LMFDB - Embedded newform 6003.2.a.q.1.15

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:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 3 x^{15} - 22 x^{14} + 68 x^{13} + 187 x^{12} - 597 x^{11} - 795 x^{10} + 2592 x^{9} + \cdots + 64 $ x^16 - 3x^15 - 22x^14 + 68x^13 + 187x^12 - 597x^11 - 795x^10 + 2592x^9 + 1860x^8 - 5877x^7 - 2496x^6 + 6612x^5 + 1842x^4 - 3011x^3 - 505x^2 + 336*x + 64
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	no (minimal twist has level 667)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.15
Root			$-2.42429$ 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)

$f(q)$	$=$	$q+2.42429 q^{2} +3.87720 q^{4} +0.275174 q^{5} -5.16310 q^{7} +4.55089 q^{8} +O(q^{10})$	\(q+2.42429 q^{2} +3.87720 q^{4} +0.275174 q^{5} -5.16310 q^{7} +4.55089 q^{8} +0.667103 q^{10} +4.59700 q^{11} +2.58445 q^{13} -12.5169 q^{14} +3.27829 q^{16} -6.77779 q^{17} -7.93422 q^{19} +1.06691 q^{20} +11.1445 q^{22} -1.00000 q^{23} -4.92428 q^{25} +6.26548 q^{26} -20.0184 q^{28} +1.00000 q^{29} -1.54451 q^{31} -1.15424 q^{32} -16.4314 q^{34} -1.42075 q^{35} -1.24464 q^{37} -19.2349 q^{38} +1.25229 q^{40} +1.39786 q^{41} -0.777148 q^{43} +17.8235 q^{44} -2.42429 q^{46} +5.57928 q^{47} +19.6576 q^{49} -11.9379 q^{50} +10.0205 q^{52} -11.5021 q^{53} +1.26498 q^{55} -23.4967 q^{56} +2.42429 q^{58} -7.31928 q^{59} +1.28650 q^{61} -3.74434 q^{62} -9.35480 q^{64} +0.711175 q^{65} +10.1934 q^{67} -26.2789 q^{68} -3.44432 q^{70} -4.16451 q^{71} -10.2878 q^{73} -3.01736 q^{74} -30.7626 q^{76} -23.7348 q^{77} -1.19088 q^{79} +0.902101 q^{80} +3.38881 q^{82} +5.53442 q^{83} -1.86507 q^{85} -1.88404 q^{86} +20.9204 q^{88} -13.4423 q^{89} -13.3438 q^{91} -3.87720 q^{92} +13.5258 q^{94} -2.18329 q^{95} +3.41413 q^{97} +47.6559 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 3 q^{2} + 21 q^{4} - 16 q^{5} + q^{7} - 9 q^{8}+O(q^{10}) $ 16 * q - 3 * q^2 + 21 * q^4 - 16 * q^5 + q^7 - 9 * q^8	$ 16 q - 3 q^{2} + 21 q^{4} - 16 q^{5} + q^{7} - 9 q^{8} - 14 q^{10} - 4 q^{11} + 15 q^{13} - 8 q^{14} + 23 q^{16} - 20 q^{17} - 4 q^{19} - 25 q^{20} + 13 q^{22} - 16 q^{23} + 30 q^{25} - 25 q^{26} - 13 q^{28} + 16 q^{29} + 19 q^{32} - 23 q^{34} - 5 q^{35} + 5 q^{37} - 38 q^{38} - 20 q^{40} - 7 q^{41} - 17 q^{43} + 21 q^{44} + 3 q^{46} - 29 q^{47} + 31 q^{49} + 44 q^{50} + 20 q^{52} - 63 q^{53} + q^{55} + 19 q^{56} - 3 q^{58} - 11 q^{59} - 33 q^{62} + 29 q^{64} - 53 q^{65} - 13 q^{67} - 63 q^{68} - 46 q^{70} + 23 q^{71} - 38 q^{73} + 47 q^{74} - 56 q^{76} - 97 q^{77} - 27 q^{79} - 8 q^{80} + 9 q^{82} - 36 q^{83} + 6 q^{85} + 11 q^{86} - 24 q^{88} + 16 q^{89} - 47 q^{91} - 21 q^{92} + 37 q^{94} + 12 q^{95} - 30 q^{97} + 27 q^{98}+O(q^{100}) $ 16 * q - 3 * q^2 + 21 * q^4 - 16 * q^5 + q^7 - 9 * q^8 - 14 * q^10 - 4 * q^11 + 15 * q^13 - 8 * q^14 + 23 * q^16 - 20 * q^17 - 4 * q^19 - 25 * q^20 + 13 * q^22 - 16 * q^23 + 30 * q^25 - 25 * q^26 - 13 * q^28 + 16 * q^29 + 19 * q^32 - 23 * q^34 - 5 * q^35 + 5 * q^37 - 38 * q^38 - 20 * q^40 - 7 * q^41 - 17 * q^43 + 21 * q^44 + 3 * q^46 - 29 * q^47 + 31 * q^49 + 44 * q^50 + 20 * q^52 - 63 * q^53 + q^55 + 19 * q^56 - 3 * q^58 - 11 * q^59 - 33 * q^62 + 29 * q^64 - 53 * q^65 - 13 * q^67 - 63 * q^68 - 46 * q^70 + 23 * q^71 - 38 * q^73 + 47 * q^74 - 56 * q^76 - 97 * q^77 - 27 * q^79 - 8 * q^80 + 9 * q^82 - 36 * q^83 + 6 * q^85 + 11 * q^86 - 24 * q^88 + 16 * q^89 - 47 * q^91 - 21 * q^92 + 37 * q^94 + 12 * q^95 - 30 * q^97 + 27 * 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 $

$2$	2.42429	1.71423	0.857117	−	0.515121i	$-0.172253\pi$
$2$	2.42429	1.71423	0.857117	+	0.515121i	$0.172253\pi$
$3$	0	0
$3$	0	0
$4$	3.87720	1.93860
$5$	0.275174	0.123062	0.0615308	−	0.998105i	$-0.480402\pi$
$5$	0.275174	0.123062	0.0615308	+	0.998105i	$0.480402\pi$
$6$	0	0
$7$	−5.16310	−1.95147	−0.975735	−	0.218956i	$-0.929735\pi$
$7$	−5.16310	−1.95147	−0.975735	+	0.218956i	$0.929735\pi$
$8$	4.55089	1.60898
$9$	0	0
$10$	0.667103	0.210957
$11$	4.59700	1.38605	0.693024	−	0.720915i	$-0.256278\pi$
$11$	4.59700	1.38605	0.693024	+	0.720915i	$0.256278\pi$
$12$	0	0
$13$	2.58445	0.716799	0.358399	−	0.933568i	$-0.383323\pi$
$13$	2.58445	0.716799	0.358399	+	0.933568i	$0.383323\pi$
$14$	−12.5169	−3.34528
$15$	0	0
$16$	3.27829	0.819573
$17$	−6.77779	−1.64386	−0.821928	−	0.569591i	$-0.807101\pi$
$17$	−6.77779	−1.64386	−0.821928	+	0.569591i	$0.807101\pi$
$18$	0	0
$19$	−7.93422	−1.82024	−0.910118	−	0.414350i	$-0.864009\pi$
$19$	−7.93422	−1.82024	−0.910118	+	0.414350i	$0.864009\pi$
$20$	1.06691	0.238567
$21$	0	0
$22$	11.1445	2.37601
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	−4.92428	−0.984856
$26$	6.26548	1.22876
$27$	0	0
$28$	−20.0184	−3.78312
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	0	0
$31$	−1.54451	−0.277402	−0.138701	−	0.990334i	$-0.544293\pi$
$31$	−1.54451	−0.277402	−0.138701	+	0.990334i	$0.544293\pi$
$32$	−1.15424	−0.204043
$33$	0	0
$34$	−16.4314	−2.81796
$35$	−1.42075	−0.240151
$36$	0	0
$37$	−1.24464	−0.204617	−0.102308	−	0.994753i	$-0.532623\pi$
$37$	−1.24464	−0.204617	−0.102308	+	0.994753i	$0.532623\pi$
$38$	−19.2349	−3.12031
$39$	0	0
$40$	1.25229	0.198004
$41$	1.39786	0.218308	0.109154	−	0.994025i	$-0.465186\pi$
$41$	1.39786	0.218308	0.109154	+	0.994025i	$0.465186\pi$
$42$	0	0
$43$	−0.777148	−0.118514	−0.0592570	−	0.998243i	$-0.518873\pi$
$43$	−0.777148	−0.118514	−0.0592570	+	0.998243i	$0.518873\pi$
$44$	17.8235	2.68699
$45$	0	0
$46$	−2.42429	−0.357443
$47$	5.57928	0.813822	0.406911	−	0.913468i	$-0.366606\pi$
$47$	5.57928	0.813822	0.406911	+	0.913468i	$0.366606\pi$
$48$	0	0
$49$	19.6576	2.80823
$50$	−11.9379	−1.68827
$51$	0	0
$52$	10.0205	1.38959
$53$	−11.5021	−1.57994	−0.789970	−	0.613145i	$-0.789904\pi$
$53$	−11.5021	−1.57994	−0.789970	+	0.613145i	$0.789904\pi$
$54$	0	0
$55$	1.26498	0.170569
$56$	−23.4967	−3.13988
$57$	0	0
$58$	2.42429	0.318325
$59$	−7.31928	−0.952889	−0.476444	−	0.879205i	$-0.658075\pi$
$59$	−7.31928	−0.952889	−0.476444	+	0.879205i	$0.658075\pi$
$60$	0	0
$61$	1.28650	0.164719	0.0823596	−	0.996603i	$-0.473754\pi$
$61$	1.28650	0.164719	0.0823596	+	0.996603i	$0.473754\pi$
$62$	−3.74434	−0.475532
$63$	0	0
$64$	−9.35480	−1.16935
$65$	0.711175	0.0882105
$66$	0	0
$67$	10.1934	1.24533	0.622663	−	0.782490i	$-0.286051\pi$
$67$	10.1934	1.24533	0.622663	+	0.782490i	$0.286051\pi$
$68$	−26.2789	−3.18678
$69$	0	0
$70$	−3.44432	−0.411675
$71$	−4.16451	−0.494236	−0.247118	−	0.968985i	$-0.579484\pi$
$71$	−4.16451	−0.494236	−0.247118	+	0.968985i	$0.579484\pi$
$72$	0	0
$73$	−10.2878	−1.20409	−0.602046	−	0.798462i	$-0.705648\pi$
$73$	−10.2878	−1.20409	−0.602046	+	0.798462i	$0.705648\pi$
$74$	−3.01736	−0.350761
$75$	0	0
$76$	−30.7626	−3.52871
$77$	−23.7348	−2.70483
$78$	0	0
$79$	−1.19088	−0.133985	−0.0669923	−	0.997753i	$-0.521340\pi$
$79$	−1.19088	−0.133985	−0.0669923	+	0.997753i	$0.521340\pi$
$80$	0.902101	0.100858
$81$	0	0
$82$	3.38881	0.374232
$83$	5.53442	0.607482	0.303741	−	0.952755i	$-0.401764\pi$
$83$	5.53442	0.607482	0.303741	+	0.952755i	$0.401764\pi$
$84$	0	0
$85$	−1.86507	−0.202296
$86$	−1.88404	−0.203161
$87$	0	0
$88$	20.9204	2.23013
$89$	−13.4423	−1.42488	−0.712441	−	0.701732i	$-0.752410\pi$
$89$	−13.4423	−1.42488	−0.712441	+	0.701732i	$0.752410\pi$
$90$	0	0
$91$	−13.3438	−1.39881
$92$	−3.87720	−0.404226
$93$	0	0
$94$	13.5258	1.39508
$95$	−2.18329	−0.224001
$96$	0	0
$97$	3.41413	0.346652	0.173326	−	0.984865i	$-0.444549\pi$
$97$	3.41413	0.346652	0.173326	+	0.984865i	$0.444549\pi$
$98$	47.6559	4.81397
$99$	0	0
$100$	−19.0924	−1.90924
$101$	6.22013	0.618926	0.309463	−	0.950912i	$-0.399851\pi$
$101$	6.22013	0.618926	0.309463	+	0.950912i	$0.399851\pi$
$102$	0	0
$103$	−5.78182	−0.569699	−0.284850	−	0.958572i	$-0.591944\pi$
$103$	−5.78182	−0.569699	−0.284850	+	0.958572i	$0.591944\pi$
$104$	11.7616	1.15332
$105$	0	0
$106$	−27.8846	−2.70839
$107$	−18.4577	−1.78437	−0.892185	−	0.451670i	$-0.850829\pi$
$107$	−18.4577	−1.78437	−0.892185	+	0.451670i	$0.850829\pi$
$108$	0	0
$109$	−0.772038	−0.0739478	−0.0369739	−	0.999316i	$-0.511772\pi$
$109$	−0.772038	−0.0739478	−0.0369739	+	0.999316i	$0.511772\pi$
$110$	3.06667	0.292396
$111$	0	0
$112$	−16.9261	−1.59937
$113$	12.7996	1.20408	0.602041	−	0.798465i	$-0.294354\pi$
$113$	12.7996	1.20408	0.602041	+	0.798465i	$0.294354\pi$
$114$	0	0
$115$	−0.275174	−0.0256601
$116$	3.87720	0.359989
$117$	0	0
$118$	−17.7441	−1.63347
$119$	34.9944	3.20793
$120$	0	0
$121$	10.1324	0.921127
$122$	3.11885	0.282367
$123$	0	0
$124$	−5.98837	−0.537772
$125$	−2.73091	−0.244260
$126$	0	0
$127$	15.1484	1.34420	0.672102	−	0.740458i	$-0.265392\pi$
$127$	15.1484	1.34420	0.672102	+	0.740458i	$0.265392\pi$
$128$	−20.3703	−1.80050
$129$	0	0
$130$	1.72410	0.151213
$131$	3.96670	0.346572	0.173286	−	0.984872i	$-0.444562\pi$
$131$	3.96670	0.346572	0.173286	+	0.984872i	$0.444562\pi$
$132$	0	0
$133$	40.9652	3.55213
$134$	24.7119	2.13478
$135$	0	0
$136$	−30.8450	−2.64494
$137$	−8.86496	−0.757385	−0.378692	−	0.925523i	$-0.623626\pi$
$137$	−8.86496	−0.757385	−0.378692	+	0.925523i	$0.623626\pi$
$138$	0	0
$139$	17.1764	1.45689	0.728443	−	0.685106i	$-0.240244\pi$
$139$	17.1764	1.45689	0.728443	+	0.685106i	$0.240244\pi$
$140$	−5.50855	−0.465557
$141$	0	0
$142$	−10.0960	−0.847237
$143$	11.8807	0.993517
$144$	0	0
$145$	0.275174	0.0228520
$146$	−24.9406	−2.06410
$147$	0	0
$148$	−4.82570	−0.396670
$149$	19.2755	1.57911	0.789557	−	0.613677i	$-0.210310\pi$
$149$	19.2755	1.57911	0.789557	+	0.613677i	$0.210310\pi$
$150$	0	0
$151$	5.14403	0.418615	0.209307	−	0.977850i	$-0.432879\pi$
$151$	5.14403	0.418615	0.209307	+	0.977850i	$0.432879\pi$
$152$	−36.1078	−2.92873
$153$	0	0
$154$	−57.5401	−4.63671
$155$	−0.425009	−0.0341376
$156$	0	0
$157$	−17.4758	−1.39472	−0.697360	−	0.716721i	$-0.745642\pi$
$157$	−17.4758	−1.39472	−0.697360	+	0.716721i	$0.745642\pi$
$158$	−2.88704	−0.229681
$159$	0	0
$160$	−0.317617	−0.0251098
$161$	5.16310	0.406909
$162$	0	0
$163$	−11.5511	−0.904750	−0.452375	−	0.891828i	$-0.649423\pi$
$163$	−11.5511	−0.904750	−0.452375	+	0.891828i	$0.649423\pi$
$164$	5.41977	0.423213
$165$	0	0
$166$	13.4171	1.04137
$167$	−16.8806	−1.30626	−0.653128	−	0.757247i	$-0.726544\pi$
$167$	−16.8806	−1.30626	−0.653128	+	0.757247i	$0.726544\pi$
$168$	0	0
$169$	−6.32059	−0.486200
$170$	−4.52149	−0.346782
$171$	0	0
$172$	−3.01316	−0.229751
$173$	2.08494	0.158515	0.0792574	−	0.996854i	$-0.474745\pi$
$173$	2.08494	0.158515	0.0792574	+	0.996854i	$0.474745\pi$
$174$	0	0
$175$	25.4246	1.92192
$176$	15.0703	1.13597
$177$	0	0
$178$	−32.5881	−2.44258
$179$	−8.87412	−0.663283	−0.331641	−	0.943406i	$-0.607602\pi$
$179$	−8.87412	−0.663283	−0.331641	+	0.943406i	$0.607602\pi$
$180$	0	0
$181$	−21.0114	−1.56176	−0.780881	−	0.624680i	$-0.785229\pi$
$181$	−21.0114	−1.56176	−0.780881	+	0.624680i	$0.785229\pi$
$182$	−32.3493	−2.39789
$183$	0	0
$184$	−4.55089	−0.335496
$185$	−0.342492	−0.0251805
$186$	0	0
$187$	−31.1575	−2.27846
$188$	21.6320	1.57768
$189$	0	0
$190$	−5.29295	−0.383991
$191$	9.11834	0.659780	0.329890	−	0.944019i	$-0.392988\pi$
$191$	9.11834	0.659780	0.329890	+	0.944019i	$0.392988\pi$
$192$	0	0
$193$	−1.41977	−0.102197	−0.0510987	−	0.998694i	$-0.516272\pi$
$193$	−1.41977	−0.102197	−0.0510987	+	0.998694i	$0.516272\pi$
$194$	8.27684	0.594243
$195$	0	0
$196$	76.2166	5.44404
$197$	2.98927	0.212977	0.106488	−	0.994314i	$-0.466039\pi$
$197$	2.98927	0.212977	0.106488	+	0.994314i	$0.466039\pi$
$198$	0	0
$199$	14.2526	1.01034	0.505170	−	0.863020i	$-0.331430\pi$
$199$	14.2526	1.01034	0.505170	+	0.863020i	$0.331430\pi$
$200$	−22.4098	−1.58462
$201$	0	0
$202$	15.0794	1.06098
$203$	−5.16310	−0.362379
$204$	0	0
$205$	0.384654	0.0268654
$206$	−14.0168	−0.976598
$207$	0	0
$208$	8.47259	0.587469
$209$	−36.4736	−2.52293
$210$	0	0
$211$	−4.99505	−0.343874	−0.171937	−	0.985108i	$-0.555002\pi$
$211$	−4.99505	−0.343874	−0.171937	+	0.985108i	$0.555002\pi$
$212$	−44.5961	−3.06287
$213$	0	0
$214$	−44.7468	−3.05883
$215$	−0.213851	−0.0145845
$216$	0	0
$217$	7.97446	0.541341
$218$	−1.87165	−0.126764
$219$	0	0
$220$	4.90457	0.330666
$221$	−17.5169	−1.17831
$222$	0	0
$223$	17.0823	1.14391	0.571956	−	0.820284i	$-0.306185\pi$
$223$	17.0823	1.14391	0.571956	+	0.820284i	$0.306185\pi$
$224$	5.95945	0.398183
$225$	0	0
$226$	31.0299	2.06408
$227$	5.65703	0.375471	0.187735	−	0.982220i	$-0.439885\pi$
$227$	5.65703	0.375471	0.187735	+	0.982220i	$0.439885\pi$
$228$	0	0
$229$	−13.6271	−0.900503	−0.450251	−	0.892902i	$-0.648666\pi$
$229$	−13.6271	−0.900503	−0.450251	+	0.892902i	$0.648666\pi$
$230$	−0.667103	−0.0439875
$231$	0	0
$232$	4.55089	0.298781
$233$	8.31854	0.544965	0.272483	−	0.962161i	$-0.412155\pi$
$233$	8.31854	0.544965	0.272483	+	0.962161i	$0.412155\pi$
$234$	0	0
$235$	1.53527	0.100150
$236$	−28.3783	−1.84727
$237$	0	0
$238$	84.8368	5.49915
$239$	−10.6654	−0.689886	−0.344943	−	0.938624i	$-0.612102\pi$
$239$	−10.6654	−0.689886	−0.344943	+	0.938624i	$0.612102\pi$
$240$	0	0
$241$	−19.2982	−1.24311	−0.621554	−	0.783371i	$-0.713498\pi$
$241$	−19.2982	−1.24311	−0.621554	+	0.783371i	$0.713498\pi$
$242$	24.5639	1.57903
$243$	0	0
$244$	4.98801	0.319325
$245$	5.40927	0.345586
$246$	0	0
$247$	−20.5056	−1.30474
$248$	−7.02889	−0.446335
$249$	0	0
$250$	−6.62052	−0.418718
$251$	−7.91467	−0.499570	−0.249785	−	0.968301i	$-0.580360\pi$
$251$	−7.91467	−0.499570	−0.249785	+	0.968301i	$0.580360\pi$
$252$	0	0
$253$	−4.59700	−0.289011
$254$	36.7242	2.30428
$255$	0	0
$256$	−30.6740	−1.91712
$257$	−5.60389	−0.349561	−0.174781	−	0.984607i	$-0.555922\pi$
$257$	−5.60389	−0.349561	−0.174781	+	0.984607i	$0.555922\pi$
$258$	0	0
$259$	6.42618	0.399304
$260$	2.75737	0.171005
$261$	0	0
$262$	9.61644	0.594106
$263$	18.0911	1.11555	0.557774	−	0.829993i	$-0.311656\pi$
$263$	18.0911	1.11555	0.557774	+	0.829993i	$0.311656\pi$
$264$	0	0
$265$	−3.16509	−0.194430
$266$	99.3117	6.08919
$267$	0	0
$268$	39.5220	2.41419
$269$	−8.37348	−0.510540	−0.255270	−	0.966870i	$-0.582164\pi$
$269$	−8.37348	−0.510540	−0.255270	+	0.966870i	$0.582164\pi$
$270$	0	0
$271$	2.69125	0.163482	0.0817408	−	0.996654i	$-0.473952\pi$
$271$	2.69125	0.163482	0.0817408	+	0.996654i	$0.473952\pi$
$272$	−22.2196	−1.34726
$273$	0	0
$274$	−21.4913	−1.29834
$275$	−22.6369	−1.36506
$276$	0	0
$277$	32.6291	1.96049	0.980247	−	0.197776i	$-0.0633718\pi$
$277$	32.6291	1.96049	0.980247	+	0.197776i	$0.0633718\pi$
$278$	41.6408	2.49745
$279$	0	0
$280$	−6.46569	−0.386399
$281$	21.2594	1.26823	0.634114	−	0.773239i	$-0.281365\pi$
$281$	21.2594	1.26823	0.634114	+	0.773239i	$0.281365\pi$
$282$	0	0
$283$	−6.43670	−0.382622	−0.191311	−	0.981529i	$-0.561274\pi$
$283$	−6.43670	−0.382622	−0.191311	+	0.981529i	$0.561274\pi$
$284$	−16.1466	−0.958127
$285$	0	0
$286$	28.8024	1.70312
$287$	−7.21727	−0.426022
$288$	0	0
$289$	28.9385	1.70226
$290$	0.667103	0.0391737
$291$	0	0
$292$	−39.8877	−2.33425
$293$	7.42249	0.433627	0.216813	−	0.976213i	$-0.430434\pi$
$293$	7.42249	0.433627	0.216813	+	0.976213i	$0.430434\pi$
$294$	0	0
$295$	−2.01408	−0.117264
$296$	−5.66420	−0.329225
$297$	0	0
$298$	46.7296	2.70697
$299$	−2.58445	−0.149463
$300$	0	0
$301$	4.01250	0.231276
$302$	12.4706	0.717604
$303$	0	0
$304$	−26.0107	−1.49181
$305$	0.354011	0.0202706
$306$	0	0
$307$	0.106672	0.00608808	0.00304404	−	0.999995i	$-0.499031\pi$
$307$	0.106672	0.00608808	0.00304404	+	0.999995i	$0.499031\pi$
$308$	−92.0245	−5.24358
$309$	0	0
$310$	−1.03035	−0.0585198
$311$	16.3006	0.924324	0.462162	−	0.886796i	$-0.347074\pi$
$311$	16.3006	0.924324	0.462162	+	0.886796i	$0.347074\pi$
$312$	0	0
$313$	1.55399	0.0878366	0.0439183	−	0.999035i	$-0.486016\pi$
$313$	1.55399	0.0878366	0.0439183	+	0.999035i	$0.486016\pi$
$314$	−42.3664	−2.39088
$315$	0	0
$316$	−4.61728	−0.259743
$317$	−32.8913	−1.84736	−0.923679	−	0.383166i	$-0.874834\pi$
$317$	−32.8913	−1.84736	−0.923679	+	0.383166i	$0.874834\pi$
$318$	0	0
$319$	4.59700	0.257383
$320$	−2.57420	−0.143902
$321$	0	0
$322$	12.5169	0.697538
$323$	53.7765	2.99220
$324$	0	0
$325$	−12.7266	−0.705943
$326$	−28.0032	−1.55095
$327$	0	0
$328$	6.36149	0.351254
$329$	−28.8064	−1.58815
$330$	0	0
$331$	21.5227	1.18299	0.591496	−	0.806308i	$-0.298537\pi$
$331$	21.5227	1.18299	0.591496	+	0.806308i	$0.298537\pi$
$332$	21.4581	1.17766
$333$	0	0
$334$	−40.9234	−2.23923
$335$	2.80497	0.153252
$336$	0	0
$337$	−30.3271	−1.65202	−0.826012	−	0.563653i	$-0.809396\pi$
$337$	−30.3271	−1.65202	−0.826012	+	0.563653i	$0.809396\pi$
$338$	−15.3230	−0.833460
$339$	0	0
$340$	−7.23127	−0.392171
$341$	−7.10011	−0.384492
$342$	0	0
$343$	−65.3526	−3.52871
$344$	−3.53672	−0.190687
$345$	0	0
$346$	5.05450	0.271732
$347$	−1.44730	−0.0776953	−0.0388476	−	0.999245i	$-0.512369\pi$
$347$	−1.44730	−0.0776953	−0.0388476	+	0.999245i	$0.512369\pi$
$348$	0	0
$349$	−0.963889	−0.0515958	−0.0257979	−	0.999667i	$-0.508213\pi$
$349$	−0.963889	−0.0515958	−0.0257979	+	0.999667i	$0.508213\pi$
$350$	61.6366	3.29462
$351$	0	0
$352$	−5.30604	−0.282813
$353$	−19.4579	−1.03564	−0.517821	−	0.855489i	$-0.673257\pi$
$353$	−19.4579	−1.03564	−0.517821	+	0.855489i	$0.673257\pi$
$354$	0	0
$355$	−1.14597	−0.0608215
$356$	−52.1186	−2.76228
$357$	0	0
$358$	−21.5135	−1.13702
$359$	22.4871	1.18682	0.593411	−	0.804900i	$-0.297781\pi$
$359$	22.4871	1.18682	0.593411	+	0.804900i	$0.297781\pi$
$360$	0	0
$361$	43.9519	2.31326
$362$	−50.9377	−2.67723
$363$	0	0
$364$	−51.7366	−2.71174
$365$	−2.83093	−0.148178
$366$	0	0
$367$	−22.9126	−1.19603	−0.598014	−	0.801486i	$-0.704043\pi$
$367$	−22.9126	−1.19603	−0.598014	+	0.801486i	$0.704043\pi$
$368$	−3.27829	−0.170893
$369$	0	0
$370$	−0.830301	−0.0431653
$371$	59.3867	3.08321
$372$	0	0
$373$	3.09309	0.160154	0.0800770	−	0.996789i	$-0.474483\pi$
$373$	3.09309	0.160154	0.0800770	+	0.996789i	$0.474483\pi$
$374$	−75.5349	−3.90582
$375$	0	0
$376$	25.3907	1.30942
$377$	2.58445	0.133106
$378$	0	0
$379$	31.6064	1.62351	0.811756	−	0.583996i	$-0.198512\pi$
$379$	31.6064	1.62351	0.811756	+	0.583996i	$0.198512\pi$
$380$	−8.46507	−0.434249
$381$	0	0
$382$	22.1055	1.13102
$383$	−14.6954	−0.750898	−0.375449	−	0.926843i	$-0.622511\pi$
$383$	−14.6954	−0.750898	−0.375449	+	0.926843i	$0.622511\pi$
$384$	0	0
$385$	−6.53120	−0.332861
$386$	−3.44195	−0.175191
$387$	0	0
$388$	13.2373	0.672020
$389$	−20.4520	−1.03696	−0.518478	−	0.855091i	$-0.673501\pi$
$389$	−20.4520	−1.03696	−0.518478	+	0.855091i	$0.673501\pi$
$390$	0	0
$391$	6.77779	0.342768
$392$	89.4597	4.51840
$393$	0	0
$394$	7.24687	0.365092
$395$	−0.327700	−0.0164884
$396$	0	0
$397$	1.93448	0.0970887	0.0485443	−	0.998821i	$-0.484542\pi$
$397$	1.93448	0.0970887	0.0485443	+	0.998821i	$0.484542\pi$
$398$	34.5525	1.73196
$399$	0	0
$400$	−16.1432	−0.807161
$401$	4.44527	0.221986	0.110993	−	0.993821i	$-0.464597\pi$
$401$	4.44527	0.221986	0.110993	+	0.993821i	$0.464597\pi$
$402$	0	0
$403$	−3.99171	−0.198841
$404$	24.1167	1.19985
$405$	0	0
$406$	−12.5169	−0.621202
$407$	−5.72159	−0.283609
$408$	0	0
$409$	7.40105	0.365958	0.182979	−	0.983117i	$-0.441426\pi$
$409$	7.40105	0.365958	0.182979	+	0.983117i	$0.441426\pi$
$410$	0.932514	0.0460536
$411$	0	0
$412$	−22.4173	−1.10442
$413$	37.7902	1.85953
$414$	0	0
$415$	1.52293	0.0747577
$416$	−2.98308	−0.146257
$417$	0	0
$418$	−88.4228	−4.32490
$419$	39.3381	1.92179	0.960896	−	0.276910i	$-0.0893103\pi$
$419$	39.3381	1.92179	0.960896	+	0.276910i	$0.0893103\pi$
$420$	0	0
$421$	34.1384	1.66380	0.831901	−	0.554924i	$-0.187253\pi$
$421$	34.1384	1.66380	0.831901	+	0.554924i	$0.187253\pi$
$422$	−12.1095	−0.589480
$423$	0	0
$424$	−52.3450	−2.54210
$425$	33.3757	1.61896
$426$	0	0
$427$	−6.64232	−0.321444
$428$	−71.5641	−3.45918
$429$	0	0
$430$	−0.518438	−0.0250013
$431$	−19.7955	−0.953518	−0.476759	−	0.879034i	$-0.658189\pi$
$431$	−19.7955	−0.953518	−0.476759	+	0.879034i	$0.658189\pi$
$432$	0	0
$433$	34.2150	1.64427	0.822134	−	0.569294i	$-0.192783\pi$
$433$	34.2150	1.64427	0.822134	+	0.569294i	$0.192783\pi$
$434$	19.3324	0.927986
$435$	0	0
$436$	−2.99335	−0.143355
$437$	7.93422	0.379545
$438$	0	0
$439$	−9.25190	−0.441569	−0.220784	−	0.975323i	$-0.570862\pi$
$439$	−9.25190	−0.441569	−0.220784	+	0.975323i	$0.570862\pi$
$440$	5.75676	0.274443
$441$	0	0
$442$	−42.4661	−2.01991
$443$	6.68002	0.317377	0.158689	−	0.987329i	$-0.449273\pi$
$443$	6.68002	0.317377	0.158689	+	0.987329i	$0.449273\pi$
$444$	0	0
$445$	−3.69898	−0.175348
$446$	41.4124	1.96093
$447$	0	0
$448$	48.2998	2.28195
$449$	5.68296	0.268195	0.134098	−	0.990968i	$-0.457186\pi$
$449$	5.68296	0.268195	0.134098	+	0.990968i	$0.457186\pi$
$450$	0	0
$451$	6.42594	0.302586
$452$	49.6265	2.33424
$453$	0	0
$454$	13.7143	0.643645
$455$	−3.67187	−0.172140
$456$	0	0
$457$	−18.4227	−0.861776	−0.430888	−	0.902405i	$-0.641800\pi$
$457$	−18.4227	−0.861776	−0.430888	+	0.902405i	$0.641800\pi$
$458$	−33.0361	−1.54367
$459$	0	0
$460$	−1.06691	−0.0497448
$461$	−29.4473	−1.37150	−0.685749	−	0.727838i	$-0.740525\pi$
$461$	−29.4473	−1.37150	−0.685749	+	0.727838i	$0.740525\pi$
$462$	0	0
$463$	32.0658	1.49022	0.745111	−	0.666941i	$-0.232397\pi$
$463$	32.0658	1.49022	0.745111	+	0.666941i	$0.232397\pi$
$464$	3.27829	0.152191
$465$	0	0
$466$	20.1666	0.934199
$467$	−14.1343	−0.654059	−0.327029	−	0.945014i	$-0.606048\pi$
$467$	−14.1343	−0.654059	−0.327029	+	0.945014i	$0.606048\pi$
$468$	0	0
$469$	−52.6298	−2.43022
$470$	3.72196	0.171681
$471$	0	0
$472$	−33.3092	−1.53318
$473$	−3.57255	−0.164266
$474$	0	0
$475$	39.0703	1.79267
$476$	135.680	6.21890
$477$	0	0
$478$	−25.8560	−1.18263
$479$	−8.84879	−0.404312	−0.202156	−	0.979353i	$-0.564795\pi$
$479$	−8.84879	−0.404312	−0.202156	+	0.979353i	$0.564795\pi$
$480$	0	0
$481$	−3.21670	−0.146669
$482$	−46.7846	−2.13098
$483$	0	0
$484$	39.2854	1.78570
$485$	0.939479	0.0426596
$486$	0	0
$487$	6.30353	0.285640	0.142820	−	0.989749i	$-0.454383\pi$
$487$	6.30353	0.285640	0.142820	+	0.989749i	$0.454383\pi$
$488$	5.85471	0.265030
$489$	0	0
$490$	13.1137	0.592415
$491$	22.2801	1.00549	0.502743	−	0.864436i	$-0.332324\pi$
$491$	22.2801	1.00549	0.502743	+	0.864436i	$0.332324\pi$
$492$	0	0
$493$	−6.77779	−0.305256
$494$	−49.7117	−2.23663
$495$	0	0
$496$	−5.06335	−0.227351
$497$	21.5018	0.964487
$498$	0	0
$499$	−5.66457	−0.253581	−0.126791	−	0.991930i	$-0.540468\pi$
$499$	−5.66457	−0.253581	−0.126791	+	0.991930i	$0.540468\pi$
$500$	−10.5883	−0.473522
$501$	0	0
$502$	−19.1875	−0.856380
$503$	−25.2240	−1.12468	−0.562340	−	0.826906i	$-0.690099\pi$
$503$	−25.2240	−1.12468	−0.562340	+	0.826906i	$0.690099\pi$
$504$	0	0
$505$	1.71162	0.0761660
$506$	−11.1445	−0.495432
$507$	0	0
$508$	58.7335	2.60588
$509$	−1.51350	−0.0670847	−0.0335423	−	0.999437i	$-0.510679\pi$
$509$	−1.51350	−0.0670847	−0.0335423	+	0.999437i	$0.510679\pi$
$510$	0	0
$511$	53.1168	2.34975
$512$	−33.6222	−1.48591
$513$	0	0
$514$	−13.5855	−0.599230
$515$	−1.59101	−0.0701081
$516$	0	0
$517$	25.6479	1.12800
$518$	15.5790	0.684500
$519$	0	0
$520$	3.23648	0.141929
$521$	−3.07057	−0.134524	−0.0672621	−	0.997735i	$-0.521426\pi$
$521$	−3.07057	−0.134524	−0.0672621	+	0.997735i	$0.521426\pi$
$522$	0	0
$523$	−8.05915	−0.352402	−0.176201	−	0.984354i	$-0.556381\pi$
$523$	−8.05915	−0.352402	−0.176201	+	0.984354i	$0.556381\pi$
$524$	15.3797	0.671865
$525$	0	0
$526$	43.8583	1.91231
$527$	10.4684	0.456009
$528$	0	0
$529$	1.00000	0.0434783
$530$	−7.67312	−0.333299
$531$	0	0
$532$	158.830	6.88617
$533$	3.61269	0.156483
$534$	0	0
$535$	−5.07908	−0.219588
$536$	46.3892	2.00371
$537$	0	0
$538$	−20.2998	−0.875186
$539$	90.3661	3.89234
$540$	0	0
$541$	43.7791	1.88221	0.941106	−	0.338112i	$-0.109788\pi$
$541$	43.7791	1.88221	0.941106	+	0.338112i	$0.109788\pi$
$542$	6.52437	0.280246
$543$	0	0
$544$	7.82319	0.335417
$545$	−0.212445	−0.00910014
$546$	0	0
$547$	−1.38785	−0.0593402	−0.0296701	−	0.999560i	$-0.509446\pi$
$547$	−1.38785	−0.0593402	−0.0296701	+	0.999560i	$0.509446\pi$
$548$	−34.3713	−1.46827
$549$	0	0
$550$	−54.8785	−2.34003
$551$	−7.93422	−0.338009
$552$	0	0
$553$	6.14864	0.261467
$554$	79.1026	3.36075
$555$	0	0
$556$	66.5965	2.82432
$557$	−22.8042	−0.966246	−0.483123	−	0.875552i	$-0.660498\pi$
$557$	−22.8042	−0.966246	−0.483123	+	0.875552i	$0.660498\pi$
$558$	0	0
$559$	−2.00850	−0.0849507
$560$	−4.65764	−0.196821
$561$	0	0
$562$	51.5390	2.17404
$563$	7.80360	0.328883	0.164441	−	0.986387i	$-0.447418\pi$
$563$	7.80360	0.328883	0.164441	+	0.986387i	$0.447418\pi$
$564$	0	0
$565$	3.52211	0.148176
$566$	−15.6045	−0.655904
$567$	0	0
$568$	−18.9522	−0.795217
$569$	37.1514	1.55747	0.778733	−	0.627356i	$-0.215863\pi$
$569$	37.1514	1.55747	0.778733	+	0.627356i	$0.215863\pi$
$570$	0	0
$571$	−22.6811	−0.949174	−0.474587	−	0.880209i	$-0.657403\pi$
$571$	−22.6811	−0.949174	−0.474587	+	0.880209i	$0.657403\pi$
$572$	46.0640	1.92603
$573$	0	0
$574$	−17.4968	−0.730302
$575$	4.92428	0.205357
$576$	0	0
$577$	−19.6189	−0.816747	−0.408374	−	0.912815i	$-0.633904\pi$
$577$	−19.6189	−0.816747	−0.408374	+	0.912815i	$0.633904\pi$
$578$	70.1553	2.91808
$579$	0	0
$580$	1.06691	0.0443009
$581$	−28.5748	−1.18548
$582$	0	0
$583$	−52.8753	−2.18987
$584$	−46.8185	−1.93736
$585$	0	0
$586$	17.9943	0.743338
$587$	−15.7582	−0.650411	−0.325206	−	0.945643i	$-0.605434\pi$
$587$	−15.7582	−0.650411	−0.325206	+	0.945643i	$0.605434\pi$
$588$	0	0
$589$	12.2545	0.504937
$590$	−4.88271	−0.201018
$591$	0	0
$592$	−4.08028	−0.167698
$593$	7.35744	0.302134	0.151067	−	0.988524i	$-0.451729\pi$
$593$	7.35744	0.302134	0.151067	+	0.988524i	$0.451729\pi$
$594$	0	0
$595$	9.62957	0.394774
$596$	74.7352	3.06127
$597$	0	0
$598$	−6.26548	−0.256214
$599$	−27.0050	−1.10340	−0.551698	−	0.834044i	$-0.686020\pi$
$599$	−27.0050	−1.10340	−0.551698	+	0.834044i	$0.686020\pi$
$600$	0	0
$601$	−23.4245	−0.955504	−0.477752	−	0.878495i	$-0.658548\pi$
$601$	−23.4245	−0.955504	−0.477752	+	0.878495i	$0.658548\pi$
$602$	9.72747	0.396462
$603$	0	0
$604$	19.9444	0.811527
$605$	2.78818	0.113355
$606$	0	0
$607$	19.8139	0.804222	0.402111	−	0.915591i	$-0.368277\pi$
$607$	19.8139	0.804222	0.402111	+	0.915591i	$0.368277\pi$
$608$	9.15799	0.371405
$609$	0	0
$610$	0.858227	0.0347486
$611$	14.4194	0.583346
$612$	0	0
$613$	7.88252	0.318372	0.159186	−	0.987249i	$-0.449113\pi$
$613$	7.88252	0.318372	0.159186	+	0.987249i	$0.449113\pi$
$614$	0.258604	0.0104364
$615$	0	0
$616$	−108.014	−4.35202
$617$	37.4119	1.50615	0.753073	−	0.657937i	$-0.228571\pi$
$617$	37.4119	1.50615	0.753073	+	0.657937i	$0.228571\pi$
$618$	0	0
$619$	−14.5529	−0.584931	−0.292465	−	0.956276i	$-0.594476\pi$
$619$	−14.5529	−0.584931	−0.292465	+	0.956276i	$0.594476\pi$
$620$	−1.64785	−0.0661791
$621$	0	0
$622$	39.5175	1.58451
$623$	69.4040	2.78061
$624$	0	0
$625$	23.8699	0.954797
$626$	3.76733	0.150573
$627$	0	0
$628$	−67.7571	−2.70380
$629$	8.43588	0.336361
$630$	0	0
$631$	−21.1886	−0.843504	−0.421752	−	0.906711i	$-0.638585\pi$
$631$	−21.1886	−0.843504	−0.421752	+	0.906711i	$0.638585\pi$
$632$	−5.41956	−0.215579
$633$	0	0
$634$	−79.7382	−3.16681
$635$	4.16845	0.165420
$636$	0	0
$637$	50.8042	2.01294
$638$	11.1445	0.441214
$639$	0	0
$640$	−5.60538	−0.221572
$641$	−25.4677	−1.00591	−0.502957	−	0.864312i	$-0.667755\pi$
$641$	−25.4677	−1.00591	−0.502957	+	0.864312i	$0.667755\pi$
$642$	0	0
$643$	36.2517	1.42963	0.714814	−	0.699314i	$-0.246511\pi$
$643$	36.2517	1.42963	0.714814	+	0.699314i	$0.246511\pi$
$644$	20.0184	0.788835
$645$	0	0
$646$	130.370	5.12934
$647$	−40.3128	−1.58486	−0.792430	−	0.609963i	$-0.791184\pi$
$647$	−40.3128	−1.58486	−0.792430	+	0.609963i	$0.791184\pi$
$648$	0	0
$649$	−33.6467	−1.32075
$650$	−30.8530	−1.21015
$651$	0	0
$652$	−44.7858	−1.75395
$653$	−28.4754	−1.11433	−0.557164	−	0.830402i	$-0.688111\pi$
$653$	−28.4754	−1.11433	−0.557164	+	0.830402i	$0.688111\pi$
$654$	0	0
$655$	1.09153	0.0426497
$656$	4.58258	0.178920
$657$	0	0
$658$	−69.8352	−2.72246
$659$	4.69239	0.182790	0.0913948	−	0.995815i	$-0.470867\pi$
$659$	4.69239	0.182790	0.0913948	+	0.995815i	$0.470867\pi$
$660$	0	0
$661$	18.8296	0.732386	0.366193	−	0.930539i	$-0.380661\pi$
$661$	18.8296	0.732386	0.366193	+	0.930539i	$0.380661\pi$
$662$	52.1773	2.02793
$663$	0	0
$664$	25.1865	0.977427
$665$	11.2726	0.437132
$666$	0	0
$667$	−1.00000	−0.0387202
$668$	−65.4493	−2.53231
$669$	0	0
$670$	6.80008	0.262710
$671$	5.91403	0.228309
$672$	0	0
$673$	−15.9078	−0.613201	−0.306600	−	0.951838i	$-0.599192\pi$
$673$	−15.9078	−0.613201	−0.306600	+	0.951838i	$0.599192\pi$
$674$	−73.5219	−2.83196
$675$	0	0
$676$	−24.5062	−0.942547
$677$	−30.7038	−1.18004	−0.590022	−	0.807387i	$-0.700881\pi$
$677$	−30.7038	−1.18004	−0.590022	+	0.807387i	$0.700881\pi$
$678$	0	0
$679$	−17.6275	−0.676481
$680$	−8.48775	−0.325490
$681$	0	0
$682$	−17.2127	−0.659110
$683$	12.1098	0.463369	0.231685	−	0.972791i	$-0.425576\pi$
$683$	12.1098	0.463369	0.231685	+	0.972791i	$0.425576\pi$
$684$	0	0
$685$	−2.43941	−0.0932051
$686$	−158.434	−6.04904
$687$	0	0
$688$	−2.54772	−0.0971308
$689$	−29.7268	−1.13250
$690$	0	0
$691$	37.0959	1.41119	0.705597	−	0.708613i	$-0.250679\pi$
$691$	37.0959	1.41119	0.705597	+	0.708613i	$0.250679\pi$
$692$	8.08372	0.307297
$693$	0	0
$694$	−3.50869	−0.133188
$695$	4.72652	0.179287
$696$	0	0
$697$	−9.47438	−0.358868
$698$	−2.33675	−0.0884473
$699$	0	0
$700$	98.5761	3.72583
$701$	−10.9045	−0.411858	−0.205929	−	0.978567i	$-0.566022\pi$
$701$	−10.9045	−0.411858	−0.205929	+	0.978567i	$0.566022\pi$
$702$	0	0
$703$	9.87522	0.372451
$704$	−43.0040	−1.62077
$705$	0	0
$706$	−47.1718	−1.77533
$707$	−32.1151	−1.20781
$708$	0	0
$709$	12.3176	0.462598	0.231299	−	0.972883i	$-0.425702\pi$
$709$	12.3176	0.462598	0.231299	+	0.972883i	$0.425702\pi$
$710$	−2.77816	−0.104262
$711$	0	0
$712$	−61.1745	−2.29261
$713$	1.54451	0.0578423
$714$	0	0
$715$	3.26927	0.122264
$716$	−34.4067	−1.28584
$717$	0	0
$718$	54.5152	2.03449
$719$	−0.321584	−0.0119931	−0.00599654	−	0.999982i	$-0.501909\pi$
$719$	−0.321584	−0.0119931	−0.00599654	+	0.999982i	$0.501909\pi$
$720$	0	0
$721$	29.8521	1.11175
$722$	106.552	3.96547
$723$	0	0
$724$	−81.4653	−3.02763
$725$	−4.92428	−0.182883
$726$	0	0
$727$	17.3701	0.644223	0.322111	−	0.946702i	$-0.395607\pi$
$727$	17.3701	0.644223	0.322111	+	0.946702i	$0.395607\pi$
$728$	−60.7262	−2.25066
$729$	0	0
$730$	−6.86300	−0.254011
$731$	5.26735	0.194820
$732$	0	0
$733$	12.0450	0.444894	0.222447	−	0.974945i	$-0.428596\pi$
$733$	12.0450	0.444894	0.222447	+	0.974945i	$0.428596\pi$
$734$	−55.5469	−2.05027
$735$	0	0
$736$	1.15424	0.0425458
$737$	46.8592	1.72608
$738$	0	0
$739$	−16.8201	−0.618737	−0.309368	−	0.950942i	$-0.600118\pi$
$739$	−16.8201	−0.618737	−0.309368	+	0.950942i	$0.600118\pi$
$740$	−1.32791	−0.0488149
$741$	0	0
$742$	143.971	5.28534
$743$	17.4044	0.638505	0.319253	−	0.947670i	$-0.396568\pi$
$743$	17.4044	0.638505	0.319253	+	0.947670i	$0.396568\pi$
$744$	0	0
$745$	5.30413	0.194328
$746$	7.49856	0.274542
$747$	0	0
$748$	−120.804	−4.41703
$749$	95.2988	3.48214
$750$	0	0
$751$	−0.100736	−0.00367590	−0.00183795	−	0.999998i	$-0.500585\pi$
$751$	−0.100736	−0.00367590	−0.00183795	+	0.999998i	$0.500585\pi$
$752$	18.2905	0.666986
$753$	0	0
$754$	6.26548	0.228175
$755$	1.41550	0.0515155
$756$	0	0
$757$	34.1198	1.24010	0.620052	−	0.784561i	$-0.287111\pi$
$757$	34.1198	1.24010	0.620052	+	0.784561i	$0.287111\pi$
$758$	76.6232	2.78308
$759$	0	0
$760$	−9.93593	−0.360414
$761$	−23.5537	−0.853822	−0.426911	−	0.904294i	$-0.640398\pi$
$761$	−23.5537	−0.853822	−0.426911	+	0.904294i	$0.640398\pi$
$762$	0	0
$763$	3.98611	0.144307
$764$	35.3536	1.27905
$765$	0	0
$766$	−35.6259	−1.28722
$767$	−18.9163	−0.683029
$768$	0	0
$769$	9.02730	0.325533	0.162766	−	0.986665i	$-0.447958\pi$
$769$	9.02730	0.325533	0.162766	+	0.986665i	$0.447958\pi$
$770$	−15.8335	−0.570602
$771$	0	0
$772$	−5.50475	−0.198120
$773$	−53.4367	−1.92198	−0.960992	−	0.276577i	$-0.910800\pi$
$773$	−53.4367	−1.92198	−0.960992	+	0.276577i	$0.910800\pi$
$774$	0	0
$775$	7.60559	0.273201
$776$	15.5373	0.557757
$777$	0	0
$778$	−49.5816	−1.77759
$779$	−11.0909	−0.397373
$780$	0	0
$781$	−19.1442	−0.685035
$782$	16.4314	0.587584
$783$	0	0
$784$	64.4434	2.30155
$785$	−4.80889	−0.171636
$786$	0	0
$787$	47.7500	1.70210	0.851052	−	0.525081i	$-0.175965\pi$
$787$	47.7500	1.70210	0.851052	+	0.525081i	$0.175965\pi$
$788$	11.5900	0.412877
$789$	0	0
$790$	−0.794440	−0.0282649
$791$	−66.0855	−2.34973
$792$	0	0
$793$	3.32489	0.118070
$794$	4.68974	0.166433
$795$	0	0
$796$	55.2602	1.95865
$797$	−6.00759	−0.212800	−0.106400	−	0.994323i	$-0.533932\pi$
$797$	−6.00759	−0.212800	−0.106400	+	0.994323i	$0.533932\pi$
$798$	0	0
$799$	−37.8152	−1.33781
$800$	5.68380	0.200953
$801$	0	0
$802$	10.7767	0.380537
$803$	−47.2928	−1.66893
$804$	0	0
$805$	1.42075	0.0500750
$806$	−9.67709	−0.340861
$807$	0	0
$808$	28.3071	0.995840
$809$	−33.4712	−1.17678	−0.588392	−	0.808576i	$-0.700239\pi$
$809$	−33.4712	−1.17678	−0.588392	+	0.808576i	$0.700239\pi$
$810$	0	0
$811$	12.2370	0.429700	0.214850	−	0.976647i	$-0.431074\pi$
$811$	12.2370	0.429700	0.214850	+	0.976647i	$0.431074\pi$
$812$	−20.0184	−0.702508
$813$	0	0
$814$	−13.8708	−0.486172
$815$	−3.17856	−0.111340
$816$	0	0
$817$	6.16607	0.215723
$818$	17.9423	0.627339
$819$	0	0
$820$	1.49138	0.0520813
$821$	13.9476	0.486774	0.243387	−	0.969929i	$-0.421742\pi$
$821$	13.9476	0.486774	0.243387	+	0.969929i	$0.421742\pi$
$822$	0	0
$823$	2.34522	0.0817493	0.0408746	−	0.999164i	$-0.486986\pi$
$823$	2.34522	0.0817493	0.0408746	+	0.999164i	$0.486986\pi$
$824$	−26.3124	−0.916636
$825$	0	0
$826$	91.6145	3.18768
$827$	−14.5485	−0.505901	−0.252950	−	0.967479i	$-0.581401\pi$
$827$	−14.5485	−0.505901	−0.252950	+	0.967479i	$0.581401\pi$
$828$	0	0
$829$	−14.6227	−0.507866	−0.253933	−	0.967222i	$-0.581724\pi$
$829$	−14.6227	−0.507866	−0.253933	+	0.967222i	$0.581724\pi$
$830$	3.69203	0.128152
$831$	0	0
$832$	−24.1770	−0.838188
$833$	−133.235	−4.61633
$834$	0	0
$835$	−4.64510	−0.160750
$836$	−141.416	−4.89096
$837$	0	0
$838$	95.3671	3.29440
$839$	−7.29576	−0.251878	−0.125939	−	0.992038i	$-0.540194\pi$
$839$	−7.29576	−0.251878	−0.125939	+	0.992038i	$0.540194\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	82.7615	2.85215
$843$	0	0
$844$	−19.3668	−0.666634
$845$	−1.73926	−0.0598325
$846$	0	0
$847$	−52.3146	−1.79755
$848$	−37.7074	−1.29488
$849$	0	0
$850$	80.9126	2.77528
$851$	1.24464	0.0426656
$852$	0	0
$853$	0.430558	0.0147420	0.00737101	−	0.999973i	$-0.497654\pi$
$853$	0.430558	0.0147420	0.00737101	+	0.999973i	$0.497654\pi$
$854$	−16.1029	−0.551031
$855$	0	0
$856$	−83.9988	−2.87102
$857$	−14.3303	−0.489513	−0.244756	−	0.969585i	$-0.578708\pi$
$857$	−14.3303	−0.489513	−0.244756	+	0.969585i	$0.578708\pi$
$858$	0	0
$859$	−28.5387	−0.973730	−0.486865	−	0.873477i	$-0.661860\pi$
$859$	−28.5387	−0.973730	−0.486865	+	0.873477i	$0.661860\pi$
$860$	−0.829144	−0.0282736
$861$	0	0
$862$	−47.9902	−1.63455
$863$	−3.69031	−0.125620	−0.0628098	−	0.998026i	$-0.520006\pi$
$863$	−3.69031	−0.125620	−0.0628098	+	0.998026i	$0.520006\pi$
$864$	0	0
$865$	0.573721	0.0195071
$866$	82.9472	2.81866
$867$	0	0
$868$	30.9186	1.04945
$869$	−5.47448	−0.185709
$870$	0	0
$871$	26.3445	0.892649
$872$	−3.51346	−0.118981
$873$	0	0
$874$	19.2349	0.650630
$875$	14.0999	0.476665
$876$	0	0
$877$	−15.6342	−0.527928	−0.263964	−	0.964532i	$-0.585030\pi$
$877$	−15.6342	−0.527928	−0.263964	+	0.964532i	$0.585030\pi$
$878$	−22.4293	−0.756953
$879$	0	0
$880$	4.14696	0.139794
$881$	−39.8258	−1.34176	−0.670882	−	0.741564i	$-0.734084\pi$
$881$	−39.8258	−1.34176	−0.670882	+	0.741564i	$0.734084\pi$
$882$	0	0
$883$	6.23215	0.209729	0.104864	−	0.994487i	$-0.466559\pi$
$883$	6.23215	0.209729	0.104864	+	0.994487i	$0.466559\pi$
$884$	−67.9165	−2.28428
$885$	0	0
$886$	16.1943	0.544059
$887$	−1.08225	−0.0363383	−0.0181691	−	0.999835i	$-0.505784\pi$
$887$	−1.08225	−0.0363383	−0.0181691	+	0.999835i	$0.505784\pi$
$888$	0	0
$889$	−78.2128	−2.62317
$890$	−8.96741	−0.300588
$891$	0	0
$892$	66.2313	2.21759
$893$	−44.2672	−1.48135
$894$	0	0
$895$	−2.44193	−0.0816247
$896$	105.174	3.51361
$897$	0	0
$898$	13.7772	0.459750
$899$	−1.54451	−0.0515123
$900$	0	0
$901$	77.9591	2.59720
$902$	15.5784	0.518703
$903$	0	0
$904$	58.2494	1.93735
$905$	−5.78179	−0.192193
$906$	0	0
$907$	27.9237	0.927192	0.463596	−	0.886047i	$-0.346559\pi$
$907$	27.9237	0.927192	0.463596	+	0.886047i	$0.346559\pi$
$908$	21.9335	0.727888
$909$	0	0
$910$	−8.90170	−0.295088
$911$	6.32742	0.209637	0.104818	−	0.994491i	$-0.466574\pi$
$911$	6.32742	0.209637	0.104818	+	0.994491i	$0.466574\pi$
$912$	0	0
$913$	25.4417	0.841998
$914$	−44.6620	−1.47729
$915$	0	0
$916$	−52.8349	−1.74572
$917$	−20.4805	−0.676324
$918$	0	0
$919$	−0.729760	−0.0240726	−0.0120363	−	0.999928i	$-0.503831\pi$
$919$	−0.729760	−0.0240726	−0.0120363	+	0.999928i	$0.503831\pi$
$920$	−1.25229	−0.0412867
$921$	0	0
$922$	−71.3889	−2.35107
$923$	−10.7630	−0.354268
$924$	0	0
$925$	6.12893	0.201518
$926$	77.7368	2.55459
$927$	0	0
$928$	−1.15424	−0.0378898
$929$	16.4689	0.540327	0.270164	−	0.962814i	$-0.412922\pi$
$929$	16.4689	0.540327	0.270164	+	0.962814i	$0.412922\pi$
$930$	0	0
$931$	−155.968	−5.11164
$932$	32.2526	1.05647
$933$	0	0
$934$	−34.2658	−1.12121
$935$	−8.57374	−0.280391
$936$	0	0
$937$	−14.5792	−0.476283	−0.238142	−	0.971230i	$-0.576538\pi$
$937$	−14.5792	−0.476283	−0.238142	+	0.971230i	$0.576538\pi$
$938$	−127.590	−4.16596
$939$	0	0
$940$	5.95257	0.194151
$941$	48.4759	1.58027	0.790134	−	0.612934i	$-0.210011\pi$
$941$	48.4759	1.58027	0.790134	+	0.612934i	$0.210011\pi$
$942$	0	0
$943$	−1.39786	−0.0455204
$944$	−23.9947	−0.780961
$945$	0	0
$946$	−8.66091	−0.281591
$947$	18.1893	0.591074	0.295537	−	0.955331i	$-0.404501\pi$
$947$	18.1893	0.591074	0.295537	+	0.955331i	$0.404501\pi$
$948$	0	0
$949$	−26.5883	−0.863091
$950$	94.7180	3.07306
$951$	0	0
$952$	159.256	5.16151
$953$	18.1827	0.588995	0.294497	−	0.955652i	$-0.404848\pi$
$953$	18.1827	0.588995	0.294497	+	0.955652i	$0.404848\pi$
$954$	0	0
$955$	2.50913	0.0811936
$956$	−41.3518	−1.33741
$957$	0	0
$958$	−21.4521	−0.693085
$959$	45.7707	1.47801
$960$	0	0
$961$	−28.6145	−0.923048
$962$	−7.79824	−0.251425
$963$	0	0
$964$	−74.8231	−2.40989
$965$	−0.390685	−0.0125766
$966$	0	0
$967$	−19.0895	−0.613876	−0.306938	−	0.951729i	$-0.599304\pi$
$967$	−19.0895	−0.613876	−0.306938	+	0.951729i	$0.599304\pi$
$968$	46.1114	1.48208
$969$	0	0
$970$	2.27757	0.0731285
$971$	−30.5549	−0.980554	−0.490277	−	0.871567i	$-0.663104\pi$
$971$	−30.5549	−0.980554	−0.490277	+	0.871567i	$0.663104\pi$
$972$	0	0
$973$	−88.6837	−2.84307
$974$	15.2816	0.489655
$975$	0	0
$976$	4.21751	0.134999
$977$	−33.4036	−1.06868	−0.534338	−	0.845271i	$-0.679439\pi$
$977$	−33.4036	−1.06868	−0.534338	+	0.845271i	$0.679439\pi$
$978$	0	0
$979$	−61.7943	−1.97495
$980$	20.9728	0.669953
$981$	0	0
$982$	54.0135	1.72364
$983$	−17.9500	−0.572517	−0.286259	−	0.958152i	$-0.592412\pi$
$983$	−17.9500	−0.572517	−0.286259	+	0.958152i	$0.592412\pi$
$984$	0	0
$985$	0.822570	0.0262093
$986$	−16.4314	−0.523281
$987$	0	0
$988$	−79.5045	−2.52937
$989$	0.777148	0.0247119
$990$	0	0
$991$	−42.9973	−1.36586	−0.682928	−	0.730486i	$-0.739294\pi$
$991$	−42.9973	−1.36586	−0.682928	+	0.730486i	$0.739294\pi$
$992$	1.78273	0.0566018
$993$	0	0
$994$	52.1266	1.65336
$995$	3.92195	0.124334
$996$	0	0
$997$	−7.34146	−0.232506	−0.116253	−	0.993220i	$-0.537088\pi$
$997$	−7.34146	−0.232506	−0.116253	+	0.993220i	$0.537088\pi$
$998$	−13.7326	−0.434698
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6003.2.a.q.1.15		16
3.2	odd	2		667.2.a.d.1.2	✓	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
667.2.a.d.1.2	✓	16	3.2	odd	2
6003.2.a.q.1.15		16	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)

\(f(q)\)	\(=\)	\(q+2.42429 q^{2} +3.87720 q^{4} +0.275174 q^{5} -5.16310 q^{7} +4.55089 q^{8} +O(q^{10})\)	\(q+2.42429 q^{2} +3.87720 q^{4} +0.275174 q^{5} -5.16310 q^{7} +4.55089 q^{8} +0.667103 q^{10} +4.59700 q^{11} +2.58445 q^{13} -12.5169 q^{14} +3.27829 q^{16} -6.77779 q^{17} -7.93422 q^{19} +1.06691 q^{20} +11.1445 q^{22} -1.00000 q^{23} -4.92428 q^{25} +6.26548 q^{26} -20.0184 q^{28} +1.00000 q^{29} -1.54451 q^{31} -1.15424 q^{32} -16.4314 q^{34} -1.42075 q^{35} -1.24464 q^{37} -19.2349 q^{38} +1.25229 q^{40} +1.39786 q^{41} -0.777148 q^{43} +17.8235 q^{44} -2.42429 q^{46} +5.57928 q^{47} +19.6576 q^{49} -11.9379 q^{50} +10.0205 q^{52} -11.5021 q^{53} +1.26498 q^{55} -23.4967 q^{56} +2.42429 q^{58} -7.31928 q^{59} +1.28650 q^{61} -3.74434 q^{62} -9.35480 q^{64} +0.711175 q^{65} +10.1934 q^{67} -26.2789 q^{68} -3.44432 q^{70} -4.16451 q^{71} -10.2878 q^{73} -3.01736 q^{74} -30.7626 q^{76} -23.7348 q^{77} -1.19088 q^{79} +0.902101 q^{80} +3.38881 q^{82} +5.53442 q^{83} -1.86507 q^{85} -1.88404 q^{86} +20.9204 q^{88} -13.4423 q^{89} -13.3438 q^{91} -3.87720 q^{92} +13.5258 q^{94} -2.18329 q^{95} +3.41413 q^{97} +47.6559 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 3 q^{2} + 21 q^{4} - 16 q^{5} + q^{7} - 9 q^{8}+O(q^{10}) \) 16 * q - 3 * q^2 + 21 * q^4 - 16 * q^5 + q^7 - 9 * q^8	\( 16 q - 3 q^{2} + 21 q^{4} - 16 q^{5} + q^{7} - 9 q^{8} - 14 q^{10} - 4 q^{11} + 15 q^{13} - 8 q^{14} + 23 q^{16} - 20 q^{17} - 4 q^{19} - 25 q^{20} + 13 q^{22} - 16 q^{23} + 30 q^{25} - 25 q^{26} - 13 q^{28} + 16 q^{29} + 19 q^{32} - 23 q^{34} - 5 q^{35} + 5 q^{37} - 38 q^{38} - 20 q^{40} - 7 q^{41} - 17 q^{43} + 21 q^{44} + 3 q^{46} - 29 q^{47} + 31 q^{49} + 44 q^{50} + 20 q^{52} - 63 q^{53} + q^{55} + 19 q^{56} - 3 q^{58} - 11 q^{59} - 33 q^{62} + 29 q^{64} - 53 q^{65} - 13 q^{67} - 63 q^{68} - 46 q^{70} + 23 q^{71} - 38 q^{73} + 47 q^{74} - 56 q^{76} - 97 q^{77} - 27 q^{79} - 8 q^{80} + 9 q^{82} - 36 q^{83} + 6 q^{85} + 11 q^{86} - 24 q^{88} + 16 q^{89} - 47 q^{91} - 21 q^{92} + 37 q^{94} + 12 q^{95} - 30 q^{97} + 27 q^{98}+O(q^{100}) \) 16 * q - 3 * q^2 + 21 * q^4 - 16 * q^5 + q^7 - 9 * q^8 - 14 * q^10 - 4 * q^11 + 15 * q^13 - 8 * q^14 + 23 * q^16 - 20 * q^17 - 4 * q^19 - 25 * q^20 + 13 * q^22 - 16 * q^23 + 30 * q^25 - 25 * q^26 - 13 * q^28 + 16 * q^29 + 19 * q^32 - 23 * q^34 - 5 * q^35 + 5 * q^37 - 38 * q^38 - 20 * q^40 - 7 * q^41 - 17 * q^43 + 21 * q^44 + 3 * q^46 - 29 * q^47 + 31 * q^49 + 44 * q^50 + 20 * q^52 - 63 * q^53 + q^55 + 19 * q^56 - 3 * q^58 - 11 * q^59 - 33 * q^62 + 29 * q^64 - 53 * q^65 - 13 * q^67 - 63 * q^68 - 46 * q^70 + 23 * q^71 - 38 * q^73 + 47 * q^74 - 56 * q^76 - 97 * q^77 - 27 * q^79 - 8 * q^80 + 9 * q^82 - 36 * q^83 + 6 * q^85 + 11 * q^86 - 24 * q^88 + 16 * q^89 - 47 * q^91 - 21 * q^92 + 37 * q^94 + 12 * q^95 - 30 * q^97 + 27 * q^98

Introduction

L-functions

Modular forms

Varieties

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Database

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