LMFDB - Embedded newform 8046.2.a.l.1.3

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} - 21 x^{10} + 116 x^{9} + 106 x^{8} - 774 x^{7} - 63 x^{6} + 2013 x^{5} - 417 x^{4} + \cdots - 375 $ x^12 - 5x^11 - 21x^10 + 116x^9 + 106x^8 - 774x^7 - 63x^6 + 2013x^5 - 417x^4 - 2249x^3 + 761x^2 + 910*x - 375
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			1.3
Root			$-1.20126$ 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)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} -1.20126 q^{5} -4.29383 q^{7} -1.00000 q^{8} +O(q^{10})$	\(q-1.00000 q^{2} +1.00000 q^{4} -1.20126 q^{5} -4.29383 q^{7} -1.00000 q^{8} +1.20126 q^{10} +1.67350 q^{11} -2.77189 q^{13} +4.29383 q^{14} +1.00000 q^{16} -5.75270 q^{17} +5.66708 q^{19} -1.20126 q^{20} -1.67350 q^{22} -3.89843 q^{23} -3.55698 q^{25} +2.77189 q^{26} -4.29383 q^{28} -3.66521 q^{29} -10.2309 q^{31} -1.00000 q^{32} +5.75270 q^{34} +5.15800 q^{35} -7.55661 q^{37} -5.66708 q^{38} +1.20126 q^{40} -1.38760 q^{41} -7.29901 q^{43} +1.67350 q^{44} +3.89843 q^{46} +5.85394 q^{47} +11.4370 q^{49} +3.55698 q^{50} -2.77189 q^{52} -0.209993 q^{53} -2.01031 q^{55} +4.29383 q^{56} +3.66521 q^{58} +1.94740 q^{59} +1.99514 q^{61} +10.2309 q^{62} +1.00000 q^{64} +3.32976 q^{65} -2.36095 q^{67} -5.75270 q^{68} -5.15800 q^{70} +9.26740 q^{71} +7.84474 q^{73} +7.55661 q^{74} +5.66708 q^{76} -7.18575 q^{77} -10.3332 q^{79} -1.20126 q^{80} +1.38760 q^{82} +3.69755 q^{83} +6.91048 q^{85} +7.29901 q^{86} -1.67350 q^{88} -12.8667 q^{89} +11.9020 q^{91} -3.89843 q^{92} -5.85394 q^{94} -6.80763 q^{95} -5.85506 q^{97} -11.4370 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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} + 10 q^{11} - q^{13} + 6 q^{14} + 12 q^{16} + 6 q^{17} - 10 q^{19} + 5 q^{20} - 10 q^{22} + 15 q^{23} + 7 q^{25} + q^{26} - 6 q^{28} + 33 q^{29} - 6 q^{31} - 12 q^{32} - 6 q^{34} + 16 q^{35} - 13 q^{37} + 10 q^{38} - 5 q^{40} + 20 q^{41} - 11 q^{43} + 10 q^{44} - 15 q^{46} + 15 q^{47} + 2 q^{49} - 7 q^{50} - q^{52} + 4 q^{53} - 17 q^{55} + 6 q^{56} - 33 q^{58} + 10 q^{59} - 12 q^{61} + 6 q^{62} + 12 q^{64} + 40 q^{65} - 19 q^{67} + 6 q^{68} - 16 q^{70} + 47 q^{71} - 2 q^{73} + 13 q^{74} - 10 q^{76} - 6 q^{77} - 15 q^{79} + 5 q^{80} - 20 q^{82} + 18 q^{83} - 25 q^{85} + 11 q^{86} - 10 q^{88} + 24 q^{89} - 3 q^{91} + 15 q^{92} - 15 q^{94} - 3 q^{95} - 25 q^{97} - 2 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 + 10 * q^11 - q^13 + 6 * q^14 + 12 * q^16 + 6 * q^17 - 10 * q^19 + 5 * q^20 - 10 * q^22 + 15 * q^23 + 7 * q^25 + q^26 - 6 * q^28 + 33 * q^29 - 6 * q^31 - 12 * q^32 - 6 * q^34 + 16 * q^35 - 13 * q^37 + 10 * q^38 - 5 * q^40 + 20 * q^41 - 11 * q^43 + 10 * q^44 - 15 * q^46 + 15 * q^47 + 2 * q^49 - 7 * q^50 - q^52 + 4 * q^53 - 17 * q^55 + 6 * q^56 - 33 * q^58 + 10 * q^59 - 12 * q^61 + 6 * q^62 + 12 * q^64 + 40 * q^65 - 19 * q^67 + 6 * q^68 - 16 * q^70 + 47 * q^71 - 2 * q^73 + 13 * q^74 - 10 * q^76 - 6 * q^77 - 15 * q^79 + 5 * q^80 - 20 * q^82 + 18 * q^83 - 25 * q^85 + 11 * q^86 - 10 * q^88 + 24 * q^89 - 3 * q^91 + 15 * q^92 - 15 * q^94 - 3 * q^95 - 25 * 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
$5$	−1.20126	−0.537219	−0.268610	−	0.963249i	$-0.586564\pi$
$5$	−1.20126	−0.537219	−0.268610	+	0.963249i	$0.586564\pi$
$6$	0	0
$7$	−4.29383	−1.62292	−0.811458	−	0.584410i	$-0.801326\pi$
$7$	−4.29383	−1.62292	−0.811458	+	0.584410i	$0.801326\pi$
$8$	−1.00000	−0.353553
$9$	0	0
$10$	1.20126	0.379871
$11$	1.67350	0.504581	0.252290	−	0.967652i	$-0.418816\pi$
$11$	1.67350	0.504581	0.252290	+	0.967652i	$0.418816\pi$
$12$	0	0
$13$	−2.77189	−0.768785	−0.384392	−	0.923170i	$-0.625589\pi$
$13$	−2.77189	−0.768785	−0.384392	+	0.923170i	$0.625589\pi$
$14$	4.29383	1.14758
$15$	0	0
$16$	1.00000	0.250000
$17$	−5.75270	−1.39524	−0.697618	−	0.716470i	$-0.745757\pi$
$17$	−5.75270	−1.39524	−0.697618	+	0.716470i	$0.745757\pi$
$18$	0	0
$19$	5.66708	1.30012	0.650058	−	0.759884i	$-0.274744\pi$
$19$	5.66708	1.30012	0.650058	+	0.759884i	$0.274744\pi$
$20$	−1.20126	−0.268610
$21$	0	0
$22$	−1.67350	−0.356792
$23$	−3.89843	−0.812878	−0.406439	−	0.913678i	$-0.633230\pi$
$23$	−3.89843	−0.812878	−0.406439	+	0.913678i	$0.633230\pi$
$24$	0	0
$25$	−3.55698	−0.711396
$26$	2.77189	0.543613
$27$	0	0
$28$	−4.29383	−0.811458
$29$	−3.66521	−0.680613	−0.340306	−	0.940315i	$-0.610531\pi$
$29$	−3.66521	−0.680613	−0.340306	+	0.940315i	$0.610531\pi$
$30$	0	0
$31$	−10.2309	−1.83752	−0.918761	−	0.394813i	$-0.870809\pi$
$31$	−10.2309	−1.83752	−0.918761	+	0.394813i	$0.870809\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	5.75270	0.986581
$35$	5.15800	0.871862
$36$	0	0
$37$	−7.55661	−1.24230	−0.621150	−	0.783692i	$-0.713334\pi$
$37$	−7.55661	−1.24230	−0.621150	+	0.783692i	$0.713334\pi$
$38$	−5.66708	−0.919322
$39$	0	0
$40$	1.20126	0.189936
$41$	−1.38760	−0.216706	−0.108353	−	0.994112i	$-0.534558\pi$
$41$	−1.38760	−0.216706	−0.108353	+	0.994112i	$0.534558\pi$
$42$	0	0
$43$	−7.29901	−1.11309	−0.556544	−	0.830818i	$-0.687873\pi$
$43$	−7.29901	−1.11309	−0.556544	+	0.830818i	$0.687873\pi$
$44$	1.67350	0.252290
$45$	0	0
$46$	3.89843	0.574792
$47$	5.85394	0.853885	0.426943	−	0.904279i	$-0.359591\pi$
$47$	5.85394	0.853885	0.426943	+	0.904279i	$0.359591\pi$
$48$	0	0
$49$	11.4370	1.63386
$50$	3.55698	0.503033
$51$	0	0
$52$	−2.77189	−0.384392
$53$	−0.209993	−0.0288448	−0.0144224	−	0.999896i	$-0.504591\pi$
$53$	−0.209993	−0.0288448	−0.0144224	+	0.999896i	$0.504591\pi$
$54$	0	0
$55$	−2.01031	−0.271070
$56$	4.29383	0.573788
$57$	0	0
$58$	3.66521	0.481266
$59$	1.94740	0.253529	0.126765	−	0.991933i	$-0.459541\pi$
$59$	1.94740	0.253529	0.126765	+	0.991933i	$0.459541\pi$
$60$	0	0
$61$	1.99514	0.255452	0.127726	−	0.991810i	$-0.459232\pi$
$61$	1.99514	0.255452	0.127726	+	0.991810i	$0.459232\pi$
$62$	10.2309	1.29932
$63$	0	0
$64$	1.00000	0.125000
$65$	3.32976	0.413006
$66$	0	0
$67$	−2.36095	−0.288436	−0.144218	−	0.989546i	$-0.546067\pi$
$67$	−2.36095	−0.288436	−0.144218	+	0.989546i	$0.546067\pi$
$68$	−5.75270	−0.697618
$69$	0	0
$70$	−5.15800	−0.616499
$71$	9.26740	1.09984	0.549919	−	0.835218i	$-0.314659\pi$
$71$	9.26740	1.09984	0.549919	+	0.835218i	$0.314659\pi$
$72$	0	0
$73$	7.84474	0.918158	0.459079	−	0.888395i	$-0.348180\pi$
$73$	7.84474	0.918158	0.459079	+	0.888395i	$0.348180\pi$
$74$	7.55661	0.878438
$75$	0	0
$76$	5.66708	0.650058
$77$	−7.18575	−0.818892
$78$	0	0
$79$	−10.3332	−1.16258	−0.581289	−	0.813697i	$-0.697452\pi$
$79$	−10.3332	−1.16258	−0.581289	+	0.813697i	$0.697452\pi$
$80$	−1.20126	−0.134305
$81$	0	0
$82$	1.38760	0.153235
$83$	3.69755	0.405859	0.202929	−	0.979193i	$-0.434954\pi$
$83$	3.69755	0.405859	0.202929	+	0.979193i	$0.434954\pi$
$84$	0	0
$85$	6.91048	0.749547
$86$	7.29901	0.787072
$87$	0	0
$88$	−1.67350	−0.178396
$89$	−12.8667	−1.36387	−0.681933	−	0.731414i	$-0.738861\pi$
$89$	−12.8667	−1.36387	−0.681933	+	0.731414i	$0.738861\pi$
$90$	0	0
$91$	11.9020	1.24767
$92$	−3.89843	−0.406439
$93$	0	0
$94$	−5.85394	−0.603788
$95$	−6.80763	−0.698448
$96$	0	0
$97$	−5.85506	−0.594492	−0.297246	−	0.954801i	$-0.596068\pi$
$97$	−5.85506	−0.594492	−0.297246	+	0.954801i	$0.596068\pi$
$98$	−11.4370	−1.15531
$99$	0	0
$100$	−3.55698	−0.355698
$101$	6.32189	0.629052	0.314526	−	0.949249i	$-0.398154\pi$
$101$	6.32189	0.629052	0.314526	+	0.949249i	$0.398154\pi$
$102$	0	0
$103$	−10.9039	−1.07439	−0.537196	−	0.843457i	$-0.680517\pi$
$103$	−10.9039	−1.07439	−0.537196	+	0.843457i	$0.680517\pi$
$104$	2.77189	0.271807
$105$	0	0
$106$	0.209993	0.0203964
$107$	−12.2897	−1.18809	−0.594043	−	0.804433i	$-0.702469\pi$
$107$	−12.2897	−1.18809	−0.594043	+	0.804433i	$0.702469\pi$
$108$	0	0
$109$	9.50134	0.910064	0.455032	−	0.890475i	$-0.349628\pi$
$109$	9.50134	0.910064	0.455032	+	0.890475i	$0.349628\pi$
$110$	2.01031	0.191676
$111$	0	0
$112$	−4.29383	−0.405729
$113$	−2.21368	−0.208246	−0.104123	−	0.994564i	$-0.533204\pi$
$113$	−2.21368	−0.208246	−0.104123	+	0.994564i	$0.533204\pi$
$114$	0	0
$115$	4.68302	0.436694
$116$	−3.66521	−0.340306
$117$	0	0
$118$	−1.94740	−0.179272
$119$	24.7012	2.26435
$120$	0	0
$121$	−8.19938	−0.745398
$122$	−1.99514	−0.180632
$123$	0	0
$124$	−10.2309	−0.918761
$125$	10.2791	0.919394
$126$	0	0
$127$	−10.9132	−0.968391	−0.484195	−	0.874960i	$-0.660888\pi$
$127$	−10.9132	−0.968391	−0.484195	+	0.874960i	$0.660888\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	−3.32976	−0.292039
$131$	7.79020	0.680633	0.340317	−	0.940311i	$-0.389466\pi$
$131$	7.79020	0.680633	0.340317	+	0.940311i	$0.389466\pi$
$132$	0	0
$133$	−24.3335	−2.10998
$134$	2.36095	0.203955
$135$	0	0
$136$	5.75270	0.493290
$137$	−8.27186	−0.706713	−0.353356	−	0.935489i	$-0.614960\pi$
$137$	−8.27186	−0.706713	−0.353356	+	0.935489i	$0.614960\pi$
$138$	0	0
$139$	−7.15365	−0.606765	−0.303382	−	0.952869i	$-0.598116\pi$
$139$	−7.15365	−0.606765	−0.303382	+	0.952869i	$0.598116\pi$
$140$	5.15800	0.435931
$141$	0	0
$142$	−9.26740	−0.777703
$143$	−4.63878	−0.387914
$144$	0	0
$145$	4.40287	0.365638
$146$	−7.84474	−0.649236
$147$	0	0
$148$	−7.55661	−0.621150
$149$	1.00000	0.0819232
$149$	1.00000	0.0819232
$150$	0	0
$151$	0.135399	0.0110186	0.00550931	−	0.999985i	$-0.498246\pi$
$151$	0.135399	0.0110186	0.00550931	+	0.999985i	$0.498246\pi$
$152$	−5.66708	−0.459661
$153$	0	0
$154$	7.18575	0.579044
$155$	12.2899	0.987152
$156$	0	0
$157$	−12.5865	−1.00451	−0.502257	−	0.864718i	$-0.667497\pi$
$157$	−12.5865	−1.00451	−0.502257	+	0.864718i	$0.667497\pi$
$158$	10.3332	0.822067
$159$	0	0
$160$	1.20126	0.0949678
$161$	16.7392	1.31923
$162$	0	0
$163$	14.2103	1.11304	0.556518	−	0.830836i	$-0.312137\pi$
$163$	14.2103	1.11304	0.556518	+	0.830836i	$0.312137\pi$
$164$	−1.38760	−0.108353
$165$	0	0
$166$	−3.69755	−0.286985
$167$	−4.15585	−0.321589	−0.160795	−	0.986988i	$-0.551406\pi$
$167$	−4.15585	−0.321589	−0.160795	+	0.986988i	$0.551406\pi$
$168$	0	0
$169$	−5.31661	−0.408970
$170$	−6.91048	−0.530010
$171$	0	0
$172$	−7.29901	−0.556544
$173$	−8.90193	−0.676801	−0.338401	−	0.941002i	$-0.609886\pi$
$173$	−8.90193	−0.676801	−0.338401	+	0.941002i	$0.609886\pi$
$174$	0	0
$175$	15.2731	1.15454
$176$	1.67350	0.126145
$177$	0	0
$178$	12.8667	0.964399
$179$	−12.9099	−0.964930	−0.482465	−	0.875915i	$-0.660258\pi$
$179$	−12.9099	−0.964930	−0.482465	+	0.875915i	$0.660258\pi$
$180$	0	0
$181$	−10.5212	−0.782035	−0.391018	−	0.920383i	$-0.627877\pi$
$181$	−10.5212	−0.782035	−0.391018	+	0.920383i	$0.627877\pi$
$182$	−11.9020	−0.882239
$183$	0	0
$184$	3.89843	0.287396
$185$	9.07744	0.667387
$186$	0	0
$187$	−9.62718	−0.704009
$188$	5.85394	0.426943
$189$	0	0
$190$	6.80763	0.493877
$191$	23.8419	1.72514	0.862568	−	0.505941i	$-0.168855\pi$
$191$	23.8419	1.72514	0.862568	+	0.505941i	$0.168855\pi$
$192$	0	0
$193$	−17.2482	−1.24155	−0.620775	−	0.783989i	$-0.713182\pi$
$193$	−17.2482	−1.24155	−0.620775	+	0.783989i	$0.713182\pi$
$194$	5.85506	0.420369
$195$	0	0
$196$	11.4370	0.816929
$197$	21.3458	1.52082	0.760412	−	0.649441i	$-0.224997\pi$
$197$	21.3458	1.52082	0.760412	+	0.649441i	$0.224997\pi$
$198$	0	0
$199$	5.58409	0.395846	0.197923	−	0.980218i	$-0.436580\pi$
$199$	5.58409	0.395846	0.197923	+	0.980218i	$0.436580\pi$
$200$	3.55698	0.251516
$201$	0	0
$202$	−6.32189	−0.444807
$203$	15.7378	1.10458
$204$	0	0
$205$	1.66686	0.116419
$206$	10.9039	0.759710
$207$	0	0
$208$	−2.77189	−0.192196
$209$	9.48388	0.656014
$210$	0	0
$211$	−8.24942	−0.567914	−0.283957	−	0.958837i	$-0.591647\pi$
$211$	−8.24942	−0.567914	−0.283957	+	0.958837i	$0.591647\pi$
$212$	−0.209993	−0.0144224
$213$	0	0
$214$	12.2897	0.840104
$215$	8.76799	0.597972
$216$	0	0
$217$	43.9297	2.98215
$218$	−9.50134	−0.643512
$219$	0	0
$220$	−2.01031	−0.135535
$221$	15.9459	1.07264
$222$	0	0
$223$	10.4684	0.701017	0.350508	−	0.936560i	$-0.386009\pi$
$223$	10.4684	0.701017	0.350508	+	0.936560i	$0.386009\pi$
$224$	4.29383	0.286894
$225$	0	0
$226$	2.21368	0.147252
$227$	25.7810	1.71114	0.855572	−	0.517684i	$-0.173206\pi$
$227$	25.7810	1.71114	0.855572	+	0.517684i	$0.173206\pi$
$228$	0	0
$229$	−28.9284	−1.91164	−0.955821	−	0.293950i	$-0.905030\pi$
$229$	−28.9284	−1.91164	−0.955821	+	0.293950i	$0.905030\pi$
$230$	−4.68302	−0.308789
$231$	0	0
$232$	3.66521	0.240633
$233$	5.51234	0.361125	0.180563	−	0.983563i	$-0.442208\pi$
$233$	5.51234	0.361125	0.180563	+	0.983563i	$0.442208\pi$
$234$	0	0
$235$	−7.03210	−0.458724
$236$	1.94740	0.126765
$237$	0	0
$238$	−24.7012	−1.60114
$239$	−20.3231	−1.31459	−0.657295	−	0.753634i	$-0.728299\pi$
$239$	−20.3231	−1.31459	−0.657295	+	0.753634i	$0.728299\pi$
$240$	0	0
$241$	−23.3767	−1.50582	−0.752911	−	0.658122i	$-0.771351\pi$
$241$	−23.3767	−1.50582	−0.752911	+	0.658122i	$0.771351\pi$
$242$	8.19938	0.527076
$243$	0	0
$244$	1.99514	0.127726
$245$	−13.7388	−0.877740
$246$	0	0
$247$	−15.7085	−0.999510
$248$	10.2309	0.649662
$249$	0	0
$250$	−10.2791	−0.650110
$251$	26.5660	1.67683	0.838416	−	0.545031i	$-0.183482\pi$
$251$	26.5660	1.67683	0.838416	+	0.545031i	$0.183482\pi$
$252$	0	0
$253$	−6.52404	−0.410163
$254$	10.9132	0.684756
$255$	0	0
$256$	1.00000	0.0625000
$257$	13.1963	0.823165	0.411582	−	0.911373i	$-0.364976\pi$
$257$	13.1963	0.823165	0.411582	+	0.911373i	$0.364976\pi$
$258$	0	0
$259$	32.4468	2.01615
$260$	3.32976	0.206503
$261$	0	0
$262$	−7.79020	−0.481280
$263$	−10.4667	−0.645406	−0.322703	−	0.946500i	$-0.604592\pi$
$263$	−10.4667	−0.645406	−0.322703	+	0.946500i	$0.604592\pi$
$264$	0	0
$265$	0.252256	0.0154960
$266$	24.3335	1.49198
$267$	0	0
$268$	−2.36095	−0.144218
$269$	27.7593	1.69251	0.846257	−	0.532775i	$-0.178851\pi$
$269$	27.7593	1.69251	0.846257	+	0.532775i	$0.178851\pi$
$270$	0	0
$271$	−26.9635	−1.63792	−0.818958	−	0.573853i	$-0.805448\pi$
$271$	−26.9635	−1.63792	−0.818958	+	0.573853i	$0.805448\pi$
$272$	−5.75270	−0.348809
$273$	0	0
$274$	8.27186	0.499721
$275$	−5.95262	−0.358957
$276$	0	0
$277$	14.7080	0.883717	0.441858	−	0.897085i	$-0.354319\pi$
$277$	14.7080	0.883717	0.441858	+	0.897085i	$0.354319\pi$
$278$	7.15365	0.429048
$279$	0	0
$280$	−5.15800	−0.308250
$281$	−6.64235	−0.396249	−0.198125	−	0.980177i	$-0.563485\pi$
$281$	−6.64235	−0.396249	−0.198125	+	0.980177i	$0.563485\pi$
$282$	0	0
$283$	12.4512	0.740149	0.370075	−	0.929002i	$-0.379332\pi$
$283$	12.4512	0.740149	0.370075	+	0.929002i	$0.379332\pi$
$284$	9.26740	0.549919
$285$	0	0
$286$	4.63878	0.274297
$287$	5.95811	0.351696
$288$	0	0
$289$	16.0936	0.946683
$290$	−4.40287	−0.258545
$291$	0	0
$292$	7.84474	0.459079
$293$	−13.0780	−0.764026	−0.382013	−	0.924157i	$-0.624769\pi$
$293$	−13.0780	−0.764026	−0.382013	+	0.924157i	$0.624769\pi$
$294$	0	0
$295$	−2.33933	−0.136201
$296$	7.55661	0.439219
$297$	0	0
$298$	−1.00000	−0.0579284
$299$	10.8060	0.624929
$300$	0	0
$301$	31.3407	1.80645
$302$	−0.135399	−0.00779135
$303$	0	0
$304$	5.66708	0.325029
$305$	−2.39668	−0.137233
$306$	0	0
$307$	31.0940	1.77463	0.887315	−	0.461164i	$-0.152568\pi$
$307$	31.0940	1.77463	0.887315	+	0.461164i	$0.152568\pi$
$308$	−7.18575	−0.409446
$309$	0	0
$310$	−12.2899	−0.698022
$311$	20.7301	1.17550	0.587748	−	0.809044i	$-0.300014\pi$
$311$	20.7301	1.17550	0.587748	+	0.809044i	$0.300014\pi$
$312$	0	0
$313$	−14.9338	−0.844107	−0.422053	−	0.906571i	$-0.638690\pi$
$313$	−14.9338	−0.844107	−0.422053	+	0.906571i	$0.638690\pi$
$314$	12.5865	0.710299
$315$	0	0
$316$	−10.3332	−0.581289
$317$	9.30324	0.522522	0.261261	−	0.965268i	$-0.415862\pi$
$317$	9.30324	0.522522	0.261261	+	0.965268i	$0.415862\pi$
$318$	0	0
$319$	−6.13375	−0.343424
$320$	−1.20126	−0.0671524
$321$	0	0
$322$	−16.7392	−0.932839
$323$	−32.6010	−1.81397
$324$	0	0
$325$	9.85957	0.546910
$326$	−14.2103	−0.787035
$327$	0	0
$328$	1.38760	0.0766173
$329$	−25.1359	−1.38578
$330$	0	0
$331$	13.1701	0.723893	0.361946	−	0.932199i	$-0.382112\pi$
$331$	13.1701	0.723893	0.361946	+	0.932199i	$0.382112\pi$
$332$	3.69755	0.202929
$333$	0	0
$334$	4.15585	0.227398
$335$	2.83611	0.154953
$336$	0	0
$337$	16.9942	0.925731	0.462866	−	0.886428i	$-0.346821\pi$
$337$	16.9942	0.925731	0.462866	+	0.886428i	$0.346821\pi$
$338$	5.31661	0.289185
$339$	0	0
$340$	6.91048	0.374774
$341$	−17.1214	−0.927178
$342$	0	0
$343$	−19.0518	−1.02870
$344$	7.29901	0.393536
$345$	0	0
$346$	8.90193	0.478571
$347$	13.8168	0.741724	0.370862	−	0.928688i	$-0.379062\pi$
$347$	13.8168	0.741724	0.370862	+	0.928688i	$0.379062\pi$
$348$	0	0
$349$	2.92340	0.156486	0.0782431	−	0.996934i	$-0.475069\pi$
$349$	2.92340	0.156486	0.0782431	+	0.996934i	$0.475069\pi$
$350$	−15.2731	−0.816380
$351$	0	0
$352$	−1.67350	−0.0891981
$353$	−20.3368	−1.08242	−0.541208	−	0.840889i	$-0.682033\pi$
$353$	−20.3368	−1.08242	−0.541208	+	0.840889i	$0.682033\pi$
$354$	0	0
$355$	−11.1325	−0.590854
$356$	−12.8667	−0.681933
$357$	0	0
$358$	12.9099	0.682308
$359$	28.1691	1.48671	0.743354	−	0.668898i	$-0.233234\pi$
$359$	28.1691	1.48671	0.743354	+	0.668898i	$0.233234\pi$
$360$	0	0
$361$	13.1158	0.690304
$362$	10.5212	0.552982
$363$	0	0
$364$	11.9020	0.623837
$365$	−9.42356	−0.493252
$366$	0	0
$367$	−6.03754	−0.315157	−0.157578	−	0.987506i	$-0.550369\pi$
$367$	−6.03754	−0.315157	−0.157578	+	0.987506i	$0.550369\pi$
$368$	−3.89843	−0.203220
$369$	0	0
$370$	−9.07744	−0.471914
$371$	0.901676	0.0468127
$372$	0	0
$373$	8.31784	0.430681	0.215341	−	0.976539i	$-0.430914\pi$
$373$	8.31784	0.430681	0.215341	+	0.976539i	$0.430914\pi$
$374$	9.62718	0.497810
$375$	0	0
$376$	−5.85394	−0.301894
$377$	10.1596	0.523245
$378$	0	0
$379$	15.4853	0.795425	0.397713	−	0.917510i	$-0.369804\pi$
$379$	15.4853	0.795425	0.397713	+	0.917510i	$0.369804\pi$
$380$	−6.80763	−0.349224
$381$	0	0
$382$	−23.8419	−1.21986
$383$	−15.0484	−0.768938	−0.384469	−	0.923138i	$-0.625615\pi$
$383$	−15.0484	−0.768938	−0.384469	+	0.923138i	$0.625615\pi$
$384$	0	0
$385$	8.63194	0.439925
$386$	17.2482	0.877908
$387$	0	0
$388$	−5.85506	−0.297246
$389$	−2.47122	−0.125296	−0.0626478	−	0.998036i	$-0.519954\pi$
$389$	−2.47122	−0.125296	−0.0626478	+	0.998036i	$0.519954\pi$
$390$	0	0
$391$	22.4265	1.13416
$392$	−11.4370	−0.577656
$393$	0	0
$394$	−21.3458	−1.07539
$395$	12.4129	0.624559
$396$	0	0
$397$	13.1443	0.659695	0.329847	−	0.944034i	$-0.393003\pi$
$397$	13.1443	0.659695	0.329847	+	0.944034i	$0.393003\pi$
$398$	−5.58409	−0.279905
$399$	0	0
$400$	−3.55698	−0.177849
$401$	36.1388	1.80469	0.902344	−	0.431018i	$-0.141845\pi$
$401$	36.1388	1.80469	0.902344	+	0.431018i	$0.141845\pi$
$402$	0	0
$403$	28.3589	1.41266
$404$	6.32189	0.314526
$405$	0	0
$406$	−15.7378	−0.781054
$407$	−12.6460	−0.626840
$408$	0	0
$409$	35.8167	1.77102	0.885512	−	0.464617i	$-0.153808\pi$
$409$	35.8167	1.77102	0.885512	+	0.464617i	$0.153808\pi$
$410$	−1.66686	−0.0823205
$411$	0	0
$412$	−10.9039	−0.537196
$413$	−8.36179	−0.411457
$414$	0	0
$415$	−4.44171	−0.218035
$416$	2.77189	0.135903
$417$	0	0
$418$	−9.48388	−0.463872
$419$	−8.65211	−0.422683	−0.211342	−	0.977412i	$-0.567783\pi$
$419$	−8.65211	−0.422683	−0.211342	+	0.977412i	$0.567783\pi$
$420$	0	0
$421$	17.5972	0.857633	0.428817	−	0.903392i	$-0.358931\pi$
$421$	17.5972	0.857633	0.428817	+	0.903392i	$0.358931\pi$
$422$	8.24942	0.401576
$423$	0	0
$424$	0.209993	0.0101982
$425$	20.4622	0.992565
$426$	0	0
$427$	−8.56680	−0.414577
$428$	−12.2897	−0.594043
$429$	0	0
$430$	−8.76799	−0.422830
$431$	−31.9670	−1.53980	−0.769898	−	0.638167i	$-0.779693\pi$
$431$	−31.9670	−1.53980	−0.769898	+	0.638167i	$0.779693\pi$
$432$	0	0
$433$	4.92015	0.236448	0.118224	−	0.992987i	$-0.462280\pi$
$433$	4.92015	0.236448	0.118224	+	0.992987i	$0.462280\pi$
$434$	−43.9297	−2.10870
$435$	0	0
$436$	9.50134	0.455032
$437$	−22.0927	−1.05684
$438$	0	0
$439$	−4.67906	−0.223319	−0.111660	−	0.993747i	$-0.535617\pi$
$439$	−4.67906	−0.223319	−0.111660	+	0.993747i	$0.535617\pi$
$440$	2.01031	0.0958379
$441$	0	0
$442$	−15.9459	−0.758468
$443$	−27.0433	−1.28487	−0.642433	−	0.766342i	$-0.722075\pi$
$443$	−27.0433	−1.28487	−0.642433	+	0.766342i	$0.722075\pi$
$444$	0	0
$445$	15.4562	0.732695
$446$	−10.4684	−0.495694
$447$	0	0
$448$	−4.29383	−0.202865
$449$	8.57230	0.404552	0.202276	−	0.979329i	$-0.435166\pi$
$449$	8.57230	0.404552	0.202276	+	0.979329i	$0.435166\pi$
$450$	0	0
$451$	−2.32215	−0.109346
$452$	−2.21368	−0.104123
$453$	0	0
$454$	−25.7810	−1.20996
$455$	−14.2974	−0.670274
$456$	0	0
$457$	28.7727	1.34593	0.672966	−	0.739673i	$-0.265020\pi$
$457$	28.7727	1.34593	0.672966	+	0.739673i	$0.265020\pi$
$458$	28.9284	1.35173
$459$	0	0
$460$	4.68302	0.218347
$461$	8.63284	0.402071	0.201036	−	0.979584i	$-0.435569\pi$
$461$	8.63284	0.402071	0.201036	+	0.979584i	$0.435569\pi$
$462$	0	0
$463$	−28.1712	−1.30923	−0.654614	−	0.755963i	$-0.727169\pi$
$463$	−28.1712	−1.30923	−0.654614	+	0.755963i	$0.727169\pi$
$464$	−3.66521	−0.170153
$465$	0	0
$466$	−5.51234	−0.255354
$467$	20.1411	0.932021	0.466011	−	0.884779i	$-0.345691\pi$
$467$	20.1411	0.932021	0.466011	+	0.884779i	$0.345691\pi$
$468$	0	0
$469$	10.1375	0.468107
$470$	7.03210	0.324367
$471$	0	0
$472$	−1.94740	−0.0896362
$473$	−12.2149	−0.561643
$474$	0	0
$475$	−20.1577	−0.924898
$476$	24.7012	1.13218
$477$	0	0
$478$	20.3231	0.929555
$479$	−28.4710	−1.30087	−0.650437	−	0.759560i	$-0.725414\pi$
$479$	−28.4710	−1.30087	−0.650437	+	0.759560i	$0.725414\pi$
$480$	0	0
$481$	20.9461	0.955061
$482$	23.3767	1.06478
$483$	0	0
$484$	−8.19938	−0.372699
$485$	7.03344	0.319372
$486$	0	0
$487$	22.6813	1.02779	0.513894	−	0.857854i	$-0.328203\pi$
$487$	22.6813	1.02779	0.513894	+	0.857854i	$0.328203\pi$
$488$	−1.99514	−0.0903158
$489$	0	0
$490$	13.7388	0.620656
$491$	19.6822	0.888246	0.444123	−	0.895966i	$-0.353515\pi$
$491$	19.6822	0.888246	0.444123	+	0.895966i	$0.353515\pi$
$492$	0	0
$493$	21.0849	0.949615
$494$	15.7085	0.706761
$495$	0	0
$496$	−10.2309	−0.459381
$497$	−39.7927	−1.78494
$498$	0	0
$499$	6.11985	0.273962	0.136981	−	0.990574i	$-0.456260\pi$
$499$	6.11985	0.273962	0.136981	+	0.990574i	$0.456260\pi$
$500$	10.2791	0.459697
$501$	0	0
$502$	−26.5660	−1.18570
$503$	−24.8603	−1.10847	−0.554233	−	0.832361i	$-0.686988\pi$
$503$	−24.8603	−1.10847	−0.554233	+	0.832361i	$0.686988\pi$
$504$	0	0
$505$	−7.59423	−0.337939
$506$	6.52404	0.290029
$507$	0	0
$508$	−10.9132	−0.484195
$509$	4.82156	0.213712	0.106856	−	0.994275i	$-0.465922\pi$
$509$	4.82156	0.213712	0.106856	+	0.994275i	$0.465922\pi$
$510$	0	0
$511$	−33.6840	−1.49009
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	−13.1963	−0.582065
$515$	13.0984	0.577184
$516$	0	0
$517$	9.79660	0.430854
$518$	−32.4468	−1.42563
$519$	0	0
$520$	−3.32976	−0.146020
$521$	24.1264	1.05700	0.528499	−	0.848934i	$-0.322755\pi$
$521$	24.1264	1.05700	0.528499	+	0.848934i	$0.322755\pi$
$522$	0	0
$523$	40.2704	1.76090	0.880451	−	0.474138i	$-0.157240\pi$
$523$	40.2704	1.76090	0.880451	+	0.474138i	$0.157240\pi$
$524$	7.79020	0.340317
$525$	0	0
$526$	10.4667	0.456371
$527$	58.8553	2.56378
$528$	0	0
$529$	−7.80226	−0.339229
$530$	−0.252256	−0.0109573
$531$	0	0
$532$	−24.3335	−1.05499
$533$	3.84627	0.166601
$534$	0	0
$535$	14.7631	0.638263
$536$	2.36095	0.101977
$537$	0	0
$538$	−27.7593	−1.19679
$539$	19.1399	0.824413
$540$	0	0
$541$	6.74172	0.289849	0.144925	−	0.989443i	$-0.453706\pi$
$541$	6.74172	0.289849	0.144925	+	0.989443i	$0.453706\pi$
$542$	26.9635	1.15818
$543$	0	0
$544$	5.75270	0.246645
$545$	−11.4136	−0.488904
$546$	0	0
$547$	−13.0462	−0.557816	−0.278908	−	0.960318i	$-0.589972\pi$
$547$	−13.0462	−0.557816	−0.278908	+	0.960318i	$0.589972\pi$
$548$	−8.27186	−0.353356
$549$	0	0
$550$	5.95262	0.253821
$551$	−20.7710	−0.884876
$552$	0	0
$553$	44.3692	1.88677
$554$	−14.7080	−0.624882
$555$	0	0
$556$	−7.15365	−0.303382
$557$	1.84618	0.0782254	0.0391127	−	0.999235i	$-0.487547\pi$
$557$	1.84618	0.0782254	0.0391127	+	0.999235i	$0.487547\pi$
$558$	0	0
$559$	20.2321	0.855725
$560$	5.15800	0.217965
$561$	0	0
$562$	6.64235	0.280191
$563$	40.7048	1.71550	0.857750	−	0.514066i	$-0.171861\pi$
$563$	40.7048	1.71550	0.857750	+	0.514066i	$0.171861\pi$
$564$	0	0
$565$	2.65921	0.111874
$566$	−12.4512	−0.523365
$567$	0	0
$568$	−9.26740	−0.388851
$569$	−20.4802	−0.858574	−0.429287	−	0.903168i	$-0.641235\pi$
$569$	−20.4802	−0.858574	−0.429287	+	0.903168i	$0.641235\pi$
$570$	0	0
$571$	1.93208	0.0808550	0.0404275	−	0.999182i	$-0.487128\pi$
$571$	1.93208	0.0808550	0.0404275	+	0.999182i	$0.487128\pi$
$572$	−4.63878	−0.193957
$573$	0	0
$574$	−5.95811	−0.248687
$575$	13.8666	0.578278
$576$	0	0
$577$	4.02426	0.167532	0.0837660	−	0.996485i	$-0.473305\pi$
$577$	4.02426	0.167532	0.0837660	+	0.996485i	$0.473305\pi$
$578$	−16.0936	−0.669406
$579$	0	0
$580$	4.40287	0.182819
$581$	−15.8767	−0.658675
$582$	0	0
$583$	−0.351425	−0.0145545
$584$	−7.84474	−0.324618
$585$	0	0
$586$	13.0780	0.540248
$587$	−5.88313	−0.242823	−0.121411	−	0.992602i	$-0.538742\pi$
$587$	−5.88313	−0.242823	−0.121411	+	0.992602i	$0.538742\pi$
$588$	0	0
$589$	−57.9793	−2.38899
$590$	2.33933	0.0963085
$591$	0	0
$592$	−7.55661	−0.310575
$593$	−25.7743	−1.05842	−0.529211	−	0.848490i	$-0.677512\pi$
$593$	−25.7743	−1.05842	−0.529211	+	0.848490i	$0.677512\pi$
$594$	0	0
$595$	−29.6725	−1.21645
$596$	1.00000	0.0409616
$597$	0	0
$598$	−10.8060	−0.441891
$599$	−24.9795	−1.02064	−0.510318	−	0.859986i	$-0.670472\pi$
$599$	−24.9795	−1.02064	−0.510318	+	0.859986i	$0.670472\pi$
$600$	0	0
$601$	10.1677	0.414748	0.207374	−	0.978262i	$-0.433508\pi$
$601$	10.1677	0.414748	0.207374	+	0.978262i	$0.433508\pi$
$602$	−31.3407	−1.27735
$603$	0	0
$604$	0.135399	0.00550931
$605$	9.84958	0.400442
$606$	0	0
$607$	−49.0887	−1.99245	−0.996224	−	0.0868242i	$-0.972328\pi$
$607$	−49.0887	−1.99245	−0.996224	+	0.0868242i	$0.972328\pi$
$608$	−5.66708	−0.229830
$609$	0	0
$610$	2.39668	0.0970387
$611$	−16.2265	−0.656454
$612$	0	0
$613$	−14.4594	−0.584009	−0.292004	−	0.956417i	$-0.594322\pi$
$613$	−14.4594	−0.584009	−0.292004	+	0.956417i	$0.594322\pi$
$614$	−31.0940	−1.25485
$615$	0	0
$616$	7.18575	0.289522
$617$	−22.0472	−0.887586	−0.443793	−	0.896129i	$-0.646367\pi$
$617$	−22.0472	−0.887586	−0.443793	+	0.896129i	$0.646367\pi$
$618$	0	0
$619$	−11.4459	−0.460051	−0.230025	−	0.973185i	$-0.573881\pi$
$619$	−11.4459	−0.460051	−0.230025	+	0.973185i	$0.573881\pi$
$620$	12.2899	0.493576
$621$	0	0
$622$	−20.7301	−0.831202
$623$	55.2474	2.21344
$624$	0	0
$625$	5.43699	0.217479
$626$	14.9338	0.596873
$627$	0	0
$628$	−12.5865	−0.502257
$629$	43.4709	1.73330
$630$	0	0
$631$	−40.5672	−1.61496	−0.807478	−	0.589897i	$-0.799168\pi$
$631$	−40.5672	−1.61496	−0.807478	+	0.589897i	$0.799168\pi$
$632$	10.3332	0.411034
$633$	0	0
$634$	−9.30324	−0.369479
$635$	13.1096	0.520238
$636$	0	0
$637$	−31.7022	−1.25609
$638$	6.13375	0.242838
$639$	0	0
$640$	1.20126	0.0474839
$641$	40.3310	1.59298	0.796489	−	0.604653i	$-0.206688\pi$
$641$	40.3310	1.59298	0.796489	+	0.604653i	$0.206688\pi$
$642$	0	0
$643$	−20.4803	−0.807662	−0.403831	−	0.914834i	$-0.632322\pi$
$643$	−20.4803	−0.807662	−0.403831	+	0.914834i	$0.632322\pi$
$644$	16.7392	0.659617
$645$	0	0
$646$	32.6010	1.28267
$647$	−33.0332	−1.29867	−0.649334	−	0.760504i	$-0.724952\pi$
$647$	−33.0332	−1.29867	−0.649334	+	0.760504i	$0.724952\pi$
$648$	0	0
$649$	3.25898	0.127926
$650$	−9.85957	−0.386724
$651$	0	0
$652$	14.2103	0.556518
$653$	−16.9821	−0.664561	−0.332281	−	0.943181i	$-0.607818\pi$
$653$	−16.9821	−0.664561	−0.332281	+	0.943181i	$0.607818\pi$
$654$	0	0
$655$	−9.35805	−0.365649
$656$	−1.38760	−0.0541766
$657$	0	0
$658$	25.1359	0.979898
$659$	−1.32800	−0.0517314	−0.0258657	−	0.999665i	$-0.508234\pi$
$659$	−1.32800	−0.0517314	−0.0258657	+	0.999665i	$0.508234\pi$
$660$	0	0
$661$	27.7686	1.08007	0.540037	−	0.841641i	$-0.318410\pi$
$661$	27.7686	1.08007	0.540037	+	0.841641i	$0.318410\pi$
$662$	−13.1701	−0.511869
$663$	0	0
$664$	−3.69755	−0.143493
$665$	29.2308	1.13352
$666$	0	0
$667$	14.2886	0.553255
$668$	−4.15585	−0.160795
$669$	0	0
$670$	−2.83611	−0.109568
$671$	3.33888	0.128896
$672$	0	0
$673$	30.5773	1.17867	0.589334	−	0.807889i	$-0.299390\pi$
$673$	30.5773	1.17867	0.589334	+	0.807889i	$0.299390\pi$
$674$	−16.9942	−0.654591
$675$	0	0
$676$	−5.31661	−0.204485
$677$	−9.42825	−0.362357	−0.181178	−	0.983450i	$-0.557991\pi$
$677$	−9.42825	−0.362357	−0.181178	+	0.983450i	$0.557991\pi$
$678$	0	0
$679$	25.1407	0.964810
$680$	−6.91048	−0.265005
$681$	0	0
$682$	17.1214	0.655614
$683$	25.8146	0.987767	0.493884	−	0.869528i	$-0.335577\pi$
$683$	25.8146	0.987767	0.493884	+	0.869528i	$0.335577\pi$
$684$	0	0
$685$	9.93664	0.379660
$686$	19.0518	0.727400
$687$	0	0
$688$	−7.29901	−0.278272
$689$	0.582079	0.0221754
$690$	0	0
$691$	−19.6482	−0.747454	−0.373727	−	0.927539i	$-0.621920\pi$
$691$	−19.6482	−0.747454	−0.373727	+	0.927539i	$0.621920\pi$
$692$	−8.90193	−0.338401
$693$	0	0
$694$	−13.8168	−0.524478
$695$	8.59338	0.325966
$696$	0	0
$697$	7.98244	0.302356
$698$	−2.92340	−0.110652
$699$	0	0
$700$	15.2731	0.577268
$701$	−2.17682	−0.0822172	−0.0411086	−	0.999155i	$-0.513089\pi$
$701$	−2.17682	−0.0822172	−0.0411086	+	0.999155i	$0.513089\pi$
$702$	0	0
$703$	−42.8239	−1.61513
$704$	1.67350	0.0630726
$705$	0	0
$706$	20.3368	0.765384
$707$	−27.1452	−1.02090
$708$	0	0
$709$	−0.337527	−0.0126761	−0.00633805	−	0.999980i	$-0.502017\pi$
$709$	−0.337527	−0.0126761	−0.00633805	+	0.999980i	$0.502017\pi$
$710$	11.1325	0.417797
$711$	0	0
$712$	12.8667	0.482200
$713$	39.8844	1.49368
$714$	0	0
$715$	5.57237	0.208395
$716$	−12.9099	−0.482465
$717$	0	0
$718$	−28.1691	−1.05126
$719$	−30.1660	−1.12500	−0.562501	−	0.826797i	$-0.690161\pi$
$719$	−30.1660	−1.12500	−0.562501	+	0.826797i	$0.690161\pi$
$720$	0	0
$721$	46.8195	1.74365
$722$	−13.1158	−0.488119
$723$	0	0
$724$	−10.5212	−0.391018
$725$	13.0371	0.484185
$726$	0	0
$727$	1.99176	0.0738705	0.0369352	−	0.999318i	$-0.488240\pi$
$727$	1.99176	0.0738705	0.0369352	+	0.999318i	$0.488240\pi$
$728$	−11.9020	−0.441119
$729$	0	0
$730$	9.42356	0.348782
$731$	41.9890	1.55302
$732$	0	0
$733$	18.5615	0.685586	0.342793	−	0.939411i	$-0.388627\pi$
$733$	18.5615	0.685586	0.342793	+	0.939411i	$0.388627\pi$
$734$	6.03754	0.222850
$735$	0	0
$736$	3.89843	0.143698
$737$	−3.95106	−0.145539
$738$	0	0
$739$	−5.18207	−0.190625	−0.0953127	−	0.995447i	$-0.530385\pi$
$739$	−5.18207	−0.190625	−0.0953127	+	0.995447i	$0.530385\pi$
$740$	9.07744	0.333693
$741$	0	0
$742$	−0.901676	−0.0331016
$743$	−33.6377	−1.23405	−0.617024	−	0.786944i	$-0.711662\pi$
$743$	−33.6377	−1.23405	−0.617024	+	0.786944i	$0.711662\pi$
$744$	0	0
$745$	−1.20126	−0.0440107
$746$	−8.31784	−0.304538
$747$	0	0
$748$	−9.62718	−0.352005
$749$	52.7698	1.92817
$750$	0	0
$751$	18.4176	0.672068	0.336034	−	0.941850i	$-0.390914\pi$
$751$	18.4176	0.672068	0.336034	+	0.941850i	$0.390914\pi$
$752$	5.85394	0.213471
$753$	0	0
$754$	−10.1596	−0.369990
$755$	−0.162649	−0.00591942
$756$	0	0
$757$	−1.84141	−0.0669272	−0.0334636	−	0.999440i	$-0.510654\pi$
$757$	−1.84141	−0.0669272	−0.0334636	+	0.999440i	$0.510654\pi$
$758$	−15.4853	−0.562450
$759$	0	0
$760$	6.80763	0.246939
$761$	14.1293	0.512187	0.256093	−	0.966652i	$-0.417565\pi$
$761$	14.1293	0.512187	0.256093	+	0.966652i	$0.417565\pi$
$762$	0	0
$763$	−40.7972	−1.47696
$764$	23.8419	0.862568
$765$	0	0
$766$	15.0484	0.543721
$767$	−5.39797	−0.194910
$768$	0	0
$769$	37.9796	1.36958	0.684790	−	0.728741i	$-0.259894\pi$
$769$	37.9796	1.36958	0.684790	+	0.728741i	$0.259894\pi$
$770$	−8.63194	−0.311074
$771$	0	0
$772$	−17.2482	−0.620775
$773$	−42.8857	−1.54249	−0.771245	−	0.636538i	$-0.780366\pi$
$773$	−42.8857	−1.54249	−0.771245	+	0.636538i	$0.780366\pi$
$774$	0	0
$775$	36.3911	1.30721
$776$	5.85506	0.210185
$777$	0	0
$778$	2.47122	0.0885974
$779$	−7.86363	−0.281744
$780$	0	0
$781$	15.5090	0.554957
$782$	−22.4265	−0.801970
$783$	0	0
$784$	11.4370	0.408464
$785$	15.1197	0.539645
$786$	0	0
$787$	−25.8911	−0.922917	−0.461458	−	0.887162i	$-0.652674\pi$
$787$	−25.8911	−0.922917	−0.461458	+	0.887162i	$0.652674\pi$
$788$	21.3458	0.760412
$789$	0	0
$790$	−12.4129	−0.441630
$791$	9.50519	0.337966
$792$	0	0
$793$	−5.53032	−0.196387
$794$	−13.1443	−0.466474
$795$	0	0
$796$	5.58409	0.197923
$797$	−20.9646	−0.742604	−0.371302	−	0.928512i	$-0.621089\pi$
$797$	−20.9646	−0.742604	−0.371302	+	0.928512i	$0.621089\pi$
$798$	0	0
$799$	−33.6760	−1.19137
$800$	3.55698	0.125758
$801$	0	0
$802$	−36.1388	−1.27611
$803$	13.1282	0.463285
$804$	0	0
$805$	−20.1081	−0.708717
$806$	−28.3589	−0.998901
$807$	0	0
$808$	−6.32189	−0.222403
$809$	10.8597	0.381805	0.190903	−	0.981609i	$-0.438859\pi$
$809$	10.8597	0.381805	0.190903	+	0.981609i	$0.438859\pi$
$810$	0	0
$811$	46.9551	1.64882	0.824408	−	0.565995i	$-0.191508\pi$
$811$	46.9551	1.64882	0.824408	+	0.565995i	$0.191508\pi$
$812$	15.7378	0.552289
$813$	0	0
$814$	12.6460	0.443243
$815$	−17.0702	−0.597944
$816$	0	0
$817$	−41.3640	−1.44714
$818$	−35.8167	−1.25230
$819$	0	0
$820$	1.66686	0.0582094
$821$	11.5063	0.401573	0.200786	−	0.979635i	$-0.435650\pi$
$821$	11.5063	0.401573	0.200786	+	0.979635i	$0.435650\pi$
$822$	0	0
$823$	−31.4829	−1.09742	−0.548712	−	0.836012i	$-0.684881\pi$
$823$	−31.4829	−1.09742	−0.548712	+	0.836012i	$0.684881\pi$
$824$	10.9039	0.379855
$825$	0	0
$826$	8.36179	0.290944
$827$	16.5022	0.573837	0.286918	−	0.957955i	$-0.407369\pi$
$827$	16.5022	0.573837	0.286918	+	0.957955i	$0.407369\pi$
$828$	0	0
$829$	29.7832	1.03441	0.517207	−	0.855860i	$-0.326972\pi$
$829$	29.7832	1.03441	0.517207	+	0.855860i	$0.326972\pi$
$830$	4.44171	0.154174
$831$	0	0
$832$	−2.77189	−0.0960981
$833$	−65.7937	−2.27962
$834$	0	0
$835$	4.99225	0.172764
$836$	9.48388	0.328007
$837$	0	0
$838$	8.65211	0.298882
$839$	20.9568	0.723507	0.361754	−	0.932274i	$-0.382178\pi$
$839$	20.9568	0.723507	0.361754	+	0.932274i	$0.382178\pi$
$840$	0	0
$841$	−15.5662	−0.536766
$842$	−17.5972	−0.606438
$843$	0	0
$844$	−8.24942	−0.283957
$845$	6.38662	0.219706
$846$	0	0
$847$	35.2068	1.20972
$848$	−0.209993	−0.00721120
$849$	0	0
$850$	−20.4622	−0.701849
$851$	29.4589	1.00984
$852$	0	0
$853$	−5.64517	−0.193287	−0.0966435	−	0.995319i	$-0.530811\pi$
$853$	−5.64517	−0.193287	−0.0966435	+	0.995319i	$0.530811\pi$
$854$	8.56680	0.293150
$855$	0	0
$856$	12.2897	0.420052
$857$	−11.8928	−0.406251	−0.203126	−	0.979153i	$-0.565110\pi$
$857$	−11.8928	−0.406251	−0.203126	+	0.979153i	$0.565110\pi$
$858$	0	0
$859$	−6.44572	−0.219925	−0.109963	−	0.993936i	$-0.535073\pi$
$859$	−6.44572	−0.219925	−0.109963	+	0.993936i	$0.535073\pi$
$860$	8.76799	0.298986
$861$	0	0
$862$	31.9670	1.08880
$863$	8.84939	0.301237	0.150618	−	0.988592i	$-0.451874\pi$
$863$	8.84939	0.301237	0.150618	+	0.988592i	$0.451874\pi$
$864$	0	0
$865$	10.6935	0.363591
$866$	−4.92015	−0.167194
$867$	0	0
$868$	43.9297	1.49107
$869$	−17.2927	−0.586615
$870$	0	0
$871$	6.54429	0.221745
$872$	−9.50134	−0.321756
$873$	0	0
$874$	22.0927	0.747297
$875$	−44.1369	−1.49210
$876$	0	0
$877$	42.5241	1.43594	0.717968	−	0.696076i	$-0.245072\pi$
$877$	42.5241	1.43594	0.717968	+	0.696076i	$0.245072\pi$
$878$	4.67906	0.157911
$879$	0	0
$880$	−2.01031	−0.0677676
$881$	−24.3006	−0.818708	−0.409354	−	0.912376i	$-0.634246\pi$
$881$	−24.3006	−0.818708	−0.409354	+	0.912376i	$0.634246\pi$
$882$	0	0
$883$	−29.0897	−0.978947	−0.489473	−	0.872018i	$-0.662811\pi$
$883$	−29.0897	−0.978947	−0.489473	+	0.872018i	$0.662811\pi$
$884$	15.9459	0.536318
$885$	0	0
$886$	27.0433	0.908538
$887$	−35.5771	−1.19456	−0.597282	−	0.802032i	$-0.703753\pi$
$887$	−35.5771	−1.19456	−0.597282	+	0.802032i	$0.703753\pi$
$888$	0	0
$889$	46.8595	1.57162
$890$	−15.4562	−0.518094
$891$	0	0
$892$	10.4684	0.350508
$893$	33.1748	1.11015
$894$	0	0
$895$	15.5081	0.518379
$896$	4.29383	0.143447
$897$	0	0
$898$	−8.57230	−0.286061
$899$	37.4984	1.25064
$900$	0	0
$901$	1.20803	0.0402453
$902$	2.32215	0.0773192
$903$	0	0
$904$	2.21368	0.0736260
$905$	12.6387	0.420124
$906$	0	0
$907$	20.9998	0.697286	0.348643	−	0.937256i	$-0.386643\pi$
$907$	20.9998	0.697286	0.348643	+	0.937256i	$0.386643\pi$
$908$	25.7810	0.855572
$909$	0	0
$910$	14.2974	0.473955
$911$	−29.5524	−0.979116	−0.489558	−	0.871971i	$-0.662842\pi$
$911$	−29.5524	−0.979116	−0.489558	+	0.871971i	$0.662842\pi$
$912$	0	0
$913$	6.18787	0.204788
$914$	−28.7727	−0.951718
$915$	0	0
$916$	−28.9284	−0.955821
$917$	−33.4498	−1.10461
$918$	0	0
$919$	−33.4974	−1.10498	−0.552488	−	0.833521i	$-0.686322\pi$
$919$	−33.4974	−1.10498	−0.552488	+	0.833521i	$0.686322\pi$
$920$	−4.68302	−0.154395
$921$	0	0
$922$	−8.63284	−0.284307
$923$	−25.6882	−0.845539
$924$	0	0
$925$	26.8787	0.883766
$926$	28.1712	0.925764
$927$	0	0
$928$	3.66521	0.120316
$929$	−29.0055	−0.951639	−0.475819	−	0.879543i	$-0.657848\pi$
$929$	−29.0055	−0.951639	−0.475819	+	0.879543i	$0.657848\pi$
$930$	0	0
$931$	64.8144	2.12421
$932$	5.51234	0.180563
$933$	0	0
$934$	−20.1411	−0.659038
$935$	11.5647	0.378207
$936$	0	0
$937$	−5.51100	−0.180037	−0.0900183	−	0.995940i	$-0.528693\pi$
$937$	−5.51100	−0.180037	−0.0900183	+	0.995940i	$0.528693\pi$
$938$	−10.1375	−0.331002
$939$	0	0
$940$	−7.03210	−0.229362
$941$	−33.4090	−1.08910	−0.544552	−	0.838727i	$-0.683300\pi$
$941$	−33.4090	−1.08910	−0.544552	+	0.838727i	$0.683300\pi$
$942$	0	0
$943$	5.40945	0.176156
$944$	1.94740	0.0633823
$945$	0	0
$946$	12.2149	0.397141
$947$	−38.5650	−1.25319	−0.626597	−	0.779344i	$-0.715553\pi$
$947$	−38.5650	−1.25319	−0.626597	+	0.779344i	$0.715553\pi$
$948$	0	0
$949$	−21.7448	−0.705866
$950$	20.1577	0.654001
$951$	0	0
$952$	−24.7012	−0.800569
$953$	34.8019	1.12734	0.563672	−	0.825999i	$-0.309388\pi$
$953$	34.8019	1.12734	0.563672	+	0.825999i	$0.309388\pi$
$954$	0	0
$955$	−28.6402	−0.926776
$956$	−20.3231	−0.657295
$957$	0	0
$958$	28.4710	0.919857
$959$	35.5180	1.14694
$960$	0	0
$961$	73.6712	2.37649
$962$	−20.9461	−0.675330
$963$	0	0
$964$	−23.3767	−0.752911
$965$	20.7195	0.666984
$966$	0	0
$967$	−17.0912	−0.549616	−0.274808	−	0.961499i	$-0.588614\pi$
$967$	−17.0912	−0.549616	−0.274808	+	0.961499i	$0.588614\pi$
$968$	8.19938	0.263538
$969$	0	0
$970$	−7.03344	−0.225830
$971$	7.44521	0.238928	0.119464	−	0.992839i	$-0.461882\pi$
$971$	7.44521	0.238928	0.119464	+	0.992839i	$0.461882\pi$
$972$	0	0
$973$	30.7166	0.984729
$974$	−22.6813	−0.726756
$975$	0	0
$976$	1.99514	0.0638629
$977$	−17.4152	−0.557161	−0.278581	−	0.960413i	$-0.589864\pi$
$977$	−17.4152	−0.557161	−0.278581	+	0.960413i	$0.589864\pi$
$978$	0	0
$979$	−21.5325	−0.688181
$980$	−13.7388	−0.438870
$981$	0	0
$982$	−19.6822	−0.628085
$983$	56.5864	1.80483	0.902413	−	0.430872i	$-0.141794\pi$
$983$	56.5864	1.80483	0.902413	+	0.430872i	$0.141794\pi$
$984$	0	0
$985$	−25.6418	−0.817016
$986$	−21.0849	−0.671479
$987$	0	0
$988$	−15.7085	−0.499755
$989$	28.4546	0.904805
$990$	0	0
$991$	−42.9138	−1.36320	−0.681601	−	0.731724i	$-0.738716\pi$
$991$	−42.9138	−1.36320	−0.681601	+	0.731724i	$0.738716\pi$
$992$	10.2309	0.324831
$993$	0	0
$994$	39.7927	1.26215
$995$	−6.70794	−0.212656
$996$	0	0
$997$	47.3180	1.49858	0.749288	−	0.662244i	$-0.230396\pi$
$997$	47.3180	1.49858	0.749288	+	0.662244i	$0.230396\pi$
$998$	−6.11985	−0.193721
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8046.2.a.l.1.3	✓	12
3.2	odd	2		8046.2.a.m.1.10	yes	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8046.2.a.l.1.3	✓	12	1.1	even	1	trivial
8046.2.a.m.1.10	yes	12	3.2	odd	2

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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} -1.20126 q^{5} -4.29383 q^{7} -1.00000 q^{8} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{4} -1.20126 q^{5} -4.29383 q^{7} -1.00000 q^{8} +1.20126 q^{10} +1.67350 q^{11} -2.77189 q^{13} +4.29383 q^{14} +1.00000 q^{16} -5.75270 q^{17} +5.66708 q^{19} -1.20126 q^{20} -1.67350 q^{22} -3.89843 q^{23} -3.55698 q^{25} +2.77189 q^{26} -4.29383 q^{28} -3.66521 q^{29} -10.2309 q^{31} -1.00000 q^{32} +5.75270 q^{34} +5.15800 q^{35} -7.55661 q^{37} -5.66708 q^{38} +1.20126 q^{40} -1.38760 q^{41} -7.29901 q^{43} +1.67350 q^{44} +3.89843 q^{46} +5.85394 q^{47} +11.4370 q^{49} +3.55698 q^{50} -2.77189 q^{52} -0.209993 q^{53} -2.01031 q^{55} +4.29383 q^{56} +3.66521 q^{58} +1.94740 q^{59} +1.99514 q^{61} +10.2309 q^{62} +1.00000 q^{64} +3.32976 q^{65} -2.36095 q^{67} -5.75270 q^{68} -5.15800 q^{70} +9.26740 q^{71} +7.84474 q^{73} +7.55661 q^{74} +5.66708 q^{76} -7.18575 q^{77} -10.3332 q^{79} -1.20126 q^{80} +1.38760 q^{82} +3.69755 q^{83} +6.91048 q^{85} +7.29901 q^{86} -1.67350 q^{88} -12.8667 q^{89} +11.9020 q^{91} -3.89843 q^{92} -5.85394 q^{94} -6.80763 q^{95} -5.85506 q^{97} -11.4370 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 12 q^{2} + 12 q^{4} + 5 q^{5} - 6 q^{7} - 12 q^{8}+O(q^{10}) \) 12 * q - 12 * q^2 + 12 * q^4 + 5 * q^5 - 6 * q^7 - 12 * q^8	\( 12 q - 12 q^{2} + 12 q^{4} + 5 q^{5} - 6 q^{7} - 12 q^{8} - 5 q^{10} + 10 q^{11} - q^{13} + 6 q^{14} + 12 q^{16} + 6 q^{17} - 10 q^{19} + 5 q^{20} - 10 q^{22} + 15 q^{23} + 7 q^{25} + q^{26} - 6 q^{28} + 33 q^{29} - 6 q^{31} - 12 q^{32} - 6 q^{34} + 16 q^{35} - 13 q^{37} + 10 q^{38} - 5 q^{40} + 20 q^{41} - 11 q^{43} + 10 q^{44} - 15 q^{46} + 15 q^{47} + 2 q^{49} - 7 q^{50} - q^{52} + 4 q^{53} - 17 q^{55} + 6 q^{56} - 33 q^{58} + 10 q^{59} - 12 q^{61} + 6 q^{62} + 12 q^{64} + 40 q^{65} - 19 q^{67} + 6 q^{68} - 16 q^{70} + 47 q^{71} - 2 q^{73} + 13 q^{74} - 10 q^{76} - 6 q^{77} - 15 q^{79} + 5 q^{80} - 20 q^{82} + 18 q^{83} - 25 q^{85} + 11 q^{86} - 10 q^{88} + 24 q^{89} - 3 q^{91} + 15 q^{92} - 15 q^{94} - 3 q^{95} - 25 q^{97} - 2 q^{98}+O(q^{100}) \) 12 * q - 12 * q^2 + 12 * q^4 + 5 * q^5 - 6 * q^7 - 12 * q^8 - 5 * q^10 + 10 * q^11 - q^13 + 6 * q^14 + 12 * q^16 + 6 * q^17 - 10 * q^19 + 5 * q^20 - 10 * q^22 + 15 * q^23 + 7 * q^25 + q^26 - 6 * q^28 + 33 * q^29 - 6 * q^31 - 12 * q^32 - 6 * q^34 + 16 * q^35 - 13 * q^37 + 10 * q^38 - 5 * q^40 + 20 * q^41 - 11 * q^43 + 10 * q^44 - 15 * q^46 + 15 * q^47 + 2 * q^49 - 7 * q^50 - q^52 + 4 * q^53 - 17 * q^55 + 6 * q^56 - 33 * q^58 + 10 * q^59 - 12 * q^61 + 6 * q^62 + 12 * q^64 + 40 * q^65 - 19 * q^67 + 6 * q^68 - 16 * q^70 + 47 * q^71 - 2 * q^73 + 13 * q^74 - 10 * q^76 - 6 * q^77 - 15 * q^79 + 5 * q^80 - 20 * q^82 + 18 * q^83 - 25 * q^85 + 11 * q^86 - 10 * q^88 + 24 * q^89 - 3 * q^91 + 15 * q^92 - 15 * q^94 - 3 * q^95 - 25 * q^97 - 2 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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