LMFDB - Embedded newform 6004.2.a.f.1.12

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6004,2,Mod(1,6004)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6004, 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("6004.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6004 = 2^{2} \cdot 19 \cdot 79 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6004.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:	$47.9421813736$
Analytic rank:	$1$
Dimension:	$25$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.12
Character	$\chi$	$=$	6004.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.0249148 q^{3} +1.33267 q^{5} -1.80071 q^{7} -2.99938 q^{9} +O(q^{10})$	$q-0.0249148 q^{3} +1.33267 q^{5} -1.80071 q^{7} -2.99938 q^{9} +2.34140 q^{11} +3.21673 q^{13} -0.0332032 q^{15} -3.19035 q^{17} -1.00000 q^{19} +0.0448642 q^{21} -0.278294 q^{23} -3.22398 q^{25} +0.149473 q^{27} +2.41313 q^{29} +1.99481 q^{31} -0.0583354 q^{33} -2.39975 q^{35} +0.199729 q^{37} -0.0801440 q^{39} +1.68407 q^{41} +1.48103 q^{43} -3.99719 q^{45} -2.83696 q^{47} -3.75745 q^{49} +0.0794867 q^{51} -2.68145 q^{53} +3.12032 q^{55} +0.0249148 q^{57} -11.7707 q^{59} +3.78542 q^{61} +5.40101 q^{63} +4.28684 q^{65} -7.60387 q^{67} +0.00693363 q^{69} -1.62350 q^{71} -1.14770 q^{73} +0.0803248 q^{75} -4.21618 q^{77} +1.00000 q^{79} +8.99441 q^{81} +16.9064 q^{83} -4.25169 q^{85} -0.0601225 q^{87} -14.4486 q^{89} -5.79239 q^{91} -0.0497003 q^{93} -1.33267 q^{95} -6.81140 q^{97} -7.02275 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 25 q + 4 q^{3} - 8 q^{5} + 2 q^{7} + 13 q^{9}+O(q^{10}) $ 25 * q + 4 * q^3 - 8 * q^5 + 2 * q^7 + 13 * q^9	$ 25 q + 4 q^{3} - 8 q^{5} + 2 q^{7} + 13 q^{9} - 3 q^{11} + q^{13} - 5 q^{15} - 13 q^{17} - 25 q^{19} - 24 q^{21} - 31 q^{23} + 21 q^{25} + 7 q^{27} - 19 q^{29} - 7 q^{31} - 30 q^{33} - q^{35} - 29 q^{37} - 26 q^{39} - 40 q^{41} - 40 q^{45} - 8 q^{47} - 9 q^{49} + 12 q^{51} - 38 q^{53} - 29 q^{55} - 4 q^{57} + 18 q^{59} - 26 q^{61} - 40 q^{63} - 70 q^{65} - 13 q^{67} + q^{69} - 47 q^{71} - 8 q^{73} + 7 q^{75} - 19 q^{77} + 25 q^{79} - 19 q^{81} - 8 q^{83} - 33 q^{85} - 50 q^{87} - 54 q^{89} - 12 q^{91} - 24 q^{93} + 8 q^{95} - 4 q^{97} - 39 q^{99}+O(q^{100}) $ 25 * q + 4 * q^3 - 8 * q^5 + 2 * q^7 + 13 * q^9 - 3 * q^11 + q^13 - 5 * q^15 - 13 * q^17 - 25 * q^19 - 24 * q^21 - 31 * q^23 + 21 * q^25 + 7 * q^27 - 19 * q^29 - 7 * q^31 - 30 * q^33 - q^35 - 29 * q^37 - 26 * q^39 - 40 * q^41 - 40 * q^45 - 8 * q^47 - 9 * q^49 + 12 * q^51 - 38 * q^53 - 29 * q^55 - 4 * q^57 + 18 * q^59 - 26 * q^61 - 40 * q^63 - 70 * q^65 - 13 * q^67 + q^69 - 47 * q^71 - 8 * q^73 + 7 * q^75 - 19 * q^77 + 25 * q^79 - 19 * q^81 - 8 * q^83 - 33 * q^85 - 50 * q^87 - 54 * q^89 - 12 * q^91 - 24 * q^93 + 8 * q^95 - 4 * q^97 - 39 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	−0.0249148	−0.0143845	−0.00719227	−	0.999974i	$-0.502289\pi$
$3$	−0.0249148	−0.0143845	−0.00719227	+	0.999974i	$0.502289\pi$
$4$	0	0
$5$	1.33267	0.595989	0.297995	−	0.954568i	$-0.403682\pi$
$5$	1.33267	0.595989	0.297995	+	0.954568i	$0.403682\pi$
$6$	0	0
$7$	−1.80071	−0.680604	−0.340302	−	0.940316i	$-0.610529\pi$
$7$	−1.80071	−0.680604	−0.340302	+	0.940316i	$0.610529\pi$
$8$	0	0
$9$	−2.99938	−0.999793
$10$	0	0
$11$	2.34140	0.705959	0.352979	−	0.935631i	$-0.385169\pi$
$11$	2.34140	0.705959	0.352979	+	0.935631i	$0.385169\pi$
$12$	0	0
$13$	3.21673	0.892159	0.446080	−	0.894993i	$-0.352820\pi$
$13$	3.21673	0.892159	0.446080	+	0.894993i	$0.352820\pi$
$14$	0	0
$15$	−0.0332032	−0.00857303
$16$	0	0
$17$	−3.19035	−0.773773	−0.386886	−	0.922127i	$-0.626449\pi$
$17$	−3.19035	−0.773773	−0.386886	+	0.922127i	$0.626449\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	0.0448642	0.00979017
$22$	0	0
$23$	−0.278294	−0.0580283	−0.0290141	−	0.999579i	$-0.509237\pi$
$23$	−0.278294	−0.0580283	−0.0290141	+	0.999579i	$0.509237\pi$
$24$	0	0
$25$	−3.22398	−0.644797
$26$	0	0
$27$	0.149473	0.0287661
$28$	0	0
$29$	2.41313	0.448107	0.224053	−	0.974577i	$-0.428071\pi$
$29$	2.41313	0.448107	0.224053	+	0.974577i	$0.428071\pi$
$30$	0	0
$31$	1.99481	0.358279	0.179140	−	0.983824i	$-0.442669\pi$
$31$	1.99481	0.358279	0.179140	+	0.983824i	$0.442669\pi$
$32$	0	0
$33$	−0.0583354	−0.0101549
$34$	0	0
$35$	−2.39975	−0.405632
$36$	0	0
$37$	0.199729	0.0328353	0.0164176	−	0.999865i	$-0.494774\pi$
$37$	0.199729	0.0328353	0.0164176	+	0.999865i	$0.494774\pi$
$38$	0	0
$39$	−0.0801440	−0.0128333
$40$	0	0
$41$	1.68407	0.263008	0.131504	−	0.991316i	$-0.458019\pi$
$41$	1.68407	0.263008	0.131504	+	0.991316i	$0.458019\pi$
$42$	0	0
$43$	1.48103	0.225855	0.112927	−	0.993603i	$-0.463977\pi$
$43$	1.48103	0.225855	0.112927	+	0.993603i	$0.463977\pi$
$44$	0	0
$45$	−3.99719	−0.595866
$46$	0	0
$47$	−2.83696	−0.413813	−0.206907	−	0.978361i	$-0.566340\pi$
$47$	−2.83696	−0.413813	−0.206907	+	0.978361i	$0.566340\pi$
$48$	0	0
$49$	−3.75745	−0.536779
$50$	0	0
$51$	0.0794867	0.0111304
$52$	0	0
$53$	−2.68145	−0.368325	−0.184163	−	0.982896i	$-0.558957\pi$
$53$	−2.68145	−0.368325	−0.184163	+	0.982896i	$0.558957\pi$
$54$	0	0
$55$	3.12032	0.420744
$56$	0	0
$57$	0.0249148	0.00330004
$58$	0	0
$59$	−11.7707	−1.53242	−0.766209	−	0.642592i	$-0.777859\pi$
$59$	−11.7707	−1.53242	−0.766209	+	0.642592i	$0.777859\pi$
$60$	0	0
$61$	3.78542	0.484674	0.242337	−	0.970192i	$-0.422086\pi$
$61$	3.78542	0.484674	0.242337	+	0.970192i	$0.422086\pi$
$62$	0	0
$63$	5.40101	0.680463
$64$	0	0
$65$	4.28684	0.531717
$66$	0	0
$67$	−7.60387	−0.928961	−0.464481	−	0.885583i	$-0.653759\pi$
$67$	−7.60387	−0.928961	−0.464481	+	0.885583i	$0.653759\pi$
$68$	0	0
$69$	0.00693363	0.000834710 0
$70$	0	0
$71$	−1.62350	−0.192674	−0.0963372	−	0.995349i	$-0.530713\pi$
$71$	−1.62350	−0.192674	−0.0963372	+	0.995349i	$0.530713\pi$
$72$	0	0
$73$	−1.14770	−0.134328	−0.0671639	−	0.997742i	$-0.521395\pi$
$73$	−1.14770	−0.134328	−0.0671639	+	0.997742i	$0.521395\pi$
$74$	0	0
$75$	0.0803248	0.00927511
$76$	0	0
$77$	−4.21618	−0.480478
$78$	0	0
$79$	1.00000	0.112509
$79$	1.00000	0.112509
$80$	0	0
$81$	8.99441	0.999379
$82$	0	0
$83$	16.9064	1.85572	0.927859	−	0.372932i	$-0.121647\pi$
$83$	16.9064	1.85572	0.927859	+	0.372932i	$0.121647\pi$
$84$	0	0
$85$	−4.25169	−0.461160
$86$	0	0
$87$	−0.0601225	−0.00644581
$88$	0	0
$89$	−14.4486	−1.53155	−0.765776	−	0.643108i	$-0.777645\pi$
$89$	−14.4486	−1.53155	−0.765776	+	0.643108i	$0.777645\pi$
$90$	0	0
$91$	−5.79239	−0.607207
$92$	0	0
$93$	−0.0497003	−0.00515368
$94$	0	0
$95$	−1.33267	−0.136729
$96$	0	0
$97$	−6.81140	−0.691593	−0.345796	−	0.938310i	$-0.612391\pi$
$97$	−6.81140	−0.691593	−0.345796	+	0.938310i	$0.612391\pi$
$98$	0	0
$99$	−7.02275	−0.705812
$100$	0	0
$101$	−3.92460	−0.390513	−0.195256	−	0.980752i	$-0.562554\pi$
$101$	−3.92460	−0.390513	−0.195256	+	0.980752i	$0.562554\pi$
$102$	0	0
$103$	−0.462340	−0.0455557	−0.0227779	−	0.999741i	$-0.507251\pi$
$103$	−0.462340	−0.0455557	−0.0227779	+	0.999741i	$0.507251\pi$
$104$	0	0
$105$	0.0597893	0.00583484
$106$	0	0
$107$	−3.55157	−0.343344	−0.171672	−	0.985154i	$-0.554917\pi$
$107$	−3.55157	−0.343344	−0.171672	+	0.985154i	$0.554917\pi$
$108$	0	0
$109$	11.0888	1.06211	0.531056	−	0.847337i	$-0.321795\pi$
$109$	11.0888	1.06211	0.531056	+	0.847337i	$0.321795\pi$
$110$	0	0
$111$	−0.00497620	−0.000472320 0
$112$	0	0
$113$	4.26351	0.401078	0.200539	−	0.979686i	$-0.435731\pi$
$113$	4.26351	0.401078	0.200539	+	0.979686i	$0.435731\pi$
$114$	0	0
$115$	−0.370875	−0.0345842
$116$	0	0
$117$	−9.64818	−0.891975
$118$	0	0
$119$	5.74488	0.526633
$120$	0	0
$121$	−5.51785	−0.501623
$122$	0	0
$123$	−0.0419582	−0.00378325
$124$	0	0
$125$	−10.9599	−0.980281
$126$	0	0
$127$	−19.9517	−1.77043	−0.885214	−	0.465185i	$-0.845988\pi$
$127$	−19.9517	−1.77043	−0.885214	+	0.465185i	$0.845988\pi$
$128$	0	0
$129$	−0.0368994	−0.00324881
$130$	0	0
$131$	−14.9259	−1.30409	−0.652043	−	0.758182i	$-0.726088\pi$
$131$	−14.9259	−1.30409	−0.652043	+	0.758182i	$0.726088\pi$
$132$	0	0
$133$	1.80071	0.156141
$134$	0	0
$135$	0.199199	0.0171443
$136$	0	0
$137$	7.54770	0.644844	0.322422	−	0.946596i	$-0.395503\pi$
$137$	7.54770	0.644844	0.322422	+	0.946596i	$0.395503\pi$
$138$	0	0
$139$	16.2642	1.37951	0.689755	−	0.724043i	$-0.257718\pi$
$139$	16.2642	1.37951	0.689755	+	0.724043i	$0.257718\pi$
$140$	0	0
$141$	0.0706822	0.00595251
$142$	0	0
$143$	7.53164	0.629828
$144$	0	0
$145$	3.21591	0.267067
$146$	0	0
$147$	0.0936160	0.00772132
$148$	0	0
$149$	−18.9110	−1.54925	−0.774623	−	0.632423i	$-0.782060\pi$
$149$	−18.9110	−1.54925	−0.774623	+	0.632423i	$0.782060\pi$
$150$	0	0
$151$	9.69759	0.789179	0.394589	−	0.918858i	$-0.370887\pi$
$151$	9.69759	0.789179	0.394589	+	0.918858i	$0.370887\pi$
$152$	0	0
$153$	9.56906	0.773613
$154$	0	0
$155$	2.65843	0.213530
$156$	0	0
$157$	−16.4909	−1.31612	−0.658058	−	0.752967i	$-0.728622\pi$
$157$	−16.4909	−1.31612	−0.658058	+	0.752967i	$0.728622\pi$
$158$	0	0
$159$	0.0668077	0.00529819
$160$	0	0
$161$	0.501126	0.0394943
$162$	0	0
$163$	−17.2803	−1.35350	−0.676749	−	0.736214i	$-0.736612\pi$
$163$	−17.2803	−1.35350	−0.676749	+	0.736214i	$0.736612\pi$
$164$	0	0
$165$	−0.0777420	−0.00605221
$166$	0	0
$167$	11.4821	0.888510	0.444255	−	0.895900i	$-0.353468\pi$
$167$	11.4821	0.888510	0.444255	+	0.895900i	$0.353468\pi$
$168$	0	0
$169$	−2.65267	−0.204051
$170$	0	0
$171$	2.99938	0.229368
$172$	0	0
$173$	24.1645	1.83719	0.918595	−	0.395201i	$-0.129325\pi$
$173$	24.1645	1.83719	0.918595	+	0.395201i	$0.129325\pi$
$174$	0	0
$175$	5.80545	0.438851
$176$	0	0
$177$	0.293265	0.0220431
$178$	0	0
$179$	−24.2862	−1.81524	−0.907618	−	0.419797i	$-0.862101\pi$
$179$	−24.2862	−1.81524	−0.907618	+	0.419797i	$0.862101\pi$
$180$	0	0
$181$	−10.8483	−0.806347	−0.403173	−	0.915124i	$-0.632093\pi$
$181$	−10.8483	−0.806347	−0.403173	+	0.915124i	$0.632093\pi$
$182$	0	0
$183$	−0.0943129	−0.00697181
$184$	0	0
$185$	0.266173	0.0195695
$186$	0	0
$187$	−7.46988	−0.546252
$188$	0	0
$189$	−0.269157	−0.0195783
$190$	0	0
$191$	14.5026	1.04937	0.524685	−	0.851297i	$-0.324183\pi$
$191$	14.5026	1.04937	0.524685	+	0.851297i	$0.324183\pi$
$192$	0	0
$193$	−23.1355	−1.66533	−0.832664	−	0.553779i	$-0.813185\pi$
$193$	−23.1355	−1.66533	−0.832664	+	0.553779i	$0.813185\pi$
$194$	0	0
$195$	−0.106806	−0.00764851
$196$	0	0
$197$	−6.04250	−0.430510	−0.215255	−	0.976558i	$-0.569058\pi$
$197$	−6.04250	−0.430510	−0.215255	+	0.976558i	$0.569058\pi$
$198$	0	0
$199$	−0.742897	−0.0526625	−0.0263313	−	0.999653i	$-0.508382\pi$
$199$	−0.742897	−0.0526625	−0.0263313	+	0.999653i	$0.508382\pi$
$200$	0	0
$201$	0.189449	0.0133627
$202$	0	0
$203$	−4.34534	−0.304983
$204$	0	0
$205$	2.24432	0.156750
$206$	0	0
$207$	0.834709	0.0580163
$208$	0	0
$209$	−2.34140	−0.161958
$210$	0	0
$211$	15.2216	1.04790	0.523950	−	0.851749i	$-0.324458\pi$
$211$	15.2216	1.04790	0.523950	+	0.851749i	$0.324458\pi$
$212$	0	0
$213$	0.0404492	0.00277153
$214$	0	0
$215$	1.97372	0.134607
$216$	0	0
$217$	−3.59208	−0.243846
$218$	0	0
$219$	0.0285946	0.00193224
$220$	0	0
$221$	−10.2625	−0.690329
$222$	0	0
$223$	−9.89420	−0.662564	−0.331282	−	0.943532i	$-0.607481\pi$
$223$	−9.89420	−0.662564	−0.331282	+	0.943532i	$0.607481\pi$
$224$	0	0
$225$	9.66995	0.644663
$226$	0	0
$227$	−21.4286	−1.42226	−0.711132	−	0.703058i	$-0.751817\pi$
$227$	−21.4286	−1.42226	−0.711132	+	0.703058i	$0.751817\pi$
$228$	0	0
$229$	−0.0640330	−0.00423142	−0.00211571	−	0.999998i	$-0.500673\pi$
$229$	−0.0640330	−0.00423142	−0.00211571	+	0.999998i	$0.500673\pi$
$230$	0	0
$231$	0.105045	0.00691146
$232$	0	0
$233$	−27.2075	−1.78242	−0.891211	−	0.453590i	$-0.850143\pi$
$233$	−27.2075	−1.78242	−0.891211	+	0.453590i	$0.850143\pi$
$234$	0	0
$235$	−3.78074	−0.246628
$236$	0	0
$237$	−0.0249148	−0.00161839
$238$	0	0
$239$	24.3162	1.57288	0.786441	−	0.617665i	$-0.211921\pi$
$239$	24.3162	1.57288	0.786441	+	0.617665i	$0.211921\pi$
$240$	0	0
$241$	−18.9540	−1.22094	−0.610469	−	0.792040i	$-0.709019\pi$
$241$	−18.9540	−1.22094	−0.610469	+	0.792040i	$0.709019\pi$
$242$	0	0
$243$	−0.672513	−0.0431417
$244$	0	0
$245$	−5.00745	−0.319914
$246$	0	0
$247$	−3.21673	−0.204675
$248$	0	0
$249$	−0.421219	−0.0266937
$250$	0	0
$251$	−13.4829	−0.851036	−0.425518	−	0.904950i	$-0.639908\pi$
$251$	−13.4829	−0.851036	−0.425518	+	0.904950i	$0.639908\pi$
$252$	0	0
$253$	−0.651597	−0.0409656
$254$	0	0
$255$	0.105930	0.00663358
$256$	0	0
$257$	−10.2027	−0.636425	−0.318213	−	0.948019i	$-0.603083\pi$
$257$	−10.2027	−0.636425	−0.318213	+	0.948019i	$0.603083\pi$
$258$	0	0
$259$	−0.359654	−0.0223478
$260$	0	0
$261$	−7.23788	−0.448014
$262$	0	0
$263$	30.7069	1.89347	0.946734	−	0.322018i	$-0.104361\pi$
$263$	30.7069	1.89347	0.946734	+	0.322018i	$0.104361\pi$
$264$	0	0
$265$	−3.57349	−0.219518
$266$	0	0
$267$	0.359984	0.0220307
$268$	0	0
$269$	2.08089	0.126874	0.0634371	−	0.997986i	$-0.479794\pi$
$269$	2.08089	0.126874	0.0634371	+	0.997986i	$0.479794\pi$
$270$	0	0
$271$	−0.379437	−0.0230491	−0.0115246	−	0.999934i	$-0.503668\pi$
$271$	−0.379437	−0.0230491	−0.0115246	+	0.999934i	$0.503668\pi$
$272$	0	0
$273$	0.144316	0.00873440
$274$	0	0
$275$	−7.54864	−0.455200
$276$	0	0
$277$	5.48721	0.329695	0.164847	−	0.986319i	$-0.447287\pi$
$277$	5.48721	0.329695	0.164847	+	0.986319i	$0.447287\pi$
$278$	0	0
$279$	−5.98320	−0.358205
$280$	0	0
$281$	3.17798	0.189582	0.0947911	−	0.995497i	$-0.469782\pi$
$281$	3.17798	0.189582	0.0947911	+	0.995497i	$0.469782\pi$
$282$	0	0
$283$	−8.40092	−0.499383	−0.249691	−	0.968325i	$-0.580329\pi$
$283$	−8.40092	−0.499383	−0.249691	+	0.968325i	$0.580329\pi$
$284$	0	0
$285$	0.0332032	0.00196679
$286$	0	0
$287$	−3.03252	−0.179004
$288$	0	0
$289$	−6.82168	−0.401275
$290$	0	0
$291$	0.169704	0.00994825
$292$	0	0
$293$	−24.2602	−1.41730	−0.708649	−	0.705561i	$-0.750695\pi$
$293$	−24.2602	−1.41730	−0.708649	+	0.705561i	$0.750695\pi$
$294$	0	0
$295$	−15.6865	−0.913304
$296$	0	0
$297$	0.349976	0.0203077
$298$	0	0
$299$	−0.895195	−0.0517705
$300$	0	0
$301$	−2.66690	−0.153717
$302$	0	0
$303$	0.0977805	0.00561734
$304$	0	0
$305$	5.04473	0.288860
$306$	0	0
$307$	30.4508	1.73792	0.868959	−	0.494884i	$-0.164790\pi$
$307$	30.4508	1.73792	0.868959	+	0.494884i	$0.164790\pi$
$308$	0	0
$309$	0.0115191	0.000655298 0
$310$	0	0
$311$	−25.2853	−1.43380	−0.716898	−	0.697178i	$-0.754439\pi$
$311$	−25.2853	−1.43380	−0.716898	+	0.697178i	$0.754439\pi$
$312$	0	0
$313$	21.0908	1.19212	0.596062	−	0.802939i	$-0.296731\pi$
$313$	21.0908	1.19212	0.596062	+	0.802939i	$0.296731\pi$
$314$	0	0
$315$	7.19777	0.405548
$316$	0	0
$317$	−6.41332	−0.360208	−0.180104	−	0.983648i	$-0.557643\pi$
$317$	−6.41332	−0.360208	−0.180104	+	0.983648i	$0.557643\pi$
$318$	0	0
$319$	5.65010	0.316345
$320$	0	0
$321$	0.0884866	0.00493884
$322$	0	0
$323$	3.19035	0.177516
$324$	0	0
$325$	−10.3707	−0.575262
$326$	0	0
$327$	−0.276274	−0.0152780
$328$	0	0
$329$	5.10854	0.281643
$330$	0	0
$331$	−20.2126	−1.11099	−0.555493	−	0.831521i	$-0.687470\pi$
$331$	−20.2126	−1.11099	−0.555493	+	0.831521i	$0.687470\pi$
$332$	0	0
$333$	−0.599063	−0.0328285
$334$	0	0
$335$	−10.1335	−0.553651
$336$	0	0
$337$	−8.27243	−0.450628	−0.225314	−	0.974286i	$-0.572341\pi$
$337$	−8.27243	−0.450628	−0.225314	+	0.974286i	$0.572341\pi$
$338$	0	0
$339$	−0.106224	−0.00576932
$340$	0	0
$341$	4.67066	0.252930
$342$	0	0
$343$	19.3710	1.04594
$344$	0	0
$345$	0.00924025	0.000497478 0
$346$	0	0
$347$	27.5618	1.47959	0.739796	−	0.672831i	$-0.234922\pi$
$347$	27.5618	1.47959	0.739796	+	0.672831i	$0.234922\pi$
$348$	0	0
$349$	1.13385	0.0606936	0.0303468	−	0.999539i	$-0.490339\pi$
$349$	1.13385	0.0606936	0.0303468	+	0.999539i	$0.490339\pi$
$350$	0	0
$351$	0.480814	0.0256640
$352$	0	0
$353$	21.6164	1.15053	0.575263	−	0.817969i	$-0.304900\pi$
$353$	21.6164	1.15053	0.575263	+	0.817969i	$0.304900\pi$
$354$	0	0
$355$	−2.16360	−0.114832
$356$	0	0
$357$	−0.143132	−0.00757537
$358$	0	0
$359$	23.7028	1.25099	0.625494	−	0.780229i	$-0.284898\pi$
$359$	23.7028	1.25099	0.625494	+	0.780229i	$0.284898\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	0.137476	0.00721561
$364$	0	0
$365$	−1.52950	−0.0800579
$366$	0	0
$367$	10.6324	0.555007	0.277503	−	0.960725i	$-0.410493\pi$
$367$	10.6324	0.555007	0.277503	+	0.960725i	$0.410493\pi$
$368$	0	0
$369$	−5.05117	−0.262953
$370$	0	0
$371$	4.82851	0.250684
$372$	0	0
$373$	−0.307958	−0.0159455	−0.00797273	−	0.999968i	$-0.502538\pi$
$373$	−0.307958	−0.0159455	−0.00797273	+	0.999968i	$0.502538\pi$
$374$	0	0
$375$	0.273063	0.0141009
$376$	0	0
$377$	7.76237	0.399782
$378$	0	0
$379$	−20.8171	−1.06930	−0.534651	−	0.845073i	$-0.679557\pi$
$379$	−20.8171	−1.06930	−0.534651	+	0.845073i	$0.679557\pi$
$380$	0	0
$381$	0.497092	0.0254668
$382$	0	0
$383$	−3.60305	−0.184107	−0.0920535	−	0.995754i	$-0.529343\pi$
$383$	−3.60305	−0.184107	−0.0920535	+	0.995754i	$0.529343\pi$
$384$	0	0
$385$	−5.61878	−0.286360
$386$	0	0
$387$	−4.44216	−0.225808
$388$	0	0
$389$	−36.3577	−1.84341	−0.921704	−	0.387894i	$-0.873203\pi$
$389$	−36.3577	−1.84341	−0.921704	+	0.387894i	$0.873203\pi$
$390$	0	0
$391$	0.887854	0.0449007
$392$	0	0
$393$	0.371876	0.0187587
$394$	0	0
$395$	1.33267	0.0670540
$396$	0	0
$397$	−35.3624	−1.77479	−0.887393	−	0.461014i	$-0.847486\pi$
$397$	−35.3624	−1.77479	−0.887393	+	0.461014i	$0.847486\pi$
$398$	0	0
$399$	−0.0448642	−0.00224602
$400$	0	0
$401$	−9.17494	−0.458175	−0.229087	−	0.973406i	$-0.573574\pi$
$401$	−9.17494	−0.458175	−0.229087	+	0.973406i	$0.573574\pi$
$402$	0	0
$403$	6.41677	0.319642
$404$	0	0
$405$	11.9866	0.595619
$406$	0	0
$407$	0.467646	0.0231803
$408$	0	0
$409$	1.82263	0.0901235	0.0450617	−	0.998984i	$-0.485652\pi$
$409$	1.82263	0.0901235	0.0450617	+	0.998984i	$0.485652\pi$
$410$	0	0
$411$	−0.188049	−0.00927578
$412$	0	0
$413$	21.1956	1.04297
$414$	0	0
$415$	22.5307	1.10599
$416$	0	0
$417$	−0.405218	−0.0198436
$418$	0	0
$419$	7.12494	0.348076	0.174038	−	0.984739i	$-0.444318\pi$
$419$	7.12494	0.348076	0.174038	+	0.984739i	$0.444318\pi$
$420$	0	0
$421$	10.7389	0.523381	0.261691	−	0.965152i	$-0.415720\pi$
$421$	10.7389	0.523381	0.261691	+	0.965152i	$0.415720\pi$
$422$	0	0
$423$	8.50912	0.413727
$424$	0	0
$425$	10.2856	0.498926
$426$	0	0
$427$	−6.81644	−0.329871
$428$	0	0
$429$	−0.187649	−0.00905978
$430$	0	0
$431$	−13.2986	−0.640571	−0.320286	−	0.947321i	$-0.603779\pi$
$431$	−13.2986	−0.640571	−0.320286	+	0.947321i	$0.603779\pi$
$432$	0	0
$433$	22.2763	1.07053	0.535266	−	0.844683i	$-0.320211\pi$
$433$	22.2763	1.07053	0.535266	+	0.844683i	$0.320211\pi$
$434$	0	0
$435$	−0.0801236	−0.00384163
$436$	0	0
$437$	0.278294	0.0133126
$438$	0	0
$439$	−15.0652	−0.719022	−0.359511	−	0.933141i	$-0.617056\pi$
$439$	−15.0652	−0.719022	−0.359511	+	0.933141i	$0.617056\pi$
$440$	0	0
$441$	11.2700	0.536668
$442$	0	0
$443$	11.7794	0.559655	0.279828	−	0.960050i	$-0.409723\pi$
$443$	11.7794	0.559655	0.279828	+	0.960050i	$0.409723\pi$
$444$	0	0
$445$	−19.2553	−0.912788
$446$	0	0
$447$	0.471162	0.0222852
$448$	0	0
$449$	−0.618775	−0.0292018	−0.0146009	−	0.999893i	$-0.504648\pi$
$449$	−0.618775	−0.0292018	−0.0146009	+	0.999893i	$0.504648\pi$
$450$	0	0
$451$	3.94308	0.185673
$452$	0	0
$453$	−0.241613	−0.0113520
$454$	0	0
$455$	−7.71935	−0.361889
$456$	0	0
$457$	10.4161	0.487242	0.243621	−	0.969870i	$-0.421665\pi$
$457$	10.4161	0.487242	0.243621	+	0.969870i	$0.421665\pi$
$458$	0	0
$459$	−0.476871	−0.0222584
$460$	0	0
$461$	−10.6517	−0.496101	−0.248051	−	0.968747i	$-0.579790\pi$
$461$	−10.6517	−0.496101	−0.248051	+	0.968747i	$0.579790\pi$
$462$	0	0
$463$	−3.38636	−0.157377	−0.0786886	−	0.996899i	$-0.525073\pi$
$463$	−3.38636	−0.157377	−0.0786886	+	0.996899i	$0.525073\pi$
$464$	0	0
$465$	−0.0662342	−0.00307154
$466$	0	0
$467$	6.04564	0.279759	0.139879	−	0.990169i	$-0.455329\pi$
$467$	6.04564	0.279759	0.139879	+	0.990169i	$0.455329\pi$
$468$	0	0
$469$	13.6924	0.632254
$470$	0	0
$471$	0.410867	0.0189317
$472$	0	0
$473$	3.46768	0.159444
$474$	0	0
$475$	3.22398	0.147927
$476$	0	0
$477$	8.04268	0.368249
$478$	0	0
$479$	32.7740	1.49748	0.748741	−	0.662863i	$-0.230659\pi$
$479$	32.7740	1.49748	0.748741	+	0.662863i	$0.230659\pi$
$480$	0	0
$481$	0.642474	0.0292943
$482$	0	0
$483$	−0.0124854	−0.000568107 0
$484$	0	0
$485$	−9.07736	−0.412182
$486$	0	0
$487$	28.4382	1.28866	0.644328	−	0.764749i	$-0.277137\pi$
$487$	28.4382	1.28866	0.644328	+	0.764749i	$0.277137\pi$
$488$	0	0
$489$	0.430535	0.0194694
$490$	0	0
$491$	−18.6686	−0.842504	−0.421252	−	0.906944i	$-0.638409\pi$
$491$	−18.6686	−0.842504	−0.421252	+	0.906944i	$0.638409\pi$
$492$	0	0
$493$	−7.69872	−0.346733
$494$	0	0
$495$	−9.35902	−0.420657
$496$	0	0
$497$	2.92346	0.131135
$498$	0	0
$499$	18.1207	0.811193	0.405597	−	0.914052i	$-0.367064\pi$
$499$	18.1207	0.811193	0.405597	+	0.914052i	$0.367064\pi$
$500$	0	0
$501$	−0.286073	−0.0127808
$502$	0	0
$503$	2.53831	0.113177	0.0565887	−	0.998398i	$-0.481978\pi$
$503$	2.53831	0.113177	0.0565887	+	0.998398i	$0.481978\pi$
$504$	0	0
$505$	−5.23021	−0.232741
$506$	0	0
$507$	0.0660906	0.00293519
$508$	0	0
$509$	−5.43164	−0.240753	−0.120377	−	0.992728i	$-0.538410\pi$
$509$	−5.43164	−0.240753	−0.120377	+	0.992728i	$0.538410\pi$
$510$	0	0
$511$	2.06667	0.0914240
$512$	0	0
$513$	−0.149473	−0.00659940
$514$	0	0
$515$	−0.616148	−0.0271507
$516$	0	0
$517$	−6.64246	−0.292135
$518$	0	0
$519$	−0.602052	−0.0264271
$520$	0	0
$521$	16.0659	0.703859	0.351930	−	0.936026i	$-0.385526\pi$
$521$	16.0659	0.703859	0.351930	+	0.936026i	$0.385526\pi$
$522$	0	0
$523$	−31.2657	−1.36715	−0.683577	−	0.729879i	$-0.739577\pi$
$523$	−31.2657	−1.36715	−0.683577	+	0.729879i	$0.739577\pi$
$524$	0	0
$525$	−0.144642	−0.00631267
$526$	0	0
$527$	−6.36415	−0.277227
$528$	0	0
$529$	−22.9226	−0.996633
$530$	0	0
$531$	35.3049	1.53210
$532$	0	0
$533$	5.41720	0.234645
$534$	0	0
$535$	−4.73308	−0.204629
$536$	0	0
$537$	0.605085	0.0261113
$538$	0	0
$539$	−8.79769	−0.378944
$540$	0	0
$541$	−18.3670	−0.789657	−0.394829	−	0.918755i	$-0.629196\pi$
$541$	−18.3670	−0.789657	−0.394829	+	0.918755i	$0.629196\pi$
$542$	0	0
$543$	0.270282	0.0115989
$544$	0	0
$545$	14.7777	0.633007
$546$	0	0
$547$	−33.5951	−1.43642	−0.718212	−	0.695825i	$-0.755039\pi$
$547$	−33.5951	−1.43642	−0.718212	+	0.695825i	$0.755039\pi$
$548$	0	0
$549$	−11.3539	−0.484574
$550$	0	0
$551$	−2.41313	−0.102803
$552$	0	0
$553$	−1.80071	−0.0765739
$554$	0	0
$555$	−0.00663165	−0.000281498 0
$556$	0	0
$557$	−29.8804	−1.26608	−0.633038	−	0.774121i	$-0.718192\pi$
$557$	−29.8804	−1.26608	−0.633038	+	0.774121i	$0.718192\pi$
$558$	0	0
$559$	4.76406	0.201498
$560$	0	0
$561$	0.186110	0.00785758
$562$	0	0
$563$	11.7406	0.494807	0.247403	−	0.968913i	$-0.420423\pi$
$563$	11.7406	0.494807	0.247403	+	0.968913i	$0.420423\pi$
$564$	0	0
$565$	5.68187	0.239038
$566$	0	0
$567$	−16.1963	−0.680181
$568$	0	0
$569$	−28.1359	−1.17952	−0.589759	−	0.807579i	$-0.700777\pi$
$569$	−28.1359	−1.17952	−0.589759	+	0.807579i	$0.700777\pi$
$570$	0	0
$571$	37.4421	1.56690	0.783452	−	0.621452i	$-0.213457\pi$
$571$	37.4421	1.56690	0.783452	+	0.621452i	$0.213457\pi$
$572$	0	0
$573$	−0.361328	−0.0150947
$574$	0	0
$575$	0.897215	0.0374165
$576$	0	0
$577$	17.7187	0.737640	0.368820	−	0.929501i	$-0.379762\pi$
$577$	17.7187	0.737640	0.368820	+	0.929501i	$0.379762\pi$
$578$	0	0
$579$	0.576415	0.0239550
$580$	0	0
$581$	−30.4435	−1.26301
$582$	0	0
$583$	−6.27834	−0.260022
$584$	0	0
$585$	−12.8579	−0.531607
$586$	0	0
$587$	−37.8895	−1.56387	−0.781933	−	0.623362i	$-0.785766\pi$
$587$	−37.8895	−1.56387	−0.781933	+	0.623362i	$0.785766\pi$
$588$	0	0
$589$	−1.99481	−0.0821949
$590$	0	0
$591$	0.150547	0.00619270
$592$	0	0
$593$	−31.9014	−1.31003	−0.655017	−	0.755614i	$-0.727339\pi$
$593$	−31.9014	−1.31003	−0.655017	+	0.755614i	$0.727339\pi$
$594$	0	0
$595$	7.65605	0.313867
$596$	0	0
$597$	0.0185091	0.000757527 0
$598$	0	0
$599$	29.7625	1.21606	0.608032	−	0.793913i	$-0.291959\pi$
$599$	29.7625	1.21606	0.608032	+	0.793913i	$0.291959\pi$
$600$	0	0
$601$	20.5204	0.837043	0.418522	−	0.908207i	$-0.362548\pi$
$601$	20.5204	0.837043	0.418522	+	0.908207i	$0.362548\pi$
$602$	0	0
$603$	22.8069	0.928769
$604$	0	0
$605$	−7.35348	−0.298962
$606$	0	0
$607$	22.9462	0.931358	0.465679	−	0.884954i	$-0.345810\pi$
$607$	22.9462	0.931358	0.465679	+	0.884954i	$0.345810\pi$
$608$	0	0
$609$	0.108263	0.00438704
$610$	0	0
$611$	−9.12572	−0.369187
$612$	0	0
$613$	−39.0598	−1.57761	−0.788806	−	0.614642i	$-0.789300\pi$
$613$	−39.0598	−1.57761	−0.788806	+	0.614642i	$0.789300\pi$
$614$	0	0
$615$	−0.0559166	−0.00225477
$616$	0	0
$617$	2.94891	0.118718	0.0593592	−	0.998237i	$-0.481094\pi$
$617$	2.94891	0.118718	0.0593592	+	0.998237i	$0.481094\pi$
$618$	0	0
$619$	−10.8102	−0.434499	−0.217249	−	0.976116i	$-0.569709\pi$
$619$	−10.8102	−0.434499	−0.217249	+	0.976116i	$0.569709\pi$
$620$	0	0
$621$	−0.0415975	−0.00166925
$622$	0	0
$623$	26.0178	1.04238
$624$	0	0
$625$	1.51400	0.0605599
$626$	0	0
$627$	0.0583354	0.00232969
$628$	0	0
$629$	−0.637205	−0.0254070
$630$	0	0
$631$	22.9497	0.913613	0.456806	−	0.889566i	$-0.348993\pi$
$631$	22.9497	0.913613	0.456806	+	0.889566i	$0.348993\pi$
$632$	0	0
$633$	−0.379243	−0.0150736
$634$	0	0
$635$	−26.5891	−1.05516
$636$	0	0
$637$	−12.0867	−0.478892
$638$	0	0
$639$	4.86950	0.192635
$640$	0	0
$641$	−33.2936	−1.31502	−0.657509	−	0.753446i	$-0.728390\pi$
$641$	−33.2936	−1.31502	−0.657509	+	0.753446i	$0.728390\pi$
$642$	0	0
$643$	−13.9852	−0.551522	−0.275761	−	0.961226i	$-0.588930\pi$
$643$	−13.9852	−0.551522	−0.275761	+	0.961226i	$0.588930\pi$
$644$	0	0
$645$	−0.0491749	−0.00193626
$646$	0	0
$647$	−4.96794	−0.195310	−0.0976550	−	0.995220i	$-0.531134\pi$
$647$	−4.96794	−0.195310	−0.0976550	+	0.995220i	$0.531134\pi$
$648$	0	0
$649$	−27.5600	−1.08182
$650$	0	0
$651$	0.0894957	0.00350761
$652$	0	0
$653$	22.3542	0.874787	0.437393	−	0.899270i	$-0.355902\pi$
$653$	22.3542	0.874787	0.437393	+	0.899270i	$0.355902\pi$
$654$	0	0
$655$	−19.8914	−0.777221
$656$	0	0
$657$	3.44238	0.134300
$658$	0	0
$659$	−37.5941	−1.46446	−0.732229	−	0.681059i	$-0.761520\pi$
$659$	−37.5941	−1.46446	−0.732229	+	0.681059i	$0.761520\pi$
$660$	0	0
$661$	44.8795	1.74561	0.872804	−	0.488070i	$-0.162299\pi$
$661$	44.8795	1.74561	0.872804	+	0.488070i	$0.162299\pi$
$662$	0	0
$663$	0.255687	0.00993007
$664$	0	0
$665$	2.39975	0.0930585
$666$	0	0
$667$	−0.671559	−0.0260029
$668$	0	0
$669$	0.246512	0.00953069
$670$	0	0
$671$	8.86319	0.342160
$672$	0	0
$673$	16.6382	0.641356	0.320678	−	0.947188i	$-0.396089\pi$
$673$	16.6382	0.641356	0.320678	+	0.947188i	$0.396089\pi$
$674$	0	0
$675$	−0.481899	−0.0185483
$676$	0	0
$677$	−3.51603	−0.135132	−0.0675661	−	0.997715i	$-0.521523\pi$
$677$	−3.51603	−0.135132	−0.0675661	+	0.997715i	$0.521523\pi$
$678$	0	0
$679$	12.2653	0.470701
$680$	0	0
$681$	0.533888	0.0204586
$682$	0	0
$683$	28.8703	1.10469	0.552345	−	0.833616i	$-0.313733\pi$
$683$	28.8703	1.10469	0.552345	+	0.833616i	$0.313733\pi$
$684$	0	0
$685$	10.0586	0.384320
$686$	0	0
$687$	0.00159537	6.08670e−5 0
$688$	0	0
$689$	−8.62549	−0.328605
$690$	0	0
$691$	15.4924	0.589358	0.294679	−	0.955596i	$-0.404787\pi$
$691$	15.4924	0.589358	0.294679	+	0.955596i	$0.404787\pi$
$692$	0	0
$693$	12.6459	0.480379
$694$	0	0
$695$	21.6748	0.822173
$696$	0	0
$697$	−5.37277	−0.203508
$698$	0	0
$699$	0.677868	0.0256393
$700$	0	0
$701$	16.7384	0.632200	0.316100	−	0.948726i	$-0.397627\pi$
$701$	16.7384	0.632200	0.316100	+	0.948726i	$0.397627\pi$
$702$	0	0
$703$	−0.199729	−0.00753293
$704$	0	0
$705$	0.0941962	0.00354763
$706$	0	0
$707$	7.06706	0.265784
$708$	0	0
$709$	5.12772	0.192575	0.0962877	−	0.995354i	$-0.469303\pi$
$709$	5.12772	0.192575	0.0962877	+	0.995354i	$0.469303\pi$
$710$	0	0
$711$	−2.99938	−0.112486
$712$	0	0
$713$	−0.555144	−0.0207903
$714$	0	0
$715$	10.0372	0.375370
$716$	0	0
$717$	−0.605832	−0.0226252
$718$	0	0
$719$	−44.4357	−1.65717	−0.828585	−	0.559863i	$-0.810854\pi$
$719$	−44.4357	−1.65717	−0.828585	+	0.559863i	$0.810854\pi$
$720$	0	0
$721$	0.832540	0.0310054
$722$	0	0
$723$	0.472236	0.0175626
$724$	0	0
$725$	−7.77989	−0.288938
$726$	0	0
$727$	−22.5786	−0.837395	−0.418698	−	0.908126i	$-0.637513\pi$
$727$	−22.5786	−0.837395	−0.418698	+	0.908126i	$0.637513\pi$
$728$	0	0
$729$	−26.9665	−0.998759
$730$	0	0
$731$	−4.72499	−0.174760
$732$	0	0
$733$	−36.1216	−1.33418	−0.667091	−	0.744976i	$-0.732461\pi$
$733$	−36.1216	−1.33418	−0.667091	+	0.744976i	$0.732461\pi$
$734$	0	0
$735$	0.124759	0.00460182
$736$	0	0
$737$	−17.8037	−0.655808
$738$	0	0
$739$	−29.6584	−1.09100	−0.545502	−	0.838110i	$-0.683661\pi$
$739$	−29.6584	−1.09100	−0.545502	+	0.838110i	$0.683661\pi$
$740$	0	0
$741$	0.0801440	0.00294416
$742$	0	0
$743$	13.5242	0.496154	0.248077	−	0.968740i	$-0.420201\pi$
$743$	13.5242	0.496154	0.248077	+	0.968740i	$0.420201\pi$
$744$	0	0
$745$	−25.2021	−0.923334
$746$	0	0
$747$	−50.7087	−1.85533
$748$	0	0
$749$	6.39535	0.233681
$750$	0	0
$751$	−30.2807	−1.10496	−0.552480	−	0.833526i	$-0.686318\pi$
$751$	−30.2807	−1.10496	−0.552480	+	0.833526i	$0.686318\pi$
$752$	0	0
$753$	0.335924	0.0122418
$754$	0	0
$755$	12.9237	0.470342
$756$	0	0
$757$	−11.0893	−0.403046	−0.201523	−	0.979484i	$-0.564589\pi$
$757$	−11.0893	−0.403046	−0.201523	+	0.979484i	$0.564589\pi$
$758$	0	0
$759$	0.0162344	0.000589271 0
$760$	0	0
$761$	−30.8760	−1.11925	−0.559627	−	0.828745i	$-0.689056\pi$
$761$	−30.8760	−1.11925	−0.559627	+	0.828745i	$0.689056\pi$
$762$	0	0
$763$	−19.9676	−0.722877
$764$	0	0
$765$	12.7524	0.461065
$766$	0	0
$767$	−37.8632	−1.36716
$768$	0	0
$769$	9.98385	0.360027	0.180013	−	0.983664i	$-0.442386\pi$
$769$	9.98385	0.360027	0.180013	+	0.983664i	$0.442386\pi$
$770$	0	0
$771$	0.254197	0.00915469
$772$	0	0
$773$	−32.9791	−1.18617	−0.593087	−	0.805138i	$-0.702091\pi$
$773$	−32.9791	−1.18617	−0.593087	+	0.805138i	$0.702091\pi$
$774$	0	0
$775$	−6.43125	−0.231017
$776$	0	0
$777$	0.00896069	0.000321463 0
$778$	0	0
$779$	−1.68407	−0.0603381
$780$	0	0
$781$	−3.80127	−0.136020
$782$	0	0
$783$	0.360698	0.0128903
$784$	0	0
$785$	−21.9770	−0.784391
$786$	0	0
$787$	6.88571	0.245449	0.122725	−	0.992441i	$-0.460837\pi$
$787$	6.88571	0.245449	0.122725	+	0.992441i	$0.460837\pi$
$788$	0	0
$789$	−0.765055	−0.0272367
$790$	0	0
$791$	−7.67734	−0.272975
$792$	0	0
$793$	12.1767	0.432406
$794$	0	0
$795$	0.0890327	0.00315766
$796$	0	0
$797$	29.3447	1.03944	0.519722	−	0.854336i	$-0.326036\pi$
$797$	29.3447	1.03944	0.519722	+	0.854336i	$0.326036\pi$
$798$	0	0
$799$	9.05089	0.320197
$800$	0	0
$801$	43.3369	1.53123
$802$	0	0
$803$	−2.68722	−0.0948298
$804$	0	0
$805$	0.667837	0.0235382
$806$	0	0
$807$	−0.0518449	−0.00182503
$808$	0	0
$809$	26.2473	0.922808	0.461404	−	0.887190i	$-0.347346\pi$
$809$	26.2473	0.922808	0.461404	+	0.887190i	$0.347346\pi$
$810$	0	0
$811$	14.3489	0.503860	0.251930	−	0.967746i	$-0.418935\pi$
$811$	14.3489	0.503860	0.251930	+	0.967746i	$0.418935\pi$
$812$	0	0
$813$	0.00945357	0.000331551 0
$814$	0	0
$815$	−23.0290	−0.806670
$816$	0	0
$817$	−1.48103	−0.0518146
$818$	0	0
$819$	17.3736	0.607081
$820$	0	0
$821$	41.0765	1.43358	0.716789	−	0.697290i	$-0.245611\pi$
$821$	41.0765	1.43358	0.716789	+	0.697290i	$0.245611\pi$
$822$	0	0
$823$	−7.68067	−0.267731	−0.133866	−	0.990999i	$-0.542739\pi$
$823$	−7.68067	−0.267731	−0.133866	+	0.990999i	$0.542739\pi$
$824$	0	0
$825$	0.188072	0.00654784
$826$	0	0
$827$	55.9781	1.94655	0.973275	−	0.229643i	$-0.0737559\pi$
$827$	55.9781	1.94655	0.973275	+	0.229643i	$0.0737559\pi$
$828$	0	0
$829$	1.11570	0.0387499	0.0193749	−	0.999812i	$-0.493832\pi$
$829$	1.11570	0.0387499	0.0193749	+	0.999812i	$0.493832\pi$
$830$	0	0
$831$	−0.136713	−0.00474251
$832$	0	0
$833$	11.9876	0.415345
$834$	0	0
$835$	15.3018	0.529542
$836$	0	0
$837$	0.298171	0.0103063
$838$	0	0
$839$	15.6906	0.541701	0.270850	−	0.962621i	$-0.412695\pi$
$839$	15.6906	0.541701	0.270850	+	0.962621i	$0.412695\pi$
$840$	0	0
$841$	−23.1768	−0.799201
$842$	0	0
$843$	−0.0791785	−0.00272705
$844$	0	0
$845$	−3.53514	−0.121612
$846$	0	0
$847$	9.93603	0.341406
$848$	0	0
$849$	0.209307	0.00718339
$850$	0	0
$851$	−0.0555834	−0.00190537
$852$	0	0
$853$	−20.2049	−0.691803	−0.345901	−	0.938271i	$-0.612427\pi$
$853$	−20.2049	−0.691803	−0.345901	+	0.938271i	$0.612427\pi$
$854$	0	0
$855$	3.99719	0.136701
$856$	0	0
$857$	−2.87051	−0.0980547	−0.0490274	−	0.998797i	$-0.515612\pi$
$857$	−2.87051	−0.0980547	−0.0490274	+	0.998797i	$0.515612\pi$
$858$	0	0
$859$	17.7581	0.605900	0.302950	−	0.953006i	$-0.402028\pi$
$859$	17.7581	0.605900	0.302950	+	0.953006i	$0.402028\pi$
$860$	0	0
$861$	0.0755545	0.00257489
$862$	0	0
$863$	−2.15764	−0.0734470	−0.0367235	−	0.999325i	$-0.511692\pi$
$863$	−2.15764	−0.0734470	−0.0367235	+	0.999325i	$0.511692\pi$
$864$	0	0
$865$	32.2033	1.09495
$866$	0	0
$867$	0.169961	0.00577216
$868$	0	0
$869$	2.34140	0.0794265
$870$	0	0
$871$	−24.4596	−0.828781
$872$	0	0
$873$	20.4300	0.691450
$874$	0	0
$875$	19.7355	0.667183
$876$	0	0
$877$	53.9067	1.82030	0.910151	−	0.414277i	$-0.135966\pi$
$877$	53.9067	1.82030	0.910151	+	0.414277i	$0.135966\pi$
$878$	0	0
$879$	0.604438	0.0203872
$880$	0	0
$881$	−10.7951	−0.363696	−0.181848	−	0.983327i	$-0.558208\pi$
$881$	−10.7951	−0.363696	−0.181848	+	0.983327i	$0.558208\pi$
$882$	0	0
$883$	18.5004	0.622589	0.311294	−	0.950314i	$-0.399237\pi$
$883$	18.5004	0.622589	0.311294	+	0.950314i	$0.399237\pi$
$884$	0	0
$885$	0.390826	0.0131375
$886$	0	0
$887$	22.7621	0.764275	0.382137	−	0.924105i	$-0.375188\pi$
$887$	22.7621	0.764275	0.382137	+	0.924105i	$0.375188\pi$
$888$	0	0
$889$	35.9272	1.20496
$890$	0	0
$891$	21.0595	0.705520
$892$	0	0
$893$	2.83696	0.0949352
$894$	0	0
$895$	−32.3656	−1.08186
$896$	0	0
$897$	0.0223036	0.000744695 0
$898$	0	0
$899$	4.81374	0.160547
$900$	0	0
$901$	8.55475	0.285000
$902$	0	0
$903$	0.0664451	0.00221115
$904$	0	0
$905$	−14.4572	−0.480574
$906$	0	0
$907$	−21.6114	−0.717595	−0.358797	−	0.933416i	$-0.616813\pi$
$907$	−21.6114	−0.717595	−0.358797	+	0.933416i	$0.616813\pi$
$908$	0	0
$909$	11.7714	0.390432
$910$	0	0
$911$	3.51848	0.116573	0.0582863	−	0.998300i	$-0.481436\pi$
$911$	3.51848	0.116573	0.0582863	+	0.998300i	$0.481436\pi$
$912$	0	0
$913$	39.5846	1.31006
$914$	0	0
$915$	−0.125688	−0.00415513
$916$	0	0
$917$	26.8773	0.887566
$918$	0	0
$919$	−16.6555	−0.549415	−0.274707	−	0.961528i	$-0.588581\pi$
$919$	−16.6555	−0.549415	−0.274707	+	0.961528i	$0.588581\pi$
$920$	0	0
$921$	−0.758674	−0.0249992
$922$	0	0
$923$	−5.22237	−0.171896
$924$	0	0
$925$	−0.643923	−0.0211721
$926$	0	0
$927$	1.38673	0.0455463
$928$	0	0
$929$	−47.6225	−1.56245	−0.781223	−	0.624253i	$-0.785404\pi$
$929$	−47.6225	−1.56245	−0.781223	+	0.624253i	$0.785404\pi$
$930$	0	0
$931$	3.75745	0.123145
$932$	0	0
$933$	0.629976	0.0206245
$934$	0	0
$935$	−9.95490	−0.325560
$936$	0	0
$937$	36.4306	1.19014	0.595069	−	0.803675i	$-0.297125\pi$
$937$	36.4306	1.19014	0.595069	+	0.803675i	$0.297125\pi$
$938$	0	0
$939$	−0.525473	−0.0171481
$940$	0	0
$941$	−14.6752	−0.478398	−0.239199	−	0.970971i	$-0.576885\pi$
$941$	−14.6752	−0.478398	−0.239199	+	0.970971i	$0.576885\pi$
$942$	0	0
$943$	−0.468667	−0.0152619
$944$	0	0
$945$	−0.358699	−0.0116685
$946$	0	0
$947$	47.0319	1.52833	0.764165	−	0.645021i	$-0.223151\pi$
$947$	47.0319	1.52833	0.764165	+	0.645021i	$0.223151\pi$
$948$	0	0
$949$	−3.69183	−0.119842
$950$	0	0
$951$	0.159786	0.00518142
$952$	0	0
$953$	29.9416	0.969905	0.484953	−	0.874540i	$-0.338837\pi$
$953$	29.9416	0.969905	0.484953	+	0.874540i	$0.338837\pi$
$954$	0	0
$955$	19.3272	0.625413
$956$	0	0
$957$	−0.140771	−0.00455047
$958$	0	0
$959$	−13.5912	−0.438883
$960$	0	0
$961$	−27.0207	−0.871636
$962$	0	0
$963$	10.6525	0.343273
$964$	0	0
$965$	−30.8320	−0.992517
$966$	0	0
$967$	−17.5086	−0.563037	−0.281519	−	0.959556i	$-0.590838\pi$
$967$	−17.5086	−0.563037	−0.281519	+	0.959556i	$0.590838\pi$
$968$	0	0
$969$	−0.0794867	−0.00255348
$970$	0	0
$971$	34.5165	1.10769	0.553844	−	0.832620i	$-0.313160\pi$
$971$	34.5165	1.10769	0.553844	+	0.832620i	$0.313160\pi$
$972$	0	0
$973$	−29.2870	−0.938900
$974$	0	0
$975$	0.258383	0.00827488
$976$	0	0
$977$	−17.3256	−0.554294	−0.277147	−	0.960828i	$-0.589389\pi$
$977$	−17.3256	−0.554294	−0.277147	+	0.960828i	$0.589389\pi$
$978$	0	0
$979$	−33.8300	−1.08121
$980$	0	0
$981$	−33.2594	−1.06189
$982$	0	0
$983$	31.2418	0.996458	0.498229	−	0.867045i	$-0.333984\pi$
$983$	31.2418	0.996458	0.498229	+	0.867045i	$0.333984\pi$
$984$	0	0
$985$	−8.05267	−0.256580
$986$	0	0
$987$	−0.127278	−0.00405130
$988$	0	0
$989$	−0.412161	−0.0131060
$990$	0	0
$991$	25.7988	0.819526	0.409763	−	0.912192i	$-0.365611\pi$
$991$	25.7988	0.819526	0.409763	+	0.912192i	$0.365611\pi$
$992$	0	0
$993$	0.503593	0.0159810
$994$	0	0
$995$	−0.990038	−0.0313863
$996$	0	0
$997$	−41.0357	−1.29961	−0.649807	−	0.760099i	$-0.725150\pi$
$997$	−41.0357	−1.29961	−0.649807	+	0.760099i	$0.725150\pi$
$998$	0	0
$999$	0.0298541	0.000944543 0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6004.2.a.f.1.12	✓	25

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6004.2.a.f.1.12	✓	25	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:	\( N \)	\(=\)	\( 6004 = 2^{2} \cdot 19 \cdot 79 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6004.a (trivial)

\(f(q)\)	\(=\)	\(q-0.0249148 q^{3} +1.33267 q^{5} -1.80071 q^{7} -2.99938 q^{9} +O(q^{10})\)	\(q-0.0249148 q^{3} +1.33267 q^{5} -1.80071 q^{7} -2.99938 q^{9} +2.34140 q^{11} +3.21673 q^{13} -0.0332032 q^{15} -3.19035 q^{17} -1.00000 q^{19} +0.0448642 q^{21} -0.278294 q^{23} -3.22398 q^{25} +0.149473 q^{27} +2.41313 q^{29} +1.99481 q^{31} -0.0583354 q^{33} -2.39975 q^{35} +0.199729 q^{37} -0.0801440 q^{39} +1.68407 q^{41} +1.48103 q^{43} -3.99719 q^{45} -2.83696 q^{47} -3.75745 q^{49} +0.0794867 q^{51} -2.68145 q^{53} +3.12032 q^{55} +0.0249148 q^{57} -11.7707 q^{59} +3.78542 q^{61} +5.40101 q^{63} +4.28684 q^{65} -7.60387 q^{67} +0.00693363 q^{69} -1.62350 q^{71} -1.14770 q^{73} +0.0803248 q^{75} -4.21618 q^{77} +1.00000 q^{79} +8.99441 q^{81} +16.9064 q^{83} -4.25169 q^{85} -0.0601225 q^{87} -14.4486 q^{89} -5.79239 q^{91} -0.0497003 q^{93} -1.33267 q^{95} -6.81140 q^{97} -7.02275 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 25 q + 4 q^{3} - 8 q^{5} + 2 q^{7} + 13 q^{9}+O(q^{10}) \) 25 * q + 4 * q^3 - 8 * q^5 + 2 * q^7 + 13 * q^9	\( 25 q + 4 q^{3} - 8 q^{5} + 2 q^{7} + 13 q^{9} - 3 q^{11} + q^{13} - 5 q^{15} - 13 q^{17} - 25 q^{19} - 24 q^{21} - 31 q^{23} + 21 q^{25} + 7 q^{27} - 19 q^{29} - 7 q^{31} - 30 q^{33} - q^{35} - 29 q^{37} - 26 q^{39} - 40 q^{41} - 40 q^{45} - 8 q^{47} - 9 q^{49} + 12 q^{51} - 38 q^{53} - 29 q^{55} - 4 q^{57} + 18 q^{59} - 26 q^{61} - 40 q^{63} - 70 q^{65} - 13 q^{67} + q^{69} - 47 q^{71} - 8 q^{73} + 7 q^{75} - 19 q^{77} + 25 q^{79} - 19 q^{81} - 8 q^{83} - 33 q^{85} - 50 q^{87} - 54 q^{89} - 12 q^{91} - 24 q^{93} + 8 q^{95} - 4 q^{97} - 39 q^{99}+O(q^{100}) \) 25 * q + 4 * q^3 - 8 * q^5 + 2 * q^7 + 13 * q^9 - 3 * q^11 + q^13 - 5 * q^15 - 13 * q^17 - 25 * q^19 - 24 * q^21 - 31 * q^23 + 21 * q^25 + 7 * q^27 - 19 * q^29 - 7 * q^31 - 30 * q^33 - q^35 - 29 * q^37 - 26 * q^39 - 40 * q^41 - 40 * q^45 - 8 * q^47 - 9 * q^49 + 12 * q^51 - 38 * q^53 - 29 * q^55 - 4 * q^57 + 18 * q^59 - 26 * q^61 - 40 * q^63 - 70 * q^65 - 13 * q^67 + q^69 - 47 * q^71 - 8 * q^73 + 7 * q^75 - 19 * q^77 + 25 * q^79 - 19 * q^81 - 8 * q^83 - 33 * q^85 - 50 * q^87 - 54 * q^89 - 12 * q^91 - 24 * q^93 + 8 * q^95 - 4 * q^97 - 39 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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