LMFDB - Embedded newform 8046.2.a.p.1.12

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8046,2,Mod(1,8046)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8046, 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("8046.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8046 = 2 \cdot 3^{3} \cdot 149 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8046.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:	$64.2476334663$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 5 x^{11} - 23 x^{10} + 142 x^{9} + 104 x^{8} - 1302 x^{7} + 607 x^{6} + 4323 x^{5} - 4461 x^{4} + \cdots - 553 $ x^12 - 5x^11 - 23x^10 + 142x^9 + 104x^8 - 1302x^7 + 607x^6 + 4323x^5 - 4461x^4 - 3333x^3 + 4805x^2 - 224*x - 553
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.12
Root			$3.89381$ of defining polynomial
Character	$\chi$	$=$	8046.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} +3.89381 q^{5} +0.649105 q^{7} +1.00000 q^{8} +O(q^{10})$	\(q+1.00000 q^{2} +1.00000 q^{4} +3.89381 q^{5} +0.649105 q^{7} +1.00000 q^{8} +3.89381 q^{10} +1.54726 q^{11} +0.865551 q^{13} +0.649105 q^{14} +1.00000 q^{16} +3.39744 q^{17} +1.84012 q^{19} +3.89381 q^{20} +1.54726 q^{22} -3.92153 q^{23} +10.1617 q^{25} +0.865551 q^{26} +0.649105 q^{28} +2.02618 q^{29} +5.05054 q^{31} +1.00000 q^{32} +3.39744 q^{34} +2.52749 q^{35} -5.61860 q^{37} +1.84012 q^{38} +3.89381 q^{40} +3.49270 q^{41} -4.91992 q^{43} +1.54726 q^{44} -3.92153 q^{46} +9.06074 q^{47} -6.57866 q^{49} +10.1617 q^{50} +0.865551 q^{52} -13.6906 q^{53} +6.02474 q^{55} +0.649105 q^{56} +2.02618 q^{58} +0.861320 q^{59} -6.49612 q^{61} +5.05054 q^{62} +1.00000 q^{64} +3.37029 q^{65} +10.7574 q^{67} +3.39744 q^{68} +2.52749 q^{70} -0.839750 q^{71} +9.60226 q^{73} -5.61860 q^{74} +1.84012 q^{76} +1.00434 q^{77} +11.5731 q^{79} +3.89381 q^{80} +3.49270 q^{82} -14.0794 q^{83} +13.2290 q^{85} -4.91992 q^{86} +1.54726 q^{88} -10.0233 q^{89} +0.561833 q^{91} -3.92153 q^{92} +9.06074 q^{94} +7.16507 q^{95} +6.98257 q^{97} -6.57866 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 12 q^{2} + 12 q^{4} + 5 q^{5} + 6 q^{7} + 12 q^{8}+O(q^{10}) $ 12 * q + 12 * q^2 + 12 * q^4 + 5 * q^5 + 6 * q^7 + 12 * q^8	$ 12 q + 12 q^{2} + 12 q^{4} + 5 q^{5} + 6 q^{7} + 12 q^{8} + 5 q^{10} + 6 q^{11} + 3 q^{13} + 6 q^{14} + 12 q^{16} + 6 q^{17} + 8 q^{19} + 5 q^{20} + 6 q^{22} + 11 q^{23} + 11 q^{25} + 3 q^{26} + 6 q^{28} + 29 q^{29} + 2 q^{31} + 12 q^{32} + 6 q^{34} + 4 q^{35} + 5 q^{37} + 8 q^{38} + 5 q^{40} + 22 q^{41} + 9 q^{43} + 6 q^{44} + 11 q^{46} + 15 q^{47} + 14 q^{49} + 11 q^{50} + 3 q^{52} + 12 q^{53} + 13 q^{55} + 6 q^{56} + 29 q^{58} + 34 q^{59} - 4 q^{61} + 2 q^{62} + 12 q^{64} + 12 q^{65} + q^{67} + 6 q^{68} + 4 q^{70} + 21 q^{71} - 2 q^{73} + 5 q^{74} + 8 q^{76} + 34 q^{77} + 9 q^{79} + 5 q^{80} + 22 q^{82} + 10 q^{83} + 5 q^{85} + 9 q^{86} + 6 q^{88} - 2 q^{89} + 17 q^{91} + 11 q^{92} + 15 q^{94} + 69 q^{95} - 13 q^{97} + 14 q^{98}+O(q^{100}) $ 12 * q + 12 * q^2 + 12 * q^4 + 5 * q^5 + 6 * q^7 + 12 * q^8 + 5 * q^10 + 6 * q^11 + 3 * q^13 + 6 * q^14 + 12 * q^16 + 6 * q^17 + 8 * q^19 + 5 * q^20 + 6 * q^22 + 11 * q^23 + 11 * q^25 + 3 * q^26 + 6 * q^28 + 29 * q^29 + 2 * q^31 + 12 * q^32 + 6 * q^34 + 4 * q^35 + 5 * q^37 + 8 * q^38 + 5 * q^40 + 22 * q^41 + 9 * q^43 + 6 * q^44 + 11 * q^46 + 15 * q^47 + 14 * q^49 + 11 * q^50 + 3 * q^52 + 12 * q^53 + 13 * q^55 + 6 * q^56 + 29 * q^58 + 34 * q^59 - 4 * q^61 + 2 * q^62 + 12 * q^64 + 12 * q^65 + q^67 + 6 * q^68 + 4 * q^70 + 21 * q^71 - 2 * q^73 + 5 * q^74 + 8 * q^76 + 34 * q^77 + 9 * q^79 + 5 * q^80 + 22 * q^82 + 10 * q^83 + 5 * q^85 + 9 * q^86 + 6 * q^88 - 2 * q^89 + 17 * q^91 + 11 * q^92 + 15 * q^94 + 69 * q^95 - 13 * 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
$5$	3.89381	1.74136	0.870681	−	0.491848i	$-0.163678\pi$
$5$	3.89381	1.74136	0.870681	+	0.491848i	$0.163678\pi$
$6$	0	0
$7$	0.649105	0.245338	0.122669	−	0.992448i	$-0.460855\pi$
$7$	0.649105	0.245338	0.122669	+	0.992448i	$0.460855\pi$
$8$	1.00000	0.353553
$9$	0	0
$10$	3.89381	1.23133
$11$	1.54726	0.466517	0.233259	−	0.972415i	$-0.425061\pi$
$11$	1.54726	0.466517	0.233259	+	0.972415i	$0.425061\pi$
$12$	0	0
$13$	0.865551	0.240061	0.120030	−	0.992770i	$-0.461701\pi$
$13$	0.865551	0.240061	0.120030	+	0.992770i	$0.461701\pi$
$14$	0.649105	0.173481
$15$	0	0
$16$	1.00000	0.250000
$17$	3.39744	0.824000	0.412000	−	0.911184i	$-0.364830\pi$
$17$	3.39744	0.824000	0.412000	+	0.911184i	$0.364830\pi$
$18$	0	0
$19$	1.84012	0.422153	0.211076	−	0.977470i	$-0.432303\pi$
$19$	1.84012	0.422153	0.211076	+	0.977470i	$0.432303\pi$
$20$	3.89381	0.870681
$21$	0	0
$22$	1.54726	0.329877
$23$	−3.92153	−0.817696	−0.408848	−	0.912602i	$-0.634069\pi$
$23$	−3.92153	−0.817696	−0.408848	+	0.912602i	$0.634069\pi$
$24$	0	0
$25$	10.1617	2.03234
$26$	0.865551	0.169749
$27$	0	0
$28$	0.649105	0.122669
$29$	2.02618	0.376252	0.188126	−	0.982145i	$-0.439759\pi$
$29$	2.02618	0.376252	0.188126	+	0.982145i	$0.439759\pi$
$30$	0	0
$31$	5.05054	0.907104	0.453552	−	0.891230i	$-0.350157\pi$
$31$	5.05054	0.907104	0.453552	+	0.891230i	$0.350157\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	3.39744	0.582656
$35$	2.52749	0.427223
$36$	0	0
$37$	−5.61860	−0.923692	−0.461846	−	0.886960i	$-0.652813\pi$
$37$	−5.61860	−0.923692	−0.461846	+	0.886960i	$0.652813\pi$
$38$	1.84012	0.298507
$39$	0	0
$40$	3.89381	0.615665
$41$	3.49270	0.545468	0.272734	−	0.962089i	$-0.412072\pi$
$41$	3.49270	0.545468	0.272734	+	0.962089i	$0.412072\pi$
$42$	0	0
$43$	−4.91992	−0.750281	−0.375141	−	0.926968i	$-0.622406\pi$
$43$	−4.91992	−0.750281	−0.375141	+	0.926968i	$0.622406\pi$
$44$	1.54726	0.233259
$45$	0	0
$46$	−3.92153	−0.578198
$47$	9.06074	1.32164	0.660822	−	0.750542i	$-0.270208\pi$
$47$	9.06074	1.32164	0.660822	+	0.750542i	$0.270208\pi$
$48$	0	0
$49$	−6.57866	−0.939809
$50$	10.1617	1.43708
$51$	0	0
$52$	0.865551	0.120030
$53$	−13.6906	−1.88055	−0.940274	−	0.340418i	$-0.889431\pi$
$53$	−13.6906	−1.88055	−0.940274	+	0.340418i	$0.889431\pi$
$54$	0	0
$55$	6.02474	0.812375
$56$	0.649105	0.0867403
$57$	0	0
$58$	2.02618	0.266050
$59$	0.861320	0.112134	0.0560672	−	0.998427i	$-0.482144\pi$
$59$	0.861320	0.112134	0.0560672	+	0.998427i	$0.482144\pi$
$60$	0	0
$61$	−6.49612	−0.831743	−0.415872	−	0.909423i	$-0.636523\pi$
$61$	−6.49612	−0.831743	−0.415872	+	0.909423i	$0.636523\pi$
$62$	5.05054	0.641420
$63$	0	0
$64$	1.00000	0.125000
$65$	3.37029	0.418033
$66$	0	0
$67$	10.7574	1.31422	0.657110	−	0.753794i	$-0.271779\pi$
$67$	10.7574	1.31422	0.657110	+	0.753794i	$0.271779\pi$
$68$	3.39744	0.412000
$69$	0	0
$70$	2.52749	0.302092
$71$	−0.839750	−0.0996599	−0.0498300	−	0.998758i	$-0.515868\pi$
$71$	−0.839750	−0.0996599	−0.0498300	+	0.998758i	$0.515868\pi$
$72$	0	0
$73$	9.60226	1.12386	0.561930	−	0.827185i	$-0.310059\pi$
$73$	9.60226	1.12386	0.561930	+	0.827185i	$0.310059\pi$
$74$	−5.61860	−0.653149
$75$	0	0
$76$	1.84012	0.211076
$77$	1.00434	0.114455
$78$	0	0
$79$	11.5731	1.30208	0.651039	−	0.759044i	$-0.274333\pi$
$79$	11.5731	1.30208	0.651039	+	0.759044i	$0.274333\pi$
$80$	3.89381	0.435341
$81$	0	0
$82$	3.49270	0.385704
$83$	−14.0794	−1.54542	−0.772708	−	0.634762i	$-0.781098\pi$
$83$	−14.0794	−1.54542	−0.772708	+	0.634762i	$0.781098\pi$
$84$	0	0
$85$	13.2290	1.43488
$86$	−4.91992	−0.530529
$87$	0	0
$88$	1.54726	0.164939
$89$	−10.0233	−1.06247	−0.531236	−	0.847224i	$-0.678272\pi$
$89$	−10.0233	−1.06247	−0.531236	+	0.847224i	$0.678272\pi$
$90$	0	0
$91$	0.561833	0.0588961
$92$	−3.92153	−0.408848
$93$	0	0
$94$	9.06074	0.934544
$95$	7.16507	0.735121
$96$	0	0
$97$	6.98257	0.708973	0.354486	−	0.935061i	$-0.384656\pi$
$97$	6.98257	0.708973	0.354486	+	0.935061i	$0.384656\pi$
$98$	−6.57866	−0.664545
$99$	0	0
$100$	10.1617	1.01617
$101$	−11.9644	−1.19050	−0.595251	−	0.803539i	$-0.702948\pi$
$101$	−11.9644	−1.19050	−0.595251	+	0.803539i	$0.702948\pi$
$102$	0	0
$103$	−19.8650	−1.95736	−0.978680	−	0.205392i	$-0.934153\pi$
$103$	−19.8650	−1.95736	−0.978680	+	0.205392i	$0.934153\pi$
$104$	0.865551	0.0848743
$105$	0	0
$106$	−13.6906	−1.32975
$107$	13.7141	1.32579	0.662894	−	0.748713i	$-0.269328\pi$
$107$	13.7141	1.32579	0.662894	+	0.748713i	$0.269328\pi$
$108$	0	0
$109$	6.65657	0.637584	0.318792	−	0.947825i	$-0.396723\pi$
$109$	6.65657	0.637584	0.318792	+	0.947825i	$0.396723\pi$
$110$	6.02474	0.574436
$111$	0	0
$112$	0.649105	0.0613346
$113$	−10.7995	−1.01593	−0.507964	−	0.861378i	$-0.669602\pi$
$113$	−10.7995	−1.01593	−0.507964	+	0.861378i	$0.669602\pi$
$114$	0	0
$115$	−15.2697	−1.42391
$116$	2.02618	0.188126
$117$	0	0
$118$	0.861320	0.0792910
$119$	2.20529	0.202159
$120$	0	0
$121$	−8.60598	−0.782362
$122$	−6.49612	−0.588131
$123$	0	0
$124$	5.05054	0.453552
$125$	20.0987	1.79769
$126$	0	0
$127$	−2.45779	−0.218093	−0.109047	−	0.994037i	$-0.534780\pi$
$127$	−2.45779	−0.218093	−0.109047	+	0.994037i	$0.534780\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	3.37029	0.295594
$131$	−7.72291	−0.674754	−0.337377	−	0.941370i	$-0.609540\pi$
$131$	−7.72291	−0.674754	−0.337377	+	0.941370i	$0.609540\pi$
$132$	0	0
$133$	1.19443	0.103570
$134$	10.7574	0.929294
$135$	0	0
$136$	3.39744	0.291328
$137$	9.93733	0.849004	0.424502	−	0.905427i	$-0.360449\pi$
$137$	9.93733	0.849004	0.424502	+	0.905427i	$0.360449\pi$
$138$	0	0
$139$	11.2968	0.958178	0.479089	−	0.877766i	$-0.340967\pi$
$139$	11.2968	0.958178	0.479089	+	0.877766i	$0.340967\pi$
$140$	2.52749	0.213612
$141$	0	0
$142$	−0.839750	−0.0704702
$143$	1.33923	0.111992
$144$	0	0
$145$	7.88954	0.655191
$146$	9.60226	0.794688
$147$	0	0
$148$	−5.61860	−0.461846
$149$	−1.00000	−0.0819232
$149$	−1.00000	−0.0819232
$150$	0	0
$151$	−15.5422	−1.26480	−0.632402	−	0.774641i	$-0.717931\pi$
$151$	−15.5422	−1.26480	−0.632402	+	0.774641i	$0.717931\pi$
$152$	1.84012	0.149253
$153$	0	0
$154$	1.00434	0.0809316
$155$	19.6658	1.57960
$156$	0	0
$157$	20.2751	1.61813	0.809063	−	0.587722i	$-0.199975\pi$
$157$	20.2751	1.61813	0.809063	+	0.587722i	$0.199975\pi$
$158$	11.5731	0.920709
$159$	0	0
$160$	3.89381	0.307832
$161$	−2.54549	−0.200612
$162$	0	0
$163$	3.07431	0.240799	0.120399	−	0.992726i	$-0.461582\pi$
$163$	3.07431	0.240799	0.120399	+	0.992726i	$0.461582\pi$
$164$	3.49270	0.272734
$165$	0	0
$166$	−14.0794	−1.09277
$167$	0.216671	0.0167665	0.00838324	−	0.999965i	$-0.497332\pi$
$167$	0.216671	0.0167665	0.00838324	+	0.999965i	$0.497332\pi$
$168$	0	0
$169$	−12.2508	−0.942371
$170$	13.2290	1.01462
$171$	0	0
$172$	−4.91992	−0.375141
$173$	17.6531	1.34214	0.671071	−	0.741393i	$-0.265834\pi$
$173$	17.6531	1.34214	0.671071	+	0.741393i	$0.265834\pi$
$174$	0	0
$175$	6.59602	0.498612
$176$	1.54726	0.116629
$177$	0	0
$178$	−10.0233	−0.751281
$179$	5.95874	0.445377	0.222688	−	0.974890i	$-0.428517\pi$
$179$	5.95874	0.445377	0.222688	+	0.974890i	$0.428517\pi$
$180$	0	0
$181$	0.631864	0.0469661	0.0234830	−	0.999724i	$-0.492524\pi$
$181$	0.631864	0.0469661	0.0234830	+	0.999724i	$0.492524\pi$
$182$	0.561833	0.0416458
$183$	0	0
$184$	−3.92153	−0.289099
$185$	−21.8777	−1.60848
$186$	0	0
$187$	5.25673	0.384410
$188$	9.06074	0.660822
$189$	0	0
$190$	7.16507	0.519809
$191$	21.1967	1.53374	0.766868	−	0.641805i	$-0.221814\pi$
$191$	21.1967	1.53374	0.766868	+	0.641805i	$0.221814\pi$
$192$	0	0
$193$	−17.4165	−1.25367	−0.626834	−	0.779153i	$-0.715649\pi$
$193$	−17.4165	−1.25367	−0.626834	+	0.779153i	$0.715649\pi$
$194$	6.98257	0.501320
$195$	0	0
$196$	−6.57866	−0.469905
$197$	−19.4588	−1.38638	−0.693191	−	0.720754i	$-0.743796\pi$
$197$	−19.4588	−1.38638	−0.693191	+	0.720754i	$0.743796\pi$
$198$	0	0
$199$	18.7195	1.32699	0.663497	−	0.748179i	$-0.269072\pi$
$199$	18.7195	1.32699	0.663497	+	0.748179i	$0.269072\pi$
$200$	10.1617	0.718542
$201$	0	0
$202$	−11.9644	−0.841813
$203$	1.31520	0.0923091
$204$	0	0
$205$	13.5999	0.949857
$206$	−19.8650	−1.38406
$207$	0	0
$208$	0.865551	0.0600152
$209$	2.84715	0.196941
$210$	0	0
$211$	6.00013	0.413066	0.206533	−	0.978440i	$-0.433782\pi$
$211$	6.00013	0.413066	0.206533	+	0.978440i	$0.433782\pi$
$212$	−13.6906	−0.940274
$213$	0	0
$214$	13.7141	0.937473
$215$	−19.1572	−1.30651
$216$	0	0
$217$	3.27833	0.222548
$218$	6.65657	0.450840
$219$	0	0
$220$	6.02474	0.406188
$221$	2.94066	0.197810
$222$	0	0
$223$	−7.48223	−0.501047	−0.250524	−	0.968111i	$-0.580603\pi$
$223$	−7.48223	−0.501047	−0.250524	+	0.968111i	$0.580603\pi$
$224$	0.649105	0.0433701
$225$	0	0
$226$	−10.7995	−0.718370
$227$	1.86337	0.123676	0.0618382	−	0.998086i	$-0.480304\pi$
$227$	1.86337	0.123676	0.0618382	+	0.998086i	$0.480304\pi$
$228$	0	0
$229$	−8.98233	−0.593569	−0.296784	−	0.954945i	$-0.595914\pi$
$229$	−8.98233	−0.593569	−0.296784	+	0.954945i	$0.595914\pi$
$230$	−15.2697	−1.00685
$231$	0	0
$232$	2.02618	0.133025
$233$	−21.0086	−1.37632	−0.688161	−	0.725558i	$-0.741582\pi$
$233$	−21.0086	−1.37632	−0.688161	+	0.725558i	$0.741582\pi$
$234$	0	0
$235$	35.2807	2.30146
$236$	0.861320	0.0560672
$237$	0	0
$238$	2.20529	0.142948
$239$	−11.4102	−0.738065	−0.369033	−	0.929416i	$-0.620311\pi$
$239$	−11.4102	−0.738065	−0.369033	+	0.929416i	$0.620311\pi$
$240$	0	0
$241$	−23.9632	−1.54360	−0.771802	−	0.635862i	$-0.780645\pi$
$241$	−23.9632	−1.54360	−0.771802	+	0.635862i	$0.780645\pi$
$242$	−8.60598	−0.553213
$243$	0	0
$244$	−6.49612	−0.415872
$245$	−25.6160	−1.63655
$246$	0	0
$247$	1.59272	0.101342
$248$	5.05054	0.320710
$249$	0	0
$250$	20.0987	1.27116
$251$	30.3598	1.91629	0.958147	−	0.286275i	$-0.0924171\pi$
$251$	30.3598	1.91629	0.958147	+	0.286275i	$0.0924171\pi$
$252$	0	0
$253$	−6.06764	−0.381469
$254$	−2.45779	−0.154215
$255$	0	0
$256$	1.00000	0.0625000
$257$	8.82453	0.550459	0.275229	−	0.961379i	$-0.411246\pi$
$257$	8.82453	0.550459	0.275229	+	0.961379i	$0.411246\pi$
$258$	0	0
$259$	−3.64706	−0.226617
$260$	3.37029	0.209016
$261$	0	0
$262$	−7.72291	−0.477123
$263$	15.1382	0.933459	0.466730	−	0.884400i	$-0.345432\pi$
$263$	15.1382	0.933459	0.466730	+	0.884400i	$0.345432\pi$
$264$	0	0
$265$	−53.3085	−3.27472
$266$	1.19443	0.0732353
$267$	0	0
$268$	10.7574	0.657110
$269$	1.28935	0.0786131	0.0393066	−	0.999227i	$-0.487485\pi$
$269$	1.28935	0.0786131	0.0393066	+	0.999227i	$0.487485\pi$
$270$	0	0
$271$	5.99315	0.364058	0.182029	−	0.983293i	$-0.441734\pi$
$271$	5.99315	0.364058	0.182029	+	0.983293i	$0.441734\pi$
$272$	3.39744	0.206000
$273$	0	0
$274$	9.93733	0.600336
$275$	15.7228	0.948123
$276$	0	0
$277$	−23.0526	−1.38509	−0.692547	−	0.721373i	$-0.743512\pi$
$277$	−23.0526	−1.38509	−0.692547	+	0.721373i	$0.743512\pi$
$278$	11.2968	0.677535
$279$	0	0
$280$	2.52749	0.151046
$281$	13.6179	0.812377	0.406188	−	0.913789i	$-0.366858\pi$
$281$	13.6179	0.812377	0.406188	+	0.913789i	$0.366858\pi$
$282$	0	0
$283$	10.2171	0.607346	0.303673	−	0.952776i	$-0.401787\pi$
$283$	10.2171	0.607346	0.303673	+	0.952776i	$0.401787\pi$
$284$	−0.839750	−0.0498300
$285$	0	0
$286$	1.33923	0.0791906
$287$	2.26713	0.133824
$288$	0	0
$289$	−5.45742	−0.321024
$290$	7.88954	0.463290
$291$	0	0
$292$	9.60226	0.561930
$293$	4.57955	0.267540	0.133770	−	0.991012i	$-0.457292\pi$
$293$	4.57955	0.267540	0.133770	+	0.991012i	$0.457292\pi$
$294$	0	0
$295$	3.35381	0.195267
$296$	−5.61860	−0.326575
$297$	0	0
$298$	−1.00000	−0.0579284
$299$	−3.39429	−0.196297
$300$	0	0
$301$	−3.19354	−0.184073
$302$	−15.5422	−0.894351
$303$	0	0
$304$	1.84012	0.105538
$305$	−25.2946	−1.44837
$306$	0	0
$307$	−6.89999	−0.393803	−0.196902	−	0.980423i	$-0.563088\pi$
$307$	−6.89999	−0.393803	−0.196902	+	0.980423i	$0.563088\pi$
$308$	1.00434	0.0572273
$309$	0	0
$310$	19.6658	1.11694
$311$	−11.0770	−0.628121	−0.314061	−	0.949403i	$-0.601689\pi$
$311$	−11.0770	−0.628121	−0.314061	+	0.949403i	$0.601689\pi$
$312$	0	0
$313$	−5.18106	−0.292851	−0.146426	−	0.989222i	$-0.546777\pi$
$313$	−5.18106	−0.292851	−0.146426	+	0.989222i	$0.546777\pi$
$314$	20.2751	1.14419
$315$	0	0
$316$	11.5731	0.651039
$317$	−0.540504	−0.0303577	−0.0151789	−	0.999885i	$-0.504832\pi$
$317$	−0.540504	−0.0303577	−0.0151789	+	0.999885i	$0.504832\pi$
$318$	0	0
$319$	3.13503	0.175528
$320$	3.89381	0.217670
$321$	0	0
$322$	−2.54549	−0.141854
$323$	6.25170	0.347854
$324$	0	0
$325$	8.79549	0.487886
$326$	3.07431	0.170271
$327$	0	0
$328$	3.49270	0.192852
$329$	5.88137	0.324250
$330$	0	0
$331$	−27.0454	−1.48655	−0.743275	−	0.668986i	$-0.766729\pi$
$331$	−27.0454	−1.48655	−0.743275	+	0.668986i	$0.766729\pi$
$332$	−14.0794	−0.772708
$333$	0	0
$334$	0.216671	0.0118557
$335$	41.8871	2.28853
$336$	0	0
$337$	−17.3293	−0.943986	−0.471993	−	0.881602i	$-0.656465\pi$
$337$	−17.3293	−0.943986	−0.471993	+	0.881602i	$0.656465\pi$
$338$	−12.2508	−0.666357
$339$	0	0
$340$	13.2290	0.717441
$341$	7.81451	0.423180
$342$	0	0
$343$	−8.81397	−0.475910
$344$	−4.91992	−0.265264
$345$	0	0
$346$	17.6531	0.949038
$347$	−9.31515	−0.500064	−0.250032	−	0.968238i	$-0.580441\pi$
$347$	−9.31515	−0.500064	−0.250032	+	0.968238i	$0.580441\pi$
$348$	0	0
$349$	22.7466	1.21760	0.608798	−	0.793325i	$-0.291652\pi$
$349$	22.7466	1.21760	0.608798	+	0.793325i	$0.291652\pi$
$350$	6.59602	0.352572
$351$	0	0
$352$	1.54726	0.0824693
$353$	−11.1814	−0.595126	−0.297563	−	0.954702i	$-0.596174\pi$
$353$	−11.1814	−0.595126	−0.297563	+	0.954702i	$0.596174\pi$
$354$	0	0
$355$	−3.26982	−0.173544
$356$	−10.0233	−0.531236
$357$	0	0
$358$	5.95874	0.314929
$359$	−16.7948	−0.886394	−0.443197	−	0.896424i	$-0.646156\pi$
$359$	−16.7948	−0.886394	−0.443197	+	0.896424i	$0.646156\pi$
$360$	0	0
$361$	−15.6140	−0.821787
$362$	0.631864	0.0332100
$363$	0	0
$364$	0.561833	0.0294481
$365$	37.3893	1.95705
$366$	0	0
$367$	−16.2821	−0.849921	−0.424961	−	0.905212i	$-0.639712\pi$
$367$	−16.2821	−0.849921	−0.424961	+	0.905212i	$0.639712\pi$
$368$	−3.92153	−0.204424
$369$	0	0
$370$	−21.8777	−1.13737
$371$	−8.88663	−0.461371
$372$	0	0
$373$	−31.4387	−1.62784	−0.813918	−	0.580980i	$-0.802669\pi$
$373$	−31.4387	−1.62784	−0.813918	+	0.580980i	$0.802669\pi$
$374$	5.25673	0.271819
$375$	0	0
$376$	9.06074	0.467272
$377$	1.75376	0.0903233
$378$	0	0
$379$	−6.87282	−0.353033	−0.176517	−	0.984298i	$-0.556483\pi$
$379$	−6.87282	−0.353033	−0.176517	+	0.984298i	$0.556483\pi$
$380$	7.16507	0.367560
$381$	0	0
$382$	21.1967	1.08452
$383$	14.3147	0.731448	0.365724	−	0.930723i	$-0.380821\pi$
$383$	14.3147	0.731448	0.365724	+	0.930723i	$0.380821\pi$
$384$	0	0
$385$	3.91069	0.199307
$386$	−17.4165	−0.886477
$387$	0	0
$388$	6.98257	0.354486
$389$	−2.21328	−0.112218	−0.0561089	−	0.998425i	$-0.517869\pi$
$389$	−2.21328	−0.112218	−0.0561089	+	0.998425i	$0.517869\pi$
$390$	0	0
$391$	−13.3232	−0.673781
$392$	−6.57866	−0.332273
$393$	0	0
$394$	−19.4588	−0.980321
$395$	45.0635	2.26739
$396$	0	0
$397$	13.8818	0.696709	0.348355	−	0.937363i	$-0.386741\pi$
$397$	13.8818	0.696709	0.348355	+	0.937363i	$0.386741\pi$
$398$	18.7195	0.938326
$399$	0	0
$400$	10.1617	0.508086
$401$	6.95106	0.347120	0.173560	−	0.984823i	$-0.444473\pi$
$401$	6.95106	0.347120	0.173560	+	0.984823i	$0.444473\pi$
$402$	0	0
$403$	4.37150	0.217760
$404$	−11.9644	−0.595251
$405$	0	0
$406$	1.31520	0.0652724
$407$	−8.69345	−0.430918
$408$	0	0
$409$	−6.92805	−0.342570	−0.171285	−	0.985222i	$-0.554792\pi$
$409$	−6.92805	−0.342570	−0.171285	+	0.985222i	$0.554792\pi$
$410$	13.5999	0.671651
$411$	0	0
$412$	−19.8650	−0.978680
$413$	0.559087	0.0275109
$414$	0	0
$415$	−54.8225	−2.69113
$416$	0.865551	0.0424371
$417$	0	0
$418$	2.84715	0.139259
$419$	3.56230	0.174030	0.0870148	−	0.996207i	$-0.472267\pi$
$419$	3.56230	0.174030	0.0870148	+	0.996207i	$0.472267\pi$
$420$	0	0
$421$	30.7465	1.49849	0.749245	−	0.662293i	$-0.230416\pi$
$421$	30.7465	1.49849	0.749245	+	0.662293i	$0.230416\pi$
$422$	6.00013	0.292082
$423$	0	0
$424$	−13.6906	−0.664874
$425$	34.5238	1.67465
$426$	0	0
$427$	−4.21666	−0.204059
$428$	13.7141	0.662894
$429$	0	0
$430$	−19.1572	−0.923843
$431$	40.5944	1.95537	0.977683	−	0.210086i	$-0.0673745\pi$
$431$	40.5944	1.95537	0.977683	+	0.210086i	$0.0673745\pi$
$432$	0	0
$433$	31.4561	1.51169	0.755843	−	0.654753i	$-0.227227\pi$
$433$	31.4561	1.51169	0.755843	+	0.654753i	$0.227227\pi$
$434$	3.27833	0.157365
$435$	0	0
$436$	6.65657	0.318792
$437$	−7.21609	−0.345193
$438$	0	0
$439$	13.9919	0.667796	0.333898	−	0.942609i	$-0.391636\pi$
$439$	13.9919	0.667796	0.333898	+	0.942609i	$0.391636\pi$
$440$	6.02474	0.287218
$441$	0	0
$442$	2.94066	0.139873
$443$	−29.1579	−1.38533	−0.692667	−	0.721258i	$-0.743564\pi$
$443$	−29.1579	−1.38533	−0.692667	+	0.721258i	$0.743564\pi$
$444$	0	0
$445$	−39.0289	−1.85015
$446$	−7.48223	−0.354294
$447$	0	0
$448$	0.649105	0.0306673
$449$	2.20608	0.104112	0.0520558	−	0.998644i	$-0.483423\pi$
$449$	2.20608	0.104112	0.0520558	+	0.998644i	$0.483423\pi$
$450$	0	0
$451$	5.40412	0.254470
$452$	−10.7995	−0.507964
$453$	0	0
$454$	1.86337	0.0874524
$455$	2.18767	0.102560
$456$	0	0
$457$	−12.1774	−0.569636	−0.284818	−	0.958582i	$-0.591933\pi$
$457$	−12.1774	−0.569636	−0.284818	+	0.958582i	$0.591933\pi$
$458$	−8.98233	−0.419717
$459$	0	0
$460$	−15.2697	−0.711953
$461$	−0.564708	−0.0263011	−0.0131505	−	0.999914i	$-0.504186\pi$
$461$	−0.564708	−0.0263011	−0.0131505	+	0.999914i	$0.504186\pi$
$462$	0	0
$463$	−5.44498	−0.253050	−0.126525	−	0.991963i	$-0.540382\pi$
$463$	−5.44498	−0.253050	−0.126525	+	0.991963i	$0.540382\pi$
$464$	2.02618	0.0940630
$465$	0	0
$466$	−21.0086	−0.973206
$467$	−1.75744	−0.0813245	−0.0406623	−	0.999173i	$-0.512947\pi$
$467$	−1.75744	−0.0813245	−0.0406623	+	0.999173i	$0.512947\pi$
$468$	0	0
$469$	6.98265	0.322429
$470$	35.2807	1.62738
$471$	0	0
$472$	0.861320	0.0396455
$473$	−7.61241	−0.350019
$474$	0	0
$475$	18.6988	0.857959
$476$	2.20529	0.101079
$477$	0	0
$478$	−11.4102	−0.521891
$479$	38.8299	1.77418	0.887092	−	0.461592i	$-0.152722\pi$
$479$	38.8299	1.77418	0.887092	+	0.461592i	$0.152722\pi$
$480$	0	0
$481$	−4.86319	−0.221742
$482$	−23.9632	−1.09149
$483$	0	0
$484$	−8.60598	−0.391181
$485$	27.1888	1.23458
$486$	0	0
$487$	−34.7151	−1.57309	−0.786546	−	0.617531i	$-0.788133\pi$
$487$	−34.7151	−1.57309	−0.786546	+	0.617531i	$0.788133\pi$
$488$	−6.49612	−0.294066
$489$	0	0
$490$	−25.6160	−1.15721
$491$	19.6488	0.886739	0.443369	−	0.896339i	$-0.353783\pi$
$491$	19.6488	0.886739	0.443369	+	0.896339i	$0.353783\pi$
$492$	0	0
$493$	6.88381	0.310031
$494$	1.59272	0.0716598
$495$	0	0
$496$	5.05054	0.226776
$497$	−0.545085	−0.0244504
$498$	0	0
$499$	−15.3738	−0.688225	−0.344112	−	0.938929i	$-0.611820\pi$
$499$	−15.3738	−0.688225	−0.344112	+	0.938929i	$0.611820\pi$
$500$	20.0987	0.898843
$501$	0	0
$502$	30.3598	1.35503
$503$	20.9101	0.932337	0.466169	−	0.884696i	$-0.345634\pi$
$503$	20.9101	0.932337	0.466169	+	0.884696i	$0.345634\pi$
$504$	0	0
$505$	−46.5871	−2.07310
$506$	−6.06764	−0.269739
$507$	0	0
$508$	−2.45779	−0.109047
$509$	19.7212	0.874127	0.437063	−	0.899431i	$-0.356019\pi$
$509$	19.7212	0.874127	0.437063	+	0.899431i	$0.356019\pi$
$510$	0	0
$511$	6.23287	0.275726
$512$	1.00000	0.0441942
$513$	0	0
$514$	8.82453	0.389233
$515$	−77.3506	−3.40847
$516$	0	0
$517$	14.0193	0.616570
$518$	−3.64706	−0.160243
$519$	0	0
$520$	3.37029	0.147797
$521$	−13.3498	−0.584867	−0.292434	−	0.956286i	$-0.594465\pi$
$521$	−13.3498	−0.584867	−0.292434	+	0.956286i	$0.594465\pi$
$522$	0	0
$523$	24.8707	1.08752	0.543760	−	0.839241i	$-0.317000\pi$
$523$	24.8707	1.08752	0.543760	+	0.839241i	$0.317000\pi$
$524$	−7.72291	−0.337377
$525$	0	0
$526$	15.1382	0.660056
$527$	17.1589	0.747454
$528$	0	0
$529$	−7.62158	−0.331373
$530$	−53.3085	−2.31557
$531$	0	0
$532$	1.19443	0.0517851
$533$	3.02311	0.130945
$534$	0	0
$535$	53.3999	2.30868
$536$	10.7574	0.464647
$537$	0	0
$538$	1.28935	0.0555879
$539$	−10.1789	−0.438437
$540$	0	0
$541$	23.0386	0.990506	0.495253	−	0.868749i	$-0.335075\pi$
$541$	23.0386	0.990506	0.495253	+	0.868749i	$0.335075\pi$
$542$	5.99315	0.257428
$543$	0	0
$544$	3.39744	0.145664
$545$	25.9194	1.11026
$546$	0	0
$547$	−17.8509	−0.763248	−0.381624	−	0.924318i	$-0.624635\pi$
$547$	−17.8509	−0.763248	−0.381624	+	0.924318i	$0.624635\pi$
$548$	9.93733	0.424502
$549$	0	0
$550$	15.7228	0.670424
$551$	3.72841	0.158836
$552$	0	0
$553$	7.51217	0.319450
$554$	−23.0526	−0.979410
$555$	0	0
$556$	11.2968	0.479089
$557$	−13.2149	−0.559933	−0.279967	−	0.960010i	$-0.590323\pi$
$557$	−13.2149	−0.559933	−0.279967	+	0.960010i	$0.590323\pi$
$558$	0	0
$559$	−4.25844	−0.180113
$560$	2.52749	0.106806
$561$	0	0
$562$	13.6179	0.574437
$563$	41.3712	1.74359	0.871794	−	0.489873i	$-0.162957\pi$
$563$	41.3712	1.74359	0.871794	+	0.489873i	$0.162957\pi$
$564$	0	0
$565$	−42.0510	−1.76910
$566$	10.2171	0.429459
$567$	0	0
$568$	−0.839750	−0.0352351
$569$	4.88936	0.204973	0.102486	−	0.994734i	$-0.467320\pi$
$569$	4.88936	0.204973	0.102486	+	0.994734i	$0.467320\pi$
$570$	0	0
$571$	−27.6263	−1.15612	−0.578062	−	0.815993i	$-0.696191\pi$
$571$	−27.6263	−1.15612	−0.578062	+	0.815993i	$0.696191\pi$
$572$	1.33923	0.0559962
$573$	0	0
$574$	2.26713	0.0946281
$575$	−39.8495	−1.66184
$576$	0	0
$577$	41.9110	1.74478	0.872389	−	0.488812i	$-0.162570\pi$
$577$	41.9110	1.74478	0.872389	+	0.488812i	$0.162570\pi$
$578$	−5.45742	−0.226999
$579$	0	0
$580$	7.88954	0.327595
$581$	−9.13901	−0.379150
$582$	0	0
$583$	−21.1829	−0.877308
$584$	9.60226	0.397344
$585$	0	0
$586$	4.57955	0.189179
$587$	12.5012	0.515982	0.257991	−	0.966147i	$-0.416940\pi$
$587$	12.5012	0.515982	0.257991	+	0.966147i	$0.416940\pi$
$588$	0	0
$589$	9.29361	0.382936
$590$	3.35381	0.138074
$591$	0	0
$592$	−5.61860	−0.230923
$593$	−23.8209	−0.978207	−0.489104	−	0.872226i	$-0.662676\pi$
$593$	−23.8209	−0.978207	−0.489104	+	0.872226i	$0.662676\pi$
$594$	0	0
$595$	8.58698	0.352032
$596$	−1.00000	−0.0409616
$597$	0	0
$598$	−3.39429	−0.138803
$599$	22.6976	0.927400	0.463700	−	0.885992i	$-0.346522\pi$
$599$	22.6976	0.927400	0.463700	+	0.885992i	$0.346522\pi$
$600$	0	0
$601$	3.76342	0.153513	0.0767566	−	0.997050i	$-0.475544\pi$
$601$	3.76342	0.153513	0.0767566	+	0.997050i	$0.475544\pi$
$602$	−3.19354	−0.130159
$603$	0	0
$604$	−15.5422	−0.632402
$605$	−33.5100	−1.36238
$606$	0	0
$607$	−36.2181	−1.47005	−0.735023	−	0.678042i	$-0.762829\pi$
$607$	−36.2181	−1.47005	−0.735023	+	0.678042i	$0.762829\pi$
$608$	1.84012	0.0746267
$609$	0	0
$610$	−25.2946	−1.02415
$611$	7.84253	0.317275
$612$	0	0
$613$	14.1825	0.572825	0.286413	−	0.958106i	$-0.407537\pi$
$613$	14.1825	0.572825	0.286413	+	0.958106i	$0.407537\pi$
$614$	−6.89999	−0.278461
$615$	0	0
$616$	1.00434	0.0404658
$617$	33.8174	1.36144	0.680718	−	0.732545i	$-0.261668\pi$
$617$	33.8174	1.36144	0.680718	+	0.732545i	$0.261668\pi$
$618$	0	0
$619$	8.26809	0.332322	0.166161	−	0.986099i	$-0.446863\pi$
$619$	8.26809	0.332322	0.166161	+	0.986099i	$0.446863\pi$
$620$	19.6658	0.789799
$621$	0	0
$622$	−11.0770	−0.444149
$623$	−6.50620	−0.260665
$624$	0	0
$625$	27.4519	1.09808
$626$	−5.18106	−0.207077
$627$	0	0
$628$	20.2751	0.809063
$629$	−19.0888	−0.761122
$630$	0	0
$631$	35.1380	1.39882	0.699412	−	0.714719i	$-0.253445\pi$
$631$	35.1380	1.39882	0.699412	+	0.714719i	$0.253445\pi$
$632$	11.5731	0.460354
$633$	0	0
$634$	−0.540504	−0.0214661
$635$	−9.57015	−0.379780
$636$	0	0
$637$	−5.69417	−0.225611
$638$	3.13503	0.124117
$639$	0	0
$640$	3.89381	0.153916
$641$	−16.7776	−0.662674	−0.331337	−	0.943512i	$-0.607500\pi$
$641$	−16.7776	−0.662674	−0.331337	+	0.943512i	$0.607500\pi$
$642$	0	0
$643$	2.47892	0.0977591	0.0488796	−	0.998805i	$-0.484435\pi$
$643$	2.47892	0.0977591	0.0488796	+	0.998805i	$0.484435\pi$
$644$	−2.54549	−0.100306
$645$	0	0
$646$	6.25170	0.245970
$647$	−11.6630	−0.458520	−0.229260	−	0.973365i	$-0.573631\pi$
$647$	−11.6630	−0.458520	−0.229260	+	0.973365i	$0.573631\pi$
$648$	0	0
$649$	1.33269	0.0523126
$650$	8.79549	0.344987
$651$	0	0
$652$	3.07431	0.120399
$653$	15.6073	0.610759	0.305380	−	0.952231i	$-0.401217\pi$
$653$	15.6073	0.610759	0.305380	+	0.952231i	$0.401217\pi$
$654$	0	0
$655$	−30.0715	−1.17499
$656$	3.49270	0.136367
$657$	0	0
$658$	5.88137	0.229280
$659$	28.0535	1.09281	0.546405	−	0.837521i	$-0.315996\pi$
$659$	28.0535	1.09281	0.546405	+	0.837521i	$0.315996\pi$
$660$	0	0
$661$	17.4202	0.677567	0.338783	−	0.940864i	$-0.389985\pi$
$661$	17.4202	0.677567	0.338783	+	0.940864i	$0.389985\pi$
$662$	−27.0454	−1.05115
$663$	0	0
$664$	−14.0794	−0.546387
$665$	4.65088	0.180353
$666$	0	0
$667$	−7.94572	−0.307660
$668$	0.216671	0.00838324
$669$	0	0
$670$	41.8871	1.61824
$671$	−10.0512	−0.388023
$672$	0	0
$673$	−5.28678	−0.203790	−0.101895	−	0.994795i	$-0.532491\pi$
$673$	−5.28678	−0.203790	−0.101895	+	0.994795i	$0.532491\pi$
$674$	−17.3293	−0.667499
$675$	0	0
$676$	−12.2508	−0.471185
$677$	−9.96692	−0.383060	−0.191530	−	0.981487i	$-0.561345\pi$
$677$	−9.96692	−0.383060	−0.191530	+	0.981487i	$0.561345\pi$
$678$	0	0
$679$	4.53242	0.173938
$680$	13.2290	0.507308
$681$	0	0
$682$	7.81451	0.299233
$683$	−45.7459	−1.75042	−0.875210	−	0.483743i	$-0.839277\pi$
$683$	−45.7459	−1.75042	−0.875210	+	0.483743i	$0.839277\pi$
$684$	0	0
$685$	38.6940	1.47842
$686$	−8.81397	−0.336519
$687$	0	0
$688$	−4.91992	−0.187570
$689$	−11.8499	−0.451446
$690$	0	0
$691$	−32.0717	−1.22007	−0.610033	−	0.792376i	$-0.708844\pi$
$691$	−32.0717	−1.22007	−0.610033	+	0.792376i	$0.708844\pi$
$692$	17.6531	0.671071
$693$	0	0
$694$	−9.31515	−0.353598
$695$	43.9874	1.66854
$696$	0	0
$697$	11.8662	0.449465
$698$	22.7466	0.860970
$699$	0	0
$700$	6.59602	0.249306
$701$	1.46513	0.0553370	0.0276685	−	0.999617i	$-0.491192\pi$
$701$	1.46513	0.0553370	0.0276685	+	0.999617i	$0.491192\pi$
$702$	0	0
$703$	−10.3389	−0.389939
$704$	1.54726	0.0583146
$705$	0	0
$706$	−11.1814	−0.420817
$707$	−7.76615	−0.292076
$708$	0	0
$709$	−30.8427	−1.15832	−0.579160	−	0.815214i	$-0.696619\pi$
$709$	−30.8427	−1.15832	−0.579160	+	0.815214i	$0.696619\pi$
$710$	−3.26982	−0.122714
$711$	0	0
$712$	−10.0233	−0.375641
$713$	−19.8059	−0.741736
$714$	0	0
$715$	5.21472	0.195019
$716$	5.95874	0.222688
$717$	0	0
$718$	−16.7948	−0.626775
$719$	43.5158	1.62286	0.811432	−	0.584447i	$-0.198688\pi$
$719$	43.5158	1.62286	0.811432	+	0.584447i	$0.198688\pi$
$720$	0	0
$721$	−12.8945	−0.480216
$722$	−15.6140	−0.581091
$723$	0	0
$724$	0.631864	0.0234830
$725$	20.5895	0.764673
$726$	0	0
$727$	−14.5588	−0.539955	−0.269978	−	0.962867i	$-0.587016\pi$
$727$	−14.5588	−0.539955	−0.269978	+	0.962867i	$0.587016\pi$
$728$	0.561833	0.0208229
$729$	0	0
$730$	37.3893	1.38384
$731$	−16.7151	−0.618231
$732$	0	0
$733$	−20.3099	−0.750164	−0.375082	−	0.926992i	$-0.622385\pi$
$733$	−20.3099	−0.750164	−0.375082	+	0.926992i	$0.622385\pi$
$734$	−16.2821	−0.600985
$735$	0	0
$736$	−3.92153	−0.144550
$737$	16.6445	0.613106
$738$	0	0
$739$	7.01818	0.258168	0.129084	−	0.991634i	$-0.458796\pi$
$739$	7.01818	0.258168	0.129084	+	0.991634i	$0.458796\pi$
$740$	−21.8777	−0.804242
$741$	0	0
$742$	−8.88663	−0.326238
$743$	−11.6216	−0.426355	−0.213178	−	0.977013i	$-0.568381\pi$
$743$	−11.6216	−0.426355	−0.213178	+	0.977013i	$0.568381\pi$
$744$	0	0
$745$	−3.89381	−0.142658
$746$	−31.4387	−1.15105
$747$	0	0
$748$	5.25673	0.192205
$749$	8.90186	0.325267
$750$	0	0
$751$	15.9759	0.582968	0.291484	−	0.956576i	$-0.405851\pi$
$751$	15.9759	0.582968	0.291484	+	0.956576i	$0.405851\pi$
$752$	9.06074	0.330411
$753$	0	0
$754$	1.75376	0.0638682
$755$	−60.5182	−2.20248
$756$	0	0
$757$	29.5583	1.07431	0.537157	−	0.843482i	$-0.319498\pi$
$757$	29.5583	1.07431	0.537157	+	0.843482i	$0.319498\pi$
$758$	−6.87282	−0.249632
$759$	0	0
$760$	7.16507	0.259904
$761$	13.5691	0.491880	0.245940	−	0.969285i	$-0.420903\pi$
$761$	13.5691	0.491880	0.245940	+	0.969285i	$0.420903\pi$
$762$	0	0
$763$	4.32081	0.156424
$764$	21.1967	0.766868
$765$	0	0
$766$	14.3147	0.517212
$767$	0.745517	0.0269190
$768$	0	0
$769$	−51.6950	−1.86417	−0.932084	−	0.362242i	$-0.882012\pi$
$769$	−51.6950	−1.86417	−0.932084	+	0.362242i	$0.882012\pi$
$770$	3.91069	0.140931
$771$	0	0
$772$	−17.4165	−0.626834
$773$	−28.0510	−1.00892	−0.504462	−	0.863434i	$-0.668309\pi$
$773$	−28.0510	−1.00892	−0.504462	+	0.863434i	$0.668309\pi$
$774$	0	0
$775$	51.3222	1.84355
$776$	6.98257	0.250660
$777$	0	0
$778$	−2.21328	−0.0793500
$779$	6.42699	0.230271
$780$	0	0
$781$	−1.29931	−0.0464931
$782$	−13.3232	−0.476435
$783$	0	0
$784$	−6.57866	−0.234952
$785$	78.9471	2.81774
$786$	0	0
$787$	2.32867	0.0830080	0.0415040	−	0.999138i	$-0.486785\pi$
$787$	2.32867	0.0830080	0.0415040	+	0.999138i	$0.486785\pi$
$788$	−19.4588	−0.693191
$789$	0	0
$790$	45.0635	1.60329
$791$	−7.00999	−0.249246
$792$	0	0
$793$	−5.62273	−0.199669
$794$	13.8818	0.492648
$795$	0	0
$796$	18.7195	0.663497
$797$	−22.4899	−0.796631	−0.398316	−	0.917248i	$-0.630405\pi$
$797$	−22.4899	−0.796631	−0.398316	+	0.917248i	$0.630405\pi$
$798$	0	0
$799$	30.7833	1.08903
$800$	10.1617	0.359271
$801$	0	0
$802$	6.95106	0.245451
$803$	14.8572	0.524299
$804$	0	0
$805$	−9.91162	−0.349339
$806$	4.37150	0.153980
$807$	0	0
$808$	−11.9644	−0.420906
$809$	8.96624	0.315236	0.157618	−	0.987500i	$-0.449618\pi$
$809$	8.96624	0.315236	0.157618	+	0.987500i	$0.449618\pi$
$810$	0	0
$811$	−23.1902	−0.814317	−0.407158	−	0.913358i	$-0.633480\pi$
$811$	−23.1902	−0.814317	−0.407158	+	0.913358i	$0.633480\pi$
$812$	1.31520	0.0461545
$813$	0	0
$814$	−8.69345	−0.304705
$815$	11.9708	0.419318
$816$	0	0
$817$	−9.05325	−0.316733
$818$	−6.92805	−0.242234
$819$	0	0
$820$	13.5999	0.474929
$821$	5.98901	0.209018	0.104509	−	0.994524i	$-0.466673\pi$
$821$	5.98901	0.209018	0.104509	+	0.994524i	$0.466673\pi$
$822$	0	0
$823$	−6.07505	−0.211763	−0.105881	−	0.994379i	$-0.533766\pi$
$823$	−6.07505	−0.211763	−0.105881	+	0.994379i	$0.533766\pi$
$824$	−19.8650	−0.692031
$825$	0	0
$826$	0.559087	0.0194531
$827$	0.890259	0.0309573	0.0154787	−	0.999880i	$-0.495073\pi$
$827$	0.890259	0.0309573	0.0154787	+	0.999880i	$0.495073\pi$
$828$	0	0
$829$	−9.96759	−0.346189	−0.173094	−	0.984905i	$-0.555377\pi$
$829$	−9.96759	−0.346189	−0.173094	+	0.984905i	$0.555377\pi$
$830$	−54.8225	−1.90292
$831$	0	0
$832$	0.865551	0.0300076
$833$	−22.3506	−0.774402
$834$	0	0
$835$	0.843673	0.0291965
$836$	2.84715	0.0984707
$837$	0	0
$838$	3.56230	0.123058
$839$	−15.0527	−0.519676	−0.259838	−	0.965652i	$-0.583669\pi$
$839$	−15.0527	−0.519676	−0.259838	+	0.965652i	$0.583669\pi$
$840$	0	0
$841$	−24.8946	−0.858435
$842$	30.7465	1.05959
$843$	0	0
$844$	6.00013	0.206533
$845$	−47.7023	−1.64101
$846$	0	0
$847$	−5.58618	−0.191943
$848$	−13.6906	−0.470137
$849$	0	0
$850$	34.5238	1.18416
$851$	22.0335	0.755300
$852$	0	0
$853$	−6.92234	−0.237016	−0.118508	−	0.992953i	$-0.537811\pi$
$853$	−6.92234	−0.237016	−0.118508	+	0.992953i	$0.537811\pi$
$854$	−4.21666	−0.144291
$855$	0	0
$856$	13.7141	0.468737
$857$	−18.8477	−0.643824	−0.321912	−	0.946770i	$-0.604325\pi$
$857$	−18.8477	−0.643824	−0.321912	+	0.946770i	$0.604325\pi$
$858$	0	0
$859$	34.8422	1.18880	0.594400	−	0.804169i	$-0.297390\pi$
$859$	34.8422	1.18880	0.594400	+	0.804169i	$0.297390\pi$
$860$	−19.1572	−0.653256
$861$	0	0
$862$	40.5944	1.38265
$863$	12.2260	0.416179	0.208089	−	0.978110i	$-0.433275\pi$
$863$	12.2260	0.416179	0.208089	+	0.978110i	$0.433275\pi$
$864$	0	0
$865$	68.7379	2.33716
$866$	31.4561	1.06892
$867$	0	0
$868$	3.27833	0.111274
$869$	17.9067	0.607442
$870$	0	0
$871$	9.31104	0.315493
$872$	6.65657	0.225420
$873$	0	0
$874$	−7.21609	−0.244088
$875$	13.0462	0.441041
$876$	0	0
$877$	17.9067	0.604666	0.302333	−	0.953202i	$-0.402235\pi$
$877$	17.9067	0.604666	0.302333	+	0.953202i	$0.402235\pi$
$878$	13.9919	0.472203
$879$	0	0
$880$	6.02474	0.203094
$881$	50.7864	1.71104	0.855519	−	0.517771i	$-0.173238\pi$
$881$	50.7864	1.71104	0.855519	+	0.517771i	$0.173238\pi$
$882$	0	0
$883$	−43.9455	−1.47888	−0.739442	−	0.673220i	$-0.764911\pi$
$883$	−43.9455	−1.47888	−0.739442	+	0.673220i	$0.764911\pi$
$884$	2.94066	0.0989050
$885$	0	0
$886$	−29.1579	−0.979579
$887$	38.3374	1.28725	0.643623	−	0.765343i	$-0.277431\pi$
$887$	38.3374	1.28725	0.643623	+	0.765343i	$0.277431\pi$
$888$	0	0
$889$	−1.59536	−0.0535067
$890$	−39.0289	−1.30825
$891$	0	0
$892$	−7.48223	−0.250524
$893$	16.6729	0.557936
$894$	0	0
$895$	23.2022	0.775563
$896$	0.649105	0.0216851
$897$	0	0
$898$	2.20608	0.0736180
$899$	10.2333	0.341300
$900$	0	0
$901$	−46.5130	−1.54957
$902$	5.40412	0.179938
$903$	0	0
$904$	−10.7995	−0.359185
$905$	2.46036	0.0817850
$906$	0	0
$907$	−31.8733	−1.05834	−0.529168	−	0.848517i	$-0.677496\pi$
$907$	−31.8733	−1.05834	−0.529168	+	0.848517i	$0.677496\pi$
$908$	1.86337	0.0618382
$909$	0	0
$910$	2.18767	0.0725205
$911$	−26.6954	−0.884459	−0.442229	−	0.896902i	$-0.645812\pi$
$911$	−26.6954	−0.884459	−0.442229	+	0.896902i	$0.645812\pi$
$912$	0	0
$913$	−21.7845	−0.720963
$914$	−12.1774	−0.402793
$915$	0	0
$916$	−8.98233	−0.296784
$917$	−5.01298	−0.165543
$918$	0	0
$919$	−28.1015	−0.926984	−0.463492	−	0.886101i	$-0.653404\pi$
$919$	−28.1015	−0.926984	−0.463492	+	0.886101i	$0.653404\pi$
$920$	−15.2697	−0.503427
$921$	0	0
$922$	−0.564708	−0.0185977
$923$	−0.726846	−0.0239244
$924$	0	0
$925$	−57.0946	−1.87726
$926$	−5.44498	−0.178933
$927$	0	0
$928$	2.02618	0.0665126
$929$	−12.8554	−0.421773	−0.210886	−	0.977511i	$-0.567635\pi$
$929$	−12.8554	−0.421773	−0.210886	+	0.977511i	$0.567635\pi$
$930$	0	0
$931$	−12.1055	−0.396743
$932$	−21.0086	−0.688161
$933$	0	0
$934$	−1.75744	−0.0575051
$935$	20.4687	0.669397
$936$	0	0
$937$	36.1225	1.18007	0.590035	−	0.807377i	$-0.299114\pi$
$937$	36.1225	1.18007	0.590035	+	0.807377i	$0.299114\pi$
$938$	6.98265	0.227992
$939$	0	0
$940$	35.2807	1.15073
$941$	8.08620	0.263603	0.131801	−	0.991276i	$-0.457924\pi$
$941$	8.08620	0.263603	0.131801	+	0.991276i	$0.457924\pi$
$942$	0	0
$943$	−13.6967	−0.446027
$944$	0.861320	0.0280336
$945$	0	0
$946$	−7.61241	−0.247501
$947$	−2.93075	−0.0952367	−0.0476184	−	0.998866i	$-0.515163\pi$
$947$	−2.93075	−0.0952367	−0.0476184	+	0.998866i	$0.515163\pi$
$948$	0	0
$949$	8.31124	0.269794
$950$	18.6988	0.606669
$951$	0	0
$952$	2.20529	0.0714739
$953$	57.2817	1.85554	0.927768	−	0.373158i	$-0.121725\pi$
$953$	57.2817	1.85554	0.927768	+	0.373158i	$0.121725\pi$
$954$	0	0
$955$	82.5357	2.67079
$956$	−11.4102	−0.369033
$957$	0	0
$958$	38.8299	1.25454
$959$	6.45037	0.208293
$960$	0	0
$961$	−5.49202	−0.177162
$962$	−4.86319	−0.156795
$963$	0	0
$964$	−23.9632	−0.771802
$965$	−67.8165	−2.18309
$966$	0	0
$967$	27.3968	0.881023	0.440511	−	0.897747i	$-0.354797\pi$
$967$	27.3968	0.881023	0.440511	+	0.897747i	$0.354797\pi$
$968$	−8.60598	−0.276607
$969$	0	0
$970$	27.1888	0.872979
$971$	1.88652	0.0605414	0.0302707	−	0.999542i	$-0.490363\pi$
$971$	1.88652	0.0605414	0.0302707	+	0.999542i	$0.490363\pi$
$972$	0	0
$973$	7.33278	0.235078
$974$	−34.7151	−1.11234
$975$	0	0
$976$	−6.49612	−0.207936
$977$	16.3388	0.522724	0.261362	−	0.965241i	$-0.415828\pi$
$977$	16.3388	0.522724	0.261362	+	0.965241i	$0.415828\pi$
$978$	0	0
$979$	−15.5087	−0.495661
$980$	−25.6160	−0.818274
$981$	0	0
$982$	19.6488	0.627019
$983$	−35.2901	−1.12558	−0.562790	−	0.826600i	$-0.690272\pi$
$983$	−35.2901	−1.12558	−0.562790	+	0.826600i	$0.690272\pi$
$984$	0	0
$985$	−75.7688	−2.41420
$986$	6.88381	0.219225
$987$	0	0
$988$	1.59272	0.0506711
$989$	19.2936	0.613502
$990$	0	0
$991$	−18.4151	−0.584976	−0.292488	−	0.956269i	$-0.594483\pi$
$991$	−18.4151	−0.584976	−0.292488	+	0.956269i	$0.594483\pi$
$992$	5.05054	0.160355
$993$	0	0
$994$	−0.545085	−0.0172891
$995$	72.8903	2.31078
$996$	0	0
$997$	−29.3202	−0.928580	−0.464290	−	0.885683i	$-0.653691\pi$
$997$	−29.3202	−0.928580	−0.464290	+	0.885683i	$0.653691\pi$
$998$	−15.3738	−0.486648
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8046.2.a.p.1.12	yes	12
3.2	odd	2		8046.2.a.i.1.1	✓	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8046.2.a.i.1.1	✓	12	3.2	odd	2
8046.2.a.p.1.12	yes	12	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:	\( N \)	\(=\)	\( 8046 = 2 \cdot 3^{3} \cdot 149 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8046.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} +3.89381 q^{5} +0.649105 q^{7} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} +3.89381 q^{5} +0.649105 q^{7} +1.00000 q^{8} +3.89381 q^{10} +1.54726 q^{11} +0.865551 q^{13} +0.649105 q^{14} +1.00000 q^{16} +3.39744 q^{17} +1.84012 q^{19} +3.89381 q^{20} +1.54726 q^{22} -3.92153 q^{23} +10.1617 q^{25} +0.865551 q^{26} +0.649105 q^{28} +2.02618 q^{29} +5.05054 q^{31} +1.00000 q^{32} +3.39744 q^{34} +2.52749 q^{35} -5.61860 q^{37} +1.84012 q^{38} +3.89381 q^{40} +3.49270 q^{41} -4.91992 q^{43} +1.54726 q^{44} -3.92153 q^{46} +9.06074 q^{47} -6.57866 q^{49} +10.1617 q^{50} +0.865551 q^{52} -13.6906 q^{53} +6.02474 q^{55} +0.649105 q^{56} +2.02618 q^{58} +0.861320 q^{59} -6.49612 q^{61} +5.05054 q^{62} +1.00000 q^{64} +3.37029 q^{65} +10.7574 q^{67} +3.39744 q^{68} +2.52749 q^{70} -0.839750 q^{71} +9.60226 q^{73} -5.61860 q^{74} +1.84012 q^{76} +1.00434 q^{77} +11.5731 q^{79} +3.89381 q^{80} +3.49270 q^{82} -14.0794 q^{83} +13.2290 q^{85} -4.91992 q^{86} +1.54726 q^{88} -10.0233 q^{89} +0.561833 q^{91} -3.92153 q^{92} +9.06074 q^{94} +7.16507 q^{95} +6.98257 q^{97} -6.57866 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 12 q^{2} + 12 q^{4} + 5 q^{5} + 6 q^{7} + 12 q^{8}+O(q^{10}) \) 12 * q + 12 * q^2 + 12 * q^4 + 5 * q^5 + 6 * q^7 + 12 * q^8	\( 12 q + 12 q^{2} + 12 q^{4} + 5 q^{5} + 6 q^{7} + 12 q^{8} + 5 q^{10} + 6 q^{11} + 3 q^{13} + 6 q^{14} + 12 q^{16} + 6 q^{17} + 8 q^{19} + 5 q^{20} + 6 q^{22} + 11 q^{23} + 11 q^{25} + 3 q^{26} + 6 q^{28} + 29 q^{29} + 2 q^{31} + 12 q^{32} + 6 q^{34} + 4 q^{35} + 5 q^{37} + 8 q^{38} + 5 q^{40} + 22 q^{41} + 9 q^{43} + 6 q^{44} + 11 q^{46} + 15 q^{47} + 14 q^{49} + 11 q^{50} + 3 q^{52} + 12 q^{53} + 13 q^{55} + 6 q^{56} + 29 q^{58} + 34 q^{59} - 4 q^{61} + 2 q^{62} + 12 q^{64} + 12 q^{65} + q^{67} + 6 q^{68} + 4 q^{70} + 21 q^{71} - 2 q^{73} + 5 q^{74} + 8 q^{76} + 34 q^{77} + 9 q^{79} + 5 q^{80} + 22 q^{82} + 10 q^{83} + 5 q^{85} + 9 q^{86} + 6 q^{88} - 2 q^{89} + 17 q^{91} + 11 q^{92} + 15 q^{94} + 69 q^{95} - 13 q^{97} + 14 q^{98}+O(q^{100}) \) 12 * q + 12 * q^2 + 12 * q^4 + 5 * q^5 + 6 * q^7 + 12 * q^8 + 5 * q^10 + 6 * q^11 + 3 * q^13 + 6 * q^14 + 12 * q^16 + 6 * q^17 + 8 * q^19 + 5 * q^20 + 6 * q^22 + 11 * q^23 + 11 * q^25 + 3 * q^26 + 6 * q^28 + 29 * q^29 + 2 * q^31 + 12 * q^32 + 6 * q^34 + 4 * q^35 + 5 * q^37 + 8 * q^38 + 5 * q^40 + 22 * q^41 + 9 * q^43 + 6 * q^44 + 11 * q^46 + 15 * q^47 + 14 * q^49 + 11 * q^50 + 3 * q^52 + 12 * q^53 + 13 * q^55 + 6 * q^56 + 29 * q^58 + 34 * q^59 - 4 * q^61 + 2 * q^62 + 12 * q^64 + 12 * q^65 + q^67 + 6 * q^68 + 4 * q^70 + 21 * q^71 - 2 * q^73 + 5 * q^74 + 8 * q^76 + 34 * q^77 + 9 * q^79 + 5 * q^80 + 22 * q^82 + 10 * q^83 + 5 * q^85 + 9 * q^86 + 6 * q^88 - 2 * q^89 + 17 * q^91 + 11 * q^92 + 15 * q^94 + 69 * q^95 - 13 * q^97 + 14 * q^98

Introduction

L-functions

Modular forms

Varieties

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Database

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