LMFDB - Embedded newform 1300.2.d.d.649.2

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			649.2
Root			$-2.26180i$ of defining polynomial
Character	$\chi$	$=$	1300.649
Dual form			1300.2.d.d.649.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.26180i q^{3} +1.11575 q^{7} -2.11575 q^{9} +O(q^{10})$	$q-2.26180i q^{3} +1.11575 q^{7} -2.11575 q^{9} +5.37755i q^{11} +(-1.26180 - 3.37755i) q^{13} -7.90116i q^{19} -2.52360i q^{21} -6.49330i q^{23} -2.00000i q^{27} -3.63935 q^{29} -3.14605i q^{31} +12.1630 q^{33} +7.40786 q^{37} +(-7.63935 + 2.85395i) q^{39} -4.75510i q^{41} -4.78541i q^{43} -6.16296 q^{47} -5.75510 q^{49} +0.292106i q^{53} -17.8709 q^{57} +11.3776i q^{59} -5.34725 q^{61} -2.36065 q^{63} +11.8709 q^{67} -14.6866 q^{69} +3.43816i q^{71} -4.59214 q^{73} +6.00000i q^{77} +10.8157 q^{79} -10.8709 q^{81} +7.40786 q^{83} +8.23150i q^{87} -14.8157i q^{89} +(-1.40786 - 3.76850i) q^{91} -7.11575 q^{93} -3.76850 q^{97} -11.3776i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$		−	2.26180i		−	1.30585i	−0.757422	−	0.652926i	$-0.773541\pi$
$3$		−	2.26180i		−	1.30585i	0.757422	−	0.652926i	$-0.226459\pi$
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	1.11575			0.421714			0.210857	−	0.977517i	$-0.432375\pi$
$7$	1.11575			0.421714			0.210857	+	0.977517i	$0.432375\pi$
$8$		0			0
$9$	−2.11575			−0.705250
$10$		0			0
$11$			5.37755i			1.62139i	0.585467	+	0.810696i	$0.300911\pi$
$11$			5.37755i			1.62139i	−0.585467	+	0.810696i	$0.699089\pi$
$12$		0			0
$13$	−1.26180	−	3.37755i	−0.349961	−	0.936764i
$13$	−1.26180	−	3.37755i	−0.349961	−	0.936764i
$14$		0			0
$15$		0			0
$16$		0			0
$17$		0			0		1.00000			$0$
$17$		0			0		−1.00000			$\pi$
$18$		0			0
$19$		−	7.90116i		−	1.81265i	−0.422582	−	0.906325i	$-0.638876\pi$
$19$		−	7.90116i		−	1.81265i	0.422582	−	0.906325i	$-0.361124\pi$
$20$		0			0
$21$		−	2.52360i		−	0.550696i
$22$		0			0
$23$		−	6.49330i		−	1.35395i	−0.736007	−	0.676973i	$-0.763291\pi$
$23$		−	6.49330i		−	1.35395i	0.736007	−	0.676973i	$-0.236709\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$		−	2.00000i		−	0.384900i
$28$		0			0
$29$	−3.63935			−0.675811			−0.337906	−	0.941180i	$-0.609718\pi$
$29$	−3.63935			−0.675811			−0.337906	+	0.941180i	$0.609718\pi$
$30$		0			0
$31$		−	3.14605i		−	0.565048i	−0.959260	−	0.282524i	$-0.908828\pi$
$31$		−	3.14605i		−	0.565048i	0.959260	−	0.282524i	$-0.0911716\pi$
$32$		0			0
$33$	12.1630			2.11730
$34$		0			0
$35$		0			0
$36$		0			0
$37$	7.40786			1.21784			0.608922	−	0.793230i	$-0.291602\pi$
$37$	7.40786			1.21784			0.608922	+	0.793230i	$0.291602\pi$
$38$		0			0
$39$	−7.63935	+	2.85395i	−1.22328	+	0.456997i
$40$		0			0
$41$		−	4.75510i		−	0.742622i	−0.928508	−	0.371311i	$-0.878908\pi$
$41$		−	4.75510i		−	0.742622i	0.928508	−	0.371311i	$-0.121092\pi$
$42$		0			0
$43$		−	4.78541i		−	0.729768i	−0.931053	−	0.364884i	$-0.881109\pi$
$43$		−	4.78541i		−	0.729768i	0.931053	−	0.364884i	$-0.118891\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−6.16296			−0.898960			−0.449480	−	0.893290i	$-0.648391\pi$
$47$	−6.16296			−0.898960			−0.449480	+	0.893290i	$0.648391\pi$
$48$		0			0
$49$	−5.75510			−0.822158
$50$		0			0
$51$		0			0
$52$		0			0
$53$			0.292106i			0.0401238i	0.999799	+	0.0200619i	$0.00638633\pi$
$53$			0.292106i			0.0401238i	−0.999799	+	0.0200619i	$0.993614\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	−17.8709			−2.36705
$58$		0			0
$59$			11.3776i			1.48123i	0.671929	+	0.740616i	$0.265466\pi$
$59$			11.3776i			1.48123i	−0.671929	+	0.740616i	$0.734534\pi$
$60$		0			0
$61$	−5.34725			−0.684645			−0.342322	−	0.939583i	$-0.611214\pi$
$61$	−5.34725			−0.684645			−0.342322	+	0.939583i	$0.611214\pi$
$62$		0			0
$63$	−2.36065			−0.297413
$64$		0			0
$65$		0			0
$66$		0			0
$67$	11.8709			1.45026			0.725128	−	0.688614i	$-0.241781\pi$
$67$	11.8709			1.45026			0.725128	+	0.688614i	$0.241781\pi$
$68$		0			0
$69$	−14.6866			−1.76805
$70$		0			0
$71$			3.43816i			0.408034i	0.978967	+	0.204017i	$0.0653998\pi$
$71$			3.43816i			0.408034i	−0.978967	+	0.204017i	$0.934600\pi$
$72$		0			0
$73$	−4.59214			−0.537470			−0.268735	−	0.963214i	$-0.586606\pi$
$73$	−4.59214			−0.537470			−0.268735	+	0.963214i	$0.586606\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$			6.00000i			0.683763i
$78$		0			0
$79$	10.8157			1.21686			0.608431	−	0.793607i	$-0.291799\pi$
$79$	10.8157			1.21686			0.608431	+	0.793607i	$0.291799\pi$
$80$		0			0
$81$	−10.8709			−1.20787
$82$		0			0
$83$	7.40786			0.813118			0.406559	−	0.913625i	$-0.366729\pi$
$83$	7.40786			0.813118			0.406559	+	0.913625i	$0.366729\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$			8.23150i			0.882509i
$88$		0			0
$89$		−	14.8157i		−	1.57046i	−0.619203	−	0.785231i	$-0.712544\pi$
$89$		−	14.8157i		−	1.57046i	0.619203	−	0.785231i	$-0.287456\pi$
$90$		0			0
$91$	−1.40786	−	3.76850i	−0.147583	−	0.395046i
$92$		0			0
$93$	−7.11575			−0.737869
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−3.76850			−0.382633			−0.191317	−	0.981528i	$-0.561276\pi$
$97$	−3.76850			−0.382633			−0.191317	+	0.981528i	$0.561276\pi$
$98$		0			0
$99$		−	11.3776i		−	1.14349i
$100$		0			0
$101$	−0.292106			−0.0290656			−0.0145328	−	0.999894i	$-0.504626\pi$
$101$	−0.292106			−0.0290656			−0.0145328	+	0.999894i	$0.504626\pi$
$102$		0			0
$103$			4.03030i			0.397118i	0.980089	+	0.198559i	$0.0636261\pi$
$103$			4.03030i			0.397118i	−0.980089	+	0.198559i	$0.936374\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	5.50670i		−	0.532353i	−0.963924	−	0.266176i	$-0.914240\pi$
$107$		−	5.50670i		−	0.532353i	0.963924	−	0.266176i	$-0.0857603\pi$
$108$		0			0
$109$		−	1.27871i		−	0.122478i	−0.998123	−	0.0612390i	$-0.980495\pi$
$109$		−	1.27871i		−	0.122478i	0.998123	−	0.0612390i	$-0.0195052\pi$
$110$		0			0
$111$		−	16.7551i		−	1.59032i
$112$		0			0
$113$			19.2787i			1.81359i	0.421574	+	0.906794i	$0.361478\pi$
$113$			19.2787i			1.81359i	−0.421574	+	0.906794i	$0.638522\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	2.66966	+	7.14605i	0.246810	+	0.660653i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−17.9181			−1.62891
$122$		0			0
$123$	−10.7551			−0.969755
$124$		0			0
$125$		0			0
$126$		0			0
$127$		−	7.96970i		−	0.707196i	−0.935397	−	0.353598i	$-0.884958\pi$
$127$		−	7.96970i		−	0.707196i	0.935397	−	0.353598i	$-0.115042\pi$
$128$		0			0
$129$	−10.8236			−0.952969
$130$		0			0
$131$	7.27871			0.635944			0.317972	−	0.948100i	$-0.396998\pi$
$131$	7.27871			0.635944			0.317972	+	0.948100i	$0.396998\pi$
$132$		0			0
$133$		−	8.81571i		−	0.764419i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	4.75510			0.406256			0.203128	−	0.979152i	$-0.434889\pi$
$137$	4.75510			0.406256			0.203128	+	0.979152i	$0.434889\pi$
$138$		0			0
$139$	16.3259			1.38475			0.692373	−	0.721540i	$-0.256565\pi$
$139$	16.3259			1.38475			0.692373	+	0.721540i	$0.256565\pi$
$140$		0			0
$141$			13.9394i			1.17391i
$142$		0			0
$143$	18.1630	−	6.78541i	1.51886	−	0.567424i
$144$		0			0
$145$		0			0
$146$		0			0
$147$			13.0169i			1.07362i
$148$		0			0
$149$			8.81571i			0.722211i	0.932525	+	0.361106i	$0.117601\pi$
$149$			8.81571i			0.722211i	−0.932525	+	0.361106i	$0.882399\pi$
$150$		0			0
$151$			19.9012i			1.61953i	0.586752	+	0.809767i	$0.300406\pi$
$151$			19.9012i			1.61953i	−0.586752	+	0.809767i	$0.699594\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$		−	18.7551i		−	1.49682i	−0.663236	−	0.748410i	$-0.730818\pi$
$157$		−	18.7551i		−	1.49682i	0.663236	−	0.748410i	$-0.269182\pi$
$158$		0			0
$159$	0.660685			0.0523958
$160$		0			0
$161$		−	7.24490i		−	0.570978i
$162$		0			0
$163$	−13.4417			−1.05283			−0.526416	−	0.850227i	$-0.676465\pi$
$163$	−13.4417			−1.05283			−0.526416	+	0.850227i	$0.676465\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−4.91806			−0.380571			−0.190286	−	0.981729i	$-0.560941\pi$
$167$	−4.91806			−0.380571			−0.190286	+	0.981729i	$0.560941\pi$
$168$		0			0
$169$	−9.81571	+	8.52360i	−0.755055	+	0.655662i
$170$		0			0
$171$			16.7169i			1.27837i
$172$		0			0
$173$		−	19.5708i		−	1.48794i	−0.668212	−	0.743971i	$-0.732940\pi$
$173$		−	19.5708i		−	1.48794i	0.668212	−	0.743971i	$-0.267060\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	25.7338			1.93427
$178$		0			0
$179$	12.9866			0.970664			0.485332	−	0.874330i	$-0.338699\pi$
$179$	12.9866			0.970664			0.485332	+	0.874330i	$0.338699\pi$
$180$		0			0
$181$	9.40786			0.699280			0.349640	−	0.936884i	$-0.386304\pi$
$181$	9.40786			0.699280			0.349640	+	0.936884i	$0.386304\pi$
$182$		0			0
$183$			12.0944i			0.894045i
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$		−	2.23150i		−	0.162318i
$190$		0			0
$191$	26.5574			1.92163			0.960814	−	0.277195i	$-0.0894049\pi$
$191$	26.5574			1.92163			0.960814	+	0.277195i	$0.0894049\pi$
$192$		0			0
$193$	−21.8023			−1.56936			−0.784682	−	0.619898i	$-0.787174\pi$
$193$	−21.8023			−1.56936			−0.784682	+	0.619898i	$0.787174\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−7.57081			−0.539398			−0.269699	−	0.962945i	$-0.586924\pi$
$197$	−7.57081			−0.539398			−0.269699	+	0.962945i	$0.586924\pi$
$198$		0			0
$199$	−17.5708			−1.24556			−0.622781	−	0.782396i	$-0.713997\pi$
$199$	−17.5708			−1.24556			−0.622781	+	0.782396i	$0.713997\pi$
$200$		0			0
$201$		−	26.8495i		−	1.89382i
$202$		0			0
$203$	−4.06061			−0.284999
$204$		0			0
$205$		0			0
$206$		0			0
$207$			13.7382i			0.954871i
$208$		0			0
$209$	42.4889			2.93902
$210$		0			0
$211$	−0.0606069			−0.00417235			−0.00208618	−	0.999998i	$-0.500664\pi$
$211$	−0.0606069			−0.00417235			−0.00208618	+	0.999998i	$0.500664\pi$
$212$		0			0
$213$	7.77643			0.532833
$214$		0			0
$215$		0			0
$216$		0			0
$217$		−	3.51021i		−	0.238288i
$218$		0			0
$219$			10.3865i			0.701856i
$220$		0			0
$221$		0			0
$222$		0			0
$223$	22.6260			1.51515			0.757573	−	0.652750i	$-0.226385\pi$
$223$	22.6260			1.51515			0.757573	+	0.652750i	$0.226385\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	4.59214			0.304791			0.152396	−	0.988320i	$-0.451301\pi$
$227$	4.59214			0.304791			0.152396	+	0.988320i	$0.451301\pi$
$228$		0			0
$229$			0.986602i			0.0651965i	0.999469	+	0.0325983i	$0.0103782\pi$
$229$			0.986602i			0.0651965i	−0.999469	+	0.0325983i	$0.989622\pi$
$230$		0			0
$231$	13.5708			0.892894
$232$		0			0
$233$			13.2787i			0.869917i	0.900450	+	0.434959i	$0.143237\pi$
$233$			13.2787i			0.869917i	−0.900450	+	0.434959i	$0.856763\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		−	24.4630i		−	1.58904i
$238$		0			0
$239$			25.6429i			1.65870i	0.558730	+	0.829349i	$0.311289\pi$
$239$			25.6429i			1.65870i	−0.558730	+	0.829349i	$0.688711\pi$
$240$		0			0
$241$			14.5236i			0.935548i	0.883848	+	0.467774i	$0.154944\pi$
$241$			14.5236i			0.935548i	−0.883848	+	0.467774i	$0.845056\pi$
$242$		0			0
$243$			18.5877i			1.19240i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−26.6866	+	9.96970i	−1.69803	+	0.634357i
$248$		0			0
$249$		−	16.7551i		−	1.06181i
$250$		0			0
$251$	20.2653			1.27914			0.639568	−	0.768735i	$-0.279113\pi$
$251$	20.2653			1.27914			0.639568	+	0.768735i	$0.279113\pi$
$252$		0			0
$253$	34.9181			2.19528
$254$		0			0
$255$		0			0
$256$		0			0
$257$		−	6.58421i		−	0.410712i	−0.978687	−	0.205356i	$-0.934165\pi$
$257$		−	6.58421i		−	0.410712i	0.978687	−	0.205356i	$-0.0658352\pi$
$258$		0			0
$259$	8.26531			0.513581
$260$		0			0
$261$	7.69996			0.476616
$262$		0			0
$263$		−	18.7854i		−	1.15836i	−0.815200	−	0.579179i	$-0.803373\pi$
$263$		−	18.7854i		−	1.15836i	0.815200	−	0.579179i	$-0.196627\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	−33.5102			−2.05079
$268$		0			0
$269$	14.8495			0.905391			0.452696	−	0.891665i	$-0.350462\pi$
$269$	14.8495			0.905391			0.452696	+	0.891665i	$0.350462\pi$
$270$		0			0
$271$		−	20.8878i		−	1.26884i	−0.772988	−	0.634420i	$-0.781239\pi$
$271$		−	20.8878i		−	1.26884i	0.772988	−	0.634420i	$-0.218761\pi$
$272$		0			0
$273$	−8.52360	+	3.18429i	−0.515872	+	0.192722i
$274$		0			0
$275$		0			0
$276$		0			0
$277$		−	22.2653i		−	1.33779i	−0.743356	−	0.668896i	$-0.766767\pi$
$277$		−	22.2653i		−	1.33779i	0.743356	−	0.668896i	$-0.233233\pi$
$278$		0			0
$279$			6.65626i			0.398500i
$280$		0			0
$281$			7.57081i			0.451637i	0.974169	+	0.225818i	$0.0725056\pi$
$281$			7.57081i			0.451637i	−0.974169	+	0.225818i	$0.927494\pi$
$282$		0			0
$283$			0.724800i			0.0430849i	0.999768	+	0.0215424i	$0.00685770\pi$
$283$			0.724800i			0.0430849i	−0.999768	+	0.0215424i	$0.993142\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		−	5.30550i		−	0.313174i
$288$		0			0
$289$	17.0000			1.00000
$290$		0			0
$291$			8.52360i			0.499663i
$292$		0			0
$293$	−10.2236			−0.597267			−0.298634	−	0.954368i	$-0.596531\pi$
$293$	−10.2236			−0.597267			−0.298634	+	0.954368i	$0.596531\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	10.7551			0.624074
$298$		0			0
$299$	−21.9315	+	8.19326i	−1.26833	+	0.473829i
$300$		0			0
$301$		−	5.33931i		−	0.307753i
$302$		0			0
$303$			0.660685i			0.0379554i
$304$		0			0
$305$		0			0
$306$		0			0
$307$	21.6394			1.23502			0.617512	−	0.786562i	$-0.288141\pi$
$307$	21.6394			1.23502			0.617512	+	0.786562i	$0.288141\pi$
$308$		0			0
$309$	9.11575			0.518577
$310$		0			0
$311$	0.986602			0.0559451			0.0279725	−	0.999609i	$-0.491095\pi$
$311$	0.986602			0.0559451			0.0279725	+	0.999609i	$0.491095\pi$
$312$		0			0
$313$			25.8361i			1.46034i	0.683263	+	0.730172i	$0.260560\pi$
$313$			25.8361i			1.46034i	−0.683263	+	0.730172i	$0.739440\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	26.4283			1.48436			0.742180	−	0.670201i	$-0.233792\pi$
$317$	26.4283			1.48436			0.742180	+	0.670201i	$0.233792\pi$
$318$		0			0
$319$		−	19.5708i		−	1.09576i
$320$		0			0
$321$	−12.4551			−0.695174
$322$		0			0
$323$		0			0
$324$		0			0
$325$		0			0
$326$		0			0
$327$	−2.89218			−0.159938
$328$		0			0
$329$	−6.87632			−0.379104
$330$		0			0
$331$			14.8878i			0.818305i	0.912466	+	0.409153i	$0.134176\pi$
$331$			14.8878i			0.818305i	−0.912466	+	0.409153i	$0.865824\pi$
$332$		0			0
$333$	−15.6732			−0.858884
$334$		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
$339$	43.6046			2.36828
$340$		0			0
$341$	16.9181			0.916164
$342$		0			0
$343$	−14.2315			−0.768429
$344$		0			0
$345$		0			0
$346$		0			0
$347$			26.0641i			1.39919i	0.714537	+	0.699597i	$0.246637\pi$
$347$			26.0641i			1.39919i	−0.714537	+	0.699597i	$0.753363\pi$
$348$		0			0
$349$			10.7551i			0.575707i	0.957674	+	0.287854i	$0.0929417\pi$
$349$			10.7551i			0.575707i	−0.957674	+	0.287854i	$0.907058\pi$
$350$		0			0
$351$	−6.75510	+	2.52360i	−0.360561	+	0.134700i
$352$		0			0
$353$	−35.6126			−1.89547			−0.947733	−	0.319066i	$-0.896631\pi$
$353$	−35.6126			−1.89547			−0.947733	+	0.319066i	$0.896631\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		−	6.07205i		−	0.320470i	−0.987079	−	0.160235i	$-0.948775\pi$
$359$		−	6.07205i		−	0.320470i	0.987079	−	0.160235i	$-0.0512253\pi$
$360$		0			0
$361$	−43.4283			−2.28570
$362$		0			0
$363$			40.5271i			2.12712i
$364$		0			0
$365$		0			0
$366$		0			0
$367$		−	26.7854i		−	1.39819i	−0.715030	−	0.699093i	$-0.753587\pi$
$367$		−	26.7854i		−	1.39819i	0.715030	−	0.699093i	$-0.246413\pi$
$368$		0			0
$369$			10.0606i			0.523734i
$370$		0			0
$371$			0.325917i			0.0169208i
$372$		0			0
$373$			10.8763i			0.563154i	0.959539	+	0.281577i	$0.0908575\pi$
$373$			10.8763i			0.563154i	−0.959539	+	0.281577i	$0.909142\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	4.59214	+	12.2921i	0.236507	+	0.633076i
$378$		0			0
$379$			5.96176i			0.306235i	0.988208	+	0.153118i	$0.0489313\pi$
$379$			5.96176i			0.306235i	−0.988208	+	0.153118i	$0.951069\pi$
$380$		0			0
$381$	−18.0259			−0.923494
$382$		0			0
$383$	−19.4079			−0.991695			−0.495848	−	0.868410i	$-0.665143\pi$
$383$	−19.4079			−0.991695			−0.495848	+	0.868410i	$0.665143\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$			10.1247i			0.514669i
$388$		0			0
$389$	8.55742			0.433878			0.216939	−	0.976185i	$-0.430393\pi$
$389$	8.55742			0.433878			0.216939	+	0.976185i	$0.430393\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		−	16.4630i		−	0.830448i
$394$		0			0
$395$		0			0
$396$		0			0
$397$	14.9449			0.750061			0.375030	−	0.927012i	$-0.377632\pi$
$397$	14.9449			0.750061			0.375030	+	0.927012i	$0.377632\pi$
$398$		0			0
$399$	−19.9394			−0.998218
$400$		0			0
$401$			27.1416i			1.35539i	0.735344	+	0.677694i	$0.237021\pi$
$401$			27.1416i			1.35539i	−0.735344	+	0.677694i	$0.762979\pi$
$402$		0			0
$403$	−10.6260	+	3.96970i	−0.529317	+	0.197745i
$404$		0			0
$405$		0			0
$406$		0			0
$407$			39.8361i			1.97460i
$408$		0			0
$409$			33.2181i			1.64253i	0.570547	+	0.821265i	$0.306731\pi$
$409$			33.2181i			1.64253i	−0.570547	+	0.821265i	$0.693269\pi$
$410$		0			0
$411$		−	10.7551i		−	0.530510i
$412$		0			0
$413$			12.6945i			0.624655i
$414$		0			0
$415$		0			0
$416$		0			0
$417$		−	36.9260i		−	1.80827i
$418$		0			0
$419$	17.7079			0.865087			0.432544	−	0.901613i	$-0.357616\pi$
$419$	17.7079			0.865087			0.432544	+	0.901613i	$0.357616\pi$
$420$		0			0
$421$		−	22.2047i		−	1.08219i	−0.840961	−	0.541096i	$-0.818010\pi$
$421$		−	22.2047i		−	1.08219i	0.840961	−	0.541096i	$-0.181990\pi$
$422$		0			0
$423$	13.0393			0.633991
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−5.96619			−0.288724
$428$		0			0
$429$	−15.3472	−	41.0810i	−0.740972	−	1.98341i
$430$		0			0
$431$		−	10.6831i		−	0.514585i	−0.966334	−	0.257292i	$-0.917170\pi$
$431$		−	10.6831i		−	0.514585i	0.966334	−	0.257292i	$-0.0828303\pi$
$432$		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
$437$	−51.3046			−2.45423
$438$		0			0
$439$	5.24490			0.250325			0.125163	−	0.992136i	$-0.460055\pi$
$439$	5.24490			0.250325			0.125163	+	0.992136i	$0.460055\pi$
$440$		0			0
$441$	12.1764			0.579826
$442$		0			0
$443$			25.4799i			1.21059i	0.796002	+	0.605293i	$0.206944\pi$
$443$			25.4799i			1.21059i	−0.796002	+	0.605293i	$0.793056\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	19.9394			0.943101
$448$		0			0
$449$		−	13.9394i		−	0.657841i	−0.944358	−	0.328920i	$-0.893315\pi$
$449$		−	13.9394i		−	0.657841i	0.944358	−	0.328920i	$-0.106685\pi$
$450$		0			0
$451$	25.5708			1.20408
$452$		0			0
$453$	45.0125			2.11487
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−12.6945			−0.593823			−0.296912	−	0.954905i	$-0.595957\pi$
$457$	−12.6945			−0.593823			−0.296912	+	0.954905i	$0.595957\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$			23.0810i			1.07499i	0.843267	+	0.537495i	$0.180629\pi$
$461$			23.0810i			1.07499i	−0.843267	+	0.537495i	$0.819371\pi$
$462$		0			0
$463$	6.16296			0.286417			0.143208	−	0.989693i	$-0.454258\pi$
$463$	6.16296			0.286417			0.143208	+	0.989693i	$0.454258\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$			23.5067i			1.08776i	0.839163	+	0.543880i	$0.183045\pi$
$467$			23.5067i			1.08776i	−0.839163	+	0.543880i	$0.816955\pi$
$468$		0			0
$469$	13.2449			0.611593
$470$		0			0
$471$	−42.4203			−1.95463
$472$		0			0
$473$	25.7338			1.18324
$474$		0			0
$475$		0			0
$476$		0			0
$477$		−	0.618022i		−	0.0282973i
$478$		0			0
$479$		−	33.2137i		−	1.51757i	−0.651340	−	0.758786i	$-0.725793\pi$
$479$		−	33.2137i		−	1.51757i	0.651340	−	0.758786i	$-0.274207\pi$
$480$		0			0
$481$	−9.34725	−	25.0204i	−0.426198	−	1.14083i
$482$		0			0
$483$	−16.3865			−0.745613
$484$		0			0
$485$		0			0
$486$		0			0
$487$	18.7472			0.849515			0.424758	−	0.905307i	$-0.360359\pi$
$487$	18.7472			0.849515			0.424758	+	0.905307i	$0.360359\pi$
$488$		0			0
$489$			30.4024i			1.37484i
$490$		0			0
$491$	26.5574			1.19852			0.599260	−	0.800555i	$-0.295462\pi$
$491$	26.5574			1.19852			0.599260	+	0.800555i	$0.295462\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$			3.83612i			0.172074i
$498$		0			0
$499$		−	26.8539i		−	1.20215i	−0.799193	−	0.601074i	$-0.794740\pi$
$499$		−	26.8539i		−	1.20215i	0.799193	−	0.601074i	$-0.205260\pi$
$500$		0			0
$501$			11.1237i			0.496970i
$502$		0			0
$503$		−	20.4665i		−	0.912556i	−0.889837	−	0.456278i	$-0.849182\pi$
$503$		−	20.4665i		−	0.912556i	0.889837	−	0.456278i	$-0.150818\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	19.2787	+	22.2012i	0.856197	+	0.985990i
$508$		0			0
$509$			26.9598i			1.19497i	0.801879	+	0.597486i	$0.203834\pi$
$509$			26.9598i			1.19497i	−0.801879	+	0.597486i	$0.796166\pi$
$510$		0			0
$511$	−5.12368			−0.226658
$512$		0			0
$513$	−15.8023			−0.697689
$514$		0			0
$515$		0			0
$516$		0			0
$517$		−	33.1416i		−	1.45757i
$518$		0			0
$519$	−44.2653			−1.94303
$520$		0			0
$521$	−8.36065			−0.366287			−0.183143	−	0.983086i	$-0.558627\pi$
$521$	−8.36065			−0.366287			−0.183143	+	0.983086i	$0.558627\pi$
$522$		0			0
$523$		−	18.7248i		−	0.818778i	−0.912360	−	0.409389i	$-0.865742\pi$
$523$		−	18.7248i		−	0.818778i	0.912360	−	0.409389i	$-0.134258\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−19.1630			−0.833172
$530$		0			0
$531$		−	24.0720i		−	1.04464i
$532$		0			0
$533$	−16.0606	+	6.00000i	−0.695662	+	0.259889i
$534$		0			0
$535$		0			0
$536$		0			0
$537$		−	29.3731i		−	1.26754i
$538$		0			0
$539$		−	30.9484i		−	1.33304i
$540$		0			0
$541$		−	42.9439i		−	1.84630i	−0.384435	−	0.923152i	$-0.625604\pi$
$541$		−	42.9439i		−	1.84630i	0.384435	−	0.923152i	$-0.374396\pi$
$542$		0			0
$543$		−	21.2787i		−	0.913157i
$544$		0			0
$545$		0			0
$546$		0			0
$547$		−	14.0909i		−	0.602484i	−0.953548	−	0.301242i	$-0.902599\pi$
$547$		−	14.0909i		−	0.602484i	0.953548	−	0.301242i	$-0.0974012\pi$
$548$		0			0
$549$	11.3134			0.482846
$550$		0			0
$551$			28.7551i			1.22501i
$552$		0			0
$553$	12.0676			0.513167
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−0.713359			−0.0302260			−0.0151130	−	0.999886i	$-0.504811\pi$
$557$	−0.713359			−0.0302260			−0.0151130	+	0.999886i	$0.504811\pi$
$558$		0			0
$559$	−16.1630	+	6.03824i	−0.683620	+	0.255390i
$560$		0			0
$561$		0			0
$562$		0			0
$563$		−	12.2012i		−	0.514219i	−0.966382	−	0.257110i	$-0.917230\pi$
$563$		−	12.2012i		−	0.514219i	0.966382	−	0.257110i	$-0.0827701\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	−12.1291			−0.509376
$568$		0			0
$569$	−16.6260			−0.696996			−0.348498	−	0.937309i	$-0.613308\pi$
$569$	−16.6260			−0.696996			−0.348498	+	0.937309i	$0.613308\pi$
$570$		0			0
$571$	−20.3259			−0.850613			−0.425307	−	0.905049i	$-0.639834\pi$
$571$	−20.3259			−0.850613			−0.425307	+	0.905049i	$0.639834\pi$
$572$		0			0
$573$		−	60.0676i		−	2.50936i
$574$		0			0
$575$		0			0
$576$		0			0
$577$	−19.1496			−0.797207			−0.398603	−	0.917123i	$-0.630505\pi$
$577$	−19.1496			−0.797207			−0.398603	+	0.917123i	$0.630505\pi$
$578$		0			0
$579$			49.3125i			2.04936i
$580$		0			0
$581$	8.26531			0.342903
$582$		0			0
$583$	−1.57081			−0.0650564
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−30.4889			−1.25841			−0.629205	−	0.777239i	$-0.716620\pi$
$587$	−30.4889			−1.25841			−0.629205	+	0.777239i	$0.716620\pi$
$588$		0			0
$589$	−24.8575			−1.02423
$590$		0			0
$591$			17.1237i			0.704374i
$592$		0			0
$593$	42.3259			1.73812			0.869059	−	0.494709i	$-0.164725\pi$
$593$	42.3259			1.73812			0.869059	+	0.494709i	$0.164725\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$			39.7417i			1.62652i
$598$		0			0
$599$	−15.5440			−0.635111			−0.317556	−	0.948240i	$-0.602862\pi$
$599$	−15.5440			−0.635111			−0.317556	+	0.948240i	$0.602862\pi$
$600$		0			0
$601$	25.1416			1.02555			0.512774	−	0.858524i	$-0.328618\pi$
$601$	25.1416			1.02555			0.512774	+	0.858524i	$0.328618\pi$
$602$		0			0
$603$	−25.1157			−1.02279
$604$		0			0
$605$		0			0
$606$		0			0
$607$			10.0909i			0.409577i	0.978806	+	0.204789i	$0.0656507\pi$
$607$			10.0909i			0.409577i	−0.978806	+	0.204789i	$0.934349\pi$
$608$		0			0
$609$			9.18429i			0.372166i
$610$		0			0
$611$	7.77643	+	20.8157i	0.314601	+	0.842113i
$612$		0			0
$613$	37.0204			1.49524			0.747620	−	0.664127i	$-0.231196\pi$
$613$	37.0204			1.49524			0.747620	+	0.664127i	$0.231196\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−8.63389			−0.347587			−0.173794	−	0.984782i	$-0.555603\pi$
$617$	−8.63389			−0.347587			−0.173794	+	0.984782i	$0.555603\pi$
$618$		0			0
$619$			9.47197i			0.380711i	0.981715	+	0.190355i	$0.0609640\pi$
$619$			9.47197i			0.380711i	−0.981715	+	0.190355i	$0.939036\pi$
$620$		0			0
$621$	−12.9866			−0.521134
$622$		0			0
$623$		−	16.5306i		−	0.662285i
$624$		0			0
$625$		0			0
$626$		0			0
$627$		−	96.1014i		−	3.83792i
$628$		0			0
$629$		0			0
$630$		0			0
$631$			18.0382i			0.718091i	0.933320	+	0.359045i	$0.116898\pi$
$631$			18.0382i			0.718091i	−0.933320	+	0.359045i	$0.883102\pi$
$632$		0			0
$633$			0.137081i			0.00544847i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	7.26180	+	19.4382i	0.287723	+	0.770168i
$638$		0			0
$639$		−	7.27428i		−	0.287766i
$640$		0			0
$641$	17.4158			0.687882			0.343941	−	0.938991i	$-0.388238\pi$
$641$	17.4158			0.687882			0.343941	+	0.938991i	$0.388238\pi$
$642$		0			0
$643$	15.3472			0.605236			0.302618	−	0.953112i	$-0.402139\pi$
$643$	15.3472			0.605236			0.302618	+	0.953112i	$0.402139\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			3.05072i			0.119936i	0.998200	+	0.0599680i	$0.0190999\pi$
$647$			3.05072i			0.119936i	−0.998200	+	0.0599680i	$0.980900\pi$
$648$		0			0
$649$	−61.1834			−2.40166
$650$		0			0
$651$	−7.93939			−0.311169
$652$		0			0
$653$		−	25.5708i		−	1.00066i	−0.865834	−	0.500332i	$-0.833211\pi$
$653$		−	25.5708i		−	1.00066i	0.865834	−	0.500332i	$-0.166789\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	9.71583			0.379051
$658$		0			0
$659$	−6.29211			−0.245106			−0.122553	−	0.992462i	$-0.539108\pi$
$659$	−6.29211			−0.245106			−0.122553	+	0.992462i	$0.539108\pi$
$660$		0			0
$661$		−	39.5102i		−	1.53677i	−0.639989	−	0.768384i	$-0.721061\pi$
$661$		−	39.5102i		−	1.53677i	0.639989	−	0.768384i	$-0.278939\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$			23.6314i			0.915012i
$668$		0			0
$669$		−	51.1754i		−	1.97856i
$670$		0			0
$671$		−	28.7551i		−	1.11008i
$672$		0			0
$673$			1.30550i			0.0503235i	0.999683	+	0.0251617i	$0.00801007\pi$
$673$			1.30550i			0.0503235i	−0.999683	+	0.0251617i	$0.991990\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		−	5.01340i		−	0.192681i	−0.995348	−	0.0963403i	$-0.969286\pi$
$677$		−	5.01340i		−	0.192681i	0.995348	−	0.0963403i	$-0.0307137\pi$
$678$		0			0
$679$	−4.20470			−0.161362
$680$		0			0
$681$		−	10.3865i		−	0.398012i
$682$		0			0
$683$	19.4079			0.742621			0.371310	−	0.928509i	$-0.378909\pi$
$683$	19.4079			0.742621			0.371310	+	0.928509i	$0.378909\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	2.23150			0.0851370
$688$		0			0
$689$	0.986602	−	0.368580i	0.0375865	−	0.0140418i
$690$		0			0
$691$			2.56184i			0.0974570i	0.998812	+	0.0487285i	$0.0155169\pi$
$691$			2.56184i			0.0974570i	−0.998812	+	0.0487285i	$0.984483\pi$
$692$		0			0
$693$		−	12.6945i		−	0.482224i
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$	30.0338			1.13598
$700$		0			0
$701$	38.8495			1.46733			0.733663	−	0.679513i	$-0.237809\pi$
$701$	38.8495			1.46733			0.733663	+	0.679513i	$0.237809\pi$
$702$		0			0
$703$		−	58.5306i		−	2.20752i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−0.325917			−0.0122574
$708$		0			0
$709$		−	35.3393i		−	1.32720i	−0.748089	−	0.663598i	$-0.769029\pi$
$709$		−	35.3393i		−	1.32720i	0.748089	−	0.663598i	$-0.230971\pi$
$710$		0			0
$711$	−22.8833			−0.858192
$712$		0			0
$713$	−20.4283			−0.765045
$714$		0			0
$715$		0			0
$716$		0			0
$717$	57.9991			2.16602
$718$		0			0
$719$	−6.11028			−0.227875			−0.113938	−	0.993488i	$-0.536346\pi$
$719$	−6.11028			−0.227875			−0.113938	+	0.993488i	$0.536346\pi$
$720$		0			0
$721$			4.49681i			0.167470i
$722$		0			0
$723$	32.8495			1.22169
$724$		0			0
$725$		0			0
$726$		0			0
$727$		−	1.05072i		−	0.0389689i	−0.999810	−	0.0194845i	$-0.993798\pi$
$727$		−	1.05072i		−	0.0389689i	0.999810	−	0.0194845i	$-0.00620249\pi$
$728$		0			0
$729$	9.42919			0.349229
$730$		0			0
$731$		0			0
$732$		0			0
$733$	−31.5708			−1.16609			−0.583047	−	0.812438i	$-0.698140\pi$
$733$	−31.5708			−1.16609			−0.583047	+	0.812438i	$0.698140\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			63.8361i			2.35143i
$738$		0			0
$739$			31.8673i			1.17226i	0.810217	+	0.586130i	$0.199349\pi$
$739$			31.8673i			1.17226i	−0.810217	+	0.586130i	$0.800651\pi$
$740$		0			0
$741$	22.5495	+	60.3597i	0.828376	+	2.21737i
$742$		0			0
$743$	39.8550			1.46214			0.731069	−	0.682304i	$-0.239022\pi$
$743$	39.8550			1.46214			0.731069	+	0.682304i	$0.239022\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	−15.6732			−0.573451
$748$		0			0
$749$		−	6.14410i		−	0.224500i
$750$		0			0
$751$	9.24490			0.337351			0.168676	−	0.985672i	$-0.446051\pi$
$751$	9.24490			0.337351			0.168676	+	0.985672i	$0.446051\pi$
$752$		0			0
$753$		−	45.8361i		−	1.67036i
$754$		0			0
$755$		0			0
$756$		0			0
$757$			22.5912i			0.821092i	0.911840	+	0.410546i	$0.134662\pi$
$757$			22.5912i			0.821092i	−0.911840	+	0.410546i	$0.865338\pi$
$758$		0			0
$759$		−	78.9778i		−	2.86671i
$760$		0			0
$761$		−	16.2047i		−	0.587420i	−0.955895	−	0.293710i	$-0.905110\pi$
$761$		−	16.2047i		−	0.587420i	0.955895	−	0.293710i	$-0.0948900\pi$
$762$		0			0
$763$		−	1.42672i		−	0.0516506i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	38.4283	−	14.3562i	1.38756	−	0.518373i
$768$		0			0
$769$		−	16.7889i		−	0.605424i	−0.953082	−	0.302712i	$-0.902108\pi$
$769$		−	16.7889i		−	0.605424i	0.953082	−	0.302712i	$-0.0978920\pi$
$770$		0			0
$771$	−14.8922			−0.536329
$772$		0			0
$773$	−16.5921			−0.596778			−0.298389	−	0.954444i	$-0.596449\pi$
$773$	−16.5921			−0.596778			−0.298389	+	0.954444i	$0.596449\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		−	18.6945i		−	0.670661i
$778$		0			0
$779$	−37.5708			−1.34611
$780$		0			0
$781$	−18.4889			−0.661584
$782$		0			0
$783$			7.27871i			0.260120i
$784$		0			0
$785$		0			0
$786$		0			0
$787$	31.1496			1.11036			0.555181	−	0.831730i	$-0.312649\pi$
$787$	31.1496			1.11036			0.555181	+	0.831730i	$0.312649\pi$
$788$		0			0
$789$	−42.4889			−1.51264
$790$		0			0
$791$			21.5102i			0.764815i
$792$		0			0
$793$	6.74717	+	18.0606i	0.239599	+	0.641351i
$794$		0			0
$795$		0			0
$796$		0			0
$797$			0.402391i			0.0142534i	0.999975	+	0.00712670i	$0.00226852\pi$
$797$			0.402391i			0.0142534i	−0.999975	+	0.00712670i	$0.997731\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$			31.3463i			1.10757i
$802$		0			0
$803$		−	24.6945i		−	0.871450i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		−	33.5867i		−	1.18231i
$808$		0			0
$809$	55.3652			1.94654			0.973268	−	0.229671i	$-0.0737651\pi$
$809$	55.3652			1.94654			0.973268	+	0.229671i	$0.0737651\pi$
$810$		0			0
$811$			27.1461i			0.953227i	0.879113	+	0.476613i	$0.158136\pi$
$811$			27.1461i			0.953227i	−0.879113	+	0.476613i	$0.841864\pi$
$812$		0			0
$813$	−47.2440			−1.65692
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−37.8102			−1.32281
$818$		0			0
$819$	2.97867	+	7.97320i	0.104083	+	0.278606i
$820$		0			0
$821$			27.8361i			0.971487i	0.874101	+	0.485744i	$0.161451\pi$
$821$			27.8361i			0.971487i	−0.874101	+	0.485744i	$0.838549\pi$
$822$		0			0
$823$			25.2146i			0.878925i	0.898261	+	0.439463i	$0.144831\pi$
$823$			25.2146i			0.878925i	−0.898261	+	0.439463i	$0.855169\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	1.03928			0.0361391			0.0180696	−	0.999837i	$-0.494248\pi$
$827$	1.03928			0.0361391			0.0180696	+	0.999837i	$0.494248\pi$
$828$		0			0
$829$	−6.97867			−0.242379			−0.121190	−	0.992629i	$-0.538671\pi$
$829$	−6.97867			−0.242379			−0.121190	+	0.992629i	$0.538671\pi$
$830$		0			0
$831$	−50.3597			−1.74696
$832$		0			0
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$	−6.29211			−0.217487
$838$		0			0
$839$			13.3169i			0.459752i	0.973220	+	0.229876i	$0.0738321\pi$
$839$			13.3169i			0.459752i	−0.973220	+	0.229876i	$0.926168\pi$
$840$		0			0
$841$	−15.7551			−0.543279
$842$		0			0
$843$	17.1237			0.589771
$844$		0			0
$845$		0			0
$846$		0			0
$847$	−19.9921			−0.686936
$848$		0			0
$849$	1.63935			0.0562625
$850$		0			0
$851$		−	48.1014i		−	1.64890i
$852$		0			0
$853$	−30.5653			−1.04654			−0.523269	−	0.852168i	$-0.675288\pi$
$853$	−30.5653			−1.04654			−0.523269	+	0.852168i	$0.675288\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$			52.4203i			1.79064i	0.445419	+	0.895322i	$0.353055\pi$
$857$			52.4203i			1.79064i	−0.445419	+	0.895322i	$0.646945\pi$
$858$		0			0
$859$	−19.1416			−0.653104			−0.326552	−	0.945179i	$-0.605887\pi$
$859$	−19.1416			−0.653104			−0.326552	+	0.945179i	$0.605887\pi$
$860$		0			0
$861$	−12.0000			−0.408959
$862$		0			0
$863$	−36.8575			−1.25464			−0.627321	−	0.778761i	$-0.715849\pi$
$863$	−36.8575			−1.25464			−0.627321	+	0.778761i	$0.715849\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		−	38.4506i		−	1.30585i
$868$		0			0
$869$			58.1620i			1.97301i
$870$		0			0
$871$	−14.9787	−	40.0944i	−0.507533	−	1.35855i
$872$		0			0
$873$	7.97320			0.269852
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−37.5370			−1.26753			−0.633767	−	0.773524i	$-0.718492\pi$
$877$	−37.5370			−1.26753			−0.633767	+	0.773524i	$0.718492\pi$
$878$		0			0
$879$			23.1237i			0.779942i
$880$		0			0
$881$	−15.8212			−0.533029			−0.266514	−	0.963831i	$-0.585872\pi$
$881$	−15.8212			−0.533029			−0.266514	+	0.963831i	$0.585872\pi$
$882$		0			0
$883$			49.0507i			1.65069i	0.564630	+	0.825344i	$0.309019\pi$
$883$			49.0507i			1.65069i	−0.564630	+	0.825344i	$0.690981\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		−	12.0909i		−	0.405973i	−0.979182	−	0.202987i	$-0.934935\pi$
$887$		−	12.0909i		−	0.405973i	0.979182	−	0.202987i	$-0.0650648\pi$
$888$		0			0
$889$		−	8.89218i		−	0.298234i
$890$		0			0
$891$		−	58.4586i		−	1.95844i
$892$		0			0
$893$			48.6945i			1.62950i
$894$		0			0
$895$		0			0
$896$		0			0
$897$	18.5315	+	49.6046i	0.618750	+	1.65625i
$898$		0			0
$899$			11.4496i			0.381866i
$900$		0			0
$901$		0			0
$902$		0			0
$903$	−12.0765			−0.401880
$904$		0			0
$905$		0			0
$906$		0			0
$907$		−	9.54051i		−	0.316787i	−0.987376	−	0.158394i	$-0.949368\pi$
$907$		−	9.54051i		−	0.316787i	0.987376	−	0.158394i	$-0.0506315\pi$
$908$		0			0
$909$	0.618022			0.0204985
$910$		0			0
$911$	−26.7392			−0.885910			−0.442955	−	0.896544i	$-0.646070\pi$
$911$	−26.7392			−0.885910			−0.442955	+	0.896544i	$0.646070\pi$
$912$		0			0
$913$			39.8361i			1.31838i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	8.12121			0.268186
$918$		0			0
$919$	−51.0810			−1.68501			−0.842504	−	0.538691i	$-0.818919\pi$
$919$	−51.0810			−1.68501			−0.842504	+	0.538691i	$0.818919\pi$
$920$		0			0
$921$		−	48.9439i		−	1.61276i
$922$		0			0
$923$	11.6126	−	4.33828i	0.382232	−	0.142796i
$924$		0			0
$925$		0			0
$926$		0			0
$927$		−	8.52711i		−	0.280067i
$928$		0			0
$929$		−	11.1237i		−	0.364956i	−0.983210	−	0.182478i	$-0.941588\pi$
$929$		−	11.1237i		−	0.364956i	0.983210	−	0.182478i	$-0.0584119\pi$
$930$		0			0
$931$			45.4720i			1.49028i
$932$		0			0
$933$		−	2.23150i		−	0.0730560i
$934$		0			0
$935$		0			0
$936$		0			0
$937$		−	15.3055i		−	0.500009i	−0.968245	−	0.250005i	$-0.919568\pi$
$937$		−	15.3055i		−	0.500009i	0.968245	−	0.250005i	$-0.0804321\pi$
$938$		0			0
$939$	58.4362			1.90699
$940$		0			0
$941$			28.7124i			0.935999i	0.883729	+	0.467999i	$0.155025\pi$
$941$			28.7124i			0.935999i	−0.883729	+	0.467999i	$0.844975\pi$
$942$		0			0
$943$	−30.8763			−1.00547
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−14.1024			−0.458265			−0.229132	−	0.973395i	$-0.573589\pi$
$947$	−14.1024			−0.458265			−0.229132	+	0.973395i	$0.573589\pi$
$948$		0			0
$949$	5.79438	+	15.5102i	0.188093	+	0.503483i
$950$		0			0
$951$		−	59.7755i		−	1.93835i
$952$		0			0
$953$			15.8361i			0.512982i	0.966547	+	0.256491i	$0.0825665\pi$
$953$			15.8361i			0.512982i	−0.966547	+	0.256491i	$0.917434\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$	−44.2653			−1.43089
$958$		0			0
$959$	5.30550			0.171324
$960$		0			0
$961$	21.1024			0.680721
$962$		0			0
$963$			11.6508i			0.375442i
$964$		0			0
$965$		0			0
$966$		0			0
$967$	0.713359			0.0229401			0.0114700	−	0.999934i	$-0.496349\pi$
$967$	0.713359			0.0229401			0.0114700	+	0.999934i	$0.496349\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−11.8182			−0.379263			−0.189632	−	0.981855i	$-0.560729\pi$
$971$	−11.8182			−0.379263			−0.189632	+	0.981855i	$0.560729\pi$
$972$		0			0
$973$	18.2156			0.583966
$974$		0			0
$975$		0			0
$976$		0			0
$977$	16.9181			0.541257			0.270628	−	0.962684i	$-0.412769\pi$
$977$	16.9181			0.541257			0.270628	+	0.962684i	$0.412769\pi$
$978$		0			0
$979$	79.6722			2.54634
$980$		0			0
$981$			2.70543i			0.0863776i
$982$		0			0
$983$	12.5315			0.399694			0.199847	−	0.979827i	$-0.435955\pi$
$983$	12.5315			0.399694			0.199847	+	0.979827i	$0.435955\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$			15.5529i			0.495053i
$988$		0			0
$989$	−31.0731			−0.988067
$990$		0			0
$991$	−49.6543			−1.57732			−0.788660	−	0.614829i	$-0.789225\pi$
$991$	−49.6543			−1.57732			−0.788660	+	0.614829i	$0.789225\pi$
$992$		0			0
$993$	33.6732			1.06859
$994$		0			0
$995$		0			0
$996$		0			0
$997$			54.5912i			1.72892i	0.502700	+	0.864461i	$0.332340\pi$
$997$			54.5912i			1.72892i	−0.502700	+	0.864461i	$0.667660\pi$
$998$		0			0
$999$		−	14.8157i		−	0.468748i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1300.2.d.d.649.2		6
5.2	odd	4		260.2.f.a.181.1	✓	6
5.3	odd	4		1300.2.f.e.701.5		6
5.4	even	2		1300.2.d.c.649.5		6
13.12	even	2		1300.2.d.c.649.2		6
15.2	even	4		2340.2.c.d.181.5		6
20.7	even	4		1040.2.k.c.961.5		6
65.12	odd	4		260.2.f.a.181.2	yes	6
65.38	odd	4		1300.2.f.e.701.6		6
65.47	even	4		3380.2.a.n.1.1		3
65.57	even	4		3380.2.a.m.1.1		3
65.64	even	2	inner	1300.2.d.d.649.5		6
195.77	even	4		2340.2.c.d.181.2		6
260.207	even	4		1040.2.k.c.961.6		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
260.2.f.a.181.1	✓	6	5.2	odd	4
260.2.f.a.181.2	yes	6	65.12	odd	4
1040.2.k.c.961.5		6	20.7	even	4
1040.2.k.c.961.6		6	260.207	even	4
1300.2.d.c.649.2		6	13.12	even	2
1300.2.d.c.649.5		6	5.4	even	2
1300.2.d.d.649.2		6	1.1	even	1	trivial
1300.2.d.d.649.5		6	65.64	even	2	inner
1300.2.f.e.701.5		6	5.3	odd	4
1300.2.f.e.701.6		6	65.38	odd	4
2340.2.c.d.181.2		6	195.77	even	4
2340.2.c.d.181.5		6	15.2	even	4
3380.2.a.m.1.1		3	65.57	even	4
3380.2.a.n.1.1		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)

\(f(q)\)	\(=\)	\(q-2.26180i q^{3} +1.11575 q^{7} -2.11575 q^{9} +O(q^{10})\)	\(q-2.26180i q^{3} +1.11575 q^{7} -2.11575 q^{9} +5.37755i q^{11} +(-1.26180 - 3.37755i) q^{13} -7.90116i q^{19} -2.52360i q^{21} -6.49330i q^{23} -2.00000i q^{27} -3.63935 q^{29} -3.14605i q^{31} +12.1630 q^{33} +7.40786 q^{37} +(-7.63935 + 2.85395i) q^{39} -4.75510i q^{41} -4.78541i q^{43} -6.16296 q^{47} -5.75510 q^{49} +0.292106i q^{53} -17.8709 q^{57} +11.3776i q^{59} -5.34725 q^{61} -2.36065 q^{63} +11.8709 q^{67} -14.6866 q^{69} +3.43816i q^{71} -4.59214 q^{73} +6.00000i q^{77} +10.8157 q^{79} -10.8709 q^{81} +7.40786 q^{83} +8.23150i q^{87} -14.8157i q^{89} +(-1.40786 - 3.76850i) q^{91} -7.11575 q^{93} -3.76850 q^{97} -11.3776i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 6 q^{9}+O(q^{10}) \) 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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