LMFDB - Embedded newform 6003.2.a.f.1.2

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:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 2001)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.618034$ 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)

$f(q)$	$=$	$q+1.00000 q^{2} -1.00000 q^{4} +2.00000 q^{5} -3.00000 q^{8} +O(q^{10})$	$q+1.00000 q^{2} -1.00000 q^{4} +2.00000 q^{5} -3.00000 q^{8} +2.00000 q^{10} +6.47214 q^{11} -2.00000 q^{13} -1.00000 q^{16} -4.47214 q^{17} -6.47214 q^{19} -2.00000 q^{20} +6.47214 q^{22} +1.00000 q^{23} -1.00000 q^{25} -2.00000 q^{26} -1.00000 q^{29} -8.00000 q^{31} +5.00000 q^{32} -4.47214 q^{34} -4.47214 q^{37} -6.47214 q^{38} -6.00000 q^{40} +10.9443 q^{41} +6.47214 q^{43} -6.47214 q^{44} +1.00000 q^{46} -12.9443 q^{47} -7.00000 q^{49} -1.00000 q^{50} +2.00000 q^{52} +2.00000 q^{53} +12.9443 q^{55} -1.00000 q^{58} +4.00000 q^{59} -12.4721 q^{61} -8.00000 q^{62} +7.00000 q^{64} -4.00000 q^{65} -4.00000 q^{67} +4.47214 q^{68} -8.00000 q^{71} -10.9443 q^{73} -4.47214 q^{74} +6.47214 q^{76} +10.4721 q^{79} -2.00000 q^{80} +10.9443 q^{82} -4.00000 q^{83} -8.94427 q^{85} +6.47214 q^{86} -19.4164 q^{88} +13.4164 q^{89} -1.00000 q^{92} -12.9443 q^{94} -12.9443 q^{95} +17.4164 q^{97} -7.00000 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} - 2 q^{4} + 4 q^{5} - 6 q^{8}+O(q^{10}) $ 2 * q + 2 * q^2 - 2 * q^4 + 4 * q^5 - 6 * q^8	$ 2 q + 2 q^{2} - 2 q^{4} + 4 q^{5} - 6 q^{8} + 4 q^{10} + 4 q^{11} - 4 q^{13} - 2 q^{16} - 4 q^{19} - 4 q^{20} + 4 q^{22} + 2 q^{23} - 2 q^{25} - 4 q^{26} - 2 q^{29} - 16 q^{31} + 10 q^{32} - 4 q^{38} - 12 q^{40} + 4 q^{41} + 4 q^{43} - 4 q^{44} + 2 q^{46} - 8 q^{47} - 14 q^{49} - 2 q^{50} + 4 q^{52} + 4 q^{53} + 8 q^{55} - 2 q^{58} + 8 q^{59} - 16 q^{61} - 16 q^{62} + 14 q^{64} - 8 q^{65} - 8 q^{67} - 16 q^{71} - 4 q^{73} + 4 q^{76} + 12 q^{79} - 4 q^{80} + 4 q^{82} - 8 q^{83} + 4 q^{86} - 12 q^{88} - 2 q^{92} - 8 q^{94} - 8 q^{95} + 8 q^{97} - 14 q^{98}+O(q^{100}) $ 2 * q + 2 * q^2 - 2 * q^4 + 4 * q^5 - 6 * q^8 + 4 * q^10 + 4 * q^11 - 4 * q^13 - 2 * q^16 - 4 * q^19 - 4 * q^20 + 4 * q^22 + 2 * q^23 - 2 * q^25 - 4 * q^26 - 2 * q^29 - 16 * q^31 + 10 * q^32 - 4 * q^38 - 12 * q^40 + 4 * q^41 + 4 * q^43 - 4 * q^44 + 2 * q^46 - 8 * q^47 - 14 * q^49 - 2 * q^50 + 4 * q^52 + 4 * q^53 + 8 * q^55 - 2 * q^58 + 8 * q^59 - 16 * q^61 - 16 * q^62 + 14 * q^64 - 8 * q^65 - 8 * q^67 - 16 * q^71 - 4 * q^73 + 4 * q^76 + 12 * q^79 - 4 * q^80 + 4 * q^82 - 8 * q^83 + 4 * q^86 - 12 * q^88 - 2 * q^92 - 8 * q^94 - 8 * q^95 + 8 * q^97 - 14 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	1.00000	0.707107	0.353553	−	0.935414i	$-0.384973\pi$
$2$	1.00000	0.707107	0.353553	+	0.935414i	$0.384973\pi$
$3$	0	0
$3$	0	0
$4$	−1.00000	−0.500000
$5$	2.00000	0.894427	0.447214	−	0.894427i	$-0.352416\pi$
$5$	2.00000	0.894427	0.447214	+	0.894427i	$0.352416\pi$
$6$	0	0
$7$	0	0		−	1.00000i	$-0.5\pi$
$7$	0	0			1.00000i	$0.5\pi$
$8$	−3.00000	−1.06066
$9$	0	0
$10$	2.00000	0.632456
$11$	6.47214	1.95142	0.975711	−	0.219061i	$-0.0702993\pi$
$11$	6.47214	1.95142	0.975711	+	0.219061i	$0.0702993\pi$
$12$	0	0
$13$	−2.00000	−0.554700	−0.277350	−	0.960769i	$-0.589456\pi$
$13$	−2.00000	−0.554700	−0.277350	+	0.960769i	$0.589456\pi$
$14$	0	0
$15$	0	0
$16$	−1.00000	−0.250000
$17$	−4.47214	−1.08465	−0.542326	−	0.840168i	$-0.682456\pi$
$17$	−4.47214	−1.08465	−0.542326	+	0.840168i	$0.682456\pi$
$18$	0	0
$19$	−6.47214	−1.48481	−0.742405	−	0.669951i	$-0.766315\pi$
$19$	−6.47214	−1.48481	−0.742405	+	0.669951i	$0.766315\pi$
$20$	−2.00000	−0.447214
$21$	0	0
$22$	6.47214	1.37986
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	−1.00000	−0.200000
$26$	−2.00000	−0.392232
$27$	0	0
$28$	0	0
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	0	0
$31$	−8.00000	−1.43684	−0.718421	−	0.695608i	$-0.755135\pi$
$31$	−8.00000	−1.43684	−0.718421	+	0.695608i	$0.755135\pi$
$32$	5.00000	0.883883
$33$	0	0
$34$	−4.47214	−0.766965
$35$	0	0
$36$	0	0
$37$	−4.47214	−0.735215	−0.367607	−	0.929981i	$-0.619823\pi$
$37$	−4.47214	−0.735215	−0.367607	+	0.929981i	$0.619823\pi$
$38$	−6.47214	−1.04992
$39$	0	0
$40$	−6.00000	−0.948683
$41$	10.9443	1.70921	0.854604	−	0.519280i	$-0.173800\pi$
$41$	10.9443	1.70921	0.854604	+	0.519280i	$0.173800\pi$
$42$	0	0
$43$	6.47214	0.986991	0.493496	−	0.869748i	$-0.335719\pi$
$43$	6.47214	0.986991	0.493496	+	0.869748i	$0.335719\pi$
$44$	−6.47214	−0.975711
$45$	0	0
$46$	1.00000	0.147442
$47$	−12.9443	−1.88812	−0.944058	−	0.329779i	$-0.893026\pi$
$47$	−12.9443	−1.88812	−0.944058	+	0.329779i	$0.893026\pi$
$48$	0	0
$49$	−7.00000	−1.00000
$50$	−1.00000	−0.141421
$51$	0	0
$52$	2.00000	0.277350
$53$	2.00000	0.274721	0.137361	−	0.990521i	$-0.456138\pi$
$53$	2.00000	0.274721	0.137361	+	0.990521i	$0.456138\pi$
$54$	0	0
$55$	12.9443	1.74541
$56$	0	0
$57$	0	0
$58$	−1.00000	−0.131306
$59$	4.00000	0.520756	0.260378	−	0.965507i	$-0.416153\pi$
$59$	4.00000	0.520756	0.260378	+	0.965507i	$0.416153\pi$
$60$	0	0
$61$	−12.4721	−1.59689	−0.798447	−	0.602066i	$-0.794345\pi$
$61$	−12.4721	−1.59689	−0.798447	+	0.602066i	$0.794345\pi$
$62$	−8.00000	−1.01600
$63$	0	0
$64$	7.00000	0.875000
$65$	−4.00000	−0.496139
$66$	0	0
$67$	−4.00000	−0.488678	−0.244339	−	0.969690i	$-0.578571\pi$
$67$	−4.00000	−0.488678	−0.244339	+	0.969690i	$0.578571\pi$
$68$	4.47214	0.542326
$69$	0	0
$70$	0	0
$71$	−8.00000	−0.949425	−0.474713	−	0.880141i	$-0.657448\pi$
$71$	−8.00000	−0.949425	−0.474713	+	0.880141i	$0.657448\pi$
$72$	0	0
$73$	−10.9443	−1.28093	−0.640465	−	0.767987i	$-0.721258\pi$
$73$	−10.9443	−1.28093	−0.640465	+	0.767987i	$0.721258\pi$
$74$	−4.47214	−0.519875
$75$	0	0
$76$	6.47214	0.742405
$77$	0	0
$78$	0	0
$79$	10.4721	1.17821	0.589104	−	0.808057i	$-0.299481\pi$
$79$	10.4721	1.17821	0.589104	+	0.808057i	$0.299481\pi$
$80$	−2.00000	−0.223607
$81$	0	0
$82$	10.9443	1.20859
$83$	−4.00000	−0.439057	−0.219529	−	0.975606i	$-0.570452\pi$
$83$	−4.00000	−0.439057	−0.219529	+	0.975606i	$0.570452\pi$
$84$	0	0
$85$	−8.94427	−0.970143
$86$	6.47214	0.697908
$87$	0	0
$88$	−19.4164	−2.06980
$89$	13.4164	1.42214	0.711068	−	0.703123i	$-0.248212\pi$
$89$	13.4164	1.42214	0.711068	+	0.703123i	$0.248212\pi$
$90$	0	0
$91$	0	0
$92$	−1.00000	−0.104257
$93$	0	0
$94$	−12.9443	−1.33510
$95$	−12.9443	−1.32805
$96$	0	0
$97$	17.4164	1.76837	0.884184	−	0.467139i	$-0.154715\pi$
$97$	17.4164	1.76837	0.884184	+	0.467139i	$0.154715\pi$
$98$	−7.00000	−0.707107
$99$	0	0
$100$	1.00000	0.100000
$101$	−1.05573	−0.105049	−0.0525244	−	0.998620i	$-0.516727\pi$
$101$	−1.05573	−0.105049	−0.0525244	+	0.998620i	$0.516727\pi$
$102$	0	0
$103$	4.94427	0.487174	0.243587	−	0.969879i	$-0.421676\pi$
$103$	4.94427	0.487174	0.243587	+	0.969879i	$0.421676\pi$
$104$	6.00000	0.588348
$105$	0	0
$106$	2.00000	0.194257
$107$	−16.9443	−1.63806	−0.819032	−	0.573747i	$-0.805489\pi$
$107$	−16.9443	−1.63806	−0.819032	+	0.573747i	$0.805489\pi$
$108$	0	0
$109$	−6.94427	−0.665141	−0.332570	−	0.943078i	$-0.607916\pi$
$109$	−6.94427	−0.665141	−0.332570	+	0.943078i	$0.607916\pi$
$110$	12.9443	1.23419
$111$	0	0
$112$	0	0
$113$	0.472136	0.0444148	0.0222074	−	0.999753i	$-0.492931\pi$
$113$	0.472136	0.0444148	0.0222074	+	0.999753i	$0.492931\pi$
$114$	0	0
$115$	2.00000	0.186501
$116$	1.00000	0.0928477
$117$	0	0
$118$	4.00000	0.368230
$119$	0	0
$120$	0	0
$121$	30.8885	2.80805
$122$	−12.4721	−1.12917
$123$	0	0
$124$	8.00000	0.718421
$125$	−12.0000	−1.07331
$126$	0	0
$127$	−17.8885	−1.58735	−0.793676	−	0.608341i	$-0.791835\pi$
$127$	−17.8885	−1.58735	−0.793676	+	0.608341i	$0.791835\pi$
$128$	−3.00000	−0.265165
$129$	0	0
$130$	−4.00000	−0.350823
$131$	−12.0000	−1.04844	−0.524222	−	0.851581i	$-0.675644\pi$
$131$	−12.0000	−1.04844	−0.524222	+	0.851581i	$0.675644\pi$
$132$	0	0
$133$	0	0
$134$	−4.00000	−0.345547
$135$	0	0
$136$	13.4164	1.15045
$137$	−12.4721	−1.06557	−0.532783	−	0.846252i	$-0.678854\pi$
$137$	−12.4721	−1.06557	−0.532783	+	0.846252i	$0.678854\pi$
$138$	0	0
$139$	16.9443	1.43719	0.718597	−	0.695427i	$-0.244785\pi$
$139$	16.9443	1.43719	0.718597	+	0.695427i	$0.244785\pi$
$140$	0	0
$141$	0	0
$142$	−8.00000	−0.671345
$143$	−12.9443	−1.08245
$144$	0	0
$145$	−2.00000	−0.166091
$146$	−10.9443	−0.905754
$147$	0	0
$148$	4.47214	0.367607
$149$	6.94427	0.568897	0.284448	−	0.958691i	$-0.408190\pi$
$149$	6.94427	0.568897	0.284448	+	0.958691i	$0.408190\pi$
$150$	0	0
$151$	3.05573	0.248672	0.124336	−	0.992240i	$-0.460320\pi$
$151$	3.05573	0.248672	0.124336	+	0.992240i	$0.460320\pi$
$152$	19.4164	1.57488
$153$	0	0
$154$	0	0
$155$	−16.0000	−1.28515
$156$	0	0
$157$	−12.4721	−0.995385	−0.497692	−	0.867354i	$-0.665819\pi$
$157$	−12.4721	−0.995385	−0.497692	+	0.867354i	$0.665819\pi$
$158$	10.4721	0.833118
$159$	0	0
$160$	10.0000	0.790569
$161$	0	0
$162$	0	0
$163$	16.9443	1.32718	0.663589	−	0.748097i	$-0.269032\pi$
$163$	16.9443	1.32718	0.663589	+	0.748097i	$0.269032\pi$
$164$	−10.9443	−0.854604
$165$	0	0
$166$	−4.00000	−0.310460
$167$	−3.05573	−0.236459	−0.118230	−	0.992986i	$-0.537722\pi$
$167$	−3.05573	−0.236459	−0.118230	+	0.992986i	$0.537722\pi$
$168$	0	0
$169$	−9.00000	−0.692308
$170$	−8.94427	−0.685994
$171$	0	0
$172$	−6.47214	−0.493496
$173$	2.00000	0.152057	0.0760286	−	0.997106i	$-0.475776\pi$
$173$	2.00000	0.152057	0.0760286	+	0.997106i	$0.475776\pi$
$174$	0	0
$175$	0	0
$176$	−6.47214	−0.487856
$177$	0	0
$178$	13.4164	1.00560
$179$	−8.94427	−0.668526	−0.334263	−	0.942480i	$-0.608487\pi$
$179$	−8.94427	−0.668526	−0.334263	+	0.942480i	$0.608487\pi$
$180$	0	0
$181$	−10.0000	−0.743294	−0.371647	−	0.928374i	$-0.621207\pi$
$181$	−10.0000	−0.743294	−0.371647	+	0.928374i	$0.621207\pi$
$182$	0	0
$183$	0	0
$184$	−3.00000	−0.221163
$185$	−8.94427	−0.657596
$186$	0	0
$187$	−28.9443	−2.11661
$188$	12.9443	0.944058
$189$	0	0
$190$	−12.9443	−0.939076
$191$	−2.47214	−0.178877	−0.0894387	−	0.995992i	$-0.528507\pi$
$191$	−2.47214	−0.178877	−0.0894387	+	0.995992i	$0.528507\pi$
$192$	0	0
$193$	22.9443	1.65156	0.825782	−	0.563989i	$-0.190734\pi$
$193$	22.9443	1.65156	0.825782	+	0.563989i	$0.190734\pi$
$194$	17.4164	1.25043
$195$	0	0
$196$	7.00000	0.500000
$197$	10.0000	0.712470	0.356235	−	0.934396i	$-0.384060\pi$
$197$	10.0000	0.712470	0.356235	+	0.934396i	$0.384060\pi$
$198$	0	0
$199$	4.94427	0.350490	0.175245	−	0.984525i	$-0.443928\pi$
$199$	4.94427	0.350490	0.175245	+	0.984525i	$0.443928\pi$
$200$	3.00000	0.212132
$201$	0	0
$202$	−1.05573	−0.0742808
$203$	0	0
$204$	0	0
$205$	21.8885	1.52876
$206$	4.94427	0.344484
$207$	0	0
$208$	2.00000	0.138675
$209$	−41.8885	−2.89749
$210$	0	0
$211$	0.944272	0.0650064	0.0325032	−	0.999472i	$-0.489652\pi$
$211$	0.944272	0.0650064	0.0325032	+	0.999472i	$0.489652\pi$
$212$	−2.00000	−0.137361
$213$	0	0
$214$	−16.9443	−1.15829
$215$	12.9443	0.882792
$216$	0	0
$217$	0	0
$218$	−6.94427	−0.470325
$219$	0	0
$220$	−12.9443	−0.872703
$221$	8.94427	0.601657
$222$	0	0
$223$	−9.88854	−0.662186	−0.331093	−	0.943598i	$-0.607417\pi$
$223$	−9.88854	−0.662186	−0.331093	+	0.943598i	$0.607417\pi$
$224$	0	0
$225$	0	0
$226$	0.472136	0.0314060
$227$	−13.8885	−0.921815	−0.460908	−	0.887448i	$-0.652476\pi$
$227$	−13.8885	−0.921815	−0.460908	+	0.887448i	$0.652476\pi$
$228$	0	0
$229$	−9.41641	−0.622254	−0.311127	−	0.950368i	$-0.600706\pi$
$229$	−9.41641	−0.622254	−0.311127	+	0.950368i	$0.600706\pi$
$230$	2.00000	0.131876
$231$	0	0
$232$	3.00000	0.196960
$233$	15.8885	1.04089	0.520447	−	0.853894i	$-0.325765\pi$
$233$	15.8885	1.04089	0.520447	+	0.853894i	$0.325765\pi$
$234$	0	0
$235$	−25.8885	−1.68878
$236$	−4.00000	−0.260378
$237$	0	0
$238$	0	0
$239$	4.94427	0.319818	0.159909	−	0.987132i	$-0.448880\pi$
$239$	4.94427	0.319818	0.159909	+	0.987132i	$0.448880\pi$
$240$	0	0
$241$	−9.05573	−0.583331	−0.291665	−	0.956520i	$-0.594209\pi$
$241$	−9.05573	−0.583331	−0.291665	+	0.956520i	$0.594209\pi$
$242$	30.8885	1.98559
$243$	0	0
$244$	12.4721	0.798447
$245$	−14.0000	−0.894427
$246$	0	0
$247$	12.9443	0.823624
$248$	24.0000	1.52400
$249$	0	0
$250$	−12.0000	−0.758947
$251$	16.3607	1.03268	0.516338	−	0.856385i	$-0.327295\pi$
$251$	16.3607	1.03268	0.516338	+	0.856385i	$0.327295\pi$
$252$	0	0
$253$	6.47214	0.406900
$254$	−17.8885	−1.12243
$255$	0	0
$256$	−17.0000	−1.06250
$257$	−11.8885	−0.741587	−0.370793	−	0.928715i	$-0.620914\pi$
$257$	−11.8885	−0.741587	−0.370793	+	0.928715i	$0.620914\pi$
$258$	0	0
$259$	0	0
$260$	4.00000	0.248069
$261$	0	0
$262$	−12.0000	−0.741362
$263$	−10.4721	−0.645740	−0.322870	−	0.946443i	$-0.604648\pi$
$263$	−10.4721	−0.645740	−0.322870	+	0.946443i	$0.604648\pi$
$264$	0	0
$265$	4.00000	0.245718
$266$	0	0
$267$	0	0
$268$	4.00000	0.244339
$269$	−14.0000	−0.853595	−0.426798	−	0.904347i	$-0.640358\pi$
$269$	−14.0000	−0.853595	−0.426798	+	0.904347i	$0.640358\pi$
$270$	0	0
$271$	−17.8885	−1.08665	−0.543326	−	0.839522i	$-0.682835\pi$
$271$	−17.8885	−1.08665	−0.543326	+	0.839522i	$0.682835\pi$
$272$	4.47214	0.271163
$273$	0	0
$274$	−12.4721	−0.753469
$275$	−6.47214	−0.390284
$276$	0	0
$277$	−19.8885	−1.19499	−0.597493	−	0.801874i	$-0.703837\pi$
$277$	−19.8885	−1.19499	−0.597493	+	0.801874i	$0.703837\pi$
$278$	16.9443	1.01625
$279$	0	0
$280$	0	0
$281$	−13.0557	−0.778839	−0.389420	−	0.921060i	$-0.627324\pi$
$281$	−13.0557	−0.778839	−0.389420	+	0.921060i	$0.627324\pi$
$282$	0	0
$283$	−0.944272	−0.0561311	−0.0280656	−	0.999606i	$-0.508935\pi$
$283$	−0.944272	−0.0561311	−0.0280656	+	0.999606i	$0.508935\pi$
$284$	8.00000	0.474713
$285$	0	0
$286$	−12.9443	−0.765411
$287$	0	0
$288$	0	0
$289$	3.00000	0.176471
$290$	−2.00000	−0.117444
$291$	0	0
$292$	10.9443	0.640465
$293$	−13.4164	−0.783795	−0.391897	−	0.920009i	$-0.628181\pi$
$293$	−13.4164	−0.783795	−0.391897	+	0.920009i	$0.628181\pi$
$294$	0	0
$295$	8.00000	0.465778
$296$	13.4164	0.779813
$297$	0	0
$298$	6.94427	0.402271
$299$	−2.00000	−0.115663
$300$	0	0
$301$	0	0
$302$	3.05573	0.175837
$303$	0	0
$304$	6.47214	0.371202
$305$	−24.9443	−1.42830
$306$	0	0
$307$	−24.9443	−1.42364	−0.711822	−	0.702360i	$-0.752130\pi$
$307$	−24.9443	−1.42364	−0.711822	+	0.702360i	$0.752130\pi$
$308$	0	0
$309$	0	0
$310$	−16.0000	−0.908739
$311$	−4.94427	−0.280364	−0.140182	−	0.990126i	$-0.544769\pi$
$311$	−4.94427	−0.280364	−0.140182	+	0.990126i	$0.544769\pi$
$312$	0	0
$313$	5.05573	0.285767	0.142883	−	0.989740i	$-0.454363\pi$
$313$	5.05573	0.285767	0.142883	+	0.989740i	$0.454363\pi$
$314$	−12.4721	−0.703843
$315$	0	0
$316$	−10.4721	−0.589104
$317$	2.00000	0.112331	0.0561656	−	0.998421i	$-0.482113\pi$
$317$	2.00000	0.112331	0.0561656	+	0.998421i	$0.482113\pi$
$318$	0	0
$319$	−6.47214	−0.362370
$320$	14.0000	0.782624
$321$	0	0
$322$	0	0
$323$	28.9443	1.61050
$324$	0	0
$325$	2.00000	0.110940
$326$	16.9443	0.938456
$327$	0	0
$328$	−32.8328	−1.81289
$329$	0	0
$330$	0	0
$331$	−5.88854	−0.323664	−0.161832	−	0.986818i	$-0.551740\pi$
$331$	−5.88854	−0.323664	−0.161832	+	0.986818i	$0.551740\pi$
$332$	4.00000	0.219529
$333$	0	0
$334$	−3.05573	−0.167202
$335$	−8.00000	−0.437087
$336$	0	0
$337$	−8.47214	−0.461507	−0.230753	−	0.973012i	$-0.574119\pi$
$337$	−8.47214	−0.461507	−0.230753	+	0.973012i	$0.574119\pi$
$338$	−9.00000	−0.489535
$339$	0	0
$340$	8.94427	0.485071
$341$	−51.7771	−2.80389
$342$	0	0
$343$	0	0
$344$	−19.4164	−1.04686
$345$	0	0
$346$	2.00000	0.107521
$347$	29.8885	1.60450	0.802251	−	0.596987i	$-0.203636\pi$
$347$	29.8885	1.60450	0.802251	+	0.596987i	$0.203636\pi$
$348$	0	0
$349$	7.88854	0.422264	0.211132	−	0.977458i	$-0.432285\pi$
$349$	7.88854	0.422264	0.211132	+	0.977458i	$0.432285\pi$
$350$	0	0
$351$	0	0
$352$	32.3607	1.72483
$353$	−2.00000	−0.106449	−0.0532246	−	0.998583i	$-0.516950\pi$
$353$	−2.00000	−0.106449	−0.0532246	+	0.998583i	$0.516950\pi$
$354$	0	0
$355$	−16.0000	−0.849192
$356$	−13.4164	−0.711068
$357$	0	0
$358$	−8.94427	−0.472719
$359$	−21.5279	−1.13620	−0.568099	−	0.822960i	$-0.692321\pi$
$359$	−21.5279	−1.13620	−0.568099	+	0.822960i	$0.692321\pi$
$360$	0	0
$361$	22.8885	1.20466
$362$	−10.0000	−0.525588
$363$	0	0
$364$	0	0
$365$	−21.8885	−1.14570
$366$	0	0
$367$	10.4721	0.546641	0.273321	−	0.961923i	$-0.411878\pi$
$367$	10.4721	0.546641	0.273321	+	0.961923i	$0.411878\pi$
$368$	−1.00000	−0.0521286
$369$	0	0
$370$	−8.94427	−0.464991
$371$	0	0
$372$	0	0
$373$	6.00000	0.310668	0.155334	−	0.987862i	$-0.450355\pi$
$373$	6.00000	0.310668	0.155334	+	0.987862i	$0.450355\pi$
$374$	−28.9443	−1.49667
$375$	0	0
$376$	38.8328	2.00265
$377$	2.00000	0.103005
$378$	0	0
$379$	1.52786	0.0784811	0.0392406	−	0.999230i	$-0.487506\pi$
$379$	1.52786	0.0784811	0.0392406	+	0.999230i	$0.487506\pi$
$380$	12.9443	0.664027
$381$	0	0
$382$	−2.47214	−0.126485
$383$	−36.9443	−1.88776	−0.943882	−	0.330283i	$-0.892856\pi$
$383$	−36.9443	−1.88776	−0.943882	+	0.330283i	$0.892856\pi$
$384$	0	0
$385$	0	0
$386$	22.9443	1.16783
$387$	0	0
$388$	−17.4164	−0.884184
$389$	−13.4164	−0.680239	−0.340119	−	0.940382i	$-0.610468\pi$
$389$	−13.4164	−0.680239	−0.340119	+	0.940382i	$0.610468\pi$
$390$	0	0
$391$	−4.47214	−0.226166
$392$	21.0000	1.06066
$393$	0	0
$394$	10.0000	0.503793
$395$	20.9443	1.05382
$396$	0	0
$397$	−2.00000	−0.100377	−0.0501886	−	0.998740i	$-0.515982\pi$
$397$	−2.00000	−0.100377	−0.0501886	+	0.998740i	$0.515982\pi$
$398$	4.94427	0.247834
$399$	0	0
$400$	1.00000	0.0500000
$401$	10.9443	0.546531	0.273265	−	0.961939i	$-0.411896\pi$
$401$	10.9443	0.546531	0.273265	+	0.961939i	$0.411896\pi$
$402$	0	0
$403$	16.0000	0.797017
$404$	1.05573	0.0525244
$405$	0	0
$406$	0	0
$407$	−28.9443	−1.43471
$408$	0	0
$409$	−26.9443	−1.33231	−0.666154	−	0.745814i	$-0.732061\pi$
$409$	−26.9443	−1.33231	−0.666154	+	0.745814i	$0.732061\pi$
$410$	21.8885	1.08100
$411$	0	0
$412$	−4.94427	−0.243587
$413$	0	0
$414$	0	0
$415$	−8.00000	−0.392705
$416$	−10.0000	−0.490290
$417$	0	0
$418$	−41.8885	−2.04884
$419$	−13.8885	−0.678500	−0.339250	−	0.940696i	$-0.610173\pi$
$419$	−13.8885	−0.678500	−0.339250	+	0.940696i	$0.610173\pi$
$420$	0	0
$421$	−20.4721	−0.997751	−0.498875	−	0.866674i	$-0.666253\pi$
$421$	−20.4721	−0.997751	−0.498875	+	0.866674i	$0.666253\pi$
$422$	0.944272	0.0459664
$423$	0	0
$424$	−6.00000	−0.291386
$425$	4.47214	0.216930
$426$	0	0
$427$	0	0
$428$	16.9443	0.819032
$429$	0	0
$430$	12.9443	0.624228
$431$	0	0		−	1.00000i	$-0.5\pi$
$431$	0	0			1.00000i	$0.5\pi$
$432$	0	0
$433$	38.3607	1.84350	0.921748	−	0.387789i	$-0.126761\pi$
$433$	38.3607	1.84350	0.921748	+	0.387789i	$0.126761\pi$
$434$	0	0
$435$	0	0
$436$	6.94427	0.332570
$437$	−6.47214	−0.309604
$438$	0	0
$439$	8.00000	0.381819	0.190910	−	0.981608i	$-0.438856\pi$
$439$	8.00000	0.381819	0.190910	+	0.981608i	$0.438856\pi$
$440$	−38.8328	−1.85128
$441$	0	0
$442$	8.94427	0.425436
$443$	26.8328	1.27487	0.637433	−	0.770506i	$-0.279996\pi$
$443$	26.8328	1.27487	0.637433	+	0.770506i	$0.279996\pi$
$444$	0	0
$445$	26.8328	1.27200
$446$	−9.88854	−0.468236
$447$	0	0
$448$	0	0
$449$	9.05573	0.427366	0.213683	−	0.976903i	$-0.431454\pi$
$449$	9.05573	0.427366	0.213683	+	0.976903i	$0.431454\pi$
$450$	0	0
$451$	70.8328	3.33539
$452$	−0.472136	−0.0222074
$453$	0	0
$454$	−13.8885	−0.651822
$455$	0	0
$456$	0	0
$457$	−6.00000	−0.280668	−0.140334	−	0.990104i	$-0.544818\pi$
$457$	−6.00000	−0.280668	−0.140334	+	0.990104i	$0.544818\pi$
$458$	−9.41641	−0.440000
$459$	0	0
$460$	−2.00000	−0.0932505
$461$	18.0000	0.838344	0.419172	−	0.907907i	$-0.362320\pi$
$461$	18.0000	0.838344	0.419172	+	0.907907i	$0.362320\pi$
$462$	0	0
$463$	−4.94427	−0.229780	−0.114890	−	0.993378i	$-0.536652\pi$
$463$	−4.94427	−0.229780	−0.114890	+	0.993378i	$0.536652\pi$
$464$	1.00000	0.0464238
$465$	0	0
$466$	15.8885	0.736023
$467$	25.5279	1.18129	0.590644	−	0.806932i	$-0.298874\pi$
$467$	25.5279	1.18129	0.590644	+	0.806932i	$0.298874\pi$
$468$	0	0
$469$	0	0
$470$	−25.8885	−1.19415
$471$	0	0
$472$	−12.0000	−0.552345
$473$	41.8885	1.92604
$474$	0	0
$475$	6.47214	0.296962
$476$	0	0
$477$	0	0
$478$	4.94427	0.226146
$479$	38.2492	1.74765	0.873826	−	0.486239i	$-0.161632\pi$
$479$	38.2492	1.74765	0.873826	+	0.486239i	$0.161632\pi$
$480$	0	0
$481$	8.94427	0.407824
$482$	−9.05573	−0.412477
$483$	0	0
$484$	−30.8885	−1.40402
$485$	34.8328	1.58168
$486$	0	0
$487$	−28.9443	−1.31159	−0.655795	−	0.754939i	$-0.727666\pi$
$487$	−28.9443	−1.31159	−0.655795	+	0.754939i	$0.727666\pi$
$488$	37.4164	1.69376
$489$	0	0
$490$	−14.0000	−0.632456
$491$	37.8885	1.70989	0.854943	−	0.518722i	$-0.173592\pi$
$491$	37.8885	1.70989	0.854943	+	0.518722i	$0.173592\pi$
$492$	0	0
$493$	4.47214	0.201415
$494$	12.9443	0.582390
$495$	0	0
$496$	8.00000	0.359211
$497$	0	0
$498$	0	0
$499$	15.0557	0.673987	0.336993	−	0.941507i	$-0.390590\pi$
$499$	15.0557	0.673987	0.336993	+	0.941507i	$0.390590\pi$
$500$	12.0000	0.536656
$501$	0	0
$502$	16.3607	0.730213
$503$	−26.4721	−1.18033	−0.590167	−	0.807281i	$-0.700938\pi$
$503$	−26.4721	−1.18033	−0.590167	+	0.807281i	$0.700938\pi$
$504$	0	0
$505$	−2.11146	−0.0939586
$506$	6.47214	0.287722
$507$	0	0
$508$	17.8885	0.793676
$509$	37.7771	1.67444	0.837220	−	0.546866i	$-0.184179\pi$
$509$	37.7771	1.67444	0.837220	+	0.546866i	$0.184179\pi$
$510$	0	0
$511$	0	0
$512$	−11.0000	−0.486136
$513$	0	0
$514$	−11.8885	−0.524381
$515$	9.88854	0.435741
$516$	0	0
$517$	−83.7771	−3.68451
$518$	0	0
$519$	0	0
$520$	12.0000	0.526235
$521$	−6.94427	−0.304234	−0.152117	−	0.988362i	$-0.548609\pi$
$521$	−6.94427	−0.304234	−0.152117	+	0.988362i	$0.548609\pi$
$522$	0	0
$523$	−21.8885	−0.957119	−0.478560	−	0.878055i	$-0.658841\pi$
$523$	−21.8885	−0.957119	−0.478560	+	0.878055i	$0.658841\pi$
$524$	12.0000	0.524222
$525$	0	0
$526$	−10.4721	−0.456607
$527$	35.7771	1.55847
$528$	0	0
$529$	1.00000	0.0434783
$530$	4.00000	0.173749
$531$	0	0
$532$	0	0
$533$	−21.8885	−0.948098
$534$	0	0
$535$	−33.8885	−1.46513
$536$	12.0000	0.518321
$537$	0	0
$538$	−14.0000	−0.603583
$539$	−45.3050	−1.95142
$540$	0	0
$541$	44.8328	1.92751	0.963757	−	0.266783i	$-0.0859607\pi$
$541$	44.8328	1.92751	0.963757	+	0.266783i	$0.0859607\pi$
$542$	−17.8885	−0.768379
$543$	0	0
$544$	−22.3607	−0.958706
$545$	−13.8885	−0.594920
$546$	0	0
$547$	29.8885	1.27794	0.638971	−	0.769231i	$-0.279360\pi$
$547$	29.8885	1.27794	0.638971	+	0.769231i	$0.279360\pi$
$548$	12.4721	0.532783
$549$	0	0
$550$	−6.47214	−0.275973
$551$	6.47214	0.275722
$552$	0	0
$553$	0	0
$554$	−19.8885	−0.844983
$555$	0	0
$556$	−16.9443	−0.718597
$557$	19.8885	0.842705	0.421352	−	0.906897i	$-0.361556\pi$
$557$	19.8885	0.842705	0.421352	+	0.906897i	$0.361556\pi$
$558$	0	0
$559$	−12.9443	−0.547484
$560$	0	0
$561$	0	0
$562$	−13.0557	−0.550723
$563$	19.4164	0.818304	0.409152	−	0.912466i	$-0.365825\pi$
$563$	19.4164	0.818304	0.409152	+	0.912466i	$0.365825\pi$
$564$	0	0
$565$	0.944272	0.0397258
$566$	−0.944272	−0.0396907
$567$	0	0
$568$	24.0000	1.00702
$569$	−23.5279	−0.986339	−0.493170	−	0.869933i	$-0.664162\pi$
$569$	−23.5279	−0.986339	−0.493170	+	0.869933i	$0.664162\pi$
$570$	0	0
$571$	−7.05573	−0.295273	−0.147637	−	0.989042i	$-0.547167\pi$
$571$	−7.05573	−0.295273	−0.147637	+	0.989042i	$0.547167\pi$
$572$	12.9443	0.541227
$573$	0	0
$574$	0	0
$575$	−1.00000	−0.0417029
$576$	0	0
$577$	−9.05573	−0.376995	−0.188497	−	0.982074i	$-0.560362\pi$
$577$	−9.05573	−0.376995	−0.188497	+	0.982074i	$0.560362\pi$
$578$	3.00000	0.124784
$579$	0	0
$580$	2.00000	0.0830455
$581$	0	0
$582$	0	0
$583$	12.9443	0.536097
$584$	32.8328	1.35863
$585$	0	0
$586$	−13.4164	−0.554227
$587$	8.94427	0.369170	0.184585	−	0.982817i	$-0.440906\pi$
$587$	8.94427	0.369170	0.184585	+	0.982817i	$0.440906\pi$
$588$	0	0
$589$	51.7771	2.13344
$590$	8.00000	0.329355
$591$	0	0
$592$	4.47214	0.183804
$593$	−11.8885	−0.488204	−0.244102	−	0.969750i	$-0.578493\pi$
$593$	−11.8885	−0.488204	−0.244102	+	0.969750i	$0.578493\pi$
$594$	0	0
$595$	0	0
$596$	−6.94427	−0.284448
$597$	0	0
$598$	−2.00000	−0.0817861
$599$	20.9443	0.855760	0.427880	−	0.903836i	$-0.359261\pi$
$599$	20.9443	0.855760	0.427880	+	0.903836i	$0.359261\pi$
$600$	0	0
$601$	14.9443	0.609590	0.304795	−	0.952418i	$-0.401412\pi$
$601$	14.9443	0.609590	0.304795	+	0.952418i	$0.401412\pi$
$602$	0	0
$603$	0	0
$604$	−3.05573	−0.124336
$605$	61.7771	2.51160
$606$	0	0
$607$	22.8328	0.926755	0.463378	−	0.886161i	$-0.346637\pi$
$607$	22.8328	0.926755	0.463378	+	0.886161i	$0.346637\pi$
$608$	−32.3607	−1.31240
$609$	0	0
$610$	−24.9443	−1.00996
$611$	25.8885	1.04734
$612$	0	0
$613$	−8.83282	−0.356754	−0.178377	−	0.983962i	$-0.557085\pi$
$613$	−8.83282	−0.356754	−0.178377	+	0.983962i	$0.557085\pi$
$614$	−24.9443	−1.00667
$615$	0	0
$616$	0	0
$617$	8.47214	0.341075	0.170538	−	0.985351i	$-0.445450\pi$
$617$	8.47214	0.341075	0.170538	+	0.985351i	$0.445450\pi$
$618$	0	0
$619$	27.4164	1.10196	0.550979	−	0.834519i	$-0.314254\pi$
$619$	27.4164	1.10196	0.550979	+	0.834519i	$0.314254\pi$
$620$	16.0000	0.642575
$621$	0	0
$622$	−4.94427	−0.198247
$623$	0	0
$624$	0	0
$625$	−19.0000	−0.760000
$626$	5.05573	0.202068
$627$	0	0
$628$	12.4721	0.497692
$629$	20.0000	0.797452
$630$	0	0
$631$	36.9443	1.47073	0.735364	−	0.677672i	$-0.237011\pi$
$631$	36.9443	1.47073	0.735364	+	0.677672i	$0.237011\pi$
$632$	−31.4164	−1.24968
$633$	0	0
$634$	2.00000	0.0794301
$635$	−35.7771	−1.41977
$636$	0	0
$637$	14.0000	0.554700
$638$	−6.47214	−0.256234
$639$	0	0
$640$	−6.00000	−0.237171
$641$	32.4721	1.28257	0.641286	−	0.767302i	$-0.278401\pi$
$641$	32.4721	1.28257	0.641286	+	0.767302i	$0.278401\pi$
$642$	0	0
$643$	21.8885	0.863200	0.431600	−	0.902065i	$-0.357949\pi$
$643$	21.8885	0.863200	0.431600	+	0.902065i	$0.357949\pi$
$644$	0	0
$645$	0	0
$646$	28.9443	1.13880
$647$	−43.7771	−1.72105	−0.860527	−	0.509404i	$-0.829866\pi$
$647$	−43.7771	−1.72105	−0.860527	+	0.509404i	$0.829866\pi$
$648$	0	0
$649$	25.8885	1.01621
$650$	2.00000	0.0784465
$651$	0	0
$652$	−16.9443	−0.663589
$653$	13.0557	0.510910	0.255455	−	0.966821i	$-0.417775\pi$
$653$	13.0557	0.510910	0.255455	+	0.966821i	$0.417775\pi$
$654$	0	0
$655$	−24.0000	−0.937758
$656$	−10.9443	−0.427302
$657$	0	0
$658$	0	0
$659$	35.4164	1.37963	0.689814	−	0.723987i	$-0.257692\pi$
$659$	35.4164	1.37963	0.689814	+	0.723987i	$0.257692\pi$
$660$	0	0
$661$	17.0557	0.663391	0.331695	−	0.943387i	$-0.392379\pi$
$661$	17.0557	0.663391	0.331695	+	0.943387i	$0.392379\pi$
$662$	−5.88854	−0.228865
$663$	0	0
$664$	12.0000	0.465690
$665$	0	0
$666$	0	0
$667$	−1.00000	−0.0387202
$668$	3.05573	0.118230
$669$	0	0
$670$	−8.00000	−0.309067
$671$	−80.7214	−3.11621
$672$	0	0
$673$	34.0000	1.31060	0.655302	−	0.755367i	$-0.272541\pi$
$673$	34.0000	1.31060	0.655302	+	0.755367i	$0.272541\pi$
$674$	−8.47214	−0.326334
$675$	0	0
$676$	9.00000	0.346154
$677$	−19.5279	−0.750517	−0.375258	−	0.926920i	$-0.622446\pi$
$677$	−19.5279	−0.750517	−0.375258	+	0.926920i	$0.622446\pi$
$678$	0	0
$679$	0	0
$680$	26.8328	1.02899
$681$	0	0
$682$	−51.7771	−1.98265
$683$	−32.9443	−1.26058	−0.630289	−	0.776361i	$-0.717064\pi$
$683$	−32.9443	−1.26058	−0.630289	+	0.776361i	$0.717064\pi$
$684$	0	0
$685$	−24.9443	−0.953072
$686$	0	0
$687$	0	0
$688$	−6.47214	−0.246748
$689$	−4.00000	−0.152388
$690$	0	0
$691$	−42.8328	−1.62944	−0.814719	−	0.579857i	$-0.803109\pi$
$691$	−42.8328	−1.62944	−0.814719	+	0.579857i	$0.803109\pi$
$692$	−2.00000	−0.0760286
$693$	0	0
$694$	29.8885	1.13455
$695$	33.8885	1.28547
$696$	0	0
$697$	−48.9443	−1.85390
$698$	7.88854	0.298586
$699$	0	0
$700$	0	0
$701$	−42.9443	−1.62198	−0.810991	−	0.585058i	$-0.801072\pi$
$701$	−42.9443	−1.62198	−0.810991	+	0.585058i	$0.801072\pi$
$702$	0	0
$703$	28.9443	1.09165
$704$	45.3050	1.70749
$705$	0	0
$706$	−2.00000	−0.0752710
$707$	0	0
$708$	0	0
$709$	−35.8885	−1.34782	−0.673911	−	0.738812i	$-0.735387\pi$
$709$	−35.8885	−1.34782	−0.673911	+	0.738812i	$0.735387\pi$
$710$	−16.0000	−0.600469
$711$	0	0
$712$	−40.2492	−1.50840
$713$	−8.00000	−0.299602
$714$	0	0
$715$	−25.8885	−0.968177
$716$	8.94427	0.334263
$717$	0	0
$718$	−21.5279	−0.803413
$719$	11.0557	0.412309	0.206155	−	0.978519i	$-0.433905\pi$
$719$	11.0557	0.412309	0.206155	+	0.978519i	$0.433905\pi$
$720$	0	0
$721$	0	0
$722$	22.8885	0.851823
$723$	0	0
$724$	10.0000	0.371647
$725$	1.00000	0.0371391
$726$	0	0
$727$	3.63932	0.134975	0.0674875	−	0.997720i	$-0.478502\pi$
$727$	3.63932	0.134975	0.0674875	+	0.997720i	$0.478502\pi$
$728$	0	0
$729$	0	0
$730$	−21.8885	−0.810131
$731$	−28.9443	−1.07054
$732$	0	0
$733$	2.36068	0.0871937	0.0435968	−	0.999049i	$-0.486118\pi$
$733$	2.36068	0.0871937	0.0435968	+	0.999049i	$0.486118\pi$
$734$	10.4721	0.386534
$735$	0	0
$736$	5.00000	0.184302
$737$	−25.8885	−0.953617
$738$	0	0
$739$	48.9443	1.80044	0.900222	−	0.435431i	$-0.143404\pi$
$739$	48.9443	1.80044	0.900222	+	0.435431i	$0.143404\pi$
$740$	8.94427	0.328798
$741$	0	0
$742$	0	0
$743$	−46.2492	−1.69672	−0.848360	−	0.529420i	$-0.822409\pi$
$743$	−46.2492	−1.69672	−0.848360	+	0.529420i	$0.822409\pi$
$744$	0	0
$745$	13.8885	0.508837
$746$	6.00000	0.219676
$747$	0	0
$748$	28.9443	1.05831
$749$	0	0
$750$	0	0
$751$	42.4721	1.54983	0.774915	−	0.632065i	$-0.217793\pi$
$751$	42.4721	1.54983	0.774915	+	0.632065i	$0.217793\pi$
$752$	12.9443	0.472029
$753$	0	0
$754$	2.00000	0.0728357
$755$	6.11146	0.222419
$756$	0	0
$757$	−15.5279	−0.564370	−0.282185	−	0.959360i	$-0.591059\pi$
$757$	−15.5279	−0.564370	−0.282185	+	0.959360i	$0.591059\pi$
$758$	1.52786	0.0554945
$759$	0	0
$760$	38.8328	1.40861
$761$	25.7771	0.934419	0.467209	−	0.884147i	$-0.345259\pi$
$761$	25.7771	0.934419	0.467209	+	0.884147i	$0.345259\pi$
$762$	0	0
$763$	0	0
$764$	2.47214	0.0894387
$765$	0	0
$766$	−36.9443	−1.33485
$767$	−8.00000	−0.288863
$768$	0	0
$769$	−39.3050	−1.41737	−0.708686	−	0.705524i	$-0.750712\pi$
$769$	−39.3050	−1.41737	−0.708686	+	0.705524i	$0.750712\pi$
$770$	0	0
$771$	0	0
$772$	−22.9443	−0.825782
$773$	33.4164	1.20190	0.600952	−	0.799285i	$-0.294788\pi$
$773$	33.4164	1.20190	0.600952	+	0.799285i	$0.294788\pi$
$774$	0	0
$775$	8.00000	0.287368
$776$	−52.2492	−1.87564
$777$	0	0
$778$	−13.4164	−0.481002
$779$	−70.8328	−2.53785
$780$	0	0
$781$	−51.7771	−1.85273
$782$	−4.47214	−0.159923
$783$	0	0
$784$	7.00000	0.250000
$785$	−24.9443	−0.890299
$786$	0	0
$787$	−45.8885	−1.63575	−0.817875	−	0.575396i	$-0.804848\pi$
$787$	−45.8885	−1.63575	−0.817875	+	0.575396i	$0.804848\pi$
$788$	−10.0000	−0.356235
$789$	0	0
$790$	20.9443	0.745164
$791$	0	0
$792$	0	0
$793$	24.9443	0.885797
$794$	−2.00000	−0.0709773
$795$	0	0
$796$	−4.94427	−0.175245
$797$	5.63932	0.199755	0.0998775	−	0.995000i	$-0.468155\pi$
$797$	5.63932	0.199755	0.0998775	+	0.995000i	$0.468155\pi$
$798$	0	0
$799$	57.8885	2.04795
$800$	−5.00000	−0.176777
$801$	0	0
$802$	10.9443	0.386456
$803$	−70.8328	−2.49964
$804$	0	0
$805$	0	0
$806$	16.0000	0.563576
$807$	0	0
$808$	3.16718	0.111421
$809$	6.00000	0.210949	0.105474	−	0.994422i	$-0.466364\pi$
$809$	6.00000	0.210949	0.105474	+	0.994422i	$0.466364\pi$
$810$	0	0
$811$	12.0000	0.421377	0.210688	−	0.977553i	$-0.432429\pi$
$811$	12.0000	0.421377	0.210688	+	0.977553i	$0.432429\pi$
$812$	0	0
$813$	0	0
$814$	−28.9443	−1.01450
$815$	33.8885	1.18706
$816$	0	0
$817$	−41.8885	−1.46549
$818$	−26.9443	−0.942084
$819$	0	0
$820$	−21.8885	−0.764381
$821$	3.88854	0.135711	0.0678556	−	0.997695i	$-0.478384\pi$
$821$	3.88854	0.135711	0.0678556	+	0.997695i	$0.478384\pi$
$822$	0	0
$823$	27.0557	0.943103	0.471552	−	0.881838i	$-0.343694\pi$
$823$	27.0557	0.943103	0.471552	+	0.881838i	$0.343694\pi$
$824$	−14.8328	−0.516726
$825$	0	0
$826$	0	0
$827$	−50.2492	−1.74734	−0.873668	−	0.486522i	$-0.838265\pi$
$827$	−50.2492	−1.74734	−0.873668	+	0.486522i	$0.838265\pi$
$828$	0	0
$829$	2.94427	0.102259	0.0511294	−	0.998692i	$-0.483718\pi$
$829$	2.94427	0.102259	0.0511294	+	0.998692i	$0.483718\pi$
$830$	−8.00000	−0.277684
$831$	0	0
$832$	−14.0000	−0.485363
$833$	31.3050	1.08465
$834$	0	0
$835$	−6.11146	−0.211496
$836$	41.8885	1.44875
$837$	0	0
$838$	−13.8885	−0.479772
$839$	4.36068	0.150547	0.0752737	−	0.997163i	$-0.476017\pi$
$839$	4.36068	0.150547	0.0752737	+	0.997163i	$0.476017\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	−20.4721	−0.705516
$843$	0	0
$844$	−0.944272	−0.0325032
$845$	−18.0000	−0.619219
$846$	0	0
$847$	0	0
$848$	−2.00000	−0.0686803
$849$	0	0
$850$	4.47214	0.153393
$851$	−4.47214	−0.153303
$852$	0	0
$853$	25.7771	0.882591	0.441295	−	0.897362i	$-0.354519\pi$
$853$	25.7771	0.882591	0.441295	+	0.897362i	$0.354519\pi$
$854$	0	0
$855$	0	0
$856$	50.8328	1.73743
$857$	−29.7771	−1.01717	−0.508583	−	0.861013i	$-0.669830\pi$
$857$	−29.7771	−1.01717	−0.508583	+	0.861013i	$0.669830\pi$
$858$	0	0
$859$	−37.8885	−1.29274	−0.646370	−	0.763024i	$-0.723714\pi$
$859$	−37.8885	−1.29274	−0.646370	+	0.763024i	$0.723714\pi$
$860$	−12.9443	−0.441396
$861$	0	0
$862$	0	0
$863$	30.8328	1.04956	0.524781	−	0.851238i	$-0.324147\pi$
$863$	30.8328	1.04956	0.524781	+	0.851238i	$0.324147\pi$
$864$	0	0
$865$	4.00000	0.136004
$866$	38.3607	1.30355
$867$	0	0
$868$	0	0
$869$	67.7771	2.29918
$870$	0	0
$871$	8.00000	0.271070
$872$	20.8328	0.705488
$873$	0	0
$874$	−6.47214	−0.218923
$875$	0	0
$876$	0	0
$877$	39.8885	1.34694	0.673470	−	0.739214i	$-0.264803\pi$
$877$	39.8885	1.34694	0.673470	+	0.739214i	$0.264803\pi$
$878$	8.00000	0.269987
$879$	0	0
$880$	−12.9443	−0.436351
$881$	−20.4721	−0.689724	−0.344862	−	0.938653i	$-0.612074\pi$
$881$	−20.4721	−0.689724	−0.344862	+	0.938653i	$0.612074\pi$
$882$	0	0
$883$	−16.9443	−0.570220	−0.285110	−	0.958495i	$-0.592030\pi$
$883$	−16.9443	−0.570220	−0.285110	+	0.958495i	$0.592030\pi$
$884$	−8.94427	−0.300828
$885$	0	0
$886$	26.8328	0.901466
$887$	25.8885	0.869252	0.434626	−	0.900611i	$-0.356881\pi$
$887$	25.8885	0.869252	0.434626	+	0.900611i	$0.356881\pi$
$888$	0	0
$889$	0	0
$890$	26.8328	0.899438
$891$	0	0
$892$	9.88854	0.331093
$893$	83.7771	2.80349
$894$	0	0
$895$	−17.8885	−0.597948
$896$	0	0
$897$	0	0
$898$	9.05573	0.302194
$899$	8.00000	0.266815
$900$	0	0
$901$	−8.94427	−0.297977
$902$	70.8328	2.35847
$903$	0	0
$904$	−1.41641	−0.0471090
$905$	−20.0000	−0.664822
$906$	0	0
$907$	−25.5279	−0.847639	−0.423819	−	0.905747i	$-0.639311\pi$
$907$	−25.5279	−0.847639	−0.423819	+	0.905747i	$0.639311\pi$
$908$	13.8885	0.460908
$909$	0	0
$910$	0	0
$911$	−28.3607	−0.939631	−0.469816	−	0.882765i	$-0.655680\pi$
$911$	−28.3607	−0.939631	−0.469816	+	0.882765i	$0.655680\pi$
$912$	0	0
$913$	−25.8885	−0.856786
$914$	−6.00000	−0.198462
$915$	0	0
$916$	9.41641	0.311127
$917$	0	0
$918$	0	0
$919$	0	0		−	1.00000i	$-0.5\pi$
$919$	0	0			1.00000i	$0.5\pi$
$920$	−6.00000	−0.197814
$921$	0	0
$922$	18.0000	0.592798
$923$	16.0000	0.526646
$924$	0	0
$925$	4.47214	0.147043
$926$	−4.94427	−0.162479
$927$	0	0
$928$	−5.00000	−0.164133
$929$	−37.7771	−1.23943	−0.619713	−	0.784828i	$-0.712751\pi$
$929$	−37.7771	−1.23943	−0.619713	+	0.784828i	$0.712751\pi$
$930$	0	0
$931$	45.3050	1.48481
$932$	−15.8885	−0.520447
$933$	0	0
$934$	25.5279	0.835297
$935$	−57.8885	−1.89316
$936$	0	0
$937$	0.111456	0.00364111	0.00182056	−	0.999998i	$-0.499420\pi$
$937$	0.111456	0.00364111	0.00182056	+	0.999998i	$0.499420\pi$
$938$	0	0
$939$	0	0
$940$	25.8885	0.844391
$941$	−6.00000	−0.195594	−0.0977972	−	0.995206i	$-0.531180\pi$
$941$	−6.00000	−0.195594	−0.0977972	+	0.995206i	$0.531180\pi$
$942$	0	0
$943$	10.9443	0.356395
$944$	−4.00000	−0.130189
$945$	0	0
$946$	41.8885	1.36191
$947$	45.8885	1.49118	0.745589	−	0.666406i	$-0.232168\pi$
$947$	45.8885	1.49118	0.745589	+	0.666406i	$0.232168\pi$
$948$	0	0
$949$	21.8885	0.710532
$950$	6.47214	0.209984
$951$	0	0
$952$	0	0
$953$	12.8328	0.415696	0.207848	−	0.978161i	$-0.433354\pi$
$953$	12.8328	0.415696	0.207848	+	0.978161i	$0.433354\pi$
$954$	0	0
$955$	−4.94427	−0.159993
$956$	−4.94427	−0.159909
$957$	0	0
$958$	38.2492	1.23578
$959$	0	0
$960$	0	0
$961$	33.0000	1.06452
$962$	8.94427	0.288375
$963$	0	0
$964$	9.05573	0.291665
$965$	45.8885	1.47720
$966$	0	0
$967$	−56.7214	−1.82404	−0.912018	−	0.410150i	$-0.865476\pi$
$967$	−56.7214	−1.82404	−0.912018	+	0.410150i	$0.865476\pi$
$968$	−92.6656	−2.97839
$969$	0	0
$970$	34.8328	1.11841
$971$	−8.36068	−0.268307	−0.134153	−	0.990961i	$-0.542832\pi$
$971$	−8.36068	−0.268307	−0.134153	+	0.990961i	$0.542832\pi$
$972$	0	0
$973$	0	0
$974$	−28.9443	−0.927434
$975$	0	0
$976$	12.4721	0.399223
$977$	−35.8885	−1.14818	−0.574088	−	0.818794i	$-0.694643\pi$
$977$	−35.8885	−1.14818	−0.574088	+	0.818794i	$0.694643\pi$
$978$	0	0
$979$	86.8328	2.77519
$980$	14.0000	0.447214
$981$	0	0
$982$	37.8885	1.20907
$983$	36.3607	1.15973	0.579863	−	0.814714i	$-0.303106\pi$
$983$	36.3607	1.15973	0.579863	+	0.814714i	$0.303106\pi$
$984$	0	0
$985$	20.0000	0.637253
$986$	4.47214	0.142422
$987$	0	0
$988$	−12.9443	−0.411812
$989$	6.47214	0.205802
$990$	0	0
$991$	0	0		−	1.00000i	$-0.5\pi$
$991$	0	0			1.00000i	$0.5\pi$
$992$	−40.0000	−1.27000
$993$	0	0
$994$	0	0
$995$	9.88854	0.313488
$996$	0	0
$997$	1.05573	0.0334352	0.0167176	−	0.999860i	$-0.494678\pi$
$997$	1.05573	0.0334352	0.0167176	+	0.999860i	$0.494678\pi$
$998$	15.0557	0.476581
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6003.2.a.f.1.2		2
3.2	odd	2		2001.2.a.d.1.1	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2001.2.a.d.1.1	✓	2	3.2	odd	2
6003.2.a.f.1.2		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:	\( N \)	\(=\)	\( 6003 = 3^{2} \cdot 23 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6003.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -1.00000 q^{4} +2.00000 q^{5} -3.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} -1.00000 q^{4} +2.00000 q^{5} -3.00000 q^{8} +2.00000 q^{10} +6.47214 q^{11} -2.00000 q^{13} -1.00000 q^{16} -4.47214 q^{17} -6.47214 q^{19} -2.00000 q^{20} +6.47214 q^{22} +1.00000 q^{23} -1.00000 q^{25} -2.00000 q^{26} -1.00000 q^{29} -8.00000 q^{31} +5.00000 q^{32} -4.47214 q^{34} -4.47214 q^{37} -6.47214 q^{38} -6.00000 q^{40} +10.9443 q^{41} +6.47214 q^{43} -6.47214 q^{44} +1.00000 q^{46} -12.9443 q^{47} -7.00000 q^{49} -1.00000 q^{50} +2.00000 q^{52} +2.00000 q^{53} +12.9443 q^{55} -1.00000 q^{58} +4.00000 q^{59} -12.4721 q^{61} -8.00000 q^{62} +7.00000 q^{64} -4.00000 q^{65} -4.00000 q^{67} +4.47214 q^{68} -8.00000 q^{71} -10.9443 q^{73} -4.47214 q^{74} +6.47214 q^{76} +10.4721 q^{79} -2.00000 q^{80} +10.9443 q^{82} -4.00000 q^{83} -8.94427 q^{85} +6.47214 q^{86} -19.4164 q^{88} +13.4164 q^{89} -1.00000 q^{92} -12.9443 q^{94} -12.9443 q^{95} +17.4164 q^{97} -7.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} - 2 q^{4} + 4 q^{5} - 6 q^{8}+O(q^{10}) \) 2 * q + 2 * q^2 - 2 * q^4 + 4 * q^5 - 6 * q^8	\( 2 q + 2 q^{2} - 2 q^{4} + 4 q^{5} - 6 q^{8} + 4 q^{10} + 4 q^{11} - 4 q^{13} - 2 q^{16} - 4 q^{19} - 4 q^{20} + 4 q^{22} + 2 q^{23} - 2 q^{25} - 4 q^{26} - 2 q^{29} - 16 q^{31} + 10 q^{32} - 4 q^{38} - 12 q^{40} + 4 q^{41} + 4 q^{43} - 4 q^{44} + 2 q^{46} - 8 q^{47} - 14 q^{49} - 2 q^{50} + 4 q^{52} + 4 q^{53} + 8 q^{55} - 2 q^{58} + 8 q^{59} - 16 q^{61} - 16 q^{62} + 14 q^{64} - 8 q^{65} - 8 q^{67} - 16 q^{71} - 4 q^{73} + 4 q^{76} + 12 q^{79} - 4 q^{80} + 4 q^{82} - 8 q^{83} + 4 q^{86} - 12 q^{88} - 2 q^{92} - 8 q^{94} - 8 q^{95} + 8 q^{97} - 14 q^{98}+O(q^{100}) \) 2 * q + 2 * q^2 - 2 * q^4 + 4 * q^5 - 6 * q^8 + 4 * q^10 + 4 * q^11 - 4 * q^13 - 2 * q^16 - 4 * q^19 - 4 * q^20 + 4 * q^22 + 2 * q^23 - 2 * q^25 - 4 * q^26 - 2 * q^29 - 16 * q^31 + 10 * q^32 - 4 * q^38 - 12 * q^40 + 4 * q^41 + 4 * q^43 - 4 * q^44 + 2 * q^46 - 8 * q^47 - 14 * q^49 - 2 * q^50 + 4 * q^52 + 4 * q^53 + 8 * q^55 - 2 * q^58 + 8 * q^59 - 16 * q^61 - 16 * q^62 + 14 * q^64 - 8 * q^65 - 8 * q^67 - 16 * q^71 - 4 * q^73 + 4 * q^76 + 12 * q^79 - 4 * q^80 + 4 * q^82 - 8 * q^83 + 4 * q^86 - 12 * q^88 - 2 * q^92 - 8 * q^94 - 8 * q^95 + 8 * q^97 - 14 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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