LMFDB - Embedded newform 6003.2.a.s.1.17

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			1.17
Root			$-1.97506$ 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)

$f(q)$	$=$	$q+1.97506 q^{2} +1.90086 q^{4} +3.88400 q^{5} +2.64382 q^{7} -0.195801 q^{8} +O(q^{10})$	\(q+1.97506 q^{2} +1.90086 q^{4} +3.88400 q^{5} +2.64382 q^{7} -0.195801 q^{8} +7.67114 q^{10} -4.76488 q^{11} +3.33092 q^{13} +5.22171 q^{14} -4.18845 q^{16} +7.28945 q^{17} +5.43957 q^{19} +7.38296 q^{20} -9.41093 q^{22} -1.00000 q^{23} +10.0855 q^{25} +6.57877 q^{26} +5.02554 q^{28} -1.00000 q^{29} +6.87216 q^{31} -7.88083 q^{32} +14.3971 q^{34} +10.2686 q^{35} -9.05053 q^{37} +10.7435 q^{38} -0.760490 q^{40} -5.39284 q^{41} -10.4237 q^{43} -9.05739 q^{44} -1.97506 q^{46} -8.49635 q^{47} -0.0102125 q^{49} +19.9194 q^{50} +6.33163 q^{52} +7.62772 q^{53} -18.5068 q^{55} -0.517662 q^{56} -1.97506 q^{58} +10.2552 q^{59} +0.452237 q^{61} +13.5729 q^{62} -7.18823 q^{64} +12.9373 q^{65} +12.0887 q^{67} +13.8562 q^{68} +20.2811 q^{70} -12.0747 q^{71} +6.12294 q^{73} -17.8753 q^{74} +10.3399 q^{76} -12.5975 q^{77} -9.83566 q^{79} -16.2679 q^{80} -10.6512 q^{82} +10.0824 q^{83} +28.3122 q^{85} -20.5874 q^{86} +0.932967 q^{88} +0.490243 q^{89} +8.80636 q^{91} -1.90086 q^{92} -16.7808 q^{94} +21.1273 q^{95} -0.789164 q^{97} -0.0201703 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	1.97506	1.39658	0.698289	−	0.715816i	$-0.253945\pi$
$2$	1.97506	1.39658	0.698289	+	0.715816i	$0.253945\pi$
$3$	0	0
$3$	0	0
$4$	1.90086	0.950432
$5$	3.88400	1.73698	0.868489	−	0.495708i	$-0.165091\pi$
$5$	3.88400	1.73698	0.868489	+	0.495708i	$0.165091\pi$
$6$	0	0
$7$	2.64382	0.999270	0.499635	−	0.866236i	$-0.333467\pi$
$7$	2.64382	0.999270	0.499635	+	0.866236i	$0.333467\pi$
$8$	−0.195801	−0.0692260
$9$	0	0
$10$	7.67114	2.42583
$11$	−4.76488	−1.43667	−0.718333	−	0.695700i	$-0.755094\pi$
$11$	−4.76488	−1.43667	−0.718333	+	0.695700i	$0.755094\pi$
$12$	0	0
$13$	3.33092	0.923832	0.461916	−	0.886924i	$-0.347162\pi$
$13$	3.33092	0.923832	0.461916	+	0.886924i	$0.347162\pi$
$14$	5.22171	1.39556
$15$	0	0
$16$	−4.18845	−1.04711
$17$	7.28945	1.76795	0.883975	−	0.467533i	$-0.154857\pi$
$17$	7.28945	1.76795	0.883975	+	0.467533i	$0.154857\pi$
$18$	0	0
$19$	5.43957	1.24792	0.623962	−	0.781455i	$-0.285522\pi$
$19$	5.43957	1.24792	0.623962	+	0.781455i	$0.285522\pi$
$20$	7.38296	1.65088
$21$	0	0
$22$	−9.41093	−2.00642
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	10.0855	2.01710
$26$	6.57877	1.29020
$27$	0	0
$28$	5.02554	0.949738
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	0	0
$31$	6.87216	1.23428	0.617138	−	0.786855i	$-0.288292\pi$
$31$	6.87216	1.23428	0.617138	+	0.786855i	$0.288292\pi$
$32$	−7.88083	−1.39315
$33$	0	0
$34$	14.3971	2.46908
$35$	10.2686	1.73571
$36$	0	0
$37$	−9.05053	−1.48790	−0.743949	−	0.668236i	$-0.767050\pi$
$37$	−9.05053	−1.48790	−0.743949	+	0.668236i	$0.767050\pi$
$38$	10.7435	1.74282
$39$	0	0
$40$	−0.760490	−0.120244
$41$	−5.39284	−0.842220	−0.421110	−	0.907010i	$-0.638359\pi$
$41$	−5.39284	−0.842220	−0.421110	+	0.907010i	$0.638359\pi$
$42$	0	0
$43$	−10.4237	−1.58960	−0.794799	−	0.606873i	$-0.792424\pi$
$43$	−10.4237	−1.58960	−0.794799	+	0.606873i	$0.792424\pi$
$44$	−9.05739	−1.36545
$45$	0	0
$46$	−1.97506	−0.291207
$47$	−8.49635	−1.23932	−0.619660	−	0.784870i	$-0.712730\pi$
$47$	−8.49635	−1.23932	−0.619660	+	0.784870i	$0.712730\pi$
$48$	0	0
$49$	−0.0102125	−0.00145893
$50$	19.9194	2.81703
$51$	0	0
$52$	6.33163	0.878039
$53$	7.62772	1.04775	0.523874	−	0.851796i	$-0.324486\pi$
$53$	7.62772	1.04775	0.523874	+	0.851796i	$0.324486\pi$
$54$	0	0
$55$	−18.5068	−2.49546
$56$	−0.517662	−0.0691755
$57$	0	0
$58$	−1.97506	−0.259338
$59$	10.2552	1.33511	0.667554	−	0.744562i	$-0.267341\pi$
$59$	10.2552	1.33511	0.667554	+	0.744562i	$0.267341\pi$
$60$	0	0
$61$	0.452237	0.0579030	0.0289515	−	0.999581i	$-0.490783\pi$
$61$	0.452237	0.0579030	0.0289515	+	0.999581i	$0.490783\pi$
$62$	13.5729	1.72376
$63$	0	0
$64$	−7.18823	−0.898528
$65$	12.9373	1.60468
$66$	0	0
$67$	12.0887	1.47687	0.738435	−	0.674325i	$-0.235565\pi$
$67$	12.0887	1.47687	0.738435	+	0.674325i	$0.235565\pi$
$68$	13.8562	1.68032
$69$	0	0
$70$	20.2811	2.42406
$71$	−12.0747	−1.43300	−0.716501	−	0.697586i	$-0.754258\pi$
$71$	−12.0747	−1.43300	−0.716501	+	0.697586i	$0.754258\pi$
$72$	0	0
$73$	6.12294	0.716636	0.358318	−	0.933600i	$-0.383350\pi$
$73$	6.12294	0.716636	0.358318	+	0.933600i	$0.383350\pi$
$74$	−17.8753	−2.07797
$75$	0	0
$76$	10.3399	1.18607
$77$	−12.5975	−1.43562
$78$	0	0
$79$	−9.83566	−1.10660	−0.553299	−	0.832983i	$-0.686631\pi$
$79$	−9.83566	−1.10660	−0.553299	+	0.832983i	$0.686631\pi$
$80$	−16.2679	−1.81881
$81$	0	0
$82$	−10.6512	−1.17623
$83$	10.0824	1.10668	0.553341	−	0.832955i	$-0.313352\pi$
$83$	10.0824	1.10668	0.553341	+	0.832955i	$0.313352\pi$
$84$	0	0
$85$	28.3122	3.07089
$86$	−20.5874	−2.22000
$87$	0	0
$88$	0.932967	0.0994546
$89$	0.490243	0.0519657	0.0259828	−	0.999662i	$-0.491728\pi$
$89$	0.490243	0.0519657	0.0259828	+	0.999662i	$0.491728\pi$
$90$	0	0
$91$	8.80636	0.923157
$92$	−1.90086	−0.198179
$93$	0	0
$94$	−16.7808	−1.73081
$95$	21.1273	2.16762
$96$	0	0
$97$	−0.789164	−0.0801274	−0.0400637	−	0.999197i	$-0.512756\pi$
$97$	−0.789164	−0.0801274	−0.0400637	+	0.999197i	$0.512756\pi$
$98$	−0.0201703	−0.00203751
$99$	0	0
$100$	19.1711	1.91711
$101$	−5.14905	−0.512350	−0.256175	−	0.966630i	$-0.582462\pi$
$101$	−5.14905	−0.512350	−0.256175	+	0.966630i	$0.582462\pi$
$102$	0	0
$103$	−4.12758	−0.406702	−0.203351	−	0.979106i	$-0.565183\pi$
$103$	−4.12758	−0.406702	−0.203351	+	0.979106i	$0.565183\pi$
$104$	−0.652197	−0.0639532
$105$	0	0
$106$	15.0652	1.46326
$107$	−11.9691	−1.15710	−0.578549	−	0.815648i	$-0.696381\pi$
$107$	−11.9691	−1.15710	−0.578549	+	0.815648i	$0.696381\pi$
$108$	0	0
$109$	13.1260	1.25724	0.628622	−	0.777711i	$-0.283619\pi$
$109$	13.1260	1.25724	0.628622	+	0.777711i	$0.283619\pi$
$110$	−36.5521	−3.48510
$111$	0	0
$112$	−11.0735	−1.04635
$113$	2.19956	0.206917	0.103459	−	0.994634i	$-0.467009\pi$
$113$	2.19956	0.206917	0.103459	+	0.994634i	$0.467009\pi$
$114$	0	0
$115$	−3.88400	−0.362185
$116$	−1.90086	−0.176491
$117$	0	0
$118$	20.2545	1.86458
$119$	19.2720	1.76666
$120$	0	0
$121$	11.7041	1.06401
$122$	0.893196	0.0808661
$123$	0	0
$124$	13.0630	1.17309
$125$	19.7520	1.76667
$126$	0	0
$127$	11.8159	1.04849	0.524245	−	0.851567i	$-0.324347\pi$
$127$	11.8159	1.04849	0.524245	+	0.851567i	$0.324347\pi$
$128$	1.56448	0.138282
$129$	0	0
$130$	25.5520	2.24106
$131$	−13.0941	−1.14403	−0.572017	−	0.820242i	$-0.693839\pi$
$131$	−13.0941	−1.14403	−0.572017	+	0.820242i	$0.693839\pi$
$132$	0	0
$133$	14.3813	1.24701
$134$	23.8759	2.06256
$135$	0	0
$136$	−1.42728	−0.122388
$137$	−7.92555	−0.677126	−0.338563	−	0.940944i	$-0.609941\pi$
$137$	−7.92555	−0.677126	−0.338563	+	0.940944i	$0.609941\pi$
$138$	0	0
$139$	3.89433	0.330313	0.165157	−	0.986267i	$-0.447187\pi$
$139$	3.89433	0.330313	0.165157	+	0.986267i	$0.447187\pi$
$140$	19.5192	1.64968
$141$	0	0
$142$	−23.8482	−2.00130
$143$	−15.8714	−1.32724
$144$	0	0
$145$	−3.88400	−0.322549
$146$	12.0932	1.00084
$147$	0	0
$148$	−17.2038	−1.41415
$149$	−5.40852	−0.443083	−0.221542	−	0.975151i	$-0.571109\pi$
$149$	−5.40852	−0.443083	−0.221542	+	0.975151i	$0.571109\pi$
$150$	0	0
$151$	9.81356	0.798616	0.399308	−	0.916817i	$-0.369250\pi$
$151$	9.81356	0.798616	0.399308	+	0.916817i	$0.369250\pi$
$152$	−1.06507	−0.0863887
$153$	0	0
$154$	−24.8808	−2.00495
$155$	26.6915	2.14391
$156$	0	0
$157$	17.1417	1.36806	0.684030	−	0.729454i	$-0.260226\pi$
$157$	17.1417	1.36806	0.684030	+	0.729454i	$0.260226\pi$
$158$	−19.4260	−1.54545
$159$	0	0
$160$	−30.6092	−2.41987
$161$	−2.64382	−0.208362
$162$	0	0
$163$	−9.30053	−0.728474	−0.364237	−	0.931306i	$-0.618670\pi$
$163$	−9.30053	−0.728474	−0.364237	+	0.931306i	$0.618670\pi$
$164$	−10.2510	−0.800472
$165$	0	0
$166$	19.9133	1.54557
$167$	16.9187	1.30921	0.654603	−	0.755972i	$-0.272836\pi$
$167$	16.9187	1.30921	0.654603	+	0.755972i	$0.272836\pi$
$168$	0	0
$169$	−1.90496	−0.146535
$170$	55.9184	4.28874
$171$	0	0
$172$	−19.8140	−1.51080
$173$	−3.35028	−0.254717	−0.127359	−	0.991857i	$-0.540650\pi$
$173$	−3.35028	−0.254717	−0.127359	+	0.991857i	$0.540650\pi$
$174$	0	0
$175$	26.6642	2.01562
$176$	19.9574	1.50435
$177$	0	0
$178$	0.968260	0.0725742
$179$	−9.84156	−0.735593	−0.367796	−	0.929906i	$-0.619888\pi$
$179$	−9.84156	−0.735593	−0.367796	+	0.929906i	$0.619888\pi$
$180$	0	0
$181$	−3.83355	−0.284945	−0.142473	−	0.989799i	$-0.545505\pi$
$181$	−3.83355	−0.284945	−0.142473	+	0.989799i	$0.545505\pi$
$182$	17.3931	1.28926
$183$	0	0
$184$	0.195801	0.0144346
$185$	−35.1523	−2.58445
$186$	0	0
$187$	−34.7333	−2.53995
$188$	−16.1504	−1.17789
$189$	0	0
$190$	41.7277	3.02725
$191$	−25.4250	−1.83969	−0.919843	−	0.392287i	$-0.871684\pi$
$191$	−25.4250	−1.83969	−0.919843	+	0.392287i	$0.871684\pi$
$192$	0	0
$193$	24.0004	1.72759	0.863795	−	0.503844i	$-0.168081\pi$
$193$	24.0004	1.72759	0.863795	+	0.503844i	$0.168081\pi$
$194$	−1.55865	−0.111904
$195$	0	0
$196$	−0.0194126	−0.00138661
$197$	−23.4966	−1.67406	−0.837032	−	0.547154i	$-0.815711\pi$
$197$	−23.4966	−1.67406	−0.837032	+	0.547154i	$0.815711\pi$
$198$	0	0
$199$	−6.12765	−0.434378	−0.217189	−	0.976130i	$-0.569689\pi$
$199$	−6.12765	−0.434378	−0.217189	+	0.976130i	$0.569689\pi$
$200$	−1.97474	−0.139635
$201$	0	0
$202$	−10.1697	−0.715536
$203$	−2.64382	−0.185560
$204$	0	0
$205$	−20.9458	−1.46292
$206$	−8.15222	−0.567992
$207$	0	0
$208$	−13.9514	−0.967354
$209$	−25.9189	−1.79285
$210$	0	0
$211$	−7.19073	−0.495030	−0.247515	−	0.968884i	$-0.579614\pi$
$211$	−7.19073	−0.495030	−0.247515	+	0.968884i	$0.579614\pi$
$212$	14.4992	0.995812
$213$	0	0
$214$	−23.6397	−1.61598
$215$	−40.4856	−2.76110
$216$	0	0
$217$	18.1688	1.23338
$218$	25.9247	1.75584
$219$	0	0
$220$	−35.1789	−2.37176
$221$	24.2806	1.63329
$222$	0	0
$223$	−8.86810	−0.593852	−0.296926	−	0.954900i	$-0.595961\pi$
$223$	−8.86810	−0.593852	−0.296926	+	0.954900i	$0.595961\pi$
$224$	−20.8355	−1.39213
$225$	0	0
$226$	4.34427	0.288976
$227$	−10.6725	−0.708359	−0.354180	−	0.935177i	$-0.615240\pi$
$227$	−10.6725	−0.708359	−0.354180	+	0.935177i	$0.615240\pi$
$228$	0	0
$229$	4.91406	0.324730	0.162365	−	0.986731i	$-0.448088\pi$
$229$	4.91406	0.324730	0.162365	+	0.986731i	$0.448088\pi$
$230$	−7.67114	−0.505820
$231$	0	0
$232$	0.195801	0.0128549
$233$	22.8105	1.49436	0.747182	−	0.664619i	$-0.231406\pi$
$233$	22.8105	1.49436	0.747182	+	0.664619i	$0.231406\pi$
$234$	0	0
$235$	−32.9998	−2.15267
$236$	19.4936	1.26893
$237$	0	0
$238$	38.0633	2.46728
$239$	22.1209	1.43088	0.715441	−	0.698673i	$-0.246226\pi$
$239$	22.1209	1.43088	0.715441	+	0.698673i	$0.246226\pi$
$240$	0	0
$241$	−5.77198	−0.371806	−0.185903	−	0.982568i	$-0.559521\pi$
$241$	−5.77198	−0.371806	−0.185903	+	0.982568i	$0.559521\pi$
$242$	23.1163	1.48597
$243$	0	0
$244$	0.859641	0.0550329
$245$	−0.0396654	−0.00253413
$246$	0	0
$247$	18.1188	1.15287
$248$	−1.34557	−0.0854440
$249$	0	0
$250$	39.0114	2.46730
$251$	18.2189	1.14997	0.574984	−	0.818165i	$-0.305008\pi$
$251$	18.2189	1.14997	0.574984	+	0.818165i	$0.305008\pi$
$252$	0	0
$253$	4.76488	0.299565
$254$	23.3371	1.46430
$255$	0	0
$256$	17.4664	1.09165
$257$	−6.94804	−0.433407	−0.216703	−	0.976237i	$-0.569530\pi$
$257$	−6.94804	−0.433407	−0.216703	+	0.976237i	$0.569530\pi$
$258$	0	0
$259$	−23.9280	−1.48681
$260$	24.5921	1.52513
$261$	0	0
$262$	−25.8616	−1.59773
$263$	−7.20446	−0.444246	−0.222123	−	0.975019i	$-0.571299\pi$
$263$	−7.20446	−0.444246	−0.222123	+	0.975019i	$0.571299\pi$
$264$	0	0
$265$	29.6261	1.81992
$266$	28.4038	1.74155
$267$	0	0
$268$	22.9790	1.40366
$269$	4.24647	0.258912	0.129456	−	0.991585i	$-0.458677\pi$
$269$	4.24647	0.258912	0.129456	+	0.991585i	$0.458677\pi$
$270$	0	0
$271$	19.9356	1.21100	0.605501	−	0.795845i	$-0.292973\pi$
$271$	19.9356	1.21100	0.605501	+	0.795845i	$0.292973\pi$
$272$	−30.5315	−1.85124
$273$	0	0
$274$	−15.6534	−0.945659
$275$	−48.0561	−2.89789
$276$	0	0
$277$	−2.38195	−0.143117	−0.0715586	−	0.997436i	$-0.522797\pi$
$277$	−2.38195	−0.143117	−0.0715586	+	0.997436i	$0.522797\pi$
$278$	7.69154	0.461308
$279$	0	0
$280$	−2.01060	−0.120156
$281$	−13.4801	−0.804155	−0.402078	−	0.915606i	$-0.631712\pi$
$281$	−13.4801	−0.804155	−0.402078	+	0.915606i	$0.631712\pi$
$282$	0	0
$283$	7.65647	0.455130	0.227565	−	0.973763i	$-0.426924\pi$
$283$	7.65647	0.455130	0.227565	+	0.973763i	$0.426924\pi$
$284$	−22.9523	−1.36197
$285$	0	0
$286$	−31.3471	−1.85359
$287$	−14.2577	−0.841605
$288$	0	0
$289$	36.1361	2.12565
$290$	−7.67114	−0.450465
$291$	0	0
$292$	11.6389	0.681113
$293$	−25.5856	−1.49473	−0.747364	−	0.664415i	$-0.768681\pi$
$293$	−25.5856	−1.49473	−0.747364	+	0.664415i	$0.768681\pi$
$294$	0	0
$295$	39.8310	2.31905
$296$	1.77210	0.103001
$297$	0	0
$298$	−10.6821	−0.618800
$299$	−3.33092	−0.192632
$300$	0	0
$301$	−27.5584	−1.58844
$302$	19.3824	1.11533
$303$	0	0
$304$	−22.7833	−1.30671
$305$	1.75649	0.100576
$306$	0	0
$307$	−9.19667	−0.524882	−0.262441	−	0.964948i	$-0.584527\pi$
$307$	−9.19667	−0.524882	−0.262441	+	0.964948i	$0.584527\pi$
$308$	−23.9461	−1.36446
$309$	0	0
$310$	52.7173	2.99414
$311$	10.3312	0.585831	0.292916	−	0.956138i	$-0.405374\pi$
$311$	10.3312	0.585831	0.292916	+	0.956138i	$0.405374\pi$
$312$	0	0
$313$	−17.7542	−1.00353	−0.501763	−	0.865005i	$-0.667315\pi$
$313$	−17.7542	−1.00353	−0.501763	+	0.865005i	$0.667315\pi$
$314$	33.8560	1.91060
$315$	0	0
$316$	−18.6962	−1.05175
$317$	−2.01941	−0.113421	−0.0567106	−	0.998391i	$-0.518061\pi$
$317$	−2.01941	−0.113421	−0.0567106	+	0.998391i	$0.518061\pi$
$318$	0	0
$319$	4.76488	0.266782
$320$	−27.9191	−1.56072
$321$	0	0
$322$	−5.22171	−0.290994
$323$	39.6515	2.20627
$324$	0	0
$325$	33.5939	1.86346
$326$	−18.3691	−1.01737
$327$	0	0
$328$	1.05592	0.0583035
$329$	−22.4628	−1.23842
$330$	0	0
$331$	−19.2584	−1.05854	−0.529268	−	0.848455i	$-0.677533\pi$
$331$	−19.2584	−1.05854	−0.529268	+	0.848455i	$0.677533\pi$
$332$	19.1652	1.05183
$333$	0	0
$334$	33.4154	1.82841
$335$	46.9525	2.56529
$336$	0	0
$337$	1.62764	0.0886634	0.0443317	−	0.999017i	$-0.485884\pi$
$337$	1.62764	0.0886634	0.0443317	+	0.999017i	$0.485884\pi$
$338$	−3.76241	−0.204648
$339$	0	0
$340$	53.8177	2.91867
$341$	−32.7450	−1.77324
$342$	0	0
$343$	−18.5337	−1.00073
$344$	2.04096	0.110041
$345$	0	0
$346$	−6.61700	−0.355732
$347$	−24.2893	−1.30392	−0.651959	−	0.758254i	$-0.726053\pi$
$347$	−24.2893	−1.30392	−0.651959	+	0.758254i	$0.726053\pi$
$348$	0	0
$349$	0.452922	0.0242443	0.0121222	−	0.999927i	$-0.496141\pi$
$349$	0.452922	0.0242443	0.0121222	+	0.999927i	$0.496141\pi$
$350$	52.6634	2.81498
$351$	0	0
$352$	37.5512	2.00149
$353$	2.13217	0.113484	0.0567421	−	0.998389i	$-0.481929\pi$
$353$	2.13217	0.113484	0.0567421	+	0.998389i	$0.481929\pi$
$354$	0	0
$355$	−46.8981	−2.48909
$356$	0.931886	0.0493898
$357$	0	0
$358$	−19.4377	−1.02731
$359$	−4.24403	−0.223991	−0.111996	−	0.993709i	$-0.535724\pi$
$359$	−4.24403	−0.223991	−0.111996	+	0.993709i	$0.535724\pi$
$360$	0	0
$361$	10.5889	0.557313
$362$	−7.57148	−0.397948
$363$	0	0
$364$	16.7397	0.877398
$365$	23.7815	1.24478
$366$	0	0
$367$	−18.1283	−0.946288	−0.473144	−	0.880985i	$-0.656881\pi$
$367$	−18.1283	−0.946288	−0.473144	+	0.880985i	$0.656881\pi$
$368$	4.18845	0.218338
$369$	0	0
$370$	−69.4279	−3.60938
$371$	20.1663	1.04698
$372$	0	0
$373$	7.46611	0.386581	0.193290	−	0.981142i	$-0.438084\pi$
$373$	7.46611	0.386581	0.193290	+	0.981142i	$0.438084\pi$
$374$	−68.6005	−3.54725
$375$	0	0
$376$	1.66359	0.0857932
$377$	−3.33092	−0.171551
$378$	0	0
$379$	−30.6908	−1.57648	−0.788239	−	0.615369i	$-0.789007\pi$
$379$	−30.6908	−1.57648	−0.788239	+	0.615369i	$0.789007\pi$
$380$	40.1601	2.06017
$381$	0	0
$382$	−50.2158	−2.56927
$383$	16.3891	0.837442	0.418721	−	0.908115i	$-0.362479\pi$
$383$	16.3891	0.837442	0.418721	+	0.908115i	$0.362479\pi$
$384$	0	0
$385$	−48.9287	−2.49364
$386$	47.4023	2.41271
$387$	0	0
$388$	−1.50009	−0.0761556
$389$	−0.876026	−0.0444163	−0.0222081	−	0.999753i	$-0.507070\pi$
$389$	−0.876026	−0.0444163	−0.0222081	+	0.999753i	$0.507070\pi$
$390$	0	0
$391$	−7.28945	−0.368643
$392$	0.00199961	0.000100996 0
$393$	0	0
$394$	−46.4072	−2.33796
$395$	−38.2017	−1.92214
$396$	0	0
$397$	6.14689	0.308503	0.154252	−	0.988032i	$-0.450703\pi$
$397$	6.14689	0.308503	0.154252	+	0.988032i	$0.450703\pi$
$398$	−12.1025	−0.606643
$399$	0	0
$400$	−42.2425	−2.11212
$401$	−8.06278	−0.402636	−0.201318	−	0.979526i	$-0.564523\pi$
$401$	−8.06278	−0.402636	−0.201318	+	0.979526i	$0.564523\pi$
$402$	0	0
$403$	22.8906	1.14026
$404$	−9.78764	−0.486953
$405$	0	0
$406$	−5.22171	−0.259149
$407$	43.1247	2.13761
$408$	0	0
$409$	3.21546	0.158994	0.0794970	−	0.996835i	$-0.474669\pi$
$409$	3.21546	0.158994	0.0794970	+	0.996835i	$0.474669\pi$
$410$	−41.3692	−2.04308
$411$	0	0
$412$	−7.84596	−0.386543
$413$	27.1128	1.33413
$414$	0	0
$415$	39.1599	1.92228
$416$	−26.2504	−1.28703
$417$	0	0
$418$	−51.1914	−2.50385
$419$	−29.0993	−1.42160	−0.710798	−	0.703396i	$-0.751666\pi$
$419$	−29.0993	−1.42160	−0.710798	+	0.703396i	$0.751666\pi$
$420$	0	0
$421$	13.9597	0.680352	0.340176	−	0.940362i	$-0.389513\pi$
$421$	13.9597	0.680352	0.340176	+	0.940362i	$0.389513\pi$
$422$	−14.2021	−0.691348
$423$	0	0
$424$	−1.49351	−0.0725314
$425$	73.5176	3.56613
$426$	0	0
$427$	1.19563	0.0578608
$428$	−22.7516	−1.09974
$429$	0	0
$430$	−79.9616	−3.85609
$431$	−13.7446	−0.662055	−0.331028	−	0.943621i	$-0.607395\pi$
$431$	−13.7446	−0.662055	−0.331028	+	0.943621i	$0.607395\pi$
$432$	0	0
$433$	−24.8189	−1.19272	−0.596361	−	0.802716i	$-0.703387\pi$
$433$	−24.8189	−1.19272	−0.596361	+	0.802716i	$0.703387\pi$
$434$	35.8844	1.72251
$435$	0	0
$436$	24.9507	1.19492
$437$	−5.43957	−0.260210
$438$	0	0
$439$	−15.9294	−0.760268	−0.380134	−	0.924932i	$-0.624122\pi$
$439$	−15.9294	−0.760268	−0.380134	+	0.924932i	$0.624122\pi$
$440$	3.62365	0.172751
$441$	0	0
$442$	47.9556	2.28102
$443$	37.1050	1.76291	0.881456	−	0.472265i	$-0.156564\pi$
$443$	37.1050	1.76291	0.881456	+	0.472265i	$0.156564\pi$
$444$	0	0
$445$	1.90411	0.0902633
$446$	−17.5150	−0.829361
$447$	0	0
$448$	−19.0044	−0.897873
$449$	13.0638	0.616518	0.308259	−	0.951303i	$-0.400254\pi$
$449$	13.0638	0.616518	0.308259	+	0.951303i	$0.400254\pi$
$450$	0	0
$451$	25.6962	1.20999
$452$	4.18107	0.196661
$453$	0	0
$454$	−21.0788	−0.989279
$455$	34.2039	1.60350
$456$	0	0
$457$	20.3373	0.951341	0.475670	−	0.879624i	$-0.342205\pi$
$457$	20.3373	0.951341	0.475670	+	0.879624i	$0.342205\pi$
$458$	9.70557	0.453512
$459$	0	0
$460$	−7.38296	−0.344232
$461$	25.0320	1.16585	0.582927	−	0.812524i	$-0.301907\pi$
$461$	25.0320	1.16585	0.582927	+	0.812524i	$0.301907\pi$
$462$	0	0
$463$	−22.0134	−1.02305	−0.511524	−	0.859269i	$-0.670919\pi$
$463$	−22.0134	−1.02305	−0.511524	+	0.859269i	$0.670919\pi$
$464$	4.18845	0.194444
$465$	0	0
$466$	45.0521	2.08700
$467$	−11.1367	−0.515345	−0.257672	−	0.966232i	$-0.582956\pi$
$467$	−11.1367	−0.515345	−0.257672	+	0.966232i	$0.582956\pi$
$468$	0	0
$469$	31.9603	1.47579
$470$	−65.1767	−3.00638
$471$	0	0
$472$	−2.00797	−0.0924241
$473$	49.6676	2.28372
$474$	0	0
$475$	54.8607	2.51718
$476$	36.6334	1.67909
$477$	0	0
$478$	43.6901	1.99834
$479$	−26.1947	−1.19687	−0.598433	−	0.801173i	$-0.704210\pi$
$479$	−26.1947	−1.19687	−0.598433	+	0.801173i	$0.704210\pi$
$480$	0	0
$481$	−30.1466	−1.37457
$482$	−11.4000	−0.519256
$483$	0	0
$484$	22.2479	1.01127
$485$	−3.06511	−0.139180
$486$	0	0
$487$	41.1512	1.86474	0.932369	−	0.361508i	$-0.117738\pi$
$487$	41.1512	1.86474	0.932369	+	0.361508i	$0.117738\pi$
$488$	−0.0885483	−0.00400839
$489$	0	0
$490$	−0.0783415	−0.00353911
$491$	−23.0875	−1.04192	−0.520962	−	0.853580i	$-0.674427\pi$
$491$	−23.0875	−1.04192	−0.520962	+	0.853580i	$0.674427\pi$
$492$	0	0
$493$	−7.28945	−0.328300
$494$	35.7857	1.61008
$495$	0	0
$496$	−28.7837	−1.29242
$497$	−31.9233	−1.43196
$498$	0	0
$499$	−0.628061	−0.0281159	−0.0140579	−	0.999901i	$-0.504475\pi$
$499$	−0.628061	−0.0281159	−0.0140579	+	0.999901i	$0.504475\pi$
$500$	37.5459	1.67910
$501$	0	0
$502$	35.9835	1.60602
$503$	−35.0299	−1.56190	−0.780952	−	0.624591i	$-0.785266\pi$
$503$	−35.0299	−1.56190	−0.780952	+	0.624591i	$0.785266\pi$
$504$	0	0
$505$	−19.9989	−0.889940
$506$	9.41093	0.418367
$507$	0	0
$508$	22.4604	0.996519
$509$	−2.65516	−0.117688	−0.0588439	−	0.998267i	$-0.518741\pi$
$509$	−2.65516	−0.117688	−0.0588439	+	0.998267i	$0.518741\pi$
$510$	0	0
$511$	16.1879	0.716113
$512$	31.3682	1.38629
$513$	0	0
$514$	−13.7228	−0.605287
$515$	−16.0315	−0.706433
$516$	0	0
$517$	40.4841	1.78049
$518$	−47.2592	−2.07645
$519$	0	0
$520$	−2.53313	−0.111085
$521$	8.23178	0.360641	0.180321	−	0.983608i	$-0.442287\pi$
$521$	8.23178	0.360641	0.180321	+	0.983608i	$0.442287\pi$
$522$	0	0
$523$	9.52821	0.416640	0.208320	−	0.978061i	$-0.433201\pi$
$523$	9.52821	0.416640	0.208320	+	0.978061i	$0.433201\pi$
$524$	−24.8900	−1.08733
$525$	0	0
$526$	−14.2293	−0.620425
$527$	50.0942	2.18214
$528$	0	0
$529$	1.00000	0.0434783
$530$	58.5133	2.54165
$531$	0	0
$532$	27.3368	1.18520
$533$	−17.9631	−0.778069
$534$	0	0
$535$	−46.4881	−2.00985
$536$	−2.36697	−0.102238
$537$	0	0
$538$	8.38704	0.361591
$539$	0.0486613	0.00209599
$540$	0	0
$541$	44.6497	1.91964	0.959821	−	0.280612i	$-0.0905375\pi$
$541$	44.6497	1.91964	0.959821	+	0.280612i	$0.0905375\pi$
$542$	39.3740	1.69126
$543$	0	0
$544$	−57.4469	−2.46302
$545$	50.9814	2.18381
$546$	0	0
$547$	−35.0805	−1.49993	−0.749967	−	0.661476i	$-0.769931\pi$
$547$	−35.0805	−1.49993	−0.749967	+	0.661476i	$0.769931\pi$
$548$	−15.0654	−0.643562
$549$	0	0
$550$	−94.9137	−4.04713
$551$	−5.43957	−0.231734
$552$	0	0
$553$	−26.0037	−1.10579
$554$	−4.70449	−0.199874
$555$	0	0
$556$	7.40260	0.313940
$557$	23.6013	1.00002	0.500009	−	0.866020i	$-0.333330\pi$
$557$	23.6013	1.00002	0.500009	+	0.866020i	$0.333330\pi$
$558$	0	0
$559$	−34.7205	−1.46852
$560$	−43.0095	−1.81748
$561$	0	0
$562$	−26.6240	−1.12307
$563$	−5.46440	−0.230297	−0.115149	−	0.993348i	$-0.536734\pi$
$563$	−5.46440	−0.230297	−0.115149	+	0.993348i	$0.536734\pi$
$564$	0	0
$565$	8.54310	0.359411
$566$	15.1220	0.635625
$567$	0	0
$568$	2.36423	0.0992010
$569$	−18.3374	−0.768744	−0.384372	−	0.923178i	$-0.625582\pi$
$569$	−18.3374	−0.768744	−0.384372	+	0.923178i	$0.625582\pi$
$570$	0	0
$571$	−44.5394	−1.86391	−0.931957	−	0.362569i	$-0.881900\pi$
$571$	−44.5394	−1.86391	−0.931957	+	0.362569i	$0.881900\pi$
$572$	−30.1695	−1.26145
$573$	0	0
$574$	−28.1598	−1.17537
$575$	−10.0855	−0.420593
$576$	0	0
$577$	−9.61196	−0.400151	−0.200076	−	0.979780i	$-0.564119\pi$
$577$	−9.61196	−0.400151	−0.200076	+	0.979780i	$0.564119\pi$
$578$	71.3709	2.96864
$579$	0	0
$580$	−7.38296	−0.306561
$581$	26.6560	1.10588
$582$	0	0
$583$	−36.3452	−1.50526
$584$	−1.19888	−0.0496098
$585$	0	0
$586$	−50.5332	−2.08751
$587$	26.4163	1.09032	0.545158	−	0.838333i	$-0.316470\pi$
$587$	26.4163	1.09032	0.545158	+	0.838333i	$0.316470\pi$
$588$	0	0
$589$	37.3816	1.54028
$590$	78.6687	3.23874
$591$	0	0
$592$	37.9077	1.55800
$593$	14.0020	0.574993	0.287496	−	0.957782i	$-0.407177\pi$
$593$	14.0020	0.574993	0.287496	+	0.957782i	$0.407177\pi$
$594$	0	0
$595$	74.8525	3.06865
$596$	−10.2809	−0.421120
$597$	0	0
$598$	−6.57877	−0.269026
$599$	−27.8770	−1.13902	−0.569512	−	0.821983i	$-0.692868\pi$
$599$	−27.8770	−1.13902	−0.569512	+	0.821983i	$0.692868\pi$
$600$	0	0
$601$	−7.86933	−0.320997	−0.160498	−	0.987036i	$-0.551310\pi$
$601$	−7.86933	−0.320997	−0.160498	+	0.987036i	$0.551310\pi$
$602$	−54.4294	−2.21838
$603$	0	0
$604$	18.6542	0.759030
$605$	45.4587	1.84816
$606$	0	0
$607$	−42.6180	−1.72981	−0.864906	−	0.501933i	$-0.832622\pi$
$607$	−42.6180	−1.72981	−0.864906	+	0.501933i	$0.832622\pi$
$608$	−42.8683	−1.73854
$609$	0	0
$610$	3.46917	0.140463
$611$	−28.3007	−1.14492
$612$	0	0
$613$	−3.58163	−0.144660	−0.0723302	−	0.997381i	$-0.523044\pi$
$613$	−3.58163	−0.144660	−0.0723302	+	0.997381i	$0.523044\pi$
$614$	−18.1640	−0.733039
$615$	0	0
$616$	2.46660	0.0993820
$617$	−25.1171	−1.01118	−0.505589	−	0.862775i	$-0.668725\pi$
$617$	−25.1171	−1.01118	−0.505589	+	0.862775i	$0.668725\pi$
$618$	0	0
$619$	−1.91575	−0.0770004	−0.0385002	−	0.999259i	$-0.512258\pi$
$619$	−1.91575	−0.0770004	−0.0385002	+	0.999259i	$0.512258\pi$
$620$	50.7369	2.03764
$621$	0	0
$622$	20.4048	0.818160
$623$	1.29612	0.0519278
$624$	0	0
$625$	26.2895	1.05158
$626$	−35.0656	−1.40150
$627$	0	0
$628$	32.5841	1.30025
$629$	−65.9734	−2.63053
$630$	0	0
$631$	41.3427	1.64583	0.822913	−	0.568168i	$-0.192348\pi$
$631$	41.3427	1.64583	0.822913	+	0.568168i	$0.192348\pi$
$632$	1.92583	0.0766054
$633$	0	0
$634$	−3.98845	−0.158402
$635$	45.8930	1.82121
$636$	0	0
$637$	−0.0340170	−0.00134780
$638$	9.41093	0.372582
$639$	0	0
$640$	6.07645	0.240193
$641$	21.8011	0.861090	0.430545	−	0.902569i	$-0.358321\pi$
$641$	21.8011	0.861090	0.430545	+	0.902569i	$0.358321\pi$
$642$	0	0
$643$	−2.00022	−0.0788810	−0.0394405	−	0.999222i	$-0.512558\pi$
$643$	−2.00022	−0.0788810	−0.0394405	+	0.999222i	$0.512558\pi$
$644$	−5.02554	−0.198034
$645$	0	0
$646$	78.3141	3.08123
$647$	−47.7606	−1.87766	−0.938831	−	0.344379i	$-0.888090\pi$
$647$	−47.7606	−1.87766	−0.938831	+	0.344379i	$0.888090\pi$
$648$	0	0
$649$	−48.8646	−1.91810
$650$	66.3501	2.60246
$651$	0	0
$652$	−17.6790	−0.692365
$653$	20.8681	0.816631	0.408316	−	0.912841i	$-0.366116\pi$
$653$	20.8681	0.816631	0.408316	+	0.912841i	$0.366116\pi$
$654$	0	0
$655$	−50.8574	−1.98716
$656$	22.5876	0.881898
$657$	0	0
$658$	−44.3654	−1.72954
$659$	38.0743	1.48316	0.741582	−	0.670863i	$-0.234076\pi$
$659$	38.0743	1.48316	0.741582	+	0.670863i	$0.234076\pi$
$660$	0	0
$661$	24.0049	0.933684	0.466842	−	0.884341i	$-0.345392\pi$
$661$	24.0049	0.933684	0.466842	+	0.884341i	$0.345392\pi$
$662$	−38.0364	−1.47833
$663$	0	0
$664$	−1.97413	−0.0766112
$665$	55.8568	2.16603
$666$	0	0
$667$	1.00000	0.0387202
$668$	32.1601	1.24431
$669$	0	0
$670$	92.7341	3.58263
$671$	−2.15486	−0.0831873
$672$	0	0
$673$	−4.39075	−0.169251	−0.0846255	−	0.996413i	$-0.526969\pi$
$673$	−4.39075	−0.169251	−0.0846255	+	0.996413i	$0.526969\pi$
$674$	3.21469	0.123825
$675$	0	0
$676$	−3.62107	−0.139272
$677$	11.7070	0.449938	0.224969	−	0.974366i	$-0.427772\pi$
$677$	11.7070	0.449938	0.224969	+	0.974366i	$0.427772\pi$
$678$	0	0
$679$	−2.08641	−0.0800690
$680$	−5.54356	−0.212586
$681$	0	0
$682$	−64.6734	−2.47647
$683$	29.3902	1.12458	0.562292	−	0.826939i	$-0.309920\pi$
$683$	29.3902	1.12458	0.562292	+	0.826939i	$0.309920\pi$
$684$	0	0
$685$	−30.7829	−1.17615
$686$	−36.6053	−1.39760
$687$	0	0
$688$	43.6590	1.66449
$689$	25.4073	0.967942
$690$	0	0
$691$	−16.9740	−0.645723	−0.322861	−	0.946446i	$-0.604645\pi$
$691$	−16.9740	−0.645723	−0.322861	+	0.946446i	$0.604645\pi$
$692$	−6.36842	−0.242091
$693$	0	0
$694$	−47.9728	−1.82102
$695$	15.1256	0.573747
$696$	0	0
$697$	−39.3108	−1.48900
$698$	0.894547	0.0338591
$699$	0	0
$700$	50.6850	1.91571
$701$	43.4787	1.64217	0.821084	−	0.570808i	$-0.193370\pi$
$701$	43.4787	1.64217	0.821084	+	0.570808i	$0.193370\pi$
$702$	0	0
$703$	−49.2310	−1.85678
$704$	34.2510	1.29088
$705$	0	0
$706$	4.21117	0.158490
$707$	−13.6132	−0.511976
$708$	0	0
$709$	9.31290	0.349753	0.174877	−	0.984590i	$-0.444047\pi$
$709$	9.31290	0.349753	0.174877	+	0.984590i	$0.444047\pi$
$710$	−92.6267	−3.47622
$711$	0	0
$712$	−0.0959900	−0.00359738
$713$	−6.87216	−0.257364
$714$	0	0
$715$	−61.6447	−2.30538
$716$	−18.7075	−0.699131
$717$	0	0
$718$	−8.38221	−0.312821
$719$	−33.4838	−1.24873	−0.624367	−	0.781131i	$-0.714643\pi$
$719$	−33.4838	−1.24873	−0.624367	+	0.781131i	$0.714643\pi$
$720$	0	0
$721$	−10.9126	−0.406406
$722$	20.9138	0.778331
$723$	0	0
$724$	−7.28705	−0.270821
$725$	−10.0855	−0.374565
$726$	0	0
$727$	−13.2045	−0.489726	−0.244863	−	0.969558i	$-0.578743\pi$
$727$	−13.2045	−0.489726	−0.244863	+	0.969558i	$0.578743\pi$
$728$	−1.72429	−0.0639065
$729$	0	0
$730$	46.9699	1.73843
$731$	−75.9829	−2.81033
$732$	0	0
$733$	6.77920	0.250395	0.125198	−	0.992132i	$-0.460043\pi$
$733$	6.77920	0.250395	0.125198	+	0.992132i	$0.460043\pi$
$734$	−35.8044	−1.32156
$735$	0	0
$736$	7.88083	0.290491
$737$	−57.6012	−2.12177
$738$	0	0
$739$	−32.2217	−1.18530	−0.592648	−	0.805462i	$-0.701917\pi$
$739$	−32.2217	−1.18530	−0.592648	+	0.805462i	$0.701917\pi$
$740$	−66.8197	−2.45634
$741$	0	0
$742$	39.8297	1.46219
$743$	35.2349	1.29264	0.646321	−	0.763065i	$-0.276307\pi$
$743$	35.2349	1.29264	0.646321	+	0.763065i	$0.276307\pi$
$744$	0	0
$745$	−21.0067	−0.769626
$746$	14.7460	0.539890
$747$	0	0
$748$	−66.0234	−2.41405
$749$	−31.6442	−1.15625
$750$	0	0
$751$	−11.3638	−0.414670	−0.207335	−	0.978270i	$-0.566479\pi$
$751$	−11.3638	−0.414670	−0.207335	+	0.978270i	$0.566479\pi$
$752$	35.5865	1.29771
$753$	0	0
$754$	−6.57877	−0.239585
$755$	38.1159	1.38718
$756$	0	0
$757$	−18.9819	−0.689910	−0.344955	−	0.938619i	$-0.612106\pi$
$757$	−18.9819	−0.689910	−0.344955	+	0.938619i	$0.612106\pi$
$758$	−60.6161	−2.20168
$759$	0	0
$760$	−4.13674	−0.150055
$761$	−3.79127	−0.137434	−0.0687168	−	0.997636i	$-0.521890\pi$
$761$	−3.79127	−0.137434	−0.0687168	+	0.997636i	$0.521890\pi$
$762$	0	0
$763$	34.7028	1.25633
$764$	−48.3294	−1.74850
$765$	0	0
$766$	32.3694	1.16955
$767$	34.1591	1.23341
$768$	0	0
$769$	47.9385	1.72871	0.864354	−	0.502884i	$-0.167728\pi$
$769$	47.9385	1.72871	0.864354	+	0.502884i	$0.167728\pi$
$770$	−96.6371	−3.48256
$771$	0	0
$772$	45.6216	1.64196
$773$	48.7564	1.75365	0.876823	−	0.480812i	$-0.159658\pi$
$773$	48.7564	1.75365	0.876823	+	0.480812i	$0.159658\pi$
$774$	0	0
$775$	69.3090	2.48965
$776$	0.154519	0.00554690
$777$	0	0
$778$	−1.73020	−0.0620308
$779$	−29.3347	−1.05103
$780$	0	0
$781$	57.5345	2.05875
$782$	−14.3971	−0.514839
$783$	0	0
$784$	0.0427745	0.00152766
$785$	66.5786	2.37629
$786$	0	0
$787$	7.87475	0.280705	0.140352	−	0.990102i	$-0.455176\pi$
$787$	7.87475	0.280705	0.140352	+	0.990102i	$0.455176\pi$
$788$	−44.6638	−1.59108
$789$	0	0
$790$	−75.4507	−2.68442
$791$	5.81525	0.206766
$792$	0	0
$793$	1.50637	0.0534926
$794$	12.1405	0.430849
$795$	0	0
$796$	−11.6478	−0.412846
$797$	5.88439	0.208436	0.104218	−	0.994554i	$-0.466766\pi$
$797$	5.88439	0.208436	0.104218	+	0.994554i	$0.466766\pi$
$798$	0	0
$799$	−61.9337	−2.19106
$800$	−79.4819	−2.81011
$801$	0	0
$802$	−15.9245	−0.562313
$803$	−29.1751	−1.02957
$804$	0	0
$805$	−10.2686	−0.361921
$806$	45.2104	1.59247
$807$	0	0
$808$	1.00819	0.0354679
$809$	3.36443	0.118287	0.0591436	−	0.998249i	$-0.481163\pi$
$809$	3.36443	0.118287	0.0591436	+	0.998249i	$0.481163\pi$
$810$	0	0
$811$	39.9912	1.40428	0.702141	−	0.712038i	$-0.252228\pi$
$811$	39.9912	1.40428	0.702141	+	0.712038i	$0.252228\pi$
$812$	−5.02554	−0.176362
$813$	0	0
$814$	85.1739	2.98534
$815$	−36.1233	−1.26534
$816$	0	0
$817$	−56.7004	−1.98370
$818$	6.35072	0.222048
$819$	0	0
$820$	−39.8151	−1.39040
$821$	−41.2944	−1.44119	−0.720593	−	0.693359i	$-0.756130\pi$
$821$	−41.2944	−1.44119	−0.720593	+	0.693359i	$0.756130\pi$
$822$	0	0
$823$	−12.7752	−0.445314	−0.222657	−	0.974897i	$-0.571473\pi$
$823$	−12.7752	−0.445314	−0.222657	+	0.974897i	$0.571473\pi$
$824$	0.808183	0.0281544
$825$	0	0
$826$	53.5494	1.86322
$827$	0.0598083	0.00207974	0.00103987	−	0.999999i	$-0.499669\pi$
$827$	0.0598083	0.00207974	0.00103987	+	0.999999i	$0.499669\pi$
$828$	0	0
$829$	31.8700	1.10689	0.553446	−	0.832885i	$-0.313313\pi$
$829$	31.8700	1.10689	0.553446	+	0.832885i	$0.313313\pi$
$830$	77.3432	2.68462
$831$	0	0
$832$	−23.9434	−0.830089
$833$	−0.0744434	−0.00257931
$834$	0	0
$835$	65.7122	2.27406
$836$	−49.2683	−1.70398
$837$	0	0
$838$	−57.4730	−1.98537
$839$	10.1762	0.351321	0.175661	−	0.984451i	$-0.443794\pi$
$839$	10.1762	0.351321	0.175661	+	0.984451i	$0.443794\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	27.5712	0.950165
$843$	0	0
$844$	−13.6686	−0.470492
$845$	−7.39886	−0.254529
$846$	0	0
$847$	30.9435	1.06323
$848$	−31.9483	−1.09711
$849$	0	0
$850$	145.202	4.98037
$851$	9.05053	0.310248
$852$	0	0
$853$	−24.6152	−0.842807	−0.421404	−	0.906873i	$-0.638462\pi$
$853$	−24.6152	−0.842807	−0.421404	+	0.906873i	$0.638462\pi$
$854$	2.36145	0.0808071
$855$	0	0
$856$	2.34356	0.0801012
$857$	−21.6841	−0.740715	−0.370357	−	0.928889i	$-0.620765\pi$
$857$	−21.6841	−0.740715	−0.370357	+	0.928889i	$0.620765\pi$
$858$	0	0
$859$	−24.9541	−0.851422	−0.425711	−	0.904859i	$-0.639976\pi$
$859$	−24.9541	−0.851422	−0.425711	+	0.904859i	$0.639976\pi$
$860$	−76.9576	−2.62423
$861$	0	0
$862$	−27.1465	−0.924612
$863$	−24.6724	−0.839859	−0.419930	−	0.907557i	$-0.637945\pi$
$863$	−24.6724	−0.839859	−0.419930	+	0.907557i	$0.637945\pi$
$864$	0	0
$865$	−13.0125	−0.442438
$866$	−49.0189	−1.66573
$867$	0	0
$868$	34.5363	1.17224
$869$	46.8657	1.58981
$870$	0	0
$871$	40.2665	1.36438
$872$	−2.57008	−0.0870339
$873$	0	0
$874$	−10.7435	−0.363404
$875$	52.2208	1.76538
$876$	0	0
$877$	−38.5718	−1.30248	−0.651239	−	0.758873i	$-0.725750\pi$
$877$	−38.5718	−1.30248	−0.651239	+	0.758873i	$0.725750\pi$
$878$	−31.4615	−1.06177
$879$	0	0
$880$	77.5148	2.61302
$881$	4.63783	0.156253	0.0781263	−	0.996943i	$-0.475106\pi$
$881$	4.63783	0.156253	0.0781263	+	0.996943i	$0.475106\pi$
$882$	0	0
$883$	2.57549	0.0866721	0.0433360	−	0.999061i	$-0.486201\pi$
$883$	2.57549	0.0866721	0.0433360	+	0.999061i	$0.486201\pi$
$884$	46.1541	1.55233
$885$	0	0
$886$	73.2847	2.46205
$887$	7.91018	0.265598	0.132799	−	0.991143i	$-0.457604\pi$
$887$	7.91018	0.265598	0.132799	+	0.991143i	$0.457604\pi$
$888$	0	0
$889$	31.2391	1.04773
$890$	3.76073	0.126060
$891$	0	0
$892$	−16.8571	−0.564416
$893$	−46.2165	−1.54658
$894$	0	0
$895$	−38.2247	−1.27771
$896$	4.13621	0.138181
$897$	0	0
$898$	25.8017	0.861015
$899$	−6.87216	−0.229199
$900$	0	0
$901$	55.6018	1.85237
$902$	50.7516	1.68984
$903$	0	0
$904$	−0.430676	−0.0143241
$905$	−14.8895	−0.494944
$906$	0	0
$907$	−20.5276	−0.681606	−0.340803	−	0.940135i	$-0.610699\pi$
$907$	−20.5276	−0.681606	−0.340803	+	0.940135i	$0.610699\pi$
$908$	−20.2870	−0.673247
$909$	0	0
$910$	67.5548	2.23942
$911$	−49.7236	−1.64742	−0.823708	−	0.567014i	$-0.808099\pi$
$911$	−49.7236	−1.64742	−0.823708	+	0.567014i	$0.808099\pi$
$912$	0	0
$913$	−48.0412	−1.58993
$914$	40.1675	1.32862
$915$	0	0
$916$	9.34096	0.308634
$917$	−34.6184	−1.14320
$918$	0	0
$919$	45.2858	1.49384	0.746921	−	0.664913i	$-0.231531\pi$
$919$	45.2858	1.49384	0.746921	+	0.664913i	$0.231531\pi$
$920$	0.760490	0.0250726
$921$	0	0
$922$	49.4396	1.62821
$923$	−40.2199	−1.32385
$924$	0	0
$925$	−91.2789	−3.00123
$926$	−43.4777	−1.42877
$927$	0	0
$928$	7.88083	0.258701
$929$	−19.7525	−0.648057	−0.324028	−	0.946047i	$-0.605037\pi$
$929$	−19.7525	−0.648057	−0.324028	+	0.946047i	$0.605037\pi$
$930$	0	0
$931$	−0.0555516	−0.00182063
$932$	43.3596	1.42029
$933$	0	0
$934$	−21.9957	−0.719720
$935$	−134.904	−4.41185
$936$	0	0
$937$	20.5188	0.670321	0.335160	−	0.942161i	$-0.391210\pi$
$937$	20.5188	0.670321	0.335160	+	0.942161i	$0.391210\pi$
$938$	63.1236	2.06106
$939$	0	0
$940$	−62.7282	−2.04597
$941$	48.6773	1.58683	0.793417	−	0.608678i	$-0.208300\pi$
$941$	48.6773	1.58683	0.793417	+	0.608678i	$0.208300\pi$
$942$	0	0
$943$	5.39284	0.175615
$944$	−42.9532	−1.39801
$945$	0	0
$946$	98.0966	3.18939
$947$	52.1168	1.69357	0.846785	−	0.531936i	$-0.178535\pi$
$947$	52.1168	1.69357	0.846785	+	0.531936i	$0.178535\pi$
$948$	0	0
$949$	20.3950	0.662051
$950$	108.353	3.51544
$951$	0	0
$952$	−3.77347	−0.122299
$953$	−49.9080	−1.61668	−0.808339	−	0.588717i	$-0.799633\pi$
$953$	−49.9080	−1.61668	−0.808339	+	0.588717i	$0.799633\pi$
$954$	0	0
$955$	−98.7506	−3.19550
$956$	42.0488	1.35996
$957$	0	0
$958$	−51.7361	−1.67152
$959$	−20.9537	−0.676632
$960$	0	0
$961$	16.2265	0.523437
$962$	−59.5414	−1.91969
$963$	0	0
$964$	−10.9717	−0.353376
$965$	93.2178	3.00079
$966$	0	0
$967$	11.0712	0.356027	0.178014	−	0.984028i	$-0.443033\pi$
$967$	11.0712	0.356027	0.178014	+	0.984028i	$0.443033\pi$
$968$	−2.29167	−0.0736570
$969$	0	0
$970$	−6.05378	−0.194375
$971$	0.191618	0.00614932	0.00307466	−	0.999995i	$-0.499021\pi$
$971$	0.191618	0.00614932	0.00307466	+	0.999995i	$0.499021\pi$
$972$	0	0
$973$	10.2959	0.330072
$974$	81.2761	2.60425
$975$	0	0
$976$	−1.89417	−0.0606309
$977$	−9.08511	−0.290659	−0.145329	−	0.989383i	$-0.546424\pi$
$977$	−9.08511	−0.290659	−0.145329	+	0.989383i	$0.546424\pi$
$978$	0	0
$979$	−2.33595	−0.0746573
$980$	−0.0753984	−0.00240851
$981$	0	0
$982$	−45.5992	−1.45513
$983$	1.42571	0.0454730	0.0227365	−	0.999741i	$-0.492762\pi$
$983$	1.42571	0.0454730	0.0227365	+	0.999741i	$0.492762\pi$
$984$	0	0
$985$	−91.2609	−2.90781
$986$	−14.3971	−0.458497
$987$	0	0
$988$	34.4413	1.09573
$989$	10.4237	0.331454
$990$	0	0
$991$	44.4287	1.41132	0.705662	−	0.708549i	$-0.250650\pi$
$991$	44.4287	1.41132	0.705662	+	0.708549i	$0.250650\pi$
$992$	−54.1583	−1.71953
$993$	0	0
$994$	−63.0505	−1.99984
$995$	−23.7998	−0.754505
$996$	0	0
$997$	−5.76382	−0.182542	−0.0912709	−	0.995826i	$-0.529093\pi$
$997$	−5.76382	−0.182542	−0.0912709	+	0.995826i	$0.529093\pi$
$998$	−1.24046	−0.0392660
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6003.2.a.s.1.17		20
3.2	odd	2		2001.2.a.o.1.4	✓	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2001.2.a.o.1.4	✓	20	3.2	odd	2
6003.2.a.s.1.17		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)

\(f(q)\)	\(=\)	\(q+1.97506 q^{2} +1.90086 q^{4} +3.88400 q^{5} +2.64382 q^{7} -0.195801 q^{8} +O(q^{10})\)	\(q+1.97506 q^{2} +1.90086 q^{4} +3.88400 q^{5} +2.64382 q^{7} -0.195801 q^{8} +7.67114 q^{10} -4.76488 q^{11} +3.33092 q^{13} +5.22171 q^{14} -4.18845 q^{16} +7.28945 q^{17} +5.43957 q^{19} +7.38296 q^{20} -9.41093 q^{22} -1.00000 q^{23} +10.0855 q^{25} +6.57877 q^{26} +5.02554 q^{28} -1.00000 q^{29} +6.87216 q^{31} -7.88083 q^{32} +14.3971 q^{34} +10.2686 q^{35} -9.05053 q^{37} +10.7435 q^{38} -0.760490 q^{40} -5.39284 q^{41} -10.4237 q^{43} -9.05739 q^{44} -1.97506 q^{46} -8.49635 q^{47} -0.0102125 q^{49} +19.9194 q^{50} +6.33163 q^{52} +7.62772 q^{53} -18.5068 q^{55} -0.517662 q^{56} -1.97506 q^{58} +10.2552 q^{59} +0.452237 q^{61} +13.5729 q^{62} -7.18823 q^{64} +12.9373 q^{65} +12.0887 q^{67} +13.8562 q^{68} +20.2811 q^{70} -12.0747 q^{71} +6.12294 q^{73} -17.8753 q^{74} +10.3399 q^{76} -12.5975 q^{77} -9.83566 q^{79} -16.2679 q^{80} -10.6512 q^{82} +10.0824 q^{83} +28.3122 q^{85} -20.5874 q^{86} +0.932967 q^{88} +0.490243 q^{89} +8.80636 q^{91} -1.90086 q^{92} -16.7808 q^{94} +21.1273 q^{95} -0.789164 q^{97} -0.0201703 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 2 q^{2} + 30 q^{4} + q^{5} + 9 q^{7} - 6 q^{8}+O(q^{10}) \) 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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