LMFDB - Embedded newform 1300.2.d.d.649.3

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			649.3
Root			$-0.339877i$ of defining polynomial
Character	$\chi$	$=$	1300.649
Dual form			1300.2.d.d.649.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.339877i q^{3} -3.88448 q^{7} +2.88448 q^{9} +O(q^{10})$	$q-0.339877i q^{3} -3.88448 q^{7} +2.88448 q^{9} -1.54461i q^{11} +(0.660123 + 3.54461i) q^{13} +2.86485i q^{19} +1.32025i q^{21} +5.42909i q^{23} -2.00000i q^{27} +5.20473 q^{29} -6.22436i q^{31} -0.524976 q^{33} +8.56424 q^{37} +(1.20473 - 0.224361i) q^{39} +9.08921i q^{41} +0.980369i q^{43} +6.52498 q^{47} +8.08921 q^{49} +6.44872i q^{53} +0.973697 q^{57} +4.45539i q^{59} +9.65345 q^{61} -11.2047 q^{63} -6.97370 q^{67} +1.84522 q^{69} +12.6731i q^{71} -3.43576 q^{73} +6.00000i q^{77} +13.1285 q^{79} +7.97370 q^{81} +8.56424 q^{83} -1.76897i q^{87} -17.1285i q^{89} +(-2.56424 - 13.7690i) q^{91} -2.11552 q^{93} -13.7690 q^{97} -4.45539i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$		−	0.339877i		−	0.196228i	−0.995175	−	0.0981140i	$-0.968719\pi$
$3$		−	0.339877i		−	0.196228i	0.995175	−	0.0981140i	$-0.0312810\pi$
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	−3.88448			−1.46820			−0.734098	−	0.679043i	$-0.762395\pi$
$7$	−3.88448			−1.46820			−0.734098	+	0.679043i	$0.762395\pi$
$8$		0			0
$9$	2.88448			0.961495
$10$		0			0
$11$		−	1.54461i		−	0.465716i	−0.972511	−	0.232858i	$-0.925192\pi$
$11$		−	1.54461i		−	0.465716i	0.972511	−	0.232858i	$-0.0748078\pi$
$12$		0			0
$13$	0.660123	+	3.54461i	0.183085	+	0.983097i
$13$	0.660123	+	3.54461i	0.183085	+	0.983097i
$14$		0			0
$15$		0			0
$16$		0			0
$17$		0			0		1.00000			$0$
$17$		0			0		−1.00000			$\pi$
$18$		0			0
$19$			2.86485i			0.657242i	0.944462	+	0.328621i	$0.106584\pi$
$19$			2.86485i			0.657242i	−0.944462	+	0.328621i	$0.893416\pi$
$20$		0			0
$21$			1.32025i			0.288101i
$22$		0			0
$23$			5.42909i			1.13204i	0.824390	+	0.566022i	$0.191518\pi$
$23$			5.42909i			1.13204i	−0.824390	+	0.566022i	$0.808482\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$		−	2.00000i		−	0.384900i
$28$		0			0
$29$	5.20473			0.966494			0.483247	−	0.875484i	$-0.339457\pi$
$29$	5.20473			0.966494			0.483247	+	0.875484i	$0.339457\pi$
$30$		0			0
$31$		−	6.22436i		−	1.11793i	−0.829192	−	0.558964i	$-0.811199\pi$
$31$		−	6.22436i		−	1.11793i	0.829192	−	0.558964i	$-0.188801\pi$
$32$		0			0
$33$	−0.524976			−0.0913866
$34$		0			0
$35$		0			0
$36$		0			0
$37$	8.56424			1.40795			0.703976	−	0.710224i	$-0.251406\pi$
$37$	8.56424			1.40795			0.703976	+	0.710224i	$0.251406\pi$
$38$		0			0
$39$	1.20473	−	0.224361i	0.192911	−	0.0359264i
$40$		0			0
$41$			9.08921i			1.41950i	0.704455	+	0.709748i	$0.251191\pi$
$41$			9.08921i			1.41950i	−0.704455	+	0.709748i	$0.748809\pi$
$42$		0			0
$43$			0.980369i			0.149505i	0.997202	+	0.0747525i	$0.0238167\pi$
$43$			0.980369i			0.149505i	−0.997202	+	0.0747525i	$0.976183\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	6.52498			0.951766			0.475883	−	0.879509i	$-0.342129\pi$
$47$	6.52498			0.951766			0.475883	+	0.879509i	$0.342129\pi$
$48$		0			0
$49$	8.08921			1.15560
$50$		0			0
$51$		0			0
$52$		0			0
$53$			6.44872i			0.885800i	0.896571	+	0.442900i	$0.146050\pi$
$53$			6.44872i			0.885800i	−0.896571	+	0.442900i	$0.853950\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	0.973697			0.128969
$58$		0			0
$59$			4.45539i			0.580043i	0.957020	+	0.290021i	$0.0936624\pi$
$59$			4.45539i			0.580043i	−0.957020	+	0.290021i	$0.906338\pi$
$60$		0			0
$61$	9.65345			1.23600			0.617999	−	0.786179i	$-0.287944\pi$
$61$	9.65345			1.23600			0.617999	+	0.786179i	$0.287944\pi$
$62$		0			0
$63$	−11.2047			−1.41166
$64$		0			0
$65$		0			0
$66$		0			0
$67$	−6.97370			−0.851973			−0.425986	−	0.904730i	$-0.640073\pi$
$67$	−6.97370			−0.851973			−0.425986	+	0.904730i	$0.640073\pi$
$68$		0			0
$69$	1.84522			0.222139
$70$		0			0
$71$			12.6731i			1.50402i	0.659153	+	0.752009i	$0.270915\pi$
$71$			12.6731i			1.50402i	−0.659153	+	0.752009i	$0.729085\pi$
$72$		0			0
$73$	−3.43576			−0.402126			−0.201063	−	0.979578i	$-0.564440\pi$
$73$	−3.43576			−0.402126			−0.201063	+	0.979578i	$0.564440\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$			6.00000i			0.683763i
$78$		0			0
$79$	13.1285			1.47707			0.738534	−	0.674216i	$-0.235518\pi$
$79$	13.1285			1.47707			0.738534	+	0.674216i	$0.235518\pi$
$80$		0			0
$81$	7.97370			0.885966
$82$		0			0
$83$	8.56424			0.940047			0.470024	−	0.882654i	$-0.344245\pi$
$83$	8.56424			0.940047			0.470024	+	0.882654i	$0.344245\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		−	1.76897i		−	0.189653i
$88$		0			0
$89$		−	17.1285i		−	1.81561i	−0.419387	−	0.907807i	$-0.637755\pi$
$89$		−	17.1285i		−	1.81561i	0.419387	−	0.907807i	$-0.362245\pi$
$90$		0			0
$91$	−2.56424	−	13.7690i	−0.268805	−	1.44338i
$92$		0			0
$93$	−2.11552			−0.219369
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−13.7690			−1.39803			−0.699013	−	0.715109i	$-0.746377\pi$
$97$	−13.7690			−1.39803			−0.699013	+	0.715109i	$0.746377\pi$
$98$		0			0
$99$		−	4.45539i		−	0.447784i
$100$		0			0
$101$	−6.44872			−0.641672			−0.320836	−	0.947135i	$-0.603964\pi$
$101$	−6.44872			−0.641672			−0.320836	+	0.947135i	$0.603964\pi$
$102$		0			0
$103$			12.1088i			1.19312i	0.802569	+	0.596560i	$0.203466\pi$
$103$			12.1088i			1.19312i	−0.802569	+	0.596560i	$0.796534\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	17.4291i		−	1.68493i	−0.538748	−	0.842467i	$-0.681103\pi$
$107$		−	17.4291i		−	1.68493i	0.538748	−	0.842467i	$-0.318897\pi$
$108$		0			0
$109$			16.4095i			1.57174i	0.618391	+	0.785871i	$0.287785\pi$
$109$			16.4095i			1.57174i	−0.618391	+	0.785871i	$0.712215\pi$
$110$		0			0
$111$		−	2.91079i		−	0.276280i
$112$		0			0
$113$			1.59054i			0.149625i	0.997198	+	0.0748127i	$0.0238359\pi$
$113$			1.59054i			0.149625i	−0.997198	+	0.0748127i	$0.976164\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	1.90411	+	10.2244i	0.176035	+	0.945242i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	8.61419			0.783108
$122$		0			0
$123$	3.08921			0.278545
$124$		0			0
$125$		0			0
$126$		0			0
$127$			0.108844i			0.00965837i	0.999988	+	0.00482918i	$0.00153718\pi$
$127$			0.108844i			0.00965837i	−0.999988	+	0.00482918i	$0.998463\pi$
$128$		0			0
$129$	0.333205			0.0293371
$130$		0			0
$131$	−10.4095			−0.909479			−0.454739	−	0.890625i	$-0.650268\pi$
$131$	−10.4095			−0.909479			−0.454739	+	0.890625i	$0.650268\pi$
$132$		0			0
$133$		−	11.1285i		−	0.964961i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−9.08921			−0.776544			−0.388272	−	0.921545i	$-0.626928\pi$
$137$	−9.08921			−0.776544			−0.388272	+	0.921545i	$0.626928\pi$
$138$		0			0
$139$	−9.04995			−0.767607			−0.383803	−	0.923415i	$-0.625386\pi$
$139$	−9.04995			−0.767607			−0.383803	+	0.923415i	$0.625386\pi$
$140$		0			0
$141$		−	2.21769i		−	0.186763i
$142$		0			0
$143$	5.47502	−	1.01963i	0.457844	−	0.0852658i
$144$		0			0
$145$		0			0
$146$		0			0
$147$		−	2.74934i		−	0.226761i
$148$		0			0
$149$			11.1285i			0.911680i	0.890062	+	0.455840i	$0.150661\pi$
$149$			11.1285i			0.911680i	−0.890062	+	0.455840i	$0.849339\pi$
$150$		0			0
$151$			9.13515i			0.743408i	0.928351	+	0.371704i	$0.121226\pi$
$151$			9.13515i			0.743408i	−0.928351	+	0.371704i	$0.878774\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$		−	4.91079i		−	0.391923i	−0.980612	−	0.195962i	$-0.937217\pi$
$157$		−	4.91079i		−	0.391923i	0.980612	−	0.195962i	$-0.0627828\pi$
$158$		0			0
$159$	2.19177			0.173819
$160$		0			0
$161$		−	21.0892i		−	1.66206i
$162$		0			0
$163$	16.9344			1.32641			0.663204	−	0.748439i	$-0.269196\pi$
$163$	16.9344			1.32641			0.663204	+	0.748439i	$0.269196\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	21.6142			1.67256			0.836278	−	0.548306i	$-0.184727\pi$
$167$	21.6142			1.67256			0.836278	+	0.548306i	$0.184727\pi$
$168$		0			0
$169$	−12.1285	+	4.67975i	−0.932960	+	0.359981i
$170$		0			0
$171$			8.26362i			0.631935i
$172$		0			0
$173$		−	8.03926i		−	0.611214i	−0.952158	−	0.305607i	$-0.901141\pi$
$173$		−	8.03926i		−	0.611214i	0.952158	−	0.305607i	$-0.0988593\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	1.51429			0.113821
$178$		0			0
$179$	−10.8582			−0.811579			−0.405789	−	0.913967i	$-0.633003\pi$
$179$	−10.8582			−0.811579			−0.405789	+	0.913967i	$0.633003\pi$
$180$		0			0
$181$	10.5642			0.785234			0.392617	−	0.919702i	$-0.371570\pi$
$181$	10.5642			0.785234			0.392617	+	0.919702i	$0.371570\pi$
$182$		0			0
$183$		−	3.28098i		−	0.242537i
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$			7.76897i			0.565109i
$190$		0			0
$191$	−8.81892			−0.638115			−0.319057	−	0.947735i	$-0.603366\pi$
$191$	−8.81892			−0.638115			−0.319057	+	0.947735i	$0.603366\pi$
$192$		0			0
$193$	−0.270294			−0.0194562			−0.00972809	−	0.999953i	$-0.503097\pi$
$193$	−0.270294			−0.0194562			−0.00972809	+	0.999953i	$0.503097\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	3.96074			0.282191			0.141095	−	0.989996i	$-0.454938\pi$
$197$	3.96074			0.282191			0.141095	+	0.989996i	$0.454938\pi$
$198$		0			0
$199$	−6.03926			−0.428112			−0.214056	−	0.976821i	$-0.568667\pi$
$199$	−6.03926			−0.428112			−0.214056	+	0.976821i	$0.568667\pi$
$200$		0			0
$201$			2.37020i			0.167181i
$202$		0			0
$203$	−20.2177			−1.41900
$204$		0			0
$205$		0			0
$206$		0			0
$207$			15.6601i			1.08845i
$208$		0			0
$209$	4.42507			0.306089
$210$		0			0
$211$	−16.2177			−1.11647			−0.558236	−	0.829682i	$-0.688522\pi$
$211$	−16.2177			−1.11647			−0.558236	+	0.829682i	$0.688522\pi$
$212$		0			0
$213$	4.30729			0.295130
$214$		0			0
$215$		0			0
$216$		0			0
$217$			24.1784i			1.64134i
$218$		0			0
$219$			1.16774i			0.0789083i
$220$		0			0
$221$		0			0
$222$		0			0
$223$	−10.0629			−0.673862			−0.336931	−	0.941529i	$-0.609389\pi$
$223$	−10.0629			−0.673862			−0.336931	+	0.941529i	$0.609389\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	3.43576			0.228040			0.114020	−	0.993478i	$-0.463627\pi$
$227$	3.43576			0.228040			0.114020	+	0.993478i	$0.463627\pi$
$228$		0			0
$229$		−	22.8582i		−	1.51051i	−0.655430	−	0.755256i	$-0.727513\pi$
$229$		−	22.8582i		−	1.51051i	0.655430	−	0.755256i	$-0.272487\pi$
$230$		0			0
$231$	2.03926			0.134174
$232$		0			0
$233$		−	4.40946i		−	0.288873i	−0.989514	−	0.144437i	$-0.953863\pi$
$233$		−	4.40946i		−	0.288873i	0.989514	−	0.144437i	$-0.0461370\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		−	4.46207i		−	0.289842i
$238$		0			0
$239$		−	22.8122i		−	1.47560i	−0.675018	−	0.737801i	$-0.735864\pi$
$239$		−	22.8122i		−	1.47560i	0.675018	−	0.737801i	$-0.264136\pi$
$240$		0			0
$241$			10.6798i			0.687943i	0.938980	+	0.343972i	$0.111772\pi$
$241$			10.6798i			0.687943i	−0.938980	+	0.343972i	$0.888228\pi$
$242$		0			0
$243$		−	8.71008i		−	0.558752i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−10.1548	+	1.89116i	−0.646133	+	0.120331i
$248$		0			0
$249$		−	2.91079i		−	0.184464i
$250$		0			0
$251$	−21.2676			−1.34240			−0.671201	−	0.741276i	$-0.734221\pi$
$251$	−21.2676			−1.34240			−0.671201	+	0.741276i	$0.734221\pi$
$252$		0			0
$253$	8.38581			0.527211
$254$		0			0
$255$		0			0
$256$		0			0
$257$		−	18.8974i		−	1.17879i	−0.807845	−	0.589395i	$-0.799366\pi$
$257$		−	18.8974i		−	1.17879i	0.807845	−	0.589395i	$-0.200634\pi$
$258$		0			0
$259$	−33.2676			−2.06715
$260$		0			0
$261$	15.0130			0.929279
$262$		0			0
$263$		−	13.0196i		−	0.802825i	−0.915897	−	0.401412i	$-0.868519\pi$
$263$		−	13.0196i		−	0.802825i	0.915897	−	0.401412i	$-0.131481\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	−5.82157			−0.356274
$268$		0			0
$269$	−14.3702			−0.876166			−0.438083	−	0.898934i	$-0.644342\pi$
$269$	−14.3702			−0.876166			−0.438083	+	0.898934i	$0.644342\pi$
$270$		0			0
$271$			13.7230i			0.833615i	0.908995	+	0.416807i	$0.136851\pi$
$271$			13.7230i			0.833615i	−0.908995	+	0.416807i	$0.863149\pi$
$272$		0			0
$273$	−4.67975	+	0.871525i	−0.283232	+	0.0527471i
$274$		0			0
$275$		0			0
$276$		0			0
$277$			19.2676i			1.15768i	0.815441	+	0.578840i	$0.196494\pi$
$277$			19.2676i			1.15768i	−0.815441	+	0.578840i	$0.803506\pi$
$278$		0			0
$279$		−	17.9541i		−	1.07488i
$280$		0			0
$281$		−	3.96074i		−	0.236278i	−0.992997	−	0.118139i	$-0.962307\pi$
$281$		−	3.96074i		−	0.236278i	0.992997	−	0.118139i	$-0.0376928\pi$
$282$		0			0
$283$		−	21.1981i		−	1.26009i	−0.776557	−	0.630047i	$-0.783036\pi$
$283$		−	21.1981i		−	1.26009i	0.776557	−	0.630047i	$-0.216964\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		−	35.3069i		−	2.08410i
$288$		0			0
$289$	17.0000			1.00000
$290$		0			0
$291$			4.67975i			0.274332i
$292$		0			0
$293$	−13.6927			−0.799937			−0.399968	−	0.916529i	$-0.630979\pi$
$293$	−13.6927			−0.799937			−0.399968	+	0.916529i	$0.630979\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	−3.08921			−0.179254
$298$		0			0
$299$	−19.2440	+	3.58387i	−1.11291	+	0.207260i
$300$		0			0
$301$		−	3.80823i		−	0.219503i
$302$		0			0
$303$			2.19177i			0.125914i
$304$		0			0
$305$		0			0
$306$		0			0
$307$	12.7953			0.730265			0.365132	−	0.930956i	$-0.381024\pi$
$307$	12.7953			0.730265			0.365132	+	0.930956i	$0.381024\pi$
$308$		0			0
$309$	4.11552			0.234124
$310$		0			0
$311$	−22.8582			−1.29617			−0.648084	−	0.761569i	$-0.724430\pi$
$311$	−22.8582			−1.29617			−0.648084	+	0.761569i	$0.724430\pi$
$312$		0			0
$313$		−	27.2284i		−	1.53904i	−0.638623	−	0.769520i	$-0.720496\pi$
$313$		−	27.2284i		−	1.53904i	0.638623	−	0.769520i	$-0.279504\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−27.7926			−1.56099			−0.780494	−	0.625163i	$-0.785033\pi$
$317$	−27.7926			−1.56099			−0.780494	+	0.625163i	$0.785033\pi$
$318$		0			0
$319$		−	8.03926i		−	0.450112i
$320$		0			0
$321$	−5.92375			−0.330631
$322$		0			0
$323$		0			0
$324$		0			0
$325$		0			0
$326$		0			0
$327$	5.57720			0.308420
$328$		0			0
$329$	−25.3462			−1.39738
$330$		0			0
$331$		−	19.7230i		−	1.08408i	−0.840354	−	0.542038i	$-0.817653\pi$
$331$		−	19.7230i		−	1.08408i	0.840354	−	0.542038i	$-0.182347\pi$
$332$		0			0
$333$	24.7034			1.35374
$334$		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
$339$	0.540588			0.0293607
$340$		0			0
$341$	−9.61419			−0.520638
$342$		0			0
$343$	−4.23103			−0.228454
$344$		0			0
$345$		0			0
$346$		0			0
$347$			2.61017i			0.140121i	0.997543	+	0.0700607i	$0.0223193\pi$
$347$			2.61017i			0.140121i	−0.997543	+	0.0700607i	$0.977681\pi$
$348$		0			0
$349$		−	3.08921i		−	0.165362i	−0.996576	−	0.0826809i	$-0.973652\pi$
$349$		−	3.08921i		−	0.165362i	0.996576	−	0.0826809i	$-0.0263482\pi$
$350$		0			0
$351$	7.08921	−	1.32025i	0.378394	−	0.0704695i
$352$		0			0
$353$	20.9211			1.11352			0.556759	−	0.830674i	$-0.312045\pi$
$353$	20.9211			1.11352			0.556759	+	0.830674i	$0.312045\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$			30.8515i			1.62828i	0.580668	+	0.814140i	$0.302791\pi$
$359$			30.8515i			1.62828i	−0.580668	+	0.814140i	$0.697209\pi$
$360$		0			0
$361$	10.7926			0.568032
$362$		0			0
$363$		−	2.92776i		−	0.153668i
$364$		0			0
$365$		0			0
$366$		0			0
$367$		−	21.0196i		−	1.09722i	−0.836080	−	0.548608i	$-0.815158\pi$
$367$		−	21.0196i		−	1.09722i	0.836080	−	0.548608i	$-0.184842\pi$
$368$		0			0
$369$			26.2177i			1.36484i
$370$		0			0
$371$		−	25.0500i		−	1.30053i
$372$		0			0
$373$			29.3462i			1.51949i	0.650223	+	0.759743i	$0.274675\pi$
$373$			29.3462i			1.51949i	−0.650223	+	0.759743i	$0.725325\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	3.43576	+	18.4487i	0.176951	+	0.950157i
$378$		0			0
$379$			11.3528i			0.583156i	0.956547	+	0.291578i	$0.0941803\pi$
$379$			11.3528i			0.583156i	−0.956547	+	0.291578i	$0.905820\pi$
$380$		0			0
$381$	0.0369937			0.00189524
$382$		0			0
$383$	−20.5642			−1.05078			−0.525392	−	0.850860i	$-0.676081\pi$
$383$	−20.5642			−1.05078			−0.525392	+	0.850860i	$0.676081\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$			2.82786i			0.143748i
$388$		0			0
$389$	−26.8189			−1.35977			−0.679887	−	0.733317i	$-0.737971\pi$
$389$	−26.8189			−1.35977			−0.679887	+	0.733317i	$0.737971\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$			3.53793i			0.178465i
$394$		0			0
$395$		0			0
$396$		0			0
$397$	36.1022			1.81192			0.905958	−	0.423367i	$-0.139152\pi$
$397$	36.1022			1.81192			0.905958	+	0.423367i	$0.139152\pi$
$398$		0			0
$399$	−3.78231			−0.189352
$400$		0			0
$401$			4.07852i			0.203672i	0.994801	+	0.101836i	$0.0324716\pi$
$401$			4.07852i			0.203672i	−0.994801	+	0.101836i	$0.967528\pi$
$402$		0			0
$403$	22.0629	−	4.10884i	1.09903	−	0.204676i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	13.2284i		−	0.655706i
$408$		0			0
$409$		−	0.627148i		−	0.0310105i	−0.999880	−	0.0155052i	$-0.995064\pi$
$409$		−	0.627148i		−	0.0310105i	0.999880	−	0.0155052i	$-0.00493567\pi$
$410$		0			0
$411$			3.08921i			0.152380i
$412$		0			0
$413$		−	17.3069i		−	0.851617i
$414$		0			0
$415$		0			0
$416$		0			0
$417$			3.07587i			0.150626i
$418$		0			0
$419$	11.5513			0.564317			0.282158	−	0.959368i	$-0.408950\pi$
$419$	11.5513			0.564317			0.282158	+	0.959368i	$0.408950\pi$
$420$		0			0
$421$			35.4853i			1.72945i	0.502246	+	0.864725i	$0.332507\pi$
$421$			35.4853i			1.72945i	−0.502246	+	0.864725i	$0.667493\pi$
$422$		0			0
$423$	18.8212			0.915117
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−37.4987			−1.81469
$428$		0			0
$429$	−0.346549	−	1.86083i	−0.0167315	−	0.0898419i
$430$		0			0
$431$		−	33.7623i		−	1.62627i	−0.582073	−	0.813136i	$-0.697758\pi$
$431$		−	33.7623i		−	1.62627i	0.582073	−	0.813136i	$-0.302242\pi$
$432$		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
$437$	−15.5535			−0.744027
$438$		0			0
$439$	19.0892			0.911078			0.455539	−	0.890216i	$-0.349446\pi$
$439$	19.0892			0.911078			0.455539	+	0.890216i	$0.349446\pi$
$440$		0			0
$441$	23.3332			1.11110
$442$		0			0
$443$		−	10.2873i		−	0.488763i	−0.969679	−	0.244382i	$-0.921415\pi$
$443$		−	10.2873i		−	0.488763i	0.969679	−	0.244382i	$-0.0785849\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	3.78231			0.178897
$448$		0			0
$449$			2.21769i			0.104659i	0.998630	+	0.0523296i	$0.0166646\pi$
$449$			2.21769i			0.104659i	−0.998630	+	0.0523296i	$0.983335\pi$
$450$		0			0
$451$	14.0393			0.661083
$452$		0			0
$453$	3.10483			0.145877
$454$		0			0
$455$		0			0
$456$		0			0
$457$	17.3069			0.809583			0.404791	−	0.914409i	$-0.367344\pi$
$457$	17.3069			0.809583			0.404791	+	0.914409i	$0.367344\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		−	16.1392i		−	0.751676i	−0.926685	−	0.375838i	$-0.877355\pi$
$461$		−	16.1392i		−	0.751676i	0.926685	−	0.375838i	$-0.122645\pi$
$462$		0			0
$463$	−6.52498			−0.303241			−0.151621	−	0.988439i	$-0.548449\pi$
$463$	−6.52498			−0.303241			−0.151621	+	0.988439i	$0.548449\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$			35.4291i			1.63946i	0.572748	+	0.819731i	$0.305877\pi$
$467$			35.4291i			1.63946i	−0.572748	+	0.819731i	$0.694123\pi$
$468$		0			0
$469$	27.0892			1.25086
$470$		0			0
$471$	−1.66906			−0.0769064
$472$		0			0
$473$	1.51429			0.0696269
$474$		0			0
$475$		0			0
$476$		0			0
$477$			18.6012i			0.851692i
$478$		0			0
$479$			26.7730i			1.22329i	0.791133	+	0.611644i	$0.209492\pi$
$479$			26.7730i			1.22329i	−0.791133	+	0.611644i	$0.790508\pi$
$480$		0			0
$481$	5.65345	+	30.3569i	0.257775	+	1.38415i
$482$		0			0
$483$	−7.16774			−0.326143
$484$		0			0
$485$		0			0
$486$		0			0
$487$	18.3725			0.832536			0.416268	−	0.909242i	$-0.363338\pi$
$487$	18.3725			0.832536			0.416268	+	0.909242i	$0.363338\pi$
$488$		0			0
$489$		−	5.75562i		−	0.260278i
$490$		0			0
$491$	−8.81892			−0.397992			−0.198996	−	0.980000i	$-0.563768\pi$
$491$	−8.81892			−0.397992			−0.198996	+	0.980000i	$0.563768\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		−	49.2284i		−	2.20819i
$498$		0			0
$499$		−	23.7756i		−	1.06434i	−0.846636	−	0.532172i	$-0.821376\pi$
$499$		−	23.7756i		−	1.06434i	0.846636	−	0.532172i	$-0.178624\pi$
$500$		0			0
$501$		−	7.34616i		−	0.328202i
$502$		0			0
$503$			39.1455i			1.74541i	0.488248	+	0.872705i	$0.337636\pi$
$503$			39.1455i			1.74541i	−0.488248	+	0.872705i	$0.662364\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	1.59054	+	4.12219i	0.0706384	+	0.183073i
$508$		0			0
$509$		−	44.5745i		−	1.97573i	−0.155309	−	0.987866i	$-0.549637\pi$
$509$		−	44.5745i		−	1.97573i	0.155309	−	0.987866i	$-0.450363\pi$
$510$		0			0
$511$	13.3462			0.590400
$512$		0			0
$513$	5.72971			0.252973
$514$		0			0
$515$		0			0
$516$		0			0
$517$		−	10.0785i		−	0.443253i
$518$		0			0
$519$	−2.73236			−0.119937
$520$		0			0
$521$	−17.2047			−0.753753			−0.376876	−	0.926264i	$-0.623002\pi$
$521$	−17.2047			−0.753753			−0.376876	+	0.926264i	$0.623002\pi$
$522$		0			0
$523$			3.19806i			0.139841i	0.997553	+	0.0699207i	$0.0222746\pi$
$523$			3.19806i			0.139841i	−0.997553	+	0.0699207i	$0.977725\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−6.47502			−0.281523
$530$		0			0
$531$			12.8515i			0.557708i
$532$		0			0
$533$	−32.2177	+	6.00000i	−1.39550	+	0.259889i
$534$		0			0
$535$		0			0
$536$		0			0
$537$			3.69044i			0.159254i
$538$		0			0
$539$		−	12.4947i		−	0.538183i
$540$		0			0
$541$			1.65118i			0.0709899i	0.999370	+	0.0354950i	$0.0113008\pi$
$541$			1.65118i			0.0709899i	−0.999370	+	0.0354950i	$0.988699\pi$
$542$		0			0
$543$		−	3.59054i		−	0.154085i
$544$		0			0
$545$		0			0
$546$		0			0
$547$		−	38.3265i		−	1.63872i	−0.573276	−	0.819362i	$-0.694328\pi$
$547$		−	38.3265i		−	1.63872i	0.573276	−	0.819362i	$-0.305672\pi$
$548$		0			0
$549$	27.8452			1.18841
$550$		0			0
$551$			14.9108i			0.635221i
$552$		0			0
$553$	−50.9973			−2.16863
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−31.8711			−1.35042			−0.675212	−	0.737624i	$-0.735948\pi$
$557$	−31.8711			−1.35042			−0.675212	+	0.737624i	$0.735948\pi$
$558$		0			0
$559$	−3.47502	+	0.647164i	−0.146978	+	0.0273721i
$560$		0			0
$561$		0			0
$562$		0			0
$563$			5.87781i			0.247720i	0.992300	+	0.123860i	$0.0395274\pi$
$563$			5.87781i			0.247720i	−0.992300	+	0.123860i	$0.960473\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	−30.9737			−1.30077
$568$		0			0
$569$	16.0629			0.673392			0.336696	−	0.941613i	$-0.390690\pi$
$569$	16.0629			0.673392			0.336696	+	0.941613i	$0.390690\pi$
$570$		0			0
$571$	5.04995			0.211334			0.105667	−	0.994402i	$-0.466302\pi$
$571$	5.04995			0.211334			0.105667	+	0.994402i	$0.466302\pi$
$572$		0			0
$573$			2.99735i			0.125216i
$574$		0			0
$575$		0			0
$576$		0			0
$577$	17.3832			0.723670			0.361835	−	0.932242i	$-0.382150\pi$
$577$	17.3832			0.723670			0.361835	+	0.932242i	$0.382150\pi$
$578$		0			0
$579$			0.0918667i			0.00381785i
$580$		0			0
$581$	−33.2676			−1.38017
$582$		0			0
$583$	9.96074			0.412532
$584$		0			0
$585$		0			0
$586$		0			0
$587$	7.57493			0.312651			0.156325	−	0.987706i	$-0.450035\pi$
$587$	7.57493			0.312651			0.156325	+	0.987706i	$0.450035\pi$
$588$		0			0
$589$	17.8319			0.734750
$590$		0			0
$591$		−	1.34616i		−	0.0553738i
$592$		0			0
$593$	16.9500			0.696055			0.348028	−	0.937484i	$-0.386852\pi$
$593$	16.9500			0.696055			0.348028	+	0.937484i	$0.386852\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$			2.05261i			0.0840075i
$598$		0			0
$599$	43.6771			1.78460			0.892299	−	0.451445i	$-0.149091\pi$
$599$	43.6771			1.78460			0.892299	+	0.451445i	$0.149091\pi$
$600$		0			0
$601$	2.07852			0.0847847			0.0423924	−	0.999101i	$-0.486502\pi$
$601$	2.07852			0.0847847			0.0423924	+	0.999101i	$0.486502\pi$
$602$		0			0
$603$	−20.1155			−0.819167
$604$		0			0
$605$		0			0
$606$		0			0
$607$			34.3265i			1.39327i	0.717425	+	0.696635i	$0.245320\pi$
$607$			34.3265i			1.39327i	−0.717425	+	0.696635i	$0.754680\pi$
$608$		0			0
$609$			6.87153i			0.278448i
$610$		0			0
$611$	4.30729	+	23.1285i	0.174254	+	0.935678i
$612$		0			0
$613$	−18.3569			−0.741426			−0.370713	−	0.928747i	$-0.620887\pi$
$613$	−18.3569			−0.741426			−0.370713	+	0.928747i	$0.620887\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	37.5246			1.51068			0.755342	−	0.655331i	$-0.227471\pi$
$617$	37.5246			1.51068			0.755342	+	0.655331i	$0.227471\pi$
$618$		0			0
$619$		−	12.8256i		−	0.515504i	−0.966211	−	0.257752i	$-0.917018\pi$
$619$		−	12.8256i		−	0.515504i	0.966211	−	0.257752i	$-0.0829818\pi$
$620$		0			0
$621$	10.8582			0.435724
$622$		0			0
$623$			66.5353i			2.66568i
$624$		0			0
$625$		0			0
$626$		0			0
$627$		−	1.50398i		−	0.0600632i
$628$		0			0
$629$		0			0
$630$		0			0
$631$			12.6472i			0.503476i	0.967795	+	0.251738i	$0.0810021\pi$
$631$			12.6472i			0.503476i	−0.967795	+	0.251738i	$0.918998\pi$
$632$		0			0
$633$			5.51202i			0.219083i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	5.33988	+	28.6731i	0.211574	+	1.13607i
$638$		0			0
$639$			36.5553i			1.44611i
$640$		0			0
$641$	5.10256			0.201539			0.100769	−	0.994910i	$-0.467870\pi$
$641$	5.10256			0.201539			0.100769	+	0.994910i	$0.467870\pi$
$642$		0			0
$643$	0.346549			0.0136666			0.00683328	−	0.999977i	$-0.497825\pi$
$643$	0.346549			0.0136666			0.00683328	+	0.999977i	$0.497825\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	44.2480i		−	1.73957i	−0.493432	−	0.869784i	$-0.664258\pi$
$647$		−	44.2480i		−	1.73957i	0.493432	−	0.869784i	$-0.335742\pi$
$648$		0			0
$649$	6.88183			0.270135
$650$		0			0
$651$	8.21769			0.322077
$652$		0			0
$653$		−	14.0393i		−	0.549399i	−0.961530	−	0.274699i	$-0.911422\pi$
$653$		−	14.0393i		−	0.549399i	0.961530	−	0.274699i	$-0.0885783\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	−9.91040			−0.386642
$658$		0			0
$659$	−12.4487			−0.484933			−0.242467	−	0.970160i	$-0.577957\pi$
$659$	−12.4487			−0.484933			−0.242467	+	0.970160i	$0.577957\pi$
$660$		0			0
$661$		−	11.8216i		−	0.459806i	−0.973214	−	0.229903i	$-0.926159\pi$
$661$		−	11.8216i		−	0.459806i	0.973214	−	0.229903i	$-0.0738409\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$			28.2569i			1.09411i
$668$		0			0
$669$			3.42015i			0.132231i
$670$		0			0
$671$		−	14.9108i		−	0.575625i
$672$		0			0
$673$			31.3069i			1.20679i	0.797442	+	0.603396i	$0.206186\pi$
$673$			31.3069i			1.20679i	−0.797442	+	0.603396i	$0.793814\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		−	28.8582i		−	1.10911i	−0.832147	−	0.554555i	$-0.812889\pi$
$677$		−	28.8582i		−	1.10911i	0.832147	−	0.554555i	$-0.187111\pi$
$678$		0			0
$679$	53.4853			2.05258
$680$		0			0
$681$		−	1.16774i		−	0.0447478i
$682$		0			0
$683$	20.5642			0.786869			0.393434	−	0.919353i	$-0.371287\pi$
$683$	20.5642			0.786869			0.393434	+	0.919353i	$0.371287\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	−7.76897			−0.296405
$688$		0			0
$689$	−22.8582	+	4.25695i	−0.870827	+	0.162177i
$690$		0			0
$691$		−	6.67308i		−	0.253856i	−0.991912	−	0.126928i	$-0.959488\pi$
$691$		−	6.67308i		−	0.253856i	0.991912	−	0.126928i	$-0.0405117\pi$
$692$		0			0
$693$			17.3069i			0.657435i
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$	−1.49867			−0.0566850
$700$		0			0
$701$	9.62980			0.363713			0.181856	−	0.983325i	$-0.441789\pi$
$701$	9.62980			0.363713			0.181856	+	0.983325i	$0.441789\pi$
$702$		0			0
$703$			24.5353i			0.925366i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	25.0500			0.942100
$708$		0			0
$709$		−	33.8082i		−	1.26969i	−0.772638	−	0.634847i	$-0.781063\pi$
$709$		−	33.8082i		−	1.26969i	0.772638	−	0.634847i	$-0.218937\pi$
$710$		0			0
$711$	37.8689			1.42019
$712$		0			0
$713$	33.7926			1.26554
$714$		0			0
$715$		0			0
$716$		0			0
$717$	−7.75336			−0.289554
$718$		0			0
$719$	36.2043			1.35019			0.675097	−	0.737729i	$-0.264102\pi$
$719$	36.2043			1.35019			0.675097	+	0.737729i	$0.264102\pi$
$720$		0			0
$721$		−	47.0366i		−	1.75173i
$722$		0			0
$723$	3.62980			0.134994
$724$		0			0
$725$		0			0
$726$		0			0
$727$			46.2480i			1.71524i	0.514281	+	0.857622i	$0.328059\pi$
$727$			46.2480i			1.71524i	−0.514281	+	0.857622i	$0.671941\pi$
$728$		0			0
$729$	20.9607			0.776324
$730$		0			0
$731$		0			0
$732$		0			0
$733$	−20.0393			−0.740167			−0.370084	−	0.928998i	$-0.620671\pi$
$733$	−20.0393			−0.740167			−0.370084	+	0.928998i	$0.620671\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			10.7716i			0.396778i
$738$		0			0
$739$			52.6338i			1.93617i	0.250629	+	0.968083i	$0.419363\pi$
$739$			52.6338i			1.93617i	−0.250629	+	0.968083i	$0.580637\pi$
$740$		0			0
$741$	0.642760	+	3.45137i	0.0236124	+	0.126789i
$742$		0			0
$743$	47.9497			1.75910			0.879551	−	0.475804i	$-0.157843\pi$
$743$	47.9497			1.75910			0.879551	+	0.475804i	$0.157843\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	24.7034			0.903850
$748$		0			0
$749$			67.7030i			2.47381i
$750$		0			0
$751$	23.0892			0.842537			0.421269	−	0.906936i	$-0.361585\pi$
$751$	23.0892			0.842537			0.421269	+	0.906936i	$0.361585\pi$
$752$		0			0
$753$			7.22838i			0.263417i
$754$		0			0
$755$		0			0
$756$		0			0
$757$		−	44.3176i		−	1.61075i	−0.592765	−	0.805375i	$-0.701964\pi$
$757$		−	44.3176i		−	1.61075i	0.592765	−	0.805375i	$-0.298036\pi$
$758$		0			0
$759$		−	2.85014i		−	0.103454i
$760$		0			0
$761$			41.4853i			1.50384i	0.659253	+	0.751921i	$0.270873\pi$
$761$			41.4853i			1.50384i	−0.659253	+	0.751921i	$0.729127\pi$
$762$		0			0
$763$		−	63.7423i		−	2.30763i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−15.7926	+	2.94111i	−0.570238	+	0.106197i
$768$		0			0
$769$			28.5879i			1.03091i	0.856918	+	0.515453i	$0.172376\pi$
$769$			28.5879i			1.03091i	−0.856918	+	0.515453i	$0.827624\pi$
$770$		0			0
$771$	−6.42280			−0.231312
$772$		0			0
$773$	−15.4358			−0.555186			−0.277593	−	0.960699i	$-0.589537\pi$
$773$	−15.4358			−0.555186			−0.277593	+	0.960699i	$0.589537\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$			11.3069i			0.405633i
$778$		0			0
$779$	−26.0393			−0.932953
$780$		0			0
$781$	19.5749			0.700446
$782$		0			0
$783$		−	10.4095i		−	0.372004i
$784$		0			0
$785$		0			0
$786$		0			0
$787$	−5.38316			−0.191889			−0.0959444	−	0.995387i	$-0.530587\pi$
$787$	−5.38316			−0.191889			−0.0959444	+	0.995387i	$0.530587\pi$
$788$		0			0
$789$	−4.42507			−0.157537
$790$		0			0
$791$		−	6.17843i		−	0.219680i
$792$		0			0
$793$	6.37247	+	34.2177i	0.226293	+	1.21511i
$794$		0			0
$795$		0			0
$796$		0			0
$797$		−	35.7556i		−	1.26653i	−0.773935	−	0.633265i	$-0.781714\pi$
$797$		−	35.7556i		−	1.26653i	0.773935	−	0.633265i	$-0.218286\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		−	49.4068i		−	1.74570i
$802$		0			0
$803$			5.30690i			0.187277i
$804$		0			0
$805$		0			0
$806$		0			0
$807$			4.88410i			0.171928i
$808$		0			0
$809$	35.7712			1.25765			0.628825	−	0.777547i	$-0.283536\pi$
$809$	35.7712			1.25765			0.628825	+	0.777547i	$0.283536\pi$
$810$		0			0
$811$			30.2244i			1.06132i	0.847585	+	0.530660i	$0.178056\pi$
$811$			30.2244i			1.06132i	−0.847585	+	0.530660i	$0.821944\pi$
$812$		0			0
$813$	4.66414			0.163579
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−2.80861			−0.0982610
$818$		0			0
$819$	−7.39650	−	39.7164i	−0.258455	−	1.38780i
$820$		0			0
$821$		−	25.2284i		−	0.880477i	−0.897881	−	0.440238i	$-0.854894\pi$
$821$		−	25.2284i		−	0.880477i	0.897881	−	0.440238i	$-0.145106\pi$
$822$		0			0
$823$			30.9804i			1.07991i	0.841695	+	0.539954i	$0.181558\pi$
$823$			30.9804i			1.07991i	−0.841695	+	0.539954i	$0.818442\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	6.82119			0.237196			0.118598	−	0.992942i	$-0.462160\pi$
$827$	6.82119			0.237196			0.118598	+	0.992942i	$0.462160\pi$
$828$		0			0
$829$	3.39650			0.117965			0.0589827	−	0.998259i	$-0.481214\pi$
$829$	3.39650			0.117965			0.0589827	+	0.998259i	$0.481214\pi$
$830$		0			0
$831$	6.54863			0.227169
$832$		0			0
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$	−12.4487			−0.430291
$838$		0			0
$839$		−	9.76230i		−	0.337032i	−0.985699	−	0.168516i	$-0.946103\pi$
$839$		−	9.76230i		−	0.337032i	0.985699	−	0.168516i	$-0.0538975\pi$
$840$		0			0
$841$	−1.91079			−0.0658892
$842$		0			0
$843$	−1.34616			−0.0463643
$844$		0			0
$845$		0			0
$846$		0			0
$847$	−33.4617			−1.14976
$848$		0			0
$849$	−7.20473			−0.247266
$850$		0			0
$851$			46.4960i			1.59386i
$852$		0			0
$853$	18.2806			0.625916			0.312958	−	0.949767i	$-0.398680\pi$
$853$	18.2806			0.625916			0.312958	+	0.949767i	$0.398680\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$			11.6691i			0.398608i	0.979938	+	0.199304i	$0.0638681\pi$
$857$			11.6691i			0.398608i	−0.979938	+	0.199304i	$0.936132\pi$
$858$		0			0
$859$	3.92148			0.133799			0.0668995	−	0.997760i	$-0.478689\pi$
$859$	3.92148			0.133799			0.0668995	+	0.997760i	$0.478689\pi$
$860$		0			0
$861$	−12.0000			−0.408959
$862$		0			0
$863$	5.83188			0.198519			0.0992597	−	0.995062i	$-0.468353\pi$
$863$	5.83188			0.198519			0.0992597	+	0.995062i	$0.468353\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		−	5.77791i		−	0.196228i
$868$		0			0
$869$		−	20.2783i		−	0.687895i
$870$		0			0
$871$	−4.60350	−	24.7190i	−0.155984	−	0.837572i
$872$		0			0
$873$	−39.7164			−1.34420
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−57.5379			−1.94292			−0.971459	−	0.237208i	$-0.923768\pi$
$877$	−57.5379			−1.94292			−0.971459	+	0.237208i	$0.923768\pi$
$878$		0			0
$879$			4.65384i			0.156970i
$880$		0			0
$881$	−55.4483			−1.86810			−0.934051	−	0.357140i	$-0.883752\pi$
$881$	−55.4483			−1.86810			−0.934051	+	0.357140i	$0.883752\pi$
$882$		0			0
$883$			1.75199i			0.0589591i	0.999565	+	0.0294796i	$0.00938500\pi$
$883$			1.75199i			0.0589591i	−0.999565	+	0.0294796i	$0.990615\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		−	36.3265i		−	1.21973i	−0.792507	−	0.609863i	$-0.791225\pi$
$887$		−	36.3265i		−	1.21973i	0.792507	−	0.609863i	$-0.208775\pi$
$888$		0			0
$889$		−	0.422804i		−	0.0141804i
$890$		0			0
$891$		−	12.3162i		−	0.412609i
$892$		0			0
$893$			18.6931i			0.625541i
$894$		0			0
$895$		0			0
$896$		0			0
$897$	1.21807	+	6.54059i	0.0406703	+	0.218384i
$898$		0			0
$899$		−	32.3961i		−	1.08047i
$900$		0			0
$901$		0			0
$902$		0			0
$903$	−1.29433			−0.0430726
$904$		0			0
$905$		0			0
$906$		0			0
$907$			10.0696i			0.334355i	0.985927	+	0.167178i	$0.0534653\pi$
$907$			10.0696i			0.334355i	−0.985927	+	0.167178i	$0.946535\pi$
$908$		0			0
$909$	−18.6012			−0.616964
$910$		0			0
$911$	−39.8341			−1.31976			−0.659882	−	0.751369i	$-0.729394\pi$
$911$	−39.8341			−1.31976			−0.659882	+	0.751369i	$0.729394\pi$
$912$		0			0
$913$		−	13.2284i		−	0.437795i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	40.4354			1.33529
$918$		0			0
$919$	−11.8608			−0.391253			−0.195626	−	0.980678i	$-0.562674\pi$
$919$	−11.8608			−0.391253			−0.195626	+	0.980678i	$0.562674\pi$
$920$		0			0
$921$		−	4.34882i		−	0.143298i
$922$		0			0
$923$	−44.9211	+	8.36579i	−1.47860	+	0.275363i
$924$		0			0
$925$		0			0
$926$		0			0
$927$			34.9278i			1.14718i
$928$		0			0
$929$			7.34616i			0.241020i	0.992712	+	0.120510i	$0.0384530\pi$
$929$			7.34616i			0.241020i	−0.992712	+	0.120510i	$0.961547\pi$
$930$		0			0
$931$			23.1744i			0.759511i
$932$		0			0
$933$			7.76897i			0.254345i
$934$		0			0
$935$		0			0
$936$		0			0
$937$		−	45.3069i		−	1.48011i	−0.672545	−	0.740056i	$-0.734799\pi$
$937$		−	45.3069i		−	1.48011i	0.672545	−	0.740056i	$-0.265201\pi$
$938$		0			0
$939$	−9.25430			−0.302003
$940$		0			0
$941$		−	5.88222i		−	0.191755i	−0.995393	−	0.0958774i	$-0.969434\pi$
$941$		−	5.88222i		−	0.191755i	0.995393	−	0.0958774i	$-0.0305657\pi$
$942$		0			0
$943$	−49.3462			−1.60693
$944$		0			0
$945$		0			0
$946$		0			0
$947$	14.7427			0.479072			0.239536	−	0.970887i	$-0.423005\pi$
$947$	14.7427			0.479072			0.239536	+	0.970887i	$0.423005\pi$
$948$		0			0
$949$	−2.26803	−	12.1784i	−0.0736232	−	0.395328i
$950$		0			0
$951$			9.44607i			0.306310i
$952$		0			0
$953$		−	37.2284i		−	1.20595i	−0.797762	−	0.602973i	$-0.793983\pi$
$953$		−	37.2284i		−	1.20595i	0.797762	−	0.602973i	$-0.206017\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$	−2.73236			−0.0883246
$958$		0			0
$959$	35.3069			1.14012
$960$		0			0
$961$	−7.74266			−0.249763
$962$		0			0
$963$		−	50.2739i		−	1.62005i
$964$		0			0
$965$		0			0
$966$		0			0
$967$	31.8711			1.02491			0.512453	−	0.858715i	$-0.328737\pi$
$967$	31.8711			1.02491			0.512453	+	0.858715i	$0.328737\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	36.6531			1.17625			0.588126	−	0.808769i	$-0.299866\pi$
$971$	36.6531			1.17625			0.588126	+	0.808769i	$0.299866\pi$
$972$		0			0
$973$	35.1544			1.12700
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−9.61419			−0.307585			−0.153793	−	0.988103i	$-0.549149\pi$
$977$	−9.61419			−0.307585			−0.153793	+	0.988103i	$0.549149\pi$
$978$		0			0
$979$	−26.4568			−0.845562
$980$		0			0
$981$			47.3328i			1.51122i
$982$		0			0
$983$	−4.78193			−0.152520			−0.0762599	−	0.997088i	$-0.524298\pi$
$983$	−4.78193			−0.152520			−0.0762599	+	0.997088i	$0.524298\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$			8.61458i			0.274205i
$988$		0			0
$989$	−5.32251			−0.169246
$990$		0			0
$991$	51.8814			1.64807			0.824034	−	0.566540i	$-0.191718\pi$
$991$	51.8814			1.64807			0.824034	+	0.566540i	$0.191718\pi$
$992$		0			0
$993$	−6.70340			−0.212726
$994$		0			0
$995$		0			0
$996$		0			0
$997$		−	12.3176i		−	0.390102i	−0.980793	−	0.195051i	$-0.937513\pi$
$997$		−	12.3176i		−	0.390102i	0.980793	−	0.195051i	$-0.0624873\pi$
$998$		0			0
$999$		−	17.1285i		−	0.541921i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1300.2.d.d.649.3		6
5.2	odd	4		260.2.f.a.181.3	✓	6
5.3	odd	4		1300.2.f.e.701.4		6
5.4	even	2		1300.2.d.c.649.4		6
13.12	even	2		1300.2.d.c.649.3		6
15.2	even	4		2340.2.c.d.181.4		6
20.7	even	4		1040.2.k.c.961.3		6
65.12	odd	4		260.2.f.a.181.4	yes	6
65.38	odd	4		1300.2.f.e.701.3		6
65.47	even	4		3380.2.a.n.1.2		3
65.57	even	4		3380.2.a.m.1.2		3
65.64	even	2	inner	1300.2.d.d.649.4		6
195.77	even	4		2340.2.c.d.181.3		6
260.207	even	4		1040.2.k.c.961.4		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
260.2.f.a.181.3	✓	6	5.2	odd	4
260.2.f.a.181.4	yes	6	65.12	odd	4
1040.2.k.c.961.3		6	20.7	even	4
1040.2.k.c.961.4		6	260.207	even	4
1300.2.d.c.649.3		6	13.12	even	2
1300.2.d.c.649.4		6	5.4	even	2
1300.2.d.d.649.3		6	1.1	even	1	trivial
1300.2.d.d.649.4		6	65.64	even	2	inner
1300.2.f.e.701.3		6	65.38	odd	4
1300.2.f.e.701.4		6	5.3	odd	4
2340.2.c.d.181.3		6	195.77	even	4
2340.2.c.d.181.4		6	15.2	even	4
3380.2.a.m.1.2		3	65.57	even	4
3380.2.a.n.1.2		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)

\(f(q)\)	\(=\)	\(q-0.339877i q^{3} -3.88448 q^{7} +2.88448 q^{9} +O(q^{10})\)	\(q-0.339877i q^{3} -3.88448 q^{7} +2.88448 q^{9} -1.54461i q^{11} +(0.660123 + 3.54461i) q^{13} +2.86485i q^{19} +1.32025i q^{21} +5.42909i q^{23} -2.00000i q^{27} +5.20473 q^{29} -6.22436i q^{31} -0.524976 q^{33} +8.56424 q^{37} +(1.20473 - 0.224361i) q^{39} +9.08921i q^{41} +0.980369i q^{43} +6.52498 q^{47} +8.08921 q^{49} +6.44872i q^{53} +0.973697 q^{57} +4.45539i q^{59} +9.65345 q^{61} -11.2047 q^{63} -6.97370 q^{67} +1.84522 q^{69} +12.6731i q^{71} -3.43576 q^{73} +6.00000i q^{77} +13.1285 q^{79} +7.97370 q^{81} +8.56424 q^{83} -1.76897i q^{87} -17.1285i q^{89} +(-2.56424 - 13.7690i) q^{91} -2.11552 q^{93} -13.7690 q^{97} -4.45539i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 6 q^{9}+O(q^{10}) \) 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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