LMFDB - Embedded newform 2340.2.c.d.181.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2340,2,Mod(181,2340)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2340, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("2340.181");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2340 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2340.c (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:	$18.6849940730$
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:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 260)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			181.4
Root			$0.339877i$ of defining polynomial
Character	$\chi$	$=$	2340.181
Dual form			2340.2.c.d.181.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{5} -3.88448i q^{7} +O(q^{10})$	$q+1.00000i q^{5} -3.88448i q^{7} +1.54461i q^{11} +(3.54461 - 0.660123i) q^{13} -2.86485i q^{19} -5.42909 q^{23} -1.00000 q^{25} +5.20473 q^{29} -6.22436i q^{31} +3.88448 q^{35} +8.56424i q^{37} -9.08921i q^{41} +0.980369 q^{43} -6.52498i q^{47} -8.08921 q^{49} -6.44872 q^{53} -1.54461 q^{55} +4.45539i q^{59} +9.65345 q^{61} +(0.660123 + 3.54461i) q^{65} -6.97370i q^{67} -12.6731i q^{71} +3.43576i q^{73} +6.00000 q^{77} -13.1285 q^{79} +8.56424i q^{83} -17.1285i q^{89} +(-2.56424 - 13.7690i) q^{91} +2.86485 q^{95} -13.7690i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q+O(q^{10}) $ 6 * q	$ 6 q + 12 q^{23} - 6 q^{25} + 12 q^{29} + 12 q^{43} - 6 q^{49} + 12 q^{53} + 12 q^{55} - 12 q^{61} + 6 q^{65} + 36 q^{77} - 24 q^{79} + 12 q^{91}+O(q^{100}) $ 6 * q + 12 * q^23 - 6 * q^25 + 12 * q^29 + 12 * q^43 - 6 * q^49 + 12 * q^53 + 12 * q^55 - 12 * q^61 + 6 * q^65 + 36 * q^77 - 24 * q^79 + 12 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2340\mathbb{Z}\right)^\times$.

$n$	$937$	$1081$	$1171$	$2081$
$\chi(n)$	$1$	$-1$	$1$	$1$

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			0
$3$		0			0
$4$		0			0
$5$			1.00000i			0.447214i
$5$			1.00000i			0.447214i
$6$		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
$9$		0			0
$10$		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	$-0.893416\pi$
$19$		−	2.86485i		−	0.657242i	0.944462	−	0.328621i	$-0.106584\pi$
$20$		0			0
$21$		0			0
$22$		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
$25$	−1.00000			−0.200000
$26$		0			0
$27$		0			0
$28$		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			0
$34$		0			0
$35$	3.88448			0.656598
$36$		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
$39$		0			0
$40$		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	$-0.476183\pi$
$43$	0.980369			0.149505			0.0747525	+	0.997202i	$0.476183\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		−	6.52498i		−	0.951766i	−0.879509	−	0.475883i	$-0.842129\pi$
$47$		−	6.52498i		−	0.951766i	0.879509	−	0.475883i	$-0.157871\pi$
$48$		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
$55$	−1.54461			−0.208275
$56$		0			0
$57$		0			0
$58$		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
$63$		0			0
$64$		0			0
$65$	0.660123	+	3.54461i	0.0818782	+	0.439654i
$66$		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
$69$		0			0
$70$		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
$81$		0			0
$82$		0			0
$83$			8.56424i			0.940047i	0.882654	+	0.470024i	$0.155755\pi$
$83$			8.56424i			0.940047i	−0.882654	+	0.470024i	$0.844245\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		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
$93$		0			0
$94$		0			0
$95$	2.86485			0.293928
$96$		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
$99$		0			0
$100$		0			0
$101$	6.44872			0.641672			0.320836	−	0.947135i	$-0.396036\pi$
$101$	6.44872			0.641672			0.320836	+	0.947135i	$0.396036\pi$
$102$		0			0
$103$	12.1088			1.19312			0.596560	−	0.802569i	$-0.296534\pi$
$103$	12.1088			1.19312			0.596560	+	0.802569i	$0.296534\pi$
$104$		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	$-0.712215\pi$
$109$		−	16.4095i		−	1.57174i	0.618391	−	0.785871i	$-0.287785\pi$
$110$		0			0
$111$		0			0
$112$		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
$115$		−	5.42909i		−	0.506265i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	8.61419			0.783108
$122$		0			0
$123$		0			0
$124$		0			0
$125$		−	1.00000i		−	0.0894427i
$126$		0			0
$127$	−0.108844			−0.00965837			−0.00482918	−	0.999988i	$-0.501537\pi$
$127$	−0.108844			−0.00965837			−0.00482918	+	0.999988i	$0.501537\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	10.4095			0.909479			0.454739	−	0.890625i	$-0.349732\pi$
$131$	10.4095			0.909479			0.454739	+	0.890625i	$0.349732\pi$
$132$		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	$0.126928\pi$
$137$			9.08921i			0.776544i	−0.921545	+	0.388272i	$0.873072\pi$
$138$		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
$141$		0			0
$142$		0			0
$143$	1.01963	+	5.47502i	0.0852658	+	0.457844i
$144$		0			0
$145$			5.20473i			0.432229i
$146$		0			0
$147$		0			0
$148$		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
$155$	6.22436			0.499953
$156$		0			0
$157$	4.91079			0.391923			0.195962	−	0.980612i	$-0.437217\pi$
$157$	4.91079			0.391923			0.195962	+	0.980612i	$0.437217\pi$
$158$		0			0
$159$		0			0
$160$		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	$-0.684727\pi$
$167$		−	21.6142i		−	1.67256i	0.548306	−	0.836278i	$-0.315273\pi$
$168$		0			0
$169$	12.1285	−	4.67975i	0.932960	−	0.359981i
$170$		0			0
$171$		0			0
$172$		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
$175$			3.88448i			0.293639i
$176$		0			0
$177$		0			0
$178$		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
$183$		0			0
$184$		0			0
$185$	−8.56424			−0.629655
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$	8.81892			0.638115			0.319057	−	0.947735i	$-0.396634\pi$
$191$	8.81892			0.638115			0.319057	+	0.947735i	$0.396634\pi$
$192$		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	$-0.954938\pi$
$197$		−	3.96074i		−	0.282191i	0.989996	−	0.141095i	$-0.0450625\pi$
$198$		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
$201$		0			0
$202$		0			0
$203$		−	20.2177i		−	1.41900i
$204$		0			0
$205$	9.08921			0.634818
$206$		0			0
$207$		0			0
$208$		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
$213$		0			0
$214$		0			0
$215$			0.980369i			0.0668606i
$216$		0			0
$217$	−24.1784			−1.64134
$218$		0			0
$219$		0			0
$220$		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	$-0.963627\pi$
$227$		−	3.43576i		−	0.228040i	0.993478	−	0.114020i	$-0.0363727\pi$
$228$		0			0
$229$			22.8582i			1.51051i	0.655430	+	0.755256i	$0.272487\pi$
$229$			22.8582i			1.51051i	−0.655430	+	0.755256i	$0.727513\pi$
$230$		0			0
$231$		0			0
$232$		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
$235$	6.52498			0.425643
$236$		0			0
$237$		0			0
$238$		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
$243$		0			0
$244$		0			0
$245$		−	8.08921i		−	0.516801i
$246$		0			0
$247$	−1.89116	−	10.1548i	−0.120331	−	0.646133i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	21.2676			1.34240			0.671201	−	0.741276i	$-0.265779\pi$
$251$	21.2676			1.34240			0.671201	+	0.741276i	$0.265779\pi$
$252$		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
$261$		0			0
$262$		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
$265$		−	6.44872i		−	0.396142i
$266$		0			0
$267$		0			0
$268$		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
$273$		0			0
$274$		0			0
$275$		−	1.54461i		−	0.0931433i
$276$		0			0
$277$	−19.2676			−1.15768			−0.578840	−	0.815441i	$-0.696494\pi$
$277$	−19.2676			−1.15768			−0.578840	+	0.815441i	$0.696494\pi$
$278$		0			0
$279$		0			0
$280$		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	$-0.716964\pi$
$283$	−21.1981			−1.26009			−0.630047	+	0.776557i	$0.716964\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−35.3069			−2.08410
$288$		0			0
$289$	−17.0000			−1.00000
$290$		0			0
$291$		0			0
$292$		0			0
$293$		−	13.6927i		−	0.799937i	−0.916529	−	0.399968i	$-0.869021\pi$
$293$		−	13.6927i		−	0.799937i	0.916529	−	0.399968i	$-0.130979\pi$
$294$		0			0
$295$	−4.45539			−0.259403
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−19.2440	+	3.58387i	−1.11291	+	0.207260i
$300$		0			0
$301$		−	3.80823i		−	0.219503i
$302$		0			0
$303$		0			0
$304$		0			0
$305$			9.65345i			0.552755i
$306$		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
$309$		0			0
$310$		0			0
$311$	22.8582			1.29617			0.648084	−	0.761569i	$-0.275570\pi$
$311$	22.8582			1.29617			0.648084	+	0.761569i	$0.275570\pi$
$312$		0			0
$313$	−27.2284			−1.53904			−0.769520	−	0.638623i	$-0.779504\pi$
$313$	−27.2284			−1.53904			−0.769520	+	0.638623i	$0.779504\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$			27.7926i			1.56099i	0.625163	+	0.780494i	$0.285033\pi$
$317$			27.7926i			1.56099i	−0.625163	+	0.780494i	$0.714967\pi$
$318$		0			0
$319$			8.03926i			0.450112i
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$	−3.54461	+	0.660123i	−0.196619	+	0.0366170i
$326$		0			0
$327$		0			0
$328$		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
$333$		0			0
$334$		0			0
$335$	6.97370			0.381014
$336$		0			0
$337$	−20.0000			−1.08947			−0.544735	−	0.838608i	$-0.683370\pi$
$337$	−20.0000			−1.08947			−0.544735	+	0.838608i	$0.683370\pi$
$338$		0			0
$339$		0			0
$340$		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	$0.0263482\pi$
$349$			3.08921i			0.165362i	−0.996576	+	0.0826809i	$0.973652\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$			20.9211i			1.11352i	0.830674	+	0.556759i	$0.187955\pi$
$353$			20.9211i			1.11352i	−0.830674	+	0.556759i	$0.812045\pi$
$354$		0			0
$355$	12.6731			0.672617
$356$		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
$363$		0			0
$364$		0			0
$365$	−3.43576			−0.179836
$366$		0			0
$367$	21.0196			1.09722			0.548608	−	0.836080i	$-0.315158\pi$
$367$	21.0196			1.09722			0.548608	+	0.836080i	$0.315158\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$			25.0500i			1.30053i
$372$		0			0
$373$	29.3462			1.51949			0.759743	−	0.650223i	$-0.225325\pi$
$373$	29.3462			1.51949			0.759743	+	0.650223i	$0.225325\pi$
$374$		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	$-0.905820\pi$
$379$		−	11.3528i		−	0.583156i	0.956547	−	0.291578i	$-0.0941803\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		−	20.5642i		−	1.05078i	−0.850860	−	0.525392i	$-0.823919\pi$
$383$		−	20.5642i		−	1.05078i	0.850860	−	0.525392i	$-0.176081\pi$
$384$		0			0
$385$			6.00000i			0.305788i
$386$		0			0
$387$		0			0
$388$		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
$393$		0			0
$394$		0			0
$395$		−	13.1285i		−	0.660565i
$396$		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
$399$		0			0
$400$		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	$0.00493567\pi$
$409$			0.627148i			0.0310105i	−0.999880	+	0.0155052i	$0.995064\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	17.3069			0.851617
$414$		0			0
$415$	−8.56424			−0.420402
$416$		0			0
$417$		0			0
$418$		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
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$		−	37.4987i		−	1.81469i
$428$		0			0
$429$		0			0
$430$		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	$-0.515303\pi$
$433$	−2.00000			−0.0961139			−0.0480569	+	0.998845i	$0.515303\pi$
$434$		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
$441$		0			0
$442$		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
$445$	17.1285			0.811968
$446$		0			0
$447$		0			0
$448$		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
$453$		0			0
$454$		0			0
$455$	13.7690	−	2.56424i	0.645499	−	0.120213i
$456$		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
$471$		0			0
$472$		0			0
$473$			1.51429i			0.0696269i
$474$		0			0
$475$			2.86485i			0.131448i
$476$		0			0
$477$		0			0
$478$		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
$483$		0			0
$484$		0			0
$485$	13.7690			0.625217
$486$		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
$489$		0			0
$490$		0			0
$491$	8.81892			0.397992			0.198996	−	0.980000i	$-0.436232\pi$
$491$	8.81892			0.397992			0.198996	+	0.980000i	$0.436232\pi$
$492$		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	$0.178624\pi$
$499$			23.7756i			1.06434i	−0.846636	+	0.532172i	$0.821376\pi$
$500$		0			0
$501$		0			0
$502$		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
$505$			6.44872i			0.286964i
$506$		0			0
$507$		0			0
$508$		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
$513$		0			0
$514$		0			0
$515$			12.1088i			0.533579i
$516$		0			0
$517$	10.0785			0.443253
$518$		0			0
$519$		0			0
$520$		0			0
$521$	17.2047			0.753753			0.376876	−	0.926264i	$-0.376998\pi$
$521$	17.2047			0.753753			0.376876	+	0.926264i	$0.376998\pi$
$522$		0			0
$523$	3.19806			0.139841			0.0699207	−	0.997553i	$-0.477725\pi$
$523$	3.19806			0.139841			0.0699207	+	0.997553i	$0.477725\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	6.47502			0.281523
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−6.00000	−	32.2177i	−0.259889	−	1.39550i
$534$		0			0
$535$		−	17.4291i		−	0.753525i
$536$		0			0
$537$		0			0
$538$		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
$543$		0			0
$544$		0			0
$545$	16.4095			0.702904
$546$		0			0
$547$	38.3265			1.63872			0.819362	−	0.573276i	$-0.194328\pi$
$547$	38.3265			1.63872			0.819362	+	0.573276i	$0.194328\pi$
$548$		0			0
$549$		0			0
$550$		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	$0.235948\pi$
$557$			31.8711i			1.35042i	−0.737624	+	0.675212i	$0.764052\pi$
$558$		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
$565$		−	1.59054i		−	0.0669145i
$566$		0			0
$567$		0			0
$568$		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
$573$		0			0
$574$		0			0
$575$	5.42909			0.226409
$576$		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			0
$580$		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	$-0.950035\pi$
$587$		−	7.57493i		−	0.312651i	0.987706	−	0.156325i	$-0.0499648\pi$
$588$		0			0
$589$	−17.8319			−0.734750
$590$		0			0
$591$		0			0
$592$		0			0
$593$			16.9500i			0.696055i	0.937484	+	0.348028i	$0.113148\pi$
$593$			16.9500i			0.696055i	−0.937484	+	0.348028i	$0.886852\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		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
$603$		0			0
$604$		0			0
$605$			8.61419i			0.350217i
$606$		0			0
$607$	−34.3265			−1.39327			−0.696635	−	0.717425i	$-0.745320\pi$
$607$	−34.3265			−1.39327			−0.696635	+	0.717425i	$0.745320\pi$
$608$		0			0
$609$		0			0
$610$		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	$-0.727471\pi$
$617$		−	37.5246i		−	1.51068i	0.655331	−	0.755342i	$-0.272529\pi$
$618$		0			0
$619$			12.8256i			0.515504i	0.966211	+	0.257752i	$0.0829818\pi$
$619$			12.8256i			0.515504i	−0.966211	+	0.257752i	$0.917018\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−66.5353			−2.66568
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$		0			0
$628$		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
$633$		0			0
$634$		0			0
$635$		−	0.108844i		−	0.00431935i
$636$		0			0
$637$	−28.6731	+	5.33988i	−1.13607	+	0.211574i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−5.10256			−0.201539			−0.100769	−	0.994910i	$-0.532130\pi$
$641$	−5.10256			−0.201539			−0.100769	+	0.994910i	$0.532130\pi$
$642$		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
$651$		0			0
$652$		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
$655$			10.4095i			0.406731i
$656$		0			0
$657$		0			0
$658$		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
$665$		−	11.1285i		−	0.431544i
$666$		0			0
$667$	−28.2569			−1.09411
$668$		0			0
$669$		0			0
$670$		0			0
$671$			14.9108i			0.575625i
$672$		0			0
$673$	31.3069			1.20679			0.603396	−	0.797442i	$-0.293814\pi$
$673$	31.3069			1.20679			0.603396	+	0.797442i	$0.293814\pi$
$674$		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
$681$		0			0
$682$		0			0
$683$			20.5642i			0.786869i	0.919353	+	0.393434i	$0.128713\pi$
$683$			20.5642i			0.786869i	−0.919353	+	0.393434i	$0.871287\pi$
$684$		0			0
$685$	−9.08921			−0.347281
$686$		0			0
$687$		0			0
$688$		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
$693$		0			0
$694$		0			0
$695$			9.04995i			0.343284i
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−9.62980			−0.363713			−0.181856	−	0.983325i	$-0.558211\pi$
$701$	−9.62980			−0.363713			−0.181856	+	0.983325i	$0.558211\pi$
$702$		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	$0.218937\pi$
$709$			33.8082i			1.26969i	−0.772638	+	0.634847i	$0.781063\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$			33.7926i			1.26554i
$714$		0			0
$715$	−5.47502	+	1.01963i	−0.204754	+	0.0381320i
$716$		0			0
$717$		0			0
$718$		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
$723$		0			0
$724$		0			0
$725$	−5.20473			−0.193299
$726$		0			0
$727$	−46.2480			−1.71524			−0.857622	−	0.514281i	$-0.828059\pi$
$727$	−46.2480			−1.71524			−0.857622	+	0.514281i	$0.828059\pi$
$728$		0			0
$729$		0			0
$730$		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	$-0.580637\pi$
$739$		−	52.6338i		−	1.93617i	0.250629	−	0.968083i	$-0.419363\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			47.9497i			1.75910i	0.475804	+	0.879551i	$0.342157\pi$
$743$			47.9497i			1.75910i	−0.475804	+	0.879551i	$0.657843\pi$
$744$		0			0
$745$	−11.1285			−0.407716
$746$		0			0
$747$		0			0
$748$		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
$753$		0			0
$754$		0			0
$755$	−9.13515			−0.332462
$756$		0			0
$757$	44.3176			1.61075			0.805375	−	0.592765i	$-0.201964\pi$
$757$	44.3176			1.61075			0.805375	+	0.592765i	$0.201964\pi$
$758$		0			0
$759$		0			0
$760$		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	$-0.827624\pi$
$769$		−	28.5879i		−	1.03091i	0.856918	−	0.515453i	$-0.172376\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		−	15.4358i		−	0.555186i	−0.960699	−	0.277593i	$-0.910463\pi$
$773$		−	15.4358i		−	0.555186i	0.960699	−	0.277593i	$-0.0895366\pi$
$774$		0			0
$775$			6.22436i			0.223586i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−26.0393			−0.932953
$780$		0			0
$781$	19.5749			0.700446
$782$		0			0
$783$		0			0
$784$		0			0
$785$			4.91079i			0.175273i
$786$		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
$789$		0			0
$790$		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
$801$		0			0
$802$		0			0
$803$	−5.30690			−0.187277
$804$		0			0
$805$	−21.0892			−0.743297
$806$		0			0
$807$		0			0
$808$		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
$813$		0			0
$814$		0			0
$815$	16.9344			0.593187
$816$		0			0
$817$		−	2.80861i		−	0.0982610i
$818$		0			0
$819$		0			0
$820$		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	$-0.318442\pi$
$823$	30.9804			1.07991			0.539954	+	0.841695i	$0.318442\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	6.82119i		−	0.237196i	−0.992942	−	0.118598i	$-0.962160\pi$
$827$		−	6.82119i		−	0.237196i	0.992942	−	0.118598i	$-0.0378400\pi$
$828$		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
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$	21.6142			0.747990
$836$		0			0
$837$		0			0
$838$		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
$843$		0			0
$844$		0			0
$845$	4.67975	+	12.1285i	0.160988	+	0.417232i
$846$		0			0
$847$		−	33.4617i		−	1.14976i
$848$		0			0
$849$		0			0
$850$		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
$861$		0			0
$862$		0			0
$863$			5.83188i			0.198519i	0.995062	+	0.0992597i	$0.0316475\pi$
$863$			5.83188i			0.198519i	−0.995062	+	0.0992597i	$0.968353\pi$
$864$		0			0
$865$			8.03926i			0.273343i
$866$		0			0
$867$		0			0
$868$		0			0
$869$		−	20.2783i		−	0.687895i
$870$		0			0
$871$	−4.60350	−	24.7190i	−0.155984	−	0.837572i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−3.88448			−0.131320
$876$		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
$879$		0			0
$880$		0			0
$881$	55.4483			1.86810			0.934051	−	0.357140i	$-0.116248\pi$
$881$	55.4483			1.86810			0.934051	+	0.357140i	$0.116248\pi$
$882$		0			0
$883$	1.75199			0.0589591			0.0294796	−	0.999565i	$-0.490615\pi$
$883$	1.75199			0.0589591			0.0294796	+	0.999565i	$0.490615\pi$
$884$		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
$891$		0			0
$892$		0			0
$893$	−18.6931			−0.625541
$894$		0			0
$895$		−	10.8582i		−	0.362949i
$896$		0			0
$897$		0			0
$898$		0			0
$899$		−	32.3961i		−	1.08047i
$900$		0			0
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$			10.5642i			0.351167i
$906$		0			0
$907$	−10.0696			−0.334355			−0.167178	−	0.985927i	$-0.553465\pi$
$907$	−10.0696			−0.334355			−0.167178	+	0.985927i	$0.553465\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	39.8341			1.31976			0.659882	−	0.751369i	$-0.270606\pi$
$911$	39.8341			1.31976			0.659882	+	0.751369i	$0.270606\pi$
$912$		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
$921$		0			0
$922$		0			0
$923$	−8.36579	−	44.9211i	−0.275363	−	1.47860i
$924$		0			0
$925$		−	8.56424i		−	0.281590i
$926$		0			0
$927$		0			0
$928$		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
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	45.3069			1.48011			0.740056	−	0.672545i	$-0.234799\pi$
$937$	45.3069			1.48011			0.740056	+	0.672545i	$0.234799\pi$
$938$		0			0
$939$		0			0
$940$		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	$-0.923005\pi$
$947$		−	14.7427i		−	0.479072i	0.970887	−	0.239536i	$-0.0769954\pi$
$948$		0			0
$949$	2.26803	+	12.1784i	0.0736232	+	0.395328i
$950$		0			0
$951$		0			0
$952$		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
$955$			8.81892i			0.285374i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	35.3069			1.14012
$960$		0			0
$961$	−7.74266			−0.249763
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−0.270294			−0.00870107
$966$		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	$-0.700134\pi$
$971$	−36.6531			−1.17625			−0.588126	+	0.808769i	$0.700134\pi$
$972$		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	$0.0491488\pi$
$977$			9.61419i			0.307585i	−0.988103	+	0.153793i	$0.950851\pi$
$978$		0			0
$979$	26.4568			0.845562
$980$		0			0
$981$		0			0
$982$		0			0
$983$		−	4.78193i		−	0.152520i	−0.997088	−	0.0762599i	$-0.975702\pi$
$983$		−	4.78193i		−	0.152520i	0.997088	−	0.0762599i	$-0.0242979\pi$
$984$		0			0
$985$	3.96074			0.126200
$986$		0			0
$987$		0			0
$988$		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
$993$		0			0
$994$		0			0
$995$			6.03926i			0.191457i
$996$		0			0
$997$	12.3176			0.390102			0.195051	−	0.980793i	$-0.437513\pi$
$997$	12.3176			0.390102			0.195051	+	0.980793i	$0.437513\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2340.2.c.d.181.4		6
3.2	odd	2		260.2.f.a.181.3	✓	6
12.11	even	2		1040.2.k.c.961.3		6
13.12	even	2	inner	2340.2.c.d.181.3		6
15.2	even	4		1300.2.d.c.649.4		6
15.8	even	4		1300.2.d.d.649.3		6
15.14	odd	2		1300.2.f.e.701.4		6
39.5	even	4		3380.2.a.m.1.2		3
39.8	even	4		3380.2.a.n.1.2		3
39.38	odd	2		260.2.f.a.181.4	yes	6
156.155	even	2		1040.2.k.c.961.4		6
195.38	even	4		1300.2.d.c.649.3		6
195.77	even	4		1300.2.d.d.649.4		6
195.194	odd	2		1300.2.f.e.701.3		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
260.2.f.a.181.3	✓	6	3.2	odd	2
260.2.f.a.181.4	yes	6	39.38	odd	2
1040.2.k.c.961.3		6	12.11	even	2
1040.2.k.c.961.4		6	156.155	even	2
1300.2.d.c.649.3		6	195.38	even	4
1300.2.d.c.649.4		6	15.2	even	4
1300.2.d.d.649.3		6	15.8	even	4
1300.2.d.d.649.4		6	195.77	even	4
1300.2.f.e.701.3		6	195.194	odd	2
1300.2.f.e.701.4		6	15.14	odd	2
2340.2.c.d.181.3		6	13.12	even	2	inner
2340.2.c.d.181.4		6	1.1	even	1	trivial
3380.2.a.m.1.2		3	39.5	even	4
3380.2.a.n.1.2		3	39.8	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:	\( N \)	\(=\)	\( 2340 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2340.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{5} -3.88448i q^{7} +O(q^{10})\)	\(q+1.00000i q^{5} -3.88448i q^{7} +1.54461i q^{11} +(3.54461 - 0.660123i) q^{13} -2.86485i q^{19} -5.42909 q^{23} -1.00000 q^{25} +5.20473 q^{29} -6.22436i q^{31} +3.88448 q^{35} +8.56424i q^{37} -9.08921i q^{41} +0.980369 q^{43} -6.52498i q^{47} -8.08921 q^{49} -6.44872 q^{53} -1.54461 q^{55} +4.45539i q^{59} +9.65345 q^{61} +(0.660123 + 3.54461i) q^{65} -6.97370i q^{67} -12.6731i q^{71} +3.43576i q^{73} +6.00000 q^{77} -13.1285 q^{79} +8.56424i q^{83} -17.1285i q^{89} +(-2.56424 - 13.7690i) q^{91} +2.86485 q^{95} -13.7690i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q+O(q^{10}) \) 6 * q	\( 6 q + 12 q^{23} - 6 q^{25} + 12 q^{29} + 12 q^{43} - 6 q^{49} + 12 q^{53} + 12 q^{55} - 12 q^{61} + 6 q^{65} + 36 q^{77} - 24 q^{79} + 12 q^{91}+O(q^{100}) \) 6 * q + 12 * q^23 - 6 * q^25 + 12 * q^29 + 12 * q^43 - 6 * q^49 + 12 * q^53 + 12 * q^55 - 12 * q^61 + 6 * q^65 + 36 * q^77 - 24 * q^79 + 12 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more