LMFDB - Embedded newform 4014.2.a.l.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4014,2,Mod(1,4014)]

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([0, 0]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("4014.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4014 = 2 \cdot 3^{2} \cdot 223 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4014.a (trivial)

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:	yes
Analytic conductor:	$32.0519513713$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{13}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 3 $ x^2 - x - 3
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.30278$ of defining polynomial
Character	$\chi$	$=$	4014.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} -4.60555 q^{7} +1.00000 q^{8} +O(q^{10})$	$q+1.00000 q^{2} +1.00000 q^{4} -4.60555 q^{7} +1.00000 q^{8} +2.69722 q^{11} +1.69722 q^{13} -4.60555 q^{14} +1.00000 q^{16} -2.60555 q^{17} +4.30278 q^{19} +2.69722 q^{22} -4.00000 q^{23} -5.00000 q^{25} +1.69722 q^{26} -4.60555 q^{28} +5.30278 q^{29} +6.60555 q^{31} +1.00000 q^{32} -2.60555 q^{34} +4.60555 q^{37} +4.30278 q^{38} -4.00000 q^{41} +3.90833 q^{43} +2.69722 q^{44} -4.00000 q^{46} +0.697224 q^{47} +14.2111 q^{49} -5.00000 q^{50} +1.69722 q^{52} +6.90833 q^{53} -4.60555 q^{56} +5.30278 q^{58} +0.302776 q^{59} -2.69722 q^{61} +6.60555 q^{62} +1.00000 q^{64} +4.60555 q^{67} -2.60555 q^{68} +11.2111 q^{71} -8.90833 q^{73} +4.60555 q^{74} +4.30278 q^{76} -12.4222 q^{77} +8.51388 q^{79} -4.00000 q^{82} -5.21110 q^{83} +3.90833 q^{86} +2.69722 q^{88} +17.2111 q^{89} -7.81665 q^{91} -4.00000 q^{92} +0.697224 q^{94} -0.788897 q^{97} +14.2111 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} + 2 q^{4} - 2 q^{7} + 2 q^{8}+O(q^{10}) $ 2 * q + 2 * q^2 + 2 * q^4 - 2 * q^7 + 2 * q^8	$ 2 q + 2 q^{2} + 2 q^{4} - 2 q^{7} + 2 q^{8} + 9 q^{11} + 7 q^{13} - 2 q^{14} + 2 q^{16} + 2 q^{17} + 5 q^{19} + 9 q^{22} - 8 q^{23} - 10 q^{25} + 7 q^{26} - 2 q^{28} + 7 q^{29} + 6 q^{31} + 2 q^{32} + 2 q^{34} + 2 q^{37} + 5 q^{38} - 8 q^{41} - 3 q^{43} + 9 q^{44} - 8 q^{46} + 5 q^{47} + 14 q^{49} - 10 q^{50} + 7 q^{52} + 3 q^{53} - 2 q^{56} + 7 q^{58} - 3 q^{59} - 9 q^{61} + 6 q^{62} + 2 q^{64} + 2 q^{67} + 2 q^{68} + 8 q^{71} - 7 q^{73} + 2 q^{74} + 5 q^{76} + 4 q^{77} - q^{79} - 8 q^{82} + 4 q^{83} - 3 q^{86} + 9 q^{88} + 20 q^{89} + 6 q^{91} - 8 q^{92} + 5 q^{94} - 16 q^{97} + 14 q^{98}+O(q^{100}) $ 2 * q + 2 * q^2 + 2 * q^4 - 2 * q^7 + 2 * q^8 + 9 * q^11 + 7 * q^13 - 2 * q^14 + 2 * q^16 + 2 * q^17 + 5 * q^19 + 9 * q^22 - 8 * q^23 - 10 * q^25 + 7 * q^26 - 2 * q^28 + 7 * q^29 + 6 * q^31 + 2 * q^32 + 2 * q^34 + 2 * q^37 + 5 * q^38 - 8 * q^41 - 3 * q^43 + 9 * q^44 - 8 * q^46 + 5 * q^47 + 14 * q^49 - 10 * q^50 + 7 * q^52 + 3 * q^53 - 2 * q^56 + 7 * q^58 - 3 * q^59 - 9 * q^61 + 6 * q^62 + 2 * q^64 + 2 * q^67 + 2 * q^68 + 8 * q^71 - 7 * q^73 + 2 * q^74 + 5 * q^76 + 4 * q^77 - q^79 - 8 * q^82 + 4 * q^83 - 3 * q^86 + 9 * q^88 + 20 * q^89 + 6 * q^91 - 8 * q^92 + 5 * q^94 - 16 * q^97 + 14 * q^98

Display 100 coefficients 10 coefficients

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 $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	0	0		−	1.00000i	$-0.5\pi$
$5$	0	0			1.00000i	$0.5\pi$
$6$	0	0
$7$	−4.60555	−1.74073	−0.870367	−	0.492403i	$-0.836119\pi$
$7$	−4.60555	−1.74073	−0.870367	+	0.492403i	$0.836119\pi$
$8$	1.00000	0.353553
$9$	0	0
$10$	0	0
$11$	2.69722	0.813244	0.406622	−	0.913597i	$-0.366707\pi$
$11$	2.69722	0.813244	0.406622	+	0.913597i	$0.366707\pi$
$12$	0	0
$13$	1.69722	0.470725	0.235363	−	0.971908i	$-0.424372\pi$
$13$	1.69722	0.470725	0.235363	+	0.971908i	$0.424372\pi$
$14$	−4.60555	−1.23089
$15$	0	0
$16$	1.00000	0.250000
$17$	−2.60555	−0.631939	−0.315970	−	0.948769i	$-0.602330\pi$
$17$	−2.60555	−0.631939	−0.315970	+	0.948769i	$0.602330\pi$
$18$	0	0
$19$	4.30278	0.987124	0.493562	−	0.869710i	$-0.335695\pi$
$19$	4.30278	0.987124	0.493562	+	0.869710i	$0.335695\pi$
$20$	0	0
$21$	0	0
$22$	2.69722	0.575050
$23$	−4.00000	−0.834058	−0.417029	−	0.908893i	$-0.636929\pi$
$23$	−4.00000	−0.834058	−0.417029	+	0.908893i	$0.636929\pi$
$24$	0	0
$25$	−5.00000	−1.00000
$26$	1.69722	0.332853
$27$	0	0
$28$	−4.60555	−0.870367
$29$	5.30278	0.984701	0.492350	−	0.870397i	$-0.336138\pi$
$29$	5.30278	0.984701	0.492350	+	0.870397i	$0.336138\pi$
$30$	0	0
$31$	6.60555	1.18639	0.593196	−	0.805058i	$-0.297866\pi$
$31$	6.60555	1.18639	0.593196	+	0.805058i	$0.297866\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	−2.60555	−0.446848
$35$	0	0
$36$	0	0
$37$	4.60555	0.757148	0.378574	−	0.925571i	$-0.376415\pi$
$37$	4.60555	0.757148	0.378574	+	0.925571i	$0.376415\pi$
$38$	4.30278	0.698002
$39$	0	0
$40$	0	0
$41$	−4.00000	−0.624695	−0.312348	−	0.949968i	$-0.601115\pi$
$41$	−4.00000	−0.624695	−0.312348	+	0.949968i	$0.601115\pi$
$42$	0	0
$43$	3.90833	0.596014	0.298007	−	0.954564i	$-0.403678\pi$
$43$	3.90833	0.596014	0.298007	+	0.954564i	$0.403678\pi$
$44$	2.69722	0.406622
$45$	0	0
$46$	−4.00000	−0.589768
$47$	0.697224	0.101701	0.0508503	−	0.998706i	$-0.483807\pi$
$47$	0.697224	0.101701	0.0508503	+	0.998706i	$0.483807\pi$
$48$	0	0
$49$	14.2111	2.03016
$50$	−5.00000	−0.707107
$51$	0	0
$52$	1.69722	0.235363
$53$	6.90833	0.948932	0.474466	−	0.880274i	$-0.342641\pi$
$53$	6.90833	0.948932	0.474466	+	0.880274i	$0.342641\pi$
$54$	0	0
$55$	0	0
$56$	−4.60555	−0.615443
$57$	0	0
$58$	5.30278	0.696289
$59$	0.302776	0.0394180	0.0197090	−	0.999806i	$-0.493726\pi$
$59$	0.302776	0.0394180	0.0197090	+	0.999806i	$0.493726\pi$
$60$	0	0
$61$	−2.69722	−0.345344	−0.172672	−	0.984979i	$-0.555240\pi$
$61$	−2.69722	−0.345344	−0.172672	+	0.984979i	$0.555240\pi$
$62$	6.60555	0.838906
$63$	0	0
$64$	1.00000	0.125000
$65$	0	0
$66$	0	0
$67$	4.60555	0.562658	0.281329	−	0.959611i	$-0.409225\pi$
$67$	4.60555	0.562658	0.281329	+	0.959611i	$0.409225\pi$
$68$	−2.60555	−0.315970
$69$	0	0
$70$	0	0
$71$	11.2111	1.33051	0.665257	−	0.746615i	$-0.268322\pi$
$71$	11.2111	1.33051	0.665257	+	0.746615i	$0.268322\pi$
$72$	0	0
$73$	−8.90833	−1.04264	−0.521320	−	0.853361i	$-0.674560\pi$
$73$	−8.90833	−1.04264	−0.521320	+	0.853361i	$0.674560\pi$
$74$	4.60555	0.535384
$75$	0	0
$76$	4.30278	0.493562
$77$	−12.4222	−1.41564
$78$	0	0
$79$	8.51388	0.957886	0.478943	−	0.877846i	$-0.341020\pi$
$79$	8.51388	0.957886	0.478943	+	0.877846i	$0.341020\pi$
$80$	0	0
$81$	0	0
$82$	−4.00000	−0.441726
$83$	−5.21110	−0.571993	−0.285996	−	0.958231i	$-0.592325\pi$
$83$	−5.21110	−0.571993	−0.285996	+	0.958231i	$0.592325\pi$
$84$	0	0
$85$	0	0
$86$	3.90833	0.421446
$87$	0	0
$88$	2.69722	0.287525
$89$	17.2111	1.82437	0.912187	−	0.409775i	$-0.134393\pi$
$89$	17.2111	1.82437	0.912187	+	0.409775i	$0.134393\pi$
$90$	0	0
$91$	−7.81665	−0.819408
$92$	−4.00000	−0.417029
$93$	0	0
$94$	0.697224	0.0719132
$95$	0	0
$96$	0	0
$97$	−0.788897	−0.0801004	−0.0400502	−	0.999198i	$-0.512752\pi$
$97$	−0.788897	−0.0801004	−0.0400502	+	0.999198i	$0.512752\pi$
$98$	14.2111	1.43554
$99$	0	0
$100$	−5.00000	−0.500000
$101$	10.9083	1.08542	0.542710	−	0.839920i	$-0.317398\pi$
$101$	10.9083	1.08542	0.542710	+	0.839920i	$0.317398\pi$
$102$	0	0
$103$	−2.90833	−0.286566	−0.143283	−	0.989682i	$-0.545766\pi$
$103$	−2.90833	−0.286566	−0.143283	+	0.989682i	$0.545766\pi$
$104$	1.69722	0.166427
$105$	0	0
$106$	6.90833	0.670996
$107$	0.908327	0.0878113	0.0439056	−	0.999036i	$-0.486020\pi$
$107$	0.908327	0.0878113	0.0439056	+	0.999036i	$0.486020\pi$
$108$	0	0
$109$	−5.39445	−0.516694	−0.258347	−	0.966052i	$-0.583178\pi$
$109$	−5.39445	−0.516694	−0.258347	+	0.966052i	$0.583178\pi$
$110$	0	0
$111$	0	0
$112$	−4.60555	−0.435184
$113$	15.6972	1.47667	0.738335	−	0.674434i	$-0.235612\pi$
$113$	15.6972	1.47667	0.738335	+	0.674434i	$0.235612\pi$
$114$	0	0
$115$	0	0
$116$	5.30278	0.492350
$117$	0	0
$118$	0.302776	0.0278728
$119$	12.0000	1.10004
$120$	0	0
$121$	−3.72498	−0.338635
$122$	−2.69722	−0.244195
$123$	0	0
$124$	6.60555	0.593196
$125$	0	0
$126$	0	0
$127$	7.21110	0.639882	0.319941	−	0.947437i	$-0.396337\pi$
$127$	7.21110	0.639882	0.319941	+	0.947437i	$0.396337\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	0	0
$131$	16.4222	1.43481	0.717407	−	0.696654i	$-0.245329\pi$
$131$	16.4222	1.43481	0.717407	+	0.696654i	$0.245329\pi$
$132$	0	0
$133$	−19.8167	−1.71832
$134$	4.60555	0.397859
$135$	0	0
$136$	−2.60555	−0.223424
$137$	7.30278	0.623918	0.311959	−	0.950096i	$-0.399015\pi$
$137$	7.30278	0.623918	0.311959	+	0.950096i	$0.399015\pi$
$138$	0	0
$139$	−11.9083	−1.01005	−0.505026	−	0.863104i	$-0.668517\pi$
$139$	−11.9083	−1.01005	−0.505026	+	0.863104i	$0.668517\pi$
$140$	0	0
$141$	0	0
$142$	11.2111	0.940815
$143$	4.57779	0.382814
$144$	0	0
$145$	0	0
$146$	−8.90833	−0.737258
$147$	0	0
$148$	4.60555	0.378574
$149$	5.21110	0.426910	0.213455	−	0.976953i	$-0.431528\pi$
$149$	5.21110	0.426910	0.213455	+	0.976953i	$0.431528\pi$
$150$	0	0
$151$	0.486122	0.0395600	0.0197800	−	0.999804i	$-0.493703\pi$
$151$	0.486122	0.0395600	0.0197800	+	0.999804i	$0.493703\pi$
$152$	4.30278	0.349001
$153$	0	0
$154$	−12.4222	−1.00101
$155$	0	0
$156$	0	0
$157$	2.69722	0.215262	0.107631	−	0.994191i	$-0.465674\pi$
$157$	2.69722	0.215262	0.107631	+	0.994191i	$0.465674\pi$
$158$	8.51388	0.677328
$159$	0	0
$160$	0	0
$161$	18.4222	1.45187
$162$	0	0
$163$	8.60555	0.674039	0.337019	−	0.941498i	$-0.390581\pi$
$163$	8.60555	0.674039	0.337019	+	0.941498i	$0.390581\pi$
$164$	−4.00000	−0.312348
$165$	0	0
$166$	−5.21110	−0.404460
$167$	2.60555	0.201624	0.100812	−	0.994906i	$-0.467856\pi$
$167$	2.60555	0.201624	0.100812	+	0.994906i	$0.467856\pi$
$168$	0	0
$169$	−10.1194	−0.778418
$170$	0	0
$171$	0	0
$172$	3.90833	0.298007
$173$	−13.0278	−0.990482	−0.495241	−	0.868756i	$-0.664920\pi$
$173$	−13.0278	−0.990482	−0.495241	+	0.868756i	$0.664920\pi$
$174$	0	0
$175$	23.0278	1.74073
$176$	2.69722	0.203311
$177$	0	0
$178$	17.2111	1.29003
$179$	13.2111	0.987444	0.493722	−	0.869620i	$-0.335636\pi$
$179$	13.2111	0.987444	0.493722	+	0.869620i	$0.335636\pi$
$180$	0	0
$181$	1.39445	0.103649	0.0518243	−	0.998656i	$-0.483496\pi$
$181$	1.39445	0.103649	0.0518243	+	0.998656i	$0.483496\pi$
$182$	−7.81665	−0.579409
$183$	0	0
$184$	−4.00000	−0.294884
$185$	0	0
$186$	0	0
$187$	−7.02776	−0.513920
$188$	0.697224	0.0508503
$189$	0	0
$190$	0	0
$191$	−7.81665	−0.565593	−0.282797	−	0.959180i	$-0.591262\pi$
$191$	−7.81665	−0.565593	−0.282797	+	0.959180i	$0.591262\pi$
$192$	0	0
$193$	8.42221	0.606244	0.303122	−	0.952952i	$-0.401971\pi$
$193$	8.42221	0.606244	0.303122	+	0.952952i	$0.401971\pi$
$194$	−0.788897	−0.0566395
$195$	0	0
$196$	14.2111	1.01508
$197$	−5.30278	−0.377807	−0.188904	−	0.981996i	$-0.560493\pi$
$197$	−5.30278	−0.377807	−0.188904	+	0.981996i	$0.560493\pi$
$198$	0	0
$199$	10.4222	0.738811	0.369405	−	0.929268i	$-0.379561\pi$
$199$	10.4222	0.738811	0.369405	+	0.929268i	$0.379561\pi$
$200$	−5.00000	−0.353553
$201$	0	0
$202$	10.9083	0.767507
$203$	−24.4222	−1.71410
$204$	0	0
$205$	0	0
$206$	−2.90833	−0.202633
$207$	0	0
$208$	1.69722	0.117681
$209$	11.6056	0.802773
$210$	0	0
$211$	5.30278	0.365058	0.182529	−	0.983200i	$-0.441572\pi$
$211$	5.30278	0.365058	0.182529	+	0.983200i	$0.441572\pi$
$212$	6.90833	0.474466
$213$	0	0
$214$	0.908327	0.0620919
$215$	0	0
$216$	0	0
$217$	−30.4222	−2.06519
$218$	−5.39445	−0.365358
$219$	0	0
$220$	0	0
$221$	−4.42221	−0.297470
$222$	0	0
$223$	−1.00000	−0.0669650
$223$	−1.00000	−0.0669650
$224$	−4.60555	−0.307721
$225$	0	0
$226$	15.6972	1.04416
$227$	19.0278	1.26292	0.631458	−	0.775410i	$-0.282457\pi$
$227$	19.0278	1.26292	0.631458	+	0.775410i	$0.282457\pi$
$228$	0	0
$229$	−10.0000	−0.660819	−0.330409	−	0.943838i	$-0.607187\pi$
$229$	−10.0000	−0.660819	−0.330409	+	0.943838i	$0.607187\pi$
$230$	0	0
$231$	0	0
$232$	5.30278	0.348144
$233$	−4.69722	−0.307725	−0.153863	−	0.988092i	$-0.549171\pi$
$233$	−4.69722	−0.307725	−0.153863	+	0.988092i	$0.549171\pi$
$234$	0	0
$235$	0	0
$236$	0.302776	0.0197090
$237$	0	0
$238$	12.0000	0.777844
$239$	−8.72498	−0.564372	−0.282186	−	0.959360i	$-0.591060\pi$
$239$	−8.72498	−0.564372	−0.282186	+	0.959360i	$0.591060\pi$
$240$	0	0
$241$	−4.51388	−0.290764	−0.145382	−	0.989376i	$-0.546441\pi$
$241$	−4.51388	−0.290764	−0.145382	+	0.989376i	$0.546441\pi$
$242$	−3.72498	−0.239451
$243$	0	0
$244$	−2.69722	−0.172672
$245$	0	0
$246$	0	0
$247$	7.30278	0.464664
$248$	6.60555	0.419453
$249$	0	0
$250$	0	0
$251$	−22.4222	−1.41528	−0.707639	−	0.706575i	$-0.750239\pi$
$251$	−22.4222	−1.41528	−0.707639	+	0.706575i	$0.750239\pi$
$252$	0	0
$253$	−10.7889	−0.678292
$254$	7.21110	0.452465
$255$	0	0
$256$	1.00000	0.0625000
$257$	6.00000	0.374270	0.187135	−	0.982334i	$-0.440080\pi$
$257$	6.00000	0.374270	0.187135	+	0.982334i	$0.440080\pi$
$258$	0	0
$259$	−21.2111	−1.31799
$260$	0	0
$261$	0	0
$262$	16.4222	1.01457
$263$	−7.39445	−0.455961	−0.227981	−	0.973666i	$-0.573212\pi$
$263$	−7.39445	−0.455961	−0.227981	+	0.973666i	$0.573212\pi$
$264$	0	0
$265$	0	0
$266$	−19.8167	−1.21504
$267$	0	0
$268$	4.60555	0.281329
$269$	15.0278	0.916258	0.458129	−	0.888886i	$-0.348520\pi$
$269$	15.0278	0.916258	0.458129	+	0.888886i	$0.348520\pi$
$270$	0	0
$271$	15.1194	0.918440	0.459220	−	0.888323i	$-0.348129\pi$
$271$	15.1194	0.918440	0.459220	+	0.888323i	$0.348129\pi$
$272$	−2.60555	−0.157985
$273$	0	0
$274$	7.30278	0.441177
$275$	−13.4861	−0.813244
$276$	0	0
$277$	−32.4222	−1.94806	−0.974031	−	0.226416i	$-0.927299\pi$
$277$	−32.4222	−1.94806	−0.974031	+	0.226416i	$0.927299\pi$
$278$	−11.9083	−0.714214
$279$	0	0
$280$	0	0
$281$	14.6056	0.871294	0.435647	−	0.900118i	$-0.356520\pi$
$281$	14.6056	0.871294	0.435647	+	0.900118i	$0.356520\pi$
$282$	0	0
$283$	−15.6972	−0.933103	−0.466552	−	0.884494i	$-0.654504\pi$
$283$	−15.6972	−0.933103	−0.466552	+	0.884494i	$0.654504\pi$
$284$	11.2111	0.665257
$285$	0	0
$286$	4.57779	0.270691
$287$	18.4222	1.08743
$288$	0	0
$289$	−10.2111	−0.600653
$290$	0	0
$291$	0	0
$292$	−8.90833	−0.521320
$293$	−20.2389	−1.18237	−0.591183	−	0.806537i	$-0.701339\pi$
$293$	−20.2389	−1.18237	−0.591183	+	0.806537i	$0.701339\pi$
$294$	0	0
$295$	0	0
$296$	4.60555	0.267692
$297$	0	0
$298$	5.21110	0.301871
$299$	−6.78890	−0.392612
$300$	0	0
$301$	−18.0000	−1.03750
$302$	0.486122	0.0279732
$303$	0	0
$304$	4.30278	0.246781
$305$	0	0
$306$	0	0
$307$	33.0278	1.88499	0.942497	−	0.334215i	$-0.108471\pi$
$307$	33.0278	1.88499	0.942497	+	0.334215i	$0.108471\pi$
$308$	−12.4222	−0.707821
$309$	0	0
$310$	0	0
$311$	10.6056	0.601386	0.300693	−	0.953721i	$-0.402782\pi$
$311$	10.6056	0.601386	0.300693	+	0.953721i	$0.402782\pi$
$312$	0	0
$313$	−20.0000	−1.13047	−0.565233	−	0.824931i	$-0.691214\pi$
$313$	−20.0000	−1.13047	−0.565233	+	0.824931i	$0.691214\pi$
$314$	2.69722	0.152213
$315$	0	0
$316$	8.51388	0.478943
$317$	−9.69722	−0.544650	−0.272325	−	0.962205i	$-0.587793\pi$
$317$	−9.69722	−0.544650	−0.272325	+	0.962205i	$0.587793\pi$
$318$	0	0
$319$	14.3028	0.800802
$320$	0	0
$321$	0	0
$322$	18.4222	1.02663
$323$	−11.2111	−0.623802
$324$	0	0
$325$	−8.48612	−0.470725
$326$	8.60555	0.476617
$327$	0	0
$328$	−4.00000	−0.220863
$329$	−3.21110	−0.177034
$330$	0	0
$331$	−19.2111	−1.05594	−0.527969	−	0.849264i	$-0.677046\pi$
$331$	−19.2111	−1.05594	−0.527969	+	0.849264i	$0.677046\pi$
$332$	−5.21110	−0.285996
$333$	0	0
$334$	2.60555	0.142569
$335$	0	0
$336$	0	0
$337$	5.21110	0.283867	0.141933	−	0.989876i	$-0.454668\pi$
$337$	5.21110	0.283867	0.141933	+	0.989876i	$0.454668\pi$
$338$	−10.1194	−0.550424
$339$	0	0
$340$	0	0
$341$	17.8167	0.964826
$342$	0	0
$343$	−33.2111	−1.79323
$344$	3.90833	0.210723
$345$	0	0
$346$	−13.0278	−0.700377
$347$	−10.4222	−0.559493	−0.279747	−	0.960074i	$-0.590250\pi$
$347$	−10.4222	−0.559493	−0.279747	+	0.960074i	$0.590250\pi$
$348$	0	0
$349$	−25.3944	−1.35933	−0.679667	−	0.733521i	$-0.737876\pi$
$349$	−25.3944	−1.35933	−0.679667	+	0.733521i	$0.737876\pi$
$350$	23.0278	1.23089
$351$	0	0
$352$	2.69722	0.143763
$353$	15.3944	0.819364	0.409682	−	0.912228i	$-0.365640\pi$
$353$	15.3944	0.819364	0.409682	+	0.912228i	$0.365640\pi$
$354$	0	0
$355$	0	0
$356$	17.2111	0.912187
$357$	0	0
$358$	13.2111	0.698228
$359$	23.5139	1.24102	0.620508	−	0.784201i	$-0.286927\pi$
$359$	23.5139	1.24102	0.620508	+	0.784201i	$0.286927\pi$
$360$	0	0
$361$	−0.486122	−0.0255854
$362$	1.39445	0.0732906
$363$	0	0
$364$	−7.81665	−0.409704
$365$	0	0
$366$	0	0
$367$	8.00000	0.417597	0.208798	−	0.977959i	$-0.433045\pi$
$367$	8.00000	0.417597	0.208798	+	0.977959i	$0.433045\pi$
$368$	−4.00000	−0.208514
$369$	0	0
$370$	0	0
$371$	−31.8167	−1.65184
$372$	0	0
$373$	−26.8444	−1.38995	−0.694975	−	0.719033i	$-0.744585\pi$
$373$	−26.8444	−1.38995	−0.694975	+	0.719033i	$0.744585\pi$
$374$	−7.02776	−0.363397
$375$	0	0
$376$	0.697224	0.0359566
$377$	9.00000	0.463524
$378$	0	0
$379$	−3.33053	−0.171078	−0.0855390	−	0.996335i	$-0.527261\pi$
$379$	−3.33053	−0.171078	−0.0855390	+	0.996335i	$0.527261\pi$
$380$	0	0
$381$	0	0
$382$	−7.81665	−0.399935
$383$	9.81665	0.501608	0.250804	−	0.968038i	$-0.419305\pi$
$383$	9.81665	0.501608	0.250804	+	0.968038i	$0.419305\pi$
$384$	0	0
$385$	0	0
$386$	8.42221	0.428679
$387$	0	0
$388$	−0.788897	−0.0400502
$389$	−24.9361	−1.26431	−0.632155	−	0.774842i	$-0.717829\pi$
$389$	−24.9361	−1.26431	−0.632155	+	0.774842i	$0.717829\pi$
$390$	0	0
$391$	10.4222	0.527074
$392$	14.2111	0.717769
$393$	0	0
$394$	−5.30278	−0.267150
$395$	0	0
$396$	0	0
$397$	−1.48612	−0.0745863	−0.0372932	−	0.999304i	$-0.511874\pi$
$397$	−1.48612	−0.0745863	−0.0372932	+	0.999304i	$0.511874\pi$
$398$	10.4222	0.522418
$399$	0	0
$400$	−5.00000	−0.250000
$401$	−39.0278	−1.94895	−0.974477	−	0.224489i	$-0.927929\pi$
$401$	−39.0278	−1.94895	−0.974477	+	0.224489i	$0.927929\pi$
$402$	0	0
$403$	11.2111	0.558465
$404$	10.9083	0.542710
$405$	0	0
$406$	−24.4222	−1.21205
$407$	12.4222	0.615746
$408$	0	0
$409$	−26.2389	−1.29743	−0.648714	−	0.761032i	$-0.724693\pi$
$409$	−26.2389	−1.29743	−0.648714	+	0.761032i	$0.724693\pi$
$410$	0	0
$411$	0	0
$412$	−2.90833	−0.143283
$413$	−1.39445	−0.0686163
$414$	0	0
$415$	0	0
$416$	1.69722	0.0832133
$417$	0	0
$418$	11.6056	0.567646
$419$	10.6056	0.518115	0.259058	−	0.965862i	$-0.416588\pi$
$419$	10.6056	0.518115	0.259058	+	0.965862i	$0.416588\pi$
$420$	0	0
$421$	12.7250	0.620178	0.310089	−	0.950708i	$-0.399641\pi$
$421$	12.7250	0.620178	0.310089	+	0.950708i	$0.399641\pi$
$422$	5.30278	0.258135
$423$	0	0
$424$	6.90833	0.335498
$425$	13.0278	0.631939
$426$	0	0
$427$	12.4222	0.601153
$428$	0.908327	0.0439056
$429$	0	0
$430$	0	0
$431$	4.18335	0.201505	0.100752	−	0.994912i	$-0.467875\pi$
$431$	4.18335	0.201505	0.100752	+	0.994912i	$0.467875\pi$
$432$	0	0
$433$	−32.5416	−1.56385	−0.781926	−	0.623372i	$-0.785762\pi$
$433$	−32.5416	−1.56385	−0.781926	+	0.623372i	$0.785762\pi$
$434$	−30.4222	−1.46031
$435$	0	0
$436$	−5.39445	−0.258347
$437$	−17.2111	−0.823319
$438$	0	0
$439$	27.6333	1.31887	0.659433	−	0.751763i	$-0.270796\pi$
$439$	27.6333	1.31887	0.659433	+	0.751763i	$0.270796\pi$
$440$	0	0
$441$	0	0
$442$	−4.42221	−0.210343
$443$	−1.39445	−0.0662523	−0.0331261	−	0.999451i	$-0.510546\pi$
$443$	−1.39445	−0.0662523	−0.0331261	+	0.999451i	$0.510546\pi$
$444$	0	0
$445$	0	0
$446$	−1.00000	−0.0473514
$447$	0	0
$448$	−4.60555	−0.217592
$449$	−15.2111	−0.717856	−0.358928	−	0.933365i	$-0.616858\pi$
$449$	−15.2111	−0.717856	−0.358928	+	0.933365i	$0.616858\pi$
$450$	0	0
$451$	−10.7889	−0.508029
$452$	15.6972	0.738335
$453$	0	0
$454$	19.0278	0.893017
$455$	0	0
$456$	0	0
$457$	31.8167	1.48832	0.744160	−	0.668001i	$-0.232850\pi$
$457$	31.8167	1.48832	0.744160	+	0.668001i	$0.232850\pi$
$458$	−10.0000	−0.467269
$459$	0	0
$460$	0	0
$461$	−4.72498	−0.220064	−0.110032	−	0.993928i	$-0.535095\pi$
$461$	−4.72498	−0.220064	−0.110032	+	0.993928i	$0.535095\pi$
$462$	0	0
$463$	−11.2111	−0.521024	−0.260512	−	0.965471i	$-0.583891\pi$
$463$	−11.2111	−0.521024	−0.260512	+	0.965471i	$0.583891\pi$
$464$	5.30278	0.246175
$465$	0	0
$466$	−4.69722	−0.217595
$467$	−12.9083	−0.597326	−0.298663	−	0.954359i	$-0.596541\pi$
$467$	−12.9083	−0.597326	−0.298663	+	0.954359i	$0.596541\pi$
$468$	0	0
$469$	−21.2111	−0.979438
$470$	0	0
$471$	0	0
$472$	0.302776	0.0139364
$473$	10.5416	0.484705
$474$	0	0
$475$	−21.5139	−0.987124
$476$	12.0000	0.550019
$477$	0	0
$478$	−8.72498	−0.399071
$479$	26.0917	1.19216	0.596079	−	0.802925i	$-0.296724\pi$
$479$	26.0917	1.19216	0.596079	+	0.802925i	$0.296724\pi$
$480$	0	0
$481$	7.81665	0.356409
$482$	−4.51388	−0.205602
$483$	0	0
$484$	−3.72498	−0.169317
$485$	0	0
$486$	0	0
$487$	−4.00000	−0.181257	−0.0906287	−	0.995885i	$-0.528888\pi$
$487$	−4.00000	−0.181257	−0.0906287	+	0.995885i	$0.528888\pi$
$488$	−2.69722	−0.122098
$489$	0	0
$490$	0	0
$491$	32.9361	1.48638	0.743192	−	0.669078i	$-0.233311\pi$
$491$	32.9361	1.48638	0.743192	+	0.669078i	$0.233311\pi$
$492$	0	0
$493$	−13.8167	−0.622271
$494$	7.30278	0.328567
$495$	0	0
$496$	6.60555	0.296598
$497$	−51.6333	−2.31607
$498$	0	0
$499$	−2.69722	−0.120744	−0.0603722	−	0.998176i	$-0.519229\pi$
$499$	−2.69722	−0.120744	−0.0603722	+	0.998176i	$0.519229\pi$
$500$	0	0
$501$	0	0
$502$	−22.4222	−1.00075
$503$	−12.0000	−0.535054	−0.267527	−	0.963550i	$-0.586206\pi$
$503$	−12.0000	−0.535054	−0.267527	+	0.963550i	$0.586206\pi$
$504$	0	0
$505$	0	0
$506$	−10.7889	−0.479625
$507$	0	0
$508$	7.21110	0.319941
$509$	40.7527	1.80633	0.903167	−	0.429290i	$-0.141236\pi$
$509$	40.7527	1.80633	0.903167	+	0.429290i	$0.141236\pi$
$510$	0	0
$511$	41.0278	1.81496
$512$	1.00000	0.0441942
$513$	0	0
$514$	6.00000	0.264649
$515$	0	0
$516$	0	0
$517$	1.88057	0.0827074
$518$	−21.2111	−0.931962
$519$	0	0
$520$	0	0
$521$	6.00000	0.262865	0.131432	−	0.991325i	$-0.458042\pi$
$521$	6.00000	0.262865	0.131432	+	0.991325i	$0.458042\pi$
$522$	0	0
$523$	−34.4222	−1.50518	−0.752589	−	0.658491i	$-0.771195\pi$
$523$	−34.4222	−1.50518	−0.752589	+	0.658491i	$0.771195\pi$
$524$	16.4222	0.717407
$525$	0	0
$526$	−7.39445	−0.322413
$527$	−17.2111	−0.749727
$528$	0	0
$529$	−7.00000	−0.304348
$530$	0	0
$531$	0	0
$532$	−19.8167	−0.859161
$533$	−6.78890	−0.294060
$534$	0	0
$535$	0	0
$536$	4.60555	0.198930
$537$	0	0
$538$	15.0278	0.647893
$539$	38.3305	1.65101
$540$	0	0
$541$	40.7250	1.75090	0.875452	−	0.483305i	$-0.160564\pi$
$541$	40.7250	1.75090	0.875452	+	0.483305i	$0.160564\pi$
$542$	15.1194	0.649435
$543$	0	0
$544$	−2.60555	−0.111712
$545$	0	0
$546$	0	0
$547$	−22.7250	−0.971650	−0.485825	−	0.874056i	$-0.661481\pi$
$547$	−22.7250	−0.971650	−0.485825	+	0.874056i	$0.661481\pi$
$548$	7.30278	0.311959
$549$	0	0
$550$	−13.4861	−0.575050
$551$	22.8167	0.972022
$552$	0	0
$553$	−39.2111	−1.66743
$554$	−32.4222	−1.37749
$555$	0	0
$556$	−11.9083	−0.505026
$557$	9.57779	0.405824	0.202912	−	0.979197i	$-0.434959\pi$
$557$	9.57779	0.405824	0.202912	+	0.979197i	$0.434959\pi$
$558$	0	0
$559$	6.63331	0.280559
$560$	0	0
$561$	0	0
$562$	14.6056	0.616098
$563$	−11.5416	−0.486422	−0.243211	−	0.969973i	$-0.578201\pi$
$563$	−11.5416	−0.486422	−0.243211	+	0.969973i	$0.578201\pi$
$564$	0	0
$565$	0	0
$566$	−15.6972	−0.659804
$567$	0	0
$568$	11.2111	0.470407
$569$	20.5416	0.861150	0.430575	−	0.902555i	$-0.358311\pi$
$569$	20.5416	0.861150	0.430575	+	0.902555i	$0.358311\pi$
$570$	0	0
$571$	8.00000	0.334790	0.167395	−	0.985890i	$-0.446465\pi$
$571$	8.00000	0.334790	0.167395	+	0.985890i	$0.446465\pi$
$572$	4.57779	0.191407
$573$	0	0
$574$	18.4222	0.768928
$575$	20.0000	0.834058
$576$	0	0
$577$	39.9638	1.66372	0.831858	−	0.554988i	$-0.187277\pi$
$577$	39.9638	1.66372	0.831858	+	0.554988i	$0.187277\pi$
$578$	−10.2111	−0.424726
$579$	0	0
$580$	0	0
$581$	24.0000	0.995688
$582$	0	0
$583$	18.6333	0.771713
$584$	−8.90833	−0.368629
$585$	0	0
$586$	−20.2389	−0.836060
$587$	31.9361	1.31814	0.659072	−	0.752080i	$-0.270949\pi$
$587$	31.9361	1.31814	0.659072	+	0.752080i	$0.270949\pi$
$588$	0	0
$589$	28.4222	1.17112
$590$	0	0
$591$	0	0
$592$	4.60555	0.189287
$593$	−3.57779	−0.146922	−0.0734612	−	0.997298i	$-0.523405\pi$
$593$	−3.57779	−0.146922	−0.0734612	+	0.997298i	$0.523405\pi$
$594$	0	0
$595$	0	0
$596$	5.21110	0.213455
$597$	0	0
$598$	−6.78890	−0.277619
$599$	35.1194	1.43494	0.717470	−	0.696589i	$-0.245300\pi$
$599$	35.1194	1.43494	0.717470	+	0.696589i	$0.245300\pi$
$600$	0	0
$601$	0.183346	0.00747885	0.00373942	−	0.999993i	$-0.498810\pi$
$601$	0.183346	0.00747885	0.00373942	+	0.999993i	$0.498810\pi$
$602$	−18.0000	−0.733625
$603$	0	0
$604$	0.486122	0.0197800
$605$	0	0
$606$	0	0
$607$	−24.8444	−1.00840	−0.504202	−	0.863586i	$-0.668213\pi$
$607$	−24.8444	−1.00840	−0.504202	+	0.863586i	$0.668213\pi$
$608$	4.30278	0.174501
$609$	0	0
$610$	0	0
$611$	1.18335	0.0478731
$612$	0	0
$613$	18.5139	0.747768	0.373884	−	0.927475i	$-0.378026\pi$
$613$	18.5139	0.747768	0.373884	+	0.927475i	$0.378026\pi$
$614$	33.0278	1.33289
$615$	0	0
$616$	−12.4222	−0.500505
$617$	−10.4222	−0.419582	−0.209791	−	0.977746i	$-0.567278\pi$
$617$	−10.4222	−0.419582	−0.209791	+	0.977746i	$0.567278\pi$
$618$	0	0
$619$	−18.4222	−0.740451	−0.370225	−	0.928942i	$-0.620720\pi$
$619$	−18.4222	−0.740451	−0.370225	+	0.928942i	$0.620720\pi$
$620$	0	0
$621$	0	0
$622$	10.6056	0.425244
$623$	−79.2666	−3.17575
$624$	0	0
$625$	25.0000	1.00000
$626$	−20.0000	−0.799361
$627$	0	0
$628$	2.69722	0.107631
$629$	−12.0000	−0.478471
$630$	0	0
$631$	8.88057	0.353530	0.176765	−	0.984253i	$-0.443437\pi$
$631$	8.88057	0.353530	0.176765	+	0.984253i	$0.443437\pi$
$632$	8.51388	0.338664
$633$	0	0
$634$	−9.69722	−0.385126
$635$	0	0
$636$	0	0
$637$	24.1194	0.955647
$638$	14.3028	0.566252
$639$	0	0
$640$	0	0
$641$	−22.4861	−0.888148	−0.444074	−	0.895990i	$-0.646467\pi$
$641$	−22.4861	−0.888148	−0.444074	+	0.895990i	$0.646467\pi$
$642$	0	0
$643$	26.6972	1.05284	0.526418	−	0.850226i	$-0.323535\pi$
$643$	26.6972	1.05284	0.526418	+	0.850226i	$0.323535\pi$
$644$	18.4222	0.725937
$645$	0	0
$646$	−11.2111	−0.441095
$647$	34.9083	1.37239	0.686194	−	0.727419i	$-0.259280\pi$
$647$	34.9083	1.37239	0.686194	+	0.727419i	$0.259280\pi$
$648$	0	0
$649$	0.816654	0.0320565
$650$	−8.48612	−0.332853
$651$	0	0
$652$	8.60555	0.337019
$653$	35.6333	1.39444	0.697220	−	0.716858i	$-0.254420\pi$
$653$	35.6333	1.39444	0.697220	+	0.716858i	$0.254420\pi$
$654$	0	0
$655$	0	0
$656$	−4.00000	−0.156174
$657$	0	0
$658$	−3.21110	−0.125182
$659$	21.6333	0.842714	0.421357	−	0.906895i	$-0.361554\pi$
$659$	21.6333	0.842714	0.421357	+	0.906895i	$0.361554\pi$
$660$	0	0
$661$	8.93608	0.347573	0.173787	−	0.984783i	$-0.444400\pi$
$661$	8.93608	0.347573	0.173787	+	0.984783i	$0.444400\pi$
$662$	−19.2111	−0.746661
$663$	0	0
$664$	−5.21110	−0.202230
$665$	0	0
$666$	0	0
$667$	−21.2111	−0.821297
$668$	2.60555	0.100812
$669$	0	0
$670$	0	0
$671$	−7.27502	−0.280849
$672$	0	0
$673$	−14.1194	−0.544264	−0.272132	−	0.962260i	$-0.587729\pi$
$673$	−14.1194	−0.544264	−0.272132	+	0.962260i	$0.587729\pi$
$674$	5.21110	0.200724
$675$	0	0
$676$	−10.1194	−0.389209
$677$	−35.2111	−1.35327	−0.676636	−	0.736317i	$-0.736563\pi$
$677$	−35.2111	−1.35327	−0.676636	+	0.736317i	$0.736563\pi$
$678$	0	0
$679$	3.63331	0.139434
$680$	0	0
$681$	0	0
$682$	17.8167	0.682235
$683$	−24.1833	−0.925350	−0.462675	−	0.886528i	$-0.653110\pi$
$683$	−24.1833	−0.925350	−0.462675	+	0.886528i	$0.653110\pi$
$684$	0	0
$685$	0	0
$686$	−33.2111	−1.26801
$687$	0	0
$688$	3.90833	0.149004
$689$	11.7250	0.446686
$690$	0	0
$691$	−21.8167	−0.829945	−0.414972	−	0.909834i	$-0.636209\pi$
$691$	−21.8167	−0.829945	−0.414972	+	0.909834i	$0.636209\pi$
$692$	−13.0278	−0.495241
$693$	0	0
$694$	−10.4222	−0.395621
$695$	0	0
$696$	0	0
$697$	10.4222	0.394769
$698$	−25.3944	−0.961194
$699$	0	0
$700$	23.0278	0.870367
$701$	−6.00000	−0.226617	−0.113308	−	0.993560i	$-0.536145\pi$
$701$	−6.00000	−0.226617	−0.113308	+	0.993560i	$0.536145\pi$
$702$	0	0
$703$	19.8167	0.747399
$704$	2.69722	0.101655
$705$	0	0
$706$	15.3944	0.579378
$707$	−50.2389	−1.88943
$708$	0	0
$709$	−32.4222	−1.21764	−0.608821	−	0.793308i	$-0.708357\pi$
$709$	−32.4222	−1.21764	−0.608821	+	0.793308i	$0.708357\pi$
$710$	0	0
$711$	0	0
$712$	17.2111	0.645013
$713$	−26.4222	−0.989519
$714$	0	0
$715$	0	0
$716$	13.2111	0.493722
$717$	0	0
$718$	23.5139	0.877530
$719$	6.78890	0.253183	0.126592	−	0.991955i	$-0.459596\pi$
$719$	6.78890	0.253183	0.126592	+	0.991955i	$0.459596\pi$
$720$	0	0
$721$	13.3944	0.498835
$722$	−0.486122	−0.0180916
$723$	0	0
$724$	1.39445	0.0518243
$725$	−26.5139	−0.984701
$726$	0	0
$727$	−35.8167	−1.32837	−0.664183	−	0.747570i	$-0.731220\pi$
$727$	−35.8167	−1.32837	−0.664183	+	0.747570i	$0.731220\pi$
$728$	−7.81665	−0.289704
$729$	0	0
$730$	0	0
$731$	−10.1833	−0.376645
$732$	0	0
$733$	−8.42221	−0.311081	−0.155541	−	0.987829i	$-0.549712\pi$
$733$	−8.42221	−0.311081	−0.155541	+	0.987829i	$0.549712\pi$
$734$	8.00000	0.295285
$735$	0	0
$736$	−4.00000	−0.147442
$737$	12.4222	0.457578
$738$	0	0
$739$	−21.4500	−0.789050	−0.394525	−	0.918885i	$-0.629091\pi$
$739$	−21.4500	−0.789050	−0.394525	+	0.918885i	$0.629091\pi$
$740$	0	0
$741$	0	0
$742$	−31.8167	−1.16803
$743$	−29.9361	−1.09825	−0.549124	−	0.835741i	$-0.685039\pi$
$743$	−29.9361	−1.09825	−0.549124	+	0.835741i	$0.685039\pi$
$744$	0	0
$745$	0	0
$746$	−26.8444	−0.982844
$747$	0	0
$748$	−7.02776	−0.256960
$749$	−4.18335	−0.152856
$750$	0	0
$751$	−21.4500	−0.782720	−0.391360	−	0.920238i	$-0.627995\pi$
$751$	−21.4500	−0.782720	−0.391360	+	0.920238i	$0.627995\pi$
$752$	0.697224	0.0254252
$753$	0	0
$754$	9.00000	0.327761
$755$	0	0
$756$	0	0
$757$	33.3305	1.21142	0.605709	−	0.795686i	$-0.292889\pi$
$757$	33.3305	1.21142	0.605709	+	0.795686i	$0.292889\pi$
$758$	−3.33053	−0.120970
$759$	0	0
$760$	0	0
$761$	−36.9083	−1.33793	−0.668963	−	0.743296i	$-0.733262\pi$
$761$	−36.9083	−1.33793	−0.668963	+	0.743296i	$0.733262\pi$
$762$	0	0
$763$	24.8444	0.899428
$764$	−7.81665	−0.282797
$765$	0	0
$766$	9.81665	0.354690
$767$	0.513878	0.0185551
$768$	0	0
$769$	−8.88057	−0.320242	−0.160121	−	0.987097i	$-0.551188\pi$
$769$	−8.88057	−0.320242	−0.160121	+	0.987097i	$0.551188\pi$
$770$	0	0
$771$	0	0
$772$	8.42221	0.303122
$773$	22.2389	0.799876	0.399938	−	0.916542i	$-0.369032\pi$
$773$	22.2389	0.799876	0.399938	+	0.916542i	$0.369032\pi$
$774$	0	0
$775$	−33.0278	−1.18639
$776$	−0.788897	−0.0283198
$777$	0	0
$778$	−24.9361	−0.894002
$779$	−17.2111	−0.616652
$780$	0	0
$781$	30.2389	1.08203
$782$	10.4222	0.372697
$783$	0	0
$784$	14.2111	0.507539
$785$	0	0
$786$	0	0
$787$	14.7889	0.527167	0.263584	−	0.964637i	$-0.415096\pi$
$787$	14.7889	0.527167	0.263584	+	0.964637i	$0.415096\pi$
$788$	−5.30278	−0.188904
$789$	0	0
$790$	0	0
$791$	−72.2944	−2.57049
$792$	0	0
$793$	−4.57779	−0.162562
$794$	−1.48612	−0.0527405
$795$	0	0
$796$	10.4222	0.369405
$797$	16.4222	0.581704	0.290852	−	0.956768i	$-0.406061\pi$
$797$	16.4222	0.581704	0.290852	+	0.956768i	$0.406061\pi$
$798$	0	0
$799$	−1.81665	−0.0642686
$800$	−5.00000	−0.176777
$801$	0	0
$802$	−39.0278	−1.37812
$803$	−24.0278	−0.847921
$804$	0	0
$805$	0	0
$806$	11.2111	0.394894
$807$	0	0
$808$	10.9083	0.383754
$809$	−42.8444	−1.50633	−0.753165	−	0.657832i	$-0.771474\pi$
$809$	−42.8444	−1.50633	−0.753165	+	0.657832i	$0.771474\pi$
$810$	0	0
$811$	−46.2389	−1.62367	−0.811833	−	0.583890i	$-0.801530\pi$
$811$	−46.2389	−1.62367	−0.811833	+	0.583890i	$0.801530\pi$
$812$	−24.4222	−0.857051
$813$	0	0
$814$	12.4222	0.435398
$815$	0	0
$816$	0	0
$817$	16.8167	0.588340
$818$	−26.2389	−0.917420
$819$	0	0
$820$	0	0
$821$	−55.5416	−1.93842	−0.969208	−	0.246243i	$-0.920804\pi$
$821$	−55.5416	−1.93842	−0.969208	+	0.246243i	$0.920804\pi$
$822$	0	0
$823$	35.2666	1.22932	0.614658	−	0.788793i	$-0.289294\pi$
$823$	35.2666	1.22932	0.614658	+	0.788793i	$0.289294\pi$
$824$	−2.90833	−0.101316
$825$	0	0
$826$	−1.39445	−0.0485191
$827$	0.0639167	0.00222260	0.00111130	−	0.999999i	$-0.499646\pi$
$827$	0.0639167	0.00222260	0.00111130	+	0.999999i	$0.499646\pi$
$828$	0	0
$829$	−33.4861	−1.16302	−0.581511	−	0.813539i	$-0.697538\pi$
$829$	−33.4861	−1.16302	−0.581511	+	0.813539i	$0.697538\pi$
$830$	0	0
$831$	0	0
$832$	1.69722	0.0588407
$833$	−37.0278	−1.28294
$834$	0	0
$835$	0	0
$836$	11.6056	0.401386
$837$	0	0
$838$	10.6056	0.366363
$839$	−9.39445	−0.324332	−0.162166	−	0.986763i	$-0.551848\pi$
$839$	−9.39445	−0.324332	−0.162166	+	0.986763i	$0.551848\pi$
$840$	0	0
$841$	−0.880571	−0.0303645
$842$	12.7250	0.438532
$843$	0	0
$844$	5.30278	0.182529
$845$	0	0
$846$	0	0
$847$	17.1556	0.589473
$848$	6.90833	0.237233
$849$	0	0
$850$	13.0278	0.446848
$851$	−18.4222	−0.631505
$852$	0	0
$853$	2.88057	0.0986289	0.0493144	−	0.998783i	$-0.484296\pi$
$853$	2.88057	0.0986289	0.0493144	+	0.998783i	$0.484296\pi$
$854$	12.4222	0.425079
$855$	0	0
$856$	0.908327	0.0310460
$857$	−14.6056	−0.498916	−0.249458	−	0.968386i	$-0.580252\pi$
$857$	−14.6056	−0.498916	−0.249458	+	0.968386i	$0.580252\pi$
$858$	0	0
$859$	−18.9722	−0.647325	−0.323662	−	0.946173i	$-0.604914\pi$
$859$	−18.9722	−0.647325	−0.323662	+	0.946173i	$0.604914\pi$
$860$	0	0
$861$	0	0
$862$	4.18335	0.142485
$863$	20.7889	0.707662	0.353831	−	0.935309i	$-0.384879\pi$
$863$	20.7889	0.707662	0.353831	+	0.935309i	$0.384879\pi$
$864$	0	0
$865$	0	0
$866$	−32.5416	−1.10581
$867$	0	0
$868$	−30.4222	−1.03260
$869$	22.9638	0.778995
$870$	0	0
$871$	7.81665	0.264857
$872$	−5.39445	−0.182679
$873$	0	0
$874$	−17.2111	−0.582174
$875$	0	0
$876$	0	0
$877$	14.8806	0.502481	0.251241	−	0.967925i	$-0.419161\pi$
$877$	14.8806	0.502481	0.251241	+	0.967925i	$0.419161\pi$
$878$	27.6333	0.932579
$879$	0	0
$880$	0	0
$881$	33.2111	1.11891	0.559455	−	0.828861i	$-0.311010\pi$
$881$	33.2111	1.11891	0.559455	+	0.828861i	$0.311010\pi$
$882$	0	0
$883$	−40.0555	−1.34798	−0.673988	−	0.738743i	$-0.735420\pi$
$883$	−40.0555	−1.34798	−0.673988	+	0.738743i	$0.735420\pi$
$884$	−4.42221	−0.148735
$885$	0	0
$886$	−1.39445	−0.0468474
$887$	−32.7250	−1.09880	−0.549399	−	0.835560i	$-0.685143\pi$
$887$	−32.7250	−1.09880	−0.549399	+	0.835560i	$0.685143\pi$
$888$	0	0
$889$	−33.2111	−1.11386
$890$	0	0
$891$	0	0
$892$	−1.00000	−0.0334825
$893$	3.00000	0.100391
$894$	0	0
$895$	0	0
$896$	−4.60555	−0.153861
$897$	0	0
$898$	−15.2111	−0.507601
$899$	35.0278	1.16824
$900$	0	0
$901$	−18.0000	−0.599667
$902$	−10.7889	−0.359231
$903$	0	0
$904$	15.6972	0.522082
$905$	0	0
$906$	0	0
$907$	−11.3305	−0.376224	−0.188112	−	0.982148i	$-0.560237\pi$
$907$	−11.3305	−0.376224	−0.188112	+	0.982148i	$0.560237\pi$
$908$	19.0278	0.631458
$909$	0	0
$910$	0	0
$911$	38.4222	1.27298	0.636492	−	0.771283i	$-0.280385\pi$
$911$	38.4222	1.27298	0.636492	+	0.771283i	$0.280385\pi$
$912$	0	0
$913$	−14.0555	−0.465170
$914$	31.8167	1.05240
$915$	0	0
$916$	−10.0000	−0.330409
$917$	−75.6333	−2.49763
$918$	0	0
$919$	46.5416	1.53527	0.767633	−	0.640889i	$-0.221434\pi$
$919$	46.5416	1.53527	0.767633	+	0.640889i	$0.221434\pi$
$920$	0	0
$921$	0	0
$922$	−4.72498	−0.155609
$923$	19.0278	0.626306
$924$	0	0
$925$	−23.0278	−0.757148
$926$	−11.2111	−0.368420
$927$	0	0
$928$	5.30278	0.174072
$929$	−59.0278	−1.93664	−0.968319	−	0.249717i	$-0.919662\pi$
$929$	−59.0278	−1.93664	−0.968319	+	0.249717i	$0.919662\pi$
$930$	0	0
$931$	61.1472	2.00402
$932$	−4.69722	−0.153863
$933$	0	0
$934$	−12.9083	−0.422373
$935$	0	0
$936$	0	0
$937$	−39.2111	−1.28097	−0.640485	−	0.767970i	$-0.721267\pi$
$937$	−39.2111	−1.28097	−0.640485	+	0.767970i	$0.721267\pi$
$938$	−21.2111	−0.692567
$939$	0	0
$940$	0	0
$941$	23.8806	0.778484	0.389242	−	0.921135i	$-0.372737\pi$
$941$	23.8806	0.778484	0.389242	+	0.921135i	$0.372737\pi$
$942$	0	0
$943$	16.0000	0.521032
$944$	0.302776	0.00985451
$945$	0	0
$946$	10.5416	0.342738
$947$	−4.18335	−0.135940	−0.0679702	−	0.997687i	$-0.521652\pi$
$947$	−4.18335	−0.135940	−0.0679702	+	0.997687i	$0.521652\pi$
$948$	0	0
$949$	−15.1194	−0.490797
$950$	−21.5139	−0.698002
$951$	0	0
$952$	12.0000	0.388922
$953$	−28.0555	−0.908807	−0.454404	−	0.890796i	$-0.650148\pi$
$953$	−28.0555	−0.908807	−0.454404	+	0.890796i	$0.650148\pi$
$954$	0	0
$955$	0	0
$956$	−8.72498	−0.282186
$957$	0	0
$958$	26.0917	0.842984
$959$	−33.6333	−1.08608
$960$	0	0
$961$	12.6333	0.407526
$962$	7.81665	0.252019
$963$	0	0
$964$	−4.51388	−0.145382
$965$	0	0
$966$	0	0
$967$	22.9083	0.736682	0.368341	−	0.929691i	$-0.379926\pi$
$967$	22.9083	0.736682	0.368341	+	0.929691i	$0.379926\pi$
$968$	−3.72498	−0.119725
$969$	0	0
$970$	0	0
$971$	−4.27502	−0.137192	−0.0685959	−	0.997645i	$-0.521852\pi$
$971$	−4.27502	−0.137192	−0.0685959	+	0.997645i	$0.521852\pi$
$972$	0	0
$973$	54.8444	1.75823
$974$	−4.00000	−0.128168
$975$	0	0
$976$	−2.69722	−0.0863360
$977$	1.72498	0.0551870	0.0275935	−	0.999619i	$-0.491216\pi$
$977$	1.72498	0.0551870	0.0275935	+	0.999619i	$0.491216\pi$
$978$	0	0
$979$	46.4222	1.48366
$980$	0	0
$981$	0	0
$982$	32.9361	1.05103
$983$	−51.2111	−1.63338	−0.816690	−	0.577076i	$-0.804194\pi$
$983$	−51.2111	−1.63338	−0.816690	+	0.577076i	$0.804194\pi$
$984$	0	0
$985$	0	0
$986$	−13.8167	−0.440012
$987$	0	0
$988$	7.30278	0.232332
$989$	−15.6333	−0.497110
$990$	0	0
$991$	56.1194	1.78269	0.891346	−	0.453323i	$-0.149762\pi$
$991$	56.1194	1.78269	0.891346	+	0.453323i	$0.149762\pi$
$992$	6.60555	0.209726
$993$	0	0
$994$	−51.6333	−1.63771
$995$	0	0
$996$	0	0
$997$	−9.81665	−0.310897	−0.155448	−	0.987844i	$-0.549682\pi$
$997$	−9.81665	−0.310897	−0.155448	+	0.987844i	$0.549682\pi$
$998$	−2.69722	−0.0853791
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4014.2.a.l.1.1	yes	2
3.2	odd	2		4014.2.a.j.1.1	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4014.2.a.j.1.1	✓	2	3.2	odd	2
4014.2.a.l.1.1	yes	2	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 4014 = 2 \cdot 3^{2} \cdot 223 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4014.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} -4.60555 q^{7} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} -4.60555 q^{7} +1.00000 q^{8} +2.69722 q^{11} +1.69722 q^{13} -4.60555 q^{14} +1.00000 q^{16} -2.60555 q^{17} +4.30278 q^{19} +2.69722 q^{22} -4.00000 q^{23} -5.00000 q^{25} +1.69722 q^{26} -4.60555 q^{28} +5.30278 q^{29} +6.60555 q^{31} +1.00000 q^{32} -2.60555 q^{34} +4.60555 q^{37} +4.30278 q^{38} -4.00000 q^{41} +3.90833 q^{43} +2.69722 q^{44} -4.00000 q^{46} +0.697224 q^{47} +14.2111 q^{49} -5.00000 q^{50} +1.69722 q^{52} +6.90833 q^{53} -4.60555 q^{56} +5.30278 q^{58} +0.302776 q^{59} -2.69722 q^{61} +6.60555 q^{62} +1.00000 q^{64} +4.60555 q^{67} -2.60555 q^{68} +11.2111 q^{71} -8.90833 q^{73} +4.60555 q^{74} +4.30278 q^{76} -12.4222 q^{77} +8.51388 q^{79} -4.00000 q^{82} -5.21110 q^{83} +3.90833 q^{86} +2.69722 q^{88} +17.2111 q^{89} -7.81665 q^{91} -4.00000 q^{92} +0.697224 q^{94} -0.788897 q^{97} +14.2111 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + 2 q^{4} - 2 q^{7} + 2 q^{8}+O(q^{10}) \) 2 * q + 2 * q^2 + 2 * q^4 - 2 * q^7 + 2 * q^8	\( 2 q + 2 q^{2} + 2 q^{4} - 2 q^{7} + 2 q^{8} + 9 q^{11} + 7 q^{13} - 2 q^{14} + 2 q^{16} + 2 q^{17} + 5 q^{19} + 9 q^{22} - 8 q^{23} - 10 q^{25} + 7 q^{26} - 2 q^{28} + 7 q^{29} + 6 q^{31} + 2 q^{32} + 2 q^{34} + 2 q^{37} + 5 q^{38} - 8 q^{41} - 3 q^{43} + 9 q^{44} - 8 q^{46} + 5 q^{47} + 14 q^{49} - 10 q^{50} + 7 q^{52} + 3 q^{53} - 2 q^{56} + 7 q^{58} - 3 q^{59} - 9 q^{61} + 6 q^{62} + 2 q^{64} + 2 q^{67} + 2 q^{68} + 8 q^{71} - 7 q^{73} + 2 q^{74} + 5 q^{76} + 4 q^{77} - q^{79} - 8 q^{82} + 4 q^{83} - 3 q^{86} + 9 q^{88} + 20 q^{89} + 6 q^{91} - 8 q^{92} + 5 q^{94} - 16 q^{97} + 14 q^{98}+O(q^{100}) \) 2 * q + 2 * q^2 + 2 * q^4 - 2 * q^7 + 2 * q^8 + 9 * q^11 + 7 * q^13 - 2 * q^14 + 2 * q^16 + 2 * q^17 + 5 * q^19 + 9 * q^22 - 8 * q^23 - 10 * q^25 + 7 * q^26 - 2 * q^28 + 7 * q^29 + 6 * q^31 + 2 * q^32 + 2 * q^34 + 2 * q^37 + 5 * q^38 - 8 * q^41 - 3 * q^43 + 9 * q^44 - 8 * q^46 + 5 * q^47 + 14 * q^49 - 10 * q^50 + 7 * q^52 + 3 * q^53 - 2 * q^56 + 7 * q^58 - 3 * q^59 - 9 * q^61 + 6 * q^62 + 2 * q^64 + 2 * q^67 + 2 * q^68 + 8 * q^71 - 7 * q^73 + 2 * q^74 + 5 * q^76 + 4 * q^77 - q^79 - 8 * q^82 + 4 * q^83 - 3 * q^86 + 9 * q^88 + 20 * q^89 + 6 * q^91 - 8 * q^92 + 5 * q^94 - 16 * q^97 + 14 * q^98

Introduction

L-functions

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Varieties

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Database

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