LMFDB - Embedded newform 1300.2.f.e.701.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("1300.701");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

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

Embedding invariants

Embedding label			701.3
Root			$-0.339877i$ of defining polynomial
Character	$\chi$	$=$	1300.701
Dual form			1300.2.f.e.701.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.339877 q^{3} -3.88448i q^{7} -2.88448 q^{9} +O(q^{10})$	$q+0.339877 q^{3} -3.88448i q^{7} -2.88448 q^{9} +1.54461i q^{11} +(-3.54461 - 0.660123i) q^{13} +2.86485i q^{19} -1.32025i q^{21} -5.42909 q^{23} -2.00000 q^{27} -5.20473 q^{29} +6.22436i q^{31} +0.524976i q^{33} +8.56424i q^{37} +(-1.20473 - 0.224361i) q^{39} -9.08921i q^{41} -0.980369 q^{43} +6.52498i q^{47} -8.08921 q^{49} -6.44872 q^{53} +0.973697i q^{57} +4.45539i q^{59} +9.65345 q^{61} +11.2047i q^{63} -6.97370i q^{67} -1.84522 q^{69} -12.6731i q^{71} +3.43576i q^{73} +6.00000 q^{77} -13.1285 q^{79} +7.97370 q^{81} -8.56424i q^{83} -1.76897 q^{87} -17.1285i q^{89} +(-2.56424 + 13.7690i) q^{91} +2.11552i q^{93} -13.7690i q^{97} -4.45539i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 6 q^{9}+O(q^{10}) $ 6 * q + 6 * q^9	$ 6 q + 6 q^{9} + 12 q^{23} - 12 q^{27} - 12 q^{29} + 12 q^{39} - 12 q^{43} - 6 q^{49} + 12 q^{53} - 12 q^{61} + 36 q^{77} - 24 q^{79} - 18 q^{81} + 36 q^{87} + 12 q^{91}+O(q^{100}) $ 6 * q + 6 * q^9 + 12 * q^23 - 12 * q^27 - 12 * q^29 + 12 * q^39 - 12 * q^43 - 6 * q^49 + 12 * q^53 - 12 * q^61 + 36 * q^77 - 24 * q^79 - 18 * q^81 + 36 * q^87 + 12 * q^91

Display 100 coefficients 10 coefficients

Character values

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

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

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$	0.339877			0.196228			0.0981140	−	0.995175i	$-0.468719\pi$
$3$	0.339877			0.196228			0.0981140	+	0.995175i	$0.468719\pi$
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$		−	3.88448i		−	1.46820i	−0.679043	−	0.734098i	$-0.737605\pi$
$7$		−	3.88448i		−	1.46820i	0.679043	−	0.734098i	$-0.262395\pi$
$8$		0			0
$9$	−2.88448			−0.961495
$10$		0			0
$11$			1.54461i			0.465716i	0.972511	+	0.232858i	$0.0748078\pi$
$11$			1.54461i			0.465716i	−0.972511	+	0.232858i	$0.925192\pi$
$12$		0			0
$13$	−3.54461	−	0.660123i	−0.983097	−	0.183085i
$13$	−3.54461	−	0.660123i	−0.983097	−	0.183085i
$14$		0			0
$15$		0			0
$16$		0			0
$17$		0			0			−	1.00000i	$-0.5\pi$
$17$		0			0				1.00000i	$0.5\pi$
$18$		0			0
$19$			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
$23$	−5.42909			−1.13204			−0.566022	−	0.824390i	$-0.691518\pi$
$23$	−5.42909			−1.13204			−0.566022	+	0.824390i	$0.691518\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$	−2.00000			−0.384900
$28$		0			0
$29$	−5.20473			−0.966494			−0.483247	−	0.875484i	$-0.660543\pi$
$29$	−5.20473			−0.966494			−0.483247	+	0.875484i	$0.660543\pi$
$30$		0			0
$31$			6.22436i			1.11793i	0.829192	+	0.558964i	$0.188801\pi$
$31$			6.22436i			1.11793i	−0.829192	+	0.558964i	$0.811199\pi$
$32$		0			0
$33$			0.524976i			0.0913866i
$34$		0			0
$35$		0			0
$36$		0			0
$37$			8.56424i			1.40795i	0.710224	+	0.703976i	$0.248594\pi$
$37$			8.56424i			1.40795i	−0.710224	+	0.703976i	$0.751406\pi$
$38$		0			0
$39$	−1.20473	−	0.224361i	−0.192911	−	0.0359264i
$40$		0			0
$41$		−	9.08921i		−	1.41950i	−0.704455	−	0.709748i	$-0.748809\pi$
$41$		−	9.08921i		−	1.41950i	0.704455	−	0.709748i	$-0.251191\pi$
$42$		0			0
$43$	−0.980369			−0.149505			−0.0747525	−	0.997202i	$-0.523817\pi$
$43$	−0.980369			−0.149505			−0.0747525	+	0.997202i	$0.523817\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$			6.52498i			0.951766i	0.879509	+	0.475883i	$0.157871\pi$
$47$			6.52498i			0.951766i	−0.879509	+	0.475883i	$0.842129\pi$
$48$		0			0
$49$	−8.08921			−1.15560
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−6.44872			−0.885800			−0.442900	−	0.896571i	$-0.646050\pi$
$53$	−6.44872			−0.885800			−0.442900	+	0.896571i	$0.646050\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$			0.973697i			0.128969i
$58$		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
$63$			11.2047i			1.41166i
$64$		0			0
$65$		0			0
$66$		0			0
$67$		−	6.97370i		−	0.851973i	−0.904730	−	0.425986i	$-0.859927\pi$
$67$		−	6.97370i		−	0.851973i	0.904730	−	0.425986i	$-0.140073\pi$
$68$		0			0
$69$	−1.84522			−0.222139
$70$		0			0
$71$		−	12.6731i		−	1.50402i	−0.659153	−	0.752009i	$-0.729085\pi$
$71$		−	12.6731i		−	1.50402i	0.659153	−	0.752009i	$-0.270915\pi$
$72$		0			0
$73$			3.43576i			0.402126i	0.979578	+	0.201063i	$0.0644395\pi$
$73$			3.43576i			0.402126i	−0.979578	+	0.201063i	$0.935560\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	6.00000			0.683763
$78$		0			0
$79$	−13.1285			−1.47707			−0.738534	−	0.674216i	$-0.764482\pi$
$79$	−13.1285			−1.47707			−0.738534	+	0.674216i	$0.764482\pi$
$80$		0			0
$81$	7.97370			0.885966
$82$		0			0
$83$		−	8.56424i		−	0.940047i	−0.882654	−	0.470024i	$-0.844245\pi$
$83$		−	8.56424i		−	0.940047i	0.882654	−	0.470024i	$-0.155755\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	−1.76897			−0.189653
$88$		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
$93$			2.11552i			0.219369i
$94$		0			0
$95$		0			0
$96$		0			0
$97$		−	13.7690i		−	1.39803i	−0.715109	−	0.699013i	$-0.753623\pi$
$97$		−	13.7690i		−	1.39803i	0.715109	−	0.699013i	$-0.246377\pi$
$98$		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
$103$	−12.1088			−1.19312			−0.596560	−	0.802569i	$-0.703466\pi$
$103$	−12.1088			−1.19312			−0.596560	+	0.802569i	$0.703466\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−17.4291			−1.68493			−0.842467	−	0.538748i	$-0.818897\pi$
$107$	−17.4291			−1.68493			−0.842467	+	0.538748i	$0.818897\pi$
$108$		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
$113$	−1.59054			−0.149625			−0.0748127	−	0.997198i	$-0.523836\pi$
$113$	−1.59054			−0.149625			−0.0748127	+	0.997198i	$0.523836\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	10.2244	+	1.90411i	0.945242	+	0.176035i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	8.61419			0.783108
$122$		0			0
$123$		−	3.08921i		−	0.278545i
$124$		0			0
$125$		0			0
$126$		0			0
$127$	0.108844			0.00965837			0.00482918	−	0.999988i	$-0.498463\pi$
$127$	0.108844			0.00965837			0.00482918	+	0.999988i	$0.498463\pi$
$128$		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
$133$	11.1285			0.964961
$134$		0			0
$135$		0			0
$136$		0			0
$137$		−	9.08921i		−	0.776544i	−0.921545	−	0.388272i	$-0.873072\pi$
$137$		−	9.08921i		−	0.776544i	0.921545	−	0.388272i	$-0.126928\pi$
$138$		0			0
$139$	9.04995			0.767607			0.383803	−	0.923415i	$-0.374614\pi$
$139$	9.04995			0.767607			0.383803	+	0.923415i	$0.374614\pi$
$140$		0			0
$141$			2.21769i			0.186763i
$142$		0			0
$143$	1.01963	−	5.47502i	0.0852658	−	0.457844i
$144$		0			0
$145$		0			0
$146$		0			0
$147$	−2.74934			−0.226761
$148$		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	$-0.878774\pi$
$151$		−	9.13515i		−	0.743408i	0.928351	−	0.371704i	$-0.121226\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−4.91079			−0.391923			−0.195962	−	0.980612i	$-0.562783\pi$
$157$	−4.91079			−0.391923			−0.195962	+	0.980612i	$0.562783\pi$
$158$		0			0
$159$	−2.19177			−0.173819
$160$		0			0
$161$			21.0892i			1.66206i
$162$		0			0
$163$		−	16.9344i		−	1.32641i	−0.748439	−	0.663204i	$-0.769196\pi$
$163$		−	16.9344i		−	1.32641i	0.748439	−	0.663204i	$-0.230804\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			21.6142i			1.67256i	0.548306	+	0.836278i	$0.315273\pi$
$167$			21.6142i			1.67256i	−0.548306	+	0.836278i	$0.684727\pi$
$168$		0			0
$169$	12.1285	+	4.67975i	0.932960	+	0.359981i
$170$		0			0
$171$		−	8.26362i		−	0.631935i
$172$		0			0
$173$	8.03926			0.611214			0.305607	−	0.952158i	$-0.401141\pi$
$173$	8.03926			0.611214			0.305607	+	0.952158i	$0.401141\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$			1.51429i			0.113821i
$178$		0			0
$179$	10.8582			0.811579			0.405789	−	0.913967i	$-0.366997\pi$
$179$	10.8582			0.811579			0.405789	+	0.913967i	$0.366997\pi$
$180$		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
$183$	3.28098			0.242537
$184$		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
$193$			0.270294i			0.0194562i	0.999953	+	0.00972809i	$0.00309660\pi$
$193$			0.270294i			0.0194562i	−0.999953	+	0.00972809i	$0.996903\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			3.96074i			0.282191i	0.989996	+	0.141095i	$0.0450625\pi$
$197$			3.96074i			0.282191i	−0.989996	+	0.141095i	$0.954938\pi$
$198$		0			0
$199$	6.03926			0.428112			0.214056	−	0.976821i	$-0.431333\pi$
$199$	6.03926			0.428112			0.214056	+	0.976821i	$0.431333\pi$
$200$		0			0
$201$		−	2.37020i		−	0.167181i
$202$		0			0
$203$			20.2177i			1.41900i
$204$		0			0
$205$		0			0
$206$		0			0
$207$	15.6601			1.08845
$208$		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
$213$		−	4.30729i		−	0.295130i
$214$		0			0
$215$		0			0
$216$		0			0
$217$	24.1784			1.64134
$218$		0			0
$219$			1.16774i			0.0789083i
$220$		0			0
$221$		0			0
$222$		0			0
$223$			10.0629i			0.673862i	0.941529	+	0.336931i	$0.109389\pi$
$223$			10.0629i			0.673862i	−0.941529	+	0.336931i	$0.890611\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$			3.43576i			0.228040i	0.993478	+	0.114020i	$0.0363727\pi$
$227$			3.43576i			0.228040i	−0.993478	+	0.114020i	$0.963627\pi$
$228$		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
$233$	4.40946			0.288873			0.144437	−	0.989514i	$-0.453863\pi$
$233$	4.40946			0.288873			0.144437	+	0.989514i	$0.453863\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	−4.46207			−0.289842
$238$		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	$-0.888228\pi$
$241$		−	10.6798i		−	0.687943i	0.938980	−	0.343972i	$-0.111772\pi$
$242$		0			0
$243$	8.71008			0.558752
$244$		0			0
$245$		0			0
$246$		0			0
$247$	1.89116	−	10.1548i	0.120331	−	0.646133i
$248$		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
$253$		−	8.38581i		−	0.527211i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−18.8974			−1.17879			−0.589395	−	0.807845i	$-0.700634\pi$
$257$	−18.8974			−1.17879			−0.589395	+	0.807845i	$0.700634\pi$
$258$		0			0
$259$	33.2676			2.06715
$260$		0			0
$261$	15.0130			0.929279
$262$		0			0
$263$	13.0196			0.802825			0.401412	−	0.915897i	$-0.368519\pi$
$263$	13.0196			0.802825			0.401412	+	0.915897i	$0.368519\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		−	5.82157i		−	0.356274i
$268$		0			0
$269$	14.3702			0.876166			0.438083	−	0.898934i	$-0.355658\pi$
$269$	14.3702			0.876166			0.438083	+	0.898934i	$0.355658\pi$
$270$		0			0
$271$		−	13.7230i		−	0.833615i	−0.908995	−	0.416807i	$-0.863149\pi$
$271$		−	13.7230i		−	0.833615i	0.908995	−	0.416807i	$-0.136851\pi$
$272$		0			0
$273$	−0.871525	+	4.67975i	−0.0527471	+	0.283232i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	19.2676			1.15768			0.578840	−	0.815441i	$-0.303506\pi$
$277$	19.2676			1.15768			0.578840	+	0.815441i	$0.303506\pi$
$278$		0			0
$279$		−	17.9541i		−	1.07488i
$280$		0			0
$281$			3.96074i			0.236278i	0.992997	+	0.118139i	$0.0376928\pi$
$281$			3.96074i			0.236278i	−0.992997	+	0.118139i	$0.962307\pi$
$282$		0			0
$283$	21.1981			1.26009			0.630047	−	0.776557i	$-0.283036\pi$
$283$	21.1981			1.26009			0.630047	+	0.776557i	$0.283036\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−35.3069			−2.08410
$288$		0			0
$289$	−17.0000			−1.00000
$290$		0			0
$291$		−	4.67975i		−	0.274332i
$292$		0			0
$293$			13.6927i			0.799937i	0.916529	+	0.399968i	$0.130979\pi$
$293$			13.6927i			0.799937i	−0.916529	+	0.399968i	$0.869021\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		−	3.08921i		−	0.179254i
$298$		0			0
$299$	19.2440	+	3.58387i	1.11291	+	0.207260i
$300$		0			0
$301$			3.80823i			0.219503i
$302$		0			0
$303$	−2.19177			−0.125914
$304$		0			0
$305$		0			0
$306$		0			0
$307$			12.7953i			0.730265i	0.930956	+	0.365132i	$0.118976\pi$
$307$			12.7953i			0.730265i	−0.930956	+	0.365132i	$0.881024\pi$
$308$		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
$313$	27.2284			1.53904			0.769520	−	0.638623i	$-0.220496\pi$
$313$	27.2284			1.53904			0.769520	+	0.638623i	$0.220496\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	27.7926i		−	1.56099i	−0.625163	−	0.780494i	$-0.714967\pi$
$317$		−	27.7926i		−	1.56099i	0.625163	−	0.780494i	$-0.285033\pi$
$318$		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
$327$			5.57720i			0.308420i
$328$		0			0
$329$	25.3462			1.39738
$330$		0			0
$331$			19.7230i			1.08408i	0.840354	+	0.542038i	$0.182347\pi$
$331$			19.7230i			1.08408i	−0.840354	+	0.542038i	$0.817653\pi$
$332$		0			0
$333$		−	24.7034i		−	1.35374i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	20.0000			1.08947			0.544735	−	0.838608i	$-0.316630\pi$
$337$	20.0000			1.08947			0.544735	+	0.838608i	$0.316630\pi$
$338$		0			0
$339$	−0.540588			−0.0293607
$340$		0			0
$341$	−9.61419			−0.520638
$342$		0			0
$343$			4.23103i			0.228454i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	2.61017			0.140121			0.0700607	−	0.997543i	$-0.477681\pi$
$347$	2.61017			0.140121			0.0700607	+	0.997543i	$0.477681\pi$
$348$		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
$353$		−	20.9211i		−	1.11352i	−0.830674	−	0.556759i	$-0.812045\pi$
$353$		−	20.9211i		−	1.11352i	0.830674	−	0.556759i	$-0.187955\pi$
$354$		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
$363$	2.92776			0.153668
$364$		0			0
$365$		0			0
$366$		0			0
$367$	−21.0196			−1.09722			−0.548608	−	0.836080i	$-0.684842\pi$
$367$	−21.0196			−1.09722			−0.548608	+	0.836080i	$0.684842\pi$
$368$		0			0
$369$			26.2177i			1.36484i
$370$		0			0
$371$			25.0500i			1.30053i
$372$		0			0
$373$	−29.3462			−1.51949			−0.759743	−	0.650223i	$-0.774675\pi$
$373$	−29.3462			−1.51949			−0.759743	+	0.650223i	$0.774675\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	18.4487	+	3.43576i	0.950157	+	0.176951i
$378$		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
$383$			20.5642i			1.05078i	0.850860	+	0.525392i	$0.176081\pi$
$383$			20.5642i			1.05078i	−0.850860	+	0.525392i	$0.823919\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	2.82786			0.143748
$388$		0			0
$389$	26.8189			1.35977			0.679887	−	0.733317i	$-0.262029\pi$
$389$	26.8189			1.35977			0.679887	+	0.733317i	$0.262029\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$	−3.53793			−0.178465
$394$		0			0
$395$		0			0
$396$		0			0
$397$			36.1022i			1.81192i	0.423367	+	0.905958i	$0.360848\pi$
$397$			36.1022i			1.81192i	−0.423367	+	0.905958i	$0.639152\pi$
$398$		0			0
$399$	3.78231			0.189352
$400$		0			0
$401$		−	4.07852i		−	0.203672i	−0.994801	−	0.101836i	$-0.967528\pi$
$401$		−	4.07852i		−	0.203672i	0.994801	−	0.101836i	$-0.0324716\pi$
$402$		0			0
$403$	4.10884	−	22.0629i	0.204676	−	1.09903i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−13.2284			−0.655706
$408$		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
$413$	17.3069			0.851617
$414$		0			0
$415$		0			0
$416$		0			0
$417$	3.07587			0.150626
$418$		0			0
$419$	−11.5513			−0.564317			−0.282158	−	0.959368i	$-0.591050\pi$
$419$	−11.5513			−0.564317			−0.282158	+	0.959368i	$0.591050\pi$
$420$		0			0
$421$		−	35.4853i		−	1.72945i	−0.502246	−	0.864725i	$-0.667493\pi$
$421$		−	35.4853i		−	1.72945i	0.502246	−	0.864725i	$-0.332507\pi$
$422$		0			0
$423$		−	18.8212i		−	0.915117i
$424$		0			0
$425$		0			0
$426$		0			0
$427$		−	37.4987i		−	1.81469i
$428$		0			0
$429$	0.346549	−	1.86083i	0.0167315	−	0.0898419i
$430$		0			0
$431$			33.7623i			1.62627i	0.582073	+	0.813136i	$0.302242\pi$
$431$			33.7623i			1.62627i	−0.582073	+	0.813136i	$0.697758\pi$
$432$		0			0
$433$	2.00000			0.0961139			0.0480569	−	0.998845i	$-0.484697\pi$
$433$	2.00000			0.0961139			0.0480569	+	0.998845i	$0.484697\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		−	15.5535i		−	0.744027i
$438$		0			0
$439$	−19.0892			−0.911078			−0.455539	−	0.890216i	$-0.650554\pi$
$439$	−19.0892			−0.911078			−0.455539	+	0.890216i	$0.650554\pi$
$440$		0			0
$441$	23.3332			1.11110
$442$		0			0
$443$	10.2873			0.488763			0.244382	−	0.969679i	$-0.421415\pi$
$443$	10.2873			0.488763			0.244382	+	0.969679i	$0.421415\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$			3.78231i			0.178897i
$448$		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
$453$		−	3.10483i		−	0.145877i
$454$		0			0
$455$		0			0
$456$		0			0
$457$			17.3069i			0.809583i	0.914409	+	0.404791i	$0.132656\pi$
$457$			17.3069i			0.809583i	−0.914409	+	0.404791i	$0.867344\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$			16.1392i			0.751676i	0.926685	+	0.375838i	$0.122645\pi$
$461$			16.1392i			0.751676i	−0.926685	+	0.375838i	$0.877355\pi$
$462$		0			0
$463$			6.52498i			0.303241i	0.988439	+	0.151621i	$0.0484492\pi$
$463$			6.52498i			0.303241i	−0.988439	+	0.151621i	$0.951551\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	35.4291			1.63946			0.819731	−	0.572748i	$-0.194123\pi$
$467$	35.4291			1.63946			0.819731	+	0.572748i	$0.194123\pi$
$468$		0			0
$469$	−27.0892			−1.25086
$470$		0			0
$471$	−1.66906			−0.0769064
$472$		0			0
$473$		−	1.51429i		−	0.0696269i
$474$		0			0
$475$		0			0
$476$		0			0
$477$	18.6012			0.851692
$478$		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
$483$			7.16774i			0.326143i
$484$		0			0
$485$		0			0
$486$		0			0
$487$			18.3725i			0.832536i	0.909242	+	0.416268i	$0.136662\pi$
$487$			18.3725i			0.832536i	−0.909242	+	0.416268i	$0.863338\pi$
$488$		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
$497$	−49.2284			−2.20819
$498$		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
$503$	−39.1455			−1.74541			−0.872705	−	0.488248i	$-0.837636\pi$
$503$	−39.1455			−1.74541			−0.872705	+	0.488248i	$0.837636\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	4.12219	+	1.59054i	0.183073	+	0.0706384i
$508$		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
$513$		−	5.72971i		−	0.252973i
$514$		0			0
$515$		0			0
$516$		0			0
$517$	−10.0785			−0.443253
$518$		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
$523$	−3.19806			−0.139841			−0.0699207	−	0.997553i	$-0.522275\pi$
$523$	−3.19806			−0.139841			−0.0699207	+	0.997553i	$0.522275\pi$
$524$		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
$533$	−6.00000	+	32.2177i	−0.259889	+	1.39550i
$534$		0			0
$535$		0			0
$536$		0			0
$537$	3.69044			0.159254
$538$		0			0
$539$		−	12.4947i		−	0.538183i
$540$		0			0
$541$		−	1.65118i		−	0.0709899i	−0.999370	−	0.0354950i	$-0.988699\pi$
$541$		−	1.65118i		−	0.0709899i	0.999370	−	0.0354950i	$-0.0113008\pi$
$542$		0			0
$543$	3.59054			0.154085
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−38.3265			−1.63872			−0.819362	−	0.573276i	$-0.805672\pi$
$547$	−38.3265			−1.63872			−0.819362	+	0.573276i	$0.805672\pi$
$548$		0			0
$549$	−27.8452			−1.18841
$550$		0			0
$551$		−	14.9108i		−	0.635221i
$552$		0			0
$553$			50.9973i			2.16863i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		−	31.8711i		−	1.35042i	−0.737624	−	0.675212i	$-0.764052\pi$
$557$		−	31.8711i		−	1.35042i	0.737624	−	0.675212i	$-0.235948\pi$
$558$		0			0
$559$	3.47502	+	0.647164i	0.146978	+	0.0273721i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−5.87781			−0.247720			−0.123860	−	0.992300i	$-0.539527\pi$
$563$	−5.87781			−0.247720			−0.123860	+	0.992300i	$0.539527\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		−	30.9737i		−	1.30077i
$568$		0			0
$569$	−16.0629			−0.673392			−0.336696	−	0.941613i	$-0.609310\pi$
$569$	−16.0629			−0.673392			−0.336696	+	0.941613i	$0.609310\pi$
$570$		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
$573$	−2.99735			−0.125216
$574$		0			0
$575$		0			0
$576$		0			0
$577$			17.3832i			0.723670i	0.932242	+	0.361835i	$0.117850\pi$
$577$			17.3832i			0.723670i	−0.932242	+	0.361835i	$0.882150\pi$
$578$		0			0
$579$			0.0918667i			0.00381785i
$580$		0			0
$581$	−33.2676			−1.38017
$582$		0			0
$583$		−	9.96074i		−	0.412532i
$584$		0			0
$585$		0			0
$586$		0			0
$587$			7.57493i			0.312651i	0.987706	+	0.156325i	$0.0499648\pi$
$587$			7.57493i			0.312651i	−0.987706	+	0.156325i	$0.950035\pi$
$588$		0			0
$589$	−17.8319			−0.734750
$590$		0			0
$591$			1.34616i			0.0553738i
$592$		0			0
$593$		−	16.9500i		−	0.696055i	−0.937484	−	0.348028i	$-0.886852\pi$
$593$		−	16.9500i		−	0.696055i	0.937484	−	0.348028i	$-0.113148\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	2.05261			0.0840075
$598$		0			0
$599$	−43.6771			−1.78460			−0.892299	−	0.451445i	$-0.850909\pi$
$599$	−43.6771			−1.78460			−0.892299	+	0.451445i	$0.850909\pi$
$600$		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
$603$			20.1155i			0.819167i
$604$		0			0
$605$		0			0
$606$		0			0
$607$	34.3265			1.39327			0.696635	−	0.717425i	$-0.254680\pi$
$607$	34.3265			1.39327			0.696635	+	0.717425i	$0.254680\pi$
$608$		0			0
$609$			6.87153i			0.278448i
$610$		0			0
$611$	4.30729	−	23.1285i	0.174254	−	0.935678i
$612$		0			0
$613$			18.3569i			0.741426i	0.928747	+	0.370713i	$0.120887\pi$
$613$			18.3569i			0.741426i	−0.928747	+	0.370713i	$0.879113\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			37.5246i			1.51068i	0.655331	+	0.755342i	$0.272529\pi$
$617$			37.5246i			1.51068i	−0.655331	+	0.755342i	$0.727471\pi$
$618$		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
$623$	−66.5353			−2.66568
$624$		0			0
$625$		0			0
$626$		0			0
$627$	−1.50398			−0.0600632
$628$		0			0
$629$		0			0
$630$		0			0
$631$		−	12.6472i		−	0.503476i	−0.967795	−	0.251738i	$-0.918998\pi$
$631$		−	12.6472i		−	0.503476i	0.967795	−	0.251738i	$-0.0810021\pi$
$632$		0			0
$633$	−5.51202			−0.219083
$634$		0			0
$635$		0			0
$636$		0			0
$637$	28.6731	+	5.33988i	1.13607	+	0.211574i
$638$		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
$643$		−	0.346549i		−	0.0136666i	−0.999977	−	0.00683328i	$-0.997825\pi$
$643$		−	0.346549i		−	0.0136666i	0.999977	−	0.00683328i	$-0.00217512\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−44.2480			−1.73957			−0.869784	−	0.493432i	$-0.835742\pi$
$647$	−44.2480			−1.73957			−0.869784	+	0.493432i	$0.835742\pi$
$648$		0			0
$649$	−6.88183			−0.270135
$650$		0			0
$651$	8.21769			0.322077
$652$		0			0
$653$	14.0393			0.549399			0.274699	−	0.961530i	$-0.411422\pi$
$653$	14.0393			0.549399			0.274699	+	0.961530i	$0.411422\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		−	9.91040i		−	0.386642i
$658$		0			0
$659$	12.4487			0.484933			0.242467	−	0.970160i	$-0.422043\pi$
$659$	12.4487			0.484933			0.242467	+	0.970160i	$0.422043\pi$
$660$		0			0
$661$			11.8216i			0.459806i	0.973214	+	0.229903i	$0.0738409\pi$
$661$			11.8216i			0.459806i	−0.973214	+	0.229903i	$0.926159\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	28.2569			1.09411
$668$		0			0
$669$			3.42015i			0.132231i
$670$		0			0
$671$			14.9108i			0.575625i
$672$		0			0
$673$	−31.3069			−1.20679			−0.603396	−	0.797442i	$-0.706186\pi$
$673$	−31.3069			−1.20679			−0.603396	+	0.797442i	$0.706186\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−28.8582			−1.10911			−0.554555	−	0.832147i	$-0.687111\pi$
$677$	−28.8582			−1.10911			−0.554555	+	0.832147i	$0.687111\pi$
$678$		0			0
$679$	−53.4853			−2.05258
$680$		0			0
$681$			1.16774i			0.0447478i
$682$		0			0
$683$		−	20.5642i		−	0.786869i	−0.919353	−	0.393434i	$-0.871287\pi$
$683$		−	20.5642i		−	0.786869i	0.919353	−	0.393434i	$-0.128713\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		−	7.76897i		−	0.296405i
$688$		0			0
$689$	22.8582	+	4.25695i	0.870827	+	0.162177i
$690$		0			0
$691$			6.67308i			0.253856i	0.991912	+	0.126928i	$0.0405117\pi$
$691$			6.67308i			0.253856i	−0.991912	+	0.126928i	$0.959488\pi$
$692$		0			0
$693$	−17.3069			−0.657435
$694$		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
$703$	−24.5353			−0.925366
$704$		0			0
$705$		0			0
$706$		0			0
$707$			25.0500i			0.942100i
$708$		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
$713$		−	33.7926i		−	1.26554i
$714$		0			0
$715$		0			0
$716$		0			0
$717$		−	7.75336i		−	0.289554i
$718$		0			0
$719$	−36.2043			−1.35019			−0.675097	−	0.737729i	$-0.735898\pi$
$719$	−36.2043			−1.35019			−0.675097	+	0.737729i	$0.735898\pi$
$720$		0			0
$721$			47.0366i			1.75173i
$722$		0			0
$723$		−	3.62980i		−	0.134994i
$724$		0			0
$725$		0			0
$726$		0			0
$727$	46.2480			1.71524			0.857622	−	0.514281i	$-0.171941\pi$
$727$	46.2480			1.71524			0.857622	+	0.514281i	$0.171941\pi$
$728$		0			0
$729$	−20.9607			−0.776324
$730$		0			0
$731$		0			0
$732$		0			0
$733$			20.0393i			0.740167i	0.928998	+	0.370084i	$0.120671\pi$
$733$			20.0393i			0.740167i	−0.928998	+	0.370084i	$0.879329\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	10.7716			0.396778
$738$		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
$743$		−	47.9497i		−	1.75910i	−0.475804	−	0.879551i	$-0.657843\pi$
$743$		−	47.9497i		−	1.75910i	0.475804	−	0.879551i	$-0.342157\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$			24.7034i			0.903850i
$748$		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
$753$	−7.22838			−0.263417
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−44.3176			−1.61075			−0.805375	−	0.592765i	$-0.798036\pi$
$757$	−44.3176			−1.61075			−0.805375	+	0.592765i	$0.798036\pi$
$758$		0			0
$759$		−	2.85014i		−	0.103454i
$760$		0			0
$761$		−	41.4853i		−	1.50384i	−0.659253	−	0.751921i	$-0.729127\pi$
$761$		−	41.4853i		−	1.50384i	0.659253	−	0.751921i	$-0.270873\pi$
$762$		0			0
$763$	63.7423			2.30763
$764$		0			0
$765$		0			0
$766$		0			0
$767$	2.94111	−	15.7926i	0.106197	−	0.570238i
$768$		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
$773$			15.4358i			0.555186i	0.960699	+	0.277593i	$0.0895366\pi$
$773$			15.4358i			0.555186i	−0.960699	+	0.277593i	$0.910463\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	11.3069			0.405633
$778$		0			0
$779$	26.0393			0.932953
$780$		0			0
$781$	19.5749			0.700446
$782$		0			0
$783$	10.4095			0.372004
$784$		0			0
$785$		0			0
$786$		0			0
$787$		−	5.38316i		−	0.191889i	−0.995387	−	0.0959444i	$-0.969413\pi$
$787$		−	5.38316i		−	0.191889i	0.995387	−	0.0959444i	$-0.0305871\pi$
$788$		0			0
$789$	4.42507			0.157537
$790$		0			0
$791$			6.17843i			0.219680i
$792$		0			0
$793$	−34.2177	−	6.37247i	−1.21511	−	0.226293i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−35.7556			−1.26653			−0.633265	−	0.773935i	$-0.718286\pi$
$797$	−35.7556			−1.26653			−0.633265	+	0.773935i	$0.718286\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$			49.4068i			1.74570i
$802$		0			0
$803$	−5.30690			−0.187277
$804$		0			0
$805$		0			0
$806$		0			0
$807$	4.88410			0.171928
$808$		0			0
$809$	−35.7712			−1.25765			−0.628825	−	0.777547i	$-0.716464\pi$
$809$	−35.7712			−1.25765			−0.628825	+	0.777547i	$0.716464\pi$
$810$		0			0
$811$		−	30.2244i		−	1.06132i	−0.847585	−	0.530660i	$-0.821944\pi$
$811$		−	30.2244i		−	1.06132i	0.847585	−	0.530660i	$-0.178056\pi$
$812$		0			0
$813$		−	4.66414i		−	0.163579i
$814$		0			0
$815$		0			0
$816$		0			0
$817$		−	2.80861i		−	0.0982610i
$818$		0			0
$819$	7.39650	−	39.7164i	0.258455	−	1.38780i
$820$		0			0
$821$			25.2284i			0.880477i	0.897881	+	0.440238i	$0.145106\pi$
$821$			25.2284i			0.880477i	−0.897881	+	0.440238i	$0.854894\pi$
$822$		0			0
$823$	−30.9804			−1.07991			−0.539954	−	0.841695i	$-0.681558\pi$
$823$	−30.9804			−1.07991			−0.539954	+	0.841695i	$0.681558\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			6.82119i			0.237196i	0.992942	+	0.118598i	$0.0378400\pi$
$827$			6.82119i			0.237196i	−0.992942	+	0.118598i	$0.962160\pi$
$828$		0			0
$829$	−3.39650			−0.117965			−0.0589827	−	0.998259i	$-0.518786\pi$
$829$	−3.39650			−0.117965			−0.0589827	+	0.998259i	$0.518786\pi$
$830$		0			0
$831$	6.54863			0.227169
$832$		0			0
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$		−	12.4487i		−	0.430291i
$838$		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
$843$			1.34616i			0.0463643i
$844$		0			0
$845$		0			0
$846$		0			0
$847$		−	33.4617i		−	1.14976i
$848$		0			0
$849$	7.20473			0.247266
$850$		0			0
$851$		−	46.4960i		−	1.59386i
$852$		0			0
$853$		−	18.2806i		−	0.625916i	−0.949767	−	0.312958i	$-0.898680\pi$
$853$		−	18.2806i		−	0.625916i	0.949767	−	0.312958i	$-0.101320\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	11.6691			0.398608			0.199304	−	0.979938i	$-0.436132\pi$
$857$	11.6691			0.398608			0.199304	+	0.979938i	$0.436132\pi$
$858$		0			0
$859$	−3.92148			−0.133799			−0.0668995	−	0.997760i	$-0.521311\pi$
$859$	−3.92148			−0.133799			−0.0668995	+	0.997760i	$0.521311\pi$
$860$		0			0
$861$	−12.0000			−0.408959
$862$		0			0
$863$		−	5.83188i		−	0.198519i	−0.995062	−	0.0992597i	$-0.968353\pi$
$863$		−	5.83188i		−	0.198519i	0.995062	−	0.0992597i	$-0.0316475\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	−5.77791			−0.196228
$868$		0			0
$869$		−	20.2783i		−	0.687895i
$870$		0			0
$871$	−4.60350	+	24.7190i	−0.155984	+	0.837572i
$872$		0			0
$873$			39.7164i			1.34420i
$874$		0			0
$875$		0			0
$876$		0			0
$877$		−	57.5379i		−	1.94292i	−0.237208	−	0.971459i	$-0.576232\pi$
$877$		−	57.5379i		−	1.94292i	0.237208	−	0.971459i	$-0.423768\pi$
$878$		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
$883$	−1.75199			−0.0589591			−0.0294796	−	0.999565i	$-0.509385\pi$
$883$	−1.75199			−0.0589591			−0.0294796	+	0.999565i	$0.509385\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−36.3265			−1.21973			−0.609863	−	0.792507i	$-0.708775\pi$
$887$	−36.3265			−1.21973			−0.609863	+	0.792507i	$0.708775\pi$
$888$		0			0
$889$		−	0.422804i		−	0.0141804i
$890$		0			0
$891$			12.3162i			0.412609i
$892$		0			0
$893$	−18.6931			−0.625541
$894$		0			0
$895$		0			0
$896$		0			0
$897$	6.54059	+	1.21807i	0.218384	+	0.0406703i
$898$		0			0
$899$		−	32.3961i		−	1.08047i
$900$		0			0
$901$		0			0
$902$		0			0
$903$			1.29433i			0.0430726i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	10.0696			0.334355			0.167178	−	0.985927i	$-0.446535\pi$
$907$	10.0696			0.334355			0.167178	+	0.985927i	$0.446535\pi$
$908$		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
$913$	13.2284			0.437795
$914$		0			0
$915$		0			0
$916$		0			0
$917$			40.4354i			1.33529i
$918$		0			0
$919$	11.8608			0.391253			0.195626	−	0.980678i	$-0.437326\pi$
$919$	11.8608			0.391253			0.195626	+	0.980678i	$0.437326\pi$
$920$		0			0
$921$			4.34882i			0.143298i
$922$		0			0
$923$	−8.36579	+	44.9211i	−0.275363	+	1.47860i
$924$		0			0
$925$		0			0
$926$		0			0
$927$	34.9278			1.14718
$928$		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
$933$	−7.76897			−0.254345
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−45.3069			−1.48011			−0.740056	−	0.672545i	$-0.765201\pi$
$937$	−45.3069			−1.48011			−0.740056	+	0.672545i	$0.765201\pi$
$938$		0			0
$939$	9.25430			0.302003
$940$		0			0
$941$			5.88222i			0.191755i	0.995393	+	0.0958774i	$0.0305657\pi$
$941$			5.88222i			0.191755i	−0.995393	+	0.0958774i	$0.969434\pi$
$942$		0			0
$943$			49.3462i			1.60693i
$944$		0			0
$945$		0			0
$946$		0			0
$947$			14.7427i			0.479072i	0.970887	+	0.239536i	$0.0769954\pi$
$947$			14.7427i			0.479072i	−0.970887	+	0.239536i	$0.923005\pi$
$948$		0			0
$949$	2.26803	−	12.1784i	0.0736232	−	0.395328i
$950$		0			0
$951$		−	9.44607i		−	0.306310i
$952$		0			0
$953$	37.2284			1.20595			0.602973	−	0.797762i	$-0.293983\pi$
$953$	37.2284			1.20595			0.602973	+	0.797762i	$0.293983\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		−	2.73236i		−	0.0883246i
$958$		0			0
$959$	−35.3069			−1.14012
$960$		0			0
$961$	−7.74266			−0.249763
$962$		0			0
$963$	50.2739			1.62005
$964$		0			0
$965$		0			0
$966$		0			0
$967$			31.8711i			1.02491i	0.858715	+	0.512453i	$0.171263\pi$
$967$			31.8711i			1.02491i	−0.858715	+	0.512453i	$0.828737\pi$
$968$		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
$973$		−	35.1544i		−	1.12700i
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	9.61419i		−	0.307585i	−0.988103	−	0.153793i	$-0.950851\pi$
$977$		−	9.61419i		−	0.307585i	0.988103	−	0.153793i	$-0.0491488\pi$
$978$		0			0
$979$	26.4568			0.845562
$980$		0			0
$981$		−	47.3328i		−	1.51122i
$982$		0			0
$983$			4.78193i			0.152520i	0.997088	+	0.0762599i	$0.0242979\pi$
$983$			4.78193i			0.152520i	−0.997088	+	0.0762599i	$0.975702\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	8.61458			0.274205
$988$		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
$993$			6.70340i			0.212726i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−12.3176			−0.390102			−0.195051	−	0.980793i	$-0.562487\pi$
$997$	−12.3176			−0.390102			−0.195051	+	0.980793i	$0.562487\pi$
$998$		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.f.e.701.3		6
5.2	odd	4		1300.2.d.c.649.3		6
5.3	odd	4		1300.2.d.d.649.4		6
5.4	even	2		260.2.f.a.181.4	yes	6
13.12	even	2	inner	1300.2.f.e.701.4		6
15.14	odd	2		2340.2.c.d.181.3		6
20.19	odd	2		1040.2.k.c.961.4		6
65.12	odd	4		1300.2.d.d.649.3		6
65.34	odd	4		3380.2.a.m.1.2		3
65.38	odd	4		1300.2.d.c.649.4		6
65.44	odd	4		3380.2.a.n.1.2		3
65.64	even	2		260.2.f.a.181.3	✓	6
195.194	odd	2		2340.2.c.d.181.4		6
260.259	odd	2		1040.2.k.c.961.3		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
260.2.f.a.181.3	✓	6	65.64	even	2
260.2.f.a.181.4	yes	6	5.4	even	2
1040.2.k.c.961.3		6	260.259	odd	2
1040.2.k.c.961.4		6	20.19	odd	2
1300.2.d.c.649.3		6	5.2	odd	4
1300.2.d.c.649.4		6	65.38	odd	4
1300.2.d.d.649.3		6	65.12	odd	4
1300.2.d.d.649.4		6	5.3	odd	4
1300.2.f.e.701.3		6	1.1	even	1	trivial
1300.2.f.e.701.4		6	13.12	even	2	inner
2340.2.c.d.181.3		6	15.14	odd	2
2340.2.c.d.181.4		6	195.194	odd	2
3380.2.a.m.1.2		3	65.34	odd	4
3380.2.a.n.1.2		3	65.44	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

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

\(f(q)\)	\(=\)	\(q+0.339877 q^{3} -3.88448i q^{7} -2.88448 q^{9} +O(q^{10})\)	\(q+0.339877 q^{3} -3.88448i q^{7} -2.88448 q^{9} +1.54461i q^{11} +(-3.54461 - 0.660123i) q^{13} +2.86485i q^{19} -1.32025i q^{21} -5.42909 q^{23} -2.00000 q^{27} -5.20473 q^{29} +6.22436i q^{31} +0.524976i q^{33} +8.56424i q^{37} +(-1.20473 - 0.224361i) q^{39} -9.08921i q^{41} -0.980369 q^{43} +6.52498i q^{47} -8.08921 q^{49} -6.44872 q^{53} +0.973697i q^{57} +4.45539i q^{59} +9.65345 q^{61} +11.2047i q^{63} -6.97370i q^{67} -1.84522 q^{69} -12.6731i q^{71} +3.43576i q^{73} +6.00000 q^{77} -13.1285 q^{79} +7.97370 q^{81} -8.56424i q^{83} -1.76897 q^{87} -17.1285i q^{89} +(-2.56424 + 13.7690i) q^{91} +2.11552i q^{93} -13.7690i q^{97} -4.45539i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 6 q^{9}+O(q^{10}) \) 6 * q + 6 * q^9	\( 6 q + 6 q^{9} + 12 q^{23} - 12 q^{27} - 12 q^{29} + 12 q^{39} - 12 q^{43} - 6 q^{49} + 12 q^{53} - 12 q^{61} + 36 q^{77} - 24 q^{79} - 18 q^{81} + 36 q^{87} + 12 q^{91}+O(q^{100}) \) 6 * q + 6 * q^9 + 12 * q^23 - 12 * q^27 - 12 * q^29 + 12 * q^39 - 12 * q^43 - 6 * q^49 + 12 * q^53 - 12 * q^61 + 36 * q^77 - 24 * q^79 - 18 * q^81 + 36 * q^87 + 12 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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