LMFDB - Embedded newform 4014.2.a.r.1.5

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:	$1$
Dimension:	$5$
Coefficient field:	5.5.356173.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 2x^{4} - 7x^{3} + 9x^{2} + 14x - 4 $ x^5 - 2x^4 - 7x^3 + 9x^2 + 14x - 4
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1338)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$2.23241$ 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)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} +3.02013 q^{5} -3.77138 q^{7} +1.00000 q^{8} +O(q^{10})$	\(q+1.00000 q^{2} +1.00000 q^{4} +3.02013 q^{5} -3.77138 q^{7} +1.00000 q^{8} +3.02013 q^{10} -1.20245 q^{11} -1.04630 q^{13} -3.77138 q^{14} +1.00000 q^{16} -3.48116 q^{17} -4.38932 q^{19} +3.02013 q^{20} -1.20245 q^{22} -6.93812 q^{23} +4.12120 q^{25} -1.04630 q^{26} -3.77138 q^{28} +1.44714 q^{29} -3.28522 q^{31} +1.00000 q^{32} -3.48116 q^{34} -11.3901 q^{35} -5.93312 q^{37} -4.38932 q^{38} +3.02013 q^{40} -5.58831 q^{41} +9.61326 q^{43} -1.20245 q^{44} -6.93812 q^{46} -1.67635 q^{47} +7.22334 q^{49} +4.12120 q^{50} -1.04630 q^{52} -7.89183 q^{53} -3.63156 q^{55} -3.77138 q^{56} +1.44714 q^{58} -13.9770 q^{59} +10.0775 q^{61} -3.28522 q^{62} +1.00000 q^{64} -3.15995 q^{65} +7.69317 q^{67} -3.48116 q^{68} -11.3901 q^{70} -2.66924 q^{71} +3.48767 q^{73} -5.93312 q^{74} -4.38932 q^{76} +4.53491 q^{77} -7.37800 q^{79} +3.02013 q^{80} -5.58831 q^{82} +2.11619 q^{83} -10.5136 q^{85} +9.61326 q^{86} -1.20245 q^{88} +6.47710 q^{89} +3.94598 q^{91} -6.93812 q^{92} -1.67635 q^{94} -13.2563 q^{95} +8.21487 q^{97} +7.22334 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 5 q^{2} + 5 q^{4} - 5 q^{5} - q^{7} + 5 q^{8}+O(q^{10}) $ 5 * q + 5 * q^2 + 5 * q^4 - 5 * q^5 - q^7 + 5 * q^8	$ 5 q + 5 q^{2} + 5 q^{4} - 5 q^{5} - q^{7} + 5 q^{8} - 5 q^{10} - 9 q^{11} - q^{14} + 5 q^{16} - 6 q^{17} - 4 q^{19} - 5 q^{20} - 9 q^{22} - 16 q^{23} + 8 q^{25} - q^{28} - 8 q^{29} - q^{31} + 5 q^{32} - 6 q^{34} - 22 q^{35} - 2 q^{37} - 4 q^{38} - 5 q^{40} - 4 q^{41} + 3 q^{43} - 9 q^{44} - 16 q^{46} - 18 q^{47} + 2 q^{49} + 8 q^{50} - 26 q^{53} + q^{55} - q^{56} - 8 q^{58} - 21 q^{59} - 20 q^{61} - q^{62} + 5 q^{64} + 3 q^{65} - 5 q^{67} - 6 q^{68} - 22 q^{70} - 17 q^{71} + 5 q^{73} - 2 q^{74} - 4 q^{76} - 2 q^{77} - 21 q^{79} - 5 q^{80} - 4 q^{82} - 11 q^{83} - 12 q^{85} + 3 q^{86} - 9 q^{88} + 5 q^{89} - 10 q^{91} - 16 q^{92} - 18 q^{94} - 10 q^{95} - 11 q^{97} + 2 q^{98}+O(q^{100}) $ 5 * q + 5 * q^2 + 5 * q^4 - 5 * q^5 - q^7 + 5 * q^8 - 5 * q^10 - 9 * q^11 - q^14 + 5 * q^16 - 6 * q^17 - 4 * q^19 - 5 * q^20 - 9 * q^22 - 16 * q^23 + 8 * q^25 - q^28 - 8 * q^29 - q^31 + 5 * q^32 - 6 * q^34 - 22 * q^35 - 2 * q^37 - 4 * q^38 - 5 * q^40 - 4 * q^41 + 3 * q^43 - 9 * q^44 - 16 * q^46 - 18 * q^47 + 2 * q^49 + 8 * q^50 - 26 * q^53 + q^55 - q^56 - 8 * q^58 - 21 * q^59 - 20 * q^61 - q^62 + 5 * q^64 + 3 * q^65 - 5 * q^67 - 6 * q^68 - 22 * q^70 - 17 * q^71 + 5 * q^73 - 2 * q^74 - 4 * q^76 - 2 * q^77 - 21 * q^79 - 5 * q^80 - 4 * q^82 - 11 * q^83 - 12 * q^85 + 3 * q^86 - 9 * q^88 + 5 * q^89 - 10 * q^91 - 16 * q^92 - 18 * q^94 - 10 * q^95 - 11 * q^97 + 2 * 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
$5$	3.02013	1.35064	0.675322	−	0.737523i	$-0.264005\pi$
$5$	3.02013	1.35064	0.675322	+	0.737523i	$0.264005\pi$
$6$	0	0
$7$	−3.77138	−1.42545	−0.712725	−	0.701444i	$-0.752539\pi$
$7$	−3.77138	−1.42545	−0.712725	+	0.701444i	$0.752539\pi$
$8$	1.00000	0.353553
$9$	0	0
$10$	3.02013	0.955050
$11$	−1.20245	−0.362553	−0.181276	−	0.983432i	$-0.558023\pi$
$11$	−1.20245	−0.362553	−0.181276	+	0.983432i	$0.558023\pi$
$12$	0	0
$13$	−1.04630	−0.290190	−0.145095	−	0.989418i	$-0.546349\pi$
$13$	−1.04630	−0.290190	−0.145095	+	0.989418i	$0.546349\pi$
$14$	−3.77138	−1.00794
$15$	0	0
$16$	1.00000	0.250000
$17$	−3.48116	−0.844305	−0.422153	−	0.906525i	$-0.638725\pi$
$17$	−3.48116	−0.844305	−0.422153	+	0.906525i	$0.638725\pi$
$18$	0	0
$19$	−4.38932	−1.00698	−0.503490	−	0.864001i	$-0.667951\pi$
$19$	−4.38932	−1.00698	−0.503490	+	0.864001i	$0.667951\pi$
$20$	3.02013	0.675322
$21$	0	0
$22$	−1.20245	−0.256364
$23$	−6.93812	−1.44670	−0.723349	−	0.690482i	$-0.757398\pi$
$23$	−6.93812	−1.44670	−0.723349	+	0.690482i	$0.757398\pi$
$24$	0	0
$25$	4.12120	0.824240
$26$	−1.04630	−0.205196
$27$	0	0
$28$	−3.77138	−0.712725
$29$	1.44714	0.268727	0.134363	−	0.990932i	$-0.457101\pi$
$29$	1.44714	0.268727	0.134363	+	0.990932i	$0.457101\pi$
$30$	0	0
$31$	−3.28522	−0.590042	−0.295021	−	0.955491i	$-0.595327\pi$
$31$	−3.28522	−0.590042	−0.295021	+	0.955491i	$0.595327\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	−3.48116	−0.597014
$35$	−11.3901	−1.92528
$36$	0	0
$37$	−5.93312	−0.975398	−0.487699	−	0.873012i	$-0.662164\pi$
$37$	−5.93312	−0.975398	−0.487699	+	0.873012i	$0.662164\pi$
$38$	−4.38932	−0.712042
$39$	0	0
$40$	3.02013	0.477525
$41$	−5.58831	−0.872747	−0.436374	−	0.899766i	$-0.643737\pi$
$41$	−5.58831	−0.872747	−0.436374	+	0.899766i	$0.643737\pi$
$42$	0	0
$43$	9.61326	1.46601	0.733004	−	0.680224i	$-0.238118\pi$
$43$	9.61326	1.46601	0.733004	+	0.680224i	$0.238118\pi$
$44$	−1.20245	−0.181276
$45$	0	0
$46$	−6.93812	−1.02297
$47$	−1.67635	−0.244520	−0.122260	−	0.992498i	$-0.539014\pi$
$47$	−1.67635	−0.244520	−0.122260	+	0.992498i	$0.539014\pi$
$48$	0	0
$49$	7.22334	1.03191
$50$	4.12120	0.582826
$51$	0	0
$52$	−1.04630	−0.145095
$53$	−7.89183	−1.08403	−0.542013	−	0.840370i	$-0.682338\pi$
$53$	−7.89183	−1.08403	−0.542013	+	0.840370i	$0.682338\pi$
$54$	0	0
$55$	−3.63156	−0.489680
$56$	−3.77138	−0.503972
$57$	0	0
$58$	1.44714	0.190018
$59$	−13.9770	−1.81966	−0.909828	−	0.414986i	$-0.863786\pi$
$59$	−13.9770	−1.81966	−0.909828	+	0.414986i	$0.863786\pi$
$60$	0	0
$61$	10.0775	1.29029	0.645145	−	0.764060i	$-0.276797\pi$
$61$	10.0775	1.29029	0.645145	+	0.764060i	$0.276797\pi$
$62$	−3.28522	−0.417223
$63$	0	0
$64$	1.00000	0.125000
$65$	−3.15995	−0.391944
$66$	0	0
$67$	7.69317	0.939871	0.469935	−	0.882701i	$-0.344277\pi$
$67$	7.69317	0.939871	0.469935	+	0.882701i	$0.344277\pi$
$68$	−3.48116	−0.422153
$69$	0	0
$70$	−11.3901	−1.36138
$71$	−2.66924	−0.316781	−0.158390	−	0.987377i	$-0.550630\pi$
$71$	−2.66924	−0.316781	−0.158390	+	0.987377i	$0.550630\pi$
$72$	0	0
$73$	3.48767	0.408201	0.204100	−	0.978950i	$-0.434573\pi$
$73$	3.48767	0.408201	0.204100	+	0.978950i	$0.434573\pi$
$74$	−5.93312	−0.689711
$75$	0	0
$76$	−4.38932	−0.503490
$77$	4.53491	0.516801
$78$	0	0
$79$	−7.37800	−0.830089	−0.415045	−	0.909801i	$-0.636234\pi$
$79$	−7.37800	−0.830089	−0.415045	+	0.909801i	$0.636234\pi$
$80$	3.02013	0.337661
$81$	0	0
$82$	−5.58831	−0.617125
$83$	2.11619	0.232282	0.116141	−	0.993233i	$-0.462948\pi$
$83$	2.11619	0.232282	0.116141	+	0.993233i	$0.462948\pi$
$84$	0	0
$85$	−10.5136	−1.14036
$86$	9.61326	1.03662
$87$	0	0
$88$	−1.20245	−0.128182
$89$	6.47710	0.686571	0.343285	−	0.939231i	$-0.388460\pi$
$89$	6.47710	0.686571	0.343285	+	0.939231i	$0.388460\pi$
$90$	0	0
$91$	3.94598	0.413652
$92$	−6.93812	−0.723349
$93$	0	0
$94$	−1.67635	−0.172902
$95$	−13.2563	−1.36007
$96$	0	0
$97$	8.21487	0.834093	0.417047	−	0.908885i	$-0.363065\pi$
$97$	8.21487	0.834093	0.417047	+	0.908885i	$0.363065\pi$
$98$	7.22334	0.729668
$99$	0	0
$100$	4.12120	0.412120
$101$	1.83630	0.182719	0.0913594	−	0.995818i	$-0.470879\pi$
$101$	1.83630	0.182719	0.0913594	+	0.995818i	$0.470879\pi$
$102$	0	0
$103$	1.86823	0.184082	0.0920410	−	0.995755i	$-0.470661\pi$
$103$	1.86823	0.184082	0.0920410	+	0.995755i	$0.470661\pi$
$104$	−1.04630	−0.102598
$105$	0	0
$106$	−7.89183	−0.766522
$107$	0.540324	0.0522351	0.0261176	−	0.999659i	$-0.491686\pi$
$107$	0.540324	0.0522351	0.0261176	+	0.999659i	$0.491686\pi$
$108$	0	0
$109$	−2.22693	−0.213301	−0.106650	−	0.994297i	$-0.534013\pi$
$109$	−2.22693	−0.213301	−0.106650	+	0.994297i	$0.534013\pi$
$110$	−3.63156	−0.346256
$111$	0	0
$112$	−3.77138	−0.356362
$113$	11.0410	1.03865	0.519326	−	0.854576i	$-0.326183\pi$
$113$	11.0410	1.03865	0.519326	+	0.854576i	$0.326183\pi$
$114$	0	0
$115$	−20.9541	−1.95398
$116$	1.44714	0.134363
$117$	0	0
$118$	−13.9770	−1.28669
$119$	13.1288	1.20351
$120$	0	0
$121$	−9.55411	−0.868555
$122$	10.0775	0.912373
$123$	0	0
$124$	−3.28522	−0.295021
$125$	−2.65409	−0.237389
$126$	0	0
$127$	−18.1860	−1.61375	−0.806873	−	0.590725i	$-0.798842\pi$
$127$	−18.1860	−1.61375	−0.806873	+	0.590725i	$0.798842\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	−3.15995	−0.277146
$131$	−13.8555	−1.21056	−0.605280	−	0.796012i	$-0.706939\pi$
$131$	−13.8555	−1.21056	−0.605280	+	0.796012i	$0.706939\pi$
$132$	0	0
$133$	16.5538	1.43540
$134$	7.69317	0.664589
$135$	0	0
$136$	−3.48116	−0.298507
$137$	−13.6133	−1.16306	−0.581530	−	0.813525i	$-0.697546\pi$
$137$	−13.6133	−1.16306	−0.581530	+	0.813525i	$0.697546\pi$
$138$	0	0
$139$	11.0528	0.937488	0.468744	−	0.883334i	$-0.344707\pi$
$139$	11.0528	0.937488	0.468744	+	0.883334i	$0.344707\pi$
$140$	−11.3901	−0.962638
$141$	0	0
$142$	−2.66924	−0.223998
$143$	1.25812	0.105209
$144$	0	0
$145$	4.37054	0.362954
$146$	3.48767	0.288641
$147$	0	0
$148$	−5.93312	−0.487699
$149$	13.1615	1.07823	0.539115	−	0.842232i	$-0.318759\pi$
$149$	13.1615	1.07823	0.539115	+	0.842232i	$0.318759\pi$
$150$	0	0
$151$	10.5025	0.854683	0.427342	−	0.904090i	$-0.359450\pi$
$151$	10.5025	0.854683	0.427342	+	0.904090i	$0.359450\pi$
$152$	−4.38932	−0.356021
$153$	0	0
$154$	4.53491	0.365433
$155$	−9.92179	−0.796937
$156$	0	0
$157$	17.9254	1.43060	0.715301	−	0.698816i	$-0.246289\pi$
$157$	17.9254	1.43060	0.715301	+	0.698816i	$0.246289\pi$
$158$	−7.37800	−0.586962
$159$	0	0
$160$	3.02013	0.238762
$161$	26.1663	2.06220
$162$	0	0
$163$	10.0729	0.788973	0.394487	−	0.918902i	$-0.370922\pi$
$163$	10.0729	0.788973	0.394487	+	0.918902i	$0.370922\pi$
$164$	−5.58831	−0.436374
$165$	0	0
$166$	2.11619	0.164248
$167$	8.97670	0.694638	0.347319	−	0.937747i	$-0.387092\pi$
$167$	8.97670	0.694638	0.347319	+	0.937747i	$0.387092\pi$
$168$	0	0
$169$	−11.9053	−0.915790
$170$	−10.5136	−0.806354
$171$	0	0
$172$	9.61326	0.733004
$173$	−22.5083	−1.71127	−0.855637	−	0.517577i	$-0.826834\pi$
$173$	−22.5083	−1.71127	−0.855637	+	0.517577i	$0.826834\pi$
$174$	0	0
$175$	−15.5426	−1.17491
$176$	−1.20245	−0.0906382
$177$	0	0
$178$	6.47710	0.485479
$179$	18.4992	1.38270	0.691348	−	0.722522i	$-0.257017\pi$
$179$	18.4992	1.38270	0.691348	+	0.722522i	$0.257017\pi$
$180$	0	0
$181$	19.2505	1.43088	0.715438	−	0.698676i	$-0.246227\pi$
$181$	19.2505	1.43088	0.715438	+	0.698676i	$0.246227\pi$
$182$	3.94598	0.292496
$183$	0	0
$184$	−6.93812	−0.511485
$185$	−17.9188	−1.31742
$186$	0	0
$187$	4.18593	0.306105
$188$	−1.67635	−0.122260
$189$	0	0
$190$	−13.2563	−0.961716
$191$	−13.2214	−0.956666	−0.478333	−	0.878179i	$-0.658759\pi$
$191$	−13.2214	−0.956666	−0.478333	+	0.878179i	$0.658759\pi$
$192$	0	0
$193$	12.7010	0.914240	0.457120	−	0.889405i	$-0.348881\pi$
$193$	12.7010	0.914240	0.457120	+	0.889405i	$0.348881\pi$
$194$	8.21487	0.589793
$195$	0	0
$196$	7.22334	0.515953
$197$	6.80378	0.484749	0.242375	−	0.970183i	$-0.422074\pi$
$197$	6.80378	0.484749	0.242375	+	0.970183i	$0.422074\pi$
$198$	0	0
$199$	−18.3936	−1.30389	−0.651944	−	0.758267i	$-0.726046\pi$
$199$	−18.3936	−1.30389	−0.651944	+	0.758267i	$0.726046\pi$
$200$	4.12120	0.291413
$201$	0	0
$202$	1.83630	0.129202
$203$	−5.45771	−0.383056
$204$	0	0
$205$	−16.8774	−1.17877
$206$	1.86823	0.130166
$207$	0	0
$208$	−1.04630	−0.0725476
$209$	5.27795	0.365084
$210$	0	0
$211$	−23.9426	−1.64827	−0.824137	−	0.566391i	$-0.808339\pi$
$211$	−23.9426	−1.64827	−0.824137	+	0.566391i	$0.808339\pi$
$212$	−7.89183	−0.542013
$213$	0	0
$214$	0.540324	0.0369358
$215$	29.0333	1.98006
$216$	0	0
$217$	12.3898	0.841075
$218$	−2.22693	−0.150827
$219$	0	0
$220$	−3.63156	−0.244840
$221$	3.64232	0.245009
$222$	0	0
$223$	−1.00000	−0.0669650
$223$	−1.00000	−0.0669650
$224$	−3.77138	−0.251986
$225$	0	0
$226$	11.0410	0.734438
$227$	−18.9100	−1.25510	−0.627549	−	0.778577i	$-0.715942\pi$
$227$	−18.9100	−1.25510	−0.627549	+	0.778577i	$0.715942\pi$
$228$	0	0
$229$	−9.36817	−0.619066	−0.309533	−	0.950889i	$-0.600173\pi$
$229$	−9.36817	−0.619066	−0.309533	+	0.950889i	$0.600173\pi$
$230$	−20.9541	−1.38167
$231$	0	0
$232$	1.44714	0.0950092
$233$	22.4238	1.46903	0.734517	−	0.678590i	$-0.237409\pi$
$233$	22.4238	1.46903	0.734517	+	0.678590i	$0.237409\pi$
$234$	0	0
$235$	−5.06279	−0.330260
$236$	−13.9770	−0.909828
$237$	0	0
$238$	13.1288	0.851013
$239$	−12.7281	−0.823313	−0.411657	−	0.911339i	$-0.635050\pi$
$239$	−12.7281	−0.823313	−0.411657	+	0.911339i	$0.635050\pi$
$240$	0	0
$241$	−14.7087	−0.947472	−0.473736	−	0.880667i	$-0.657095\pi$
$241$	−14.7087	−0.947472	−0.473736	+	0.880667i	$0.657095\pi$
$242$	−9.55411	−0.614161
$243$	0	0
$244$	10.0775	0.645145
$245$	21.8154	1.39374
$246$	0	0
$247$	4.59253	0.292216
$248$	−3.28522	−0.208611
$249$	0	0
$250$	−2.65409	−0.167860
$251$	14.4741	0.913597	0.456799	−	0.889570i	$-0.348996\pi$
$251$	14.4741	0.913597	0.456799	+	0.889570i	$0.348996\pi$
$252$	0	0
$253$	8.34276	0.524505
$254$	−18.1860	−1.14109
$255$	0	0
$256$	1.00000	0.0625000
$257$	4.56495	0.284754	0.142377	−	0.989813i	$-0.454526\pi$
$257$	4.56495	0.284754	0.142377	+	0.989813i	$0.454526\pi$
$258$	0	0
$259$	22.3761	1.39038
$260$	−3.15995	−0.195972
$261$	0	0
$262$	−13.8555	−0.855996
$263$	−27.4213	−1.69087	−0.845434	−	0.534080i	$-0.820658\pi$
$263$	−27.4213	−1.69087	−0.845434	+	0.534080i	$0.820658\pi$
$264$	0	0
$265$	−23.8344	−1.46413
$266$	16.5538	1.01498
$267$	0	0
$268$	7.69317	0.469935
$269$	−3.32320	−0.202619	−0.101309	−	0.994855i	$-0.532303\pi$
$269$	−3.32320	−0.202619	−0.101309	+	0.994855i	$0.532303\pi$
$270$	0	0
$271$	7.89850	0.479800	0.239900	−	0.970798i	$-0.422885\pi$
$271$	7.89850	0.479800	0.239900	+	0.970798i	$0.422885\pi$
$272$	−3.48116	−0.211076
$273$	0	0
$274$	−13.6133	−0.822407
$275$	−4.95555	−0.298831
$276$	0	0
$277$	−23.2990	−1.39990	−0.699952	−	0.714190i	$-0.746795\pi$
$277$	−23.2990	−1.39990	−0.699952	+	0.714190i	$0.746795\pi$
$278$	11.0528	0.662904
$279$	0	0
$280$	−11.3901	−0.680688
$281$	6.05838	0.361413	0.180706	−	0.983537i	$-0.442162\pi$
$281$	6.05838	0.361413	0.180706	+	0.983537i	$0.442162\pi$
$282$	0	0
$283$	20.8294	1.23818	0.619088	−	0.785321i	$-0.287502\pi$
$283$	20.8294	1.23818	0.619088	+	0.785321i	$0.287502\pi$
$284$	−2.66924	−0.158390
$285$	0	0
$286$	1.25812	0.0743942
$287$	21.0757	1.24406
$288$	0	0
$289$	−4.88152	−0.287148
$290$	4.37054	0.256647
$291$	0	0
$292$	3.48767	0.204100
$293$	12.3205	0.719771	0.359886	−	0.932996i	$-0.382816\pi$
$293$	12.3205	0.719771	0.359886	+	0.932996i	$0.382816\pi$
$294$	0	0
$295$	−42.2125	−2.45771
$296$	−5.93312	−0.344855
$297$	0	0
$298$	13.1615	0.762423
$299$	7.25933	0.419818
$300$	0	0
$301$	−36.2553	−2.08972
$302$	10.5025	0.604352
$303$	0	0
$304$	−4.38932	−0.251745
$305$	30.4354	1.74272
$306$	0	0
$307$	−4.95093	−0.282565	−0.141282	−	0.989969i	$-0.545123\pi$
$307$	−4.95093	−0.282565	−0.141282	+	0.989969i	$0.545123\pi$
$308$	4.53491	0.258400
$309$	0	0
$310$	−9.92179	−0.563520
$311$	−14.6555	−0.831037	−0.415518	−	0.909585i	$-0.636400\pi$
$311$	−14.6555	−0.831037	−0.415518	+	0.909585i	$0.636400\pi$
$312$	0	0
$313$	27.3865	1.54798	0.773988	−	0.633200i	$-0.218259\pi$
$313$	27.3865	1.54798	0.773988	+	0.633200i	$0.218259\pi$
$314$	17.9254	1.01159
$315$	0	0
$316$	−7.37800	−0.415045
$317$	17.2101	0.966614	0.483307	−	0.875451i	$-0.339436\pi$
$317$	17.2101	0.966614	0.483307	+	0.875451i	$0.339436\pi$
$318$	0	0
$319$	−1.74011	−0.0974276
$320$	3.02013	0.168831
$321$	0	0
$322$	26.1663	1.45819
$323$	15.2799	0.850199
$324$	0	0
$325$	−4.31200	−0.239186
$326$	10.0729	0.557888
$327$	0	0
$328$	−5.58831	−0.308563
$329$	6.32215	0.348552
$330$	0	0
$331$	−7.49987	−0.412230	−0.206115	−	0.978528i	$-0.566082\pi$
$331$	−7.49987	−0.412230	−0.206115	+	0.978528i	$0.566082\pi$
$332$	2.11619	0.116141
$333$	0	0
$334$	8.97670	0.491183
$335$	23.2344	1.26943
$336$	0	0
$337$	2.60559	0.141935	0.0709677	−	0.997479i	$-0.477391\pi$
$337$	2.60559	0.141935	0.0709677	+	0.997479i	$0.477391\pi$
$338$	−11.9053	−0.647561
$339$	0	0
$340$	−10.5136	−0.570178
$341$	3.95031	0.213922
$342$	0	0
$343$	−0.842304	−0.0454801
$344$	9.61326	0.518312
$345$	0	0
$346$	−22.5083	−1.21005
$347$	6.30819	0.338641	0.169321	−	0.985561i	$-0.445843\pi$
$347$	6.30819	0.338641	0.169321	+	0.985561i	$0.445843\pi$
$348$	0	0
$349$	7.33681	0.392730	0.196365	−	0.980531i	$-0.437086\pi$
$349$	7.33681	0.392730	0.196365	+	0.980531i	$0.437086\pi$
$350$	−15.5426	−0.830789
$351$	0	0
$352$	−1.20245	−0.0640909
$353$	−24.8373	−1.32195	−0.660977	−	0.750406i	$-0.729858\pi$
$353$	−24.8373	−1.32195	−0.660977	+	0.750406i	$0.729858\pi$
$354$	0	0
$355$	−8.06147	−0.427858
$356$	6.47710	0.343285
$357$	0	0
$358$	18.4992	0.977714
$359$	3.12865	0.165124	0.0825620	−	0.996586i	$-0.473690\pi$
$359$	3.12865	0.165124	0.0825620	+	0.996586i	$0.473690\pi$
$360$	0	0
$361$	0.266169	0.0140089
$362$	19.2505	1.01178
$363$	0	0
$364$	3.94598	0.206826
$365$	10.5332	0.551334
$366$	0	0
$367$	10.0031	0.522155	0.261078	−	0.965318i	$-0.415922\pi$
$367$	10.0031	0.522155	0.261078	+	0.965318i	$0.415922\pi$
$368$	−6.93812	−0.361675
$369$	0	0
$370$	−17.9188	−0.931554
$371$	29.7631	1.54522
$372$	0	0
$373$	18.1472	0.939628	0.469814	−	0.882765i	$-0.344321\pi$
$373$	18.1472	0.939628	0.469814	+	0.882765i	$0.344321\pi$
$374$	4.18593	0.216449
$375$	0	0
$376$	−1.67635	−0.0864510
$377$	−1.51413	−0.0779818
$378$	0	0
$379$	−7.77655	−0.399455	−0.199727	−	0.979852i	$-0.564006\pi$
$379$	−7.77655	−0.399455	−0.199727	+	0.979852i	$0.564006\pi$
$380$	−13.2563	−0.680036
$381$	0	0
$382$	−13.2214	−0.676465
$383$	−29.0604	−1.48492	−0.742458	−	0.669893i	$-0.766340\pi$
$383$	−29.0604	−1.48492	−0.742458	+	0.669893i	$0.766340\pi$
$384$	0	0
$385$	13.6960	0.698014
$386$	12.7010	0.646466
$387$	0	0
$388$	8.21487	0.417047
$389$	9.50596	0.481972	0.240986	−	0.970529i	$-0.422529\pi$
$389$	9.50596	0.481972	0.240986	+	0.970529i	$0.422529\pi$
$390$	0	0
$391$	24.1527	1.22146
$392$	7.22334	0.364834
$393$	0	0
$394$	6.80378	0.342769
$395$	−22.2825	−1.12116
$396$	0	0
$397$	21.6620	1.08718	0.543592	−	0.839350i	$-0.317064\pi$
$397$	21.6620	1.08718	0.543592	+	0.839350i	$0.317064\pi$
$398$	−18.3936	−0.921988
$399$	0	0
$400$	4.12120	0.206060
$401$	−20.1045	−1.00397	−0.501985	−	0.864876i	$-0.667397\pi$
$401$	−20.1045	−1.00397	−0.501985	+	0.864876i	$0.667397\pi$
$402$	0	0
$403$	3.43731	0.171225
$404$	1.83630	0.0913594
$405$	0	0
$406$	−5.45771	−0.270862
$407$	7.13429	0.353634
$408$	0	0
$409$	−36.8815	−1.82367	−0.911836	−	0.410555i	$-0.865335\pi$
$409$	−36.8815	−1.82367	−0.911836	+	0.410555i	$0.865335\pi$
$410$	−16.8774	−0.833517
$411$	0	0
$412$	1.86823	0.0920410
$413$	52.7128	2.59383
$414$	0	0
$415$	6.39118	0.313731
$416$	−1.04630	−0.0512989
$417$	0	0
$418$	5.27795	0.258153
$419$	−33.9553	−1.65883	−0.829413	−	0.558636i	$-0.811325\pi$
$419$	−33.9553	−1.65883	−0.829413	+	0.558636i	$0.811325\pi$
$420$	0	0
$421$	−36.8261	−1.79479	−0.897397	−	0.441225i	$-0.854544\pi$
$421$	−36.8261	−1.79479	−0.897397	+	0.441225i	$0.854544\pi$
$422$	−23.9426	−1.16551
$423$	0	0
$424$	−7.89183	−0.383261
$425$	−14.3466	−0.695910
$426$	0	0
$427$	−38.0061	−1.83924
$428$	0.540324	0.0261176
$429$	0	0
$430$	29.0333	1.40011
$431$	1.58581	0.0763860	0.0381930	−	0.999270i	$-0.487840\pi$
$431$	1.58581	0.0763860	0.0381930	+	0.999270i	$0.487840\pi$
$432$	0	0
$433$	−2.99805	−0.144077	−0.0720386	−	0.997402i	$-0.522950\pi$
$433$	−2.99805	−0.144077	−0.0720386	+	0.997402i	$0.522950\pi$
$434$	12.3898	0.594730
$435$	0	0
$436$	−2.22693	−0.106650
$437$	30.4537	1.45680
$438$	0	0
$439$	17.2951	0.825448	0.412724	−	0.910856i	$-0.364577\pi$
$439$	17.2951	0.825448	0.412724	+	0.910856i	$0.364577\pi$
$440$	−3.63156	−0.173128
$441$	0	0
$442$	3.64232	0.173248
$443$	−22.8314	−1.08475	−0.542377	−	0.840135i	$-0.682476\pi$
$443$	−22.8314	−1.08475	−0.542377	+	0.840135i	$0.682476\pi$
$444$	0	0
$445$	19.5617	0.927313
$446$	−1.00000	−0.0473514
$447$	0	0
$448$	−3.77138	−0.178181
$449$	−28.4260	−1.34150	−0.670752	−	0.741682i	$-0.734028\pi$
$449$	−28.4260	−1.34150	−0.670752	+	0.741682i	$0.734028\pi$
$450$	0	0
$451$	6.71967	0.316417
$452$	11.0410	0.519326
$453$	0	0
$454$	−18.9100	−0.887489
$455$	11.9174	0.558696
$456$	0	0
$457$	−4.12755	−0.193079	−0.0965393	−	0.995329i	$-0.530777\pi$
$457$	−4.12755	−0.193079	−0.0965393	+	0.995329i	$0.530777\pi$
$458$	−9.36817	−0.437746
$459$	0	0
$460$	−20.9541	−0.976988
$461$	23.7885	1.10794	0.553970	−	0.832537i	$-0.313112\pi$
$461$	23.7885	1.10794	0.553970	+	0.832537i	$0.313112\pi$
$462$	0	0
$463$	−5.25289	−0.244123	−0.122061	−	0.992523i	$-0.538950\pi$
$463$	−5.25289	−0.244123	−0.122061	+	0.992523i	$0.538950\pi$
$464$	1.44714	0.0671816
$465$	0	0
$466$	22.4238	1.03876
$467$	−28.3361	−1.31124	−0.655620	−	0.755091i	$-0.727592\pi$
$467$	−28.3361	−1.31124	−0.655620	+	0.755091i	$0.727592\pi$
$468$	0	0
$469$	−29.0139	−1.33974
$470$	−5.06279	−0.233529
$471$	0	0
$472$	−13.9770	−0.643345
$473$	−11.5595	−0.531506
$474$	0	0
$475$	−18.0893	−0.829993
$476$	13.1288	0.601757
$477$	0	0
$478$	−12.7281	−0.582170
$479$	−12.1657	−0.555865	−0.277932	−	0.960601i	$-0.589649\pi$
$479$	−12.1657	−0.555865	−0.277932	+	0.960601i	$0.589649\pi$
$480$	0	0
$481$	6.20780	0.283051
$482$	−14.7087	−0.669964
$483$	0	0
$484$	−9.55411	−0.434278
$485$	24.8100	1.12656
$486$	0	0
$487$	−18.2502	−0.826994	−0.413497	−	0.910505i	$-0.635693\pi$
$487$	−18.2502	−0.826994	−0.413497	+	0.910505i	$0.635693\pi$
$488$	10.0775	0.456187
$489$	0	0
$490$	21.8154	0.985521
$491$	5.52096	0.249157	0.124579	−	0.992210i	$-0.460242\pi$
$491$	5.52096	0.249157	0.124579	+	0.992210i	$0.460242\pi$
$492$	0	0
$493$	−5.03771	−0.226887
$494$	4.59253	0.206628
$495$	0	0
$496$	−3.28522	−0.147511
$497$	10.0667	0.451555
$498$	0	0
$499$	32.2553	1.44394	0.721972	−	0.691922i	$-0.243236\pi$
$499$	32.2553	1.44394	0.721972	+	0.691922i	$0.243236\pi$
$500$	−2.65409	−0.118695
$501$	0	0
$502$	14.4741	0.646011
$503$	12.8314	0.572125	0.286063	−	0.958211i	$-0.407653\pi$
$503$	12.8314	0.572125	0.286063	+	0.958211i	$0.407653\pi$
$504$	0	0
$505$	5.54587	0.246788
$506$	8.34276	0.370881
$507$	0	0
$508$	−18.1860	−0.806873
$509$	−4.24927	−0.188345	−0.0941727	−	0.995556i	$-0.530021\pi$
$509$	−4.24927	−0.188345	−0.0941727	+	0.995556i	$0.530021\pi$
$510$	0	0
$511$	−13.1533	−0.581869
$512$	1.00000	0.0441942
$513$	0	0
$514$	4.56495	0.201351
$515$	5.64230	0.248629
$516$	0	0
$517$	2.01573	0.0886516
$518$	22.3761	0.983148
$519$	0	0
$520$	−3.15995	−0.138573
$521$	26.2874	1.15167	0.575836	−	0.817565i	$-0.304677\pi$
$521$	26.2874	1.15167	0.575836	+	0.817565i	$0.304677\pi$
$522$	0	0
$523$	7.47399	0.326815	0.163407	−	0.986559i	$-0.447752\pi$
$523$	7.47399	0.326815	0.163407	+	0.986559i	$0.447752\pi$
$524$	−13.8555	−0.605280
$525$	0	0
$526$	−27.4213	−1.19562
$527$	11.4364	0.498176
$528$	0	0
$529$	25.1376	1.09294
$530$	−23.8344	−1.03530
$531$	0	0
$532$	16.5538	0.717700
$533$	5.84703	0.253263
$534$	0	0
$535$	1.63185	0.0705511
$536$	7.69317	0.332294
$537$	0	0
$538$	−3.32320	−0.143273
$539$	−8.68572	−0.374120
$540$	0	0
$541$	−14.4483	−0.621181	−0.310590	−	0.950544i	$-0.600527\pi$
$541$	−14.4483	−0.621181	−0.310590	+	0.950544i	$0.600527\pi$
$542$	7.89850	0.339269
$543$	0	0
$544$	−3.48116	−0.149254
$545$	−6.72562	−0.288094
$546$	0	0
$547$	11.1792	0.477986	0.238993	−	0.971021i	$-0.423183\pi$
$547$	11.1792	0.477986	0.238993	+	0.971021i	$0.423183\pi$
$548$	−13.6133	−0.581530
$549$	0	0
$550$	−4.95555	−0.211305
$551$	−6.35195	−0.270602
$552$	0	0
$553$	27.8253	1.18325
$554$	−23.2990	−0.989881
$555$	0	0
$556$	11.0528	0.468744
$557$	−7.03110	−0.297917	−0.148959	−	0.988843i	$-0.547592\pi$
$557$	−7.03110	−0.297917	−0.148959	+	0.988843i	$0.547592\pi$
$558$	0	0
$559$	−10.0583	−0.425421
$560$	−11.3901	−0.481319
$561$	0	0
$562$	6.05838	0.255557
$563$	25.3616	1.06886	0.534432	−	0.845212i	$-0.320526\pi$
$563$	25.3616	1.06886	0.534432	+	0.845212i	$0.320526\pi$
$564$	0	0
$565$	33.3453	1.40285
$566$	20.8294	0.875523
$567$	0	0
$568$	−2.66924	−0.111999
$569$	−29.5466	−1.23866	−0.619329	−	0.785131i	$-0.712595\pi$
$569$	−29.5466	−1.23866	−0.619329	+	0.785131i	$0.712595\pi$
$570$	0	0
$571$	29.4674	1.23317	0.616586	−	0.787288i	$-0.288515\pi$
$571$	29.4674	1.23317	0.616586	+	0.787288i	$0.288515\pi$
$572$	1.25812	0.0526047
$573$	0	0
$574$	21.0757	0.879681
$575$	−28.5934	−1.19243
$576$	0	0
$577$	−0.367741	−0.0153093	−0.00765464	−	0.999971i	$-0.502437\pi$
$577$	−0.367741	−0.0153093	−0.00765464	+	0.999971i	$0.502437\pi$
$578$	−4.88152	−0.203045
$579$	0	0
$580$	4.37054	0.181477
$581$	−7.98097	−0.331107
$582$	0	0
$583$	9.48954	0.393017
$584$	3.48767	0.144321
$585$	0	0
$586$	12.3205	0.508955
$587$	−31.4369	−1.29754	−0.648770	−	0.760985i	$-0.724716\pi$
$587$	−31.4369	−1.29754	−0.648770	+	0.760985i	$0.724716\pi$
$588$	0	0
$589$	14.4199	0.594161
$590$	−42.2125	−1.73786
$591$	0	0
$592$	−5.93312	−0.243850
$593$	39.8376	1.63593	0.817966	−	0.575266i	$-0.195102\pi$
$593$	39.8376	1.63593	0.817966	+	0.575266i	$0.195102\pi$
$594$	0	0
$595$	39.6507	1.62552
$596$	13.1615	0.539115
$597$	0	0
$598$	7.25933	0.296856
$599$	16.6473	0.680191	0.340096	−	0.940391i	$-0.389541\pi$
$599$	16.6473	0.680191	0.340096	+	0.940391i	$0.389541\pi$
$600$	0	0
$601$	−41.2749	−1.68364	−0.841818	−	0.539761i	$-0.818515\pi$
$601$	−41.2749	−1.68364	−0.841818	+	0.539761i	$0.818515\pi$
$602$	−36.2553	−1.47766
$603$	0	0
$604$	10.5025	0.427342
$605$	−28.8547	−1.17311
$606$	0	0
$607$	−46.5136	−1.88793	−0.943965	−	0.330045i	$-0.892936\pi$
$607$	−46.5136	−1.88793	−0.943965	+	0.330045i	$0.892936\pi$
$608$	−4.38932	−0.178011
$609$	0	0
$610$	30.4354	1.23229
$611$	1.75396	0.0709575
$612$	0	0
$613$	15.8555	0.640398	0.320199	−	0.947350i	$-0.396250\pi$
$613$	15.8555	0.640398	0.320199	+	0.947350i	$0.396250\pi$
$614$	−4.95093	−0.199803
$615$	0	0
$616$	4.53491	0.182717
$617$	16.3571	0.658513	0.329256	−	0.944241i	$-0.393202\pi$
$617$	16.3571	0.658513	0.329256	+	0.944241i	$0.393202\pi$
$618$	0	0
$619$	7.77228	0.312394	0.156197	−	0.987726i	$-0.450077\pi$
$619$	7.77228	0.312394	0.156197	+	0.987726i	$0.450077\pi$
$620$	−9.92179	−0.398469
$621$	0	0
$622$	−14.6555	−0.587632
$623$	−24.4276	−0.978672
$624$	0	0
$625$	−28.6217	−1.14487
$626$	27.3865	1.09458
$627$	0	0
$628$	17.9254	0.715301
$629$	20.6541	0.823534
$630$	0	0
$631$	21.8810	0.871068	0.435534	−	0.900172i	$-0.356560\pi$
$631$	21.8810	0.871068	0.435534	+	0.900172i	$0.356560\pi$
$632$	−7.37800	−0.293481
$633$	0	0
$634$	17.2101	0.683499
$635$	−54.9241	−2.17960
$636$	0	0
$637$	−7.55775	−0.299449
$638$	−1.74011	−0.0688917
$639$	0	0
$640$	3.02013	0.119381
$641$	−31.3568	−1.23852	−0.619260	−	0.785186i	$-0.712567\pi$
$641$	−31.3568	−1.23852	−0.619260	+	0.785186i	$0.712567\pi$
$642$	0	0
$643$	13.8201	0.545012	0.272506	−	0.962154i	$-0.412148\pi$
$643$	13.8201	0.545012	0.272506	+	0.962154i	$0.412148\pi$
$644$	26.1663	1.03110
$645$	0	0
$646$	15.2799	0.601181
$647$	−5.48150	−0.215500	−0.107750	−	0.994178i	$-0.534365\pi$
$647$	−5.48150	−0.215500	−0.107750	+	0.994178i	$0.534365\pi$
$648$	0	0
$649$	16.8067	0.659721
$650$	−4.31200	−0.169130
$651$	0	0
$652$	10.0729	0.394487
$653$	−0.281538	−0.0110174	−0.00550872	−	0.999985i	$-0.501753\pi$
$653$	−0.281538	−0.0110174	−0.00550872	+	0.999985i	$0.501753\pi$
$654$	0	0
$655$	−41.8454	−1.63504
$656$	−5.58831	−0.218187
$657$	0	0
$658$	6.32215	0.246463
$659$	−19.5223	−0.760481	−0.380241	−	0.924888i	$-0.624159\pi$
$659$	−19.5223	−0.760481	−0.380241	+	0.924888i	$0.624159\pi$
$660$	0	0
$661$	1.06418	0.0413919	0.0206959	−	0.999786i	$-0.493412\pi$
$661$	1.06418	0.0413919	0.0206959	+	0.999786i	$0.493412\pi$
$662$	−7.49987	−0.291491
$663$	0	0
$664$	2.11619	0.0821242
$665$	49.9948	1.93871
$666$	0	0
$667$	−10.0404	−0.388766
$668$	8.97670	0.347319
$669$	0	0
$670$	23.2344	0.897623
$671$	−12.1177	−0.467799
$672$	0	0
$673$	4.31488	0.166326	0.0831631	−	0.996536i	$-0.473498\pi$
$673$	4.31488	0.166326	0.0831631	+	0.996536i	$0.473498\pi$
$674$	2.60559	0.100364
$675$	0	0
$676$	−11.9053	−0.457895
$677$	−3.50610	−0.134750	−0.0673752	−	0.997728i	$-0.521462\pi$
$677$	−3.50610	−0.134750	−0.0673752	+	0.997728i	$0.521462\pi$
$678$	0	0
$679$	−30.9814	−1.18896
$680$	−10.5136	−0.403177
$681$	0	0
$682$	3.95031	0.151265
$683$	−10.4697	−0.400610	−0.200305	−	0.979734i	$-0.564193\pi$
$683$	−10.4697	−0.400610	−0.200305	+	0.979734i	$0.564193\pi$
$684$	0	0
$685$	−41.1139	−1.57088
$686$	−0.842304	−0.0321593
$687$	0	0
$688$	9.61326	0.366502
$689$	8.25719	0.314574
$690$	0	0
$691$	38.5631	1.46701	0.733504	−	0.679685i	$-0.237883\pi$
$691$	38.5631	1.46701	0.733504	+	0.679685i	$0.237883\pi$
$692$	−22.5083	−0.855637
$693$	0	0
$694$	6.30819	0.239456
$695$	33.3810	1.26621
$696$	0	0
$697$	19.4538	0.736865
$698$	7.33681	0.277702
$699$	0	0
$700$	−15.5426	−0.587456
$701$	−23.1419	−0.874056	−0.437028	−	0.899448i	$-0.643969\pi$
$701$	−23.1419	−0.874056	−0.437028	+	0.899448i	$0.643969\pi$
$702$	0	0
$703$	26.0424	0.982207
$704$	−1.20245	−0.0453191
$705$	0	0
$706$	−24.8373	−0.934763
$707$	−6.92539	−0.260456
$708$	0	0
$709$	12.8373	0.482113	0.241057	−	0.970511i	$-0.422506\pi$
$709$	12.8373	0.482113	0.241057	+	0.970511i	$0.422506\pi$
$710$	−8.06147	−0.302542
$711$	0	0
$712$	6.47710	0.242739
$713$	22.7932	0.853614
$714$	0	0
$715$	3.79969	0.142100
$716$	18.4992	0.691348
$717$	0	0
$718$	3.12865	0.116760
$719$	−24.3201	−0.906987	−0.453494	−	0.891260i	$-0.649823\pi$
$719$	−24.3201	−0.906987	−0.453494	+	0.891260i	$0.649823\pi$
$720$	0	0
$721$	−7.04581	−0.262400
$722$	0.266169	0.00990579
$723$	0	0
$724$	19.2505	0.715438
$725$	5.96394	0.221495
$726$	0	0
$727$	4.96463	0.184128	0.0920640	−	0.995753i	$-0.470654\pi$
$727$	4.96463	0.184128	0.0920640	+	0.995753i	$0.470654\pi$
$728$	3.94598	0.146248
$729$	0	0
$730$	10.5332	0.389852
$731$	−33.4653	−1.23776
$732$	0	0
$733$	9.90002	0.365666	0.182833	−	0.983144i	$-0.441473\pi$
$733$	9.90002	0.365666	0.182833	+	0.983144i	$0.441473\pi$
$734$	10.0031	0.369219
$735$	0	0
$736$	−6.93812	−0.255743
$737$	−9.25067	−0.340753
$738$	0	0
$739$	−2.33921	−0.0860493	−0.0430247	−	0.999074i	$-0.513699\pi$
$739$	−2.33921	−0.0860493	−0.0430247	+	0.999074i	$0.513699\pi$
$740$	−17.9188	−0.658708
$741$	0	0
$742$	29.7631	1.09264
$743$	−28.5828	−1.04860	−0.524301	−	0.851533i	$-0.675673\pi$
$743$	−28.5828	−1.04860	−0.524301	+	0.851533i	$0.675673\pi$
$744$	0	0
$745$	39.7494	1.45630
$746$	18.1472	0.664418
$747$	0	0
$748$	4.18593	0.153053
$749$	−2.03777	−0.0744585
$750$	0	0
$751$	2.12757	0.0776362	0.0388181	−	0.999246i	$-0.487641\pi$
$751$	2.12757	0.0776362	0.0388181	+	0.999246i	$0.487641\pi$
$752$	−1.67635	−0.0611301
$753$	0	0
$754$	−1.51413	−0.0551415
$755$	31.7190	1.15437
$756$	0	0
$757$	22.6613	0.823638	0.411819	−	0.911266i	$-0.364894\pi$
$757$	22.6613	0.823638	0.411819	+	0.911266i	$0.364894\pi$
$758$	−7.77655	−0.282457
$759$	0	0
$760$	−13.2563	−0.480858
$761$	28.4967	1.03300	0.516502	−	0.856286i	$-0.327234\pi$
$761$	28.4967	1.03300	0.516502	+	0.856286i	$0.327234\pi$
$762$	0	0
$763$	8.39860	0.304050
$764$	−13.2214	−0.478333
$765$	0	0
$766$	−29.0604	−1.04999
$767$	14.6241	0.528046
$768$	0	0
$769$	44.0265	1.58764	0.793819	−	0.608154i	$-0.208090\pi$
$769$	44.0265	1.58764	0.793819	+	0.608154i	$0.208090\pi$
$770$	13.6960	0.493570
$771$	0	0
$772$	12.7010	0.457120
$773$	49.3490	1.77496	0.887481	−	0.460845i	$-0.152453\pi$
$773$	49.3490	1.77496	0.887481	+	0.460845i	$0.152453\pi$
$774$	0	0
$775$	−13.5390	−0.486336
$776$	8.21487	0.294896
$777$	0	0
$778$	9.50596	0.340805
$779$	24.5289	0.878839
$780$	0	0
$781$	3.20964	0.114850
$782$	24.1527	0.863700
$783$	0	0
$784$	7.22334	0.257976
$785$	54.1371	1.93224
$786$	0	0
$787$	34.8197	1.24119	0.620594	−	0.784132i	$-0.286892\pi$
$787$	34.8197	1.24119	0.620594	+	0.784132i	$0.286892\pi$
$788$	6.80378	0.242375
$789$	0	0
$790$	−22.2825	−0.792777
$791$	−41.6399	−1.48055
$792$	0	0
$793$	−10.5440	−0.374430
$794$	21.6620	0.768755
$795$	0	0
$796$	−18.3936	−0.651944
$797$	−3.93026	−0.139217	−0.0696086	−	0.997574i	$-0.522175\pi$
$797$	−3.93026	−0.139217	−0.0696086	+	0.997574i	$0.522175\pi$
$798$	0	0
$799$	5.83564	0.206450
$800$	4.12120	0.145706
$801$	0	0
$802$	−20.1045	−0.709914
$803$	−4.19375	−0.147994
$804$	0	0
$805$	79.0258	2.78529
$806$	3.43731	0.121074
$807$	0	0
$808$	1.83630	0.0646008
$809$	45.9514	1.61556	0.807782	−	0.589481i	$-0.200668\pi$
$809$	45.9514	1.61556	0.807782	+	0.589481i	$0.200668\pi$
$810$	0	0
$811$	27.2715	0.957630	0.478815	−	0.877916i	$-0.341066\pi$
$811$	27.2715	0.957630	0.478815	+	0.877916i	$0.341066\pi$
$812$	−5.45771	−0.191528
$813$	0	0
$814$	7.13429	0.250057
$815$	30.4216	1.06562
$816$	0	0
$817$	−42.1957	−1.47624
$818$	−36.8815	−1.28953
$819$	0	0
$820$	−16.8774	−0.589386
$821$	32.9435	1.14974	0.574868	−	0.818246i	$-0.305053\pi$
$821$	32.9435	1.14974	0.574868	+	0.818246i	$0.305053\pi$
$822$	0	0
$823$	37.5067	1.30740	0.653700	−	0.756754i	$-0.273216\pi$
$823$	37.5067	1.30740	0.653700	+	0.756754i	$0.273216\pi$
$824$	1.86823	0.0650828
$825$	0	0
$826$	52.7128	1.83411
$827$	−5.28182	−0.183667	−0.0918334	−	0.995774i	$-0.529273\pi$
$827$	−5.28182	−0.183667	−0.0918334	+	0.995774i	$0.529273\pi$
$828$	0	0
$829$	−21.4341	−0.744437	−0.372218	−	0.928145i	$-0.621403\pi$
$829$	−21.4341	−0.744437	−0.372218	+	0.928145i	$0.621403\pi$
$830$	6.39118	0.221841
$831$	0	0
$832$	−1.04630	−0.0362738
$833$	−25.1456	−0.871244
$834$	0	0
$835$	27.1108	0.938209
$836$	5.27795	0.182542
$837$	0	0
$838$	−33.9553	−1.17297
$839$	−9.17185	−0.316647	−0.158324	−	0.987387i	$-0.550609\pi$
$839$	−9.17185	−0.316647	−0.158324	+	0.987387i	$0.550609\pi$
$840$	0	0
$841$	−26.9058	−0.927786
$842$	−36.8261	−1.26911
$843$	0	0
$844$	−23.9426	−0.824137
$845$	−35.9555	−1.23691
$846$	0	0
$847$	36.0322	1.23808
$848$	−7.89183	−0.271006
$849$	0	0
$850$	−14.3466	−0.492083
$851$	41.1647	1.41111
$852$	0	0
$853$	−6.81667	−0.233398	−0.116699	−	0.993167i	$-0.537231\pi$
$853$	−6.81667	−0.233398	−0.116699	+	0.993167i	$0.537231\pi$
$854$	−38.0061	−1.30054
$855$	0	0
$856$	0.540324	0.0184679
$857$	7.61268	0.260044	0.130022	−	0.991511i	$-0.458495\pi$
$857$	7.61268	0.260044	0.130022	+	0.991511i	$0.458495\pi$
$858$	0	0
$859$	18.6435	0.636109	0.318055	−	0.948072i	$-0.396970\pi$
$859$	18.6435	0.636109	0.318055	+	0.948072i	$0.396970\pi$
$860$	29.0333	0.990028
$861$	0	0
$862$	1.58581	0.0540131
$863$	39.6210	1.34872	0.674358	−	0.738405i	$-0.264421\pi$
$863$	39.6210	1.34872	0.674358	+	0.738405i	$0.264421\pi$
$864$	0	0
$865$	−67.9780	−2.31132
$866$	−2.99805	−0.101878
$867$	0	0
$868$	12.3898	0.420538
$869$	8.87169	0.300951
$870$	0	0
$871$	−8.04934	−0.272741
$872$	−2.22693	−0.0754133
$873$	0	0
$874$	30.4537	1.03011
$875$	10.0096	0.338386
$876$	0	0
$877$	43.4068	1.46574	0.732871	−	0.680367i	$-0.238180\pi$
$877$	43.4068	1.46574	0.732871	+	0.680367i	$0.238180\pi$
$878$	17.2951	0.583680
$879$	0	0
$880$	−3.63156	−0.122420
$881$	−16.6130	−0.559706	−0.279853	−	0.960043i	$-0.590286\pi$
$881$	−16.6130	−0.559706	−0.279853	+	0.960043i	$0.590286\pi$
$882$	0	0
$883$	41.3726	1.39230	0.696149	−	0.717897i	$-0.254895\pi$
$883$	41.3726	1.39230	0.696149	+	0.717897i	$0.254895\pi$
$884$	3.64232	0.122505
$885$	0	0
$886$	−22.8314	−0.767037
$887$	−14.7151	−0.494085	−0.247042	−	0.969005i	$-0.579459\pi$
$887$	−14.7151	−0.494085	−0.247042	+	0.969005i	$0.579459\pi$
$888$	0	0
$889$	68.5864	2.30031
$890$	19.5617	0.655709
$891$	0	0
$892$	−1.00000	−0.0334825
$893$	7.35803	0.246227
$894$	0	0
$895$	55.8701	1.86753
$896$	−3.77138	−0.125993
$897$	0	0
$898$	−28.4260	−0.948587
$899$	−4.75416	−0.158560
$900$	0	0
$901$	27.4727	0.915249
$902$	6.71967	0.223741
$903$	0	0
$904$	11.0410	0.367219
$905$	58.1390	1.93260
$906$	0	0
$907$	−21.9360	−0.728374	−0.364187	−	0.931326i	$-0.618653\pi$
$907$	−21.9360	−0.728374	−0.364187	+	0.931326i	$0.618653\pi$
$908$	−18.9100	−0.627549
$909$	0	0
$910$	11.9174	0.395058
$911$	30.5867	1.01338	0.506691	−	0.862128i	$-0.330869\pi$
$911$	30.5867	1.01338	0.506691	+	0.862128i	$0.330869\pi$
$912$	0	0
$913$	−2.54462	−0.0842146
$914$	−4.12755	−0.136527
$915$	0	0
$916$	−9.36817	−0.309533
$917$	52.2544	1.72559
$918$	0	0
$919$	−45.1531	−1.48946	−0.744732	−	0.667363i	$-0.767423\pi$
$919$	−45.1531	−1.48946	−0.744732	+	0.667363i	$0.767423\pi$
$920$	−20.9541	−0.690835
$921$	0	0
$922$	23.7885	0.783432
$923$	2.79282	0.0919268
$924$	0	0
$925$	−24.4516	−0.803962
$926$	−5.25289	−0.172621
$927$	0	0
$928$	1.44714	0.0475046
$929$	2.38827	0.0783565	0.0391782	−	0.999232i	$-0.487526\pi$
$929$	2.38827	0.0783565	0.0391782	+	0.999232i	$0.487526\pi$
$930$	0	0
$931$	−31.7056	−1.03911
$932$	22.4238	0.734517
$933$	0	0
$934$	−28.3361	−0.927186
$935$	12.6421	0.413439
$936$	0	0
$937$	−31.0450	−1.01419	−0.507097	−	0.861889i	$-0.669282\pi$
$937$	−31.0450	−1.01419	−0.507097	+	0.861889i	$0.669282\pi$
$938$	−29.0139	−0.947338
$939$	0	0
$940$	−5.06279	−0.165130
$941$	−15.1571	−0.494106	−0.247053	−	0.969002i	$-0.579462\pi$
$941$	−15.1571	−0.494106	−0.247053	+	0.969002i	$0.579462\pi$
$942$	0	0
$943$	38.7724	1.26260
$944$	−13.9770	−0.454914
$945$	0	0
$946$	−11.5595	−0.375831
$947$	40.5902	1.31900	0.659502	−	0.751703i	$-0.270767\pi$
$947$	40.5902	1.31900	0.659502	+	0.751703i	$0.270767\pi$
$948$	0	0
$949$	−3.64913	−0.118456
$950$	−18.0893	−0.586894
$951$	0	0
$952$	13.1288	0.425507
$953$	9.71509	0.314703	0.157351	−	0.987543i	$-0.449705\pi$
$953$	9.71509	0.314703	0.157351	+	0.987543i	$0.449705\pi$
$954$	0	0
$955$	−39.9303	−1.29212
$956$	−12.7281	−0.411657
$957$	0	0
$958$	−12.1657	−0.393056
$959$	51.3408	1.65788
$960$	0	0
$961$	−20.2074	−0.651850
$962$	6.20780	0.200147
$963$	0	0
$964$	−14.7087	−0.473736
$965$	38.3588	1.23481
$966$	0	0
$967$	23.7630	0.764166	0.382083	−	0.924128i	$-0.375207\pi$
$967$	23.7630	0.764166	0.382083	+	0.924128i	$0.375207\pi$
$968$	−9.55411	−0.307081
$969$	0	0
$970$	24.8100	0.796600
$971$	−9.20942	−0.295544	−0.147772	−	0.989021i	$-0.547210\pi$
$971$	−9.20942	−0.295544	−0.147772	+	0.989021i	$0.547210\pi$
$972$	0	0
$973$	−41.6844	−1.33634
$974$	−18.2502	−0.584773
$975$	0	0
$976$	10.0775	0.322573
$977$	−22.2339	−0.711325	−0.355662	−	0.934615i	$-0.615745\pi$
$977$	−22.2339	−0.711325	−0.355662	+	0.934615i	$0.615745\pi$
$978$	0	0
$979$	−7.78840	−0.248918
$980$	21.8154	0.696869
$981$	0	0
$982$	5.52096	0.176181
$983$	−22.8407	−0.728506	−0.364253	−	0.931300i	$-0.618676\pi$
$983$	−22.8407	−0.728506	−0.364253	+	0.931300i	$0.618676\pi$
$984$	0	0
$985$	20.5483	0.654724
$986$	−5.03771	−0.160433
$987$	0	0
$988$	4.59253	0.146108
$989$	−66.6980	−2.12087
$990$	0	0
$991$	30.1784	0.958647	0.479323	−	0.877638i	$-0.340882\pi$
$991$	30.1784	0.958647	0.479323	+	0.877638i	$0.340882\pi$
$992$	−3.28522	−0.104306
$993$	0	0
$994$	10.0667	0.319298
$995$	−55.5511	−1.76109
$996$	0	0
$997$	15.3755	0.486947	0.243473	−	0.969908i	$-0.421713\pi$
$997$	15.3755	0.486947	0.243473	+	0.969908i	$0.421713\pi$
$998$	32.2553	1.02102
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4014.2.a.r.1.5		5
3.2	odd	2		1338.2.a.h.1.1	✓	5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1338.2.a.h.1.1	✓	5	3.2	odd	2
4014.2.a.r.1.5		5	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} +3.02013 q^{5} -3.77138 q^{7} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} +3.02013 q^{5} -3.77138 q^{7} +1.00000 q^{8} +3.02013 q^{10} -1.20245 q^{11} -1.04630 q^{13} -3.77138 q^{14} +1.00000 q^{16} -3.48116 q^{17} -4.38932 q^{19} +3.02013 q^{20} -1.20245 q^{22} -6.93812 q^{23} +4.12120 q^{25} -1.04630 q^{26} -3.77138 q^{28} +1.44714 q^{29} -3.28522 q^{31} +1.00000 q^{32} -3.48116 q^{34} -11.3901 q^{35} -5.93312 q^{37} -4.38932 q^{38} +3.02013 q^{40} -5.58831 q^{41} +9.61326 q^{43} -1.20245 q^{44} -6.93812 q^{46} -1.67635 q^{47} +7.22334 q^{49} +4.12120 q^{50} -1.04630 q^{52} -7.89183 q^{53} -3.63156 q^{55} -3.77138 q^{56} +1.44714 q^{58} -13.9770 q^{59} +10.0775 q^{61} -3.28522 q^{62} +1.00000 q^{64} -3.15995 q^{65} +7.69317 q^{67} -3.48116 q^{68} -11.3901 q^{70} -2.66924 q^{71} +3.48767 q^{73} -5.93312 q^{74} -4.38932 q^{76} +4.53491 q^{77} -7.37800 q^{79} +3.02013 q^{80} -5.58831 q^{82} +2.11619 q^{83} -10.5136 q^{85} +9.61326 q^{86} -1.20245 q^{88} +6.47710 q^{89} +3.94598 q^{91} -6.93812 q^{92} -1.67635 q^{94} -13.2563 q^{95} +8.21487 q^{97} +7.22334 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 5 q^{2} + 5 q^{4} - 5 q^{5} - q^{7} + 5 q^{8}+O(q^{10}) \) 5 * q + 5 * q^2 + 5 * q^4 - 5 * q^5 - q^7 + 5 * q^8	\( 5 q + 5 q^{2} + 5 q^{4} - 5 q^{5} - q^{7} + 5 q^{8} - 5 q^{10} - 9 q^{11} - q^{14} + 5 q^{16} - 6 q^{17} - 4 q^{19} - 5 q^{20} - 9 q^{22} - 16 q^{23} + 8 q^{25} - q^{28} - 8 q^{29} - q^{31} + 5 q^{32} - 6 q^{34} - 22 q^{35} - 2 q^{37} - 4 q^{38} - 5 q^{40} - 4 q^{41} + 3 q^{43} - 9 q^{44} - 16 q^{46} - 18 q^{47} + 2 q^{49} + 8 q^{50} - 26 q^{53} + q^{55} - q^{56} - 8 q^{58} - 21 q^{59} - 20 q^{61} - q^{62} + 5 q^{64} + 3 q^{65} - 5 q^{67} - 6 q^{68} - 22 q^{70} - 17 q^{71} + 5 q^{73} - 2 q^{74} - 4 q^{76} - 2 q^{77} - 21 q^{79} - 5 q^{80} - 4 q^{82} - 11 q^{83} - 12 q^{85} + 3 q^{86} - 9 q^{88} + 5 q^{89} - 10 q^{91} - 16 q^{92} - 18 q^{94} - 10 q^{95} - 11 q^{97} + 2 q^{98}+O(q^{100}) \) 5 * q + 5 * q^2 + 5 * q^4 - 5 * q^5 - q^7 + 5 * q^8 - 5 * q^10 - 9 * q^11 - q^14 + 5 * q^16 - 6 * q^17 - 4 * q^19 - 5 * q^20 - 9 * q^22 - 16 * q^23 + 8 * q^25 - q^28 - 8 * q^29 - q^31 + 5 * q^32 - 6 * q^34 - 22 * q^35 - 2 * q^37 - 4 * q^38 - 5 * q^40 - 4 * q^41 + 3 * q^43 - 9 * q^44 - 16 * q^46 - 18 * q^47 + 2 * q^49 + 8 * q^50 - 26 * q^53 + q^55 - q^56 - 8 * q^58 - 21 * q^59 - 20 * q^61 - q^62 + 5 * q^64 + 3 * q^65 - 5 * q^67 - 6 * q^68 - 22 * q^70 - 17 * q^71 + 5 * q^73 - 2 * q^74 - 4 * q^76 - 2 * q^77 - 21 * q^79 - 5 * q^80 - 4 * q^82 - 11 * q^83 - 12 * q^85 + 3 * q^86 - 9 * q^88 + 5 * q^89 - 10 * q^91 - 16 * q^92 - 18 * q^94 - 10 * q^95 - 11 * q^97 + 2 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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