LMFDB - Embedded newform 3456.2.s.d.2303.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3456,2,Mod(1151,3456)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3456, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([3, 0, 5]))

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

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

chi := DirichletCharacter("3456.1151");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3456 = 2^{7} \cdot 3^{3} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3456.s (of order $6$, 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:	$27.5962989386$
Analytic rank:	$0$
Dimension:	$24$
Relative dimension:	$12$ over $\Q(\zeta_{6})$
Twist minimal:	no (minimal twist has level 1152)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			2303.3
Character	$\chi$	$=$	3456.2303
Dual form			3456.2.s.d.1151.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.94279 + 1.12167i) q^{5} +(3.69568 + 2.13370i) q^{7} +O(q^{10})$	$q+(-1.94279 + 1.12167i) q^{5} +(3.69568 + 2.13370i) q^{7} +(0.430411 - 0.745494i) q^{11} +(-1.00136 - 1.73441i) q^{13} -3.56030i q^{17} +5.87695i q^{19} +(2.64355 + 4.57877i) q^{23} +(0.0163009 - 0.0282341i) q^{25} +(6.29707 + 3.63561i) q^{29} +(3.48584 - 2.01255i) q^{31} -9.57327 q^{35} -5.11472 q^{37} +(7.52515 - 4.34465i) q^{41} +(-8.37531 - 4.83549i) q^{43} +(2.35043 - 4.07106i) q^{47} +(5.60538 + 9.70881i) q^{49} +9.60856i q^{53} +1.93112i q^{55} +(0.872053 + 1.51044i) q^{59} +(2.44953 - 4.24271i) q^{61} +(3.89088 + 2.24640i) q^{65} +(9.93455 - 5.73571i) q^{67} -1.67196 q^{71} -16.2374 q^{73} +(3.18133 - 1.83674i) q^{77} +(13.0188 + 7.51641i) q^{79} +(-5.53430 + 9.58570i) q^{83} +(3.99349 + 6.91693i) q^{85} +16.1380i q^{89} -8.54642i q^{91} +(-6.59202 - 11.4177i) q^{95} +(-5.14699 + 8.91485i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q+O(q^{10}) $ 24 * q	$ 24 q + 6 q^{11} + 12 q^{25} + 24 q^{29} + 36 q^{31} - 12 q^{41} - 42 q^{43} + 12 q^{49} - 6 q^{59} + 54 q^{67} + 48 q^{77} - 60 q^{79} - 36 q^{83}+O(q^{100}) $ 24 * q + 6 * q^11 + 12 * q^25 + 24 * q^29 + 36 * q^31 - 12 * q^41 - 42 * q^43 + 12 * q^49 - 6 * q^59 + 54 * q^67 + 48 * q^77 - 60 * q^79 - 36 * q^83

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3456\mathbb{Z}\right)^\times$.

$n$	$2053$	$2431$	$2945$
$\chi(n)$	$1$	$-1$	$e\left(\frac{1}{6}\right)$

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			0
$3$		0			0
$4$		0			0
$5$	−1.94279	+	1.12167i	−0.868844	+	0.501627i	−0.866964	−	0.498371i	$-0.833932\pi$
$5$	−1.94279	+	1.12167i	−0.868844	+	0.501627i	−0.00188023	+	0.999998i	$0.500598\pi$
$6$		0			0
$7$	3.69568	+	2.13370i	1.39684	+	0.806464i	0.994060	−	0.108834i	$-0.0347117\pi$
$7$	3.69568	+	2.13370i	1.39684	+	0.806464i	0.402777	+	0.915298i	$0.368045\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	0.430411	−	0.745494i	0.129774	−	0.224775i	−0.793815	−	0.608159i	$-0.791908\pi$
$11$	0.430411	−	0.745494i	0.129774	−	0.224775i	0.923589	+	0.383384i	$0.125242\pi$
$12$		0			0
$13$	−1.00136	−	1.73441i	−0.277727	−	0.481038i	0.693092	−	0.720849i	$-0.256248\pi$
$13$	−1.00136	−	1.73441i	−0.277727	−	0.481038i	−0.970820	+	0.239811i	$0.922915\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		−	3.56030i		−	0.863499i	−0.901993	−	0.431750i	$-0.857896\pi$
$17$		−	3.56030i		−	0.863499i	0.901993	−	0.431750i	$-0.142104\pi$
$18$		0			0
$19$			5.87695i			1.34827i	0.738610	+	0.674133i	$0.235483\pi$
$19$			5.87695i			1.34827i	−0.738610	+	0.674133i	$0.764517\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	2.64355	+	4.57877i	0.551219	+	0.954739i	0.998187	+	0.0601894i	$0.0191705\pi$
$23$	2.64355	+	4.57877i	0.551219	+	0.954739i	−0.446968	+	0.894550i	$0.647496\pi$
$24$		0			0
$25$	0.0163009	−	0.0282341i	0.00326019	−	0.00564681i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	6.29707	+	3.63561i	1.16934	+	0.675117i	0.953524	−	0.301318i	$-0.0974266\pi$
$29$	6.29707	+	3.63561i	1.16934	+	0.675117i	0.215813	+	0.976435i	$0.430760\pi$
$30$		0			0
$31$	3.48584	−	2.01255i	0.626076	−	0.361465i	−0.153155	−	0.988202i	$-0.548943\pi$
$31$	3.48584	−	2.01255i	0.626076	−	0.361465i	0.779231	+	0.626737i	$0.215610\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−9.57327			−1.61818
$36$		0			0
$37$	−5.11472			−0.840854			−0.420427	−	0.907326i	$-0.638120\pi$
$37$	−5.11472			−0.840854			−0.420427	+	0.907326i	$0.638120\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	7.52515	−	4.34465i	1.17523	−	0.678520i	0.220325	−	0.975427i	$-0.429288\pi$
$41$	7.52515	−	4.34465i	1.17523	−	0.678520i	0.954907	+	0.296906i	$0.0959550\pi$
$42$		0			0
$43$	−8.37531	−	4.83549i	−1.27722	−	0.737405i	−0.300886	−	0.953660i	$-0.597282\pi$
$43$	−8.37531	−	4.83549i	−1.27722	−	0.737405i	−0.976337	+	0.216255i	$0.930616\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	2.35043	−	4.07106i	0.342845	−	0.593825i	−0.642115	−	0.766609i	$-0.721943\pi$
$47$	2.35043	−	4.07106i	0.342845	−	0.593825i	0.984960	+	0.172783i	$0.0552760\pi$
$48$		0			0
$49$	5.60538	+	9.70881i	0.800769	+	1.38697i
$50$		0			0
$51$		0			0
$52$		0			0
$53$			9.60856i			1.31984i	0.751337	+	0.659918i	$0.229409\pi$
$53$			9.60856i			1.31984i	−0.751337	+	0.659918i	$0.770591\pi$
$54$		0			0
$55$			1.93112i			0.260393i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	0.872053	+	1.51044i	0.113532	+	0.196643i	0.917192	−	0.398446i	$-0.130450\pi$
$59$	0.872053	+	1.51044i	0.113532	+	0.196643i	−0.803660	+	0.595088i	$0.797117\pi$
$60$		0			0
$61$	2.44953	−	4.24271i	0.313630	−	0.543224i	−0.665515	−	0.746384i	$-0.731788\pi$
$61$	2.44953	−	4.24271i	0.313630	−	0.543224i	0.979145	+	0.203161i	$0.0651214\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	3.89088	+	2.24640i	0.482604	+	0.278631i
$66$		0			0
$67$	9.93455	−	5.73571i	1.21370	−	0.700729i	0.250135	−	0.968211i	$-0.419525\pi$
$67$	9.93455	−	5.73571i	1.21370	−	0.700729i	0.963563	+	0.267482i	$0.0861916\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−1.67196			−0.198426			−0.0992128	−	0.995066i	$-0.531632\pi$
$71$	−1.67196			−0.198426			−0.0992128	+	0.995066i	$0.531632\pi$
$72$		0			0
$73$	−16.2374			−1.90044			−0.950221	−	0.311577i	$-0.899143\pi$
$73$	−16.2374			−1.90044			−0.950221	+	0.311577i	$0.899143\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	3.18133	−	1.83674i	0.362546	−	0.209316i
$78$		0			0
$79$	13.0188	+	7.51641i	1.46473	+	0.845662i	0.999224	−	0.0393822i	$-0.0125390\pi$
$79$	13.0188	+	7.51641i	1.46473	+	0.845662i	0.465506	+	0.885045i	$0.345872\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−5.53430	+	9.58570i	−0.607469	+	1.05217i	0.384187	+	0.923255i	$0.374482\pi$
$83$	−5.53430	+	9.58570i	−0.607469	+	1.05217i	−0.991656	+	0.128912i	$0.958852\pi$
$84$		0			0
$85$	3.99349	+	6.91693i	0.433155	+	0.750246i
$86$		0			0
$87$		0			0
$88$		0			0
$89$			16.1380i			1.71063i	0.518112	+	0.855313i	$0.326635\pi$
$89$			16.1380i			1.71063i	−0.518112	+	0.855313i	$0.673365\pi$
$90$		0			0
$91$		−	8.54642i		−	0.895909i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−6.59202	−	11.4177i	−0.676327	−	1.17143i
$96$		0			0
$97$	−5.14699	+	8.91485i	−0.522598	+	0.905166i	0.477057	+	0.878873i	$0.341704\pi$
$97$	−5.14699	+	8.91485i	−0.522598	+	0.905166i	−0.999654	+	0.0262932i	$0.991630\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	5.77965	+	3.33688i	0.575096	+	0.332032i	0.759182	−	0.650878i	$-0.225599\pi$
$101$	5.77965	+	3.33688i	0.575096	+	0.332032i	−0.184086	+	0.982910i	$0.558932\pi$
$102$		0			0
$103$	6.93199	−	4.00219i	0.683029	−	0.394347i	−0.117966	−	0.993018i	$-0.537637\pi$
$103$	6.93199	−	4.00219i	0.683029	−	0.394347i	0.800995	+	0.598670i	$0.204304\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−9.86383			−0.953573			−0.476786	−	0.879019i	$-0.658198\pi$
$107$	−9.86383			−0.953573			−0.476786	+	0.879019i	$0.658198\pi$
$108$		0			0
$109$	−1.25969			−0.120657			−0.0603283	−	0.998179i	$-0.519215\pi$
$109$	−1.25969			−0.120657			−0.0603283	+	0.998179i	$0.519215\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−8.35293	+	4.82257i	−0.785778	+	0.453669i	−0.838474	−	0.544942i	$-0.816552\pi$
$113$	−8.35293	+	4.82257i	−0.785778	+	0.453669i	0.0526963	+	0.998611i	$0.483218\pi$
$114$		0			0
$115$	−10.2718	−	5.93041i	−0.957847	−	0.553013i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	7.59662	−	13.1577i	0.696381	−	1.20617i
$120$		0			0
$121$	5.12949	+	8.88454i	0.466317	+	0.807686i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		−	11.1436i		−	0.996713i
$126$		0			0
$127$			2.22426i			0.197371i	0.995119	+	0.0986854i	$0.0314637\pi$
$127$			2.22426i			0.197371i	−0.995119	+	0.0986854i	$0.968536\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−3.79353	−	6.57059i	−0.331442	−	0.574075i	0.651353	−	0.758775i	$-0.274202\pi$
$131$	−3.79353	−	6.57059i	−0.331442	−	0.574075i	−0.982795	+	0.184700i	$0.940869\pi$
$132$		0			0
$133$	−12.5397	+	21.7194i	−1.08733	+	1.88331i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−2.40826	−	1.39041i	−0.205751	−	0.118791i	0.393584	−	0.919289i	$-0.371235\pi$
$137$	−2.40826	−	1.39041i	−0.205751	−	0.118791i	−0.599335	+	0.800498i	$0.704568\pi$
$138$		0			0
$139$	−6.39258	+	3.69076i	−0.542211	+	0.313046i	−0.745975	−	0.665974i	$-0.768016\pi$
$139$	−6.39258	+	3.69076i	−0.542211	+	0.313046i	0.203763	+	0.979020i	$0.434683\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−1.72399			−0.144167
$144$		0			0
$145$	−16.3119			−1.35463
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−10.6769	+	6.16433i	−0.874688	+	0.505001i	−0.868903	−	0.494982i	$-0.835175\pi$
$149$	−10.6769	+	6.16433i	−0.874688	+	0.505001i	−0.00578453	+	0.999983i	$0.501841\pi$
$150$		0			0
$151$	−17.1089	−	9.87784i	−1.39230	−	0.803847i	−0.398734	−	0.917067i	$-0.630550\pi$
$151$	−17.1089	−	9.87784i	−1.39230	−	0.803847i	−0.993570	+	0.113219i	$0.963884\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−4.51485	+	7.81995i	−0.362642	+	0.628114i
$156$		0			0
$157$	1.79552	+	3.10994i	0.143298	+	0.248200i	0.928737	−	0.370740i	$-0.120896\pi$
$157$	1.79552	+	3.10994i	0.143298	+	0.248200i	−0.785438	+	0.618940i	$0.787562\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$			22.5622i			1.77815i
$162$		0			0
$163$			3.80217i			0.297809i	0.988852	+	0.148905i	$0.0475747\pi$
$163$			3.80217i			0.297809i	−0.988852	+	0.148905i	$0.952425\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	3.21229	+	5.56386i	0.248575	+	0.430544i	0.963131	−	0.269034i	$-0.0867045\pi$
$167$	3.21229	+	5.56386i	0.248575	+	0.430544i	−0.714556	+	0.699578i	$0.753371\pi$
$168$		0			0
$169$	4.49455	−	7.78480i	0.345735	−	0.598831i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	16.7183	+	9.65232i	1.27107	+	0.733852i	0.975189	−	0.221373i	$-0.0710539\pi$
$173$	16.7183	+	9.65232i	1.27107	+	0.733852i	0.295880	+	0.955225i	$0.404387\pi$
$174$		0			0
$175$	0.120486	−	0.0695628i	0.00910790	−	0.00525845i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	0.956658			0.0715040			0.0357520	−	0.999361i	$-0.488617\pi$
$179$	0.956658			0.0715040			0.0357520	+	0.999361i	$0.488617\pi$
$180$		0			0
$181$	14.4784			1.07617			0.538085	−	0.842891i	$-0.319148\pi$
$181$	14.4784			1.07617			0.538085	+	0.842891i	$0.319148\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	9.93685	−	5.73704i	0.730572	−	0.421796i
$186$		0			0
$187$	−2.65418	−	1.53239i	−0.194093	−	0.112060i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	0.756371	−	1.31007i	0.0547291	−	0.0947936i	−0.837363	−	0.546647i	$-0.815904\pi$
$191$	0.756371	−	1.31007i	0.0547291	−	0.0947936i	0.892092	+	0.451854i	$0.149237\pi$
$192$		0			0
$193$	8.82882	+	15.2920i	0.635512	+	1.10074i	0.986406	+	0.164325i	$0.0525445\pi$
$193$	8.82882	+	15.2920i	0.635512	+	1.10074i	−0.350894	+	0.936415i	$0.614122\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			17.9759i			1.28073i	0.768072	+	0.640363i	$0.221216\pi$
$197$			17.9759i			1.28073i	−0.768072	+	0.640363i	$0.778784\pi$
$198$		0			0
$199$			13.2622i			0.940132i	0.882631	+	0.470066i	$0.155770\pi$
$199$			13.2622i			0.940132i	−0.882631	+	0.470066i	$0.844230\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	15.5146	+	26.8722i	1.08891	+	1.88606i
$204$		0			0
$205$	−9.74655	+	16.8815i	−0.680729	+	1.17906i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	4.38124	+	2.52951i	0.303056	+	0.174970i
$210$		0			0
$211$	−7.09829	+	4.09820i	−0.488666	+	0.282132i	−0.724021	−	0.689778i	$-0.757708\pi$
$211$	−7.09829	+	4.09820i	−0.488666	+	0.282132i	0.235355	+	0.971910i	$0.424375\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	21.6953			1.47961
$216$		0			0
$217$	17.1768			1.16603
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−6.17501	+	3.56514i	−0.415376	+	0.239817i
$222$		0			0
$223$	2.02793	+	1.17083i	0.135800	+	0.0784044i	0.566361	−	0.824157i	$-0.308351\pi$
$223$	2.02793	+	1.17083i	0.135800	+	0.0784044i	−0.430561	+	0.902562i	$0.641684\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	7.07268	−	12.2502i	0.469430	−	0.813077i	−0.529959	−	0.848023i	$-0.677793\pi$
$227$	7.07268	−	12.2502i	0.469430	−	0.813077i	0.999389	+	0.0349462i	$0.0111260\pi$
$228$		0			0
$229$	0.764598	+	1.32432i	0.0505261	+	0.0875137i	0.890182	−	0.455605i	$-0.150577\pi$
$229$	0.764598	+	1.32432i	0.0505261	+	0.0875137i	−0.839656	+	0.543118i	$0.817244\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	16.8523i		−	1.10403i	−0.833833	−	0.552016i	$-0.813859\pi$
$233$		−	16.8523i		−	1.10403i	0.833833	−	0.552016i	$-0.186141\pi$
$234$		0			0
$235$			10.5456i			0.687922i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	8.43121	+	14.6033i	0.545370	+	0.944608i	0.998584	+	0.0532060i	$0.0169440\pi$
$239$	8.43121	+	14.6033i	0.545370	+	0.944608i	−0.453214	+	0.891402i	$0.649723\pi$
$240$		0			0
$241$	−1.62207	+	2.80950i	−0.104487	+	0.180976i	−0.913528	−	0.406775i	$-0.866653\pi$
$241$	−1.62207	+	2.80950i	−0.104487	+	0.180976i	0.809042	+	0.587751i	$0.199987\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−21.7802	−	12.5748i	−1.39149	−	0.803375i
$246$		0			0
$247$	10.1930	−	5.88495i	0.648567	−	0.374450i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	3.73846			0.235970			0.117985	−	0.993015i	$-0.462357\pi$
$251$	3.73846			0.235970			0.117985	+	0.993015i	$0.462357\pi$
$252$		0			0
$253$	4.55126			0.286135
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−16.7031	+	9.64355i	−1.04191	+	0.601548i	−0.920374	−	0.391039i	$-0.872116\pi$
$257$	−16.7031	+	9.64355i	−1.04191	+	0.601548i	−0.121538	+	0.992587i	$0.538782\pi$
$258$		0			0
$259$	−18.9024	−	10.9133i	−1.17454	−	0.678119i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	8.02755	−	13.9041i	0.495000	−	0.857365i	−0.504983	−	0.863129i	$-0.668501\pi$
$263$	8.02755	−	13.9041i	0.495000	−	0.857365i	0.999983	+	0.00576384i	$0.00183470\pi$
$264$		0			0
$265$	−10.7777	−	18.6675i	−0.662066	−	1.14673i
$266$		0			0
$267$		0			0
$268$		0			0
$269$		−	0.963696i		−	0.0587576i	−0.999568	−	0.0293788i	$-0.990647\pi$
$269$		−	0.963696i		−	0.0587576i	0.999568	−	0.0293788i	$-0.00935291\pi$
$270$		0			0
$271$			25.1446i			1.52742i	0.645557	+	0.763712i	$0.276625\pi$
$271$			25.1446i			1.52742i	−0.645557	+	0.763712i	$0.723375\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−0.0140322	−	0.0243045i	−0.000846175	−	0.00146562i
$276$		0			0
$277$	−15.3113	+	26.5199i	−0.919965	+	1.59343i	−0.120501	+	0.992713i	$0.538450\pi$
$277$	−15.3113	+	26.5199i	−0.919965	+	1.59343i	−0.799464	+	0.600714i	$0.794883\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	26.9501	+	15.5596i	1.60771	+	0.928211i	0.989882	+	0.141896i	$0.0453199\pi$
$281$	26.9501	+	15.5596i	1.60771	+	0.928211i	0.617826	+	0.786314i	$0.288013\pi$
$282$		0			0
$283$	−5.65268	+	3.26358i	−0.336017	+	0.194000i	−0.658509	−	0.752572i	$-0.728813\pi$
$283$	−5.65268	+	3.26358i	−0.336017	+	0.194000i	0.322492	+	0.946572i	$0.395479\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	37.0808			2.18881
$288$		0			0
$289$	4.32427			0.254369
$290$		0			0
$291$		0			0
$292$		0			0
$293$	21.5816	−	12.4602i	1.26081	−	0.727931i	0.287581	−	0.957756i	$-0.407149\pi$
$293$	21.5816	−	12.4602i	1.26081	−	0.727931i	0.973232	+	0.229825i	$0.0738155\pi$
$294$		0			0
$295$	−3.38844	−	1.95632i	−0.197283	−	0.113901i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	5.29430	−	9.17000i	0.306177	−	0.530315i
$300$		0			0
$301$	−20.6350	−	35.7409i	−1.18938	−	2.06007i
$302$		0			0
$303$		0			0
$304$		0			0
$305$			10.9903i			0.629303i
$306$		0			0
$307$			0.330424i			0.0188583i	0.999956	+	0.00942915i	$0.00300143\pi$
$307$			0.330424i			0.0188583i	−0.999956	+	0.00942915i	$0.996999\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−2.72247	−	4.71546i	−0.154377	−	0.267389i	0.778455	−	0.627701i	$-0.216004\pi$
$311$	−2.72247	−	4.71546i	−0.154377	−	0.267389i	−0.932832	+	0.360311i	$0.882670\pi$
$312$		0			0
$313$	5.34114	−	9.25113i	0.301899	−	0.522905i	−0.674667	−	0.738122i	$-0.735713\pi$
$313$	5.34114	−	9.25113i	0.301899	−	0.522905i	0.976566	+	0.215218i	$0.0690461\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	2.32295	+	1.34116i	0.130470	+	0.0753269i	0.563814	−	0.825901i	$-0.309333\pi$
$317$	2.32295	+	1.34116i	0.130470	+	0.0753269i	−0.433344	+	0.901228i	$0.642667\pi$
$318$		0			0
$319$	5.42066	−	3.12962i	0.303499	−	0.175225i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	20.9237			1.16423
$324$		0			0
$325$	−0.0652925			−0.00362178
$326$		0			0
$327$		0			0
$328$		0			0
$329$	17.3729	−	10.0302i	0.957798	−	0.552985i
$330$		0			0
$331$	−5.93966	−	3.42927i	−0.326473	−	0.188490i	0.327801	−	0.944747i	$-0.393692\pi$
$331$	−5.93966	−	3.42927i	−0.326473	−	0.188490i	−0.654274	+	0.756257i	$0.727026\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−12.8672	+	22.2866i	−0.703010	+	1.21765i
$336$		0			0
$337$	1.63517	+	2.83220i	0.0890734	+	0.154280i	0.907120	−	0.420873i	$-0.138276\pi$
$337$	1.63517	+	2.83220i	0.0890734	+	0.154280i	−0.818046	+	0.575152i	$0.804943\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		−	3.46490i		−	0.187635i
$342$		0			0
$343$			17.9690i			0.970237i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−13.6766	−	23.6886i	−0.734200	−	1.27167i	−0.955073	−	0.296369i	$-0.904224\pi$
$347$	−13.6766	−	23.6886i	−0.734200	−	1.27167i	0.220873	−	0.975303i	$-0.429109\pi$
$348$		0			0
$349$	−3.38337	+	5.86017i	−0.181108	+	0.313688i	−0.942258	−	0.334888i	$-0.891302\pi$
$349$	−3.38337	+	5.86017i	−0.181108	+	0.313688i	0.761150	+	0.648576i	$0.224635\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−4.17042	−	2.40780i	−0.221969	−	0.128154i	0.384892	−	0.922961i	$-0.374239\pi$
$353$	−4.17042	−	2.40780i	−0.221969	−	0.128154i	−0.606862	+	0.794807i	$0.707572\pi$
$354$		0			0
$355$	3.24828	−	1.87540i	0.172401	−	0.0995357i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−21.7135			−1.14599			−0.572996	−	0.819558i	$-0.694219\pi$
$359$	−21.7135			−1.14599			−0.572996	+	0.819558i	$0.694219\pi$
$360$		0			0
$361$	−15.5386			−0.817821
$362$		0			0
$363$		0			0
$364$		0			0
$365$	31.5459	−	18.2130i	1.65119	−	0.953314i
$366$		0			0
$367$	−13.9374	−	8.04675i	−0.727526	−	0.420037i	0.0899907	−	0.995943i	$-0.471316\pi$
$367$	−13.9374	−	8.04675i	−0.727526	−	0.420037i	−0.817516	+	0.575906i	$0.804650\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−20.5018	+	35.5102i	−1.06440	+	1.84360i
$372$		0			0
$373$	5.74639	+	9.95304i	0.297537	+	0.515349i	0.975572	−	0.219681i	$-0.0705015\pi$
$373$	5.74639	+	9.95304i	0.297537	+	0.515349i	−0.678035	+	0.735030i	$0.737168\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	14.5622i		−	0.749994i
$378$		0			0
$379$			6.94665i			0.356825i	0.983956	+	0.178413i	$0.0570962\pi$
$379$			6.94665i			0.356825i	−0.983956	+	0.178413i	$0.942904\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−10.3338	−	17.8986i	−0.528030	−	0.914576i	−0.999466	−	0.0326750i	$-0.989597\pi$
$383$	−10.3338	−	17.8986i	−0.528030	−	0.914576i	0.471436	−	0.881900i	$-0.343736\pi$
$384$		0			0
$385$	−4.12044	+	7.13682i	−0.209997	+	0.363726i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	9.75581	+	5.63252i	0.494639	+	0.285580i	0.726497	−	0.687170i	$-0.241147\pi$
$389$	9.75581	+	5.63252i	0.494639	+	0.285580i	−0.231858	+	0.972750i	$0.574480\pi$
$390$		0			0
$391$	16.3018	−	9.41184i	0.824417	−	0.475977i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−33.7238			−1.69683
$396$		0			0
$397$	7.14785			0.358740			0.179370	−	0.983782i	$-0.442594\pi$
$397$	7.14785			0.358740			0.179370	+	0.983782i	$0.442594\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−12.8828	+	7.43788i	−0.643335	+	0.371430i	−0.785898	−	0.618356i	$-0.787799\pi$
$401$	−12.8828	+	7.43788i	−0.643335	+	0.371430i	0.142563	+	0.989786i	$0.454466\pi$
$402$		0			0
$403$	−6.98117	−	4.03058i	−0.347757	−	0.200777i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−2.20143	+	3.81299i	−0.109121	+	0.189003i
$408$		0			0
$409$	−12.7714	−	22.1207i	−0.631505	−	1.09380i	−0.987244	−	0.159214i	$-0.949104\pi$
$409$	−12.7714	−	22.1207i	−0.631505	−	1.09380i	0.355739	−	0.934585i	$-0.384229\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$			7.44281i			0.366237i
$414$		0			0
$415$		−	24.8307i		−	1.21889i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−7.47805	−	12.9524i	−0.365327	−	0.632765i	0.623502	−	0.781822i	$-0.285709\pi$
$419$	−7.47805	−	12.9524i	−0.365327	−	0.632765i	−0.988829	+	0.149057i	$0.952376\pi$
$420$		0			0
$421$	4.93456	−	8.54690i	0.240496	−	0.416551i	−0.720360	−	0.693600i	$-0.756023\pi$
$421$	4.93456	−	8.54690i	0.240496	−	0.416551i	0.960856	+	0.277050i	$0.0893567\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−0.100522	−	0.0580362i	−0.00487602	−	0.00281517i
$426$		0			0
$427$	18.1054	−	10.4532i	0.876181	−	0.505863i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−6.09208			−0.293445			−0.146723	−	0.989178i	$-0.546872\pi$
$431$	−6.09208			−0.293445			−0.146723	+	0.989178i	$0.546872\pi$
$432$		0			0
$433$	−10.7593			−0.517060			−0.258530	−	0.966003i	$-0.583238\pi$
$433$	−10.7593			−0.517060			−0.258530	+	0.966003i	$0.583238\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−26.9092	+	15.5360i	−1.28724	+	0.743190i
$438$		0			0
$439$	0.133374	+	0.0770034i	0.00636559	+	0.00367517i	0.503179	−	0.864182i	$-0.332163\pi$
$439$	0.133374	+	0.0770034i	0.00636559	+	0.00367517i	−0.496814	+	0.867857i	$0.665497\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	6.91067	−	11.9696i	0.328336	−	0.568694i	−0.653846	−	0.756628i	$-0.726846\pi$
$443$	6.91067	−	11.9696i	0.328336	−	0.568694i	0.982182	+	0.187933i	$0.0601789\pi$
$444$		0			0
$445$	−18.1016	−	31.3528i	−0.858097	−	1.48627i
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	16.0661i		−	0.758207i	−0.925354	−	0.379103i	$-0.876232\pi$
$449$		−	16.0661i		−	0.758207i	0.925354	−	0.379103i	$-0.123768\pi$
$450$		0			0
$451$		−	7.47994i		−	0.352217i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	9.58629	+	16.6039i	0.449412	+	0.778405i
$456$		0			0
$457$	17.5473	−	30.3928i	0.820828	−	1.42172i	−0.0842377	−	0.996446i	$-0.526846\pi$
$457$	17.5473	−	30.3928i	0.820828	−	1.42172i	0.905066	−	0.425271i	$-0.139821\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−13.8809	−	8.01415i	−0.646499	−	0.373256i	0.140615	−	0.990064i	$-0.455092\pi$
$461$	−13.8809	−	8.01415i	−0.646499	−	0.373256i	−0.787114	+	0.616808i	$0.788425\pi$
$462$		0			0
$463$	20.5465	−	11.8625i	0.954877	−	0.551299i	0.0602848	−	0.998181i	$-0.480799\pi$
$463$	20.5465	−	11.8625i	0.954877	−	0.551299i	0.894593	+	0.446882i	$0.147466\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	30.2580			1.40017			0.700087	−	0.714057i	$-0.253144\pi$
$467$	30.2580			1.40017			0.700087	+	0.714057i	$0.253144\pi$
$468$		0			0
$469$	48.9532			2.26045
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−7.20965	+	4.16250i	−0.331500	+	0.191392i
$474$		0			0
$475$	0.165930	+	0.0957999i	0.00761341	+	0.00439560i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	13.1141	−	22.7142i	0.599197	−	1.03784i	−0.393742	−	0.919221i	$-0.628820\pi$
$479$	13.1141	−	22.7142i	0.599197	−	1.03784i	0.992940	−	0.118620i	$-0.0378469\pi$
$480$		0			0
$481$	5.12168	+	8.87100i	0.233528	+	0.404483i
$482$		0			0
$483$		0			0
$484$		0			0
$485$		−	23.0930i		−	1.04860i
$486$		0			0
$487$			19.1150i			0.866181i	0.901350	+	0.433091i	$0.142577\pi$
$487$			19.1150i			0.866181i	−0.901350	+	0.433091i	$0.857423\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−1.86051	−	3.22249i	−0.0839634	−	0.145429i	0.820986	−	0.570949i	$-0.193425\pi$
$491$	−1.86051	−	3.22249i	−0.0839634	−	0.145429i	−0.904949	+	0.425520i	$0.860091\pi$
$492$		0			0
$493$	12.9439	−	22.4195i	0.582963	−	1.00972i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−6.17905	−	3.56748i	−0.277168	−	0.160023i
$498$		0			0
$499$	−1.72264	+	0.994567i	−0.0771160	+	0.0445230i	−0.538062	−	0.842905i	$-0.680843\pi$
$499$	−1.72264	+	0.994567i	−0.0771160	+	0.0445230i	0.460946	+	0.887428i	$0.347510\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−42.1350			−1.87871			−0.939354	−	0.342948i	$-0.888575\pi$
$503$	−42.1350			−1.87871			−0.939354	+	0.342948i	$0.888575\pi$
$504$		0			0
$505$	−14.9716			−0.666226
$506$		0			0
$507$		0			0
$508$		0			0
$509$	19.4295	−	11.2176i	0.861197	−	0.497212i	−0.00321586	−	0.999995i	$-0.501024\pi$
$509$	19.4295	−	11.2176i	0.861197	−	0.497212i	0.864413	+	0.502782i	$0.167690\pi$
$510$		0			0
$511$	−60.0082	−	34.6458i	−2.65461	−	1.53264i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−8.97829	+	15.5509i	−0.395631	+	0.685253i
$516$		0			0
$517$	−2.02330	−	3.50446i	−0.0889847	−	0.154126i
$518$		0			0
$519$		0			0
$520$		0			0
$521$			1.67553i			0.0734065i	0.999326	+	0.0367032i	$0.0116856\pi$
$521$			1.67553i			0.0734065i	−0.999326	+	0.0367032i	$0.988314\pi$
$522$		0			0
$523$		−	5.59296i		−	0.244563i	−0.992495	−	0.122281i	$-0.960979\pi$
$523$		−	5.59296i		−	0.244563i	0.992495	−	0.122281i	$-0.0390211\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−7.16529	−	12.4106i	−0.312125	−	0.540616i
$528$		0			0
$529$	−2.47675	+	4.28986i	−0.107685	+	0.186516i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−15.0708	−	8.70112i	−0.652788	−	0.376887i
$534$		0			0
$535$	19.1634	−	11.0640i	0.828506	−	0.478338i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	9.65048			0.415676
$540$		0			0
$541$	17.2602			0.742072			0.371036	−	0.928618i	$-0.379003\pi$
$541$	17.2602			0.742072			0.371036	+	0.928618i	$0.379003\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	2.44732	−	1.41296i	0.104832	−	0.0605246i
$546$		0			0
$547$	26.3909	+	15.2368i	1.12839	+	0.651479i	0.943531	−	0.331285i	$-0.107482\pi$
$547$	26.3909	+	15.2368i	1.12839	+	0.651479i	0.184864	+	0.982764i	$0.440816\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−21.3663	+	37.0076i	−0.910237	+	1.57658i
$552$		0			0
$553$	32.0756	+	55.5566i	1.36399	+	2.36251i
$554$		0			0
$555$		0			0
$556$		0			0
$557$			8.77002i			0.371598i	0.982588	+	0.185799i	$0.0594873\pi$
$557$			8.77002i			0.371598i	−0.982588	+	0.185799i	$0.940513\pi$
$558$		0			0
$559$			19.3683i			0.819190i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−7.66047	−	13.2683i	−0.322850	−	0.559193i	0.658225	−	0.752822i	$-0.271308\pi$
$563$	−7.66047	−	13.2683i	−0.322850	−	0.559193i	−0.981075	+	0.193628i	$0.937974\pi$
$564$		0			0
$565$	10.8187	−	18.7385i	0.455146	−	0.788335i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−19.6228	−	11.3292i	−0.822632	−	0.474947i	0.0286914	−	0.999588i	$-0.490866\pi$
$569$	−19.6228	−	11.3292i	−0.822632	−	0.474947i	−0.851323	+	0.524642i	$0.824199\pi$
$570$		0			0
$571$	34.9763	−	20.1936i	1.46371	−	0.845075i	0.464533	−	0.885556i	$-0.346222\pi$
$571$	34.9763	−	20.1936i	1.46371	−	0.845075i	0.999180	+	0.0404804i	$0.0128888\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	0.172370			0.00718831
$576$		0			0
$577$	−4.49352			−0.187068			−0.0935339	−	0.995616i	$-0.529816\pi$
$577$	−4.49352			−0.187068			−0.0935339	+	0.995616i	$0.529816\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−40.9061	+	23.6171i	−1.69707	+	0.979804i
$582$		0			0
$583$	7.16312	+	4.13563i	0.296666	+	0.171280i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	20.1321	−	34.8698i	0.830941	−	1.43923i	−0.0663525	−	0.997796i	$-0.521136\pi$
$587$	20.1321	−	34.8698i	0.830941	−	1.43923i	0.897293	−	0.441435i	$-0.145530\pi$
$588$		0			0
$589$	11.8277	+	20.4861i	0.487351	+	0.844117i
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	6.05257i		−	0.248549i	−0.992248	−	0.124275i	$-0.960340\pi$
$593$		−	6.05257i		−	0.248549i	0.992248	−	0.124275i	$-0.0396604\pi$
$594$		0			0
$595$			34.0837i			1.39730i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−12.9490	−	22.4284i	−0.529084	−	0.916400i	−0.999425	−	0.0339151i	$-0.989202\pi$
$599$	−12.9490	−	22.4284i	−0.529084	−	0.916400i	0.470341	−	0.882485i	$-0.344131\pi$
$600$		0			0
$601$	−3.94258	+	6.82875i	−0.160821	+	0.278550i	−0.935163	−	0.354217i	$-0.884748\pi$
$601$	−3.94258	+	6.82875i	−0.160821	+	0.278550i	0.774342	+	0.632767i	$0.218081\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−19.9311	−	11.5072i	−0.810314	−	0.467835i
$606$		0			0
$607$	−18.8189	+	10.8651i	−0.763838	+	0.441002i	−0.830672	−	0.556762i	$-0.812043\pi$
$607$	−18.8189	+	10.8651i	−0.763838	+	0.441002i	0.0668344	+	0.997764i	$0.478710\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−9.41450			−0.380870
$612$		0			0
$613$	−41.3513			−1.67016			−0.835081	−	0.550127i	$-0.814579\pi$
$613$	−41.3513			−1.67016			−0.835081	+	0.550127i	$0.814579\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−16.3660	+	9.44893i	−0.658871	+	0.380400i	−0.791847	−	0.610720i	$-0.790880\pi$
$617$	−16.3660	+	9.44893i	−0.658871	+	0.380400i	0.132975	+	0.991119i	$0.457547\pi$
$618$		0			0
$619$	27.9868	+	16.1582i	1.12489	+	0.649453i	0.942644	−	0.333800i	$-0.108331\pi$
$619$	27.9868	+	16.1582i	1.12489	+	0.649453i	0.182242	+	0.983254i	$0.441664\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−34.4337	+	59.6410i	−1.37956	+	2.38946i
$624$		0			0
$625$	12.5810	+	21.7909i	0.503239	+	0.871635i
$626$		0			0
$627$		0			0
$628$		0			0
$629$			18.2099i			0.726077i
$630$		0			0
$631$		−	26.1886i		−	1.04255i	−0.853388	−	0.521276i	$-0.825456\pi$
$631$		−	26.1886i		−	1.04255i	0.853388	−	0.521276i	$-0.174544\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−2.49489	−	4.32127i	−0.0990066	−	0.171484i
$636$		0			0
$637$	11.2260	−	19.4440i	0.444791	−	0.770401i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−42.2595	−	24.3985i	−1.66915	−	0.963683i	−0.968098	−	0.250572i	$-0.919381\pi$
$641$	−42.2595	−	24.3985i	−1.66915	−	0.963683i	−0.701050	−	0.713112i	$-0.747285\pi$
$642$		0			0
$643$	36.8893	−	21.2980i	1.45477	−	0.839913i	0.456026	−	0.889967i	$-0.349273\pi$
$643$	36.8893	−	21.2980i	1.45477	−	0.839913i	0.998747	+	0.0500536i	$0.0159392\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−9.95266			−0.391279			−0.195640	−	0.980676i	$-0.562678\pi$
$647$	−9.95266			−0.391279			−0.195640	+	0.980676i	$0.562678\pi$
$648$		0			0
$649$	1.50137			0.0589338
$650$		0			0
$651$		0			0
$652$		0			0
$653$	1.63982	−	0.946748i	0.0641710	−	0.0370491i	−0.467571	−	0.883955i	$-0.654871\pi$
$653$	1.63982	−	0.946748i	0.0641710	−	0.0370491i	0.531742	+	0.846906i	$0.321538\pi$
$654$		0			0
$655$	14.7401	+	8.51020i	0.575943	+	0.332521i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	15.0835	−	26.1253i	0.587569	−	1.01770i	−0.406981	−	0.913437i	$-0.633418\pi$
$659$	15.0835	−	26.1253i	0.587569	−	1.01770i	0.994550	−	0.104262i	$-0.0332482\pi$
$660$		0			0
$661$	−13.7300	−	23.7810i	−0.534034	−	0.924974i	−0.999209	−	0.0397553i	$-0.987342\pi$
$661$	−13.7300	−	23.7810i	−0.534034	−	0.924974i	0.465176	−	0.885218i	$-0.345991\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		−	56.2617i		−	2.18173i
$666$		0			0
$667$			38.4438i			1.48855i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−2.10861	−	3.65222i	−0.0814021	−	0.140993i
$672$		0			0
$673$	21.7401	−	37.6550i	0.838021	−	1.45149i	−0.0535258	−	0.998566i	$-0.517046\pi$
$673$	21.7401	−	37.6550i	0.838021	−	1.45149i	0.891547	−	0.452929i	$-0.149621\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	27.9492	+	16.1365i	1.07417	+	0.620174i	0.929319	−	0.369279i	$-0.120395\pi$
$677$	27.9492	+	16.1365i	1.07417	+	0.620174i	0.144855	+	0.989453i	$0.453729\pi$
$678$		0			0
$679$	−38.0433	+	21.9643i	−1.45997	+	0.842913i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−0.967844			−0.0370335			−0.0185168	−	0.999829i	$-0.505894\pi$
$683$	−0.967844			−0.0370335			−0.0185168	+	0.999829i	$0.505894\pi$
$684$		0			0
$685$	6.23833			0.238354
$686$		0			0
$687$		0			0
$688$		0			0
$689$	16.6651	−	9.62163i	0.634892	−	0.366555i
$690$		0			0
$691$	−2.62703	−	1.51672i	−0.0999369	−	0.0576986i	0.449199	−	0.893432i	$-0.351710\pi$
$691$	−2.62703	−	1.51672i	−0.0999369	−	0.0576986i	−0.549135	+	0.835733i	$0.685043\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	8.27965	−	14.3408i	0.314065	−	0.543976i
$696$		0			0
$697$	−15.4682	−	26.7918i	−0.585902	−	1.01481i
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	19.4454i		−	0.734442i	−0.930134	−	0.367221i	$-0.880309\pi$
$701$		−	19.4454i		−	0.734442i	0.930134	−	0.367221i	$-0.119691\pi$
$702$		0			0
$703$		−	30.0590i		−	1.13370i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	14.2398	+	24.6641i	0.535544	+	0.927589i
$708$		0			0
$709$	−21.6255	+	37.4565i	−0.812163	+	1.40671i	0.0991843	+	0.995069i	$0.468377\pi$
$709$	−21.6255	+	37.4565i	−0.812163	+	1.40671i	−0.911347	+	0.411638i	$0.864957\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	18.4300	+	10.6406i	0.690210	+	0.398493i
$714$		0			0
$715$	3.34935	−	1.93375i	0.125259	−	0.0723181i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	24.6917			0.920844			0.460422	−	0.887700i	$-0.347698\pi$
$719$	24.6917			0.920844			0.460422	+	0.887700i	$0.347698\pi$
$720$		0			0
$721$	34.1579			1.27211
$722$		0			0
$723$		0			0
$724$		0			0
$725$	0.205296	−	0.118528i	0.00762452	−	0.00440202i
$726$		0			0
$727$	14.1473	+	8.16793i	0.524693	+	0.302932i	0.738853	−	0.673867i	$-0.235368\pi$
$727$	14.1473	+	8.16793i	0.524693	+	0.302932i	−0.214159	+	0.976799i	$0.568701\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−17.2158	+	29.8186i	−0.636749	+	1.10288i
$732$		0			0
$733$	2.25105	+	3.89893i	0.0831444	+	0.144010i	0.904599	−	0.426264i	$-0.140170\pi$
$733$	2.25105	+	3.89893i	0.0831444	+	0.144010i	−0.821455	+	0.570274i	$0.806837\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	9.87486i		−	0.363745i
$738$		0			0
$739$			29.2715i			1.07677i	0.842700	+	0.538384i	$0.180965\pi$
$739$			29.2715i			1.07677i	−0.842700	+	0.538384i	$0.819035\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	1.64312	+	2.84596i	0.0602801	+	0.104408i	0.894591	−	0.446887i	$-0.147467\pi$
$743$	1.64312	+	2.84596i	0.0602801	+	0.104408i	−0.834311	+	0.551295i	$0.814134\pi$
$744$		0			0
$745$	13.8287	−	23.9520i	0.506645	−	0.877535i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−36.4536	−	21.0465i	−1.33199	−	0.769022i
$750$		0			0
$751$	−0.0725619	+	0.0418936i	−0.00264782	+	0.00152872i	−0.501323	−	0.865260i	$-0.667153\pi$
$751$	−0.0725619	+	0.0418936i	−0.00264782	+	0.00152872i	0.498676	+	0.866789i	$0.333820\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	44.3188			1.61293
$756$		0			0
$757$	−15.2666			−0.554875			−0.277437	−	0.960744i	$-0.589485\pi$
$757$	−15.2666			−0.554875			−0.277437	+	0.960744i	$0.589485\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−7.03988	+	4.06448i	−0.255196	+	0.147337i	−0.622141	−	0.782905i	$-0.713737\pi$
$761$	−7.03988	+	4.06448i	−0.255196	+	0.147337i	0.366945	+	0.930242i	$0.380404\pi$
$762$		0			0
$763$	−4.65542	−	2.68781i	−0.168537	−	0.0973052i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	1.74648	−	3.02499i	0.0630617	−	0.109226i
$768$		0			0
$769$	0.109508	+	0.189673i	0.00394895	+	0.00683978i	0.867993	−	0.496576i	$-0.165410\pi$
$769$	0.109508	+	0.189673i	0.00394895	+	0.00683978i	−0.864044	+	0.503416i	$0.832076\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		−	3.33995i		−	0.120130i	−0.998194	−	0.0600649i	$-0.980869\pi$
$773$		−	3.33995i		−	0.120130i	0.998194	−	0.0600649i	$-0.0191308\pi$
$774$		0			0
$775$		−	0.131226i		−	0.00471378i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	25.5333	+	44.2250i	0.914826	+	1.58452i
$780$		0			0
$781$	−0.719632	+	1.24644i	−0.0257505	+	0.0446011i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−6.97666	−	4.02798i	−0.249008	−	0.143765i
$786$		0			0
$787$	35.2099	−	20.3284i	1.25510	−	0.724631i	0.282980	−	0.959126i	$-0.408677\pi$
$787$	35.2099	−	20.3284i	1.25510	−	0.724631i	0.972117	+	0.234495i	$0.0753437\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−41.1597			−1.46347
$792$		0			0
$793$	−9.81146			−0.348415
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−39.3083	+	22.6947i	−1.39237	+	0.803887i	−0.993578	−	0.113153i	$-0.963905\pi$
$797$	−39.3083	+	22.6947i	−1.39237	+	0.803887i	−0.398796	+	0.917040i	$0.630572\pi$
$798$		0			0
$799$	−14.4942	−	8.36823i	−0.512768	−	0.296047i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−6.98875	+	12.1049i	−0.246628	+	0.427172i
$804$		0			0
$805$	−25.3075	−	43.8338i	−0.891971	−	1.54494i
$806$		0			0
$807$		0			0
$808$		0			0
$809$			13.7643i			0.483927i	0.970285	+	0.241963i	$0.0777914\pi$
$809$			13.7643i			0.483927i	−0.970285	+	0.241963i	$0.922209\pi$
$810$		0			0
$811$			26.6907i			0.937237i	0.883401	+	0.468618i	$0.155248\pi$
$811$			26.6907i			0.937237i	−0.883401	+	0.468618i	$0.844752\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−4.26479	−	7.38684i	−0.149389	−	0.258750i
$816$		0			0
$817$	28.4179	−	49.2213i	0.994218	−	1.72204i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−3.52515	−	2.03525i	−0.123029	−	0.0710306i	0.437223	−	0.899353i	$-0.355962\pi$
$821$	−3.52515	−	2.03525i	−0.123029	−	0.0710306i	−0.560251	+	0.828323i	$0.689295\pi$
$822$		0			0
$823$	−36.4410	+	21.0392i	−1.27025	+	0.733382i	−0.975036	−	0.222048i	$-0.928726\pi$
$823$	−36.4410	+	21.0392i	−1.27025	+	0.733382i	−0.295219	+	0.955430i	$0.595392\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	6.32300			0.219872			0.109936	−	0.993939i	$-0.464935\pi$
$827$	6.32300			0.219872			0.109936	+	0.993939i	$0.464935\pi$
$828$		0			0
$829$	28.8344			1.00146			0.500731	−	0.865603i	$-0.333065\pi$
$829$	28.8344			1.00146			0.500731	+	0.865603i	$0.333065\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	34.5663	−	19.9568i	1.19765	−	0.691463i
$834$		0			0
$835$	−12.4817	−	7.20629i	−0.431946	−	0.249384i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	23.5240	−	40.7448i	0.812139	−	1.40667i	−0.0992245	−	0.995065i	$-0.531636\pi$
$839$	23.5240	−	40.7448i	0.812139	−	1.40667i	0.911364	−	0.411602i	$-0.135030\pi$
$840$		0			0
$841$	11.9354	+	20.6727i	0.411565	+	0.712852i
$842$		0			0
$843$		0			0
$844$		0			0
$845$			20.1657i			0.693721i
$846$		0			0
$847$			43.7793i			1.50427i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−13.5210	−	23.4191i	−0.463495	−	0.802797i
$852$		0			0
$853$	23.8961	−	41.3893i	0.818187	−	1.41714i	−0.0888293	−	0.996047i	$-0.528313\pi$
$853$	23.8961	−	41.3893i	0.818187	−	1.41714i	0.907017	−	0.421095i	$-0.138354\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−30.8514	−	17.8121i	−1.05386	−	0.608449i	−0.130136	−	0.991496i	$-0.541541\pi$
$857$	−30.8514	−	17.8121i	−1.05386	−	0.608449i	−0.923729	+	0.383047i	$0.874875\pi$
$858$		0			0
$859$	17.2603	−	9.96526i	0.588915	−	0.340010i	−0.175753	−	0.984434i	$-0.556236\pi$
$859$	17.2603	−	9.96526i	0.588915	−	0.340010i	0.764668	+	0.644424i	$0.222903\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	1.95867			0.0666738			0.0333369	−	0.999444i	$-0.489387\pi$
$863$	1.95867			0.0666738			0.0333369	+	0.999444i	$0.489387\pi$
$864$		0			0
$865$	−43.3070			−1.47248
$866$		0			0
$867$		0			0
$868$		0			0
$869$	11.2069	−	6.47030i	0.380167	−	0.219490i
$870$		0			0
$871$	−19.8961	−	11.4870i	−0.674154	−	0.389223i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	23.7771	−	41.1832i	0.803814	−	1.39225i
$876$		0			0
$877$	−12.0233	−	20.8250i	−0.406000	−	0.703212i	0.588438	−	0.808543i	$-0.299743\pi$
$877$	−12.0233	−	20.8250i	−0.406000	−	0.703212i	−0.994437	+	0.105331i	$0.966410\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	2.14441i		−	0.0722469i	−0.999347	−	0.0361234i	$-0.988499\pi$
$881$		−	2.14441i		−	0.0722469i	0.999347	−	0.0361234i	$-0.0115010\pi$
$882$		0			0
$883$		−	24.4698i		−	0.823476i	−0.911302	−	0.411738i	$-0.864922\pi$
$883$		−	24.4698i		−	0.823476i	0.911302	−	0.411738i	$-0.135078\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	28.7592	+	49.8123i	0.965638	+	1.67253i	0.707891	+	0.706322i	$0.249647\pi$
$887$	28.7592	+	49.8123i	0.965638	+	1.67253i	0.257747	+	0.966212i	$0.417020\pi$
$888$		0			0
$889$	−4.74590	+	8.22014i	−0.159172	+	0.275695i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	23.9254	+	13.8134i	0.800634	+	0.462247i
$894$		0			0
$895$	−1.85859	+	1.07306i	−0.0621258	+	0.0358683i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	29.2675			0.976124
$900$		0			0
$901$	34.2093			1.13968
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−28.1285	+	16.2400i	−0.935024	+	0.539836i
$906$		0			0
$907$	−13.2635	−	7.65770i	−0.440408	−	0.254270i	0.263363	−	0.964697i	$-0.415168\pi$
$907$	−13.2635	−	7.65770i	−0.440408	−	0.254270i	−0.703771	+	0.710427i	$0.748502\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−27.8373	+	48.2156i	−0.922291	+	1.59745i	−0.126430	+	0.991976i	$0.540352\pi$
$911$	−27.8373	+	48.2156i	−0.922291	+	1.59745i	−0.795861	+	0.605479i	$0.792981\pi$
$912$		0			0
$913$	4.76405	+	8.25158i	0.157667	+	0.273088i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		−	32.3771i		−	1.06918i
$918$		0			0
$919$		−	38.2475i		−	1.26167i	−0.775917	−	0.630834i	$-0.782713\pi$
$919$		−	38.2475i		−	1.26167i	0.775917	−	0.630834i	$-0.217287\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	1.67424	+	2.89987i	0.0551082	+	0.0954502i
$924$		0			0
$925$	−0.0833747	+	0.144409i	−0.00274134	+	0.00474815i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	39.7519	+	22.9508i	1.30422	+	0.752990i	0.981125	−	0.193377i	$-0.0619440\pi$
$929$	39.7519	+	22.9508i	1.30422	+	0.752990i	0.323093	+	0.946367i	$0.395277\pi$
$930$		0			0
$931$	−57.0582	+	32.9426i	−1.87001	+	1.07965i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	6.87538			0.224849
$936$		0			0
$937$	−4.65199			−0.151974			−0.0759870	−	0.997109i	$-0.524211\pi$
$937$	−4.65199			−0.151974			−0.0759870	+	0.997109i	$0.524211\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	48.9494	−	28.2610i	1.59571	−	0.921281i	0.603404	−	0.797436i	$-0.293811\pi$
$941$	48.9494	−	28.2610i	1.59571	−	0.921281i	0.992302	−	0.123845i	$-0.0395226\pi$
$942$		0			0
$943$	39.7863	+	22.9706i	1.29562	+	0.748026i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	11.5631	−	20.0279i	0.375751	−	0.650820i	−0.614688	−	0.788770i	$-0.710718\pi$
$947$	11.5631	−	20.0279i	0.375751	−	0.650820i	0.990439	+	0.137951i	$0.0440515\pi$
$948$		0			0
$949$	16.2595	+	28.1622i	0.527805	+	0.914185i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	2.30527i		−	0.0746751i	−0.999303	−	0.0373376i	$-0.988112\pi$
$953$		−	2.30527i		−	0.0746751i	0.999303	−	0.0373376i	$-0.0118877\pi$
$954$		0			0
$955$			3.39360i			0.109814i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−5.93344	−	10.2770i	−0.191601	−	0.331862i
$960$		0			0
$961$	−7.39927	+	12.8159i	−0.238686	+	0.413416i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−34.3052	−	19.8061i	−1.10432	−	0.637581i
$966$		0			0
$967$	14.2255	−	8.21309i	0.457461	−	0.264115i	−0.253515	−	0.967331i	$-0.581587\pi$
$967$	14.2255	−	8.21309i	0.457461	−	0.264115i	0.710976	+	0.703216i	$0.248253\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−27.1173			−0.870236			−0.435118	−	0.900373i	$-0.643293\pi$
$971$	−27.1173			−0.870236			−0.435118	+	0.900373i	$0.643293\pi$
$972$		0			0
$973$	−31.4999			−1.00984
$974$		0			0
$975$		0			0
$976$		0			0
$977$	6.94558	−	4.01003i	0.222209	−	0.128292i	−0.384764	−	0.923015i	$-0.625717\pi$
$977$	6.94558	−	4.01003i	0.222209	−	0.128292i	0.606973	+	0.794723i	$0.292384\pi$
$978$		0			0
$979$	12.0308	+	6.94598i	0.384506	+	0.221994i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	23.4695	−	40.6503i	0.748559	−	1.29654i	−0.199954	−	0.979805i	$-0.564079\pi$
$983$	23.4695	−	40.6503i	0.748559	−	1.29654i	0.948513	−	0.316738i	$-0.102587\pi$
$984$		0			0
$985$	−20.1630	−	34.9234i	−0.642448	−	1.11275i
$986$		0			0
$987$		0			0
$988$		0			0
$989$		−	51.1315i		−	1.62589i
$990$		0			0
$991$		−	5.91939i		−	0.188035i	−0.995571	−	0.0940177i	$-0.970029\pi$
$991$		−	5.91939i		−	0.188035i	0.995571	−	0.0940177i	$-0.0299710\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−14.8758	−	25.7657i	−0.471596	−	0.816828i
$996$		0			0
$997$	14.0957	−	24.4144i	0.446415	−	0.773213i	−0.551735	−	0.834020i	$-0.686034\pi$
$997$	14.0957	−	24.4144i	0.446415	−	0.773213i	0.998150	+	0.0608067i	$0.0193673\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3456.2.s.d.2303.3		24
3.2	odd	2		1152.2.s.a.767.12	yes	24
4.3	odd	2		3456.2.s.b.2303.3		24
8.3	odd	2		3456.2.s.c.2303.10		24
8.5	even	2		3456.2.s.a.2303.10		24
9.4	even	3		1152.2.s.d.383.1	yes	24
9.5	odd	6		3456.2.s.b.1151.3		24
12.11	even	2		1152.2.s.d.767.1	yes	24
24.5	odd	2		1152.2.s.c.767.1	yes	24
24.11	even	2		1152.2.s.b.767.12	yes	24
36.23	even	6	inner	3456.2.s.d.1151.3		24
36.31	odd	6		1152.2.s.a.383.12	✓	24
72.5	odd	6		3456.2.s.c.1151.10		24
72.13	even	6		1152.2.s.b.383.12	yes	24
72.59	even	6		3456.2.s.a.1151.10		24
72.67	odd	6		1152.2.s.c.383.1	yes	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1152.2.s.a.383.12	✓	24	36.31	odd	6
1152.2.s.a.767.12	yes	24	3.2	odd	2
1152.2.s.b.383.12	yes	24	72.13	even	6
1152.2.s.b.767.12	yes	24	24.11	even	2
1152.2.s.c.383.1	yes	24	72.67	odd	6
1152.2.s.c.767.1	yes	24	24.5	odd	2
1152.2.s.d.383.1	yes	24	9.4	even	3
1152.2.s.d.767.1	yes	24	12.11	even	2
3456.2.s.a.1151.10		24	72.59	even	6
3456.2.s.a.2303.10		24	8.5	even	2
3456.2.s.b.1151.3		24	9.5	odd	6
3456.2.s.b.2303.3		24	4.3	odd	2
3456.2.s.c.1151.10		24	72.5	odd	6
3456.2.s.c.2303.10		24	8.3	odd	2
3456.2.s.d.1151.3		24	36.23	even	6	inner
3456.2.s.d.2303.3		24	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 3456 = 2^{7} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3456.s (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-1.94279 + 1.12167i) q^{5} +(3.69568 + 2.13370i) q^{7} +O(q^{10})\)	\(q+(-1.94279 + 1.12167i) q^{5} +(3.69568 + 2.13370i) q^{7} +(0.430411 - 0.745494i) q^{11} +(-1.00136 - 1.73441i) q^{13} -3.56030i q^{17} +5.87695i q^{19} +(2.64355 + 4.57877i) q^{23} +(0.0163009 - 0.0282341i) q^{25} +(6.29707 + 3.63561i) q^{29} +(3.48584 - 2.01255i) q^{31} -9.57327 q^{35} -5.11472 q^{37} +(7.52515 - 4.34465i) q^{41} +(-8.37531 - 4.83549i) q^{43} +(2.35043 - 4.07106i) q^{47} +(5.60538 + 9.70881i) q^{49} +9.60856i q^{53} +1.93112i q^{55} +(0.872053 + 1.51044i) q^{59} +(2.44953 - 4.24271i) q^{61} +(3.89088 + 2.24640i) q^{65} +(9.93455 - 5.73571i) q^{67} -1.67196 q^{71} -16.2374 q^{73} +(3.18133 - 1.83674i) q^{77} +(13.0188 + 7.51641i) q^{79} +(-5.53430 + 9.58570i) q^{83} +(3.99349 + 6.91693i) q^{85} +16.1380i q^{89} -8.54642i q^{91} +(-6.59202 - 11.4177i) q^{95} +(-5.14699 + 8.91485i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q+O(q^{10}) \) 24 * q	\( 24 q + 6 q^{11} + 12 q^{25} + 24 q^{29} + 36 q^{31} - 12 q^{41} - 42 q^{43} + 12 q^{49} - 6 q^{59} + 54 q^{67} + 48 q^{77} - 60 q^{79} - 36 q^{83}+O(q^{100}) \) 24 * q + 6 * q^11 + 12 * q^25 + 24 * q^29 + 36 * q^31 - 12 * q^41 - 42 * q^43 + 12 * q^49 - 6 * q^59 + 54 * q^67 + 48 * q^77 - 60 * q^79 - 36 * q^83

Introduction

L-functions

Modular forms

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Database

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