LMFDB - Embedded newform 6003.2.a.i.1.5

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:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - x^{6} - 9x^{5} + 10x^{4} + 19x^{3} - 20x^{2} - 5x + 1 $ x^7 - x^6 - 9x^5 + 10x^4 + 19x^3 - 20x^2 - 5*x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 2001)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-0.325238$ 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)

$f(q)$	$=$	$q+1.49169 q^{2} +0.225141 q^{4} +2.94997 q^{5} +1.89422 q^{7} -2.64754 q^{8} +O(q^{10})$	\(q+1.49169 q^{2} +0.225141 q^{4} +2.94997 q^{5} +1.89422 q^{7} -2.64754 q^{8} +4.40045 q^{10} -1.03260 q^{11} -6.97182 q^{13} +2.82559 q^{14} -4.39959 q^{16} -4.27786 q^{17} +1.77517 q^{19} +0.664159 q^{20} -1.54032 q^{22} +1.00000 q^{23} +3.70234 q^{25} -10.3998 q^{26} +0.426466 q^{28} -1.00000 q^{29} -6.36283 q^{31} -1.26775 q^{32} -6.38124 q^{34} +5.58790 q^{35} +3.93181 q^{37} +2.64801 q^{38} -7.81017 q^{40} -4.83387 q^{41} -5.21503 q^{43} -0.232480 q^{44} +1.49169 q^{46} -3.22335 q^{47} -3.41193 q^{49} +5.52274 q^{50} -1.56964 q^{52} -2.07256 q^{53} -3.04614 q^{55} -5.01502 q^{56} -1.49169 q^{58} -0.733990 q^{59} +11.8957 q^{61} -9.49137 q^{62} +6.90809 q^{64} -20.5667 q^{65} -3.89150 q^{67} -0.963120 q^{68} +8.33541 q^{70} +3.42654 q^{71} -15.7077 q^{73} +5.86504 q^{74} +0.399664 q^{76} -1.95597 q^{77} -15.1271 q^{79} -12.9787 q^{80} -7.21063 q^{82} -17.1374 q^{83} -12.6196 q^{85} -7.77921 q^{86} +2.73385 q^{88} +10.7253 q^{89} -13.2062 q^{91} +0.225141 q^{92} -4.80824 q^{94} +5.23671 q^{95} -11.6723 q^{97} -5.08954 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q + q^{2} + 7 q^{4} + 5 q^{5} - 5 q^{7} - 3 q^{8}+O(q^{10}) $ 7 * q + q^2 + 7 * q^4 + 5 * q^5 - 5 * q^7 - 3 * q^8	$ 7 q + q^{2} + 7 q^{4} + 5 q^{5} - 5 q^{7} - 3 q^{8} - 11 q^{10} + 12 q^{11} - 13 q^{13} + 3 q^{14} - 13 q^{16} + 12 q^{17} - 5 q^{19} + 8 q^{20} - q^{22} + 7 q^{23} - 4 q^{25} - 2 q^{26} - 21 q^{28} - 7 q^{29} - 8 q^{31} + 5 q^{32} - 28 q^{34} - 5 q^{35} - 24 q^{37} + 6 q^{38} - 20 q^{40} - 9 q^{41} - q^{43} + 23 q^{44} + q^{46} - 27 q^{47} - 14 q^{49} - 7 q^{50} - 9 q^{52} + q^{53} - 11 q^{55} + 20 q^{56} - q^{58} - 8 q^{59} + q^{61} + 3 q^{64} - 12 q^{65} - 16 q^{67} - 15 q^{68} + 40 q^{70} + 13 q^{71} - 23 q^{73} + 8 q^{74} - 2 q^{76} - 13 q^{77} - 44 q^{79} - 30 q^{80} - 10 q^{82} - 21 q^{83} - 6 q^{86} + 21 q^{88} + 5 q^{89} - 18 q^{91} + 7 q^{92} + 28 q^{94} - 9 q^{95} - 55 q^{97} - 36 q^{98}+O(q^{100}) $ 7 * q + q^2 + 7 * q^4 + 5 * q^5 - 5 * q^7 - 3 * q^8 - 11 * q^10 + 12 * q^11 - 13 * q^13 + 3 * q^14 - 13 * q^16 + 12 * q^17 - 5 * q^19 + 8 * q^20 - q^22 + 7 * q^23 - 4 * q^25 - 2 * q^26 - 21 * q^28 - 7 * q^29 - 8 * q^31 + 5 * q^32 - 28 * q^34 - 5 * q^35 - 24 * q^37 + 6 * q^38 - 20 * q^40 - 9 * q^41 - q^43 + 23 * q^44 + q^46 - 27 * q^47 - 14 * q^49 - 7 * q^50 - 9 * q^52 + q^53 - 11 * q^55 + 20 * q^56 - q^58 - 8 * q^59 + q^61 + 3 * q^64 - 12 * q^65 - 16 * q^67 - 15 * q^68 + 40 * q^70 + 13 * q^71 - 23 * q^73 + 8 * q^74 - 2 * q^76 - 13 * q^77 - 44 * q^79 - 30 * q^80 - 10 * q^82 - 21 * q^83 - 6 * q^86 + 21 * q^88 + 5 * q^89 - 18 * q^91 + 7 * q^92 + 28 * q^94 - 9 * q^95 - 55 * q^97 - 36 * 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 $

$2$	1.49169	1.05478	0.527392	−	0.849622i	$-0.323170\pi$
$2$	1.49169	1.05478	0.527392	+	0.849622i	$0.323170\pi$
$3$	0	0
$3$	0	0
$4$	0.225141	0.112570
$5$	2.94997	1.31927	0.659634	−	0.751587i	$-0.270711\pi$
$5$	2.94997	1.31927	0.659634	+	0.751587i	$0.270711\pi$
$6$	0	0
$7$	1.89422	0.715948	0.357974	−	0.933732i	$-0.383468\pi$
$7$	1.89422	0.715948	0.357974	+	0.933732i	$0.383468\pi$
$8$	−2.64754	−0.936047
$9$	0	0
$10$	4.40045	1.39154
$11$	−1.03260	−0.311340	−0.155670	−	0.987809i	$-0.549754\pi$
$11$	−1.03260	−0.311340	−0.155670	+	0.987809i	$0.549754\pi$
$12$	0	0
$13$	−6.97182	−1.93364	−0.966818	−	0.255466i	$-0.917771\pi$
$13$	−6.97182	−1.93364	−0.966818	+	0.255466i	$0.917771\pi$
$14$	2.82559	0.755171
$15$	0	0
$16$	−4.39959	−1.09990
$17$	−4.27786	−1.03753	−0.518766	−	0.854916i	$-0.673608\pi$
$17$	−4.27786	−1.03753	−0.518766	+	0.854916i	$0.673608\pi$
$18$	0	0
$19$	1.77517	0.407252	0.203626	−	0.979049i	$-0.434727\pi$
$19$	1.77517	0.407252	0.203626	+	0.979049i	$0.434727\pi$
$20$	0.664159	0.148511
$21$	0	0
$22$	−1.54032	−0.328397
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	3.70234	0.740468
$26$	−10.3998	−2.03957
$27$	0	0
$28$	0.426466	0.0805946
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	0	0
$31$	−6.36283	−1.14280	−0.571399	−	0.820673i	$-0.693599\pi$
$31$	−6.36283	−1.14280	−0.571399	+	0.820673i	$0.693599\pi$
$32$	−1.26775	−0.224109
$33$	0	0
$34$	−6.38124	−1.09437
$35$	5.58790	0.944527
$36$	0	0
$37$	3.93181	0.646385	0.323193	−	0.946333i	$-0.395244\pi$
$37$	3.93181	0.646385	0.323193	+	0.946333i	$0.395244\pi$
$38$	2.64801	0.429564
$39$	0	0
$40$	−7.81017	−1.23490
$41$	−4.83387	−0.754923	−0.377462	−	0.926025i	$-0.623203\pi$
$41$	−4.83387	−0.754923	−0.377462	+	0.926025i	$0.623203\pi$
$42$	0	0
$43$	−5.21503	−0.795284	−0.397642	−	0.917541i	$-0.630171\pi$
$43$	−5.21503	−0.795284	−0.397642	+	0.917541i	$0.630171\pi$
$44$	−0.232480	−0.0350477
$45$	0	0
$46$	1.49169	0.219938
$47$	−3.22335	−0.470174	−0.235087	−	0.971974i	$-0.575537\pi$
$47$	−3.22335	−0.470174	−0.235087	+	0.971974i	$0.575537\pi$
$48$	0	0
$49$	−3.41193	−0.487419
$50$	5.52274	0.781034
$51$	0	0
$52$	−1.56964	−0.217670
$53$	−2.07256	−0.284688	−0.142344	−	0.989817i	$-0.545464\pi$
$53$	−2.07256	−0.284688	−0.142344	+	0.989817i	$0.545464\pi$
$54$	0	0
$55$	−3.04614	−0.410741
$56$	−5.01502	−0.670161
$57$	0	0
$58$	−1.49169	−0.195869
$59$	−0.733990	−0.0955574	−0.0477787	−	0.998858i	$-0.515214\pi$
$59$	−0.733990	−0.0955574	−0.0477787	+	0.998858i	$0.515214\pi$
$60$	0	0
$61$	11.8957	1.52309	0.761545	−	0.648113i	$-0.224441\pi$
$61$	11.8957	1.52309	0.761545	+	0.648113i	$0.224441\pi$
$62$	−9.49137	−1.20541
$63$	0	0
$64$	6.90809	0.863512
$65$	−20.5667	−2.55098
$66$	0	0
$67$	−3.89150	−0.475423	−0.237711	−	0.971336i	$-0.576397\pi$
$67$	−3.89150	−0.475423	−0.237711	+	0.971336i	$0.576397\pi$
$68$	−0.963120	−0.116795
$69$	0	0
$70$	8.33541	0.996272
$71$	3.42654	0.406655	0.203328	−	0.979111i	$-0.434824\pi$
$71$	3.42654	0.406655	0.203328	+	0.979111i	$0.434824\pi$
$72$	0	0
$73$	−15.7077	−1.83845	−0.919225	−	0.393733i	$-0.871183\pi$
$73$	−15.7077	−1.83845	−0.919225	+	0.393733i	$0.871183\pi$
$74$	5.86504	0.681797
$75$	0	0
$76$	0.399664	0.0458446
$77$	−1.95597	−0.222903
$78$	0	0
$79$	−15.1271	−1.70193	−0.850964	−	0.525224i	$-0.823982\pi$
$79$	−15.1271	−1.70193	−0.850964	+	0.525224i	$0.823982\pi$
$80$	−12.9787	−1.45106
$81$	0	0
$82$	−7.21063	−0.796281
$83$	−17.1374	−1.88107	−0.940535	−	0.339697i	$-0.889676\pi$
$83$	−17.1374	−1.88107	−0.940535	+	0.339697i	$0.889676\pi$
$84$	0	0
$85$	−12.6196	−1.36878
$86$	−7.77921	−0.838854
$87$	0	0
$88$	2.73385	0.291429
$89$	10.7253	1.13688	0.568442	−	0.822724i	$-0.307546\pi$
$89$	10.7253	1.13688	0.568442	+	0.822724i	$0.307546\pi$
$90$	0	0
$91$	−13.2062	−1.38438
$92$	0.225141	0.0234726
$93$	0	0
$94$	−4.80824	−0.495932
$95$	5.23671	0.537275
$96$	0	0
$97$	−11.6723	−1.18515	−0.592573	−	0.805517i	$-0.701888\pi$
$97$	−11.6723	−1.18515	−0.592573	+	0.805517i	$0.701888\pi$
$98$	−5.08954	−0.514122
$99$	0	0
$100$	0.833548	0.0833548
$101$	19.1067	1.90119	0.950596	−	0.310431i	$-0.100473\pi$
$101$	19.1067	1.90119	0.950596	+	0.310431i	$0.100473\pi$
$102$	0	0
$103$	7.98721	0.787003	0.393502	−	0.919324i	$-0.371264\pi$
$103$	7.98721	0.787003	0.393502	+	0.919324i	$0.371264\pi$
$104$	18.4582	1.80997
$105$	0	0
$106$	−3.09162	−0.300285
$107$	11.4872	1.11051	0.555256	−	0.831680i	$-0.312620\pi$
$107$	11.4872	1.11051	0.555256	+	0.831680i	$0.312620\pi$
$108$	0	0
$109$	0.434587	0.0416258	0.0208129	−	0.999783i	$-0.493375\pi$
$109$	0.434587	0.0416258	0.0208129	+	0.999783i	$0.493375\pi$
$110$	−4.54389	−0.433243
$111$	0	0
$112$	−8.33380	−0.787470
$113$	15.8674	1.49268	0.746338	−	0.665567i	$-0.231810\pi$
$113$	15.8674	1.49268	0.746338	+	0.665567i	$0.231810\pi$
$114$	0	0
$115$	2.94997	0.275086
$116$	−0.225141	−0.0209038
$117$	0	0
$118$	−1.09489	−0.100792
$119$	−8.10320	−0.742819
$120$	0	0
$121$	−9.93374	−0.903067
$122$	17.7447	1.60653
$123$	0	0
$124$	−1.43253	−0.128645
$125$	−3.82807	−0.342393
$126$	0	0
$127$	13.0341	1.15659	0.578296	−	0.815827i	$-0.303718\pi$
$127$	13.0341	1.15659	0.578296	+	0.815827i	$0.303718\pi$
$128$	12.8402	1.13493
$129$	0	0
$130$	−30.6791	−2.69074
$131$	−9.42367	−0.823350	−0.411675	−	0.911331i	$-0.635056\pi$
$131$	−9.42367	−0.823350	−0.411675	+	0.911331i	$0.635056\pi$
$132$	0	0
$133$	3.36257	0.291572
$134$	−5.80492	−0.501469
$135$	0	0
$136$	11.3258	0.971179
$137$	0.347023	0.0296482	0.0148241	−	0.999890i	$-0.495281\pi$
$137$	0.347023	0.0296482	0.0148241	+	0.999890i	$0.495281\pi$
$138$	0	0
$139$	−9.23248	−0.783089	−0.391544	−	0.920159i	$-0.628059\pi$
$139$	−9.23248	−0.783089	−0.391544	+	0.920159i	$0.628059\pi$
$140$	1.25806	0.106326
$141$	0	0
$142$	5.11133	0.428934
$143$	7.19909	0.602018
$144$	0	0
$145$	−2.94997	−0.244982
$146$	−23.4311	−1.93917
$147$	0	0
$148$	0.885211	0.0727639
$149$	10.2421	0.839067	0.419534	−	0.907740i	$-0.362194\pi$
$149$	10.2421	0.839067	0.419534	+	0.907740i	$0.362194\pi$
$150$	0	0
$151$	−15.9681	−1.29946	−0.649732	−	0.760164i	$-0.725119\pi$
$151$	−15.9681	−1.29946	−0.649732	+	0.760164i	$0.725119\pi$
$152$	−4.69984	−0.381207
$153$	0	0
$154$	−2.91770	−0.235115
$155$	−18.7702	−1.50766
$156$	0	0
$157$	−17.2552	−1.37711	−0.688556	−	0.725183i	$-0.741755\pi$
$157$	−17.2552	−1.37711	−0.688556	+	0.725183i	$0.741755\pi$
$158$	−22.5649	−1.79517
$159$	0	0
$160$	−3.73983	−0.295660
$161$	1.89422	0.149285
$162$	0	0
$163$	4.35221	0.340891	0.170446	−	0.985367i	$-0.445479\pi$
$163$	4.35221	0.340891	0.170446	+	0.985367i	$0.445479\pi$
$164$	−1.08830	−0.0849820
$165$	0	0
$166$	−25.5636	−1.98412
$167$	9.54731	0.738793	0.369396	−	0.929272i	$-0.379564\pi$
$167$	9.54731	0.738793	0.369396	+	0.929272i	$0.379564\pi$
$168$	0	0
$169$	35.6063	2.73895
$170$	−18.8245	−1.44377
$171$	0	0
$172$	−1.17412	−0.0895255
$173$	14.8449	1.12864	0.564318	−	0.825557i	$-0.309139\pi$
$173$	14.8449	1.12864	0.564318	+	0.825557i	$0.309139\pi$
$174$	0	0
$175$	7.01304	0.530136
$176$	4.54301	0.342442
$177$	0	0
$178$	15.9989	1.19917
$179$	−4.86946	−0.363961	−0.181980	−	0.983302i	$-0.558251\pi$
$179$	−4.86946	−0.363961	−0.181980	+	0.983302i	$0.558251\pi$
$180$	0	0
$181$	−6.58552	−0.489498	−0.244749	−	0.969586i	$-0.578706\pi$
$181$	−6.58552	−0.489498	−0.244749	+	0.969586i	$0.578706\pi$
$182$	−19.6995	−1.46023
$183$	0	0
$184$	−2.64754	−0.195179
$185$	11.5987	0.852755
$186$	0	0
$187$	4.41731	0.323025
$188$	−0.725708	−0.0529277
$189$	0	0
$190$	7.81155	0.566709
$191$	10.1273	0.732783	0.366392	−	0.930461i	$-0.380593\pi$
$191$	10.1273	0.732783	0.366392	+	0.930461i	$0.380593\pi$
$192$	0	0
$193$	−21.1204	−1.52028	−0.760139	−	0.649761i	$-0.774869\pi$
$193$	−21.1204	−1.52028	−0.760139	+	0.649761i	$0.774869\pi$
$194$	−17.4115	−1.25007
$195$	0	0
$196$	−0.768165	−0.0548689
$197$	13.9049	0.990682	0.495341	−	0.868699i	$-0.335043\pi$
$197$	13.9049	0.990682	0.495341	+	0.868699i	$0.335043\pi$
$198$	0	0
$199$	12.3190	0.873268	0.436634	−	0.899639i	$-0.356170\pi$
$199$	12.3190	0.873268	0.436634	+	0.899639i	$0.356170\pi$
$200$	−9.80209	−0.693113
$201$	0	0
$202$	28.5013	2.00535
$203$	−1.89422	−0.132948
$204$	0	0
$205$	−14.2598	−0.995946
$206$	11.9145	0.830119
$207$	0	0
$208$	30.6732	2.12680
$209$	−1.83304	−0.126794
$210$	0	0
$211$	15.1301	1.04160	0.520798	−	0.853680i	$-0.325634\pi$
$211$	15.1301	1.04160	0.520798	+	0.853680i	$0.325634\pi$
$212$	−0.466619	−0.0320475
$213$	0	0
$214$	17.1354	1.17135
$215$	−15.3842	−1.04919
$216$	0	0
$217$	−12.0526	−0.818183
$218$	0.648269	0.0439063
$219$	0	0
$220$	−0.685810	−0.0462373
$221$	29.8245	2.00621
$222$	0	0
$223$	−23.4343	−1.56928	−0.784639	−	0.619953i	$-0.787152\pi$
$223$	−23.4343	−1.56928	−0.784639	+	0.619953i	$0.787152\pi$
$224$	−2.40140	−0.160450
$225$	0	0
$226$	23.6692	1.57445
$227$	−12.2912	−0.815792	−0.407896	−	0.913028i	$-0.633737\pi$
$227$	−12.2912	−0.815792	−0.407896	+	0.913028i	$0.633737\pi$
$228$	0	0
$229$	6.81404	0.450285	0.225142	−	0.974326i	$-0.427715\pi$
$229$	6.81404	0.450285	0.225142	+	0.974326i	$0.427715\pi$
$230$	4.40045	0.290157
$231$	0	0
$232$	2.64754	0.173820
$233$	4.79553	0.314166	0.157083	−	0.987585i	$-0.449791\pi$
$233$	4.79553	0.314166	0.157083	+	0.987585i	$0.449791\pi$
$234$	0	0
$235$	−9.50879	−0.620285
$236$	−0.165251	−0.0107569
$237$	0	0
$238$	−12.0875	−0.783514
$239$	16.9813	1.09843	0.549216	−	0.835681i	$-0.314927\pi$
$239$	16.9813	1.09843	0.549216	+	0.835681i	$0.314927\pi$
$240$	0	0
$241$	−14.1169	−0.909347	−0.454674	−	0.890658i	$-0.650244\pi$
$241$	−14.1169	−0.909347	−0.454674	+	0.890658i	$0.650244\pi$
$242$	−14.8181	−0.952541
$243$	0	0
$244$	2.67821	0.171455
$245$	−10.0651	−0.643036
$246$	0	0
$247$	−12.3762	−0.787478
$248$	16.8458	1.06971
$249$	0	0
$250$	−5.71029	−0.361150
$251$	−14.7368	−0.930176	−0.465088	−	0.885264i	$-0.653977\pi$
$251$	−14.7368	−0.930176	−0.465088	+	0.885264i	$0.653977\pi$
$252$	0	0
$253$	−1.03260	−0.0649189
$254$	19.4429	1.21996
$255$	0	0
$256$	5.33748	0.333592
$257$	−17.7855	−1.10943	−0.554715	−	0.832040i	$-0.687173\pi$
$257$	−17.7855	−1.10943	−0.554715	+	0.832040i	$0.687173\pi$
$258$	0	0
$259$	7.44771	0.462778
$260$	−4.63040	−0.287165
$261$	0	0
$262$	−14.0572	−0.868457
$263$	0.833449	0.0513927	0.0256963	−	0.999670i	$-0.491820\pi$
$263$	0.833449	0.0513927	0.0256963	+	0.999670i	$0.491820\pi$
$264$	0	0
$265$	−6.11401	−0.375580
$266$	5.01591	0.307545
$267$	0	0
$268$	−0.876137	−0.0535186
$269$	5.15392	0.314240	0.157120	−	0.987580i	$-0.449779\pi$
$269$	5.15392	0.314240	0.157120	+	0.987580i	$0.449779\pi$
$270$	0	0
$271$	−4.08378	−0.248072	−0.124036	−	0.992278i	$-0.539584\pi$
$271$	−4.08378	−0.248072	−0.124036	+	0.992278i	$0.539584\pi$
$272$	18.8208	1.14118
$273$	0	0
$274$	0.517651	0.0312724
$275$	−3.82303	−0.230537
$276$	0	0
$277$	25.9792	1.56094	0.780470	−	0.625193i	$-0.214980\pi$
$277$	25.9792	1.56094	0.780470	+	0.625193i	$0.214980\pi$
$278$	−13.7720	−0.825990
$279$	0	0
$280$	−14.7942	−0.884122
$281$	−30.6808	−1.83026	−0.915132	−	0.403155i	$-0.867914\pi$
$281$	−30.6808	−1.83026	−0.915132	+	0.403155i	$0.867914\pi$
$282$	0	0
$283$	22.1309	1.31555	0.657774	−	0.753215i	$-0.271498\pi$
$283$	22.1309	1.31555	0.657774	+	0.753215i	$0.271498\pi$
$284$	0.771454	0.0457773
$285$	0	0
$286$	10.7388	0.635000
$287$	−9.15641	−0.540486
$288$	0	0
$289$	1.30005	0.0764735
$290$	−4.40045	−0.258403
$291$	0	0
$292$	−3.53645	−0.206955
$293$	−6.90720	−0.403523	−0.201761	−	0.979435i	$-0.564667\pi$
$293$	−6.90720	−0.403523	−0.201761	+	0.979435i	$0.564667\pi$
$294$	0	0
$295$	−2.16525	−0.126066
$296$	−10.4096	−0.605047
$297$	0	0
$298$	15.2781	0.885035
$299$	−6.97182	−0.403191
$300$	0	0
$301$	−9.87841	−0.569382
$302$	−23.8194	−1.37065
$303$	0	0
$304$	−7.81004	−0.447936
$305$	35.0920	2.00936
$306$	0	0
$307$	−7.73971	−0.441729	−0.220864	−	0.975305i	$-0.570888\pi$
$307$	−7.73971	−0.441729	−0.220864	+	0.975305i	$0.570888\pi$
$308$	−0.440368	−0.0250923
$309$	0	0
$310$	−27.9993	−1.59025
$311$	29.3141	1.66225	0.831124	−	0.556087i	$-0.187698\pi$
$311$	29.3141	1.66225	0.831124	+	0.556087i	$0.187698\pi$
$312$	0	0
$313$	−28.7074	−1.62264	−0.811320	−	0.584602i	$-0.801251\pi$
$313$	−28.7074	−1.62264	−0.811320	+	0.584602i	$0.801251\pi$
$314$	−25.7394	−1.45256
$315$	0	0
$316$	−3.40572	−0.191587
$317$	14.0792	0.790764	0.395382	−	0.918517i	$-0.370612\pi$
$317$	14.0792	0.790764	0.395382	+	0.918517i	$0.370612\pi$
$318$	0	0
$319$	1.03260	0.0578144
$320$	20.3787	1.13920
$321$	0	0
$322$	2.82559	0.157464
$323$	−7.59393	−0.422538
$324$	0	0
$325$	−25.8121	−1.43180
$326$	6.49215	0.359567
$327$	0	0
$328$	12.7979	0.706644
$329$	−6.10573	−0.336620
$330$	0	0
$331$	−7.45120	−0.409555	−0.204777	−	0.978809i	$-0.565647\pi$
$331$	−7.45120	−0.409555	−0.204777	+	0.978809i	$0.565647\pi$
$332$	−3.85832	−0.211753
$333$	0	0
$334$	14.2416	0.779267
$335$	−11.4798	−0.627210
$336$	0	0
$337$	9.65865	0.526140	0.263070	−	0.964777i	$-0.415265\pi$
$337$	9.65865	0.526140	0.263070	+	0.964777i	$0.415265\pi$
$338$	53.1136	2.88900
$339$	0	0
$340$	−2.84118	−0.154085
$341$	6.57024	0.355799
$342$	0	0
$343$	−19.7225	−1.06491
$344$	13.8070	0.744423
$345$	0	0
$346$	22.1440	1.19047
$347$	13.1093	0.703746	0.351873	−	0.936048i	$-0.385545\pi$
$347$	13.1093	0.703746	0.351873	+	0.936048i	$0.385545\pi$
$348$	0	0
$349$	15.6626	0.838401	0.419201	−	0.907894i	$-0.362310\pi$
$349$	15.6626	0.838401	0.419201	+	0.907894i	$0.362310\pi$
$350$	10.4613	0.559180
$351$	0	0
$352$	1.30908	0.0697740
$353$	−24.4088	−1.29915	−0.649574	−	0.760298i	$-0.725053\pi$
$353$	−24.4088	−1.29915	−0.649574	+	0.760298i	$0.725053\pi$
$354$	0	0
$355$	10.1082	0.536487
$356$	2.41471	0.127979
$357$	0	0
$358$	−7.26373	−0.383900
$359$	28.3858	1.49815	0.749074	−	0.662487i	$-0.230499\pi$
$359$	28.3858	1.49815	0.749074	+	0.662487i	$0.230499\pi$
$360$	0	0
$361$	−15.8488	−0.834145
$362$	−9.82356	−0.516315
$363$	0	0
$364$	−2.97325	−0.155841
$365$	−46.3373	−2.42541
$366$	0	0
$367$	−12.2491	−0.639398	−0.319699	−	0.947519i	$-0.603582\pi$
$367$	−12.2491	−0.639398	−0.319699	+	0.947519i	$0.603582\pi$
$368$	−4.39959	−0.229345
$369$	0	0
$370$	17.3017	0.899473
$371$	−3.92589	−0.203822
$372$	0	0
$373$	−17.4400	−0.903009	−0.451505	−	0.892269i	$-0.649113\pi$
$373$	−17.4400	−0.903009	−0.451505	+	0.892269i	$0.649113\pi$
$374$	6.58925	0.340722
$375$	0	0
$376$	8.53395	0.440105
$377$	6.97182	0.359067
$378$	0	0
$379$	20.8320	1.07007	0.535034	−	0.844831i	$-0.320299\pi$
$379$	20.8320	1.07007	0.535034	+	0.844831i	$0.320299\pi$
$380$	1.17900	0.0604813
$381$	0	0
$382$	15.1067	0.772928
$383$	−18.4387	−0.942175	−0.471087	−	0.882087i	$-0.656138\pi$
$383$	−18.4387	−0.942175	−0.471087	+	0.882087i	$0.656138\pi$
$384$	0	0
$385$	−5.77005	−0.294069
$386$	−31.5050	−1.60356
$387$	0	0
$388$	−2.62792	−0.133412
$389$	−10.0238	−0.508227	−0.254113	−	0.967174i	$-0.581784\pi$
$389$	−10.0238	−0.508227	−0.254113	+	0.967174i	$0.581784\pi$
$390$	0	0
$391$	−4.27786	−0.216340
$392$	9.03322	0.456247
$393$	0	0
$394$	20.7418	1.04496
$395$	−44.6244	−2.24530
$396$	0	0
$397$	−14.5964	−0.732572	−0.366286	−	0.930502i	$-0.619371\pi$
$397$	−14.5964	−0.732572	−0.366286	+	0.930502i	$0.619371\pi$
$398$	18.3761	0.921110
$399$	0	0
$400$	−16.2888	−0.814439
$401$	−5.26010	−0.262677	−0.131339	−	0.991338i	$-0.541927\pi$
$401$	−5.26010	−0.262677	−0.131339	+	0.991338i	$0.541927\pi$
$402$	0	0
$403$	44.3605	2.20975
$404$	4.30171	0.214018
$405$	0	0
$406$	−2.82559	−0.140232
$407$	−4.05998	−0.201246
$408$	0	0
$409$	38.4905	1.90323	0.951616	−	0.307290i	$-0.0994223\pi$
$409$	38.4905	1.90323	0.951616	+	0.307290i	$0.0994223\pi$
$410$	−21.2712	−1.05051
$411$	0	0
$412$	1.79825	0.0885933
$413$	−1.39034	−0.0684141
$414$	0	0
$415$	−50.5548	−2.48164
$416$	8.83854	0.433345
$417$	0	0
$418$	−2.73433	−0.133740
$419$	−22.8708	−1.11731	−0.558655	−	0.829400i	$-0.688683\pi$
$419$	−22.8708	−1.11731	−0.558655	+	0.829400i	$0.688683\pi$
$420$	0	0
$421$	−17.1008	−0.833444	−0.416722	−	0.909034i	$-0.636821\pi$
$421$	−17.1008	−0.833444	−0.416722	+	0.909034i	$0.636821\pi$
$422$	22.5694	1.09866
$423$	0	0
$424$	5.48720	0.266482
$425$	−15.8381	−0.768259
$426$	0	0
$427$	22.5331	1.09045
$428$	2.58624	0.125011
$429$	0	0
$430$	−22.9484	−1.10667
$431$	−34.8461	−1.67848	−0.839239	−	0.543763i	$-0.816999\pi$
$431$	−34.8461	−1.67848	−0.839239	+	0.543763i	$0.816999\pi$
$432$	0	0
$433$	18.5729	0.892556	0.446278	−	0.894894i	$-0.352749\pi$
$433$	18.5729	0.892556	0.446278	+	0.894894i	$0.352749\pi$
$434$	−17.9787	−0.863007
$435$	0	0
$436$	0.0978432	0.00468584
$437$	1.77517	0.0849180
$438$	0	0
$439$	−38.7264	−1.84831	−0.924154	−	0.382019i	$-0.875229\pi$
$439$	−38.7264	−1.84831	−0.924154	+	0.382019i	$0.875229\pi$
$440$	8.06477	0.384473
$441$	0	0
$442$	44.4889	2.11612
$443$	−3.58006	−0.170094	−0.0850468	−	0.996377i	$-0.527104\pi$
$443$	−3.58006	−0.170094	−0.0850468	+	0.996377i	$0.527104\pi$
$444$	0	0
$445$	31.6394	1.49985
$446$	−34.9567	−1.65525
$447$	0	0
$448$	13.0855	0.618229
$449$	−5.45892	−0.257622	−0.128811	−	0.991669i	$-0.541116\pi$
$449$	−5.45892	−0.257622	−0.128811	+	0.991669i	$0.541116\pi$
$450$	0	0
$451$	4.99144	0.235038
$452$	3.57239	0.168031
$453$	0	0
$454$	−18.3346	−0.860485
$455$	−38.9578	−1.82637
$456$	0	0
$457$	−4.09776	−0.191685	−0.0958426	−	0.995397i	$-0.530555\pi$
$457$	−4.09776	−0.191685	−0.0958426	+	0.995397i	$0.530555\pi$
$458$	10.1644	0.474953
$459$	0	0
$460$	0.664159	0.0309666
$461$	16.6933	0.777485	0.388743	−	0.921346i	$-0.372910\pi$
$461$	16.6933	0.777485	0.388743	+	0.921346i	$0.372910\pi$
$462$	0	0
$463$	14.0355	0.652284	0.326142	−	0.945321i	$-0.394251\pi$
$463$	14.0355	0.652284	0.326142	+	0.945321i	$0.394251\pi$
$464$	4.39959	0.204246
$465$	0	0
$466$	7.15345	0.331377
$467$	−32.8549	−1.52034	−0.760171	−	0.649723i	$-0.774885\pi$
$467$	−32.8549	−1.52034	−0.760171	+	0.649723i	$0.774885\pi$
$468$	0	0
$469$	−7.37136	−0.340378
$470$	−14.1842	−0.654267
$471$	0	0
$472$	1.94327	0.0894462
$473$	5.38503	0.247604
$474$	0	0
$475$	6.57229	0.301557
$476$	−1.82436	−0.0836195
$477$	0	0
$478$	25.3309	1.15861
$479$	7.31704	0.334324	0.167162	−	0.985929i	$-0.446540\pi$
$479$	7.31704	0.334324	0.167162	+	0.985929i	$0.446540\pi$
$480$	0	0
$481$	−27.4119	−1.24987
$482$	−21.0580	−0.959165
$483$	0	0
$484$	−2.23649	−0.101659
$485$	−34.4330	−1.56352
$486$	0	0
$487$	38.0803	1.72558	0.862791	−	0.505561i	$-0.168714\pi$
$487$	38.0803	1.72558	0.862791	+	0.505561i	$0.168714\pi$
$488$	−31.4944	−1.42568
$489$	0	0
$490$	−15.0140	−0.678264
$491$	−31.5128	−1.42215	−0.711075	−	0.703116i	$-0.751791\pi$
$491$	−31.5128	−1.42215	−0.711075	+	0.703116i	$0.751791\pi$
$492$	0	0
$493$	4.27786	0.192665
$494$	−18.4614	−0.830620
$495$	0	0
$496$	27.9939	1.25696
$497$	6.49061	0.291144
$498$	0	0
$499$	20.8639	0.933998	0.466999	−	0.884258i	$-0.345335\pi$
$499$	20.8639	0.933998	0.466999	+	0.884258i	$0.345335\pi$
$500$	−0.861854	−0.0385433
$501$	0	0
$502$	−21.9827	−0.981135
$503$	11.1822	0.498590	0.249295	−	0.968428i	$-0.419801\pi$
$503$	11.1822	0.498590	0.249295	+	0.968428i	$0.419801\pi$
$504$	0	0
$505$	56.3644	2.50818
$506$	−1.54032	−0.0684754
$507$	0	0
$508$	2.93452	0.130198
$509$	13.5566	0.600884	0.300442	−	0.953800i	$-0.402866\pi$
$509$	13.5566	0.600884	0.300442	+	0.953800i	$0.402866\pi$
$510$	0	0
$511$	−29.7539	−1.31623
$512$	−17.7186	−0.783060
$513$	0	0
$514$	−26.5305	−1.17021
$515$	23.5621	1.03827
$516$	0	0
$517$	3.32842	0.146384
$518$	11.1097	0.488131
$519$	0	0
$520$	54.4511	2.38784
$521$	17.8446	0.781786	0.390893	−	0.920436i	$-0.372166\pi$
$521$	17.8446	0.781786	0.390893	+	0.920436i	$0.372166\pi$
$522$	0	0
$523$	−15.8972	−0.695135	−0.347568	−	0.937655i	$-0.612992\pi$
$523$	−15.8972	−0.695135	−0.347568	+	0.937655i	$0.612992\pi$
$524$	−2.12165	−0.0926849
$525$	0	0
$526$	1.24325	0.0542082
$527$	27.2193	1.18569
$528$	0	0
$529$	1.00000	0.0434783
$530$	−9.12020	−0.396156
$531$	0	0
$532$	0.757051	0.0328223
$533$	33.7009	1.45975
$534$	0	0
$535$	33.8870	1.46506
$536$	10.3029	0.445018
$537$	0	0
$538$	7.68805	0.331456
$539$	3.52315	0.151753
$540$	0	0
$541$	−1.83989	−0.0791031	−0.0395515	−	0.999218i	$-0.512593\pi$
$541$	−1.83989	−0.0791031	−0.0395515	+	0.999218i	$0.512593\pi$
$542$	−6.09174	−0.261663
$543$	0	0
$544$	5.42325	0.232520
$545$	1.28202	0.0549156
$546$	0	0
$547$	16.3989	0.701166	0.350583	−	0.936532i	$-0.385984\pi$
$547$	16.3989	0.701166	0.350583	+	0.936532i	$0.385984\pi$
$548$	0.0781290	0.00333751
$549$	0	0
$550$	−5.70277	−0.243167
$551$	−1.77517	−0.0756249
$552$	0	0
$553$	−28.6540	−1.21849
$554$	38.7530	1.64646
$555$	0	0
$556$	−2.07861	−0.0881526
$557$	5.73609	0.243046	0.121523	−	0.992589i	$-0.461222\pi$
$557$	5.73609	0.243046	0.121523	+	0.992589i	$0.461222\pi$
$558$	0	0
$559$	36.3583	1.53779
$560$	−24.5845	−1.03888
$561$	0	0
$562$	−45.7663	−1.93053
$563$	43.4648	1.83182	0.915911	−	0.401380i	$-0.131469\pi$
$563$	43.4648	1.83182	0.915911	+	0.401380i	$0.131469\pi$
$564$	0	0
$565$	46.8083	1.96924
$566$	33.0125	1.38762
$567$	0	0
$568$	−9.07190	−0.380648
$569$	−4.88842	−0.204933	−0.102467	−	0.994736i	$-0.532673\pi$
$569$	−4.88842	−0.204933	−0.102467	+	0.994736i	$0.532673\pi$
$570$	0	0
$571$	−30.1287	−1.26085	−0.630423	−	0.776252i	$-0.717119\pi$
$571$	−30.1287	−1.26085	−0.630423	+	0.776252i	$0.717119\pi$
$572$	1.62081	0.0677695
$573$	0	0
$574$	−13.6585	−0.570096
$575$	3.70234	0.154398
$576$	0	0
$577$	−3.64163	−0.151603	−0.0758016	−	0.997123i	$-0.524152\pi$
$577$	−3.64163	−0.151603	−0.0758016	+	0.997123i	$0.524152\pi$
$578$	1.93927	0.0806630
$579$	0	0
$580$	−0.664159	−0.0275777
$581$	−32.4619	−1.34675
$582$	0	0
$583$	2.14013	0.0886349
$584$	41.5868	1.72088
$585$	0	0
$586$	−10.3034	−0.425630
$587$	−32.0472	−1.32273	−0.661364	−	0.750065i	$-0.730022\pi$
$587$	−32.0472	−1.32273	−0.661364	+	0.750065i	$0.730022\pi$
$588$	0	0
$589$	−11.2951	−0.465407
$590$	−3.22988	−0.132972
$591$	0	0
$592$	−17.2984	−0.710958
$593$	−6.87029	−0.282129	−0.141065	−	0.990000i	$-0.545053\pi$
$593$	−6.87029	−0.282129	−0.141065	+	0.990000i	$0.545053\pi$
$594$	0	0
$595$	−23.9042	−0.979977
$596$	2.30592	0.0944542
$597$	0	0
$598$	−10.3998	−0.425280
$599$	−0.0766761	−0.00313290	−0.00156645	−	0.999999i	$-0.500499\pi$
$599$	−0.0766761	−0.00313290	−0.00156645	+	0.999999i	$0.500499\pi$
$600$	0	0
$601$	−24.5304	−1.00062	−0.500308	−	0.865848i	$-0.666780\pi$
$601$	−24.5304	−1.00062	−0.500308	+	0.865848i	$0.666780\pi$
$602$	−14.7355	−0.600575
$603$	0	0
$604$	−3.59507	−0.146281
$605$	−29.3043	−1.19139
$606$	0	0
$607$	45.8348	1.86038	0.930189	−	0.367081i	$-0.119643\pi$
$607$	45.8348	1.86038	0.930189	+	0.367081i	$0.119643\pi$
$608$	−2.25048	−0.0912688
$609$	0	0
$610$	52.3464	2.11944
$611$	22.4726	0.909145
$612$	0	0
$613$	17.1955	0.694518	0.347259	−	0.937769i	$-0.387112\pi$
$613$	17.1955	0.694518	0.347259	+	0.937769i	$0.387112\pi$
$614$	−11.5453	−0.465929
$615$	0	0
$616$	5.17851	0.208648
$617$	−45.2287	−1.82084	−0.910420	−	0.413686i	$-0.864241\pi$
$617$	−45.2287	−1.82084	−0.910420	+	0.413686i	$0.864241\pi$
$618$	0	0
$619$	−20.5688	−0.826729	−0.413364	−	0.910566i	$-0.635646\pi$
$619$	−20.5688	−0.826729	−0.413364	+	0.910566i	$0.635646\pi$
$620$	−4.22593	−0.169717
$621$	0	0
$622$	43.7275	1.75331
$623$	20.3161	0.813949
$624$	0	0
$625$	−29.8044	−1.19218
$626$	−42.8226	−1.71154
$627$	0	0
$628$	−3.88484	−0.155022
$629$	−16.8197	−0.670646
$630$	0	0
$631$	−42.4889	−1.69145	−0.845727	−	0.533615i	$-0.820833\pi$
$631$	−42.4889	−1.69145	−0.845727	+	0.533615i	$0.820833\pi$
$632$	40.0495	1.59308
$633$	0	0
$634$	21.0017	0.834086
$635$	38.4503	1.52585
$636$	0	0
$637$	23.7874	0.942490
$638$	1.54032	0.0609817
$639$	0	0
$640$	37.8784	1.49727
$641$	28.4650	1.12430	0.562150	−	0.827035i	$-0.309974\pi$
$641$	28.4650	1.12430	0.562150	+	0.827035i	$0.309974\pi$
$642$	0	0
$643$	9.66714	0.381235	0.190617	−	0.981664i	$-0.438951\pi$
$643$	9.66714	0.381235	0.190617	+	0.981664i	$0.438951\pi$
$644$	0.426466	0.0168051
$645$	0	0
$646$	−11.3278	−0.445686
$647$	−5.00023	−0.196579	−0.0982896	−	0.995158i	$-0.531337\pi$
$647$	−5.00023	−0.196579	−0.0982896	+	0.995158i	$0.531337\pi$
$648$	0	0
$649$	0.757917	0.0297508
$650$	−38.5036	−1.51024
$651$	0	0
$652$	0.979860	0.0383743
$653$	−49.4018	−1.93324	−0.966621	−	0.256211i	$-0.917526\pi$
$653$	−49.4018	−1.93324	−0.966621	+	0.256211i	$0.917526\pi$
$654$	0	0
$655$	−27.7996	−1.08622
$656$	21.2670	0.830339
$657$	0	0
$658$	−9.10786	−0.355062
$659$	25.2486	0.983547	0.491774	−	0.870723i	$-0.336349\pi$
$659$	25.2486	0.983547	0.491774	+	0.870723i	$0.336349\pi$
$660$	0	0
$661$	7.87807	0.306421	0.153211	−	0.988194i	$-0.451039\pi$
$661$	7.87807	0.306421	0.153211	+	0.988194i	$0.451039\pi$
$662$	−11.1149	−0.431992
$663$	0	0
$664$	45.3719	1.76077
$665$	9.91948	0.384661
$666$	0	0
$667$	−1.00000	−0.0387202
$668$	2.14949	0.0831662
$669$	0	0
$670$	−17.1244	−0.661572
$671$	−12.2835	−0.474199
$672$	0	0
$673$	9.40658	0.362597	0.181299	−	0.983428i	$-0.441970\pi$
$673$	9.40658	0.362597	0.181299	+	0.983428i	$0.441970\pi$
$674$	14.4077	0.554964
$675$	0	0
$676$	8.01644	0.308325
$677$	10.4280	0.400780	0.200390	−	0.979716i	$-0.435779\pi$
$677$	10.4280	0.400780	0.200390	+	0.979716i	$0.435779\pi$
$678$	0	0
$679$	−22.1100	−0.848502
$680$	33.4108	1.28125
$681$	0	0
$682$	9.80077	0.375291
$683$	4.62535	0.176984	0.0884921	−	0.996077i	$-0.471795\pi$
$683$	4.62535	0.176984	0.0884921	+	0.996077i	$0.471795\pi$
$684$	0	0
$685$	1.02371	0.0391139
$686$	−29.4198	−1.12326
$687$	0	0
$688$	22.9440	0.874732
$689$	14.4495	0.550484
$690$	0	0
$691$	34.8517	1.32582	0.662910	−	0.748699i	$-0.269321\pi$
$691$	34.8517	1.32582	0.662910	+	0.748699i	$0.269321\pi$
$692$	3.34219	0.127051
$693$	0	0
$694$	19.5551	0.742300
$695$	−27.2356	−1.03310
$696$	0	0
$697$	20.6786	0.783257
$698$	23.3638	0.884333
$699$	0	0
$700$	1.57892	0.0596777
$701$	2.09402	0.0790899	0.0395450	−	0.999218i	$-0.487409\pi$
$701$	2.09402	0.0790899	0.0395450	+	0.999218i	$0.487409\pi$
$702$	0	0
$703$	6.97964	0.263242
$704$	−7.13329	−0.268846
$705$	0	0
$706$	−36.4103	−1.37032
$707$	36.1924	1.36115
$708$	0	0
$709$	24.2955	0.912437	0.456218	−	0.889868i	$-0.349204\pi$
$709$	24.2955	0.912437	0.456218	+	0.889868i	$0.349204\pi$
$710$	15.0783	0.565878
$711$	0	0
$712$	−28.3958	−1.06418
$713$	−6.36283	−0.238290
$714$	0	0
$715$	21.2371	0.794224
$716$	−1.09631	−0.0409712
$717$	0	0
$718$	42.3429	1.58022
$719$	−22.2755	−0.830736	−0.415368	−	0.909653i	$-0.636347\pi$
$719$	−22.2755	−0.830736	−0.415368	+	0.909653i	$0.636347\pi$
$720$	0	0
$721$	15.1295	0.563453
$722$	−23.6415	−0.879844
$723$	0	0
$724$	−1.48267	−0.0551030
$725$	−3.70234	−0.137501
$726$	0	0
$727$	−7.88909	−0.292590	−0.146295	−	0.989241i	$-0.546735\pi$
$727$	−7.88909	−0.292590	−0.146295	+	0.989241i	$0.546735\pi$
$728$	34.9639	1.29585
$729$	0	0
$730$	−69.1210	−2.55828
$731$	22.3091	0.825133
$732$	0	0
$733$	31.2988	1.15605	0.578023	−	0.816020i	$-0.303824\pi$
$733$	31.2988	1.15605	0.578023	+	0.816020i	$0.303824\pi$
$734$	−18.2719	−0.674427
$735$	0	0
$736$	−1.26775	−0.0467299
$737$	4.01836	0.148018
$738$	0	0
$739$	35.1270	1.29217	0.646083	−	0.763267i	$-0.276406\pi$
$739$	35.1270	1.29217	0.646083	+	0.763267i	$0.276406\pi$
$740$	2.61135	0.0959951
$741$	0	0
$742$	−5.85621	−0.214988
$743$	25.7881	0.946072	0.473036	−	0.881043i	$-0.343158\pi$
$743$	25.7881	0.946072	0.473036	+	0.881043i	$0.343158\pi$
$744$	0	0
$745$	30.2140	1.10695
$746$	−26.0151	−0.952480
$747$	0	0
$748$	0.994516	0.0363631
$749$	21.7593	0.795068
$750$	0	0
$751$	−34.9502	−1.27535	−0.637676	−	0.770305i	$-0.720104\pi$
$751$	−34.9502	−1.27535	−0.637676	+	0.770305i	$0.720104\pi$
$752$	14.1814	0.517143
$753$	0	0
$754$	10.3998	0.378739
$755$	−47.1054	−1.71434
$756$	0	0
$757$	−35.0761	−1.27486	−0.637432	−	0.770507i	$-0.720003\pi$
$757$	−35.0761	−1.27486	−0.637432	+	0.770507i	$0.720003\pi$
$758$	31.0749	1.12869
$759$	0	0
$760$	−13.8644	−0.502915
$761$	29.0536	1.05319	0.526596	−	0.850116i	$-0.323468\pi$
$761$	29.0536	1.05319	0.526596	+	0.850116i	$0.323468\pi$
$762$	0	0
$763$	0.823203	0.0298019
$764$	2.28006	0.0824897
$765$	0	0
$766$	−27.5049	−0.993792
$767$	5.11725	0.184773
$768$	0	0
$769$	29.9107	1.07861	0.539303	−	0.842112i	$-0.318688\pi$
$769$	29.9107	1.07861	0.539303	+	0.842112i	$0.318688\pi$
$770$	−8.60713	−0.310180
$771$	0	0
$772$	−4.75506	−0.171138
$773$	−27.5941	−0.992491	−0.496245	−	0.868182i	$-0.665288\pi$
$773$	−27.5941	−0.992491	−0.496245	+	0.868182i	$0.665288\pi$
$774$	0	0
$775$	−23.5573	−0.846205
$776$	30.9030	1.10935
$777$	0	0
$778$	−14.9524	−0.536070
$779$	−8.58095	−0.307444
$780$	0	0
$781$	−3.53824	−0.126608
$782$	−6.38124	−0.228193
$783$	0	0
$784$	15.0111	0.536111
$785$	−50.9023	−1.81678
$786$	0	0
$787$	−0.756025	−0.0269494	−0.0134747	−	0.999909i	$-0.504289\pi$
$787$	−0.756025	−0.0269494	−0.0134747	+	0.999909i	$0.504289\pi$
$788$	3.13056	0.111522
$789$	0	0
$790$	−66.5659	−2.36831
$791$	30.0563	1.06868
$792$	0	0
$793$	−82.9348	−2.94510
$794$	−21.7733	−0.772706
$795$	0	0
$796$	2.77350	0.0983042
$797$	−35.5472	−1.25915	−0.629574	−	0.776941i	$-0.716770\pi$
$797$	−35.5472	−1.25915	−0.629574	+	0.776941i	$0.716770\pi$
$798$	0	0
$799$	13.7890	0.487821
$800$	−4.69364	−0.165945
$801$	0	0
$802$	−7.84645	−0.277068
$803$	16.2198	0.572383
$804$	0	0
$805$	5.58790	0.196947
$806$	66.1722	2.33081
$807$	0	0
$808$	−50.5859	−1.77960
$809$	−3.31740	−0.116634	−0.0583168	−	0.998298i	$-0.518573\pi$
$809$	−3.31740	−0.116634	−0.0583168	+	0.998298i	$0.518573\pi$
$810$	0	0
$811$	47.0806	1.65322	0.826611	−	0.562773i	$-0.190266\pi$
$811$	47.0806	1.65322	0.826611	+	0.562773i	$0.190266\pi$
$812$	−0.426466	−0.0149660
$813$	0	0
$814$	−6.05623	−0.212271
$815$	12.8389	0.449727
$816$	0	0
$817$	−9.25757	−0.323881
$818$	57.4159	2.00750
$819$	0	0
$820$	−3.21046	−0.112114
$821$	40.1843	1.40244	0.701221	−	0.712944i	$-0.252639\pi$
$821$	40.1843	1.40244	0.701221	+	0.712944i	$0.252639\pi$
$822$	0	0
$823$	−12.5502	−0.437472	−0.218736	−	0.975784i	$-0.570193\pi$
$823$	−12.5502	−0.437472	−0.218736	+	0.975784i	$0.570193\pi$
$824$	−21.1465	−0.736672
$825$	0	0
$826$	−2.07396	−0.0721621
$827$	2.46442	0.0856961	0.0428481	−	0.999082i	$-0.486357\pi$
$827$	2.46442	0.0856961	0.0428481	+	0.999082i	$0.486357\pi$
$828$	0	0
$829$	−18.5052	−0.642712	−0.321356	−	0.946958i	$-0.604139\pi$
$829$	−18.5052	−0.642712	−0.321356	+	0.946958i	$0.604139\pi$
$830$	−75.4121	−2.61759
$831$	0	0
$832$	−48.1620	−1.66972
$833$	14.5957	0.505713
$834$	0	0
$835$	28.1643	0.974666
$836$	−0.412692	−0.0142733
$837$	0	0
$838$	−34.1161	−1.17852
$839$	11.8023	0.407459	0.203730	−	0.979027i	$-0.434694\pi$
$839$	11.8023	0.407459	0.203730	+	0.979027i	$0.434694\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	−25.5092	−0.879104
$843$	0	0
$844$	3.40640	0.117253
$845$	105.038	3.61341
$846$	0	0
$847$	−18.8167	−0.646549
$848$	9.11844	0.313128
$849$	0	0
$850$	−23.6255	−0.810348
$851$	3.93181	0.134781
$852$	0	0
$853$	−3.37812	−0.115665	−0.0578323	−	0.998326i	$-0.518419\pi$
$853$	−3.37812	−0.115665	−0.0578323	+	0.998326i	$0.518419\pi$
$854$	33.6124	1.15019
$855$	0	0
$856$	−30.4129	−1.03949
$857$	3.82113	0.130527	0.0652636	−	0.997868i	$-0.479211\pi$
$857$	3.82113	0.130527	0.0652636	+	0.997868i	$0.479211\pi$
$858$	0	0
$859$	22.5475	0.769310	0.384655	−	0.923061i	$-0.374321\pi$
$859$	22.5475	0.769310	0.384655	+	0.923061i	$0.374321\pi$
$860$	−3.46361	−0.118108
$861$	0	0
$862$	−51.9796	−1.77043
$863$	32.9670	1.12221	0.561105	−	0.827745i	$-0.310376\pi$
$863$	32.9670	1.12221	0.561105	+	0.827745i	$0.310376\pi$
$864$	0	0
$865$	43.7920	1.48897
$866$	27.7050	0.941454
$867$	0	0
$868$	−2.71353	−0.0921033
$869$	15.6202	0.529878
$870$	0	0
$871$	27.1309	0.919295
$872$	−1.15059	−0.0389637
$873$	0	0
$874$	2.64801	0.0895702
$875$	−7.25120	−0.245135
$876$	0	0
$877$	−11.5918	−0.391426	−0.195713	−	0.980661i	$-0.562702\pi$
$877$	−11.5918	−0.391426	−0.195713	+	0.980661i	$0.562702\pi$
$878$	−57.7678	−1.94957
$879$	0	0
$880$	13.4018	0.451773
$881$	−4.62651	−0.155871	−0.0779356	−	0.996958i	$-0.524833\pi$
$881$	−4.62651	−0.155871	−0.0779356	+	0.996958i	$0.524833\pi$
$882$	0	0
$883$	−34.4453	−1.15918	−0.579588	−	0.814909i	$-0.696787\pi$
$883$	−34.4453	−1.15918	−0.579588	+	0.814909i	$0.696787\pi$
$884$	6.71470	0.225840
$885$	0	0
$886$	−5.34033	−0.179412
$887$	−10.2165	−0.343037	−0.171519	−	0.985181i	$-0.554867\pi$
$887$	−10.2165	−0.343037	−0.171519	+	0.985181i	$0.554867\pi$
$888$	0	0
$889$	24.6895	0.828060
$890$	47.1963	1.58202
$891$	0	0
$892$	−5.27602	−0.176654
$893$	−5.72200	−0.191479
$894$	0	0
$895$	−14.3648	−0.480162
$896$	24.3222	0.812549
$897$	0	0
$898$	−8.14302	−0.271736
$899$	6.36283	0.212212
$900$	0	0
$901$	8.86613	0.295374
$902$	7.44569	0.247914
$903$	0	0
$904$	−42.0095	−1.39721
$905$	−19.4271	−0.645779
$906$	0	0
$907$	−26.8200	−0.890543	−0.445271	−	0.895396i	$-0.646893\pi$
$907$	−26.8200	−0.890543	−0.445271	+	0.895396i	$0.646893\pi$
$908$	−2.76724	−0.0918341
$909$	0	0
$910$	−58.1130	−1.92643
$911$	49.8480	1.65154	0.825768	−	0.564010i	$-0.190742\pi$
$911$	49.8480	1.65154	0.825768	+	0.564010i	$0.190742\pi$
$912$	0	0
$913$	17.6960	0.585653
$914$	−6.11259	−0.202187
$915$	0	0
$916$	1.53412	0.0506887
$917$	−17.8505	−0.589476
$918$	0	0
$919$	−2.73771	−0.0903086	−0.0451543	−	0.998980i	$-0.514378\pi$
$919$	−2.73771	−0.0903086	−0.0451543	+	0.998980i	$0.514378\pi$
$920$	−7.81017	−0.257494
$921$	0	0
$922$	24.9013	0.820080
$923$	−23.8892	−0.786323
$924$	0	0
$925$	14.5569	0.478627
$926$	20.9366	0.688019
$927$	0	0
$928$	1.26775	0.0416160
$929$	−59.1266	−1.93988	−0.969941	−	0.243342i	$-0.921756\pi$
$929$	−59.1266	−1.93988	−0.969941	+	0.243342i	$0.921756\pi$
$930$	0	0
$931$	−6.05676	−0.198502
$932$	1.07967	0.0353658
$933$	0	0
$934$	−49.0093	−1.60363
$935$	13.0309	0.426157
$936$	0	0
$937$	12.0668	0.394206	0.197103	−	0.980383i	$-0.436847\pi$
$937$	12.0668	0.394206	0.197103	+	0.980383i	$0.436847\pi$
$938$	−10.9958	−0.359025
$939$	0	0
$940$	−2.14082	−0.0698258
$941$	19.9795	0.651312	0.325656	−	0.945488i	$-0.394415\pi$
$941$	19.9795	0.651312	0.325656	+	0.945488i	$0.394415\pi$
$942$	0	0
$943$	−4.83387	−0.157412
$944$	3.22926	0.105103
$945$	0	0
$946$	8.03280	0.261169
$947$	−50.1300	−1.62901	−0.814503	−	0.580159i	$-0.802991\pi$
$947$	−50.1300	−1.62901	−0.814503	+	0.580159i	$0.802991\pi$
$948$	0	0
$949$	109.511	3.55489
$950$	9.80382	0.318078
$951$	0	0
$952$	21.4536	0.695314
$953$	60.8658	1.97164	0.985819	−	0.167813i	$-0.0536704\pi$
$953$	60.8658	1.97164	0.985819	+	0.167813i	$0.0536704\pi$
$954$	0	0
$955$	29.8752	0.966737
$956$	3.82319	0.123651
$957$	0	0
$958$	10.9148	0.352640
$959$	0.657338	0.0212265
$960$	0	0
$961$	9.48556	0.305986
$962$	−40.8900	−1.31835
$963$	0	0
$964$	−3.17828	−0.102366
$965$	−62.3045	−2.00565
$966$	0	0
$967$	33.4233	1.07482	0.537411	−	0.843321i	$-0.319402\pi$
$967$	33.4233	1.07482	0.537411	+	0.843321i	$0.319402\pi$
$968$	26.3000	0.845313
$969$	0	0
$970$	−51.3635	−1.64918
$971$	47.0356	1.50944	0.754721	−	0.656045i	$-0.227772\pi$
$971$	47.0356	1.50944	0.754721	+	0.656045i	$0.227772\pi$
$972$	0	0
$973$	−17.4883	−0.560651
$974$	56.8040	1.82012
$975$	0	0
$976$	−52.3363	−1.67524
$977$	10.7185	0.342916	0.171458	−	0.985191i	$-0.445152\pi$
$977$	10.7185	0.342916	0.171458	+	0.985191i	$0.445152\pi$
$978$	0	0
$979$	−11.0750	−0.353957
$980$	−2.26607	−0.0723868
$981$	0	0
$982$	−47.0073	−1.50006
$983$	18.5204	0.590709	0.295354	−	0.955388i	$-0.404562\pi$
$983$	18.5204	0.590709	0.295354	+	0.955388i	$0.404562\pi$
$984$	0	0
$985$	41.0190	1.30698
$986$	6.38124	0.203220
$987$	0	0
$988$	−2.78639	−0.0886467
$989$	−5.21503	−0.165828
$990$	0	0
$991$	30.4333	0.966744	0.483372	−	0.875415i	$-0.339412\pi$
$991$	30.4333	0.966744	0.483372	+	0.875415i	$0.339412\pi$
$992$	8.06648	0.256111
$993$	0	0
$994$	9.68199	0.307094
$995$	36.3406	1.15207
$996$	0	0
$997$	−9.96784	−0.315685	−0.157842	−	0.987464i	$-0.550454\pi$
$997$	−9.96784	−0.315685	−0.157842	+	0.987464i	$0.550454\pi$
$998$	31.1226	0.985167
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6003.2.a.i.1.5		7
3.2	odd	2		2001.2.a.j.1.3	✓	7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2001.2.a.j.1.3	✓	7	3.2	odd	2
6003.2.a.i.1.5		7	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)

\(f(q)\)	\(=\)	\(q+1.49169 q^{2} +0.225141 q^{4} +2.94997 q^{5} +1.89422 q^{7} -2.64754 q^{8} +O(q^{10})\)	\(q+1.49169 q^{2} +0.225141 q^{4} +2.94997 q^{5} +1.89422 q^{7} -2.64754 q^{8} +4.40045 q^{10} -1.03260 q^{11} -6.97182 q^{13} +2.82559 q^{14} -4.39959 q^{16} -4.27786 q^{17} +1.77517 q^{19} +0.664159 q^{20} -1.54032 q^{22} +1.00000 q^{23} +3.70234 q^{25} -10.3998 q^{26} +0.426466 q^{28} -1.00000 q^{29} -6.36283 q^{31} -1.26775 q^{32} -6.38124 q^{34} +5.58790 q^{35} +3.93181 q^{37} +2.64801 q^{38} -7.81017 q^{40} -4.83387 q^{41} -5.21503 q^{43} -0.232480 q^{44} +1.49169 q^{46} -3.22335 q^{47} -3.41193 q^{49} +5.52274 q^{50} -1.56964 q^{52} -2.07256 q^{53} -3.04614 q^{55} -5.01502 q^{56} -1.49169 q^{58} -0.733990 q^{59} +11.8957 q^{61} -9.49137 q^{62} +6.90809 q^{64} -20.5667 q^{65} -3.89150 q^{67} -0.963120 q^{68} +8.33541 q^{70} +3.42654 q^{71} -15.7077 q^{73} +5.86504 q^{74} +0.399664 q^{76} -1.95597 q^{77} -15.1271 q^{79} -12.9787 q^{80} -7.21063 q^{82} -17.1374 q^{83} -12.6196 q^{85} -7.77921 q^{86} +2.73385 q^{88} +10.7253 q^{89} -13.2062 q^{91} +0.225141 q^{92} -4.80824 q^{94} +5.23671 q^{95} -11.6723 q^{97} -5.08954 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q + q^{2} + 7 q^{4} + 5 q^{5} - 5 q^{7} - 3 q^{8}+O(q^{10}) \) 7 * q + q^2 + 7 * q^4 + 5 * q^5 - 5 * q^7 - 3 * q^8	\( 7 q + q^{2} + 7 q^{4} + 5 q^{5} - 5 q^{7} - 3 q^{8} - 11 q^{10} + 12 q^{11} - 13 q^{13} + 3 q^{14} - 13 q^{16} + 12 q^{17} - 5 q^{19} + 8 q^{20} - q^{22} + 7 q^{23} - 4 q^{25} - 2 q^{26} - 21 q^{28} - 7 q^{29} - 8 q^{31} + 5 q^{32} - 28 q^{34} - 5 q^{35} - 24 q^{37} + 6 q^{38} - 20 q^{40} - 9 q^{41} - q^{43} + 23 q^{44} + q^{46} - 27 q^{47} - 14 q^{49} - 7 q^{50} - 9 q^{52} + q^{53} - 11 q^{55} + 20 q^{56} - q^{58} - 8 q^{59} + q^{61} + 3 q^{64} - 12 q^{65} - 16 q^{67} - 15 q^{68} + 40 q^{70} + 13 q^{71} - 23 q^{73} + 8 q^{74} - 2 q^{76} - 13 q^{77} - 44 q^{79} - 30 q^{80} - 10 q^{82} - 21 q^{83} - 6 q^{86} + 21 q^{88} + 5 q^{89} - 18 q^{91} + 7 q^{92} + 28 q^{94} - 9 q^{95} - 55 q^{97} - 36 q^{98}+O(q^{100}) \) 7 * q + q^2 + 7 * q^4 + 5 * q^5 - 5 * q^7 - 3 * q^8 - 11 * q^10 + 12 * q^11 - 13 * q^13 + 3 * q^14 - 13 * q^16 + 12 * q^17 - 5 * q^19 + 8 * q^20 - q^22 + 7 * q^23 - 4 * q^25 - 2 * q^26 - 21 * q^28 - 7 * q^29 - 8 * q^31 + 5 * q^32 - 28 * q^34 - 5 * q^35 - 24 * q^37 + 6 * q^38 - 20 * q^40 - 9 * q^41 - q^43 + 23 * q^44 + q^46 - 27 * q^47 - 14 * q^49 - 7 * q^50 - 9 * q^52 + q^53 - 11 * q^55 + 20 * q^56 - q^58 - 8 * q^59 + q^61 + 3 * q^64 - 12 * q^65 - 16 * q^67 - 15 * q^68 + 40 * q^70 + 13 * q^71 - 23 * q^73 + 8 * q^74 - 2 * q^76 - 13 * q^77 - 44 * q^79 - 30 * q^80 - 10 * q^82 - 21 * q^83 - 6 * q^86 + 21 * q^88 + 5 * q^89 - 18 * q^91 + 7 * q^92 + 28 * q^94 - 9 * q^95 - 55 * q^97 - 36 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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