LMFDB - Embedded newform 4014.2.d.a.4013.13

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4014,2,Mod(4013,4014)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4014, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("4014.4013");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4014 = 2 \cdot 3^{2} \cdot 223 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4014.d (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:	$32.0519513713$
Analytic rank:	$0$
Dimension:	$72$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			4013.13
Character	$\chi$	$=$	4014.4013
Dual form			4014.2.d.a.4013.14

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{2} -1.00000 q^{4} +2.29842 q^{5} +2.05558 q^{7} +1.00000i q^{8} +O(q^{10})$	\(q-1.00000i q^{2} -1.00000 q^{4} +2.29842 q^{5} +2.05558 q^{7} +1.00000i q^{8} -2.29842i q^{10} +0.573894 q^{11} -5.56690i q^{13} -2.05558i q^{14} +1.00000 q^{16} -5.74146i q^{17} +4.17599 q^{19} -2.29842 q^{20} -0.573894i q^{22} +2.06070 q^{23} +0.282734 q^{25} -5.56690 q^{26} -2.05558 q^{28} -3.14750i q^{29} +2.78812 q^{31} -1.00000i q^{32} -5.74146 q^{34} +4.72459 q^{35} +0.611051 q^{37} -4.17599i q^{38} +2.29842i q^{40} -1.16658i q^{41} -9.54102 q^{43} -0.573894 q^{44} -2.06070i q^{46} +6.02089i q^{47} -2.77458 q^{49} -0.282734i q^{50} +5.56690i q^{52} +6.82999i q^{53} +1.31905 q^{55} +2.05558i q^{56} -3.14750 q^{58} -1.99536 q^{59} -14.1655i q^{61} -2.78812i q^{62} -1.00000 q^{64} -12.7951i q^{65} +6.44119i q^{67} +5.74146i q^{68} -4.72459i q^{70} -0.0503915 q^{71} -0.158279 q^{73} -0.611051i q^{74} -4.17599 q^{76} +1.17969 q^{77} +3.77417i q^{79} +2.29842 q^{80} -1.16658 q^{82} +3.37366i q^{83} -13.1963i q^{85} +9.54102i q^{86} +0.573894i q^{88} +8.91850i q^{89} -11.4432i q^{91} -2.06070 q^{92} +6.02089 q^{94} +9.59817 q^{95} -9.51962i q^{97} +2.77458i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 72 q - 72 q^{4} + 16 q^{7}+O(q^{10}) $ 72 * q - 72 * q^4 + 16 * q^7	$ 72 q - 72 q^{4} + 16 q^{7} + 72 q^{16} - 40 q^{19} + 96 q^{25} - 16 q^{28} - 24 q^{37} - 8 q^{43} + 56 q^{49} + 40 q^{58} - 72 q^{64} - 32 q^{73} + 40 q^{76} + 16 q^{82}+O(q^{100}) $ 72 * q - 72 * q^4 + 16 * q^7 + 72 * q^16 - 40 * q^19 + 96 * q^25 - 16 * q^28 - 24 * q^37 - 8 * q^43 + 56 * q^49 + 40 * q^58 - 72 * q^64 - 32 * q^73 + 40 * q^76 + 16 * q^82

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4014\mathbb{Z}\right)^\times$.

$n$	$893$	$2233$
$\chi(n)$	$-1$	$-1$

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 $

$2$		−	1.00000i		−	0.707107i
$2$		−	1.00000i		−	0.707107i
$3$		0			0
$3$		0			0
$4$	−1.00000			−0.500000
$5$	2.29842			1.02788			0.513942	−	0.857825i	$-0.328185\pi$
$5$	2.29842			1.02788			0.513942	+	0.857825i	$0.328185\pi$
$6$		0			0
$7$	2.05558			0.776937			0.388468	−	0.921462i	$-0.373004\pi$
$7$	2.05558			0.776937			0.388468	+	0.921462i	$0.373004\pi$
$8$			1.00000i			0.353553i
$9$		0			0
$10$		−	2.29842i		−	0.726824i
$11$	0.573894			0.173036			0.0865178	−	0.996250i	$-0.472426\pi$
$11$	0.573894			0.173036			0.0865178	+	0.996250i	$0.472426\pi$
$12$		0			0
$13$		−	5.56690i		−	1.54398i	−0.635635	−	0.771990i	$-0.719262\pi$
$13$		−	5.56690i		−	1.54398i	0.635635	−	0.771990i	$-0.280738\pi$
$14$		−	2.05558i		−	0.549377i
$15$		0			0
$16$	1.00000			0.250000
$17$		−	5.74146i		−	1.39251i	−0.717795	−	0.696255i	$-0.754848\pi$
$17$		−	5.74146i		−	1.39251i	0.717795	−	0.696255i	$-0.245152\pi$
$18$		0			0
$19$	4.17599			0.958037			0.479018	−	0.877805i	$-0.340993\pi$
$19$	4.17599			0.958037			0.479018	+	0.877805i	$0.340993\pi$
$20$	−2.29842			−0.513942
$21$		0			0
$22$		−	0.573894i		−	0.122355i
$23$	2.06070			0.429685			0.214843	−	0.976649i	$-0.431076\pi$
$23$	2.06070			0.429685			0.214843	+	0.976649i	$0.431076\pi$
$24$		0			0
$25$	0.282734			0.0565469
$26$	−5.56690			−1.09176
$27$		0			0
$28$	−2.05558			−0.388468
$29$		−	3.14750i		−	0.584477i	−0.956346	−	0.292238i	$-0.905600\pi$
$29$		−	3.14750i		−	0.584477i	0.956346	−	0.292238i	$-0.0944001\pi$
$30$		0			0
$31$	2.78812			0.500761			0.250381	−	0.968147i	$-0.419444\pi$
$31$	2.78812			0.500761			0.250381	+	0.968147i	$0.419444\pi$
$32$		−	1.00000i		−	0.176777i
$33$		0			0
$34$	−5.74146			−0.984653
$35$	4.72459			0.798601
$36$		0			0
$37$	0.611051			0.100456			0.0502281	−	0.998738i	$-0.484005\pi$
$37$	0.611051			0.100456			0.0502281	+	0.998738i	$0.484005\pi$
$38$		−	4.17599i		−	0.677434i
$39$		0			0
$40$			2.29842i			0.363412i
$41$		−	1.16658i		−	0.182189i	−0.995842	−	0.0910945i	$-0.970963\pi$
$41$		−	1.16658i		−	0.182189i	0.995842	−	0.0910945i	$-0.0290365\pi$
$42$		0			0
$43$	−9.54102			−1.45499			−0.727496	−	0.686112i	$-0.759316\pi$
$43$	−9.54102			−1.45499			−0.727496	+	0.686112i	$0.759316\pi$
$44$	−0.573894			−0.0865178
$45$		0			0
$46$		−	2.06070i		−	0.303833i
$47$			6.02089i			0.878237i	0.898429	+	0.439118i	$0.144709\pi$
$47$			6.02089i			0.878237i	−0.898429	+	0.439118i	$0.855291\pi$
$48$		0			0
$49$	−2.77458			−0.396369
$50$		−	0.282734i		−	0.0399847i
$51$		0			0
$52$			5.56690i			0.771990i
$53$			6.82999i			0.938171i	0.883153	+	0.469086i	$0.155416\pi$
$53$			6.82999i			0.938171i	−0.883153	+	0.469086i	$0.844584\pi$
$54$		0			0
$55$	1.31905			0.177861
$56$			2.05558i			0.274689i
$57$		0			0
$58$	−3.14750			−0.413287
$59$	−1.99536			−0.259773			−0.129887	−	0.991529i	$-0.541461\pi$
$59$	−1.99536			−0.259773			−0.129887	+	0.991529i	$0.541461\pi$
$60$		0			0
$61$		−	14.1655i		−	1.81371i	−0.421443	−	0.906855i	$-0.638476\pi$
$61$		−	14.1655i		−	1.81371i	0.421443	−	0.906855i	$-0.361524\pi$
$62$		−	2.78812i		−	0.354092i
$63$		0			0
$64$	−1.00000			−0.125000
$65$		−	12.7951i		−	1.58703i
$66$		0			0
$67$			6.44119i			0.786917i	0.919342	+	0.393458i	$0.128721\pi$
$67$			6.44119i			0.786917i	−0.919342	+	0.393458i	$0.871279\pi$
$68$			5.74146i			0.696255i
$69$		0			0
$70$		−	4.72459i		−	0.564697i
$71$	−0.0503915			−0.00598037			−0.00299018	−	0.999996i	$-0.500952\pi$
$71$	−0.0503915			−0.00598037			−0.00299018	+	0.999996i	$0.500952\pi$
$72$		0			0
$73$	−0.158279			−0.0185252			−0.00926259	−	0.999957i	$-0.502948\pi$
$73$	−0.158279			−0.0185252			−0.00926259	+	0.999957i	$0.502948\pi$
$74$		−	0.611051i		−	0.0710332i
$75$		0			0
$76$	−4.17599			−0.479018
$77$	1.17969			0.134438
$78$		0			0
$79$			3.77417i			0.424628i	0.977202	+	0.212314i	$0.0680999\pi$
$79$			3.77417i			0.424628i	−0.977202	+	0.212314i	$0.931900\pi$
$80$	2.29842			0.256971
$81$		0			0
$82$	−1.16658			−0.128827
$83$			3.37366i			0.370308i	0.982710	+	0.185154i	$0.0592783\pi$
$83$			3.37366i			0.370308i	−0.982710	+	0.185154i	$0.940722\pi$
$84$		0			0
$85$		−	13.1963i		−	1.43134i
$86$			9.54102i			1.02883i
$87$		0			0
$88$			0.573894i			0.0611773i
$89$			8.91850i			0.945359i	0.881234	+	0.472680i	$0.156713\pi$
$89$			8.91850i			0.945359i	−0.881234	+	0.472680i	$0.843287\pi$
$90$		0			0
$91$		−	11.4432i		−	1.19957i
$92$	−2.06070			−0.214843
$93$		0			0
$94$	6.02089			0.621007
$95$	9.59817			0.984751
$96$		0			0
$97$		−	9.51962i		−	0.966571i	−0.875463	−	0.483285i	$-0.839443\pi$
$97$		−	9.51962i		−	0.966571i	0.875463	−	0.483285i	$-0.160557\pi$
$98$			2.77458i			0.280275i
$99$		0			0
$100$	−0.282734			−0.0282734
$101$			5.99617i			0.596641i	0.954466	+	0.298321i	$0.0964265\pi$
$101$			5.99617i			0.596641i	−0.954466	+	0.298321i	$0.903574\pi$
$102$		0			0
$103$		−	12.1527i		−	1.19744i	−0.800957	−	0.598721i	$-0.795676\pi$
$103$		−	12.1527i		−	1.19744i	0.800957	−	0.598721i	$-0.204324\pi$
$104$	5.56690			0.545879
$105$		0			0
$106$	6.82999			0.663387
$107$	−15.0973			−1.45951			−0.729755	−	0.683709i	$-0.760366\pi$
$107$	−15.0973			−1.45951			−0.729755	+	0.683709i	$0.760366\pi$
$108$		0			0
$109$	9.61079			0.920547			0.460273	−	0.887777i	$-0.347751\pi$
$109$	9.61079			0.920547			0.460273	+	0.887777i	$0.347751\pi$
$110$		−	1.31905i		−	0.125766i
$111$		0			0
$112$	2.05558			0.194234
$113$	7.55260			0.710489			0.355245	−	0.934773i	$-0.384398\pi$
$113$	7.55260			0.710489			0.355245	+	0.934773i	$0.384398\pi$
$114$		0			0
$115$	4.73635			0.441667
$116$			3.14750i			0.292238i
$117$		0			0
$118$			1.99536i			0.183687i
$119$		−	11.8020i		−	1.08189i
$120$		0			0
$121$	−10.6706			−0.970059
$122$	−14.1655			−1.28249
$123$		0			0
$124$	−2.78812			−0.250381
$125$	−10.8423			−0.969761
$126$		0			0
$127$	5.15057			0.457039			0.228520	−	0.973539i	$-0.426611\pi$
$127$	5.15057			0.457039			0.228520	+	0.973539i	$0.426611\pi$
$128$			1.00000i			0.0883883i
$129$		0			0
$130$	−12.7951			−1.12220
$131$		−	4.12258i		−	0.360191i	−0.983649	−	0.180096i	$-0.942359\pi$
$131$		−	4.12258i		−	0.360191i	0.983649	−	0.180096i	$-0.0576408\pi$
$132$		0			0
$133$	8.58408			0.744334
$134$	6.44119			0.556434
$135$		0			0
$136$	5.74146			0.492326
$137$	19.9960			1.70838			0.854188	−	0.519964i	$-0.174055\pi$
$137$	19.9960			1.70838			0.854188	+	0.519964i	$0.174055\pi$
$138$		0			0
$139$	17.2958			1.46701			0.733503	−	0.679686i	$-0.237884\pi$
$139$	17.2958			1.46701			0.733503	+	0.679686i	$0.237884\pi$
$140$	−4.72459			−0.399301
$141$		0			0
$142$			0.0503915i			0.00422876i
$143$		−	3.19481i		−	0.267163i
$144$		0			0
$145$		−	7.23428i		−	0.600775i
$146$			0.158279i			0.0130993i
$147$		0			0
$148$	−0.611051			−0.0502281
$149$	−7.18737			−0.588812			−0.294406	−	0.955680i	$-0.595122\pi$
$149$	−7.18737			−0.588812			−0.294406	+	0.955680i	$0.595122\pi$
$150$		0			0
$151$		−	19.2437i		−	1.56603i	−0.622000	−	0.783017i	$-0.713680\pi$
$151$		−	19.2437i		−	1.56603i	0.622000	−	0.783017i	$-0.286320\pi$
$152$			4.17599i			0.338717i
$153$		0			0
$154$		−	1.17969i		−	0.0950619i
$155$	6.40827			0.514725
$156$		0			0
$157$			2.53661i			0.202444i	0.994864	+	0.101222i	$0.0322752\pi$
$157$			2.53661i			0.202444i	−0.994864	+	0.101222i	$0.967725\pi$
$158$	3.77417			0.300257
$159$		0			0
$160$		−	2.29842i		−	0.181706i
$161$	4.23593			0.333838
$162$		0			0
$163$			0.731953i			0.0573310i	0.999589	+	0.0286655i	$0.00912576\pi$
$163$			0.731953i			0.0573310i	−0.999589	+	0.0286655i	$0.990874\pi$
$164$			1.16658i			0.0910945i
$165$		0			0
$166$	3.37366			0.261847
$167$	6.96980			0.539339			0.269670	−	0.962953i	$-0.413086\pi$
$167$	6.96980			0.539339			0.269670	+	0.962953i	$0.413086\pi$
$168$		0			0
$169$	−17.9903			−1.38387
$170$	−13.1963			−1.01211
$171$		0			0
$172$	9.54102			0.727496
$173$	18.9794			1.44298			0.721488	−	0.692427i	$-0.243458\pi$
$173$	18.9794			1.44298			0.721488	+	0.692427i	$0.243458\pi$
$174$		0			0
$175$	0.581184			0.0439334
$176$	0.573894			0.0432589
$177$		0			0
$178$	8.91850			0.668470
$179$			6.60748i			0.493866i	0.969033	+	0.246933i	$0.0794228\pi$
$179$			6.60748i			0.493866i	−0.969033	+	0.246933i	$0.920577\pi$
$180$		0			0
$181$	12.1359			0.902052			0.451026	−	0.892511i	$-0.351058\pi$
$181$	12.1359			0.902052			0.451026	+	0.892511i	$0.351058\pi$
$182$	−11.4432			−0.848227
$183$		0			0
$184$			2.06070i			0.151917i
$185$	1.40445			0.103257
$186$		0			0
$187$		−	3.29499i		−	0.240954i
$188$		−	6.02089i		−	0.439118i
$189$		0			0
$190$		−	9.59817i		−	0.696324i
$191$	3.80003			0.274961			0.137480	−	0.990504i	$-0.456100\pi$
$191$	3.80003			0.274961			0.137480	+	0.990504i	$0.456100\pi$
$192$		0			0
$193$		−	16.9361i		−	1.21909i	−0.792751	−	0.609545i	$-0.791352\pi$
$193$		−	16.9361i		−	1.21909i	0.792751	−	0.609545i	$-0.208648\pi$
$194$	−9.51962			−0.683469
$195$		0			0
$196$	2.77458			0.198185
$197$		−	5.10372i		−	0.363625i	−0.983333	−	0.181812i	$-0.941804\pi$
$197$		−	5.10372i		−	0.363625i	0.983333	−	0.181812i	$-0.0581964\pi$
$198$		0			0
$199$	16.5465			1.17295			0.586475	−	0.809967i	$-0.300515\pi$
$199$	16.5465			1.17295			0.586475	+	0.809967i	$0.300515\pi$
$200$			0.282734i			0.0199923i
$201$		0			0
$202$	5.99617			0.421889
$203$		−	6.46995i		−	0.454101i
$204$		0			0
$205$		−	2.68129i		−	0.187269i
$206$	−12.1527			−0.846720
$207$		0			0
$208$		−	5.56690i		−	0.385995i
$209$	2.39657			0.165775
$210$		0			0
$211$	5.75275			0.396036			0.198018	−	0.980198i	$-0.436550\pi$
$211$	5.75275			0.396036			0.198018	+	0.980198i	$0.436550\pi$
$212$		−	6.82999i		−	0.469086i
$213$		0			0
$214$			15.0973i			1.03203i
$215$	−21.9293			−1.49556
$216$		0			0
$217$	5.73121			0.389060
$218$		−	9.61079i		−	0.650925i
$219$		0			0
$220$	−1.31905			−0.0889303
$221$	−31.9621			−2.15001
$222$		0			0
$223$	2.79911	+	14.6685i	0.187442	+	0.982276i
$223$	2.79911	+	14.6685i	0.187442	+	0.982276i
$224$		−	2.05558i		−	0.137344i
$225$		0			0
$226$		−	7.55260i		−	0.502392i
$227$		−	3.42425i		−	0.227276i	−0.993522	−	0.113638i	$-0.963750\pi$
$227$		−	3.42425i		−	0.227276i	0.993522	−	0.113638i	$-0.0362503\pi$
$228$		0			0
$229$		−	2.30551i		−	0.152352i	−0.997094	−	0.0761760i	$-0.975729\pi$
$229$		−	2.30551i		−	0.152352i	0.997094	−	0.0761760i	$-0.0242711\pi$
$230$		−	4.73635i		−	0.312305i
$231$		0			0
$232$	3.14750			0.206644
$233$	−19.6834			−1.28950			−0.644752	−	0.764392i	$-0.723039\pi$
$233$	−19.6834			−1.28950			−0.644752	+	0.764392i	$0.723039\pi$
$234$		0			0
$235$			13.8385i			0.902726i
$236$	1.99536			0.129887
$237$		0			0
$238$	−11.8020			−0.765013
$239$			29.8970i			1.93388i	0.255008	+	0.966939i	$0.417922\pi$
$239$			29.8970i			1.93388i	−0.255008	+	0.966939i	$0.582078\pi$
$240$		0			0
$241$	−25.4297			−1.63807			−0.819037	−	0.573741i	$-0.805492\pi$
$241$	−25.4297			−1.63807			−0.819037	+	0.573741i	$0.805492\pi$
$242$			10.6706i			0.685935i
$243$		0			0
$244$			14.1655i			0.906855i
$245$	−6.37716			−0.407422
$246$		0			0
$247$		−	23.2473i		−	1.47919i
$248$			2.78812i			0.177046i
$249$		0			0
$250$			10.8423i			0.685725i
$251$			17.3937i			1.09788i	0.835862	+	0.548939i	$0.184968\pi$
$251$			17.3937i			1.09788i	−0.835862	+	0.548939i	$0.815032\pi$
$252$		0			0
$253$	1.18262			0.0743508
$254$		−	5.15057i		−	0.323176i
$255$		0			0
$256$	1.00000			0.0625000
$257$		−	4.90233i		−	0.305799i	−0.988242	−	0.152900i	$-0.951139\pi$
$257$		−	4.90233i		−	0.305799i	0.988242	−	0.152900i	$-0.0488611\pi$
$258$		0			0
$259$	1.25606			0.0780481
$260$			12.7951i			0.793516i
$261$		0			0
$262$	−4.12258			−0.254694
$263$	−13.0032			−0.801814			−0.400907	−	0.916119i	$-0.631305\pi$
$263$	−13.0032			−0.801814			−0.400907	+	0.916119i	$0.631305\pi$
$264$		0			0
$265$			15.6982i			0.964332i
$266$		−	8.58408i		−	0.526324i
$267$		0			0
$268$		−	6.44119i		−	0.393458i
$269$	−25.2677			−1.54060			−0.770299	−	0.637682i	$-0.779893\pi$
$269$	−25.2677			−1.54060			−0.770299	+	0.637682i	$0.779893\pi$
$270$		0			0
$271$		−	25.4047i		−	1.54323i	−0.636093	−	0.771613i	$-0.719450\pi$
$271$		−	25.4047i		−	1.54323i	0.636093	−	0.771613i	$-0.280550\pi$
$272$		−	5.74146i		−	0.348127i
$273$		0			0
$274$		−	19.9960i		−	1.20800i
$275$	0.162260			0.00978463
$276$		0			0
$277$		−	9.06338i		−	0.544566i	−0.962217	−	0.272283i	$-0.912221\pi$
$277$		−	9.06338i		−	0.544566i	0.962217	−	0.272283i	$-0.0877787\pi$
$278$		−	17.2958i		−	1.03733i
$279$		0			0
$280$			4.72459i			0.282348i
$281$			23.4026i			1.39608i	0.716056	+	0.698042i	$0.245945\pi$
$281$			23.4026i			1.39608i	−0.716056	+	0.698042i	$0.754055\pi$
$282$		0			0
$283$	22.1892			1.31901			0.659504	−	0.751701i	$-0.270766\pi$
$283$	22.1892			1.31901			0.659504	+	0.751701i	$0.270766\pi$
$284$	0.0503915			0.00299018
$285$		0			0
$286$	−3.19481			−0.188913
$287$		−	2.39800i		−	0.141549i
$288$		0			0
$289$	−15.9644			−0.939082
$290$	−7.23428			−0.424812
$291$		0			0
$292$	0.158279			0.00926259
$293$	18.0925			1.05697			0.528487	−	0.848941i	$-0.322760\pi$
$293$	18.0925			1.05697			0.528487	+	0.848941i	$0.322760\pi$
$294$		0			0
$295$	−4.58617			−0.267017
$296$			0.611051i			0.0355166i
$297$		0			0
$298$			7.18737i			0.416353i
$299$		−	11.4717i		−	0.663425i
$300$		0			0
$301$	−19.6123			−1.13044
$302$	−19.2437			−1.10735
$303$		0			0
$304$	4.17599			0.239509
$305$		−	32.5583i		−	1.86428i
$306$		0			0
$307$		−	19.3483i		−	1.10426i	−0.833757	−	0.552132i	$-0.813814\pi$
$307$		−	19.3483i		−	1.10426i	0.833757	−	0.552132i	$-0.186186\pi$
$308$	−1.17969			−0.0672189
$309$		0			0
$310$		−	6.40827i		−	0.363965i
$311$	4.22237			0.239429			0.119714	−	0.992808i	$-0.461802\pi$
$311$	4.22237			0.239429			0.119714	+	0.992808i	$0.461802\pi$
$312$		0			0
$313$		−	18.6943i		−	1.05666i	−0.849038	−	0.528331i	$-0.822818\pi$
$313$		−	18.6943i		−	1.05666i	0.849038	−	0.528331i	$-0.177182\pi$
$314$	2.53661			0.143149
$315$		0			0
$316$		−	3.77417i		−	0.212314i
$317$			26.7226i			1.50089i	0.660932	+	0.750445i	$0.270161\pi$
$317$			26.7226i			1.50089i	−0.660932	+	0.750445i	$0.729839\pi$
$318$		0			0
$319$		−	1.80633i		−	0.101135i
$320$	−2.29842			−0.128486
$321$		0			0
$322$		−	4.23593i		−	0.236059i
$323$		−	23.9763i		−	1.33408i
$324$		0			0
$325$		−	1.57395i		−	0.0873072i
$326$	0.731953			0.0405391
$327$		0			0
$328$	1.16658			0.0644135
$329$			12.3764i			0.682334i
$330$		0			0
$331$			2.01859i			0.110952i	0.998460	+	0.0554758i	$0.0176676\pi$
$331$			2.01859i			0.110952i	−0.998460	+	0.0554758i	$0.982332\pi$
$332$		−	3.37366i		−	0.185154i
$333$		0			0
$334$		−	6.96980i		−	0.381370i
$335$			14.8046i			0.808860i
$336$		0			0
$337$			12.2132i			0.665293i	0.943052	+	0.332646i	$0.107942\pi$
$337$			12.2132i			0.665293i	−0.943052	+	0.332646i	$0.892058\pi$
$338$			17.9903i			0.978545i
$339$		0			0
$340$			13.1963i			0.715670i
$341$	1.60009			0.0866495
$342$		0			0
$343$	−20.0925			−1.08489
$344$		−	9.54102i		−	0.514417i
$345$		0			0
$346$		−	18.9794i		−	1.02034i
$347$		−	27.1269i		−	1.45625i	−0.685445	−	0.728124i	$-0.740392\pi$
$347$		−	27.1269i		−	1.45625i	0.685445	−	0.728124i	$-0.259608\pi$
$348$		0			0
$349$	5.01163			0.268266			0.134133	−	0.990963i	$-0.457175\pi$
$349$	5.01163			0.268266			0.134133	+	0.990963i	$0.457175\pi$
$350$		−	0.581184i		−	0.0310656i
$351$		0			0
$352$		−	0.573894i		−	0.0305887i
$353$			21.8446i			1.16267i	0.813665	+	0.581334i	$0.197469\pi$
$353$			21.8446i			1.16267i	−0.813665	+	0.581334i	$0.802531\pi$
$354$		0			0
$355$	−0.115821			−0.00614713
$356$		−	8.91850i		−	0.472680i
$357$		0			0
$358$	6.60748			0.349216
$359$		−	17.1656i		−	0.905966i	−0.891519	−	0.452983i	$-0.850360\pi$
$359$		−	17.1656i		−	0.905966i	0.891519	−	0.452983i	$-0.149640\pi$
$360$		0			0
$361$	−1.56114			−0.0821655
$362$		−	12.1359i		−	0.637847i
$363$		0			0
$364$			11.4432i			0.599787i
$365$	−0.363792			−0.0190417
$366$		0			0
$367$	−14.9021			−0.777885			−0.388942	−	0.921262i	$-0.627160\pi$
$367$	−14.9021			−0.777885			−0.388942	+	0.921262i	$0.627160\pi$
$368$	2.06070			0.107421
$369$		0			0
$370$		−	1.40445i		−	0.0730139i
$371$			14.0396i			0.728900i
$372$		0			0
$373$		−	4.47759i		−	0.231841i	−0.993258	−	0.115920i	$-0.963018\pi$
$373$		−	4.47759i		−	0.231841i	0.993258	−	0.115920i	$-0.0369818\pi$
$374$	−3.29499			−0.170380
$375$		0			0
$376$	−6.02089			−0.310504
$377$	−17.5218			−0.902420
$378$		0			0
$379$	−9.65998			−0.496200			−0.248100	−	0.968734i	$-0.579806\pi$
$379$	−9.65998			−0.496200			−0.248100	+	0.968734i	$0.579806\pi$
$380$	−9.59817			−0.492376
$381$		0			0
$382$		−	3.80003i		−	0.194427i
$383$	24.1856			1.23583			0.617914	−	0.786246i	$-0.287978\pi$
$383$	24.1856			1.23583			0.617914	+	0.786246i	$0.287978\pi$
$384$		0			0
$385$	2.71142			0.138187
$386$	−16.9361			−0.862027
$387$		0			0
$388$			9.51962i			0.483285i
$389$		−	9.66092i		−	0.489828i	−0.969545	−	0.244914i	$-0.921240\pi$
$389$		−	9.66092i		−	0.489828i	0.969545	−	0.244914i	$-0.0787598\pi$
$390$		0			0
$391$		−	11.8314i		−	0.598340i
$392$		−	2.77458i		−	0.140138i
$393$		0			0
$394$	−5.10372			−0.257122
$395$			8.67463i			0.436468i
$396$		0			0
$397$			10.7987i			0.541971i	0.962583	+	0.270986i	$0.0873496\pi$
$397$			10.7987i			0.541971i	−0.962583	+	0.270986i	$0.912650\pi$
$398$		−	16.5465i		−	0.829401i
$399$		0			0
$400$	0.282734			0.0141367
$401$			18.3928i			0.918492i	0.888309	+	0.459246i	$0.151880\pi$
$401$			18.3928i			0.918492i	−0.888309	+	0.459246i	$0.848120\pi$
$402$		0			0
$403$		−	15.5212i		−	0.773165i
$404$		−	5.99617i		−	0.298321i
$405$		0			0
$406$	−6.46995			−0.321098
$407$	0.350678			0.0173825
$408$		0			0
$409$		−	12.4348i		−	0.614863i	−0.951570	−	0.307432i	$-0.900530\pi$
$409$		−	12.4348i		−	0.614863i	0.951570	−	0.307432i	$-0.0994695\pi$
$410$	−2.68129			−0.132419
$411$		0			0
$412$			12.1527i			0.598721i
$413$	−4.10162			−0.201827
$414$		0			0
$415$			7.75409i			0.380634i
$416$	−5.56690			−0.272940
$417$		0			0
$418$		−	2.39657i		−	0.117220i
$419$		−	0.593093i		−	0.0289745i	−0.999895	−	0.0144873i	$-0.995388\pi$
$419$		−	0.593093i		−	0.0289745i	0.999895	−	0.0144873i	$-0.00461160\pi$
$420$		0			0
$421$			20.2952i			0.989128i	0.869141	+	0.494564i	$0.164672\pi$
$421$			20.2952i			0.989128i	−0.869141	+	0.494564i	$0.835328\pi$
$422$		−	5.75275i		−	0.280040i
$423$		0			0
$424$	−6.82999			−0.331694
$425$		−	1.62331i		−	0.0787421i
$426$		0			0
$427$		−	29.1184i		−	1.40914i
$428$	15.0973			0.729755
$429$		0			0
$430$			21.9293i			1.05752i
$431$	31.7299			1.52838			0.764188	−	0.644994i	$-0.223140\pi$
$431$	31.7299			1.52838			0.764188	+	0.644994i	$0.223140\pi$
$432$		0			0
$433$	13.9906			0.672344			0.336172	−	0.941801i	$-0.390868\pi$
$433$	13.9906			0.672344			0.336172	+	0.941801i	$0.390868\pi$
$434$		−	5.73121i		−	0.275107i
$435$		0			0
$436$	−9.61079			−0.460273
$437$	8.60544			0.411654
$438$		0			0
$439$			1.26903i			0.0605673i	0.999541	+	0.0302836i	$0.00964105\pi$
$439$			1.26903i			0.0605673i	−0.999541	+	0.0302836i	$0.990359\pi$
$440$			1.31905i			0.0628832i
$441$		0			0
$442$			31.9621i			1.52028i
$443$			15.2558i			0.724827i	0.932017	+	0.362413i	$0.118047\pi$
$443$			15.2558i			0.724827i	−0.932017	+	0.362413i	$0.881953\pi$
$444$		0			0
$445$			20.4985i			0.971720i
$446$	14.6685	−	2.79911i	0.694574	−	0.132542i
$447$		0			0
$448$	−2.05558			−0.0971171
$449$	4.87862			0.230236			0.115118	−	0.993352i	$-0.463275\pi$
$449$	4.87862			0.230236			0.115118	+	0.993352i	$0.463275\pi$
$450$		0			0
$451$		−	0.669493i		−	0.0315252i
$452$	−7.55260			−0.355245
$453$		0			0
$454$	−3.42425			−0.160708
$455$		−	26.3013i		−	1.23302i
$456$		0			0
$457$			31.9179i			1.49306i	0.665353	+	0.746529i	$0.268281\pi$
$457$			31.9179i			1.49306i	−0.665353	+	0.746529i	$0.731719\pi$
$458$	−2.30551			−0.107729
$459$		0			0
$460$	−4.73635			−0.220833
$461$		−	4.76109i		−	0.221746i	−0.993835	−	0.110873i	$-0.964635\pi$
$461$		−	4.76109i		−	0.221746i	0.993835	−	0.110873i	$-0.0353647\pi$
$462$		0			0
$463$	33.0970			1.53815			0.769073	−	0.639161i	$-0.220718\pi$
$463$	33.0970			1.53815			0.769073	+	0.639161i	$0.220718\pi$
$464$		−	3.14750i		−	0.146119i
$465$		0			0
$466$			19.6834i			0.911817i
$467$	4.79469			0.221872			0.110936	−	0.993828i	$-0.464615\pi$
$467$	4.79469			0.221872			0.110936	+	0.993828i	$0.464615\pi$
$468$		0			0
$469$			13.2404i			0.611385i
$470$	13.8385			0.638324
$471$		0			0
$472$		−	1.99536i		−	0.0918437i
$473$	−5.47554			−0.251765
$474$		0			0
$475$	1.18070			0.0541740
$476$			11.8020i			0.540946i
$477$		0			0
$478$	29.8970			1.36746
$479$		−	21.0657i		−	0.962516i	−0.876579	−	0.481258i	$-0.840180\pi$
$479$		−	21.0657i		−	0.962516i	0.876579	−	0.481258i	$-0.159820\pi$
$480$		0			0
$481$		−	3.40166i		−	0.155102i
$482$			25.4297i			1.15829i
$483$		0			0
$484$	10.6706			0.485029
$485$		−	21.8801i		−	0.993523i
$486$		0			0
$487$	15.3348			0.694888			0.347444	−	0.937701i	$-0.387050\pi$
$487$	15.3348			0.694888			0.347444	+	0.937701i	$0.387050\pi$
$488$	14.1655			0.641243
$489$		0			0
$490$			6.37716i			0.288091i
$491$	12.7519			0.575487			0.287744	−	0.957707i	$-0.407095\pi$
$491$	12.7519			0.575487			0.287744	+	0.957707i	$0.407095\pi$
$492$		0			0
$493$	−18.0713			−0.813889
$494$	−23.2473			−1.04594
$495$		0			0
$496$	2.78812			0.125190
$497$	−0.103584			−0.00464637
$498$		0			0
$499$	−19.5666			−0.875923			−0.437962	−	0.898994i	$-0.644299\pi$
$499$	−19.5666			−0.875923			−0.437962	+	0.898994i	$0.644299\pi$
$500$	10.8423			0.484880
$501$		0			0
$502$	17.3937			0.776317
$503$	−4.83916			−0.215767			−0.107884	−	0.994164i	$-0.534407\pi$
$503$	−4.83916			−0.215767			−0.107884	+	0.994164i	$0.534407\pi$
$504$		0			0
$505$			13.7817i			0.613279i
$506$		−	1.18262i		−	0.0525740i
$507$		0			0
$508$	−5.15057			−0.228520
$509$			18.1515i			0.804549i	0.915519	+	0.402275i	$0.131780\pi$
$509$			18.1515i			0.804549i	−0.915519	+	0.402275i	$0.868220\pi$
$510$		0			0
$511$	−0.325356			−0.0143929
$512$		−	1.00000i		−	0.0441942i
$513$		0			0
$514$	−4.90233			−0.216233
$515$		−	27.9320i		−	1.23083i
$516$		0			0
$517$			3.45535i			0.151966i
$518$		−	1.25606i		−	0.0551883i
$519$		0			0
$520$	12.7951			0.561101
$521$	−4.91449			−0.215308			−0.107654	−	0.994188i	$-0.534334\pi$
$521$	−4.91449			−0.215308			−0.107654	+	0.994188i	$0.534334\pi$
$522$		0			0
$523$		−	14.4996i		−	0.634024i	−0.948422	−	0.317012i	$-0.897320\pi$
$523$		−	14.4996i		−	0.634024i	0.948422	−	0.317012i	$-0.102680\pi$
$524$			4.12258i			0.180096i
$525$		0			0
$526$			13.0032i			0.566968i
$527$		−	16.0079i		−	0.697315i
$528$		0			0
$529$	−18.7535			−0.815371
$530$	15.6982			0.681885
$531$		0			0
$532$	−8.58408			−0.372167
$533$	−6.49422			−0.281296
$534$		0			0
$535$	−34.6999			−1.50021
$536$	−6.44119			−0.278217
$537$		0			0
$538$			25.2677i			1.08937i
$539$	−1.59232			−0.0685860
$540$		0			0
$541$			13.0502i			0.561070i	0.959844	+	0.280535i	$0.0905119\pi$
$541$			13.0502i			0.561070i	−0.959844	+	0.280535i	$0.909488\pi$
$542$	−25.4047			−1.09122
$543$		0			0
$544$	−5.74146			−0.246163
$545$	22.0896			0.946216
$546$		0			0
$547$	−31.4676			−1.34546			−0.672729	−	0.739889i	$-0.734878\pi$
$547$	−31.4676			−1.34546			−0.672729	+	0.739889i	$0.734878\pi$
$548$	−19.9960			−0.854188
$549$		0			0
$550$		−	0.162260i		−	0.00691878i
$551$		−	13.1439i		−	0.559950i
$552$		0			0
$553$			7.75812i			0.329909i
$554$	−9.06338			−0.385066
$555$		0			0
$556$	−17.2958			−0.733503
$557$	9.01148			0.381829			0.190914	−	0.981607i	$-0.438855\pi$
$557$	9.01148			0.381829			0.190914	+	0.981607i	$0.438855\pi$
$558$		0			0
$559$			53.1139i			2.24648i
$560$	4.72459			0.199650
$561$		0			0
$562$	23.4026			0.987181
$563$	19.4520			0.819804			0.409902	−	0.912130i	$-0.365563\pi$
$563$	19.4520			0.819804			0.409902	+	0.912130i	$0.365563\pi$
$564$		0			0
$565$	17.3591			0.730301
$566$		−	22.1892i		−	0.932680i
$567$		0			0
$568$		−	0.0503915i		−	0.00211438i
$569$	−20.0444			−0.840306			−0.420153	−	0.907453i	$-0.638024\pi$
$569$	−20.0444			−0.840306			−0.420153	+	0.907453i	$0.638024\pi$
$570$		0			0
$571$			11.4368i			0.478614i	0.970944	+	0.239307i	$0.0769202\pi$
$571$			11.4368i			0.478614i	−0.970944	+	0.239307i	$0.923080\pi$
$572$			3.19481i			0.133582i
$573$		0			0
$574$	−2.39800			−0.100091
$575$	0.582630			0.0242974
$576$		0			0
$577$	24.5788			1.02323			0.511615	−	0.859215i	$-0.329047\pi$
$577$	24.5788			1.02323			0.511615	+	0.859215i	$0.329047\pi$
$578$			15.9644i			0.664031i
$579$		0			0
$580$			7.23428i			0.300387i
$581$			6.93484i			0.287706i
$582$		0			0
$583$			3.91969i			0.162337i
$584$		−	0.158279i		−	0.00654964i
$585$		0			0
$586$		−	18.0925i		−	0.747394i
$587$	22.7676			0.939719			0.469860	−	0.882741i	$-0.344305\pi$
$587$	22.7676			0.939719			0.469860	+	0.882741i	$0.344305\pi$
$588$		0			0
$589$	11.6432			0.479748
$590$			4.58617i			0.188810i
$591$		0			0
$592$	0.611051			0.0251140
$593$	6.60487			0.271229			0.135615	−	0.990762i	$-0.456699\pi$
$593$	6.60487			0.271229			0.135615	+	0.990762i	$0.456699\pi$
$594$		0			0
$595$		−	27.1261i		−	1.11206i
$596$	7.18737			0.294406
$597$		0			0
$598$	−11.4717			−0.469112
$599$			26.3843i			1.07803i	0.842295	+	0.539017i	$0.181204\pi$
$599$			26.3843i			1.07803i	−0.842295	+	0.539017i	$0.818796\pi$
$600$		0			0
$601$			12.7513i			0.520135i	0.965590	+	0.260068i	$0.0837448\pi$
$601$			12.7513i			0.520135i	−0.965590	+	0.260068i	$0.916255\pi$
$602$			19.6123i			0.799339i
$603$		0			0
$604$			19.2437i			0.783017i
$605$	−24.5256			−0.997108
$606$		0			0
$607$			14.5815i			0.591844i	0.955212	+	0.295922i	$0.0956268\pi$
$607$			14.5815i			0.591844i	−0.955212	+	0.295922i	$0.904373\pi$
$608$		−	4.17599i		−	0.169359i
$609$		0			0
$610$	−32.5583			−1.31825
$611$	33.5177			1.35598
$612$		0			0
$613$		−	9.68272i		−	0.391081i	−0.980696	−	0.195541i	$-0.937354\pi$
$613$		−	9.68272i		−	0.391081i	0.980696	−	0.195541i	$-0.0626461\pi$
$614$	−19.3483			−0.780833
$615$		0			0
$616$			1.17969i			0.0475309i
$617$		−	17.1345i		−	0.689809i	−0.938638	−	0.344905i	$-0.887911\pi$
$617$		−	17.1345i		−	0.689809i	0.938638	−	0.344905i	$-0.112089\pi$
$618$		0			0
$619$			9.04916i			0.363717i	0.983325	+	0.181858i	$0.0582112\pi$
$619$			9.04916i			0.363717i	−0.983325	+	0.181858i	$0.941789\pi$
$620$	−6.40827			−0.257362
$621$		0			0
$622$		−	4.22237i		−	0.169302i
$623$			18.3327i			0.734485i
$624$		0			0
$625$	−26.3337			−1.05335
$626$	−18.6943			−0.747173
$627$		0			0
$628$		−	2.53661i		−	0.101222i
$629$		−	3.50833i		−	0.139886i
$630$		0			0
$631$			2.23782i			0.0890861i	0.999007	+	0.0445431i	$0.0141832\pi$
$631$			2.23782i			0.0890861i	−0.999007	+	0.0445431i	$0.985817\pi$
$632$	−3.77417			−0.150129
$633$		0			0
$634$	26.7226			1.06129
$635$	11.8382			0.469784
$636$		0			0
$637$			15.4458i			0.611986i
$638$	−1.80633			−0.0715135
$639$		0			0
$640$			2.29842i			0.0908530i
$641$	−13.1675			−0.520085			−0.260043	−	0.965597i	$-0.583737\pi$
$641$	−13.1675			−0.520085			−0.260043	+	0.965597i	$0.583737\pi$
$642$		0			0
$643$	11.5968			0.457333			0.228666	−	0.973505i	$-0.426563\pi$
$643$	11.5968			0.457333			0.228666	+	0.973505i	$0.426563\pi$
$644$	−4.23593			−0.166919
$645$		0			0
$646$	−23.9763			−0.943334
$647$		−	16.0490i		−	0.630951i	−0.948934	−	0.315475i	$-0.897836\pi$
$647$		−	16.0490i		−	0.630951i	0.948934	−	0.315475i	$-0.102164\pi$
$648$		0			0
$649$	−1.14512			−0.0449500
$650$	−1.57395			−0.0617355
$651$		0			0
$652$		−	0.731953i		−	0.0286655i
$653$	15.4365			0.604077			0.302039	−	0.953296i	$-0.402333\pi$
$653$	15.4365			0.604077			0.302039	+	0.953296i	$0.402333\pi$
$654$		0			0
$655$		−	9.47542i		−	0.370235i
$656$		−	1.16658i		−	0.0455473i
$657$		0			0
$658$	12.3764			0.482483
$659$		−	7.18458i		−	0.279872i	−0.990161	−	0.139936i	$-0.955310\pi$
$659$		−	7.18458i		−	0.279872i	0.990161	−	0.139936i	$-0.0446896\pi$
$660$		0			0
$661$		−	10.6665i		−	0.414877i	−0.978248	−	0.207439i	$-0.933487\pi$
$661$		−	10.6665i		−	0.414877i	0.978248	−	0.207439i	$-0.0665127\pi$
$662$	2.01859			0.0784546
$663$		0			0
$664$	−3.37366			−0.130924
$665$	19.7298			0.765090
$666$		0			0
$667$		−	6.48605i		−	0.251141i
$668$	−6.96980			−0.269670
$669$		0			0
$670$	14.8046			0.571950
$671$		−	8.12952i		−	0.313836i
$672$		0			0
$673$	8.15548			0.314371			0.157185	−	0.987569i	$-0.449758\pi$
$673$	8.15548			0.314371			0.157185	+	0.987569i	$0.449758\pi$
$674$	12.2132			0.470433
$675$		0			0
$676$	17.9903			0.691936
$677$			32.2397i			1.23907i	0.784968	+	0.619536i	$0.212679\pi$
$677$			32.2397i			1.23907i	−0.784968	+	0.619536i	$0.787321\pi$
$678$		0			0
$679$		−	19.5684i		−	0.750965i
$680$	13.1963			0.506055
$681$		0			0
$682$		−	1.60009i		−	0.0612705i
$683$			44.5137i			1.70327i	0.524136	+	0.851635i	$0.324388\pi$
$683$			44.5137i			1.70327i	−0.524136	+	0.851635i	$0.675612\pi$
$684$		0			0
$685$	45.9593			1.75601
$686$			20.0925i			0.767134i
$687$		0			0
$688$	−9.54102			−0.363748
$689$	38.0218			1.44852
$690$		0			0
$691$			45.7153i			1.73909i	0.493853	+	0.869546i	$0.335588\pi$
$691$			45.7153i			1.73909i	−0.493853	+	0.869546i	$0.664412\pi$
$692$	−18.9794			−0.721488
$693$		0			0
$694$	−27.1269			−1.02972
$695$	39.7529			1.50791
$696$		0			0
$697$	−6.69787			−0.253700
$698$		−	5.01163i		−	0.189693i
$699$		0			0
$700$	−0.581184			−0.0219667
$701$			2.02374i			0.0764355i	0.999269	+	0.0382178i	$0.0121681\pi$
$701$			2.02374i			0.0764355i	−0.999269	+	0.0382178i	$0.987832\pi$
$702$		0			0
$703$	2.55174			0.0962407
$704$	−0.573894			−0.0216295
$705$		0			0
$706$	21.8446			0.822131
$707$			12.3256i			0.463553i
$708$		0			0
$709$			33.0662i			1.24183i	0.783879	+	0.620914i	$0.213238\pi$
$709$			33.0662i			1.24183i	−0.783879	+	0.620914i	$0.786762\pi$
$710$			0.115821i			0.00434668i
$711$		0			0
$712$	−8.91850			−0.334235
$713$	5.74547			0.215170
$714$		0			0
$715$		−	7.34301i		−	0.274613i
$716$		−	6.60748i		−	0.246933i
$717$		0			0
$718$	−17.1656			−0.640615
$719$			22.6224i			0.843672i	0.906672	+	0.421836i	$0.138614\pi$
$719$			22.6224i			0.843672i	−0.906672	+	0.421836i	$0.861386\pi$
$720$		0			0
$721$		−	24.9809i		−	0.930337i
$722$			1.56114i			0.0580998i
$723$		0			0
$724$	−12.1359			−0.451026
$725$		−	0.889908i		−	0.0330503i
$726$		0			0
$727$	44.0215			1.63267			0.816334	−	0.577580i	$-0.196003\pi$
$727$	44.0215			1.63267			0.816334	+	0.577580i	$0.196003\pi$
$728$	11.4432			0.424114
$729$		0			0
$730$			0.363792i			0.0134645i
$731$			54.7794i			2.02609i
$732$		0			0
$733$	−3.97676			−0.146885			−0.0734425	−	0.997299i	$-0.523399\pi$
$733$	−3.97676			−0.146885			−0.0734425	+	0.997299i	$0.523399\pi$
$734$			14.9021i			0.550048i
$735$		0			0
$736$		−	2.06070i		−	0.0759583i
$737$			3.69656i			0.136165i
$738$		0			0
$739$			32.7168i			1.20351i	0.798682	+	0.601753i	$0.205531\pi$
$739$			32.7168i			1.20351i	−0.798682	+	0.601753i	$0.794469\pi$
$740$	−1.40445			−0.0516287
$741$		0			0
$742$	14.0396			0.515410
$743$			12.0003i			0.440250i	0.975472	+	0.220125i	$0.0706465\pi$
$743$			12.0003i			0.440250i	−0.975472	+	0.220125i	$0.929353\pi$
$744$		0			0
$745$	−16.5196			−0.605231
$746$	−4.47759			−0.163936
$747$		0			0
$748$			3.29499i			0.120477i
$749$	−31.0337			−1.13395
$750$		0			0
$751$	−44.2429			−1.61444			−0.807222	−	0.590247i	$-0.799030\pi$
$751$	−44.2429			−1.61444			−0.807222	+	0.590247i	$0.799030\pi$
$752$			6.02089i			0.219559i
$753$		0			0
$754$			17.5218i			0.638107i
$755$		−	44.2302i		−	1.60970i
$756$		0			0
$757$		−	27.8975i		−	1.01395i	−0.861960	−	0.506976i	$-0.830763\pi$
$757$		−	27.8975i		−	1.01395i	0.861960	−	0.506976i	$-0.169237\pi$
$758$			9.65998i			0.350866i
$759$		0			0
$760$			9.59817i			0.348162i
$761$	37.2469			1.35020			0.675099	−	0.737727i	$-0.264101\pi$
$761$	37.2469			1.35020			0.675099	+	0.737727i	$0.264101\pi$
$762$		0			0
$763$	19.7558			0.715207
$764$	−3.80003			−0.137480
$765$		0			0
$766$		−	24.1856i		−	0.873862i
$767$			11.1079i			0.401085i
$768$		0			0
$769$	29.6601			1.06957			0.534785	−	0.844988i	$-0.320392\pi$
$769$	29.6601			1.06957			0.534785	+	0.844988i	$0.320392\pi$
$770$		−	2.71142i		−	0.0977126i
$771$		0			0
$772$			16.9361i			0.609545i
$773$	12.2136			0.439292			0.219646	−	0.975580i	$-0.429510\pi$
$773$	12.2136			0.439292			0.219646	+	0.975580i	$0.429510\pi$
$774$		0			0
$775$	0.788298			0.0283165
$776$	9.51962			0.341734
$777$		0			0
$778$	−9.66092			−0.346361
$779$		−	4.87162i		−	0.174544i
$780$		0			0
$781$	−0.0289194			−0.00103482
$782$	−11.8314			−0.423091
$783$		0			0
$784$	−2.77458			−0.0990923
$785$			5.83020i			0.208089i
$786$		0			0
$787$		−	26.5060i		−	0.944836i	−0.881375	−	0.472418i	$-0.843381\pi$
$787$		−	26.5060i		−	0.944836i	0.881375	−	0.472418i	$-0.156619\pi$
$788$			5.10372i			0.181812i
$789$		0			0
$790$	8.67463			0.308630
$791$	15.5250			0.552005
$792$		0			0
$793$	−78.8580			−2.80033
$794$	10.7987			0.383232
$795$		0			0
$796$	−16.5465			−0.586475
$797$		−	3.63008i		−	0.128584i	−0.997931	−	0.0642920i	$-0.979521\pi$
$797$		−	3.63008i		−	0.128584i	0.997931	−	0.0642920i	$-0.0204789\pi$
$798$		0			0
$799$	34.5687			1.22295
$800$		−	0.282734i		−	0.00999617i
$801$		0			0
$802$	18.3928			0.649472
$803$	−0.0908355			−0.00320552
$804$		0			0
$805$	9.73595			0.343147
$806$	−15.5212			−0.546710
$807$		0			0
$808$	−5.99617			−0.210945
$809$	−39.6296			−1.39330			−0.696652	−	0.717409i	$-0.745328\pi$
$809$	−39.6296			−1.39330			−0.696652	+	0.717409i	$0.745328\pi$
$810$		0			0
$811$		−	55.6661i		−	1.95470i	−0.211631	−	0.977350i	$-0.567877\pi$
$811$		−	55.6661i		−	1.95470i	0.211631	−	0.977350i	$-0.432123\pi$
$812$			6.46995i			0.227051i
$813$		0			0
$814$		−	0.350678i		−	0.0122913i
$815$			1.68233i			0.0589296i
$816$		0			0
$817$	−39.8432			−1.39394
$818$	−12.4348			−0.434774
$819$		0			0
$820$			2.68129i			0.0936346i
$821$		−	2.41117i		−	0.0841503i	−0.999114	−	0.0420752i	$-0.986603\pi$
$821$		−	2.41117i		−	0.0841503i	0.999114	−	0.0420752i	$-0.0133969\pi$
$822$		0			0
$823$			23.8371i			0.830909i	0.909614	+	0.415455i	$0.136377\pi$
$823$			23.8371i			0.830909i	−0.909614	+	0.415455i	$0.863623\pi$
$824$	12.1527			0.423360
$825$		0			0
$826$			4.10162i			0.142714i
$827$	−9.39474			−0.326687			−0.163344	−	0.986569i	$-0.552228\pi$
$827$	−9.39474			−0.326687			−0.163344	+	0.986569i	$0.552228\pi$
$828$		0			0
$829$			20.9559i			0.727827i	0.931433	+	0.363914i	$0.118560\pi$
$829$			20.9559i			0.727827i	−0.931433	+	0.363914i	$0.881440\pi$
$830$	7.75409			0.269149
$831$		0			0
$832$			5.56690i			0.192997i
$833$			15.9302i			0.551948i
$834$		0			0
$835$	16.0195			0.554378
$836$	−2.39657			−0.0828873
$837$		0			0
$838$	−0.593093			−0.0204881
$839$	30.0123			1.03614			0.518070	−	0.855338i	$-0.326651\pi$
$839$	30.0123			1.03614			0.518070	+	0.855338i	$0.326651\pi$
$840$		0			0
$841$	19.0932			0.658387
$842$	20.2952			0.699419
$843$		0			0
$844$	−5.75275			−0.198018
$845$	−41.3493			−1.42246
$846$		0			0
$847$	−21.9344			−0.753674
$848$			6.82999i			0.234543i
$849$		0			0
$850$	−1.62331			−0.0556791
$851$	1.25919			0.0431645
$852$		0			0
$853$			33.7315i			1.15495i	0.816410	+	0.577473i	$0.195961\pi$
$853$			33.7315i			1.15495i	−0.816410	+	0.577473i	$0.804039\pi$
$854$	−29.1184			−0.996411
$855$		0			0
$856$		−	15.0973i		−	0.516015i
$857$		−	39.2212i		−	1.33977i	−0.742464	−	0.669886i	$-0.766343\pi$
$857$		−	39.2212i		−	1.33977i	0.742464	−	0.669886i	$-0.233657\pi$
$858$		0			0
$859$			41.8288i			1.42718i	0.700563	+	0.713590i	$0.252932\pi$
$859$			41.8288i			1.42718i	−0.700563	+	0.713590i	$0.747068\pi$
$860$	21.9293			0.747782
$861$		0			0
$862$		−	31.7299i		−	1.08072i
$863$	25.1998			0.857812			0.428906	−	0.903349i	$-0.358899\pi$
$863$	25.1998			0.857812			0.428906	+	0.903349i	$0.358899\pi$
$864$		0			0
$865$	43.6226			1.48321
$866$		−	13.9906i		−	0.475419i
$867$		0			0
$868$	−5.73121			−0.194530
$869$			2.16598i			0.0734757i
$870$		0			0
$871$	35.8574			1.21498
$872$			9.61079i			0.325462i
$873$		0			0
$874$		−	8.60544i		−	0.291083i
$875$	−22.2871			−0.753443
$876$		0			0
$877$			22.7485i			0.768161i	0.923300	+	0.384080i	$0.125481\pi$
$877$			22.7485i			0.768161i	−0.923300	+	0.384080i	$0.874519\pi$
$878$	1.26903			0.0428275
$879$		0			0
$880$	1.31905			0.0444652
$881$		−	0.622341i		−	0.0209672i	−0.999945	−	0.0104836i	$-0.996663\pi$
$881$		−	0.622341i		−	0.0209672i	0.999945	−	0.0104836i	$-0.00333709\pi$
$882$		0			0
$883$			31.7943i			1.06996i	0.844864	+	0.534981i	$0.179681\pi$
$883$			31.7943i			1.06996i	−0.844864	+	0.534981i	$0.820319\pi$
$884$	31.9621			1.07500
$885$		0			0
$886$	15.2558			0.512530
$887$		−	35.2267i		−	1.18280i	−0.806380	−	0.591398i	$-0.798576\pi$
$887$		−	35.2267i		−	1.18280i	0.806380	−	0.591398i	$-0.201424\pi$
$888$		0			0
$889$	10.5874			0.355091
$890$	20.4985			0.687110
$891$		0			0
$892$	−2.79911	−	14.6685i	−0.0937211	−	0.491138i
$893$			25.1431i			0.841383i
$894$		0			0
$895$			15.1868i			0.507637i
$896$			2.05558i			0.0686722i
$897$		0			0
$898$		−	4.87862i		−	0.162802i
$899$		−	8.77562i		−	0.292683i
$900$		0			0
$901$	39.2141			1.30641
$902$	−0.669493			−0.0222917
$903$		0			0
$904$			7.55260i			0.251196i
$905$	27.8933			0.927206
$906$		0			0
$907$	−16.4251			−0.545387			−0.272693	−	0.962101i	$-0.587914\pi$
$907$	−16.4251			−0.545387			−0.272693	+	0.962101i	$0.587914\pi$
$908$			3.42425i			0.113638i
$909$		0			0
$910$	−26.3013			−0.871880
$911$			13.4933i			0.447054i	0.974698	+	0.223527i	$0.0717571\pi$
$911$			13.4933i			0.447054i	−0.974698	+	0.223527i	$0.928243\pi$
$912$		0			0
$913$			1.93613i			0.0640764i
$914$	31.9179			1.05575
$915$		0			0
$916$			2.30551i			0.0761760i
$917$		−	8.47430i		−	0.279846i
$918$		0			0
$919$		−	9.58044i		−	0.316029i	−0.987437	−	0.158015i	$-0.949491\pi$
$919$		−	9.58044i		−	0.316029i	0.987437	−	0.158015i	$-0.0505094\pi$
$920$			4.73635i			0.156153i
$921$		0			0
$922$	−4.76109			−0.156798
$923$			0.280524i			0.00923356i
$924$		0			0
$925$	0.172765			0.00568048
$926$		−	33.0970i		−	1.08763i
$927$		0			0
$928$	−3.14750			−0.103322
$929$			45.1222i			1.48041i	0.672381	+	0.740206i	$0.265272\pi$
$929$			45.1222i			1.48041i	−0.672381	+	0.740206i	$0.734728\pi$
$930$		0			0
$931$	−11.5866			−0.379736
$932$	19.6834			0.644752
$933$		0			0
$934$		−	4.79469i		−	0.156887i
$935$		−	7.57328i		−	0.247673i
$936$		0			0
$937$			26.4298i			0.863424i	0.902011	+	0.431712i	$0.142090\pi$
$937$			26.4298i			0.863424i	−0.902011	+	0.431712i	$0.857910\pi$
$938$	13.2404			0.432314
$939$		0			0
$940$		−	13.8385i		−	0.451363i
$941$		−	35.8595i		−	1.16899i	−0.811398	−	0.584494i	$-0.801293\pi$
$941$		−	35.8595i		−	1.16899i	0.811398	−	0.584494i	$-0.198707\pi$
$942$		0			0
$943$		−	2.40397i		−	0.0782839i
$944$	−1.99536			−0.0649433
$945$		0			0
$946$			5.47554i			0.178025i
$947$			3.01303i			0.0979104i	0.998801	+	0.0489552i	$0.0155892\pi$
$947$			3.01303i			0.0979104i	−0.998801	+	0.0489552i	$0.984411\pi$
$948$		0			0
$949$			0.881124i			0.0286025i
$950$		−	1.18070i		−	0.0383068i
$951$		0			0
$952$	11.8020			0.382507
$953$	−54.5051			−1.76559			−0.882797	−	0.469755i	$-0.844342\pi$
$953$	−54.5051			−1.76559			−0.882797	+	0.469755i	$0.844342\pi$
$954$		0			0
$955$	8.73407			0.282628
$956$		−	29.8970i		−	0.966939i
$957$		0			0
$958$	−21.0657			−0.680601
$959$	41.1035			1.32730
$960$		0			0
$961$	−23.2264			−0.749238
$962$	−3.40166			−0.109674
$963$		0			0
$964$	25.4297			0.819037
$965$		−	38.9264i		−	1.25308i
$966$		0			0
$967$		−	49.4002i		−	1.58860i	−0.607524	−	0.794302i	$-0.707837\pi$
$967$		−	49.4002i		−	1.58860i	0.607524	−	0.794302i	$-0.292163\pi$
$968$		−	10.6706i		−	0.342968i
$969$		0			0
$970$	−21.8801			−0.702527
$971$	4.20429			0.134922			0.0674611	−	0.997722i	$-0.478510\pi$
$971$	4.20429			0.134922			0.0674611	+	0.997722i	$0.478510\pi$
$972$		0			0
$973$	35.5528			1.13977
$974$		−	15.3348i		−	0.491360i
$975$		0			0
$976$		−	14.1655i		−	0.453428i
$977$	−33.2141			−1.06261			−0.531307	−	0.847180i	$-0.678299\pi$
$977$	−33.2141			−1.06261			−0.531307	+	0.847180i	$0.678299\pi$
$978$		0			0
$979$			5.11828i			0.163581i
$980$	6.37716			0.203711
$981$		0			0
$982$		−	12.7519i		−	0.406931i
$983$	−5.12760			−0.163545			−0.0817725	−	0.996651i	$-0.526058\pi$
$983$	−5.12760			−0.163545			−0.0817725	+	0.996651i	$0.526058\pi$
$984$		0			0
$985$		−	11.7305i		−	0.373764i
$986$			18.0713i			0.575507i
$987$		0			0
$988$			23.2473i			0.739594i
$989$	−19.6612			−0.625188
$990$		0			0
$991$		−	0.444200i		−	0.0141105i	−0.999975	−	0.00705523i	$-0.997754\pi$
$991$		−	0.444200i		−	0.0141105i	0.999975	−	0.00705523i	$-0.00224577\pi$
$992$		−	2.78812i		−	0.0885229i
$993$		0			0
$994$			0.103584i			0.00328548i
$995$	38.0308			1.20566
$996$		0			0
$997$	−8.42401			−0.266791			−0.133396	−	0.991063i	$-0.542588\pi$
$997$	−8.42401			−0.266791			−0.133396	+	0.991063i	$0.542588\pi$
$998$			19.5666i			0.619371i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4014.2.d.a.4013.13	✓	72
3.2	odd	2	inner	4014.2.d.a.4013.62	yes	72
223.222	odd	2	inner	4014.2.d.a.4013.61	yes	72
669.668	even	2	inner	4014.2.d.a.4013.14	yes	72

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4014.2.d.a.4013.13	✓	72	1.1	even	1	trivial
4014.2.d.a.4013.14	yes	72	669.668	even	2	inner
4014.2.d.a.4013.61	yes	72	223.222	odd	2	inner
4014.2.d.a.4013.62	yes	72	3.2	odd	2	inner

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:	\( N \)	\(=\)	\( 4014 = 2 \cdot 3^{2} \cdot 223 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4014.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} -1.00000 q^{4} +2.29842 q^{5} +2.05558 q^{7} +1.00000i q^{8} +O(q^{10})\)	\(q-1.00000i q^{2} -1.00000 q^{4} +2.29842 q^{5} +2.05558 q^{7} +1.00000i q^{8} -2.29842i q^{10} +0.573894 q^{11} -5.56690i q^{13} -2.05558i q^{14} +1.00000 q^{16} -5.74146i q^{17} +4.17599 q^{19} -2.29842 q^{20} -0.573894i q^{22} +2.06070 q^{23} +0.282734 q^{25} -5.56690 q^{26} -2.05558 q^{28} -3.14750i q^{29} +2.78812 q^{31} -1.00000i q^{32} -5.74146 q^{34} +4.72459 q^{35} +0.611051 q^{37} -4.17599i q^{38} +2.29842i q^{40} -1.16658i q^{41} -9.54102 q^{43} -0.573894 q^{44} -2.06070i q^{46} +6.02089i q^{47} -2.77458 q^{49} -0.282734i q^{50} +5.56690i q^{52} +6.82999i q^{53} +1.31905 q^{55} +2.05558i q^{56} -3.14750 q^{58} -1.99536 q^{59} -14.1655i q^{61} -2.78812i q^{62} -1.00000 q^{64} -12.7951i q^{65} +6.44119i q^{67} +5.74146i q^{68} -4.72459i q^{70} -0.0503915 q^{71} -0.158279 q^{73} -0.611051i q^{74} -4.17599 q^{76} +1.17969 q^{77} +3.77417i q^{79} +2.29842 q^{80} -1.16658 q^{82} +3.37366i q^{83} -13.1963i q^{85} +9.54102i q^{86} +0.573894i q^{88} +8.91850i q^{89} -11.4432i q^{91} -2.06070 q^{92} +6.02089 q^{94} +9.59817 q^{95} -9.51962i q^{97} +2.77458i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 72 q - 72 q^{4} + 16 q^{7}+O(q^{10}) \) 72 * q - 72 * q^4 + 16 * q^7	\( 72 q - 72 q^{4} + 16 q^{7} + 72 q^{16} - 40 q^{19} + 96 q^{25} - 16 q^{28} - 24 q^{37} - 8 q^{43} + 56 q^{49} + 40 q^{58} - 72 q^{64} - 32 q^{73} + 40 q^{76} + 16 q^{82}+O(q^{100}) \) 72 * q - 72 * q^4 + 16 * q^7 + 72 * q^16 - 40 * q^19 + 96 * q^25 - 16 * q^28 - 24 * q^37 - 8 * q^43 + 56 * q^49 + 40 * q^58 - 72 * q^64 - 32 * q^73 + 40 * q^76 + 16 * q^82

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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