LMFDB - Embedded newform 6660.2.f.c.5329.12

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [6660,2,Mod(5329,6660)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(6660, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("6660.5329"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 6660 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 37 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6660.f (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [18,0,0,0,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	no
Analytic conductor:	$53.1803677462$
Analytic rank:	$0$
Dimension:	$18$
Coefficient field:	$\mathbb{Q}[x]/(x^{18} + \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{18} + 26 x^{16} + 283 x^{14} + 1674 x^{12} + 5841 x^{10} + 12196 x^{8} + 14736 x^{6} + 9408 x^{4} + \cdots + 144 $ x^18 + 26x^16 + 283x^14 + 1674x^12 + 5841x^10 + 12196x^8 + 14736x^6 + 9408x^4 + 2592x^2 + 144
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{4}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 740)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			5329.12
Root			$-2.40934i$ of defining polynomial
Character	$\chi$	$=$	6660.5329
Dual form			6660.2.f.c.5329.11

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.672341 + 2.13259i) q^{5} +3.62709i q^{7} -0.391667 q^{11} -2.13212i q^{13} +2.05913i q^{17} +2.97636 q^{19} -7.43110i q^{23} +(-4.09591 + 2.86766i) q^{25} -9.88431 q^{29} +3.27376 q^{31} +(-7.73512 + 2.43864i) q^{35} -1.00000i q^{37} -6.67260 q^{41} +4.73480i q^{43} +10.5345i q^{47} -6.15580 q^{49} +1.34461i q^{53} +(-0.263334 - 0.835266i) q^{55} -13.7792 q^{59} +3.69839 q^{61} +(4.54694 - 1.43351i) q^{65} +2.89753i q^{67} +7.50571 q^{71} +11.0649i q^{73} -1.42061i q^{77} -16.0380 q^{79} +7.88415i q^{83} +(-4.39129 + 1.38444i) q^{85} +9.88390 q^{89} +7.73338 q^{91} +(2.00113 + 6.34737i) q^{95} -5.75855i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 18 q - 2 q^{5} + 4 q^{19} - 2 q^{25} + 4 q^{29} + 8 q^{31} + 2 q^{35} + 4 q^{41} + 6 q^{49} - 6 q^{55} - 8 q^{59} + 12 q^{65} + 24 q^{71} - 24 q^{79} + 36 q^{91} - 24 q^{95}+O(q^{100}) $ 18 * q - 2 * q^5 + 4 * q^19 - 2 * q^25 + 4 * q^29 + 8 * q^31 + 2 * q^35 + 4 * q^41 + 6 * q^49 - 6 * q^55 - 8 * q^59 + 12 * q^65 + 24 * q^71 - 24 * q^79 + 36 * q^91 - 24 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/6660\mathbb{Z}\right)^\times$.

$n$	$3331$	$3701$	$3961$	$3997$
$\chi(n)$	$1$	$1$	$1$	$-1$

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
$5$	0.672341	+	2.13259i	0.300680	+	0.953725i
$5$	0.672341	+	2.13259i	0.300680	+	0.953725i
$6$		0			0
$7$			3.62709i			1.37091i	0.728114	+	0.685456i	$0.240397\pi$
$7$			3.62709i			1.37091i	−0.728114	+	0.685456i	$0.759603\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−0.391667			−0.118092			−0.0590460	−	0.998255i	$-0.518806\pi$
$11$	−0.391667			−0.118092			−0.0590460	+	0.998255i	$0.518806\pi$
$12$		0			0
$13$		−	2.13212i		−	0.591342i	−0.955290	−	0.295671i	$-0.904457\pi$
$13$		−	2.13212i		−	0.591342i	0.955290	−	0.295671i	$-0.0955433\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$			2.05913i			0.499412i	0.968322	+	0.249706i	$0.0803340\pi$
$17$			2.05913i			0.499412i	−0.968322	+	0.249706i	$0.919666\pi$
$18$		0			0
$19$	2.97636			0.682824			0.341412	−	0.939914i	$-0.389095\pi$
$19$	2.97636			0.682824			0.341412	+	0.939914i	$0.389095\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		−	7.43110i		−	1.54949i	−0.632273	−	0.774745i	$-0.717878\pi$
$23$		−	7.43110i		−	1.54949i	0.632273	−	0.774745i	$-0.282122\pi$
$24$		0			0
$25$	−4.09591	+	2.86766i	−0.819183	+	0.573532i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−9.88431			−1.83547			−0.917735	−	0.397193i	$-0.869984\pi$
$29$	−9.88431			−1.83547			−0.917735	+	0.397193i	$0.869984\pi$
$30$		0			0
$31$	3.27376			0.587984			0.293992	−	0.955808i	$-0.405016\pi$
$31$	3.27376			0.587984			0.293992	+	0.955808i	$0.405016\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−7.73512	+	2.43864i	−1.30747	+	0.412206i
$36$		0			0
$37$		−	1.00000i		−	0.164399i
$37$		−	1.00000i		−	0.164399i
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−6.67260			−1.04208			−0.521042	−	0.853531i	$-0.674457\pi$
$41$	−6.67260			−1.04208			−0.521042	+	0.853531i	$0.674457\pi$
$42$		0			0
$43$			4.73480i			0.722050i	0.932556	+	0.361025i	$0.117573\pi$
$43$			4.73480i			0.722050i	−0.932556	+	0.361025i	$0.882427\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$			10.5345i			1.53662i	0.640080	+	0.768308i	$0.278901\pi$
$47$			10.5345i			1.53662i	−0.640080	+	0.768308i	$0.721099\pi$
$48$		0			0
$49$	−6.15580			−0.879401
$50$		0			0
$51$		0			0
$52$		0			0
$53$			1.34461i			0.184697i	0.995727	+	0.0923485i	$0.0294374\pi$
$53$			1.34461i			0.184697i	−0.995727	+	0.0923485i	$0.970563\pi$
$54$		0			0
$55$	−0.263334	−	0.835266i	−0.0355079	−	0.112627i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−13.7792			−1.79390			−0.896952	−	0.442129i	$-0.854223\pi$
$59$	−13.7792			−1.79390			−0.896952	+	0.442129i	$0.854223\pi$
$60$		0			0
$61$	3.69839			0.473530			0.236765	−	0.971567i	$-0.423913\pi$
$61$	3.69839			0.473530			0.236765	+	0.971567i	$0.423913\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	4.54694	−	1.43351i	0.563978	−	0.177805i
$66$		0			0
$67$			2.89753i			0.353990i	0.984212	+	0.176995i	$0.0566376\pi$
$67$			2.89753i			0.353990i	−0.984212	+	0.176995i	$0.943362\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	7.50571			0.890763			0.445382	−	0.895341i	$-0.353068\pi$
$71$	7.50571			0.890763			0.445382	+	0.895341i	$0.353068\pi$
$72$		0			0
$73$			11.0649i			1.29505i	0.762043	+	0.647526i	$0.224196\pi$
$73$			11.0649i			1.29505i	−0.762043	+	0.647526i	$0.775804\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		−	1.42061i		−	0.161894i
$78$		0			0
$79$	−16.0380			−1.80442			−0.902209	−	0.431298i	$-0.858056\pi$
$79$	−16.0380			−1.80442			−0.902209	+	0.431298i	$0.858056\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$			7.88415i			0.865398i	0.901539	+	0.432699i	$0.142439\pi$
$83$			7.88415i			0.865398i	−0.901539	+	0.432699i	$0.857561\pi$
$84$		0			0
$85$	−4.39129	+	1.38444i	−0.476302	+	0.150163i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	9.88390			1.04769			0.523845	−	0.851813i	$-0.324497\pi$
$89$	9.88390			1.04769			0.523845	+	0.851813i	$0.324497\pi$
$90$		0			0
$91$	7.73338			0.810679
$92$		0			0
$93$		0			0
$94$		0			0
$95$	2.00113	+	6.34737i	0.205312	+	0.651226i
$96$		0			0
$97$		−	5.75855i		−	0.584693i	−0.956313	−	0.292346i	$-0.905564\pi$
$97$		−	5.75855i		−	0.584693i	0.956313	−	0.292346i	$-0.0944360\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	6.24151			0.621053			0.310527	−	0.950565i	$-0.399495\pi$
$101$	6.24151			0.621053			0.310527	+	0.950565i	$0.399495\pi$
$102$		0			0
$103$			5.06627i			0.499194i	0.968350	+	0.249597i	$0.0802982\pi$
$103$			5.06627i			0.499194i	−0.968350	+	0.249597i	$0.919702\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	0.922468i		−	0.0891783i	−0.999005	−	0.0445892i	$-0.985802\pi$
$107$		−	0.922468i		−	0.0891783i	0.999005	−	0.0445892i	$-0.0141979\pi$
$108$		0			0
$109$	−18.6311			−1.78454			−0.892269	−	0.451504i	$-0.850888\pi$
$109$	−18.6311			−1.78454			−0.892269	+	0.451504i	$0.850888\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		−	11.9336i		−	1.12262i	−0.827607	−	0.561308i	$-0.810298\pi$
$113$		−	11.9336i		−	1.12262i	0.827607	−	0.561308i	$-0.189702\pi$
$114$		0			0
$115$	15.8475	−	4.99623i	1.47779	−	0.465901i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−7.46865			−0.684650
$120$		0			0
$121$	−10.8466			−0.986054
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−8.86941	−	6.80688i	−0.793304	−	0.608826i
$126$		0			0
$127$		−	18.5885i		−	1.64946i	−0.565523	−	0.824732i	$-0.691326\pi$
$127$		−	18.5885i		−	1.64946i	0.565523	−	0.824732i	$-0.308674\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−6.01448			−0.525487			−0.262744	−	0.964866i	$-0.584627\pi$
$131$	−6.01448			−0.525487			−0.262744	+	0.964866i	$0.584627\pi$
$132$		0			0
$133$			10.7955i			0.936092i
$134$		0			0
$135$		0			0
$136$		0			0
$137$			21.8273i			1.86483i	0.361390	+	0.932415i	$0.382302\pi$
$137$			21.8273i			1.86483i	−0.361390	+	0.932415i	$0.617698\pi$
$138$		0			0
$139$	−6.53555			−0.554338			−0.277169	−	0.960821i	$-0.589396\pi$
$139$	−6.53555			−0.554338			−0.277169	+	0.960821i	$0.589396\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$			0.835079i			0.0698328i
$144$		0			0
$145$	−6.64563	−	21.0792i	−0.551889	−	1.75053i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	15.0837			1.23571			0.617853	−	0.786294i	$-0.288003\pi$
$149$	15.0837			1.23571			0.617853	+	0.786294i	$0.288003\pi$
$150$		0			0
$151$	−9.67034			−0.786961			−0.393480	−	0.919333i	$-0.628729\pi$
$151$	−9.67034			−0.786961			−0.393480	+	0.919333i	$0.628729\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	2.20108	+	6.98159i	0.176795	+	0.560775i
$156$		0			0
$157$		−	7.55190i		−	0.602707i	−0.953512	−	0.301354i	$-0.902562\pi$
$157$		−	7.55190i		−	0.602707i	0.953512	−	0.301354i	$-0.0974384\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	26.9533			2.12422
$162$		0			0
$163$		−	14.5651i		−	1.14083i	−0.821358	−	0.570413i	$-0.806783\pi$
$163$		−	14.5651i		−	1.14083i	0.821358	−	0.570413i	$-0.193217\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	21.4979i		−	1.66356i	−0.555107	−	0.831779i	$-0.687323\pi$
$167$		−	21.4979i		−	1.66356i	0.555107	−	0.831779i	$-0.312677\pi$
$168$		0			0
$169$	8.45408			0.650314
$170$		0			0
$171$		0			0
$172$		0			0
$173$		−	5.52812i		−	0.420295i	−0.977670	−	0.210148i	$-0.932606\pi$
$173$		−	5.52812i		−	0.420295i	0.977670	−	0.210148i	$-0.0673945\pi$
$174$		0			0
$175$	−10.4013	−	14.8563i	−0.786263	−	1.12303i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−19.6456			−1.46838			−0.734190	−	0.678944i	$-0.762438\pi$
$179$	−19.6456			−1.46838			−0.734190	+	0.678944i	$0.762438\pi$
$180$		0			0
$181$	−15.1017			−1.12250			−0.561250	−	0.827646i	$-0.689679\pi$
$181$	−15.1017			−1.12250			−0.561250	+	0.827646i	$0.689679\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	2.13259	−	0.672341i	0.156791	−	0.0494315i
$186$		0			0
$187$		−	0.806493i		−	0.0589766i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−2.66075			−0.192525			−0.0962625	−	0.995356i	$-0.530689\pi$
$191$	−2.66075			−0.192525			−0.0962625	+	0.995356i	$0.530689\pi$
$192$		0			0
$193$			5.69658i			0.410049i	0.978757	+	0.205024i	$0.0657274\pi$
$193$			5.69658i			0.410049i	−0.978757	+	0.205024i	$0.934273\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			7.03316i			0.501092i	0.968105	+	0.250546i	$0.0806101\pi$
$197$			7.03316i			0.501092i	−0.968105	+	0.250546i	$0.919390\pi$
$198$		0			0
$199$	−14.2740			−1.01186			−0.505930	−	0.862575i	$-0.668850\pi$
$199$	−14.2740			−1.01186			−0.505930	+	0.862575i	$0.668850\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		−	35.8513i		−	2.51627i
$204$		0			0
$205$	−4.48626	−	14.2299i	−0.313334	−	0.993862i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−1.16574			−0.0806360
$210$		0			0
$211$	−18.6345			−1.28285			−0.641426	−	0.767185i	$-0.721657\pi$
$211$	−18.6345			−1.28285			−0.641426	+	0.767185i	$0.721657\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−10.0974	+	3.18340i	−0.688637	+	0.217106i
$216$		0			0
$217$			11.8742i			0.806075i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	4.39030			0.295324
$222$		0			0
$223$			1.71850i			0.115079i	0.998343	+	0.0575395i	$0.0183255\pi$
$223$			1.71850i			0.115079i	−0.998343	+	0.0575395i	$0.981674\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$			3.80424i			0.252497i	0.991999	+	0.126248i	$0.0402936\pi$
$227$			3.80424i			0.252497i	−0.991999	+	0.126248i	$0.959706\pi$
$228$		0			0
$229$	13.2332			0.874475			0.437238	−	0.899346i	$-0.355957\pi$
$229$	13.2332			0.874475			0.437238	+	0.899346i	$0.355957\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			4.07618i			0.267040i	0.991046	+	0.133520i	$0.0426280\pi$
$233$			4.07618i			0.267040i	−0.991046	+	0.133520i	$0.957372\pi$
$234$		0			0
$235$	−22.4658	+	7.08278i	−1.46551	+	0.462030i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−5.20603			−0.336750			−0.168375	−	0.985723i	$-0.553852\pi$
$239$	−5.20603			−0.336750			−0.168375	+	0.985723i	$0.553852\pi$
$240$		0			0
$241$	14.7128			0.947732			0.473866	−	0.880597i	$-0.342858\pi$
$241$	14.7128			0.947732			0.473866	+	0.880597i	$0.342858\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−4.13880	−	13.1278i	−0.264418	−	0.838706i
$246$		0			0
$247$		−	6.34594i		−	0.403783i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	2.25931			0.142607			0.0713033	−	0.997455i	$-0.477284\pi$
$251$	2.25931			0.142607			0.0713033	+	0.997455i	$0.477284\pi$
$252$		0			0
$253$			2.91051i			0.182982i
$254$		0			0
$255$		0			0
$256$		0			0
$257$			5.71354i			0.356401i	0.983994	+	0.178200i	$0.0570275\pi$
$257$			5.71354i			0.356401i	−0.983994	+	0.178200i	$0.942973\pi$
$258$		0			0
$259$	3.62709			0.225377
$260$		0			0
$261$		0			0
$262$		0			0
$263$		−	5.44181i		−	0.335556i	−0.985825	−	0.167778i	$-0.946341\pi$
$263$		−	5.44181i		−	0.335556i	0.985825	−	0.167778i	$-0.0536592\pi$
$264$		0			0
$265$	−2.86752	+	0.904039i	−0.176150	+	0.0555347i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	12.7881			0.779703			0.389852	−	0.920878i	$-0.372526\pi$
$269$	12.7881			0.779703			0.389852	+	0.920878i	$0.372526\pi$
$270$		0			0
$271$	11.8653			0.720766			0.360383	−	0.932805i	$-0.382646\pi$
$271$	11.8653			0.720766			0.360383	+	0.932805i	$0.382646\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	1.60423	−	1.12317i	0.0967390	−	0.0677296i
$276$		0			0
$277$			3.92006i			0.235534i	0.993041	+	0.117767i	$0.0375736\pi$
$277$			3.92006i			0.235534i	−0.993041	+	0.117767i	$0.962426\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−1.34526			−0.0802513			−0.0401257	−	0.999195i	$-0.512776\pi$
$281$	−1.34526			−0.0802513			−0.0401257	+	0.999195i	$0.512776\pi$
$282$		0			0
$283$		−	4.26021i		−	0.253243i	−0.991951	−	0.126621i	$-0.959587\pi$
$283$		−	4.26021i		−	0.253243i	0.991951	−	0.126621i	$-0.0404133\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		−	24.2021i		−	1.42861i
$288$		0			0
$289$	12.7600			0.750587
$290$		0			0
$291$		0			0
$292$		0			0
$293$		−	11.2397i		−	0.656631i	−0.944568	−	0.328315i	$-0.893519\pi$
$293$		−	11.2397i		−	0.656631i	0.944568	−	0.328315i	$-0.106481\pi$
$294$		0			0
$295$	−9.26434	−	29.3855i	−0.539391	−	1.71089i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−15.8440			−0.916280
$300$		0			0
$301$	−17.1735			−0.989867
$302$		0			0
$303$		0			0
$304$		0			0
$305$	2.48658	+	7.88716i	0.142381	+	0.451618i
$306$		0			0
$307$		−	5.44256i		−	0.310623i	−0.987866	−	0.155312i	$-0.950362\pi$
$307$		−	5.44256i		−	0.310623i	0.987866	−	0.155312i	$-0.0496381\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	13.7705			0.780856			0.390428	−	0.920633i	$-0.372327\pi$
$311$	13.7705			0.780856			0.390428	+	0.920633i	$0.372327\pi$
$312$		0			0
$313$		−	30.9562i		−	1.74975i	−0.484352	−	0.874873i	$-0.660944\pi$
$313$		−	30.9562i		−	1.74975i	0.484352	−	0.874873i	$-0.339056\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$			25.4057i			1.42693i	0.700692	+	0.713464i	$0.252875\pi$
$317$			25.4057i			1.42693i	−0.700692	+	0.713464i	$0.747125\pi$
$318$		0			0
$319$	3.87136			0.216754
$320$		0			0
$321$		0			0
$322$		0			0
$323$			6.12871i			0.341011i
$324$		0			0
$325$	6.11419	+	8.73296i	0.339154	+	0.484418i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−38.2096			−2.10657
$330$		0			0
$331$	5.42478			0.298173			0.149086	−	0.988824i	$-0.452367\pi$
$331$	5.42478			0.298173			0.149086	+	0.988824i	$0.452367\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−6.17926	+	1.94813i	−0.337609	+	0.106438i
$336$		0			0
$337$		−	24.1817i		−	1.31726i	−0.752466	−	0.658631i	$-0.771136\pi$
$337$		−	24.1817i		−	1.31726i	0.752466	−	0.658631i	$-0.228864\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−1.28222			−0.0694362
$342$		0			0
$343$			3.06198i			0.165331i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		−	11.7883i		−	0.632831i	−0.948621	−	0.316416i	$-0.897521\pi$
$347$		−	11.7883i		−	0.632831i	0.948621	−	0.316416i	$-0.102479\pi$
$348$		0			0
$349$	19.9490			1.06784			0.533922	−	0.845533i	$-0.320717\pi$
$349$	19.9490			1.06784			0.533922	+	0.845533i	$0.320717\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		−	24.5123i		−	1.30466i	−0.757936	−	0.652329i	$-0.773792\pi$
$353$		−	24.5123i		−	1.30466i	0.757936	−	0.652329i	$-0.226208\pi$
$354$		0			0
$355$	5.04639	+	16.0066i	0.267835	+	0.849543i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	26.9812			1.42401			0.712007	−	0.702172i	$-0.247786\pi$
$359$	26.9812			1.42401			0.712007	+	0.702172i	$0.247786\pi$
$360$		0			0
$361$	−10.1413			−0.533751
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−23.5970	+	7.43941i	−1.23512	+	0.389396i
$366$		0			0
$367$			0.444307i			0.0231927i	0.999933	+	0.0115963i	$0.00369131\pi$
$367$			0.444307i			0.0231927i	−0.999933	+	0.0115963i	$0.996309\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−4.87704			−0.253203
$372$		0			0
$373$			25.5341i			1.32211i	0.750339	+	0.661053i	$0.229890\pi$
$373$			25.5341i			1.32211i	−0.750339	+	0.661053i	$0.770110\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			21.0745i			1.08539i
$378$		0			0
$379$	12.3552			0.634644			0.317322	−	0.948318i	$-0.397216\pi$
$379$	12.3552			0.634644			0.317322	+	0.948318i	$0.397216\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$			26.3246i			1.34512i	0.740041	+	0.672562i	$0.234806\pi$
$383$			26.3246i			1.34512i	−0.740041	+	0.672562i	$0.765194\pi$
$384$		0			0
$385$	3.02959	−	0.955136i	0.154402	−	0.0486782i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	14.6564			0.743111			0.371555	−	0.928411i	$-0.378825\pi$
$389$	14.6564			0.743111			0.371555	+	0.928411i	$0.378825\pi$
$390$		0			0
$391$	15.3016			0.773835
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−10.7830	−	34.2026i	−0.542553	−	1.72092i
$396$		0			0
$397$			30.0764i			1.50949i	0.656019	+	0.754745i	$0.272239\pi$
$397$			30.0764i			1.50949i	−0.656019	+	0.754745i	$0.727761\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−4.66736			−0.233077			−0.116538	−	0.993186i	$-0.537180\pi$
$401$	−4.66736			−0.233077			−0.116538	+	0.993186i	$0.537180\pi$
$402$		0			0
$403$		−	6.98003i		−	0.347700i
$404$		0			0
$405$		0			0
$406$		0			0
$407$			0.391667i			0.0194142i
$408$		0			0
$409$	1.86403			0.0921701			0.0460851	−	0.998938i	$-0.485325\pi$
$409$	1.86403			0.0921701			0.0460851	+	0.998938i	$0.485325\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		−	49.9786i		−	2.45928i
$414$		0			0
$415$	−16.8137	+	5.30084i	−0.825351	+	0.260208i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	2.75283			0.134485			0.0672424	−	0.997737i	$-0.478580\pi$
$419$	2.75283			0.134485			0.0672424	+	0.997737i	$0.478580\pi$
$420$		0			0
$421$	7.33398			0.357436			0.178718	−	0.983900i	$-0.442805\pi$
$421$	7.33398			0.357436			0.178718	+	0.983900i	$0.442805\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−5.90489	−	8.43402i	−0.286429	−	0.409110i
$426$		0			0
$427$			13.4144i			0.649168i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−0.803161			−0.0386869			−0.0193434	−	0.999813i	$-0.506158\pi$
$431$	−0.803161			−0.0386869			−0.0193434	+	0.999813i	$0.506158\pi$
$432$		0			0
$433$			25.9277i			1.24601i	0.782220	+	0.623003i	$0.214087\pi$
$433$			25.9277i			1.24601i	−0.782220	+	0.623003i	$0.785913\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		−	22.1176i		−	1.05803i
$438$		0			0
$439$	−34.5941			−1.65109			−0.825544	−	0.564337i	$-0.809132\pi$
$439$	−34.5941			−1.65109			−0.825544	+	0.564337i	$0.809132\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$		−	1.06316i		−	0.0505123i	−0.999681	−	0.0252562i	$-0.991960\pi$
$443$		−	1.06316i		−	0.0505123i	0.999681	−	0.0252562i	$-0.00804014\pi$
$444$		0			0
$445$	6.64535	+	21.0783i	0.315020	+	0.999209i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−30.4830			−1.43858			−0.719291	−	0.694709i	$-0.755533\pi$
$449$	−30.4830			−1.43858			−0.719291	+	0.694709i	$0.755533\pi$
$450$		0			0
$451$	2.61344			0.123062
$452$		0			0
$453$		0			0
$454$		0			0
$455$	5.19947	+	16.4922i	0.243755	+	0.773164i
$456$		0			0
$457$		−	21.4264i		−	1.00228i	−0.865365	−	0.501141i	$-0.832914\pi$
$457$		−	21.4264i		−	1.00228i	0.865365	−	0.501141i	$-0.167086\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−15.9224			−0.741582			−0.370791	−	0.928716i	$-0.620913\pi$
$461$	−15.9224			−0.741582			−0.370791	+	0.928716i	$0.620913\pi$
$462$		0			0
$463$			20.5439i			0.954754i	0.878698	+	0.477377i	$0.158412\pi$
$463$			20.5439i			0.954754i	−0.878698	+	0.477377i	$0.841588\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$			5.66273i			0.262040i	0.991380	+	0.131020i	$0.0418252\pi$
$467$			5.66273i			0.262040i	−0.991380	+	0.131020i	$0.958175\pi$
$468$		0			0
$469$	−10.5096			−0.485289
$470$		0			0
$471$		0			0
$472$		0			0
$473$		−	1.85446i		−	0.0852683i
$474$		0			0
$475$	−12.1909	+	8.53520i	−0.559358	+	0.391622i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−12.0062			−0.548579			−0.274290	−	0.961647i	$-0.588443\pi$
$479$	−12.0062			−0.548579			−0.274290	+	0.961647i	$0.588443\pi$
$480$		0			0
$481$	−2.13212			−0.0972161
$482$		0			0
$483$		0			0
$484$		0			0
$485$	12.2807	−	3.87171i	0.557636	−	0.175805i
$486$		0			0
$487$			22.3755i			1.01393i	0.861966	+	0.506966i	$0.169233\pi$
$487$			22.3755i			1.01393i	−0.861966	+	0.506966i	$0.830767\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−37.0235			−1.67085			−0.835424	−	0.549606i	$-0.814778\pi$
$491$	−37.0235			−1.67085			−0.835424	+	0.549606i	$0.814778\pi$
$492$		0			0
$493$		−	20.3531i		−	0.916656i
$494$		0			0
$495$		0			0
$496$		0			0
$497$			27.2239i			1.22116i
$498$		0			0
$499$	−18.3283			−0.820487			−0.410243	−	0.911976i	$-0.634556\pi$
$499$	−18.3283			−0.820487			−0.410243	+	0.911976i	$0.634556\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$			28.0672i			1.25146i	0.780041	+	0.625728i	$0.215198\pi$
$503$			28.0672i			1.25146i	−0.780041	+	0.625728i	$0.784802\pi$
$504$		0			0
$505$	4.19642	+	13.3106i	0.186738	+	0.592314i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	38.0600			1.68698			0.843489	−	0.537146i	$-0.180498\pi$
$509$	38.0600			1.68698			0.843489	+	0.537146i	$0.180498\pi$
$510$		0			0
$511$	−40.1335			−1.77540
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−10.8043	+	3.40626i	−0.476094	+	0.150098i
$516$		0			0
$517$		−	4.12602i		−	0.181462i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	13.0268			0.570715			0.285358	−	0.958421i	$-0.407888\pi$
$521$	13.0268			0.570715			0.285358	+	0.958421i	$0.407888\pi$
$522$		0			0
$523$		−	30.0313i		−	1.31318i	−0.754250	−	0.656588i	$-0.771999\pi$
$523$		−	30.0313i		−	1.31318i	0.754250	−	0.656588i	$-0.228001\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$			6.74109i			0.293646i
$528$		0			0
$529$	−32.2212			−1.40092
$530$		0			0
$531$		0			0
$532$		0			0
$533$			14.2267i			0.616229i
$534$		0			0
$535$	1.96725	−	0.620213i	0.0850516	−	0.0268142i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	2.41102			0.103850
$540$		0			0
$541$	−0.0480448			−0.00206561			−0.00103280	−	0.999999i	$-0.500329\pi$
$541$	−0.0480448			−0.00206561			−0.00103280	+	0.999999i	$0.500329\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−12.5265	−	39.7326i	−0.536575	−	1.70196i
$546$		0			0
$547$			17.3072i			0.740004i	0.929031	+	0.370002i	$0.120643\pi$
$547$			17.3072i			0.740004i	−0.929031	+	0.370002i	$0.879357\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−29.4193			−1.25330
$552$		0			0
$553$		−	58.1714i		−	2.47370i
$554$		0			0
$555$		0			0
$556$		0			0
$557$			38.0098i			1.61053i	0.592917	+	0.805264i	$0.297976\pi$
$557$			38.0098i			1.61053i	−0.592917	+	0.805264i	$0.702024\pi$
$558$		0			0
$559$	10.0951			0.426979
$560$		0			0
$561$		0			0
$562$		0			0
$563$		−	5.50462i		−	0.231992i	−0.993250	−	0.115996i	$-0.962994\pi$
$563$		−	5.50462i		−	0.231992i	0.993250	−	0.115996i	$-0.0370060\pi$
$564$		0			0
$565$	25.4495	−	8.02343i	1.07067	−	0.337548i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−6.97732			−0.292504			−0.146252	−	0.989247i	$-0.546721\pi$
$569$	−6.97732			−0.292504			−0.146252	+	0.989247i	$0.546721\pi$
$570$		0			0
$571$	−33.9520			−1.42085			−0.710424	−	0.703774i	$-0.751497\pi$
$571$	−33.9520			−1.42085			−0.710424	+	0.703774i	$0.751497\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	21.3099	+	30.4371i	0.888683	+	1.26932i
$576$		0			0
$577$			29.4116i			1.22442i	0.790695	+	0.612211i	$0.209720\pi$
$577$			29.4116i			1.22442i	−0.790695	+	0.612211i	$0.790280\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−28.5965			−1.18638
$582$		0			0
$583$		−	0.526641i		−	0.0218112i
$584$		0			0
$585$		0			0
$586$		0			0
$587$			11.1028i			0.458261i	0.973396	+	0.229131i	$0.0735883\pi$
$587$			11.1028i			0.458261i	−0.973396	+	0.229131i	$0.926412\pi$
$588$		0			0
$589$	9.74388			0.401490
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	6.65107i		−	0.273127i	−0.990631	−	0.136563i	$-0.956394\pi$
$593$		−	6.65107i		−	0.273127i	0.990631	−	0.136563i	$-0.0436058\pi$
$594$		0			0
$595$	−5.02148	−	15.9276i	−0.205861	−	0.652968i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	17.7417			0.724906			0.362453	−	0.932002i	$-0.381939\pi$
$599$	17.7417			0.724906			0.362453	+	0.932002i	$0.381939\pi$
$600$		0			0
$601$	27.7545			1.13213			0.566064	−	0.824361i	$-0.308466\pi$
$601$	27.7545			1.13213			0.566064	+	0.824361i	$0.308466\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−7.29261	−	23.1314i	−0.296487	−	0.940425i
$606$		0			0
$607$			23.0209i			0.934390i	0.884154	+	0.467195i	$0.154735\pi$
$607$			23.0209i			0.934390i	−0.884154	+	0.467195i	$0.845265\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	22.4608			0.908666
$612$		0			0
$613$			17.5217i			0.707696i	0.935303	+	0.353848i	$0.115127\pi$
$613$			17.5217i			0.707696i	−0.935303	+	0.353848i	$0.884873\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	18.1871i		−	0.732187i	−0.930578	−	0.366093i	$-0.880695\pi$
$617$		−	18.1871i		−	0.732187i	0.930578	−	0.366093i	$-0.119305\pi$
$618$		0			0
$619$	−30.1269			−1.21090			−0.605451	−	0.795883i	$-0.707007\pi$
$619$	−30.1269			−1.21090			−0.605451	+	0.795883i	$0.707007\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$			35.8498i			1.43629i
$624$		0			0
$625$	8.55303	−	23.4914i	0.342121	−	0.939656i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	2.05913			0.0821029
$630$		0			0
$631$	−41.4773			−1.65119			−0.825593	−	0.564266i	$-0.809159\pi$
$631$	−41.4773			−1.65119			−0.825593	+	0.564266i	$0.809159\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	39.6418	−	12.4978i	1.57314	−	0.495961i
$636$		0			0
$637$			13.1249i			0.520027i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	26.1370			1.03235			0.516175	−	0.856483i	$-0.327356\pi$
$641$	26.1370			1.03235			0.516175	+	0.856483i	$0.327356\pi$
$642$		0			0
$643$			37.1020i			1.46316i	0.681755	+	0.731580i	$0.261217\pi$
$643$			37.1020i			1.46316i	−0.681755	+	0.731580i	$0.738783\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			4.70071i			0.184804i	0.995722	+	0.0924020i	$0.0294545\pi$
$647$			4.70071i			0.184804i	−0.995722	+	0.0924020i	$0.970546\pi$
$648$		0			0
$649$	5.39687			0.211846
$650$		0			0
$651$		0			0
$652$		0			0
$653$			39.6121i			1.55014i	0.631874	+	0.775071i	$0.282286\pi$
$653$			39.6121i			1.55014i	−0.631874	+	0.775071i	$0.717714\pi$
$654$		0			0
$655$	−4.04378	−	12.8264i	−0.158004	−	0.501171i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−8.14782			−0.317394			−0.158697	−	0.987327i	$-0.550729\pi$
$659$	−8.14782			−0.317394			−0.158697	+	0.987327i	$0.550729\pi$
$660$		0			0
$661$	19.5663			0.761039			0.380520	−	0.924773i	$-0.375745\pi$
$661$	19.5663			0.761039			0.380520	+	0.924773i	$0.375745\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−23.0225	+	7.25828i	−0.892774	+	0.281464i
$666$		0			0
$667$			73.4513i			2.84404i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−1.44854			−0.0559201
$672$		0			0
$673$		−	25.8110i		−	0.994939i	−0.867481	−	0.497470i	$-0.834263\pi$
$673$		−	25.8110i		−	0.994939i	0.867481	−	0.497470i	$-0.165737\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$			15.5145i			0.596273i	0.954523	+	0.298136i	$0.0963650\pi$
$677$			15.5145i			0.596273i	−0.954523	+	0.298136i	$0.903635\pi$
$678$		0			0
$679$	20.8868			0.801562
$680$		0			0
$681$		0			0
$682$		0			0
$683$			49.9137i			1.90990i	0.296773	+	0.954948i	$0.404090\pi$
$683$			49.9137i			1.90990i	−0.296773	+	0.954948i	$0.595910\pi$
$684$		0			0
$685$	−46.5487	+	14.6754i	−1.77853	+	0.560717i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	2.86687			0.109219
$690$		0			0
$691$	−17.7800			−0.676384			−0.338192	−	0.941077i	$-0.609815\pi$
$691$	−17.7800			−0.676384			−0.338192	+	0.941077i	$0.609815\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−4.39412	−	13.9377i	−0.166678	−	0.528686i
$696$		0			0
$697$		−	13.7397i		−	0.520430i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−34.6823			−1.30993			−0.654965	−	0.755659i	$-0.727317\pi$
$701$	−34.6823			−1.30993			−0.654965	+	0.755659i	$0.727317\pi$
$702$		0			0
$703$		−	2.97636i		−	0.112256i
$704$		0			0
$705$		0			0
$706$		0			0
$707$			22.6385i			0.851409i
$708$		0			0
$709$	30.0916			1.13011			0.565057	−	0.825052i	$-0.308854\pi$
$709$	30.0916			1.13011			0.565057	+	0.825052i	$0.308854\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		−	24.3276i		−	0.911076i
$714$		0			0
$715$	−1.78088	+	0.561458i	−0.0666013	+	0.0209973i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	24.5123			0.914155			0.457078	−	0.889427i	$-0.348896\pi$
$719$	24.5123			0.914155			0.457078	+	0.889427i	$0.348896\pi$
$720$		0			0
$721$	−18.3758			−0.684351
$722$		0			0
$723$		0			0
$724$		0			0
$725$	40.4853	−	28.3449i	1.50359	−	1.05270i
$726$		0			0
$727$		−	16.7349i		−	0.620661i	−0.950629	−	0.310331i	$-0.899560\pi$
$727$		−	16.7349i		−	0.620661i	0.950629	−	0.310331i	$-0.100440\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−9.74956			−0.360600
$732$		0			0
$733$		−	46.6910i		−	1.72457i	−0.506422	−	0.862286i	$-0.669032\pi$
$733$		−	46.6910i		−	1.72457i	0.506422	−	0.862286i	$-0.330968\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	1.13487i		−	0.0418033i
$738$		0			0
$739$	−19.1792			−0.705519			−0.352760	−	0.935714i	$-0.614757\pi$
$739$	−19.1792			−0.705519			−0.352760	+	0.935714i	$0.614757\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	44.2735i		−	1.62424i	−0.583493	−	0.812119i	$-0.698314\pi$
$743$		−	44.2735i		−	1.62424i	0.583493	−	0.812119i	$-0.301686\pi$
$744$		0			0
$745$	10.1414	+	32.1674i	0.371552	+	1.17852i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	3.34588			0.122256
$750$		0			0
$751$	−11.0139			−0.401901			−0.200951	−	0.979601i	$-0.564403\pi$
$751$	−11.0139			−0.401901			−0.200951	+	0.979601i	$0.564403\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−6.50177	−	20.6229i	−0.236623	−	0.750544i
$756$		0			0
$757$		−	33.2491i		−	1.20846i	−0.796811	−	0.604229i	$-0.793481\pi$
$757$		−	33.2491i		−	1.20846i	0.796811	−	0.604229i	$-0.206519\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	32.7798			1.18827			0.594133	−	0.804367i	$-0.297495\pi$
$761$	32.7798			1.18827			0.594133	+	0.804367i	$0.297495\pi$
$762$		0			0
$763$		−	67.5768i		−	2.44645i
$764$		0			0
$765$		0			0
$766$		0			0
$767$			29.3789i			1.06081i
$768$		0			0
$769$	−32.8054			−1.18299			−0.591496	−	0.806308i	$-0.701463\pi$
$769$	−32.8054			−1.18299			−0.591496	+	0.806308i	$0.701463\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$			36.6815i			1.31934i	0.751554	+	0.659672i	$0.229305\pi$
$773$			36.6815i			1.31934i	−0.751554	+	0.659672i	$0.770695\pi$
$774$		0			0
$775$	−13.4090	+	9.38803i	−0.481667	+	0.337228i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−19.8601			−0.711560
$780$		0			0
$781$	−2.93974			−0.105192
$782$		0			0
$783$		0			0
$784$		0			0
$785$	16.1051	−	5.07745i	0.574817	−	0.181222i
$786$		0			0
$787$			2.99903i			0.106904i	0.998570	+	0.0534519i	$0.0170224\pi$
$787$			2.99903i			0.106904i	−0.998570	+	0.0534519i	$0.982978\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	43.2842			1.53901
$792$		0			0
$793$		−	7.88539i		−	0.280018i
$794$		0			0
$795$		0			0
$796$		0			0
$797$			44.1996i			1.56563i	0.622255	+	0.782815i	$0.286217\pi$
$797$			44.1996i			1.56563i	−0.622255	+	0.782815i	$0.713783\pi$
$798$		0			0
$799$	−21.6919			−0.767405
$800$		0			0
$801$		0			0
$802$		0			0
$803$		−	4.33377i		−	0.152935i
$804$		0			0
$805$	18.1218	+	57.4804i	0.638710	+	2.02592i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−14.2774			−0.501966			−0.250983	−	0.967992i	$-0.580754\pi$
$809$	−14.2774			−0.501966			−0.250983	+	0.967992i	$0.580754\pi$
$810$		0			0
$811$	40.2885			1.41472			0.707359	−	0.706854i	$-0.249886\pi$
$811$	40.2885			1.41472			0.707359	+	0.706854i	$0.249886\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	31.0614	−	9.79270i	1.08803	−	0.343023i
$816$		0			0
$817$			14.0925i			0.493033i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−30.5173			−1.06506			−0.532531	−	0.846411i	$-0.678759\pi$
$821$	−30.5173			−1.06506			−0.532531	+	0.846411i	$0.678759\pi$
$822$		0			0
$823$		−	21.8216i		−	0.760654i	−0.924852	−	0.380327i	$-0.875811\pi$
$823$		−	21.8216i		−	0.760654i	0.924852	−	0.380327i	$-0.124189\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	46.0668i		−	1.60190i	−0.598732	−	0.800949i	$-0.704329\pi$
$827$		−	46.0668i		−	1.60190i	0.598732	−	0.800949i	$-0.295671\pi$
$828$		0			0
$829$	28.4791			0.989119			0.494560	−	0.869144i	$-0.335329\pi$
$829$	28.4791			0.989119			0.494560	+	0.869144i	$0.335329\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		−	12.6756i		−	0.439183i
$834$		0			0
$835$	45.8463	−	14.4539i	1.58658	−	0.500199i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−1.01118			−0.0349098			−0.0174549	−	0.999848i	$-0.505556\pi$
$839$	−1.01118			−0.0349098			−0.0174549	+	0.999848i	$0.505556\pi$
$840$		0			0
$841$	68.6995			2.36895
$842$		0			0
$843$		0			0
$844$		0			0
$845$	5.68403	+	18.0291i	0.195537	+	0.620221i
$846$		0			0
$847$		−	39.3416i		−	1.35179i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−7.43110			−0.254735
$852$		0			0
$853$			4.06708i			0.139254i	0.997573	+	0.0696271i	$0.0221809\pi$
$853$			4.06708i			0.139254i	−0.997573	+	0.0696271i	$0.977819\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	38.5658i		−	1.31738i	−0.752413	−	0.658691i	$-0.771110\pi$
$857$		−	38.5658i		−	1.31738i	0.752413	−	0.658691i	$-0.228890\pi$
$858$		0			0
$859$	−3.74753			−0.127864			−0.0639320	−	0.997954i	$-0.520364\pi$
$859$	−3.74753			−0.127864			−0.0639320	+	0.997954i	$0.520364\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	2.91226i		−	0.0991344i	−0.998771	−	0.0495672i	$-0.984216\pi$
$863$		−	2.91226i		−	0.0991344i	0.998771	−	0.0495672i	$-0.0157842\pi$
$864$		0			0
$865$	11.7892	−	3.71678i	0.400846	−	0.126374i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	6.28156			0.213087
$870$		0			0
$871$	6.17787			0.209329
$872$		0			0
$873$		0			0
$874$		0			0
$875$	24.6892	−	32.1702i	0.834646	−	1.08755i
$876$		0			0
$877$			9.49885i			0.320753i	0.987056	+	0.160377i	$0.0512709\pi$
$877$			9.49885i			0.320753i	−0.987056	+	0.160377i	$0.948729\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	24.9892			0.841907			0.420954	−	0.907082i	$-0.361696\pi$
$881$	24.9892			0.841907			0.420954	+	0.907082i	$0.361696\pi$
$882$		0			0
$883$		−	45.0505i		−	1.51607i	−0.652213	−	0.758036i	$-0.726159\pi$
$883$		−	45.0505i		−	1.51607i	0.652213	−	0.758036i	$-0.273841\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$			54.1959i			1.81972i	0.414915	+	0.909860i	$0.363811\pi$
$887$			54.1959i			1.81972i	−0.414915	+	0.909860i	$0.636189\pi$
$888$		0			0
$889$	67.4223			2.26127
$890$		0			0
$891$		0			0
$892$		0			0
$893$			31.3545i			1.04924i
$894$		0			0
$895$	−13.2085	−	41.8961i	−0.441513	−	1.40043i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−32.3588			−1.07923
$900$		0			0
$901$	−2.76873			−0.0922399
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−10.1535	−	32.2058i	−0.337513	−	1.07056i
$906$		0			0
$907$			4.36042i			0.144785i	0.997376	+	0.0723927i	$0.0230635\pi$
$907$			4.36042i			0.144785i	−0.997376	+	0.0723927i	$0.976937\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	15.3187			0.507531			0.253766	−	0.967266i	$-0.418331\pi$
$911$	15.3187			0.507531			0.253766	+	0.967266i	$0.418331\pi$
$912$		0			0
$913$		−	3.08796i		−	0.102197i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		−	21.8151i		−	0.720397i
$918$		0			0
$919$	−19.8290			−0.654097			−0.327049	−	0.945007i	$-0.606054\pi$
$919$	−19.8290			−0.654097			−0.327049	+	0.945007i	$0.606054\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		−	16.0030i		−	0.526746i
$924$		0			0
$925$	2.86766	+	4.09591i	0.0942881	+	0.134673i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	41.6818			1.36754			0.683768	−	0.729699i	$-0.260340\pi$
$929$	41.6818			1.36754			0.683768	+	0.729699i	$0.260340\pi$
$930$		0			0
$931$	−18.3219			−0.600476
$932$		0			0
$933$		0			0
$934$		0			0
$935$	1.71992	−	0.542238i	0.0562474	−	0.0177331i
$936$		0			0
$937$			24.0501i			0.785682i	0.919607	+	0.392841i	$0.128508\pi$
$937$			24.0501i			0.785682i	−0.919607	+	0.392841i	$0.871492\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−43.5451			−1.41953			−0.709764	−	0.704439i	$-0.751199\pi$
$941$	−43.5451			−1.41953			−0.709764	+	0.704439i	$0.751199\pi$
$942$		0			0
$943$			49.5847i			1.61470i
$944$		0			0
$945$		0			0
$946$		0			0
$947$		−	51.9175i		−	1.68709i	−0.537056	−	0.843546i	$-0.680464\pi$
$947$		−	51.9175i		−	1.68709i	0.537056	−	0.843546i	$-0.319536\pi$
$948$		0			0
$949$	23.5917			0.765819
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	24.2220i		−	0.784628i	−0.919831	−	0.392314i	$-0.871675\pi$
$953$		−	24.2220i		−	0.784628i	0.919831	−	0.392314i	$-0.128325\pi$
$954$		0			0
$955$	−1.78893	−	5.67430i	−0.0578884	−	0.183616i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−79.1696			−2.55652
$960$		0			0
$961$	−20.2825			−0.654275
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−12.1485	+	3.83005i	−0.391074	+	0.123294i
$966$		0			0
$967$			41.1738i			1.32406i	0.749477	+	0.662031i	$0.230305\pi$
$967$			41.1738i			1.32406i	−0.749477	+	0.662031i	$0.769695\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	11.7115			0.375839			0.187919	−	0.982184i	$-0.439826\pi$
$971$	11.7115			0.375839			0.187919	+	0.982184i	$0.439826\pi$
$972$		0			0
$973$		−	23.7050i		−	0.759949i
$974$		0			0
$975$		0			0
$976$		0			0
$977$			46.1536i			1.47658i	0.674482	+	0.738292i	$0.264367\pi$
$977$			46.1536i			1.47658i	−0.674482	+	0.738292i	$0.735633\pi$
$978$		0			0
$979$	−3.87119			−0.123724
$980$		0			0
$981$		0			0
$982$		0			0
$983$			1.18675i			0.0378513i	0.999821	+	0.0189257i	$0.00602458\pi$
$983$			1.18675i			0.0378513i	−0.999821	+	0.0189257i	$0.993975\pi$
$984$		0			0
$985$	−14.9989	+	4.72868i	−0.477904	+	0.150668i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	35.1847			1.11881
$990$		0			0
$991$	−44.7223			−1.42065			−0.710325	−	0.703874i	$-0.751452\pi$
$991$	−44.7223			−1.42065			−0.710325	+	0.703874i	$0.751452\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−9.59702	−	30.4407i	−0.304246	−	0.965036i
$996$		0			0
$997$		−	48.4016i		−	1.53289i	−0.642308	−	0.766446i	$-0.722023\pi$
$997$		−	48.4016i		−	1.53289i	0.642308	−	0.766446i	$-0.277977\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6660.2.f.c.5329.12		18
3.2	odd	2		740.2.d.a.149.11	yes	18
5.4	even	2	inner	6660.2.f.c.5329.11		18
15.2	even	4		3700.2.a.p.1.6		9
15.8	even	4		3700.2.a.o.1.4		9
15.14	odd	2		740.2.d.a.149.8	✓	18

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
740.2.d.a.149.8	✓	18	15.14	odd	2
740.2.d.a.149.11	yes	18	3.2	odd	2
3700.2.a.o.1.4		9	15.8	even	4
3700.2.a.p.1.6		9	15.2	even	4
6660.2.f.c.5329.11		18	5.4	even	2	inner
6660.2.f.c.5329.12		18	1.1	even	1	trivial

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 6660 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6660.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(0.672341 + 2.13259i) q^{5} +3.62709i q^{7} -0.391667 q^{11} -2.13212i q^{13} +2.05913i q^{17} +2.97636 q^{19} -7.43110i q^{23} +(-4.09591 + 2.86766i) q^{25} -9.88431 q^{29} +3.27376 q^{31} +(-7.73512 + 2.43864i) q^{35} -1.00000i q^{37} -6.67260 q^{41} +4.73480i q^{43} +10.5345i q^{47} -6.15580 q^{49} +1.34461i q^{53} +(-0.263334 - 0.835266i) q^{55} -13.7792 q^{59} +3.69839 q^{61} +(4.54694 - 1.43351i) q^{65} +2.89753i q^{67} +7.50571 q^{71} +11.0649i q^{73} -1.42061i q^{77} -16.0380 q^{79} +7.88415i q^{83} +(-4.39129 + 1.38444i) q^{85} +9.88390 q^{89} +7.73338 q^{91} +(2.00113 + 6.34737i) q^{95} -5.75855i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q - 2 q^{5} + 4 q^{19} - 2 q^{25} + 4 q^{29} + 8 q^{31} + 2 q^{35} + 4 q^{41} + 6 q^{49} - 6 q^{55} - 8 q^{59} + 12 q^{65} + 24 q^{71} - 24 q^{79} + 36 q^{91} - 24 q^{95}+O(q^{100}) \) 18 * q - 2 * q^5 + 4 * q^19 - 2 * q^25 + 4 * q^29 + 8 * q^31 + 2 * q^35 + 4 * q^41 + 6 * q^49 - 6 * q^55 - 8 * q^59 + 12 * q^65 + 24 * q^71 - 24 * q^79 + 36 * q^91 - 24 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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