356200i
$170$		0			0
$171$	3.47542	+	3.47542i	0.265772	+	0.265772i
$172$		0			0
$173$	8.30745	+	8.30745i	0.631603	+	0.631603i	0.948470	−	0.316867i	$-0.102631\pi$
$173$	8.30745	+	8.30745i	0.631603	+	0.631603i	−0.316867	+	0.948470i	$0.602631\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	11.3553			0.853518
$178$		0			0
$179$	−13.3231			−0.995814			−0.497907	−	0.867231i	$-0.665898\pi$
$179$	−13.3231			−0.995814			−0.497907	+	0.867231i	$0.665898\pi$
$180$		0			0
$181$		−	5.32420i		−	0.395745i	−0.980228	−	0.197873i	$-0.936597\pi$
$181$		−	5.32420i		−	0.395745i	0.980228	−	0.197873i	$-0.0634032\pi$
$182$		0			0
$183$	10.2023	−	10.2023i	0.754177	−	0.754177i
$184$		0			0
$185$		0			0
$186$		0			0
$187$			3.11244i			0.227604i
$188$		0			0
$189$	−0.429770	−	0.429770i	−0.0312612	−	0.0312612i
$190$		0			0
$191$	16.9803			1.22865			0.614325	−	0.789053i	$-0.289429\pi$
$191$	16.9803			1.22865			0.614325	+	0.789053i	$0.289429\pi$
$192$		0			0
$193$		−	6.93699i		−	0.499336i	−0.968332	−	0.249668i	$-0.919679\pi$
$193$		−	6.93699i		−	0.499336i	0.968332	−	0.249668i	$-0.0803214\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			0.868198i			0.0618565i	0.999522	+	0.0309283i	$0.00984634\pi$
$197$			0.868198i			0.0618565i	−0.999522	+	0.0309283i	$0.990154\pi$
$198$		0			0
$199$	−14.8231			−1.05078			−0.525392	−	0.850860i	$-0.676081\pi$
$199$	−14.8231			−1.05078			−0.525392	+	0.850860i	$0.676081\pi$
$200$		0			0
$201$	−6.25837	−	6.25837i	−0.441431	−	0.441431i
$202$		0			0
$203$			4.13152i			0.289976i
$204$		0			0
$205$		0			0
$206$		0			0
$207$	6.00343	−	6.00343i	0.417267	−	0.417267i
$208$		0			0
$209$		−	9.93811i		−	0.687434i
$210$		0			0
$211$	10.3890			0.715209			0.357604	−	0.933873i	$-0.383594\pi$
$211$	10.3890			0.715209			0.357604	+	0.933873i	$0.383594\pi$
$212$		0			0
$213$	−9.42084			−0.645505
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−1.83230	−	1.83230i	−0.124385	−	0.124385i
$218$		0			0
$219$	−7.01471	−	7.01471i	−0.474010	−	0.474010i
$220$		0			0
$221$	−4.57023	−	3.14885i	−0.307427	−	0.211815i
$222$		0			0
$223$	3.46196			0.231830			0.115915	−	0.993259i	$-0.463020\pi$
$223$	3.46196			0.231830			0.115915	+	0.993259i	$0.463020\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$			16.2675i			1.07971i	0.841757	+	0.539856i	$0.181521\pi$
$227$			16.2675i			1.07971i	−0.841757	+	0.539856i	$0.818479\pi$
$228$		0			0
$229$	−10.4155	−	10.4155i	−0.688277	−	0.688277i	0.273574	−	0.961851i	$-0.411794\pi$
$229$	−10.4155	−	10.4155i	−0.688277	−	0.688277i	−0.961851	+	0.273574i	$0.911794\pi$
$230$		0			0
$231$			1.22894i			0.0808586i
$232$		0			0
$233$	−8.52489	−	8.52489i	−0.558484	−	0.558484i	0.370391	−	0.928876i	$-0.379224\pi$
$233$	−8.52489	−	8.52489i	−0.558484	−	0.558484i	−0.928876	+	0.370391i	$0.879224\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	8.11575	+	8.11575i	0.527174	+	0.527174i
$238$		0			0
$239$	−1.52108	+	1.52108i	−0.0983904	+	0.0983904i	−0.754589	−	0.656198i	$-0.772164\pi$
$239$	−1.52108	+	1.52108i	−0.0983904	+	0.0983904i	0.656198	+	0.754589i	$0.272164\pi$
$240$		0			0
$241$	−4.34968	+	4.34968i	−0.280187	+	0.280187i	−0.833184	−	0.552996i	$-0.813484\pi$
$241$	−4.34968	+	4.34968i	−0.280187	+	0.280187i	0.552996	+	0.833184i	$0.313484\pi$
$242$		0			0
$243$	0.707107	−	0.707107i	0.0453609	−	0.0453609i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	14.5929	+	10.0544i	0.928523	+	0.639745i
$248$		0			0
$249$	7.25837	−	7.25837i	0.459980	−	0.459980i
$250$		0			0
$251$			7.29761i			0.460621i	0.973117	+	0.230310i	$0.0739742\pi$
$251$			7.29761i			0.460621i	−0.973117	+	0.230310i	$0.926026\pi$
$252$		0			0
$253$	−17.1671			−1.07928
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−7.87255	+	7.87255i	−0.491076	+	0.491076i	−0.908645	−	0.417569i	$-0.862882\pi$
$257$	−7.87255	+	7.87255i	−0.491076	+	0.491076i	0.417569	+	0.908645i	$0.362882\pi$
$258$		0			0
$259$	−5.14329			−0.319588
$260$		0			0
$261$	−6.79765			−0.420764
$262$		0			0
$263$	−6.78491	+	6.78491i	−0.418375	+	0.418375i	−0.884643	−	0.466268i	$-0.845598\pi$
$263$	−6.78491	+	6.78491i	−0.418375	+	0.418375i	0.466268	+	0.884643i	$0.345598\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	5.16019			0.315798
$268$		0			0
$269$			0.0254701i			0.00155294i	1.00000		0.000776470i	$0.000247158\pi$
$269$			0.0254701i			0.00155294i	−1.00000		0.000776470i	$0.999753\pi$
$270$		0			0
$271$	−16.3910	+	16.3910i	−0.995681	+	0.995681i	−0.999991	−	0.00430939i	$-0.998628\pi$
$271$	−16.3910	+	16.3910i	−0.995681	+	0.995681i	0.00430939	+	0.999991i	$0.498628\pi$
$272$		0			0
$273$	−1.80455	−	1.24332i	−0.109216	−	0.0752492i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−13.7219	+	13.7219i	−0.824469	+	0.824469i	−0.986745	−	0.162276i	$-0.948116\pi$
$277$	−13.7219	+	13.7219i	−0.824469	+	0.824469i	0.162276	+	0.986745i	$0.448116\pi$
$278$		0			0
$279$	3.01471	−	3.01471i	0.180486	−	0.180486i
$280$		0			0
$281$	−8.12073	+	8.12073i	−0.484442	+	0.484442i	−0.906547	−	0.422105i	$-0.861291\pi$
$281$	−8.12073	+	8.12073i	−0.484442	+	0.484442i	0.422105	+	0.906547i	$0.361291\pi$
$282$		0			0
$283$	−10.7475	−	10.7475i	−0.638874	−	0.638874i	0.311403	−	0.950278i	$-0.399201\pi$
$283$	−10.7475	−	10.7475i	−0.638874	−	0.638874i	−0.950278	+	0.311403i	$0.899201\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	2.74011	+	2.74011i	0.161744	+	0.161744i
$288$		0			0
$289$			14.6306i			0.860623i
$290$		0			0
$291$	0.905194	+	0.905194i	0.0530634	+	0.0530634i
$292$		0			0
$293$			15.4369i			0.901835i	0.892566	+	0.450918i	$0.148903\pi$
$293$			15.4369i			0.901835i	−0.892566	+	0.450918i	$0.851097\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	−2.02200			−0.117328
$298$		0			0
$299$	17.3679	−	25.2077i	1.00441	−	1.45780i
$300$		0			0
$301$	−1.22894	−	1.22894i	−0.0708351	−	0.0708351i
$302$		0			0
$303$	−9.78622	−	9.78622i	−0.562204	−	0.562204i
$304$		0			0
$305$		0			0
$306$		0			0
$307$	−23.0546			−1.31579			−0.657897	−	0.753108i	$-0.728554\pi$
$307$	−23.0546			−1.31579			−0.657897	+	0.753108i	$0.728554\pi$
$308$		0			0
$309$	0.921426			0.0524181
$310$		0			0
$311$			2.49995i			0.141759i	0.997485	+	0.0708797i	$0.0225806\pi$
$311$			2.49995i			0.141759i	−0.997485	+	0.0708797i	$0.977419\pi$
$312$		0			0
$313$	7.68165	−	7.68165i	0.434192	−	0.434192i	−0.455859	−	0.890052i	$-0.650668\pi$
$313$	7.68165	−	7.68165i	0.434192	−	0.434192i	0.890052	+	0.455859i	$0.150668\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	25.8032i		−	1.44925i	−0.689142	−	0.724627i	$-0.742012\pi$
$317$		−	25.8032i		−	1.44925i	0.689142	−	0.724627i	$-0.257988\pi$
$318$		0			0
$319$	9.71908	+	9.71908i	0.544164	+	0.544164i
$320$		0			0
$321$	−3.69248			−0.206094
$322$		0			0
$323$			7.56558i			0.420960i
$324$		0			0
$325$		0			0
$326$		0			0
$327$			5.24156i			0.289859i
$328$		0			0
$329$	3.04216			0.167720
$330$		0			0
$331$	7.29563	+	7.29563i	0.401004	+	0.401004i	0.878587	−	0.477583i	$-0.158487\pi$
$331$	7.29563	+	7.29563i	0.401004	+	0.401004i	−0.477583	+	0.878587i	$0.658487\pi$
$332$		0			0
$333$		−	8.46233i		−	0.463733i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	24.6910	−	24.6910i	1.34501	−	1.34501i	0.454009	−	0.890997i	$-0.349994\pi$
$337$	24.6910	−	24.6910i	1.34501	−	1.34501i	0.890997	−	0.454009i	$-0.150006\pi$
$338$		0			0
$339$			4.66993i			0.253636i
$340$		0			0
$341$	−8.62069			−0.466836
$342$		0			0
$343$	8.28449			0.447320
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−14.8361	−	14.8361i	−0.796443	−	0.796443i	0.186090	−	0.982533i	$-0.440418\pi$
$347$	−14.8361	−	14.8361i	−0.796443	−	0.796443i	−0.982533	+	0.186090i	$0.940418\pi$
$348$		0			0
$349$	−15.3103	−	15.3103i	−0.819544	−	0.819544i	0.166498	−	0.986042i	$-0.446754\pi$
$349$	−15.3103	−	15.3103i	−0.819544	−	0.819544i	−0.986042	+	0.166498i	$0.946754\pi$
$350$		0			0
$351$	2.04565	−	2.96906i	0.109189	−	0.158477i
$352$		0			0
$353$	29.9248			1.59274			0.796369	−	0.604810i	$-0.206751\pi$
$353$	29.9248			1.59274			0.796369	+	0.604810i	$0.206751\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		−	0.935558i		−	0.0495150i
$358$		0			0
$359$	−13.9636	−	13.9636i	−0.736970	−	0.736970i	0.235021	−	0.971990i	$-0.424484\pi$
$359$	−13.9636	−	13.9636i	−0.736970	−	0.736970i	−0.971990	+	0.235021i	$0.924484\pi$
$360$		0			0
$361$		−	5.15715i		−	0.271429i
$362$		0			0
$363$	−4.88718	−	4.88718i	−0.256510	−	0.256510i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	4.91994	+	4.91994i	0.256819	+	0.256819i	0.823759	−	0.566940i	$-0.191873\pi$
$367$	4.91994	+	4.91994i	0.256819	+	0.256819i	−0.566940	+	0.823759i	$0.691873\pi$
$368$		0			0
$369$	−4.50834	+	4.50834i	−0.234695	+	0.234695i
$370$		0			0
$371$	2.72343	−	2.72343i	0.141393	−	0.141393i
$372$		0			0
$373$	−3.38966	+	3.38966i	−0.175510	+	0.175510i	−0.789395	−	0.613885i	$-0.789606\pi$
$373$	−3.38966	+	3.38966i	−0.175510	+	0.175510i	0.613885	+	0.789395i	$0.289606\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−24.1040	+	4.43849i	−1.24142	+	0.228594i
$378$		0			0
$379$	0.344783	−	0.344783i	0.0177103	−	0.0177103i	−0.698196	−	0.715906i	$-0.746014\pi$
$379$	0.344783	−	0.344783i	0.0177103	−	0.0177103i	0.715906	+	0.698196i	$0.246014\pi$
$380$		0			0
$381$			19.9017i			1.01959i
$382$		0			0
$383$	1.82951			0.0934834			0.0467417	−	0.998907i	$-0.485116\pi$
$383$	1.82951			0.0934834			0.0467417	+	0.998907i	$0.485116\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	2.02200	−	2.02200i	0.102784	−	0.102784i
$388$		0			0
$389$	−32.3565			−1.64054			−0.820271	−	0.571975i	$-0.806177\pi$
$389$	−32.3565			−1.64054			−0.820271	+	0.571975i	$0.806177\pi$
$390$		0			0
$391$	13.0688			0.660915
$392$		0			0
$393$	11.0713	−	11.0713i	0.558474	−	0.558474i
$394$		0			0
$395$		0			0
$396$		0			0
$397$	38.1329			1.91384			0.956918	−	0.290359i	$-0.0937746\pi$
$397$	38.1329			1.91384			0.956918	+	0.290359i	$0.0937746\pi$
$398$		0			0
$399$			2.98726i			0.149550i
$400$		0			0
$401$	−6.85954	+	6.85954i	−0.342549	+	0.342549i	−0.857325	−	0.514776i	$-0.827875\pi$
$401$	−6.85954	+	6.85954i	−0.342549	+	0.342549i	0.514776	+	0.857325i	$0.327875\pi$
$402$		0			0
$403$	8.72154	−	12.6584i	0.434451	−	0.630560i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−12.0992	+	12.0992i	−0.599734	+	0.599734i
$408$		0			0
$409$	6.04216	−	6.04216i	0.298765	−	0.298765i	−0.541765	−	0.840530i	$-0.682244\pi$
$409$	6.04216	−	6.04216i	0.298765	−	0.298765i	0.840530	+	0.541765i	$0.182244\pi$
$410$		0			0
$411$	12.8890	−	12.8890i	0.635766	−	0.635766i
$412$		0			0
$413$	4.88017	+	4.88017i	0.240138	+	0.240138i
$414$		0			0
$415$		0			0
$416$		0			0
$417$	13.5948	+	13.5948i	0.665739	+	0.665739i
$418$		0			0
$419$			4.56184i			0.222861i	0.993772	+	0.111430i	$0.0355432\pi$
$419$			4.56184i			0.222861i	−0.993772	+	0.111430i	$0.964457\pi$
$420$		0			0
$421$	−16.1080	−	16.1080i	−0.785056	−	0.785056i	0.195623	−	0.980679i	$-0.437327\pi$
$421$	−16.1080	−	16.1080i	−0.785056	−	0.785056i	−0.980679	+	0.195623i	$0.937327\pi$
$422$		0			0
$423$			5.00531i			0.243366i
$424$		0			0
$425$		0			0
$426$		0			0
$427$	8.76929			0.424376
$428$		0			0
$429$	−7.16988	+	1.32025i	−0.346165	+	0.0637424i
$430$		0			0
$431$	2.79765	+	2.79765i	0.134758	+	0.134758i	0.771268	−	0.636510i	$-0.219623\pi$
$431$	2.79765	+	2.79765i	0.134758	+	0.134758i	−0.636510	+	0.771268i	$0.719623\pi$
$432$		0			0
$433$	−9.45845	−	9.45845i	−0.454544	−	0.454544i	0.442315	−	0.896860i	$-0.354157\pi$
$433$	−9.45845	−	9.45845i	−0.454544	−	0.454544i	−0.896860	+	0.442315i	$0.854157\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−41.7289			−1.99617
$438$		0			0
$439$	17.3791			0.829459			0.414730	−	0.909945i	$-0.363876\pi$
$439$	17.3791			0.829459			0.414730	+	0.909945i	$0.363876\pi$
$440$		0			0
$441$			6.63060i			0.315743i
$442$		0			0
$443$	6.25770	−	6.25770i	0.297312	−	0.297312i	−0.542648	−	0.839960i	$-0.682578\pi$
$443$	6.25770	−	6.25770i	0.297312	−	0.297312i	0.839960	+	0.542648i	$0.182578\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		−	17.6380i		−	0.834247i
$448$		0			0
$449$	26.6910	+	26.6910i	1.25962	+	1.25962i	0.951272	+	0.308352i	$0.0997775\pi$
$449$	26.6910	+	26.6910i	1.25962	+	1.25962i	0.308352	+	0.951272i	$0.400223\pi$
$450$		0			0
$451$	12.8918			0.607051
$452$		0			0
$453$		−	8.74234i		−	0.410751i
$454$		0			0
$455$		0			0
$456$		0			0
$457$			0.332717i			0.0155638i	0.999970	+	0.00778192i	$0.00247709\pi$
$457$			0.332717i			0.0155638i	−0.999970	+	0.00778192i	$0.997523\pi$
$458$		0			0
$459$	1.53929			0.0718478
$460$		0			0
$461$	−18.9268	−	18.9268i	−0.881508	−	0.881508i	0.112180	−	0.993688i	$-0.464217\pi$
$461$	−18.9268	−	18.9268i	−0.881508	−	0.881508i	−0.993688	+	0.112180i	$0.964217\pi$
$462$		0			0
$463$			19.1037i			0.887823i	0.896071	+	0.443912i	$0.146410\pi$
$463$			19.1037i			0.887823i	−0.896071	+	0.443912i	$0.853590\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−14.1330	+	14.1330i	−0.653999	+	0.653999i	−0.953954	−	0.299954i	$-0.903029\pi$
$467$	−14.1330	+	14.1330i	−0.653999	+	0.653999i	0.299954	+	0.953954i	$0.403029\pi$
$468$		0			0
$469$		−	5.37931i		−	0.248394i
$470$		0			0
$471$	11.0422			0.508796
$472$		0			0
$473$	−5.78199			−0.265856
$474$		0			0
$475$		0			0
$476$		0			0
$477$	4.48089	+	4.48089i	0.205166	+	0.205166i
$478$		0			0
$479$	18.5378	+	18.5378i	0.847012	+	0.847012i	0.989759	−	0.142747i	$-0.0455935\pi$
$479$	18.5378	+	18.5378i	0.847012	+	0.847012i	−0.142747	+	0.989759i	$0.545594\pi$
$480$		0			0
$481$	−5.52543	−	30.0069i	−0.251938	−	1.36820i
$482$		0			0
$483$	5.16019			0.234797
$484$		0			0
$485$		0			0
$486$		0			0
$487$			5.83134i			0.264243i	0.991234	+	0.132122i	$0.0421789\pi$
$487$			5.83134i			0.264243i	−0.991234	+	0.132122i	$0.957821\pi$
$488$		0			0
$489$	10.4592	+	10.4592i	0.472981	+	0.472981i
$490$		0			0
$491$		−	9.66589i		−	0.436215i	−0.975925	−	0.218108i	$-0.930012\pi$
$491$		−	9.66589i		−	0.436215i	0.975925	−	0.218108i	$-0.0699884\pi$
$492$		0			0
$493$	−7.39884	−	7.39884i	−0.333227	−	0.333227i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−4.04879	−	4.04879i	−0.181613	−	0.181613i
$498$		0			0
$499$	18.5176	−	18.5176i	0.828961	−	0.828961i	−0.158412	−	0.987373i	$-0.550637\pi$
$499$	18.5176	−	18.5176i	0.828961	−	0.828961i	0.987373	+	0.158412i	$0.0506375\pi$
$500$		0			0
$501$	17.9675	−	17.9675i	0.802730	−	0.802730i
$502$		0			0
$503$	−30.8125	+	30.8125i	−1.37386	+	1.37386i	−0.519225	+	0.854638i	$0.673779\pi$
$503$	−30.8125	+	30.8125i	−1.37386	+	1.37386i	−0.854638	+	0.519225i	$0.826221\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	5.31513	−	11.8638i	0.236053	−	0.526889i
$508$		0			0
$509$	−4.57023	+	4.57023i	−0.202572	+	0.202572i	−0.801101	−	0.598529i	$-0.795752\pi$
$509$	−4.57023	+	4.57023i	−0.202572	+	0.202572i	0.598529	+	0.801101i	$0.295752\pi$
$510$		0			0
$511$		−	6.02942i		−	0.266726i
$512$		0			0
$513$	−4.91499			−0.217002
$514$		0			0
$515$		0			0
$516$		0			0
$517$	7.15644	−	7.15644i	0.314740	−	0.314740i
$518$		0			0
$519$	−11.7485			−0.515702
$520$		0			0
$521$	−3.62382			−0.158762			−0.0793812	−	0.996844i	$-0.525294\pi$
$521$	−3.62382			−0.158762			−0.0793812	+	0.996844i	$0.525294\pi$
$522$		0			0
$523$	12.0506	−	12.0506i	0.526937	−	0.526937i	−0.392721	−	0.919658i	$-0.628466\pi$
$523$	12.0506	−	12.0506i	0.526937	−	0.526937i	0.919658	+	0.392721i	$0.128466\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	6.56267			0.285874
$528$		0			0
$529$			49.0824i			2.13402i
$530$		0			0
$531$	−8.02942	+	8.02942i	−0.348447	+	0.348447i
$532$		0			0
$533$	−13.0426	+	18.9300i	−0.564938	+	0.819949i
$534$		0			0
$535$		0			0
$536$		0			0
$537$	9.42084	−	9.42084i	0.406539	−	0.406539i
$538$		0			0
$539$	9.48023	−	9.48023i	0.408342	−	0.408342i
$540$		0			0
$541$	−11.7318	+	11.7318i	−0.504390	+	0.504390i	−0.912799	−	0.408409i	$-0.866084\pi$
$541$	−11.7318	+	11.7318i	−0.504390	+	0.504390i	0.408409	+	0.912799i	$0.366084\pi$
$542$		0			0
$543$	3.76478	+	3.76478i	0.161562	+	0.161562i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−0.353521	−	0.353521i	−0.0151155	−	0.0151155i	0.699509	−	0.714624i	$-0.253402\pi$
$547$	−0.353521	−	0.353521i	−0.0151155	−	0.0151155i	−0.714624	+	0.699509i	$0.753402\pi$
$548$		0			0
$549$			14.4282i			0.615783i
$550$		0			0
$551$	23.6247	+	23.6247i	1.00645	+	1.00645i
$552$		0			0
$553$			6.97581i			0.296641i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−7.39884			−0.313499			−0.156749	−	0.987638i	$-0.550102\pi$
$557$	−7.39884			−0.313499			−0.156749	+	0.987638i	$0.550102\pi$
$558$		0			0
$559$	5.84963	−	8.49014i	0.247413	−	0.359094i
$560$		0			0
$561$	−2.20083	−	2.20083i	−0.0929189	−	0.0929189i
$562$		0			0
$563$	−2.79161	−	2.79161i	−0.117652	−	0.117652i	0.645829	−	0.763482i	$-0.276512\pi$
$563$	−2.79161	−	2.79161i	−0.117652	−	0.117652i	−0.763482	+	0.645829i	$0.776512\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	0.607786			0.0255246
$568$		0			0
$569$	5.22478			0.219034			0.109517	−	0.993985i	$-0.465070\pi$
$569$	5.22478			0.219034			0.109517	+	0.993985i	$0.465070\pi$
$570$		0			0
$571$			41.9981i			1.75757i	0.477222	+	0.878783i	$0.341644\pi$
$571$			41.9981i			1.75757i	−0.477222	+	0.878783i	$0.658356\pi$
$572$		0			0
$573$	−12.0069	+	12.0069i	−0.501594	+	0.501594i
$574$		0			0
$575$		0			0
$576$		0			0
$577$			5.32973i			0.221879i	0.993827	+	0.110940i	$0.0353860\pi$
$577$			5.32973i			0.221879i	−0.993827	+	0.110940i	$0.964614\pi$
$578$		0			0
$579$	4.90519	+	4.90519i	0.203853	+	0.203853i
$580$		0			0
$581$	6.23885			0.258831
$582$		0			0
$583$		−	12.8133i		−	0.530672i
$584$		0			0
$585$		0			0
$586$		0			0
$587$			16.5912i			0.684791i	0.939556	+	0.342395i	$0.111238\pi$
$587$			16.5912i			0.684791i	−0.939556	+	0.342395i	$0.888762\pi$
$588$		0			0
$589$	−20.9548			−0.863428
$590$		0			0
$591$	−0.613909	−	0.613909i	−0.0252528	−	0.0252528i
$592$		0			0
$593$			31.2850i			1.28472i	0.766402	+	0.642361i	$0.222045\pi$
$593$			31.2850i			1.28472i	−0.766402	+	0.642361i	$0.777955\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	10.4815	−	10.4815i	0.428981	−	0.428981i
$598$		0			0
$599$		−	13.8485i		−	0.565834i	−0.959144	−	0.282917i	$-0.908698\pi$
$599$		−	13.8485i		−	0.565834i	0.959144	−	0.282917i	$-0.0913022\pi$
$600$		0			0
$601$	15.9647			0.651214			0.325607	−	0.945505i	$-0.394431\pi$
$601$	15.9647			0.651214			0.325607	+	0.945505i	$0.394431\pi$
$602$		0			0
$603$	8.85067			0.360427
$604$		0			0
$605$		0			0
$606$		0			0
$607$	3.41605	+	3.41605i	0.138653	+	0.138653i	0.773027	−	0.634373i	$-0.218742\pi$
$607$	3.41605	+	3.41605i	0.138653	+	0.138653i	−0.634373	+	0.773027i	$0.718742\pi$
$608$		0			0
$609$	−2.92143	−	2.92143i	−0.118382	−	0.118382i
$610$		0			0
$611$	3.26819	+	17.7485i	0.132217	+	0.718028i
$612$		0			0
$613$	24.4340			0.986879			0.493439	−	0.869780i	$-0.335740\pi$
$613$	24.4340			0.986879			0.493439	+	0.869780i	$0.335740\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			6.27079i			0.252452i	0.992001	+	0.126226i	$0.0402865\pi$
$617$			6.27079i			0.252452i	−0.992001	+	0.126226i	$0.959713\pi$
$618$		0			0
$619$	29.4696	+	29.4696i	1.18448	+	1.18448i	0.978571	+	0.205911i	$0.0660157\pi$
$619$	29.4696	+	29.4696i	1.18448	+	1.18448i	0.205911	+	0.978571i	$0.433984\pi$
$620$		0			0
$621$			8.49014i			0.340697i
$622$		0			0
$623$	2.21769	+	2.21769i	0.0888500	+	0.0888500i
$624$		0			0
$625$		0			0
$626$		0			0
$627$	7.02731	+	7.02731i	0.280644	+	0.280644i
$628$		0			0
$629$	9.21074	−	9.21074i	0.367256	−	0.367256i
$630$		0			0
$631$	29.5343	−	29.5343i	1.17574	−	1.17574i	0.194922	−	0.980819i	$-0.437555\pi$
$631$	29.5343	−	29.5343i	1.17574	−	1.17574i	0.980819	−	0.194922i	$-0.0624453\pi$
$632$		0			0
$633$	−7.34614	+	7.34614i	−0.291983	+	0.291983i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	4.32941	+	23.5117i	0.171537	+	0.931566i
$638$		0			0
$639$	6.66154	−	6.66154i	0.263526	−	0.263526i
$640$		0			0
$641$		−	46.7916i		−	1.84816i	−0.382204	−	0.924078i	$-0.624835\pi$
$641$		−	46.7916i		−	1.84816i	0.382204	−	0.924078i	$-0.375165\pi$
$642$		0			0
$643$	34.5905			1.36412			0.682058	−	0.731298i	$-0.261085\pi$
$643$	34.5905			1.36412			0.682058	+	0.731298i	$0.261085\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−7.26061	+	7.26061i	−0.285444	+	0.285444i	−0.835276	−	0.549831i	$-0.814692\pi$
$647$	−7.26061	+	7.26061i	−0.285444	+	0.285444i	0.549831	+	0.835276i	$0.314692\pi$
$648$		0			0
$649$	22.9605			0.901277
$650$		0			0
$651$	2.59126			0.101560
$652$		0			0
$653$	19.2738	−	19.2738i	0.754241	−	0.754241i	−0.221027	−	0.975268i	$-0.570941\pi$
$653$	19.2738	−	19.2738i	0.754241	−	0.754241i	0.975268	+	0.221027i	$0.0709407\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	9.92030			0.387028
$658$		0			0
$659$			15.9715i			0.622161i	0.950384	+	0.311080i	$0.100691\pi$
$659$			15.9715i			0.622161i	−0.950384	+	0.311080i	$0.899309\pi$
$660$		0			0
$661$	24.1346	−	24.1346i	0.938727	−	0.938727i	−0.0595013	−	0.998228i	$-0.518951\pi$
$661$	24.1346	−	24.1346i	0.938727	−	0.938727i	0.998228	+	0.0595013i	$0.0189511\pi$
$662$		0			0
$663$	5.45821	−	1.00507i	0.211979	−	0.0390336i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	40.8093	−	40.8093i	1.58014	−	1.58014i
$668$		0			0
$669$	−2.44798	+	2.44798i	−0.0946443	+	0.0946443i
$670$		0			0
$671$	20.6291	−	20.6291i	0.796377	−	0.796377i
$672$		0			0
$673$	−28.3904	−	28.3904i	−1.09437	−	1.09437i	−0.995056	−	0.0993152i	$-0.968335\pi$
$673$	−28.3904	−	28.3904i	−1.09437	−	1.09437i	−0.0993152	−	0.995056i	$-0.531665\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−23.0184	−	23.0184i	−0.884669	−	0.884669i	0.109336	−	0.994005i	$-0.465128\pi$
$677$	−23.0184	−	23.0184i	−0.884669	−	0.884669i	−0.994005	+	0.109336i	$0.965128\pi$
$678$		0			0
$679$			0.778050i			0.0298588i
$680$		0			0
$681$	−11.5029	−	11.5029i	−0.440791	−	0.440791i
$682$		0			0
$683$			29.2773i			1.12026i	0.828403	+	0.560132i	$0.189250\pi$
$683$			29.2773i			1.12026i	−0.828403	+	0.560132i	$0.810750\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	14.7298			0.561975
$688$		0			0
$689$	18.8147	+	12.9632i	0.716784	+	0.493858i
$690$		0			0
$691$	−3.74653	−	3.74653i	−0.142525	−	0.142525i	0.632244	−	0.774769i	$-0.282134\pi$
$691$	−3.74653	−	3.74653i	−0.142525	−	0.142525i	−0.774769	+	0.632244i	$0.782134\pi$
$692$		0			0
$693$	−0.868995	−	0.868995i	−0.0330104	−	0.0330104i
$694$		0			0
$695$		0			0
$696$		0			0
$697$	−9.81413			−0.371736
$698$		0			0
$699$	12.0560			0.456001
$700$		0			0
$701$		−	17.0786i		−	0.645049i	−0.946561	−	0.322524i	$-0.895469\pi$
$701$		−	17.0786i		−	0.645049i	0.946561	−	0.322524i	$-0.104531\pi$
$702$		0			0
$703$	−29.4102	+	29.4102i	−1.10923	+	1.10923i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		−	8.41165i		−	0.316353i
$708$		0			0
$709$	12.6012	+	12.6012i	0.473247	+	0.473247i	0.902964	−	0.429717i	$-0.141387\pi$
$709$	12.6012	+	12.6012i	0.473247	+	0.473247i	−0.429717	+	0.902964i	$0.641387\pi$
$710$		0			0
$711$	−11.4774			−0.430436
$712$		0			0
$713$			36.1972i			1.35560i
$714$		0			0
$715$		0			0
$716$		0			0
$717$		−	2.15113i		−	0.0803354i
$718$		0			0
$719$	−11.1150			−0.414519			−0.207260	−	0.978286i	$-0.566454\pi$
$719$	−11.1150			−0.414519			−0.207260	+	0.978286i	$0.566454\pi$
$720$		0			0
$721$	0.396001	+	0.396001i	0.0147479	+	0.0147479i
$722$		0			0
$723$		−	6.15137i		−	0.228772i
$724$		0			0
$725$		0			0
$726$		0			0
$727$	14.9048	−	14.9048i	0.552789	−	0.552789i	−0.374456	−	0.927245i	$-0.622171\pi$
$727$	14.9048	−	14.9048i	0.552789	−	0.552789i	0.927245	+	0.374456i	$0.122171\pi$
$728$		0			0
$729$			1.00000i			0.0370370i
$730$		0			0
$731$	4.40165			0.162801
$732$		0			0
$733$	−19.2675			−0.711662			−0.355831	−	0.934550i	$-0.615802\pi$
$733$	−19.2675			−0.711662			−0.355831	+	0.934550i	$0.615802\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−12.6544	−	12.6544i	−0.466131	−	0.466131i
$738$		0			0
$739$	1.14244	+	1.14244i	0.0420252	+	0.0420252i	0.727807	−	0.685782i	$-0.240540\pi$
$739$	1.14244	+	1.14244i	0.0420252	+	0.0420252i	−0.685782	+	0.727807i	$0.740540\pi$
$740$		0			0
$741$	−17.4282	+	3.20921i	−0.640243	+	0.117893i
$742$		0			0
$743$	−21.7133			−0.796584			−0.398292	−	0.917259i	$-0.630397\pi$
$743$	−21.7133			−0.796584			−0.398292	+	0.917259i	$0.630397\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$			10.2649i			0.375572i
$748$		0			0
$749$	−1.58692	−	1.58692i	−0.0579847	−	0.0579847i
$750$		0			0
$751$			6.01377i			0.219446i	0.993962	+	0.109723i	$0.0349963\pi$
$751$			6.01377i			0.219446i	−0.993962	+	0.109723i	$0.965004\pi$
$752$		0			0
$753$	−5.16019	−	5.16019i	−0.188048	−	0.188048i
$754$		0			0
$755$		0			0
$756$		0			0
$757$	33.4215	+	33.4215i	1.21473	+	1.21473i	0.969455	+	0.245271i	$0.0788768\pi$
$757$	33.4215	+	33.4215i	1.21473	+	1.21473i	0.245271	+	0.969455i	$0.421123\pi$
$758$		0			0
$759$	12.1389	−	12.1389i	0.440616	−	0.440616i
$760$		0			0
$761$	14.7103	−	14.7103i	0.533248	−	0.533248i	−0.388290	−	0.921537i	$-0.626934\pi$
$761$	14.7103	−	14.7103i	0.533248	−	0.533248i	0.921537	+	0.388290i	$0.126934\pi$
$762$		0			0
$763$	−2.25267	+	2.25267i	−0.0815520	+	0.0815520i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−23.2291	+	33.7146i	−0.838753	+	1.21736i
$768$		0			0
$769$	−10.2623	+	10.2623i	−0.370069	+	0.370069i	−0.867502	−	0.497433i	$-0.834276\pi$
$769$	−10.2623	+	10.2623i	−0.370069	+	0.370069i	0.497433	+	0.867502i	$0.334276\pi$
$770$		0			0
$771$		−	11.1335i		−	0.400962i
$772$		0			0
$773$	51.3324			1.84630			0.923149	−	0.384443i	$-0.125606\pi$
$773$	51.3324			1.84630			0.923149	+	0.384443i	$0.125606\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	3.63685	−	3.63685i	0.130471	−	0.130471i
$778$		0			0
$779$	31.3368			1.12276
$780$		0			0
$781$	−19.0489			−0.681624
$782$		0			0
$783$	4.80667	−	4.80667i	0.171776	−	0.171776i
$784$		0			0
$785$		0			0
$786$		0			0
$787$	−12.9779			−0.462612			−0.231306	−	0.972881i	$-0.574300\pi$
$787$	−12.9779			−0.462612			−0.231306	+	0.972881i	$0.574300\pi$
$788$		0			0
$789$		−	9.59531i		−	0.341602i
$790$		0			0
$791$	−2.00699	+	2.00699i	−0.0713605	+	0.0713605i
$792$		0			0
$793$	9.42084	+	51.1617i	0.334544	+	1.81680i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	13.9651	−	13.9651i	0.494669	−	0.494669i	−0.415105	−	0.909774i	$-0.636255\pi$
$797$	13.9651	−	13.9651i	0.494669	−	0.494669i	0.909774	+	0.415105i	$0.136255\pi$
$798$		0			0
$799$	−5.44798	+	5.44798i	−0.192736	+	0.192736i
$800$		0			0
$801$	−3.64880	+	3.64880i	−0.128924	+	0.128924i
$802$		0			0
$803$	−14.1837	−	14.1837i	−0.500533	−	0.500533i
$804$		0			0
$805$		0			0
$806$		0			0
$807$	−0.0180101	−	0.0180101i	−0.000633985	−	0.000633985i
$808$		0			0
$809$			28.3056i			0.995172i	0.867415	+	0.497586i	$0.165780\pi$
$809$			28.3056i			0.995172i	−0.867415	+	0.497586i	$0.834220\pi$
$810$		0			0
$811$	−4.95869	−	4.95869i	−0.174123	−	0.174123i	0.614665	−	0.788788i	$-0.289291\pi$
$811$	−4.95869	−	4.95869i	−0.174123	−	0.174123i	−0.788788	+	0.614665i	$0.789291\pi$
$812$		0			0
$813$		−	23.1804i		−	0.812970i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−14.0546			−0.491709
$818$		0			0
$819$	2.15517	−	0.396850i	0.0753078	−	0.0138671i
$820$		0			0
$821$	−6.69400	−	6.69400i	−0.233622	−	0.233622i	0.580581	−	0.814203i	$-0.302826\pi$
$821$	−6.69400	−	6.69400i	−0.233622	−	0.233622i	−0.814203	+	0.580581i	$0.802826\pi$
$822$		0			0
$823$	8.55264	+	8.55264i	0.298126	+	0.298126i	0.840280	−	0.542153i	$-0.182391\pi$
$823$	8.55264	+	8.55264i	0.298126	+	0.298126i	−0.542153	+	0.840280i	$0.682391\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−45.9064			−1.59632			−0.798161	−	0.602445i	$-0.794193\pi$
$827$	−45.9064			−1.59632			−0.798161	+	0.602445i	$0.794193\pi$
$828$		0			0
$829$	7.54211			0.261948			0.130974	−	0.991386i	$-0.458189\pi$
$829$	7.54211			0.261948			0.130974	+	0.991386i	$0.458189\pi$
$830$		0			0
$831$		−	19.4057i		−	0.673176i
$832$		0			0
$833$	−7.21701	+	7.21701i	−0.250054	+	0.250054i
$834$		0			0
$835$		0			0
$836$		0			0
$837$			4.26345i			0.147366i
$838$		0			0
$839$	34.7736	+	34.7736i	1.20052	+	1.20052i	0.974009	+	0.226508i	$0.0727309\pi$
$839$	34.7736	+	34.7736i	1.20052	+	1.20052i	0.226508	+	0.974009i	$0.427269\pi$
$840$		0			0
$841$	−17.2081			−0.593382
$842$		0			0
$843$		−	11.4844i		−	0.395545i
$844$		0			0
$845$		0			0
$846$		0			0
$847$		−	4.20073i		−	0.144339i
$848$		0			0
$849$	15.1993			0.521639
$850$		0			0
$851$	50.8030	+	50.8030i	1.74150	+	1.74150i
$852$		0			0
$853$		−	27.3333i		−	0.935875i	−0.883762	−	0.467938i	$-0.844997\pi$
$853$		−	27.3333i		−	0.935875i	0.883762	−	0.467938i	$-0.155003\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	26.8540	−	26.8540i	0.917317	−	0.917317i	−0.0795168	−	0.996834i	$-0.525338\pi$
$857$	26.8540	−	26.8540i	0.917317	−	0.917317i	0.996834	+	0.0795168i	$0.0253377\pi$
$858$		0			0
$859$			13.8704i			0.473251i	0.971601	+	0.236625i	$0.0760414\pi$
$859$			13.8704i			0.473251i	−0.971601	+	0.236625i	$0.923959\pi$
$860$		0			0
$861$	−3.87510			−0.132063
$862$		0			0
$863$	−21.4412			−0.729866			−0.364933	−	0.931034i	$-0.618908\pi$
$863$	−21.4412			−0.729866			−0.364933	+	0.931034i	$0.618908\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	−10.3454	−	10.3454i	−0.351348	−	0.351348i
$868$		0			0
$869$	16.4100	+	16.4100i	0.556673	+	0.556673i
$870$		0			0
$871$	31.3839	−	5.77899i	1.06340	−	0.195814i
$872$		0			0
$873$	−1.28014			−0.0433261
$874$		0			0
$875$		0			0
$876$		0			0
$877$			29.7193i			1.00355i	0.864998	+	0.501775i	$0.167319\pi$
$877$			29.7193i			1.00355i	−0.864998	+	0.501775i	$0.832681\pi$
$878$		0			0
$879$	−10.9156	−	10.9156i	−0.368173	−	0.368173i
$880$		0			0
$881$		−	42.0303i		−	1.41604i	−0.706193	−	0.708019i	$-0.749589\pi$
$881$		−	42.0303i		−	1.41604i	0.706193	−	0.708019i	$-0.250411\pi$
$882$		0			0
$883$	−2.98546	−	2.98546i	−0.100469	−	0.100469i	0.655086	−	0.755554i	$-0.272632\pi$
$883$	−2.98546	−	2.98546i	−0.100469	−	0.100469i	−0.755554	+	0.655086i	$0.772632\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	27.2346	+	27.2346i	0.914450	+	0.914450i	0.996618	−	0.0821685i	$-0.0261846\pi$
$887$	27.2346	+	27.2346i	0.914450	+	0.914450i	−0.0821685	+	0.996618i	$0.526185\pi$
$888$		0			0
$889$	−8.55315	+	8.55315i	−0.286863	+	0.286863i
$890$		0			0
$891$	1.42977	−	1.42977i	0.0478991	−	0.0478991i
$892$		0			0
$893$	17.3956	−	17.3956i	0.582120	−	0.582120i
$894$		0			0
$895$		0			0
$896$		0			0
$897$	5.54358	+	30.1055i	0.185095	+	1.00519i
$898$		0			0
$899$	20.4930	−	20.4930i	0.683479	−	0.683479i
$900$		0			0
$901$			9.75437i			0.324965i
$902$		0			0
$903$	1.73799			0.0578367
$904$		0			0
$905$		0			0
$906$		0			0
$907$	12.9204	−	12.9204i	0.429015	−	0.429015i	−0.459277	−	0.888293i	$-0.651892\pi$
$907$	12.9204	−	12.9204i	0.429015	−	0.429015i	0.888293	+	0.459277i	$0.151892\pi$
$908$		0			0
$909$	13.8398			0.459038
$910$		0			0
$911$	−27.6040			−0.914561			−0.457281	−	0.889322i	$-0.651177\pi$
$911$	−27.6040			−0.914561			−0.457281	+	0.889322i	$0.651177\pi$
$912$		0			0
$913$	14.6764	−	14.6764i	0.485718	−	0.485718i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	9.51623			0.314254
$918$		0			0
$919$		−	14.4831i		−	0.477755i	−0.971050	−	0.238877i	$-0.923221\pi$
$919$		−	14.4831i		−	0.477755i	0.971050	−	0.238877i	$-0.0767794\pi$
$920$		0			0
$921$	16.3020	−	16.3020i	0.537171	−	0.537171i
$922$		0			0
$923$	19.2718	−	27.9710i	0.634338	−	0.920677i
$924$		0			0
$925$		0			0
$926$		0			0
$927$	−0.651547	+	0.651547i	−0.0213996	+	0.0213996i
$928$		0			0
$929$	−35.3394	+	35.3394i	−1.15945	+	1.15945i	−0.174853	+	0.984595i	$0.555945\pi$
$929$	−35.3394	+	35.3394i	−1.15945	+	1.15945i	−0.984595	+	0.174853i	$0.944055\pi$
$930$		0			0
$931$	23.0441	−	23.0441i	0.755241	−	0.755241i
$932$		0			0
$933$	−1.76774	−	1.76774i	−0.0578730	−	0.0578730i
$934$		0			0
$935$		0			0
$936$		0			0
$937$	16.4509	+	16.4509i	0.537429	+	0.537429i	0.922773	−	0.385344i	$-0.125917\pi$
$937$	16.4509	+	16.4509i	0.537429	+	0.537429i	−0.385344	+	0.922773i	$0.625917\pi$
$938$		0			0
$939$			10.8635i			0.354517i
$940$		0			0
$941$	−9.66558	−	9.66558i	−0.315089	−	0.315089i	0.531788	−	0.846877i	$-0.321520\pi$
$941$	−9.66558	−	9.66558i	−0.315089	−	0.315089i	−0.846877	+	0.531788i	$0.821520\pi$
$942$		0			0
$943$		−	54.1311i		−	1.76275i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	0.129130			0.00419617			0.00209808	−	0.999998i	$-0.499332\pi$
$947$	0.129130			0.00419617			0.00209808	+	0.999998i	$0.499332\pi$
$948$		0			0
$949$	35.1768	−	6.47740i	1.14189	−	0.210265i
$950$		0			0
$951$	18.2456	+	18.2456i	0.591655	+	0.591655i
$952$		0			0
$953$	2.86804	+	2.86804i	0.0929049	+	0.0929049i	0.752032	−	0.659127i	$-0.229074\pi$
$953$	2.86804	+	2.86804i	0.0929049	+	0.0929049i	−0.659127	+	0.752032i	$0.729074\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$	−13.7449			−0.444308
$958$		0			0
$959$	11.0786			0.357746
$960$		0			0
$961$		−	12.8230i		−	0.413646i
$962$		0			0
$963$	2.61098	−	2.61098i	0.0841376	−	0.0841376i
$964$		0			0
$965$		0			0
$966$		0			0
$967$		−	53.4996i		−	1.72043i	−0.509932	−	0.860215i	$-0.670329\pi$
$967$		−	53.4996i		−	1.72043i	0.509932	−	0.860215i	$-0.329671\pi$
$968$		0			0
$969$	−5.34968	−	5.34968i	−0.171856	−	0.171856i
$970$		0			0
$971$	17.7023			0.568094			0.284047	−	0.958810i	$-0.408323\pi$
$971$	17.7023			0.568094			0.284047	+	0.958810i	$0.408323\pi$
$972$		0			0
$973$			11.6852i			0.374612i
$974$		0			0
$975$		0			0
$976$		0			0
$977$			25.6266i			0.819867i	0.912115	+	0.409934i	$0.134448\pi$
$977$			25.6266i			0.819867i	−0.912115	+	0.409934i	$0.865552\pi$
$978$		0			0
$979$	10.4339			0.333469
$980$		0			0
$981$	−3.70634	−	3.70634i	−0.118334	−	0.118334i
$982$		0			0
$983$			59.1605i			1.88693i	0.331475	+	0.943464i	$0.392454\pi$
$983$			59.1605i			1.88693i	−0.331475	+	0.943464i	$0.607546\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	−2.15113	+	2.15113i	−0.0684712	+	0.0684712i
$988$		0			0
$989$			24.2779i			0.771992i
$990$		0			0
$991$	−4.50691			−0.143167			−0.0715834	−	0.997435i	$-0.522805\pi$
$991$	−4.50691			−0.143167			−0.0715834	+	0.997435i	$0.522805\pi$
$992$		0			0
$993$	−10.3176			−0.327419
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−34.2885	−	34.2885i	−1.08593	−	1.08593i	−0.995943	−	0.0899845i	$-0.971318\pi$
$997$	−34.2885	−	34.2885i	−1.08593	−	1.08593i	−0.0899845	−	0.995943i	$-0.528682\pi$
$998$		0			0
$999$	5.98377	+	5.98377i	0.189318	+	0.189318i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3900.2.bm.a.2257.2	yes	16
5.2	odd	4		3900.2.r.a.3193.6	yes	16
5.3	odd	4		3900.2.r.a.3193.3	yes	16
5.4	even	2	inner	3900.2.bm.a.2257.7	yes	16
13.5	odd	4		3900.2.r.a.1357.2	✓	16
65.18	even	4	inner	3900.2.bm.a.2293.2	yes	16
65.44	odd	4		3900.2.r.a.1357.7	yes	16
65.57	even	4	inner	3900.2.bm.a.2293.7	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3900.2.r.a.1357.2	✓	16	13.5	odd	4
3900.2.r.a.1357.7	yes	16	65.44	odd	4
3900.2.r.a.3193.3	yes	16	5.3	odd	4
3900.2.r.a.3193.6	yes	16	5.2	odd	4
3900.2.bm.a.2257.2	yes	16	1.1	even	1	trivial
3900.2.bm.a.2257.7	yes	16	5.4	even	2	inner
3900.2.bm.a.2293.2	yes	16	65.18	even	4	inner
3900.2.bm.a.2293.7	yes	16	65.57	even	4	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:	\( N \)	\(=\)	\( 3900 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3900.bm (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.707107 + 0.707107i) q^{3} -0.607786 q^{7} -1.00000i q^{9} +O(q^{10})\)	\(q+(-0.707107 + 0.707107i) q^{3} -0.607786 q^{7} -1.00000i q^{9} +(-1.42977 + 1.42977i) q^{11} +(-0.652944 - 3.54594i) q^{13} +(1.08844 - 1.08844i) q^{17} +(-3.47542 + 3.47542i) q^{19} +(0.429770 - 0.429770i) q^{21} +(6.00343 + 6.00343i) q^{23} +(0.707107 + 0.707107i) q^{27} -6.79765i q^{29} +(3.01471 + 3.01471i) q^{31} -2.02200i q^{33} +8.46233 q^{37} +(2.96906 + 2.04565i) q^{39} +(-4.50834 - 4.50834i) q^{41} +(2.02200 + 2.02200i) q^{43} -5.00531 q^{47} -6.63060 q^{49} +1.53929i q^{51} +(-4.48089 + 4.48089i) q^{53} -4.91499i q^{57} +(-8.02942 - 8.02942i) q^{59} -14.4282 q^{61} +0.607786i q^{63} +8.85067i q^{67} -8.49014 q^{69} +(6.66154 + 6.66154i) q^{71} +9.92030i q^{73} +(0.868995 - 0.868995i) q^{77} -11.4774i q^{79} -1.00000 q^{81} -10.2649 q^{83} +(4.80667 + 4.80667i) q^{87} +(-3.64880 - 3.64880i) q^{89} +(0.396850 + 2.15517i) q^{91} -4.26345 q^{93} -1.28014i q^{97} +(1.42977 + 1.42977i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q+O(q^{10}) \) 16 * q	\( 16 q - 16 q^{11} - 8 q^{19} - 32 q^{31} + 8 q^{39} + 8 q^{49} + 32 q^{59} - 24 q^{61} - 8 q^{69} + 32 q^{71} - 16 q^{81} + 24 q^{91} + 16 q^{99}+O(q^{100}) \) 16 * q - 16 * q^11 - 8 * q^19 - 32 * q^31 + 8 * q^39 + 8 * q^49 + 32 * q^59 - 24 * q^61 - 8 * q^69 + 32 * q^71 - 16 * q^81 + 24 * q^91 + 16 * q^99

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