LMFDB - Embedded newform 4003.2.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4003,2,Mod(1,4003)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4003, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("4003.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4003 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4003.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:	$31.9641159291$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.41421$ of defining polynomial
Character	$\chi$	$=$	4003.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.41421 q^{2} -1.41421 q^{3} -1.41421 q^{5} +2.00000 q^{6} -1.00000 q^{7} +2.82843 q^{8} -1.00000 q^{9} +O(q^{10})$	\(q-1.41421 q^{2} -1.41421 q^{3} -1.41421 q^{5} +2.00000 q^{6} -1.00000 q^{7} +2.82843 q^{8} -1.00000 q^{9} +2.00000 q^{10} +0.828427 q^{11} +1.17157 q^{13} +1.41421 q^{14} +2.00000 q^{15} -4.00000 q^{16} +4.65685 q^{17} +1.41421 q^{18} -4.65685 q^{19} +1.41421 q^{21} -1.17157 q^{22} -8.82843 q^{23} -4.00000 q^{24} -3.00000 q^{25} -1.65685 q^{26} +5.65685 q^{27} -1.17157 q^{29} -2.82843 q^{30} +10.4853 q^{31} -1.17157 q^{33} -6.58579 q^{34} +1.41421 q^{35} -4.17157 q^{37} +6.58579 q^{38} -1.65685 q^{39} -4.00000 q^{40} +5.17157 q^{41} -2.00000 q^{42} +1.41421 q^{45} +12.4853 q^{46} -3.07107 q^{47} +5.65685 q^{48} -6.00000 q^{49} +4.24264 q^{50} -6.58579 q^{51} +9.31371 q^{53} -8.00000 q^{54} -1.17157 q^{55} -2.82843 q^{56} +6.58579 q^{57} +1.65685 q^{58} -8.82843 q^{59} +7.17157 q^{61} -14.8284 q^{62} +1.00000 q^{63} +8.00000 q^{64} -1.65685 q^{65} +1.65685 q^{66} +12.6569 q^{67} +12.4853 q^{69} -2.00000 q^{70} +6.00000 q^{71} -2.82843 q^{72} -3.00000 q^{73} +5.89949 q^{74} +4.24264 q^{75} -0.828427 q^{77} +2.34315 q^{78} +11.4853 q^{79} +5.65685 q^{80} -5.00000 q^{81} -7.31371 q^{82} +17.1421 q^{83} -6.58579 q^{85} +1.65685 q^{87} +2.34315 q^{88} -11.8284 q^{89} -2.00000 q^{90} -1.17157 q^{91} -14.8284 q^{93} +4.34315 q^{94} +6.58579 q^{95} +1.75736 q^{97} +8.48528 q^{98} -0.828427 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 4 q^{6} - 2 q^{7} - 2 q^{9}+O(q^{10}) $ 2 * q + 4 * q^6 - 2 * q^7 - 2 * q^9	$ 2 q + 4 q^{6} - 2 q^{7} - 2 q^{9} + 4 q^{10} - 4 q^{11} + 8 q^{13} + 4 q^{15} - 8 q^{16} - 2 q^{17} + 2 q^{19} - 8 q^{22} - 12 q^{23} - 8 q^{24} - 6 q^{25} + 8 q^{26} - 8 q^{29} + 4 q^{31} - 8 q^{33} - 16 q^{34} - 14 q^{37} + 16 q^{38} + 8 q^{39} - 8 q^{40} + 16 q^{41} - 4 q^{42} + 8 q^{46} + 8 q^{47} - 12 q^{49} - 16 q^{51} - 4 q^{53} - 16 q^{54} - 8 q^{55} + 16 q^{57} - 8 q^{58} - 12 q^{59} + 20 q^{61} - 24 q^{62} + 2 q^{63} + 16 q^{64} + 8 q^{65} - 8 q^{66} + 14 q^{67} + 8 q^{69} - 4 q^{70} + 12 q^{71} - 6 q^{73} - 8 q^{74} + 4 q^{77} + 16 q^{78} + 6 q^{79} - 10 q^{81} + 8 q^{82} + 6 q^{83} - 16 q^{85} - 8 q^{87} + 16 q^{88} - 18 q^{89} - 4 q^{90} - 8 q^{91} - 24 q^{93} + 20 q^{94} + 16 q^{95} + 12 q^{97} + 4 q^{99}+O(q^{100}) $ 2 * q + 4 * q^6 - 2 * q^7 - 2 * q^9 + 4 * q^10 - 4 * q^11 + 8 * q^13 + 4 * q^15 - 8 * q^16 - 2 * q^17 + 2 * q^19 - 8 * q^22 - 12 * q^23 - 8 * q^24 - 6 * q^25 + 8 * q^26 - 8 * q^29 + 4 * q^31 - 8 * q^33 - 16 * q^34 - 14 * q^37 + 16 * q^38 + 8 * q^39 - 8 * q^40 + 16 * q^41 - 4 * q^42 + 8 * q^46 + 8 * q^47 - 12 * q^49 - 16 * q^51 - 4 * q^53 - 16 * q^54 - 8 * q^55 + 16 * q^57 - 8 * q^58 - 12 * q^59 + 20 * q^61 - 24 * q^62 + 2 * q^63 + 16 * q^64 + 8 * q^65 - 8 * q^66 + 14 * q^67 + 8 * q^69 - 4 * q^70 + 12 * q^71 - 6 * q^73 - 8 * q^74 + 4 * q^77 + 16 * q^78 + 6 * q^79 - 10 * q^81 + 8 * q^82 + 6 * q^83 - 16 * q^85 - 8 * q^87 + 16 * q^88 - 18 * q^89 - 4 * q^90 - 8 * q^91 - 24 * q^93 + 20 * q^94 + 16 * q^95 + 12 * q^97 + 4 * q^99

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 $

$2$	−1.41421	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$2$	−1.41421	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$3$	−1.41421	−0.816497	−0.408248	−	0.912871i	$-0.633860\pi$
$3$	−1.41421	−0.816497	−0.408248	+	0.912871i	$0.633860\pi$
$4$	0	0
$5$	−1.41421	−0.632456	−0.316228	−	0.948683i	$-0.602416\pi$
$5$	−1.41421	−0.632456	−0.316228	+	0.948683i	$0.602416\pi$
$6$	2.00000	0.816497
$7$	−1.00000	−0.377964	−0.188982	−	0.981981i	$-0.560519\pi$
$7$	−1.00000	−0.377964	−0.188982	+	0.981981i	$0.560519\pi$
$8$	2.82843	1.00000
$9$	−1.00000	−0.333333
$10$	2.00000	0.632456
$11$	0.828427	0.249780	0.124890	−	0.992171i	$-0.460142\pi$
$11$	0.828427	0.249780	0.124890	+	0.992171i	$0.460142\pi$
$12$	0	0
$13$	1.17157	0.324936	0.162468	−	0.986714i	$-0.448055\pi$
$13$	1.17157	0.324936	0.162468	+	0.986714i	$0.448055\pi$
$14$	1.41421	0.377964
$15$	2.00000	0.516398
$16$	−4.00000	−1.00000
$17$	4.65685	1.12945	0.564727	−	0.825278i	$-0.308982\pi$
$17$	4.65685	1.12945	0.564727	+	0.825278i	$0.308982\pi$
$18$	1.41421	0.333333
$19$	−4.65685	−1.06836	−0.534178	−	0.845372i	$-0.679379\pi$
$19$	−4.65685	−1.06836	−0.534178	+	0.845372i	$0.679379\pi$
$20$	0	0
$21$	1.41421	0.308607
$22$	−1.17157	−0.249780
$23$	−8.82843	−1.84085	−0.920427	−	0.390914i	$-0.872159\pi$
$23$	−8.82843	−1.84085	−0.920427	+	0.390914i	$0.872159\pi$
$24$	−4.00000	−0.816497
$25$	−3.00000	−0.600000
$26$	−1.65685	−0.324936
$27$	5.65685	1.08866
$28$	0	0
$29$	−1.17157	−0.217556	−0.108778	−	0.994066i	$-0.534694\pi$
$29$	−1.17157	−0.217556	−0.108778	+	0.994066i	$0.534694\pi$
$30$	−2.82843	−0.516398
$31$	10.4853	1.88321	0.941606	−	0.336717i	$-0.109316\pi$
$31$	10.4853	1.88321	0.941606	+	0.336717i	$0.109316\pi$
$32$	0	0
$33$	−1.17157	−0.203945
$34$	−6.58579	−1.12945
$35$	1.41421	0.239046
$36$	0	0
$37$	−4.17157	−0.685802	−0.342901	−	0.939371i	$-0.611410\pi$
$37$	−4.17157	−0.685802	−0.342901	+	0.939371i	$0.611410\pi$
$38$	6.58579	1.06836
$39$	−1.65685	−0.265309
$40$	−4.00000	−0.632456
$41$	5.17157	0.807664	0.403832	−	0.914833i	$-0.367678\pi$
$41$	5.17157	0.807664	0.403832	+	0.914833i	$0.367678\pi$
$42$	−2.00000	−0.308607
$43$	0	0		−	1.00000i	$-0.5\pi$
$43$	0	0			1.00000i	$0.5\pi$
$44$	0	0
$45$	1.41421	0.210819
$46$	12.4853	1.84085
$47$	−3.07107	−0.447961	−0.223981	−	0.974594i	$-0.571905\pi$
$47$	−3.07107	−0.447961	−0.223981	+	0.974594i	$0.571905\pi$
$48$	5.65685	0.816497
$49$	−6.00000	−0.857143
$50$	4.24264	0.600000
$51$	−6.58579	−0.922195
$52$	0	0
$53$	9.31371	1.27934	0.639668	−	0.768651i	$-0.279072\pi$
$53$	9.31371	1.27934	0.639668	+	0.768651i	$0.279072\pi$
$54$	−8.00000	−1.08866
$55$	−1.17157	−0.157975
$56$	−2.82843	−0.377964
$57$	6.58579	0.872309
$58$	1.65685	0.217556
$59$	−8.82843	−1.14936	−0.574682	−	0.818377i	$-0.694874\pi$
$59$	−8.82843	−1.14936	−0.574682	+	0.818377i	$0.694874\pi$
$60$	0	0
$61$	7.17157	0.918226	0.459113	−	0.888378i	$-0.348167\pi$
$61$	7.17157	0.918226	0.459113	+	0.888378i	$0.348167\pi$
$62$	−14.8284	−1.88321
$63$	1.00000	0.125988
$64$	8.00000	1.00000
$65$	−1.65685	−0.205507
$66$	1.65685	0.203945
$67$	12.6569	1.54628	0.773140	−	0.634235i	$-0.218685\pi$
$67$	12.6569	1.54628	0.773140	+	0.634235i	$0.218685\pi$
$68$	0	0
$69$	12.4853	1.50305
$70$	−2.00000	−0.239046
$71$	6.00000	0.712069	0.356034	−	0.934473i	$-0.384129\pi$
$71$	6.00000	0.712069	0.356034	+	0.934473i	$0.384129\pi$
$72$	−2.82843	−0.333333
$73$	−3.00000	−0.351123	−0.175562	−	0.984468i	$-0.556174\pi$
$73$	−3.00000	−0.351123	−0.175562	+	0.984468i	$0.556174\pi$
$74$	5.89949	0.685802
$75$	4.24264	0.489898
$76$	0	0
$77$	−0.828427	−0.0944080
$78$	2.34315	0.265309
$79$	11.4853	1.29220	0.646098	−	0.763255i	$-0.276400\pi$
$79$	11.4853	1.29220	0.646098	+	0.763255i	$0.276400\pi$
$80$	5.65685	0.632456
$81$	−5.00000	−0.555556
$82$	−7.31371	−0.807664
$83$	17.1421	1.88159	0.940797	−	0.338971i	$-0.110079\pi$
$83$	17.1421	1.88159	0.940797	+	0.338971i	$0.110079\pi$
$84$	0	0
$85$	−6.58579	−0.714329
$86$	0	0
$87$	1.65685	0.177633
$88$	2.34315	0.249780
$89$	−11.8284	−1.25381	−0.626905	−	0.779095i	$-0.715679\pi$
$89$	−11.8284	−1.25381	−0.626905	+	0.779095i	$0.715679\pi$
$90$	−2.00000	−0.210819
$91$	−1.17157	−0.122814
$92$	0	0
$93$	−14.8284	−1.53764
$94$	4.34315	0.447961
$95$	6.58579	0.675687
$96$	0	0
$97$	1.75736	0.178433	0.0892164	−	0.996012i	$-0.471564\pi$
$97$	1.75736	0.178433	0.0892164	+	0.996012i	$0.471564\pi$
$98$	8.48528	0.857143
$99$	−0.828427	−0.0832601
$100$	0	0
$101$	9.82843	0.977965	0.488983	−	0.872294i	$-0.337368\pi$
$101$	9.82843	0.977965	0.488983	+	0.872294i	$0.337368\pi$
$102$	9.31371	0.922195
$103$	−13.4853	−1.32874	−0.664372	−	0.747402i	$-0.731301\pi$
$103$	−13.4853	−1.32874	−0.664372	+	0.747402i	$0.731301\pi$
$104$	3.31371	0.324936
$105$	−2.00000	−0.195180
$106$	−13.1716	−1.27934
$107$	8.48528	0.820303	0.410152	−	0.912017i	$-0.365476\pi$
$107$	8.48528	0.820303	0.410152	+	0.912017i	$0.365476\pi$
$108$	0	0
$109$	−10.4853	−1.00431	−0.502154	−	0.864778i	$-0.667459\pi$
$109$	−10.4853	−1.00431	−0.502154	+	0.864778i	$0.667459\pi$
$110$	1.65685	0.157975
$111$	5.89949	0.559955
$112$	4.00000	0.377964
$113$	15.0711	1.41777	0.708883	−	0.705326i	$-0.249199\pi$
$113$	15.0711	1.41777	0.708883	+	0.705326i	$0.249199\pi$
$114$	−9.31371	−0.872309
$115$	12.4853	1.16426
$116$	0	0
$117$	−1.17157	−0.108312
$118$	12.4853	1.14936
$119$	−4.65685	−0.426893
$120$	5.65685	0.516398
$121$	−10.3137	−0.937610
$122$	−10.1421	−0.918226
$123$	−7.31371	−0.659455
$124$	0	0
$125$	11.3137	1.01193
$126$	−1.41421	−0.125988
$127$	18.7990	1.66814	0.834070	−	0.551658i	$-0.186005\pi$
$127$	18.7990	1.66814	0.834070	+	0.551658i	$0.186005\pi$
$128$	−11.3137	−1.00000
$129$	0	0
$130$	2.34315	0.205507
$131$	0.514719	0.0449712	0.0224856	−	0.999747i	$-0.492842\pi$
$131$	0.514719	0.0449712	0.0224856	+	0.999747i	$0.492842\pi$
$132$	0	0
$133$	4.65685	0.403800
$134$	−17.8995	−1.54628
$135$	−8.00000	−0.688530
$136$	13.1716	1.12945
$137$	−3.00000	−0.256307	−0.128154	−	0.991754i	$-0.540905\pi$
$137$	−3.00000	−0.256307	−0.128154	+	0.991754i	$0.540905\pi$
$138$	−17.6569	−1.50305
$139$	−4.34315	−0.368381	−0.184190	−	0.982891i	$-0.558966\pi$
$139$	−4.34315	−0.368381	−0.184190	+	0.982891i	$0.558966\pi$
$140$	0	0
$141$	4.34315	0.365759
$142$	−8.48528	−0.712069
$143$	0.970563	0.0811625
$144$	4.00000	0.333333
$145$	1.65685	0.137594
$146$	4.24264	0.351123
$147$	8.48528	0.699854
$148$	0	0
$149$	7.14214	0.585107	0.292553	−	0.956249i	$-0.405495\pi$
$149$	7.14214	0.585107	0.292553	+	0.956249i	$0.405495\pi$
$150$	−6.00000	−0.489898
$151$	−8.17157	−0.664993	−0.332497	−	0.943104i	$-0.607891\pi$
$151$	−8.17157	−0.664993	−0.332497	+	0.943104i	$0.607891\pi$
$152$	−13.1716	−1.06836
$153$	−4.65685	−0.376484
$154$	1.17157	0.0944080
$155$	−14.8284	−1.19105
$156$	0	0
$157$	−15.3137	−1.22217	−0.611083	−	0.791566i	$-0.709266\pi$
$157$	−15.3137	−1.22217	−0.611083	+	0.791566i	$0.709266\pi$
$158$	−16.2426	−1.29220
$159$	−13.1716	−1.04457
$160$	0	0
$161$	8.82843	0.695778
$162$	7.07107	0.555556
$163$	−2.00000	−0.156652	−0.0783260	−	0.996928i	$-0.524958\pi$
$163$	−2.00000	−0.156652	−0.0783260	+	0.996928i	$0.524958\pi$
$164$	0	0
$165$	1.65685	0.128986
$166$	−24.2426	−1.88159
$167$	−21.8995	−1.69463	−0.847317	−	0.531088i	$-0.821783\pi$
$167$	−21.8995	−1.69463	−0.847317	+	0.531088i	$0.821783\pi$
$168$	4.00000	0.308607
$169$	−11.6274	−0.894417
$170$	9.31371	0.714329
$171$	4.65685	0.356119
$172$	0	0
$173$	−23.3137	−1.77251	−0.886254	−	0.463199i	$-0.846701\pi$
$173$	−23.3137	−1.77251	−0.886254	+	0.463199i	$0.846701\pi$
$174$	−2.34315	−0.177633
$175$	3.00000	0.226779
$176$	−3.31371	−0.249780
$177$	12.4853	0.938451
$178$	16.7279	1.25381
$179$	9.65685	0.721787	0.360894	−	0.932607i	$-0.382472\pi$
$179$	9.65685	0.721787	0.360894	+	0.932607i	$0.382472\pi$
$180$	0	0
$181$	−3.31371	−0.246306	−0.123153	−	0.992388i	$-0.539301\pi$
$181$	−3.31371	−0.246306	−0.123153	+	0.992388i	$0.539301\pi$
$182$	1.65685	0.122814
$183$	−10.1421	−0.749728
$184$	−24.9706	−1.84085
$185$	5.89949	0.433739
$186$	20.9706	1.53764
$187$	3.85786	0.282115
$188$	0	0
$189$	−5.65685	−0.411476
$190$	−9.31371	−0.675687
$191$	1.34315	0.0971866	0.0485933	−	0.998819i	$-0.484526\pi$
$191$	1.34315	0.0971866	0.0485933	+	0.998819i	$0.484526\pi$
$192$	−11.3137	−0.816497
$193$	10.3137	0.742397	0.371198	−	0.928554i	$-0.378947\pi$
$193$	10.3137	0.742397	0.371198	+	0.928554i	$0.378947\pi$
$194$	−2.48528	−0.178433
$195$	2.34315	0.167796
$196$	0	0
$197$	−2.65685	−0.189293	−0.0946465	−	0.995511i	$-0.530172\pi$
$197$	−2.65685	−0.189293	−0.0946465	+	0.995511i	$0.530172\pi$
$198$	1.17157	0.0832601
$199$	4.10051	0.290677	0.145339	−	0.989382i	$-0.453573\pi$
$199$	4.10051	0.290677	0.145339	+	0.989382i	$0.453573\pi$
$200$	−8.48528	−0.600000
$201$	−17.8995	−1.26253
$202$	−13.8995	−0.977965
$203$	1.17157	0.0822283
$204$	0	0
$205$	−7.31371	−0.510812
$206$	19.0711	1.32874
$207$	8.82843	0.613618
$208$	−4.68629	−0.324936
$209$	−3.85786	−0.266854
$210$	2.82843	0.195180
$211$	13.3137	0.916553	0.458277	−	0.888810i	$-0.348467\pi$
$211$	13.3137	0.916553	0.458277	+	0.888810i	$0.348467\pi$
$212$	0	0
$213$	−8.48528	−0.581402
$214$	−12.0000	−0.820303
$215$	0	0
$216$	16.0000	1.08866
$217$	−10.4853	−0.711787
$218$	14.8284	1.00431
$219$	4.24264	0.286691
$220$	0	0
$221$	5.45584	0.367000
$222$	−8.34315	−0.559955
$223$	26.7279	1.78983	0.894917	−	0.446233i	$-0.147235\pi$
$223$	26.7279	1.78983	0.894917	+	0.446233i	$0.147235\pi$
$224$	0	0
$225$	3.00000	0.200000
$226$	−21.3137	−1.41777
$227$	−18.5858	−1.23358	−0.616791	−	0.787127i	$-0.711568\pi$
$227$	−18.5858	−1.23358	−0.616791	+	0.787127i	$0.711568\pi$
$228$	0	0
$229$	26.4853	1.75020	0.875098	−	0.483945i	$-0.160797\pi$
$229$	26.4853	1.75020	0.875098	+	0.483945i	$0.160797\pi$
$230$	−17.6569	−1.16426
$231$	1.17157	0.0770838
$232$	−3.31371	−0.217556
$233$	−11.6569	−0.763666	−0.381833	−	0.924231i	$-0.624707\pi$
$233$	−11.6569	−0.763666	−0.381833	+	0.924231i	$0.624707\pi$
$234$	1.65685	0.108312
$235$	4.34315	0.283316
$236$	0	0
$237$	−16.2426	−1.05507
$238$	6.58579	0.426893
$239$	−2.65685	−0.171858	−0.0859288	−	0.996301i	$-0.527386\pi$
$239$	−2.65685	−0.171858	−0.0859288	+	0.996301i	$0.527386\pi$
$240$	−8.00000	−0.516398
$241$	−5.48528	−0.353338	−0.176669	−	0.984270i	$-0.556532\pi$
$241$	−5.48528	−0.353338	−0.176669	+	0.984270i	$0.556532\pi$
$242$	14.5858	0.937610
$243$	−9.89949	−0.635053
$244$	0	0
$245$	8.48528	0.542105
$246$	10.3431	0.659455
$247$	−5.45584	−0.347147
$248$	29.6569	1.88321
$249$	−24.2426	−1.53631
$250$	−16.0000	−1.01193
$251$	−14.4853	−0.914303	−0.457151	−	0.889389i	$-0.651130\pi$
$251$	−14.4853	−0.914303	−0.457151	+	0.889389i	$0.651130\pi$
$252$	0	0
$253$	−7.31371	−0.459809
$254$	−26.5858	−1.66814
$255$	9.31371	0.583247
$256$	0	0
$257$	16.9706	1.05859	0.529297	−	0.848436i	$-0.322456\pi$
$257$	16.9706	1.05859	0.529297	+	0.848436i	$0.322456\pi$
$258$	0	0
$259$	4.17157	0.259209
$260$	0	0
$261$	1.17157	0.0725185
$262$	−0.727922	−0.0449712
$263$	−10.3137	−0.635971	−0.317985	−	0.948096i	$-0.603006\pi$
$263$	−10.3137	−0.635971	−0.317985	+	0.948096i	$0.603006\pi$
$264$	−3.31371	−0.203945
$265$	−13.1716	−0.809123
$266$	−6.58579	−0.403800
$267$	16.7279	1.02373
$268$	0	0
$269$	−29.3137	−1.78729	−0.893644	−	0.448776i	$-0.851860\pi$
$269$	−29.3137	−1.78729	−0.893644	+	0.448776i	$0.851860\pi$
$270$	11.3137	0.688530
$271$	28.6569	1.74078	0.870390	−	0.492363i	$-0.163867\pi$
$271$	28.6569	1.74078	0.870390	+	0.492363i	$0.163867\pi$
$272$	−18.6274	−1.12945
$273$	1.65685	0.100277
$274$	4.24264	0.256307
$275$	−2.48528	−0.149868
$276$	0	0
$277$	2.48528	0.149326	0.0746630	−	0.997209i	$-0.476212\pi$
$277$	2.48528	0.149326	0.0746630	+	0.997209i	$0.476212\pi$
$278$	6.14214	0.368381
$279$	−10.4853	−0.627737
$280$	4.00000	0.239046
$281$	9.82843	0.586315	0.293157	−	0.956064i	$-0.405294\pi$
$281$	9.82843	0.586315	0.293157	+	0.956064i	$0.405294\pi$
$282$	−6.14214	−0.365759
$283$	−12.8284	−0.762571	−0.381285	−	0.924457i	$-0.624518\pi$
$283$	−12.8284	−0.762571	−0.381285	+	0.924457i	$0.624518\pi$
$284$	0	0
$285$	−9.31371	−0.551696
$286$	−1.37258	−0.0811625
$287$	−5.17157	−0.305268
$288$	0	0
$289$	4.68629	0.275664
$290$	−2.34315	−0.137594
$291$	−2.48528	−0.145690
$292$	0	0
$293$	−8.68629	−0.507459	−0.253729	−	0.967275i	$-0.581657\pi$
$293$	−8.68629	−0.507459	−0.253729	+	0.967275i	$0.581657\pi$
$294$	−12.0000	−0.699854
$295$	12.4853	0.726921
$296$	−11.7990	−0.685802
$297$	4.68629	0.271926
$298$	−10.1005	−0.585107
$299$	−10.3431	−0.598160
$300$	0	0
$301$	0	0
$302$	11.5563	0.664993
$303$	−13.8995	−0.798505
$304$	18.6274	1.06836
$305$	−10.1421	−0.580737
$306$	6.58579	0.376484
$307$	−25.3137	−1.44473	−0.722365	−	0.691512i	$-0.756945\pi$
$307$	−25.3137	−1.44473	−0.722365	+	0.691512i	$0.756945\pi$
$308$	0	0
$309$	19.0711	1.08492
$310$	20.9706	1.19105
$311$	−13.9706	−0.792198	−0.396099	−	0.918208i	$-0.629636\pi$
$311$	−13.9706	−0.792198	−0.396099	+	0.918208i	$0.629636\pi$
$312$	−4.68629	−0.265309
$313$	3.48528	0.197000	0.0984999	−	0.995137i	$-0.468596\pi$
$313$	3.48528	0.197000	0.0984999	+	0.995137i	$0.468596\pi$
$314$	21.6569	1.22217
$315$	−1.41421	−0.0796819
$316$	0	0
$317$	1.75736	0.0987031	0.0493516	−	0.998781i	$-0.484285\pi$
$317$	1.75736	0.0987031	0.0493516	+	0.998781i	$0.484285\pi$
$318$	18.6274	1.04457
$319$	−0.970563	−0.0543411
$320$	−11.3137	−0.632456
$321$	−12.0000	−0.669775
$322$	−12.4853	−0.695778
$323$	−21.6863	−1.20666
$324$	0	0
$325$	−3.51472	−0.194962
$326$	2.82843	0.156652
$327$	14.8284	0.820014
$328$	14.6274	0.807664
$329$	3.07107	0.169313
$330$	−2.34315	−0.128986
$331$	−2.48528	−0.136603	−0.0683017	−	0.997665i	$-0.521758\pi$
$331$	−2.48528	−0.136603	−0.0683017	+	0.997665i	$0.521758\pi$
$332$	0	0
$333$	4.17157	0.228601
$334$	30.9706	1.69463
$335$	−17.8995	−0.977954
$336$	−5.65685	−0.308607
$337$	−7.31371	−0.398403	−0.199202	−	0.979959i	$-0.563835\pi$
$337$	−7.31371	−0.398403	−0.199202	+	0.979959i	$0.563835\pi$
$338$	16.4437	0.894417
$339$	−21.3137	−1.15760
$340$	0	0
$341$	8.68629	0.470389
$342$	−6.58579	−0.356119
$343$	13.0000	0.701934
$344$	0	0
$345$	−17.6569	−0.950613
$346$	32.9706	1.77251
$347$	−25.2843	−1.35733	−0.678665	−	0.734448i	$-0.737441\pi$
$347$	−25.2843	−1.35733	−0.678665	+	0.734448i	$0.737441\pi$
$348$	0	0
$349$	13.3431	0.714242	0.357121	−	0.934058i	$-0.383758\pi$
$349$	13.3431	0.714242	0.357121	+	0.934058i	$0.383758\pi$
$350$	−4.24264	−0.226779
$351$	6.62742	0.353745
$352$	0	0
$353$	−26.6569	−1.41880	−0.709401	−	0.704806i	$-0.751034\pi$
$353$	−26.6569	−1.41880	−0.709401	+	0.704806i	$0.751034\pi$
$354$	−17.6569	−0.938451
$355$	−8.48528	−0.450352
$356$	0	0
$357$	6.58579	0.348557
$358$	−13.6569	−0.721787
$359$	−5.89949	−0.311363	−0.155682	−	0.987807i	$-0.549757\pi$
$359$	−5.89949	−0.311363	−0.155682	+	0.987807i	$0.549757\pi$
$360$	4.00000	0.210819
$361$	2.68629	0.141384
$362$	4.68629	0.246306
$363$	14.5858	0.765555
$364$	0	0
$365$	4.24264	0.222070
$366$	14.3431	0.749728
$367$	0.828427	0.0432435	0.0216218	−	0.999766i	$-0.493117\pi$
$367$	0.828427	0.0432435	0.0216218	+	0.999766i	$0.493117\pi$
$368$	35.3137	1.84085
$369$	−5.17157	−0.269221
$370$	−8.34315	−0.433739
$371$	−9.31371	−0.483544
$372$	0	0
$373$	0.485281	0.0251269	0.0125635	−	0.999921i	$-0.496001\pi$
$373$	0.485281	0.0251269	0.0125635	+	0.999921i	$0.496001\pi$
$374$	−5.45584	−0.282115
$375$	−16.0000	−0.826236
$376$	−8.68629	−0.447961
$377$	−1.37258	−0.0706916
$378$	8.00000	0.411476
$379$	23.2132	1.19238	0.596191	−	0.802843i	$-0.296680\pi$
$379$	23.2132	1.19238	0.596191	+	0.802843i	$0.296680\pi$
$380$	0	0
$381$	−26.5858	−1.36203
$382$	−1.89949	−0.0971866
$383$	−23.5563	−1.20367	−0.601837	−	0.798619i	$-0.705564\pi$
$383$	−23.5563	−1.20367	−0.601837	+	0.798619i	$0.705564\pi$
$384$	16.0000	0.816497
$385$	1.17157	0.0597089
$386$	−14.5858	−0.742397
$387$	0	0
$388$	0	0
$389$	−34.6274	−1.75568	−0.877840	−	0.478954i	$-0.841016\pi$
$389$	−34.6274	−1.75568	−0.877840	+	0.478954i	$0.841016\pi$
$390$	−3.31371	−0.167796
$391$	−41.1127	−2.07916
$392$	−16.9706	−0.857143
$393$	−0.727922	−0.0367188
$394$	3.75736	0.189293
$395$	−16.2426	−0.817256
$396$	0	0
$397$	−15.0000	−0.752828	−0.376414	−	0.926451i	$-0.622843\pi$
$397$	−15.0000	−0.752828	−0.376414	+	0.926451i	$0.622843\pi$
$398$	−5.79899	−0.290677
$399$	−6.58579	−0.329702
$400$	12.0000	0.600000
$401$	−25.4558	−1.27120	−0.635602	−	0.772017i	$-0.719248\pi$
$401$	−25.4558	−1.27120	−0.635602	+	0.772017i	$0.719248\pi$
$402$	25.3137	1.26253
$403$	12.2843	0.611923
$404$	0	0
$405$	7.07107	0.351364
$406$	−1.65685	−0.0822283
$407$	−3.45584	−0.171300
$408$	−18.6274	−0.922195
$409$	−7.55635	−0.373637	−0.186819	−	0.982394i	$-0.559818\pi$
$409$	−7.55635	−0.373637	−0.186819	+	0.982394i	$0.559818\pi$
$410$	10.3431	0.510812
$411$	4.24264	0.209274
$412$	0	0
$413$	8.82843	0.434418
$414$	−12.4853	−0.613618
$415$	−24.2426	−1.19002
$416$	0	0
$417$	6.14214	0.300782
$418$	5.45584	0.266854
$419$	−8.68629	−0.424353	−0.212177	−	0.977231i	$-0.568055\pi$
$419$	−8.68629	−0.424353	−0.212177	+	0.977231i	$0.568055\pi$
$420$	0	0
$421$	9.51472	0.463719	0.231860	−	0.972749i	$-0.425519\pi$
$421$	9.51472	0.463719	0.231860	+	0.972749i	$0.425519\pi$
$422$	−18.8284	−0.916553
$423$	3.07107	0.149320
$424$	26.3431	1.27934
$425$	−13.9706	−0.677672
$426$	12.0000	0.581402
$427$	−7.17157	−0.347057
$428$	0	0
$429$	−1.37258	−0.0662689
$430$	0	0
$431$	−18.0000	−0.867029	−0.433515	−	0.901146i	$-0.642727\pi$
$431$	−18.0000	−0.867029	−0.433515	+	0.901146i	$0.642727\pi$
$432$	−22.6274	−1.08866
$433$	−4.02944	−0.193642	−0.0968212	−	0.995302i	$-0.530868\pi$
$433$	−4.02944	−0.193642	−0.0968212	+	0.995302i	$0.530868\pi$
$434$	14.8284	0.711787
$435$	−2.34315	−0.112345
$436$	0	0
$437$	41.1127	1.96669
$438$	−6.00000	−0.286691
$439$	1.51472	0.0722936	0.0361468	−	0.999346i	$-0.488492\pi$
$439$	1.51472	0.0722936	0.0361468	+	0.999346i	$0.488492\pi$
$440$	−3.31371	−0.157975
$441$	6.00000	0.285714
$442$	−7.71573	−0.367000
$443$	−1.02944	−0.0489100	−0.0244550	−	0.999701i	$-0.507785\pi$
$443$	−1.02944	−0.0489100	−0.0244550	+	0.999701i	$0.507785\pi$
$444$	0	0
$445$	16.7279	0.792980
$446$	−37.7990	−1.78983
$447$	−10.1005	−0.477737
$448$	−8.00000	−0.377964
$449$	8.04163	0.379508	0.189754	−	0.981832i	$-0.439231\pi$
$449$	8.04163	0.379508	0.189754	+	0.981832i	$0.439231\pi$
$450$	−4.24264	−0.200000
$451$	4.28427	0.201738
$452$	0	0
$453$	11.5563	0.542965
$454$	26.2843	1.23358
$455$	1.65685	0.0776745
$456$	18.6274	0.872309
$457$	20.6274	0.964910	0.482455	−	0.875921i	$-0.339745\pi$
$457$	20.6274	0.964910	0.482455	+	0.875921i	$0.339745\pi$
$458$	−37.4558	−1.75020
$459$	26.3431	1.22959
$460$	0	0
$461$	−23.6569	−1.10181	−0.550905	−	0.834568i	$-0.685717\pi$
$461$	−23.6569	−1.10181	−0.550905	+	0.834568i	$0.685717\pi$
$462$	−1.65685	−0.0770838
$463$	15.2132	0.707018	0.353509	−	0.935431i	$-0.384988\pi$
$463$	15.2132	0.707018	0.353509	+	0.935431i	$0.384988\pi$
$464$	4.68629	0.217556
$465$	20.9706	0.972487
$466$	16.4853	0.763666
$467$	9.51472	0.440289	0.220144	−	0.975467i	$-0.429347\pi$
$467$	9.51472	0.440289	0.220144	+	0.975467i	$0.429347\pi$
$468$	0	0
$469$	−12.6569	−0.584439
$470$	−6.14214	−0.283316
$471$	21.6569	0.997895
$472$	−24.9706	−1.14936
$473$	0	0
$474$	22.9706	1.05507
$475$	13.9706	0.641013
$476$	0	0
$477$	−9.31371	−0.426445
$478$	3.75736	0.171858
$479$	34.4558	1.57433	0.787164	−	0.616744i	$-0.211549\pi$
$479$	34.4558	1.57433	0.787164	+	0.616744i	$0.211549\pi$
$480$	0	0
$481$	−4.88730	−0.222842
$482$	7.75736	0.353338
$483$	−12.4853	−0.568100
$484$	0	0
$485$	−2.48528	−0.112851
$486$	14.0000	0.635053
$487$	−27.9411	−1.26613	−0.633067	−	0.774097i	$-0.718204\pi$
$487$	−27.9411	−1.26613	−0.633067	+	0.774097i	$0.718204\pi$
$488$	20.2843	0.918226
$489$	2.82843	0.127906
$490$	−12.0000	−0.542105
$491$	−23.8995	−1.07857	−0.539285	−	0.842124i	$-0.681305\pi$
$491$	−23.8995	−1.07857	−0.539285	+	0.842124i	$0.681305\pi$
$492$	0	0
$493$	−5.45584	−0.245719
$494$	7.71573	0.347147
$495$	1.17157	0.0526583
$496$	−41.9411	−1.88321
$497$	−6.00000	−0.269137
$498$	34.2843	1.53631
$499$	−36.6569	−1.64099	−0.820493	−	0.571656i	$-0.806301\pi$
$499$	−36.6569	−1.64099	−0.820493	+	0.571656i	$0.806301\pi$
$500$	0	0
$501$	30.9706	1.38366
$502$	20.4853	0.914303
$503$	7.97056	0.355390	0.177695	−	0.984086i	$-0.443136\pi$
$503$	7.97056	0.355390	0.177695	+	0.984086i	$0.443136\pi$
$504$	2.82843	0.125988
$505$	−13.8995	−0.618519
$506$	10.3431	0.459809
$507$	16.4437	0.730288
$508$	0	0
$509$	−26.4853	−1.17394	−0.586970	−	0.809609i	$-0.699679\pi$
$509$	−26.4853	−1.17394	−0.586970	+	0.809609i	$0.699679\pi$
$510$	−13.1716	−0.583247
$511$	3.00000	0.132712
$512$	22.6274	1.00000
$513$	−26.3431	−1.16308
$514$	−24.0000	−1.05859
$515$	19.0711	0.840372
$516$	0	0
$517$	−2.54416	−0.111892
$518$	−5.89949	−0.259209
$519$	32.9706	1.44725
$520$	−4.68629	−0.205507
$521$	22.5858	0.989501	0.494751	−	0.869035i	$-0.335259\pi$
$521$	22.5858	0.989501	0.494751	+	0.869035i	$0.335259\pi$
$522$	−1.65685	−0.0725185
$523$	1.02944	0.0450141	0.0225071	−	0.999747i	$-0.492835\pi$
$523$	1.02944	0.0450141	0.0225071	+	0.999747i	$0.492835\pi$
$524$	0	0
$525$	−4.24264	−0.185164
$526$	14.5858	0.635971
$527$	48.8284	2.12700
$528$	4.68629	0.203945
$529$	54.9411	2.38874
$530$	18.6274	0.809123
$531$	8.82843	0.383121
$532$	0	0
$533$	6.05887	0.262439
$534$	−23.6569	−1.02373
$535$	−12.0000	−0.518805
$536$	35.7990	1.54628
$537$	−13.6569	−0.589337
$538$	41.4558	1.78729
$539$	−4.97056	−0.214097
$540$	0	0
$541$	0.485281	0.0208639	0.0104319	−	0.999946i	$-0.496679\pi$
$541$	0.485281	0.0208639	0.0104319	+	0.999946i	$0.496679\pi$
$542$	−40.5269	−1.74078
$543$	4.68629	0.201108
$544$	0	0
$545$	14.8284	0.635180
$546$	−2.34315	−0.100277
$547$	22.4853	0.961401	0.480701	−	0.876885i	$-0.340382\pi$
$547$	22.4853	0.961401	0.480701	+	0.876885i	$0.340382\pi$
$548$	0	0
$549$	−7.17157	−0.306075
$550$	3.51472	0.149868
$551$	5.45584	0.232427
$552$	35.3137	1.50305
$553$	−11.4853	−0.488404
$554$	−3.51472	−0.149326
$555$	−8.34315	−0.354147
$556$	0	0
$557$	−0.171573	−0.00726978	−0.00363489	−	0.999993i	$-0.501157\pi$
$557$	−0.171573	−0.00726978	−0.00363489	+	0.999993i	$0.501157\pi$
$558$	14.8284	0.627737
$559$	0	0
$560$	−5.65685	−0.239046
$561$	−5.45584	−0.230346
$562$	−13.8995	−0.586315
$563$	−9.89949	−0.417214	−0.208607	−	0.978000i	$-0.566893\pi$
$563$	−9.89949	−0.417214	−0.208607	+	0.978000i	$0.566893\pi$
$564$	0	0
$565$	−21.3137	−0.896674
$566$	18.1421	0.762571
$567$	5.00000	0.209980
$568$	16.9706	0.712069
$569$	16.1127	0.675479	0.337740	−	0.941240i	$-0.390338\pi$
$569$	16.1127	0.675479	0.337740	+	0.941240i	$0.390338\pi$
$570$	13.1716	0.551696
$571$	−16.8284	−0.704248	−0.352124	−	0.935953i	$-0.614540\pi$
$571$	−16.8284	−0.704248	−0.352124	+	0.935953i	$0.614540\pi$
$572$	0	0
$573$	−1.89949	−0.0793525
$574$	7.31371	0.305268
$575$	26.4853	1.10451
$576$	−8.00000	−0.333333
$577$	19.9706	0.831385	0.415693	−	0.909505i	$-0.363539\pi$
$577$	19.9706	0.831385	0.415693	+	0.909505i	$0.363539\pi$
$578$	−6.62742	−0.275664
$579$	−14.5858	−0.606165
$580$	0	0
$581$	−17.1421	−0.711176
$582$	3.51472	0.145690
$583$	7.71573	0.319553
$584$	−8.48528	−0.351123
$585$	1.65685	0.0685025
$586$	12.2843	0.507459
$587$	−32.8701	−1.35669	−0.678346	−	0.734742i	$-0.737303\pi$
$587$	−32.8701	−1.35669	−0.678346	+	0.734742i	$0.737303\pi$
$588$	0	0
$589$	−48.8284	−2.01194
$590$	−17.6569	−0.726921
$591$	3.75736	0.154557
$592$	16.6863	0.685802
$593$	−28.4558	−1.16854	−0.584271	−	0.811559i	$-0.698619\pi$
$593$	−28.4558	−1.16854	−0.584271	+	0.811559i	$0.698619\pi$
$594$	−6.62742	−0.271926
$595$	6.58579	0.269991
$596$	0	0
$597$	−5.79899	−0.237337
$598$	14.6274	0.598160
$599$	15.6569	0.639722	0.319861	−	0.947465i	$-0.396364\pi$
$599$	15.6569	0.639722	0.319861	+	0.947465i	$0.396364\pi$
$600$	12.0000	0.489898
$601$	−2.68629	−0.109576	−0.0547881	−	0.998498i	$-0.517448\pi$
$601$	−2.68629	−0.109576	−0.0547881	+	0.998498i	$0.517448\pi$
$602$	0	0
$603$	−12.6569	−0.515427
$604$	0	0
$605$	14.5858	0.592997
$606$	19.6569	0.798505
$607$	−14.7279	−0.597788	−0.298894	−	0.954286i	$-0.596618\pi$
$607$	−14.7279	−0.597788	−0.298894	+	0.954286i	$0.596618\pi$
$608$	0	0
$609$	−1.65685	−0.0671391
$610$	14.3431	0.580737
$611$	−3.59798	−0.145559
$612$	0	0
$613$	−31.2132	−1.26069	−0.630345	−	0.776315i	$-0.717086\pi$
$613$	−31.2132	−1.26069	−0.630345	+	0.776315i	$0.717086\pi$
$614$	35.7990	1.44473
$615$	10.3431	0.417076
$616$	−2.34315	−0.0944080
$617$	−16.6863	−0.671765	−0.335882	−	0.941904i	$-0.609034\pi$
$617$	−16.6863	−0.671765	−0.335882	+	0.941904i	$0.609034\pi$
$618$	−26.9706	−1.08492
$619$	43.5563	1.75068	0.875339	−	0.483510i	$-0.160638\pi$
$619$	43.5563	1.75068	0.875339	+	0.483510i	$0.160638\pi$
$620$	0	0
$621$	−49.9411	−2.00407
$622$	19.7574	0.792198
$623$	11.8284	0.473896
$624$	6.62742	0.265309
$625$	−1.00000	−0.0400000
$626$	−4.92893	−0.197000
$627$	5.45584	0.217885
$628$	0	0
$629$	−19.4264	−0.774582
$630$	2.00000	0.0796819
$631$	−6.79899	−0.270664	−0.135332	−	0.990800i	$-0.543210\pi$
$631$	−6.79899	−0.270664	−0.135332	+	0.990800i	$0.543210\pi$
$632$	32.4853	1.29220
$633$	−18.8284	−0.748363
$634$	−2.48528	−0.0987031
$635$	−26.5858	−1.05502
$636$	0	0
$637$	−7.02944	−0.278516
$638$	1.37258	0.0543411
$639$	−6.00000	−0.237356
$640$	16.0000	0.632456
$641$	−32.3137	−1.27631	−0.638157	−	0.769906i	$-0.720303\pi$
$641$	−32.3137	−1.27631	−0.638157	+	0.769906i	$0.720303\pi$
$642$	16.9706	0.669775
$643$	−8.62742	−0.340232	−0.170116	−	0.985424i	$-0.554414\pi$
$643$	−8.62742	−0.340232	−0.170116	+	0.985424i	$0.554414\pi$
$644$	0	0
$645$	0	0
$646$	30.6690	1.20666
$647$	31.7990	1.25015	0.625074	−	0.780566i	$-0.285069\pi$
$647$	31.7990	1.25015	0.625074	+	0.780566i	$0.285069\pi$
$648$	−14.1421	−0.555556
$649$	−7.31371	−0.287088
$650$	4.97056	0.194962
$651$	14.8284	0.581172
$652$	0	0
$653$	−22.6274	−0.885479	−0.442740	−	0.896650i	$-0.645993\pi$
$653$	−22.6274	−0.885479	−0.442740	+	0.896650i	$0.645993\pi$
$654$	−20.9706	−0.820014
$655$	−0.727922	−0.0284423
$656$	−20.6863	−0.807664
$657$	3.00000	0.117041
$658$	−4.34315	−0.169313
$659$	8.82843	0.343907	0.171953	−	0.985105i	$-0.444992\pi$
$659$	8.82843	0.343907	0.171953	+	0.985105i	$0.444992\pi$
$660$	0	0
$661$	−16.0000	−0.622328	−0.311164	−	0.950356i	$-0.600719\pi$
$661$	−16.0000	−0.622328	−0.311164	+	0.950356i	$0.600719\pi$
$662$	3.51472	0.136603
$663$	−7.71573	−0.299654
$664$	48.4853	1.88159
$665$	−6.58579	−0.255386
$666$	−5.89949	−0.228601
$667$	10.3431	0.400488
$668$	0	0
$669$	−37.7990	−1.46139
$670$	25.3137	0.977954
$671$	5.94113	0.229355
$672$	0	0
$673$	−25.6985	−0.990604	−0.495302	−	0.868721i	$-0.664943\pi$
$673$	−25.6985	−0.990604	−0.495302	+	0.868721i	$0.664943\pi$
$674$	10.3431	0.398403
$675$	−16.9706	−0.653197
$676$	0	0
$677$	14.6569	0.563309	0.281654	−	0.959516i	$-0.409117\pi$
$677$	14.6569	0.563309	0.281654	+	0.959516i	$0.409117\pi$
$678$	30.1421	1.15760
$679$	−1.75736	−0.0674413
$680$	−18.6274	−0.714329
$681$	26.2843	1.00722
$682$	−12.2843	−0.470389
$683$	12.0000	0.459167	0.229584	−	0.973289i	$-0.426264\pi$
$683$	12.0000	0.459167	0.229584	+	0.973289i	$0.426264\pi$
$684$	0	0
$685$	4.24264	0.162103
$686$	−18.3848	−0.701934
$687$	−37.4558	−1.42903
$688$	0	0
$689$	10.9117	0.415702
$690$	24.9706	0.950613
$691$	43.9706	1.67272	0.836360	−	0.548181i	$-0.184680\pi$
$691$	43.9706	1.67272	0.836360	+	0.548181i	$0.184680\pi$
$692$	0	0
$693$	0.828427	0.0314693
$694$	35.7574	1.35733
$695$	6.14214	0.232984
$696$	4.68629	0.177633
$697$	24.0833	0.912219
$698$	−18.8701	−0.714242
$699$	16.4853	0.623531
$700$	0	0
$701$	11.6274	0.439161	0.219581	−	0.975594i	$-0.429531\pi$
$701$	11.6274	0.439161	0.219581	+	0.975594i	$0.429531\pi$
$702$	−9.37258	−0.353745
$703$	19.4264	0.732681
$704$	6.62742	0.249780
$705$	−6.14214	−0.231326
$706$	37.6985	1.41880
$707$	−9.82843	−0.369636
$708$	0	0
$709$	−11.9706	−0.449564	−0.224782	−	0.974409i	$-0.572167\pi$
$709$	−11.9706	−0.449564	−0.224782	+	0.974409i	$0.572167\pi$
$710$	12.0000	0.450352
$711$	−11.4853	−0.430732
$712$	−33.4558	−1.25381
$713$	−92.5685	−3.46672
$714$	−9.31371	−0.348557
$715$	−1.37258	−0.0513317
$716$	0	0
$717$	3.75736	0.140321
$718$	8.34315	0.311363
$719$	10.0294	0.374035	0.187017	−	0.982357i	$-0.440118\pi$
$719$	10.0294	0.374035	0.187017	+	0.982357i	$0.440118\pi$
$720$	−5.65685	−0.210819
$721$	13.4853	0.502218
$722$	−3.79899	−0.141384
$723$	7.75736	0.288499
$724$	0	0
$725$	3.51472	0.130533
$726$	−20.6274	−0.765555
$727$	−8.72792	−0.323701	−0.161850	−	0.986815i	$-0.551746\pi$
$727$	−8.72792	−0.323701	−0.161850	+	0.986815i	$0.551746\pi$
$728$	−3.31371	−0.122814
$729$	29.0000	1.07407
$730$	−6.00000	−0.222070
$731$	0	0
$732$	0	0
$733$	−48.8284	−1.80352	−0.901760	−	0.432238i	$-0.857724\pi$
$733$	−48.8284	−1.80352	−0.901760	+	0.432238i	$0.857724\pi$
$734$	−1.17157	−0.0432435
$735$	−12.0000	−0.442627
$736$	0	0
$737$	10.4853	0.386230
$738$	7.31371	0.269221
$739$	2.17157	0.0798826	0.0399413	−	0.999202i	$-0.487283\pi$
$739$	2.17157	0.0798826	0.0399413	+	0.999202i	$0.487283\pi$
$740$	0	0
$741$	7.71573	0.283444
$742$	13.1716	0.483544
$743$	−39.4142	−1.44597	−0.722984	−	0.690865i	$-0.757230\pi$
$743$	−39.4142	−1.44597	−0.722984	+	0.690865i	$0.757230\pi$
$744$	−41.9411	−1.53764
$745$	−10.1005	−0.370054
$746$	−0.686292	−0.0251269
$747$	−17.1421	−0.627198
$748$	0	0
$749$	−8.48528	−0.310045
$750$	22.6274	0.826236
$751$	30.5147	1.11350	0.556749	−	0.830681i	$-0.312049\pi$
$751$	30.5147	1.11350	0.556749	+	0.830681i	$0.312049\pi$
$752$	12.2843	0.447961
$753$	20.4853	0.746525
$754$	1.94113	0.0706916
$755$	11.5563	0.420579
$756$	0	0
$757$	48.7990	1.77363	0.886815	−	0.462125i	$-0.152913\pi$
$757$	48.7990	1.77363	0.886815	+	0.462125i	$0.152913\pi$
$758$	−32.8284	−1.19238
$759$	10.3431	0.375432
$760$	18.6274	0.675687
$761$	22.9706	0.832682	0.416341	−	0.909208i	$-0.363312\pi$
$761$	22.9706	0.832682	0.416341	+	0.909208i	$0.363312\pi$
$762$	37.5980	1.36203
$763$	10.4853	0.379593
$764$	0	0
$765$	6.58579	0.238110
$766$	33.3137	1.20367
$767$	−10.3431	−0.373469
$768$	0	0
$769$	44.1838	1.59331	0.796654	−	0.604436i	$-0.206601\pi$
$769$	44.1838	1.59331	0.796654	+	0.604436i	$0.206601\pi$
$770$	−1.65685	−0.0597089
$771$	−24.0000	−0.864339
$772$	0	0
$773$	−48.3848	−1.74028	−0.870140	−	0.492805i	$-0.835972\pi$
$773$	−48.3848	−1.74028	−0.870140	+	0.492805i	$0.835972\pi$
$774$	0	0
$775$	−31.4558	−1.12993
$776$	4.97056	0.178433
$777$	−5.89949	−0.211643
$778$	48.9706	1.75568
$779$	−24.0833	−0.862872
$780$	0	0
$781$	4.97056	0.177861
$782$	58.1421	2.07916
$783$	−6.62742	−0.236845
$784$	24.0000	0.857143
$785$	21.6569	0.772966
$786$	1.02944	0.0367188
$787$	−38.9411	−1.38810	−0.694051	−	0.719926i	$-0.744176\pi$
$787$	−38.9411	−1.38810	−0.694051	+	0.719926i	$0.744176\pi$
$788$	0	0
$789$	14.5858	0.519268
$790$	22.9706	0.817256
$791$	−15.0711	−0.535865
$792$	−2.34315	−0.0832601
$793$	8.40202	0.298364
$794$	21.2132	0.752828
$795$	18.6274	0.660646
$796$	0	0
$797$	−45.0711	−1.59650	−0.798250	−	0.602327i	$-0.794240\pi$
$797$	−45.0711	−1.59650	−0.798250	+	0.602327i	$0.794240\pi$
$798$	9.31371	0.329702
$799$	−14.3015	−0.505951
$800$	0	0
$801$	11.8284	0.417937
$802$	36.0000	1.27120
$803$	−2.48528	−0.0877037
$804$	0	0
$805$	−12.4853	−0.440048
$806$	−17.3726	−0.611923
$807$	41.4558	1.45931
$808$	27.7990	0.977965
$809$	34.9706	1.22950	0.614750	−	0.788722i	$-0.289257\pi$
$809$	34.9706	1.22950	0.614750	+	0.788722i	$0.289257\pi$
$810$	−10.0000	−0.351364
$811$	22.8284	0.801614	0.400807	−	0.916162i	$-0.368730\pi$
$811$	22.8284	0.801614	0.400807	+	0.916162i	$0.368730\pi$
$812$	0	0
$813$	−40.5269	−1.42134
$814$	4.88730	0.171300
$815$	2.82843	0.0990755
$816$	26.3431	0.922195
$817$	0	0
$818$	10.6863	0.373637
$819$	1.17157	0.0409381
$820$	0	0
$821$	24.5858	0.858050	0.429025	−	0.903293i	$-0.358857\pi$
$821$	24.5858	0.858050	0.429025	+	0.903293i	$0.358857\pi$
$822$	−6.00000	−0.209274
$823$	38.7279	1.34997	0.674985	−	0.737831i	$-0.264150\pi$
$823$	38.7279	1.34997	0.674985	+	0.737831i	$0.264150\pi$
$824$	−38.1421	−1.32874
$825$	3.51472	0.122367
$826$	−12.4853	−0.434418
$827$	−5.48528	−0.190742	−0.0953710	−	0.995442i	$-0.530404\pi$
$827$	−5.48528	−0.190742	−0.0953710	+	0.995442i	$0.530404\pi$
$828$	0	0
$829$	2.72792	0.0947446	0.0473723	−	0.998877i	$-0.484915\pi$
$829$	2.72792	0.0947446	0.0473723	+	0.998877i	$0.484915\pi$
$830$	34.2843	1.19002
$831$	−3.51472	−0.121924
$832$	9.37258	0.324936
$833$	−27.9411	−0.968103
$834$	−8.68629	−0.300782
$835$	30.9706	1.07178
$836$	0	0
$837$	59.3137	2.05018
$838$	12.2843	0.424353
$839$	−45.7990	−1.58116	−0.790578	−	0.612361i	$-0.790220\pi$
$839$	−45.7990	−1.58116	−0.790578	+	0.612361i	$0.790220\pi$
$840$	−5.65685	−0.195180
$841$	−27.6274	−0.952670
$842$	−13.4558	−0.463719
$843$	−13.8995	−0.478724
$844$	0	0
$845$	16.4437	0.565679
$846$	−4.34315	−0.149320
$847$	10.3137	0.354383
$848$	−37.2548	−1.27934
$849$	18.1421	0.622636
$850$	19.7574	0.677672
$851$	36.8284	1.26246
$852$	0	0
$853$	−31.3137	−1.07216	−0.536080	−	0.844167i	$-0.680096\pi$
$853$	−31.3137	−1.07216	−0.536080	+	0.844167i	$0.680096\pi$
$854$	10.1421	0.347057
$855$	−6.58579	−0.225229
$856$	24.0000	0.820303
$857$	−26.6569	−0.910581	−0.455290	−	0.890343i	$-0.650465\pi$
$857$	−26.6569	−0.910581	−0.455290	+	0.890343i	$0.650465\pi$
$858$	1.94113	0.0662689
$859$	38.7696	1.32280	0.661400	−	0.750033i	$-0.269963\pi$
$859$	38.7696	1.32280	0.661400	+	0.750033i	$0.269963\pi$
$860$	0	0
$861$	7.31371	0.249251
$862$	25.4558	0.867029
$863$	−5.11270	−0.174038	−0.0870191	−	0.996207i	$-0.527734\pi$
$863$	−5.11270	−0.174038	−0.0870191	+	0.996207i	$0.527734\pi$
$864$	0	0
$865$	32.9706	1.12103
$866$	5.69848	0.193642
$867$	−6.62742	−0.225079
$868$	0	0
$869$	9.51472	0.322765
$870$	3.31371	0.112345
$871$	14.8284	0.502442
$872$	−29.6569	−1.00431
$873$	−1.75736	−0.0594776
$874$	−58.1421	−1.96669
$875$	−11.3137	−0.382473
$876$	0	0
$877$	16.6569	0.562462	0.281231	−	0.959640i	$-0.409257\pi$
$877$	16.6569	0.562462	0.281231	+	0.959640i	$0.409257\pi$
$878$	−2.14214	−0.0722936
$879$	12.2843	0.414338
$880$	4.68629	0.157975
$881$	−1.07107	−0.0360852	−0.0180426	−	0.999837i	$-0.505743\pi$
$881$	−1.07107	−0.0360852	−0.0180426	+	0.999837i	$0.505743\pi$
$882$	−8.48528	−0.285714
$883$	9.45584	0.318214	0.159107	−	0.987261i	$-0.449138\pi$
$883$	9.45584	0.318214	0.159107	+	0.987261i	$0.449138\pi$
$884$	0	0
$885$	−17.6569	−0.593529
$886$	1.45584	0.0489100
$887$	−17.3431	−0.582326	−0.291163	−	0.956673i	$-0.594042\pi$
$887$	−17.3431	−0.582326	−0.291163	+	0.956673i	$0.594042\pi$
$888$	16.6863	0.559955
$889$	−18.7990	−0.630498
$890$	−23.6569	−0.792980
$891$	−4.14214	−0.138767
$892$	0	0
$893$	14.3015	0.478582
$894$	14.2843	0.477737
$895$	−13.6569	−0.456498
$896$	11.3137	0.377964
$897$	14.6274	0.488395
$898$	−11.3726	−0.379508
$899$	−12.2843	−0.409703
$900$	0	0
$901$	43.3726	1.44495
$902$	−6.05887	−0.201738
$903$	0	0
$904$	42.6274	1.41777
$905$	4.68629	0.155778
$906$	−16.3431	−0.542965
$907$	36.5269	1.21286	0.606428	−	0.795138i	$-0.292602\pi$
$907$	36.5269	1.21286	0.606428	+	0.795138i	$0.292602\pi$
$908$	0	0
$909$	−9.82843	−0.325988
$910$	−2.34315	−0.0776745
$911$	−29.6863	−0.983551	−0.491775	−	0.870722i	$-0.663652\pi$
$911$	−29.6863	−0.983551	−0.491775	+	0.870722i	$0.663652\pi$
$912$	−26.3431	−0.872309
$913$	14.2010	0.469985
$914$	−29.1716	−0.964910
$915$	14.3431	0.474170
$916$	0	0
$917$	−0.514719	−0.0169975
$918$	−37.2548	−1.22959
$919$	−6.78680	−0.223876	−0.111938	−	0.993715i	$-0.535706\pi$
$919$	−6.78680	−0.223876	−0.111938	+	0.993715i	$0.535706\pi$
$920$	35.3137	1.16426
$921$	35.7990	1.17962
$922$	33.4558	1.10181
$923$	7.02944	0.231377
$924$	0	0
$925$	12.5147	0.411481
$926$	−21.5147	−0.707018
$927$	13.4853	0.442915
$928$	0	0
$929$	−3.00000	−0.0984268	−0.0492134	−	0.998788i	$-0.515671\pi$
$929$	−3.00000	−0.0984268	−0.0492134	+	0.998788i	$0.515671\pi$
$930$	−29.6569	−0.972487
$931$	27.9411	0.915733
$932$	0	0
$933$	19.7574	0.646827
$934$	−13.4558	−0.440289
$935$	−5.45584	−0.178425
$936$	−3.31371	−0.108312
$937$	−48.4853	−1.58395	−0.791973	−	0.610557i	$-0.790946\pi$
$937$	−48.4853	−1.58395	−0.791973	+	0.610557i	$0.790946\pi$
$938$	17.8995	0.584439
$939$	−4.92893	−0.160850
$940$	0	0
$941$	36.2843	1.18283	0.591417	−	0.806366i	$-0.298569\pi$
$941$	36.2843	1.18283	0.591417	+	0.806366i	$0.298569\pi$
$942$	−30.6274	−0.997895
$943$	−45.6569	−1.48679
$944$	35.3137	1.14936
$945$	8.00000	0.260240
$946$	0	0
$947$	20.1838	0.655884	0.327942	−	0.944698i	$-0.393645\pi$
$947$	20.1838	0.655884	0.327942	+	0.944698i	$0.393645\pi$
$948$	0	0
$949$	−3.51472	−0.114093
$950$	−19.7574	−0.641013
$951$	−2.48528	−0.0805908
$952$	−13.1716	−0.426893
$953$	24.0416	0.778785	0.389392	−	0.921072i	$-0.372685\pi$
$953$	24.0416	0.778785	0.389392	+	0.921072i	$0.372685\pi$
$954$	13.1716	0.426445
$955$	−1.89949	−0.0614662
$956$	0	0
$957$	1.37258	0.0443693
$958$	−48.7279	−1.57433
$959$	3.00000	0.0968751
$960$	16.0000	0.516398
$961$	78.9411	2.54649
$962$	6.91169	0.222842
$963$	−8.48528	−0.273434
$964$	0	0
$965$	−14.5858	−0.469533
$966$	17.6569	0.568100
$967$	4.72792	0.152040	0.0760199	−	0.997106i	$-0.475779\pi$
$967$	4.72792	0.152040	0.0760199	+	0.997106i	$0.475779\pi$
$968$	−29.1716	−0.937610
$969$	30.6690	0.985232
$970$	3.51472	0.112851
$971$	20.6274	0.661965	0.330983	−	0.943637i	$-0.392620\pi$
$971$	20.6274	0.661965	0.330983	+	0.943637i	$0.392620\pi$
$972$	0	0
$973$	4.34315	0.139235
$974$	39.5147	1.26613
$975$	4.97056	0.159185
$976$	−28.6863	−0.918226
$977$	−19.0294	−0.608806	−0.304403	−	0.952543i	$-0.598457\pi$
$977$	−19.0294	−0.608806	−0.304403	+	0.952543i	$0.598457\pi$
$978$	−4.00000	−0.127906
$979$	−9.79899	−0.313177
$980$	0	0
$981$	10.4853	0.334769
$982$	33.7990	1.07857
$983$	−12.5147	−0.399158	−0.199579	−	0.979882i	$-0.563957\pi$
$983$	−12.5147	−0.399158	−0.199579	+	0.979882i	$0.563957\pi$
$984$	−20.6863	−0.659455
$985$	3.75736	0.119719
$986$	7.71573	0.245719
$987$	−4.34315	−0.138244
$988$	0	0
$989$	0	0
$990$	−1.65685	−0.0526583
$991$	−20.4558	−0.649801	−0.324901	−	0.945748i	$-0.605331\pi$
$991$	−20.4558	−0.649801	−0.324901	+	0.945748i	$0.605331\pi$
$992$	0	0
$993$	3.51472	0.111536
$994$	8.48528	0.269137
$995$	−5.79899	−0.183840
$996$	0	0
$997$	−4.38478	−0.138867	−0.0694336	−	0.997587i	$-0.522119\pi$
$997$	−4.38478	−0.138867	−0.0694336	+	0.997587i	$0.522119\pi$
$998$	51.8406	1.64099
$999$	−23.5980	−0.746607

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4003.2.a.a.1.1	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4003.2.a.a.1.1	✓	2	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:	\( N \)	\(=\)	\( 4003 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4003.a (trivial)

\(f(q)\)	\(=\)	\(q-1.41421 q^{2} -1.41421 q^{3} -1.41421 q^{5} +2.00000 q^{6} -1.00000 q^{7} +2.82843 q^{8} -1.00000 q^{9} +O(q^{10})\)	\(q-1.41421 q^{2} -1.41421 q^{3} -1.41421 q^{5} +2.00000 q^{6} -1.00000 q^{7} +2.82843 q^{8} -1.00000 q^{9} +2.00000 q^{10} +0.828427 q^{11} +1.17157 q^{13} +1.41421 q^{14} +2.00000 q^{15} -4.00000 q^{16} +4.65685 q^{17} +1.41421 q^{18} -4.65685 q^{19} +1.41421 q^{21} -1.17157 q^{22} -8.82843 q^{23} -4.00000 q^{24} -3.00000 q^{25} -1.65685 q^{26} +5.65685 q^{27} -1.17157 q^{29} -2.82843 q^{30} +10.4853 q^{31} -1.17157 q^{33} -6.58579 q^{34} +1.41421 q^{35} -4.17157 q^{37} +6.58579 q^{38} -1.65685 q^{39} -4.00000 q^{40} +5.17157 q^{41} -2.00000 q^{42} +1.41421 q^{45} +12.4853 q^{46} -3.07107 q^{47} +5.65685 q^{48} -6.00000 q^{49} +4.24264 q^{50} -6.58579 q^{51} +9.31371 q^{53} -8.00000 q^{54} -1.17157 q^{55} -2.82843 q^{56} +6.58579 q^{57} +1.65685 q^{58} -8.82843 q^{59} +7.17157 q^{61} -14.8284 q^{62} +1.00000 q^{63} +8.00000 q^{64} -1.65685 q^{65} +1.65685 q^{66} +12.6569 q^{67} +12.4853 q^{69} -2.00000 q^{70} +6.00000 q^{71} -2.82843 q^{72} -3.00000 q^{73} +5.89949 q^{74} +4.24264 q^{75} -0.828427 q^{77} +2.34315 q^{78} +11.4853 q^{79} +5.65685 q^{80} -5.00000 q^{81} -7.31371 q^{82} +17.1421 q^{83} -6.58579 q^{85} +1.65685 q^{87} +2.34315 q^{88} -11.8284 q^{89} -2.00000 q^{90} -1.17157 q^{91} -14.8284 q^{93} +4.34315 q^{94} +6.58579 q^{95} +1.75736 q^{97} +8.48528 q^{98} -0.828427 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 4 q^{6} - 2 q^{7} - 2 q^{9}+O(q^{10}) \) 2 * q + 4 * q^6 - 2 * q^7 - 2 * q^9	\( 2 q + 4 q^{6} - 2 q^{7} - 2 q^{9} + 4 q^{10} - 4 q^{11} + 8 q^{13} + 4 q^{15} - 8 q^{16} - 2 q^{17} + 2 q^{19} - 8 q^{22} - 12 q^{23} - 8 q^{24} - 6 q^{25} + 8 q^{26} - 8 q^{29} + 4 q^{31} - 8 q^{33} - 16 q^{34} - 14 q^{37} + 16 q^{38} + 8 q^{39} - 8 q^{40} + 16 q^{41} - 4 q^{42} + 8 q^{46} + 8 q^{47} - 12 q^{49} - 16 q^{51} - 4 q^{53} - 16 q^{54} - 8 q^{55} + 16 q^{57} - 8 q^{58} - 12 q^{59} + 20 q^{61} - 24 q^{62} + 2 q^{63} + 16 q^{64} + 8 q^{65} - 8 q^{66} + 14 q^{67} + 8 q^{69} - 4 q^{70} + 12 q^{71} - 6 q^{73} - 8 q^{74} + 4 q^{77} + 16 q^{78} + 6 q^{79} - 10 q^{81} + 8 q^{82} + 6 q^{83} - 16 q^{85} - 8 q^{87} + 16 q^{88} - 18 q^{89} - 4 q^{90} - 8 q^{91} - 24 q^{93} + 20 q^{94} + 16 q^{95} + 12 q^{97} + 4 q^{99}+O(q^{100}) \) 2 * q + 4 * q^6 - 2 * q^7 - 2 * q^9 + 4 * q^10 - 4 * q^11 + 8 * q^13 + 4 * q^15 - 8 * q^16 - 2 * q^17 + 2 * q^19 - 8 * q^22 - 12 * q^23 - 8 * q^24 - 6 * q^25 + 8 * q^26 - 8 * q^29 + 4 * q^31 - 8 * q^33 - 16 * q^34 - 14 * q^37 + 16 * q^38 + 8 * q^39 - 8 * q^40 + 16 * q^41 - 4 * q^42 + 8 * q^46 + 8 * q^47 - 12 * q^49 - 16 * q^51 - 4 * q^53 - 16 * q^54 - 8 * q^55 + 16 * q^57 - 8 * q^58 - 12 * q^59 + 20 * q^61 - 24 * q^62 + 2 * q^63 + 16 * q^64 + 8 * q^65 - 8 * q^66 + 14 * q^67 + 8 * q^69 - 4 * q^70 + 12 * q^71 - 6 * q^73 - 8 * q^74 + 4 * q^77 + 16 * q^78 + 6 * q^79 - 10 * q^81 + 8 * q^82 + 6 * q^83 - 16 * q^85 - 8 * q^87 + 16 * q^88 - 18 * q^89 - 4 * q^90 - 8 * q^91 - 24 * q^93 + 20 * q^94 + 16 * q^95 + 12 * q^97 + 4 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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