LMFDB - Embedded newform 6003.2.a.s.1.12

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6003,2,Mod(1,6003)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6003, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("6003.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6003 = 3^{2} \cdot 23 \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6003.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:	$47.9341963334$
Analytic rank:	$0$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - 2 x^{19} - 33 x^{18} + 64 x^{17} + 453 x^{16} - 846 x^{15} - 3353 x^{14} + 5985 x^{13} + \cdots - 256 $ x^20 - 2x^19 - 33x^18 + 64x^17 + 453x^16 - 846x^15 - 3353x^14 + 5985x^13 + 14484x^12 - 24566x^11 - 36791x^10 + 59410x^9 + 52109x^8 - 82362x^7 - 34967x^6 + 60661x^5 + 5201x^4 - 19624x^3 + 2768x^2 + 1408*x - 256
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{7} $
Twist minimal:	no (minimal twist has level 2001)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.12
Root			$-0.302778$ of defining polynomial
Character	$\chi$	$=$	6003.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.302778 q^{2} -1.90833 q^{4} -3.34986 q^{5} +0.428154 q^{7} -1.18336 q^{8} +O(q^{10})$	\(q+0.302778 q^{2} -1.90833 q^{4} -3.34986 q^{5} +0.428154 q^{7} -1.18336 q^{8} -1.01426 q^{10} -4.01497 q^{11} +5.31187 q^{13} +0.129636 q^{14} +3.45836 q^{16} -1.05018 q^{17} -3.12533 q^{19} +6.39262 q^{20} -1.21565 q^{22} -1.00000 q^{23} +6.22156 q^{25} +1.60832 q^{26} -0.817056 q^{28} -1.00000 q^{29} -10.3617 q^{31} +3.41383 q^{32} -0.317973 q^{34} -1.43425 q^{35} -10.3543 q^{37} -0.946281 q^{38} +3.96408 q^{40} -7.14375 q^{41} +3.22443 q^{43} +7.66187 q^{44} -0.302778 q^{46} -10.8487 q^{47} -6.81668 q^{49} +1.88375 q^{50} -10.1368 q^{52} +9.70877 q^{53} +13.4496 q^{55} -0.506658 q^{56} -0.302778 q^{58} +4.68976 q^{59} -4.20131 q^{61} -3.13730 q^{62} -5.88308 q^{64} -17.7940 q^{65} -3.89854 q^{67} +2.00409 q^{68} -0.434261 q^{70} -13.7035 q^{71} +8.38069 q^{73} -3.13506 q^{74} +5.96414 q^{76} -1.71902 q^{77} +8.80707 q^{79} -11.5850 q^{80} -2.16297 q^{82} -7.18985 q^{83} +3.51797 q^{85} +0.976287 q^{86} +4.75114 q^{88} +10.2900 q^{89} +2.27430 q^{91} +1.90833 q^{92} -3.28475 q^{94} +10.4694 q^{95} -3.31072 q^{97} -2.06394 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q - 2 q^{2} + 30 q^{4} + q^{5} + 9 q^{7} - 6 q^{8}+O(q^{10}) $ 20 * q - 2 * q^2 + 30 * q^4 + q^5 + 9 * q^7 - 6 * q^8	$ 20 q - 2 q^{2} + 30 q^{4} + q^{5} + 9 q^{7} - 6 q^{8} + 7 q^{10} + 21 q^{13} + q^{14} + 58 q^{16} + 4 q^{17} + 7 q^{19} + 20 q^{20} + 7 q^{22} - 20 q^{23} + 47 q^{25} - 8 q^{26} + 11 q^{28} - 20 q^{29} + 28 q^{31} - 14 q^{32} + 16 q^{34} - 9 q^{35} + 14 q^{37} + 20 q^{38} + 34 q^{40} - 7 q^{41} + 3 q^{43} + q^{44} + 2 q^{46} - 3 q^{47} + 35 q^{49} + 24 q^{50} + 73 q^{52} + 19 q^{53} + 29 q^{55} + 30 q^{56} + 2 q^{58} - 20 q^{59} + 15 q^{61} - 12 q^{62} + 82 q^{64} + 28 q^{65} + 20 q^{67} + 23 q^{68} - 24 q^{70} - 63 q^{71} + 19 q^{73} - 16 q^{74} - 44 q^{76} + 7 q^{77} + 32 q^{79} + 56 q^{80} - 20 q^{82} + 21 q^{83} + 4 q^{85} + 6 q^{86} + 55 q^{88} + 13 q^{89} + 70 q^{91} - 30 q^{92} - 12 q^{94} - 9 q^{95} - 9 q^{97} - 31 q^{98}+O(q^{100}) $ 20 * q - 2 * q^2 + 30 * q^4 + q^5 + 9 * q^7 - 6 * q^8 + 7 * q^10 + 21 * q^13 + q^14 + 58 * q^16 + 4 * q^17 + 7 * q^19 + 20 * q^20 + 7 * q^22 - 20 * q^23 + 47 * q^25 - 8 * q^26 + 11 * q^28 - 20 * q^29 + 28 * q^31 - 14 * q^32 + 16 * q^34 - 9 * q^35 + 14 * q^37 + 20 * q^38 + 34 * q^40 - 7 * q^41 + 3 * q^43 + q^44 + 2 * q^46 - 3 * q^47 + 35 * q^49 + 24 * q^50 + 73 * q^52 + 19 * q^53 + 29 * q^55 + 30 * q^56 + 2 * q^58 - 20 * q^59 + 15 * q^61 - 12 * q^62 + 82 * q^64 + 28 * q^65 + 20 * q^67 + 23 * q^68 - 24 * q^70 - 63 * q^71 + 19 * q^73 - 16 * q^74 - 44 * q^76 + 7 * q^77 + 32 * q^79 + 56 * q^80 - 20 * q^82 + 21 * q^83 + 4 * q^85 + 6 * q^86 + 55 * q^88 + 13 * q^89 + 70 * q^91 - 30 * q^92 - 12 * q^94 - 9 * q^95 - 9 * q^97 - 31 * 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 $

$2$	0.302778	0.214097	0.107048	−	0.994254i	$-0.465860\pi$
$2$	0.302778	0.214097	0.107048	+	0.994254i	$0.465860\pi$
$3$	0	0
$3$	0	0
$4$	−1.90833	−0.954163
$5$	−3.34986	−1.49810	−0.749051	−	0.662512i	$-0.769490\pi$
$5$	−3.34986	−1.49810	−0.749051	+	0.662512i	$0.769490\pi$
$6$	0	0
$7$	0.428154	0.161827	0.0809134	−	0.996721i	$-0.474216\pi$
$7$	0.428154	0.161827	0.0809134	+	0.996721i	$0.474216\pi$
$8$	−1.18336	−0.418379
$9$	0	0
$10$	−1.01426	−0.320739
$11$	−4.01497	−1.21056	−0.605279	−	0.796013i	$-0.706939\pi$
$11$	−4.01497	−1.21056	−0.605279	+	0.796013i	$0.706939\pi$
$12$	0	0
$13$	5.31187	1.47325	0.736624	−	0.676302i	$-0.236419\pi$
$13$	5.31187	1.47325	0.736624	+	0.676302i	$0.236419\pi$
$14$	0.129636	0.0346466
$15$	0	0
$16$	3.45836	0.864589
$17$	−1.05018	−0.254707	−0.127354	−	0.991857i	$-0.540648\pi$
$17$	−1.05018	−0.254707	−0.127354	+	0.991857i	$0.540648\pi$
$18$	0	0
$19$	−3.12533	−0.717000	−0.358500	−	0.933530i	$-0.616712\pi$
$19$	−3.12533	−0.717000	−0.358500	+	0.933530i	$0.616712\pi$
$20$	6.39262	1.42943
$21$	0	0
$22$	−1.21565	−0.259176
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	6.22156	1.24431
$26$	1.60832	0.315418
$27$	0	0
$28$	−0.817056	−0.154409
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	0	0
$31$	−10.3617	−1.86102	−0.930508	−	0.366271i	$-0.880634\pi$
$31$	−10.3617	−1.86102	−0.930508	+	0.366271i	$0.880634\pi$
$32$	3.41383	0.603485
$33$	0	0
$34$	−0.317973	−0.0545319
$35$	−1.43425	−0.242433
$36$	0	0
$37$	−10.3543	−1.70224	−0.851119	−	0.524972i	$-0.824076\pi$
$37$	−10.3543	−1.70224	−0.851119	+	0.524972i	$0.824076\pi$
$38$	−0.946281	−0.153507
$39$	0	0
$40$	3.96408	0.626775
$41$	−7.14375	−1.11567	−0.557833	−	0.829953i	$-0.688367\pi$
$41$	−7.14375	−1.11567	−0.557833	+	0.829953i	$0.688367\pi$
$42$	0	0
$43$	3.22443	0.491721	0.245861	−	0.969305i	$-0.420929\pi$
$43$	3.22443	0.491721	0.245861	+	0.969305i	$0.420929\pi$
$44$	7.66187	1.15507
$45$	0	0
$46$	−0.302778	−0.0446422
$47$	−10.8487	−1.58244	−0.791222	−	0.611529i	$-0.790555\pi$
$47$	−10.8487	−1.58244	−0.791222	+	0.611529i	$0.790555\pi$
$48$	0	0
$49$	−6.81668	−0.973812
$50$	1.88375	0.266403
$51$	0	0
$52$	−10.1368	−1.40572
$53$	9.70877	1.33360	0.666801	−	0.745236i	$-0.267663\pi$
$53$	9.70877	1.33360	0.666801	+	0.745236i	$0.267663\pi$
$54$	0	0
$55$	13.4496	1.81354
$56$	−0.506658	−0.0677050
$57$	0	0
$58$	−0.302778	−0.0397567
$59$	4.68976	0.610555	0.305277	−	0.952263i	$-0.401251\pi$
$59$	4.68976	0.610555	0.305277	+	0.952263i	$0.401251\pi$
$60$	0	0
$61$	−4.20131	−0.537923	−0.268962	−	0.963151i	$-0.586680\pi$
$61$	−4.20131	−0.537923	−0.268962	+	0.963151i	$0.586680\pi$
$62$	−3.13730	−0.398437
$63$	0	0
$64$	−5.88308	−0.735385
$65$	−17.7940	−2.20708
$66$	0	0
$67$	−3.89854	−0.476283	−0.238141	−	0.971231i	$-0.576538\pi$
$67$	−3.89854	−0.476283	−0.238141	+	0.971231i	$0.576538\pi$
$68$	2.00409	0.243032
$69$	0	0
$70$	−0.434261	−0.0519041
$71$	−13.7035	−1.62630	−0.813152	−	0.582052i	$-0.802250\pi$
$71$	−13.7035	−1.62630	−0.813152	+	0.582052i	$0.802250\pi$
$72$	0	0
$73$	8.38069	0.980886	0.490443	−	0.871473i	$-0.336835\pi$
$73$	8.38069	0.980886	0.490443	+	0.871473i	$0.336835\pi$
$74$	−3.13506	−0.364443
$75$	0	0
$76$	5.96414	0.684134
$77$	−1.71902	−0.195901
$78$	0	0
$79$	8.80707	0.990873	0.495437	−	0.868644i	$-0.335008\pi$
$79$	8.80707	0.990873	0.495437	+	0.868644i	$0.335008\pi$
$80$	−11.5850	−1.29524
$81$	0	0
$82$	−2.16297	−0.238860
$83$	−7.18985	−0.789189	−0.394594	−	0.918855i	$-0.629115\pi$
$83$	−7.18985	−0.789189	−0.394594	+	0.918855i	$0.629115\pi$
$84$	0	0
$85$	3.51797	0.381577
$86$	0.976287	0.105276
$87$	0	0
$88$	4.75114	0.506473
$89$	10.2900	1.09074	0.545368	−	0.838197i	$-0.316390\pi$
$89$	10.2900	1.09074	0.545368	+	0.838197i	$0.316390\pi$
$90$	0	0
$91$	2.27430	0.238411
$92$	1.90833	0.198957
$93$	0	0
$94$	−3.28475	−0.338796
$95$	10.4694	1.07414
$96$	0	0
$97$	−3.31072	−0.336153	−0.168076	−	0.985774i	$-0.553756\pi$
$97$	−3.31072	−0.336153	−0.168076	+	0.985774i	$0.553756\pi$
$98$	−2.06394	−0.208490
$99$	0	0
$100$	−11.8728	−1.18728
$101$	4.08415	0.406388	0.203194	−	0.979138i	$-0.434868\pi$
$101$	4.08415	0.406388	0.203194	+	0.979138i	$0.434868\pi$
$102$	0	0
$103$	14.4370	1.42252	0.711260	−	0.702929i	$-0.248125\pi$
$103$	14.4370	1.42252	0.711260	+	0.702929i	$0.248125\pi$
$104$	−6.28584	−0.616377
$105$	0	0
$106$	2.93960	0.285520
$107$	−6.88495	−0.665593	−0.332796	−	0.942999i	$-0.607992\pi$
$107$	−6.88495	−0.665593	−0.332796	+	0.942999i	$0.607992\pi$
$108$	0	0
$109$	−15.2270	−1.45848	−0.729240	−	0.684257i	$-0.760126\pi$
$109$	−15.2270	−1.45848	−0.729240	+	0.684257i	$0.760126\pi$
$110$	4.07224	0.388273
$111$	0	0
$112$	1.48071	0.139914
$113$	5.66251	0.532684	0.266342	−	0.963879i	$-0.414185\pi$
$113$	5.66251	0.532684	0.266342	+	0.963879i	$0.414185\pi$
$114$	0	0
$115$	3.34986	0.312376
$116$	1.90833	0.177184
$117$	0	0
$118$	1.41996	0.130718
$119$	−0.449640	−0.0412184
$120$	0	0
$121$	5.11998	0.465453
$122$	−1.27207	−0.115168
$123$	0	0
$124$	19.7735	1.77571
$125$	−4.09204	−0.366003
$126$	0	0
$127$	6.52259	0.578786	0.289393	−	0.957210i	$-0.406546\pi$
$127$	6.52259	0.578786	0.289393	+	0.957210i	$0.406546\pi$
$128$	−8.60892	−0.760928
$129$	0	0
$130$	−5.38765	−0.472528
$131$	5.30306	0.463331	0.231665	−	0.972796i	$-0.425583\pi$
$131$	5.30306	0.463331	0.231665	+	0.972796i	$0.425583\pi$
$132$	0	0
$133$	−1.33812	−0.116030
$134$	−1.18039	−0.101970
$135$	0	0
$136$	1.24274	0.106564
$137$	13.1674	1.12496	0.562482	−	0.826810i	$-0.309847\pi$
$137$	13.1674	1.12496	0.562482	+	0.826810i	$0.309847\pi$
$138$	0	0
$139$	−2.28705	−0.193985	−0.0969925	−	0.995285i	$-0.530922\pi$
$139$	−2.28705	−0.193985	−0.0969925	+	0.995285i	$0.530922\pi$
$140$	2.73702	0.231321
$141$	0	0
$142$	−4.14911	−0.348186
$143$	−21.3270	−1.78345
$144$	0	0
$145$	3.34986	0.278191
$146$	2.53749	0.210004
$147$	0	0
$148$	19.7594	1.62421
$149$	11.9584	0.979672	0.489836	−	0.871815i	$-0.337057\pi$
$149$	11.9584	0.979672	0.489836	+	0.871815i	$0.337057\pi$
$150$	0	0
$151$	24.0005	1.95313	0.976567	−	0.215213i	$-0.0690446\pi$
$151$	24.0005	1.95313	0.976567	+	0.215213i	$0.0690446\pi$
$152$	3.69838	0.299978
$153$	0	0
$154$	−0.520483	−0.0419417
$155$	34.7102	2.78799
$156$	0	0
$157$	2.54328	0.202976	0.101488	−	0.994837i	$-0.467640\pi$
$157$	2.54328	0.202976	0.101488	+	0.994837i	$0.467640\pi$
$158$	2.66659	0.212143
$159$	0	0
$160$	−11.4358	−0.904082
$161$	−0.428154	−0.0337432
$162$	0	0
$163$	−20.4369	−1.60074	−0.800372	−	0.599504i	$-0.795365\pi$
$163$	−20.4369	−1.60074	−0.800372	+	0.599504i	$0.795365\pi$
$164$	13.6326	1.06453
$165$	0	0
$166$	−2.17693	−0.168963
$167$	−12.3398	−0.954885	−0.477443	−	0.878663i	$-0.658436\pi$
$167$	−12.3398	−0.954885	−0.477443	+	0.878663i	$0.658436\pi$
$168$	0	0
$169$	15.2160	1.17046
$170$	1.06516	0.0816944
$171$	0	0
$172$	−6.15326	−0.469182
$173$	11.3271	0.861186	0.430593	−	0.902546i	$-0.358304\pi$
$173$	11.3271	0.861186	0.430593	+	0.902546i	$0.358304\pi$
$174$	0	0
$175$	2.66378	0.201363
$176$	−13.8852	−1.04664
$177$	0	0
$178$	3.11558	0.233523
$179$	−12.6720	−0.947153	−0.473577	−	0.880753i	$-0.657037\pi$
$179$	−12.6720	−0.947153	−0.473577	+	0.880753i	$0.657037\pi$
$180$	0	0
$181$	21.2565	1.57998	0.789992	−	0.613117i	$-0.210084\pi$
$181$	21.2565	1.57998	0.789992	+	0.613117i	$0.210084\pi$
$182$	0.688608	0.0510430
$183$	0	0
$184$	1.18336	0.0872382
$185$	34.6855	2.55013
$186$	0	0
$187$	4.21646	0.308338
$188$	20.7028	1.50991
$189$	0	0
$190$	3.16991	0.229969
$191$	11.9418	0.864078	0.432039	−	0.901855i	$-0.357794\pi$
$191$	11.9418	0.864078	0.432039	+	0.901855i	$0.357794\pi$
$192$	0	0
$193$	9.05494	0.651789	0.325894	−	0.945406i	$-0.394335\pi$
$193$	9.05494	0.651789	0.325894	+	0.945406i	$0.394335\pi$
$194$	−1.00241	−0.0719692
$195$	0	0
$196$	13.0085	0.929175
$197$	−6.84424	−0.487632	−0.243816	−	0.969822i	$-0.578399\pi$
$197$	−6.84424	−0.487632	−0.243816	+	0.969822i	$0.578399\pi$
$198$	0	0
$199$	12.4741	0.884269	0.442134	−	0.896949i	$-0.354221\pi$
$199$	12.4741	0.884269	0.442134	+	0.896949i	$0.354221\pi$
$200$	−7.36232	−0.520594
$201$	0	0
$202$	1.23659	0.0870063
$203$	−0.428154	−0.0300505
$204$	0	0
$205$	23.9305	1.67138
$206$	4.37121	0.304556
$207$	0	0
$208$	18.3704	1.27375
$209$	12.5481	0.867970
$210$	0	0
$211$	−16.1997	−1.11524	−0.557618	−	0.830098i	$-0.688285\pi$
$211$	−16.1997	−1.11524	−0.557618	+	0.830098i	$0.688285\pi$
$212$	−18.5275	−1.27247
$213$	0	0
$214$	−2.08461	−0.142501
$215$	−10.8014	−0.736649
$216$	0	0
$217$	−4.43640	−0.301162
$218$	−4.61040	−0.312256
$219$	0	0
$220$	−25.6662	−1.73041
$221$	−5.57845	−0.375247
$222$	0	0
$223$	−4.70284	−0.314926	−0.157463	−	0.987525i	$-0.550331\pi$
$223$	−4.70284	−0.314926	−0.157463	+	0.987525i	$0.550331\pi$
$224$	1.46164	0.0976601
$225$	0	0
$226$	1.71448	0.114046
$227$	27.4304	1.82062	0.910309	−	0.413928i	$-0.135844\pi$
$227$	27.4304	1.82062	0.910309	+	0.413928i	$0.135844\pi$
$228$	0	0
$229$	0.510879	0.0337598	0.0168799	−	0.999858i	$-0.494627\pi$
$229$	0.510879	0.0337598	0.0168799	+	0.999858i	$0.494627\pi$
$230$	1.01426	0.0668786
$231$	0	0
$232$	1.18336	0.0776911
$233$	−11.3044	−0.740577	−0.370289	−	0.928917i	$-0.620741\pi$
$233$	−11.3044	−0.740577	−0.370289	+	0.928917i	$0.620741\pi$
$234$	0	0
$235$	36.3416	2.37066
$236$	−8.94959	−0.582569
$237$	0	0
$238$	−0.136141	−0.00882473
$239$	10.3539	0.669737	0.334868	−	0.942265i	$-0.391308\pi$
$239$	10.3539	0.669737	0.334868	+	0.942265i	$0.391308\pi$
$240$	0	0
$241$	−3.01550	−0.194246	−0.0971228	−	0.995272i	$-0.530964\pi$
$241$	−3.01550	−0.194246	−0.0971228	+	0.995272i	$0.530964\pi$
$242$	1.55022	0.0996518
$243$	0	0
$244$	8.01748	0.513266
$245$	22.8349	1.45887
$246$	0	0
$247$	−16.6014	−1.05632
$248$	12.2616	0.778611
$249$	0	0
$250$	−1.23898	−0.0783601
$251$	−16.0714	−1.01442	−0.507208	−	0.861823i	$-0.669323\pi$
$251$	−16.0714	−1.01442	−0.507208	+	0.861823i	$0.669323\pi$
$252$	0	0
$253$	4.01497	0.252419
$254$	1.97490	0.123916
$255$	0	0
$256$	9.15957	0.572473
$257$	−19.4598	−1.21387	−0.606935	−	0.794752i	$-0.707601\pi$
$257$	−19.4598	−1.21387	−0.606935	+	0.794752i	$0.707601\pi$
$258$	0	0
$259$	−4.43324	−0.275468
$260$	33.9568	2.10591
$261$	0	0
$262$	1.60565	0.0991975
$263$	−28.1751	−1.73735	−0.868675	−	0.495382i	$-0.835028\pi$
$263$	−28.1751	−1.73735	−0.868675	+	0.495382i	$0.835028\pi$
$264$	0	0
$265$	−32.5230	−1.99787
$266$	−0.405154	−0.0248416
$267$	0	0
$268$	7.43969	0.454451
$269$	12.0815	0.736623	0.368312	−	0.929702i	$-0.379936\pi$
$269$	12.0815	0.736623	0.368312	+	0.929702i	$0.379936\pi$
$270$	0	0
$271$	−29.4534	−1.78917	−0.894583	−	0.446903i	$-0.852527\pi$
$271$	−29.4534	−1.78917	−0.894583	+	0.446903i	$0.852527\pi$
$272$	−3.63191	−0.220217
$273$	0	0
$274$	3.98679	0.240851
$275$	−24.9794	−1.50631
$276$	0	0
$277$	10.2385	0.615172	0.307586	−	0.951520i	$-0.400479\pi$
$277$	10.2385	0.615172	0.307586	+	0.951520i	$0.400479\pi$
$278$	−0.692469	−0.0415315
$279$	0	0
$280$	1.69723	0.101429
$281$	16.2447	0.969077	0.484538	−	0.874770i	$-0.338988\pi$
$281$	16.2447	0.969077	0.484538	+	0.874770i	$0.338988\pi$
$282$	0	0
$283$	18.9453	1.12618	0.563091	−	0.826395i	$-0.309612\pi$
$283$	18.9453	1.12618	0.563091	+	0.826395i	$0.309612\pi$
$284$	26.1507	1.55176
$285$	0	0
$286$	−6.45736	−0.381831
$287$	−3.05862	−0.180545
$288$	0	0
$289$	−15.8971	−0.935124
$290$	1.01426	0.0595597
$291$	0	0
$292$	−15.9931	−0.935924
$293$	12.8758	0.752213	0.376106	−	0.926577i	$-0.377263\pi$
$293$	12.8758	0.752213	0.376106	+	0.926577i	$0.377263\pi$
$294$	0	0
$295$	−15.7100	−0.914674
$296$	12.2528	0.712182
$297$	0	0
$298$	3.62075	0.209744
$299$	−5.31187	−0.307194
$300$	0	0
$301$	1.38055	0.0795737
$302$	7.26683	0.418159
$303$	0	0
$304$	−10.8085	−0.619910
$305$	14.0738	0.805864
$306$	0	0
$307$	−19.1819	−1.09477	−0.547385	−	0.836881i	$-0.684377\pi$
$307$	−19.1819	−1.09477	−0.547385	+	0.836881i	$0.684377\pi$
$308$	3.28046	0.186921
$309$	0	0
$310$	10.5095	0.596900
$311$	25.1290	1.42493	0.712466	−	0.701707i	$-0.247578\pi$
$311$	25.1290	1.42493	0.712466	+	0.701707i	$0.247578\pi$
$312$	0	0
$313$	−1.00877	−0.0570190	−0.0285095	−	0.999594i	$-0.509076\pi$
$313$	−1.00877	−0.0570190	−0.0285095	+	0.999594i	$0.509076\pi$
$314$	0.770051	0.0434565
$315$	0	0
$316$	−16.8068	−0.945454
$317$	3.08943	0.173520	0.0867599	−	0.996229i	$-0.472349\pi$
$317$	3.08943	0.173520	0.0867599	+	0.996229i	$0.472349\pi$
$318$	0	0
$319$	4.01497	0.224795
$320$	19.7075	1.10168
$321$	0	0
$322$	−0.129636	−0.00722431
$323$	3.28217	0.182625
$324$	0	0
$325$	33.0481	1.83318
$326$	−6.18786	−0.342714
$327$	0	0
$328$	8.45359	0.466772
$329$	−4.64490	−0.256082
$330$	0	0
$331$	14.8847	0.818139	0.409069	−	0.912503i	$-0.365853\pi$
$331$	14.8847	0.818139	0.409069	+	0.912503i	$0.365853\pi$
$332$	13.7206	0.753015
$333$	0	0
$334$	−3.73623	−0.204438
$335$	13.0596	0.713520
$336$	0	0
$337$	23.9721	1.30584	0.652921	−	0.757426i	$-0.273543\pi$
$337$	23.9721	1.30584	0.652921	+	0.757426i	$0.273543\pi$
$338$	4.60708	0.250592
$339$	0	0
$340$	−6.71343	−0.364087
$341$	41.6019	2.25287
$342$	0	0
$343$	−5.91566	−0.319416
$344$	−3.81565	−0.205726
$345$	0	0
$346$	3.42961	0.184377
$347$	29.4423	1.58055	0.790273	−	0.612755i	$-0.209939\pi$
$347$	29.4423	1.58055	0.790273	+	0.612755i	$0.209939\pi$
$348$	0	0
$349$	−4.84185	−0.259178	−0.129589	−	0.991568i	$-0.541366\pi$
$349$	−4.84185	−0.259178	−0.129589	+	0.991568i	$0.541366\pi$
$350$	0.806535	0.0431111
$351$	0	0
$352$	−13.7064	−0.730554
$353$	−11.1009	−0.590839	−0.295420	−	0.955368i	$-0.595459\pi$
$353$	−11.1009	−0.590839	−0.295420	+	0.955368i	$0.595459\pi$
$354$	0	0
$355$	45.9047	2.43637
$356$	−19.6366	−1.04074
$357$	0	0
$358$	−3.83682	−0.202782
$359$	−21.0718	−1.11213	−0.556063	−	0.831140i	$-0.687689\pi$
$359$	−21.0718	−1.11213	−0.556063	+	0.831140i	$0.687689\pi$
$360$	0	0
$361$	−9.23232	−0.485912
$362$	6.43601	0.338269
$363$	0	0
$364$	−4.34010	−0.227483
$365$	−28.0741	−1.46947
$366$	0	0
$367$	−31.5820	−1.64857	−0.824283	−	0.566178i	$-0.808421\pi$
$367$	−31.5820	−1.64857	−0.824283	+	0.566178i	$0.808421\pi$
$368$	−3.45836	−0.180279
$369$	0	0
$370$	10.5020	0.545974
$371$	4.15684	0.215813
$372$	0	0
$373$	−6.16681	−0.319305	−0.159653	−	0.987173i	$-0.551037\pi$
$373$	−6.16681	−0.319305	−0.159653	+	0.987173i	$0.551037\pi$
$374$	1.27665	0.0660141
$375$	0	0
$376$	12.8379	0.662062
$377$	−5.31187	−0.273575
$378$	0	0
$379$	12.6195	0.648219	0.324109	−	0.946020i	$-0.394935\pi$
$379$	12.6195	0.648219	0.324109	+	0.946020i	$0.394935\pi$
$380$	−19.9790	−1.02490
$381$	0	0
$382$	3.61572	0.184996
$383$	5.45898	0.278941	0.139470	−	0.990226i	$-0.455460\pi$
$383$	5.45898	0.278941	0.139470	+	0.990226i	$0.455460\pi$
$384$	0	0
$385$	5.75849	0.293480
$386$	2.74164	0.139546
$387$	0	0
$388$	6.31794	0.320745
$389$	17.4154	0.882996	0.441498	−	0.897262i	$-0.354447\pi$
$389$	17.4154	0.882996	0.441498	+	0.897262i	$0.354447\pi$
$390$	0	0
$391$	1.05018	0.0531101
$392$	8.06656	0.407423
$393$	0	0
$394$	−2.07229	−0.104400
$395$	−29.5025	−1.48443
$396$	0	0
$397$	−29.5353	−1.48234	−0.741168	−	0.671320i	$-0.765728\pi$
$397$	−29.5353	−1.48234	−0.741168	+	0.671320i	$0.765728\pi$
$398$	3.77690	0.189319
$399$	0	0
$400$	21.5164	1.07582
$401$	19.0745	0.952537	0.476268	−	0.879300i	$-0.341989\pi$
$401$	19.0745	0.952537	0.476268	+	0.879300i	$0.341989\pi$
$402$	0	0
$403$	−55.0401	−2.74174
$404$	−7.79389	−0.387760
$405$	0	0
$406$	−0.129636	−0.00643371
$407$	41.5723	2.06066
$408$	0	0
$409$	33.1090	1.63714	0.818568	−	0.574410i	$-0.194768\pi$
$409$	33.1090	1.63714	0.818568	+	0.574410i	$0.194768\pi$
$410$	7.24565	0.357837
$411$	0	0
$412$	−27.5505	−1.35731
$413$	2.00794	0.0988042
$414$	0	0
$415$	24.0850	1.18229
$416$	18.1338	0.889084
$417$	0	0
$418$	3.79929	0.185829
$419$	−17.5578	−0.857754	−0.428877	−	0.903363i	$-0.641091\pi$
$419$	−17.5578	−0.857754	−0.428877	+	0.903363i	$0.641091\pi$
$420$	0	0
$421$	−4.35212	−0.212109	−0.106055	−	0.994360i	$-0.533822\pi$
$421$	−4.35212	−0.212109	−0.106055	+	0.994360i	$0.533822\pi$
$422$	−4.90493	−0.238768
$423$	0	0
$424$	−11.4889	−0.557952
$425$	−6.53378	−0.316935
$426$	0	0
$427$	−1.79881	−0.0870504
$428$	13.1387	0.635084
$429$	0	0
$430$	−3.27043	−0.157714
$431$	−0.733390	−0.0353261	−0.0176631	−	0.999844i	$-0.505623\pi$
$431$	−0.733390	−0.0353261	−0.0176631	+	0.999844i	$0.505623\pi$
$432$	0	0
$433$	21.2117	1.01937	0.509686	−	0.860361i	$-0.329762\pi$
$433$	21.2117	1.01937	0.509686	+	0.860361i	$0.329762\pi$
$434$	−1.34325	−0.0644778
$435$	0	0
$436$	29.0580	1.39163
$437$	3.12533	0.149505
$438$	0	0
$439$	−25.0957	−1.19775	−0.598877	−	0.800841i	$-0.704386\pi$
$439$	−25.0957	−1.19775	−0.598877	+	0.800841i	$0.704386\pi$
$440$	−15.9156	−0.758748
$441$	0	0
$442$	−1.68903	−0.0803391
$443$	−35.2387	−1.67424	−0.837121	−	0.547017i	$-0.815763\pi$
$443$	−35.2387	−1.67424	−0.837121	+	0.547017i	$0.815763\pi$
$444$	0	0
$445$	−34.4700	−1.63403
$446$	−1.42392	−0.0674245
$447$	0	0
$448$	−2.51886	−0.119005
$449$	11.4866	0.542086	0.271043	−	0.962567i	$-0.412631\pi$
$449$	11.4866	0.542086	0.271043	+	0.962567i	$0.412631\pi$
$450$	0	0
$451$	28.6819	1.35058
$452$	−10.8059	−0.508267
$453$	0	0
$454$	8.30532	0.389788
$455$	−7.61858	−0.357164
$456$	0	0
$457$	10.2250	0.478307	0.239154	−	0.970982i	$-0.423130\pi$
$457$	10.2250	0.478307	0.239154	+	0.970982i	$0.423130\pi$
$458$	0.154683	0.00722786
$459$	0	0
$460$	−6.39262	−0.298057
$461$	−8.68719	−0.404603	−0.202301	−	0.979323i	$-0.564842\pi$
$461$	−8.68719	−0.404603	−0.202301	+	0.979323i	$0.564842\pi$
$462$	0	0
$463$	−11.5538	−0.536950	−0.268475	−	0.963287i	$-0.586520\pi$
$463$	−11.5538	−0.536950	−0.268475	+	0.963287i	$0.586520\pi$
$464$	−3.45836	−0.160550
$465$	0	0
$466$	−3.42273	−0.158555
$467$	33.0201	1.52799	0.763994	−	0.645223i	$-0.223236\pi$
$467$	33.0201	1.52799	0.763994	+	0.645223i	$0.223236\pi$
$468$	0	0
$469$	−1.66917	−0.0770753
$470$	11.0034	0.507551
$471$	0	0
$472$	−5.54966	−0.255444
$473$	−12.9460	−0.595257
$474$	0	0
$475$	−19.4444	−0.892171
$476$	0.858060	0.0393291
$477$	0	0
$478$	3.13493	0.143388
$479$	18.5401	0.847118	0.423559	−	0.905869i	$-0.360781\pi$
$479$	18.5401	0.847118	0.423559	+	0.905869i	$0.360781\pi$
$480$	0	0
$481$	−55.0008	−2.50782
$482$	−0.913029	−0.0415873
$483$	0	0
$484$	−9.77059	−0.444118
$485$	11.0905	0.503592
$486$	0	0
$487$	7.53967	0.341655	0.170828	−	0.985301i	$-0.445356\pi$
$487$	7.53967	0.341655	0.170828	+	0.985301i	$0.445356\pi$
$488$	4.97165	0.225056
$489$	0	0
$490$	6.91392	0.312339
$491$	−1.90597	−0.0860154	−0.0430077	−	0.999075i	$-0.513694\pi$
$491$	−1.90597	−0.0860154	−0.0430077	+	0.999075i	$0.513694\pi$
$492$	0	0
$493$	1.05018	0.0472979
$494$	−5.02653	−0.226154
$495$	0	0
$496$	−35.8345	−1.60901
$497$	−5.86719	−0.263180
$498$	0	0
$499$	9.26667	0.414833	0.207417	−	0.978253i	$-0.433494\pi$
$499$	9.26667	0.414833	0.207417	+	0.978253i	$0.433494\pi$
$500$	7.80895	0.349227
$501$	0	0
$502$	−4.86606	−0.217183
$503$	16.4993	0.735667	0.367833	−	0.929892i	$-0.380100\pi$
$503$	16.4993	0.735667	0.367833	+	0.929892i	$0.380100\pi$
$504$	0	0
$505$	−13.6813	−0.608811
$506$	1.21565	0.0540420
$507$	0	0
$508$	−12.4472	−0.552256
$509$	19.3897	0.859434	0.429717	−	0.902964i	$-0.358613\pi$
$509$	19.3897	0.859434	0.429717	+	0.902964i	$0.358613\pi$
$510$	0	0
$511$	3.58822	0.158734
$512$	19.9912	0.883493
$513$	0	0
$514$	−5.89201	−0.259885
$515$	−48.3619	−2.13108
$516$	0	0
$517$	43.5571	1.91564
$518$	−1.34229	−0.0589767
$519$	0	0
$520$	21.0567	0.923396
$521$	−19.8726	−0.870633	−0.435316	−	0.900277i	$-0.643364\pi$
$521$	−19.8726	−0.870633	−0.435316	+	0.900277i	$0.643364\pi$
$522$	0	0
$523$	5.32708	0.232937	0.116468	−	0.993194i	$-0.462843\pi$
$523$	5.32708	0.232937	0.116468	+	0.993194i	$0.462843\pi$
$524$	−10.1200	−0.442093
$525$	0	0
$526$	−8.53080	−0.371961
$527$	10.8817	0.474014
$528$	0	0
$529$	1.00000	0.0434783
$530$	−9.84726	−0.427738
$531$	0	0
$532$	2.55357	0.110711
$533$	−37.9467	−1.64365
$534$	0	0
$535$	23.0636	0.997127
$536$	4.61336	0.199267
$537$	0	0
$538$	3.65802	0.157709
$539$	27.3688	1.17886
$540$	0	0
$541$	−39.5108	−1.69870	−0.849351	−	0.527828i	$-0.823007\pi$
$541$	−39.5108	−1.69870	−0.849351	+	0.527828i	$0.823007\pi$
$542$	−8.91784	−0.383054
$543$	0	0
$544$	−3.58515	−0.153712
$545$	51.0083	2.18495
$546$	0	0
$547$	25.8008	1.10316	0.551581	−	0.834121i	$-0.314025\pi$
$547$	25.8008	1.10316	0.551581	+	0.834121i	$0.314025\pi$
$548$	−25.1276	−1.07340
$549$	0	0
$550$	−7.56321	−0.322496
$551$	3.12533	0.133143
$552$	0	0
$553$	3.77078	0.160350
$554$	3.09999	0.131706
$555$	0	0
$556$	4.36443	0.185093
$557$	22.5345	0.954818	0.477409	−	0.878681i	$-0.341576\pi$
$557$	22.5345	0.954818	0.477409	+	0.878681i	$0.341576\pi$
$558$	0	0
$559$	17.1278	0.724428
$560$	−4.96016	−0.209605
$561$	0	0
$562$	4.91854	0.207476
$563$	4.25638	0.179385	0.0896926	−	0.995969i	$-0.471412\pi$
$563$	4.25638	0.179385	0.0896926	+	0.995969i	$0.471412\pi$
$564$	0	0
$565$	−18.9686	−0.798015
$566$	5.73623	0.241112
$567$	0	0
$568$	16.2161	0.680412
$569$	41.4992	1.73974	0.869868	−	0.493284i	$-0.164204\pi$
$569$	41.4992	1.73974	0.869868	+	0.493284i	$0.164204\pi$
$570$	0	0
$571$	20.8965	0.874490	0.437245	−	0.899343i	$-0.355954\pi$
$571$	20.8965	0.874490	0.437245	+	0.899343i	$0.355954\pi$
$572$	40.6989	1.70171
$573$	0	0
$574$	−0.926084	−0.0386540
$575$	−6.22156	−0.259457
$576$	0	0
$577$	10.2811	0.428007	0.214004	−	0.976833i	$-0.431350\pi$
$577$	10.2811	0.428007	0.214004	+	0.976833i	$0.431350\pi$
$578$	−4.81330	−0.200207
$579$	0	0
$580$	−6.39262	−0.265439
$581$	−3.07836	−0.127712
$582$	0	0
$583$	−38.9804	−1.61440
$584$	−9.91734	−0.410382
$585$	0	0
$586$	3.89851	0.161046
$587$	11.8524	0.489201	0.244601	−	0.969624i	$-0.421343\pi$
$587$	11.8524	0.489201	0.244601	+	0.969624i	$0.421343\pi$
$588$	0	0
$589$	32.3837	1.33435
$590$	−4.75666	−0.195829
$591$	0	0
$592$	−35.8089	−1.47174
$593$	−6.17813	−0.253705	−0.126853	−	0.991922i	$-0.540488\pi$
$593$	−6.17813	−0.253705	−0.126853	+	0.991922i	$0.540488\pi$
$594$	0	0
$595$	1.50623	0.0617495
$596$	−22.8205	−0.934766
$597$	0	0
$598$	−1.60832	−0.0657691
$599$	−18.0981	−0.739467	−0.369734	−	0.929138i	$-0.620551\pi$
$599$	−18.0981	−0.739467	−0.369734	+	0.929138i	$0.620551\pi$
$600$	0	0
$601$	−6.20128	−0.252956	−0.126478	−	0.991969i	$-0.540367\pi$
$601$	−6.20128	−0.252956	−0.126478	+	0.991969i	$0.540367\pi$
$602$	0.418001	0.0170364
$603$	0	0
$604$	−45.8008	−1.86361
$605$	−17.1512	−0.697296
$606$	0	0
$607$	13.0037	0.527804	0.263902	−	0.964550i	$-0.414990\pi$
$607$	13.0037	0.527804	0.263902	+	0.964550i	$0.414990\pi$
$608$	−10.6693	−0.432698
$609$	0	0
$610$	4.26124	0.172533
$611$	−57.6269	−2.33133
$612$	0	0
$613$	−26.0974	−1.05406	−0.527031	−	0.849846i	$-0.676695\pi$
$613$	−26.0974	−1.05406	−0.527031	+	0.849846i	$0.676695\pi$
$614$	−5.80787	−0.234386
$615$	0	0
$616$	2.03422	0.0819609
$617$	4.77204	0.192115	0.0960576	−	0.995376i	$-0.469377\pi$
$617$	4.77204	0.192115	0.0960576	+	0.995376i	$0.469377\pi$
$618$	0	0
$619$	7.79517	0.313314	0.156657	−	0.987653i	$-0.449928\pi$
$619$	7.79517	0.313314	0.156657	+	0.987653i	$0.449928\pi$
$620$	−66.2384	−2.66020
$621$	0	0
$622$	7.60850	0.305073
$623$	4.40569	0.176510
$624$	0	0
$625$	−17.4000	−0.696001
$626$	−0.305434	−0.0122076
$627$	0	0
$628$	−4.85341	−0.193672
$629$	10.8739	0.433572
$630$	0	0
$631$	13.9161	0.553990	0.276995	−	0.960871i	$-0.410661\pi$
$631$	13.9161	0.553990	0.276995	+	0.960871i	$0.410661\pi$
$632$	−10.4219	−0.414561
$633$	0	0
$634$	0.935413	0.0371500
$635$	−21.8498	−0.867082
$636$	0	0
$637$	−36.2094	−1.43467
$638$	1.21565	0.0481279
$639$	0	0
$640$	28.8387	1.13995
$641$	7.91467	0.312611	0.156305	−	0.987709i	$-0.450042\pi$
$641$	7.91467	0.312611	0.156305	+	0.987709i	$0.450042\pi$
$642$	0	0
$643$	−13.3238	−0.525439	−0.262720	−	0.964872i	$-0.584619\pi$
$643$	−13.3238	−0.525439	−0.262720	+	0.964872i	$0.584619\pi$
$644$	0.817056	0.0321965
$645$	0	0
$646$	0.993770	0.0390994
$647$	−1.47922	−0.0581543	−0.0290771	−	0.999577i	$-0.509257\pi$
$647$	−1.47922	−0.0581543	−0.0290771	+	0.999577i	$0.509257\pi$
$648$	0	0
$649$	−18.8292	−0.739113
$650$	10.0063	0.392478
$651$	0	0
$652$	39.0003	1.52737
$653$	−28.7736	−1.12600	−0.562998	−	0.826458i	$-0.690352\pi$
$653$	−28.7736	−1.12600	−0.562998	+	0.826458i	$0.690352\pi$
$654$	0	0
$655$	−17.7645	−0.694117
$656$	−24.7056	−0.964592
$657$	0	0
$658$	−1.40638	−0.0548262
$659$	18.2082	0.709293	0.354646	−	0.935001i	$-0.384601\pi$
$659$	18.2082	0.709293	0.354646	+	0.935001i	$0.384601\pi$
$660$	0	0
$661$	13.9217	0.541493	0.270746	−	0.962651i	$-0.412729\pi$
$661$	13.9217	0.541493	0.270746	+	0.962651i	$0.412729\pi$
$662$	4.50677	0.175161
$663$	0	0
$664$	8.50816	0.330180
$665$	4.48252	0.173824
$666$	0	0
$667$	1.00000	0.0387202
$668$	23.5484	0.911116
$669$	0	0
$670$	3.95415	0.152762
$671$	16.8682	0.651188
$672$	0	0
$673$	−29.7330	−1.14612	−0.573062	−	0.819512i	$-0.694245\pi$
$673$	−29.7330	−1.14612	−0.573062	+	0.819512i	$0.694245\pi$
$674$	7.25822	0.279576
$675$	0	0
$676$	−29.0371	−1.11681
$677$	−0.854675	−0.0328478	−0.0164239	−	0.999865i	$-0.505228\pi$
$677$	−0.854675	−0.0328478	−0.0164239	+	0.999865i	$0.505228\pi$
$678$	0	0
$679$	−1.41750	−0.0543986
$680$	−4.16301	−0.159644
$681$	0	0
$682$	12.5962	0.482332
$683$	17.6454	0.675183	0.337591	−	0.941293i	$-0.390388\pi$
$683$	17.6454	0.675183	0.337591	+	0.941293i	$0.390388\pi$
$684$	0	0
$685$	−44.1088	−1.68531
$686$	−1.79113	−0.0683858
$687$	0	0
$688$	11.1512	0.425137
$689$	51.5718	1.96473
$690$	0	0
$691$	−27.0772	−1.03007	−0.515033	−	0.857170i	$-0.672220\pi$
$691$	−27.0772	−1.03007	−0.515033	+	0.857170i	$0.672220\pi$
$692$	−21.6159	−0.821712
$693$	0	0
$694$	8.91449	0.338389
$695$	7.66129	0.290609
$696$	0	0
$697$	7.50225	0.284168
$698$	−1.46601	−0.0554891
$699$	0	0
$700$	−5.08336	−0.192133
$701$	41.8074	1.57904	0.789522	−	0.613722i	$-0.210328\pi$
$701$	41.8074	1.57904	0.789522	+	0.613722i	$0.210328\pi$
$702$	0	0
$703$	32.3606	1.22050
$704$	23.6204	0.890227
$705$	0	0
$706$	−3.36110	−0.126497
$707$	1.74864	0.0657645
$708$	0	0
$709$	28.6271	1.07511	0.537557	−	0.843228i	$-0.319347\pi$
$709$	28.6271	1.07511	0.537557	+	0.843228i	$0.319347\pi$
$710$	13.8989	0.521618
$711$	0	0
$712$	−12.1767	−0.456342
$713$	10.3617	0.388049
$714$	0	0
$715$	71.4425	2.67180
$716$	24.1824	0.903738
$717$	0	0
$718$	−6.38008	−0.238102
$719$	32.5891	1.21537	0.607685	−	0.794178i	$-0.292098\pi$
$719$	32.5891	1.21537	0.607685	+	0.794178i	$0.292098\pi$
$720$	0	0
$721$	6.18125	0.230202
$722$	−2.79535	−0.104032
$723$	0	0
$724$	−40.5644	−1.50756
$725$	−6.22156	−0.231063
$726$	0	0
$727$	46.4493	1.72271	0.861354	−	0.508005i	$-0.169617\pi$
$727$	46.4493	1.72271	0.861354	+	0.508005i	$0.169617\pi$
$728$	−2.69130	−0.0997464
$729$	0	0
$730$	−8.50024	−0.314608
$731$	−3.38625	−0.125245
$732$	0	0
$733$	−49.8584	−1.84156	−0.920781	−	0.390079i	$-0.872448\pi$
$733$	−49.8584	−1.84156	−0.920781	+	0.390079i	$0.872448\pi$
$734$	−9.56233	−0.352952
$735$	0	0
$736$	−3.41383	−0.125835
$737$	15.6525	0.576568
$738$	0	0
$739$	−42.1893	−1.55196	−0.775979	−	0.630759i	$-0.782744\pi$
$739$	−42.1893	−1.55196	−0.775979	+	0.630759i	$0.782744\pi$
$740$	−66.1912	−2.43324
$741$	0	0
$742$	1.25860	0.0462047
$743$	−9.91569	−0.363771	−0.181886	−	0.983320i	$-0.558220\pi$
$743$	−9.91569	−0.363771	−0.181886	+	0.983320i	$0.558220\pi$
$744$	0	0
$745$	−40.0590	−1.46765
$746$	−1.86718	−0.0683622
$747$	0	0
$748$	−8.04637	−0.294205
$749$	−2.94781	−0.107711
$750$	0	0
$751$	−18.1042	−0.660632	−0.330316	−	0.943870i	$-0.607155\pi$
$751$	−18.1042	−0.660632	−0.330316	+	0.943870i	$0.607155\pi$
$752$	−37.5186	−1.36816
$753$	0	0
$754$	−1.60832	−0.0585716
$755$	−80.3983	−2.92600
$756$	0	0
$757$	−9.06824	−0.329591	−0.164795	−	0.986328i	$-0.552696\pi$
$757$	−9.06824	−0.329591	−0.164795	+	0.986328i	$0.552696\pi$
$758$	3.82090	0.138781
$759$	0	0
$760$	−12.3890	−0.449398
$761$	53.3930	1.93549	0.967747	−	0.251924i	$-0.0810632\pi$
$761$	53.3930	1.93549	0.967747	+	0.251924i	$0.0810632\pi$
$762$	0	0
$763$	−6.51949	−0.236021
$764$	−22.7888	−0.824471
$765$	0	0
$766$	1.65286	0.0597203
$767$	24.9114	0.899499
$768$	0	0
$769$	21.6662	0.781305	0.390652	−	0.920538i	$-0.372249\pi$
$769$	21.6662	0.781305	0.390652	+	0.920538i	$0.372249\pi$
$770$	1.74354	0.0628330
$771$	0	0
$772$	−17.2798	−0.621912
$773$	−26.9885	−0.970710	−0.485355	−	0.874317i	$-0.661310\pi$
$773$	−26.9885	−0.970710	−0.485355	+	0.874317i	$0.661310\pi$
$774$	0	0
$775$	−64.4659	−2.31568
$776$	3.91776	0.140639
$777$	0	0
$778$	5.27301	0.189046
$779$	22.3266	0.799932
$780$	0	0
$781$	55.0190	1.96874
$782$	0.317973	0.0113707
$783$	0	0
$784$	−23.5745	−0.841947
$785$	−8.51964	−0.304079
$786$	0	0
$787$	40.2357	1.43425	0.717125	−	0.696945i	$-0.245458\pi$
$787$	40.2357	1.43425	0.717125	+	0.696945i	$0.245458\pi$
$788$	13.0610	0.465280
$789$	0	0
$790$	−8.93270	−0.317811
$791$	2.42442	0.0862025
$792$	0	0
$793$	−22.3169	−0.792495
$794$	−8.94265	−0.317363
$795$	0	0
$796$	−23.8047	−0.843736
$797$	−25.7410	−0.911792	−0.455896	−	0.890033i	$-0.650681\pi$
$797$	−25.7410	−0.911792	−0.455896	+	0.890033i	$0.650681\pi$
$798$	0	0
$799$	11.3931	0.403060
$800$	21.2393	0.750923
$801$	0	0
$802$	5.77535	0.203935
$803$	−33.6482	−1.18742
$804$	0	0
$805$	1.43425	0.0505508
$806$	−16.6649	−0.586997
$807$	0	0
$808$	−4.83300	−0.170024
$809$	37.9813	1.33535	0.667675	−	0.744453i	$-0.267289\pi$
$809$	37.9813	1.33535	0.667675	+	0.744453i	$0.267289\pi$
$810$	0	0
$811$	41.0679	1.44209	0.721045	−	0.692888i	$-0.243662\pi$
$811$	41.0679	1.44209	0.721045	+	0.692888i	$0.243662\pi$
$812$	0.817056	0.0286731
$813$	0	0
$814$	12.5872	0.441180
$815$	68.4608	2.39808
$816$	0	0
$817$	−10.0774	−0.352564
$818$	10.0247	0.350505
$819$	0	0
$820$	−45.6673	−1.59477
$821$	−43.9349	−1.53334	−0.766670	−	0.642042i	$-0.778088\pi$
$821$	−43.9349	−1.53334	−0.766670	+	0.642042i	$0.778088\pi$
$822$	0	0
$823$	−7.58709	−0.264469	−0.132235	−	0.991218i	$-0.542215\pi$
$823$	−7.58709	−0.264469	−0.132235	+	0.991218i	$0.542215\pi$
$824$	−17.0841	−0.595153
$825$	0	0
$826$	0.607960	0.0211536
$827$	−32.5844	−1.13307	−0.566536	−	0.824037i	$-0.691717\pi$
$827$	−32.5844	−1.13307	−0.566536	+	0.824037i	$0.691717\pi$
$828$	0	0
$829$	52.0958	1.80936	0.904681	−	0.426089i	$-0.140109\pi$
$829$	52.0958	1.80936	0.904681	+	0.426089i	$0.140109\pi$
$830$	7.29241	0.253123
$831$	0	0
$832$	−31.2502	−1.08341
$833$	7.15877	0.248037
$834$	0	0
$835$	41.3367	1.43052
$836$	−23.9459	−0.828185
$837$	0	0
$838$	−5.31611	−0.183642
$839$	−37.5788	−1.29736	−0.648682	−	0.761059i	$-0.724680\pi$
$839$	−37.5788	−1.29736	−0.648682	+	0.761059i	$0.724680\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	−1.31773	−0.0454119
$843$	0	0
$844$	30.9144	1.06412
$845$	−50.9715	−1.75347
$846$	0	0
$847$	2.19214	0.0753227
$848$	33.5764	1.15302
$849$	0	0
$850$	−1.97829	−0.0678547
$851$	10.3543	0.354941
$852$	0	0
$853$	−1.37770	−0.0471717	−0.0235858	−	0.999722i	$-0.507508\pi$
$853$	−1.37770	−0.0471717	−0.0235858	+	0.999722i	$0.507508\pi$
$854$	−0.544640	−0.0186372
$855$	0	0
$856$	8.14734	0.278470
$857$	−10.7773	−0.368145	−0.184072	−	0.982913i	$-0.558928\pi$
$857$	−10.7773	−0.368145	−0.184072	+	0.982913i	$0.558928\pi$
$858$	0	0
$859$	−43.8307	−1.49549	−0.747743	−	0.663988i	$-0.768862\pi$
$859$	−43.8307	−1.49549	−0.747743	+	0.663988i	$0.768862\pi$
$860$	20.6126	0.702883
$861$	0	0
$862$	−0.222054	−0.00756320
$863$	−31.7238	−1.07989	−0.539946	−	0.841700i	$-0.681555\pi$
$863$	−31.7238	−1.07989	−0.539946	+	0.841700i	$0.681555\pi$
$864$	0	0
$865$	−37.9443	−1.29015
$866$	6.42245	0.218244
$867$	0	0
$868$	8.46609	0.287358
$869$	−35.3601	−1.19951
$870$	0	0
$871$	−20.7086	−0.701683
$872$	18.0189	0.610199
$873$	0	0
$874$	0.946281	0.0320084
$875$	−1.75202	−0.0592292
$876$	0	0
$877$	58.0758	1.96108	0.980540	−	0.196319i	$-0.0628988\pi$
$877$	58.0758	1.96108	0.980540	+	0.196319i	$0.0628988\pi$
$878$	−7.59844	−0.256435
$879$	0	0
$880$	46.5134	1.56797
$881$	−50.1319	−1.68899	−0.844493	−	0.535567i	$-0.820098\pi$
$881$	−50.1319	−1.68899	−0.844493	+	0.535567i	$0.820098\pi$
$882$	0	0
$883$	10.4412	0.351375	0.175687	−	0.984446i	$-0.443785\pi$
$883$	10.4412	0.351375	0.175687	+	0.984446i	$0.443785\pi$
$884$	10.6455	0.358047
$885$	0	0
$886$	−10.6695	−0.358450
$887$	−33.4645	−1.12363	−0.561815	−	0.827263i	$-0.689897\pi$
$887$	−33.4645	−1.12363	−0.561815	+	0.827263i	$0.689897\pi$
$888$	0	0
$889$	2.79267	0.0936632
$890$	−10.4368	−0.349841
$891$	0	0
$892$	8.97456	0.300490
$893$	33.9057	1.13461
$894$	0	0
$895$	42.4496	1.41893
$896$	−3.68594	−0.123139
$897$	0	0
$898$	3.47789	0.116059
$899$	10.3617	0.345582
$900$	0	0
$901$	−10.1960	−0.339678
$902$	8.68426	0.289154
$903$	0	0
$904$	−6.70076	−0.222864
$905$	−71.2064	−2.36698
$906$	0	0
$907$	−38.3696	−1.27404	−0.637021	−	0.770846i	$-0.719834\pi$
$907$	−38.3696	−1.27404	−0.637021	+	0.770846i	$0.719834\pi$
$908$	−52.3461	−1.73717
$909$	0	0
$910$	−2.30674	−0.0764677
$911$	30.9361	1.02496	0.512480	−	0.858699i	$-0.328727\pi$
$911$	30.9361	1.02496	0.512480	+	0.858699i	$0.328727\pi$
$912$	0	0
$913$	28.8670	0.955360
$914$	3.09592	0.102404
$915$	0	0
$916$	−0.974923	−0.0322124
$917$	2.27052	0.0749793
$918$	0	0
$919$	−15.7028	−0.517986	−0.258993	−	0.965879i	$-0.583391\pi$
$919$	−15.7028	−0.517986	−0.258993	+	0.965879i	$0.583391\pi$
$920$	−3.96408	−0.130692
$921$	0	0
$922$	−2.63029	−0.0866241
$923$	−72.7911	−2.39595
$924$	0	0
$925$	−64.4200	−2.11811
$926$	−3.49824	−0.114959
$927$	0	0
$928$	−3.41383	−0.112064
$929$	−40.9864	−1.34472	−0.672360	−	0.740224i	$-0.734719\pi$
$929$	−40.9864	−1.34472	−0.672360	+	0.740224i	$0.734719\pi$
$930$	0	0
$931$	21.3044	0.698223
$932$	21.5725	0.706631
$933$	0	0
$934$	9.99777	0.327137
$935$	−14.1245	−0.461922
$936$	0	0
$937$	−28.1083	−0.918260	−0.459130	−	0.888369i	$-0.651839\pi$
$937$	−28.1083	−0.918260	−0.459130	+	0.888369i	$0.651839\pi$
$938$	−0.505390	−0.0165016
$939$	0	0
$940$	−69.3515	−2.26200
$941$	3.64388	0.118787	0.0593935	−	0.998235i	$-0.481083\pi$
$941$	3.64388	0.118787	0.0593935	+	0.998235i	$0.481083\pi$
$942$	0	0
$943$	7.14375	0.232632
$944$	16.2189	0.527879
$945$	0	0
$946$	−3.91976	−0.127443
$947$	19.1649	0.622775	0.311388	−	0.950283i	$-0.399206\pi$
$947$	19.1649	0.622775	0.311388	+	0.950283i	$0.399206\pi$
$948$	0	0
$949$	44.5172	1.44509
$950$	−5.88734	−0.191011
$951$	0	0
$952$	0.532084	0.0172450
$953$	−31.6253	−1.02445	−0.512223	−	0.858853i	$-0.671178\pi$
$953$	−31.6253	−1.02445	−0.512223	+	0.858853i	$0.671178\pi$
$954$	0	0
$955$	−40.0033	−1.29448
$956$	−19.7586	−0.639038
$957$	0	0
$958$	5.61353	0.181365
$959$	5.63765	0.182049
$960$	0	0
$961$	76.3649	2.46338
$962$	−16.6531	−0.536916
$963$	0	0
$964$	5.75456	0.185342
$965$	−30.3328	−0.976446
$966$	0	0
$967$	−27.6891	−0.890421	−0.445211	−	0.895426i	$-0.646871\pi$
$967$	−27.6891	−0.890421	−0.445211	+	0.895426i	$0.646871\pi$
$968$	−6.05876	−0.194736
$969$	0	0
$970$	3.35795	0.107817
$971$	35.9385	1.15332	0.576661	−	0.816984i	$-0.304356\pi$
$971$	35.9385	1.15332	0.576661	+	0.816984i	$0.304356\pi$
$972$	0	0
$973$	−0.979208	−0.0313920
$974$	2.28285	0.0731472
$975$	0	0
$976$	−14.5296	−0.465083
$977$	−38.4441	−1.22993	−0.614967	−	0.788553i	$-0.710831\pi$
$977$	−38.4441	−1.22993	−0.614967	+	0.788553i	$0.710831\pi$
$978$	0	0
$979$	−41.3140	−1.32040
$980$	−43.5765	−1.39200
$981$	0	0
$982$	−0.577088	−0.0184156
$983$	17.4592	0.556863	0.278431	−	0.960456i	$-0.410186\pi$
$983$	17.4592	0.556863	0.278431	+	0.960456i	$0.410186\pi$
$984$	0	0
$985$	22.9272	0.730522
$986$	0.317973	0.0101263
$987$	0	0
$988$	31.6808	1.00790
$989$	−3.22443	−0.102531
$990$	0	0
$991$	42.9628	1.36476	0.682380	−	0.730998i	$-0.260945\pi$
$991$	42.9628	1.36476	0.682380	+	0.730998i	$0.260945\pi$
$992$	−35.3731	−1.12310
$993$	0	0
$994$	−1.77646	−0.0563458
$995$	−41.7866	−1.32473
$996$	0	0
$997$	−16.7387	−0.530121	−0.265061	−	0.964232i	$-0.585392\pi$
$997$	−16.7387	−0.530121	−0.265061	+	0.964232i	$0.585392\pi$
$998$	2.80575	0.0888144
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6003.2.a.s.1.12		20
3.2	odd	2		2001.2.a.o.1.9	✓	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2001.2.a.o.1.9	✓	20	3.2	odd	2
6003.2.a.s.1.12		20	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:	\( N \)	\(=\)	\( 6003 = 3^{2} \cdot 23 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6003.a (trivial)

\(f(q)\)	\(=\)	\(q+0.302778 q^{2} -1.90833 q^{4} -3.34986 q^{5} +0.428154 q^{7} -1.18336 q^{8} +O(q^{10})\)	\(q+0.302778 q^{2} -1.90833 q^{4} -3.34986 q^{5} +0.428154 q^{7} -1.18336 q^{8} -1.01426 q^{10} -4.01497 q^{11} +5.31187 q^{13} +0.129636 q^{14} +3.45836 q^{16} -1.05018 q^{17} -3.12533 q^{19} +6.39262 q^{20} -1.21565 q^{22} -1.00000 q^{23} +6.22156 q^{25} +1.60832 q^{26} -0.817056 q^{28} -1.00000 q^{29} -10.3617 q^{31} +3.41383 q^{32} -0.317973 q^{34} -1.43425 q^{35} -10.3543 q^{37} -0.946281 q^{38} +3.96408 q^{40} -7.14375 q^{41} +3.22443 q^{43} +7.66187 q^{44} -0.302778 q^{46} -10.8487 q^{47} -6.81668 q^{49} +1.88375 q^{50} -10.1368 q^{52} +9.70877 q^{53} +13.4496 q^{55} -0.506658 q^{56} -0.302778 q^{58} +4.68976 q^{59} -4.20131 q^{61} -3.13730 q^{62} -5.88308 q^{64} -17.7940 q^{65} -3.89854 q^{67} +2.00409 q^{68} -0.434261 q^{70} -13.7035 q^{71} +8.38069 q^{73} -3.13506 q^{74} +5.96414 q^{76} -1.71902 q^{77} +8.80707 q^{79} -11.5850 q^{80} -2.16297 q^{82} -7.18985 q^{83} +3.51797 q^{85} +0.976287 q^{86} +4.75114 q^{88} +10.2900 q^{89} +2.27430 q^{91} +1.90833 q^{92} -3.28475 q^{94} +10.4694 q^{95} -3.31072 q^{97} -2.06394 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 2 q^{2} + 30 q^{4} + q^{5} + 9 q^{7} - 6 q^{8}+O(q^{10}) \) 20 * q - 2 * q^2 + 30 * q^4 + q^5 + 9 * q^7 - 6 * q^8	\( 20 q - 2 q^{2} + 30 q^{4} + q^{5} + 9 q^{7} - 6 q^{8} + 7 q^{10} + 21 q^{13} + q^{14} + 58 q^{16} + 4 q^{17} + 7 q^{19} + 20 q^{20} + 7 q^{22} - 20 q^{23} + 47 q^{25} - 8 q^{26} + 11 q^{28} - 20 q^{29} + 28 q^{31} - 14 q^{32} + 16 q^{34} - 9 q^{35} + 14 q^{37} + 20 q^{38} + 34 q^{40} - 7 q^{41} + 3 q^{43} + q^{44} + 2 q^{46} - 3 q^{47} + 35 q^{49} + 24 q^{50} + 73 q^{52} + 19 q^{53} + 29 q^{55} + 30 q^{56} + 2 q^{58} - 20 q^{59} + 15 q^{61} - 12 q^{62} + 82 q^{64} + 28 q^{65} + 20 q^{67} + 23 q^{68} - 24 q^{70} - 63 q^{71} + 19 q^{73} - 16 q^{74} - 44 q^{76} + 7 q^{77} + 32 q^{79} + 56 q^{80} - 20 q^{82} + 21 q^{83} + 4 q^{85} + 6 q^{86} + 55 q^{88} + 13 q^{89} + 70 q^{91} - 30 q^{92} - 12 q^{94} - 9 q^{95} - 9 q^{97} - 31 q^{98}+O(q^{100}) \) 20 * q - 2 * q^2 + 30 * q^4 + q^5 + 9 * q^7 - 6 * q^8 + 7 * q^10 + 21 * q^13 + q^14 + 58 * q^16 + 4 * q^17 + 7 * q^19 + 20 * q^20 + 7 * q^22 - 20 * q^23 + 47 * q^25 - 8 * q^26 + 11 * q^28 - 20 * q^29 + 28 * q^31 - 14 * q^32 + 16 * q^34 - 9 * q^35 + 14 * q^37 + 20 * q^38 + 34 * q^40 - 7 * q^41 + 3 * q^43 + q^44 + 2 * q^46 - 3 * q^47 + 35 * q^49 + 24 * q^50 + 73 * q^52 + 19 * q^53 + 29 * q^55 + 30 * q^56 + 2 * q^58 - 20 * q^59 + 15 * q^61 - 12 * q^62 + 82 * q^64 + 28 * q^65 + 20 * q^67 + 23 * q^68 - 24 * q^70 - 63 * q^71 + 19 * q^73 - 16 * q^74 - 44 * q^76 + 7 * q^77 + 32 * q^79 + 56 * q^80 - 20 * q^82 + 21 * q^83 + 4 * q^85 + 6 * q^86 + 55 * q^88 + 13 * q^89 + 70 * q^91 - 30 * q^92 - 12 * q^94 - 9 * q^95 - 9 * q^97 - 31 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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