LMFDB - Embedded newform 4014.2.d.a.4013.19

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			4013.19
Character	$\chi$	$=$	4014.4013
Dual form			4014.2.d.a.4013.20

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{2} -1.00000 q^{4} -1.71459 q^{5} +2.72316 q^{7} +1.00000i q^{8} +O(q^{10})$	\(q-1.00000i q^{2} -1.00000 q^{4} -1.71459 q^{5} +2.72316 q^{7} +1.00000i q^{8} +1.71459i q^{10} +5.69509 q^{11} +2.15531i q^{13} -2.72316i q^{14} +1.00000 q^{16} -5.09797i q^{17} -6.22079 q^{19} +1.71459 q^{20} -5.69509i q^{22} +8.90450 q^{23} -2.06019 q^{25} +2.15531 q^{26} -2.72316 q^{28} +4.97921i q^{29} +5.24366 q^{31} -1.00000i q^{32} -5.09797 q^{34} -4.66910 q^{35} +10.4615 q^{37} +6.22079i q^{38} -1.71459i q^{40} +8.23537i q^{41} -8.34624 q^{43} -5.69509 q^{44} -8.90450i q^{46} +3.32674i q^{47} +0.415600 q^{49} +2.06019i q^{50} -2.15531i q^{52} +6.09267i q^{53} -9.76473 q^{55} +2.72316i q^{56} +4.97921 q^{58} -9.22605 q^{59} -14.7558i q^{61} -5.24366i q^{62} -1.00000 q^{64} -3.69547i q^{65} -9.61141i q^{67} +5.09797i q^{68} +4.66910i q^{70} +13.8983 q^{71} -2.73390 q^{73} -10.4615i q^{74} +6.22079 q^{76} +15.5086 q^{77} +14.2075i q^{79} -1.71459 q^{80} +8.23537 q^{82} -10.3344i q^{83} +8.74093i q^{85} +8.34624i q^{86} +5.69509i q^{88} +14.3149i q^{89} +5.86925i q^{91} -8.90450 q^{92} +3.32674 q^{94} +10.6661 q^{95} +9.02597i q^{97} -0.415600i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	−1.71459			−0.766787			−0.383394	−	0.923585i	$-0.625245\pi$
$5$	−1.71459			−0.766787			−0.383394	+	0.923585i	$0.625245\pi$
$6$		0			0
$7$	2.72316			1.02926			0.514629	−	0.857413i	$-0.327930\pi$
$7$	2.72316			1.02926			0.514629	+	0.857413i	$0.327930\pi$
$8$			1.00000i			0.353553i
$9$		0			0
$10$			1.71459i			0.542200i
$11$	5.69509			1.71713			0.858566	−	0.512702i	$-0.171356\pi$
$11$	5.69509			1.71713			0.858566	+	0.512702i	$0.171356\pi$
$12$		0			0
$13$			2.15531i			0.597775i	0.954288	+	0.298887i	$0.0966155\pi$
$13$			2.15531i			0.597775i	−0.954288	+	0.298887i	$0.903384\pi$
$14$		−	2.72316i		−	0.727795i
$15$		0			0
$16$	1.00000			0.250000
$17$		−	5.09797i		−	1.23644i	−0.786005	−	0.618220i	$-0.787854\pi$
$17$		−	5.09797i		−	1.23644i	0.786005	−	0.618220i	$-0.212146\pi$
$18$		0			0
$19$	−6.22079			−1.42715			−0.713573	−	0.700581i	$-0.752924\pi$
$19$	−6.22079			−1.42715			−0.713573	+	0.700581i	$0.752924\pi$
$20$	1.71459			0.383394
$21$		0			0
$22$		−	5.69509i		−	1.21420i
$23$	8.90450			1.85672			0.928359	−	0.371686i	$-0.121220\pi$
$23$	8.90450			1.85672			0.928359	+	0.371686i	$0.121220\pi$
$24$		0			0
$25$	−2.06019			−0.412037
$26$	2.15531			0.422691
$27$		0			0
$28$	−2.72316			−0.514629
$29$			4.97921i			0.924616i	0.886719	+	0.462308i	$0.152979\pi$
$29$			4.97921i			0.924616i	−0.886719	+	0.462308i	$0.847021\pi$
$30$		0			0
$31$	5.24366			0.941789			0.470895	−	0.882190i	$-0.343931\pi$
$31$	5.24366			0.941789			0.470895	+	0.882190i	$0.343931\pi$
$32$		−	1.00000i		−	0.176777i
$33$		0			0
$34$	−5.09797			−0.874295
$35$	−4.66910			−0.789222
$36$		0			0
$37$	10.4615			1.71986			0.859929	−	0.510414i	$-0.170508\pi$
$37$	10.4615			1.71986			0.859929	+	0.510414i	$0.170508\pi$
$38$			6.22079i			1.00914i
$39$		0			0
$40$		−	1.71459i		−	0.271100i
$41$			8.23537i			1.28615i	0.765804	+	0.643074i	$0.222341\pi$
$41$			8.23537i			1.28615i	−0.765804	+	0.643074i	$0.777659\pi$
$42$		0			0
$43$	−8.34624			−1.27279			−0.636395	−	0.771363i	$-0.719575\pi$
$43$	−8.34624			−1.27279			−0.636395	+	0.771363i	$0.719575\pi$
$44$	−5.69509			−0.858566
$45$		0			0
$46$		−	8.90450i		−	1.31290i
$47$			3.32674i			0.485255i	0.970119	+	0.242628i	$0.0780093\pi$
$47$			3.32674i			0.485255i	−0.970119	+	0.242628i	$0.921991\pi$
$48$		0			0
$49$	0.415600			0.0593715
$50$			2.06019i			0.291354i
$51$		0			0
$52$		−	2.15531i		−	0.298887i
$53$			6.09267i			0.836893i	0.908242	+	0.418446i	$0.137425\pi$
$53$			6.09267i			0.836893i	−0.908242	+	0.418446i	$0.862575\pi$
$54$		0			0
$55$	−9.76473			−1.31668
$56$			2.72316i			0.363898i
$57$		0			0
$58$	4.97921			0.653802
$59$	−9.22605			−1.20113			−0.600565	−	0.799576i	$-0.705058\pi$
$59$	−9.22605			−1.20113			−0.600565	+	0.799576i	$0.705058\pi$
$60$		0			0
$61$		−	14.7558i		−	1.88928i	−0.328103	−	0.944642i	$-0.606409\pi$
$61$		−	14.7558i		−	1.88928i	0.328103	−	0.944642i	$-0.393591\pi$
$62$		−	5.24366i		−	0.665945i
$63$		0			0
$64$	−1.00000			−0.125000
$65$		−	3.69547i		−	0.458366i
$66$		0			0
$67$		−	9.61141i		−	1.17422i	−0.809507	−	0.587110i	$-0.800266\pi$
$67$		−	9.61141i		−	1.17422i	0.809507	−	0.587110i	$-0.199734\pi$
$68$			5.09797i			0.618220i
$69$		0			0
$70$			4.66910i			0.558064i
$71$	13.8983			1.64942			0.824710	−	0.565556i	$-0.191338\pi$
$71$	13.8983			1.64942			0.824710	+	0.565556i	$0.191338\pi$
$72$		0			0
$73$	−2.73390			−0.319978			−0.159989	−	0.987119i	$-0.551146\pi$
$73$	−2.73390			−0.319978			−0.159989	+	0.987119i	$0.551146\pi$
$74$		−	10.4615i		−	1.21612i
$75$		0			0
$76$	6.22079			0.713573
$77$	15.5086			1.76737
$78$		0			0
$79$			14.2075i			1.59847i	0.601021	+	0.799233i	$0.294761\pi$
$79$			14.2075i			1.59847i	−0.601021	+	0.799233i	$0.705239\pi$
$80$	−1.71459			−0.191697
$81$		0			0
$82$	8.23537			0.909444
$83$		−	10.3344i		−	1.13435i	−0.823598	−	0.567174i	$-0.808037\pi$
$83$		−	10.3344i		−	1.13435i	0.823598	−	0.567174i	$-0.191963\pi$
$84$		0			0
$85$			8.74093i			0.948087i
$86$			8.34624i			0.899998i
$87$		0			0
$88$			5.69509i			0.607098i
$89$			14.3149i			1.51738i	0.651451	+	0.758691i	$0.274161\pi$
$89$			14.3149i			1.51738i	−0.651451	+	0.758691i	$0.725839\pi$
$90$		0			0
$91$			5.86925i			0.615265i
$92$	−8.90450			−0.928359
$93$		0			0
$94$	3.32674			0.343127
$95$	10.6661			1.09432
$96$		0			0
$97$			9.02597i			0.916449i	0.888837	+	0.458224i	$0.151514\pi$
$97$			9.02597i			0.916449i	−0.888837	+	0.458224i	$0.848486\pi$
$98$		−	0.415600i		−	0.0419820i
$99$		0			0
$100$	2.06019			0.206019
$101$			10.0918i			1.00417i	0.864817	+	0.502087i	$0.167434\pi$
$101$			10.0918i			1.00417i	−0.864817	+	0.502087i	$0.832566\pi$
$102$		0			0
$103$		−	1.81190i		−	0.178532i	−0.996008	−	0.0892659i	$-0.971548\pi$
$103$		−	1.81190i		−	0.178532i	0.996008	−	0.0892659i	$-0.0284521\pi$
$104$	−2.15531			−0.211345
$105$		0			0
$106$	6.09267			0.591773
$107$	15.3989			1.48866			0.744332	−	0.667809i	$-0.232768\pi$
$107$	15.3989			1.48866			0.744332	+	0.667809i	$0.232768\pi$
$108$		0			0
$109$	1.26171			0.120850			0.0604252	−	0.998173i	$-0.480754\pi$
$109$	1.26171			0.120850			0.0604252	+	0.998173i	$0.480754\pi$
$110$			9.76473i			0.931030i
$111$		0			0
$112$	2.72316			0.257314
$113$	2.41440			0.227127			0.113564	−	0.993531i	$-0.463773\pi$
$113$	2.41440			0.227127			0.113564	+	0.993531i	$0.463773\pi$
$114$		0			0
$115$	−15.2676			−1.42371
$116$		−	4.97921i		−	0.462308i
$117$		0			0
$118$			9.22605i			0.849327i
$119$		−	13.8826i		−	1.27262i
$120$		0			0
$121$	21.4340			1.94855
$122$	−14.7558			−1.33593
$123$		0			0
$124$	−5.24366			−0.470895
$125$	12.1053			1.08273
$126$		0			0
$127$	4.80610			0.426473			0.213236	−	0.977001i	$-0.431600\pi$
$127$	4.80610			0.426473			0.213236	+	0.977001i	$0.431600\pi$
$128$			1.00000i			0.0883883i
$129$		0			0
$130$	−3.69547			−0.324114
$131$		−	10.5911i		−	0.925346i	−0.886529	−	0.462673i	$-0.846890\pi$
$131$		−	10.5911i		−	0.925346i	0.886529	−	0.462673i	$-0.153110\pi$
$132$		0			0
$133$	−16.9402			−1.46890
$134$	−9.61141			−0.830299
$135$		0			0
$136$	5.09797			0.437148
$137$	10.9631			0.936638			0.468319	−	0.883559i	$-0.344860\pi$
$137$	10.9631			0.936638			0.468319	+	0.883559i	$0.344860\pi$
$138$		0			0
$139$	−6.45147			−0.547206			−0.273603	−	0.961843i	$-0.588216\pi$
$139$	−6.45147			−0.547206			−0.273603	+	0.961843i	$0.588216\pi$
$140$	4.66910			0.394611
$141$		0			0
$142$		−	13.8983i		−	1.16632i
$143$			12.2747i			1.02646i
$144$		0			0
$145$		−	8.53730i		−	0.708984i
$146$			2.73390i			0.226259i
$147$		0			0
$148$	−10.4615			−0.859929
$149$	2.05653			0.168478			0.0842389	−	0.996446i	$-0.473154\pi$
$149$	2.05653			0.168478			0.0842389	+	0.996446i	$0.473154\pi$
$150$		0			0
$151$			1.91486i			0.155829i	0.996960	+	0.0779144i	$0.0248261\pi$
$151$			1.91486i			0.155829i	−0.996960	+	0.0779144i	$0.975174\pi$
$152$		−	6.22079i		−	0.504572i
$153$		0			0
$154$		−	15.5086i		−	1.24972i
$155$	−8.99072			−0.722152
$156$		0			0
$157$			5.93837i			0.473933i	0.971518	+	0.236967i	$0.0761532\pi$
$157$			5.93837i			0.473933i	−0.971518	+	0.236967i	$0.923847\pi$
$158$	14.2075			1.13029
$159$		0			0
$160$			1.71459i			0.135550i
$161$	24.2484			1.91104
$162$		0			0
$163$		−	7.42302i		−	0.581416i	−0.956812	−	0.290708i	$-0.906109\pi$
$163$		−	7.42302i		−	0.581416i	0.956812	−	0.290708i	$-0.0938908\pi$
$164$		−	8.23537i		−	0.643074i
$165$		0			0
$166$	−10.3344			−0.802105
$167$	17.3380			1.34166			0.670828	−	0.741613i	$-0.265939\pi$
$167$	17.3380			1.34166			0.670828	+	0.741613i	$0.265939\pi$
$168$		0			0
$169$	8.35465			0.642665
$170$	8.74093			0.670399
$171$		0			0
$172$	8.34624			0.636395
$173$	17.7439			1.34904			0.674521	−	0.738256i	$-0.264350\pi$
$173$	17.7439			1.34904			0.674521	+	0.738256i	$0.264350\pi$
$174$		0			0
$175$	−5.61022			−0.424093
$176$	5.69509			0.429283
$177$		0			0
$178$	14.3149			1.07295
$179$		−	23.7177i		−	1.77274i	−0.462975	−	0.886371i	$-0.653218\pi$
$179$		−	23.7177i		−	1.77274i	0.462975	−	0.886371i	$-0.346782\pi$
$180$		0			0
$181$	6.96204			0.517484			0.258742	−	0.965946i	$-0.416692\pi$
$181$	6.96204			0.517484			0.258742	+	0.965946i	$0.416692\pi$
$182$	5.86925			0.435058
$183$		0			0
$184$			8.90450i			0.656449i
$185$	−17.9371			−1.31876
$186$		0			0
$187$		−	29.0334i		−	2.12313i
$188$		−	3.32674i		−	0.242628i
$189$		0			0
$190$		−	10.6661i		−	0.773799i
$191$	−16.1236			−1.16666			−0.583332	−	0.812234i	$-0.698251\pi$
$191$	−16.1236			−1.16666			−0.583332	+	0.812234i	$0.698251\pi$
$192$		0			0
$193$		−	4.74952i		−	0.341878i	−0.985282	−	0.170939i	$-0.945320\pi$
$193$		−	4.74952i		−	0.341878i	0.985282	−	0.170939i	$-0.0546801\pi$
$194$	9.02597			0.648027
$195$		0			0
$196$	−0.415600			−0.0296857
$197$			7.80020i			0.555741i	0.960618	+	0.277871i	$0.0896287\pi$
$197$			7.80020i			0.555741i	−0.960618	+	0.277871i	$0.910371\pi$
$198$		0			0
$199$	−2.07044			−0.146770			−0.0733848	−	0.997304i	$-0.523380\pi$
$199$	−2.07044			−0.146770			−0.0733848	+	0.997304i	$0.523380\pi$
$200$		−	2.06019i		−	0.145677i
$201$		0			0
$202$	10.0918			0.710058
$203$			13.5592i			0.951669i
$204$		0			0
$205$		−	14.1203i		−	0.986202i
$206$	−1.81190			−0.126241
$207$		0			0
$208$			2.15531i			0.149444i
$209$	−35.4279			−2.45060
$210$		0			0
$211$	−6.22233			−0.428363			−0.214182	−	0.976794i	$-0.568708\pi$
$211$	−6.22233			−0.428363			−0.214182	+	0.976794i	$0.568708\pi$
$212$		−	6.09267i		−	0.418446i
$213$		0			0
$214$		−	15.3989i		−	1.05264i
$215$	14.3104			0.975959
$216$		0			0
$217$	14.2793			0.969344
$218$		−	1.26171i		−	0.0854541i
$219$		0			0
$220$	9.76473			0.658338
$221$	10.9877			0.739113
$222$		0			0
$223$	−3.68568	−	14.4712i	−0.246812	−	0.969064i
$223$	−3.68568	−	14.4712i	−0.246812	−	0.969064i
$224$		−	2.72316i		−	0.181949i
$225$		0			0
$226$		−	2.41440i		−	0.160603i
$227$		−	11.1613i		−	0.740800i	−0.928872	−	0.370400i	$-0.879221\pi$
$227$		−	11.1613i		−	0.740800i	0.928872	−	0.370400i	$-0.120779\pi$
$228$		0			0
$229$			8.01853i			0.529880i	0.964265	+	0.264940i	$0.0853521\pi$
$229$			8.01853i			0.529880i	−0.964265	+	0.264940i	$0.914648\pi$
$230$			15.2676i			1.00671i
$231$		0			0
$232$	−4.97921			−0.326901
$233$	1.26766			0.0830469			0.0415235	−	0.999138i	$-0.486779\pi$
$233$	1.26766			0.0830469			0.0415235	+	0.999138i	$0.486779\pi$
$234$		0			0
$235$		−	5.70399i		−	0.372088i
$236$	9.22605			0.600565
$237$		0			0
$238$	−13.8826			−0.899875
$239$		−	24.4917i		−	1.58423i	−0.610369	−	0.792117i	$-0.708979\pi$
$239$		−	24.4917i		−	1.58423i	0.610369	−	0.792117i	$-0.291021\pi$
$240$		0			0
$241$	−15.7720			−1.01597			−0.507983	−	0.861367i	$-0.669609\pi$
$241$	−15.7720			−1.01597			−0.507983	+	0.861367i	$0.669609\pi$
$242$		−	21.4340i		−	1.37783i
$243$		0			0
$244$			14.7558i			0.944642i
$245$	−0.712584			−0.0455253
$246$		0			0
$247$		−	13.4077i		−	0.853112i
$248$			5.24366i			0.332973i
$249$		0			0
$250$		−	12.1053i		−	0.765607i
$251$		−	9.13742i		−	0.576749i	−0.957518	−	0.288374i	$-0.906885\pi$
$251$		−	9.13742i		−	0.576749i	0.957518	−	0.288374i	$-0.0931147\pi$
$252$		0			0
$253$	50.7119			3.18823
$254$		−	4.80610i		−	0.301562i
$255$		0			0
$256$	1.00000			0.0625000
$257$		−	16.8435i		−	1.05067i	−0.850895	−	0.525336i	$-0.823940\pi$
$257$		−	16.8435i		−	1.05067i	0.850895	−	0.525336i	$-0.176060\pi$
$258$		0			0
$259$	28.4883			1.77018
$260$			3.69547i			0.229183i
$261$		0			0
$262$	−10.5911			−0.654318
$263$	−5.58731			−0.344529			−0.172264	−	0.985051i	$-0.555108\pi$
$263$	−5.58731			−0.344529			−0.172264	+	0.985051i	$0.555108\pi$
$264$		0			0
$265$		−	10.4464i		−	0.641719i
$266$			16.9402i			1.03867i
$267$		0			0
$268$			9.61141i			0.587110i
$269$	15.9206			0.970698			0.485349	−	0.874320i	$-0.338693\pi$
$269$	15.9206			0.970698			0.485349	+	0.874320i	$0.338693\pi$
$270$		0			0
$271$			22.3301i			1.35646i	0.734851	+	0.678229i	$0.237252\pi$
$271$			22.3301i			1.35646i	−0.734851	+	0.678229i	$0.762748\pi$
$272$		−	5.09797i		−	0.309110i
$273$		0			0
$274$		−	10.9631i		−	0.662303i
$275$	−11.7329			−0.707523
$276$		0			0
$277$		−	23.9138i		−	1.43684i	−0.695608	−	0.718422i	$-0.744865\pi$
$277$		−	23.9138i		−	1.43684i	0.695608	−	0.718422i	$-0.255135\pi$
$278$			6.45147i			0.386933i
$279$		0			0
$280$		−	4.66910i		−	0.279032i
$281$		−	9.59757i		−	0.572543i	−0.958149	−	0.286271i	$-0.907584\pi$
$281$		−	9.59757i		−	0.572543i	0.958149	−	0.286271i	$-0.0924159\pi$
$282$		0			0
$283$	24.5979			1.46220			0.731098	−	0.682272i	$-0.239008\pi$
$283$	24.5979			1.46220			0.731098	+	0.682272i	$0.239008\pi$
$284$	−13.8983			−0.824710
$285$		0			0
$286$	12.2747			0.725816
$287$			22.4262i			1.32378i
$288$		0			0
$289$	−8.98935			−0.528785
$290$	−8.53730			−0.501327
$291$		0			0
$292$	2.73390			0.159989
$293$	−5.79992			−0.338835			−0.169417	−	0.985544i	$-0.554189\pi$
$293$	−5.79992			−0.338835			−0.169417	+	0.985544i	$0.554189\pi$
$294$		0			0
$295$	15.8189			0.921011
$296$			10.4615i			0.608062i
$297$		0			0
$298$		−	2.05653i		−	0.119132i
$299$			19.1919i			1.10990i
$300$		0			0
$301$	−22.7282			−1.31003
$302$	1.91486			0.110188
$303$		0			0
$304$	−6.22079			−0.356787
$305$			25.3001i			1.44868i
$306$		0			0
$307$			6.81333i			0.388857i	0.980917	+	0.194429i	$0.0622852\pi$
$307$			6.81333i			0.388857i	−0.980917	+	0.194429i	$0.937715\pi$
$308$	−15.5086			−0.883686
$309$		0			0
$310$			8.99072i			0.510638i
$311$	10.1190			0.573796			0.286898	−	0.957961i	$-0.407376\pi$
$311$	10.1190			0.573796			0.286898	+	0.957961i	$0.407376\pi$
$312$		0			0
$313$		−	10.3859i		−	0.587047i	−0.955952	−	0.293523i	$-0.905172\pi$
$313$		−	10.3859i		−	0.587047i	0.955952	−	0.293523i	$-0.0948279\pi$
$314$	5.93837			0.335121
$315$		0			0
$316$		−	14.2075i		−	0.799233i
$317$		−	27.9548i		−	1.57010i	−0.619433	−	0.785050i	$-0.712637\pi$
$317$		−	27.9548i		−	1.57010i	0.619433	−	0.785050i	$-0.287363\pi$
$318$		0			0
$319$			28.3570i			1.58769i
$320$	1.71459			0.0958484
$321$		0			0
$322$		−	24.2484i		−	1.35131i
$323$			31.7134i			1.76458i
$324$		0			0
$325$		−	4.44034i		−	0.246306i
$326$	−7.42302			−0.411123
$327$		0			0
$328$	−8.23537			−0.454722
$329$			9.05925i			0.499453i
$330$		0			0
$331$		−	14.9923i		−	0.824050i	−0.911173	−	0.412025i	$-0.864822\pi$
$331$		−	14.9923i		−	0.824050i	0.911173	−	0.412025i	$-0.135178\pi$
$332$			10.3344i			0.567174i
$333$		0			0
$334$		−	17.3380i		−	0.948694i
$335$			16.4796i			0.900377i
$336$		0			0
$337$			10.9355i			0.595694i	0.954614	+	0.297847i	$0.0962686\pi$
$337$			10.9355i			0.595694i	−0.954614	+	0.297847i	$0.903731\pi$
$338$		−	8.35465i		−	0.454433i
$339$		0			0
$340$		−	8.74093i		−	0.474043i
$341$	29.8631			1.61718
$342$		0			0
$343$	−17.9304			−0.968149
$344$		−	8.34624i		−	0.449999i
$345$		0			0
$346$		−	17.7439i		−	0.953916i
$347$			24.5520i			1.31802i	0.752134	+	0.659011i	$0.229025\pi$
$347$			24.5520i			1.31802i	−0.752134	+	0.659011i	$0.770975\pi$
$348$		0			0
$349$	−24.7650			−1.32564			−0.662820	−	0.748779i	$-0.730641\pi$
$349$	−24.7650			−1.32564			−0.662820	+	0.748779i	$0.730641\pi$
$350$			5.61022i			0.299879i
$351$		0			0
$352$		−	5.69509i		−	0.303549i
$353$			13.0320i			0.693625i	0.937934	+	0.346813i	$0.112736\pi$
$353$			13.0320i			0.693625i	−0.937934	+	0.346813i	$0.887264\pi$
$354$		0			0
$355$	−23.8298			−1.26475
$356$		−	14.3149i		−	0.758691i
$357$		0			0
$358$	−23.7177			−1.25352
$359$			23.6647i			1.24897i	0.781036	+	0.624487i	$0.214692\pi$
$359$			23.6647i			1.24897i	−0.781036	+	0.624487i	$0.785308\pi$
$360$		0			0
$361$	19.6982			1.03675
$362$		−	6.96204i		−	0.365916i
$363$		0			0
$364$		−	5.86925i		−	0.307632i
$365$	4.68751			0.245355
$366$		0			0
$367$	−27.1296			−1.41615			−0.708076	−	0.706136i	$-0.750437\pi$
$367$	−27.1296			−1.41615			−0.708076	+	0.706136i	$0.750437\pi$
$368$	8.90450			0.464179
$369$		0			0
$370$			17.9371i			0.932508i
$371$			16.5913i			0.861378i
$372$		0			0
$373$			14.9918i			0.776246i	0.921608	+	0.388123i	$0.126876\pi$
$373$			14.9918i			0.776246i	−0.921608	+	0.388123i	$0.873124\pi$
$374$	−29.0334			−1.50128
$375$		0			0
$376$	−3.32674			−0.171564
$377$	−10.7317			−0.552713
$378$		0			0
$379$	31.8803			1.63758			0.818789	−	0.574094i	$-0.194646\pi$
$379$	31.8803			1.63758			0.818789	+	0.574094i	$0.194646\pi$
$380$	−10.6661			−0.547159
$381$		0			0
$382$			16.1236i			0.824955i
$383$	13.7294			0.701539			0.350770	−	0.936462i	$-0.385920\pi$
$383$	13.7294			0.701539			0.350770	+	0.936462i	$0.385920\pi$
$384$		0			0
$385$	−26.5909			−1.35520
$386$	−4.74952			−0.241744
$387$		0			0
$388$		−	9.02597i		−	0.458224i
$389$		−	1.16792i		−	0.0592159i	−0.999562	−	0.0296079i	$-0.990574\pi$
$389$		−	1.16792i		−	0.0592159i	0.999562	−	0.0296079i	$-0.00942588\pi$
$390$		0			0
$391$		−	45.3949i		−	2.29572i
$392$			0.415600i			0.0209910i
$393$		0			0
$394$	7.80020			0.392968
$395$		−	24.3600i		−	1.22568i
$396$		0			0
$397$		−	3.76726i		−	0.189073i	−0.995521	−	0.0945367i	$-0.969863\pi$
$397$		−	3.76726i		−	0.189073i	0.995521	−	0.0945367i	$-0.0301370\pi$
$398$			2.07044i			0.103782i
$399$		0			0
$400$	−2.06019			−0.103009
$401$			16.8892i			0.843406i	0.906734	+	0.421703i	$0.138568\pi$
$401$			16.8892i			0.843406i	−0.906734	+	0.421703i	$0.861432\pi$
$402$		0			0
$403$			11.3017i			0.562978i
$404$		−	10.0918i		−	0.502087i
$405$		0			0
$406$	13.5592			0.672931
$407$	59.5791			2.95322
$408$		0			0
$409$			28.6135i			1.41485i	0.706790	+	0.707424i	$0.250143\pi$
$409$			28.6135i			1.41485i	−0.706790	+	0.707424i	$0.749857\pi$
$410$	−14.1203			−0.697350
$411$		0			0
$412$			1.81190i			0.0892659i
$413$	−25.1240			−1.23627
$414$		0			0
$415$			17.7192i			0.869803i
$416$	2.15531			0.105673
$417$		0			0
$418$			35.4279i			1.73284i
$419$		−	27.2582i		−	1.33165i	−0.746107	−	0.665826i	$-0.768079\pi$
$419$		−	27.2582i		−	1.33165i	0.746107	−	0.665826i	$-0.231921\pi$
$420$		0			0
$421$		−	16.6079i		−	0.809418i	−0.914446	−	0.404709i	$-0.867373\pi$
$421$		−	16.6079i		−	0.809418i	0.914446	−	0.404709i	$-0.132627\pi$
$422$			6.22233i			0.302898i
$423$		0			0
$424$	−6.09267			−0.295886
$425$			10.5028i			0.509460i
$426$		0			0
$427$		−	40.1823i		−	1.94456i
$428$	−15.3989			−0.744332
$429$		0			0
$430$		−	14.3104i		−	0.690107i
$431$	29.1501			1.40411			0.702056	−	0.712122i	$-0.252266\pi$
$431$	29.1501			1.40411			0.702056	+	0.712122i	$0.252266\pi$
$432$		0			0
$433$	−5.94279			−0.285592			−0.142796	−	0.989752i	$-0.545609\pi$
$433$	−5.94279			−0.285592			−0.142796	+	0.989752i	$0.545609\pi$
$434$		−	14.2793i		−	0.685430i
$435$		0			0
$436$	−1.26171			−0.0604252
$437$	−55.3930			−2.64981
$438$		0			0
$439$			24.6887i			1.17833i	0.808014	+	0.589164i	$0.200543\pi$
$439$			24.6887i			1.17833i	−0.808014	+	0.589164i	$0.799457\pi$
$440$		−	9.76473i		−	0.465515i
$441$		0			0
$442$		−	10.9877i		−	0.522632i
$443$			9.92339i			0.471475i	0.971817	+	0.235737i	$0.0757505\pi$
$443$			9.92339i			0.471475i	−0.971817	+	0.235737i	$0.924250\pi$
$444$		0			0
$445$		−	24.5442i		−	1.16351i
$446$	−14.4712	+	3.68568i	−0.685231	+	0.174522i
$447$		0			0
$448$	−2.72316			−0.128657
$449$	7.36039			0.347358			0.173679	−	0.984802i	$-0.444434\pi$
$449$	7.36039			0.347358			0.173679	+	0.984802i	$0.444434\pi$
$450$		0			0
$451$			46.9011i			2.20849i
$452$	−2.41440			−0.113564
$453$		0			0
$454$	−11.1613			−0.523825
$455$		−	10.0633i		−	0.471777i
$456$		0			0
$457$			38.5380i			1.80273i	0.433058	+	0.901366i	$0.357434\pi$
$457$			38.5380i			1.80273i	−0.433058	+	0.901366i	$0.642566\pi$
$458$	8.01853			0.374681
$459$		0			0
$460$	15.2676			0.711854
$461$			3.14232i			0.146353i	0.997319	+	0.0731763i	$0.0233136\pi$
$461$			3.14232i			0.146353i	−0.997319	+	0.0731763i	$0.976686\pi$
$462$		0			0
$463$	12.2547			0.569524			0.284762	−	0.958598i	$-0.408085\pi$
$463$	12.2547			0.569524			0.284762	+	0.958598i	$0.408085\pi$
$464$			4.97921i			0.231154i
$465$		0			0
$466$		−	1.26766i		−	0.0587230i
$467$	3.91394			0.181115			0.0905577	−	0.995891i	$-0.471135\pi$
$467$	3.91394			0.181115			0.0905577	+	0.995891i	$0.471135\pi$
$468$		0			0
$469$		−	26.1734i		−	1.20858i
$470$	−5.70399			−0.263106
$471$		0			0
$472$		−	9.22605i		−	0.424663i
$473$	−47.5326			−2.18555
$474$		0			0
$475$	12.8160			0.588038
$476$			13.8826i			0.636308i
$477$		0			0
$478$	−24.4917			−1.12022
$479$			26.6347i			1.21697i	0.793564	+	0.608486i	$0.208223\pi$
$479$			26.6347i			1.21697i	−0.793564	+	0.608486i	$0.791777\pi$
$480$		0			0
$481$			22.5477i			1.02809i
$482$			15.7720i			0.718396i
$483$		0			0
$484$	−21.4340			−0.974273
$485$		−	15.4758i		−	0.702721i
$486$		0			0
$487$	−10.7806			−0.488515			−0.244258	−	0.969710i	$-0.578544\pi$
$487$	−10.7806			−0.488515			−0.244258	+	0.969710i	$0.578544\pi$
$488$	14.7558			0.667963
$489$		0			0
$490$			0.712584i			0.0321912i
$491$	−37.6216			−1.69784			−0.848920	−	0.528522i	$-0.822746\pi$
$491$	−37.6216			−1.69784			−0.848920	+	0.528522i	$0.822746\pi$
$492$		0			0
$493$	25.3839			1.14323
$494$	−13.4077			−0.603241
$495$		0			0
$496$	5.24366			0.235447
$497$	37.8472			1.69768
$498$		0			0
$499$	38.7196			1.73333			0.866664	−	0.498893i	$-0.166260\pi$
$499$	38.7196			1.73333			0.866664	+	0.498893i	$0.166260\pi$
$500$	−12.1053			−0.541366
$501$		0			0
$502$	−9.13742			−0.407823
$503$	−15.2399			−0.679514			−0.339757	−	0.940513i	$-0.610345\pi$
$503$	−15.2399			−0.679514			−0.339757	+	0.940513i	$0.610345\pi$
$504$		0			0
$505$		−	17.3033i		−	0.769988i
$506$		−	50.7119i		−	2.25442i
$507$		0			0
$508$	−4.80610			−0.213236
$509$		−	29.7847i		−	1.32018i	−0.751185	−	0.660091i	$-0.770518\pi$
$509$		−	29.7847i		−	1.32018i	0.751185	−	0.660091i	$-0.229482\pi$
$510$		0			0
$511$	−7.44484			−0.329340
$512$		−	1.00000i		−	0.0441942i
$513$		0			0
$514$	−16.8435			−0.742937
$515$			3.10666i			0.136896i
$516$		0			0
$517$			18.9461i			0.833248i
$518$		−	28.4883i		−	1.25170i
$519$		0			0
$520$	3.69547			0.162057
$521$	7.47400			0.327442			0.163721	−	0.986507i	$-0.447650\pi$
$521$	7.47400			0.327442			0.163721	+	0.986507i	$0.447650\pi$
$522$		0			0
$523$		−	14.6326i		−	0.639840i	−0.947445	−	0.319920i	$-0.896344\pi$
$523$		−	14.6326i		−	0.639840i	0.947445	−	0.319920i	$-0.103656\pi$
$524$			10.5911i			0.462673i
$525$		0			0
$526$			5.58731i			0.243618i
$527$		−	26.7320i		−	1.16447i
$528$		0			0
$529$	56.2902			2.44740
$530$	−10.4464			−0.453764
$531$		0			0
$532$	16.9402			0.734451
$533$	−17.7498			−0.768827
$534$		0			0
$535$	−26.4027			−1.14149
$536$	9.61141			0.415150
$537$		0			0
$538$		−	15.9206i		−	0.686387i
$539$	2.36688			0.101949
$540$		0			0
$541$			43.8037i			1.88327i	0.336639	+	0.941634i	$0.390710\pi$
$541$			43.8037i			1.88327i	−0.336639	+	0.941634i	$0.609290\pi$
$542$	22.3301			0.959160
$543$		0			0
$544$	−5.09797			−0.218574
$545$	−2.16332			−0.0926665
$546$		0			0
$547$	10.9414			0.467822			0.233911	−	0.972258i	$-0.424848\pi$
$547$	10.9414			0.467822			0.233911	+	0.972258i	$0.424848\pi$
$548$	−10.9631			−0.468319
$549$		0			0
$550$			11.7329i			0.500294i
$551$		−	30.9746i		−	1.31956i
$552$		0			0
$553$			38.6893i			1.64523i
$554$	−23.9138			−1.01600
$555$		0			0
$556$	6.45147			0.273603
$557$	−13.0992			−0.555032			−0.277516	−	0.960721i	$-0.589511\pi$
$557$	−13.0992			−0.555032			−0.277516	+	0.960721i	$0.589511\pi$
$558$		0			0
$559$		−	17.9887i		−	0.760842i
$560$	−4.66910			−0.197305
$561$		0			0
$562$	−9.59757			−0.404849
$563$	16.8641			0.710738			0.355369	−	0.934726i	$-0.384355\pi$
$563$	16.8641			0.710738			0.355369	+	0.934726i	$0.384355\pi$
$564$		0			0
$565$	−4.13970			−0.174158
$566$		−	24.5979i		−	1.03393i
$567$		0			0
$568$			13.8983i			0.583158i
$569$	−2.90859			−0.121934			−0.0609672	−	0.998140i	$-0.519419\pi$
$569$	−2.90859			−0.121934			−0.0609672	+	0.998140i	$0.519419\pi$
$570$		0			0
$571$		−	36.3865i		−	1.52273i	−0.648325	−	0.761364i	$-0.724530\pi$
$571$		−	36.3865i		−	1.52273i	0.648325	−	0.761364i	$-0.275470\pi$
$572$		−	12.2747i		−	0.513230i
$573$		0			0
$574$	22.4262			0.936052
$575$	−18.3449			−0.765037
$576$		0			0
$577$	−30.2202			−1.25808			−0.629042	−	0.777371i	$-0.716553\pi$
$577$	−30.2202			−1.25808			−0.629042	+	0.777371i	$0.716553\pi$
$578$			8.98935i			0.373907i
$579$		0			0
$580$			8.53730i			0.354492i
$581$		−	28.1422i		−	1.16754i
$582$		0			0
$583$			34.6983i			1.43706i
$584$		−	2.73390i		−	0.113129i
$585$		0			0
$586$			5.79992i			0.239592i
$587$	−19.2270			−0.793582			−0.396791	−	0.917909i	$-0.629876\pi$
$587$	−19.2270			−0.793582			−0.396791	+	0.917909i	$0.629876\pi$
$588$		0			0
$589$	−32.6197			−1.34407
$590$		−	15.8189i		−	0.651253i
$591$		0			0
$592$	10.4615			0.429964
$593$	42.6943			1.75325			0.876623	−	0.481178i	$-0.159791\pi$
$593$	42.6943			1.75325			0.876623	+	0.481178i	$0.159791\pi$
$594$		0			0
$595$			23.8029i			0.975826i
$596$	−2.05653			−0.0842389
$597$		0			0
$598$	19.1919			0.784817
$599$			6.81563i			0.278479i	0.990259	+	0.139240i	$0.0444658\pi$
$599$			6.81563i			0.278479i	−0.990259	+	0.139240i	$0.955534\pi$
$600$		0			0
$601$			0.694341i			0.0283228i	0.999900	+	0.0141614i	$0.00450786\pi$
$601$			0.694341i			0.0283228i	−0.999900	+	0.0141614i	$0.995492\pi$
$602$			22.7282i			0.926330i
$603$		0			0
$604$		−	1.91486i		−	0.0779144i
$605$	−36.7505			−1.49412
$606$		0			0
$607$			5.27000i			0.213903i	0.994264	+	0.106951i	$0.0341089\pi$
$607$			5.27000i			0.213903i	−0.994264	+	0.106951i	$0.965891\pi$
$608$			6.22079i			0.252286i
$609$		0			0
$610$	25.3001			1.02437
$611$	−7.17016			−0.290073
$612$		0			0
$613$		−	17.3462i		−	0.700608i	−0.936636	−	0.350304i	$-0.886078\pi$
$613$		−	17.3462i		−	0.700608i	0.936636	−	0.350304i	$-0.113922\pi$
$614$	6.81333			0.274963
$615$		0			0
$616$			15.5086i			0.624860i
$617$			32.0140i			1.28883i	0.764675	+	0.644417i	$0.222900\pi$
$617$			32.0140i			1.28883i	−0.764675	+	0.644417i	$0.777100\pi$
$618$		0			0
$619$			8.71506i			0.350288i	0.984543	+	0.175144i	$0.0560391\pi$
$619$			8.71506i			0.350288i	−0.984543	+	0.175144i	$0.943961\pi$
$620$	8.99072			0.361076
$621$		0			0
$622$		−	10.1190i		−	0.405735i
$623$			38.9819i			1.56178i
$624$		0			0
$625$	−10.4547			−0.418188
$626$	−10.3859			−0.415105
$627$		0			0
$628$		−	5.93837i		−	0.236967i
$629$		−	53.3324i		−	2.12650i
$630$		0			0
$631$			18.3094i			0.728887i	0.931225	+	0.364444i	$0.118741\pi$
$631$			18.3094i			0.728887i	−0.931225	+	0.364444i	$0.881259\pi$
$632$	−14.2075			−0.565143
$633$		0			0
$634$	−27.9548			−1.11023
$635$	−8.24049			−0.327014
$636$		0			0
$637$			0.895747i			0.0354908i
$638$	28.3570			1.12267
$639$		0			0
$640$		−	1.71459i		−	0.0677751i
$641$	−50.3046			−1.98691			−0.993456	−	0.114211i	$-0.963566\pi$
$641$	−50.3046			−1.98691			−0.993456	+	0.114211i	$0.963566\pi$
$642$		0			0
$643$	7.19229			0.283636			0.141818	−	0.989893i	$-0.454705\pi$
$643$	7.19229			0.283636			0.141818	+	0.989893i	$0.454705\pi$
$644$	−24.2484			−0.955520
$645$		0			0
$646$	31.7134			1.24775
$647$			7.66769i			0.301448i	0.988576	+	0.150724i	$0.0481605\pi$
$647$			7.66769i			0.301448i	−0.988576	+	0.150724i	$0.951840\pi$
$648$		0			0
$649$	−52.5431			−2.06250
$650$	−4.44034			−0.174164
$651$		0			0
$652$			7.42302i			0.290708i
$653$	2.42193			0.0947774			0.0473887	−	0.998877i	$-0.484910\pi$
$653$	2.42193			0.0947774			0.0473887	+	0.998877i	$0.484910\pi$
$654$		0			0
$655$			18.1593i			0.709543i
$656$			8.23537i			0.321537i
$657$		0			0
$658$	9.05925			0.353166
$659$			34.7281i			1.35281i	0.736528	+	0.676407i	$0.236464\pi$
$659$			34.7281i			1.35281i	−0.736528	+	0.676407i	$0.763536\pi$
$660$		0			0
$661$			32.0888i			1.24811i	0.781380	+	0.624055i	$0.214516\pi$
$661$			32.0888i			1.24811i	−0.781380	+	0.624055i	$0.785484\pi$
$662$	−14.9923			−0.582691
$663$		0			0
$664$	10.3344			0.401053
$665$	29.0455			1.12633
$666$		0			0
$667$			44.3374i			1.71675i
$668$	−17.3380			−0.670828
$669$		0			0
$670$	16.4796			0.636663
$671$		−	84.0354i		−	3.24415i
$672$		0			0
$673$	−29.0922			−1.12142			−0.560711	−	0.828012i	$-0.689472\pi$
$673$	−29.0922			−1.12142			−0.560711	+	0.828012i	$0.689472\pi$
$674$	10.9355			0.421220
$675$		0			0
$676$	−8.35465			−0.321333
$677$			18.9968i			0.730108i	0.930986	+	0.365054i	$0.118949\pi$
$677$			18.9968i			0.730108i	−0.930986	+	0.365054i	$0.881051\pi$
$678$		0			0
$679$			24.5792i			0.943262i
$680$	−8.74093			−0.335199
$681$		0			0
$682$		−	29.8631i		−	1.14352i
$683$		−	9.90389i		−	0.378962i	−0.981884	−	0.189481i	$-0.939320\pi$
$683$		−	9.90389i		−	0.378962i	0.981884	−	0.189481i	$-0.0606805\pi$
$684$		0			0
$685$	−18.7971			−0.718202
$686$			17.9304i			0.684585i
$687$		0			0
$688$	−8.34624			−0.318198
$689$	−13.1316			−0.500274
$690$		0			0
$691$			11.5659i			0.439989i	0.975501	+	0.219995i	$0.0706040\pi$
$691$			11.5659i			0.439989i	−0.975501	+	0.219995i	$0.929396\pi$
$692$	−17.7439			−0.674521
$693$		0			0
$694$	24.5520			0.931982
$695$	11.0616			0.419591
$696$		0			0
$697$	41.9837			1.59025
$698$			24.7650i			0.937369i
$699$		0			0
$700$	5.61022			0.212046
$701$		−	14.8527i		−	0.560980i	−0.959857	−	0.280490i	$-0.909503\pi$
$701$		−	14.8527i		−	0.560980i	0.959857	−	0.280490i	$-0.0904969\pi$
$702$		0			0
$703$	−65.0787			−2.45449
$704$	−5.69509			−0.214642
$705$		0			0
$706$	13.0320			0.490467
$707$			27.4816i			1.03355i
$708$		0			0
$709$		−	26.2674i		−	0.986495i	−0.869889	−	0.493247i	$-0.835810\pi$
$709$		−	26.2674i		−	0.986495i	0.869889	−	0.493247i	$-0.164190\pi$
$710$			23.8298i			0.894316i
$711$		0			0
$712$	−14.3149			−0.536475
$713$	46.6922			1.74864
$714$		0			0
$715$		−	21.0460i		−	0.787076i
$716$			23.7177i			0.886371i
$717$		0			0
$718$	23.6647			0.883157
$719$			30.2566i			1.12838i	0.825645	+	0.564190i	$0.190811\pi$
$719$			30.2566i			1.12838i	−0.825645	+	0.564190i	$0.809189\pi$
$720$		0			0
$721$		−	4.93410i		−	0.183755i
$722$		−	19.6982i		−	0.733090i
$723$		0			0
$724$	−6.96204			−0.258742
$725$		−	10.2581i		−	0.380977i
$726$		0			0
$727$	−27.7706			−1.02996			−0.514978	−	0.857204i	$-0.672200\pi$
$727$	−27.7706			−1.02996			−0.514978	+	0.857204i	$0.672200\pi$
$728$	−5.86925			−0.217529
$729$		0			0
$730$		−	4.68751i		−	0.173492i
$731$			42.5489i			1.57373i
$732$		0			0
$733$	−15.8527			−0.585535			−0.292767	−	0.956184i	$-0.594576\pi$
$733$	−15.8527			−0.585535			−0.292767	+	0.956184i	$0.594576\pi$
$734$			27.1296i			1.00137i
$735$		0			0
$736$		−	8.90450i		−	0.328224i
$737$		−	54.7378i		−	2.01629i
$738$		0			0
$739$			25.5906i			0.941365i	0.882303	+	0.470683i	$0.155992\pi$
$739$			25.5906i			0.941365i	−0.882303	+	0.470683i	$0.844008\pi$
$740$	17.9371			0.659382
$741$		0			0
$742$	16.5913			0.609086
$743$		−	28.5801i		−	1.04850i	−0.851563	−	0.524252i	$-0.824345\pi$
$743$		−	28.5801i		−	1.04850i	0.851563	−	0.524252i	$-0.175655\pi$
$744$		0			0
$745$	−3.52611			−0.129187
$746$	14.9918			0.548889
$747$		0			0
$748$			29.0334i			1.06157i
$749$	41.9336			1.53222
$750$		0			0
$751$	−49.1134			−1.79217			−0.896086	−	0.443880i	$-0.853602\pi$
$751$	−49.1134			−1.79217			−0.896086	+	0.443880i	$0.853602\pi$
$752$			3.32674i			0.121314i
$753$		0			0
$754$			10.7317i			0.390827i
$755$		−	3.28319i		−	0.119488i
$756$		0			0
$757$		−	43.7942i		−	1.59173i	−0.605475	−	0.795864i	$-0.707017\pi$
$757$		−	43.7942i		−	1.59173i	0.605475	−	0.795864i	$-0.292983\pi$
$758$		−	31.8803i		−	1.15794i
$759$		0			0
$760$			10.6661i			0.386900i
$761$	−40.0949			−1.45344			−0.726720	−	0.686934i	$-0.758956\pi$
$761$	−40.0949			−1.45344			−0.726720	+	0.686934i	$0.758956\pi$
$762$		0			0
$763$	3.43585			0.124386
$764$	16.1236			0.583332
$765$		0			0
$766$		−	13.7294i		−	0.496063i
$767$		−	19.8850i		−	0.718005i
$768$		0			0
$769$	−26.6450			−0.960843			−0.480421	−	0.877038i	$-0.659516\pi$
$769$	−26.6450			−0.960843			−0.480421	+	0.877038i	$0.659516\pi$
$770$			26.5909i			0.958270i
$771$		0			0
$772$			4.74952i			0.170939i
$773$	−33.0715			−1.18950			−0.594749	−	0.803911i	$-0.702749\pi$
$773$	−33.0715			−1.18950			−0.594749	+	0.803911i	$0.702749\pi$
$774$		0			0
$775$	−10.8029			−0.388052
$776$	−9.02597			−0.324014
$777$		0			0
$778$	−1.16792			−0.0418720
$779$		−	51.2304i		−	1.83552i
$780$		0			0
$781$	79.1518			2.83227
$782$	−45.3949			−1.62332
$783$		0			0
$784$	0.415600			0.0148429
$785$		−	10.1819i		−	0.363406i
$786$		0			0
$787$			36.3758i			1.29666i	0.761361	+	0.648329i	$0.224532\pi$
$787$			36.3758i			1.29666i	−0.761361	+	0.648329i	$0.775468\pi$
$788$		−	7.80020i		−	0.277871i
$789$		0			0
$790$	−24.3600			−0.866689
$791$	6.57479			0.233773
$792$		0			0
$793$	31.8032			1.12937
$794$	−3.76726			−0.133695
$795$		0			0
$796$	2.07044			0.0733848
$797$		−	7.25466i		−	0.256973i	−0.991711	−	0.128487i	$-0.958988\pi$
$797$		−	7.25466i		−	0.256973i	0.991711	−	0.128487i	$-0.0410119\pi$
$798$		0			0
$799$	16.9597			0.599989
$800$			2.06019i			0.0728386i
$801$		0			0
$802$	16.8892			0.596378
$803$	−15.5698			−0.549445
$804$		0			0
$805$	−41.5760			−1.46536
$806$	11.3017			0.398086
$807$		0			0
$808$	−10.0918			−0.355029
$809$	−14.2560			−0.501215			−0.250607	−	0.968089i	$-0.580630\pi$
$809$	−14.2560			−0.501215			−0.250607	+	0.968089i	$0.580630\pi$
$810$		0			0
$811$		−	10.2537i		−	0.360058i	−0.983661	−	0.180029i	$-0.942381\pi$
$811$		−	10.2537i		−	0.360058i	0.983661	−	0.180029i	$-0.0576191\pi$
$812$		−	13.5592i		−	0.475834i
$813$		0			0
$814$		−	59.5791i		−	2.08825i
$815$			12.7274i			0.445822i
$816$		0			0
$817$	51.9202			1.81646
$818$	28.6135			1.00045
$819$		0			0
$820$			14.1203i			0.493101i
$821$			39.2527i			1.36993i	0.728576	+	0.684965i	$0.240182\pi$
$821$			39.2527i			1.36993i	−0.728576	+	0.684965i	$0.759818\pi$
$822$		0			0
$823$		−	29.0943i		−	1.01416i	−0.861898	−	0.507082i	$-0.830724\pi$
$823$		−	29.0943i		−	1.01416i	0.861898	−	0.507082i	$-0.169276\pi$
$824$	1.81190			0.0631206
$825$		0			0
$826$			25.1240i			0.874176i
$827$	19.6811			0.684378			0.342189	−	0.939631i	$-0.388832\pi$
$827$	19.6811			0.684378			0.342189	+	0.939631i	$0.388832\pi$
$828$		0			0
$829$		−	13.5708i		−	0.471335i	−0.971834	−	0.235667i	$-0.924272\pi$
$829$		−	13.5708i		−	0.471335i	0.971834	−	0.235667i	$-0.0757276\pi$
$830$	17.7192			0.615044
$831$		0			0
$832$		−	2.15531i		−	0.0747219i
$833$		−	2.11872i		−	0.0734093i
$834$		0			0
$835$	−29.7276			−1.02876
$836$	35.4279			1.22530
$837$		0			0
$838$	−27.2582			−0.941620
$839$	14.4254			0.498020			0.249010	−	0.968501i	$-0.419895\pi$
$839$	14.4254			0.498020			0.249010	+	0.968501i	$0.419895\pi$
$840$		0			0
$841$	4.20745			0.145085
$842$	−16.6079			−0.572345
$843$		0			0
$844$	6.22233			0.214182
$845$	−14.3248			−0.492787
$846$		0			0
$847$	58.3682			2.00556
$848$			6.09267i			0.209223i
$849$		0			0
$850$	10.5028			0.360242
$851$	93.1543			3.19329
$852$		0			0
$853$		−	36.4130i		−	1.24676i	−0.781920	−	0.623379i	$-0.785760\pi$
$853$		−	36.4130i		−	1.24676i	0.781920	−	0.623379i	$-0.214240\pi$
$854$	−40.1823			−1.37501
$855$		0			0
$856$			15.3989i			0.526322i
$857$			21.8809i			0.747439i	0.927542	+	0.373719i	$0.121918\pi$
$857$			21.8809i			0.747439i	−0.927542	+	0.373719i	$0.878082\pi$
$858$		0			0
$859$			51.3957i			1.75360i	0.480858	+	0.876799i	$0.340325\pi$
$859$			51.3957i			1.75360i	−0.480858	+	0.876799i	$0.659675\pi$
$860$	−14.3104			−0.487980
$861$		0			0
$862$		−	29.1501i		−	0.992857i
$863$	0.421982			0.0143644			0.00718221	−	0.999974i	$-0.497714\pi$
$863$	0.421982			0.0143644			0.00718221	+	0.999974i	$0.497714\pi$
$864$		0			0
$865$	−30.4234			−1.03443
$866$			5.94279i			0.201944i
$867$		0			0
$868$	−14.2793			−0.484672
$869$			80.9128i			2.74478i
$870$		0			0
$871$	20.7155			0.701920
$872$			1.26171i			0.0427270i
$873$		0			0
$874$			55.3930i			1.87370i
$875$	32.9647			1.11441
$876$		0			0
$877$			22.2237i			0.750440i	0.926936	+	0.375220i	$0.122433\pi$
$877$			22.2237i			0.750440i	−0.926936	+	0.375220i	$0.877567\pi$
$878$	24.6887			0.833204
$879$		0			0
$880$	−9.76473			−0.329169
$881$		−	38.2018i		−	1.28705i	−0.765424	−	0.643526i	$-0.777471\pi$
$881$		−	38.2018i		−	1.28705i	0.765424	−	0.643526i	$-0.222529\pi$
$882$		0			0
$883$		−	14.9675i		−	0.503697i	−0.967767	−	0.251848i	$-0.918962\pi$
$883$		−	14.9675i		−	0.503697i	0.967767	−	0.251848i	$-0.0810384\pi$
$884$	−10.9877			−0.369557
$885$		0			0
$886$	9.92339			0.333383
$887$			47.8003i			1.60498i	0.596668	+	0.802488i	$0.296491\pi$
$887$			47.8003i			1.60498i	−0.596668	+	0.802488i	$0.703509\pi$
$888$		0			0
$889$	13.0878			0.438950
$890$	−24.5442			−0.822725
$891$		0			0
$892$	3.68568	+	14.4712i	0.123406	+	0.484532i
$893$		−	20.6950i		−	0.692530i
$894$		0			0
$895$			40.6661i			1.35932i
$896$			2.72316i			0.0909744i
$897$		0			0
$898$		−	7.36039i		−	0.245619i
$899$			26.1093i			0.870794i
$900$		0			0
$901$	31.0603			1.03477
$902$	46.9011			1.56164
$903$		0			0
$904$			2.41440i			0.0803017i
$905$	−11.9370			−0.396800
$906$		0			0
$907$	−20.3547			−0.675868			−0.337934	−	0.941170i	$-0.609728\pi$
$907$	−20.3547			−0.675868			−0.337934	+	0.941170i	$0.609728\pi$
$908$			11.1613i			0.370400i
$909$		0			0
$910$	−10.0633			−0.333597
$911$		−	51.1890i		−	1.69597i	−0.530022	−	0.847984i	$-0.677816\pi$
$911$		−	51.1890i		−	1.69597i	0.530022	−	0.847984i	$-0.322184\pi$
$912$		0			0
$913$		−	58.8553i		−	1.94783i
$914$	38.5380			1.27472
$915$		0			0
$916$		−	8.01853i		−	0.264940i
$917$		−	28.8412i		−	0.952419i
$918$		0			0
$919$		−	16.1875i		−	0.533976i	−0.963700	−	0.266988i	$-0.913972\pi$
$919$		−	16.1875i		−	0.533976i	0.963700	−	0.266988i	$-0.0860284\pi$
$920$		−	15.2676i		−	0.503356i
$921$		0			0
$922$	3.14232			0.103487
$923$			29.9550i			0.985982i
$924$		0			0
$925$	−21.5526			−0.708646
$926$		−	12.2547i		−	0.402714i
$927$		0			0
$928$	4.97921			0.163451
$929$		−	42.1013i		−	1.38130i	−0.723190	−	0.690649i	$-0.757325\pi$
$929$		−	42.1013i		−	1.38130i	0.723190	−	0.690649i	$-0.242675\pi$
$930$		0			0
$931$	−2.58536			−0.0847318
$932$	−1.26766			−0.0415235
$933$		0			0
$934$		−	3.91394i		−	0.128068i
$935$			49.7803i			1.62799i
$936$		0			0
$937$		−	22.5094i		−	0.735351i	−0.929954	−	0.367676i	$-0.880154\pi$
$937$		−	22.5094i		−	0.735351i	0.929954	−	0.367676i	$-0.119846\pi$
$938$	−26.1734			−0.854592
$939$		0			0
$940$			5.70399i			0.186044i
$941$			15.1957i			0.495365i	0.968841	+	0.247683i	$0.0796690\pi$
$941$			15.1957i			0.495365i	−0.968841	+	0.247683i	$0.920331\pi$
$942$		0			0
$943$			73.3318i			2.38801i
$944$	−9.22605			−0.300282
$945$		0			0
$946$			47.5326i			1.54542i
$947$		−	19.3168i		−	0.627712i	−0.949471	−	0.313856i	$-0.898379\pi$
$947$		−	19.3168i		−	0.627712i	0.949471	−	0.313856i	$-0.101621\pi$
$948$		0			0
$949$		−	5.89239i		−	0.191275i
$950$		−	12.8160i		−	0.415805i
$951$		0			0
$952$	13.8826			0.449938
$953$	18.0791			0.585640			0.292820	−	0.956168i	$-0.405406\pi$
$953$	18.0791			0.585640			0.292820	+	0.956168i	$0.405406\pi$
$954$		0			0
$955$	27.6453			0.894582
$956$			24.4917i			0.792117i
$957$		0			0
$958$	26.6347			0.860530
$959$	29.8542			0.964042
$960$		0			0
$961$	−3.50403			−0.113033
$962$	22.5477			0.726968
$963$		0			0
$964$	15.7720			0.507983
$965$			8.14347i			0.262148i
$966$		0			0
$967$			16.4097i			0.527701i	0.964564	+	0.263851i	$0.0849926\pi$
$967$			16.4097i			0.527701i	−0.964564	+	0.263851i	$0.915007\pi$
$968$			21.4340i			0.688915i
$969$		0			0
$970$	−15.4758			−0.496899
$971$	−10.4129			−0.334166			−0.167083	−	0.985943i	$-0.553435\pi$
$971$	−10.4129			−0.334166			−0.167083	+	0.985943i	$0.553435\pi$
$972$		0			0
$973$	−17.5684			−0.563216
$974$			10.7806i			0.345433i
$975$		0			0
$976$		−	14.7558i		−	0.472321i
$977$	−55.0812			−1.76220			−0.881102	−	0.472927i	$-0.843198\pi$
$977$	−55.0812			−1.76220			−0.881102	+	0.472927i	$0.843198\pi$
$978$		0			0
$979$			81.5248i			2.60555i
$980$	0.712584			0.0227626
$981$		0			0
$982$			37.6216i			1.20055i
$983$	−9.13960			−0.291508			−0.145754	−	0.989321i	$-0.546561\pi$
$983$	−9.13960			−0.291508			−0.145754	+	0.989321i	$0.546561\pi$
$984$		0			0
$985$		−	13.3741i		−	0.426135i
$986$		−	25.3839i		−	0.808388i
$987$		0			0
$988$			13.4077i			0.426556i
$989$	−74.3191			−2.36321
$990$		0			0
$991$		−	6.37914i		−	0.202640i	−0.994854	−	0.101320i	$-0.967693\pi$
$991$		−	6.37914i		−	0.202640i	0.994854	−	0.101320i	$-0.0323066\pi$
$992$		−	5.24366i		−	0.166486i
$993$		0			0
$994$		−	37.8472i		−	1.20044i
$995$	3.54995			0.112541
$996$		0			0
$997$	−32.3280			−1.02384			−0.511919	−	0.859034i	$-0.671065\pi$
$997$	−32.3280			−1.02384			−0.511919	+	0.859034i	$0.671065\pi$
$998$		−	38.7196i		−	1.22565i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4014.2.d.a.4013.19	✓	72
3.2	odd	2	inner	4014.2.d.a.4013.50	yes	72
223.222	odd	2	inner	4014.2.d.a.4013.49	yes	72
669.668	even	2	inner	4014.2.d.a.4013.20	yes	72

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4014.2.d.a.4013.19	✓	72	1.1	even	1	trivial
4014.2.d.a.4013.20	yes	72	669.668	even	2	inner
4014.2.d.a.4013.49	yes	72	223.222	odd	2	inner
4014.2.d.a.4013.50	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)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} -1.00000 q^{4} -1.71459 q^{5} +2.72316 q^{7} +1.00000i q^{8} +O(q^{10})\)	\(q-1.00000i q^{2} -1.00000 q^{4} -1.71459 q^{5} +2.72316 q^{7} +1.00000i q^{8} +1.71459i q^{10} +5.69509 q^{11} +2.15531i q^{13} -2.72316i q^{14} +1.00000 q^{16} -5.09797i q^{17} -6.22079 q^{19} +1.71459 q^{20} -5.69509i q^{22} +8.90450 q^{23} -2.06019 q^{25} +2.15531 q^{26} -2.72316 q^{28} +4.97921i q^{29} +5.24366 q^{31} -1.00000i q^{32} -5.09797 q^{34} -4.66910 q^{35} +10.4615 q^{37} +6.22079i q^{38} -1.71459i q^{40} +8.23537i q^{41} -8.34624 q^{43} -5.69509 q^{44} -8.90450i q^{46} +3.32674i q^{47} +0.415600 q^{49} +2.06019i q^{50} -2.15531i q^{52} +6.09267i q^{53} -9.76473 q^{55} +2.72316i q^{56} +4.97921 q^{58} -9.22605 q^{59} -14.7558i q^{61} -5.24366i q^{62} -1.00000 q^{64} -3.69547i q^{65} -9.61141i q^{67} +5.09797i q^{68} +4.66910i q^{70} +13.8983 q^{71} -2.73390 q^{73} -10.4615i q^{74} +6.22079 q^{76} +15.5086 q^{77} +14.2075i q^{79} -1.71459 q^{80} +8.23537 q^{82} -10.3344i q^{83} +8.74093i q^{85} +8.34624i q^{86} +5.69509i q^{88} +14.3149i q^{89} +5.86925i q^{91} -8.90450 q^{92} +3.32674 q^{94} +10.6661 q^{95} +9.02597i q^{97} -0.415600i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 72 q - 72 q^{4} + 16 q^{7}+O(q^{10}) \) 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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