LMFDB - Embedded newform 6003.2.a.o.1.8

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:	$1$
Dimension:	$13$
Coefficient field:	$\mathbb{Q}[x]/(x^{13} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{13} - 4 x^{12} - 11 x^{11} + 58 x^{10} + 24 x^{9} - 298 x^{8} + 97 x^{7} + 641 x^{6} - 402 x^{5} + \cdots - 40 $ x^13 - 4x^12 - 11x^11 + 58x^10 + 24x^9 - 298x^8 + 97x^7 + 641x^6 - 402x^5 - 547x^4 + 352x^3 + 219x^2 - 88x - 40
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 667)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Root			$-0.397523$ 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)

$f(q)$	$=$	$q+0.397523 q^{2} -1.84198 q^{4} +3.16986 q^{5} -1.67469 q^{7} -1.52727 q^{8} +O(q^{10})$	\(q+0.397523 q^{2} -1.84198 q^{4} +3.16986 q^{5} -1.67469 q^{7} -1.52727 q^{8} +1.26009 q^{10} -1.35258 q^{11} -0.657616 q^{13} -0.665727 q^{14} +3.07682 q^{16} -4.41950 q^{17} -0.226826 q^{19} -5.83880 q^{20} -0.537681 q^{22} +1.00000 q^{23} +5.04800 q^{25} -0.261418 q^{26} +3.08473 q^{28} -1.00000 q^{29} +9.64062 q^{31} +4.27766 q^{32} -1.75685 q^{34} -5.30852 q^{35} +2.51972 q^{37} -0.0901685 q^{38} -4.84124 q^{40} +7.69417 q^{41} +1.41633 q^{43} +2.49141 q^{44} +0.397523 q^{46} -1.67270 q^{47} -4.19543 q^{49} +2.00670 q^{50} +1.21131 q^{52} -7.39704 q^{53} -4.28747 q^{55} +2.55771 q^{56} -0.397523 q^{58} -6.61560 q^{59} +0.912075 q^{61} +3.83237 q^{62} -4.45318 q^{64} -2.08455 q^{65} -3.75542 q^{67} +8.14060 q^{68} -2.11026 q^{70} -14.6425 q^{71} -3.28531 q^{73} +1.00165 q^{74} +0.417807 q^{76} +2.26514 q^{77} -14.7355 q^{79} +9.75309 q^{80} +3.05861 q^{82} +10.3184 q^{83} -14.0092 q^{85} +0.563026 q^{86} +2.06576 q^{88} -14.5878 q^{89} +1.10130 q^{91} -1.84198 q^{92} -0.664939 q^{94} -0.719005 q^{95} -9.17309 q^{97} -1.66778 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 13 q - 4 q^{2} + 12 q^{4} - 16 q^{5} + q^{7} - 6 q^{8}+O(q^{10}) $ 13 * q - 4 * q^2 + 12 * q^4 - 16 * q^5 + q^7 - 6 * q^8	$ 13 q - 4 q^{2} + 12 q^{4} - 16 q^{5} + q^{7} - 6 q^{8} + 10 q^{10} - 10 q^{11} + 7 q^{13} + 12 q^{14} + 2 q^{16} - 26 q^{17} - 25 q^{20} - 15 q^{22} + 13 q^{23} + 19 q^{25} + 15 q^{26} + 5 q^{28} - 13 q^{29} - 6 q^{31} - 16 q^{32} + 11 q^{34} - q^{35} + 15 q^{37} - 8 q^{38} + 14 q^{40} - 9 q^{41} + q^{43} - 29 q^{44} - 4 q^{46} - 15 q^{47} + 4 q^{49} - 31 q^{50} - 8 q^{52} - 43 q^{53} - 3 q^{55} + 5 q^{56} + 4 q^{58} + 9 q^{59} + 20 q^{61} - 11 q^{62} - 16 q^{64} + 25 q^{65} + q^{67} - 21 q^{68} - 2 q^{70} - 17 q^{71} + 26 q^{73} - 11 q^{74} + 8 q^{76} - 17 q^{77} + 5 q^{79} - 10 q^{80} - 25 q^{82} - 4 q^{83} + 20 q^{85} + 13 q^{86} - 32 q^{88} - 48 q^{89} - 9 q^{91} + 12 q^{92} - 65 q^{94} - 8 q^{95} + 30 q^{97} - 2 q^{98}+O(q^{100}) $ 13 * q - 4 * q^2 + 12 * q^4 - 16 * q^5 + q^7 - 6 * q^8 + 10 * q^10 - 10 * q^11 + 7 * q^13 + 12 * q^14 + 2 * q^16 - 26 * q^17 - 25 * q^20 - 15 * q^22 + 13 * q^23 + 19 * q^25 + 15 * q^26 + 5 * q^28 - 13 * q^29 - 6 * q^31 - 16 * q^32 + 11 * q^34 - q^35 + 15 * q^37 - 8 * q^38 + 14 * q^40 - 9 * q^41 + q^43 - 29 * q^44 - 4 * q^46 - 15 * q^47 + 4 * q^49 - 31 * q^50 - 8 * q^52 - 43 * q^53 - 3 * q^55 + 5 * q^56 + 4 * q^58 + 9 * q^59 + 20 * q^61 - 11 * q^62 - 16 * q^64 + 25 * q^65 + q^67 - 21 * q^68 - 2 * q^70 - 17 * q^71 + 26 * q^73 - 11 * q^74 + 8 * q^76 - 17 * q^77 + 5 * q^79 - 10 * q^80 - 25 * q^82 - 4 * q^83 + 20 * q^85 + 13 * q^86 - 32 * q^88 - 48 * q^89 - 9 * q^91 + 12 * q^92 - 65 * q^94 - 8 * q^95 + 30 * 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 $

$2$	0.397523	0.281091	0.140546	−	0.990074i	$-0.455114\pi$
$2$	0.397523	0.281091	0.140546	+	0.990074i	$0.455114\pi$
$3$	0	0
$3$	0	0
$4$	−1.84198	−0.920988
$5$	3.16986	1.41760	0.708802	−	0.705408i	$-0.249236\pi$
$5$	3.16986	1.41760	0.708802	+	0.705408i	$0.249236\pi$
$6$	0	0
$7$	−1.67469	−0.632972	−0.316486	−	0.948597i	$-0.602503\pi$
$7$	−1.67469	−0.632972	−0.316486	+	0.948597i	$0.602503\pi$
$8$	−1.52727	−0.539973
$9$	0	0
$10$	1.26009	0.398476
$11$	−1.35258	−0.407817	−0.203909	−	0.978990i	$-0.565365\pi$
$11$	−1.35258	−0.407817	−0.203909	+	0.978990i	$0.565365\pi$
$12$	0	0
$13$	−0.657616	−0.182390	−0.0911949	−	0.995833i	$-0.529069\pi$
$13$	−0.657616	−0.182390	−0.0911949	+	0.995833i	$0.529069\pi$
$14$	−0.665727	−0.177923
$15$	0	0
$16$	3.07682	0.769206
$17$	−4.41950	−1.07189	−0.535943	−	0.844254i	$-0.680044\pi$
$17$	−4.41950	−1.07189	−0.535943	+	0.844254i	$0.680044\pi$
$18$	0	0
$19$	−0.226826	−0.0520374	−0.0260187	−	0.999661i	$-0.508283\pi$
$19$	−0.226826	−0.0520374	−0.0260187	+	0.999661i	$0.508283\pi$
$20$	−5.83880	−1.30560
$21$	0	0
$22$	−0.537681	−0.114634
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	5.04800	1.00960
$26$	−0.261418	−0.0512682
$27$	0	0
$28$	3.08473	0.582959
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	0	0
$31$	9.64062	1.73151	0.865754	−	0.500470i	$-0.166840\pi$
$31$	9.64062	1.73151	0.865754	+	0.500470i	$0.166840\pi$
$32$	4.27766	0.756190
$33$	0	0
$34$	−1.75685	−0.301298
$35$	−5.30852	−0.897303
$36$	0	0
$37$	2.51972	0.414240	0.207120	−	0.978316i	$-0.433591\pi$
$37$	2.51972	0.414240	0.207120	+	0.978316i	$0.433591\pi$
$38$	−0.0901685	−0.0146273
$39$	0	0
$40$	−4.84124	−0.765468
$41$	7.69417	1.20163	0.600814	−	0.799389i	$-0.294843\pi$
$41$	7.69417	1.20163	0.600814	+	0.799389i	$0.294843\pi$
$42$	0	0
$43$	1.41633	0.215989	0.107995	−	0.994151i	$-0.465557\pi$
$43$	1.41633	0.215989	0.107995	+	0.994151i	$0.465557\pi$
$44$	2.49141	0.375595
$45$	0	0
$46$	0.397523	0.0586116
$47$	−1.67270	−0.243989	−0.121995	−	0.992531i	$-0.538929\pi$
$47$	−1.67270	−0.243989	−0.121995	+	0.992531i	$0.538929\pi$
$48$	0	0
$49$	−4.19543	−0.599347
$50$	2.00670	0.283790
$51$	0	0
$52$	1.21131	0.167979
$53$	−7.39704	−1.01606	−0.508031	−	0.861339i	$-0.669626\pi$
$53$	−7.39704	−1.01606	−0.508031	+	0.861339i	$0.669626\pi$
$54$	0	0
$55$	−4.28747	−0.578123
$56$	2.55771	0.341788
$57$	0	0
$58$	−0.397523	−0.0521974
$59$	−6.61560	−0.861278	−0.430639	−	0.902524i	$-0.641712\pi$
$59$	−6.61560	−0.861278	−0.430639	+	0.902524i	$0.641712\pi$
$60$	0	0
$61$	0.912075	0.116779	0.0583896	−	0.998294i	$-0.481403\pi$
$61$	0.912075	0.116779	0.0583896	+	0.998294i	$0.481403\pi$
$62$	3.83237	0.486712
$63$	0	0
$64$	−4.45318	−0.556647
$65$	−2.08455	−0.258556
$66$	0	0
$67$	−3.75542	−0.458798	−0.229399	−	0.973332i	$-0.573676\pi$
$67$	−3.75542	−0.458798	−0.229399	+	0.973332i	$0.573676\pi$
$68$	8.14060	0.987193
$69$	0	0
$70$	−2.11026	−0.252224
$71$	−14.6425	−1.73774	−0.868872	−	0.495036i	$-0.835155\pi$
$71$	−14.6425	−1.73774	−0.868872	+	0.495036i	$0.835155\pi$
$72$	0	0
$73$	−3.28531	−0.384516	−0.192258	−	0.981344i	$-0.561581\pi$
$73$	−3.28531	−0.384516	−0.192258	+	0.981344i	$0.561581\pi$
$74$	1.00165	0.116439
$75$	0	0
$76$	0.417807	0.0479258
$77$	2.26514	0.258137
$78$	0	0
$79$	−14.7355	−1.65788	−0.828939	−	0.559340i	$-0.811055\pi$
$79$	−14.7355	−1.65788	−0.828939	+	0.559340i	$0.811055\pi$
$80$	9.75309	1.09043
$81$	0	0
$82$	3.05861	0.337767
$83$	10.3184	1.13259	0.566295	−	0.824203i	$-0.308376\pi$
$83$	10.3184	1.13259	0.566295	+	0.824203i	$0.308376\pi$
$84$	0	0
$85$	−14.0092	−1.51951
$86$	0.563026	0.0607127
$87$	0	0
$88$	2.06576	0.220210
$89$	−14.5878	−1.54630	−0.773152	−	0.634221i	$-0.781321\pi$
$89$	−14.5878	−1.54630	−0.773152	+	0.634221i	$0.781321\pi$
$90$	0	0
$91$	1.10130	0.115448
$92$	−1.84198	−0.192039
$93$	0	0
$94$	−0.664939	−0.0685832
$95$	−0.719005	−0.0737683
$96$	0	0
$97$	−9.17309	−0.931386	−0.465693	−	0.884946i	$-0.654195\pi$
$97$	−9.17309	−0.931386	−0.465693	+	0.884946i	$0.654195\pi$
$98$	−1.66778	−0.168471
$99$	0	0
$100$	−9.29829	−0.929829
$101$	−0.636772	−0.0633612	−0.0316806	−	0.999498i	$-0.510086\pi$
$101$	−0.636772	−0.0633612	−0.0316806	+	0.999498i	$0.510086\pi$
$102$	0	0
$103$	12.3245	1.21437	0.607183	−	0.794562i	$-0.292299\pi$
$103$	12.3245	1.21437	0.607183	+	0.794562i	$0.292299\pi$
$104$	1.00436	0.0984856
$105$	0	0
$106$	−2.94050	−0.285606
$107$	−17.0007	−1.64352	−0.821760	−	0.569834i	$-0.807008\pi$
$107$	−17.0007	−1.64352	−0.821760	+	0.569834i	$0.807008\pi$
$108$	0	0
$109$	2.98969	0.286360	0.143180	−	0.989697i	$-0.454267\pi$
$109$	2.98969	0.286360	0.143180	+	0.989697i	$0.454267\pi$
$110$	−1.70437	−0.162505
$111$	0	0
$112$	−5.15271	−0.486886
$113$	−3.83318	−0.360595	−0.180298	−	0.983612i	$-0.557706\pi$
$113$	−3.83318	−0.360595	−0.180298	+	0.983612i	$0.557706\pi$
$114$	0	0
$115$	3.16986	0.295591
$116$	1.84198	0.171023
$117$	0	0
$118$	−2.62986	−0.242098
$119$	7.40127	0.678473
$120$	0	0
$121$	−9.17054	−0.833685
$122$	0.362571	0.0328257
$123$	0	0
$124$	−17.7578	−1.59470
$125$	0.152151	0.0136088
$126$	0	0
$127$	9.31467	0.826543	0.413271	−	0.910608i	$-0.364386\pi$
$127$	9.31467	0.826543	0.413271	+	0.910608i	$0.364386\pi$
$128$	−10.3256	−0.912659
$129$	0	0
$130$	−0.828657	−0.0726780
$131$	−4.17017	−0.364349	−0.182175	−	0.983266i	$-0.558314\pi$
$131$	−4.17017	−0.364349	−0.182175	+	0.983266i	$0.558314\pi$
$132$	0	0
$133$	0.379862	0.0329382
$134$	−1.49287	−0.128964
$135$	0	0
$136$	6.74979	0.578789
$137$	−3.55809	−0.303988	−0.151994	−	0.988381i	$-0.548570\pi$
$137$	−3.55809	−0.303988	−0.151994	+	0.988381i	$0.548570\pi$
$138$	0	0
$139$	0.173896	0.0147496	0.00737482	−	0.999973i	$-0.497652\pi$
$139$	0.173896	0.0147496	0.00737482	+	0.999973i	$0.497652\pi$
$140$	9.77816	0.826405
$141$	0	0
$142$	−5.82073	−0.488465
$143$	0.889476	0.0743817
$144$	0	0
$145$	−3.16986	−0.263242
$146$	−1.30599	−0.108084
$147$	0	0
$148$	−4.64126	−0.381510
$149$	−3.06023	−0.250704	−0.125352	−	0.992112i	$-0.540006\pi$
$149$	−3.06023	−0.250704	−0.125352	+	0.992112i	$0.540006\pi$
$150$	0	0
$151$	11.2536	0.915808	0.457904	−	0.889002i	$-0.348600\pi$
$151$	11.2536	0.915808	0.457904	+	0.889002i	$0.348600\pi$
$152$	0.346425	0.0280988
$153$	0	0
$154$	0.900446	0.0725600
$155$	30.5594	2.45459
$156$	0	0
$157$	0.269281	0.0214909	0.0107455	−	0.999942i	$-0.496580\pi$
$157$	0.269281	0.0214909	0.0107455	+	0.999942i	$0.496580\pi$
$158$	−5.85772	−0.466015
$159$	0	0
$160$	13.5596	1.07198
$161$	−1.67469	−0.131984
$162$	0	0
$163$	5.88692	0.461099	0.230550	−	0.973061i	$-0.425948\pi$
$163$	5.88692	0.461099	0.230550	+	0.973061i	$0.425948\pi$
$164$	−14.1725	−1.10668
$165$	0	0
$166$	4.10180	0.318361
$167$	10.6448	0.823721	0.411861	−	0.911247i	$-0.364879\pi$
$167$	10.6448	0.823721	0.411861	+	0.911247i	$0.364879\pi$
$168$	0	0
$169$	−12.5675	−0.966734
$170$	−5.56897	−0.427121
$171$	0	0
$172$	−2.60885	−0.198923
$173$	1.05240	0.0800126	0.0400063	−	0.999199i	$-0.487262\pi$
$173$	1.05240	0.0800126	0.0400063	+	0.999199i	$0.487262\pi$
$174$	0	0
$175$	−8.45381	−0.639048
$176$	−4.16164	−0.313695
$177$	0	0
$178$	−5.79899	−0.434653
$179$	6.96566	0.520638	0.260319	−	0.965523i	$-0.416172\pi$
$179$	6.96566	0.520638	0.260319	+	0.965523i	$0.416172\pi$
$180$	0	0
$181$	15.5790	1.15798	0.578989	−	0.815335i	$-0.303447\pi$
$181$	15.5790	1.15798	0.578989	+	0.815335i	$0.303447\pi$
$182$	0.437793	0.0324513
$183$	0	0
$184$	−1.52727	−0.112592
$185$	7.98716	0.587228
$186$	0	0
$187$	5.97771	0.437133
$188$	3.08108	0.224711
$189$	0	0
$190$	−0.285821	−0.0207356
$191$	−5.98633	−0.433156	−0.216578	−	0.976265i	$-0.569489\pi$
$191$	−5.98633	−0.433156	−0.216578	+	0.976265i	$0.569489\pi$
$192$	0	0
$193$	−23.2496	−1.67354	−0.836770	−	0.547555i	$-0.815559\pi$
$193$	−23.2496	−1.67354	−0.836770	+	0.547555i	$0.815559\pi$
$194$	−3.64652	−0.261805
$195$	0	0
$196$	7.72787	0.551991
$197$	−24.4309	−1.74063	−0.870315	−	0.492495i	$-0.836085\pi$
$197$	−24.4309	−1.74063	−0.870315	+	0.492495i	$0.836085\pi$
$198$	0	0
$199$	−4.61601	−0.327220	−0.163610	−	0.986525i	$-0.552314\pi$
$199$	−4.61601	−0.327220	−0.163610	+	0.986525i	$0.552314\pi$
$200$	−7.70968	−0.545157
$201$	0	0
$202$	−0.253132	−0.0178103
$203$	1.67469	0.117540
$204$	0	0
$205$	24.3894	1.70343
$206$	4.89926	0.341348
$207$	0	0
$208$	−2.02337	−0.140295
$209$	0.306799	0.0212217
$210$	0	0
$211$	−28.4009	−1.95520	−0.977599	−	0.210477i	$-0.932498\pi$
$211$	−28.4009	−1.95520	−0.977599	+	0.210477i	$0.932498\pi$
$212$	13.6252	0.935780
$213$	0	0
$214$	−6.75818	−0.461979
$215$	4.48958	0.306187
$216$	0	0
$217$	−16.1450	−1.09600
$218$	1.18847	0.0804934
$219$	0	0
$220$	7.89742	0.532444
$221$	2.90633	0.195501
$222$	0	0
$223$	−15.4282	−1.03315	−0.516573	−	0.856243i	$-0.672793\pi$
$223$	−15.4282	−1.03315	−0.516573	+	0.856243i	$0.672793\pi$
$224$	−7.16374	−0.478647
$225$	0	0
$226$	−1.52378	−0.101360
$227$	14.9603	0.992950	0.496475	−	0.868051i	$-0.334627\pi$
$227$	14.9603	0.992950	0.496475	+	0.868051i	$0.334627\pi$
$228$	0	0
$229$	9.10667	0.601786	0.300893	−	0.953658i	$-0.402715\pi$
$229$	9.10667	0.601786	0.300893	+	0.953658i	$0.402715\pi$
$230$	1.26009	0.0830880
$231$	0	0
$232$	1.52727	0.100271
$233$	8.00051	0.524131	0.262065	−	0.965050i	$-0.415596\pi$
$233$	8.00051	0.524131	0.262065	+	0.965050i	$0.415596\pi$
$234$	0	0
$235$	−5.30224	−0.345880
$236$	12.1858	0.793226
$237$	0	0
$238$	2.94218	0.190713
$239$	4.11046	0.265884	0.132942	−	0.991124i	$-0.457558\pi$
$239$	4.11046	0.265884	0.132942	+	0.991124i	$0.457558\pi$
$240$	0	0
$241$	26.8825	1.73165	0.865827	−	0.500343i	$-0.166793\pi$
$241$	26.8825	1.73165	0.865827	+	0.500343i	$0.166793\pi$
$242$	−3.64550	−0.234342
$243$	0	0
$244$	−1.68002	−0.107552
$245$	−13.2989	−0.849636
$246$	0	0
$247$	0.149164	0.00949108
$248$	−14.7239	−0.934968
$249$	0	0
$250$	0.0604835	0.00382531
$251$	−2.97370	−0.187698	−0.0938492	−	0.995586i	$-0.529917\pi$
$251$	−2.97370	−0.187698	−0.0938492	+	0.995586i	$0.529917\pi$
$252$	0	0
$253$	−1.35258	−0.0850357
$254$	3.70280	0.232334
$255$	0	0
$256$	4.80170	0.300106
$257$	−22.6761	−1.41450	−0.707248	−	0.706965i	$-0.750064\pi$
$257$	−22.6761	−1.41450	−0.707248	+	0.706965i	$0.750064\pi$
$258$	0	0
$259$	−4.21974	−0.262202
$260$	3.83969	0.238127
$261$	0	0
$262$	−1.65774	−0.102415
$263$	−25.7248	−1.58626	−0.793130	−	0.609053i	$-0.791550\pi$
$263$	−25.7248	−1.58626	−0.793130	+	0.609053i	$0.791550\pi$
$264$	0	0
$265$	−23.4476	−1.44037
$266$	0.151004	0.00925864
$267$	0	0
$268$	6.91740	0.422547
$269$	−20.1571	−1.22900	−0.614500	−	0.788917i	$-0.710642\pi$
$269$	−20.1571	−1.22900	−0.614500	+	0.788917i	$0.710642\pi$
$270$	0	0
$271$	−2.32005	−0.140933	−0.0704664	−	0.997514i	$-0.522449\pi$
$271$	−2.32005	−0.140933	−0.0704664	+	0.997514i	$0.522449\pi$
$272$	−13.5980	−0.824500
$273$	0	0
$274$	−1.41443	−0.0854485
$275$	−6.82780	−0.411732
$276$	0	0
$277$	8.38015	0.503514	0.251757	−	0.967790i	$-0.418992\pi$
$277$	8.38015	0.503514	0.251757	+	0.967790i	$0.418992\pi$
$278$	0.0691276	0.00414600
$279$	0	0
$280$	8.10756	0.484520
$281$	−13.7243	−0.818724	−0.409362	−	0.912372i	$-0.634249\pi$
$281$	−13.7243	−0.818724	−0.409362	+	0.912372i	$0.634249\pi$
$282$	0	0
$283$	17.4806	1.03912	0.519558	−	0.854435i	$-0.326097\pi$
$283$	17.4806	1.03912	0.519558	+	0.854435i	$0.326097\pi$
$284$	26.9711	1.60044
$285$	0	0
$286$	0.353587	0.0209081
$287$	−12.8853	−0.760596
$288$	0	0
$289$	2.53195	0.148938
$290$	−1.26009	−0.0739952
$291$	0	0
$292$	6.05145	0.354135
$293$	−25.2785	−1.47679	−0.738393	−	0.674370i	$-0.764415\pi$
$293$	−25.2785	−1.47679	−0.738393	+	0.674370i	$0.764415\pi$
$294$	0	0
$295$	−20.9705	−1.22095
$296$	−3.84831	−0.223678
$297$	0	0
$298$	−1.21651	−0.0704707
$299$	−0.657616	−0.0380309
$300$	0	0
$301$	−2.37192	−0.136715
$302$	4.47358	0.257426
$303$	0	0
$304$	−0.697902	−0.0400274
$305$	2.89115	0.165547
$306$	0	0
$307$	0.488401	0.0278745	0.0139373	−	0.999903i	$-0.495563\pi$
$307$	0.488401	0.0278745	0.0139373	+	0.999903i	$0.495563\pi$
$308$	−4.17233	−0.237741
$309$	0	0
$310$	12.1481	0.689965
$311$	−5.22317	−0.296179	−0.148089	−	0.988974i	$-0.547312\pi$
$311$	−5.22317	−0.296179	−0.148089	+	0.988974i	$0.547312\pi$
$312$	0	0
$313$	31.4710	1.77885	0.889424	−	0.457083i	$-0.151106\pi$
$313$	31.4710	1.77885	0.889424	+	0.457083i	$0.151106\pi$
$314$	0.107045	0.00604092
$315$	0	0
$316$	27.1425	1.52688
$317$	−23.1945	−1.30273	−0.651366	−	0.758764i	$-0.725804\pi$
$317$	−23.1945	−1.30273	−0.651366	+	0.758764i	$0.725804\pi$
$318$	0	0
$319$	1.35258	0.0757297
$320$	−14.1159	−0.789105
$321$	0	0
$322$	−0.665727	−0.0370995
$323$	1.00245	0.0557781
$324$	0	0
$325$	−3.31964	−0.184141
$326$	2.34019	0.129611
$327$	0	0
$328$	−11.7511	−0.648847
$329$	2.80126	0.154438
$330$	0	0
$331$	−8.40340	−0.461893	−0.230946	−	0.972966i	$-0.574182\pi$
$331$	−8.40340	−0.461893	−0.230946	+	0.972966i	$0.574182\pi$
$332$	−19.0062	−1.04310
$333$	0	0
$334$	4.23157	0.231541
$335$	−11.9042	−0.650394
$336$	0	0
$337$	9.69433	0.528084	0.264042	−	0.964511i	$-0.414944\pi$
$337$	9.69433	0.528084	0.264042	+	0.964511i	$0.414944\pi$
$338$	−4.99589	−0.271741
$339$	0	0
$340$	25.8046	1.39945
$341$	−13.0397	−0.706138
$342$	0	0
$343$	18.7488	1.01234
$344$	−2.16313	−0.116628
$345$	0	0
$346$	0.418354	0.0224908
$347$	−14.0015	−0.751642	−0.375821	−	0.926692i	$-0.622639\pi$
$347$	−14.0015	−0.751642	−0.375821	+	0.926692i	$0.622639\pi$
$348$	0	0
$349$	−0.389464	−0.0208475	−0.0104238	−	0.999946i	$-0.503318\pi$
$349$	−0.389464	−0.0208475	−0.0104238	+	0.999946i	$0.503318\pi$
$350$	−3.36059	−0.179631
$351$	0	0
$352$	−5.78586	−0.308387
$353$	−17.2560	−0.918446	−0.459223	−	0.888321i	$-0.651872\pi$
$353$	−17.2560	−0.918446	−0.459223	+	0.888321i	$0.651872\pi$
$354$	0	0
$355$	−46.4146	−2.46343
$356$	26.8704	1.42413
$357$	0	0
$358$	2.76901	0.146347
$359$	30.6718	1.61879	0.809397	−	0.587262i	$-0.199794\pi$
$359$	30.6718	1.61879	0.809397	+	0.587262i	$0.199794\pi$
$360$	0	0
$361$	−18.9486	−0.997292
$362$	6.19301	0.325498
$363$	0	0
$364$	−2.02857	−0.106326
$365$	−10.4140	−0.545091
$366$	0	0
$367$	−9.11035	−0.475557	−0.237778	−	0.971319i	$-0.576419\pi$
$367$	−9.11035	−0.475557	−0.237778	+	0.971319i	$0.576419\pi$
$368$	3.07682	0.160390
$369$	0	0
$370$	3.17508	0.165065
$371$	12.3877	0.643138
$372$	0	0
$373$	5.35362	0.277200	0.138600	−	0.990348i	$-0.455740\pi$
$373$	5.35362	0.277200	0.138600	+	0.990348i	$0.455740\pi$
$374$	2.37628	0.122874
$375$	0	0
$376$	2.55468	0.131748
$377$	0.657616	0.0338689
$378$	0	0
$379$	−11.6032	−0.596016	−0.298008	−	0.954563i	$-0.596322\pi$
$379$	−11.6032	−0.596016	−0.298008	+	0.954563i	$0.596322\pi$
$380$	1.32439	0.0679397
$381$	0	0
$382$	−2.37971	−0.121756
$383$	8.31496	0.424874	0.212437	−	0.977175i	$-0.431860\pi$
$383$	8.31496	0.424874	0.212437	+	0.977175i	$0.431860\pi$
$384$	0	0
$385$	7.18017	0.365936
$386$	−9.24224	−0.470418
$387$	0	0
$388$	16.8966	0.857795
$389$	2.26821	0.115003	0.0575014	−	0.998345i	$-0.481687\pi$
$389$	2.26821	0.115003	0.0575014	+	0.998345i	$0.481687\pi$
$390$	0	0
$391$	−4.41950	−0.223504
$392$	6.40757	0.323631
$393$	0	0
$394$	−9.71186	−0.489276
$395$	−46.7096	−2.35021
$396$	0	0
$397$	−12.9717	−0.651032	−0.325516	−	0.945537i	$-0.605538\pi$
$397$	−12.9717	−0.651032	−0.325516	+	0.945537i	$0.605538\pi$
$398$	−1.83497	−0.0919788
$399$	0	0
$400$	15.5318	0.776590
$401$	0.997267	0.0498011	0.0249006	−	0.999690i	$-0.492073\pi$
$401$	0.997267	0.0498011	0.0249006	+	0.999690i	$0.492073\pi$
$402$	0	0
$403$	−6.33983	−0.315809
$404$	1.17292	0.0583548
$405$	0	0
$406$	0.665727	0.0330395
$407$	−3.40812	−0.168934
$408$	0	0
$409$	15.6229	0.772502	0.386251	−	0.922394i	$-0.373770\pi$
$409$	15.6229	0.772502	0.386251	+	0.922394i	$0.373770\pi$
$410$	9.69536	0.478820
$411$	0	0
$412$	−22.7014	−1.11842
$413$	11.0791	0.545165
$414$	0	0
$415$	32.7078	1.60556
$416$	−2.81306	−0.137921
$417$	0	0
$418$	0.121960	0.00596524
$419$	−14.2200	−0.694693	−0.347347	−	0.937737i	$-0.612917\pi$
$419$	−14.2200	−0.694693	−0.347347	+	0.937737i	$0.612917\pi$
$420$	0	0
$421$	32.2664	1.57257	0.786284	−	0.617866i	$-0.212002\pi$
$421$	32.2664	1.57257	0.786284	+	0.617866i	$0.212002\pi$
$422$	−11.2900	−0.549589
$423$	0	0
$424$	11.2973	0.548646
$425$	−22.3096	−1.08218
$426$	0	0
$427$	−1.52744	−0.0739180
$428$	31.3149	1.51366
$429$	0	0
$430$	1.78471	0.0860665
$431$	35.5584	1.71279	0.856393	−	0.516325i	$-0.172700\pi$
$431$	35.5584	1.71279	0.856393	+	0.516325i	$0.172700\pi$
$432$	0	0
$433$	4.61554	0.221809	0.110904	−	0.993831i	$-0.464625\pi$
$433$	4.61554	0.221809	0.110904	+	0.993831i	$0.464625\pi$
$434$	−6.41802	−0.308075
$435$	0	0
$436$	−5.50693	−0.263734
$437$	−0.226826	−0.0108505
$438$	0	0
$439$	−2.30348	−0.109939	−0.0549695	−	0.998488i	$-0.517506\pi$
$439$	−2.30348	−0.109939	−0.0549695	+	0.998488i	$0.517506\pi$
$440$	6.54815	0.312171
$441$	0	0
$442$	1.15533	0.0549537
$443$	−8.59573	−0.408395	−0.204198	−	0.978930i	$-0.565459\pi$
$443$	−8.59573	−0.408395	−0.204198	+	0.978930i	$0.565459\pi$
$444$	0	0
$445$	−46.2413	−2.19205
$446$	−6.13306	−0.290409
$447$	0	0
$448$	7.45767	0.352342
$449$	16.1317	0.761303	0.380651	−	0.924719i	$-0.375700\pi$
$449$	16.1317	0.761303	0.380651	+	0.924719i	$0.375700\pi$
$450$	0	0
$451$	−10.4070	−0.490044
$452$	7.06062	0.332104
$453$	0	0
$454$	5.94707	0.279110
$455$	3.49097	0.163659
$456$	0	0
$457$	−27.4394	−1.28356	−0.641780	−	0.766889i	$-0.721804\pi$
$457$	−27.4394	−1.28356	−0.641780	+	0.766889i	$0.721804\pi$
$458$	3.62011	0.169157
$459$	0	0
$460$	−5.83880	−0.272235
$461$	17.7923	0.828669	0.414334	−	0.910125i	$-0.364014\pi$
$461$	17.7923	0.828669	0.414334	+	0.910125i	$0.364014\pi$
$462$	0	0
$463$	−3.23767	−0.150467	−0.0752336	−	0.997166i	$-0.523970\pi$
$463$	−3.23767	−0.150467	−0.0752336	+	0.997166i	$0.523970\pi$
$464$	−3.07682	−0.142838
$465$	0	0
$466$	3.18039	0.147329
$467$	−16.3389	−0.756075	−0.378038	−	0.925790i	$-0.623401\pi$
$467$	−16.3389	−0.756075	−0.378038	+	0.925790i	$0.623401\pi$
$468$	0	0
$469$	6.28916	0.290406
$470$	−2.10776	−0.0972238
$471$	0	0
$472$	10.1038	0.465067
$473$	−1.91570	−0.0880840
$474$	0	0
$475$	−1.14502	−0.0525369
$476$	−13.6330	−0.624865
$477$	0	0
$478$	1.63401	0.0747377
$479$	−7.83185	−0.357846	−0.178923	−	0.983863i	$-0.557261\pi$
$479$	−7.83185	−0.357846	−0.178923	+	0.983863i	$0.557261\pi$
$480$	0	0
$481$	−1.65701	−0.0755531
$482$	10.6864	0.486753
$483$	0	0
$484$	16.8919	0.767814
$485$	−29.0774	−1.32034
$486$	0	0
$487$	38.3921	1.73971	0.869856	−	0.493305i	$-0.164211\pi$
$487$	38.3921	1.73971	0.869856	+	0.493305i	$0.164211\pi$
$488$	−1.39299	−0.0630577
$489$	0	0
$490$	−5.28663	−0.238825
$491$	32.4833	1.46595	0.732975	−	0.680256i	$-0.238131\pi$
$491$	32.4833	1.46595	0.732975	+	0.680256i	$0.238131\pi$
$492$	0	0
$493$	4.41950	0.199044
$494$	0.0592962	0.00266786
$495$	0	0
$496$	29.6625	1.33189
$497$	24.5216	1.09994
$498$	0	0
$499$	−8.26214	−0.369864	−0.184932	−	0.982751i	$-0.559207\pi$
$499$	−8.26214	−0.369864	−0.184932	+	0.982751i	$0.559207\pi$
$500$	−0.280258	−0.0125335
$501$	0	0
$502$	−1.18212	−0.0527604
$503$	−0.275633	−0.0122899	−0.00614493	−	0.999981i	$-0.501956\pi$
$503$	−0.275633	−0.0122899	−0.00614493	+	0.999981i	$0.501956\pi$
$504$	0	0
$505$	−2.01848	−0.0898210
$506$	−0.537681	−0.0239028
$507$	0	0
$508$	−17.1574	−0.761236
$509$	−2.03889	−0.0903720	−0.0451860	−	0.998979i	$-0.514388\pi$
$509$	−2.03889	−0.0903720	−0.0451860	+	0.998979i	$0.514388\pi$
$510$	0	0
$511$	5.50186	0.243388
$512$	22.5599	0.997016
$513$	0	0
$514$	−9.01428	−0.397603
$515$	39.0668	1.72149
$516$	0	0
$517$	2.26246	0.0995029
$518$	−1.67745	−0.0737027
$519$	0	0
$520$	3.18368	0.139614
$521$	36.2346	1.58747	0.793734	−	0.608265i	$-0.208134\pi$
$521$	36.2346	1.58747	0.793734	+	0.608265i	$0.208134\pi$
$522$	0	0
$523$	−23.2807	−1.01800	−0.508998	−	0.860768i	$-0.669984\pi$
$523$	−23.2807	−1.01800	−0.508998	+	0.860768i	$0.669984\pi$
$524$	7.68134	0.335561
$525$	0	0
$526$	−10.2262	−0.445884
$527$	−42.6067	−1.85598
$528$	0	0
$529$	1.00000	0.0434783
$530$	−9.32095	−0.404876
$531$	0	0
$532$	−0.699696	−0.0303357
$533$	−5.05981	−0.219165
$534$	0	0
$535$	−53.8898	−2.32986
$536$	5.73557	0.247739
$537$	0	0
$538$	−8.01291	−0.345461
$539$	5.67463	0.244424
$540$	0	0
$541$	−26.0846	−1.12146	−0.560732	−	0.827997i	$-0.689480\pi$
$541$	−26.0846	−1.12146	−0.560732	+	0.827997i	$0.689480\pi$
$542$	−0.922272	−0.0396150
$543$	0	0
$544$	−18.9051	−0.810549
$545$	9.47689	0.405945
$546$	0	0
$547$	10.3897	0.444230	0.222115	−	0.975020i	$-0.428704\pi$
$547$	10.3897	0.444230	0.222115	+	0.975020i	$0.428704\pi$
$548$	6.55392	0.279970
$549$	0	0
$550$	−2.71421	−0.115734
$551$	0.226826	0.00966309
$552$	0	0
$553$	24.6774	1.04939
$554$	3.33130	0.141534
$555$	0	0
$556$	−0.320312	−0.0135842
$557$	7.93520	0.336225	0.168113	−	0.985768i	$-0.446233\pi$
$557$	7.93520	0.336225	0.168113	+	0.985768i	$0.446233\pi$
$558$	0	0
$559$	−0.931404	−0.0393942
$560$	−16.3334	−0.690211
$561$	0	0
$562$	−5.45574	−0.230136
$563$	−30.9842	−1.30583	−0.652915	−	0.757431i	$-0.726454\pi$
$563$	−30.9842	−1.30583	−0.652915	+	0.757431i	$0.726454\pi$
$564$	0	0
$565$	−12.1506	−0.511181
$566$	6.94896	0.292087
$567$	0	0
$568$	22.3631	0.938336
$569$	−36.3560	−1.52412	−0.762061	−	0.647505i	$-0.775813\pi$
$569$	−36.3560	−1.52412	−0.762061	+	0.647505i	$0.775813\pi$
$570$	0	0
$571$	2.76541	0.115729	0.0578644	−	0.998324i	$-0.481571\pi$
$571$	2.76541	0.115729	0.0578644	+	0.998324i	$0.481571\pi$
$572$	−1.63839	−0.0685046
$573$	0	0
$574$	−5.12221	−0.213797
$575$	5.04800	0.210516
$576$	0	0
$577$	−17.1204	−0.712730	−0.356365	−	0.934347i	$-0.615984\pi$
$577$	−17.1204	−0.712730	−0.356365	+	0.934347i	$0.615984\pi$
$578$	1.00651	0.0418652
$579$	0	0
$580$	5.83880	0.242443
$581$	−17.2801	−0.716898
$582$	0	0
$583$	10.0051	0.414367
$584$	5.01757	0.207628
$585$	0	0
$586$	−10.0488	−0.415112
$587$	28.2800	1.16724	0.583621	−	0.812026i	$-0.301635\pi$
$587$	28.2800	1.16724	0.583621	+	0.812026i	$0.301635\pi$
$588$	0	0
$589$	−2.18674	−0.0901031
$590$	−8.33627	−0.343199
$591$	0	0
$592$	7.75274	0.318635
$593$	−36.0830	−1.48175	−0.740876	−	0.671642i	$-0.765589\pi$
$593$	−36.0830	−1.48175	−0.740876	+	0.671642i	$0.765589\pi$
$594$	0	0
$595$	23.4610	0.961806
$596$	5.63687	0.230895
$597$	0	0
$598$	−0.261418	−0.0106902
$599$	42.6659	1.74328	0.871641	−	0.490145i	$-0.163056\pi$
$599$	42.6659	1.74328	0.871641	+	0.490145i	$0.163056\pi$
$600$	0	0
$601$	11.0225	0.449615	0.224808	−	0.974403i	$-0.427825\pi$
$601$	11.0225	0.449615	0.224808	+	0.974403i	$0.427825\pi$
$602$	−0.942892	−0.0384294
$603$	0	0
$604$	−20.7289	−0.843447
$605$	−29.0693	−1.18184
$606$	0	0
$607$	1.73655	0.0704844	0.0352422	−	0.999379i	$-0.488780\pi$
$607$	1.73655	0.0704844	0.0352422	+	0.999379i	$0.488780\pi$
$608$	−0.970282	−0.0393501
$609$	0	0
$610$	1.14930	0.0465338
$611$	1.10000	0.0445011
$612$	0	0
$613$	−35.0712	−1.41651	−0.708257	−	0.705955i	$-0.750518\pi$
$613$	−35.0712	−1.41651	−0.708257	+	0.705955i	$0.750518\pi$
$614$	0.194151	0.00783529
$615$	0	0
$616$	−3.45949	−0.139387
$617$	32.4852	1.30780	0.653902	−	0.756580i	$-0.273131\pi$
$617$	32.4852	1.30780	0.653902	+	0.756580i	$0.273131\pi$
$618$	0	0
$619$	31.7143	1.27470	0.637352	−	0.770573i	$-0.280030\pi$
$619$	31.7143	1.27470	0.637352	+	0.770573i	$0.280030\pi$
$620$	−56.2897	−2.26065
$621$	0	0
$622$	−2.07633	−0.0832534
$623$	24.4300	0.978767
$624$	0	0
$625$	−24.7577	−0.990308
$626$	12.5105	0.500019
$627$	0	0
$628$	−0.496008	−0.0197929
$629$	−11.1359	−0.444017
$630$	0	0
$631$	1.92400	0.0765931	0.0382965	−	0.999266i	$-0.487807\pi$
$631$	1.92400	0.0765931	0.0382965	+	0.999266i	$0.487807\pi$
$632$	22.5052	0.895209
$633$	0	0
$634$	−9.22034	−0.366187
$635$	29.5262	1.17171
$636$	0	0
$637$	2.75898	0.109315
$638$	0.537681	0.0212870
$639$	0	0
$640$	−32.7306	−1.29379
$641$	−41.4326	−1.63649	−0.818245	−	0.574869i	$-0.805053\pi$
$641$	−41.4326	−1.63649	−0.818245	+	0.574869i	$0.805053\pi$
$642$	0	0
$643$	−13.2866	−0.523972	−0.261986	−	0.965072i	$-0.584377\pi$
$643$	−13.2866	−0.523972	−0.261986	+	0.965072i	$0.584377\pi$
$644$	3.08473	0.121555
$645$	0	0
$646$	0.398499	0.0156787
$647$	33.4357	1.31449	0.657246	−	0.753676i	$-0.271721\pi$
$647$	33.4357	1.31449	0.657246	+	0.753676i	$0.271721\pi$
$648$	0	0
$649$	8.94810	0.351244
$650$	−1.31964	−0.0517604
$651$	0	0
$652$	−10.8436	−0.424667
$653$	−40.6149	−1.58938	−0.794692	−	0.607013i	$-0.792368\pi$
$653$	−40.6149	−1.58938	−0.794692	+	0.607013i	$0.792368\pi$
$654$	0	0
$655$	−13.2188	−0.516502
$656$	23.6736	0.924299
$657$	0	0
$658$	1.11356	0.0434113
$659$	37.2253	1.45009	0.725045	−	0.688701i	$-0.241819\pi$
$659$	37.2253	1.45009	0.725045	+	0.688701i	$0.241819\pi$
$660$	0	0
$661$	14.0154	0.545137	0.272569	−	0.962136i	$-0.412127\pi$
$661$	14.0154	0.545137	0.272569	+	0.962136i	$0.412127\pi$
$662$	−3.34055	−0.129834
$663$	0	0
$664$	−15.7590	−0.611568
$665$	1.20411	0.0466933
$666$	0	0
$667$	−1.00000	−0.0387202
$668$	−19.6075	−0.758637
$669$	0	0
$670$	−4.73218	−0.182820
$671$	−1.23365	−0.0476246
$672$	0	0
$673$	−3.86624	−0.149033	−0.0745164	−	0.997220i	$-0.523741\pi$
$673$	−3.86624	−0.149033	−0.0745164	+	0.997220i	$0.523741\pi$
$674$	3.85372	0.148440
$675$	0	0
$676$	23.1491	0.890350
$677$	−4.47723	−0.172074	−0.0860369	−	0.996292i	$-0.527420\pi$
$677$	−4.47723	−0.172074	−0.0860369	+	0.996292i	$0.527420\pi$
$678$	0	0
$679$	15.3620	0.589541
$680$	21.3959	0.820494
$681$	0	0
$682$	−5.18358	−0.198489
$683$	−26.2668	−1.00507	−0.502536	−	0.864556i	$-0.667599\pi$
$683$	−26.2668	−1.00507	−0.502536	+	0.864556i	$0.667599\pi$
$684$	0	0
$685$	−11.2787	−0.430935
$686$	7.45310	0.284561
$687$	0	0
$688$	4.35781	0.166140
$689$	4.86441	0.185319
$690$	0	0
$691$	−50.2708	−1.91239	−0.956196	−	0.292727i	$-0.905437\pi$
$691$	−50.2708	−1.91239	−0.956196	+	0.292727i	$0.905437\pi$
$692$	−1.93850	−0.0736906
$693$	0	0
$694$	−5.56594	−0.211280
$695$	0.551225	0.0209092
$696$	0	0
$697$	−34.0044	−1.28801
$698$	−0.154821	−0.00586006
$699$	0	0
$700$	15.5717	0.588556
$701$	1.25281	0.0473180	0.0236590	−	0.999720i	$-0.492468\pi$
$701$	1.25281	0.0473180	0.0236590	+	0.999720i	$0.492468\pi$
$702$	0	0
$703$	−0.571537	−0.0215559
$704$	6.02326	0.227010
$705$	0	0
$706$	−6.85967	−0.258167
$707$	1.06639	0.0401058
$708$	0	0
$709$	−10.1837	−0.382457	−0.191228	−	0.981546i	$-0.561247\pi$
$709$	−10.1837	−0.382457	−0.191228	+	0.981546i	$0.561247\pi$
$710$	−18.4509	−0.692450
$711$	0	0
$712$	22.2796	0.834963
$713$	9.64062	0.361044
$714$	0	0
$715$	2.81951	0.105444
$716$	−12.8306	−0.479501
$717$	0	0
$718$	12.1927	0.455029
$719$	1.72692	0.0644032	0.0322016	−	0.999481i	$-0.489748\pi$
$719$	1.72692	0.0644032	0.0322016	+	0.999481i	$0.489748\pi$
$720$	0	0
$721$	−20.6396	−0.768660
$722$	−7.53249	−0.280330
$723$	0	0
$724$	−28.6961	−1.06648
$725$	−5.04800	−0.187478
$726$	0	0
$727$	−4.32804	−0.160518	−0.0802592	−	0.996774i	$-0.525575\pi$
$727$	−4.32804	−0.160518	−0.0802592	+	0.996774i	$0.525575\pi$
$728$	−1.68199	−0.0623386
$729$	0	0
$730$	−4.13979	−0.153221
$731$	−6.25949	−0.231516
$732$	0	0
$733$	44.0511	1.62707	0.813533	−	0.581519i	$-0.197541\pi$
$733$	44.0511	1.62707	0.813533	+	0.581519i	$0.197541\pi$
$734$	−3.62158	−0.133675
$735$	0	0
$736$	4.27766	0.157677
$737$	5.07950	0.187106
$738$	0	0
$739$	−36.1834	−1.33103	−0.665514	−	0.746385i	$-0.731788\pi$
$739$	−36.1834	−1.33103	−0.665514	+	0.746385i	$0.731788\pi$
$740$	−14.7121	−0.540829
$741$	0	0
$742$	4.92441	0.180781
$743$	14.8073	0.543227	0.271614	−	0.962406i	$-0.412443\pi$
$743$	14.8073	0.543227	0.271614	+	0.962406i	$0.412443\pi$
$744$	0	0
$745$	−9.70050	−0.355399
$746$	2.12819	0.0779185
$747$	0	0
$748$	−11.0108	−0.402594
$749$	28.4708	1.04030
$750$	0	0
$751$	−30.7780	−1.12311	−0.561553	−	0.827441i	$-0.689796\pi$
$751$	−30.7780	−1.12311	−0.561553	+	0.827441i	$0.689796\pi$
$752$	−5.14662	−0.187678
$753$	0	0
$754$	0.261418	0.00952027
$755$	35.6724	1.29825
$756$	0	0
$757$	15.6080	0.567281	0.283640	−	0.958931i	$-0.408458\pi$
$757$	15.6080	0.567281	0.283640	+	0.958931i	$0.408458\pi$
$758$	−4.61254	−0.167535
$759$	0	0
$760$	1.09812	0.0398329
$761$	21.6974	0.786529	0.393264	−	0.919425i	$-0.371346\pi$
$761$	21.6974	0.786529	0.393264	+	0.919425i	$0.371346\pi$
$762$	0	0
$763$	−5.00679	−0.181258
$764$	11.0267	0.398931
$765$	0	0
$766$	3.30539	0.119429
$767$	4.35052	0.157088
$768$	0	0
$769$	−41.3361	−1.49062	−0.745309	−	0.666720i	$-0.767698\pi$
$769$	−41.3361	−1.49062	−0.745309	+	0.666720i	$0.767698\pi$
$770$	2.85429	0.102861
$771$	0	0
$772$	42.8251	1.54131
$773$	−9.27907	−0.333745	−0.166873	−	0.985978i	$-0.553367\pi$
$773$	−9.27907	−0.333745	−0.166873	+	0.985978i	$0.553367\pi$
$774$	0	0
$775$	48.6659	1.74813
$776$	14.0098	0.502924
$777$	0	0
$778$	0.901666	0.0323263
$779$	−1.74523	−0.0625295
$780$	0	0
$781$	19.8051	0.708682
$782$	−1.75685	−0.0628249
$783$	0	0
$784$	−12.9086	−0.461021
$785$	0.853582	0.0304656
$786$	0	0
$787$	33.1118	1.18031	0.590153	−	0.807291i	$-0.299067\pi$
$787$	33.1118	1.18031	0.590153	+	0.807291i	$0.299067\pi$
$788$	45.0011	1.60310
$789$	0	0
$790$	−18.5681	−0.660625
$791$	6.41937	0.228247
$792$	0	0
$793$	−0.599795	−0.0212993
$794$	−5.15656	−0.182999
$795$	0	0
$796$	8.50257	0.301366
$797$	−21.4212	−0.758779	−0.379390	−	0.925237i	$-0.623866\pi$
$797$	−21.4212	−0.758779	−0.379390	+	0.925237i	$0.623866\pi$
$798$	0	0
$799$	7.39251	0.261528
$800$	21.5936	0.763450
$801$	0	0
$802$	0.396437	0.0139987
$803$	4.44363	0.156812
$804$	0	0
$805$	−5.30852	−0.187101
$806$	−2.52023	−0.0887713
$807$	0	0
$808$	0.972525	0.0342133
$809$	22.5673	0.793423	0.396712	−	0.917943i	$-0.370151\pi$
$809$	22.5673	0.793423	0.396712	+	0.917943i	$0.370151\pi$
$810$	0	0
$811$	24.3804	0.856112	0.428056	−	0.903752i	$-0.359199\pi$
$811$	24.3804	0.856112	0.428056	+	0.903752i	$0.359199\pi$
$812$	−3.08473	−0.108253
$813$	0	0
$814$	−1.35481	−0.0474859
$815$	18.6607	0.653656
$816$	0	0
$817$	−0.321261	−0.0112395
$818$	6.21047	0.217144
$819$	0	0
$820$	−44.9247	−1.56884
$821$	−36.5337	−1.27504	−0.637518	−	0.770436i	$-0.720039\pi$
$821$	−36.5337	−1.27504	−0.637518	+	0.770436i	$0.720039\pi$
$822$	0	0
$823$	−40.5937	−1.41501	−0.707503	−	0.706710i	$-0.750178\pi$
$823$	−40.5937	−1.41501	−0.707503	+	0.706710i	$0.750178\pi$
$824$	−18.8229	−0.655725
$825$	0	0
$826$	4.40418	0.153241
$827$	−29.0707	−1.01089	−0.505444	−	0.862860i	$-0.668671\pi$
$827$	−29.0707	−1.01089	−0.505444	+	0.862860i	$0.668671\pi$
$828$	0	0
$829$	35.3536	1.22788	0.613940	−	0.789352i	$-0.289584\pi$
$829$	35.3536	1.22788	0.613940	+	0.789352i	$0.289584\pi$
$830$	13.0021	0.451310
$831$	0	0
$832$	2.92848	0.101527
$833$	18.5417	0.642431
$834$	0	0
$835$	33.7426	1.16771
$836$	−0.565116	−0.0195449
$837$	0	0
$838$	−5.65279	−0.195272
$839$	11.1677	0.385553	0.192776	−	0.981243i	$-0.438251\pi$
$839$	11.1677	0.385553	0.192776	+	0.981243i	$0.438251\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	12.8266	0.442035
$843$	0	0
$844$	52.3137	1.80071
$845$	−39.8373	−1.37045
$846$	0	0
$847$	15.3578	0.527699
$848$	−22.7594	−0.781560
$849$	0	0
$850$	−8.86859	−0.304190
$851$	2.51972	0.0863749
$852$	0	0
$853$	36.4163	1.24687	0.623436	−	0.781875i	$-0.285736\pi$
$853$	36.4163	1.24687	0.623436	+	0.781875i	$0.285736\pi$
$854$	−0.607193	−0.0207777
$855$	0	0
$856$	25.9647	0.887457
$857$	−17.1591	−0.586144	−0.293072	−	0.956090i	$-0.594678\pi$
$857$	−17.1591	−0.586144	−0.293072	+	0.956090i	$0.594678\pi$
$858$	0	0
$859$	35.2599	1.20305	0.601525	−	0.798854i	$-0.294560\pi$
$859$	35.2599	1.20305	0.601525	+	0.798854i	$0.294560\pi$
$860$	−8.26970	−0.281994
$861$	0	0
$862$	14.1353	0.481449
$863$	−4.02541	−0.137027	−0.0685133	−	0.997650i	$-0.521826\pi$
$863$	−4.02541	−0.137027	−0.0685133	+	0.997650i	$0.521826\pi$
$864$	0	0
$865$	3.33596	0.113426
$866$	1.83479	0.0623485
$867$	0	0
$868$	29.7387	1.00940
$869$	19.9309	0.676111
$870$	0	0
$871$	2.46963	0.0836801
$872$	−4.56607	−0.154627
$873$	0	0
$874$	−0.0901685	−0.00304999
$875$	−0.254805	−0.00861398
$876$	0	0
$877$	−35.1631	−1.18737	−0.593687	−	0.804696i	$-0.702328\pi$
$877$	−35.1631	−1.18737	−0.593687	+	0.804696i	$0.702328\pi$
$878$	−0.915686	−0.0309029
$879$	0	0
$880$	−13.1918	−0.444696
$881$	11.8258	0.398422	0.199211	−	0.979957i	$-0.436162\pi$
$881$	11.8258	0.398422	0.199211	+	0.979957i	$0.436162\pi$
$882$	0	0
$883$	27.9925	0.942023	0.471012	−	0.882127i	$-0.343889\pi$
$883$	27.9925	0.942023	0.471012	+	0.882127i	$0.343889\pi$
$884$	−5.35339	−0.180054
$885$	0	0
$886$	−3.41700	−0.114796
$887$	22.6326	0.759927	0.379963	−	0.925001i	$-0.375937\pi$
$887$	22.6326	0.759927	0.379963	+	0.925001i	$0.375937\pi$
$888$	0	0
$889$	−15.5991	−0.523178
$890$	−18.3820	−0.616165
$891$	0	0
$892$	28.4183	0.951515
$893$	0.379412	0.0126965
$894$	0	0
$895$	22.0801	0.738058
$896$	17.2921	0.577688
$897$	0	0
$898$	6.41273	0.213996
$899$	−9.64062	−0.321533
$900$	0	0
$901$	32.6912	1.08910
$902$	−4.13701	−0.137747
$903$	0	0
$904$	5.85432	0.194712
$905$	49.3832	1.64155
$906$	0	0
$907$	51.0746	1.69590	0.847952	−	0.530073i	$-0.177836\pi$
$907$	51.0746	1.69590	0.847952	+	0.530073i	$0.177836\pi$
$908$	−27.5565	−0.914495
$909$	0	0
$910$	1.38774	0.0460031
$911$	36.9765	1.22509	0.612543	−	0.790437i	$-0.290147\pi$
$911$	36.9765	1.22509	0.612543	+	0.790437i	$0.290147\pi$
$912$	0	0
$913$	−13.9564	−0.461890
$914$	−10.9078	−0.360798
$915$	0	0
$916$	−16.7743	−0.554237
$917$	6.98372	0.230623
$918$	0	0
$919$	−31.0396	−1.02390	−0.511951	−	0.859014i	$-0.671077\pi$
$919$	−31.0396	−1.02390	−0.511951	+	0.859014i	$0.671077\pi$
$920$	−4.84124	−0.159611
$921$	0	0
$922$	7.07284	0.232932
$923$	9.62914	0.316947
$924$	0	0
$925$	12.7196	0.418216
$926$	−1.28705	−0.0422950
$927$	0	0
$928$	−4.27766	−0.140421
$929$	52.2135	1.71307	0.856535	−	0.516090i	$-0.172613\pi$
$929$	52.2135	1.71307	0.856535	+	0.516090i	$0.172613\pi$
$930$	0	0
$931$	0.951630	0.0311884
$932$	−14.7367	−0.482718
$933$	0	0
$934$	−6.49510	−0.212526
$935$	18.9485	0.619682
$936$	0	0
$937$	12.6887	0.414520	0.207260	−	0.978286i	$-0.433545\pi$
$937$	12.6887	0.414520	0.207260	+	0.978286i	$0.433545\pi$
$938$	2.50009	0.0816307
$939$	0	0
$940$	9.76659	0.318551
$941$	−41.5030	−1.35296	−0.676479	−	0.736462i	$-0.736495\pi$
$941$	−41.5030	−1.35296	−0.676479	+	0.736462i	$0.736495\pi$
$942$	0	0
$943$	7.69417	0.250557
$944$	−20.3550	−0.662500
$945$	0	0
$946$	−0.761536	−0.0247597
$947$	40.5269	1.31695	0.658474	−	0.752604i	$-0.271202\pi$
$947$	40.5269	1.31695	0.658474	+	0.752604i	$0.271202\pi$
$948$	0	0
$949$	2.16047	0.0701318
$950$	−0.455170	−0.0147677
$951$	0	0
$952$	−11.3038	−0.366357
$953$	−24.8116	−0.803728	−0.401864	−	0.915699i	$-0.631637\pi$
$953$	−24.8116	−0.803728	−0.401864	+	0.915699i	$0.631637\pi$
$954$	0	0
$955$	−18.9758	−0.614043
$956$	−7.57137	−0.244876
$957$	0	0
$958$	−3.11334	−0.100588
$959$	5.95869	0.192416
$960$	0	0
$961$	61.9416	1.99812
$962$	−0.658700	−0.0212373
$963$	0	0
$964$	−49.5169	−1.59483
$965$	−73.6978	−2.37242
$966$	0	0
$967$	40.2551	1.29452	0.647259	−	0.762270i	$-0.275915\pi$
$967$	40.2551	1.29452	0.647259	+	0.762270i	$0.275915\pi$
$968$	14.0059	0.450168
$969$	0	0
$970$	−11.5589	−0.371135
$971$	28.3848	0.910910	0.455455	−	0.890259i	$-0.349477\pi$
$971$	28.3848	0.910910	0.455455	+	0.890259i	$0.349477\pi$
$972$	0	0
$973$	−0.291221	−0.00933611
$974$	15.2618	0.489018
$975$	0	0
$976$	2.80629	0.0898273
$977$	−49.2960	−1.57712	−0.788560	−	0.614958i	$-0.789173\pi$
$977$	−49.2960	−1.57712	−0.788560	+	0.614958i	$0.789173\pi$
$978$	0	0
$979$	19.7311	0.630609
$980$	24.4963	0.782504
$981$	0	0
$982$	12.9129	0.412066
$983$	6.95652	0.221878	0.110939	−	0.993827i	$-0.464614\pi$
$983$	6.95652	0.221878	0.110939	+	0.993827i	$0.464614\pi$
$984$	0	0
$985$	−77.4425	−2.46752
$986$	1.75685	0.0559496
$987$	0	0
$988$	−0.274757	−0.00874117
$989$	1.41633	0.0450368
$990$	0	0
$991$	−18.9477	−0.601894	−0.300947	−	0.953641i	$-0.597303\pi$
$991$	−18.9477	−0.601894	−0.300947	+	0.953641i	$0.597303\pi$
$992$	41.2393	1.30935
$993$	0	0
$994$	9.74790	0.309185
$995$	−14.6321	−0.463868
$996$	0	0
$997$	−27.6170	−0.874640	−0.437320	−	0.899306i	$-0.644072\pi$
$997$	−27.6170	−0.874640	−0.437320	+	0.899306i	$0.644072\pi$
$998$	−3.28439	−0.103966
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6003.2.a.o.1.8		13
3.2	odd	2		667.2.a.c.1.6	✓	13

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
667.2.a.c.1.6	✓	13	3.2	odd	2
6003.2.a.o.1.8		13	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)

\(f(q)\)	\(=\)	\(q+0.397523 q^{2} -1.84198 q^{4} +3.16986 q^{5} -1.67469 q^{7} -1.52727 q^{8} +O(q^{10})\)	\(q+0.397523 q^{2} -1.84198 q^{4} +3.16986 q^{5} -1.67469 q^{7} -1.52727 q^{8} +1.26009 q^{10} -1.35258 q^{11} -0.657616 q^{13} -0.665727 q^{14} +3.07682 q^{16} -4.41950 q^{17} -0.226826 q^{19} -5.83880 q^{20} -0.537681 q^{22} +1.00000 q^{23} +5.04800 q^{25} -0.261418 q^{26} +3.08473 q^{28} -1.00000 q^{29} +9.64062 q^{31} +4.27766 q^{32} -1.75685 q^{34} -5.30852 q^{35} +2.51972 q^{37} -0.0901685 q^{38} -4.84124 q^{40} +7.69417 q^{41} +1.41633 q^{43} +2.49141 q^{44} +0.397523 q^{46} -1.67270 q^{47} -4.19543 q^{49} +2.00670 q^{50} +1.21131 q^{52} -7.39704 q^{53} -4.28747 q^{55} +2.55771 q^{56} -0.397523 q^{58} -6.61560 q^{59} +0.912075 q^{61} +3.83237 q^{62} -4.45318 q^{64} -2.08455 q^{65} -3.75542 q^{67} +8.14060 q^{68} -2.11026 q^{70} -14.6425 q^{71} -3.28531 q^{73} +1.00165 q^{74} +0.417807 q^{76} +2.26514 q^{77} -14.7355 q^{79} +9.75309 q^{80} +3.05861 q^{82} +10.3184 q^{83} -14.0092 q^{85} +0.563026 q^{86} +2.06576 q^{88} -14.5878 q^{89} +1.10130 q^{91} -1.84198 q^{92} -0.664939 q^{94} -0.719005 q^{95} -9.17309 q^{97} -1.66778 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 13 q - 4 q^{2} + 12 q^{4} - 16 q^{5} + q^{7} - 6 q^{8}+O(q^{10}) \) 13 * q - 4 * q^2 + 12 * q^4 - 16 * q^5 + q^7 - 6 * q^8	\( 13 q - 4 q^{2} + 12 q^{4} - 16 q^{5} + q^{7} - 6 q^{8} + 10 q^{10} - 10 q^{11} + 7 q^{13} + 12 q^{14} + 2 q^{16} - 26 q^{17} - 25 q^{20} - 15 q^{22} + 13 q^{23} + 19 q^{25} + 15 q^{26} + 5 q^{28} - 13 q^{29} - 6 q^{31} - 16 q^{32} + 11 q^{34} - q^{35} + 15 q^{37} - 8 q^{38} + 14 q^{40} - 9 q^{41} + q^{43} - 29 q^{44} - 4 q^{46} - 15 q^{47} + 4 q^{49} - 31 q^{50} - 8 q^{52} - 43 q^{53} - 3 q^{55} + 5 q^{56} + 4 q^{58} + 9 q^{59} + 20 q^{61} - 11 q^{62} - 16 q^{64} + 25 q^{65} + q^{67} - 21 q^{68} - 2 q^{70} - 17 q^{71} + 26 q^{73} - 11 q^{74} + 8 q^{76} - 17 q^{77} + 5 q^{79} - 10 q^{80} - 25 q^{82} - 4 q^{83} + 20 q^{85} + 13 q^{86} - 32 q^{88} - 48 q^{89} - 9 q^{91} + 12 q^{92} - 65 q^{94} - 8 q^{95} + 30 q^{97} - 2 q^{98}+O(q^{100}) \) 13 * q - 4 * q^2 + 12 * q^4 - 16 * q^5 + q^7 - 6 * q^8 + 10 * q^10 - 10 * q^11 + 7 * q^13 + 12 * q^14 + 2 * q^16 - 26 * q^17 - 25 * q^20 - 15 * q^22 + 13 * q^23 + 19 * q^25 + 15 * q^26 + 5 * q^28 - 13 * q^29 - 6 * q^31 - 16 * q^32 + 11 * q^34 - q^35 + 15 * q^37 - 8 * q^38 + 14 * q^40 - 9 * q^41 + q^43 - 29 * q^44 - 4 * q^46 - 15 * q^47 + 4 * q^49 - 31 * q^50 - 8 * q^52 - 43 * q^53 - 3 * q^55 + 5 * q^56 + 4 * q^58 + 9 * q^59 + 20 * q^61 - 11 * q^62 - 16 * q^64 + 25 * q^65 + q^67 - 21 * q^68 - 2 * q^70 - 17 * q^71 + 26 * q^73 - 11 * q^74 + 8 * q^76 - 17 * q^77 + 5 * q^79 - 10 * q^80 - 25 * q^82 - 4 * q^83 + 20 * q^85 + 13 * q^86 - 32 * q^88 - 48 * q^89 - 9 * q^91 + 12 * q^92 - 65 * q^94 - 8 * q^95 + 30 * q^97 - 2 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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