LMFDB - Embedded newform 2646.2.d.e.2645.13

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2646,2,Mod(2645,2646)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2646, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2646.2645");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2646 = 2 \cdot 3^{3} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2646.d (of order $2$, degree $1$, 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:	$21.1284163748$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\Q(\zeta_{48})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - x^{8} + 1 $ x^16 - x^8 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{14} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2645.13
Root			$-0.991445 + 0.130526i$ of defining polynomial
Character	$\chi$	$=$	2646.2645
Dual form			2646.2.d.e.2645.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{2} -1.00000 q^{4} +0.601731 q^{5} -1.00000i q^{8} +O(q^{10})$	$q+1.00000i q^{2} -1.00000 q^{4} +0.601731 q^{5} -1.00000i q^{8} +0.601731i q^{10} -1.20041i q^{11} -0.649659i q^{13} +1.00000 q^{16} +0.564050 q^{17} +0.249245i q^{19} -0.601731 q^{20} +1.20041 q^{22} +3.95367i q^{23} -4.63792 q^{25} +0.649659 q^{26} -1.37199i q^{29} +7.11169i q^{31} +1.00000i q^{32} +0.564050i q^{34} +5.17738 q^{37} -0.249245 q^{38} -0.601731i q^{40} +7.32326 q^{41} +9.36414 q^{43} +1.20041i q^{44} -3.95367 q^{46} +9.12845 q^{47} -4.63792i q^{50} +0.649659i q^{52} -10.3178i q^{53} -0.722325i q^{55} +1.37199 q^{58} +7.26508 q^{59} -0.865467i q^{61} -7.11169 q^{62} -1.00000 q^{64} -0.390920i q^{65} -9.37779 q^{67} -0.564050 q^{68} -4.28897i q^{71} +15.7896i q^{73} +5.17738i q^{74} -0.249245i q^{76} +12.1462 q^{79} +0.601731 q^{80} +7.32326i q^{82} +5.58632 q^{83} +0.339406 q^{85} +9.36414i q^{86} -1.20041 q^{88} +7.62781 q^{89} -3.95367i q^{92} +9.12845i q^{94} +0.149978i q^{95} +13.2012i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 16 q^{4}+O(q^{10}) $ 16 * q - 16 * q^4	$ 16 q - 16 q^{4} + 16 q^{16} - 32 q^{22} + 48 q^{25} - 32 q^{37} + 16 q^{43} - 48 q^{46} + 16 q^{58} - 16 q^{64} + 16 q^{67} + 32 q^{88}+O(q^{100}) $ 16 * q - 16 * q^4 + 16 * q^16 - 32 * q^22 + 48 * q^25 - 32 * q^37 + 16 * q^43 - 48 * q^46 + 16 * q^58 - 16 * q^64 + 16 * q^67 + 32 * q^88

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2646\mathbb{Z}\right)^\times$.

$n$	$785$	$1081$
$\chi(n)$	$-1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	1.00000i	0.707107i
$2$	1.00000i	0.707107i
$3$	0	0
$3$	0	0
$4$	−1.00000	−0.500000
$5$	0.601731	0.269102	0.134551	−	0.990907i	$-0.457041\pi$
$5$	0.601731	0.269102	0.134551	+	0.990907i	$0.457041\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	− 1.00000i	− 0.353553i
$9$	0	0
$10$	0.601731i	0.190284i
$11$	− 1.20041i	− 0.361938i	−0.983489	−	0.180969i	$-0.942077\pi$
$11$	− 1.20041i	− 0.361938i	0.983489	−	0.180969i	$-0.0579234\pi$
$12$	0	0
$13$	− 0.649659i	− 0.180183i	−0.995933	−	0.0900914i	$-0.971284\pi$
$13$	− 0.649659i	− 0.180183i	0.995933	−	0.0900914i	$-0.0287159\pi$
$14$	0	0
$15$	0	0
$16$	1.00000	0.250000
$17$	0.564050	0.136802	0.0684011	−	0.997658i	$-0.478210\pi$
$17$	0.564050	0.136802	0.0684011	+	0.997658i	$0.478210\pi$
$18$	0	0
$19$	0.249245i	0.0571807i	0.999591	+	0.0285904i	$0.00910184\pi$
$19$	0.249245i	0.0571807i	−0.999591	+	0.0285904i	$0.990898\pi$
$20$	−0.601731	−0.134551
$21$	0	0
$22$	1.20041	0.255929
$23$	3.95367i	0.824397i	0.911094	+	0.412198i	$0.135239\pi$
$23$	3.95367i	0.824397i	−0.911094	+	0.412198i	$0.864761\pi$
$24$	0	0
$25$	−4.63792	−0.927584
$26$	0.649659	0.127409
$27$	0	0
$28$	0	0
$29$	− 1.37199i	− 0.254771i	−0.991853	−	0.127386i	$-0.959341\pi$
$29$	− 1.37199i	− 0.254771i	0.991853	−	0.127386i	$-0.0406586\pi$
$30$	0	0
$31$	7.11169i	1.27730i	0.769498	+	0.638649i	$0.220506\pi$
$31$	7.11169i	1.27730i	−0.769498	+	0.638649i	$0.779494\pi$
$32$	1.00000i	0.176777i
$33$	0	0
$34$	0.564050i	0.0967337i
$35$	0	0
$36$	0	0
$37$	5.17738	0.851155	0.425578	−	0.904922i	$-0.360071\pi$
$37$	5.17738	0.851155	0.425578	+	0.904922i	$0.360071\pi$
$38$	−0.249245	−0.0404329
$39$	0	0
$40$	− 0.601731i	− 0.0951420i
$41$	7.32326	1.14370	0.571850	−	0.820358i	$-0.306226\pi$
$41$	7.32326	1.14370	0.571850	+	0.820358i	$0.306226\pi$
$42$	0	0
$43$	9.36414	1.42802	0.714009	−	0.700136i	$-0.246877\pi$
$43$	9.36414	1.42802	0.714009	+	0.700136i	$0.246877\pi$
$44$	1.20041i	0.180969i
$45$	0	0
$46$	−3.95367	−0.582937
$47$	9.12845	1.33152	0.665760	−	0.746166i	$-0.268107\pi$
$47$	9.12845	1.33152	0.665760	+	0.746166i	$0.268107\pi$
$48$	0	0
$49$	0	0
$50$	− 4.63792i	− 0.655901i
$51$	0	0
$52$	0.649659i	0.0900914i
$53$	− 10.3178i	− 1.41726i	−0.705580	−	0.708630i	$-0.749313\pi$
$53$	− 10.3178i	− 1.41726i	0.705580	−	0.708630i	$-0.250687\pi$
$54$	0	0
$55$	− 0.722325i	− 0.0973983i
$56$	0	0
$57$	0	0
$58$	1.37199	0.180151
$59$	7.26508	0.945833	0.472916	−	0.881107i	$-0.343201\pi$
$59$	7.26508	0.945833	0.472916	+	0.881107i	$0.343201\pi$
$60$	0	0
$61$	− 0.865467i	− 0.110812i	−0.998464	−	0.0554059i	$-0.982355\pi$
$61$	− 0.865467i	− 0.110812i	0.998464	−	0.0554059i	$-0.0176453\pi$
$62$	−7.11169	−0.903186
$63$	0	0
$64$	−1.00000	−0.125000
$65$	− 0.390920i	− 0.0484876i
$66$	0	0
$67$	−9.37779	−1.14568	−0.572840	−	0.819668i	$-0.694158\pi$
$67$	−9.37779	−1.14568	−0.572840	+	0.819668i	$0.694158\pi$
$68$	−0.564050	−0.0684011
$69$	0	0
$70$	0	0
$71$	− 4.28897i	− 0.509007i	−0.967072	−	0.254504i	$-0.918088\pi$
$71$	− 4.28897i	− 0.509007i	0.967072	−	0.254504i	$-0.0819121\pi$
$72$	0	0
$73$	15.7896i	1.84803i	0.382356	+	0.924015i	$0.375113\pi$
$73$	15.7896i	1.84803i	−0.382356	+	0.924015i	$0.624887\pi$
$74$	5.17738i	0.601858i
$75$	0	0
$76$	− 0.249245i	− 0.0285904i
$77$	0	0
$78$	0	0
$79$	12.1462	1.36656	0.683279	−	0.730157i	$-0.260553\pi$
$79$	12.1462	1.36656	0.683279	+	0.730157i	$0.260553\pi$
$80$	0.601731	0.0672755
$81$	0	0
$82$	7.32326i	0.808718i
$83$	5.58632	0.613178	0.306589	−	0.951842i	$-0.400812\pi$
$83$	5.58632	0.613178	0.306589	+	0.951842i	$0.400812\pi$
$84$	0	0
$85$	0.339406	0.0368138
$86$	9.36414i	1.00976i
$87$	0	0
$88$	−1.20041	−0.127964
$89$	7.62781	0.808547	0.404273	−	0.914638i	$-0.367524\pi$
$89$	7.62781	0.808547	0.404273	+	0.914638i	$0.367524\pi$
$90$	0	0
$91$	0	0
$92$	− 3.95367i	− 0.412198i
$93$	0	0
$94$	9.12845i	0.941528i
$95$	0.149978i	0.0153875i
$96$	0	0
$97$	13.2012i	1.34038i	0.742191	+	0.670189i	$0.233787\pi$
$97$	13.2012i	1.34038i	−0.742191	+	0.670189i	$0.766213\pi$
$98$	0	0
$99$	0	0
$100$	4.63792	0.463792
$101$	−5.20842	−0.518257	−0.259129	−	0.965843i	$-0.583435\pi$
$101$	−5.20842	−0.518257	−0.259129	+	0.965843i	$0.583435\pi$
$102$	0	0
$103$	6.31086i	0.621828i	0.950438	+	0.310914i	$0.100635\pi$
$103$	6.31086i	0.621828i	−0.950438	+	0.310914i	$0.899365\pi$
$104$	−0.649659	−0.0637043
$105$	0	0
$106$	10.3178	1.00215
$107$	− 14.7551i	− 1.42643i	−0.700948	−	0.713213i	$-0.747239\pi$
$107$	− 14.7551i	− 1.42643i	0.700948	−	0.713213i	$-0.252761\pi$
$108$	0	0
$109$	−1.94812	−0.186596	−0.0932982	−	0.995638i	$-0.529741\pi$
$109$	−1.94812	−0.186596	−0.0932982	+	0.995638i	$0.529741\pi$
$110$	0.722325	0.0688710
$111$	0	0
$112$	0	0
$113$	8.35305i	0.785789i	0.919584	+	0.392894i	$0.128526\pi$
$113$	8.35305i	0.785789i	−0.919584	+	0.392894i	$0.871474\pi$
$114$	0	0
$115$	2.37904i	0.221847i
$116$	1.37199i	0.127386i
$117$	0	0
$118$	7.26508i	0.668805i
$119$	0	0
$120$	0	0
$121$	9.55901	0.869001
$122$	0.865467	0.0783557
$123$	0	0
$124$	− 7.11169i	− 0.638649i
$125$	−5.79943	−0.518717
$126$	0	0
$127$	1.09410	0.0970861	0.0485431	−	0.998821i	$-0.484542\pi$
$127$	1.09410	0.0970861	0.0485431	+	0.998821i	$0.484542\pi$
$128$	− 1.00000i	− 0.0883883i
$129$	0	0
$130$	0.390920	0.0342859
$131$	−6.60896	−0.577427	−0.288714	−	0.957415i	$-0.593228\pi$
$131$	−6.60896	−0.577427	−0.288714	+	0.957415i	$0.593228\pi$
$132$	0	0
$133$	0	0
$134$	− 9.37779i	− 0.810117i
$135$	0	0
$136$	− 0.564050i	− 0.0483669i
$137$	− 9.61488i	− 0.821455i	−0.911758	−	0.410727i	$-0.865275\pi$
$137$	− 9.61488i	− 0.821455i	0.911758	−	0.410727i	$-0.134725\pi$
$138$	0	0
$139$	1.95858i	0.166125i	0.996544	+	0.0830623i	$0.0264701\pi$
$139$	1.95858i	0.166125i	−0.996544	+	0.0830623i	$0.973530\pi$
$140$	0	0
$141$	0	0
$142$	4.28897	0.359923
$143$	−0.779858	−0.0652150
$144$	0	0
$145$	− 0.825566i	− 0.0685595i
$146$	−15.7896	−1.30675
$147$	0	0
$148$	−5.17738	−0.425578
$149$	16.4159i	1.34484i	0.740168	+	0.672422i	$0.234746\pi$
$149$	16.4159i	1.34484i	−0.740168	+	0.672422i	$0.765254\pi$
$150$	0	0
$151$	−7.35039	−0.598166	−0.299083	−	0.954227i	$-0.596681\pi$
$151$	−7.35039	−0.598166	−0.299083	+	0.954227i	$0.596681\pi$
$152$	0.249245	0.0202164
$153$	0	0
$154$	0	0
$155$	4.27932i	0.343724i
$156$	0	0
$157$	13.3202i	1.06307i	0.847036	+	0.531535i	$0.178385\pi$
$157$	13.3202i	1.06307i	−0.847036	+	0.531535i	$0.821615\pi$
$158$	12.1462i	0.966303i
$159$	0	0
$160$	0.601731i	0.0475710i
$161$	0	0
$162$	0	0
$163$	11.5302	0.903113	0.451556	−	0.892243i	$-0.350869\pi$
$163$	11.5302	0.903113	0.451556	+	0.892243i	$0.350869\pi$
$164$	−7.32326	−0.571850
$165$	0	0
$166$	5.58632i	0.433582i
$167$	25.7937	1.99598	0.997990	−	0.0633754i	$-0.0201865\pi$
$167$	25.7937	1.99598	0.997990	+	0.0633754i	$0.0201865\pi$
$168$	0	0
$169$	12.5779	0.967534
$170$	0.339406i	0.0260313i
$171$	0	0
$172$	−9.36414	−0.714009
$173$	4.04800	0.307764	0.153882	−	0.988089i	$-0.450822\pi$
$173$	4.04800	0.307764	0.153882	+	0.988089i	$0.450822\pi$
$174$	0	0
$175$	0	0
$176$	− 1.20041i	− 0.0904845i
$177$	0	0
$178$	7.62781i	0.571729i
$179$	− 8.94172i	− 0.668336i	−0.942514	−	0.334168i	$-0.891545\pi$
$179$	− 8.94172i	− 0.668336i	0.942514	−	0.334168i	$-0.108455\pi$
$180$	0	0
$181$	1.77776i	0.132140i	0.997815	+	0.0660699i	$0.0210460\pi$
$181$	1.77776i	0.132140i	−0.997815	+	0.0660699i	$0.978954\pi$
$182$	0	0
$183$	0	0
$184$	3.95367	0.291468
$185$	3.11539	0.229048
$186$	0	0
$187$	− 0.677092i	− 0.0495139i
$188$	−9.12845	−0.665760
$189$	0	0
$190$	−0.149978	−0.0108806
$191$	− 18.7745i	− 1.35848i	−0.733918	−	0.679238i	$-0.762310\pi$
$191$	− 18.7745i	− 1.35848i	0.733918	−	0.679238i	$-0.237690\pi$
$192$	0	0
$193$	−7.34315	−0.528571	−0.264286	−	0.964444i	$-0.585136\pi$
$193$	−7.34315	−0.528571	−0.264286	+	0.964444i	$0.585136\pi$
$194$	−13.2012	−0.947790
$195$	0	0
$196$	0	0
$197$	14.8593i	1.05868i	0.848409	+	0.529342i	$0.177561\pi$
$197$	14.8593i	1.05868i	−0.848409	+	0.529342i	$0.822439\pi$
$198$	0	0
$199$	14.9366i	1.05883i	0.848364	+	0.529413i	$0.177588\pi$
$199$	14.9366i	1.05883i	−0.848364	+	0.529413i	$0.822412\pi$
$200$	4.63792i	0.327950i
$201$	0	0
$202$	− 5.20842i	− 0.366463i
$203$	0	0
$204$	0	0
$205$	4.40663	0.307772
$206$	−6.31086	−0.439699
$207$	0	0
$208$	− 0.649659i	− 0.0450457i
$209$	0.299197	0.0206959
$210$	0	0
$211$	−6.38537	−0.439587	−0.219794	−	0.975546i	$-0.570538\pi$
$211$	−6.38537	−0.439587	−0.219794	+	0.975546i	$0.570538\pi$
$212$	10.3178i	0.708630i
$213$	0	0
$214$	14.7551	1.00864
$215$	5.63469	0.384283
$216$	0	0
$217$	0	0
$218$	− 1.94812i	− 0.131944i
$219$	0	0
$220$	0.722325i	0.0486991i
$221$	− 0.366440i	− 0.0246494i
$222$	0	0
$223$	13.1867i	0.883047i	0.897250	+	0.441523i	$0.145562\pi$
$223$	13.1867i	0.883047i	−0.897250	+	0.441523i	$0.854438\pi$
$224$	0	0
$225$	0	0
$226$	−8.35305	−0.555637
$227$	6.70481	0.445014	0.222507	−	0.974931i	$-0.428576\pi$
$227$	6.70481	0.445014	0.222507	+	0.974931i	$0.428576\pi$
$228$	0	0
$229$	14.7600i	0.975368i	0.873020	+	0.487684i	$0.162158\pi$
$229$	14.7600i	0.975368i	−0.873020	+	0.487684i	$0.837842\pi$
$230$	−2.37904	−0.156870
$231$	0	0
$232$	−1.37199	−0.0900753
$233$	− 20.9327i	− 1.37135i	−0.727910	−	0.685673i	$-0.759508\pi$
$233$	− 20.9327i	− 1.37135i	0.727910	−	0.685673i	$-0.240492\pi$
$234$	0	0
$235$	5.49287	0.358315
$236$	−7.26508	−0.472916
$237$	0	0
$238$	0	0
$239$	15.3681i	0.994082i	0.867727	+	0.497041i	$0.165580\pi$
$239$	15.3681i	0.994082i	−0.867727	+	0.497041i	$0.834420\pi$
$240$	0	0
$241$	− 13.4328i	− 0.865283i	−0.901566	−	0.432641i	$-0.857582\pi$
$241$	− 13.4328i	− 0.865283i	0.901566	−	0.432641i	$-0.142418\pi$
$242$	9.55901i	0.614476i
$243$	0	0
$244$	0.865467i	0.0554059i
$245$	0	0
$246$	0	0
$247$	0.161924	0.0103030
$248$	7.11169	0.451593
$249$	0	0
$250$	− 5.79943i	− 0.366788i
$251$	−3.61325	−0.228066	−0.114033	−	0.993477i	$-0.536377\pi$
$251$	−3.61325	−0.228066	−0.114033	+	0.993477i	$0.536377\pi$
$252$	0	0
$253$	4.74603	0.298381
$254$	1.09410i	0.0686502i
$255$	0	0
$256$	1.00000	0.0625000
$257$	−31.3932	−1.95825	−0.979126	−	0.203254i	$-0.934848\pi$
$257$	−31.3932	−1.95825	−0.979126	+	0.203254i	$0.934848\pi$
$258$	0	0
$259$	0	0
$260$	0.390920i	0.0242438i
$261$	0	0
$262$	− 6.60896i	− 0.408303i
$263$	3.96187i	0.244300i	0.992512	+	0.122150i	$0.0389788\pi$
$263$	3.96187i	0.244300i	−0.992512	+	0.122150i	$0.961021\pi$
$264$	0	0
$265$	− 6.20854i	− 0.381388i
$266$	0	0
$267$	0	0
$268$	9.37779	0.572840
$269$	−21.5509	−1.31398	−0.656990	−	0.753899i	$-0.728171\pi$
$269$	−21.5509	−1.31398	−0.656990	+	0.753899i	$0.728171\pi$
$270$	0	0
$271$	− 6.18603i	− 0.375774i	−0.982191	−	0.187887i	$-0.939836\pi$
$271$	− 6.18603i	− 0.375774i	0.982191	−	0.187887i	$-0.0601639\pi$
$272$	0.564050	0.0342005
$273$	0	0
$274$	9.61488	0.580856
$275$	5.56742i	0.335728i
$276$	0	0
$277$	−11.1419	−0.669451	−0.334725	−	0.942316i	$-0.608644\pi$
$277$	−11.1419	−0.669451	−0.334725	+	0.942316i	$0.608644\pi$
$278$	−1.95858	−0.117468
$279$	0	0
$280$	0	0
$281$	15.4122i	0.919413i	0.888071	+	0.459707i	$0.152045\pi$
$281$	15.4122i	0.919413i	−0.888071	+	0.459707i	$0.847955\pi$
$282$	0	0
$283$	− 0.871470i	− 0.0518035i	−0.999664	−	0.0259018i	$-0.991754\pi$
$283$	− 0.871470i	− 0.0518035i	0.999664	−	0.0259018i	$-0.00824571\pi$
$284$	4.28897i	0.254504i
$285$	0	0
$286$	− 0.779858i	− 0.0461140i
$287$	0	0
$288$	0	0
$289$	−16.6818	−0.981285
$290$	0.825566	0.0484789
$291$	0	0
$292$	− 15.7896i	− 0.924015i
$293$	26.3170	1.53745	0.768727	−	0.639577i	$-0.220891\pi$
$293$	26.3170	1.53745	0.768727	+	0.639577i	$0.220891\pi$
$294$	0	0
$295$	4.37162	0.254526
$296$	− 5.17738i	− 0.300929i
$297$	0	0
$298$	−16.4159	−0.950948
$299$	2.56853	0.148542
$300$	0	0
$301$	0	0
$302$	− 7.35039i	− 0.422967i
$303$	0	0
$304$	0.249245i	0.0142952i
$305$	− 0.520778i	− 0.0298197i
$306$	0	0
$307$	− 25.0183i	− 1.42787i	−0.700211	−	0.713936i	$-0.746911\pi$
$307$	− 25.0183i	− 1.42787i	0.700211	−	0.713936i	$-0.253089\pi$
$308$	0	0
$309$	0	0
$310$	−4.27932	−0.243049
$311$	27.2433	1.54483	0.772413	−	0.635120i	$-0.219050\pi$
$311$	27.2433	1.54483	0.772413	+	0.635120i	$0.219050\pi$
$312$	0	0
$313$	0.936585i	0.0529389i	0.999650	+	0.0264694i	$0.00842647\pi$
$313$	0.936585i	0.0529389i	−0.999650	+	0.0264694i	$0.991574\pi$
$314$	−13.3202	−0.751704
$315$	0	0
$316$	−12.1462	−0.683279
$317$	− 8.93218i	− 0.501681i	−0.968028	−	0.250841i	$-0.919293\pi$
$317$	− 8.93218i	− 0.501681i	0.968028	−	0.250841i	$-0.0807070\pi$
$318$	0	0
$319$	−1.64695	−0.0922114
$320$	−0.601731	−0.0336378
$321$	0	0
$322$	0	0
$323$	0.140587i	0.00782245i
$324$	0	0
$325$	3.01306i	0.167135i
$326$	11.5302i	0.638597i
$327$	0	0
$328$	− 7.32326i	− 0.404359i
$329$	0	0
$330$	0	0
$331$	−13.6805	−0.751949	−0.375974	−	0.926630i	$-0.622692\pi$
$331$	−13.6805	−0.751949	−0.375974	+	0.926630i	$0.622692\pi$
$332$	−5.58632	−0.306589
$333$	0	0
$334$	25.7937i	1.41137i
$335$	−5.64290	−0.308305
$336$	0	0
$337$	−11.3662	−0.619157	−0.309578	−	0.950874i	$-0.600188\pi$
$337$	−11.3662	−0.619157	−0.309578	+	0.950874i	$0.600188\pi$
$338$	12.5779i	0.684150i
$339$	0	0
$340$	−0.339406	−0.0184069
$341$	8.53696	0.462303
$342$	0	0
$343$	0	0
$344$	− 9.36414i	− 0.504881i
$345$	0	0
$346$	4.04800i	0.217622i
$347$	15.1885i	0.815360i	0.913125	+	0.407680i	$0.133662\pi$
$347$	15.1885i	0.815360i	−0.913125	+	0.407680i	$0.866338\pi$
$348$	0	0
$349$	1.64664i	0.0881425i	0.999028	+	0.0440713i	$0.0140329\pi$
$349$	1.64664i	0.0881425i	−0.999028	+	0.0440713i	$0.985967\pi$
$350$	0	0
$351$	0	0
$352$	1.20041	0.0639822
$353$	−18.2235	−0.969937	−0.484968	−	0.874532i	$-0.661169\pi$
$353$	−18.2235	−0.969937	−0.484968	+	0.874532i	$0.661169\pi$
$354$	0	0
$355$	− 2.58081i	− 0.136975i
$356$	−7.62781	−0.404273
$357$	0	0
$358$	8.94172	0.472585
$359$	− 27.7530i	− 1.46475i	−0.680902	−	0.732374i	$-0.738412\pi$
$359$	− 27.7530i	− 1.46475i	0.680902	−	0.732374i	$-0.261588\pi$
$360$	0	0
$361$	18.9379	0.996730
$362$	−1.77776	−0.0934369
$363$	0	0
$364$	0	0
$365$	9.50107i	0.497309i
$366$	0	0
$367$	− 30.5271i	− 1.59350i	−0.604309	−	0.796750i	$-0.706551\pi$
$367$	− 30.5271i	− 1.59350i	0.604309	−	0.796750i	$-0.293449\pi$
$368$	3.95367i	0.206099i
$369$	0	0
$370$	3.11539i	0.161961i
$371$	0	0
$372$	0	0
$373$	−12.2467	−0.634112	−0.317056	−	0.948407i	$-0.602694\pi$
$373$	−12.2467	−0.634112	−0.317056	+	0.948407i	$0.602694\pi$
$374$	0.677092	0.0350116
$375$	0	0
$376$	− 9.12845i	− 0.470764i
$377$	−0.891322	−0.0459054
$378$	0	0
$379$	3.76316	0.193301	0.0966503	−	0.995318i	$-0.469187\pi$
$379$	3.76316	0.193301	0.0966503	+	0.995318i	$0.469187\pi$
$380$	− 0.149978i	− 0.00769373i
$381$	0	0
$382$	18.7745	0.960588
$383$	−25.3157	−1.29357	−0.646787	−	0.762671i	$-0.723888\pi$
$383$	−25.3157	−1.29357	−0.646787	+	0.762671i	$0.723888\pi$
$384$	0	0
$385$	0	0
$386$	− 7.34315i	− 0.373756i
$387$	0	0
$388$	− 13.2012i	− 0.670189i
$389$	− 30.7667i	− 1.55993i	−0.625822	−	0.779966i	$-0.715236\pi$
$389$	− 30.7667i	− 1.55993i	0.625822	−	0.779966i	$-0.284764\pi$
$390$	0	0
$391$	2.23007i	0.112779i
$392$	0	0
$393$	0	0
$394$	−14.8593	−0.748602
$395$	7.30876	0.367744
$396$	0	0
$397$	− 34.8793i	− 1.75054i	−0.483631	−	0.875272i	$-0.660682\pi$
$397$	− 34.8793i	− 1.75054i	0.483631	−	0.875272i	$-0.339318\pi$
$398$	−14.9366	−0.748703
$399$	0	0
$400$	−4.63792	−0.231896
$401$	31.8975i	1.59288i	0.604716	+	0.796441i	$0.293287\pi$
$401$	31.8975i	1.59288i	−0.604716	+	0.796441i	$0.706713\pi$
$402$	0	0
$403$	4.62017	0.230147
$404$	5.20842	0.259129
$405$	0	0
$406$	0	0
$407$	− 6.21499i	− 0.308065i
$408$	0	0
$409$	− 26.3923i	− 1.30502i	−0.757782	−	0.652508i	$-0.773717\pi$
$409$	− 26.3923i	− 1.30502i	0.757782	−	0.652508i	$-0.226283\pi$
$410$	4.40663i	0.217628i
$411$	0	0
$412$	− 6.31086i	− 0.310914i
$413$	0	0
$414$	0	0
$415$	3.36146	0.165008
$416$	0.649659	0.0318521
$417$	0	0
$418$	0.299197i	0.0146342i
$419$	23.9692	1.17097	0.585485	−	0.810683i	$-0.300904\pi$
$419$	23.9692	1.17097	0.585485	+	0.810683i	$0.300904\pi$
$420$	0	0
$421$	−15.5279	−0.756782	−0.378391	−	0.925646i	$-0.623523\pi$
$421$	−15.5279	−0.756782	−0.378391	+	0.925646i	$0.623523\pi$
$422$	− 6.38537i	− 0.310835i
$423$	0	0
$424$	−10.3178	−0.501077
$425$	−2.61602	−0.126896
$426$	0	0
$427$	0	0
$428$	14.7551i	0.713213i
$429$	0	0
$430$	5.63469i	0.271729i
$431$	15.3414i	0.738967i	0.929237	+	0.369484i	$0.120465\pi$
$431$	15.3414i	0.738967i	−0.929237	+	0.369484i	$0.879535\pi$
$432$	0	0
$433$	− 13.1662i	− 0.632727i	−0.948638	−	0.316364i	$-0.897538\pi$
$433$	− 13.1662i	− 0.632727i	0.948638	−	0.316364i	$-0.102462\pi$
$434$	0	0
$435$	0	0
$436$	1.94812	0.0932982
$437$	−0.985432	−0.0471396
$438$	0	0
$439$	19.7473i	0.942486i	0.882003	+	0.471243i	$0.156195\pi$
$439$	19.7473i	0.942486i	−0.882003	+	0.471243i	$0.843805\pi$
$440$	−0.722325	−0.0344355
$441$	0	0
$442$	0.366440	0.0174298
$443$	− 24.2237i	− 1.15090i	−0.817836	−	0.575452i	$-0.804826\pi$
$443$	− 24.2237i	− 1.15090i	0.817836	−	0.575452i	$-0.195174\pi$
$444$	0	0
$445$	4.58989	0.217582
$446$	−13.1867	−0.624408
$447$	0	0
$448$	0	0
$449$	16.9561i	0.800208i	0.916470	+	0.400104i	$0.131026\pi$
$449$	16.9561i	0.800208i	−0.916470	+	0.400104i	$0.868974\pi$
$450$	0	0
$451$	− 8.79093i	− 0.413949i
$452$	− 8.35305i	− 0.392894i
$453$	0	0
$454$	6.70481i	0.314672i
$455$	0	0
$456$	0	0
$457$	−5.24264	−0.245240	−0.122620	−	0.992454i	$-0.539130\pi$
$457$	−5.24264	−0.245240	−0.122620	+	0.992454i	$0.539130\pi$
$458$	−14.7600	−0.689689
$459$	0	0
$460$	− 2.37904i	− 0.110923i
$461$	−18.7628	−0.873869	−0.436935	−	0.899493i	$-0.643936\pi$
$461$	−18.7628	−0.873869	−0.436935	+	0.899493i	$0.643936\pi$
$462$	0	0
$463$	−3.27122	−0.152027	−0.0760133	−	0.997107i	$-0.524219\pi$
$463$	−3.27122	−0.152027	−0.0760133	+	0.997107i	$0.524219\pi$
$464$	− 1.37199i	− 0.0636928i
$465$	0	0
$466$	20.9327	0.969688
$467$	35.0751	1.62308	0.811541	−	0.584295i	$-0.198629\pi$
$467$	35.0751	1.62308	0.811541	+	0.584295i	$0.198629\pi$
$468$	0	0
$469$	0	0
$470$	5.49287i	0.253367i
$471$	0	0
$472$	− 7.26508i	− 0.334402i
$473$	− 11.2408i	− 0.516854i
$474$	0	0
$475$	− 1.15598i	− 0.0530399i
$476$	0	0
$477$	0	0
$478$	−15.3681	−0.702922
$479$	7.79415	0.356124	0.178062	−	0.984019i	$-0.443017\pi$
$479$	7.79415	0.356124	0.178062	+	0.984019i	$0.443017\pi$
$480$	0	0
$481$	− 3.36353i	− 0.153364i
$482$	13.4328	0.611847
$483$	0	0
$484$	−9.55901	−0.434500
$485$	7.94356i	0.360699i
$486$	0	0
$487$	29.1383	1.32038	0.660191	−	0.751098i	$-0.270475\pi$
$487$	29.1383	1.32038	0.660191	+	0.751098i	$0.270475\pi$
$488$	−0.865467	−0.0391779
$489$	0	0
$490$	0	0
$491$	− 19.7492i	− 0.891268i	−0.895215	−	0.445634i	$-0.852978\pi$
$491$	− 19.7492i	− 0.891268i	0.895215	−	0.445634i	$-0.147022\pi$
$492$	0	0
$493$	− 0.773868i	− 0.0348533i
$494$	0.161924i	0.00728531i
$495$	0	0
$496$	7.11169i	0.319324i
$497$	0	0
$498$	0	0
$499$	−31.6108	−1.41509	−0.707546	−	0.706667i	$-0.750198\pi$
$499$	−31.6108	−1.41509	−0.707546	+	0.706667i	$0.750198\pi$
$500$	5.79943	0.259358
$501$	0	0
$502$	− 3.61325i	− 0.161267i
$503$	−13.8811	−0.618930	−0.309465	−	0.950911i	$-0.600150\pi$
$503$	−13.8811	−0.618930	−0.309465	+	0.950911i	$0.600150\pi$
$504$	0	0
$505$	−3.13407	−0.139464
$506$	4.74603i	0.210987i
$507$	0	0
$508$	−1.09410	−0.0485431
$509$	14.6657	0.650047	0.325023	−	0.945706i	$-0.394628\pi$
$509$	14.6657	0.650047	0.325023	+	0.945706i	$0.394628\pi$
$510$	0	0
$511$	0	0
$512$	1.00000i	0.0441942i
$513$	0	0
$514$	− 31.3932i	− 1.38469i
$515$	3.79744i	0.167335i
$516$	0	0
$517$	− 10.9579i	− 0.481928i
$518$	0	0
$519$	0	0
$520$	−0.390920	−0.0171430
$521$	−39.5873	−1.73435	−0.867175	−	0.498004i	$-0.834066\pi$
$521$	−39.5873	−1.73435	−0.867175	+	0.498004i	$0.834066\pi$
$522$	0	0
$523$	1.68035i	0.0734767i	0.999325	+	0.0367384i	$0.0116968\pi$
$523$	1.68035i	0.0734767i	−0.999325	+	0.0367384i	$0.988303\pi$
$524$	6.60896	0.288714
$525$	0	0
$526$	−3.96187	−0.172746
$527$	4.01135i	0.174737i
$528$	0	0
$529$	7.36850	0.320370
$530$	6.20854	0.269682
$531$	0	0
$532$	0	0
$533$	− 4.75762i	− 0.206075i
$534$	0	0
$535$	− 8.87857i	− 0.383854i
$536$	9.37779i	0.405059i
$537$	0	0
$538$	− 21.5509i	− 0.929125i
$539$	0	0
$540$	0	0
$541$	−17.2062	−0.739753	−0.369876	−	0.929081i	$-0.620600\pi$
$541$	−17.2062	−0.739753	−0.369876	+	0.929081i	$0.620600\pi$
$542$	6.18603	0.265713
$543$	0	0
$544$	0.564050i	0.0241834i
$545$	−1.17225	−0.0502135
$546$	0	0
$547$	−43.6062	−1.86447	−0.932233	−	0.361859i	$-0.882142\pi$
$547$	−43.6062	−1.86447	−0.932233	+	0.361859i	$0.882142\pi$
$548$	9.61488i	0.410727i
$549$	0	0
$550$	−5.56742	−0.237395
$551$	0.341961	0.0145680
$552$	0	0
$553$	0	0
$554$	− 11.1419i	− 0.473373i
$555$	0	0
$556$	− 1.95858i	− 0.0830623i
$557$	− 37.4256i	− 1.58577i	−0.609369	−	0.792886i	$-0.708577\pi$
$557$	− 37.4256i	− 1.58577i	0.609369	−	0.792886i	$-0.291423\pi$
$558$	0	0
$559$	− 6.08350i	− 0.257304i
$560$	0	0
$561$	0	0
$562$	−15.4122	−0.650123
$563$	25.6620	1.08152	0.540762	−	0.841176i	$-0.318136\pi$
$563$	25.6620	1.08152	0.540762	+	0.841176i	$0.318136\pi$
$564$	0	0
$565$	5.02629i	0.211457i
$566$	0.871470	0.0366306
$567$	0	0
$568$	−4.28897	−0.179961
$569$	13.2717i	0.556380i	0.960526	+	0.278190i	$0.0897345\pi$
$569$	13.2717i	0.556380i	−0.960526	+	0.278190i	$0.910265\pi$
$570$	0	0
$571$	20.7867	0.869897	0.434948	−	0.900455i	$-0.356767\pi$
$571$	20.7867	0.869897	0.434948	+	0.900455i	$0.356767\pi$
$572$	0.779858	0.0326075
$573$	0	0
$574$	0	0
$575$	− 18.3368i	− 0.764697i
$576$	0	0
$577$	− 16.1318i	− 0.671576i	−0.941938	−	0.335788i	$-0.890997\pi$
$577$	− 16.1318i	− 0.671576i	0.941938	−	0.335788i	$-0.109003\pi$
$578$	− 16.6818i	− 0.693873i
$579$	0	0
$580$	0.825566i	0.0342798i
$581$	0	0
$582$	0	0
$583$	−12.3856	−0.512960
$584$	15.7896	0.653377
$585$	0	0
$586$	26.3170i	1.08714i
$587$	18.0600	0.745417	0.372708	−	0.927949i	$-0.378429\pi$
$587$	18.0600	0.745417	0.372708	+	0.927949i	$0.378429\pi$
$588$	0	0
$589$	−1.77255	−0.0730368
$590$	4.37162i	0.179977i
$591$	0	0
$592$	5.17738	0.212789
$593$	6.69679	0.275004	0.137502	−	0.990501i	$-0.456093\pi$
$593$	6.69679	0.275004	0.137502	+	0.990501i	$0.456093\pi$
$594$	0	0
$595$	0	0
$596$	− 16.4159i	− 0.672422i
$597$	0	0
$598$	2.56853i	0.105035i
$599$	− 3.28379i	− 0.134172i	−0.997747	−	0.0670860i	$-0.978630\pi$
$599$	− 3.28379i	− 0.134172i	0.997747	−	0.0670860i	$-0.0213702\pi$
$600$	0	0
$601$	− 4.59868i	− 0.187584i	−0.995592	−	0.0937920i	$-0.970101\pi$
$601$	− 4.59868i	− 0.187584i	0.995592	−	0.0937920i	$-0.0298989\pi$
$602$	0	0
$603$	0	0
$604$	7.35039	0.299083
$605$	5.75195	0.233850
$606$	0	0
$607$	− 27.1633i	− 1.10252i	−0.834333	−	0.551261i	$-0.814147\pi$
$607$	− 27.1633i	− 1.10252i	0.834333	−	0.551261i	$-0.185853\pi$
$608$	−0.249245	−0.0101082
$609$	0	0
$610$	0.520778	0.0210857
$611$	− 5.93037i	− 0.239917i
$612$	0	0
$613$	32.9907	1.33248	0.666241	−	0.745736i	$-0.267902\pi$
$613$	32.9907	1.33248	0.666241	+	0.745736i	$0.267902\pi$
$614$	25.0183	1.00966
$615$	0	0
$616$	0	0
$617$	20.1479i	0.811125i	0.914067	+	0.405563i	$0.132924\pi$
$617$	20.1479i	0.811125i	−0.914067	+	0.405563i	$0.867076\pi$
$618$	0	0
$619$	− 16.0966i	− 0.646975i	−0.946232	−	0.323488i	$-0.895145\pi$
$619$	− 16.0966i	− 0.646975i	0.946232	−	0.323488i	$-0.104855\pi$
$620$	− 4.27932i	− 0.171862i
$621$	0	0
$622$	27.2433i	1.09236i
$623$	0	0
$624$	0	0
$625$	19.6999	0.787996
$626$	−0.936585	−0.0374334
$627$	0	0
$628$	− 13.3202i	− 0.531535i
$629$	2.92030	0.116440
$630$	0	0
$631$	−2.81684	−0.112137	−0.0560684	−	0.998427i	$-0.517856\pi$
$631$	−2.81684	−0.112137	−0.0560684	+	0.998427i	$0.517856\pi$
$632$	− 12.1462i	− 0.483151i
$633$	0	0
$634$	8.93218	0.354742
$635$	0.658356	0.0261261
$636$	0	0
$637$	0	0
$638$	− 1.64695i	− 0.0652033i
$639$	0	0
$640$	− 0.601731i	− 0.0237855i
$641$	17.2433i	0.681068i	0.940232	+	0.340534i	$0.110608\pi$
$641$	17.2433i	0.681068i	−0.940232	+	0.340534i	$0.889392\pi$
$642$	0	0
$643$	43.9469i	1.73310i	0.499093	+	0.866549i	$0.333667\pi$
$643$	43.9469i	1.73310i	−0.499093	+	0.866549i	$0.666333\pi$
$644$	0	0
$645$	0	0
$646$	−0.140587	−0.00553131
$647$	15.4124	0.605922	0.302961	−	0.953003i	$-0.402025\pi$
$647$	15.4124	0.605922	0.302961	+	0.953003i	$0.402025\pi$
$648$	0	0
$649$	− 8.72109i	− 0.342333i
$650$	−3.01306	−0.118182
$651$	0	0
$652$	−11.5302	−0.451556
$653$	34.4687i	1.34886i	0.738337	+	0.674432i	$0.235611\pi$
$653$	34.4687i	1.34886i	−0.738337	+	0.674432i	$0.764389\pi$
$654$	0	0
$655$	−3.97681	−0.155387
$656$	7.32326	0.285925
$657$	0	0
$658$	0	0
$659$	− 16.2348i	− 0.632418i	−0.948690	−	0.316209i	$-0.897590\pi$
$659$	− 16.2348i	− 0.632418i	0.948690	−	0.316209i	$-0.102410\pi$
$660$	0	0
$661$	− 47.2554i	− 1.83802i	−0.394230	−	0.919012i	$-0.628989\pi$
$661$	− 47.2554i	− 1.83802i	0.394230	−	0.919012i	$-0.371011\pi$
$662$	− 13.6805i	− 0.531708i
$663$	0	0
$664$	− 5.58632i	− 0.216791i
$665$	0	0
$666$	0	0
$667$	5.42438	0.210033
$668$	−25.7937	−0.997990
$669$	0	0
$670$	− 5.64290i	− 0.218004i
$671$	−1.03892	−0.0401070
$672$	0	0
$673$	−16.2598	−0.626769	−0.313384	−	0.949626i	$-0.601463\pi$
$673$	−16.2598	−0.626769	−0.313384	+	0.949626i	$0.601463\pi$
$674$	− 11.3662i	− 0.437810i
$675$	0	0
$676$	−12.5779	−0.483767
$677$	12.9879	0.499165	0.249582	−	0.968354i	$-0.419707\pi$
$677$	12.9879	0.499165	0.249582	+	0.968354i	$0.419707\pi$
$678$	0	0
$679$	0	0
$680$	− 0.339406i	− 0.0130156i
$681$	0	0
$682$	8.53696i	0.326897i
$683$	30.7325i	1.17595i	0.808880	+	0.587974i	$0.200074\pi$
$683$	30.7325i	1.17595i	−0.808880	+	0.587974i	$0.799926\pi$
$684$	0	0
$685$	− 5.78557i	− 0.221055i
$686$	0	0
$687$	0	0
$688$	9.36414	0.357005
$689$	−6.70306	−0.255366
$690$	0	0
$691$	16.0781i	0.611638i	0.952090	+	0.305819i	$0.0989302\pi$
$691$	16.0781i	0.611638i	−0.952090	+	0.305819i	$0.901070\pi$
$692$	−4.04800	−0.153882
$693$	0	0
$694$	−15.1885	−0.576546
$695$	1.17854i	0.0447045i
$696$	0	0
$697$	4.13068	0.156461
$698$	−1.64664	−0.0623262
$699$	0	0
$700$	0	0
$701$	− 37.8263i	− 1.42868i	−0.699799	−	0.714339i	$-0.746727\pi$
$701$	− 37.8263i	− 1.42868i	0.699799	−	0.714339i	$-0.253273\pi$
$702$	0	0
$703$	1.29043i	0.0486697i
$704$	1.20041i	0.0452423i
$705$	0	0
$706$	− 18.2235i	− 0.685849i
$707$	0	0
$708$	0	0
$709$	−41.7113	−1.56650	−0.783251	−	0.621706i	$-0.786440\pi$
$709$	−41.7113	−1.56650	−0.783251	+	0.621706i	$0.786440\pi$
$710$	2.58081	0.0968559
$711$	0	0
$712$	− 7.62781i	− 0.285864i
$713$	−28.1173	−1.05300
$714$	0	0
$715$	−0.469265	−0.0175495
$716$	8.94172i	0.334168i
$717$	0	0
$718$	27.7530	1.03573
$719$	−2.06421	−0.0769820	−0.0384910	−	0.999259i	$-0.512255\pi$
$719$	−2.06421	−0.0769820	−0.0384910	+	0.999259i	$0.512255\pi$
$720$	0	0
$721$	0	0
$722$	18.9379i	0.704795i
$723$	0	0
$724$	− 1.77776i	− 0.0660699i
$725$	6.36316i	0.236322i
$726$	0	0
$727$	− 31.5303i	− 1.16939i	−0.811251	−	0.584697i	$-0.801213\pi$
$727$	− 31.5303i	− 1.16939i	0.811251	−	0.584697i	$-0.198787\pi$
$728$	0	0
$729$	0	0
$730$	−9.50107	−0.351650
$731$	5.28184	0.195356
$732$	0	0
$733$	40.4015i	1.49226i	0.665799	+	0.746131i	$0.268091\pi$
$733$	40.4015i	1.49226i	−0.665799	+	0.746131i	$0.731909\pi$
$734$	30.5271	1.12677
$735$	0	0
$736$	−3.95367	−0.145734
$737$	11.2572i	0.414665i
$738$	0	0
$739$	40.5607	1.49205	0.746025	−	0.665918i	$-0.231960\pi$
$739$	40.5607	1.49205	0.746025	+	0.665918i	$0.231960\pi$
$740$	−3.11539	−0.114524
$741$	0	0
$742$	0	0
$743$	− 29.9301i	− 1.09803i	−0.835812	−	0.549015i	$-0.815003\pi$
$743$	− 29.9301i	− 1.09803i	0.835812	−	0.549015i	$-0.184997\pi$
$744$	0	0
$745$	9.87796i	0.361900i
$746$	− 12.2467i	− 0.448385i
$747$	0	0
$748$	0.677092i	0.0247570i
$749$	0	0
$750$	0	0
$751$	−39.8892	−1.45558	−0.727790	−	0.685800i	$-0.759452\pi$
$751$	−39.8892	−1.45558	−0.727790	+	0.685800i	$0.759452\pi$
$752$	9.12845	0.332880
$753$	0	0
$754$	− 0.891322i	− 0.0324600i
$755$	−4.42296	−0.160968
$756$	0	0
$757$	−17.9258	−0.651525	−0.325762	−	0.945452i	$-0.605621\pi$
$757$	−17.9258	−0.651525	−0.325762	+	0.945452i	$0.605621\pi$
$758$	3.76316i	0.136684i
$759$	0	0
$760$	0.149978	0.00544029
$761$	−15.6453	−0.567142	−0.283571	−	0.958951i	$-0.591519\pi$
$761$	−15.6453	−0.567142	−0.283571	+	0.958951i	$0.591519\pi$
$762$	0	0
$763$	0	0
$764$	18.7745i	0.679238i
$765$	0	0
$766$	− 25.3157i	− 0.914695i
$767$	− 4.71982i	− 0.170423i
$768$	0	0
$769$	− 2.25067i	− 0.0811613i	−0.999176	−	0.0405806i	$-0.987079\pi$
$769$	− 2.25067i	− 0.0811613i	0.999176	−	0.0405806i	$-0.0129208\pi$
$770$	0	0
$771$	0	0
$772$	7.34315	0.264286
$773$	21.9049	0.787866	0.393933	−	0.919139i	$-0.371114\pi$
$773$	21.9049	0.787866	0.393933	+	0.919139i	$0.371114\pi$
$774$	0	0
$775$	− 32.9835i	− 1.18480i
$776$	13.2012	0.473895
$777$	0	0
$778$	30.7667	1.10304
$779$	1.82528i	0.0653976i
$780$	0	0
$781$	−5.14854	−0.184229
$782$	−2.23007	−0.0797470
$783$	0	0
$784$	0	0
$785$	8.01519i	0.286075i
$786$	0	0
$787$	− 16.8406i	− 0.600301i	−0.953892	−	0.300151i	$-0.902963\pi$
$787$	− 16.8406i	− 0.600301i	0.953892	−	0.300151i	$-0.0970369\pi$
$788$	− 14.8593i	− 0.529342i
$789$	0	0
$790$	7.30876i	0.260034i
$791$	0	0
$792$	0	0
$793$	−0.562258	−0.0199664
$794$	34.8793	1.23782
$795$	0	0
$796$	− 14.9366i	− 0.529413i
$797$	−13.8756	−0.491500	−0.245750	−	0.969333i	$-0.579034\pi$
$797$	−13.8756	−0.491500	−0.245750	+	0.969333i	$0.579034\pi$
$798$	0	0
$799$	5.14890	0.182155
$800$	− 4.63792i	− 0.163975i
$801$	0	0
$802$	−31.8975	−1.12634
$803$	18.9540	0.668872
$804$	0	0
$805$	0	0
$806$	4.62017i	0.162739i
$807$	0	0
$808$	5.20842i	0.183232i
$809$	− 16.0990i	− 0.566010i	−0.959118	−	0.283005i	$-0.908669\pi$
$809$	− 16.0990i	− 0.566010i	0.959118	−	0.283005i	$-0.0913313\pi$
$810$	0	0
$811$	− 8.57333i	− 0.301050i	−0.988606	−	0.150525i	$-0.951904\pi$
$811$	− 8.57333i	− 0.301050i	0.988606	−	0.150525i	$-0.0480965\pi$
$812$	0	0
$813$	0	0
$814$	6.21499	0.217835
$815$	6.93806	0.243030
$816$	0	0
$817$	2.33397i	0.0816551i
$818$	26.3923	0.922786
$819$	0	0
$820$	−4.40663	−0.153886
$821$	41.2485i	1.43958i	0.694191	+	0.719791i	$0.255762\pi$
$821$	41.2485i	1.43958i	−0.694191	+	0.719791i	$0.744238\pi$
$822$	0	0
$823$	−25.1482	−0.876611	−0.438305	−	0.898826i	$-0.644421\pi$
$823$	−25.1482	−0.876611	−0.438305	+	0.898826i	$0.644421\pi$
$824$	6.31086	0.219849
$825$	0	0
$826$	0	0
$827$	− 12.8906i	− 0.448249i	−0.974561	−	0.224125i	$-0.928048\pi$
$827$	− 12.8906i	− 0.448249i	0.974561	−	0.224125i	$-0.0719523\pi$
$828$	0	0
$829$	24.5664i	0.853228i	0.904434	+	0.426614i	$0.140294\pi$
$829$	24.5664i	0.853228i	−0.904434	+	0.426614i	$0.859706\pi$
$830$	3.36146i	0.116678i
$831$	0	0
$832$	0.649659i	0.0225229i
$833$	0	0
$834$	0	0
$835$	15.5209	0.537122
$836$	−0.299197	−0.0103479
$837$	0	0
$838$	23.9692i	0.828001i
$839$	20.4459	0.705872	0.352936	−	0.935647i	$-0.385183\pi$
$839$	20.4459	0.705872	0.352936	+	0.935647i	$0.385183\pi$
$840$	0	0
$841$	27.1177	0.935092
$842$	− 15.5279i	− 0.535126i
$843$	0	0
$844$	6.38537	0.219794
$845$	7.56853	0.260366
$846$	0	0
$847$	0	0
$848$	− 10.3178i	− 0.354315i
$849$	0	0
$850$	− 2.61602i	− 0.0897287i
$851$	20.4696i	0.701690i
$852$	0	0
$853$	3.15935i	0.108174i	0.998536	+	0.0540870i	$0.0172248\pi$
$853$	3.15935i	0.108174i	−0.998536	+	0.0540870i	$0.982775\pi$
$854$	0	0
$855$	0	0
$856$	−14.7551	−0.504318
$857$	−38.2349	−1.30608	−0.653040	−	0.757323i	$-0.726507\pi$
$857$	−38.2349	−1.30608	−0.653040	+	0.757323i	$0.726507\pi$
$858$	0	0
$859$	− 21.9370i	− 0.748480i	−0.927332	−	0.374240i	$-0.877904\pi$
$859$	− 21.9370i	− 0.748480i	0.927332	−	0.374240i	$-0.122096\pi$
$860$	−5.63469	−0.192141
$861$	0	0
$862$	−15.3414	−0.522529
$863$	− 1.54212i	− 0.0524942i	−0.999655	−	0.0262471i	$-0.991644\pi$
$863$	− 1.54212i	− 0.0524942i	0.999655	−	0.0262471i	$-0.00835568\pi$
$864$	0	0
$865$	2.43581	0.0828200
$866$	13.1662	0.447406
$867$	0	0
$868$	0	0
$869$	− 14.5805i	− 0.494610i
$870$	0	0
$871$	6.09236i	0.206432i
$872$	1.94812i	0.0659718i
$873$	0	0
$874$	− 0.985432i	− 0.0333327i
$875$	0	0
$876$	0	0
$877$	45.3364	1.53090	0.765451	−	0.643494i	$-0.222516\pi$
$877$	45.3364	1.53090	0.765451	+	0.643494i	$0.222516\pi$
$878$	−19.7473	−0.666439
$879$	0	0
$880$	− 0.722325i	− 0.0243496i
$881$	−31.3187	−1.05516	−0.527578	−	0.849507i	$-0.676900\pi$
$881$	−31.3187	−1.05516	−0.527578	+	0.849507i	$0.676900\pi$
$882$	0	0
$883$	31.9125	1.07394	0.536971	−	0.843601i	$-0.319568\pi$
$883$	31.9125	1.07394	0.536971	+	0.843601i	$0.319568\pi$
$884$	0.366440i	0.0123247i
$885$	0	0
$886$	24.2237	0.813811
$887$	43.0139	1.44426	0.722132	−	0.691755i	$-0.243162\pi$
$887$	43.0139	1.44426	0.722132	+	0.691755i	$0.243162\pi$
$888$	0	0
$889$	0	0
$890$	4.58989i	0.153853i
$891$	0	0
$892$	− 13.1867i	− 0.441523i
$893$	2.27522i	0.0761373i
$894$	0	0
$895$	− 5.38051i	− 0.179851i
$896$	0	0
$897$	0	0
$898$	−16.9561	−0.565832
$899$	9.75714	0.325419
$900$	0	0
$901$	− 5.81976i	− 0.193884i
$902$	8.79093	0.292706
$903$	0	0
$904$	8.35305	0.277818
$905$	1.06973i	0.0355591i
$906$	0	0
$907$	−40.6686	−1.35038	−0.675189	−	0.737645i	$-0.735938\pi$
$907$	−40.6686	−1.35038	−0.675189	+	0.737645i	$0.735938\pi$
$908$	−6.70481	−0.222507
$909$	0	0
$910$	0	0
$911$	− 16.4174i	− 0.543932i	−0.962307	−	0.271966i	$-0.912326\pi$
$911$	− 16.4174i	− 0.543932i	0.962307	−	0.271966i	$-0.0876739\pi$
$912$	0	0
$913$	− 6.70589i	− 0.221932i
$914$	− 5.24264i	− 0.173411i
$915$	0	0
$916$	− 14.7600i	− 0.487684i
$917$	0	0
$918$	0	0
$919$	6.36995	0.210125	0.105063	−	0.994466i	$-0.466496\pi$
$919$	6.36995	0.210125	0.105063	+	0.994466i	$0.466496\pi$
$920$	2.37904	0.0784348
$921$	0	0
$922$	− 18.7628i	− 0.617919i
$923$	−2.78637	−0.0917144
$924$	0	0
$925$	−24.0123	−0.789518
$926$	− 3.27122i	− 0.107499i
$927$	0	0
$928$	1.37199	0.0450376
$929$	−43.2708	−1.41967	−0.709834	−	0.704369i	$-0.751230\pi$
$929$	−43.2708	−1.41967	−0.709834	+	0.704369i	$0.751230\pi$
$930$	0	0
$931$	0	0
$932$	20.9327i	0.685673i
$933$	0	0
$934$	35.0751i	1.14769i
$935$	− 0.407427i	− 0.0133243i
$936$	0	0
$937$	41.8956i	1.36867i	0.729168	+	0.684335i	$0.239907\pi$
$937$	41.8956i	1.36867i	−0.729168	+	0.684335i	$0.760093\pi$
$938$	0	0
$939$	0	0
$940$	−5.49287	−0.179158
$941$	−39.7919	−1.29718	−0.648590	−	0.761138i	$-0.724641\pi$
$941$	−39.7919	−1.29718	−0.648590	+	0.761138i	$0.724641\pi$
$942$	0	0
$943$	28.9537i	0.942863i
$944$	7.26508	0.236458
$945$	0	0
$946$	11.2408	0.365471
$947$	− 20.0515i	− 0.651587i	−0.945441	−	0.325794i	$-0.894369\pi$
$947$	− 20.0515i	− 0.651587i	0.945441	−	0.325794i	$-0.105631\pi$
$948$	0	0
$949$	10.2578	0.332983
$950$	1.15598	0.0375049
$951$	0	0
$952$	0	0
$953$	26.4300i	0.856153i	0.903743	+	0.428076i	$0.140809\pi$
$953$	26.4300i	0.856153i	−0.903743	+	0.428076i	$0.859191\pi$
$954$	0	0
$955$	− 11.2972i	− 0.365569i
$956$	− 15.3681i	− 0.497041i
$957$	0	0
$958$	7.79415i	0.251817i
$959$	0	0
$960$	0	0
$961$	−19.5762	−0.631489
$962$	3.36353	0.108444
$963$	0	0
$964$	13.4328i	0.432641i
$965$	−4.41860	−0.142240
$966$	0	0
$967$	−17.7780	−0.571702	−0.285851	−	0.958274i	$-0.592276\pi$
$967$	−17.7780	−0.571702	−0.285851	+	0.958274i	$0.592276\pi$
$968$	− 9.55901i	− 0.307238i
$969$	0	0
$970$	−7.94356	−0.255052
$971$	−46.5597	−1.49417	−0.747086	−	0.664728i	$-0.768548\pi$
$971$	−46.5597	−1.49417	−0.747086	+	0.664728i	$0.768548\pi$
$972$	0	0
$973$	0	0
$974$	29.1383i	0.933651i
$975$	0	0
$976$	− 0.865467i	− 0.0277029i
$977$	− 16.9303i	− 0.541647i	−0.962629	−	0.270824i	$-0.912704\pi$
$977$	− 16.9303i	− 0.541647i	0.962629	−	0.270824i	$-0.0872960\pi$
$978$	0	0
$979$	− 9.15652i	− 0.292644i
$980$	0	0
$981$	0	0
$982$	19.7492	0.630222
$983$	41.6087	1.32711	0.663556	−	0.748126i	$-0.269046\pi$
$983$	41.6087	1.32711	0.663556	+	0.748126i	$0.269046\pi$
$984$	0	0
$985$	8.94131i	0.284894i
$986$	0.773868	0.0246450
$987$	0	0
$988$	−0.161924	−0.00515149
$989$	37.0227i	1.17725i
$990$	0	0
$991$	−7.67254	−0.243726	−0.121863	−	0.992547i	$-0.538887\pi$
$991$	−7.67254	−0.243726	−0.121863	+	0.992547i	$0.538887\pi$
$992$	−7.11169	−0.225796
$993$	0	0
$994$	0	0
$995$	8.98780i	0.284932i
$996$	0	0
$997$	− 2.76497i	− 0.0875673i	−0.999041	−	0.0437837i	$-0.986059\pi$
$997$	− 2.76497i	− 0.0875673i	0.999041	−	0.0437837i	$-0.0139412\pi$
$998$	− 31.6108i	− 1.00062i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2646.2.d.e.2645.13	yes	16
3.2	odd	2	inner	2646.2.d.e.2645.4	✓	16
7.6	odd	2	inner	2646.2.d.e.2645.12	yes	16
21.20	even	2	inner	2646.2.d.e.2645.5	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2646.2.d.e.2645.4	✓	16	3.2	odd	2	inner
2646.2.d.e.2645.5	yes	16	21.20	even	2	inner
2646.2.d.e.2645.12	yes	16	7.6	odd	2	inner
2646.2.d.e.2645.13	yes	16	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:	\( N \)	\(=\)	\( 2646 = 2 \cdot 3^{3} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2646.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} -1.00000 q^{4} +0.601731 q^{5} -1.00000i q^{8} +O(q^{10})\)	\(q+1.00000i q^{2} -1.00000 q^{4} +0.601731 q^{5} -1.00000i q^{8} +0.601731i q^{10} -1.20041i q^{11} -0.649659i q^{13} +1.00000 q^{16} +0.564050 q^{17} +0.249245i q^{19} -0.601731 q^{20} +1.20041 q^{22} +3.95367i q^{23} -4.63792 q^{25} +0.649659 q^{26} -1.37199i q^{29} +7.11169i q^{31} +1.00000i q^{32} +0.564050i q^{34} +5.17738 q^{37} -0.249245 q^{38} -0.601731i q^{40} +7.32326 q^{41} +9.36414 q^{43} +1.20041i q^{44} -3.95367 q^{46} +9.12845 q^{47} -4.63792i q^{50} +0.649659i q^{52} -10.3178i q^{53} -0.722325i q^{55} +1.37199 q^{58} +7.26508 q^{59} -0.865467i q^{61} -7.11169 q^{62} -1.00000 q^{64} -0.390920i q^{65} -9.37779 q^{67} -0.564050 q^{68} -4.28897i q^{71} +15.7896i q^{73} +5.17738i q^{74} -0.249245i q^{76} +12.1462 q^{79} +0.601731 q^{80} +7.32326i q^{82} +5.58632 q^{83} +0.339406 q^{85} +9.36414i q^{86} -1.20041 q^{88} +7.62781 q^{89} -3.95367i q^{92} +9.12845i q^{94} +0.149978i q^{95} +13.2012i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 16 q^{4}+O(q^{10}) \) 16 * q - 16 * q^4	\( 16 q - 16 q^{4} + 16 q^{16} - 32 q^{22} + 48 q^{25} - 32 q^{37} + 16 q^{43} - 48 q^{46} + 16 q^{58} - 16 q^{64} + 16 q^{67} + 32 q^{88}+O(q^{100}) \) 16 * q - 16 * q^4 + 16 * q^16 - 32 * q^22 + 48 * q^25 - 32 * q^37 + 16 * q^43 - 48 * q^46 + 16 * q^58 - 16 * q^64 + 16 * q^67 + 32 * q^88

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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