LMFDB - Embedded newform 2520.2.k.a.1889.15

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2520,2,Mod(1889,2520)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2520, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 1, 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("2520.1889");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2520 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2520.k (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:	$20.1223013094$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1889.15
Character	$\chi$	$=$	2520.1889
Dual form			2520.2.k.a.1889.16

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.727958 - 2.11426i) q^{5} +(1.21521 + 2.35016i) q^{7} +O(q^{10})$	$q+(0.727958 - 2.11426i) q^{5} +(1.21521 + 2.35016i) q^{7} +2.91501i q^{11} +0.380654 q^{13} +7.43045i q^{17} -4.51415i q^{19} +2.63030 q^{23} +(-3.94016 - 3.07818i) q^{25} +8.44541i q^{29} -5.79961i q^{31} +(5.85347 - 0.858444i) q^{35} +9.46775i q^{37} -0.570545 q^{41} +2.65721i q^{43} +8.79311i q^{47} +(-4.04653 + 5.71188i) q^{49} -14.0197 q^{53} +(6.16307 + 2.12200i) q^{55} -6.55468 q^{59} +2.66361i q^{61} +(0.277100 - 0.804799i) q^{65} -8.87911i q^{67} -13.6005i q^{71} +6.55607 q^{73} +(-6.85074 + 3.54234i) q^{77} +6.74215 q^{79} +15.5747i q^{83} +(15.7099 + 5.40905i) q^{85} +17.4408 q^{89} +(0.462574 + 0.894598i) q^{91} +(-9.54407 - 3.28611i) q^{95} +6.88869 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q+O(q^{10}) $ 24 * q	$ 24 q - 8 q^{23} - 16 q^{25} + 4 q^{35} - 12 q^{49} + 24 q^{53} + 8 q^{65} + 4 q^{77} + 40 q^{79} + 24 q^{85} - 36 q^{91} - 48 q^{95}+O(q^{100}) $ 24 * q - 8 * q^23 - 16 * q^25 + 4 * q^35 - 12 * q^49 + 24 * q^53 + 8 * q^65 + 4 * q^77 + 40 * q^79 + 24 * q^85 - 36 * q^91 - 48 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2520\mathbb{Z}\right)^\times$.

$n$	$281$	$631$	$1081$	$1261$	$2017$
$\chi(n)$	$-1$	$1$	$-1$	$1$	$-1$

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
$5$	0.727958	−	2.11426i	0.325553	−	0.945524i
$5$	0.727958	−	2.11426i	0.325553	−	0.945524i
$6$		0			0
$7$	1.21521	+	2.35016i	0.459306	+	0.888278i
$7$	1.21521	+	2.35016i	0.459306	+	0.888278i
$8$		0			0
$9$		0			0
$10$		0			0
$11$			2.91501i			0.878908i	0.898265	+	0.439454i	$0.144828\pi$
$11$			2.91501i			0.878908i	−0.898265	+	0.439454i	$0.855172\pi$
$12$		0			0
$13$	0.380654			0.105574			0.0527872	−	0.998606i	$-0.483190\pi$
$13$	0.380654			0.105574			0.0527872	+	0.998606i	$0.483190\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$			7.43045i			1.80215i	0.433666	+	0.901074i	$0.357220\pi$
$17$			7.43045i			1.80215i	−0.433666	+	0.901074i	$0.642780\pi$
$18$		0			0
$19$		−	4.51415i		−	1.03562i	−0.855496	−	0.517809i	$-0.826748\pi$
$19$		−	4.51415i		−	1.03562i	0.855496	−	0.517809i	$-0.173252\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	2.63030			0.548456			0.274228	−	0.961665i	$-0.411578\pi$
$23$	2.63030			0.548456			0.274228	+	0.961665i	$0.411578\pi$
$24$		0			0
$25$	−3.94016	−	3.07818i	−0.788031	−	0.615636i
$26$		0			0
$27$		0			0
$28$		0			0
$29$			8.44541i			1.56827i	0.620588	+	0.784137i	$0.286894\pi$
$29$			8.44541i			1.56827i	−0.620588	+	0.784137i	$0.713106\pi$
$30$		0			0
$31$		−	5.79961i		−	1.04164i	−0.853667	−	0.520820i	$-0.825626\pi$
$31$		−	5.79961i		−	1.04164i	0.853667	−	0.520820i	$-0.174374\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	5.85347	−	0.858444i	0.989416	−	0.145104i
$36$		0			0
$37$			9.46775i			1.55649i	0.627962	+	0.778244i	$0.283889\pi$
$37$			9.46775i			1.55649i	−0.627962	+	0.778244i	$0.716111\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−0.570545			−0.0891042			−0.0445521	−	0.999007i	$-0.514186\pi$
$41$	−0.570545			−0.0891042			−0.0445521	+	0.999007i	$0.514186\pi$
$42$		0			0
$43$			2.65721i			0.405221i	0.979259	+	0.202611i	$0.0649426\pi$
$43$			2.65721i			0.405221i	−0.979259	+	0.202611i	$0.935057\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$			8.79311i			1.28261i	0.767287	+	0.641303i	$0.221606\pi$
$47$			8.79311i			1.28261i	−0.767287	+	0.641303i	$0.778394\pi$
$48$		0			0
$49$	−4.04653	+	5.71188i	−0.578076	+	0.815983i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−14.0197			−1.92576			−0.962878	−	0.269936i	$-0.912997\pi$
$53$	−14.0197			−1.92576			−0.962878	+	0.269936i	$0.912997\pi$
$54$		0			0
$55$	6.16307	+	2.12200i	0.831028	+	0.286131i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−6.55468			−0.853347			−0.426673	−	0.904406i	$-0.640315\pi$
$59$	−6.55468			−0.853347			−0.426673	+	0.904406i	$0.640315\pi$
$60$		0			0
$61$			2.66361i			0.341040i	0.985354	+	0.170520i	$0.0545448\pi$
$61$			2.66361i			0.341040i	−0.985354	+	0.170520i	$0.945455\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	0.277100	−	0.804799i	0.0343700	−	0.0998231i
$66$		0			0
$67$		−	8.87911i		−	1.08476i	−0.840135	−	0.542378i	$-0.817524\pi$
$67$		−	8.87911i		−	1.08476i	0.840135	−	0.542378i	$-0.182476\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	13.6005i		−	1.61408i	−0.590497	−	0.807040i	$-0.701068\pi$
$71$		−	13.6005i		−	1.61408i	0.590497	−	0.807040i	$-0.298932\pi$
$72$		0			0
$73$	6.55607			0.767330			0.383665	−	0.923472i	$-0.374662\pi$
$73$	6.55607			0.767330			0.383665	+	0.923472i	$0.374662\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−6.85074	+	3.54234i	−0.780715	+	0.403688i
$78$		0			0
$79$	6.74215			0.758551			0.379276	−	0.925284i	$-0.376173\pi$
$79$	6.74215			0.758551			0.379276	+	0.925284i	$0.376173\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$			15.5747i			1.70954i	0.519005	+	0.854771i	$0.326302\pi$
$83$			15.5747i			1.70954i	−0.519005	+	0.854771i	$0.673698\pi$
$84$		0			0
$85$	15.7099	+	5.40905i	1.70397	+	0.586694i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	17.4408			1.84873			0.924363	−	0.381514i	$-0.124597\pi$
$89$	17.4408			1.84873			0.924363	+	0.381514i	$0.124597\pi$
$90$		0			0
$91$	0.462574	+	0.894598i	0.0484909	+	0.0937794i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−9.54407	−	3.28611i	−0.979201	−	0.337148i
$96$		0			0
$97$	6.88869			0.699441			0.349720	−	0.936854i	$-0.386277\pi$
$97$	6.88869			0.699441			0.349720	+	0.936854i	$0.386277\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	8.60120			0.855851			0.427926	−	0.903814i	$-0.359245\pi$
$101$	8.60120			0.855851			0.427926	+	0.903814i	$0.359245\pi$
$102$		0			0
$103$	10.0268			0.987972			0.493986	−	0.869470i	$-0.335539\pi$
$103$	10.0268			0.987972			0.493986	+	0.869470i	$0.335539\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−7.31332			−0.707005			−0.353502	−	0.935434i	$-0.615009\pi$
$107$	−7.31332			−0.707005			−0.353502	+	0.935434i	$0.615009\pi$
$108$		0			0
$109$	−9.23122			−0.884191			−0.442095	−	0.896968i	$-0.645765\pi$
$109$	−9.23122			−0.884191			−0.442095	+	0.896968i	$0.645765\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	6.22792			0.585874			0.292937	−	0.956132i	$-0.405367\pi$
$113$	6.22792			0.585874			0.292937	+	0.956132i	$0.405367\pi$
$114$		0			0
$115$	1.91475	−	5.56113i	0.178551	−	0.518578i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−17.4628	+	9.02955i	−1.60081	+	0.827737i
$120$		0			0
$121$	2.50273			0.227521
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−9.37632	+	6.08971i	−0.838644	+	0.544680i
$126$		0			0
$127$			15.4529i			1.37122i	0.727968	+	0.685611i	$0.240465\pi$
$127$			15.4529i			1.37122i	−0.727968	+	0.685611i	$0.759535\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	12.0841			1.05579			0.527896	−	0.849309i	$-0.322981\pi$
$131$	12.0841			1.05579			0.527896	+	0.849309i	$0.322981\pi$
$132$		0			0
$133$	10.6090	−	5.48564i	0.919917	−	0.475665i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	10.8160			0.924077			0.462039	−	0.886860i	$-0.347118\pi$
$137$	10.8160			0.924077			0.462039	+	0.886860i	$0.347118\pi$
$138$		0			0
$139$		−	16.4141i		−	1.39223i	−0.717931	−	0.696114i	$-0.754911\pi$
$139$		−	16.4141i		−	1.39223i	0.717931	−	0.696114i	$-0.245089\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$			1.10961i			0.0927901i
$144$		0			0
$145$	17.8558	+	6.14790i	1.48284	+	0.510556i
$146$		0			0
$147$		0			0
$148$		0			0
$149$		−	2.69903i		−	0.221113i	−0.993870	−	0.110557i	$-0.964737\pi$
$149$		−	2.69903i		−	0.221113i	0.993870	−	0.110557i	$-0.0352633\pi$
$150$		0			0
$151$	5.82117			0.473720			0.236860	−	0.971544i	$-0.423882\pi$
$151$	5.82117			0.473720			0.236860	+	0.971544i	$0.423882\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−12.2618	−	4.22187i	−0.984895	−	0.339109i
$156$		0			0
$157$	−5.59936			−0.446878			−0.223439	−	0.974718i	$-0.571728\pi$
$157$	−5.59936			−0.446878			−0.223439	+	0.974718i	$0.571728\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	3.19637	+	6.18163i	0.251909	+	0.487181i
$162$		0			0
$163$		−	6.71945i		−	0.526308i	−0.964754	−	0.263154i	$-0.915237\pi$
$163$		−	6.71945i		−	0.526308i	0.964754	−	0.263154i	$-0.0847627\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			4.42108i			0.342113i	0.985261	+	0.171057i	$0.0547181\pi$
$167$			4.42108i			0.342113i	−0.985261	+	0.171057i	$0.945282\pi$
$168$		0			0
$169$	−12.8551			−0.988854
$170$		0			0
$171$		0			0
$172$		0			0
$173$			3.07730i			0.233963i	0.993134	+	0.116981i	$0.0373218\pi$
$173$			3.07730i			0.233963i	−0.993134	+	0.116981i	$0.962678\pi$
$174$		0			0
$175$	2.44611	−	13.0006i	0.184908	−	0.982756i
$176$		0			0
$177$		0			0
$178$		0			0
$179$			15.1441i			1.13192i	0.824432	+	0.565961i	$0.191495\pi$
$179$			15.1441i			1.13192i	−0.824432	+	0.565961i	$0.808505\pi$
$180$		0			0
$181$			16.4413i			1.22207i	0.791603	+	0.611036i	$0.209247\pi$
$181$			16.4413i			1.22207i	−0.791603	+	0.611036i	$0.790753\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	20.0172	+	6.89212i	1.47170	+	0.506719i
$186$		0			0
$187$	−21.6598			−1.58392
$188$		0			0
$189$		0			0
$190$		0			0
$191$			10.8136i			0.782444i	0.920296	+	0.391222i	$0.127948\pi$
$191$			10.8136i			0.782444i	−0.920296	+	0.391222i	$0.872052\pi$
$192$		0			0
$193$			19.6698i			1.41586i	0.706281	+	0.707932i	$0.250372\pi$
$193$			19.6698i			1.41586i	−0.706281	+	0.707932i	$0.749628\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	8.38817			0.597632			0.298816	−	0.954311i	$-0.403408\pi$
$197$	8.38817			0.597632			0.298816	+	0.954311i	$0.403408\pi$
$198$		0			0
$199$		−	9.97691i		−	0.707244i	−0.935388	−	0.353622i	$-0.884950\pi$
$199$		−	9.97691i		−	0.707244i	0.935388	−	0.353622i	$-0.115050\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−19.8481	+	10.2629i	−1.39306	+	0.720317i
$204$		0			0
$205$	−0.415333	+	1.20628i	−0.0290081	+	0.0842502i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	13.1588			0.910212
$210$		0			0
$211$	14.2155			0.978636			0.489318	−	0.872105i	$-0.337246\pi$
$211$	14.2155			0.978636			0.489318	+	0.872105i	$0.337246\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	5.61803	+	1.93434i	0.383146	+	0.131921i
$216$		0			0
$217$	13.6300	−	7.04773i	0.925266	−	0.478431i
$218$		0			0
$219$		0			0
$220$		0			0
$221$			2.82843i			0.190261i
$222$		0			0
$223$	−10.0049			−0.669979			−0.334989	−	0.942222i	$-0.608733\pi$
$223$	−10.0049			−0.669979			−0.334989	+	0.942222i	$0.608733\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$		−	3.40417i		−	0.225943i	−0.993598	−	0.112971i	$-0.963963\pi$
$227$		−	3.40417i		−	0.225943i	0.993598	−	0.112971i	$-0.0360369\pi$
$228$		0			0
$229$		−	28.5109i		−	1.88405i	−0.335539	−	0.942026i	$-0.608918\pi$
$229$		−	28.5109i		−	1.88405i	0.335539	−	0.942026i	$-0.391082\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−11.7391			−0.769053			−0.384526	−	0.923114i	$-0.625635\pi$
$233$	−11.7391			−0.769053			−0.384526	+	0.923114i	$0.625635\pi$
$234$		0			0
$235$	18.5909	+	6.40101i	1.21274	+	0.417556i
$236$		0			0
$237$		0			0
$238$		0			0
$239$		−	8.68841i		−	0.562006i	−0.959707	−	0.281003i	$-0.909333\pi$
$239$		−	8.68841i		−	0.562006i	0.959707	−	0.281003i	$-0.0906672\pi$
$240$		0			0
$241$			10.7916i			0.695148i	0.937653	+	0.347574i	$0.112994\pi$
$241$			10.7916i			0.695148i	−0.937653	+	0.347574i	$0.887006\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	9.13067	+	12.7134i	0.583337	+	0.812230i
$246$		0			0
$247$		−	1.71833i		−	0.109335i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−28.2397			−1.78248			−0.891238	−	0.453536i	$-0.850162\pi$
$251$	−28.2397			−1.78248			−0.891238	+	0.453536i	$0.850162\pi$
$252$		0			0
$253$			7.66734i			0.482042i
$254$		0			0
$255$		0			0
$256$		0			0
$257$		−	8.79209i		−	0.548435i	−0.961668	−	0.274218i	$-0.911581\pi$
$257$		−	8.79209i		−	0.548435i	0.961668	−	0.274218i	$-0.0884189\pi$
$258$		0			0
$259$	−22.2508	+	11.5053i	−1.38259	+	0.714904i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	15.1257			0.932693			0.466347	−	0.884602i	$-0.345570\pi$
$263$	15.1257			0.932693			0.466347	+	0.884602i	$0.345570\pi$
$264$		0			0
$265$	−10.2058	+	29.6413i	−0.626935	+	1.82085i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−12.4992			−0.762090			−0.381045	−	0.924556i	$-0.624436\pi$
$269$	−12.4992			−0.762090			−0.381045	+	0.924556i	$0.624436\pi$
$270$		0			0
$271$		−	32.3482i		−	1.96501i	−0.186225	−	0.982507i	$-0.559625\pi$
$271$		−	32.3482i		−	1.96501i	0.186225	−	0.982507i	$-0.440375\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	8.97291	−	11.4856i	0.541087	−	0.692607i
$276$		0			0
$277$		−	8.57637i		−	0.515304i	−0.966238	−	0.257652i	$-0.917051\pi$
$277$		−	8.57637i		−	0.515304i	0.966238	−	0.257652i	$-0.0829488\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			2.46377i			0.146976i	0.997296	+	0.0734882i	$0.0234131\pi$
$281$			2.46377i			0.146976i	−0.997296	+	0.0734882i	$0.976587\pi$
$282$		0			0
$283$	11.9819			0.712253			0.356126	−	0.934438i	$-0.384097\pi$
$283$	11.9819			0.712253			0.356126	+	0.934438i	$0.384097\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−0.693332	−	1.34087i	−0.0409261	−	0.0791493i
$288$		0			0
$289$	−38.2115			−2.24774
$290$		0			0
$291$		0			0
$292$		0			0
$293$		−	9.41323i		−	0.549927i	−0.961455	−	0.274964i	$-0.911334\pi$
$293$		−	9.41323i		−	0.549927i	0.961455	−	0.274964i	$-0.0886658\pi$
$294$		0			0
$295$	−4.77153	+	13.8583i	−0.277809	+	0.806860i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	1.00123			0.0579028
$300$		0			0
$301$	−6.24489	+	3.22907i	−0.359949	+	0.186121i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	5.63155	+	1.93899i	0.322462	+	0.111026i
$306$		0			0
$307$	30.6696			1.75041			0.875204	−	0.483754i	$-0.160727\pi$
$307$	30.6696			1.75041			0.875204	+	0.483754i	$0.160727\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	7.95376			0.451016			0.225508	−	0.974241i	$-0.427596\pi$
$311$	7.95376			0.451016			0.225508	+	0.974241i	$0.427596\pi$
$312$		0			0
$313$	−22.5992			−1.27738			−0.638690	−	0.769464i	$-0.720524\pi$
$313$	−22.5992			−1.27738			−0.638690	+	0.769464i	$0.720524\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−9.33670			−0.524401			−0.262201	−	0.965013i	$-0.584448\pi$
$317$	−9.33670			−0.524401			−0.262201	+	0.965013i	$0.584448\pi$
$318$		0			0
$319$	−24.6184			−1.37837
$320$		0			0
$321$		0			0
$322$		0			0
$323$	33.5422			1.86634
$324$		0			0
$325$	−1.49983	−	1.17172i	−0.0831959	−	0.0649953i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−20.6652	+	10.6855i	−1.13931	+	0.589109i
$330$		0			0
$331$	16.6878			0.917246			0.458623	−	0.888631i	$-0.348343\pi$
$331$	16.6878			0.917246			0.458623	+	0.888631i	$0.348343\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−18.7727	−	6.46361i	−1.02566	−	0.353145i
$336$		0			0
$337$		−	3.50510i		−	0.190935i	−0.995433	−	0.0954675i	$-0.969565\pi$
$337$		−	3.50510i		−	0.190935i	0.995433	−	0.0954675i	$-0.0304346\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	16.9059			0.915505
$342$		0			0
$343$	−18.3412	−	2.56888i	−0.990334	−	0.138707i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−13.2776			−0.712777			−0.356389	−	0.934338i	$-0.615992\pi$
$347$	−13.2776			−0.712777			−0.356389	+	0.934338i	$0.615992\pi$
$348$		0			0
$349$		−	4.34253i		−	0.232450i	−0.993223	−	0.116225i	$-0.962921\pi$
$349$		−	4.34253i		−	0.232450i	0.993223	−	0.116225i	$-0.0370794\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		−	1.80553i		−	0.0960985i	−0.998845	−	0.0480493i	$-0.984700\pi$
$353$		−	1.80553i		−	0.0960985i	0.998845	−	0.0480493i	$-0.0153004\pi$
$354$		0			0
$355$	−28.7549	−	9.90057i	−1.52615	−	0.525468i
$356$		0			0
$357$		0			0
$358$		0			0
$359$			19.9995i			1.05554i	0.849389	+	0.527768i	$0.176971\pi$
$359$			19.9995i			1.05554i	−0.849389	+	0.527768i	$0.823029\pi$
$360$		0			0
$361$	−1.37758			−0.0725040
$362$		0			0
$363$		0			0
$364$		0			0
$365$	4.77254	−	13.8612i	0.249806	−	0.725529i
$366$		0			0
$367$	6.99757			0.365270			0.182635	−	0.983181i	$-0.441537\pi$
$367$	6.99757			0.365270			0.182635	+	0.983181i	$0.441537\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−17.0369	−	32.9486i	−0.884511	−	1.71061i
$372$		0			0
$373$			1.32379i			0.0685430i	0.999413	+	0.0342715i	$0.0109111\pi$
$373$			1.32379i			0.0685430i	−0.999413	+	0.0342715i	$0.989089\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			3.21478i			0.165569i
$378$		0			0
$379$	−0.731204			−0.0375594			−0.0187797	−	0.999824i	$-0.505978\pi$
$379$	−0.731204			−0.0375594			−0.0187797	+	0.999824i	$0.505978\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		−	19.9628i		−	1.02005i	−0.860159	−	0.510025i	$-0.829636\pi$
$383$		−	19.9628i		−	1.02005i	0.860159	−	0.510025i	$-0.170364\pi$
$384$		0			0
$385$	2.50237	+	17.0629i	0.127533	+	0.869606i
$386$		0			0
$387$		0			0
$388$		0			0
$389$			7.52746i			0.381658i	0.981623	+	0.190829i	$0.0611175\pi$
$389$			7.52746i			0.381658i	−0.981623	+	0.190829i	$0.938882\pi$
$390$		0			0
$391$			19.5443i			0.988398i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	4.90800	−	14.2546i	0.246948	−	0.717228i
$396$		0			0
$397$	15.9313			0.799571			0.399786	−	0.916609i	$-0.369085\pi$
$397$	15.9313			0.799571			0.399786	+	0.916609i	$0.369085\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		−	27.0471i		−	1.35067i	−0.737512	−	0.675334i	$-0.763999\pi$
$401$		−	27.0471i		−	1.35067i	0.737512	−	0.675334i	$-0.236001\pi$
$402$		0			0
$403$		−	2.20764i		−	0.109970i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−27.5986			−1.36801
$408$		0			0
$409$			28.4881i			1.40865i	0.709880	+	0.704323i	$0.248749\pi$
$409$			28.4881i			1.40865i	−0.709880	+	0.704323i	$0.751251\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−7.96531	−	15.4046i	−0.391947	−	0.758009i
$414$		0			0
$415$	32.9288	+	11.3377i	1.61641	+	0.556546i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−7.88798			−0.385353			−0.192676	−	0.981262i	$-0.561717\pi$
$419$	−7.88798			−0.385353			−0.192676	+	0.981262i	$0.561717\pi$
$420$		0			0
$421$	4.92995			0.240271			0.120136	−	0.992757i	$-0.461667\pi$
$421$	4.92995			0.240271			0.120136	+	0.992757i	$0.461667\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	22.8722	−	29.2771i	1.10947	−	1.42015i
$426$		0			0
$427$	−6.25991	+	3.23684i	−0.302938	+	0.156642i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	32.7948i		−	1.57967i	−0.613321	−	0.789834i	$-0.710167\pi$
$431$		−	32.7948i		−	1.57967i	0.613321	−	0.789834i	$-0.289833\pi$
$432$		0			0
$433$	18.0271			0.866329			0.433164	−	0.901315i	$-0.357397\pi$
$433$	18.0271			0.866329			0.433164	+	0.901315i	$0.357397\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		−	11.8736i		−	0.567990i
$438$		0			0
$439$		−	18.5316i		−	0.884466i	−0.896900	−	0.442233i	$-0.854186\pi$
$439$		−	18.5316i		−	0.884466i	0.896900	−	0.442233i	$-0.145814\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	17.4637			0.829725			0.414863	−	0.909884i	$-0.363830\pi$
$443$	17.4637			0.829725			0.414863	+	0.909884i	$0.363830\pi$
$444$		0			0
$445$	12.6962	−	36.8744i	0.601858	−	1.74801i
$446$		0			0
$447$		0			0
$448$		0			0
$449$			19.6999i			0.929697i	0.885390	+	0.464849i	$0.153891\pi$
$449$			19.6999i			0.929697i	−0.885390	+	0.464849i	$0.846109\pi$
$450$		0			0
$451$		−	1.66314i		−	0.0783144i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	2.22814	−	0.326770i	0.104457	−	0.0153192i
$456$		0			0
$457$		−	11.9106i		−	0.557156i	−0.960414	−	0.278578i	$-0.910137\pi$
$457$		−	11.9106i		−	0.557156i	0.960414	−	0.278578i	$-0.0898631\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−14.6443			−0.682054			−0.341027	−	0.940053i	$-0.610775\pi$
$461$	−14.6443			−0.682054			−0.341027	+	0.940053i	$0.610775\pi$
$462$		0			0
$463$		−	38.2633i		−	1.77824i	−0.457670	−	0.889122i	$-0.651316\pi$
$463$		−	38.2633i		−	1.77824i	0.457670	−	0.889122i	$-0.348684\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		−	6.89652i		−	0.319133i	−0.987187	−	0.159567i	$-0.948990\pi$
$467$		−	6.89652i		−	0.319133i	0.987187	−	0.159567i	$-0.0510096\pi$
$468$		0			0
$469$	20.8673	−	10.7900i	0.963565	−	0.498235i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−7.74580			−0.356152
$474$		0			0
$475$	−13.8954	+	17.7865i	−0.637563	+	0.816099i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−8.44896			−0.386043			−0.193021	−	0.981195i	$-0.561829\pi$
$479$	−8.44896			−0.386043			−0.193021	+	0.981195i	$0.561829\pi$
$480$		0			0
$481$			3.60393i			0.164325i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	5.01468	−	14.5645i	0.227705	−	0.661338i
$486$		0			0
$487$		−	5.45273i		−	0.247087i	−0.992339	−	0.123543i	$-0.960574\pi$
$487$		−	5.45273i		−	0.247087i	0.992339	−	0.123543i	$-0.0394258\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$			29.7696i			1.34348i	0.740785	+	0.671742i	$0.234454\pi$
$491$			29.7696i			1.34348i	−0.740785	+	0.671742i	$0.765546\pi$
$492$		0			0
$493$	−62.7532			−2.82626
$494$		0			0
$495$		0			0
$496$		0			0
$497$	31.9633	−	16.5274i	1.43375	−	0.741356i
$498$		0			0
$499$	−24.1903			−1.08291			−0.541453	−	0.840731i	$-0.682126\pi$
$499$	−24.1903			−1.08291			−0.541453	+	0.840731i	$0.682126\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$			1.95572i			0.0872013i	0.999049	+	0.0436006i	$0.0138829\pi$
$503$			1.95572i			0.0872013i	−0.999049	+	0.0436006i	$0.986117\pi$
$504$		0			0
$505$	6.26131	−	18.1851i	0.278625	−	0.809228i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	9.78122			0.433545			0.216773	−	0.976222i	$-0.430447\pi$
$509$	9.78122			0.433545			0.216773	+	0.976222i	$0.430447\pi$
$510$		0			0
$511$	7.96700	+	15.4078i	0.352439	+	0.681603i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	7.29910	−	21.1993i	0.321637	−	0.934152i
$516$		0			0
$517$	−25.6320			−1.12729
$518$		0			0
$519$		0			0
$520$		0			0
$521$	32.2715			1.41384			0.706920	−	0.707293i	$-0.250084\pi$
$521$	32.2715			1.41384			0.706920	+	0.707293i	$0.250084\pi$
$522$		0			0
$523$	−3.86642			−0.169067			−0.0845335	−	0.996421i	$-0.526940\pi$
$523$	−3.86642			−0.169067			−0.0845335	+	0.996421i	$0.526940\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	43.0937			1.87719
$528$		0			0
$529$	−16.0815			−0.699196
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−0.217180			−0.00940712
$534$		0			0
$535$	−5.32378	+	15.4622i	−0.230167	+	0.668490i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−16.6502	−	11.7957i	−0.717174	−	0.508076i
$540$		0			0
$541$	23.1434			0.995013			0.497506	−	0.867460i	$-0.334249\pi$
$541$	23.1434			0.995013			0.497506	+	0.867460i	$0.334249\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−6.71994	+	19.5172i	−0.287851	+	0.836024i
$546$		0			0
$547$		−	32.3453i		−	1.38298i	−0.722384	−	0.691492i	$-0.756954\pi$
$547$		−	32.3453i		−	1.38298i	0.722384	−	0.691492i	$-0.243046\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	38.1239			1.62413
$552$		0			0
$553$	8.19313	+	15.8452i	0.348407	+	0.673805i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−14.1096			−0.597845			−0.298922	−	0.954277i	$-0.596627\pi$
$557$	−14.1096			−0.597845			−0.298922	+	0.954277i	$0.596627\pi$
$558$		0			0
$559$			1.01148i			0.0427810i
$560$		0			0
$561$		0			0
$562$		0			0
$563$		−	24.3636i		−	1.02680i	−0.858149	−	0.513401i	$-0.828385\pi$
$563$		−	24.3636i		−	1.02680i	0.858149	−	0.513401i	$-0.171615\pi$
$564$		0			0
$565$	4.53366	−	13.1674i	0.190733	−	0.553957i
$566$		0			0
$567$		0			0
$568$		0			0
$569$			39.9769i			1.67592i	0.545734	+	0.837958i	$0.316251\pi$
$569$			39.9769i			1.67592i	−0.545734	+	0.837958i	$0.683749\pi$
$570$		0			0
$571$	−8.49964			−0.355699			−0.177849	−	0.984058i	$-0.556914\pi$
$571$	−8.49964			−0.355699			−0.177849	+	0.984058i	$0.556914\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−10.3638	−	8.09653i	−0.432200	−	0.337649i
$576$		0			0
$577$	−5.76184			−0.239868			−0.119934	−	0.992782i	$-0.538268\pi$
$577$	−5.76184			−0.239868			−0.119934	+	0.992782i	$0.538268\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−36.6030	+	18.9265i	−1.51855	+	0.785203i
$582$		0			0
$583$		−	40.8676i		−	1.69256i
$584$		0			0
$585$		0			0
$586$		0			0
$587$		−	38.3114i		−	1.58128i	−0.612280	−	0.790641i	$-0.709748\pi$
$587$		−	38.3114i		−	1.58128i	0.612280	−	0.790641i	$-0.290252\pi$
$588$		0			0
$589$	−26.1803			−1.07874
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	40.7397i		−	1.67298i	−0.547983	−	0.836490i	$-0.684604\pi$
$593$		−	40.7397i		−	1.67298i	0.547983	−	0.836490i	$-0.315396\pi$
$594$		0			0
$595$	6.37862	+	43.4939i	0.261498	+	1.78307i
$596$		0			0
$597$		0			0
$598$		0			0
$599$			16.0028i			0.653858i	0.945049	+	0.326929i	$0.106014\pi$
$599$			16.0028i			0.653858i	−0.945049	+	0.326929i	$0.893986\pi$
$600$		0			0
$601$			16.5524i			0.675187i	0.941292	+	0.337593i	$0.109613\pi$
$601$			16.5524i			0.675187i	−0.941292	+	0.337593i	$0.890387\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	1.82188	−	5.29142i	0.0740701	−	0.215127i
$606$		0			0
$607$	−34.0003			−1.38003			−0.690015	−	0.723795i	$-0.742396\pi$
$607$	−34.0003			−1.38003			−0.690015	+	0.723795i	$0.742396\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$			3.34713i			0.135410i
$612$		0			0
$613$			34.1387i			1.37885i	0.724357	+	0.689425i	$0.242137\pi$
$613$			34.1387i			1.37885i	−0.724357	+	0.689425i	$0.757863\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−1.00333			−0.0403924			−0.0201962	−	0.999796i	$-0.506429\pi$
$617$	−1.00333			−0.0403924			−0.0201962	+	0.999796i	$0.506429\pi$
$618$		0			0
$619$		−	45.9341i		−	1.84625i	−0.384502	−	0.923124i	$-0.625627\pi$
$619$		−	45.9341i		−	1.84625i	0.384502	−	0.923124i	$-0.374373\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	21.1943	+	40.9888i	0.849131	+	1.64218i
$624$		0			0
$625$	6.04964	+	24.2570i	0.241986	+	0.970280i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−70.3496			−2.80502
$630$		0			0
$631$	−2.40734			−0.0958345			−0.0479173	−	0.998851i	$-0.515258\pi$
$631$	−2.40734			−0.0958345			−0.0479173	+	0.998851i	$0.515258\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	32.6714	+	11.2491i	1.29652	+	0.446405i
$636$		0			0
$637$	−1.54033	+	2.17425i	−0.0610300	+	0.0861469i
$638$		0			0
$639$		0			0
$640$		0			0
$641$			2.11745i			0.0836344i	0.999125	+	0.0418172i	$0.0133147\pi$
$641$			2.11745i			0.0836344i	−0.999125	+	0.0418172i	$0.986685\pi$
$642$		0			0
$643$	4.94437			0.194987			0.0974936	−	0.995236i	$-0.468917\pi$
$643$	4.94437			0.194987			0.0974936	+	0.995236i	$0.468917\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	43.9055i		−	1.72610i	−0.505115	−	0.863052i	$-0.668550\pi$
$647$		−	43.9055i		−	1.72610i	0.505115	−	0.863052i	$-0.331450\pi$
$648$		0			0
$649$		−	19.1069i		−	0.750013i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−11.8879			−0.465209			−0.232605	−	0.972571i	$-0.574725\pi$
$653$	−11.8879			−0.465209			−0.232605	+	0.972571i	$0.574725\pi$
$654$		0			0
$655$	8.79671	−	25.5489i	0.343716	−	0.998277i
$656$		0			0
$657$		0			0
$658$		0			0
$659$			22.2413i			0.866397i	0.901299	+	0.433198i	$0.142615\pi$
$659$			22.2413i			0.866397i	−0.901299	+	0.433198i	$0.857385\pi$
$660$		0			0
$661$		−	15.6615i		−	0.609161i	−0.952487	−	0.304581i	$-0.901484\pi$
$661$		−	15.6615i		−	0.609161i	0.952487	−	0.304581i	$-0.0985163\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−3.87515	−	26.4234i	−0.150272	−	1.02466i
$666$		0			0
$667$			22.2140i			0.860128i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−7.76444			−0.299743
$672$		0			0
$673$			0.600115i			0.0231327i	0.999933	+	0.0115664i	$0.00368177\pi$
$673$			0.600115i			0.0231327i	−0.999933	+	0.0115664i	$0.996318\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		−	37.1869i		−	1.42921i	−0.699529	−	0.714604i	$-0.746607\pi$
$677$		−	37.1869i		−	1.42921i	0.699529	−	0.714604i	$-0.253393\pi$
$678$		0			0
$679$	8.37120	+	16.1896i	0.321257	+	0.621298i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	28.9172			1.10648			0.553242	−	0.833020i	$-0.313390\pi$
$683$	28.9172			1.10648			0.553242	+	0.833020i	$0.313390\pi$
$684$		0			0
$685$	7.87363	−	22.8679i	0.300836	−	0.873737i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−5.33666			−0.203310
$690$		0			0
$691$			25.4317i			0.967469i	0.875215	+	0.483734i	$0.160720\pi$
$691$			25.4317i			0.967469i	−0.875215	+	0.483734i	$0.839280\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−34.7037	−	11.9488i	−1.31639	−	0.453243i
$696$		0			0
$697$		−	4.23941i		−	0.160579i
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	31.9972i		−	1.20852i	−0.796788	−	0.604259i	$-0.793469\pi$
$701$		−	31.9972i		−	1.20852i	0.796788	−	0.604259i	$-0.206531\pi$
$702$		0			0
$703$	42.7389			1.61193
$704$		0			0
$705$		0			0
$706$		0			0
$707$	10.4523	+	20.2142i	0.393098	+	0.760234i
$708$		0			0
$709$	−20.9491			−0.786759			−0.393380	−	0.919376i	$-0.628694\pi$
$709$	−20.9491			−0.786759			−0.393380	+	0.919376i	$0.628694\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		−	15.2547i		−	0.571293i
$714$		0			0
$715$	2.34600	+	0.807748i	0.0877353	+	0.0302081i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	22.4003			0.835389			0.417695	−	0.908587i	$-0.362838\pi$
$719$	22.4003			0.835389			0.417695	+	0.908587i	$0.362838\pi$
$720$		0			0
$721$	12.1847	+	23.5647i	0.453782	+	0.877594i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	25.9965	−	33.2762i	0.965485	−	1.23585i
$726$		0			0
$727$	−50.5372			−1.87432			−0.937160	−	0.348901i	$-0.886555\pi$
$727$	−50.5372			−1.87432			−0.937160	+	0.348901i	$0.886555\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−19.7443			−0.730269
$732$		0			0
$733$	−14.4051			−0.532065			−0.266033	−	0.963964i	$-0.585713\pi$
$733$	−14.4051			−0.532065			−0.266033	+	0.963964i	$0.585713\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	25.8827			0.953400
$738$		0			0
$739$	−3.14271			−0.115606			−0.0578032	−	0.998328i	$-0.518410\pi$
$739$	−3.14271			−0.115606			−0.0578032	+	0.998328i	$0.518410\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	0.382957			0.0140493			0.00702466	−	0.999975i	$-0.497764\pi$
$743$	0.382957			0.0140493			0.00702466	+	0.999975i	$0.497764\pi$
$744$		0			0
$745$	−5.70644	−	1.96478i	−0.209068	−	0.0719839i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−8.88721	−	17.1875i	−0.324732	−	0.628017i
$750$		0			0
$751$	−39.5743			−1.44409			−0.722043	−	0.691848i	$-0.756797\pi$
$751$	−39.5743			−1.44409			−0.722043	+	0.691848i	$0.756797\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	4.23757	−	12.3075i	0.154221	−	0.447914i
$756$		0			0
$757$		−	2.61073i		−	0.0948885i	−0.998874	−	0.0474442i	$-0.984892\pi$
$757$		−	2.61073i		−	0.0948885i	0.998874	−	0.0474442i	$-0.0151076\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	12.9132			0.468103			0.234051	−	0.972224i	$-0.424802\pi$
$761$	12.9132			0.468103			0.234051	+	0.972224i	$0.424802\pi$
$762$		0			0
$763$	−11.2179	−	21.6949i	−0.406114	−	0.785407i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−2.49506			−0.0900915
$768$		0			0
$769$			31.5685i			1.13839i	0.822203	+	0.569195i	$0.192745\pi$
$769$			31.5685i			1.13839i	−0.822203	+	0.569195i	$0.807255\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$			20.1893i			0.726160i	0.931758	+	0.363080i	$0.118275\pi$
$773$			20.1893i			0.726160i	−0.931758	+	0.363080i	$0.881725\pi$
$774$		0			0
$775$	−17.8522	+	22.8513i	−0.641271	+	0.820845i
$776$		0			0
$777$		0			0
$778$		0			0
$779$			2.57553i			0.0922779i
$780$		0			0
$781$	39.6455			1.41863
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−4.07610	+	11.8385i	−0.145482	+	0.422534i
$786$		0			0
$787$	−0.947805			−0.0337856			−0.0168928	−	0.999857i	$-0.505377\pi$
$787$	−0.947805			−0.0337856			−0.0168928	+	0.999857i	$0.505377\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	7.56823	+	14.6366i	0.269095	+	0.520419i
$792$		0			0
$793$			1.01391i			0.0360051i
$794$		0			0
$795$		0			0
$796$		0			0
$797$			39.7605i			1.40839i	0.710007	+	0.704195i	$0.248692\pi$
$797$			39.7605i			1.40839i	−0.710007	+	0.704195i	$0.751308\pi$
$798$		0			0
$799$	−65.3367			−2.31145
$800$		0			0
$801$		0			0
$802$		0			0
$803$			19.1110i			0.674413i
$804$		0			0
$805$	15.3964	−	2.25797i	0.542651	−	0.0795828i
$806$		0			0
$807$		0			0
$808$		0			0
$809$			12.9491i			0.455265i	0.973747	+	0.227632i	$0.0730985\pi$
$809$			12.9491i			0.455265i	−0.973747	+	0.227632i	$0.926902\pi$
$810$		0			0
$811$			26.7060i			0.937775i	0.883258	+	0.468888i	$0.155345\pi$
$811$			26.7060i			0.937775i	−0.883258	+	0.468888i	$0.844655\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−14.2066	−	4.89147i	−0.497637	−	0.171341i
$816$		0			0
$817$	11.9951			0.419654
$818$		0			0
$819$		0			0
$820$		0			0
$821$		−	38.9067i		−	1.35785i	−0.734205	−	0.678927i	$-0.762445\pi$
$821$		−	38.9067i		−	1.35785i	0.734205	−	0.678927i	$-0.237555\pi$
$822$		0			0
$823$			44.0024i			1.53383i	0.641750	+	0.766914i	$0.278209\pi$
$823$			44.0024i			1.53383i	−0.641750	+	0.766914i	$0.721791\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	6.87464			0.239055			0.119527	−	0.992831i	$-0.461862\pi$
$827$	6.87464			0.239055			0.119527	+	0.992831i	$0.461862\pi$
$828$		0			0
$829$		−	45.2222i		−	1.57063i	−0.619094	−	0.785317i	$-0.712500\pi$
$829$		−	45.2222i		−	1.57063i	0.619094	−	0.785317i	$-0.287500\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−42.4418	−	30.0675i	−1.47052	−	1.04178i
$834$		0			0
$835$	9.34729	+	3.21836i	0.323476	+	0.111376i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−50.2238			−1.73392			−0.866959	−	0.498380i	$-0.833928\pi$
$839$	−50.2238			−1.73392			−0.866959	+	0.498380i	$0.833928\pi$
$840$		0			0
$841$	−42.3250			−1.45948
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−9.35797	+	27.1790i	−0.321924	+	0.934985i
$846$		0			0
$847$	3.04134	+	5.88183i	0.104502	+	0.202102i
$848$		0			0
$849$		0			0
$850$		0			0
$851$			24.9030i			0.853665i
$852$		0			0
$853$	1.07993			0.0369762			0.0184881	−	0.999829i	$-0.494115\pi$
$853$	1.07993			0.0369762			0.0184881	+	0.999829i	$0.494115\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$			43.2118i			1.47609i	0.674753	+	0.738044i	$0.264250\pi$
$857$			43.2118i			1.47609i	−0.674753	+	0.738044i	$0.735750\pi$
$858$		0			0
$859$		−	22.0452i		−	0.752171i	−0.926585	−	0.376085i	$-0.877270\pi$
$859$		−	22.0452i		−	0.752171i	0.926585	−	0.376085i	$-0.122730\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−20.5422			−0.699264			−0.349632	−	0.936887i	$-0.613693\pi$
$863$	−20.5422			−0.699264			−0.349632	+	0.936887i	$0.613693\pi$
$864$		0			0
$865$	6.50619	+	2.24014i	0.221217	+	0.0761671i
$866$		0			0
$867$		0			0
$868$		0			0
$869$			19.6534i			0.666697i
$870$		0			0
$871$		−	3.37987i		−	0.114522i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−25.7060	−	14.6356i	−0.869022	−	0.494774i
$876$		0			0
$877$		−	32.1846i		−	1.08680i	−0.839475	−	0.543398i	$-0.817137\pi$
$877$		−	32.1846i		−	1.08680i	0.839475	−	0.543398i	$-0.182863\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−0.393359			−0.0132526			−0.00662630	−	0.999978i	$-0.502109\pi$
$881$	−0.393359			−0.0132526			−0.00662630	+	0.999978i	$0.502109\pi$
$882$		0			0
$883$			16.7863i			0.564902i	0.959282	+	0.282451i	$0.0911475\pi$
$883$			16.7863i			0.564902i	−0.959282	+	0.282451i	$0.908852\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		−	26.7032i		−	0.896604i	−0.893882	−	0.448302i	$-0.852029\pi$
$887$		−	26.7032i		−	0.896604i	0.893882	−	0.448302i	$-0.147971\pi$
$888$		0			0
$889$	−36.3168	+	18.7785i	−1.21803	+	0.629811i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	39.6934			1.32829
$894$		0			0
$895$	32.0185	+	11.0243i	1.07026	+	0.368500i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	48.9801			1.63358
$900$		0			0
$901$		−	104.173i		−	3.47050i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	34.7611	+	11.9686i	1.15550	+	0.397848i
$906$		0			0
$907$			41.9341i			1.39240i	0.717849	+	0.696199i	$0.245127\pi$
$907$			41.9341i			1.39240i	−0.717849	+	0.696199i	$0.754873\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	32.4562i		−	1.07532i	−0.843161	−	0.537661i	$-0.819308\pi$
$911$		−	32.4562i		−	1.07532i	0.843161	−	0.537661i	$-0.180692\pi$
$912$		0			0
$913$	−45.4003			−1.50253
$914$		0			0
$915$		0			0
$916$		0			0
$917$	14.6847	+	28.3996i	0.484932	+	0.937838i
$918$		0			0
$919$	−18.7809			−0.619524			−0.309762	−	0.950814i	$-0.600249\pi$
$919$	−18.7809			−0.619524			−0.309762	+	0.950814i	$0.600249\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		−	5.17707i		−	0.170405i
$924$		0			0
$925$	29.1434	−	37.3044i	0.958229	−	1.22656i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	34.3962			1.12850			0.564251	−	0.825603i	$-0.309165\pi$
$929$	34.3962			1.12850			0.564251	+	0.825603i	$0.309165\pi$
$930$		0			0
$931$	25.7843	+	18.2667i	0.845046	+	0.598666i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−15.7674	+	45.7944i	−0.515650	+	1.49764i
$936$		0			0
$937$	15.0275			0.490926			0.245463	−	0.969406i	$-0.421060\pi$
$937$	15.0275			0.490926			0.245463	+	0.969406i	$0.421060\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−38.7674			−1.26378			−0.631890	−	0.775058i	$-0.717721\pi$
$941$	−38.7674			−1.26378			−0.631890	+	0.775058i	$0.717721\pi$
$942$		0			0
$943$	−1.50071			−0.0488697
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−35.8143			−1.16381			−0.581904	−	0.813258i	$-0.697692\pi$
$947$	−35.8143			−1.16381			−0.581904	+	0.813258i	$0.697692\pi$
$948$		0			0
$949$	2.49559			0.0810104
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−36.2967			−1.17577			−0.587883	−	0.808946i	$-0.700038\pi$
$953$	−36.2967			−1.17577			−0.587883	+	0.808946i	$0.700038\pi$
$954$		0			0
$955$	22.8627	+	7.87184i	0.739820	+	0.254727i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	13.1438	+	25.4195i	0.424434	+	0.820838i
$960$		0			0
$961$	−2.63542			−0.0850137
$962$		0			0
$963$		0			0
$964$		0			0
$965$	41.5870	+	14.3188i	1.33873	+	0.460938i
$966$		0			0
$967$			35.6623i			1.14682i	0.819268	+	0.573411i	$0.194380\pi$
$967$			35.6623i			1.14682i	−0.819268	+	0.573411i	$0.805620\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	35.8220			1.14958			0.574792	−	0.818300i	$-0.305083\pi$
$971$	35.8220			1.14958			0.574792	+	0.818300i	$0.305083\pi$
$972$		0			0
$973$	38.5759	−	19.9466i	1.23669	−	0.639459i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	17.0336			0.544953			0.272477	−	0.962162i	$-0.412157\pi$
$977$	17.0336			0.544953			0.272477	+	0.962162i	$0.412157\pi$
$978$		0			0
$979$			50.8402i			1.62486i
$980$		0			0
$981$		0			0
$982$		0			0
$983$		−	3.87185i		−	0.123493i	−0.998092	−	0.0617465i	$-0.980333\pi$
$983$		−	3.87185i		−	0.123493i	0.998092	−	0.0617465i	$-0.0196670\pi$
$984$		0			0
$985$	6.10623	−	17.7347i	0.194561	−	0.565076i
$986$		0			0
$987$		0			0
$988$		0			0
$989$			6.98927i			0.222246i
$990$		0			0
$991$	33.6772			1.06979			0.534896	−	0.844918i	$-0.320351\pi$
$991$	33.6772			1.06979			0.534896	+	0.844918i	$0.320351\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−21.0937	−	7.26277i	−0.668716	−	0.230245i
$996$		0			0
$997$	−54.2789			−1.71903			−0.859515	−	0.511110i	$-0.829234\pi$
$997$	−54.2789			−1.71903			−0.859515	+	0.511110i	$0.829234\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2520.2.k.a.1889.15	yes	24
3.2	odd	2		2520.2.k.b.1889.10	yes	24
4.3	odd	2		5040.2.k.i.1889.15		24
5.4	even	2		2520.2.k.b.1889.16	yes	24
7.6	odd	2	inner	2520.2.k.a.1889.10	yes	24
12.11	even	2		5040.2.k.h.1889.10		24
15.14	odd	2	inner	2520.2.k.a.1889.9	✓	24
20.19	odd	2		5040.2.k.h.1889.16		24
21.20	even	2		2520.2.k.b.1889.15	yes	24
28.27	even	2		5040.2.k.i.1889.10		24
35.34	odd	2		2520.2.k.b.1889.9	yes	24
60.59	even	2		5040.2.k.i.1889.9		24
84.83	odd	2		5040.2.k.h.1889.15		24
105.104	even	2	inner	2520.2.k.a.1889.16	yes	24
140.139	even	2		5040.2.k.h.1889.9		24
420.419	odd	2		5040.2.k.i.1889.16		24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2520.2.k.a.1889.9	✓	24	15.14	odd	2	inner
2520.2.k.a.1889.10	yes	24	7.6	odd	2	inner
2520.2.k.a.1889.15	yes	24	1.1	even	1	trivial
2520.2.k.a.1889.16	yes	24	105.104	even	2	inner
2520.2.k.b.1889.9	yes	24	35.34	odd	2
2520.2.k.b.1889.10	yes	24	3.2	odd	2
2520.2.k.b.1889.15	yes	24	21.20	even	2
2520.2.k.b.1889.16	yes	24	5.4	even	2
5040.2.k.h.1889.9		24	140.139	even	2
5040.2.k.h.1889.10		24	12.11	even	2
5040.2.k.h.1889.15		24	84.83	odd	2
5040.2.k.h.1889.16		24	20.19	odd	2
5040.2.k.i.1889.9		24	60.59	even	2
5040.2.k.i.1889.10		24	28.27	even	2
5040.2.k.i.1889.15		24	4.3	odd	2
5040.2.k.i.1889.16		24	420.419	odd	2

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:	\( N \)	\(=\)	\( 2520 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2520.k (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(0.727958 - 2.11426i) q^{5} +(1.21521 + 2.35016i) q^{7} +O(q^{10})\)	\(q+(0.727958 - 2.11426i) q^{5} +(1.21521 + 2.35016i) q^{7} +2.91501i q^{11} +0.380654 q^{13} +7.43045i q^{17} -4.51415i q^{19} +2.63030 q^{23} +(-3.94016 - 3.07818i) q^{25} +8.44541i q^{29} -5.79961i q^{31} +(5.85347 - 0.858444i) q^{35} +9.46775i q^{37} -0.570545 q^{41} +2.65721i q^{43} +8.79311i q^{47} +(-4.04653 + 5.71188i) q^{49} -14.0197 q^{53} +(6.16307 + 2.12200i) q^{55} -6.55468 q^{59} +2.66361i q^{61} +(0.277100 - 0.804799i) q^{65} -8.87911i q^{67} -13.6005i q^{71} +6.55607 q^{73} +(-6.85074 + 3.54234i) q^{77} +6.74215 q^{79} +15.5747i q^{83} +(15.7099 + 5.40905i) q^{85} +17.4408 q^{89} +(0.462574 + 0.894598i) q^{91} +(-9.54407 - 3.28611i) q^{95} +6.88869 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q+O(q^{10}) \) 24 * q	\( 24 q - 8 q^{23} - 16 q^{25} + 4 q^{35} - 12 q^{49} + 24 q^{53} + 8 q^{65} + 4 q^{77} + 40 q^{79} + 24 q^{85} - 36 q^{91} - 48 q^{95}+O(q^{100}) \) 24 * q - 8 * q^23 - 16 * q^25 + 4 * q^35 - 12 * q^49 + 24 * q^53 + 8 * q^65 + 4 * q^77 + 40 * q^79 + 24 * q^85 - 36 * q^91 - 48 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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