LMFDB - Embedded newform 1300.2.d.d.649.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1300,2,Mod(649,1300)]

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, 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("1300.649");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1300 = 2^{2} \cdot 5^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1300.d (of order $2$, degree $1$, 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:	$10.3805522628$
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^{2} $
Twist minimal:	no (minimal twist has level 260)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			649.6
Root			$2.60168i$ of defining polynomial
Character	$\chi$	$=$	1300.649
Dual form			1300.2.d.d.649.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.60168i q^{3} +2.76873 q^{7} -3.76873 q^{9} +O(q^{10})$	$q+2.60168i q^{3} +2.76873 q^{7} -3.76873 q^{9} +2.16706i q^{11} +(3.60168 - 0.167055i) q^{13} +5.03630i q^{19} +7.20336i q^{21} -4.93579i q^{23} -2.00000i q^{27} +4.43462 q^{29} +3.37041i q^{31} -5.63798 q^{33} -3.97209 q^{37} +(0.434624 + 9.37041i) q^{39} +1.66589i q^{41} +9.80504i q^{43} +11.6380 q^{47} +0.665890 q^{49} -12.7408i q^{53} -13.1028 q^{57} +8.16706i q^{59} -10.3062 q^{61} -10.4346 q^{63} +7.10284 q^{67} +12.8413 q^{69} -16.1112i q^{71} -15.9721 q^{73} +6.00000i q^{77} -11.9442 q^{79} -6.10284 q^{81} -3.97209 q^{83} +11.5375i q^{87} +7.94419i q^{89} +(9.97209 - 0.462531i) q^{91} -8.76873 q^{93} -0.462531 q^{97} -8.16706i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 6 q^{9}+O(q^{10}) $ 6 * q - 6 * q^9	$ 6 q - 6 q^{9} + 6 q^{13} + 12 q^{29} + 12 q^{33} + 24 q^{37} - 12 q^{39} + 24 q^{47} + 6 q^{49} - 60 q^{57} - 12 q^{61} - 48 q^{63} + 24 q^{67} - 48 q^{73} + 24 q^{79} - 18 q^{81} + 24 q^{83} + 12 q^{91} - 36 q^{93} - 36 q^{97}+O(q^{100}) $ 6 * q - 6 * q^9 + 6 * q^13 + 12 * q^29 + 12 * q^33 + 24 * q^37 - 12 * q^39 + 24 * q^47 + 6 * q^49 - 60 * q^57 - 12 * q^61 - 48 * q^63 + 24 * q^67 - 48 * q^73 + 24 * q^79 - 18 * q^81 + 24 * q^83 + 12 * q^91 - 36 * q^93 - 36 * q^97

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
$3$			2.60168i			1.50208i	0.660257	+	0.751040i	$0.270448\pi$
$3$			2.60168i			1.50208i	−0.660257	+	0.751040i	$0.729552\pi$
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	2.76873			1.04648			0.523242	−	0.852184i	$-0.324723\pi$
$7$	2.76873			1.04648			0.523242	+	0.852184i	$0.324723\pi$
$8$		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
$13$	3.60168	−	0.167055i	0.998926	−	0.0463328i
$13$	3.60168	−	0.167055i	0.998926	−	0.0463328i
$14$		0			0
$15$		0			0
$16$		0			0
$17$		0			0		1.00000			$0$
$17$		0			0		−1.00000			$\pi$
$18$		0			0
$19$			5.03630i			1.15541i	0.816247	+	0.577704i	$0.196051\pi$
$19$			5.03630i			1.15541i	−0.816247	+	0.577704i	$0.803949\pi$
$20$		0			0
$21$			7.20336i			1.57190i
$22$		0			0
$23$		−	4.93579i		−	1.02918i	−0.857435	−	0.514592i	$-0.827944\pi$
$23$		−	4.93579i		−	1.02918i	0.857435	−	0.514592i	$-0.172056\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$		−	2.00000i		−	0.384900i
$28$		0			0
$29$	4.43462			0.823489			0.411744	−	0.911299i	$-0.364920\pi$
$29$	4.43462			0.823489			0.411744	+	0.911299i	$0.364920\pi$
$30$		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
$33$	−5.63798			−0.981447
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−3.97209			−0.653008			−0.326504	−	0.945196i	$-0.605871\pi$
$37$	−3.97209			−0.653008			−0.326504	+	0.945196i	$0.605871\pi$
$38$		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
$43$			9.80504i			1.49525i	0.664119	+	0.747627i	$0.268807\pi$
$43$			9.80504i			1.49525i	−0.664119	+	0.747627i	$0.731193\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	11.6380			1.69757			0.848787	−	0.528735i	$-0.177333\pi$
$47$	11.6380			1.69757			0.848787	+	0.528735i	$0.177333\pi$
$48$		0			0
$49$	0.665890			0.0951271
$50$		0			0
$51$		0			0
$52$		0			0
$53$		−	12.7408i		−	1.75009i	−0.484044	−	0.875044i	$-0.660833\pi$
$53$		−	12.7408i		−	1.75009i	0.484044	−	0.875044i	$-0.339167\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	−13.1028			−1.73551
$58$		0			0
$59$			8.16706i			1.06326i	0.846977	+	0.531630i	$0.178420\pi$
$59$			8.16706i			1.06326i	−0.846977	+	0.531630i	$0.821580\pi$
$60$		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
$63$	−10.4346			−1.31464
$64$		0			0
$65$		0			0
$66$		0			0
$67$	7.10284			0.867751			0.433875	−	0.900973i	$-0.357146\pi$
$67$	7.10284			0.867751			0.433875	+	0.900973i	$0.357146\pi$
$68$		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
$73$	−15.9721			−1.86939			−0.934696	−	0.355448i	$-0.884328\pi$
$73$	−15.9721			−1.86939			−0.934696	+	0.355448i	$0.884328\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$			6.00000i			0.683763i
$78$		0			0
$79$	−11.9442			−1.34383			−0.671913	−	0.740630i	$-0.734527\pi$
$79$	−11.9442			−1.34383			−0.671913	+	0.740630i	$0.734527\pi$
$80$		0			0
$81$	−6.10284			−0.678094
$82$		0			0
$83$	−3.97209			−0.435994			−0.217997	−	0.975949i	$-0.569952\pi$
$83$	−3.97209			−0.435994			−0.217997	+	0.975949i	$0.569952\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$			11.5375i			1.23695i
$88$		0			0
$89$			7.94419i			0.842082i	0.907042	+	0.421041i	$0.138335\pi$
$89$			7.94419i			0.842082i	−0.907042	+	0.421041i	$0.861665\pi$
$90$		0			0
$91$	9.97209	−	0.462531i	1.04536	−	0.0484865i
$92$		0			0
$93$	−8.76873			−0.909275
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−0.462531			−0.0469629			−0.0234815	−	0.999724i	$-0.507475\pi$
$97$	−0.462531			−0.0469629			−0.0234815	+	0.999724i	$0.507475\pi$
$98$		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
$103$		−	4.13915i		−	0.407842i	−0.978987	−	0.203921i	$-0.934631\pi$
$103$		−	4.13915i		−	0.407842i	0.978987	−	0.203921i	$-0.0653686\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	7.06421i		−	0.682923i	−0.939896	−	0.341462i	$-0.889078\pi$
$107$		−	7.06421i		−	0.682923i	0.939896	−	0.341462i	$-0.110922\pi$
$108$		0			0
$109$			14.8692i			1.42422i	0.702070	+	0.712108i	$0.252259\pi$
$109$			14.8692i			1.42422i	−0.702070	+	0.712108i	$0.747741\pi$
$110$		0			0
$111$		−	10.3341i		−	0.980870i
$112$		0			0
$113$			3.13075i			0.294516i	0.989098	+	0.147258i	$0.0470448\pi$
$113$			3.13075i			0.294516i	−0.989098	+	0.147258i	$0.952955\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	−13.5738	+	0.629587i	−1.25490	+	0.0582053i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	6.30387			0.573079
$122$		0			0
$123$	−4.33411			−0.390794
$124$		0			0
$125$		0			0
$126$		0			0
$127$		−	16.1391i		−	1.43212i	−0.698040	−	0.716059i	$-0.745944\pi$
$127$		−	16.1391i		−	1.43212i	0.698040	−	0.716059i	$-0.254056\pi$
$128$		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
$133$			13.9442i			1.20911i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−1.66589			−0.142327			−0.0711633	−	0.997465i	$-0.522671\pi$
$137$	−1.66589			−0.142327			−0.0711633	+	0.997465i	$0.522671\pi$
$138$		0			0
$139$	−19.2760			−1.63497			−0.817483	−	0.575953i	$-0.804631\pi$
$139$	−19.2760			−1.63497			−0.817483	+	0.575953i	$0.804631\pi$
$140$		0			0
$141$			30.2783i			2.54989i
$142$		0			0
$143$	0.362018	+	7.80504i	0.0302734	+	0.652690i
$144$		0			0
$145$		0			0
$146$		0			0
$147$			1.73243i			0.142889i
$148$		0			0
$149$		−	13.9442i		−	1.14235i	−0.820828	−	0.571176i	$-0.806487\pi$
$149$		−	13.9442i		−	1.14235i	0.820828	−	0.571176i	$-0.193513\pi$
$150$		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
$157$		−	12.3341i		−	0.984369i	−0.870491	−	0.492185i	$-0.836199\pi$
$157$		−	12.3341i		−	0.984369i	0.870491	−	0.492185i	$-0.163801\pi$
$158$		0			0
$159$	33.1475			2.62877
$160$		0			0
$161$		−	13.6659i		−	1.07702i
$162$		0			0
$163$	20.5072			1.60625			0.803125	−	0.595810i	$-0.203169\pi$
$163$	20.5072			1.60625			0.803125	+	0.595810i	$0.203169\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	19.3039			1.49378			0.746889	−	0.664948i	$-0.231547\pi$
$167$	19.3039			1.49378			0.746889	+	0.664948i	$0.231547\pi$
$168$		0			0
$169$	12.9442	−	1.20336i	0.995707	−	0.0925660i
$170$		0			0
$171$		−	18.9805i		−	1.45147i
$172$		0			0
$173$			9.61007i			0.730640i	0.930882	+	0.365320i	$0.119041\pi$
$173$			9.61007i			0.730640i	−0.930882	+	0.365320i	$0.880959\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	−21.2481			−1.59710
$178$		0			0
$179$	9.87158			0.737836			0.368918	−	0.929462i	$-0.379728\pi$
$179$	9.87158			0.737836			0.368918	+	0.929462i	$0.379728\pi$
$180$		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
$183$		−	26.8134i		−	1.98211i
$184$		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
$193$	4.07261			0.293153			0.146576	−	0.989199i	$-0.453175\pi$
$193$	4.07261			0.293153			0.146576	+	0.989199i	$0.453175\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	21.6101			1.53965			0.769827	−	0.638253i	$-0.220342\pi$
$197$	21.6101			1.53965			0.769827	+	0.638253i	$0.220342\pi$
$198$		0			0
$199$	11.6101			0.823016			0.411508	−	0.911406i	$-0.365002\pi$
$199$	11.6101			0.823016			0.411508	+	0.911406i	$0.365002\pi$
$200$		0			0
$201$			18.4793i			1.30343i
$202$		0			0
$203$	12.2783			0.861767
$204$		0			0
$205$		0			0
$206$		0			0
$207$			18.6017i			1.29291i
$208$		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
$213$	41.9163			2.87206
$214$		0			0
$215$		0			0
$216$		0			0
$217$			9.33178i			0.633482i
$218$		0			0
$219$		−	41.5543i		−	2.80798i
$220$		0			0
$221$		0			0
$222$		0			0
$223$	11.4370			0.765875			0.382938	−	0.923774i	$-0.374912\pi$
$223$	11.4370			0.765875			0.382938	+	0.923774i	$0.374912\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	15.9721			1.06011			0.530053	−	0.847965i	$-0.322172\pi$
$227$	15.9721			1.06011			0.530053	+	0.847965i	$0.322172\pi$
$228$		0			0
$229$		−	2.12842i		−	0.140650i	−0.997524	−	0.0703250i	$-0.977596\pi$
$229$		−	2.12842i		−	0.140650i	0.997524	−	0.0703250i	$-0.0224036\pi$
$230$		0			0
$231$	−15.6101			−1.02707
$232$		0			0
$233$		−	2.86925i		−	0.187971i	−0.995574	−	0.0939853i	$-0.970039\pi$
$233$		−	2.86925i		−	0.187971i	0.995574	−	0.0939853i	$-0.0299607\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		−	31.0749i		−	2.01853i
$238$		0			0
$239$			3.16939i			0.205011i	0.994732	+	0.102505i	$0.0326859\pi$
$239$			3.16939i			0.205011i	−0.994732	+	0.102505i	$0.967314\pi$
$240$		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
$243$		−	21.8776i		−	1.40345i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	0.841340	+	18.1391i	0.0535332	+	1.15417i
$248$		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
$253$	10.6961			0.672460
$254$		0			0
$255$		0			0
$256$		0			0
$257$			19.4817i			1.21523i	0.794231	+	0.607616i	$0.207874\pi$
$257$			19.4817i			1.21523i	−0.794231	+	0.607616i	$0.792126\pi$
$258$		0			0
$259$	−10.9977			−0.683362
$260$		0			0
$261$	−16.7129			−1.03450
$262$		0			0
$263$		−	4.19496i		−	0.258672i	−0.991601	−	0.129336i	$-0.958715\pi$
$263$		−	4.19496i		−	0.258672i	0.991601	−	0.129336i	$-0.0412847\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	−20.6682			−1.26487
$268$		0			0
$269$	−30.4793			−1.85836			−0.929179	−	0.369631i	$-0.879484\pi$
$269$	−30.4793			−1.85836			−0.929179	+	0.369631i	$0.879484\pi$
$270$		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
$273$	1.20336	+	25.9442i	0.0728306	+	1.57021i
$274$		0			0
$275$		0			0
$276$		0			0
$277$		−	3.00233i		−	0.180393i	−0.995924	−	0.0901963i	$-0.971251\pi$
$277$		−	3.00233i		−	0.180393i	0.995924	−	0.0901963i	$-0.0287494\pi$
$278$		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
$283$			2.47326i			0.147020i	0.997294	+	0.0735100i	$0.0234201\pi$
$283$			2.47326i			0.147020i	−0.997294	+	0.0735100i	$0.976580\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$			4.61241i			0.272262i
$288$		0			0
$289$	17.0000			1.00000
$290$		0			0
$291$		−	1.20336i		−	0.0705421i
$292$		0			0
$293$	23.9163			1.39720			0.698602	−	0.715511i	$-0.253806\pi$
$293$	23.9163			1.39720			0.698602	+	0.715511i	$0.253806\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	4.33411			0.251491
$298$		0			0
$299$	−0.824549	−	17.7771i	−0.0476849	−	1.02808i
$300$		0			0
$301$			27.1475i			1.56476i
$302$		0			0
$303$			33.1475i			1.90428i
$304$		0			0
$305$		0			0
$306$		0			0
$307$	13.5654			0.774217			0.387108	−	0.922034i	$-0.373474\pi$
$307$	13.5654			0.774217			0.387108	+	0.922034i	$0.373474\pi$
$308$		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
$313$		−	22.6077i		−	1.27787i	−0.769263	−	0.638933i	$-0.779376\pi$
$313$		−	22.6077i		−	1.27787i	0.769263	−	0.638933i	$-0.220624\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−10.6357			−0.597358			−0.298679	−	0.954354i	$-0.596546\pi$
$317$	−10.6357			−0.597358			−0.298679	+	0.954354i	$0.596546\pi$
$318$		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
$327$	−38.6850			−2.13929
$328$		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
$333$	14.9698			0.820338
$334$		0			0
$335$		0			0
$336$		0			0
$337$			20.0000i			1.08947i	0.838608	+	0.544735i	$0.183370\pi$
$337$			20.0000i			1.08947i	−0.838608	+	0.544735i	$0.816630\pi$
$338$		0			0
$339$	−8.14521			−0.442387
$340$		0			0
$341$	−7.30387			−0.395527
$342$		0			0
$343$	−17.5375			−0.946934
$344$		0			0
$345$		0			0
$346$		0			0
$347$		−	4.67429i		−	0.250929i	−0.992098	−	0.125464i	$-0.959958\pi$
$347$		−	4.67429i		−	0.250929i	0.992098	−	0.125464i	$-0.0400421\pi$
$348$		0			0
$349$			4.33411i			0.232000i	0.993249	+	0.116000i	$0.0370072\pi$
$349$			4.33411i			0.232000i	−0.993249	+	0.116000i	$0.962993\pi$
$350$		0			0
$351$	−0.334110	−	7.20336i	−0.0178335	−	0.384487i
$352$		0			0
$353$	−21.3085			−1.13414			−0.567069	−	0.823670i	$-0.691923\pi$
$353$	−21.3085			−1.13414			−0.567069	+	0.823670i	$0.691923\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		−	12.7795i		−	0.674474i	−0.941420	−	0.337237i	$-0.890508\pi$
$359$		−	12.7795i		−	0.674474i	0.941420	−	0.337237i	$-0.109492\pi$
$360$		0			0
$361$	−6.36435			−0.334966
$362$		0			0
$363$			16.4007i			0.860811i
$364$		0			0
$365$		0			0
$366$		0			0
$367$		−	12.1950i		−	0.636572i	−0.947995	−	0.318286i	$-0.896893\pi$
$367$		−	12.1950i		−	0.636572i	0.947995	−	0.318286i	$-0.103107\pi$
$368$		0			0
$369$		−	6.27830i		−	0.326835i
$370$		0			0
$371$		−	35.2760i		−	1.83144i
$372$		0			0
$373$		−	28.2225i		−	1.46130i	−0.682750	−	0.730652i	$-0.739216\pi$
$373$		−	28.2225i		−	1.46130i	0.682750	−	0.730652i	$-0.260784\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	15.9721	−	0.740827i	0.822605	−	0.0381545i
$378$		0			0
$379$		−	23.3146i		−	1.19759i	−0.800902	−	0.598795i	$-0.795646\pi$
$379$		−	23.3146i		−	1.19759i	0.800902	−	0.598795i	$-0.204354\pi$
$380$		0			0
$381$	41.9889			2.15116
$382$		0			0
$383$	−8.02791			−0.410207			−0.205103	−	0.978740i	$-0.565753\pi$
$383$	−8.02791			−0.410207			−0.205103	+	0.978740i	$0.565753\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		−	36.9526i		−	1.87841i
$388$		0			0
$389$	−23.7385			−1.20359			−0.601795	−	0.798651i	$-0.705547\pi$
$389$	−23.7385			−1.20359			−0.601795	+	0.798651i	$0.705547\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		−	23.0749i		−	1.16398i
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−3.04703			−0.152926			−0.0764630	−	0.997072i	$-0.524363\pi$
$397$	−3.04703			−0.152926			−0.0764630	+	0.997072i	$0.524363\pi$
$398$		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
$403$	0.563045	+	12.1391i	0.0280473	+	0.604694i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	8.60774i		−	0.426670i
$408$		0			0
$409$			33.4090i			1.65197i	0.563691	+	0.825986i	$0.309381\pi$
$409$			33.4090i			1.65197i	−0.563691	+	0.825986i	$0.690619\pi$
$410$		0			0
$411$		−	4.33411i		−	0.213786i
$412$		0			0
$413$			22.6124i			1.11268i
$414$		0			0
$415$		0			0
$416$		0			0
$417$		−	50.1499i		−	2.45585i
$418$		0			0
$419$	30.7408			1.50179			0.750894	−	0.660423i	$-0.229623\pi$
$419$	30.7408			1.50179			0.750894	+	0.660423i	$0.229623\pi$
$420$		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
$423$	−43.8605			−2.13257
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−28.5351			−1.38091
$428$		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
$433$		−	2.00000i		−	0.0961139i	−0.998845	−	0.0480569i	$-0.984697\pi$
$433$		−	2.00000i		−	0.0961139i	0.998845	−	0.0480569i	$-0.0153029\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	24.8581			1.18913
$438$		0			0
$439$	11.6659			0.556783			0.278391	−	0.960468i	$-0.410199\pi$
$439$	11.6659			0.556783			0.278391	+	0.960468i	$0.410199\pi$
$440$		0			0
$441$	−2.50956			−0.119503
$442$		0			0
$443$			20.8074i			0.988588i	0.869295	+	0.494294i	$0.164573\pi$
$443$			20.8074i			0.988588i	−0.869295	+	0.494294i	$0.835427\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	36.2783			1.71590
$448$		0			0
$449$		−	30.2783i		−	1.42892i	−0.699676	−	0.714461i	$-0.746672\pi$
$449$		−	30.2783i		−	1.42892i	0.699676	−	0.714461i	$-0.253328\pi$
$450$		0			0
$451$	−3.61007			−0.169992
$452$		0			0
$453$	−18.1173			−0.851225
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−22.6124			−1.05776			−0.528882	−	0.848695i	$-0.677389\pi$
$457$	−22.6124			−1.05776			−0.528882	+	0.848695i	$0.677389\pi$
$458$		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
$463$	−11.6380			−0.540863			−0.270431	−	0.962739i	$-0.587166\pi$
$463$	−11.6380			−0.540863			−0.270431	+	0.962739i	$0.587166\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$			25.0642i			1.15983i	0.814676	+	0.579917i	$0.196915\pi$
$467$			25.0642i			1.15983i	−0.814676	+	0.579917i	$0.803085\pi$
$468$		0			0
$469$	19.6659			0.908086
$470$		0			0
$471$	32.0894			1.47860
$472$		0			0
$473$	−21.2481			−0.976987
$474$		0			0
$475$		0			0
$476$		0			0
$477$			48.0168i			2.19854i
$478$		0			0
$479$			18.4407i			0.842577i	0.906927	+	0.421288i	$0.138422\pi$
$479$			18.4407i			0.842577i	−0.906927	+	0.421288i	$0.861578\pi$
$480$		0			0
$481$	−14.3062	+	0.663559i	−0.652307	+	0.0302557i
$482$		0			0
$483$	35.5543			1.61777
$484$		0			0
$485$		0			0
$486$		0			0
$487$	−25.1196			−1.13828			−0.569140	−	0.822241i	$-0.692724\pi$
$487$	−25.1196			−1.13828			−0.569140	+	0.822241i	$0.692724\pi$
$488$		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
$497$		−	44.6077i		−	2.00093i
$498$		0			0
$499$		−	33.3704i		−	1.49386i	−0.664900	−	0.746932i	$-0.731526\pi$
$499$		−	33.3704i		−	1.49386i	0.664900	−	0.746932i	$-0.268474\pi$
$500$		0			0
$501$			50.2225i			2.24377i
$502$		0			0
$503$		−	12.6789i		−	0.565326i	−0.959219	−	0.282663i	$-0.908782\pi$
$503$		−	12.6789i		−	0.565326i	0.959219	−	0.282663i	$-0.0912179\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	3.13075	+	33.6766i	0.139042	+	1.49563i
$508$		0			0
$509$			17.6147i			0.780759i	0.920654	+	0.390380i	$0.127656\pi$
$509$			17.6147i			0.780759i	−0.920654	+	0.390380i	$0.872344\pi$
$510$		0			0
$511$	−44.2225			−1.95629
$512$		0			0
$513$	10.0726			0.444716
$514$		0			0
$515$		0			0
$516$		0			0
$517$			25.2201i			1.10918i
$518$		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
$523$		−	20.4733i		−	0.895233i	−0.894226	−	0.447617i	$-0.852273\pi$
$523$		−	20.4733i		−	0.895233i	0.894226	−	0.447617i	$-0.147727\pi$
$524$		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
$533$	0.278295	+	6.00000i	0.0120543	+	0.259889i
$534$		0			0
$535$		0			0
$536$		0			0
$537$			25.6827i			1.10829i
$538$		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
$543$		−	5.13075i		−	0.220182i
$544$		0			0
$545$		0			0
$546$		0			0
$547$			10.4174i			0.445418i	0.974885	+	0.222709i	$0.0714899\pi$
$547$			10.4174i			0.445418i	−0.974885	+	0.222709i	$0.928510\pi$
$548$		0			0
$549$	38.8413			1.65771
$550$		0			0
$551$			22.3341i			0.951465i
$552$		0			0
$553$	−33.0703			−1.40629
$554$		0			0
$555$		0			0
$556$		0			0
$557$	20.5845			0.872193			0.436097	−	0.899900i	$-0.356361\pi$
$557$	20.5845			0.872193			0.436097	+	0.899900i	$0.356361\pi$
$558$		0			0
$559$	1.63798	+	35.3146i	0.0692793	+	1.49365i
$560$		0			0
$561$		0			0
$562$		0			0
$563$		−	23.6766i		−	0.997850i	−0.866645	−	0.498925i	$-0.833728\pi$
$563$		−	23.6766i		−	0.997850i	0.866645	−	0.498925i	$-0.166272\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	−16.8972			−0.709614
$568$		0			0
$569$	−5.43695			−0.227929			−0.113965	−	0.993485i	$-0.536355\pi$
$569$	−5.43695			−0.227929			−0.113965	+	0.993485i	$0.536355\pi$
$570$		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
$573$		−	14.9297i		−	0.623698i
$574$		0			0
$575$		0			0
$576$		0			0
$577$	1.76640			0.0735363			0.0367682	−	0.999324i	$-0.488294\pi$
$577$	1.76640			0.0735363			0.0367682	+	0.999324i	$0.488294\pi$
$578$		0			0
$579$			10.5956i			0.440339i
$580$		0			0
$581$	−10.9977			−0.456260
$582$		0			0
$583$	27.6101			1.14349
$584$		0			0
$585$		0			0
$586$		0			0
$587$	22.9139			0.945760			0.472880	−	0.881127i	$-0.343214\pi$
$587$	22.9139			0.945760			0.472880	+	0.881127i	$0.343214\pi$
$588$		0			0
$589$	−16.9744			−0.699419
$590$		0			0
$591$			56.2225i			2.31268i
$592$		0			0
$593$	6.72404			0.276123			0.138062	−	0.990424i	$-0.455913\pi$
$593$	6.72404			0.276123			0.138062	+	0.990424i	$0.455913\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$			30.2057i			1.23624i
$598$		0			0
$599$	19.8669			0.811740			0.405870	−	0.913931i	$-0.366969\pi$
$599$	19.8669			0.811740			0.405870	+	0.913931i	$0.366969\pi$
$600$		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
$603$	−26.7687			−1.09011
$604$		0			0
$605$		0			0
$606$		0			0
$607$		−	14.4174i		−	0.585186i	−0.956237	−	0.292593i	$-0.905482\pi$
$607$		−	14.4174i		−	0.585186i	0.956237	−	0.292593i	$-0.0945181\pi$
$608$		0			0
$609$			31.9442i			1.29444i
$610$		0			0
$611$	41.9163	−	1.94419i	1.69575	−	0.0786533i
$612$		0			0
$613$	11.3364			0.457875			0.228937	−	0.973441i	$-0.426475\pi$
$613$	11.3364			0.457875			0.228937	+	0.973441i	$0.426475\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−34.8907			−1.40465			−0.702323	−	0.711858i	$-0.747854\pi$
$617$	−34.8907			−1.40465			−0.702323	+	0.711858i	$0.747854\pi$
$618$		0			0
$619$		−	32.6464i		−	1.31217i	−0.754688	−	0.656084i	$-0.772212\pi$
$619$		−	32.6464i		−	1.31217i	0.754688	−	0.656084i	$-0.227788\pi$
$620$		0			0
$621$	−9.87158			−0.396133
$622$		0			0
$623$			21.9953i			0.881225i
$624$		0			0
$625$		0			0
$626$		0			0
$627$		−	28.3946i		−	1.13397i
$628$		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
$633$			42.3509i			1.68330i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	2.39832	−	0.111240i	0.0950249	−	0.00440750i
$638$		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
$643$	20.3062			0.800798			0.400399	−	0.916341i	$-0.368871\pi$
$643$	20.3062			0.800798			0.400399	+	0.916341i	$0.368871\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	30.8027i		−	1.21098i	−0.795853	−	0.605490i	$-0.792977\pi$
$647$		−	30.8027i		−	1.21098i	0.795853	−	0.605490i	$-0.207023\pi$
$648$		0			0
$649$	−17.6985			−0.694725
$650$		0			0
$651$	−24.2783			−0.951541
$652$		0			0
$653$			3.61007i			0.141273i	0.997502	+	0.0706366i	$0.0225031\pi$
$653$			3.61007i			0.141273i	−0.997502	+	0.0706366i	$0.977497\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	60.1946			2.34841
$658$		0			0
$659$	6.74083			0.262585			0.131293	−	0.991344i	$-0.458087\pi$
$659$	6.74083			0.262585			0.131293	+	0.991344i	$0.458087\pi$
$660$		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
$667$		−	21.8884i		−	0.847521i
$668$		0			0
$669$			29.7553i			1.15041i
$670$		0			0
$671$		−	22.3341i		−	0.862199i
$672$		0			0
$673$		−	8.61241i		−	0.331984i	−0.986127	−	0.165992i	$-0.946917\pi$
$673$		−	8.61241i		−	0.331984i	0.986127	−	0.165992i	$-0.0530826\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		−	8.12842i		−	0.312401i	−0.987725	−	0.156200i	$-0.950075\pi$
$677$		−	8.12842i		−	0.312401i	0.987725	−	0.156200i	$-0.0499245\pi$
$678$		0			0
$679$	−1.28063			−0.0491459
$680$		0			0
$681$			41.5543i			1.59236i
$682$		0			0
$683$	8.02791			0.307179			0.153590	−	0.988135i	$-0.450917\pi$
$683$	8.02791			0.307179			0.153590	+	0.988135i	$0.450917\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	5.53747			0.211268
$688$		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
$693$		−	22.6124i		−	0.858974i
$694$		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
$703$		−	20.0047i		−	0.754490i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	35.2760			1.32669
$708$		0			0
$709$		−	2.85246i		−	0.107126i	−0.998564	−	0.0535631i	$-0.982942\pi$
$709$		−	2.85246i		−	0.107126i	0.998564	−	0.0535631i	$-0.0170578\pi$
$710$		0			0
$711$	45.0145			1.68817
$712$		0			0
$713$	16.6357			0.623010
$714$		0			0
$715$		0			0
$716$		0			0
$717$	−8.24573			−0.307942
$718$		0			0
$719$	−42.0941			−1.56984			−0.784922	−	0.619595i	$-0.787297\pi$
$719$	−42.0941			−1.56984			−0.784922	+	0.619595i	$0.787297\pi$
$720$		0			0
$721$		−	11.4602i		−	0.426800i
$722$		0			0
$723$	−12.4793			−0.464111
$724$		0			0
$725$		0			0
$726$		0			0
$727$			32.8027i			1.21659i	0.793713	+	0.608293i	$0.208145\pi$
$727$			32.8027i			1.21659i	−0.793713	+	0.608293i	$0.791855\pi$
$728$		0			0
$729$	38.6101			1.43000
$730$		0			0
$731$		0			0
$732$		0			0
$733$	−2.38993			−0.0882739			−0.0441370	−	0.999025i	$-0.514054\pi$
$733$	−2.38993			−0.0882739			−0.0441370	+	0.999025i	$0.514054\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			15.3923i			0.566981i
$738$		0			0
$739$			41.4988i			1.52656i	0.646068	+	0.763280i	$0.276412\pi$
$739$			41.4988i			1.52656i	−0.646068	+	0.763280i	$0.723588\pi$
$740$		0			0
$741$	−47.1922	+	2.18890i	−1.73365	+	0.0804112i
$742$		0			0
$743$	−39.8046			−1.46029			−0.730145	−	0.683292i	$-0.760548\pi$
$743$	−39.8046			−1.46029			−0.730145	+	0.683292i	$0.760548\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	14.9698			0.547715
$748$		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
$753$			2.60774i			0.0950315i
$754$		0			0
$755$		0			0
$756$		0			0
$757$		−	32.2736i		−	1.17301i	−0.809947	−	0.586503i	$-0.800504\pi$
$757$		−	32.2736i		−	1.17301i	0.809947	−	0.586503i	$-0.199496\pi$
$758$		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
$763$			41.1690i			1.49042i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	1.36435	+	29.4151i	0.0492638	+	1.06212i
$768$		0			0
$769$			12.2010i			0.439980i	0.975502	+	0.219990i	$0.0706025\pi$
$769$			12.2010i			0.439980i	−0.975502	+	0.219990i	$0.929397\pi$
$770$		0			0
$771$	−50.6850			−1.82538
$772$		0			0
$773$	−27.9721			−1.00609			−0.503043	−	0.864261i	$-0.667786\pi$
$773$	−27.9721			−1.00609			−0.503043	+	0.864261i	$0.667786\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		−	28.6124i		−	1.02646i
$778$		0			0
$779$	−8.38993			−0.300600
$780$		0			0
$781$	34.9139			1.24932
$782$		0			0
$783$		−	8.86925i		−	0.316961i
$784$		0			0
$785$		0			0
$786$		0			0
$787$	10.2336			0.364788			0.182394	−	0.983225i	$-0.441615\pi$
$787$	10.2336			0.364788			0.182394	+	0.983225i	$0.441615\pi$
$788$		0			0
$789$	10.9139			0.388547
$790$		0			0
$791$			8.66822i			0.308206i
$792$		0			0
$793$	−37.1196	+	1.72170i	−1.31816	+	0.0611395i
$794$		0			0
$795$		0			0
$796$		0			0
$797$			23.3532i			0.827214i	0.910456	+	0.413607i	$0.135731\pi$
$797$			23.3532i			0.827214i	−0.910456	+	0.413607i	$0.864269\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		−	29.9395i		−	1.05786i
$802$		0			0
$803$		−	34.6124i		−	1.22145i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		−	79.2974i		−	2.79140i
$808$		0			0
$809$	−37.1364			−1.30565			−0.652824	−	0.757510i	$-0.726416\pi$
$809$	−37.1364			−1.30565			−0.652824	+	0.757510i	$0.726416\pi$
$810$		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
$813$	12.5798			0.441194
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−49.3811			−1.72763
$818$		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
$823$			39.8050i			1.38752i	0.720208	+	0.693758i	$0.244046\pi$
$823$			39.8050i			1.38752i	−0.720208	+	0.693758i	$0.755954\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−55.8605			−1.94246			−0.971229	−	0.238146i	$-0.923460\pi$
$827$	−55.8605			−1.94246			−0.971229	+	0.238146i	$0.923460\pi$
$828$		0			0
$829$	33.5822			1.16636			0.583178	−	0.812344i	$-0.301809\pi$
$829$	33.5822			1.16636			0.583178	+	0.812344i	$0.301809\pi$
$830$		0			0
$831$	7.81110			0.270964
$832$		0			0
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$	6.74083			0.232997
$838$		0			0
$839$			26.4454i			0.912995i	0.889725	+	0.456497i	$0.150896\pi$
$839$			26.4454i			0.912995i	−0.889725	+	0.456497i	$0.849104\pi$
$840$		0			0
$841$	−9.33411			−0.321866
$842$		0			0
$843$	56.2225			1.93641
$844$		0			0
$845$		0			0
$846$		0			0
$847$	17.4537			0.599718
$848$		0			0
$849$	−6.43462			−0.220836
$850$		0			0
$851$			19.6054i			0.672065i
$852$		0			0
$853$	−35.7153			−1.22287			−0.611433	−	0.791296i	$-0.709407\pi$
$853$	−35.7153			−1.22287			−0.611433	+	0.791296i	$0.709407\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	22.0894i		−	0.754559i	−0.926099	−	0.377280i	$-0.876860\pi$
$857$		−	22.0894i		−	0.754559i	0.926099	−	0.377280i	$-0.123140\pi$
$858$		0			0
$859$	39.2201			1.33817			0.669087	−	0.743184i	$-0.266685\pi$
$859$	39.2201			1.33817			0.669087	+	0.743184i	$0.266685\pi$
$860$		0			0
$861$	−12.0000			−0.408959
$862$		0			0
$863$	−28.9744			−0.986301			−0.493150	−	0.869944i	$-0.664155\pi$
$863$	−28.9744			−0.986301			−0.493150	+	0.869944i	$0.664155\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$			44.2285i			1.50208i
$868$		0			0
$869$		−	25.8837i		−	0.878045i
$870$		0			0
$871$	25.5822	−	1.18657i	0.866819	−	0.0402053i
$872$		0			0
$873$	1.74316			0.0589970
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−30.9251			−1.04427			−0.522133	−	0.852864i	$-0.674863\pi$
$877$	−30.9251			−1.04427			−0.522133	+	0.852864i	$0.674863\pi$
$878$		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
$883$			15.1973i			0.511430i	0.966752	+	0.255715i	$0.0823108\pi$
$883$			15.1973i			0.511430i	−0.966752	+	0.255715i	$0.917689\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$			12.4174i			0.416937i	0.978029	+	0.208468i	$0.0668479\pi$
$887$			12.4174i			0.416937i	−0.978029	+	0.208468i	$0.933152\pi$
$888$		0			0
$889$		−	44.6850i		−	1.49869i
$890$		0			0
$891$		−	13.2252i		−	0.443061i
$892$		0			0
$893$			58.6124i			1.96139i
$894$		0			0
$895$		0			0
$896$		0			0
$897$	46.2504	−	2.14521i	1.54426	−	0.0716266i
$898$		0			0
$899$			14.9465i			0.498494i
$900$		0			0
$901$		0			0
$902$		0			0
$903$	−70.6292			−2.35039
$904$		0			0
$905$		0			0
$906$		0			0
$907$			11.4709i			0.380886i	0.981698	+	0.190443i	$0.0609924\pi$
$907$			11.4709i			0.380886i	−0.981698	+	0.190443i	$0.939008\pi$
$908$		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
$913$		−	8.60774i		−	0.284875i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−24.5566			−0.810930
$918$		0			0
$919$	−9.05815			−0.298801			−0.149400	−	0.988777i	$-0.547734\pi$
$919$	−9.05815			−0.298801			−0.149400	+	0.988777i	$0.547734\pi$
$920$		0			0
$921$			35.2928i			1.16294i
$922$		0			0
$923$	−2.69147	−	58.0275i	−0.0885907	−	1.91000i
$924$		0			0
$925$		0			0
$926$		0			0
$927$			15.5993i			0.512350i
$928$		0			0
$929$		−	50.2225i		−	1.64775i	−0.566774	−	0.823873i	$-0.691809\pi$
$929$		−	50.2225i		−	1.64775i	0.566774	−	0.823873i	$-0.308191\pi$
$930$		0			0
$931$			3.35362i			0.109911i
$932$		0			0
$933$		−	5.53747i		−	0.181289i
$934$		0			0
$935$		0			0
$936$		0			0
$937$		−	5.38759i		−	0.176005i	−0.996120	−	0.0880025i	$-0.971952\pi$
$937$		−	5.38759i		−	0.176005i	0.996120	−	0.0880025i	$-0.0280484\pi$
$938$		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
$943$	8.22248			0.267761
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−12.6403			−0.410755			−0.205377	−	0.978683i	$-0.565842\pi$
$947$	−12.6403			−0.410755			−0.205377	+	0.978683i	$0.565842\pi$
$948$		0			0
$949$	−57.5264	+	2.66822i	−1.86738	+	0.0866141i
$950$		0			0
$951$		−	27.6706i		−	0.897279i
$952$		0			0
$953$		−	32.6077i		−	1.05627i	−0.849161	−	0.528134i	$-0.822892\pi$
$953$		−	32.6077i		−	1.05627i	0.849161	−	0.528134i	$-0.177108\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$	−25.0023			−0.808211
$958$		0			0
$959$	−4.61241			−0.148942
$960$		0			0
$961$	19.6403			0.633558
$962$		0			0
$963$			26.6231i			0.857918i
$964$		0			0
$965$		0			0
$966$		0			0
$967$	−20.5845			−0.661953			−0.330976	−	0.943639i	$-0.607378\pi$
$967$	−20.5845			−0.661953			−0.330976	+	0.943639i	$0.607378\pi$
$968$		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
$973$	−53.3700			−1.71096
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−7.30387			−0.233672			−0.116836	−	0.993151i	$-0.537275\pi$
$977$	−7.30387			−0.233672			−0.116836	+	0.993151i	$0.537275\pi$
$978$		0			0
$979$	−17.2155			−0.550209
$980$		0			0
$981$		−	56.0382i		−	1.78916i
$982$		0			0
$983$	40.2504			1.28379			0.641894	−	0.766793i	$-0.278149\pi$
$983$	40.2504			1.28379			0.641894	+	0.766793i	$0.278149\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$			83.8326i			2.66842i
$988$		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
$993$	3.03024			0.0961617
$994$		0			0
$995$		0			0
$996$		0			0
$997$		−	0.273633i		−	0.00866606i	−0.999991	−	0.00433303i	$-0.998621\pi$
$997$		−	0.273633i		−	0.00866606i	0.999991	−	0.00433303i	$-0.00137925\pi$
$998$		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.d.d.649.6		6
5.2	odd	4		260.2.f.a.181.5	✓	6
5.3	odd	4		1300.2.f.e.701.1		6
5.4	even	2		1300.2.d.c.649.1		6
13.12	even	2		1300.2.d.c.649.6		6
15.2	even	4		2340.2.c.d.181.6		6
20.7	even	4		1040.2.k.c.961.1		6
65.12	odd	4		260.2.f.a.181.6	yes	6
65.38	odd	4		1300.2.f.e.701.2		6
65.47	even	4		3380.2.a.n.1.3		3
65.57	even	4		3380.2.a.m.1.3		3
65.64	even	2	inner	1300.2.d.d.649.1		6
195.77	even	4		2340.2.c.d.181.1		6
260.207	even	4		1040.2.k.c.961.2		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
260.2.f.a.181.5	✓	6	5.2	odd	4
260.2.f.a.181.6	yes	6	65.12	odd	4
1040.2.k.c.961.1		6	20.7	even	4
1040.2.k.c.961.2		6	260.207	even	4
1300.2.d.c.649.1		6	5.4	even	2
1300.2.d.c.649.6		6	13.12	even	2
1300.2.d.d.649.1		6	65.64	even	2	inner
1300.2.d.d.649.6		6	1.1	even	1	trivial
1300.2.f.e.701.1		6	5.3	odd	4
1300.2.f.e.701.2		6	65.38	odd	4
2340.2.c.d.181.1		6	195.77	even	4
2340.2.c.d.181.6		6	15.2	even	4
3380.2.a.m.1.3		3	65.57	even	4
3380.2.a.n.1.3		3	65.47	even	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.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+2.60168i q^{3} +2.76873 q^{7} -3.76873 q^{9} +O(q^{10})\)	\(q+2.60168i q^{3} +2.76873 q^{7} -3.76873 q^{9} +2.16706i q^{11} +(3.60168 - 0.167055i) q^{13} +5.03630i q^{19} +7.20336i q^{21} -4.93579i q^{23} -2.00000i q^{27} +4.43462 q^{29} +3.37041i q^{31} -5.63798 q^{33} -3.97209 q^{37} +(0.434624 + 9.37041i) q^{39} +1.66589i q^{41} +9.80504i q^{43} +11.6380 q^{47} +0.665890 q^{49} -12.7408i q^{53} -13.1028 q^{57} +8.16706i q^{59} -10.3062 q^{61} -10.4346 q^{63} +7.10284 q^{67} +12.8413 q^{69} -16.1112i q^{71} -15.9721 q^{73} +6.00000i q^{77} -11.9442 q^{79} -6.10284 q^{81} -3.97209 q^{83} +11.5375i q^{87} +7.94419i q^{89} +(9.97209 - 0.462531i) q^{91} -8.76873 q^{93} -0.462531 q^{97} -8.16706i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 6 q^{9}+O(q^{10}) \) 6 * q - 6 * q^9	\( 6 q - 6 q^{9} + 6 q^{13} + 12 q^{29} + 12 q^{33} + 24 q^{37} - 12 q^{39} + 24 q^{47} + 6 q^{49} - 60 q^{57} - 12 q^{61} - 48 q^{63} + 24 q^{67} - 48 q^{73} + 24 q^{79} - 18 q^{81} + 24 q^{83} + 12 q^{91} - 36 q^{93} - 36 q^{97}+O(q^{100}) \) 6 * q - 6 * q^9 + 6 * q^13 + 12 * q^29 + 12 * q^33 + 24 * q^37 - 12 * q^39 + 24 * q^47 + 6 * q^49 - 60 * q^57 - 12 * q^61 - 48 * q^63 + 24 * q^67 - 48 * q^73 + 24 * q^79 - 18 * q^81 + 24 * q^83 + 12 * q^91 - 36 * q^93 - 36 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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