LMFDB - Embedded newform 1300.2.f.e.701.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1300,2,Mod(701,1300)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1300, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1300.701");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1300 = 2^{2} \cdot 5^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1300.f (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:	$10.3805522628$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.9144576.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} + 12x^{4} + 36x^{2} + 4 $ x^6 + 12x^4 + 36x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2\cdot 3 $
Twist minimal:	no (minimal twist has level 260)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			701.1
Root			$-2.60168i$ of defining polynomial
Character	$\chi$	$=$	1300.701
Dual form			1300.2.f.e.701.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.60168 q^{3} -2.76873i q^{7} +3.76873 q^{9} +O(q^{10})$	$q-2.60168 q^{3} -2.76873i q^{7} +3.76873 q^{9} +2.16706i q^{11} +(0.167055 + 3.60168i) q^{13} -5.03630i q^{19} +7.20336i q^{21} +4.93579 q^{23} -2.00000 q^{27} -4.43462 q^{29} +3.37041i q^{31} -5.63798i q^{33} +3.97209i q^{37} +(-0.434624 - 9.37041i) q^{39} +1.66589i q^{41} -9.80504 q^{43} -11.6380i q^{47} -0.665890 q^{49} +12.7408 q^{53} +13.1028i q^{57} -8.16706i q^{59} -10.3062 q^{61} -10.4346i q^{63} -7.10284i q^{67} -12.8413 q^{69} -16.1112i q^{71} -15.9721i q^{73} +6.00000 q^{77} +11.9442 q^{79} -6.10284 q^{81} -3.97209i q^{83} +11.5375 q^{87} -7.94419i q^{89} +(9.97209 - 0.462531i) q^{91} -8.76873i q^{93} +0.462531i q^{97} +8.16706i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 6 q^{9}+O(q^{10}) $ 6 * q + 6 * q^9	$ 6 q + 6 q^{9} + 12 q^{23} - 12 q^{27} - 12 q^{29} + 12 q^{39} - 12 q^{43} - 6 q^{49} + 12 q^{53} - 12 q^{61} + 36 q^{77} - 24 q^{79} - 18 q^{81} + 36 q^{87} + 12 q^{91}+O(q^{100}) $ 6 * q + 6 * q^9 + 12 * q^23 - 12 * q^27 - 12 * q^29 + 12 * q^39 - 12 * q^43 - 6 * q^49 + 12 * q^53 - 12 * q^61 + 36 * q^77 - 24 * q^79 - 18 * q^81 + 36 * q^87 + 12 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1300\mathbb{Z}\right)^\times$.

$n$	$301$	$651$	$677$
$\chi(n)$	$-1$	$1$	$1$

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 $

$2$		0			0
$2$		0			0
$3$	−2.60168			−1.50208			−0.751040	−	0.660257i	$-0.770448\pi$
$3$	−2.60168			−1.50208			−0.751040	+	0.660257i	$0.770448\pi$
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$		−	2.76873i		−	1.04648i	−0.852184	−	0.523242i	$-0.824723\pi$
$7$		−	2.76873i		−	1.04648i	0.852184	−	0.523242i	$-0.175277\pi$
$8$		0			0
$9$	3.76873			1.25624
$10$		0			0
$11$			2.16706i			0.653392i	0.945130	+	0.326696i	$0.105935\pi$
$11$			2.16706i			0.653392i	−0.945130	+	0.326696i	$0.894065\pi$
$12$		0			0
$13$	0.167055	+	3.60168i	0.0463328	+	0.998926i
$13$	0.167055	+	3.60168i	0.0463328	+	0.998926i
$14$		0			0
$15$		0			0
$16$		0			0
$17$		0			0			−	1.00000i	$-0.5\pi$
$17$		0			0				1.00000i	$0.5\pi$
$18$		0			0
$19$		−	5.03630i		−	1.15541i	−0.816247	−	0.577704i	$-0.803949\pi$
$19$		−	5.03630i		−	1.15541i	0.816247	−	0.577704i	$-0.196051\pi$
$20$		0			0
$21$			7.20336i			1.57190i
$22$		0			0
$23$	4.93579			1.02918			0.514592	−	0.857435i	$-0.327944\pi$
$23$	4.93579			1.02918			0.514592	+	0.857435i	$0.327944\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$	−2.00000			−0.384900
$28$		0			0
$29$	−4.43462			−0.823489			−0.411744	−	0.911299i	$-0.635080\pi$
$29$	−4.43462			−0.823489			−0.411744	+	0.911299i	$0.635080\pi$
$30$		0			0
$31$			3.37041i			0.605344i	0.953095	+	0.302672i	$0.0978787\pi$
$31$			3.37041i			0.605344i	−0.953095	+	0.302672i	$0.902121\pi$
$32$		0			0
$33$		−	5.63798i		−	0.981447i
$34$		0			0
$35$		0			0
$36$		0			0
$37$			3.97209i			0.653008i	0.945196	+	0.326504i	$0.105871\pi$
$37$			3.97209i			0.653008i	−0.945196	+	0.326504i	$0.894129\pi$
$38$		0			0
$39$	−0.434624	−	9.37041i	−0.0695955	−	1.50047i
$40$		0			0
$41$			1.66589i			0.260168i	0.991503	+	0.130084i	$0.0415247\pi$
$41$			1.66589i			0.260168i	−0.991503	+	0.130084i	$0.958475\pi$
$42$		0			0
$43$	−9.80504			−1.49525			−0.747627	−	0.664119i	$-0.768807\pi$
$43$	−9.80504			−1.49525			−0.747627	+	0.664119i	$0.768807\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		−	11.6380i		−	1.69757i	−0.528735	−	0.848787i	$-0.677333\pi$
$47$		−	11.6380i		−	1.69757i	0.528735	−	0.848787i	$-0.322667\pi$
$48$		0			0
$49$	−0.665890			−0.0951271
$50$		0			0
$51$		0			0
$52$		0			0
$53$	12.7408			1.75009			0.875044	−	0.484044i	$-0.160833\pi$
$53$	12.7408			1.75009			0.875044	+	0.484044i	$0.160833\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$			13.1028i			1.73551i
$58$		0			0
$59$		−	8.16706i		−	1.06326i	−0.846977	−	0.531630i	$-0.821580\pi$
$59$		−	8.16706i		−	1.06326i	0.846977	−	0.531630i	$-0.178420\pi$
$60$		0			0
$61$	−10.3062			−1.31957			−0.659787	−	0.751453i	$-0.729354\pi$
$61$	−10.3062			−1.31957			−0.659787	+	0.751453i	$0.729354\pi$
$62$		0			0
$63$		−	10.4346i		−	1.31464i
$64$		0			0
$65$		0			0
$66$		0			0
$67$		−	7.10284i		−	0.867751i	−0.900973	−	0.433875i	$-0.857146\pi$
$67$		−	7.10284i		−	0.867751i	0.900973	−	0.433875i	$-0.142854\pi$
$68$		0			0
$69$	−12.8413			−1.54592
$70$		0			0
$71$		−	16.1112i		−	1.91205i	−0.293281	−	0.956026i	$-0.594747\pi$
$71$		−	16.1112i		−	1.91205i	0.293281	−	0.956026i	$-0.405253\pi$
$72$		0			0
$73$		−	15.9721i		−	1.86939i	−0.355448	−	0.934696i	$-0.615672\pi$
$73$		−	15.9721i		−	1.86939i	0.355448	−	0.934696i	$-0.384328\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	6.00000			0.683763
$78$		0			0
$79$	11.9442			1.34383			0.671913	−	0.740630i	$-0.265473\pi$
$79$	11.9442			1.34383			0.671913	+	0.740630i	$0.265473\pi$
$80$		0			0
$81$	−6.10284			−0.678094
$82$		0			0
$83$		−	3.97209i		−	0.435994i	−0.975949	−	0.217997i	$-0.930048\pi$
$83$		−	3.97209i		−	0.435994i	0.975949	−	0.217997i	$-0.0699523\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	11.5375			1.23695
$88$		0			0
$89$		−	7.94419i		−	0.842082i	−0.907042	−	0.421041i	$-0.861665\pi$
$89$		−	7.94419i		−	0.842082i	0.907042	−	0.421041i	$-0.138335\pi$
$90$		0			0
$91$	9.97209	−	0.462531i	1.04536	−	0.0484865i
$92$		0			0
$93$		−	8.76873i		−	0.909275i
$94$		0			0
$95$		0			0
$96$		0			0
$97$			0.462531i			0.0469629i	0.999724	+	0.0234815i	$0.00747507\pi$
$97$			0.462531i			0.0469629i	−0.999724	+	0.0234815i	$0.992525\pi$
$98$		0			0
$99$			8.16706i			0.820820i
$100$		0			0
$101$	12.7408			1.26776			0.633880	−	0.773432i	$-0.281461\pi$
$101$	12.7408			1.26776			0.633880	+	0.773432i	$0.281461\pi$
$102$		0			0
$103$	4.13915			0.407842			0.203921	−	0.978987i	$-0.434631\pi$
$103$	4.13915			0.407842			0.203921	+	0.978987i	$0.434631\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−7.06421			−0.682923			−0.341462	−	0.939896i	$-0.610922\pi$
$107$	−7.06421			−0.682923			−0.341462	+	0.939896i	$0.610922\pi$
$108$		0			0
$109$		−	14.8692i		−	1.42422i	−0.702070	−	0.712108i	$-0.747741\pi$
$109$		−	14.8692i		−	1.42422i	0.702070	−	0.712108i	$-0.252259\pi$
$110$		0			0
$111$		−	10.3341i		−	0.980870i
$112$		0			0
$113$	−3.13075			−0.294516			−0.147258	−	0.989098i	$-0.547045\pi$
$113$	−3.13075			−0.294516			−0.147258	+	0.989098i	$0.547045\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	0.629587	+	13.5738i	0.0582053	+	1.25490i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	6.30387			0.573079
$122$		0			0
$123$		−	4.33411i		−	0.390794i
$124$		0			0
$125$		0			0
$126$		0			0
$127$	−16.1391			−1.43212			−0.716059	−	0.698040i	$-0.754056\pi$
$127$	−16.1391			−1.43212			−0.716059	+	0.698040i	$0.754056\pi$
$128$		0			0
$129$	25.5096			2.24599
$130$		0			0
$131$	−8.86925			−0.774910			−0.387455	−	0.921889i	$-0.626646\pi$
$131$	−8.86925			−0.774910			−0.387455	+	0.921889i	$0.626646\pi$
$132$		0			0
$133$	−13.9442			−1.20911
$134$		0			0
$135$		0			0
$136$		0			0
$137$			1.66589i			0.142327i	0.997465	+	0.0711633i	$0.0226711\pi$
$137$			1.66589i			0.142327i	−0.997465	+	0.0711633i	$0.977329\pi$
$138$		0			0
$139$	19.2760			1.63497			0.817483	−	0.575953i	$-0.195369\pi$
$139$	19.2760			1.63497			0.817483	+	0.575953i	$0.195369\pi$
$140$		0			0
$141$			30.2783i			2.54989i
$142$		0			0
$143$	−7.80504	+	0.362018i	−0.652690	+	0.0302734i
$144$		0			0
$145$		0			0
$146$		0			0
$147$	1.73243			0.142889
$148$		0			0
$149$			13.9442i			1.14235i	0.820828	+	0.571176i	$0.193513\pi$
$149$			13.9442i			1.14235i	−0.820828	+	0.571176i	$0.806487\pi$
$150$		0			0
$151$			6.96370i			0.566698i	0.959017	+	0.283349i	$0.0914454\pi$
$151$			6.96370i			0.566698i	−0.959017	+	0.283349i	$0.908555\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−12.3341			−0.984369			−0.492185	−	0.870491i	$-0.663801\pi$
$157$	−12.3341			−0.984369			−0.492185	+	0.870491i	$0.663801\pi$
$158$		0			0
$159$	−33.1475			−2.62877
$160$		0			0
$161$		−	13.6659i		−	1.07702i
$162$		0			0
$163$			20.5072i			1.60625i	0.595810	+	0.803125i	$0.296831\pi$
$163$			20.5072i			1.60625i	−0.595810	+	0.803125i	$0.703169\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	19.3039i		−	1.49378i	−0.664948	−	0.746889i	$-0.731547\pi$
$167$		−	19.3039i		−	1.49378i	0.664948	−	0.746889i	$-0.268453\pi$
$168$		0			0
$169$	−12.9442	+	1.20336i	−0.995707	+	0.0925660i
$170$		0			0
$171$		−	18.9805i		−	1.45147i
$172$		0			0
$173$	−9.61007			−0.730640			−0.365320	−	0.930882i	$-0.619041\pi$
$173$	−9.61007			−0.730640			−0.365320	+	0.930882i	$0.619041\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$			21.2481i			1.59710i
$178$		0			0
$179$	−9.87158			−0.737836			−0.368918	−	0.929462i	$-0.620272\pi$
$179$	−9.87158			−0.737836			−0.368918	+	0.929462i	$0.620272\pi$
$180$		0			0
$181$	−1.97209			−0.146584			−0.0732922	−	0.997311i	$-0.523351\pi$
$181$	−1.97209			−0.146584			−0.0732922	+	0.997311i	$0.523351\pi$
$182$		0			0
$183$	26.8134			1.98211
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$			5.53747i			0.402792i
$190$		0			0
$191$	−5.73850			−0.415223			−0.207611	−	0.978211i	$-0.566569\pi$
$191$	−5.73850			−0.415223			−0.207611	+	0.978211i	$0.566569\pi$
$192$		0			0
$193$			4.07261i			0.293153i	0.989199	+	0.146576i	$0.0468254\pi$
$193$			4.07261i			0.293153i	−0.989199	+	0.146576i	$0.953175\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	21.6101i		−	1.53965i	−0.638253	−	0.769827i	$-0.720342\pi$
$197$		−	21.6101i		−	1.53965i	0.638253	−	0.769827i	$-0.279658\pi$
$198$		0			0
$199$	−11.6101			−0.823016			−0.411508	−	0.911406i	$-0.634998\pi$
$199$	−11.6101			−0.823016			−0.411508	+	0.911406i	$0.634998\pi$
$200$		0			0
$201$			18.4793i			1.30343i
$202$		0			0
$203$			12.2783i			0.861767i
$204$		0			0
$205$		0			0
$206$		0			0
$207$	18.6017			1.29291
$208$		0			0
$209$	10.9139			0.754933
$210$		0			0
$211$	16.2783			1.12064			0.560322	−	0.828275i	$-0.310677\pi$
$211$	16.2783			1.12064			0.560322	+	0.828275i	$0.310677\pi$
$212$		0			0
$213$			41.9163i			2.87206i
$214$		0			0
$215$		0			0
$216$		0			0
$217$	9.33178			0.633482
$218$		0			0
$219$			41.5543i			2.80798i
$220$		0			0
$221$		0			0
$222$		0			0
$223$			11.4370i			0.765875i	0.923774	+	0.382938i	$0.125088\pi$
$223$			11.4370i			0.765875i	−0.923774	+	0.382938i	$0.874912\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$		−	15.9721i		−	1.06011i	−0.847965	−	0.530053i	$-0.822172\pi$
$227$		−	15.9721i		−	1.06011i	0.847965	−	0.530053i	$-0.177828\pi$
$228$		0			0
$229$			2.12842i			0.140650i	0.997524	+	0.0703250i	$0.0224036\pi$
$229$			2.12842i			0.140650i	−0.997524	+	0.0703250i	$0.977596\pi$
$230$		0			0
$231$	−15.6101			−1.02707
$232$		0			0
$233$	2.86925			0.187971			0.0939853	−	0.995574i	$-0.470039\pi$
$233$	2.86925			0.187971			0.0939853	+	0.995574i	$0.470039\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	−31.0749			−2.01853
$238$		0			0
$239$		−	3.16939i		−	0.205011i	−0.994732	−	0.102505i	$-0.967314\pi$
$239$		−	3.16939i		−	0.205011i	0.994732	−	0.102505i	$-0.0326859\pi$
$240$		0			0
$241$			4.79664i			0.308979i	0.987994	+	0.154489i	$0.0493733\pi$
$241$			4.79664i			0.308979i	−0.987994	+	0.154489i	$0.950627\pi$
$242$		0			0
$243$	21.8776			1.40345
$244$		0			0
$245$		0			0
$246$		0			0
$247$	18.1391	−	0.841340i	1.15417	−	0.0535332i
$248$		0			0
$249$			10.3341i			0.654898i
$250$		0			0
$251$	1.00233			0.0632666			0.0316333	−	0.999500i	$-0.489929\pi$
$251$	1.00233			0.0632666			0.0316333	+	0.999500i	$0.489929\pi$
$252$		0			0
$253$			10.6961i			0.672460i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	19.4817			1.21523			0.607616	−	0.794231i	$-0.292126\pi$
$257$	19.4817			1.21523			0.607616	+	0.794231i	$0.292126\pi$
$258$		0			0
$259$	10.9977			0.683362
$260$		0			0
$261$	−16.7129			−1.03450
$262$		0			0
$263$	4.19496			0.258672			0.129336	−	0.991601i	$-0.458715\pi$
$263$	4.19496			0.258672			0.129336	+	0.991601i	$0.458715\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$			20.6682i			1.26487i
$268$		0			0
$269$	30.4793			1.85836			0.929179	−	0.369631i	$-0.120516\pi$
$269$	30.4793			1.85836			0.929179	+	0.369631i	$0.120516\pi$
$270$		0			0
$271$		−	4.83528i		−	0.293722i	−0.989157	−	0.146861i	$-0.953083\pi$
$271$		−	4.83528i		−	0.293722i	0.989157	−	0.146861i	$-0.0469170\pi$
$272$		0			0
$273$	−25.9442	+	1.20336i	−1.57021	+	0.0728306i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−3.00233			−0.180393			−0.0901963	−	0.995924i	$-0.528749\pi$
$277$	−3.00233			−0.180393			−0.0901963	+	0.995924i	$0.528749\pi$
$278$		0			0
$279$			12.7022i			0.760460i
$280$		0			0
$281$		−	21.6101i		−	1.28915i	−0.764542	−	0.644574i	$-0.777035\pi$
$281$		−	21.6101i		−	1.28915i	0.764542	−	0.644574i	$-0.222965\pi$
$282$		0			0
$283$	−2.47326			−0.147020			−0.0735100	−	0.997294i	$-0.523420\pi$
$283$	−2.47326			−0.147020			−0.0735100	+	0.997294i	$0.523420\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	4.61241			0.272262
$288$		0			0
$289$	−17.0000			−1.00000
$290$		0			0
$291$		−	1.20336i		−	0.0705421i
$292$		0			0
$293$			23.9163i			1.39720i	0.715511	+	0.698602i	$0.246194\pi$
$293$			23.9163i			1.39720i	−0.715511	+	0.698602i	$0.753806\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		−	4.33411i		−	0.251491i
$298$		0			0
$299$	0.824549	+	17.7771i	0.0476849	+	1.02808i
$300$		0			0
$301$			27.1475i			1.56476i
$302$		0			0
$303$	−33.1475			−1.90428
$304$		0			0
$305$		0			0
$306$		0			0
$307$		−	13.5654i		−	0.774217i	−0.922034	−	0.387108i	$-0.873474\pi$
$307$		−	13.5654i		−	0.774217i	0.922034	−	0.387108i	$-0.126526\pi$
$308$		0			0
$309$	−10.7687			−0.612612
$310$		0			0
$311$	−2.12842			−0.120692			−0.0603458	−	0.998178i	$-0.519220\pi$
$311$	−2.12842			−0.120692			−0.0603458	+	0.998178i	$0.519220\pi$
$312$		0			0
$313$	22.6077			1.27787			0.638933	−	0.769263i	$-0.279376\pi$
$313$	22.6077			1.27787			0.638933	+	0.769263i	$0.279376\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$			10.6357i			0.597358i	0.954354	+	0.298679i	$0.0965459\pi$
$317$			10.6357i			0.597358i	−0.954354	+	0.298679i	$0.903454\pi$
$318$		0			0
$319$		−	9.61007i		−	0.538061i
$320$		0			0
$321$	18.3788			1.02581
$322$		0			0
$323$		0			0
$324$		0			0
$325$		0			0
$326$		0			0
$327$			38.6850i			2.13929i
$328$		0			0
$329$	−32.2225			−1.77648
$330$		0			0
$331$		−	1.16472i		−	0.0640190i	−0.999488	−	0.0320095i	$-0.989809\pi$
$331$		−	1.16472i		−	0.0640190i	0.999488	−	0.0320095i	$-0.0101907\pi$
$332$		0			0
$333$			14.9698i			0.820338i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	20.0000			1.08947			0.544735	−	0.838608i	$-0.316630\pi$
$337$	20.0000			1.08947			0.544735	+	0.838608i	$0.316630\pi$
$338$		0			0
$339$	8.14521			0.442387
$340$		0			0
$341$	−7.30387			−0.395527
$342$		0			0
$343$		−	17.5375i		−	0.946934i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−4.67429			−0.250929			−0.125464	−	0.992098i	$-0.540042\pi$
$347$	−4.67429			−0.250929			−0.125464	+	0.992098i	$0.540042\pi$
$348$		0			0
$349$		−	4.33411i		−	0.232000i	−0.993249	−	0.116000i	$-0.962993\pi$
$349$		−	4.33411i		−	0.232000i	0.993249	−	0.116000i	$-0.0370072\pi$
$350$		0			0
$351$	−0.334110	−	7.20336i	−0.0178335	−	0.384487i
$352$		0			0
$353$		−	21.3085i		−	1.13414i	−0.823670	−	0.567069i	$-0.808077\pi$
$353$		−	21.3085i		−	1.13414i	0.823670	−	0.567069i	$-0.191923\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$			12.7795i			0.674474i	0.941420	+	0.337237i	$0.109492\pi$
$359$			12.7795i			0.674474i	−0.941420	+	0.337237i	$0.890508\pi$
$360$		0			0
$361$	−6.36435			−0.334966
$362$		0			0
$363$	−16.4007			−0.860811
$364$		0			0
$365$		0			0
$366$		0			0
$367$	−12.1950			−0.636572			−0.318286	−	0.947995i	$-0.603107\pi$
$367$	−12.1950			−0.636572			−0.318286	+	0.947995i	$0.603107\pi$
$368$		0			0
$369$			6.27830i			0.326835i
$370$		0			0
$371$		−	35.2760i		−	1.83144i
$372$		0			0
$373$	28.2225			1.46130			0.730652	−	0.682750i	$-0.239216\pi$
$373$	28.2225			1.46130			0.730652	+	0.682750i	$0.239216\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−0.740827	−	15.9721i	−0.0381545	−	0.822605i
$378$		0			0
$379$			23.3146i			1.19759i	0.800902	+	0.598795i	$0.204354\pi$
$379$			23.3146i			1.19759i	−0.800902	+	0.598795i	$0.795646\pi$
$380$		0			0
$381$	41.9889			2.15116
$382$		0			0
$383$		−	8.02791i		−	0.410207i	−0.978740	−	0.205103i	$-0.934247\pi$
$383$		−	8.02791i		−	0.410207i	0.978740	−	0.205103i	$-0.0657531\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−36.9526			−1.87841
$388$		0			0
$389$	23.7385			1.20359			0.601795	−	0.798651i	$-0.294453\pi$
$389$	23.7385			1.20359			0.601795	+	0.798651i	$0.294453\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$	23.0749			1.16398
$394$		0			0
$395$		0			0
$396$		0			0
$397$			3.04703i			0.152926i	0.997072	+	0.0764630i	$0.0243627\pi$
$397$			3.04703i			0.152926i	−0.997072	+	0.0764630i	$0.975637\pi$
$398$		0			0
$399$	36.2783			1.81619
$400$		0			0
$401$		−	31.2201i		−	1.55906i	−0.626365	−	0.779530i	$-0.715458\pi$
$401$		−	31.2201i		−	1.55906i	0.626365	−	0.779530i	$-0.284542\pi$
$402$		0			0
$403$	−12.1391	+	0.563045i	−0.604694	+	0.0280473i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−8.60774			−0.426670
$408$		0			0
$409$		−	33.4090i		−	1.65197i	−0.563691	−	0.825986i	$-0.690619\pi$
$409$		−	33.4090i		−	1.65197i	0.563691	−	0.825986i	$-0.309381\pi$
$410$		0			0
$411$		−	4.33411i		−	0.213786i
$412$		0			0
$413$	−22.6124			−1.11268
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−50.1499			−2.45585
$418$		0			0
$419$	−30.7408			−1.50179			−0.750894	−	0.660423i	$-0.770377\pi$
$419$	−30.7408			−1.50179			−0.750894	+	0.660423i	$0.770377\pi$
$420$		0			0
$421$		−	19.2806i		−	0.939680i	−0.882752	−	0.469840i	$-0.844312\pi$
$421$		−	19.2806i		−	0.939680i	0.882752	−	0.469840i	$-0.155688\pi$
$422$		0			0
$423$		−	43.8605i		−	2.13257i
$424$		0			0
$425$		0			0
$426$		0			0
$427$			28.5351i			1.38091i
$428$		0			0
$429$	20.3062	−	0.941854i	0.980393	−	0.0454731i
$430$		0			0
$431$			2.44535i			0.117788i	0.998264	+	0.0588942i	$0.0187575\pi$
$431$			2.44535i			0.117788i	−0.998264	+	0.0588942i	$0.981243\pi$
$432$		0			0
$433$	2.00000			0.0961139			0.0480569	−	0.998845i	$-0.484697\pi$
$433$	2.00000			0.0961139			0.0480569	+	0.998845i	$0.484697\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		−	24.8581i		−	1.18913i
$438$		0			0
$439$	−11.6659			−0.556783			−0.278391	−	0.960468i	$-0.589801\pi$
$439$	−11.6659			−0.556783			−0.278391	+	0.960468i	$0.589801\pi$
$440$		0			0
$441$	−2.50956			−0.119503
$442$		0			0
$443$	−20.8074			−0.988588			−0.494294	−	0.869295i	$-0.664573\pi$
$443$	−20.8074			−0.988588			−0.494294	+	0.869295i	$0.664573\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		−	36.2783i		−	1.71590i
$448$		0			0
$449$			30.2783i			1.42892i	0.699676	+	0.714461i	$0.253328\pi$
$449$			30.2783i			1.42892i	−0.699676	+	0.714461i	$0.746672\pi$
$450$		0			0
$451$	−3.61007			−0.169992
$452$		0			0
$453$		−	18.1173i		−	0.851225i
$454$		0			0
$455$		0			0
$456$		0			0
$457$			22.6124i			1.05776i	0.848695	+	0.528882i	$0.177389\pi$
$457$			22.6124i			1.05776i	−0.848695	+	0.528882i	$0.822611\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		−	18.9419i		−	0.882210i	−0.897455	−	0.441105i	$-0.854587\pi$
$461$		−	18.9419i		−	0.882210i	0.897455	−	0.441105i	$-0.145413\pi$
$462$		0			0
$463$		−	11.6380i		−	0.540863i	−0.962739	−	0.270431i	$-0.912834\pi$
$463$		−	11.6380i		−	0.540863i	0.962739	−	0.270431i	$-0.0871663\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	25.0642			1.15983			0.579917	−	0.814676i	$-0.303085\pi$
$467$	25.0642			1.15983			0.579917	+	0.814676i	$0.303085\pi$
$468$		0			0
$469$	−19.6659			−0.908086
$470$		0			0
$471$	32.0894			1.47860
$472$		0			0
$473$		−	21.2481i		−	0.976987i
$474$		0			0
$475$		0			0
$476$		0			0
$477$	48.0168			2.19854
$478$		0			0
$479$		−	18.4407i		−	0.842577i	−0.906927	−	0.421288i	$-0.861578\pi$
$479$		−	18.4407i		−	0.842577i	0.906927	−	0.421288i	$-0.138422\pi$
$480$		0			0
$481$	−14.3062	+	0.663559i	−0.652307	+	0.0302557i
$482$		0			0
$483$			35.5543i			1.61777i
$484$		0			0
$485$		0			0
$486$		0			0
$487$			25.1196i			1.13828i	0.822241	+	0.569140i	$0.192724\pi$
$487$			25.1196i			1.13828i	−0.822241	+	0.569140i	$0.807276\pi$
$488$		0			0
$489$		−	53.3532i		−	2.41272i
$490$		0			0
$491$	−5.73850			−0.258975			−0.129487	−	0.991581i	$-0.541333\pi$
$491$	−5.73850			−0.258975			−0.129487	+	0.991581i	$0.541333\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−44.6077			−2.00093
$498$		0			0
$499$			33.3704i			1.49386i	0.664900	+	0.746932i	$0.268474\pi$
$499$			33.3704i			1.49386i	−0.664900	+	0.746932i	$0.731526\pi$
$500$		0			0
$501$			50.2225i			2.24377i
$502$		0			0
$503$	12.6789			0.565326			0.282663	−	0.959219i	$-0.408782\pi$
$503$	12.6789			0.565326			0.282663	+	0.959219i	$0.408782\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	33.6766	−	3.13075i	1.49563	−	0.139042i
$508$		0			0
$509$		−	17.6147i		−	0.780759i	−0.920654	−	0.390380i	$-0.872344\pi$
$509$		−	17.6147i		−	0.780759i	0.920654	−	0.390380i	$-0.127656\pi$
$510$		0			0
$511$	−44.2225			−1.95629
$512$		0			0
$513$			10.0726i			0.444716i
$514$		0			0
$515$		0			0
$516$		0			0
$517$	25.2201			1.10918
$518$		0			0
$519$	25.0023			1.09748
$520$		0			0
$521$	−16.4346			−0.720014			−0.360007	−	0.932950i	$-0.617226\pi$
$521$	−16.4346			−0.720014			−0.360007	+	0.932950i	$0.617226\pi$
$522$		0			0
$523$	20.4733			0.895233			0.447617	−	0.894226i	$-0.352273\pi$
$523$	20.4733			0.895233			0.447617	+	0.894226i	$0.352273\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	1.36202			0.0592182
$530$		0			0
$531$		−	30.7795i		−	1.33571i
$532$		0			0
$533$	−6.00000	+	0.278295i	−0.259889	+	0.0120543i
$534$		0			0
$535$		0			0
$536$		0			0
$537$	25.6827			1.10829
$538$		0			0
$539$		−	1.44302i		−	0.0621553i
$540$		0			0
$541$			41.2928i			1.77531i	0.460505	+	0.887657i	$0.347668\pi$
$541$			41.2928i			1.77531i	−0.460505	+	0.887657i	$0.652332\pi$
$542$		0			0
$543$	5.13075			0.220182
$544$		0			0
$545$		0			0
$546$		0			0
$547$	10.4174			0.445418			0.222709	−	0.974885i	$-0.428510\pi$
$547$	10.4174			0.445418			0.222709	+	0.974885i	$0.428510\pi$
$548$		0			0
$549$	−38.8413			−1.65771
$550$		0			0
$551$			22.3341i			0.951465i
$552$		0			0
$553$		−	33.0703i		−	1.40629i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		−	20.5845i		−	0.872193i	−0.899900	−	0.436097i	$-0.856361\pi$
$557$		−	20.5845i		−	0.872193i	0.899900	−	0.436097i	$-0.143639\pi$
$558$		0			0
$559$	−1.63798	−	35.3146i	−0.0692793	−	1.49365i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	23.6766			0.997850			0.498925	−	0.866645i	$-0.333728\pi$
$563$	23.6766			0.997850			0.498925	+	0.866645i	$0.333728\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$			16.8972i			0.709614i
$568$		0			0
$569$	5.43695			0.227929			0.113965	−	0.993485i	$-0.463645\pi$
$569$	5.43695			0.227929			0.113965	+	0.993485i	$0.463645\pi$
$570$		0			0
$571$	15.2760			0.639279			0.319640	−	0.947539i	$-0.396438\pi$
$571$	15.2760			0.639279			0.319640	+	0.947539i	$0.396438\pi$
$572$		0			0
$573$	14.9297			0.623698
$574$		0			0
$575$		0			0
$576$		0			0
$577$		−	1.76640i		−	0.0735363i	−0.999324	−	0.0367682i	$-0.988294\pi$
$577$		−	1.76640i		−	0.0735363i	0.999324	−	0.0367682i	$-0.0117063\pi$
$578$		0			0
$579$		−	10.5956i		−	0.440339i
$580$		0			0
$581$	−10.9977			−0.456260
$582$		0			0
$583$			27.6101i			1.14349i
$584$		0			0
$585$		0			0
$586$		0			0
$587$		−	22.9139i		−	0.945760i	−0.881127	−	0.472880i	$-0.843214\pi$
$587$		−	22.9139i		−	0.945760i	0.881127	−	0.472880i	$-0.156786\pi$
$588$		0			0
$589$	16.9744			0.699419
$590$		0			0
$591$			56.2225i			2.31268i
$592$		0			0
$593$			6.72404i			0.276123i	0.990424	+	0.138062i	$0.0440872\pi$
$593$			6.72404i			0.276123i	−0.990424	+	0.138062i	$0.955913\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	30.2057			1.23624
$598$		0			0
$599$	−19.8669			−0.811740			−0.405870	−	0.913931i	$-0.633031\pi$
$599$	−19.8669			−0.811740			−0.405870	+	0.913931i	$0.633031\pi$
$600$		0			0
$601$	−33.2201			−1.35508			−0.677539	−	0.735487i	$-0.736954\pi$
$601$	−33.2201			−1.35508			−0.677539	+	0.735487i	$0.736954\pi$
$602$		0			0
$603$		−	26.7687i		−	1.09011i
$604$		0			0
$605$		0			0
$606$		0			0
$607$	−14.4174			−0.585186			−0.292593	−	0.956237i	$-0.594518\pi$
$607$	−14.4174			−0.585186			−0.292593	+	0.956237i	$0.594518\pi$
$608$		0			0
$609$		−	31.9442i		−	1.29444i
$610$		0			0
$611$	41.9163	−	1.94419i	1.69575	−	0.0786533i
$612$		0			0
$613$			11.3364i			0.457875i	0.973441	+	0.228937i	$0.0735251\pi$
$613$			11.3364i			0.457875i	−0.973441	+	0.228937i	$0.926475\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			34.8907i			1.40465i	0.711858	+	0.702323i	$0.247854\pi$
$617$			34.8907i			1.40465i	−0.711858	+	0.702323i	$0.752146\pi$
$618$		0			0
$619$			32.6464i			1.31217i	0.754688	+	0.656084i	$0.227788\pi$
$619$			32.6464i			1.31217i	−0.754688	+	0.656084i	$0.772212\pi$
$620$		0			0
$621$	−9.87158			−0.396133
$622$		0			0
$623$	−21.9953			−0.881225
$624$		0			0
$625$		0			0
$626$		0			0
$627$	−28.3946			−1.13397
$628$		0			0
$629$		0			0
$630$		0			0
$631$			47.3146i			1.88356i	0.336224	+	0.941782i	$0.390850\pi$
$631$			47.3146i			1.88356i	−0.336224	+	0.941782i	$0.609150\pi$
$632$		0			0
$633$	−42.3509			−1.68330
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−0.111240	−	2.39832i	−0.00440750	−	0.0950249i
$638$		0			0
$639$		−	60.7190i		−	2.40201i
$640$		0			0
$641$	43.4817			1.71742			0.858711	−	0.512460i	$-0.171266\pi$
$641$	43.4817			1.71742			0.858711	+	0.512460i	$0.171266\pi$
$642$		0			0
$643$			20.3062i			0.800798i	0.916341	+	0.400399i	$0.131129\pi$
$643$			20.3062i			0.800798i	−0.916341	+	0.400399i	$0.868871\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−30.8027			−1.21098			−0.605490	−	0.795853i	$-0.707023\pi$
$647$	−30.8027			−1.21098			−0.605490	+	0.795853i	$0.707023\pi$
$648$		0			0
$649$	17.6985			0.694725
$650$		0			0
$651$	−24.2783			−0.951541
$652$		0			0
$653$	−3.61007			−0.141273			−0.0706366	−	0.997502i	$-0.522503\pi$
$653$	−3.61007			−0.141273			−0.0706366	+	0.997502i	$0.522503\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		−	60.1946i		−	2.34841i
$658$		0			0
$659$	−6.74083			−0.262585			−0.131293	−	0.991344i	$-0.541913\pi$
$659$	−6.74083			−0.262585			−0.131293	+	0.991344i	$0.541913\pi$
$660$		0			0
$661$		−	26.6682i		−	1.03727i	−0.854995	−	0.518637i	$-0.826440\pi$
$661$		−	26.6682i		−	1.03727i	0.854995	−	0.518637i	$-0.173560\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−21.8884			−0.847521
$668$		0			0
$669$		−	29.7553i		−	1.15041i
$670$		0			0
$671$		−	22.3341i		−	0.862199i
$672$		0			0
$673$	8.61241			0.331984			0.165992	−	0.986127i	$-0.446917\pi$
$673$	8.61241			0.331984			0.165992	+	0.986127i	$0.446917\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−8.12842			−0.312401			−0.156200	−	0.987725i	$-0.549925\pi$
$677$	−8.12842			−0.312401			−0.156200	+	0.987725i	$0.549925\pi$
$678$		0			0
$679$	1.28063			0.0491459
$680$		0			0
$681$			41.5543i			1.59236i
$682$		0			0
$683$			8.02791i			0.307179i	0.988135	+	0.153590i	$0.0490834\pi$
$683$			8.02791i			0.307179i	−0.988135	+	0.153590i	$0.950917\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		−	5.53747i		−	0.211268i
$688$		0			0
$689$	2.12842	+	45.8884i	0.0810864	+	1.74821i
$690$		0			0
$691$			22.1112i			0.841151i	0.907257	+	0.420576i	$0.138172\pi$
$691$			22.1112i			0.841151i	−0.907257	+	0.420576i	$0.861828\pi$
$692$		0			0
$693$	22.6124			0.858974
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$	−7.46486			−0.282347
$700$		0			0
$701$	−6.47932			−0.244721			−0.122360	−	0.992486i	$-0.539046\pi$
$701$	−6.47932			−0.244721			−0.122360	+	0.992486i	$0.539046\pi$
$702$		0			0
$703$	20.0047			0.754490
$704$		0			0
$705$		0			0
$706$		0			0
$707$		−	35.2760i		−	1.32669i
$708$		0			0
$709$			2.85246i			0.107126i	0.998564	+	0.0535631i	$0.0170578\pi$
$709$			2.85246i			0.107126i	−0.998564	+	0.0535631i	$0.982942\pi$
$710$		0			0
$711$	45.0145			1.68817
$712$		0			0
$713$			16.6357i			0.623010i
$714$		0			0
$715$		0			0
$716$		0			0
$717$			8.24573i			0.307942i
$718$		0			0
$719$	42.0941			1.56984			0.784922	−	0.619595i	$-0.212703\pi$
$719$	42.0941			1.56984			0.784922	+	0.619595i	$0.212703\pi$
$720$		0			0
$721$		−	11.4602i		−	0.426800i
$722$		0			0
$723$		−	12.4793i		−	0.464111i
$724$		0			0
$725$		0			0
$726$		0			0
$727$	32.8027			1.21659			0.608293	−	0.793713i	$-0.291855\pi$
$727$	32.8027			1.21659			0.608293	+	0.793713i	$0.291855\pi$
$728$		0			0
$729$	−38.6101			−1.43000
$730$		0			0
$731$		0			0
$732$		0			0
$733$		−	2.38993i		−	0.0882739i	−0.999025	−	0.0441370i	$-0.985946\pi$
$733$		−	2.38993i		−	0.0882739i	0.999025	−	0.0441370i	$-0.0140538\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	15.3923			0.566981
$738$		0			0
$739$		−	41.4988i		−	1.52656i	−0.646068	−	0.763280i	$-0.723588\pi$
$739$		−	41.4988i		−	1.52656i	0.646068	−	0.763280i	$-0.276412\pi$
$740$		0			0
$741$	−47.1922	+	2.18890i	−1.73365	+	0.0804112i
$742$		0			0
$743$		−	39.8046i		−	1.46029i	−0.683292	−	0.730145i	$-0.739452\pi$
$743$		−	39.8046i		−	1.46029i	0.683292	−	0.730145i	$-0.260548\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		−	14.9698i		−	0.547715i
$748$		0			0
$749$			19.5589i			0.714667i
$750$		0			0
$751$	15.6659			0.571656			0.285828	−	0.958281i	$-0.407731\pi$
$751$	15.6659			0.571656			0.285828	+	0.958281i	$0.407731\pi$
$752$		0			0
$753$	−2.60774			−0.0950315
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−32.2736			−1.17301			−0.586503	−	0.809947i	$-0.699496\pi$
$757$	−32.2736			−1.17301			−0.586503	+	0.809947i	$0.699496\pi$
$758$		0			0
$759$		−	27.8279i		−	1.01009i
$760$		0			0
$761$		−	13.2806i		−	0.481422i	−0.970597	−	0.240711i	$-0.922619\pi$
$761$		−	13.2806i		−	0.481422i	0.970597	−	0.240711i	$-0.0773807\pi$
$762$		0			0
$763$	−41.1690			−1.49042
$764$		0			0
$765$		0			0
$766$		0			0
$767$	29.4151	−	1.36435i	1.06212	−	0.0492638i
$768$		0			0
$769$		−	12.2010i		−	0.439980i	−0.975502	−	0.219990i	$-0.929397\pi$
$769$		−	12.2010i		−	0.439980i	0.975502	−	0.219990i	$-0.0706025\pi$
$770$		0			0
$771$	−50.6850			−1.82538
$772$		0			0
$773$		−	27.9721i		−	1.00609i	−0.864261	−	0.503043i	$-0.832214\pi$
$773$		−	27.9721i		−	1.00609i	0.864261	−	0.503043i	$-0.167786\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	−28.6124			−1.02646
$778$		0			0
$779$	8.38993			0.300600
$780$		0			0
$781$	34.9139			1.24932
$782$		0			0
$783$	8.86925			0.316961
$784$		0			0
$785$		0			0
$786$		0			0
$787$		−	10.2336i		−	0.364788i	−0.983225	−	0.182394i	$-0.941615\pi$
$787$		−	10.2336i		−	0.364788i	0.983225	−	0.182394i	$-0.0583847\pi$
$788$		0			0
$789$	−10.9139			−0.388547
$790$		0			0
$791$			8.66822i			0.308206i
$792$		0			0
$793$	−1.72170	−	37.1196i	−0.0611395	−	1.31816i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	23.3532			0.827214			0.413607	−	0.910456i	$-0.364269\pi$
$797$	23.3532			0.827214			0.413607	+	0.910456i	$0.364269\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		−	29.9395i		−	1.05786i
$802$		0			0
$803$	34.6124			1.22145
$804$		0			0
$805$		0			0
$806$		0			0
$807$	−79.2974			−2.79140
$808$		0			0
$809$	37.1364			1.30565			0.652824	−	0.757510i	$-0.273584\pi$
$809$	37.1364			1.30565			0.652824	+	0.757510i	$0.273584\pi$
$810$		0			0
$811$			20.6296i			0.724403i	0.932100	+	0.362201i	$0.117975\pi$
$811$			20.6296i			0.724403i	−0.932100	+	0.362201i	$0.882025\pi$
$812$		0			0
$813$			12.5798i			0.441194i
$814$		0			0
$815$		0			0
$816$		0			0
$817$			49.3811i			1.72763i
$818$		0			0
$819$	37.5822	−	1.74316i	1.31323	−	0.0609109i
$820$		0			0
$821$		−	20.6077i		−	0.719215i	−0.933104	−	0.359608i	$-0.882911\pi$
$821$		−	20.6077i		−	0.719215i	0.933104	−	0.359608i	$-0.117089\pi$
$822$		0			0
$823$	−39.8050			−1.38752			−0.693758	−	0.720208i	$-0.744046\pi$
$823$	−39.8050			−1.38752			−0.693758	+	0.720208i	$0.744046\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			55.8605i			1.94246i	0.238146	+	0.971229i	$0.423460\pi$
$827$			55.8605i			1.94246i	−0.238146	+	0.971229i	$0.576540\pi$
$828$		0			0
$829$	−33.5822			−1.16636			−0.583178	−	0.812344i	$-0.698191\pi$
$829$	−33.5822			−1.16636			−0.583178	+	0.812344i	$0.698191\pi$
$830$		0			0
$831$	7.81110			0.270964
$832$		0			0
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$		−	6.74083i		−	0.232997i
$838$		0			0
$839$		−	26.4454i		−	0.912995i	−0.889725	−	0.456497i	$-0.849104\pi$
$839$		−	26.4454i		−	0.912995i	0.889725	−	0.456497i	$-0.150896\pi$
$840$		0			0
$841$	−9.33411			−0.321866
$842$		0			0
$843$			56.2225i			1.93641i
$844$		0			0
$845$		0			0
$846$		0			0
$847$		−	17.4537i		−	0.599718i
$848$		0			0
$849$	6.43462			0.220836
$850$		0			0
$851$			19.6054i			0.672065i
$852$		0			0
$853$		−	35.7153i		−	1.22287i	−0.791296	−	0.611433i	$-0.790593\pi$
$853$		−	35.7153i		−	1.22287i	0.791296	−	0.611433i	$-0.209407\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−22.0894			−0.754559			−0.377280	−	0.926099i	$-0.623140\pi$
$857$	−22.0894			−0.754559			−0.377280	+	0.926099i	$0.623140\pi$
$858$		0			0
$859$	−39.2201			−1.33817			−0.669087	−	0.743184i	$-0.733315\pi$
$859$	−39.2201			−1.33817			−0.669087	+	0.743184i	$0.733315\pi$
$860$		0			0
$861$	−12.0000			−0.408959
$862$		0			0
$863$		−	28.9744i		−	0.986301i	−0.869944	−	0.493150i	$-0.835845\pi$
$863$		−	28.9744i		−	0.986301i	0.869944	−	0.493150i	$-0.164155\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	44.2285			1.50208
$868$		0			0
$869$			25.8837i			0.878045i
$870$		0			0
$871$	25.5822	−	1.18657i	0.866819	−	0.0402053i
$872$		0			0
$873$			1.74316i			0.0589970i
$874$		0			0
$875$		0			0
$876$		0			0
$877$			30.9251i			1.04427i	0.852864	+	0.522133i	$0.174863\pi$
$877$			30.9251i			1.04427i	−0.852864	+	0.522133i	$0.825137\pi$
$878$		0			0
$879$		−	62.2225i		−	2.09871i
$880$		0			0
$881$	41.2695			1.39041			0.695203	−	0.718814i	$-0.255315\pi$
$881$	41.2695			1.39041			0.695203	+	0.718814i	$0.255315\pi$
$882$		0			0
$883$	−15.1973			−0.511430			−0.255715	−	0.966752i	$-0.582311\pi$
$883$	−15.1973			−0.511430			−0.255715	+	0.966752i	$0.582311\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	12.4174			0.416937			0.208468	−	0.978029i	$-0.433152\pi$
$887$	12.4174			0.416937			0.208468	+	0.978029i	$0.433152\pi$
$888$		0			0
$889$			44.6850i			1.49869i
$890$		0			0
$891$		−	13.2252i		−	0.443061i
$892$		0			0
$893$	−58.6124			−1.96139
$894$		0			0
$895$		0			0
$896$		0			0
$897$	−2.14521	−	46.2504i	−0.0716266	−	1.54426i
$898$		0			0
$899$		−	14.9465i		−	0.498494i
$900$		0			0
$901$		0			0
$902$		0			0
$903$		−	70.6292i		−	2.35039i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	11.4709			0.380886			0.190443	−	0.981698i	$-0.439008\pi$
$907$	11.4709			0.380886			0.190443	+	0.981698i	$0.439008\pi$
$908$		0			0
$909$	48.0168			1.59262
$910$		0			0
$911$	54.5734			1.80810			0.904048	−	0.427430i	$-0.140581\pi$
$911$	54.5734			1.80810			0.904048	+	0.427430i	$0.140581\pi$
$912$		0			0
$913$	8.60774			0.284875
$914$		0			0
$915$		0			0
$916$		0			0
$917$			24.5566i			0.810930i
$918$		0			0
$919$	9.05815			0.298801			0.149400	−	0.988777i	$-0.452266\pi$
$919$	9.05815			0.298801			0.149400	+	0.988777i	$0.452266\pi$
$920$		0			0
$921$			35.2928i			1.16294i
$922$		0			0
$923$	58.0275	−	2.69147i	1.91000	−	0.0885907i
$924$		0			0
$925$		0			0
$926$		0			0
$927$	15.5993			0.512350
$928$		0			0
$929$			50.2225i			1.64775i	0.566774	+	0.823873i	$0.308191\pi$
$929$			50.2225i			1.64775i	−0.566774	+	0.823873i	$0.691809\pi$
$930$		0			0
$931$			3.35362i			0.109911i
$932$		0			0
$933$	5.53747			0.181289
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−5.38759			−0.176005			−0.0880025	−	0.996120i	$-0.528048\pi$
$937$	−5.38759			−0.176005			−0.0880025	+	0.996120i	$0.528048\pi$
$938$		0			0
$939$	−58.8181			−1.91946
$940$		0			0
$941$		−	58.8302i		−	1.91781i	−0.283726	−	0.958905i	$-0.591571\pi$
$941$		−	58.8302i		−	1.91781i	0.283726	−	0.958905i	$-0.408429\pi$
$942$		0			0
$943$			8.22248i			0.267761i
$944$		0			0
$945$		0			0
$946$		0			0
$947$			12.6403i			0.410755i	0.978683	+	0.205377i	$0.0658422\pi$
$947$			12.6403i			0.410755i	−0.978683	+	0.205377i	$0.934158\pi$
$948$		0			0
$949$	57.5264	−	2.66822i	1.86738	−	0.0866141i
$950$		0			0
$951$		−	27.6706i		−	0.897279i
$952$		0			0
$953$	32.6077			1.05627			0.528134	−	0.849161i	$-0.322892\pi$
$953$	32.6077			1.05627			0.528134	+	0.849161i	$0.322892\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$			25.0023i			0.808211i
$958$		0			0
$959$	4.61241			0.148942
$960$		0			0
$961$	19.6403			0.633558
$962$		0			0
$963$	−26.6231			−0.857918
$964$		0			0
$965$		0			0
$966$		0			0
$967$			20.5845i			0.661953i	0.943639	+	0.330976i	$0.107378\pi$
$967$			20.5845i			0.661953i	−0.943639	+	0.330976i	$0.892622\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−60.8349			−1.95228			−0.976142	−	0.217132i	$-0.930330\pi$
$971$	−60.8349			−1.95228			−0.976142	+	0.217132i	$0.930330\pi$
$972$		0			0
$973$		−	53.3700i		−	1.71096i
$974$		0			0
$975$		0			0
$976$		0			0
$977$			7.30387i			0.233672i	0.993151	+	0.116836i	$0.0372751\pi$
$977$			7.30387i			0.233672i	−0.993151	+	0.116836i	$0.962725\pi$
$978$		0			0
$979$	17.2155			0.550209
$980$		0			0
$981$		−	56.0382i		−	1.78916i
$982$		0			0
$983$			40.2504i			1.28379i	0.766793	+	0.641894i	$0.221851\pi$
$983$			40.2504i			1.28379i	−0.766793	+	0.641894i	$0.778149\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	83.8326			2.66842
$988$		0			0
$989$	−48.3956			−1.53889
$990$		0			0
$991$	−50.2271			−1.59552			−0.797759	−	0.602977i	$-0.793981\pi$
$991$	−50.2271			−1.59552			−0.797759	+	0.602977i	$0.793981\pi$
$992$		0			0
$993$			3.03024i			0.0961617i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−0.273633			−0.00866606			−0.00433303	−	0.999991i	$-0.501379\pi$
$997$	−0.273633			−0.00866606			−0.00433303	+	0.999991i	$0.501379\pi$
$998$		0			0
$999$		−	7.94419i		−	0.251343i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1300.2.f.e.701.1		6
5.2	odd	4		1300.2.d.d.649.6		6
5.3	odd	4		1300.2.d.c.649.1		6
5.4	even	2		260.2.f.a.181.5	✓	6
13.12	even	2	inner	1300.2.f.e.701.2		6
15.14	odd	2		2340.2.c.d.181.6		6
20.19	odd	2		1040.2.k.c.961.1		6
65.12	odd	4		1300.2.d.c.649.6		6
65.34	odd	4		3380.2.a.n.1.3		3
65.38	odd	4		1300.2.d.d.649.1		6
65.44	odd	4		3380.2.a.m.1.3		3
65.64	even	2		260.2.f.a.181.6	yes	6
195.194	odd	2		2340.2.c.d.181.1		6
260.259	odd	2		1040.2.k.c.961.2		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
260.2.f.a.181.5	✓	6	5.4	even	2
260.2.f.a.181.6	yes	6	65.64	even	2
1040.2.k.c.961.1		6	20.19	odd	2
1040.2.k.c.961.2		6	260.259	odd	2
1300.2.d.c.649.1		6	5.3	odd	4
1300.2.d.c.649.6		6	65.12	odd	4
1300.2.d.d.649.1		6	65.38	odd	4
1300.2.d.d.649.6		6	5.2	odd	4
1300.2.f.e.701.1		6	1.1	even	1	trivial
1300.2.f.e.701.2		6	13.12	even	2	inner
2340.2.c.d.181.1		6	195.194	odd	2
2340.2.c.d.181.6		6	15.14	odd	2
3380.2.a.m.1.3		3	65.44	odd	4
3380.2.a.n.1.3		3	65.34	odd	4

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:	\( N \)	\(=\)	\( 1300 = 2^{2} \cdot 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1300.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-2.60168 q^{3} -2.76873i q^{7} +3.76873 q^{9} +O(q^{10})\)	\(q-2.60168 q^{3} -2.76873i q^{7} +3.76873 q^{9} +2.16706i q^{11} +(0.167055 + 3.60168i) q^{13} -5.03630i q^{19} +7.20336i q^{21} +4.93579 q^{23} -2.00000 q^{27} -4.43462 q^{29} +3.37041i q^{31} -5.63798i q^{33} +3.97209i q^{37} +(-0.434624 - 9.37041i) q^{39} +1.66589i q^{41} -9.80504 q^{43} -11.6380i q^{47} -0.665890 q^{49} +12.7408 q^{53} +13.1028i q^{57} -8.16706i q^{59} -10.3062 q^{61} -10.4346i q^{63} -7.10284i q^{67} -12.8413 q^{69} -16.1112i q^{71} -15.9721i q^{73} +6.00000 q^{77} +11.9442 q^{79} -6.10284 q^{81} -3.97209i q^{83} +11.5375 q^{87} -7.94419i q^{89} +(9.97209 - 0.462531i) q^{91} -8.76873i q^{93} +0.462531i q^{97} +8.16706i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 6 q^{9}+O(q^{10}) \) 6 * q + 6 * q^9	\( 6 q + 6 q^{9} + 12 q^{23} - 12 q^{27} - 12 q^{29} + 12 q^{39} - 12 q^{43} - 6 q^{49} + 12 q^{53} - 12 q^{61} + 36 q^{77} - 24 q^{79} - 18 q^{81} + 36 q^{87} + 12 q^{91}+O(q^{100}) \) 6 * q + 6 * q^9 + 12 * q^23 - 12 * q^27 - 12 * q^29 + 12 * q^39 - 12 * q^43 - 6 * q^49 + 12 * q^53 - 12 * q^61 + 36 * q^77 - 24 * q^79 - 18 * q^81 + 36 * q^87 + 12 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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