LMFDB - Embedded newform 1305.4.a.m.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1305,4,Mod(1,1305)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1305, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1305.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1305 = 3^{2} \cdot 5 \cdot 29 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1305.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:	$76.9974925575$
Analytic rank:	$1$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - x^{6} - 35x^{5} + 18x^{4} + 329x^{3} - 167x^{2} - 767x + 638 $ x^7 - x^6 - 35x^5 + 18x^4 + 329x^3 - 167x^2 - 767*x + 638
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 435)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$1.26184$ of defining polynomial
Character	$\chi$	$=$	1305.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.26184 q^{2} -6.40776 q^{4} +5.00000 q^{5} -13.4311 q^{7} -18.1803 q^{8} +6.30920 q^{10} +1.30869 q^{11} +85.6641 q^{13} -16.9479 q^{14} +28.3214 q^{16} -67.9685 q^{17} -63.3429 q^{19} -32.0388 q^{20} +1.65136 q^{22} +139.990 q^{23} +25.0000 q^{25} +108.094 q^{26} +86.0634 q^{28} +29.0000 q^{29} -170.385 q^{31} +181.179 q^{32} -85.7654 q^{34} -67.1557 q^{35} -405.970 q^{37} -79.9286 q^{38} -90.9015 q^{40} +447.863 q^{41} +407.779 q^{43} -8.38576 q^{44} +176.645 q^{46} +178.857 q^{47} -162.605 q^{49} +31.5460 q^{50} -548.915 q^{52} -93.3357 q^{53} +6.54344 q^{55} +244.182 q^{56} +36.5934 q^{58} -279.796 q^{59} -793.549 q^{61} -214.999 q^{62} +2.04805 q^{64} +428.321 q^{65} +460.471 q^{67} +435.526 q^{68} -84.7397 q^{70} -803.153 q^{71} -150.833 q^{73} -512.269 q^{74} +405.886 q^{76} -17.5772 q^{77} -313.814 q^{79} +141.607 q^{80} +565.132 q^{82} -250.514 q^{83} -339.842 q^{85} +514.552 q^{86} -23.7923 q^{88} +97.2972 q^{89} -1150.57 q^{91} -897.024 q^{92} +225.689 q^{94} -316.714 q^{95} -1342.23 q^{97} -205.181 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q + q^{2} + 15 q^{4} + 35 q^{5} - 37 q^{7} + 36 q^{8} + 5 q^{10} + 11 q^{11} - 133 q^{13} + 75 q^{14} - 53 q^{16} - 21 q^{17} - 170 q^{19} + 75 q^{20} - 369 q^{22} + 68 q^{23} + 175 q^{25} - 181 q^{26}+ \cdots - 1068 q^{98}+O(q^{100}) $ 7 * q + q^2 + 15 * q^4 + 35 * q^5 - 37 * q^7 + 36 * q^8 + 5 * q^10 + 11 * q^11 - 133 * q^13 + 75 * q^14 - 53 * q^16 - 21 * q^17 - 170 * q^19 + 75 * q^20 - 369 * q^22 + 68 * q^23 + 175 * q^25 - 181 * q^26 - 637 * q^28 + 203 * q^29 - 480 * q^31 + 779 * q^32 - 897 * q^34 - 185 * q^35 - 1032 * q^37 + 194 * q^38 + 180 * q^40 + 638 * q^41 - 512 * q^43 - 625 * q^44 + 16 * q^46 + 111 * q^47 + 178 * q^49 + 25 * q^50 - 1263 * q^52 - 410 * q^53 + 55 * q^55 - 1174 * q^56 + 29 * q^58 + 426 * q^59 - 1192 * q^61 - 460 * q^62 + 390 * q^64 - 665 * q^65 - 1671 * q^67 - 1509 * q^68 + 375 * q^70 + 1324 * q^71 - 852 * q^73 - 1780 * q^74 - 564 * q^76 + 2107 * q^77 + 366 * q^79 - 265 * q^80 - 318 * q^82 - 470 * q^83 - 105 * q^85 + 2196 * q^86 - 2518 * q^88 - 51 * q^89 - 1297 * q^91 + 684 * q^92 - 1837 * q^94 - 850 * q^95 - 3322 * q^97 - 1068 * q^98

Coefficient data

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

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

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

$2$	1.26184	0.446128	0.223064	−	0.974804i	$-0.428394\pi$
$2$	1.26184	0.446128	0.223064	+	0.974804i	$0.428394\pi$
$3$	0	0
$3$	0	0
$4$	−6.40776	−0.800970
$5$	5.00000	0.447214
$5$	5.00000	0.447214
$6$	0	0
$7$	−13.4311	−0.725213	−0.362606	−	0.931942i	$-0.618113\pi$
$7$	−13.4311	−0.725213	−0.362606	+	0.931942i	$0.618113\pi$
$8$	−18.1803	−0.803463
$9$	0	0
$10$	6.30920	0.199514
$11$	1.30869	0.0358713	0.0179357	−	0.999839i	$-0.494291\pi$
$11$	1.30869	0.0358713	0.0179357	+	0.999839i	$0.494291\pi$
$12$	0	0
$13$	85.6641	1.82761	0.913806	−	0.406151i	$-0.133129\pi$
$13$	85.6641	1.82761	0.913806	+	0.406151i	$0.133129\pi$
$14$	−16.9479	−0.323538
$15$	0	0
$16$	28.3214	0.442523
$17$	−67.9685	−0.969693	−0.484846	−	0.874599i	$-0.661124\pi$
$17$	−67.9685	−0.969693	−0.484846	+	0.874599i	$0.661124\pi$
$18$	0	0
$19$	−63.3429	−0.764834	−0.382417	−	0.923990i	$-0.624908\pi$
$19$	−63.3429	−0.764834	−0.382417	+	0.923990i	$0.624908\pi$
$20$	−32.0388	−0.358205
$21$	0	0
$22$	1.65136	0.0160032
$23$	139.990	1.26913	0.634565	−	0.772870i	$-0.281179\pi$
$23$	139.990	1.26913	0.634565	+	0.772870i	$0.281179\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	108.094	0.815349
$27$	0	0
$28$	86.0634	0.580874
$29$	29.0000	0.185695
$29$	29.0000	0.185695
$30$	0	0
$31$	−170.385	−0.987164	−0.493582	−	0.869699i	$-0.664313\pi$
$31$	−170.385	−0.987164	−0.493582	+	0.869699i	$0.664313\pi$
$32$	181.179	1.00088
$33$	0	0
$34$	−85.7654	−0.432607
$35$	−67.1557	−0.324325
$36$	0	0
$37$	−405.970	−1.80381	−0.901905	−	0.431934i	$-0.857831\pi$
$37$	−405.970	−1.80381	−0.901905	+	0.431934i	$0.857831\pi$
$38$	−79.9286	−0.341214
$39$	0	0
$40$	−90.9015	−0.359320
$41$	447.863	1.70596	0.852982	−	0.521940i	$-0.174792\pi$
$41$	447.863	1.70596	0.852982	+	0.521940i	$0.174792\pi$
$42$	0	0
$43$	407.779	1.44618	0.723090	−	0.690754i	$-0.242721\pi$
$43$	407.779	1.44618	0.723090	+	0.690754i	$0.242721\pi$
$44$	−8.38576	−0.0287318
$45$	0	0
$46$	176.645	0.566194
$47$	178.857	0.555086	0.277543	−	0.960713i	$-0.410480\pi$
$47$	178.857	0.555086	0.277543	+	0.960713i	$0.410480\pi$
$48$	0	0
$49$	−162.605	−0.474066
$50$	31.5460	0.0892256
$51$	0	0
$52$	−548.915	−1.46386
$53$	−93.3357	−0.241899	−0.120949	−	0.992659i	$-0.538594\pi$
$53$	−93.3357	−0.241899	−0.120949	+	0.992659i	$0.538594\pi$
$54$	0	0
$55$	6.54344	0.0160421
$56$	244.182	0.582682
$57$	0	0
$58$	36.5934	0.0828439
$59$	−279.796	−0.617396	−0.308698	−	0.951160i	$-0.599893\pi$
$59$	−279.796	−0.617396	−0.308698	+	0.951160i	$0.599893\pi$
$60$	0	0
$61$	−793.549	−1.66563	−0.832816	−	0.553550i	$-0.813273\pi$
$61$	−793.549	−1.66563	−0.832816	+	0.553550i	$0.813273\pi$
$62$	−214.999	−0.440402
$63$	0	0
$64$	2.04805	0.00400010
$65$	428.321	0.817333
$66$	0	0
$67$	460.471	0.839635	0.419818	−	0.907609i	$-0.362094\pi$
$67$	460.471	0.839635	0.419818	+	0.907609i	$0.362094\pi$
$68$	435.526	0.776695
$69$	0	0
$70$	−84.7397	−0.144690
$71$	−803.153	−1.34249	−0.671244	−	0.741236i	$-0.734240\pi$
$71$	−803.153	−1.34249	−0.671244	+	0.741236i	$0.734240\pi$
$72$	0	0
$73$	−150.833	−0.241832	−0.120916	−	0.992663i	$-0.538583\pi$
$73$	−150.833	−0.241832	−0.120916	+	0.992663i	$0.538583\pi$
$74$	−512.269	−0.804730
$75$	0	0
$76$	405.886	0.612609
$77$	−17.5772	−0.0260143
$78$	0	0
$79$	−313.814	−0.446922	−0.223461	−	0.974713i	$-0.571735\pi$
$79$	−313.814	−0.446922	−0.223461	+	0.974713i	$0.571735\pi$
$80$	141.607	0.197902
$81$	0	0
$82$	565.132	0.761078
$83$	−250.514	−0.331295	−0.165648	−	0.986185i	$-0.552971\pi$
$83$	−250.514	−0.331295	−0.165648	+	0.986185i	$0.552971\pi$
$84$	0	0
$85$	−339.842	−0.433660
$86$	514.552	0.645181
$87$	0	0
$88$	−23.7923	−0.0288213
$89$	97.2972	0.115882	0.0579409	−	0.998320i	$-0.481547\pi$
$89$	97.2972	0.115882	0.0579409	+	0.998320i	$0.481547\pi$
$90$	0	0
$91$	−1150.57	−1.32541
$92$	−897.024	−1.01653
$93$	0	0
$94$	225.689	0.247639
$95$	−316.714	−0.342044
$96$	0	0
$97$	−1342.23	−1.40497	−0.702487	−	0.711697i	$-0.747927\pi$
$97$	−1342.23	−1.40497	−0.702487	+	0.711697i	$0.747927\pi$
$98$	−205.181	−0.211494
$99$	0	0
$100$	−160.194	−0.160194
$101$	−73.7087	−0.0726167	−0.0363084	−	0.999341i	$-0.511560\pi$
$101$	−73.7087	−0.0726167	−0.0363084	+	0.999341i	$0.511560\pi$
$102$	0	0
$103$	−1307.09	−1.25040	−0.625199	−	0.780465i	$-0.714982\pi$
$103$	−1307.09	−1.25040	−0.625199	+	0.780465i	$0.714982\pi$
$104$	−1557.40	−1.46842
$105$	0	0
$106$	−117.775	−0.107918
$107$	−845.454	−0.763861	−0.381930	−	0.924191i	$-0.624741\pi$
$107$	−845.454	−0.763861	−0.381930	+	0.924191i	$0.624741\pi$
$108$	0	0
$109$	−1502.99	−1.32074	−0.660370	−	0.750941i	$-0.729600\pi$
$109$	−1502.99	−1.32074	−0.660370	+	0.750941i	$0.729600\pi$
$110$	8.25678	0.00715685
$111$	0	0
$112$	−380.389	−0.320923
$113$	317.218	0.264083	0.132042	−	0.991244i	$-0.457847\pi$
$113$	317.218	0.264083	0.132042	+	0.991244i	$0.457847\pi$
$114$	0	0
$115$	699.951	0.567572
$116$	−185.825	−0.148736
$117$	0	0
$118$	−353.058	−0.275438
$119$	912.893	0.703234
$120$	0	0
$121$	−1329.29	−0.998713
$122$	−1001.33	−0.743085
$123$	0	0
$124$	1091.79	0.790689
$125$	125.000	0.0894427
$126$	0	0
$127$	−1936.01	−1.35270	−0.676349	−	0.736581i	$-0.736439\pi$
$127$	−1936.01	−1.35270	−0.676349	+	0.736581i	$0.736439\pi$
$128$	−1446.85	−0.999100
$129$	0	0
$130$	540.472	0.364635
$131$	1234.54	0.823379	0.411689	−	0.911324i	$-0.364939\pi$
$131$	1234.54	0.823379	0.411689	+	0.911324i	$0.364939\pi$
$132$	0	0
$133$	850.767	0.554668
$134$	581.042	0.374585
$135$	0	0
$136$	1235.69	0.779112
$137$	886.190	0.552645	0.276322	−	0.961065i	$-0.410884\pi$
$137$	886.190	0.552645	0.276322	+	0.961065i	$0.410884\pi$
$138$	0	0
$139$	1582.12	0.965421	0.482711	−	0.875780i	$-0.339652\pi$
$139$	1582.12	0.965421	0.482711	+	0.875780i	$0.339652\pi$
$140$	430.317	0.259775
$141$	0	0
$142$	−1013.45	−0.598922
$143$	112.108	0.0655588
$144$	0	0
$145$	145.000	0.0830455
$146$	−190.328	−0.107888
$147$	0	0
$148$	2601.36	1.44480
$149$	1649.57	0.906969	0.453485	−	0.891264i	$-0.350181\pi$
$149$	1649.57	0.906969	0.453485	+	0.891264i	$0.350181\pi$
$150$	0	0
$151$	343.852	0.185313	0.0926565	−	0.995698i	$-0.470464\pi$
$151$	343.852	0.185313	0.0926565	+	0.995698i	$0.470464\pi$
$152$	1151.59	0.614516
$153$	0	0
$154$	−22.1796	−0.0116057
$155$	−851.926	−0.441473
$156$	0	0
$157$	2493.83	1.26770	0.633851	−	0.773455i	$-0.281473\pi$
$157$	2493.83	1.26770	0.633851	+	0.773455i	$0.281473\pi$
$158$	−395.983	−0.199384
$159$	0	0
$160$	905.897	0.447609
$161$	−1880.23	−0.920389
$162$	0	0
$163$	−1915.12	−0.920270	−0.460135	−	0.887849i	$-0.652199\pi$
$163$	−1915.12	−0.920270	−0.460135	+	0.887849i	$0.652199\pi$
$164$	−2869.80	−1.36643
$165$	0	0
$166$	−316.109	−0.147800
$167$	−99.5806	−0.0461424	−0.0230712	−	0.999734i	$-0.507344\pi$
$167$	−99.5806	−0.0461424	−0.0230712	+	0.999734i	$0.507344\pi$
$168$	0	0
$169$	5141.34	2.34016
$170$	−428.827	−0.193468
$171$	0	0
$172$	−2612.95	−1.15835
$173$	−2139.30	−0.940160	−0.470080	−	0.882624i	$-0.655775\pi$
$173$	−2139.30	−0.940160	−0.470080	+	0.882624i	$0.655775\pi$
$174$	0	0
$175$	−335.778	−0.145043
$176$	37.0640	0.0158739
$177$	0	0
$178$	122.773	0.0516981
$179$	−1863.54	−0.778142	−0.389071	−	0.921208i	$-0.627204\pi$
$179$	−1863.54	−0.778142	−0.389071	+	0.921208i	$0.627204\pi$
$180$	0	0
$181$	−3008.76	−1.23558	−0.617788	−	0.786345i	$-0.711971\pi$
$181$	−3008.76	−1.23558	−0.617788	+	0.786345i	$0.711971\pi$
$182$	−1451.83	−0.591301
$183$	0	0
$184$	−2545.06	−1.01970
$185$	−2029.85	−0.806689
$186$	0	0
$187$	−88.9496	−0.0347842
$188$	−1146.07	−0.444607
$189$	0	0
$190$	−399.643	−0.152596
$191$	−5175.24	−1.96056	−0.980280	−	0.197611i	$-0.936682\pi$
$191$	−5175.24	−1.96056	−0.980280	+	0.197611i	$0.936682\pi$
$192$	0	0
$193$	−3407.89	−1.27101	−0.635505	−	0.772097i	$-0.719208\pi$
$193$	−3407.89	−1.27101	−0.635505	+	0.772097i	$0.719208\pi$
$194$	−1693.68	−0.626798
$195$	0	0
$196$	1041.93	0.379713
$197$	2662.23	0.962823	0.481412	−	0.876495i	$-0.340124\pi$
$197$	2662.23	0.962823	0.481412	+	0.876495i	$0.340124\pi$
$198$	0	0
$199$	1124.12	0.400437	0.200218	−	0.979751i	$-0.435835\pi$
$199$	1124.12	0.400437	0.200218	+	0.979751i	$0.435835\pi$
$200$	−454.507	−0.160693
$201$	0	0
$202$	−93.0086	−0.0323964
$203$	−389.503	−0.134669
$204$	0	0
$205$	2239.32	0.762930
$206$	−1649.33	−0.557837
$207$	0	0
$208$	2426.13	0.808759
$209$	−82.8961	−0.0274356
$210$	0	0
$211$	109.248	0.0356442	0.0178221	−	0.999841i	$-0.494327\pi$
$211$	109.248	0.0356442	0.0178221	+	0.999841i	$0.494327\pi$
$212$	598.072	0.193754
$213$	0	0
$214$	−1066.83	−0.340780
$215$	2038.89	0.646751
$216$	0	0
$217$	2288.47	0.715904
$218$	−1896.54	−0.589219
$219$	0	0
$220$	−41.9288	−0.0128493
$221$	−5822.46	−1.77222
$222$	0	0
$223$	−4655.65	−1.39805	−0.699026	−	0.715097i	$-0.746383\pi$
$223$	−4655.65	−1.39805	−0.699026	+	0.715097i	$0.746383\pi$
$224$	−2433.45	−0.725854
$225$	0	0
$226$	400.279	0.117815
$227$	−3715.56	−1.08639	−0.543195	−	0.839606i	$-0.682786\pi$
$227$	−3715.56	−1.08639	−0.543195	+	0.839606i	$0.682786\pi$
$228$	0	0
$229$	2437.74	0.703451	0.351726	−	0.936103i	$-0.385595\pi$
$229$	2437.74	0.703451	0.351726	+	0.936103i	$0.385595\pi$
$230$	883.227	0.253210
$231$	0	0
$232$	−527.228	−0.149199
$233$	2704.29	0.760360	0.380180	−	0.924912i	$-0.375862\pi$
$233$	2704.29	0.760360	0.380180	+	0.924912i	$0.375862\pi$
$234$	0	0
$235$	894.287	0.248242
$236$	1792.87	0.494516
$237$	0	0
$238$	1151.93	0.313732
$239$	−6018.67	−1.62893	−0.814467	−	0.580210i	$-0.802970\pi$
$239$	−6018.67	−1.62893	−0.814467	+	0.580210i	$0.802970\pi$
$240$	0	0
$241$	−1244.90	−0.332743	−0.166372	−	0.986063i	$-0.553205\pi$
$241$	−1244.90	−0.332743	−0.166372	+	0.986063i	$0.553205\pi$
$242$	−1677.35	−0.445554
$243$	0	0
$244$	5084.87	1.33412
$245$	−813.024	−0.212009
$246$	0	0
$247$	−5426.21	−1.39782
$248$	3097.65	0.793150
$249$	0	0
$250$	157.730	0.0399029
$251$	−2477.48	−0.623016	−0.311508	−	0.950244i	$-0.600834\pi$
$251$	−2477.48	−0.623016	−0.311508	+	0.950244i	$0.600834\pi$
$252$	0	0
$253$	183.204	0.0455254
$254$	−2442.93	−0.603477
$255$	0	0
$256$	−1842.08	−0.449727
$257$	4778.08	1.15972	0.579860	−	0.814716i	$-0.303107\pi$
$257$	4778.08	1.15972	0.579860	+	0.814716i	$0.303107\pi$
$258$	0	0
$259$	5452.63	1.30815
$260$	−2744.58	−0.654659
$261$	0	0
$262$	1557.80	0.367332
$263$	1571.56	0.368467	0.184233	−	0.982883i	$-0.441020\pi$
$263$	1571.56	0.368467	0.184233	+	0.982883i	$0.441020\pi$
$264$	0	0
$265$	−466.678	−0.108180
$266$	1073.53	0.247453
$267$	0	0
$268$	−2950.59	−0.672522
$269$	−4946.58	−1.12118	−0.560591	−	0.828093i	$-0.689426\pi$
$269$	−4946.58	−1.12118	−0.560591	+	0.828093i	$0.689426\pi$
$270$	0	0
$271$	−914.942	−0.205088	−0.102544	−	0.994728i	$-0.532698\pi$
$271$	−914.942	−0.205088	−0.102544	+	0.994728i	$0.532698\pi$
$272$	−1924.97	−0.429111
$273$	0	0
$274$	1118.23	0.246550
$275$	32.7172	0.00717426
$276$	0	0
$277$	−2441.56	−0.529599	−0.264799	−	0.964304i	$-0.585306\pi$
$277$	−2441.56	−0.529599	−0.264799	+	0.964304i	$0.585306\pi$
$278$	1996.38	0.430701
$279$	0	0
$280$	1220.91	0.260583
$281$	5311.04	1.12751	0.563755	−	0.825942i	$-0.309356\pi$
$281$	5311.04	1.12751	0.563755	+	0.825942i	$0.309356\pi$
$282$	0	0
$283$	669.167	0.140558	0.0702789	−	0.997527i	$-0.477611\pi$
$283$	669.167	0.140558	0.0702789	+	0.997527i	$0.477611\pi$
$284$	5146.41	1.07529
$285$	0	0
$286$	141.462	0.0292476
$287$	−6015.31	−1.23719
$288$	0	0
$289$	−293.286	−0.0596960
$290$	182.967	0.0370489
$291$	0	0
$292$	966.504	0.193700
$293$	7468.02	1.48903	0.744516	−	0.667604i	$-0.232680\pi$
$293$	7468.02	1.48903	0.744516	+	0.667604i	$0.232680\pi$
$294$	0	0
$295$	−1398.98	−0.276108
$296$	7380.65	1.44930
$297$	0	0
$298$	2081.50	0.404624
$299$	11992.1	2.31948
$300$	0	0
$301$	−5476.93	−1.04879
$302$	433.886	0.0826733
$303$	0	0
$304$	−1793.96	−0.338456
$305$	−3967.74	−0.744893
$306$	0	0
$307$	−9927.63	−1.84560	−0.922801	−	0.385278i	$-0.874106\pi$
$307$	−9927.63	−1.84560	−0.922801	+	0.385278i	$0.874106\pi$
$308$	112.630	0.0208367
$309$	0	0
$310$	−1075.00	−0.196954
$311$	−3131.42	−0.570953	−0.285477	−	0.958386i	$-0.592152\pi$
$311$	−3131.42	−0.570953	−0.285477	+	0.958386i	$0.592152\pi$
$312$	0	0
$313$	8434.75	1.52320	0.761598	−	0.648050i	$-0.224415\pi$
$313$	8434.75	1.52320	0.761598	+	0.648050i	$0.224415\pi$
$314$	3146.82	0.565558
$315$	0	0
$316$	2010.84	0.357971
$317$	9162.80	1.62345	0.811725	−	0.584039i	$-0.198529\pi$
$317$	9162.80	1.62345	0.811725	+	0.584039i	$0.198529\pi$
$318$	0	0
$319$	37.9520	0.00666114
$320$	10.2403	0.00178890
$321$	0	0
$322$	−2372.55	−0.410611
$323$	4305.32	0.741654
$324$	0	0
$325$	2141.60	0.365522
$326$	−2416.58	−0.410558
$327$	0	0
$328$	−8142.29	−1.37068
$329$	−2402.26	−0.402555
$330$	0	0
$331$	5194.87	0.862647	0.431323	−	0.902197i	$-0.358047\pi$
$331$	5194.87	0.862647	0.431323	+	0.902197i	$0.358047\pi$
$332$	1605.23	0.265357
$333$	0	0
$334$	−125.655	−0.0205854
$335$	2302.36	0.375496
$336$	0	0
$337$	−3217.15	−0.520027	−0.260014	−	0.965605i	$-0.583727\pi$
$337$	−3217.15	−0.520027	−0.260014	+	0.965605i	$0.583727\pi$
$338$	6487.55	1.04401
$339$	0	0
$340$	2177.63	0.347348
$341$	−222.981	−0.0354109
$342$	0	0
$343$	6790.84	1.06901
$344$	−7413.54	−1.16195
$345$	0	0
$346$	−2699.45	−0.419432
$347$	609.761	0.0943334	0.0471667	−	0.998887i	$-0.484981\pi$
$347$	609.761	0.0943334	0.0471667	+	0.998887i	$0.484981\pi$
$348$	0	0
$349$	4406.40	0.675843	0.337921	−	0.941174i	$-0.390276\pi$
$349$	4406.40	0.675843	0.337921	+	0.941174i	$0.390276\pi$
$350$	−423.699	−0.0647075
$351$	0	0
$352$	237.108	0.0359031
$353$	−2480.09	−0.373943	−0.186971	−	0.982365i	$-0.559867\pi$
$353$	−2480.09	−0.373943	−0.186971	+	0.982365i	$0.559867\pi$
$354$	0	0
$355$	−4015.76	−0.600379
$356$	−623.457	−0.0928178
$357$	0	0
$358$	−2351.49	−0.347151
$359$	−6578.82	−0.967177	−0.483589	−	0.875295i	$-0.660667\pi$
$359$	−6578.82	−0.967177	−0.483589	+	0.875295i	$0.660667\pi$
$360$	0	0
$361$	−2846.68	−0.415028
$362$	−3796.57	−0.551225
$363$	0	0
$364$	7372.55	1.06161
$365$	−754.167	−0.108150
$366$	0	0
$367$	−731.114	−0.103989	−0.0519943	−	0.998647i	$-0.516558\pi$
$367$	−731.114	−0.103989	−0.0519943	+	0.998647i	$0.516558\pi$
$368$	3964.72	0.561618
$369$	0	0
$370$	−2561.34	−0.359886
$371$	1253.60	0.175428
$372$	0	0
$373$	−1673.03	−0.232242	−0.116121	−	0.993235i	$-0.537046\pi$
$373$	−1673.03	−0.232242	−0.116121	+	0.993235i	$0.537046\pi$
$374$	−112.240	−0.0155182
$375$	0	0
$376$	−3251.68	−0.445991
$377$	2484.26	0.339379
$378$	0	0
$379$	3760.44	0.509659	0.254830	−	0.966986i	$-0.417981\pi$
$379$	3760.44	0.509659	0.254830	+	0.966986i	$0.417981\pi$
$380$	2029.43	0.273967
$381$	0	0
$382$	−6530.33	−0.874661
$383$	12540.8	1.67312	0.836562	−	0.547872i	$-0.184562\pi$
$383$	12540.8	1.67312	0.836562	+	0.547872i	$0.184562\pi$
$384$	0	0
$385$	−87.8859	−0.0116340
$386$	−4300.21	−0.567033
$387$	0	0
$388$	8600.66	1.12534
$389$	−5779.21	−0.753259	−0.376629	−	0.926364i	$-0.622917\pi$
$389$	−5779.21	−0.753259	−0.376629	+	0.926364i	$0.622917\pi$
$390$	0	0
$391$	−9514.92	−1.23067
$392$	2956.20	0.380895
$393$	0	0
$394$	3359.31	0.429542
$395$	−1569.07	−0.199869
$396$	0	0
$397$	14394.6	1.81976	0.909881	−	0.414869i	$-0.136173\pi$
$397$	14394.6	1.81976	0.909881	+	0.414869i	$0.136173\pi$
$398$	1418.46	0.178646
$399$	0	0
$400$	708.036	0.0885045
$401$	−8620.31	−1.07351	−0.536756	−	0.843738i	$-0.680350\pi$
$401$	−8620.31	−1.07351	−0.536756	+	0.843738i	$0.680350\pi$
$402$	0	0
$403$	−14595.9	−1.80415
$404$	472.308	0.0581638
$405$	0	0
$406$	−491.490	−0.0600795
$407$	−531.288	−0.0647051
$408$	0	0
$409$	−14058.0	−1.69957	−0.849783	−	0.527133i	$-0.823267\pi$
$409$	−14058.0	−1.69957	−0.849783	+	0.527133i	$0.823267\pi$
$410$	2825.66	0.340365
$411$	0	0
$412$	8375.49	1.00153
$413$	3757.98	0.447744
$414$	0	0
$415$	−1252.57	−0.148160
$416$	15520.6	1.82923
$417$	0	0
$418$	−104.602	−0.0122398
$419$	308.722	0.0359954	0.0179977	−	0.999838i	$-0.494271\pi$
$419$	308.722	0.0359954	0.0179977	+	0.999838i	$0.494271\pi$
$420$	0	0
$421$	7341.48	0.849886	0.424943	−	0.905220i	$-0.360294\pi$
$421$	7341.48	0.849886	0.424943	+	0.905220i	$0.360294\pi$
$422$	137.853	0.0159019
$423$	0	0
$424$	1696.87	0.194357
$425$	−1699.21	−0.193939
$426$	0	0
$427$	10658.3	1.20794
$428$	5417.47	0.611830
$429$	0	0
$430$	2572.76	0.288534
$431$	16660.8	1.86201	0.931003	−	0.365013i	$-0.118935\pi$
$431$	16660.8	1.86201	0.931003	+	0.365013i	$0.118935\pi$
$432$	0	0
$433$	10091.5	1.12001	0.560006	−	0.828488i	$-0.310799\pi$
$433$	10091.5	1.12001	0.560006	+	0.828488i	$0.310799\pi$
$434$	2887.68	0.319385
$435$	0	0
$436$	9630.82	1.05787
$437$	−8867.38	−0.970674
$438$	0	0
$439$	5149.74	0.559871	0.279936	−	0.960019i	$-0.409687\pi$
$439$	5149.74	0.559871	0.279936	+	0.960019i	$0.409687\pi$
$440$	−118.962	−0.0128893
$441$	0	0
$442$	−7347.02	−0.790638
$443$	−15820.5	−1.69674	−0.848371	−	0.529402i	$-0.822416\pi$
$443$	−15820.5	−1.69674	−0.848371	+	0.529402i	$0.822416\pi$
$444$	0	0
$445$	486.486	0.0518239
$446$	−5874.69	−0.623710
$447$	0	0
$448$	−27.5076	−0.00290092
$449$	13903.0	1.46130	0.730651	−	0.682751i	$-0.239217\pi$
$449$	13903.0	1.46130	0.730651	+	0.682751i	$0.239217\pi$
$450$	0	0
$451$	586.114	0.0611952
$452$	−2032.66	−0.211523
$453$	0	0
$454$	−4688.45	−0.484669
$455$	−5752.83	−0.592740
$456$	0	0
$457$	4315.24	0.441704	0.220852	−	0.975307i	$-0.429116\pi$
$457$	4315.24	0.441704	0.220852	+	0.975307i	$0.429116\pi$
$458$	3076.04	0.313829
$459$	0	0
$460$	−4485.12	−0.454608
$461$	−5936.08	−0.599720	−0.299860	−	0.953983i	$-0.596940\pi$
$461$	−5936.08	−0.599720	−0.299860	+	0.953983i	$0.596940\pi$
$462$	0	0
$463$	−12894.4	−1.29428	−0.647141	−	0.762370i	$-0.724036\pi$
$463$	−12894.4	−1.29428	−0.647141	+	0.762370i	$0.724036\pi$
$464$	821.322	0.0821744
$465$	0	0
$466$	3412.38	0.339218
$467$	−12653.8	−1.25385	−0.626924	−	0.779081i	$-0.715686\pi$
$467$	−12653.8	−1.25385	−0.626924	+	0.779081i	$0.715686\pi$
$468$	0	0
$469$	−6184.65	−0.608914
$470$	1128.45	0.110748
$471$	0	0
$472$	5086.78	0.496055
$473$	533.656	0.0518764
$474$	0	0
$475$	−1583.57	−0.152967
$476$	−5849.60	−0.563269
$477$	0	0
$478$	−7594.60	−0.726713
$479$	−8991.57	−0.857694	−0.428847	−	0.903377i	$-0.641080\pi$
$479$	−8991.57	−0.857694	−0.428847	+	0.903377i	$0.641080\pi$
$480$	0	0
$481$	−34777.0	−3.29667
$482$	−1570.87	−0.148446
$483$	0	0
$484$	8517.75	0.799939
$485$	−6711.13	−0.628323
$486$	0	0
$487$	7671.39	0.713806	0.356903	−	0.934141i	$-0.383833\pi$
$487$	7671.39	0.713806	0.356903	+	0.934141i	$0.383833\pi$
$488$	14426.9	1.33827
$489$	0	0
$490$	−1025.91	−0.0945831
$491$	−10322.9	−0.948809	−0.474405	−	0.880307i	$-0.657337\pi$
$491$	−10322.9	−0.948809	−0.474405	+	0.880307i	$0.657337\pi$
$492$	0	0
$493$	−1971.09	−0.180067
$494$	−6847.01	−0.623607
$495$	0	0
$496$	−4825.56	−0.436843
$497$	10787.3	0.973590
$498$	0	0
$499$	−4385.18	−0.393402	−0.196701	−	0.980464i	$-0.563023\pi$
$499$	−4385.18	−0.393402	−0.196701	+	0.980464i	$0.563023\pi$
$500$	−800.970	−0.0716409
$501$	0	0
$502$	−3126.18	−0.277945
$503$	−8327.07	−0.738143	−0.369071	−	0.929401i	$-0.620324\pi$
$503$	−8327.07	−0.738143	−0.369071	+	0.929401i	$0.620324\pi$
$504$	0	0
$505$	−368.544	−0.0324752
$506$	231.174	0.0203101
$507$	0	0
$508$	12405.5	1.08347
$509$	20580.8	1.79220	0.896098	−	0.443856i	$-0.146390\pi$
$509$	20580.8	1.79220	0.896098	+	0.443856i	$0.146390\pi$
$510$	0	0
$511$	2025.86	0.175379
$512$	9250.40	0.798465
$513$	0	0
$514$	6029.17	0.517384
$515$	−6535.43	−0.559195
$516$	0	0
$517$	234.069	0.0199117
$518$	6880.35	0.583601
$519$	0	0
$520$	−7786.99	−0.656697
$521$	15109.3	1.27054	0.635270	−	0.772290i	$-0.280889\pi$
$521$	15109.3	1.27054	0.635270	+	0.772290i	$0.280889\pi$
$522$	0	0
$523$	−507.100	−0.0423976	−0.0211988	−	0.999775i	$-0.506748\pi$
$523$	−507.100	−0.0423976	−0.0211988	+	0.999775i	$0.506748\pi$
$524$	−7910.66	−0.659502
$525$	0	0
$526$	1983.06	0.164383
$527$	11580.8	0.957246
$528$	0	0
$529$	7430.26	0.610690
$530$	−588.874	−0.0482623
$531$	0	0
$532$	−5451.51	−0.444272
$533$	38365.8	3.11784
$534$	0	0
$535$	−4227.27	−0.341609
$536$	−8371.51	−0.674616
$537$	0	0
$538$	−6241.79	−0.500191
$539$	−212.799	−0.0170054
$540$	0	0
$541$	8249.02	0.655551	0.327776	−	0.944756i	$-0.393701\pi$
$541$	8249.02	0.655551	0.327776	+	0.944756i	$0.393701\pi$
$542$	−1154.51	−0.0914954
$543$	0	0
$544$	−12314.5	−0.970551
$545$	−7514.96	−0.590653
$546$	0	0
$547$	−19432.6	−1.51897	−0.759486	−	0.650524i	$-0.774550\pi$
$547$	−19432.6	−1.51897	−0.759486	+	0.650524i	$0.774550\pi$
$548$	−5678.49	−0.442652
$549$	0	0
$550$	41.2839	0.00320064
$551$	−1836.94	−0.142026
$552$	0	0
$553$	4214.87	0.324113
$554$	−3080.85	−0.236269
$555$	0	0
$556$	−10137.8	−0.773273
$557$	−18807.6	−1.43071	−0.715354	−	0.698762i	$-0.753735\pi$
$557$	−18807.6	−1.43071	−0.715354	+	0.698762i	$0.753735\pi$
$558$	0	0
$559$	34932.0	2.64305
$560$	−1901.94	−0.143521
$561$	0	0
$562$	6701.69	0.503013
$563$	−7337.15	−0.549243	−0.274622	−	0.961552i	$-0.588553\pi$
$563$	−7337.15	−0.549243	−0.274622	+	0.961552i	$0.588553\pi$
$564$	0	0
$565$	1586.09	0.118102
$566$	844.382	0.0627068
$567$	0	0
$568$	14601.6	1.07864
$569$	−5459.12	−0.402211	−0.201106	−	0.979570i	$-0.564453\pi$
$569$	−5459.12	−0.402211	−0.201106	+	0.979570i	$0.564453\pi$
$570$	0	0
$571$	5306.93	0.388946	0.194473	−	0.980908i	$-0.437700\pi$
$571$	5306.93	0.388946	0.194473	+	0.980908i	$0.437700\pi$
$572$	−718.359	−0.0525107
$573$	0	0
$574$	−7590.36	−0.551944
$575$	3499.76	0.253826
$576$	0	0
$577$	16127.1	1.16357	0.581785	−	0.813343i	$-0.302355\pi$
$577$	16127.1	1.16357	0.581785	+	0.813343i	$0.302355\pi$
$578$	−370.081	−0.0266320
$579$	0	0
$580$	−929.125	−0.0665169
$581$	3364.69	0.240260
$582$	0	0
$583$	−122.147	−0.00867723
$584$	2742.19	0.194303
$585$	0	0
$586$	9423.45	0.664299
$587$	−9109.13	−0.640501	−0.320250	−	0.947333i	$-0.603767\pi$
$587$	−9109.13	−0.640501	−0.320250	+	0.947333i	$0.603767\pi$
$588$	0	0
$589$	10792.7	0.755017
$590$	−1765.29	−0.123180
$591$	0	0
$592$	−11497.6	−0.798227
$593$	−28352.1	−1.96337	−0.981687	−	0.190500i	$-0.938989\pi$
$593$	−28352.1	−1.96337	−0.981687	+	0.190500i	$0.938989\pi$
$594$	0	0
$595$	4564.47	0.314496
$596$	−10570.1	−0.726455
$597$	0	0
$598$	15132.2	1.03478
$599$	15496.1	1.05702	0.528508	−	0.848928i	$-0.322752\pi$
$599$	15496.1	1.05702	0.528508	+	0.848928i	$0.322752\pi$
$600$	0	0
$601$	7637.92	0.518398	0.259199	−	0.965824i	$-0.416541\pi$
$601$	7637.92	0.518398	0.259199	+	0.965824i	$0.416541\pi$
$602$	−6911.01	−0.467894
$603$	0	0
$604$	−2203.32	−0.148430
$605$	−6646.44	−0.446638
$606$	0	0
$607$	−12012.8	−0.803268	−0.401634	−	0.915800i	$-0.631558\pi$
$607$	−12012.8	−0.803268	−0.401634	+	0.915800i	$0.631558\pi$
$608$	−11476.4	−0.765511
$609$	0	0
$610$	−5006.66	−0.332318
$611$	15321.7	1.01448
$612$	0	0
$613$	−6113.47	−0.402807	−0.201404	−	0.979508i	$-0.564550\pi$
$613$	−6113.47	−0.402807	−0.201404	+	0.979508i	$0.564550\pi$
$614$	−12527.1	−0.823374
$615$	0	0
$616$	319.558	0.0209016
$617$	28775.9	1.87759	0.938797	−	0.344472i	$-0.111942\pi$
$617$	28775.9	1.87759	0.938797	+	0.344472i	$0.111942\pi$
$618$	0	0
$619$	−7092.81	−0.460556	−0.230278	−	0.973125i	$-0.573964\pi$
$619$	−7092.81	−0.460556	−0.230278	+	0.973125i	$0.573964\pi$
$620$	5458.94	0.353607
$621$	0	0
$622$	−3951.35	−0.254718
$623$	−1306.81	−0.0840390
$624$	0	0
$625$	625.000	0.0400000
$626$	10643.3	0.679540
$627$	0	0
$628$	−15979.9	−1.01539
$629$	27593.1	1.74914
$630$	0	0
$631$	−2283.67	−0.144075	−0.0720377	−	0.997402i	$-0.522950\pi$
$631$	−2283.67	−0.144075	−0.0720377	+	0.997402i	$0.522950\pi$
$632$	5705.23	0.359085
$633$	0	0
$634$	11562.0	0.724267
$635$	−9680.03	−0.604945
$636$	0	0
$637$	−13929.4	−0.866409
$638$	47.8893	0.00297172
$639$	0	0
$640$	−7234.26	−0.446811
$641$	17599.2	1.08444	0.542220	−	0.840236i	$-0.317584\pi$
$641$	17599.2	1.08444	0.542220	+	0.840236i	$0.317584\pi$
$642$	0	0
$643$	9248.80	0.567242	0.283621	−	0.958936i	$-0.408464\pi$
$643$	9248.80	0.567242	0.283621	+	0.958936i	$0.408464\pi$
$644$	12048.0	0.737204
$645$	0	0
$646$	5432.63	0.330873
$647$	12201.5	0.741409	0.370704	−	0.928751i	$-0.379116\pi$
$647$	12201.5	0.741409	0.370704	+	0.928751i	$0.379116\pi$
$648$	0	0
$649$	−366.166	−0.0221468
$650$	2702.36	0.163070
$651$	0	0
$652$	12271.6	0.737108
$653$	−9769.75	−0.585482	−0.292741	−	0.956192i	$-0.594567\pi$
$653$	−9769.75	−0.585482	−0.292741	+	0.956192i	$0.594567\pi$
$654$	0	0
$655$	6172.72	0.368226
$656$	12684.1	0.754927
$657$	0	0
$658$	−3031.26	−0.179591
$659$	15386.3	0.909506	0.454753	−	0.890618i	$-0.349728\pi$
$659$	15386.3	0.909506	0.454753	+	0.890618i	$0.349728\pi$
$660$	0	0
$661$	16886.7	0.993671	0.496835	−	0.867845i	$-0.334495\pi$
$661$	16886.7	0.993671	0.496835	+	0.867845i	$0.334495\pi$
$662$	6555.10	0.384851
$663$	0	0
$664$	4554.42	0.266183
$665$	4253.83	0.248055
$666$	0	0
$667$	4059.72	0.235671
$668$	638.089	0.0369587
$669$	0	0
$670$	2905.21	0.167519
$671$	−1038.51	−0.0597484
$672$	0	0
$673$	26462.8	1.51570	0.757851	−	0.652428i	$-0.226249\pi$
$673$	26462.8	1.51570	0.757851	+	0.652428i	$0.226249\pi$
$674$	−4059.53	−0.231999
$675$	0	0
$676$	−32944.5	−1.87440
$677$	−28533.1	−1.61982	−0.809908	−	0.586556i	$-0.800483\pi$
$677$	−28533.1	−1.61982	−0.809908	+	0.586556i	$0.800483\pi$
$678$	0	0
$679$	18027.6	1.01890
$680$	6178.43	0.348430
$681$	0	0
$682$	−281.367	−0.0157978
$683$	−27990.7	−1.56813	−0.784066	−	0.620677i	$-0.786858\pi$
$683$	−27990.7	−1.56813	−0.784066	+	0.620677i	$0.786858\pi$
$684$	0	0
$685$	4430.95	0.247150
$686$	8568.96	0.476916
$687$	0	0
$688$	11548.9	0.639967
$689$	−7995.52	−0.442097
$690$	0	0
$691$	9248.47	0.509158	0.254579	−	0.967052i	$-0.418063\pi$
$691$	9248.47	0.509158	0.254579	+	0.967052i	$0.418063\pi$
$692$	13708.1	0.753040
$693$	0	0
$694$	769.421	0.0420848
$695$	7910.59	0.431749
$696$	0	0
$697$	−30440.6	−1.65426
$698$	5560.17	0.301512
$699$	0	0
$700$	2151.59	0.116175
$701$	−16805.6	−0.905474	−0.452737	−	0.891644i	$-0.649552\pi$
$701$	−16805.6	−0.905474	−0.452737	+	0.891644i	$0.649552\pi$
$702$	0	0
$703$	25715.3	1.37962
$704$	2.68026	0.000143489 0
$705$	0	0
$706$	−3129.48	−0.166826
$707$	989.991	0.0526626
$708$	0	0
$709$	28586.0	1.51421	0.757103	−	0.653296i	$-0.226614\pi$
$709$	28586.0	1.51421	0.757103	+	0.653296i	$0.226614\pi$
$710$	−5067.25	−0.267846
$711$	0	0
$712$	−1768.89	−0.0931067
$713$	−23852.3	−1.25284
$714$	0	0
$715$	560.538	0.0293188
$716$	11941.1	0.623268
$717$	0	0
$718$	−8301.42	−0.431485
$719$	37753.6	1.95824	0.979119	−	0.203289i	$-0.0651633\pi$
$719$	37753.6	1.95824	0.979119	+	0.203289i	$0.0651633\pi$
$720$	0	0
$721$	17555.6	0.906805
$722$	−3592.05	−0.185156
$723$	0	0
$724$	19279.4	0.989659
$725$	725.000	0.0371391
$726$	0	0
$727$	−10769.6	−0.549412	−0.274706	−	0.961528i	$-0.588581\pi$
$727$	−10769.6	−0.549412	−0.274706	+	0.961528i	$0.588581\pi$
$728$	20917.6	1.06492
$729$	0	0
$730$	−951.638	−0.0482489
$731$	−27716.1	−1.40235
$732$	0	0
$733$	−29083.9	−1.46554	−0.732768	−	0.680479i	$-0.761772\pi$
$733$	−29083.9	−1.46554	−0.732768	+	0.680479i	$0.761772\pi$
$734$	−922.549	−0.0463923
$735$	0	0
$736$	25363.4	1.27025
$737$	602.614	0.0301188
$738$	0	0
$739$	5594.45	0.278478	0.139239	−	0.990259i	$-0.455534\pi$
$739$	5594.45	0.278478	0.139239	+	0.990259i	$0.455534\pi$
$740$	13006.8	0.646133
$741$	0	0
$742$	1581.85	0.0782634
$743$	−8984.59	−0.443624	−0.221812	−	0.975089i	$-0.571197\pi$
$743$	−8984.59	−0.443624	−0.221812	+	0.975089i	$0.571197\pi$
$744$	0	0
$745$	8247.87	0.405609
$746$	−2111.10	−0.103610
$747$	0	0
$748$	569.967	0.0278611
$749$	11355.4	0.553962
$750$	0	0
$751$	4621.64	0.224562	0.112281	−	0.993676i	$-0.464184\pi$
$751$	4621.64	0.224562	0.112281	+	0.993676i	$0.464184\pi$
$752$	5065.50	0.245638
$753$	0	0
$754$	3134.74	0.151406
$755$	1719.26	0.0828745
$756$	0	0
$757$	−5554.76	−0.266699	−0.133349	−	0.991069i	$-0.542573\pi$
$757$	−5554.76	−0.266699	−0.133349	+	0.991069i	$0.542573\pi$
$758$	4745.07	0.227373
$759$	0	0
$760$	5757.96	0.274820
$761$	−28686.9	−1.36649	−0.683245	−	0.730189i	$-0.739432\pi$
$761$	−28686.9	−1.36649	−0.683245	+	0.730189i	$0.739432\pi$
$762$	0	0
$763$	20186.9	0.957817
$764$	33161.7	1.57035
$765$	0	0
$766$	15824.5	0.746427
$767$	−23968.5	−1.12836
$768$	0	0
$769$	26311.7	1.23384	0.616921	−	0.787025i	$-0.288380\pi$
$769$	26311.7	1.23384	0.616921	+	0.787025i	$0.288380\pi$
$770$	−110.898	−0.00519024
$771$	0	0
$772$	21836.9	1.01804
$773$	25176.4	1.17145	0.585727	−	0.810508i	$-0.300809\pi$
$773$	25176.4	1.17145	0.585727	+	0.810508i	$0.300809\pi$
$774$	0	0
$775$	−4259.63	−0.197433
$776$	24402.1	1.12884
$777$	0	0
$778$	−7292.44	−0.336050
$779$	−28369.0	−1.30478
$780$	0	0
$781$	−1051.08	−0.0481568
$782$	−12006.3	−0.549034
$783$	0	0
$784$	−4605.20	−0.209785
$785$	12469.2	0.566934
$786$	0	0
$787$	−27651.8	−1.25245	−0.626226	−	0.779641i	$-0.715401\pi$
$787$	−27651.8	−1.25245	−0.626226	+	0.779641i	$0.715401\pi$
$788$	−17058.9	−0.771192
$789$	0	0
$790$	−1979.91	−0.0891673
$791$	−4260.60	−0.191516
$792$	0	0
$793$	−67978.7	−3.04413
$794$	18163.7	0.811847
$795$	0	0
$796$	−7203.11	−0.320738
$797$	35053.5	1.55792	0.778958	−	0.627076i	$-0.215748\pi$
$797$	35053.5	1.55792	0.778958	+	0.627076i	$0.215748\pi$
$798$	0	0
$799$	−12156.7	−0.538262
$800$	4529.49	0.200177
$801$	0	0
$802$	−10877.5	−0.478923
$803$	−197.394	−0.00867482
$804$	0	0
$805$	−9401.13	−0.411610
$806$	−18417.7	−0.804883
$807$	0	0
$808$	1340.05	0.0583449
$809$	23824.8	1.03540	0.517699	−	0.855563i	$-0.326789\pi$
$809$	23824.8	1.03540	0.517699	+	0.855563i	$0.326789\pi$
$810$	0	0
$811$	19361.6	0.838322	0.419161	−	0.907912i	$-0.362324\pi$
$811$	19361.6	0.838322	0.419161	+	0.907912i	$0.362324\pi$
$812$	2495.84	0.107866
$813$	0	0
$814$	−670.401	−0.0288667
$815$	−9575.61	−0.411557
$816$	0	0
$817$	−25829.9	−1.10609
$818$	−17738.9	−0.758224
$819$	0	0
$820$	−14349.0	−0.611084
$821$	14914.6	0.634012	0.317006	−	0.948424i	$-0.397322\pi$
$821$	14914.6	0.634012	0.317006	+	0.948424i	$0.397322\pi$
$822$	0	0
$823$	7639.98	0.323588	0.161794	−	0.986825i	$-0.448272\pi$
$823$	7639.98	0.323588	0.161794	+	0.986825i	$0.448272\pi$
$824$	23763.2	1.00465
$825$	0	0
$826$	4741.97	0.199751
$827$	−12462.1	−0.524003	−0.262001	−	0.965068i	$-0.584382\pi$
$827$	−12462.1	−0.524003	−0.262001	+	0.965068i	$0.584382\pi$
$828$	0	0
$829$	−26986.5	−1.13062	−0.565309	−	0.824880i	$-0.691243\pi$
$829$	−26986.5	−1.13062	−0.565309	+	0.824880i	$0.691243\pi$
$830$	−1580.54	−0.0660982
$831$	0	0
$832$	175.444	0.00731063
$833$	11052.0	0.459699
$834$	0	0
$835$	−497.903	−0.0206355
$836$	531.178	0.0219751
$837$	0	0
$838$	389.558	0.0160585
$839$	−23448.1	−0.964860	−0.482430	−	0.875935i	$-0.660246\pi$
$839$	−23448.1	−0.964860	−0.482430	+	0.875935i	$0.660246\pi$
$840$	0	0
$841$	841.000	0.0344828
$842$	9263.78	0.379158
$843$	0	0
$844$	−700.034	−0.0285499
$845$	25706.7	1.04655
$846$	0	0
$847$	17853.8	0.724280
$848$	−2643.40	−0.107046
$849$	0	0
$850$	−2144.13	−0.0865214
$851$	−56831.8	−2.28927
$852$	0	0
$853$	−30230.4	−1.21344	−0.606722	−	0.794914i	$-0.707516\pi$
$853$	−30230.4	−1.21344	−0.606722	+	0.794914i	$0.707516\pi$
$854$	13449.0	0.538895
$855$	0	0
$856$	15370.6	0.613734
$857$	−32088.1	−1.27900	−0.639502	−	0.768789i	$-0.720860\pi$
$857$	−32088.1	−1.27900	−0.639502	+	0.768789i	$0.720860\pi$
$858$	0	0
$859$	−43201.3	−1.71596	−0.857980	−	0.513683i	$-0.828281\pi$
$859$	−43201.3	−1.71596	−0.857980	+	0.513683i	$0.828281\pi$
$860$	−13064.7	−0.518028
$861$	0	0
$862$	21023.3	0.830692
$863$	15650.5	0.617323	0.308662	−	0.951172i	$-0.400119\pi$
$863$	15650.5	0.617323	0.308662	+	0.951172i	$0.400119\pi$
$864$	0	0
$865$	−10696.5	−0.420452
$866$	12733.8	0.499669
$867$	0	0
$868$	−14663.9	−0.573418
$869$	−410.685	−0.0160317
$870$	0	0
$871$	39445.9	1.53453
$872$	27324.9	1.06117
$873$	0	0
$874$	−11189.2	−0.433045
$875$	−1678.89	−0.0648650
$876$	0	0
$877$	−37921.1	−1.46010	−0.730049	−	0.683395i	$-0.760503\pi$
$877$	−37921.1	−1.46010	−0.730049	+	0.683395i	$0.760503\pi$
$878$	6498.14	0.249774
$879$	0	0
$880$	185.320	0.00709901
$881$	−19844.4	−0.758881	−0.379441	−	0.925216i	$-0.623884\pi$
$881$	−19844.4	−0.758881	−0.379441	+	0.925216i	$0.623884\pi$
$882$	0	0
$883$	33386.5	1.27242	0.636210	−	0.771516i	$-0.280501\pi$
$883$	33386.5	1.27242	0.636210	+	0.771516i	$0.280501\pi$
$884$	37308.9	1.41950
$885$	0	0
$886$	−19963.0	−0.756964
$887$	25988.8	0.983788	0.491894	−	0.870655i	$-0.336305\pi$
$887$	25988.8	0.983788	0.491894	+	0.870655i	$0.336305\pi$
$888$	0	0
$889$	26002.7	0.980994
$890$	613.867	0.0231201
$891$	0	0
$892$	29832.3	1.11980
$893$	−11329.3	−0.424549
$894$	0	0
$895$	−9317.69	−0.347996
$896$	19432.9	0.724560
$897$	0	0
$898$	17543.4	0.651928
$899$	−4941.17	−0.183312
$900$	0	0
$901$	6343.88	0.234568
$902$	739.582	0.0273009
$903$	0	0
$904$	−5767.12	−0.212181
$905$	−15043.8	−0.552566
$906$	0	0
$907$	−26383.8	−0.965887	−0.482943	−	0.875652i	$-0.660432\pi$
$907$	−26383.8	−0.965887	−0.482943	+	0.875652i	$0.660432\pi$
$908$	23808.4	0.870166
$909$	0	0
$910$	−7259.15	−0.264438
$911$	23448.5	0.852779	0.426390	−	0.904540i	$-0.359785\pi$
$911$	23448.5	0.852779	0.426390	+	0.904540i	$0.359785\pi$
$912$	0	0
$913$	−327.845	−0.0118840
$914$	5445.15	0.197056
$915$	0	0
$916$	−15620.4	−0.563443
$917$	−16581.3	−0.597125
$918$	0	0
$919$	−1649.45	−0.0592060	−0.0296030	−	0.999562i	$-0.509424\pi$
$919$	−1649.45	−0.0592060	−0.0296030	+	0.999562i	$0.509424\pi$
$920$	−12725.3	−0.456023
$921$	0	0
$922$	−7490.39	−0.267552
$923$	−68801.4	−2.45355
$924$	0	0
$925$	−10149.2	−0.360762
$926$	−16270.7	−0.577415
$927$	0	0
$928$	5254.20	0.185860
$929$	−48039.5	−1.69658	−0.848291	−	0.529531i	$-0.822368\pi$
$929$	−48039.5	−1.69658	−0.848291	+	0.529531i	$0.822368\pi$
$930$	0	0
$931$	10299.9	0.362582
$932$	−17328.4	−0.609026
$933$	0	0
$934$	−15967.0	−0.559376
$935$	−444.748	−0.0155559
$936$	0	0
$937$	−28647.5	−0.998796	−0.499398	−	0.866373i	$-0.666445\pi$
$937$	−28647.5	−0.998796	−0.499398	+	0.866373i	$0.666445\pi$
$938$	−7804.04	−0.271654
$939$	0	0
$940$	−5730.37	−0.198834
$941$	−9541.12	−0.330533	−0.165267	−	0.986249i	$-0.552848\pi$
$941$	−9541.12	−0.330533	−0.165267	+	0.986249i	$0.552848\pi$
$942$	0	0
$943$	62696.5	2.16509
$944$	−7924.24	−0.273212
$945$	0	0
$946$	673.388	0.0231435
$947$	−44596.6	−1.53030	−0.765151	−	0.643851i	$-0.777336\pi$
$947$	−44596.6	−1.53030	−0.765151	+	0.643851i	$0.777336\pi$
$948$	0	0
$949$	−12921.0	−0.441974
$950$	−1998.22	−0.0682428
$951$	0	0
$952$	−16596.7	−0.565022
$953$	−437.931	−0.0148856	−0.00744279	−	0.999972i	$-0.502369\pi$
$953$	−437.931	−0.0148856	−0.00744279	+	0.999972i	$0.502369\pi$
$954$	0	0
$955$	−25876.2	−0.876790
$956$	38566.2	1.30473
$957$	0	0
$958$	−11345.9	−0.382641
$959$	−11902.5	−0.400785
$960$	0	0
$961$	−759.861	−0.0255064
$962$	−43883.1	−1.47073
$963$	0	0
$964$	7977.02	0.266517
$965$	−17039.4	−0.568413
$966$	0	0
$967$	−37762.6	−1.25580	−0.627902	−	0.778292i	$-0.716086\pi$
$967$	−37762.6	−1.25580	−0.627902	+	0.778292i	$0.716086\pi$
$968$	24166.8	0.802429
$969$	0	0
$970$	−8468.38	−0.280313
$971$	−16153.0	−0.533858	−0.266929	−	0.963716i	$-0.586009\pi$
$971$	−16153.0	−0.533858	−0.266929	+	0.963716i	$0.586009\pi$
$972$	0	0
$973$	−21249.6	−0.700136
$974$	9680.07	0.318449
$975$	0	0
$976$	−22474.4	−0.737079
$977$	−33700.3	−1.10355	−0.551775	−	0.833993i	$-0.686049\pi$
$977$	−33700.3	−1.10355	−0.551775	+	0.833993i	$0.686049\pi$
$978$	0	0
$979$	127.332	0.00415683
$980$	5209.66	0.169813
$981$	0	0
$982$	−13025.8	−0.423290
$983$	7637.13	0.247799	0.123900	−	0.992295i	$-0.460460\pi$
$983$	7637.13	0.247799	0.123900	+	0.992295i	$0.460460\pi$
$984$	0	0
$985$	13311.2	0.430588
$986$	−2487.20	−0.0803331
$987$	0	0
$988$	34769.9	1.11961
$989$	57085.1	1.83539
$990$	0	0
$991$	9339.90	0.299386	0.149693	−	0.988733i	$-0.452171\pi$
$991$	9339.90	0.299386	0.149693	+	0.988733i	$0.452171\pi$
$992$	−30870.3	−0.988038
$993$	0	0
$994$	13611.8	0.434346
$995$	5620.61	0.179081
$996$	0	0
$997$	−11144.7	−0.354017	−0.177009	−	0.984209i	$-0.556642\pi$
$997$	−11144.7	−0.354017	−0.177009	+	0.984209i	$0.556642\pi$
$998$	−5533.39	−0.175508
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1305.4.a.m.1.5		7
3.2	odd	2		435.4.a.j.1.3	✓	7
15.14	odd	2		2175.4.a.m.1.5		7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
435.4.a.j.1.3	✓	7	3.2	odd	2
1305.4.a.m.1.5		7	1.1	even	1	trivial
2175.4.a.m.1.5		7	15.14	odd	2

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:	\( N \)	\(=\)	\( 1305 = 3^{2} \cdot 5 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1305.a (trivial)

\(f(q)\)	\(=\)	\(q+1.26184 q^{2} -6.40776 q^{4} +5.00000 q^{5} -13.4311 q^{7} -18.1803 q^{8} +6.30920 q^{10} +1.30869 q^{11} +85.6641 q^{13} -16.9479 q^{14} +28.3214 q^{16} -67.9685 q^{17} -63.3429 q^{19} -32.0388 q^{20} +1.65136 q^{22} +139.990 q^{23} +25.0000 q^{25} +108.094 q^{26} +86.0634 q^{28} +29.0000 q^{29} -170.385 q^{31} +181.179 q^{32} -85.7654 q^{34} -67.1557 q^{35} -405.970 q^{37} -79.9286 q^{38} -90.9015 q^{40} +447.863 q^{41} +407.779 q^{43} -8.38576 q^{44} +176.645 q^{46} +178.857 q^{47} -162.605 q^{49} +31.5460 q^{50} -548.915 q^{52} -93.3357 q^{53} +6.54344 q^{55} +244.182 q^{56} +36.5934 q^{58} -279.796 q^{59} -793.549 q^{61} -214.999 q^{62} +2.04805 q^{64} +428.321 q^{65} +460.471 q^{67} +435.526 q^{68} -84.7397 q^{70} -803.153 q^{71} -150.833 q^{73} -512.269 q^{74} +405.886 q^{76} -17.5772 q^{77} -313.814 q^{79} +141.607 q^{80} +565.132 q^{82} -250.514 q^{83} -339.842 q^{85} +514.552 q^{86} -23.7923 q^{88} +97.2972 q^{89} -1150.57 q^{91} -897.024 q^{92} +225.689 q^{94} -316.714 q^{95} -1342.23 q^{97} -205.181 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q + q^{2} + 15 q^{4} + 35 q^{5} - 37 q^{7} + 36 q^{8} + 5 q^{10} + 11 q^{11} - 133 q^{13} + 75 q^{14} - 53 q^{16} - 21 q^{17} - 170 q^{19} + 75 q^{20} - 369 q^{22} + 68 q^{23} + 175 q^{25} - 181 q^{26}+ \cdots - 1068 q^{98}+O(q^{100}) \) 7 * q + q^2 + 15 * q^4 + 35 * q^5 - 37 * q^7 + 36 * q^8 + 5 * q^10 + 11 * q^11 - 133 * q^13 + 75 * q^14 - 53 * q^16 - 21 * q^17 - 170 * q^19 + 75 * q^20 - 369 * q^22 + 68 * q^23 + 175 * q^25 - 181 * q^26 - 637 * q^28 + 203 * q^29 - 480 * q^31 + 779 * q^32 - 897 * q^34 - 185 * q^35 - 1032 * q^37 + 194 * q^38 + 180 * q^40 + 638 * q^41 - 512 * q^43 - 625 * q^44 + 16 * q^46 + 111 * q^47 + 178 * q^49 + 25 * q^50 - 1263 * q^52 - 410 * q^53 + 55 * q^55 - 1174 * q^56 + 29 * q^58 + 426 * q^59 - 1192 * q^61 - 460 * q^62 + 390 * q^64 - 665 * q^65 - 1671 * q^67 - 1509 * q^68 + 375 * q^70 + 1324 * q^71 - 852 * q^73 - 1780 * q^74 - 564 * q^76 + 2107 * q^77 + 366 * q^79 - 265 * q^80 - 318 * q^82 - 470 * q^83 - 105 * q^85 + 2196 * q^86 - 2518 * q^88 - 51 * q^89 - 1297 * q^91 + 684 * q^92 - 1837 * q^94 - 850 * q^95 - 3322 * q^97 - 1068 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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