LMFDB - Embedded newform 2156.2.i.n.1145.4

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:	$8$
Relative dimension:	$4$ over $\Q(\zeta_{3})$
Coefficient field:	8.0.7342972683264.14
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 13x^{6} + 151x^{4} + 234x^{2} + 324 $ x^8 + 13x^6 + 151x^4 + 234*x^2 + 324
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			1145.4
Root			$1.69000 - 2.92717i$ of defining polynomial
Character	$\chi$	$=$	2156.1145
Dual form			2156.2.i.n.177.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+(1.69000 - 2.92717i) q^{3} +(0.802429 + 1.38985i) q^{5} +(-4.21221 - 7.29577i) q^{9} +(-0.500000 + 0.866025i) q^{11} -4.98486 q^{13} +5.42443 q^{15} +(0.887573 - 1.53732i) q^{17} +(-3.38000 - 5.85434i) q^{19} +(0.712214 + 1.23359i) q^{23} +(1.21221 - 2.09962i) q^{25} -18.3346 q^{27} -6.00000 q^{29} +(1.69000 - 2.92717i) q^{31} +(1.69000 + 2.92717i) q^{33} +(2.71221 + 4.69769i) q^{37} +(-8.42443 + 14.5915i) q^{39} +8.19458 q^{41} -8.84886 q^{43} +(6.76001 - 11.7087i) q^{45} +(-0.887573 - 1.53732i) q^{47} +(-3.00000 - 5.19615i) q^{51} +(5.42443 - 9.39539i) q^{53} -1.60486 q^{55} -22.8489 q^{57} +(-0.0851435 + 0.147473i) q^{59} +(-5.87244 - 10.1714i) q^{61} +(-4.00000 - 6.92820i) q^{65} +(-6.71221 + 11.6259i) q^{67} +4.81458 q^{69} +5.42443 q^{71} +(2.66272 - 4.61196i) q^{73} +(-4.09729 - 7.09671i) q^{75} +(-3.00000 - 5.19615i) q^{79} +(-18.3489 + 31.7812i) q^{81} -3.55029 q^{83} +2.84886 q^{85} +(-10.1400 + 17.5630i) q^{87} +(5.95758 + 10.3188i) q^{89} +(-5.71221 - 9.89385i) q^{93} +(5.42443 - 9.39539i) q^{95} +8.36487 q^{97} +8.42443 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 14 q^{9} - 4 q^{11} + 4 q^{15} - 14 q^{23} - 10 q^{25} - 48 q^{29} + 2 q^{37} - 28 q^{39} + 8 q^{43} - 24 q^{51} + 4 q^{53} - 104 q^{57} - 32 q^{65} - 34 q^{67} + 4 q^{71} - 24 q^{79} - 68 q^{81}+ \cdots + 28 q^{99}+O(q^{100}) $ 8 * q - 14 * q^9 - 4 * q^11 + 4 * q^15 - 14 * q^23 - 10 * q^25 - 48 * q^29 + 2 * q^37 - 28 * q^39 + 8 * q^43 - 24 * q^51 + 4 * q^53 - 104 * q^57 - 32 * q^65 - 34 * q^67 + 4 * q^71 - 24 * q^79 - 68 * q^81 - 56 * q^85 - 26 * q^93 + 4 * q^95 + 28 * q^99

Character values

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

$n$	$981$	$1079$	$1277$
$\chi(n)$	$1$	$1$	$e\left(\frac{2}{3}\right)$

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
$3$	1.69000	−	2.92717i	0.975723	−	1.69000i	0.298196	−	0.954505i	$-0.403615\pi$
$3$	1.69000	−	2.92717i	0.975723	−	1.69000i	0.677528	−	0.735497i	$-0.263051\pi$
$4$		0			0
$5$	0.802429	+	1.38985i	0.358857	+	0.621559i	0.987770	−	0.155917i	$-0.0498331\pi$
$5$	0.802429	+	1.38985i	0.358857	+	0.621559i	−0.628913	+	0.777476i	$0.716500\pi$
$6$		0			0
$7$		0			0
$7$		0			0
$8$		0			0
$9$	−4.21221	−	7.29577i	−1.40407	−	2.43192i
$10$		0			0
$11$	−0.500000	+	0.866025i	−0.150756	+	0.261116i
$11$	−0.500000	+	0.866025i	−0.150756	+	0.261116i
$12$		0			0
$13$	−4.98486			−1.38255			−0.691276	−	0.722591i	$-0.742951\pi$
$13$	−4.98486			−1.38255			−0.691276	+	0.722591i	$0.742951\pi$
$14$		0			0
$15$	5.42443			1.40058
$16$		0			0
$17$	0.887573	−	1.53732i	0.215268	−	0.372855i	−0.738087	−	0.674705i	$-0.764271\pi$
$17$	0.887573	−	1.53732i	0.215268	−	0.372855i	0.953355	+	0.301850i	$0.0976042\pi$
$18$		0			0
$19$	−3.38000	−	5.85434i	−0.775426	−	1.34308i	−0.934555	−	0.355820i	$-0.884202\pi$
$19$	−3.38000	−	5.85434i	−0.775426	−	1.34308i	0.159128	−	0.987258i	$-0.449132\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	0.712214	+	1.23359i	0.148507	+	0.257222i	0.930676	−	0.365845i	$-0.119220\pi$
$23$	0.712214	+	1.23359i	0.148507	+	0.257222i	−0.782169	+	0.623066i	$0.785887\pi$
$24$		0			0
$25$	1.21221	−	2.09962i	0.242443	−	0.419923i
$26$		0			0
$27$	−18.3346			−3.52849
$28$		0			0
$29$	−6.00000			−1.11417			−0.557086	−	0.830455i	$-0.688081\pi$
$29$	−6.00000			−1.11417			−0.557086	+	0.830455i	$0.688081\pi$
$30$		0			0
$31$	1.69000	−	2.92717i	0.303533	−	0.525735i	−0.673400	−	0.739278i	$-0.735167\pi$
$31$	1.69000	−	2.92717i	0.303533	−	0.525735i	0.976934	+	0.213543i	$0.0685003\pi$
$32$		0			0
$33$	1.69000	+	2.92717i	0.294192	+	0.509555i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	2.71221	+	4.69769i	0.445885	+	0.772296i	0.998113	−	0.0613970i	$-0.0195556\pi$
$37$	2.71221	+	4.69769i	0.445885	+	0.772296i	−0.552228	+	0.833693i	$0.686222\pi$
$38$		0			0
$39$	−8.42443	+	14.5915i	−1.34899	+	2.33652i
$40$		0			0
$41$	8.19458			1.27978			0.639889	−	0.768467i	$-0.278980\pi$
$41$	8.19458			1.27978			0.639889	+	0.768467i	$0.278980\pi$
$42$		0			0
$43$	−8.84886			−1.34944			−0.674719	−	0.738075i	$-0.735735\pi$
$43$	−8.84886			−1.34944			−0.674719	+	0.738075i	$0.735735\pi$
$44$		0			0
$45$	6.76001	−	11.7087i	1.00772	−	1.74543i
$46$		0			0
$47$	−0.887573	−	1.53732i	−0.129466	−	0.224241i	0.794004	−	0.607913i	$-0.207993\pi$
$47$	−0.887573	−	1.53732i	−0.129466	−	0.224241i	−0.923470	+	0.383671i	$0.874660\pi$
$48$		0			0
$49$		0			0
$50$		0			0
$51$	−3.00000	−	5.19615i	−0.420084	−	0.727607i
$52$		0			0
$53$	5.42443	−	9.39539i	0.745103	−	1.29056i	−0.205044	−	0.978753i	$-0.565734\pi$
$53$	5.42443	−	9.39539i	0.745103	−	1.29056i	0.950147	−	0.311803i	$-0.100933\pi$
$54$		0			0
$55$	−1.60486			−0.216399
$56$		0			0
$57$	−22.8489			−3.02641
$58$		0			0
$59$	−0.0851435	+	0.147473i	−0.0110847	+	0.0191993i	−0.871515	−	0.490370i	$-0.836862\pi$
$59$	−0.0851435	+	0.147473i	−0.0110847	+	0.0191993i	0.860430	+	0.509569i	$0.170195\pi$
$60$		0			0
$61$	−5.87244	−	10.1714i	−0.751888	−	1.30231i	−0.946906	−	0.321509i	$-0.895810\pi$
$61$	−5.87244	−	10.1714i	−0.751888	−	1.30231i	0.195018	−	0.980800i	$-0.437523\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−4.00000	−	6.92820i	−0.496139	−	0.859338i
$66$		0			0
$67$	−6.71221	+	11.6259i	−0.820028	+	1.42033i	0.0856335	+	0.996327i	$0.472709\pi$
$67$	−6.71221	+	11.6259i	−0.820028	+	1.42033i	−0.905661	+	0.424003i	$0.860625\pi$
$68$		0			0
$69$	4.81458			0.579607
$70$		0			0
$71$	5.42443			0.643761			0.321881	−	0.946780i	$-0.395685\pi$
$71$	5.42443			0.643761			0.321881	+	0.946780i	$0.395685\pi$
$72$		0			0
$73$	2.66272	−	4.61196i	0.311648	−	0.539790i	−0.667072	−	0.744994i	$-0.732452\pi$
$73$	2.66272	−	4.61196i	0.311648	−	0.539790i	0.978719	+	0.205204i	$0.0657858\pi$
$74$		0			0
$75$	−4.09729	−	7.09671i	−0.473114	−	0.819458i
$76$		0			0
$77$		0			0
$78$		0			0
$79$	−3.00000	−	5.19615i	−0.337526	−	0.584613i	0.646440	−	0.762964i	$-0.276257\pi$
$79$	−3.00000	−	5.19615i	−0.337526	−	0.584613i	−0.983967	+	0.178352i	$0.942924\pi$
$80$		0			0
$81$	−18.3489	+	31.7812i	−2.03876	+	3.53124i
$82$		0			0
$83$	−3.55029			−0.389695			−0.194848	−	0.980834i	$-0.562421\pi$
$83$	−3.55029			−0.389695			−0.194848	+	0.980834i	$0.562421\pi$
$84$		0			0
$85$	2.84886			0.309002
$86$		0			0
$87$	−10.1400	+	17.5630i	−1.08712	+	1.88295i
$88$		0			0
$89$	5.95758	+	10.3188i	0.631502	+	1.09379i	0.987245	+	0.159210i	$0.0508946\pi$
$89$	5.95758	+	10.3188i	0.631502	+	1.09379i	−0.355743	+	0.934584i	$0.615772\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$	−5.71221	−	9.89385i	−0.592329	−	1.02594i
$94$		0			0
$95$	5.42443	−	9.39539i	0.556535	−	0.963946i
$96$		0			0
$97$	8.36487			0.849324			0.424662	−	0.905352i	$-0.360393\pi$
$97$	8.36487			0.849324			0.424662	+	0.905352i	$0.360393\pi$
$98$		0			0
$99$	8.42443			0.846687
$100$		0			0
$101$	5.70215	−	9.87641i	0.567385	−	0.982740i	−0.429438	−	0.903096i	$-0.641288\pi$
$101$	5.70215	−	9.87641i	0.567385	−	0.982740i	0.996823	−	0.0796435i	$-0.0253782\pi$
$102$		0			0
$103$	0.887573	+	1.53732i	0.0874552	+	0.151477i	0.906435	−	0.422346i	$-0.138793\pi$
$103$	0.887573	+	1.53732i	0.0874552	+	0.151477i	−0.818980	+	0.573823i	$0.805460\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−7.42443	−	12.8595i	−0.717747	−	1.24317i	−0.961891	−	0.273435i	$-0.911840\pi$
$107$	−7.42443	−	12.8595i	−0.717747	−	1.24317i	0.244144	−	0.969739i	$-0.421493\pi$
$108$		0			0
$109$	6.42443	−	11.1274i	0.615349	−	1.06582i	−0.374975	−	0.927035i	$-0.622349\pi$
$109$	6.42443	−	11.1274i	0.615349	−	1.06582i	0.990323	−	0.138780i	$-0.0443181\pi$
$110$		0			0
$111$	18.3346			1.74024
$112$		0			0
$113$	0.575571			0.0541452			0.0270726	−	0.999633i	$-0.491381\pi$
$113$	0.575571			0.0541452			0.0270726	+	0.999633i	$0.491381\pi$
$114$		0			0
$115$	−1.14300	+	1.97974i	−0.106586	+	0.184612i
$116$		0			0
$117$	20.9973	+	36.3684i	1.94120	+	3.36226i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−0.500000	−	0.866025i	−0.0454545	−	0.0787296i
$122$		0			0
$123$	13.8489	−	23.9869i	1.24871	−	2.16283i
$124$		0			0
$125$	11.9152			1.06572
$126$		0			0
$127$	18.8489			1.67257			0.836283	−	0.548298i	$-0.184724\pi$
$127$	18.8489			1.67257			0.836283	+	0.548298i	$0.184724\pi$
$128$		0			0
$129$	−14.9546	+	25.9021i	−1.31668	+	2.28055i
$130$		0			0
$131$	−5.15515	−	8.92898i	−0.450408	−	0.780129i	0.548004	−	0.836476i	$-0.315388\pi$
$131$	−5.15515	−	8.92898i	−0.450408	−	0.780129i	−0.998411	+	0.0563472i	$0.982055\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$	−14.7122	−	25.4823i	−1.26623	−	2.19317i
$136$		0			0
$137$	−8.71221	+	15.0900i	−0.744335	+	1.28923i	0.206170	+	0.978516i	$0.433900\pi$
$137$	−8.71221	+	15.0900i	−0.744335	+	1.28923i	−0.950505	+	0.310709i	$0.899433\pi$
$138$		0			0
$139$	6.76001			0.573376			0.286688	−	0.958024i	$-0.407446\pi$
$139$	6.76001			0.573376			0.286688	+	0.958024i	$0.407446\pi$
$140$		0			0
$141$	−6.00000			−0.505291
$142$		0			0
$143$	2.49243	−	4.31702i	0.208428	−	0.361007i
$144$		0			0
$145$	−4.81458	−	8.33909i	−0.399829	−	0.692524i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	9.84886	+	17.0587i	0.806850	+	1.39750i	0.915035	+	0.403374i	$0.132163\pi$
$149$	9.84886	+	17.0587i	0.806850	+	1.39750i	−0.108185	+	0.994131i	$0.534504\pi$
$150$		0			0
$151$	−7.00000	+	12.1244i	−0.569652	+	0.986666i	0.426948	+	0.904276i	$0.359589\pi$
$151$	−7.00000	+	12.1244i	−0.569652	+	0.986666i	−0.996600	+	0.0823900i	$0.973745\pi$
$152$		0			0
$153$	−14.9546			−1.20901
$154$		0			0
$155$	5.42443			0.435701
$156$		0			0
$157$	−4.01215	+	6.94924i	−0.320204	+	0.554610i	−0.980530	−	0.196369i	$-0.937085\pi$
$157$	−4.01215	+	6.94924i	−0.320204	+	0.554610i	0.660326	+	0.750979i	$0.270418\pi$
$158$		0			0
$159$	−18.3346	−	31.7564i	−1.45403	−	2.51845i
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−6.00000	−	10.3923i	−0.469956	−	0.813988i	0.529454	−	0.848339i	$-0.322397\pi$
$163$	−6.00000	−	10.3923i	−0.469956	−	0.813988i	−0.999410	+	0.0343508i	$0.989064\pi$
$164$		0			0
$165$	−2.71221	+	4.69769i	−0.211146	+	0.365715i
$166$		0			0
$167$	23.8303			1.84405			0.922023	−	0.387136i	$-0.126536\pi$
$167$	23.8303			1.84405			0.922023	+	0.387136i	$0.126536\pi$
$168$		0			0
$169$	11.8489			0.911451
$170$		0			0
$171$	−28.4746	+	49.3195i	−2.17751	+	3.77155i
$172$		0			0
$173$	3.92700	+	6.80177i	0.298565	+	0.517129i	0.975808	−	0.218630i	$-0.0701589\pi$
$173$	3.92700	+	6.80177i	0.298565	+	0.517129i	−0.677243	+	0.735759i	$0.736826\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	0.287786	+	0.498459i	0.0216313	+	0.0374665i
$178$		0			0
$179$	−2.13664	+	3.70077i	−0.159700	+	0.276609i	−0.934761	−	0.355278i	$-0.884386\pi$
$179$	−2.13664	+	3.70077i	−0.159700	+	0.276609i	0.775060	+	0.631887i	$0.217719\pi$
$180$		0			0
$181$	15.4654			1.14954			0.574769	−	0.818316i	$-0.305092\pi$
$181$	15.4654			1.14954			0.574769	+	0.818316i	$0.305092\pi$
$182$		0			0
$183$	−39.6977			−2.93454
$184$		0			0
$185$	−4.35272	+	7.53913i	−0.320018	+	0.554288i
$186$		0			0
$187$	0.887573	+	1.53732i	0.0649058	+	0.112420i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	6.71221	+	11.6259i	0.485679	+	0.841220i	0.999865	−	0.0164585i	$-0.00523914\pi$
$191$	6.71221	+	11.6259i	0.485679	+	0.841220i	−0.514186	+	0.857679i	$0.671906\pi$
$192$		0			0
$193$	−5.00000	+	8.66025i	−0.359908	+	0.623379i	−0.987945	−	0.154805i	$-0.950525\pi$
$193$	−5.00000	+	8.66025i	−0.359908	+	0.623379i	0.628037	+	0.778183i	$0.283859\pi$
$194$		0			0
$195$	−27.0400			−1.93638
$196$		0			0
$197$	−6.00000			−0.427482			−0.213741	−	0.976890i	$-0.568565\pi$
$197$	−6.00000			−0.427482			−0.213741	+	0.976890i	$0.568565\pi$
$198$		0			0
$199$	7.64758	−	13.2460i	0.542123	−	0.938984i	−0.456659	−	0.889642i	$-0.650954\pi$
$199$	7.64758	−	13.2460i	0.542123	−	0.938984i	0.998782	−	0.0493423i	$-0.0157125\pi$
$200$		0			0
$201$	22.6873	+	39.2956i	1.60024	+	2.77170i
$202$		0			0
$203$		0			0
$204$		0			0
$205$	6.57557	+	11.3892i	0.459258	+	0.795458i
$206$		0			0
$207$	6.00000	−	10.3923i	0.417029	−	0.722315i
$208$		0			0
$209$	6.76001			0.467600
$210$		0			0
$211$	−6.84886			−0.471495			−0.235747	−	0.971814i	$-0.575754\pi$
$211$	−6.84886			−0.471495			−0.235747	+	0.971814i	$0.575754\pi$
$212$		0			0
$213$	9.16730	−	15.8782i	0.628133	−	1.08796i
$214$		0			0
$215$	−7.10058	−	12.2986i	−0.484256	−	0.838756i
$216$		0			0
$217$		0			0
$218$		0			0
$219$	−9.00000	−	15.5885i	−0.608164	−	1.05337i
$220$		0			0
$221$	−4.42443	+	7.66334i	−0.297619	+	0.515492i
$222$		0			0
$223$	6.93030			0.464087			0.232043	−	0.972705i	$-0.425459\pi$
$223$	6.93030			0.464087			0.232043	+	0.972705i	$0.425459\pi$
$224$		0			0
$225$	−20.4244			−1.36163
$226$		0			0
$227$	3.55029	−	6.14929i	0.235641	−	0.408142i	−0.723818	−	0.689991i	$-0.757614\pi$
$227$	3.55029	−	6.14929i	0.235641	−	0.408142i	0.959459	+	0.281849i	$0.0909477\pi$
$228$		0			0
$229$	0.632142	+	1.09490i	0.0417731	+	0.0723532i	0.886156	−	0.463387i	$-0.153366\pi$
$229$	0.632142	+	1.09490i	0.0417731	+	0.0723532i	−0.844383	+	0.535740i	$0.820033\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−9.84886	−	17.0587i	−0.645220	−	1.11755i	−0.984251	−	0.176779i	$-0.943432\pi$
$233$	−9.84886	−	17.0587i	−0.645220	−	1.11755i	0.339031	−	0.940775i	$-0.389901\pi$
$234$		0			0
$235$	1.42443	−	2.46718i	0.0929195	−	0.160941i
$236$		0			0
$237$	−20.2800			−1.31733
$238$		0			0
$239$	−21.6977			−1.40351			−0.701754	−	0.712419i	$-0.747600\pi$
$239$	−21.6977			−1.40351			−0.701754	+	0.712419i	$0.747600\pi$
$240$		0			0
$241$	4.26758	−	7.39166i	0.274899	−	0.476139i	−0.695211	−	0.718806i	$-0.744689\pi$
$241$	4.26758	−	7.39166i	0.274899	−	0.476139i	0.970110	+	0.242667i	$0.0780223\pi$
$242$		0			0
$243$	34.5173	+	59.7858i	2.21429	+	3.83526i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	16.8489	+	29.1831i	1.07207	+	1.85687i
$248$		0			0
$249$	−6.00000	+	10.3923i	−0.380235	+	0.658586i
$250$		0			0
$251$	−6.58972			−0.415940			−0.207970	−	0.978135i	$-0.566686\pi$
$251$	−6.58972			−0.415940			−0.207970	+	0.978135i	$0.566686\pi$
$252$		0			0
$253$	−1.42443			−0.0895531
$254$		0			0
$255$	4.81458	−	8.33909i	0.301500	−	0.522214i
$256$		0			0
$257$	−9.96973	−	17.2681i	−0.621894	−	1.07715i	−0.989133	−	0.147026i	$-0.953030\pi$
$257$	−9.96973	−	17.2681i	−0.621894	−	1.07715i	0.367238	−	0.930127i	$-0.380303\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	25.2733	+	43.7746i	1.56438	+	2.70958i
$262$		0			0
$263$	2.57557	−	4.46102i	0.158817	−	0.275078i	−0.775626	−	0.631193i	$-0.782566\pi$
$263$	2.57557	−	4.46102i	0.158817	−	0.275078i	0.934442	+	0.356115i	$0.115899\pi$
$264$		0			0
$265$	17.4109			1.06954
$266$		0			0
$267$	40.2733			2.46469
$268$		0			0
$269$	−5.32544	+	9.22393i	−0.324698	+	0.562393i	−0.981451	−	0.191713i	$-0.938596\pi$
$269$	−5.32544	+	9.22393i	−0.324698	+	0.562393i	0.656753	+	0.754105i	$0.271929\pi$
$270$		0			0
$271$	4.98486	+	8.63404i	0.302809	+	0.524480i	0.976771	−	0.214285i	$-0.0687423\pi$
$271$	4.98486	+	8.63404i	0.302809	+	0.524480i	−0.673962	+	0.738766i	$0.735409\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	1.21221	+	2.09962i	0.0730993	+	0.126612i
$276$		0			0
$277$	−12.4244	+	21.5197i	−0.746512	+	1.29300i	0.202974	+	0.979184i	$0.434939\pi$
$277$	−12.4244	+	21.5197i	−0.746512	+	1.29300i	−0.949485	+	0.313812i	$0.898394\pi$
$278$		0			0
$279$	−28.4746			−1.70473
$280$		0			0
$281$	4.84886			0.289259			0.144629	−	0.989486i	$-0.453801\pi$
$281$	4.84886			0.289259			0.144629	+	0.989486i	$0.453801\pi$
$282$		0			0
$283$	−6.41943	+	11.1188i	−0.381596	+	0.660943i	−0.991291	−	0.131693i	$-0.957959\pi$
$283$	−6.41943	+	11.1188i	−0.381596	+	0.660943i	0.609695	+	0.792636i	$0.291292\pi$
$284$		0			0
$285$	−18.3346	−	31.7564i	−1.08605	−	1.88109i
$286$		0			0
$287$		0			0
$288$		0			0
$289$	6.92443	+	11.9935i	0.407319	+	0.705498i
$290$		0			0
$291$	14.1366	−	24.4854i	0.828705	−	1.43536i
$292$		0			0
$293$	18.1643			1.06117			0.530585	−	0.847632i	$-0.321972\pi$
$293$	18.1643			1.06117			0.530585	+	0.847632i	$0.321972\pi$
$294$		0			0
$295$	−0.273287			−0.0159114
$296$		0			0
$297$	9.16730	−	15.8782i	0.531940	−	0.921348i
$298$		0			0
$299$	−3.55029	−	6.14929i	−0.205319	−	0.355622i
$300$		0			0
$301$		0			0
$302$		0			0
$303$	−19.2733	−	33.3823i	−1.10722	−	1.91776i
$304$		0			0
$305$	9.42443	−	16.3236i	0.539641	−	0.934686i
$306$		0			0
$307$	−0.681148			−0.0388752			−0.0194376	−	0.999811i	$-0.506188\pi$
$307$	−0.681148			−0.0388752			−0.0194376	+	0.999811i	$0.506188\pi$
$308$		0			0
$309$	6.00000			0.341328
$310$		0			0
$311$	11.1979	−	19.3953i	0.634973	−	1.09981i	−0.351548	−	0.936170i	$-0.614344\pi$
$311$	11.1979	−	19.3953i	0.634973	−	1.09981i	0.986521	−	0.163636i	$-0.0523222\pi$
$312$		0			0
$313$	−9.16730	−	15.8782i	−0.518166	−	0.897490i	−0.999777	−	0.0211052i	$-0.993282\pi$
$313$	−9.16730	−	15.8782i	−0.518166	−	0.897490i	0.481611	−	0.876385i	$-0.340052\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	3.71221	+	6.42974i	0.208499	+	0.361130i	0.951242	−	0.308446i	$-0.0998090\pi$
$317$	3.71221	+	6.42974i	0.208499	+	0.361130i	−0.742743	+	0.669577i	$0.766476\pi$
$318$		0			0
$319$	3.00000	−	5.19615i	0.167968	−	0.290929i
$320$		0			0
$321$	−50.1892			−2.80129
$322$		0			0
$323$	−12.0000			−0.667698
$324$		0			0
$325$	−6.04272	+	10.4663i	−0.335190	+	0.580566i
$326$		0			0
$327$	−21.7146	−	37.6108i	−1.20082	−	2.07988i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−6.71221	−	11.6259i	−0.368937	−	0.639017i	0.620463	−	0.784236i	$-0.286945\pi$
$331$	−6.71221	−	11.6259i	−0.368937	−	0.639017i	−0.989400	+	0.145219i	$0.953611\pi$
$332$		0			0
$333$	22.8489	−	39.5754i	1.25211	−	2.16872i
$334$		0			0
$335$	−21.5443			−1.17709
$336$		0			0
$337$	15.6977			0.855109			0.427555	−	0.903990i	$-0.359375\pi$
$337$	15.6977			0.855109			0.427555	+	0.903990i	$0.359375\pi$
$338$		0			0
$339$	0.972716	−	1.68479i	0.0528307	−	0.0915055i
$340$		0			0
$341$	1.69000	+	2.92717i	0.0915187	+	0.158515i
$342$		0			0
$343$		0			0
$344$		0			0
$345$	3.86336	+	6.69153i	0.207996	+	0.360260i
$346$		0			0
$347$	5.00000	−	8.66025i	0.268414	−	0.464907i	−0.700038	−	0.714105i	$-0.746834\pi$
$347$	5.00000	−	8.66025i	0.268414	−	0.464907i	0.968452	+	0.249198i	$0.0801671\pi$
$348$		0			0
$349$	11.0637			0.592228			0.296114	−	0.955153i	$-0.404309\pi$
$349$	11.0637			0.592228			0.296114	+	0.955153i	$0.404309\pi$
$350$		0			0
$351$	91.3954			4.87833
$352$		0			0
$353$	17.8727	−	30.9565i	0.951270	−	1.64765i	0.208588	−	0.978004i	$-0.433113\pi$
$353$	17.8727	−	30.9565i	0.951270	−	1.64765i	0.742682	−	0.669644i	$-0.233553\pi$
$354$		0			0
$355$	4.35272	+	7.53913i	0.231018	+	0.400136i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−2.42443	−	4.19923i	−0.127956	−	0.221627i	0.794928	−	0.606703i	$-0.207508\pi$
$359$	−2.42443	−	4.19923i	−0.127956	−	0.221627i	−0.922885	+	0.385076i	$0.874175\pi$
$360$		0			0
$361$	−13.3489	+	23.1209i	−0.702571	+	1.21689i
$362$		0			0
$363$	−3.38000			−0.177404
$364$		0			0
$365$	8.54657			0.447348
$366$		0			0
$367$	11.8300	−	20.4902i	0.617522	−	1.06958i	−0.372415	−	0.928066i	$-0.621470\pi$
$367$	11.8300	−	20.4902i	0.617522	−	1.06958i	0.989936	−	0.141513i	$-0.0451966\pi$
$368$		0			0
$369$	−34.5173	−	59.7858i	−1.79690	−	3.11232i
$370$		0			0
$371$		0			0
$372$		0			0
$373$	−4.42443	−	7.66334i	−0.229088	−	0.396792i	0.728450	−	0.685099i	$-0.240241\pi$
$373$	−4.42443	−	7.66334i	−0.229088	−	0.396792i	−0.957538	+	0.288307i	$0.906908\pi$
$374$		0			0
$375$	20.1366	−	34.8777i	1.03985	−	1.80108i
$376$		0			0
$377$	29.9092			1.54040
$378$		0			0
$379$	26.5756			1.36510			0.682548	−	0.730841i	$-0.260872\pi$
$379$	26.5756			1.36510			0.682548	+	0.730841i	$0.260872\pi$
$380$		0			0
$381$	31.8546	−	55.1738i	1.63196	−	2.82664i
$382$		0			0
$383$	−3.29486	−	5.70687i	−0.168360	−	0.291607i	0.769484	−	0.638666i	$-0.220514\pi$
$383$	−3.29486	−	5.70687i	−0.168360	−	0.291607i	−0.937843	+	0.347059i	$0.887180\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	37.2733	+	64.5592i	1.89471	+	3.28173i
$388$		0			0
$389$	15.1366	−	26.2174i	0.767458	−	1.32928i	−0.171479	−	0.985188i	$-0.554854\pi$
$389$	15.1366	−	26.2174i	0.767458	−	1.32928i	0.938937	−	0.344089i	$-0.111812\pi$
$390$		0			0
$391$	2.52857			0.127875
$392$		0			0
$393$	−34.8489			−1.75789
$394$		0			0
$395$	4.81458	−	8.33909i	0.242248	−	0.419585i
$396$		0			0
$397$	6.76001	+	11.7087i	0.339275	+	0.587642i	0.984297	−	0.176523i	$-0.0564849\pi$
$397$	6.76001	+	11.7087i	0.339275	+	0.587642i	−0.645021	+	0.764164i	$0.723152\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−11.8489	−	20.5228i	−0.591704	−	1.02486i	−0.994003	−	0.109353i	$-0.965122\pi$
$401$	−11.8489	−	20.5228i	−0.591704	−	1.02486i	0.402299	−	0.915508i	$-0.368211\pi$
$402$		0			0
$403$	−8.42443	+	14.5915i	−0.419651	+	0.726856i
$404$		0			0
$405$	−58.8946			−2.92650
$406$		0			0
$407$	−5.42443			−0.268879
$408$		0			0
$409$	−10.8573	+	18.8054i	−0.536859	+	0.929867i	0.462212	+	0.886769i	$0.347056\pi$
$409$	−10.8573	+	18.8054i	−0.536859	+	0.929867i	−0.999071	+	0.0430974i	$0.986277\pi$
$410$		0			0
$411$	29.4473	+	51.0043i	1.45253	+	2.51585i
$412$		0			0
$413$		0			0
$414$		0			0
$415$	−2.84886	−	4.93437i	−0.139845	−	0.242219i
$416$		0			0
$417$	11.4244	−	19.7877i	0.559457	−	0.969007i
$418$		0			0
$419$	−8.87573			−0.433608			−0.216804	−	0.976215i	$-0.569563\pi$
$419$	−8.87573			−0.433608			−0.216804	+	0.976215i	$0.569563\pi$
$420$		0			0
$421$	17.1511			0.835896			0.417948	−	0.908471i	$-0.362750\pi$
$421$	17.1511			0.835896			0.417948	+	0.908471i	$0.362750\pi$
$422$		0			0
$423$	−7.47729	+	12.9511i	−0.363558	+	0.629702i
$424$		0			0
$425$	−2.15186	−	3.72713i	−0.104380	−	0.180792i
$426$		0			0
$427$		0			0
$428$		0			0
$429$	−8.42443	−	14.5915i	−0.406735	−	0.704486i
$430$		0			0
$431$	−4.42443	+	7.66334i	−0.213117	+	0.369130i	−0.952689	−	0.303948i	$-0.901695\pi$
$431$	−4.42443	+	7.66334i	−0.213117	+	0.369130i	0.739571	+	0.673078i	$0.235028\pi$
$432$		0			0
$433$	1.94543			0.0934915			0.0467458	−	0.998907i	$-0.485115\pi$
$433$	1.94543			0.0934915			0.0467458	+	0.998907i	$0.485115\pi$
$434$		0			0
$435$	−32.5466			−1.56049
$436$		0			0
$437$	4.81458	−	8.33909i	0.230312	−	0.398913i
$438$		0			0
$439$	−20.4503	−	35.4210i	−0.976040	−	1.69055i	−0.676459	−	0.736480i	$-0.736486\pi$
$439$	−20.4503	−	35.4210i	−0.976040	−	1.69055i	−0.299581	−	0.954071i	$-0.596847\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−5.28779	−	9.15871i	−0.251230	−	0.435144i	0.712635	−	0.701535i	$-0.247502\pi$
$443$	−5.28779	−	9.15871i	−0.251230	−	0.435144i	−0.963865	+	0.266392i	$0.914168\pi$
$444$		0			0
$445$	−9.56107	+	16.5603i	−0.453238	+	0.785032i
$446$		0			0
$447$	66.5784			3.14905
$448$		0			0
$449$	7.72671			0.364646			0.182323	−	0.983239i	$-0.441638\pi$
$449$	7.72671			0.364646			0.182323	+	0.983239i	$0.441638\pi$
$450$		0			0
$451$	−4.09729	+	7.09671i	−0.192934	+	0.334171i
$452$		0			0
$453$	23.6600	+	40.9804i	1.11165	+	1.92543i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	7.57557	+	13.1213i	0.354370	+	0.613787i	0.987010	−	0.160659i	$-0.0513619\pi$
$457$	7.57557	+	13.1213i	0.354370	+	0.613787i	−0.632640	+	0.774446i	$0.718029\pi$
$458$		0			0
$459$	−16.2733	+	28.1862i	−0.759572	+	1.31562i
$460$		0			0
$461$	−31.6843			−1.47569			−0.737843	−	0.674972i	$-0.764156\pi$
$461$	−31.6843			−1.47569			−0.737843	+	0.674972i	$0.764156\pi$
$462$		0			0
$463$	17.4244			0.809782			0.404891	−	0.914365i	$-0.367309\pi$
$463$	17.4244			0.809782			0.404891	+	0.914365i	$0.367309\pi$
$464$		0			0
$465$	9.16730	−	15.8782i	0.425123	−	0.736335i
$466$		0			0
$467$	−1.69000	−	2.92717i	−0.0782040	−	0.135453i	0.824271	−	0.566195i	$-0.191585\pi$
$467$	−1.69000	−	2.92717i	−0.0782040	−	0.135453i	−0.902475	+	0.430742i	$0.858252\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	13.5611	+	23.4885i	0.624861	+	1.08229i
$472$		0			0
$473$	4.42443	−	7.66334i	0.203435	−	0.352361i
$474$		0			0
$475$	−16.3892			−0.751986
$476$		0			0
$477$	−91.3954			−4.18471
$478$		0			0
$479$	−1.77515	+	3.07464i	−0.0811085	+	0.140484i	−0.903726	−	0.428111i	$-0.859179\pi$
$479$	−1.77515	+	3.07464i	−0.0811085	+	0.140484i	0.822618	+	0.568595i	$0.192513\pi$
$480$		0			0
$481$	−13.5200	−	23.4174i	−0.616460	−	1.06774i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	6.71221	+	11.6259i	0.304786	+	0.527905i
$486$		0			0
$487$	−12.1366	+	21.0213i	−0.549964	+	0.952565i	0.448313	+	0.893877i	$0.352025\pi$
$487$	−12.1366	+	21.0213i	−0.549964	+	0.952565i	−0.998276	+	0.0586884i	$0.981308\pi$
$488$		0			0
$489$	−40.5601			−1.83419
$490$		0			0
$491$	3.69772			0.166876			0.0834378	−	0.996513i	$-0.473410\pi$
$491$	3.69772			0.166876			0.0834378	+	0.996513i	$0.473410\pi$
$492$		0			0
$493$	−5.32544	+	9.22393i	−0.239846	+	0.415425i
$494$		0			0
$495$	6.76001	+	11.7087i	0.303840	+	0.526266i
$496$		0			0
$497$		0			0
$498$		0			0
$499$	2.00000	+	3.46410i	0.0895323	+	0.155074i	0.907314	−	0.420455i	$-0.138129\pi$
$499$	2.00000	+	3.46410i	0.0895323	+	0.155074i	−0.817781	+	0.575529i	$0.804796\pi$
$500$		0			0
$501$	40.2733	−	69.7554i	1.79928	−	3.11644i
$502$		0			0
$503$	30.5903			1.36396			0.681978	−	0.731373i	$-0.261120\pi$
$503$	30.5903			1.36396			0.681978	+	0.731373i	$0.261120\pi$
$504$		0			0
$505$	18.3023			0.814441
$506$		0			0
$507$	20.0246	−	34.6836i	0.889323	−	1.54035i
$508$		0			0
$509$	−20.9122	−	36.2209i	−0.926916	−	1.60546i	−0.788451	−	0.615097i	$-0.789117\pi$
$509$	−20.9122	−	36.2209i	−0.926916	−	1.60546i	−0.138464	−	0.990367i	$-0.544217\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$	61.9710	+	107.337i	2.73609	+	4.73904i
$514$		0			0
$515$	−1.42443	+	2.46718i	−0.0627678	+	0.108717i
$516$		0			0
$517$	1.77515			0.0780708
$518$		0			0
$519$	26.5466			1.16527
$520$		0			0
$521$	−15.7570	+	27.2920i	−0.690327	+	1.19568i	0.281403	+	0.959590i	$0.409200\pi$
$521$	−15.7570	+	27.2920i	−0.690327	+	1.19568i	−0.971731	+	0.236093i	$0.924133\pi$
$522$		0			0
$523$	−3.20972	−	5.55939i	−0.140351	−	0.243095i	0.787278	−	0.616599i	$-0.211490\pi$
$523$	−3.20972	−	5.55939i	−0.140351	−	0.243095i	−0.927629	+	0.373503i	$0.878156\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−3.00000	−	5.19615i	−0.130682	−	0.226348i
$528$		0			0
$529$	10.4855	−	18.1614i	0.455891	−	0.789627i
$530$		0			0
$531$	1.43457			0.0622551
$532$		0			0
$533$	−40.8489			−1.76936
$534$		0			0
$535$	11.9152	−	20.6377i	0.515137	−	0.892244i
$536$		0			0
$537$	7.22186	+	12.5086i	0.311646	+	0.539787i
$538$		0			0
$539$		0			0
$540$		0			0
$541$	5.57557	+	9.65717i	0.239713	+	0.415194i	0.960632	−	0.277825i	$-0.0896135\pi$
$541$	5.57557	+	9.65717i	0.239713	+	0.415194i	−0.720919	+	0.693019i	$0.756280\pi$
$542$		0			0
$543$	26.1366	−	45.2700i	1.12163	−	1.94272i
$544$		0			0
$545$	20.6206			0.883289
$546$		0			0
$547$	9.69772			0.414644			0.207322	−	0.978273i	$-0.433525\pi$
$547$	9.69772			0.414644			0.207322	+	0.978273i	$0.433525\pi$
$548$		0			0
$549$	−49.4719	+	85.6879i	−2.11141	+	3.65707i
$550$		0			0
$551$	20.2800	+	35.1260i	0.863958	+	1.49642i
$552$		0			0
$553$		0			0
$554$		0			0
$555$	14.7122	+	25.4823i	0.624499	+	1.08166i
$556$		0			0
$557$	−13.8489	+	23.9869i	−0.586795	+	1.01636i	0.407854	+	0.913047i	$0.366277\pi$
$557$	−13.8489	+	23.9869i	−0.586795	+	1.01636i	−0.994649	+	0.103312i	$0.967056\pi$
$558$		0			0
$559$	44.1103			1.86567
$560$		0			0
$561$	6.00000			0.253320
$562$		0			0
$563$	15.2952	−	26.4920i	0.644614	−	1.11650i	−0.339776	−	0.940506i	$-0.610351\pi$
$563$	15.2952	−	26.4920i	0.644614	−	1.11650i	0.984390	−	0.175998i	$-0.0563153\pi$
$564$		0			0
$565$	0.461855	+	0.799957i	0.0194304	+	0.0336544i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	19.0000	+	32.9090i	0.796521	+	1.37962i	0.921869	+	0.387503i	$0.126662\pi$
$569$	19.0000	+	32.9090i	0.796521	+	1.37962i	−0.125347	+	0.992113i	$0.540004\pi$
$570$		0			0
$571$	−16.8489	+	29.1831i	−0.705103	+	1.22127i	0.261552	+	0.965189i	$0.415766\pi$
$571$	−16.8489	+	29.1831i	−0.705103	+	1.22127i	−0.966654	+	0.256084i	$0.917568\pi$
$572$		0			0
$573$	45.3746			1.89555
$574$		0			0
$575$	3.45343			0.144018
$576$		0			0
$577$	−0.972716	+	1.68479i	−0.0404947	+	0.0701389i	−0.885562	−	0.464520i	$-0.846227\pi$
$577$	−0.972716	+	1.68479i	−0.0404947	+	0.0701389i	0.845068	+	0.534659i	$0.179560\pi$
$578$		0			0
$579$	16.9000	+	29.2717i	0.702341	+	1.21649i
$580$		0			0
$581$		0			0
$582$		0			0
$583$	5.42443	+	9.39539i	0.224657	+	0.389117i
$584$		0			0
$585$	−33.6977	+	58.3662i	−1.39323	+	2.41314i
$586$		0			0
$587$	−31.6843			−1.30775			−0.653876	−	0.756602i	$-0.726858\pi$
$587$	−31.6843			−1.30775			−0.653876	+	0.756602i	$0.726858\pi$
$588$		0			0
$589$	−22.8489			−0.941471
$590$		0			0
$591$	−10.1400	+	17.5630i	−0.417104	+	0.722446i
$592$		0			0
$593$	14.4076	+	24.9547i	0.591649	+	1.02477i	0.994010	+	0.109285i	$0.0348561\pi$
$593$	14.4076	+	24.9547i	0.591649	+	1.02477i	−0.402362	+	0.915481i	$0.631811\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−25.8489	−	44.7715i	−1.05792	−	1.83238i
$598$		0			0
$599$	12.8489	−	22.2549i	0.524990	−	0.909310i	−0.474586	−	0.880209i	$-0.657402\pi$
$599$	12.8489	−	22.2549i	0.524990	−	0.909310i	0.999576	−	0.0291006i	$-0.00926431\pi$
$600$		0			0
$601$	−14.2734			−0.582226			−0.291113	−	0.956689i	$-0.594026\pi$
$601$	−14.2734			−0.582226			−0.291113	+	0.956689i	$0.594026\pi$
$602$		0			0
$603$	113.093			4.60551
$604$		0			0
$605$	0.802429	−	1.38985i	0.0326234	−	0.0565054i
$606$		0			0
$607$	−1.94543	−	3.36959i	−0.0789627	−	0.136767i	0.823840	−	0.566822i	$-0.191827\pi$
$607$	−1.94543	−	3.36959i	−0.0789627	−	0.136767i	−0.902803	+	0.430055i	$0.858494\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	4.42443	+	7.66334i	0.178993	+	0.310025i
$612$		0			0
$613$	4.42443	−	7.66334i	0.178701	−	0.309519i	−0.762735	−	0.646711i	$-0.776144\pi$
$613$	4.42443	−	7.66334i	0.178701	−	0.309519i	0.941436	+	0.337192i	$0.109477\pi$
$614$		0			0
$615$	44.4509			1.79243
$616$		0			0
$617$	−14.8489			−0.597793			−0.298896	−	0.954286i	$-0.596618\pi$
$617$	−14.8489			−0.597793			−0.298896	+	0.954286i	$0.596618\pi$
$618$		0			0
$619$	4.72943	−	8.19162i	0.190092	−	0.329249i	−0.755189	−	0.655508i	$-0.772455\pi$
$619$	4.72943	−	8.19162i	0.190092	−	0.329249i	0.945281	+	0.326259i	$0.105788\pi$
$620$		0			0
$621$	−13.0582	−	22.6174i	−0.524006	−	0.907605i
$622$		0			0
$623$		0			0
$624$		0			0
$625$	3.50000	+	6.06218i	0.140000	+	0.242487i
$626$		0			0
$627$	11.4244	−	19.7877i	0.456248	−	0.790244i
$628$		0			0
$629$	9.62915			0.383939
$630$		0			0
$631$	−20.2733			−0.807067			−0.403533	−	0.914965i	$-0.632218\pi$
$631$	−20.2733			−0.807067			−0.403533	+	0.914965i	$0.632218\pi$
$632$		0			0
$633$	−11.5746	+	20.0478i	−0.460048	+	0.796827i
$634$		0			0
$635$	15.1249	+	26.1971i	0.600212	+	1.03960i
$636$		0			0
$637$		0			0
$638$		0			0
$639$	−22.8489	−	39.5754i	−0.903887	−	1.56558i
$640$		0			0
$641$	6.71221	−	11.6259i	0.265117	−	0.459195i	−0.702478	−	0.711706i	$-0.747923\pi$
$641$	6.71221	−	11.6259i	0.265117	−	0.459195i	0.967594	+	0.252511i	$0.0812562\pi$
$642$		0			0
$643$	26.1886			1.03278			0.516389	−	0.856354i	$-0.327276\pi$
$643$	26.1886			1.03278			0.516389	+	0.856354i	$0.327276\pi$
$644$		0			0
$645$	−48.0000			−1.89000
$646$		0			0
$647$	−24.6689	+	42.7278i	−0.969834	+	1.67980i	−0.273809	+	0.961784i	$0.588284\pi$
$647$	−24.6689	+	42.7278i	−0.969834	+	1.67980i	−0.696025	+	0.718017i	$0.745050\pi$
$648$		0			0
$649$	−0.0851435	−	0.147473i	−0.00334218	−	0.00578882i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−11.1366	−	19.2892i	−0.435811	−	0.754846i	0.561551	−	0.827442i	$-0.310205\pi$
$653$	−11.1366	−	19.2892i	−0.435811	−	0.754846i	−0.997361	+	0.0725962i	$0.976872\pi$
$654$		0			0
$655$	8.27329	−	14.3298i	0.323264	−	0.559910i
$656$		0			0
$657$	−44.8638			−1.75030
$658$		0			0
$659$	0.302284			0.0117753			0.00588766	−	0.999983i	$-0.498126\pi$
$659$	0.302284			0.0117753			0.00588766	+	0.999983i	$0.498126\pi$
$660$		0			0
$661$	−7.22186	+	12.5086i	−0.280898	+	0.486530i	−0.971606	−	0.236604i	$-0.923966\pi$
$661$	−7.22186	+	12.5086i	−0.280898	+	0.486530i	0.690708	+	0.723134i	$0.257299\pi$
$662$		0			0
$663$	14.9546	+	25.9021i	0.580788	+	1.00595i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−4.27329	−	7.40155i	−0.165462	−	0.286589i
$668$		0			0
$669$	11.7122	−	20.2862i	0.452820	−	0.784308i
$670$		0			0
$671$	11.7449			0.453406
$672$		0			0
$673$	15.1511			0.584034			0.292017	−	0.956413i	$-0.405674\pi$
$673$	15.1511			0.584034			0.292017	+	0.956413i	$0.405674\pi$
$674$		0			0
$675$	−22.2255	+	38.4956i	−0.855458	+	1.48170i
$676$		0			0
$677$	−15.8422	−	27.4394i	−0.608864	−	1.05458i	−0.991428	−	0.130654i	$-0.958292\pi$
$677$	−15.8422	−	27.4394i	−0.608864	−	1.05458i	0.382564	−	0.923929i	$-0.375041\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$	−12.0000	−	20.7846i	−0.459841	−	0.796468i
$682$		0			0
$683$	22.0000	−	38.1051i	0.841807	−	1.45805i	−0.0465592	−	0.998916i	$-0.514826\pi$
$683$	22.0000	−	38.1051i	0.841807	−	1.45805i	0.888366	−	0.459136i	$-0.151841\pi$
$684$		0			0
$685$	−27.9637			−1.06844
$686$		0			0
$687$	4.27329			0.163036
$688$		0			0
$689$	−27.0400	+	46.8347i	−1.03014	+	1.78426i
$690$		0			0
$691$	−1.17914	−	2.04233i	−0.0448566	−	0.0776940i	0.842725	−	0.538344i	$-0.180950\pi$
$691$	−1.17914	−	2.04233i	−0.0448566	−	0.0776940i	−0.887582	+	0.460650i	$0.847616\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	5.42443	+	9.39539i	0.205760	+	0.356387i
$696$		0			0
$697$	7.27329	−	12.5977i	0.275495	−	0.477172i
$698$		0			0
$699$	−66.5784			−2.51822
$700$		0			0
$701$	22.5466			0.851572			0.425786	−	0.904824i	$-0.359998\pi$
$701$	22.5466			0.851572			0.425786	+	0.904824i	$0.359998\pi$
$702$		0			0
$703$	18.3346	−	31.7564i	0.691502	−	1.19772i
$704$		0			0
$705$	−4.81458	−	8.33909i	−0.181327	−	0.314068i
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−7.13664	−	12.3610i	−0.268022	−	0.464228i	0.700329	−	0.713820i	$-0.253037\pi$
$709$	−7.13664	−	12.3610i	−0.268022	−	0.464228i	−0.968351	+	0.249592i	$0.919703\pi$
$710$		0			0
$711$	−25.2733	+	43.7746i	−0.947822	+	1.64168i
$712$		0			0
$713$	4.81458			0.180307
$714$		0			0
$715$	8.00000			0.299183
$716$		0			0
$717$	−36.6692	+	63.5129i	−1.36944	+	2.37193i
$718$		0			0
$719$	−6.50458	−	11.2663i	−0.242580	−	0.420161i	0.718869	−	0.695146i	$-0.244660\pi$
$719$	−6.50458	−	11.2663i	−0.242580	−	0.420161i	−0.961448	+	0.274985i	$0.911327\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$	−14.4244	−	24.9838i	−0.536450	−	0.929159i
$724$		0			0
$725$	−7.27329	+	12.5977i	−0.270123	+	0.467867i
$726$		0			0
$727$	−42.5778			−1.57912			−0.789561	−	0.613672i	$-0.789692\pi$
$727$	−42.5778			−1.57912			−0.789561	+	0.613672i	$0.789692\pi$
$728$		0			0
$729$	123.244			4.56460
$730$		0			0
$731$	−7.85401	+	13.6035i	−0.290491	+	0.503145i
$732$		0			0
$733$	22.4319	+	38.8532i	0.828540	+	1.43507i	0.899183	+	0.437573i	$0.144162\pi$
$733$	22.4319	+	38.8532i	0.828540	+	1.43507i	−0.0706426	+	0.997502i	$0.522505\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−6.71221	−	11.6259i	−0.247248	−	0.428245i
$738$		0			0
$739$	−11.4244	+	19.7877i	−0.420254	+	0.727902i	−0.995964	−	0.0897522i	$-0.971393\pi$
$739$	−11.4244	+	19.7877i	−0.420254	+	0.727902i	0.575710	+	0.817654i	$0.304726\pi$
$740$		0			0
$741$	113.898			4.18416
$742$		0			0
$743$	−5.69772			−0.209029			−0.104514	−	0.994523i	$-0.533329\pi$
$743$	−5.69772			−0.209029			−0.104514	+	0.994523i	$0.533329\pi$
$744$		0			0
$745$	−15.8060	+	27.3768i	−0.579088	+	1.00301i
$746$		0			0
$747$	14.9546	+	25.9021i	0.547160	+	0.947709i
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−18.7122	−	32.4105i	−0.682818	−	1.18268i	−0.974117	−	0.226044i	$-0.927421\pi$
$751$	−18.7122	−	32.4105i	−0.682818	−	1.18268i	0.291299	−	0.956632i	$-0.405913\pi$
$752$		0			0
$753$	−11.1366	+	19.2892i	−0.405842	+	0.702939i
$754$		0			0
$755$	−22.4680			−0.817695
$756$		0			0
$757$	11.6977			0.425161			0.212580	−	0.977144i	$-0.431813\pi$
$757$	11.6977			0.425161			0.212580	+	0.977144i	$0.431813\pi$
$758$		0			0
$759$	−2.40729	+	4.16954i	−0.0873790	+	0.151345i
$760$		0			0
$761$	−12.4622	−	21.5851i	−0.451753	−	0.782459i	0.546742	−	0.837301i	$-0.315868\pi$
$761$	−12.4622	−	21.5851i	−0.451753	−	0.782459i	−0.998495	+	0.0548421i	$0.982534\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$	−12.0000	−	20.7846i	−0.433861	−	0.751469i
$766$		0			0
$767$	0.424429	−	0.735132i	0.0153252	−	0.0265441i
$768$		0			0
$769$	14.2734			0.514713			0.257357	−	0.966316i	$-0.417148\pi$
$769$	14.2734			0.514713			0.257357	+	0.966316i	$0.417148\pi$
$770$		0			0
$771$	−67.3954			−2.42719
$772$		0			0
$773$	−8.53515	+	14.7833i	−0.306988	+	0.531719i	−0.977702	−	0.209997i	$-0.932655\pi$
$773$	−8.53515	+	14.7833i	−0.306988	+	0.531719i	0.670714	+	0.741716i	$0.265988\pi$
$774$		0			0
$775$	−4.09729	−	7.09671i	−0.147179	−	0.254922i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−27.6977	−	47.9739i	−0.992374	−	1.71884i
$780$		0			0
$781$	−2.71221	+	4.69769i	−0.0970507	+	0.168097i
$782$		0			0
$783$	110.008			3.93135
$784$		0			0
$785$	−12.8779			−0.459630
$786$		0			0
$787$	13.0092	−	22.5325i	0.463726	−	0.803198i	−0.535417	−	0.844588i	$-0.679845\pi$
$787$	13.0092	−	22.5325i	0.463726	−	0.803198i	0.999143	+	0.0413905i	$0.0131788\pi$
$788$		0			0
$789$	−8.70544	−	15.0783i	−0.309922	−	0.536801i
$790$		0			0
$791$		0			0
$792$		0			0
$793$	29.2733	+	50.7028i	1.03952	+	1.80051i
$794$		0			0
$795$	29.4244	−	50.9646i	1.04358	−	1.80753i
$796$		0			0
$797$	31.1735			1.10422			0.552110	−	0.833771i	$-0.313823\pi$
$797$	31.1735			1.10422			0.552110	+	0.833771i	$0.313823\pi$
$798$		0			0
$799$	−3.15114			−0.111479
$800$		0			0
$801$	50.1892	−	86.9302i	1.77335	−	3.07153i
$802$		0			0
$803$	2.66272	+	4.61196i	0.0939653	+	0.162753i
$804$		0			0
$805$		0			0
$806$		0			0
$807$	18.0000	+	31.1769i	0.633630	+	1.09748i
$808$		0			0
$809$	11.0000	−	19.0526i	0.386739	−	0.669852i	−0.605269	−	0.796021i	$-0.706935\pi$
$809$	11.0000	−	19.0526i	0.386739	−	0.669852i	0.992009	+	0.126168i	$0.0402680\pi$
$810$		0			0
$811$	−23.1492			−0.812877			−0.406439	−	0.913678i	$-0.633230\pi$
$811$	−23.1492			−0.812877			−0.406439	+	0.913678i	$0.633230\pi$
$812$		0			0
$813$	33.6977			1.18183
$814$		0			0
$815$	9.62915	−	16.6782i	0.337294	−	0.584211i
$816$		0			0
$817$	29.9092	+	51.8042i	1.04639	+	1.81240i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−14.6977	−	25.4572i	−0.512954	−	0.888462i	−0.999887	−	0.0150229i	$-0.995218\pi$
$821$	−14.6977	−	25.4572i	−0.512954	−	0.888462i	0.486933	−	0.873439i	$-0.338115\pi$
$822$		0			0
$823$	−20.7122	+	35.8746i	−0.721982	+	1.25051i	0.238222	+	0.971211i	$0.423436\pi$
$823$	−20.7122	+	35.8746i	−0.721982	+	1.25051i	−0.960204	+	0.279299i	$0.909898\pi$
$824$		0			0
$825$	8.19458			0.285299
$826$		0			0
$827$	−32.8489			−1.14227			−0.571133	−	0.820857i	$-0.693496\pi$
$827$	−32.8489			−1.14227			−0.571133	+	0.820857i	$0.693496\pi$
$828$		0			0
$829$	11.1127	−	19.2478i	0.385961	−	0.668504i	−0.605941	−	0.795510i	$-0.707203\pi$
$829$	11.1127	−	19.2478i	0.385961	−	0.668504i	0.991902	+	0.127005i	$0.0405366\pi$
$830$		0			0
$831$	41.9946	+	72.7368i	1.45678	+	2.52321i
$832$		0			0
$833$		0			0
$834$		0			0
$835$	19.1221	+	33.1205i	0.661749	+	1.14618i
$836$		0			0
$837$	−30.9855	+	53.6685i	−1.07102	+	1.85505i
$838$		0			0
$839$	−12.6686			−0.437368			−0.218684	−	0.975796i	$-0.570176\pi$
$839$	−12.6686			−0.437368			−0.218684	+	0.975796i	$0.570176\pi$
$840$		0			0
$841$	7.00000			0.241379
$842$		0			0
$843$	8.19458	−	14.1934i	0.282236	−	0.488848i
$844$		0			0
$845$	9.50787	+	16.4681i	0.327081	+	0.566520i
$846$		0			0
$847$		0			0
$848$		0			0
$849$	21.6977	+	37.5815i	0.744664	+	1.28980i
$850$		0			0
$851$	−3.86336	+	6.69153i	−0.132434	+	0.229383i
$852$		0			0
$853$	5.32544			0.182339			0.0911697	−	0.995835i	$-0.470939\pi$
$853$	5.32544			0.182339			0.0911697	+	0.995835i	$0.470939\pi$
$854$		0			0
$855$	−91.3954			−3.12566
$856$		0			0
$857$	−5.70215	+	9.87641i	−0.194782	+	0.337372i	−0.946829	−	0.321737i	$-0.895733\pi$
$857$	−5.70215	+	9.87641i	−0.194782	+	0.337372i	0.752047	+	0.659109i	$0.229067\pi$
$858$		0			0
$859$	−10.5657	−	18.3004i	−0.360498	−	0.624401i	0.627545	−	0.778580i	$-0.284060\pi$
$859$	−10.5657	−	18.3004i	−0.360498	−	0.624401i	−0.988043	+	0.154179i	$0.950727\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−1.15114	−	1.99384i	−0.0391853	−	0.0678710i	0.845768	−	0.533551i	$-0.179143\pi$
$863$	−1.15114	−	1.99384i	−0.0391853	−	0.0678710i	−0.884953	+	0.465680i	$0.845810\pi$
$864$		0			0
$865$	−6.30228	+	10.9159i	−0.214284	+	0.371151i
$866$		0			0
$867$	46.8092			1.58972
$868$		0			0
$869$	6.00000			0.203536
$870$		0			0
$871$	33.4595	−	57.9535i	1.13373	−	1.96368i
$872$		0			0
$873$	−35.2346	−	61.0281i	−1.19251	−	2.06549i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	7.00000	+	12.1244i	0.236373	+	0.409410i	0.959671	−	0.281126i	$-0.0907079\pi$
$877$	7.00000	+	12.1244i	0.236373	+	0.409410i	−0.723298	+	0.690536i	$0.757375\pi$
$878$		0			0
$879$	30.6977	−	53.1700i	1.03541	−	1.79338i
$880$		0			0
$881$	10.8934			0.367009			0.183505	−	0.983019i	$-0.441256\pi$
$881$	10.8934			0.367009			0.183505	+	0.983019i	$0.441256\pi$
$882$		0			0
$883$	20.0000			0.673054			0.336527	−	0.941674i	$-0.390748\pi$
$883$	20.0000			0.673054			0.336527	+	0.941674i	$0.390748\pi$
$884$		0			0
$885$	−0.461855	+	0.799957i	−0.0155251	+	0.0268902i
$886$		0			0
$887$	−9.79944	−	16.9731i	−0.329033	−	0.569902i	0.653287	−	0.757110i	$-0.273389\pi$
$887$	−9.79944	−	16.9731i	−0.329033	−	0.569902i	−0.982320	+	0.187208i	$0.940056\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	−18.3489	−	31.7812i	−0.614710	−	1.06471i
$892$		0			0
$893$	−6.00000	+	10.3923i	−0.200782	+	0.347765i
$894$		0			0
$895$	−6.85802			−0.229238
$896$		0			0
$897$	−24.0000			−0.801337
$898$		0			0
$899$	−10.1400	+	17.5630i	−0.338188	+	0.585759i
$900$		0			0
$901$	−9.62915	−	16.6782i	−0.320794	−	0.555631i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	12.4099	+	21.4946i	0.412520	+	0.714506i
$906$		0			0
$907$	9.69772	−	16.7969i	0.322007	−	0.557733i	−0.658895	−	0.752235i	$-0.728976\pi$
$907$	9.69772	−	16.7969i	0.322007	−	0.557733i	0.980902	+	0.194502i	$0.0623091\pi$
$908$		0			0
$909$	−96.0747			−3.18660
$910$		0			0
$911$	−49.6977			−1.64656			−0.823279	−	0.567636i	$-0.807858\pi$
$911$	−49.6977			−1.64656			−0.823279	+	0.567636i	$0.807858\pi$
$912$		0			0
$913$	1.77515	−	3.07464i	0.0587487	−	0.101756i
$914$		0			0
$915$	−31.8546	−	55.1738i	−1.05308	−	1.82399i
$916$		0			0
$917$		0			0
$918$		0			0
$919$	−5.15114	−	8.92204i	−0.169921	−	0.294311i	0.768471	−	0.639884i	$-0.221018\pi$
$919$	−5.15114	−	8.92204i	−0.169921	−	0.294311i	−0.938392	+	0.345573i	$0.887684\pi$
$920$		0			0
$921$	−1.15114	+	1.99384i	−0.0379314	+	0.0656992i
$922$		0			0
$923$	−27.0400			−0.890034
$924$		0			0
$925$	13.1511			0.432407
$926$		0			0
$927$	7.47729	−	12.9511i	0.245587	−	0.425368i
$928$		0			0
$929$	4.64429	+	8.04414i	0.152374	+	0.263920i	0.932100	−	0.362202i	$-0.117975\pi$
$929$	4.64429	+	8.04414i	0.152374	+	0.263920i	−0.779726	+	0.626121i	$0.784641\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$	−37.8489	−	65.5561i	−1.23912	−	2.14621i
$934$		0			0
$935$	−1.42443	+	2.46718i	−0.0465838	+	0.0806855i
$936$		0			0
$937$	−12.7666			−0.417066			−0.208533	−	0.978015i	$-0.566869\pi$
$937$	−12.7666			−0.417066			−0.208533	+	0.978015i	$0.566869\pi$
$938$		0			0
$939$	−61.9710			−2.02235
$940$		0			0
$941$	7.98816	−	13.8359i	0.260406	−	0.451037i	−0.705944	−	0.708268i	$-0.749477\pi$
$941$	7.98816	−	13.8359i	0.260406	−	0.451037i	0.966350	+	0.257231i	$0.0828101\pi$
$942$		0			0
$943$	5.83630	+	10.1088i	0.190056	+	0.329187i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	0.712214	+	1.23359i	0.0231439	+	0.0400863i	0.877365	−	0.479823i	$-0.159299\pi$
$947$	0.712214	+	1.23359i	0.0231439	+	0.0400863i	−0.854222	+	0.519909i	$0.825966\pi$
$948$		0			0
$949$	−13.2733	+	22.9900i	−0.430869	+	0.746287i
$950$		0			0
$951$	25.0946			0.813748
$952$		0			0
$953$	−4.84886			−0.157070			−0.0785350	−	0.996911i	$-0.525024\pi$
$953$	−4.84886			−0.157070			−0.0785350	+	0.996911i	$0.525024\pi$
$954$		0			0
$955$	−10.7722	+	18.6579i	−0.348579	+	0.603756i
$956$		0			0
$957$	−10.1400	−	17.5630i	−0.327780	−	0.567732i
$958$		0			0
$959$		0			0
$960$		0			0
$961$	9.78779	+	16.9529i	0.315735	+	0.546869i
$962$		0			0
$963$	−62.5466	+	108.334i	−2.01554	+	3.49101i
$964$		0			0
$965$	−16.0486			−0.516622
$966$		0			0
$967$	26.0000			0.836104			0.418052	−	0.908423i	$-0.362713\pi$
$967$	26.0000			0.836104			0.418052	+	0.908423i	$0.362713\pi$
$968$		0			0
$969$	−20.2800	+	35.1260i	−0.651488	+	1.12841i
$970$		0			0
$971$	−21.2889	−	36.8734i	−0.683193	−	1.18332i	−0.974001	−	0.226543i	$-0.927258\pi$
$971$	−21.2889	−	36.8734i	−0.683193	−	1.18332i	0.290808	−	0.956781i	$-0.406076\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$	20.4244	+	35.3761i	0.654105	+	1.13294i
$976$		0			0
$977$	0.287786	−	0.498459i	0.00920708	−	0.0159471i	−0.861385	−	0.507952i	$-0.830403\pi$
$977$	0.287786	−	0.498459i	0.00920708	−	0.0159471i	0.870592	+	0.492005i	$0.163736\pi$
$978$		0			0
$979$	−11.9152			−0.380810
$980$		0			0
$981$	−108.244			−3.45597
$982$		0			0
$983$	1.86029	−	3.22212i	0.0593340	−	0.102770i	−0.834833	−	0.550504i	$-0.814436\pi$
$983$	1.86029	−	3.22212i	0.0593340	−	0.102770i	0.894167	+	0.447734i	$0.147769\pi$
$984$		0			0
$985$	−4.81458	−	8.33909i	−0.153405	−	0.265705i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−6.30228	−	10.9159i	−0.200401	−	0.347105i
$990$		0			0
$991$	−21.6977	+	37.5815i	−0.689251	+	1.19382i	0.282830	+	0.959170i	$0.408727\pi$
$991$	−21.6977	+	37.5815i	−0.689251	+	1.19382i	−0.972081	+	0.234647i	$0.924607\pi$
$992$		0			0
$993$	−45.3746			−1.43992
$994$		0			0
$995$	24.5466			0.778179
$996$		0			0
$997$	−16.0124	+	27.7344i	−0.507119	+	0.878356i	0.492847	+	0.870116i	$0.335956\pi$
$997$	−16.0124	+	27.7344i	−0.507119	+	0.878356i	−0.999966	+	0.00824030i	$0.997377\pi$
$998$		0			0
$999$	−49.7273	−	86.1303i	−1.57330	−	2.72504i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2156.2.i.n.1145.4		8
7.2	even	3	inner	2156.2.i.n.177.4		8
7.3	odd	6		2156.2.a.m.1.4	yes	4
7.4	even	3		2156.2.a.m.1.1	✓	4
7.5	odd	6	inner	2156.2.i.n.177.1		8
7.6	odd	2	inner	2156.2.i.n.1145.1		8
28.3	even	6		8624.2.a.cx.1.1		4
28.11	odd	6		8624.2.a.cx.1.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2156.2.a.m.1.1	✓	4	7.4	even	3
2156.2.a.m.1.4	yes	4	7.3	odd	6
2156.2.i.n.177.1		8	7.5	odd	6	inner
2156.2.i.n.177.4		8	7.2	even	3	inner
2156.2.i.n.1145.1		8	7.6	odd	2	inner
2156.2.i.n.1145.4		8	1.1	even	1	trivial
8624.2.a.cx.1.1		4	28.3	even	6
8624.2.a.cx.1.4		4	28.11	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)

\(f(q)\)	\(=\)	\(q+(1.69000 - 2.92717i) q^{3} +(0.802429 + 1.38985i) q^{5} +(-4.21221 - 7.29577i) q^{9} +(-0.500000 + 0.866025i) q^{11} -4.98486 q^{13} +5.42443 q^{15} +(0.887573 - 1.53732i) q^{17} +(-3.38000 - 5.85434i) q^{19} +(0.712214 + 1.23359i) q^{23} +(1.21221 - 2.09962i) q^{25} -18.3346 q^{27} -6.00000 q^{29} +(1.69000 - 2.92717i) q^{31} +(1.69000 + 2.92717i) q^{33} +(2.71221 + 4.69769i) q^{37} +(-8.42443 + 14.5915i) q^{39} +8.19458 q^{41} -8.84886 q^{43} +(6.76001 - 11.7087i) q^{45} +(-0.887573 - 1.53732i) q^{47} +(-3.00000 - 5.19615i) q^{51} +(5.42443 - 9.39539i) q^{53} -1.60486 q^{55} -22.8489 q^{57} +(-0.0851435 + 0.147473i) q^{59} +(-5.87244 - 10.1714i) q^{61} +(-4.00000 - 6.92820i) q^{65} +(-6.71221 + 11.6259i) q^{67} +4.81458 q^{69} +5.42443 q^{71} +(2.66272 - 4.61196i) q^{73} +(-4.09729 - 7.09671i) q^{75} +(-3.00000 - 5.19615i) q^{79} +(-18.3489 + 31.7812i) q^{81} -3.55029 q^{83} +2.84886 q^{85} +(-10.1400 + 17.5630i) q^{87} +(5.95758 + 10.3188i) q^{89} +(-5.71221 - 9.89385i) q^{93} +(5.42443 - 9.39539i) q^{95} +8.36487 q^{97} +8.42443 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 14 q^{9} - 4 q^{11} + 4 q^{15} - 14 q^{23} - 10 q^{25} - 48 q^{29} + 2 q^{37} - 28 q^{39} + 8 q^{43} - 24 q^{51} + 4 q^{53} - 104 q^{57} - 32 q^{65} - 34 q^{67} + 4 q^{71} - 24 q^{79} - 68 q^{81}+ \cdots + 28 q^{99}+O(q^{100}) \) 8 * q - 14 * q^9 - 4 * q^11 + 4 * q^15 - 14 * q^23 - 10 * q^25 - 48 * q^29 + 2 * q^37 - 28 * q^39 + 8 * q^43 - 24 * q^51 + 4 * q^53 - 104 * q^57 - 32 * q^65 - 34 * q^67 + 4 * q^71 - 24 * q^79 - 68 * q^81 - 56 * q^85 - 26 * q^93 + 4 * q^95 + 28 * q^99

Introduction

L-functions

Modular forms

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