LMFDB - Embedded newform 2736.3.o.r.721.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2736,3,Mod(721,2736)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2736, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("2736.721");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 2736 = 2^{4} \cdot 3^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	2736.o (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$74.5506003290$
Analytic rank:	$0$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - 4 x^{19} + 80 x^{18} - 152 x^{17} + 4326 x^{16} - 10096 x^{15} + 70116 x^{14} - 93436 x^{13} + \cdots + 36100 $ x^20 - 4x^19 + 80x^18 - 152x^17 + 4326x^16 - 10096x^15 + 70116x^14 - 93436x^13 + 597327x^12 - 837564x^11 + 2838686x^10 - 2898276x^9 + 7652658x^8 - 6988600x^7 + 12843126x^6 - 6054124x^5 + 7434909x^4 - 621764x^3 + 3099194x^2 - 329080*x + 36100
Coefficient ring:	$\Z[a_1, \ldots, a_{29}]$
Coefficient ring index:	$ 2^{37} $
Twist minimal:	no (minimal twist has level 456)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			721.6
Root			$-0.861157 - 1.49157i$ of defining polynomial
Character	$\chi$	$=$	2736.721
Dual form			2736.3.o.r.721.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-5.26532 q^{5} +1.82069 q^{7} +O(q^{10})$	$q-5.26532 q^{5} +1.82069 q^{7} +15.7218 q^{11} +14.8565i q^{13} +17.7083 q^{17} +(-10.9243 + 15.5454i) q^{19} +44.0439 q^{23} +2.72355 q^{25} -0.665376i q^{29} -45.5774i q^{31} -9.58649 q^{35} -7.42719i q^{37} -55.1759i q^{41} -57.7656 q^{43} -59.0790 q^{47} -45.6851 q^{49} +45.8416i q^{53} -82.7805 q^{55} +98.2419i q^{59} +90.9110 q^{61} -78.2242i q^{65} -8.94893i q^{67} +39.2559i q^{71} +42.6454 q^{73} +28.6245 q^{77} +91.1359i q^{79} +40.0132 q^{83} -93.2400 q^{85} -20.1864i q^{89} +27.0490i q^{91} +(57.5201 - 81.8513i) q^{95} -120.699i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q + 16 q^{7}+O(q^{10}) $ 20 * q + 16 * q^7	$ 20 q + 16 q^{7} + 16 q^{11} - 32 q^{17} - 40 q^{19} + 64 q^{23} + 68 q^{25} - 208 q^{35} - 64 q^{43} + 48 q^{47} + 20 q^{49} + 336 q^{55} + 184 q^{61} + 104 q^{73} - 88 q^{77} + 224 q^{83} - 136 q^{85} - 320 q^{95}+O(q^{100}) $ 20 * q + 16 * q^7 + 16 * q^11 - 32 * q^17 - 40 * q^19 + 64 * q^23 + 68 * q^25 - 208 * q^35 - 64 * q^43 + 48 * q^47 + 20 * q^49 + 336 * q^55 + 184 * q^61 + 104 * q^73 - 88 * q^77 + 224 * q^83 - 136 * q^85 - 320 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2736\mathbb{Z}\right)^\times$.

$n$	$1009$	$1217$	$1711$	$2053$
$\chi(n)$	$-1$	$1$	$1$	$1$

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
$5$	−5.26532			−1.05306			−0.526532	−	0.850156i	$-0.676508\pi$
$5$	−5.26532			−1.05306			−0.526532	+	0.850156i	$0.676508\pi$
$6$		0			0
$7$	1.82069			0.260098			0.130049	−	0.991508i	$-0.458487\pi$
$7$	1.82069			0.260098			0.130049	+	0.991508i	$0.458487\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	15.7218			1.42926			0.714629	−	0.699503i	$-0.246595\pi$
$11$	15.7218			1.42926			0.714629	+	0.699503i	$0.246595\pi$
$12$		0			0
$13$			14.8565i			1.14281i	0.820669	+	0.571404i	$0.193601\pi$
$13$			14.8565i			1.14281i	−0.820669	+	0.571404i	$0.806399\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	17.7083			1.04167			0.520833	−	0.853658i	$-0.325621\pi$
$17$	17.7083			1.04167			0.520833	+	0.853658i	$0.325621\pi$
$18$		0			0
$19$	−10.9243	+	15.5454i	−0.574965	+	0.818178i
$19$	−10.9243	+	15.5454i	−0.574965	+	0.818178i
$20$		0			0
$21$		0			0
$22$		0			0
$23$	44.0439			1.91495			0.957475	−	0.288515i	$-0.0931615\pi$
$23$	44.0439			1.91495			0.957475	+	0.288515i	$0.0931615\pi$
$24$		0			0
$25$	2.72355			0.108942
$26$		0			0
$27$		0			0
$28$		0			0
$29$		−	0.665376i		−	0.0229440i	−0.999934	−	0.0114720i	$-0.996348\pi$
$29$		−	0.665376i		−	0.0229440i	0.999934	−	0.0114720i	$-0.00365173\pi$
$30$		0			0
$31$		−	45.5774i		−	1.47024i	−0.677937	−	0.735120i	$-0.737126\pi$
$31$		−	45.5774i		−	1.47024i	0.677937	−	0.735120i	$-0.262874\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−9.58649			−0.273900
$36$		0			0
$37$		−	7.42719i		−	0.200735i	−0.994950	−	0.100367i	$-0.967998\pi$
$37$		−	7.42719i		−	0.200735i	0.994950	−	0.100367i	$-0.0320018\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		−	55.1759i		−	1.34575i	−0.739754	−	0.672877i	$-0.765058\pi$
$41$		−	55.1759i		−	1.34575i	0.739754	−	0.672877i	$-0.234942\pi$
$42$		0			0
$43$	−57.7656			−1.34339			−0.671694	−	0.740829i	$-0.734433\pi$
$43$	−57.7656			−1.34339			−0.671694	+	0.740829i	$0.734433\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−59.0790			−1.25700			−0.628500	−	0.777809i	$-0.716331\pi$
$47$	−59.0790			−1.25700			−0.628500	+	0.777809i	$0.716331\pi$
$48$		0			0
$49$	−45.6851			−0.932349
$50$		0			0
$51$		0			0
$52$		0			0
$53$			45.8416i			0.864935i	0.901649	+	0.432468i	$0.142357\pi$
$53$			45.8416i			0.864935i	−0.901649	+	0.432468i	$0.857643\pi$
$54$		0			0
$55$	−82.7805			−1.50510
$56$		0			0
$57$		0			0
$58$		0			0
$59$			98.2419i			1.66512i	0.553938	+	0.832558i	$0.313125\pi$
$59$			98.2419i			1.66512i	−0.553938	+	0.832558i	$0.686875\pi$
$60$		0			0
$61$	90.9110			1.49034			0.745172	−	0.666872i	$-0.232367\pi$
$61$	90.9110			1.49034			0.745172	+	0.666872i	$0.232367\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		−	78.2242i		−	1.20345i
$66$		0			0
$67$		−	8.94893i		−	0.133566i	−0.997768	−	0.0667830i	$-0.978726\pi$
$67$		−	8.94893i		−	0.133566i	0.997768	−	0.0667830i	$-0.0212735\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			39.2559i			0.552901i	0.961028	+	0.276450i	$0.0891581\pi$
$71$			39.2559i			0.552901i	−0.961028	+	0.276450i	$0.910842\pi$
$72$		0			0
$73$	42.6454			0.584183			0.292092	−	0.956390i	$-0.405649\pi$
$73$	42.6454			0.584183			0.292092	+	0.956390i	$0.405649\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	28.6245			0.371747
$78$		0			0
$79$			91.1359i			1.15362i	0.816879	+	0.576810i	$0.195703\pi$
$79$			91.1359i			1.15362i	−0.816879	+	0.576810i	$0.804297\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	40.0132			0.482087			0.241044	−	0.970514i	$-0.422510\pi$
$83$	40.0132			0.482087			0.241044	+	0.970514i	$0.422510\pi$
$84$		0			0
$85$	−93.2400			−1.09694
$86$		0			0
$87$		0			0
$88$		0			0
$89$		−	20.1864i		−	0.226813i	−0.993549	−	0.113406i	$-0.963824\pi$
$89$		−	20.1864i		−	0.226813i	0.993549	−	0.113406i	$-0.0361762\pi$
$90$		0			0
$91$			27.0490i			0.297242i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	57.5201	−	81.8513i	0.605475	−	0.861593i
$96$		0			0
$97$		−	120.699i		−	1.24432i	−0.782890	−	0.622161i	$-0.786255\pi$
$97$		−	120.699i		−	1.24432i	0.782890	−	0.622161i	$-0.213745\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	163.735			1.62114			0.810571	−	0.585640i	$-0.199157\pi$
$101$	163.735			1.62114			0.810571	+	0.585640i	$0.199157\pi$
$102$		0			0
$103$			114.779i			1.11436i	0.830392	+	0.557179i	$0.188116\pi$
$103$			114.779i			1.11436i	−0.830392	+	0.557179i	$0.811884\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	201.869i		−	1.88662i	−0.331909	−	0.943312i	$-0.607693\pi$
$107$		−	201.869i		−	1.88662i	0.331909	−	0.943312i	$-0.392307\pi$
$108$		0			0
$109$			93.2065i			0.855106i	0.903990	+	0.427553i	$0.140624\pi$
$109$			93.2065i			0.855106i	−0.903990	+	0.427553i	$0.859376\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			44.6729i			0.395335i	0.980269	+	0.197668i	$0.0633367\pi$
$113$			44.6729i			0.395335i	−0.980269	+	0.197668i	$0.936663\pi$
$114$		0			0
$115$	−231.905			−2.01656
$116$		0			0
$117$		0			0
$118$		0			0
$119$	32.2413			0.270935
$120$		0			0
$121$	126.176			1.04278
$122$		0			0
$123$		0			0
$124$		0			0
$125$	117.293			0.938340
$126$		0			0
$127$			155.954i			1.22799i	0.789312	+	0.613993i	$0.210438\pi$
$127$			155.954i			1.22799i	−0.789312	+	0.613993i	$0.789562\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−120.237			−0.917839			−0.458919	−	0.888478i	$-0.651763\pi$
$131$	−120.237			−0.917839			−0.458919	+	0.888478i	$0.651763\pi$
$132$		0			0
$133$	−19.8898	+	28.3033i	−0.149547	+	0.212806i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	178.250			1.30109			0.650546	−	0.759467i	$-0.274540\pi$
$137$	178.250			1.30109			0.650546	+	0.759467i	$0.274540\pi$
$138$		0			0
$139$	112.217			0.807320			0.403660	−	0.914909i	$-0.367738\pi$
$139$	112.217			0.807320			0.403660	+	0.914909i	$0.367738\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$			233.572i			1.63337i
$144$		0			0
$145$			3.50342i			0.0241615i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−31.4079			−0.210791			−0.105396	−	0.994430i	$-0.533611\pi$
$149$	−31.4079			−0.210791			−0.105396	+	0.994430i	$0.533611\pi$
$150$		0			0
$151$			192.485i			1.27473i	0.770561	+	0.637366i	$0.219976\pi$
$151$			192.485i			1.27473i	−0.770561	+	0.637366i	$0.780024\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$			239.980i			1.54826i
$156$		0			0
$157$	214.459			1.36598			0.682991	−	0.730426i	$-0.260679\pi$
$157$	214.459			1.36598			0.682991	+	0.730426i	$0.260679\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	80.1901			0.498075
$162$		0			0
$163$	88.5691			0.543369			0.271684	−	0.962386i	$-0.412419\pi$
$163$	88.5691			0.543369			0.271684	+	0.962386i	$0.412419\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	86.0128i		−	0.515047i	−0.966272	−	0.257523i	$-0.917094\pi$
$167$		−	86.0128i		−	0.515047i	0.966272	−	0.257523i	$-0.0829064\pi$
$168$		0			0
$169$	−51.7155			−0.306009
$170$		0			0
$171$		0			0
$172$		0			0
$173$			321.556i			1.85871i	0.369193	+	0.929353i	$0.379634\pi$
$173$			321.556i			1.85871i	−0.369193	+	0.929353i	$0.620366\pi$
$174$		0			0
$175$	4.95874			0.0283356
$176$		0			0
$177$		0			0
$178$		0			0
$179$		−	17.2153i		−	0.0961746i	−0.998843	−	0.0480873i	$-0.984687\pi$
$179$		−	17.2153i		−	0.0961746i	0.998843	−	0.0480873i	$-0.0153126\pi$
$180$		0			0
$181$		−	211.788i		−	1.17010i	−0.810998	−	0.585049i	$-0.801075\pi$
$181$		−	211.788i		−	1.17010i	0.810998	−	0.585049i	$-0.198925\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$			39.1065i			0.211387i
$186$		0			0
$187$	278.408			1.48881
$188$		0			0
$189$		0			0
$190$		0			0
$191$	142.024			0.743583			0.371791	−	0.928316i	$-0.378744\pi$
$191$	142.024			0.743583			0.371791	+	0.928316i	$0.378744\pi$
$192$		0			0
$193$			13.2424i			0.0686134i	0.999411	+	0.0343067i	$0.0109223\pi$
$193$			13.2424i			0.0686134i	−0.999411	+	0.0343067i	$0.989078\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	23.2578			0.118060			0.0590299	−	0.998256i	$-0.481199\pi$
$197$	23.2578			0.118060			0.0590299	+	0.998256i	$0.481199\pi$
$198$		0			0
$199$	−32.6482			−0.164061			−0.0820306	−	0.996630i	$-0.526141\pi$
$199$	−32.6482			−0.164061			−0.0820306	+	0.996630i	$0.526141\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		−	1.21144i		−	0.00596769i
$204$		0			0
$205$			290.519i			1.41716i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−171.751	+	244.402i	−0.821774	+	1.16939i
$210$		0			0
$211$			14.9956i			0.0710691i	0.999368	+	0.0355345i	$0.0113134\pi$
$211$			14.9956i			0.0710691i	−0.999368	+	0.0355345i	$0.988687\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	304.154			1.41467
$216$		0			0
$217$		−	82.9822i		−	0.382407i
$218$		0			0
$219$		0			0
$220$		0			0
$221$			263.084i			1.19042i
$222$		0			0
$223$			205.551i			0.921753i	0.887464	+	0.460877i	$0.152465\pi$
$223$			205.551i			0.921753i	−0.887464	+	0.460877i	$0.847535\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$			425.810i			1.87582i	0.346883	+	0.937909i	$0.387240\pi$
$227$			425.810i			1.87582i	−0.346883	+	0.937909i	$0.612760\pi$
$228$		0			0
$229$	−143.571			−0.626949			−0.313474	−	0.949597i	$-0.601493\pi$
$229$	−143.571			−0.626949			−0.313474	+	0.949597i	$0.601493\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−158.800			−0.681545			−0.340772	−	0.940146i	$-0.610689\pi$
$233$	−158.800			−0.681545			−0.340772	+	0.940146i	$0.610689\pi$
$234$		0			0
$235$	311.070			1.32370
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−13.4507			−0.0562792			−0.0281396	−	0.999604i	$-0.508958\pi$
$239$	−13.4507			−0.0562792			−0.0281396	+	0.999604i	$0.508958\pi$
$240$		0			0
$241$			221.315i			0.918319i	0.888354	+	0.459160i	$0.151849\pi$
$241$			221.315i			0.918319i	−0.888354	+	0.459160i	$0.848151\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	240.547			0.981822
$246$		0			0
$247$	−230.950	−	162.297i	−0.935020	−	0.657075i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−383.330			−1.52721			−0.763606	−	0.645683i	$-0.776573\pi$
$251$	−383.330			−1.52721			−0.763606	+	0.645683i	$0.776573\pi$
$252$		0			0
$253$	692.451			2.73696
$254$		0			0
$255$		0			0
$256$		0			0
$257$		−	91.7299i		−	0.356926i	−0.983947	−	0.178463i	$-0.942888\pi$
$257$		−	91.7299i		−	0.356926i	0.983947	−	0.178463i	$-0.0571125\pi$
$258$		0			0
$259$		−	13.5226i		−	0.0522107i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−6.43552			−0.0244697			−0.0122348	−	0.999925i	$-0.503895\pi$
$263$	−6.43552			−0.0244697			−0.0122348	+	0.999925i	$0.503895\pi$
$264$		0			0
$265$		−	241.370i		−	0.910831i
$266$		0			0
$267$		0			0
$268$		0			0
$269$			142.556i			0.529950i	0.964255	+	0.264975i	$0.0853636\pi$
$269$			142.556i			0.529950i	−0.964255	+	0.264975i	$0.914636\pi$
$270$		0			0
$271$	−368.495			−1.35976			−0.679881	−	0.733323i	$-0.737968\pi$
$271$	−368.495			−1.35976			−0.679881	+	0.733323i	$0.737968\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	42.8193			0.155706
$276$		0			0
$277$	165.941			0.599066			0.299533	−	0.954086i	$-0.403169\pi$
$277$	165.941			0.599066			0.299533	+	0.954086i	$0.403169\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	157.476i		−	0.560411i	−0.959940	−	0.280206i	$-0.909597\pi$
$281$		−	157.476i		−	0.560411i	0.959940	−	0.280206i	$-0.0904026\pi$
$282$		0			0
$283$	272.713			0.963652			0.481826	−	0.876267i	$-0.339974\pi$
$283$	272.713			0.963652			0.481826	+	0.876267i	$0.339974\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		−	100.458i		−	0.350028i
$288$		0			0
$289$	24.5850			0.0850694
$290$		0			0
$291$		0			0
$292$		0			0
$293$		−	266.522i		−	0.909630i	−0.890586	−	0.454815i	$-0.849705\pi$
$293$		−	266.522i		−	0.909630i	0.890586	−	0.454815i	$-0.150295\pi$
$294$		0			0
$295$		−	517.274i		−	1.75347i
$296$		0			0
$297$		0			0
$298$		0			0
$299$			654.338i			2.18842i
$300$		0			0
$301$	−105.173			−0.349412
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−478.675			−1.56943
$306$		0			0
$307$		−	348.155i		−	1.13405i	−0.823699	−	0.567027i	$-0.808093\pi$
$307$		−	348.155i		−	1.13405i	0.823699	−	0.567027i	$-0.191907\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	41.9636			0.134931			0.0674656	−	0.997722i	$-0.478509\pi$
$311$	41.9636			0.134931			0.0674656	+	0.997722i	$0.478509\pi$
$312$		0			0
$313$	−100.914			−0.322407			−0.161204	−	0.986921i	$-0.551538\pi$
$313$	−100.914			−0.322407			−0.161204	+	0.986921i	$0.551538\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$			496.523i			1.56632i	0.621822	+	0.783159i	$0.286393\pi$
$317$			496.523i			1.56632i	−0.621822	+	0.783159i	$0.713607\pi$
$318$		0			0
$319$		−	10.4609i		−	0.0327929i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−193.452	+	275.283i	−0.598922	+	0.852268i
$324$		0			0
$325$			40.4625i			0.124500i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−107.564			−0.326943
$330$		0			0
$331$			49.8651i			0.150650i	0.997159	+	0.0753249i	$0.0239994\pi$
$331$			49.8651i			0.150650i	−0.997159	+	0.0753249i	$0.976001\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$			47.1189i			0.140654i
$336$		0			0
$337$			232.509i			0.689937i	0.938614	+	0.344969i	$0.112110\pi$
$337$			232.509i			0.689937i	−0.938614	+	0.344969i	$0.887890\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		−	716.561i		−	2.10135i
$342$		0			0
$343$	−172.392			−0.502600
$344$		0			0
$345$		0			0
$346$		0			0
$347$	371.792			1.07145			0.535723	−	0.844394i	$-0.320039\pi$
$347$	371.792			1.07145			0.535723	+	0.844394i	$0.320039\pi$
$348$		0			0
$349$	−377.732			−1.08233			−0.541163	−	0.840918i	$-0.682016\pi$
$349$	−377.732			−1.08233			−0.541163	+	0.840918i	$0.682016\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	354.723			1.00488			0.502441	−	0.864611i	$-0.332435\pi$
$353$	354.723			1.00488			0.502441	+	0.864611i	$0.332435\pi$
$354$		0			0
$355$		−	206.695i		−	0.582239i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	239.892			0.668222			0.334111	−	0.942534i	$-0.391564\pi$
$359$	239.892			0.668222			0.334111	+	0.942534i	$0.391564\pi$
$360$		0			0
$361$	−122.317	−	339.646i	−0.338830	−	0.940848i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−224.541			−0.615182
$366$		0			0
$367$	126.499			0.344683			0.172342	−	0.985037i	$-0.444867\pi$
$367$	126.499			0.344683			0.172342	+	0.985037i	$0.444867\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$			83.4631i			0.224968i
$372$		0			0
$373$			680.907i			1.82549i	0.408533	+	0.912743i	$0.366040\pi$
$373$			680.907i			1.82549i	−0.408533	+	0.912743i	$0.633960\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	9.88516			0.0262206
$378$		0			0
$379$			495.747i			1.30804i	0.756478	+	0.654019i	$0.226919\pi$
$379$			495.747i			1.30804i	−0.756478	+	0.654019i	$0.773081\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		−	416.620i		−	1.08778i	−0.839157	−	0.543890i	$-0.816951\pi$
$383$		−	416.620i		−	1.08778i	0.839157	−	0.543890i	$-0.183049\pi$
$384$		0			0
$385$	−150.717			−0.391473
$386$		0			0
$387$		0			0
$388$		0			0
$389$	288.566			0.741814			0.370907	−	0.928670i	$-0.379047\pi$
$389$	288.566			0.741814			0.370907	+	0.928670i	$0.379047\pi$
$390$		0			0
$391$	779.943			1.99474
$392$		0			0
$393$		0			0
$394$		0			0
$395$		−	479.859i		−	1.21483i
$396$		0			0
$397$	463.081			1.16645			0.583226	−	0.812310i	$-0.301790\pi$
$397$	463.081			1.16645			0.583226	+	0.812310i	$0.301790\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			357.965i			0.892681i	0.894863	+	0.446341i	$0.147273\pi$
$401$			357.965i			0.892681i	−0.894863	+	0.446341i	$0.852727\pi$
$402$		0			0
$403$	677.121			1.68020
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	116.769i		−	0.286902i
$408$		0			0
$409$			24.3425i			0.0595171i	0.999557	+	0.0297585i	$0.00947383\pi$
$409$			24.3425i			0.0595171i	−0.999557	+	0.0297585i	$0.990526\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$			178.868i			0.433093i
$414$		0			0
$415$	−210.682			−0.507668
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−412.807			−0.985219			−0.492609	−	0.870251i	$-0.663957\pi$
$419$	−412.807			−0.985219			−0.492609	+	0.870251i	$0.663957\pi$
$420$		0			0
$421$			297.960i			0.707745i	0.935294	+	0.353872i	$0.115135\pi$
$421$			297.960i			0.707745i	−0.935294	+	0.353872i	$0.884865\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	48.2296			0.113481
$426$		0			0
$427$	165.520			0.387636
$428$		0			0
$429$		0			0
$430$		0			0
$431$			450.233i			1.04462i	0.852754	+	0.522312i	$0.174930\pi$
$431$			450.233i			1.04462i	−0.852754	+	0.522312i	$0.825070\pi$
$432$		0			0
$433$		−	392.681i		−	0.906885i	−0.891286	−	0.453442i	$-0.850196\pi$
$433$		−	392.681i		−	0.906885i	0.891286	−	0.453442i	$-0.149804\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−481.150	+	684.679i	−1.10103	+	1.56677i
$438$		0			0
$439$		−	432.764i		−	0.985795i	−0.870087	−	0.492898i	$-0.835938\pi$
$439$		−	432.764i		−	0.985795i	0.870087	−	0.492898i	$-0.164062\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	470.774			1.06269			0.531347	−	0.847154i	$-0.321686\pi$
$443$	470.774			1.06269			0.531347	+	0.847154i	$0.321686\pi$
$444$		0			0
$445$			106.288i			0.238848i
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	455.958i		−	1.01550i	−0.861506	−	0.507748i	$-0.830478\pi$
$449$		−	455.958i		−	1.01550i	0.861506	−	0.507748i	$-0.169522\pi$
$450$		0			0
$451$		−	867.467i		−	1.92343i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		−	142.422i		−	0.313015i
$456$		0			0
$457$	−656.778			−1.43715			−0.718575	−	0.695449i	$-0.755205\pi$
$457$	−656.778			−1.43715			−0.718575	+	0.695449i	$0.755205\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	182.838			0.396612			0.198306	−	0.980140i	$-0.436456\pi$
$461$	182.838			0.396612			0.198306	+	0.980140i	$0.436456\pi$
$462$		0			0
$463$	−262.740			−0.567473			−0.283737	−	0.958902i	$-0.591574\pi$
$463$	−262.740			−0.567473			−0.283737	+	0.958902i	$0.591574\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−314.163			−0.672725			−0.336363	−	0.941733i	$-0.609197\pi$
$467$	−314.163			−0.672725			−0.336363	+	0.941733i	$0.609197\pi$
$468$		0			0
$469$		−	16.2932i		−	0.0347403i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−908.182			−1.92005
$474$		0			0
$475$	−29.7530	+	42.3387i	−0.0626380	+	0.0891340i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−104.233			−0.217606			−0.108803	−	0.994063i	$-0.534702\pi$
$479$	−104.233			−0.217606			−0.108803	+	0.994063i	$0.534702\pi$
$480$		0			0
$481$	110.342			0.229401
$482$		0			0
$483$		0			0
$484$		0			0
$485$			635.519i			1.31035i
$486$		0			0
$487$			233.674i			0.479824i	0.970795	+	0.239912i	$0.0771186\pi$
$487$			233.674i			0.479824i	−0.970795	+	0.239912i	$0.922881\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	853.107			1.73749			0.868745	−	0.495260i	$-0.164927\pi$
$491$	853.107			1.73749			0.868745	+	0.495260i	$0.164927\pi$
$492$		0			0
$493$		−	11.7827i		−	0.0239000i
$494$		0			0
$495$		0			0
$496$		0			0
$497$			71.4728i			0.143808i
$498$		0			0
$499$	340.053			0.681468			0.340734	−	0.940160i	$-0.389324\pi$
$499$	340.053			0.681468			0.340734	+	0.940160i	$0.389324\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	463.696			0.921862			0.460931	−	0.887436i	$-0.347516\pi$
$503$	463.696			0.921862			0.460931	+	0.887436i	$0.347516\pi$
$504$		0			0
$505$	−862.118			−1.70717
$506$		0			0
$507$		0			0
$508$		0			0
$509$			107.246i			0.210699i	0.994435	+	0.105350i	$0.0335962\pi$
$509$			107.246i			0.210699i	−0.994435	+	0.105350i	$0.966404\pi$
$510$		0			0
$511$	77.6439			0.151945
$512$		0			0
$513$		0			0
$514$		0			0
$515$		−	604.347i		−	1.17349i
$516$		0			0
$517$	−928.831			−1.79658
$518$		0			0
$519$		0			0
$520$		0			0
$521$			312.344i			0.599508i	0.954017	+	0.299754i	$0.0969046\pi$
$521$			312.344i			0.599508i	−0.954017	+	0.299754i	$0.903095\pi$
$522$		0			0
$523$			552.570i			1.05654i	0.849077	+	0.528269i	$0.177159\pi$
$523$			552.570i			1.05654i	−0.849077	+	0.528269i	$0.822841\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		−	807.100i		−	1.53150i
$528$		0			0
$529$	1410.86			2.66704
$530$		0			0
$531$		0			0
$532$		0			0
$533$	819.721			1.53794
$534$		0			0
$535$			1062.90i			1.98673i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−718.254			−1.33257
$540$		0			0
$541$	−875.879			−1.61900			−0.809500	−	0.587120i	$-0.800262\pi$
$541$	−875.879			−1.61900			−0.809500	+	0.587120i	$0.800262\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		−	490.762i		−	0.900480i
$546$		0			0
$547$		−	198.768i		−	0.363378i	−0.983356	−	0.181689i	$-0.941844\pi$
$547$		−	198.768i		−	0.363378i	0.983356	−	0.181689i	$-0.0581564\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	10.3435	+	7.26880i	0.0187723	+	0.0131920i
$552$		0			0
$553$			165.930i			0.300054i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−169.436			−0.304194			−0.152097	−	0.988366i	$-0.548603\pi$
$557$	−169.436			−0.304194			−0.152097	+	0.988366i	$0.548603\pi$
$558$		0			0
$559$		−	858.195i		−	1.53523i
$560$		0			0
$561$		0			0
$562$		0			0
$563$			201.520i			0.357940i	0.983855	+	0.178970i	$0.0572765\pi$
$563$			201.520i			0.357940i	−0.983855	+	0.178970i	$0.942724\pi$
$564$		0			0
$565$		−	235.217i		−	0.416313i
$566$		0			0
$567$		0			0
$568$		0			0
$569$		−	247.493i		−	0.434961i	−0.976065	−	0.217480i	$-0.930216\pi$
$569$		−	247.493i		−	0.434961i	0.976065	−	0.217480i	$-0.0697838\pi$
$570$		0			0
$571$	−558.184			−0.977555			−0.488777	−	0.872409i	$-0.662557\pi$
$571$	−558.184			−0.977555			−0.488777	+	0.872409i	$0.662557\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	119.956			0.208619
$576$		0			0
$577$	−759.677			−1.31660			−0.658299	−	0.752756i	$-0.728724\pi$
$577$	−759.677			−1.31660			−0.658299	+	0.752756i	$0.728724\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	72.8516			0.125390
$582$		0			0
$583$			720.714i			1.23622i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−304.580			−0.518875			−0.259437	−	0.965760i	$-0.583537\pi$
$587$	−304.580			−0.518875			−0.259437	+	0.965760i	$0.583537\pi$
$588$		0			0
$589$	708.519	+	497.904i	1.20292	+	0.845337i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	860.996			1.45193			0.725966	−	0.687731i	$-0.241393\pi$
$593$	860.996			1.45193			0.725966	+	0.687731i	$0.241393\pi$
$594$		0			0
$595$	−169.761			−0.285312
$596$		0			0
$597$		0			0
$598$		0			0
$599$			759.453i			1.26787i	0.773387	+	0.633934i	$0.218561\pi$
$599$			759.453i			1.26787i	−0.773387	+	0.633934i	$0.781439\pi$
$600$		0			0
$601$		−	651.604i		−	1.08420i	−0.840314	−	0.542100i	$-0.817629\pi$
$601$		−	651.604i		−	1.08420i	0.840314	−	0.542100i	$-0.182371\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−664.358			−1.09811
$606$		0			0
$607$			609.462i			1.00406i	0.864851	+	0.502028i	$0.167413\pi$
$607$			609.462i			1.00406i	−0.864851	+	0.502028i	$0.832587\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		−	877.707i		−	1.43651i
$612$		0			0
$613$	−196.796			−0.321037			−0.160519	−	0.987033i	$-0.551317\pi$
$613$	−196.796			−0.321037			−0.160519	+	0.987033i	$0.551317\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	957.322			1.55158			0.775788	−	0.630994i	$-0.217353\pi$
$617$	957.322			1.55158			0.775788	+	0.630994i	$0.217353\pi$
$618$		0			0
$619$	237.874			0.384287			0.192144	−	0.981367i	$-0.438456\pi$
$619$	237.874			0.384287			0.192144	+	0.981367i	$0.438456\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		−	36.7530i		−	0.0589936i
$624$		0			0
$625$	−685.671			−1.09707
$626$		0			0
$627$		0			0
$628$		0			0
$629$		−	131.523i		−	0.209099i
$630$		0			0
$631$	561.383			0.889672			0.444836	−	0.895612i	$-0.353262\pi$
$631$	561.383			0.889672			0.444836	+	0.895612i	$0.353262\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		−	821.148i		−	1.29315i
$636$		0			0
$637$		−	678.721i		−	1.06550i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		−	200.894i		−	0.313407i	−0.987646	−	0.156704i	$-0.949913\pi$
$641$		−	200.894i		−	0.313407i	0.987646	−	0.156704i	$-0.0500867\pi$
$642$		0			0
$643$	−900.914			−1.40111			−0.700555	−	0.713598i	$-0.747064\pi$
$643$	−900.914			−1.40111			−0.700555	+	0.713598i	$0.747064\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	192.631			0.297729			0.148864	−	0.988858i	$-0.452438\pi$
$647$	192.631			0.297729			0.148864	+	0.988858i	$0.452438\pi$
$648$		0			0
$649$			1544.54i			2.37988i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	324.749			0.497319			0.248659	−	0.968591i	$-0.420010\pi$
$653$	324.749			0.497319			0.248659	+	0.968591i	$0.420010\pi$
$654$		0			0
$655$	633.085			0.966542
$656$		0			0
$657$		0			0
$658$		0			0
$659$			751.856i			1.14090i	0.821331	+	0.570452i	$0.193232\pi$
$659$			751.856i			1.14090i	−0.821331	+	0.570452i	$0.806768\pi$
$660$		0			0
$661$		−	193.503i		−	0.292742i	−0.989230	−	0.146371i	$-0.953241\pi$
$661$		−	193.503i		−	0.292742i	0.989230	−	0.146371i	$-0.0467593\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	104.726	−	149.026i	0.157483	−	0.224099i
$666$		0			0
$667$		−	29.3057i		−	0.0439367i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	1429.29			2.13009
$672$		0			0
$673$			717.257i			1.06576i	0.846191	+	0.532880i	$0.178890\pi$
$673$			717.257i			1.06576i	−0.846191	+	0.532880i	$0.821110\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$			552.319i			0.815833i	0.913019	+	0.407916i	$0.133744\pi$
$677$			552.319i			0.815833i	−0.913019	+	0.407916i	$0.866256\pi$
$678$		0			0
$679$		−	219.755i		−	0.323646i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		−	203.228i		−	0.297552i	−0.988871	−	0.148776i	$-0.952467\pi$
$683$		−	203.228i		−	0.297552i	0.988871	−	0.148776i	$-0.0475334\pi$
$684$		0			0
$685$	−938.540			−1.37013
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−681.045			−0.988454
$690$		0			0
$691$	−1040.78			−1.50620			−0.753099	−	0.657907i	$-0.771442\pi$
$691$	−1040.78			−1.50620			−0.753099	+	0.657907i	$0.771442\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−590.860			−0.850159
$696$		0			0
$697$		−	977.074i		−	1.40183i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	433.924			0.619007			0.309503	−	0.950898i	$-0.399837\pi$
$701$	433.924			0.619007			0.309503	+	0.950898i	$0.399837\pi$
$702$		0			0
$703$	115.458	+	81.1372i	0.164237	+	0.115416i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	298.111			0.421656
$708$		0			0
$709$	620.356			0.874973			0.437487	−	0.899225i	$-0.355869\pi$
$709$	620.356			0.874973			0.437487	+	0.899225i	$0.355869\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		−	2007.41i		−	2.81544i
$714$		0			0
$715$		−	1229.83i		−	1.72004i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−246.163			−0.342369			−0.171185	−	0.985239i	$-0.554759\pi$
$719$	−246.163			−0.342369			−0.171185	+	0.985239i	$0.554759\pi$
$720$		0			0
$721$			208.976i			0.289842i
$722$		0			0
$723$		0			0
$724$		0			0
$725$		−	1.81219i		−	0.00249957i
$726$		0			0
$727$	−398.218			−0.547755			−0.273878	−	0.961765i	$-0.588306\pi$
$727$	−398.218			−0.547755			−0.273878	+	0.961765i	$0.588306\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−1022.93			−1.39936
$732$		0			0
$733$	847.993			1.15688			0.578440	−	0.815725i	$-0.303662\pi$
$733$	847.993			1.15688			0.578440	+	0.815725i	$0.303662\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	140.694i		−	0.190900i
$738$		0			0
$739$	−255.781			−0.346118			−0.173059	−	0.984911i	$-0.555365\pi$
$739$	−255.781			−0.346118			−0.173059	+	0.984911i	$0.555365\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	91.0349i		−	0.122523i	−0.998122	−	0.0612617i	$-0.980488\pi$
$743$		−	91.0349i		−	0.122523i	0.998122	−	0.0612617i	$-0.0195124\pi$
$744$		0			0
$745$	165.372			0.221976
$746$		0			0
$747$		0			0
$748$		0			0
$749$		−	367.540i		−	0.490707i
$750$		0			0
$751$			425.391i			0.566432i	0.959056	+	0.283216i	$0.0914014\pi$
$751$			425.391i			0.566432i	−0.959056	+	0.283216i	$0.908599\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		−	1013.49i		−	1.34237i
$756$		0			0
$757$	64.7355			0.0855159			0.0427579	−	0.999085i	$-0.486386\pi$
$757$	64.7355			0.0855159			0.0427579	+	0.999085i	$0.486386\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−662.440			−0.870486			−0.435243	−	0.900313i	$-0.643337\pi$
$761$	−662.440			−0.870486			−0.435243	+	0.900313i	$0.643337\pi$
$762$		0			0
$763$			169.700i			0.222411i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−1459.53			−1.90291
$768$		0			0
$769$	1352.19			1.75837			0.879186	−	0.476479i	$-0.158087\pi$
$769$	1352.19			1.75837			0.879186	+	0.476479i	$0.158087\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		−	1098.80i		−	1.42147i	−0.703457	−	0.710737i	$-0.748361\pi$
$773$		−	1098.80i		−	1.42147i	0.703457	−	0.710737i	$-0.251639\pi$
$774$		0			0
$775$		−	124.133i		−	0.160171i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	857.731	+	602.761i	1.10107	+	0.773762i
$780$		0			0
$781$			617.176i			0.790238i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−1129.20			−1.43847
$786$		0			0
$787$			848.610i			1.07828i	0.842215	+	0.539142i	$0.181252\pi$
$787$			848.610i			1.07828i	−0.842215	+	0.539142i	$0.818748\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$			81.3353i			0.102826i
$792$		0			0
$793$			1350.62i			1.70318i
$794$		0			0
$795$		0			0
$796$		0			0
$797$		−	581.820i		−	0.730013i	−0.931005	−	0.365006i	$-0.881067\pi$
$797$		−	581.820i		−	0.730013i	0.931005	−	0.365006i	$-0.118933\pi$
$798$		0			0
$799$	−1046.19			−1.30938
$800$		0			0
$801$		0			0
$802$		0			0
$803$	670.464			0.834949
$804$		0			0
$805$	−422.226			−0.524504
$806$		0			0
$807$		0			0
$808$		0			0
$809$	940.603			1.16267			0.581337	−	0.813663i	$-0.302530\pi$
$809$	940.603			1.16267			0.581337	+	0.813663i	$0.302530\pi$
$810$		0			0
$811$		−	1256.42i		−	1.54923i	−0.632436	−	0.774613i	$-0.717945\pi$
$811$		−	1256.42i		−	1.54923i	0.632436	−	0.774613i	$-0.282055\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−466.344			−0.572202
$816$		0			0
$817$	631.052	−	897.989i	0.772401	−	1.09913i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	540.562			0.658419			0.329210	−	0.944257i	$-0.393218\pi$
$821$	540.562			0.658419			0.329210	+	0.944257i	$0.393218\pi$
$822$		0			0
$823$	−1385.73			−1.68375			−0.841875	−	0.539672i	$-0.818548\pi$
$823$	−1385.73			−1.68375			−0.841875	+	0.539672i	$0.818548\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			621.992i			0.752107i	0.926598	+	0.376053i	$0.122719\pi$
$827$			621.992i			0.752107i	−0.926598	+	0.376053i	$0.877281\pi$
$828$		0			0
$829$		−	1195.71i		−	1.44235i	−0.692752	−	0.721176i	$-0.743602\pi$
$829$		−	1195.71i		−	1.44235i	0.692752	−	0.721176i	$-0.256398\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−809.007			−0.971197
$834$		0			0
$835$			452.885i			0.542377i
$836$		0			0
$837$		0			0
$838$		0			0
$839$		−	68.5165i		−	0.0816645i	−0.999166	−	0.0408322i	$-0.986999\pi$
$839$		−	68.5165i		−	0.0816645i	0.999166	−	0.0408322i	$-0.0130009\pi$
$840$		0			0
$841$	840.557			0.999474
$842$		0			0
$843$		0			0
$844$		0			0
$845$	272.299			0.322247
$846$		0			0
$847$	229.727			0.271225
$848$		0			0
$849$		0			0
$850$		0			0
$851$		−	327.122i		−	0.384397i
$852$		0			0
$853$	45.1625			0.0529455			0.0264727	−	0.999650i	$-0.491572\pi$
$853$	45.1625			0.0529455			0.0264727	+	0.999650i	$0.491572\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	1425.47i		−	1.66333i	−0.555281	−	0.831663i	$-0.687389\pi$
$857$		−	1425.47i		−	1.66333i	0.555281	−	0.831663i	$-0.312611\pi$
$858$		0			0
$859$	1119.59			1.30336			0.651681	−	0.758493i	$-0.274064\pi$
$859$	1119.59			1.30336			0.651681	+	0.758493i	$0.274064\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			401.475i			0.465208i	0.972572	+	0.232604i	$0.0747246\pi$
$863$			401.475i			0.465208i	−0.972572	+	0.232604i	$0.925275\pi$
$864$		0			0
$865$		−	1693.09i		−	1.95733i
$866$		0			0
$867$		0			0
$868$		0			0
$869$			1432.82i			1.64882i
$870$		0			0
$871$	132.950			0.152640
$872$		0			0
$873$		0			0
$874$		0			0
$875$	213.553			0.244060
$876$		0			0
$877$		−	1268.23i		−	1.44610i	−0.690797	−	0.723049i	$-0.742740\pi$
$877$		−	1268.23i		−	1.44610i	0.690797	−	0.723049i	$-0.257260\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−1653.03			−1.87631			−0.938157	−	0.346211i	$-0.887468\pi$
$881$	−1653.03			−1.87631			−0.938157	+	0.346211i	$0.887468\pi$
$882$		0			0
$883$	−228.431			−0.258699			−0.129350	−	0.991599i	$-0.541289\pi$
$883$	−228.431			−0.258699			−0.129350	+	0.991599i	$0.541289\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		−	14.3500i		−	0.0161781i	−0.999967	−	0.00808906i	$-0.997425\pi$
$887$		−	14.3500i		−	0.0161781i	0.999967	−	0.00808906i	$-0.00257486\pi$
$888$		0			0
$889$			283.944i			0.319397i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	645.400	−	918.406i	0.722732	−	1.02845i
$894$		0			0
$895$			90.6438i			0.101278i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−30.3262			−0.0337332
$900$		0			0
$901$			811.778i			0.900974i
$902$		0			0
$903$		0			0
$904$		0			0
$905$			1115.13i			1.23219i
$906$		0			0
$907$			1753.03i			1.93277i	0.257093	+	0.966387i	$0.417235\pi$
$907$			1753.03i			1.93277i	−0.257093	+	0.966387i	$0.582765\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	260.877i		−	0.286363i	−0.989696	−	0.143182i	$-0.954267\pi$
$911$		−	260.877i		−	0.286363i	0.989696	−	0.143182i	$-0.0457333\pi$
$912$		0			0
$913$	629.082			0.689027
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−218.914			−0.238728
$918$		0			0
$919$	−141.465			−0.153934			−0.0769671	−	0.997034i	$-0.524524\pi$
$919$	−141.465			−0.153934			−0.0769671	+	0.997034i	$0.524524\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−583.206			−0.631859
$924$		0			0
$925$		−	20.2283i		−	0.0218685i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−756.153			−0.813943			−0.406971	−	0.913441i	$-0.633415\pi$
$929$	−756.153			−0.813943			−0.406971	+	0.913441i	$0.633415\pi$
$930$		0			0
$931$	499.080	−	710.192i	0.536068	−	0.762827i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−1465.90			−1.56781
$936$		0			0
$937$	404.909			0.432134			0.216067	−	0.976379i	$-0.430677\pi$
$937$	404.909			0.432134			0.216067	+	0.976379i	$0.430677\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$			1087.81i			1.15602i	0.816030	+	0.578010i	$0.196171\pi$
$941$			1087.81i			1.15602i	−0.816030	+	0.578010i	$0.803829\pi$
$942$		0			0
$943$		−	2430.16i		−	2.57705i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−1301.01			−1.37382			−0.686911	−	0.726741i	$-0.741034\pi$
$947$	−1301.01			−1.37382			−0.686911	+	0.726741i	$0.741034\pi$
$948$		0			0
$949$			633.561i			0.667609i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	878.343i		−	0.921661i	−0.887488	−	0.460830i	$-0.847552\pi$
$953$		−	878.343i		−	0.921661i	0.887488	−	0.460830i	$-0.152448\pi$
$954$		0			0
$955$	−747.803			−0.783040
$956$		0			0
$957$		0			0
$958$		0			0
$959$	324.536			0.338411
$960$		0			0
$961$	−1116.30			−1.16161
$962$		0			0
$963$		0			0
$964$		0			0
$965$		−	69.7253i		−	0.0722542i
$966$		0			0
$967$	−931.268			−0.963048			−0.481524	−	0.876433i	$-0.659917\pi$
$967$	−931.268			−0.963048			−0.481524	+	0.876433i	$0.659917\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$			1074.18i			1.10626i	0.833094	+	0.553132i	$0.186567\pi$
$971$			1074.18i			1.10626i	−0.833094	+	0.553132i	$0.813433\pi$
$972$		0			0
$973$	204.313			0.209982
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	394.510i		−	0.403797i	−0.979406	−	0.201899i	$-0.935289\pi$
$977$		−	394.510i		−	0.403797i	0.979406	−	0.201899i	$-0.0647111\pi$
$978$		0			0
$979$		−	317.367i		−	0.324174i
$980$		0			0
$981$		0			0
$982$		0			0
$983$		−	1258.92i		−	1.28069i	−0.768085	−	0.640347i	$-0.778790\pi$
$983$		−	1258.92i		−	1.28069i	0.768085	−	0.640347i	$-0.221210\pi$
$984$		0			0
$985$	−122.460			−0.124324
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−2544.22			−2.57252
$990$		0			0
$991$		−	460.326i		−	0.464507i	−0.972655	−	0.232253i	$-0.925390\pi$
$991$		−	460.326i		−	0.464507i	0.972655	−	0.232253i	$-0.0746098\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	171.903			0.172767
$996$		0			0
$997$	−503.952			−0.505469			−0.252734	−	0.967536i	$-0.581330\pi$
$997$	−503.952			−0.505469			−0.252734	+	0.967536i	$0.581330\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2736.3.o.r.721.6		20
3.2	odd	2		912.3.o.e.721.8		20
4.3	odd	2		1368.3.o.c.721.6		20
12.11	even	2		456.3.o.a.265.18	yes	20
19.18	odd	2	inner	2736.3.o.r.721.5		20
57.56	even	2		912.3.o.e.721.18		20
76.75	even	2		1368.3.o.c.721.5		20
228.227	odd	2		456.3.o.a.265.8	✓	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
456.3.o.a.265.8	✓	20	228.227	odd	2
456.3.o.a.265.18	yes	20	12.11	even	2
912.3.o.e.721.8		20	3.2	odd	2
912.3.o.e.721.18		20	57.56	even	2
1368.3.o.c.721.5		20	76.75	even	2
1368.3.o.c.721.6		20	4.3	odd	2
2736.3.o.r.721.5		20	19.18	odd	2	inner
2736.3.o.r.721.6		20	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 2736 = 2^{4} \cdot 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	2736.o (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-5.26532 q^{5} +1.82069 q^{7} +O(q^{10})\)	\(q-5.26532 q^{5} +1.82069 q^{7} +15.7218 q^{11} +14.8565i q^{13} +17.7083 q^{17} +(-10.9243 + 15.5454i) q^{19} +44.0439 q^{23} +2.72355 q^{25} -0.665376i q^{29} -45.5774i q^{31} -9.58649 q^{35} -7.42719i q^{37} -55.1759i q^{41} -57.7656 q^{43} -59.0790 q^{47} -45.6851 q^{49} +45.8416i q^{53} -82.7805 q^{55} +98.2419i q^{59} +90.9110 q^{61} -78.2242i q^{65} -8.94893i q^{67} +39.2559i q^{71} +42.6454 q^{73} +28.6245 q^{77} +91.1359i q^{79} +40.0132 q^{83} -93.2400 q^{85} -20.1864i q^{89} +27.0490i q^{91} +(57.5201 - 81.8513i) q^{95} -120.699i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 16 q^{7}+O(q^{10}) \) 20 * q + 16 * q^7	\( 20 q + 16 q^{7} + 16 q^{11} - 32 q^{17} - 40 q^{19} + 64 q^{23} + 68 q^{25} - 208 q^{35} - 64 q^{43} + 48 q^{47} + 20 q^{49} + 336 q^{55} + 184 q^{61} + 104 q^{73} - 88 q^{77} + 224 q^{83} - 136 q^{85} - 320 q^{95}+O(q^{100}) \) 20 * q + 16 * q^7 + 16 * q^11 - 32 * q^17 - 40 * q^19 + 64 * q^23 + 68 * q^25 - 208 * q^35 - 64 * q^43 + 48 * q^47 + 20 * q^49 + 336 * q^55 + 184 * q^61 + 104 * q^73 - 88 * q^77 + 224 * q^83 - 136 * q^85 - 320 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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