LMFDB - Embedded newform 2156.2.i.g.177.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2156,2,Mod(177,2156)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2156, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 2, 0]))

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

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

chi := DirichletCharacter("2156.177");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2156 = 2^{2} \cdot 7^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2156.i (of order $3$, degree $2$, 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:	$17.2157466758$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{3})$
Coefficient field:	$\Q(\sqrt{2}, \sqrt{-3})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 2x^{2} + 4 $ x^4 + 2*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{9}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			177.2
Root			$0.707107 + 1.22474i$ of defining polynomial
Character	$\chi$	$=$	2156.177
Dual form			2156.2.i.g.1145.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.707107 + 1.22474i) q^{3} +(0.500000 - 0.866025i) q^{9} +(0.500000 + 0.866025i) q^{11} +1.41421 q^{13} +(-3.53553 - 6.12372i) q^{17} +(1.41421 - 2.44949i) q^{19} +(2.00000 - 3.46410i) q^{23} +(2.50000 + 4.33013i) q^{25} +5.65685 q^{27} +2.00000 q^{29} +(-2.12132 - 3.67423i) q^{31} +(-0.707107 + 1.22474i) q^{33} +(2.00000 - 3.46410i) q^{37} +(1.00000 + 1.73205i) q^{39} -1.41421 q^{41} +2.00000 q^{43} +(4.94975 - 8.57321i) q^{47} +(5.00000 - 8.66025i) q^{51} +(2.00000 + 3.46410i) q^{53} +4.00000 q^{57} +(2.12132 + 3.67423i) q^{59} +(-6.36396 + 11.0227i) q^{61} +(4.00000 + 6.92820i) q^{67} +5.65685 q^{69} +(0.707107 + 1.22474i) q^{73} +(-3.53553 + 6.12372i) q^{75} +(5.00000 - 8.66025i) q^{79} +(2.50000 + 4.33013i) q^{81} -8.48528 q^{83} +(1.41421 + 2.44949i) q^{87} +(5.65685 - 9.79796i) q^{89} +(3.00000 - 5.19615i) q^{93} -8.48528 q^{97} +1.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{9} + 2 q^{11} + 8 q^{23} + 10 q^{25} + 8 q^{29} + 8 q^{37} + 4 q^{39} + 8 q^{43} + 20 q^{51} + 8 q^{53} + 16 q^{57} + 16 q^{67} + 20 q^{79} + 10 q^{81} + 12 q^{93} + 4 q^{99}+O(q^{100}) $ 4 * q + 2 * q^9 + 2 * q^11 + 8 * q^23 + 10 * q^25 + 8 * q^29 + 8 * q^37 + 4 * q^39 + 8 * q^43 + 20 * q^51 + 8 * q^53 + 16 * q^57 + 16 * q^67 + 20 * q^79 + 10 * q^81 + 12 * q^93 + 4 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2156\mathbb{Z}\right)^\times$.

$n$	$981$	$1079$	$1277$
$\chi(n)$	$1$	$1$	$e\left(\frac{1}{3}\right)$

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	+	1.22474i	0.408248	+	0.707107i	0.994694	−	0.102882i	$-0.0328064\pi$
$3$	0.707107	+	1.22474i	0.408248	+	0.707107i	−0.586445	+	0.809989i	$0.699473\pi$
$4$		0			0
$5$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$5$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$6$		0			0
$7$		0			0
$7$		0			0
$8$		0			0
$9$	0.500000	−	0.866025i	0.166667	−	0.288675i
$10$		0			0
$11$	0.500000	+	0.866025i	0.150756	+	0.261116i
$11$	0.500000	+	0.866025i	0.150756	+	0.261116i
$12$		0			0
$13$	1.41421			0.392232			0.196116	−	0.980581i	$-0.437167\pi$
$13$	1.41421			0.392232			0.196116	+	0.980581i	$0.437167\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−3.53553	−	6.12372i	−0.857493	−	1.48522i	−0.874313	−	0.485363i	$-0.838688\pi$
$17$	−3.53553	−	6.12372i	−0.857493	−	1.48522i	0.0168199	−	0.999859i	$-0.494646\pi$
$18$		0			0
$19$	1.41421	−	2.44949i	0.324443	−	0.561951i	−0.656957	−	0.753928i	$-0.728157\pi$
$19$	1.41421	−	2.44949i	0.324443	−	0.561951i	0.981399	+	0.191977i	$0.0614899\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	2.00000	−	3.46410i	0.417029	−	0.722315i	−0.578610	−	0.815604i	$-0.696405\pi$
$23$	2.00000	−	3.46410i	0.417029	−	0.722315i	0.995639	+	0.0932891i	$0.0297381\pi$
$24$		0			0
$25$	2.50000	+	4.33013i	0.500000	+	0.866025i
$26$		0			0
$27$	5.65685			1.08866
$28$		0			0
$29$	2.00000			0.371391			0.185695	−	0.982607i	$-0.440546\pi$
$29$	2.00000			0.371391			0.185695	+	0.982607i	$0.440546\pi$
$30$		0			0
$31$	−2.12132	−	3.67423i	−0.381000	−	0.659912i	0.610205	−	0.792243i	$-0.291087\pi$
$31$	−2.12132	−	3.67423i	−0.381000	−	0.659912i	−0.991206	+	0.132331i	$0.957754\pi$
$32$		0			0
$33$	−0.707107	+	1.22474i	−0.123091	+	0.213201i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	2.00000	−	3.46410i	0.328798	−	0.569495i	−0.653476	−	0.756948i	$-0.726690\pi$
$37$	2.00000	−	3.46410i	0.328798	−	0.569495i	0.982274	+	0.187453i	$0.0600231\pi$
$38$		0			0
$39$	1.00000	+	1.73205i	0.160128	+	0.277350i
$40$		0			0
$41$	−1.41421			−0.220863			−0.110432	−	0.993884i	$-0.535223\pi$
$41$	−1.41421			−0.220863			−0.110432	+	0.993884i	$0.535223\pi$
$42$		0			0
$43$	2.00000			0.304997			0.152499	−	0.988304i	$-0.451268\pi$
$43$	2.00000			0.304997			0.152499	+	0.988304i	$0.451268\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	4.94975	−	8.57321i	0.721995	−	1.25053i	−0.238204	−	0.971215i	$-0.576559\pi$
$47$	4.94975	−	8.57321i	0.721995	−	1.25053i	0.960199	−	0.279317i	$-0.0901079\pi$
$48$		0			0
$49$		0			0
$50$		0			0
$51$	5.00000	−	8.66025i	0.700140	−	1.21268i
$52$		0			0
$53$	2.00000	+	3.46410i	0.274721	+	0.475831i	0.970065	−	0.242846i	$-0.0780811\pi$
$53$	2.00000	+	3.46410i	0.274721	+	0.475831i	−0.695344	+	0.718677i	$0.744748\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	4.00000			0.529813
$58$		0			0
$59$	2.12132	+	3.67423i	0.276172	+	0.478345i	0.970430	−	0.241382i	$-0.0776006\pi$
$59$	2.12132	+	3.67423i	0.276172	+	0.478345i	−0.694258	+	0.719726i	$0.744267\pi$
$60$		0			0
$61$	−6.36396	+	11.0227i	−0.814822	+	1.41131i	0.0946341	+	0.995512i	$0.469832\pi$
$61$	−6.36396	+	11.0227i	−0.814822	+	1.41131i	−0.909456	+	0.415800i	$0.863501\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		0			0
$66$		0			0
$67$	4.00000	+	6.92820i	0.488678	+	0.846415i	0.999915	−	0.0130248i	$-0.00414604\pi$
$67$	4.00000	+	6.92820i	0.488678	+	0.846415i	−0.511237	+	0.859440i	$0.670813\pi$
$68$		0			0
$69$	5.65685			0.681005
$70$		0			0
$71$		0			0			−	1.00000i	$-0.5\pi$
$71$		0			0				1.00000i	$0.5\pi$
$72$		0			0
$73$	0.707107	+	1.22474i	0.0827606	+	0.143346i	0.904435	−	0.426612i	$-0.140293\pi$
$73$	0.707107	+	1.22474i	0.0827606	+	0.143346i	−0.821674	+	0.569958i	$0.806960\pi$
$74$		0			0
$75$	−3.53553	+	6.12372i	−0.408248	+	0.707107i
$76$		0			0
$77$		0			0
$78$		0			0
$79$	5.00000	−	8.66025i	0.562544	−	0.974355i	−0.434730	−	0.900561i	$-0.643156\pi$
$79$	5.00000	−	8.66025i	0.562544	−	0.974355i	0.997274	−	0.0737937i	$-0.0235106\pi$
$80$		0			0
$81$	2.50000	+	4.33013i	0.277778	+	0.481125i
$82$		0			0
$83$	−8.48528			−0.931381			−0.465690	−	0.884948i	$-0.654194\pi$
$83$	−8.48528			−0.931381			−0.465690	+	0.884948i	$0.654194\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	1.41421	+	2.44949i	0.151620	+	0.262613i
$88$		0			0
$89$	5.65685	−	9.79796i	0.599625	−	1.03858i	−0.393251	−	0.919431i	$-0.628650\pi$
$89$	5.65685	−	9.79796i	0.599625	−	1.03858i	0.992876	−	0.119150i	$-0.0380171\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$	3.00000	−	5.19615i	0.311086	−	0.538816i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−8.48528			−0.861550			−0.430775	−	0.902459i	$-0.641760\pi$
$97$	−8.48528			−0.861550			−0.430775	+	0.902459i	$0.641760\pi$
$98$		0			0
$99$	1.00000			0.100504
$100$		0			0
$101$	−0.707107	−	1.22474i	−0.0703598	−	0.121867i	0.828699	−	0.559694i	$-0.189081\pi$
$101$	−0.707107	−	1.22474i	−0.0703598	−	0.121867i	−0.899059	+	0.437828i	$0.855748\pi$
$102$		0			0
$103$	3.53553	−	6.12372i	0.348367	−	0.603388i	−0.637593	−	0.770373i	$-0.720070\pi$
$103$	3.53553	−	6.12372i	0.348367	−	0.603388i	0.985959	+	0.166985i	$0.0534031\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	6.00000	−	10.3923i	0.580042	−	1.00466i	−0.415432	−	0.909624i	$-0.636370\pi$
$107$	6.00000	−	10.3923i	0.580042	−	1.00466i	0.995474	−	0.0950377i	$-0.0302972\pi$
$108$		0			0
$109$	5.00000	+	8.66025i	0.478913	+	0.829502i	0.999708	−	0.0241802i	$-0.00769755\pi$
$109$	5.00000	+	8.66025i	0.478913	+	0.829502i	−0.520794	+	0.853682i	$0.674364\pi$
$110$		0			0
$111$	5.65685			0.536925
$112$		0			0
$113$	14.0000			1.31701			0.658505	−	0.752577i	$-0.271189\pi$
$113$	14.0000			1.31701			0.658505	+	0.752577i	$0.271189\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	0.707107	−	1.22474i	0.0653720	−	0.113228i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−0.500000	+	0.866025i	−0.0454545	+	0.0787296i
$122$		0			0
$123$	−1.00000	−	1.73205i	−0.0901670	−	0.156174i
$124$		0			0
$125$		0			0
$126$		0			0
$127$		0			0			−	1.00000i	$-0.5\pi$
$127$		0			0				1.00000i	$0.5\pi$
$128$		0			0
$129$	1.41421	+	2.44949i	0.124515	+	0.215666i
$130$		0			0
$131$	−5.65685	+	9.79796i	−0.494242	+	0.856052i	−0.999978	−	0.00663646i	$-0.997888\pi$
$131$	−5.65685	+	9.79796i	−0.494242	+	0.856052i	0.505736	+	0.862688i	$0.331221\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$	4.00000	+	6.92820i	0.341743	+	0.591916i	0.984757	−	0.173939i	$-0.0556494\pi$
$137$	4.00000	+	6.92820i	0.341743	+	0.591916i	−0.643013	+	0.765855i	$0.722316\pi$
$138$		0			0
$139$	8.48528			0.719712			0.359856	−	0.933008i	$-0.382826\pi$
$139$	8.48528			0.719712			0.359856	+	0.933008i	$0.382826\pi$
$140$		0			0
$141$	14.0000			1.17901
$142$		0			0
$143$	0.707107	+	1.22474i	0.0591312	+	0.102418i
$144$		0			0
$145$		0			0
$146$		0			0
$147$		0			0
$148$		0			0
$149$	3.00000	−	5.19615i	0.245770	−	0.425685i	−0.716578	−	0.697507i	$-0.754293\pi$
$149$	3.00000	−	5.19615i	0.245770	−	0.425685i	0.962348	+	0.271821i	$0.0876260\pi$
$150$		0			0
$151$	1.00000	+	1.73205i	0.0813788	+	0.140952i	0.903842	−	0.427865i	$-0.140734\pi$
$151$	1.00000	+	1.73205i	0.0813788	+	0.140952i	−0.822464	+	0.568818i	$0.807401\pi$
$152$		0			0
$153$	−7.07107			−0.571662
$154$		0			0
$155$		0			0
$156$		0			0
$157$	5.65685	+	9.79796i	0.451466	+	0.781962i	0.998477	−	0.0551630i	$-0.0175678\pi$
$157$	5.65685	+	9.79796i	0.451466	+	0.781962i	−0.547011	+	0.837125i	$0.684235\pi$
$158$		0			0
$159$	−2.82843	+	4.89898i	−0.224309	+	0.388514i
$160$		0			0
$161$		0			0
$162$		0			0
$163$	2.00000	−	3.46410i	0.156652	−	0.271329i	−0.777007	−	0.629492i	$-0.783263\pi$
$163$	2.00000	−	3.46410i	0.156652	−	0.271329i	0.933659	+	0.358162i	$0.116597\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−11.3137			−0.875481			−0.437741	−	0.899101i	$-0.644221\pi$
$167$	−11.3137			−0.875481			−0.437741	+	0.899101i	$0.644221\pi$
$168$		0			0
$169$	−11.0000			−0.846154
$170$		0			0
$171$	−1.41421	−	2.44949i	−0.108148	−	0.187317i
$172$		0			0
$173$	−7.77817	+	13.4722i	−0.591364	+	1.02427i	0.402685	+	0.915338i	$0.368077\pi$
$173$	−7.77817	+	13.4722i	−0.591364	+	1.02427i	−0.994049	+	0.108933i	$0.965256\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	−3.00000	+	5.19615i	−0.225494	+	0.390567i
$178$		0			0
$179$	2.00000	+	3.46410i	0.149487	+	0.258919i	0.931038	−	0.364922i	$-0.118904\pi$
$179$	2.00000	+	3.46410i	0.149487	+	0.258919i	−0.781551	+	0.623841i	$0.785571\pi$
$180$		0			0
$181$	−19.7990			−1.47165			−0.735824	−	0.677173i	$-0.763205\pi$
$181$	−19.7990			−1.47165			−0.735824	+	0.677173i	$0.763205\pi$
$182$		0			0
$183$	−18.0000			−1.33060
$184$		0			0
$185$		0			0
$186$		0			0
$187$	3.53553	−	6.12372i	0.258544	−	0.447811i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	4.00000	−	6.92820i	0.289430	−	0.501307i	−0.684244	−	0.729253i	$-0.739868\pi$
$191$	4.00000	−	6.92820i	0.289430	−	0.501307i	0.973674	+	0.227946i	$0.0732010\pi$
$192$		0			0
$193$	7.00000	+	12.1244i	0.503871	+	0.872730i	0.999990	+	0.00447566i	$0.00142465\pi$
$193$	7.00000	+	12.1244i	0.503871	+	0.872730i	−0.496119	+	0.868255i	$0.665242\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	26.0000			1.85242			0.926212	−	0.377004i	$-0.123046\pi$
$197$	26.0000			1.85242			0.926212	+	0.377004i	$0.123046\pi$
$198$		0			0
$199$	12.0208	+	20.8207i	0.852133	+	1.47594i	0.879279	+	0.476306i	$0.158025\pi$
$199$	12.0208	+	20.8207i	0.852133	+	1.47594i	−0.0271465	+	0.999631i	$0.508642\pi$
$200$		0			0
$201$	−5.65685	+	9.79796i	−0.399004	+	0.691095i
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$	−2.00000	−	3.46410i	−0.139010	−	0.240772i
$208$		0			0
$209$	2.82843			0.195646
$210$		0			0
$211$	−20.0000			−1.37686			−0.688428	−	0.725304i	$-0.741699\pi$
$211$	−20.0000			−1.37686			−0.688428	+	0.725304i	$0.741699\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$		0			0
$219$	−1.00000	+	1.73205i	−0.0675737	+	0.117041i
$220$		0			0
$221$	−5.00000	−	8.66025i	−0.336336	−	0.582552i
$222$		0			0
$223$	−7.07107			−0.473514			−0.236757	−	0.971569i	$-0.576084\pi$
$223$	−7.07107			−0.473514			−0.236757	+	0.971569i	$0.576084\pi$
$224$		0			0
$225$	5.00000			0.333333
$226$		0			0
$227$	11.3137	+	19.5959i	0.750917	+	1.30063i	0.947379	+	0.320115i	$0.103722\pi$
$227$	11.3137	+	19.5959i	0.750917	+	1.30063i	−0.196461	+	0.980512i	$0.562945\pi$
$228$		0			0
$229$	7.07107	−	12.2474i	0.467269	−	0.809334i	−0.532031	−	0.846725i	$-0.678571\pi$
$229$	7.07107	−	12.2474i	0.467269	−	0.809334i	0.999301	+	0.0373904i	$0.0119045\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	1.00000	−	1.73205i	0.0655122	−	0.113470i	−0.831409	−	0.555661i	$-0.812465\pi$
$233$	1.00000	−	1.73205i	0.0655122	−	0.113470i	0.896921	+	0.442191i	$0.145799\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	14.1421			0.918630
$238$		0			0
$239$	−16.0000			−1.03495			−0.517477	−	0.855697i	$-0.673129\pi$
$239$	−16.0000			−1.03495			−0.517477	+	0.855697i	$0.673129\pi$
$240$		0			0
$241$	−13.4350	−	23.2702i	−0.865426	−	1.49896i	−0.866623	−	0.498963i	$-0.833714\pi$
$241$	−13.4350	−	23.2702i	−0.865426	−	1.49896i	0.00119700	−	0.999999i	$-0.499619\pi$
$242$		0			0
$243$	4.94975	−	8.57321i	0.317526	−	0.549972i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	2.00000	−	3.46410i	0.127257	−	0.220416i
$248$		0			0
$249$	−6.00000	−	10.3923i	−0.380235	−	0.658586i
$250$		0			0
$251$	15.5563			0.981908			0.490954	−	0.871185i	$-0.336648\pi$
$251$	15.5563			0.981908			0.490954	+	0.871185i	$0.336648\pi$
$252$		0			0
$253$	4.00000			0.251478
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−2.82843	+	4.89898i	−0.176432	+	0.305590i	−0.940656	−	0.339361i	$-0.889789\pi$
$257$	−2.82843	+	4.89898i	−0.176432	+	0.305590i	0.764224	+	0.644951i	$0.223122\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	1.00000	−	1.73205i	0.0618984	−	0.107211i
$262$		0			0
$263$	−4.00000	−	6.92820i	−0.246651	−	0.427211i	0.715944	−	0.698158i	$-0.245997\pi$
$263$	−4.00000	−	6.92820i	−0.246651	−	0.427211i	−0.962594	+	0.270947i	$0.912663\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	16.0000			0.979184
$268$		0			0
$269$	−9.89949	−	17.1464i	−0.603583	−	1.04544i	−0.992274	−	0.124068i	$-0.960406\pi$
$269$	−9.89949	−	17.1464i	−0.603583	−	1.04544i	0.388691	−	0.921368i	$-0.372927\pi$
$270$		0			0
$271$	12.7279	−	22.0454i	0.773166	−	1.33916i	−0.162653	−	0.986683i	$-0.552005\pi$
$271$	12.7279	−	22.0454i	0.773166	−	1.33916i	0.935819	−	0.352480i	$-0.114661\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−2.50000	+	4.33013i	−0.150756	+	0.261116i
$276$		0			0
$277$	1.00000	+	1.73205i	0.0600842	+	0.104069i	0.894503	−	0.447062i	$-0.147530\pi$
$277$	1.00000	+	1.73205i	0.0600842	+	0.104069i	−0.834419	+	0.551131i	$0.814196\pi$
$278$		0			0
$279$	−4.24264			−0.254000
$280$		0			0
$281$	2.00000			0.119310			0.0596550	−	0.998219i	$-0.481000\pi$
$281$	2.00000			0.119310			0.0596550	+	0.998219i	$0.481000\pi$
$282$		0			0
$283$	−14.1421	−	24.4949i	−0.840663	−	1.45607i	−0.889335	−	0.457256i	$-0.848832\pi$
$283$	−14.1421	−	24.4949i	−0.840663	−	1.45607i	0.0486726	−	0.998815i	$-0.484501\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−16.5000	+	28.5788i	−0.970588	+	1.68111i
$290$		0			0
$291$	−6.00000	−	10.3923i	−0.351726	−	0.609208i
$292$		0			0
$293$	1.41421			0.0826192			0.0413096	−	0.999146i	$-0.486847\pi$
$293$	1.41421			0.0826192			0.0413096	+	0.999146i	$0.486847\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	2.82843	+	4.89898i	0.164122	+	0.284268i
$298$		0			0
$299$	2.82843	−	4.89898i	0.163572	−	0.283315i
$300$		0			0
$301$		0			0
$302$		0			0
$303$	1.00000	−	1.73205i	0.0574485	−	0.0995037i
$304$		0			0
$305$		0			0
$306$		0			0
$307$	11.3137			0.645707			0.322854	−	0.946449i	$-0.395358\pi$
$307$	11.3137			0.645707			0.322854	+	0.946449i	$0.395358\pi$
$308$		0			0
$309$	10.0000			0.568880
$310$		0			0
$311$	−7.77817	−	13.4722i	−0.441060	−	0.763938i	0.556709	−	0.830708i	$-0.312064\pi$
$311$	−7.77817	−	13.4722i	−0.441060	−	0.763938i	−0.997768	+	0.0667698i	$0.978731\pi$
$312$		0			0
$313$	−11.3137	+	19.5959i	−0.639489	+	1.10763i	0.346057	+	0.938214i	$0.387521\pi$
$313$	−11.3137	+	19.5959i	−0.639489	+	1.10763i	−0.985545	+	0.169413i	$0.945813\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−11.0000	+	19.0526i	−0.617822	+	1.07010i	0.372061	+	0.928208i	$0.378651\pi$
$317$	−11.0000	+	19.0526i	−0.617822	+	1.07010i	−0.989882	+	0.141890i	$0.954682\pi$
$318$		0			0
$319$	1.00000	+	1.73205i	0.0559893	+	0.0969762i
$320$		0			0
$321$	16.9706			0.947204
$322$		0			0
$323$	−20.0000			−1.11283
$324$		0			0
$325$	3.53553	+	6.12372i	0.196116	+	0.339683i
$326$		0			0
$327$	−7.07107	+	12.2474i	−0.391031	+	0.677285i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−4.00000	+	6.92820i	−0.219860	+	0.380808i	−0.954765	−	0.297361i	$-0.903893\pi$
$331$	−4.00000	+	6.92820i	−0.219860	+	0.380808i	0.734905	+	0.678170i	$0.237227\pi$
$332$		0			0
$333$	−2.00000	−	3.46410i	−0.109599	−	0.189832i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−22.0000			−1.19842			−0.599208	−	0.800593i	$-0.704518\pi$
$337$	−22.0000			−1.19842			−0.599208	+	0.800593i	$0.704518\pi$
$338$		0			0
$339$	9.89949	+	17.1464i	0.537667	+	0.931266i
$340$		0			0
$341$	2.12132	−	3.67423i	0.114876	−	0.198971i
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−11.0000	−	19.0526i	−0.590511	−	1.02279i	−0.994164	−	0.107883i	$-0.965593\pi$
$347$	−11.0000	−	19.0526i	−0.590511	−	1.02279i	0.403653	−	0.914912i	$-0.367740\pi$
$348$		0			0
$349$	−4.24264			−0.227103			−0.113552	−	0.993532i	$-0.536223\pi$
$349$	−4.24264			−0.227103			−0.113552	+	0.993532i	$0.536223\pi$
$350$		0			0
$351$	8.00000			0.427008
$352$		0			0
$353$	5.65685	+	9.79796i	0.301084	+	0.521493i	0.976382	−	0.216052i	$-0.0693182\pi$
$353$	5.65685	+	9.79796i	0.301084	+	0.521493i	−0.675298	+	0.737545i	$0.735985\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$	11.0000	−	19.0526i	0.580558	−	1.00556i	−0.414855	−	0.909887i	$-0.636168\pi$
$359$	11.0000	−	19.0526i	0.580558	−	1.00556i	0.995413	−	0.0956683i	$-0.0304988\pi$
$360$		0			0
$361$	5.50000	+	9.52628i	0.289474	+	0.501383i
$362$		0			0
$363$	−1.41421			−0.0742270
$364$		0			0
$365$		0			0
$366$		0			0
$367$	−14.8492	−	25.7196i	−0.775124	−	1.34255i	−0.934725	−	0.355373i	$-0.884354\pi$
$367$	−14.8492	−	25.7196i	−0.775124	−	1.34255i	0.159601	−	0.987182i	$-0.448979\pi$
$368$		0			0
$369$	−0.707107	+	1.22474i	−0.0368105	+	0.0637577i
$370$		0			0
$371$		0			0
$372$		0			0
$373$	−19.0000	+	32.9090i	−0.983783	+	1.70396i	−0.336557	+	0.941663i	$0.609263\pi$
$373$	−19.0000	+	32.9090i	−0.983783	+	1.70396i	−0.647225	+	0.762299i	$0.724071\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	2.82843			0.145671
$378$		0			0
$379$	−24.0000			−1.23280			−0.616399	−	0.787434i	$-0.711409\pi$
$379$	−24.0000			−1.23280			−0.616399	+	0.787434i	$0.711409\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−0.707107	+	1.22474i	−0.0361315	+	0.0625815i	−0.883526	−	0.468383i	$-0.844837\pi$
$383$	−0.707107	+	1.22474i	−0.0361315	+	0.0625815i	0.847394	+	0.530964i	$0.178170\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	1.00000	−	1.73205i	0.0508329	−	0.0880451i
$388$		0			0
$389$	−13.0000	−	22.5167i	−0.659126	−	1.14164i	−0.980842	−	0.194804i	$-0.937593\pi$
$389$	−13.0000	−	22.5167i	−0.659126	−	1.14164i	0.321716	−	0.946836i	$-0.395740\pi$
$390$		0			0
$391$	−28.2843			−1.43040
$392$		0			0
$393$	−16.0000			−0.807093
$394$		0			0
$395$		0			0
$396$		0			0
$397$	11.3137	−	19.5959i	0.567819	−	0.983491i	−0.428963	−	0.903322i	$-0.641121\pi$
$397$	11.3137	−	19.5959i	0.567819	−	0.983491i	0.996781	−	0.0801687i	$-0.0255459\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−9.00000	+	15.5885i	−0.449439	+	0.778450i	−0.998350	−	0.0574304i	$-0.981709\pi$
$401$	−9.00000	+	15.5885i	−0.449439	+	0.778450i	0.548911	+	0.835881i	$0.315043\pi$
$402$		0			0
$403$	−3.00000	−	5.19615i	−0.149441	−	0.258839i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	4.00000			0.198273
$408$		0			0
$409$	6.36396	+	11.0227i	0.314678	+	0.545038i	0.979369	−	0.202080i	$-0.0647703\pi$
$409$	6.36396	+	11.0227i	0.314678	+	0.545038i	−0.664691	+	0.747118i	$0.731437\pi$
$410$		0			0
$411$	−5.65685	+	9.79796i	−0.279032	+	0.483298i
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	6.00000	+	10.3923i	0.293821	+	0.508913i
$418$		0			0
$419$	21.2132			1.03633			0.518166	−	0.855280i	$-0.326615\pi$
$419$	21.2132			1.03633			0.518166	+	0.855280i	$0.326615\pi$
$420$		0			0
$421$	32.0000			1.55958			0.779792	−	0.626038i	$-0.215325\pi$
$421$	32.0000			1.55958			0.779792	+	0.626038i	$0.215325\pi$
$422$		0			0
$423$	−4.94975	−	8.57321i	−0.240665	−	0.416844i
$424$		0			0
$425$	17.6777	−	30.6186i	0.857493	−	1.48522i
$426$		0			0
$427$		0			0
$428$		0			0
$429$	−1.00000	+	1.73205i	−0.0482805	+	0.0836242i
$430$		0			0
$431$	−15.0000	−	25.9808i	−0.722525	−	1.25145i	−0.959985	−	0.280052i	$-0.909648\pi$
$431$	−15.0000	−	25.9808i	−0.722525	−	1.25145i	0.237460	−	0.971397i	$-0.423685\pi$
$432$		0			0
$433$	25.4558			1.22333			0.611665	−	0.791117i	$-0.290500\pi$
$433$	25.4558			1.22333			0.611665	+	0.791117i	$0.290500\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−5.65685	−	9.79796i	−0.270604	−	0.468700i
$438$		0			0
$439$	−4.24264	+	7.34847i	−0.202490	+	0.350723i	−0.949330	−	0.314280i	$-0.898237\pi$
$439$	−4.24264	+	7.34847i	−0.202490	+	0.350723i	0.746840	+	0.665004i	$0.231570\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−12.0000	+	20.7846i	−0.570137	+	0.987507i	0.426414	+	0.904528i	$0.359777\pi$
$443$	−12.0000	+	20.7846i	−0.570137	+	0.987507i	−0.996551	+	0.0829786i	$0.973557\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	8.48528			0.401340
$448$		0			0
$449$	4.00000			0.188772			0.0943858	−	0.995536i	$-0.469911\pi$
$449$	4.00000			0.188772			0.0943858	+	0.995536i	$0.469911\pi$
$450$		0			0
$451$	−0.707107	−	1.22474i	−0.0332964	−	0.0576710i
$452$		0			0
$453$	−1.41421	+	2.44949i	−0.0664455	+	0.115087i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−19.0000	+	32.9090i	−0.888783	+	1.53942i	−0.0474665	+	0.998873i	$0.515115\pi$
$457$	−19.0000	+	32.9090i	−0.888783	+	1.53942i	−0.841316	+	0.540544i	$0.818219\pi$
$458$		0			0
$459$	−20.0000	−	34.6410i	−0.933520	−	1.61690i
$460$		0			0
$461$	−24.0416			−1.11973			−0.559865	−	0.828584i	$-0.689147\pi$
$461$	−24.0416			−1.11973			−0.559865	+	0.828584i	$0.689147\pi$
$462$		0			0
$463$	−20.0000			−0.929479			−0.464739	−	0.885448i	$-0.653852\pi$
$463$	−20.0000			−0.929479			−0.464739	+	0.885448i	$0.653852\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−6.36396	+	11.0227i	−0.294489	+	0.510070i	−0.974866	−	0.222792i	$-0.928483\pi$
$467$	−6.36396	+	11.0227i	−0.294489	+	0.510070i	0.680377	+	0.732862i	$0.261816\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	−8.00000	+	13.8564i	−0.368621	+	0.638470i
$472$		0			0
$473$	1.00000	+	1.73205i	0.0459800	+	0.0796398i
$474$		0			0
$475$	14.1421			0.648886
$476$		0			0
$477$	4.00000			0.183147
$478$		0			0
$479$	−12.7279	−	22.0454i	−0.581554	−	1.00728i	−0.995295	−	0.0968862i	$-0.969112\pi$
$479$	−12.7279	−	22.0454i	−0.581554	−	1.00728i	0.413742	−	0.910394i	$-0.364222\pi$
$480$		0			0
$481$	2.82843	−	4.89898i	0.128965	−	0.223374i
$482$		0			0
$483$		0			0
$484$		0			0
$485$		0			0
$486$		0			0
$487$	16.0000	+	27.7128i	0.725029	+	1.25579i	0.958962	+	0.283535i	$0.0915071\pi$
$487$	16.0000	+	27.7128i	0.725029	+	1.25579i	−0.233933	+	0.972253i	$0.575160\pi$
$488$		0			0
$489$	5.65685			0.255812
$490$		0			0
$491$	−34.0000			−1.53440			−0.767199	−	0.641409i	$-0.778350\pi$
$491$	−34.0000			−1.53440			−0.767199	+	0.641409i	$0.778350\pi$
$492$		0			0
$493$	−7.07107	−	12.2474i	−0.318465	−	0.551597i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	6.00000	−	10.3923i	0.268597	−	0.465223i	−0.699903	−	0.714238i	$-0.746773\pi$
$499$	6.00000	−	10.3923i	0.268597	−	0.465223i	0.968500	+	0.249015i	$0.0801067\pi$
$500$		0			0
$501$	−8.00000	−	13.8564i	−0.357414	−	0.619059i
$502$		0			0
$503$	−19.7990			−0.882793			−0.441397	−	0.897312i	$-0.645517\pi$
$503$	−19.7990			−0.882793			−0.441397	+	0.897312i	$0.645517\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−7.77817	−	13.4722i	−0.345441	−	0.598321i
$508$		0			0
$509$	12.7279	−	22.0454i	0.564155	−	0.977146i	−0.432972	−	0.901407i	$-0.642535\pi$
$509$	12.7279	−	22.0454i	0.564155	−	0.977146i	0.997128	−	0.0757384i	$-0.0241314\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$	8.00000	−	13.8564i	0.353209	−	0.611775i
$514$		0			0
$515$		0			0
$516$		0			0
$517$	9.89949			0.435379
$518$		0			0
$519$	−22.0000			−0.965693
$520$		0			0
$521$	18.3848	+	31.8434i	0.805452	+	1.39508i	0.915986	+	0.401211i	$0.131411\pi$
$521$	18.3848	+	31.8434i	0.805452	+	1.39508i	−0.110534	+	0.993872i	$0.535256\pi$
$522$		0			0
$523$	−8.48528	+	14.6969i	−0.371035	+	0.642652i	−0.989725	−	0.142983i	$-0.954331\pi$
$523$	−8.48528	+	14.6969i	−0.371035	+	0.642652i	0.618690	+	0.785635i	$0.287664\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−15.0000	+	25.9808i	−0.653410	+	1.13174i
$528$		0			0
$529$	3.50000	+	6.06218i	0.152174	+	0.263573i
$530$		0			0
$531$	4.24264			0.184115
$532$		0			0
$533$	−2.00000			−0.0866296
$534$		0			0
$535$		0			0
$536$		0			0
$537$	−2.82843	+	4.89898i	−0.122056	+	0.211407i
$538$		0			0
$539$		0			0
$540$		0			0
$541$	3.00000	−	5.19615i	0.128980	−	0.223400i	−0.794302	−	0.607524i	$-0.792163\pi$
$541$	3.00000	−	5.19615i	0.128980	−	0.223400i	0.923282	+	0.384124i	$0.125496\pi$
$542$		0			0
$543$	−14.0000	−	24.2487i	−0.600798	−	1.04061i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−20.0000			−0.855138			−0.427569	−	0.903983i	$-0.640630\pi$
$547$	−20.0000			−0.855138			−0.427569	+	0.903983i	$0.640630\pi$
$548$		0			0
$549$	6.36396	+	11.0227i	0.271607	+	0.470438i
$550$		0			0
$551$	2.82843	−	4.89898i	0.120495	−	0.208704i
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−15.0000	−	25.9808i	−0.635570	−	1.10084i	−0.986394	−	0.164399i	$-0.947432\pi$
$557$	−15.0000	−	25.9808i	−0.635570	−	1.10084i	0.350824	−	0.936442i	$-0.385902\pi$
$558$		0			0
$559$	2.82843			0.119630
$560$		0			0
$561$	10.0000			0.422200
$562$		0			0
$563$	−1.41421	−	2.44949i	−0.0596020	−	0.103234i	0.834685	−	0.550728i	$-0.185650\pi$
$563$	−1.41421	−	2.44949i	−0.0596020	−	0.103234i	−0.894287	+	0.447494i	$0.852316\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$	3.00000	−	5.19615i	0.125767	−	0.217834i	−0.796266	−	0.604947i	$-0.793194\pi$
$569$	3.00000	−	5.19615i	0.125767	−	0.217834i	0.922032	+	0.387113i	$0.126528\pi$
$570$		0			0
$571$	10.0000	+	17.3205i	0.418487	+	0.724841i	0.995788	−	0.0916910i	$-0.0292272\pi$
$571$	10.0000	+	17.3205i	0.418487	+	0.724841i	−0.577301	+	0.816532i	$0.695894\pi$
$572$		0			0
$573$	11.3137			0.472637
$574$		0			0
$575$	20.0000			0.834058
$576$		0			0
$577$	9.89949	+	17.1464i	0.412121	+	0.713815i	0.995122	−	0.0986569i	$-0.0314546\pi$
$577$	9.89949	+	17.1464i	0.412121	+	0.713815i	−0.583000	+	0.812472i	$0.698121\pi$
$578$		0			0
$579$	−9.89949	+	17.1464i	−0.411409	+	0.712581i
$580$		0			0
$581$		0			0
$582$		0			0
$583$	−2.00000	+	3.46410i	−0.0828315	+	0.143468i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−26.8701			−1.10905			−0.554523	−	0.832168i	$-0.687099\pi$
$587$	−26.8701			−1.10905			−0.554523	+	0.832168i	$0.687099\pi$
$588$		0			0
$589$	−12.0000			−0.494451
$590$		0			0
$591$	18.3848	+	31.8434i	0.756249	+	1.30986i
$592$		0			0
$593$	−6.36396	+	11.0227i	−0.261337	+	0.452648i	−0.966597	−	0.256300i	$-0.917497\pi$
$593$	−6.36396	+	11.0227i	−0.261337	+	0.452648i	0.705261	+	0.708948i	$0.250830\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−17.0000	+	29.4449i	−0.695764	+	1.20510i
$598$		0			0
$599$	−18.0000	−	31.1769i	−0.735460	−	1.27385i	−0.954521	−	0.298143i	$-0.903633\pi$
$599$	−18.0000	−	31.1769i	−0.735460	−	1.27385i	0.219061	−	0.975711i	$-0.429701\pi$
$600$		0			0
$601$	−12.7279			−0.519183			−0.259591	−	0.965719i	$-0.583588\pi$
$601$	−12.7279			−0.519183			−0.259591	+	0.965719i	$0.583588\pi$
$602$		0			0
$603$	8.00000			0.325785
$604$		0			0
$605$		0			0
$606$		0			0
$607$	−8.48528	+	14.6969i	−0.344407	+	0.596530i	−0.985246	−	0.171145i	$-0.945253\pi$
$607$	−8.48528	+	14.6969i	−0.344407	+	0.596530i	0.640839	+	0.767675i	$0.278587\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	7.00000	−	12.1244i	0.283190	−	0.490499i
$612$		0			0
$613$	−5.00000	−	8.66025i	−0.201948	−	0.349784i	0.747208	−	0.664590i	$-0.231394\pi$
$613$	−5.00000	−	8.66025i	−0.201948	−	0.349784i	−0.949156	+	0.314806i	$0.898061\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	32.0000			1.28827			0.644136	−	0.764911i	$-0.277217\pi$
$617$	32.0000			1.28827			0.644136	+	0.764911i	$0.277217\pi$
$618$		0			0
$619$	16.2635	+	28.1691i	0.653683	+	1.13221i	0.982222	+	0.187722i	$0.0601106\pi$
$619$	16.2635	+	28.1691i	0.653683	+	1.13221i	−0.328539	+	0.944491i	$0.606556\pi$
$620$		0			0
$621$	11.3137	−	19.5959i	0.454003	−	0.786357i
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−12.5000	+	21.6506i	−0.500000	+	0.866025i
$626$		0			0
$627$	2.00000	+	3.46410i	0.0798723	+	0.138343i
$628$		0			0
$629$	−28.2843			−1.12777
$630$		0			0
$631$	−36.0000			−1.43314			−0.716569	−	0.697517i	$-0.754288\pi$
$631$	−36.0000			−1.43314			−0.716569	+	0.697517i	$0.754288\pi$
$632$		0			0
$633$	−14.1421	−	24.4949i	−0.562099	−	0.973585i
$634$		0			0
$635$		0			0
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$	10.0000	+	17.3205i	0.394976	+	0.684119i	0.993098	−	0.117286i	$-0.0374195\pi$
$641$	10.0000	+	17.3205i	0.394976	+	0.684119i	−0.598122	+	0.801405i	$0.704086\pi$
$642$		0			0
$643$	−12.7279			−0.501940			−0.250970	−	0.967995i	$-0.580750\pi$
$643$	−12.7279			−0.501940			−0.250970	+	0.967995i	$0.580750\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−7.77817	−	13.4722i	−0.305792	−	0.529647i	0.671646	−	0.740873i	$-0.265588\pi$
$647$	−7.77817	−	13.4722i	−0.305792	−	0.529647i	−0.977437	+	0.211226i	$0.932254\pi$
$648$		0			0
$649$	−2.12132	+	3.67423i	−0.0832691	+	0.144226i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	9.00000	−	15.5885i	0.352197	−	0.610023i	−0.634437	−	0.772975i	$-0.718768\pi$
$653$	9.00000	−	15.5885i	0.352197	−	0.610023i	0.986634	+	0.162951i	$0.0521013\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	1.41421			0.0551737
$658$		0			0
$659$	38.0000			1.48027			0.740135	−	0.672458i	$-0.234762\pi$
$659$	38.0000			1.48027			0.740135	+	0.672458i	$0.234762\pi$
$660$		0			0
$661$	16.9706	+	29.3939i	0.660078	+	1.14329i	0.980595	+	0.196046i	$0.0628103\pi$
$661$	16.9706	+	29.3939i	0.660078	+	1.14329i	−0.320516	+	0.947243i	$0.603856\pi$
$662$		0			0
$663$	7.07107	−	12.2474i	0.274618	−	0.475651i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	4.00000	−	6.92820i	0.154881	−	0.268261i
$668$		0			0
$669$	−5.00000	−	8.66025i	−0.193311	−	0.334825i
$670$		0			0
$671$	−12.7279			−0.491356
$672$		0			0
$673$	−46.0000			−1.77317			−0.886585	−	0.462566i	$-0.846929\pi$
$673$	−46.0000			−1.77317			−0.886585	+	0.462566i	$0.846929\pi$
$674$		0			0
$675$	14.1421	+	24.4949i	0.544331	+	0.942809i
$676$		0			0
$677$	−17.6777	+	30.6186i	−0.679408	+	1.17677i	0.295751	+	0.955265i	$0.404430\pi$
$677$	−17.6777	+	30.6186i	−0.679408	+	1.17677i	−0.975159	+	0.221504i	$0.928903\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$	−16.0000	+	27.7128i	−0.613121	+	1.06196i
$682$		0			0
$683$	−2.00000	−	3.46410i	−0.0765279	−	0.132550i	0.825222	−	0.564809i	$-0.191050\pi$
$683$	−2.00000	−	3.46410i	−0.0765279	−	0.132550i	−0.901750	+	0.432259i	$0.857717\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	20.0000			0.763048
$688$		0			0
$689$	2.82843	+	4.89898i	0.107754	+	0.186636i
$690$		0			0
$691$	−19.0919	+	33.0681i	−0.726289	+	1.25797i	0.232152	+	0.972680i	$0.425423\pi$
$691$	−19.0919	+	33.0681i	−0.726289	+	1.25797i	−0.958441	+	0.285291i	$0.907910\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	5.00000	+	8.66025i	0.189389	+	0.328031i
$698$		0			0
$699$	2.82843			0.106981
$700$		0			0
$701$	−18.0000			−0.679851			−0.339925	−	0.940452i	$-0.610402\pi$
$701$	−18.0000			−0.679851			−0.339925	+	0.940452i	$0.610402\pi$
$702$		0			0
$703$	−5.65685	−	9.79796i	−0.213352	−	0.369537i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	5.00000	−	8.66025i	0.187779	−	0.325243i	−0.756730	−	0.653727i	$-0.773204\pi$
$709$	5.00000	−	8.66025i	0.187779	−	0.325243i	0.944509	+	0.328484i	$0.106538\pi$
$710$		0			0
$711$	−5.00000	−	8.66025i	−0.187515	−	0.324785i
$712$		0			0
$713$	−16.9706			−0.635553
$714$		0			0
$715$		0			0
$716$		0			0
$717$	−11.3137	−	19.5959i	−0.422518	−	0.731823i
$718$		0			0
$719$	21.9203	−	37.9671i	0.817490	−	1.41593i	−0.0900367	−	0.995938i	$-0.528698\pi$
$719$	21.9203	−	37.9671i	0.817490	−	1.41593i	0.907526	−	0.419995i	$-0.137968\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$	19.0000	−	32.9090i	0.706618	−	1.22390i
$724$		0			0
$725$	5.00000	+	8.66025i	0.185695	+	0.321634i
$726$		0			0
$727$	−35.3553			−1.31126			−0.655628	−	0.755084i	$-0.727596\pi$
$727$	−35.3553			−1.31126			−0.655628	+	0.755084i	$0.727596\pi$
$728$		0			0
$729$	29.0000			1.07407
$730$		0			0
$731$	−7.07107	−	12.2474i	−0.261533	−	0.452988i
$732$		0			0
$733$	−9.19239	+	15.9217i	−0.339529	+	0.588081i	−0.984344	−	0.176257i	$-0.943601\pi$
$733$	−9.19239	+	15.9217i	−0.339529	+	0.588081i	0.644815	+	0.764338i	$0.276934\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−4.00000	+	6.92820i	−0.147342	+	0.255204i
$738$		0			0
$739$	10.0000	+	17.3205i	0.367856	+	0.637145i	0.989230	−	0.146369i	$-0.0467586\pi$
$739$	10.0000	+	17.3205i	0.367856	+	0.637145i	−0.621374	+	0.783514i	$0.713425\pi$
$740$		0			0
$741$	5.65685			0.207810
$742$		0			0
$743$	16.0000			0.586983			0.293492	−	0.955962i	$-0.405183\pi$
$743$	16.0000			0.586983			0.293492	+	0.955962i	$0.405183\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	−4.24264	+	7.34847i	−0.155230	+	0.268866i
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−20.0000	+	34.6410i	−0.729810	+	1.26407i	0.227153	+	0.973859i	$0.427058\pi$
$751$	−20.0000	+	34.6410i	−0.729810	+	1.26407i	−0.956963	+	0.290209i	$0.906275\pi$
$752$		0			0
$753$	11.0000	+	19.0526i	0.400862	+	0.694314i
$754$		0			0
$755$		0			0
$756$		0			0
$757$	22.0000			0.799604			0.399802	−	0.916602i	$-0.369079\pi$
$757$	22.0000			0.799604			0.399802	+	0.916602i	$0.369079\pi$
$758$		0			0
$759$	2.82843	+	4.89898i	0.102665	+	0.177822i
$760$		0			0
$761$	6.36396	−	11.0227i	0.230693	−	0.399573i	−0.727319	−	0.686300i	$-0.759234\pi$
$761$	6.36396	−	11.0227i	0.230693	−	0.399573i	0.958012	+	0.286727i	$0.0925672\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$	3.00000	+	5.19615i	0.108324	+	0.187622i
$768$		0			0
$769$	1.41421			0.0509978			0.0254989	−	0.999675i	$-0.491883\pi$
$769$	1.41421			0.0509978			0.0254989	+	0.999675i	$0.491883\pi$
$770$		0			0
$771$	−8.00000			−0.288113
$772$		0			0
$773$	7.07107	+	12.2474i	0.254329	+	0.440510i	0.964713	−	0.263304i	$-0.0848122\pi$
$773$	7.07107	+	12.2474i	0.254329	+	0.440510i	−0.710384	+	0.703814i	$0.751479\pi$
$774$		0			0
$775$	10.6066	−	18.3712i	0.381000	−	0.659912i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−2.00000	+	3.46410i	−0.0716574	+	0.124114i
$780$		0			0
$781$		0			0
$782$		0			0
$783$	11.3137			0.404319
$784$		0			0
$785$		0			0
$786$		0			0
$787$	−1.41421	−	2.44949i	−0.0504113	−	0.0873149i	0.839719	−	0.543022i	$-0.182720\pi$
$787$	−1.41421	−	2.44949i	−0.0504113	−	0.0873149i	−0.890130	+	0.455707i	$0.849387\pi$
$788$		0			0
$789$	5.65685	−	9.79796i	0.201389	−	0.348817i
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−9.00000	+	15.5885i	−0.319599	+	0.553562i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	39.5980			1.40263			0.701316	−	0.712850i	$-0.252596\pi$
$797$	39.5980			1.40263			0.701316	+	0.712850i	$0.252596\pi$
$798$		0			0
$799$	−70.0000			−2.47642
$800$		0			0
$801$	−5.65685	−	9.79796i	−0.199875	−	0.346194i
$802$		0			0
$803$	−0.707107	+	1.22474i	−0.0249533	+	0.0432203i
$804$		0			0
$805$		0			0
$806$		0			0
$807$	14.0000	−	24.2487i	0.492823	−	0.853595i
$808$		0			0
$809$	15.0000	+	25.9808i	0.527372	+	0.913435i	0.999491	+	0.0319002i	$0.0101559\pi$
$809$	15.0000	+	25.9808i	0.527372	+	0.913435i	−0.472119	+	0.881535i	$0.656511\pi$
$810$		0			0
$811$	45.2548			1.58911			0.794556	−	0.607191i	$-0.207704\pi$
$811$	45.2548			1.58911			0.794556	+	0.607191i	$0.207704\pi$
$812$		0			0
$813$	36.0000			1.26258
$814$		0			0
$815$		0			0
$816$		0			0
$817$	2.82843	−	4.89898i	0.0989541	−	0.171394i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	27.0000	−	46.7654i	0.942306	−	1.63212i	0.181250	−	0.983437i	$-0.441986\pi$
$821$	27.0000	−	46.7654i	0.942306	−	1.63212i	0.761056	−	0.648686i	$-0.224681\pi$
$822$		0			0
$823$	14.0000	+	24.2487i	0.488009	+	0.845257i	0.999905	−	0.0137907i	$-0.00438987\pi$
$823$	14.0000	+	24.2487i	0.488009	+	0.845257i	−0.511896	+	0.859048i	$0.671057\pi$
$824$		0			0
$825$	−7.07107			−0.246183
$826$		0			0
$827$	−6.00000			−0.208640			−0.104320	−	0.994544i	$-0.533267\pi$
$827$	−6.00000			−0.208640			−0.104320	+	0.994544i	$0.533267\pi$
$828$		0			0
$829$	−5.65685	−	9.79796i	−0.196471	−	0.340297i	0.750911	−	0.660403i	$-0.229615\pi$
$829$	−5.65685	−	9.79796i	−0.196471	−	0.340297i	−0.947382	+	0.320106i	$0.896281\pi$
$830$		0			0
$831$	−1.41421	+	2.44949i	−0.0490585	+	0.0849719i
$832$		0			0
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$	−12.0000	−	20.7846i	−0.414781	−	0.718421i
$838$		0			0
$839$	35.3553			1.22060			0.610301	−	0.792170i	$-0.291049\pi$
$839$	35.3553			1.22060			0.610301	+	0.792170i	$0.291049\pi$
$840$		0			0
$841$	−25.0000			−0.862069
$842$		0			0
$843$	1.41421	+	2.44949i	0.0487081	+	0.0843649i
$844$		0			0
$845$		0			0
$846$		0			0
$847$		0			0
$848$		0			0
$849$	20.0000	−	34.6410i	0.686398	−	1.18888i
$850$		0			0
$851$	−8.00000	−	13.8564i	−0.274236	−	0.474991i
$852$		0			0
$853$	−4.24264			−0.145265			−0.0726326	−	0.997359i	$-0.523140\pi$
$853$	−4.24264			−0.145265			−0.0726326	+	0.997359i	$0.523140\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	14.8492	+	25.7196i	0.507240	+	0.878566i	0.999965	+	0.00838084i	$0.00266773\pi$
$857$	14.8492	+	25.7196i	0.507240	+	0.878566i	−0.492724	+	0.870185i	$0.663999\pi$
$858$		0			0
$859$	6.36396	−	11.0227i	0.217136	−	0.376090i	−0.736796	−	0.676116i	$-0.763662\pi$
$859$	6.36396	−	11.0227i	0.217136	−	0.376090i	0.953931	+	0.300026i	$0.0969953\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	8.00000	−	13.8564i	0.272323	−	0.471678i	−0.697133	−	0.716942i	$-0.745541\pi$
$863$	8.00000	−	13.8564i	0.272323	−	0.471678i	0.969456	+	0.245264i	$0.0788747\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	−46.6690			−1.58496
$868$		0			0
$869$	10.0000			0.339227
$870$		0			0
$871$	5.65685	+	9.79796i	0.191675	+	0.331991i
$872$		0			0
$873$	−4.24264	+	7.34847i	−0.143592	+	0.248708i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−1.00000	+	1.73205i	−0.0337676	+	0.0584872i	−0.882415	−	0.470471i	$-0.844084\pi$
$877$	−1.00000	+	1.73205i	−0.0337676	+	0.0584872i	0.848648	+	0.528958i	$0.177417\pi$
$878$		0			0
$879$	1.00000	+	1.73205i	0.0337292	+	0.0584206i
$880$		0			0
$881$	19.7990			0.667045			0.333522	−	0.942742i	$-0.391763\pi$
$881$	19.7990			0.667045			0.333522	+	0.942742i	$0.391763\pi$
$882$		0			0
$883$	36.0000			1.21150			0.605748	−	0.795656i	$-0.292874\pi$
$883$	36.0000			1.21150			0.605748	+	0.795656i	$0.292874\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−18.3848	+	31.8434i	−0.617300	+	1.06920i	0.372676	+	0.927962i	$0.378440\pi$
$887$	−18.3848	+	31.8434i	−0.617300	+	1.06920i	−0.989976	+	0.141234i	$0.954893\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	−2.50000	+	4.33013i	−0.0837532	+	0.145065i
$892$		0			0
$893$	−14.0000	−	24.2487i	−0.468492	−	0.811452i
$894$		0			0
$895$		0			0
$896$		0			0
$897$	8.00000			0.267112
$898$		0			0
$899$	−4.24264	−	7.34847i	−0.141500	−	0.245085i
$900$		0			0
$901$	14.1421	−	24.4949i	0.471143	−	0.816043i
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$	16.0000	+	27.7128i	0.531271	+	0.920189i	0.999334	+	0.0364935i	$0.0116188\pi$
$907$	16.0000	+	27.7128i	0.531271	+	0.920189i	−0.468063	+	0.883695i	$0.655048\pi$
$908$		0			0
$909$	−1.41421			−0.0469065
$910$		0			0
$911$	36.0000			1.19273			0.596367	−	0.802712i	$-0.296610\pi$
$911$	36.0000			1.19273			0.596367	+	0.802712i	$0.296610\pi$
$912$		0			0
$913$	−4.24264	−	7.34847i	−0.140411	−	0.243199i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	20.0000	−	34.6410i	0.659739	−	1.14270i	−0.320944	−	0.947098i	$-0.604000\pi$
$919$	20.0000	−	34.6410i	0.659739	−	1.14270i	0.980683	−	0.195603i	$-0.0626666\pi$
$920$		0			0
$921$	8.00000	+	13.8564i	0.263609	+	0.456584i
$922$		0			0
$923$		0			0
$924$		0			0
$925$	20.0000			0.657596
$926$		0			0
$927$	−3.53553	−	6.12372i	−0.116122	−	0.201129i
$928$		0			0
$929$	1.41421	−	2.44949i	0.0463988	−	0.0803652i	−0.841893	−	0.539644i	$-0.818559\pi$
$929$	1.41421	−	2.44949i	0.0463988	−	0.0803652i	0.888292	+	0.459279i	$0.151892\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$	11.0000	−	19.0526i	0.360124	−	0.623753i
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−55.1543			−1.80181			−0.900907	−	0.434013i	$-0.857097\pi$
$937$	−55.1543			−1.80181			−0.900907	+	0.434013i	$0.857097\pi$
$938$		0			0
$939$	−32.0000			−1.04428
$940$		0			0
$941$	13.4350	+	23.2702i	0.437969	+	0.758585i	0.997533	−	0.0702023i	$-0.0223645\pi$
$941$	13.4350	+	23.2702i	0.437969	+	0.758585i	−0.559563	+	0.828788i	$0.689031\pi$
$942$		0			0
$943$	−2.82843	+	4.89898i	−0.0921063	+	0.159533i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−10.0000	+	17.3205i	−0.324956	+	0.562841i	−0.981504	−	0.191444i	$-0.938683\pi$
$947$	−10.0000	+	17.3205i	−0.324956	+	0.562841i	0.656547	+	0.754285i	$0.272016\pi$
$948$		0			0
$949$	1.00000	+	1.73205i	0.0324614	+	0.0562247i
$950$		0			0
$951$	−31.1127			−1.00890
$952$		0			0
$953$	−34.0000			−1.10137			−0.550684	−	0.834714i	$-0.685633\pi$
$953$	−34.0000			−1.10137			−0.550684	+	0.834714i	$0.685633\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$	−1.41421	+	2.44949i	−0.0457150	+	0.0791808i
$958$		0			0
$959$		0			0
$960$		0			0
$961$	6.50000	−	11.2583i	0.209677	−	0.363172i
$962$		0			0
$963$	−6.00000	−	10.3923i	−0.193347	−	0.334887i
$964$		0			0
$965$		0			0
$966$		0			0
$967$	58.0000			1.86515			0.932577	−	0.360971i	$-0.117555\pi$
$967$	58.0000			1.86515			0.932577	+	0.360971i	$0.117555\pi$
$968$		0			0
$969$	−14.1421	−	24.4949i	−0.454311	−	0.786889i
$970$		0			0
$971$	−9.19239	+	15.9217i	−0.294998	+	0.510951i	−0.974984	−	0.222273i	$-0.928652\pi$
$971$	−9.19239	+	15.9217i	−0.294998	+	0.510951i	0.679987	+	0.733225i	$0.261986\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$	−5.00000	+	8.66025i	−0.160128	+	0.277350i
$976$		0			0
$977$	15.0000	+	25.9808i	0.479893	+	0.831198i	0.999734	−	0.0230645i	$-0.00734232\pi$
$977$	15.0000	+	25.9808i	0.479893	+	0.831198i	−0.519841	+	0.854263i	$0.674009\pi$
$978$		0			0
$979$	11.3137			0.361588
$980$		0			0
$981$	10.0000			0.319275
$982$		0			0
$983$	−12.0208	−	20.8207i	−0.383404	−	0.664076i	0.608142	−	0.793828i	$-0.291915\pi$
$983$	−12.0208	−	20.8207i	−0.383404	−	0.664076i	−0.991546	+	0.129752i	$0.958582\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$	4.00000	−	6.92820i	0.127193	−	0.220304i
$990$		0			0
$991$	20.0000	+	34.6410i	0.635321	+	1.10041i	0.986447	+	0.164080i	$0.0524655\pi$
$991$	20.0000	+	34.6410i	0.635321	+	1.10041i	−0.351126	+	0.936328i	$0.614201\pi$
$992$		0			0
$993$	−11.3137			−0.359030
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−2.12132	−	3.67423i	−0.0671829	−	0.116364i	0.830477	−	0.557052i	$-0.188068\pi$
$997$	−2.12132	−	3.67423i	−0.0671829	−	0.116364i	−0.897660	+	0.440688i	$0.854734\pi$
$998$		0			0
$999$	11.3137	−	19.5959i	0.357950	−	0.619987i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2156.2.i.g.177.2		4
7.2	even	3		2156.2.a.e.1.1	✓	2
7.3	odd	6	inner	2156.2.i.g.1145.1		4
7.4	even	3	inner	2156.2.i.g.1145.2		4
7.5	odd	6		2156.2.a.e.1.2	yes	2
7.6	odd	2	inner	2156.2.i.g.177.1		4
28.19	even	6		8624.2.a.bu.1.1		2
28.23	odd	6		8624.2.a.bu.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2156.2.a.e.1.1	✓	2	7.2	even	3
2156.2.a.e.1.2	yes	2	7.5	odd	6
2156.2.i.g.177.1		4	7.6	odd	2	inner
2156.2.i.g.177.2		4	1.1	even	1	trivial
2156.2.i.g.1145.1		4	7.3	odd	6	inner
2156.2.i.g.1145.2		4	7.4	even	3	inner
8624.2.a.bu.1.1		2	28.19	even	6
8624.2.a.bu.1.2		2	28.23	odd	6

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:	\( N \)	\(=\)	\( 2156 = 2^{2} \cdot 7^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2156.i (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(0.707107 + 1.22474i) q^{3} +(0.500000 - 0.866025i) q^{9} +(0.500000 + 0.866025i) q^{11} +1.41421 q^{13} +(-3.53553 - 6.12372i) q^{17} +(1.41421 - 2.44949i) q^{19} +(2.00000 - 3.46410i) q^{23} +(2.50000 + 4.33013i) q^{25} +5.65685 q^{27} +2.00000 q^{29} +(-2.12132 - 3.67423i) q^{31} +(-0.707107 + 1.22474i) q^{33} +(2.00000 - 3.46410i) q^{37} +(1.00000 + 1.73205i) q^{39} -1.41421 q^{41} +2.00000 q^{43} +(4.94975 - 8.57321i) q^{47} +(5.00000 - 8.66025i) q^{51} +(2.00000 + 3.46410i) q^{53} +4.00000 q^{57} +(2.12132 + 3.67423i) q^{59} +(-6.36396 + 11.0227i) q^{61} +(4.00000 + 6.92820i) q^{67} +5.65685 q^{69} +(0.707107 + 1.22474i) q^{73} +(-3.53553 + 6.12372i) q^{75} +(5.00000 - 8.66025i) q^{79} +(2.50000 + 4.33013i) q^{81} -8.48528 q^{83} +(1.41421 + 2.44949i) q^{87} +(5.65685 - 9.79796i) q^{89} +(3.00000 - 5.19615i) q^{93} -8.48528 q^{97} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{9} + 2 q^{11} + 8 q^{23} + 10 q^{25} + 8 q^{29} + 8 q^{37} + 4 q^{39} + 8 q^{43} + 20 q^{51} + 8 q^{53} + 16 q^{57} + 16 q^{67} + 20 q^{79} + 10 q^{81} + 12 q^{93} + 4 q^{99}+O(q^{100}) \) 4 * q + 2 * q^9 + 2 * q^11 + 8 * q^23 + 10 * q^25 + 8 * q^29 + 8 * q^37 + 4 * q^39 + 8 * q^43 + 20 * q^51 + 8 * q^53 + 16 * q^57 + 16 * q^67 + 20 * q^79 + 10 * q^81 + 12 * q^93 + 4 * q^99

Introduction

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