LMFDB - Embedded newform 6003.2.a.m.1.11

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:	$11$
Coefficient field:	$\mathbb{Q}[x]/(x^{11} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{11} - 2 x^{10} - 18 x^{9} + 30 x^{8} + 124 x^{7} - 152 x^{6} - 408 x^{5} + 285 x^{4} + 634 x^{3} + \cdots - 108 $ x^11 - 2x^10 - 18x^9 + 30x^8 + 124x^7 - 152x^6 - 408x^5 + 285x^4 + 634x^3 - 93x^2 - 369x - 108
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 3 $
Twist minimal:	no (minimal twist has level 2001)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.11
Root			$-2.51124$ 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)

$f(q)$	$=$	$q+2.51124 q^{2} +4.30634 q^{4} -2.07440 q^{5} -0.329384 q^{7} +5.79178 q^{8} +O(q^{10})$	\(q+2.51124 q^{2} +4.30634 q^{4} -2.07440 q^{5} -0.329384 q^{7} +5.79178 q^{8} -5.20933 q^{10} -4.89635 q^{11} +2.21503 q^{13} -0.827164 q^{14} +5.93189 q^{16} -4.14979 q^{17} +0.434775 q^{19} -8.93309 q^{20} -12.2959 q^{22} -1.00000 q^{23} -0.696851 q^{25} +5.56247 q^{26} -1.41844 q^{28} +1.00000 q^{29} +11.0142 q^{31} +3.31286 q^{32} -10.4211 q^{34} +0.683276 q^{35} +0.655818 q^{37} +1.09182 q^{38} -12.0145 q^{40} -7.37293 q^{41} +0.463287 q^{43} -21.0853 q^{44} -2.51124 q^{46} -4.14571 q^{47} -6.89151 q^{49} -1.74996 q^{50} +9.53866 q^{52} -2.30936 q^{53} +10.1570 q^{55} -1.90772 q^{56} +2.51124 q^{58} -9.22421 q^{59} -12.4374 q^{61} +27.6594 q^{62} -3.54439 q^{64} -4.59486 q^{65} -14.6596 q^{67} -17.8704 q^{68} +1.71587 q^{70} -12.1080 q^{71} +7.93152 q^{73} +1.64692 q^{74} +1.87229 q^{76} +1.61278 q^{77} -6.32788 q^{79} -12.3051 q^{80} -18.5152 q^{82} +16.4719 q^{83} +8.60834 q^{85} +1.16343 q^{86} -28.3586 q^{88} -7.88890 q^{89} -0.729595 q^{91} -4.30634 q^{92} -10.4109 q^{94} -0.901898 q^{95} -15.8429 q^{97} -17.3062 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 11 q - 2 q^{2} + 18 q^{4} - 2 q^{5} + 3 q^{7} - 18 q^{8}+O(q^{10}) $ 11 * q - 2 * q^2 + 18 * q^4 - 2 * q^5 + 3 * q^7 - 18 * q^8	$ 11 q - 2 q^{2} + 18 q^{4} - 2 q^{5} + 3 q^{7} - 18 q^{8} + 14 q^{10} - 11 q^{11} - 5 q^{13} - 17 q^{14} + 20 q^{16} - 15 q^{17} - 6 q^{19} - 21 q^{20} - 10 q^{22} - 11 q^{23} + 3 q^{25} + 5 q^{26} + 7 q^{28} + 11 q^{29} + 35 q^{31} - 28 q^{32} + 28 q^{34} - 15 q^{35} - 28 q^{37} + 2 q^{38} - q^{40} - 10 q^{41} - 6 q^{43} - 18 q^{44} + 2 q^{46} - 15 q^{47} + 22 q^{49} - 15 q^{50} - 36 q^{52} + 7 q^{53} - 12 q^{55} - 56 q^{56} - 2 q^{58} + 20 q^{59} - 20 q^{61} + 11 q^{62} + 36 q^{64} - 11 q^{65} - 39 q^{67} - 35 q^{68} + 38 q^{70} - 49 q^{71} - 3 q^{73} - 37 q^{74} - 18 q^{76} - 25 q^{77} + 41 q^{79} - 51 q^{80} - 19 q^{82} - 13 q^{83} - 62 q^{86} - 40 q^{88} - 34 q^{89} + 2 q^{91} - 18 q^{92} - 14 q^{94} - 25 q^{95} - 11 q^{97} - 53 q^{98}+O(q^{100}) $ 11 * q - 2 * q^2 + 18 * q^4 - 2 * q^5 + 3 * q^7 - 18 * q^8 + 14 * q^10 - 11 * q^11 - 5 * q^13 - 17 * q^14 + 20 * q^16 - 15 * q^17 - 6 * q^19 - 21 * q^20 - 10 * q^22 - 11 * q^23 + 3 * q^25 + 5 * q^26 + 7 * q^28 + 11 * q^29 + 35 * q^31 - 28 * q^32 + 28 * q^34 - 15 * q^35 - 28 * q^37 + 2 * q^38 - q^40 - 10 * q^41 - 6 * q^43 - 18 * q^44 + 2 * q^46 - 15 * q^47 + 22 * q^49 - 15 * q^50 - 36 * q^52 + 7 * q^53 - 12 * q^55 - 56 * q^56 - 2 * q^58 + 20 * q^59 - 20 * q^61 + 11 * q^62 + 36 * q^64 - 11 * q^65 - 39 * q^67 - 35 * q^68 + 38 * q^70 - 49 * q^71 - 3 * q^73 - 37 * q^74 - 18 * q^76 - 25 * q^77 + 41 * q^79 - 51 * q^80 - 19 * q^82 - 13 * q^83 - 62 * q^86 - 40 * q^88 - 34 * q^89 + 2 * q^91 - 18 * q^92 - 14 * q^94 - 25 * q^95 - 11 * q^97 - 53 * 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 $

$2$	2.51124	1.77572	0.887858	−	0.460117i	$-0.152192\pi$
$2$	2.51124	1.77572	0.887858	+	0.460117i	$0.152192\pi$
$3$	0	0
$3$	0	0
$4$	4.30634	2.15317
$5$	−2.07440	−0.927701	−0.463851	−	0.885913i	$-0.653533\pi$
$5$	−2.07440	−0.927701	−0.463851	+	0.885913i	$0.653533\pi$
$6$	0	0
$7$	−0.329384	−0.124496	−0.0622478	−	0.998061i	$-0.519827\pi$
$7$	−0.329384	−0.124496	−0.0622478	+	0.998061i	$0.519827\pi$
$8$	5.79178	2.04770
$9$	0	0
$10$	−5.20933	−1.64733
$11$	−4.89635	−1.47630	−0.738152	−	0.674635i	$-0.764301\pi$
$11$	−4.89635	−1.47630	−0.738152	+	0.674635i	$0.764301\pi$
$12$	0	0
$13$	2.21503	0.614338	0.307169	−	0.951655i	$-0.400618\pi$
$13$	2.21503	0.614338	0.307169	+	0.951655i	$0.400618\pi$
$14$	−0.827164	−0.221069
$15$	0	0
$16$	5.93189	1.48297
$17$	−4.14979	−1.00647	−0.503236	−	0.864149i	$-0.667857\pi$
$17$	−4.14979	−1.00647	−0.503236	+	0.864149i	$0.667857\pi$
$18$	0	0
$19$	0.434775	0.0997441	0.0498721	−	0.998756i	$-0.484119\pi$
$19$	0.434775	0.0997441	0.0498721	+	0.998756i	$0.484119\pi$
$20$	−8.93309	−1.99750
$21$	0	0
$22$	−12.2959	−2.62150
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	−0.696851	−0.139370
$26$	5.56247	1.09089
$27$	0	0
$28$	−1.41844	−0.268060
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	0	0
$31$	11.0142	1.97822	0.989108	−	0.147189i	$-0.0470227\pi$
$31$	11.0142	1.97822	0.989108	+	0.147189i	$0.0470227\pi$
$32$	3.31286	0.585636
$33$	0	0
$34$	−10.4211	−1.78721
$35$	0.683276	0.115495
$36$	0	0
$37$	0.655818	0.107816	0.0539079	−	0.998546i	$-0.482832\pi$
$37$	0.655818	0.107816	0.0539079	+	0.998546i	$0.482832\pi$
$38$	1.09182	0.177117
$39$	0	0
$40$	−12.0145	−1.89966
$41$	−7.37293	−1.15146	−0.575729	−	0.817640i	$-0.695282\pi$
$41$	−7.37293	−1.15146	−0.575729	+	0.817640i	$0.695282\pi$
$42$	0	0
$43$	0.463287	0.0706505	0.0353253	−	0.999376i	$-0.488753\pi$
$43$	0.463287	0.0706505	0.0353253	+	0.999376i	$0.488753\pi$
$44$	−21.0853	−3.17873
$45$	0	0
$46$	−2.51124	−0.370263
$47$	−4.14571	−0.604714	−0.302357	−	0.953195i	$-0.597774\pi$
$47$	−4.14571	−0.604714	−0.302357	+	0.953195i	$0.597774\pi$
$48$	0	0
$49$	−6.89151	−0.984501
$50$	−1.74996	−0.247482
$51$	0	0
$52$	9.53866	1.32277
$53$	−2.30936	−0.317215	−0.158608	−	0.987342i	$-0.550700\pi$
$53$	−2.30936	−0.317215	−0.158608	+	0.987342i	$0.550700\pi$
$54$	0	0
$55$	10.1570	1.36957
$56$	−1.90772	−0.254930
$57$	0	0
$58$	2.51124	0.329742
$59$	−9.22421	−1.20089	−0.600445	−	0.799666i	$-0.705010\pi$
$59$	−9.22421	−1.20089	−0.600445	+	0.799666i	$0.705010\pi$
$60$	0	0
$61$	−12.4374	−1.59245	−0.796226	−	0.605000i	$-0.793173\pi$
$61$	−12.4374	−1.59245	−0.796226	+	0.605000i	$0.793173\pi$
$62$	27.6594	3.51275
$63$	0	0
$64$	−3.54439	−0.443049
$65$	−4.59486	−0.569922
$66$	0	0
$67$	−14.6596	−1.79096	−0.895478	−	0.445106i	$-0.853166\pi$
$67$	−14.6596	−1.79096	−0.895478	+	0.445106i	$0.853166\pi$
$68$	−17.8704	−2.16711
$69$	0	0
$70$	1.71587	0.205086
$71$	−12.1080	−1.43695	−0.718476	−	0.695552i	$-0.755160\pi$
$71$	−12.1080	−1.43695	−0.718476	+	0.695552i	$0.755160\pi$
$72$	0	0
$73$	7.93152	0.928315	0.464157	−	0.885753i	$-0.346357\pi$
$73$	7.93152	0.928315	0.464157	+	0.885753i	$0.346357\pi$
$74$	1.64692	0.191450
$75$	0	0
$76$	1.87229	0.214766
$77$	1.61278	0.183793
$78$	0	0
$79$	−6.32788	−0.711942	−0.355971	−	0.934497i	$-0.615850\pi$
$79$	−6.32788	−0.711942	−0.355971	+	0.934497i	$0.615850\pi$
$80$	−12.3051	−1.37576
$81$	0	0
$82$	−18.5152	−2.04467
$83$	16.4719	1.80803	0.904014	−	0.427502i	$-0.140606\pi$
$83$	16.4719	1.80803	0.904014	+	0.427502i	$0.140606\pi$
$84$	0	0
$85$	8.60834	0.933706
$86$	1.16343	0.125455
$87$	0	0
$88$	−28.3586	−3.02303
$89$	−7.88890	−0.836222	−0.418111	−	0.908396i	$-0.637308\pi$
$89$	−7.88890	−0.836222	−0.418111	+	0.908396i	$0.637308\pi$
$90$	0	0
$91$	−0.729595	−0.0764823
$92$	−4.30634	−0.448967
$93$	0	0
$94$	−10.4109	−1.07380
$95$	−0.901898	−0.0925328
$96$	0	0
$97$	−15.8429	−1.60861	−0.804303	−	0.594219i	$-0.797461\pi$
$97$	−15.8429	−1.60861	−0.804303	+	0.594219i	$0.797461\pi$
$98$	−17.3062	−1.74819
$99$	0	0
$100$	−3.00088	−0.300088
$101$	12.1067	1.20466	0.602332	−	0.798246i	$-0.294239\pi$
$101$	12.1067	1.20466	0.602332	+	0.798246i	$0.294239\pi$
$102$	0	0
$103$	−11.8721	−1.16979	−0.584895	−	0.811109i	$-0.698864\pi$
$103$	−11.8721	−1.16979	−0.584895	+	0.811109i	$0.698864\pi$
$104$	12.8290	1.25798
$105$	0	0
$106$	−5.79937	−0.563284
$107$	3.91344	0.378327	0.189163	−	0.981946i	$-0.439422\pi$
$107$	3.91344	0.378327	0.189163	+	0.981946i	$0.439422\pi$
$108$	0	0
$109$	6.10250	0.584514	0.292257	−	0.956340i	$-0.405594\pi$
$109$	6.10250	0.584514	0.292257	+	0.956340i	$0.405594\pi$
$110$	25.5067	2.43197
$111$	0	0
$112$	−1.95387	−0.184624
$113$	3.69220	0.347333	0.173666	−	0.984805i	$-0.444439\pi$
$113$	3.69220	0.347333	0.173666	+	0.984805i	$0.444439\pi$
$114$	0	0
$115$	2.07440	0.193439
$116$	4.30634	0.399834
$117$	0	0
$118$	−23.1642	−2.13244
$119$	1.36688	0.125301
$120$	0	0
$121$	12.9742	1.17947
$122$	−31.2334	−2.82774
$123$	0	0
$124$	47.4311	4.25944
$125$	11.8176	1.05700
$126$	0	0
$127$	−11.0829	−0.983444	−0.491722	−	0.870752i	$-0.663632\pi$
$127$	−11.0829	−0.983444	−0.491722	+	0.870752i	$0.663632\pi$
$128$	−15.5265	−1.37237
$129$	0	0
$130$	−11.5388	−1.01202
$131$	−16.7632	−1.46461	−0.732304	−	0.680977i	$-0.761555\pi$
$131$	−16.7632	−1.46461	−0.732304	+	0.680977i	$0.761555\pi$
$132$	0	0
$133$	−0.143208	−0.0124177
$134$	−36.8138	−3.18023
$135$	0	0
$136$	−24.0347	−2.06096
$137$	−1.30482	−0.111478	−0.0557391	−	0.998445i	$-0.517752\pi$
$137$	−1.30482	−0.111478	−0.0557391	+	0.998445i	$0.517752\pi$
$138$	0	0
$139$	0.506684	0.0429764	0.0214882	−	0.999769i	$-0.493160\pi$
$139$	0.506684	0.0429764	0.0214882	+	0.999769i	$0.493160\pi$
$140$	2.94242	0.248680
$141$	0	0
$142$	−30.4061	−2.55162
$143$	−10.8455	−0.906949
$144$	0	0
$145$	−2.07440	−0.172270
$146$	19.9180	1.64842
$147$	0	0
$148$	2.82418	0.232146
$149$	12.4088	1.01657	0.508286	−	0.861188i	$-0.330279\pi$
$149$	12.4088	1.01657	0.508286	+	0.861188i	$0.330279\pi$
$150$	0	0
$151$	17.4673	1.42147	0.710736	−	0.703459i	$-0.248362\pi$
$151$	17.4673	1.42147	0.710736	+	0.703459i	$0.248362\pi$
$152$	2.51812	0.204247
$153$	0	0
$154$	4.05008	0.326365
$155$	−22.8480	−1.83519
$156$	0	0
$157$	0.289576	0.0231107	0.0115553	−	0.999933i	$-0.496322\pi$
$157$	0.289576	0.0231107	0.0115553	+	0.999933i	$0.496322\pi$
$158$	−15.8908	−1.26421
$159$	0	0
$160$	−6.87221	−0.543296
$161$	0.329384	0.0259591
$162$	0	0
$163$	16.7973	1.31566	0.657832	−	0.753165i	$-0.271474\pi$
$163$	16.7973	1.31566	0.657832	+	0.753165i	$0.271474\pi$
$164$	−31.7504	−2.47929
$165$	0	0
$166$	41.3650	3.21055
$167$	−10.5399	−0.815598	−0.407799	−	0.913072i	$-0.633704\pi$
$167$	−10.5399	−0.815598	−0.407799	+	0.913072i	$0.633704\pi$
$168$	0	0
$169$	−8.09366	−0.622589
$170$	21.6176	1.65800
$171$	0	0
$172$	1.99507	0.152123
$173$	22.2051	1.68822	0.844112	−	0.536168i	$-0.180128\pi$
$173$	22.2051	1.68822	0.844112	+	0.536168i	$0.180128\pi$
$174$	0	0
$175$	0.229532	0.0173510
$176$	−29.0446	−2.18932
$177$	0	0
$178$	−19.8109	−1.48489
$179$	−5.20296	−0.388887	−0.194444	−	0.980914i	$-0.562290\pi$
$179$	−5.20296	−0.388887	−0.194444	+	0.980914i	$0.562290\pi$
$180$	0	0
$181$	25.8911	1.92447	0.962235	−	0.272218i	$-0.0877573\pi$
$181$	25.8911	1.92447	0.962235	+	0.272218i	$0.0877573\pi$
$182$	−1.83219	−0.135811
$183$	0	0
$184$	−5.79178	−0.426976
$185$	−1.36043	−0.100021
$186$	0	0
$187$	20.3188	1.48586
$188$	−17.8529	−1.30205
$189$	0	0
$190$	−2.26488	−0.164312
$191$	18.1086	1.31029	0.655147	−	0.755501i	$-0.272607\pi$
$191$	18.1086	1.31029	0.655147	+	0.755501i	$0.272607\pi$
$192$	0	0
$193$	21.6895	1.56124	0.780622	−	0.625003i	$-0.214902\pi$
$193$	21.6895	1.56124	0.780622	+	0.625003i	$0.214902\pi$
$194$	−39.7855	−2.85643
$195$	0	0
$196$	−29.6772	−2.11980
$197$	12.8759	0.917368	0.458684	−	0.888600i	$-0.348321\pi$
$197$	12.8759	0.917368	0.458684	+	0.888600i	$0.348321\pi$
$198$	0	0
$199$	9.73022	0.689757	0.344878	−	0.938647i	$-0.387920\pi$
$199$	9.73022	0.689757	0.344878	+	0.938647i	$0.387920\pi$
$200$	−4.03601	−0.285389
$201$	0	0
$202$	30.4029	2.13914
$203$	−0.329384	−0.0231183
$204$	0	0
$205$	15.2944	1.06821
$206$	−29.8136	−2.07721
$207$	0	0
$208$	13.1393	0.911047
$209$	−2.12881	−0.147253
$210$	0	0
$211$	21.1320	1.45479	0.727394	−	0.686220i	$-0.240731\pi$
$211$	21.1320	1.45479	0.727394	+	0.686220i	$0.240731\pi$
$212$	−9.94490	−0.683018
$213$	0	0
$214$	9.82760	0.671801
$215$	−0.961043	−0.0655426
$216$	0	0
$217$	−3.62792	−0.246279
$218$	15.3249	1.03793
$219$	0	0
$220$	43.7395	2.94892
$221$	−9.19190	−0.618314
$222$	0	0
$223$	18.3905	1.23152	0.615759	−	0.787934i	$-0.288849\pi$
$223$	18.3905	1.23152	0.615759	+	0.787934i	$0.288849\pi$
$224$	−1.09120	−0.0729092
$225$	0	0
$226$	9.27201	0.616765
$227$	−20.9143	−1.38813	−0.694066	−	0.719912i	$-0.744182\pi$
$227$	−20.9143	−1.38813	−0.694066	+	0.719912i	$0.744182\pi$
$228$	0	0
$229$	6.61855	0.437366	0.218683	−	0.975796i	$-0.429824\pi$
$229$	6.61855	0.437366	0.218683	+	0.975796i	$0.429824\pi$
$230$	5.20933	0.343493
$231$	0	0
$232$	5.79178	0.380249
$233$	17.6336	1.15522	0.577608	−	0.816314i	$-0.303986\pi$
$233$	17.6336	1.15522	0.577608	+	0.816314i	$0.303986\pi$
$234$	0	0
$235$	8.59988	0.560994
$236$	−39.7226	−2.58572
$237$	0	0
$238$	3.43256	0.222500
$239$	3.61613	0.233908	0.116954	−	0.993137i	$-0.462687\pi$
$239$	3.61613	0.233908	0.116954	+	0.993137i	$0.462687\pi$
$240$	0	0
$241$	−27.5459	−1.77439	−0.887193	−	0.461398i	$-0.847348\pi$
$241$	−27.5459	−1.77439	−0.887193	+	0.461398i	$0.847348\pi$
$242$	32.5814	2.09441
$243$	0	0
$244$	−53.5599	−3.42882
$245$	14.2958	0.913323
$246$	0	0
$247$	0.963037	0.0612766
$248$	63.7921	4.05080
$249$	0	0
$250$	29.6768	1.87692
$251$	−8.21398	−0.518462	−0.259231	−	0.965815i	$-0.583469\pi$
$251$	−8.21398	−0.518462	−0.259231	+	0.965815i	$0.583469\pi$
$252$	0	0
$253$	4.89635	0.307831
$254$	−27.8317	−1.74632
$255$	0	0
$256$	−31.9022	−1.99388
$257$	−10.5170	−0.656034	−0.328017	−	0.944672i	$-0.606380\pi$
$257$	−10.5170	−0.656034	−0.328017	+	0.944672i	$0.606380\pi$
$258$	0	0
$259$	−0.216016	−0.0134226
$260$	−19.7870	−1.22714
$261$	0	0
$262$	−42.0965	−2.60073
$263$	2.77438	0.171076	0.0855378	−	0.996335i	$-0.472739\pi$
$263$	2.77438	0.171076	0.0855378	+	0.996335i	$0.472739\pi$
$264$	0	0
$265$	4.79055	0.294281
$266$	−0.359630	−0.0220503
$267$	0	0
$268$	−63.1293	−3.85623
$269$	−23.7005	−1.44505	−0.722524	−	0.691346i	$-0.757018\pi$
$269$	−23.7005	−1.44505	−0.722524	+	0.691346i	$0.757018\pi$
$270$	0	0
$271$	7.94842	0.482832	0.241416	−	0.970422i	$-0.422388\pi$
$271$	7.94842	0.482832	0.241416	+	0.970422i	$0.422388\pi$
$272$	−24.6161	−1.49257
$273$	0	0
$274$	−3.27672	−0.197954
$275$	3.41202	0.205753
$276$	0	0
$277$	−3.94280	−0.236900	−0.118450	−	0.992960i	$-0.537793\pi$
$277$	−3.94280	−0.236900	−0.118450	+	0.992960i	$0.537793\pi$
$278$	1.27241	0.0763139
$279$	0	0
$280$	3.95739	0.236499
$281$	−19.7692	−1.17933	−0.589667	−	0.807647i	$-0.700741\pi$
$281$	−19.7692	−1.17933	−0.589667	+	0.807647i	$0.700741\pi$
$282$	0	0
$283$	−19.5488	−1.16205	−0.581027	−	0.813885i	$-0.697349\pi$
$283$	−19.5488	−1.16205	−0.581027	+	0.813885i	$0.697349\pi$
$284$	−52.1411	−3.09400
$285$	0	0
$286$	−27.2358	−1.61048
$287$	2.42853	0.143352
$288$	0	0
$289$	0.220779	0.0129870
$290$	−5.20933	−0.305902
$291$	0	0
$292$	34.1558	1.99882
$293$	−10.8038	−0.631162	−0.315581	−	0.948899i	$-0.602199\pi$
$293$	−10.8038	−0.631162	−0.315581	+	0.948899i	$0.602199\pi$
$294$	0	0
$295$	19.1347	1.11407
$296$	3.79836	0.220775
$297$	0	0
$298$	31.1616	1.80514
$299$	−2.21503	−0.128098
$300$	0	0
$301$	−0.152599	−0.00879568
$302$	43.8647	2.52413
$303$	0	0
$304$	2.57904	0.147918
$305$	25.8003	1.47732
$306$	0	0
$307$	4.16574	0.237751	0.118876	−	0.992909i	$-0.462071\pi$
$307$	4.16574	0.237751	0.118876	+	0.992909i	$0.462071\pi$
$308$	6.94518	0.395738
$309$	0	0
$310$	−57.3768	−3.25879
$311$	8.17076	0.463321	0.231661	−	0.972797i	$-0.425584\pi$
$311$	8.17076	0.463321	0.231661	+	0.972797i	$0.425584\pi$
$312$	0	0
$313$	13.6902	0.773819	0.386909	−	0.922118i	$-0.373543\pi$
$313$	13.6902	0.773819	0.386909	+	0.922118i	$0.373543\pi$
$314$	0.727195	0.0410380
$315$	0	0
$316$	−27.2500	−1.53293
$317$	19.5419	1.09758	0.548792	−	0.835959i	$-0.315088\pi$
$317$	19.5419	1.09758	0.548792	+	0.835959i	$0.315088\pi$
$318$	0	0
$319$	−4.89635	−0.274143
$320$	7.35250	0.411017
$321$	0	0
$322$	0.827164	0.0460961
$323$	−1.80422	−0.100390
$324$	0	0
$325$	−1.54354	−0.0856204
$326$	42.1820	2.33625
$327$	0	0
$328$	−42.7024	−2.35785
$329$	1.36553	0.0752843
$330$	0	0
$331$	−11.9595	−0.657352	−0.328676	−	0.944443i	$-0.606602\pi$
$331$	−11.9595	−0.657352	−0.328676	+	0.944443i	$0.606602\pi$
$332$	70.9337	3.89299
$333$	0	0
$334$	−26.4681	−1.44827
$335$	30.4099	1.66147
$336$	0	0
$337$	−14.1312	−0.769773	−0.384887	−	0.922964i	$-0.625759\pi$
$337$	−14.1312	−0.769773	−0.384887	+	0.922964i	$0.625759\pi$
$338$	−20.3251	−1.10554
$339$	0	0
$340$	37.0705	2.01043
$341$	−53.9295	−2.92045
$342$	0	0
$343$	4.57565	0.247062
$344$	2.68326	0.144671
$345$	0	0
$346$	55.7624	2.99781
$347$	5.12769	0.275269	0.137635	−	0.990483i	$-0.456050\pi$
$347$	5.12769	0.275269	0.137635	+	0.990483i	$0.456050\pi$
$348$	0	0
$349$	−24.1423	−1.29231	−0.646154	−	0.763207i	$-0.723624\pi$
$349$	−24.1423	−1.29231	−0.646154	+	0.763207i	$0.723624\pi$
$350$	0.576411	0.0308104
$351$	0	0
$352$	−16.2209	−0.864577
$353$	−15.5988	−0.830241	−0.415121	−	0.909766i	$-0.636261\pi$
$353$	−15.5988	−0.830241	−0.415121	+	0.909766i	$0.636261\pi$
$354$	0	0
$355$	25.1168	1.33306
$356$	−33.9723	−1.80053
$357$	0	0
$358$	−13.0659	−0.690554
$359$	−16.0554	−0.847370	−0.423685	−	0.905810i	$-0.639264\pi$
$359$	−16.0554	−0.847370	−0.423685	+	0.905810i	$0.639264\pi$
$360$	0	0
$361$	−18.8110	−0.990051
$362$	65.0189	3.41732
$363$	0	0
$364$	−3.14189	−0.164680
$365$	−16.4532	−0.861199
$366$	0	0
$367$	24.1708	1.26170	0.630852	−	0.775903i	$-0.282706\pi$
$367$	24.1708	1.26170	0.630852	+	0.775903i	$0.282706\pi$
$368$	−5.93189	−0.309221
$369$	0	0
$370$	−3.41637	−0.177609
$371$	0.760668	0.0394919
$372$	0	0
$373$	0.418763	0.0216827	0.0108414	−	0.999941i	$-0.496549\pi$
$373$	0.418763	0.0216827	0.0108414	+	0.999941i	$0.496549\pi$
$374$	51.0255	2.63847
$375$	0	0
$376$	−24.0111	−1.23828
$377$	2.21503	0.114080
$378$	0	0
$379$	−12.9349	−0.664419	−0.332209	−	0.943206i	$-0.607794\pi$
$379$	−12.9349	−0.664419	−0.332209	+	0.943206i	$0.607794\pi$
$380$	−3.88388	−0.199239
$381$	0	0
$382$	45.4752	2.32671
$383$	−19.4317	−0.992912	−0.496456	−	0.868062i	$-0.665366\pi$
$383$	−19.4317	−0.992912	−0.496456	+	0.868062i	$0.665366\pi$
$384$	0	0
$385$	−3.34556	−0.170505
$386$	54.4676	2.77233
$387$	0	0
$388$	−68.2251	−3.46361
$389$	−12.7146	−0.644655	−0.322327	−	0.946628i	$-0.604465\pi$
$389$	−12.7146	−0.644655	−0.322327	+	0.946628i	$0.604465\pi$
$390$	0	0
$391$	4.14979	0.209864
$392$	−39.9141	−2.01597
$393$	0	0
$394$	32.3344	1.62899
$395$	13.1266	0.660470
$396$	0	0
$397$	−3.82487	−0.191965	−0.0959824	−	0.995383i	$-0.530599\pi$
$397$	−3.82487	−0.191965	−0.0959824	+	0.995383i	$0.530599\pi$
$398$	24.4349	1.22481
$399$	0	0
$400$	−4.13365	−0.206682
$401$	25.7050	1.28364	0.641822	−	0.766853i	$-0.278179\pi$
$401$	25.7050	1.28364	0.641822	+	0.766853i	$0.278179\pi$
$402$	0	0
$403$	24.3968	1.21529
$404$	52.1356	2.59385
$405$	0	0
$406$	−0.827164	−0.0410515
$407$	−3.21111	−0.159169
$408$	0	0
$409$	−3.42184	−0.169199	−0.0845996	−	0.996415i	$-0.526961\pi$
$409$	−3.42184	−0.169199	−0.0845996	+	0.996415i	$0.526961\pi$
$410$	38.4081	1.89684
$411$	0	0
$412$	−51.1252	−2.51876
$413$	3.03831	0.149505
$414$	0	0
$415$	−34.1694	−1.67731
$416$	7.33807	0.359779
$417$	0	0
$418$	−5.34595	−0.261479
$419$	6.84345	0.334324	0.167162	−	0.985929i	$-0.446540\pi$
$419$	6.84345	0.334324	0.167162	+	0.985929i	$0.446540\pi$
$420$	0	0
$421$	30.2672	1.47513	0.737567	−	0.675273i	$-0.235974\pi$
$421$	30.2672	1.47513	0.737567	+	0.675273i	$0.235974\pi$
$422$	53.0676	2.58329
$423$	0	0
$424$	−13.3753	−0.649563
$425$	2.89179	0.140272
$426$	0	0
$427$	4.09670	0.198253
$428$	16.8526	0.814602
$429$	0	0
$430$	−2.41341	−0.116385
$431$	23.8381	1.14824	0.574120	−	0.818771i	$-0.305344\pi$
$431$	23.8381	1.14824	0.574120	+	0.818771i	$0.305344\pi$
$432$	0	0
$433$	−32.3113	−1.55278	−0.776392	−	0.630250i	$-0.782952\pi$
$433$	−32.3113	−1.55278	−0.776392	+	0.630250i	$0.782952\pi$
$434$	−9.11059	−0.437322
$435$	0	0
$436$	26.2795	1.25856
$437$	−0.434775	−0.0207981
$438$	0	0
$439$	−3.14005	−0.149866	−0.0749332	−	0.997189i	$-0.523874\pi$
$439$	−3.14005	−0.149866	−0.0749332	+	0.997189i	$0.523874\pi$
$440$	58.8271	2.80447
$441$	0	0
$442$	−23.0831	−1.09795
$443$	−20.0935	−0.954673	−0.477336	−	0.878721i	$-0.658398\pi$
$443$	−20.0935	−0.954673	−0.477336	+	0.878721i	$0.658398\pi$
$444$	0	0
$445$	16.3648	0.775764
$446$	46.1830	2.18683
$447$	0	0
$448$	1.16747	0.0551577
$449$	26.5247	1.25178	0.625890	−	0.779911i	$-0.284736\pi$
$449$	26.5247	1.25178	0.625890	+	0.779911i	$0.284736\pi$
$450$	0	0
$451$	36.1004	1.69990
$452$	15.8999	0.747867
$453$	0	0
$454$	−52.5209	−2.46493
$455$	1.51347	0.0709528
$456$	0	0
$457$	24.8925	1.16442	0.582211	−	0.813038i	$-0.302188\pi$
$457$	24.8925	1.16442	0.582211	+	0.813038i	$0.302188\pi$
$458$	16.6208	0.776638
$459$	0	0
$460$	8.93309	0.416507
$461$	−14.3135	−0.666645	−0.333323	−	0.942813i	$-0.608170\pi$
$461$	−14.3135	−0.666645	−0.333323	+	0.942813i	$0.608170\pi$
$462$	0	0
$463$	−11.6147	−0.539782	−0.269891	−	0.962891i	$-0.586988\pi$
$463$	−11.6147	−0.539782	−0.269891	+	0.962891i	$0.586988\pi$
$464$	5.93189	0.275381
$465$	0	0
$466$	44.2823	2.05134
$467$	−7.84693	−0.363113	−0.181556	−	0.983381i	$-0.558113\pi$
$467$	−7.84693	−0.363113	−0.181556	+	0.983381i	$0.558113\pi$
$468$	0	0
$469$	4.82865	0.222966
$470$	21.5964	0.996167
$471$	0	0
$472$	−53.4246	−2.45907
$473$	−2.26841	−0.104302
$474$	0	0
$475$	−0.302973	−0.0139014
$476$	5.88624	0.269795
$477$	0	0
$478$	9.08098	0.415354
$479$	9.05712	0.413831	0.206915	−	0.978359i	$-0.433658\pi$
$479$	9.05712	0.413831	0.206915	+	0.978359i	$0.433658\pi$
$480$	0	0
$481$	1.45265	0.0662353
$482$	−69.1744	−3.15081
$483$	0	0
$484$	55.8713	2.53961
$485$	32.8646	1.49231
$486$	0	0
$487$	20.1934	0.915052	0.457526	−	0.889196i	$-0.348736\pi$
$487$	20.1934	0.915052	0.457526	+	0.889196i	$0.348736\pi$
$488$	−72.0350	−3.26087
$489$	0	0
$490$	35.9001	1.62180
$491$	−1.20309	−0.0542945	−0.0271472	−	0.999631i	$-0.508642\pi$
$491$	−1.20309	−0.0542945	−0.0271472	+	0.999631i	$0.508642\pi$
$492$	0	0
$493$	−4.14979	−0.186897
$494$	2.41842	0.108810
$495$	0	0
$496$	65.3353	2.93364
$497$	3.98818	0.178894
$498$	0	0
$499$	13.6638	0.611674	0.305837	−	0.952084i	$-0.401064\pi$
$499$	13.6638	0.611674	0.305837	+	0.952084i	$0.401064\pi$
$500$	50.8905	2.27589
$501$	0	0
$502$	−20.6273	−0.920642
$503$	17.6204	0.785656	0.392828	−	0.919612i	$-0.371497\pi$
$503$	17.6204	0.785656	0.392828	+	0.919612i	$0.371497\pi$
$504$	0	0
$505$	−25.1142	−1.11757
$506$	12.2959	0.546620
$507$	0	0
$508$	−47.7266	−2.11752
$509$	−38.6815	−1.71453	−0.857264	−	0.514877i	$-0.827838\pi$
$509$	−38.6815	−1.71453	−0.857264	+	0.514877i	$0.827838\pi$
$510$	0	0
$511$	−2.61252	−0.115571
$512$	−49.0610	−2.16821
$513$	0	0
$514$	−26.4108	−1.16493
$515$	24.6274	1.08521
$516$	0	0
$517$	20.2988	0.892742
$518$	−0.542470	−0.0238347
$519$	0	0
$520$	−26.6124	−1.16703
$521$	−3.48891	−0.152852	−0.0764259	−	0.997075i	$-0.524351\pi$
$521$	−3.48891	−0.152852	−0.0764259	+	0.997075i	$0.524351\pi$
$522$	0	0
$523$	1.19981	0.0524641	0.0262321	−	0.999656i	$-0.491649\pi$
$523$	1.19981	0.0524641	0.0262321	+	0.999656i	$0.491649\pi$
$524$	−72.1881	−3.15355
$525$	0	0
$526$	6.96714	0.303782
$527$	−45.7068	−1.99102
$528$	0	0
$529$	1.00000	0.0434783
$530$	12.0302	0.522560
$531$	0	0
$532$	−0.616703	−0.0267374
$533$	−16.3312	−0.707385
$534$	0	0
$535$	−8.11805	−0.350974
$536$	−84.9053	−3.66735
$537$	0	0
$538$	−59.5178	−2.56600
$539$	33.7432	1.45342
$540$	0	0
$541$	−23.3264	−1.00288	−0.501439	−	0.865193i	$-0.667196\pi$
$541$	−23.3264	−1.00288	−0.501439	+	0.865193i	$0.667196\pi$
$542$	19.9604	0.857373
$543$	0	0
$544$	−13.7477	−0.589427
$545$	−12.6590	−0.542254
$546$	0	0
$547$	4.42366	0.189142	0.0945711	−	0.995518i	$-0.469852\pi$
$547$	4.42366	0.189142	0.0945711	+	0.995518i	$0.469852\pi$
$548$	−5.61900	−0.240032
$549$	0	0
$550$	8.56842	0.365359
$551$	0.434775	0.0185220
$552$	0	0
$553$	2.08431	0.0886337
$554$	−9.90134	−0.420668
$555$	0	0
$556$	2.18196	0.0925355
$557$	−2.29640	−0.0973014	−0.0486507	−	0.998816i	$-0.515492\pi$
$557$	−2.29640	−0.0973014	−0.0486507	+	0.998816i	$0.515492\pi$
$558$	0	0
$559$	1.02619	0.0434033
$560$	4.05312	0.171276
$561$	0	0
$562$	−49.6453	−2.09416
$563$	−9.63174	−0.405929	−0.202965	−	0.979186i	$-0.565058\pi$
$563$	−9.63174	−0.405929	−0.202965	+	0.979186i	$0.565058\pi$
$564$	0	0
$565$	−7.65911	−0.322221
$566$	−49.0917	−2.06348
$567$	0	0
$568$	−70.1268	−2.94245
$569$	19.6362	0.823191	0.411596	−	0.911367i	$-0.364972\pi$
$569$	19.6362	0.823191	0.411596	+	0.911367i	$0.364972\pi$
$570$	0	0
$571$	−0.0358829	−0.00150165	−0.000750827	−	1.00000i	$-0.500239\pi$
$571$	−0.0358829	−0.00150165	−0.000750827		1.00000i	$0.500239\pi$
$572$	−46.7046	−1.95282
$573$	0	0
$574$	6.09863	0.254552
$575$	0.696851	0.0290607
$576$	0	0
$577$	20.4917	0.853082	0.426541	−	0.904468i	$-0.359732\pi$
$577$	20.4917	0.853082	0.426541	+	0.904468i	$0.359732\pi$
$578$	0.554430	0.0230613
$579$	0	0
$580$	−8.93309	−0.370926
$581$	−5.42560	−0.225092
$582$	0	0
$583$	11.3074	0.468306
$584$	45.9377	1.90091
$585$	0	0
$586$	−27.1309	−1.12077
$587$	−39.7227	−1.63953	−0.819766	−	0.572698i	$-0.805897\pi$
$587$	−39.7227	−1.63953	−0.819766	+	0.572698i	$0.805897\pi$
$588$	0	0
$589$	4.78871	0.197316
$590$	48.0519	1.97827
$591$	0	0
$592$	3.89024	0.159888
$593$	−18.9906	−0.779849	−0.389925	−	0.920847i	$-0.627499\pi$
$593$	−18.9906	−0.779849	−0.389925	+	0.920847i	$0.627499\pi$
$594$	0	0
$595$	−2.83545	−0.116242
$596$	53.4367	2.18885
$597$	0	0
$598$	−5.56247	−0.227466
$599$	15.1699	0.619823	0.309912	−	0.950765i	$-0.399700\pi$
$599$	15.1699	0.619823	0.309912	+	0.950765i	$0.399700\pi$
$600$	0	0
$601$	−15.7115	−0.640885	−0.320442	−	0.947268i	$-0.603832\pi$
$601$	−15.7115	−0.640885	−0.320442	+	0.947268i	$0.603832\pi$
$602$	−0.383214	−0.0156186
$603$	0	0
$604$	75.2203	3.06067
$605$	−26.9137	−1.09420
$606$	0	0
$607$	16.9322	0.687257	0.343628	−	0.939106i	$-0.388344\pi$
$607$	16.9322	0.687257	0.343628	+	0.939106i	$0.388344\pi$
$608$	1.44035	0.0584138
$609$	0	0
$610$	64.7907	2.62330
$611$	−9.18286	−0.371499
$612$	0	0
$613$	−41.5232	−1.67710	−0.838552	−	0.544821i	$-0.816598\pi$
$613$	−41.5232	−1.67710	−0.838552	+	0.544821i	$0.816598\pi$
$614$	10.4612	0.422179
$615$	0	0
$616$	9.34087	0.376355
$617$	15.0521	0.605975	0.302988	−	0.952994i	$-0.402016\pi$
$617$	15.0521	0.605975	0.302988	+	0.952994i	$0.402016\pi$
$618$	0	0
$619$	−33.7558	−1.35676	−0.678381	−	0.734711i	$-0.737318\pi$
$619$	−33.7558	−1.35676	−0.678381	+	0.734711i	$0.737318\pi$
$620$	−98.3912	−3.95149
$621$	0	0
$622$	20.5188	0.822727
$623$	2.59848	0.104106
$624$	0	0
$625$	−21.0301	−0.841206
$626$	34.3795	1.37408
$627$	0	0
$628$	1.24701	0.0497612
$629$	−2.72151	−0.108514
$630$	0	0
$631$	25.8463	1.02893	0.514463	−	0.857513i	$-0.327991\pi$
$631$	25.8463	1.02893	0.514463	+	0.857513i	$0.327991\pi$
$632$	−36.6497	−1.45785
$633$	0	0
$634$	49.0745	1.94900
$635$	22.9903	0.912342
$636$	0	0
$637$	−15.2649	−0.604816
$638$	−12.2959	−0.486800
$639$	0	0
$640$	32.2083	1.27315
$641$	−11.5642	−0.456757	−0.228378	−	0.973572i	$-0.573342\pi$
$641$	−11.5642	−0.456757	−0.228378	+	0.973572i	$0.573342\pi$
$642$	0	0
$643$	−19.5700	−0.771766	−0.385883	−	0.922548i	$-0.626103\pi$
$643$	−19.5700	−0.771766	−0.385883	+	0.922548i	$0.626103\pi$
$644$	1.41844	0.0558944
$645$	0	0
$646$	−4.53085	−0.178264
$647$	−7.87567	−0.309624	−0.154812	−	0.987944i	$-0.549477\pi$
$647$	−7.87567	−0.309624	−0.154812	+	0.987944i	$0.549477\pi$
$648$	0	0
$649$	45.1649	1.77288
$650$	−3.87621	−0.152038
$651$	0	0
$652$	72.3348	2.83285
$653$	11.7165	0.458503	0.229251	−	0.973367i	$-0.426372\pi$
$653$	11.7165	0.458503	0.229251	+	0.973367i	$0.426372\pi$
$654$	0	0
$655$	34.7737	1.35872
$656$	−43.7355	−1.70758
$657$	0	0
$658$	3.42919	0.133684
$659$	−23.7139	−0.923761	−0.461881	−	0.886942i	$-0.652825\pi$
$659$	−23.7139	−0.923761	−0.461881	+	0.886942i	$0.652825\pi$
$660$	0	0
$661$	9.06231	0.352483	0.176242	−	0.984347i	$-0.443606\pi$
$661$	9.06231	0.352483	0.176242	+	0.984347i	$0.443606\pi$
$662$	−30.0332	−1.16727
$663$	0	0
$664$	95.4018	3.70231
$665$	0.297071	0.0115199
$666$	0	0
$667$	−1.00000	−0.0387202
$668$	−45.3882	−1.75612
$669$	0	0
$670$	76.3667	2.95030
$671$	60.8980	2.35094
$672$	0	0
$673$	12.5398	0.483374	0.241687	−	0.970354i	$-0.422299\pi$
$673$	12.5398	0.483374	0.241687	+	0.970354i	$0.422299\pi$
$674$	−35.4868	−1.36690
$675$	0	0
$676$	−34.8541	−1.34054
$677$	13.0087	0.499964	0.249982	−	0.968250i	$-0.419575\pi$
$677$	13.0087	0.499964	0.249982	+	0.968250i	$0.419575\pi$
$678$	0	0
$679$	5.21842	0.200265
$680$	49.8577	1.91195
$681$	0	0
$682$	−135.430	−5.18589
$683$	24.1598	0.924449	0.462225	−	0.886763i	$-0.347051\pi$
$683$	24.1598	0.924449	0.462225	+	0.886763i	$0.347051\pi$
$684$	0	0
$685$	2.70672	0.103419
$686$	11.4906	0.438712
$687$	0	0
$688$	2.74817	0.104773
$689$	−5.11530	−0.194877
$690$	0	0
$691$	−8.79410	−0.334543	−0.167272	−	0.985911i	$-0.553496\pi$
$691$	−8.79410	−0.334543	−0.167272	+	0.985911i	$0.553496\pi$
$692$	95.6228	3.63503
$693$	0	0
$694$	12.8769	0.488800
$695$	−1.05107	−0.0398693
$696$	0	0
$697$	30.5961	1.15891
$698$	−60.6272	−2.29477
$699$	0	0
$700$	0.988443	0.0373596
$701$	17.1365	0.647238	0.323619	−	0.946188i	$-0.395100\pi$
$701$	17.1365	0.647238	0.323619	+	0.946188i	$0.395100\pi$
$702$	0	0
$703$	0.285133	0.0107540
$704$	17.3546	0.654075
$705$	0	0
$706$	−39.1724	−1.47427
$707$	−3.98776	−0.149975
$708$	0	0
$709$	11.6715	0.438331	0.219166	−	0.975688i	$-0.429666\pi$
$709$	11.6715	0.438331	0.219166	+	0.975688i	$0.429666\pi$
$710$	63.0744	2.36714
$711$	0	0
$712$	−45.6908	−1.71234
$713$	−11.0142	−0.412487
$714$	0	0
$715$	22.4980	0.841378
$716$	−22.4057	−0.837341
$717$	0	0
$718$	−40.3189	−1.50469
$719$	43.8644	1.63587	0.817933	−	0.575313i	$-0.195120\pi$
$719$	43.8644	1.63587	0.817933	+	0.575313i	$0.195120\pi$
$720$	0	0
$721$	3.91047	0.145634
$722$	−47.2389	−1.75805
$723$	0	0
$724$	111.496	4.14371
$725$	−0.696851	−0.0258804
$726$	0	0
$727$	50.0227	1.85524	0.927620	−	0.373526i	$-0.121851\pi$
$727$	50.0227	1.85524	0.927620	+	0.373526i	$0.121851\pi$
$728$	−4.22566	−0.156613
$729$	0	0
$730$	−41.3179	−1.52925
$731$	−1.92254	−0.0711078
$732$	0	0
$733$	−8.42563	−0.311208	−0.155604	−	0.987820i	$-0.549732\pi$
$733$	−8.42563	−0.311208	−0.155604	+	0.987820i	$0.549732\pi$
$734$	60.6987	2.24043
$735$	0	0
$736$	−3.31286	−0.122114
$737$	71.7785	2.64399
$738$	0	0
$739$	−34.3160	−1.26234	−0.631168	−	0.775646i	$-0.717424\pi$
$739$	−34.3160	−1.26234	−0.631168	+	0.775646i	$0.717424\pi$
$740$	−5.85848	−0.215362
$741$	0	0
$742$	1.91022	0.0701264
$743$	26.1566	0.959594	0.479797	−	0.877380i	$-0.340710\pi$
$743$	26.1566	0.959594	0.479797	+	0.877380i	$0.340710\pi$
$744$	0	0
$745$	−25.7409	−0.943075
$746$	1.05162	0.0385024
$747$	0	0
$748$	87.4998	3.19931
$749$	−1.28903	−0.0471000
$750$	0	0
$751$	−2.94242	−0.107370	−0.0536852	−	0.998558i	$-0.517097\pi$
$751$	−2.94242	−0.107370	−0.0536852	+	0.998558i	$0.517097\pi$
$752$	−24.5919	−0.896775
$753$	0	0
$754$	5.56247	0.202573
$755$	−36.2343	−1.31870
$756$	0	0
$757$	−30.5553	−1.11055	−0.555276	−	0.831666i	$-0.687388\pi$
$757$	−30.5553	−1.11055	−0.555276	+	0.831666i	$0.687388\pi$
$758$	−32.4826	−1.17982
$759$	0	0
$760$	−5.22360	−0.189480
$761$	−7.04354	−0.255328	−0.127664	−	0.991817i	$-0.540748\pi$
$761$	−7.04354	−0.255328	−0.127664	+	0.991817i	$0.540748\pi$
$762$	0	0
$763$	−2.01007	−0.0727694
$764$	77.9819	2.82129
$765$	0	0
$766$	−48.7977	−1.76313
$767$	−20.4319	−0.737752
$768$	0	0
$769$	−38.3769	−1.38391	−0.691954	−	0.721942i	$-0.743250\pi$
$769$	−38.3769	−1.38391	−0.691954	+	0.721942i	$0.743250\pi$
$770$	−8.40150	−0.302769
$771$	0	0
$772$	93.4024	3.36163
$773$	16.3163	0.586858	0.293429	−	0.955981i	$-0.405204\pi$
$773$	16.3163	0.586858	0.293429	+	0.955981i	$0.405204\pi$
$774$	0	0
$775$	−7.67529	−0.275705
$776$	−91.7589	−3.29395
$777$	0	0
$778$	−31.9294	−1.14472
$779$	−3.20556	−0.114851
$780$	0	0
$781$	59.2848	2.12138
$782$	10.4211	0.372659
$783$	0	0
$784$	−40.8797	−1.45999
$785$	−0.600697	−0.0214398
$786$	0	0
$787$	−34.1348	−1.21677	−0.608387	−	0.793640i	$-0.708183\pi$
$787$	−34.1348	−1.21677	−0.608387	+	0.793640i	$0.708183\pi$
$788$	55.4479	1.97525
$789$	0	0
$790$	32.9640	1.17281
$791$	−1.21615	−0.0432414
$792$	0	0
$793$	−27.5493	−0.978303
$794$	−9.60519	−0.340875
$795$	0	0
$796$	41.9016	1.48516
$797$	−2.65681	−0.0941091	−0.0470546	−	0.998892i	$-0.514983\pi$
$797$	−2.65681	−0.0941091	−0.0470546	+	0.998892i	$0.514983\pi$
$798$	0	0
$799$	17.2038	0.608628
$800$	−2.30857	−0.0816203
$801$	0	0
$802$	64.5514	2.27939
$803$	−38.8355	−1.37047
$804$	0	0
$805$	−0.683276	−0.0240823
$806$	61.2664	2.15802
$807$	0	0
$808$	70.1195	2.46679
$809$	−1.46905	−0.0516491	−0.0258246	−	0.999666i	$-0.508221\pi$
$809$	−1.46905	−0.0516491	−0.0258246	+	0.999666i	$0.508221\pi$
$810$	0	0
$811$	−10.8251	−0.380119	−0.190060	−	0.981773i	$-0.560868\pi$
$811$	−10.8251	−0.380119	−0.190060	+	0.981773i	$0.560868\pi$
$812$	−1.41844	−0.0497776
$813$	0	0
$814$	−8.06389	−0.282639
$815$	−34.8443	−1.22054
$816$	0	0
$817$	0.201425	0.00704698
$818$	−8.59308	−0.300450
$819$	0	0
$820$	65.8631	2.30004
$821$	11.6328	0.405988	0.202994	−	0.979180i	$-0.434933\pi$
$821$	11.6328	0.405988	0.202994	+	0.979180i	$0.434933\pi$
$822$	0	0
$823$	41.5811	1.44942	0.724712	−	0.689051i	$-0.241973\pi$
$823$	41.5811	1.44942	0.724712	+	0.689051i	$0.241973\pi$
$824$	−68.7604	−2.39538
$825$	0	0
$826$	7.62994	0.265479
$827$	−26.0406	−0.905521	−0.452760	−	0.891632i	$-0.649561\pi$
$827$	−26.0406	−0.905521	−0.452760	+	0.891632i	$0.649561\pi$
$828$	0	0
$829$	−7.98370	−0.277285	−0.138643	−	0.990342i	$-0.544274\pi$
$829$	−7.98370	−0.277285	−0.138643	+	0.990342i	$0.544274\pi$
$830$	−85.8077	−2.97843
$831$	0	0
$832$	−7.85092	−0.272182
$833$	28.5983	0.990873
$834$	0	0
$835$	21.8639	0.756632
$836$	−9.16737	−0.317060
$837$	0	0
$838$	17.1856	0.593666
$839$	47.9200	1.65438	0.827191	−	0.561920i	$-0.189937\pi$
$839$	47.9200	1.65438	0.827191	+	0.561920i	$0.189937\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	76.0084	2.61942
$843$	0	0
$844$	91.0017	3.13241
$845$	16.7895	0.577577
$846$	0	0
$847$	−4.27350	−0.146839
$848$	−13.6989	−0.470422
$849$	0	0
$850$	7.26198	0.249084
$851$	−0.655818	−0.0224812
$852$	0	0
$853$	−9.80607	−0.335753	−0.167877	−	0.985808i	$-0.553691\pi$
$853$	−9.80607	−0.335753	−0.167877	+	0.985808i	$0.553691\pi$
$854$	10.2878	0.352042
$855$	0	0
$856$	22.6658	0.774701
$857$	−1.34916	−0.0460864	−0.0230432	−	0.999734i	$-0.507336\pi$
$857$	−1.34916	−0.0460864	−0.0230432	+	0.999734i	$0.507336\pi$
$858$	0	0
$859$	−0.248201	−0.00846852	−0.00423426	−	0.999991i	$-0.501348\pi$
$859$	−0.248201	−0.00846852	−0.00423426	+	0.999991i	$0.501348\pi$
$860$	−4.13858	−0.141124
$861$	0	0
$862$	59.8633	2.03895
$863$	47.4175	1.61411	0.807055	−	0.590476i	$-0.201060\pi$
$863$	47.4175	1.61411	0.807055	+	0.590476i	$0.201060\pi$
$864$	0	0
$865$	−46.0623	−1.56617
$866$	−81.1416	−2.75731
$867$	0	0
$868$	−15.6231	−0.530281
$869$	30.9835	1.05104
$870$	0	0
$871$	−32.4714	−1.10025
$872$	35.3444	1.19691
$873$	0	0
$874$	−1.09182	−0.0369315
$875$	−3.89252	−0.131591
$876$	0	0
$877$	−24.3089	−0.820852	−0.410426	−	0.911894i	$-0.634620\pi$
$877$	−24.3089	−0.820852	−0.410426	+	0.911894i	$0.634620\pi$
$878$	−7.88543	−0.266120
$879$	0	0
$880$	60.2502	2.03103
$881$	27.4357	0.924332	0.462166	−	0.886793i	$-0.347072\pi$
$881$	27.4357	0.924332	0.462166	+	0.886793i	$0.347072\pi$
$882$	0	0
$883$	−24.8918	−0.837676	−0.418838	−	0.908061i	$-0.637562\pi$
$883$	−24.8918	−0.837676	−0.418838	+	0.908061i	$0.637562\pi$
$884$	−39.5835	−1.33134
$885$	0	0
$886$	−50.4597	−1.69523
$887$	−12.2114	−0.410019	−0.205009	−	0.978760i	$-0.565723\pi$
$887$	−12.2114	−0.410019	−0.205009	+	0.978760i	$0.565723\pi$
$888$	0	0
$889$	3.65052	0.122434
$890$	41.0959	1.37754
$891$	0	0
$892$	79.1958	2.65167
$893$	−1.80245	−0.0603167
$894$	0	0
$895$	10.7930	0.360771
$896$	5.11420	0.170854
$897$	0	0
$898$	66.6101	2.22281
$899$	11.0142	0.367346
$900$	0	0
$901$	9.58337	0.319268
$902$	90.6570	3.01855
$903$	0	0
$904$	21.3844	0.711235
$905$	−53.7086	−1.78533
$906$	0	0
$907$	−22.2846	−0.739949	−0.369975	−	0.929042i	$-0.620634\pi$
$907$	−22.2846	−0.739949	−0.369975	+	0.929042i	$0.620634\pi$
$908$	−90.0641	−2.98888
$909$	0	0
$910$	3.80070	0.125992
$911$	−24.4401	−0.809736	−0.404868	−	0.914375i	$-0.632682\pi$
$911$	−24.4401	−0.809736	−0.404868	+	0.914375i	$0.632682\pi$
$912$	0	0
$913$	−80.6522	−2.66920
$914$	62.5111	2.06768
$915$	0	0
$916$	28.5017	0.941723
$917$	5.52154	0.182337
$918$	0	0
$919$	−32.0653	−1.05774	−0.528869	−	0.848704i	$-0.677384\pi$
$919$	−32.0653	−1.05774	−0.528869	+	0.848704i	$0.677384\pi$
$920$	12.0145	0.396106
$921$	0	0
$922$	−35.9446	−1.18377
$923$	−26.8195	−0.882774
$924$	0	0
$925$	−0.457008	−0.0150263
$926$	−29.1674	−0.958501
$927$	0	0
$928$	3.31286	0.108750
$929$	−58.1912	−1.90919	−0.954596	−	0.297902i	$-0.903713\pi$
$929$	−58.1912	−1.90919	−0.954596	+	0.297902i	$0.903713\pi$
$930$	0	0
$931$	−2.99625	−0.0981982
$932$	75.9364	2.48738
$933$	0	0
$934$	−19.7055	−0.644785
$935$	−42.1494	−1.37843
$936$	0	0
$937$	23.5858	0.770515	0.385258	−	0.922809i	$-0.374113\pi$
$937$	23.5858	0.770515	0.385258	+	0.922809i	$0.374113\pi$
$938$	12.1259	0.395925
$939$	0	0
$940$	37.0340	1.20792
$941$	−20.4455	−0.666503	−0.333252	−	0.942838i	$-0.608146\pi$
$941$	−20.4455	−0.666503	−0.333252	+	0.942838i	$0.608146\pi$
$942$	0	0
$943$	7.37293	0.240096
$944$	−54.7170	−1.78089
$945$	0	0
$946$	−5.69653	−0.185210
$947$	−7.80239	−0.253544	−0.126772	−	0.991932i	$-0.540462\pi$
$947$	−7.80239	−0.253544	−0.126772	+	0.991932i	$0.540462\pi$
$948$	0	0
$949$	17.5685	0.570299
$950$	−0.760839	−0.0246849
$951$	0	0
$952$	7.91666	0.256580
$953$	−23.6780	−0.767007	−0.383504	−	0.923539i	$-0.625283\pi$
$953$	−23.6780	−0.767007	−0.383504	+	0.923539i	$0.625283\pi$
$954$	0	0
$955$	−37.5646	−1.21556
$956$	15.5723	0.503644
$957$	0	0
$958$	22.7446	0.734846
$959$	0.429787	0.0138786
$960$	0	0
$961$	90.3136	2.91334
$962$	3.64797	0.117615
$963$	0	0
$964$	−118.622	−3.82056
$965$	−44.9928	−1.44837
$966$	0	0
$967$	37.1140	1.19350	0.596752	−	0.802425i	$-0.296457\pi$
$967$	37.1140	1.19350	0.596752	+	0.802425i	$0.296457\pi$
$968$	75.1438	2.41521
$969$	0	0
$970$	82.5311	2.64991
$971$	−3.64698	−0.117037	−0.0585185	−	0.998286i	$-0.518638\pi$
$971$	−3.64698	−0.117037	−0.0585185	+	0.998286i	$0.518638\pi$
$972$	0	0
$973$	−0.166894	−0.00535037
$974$	50.7106	1.62487
$975$	0	0
$976$	−73.7776	−2.36156
$977$	−18.3292	−0.586403	−0.293201	−	0.956051i	$-0.594721\pi$
$977$	−18.3292	−0.586403	−0.293201	+	0.956051i	$0.594721\pi$
$978$	0	0
$979$	38.6268	1.23452
$980$	61.5624	1.96654
$981$	0	0
$982$	−3.02124	−0.0964116
$983$	11.7224	0.373885	0.186943	−	0.982371i	$-0.440142\pi$
$983$	11.7224	0.373885	0.186943	+	0.982371i	$0.440142\pi$
$984$	0	0
$985$	−26.7097	−0.851043
$986$	−10.4211	−0.331877
$987$	0	0
$988$	4.14717	0.131939
$989$	−0.463287	−0.0147317
$990$	0	0
$991$	31.0791	0.987260	0.493630	−	0.869672i	$-0.335670\pi$
$991$	31.0791	0.987260	0.493630	+	0.869672i	$0.335670\pi$
$992$	36.4886	1.15852
$993$	0	0
$994$	10.0153	0.317665
$995$	−20.1844	−0.639888
$996$	0	0
$997$	−46.6959	−1.47887	−0.739437	−	0.673226i	$-0.764908\pi$
$997$	−46.6959	−1.47887	−0.739437	+	0.673226i	$0.764908\pi$
$998$	34.3130	1.08616
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6003.2.a.m.1.11		11
3.2	odd	2		2001.2.a.l.1.1	✓	11

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2001.2.a.l.1.1	✓	11	3.2	odd	2
6003.2.a.m.1.11		11	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)

\(f(q)\)	\(=\)	\(q+2.51124 q^{2} +4.30634 q^{4} -2.07440 q^{5} -0.329384 q^{7} +5.79178 q^{8} +O(q^{10})\)	\(q+2.51124 q^{2} +4.30634 q^{4} -2.07440 q^{5} -0.329384 q^{7} +5.79178 q^{8} -5.20933 q^{10} -4.89635 q^{11} +2.21503 q^{13} -0.827164 q^{14} +5.93189 q^{16} -4.14979 q^{17} +0.434775 q^{19} -8.93309 q^{20} -12.2959 q^{22} -1.00000 q^{23} -0.696851 q^{25} +5.56247 q^{26} -1.41844 q^{28} +1.00000 q^{29} +11.0142 q^{31} +3.31286 q^{32} -10.4211 q^{34} +0.683276 q^{35} +0.655818 q^{37} +1.09182 q^{38} -12.0145 q^{40} -7.37293 q^{41} +0.463287 q^{43} -21.0853 q^{44} -2.51124 q^{46} -4.14571 q^{47} -6.89151 q^{49} -1.74996 q^{50} +9.53866 q^{52} -2.30936 q^{53} +10.1570 q^{55} -1.90772 q^{56} +2.51124 q^{58} -9.22421 q^{59} -12.4374 q^{61} +27.6594 q^{62} -3.54439 q^{64} -4.59486 q^{65} -14.6596 q^{67} -17.8704 q^{68} +1.71587 q^{70} -12.1080 q^{71} +7.93152 q^{73} +1.64692 q^{74} +1.87229 q^{76} +1.61278 q^{77} -6.32788 q^{79} -12.3051 q^{80} -18.5152 q^{82} +16.4719 q^{83} +8.60834 q^{85} +1.16343 q^{86} -28.3586 q^{88} -7.88890 q^{89} -0.729595 q^{91} -4.30634 q^{92} -10.4109 q^{94} -0.901898 q^{95} -15.8429 q^{97} -17.3062 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 11 q - 2 q^{2} + 18 q^{4} - 2 q^{5} + 3 q^{7} - 18 q^{8}+O(q^{10}) \) 11 * q - 2 * q^2 + 18 * q^4 - 2 * q^5 + 3 * q^7 - 18 * q^8	\( 11 q - 2 q^{2} + 18 q^{4} - 2 q^{5} + 3 q^{7} - 18 q^{8} + 14 q^{10} - 11 q^{11} - 5 q^{13} - 17 q^{14} + 20 q^{16} - 15 q^{17} - 6 q^{19} - 21 q^{20} - 10 q^{22} - 11 q^{23} + 3 q^{25} + 5 q^{26} + 7 q^{28} + 11 q^{29} + 35 q^{31} - 28 q^{32} + 28 q^{34} - 15 q^{35} - 28 q^{37} + 2 q^{38} - q^{40} - 10 q^{41} - 6 q^{43} - 18 q^{44} + 2 q^{46} - 15 q^{47} + 22 q^{49} - 15 q^{50} - 36 q^{52} + 7 q^{53} - 12 q^{55} - 56 q^{56} - 2 q^{58} + 20 q^{59} - 20 q^{61} + 11 q^{62} + 36 q^{64} - 11 q^{65} - 39 q^{67} - 35 q^{68} + 38 q^{70} - 49 q^{71} - 3 q^{73} - 37 q^{74} - 18 q^{76} - 25 q^{77} + 41 q^{79} - 51 q^{80} - 19 q^{82} - 13 q^{83} - 62 q^{86} - 40 q^{88} - 34 q^{89} + 2 q^{91} - 18 q^{92} - 14 q^{94} - 25 q^{95} - 11 q^{97} - 53 q^{98}+O(q^{100}) \) 11 * q - 2 * q^2 + 18 * q^4 - 2 * q^5 + 3 * q^7 - 18 * q^8 + 14 * q^10 - 11 * q^11 - 5 * q^13 - 17 * q^14 + 20 * q^16 - 15 * q^17 - 6 * q^19 - 21 * q^20 - 10 * q^22 - 11 * q^23 + 3 * q^25 + 5 * q^26 + 7 * q^28 + 11 * q^29 + 35 * q^31 - 28 * q^32 + 28 * q^34 - 15 * q^35 - 28 * q^37 + 2 * q^38 - q^40 - 10 * q^41 - 6 * q^43 - 18 * q^44 + 2 * q^46 - 15 * q^47 + 22 * q^49 - 15 * q^50 - 36 * q^52 + 7 * q^53 - 12 * q^55 - 56 * q^56 - 2 * q^58 + 20 * q^59 - 20 * q^61 + 11 * q^62 + 36 * q^64 - 11 * q^65 - 39 * q^67 - 35 * q^68 + 38 * q^70 - 49 * q^71 - 3 * q^73 - 37 * q^74 - 18 * q^76 - 25 * q^77 + 41 * q^79 - 51 * q^80 - 19 * q^82 - 13 * q^83 - 62 * q^86 - 40 * q^88 - 34 * q^89 + 2 * q^91 - 18 * q^92 - 14 * q^94 - 25 * q^95 - 11 * q^97 - 53 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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