LMFDB - Embedded newform 1859.4.a.l.1.19

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1859,4,Mod(1,1859)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1859, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("1859.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1859 = 11 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1859.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:	$109.684550701$
Analytic rank:	$1$
Dimension:	$36$
Twist minimal:	no (minimal twist has level 143)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.19
Character	$\chi$	$=$	1859.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+0.665052 q^{2} -3.21477 q^{3} -7.55771 q^{4} -10.5229 q^{5} -2.13799 q^{6} -28.9946 q^{7} -10.3467 q^{8} -16.6653 q^{9} -6.99830 q^{10} -11.0000 q^{11} +24.2963 q^{12} -19.2829 q^{14} +33.8288 q^{15} +53.5806 q^{16} -38.0338 q^{17} -11.0833 q^{18} -70.5930 q^{19} +79.5293 q^{20} +93.2109 q^{21} -7.31557 q^{22} +192.576 q^{23} +33.2622 q^{24} -14.2678 q^{25} +140.374 q^{27} +219.133 q^{28} +171.186 q^{29} +22.4979 q^{30} +168.441 q^{31} +118.407 q^{32} +35.3625 q^{33} -25.2944 q^{34} +305.108 q^{35} +125.951 q^{36} -245.943 q^{37} -46.9480 q^{38} +108.877 q^{40} -63.4575 q^{41} +61.9901 q^{42} +85.0515 q^{43} +83.1348 q^{44} +175.367 q^{45} +128.073 q^{46} +61.8204 q^{47} -172.249 q^{48} +497.686 q^{49} -9.48883 q^{50} +122.270 q^{51} -216.218 q^{53} +93.3558 q^{54} +115.752 q^{55} +299.998 q^{56} +226.940 q^{57} +113.848 q^{58} -650.371 q^{59} -255.668 q^{60} -380.057 q^{61} +112.022 q^{62} +483.202 q^{63} -349.898 q^{64} +23.5179 q^{66} -472.714 q^{67} +287.448 q^{68} -619.088 q^{69} +202.913 q^{70} +344.808 q^{71} +172.430 q^{72} +784.161 q^{73} -163.565 q^{74} +45.8677 q^{75} +533.521 q^{76} +318.940 q^{77} +408.156 q^{79} -563.825 q^{80} -1.30717 q^{81} -42.2025 q^{82} -1091.09 q^{83} -704.461 q^{84} +400.227 q^{85} +56.5636 q^{86} -550.325 q^{87} +113.813 q^{88} +662.474 q^{89} +116.628 q^{90} -1455.43 q^{92} -541.499 q^{93} +41.1138 q^{94} +742.845 q^{95} -380.652 q^{96} +1756.02 q^{97} +330.987 q^{98} +183.318 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 36 q - 4 q^{2} + 12 q^{3} + 152 q^{4} - 40 q^{5} - 98 q^{6} - 56 q^{7} - 84 q^{8} + 360 q^{9} - 56 q^{10} - 396 q^{11} + 66 q^{12} + 164 q^{14} - 120 q^{15} + 644 q^{16} + 138 q^{17} + 28 q^{18} - 498 q^{19}+ \cdots - 3960 q^{99}+O(q^{100}) $ 36 * q - 4 * q^2 + 12 * q^3 + 152 * q^4 - 40 * q^5 - 98 * q^6 - 56 * q^7 - 84 * q^8 + 360 * q^9 - 56 * q^10 - 396 * q^11 + 66 * q^12 + 164 * q^14 - 120 * q^15 + 644 * q^16 + 138 * q^17 + 28 * q^18 - 498 * q^19 - 320 * q^20 - 404 * q^21 + 44 * q^22 + 46 * q^23 - 1340 * q^24 + 818 * q^25 + 24 * q^27 - 1382 * q^28 + 262 * q^29 + 104 * q^30 - 660 * q^31 - 106 * q^32 - 132 * q^33 + 640 * q^34 + 68 * q^35 + 1960 * q^36 - 1372 * q^37 + 224 * q^38 + 108 * q^40 - 1216 * q^41 + 82 * q^42 + 436 * q^43 - 1672 * q^44 - 820 * q^45 - 2512 * q^46 + 140 * q^47 - 1944 * q^48 + 1192 * q^49 - 3484 * q^50 + 712 * q^51 + 146 * q^53 - 3078 * q^54 + 440 * q^55 - 102 * q^56 - 3656 * q^57 + 324 * q^58 - 404 * q^59 + 2944 * q^60 - 590 * q^61 + 1776 * q^62 - 4364 * q^63 + 2514 * q^64 + 1078 * q^66 - 2236 * q^67 - 1530 * q^68 + 12 * q^69 - 4852 * q^70 - 292 * q^71 - 2990 * q^72 - 1936 * q^73 + 1568 * q^74 + 4688 * q^75 - 7398 * q^76 + 616 * q^77 - 1464 * q^79 + 808 * q^80 + 3780 * q^81 - 4072 * q^82 - 4422 * q^83 + 2678 * q^84 - 3700 * q^85 + 2588 * q^86 - 1832 * q^87 + 924 * q^88 - 6856 * q^89 - 3422 * q^90 - 668 * q^92 - 840 * q^93 - 4166 * q^94 - 256 * q^95 - 18290 * q^96 - 2776 * q^97 - 4608 * q^98 - 3960 * q^99

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 $

$2$	0.665052	0.235131	0.117566	−	0.993065i	$-0.462491\pi$
$2$	0.665052	0.235131	0.117566	+	0.993065i	$0.462491\pi$
$3$	−3.21477	−0.618683	−0.309341	−	0.950951i	$-0.600109\pi$
$3$	−3.21477	−0.618683	−0.309341	+	0.950951i	$0.600109\pi$
$4$	−7.55771	−0.944713
$5$	−10.5229	−0.941200	−0.470600	−	0.882347i	$-0.655963\pi$
$5$	−10.5229	−0.941200	−0.470600	+	0.882347i	$0.655963\pi$
$6$	−2.13799	−0.145472
$7$	−28.9946	−1.56556	−0.782780	−	0.622298i	$-0.786199\pi$
$7$	−28.9946	−1.56556	−0.782780	+	0.622298i	$0.786199\pi$
$8$	−10.3467	−0.457263
$9$	−16.6653	−0.617232
$10$	−6.99830	−0.221306
$11$	−11.0000	−0.301511
$11$	−11.0000	−0.301511
$12$	24.2963	0.584478
$13$	0	0
$13$	0	0
$14$	−19.2829	−0.368112
$15$	33.8288	0.582304
$16$	53.5806	0.837196
$17$	−38.0338	−0.542620	−0.271310	−	0.962492i	$-0.587457\pi$
$17$	−38.0338	−0.542620	−0.271310	+	0.962492i	$0.587457\pi$
$18$	−11.0833	−0.145131
$19$	−70.5930	−0.852376	−0.426188	−	0.904635i	$-0.640144\pi$
$19$	−70.5930	−0.852376	−0.426188	+	0.904635i	$0.640144\pi$
$20$	79.5293	0.889164
$21$	93.2109	0.968585
$22$	−7.31557	−0.0708948
$23$	192.576	1.74587	0.872933	−	0.487840i	$-0.162215\pi$
$23$	192.576	1.74587	0.872933	+	0.487840i	$0.162215\pi$
$24$	33.2622	0.282901
$25$	−14.2678	−0.114142
$26$	0	0
$27$	140.374	1.00055
$28$	219.133	1.47901
$29$	171.186	1.09616	0.548078	−	0.836427i	$-0.315360\pi$
$29$	171.186	1.09616	0.548078	+	0.836427i	$0.315360\pi$
$30$	22.4979	0.136918
$31$	168.441	0.975900	0.487950	−	0.872872i	$-0.337745\pi$
$31$	168.441	0.975900	0.487950	+	0.872872i	$0.337745\pi$
$32$	118.407	0.654114
$33$	35.3625	0.186540
$34$	−25.2944	−0.127587
$35$	305.108	1.47351
$36$	125.951	0.583107
$37$	−245.943	−1.09278	−0.546389	−	0.837532i	$-0.683998\pi$
$37$	−245.943	−1.09278	−0.546389	+	0.837532i	$0.683998\pi$
$38$	−46.9480	−0.200420
$39$	0	0
$40$	108.877	0.430376
$41$	−63.4575	−0.241717	−0.120858	−	0.992670i	$-0.538565\pi$
$41$	−63.4575	−0.241717	−0.120858	+	0.992670i	$0.538565\pi$
$42$	61.9901	0.227745
$43$	85.0515	0.301633	0.150817	−	0.988562i	$-0.451810\pi$
$43$	85.0515	0.301633	0.150817	+	0.988562i	$0.451810\pi$
$44$	83.1348	0.284842
$45$	175.367	0.580939
$46$	128.073	0.410508
$47$	61.8204	0.191860	0.0959301	−	0.995388i	$-0.469417\pi$
$47$	61.8204	0.191860	0.0959301	+	0.995388i	$0.469417\pi$
$48$	−172.249	−0.517959
$49$	497.686	1.45098
$50$	−9.48883	−0.0268385
$51$	122.270	0.335710
$52$	0	0
$53$	−216.218	−0.560373	−0.280187	−	0.959946i	$-0.590396\pi$
$53$	−216.218	−0.560373	−0.280187	+	0.959946i	$0.590396\pi$
$54$	93.3558	0.235261
$55$	115.752	0.283782
$56$	299.998	0.715873
$57$	226.940	0.527350
$58$	113.848	0.257740
$59$	−650.371	−1.43510	−0.717551	−	0.696506i	$-0.754737\pi$
$59$	−650.371	−1.43510	−0.717551	+	0.696506i	$0.754737\pi$
$60$	−255.668	−0.550110
$61$	−380.057	−0.797726	−0.398863	−	0.917011i	$-0.630595\pi$
$61$	−380.057	−0.797726	−0.398863	+	0.917011i	$0.630595\pi$
$62$	112.022	0.229465
$63$	483.202	0.966314
$64$	−349.898	−0.683394
$65$	0	0
$66$	23.5179	0.0438614
$67$	−472.714	−0.861958	−0.430979	−	0.902362i	$-0.641832\pi$
$67$	−472.714	−0.861958	−0.430979	+	0.902362i	$0.641832\pi$
$68$	287.448	0.512621
$69$	−619.088	−1.08014
$70$	202.913	0.346467
$71$	344.808	0.576355	0.288177	−	0.957577i	$-0.406951\pi$
$71$	344.808	0.576355	0.288177	+	0.957577i	$0.406951\pi$
$72$	172.430	0.282237
$73$	784.161	1.25725	0.628624	−	0.777709i	$-0.283618\pi$
$73$	784.161	1.25725	0.628624	+	0.777709i	$0.283618\pi$
$74$	−163.565	−0.256946
$75$	45.8677	0.0706180
$76$	533.521	0.805251
$77$	318.940	0.472034
$78$	0	0
$79$	408.156	0.581280	0.290640	−	0.956832i	$-0.406132\pi$
$79$	408.156	0.581280	0.290640	+	0.956832i	$0.406132\pi$
$80$	−563.825	−0.787969
$81$	−1.30717	−0.00179310
$82$	−42.2025	−0.0568352
$83$	−1091.09	−1.44292	−0.721460	−	0.692456i	$-0.756529\pi$
$83$	−1091.09	−1.44292	−0.721460	+	0.692456i	$0.756529\pi$
$84$	−704.461	−0.915035
$85$	400.227	0.510714
$86$	56.5636	0.0709234
$87$	−550.325	−0.678172
$88$	113.813	0.137870
$89$	662.474	0.789013	0.394506	−	0.918893i	$-0.370916\pi$
$89$	662.474	0.789013	0.394506	+	0.918893i	$0.370916\pi$
$90$	116.628	0.136597
$91$	0	0
$92$	−1455.43	−1.64934
$93$	−541.499	−0.603772
$94$	41.1138	0.0451124
$95$	742.845	0.802256
$96$	−380.652	−0.404689
$97$	1756.02	1.83812	0.919058	−	0.394122i	$-0.128951\pi$
$97$	1756.02	1.83812	0.919058	+	0.394122i	$0.128951\pi$
$98$	330.987	0.341171
$99$	183.318	0.186102
$100$	107.832	0.107832
$101$	49.9908	0.0492502	0.0246251	−	0.999697i	$-0.492161\pi$
$101$	49.9908	0.0492502	0.0246251	+	0.999697i	$0.492161\pi$
$102$	81.3158	0.0789359
$103$	1388.01	1.32782	0.663908	−	0.747814i	$-0.268897\pi$
$103$	1388.01	1.32782	0.663908	+	0.747814i	$0.268897\pi$
$104$	0	0
$105$	−980.853	−0.911632
$106$	−143.796	−0.131761
$107$	−1298.08	−1.17281	−0.586404	−	0.810018i	$-0.699457\pi$
$107$	−1298.08	−1.17281	−0.586404	+	0.810018i	$0.699457\pi$
$108$	−1060.90	−0.945236
$109$	1113.79	0.978731	0.489365	−	0.872079i	$-0.337228\pi$
$109$	1113.79	0.978731	0.489365	+	0.872079i	$0.337228\pi$
$110$	76.9813	0.0667262
$111$	790.650	0.676082
$112$	−1553.55	−1.31068
$113$	371.850	0.309564	0.154782	−	0.987949i	$-0.450533\pi$
$113$	371.850	0.309564	0.154782	+	0.987949i	$0.450533\pi$
$114$	150.927	0.123997
$115$	−2026.47	−1.64321
$116$	−1293.78	−1.03555
$117$	0	0
$118$	−432.530	−0.337438
$119$	1102.77	0.849505
$120$	−350.016	−0.266266
$121$	121.000	0.0909091
$122$	−252.757	−0.187570
$123$	204.001	0.149546
$124$	−1273.03	−0.921946
$125$	1465.51	1.04863
$126$	321.355	0.227211
$127$	−15.9066	−0.0111140	−0.00555701	−	0.999985i	$-0.501769\pi$
$127$	−15.9066	−0.0111140	−0.00555701	+	0.999985i	$0.501769\pi$
$128$	−1179.96	−0.814801
$129$	−273.421	−0.186615
$130$	0	0
$131$	−1358.53	−0.906073	−0.453037	−	0.891492i	$-0.649659\pi$
$131$	−1358.53	−0.906073	−0.453037	+	0.891492i	$0.649659\pi$
$132$	−267.259	−0.176227
$133$	2046.81	1.33445
$134$	−314.379	−0.202673
$135$	−1477.14	−0.941721
$136$	393.523	0.248120
$137$	−700.027	−0.436550	−0.218275	−	0.975887i	$-0.570043\pi$
$137$	−700.027	−0.436550	−0.218275	+	0.975887i	$0.570043\pi$
$138$	−411.726	−0.253974
$139$	−2202.86	−1.34421	−0.672103	−	0.740458i	$-0.734609\pi$
$139$	−2202.86	−1.34421	−0.672103	+	0.740458i	$0.734609\pi$
$140$	−2305.92	−1.39204
$141$	−198.738	−0.118701
$142$	229.315	0.135519
$143$	0	0
$144$	−892.934	−0.516744
$145$	−1801.38	−1.03170
$146$	521.508	0.295618
$147$	−1599.95	−0.897696
$148$	1858.76	1.03236
$149$	1691.70	0.930132	0.465066	−	0.885276i	$-0.346031\pi$
$149$	1691.70	0.930132	0.465066	+	0.885276i	$0.346031\pi$
$150$	30.5044	0.0166045
$151$	−2987.30	−1.60996	−0.804978	−	0.593305i	$-0.797823\pi$
$151$	−2987.30	−1.60996	−0.804978	+	0.593305i	$0.797823\pi$
$152$	730.403	0.389760
$153$	633.843	0.334923
$154$	212.112	0.110990
$155$	−1772.49	−0.918517
$156$	0	0
$157$	−2277.76	−1.15787	−0.578934	−	0.815374i	$-0.696531\pi$
$157$	−2277.76	−1.15787	−0.578934	+	0.815374i	$0.696531\pi$
$158$	271.445	0.136677
$159$	695.090	0.346693
$160$	−1245.99	−0.615652
$161$	−5583.67	−2.73326
$162$	−0.869334	−0.000421613 0
$163$	683.066	0.328232	0.164116	−	0.986441i	$-0.447523\pi$
$163$	683.066	0.328232	0.164116	+	0.986441i	$0.447523\pi$
$164$	479.593	0.228353
$165$	−372.117	−0.175571
$166$	−725.630	−0.339276
$167$	163.801	0.0759000	0.0379500	−	0.999280i	$-0.487917\pi$
$167$	163.801	0.0759000	0.0379500	+	0.999280i	$0.487917\pi$
$168$	−964.424	−0.442898
$169$	0	0
$170$	266.172	0.120085
$171$	1176.45	0.526113
$172$	−642.794	−0.284957
$173$	2101.58	0.923584	0.461792	−	0.886988i	$-0.347207\pi$
$173$	2101.58	0.923584	0.461792	+	0.886988i	$0.347207\pi$
$174$	−365.994	−0.159460
$175$	413.689	0.178697
$176$	−589.386	−0.252424
$177$	2090.79	0.887873
$178$	440.580	0.185522
$179$	−148.591	−0.0620457	−0.0310228	−	0.999519i	$-0.509876\pi$
$179$	−148.591	−0.0620457	−0.0310228	+	0.999519i	$0.509876\pi$
$180$	−1325.38	−0.548820
$181$	2458.73	1.00970	0.504851	−	0.863206i	$-0.331547\pi$
$181$	2458.73	1.00970	0.504851	+	0.863206i	$0.331547\pi$
$182$	0	0
$183$	1221.79	0.493539
$184$	−1992.53	−0.798320
$185$	2588.04	1.02852
$186$	−360.125	−0.141966
$187$	418.372	0.163606
$188$	−467.220	−0.181253
$189$	−4070.08	−1.56643
$190$	494.031	0.188636
$191$	4516.70	1.71108	0.855541	−	0.517735i	$-0.173225\pi$
$191$	4516.70	1.71108	0.855541	+	0.517735i	$0.173225\pi$
$192$	1124.84	0.422804
$193$	1204.55	0.449252	0.224626	−	0.974445i	$-0.427884\pi$
$193$	1204.55	0.449252	0.224626	+	0.974445i	$0.427884\pi$
$194$	1167.85	0.432199
$195$	0	0
$196$	−3761.37	−1.37076
$197$	324.260	0.117272	0.0586360	−	0.998279i	$-0.481325\pi$
$197$	324.260	0.117272	0.0586360	+	0.998279i	$0.481325\pi$
$198$	121.916	0.0437585
$199$	4350.55	1.54976	0.774880	−	0.632109i	$-0.217810\pi$
$199$	4350.55	1.54976	0.774880	+	0.632109i	$0.217810\pi$
$200$	147.625	0.0521931
$201$	1519.67	0.533279
$202$	33.2465	0.0115803
$203$	−4963.48	−1.71610
$204$	−924.080	−0.317150
$205$	667.759	0.227504
$206$	923.101	0.312211
$207$	−3209.33	−1.07760
$208$	0	0
$209$	776.523	0.257001
$210$	−652.318	−0.214353
$211$	3980.03	1.29856	0.649281	−	0.760548i	$-0.275070\pi$
$211$	3980.03	1.29856	0.649281	+	0.760548i	$0.275070\pi$
$212$	1634.11	0.529392
$213$	−1108.48	−0.356581
$214$	−863.293	−0.275764
$215$	−894.991	−0.283897
$216$	−1452.40	−0.457516
$217$	−4883.88	−1.52783
$218$	740.728	0.230130
$219$	−2520.90	−0.777838
$220$	−874.822	−0.268093
$221$	0	0
$222$	525.823	0.158968
$223$	−1831.81	−0.550076	−0.275038	−	0.961433i	$-0.588690\pi$
$223$	−1831.81	−0.550076	−0.275038	+	0.961433i	$0.588690\pi$
$224$	−3433.17	−1.02406
$225$	237.777	0.0704524
$226$	247.299	0.0727881
$227$	−3554.05	−1.03916	−0.519582	−	0.854421i	$-0.673912\pi$
$227$	−3554.05	−1.03916	−0.519582	+	0.854421i	$0.673912\pi$
$228$	−1715.15	−0.498195
$229$	525.734	0.151709	0.0758547	−	0.997119i	$-0.475831\pi$
$229$	525.734	0.151709	0.0758547	+	0.997119i	$0.475831\pi$
$230$	−1347.71	−0.386370
$231$	−1025.32	−0.292039
$232$	−1771.21	−0.501231
$233$	45.3279	0.0127448	0.00637238	−	0.999980i	$-0.497972\pi$
$233$	45.3279	0.0127448	0.00637238	+	0.999980i	$0.497972\pi$
$234$	0	0
$235$	−650.532	−0.180579
$236$	4915.31	1.35576
$237$	−1312.13	−0.359628
$238$	733.402	0.199745
$239$	2128.63	0.576107	0.288053	−	0.957614i	$-0.406992\pi$
$239$	2128.63	0.576107	0.288053	+	0.957614i	$0.406992\pi$
$240$	1812.57	0.487503
$241$	−5086.06	−1.35943	−0.679713	−	0.733478i	$-0.737896\pi$
$241$	−5086.06	−1.35943	−0.679713	+	0.733478i	$0.737896\pi$
$242$	80.4713	0.0213756
$243$	−3785.89	−0.999444
$244$	2872.36	0.753622
$245$	−5237.12	−1.36566
$246$	135.671	0.0351630
$247$	0	0
$248$	−1742.81	−0.446243
$249$	3507.60	0.892710
$250$	974.638	0.246566
$251$	−1412.09	−0.355102	−0.177551	−	0.984112i	$-0.556817\pi$
$251$	−1412.09	−0.355102	−0.177551	+	0.984112i	$0.556817\pi$
$252$	−3651.90	−0.912890
$253$	−2118.34	−0.526399
$254$	−10.5787	−0.00261325
$255$	−1286.64	−0.315970
$256$	2014.45	0.491808
$257$	7034.93	1.70750	0.853749	−	0.520685i	$-0.174324\pi$
$257$	7034.93	1.70750	0.853749	+	0.520685i	$0.174324\pi$
$258$	−181.839	−0.0438791
$259$	7131.01	1.71081
$260$	0	0
$261$	−2852.86	−0.676582
$262$	−903.495	−0.213046
$263$	6518.88	1.52841	0.764204	−	0.644975i	$-0.223132\pi$
$263$	6518.88	1.52841	0.764204	+	0.644975i	$0.223132\pi$
$264$	−365.884	−0.0852978
$265$	2275.25	0.527423
$266$	1361.24	0.313770
$267$	−2129.70	−0.488148
$268$	3572.63	0.814303
$269$	−5866.44	−1.32968	−0.664839	−	0.746987i	$-0.731500\pi$
$269$	−5866.44	−1.32968	−0.664839	+	0.746987i	$0.731500\pi$
$270$	−982.377	−0.221428
$271$	−7223.83	−1.61925	−0.809624	−	0.586948i	$-0.800329\pi$
$271$	−7223.83	−1.61925	−0.809624	+	0.586948i	$0.800329\pi$
$272$	−2037.87	−0.454280
$273$	0	0
$274$	−465.554	−0.102647
$275$	156.946	0.0344153
$276$	4678.89	1.02042
$277$	5574.12	1.20908	0.604542	−	0.796573i	$-0.293356\pi$
$277$	5574.12	1.20908	0.604542	+	0.796573i	$0.293356\pi$
$278$	−1465.02	−0.316065
$279$	−2807.11	−0.602357
$280$	−3156.86	−0.673780
$281$	−83.0572	−0.0176326	−0.00881632	−	0.999961i	$-0.502806\pi$
$281$	−83.0572	−0.0176326	−0.00881632	+	0.999961i	$0.502806\pi$
$282$	−132.171	−0.0279102
$283$	4450.96	0.934920	0.467460	−	0.884014i	$-0.345169\pi$
$283$	4450.96	0.934920	0.467460	+	0.884014i	$0.345169\pi$
$284$	−2605.96	−0.544490
$285$	−2388.08	−0.496342
$286$	0	0
$287$	1839.92	0.378422
$288$	−1973.29	−0.403740
$289$	−3466.43	−0.705563
$290$	−1198.01	−0.242585
$291$	−5645.21	−1.13721
$292$	−5926.46	−1.18774
$293$	749.166	0.149374	0.0746872	−	0.997207i	$-0.476204\pi$
$293$	749.166	0.149374	0.0746872	+	0.997207i	$0.476204\pi$
$294$	−1064.05	−0.211077
$295$	6843.81	1.35072
$296$	2544.69	0.499687
$297$	−1544.11	−0.301678
$298$	1125.07	0.218703
$299$	0	0
$300$	−346.655	−0.0667137
$301$	−2466.03	−0.472225
$302$	−1986.71	−0.378551
$303$	−160.709	−0.0304703
$304$	−3782.41	−0.713606
$305$	3999.31	0.750820
$306$	421.538	0.0787508
$307$	−6250.03	−1.16192	−0.580958	−	0.813934i	$-0.697322\pi$
$307$	−6250.03	−1.16192	−0.580958	+	0.813934i	$0.697322\pi$
$308$	−2410.46	−0.445937
$309$	−4462.15	−0.821497
$310$	−1178.80	−0.215972
$311$	4650.16	0.847866	0.423933	−	0.905694i	$-0.360649\pi$
$311$	4650.16	0.847866	0.423933	+	0.905694i	$0.360649\pi$
$312$	0	0
$313$	−2500.00	−0.451465	−0.225732	−	0.974189i	$-0.572477\pi$
$313$	−2500.00	−0.451465	−0.225732	+	0.974189i	$0.572477\pi$
$314$	−1514.83	−0.272251
$315$	−5084.71	−0.909495
$316$	−3084.72	−0.549143
$317$	−9258.35	−1.64038	−0.820190	−	0.572091i	$-0.806132\pi$
$317$	−9258.35	−1.64038	−0.820190	+	0.572091i	$0.806132\pi$
$318$	462.271	0.0815184
$319$	−1883.05	−0.330503
$320$	3681.95	0.643210
$321$	4173.04	0.725596
$322$	−3713.43	−0.642675
$323$	2684.92	0.462517
$324$	9.87918	0.00169396
$325$	0	0
$326$	454.274	0.0771776
$327$	−3580.58	−0.605524
$328$	656.574	0.110528
$329$	−1792.46	−0.300369
$330$	−247.477	−0.0412823
$331$	5539.51	0.919876	0.459938	−	0.887951i	$-0.347872\pi$
$331$	5539.51	0.919876	0.459938	+	0.887951i	$0.347872\pi$
$332$	8246.12	1.36315
$333$	4098.70	0.674497
$334$	108.936	0.0178465
$335$	4974.34	0.811275
$336$	4994.29	0.810896
$337$	8246.13	1.33292	0.666462	−	0.745539i	$-0.267808\pi$
$337$	8246.13	1.33292	0.666462	+	0.745539i	$0.267808\pi$
$338$	0	0
$339$	−1195.41	−0.191522
$340$	−3024.80	−0.482479
$341$	−1852.85	−0.294245
$342$	782.400	0.123706
$343$	−4485.06	−0.706037
$344$	−880.000	−0.137926
$345$	6514.63	1.01663
$346$	1397.66	0.217164
$347$	−6695.09	−1.03577	−0.517883	−	0.855451i	$-0.673280\pi$
$347$	−6695.09	−1.03577	−0.517883	+	0.855451i	$0.673280\pi$
$348$	4159.19	0.640678
$349$	4875.65	0.747815	0.373908	−	0.927466i	$-0.378018\pi$
$349$	4875.65	0.747815	0.373908	+	0.927466i	$0.378018\pi$
$350$	275.125	0.0420173
$351$	0	0
$352$	−1302.48	−0.197223
$353$	−12313.3	−1.85658	−0.928290	−	0.371858i	$-0.878721\pi$
$353$	−12313.3	−1.85658	−0.928290	+	0.371858i	$0.878721\pi$
$354$	1390.48	0.208767
$355$	−3628.39	−0.542465
$356$	−5006.79	−0.745391
$357$	−3545.16	−0.525574
$358$	−98.8204	−0.0145889
$359$	−12095.7	−1.77824	−0.889118	−	0.457678i	$-0.848681\pi$
$359$	−12095.7	−1.77824	−0.889118	+	0.457678i	$0.848681\pi$
$360$	−1814.47	−0.265642
$361$	−1875.63	−0.273456
$362$	1635.18	0.237413
$363$	−388.987	−0.0562439
$364$	0	0
$365$	−8251.68	−1.18332
$366$	812.557	0.116047
$367$	8637.03	1.22847	0.614236	−	0.789123i	$-0.289464\pi$
$367$	8637.03	1.22847	0.614236	+	0.789123i	$0.289464\pi$
$368$	10318.3	1.46163
$369$	1057.54	0.149195
$370$	1721.18	0.241838
$371$	6269.14	0.877298
$372$	4092.49	0.570392
$373$	−13828.2	−1.91957	−0.959784	−	0.280740i	$-0.909420\pi$
$373$	−13828.2	−1.91957	−0.959784	+	0.280740i	$0.909420\pi$
$374$	278.239	0.0384689
$375$	−4711.26	−0.648770
$376$	−639.636	−0.0877306
$377$	0	0
$378$	−2706.81	−0.368316
$379$	11112.3	1.50607	0.753035	−	0.657980i	$-0.228589\pi$
$379$	11112.3	1.50607	0.753035	+	0.657980i	$0.228589\pi$
$380$	−5614.21	−0.757902
$381$	51.1359	0.00687605
$382$	3003.84	0.402329
$383$	12211.8	1.62922	0.814611	−	0.580008i	$-0.196951\pi$
$383$	12211.8	1.62922	0.814611	+	0.580008i	$0.196951\pi$
$384$	3793.29	0.504103
$385$	−3356.19	−0.444279
$386$	801.090	0.105633
$387$	−1417.40	−0.186178
$388$	−13271.5	−1.73649
$389$	−6803.65	−0.886783	−0.443392	−	0.896328i	$-0.646225\pi$
$389$	−6803.65	−0.886783	−0.443392	+	0.896328i	$0.646225\pi$
$390$	0	0
$391$	−7324.41	−0.947343
$392$	−5149.40	−0.663480
$393$	4367.37	0.560572
$394$	215.650	0.0275743
$395$	−4295.00	−0.547101
$396$	−1385.46	−0.175813
$397$	−2825.48	−0.357196	−0.178598	−	0.983922i	$-0.557156\pi$
$397$	−2825.48	−0.357196	−0.178598	+	0.983922i	$0.557156\pi$
$398$	2893.34	0.364397
$399$	−6580.04	−0.825599
$400$	−764.478	−0.0955597
$401$	13855.8	1.72550	0.862752	−	0.505627i	$-0.168739\pi$
$401$	13855.8	1.72550	0.862752	+	0.505627i	$0.168739\pi$
$402$	1010.66	0.125390
$403$	0	0
$404$	−377.816	−0.0465273
$405$	13.7552	0.00168766
$406$	−3300.97	−0.403508
$407$	2705.37	0.329485
$408$	−1265.09	−0.153508
$409$	8908.07	1.07696	0.538479	−	0.842639i	$-0.318999\pi$
$409$	8908.07	1.07696	0.538479	+	0.842639i	$0.318999\pi$
$410$	444.094	0.0534933
$411$	2250.42	0.270086
$412$	−10490.2	−1.25441
$413$	18857.2	2.24674
$414$	−2134.37	−0.253379
$415$	11481.4	1.35808
$416$	0	0
$417$	7081.70	0.831637
$418$	516.428	0.0604290
$419$	209.022	0.0243708	0.0121854	−	0.999926i	$-0.496121\pi$
$419$	209.022	0.0243708	0.0121854	+	0.999926i	$0.496121\pi$
$420$	7413.00	0.861231
$421$	6752.89	0.781747	0.390874	−	0.920444i	$-0.372173\pi$
$421$	6752.89	0.781747	0.390874	+	0.920444i	$0.372173\pi$
$422$	2646.93	0.305333
$423$	−1030.25	−0.118422
$424$	2237.14	0.256238
$425$	542.659	0.0619361
$426$	−737.196	−0.0838433
$427$	11019.6	1.24889
$428$	9810.54	1.10797
$429$	0	0
$430$	−595.216	−0.0667531
$431$	680.312	0.0760312	0.0380156	−	0.999277i	$-0.487896\pi$
$431$	680.312	0.0760312	0.0380156	+	0.999277i	$0.487896\pi$
$432$	7521.30	0.837660
$433$	−11282.8	−1.25223	−0.626116	−	0.779730i	$-0.715357\pi$
$433$	−11282.8	−1.25223	−0.626116	+	0.779730i	$0.715357\pi$
$434$	−3248.03	−0.359241
$435$	5791.03	0.638296
$436$	−8417.69	−0.924620
$437$	−13594.5	−1.48813
$438$	−1676.53	−0.182894
$439$	4730.36	0.514278	0.257139	−	0.966374i	$-0.417220\pi$
$439$	4730.36	0.514278	0.257139	+	0.966374i	$0.417220\pi$
$440$	−1197.65	−0.129763
$441$	−8294.07	−0.895591
$442$	0	0
$443$	−669.977	−0.0718546	−0.0359273	−	0.999354i	$-0.511438\pi$
$443$	−669.977	−0.0718546	−0.0359273	+	0.999354i	$0.511438\pi$
$444$	−5975.50	−0.638704
$445$	−6971.17	−0.742619
$446$	−1218.25	−0.129340
$447$	−5438.43	−0.575457
$448$	10145.1	1.06989
$449$	−1629.51	−0.171272	−0.0856361	−	0.996326i	$-0.527292\pi$
$449$	−1629.51	−0.171272	−0.0856361	+	0.996326i	$0.527292\pi$
$450$	158.134	0.0165656
$451$	698.032	0.0728804
$452$	−2810.33	−0.292449
$453$	9603.49	0.996052
$454$	−2363.62	−0.244340
$455$	0	0
$456$	−2348.08	−0.241138
$457$	−11520.8	−1.17925	−0.589626	−	0.807676i	$-0.700725\pi$
$457$	−11520.8	−1.17925	−0.589626	+	0.807676i	$0.700725\pi$
$458$	349.640	0.0356716
$459$	−5338.94	−0.542921
$460$	15315.4	1.55236
$461$	−8264.84	−0.834994	−0.417497	−	0.908678i	$-0.637093\pi$
$461$	−8264.84	−0.834994	−0.417497	+	0.908678i	$0.637093\pi$
$462$	−681.891	−0.0686676
$463$	8472.07	0.850390	0.425195	−	0.905102i	$-0.360206\pi$
$463$	8472.07	0.850390	0.425195	+	0.905102i	$0.360206\pi$
$464$	9172.26	0.917697
$465$	5698.16	0.568271
$466$	30.1454	0.00299669
$467$	6897.41	0.683456	0.341728	−	0.939799i	$-0.388988\pi$
$467$	6897.41	0.683456	0.341728	+	0.939799i	$0.388988\pi$
$468$	0	0
$469$	13706.1	1.34945
$470$	−432.638	−0.0424598
$471$	7322.48	0.716353
$472$	6729.18	0.656219
$473$	−935.566	−0.0909458
$474$	−872.633	−0.0845598
$475$	1007.21	0.0972923
$476$	−8334.44	−0.802539
$477$	3603.32	0.345880
$478$	1415.65	0.135461
$479$	−9692.29	−0.924534	−0.462267	−	0.886741i	$-0.652964\pi$
$479$	−9692.29	−0.924534	−0.462267	+	0.886741i	$0.652964\pi$
$480$	4005.58	0.380893
$481$	0	0
$482$	−3382.49	−0.319644
$483$	17950.2	1.69102
$484$	−914.482	−0.0858830
$485$	−18478.5	−1.73004
$486$	−2517.81	−0.235001
$487$	4077.43	0.379397	0.189698	−	0.981842i	$-0.439249\pi$
$487$	4077.43	0.379397	0.189698	+	0.981842i	$0.439249\pi$
$488$	3932.33	0.364771
$489$	−2195.90	−0.203071
$490$	−3482.96	−0.321110
$491$	−20477.0	−1.88211	−0.941054	−	0.338256i	$-0.890163\pi$
$491$	−20477.0	−1.88211	−0.941054	+	0.338256i	$0.890163\pi$
$492$	−1541.78	−0.141278
$493$	−6510.87	−0.594796
$494$	0	0
$495$	−1929.04	−0.175160
$496$	9025.17	0.817020
$497$	−9997.57	−0.902318
$498$	2332.73	0.209904
$499$	11763.9	1.05536	0.527679	−	0.849444i	$-0.323062\pi$
$499$	11763.9	1.05536	0.527679	+	0.849444i	$0.323062\pi$
$500$	−11075.9	−0.990656
$501$	−526.582	−0.0469580
$502$	−939.115	−0.0834955
$503$	22483.6	1.99303	0.996515	−	0.0834119i	$-0.0265817\pi$
$503$	22483.6	1.99303	0.996515	+	0.0834119i	$0.0265817\pi$
$504$	−4999.54	−0.441860
$505$	−526.050	−0.0463543
$506$	−1408.81	−0.123773
$507$	0	0
$508$	120.217	0.0104996
$509$	4666.52	0.406365	0.203183	−	0.979141i	$-0.434871\pi$
$509$	4666.52	0.406365	0.203183	+	0.979141i	$0.434871\pi$
$510$	−855.681	−0.0742945
$511$	−22736.4	−1.96830
$512$	10779.4	0.930441
$513$	−9909.40	−0.852847
$514$	4678.59	0.401486
$515$	−14606.0	−1.24974
$516$	2066.43	0.176298
$517$	−680.024	−0.0578481
$518$	4742.49	0.402265
$519$	−6756.09	−0.571405
$520$	0	0
$521$	−18591.4	−1.56335	−0.781674	−	0.623687i	$-0.785634\pi$
$521$	−18591.4	−1.56335	−0.781674	+	0.623687i	$0.785634\pi$
$522$	−1897.30	−0.159086
$523$	−5962.75	−0.498534	−0.249267	−	0.968435i	$-0.580190\pi$
$523$	−5962.75	−0.498534	−0.249267	+	0.968435i	$0.580190\pi$
$524$	10267.4	0.855979
$525$	−1329.92	−0.110557
$526$	4335.39	0.359376
$527$	−6406.45	−0.529543
$528$	1894.74	0.156170
$529$	24918.6	2.04805
$530$	1513.16	0.124014
$531$	10838.6	0.885791
$532$	−15469.2	−1.26067
$533$	0	0
$534$	−1416.36	−0.114779
$535$	13659.7	1.10385
$536$	4891.02	0.394142
$537$	477.684	0.0383866
$538$	−3901.49	−0.312649
$539$	−5474.55	−0.437487
$540$	11163.8	0.889656
$541$	10281.4	0.817068	0.408534	−	0.912743i	$-0.366040\pi$
$541$	10281.4	0.817068	0.408534	+	0.912743i	$0.366040\pi$
$542$	−4804.22	−0.380736
$543$	−7904.26	−0.624685
$544$	−4503.48	−0.354936
$545$	−11720.3	−0.921181
$546$	0	0
$547$	−2031.38	−0.158785	−0.0793926	−	0.996843i	$-0.525298\pi$
$547$	−2031.38	−0.158785	−0.0793926	+	0.996843i	$0.525298\pi$
$548$	5290.60	0.412414
$549$	6333.74	0.492382
$550$	104.377	0.00809211
$551$	−12084.6	−0.934336
$552$	6405.51	0.493907
$553$	−11834.3	−0.910029
$554$	3707.08	0.284294
$555$	−8319.96	−0.636329
$556$	16648.6	1.26989
$557$	−10240.2	−0.778979	−0.389489	−	0.921031i	$-0.627349\pi$
$557$	−10240.2	−0.778979	−0.389489	+	0.921031i	$0.627349\pi$
$558$	−1866.88	−0.141633
$559$	0	0
$560$	16347.9	1.23361
$561$	−1344.97	−0.101220
$562$	−55.2373	−0.00414599
$563$	19050.3	1.42606	0.713031	−	0.701132i	$-0.247322\pi$
$563$	19050.3	1.42606	0.713031	+	0.701132i	$0.247322\pi$
$564$	1502.01	0.112138
$565$	−3912.95	−0.291361
$566$	2960.12	0.219829
$567$	37.9008	0.00280720
$568$	−3567.62	−0.263546
$569$	6885.00	0.507266	0.253633	−	0.967301i	$-0.418374\pi$
$569$	6885.00	0.507266	0.253633	+	0.967301i	$0.418374\pi$
$570$	−1588.19	−0.116706
$571$	26902.4	1.97168	0.985839	−	0.167692i	$-0.0536313\pi$
$571$	26902.4	1.97168	0.985839	+	0.167692i	$0.0536313\pi$
$572$	0	0
$573$	−14520.1	−1.05862
$574$	1223.64	0.0889790
$575$	−2747.64	−0.199278
$576$	5831.13	0.421812
$577$	−2399.20	−0.173102	−0.0865512	−	0.996247i	$-0.527585\pi$
$577$	−2399.20	−0.173102	−0.0865512	+	0.996247i	$0.527585\pi$
$578$	−2305.36	−0.165900
$579$	−3872.36	−0.277944
$580$	13614.3	0.974662
$581$	31635.6	2.25898
$582$	−3754.36	−0.267394
$583$	2378.39	0.168959
$584$	−8113.47	−0.574893
$585$	0	0
$586$	498.234	0.0351226
$587$	−15868.7	−1.11579	−0.557897	−	0.829910i	$-0.688391\pi$
$587$	−15868.7	−1.11579	−0.557897	+	0.829910i	$0.688391\pi$
$588$	12091.9	0.848066
$589$	−11890.8	−0.831834
$590$	4551.49	0.317596
$591$	−1042.42	−0.0725542
$592$	−13177.8	−0.914869
$593$	795.575	0.0550933	0.0275467	−	0.999621i	$-0.491231\pi$
$593$	795.575	0.0550933	0.0275467	+	0.999621i	$0.491231\pi$
$594$	−1026.91	−0.0709340
$595$	−11604.4	−0.799554
$596$	−12785.4	−0.878708
$597$	−13986.0	−0.958809
$598$	0	0
$599$	11565.7	0.788917	0.394458	−	0.918914i	$-0.370932\pi$
$599$	11565.7	0.788917	0.394458	+	0.918914i	$0.370932\pi$
$600$	−474.579	−0.0322910
$601$	1121.89	0.0761446	0.0380723	−	0.999275i	$-0.487878\pi$
$601$	1121.89	0.0761446	0.0380723	+	0.999275i	$0.487878\pi$
$602$	−1640.04	−0.111035
$603$	7877.90	0.532028
$604$	22577.2	1.52095
$605$	−1273.28	−0.0855636
$606$	−106.880	−0.00716451
$607$	18983.3	1.26937	0.634684	−	0.772771i	$-0.281130\pi$
$607$	18983.3	1.26937	0.634684	+	0.772771i	$0.281130\pi$
$608$	−8358.72	−0.557551
$609$	15956.4	1.06172
$610$	2659.75	0.176541
$611$	0	0
$612$	−4790.40	−0.316406
$613$	19208.6	1.26562	0.632812	−	0.774306i	$-0.281901\pi$
$613$	19208.6	1.26562	0.632812	+	0.774306i	$0.281901\pi$
$614$	−4156.60	−0.273203
$615$	−2146.69	−0.140753
$616$	−3299.98	−0.215844
$617$	−26705.8	−1.74252	−0.871261	−	0.490819i	$-0.836698\pi$
$617$	−26705.8	−1.74252	−0.871261	+	0.490819i	$0.836698\pi$
$618$	−2967.56	−0.193160
$619$	−4297.92	−0.279076	−0.139538	−	0.990217i	$-0.544562\pi$
$619$	−4297.92	−0.279076	−0.139538	+	0.990217i	$0.544562\pi$
$620$	13396.0	0.867735
$621$	27032.7	1.74683
$622$	3092.60	0.199360
$623$	−19208.2	−1.23525
$624$	0	0
$625$	−13638.0	−0.872829
$626$	−1662.63	−0.106153
$627$	−2496.34	−0.159002
$628$	17214.7	1.09385
$629$	9354.14	0.592963
$630$	−3381.59	−0.213851
$631$	−8059.05	−0.508440	−0.254220	−	0.967146i	$-0.581819\pi$
$631$	−8059.05	−0.508440	−0.254220	+	0.967146i	$0.581819\pi$
$632$	−4223.06	−0.265798
$633$	−12794.9	−0.803398
$634$	−6157.28	−0.385705
$635$	167.384	0.0104605
$636$	−5253.29	−0.327526
$637$	0	0
$638$	−1252.33	−0.0777117
$639$	−5746.32	−0.355745
$640$	12416.6	0.766891
$641$	5838.67	0.359772	0.179886	−	0.983687i	$-0.442427\pi$
$641$	5838.67	0.359772	0.179886	+	0.983687i	$0.442427\pi$
$642$	2775.29	0.170610
$643$	−6883.41	−0.422170	−0.211085	−	0.977468i	$-0.567700\pi$
$643$	−6883.41	−0.422170	−0.211085	+	0.977468i	$0.567700\pi$
$644$	42199.7	2.58215
$645$	2877.19	0.175642
$646$	1785.61	0.108752
$647$	−17164.8	−1.04300	−0.521499	−	0.853252i	$-0.674627\pi$
$647$	−17164.8	−1.04300	−0.521499	+	0.853252i	$0.674627\pi$
$648$	13.5248	0.000819916 0
$649$	7154.08	0.432700
$650$	0	0
$651$	15700.5	0.945242
$652$	−5162.41	−0.310085
$653$	16377.2	0.981455	0.490727	−	0.871313i	$-0.336731\pi$
$653$	16377.2	0.981455	0.490727	+	0.871313i	$0.336731\pi$
$654$	−2381.27	−0.142378
$655$	14295.8	0.852796
$656$	−3400.09	−0.202365
$657$	−13068.2	−0.776014
$658$	−1192.08	−0.0706261
$659$	−20498.2	−1.21168	−0.605839	−	0.795587i	$-0.707162\pi$
$659$	−20498.2	−1.21168	−0.605839	+	0.795587i	$0.707162\pi$
$660$	2812.35	0.165865
$661$	−14666.4	−0.863020	−0.431510	−	0.902108i	$-0.642019\pi$
$661$	−14666.4	−0.863020	−0.431510	+	0.902108i	$0.642019\pi$
$662$	3684.06	0.216292
$663$	0	0
$664$	11289.1	0.659794
$665$	−21538.5	−1.25598
$666$	2725.85	0.158595
$667$	32966.4	1.91374
$668$	−1237.96	−0.0717037
$669$	5888.84	0.340322
$670$	3308.19	0.190756
$671$	4180.62	0.240523
$672$	11036.9	0.633565
$673$	−25229.1	−1.44504	−0.722520	−	0.691350i	$-0.757016\pi$
$673$	−25229.1	−1.44504	−0.722520	+	0.691350i	$0.757016\pi$
$674$	5484.10	0.313412
$675$	−2002.83	−0.114206
$676$	0	0
$677$	25265.0	1.43429	0.717143	−	0.696926i	$-0.245449\pi$
$677$	25265.0	1.43429	0.717143	+	0.696926i	$0.245449\pi$
$678$	−795.011	−0.0450327
$679$	−50915.2	−2.87768
$680$	−4141.02	−0.233531
$681$	11425.4	0.642913
$682$	−1232.24	−0.0691862
$683$	−15237.3	−0.853646	−0.426823	−	0.904335i	$-0.640367\pi$
$683$	−15237.3	−0.853646	−0.426823	+	0.904335i	$0.640367\pi$
$684$	−8891.26	−0.497026
$685$	7366.34	0.410881
$686$	−2982.80	−0.166011
$687$	−1690.11	−0.0938600
$688$	4557.11	0.252526
$689$	0	0
$690$	4332.56	0.239040
$691$	−3294.96	−0.181398	−0.0906992	−	0.995878i	$-0.528910\pi$
$691$	−3294.96	−0.181398	−0.0906992	+	0.995878i	$0.528910\pi$
$692$	−15883.1	−0.872522
$693$	−5315.23	−0.291355
$694$	−4452.58	−0.243541
$695$	23180.6	1.26517
$696$	5694.03	0.310103
$697$	2413.53	0.131161
$698$	3242.56	0.175835
$699$	−145.719	−0.00788496
$700$	−3126.54	−0.168817
$701$	747.881	0.0402954	0.0201477	−	0.999797i	$-0.493586\pi$
$701$	747.881	0.0402954	0.0201477	+	0.999797i	$0.493586\pi$
$702$	0	0
$703$	17361.8	0.931457
$704$	3848.87	0.206051
$705$	2091.31	0.111721
$706$	−8189.00	−0.436540
$707$	−1449.46	−0.0771042
$708$	−15801.6	−0.838786
$709$	5965.27	0.315981	0.157990	−	0.987441i	$-0.449499\pi$
$709$	5965.27	0.315981	0.157990	+	0.987441i	$0.449499\pi$
$710$	−2413.07	−0.127551
$711$	−6802.02	−0.358785
$712$	−6854.41	−0.360786
$713$	32437.8	1.70379
$714$	−2357.72	−0.123579
$715$	0	0
$716$	1123.00	0.0586154
$717$	−6843.05	−0.356427
$718$	−8044.27	−0.418119
$719$	−13694.8	−0.710334	−0.355167	−	0.934803i	$-0.615576\pi$
$719$	−13694.8	−0.710334	−0.355167	+	0.934803i	$0.615576\pi$
$720$	9396.29	0.486360
$721$	−40244.9	−2.07878
$722$	−1247.39	−0.0642980
$723$	16350.5	0.841054
$724$	−18582.4	−0.953879
$725$	−2442.45	−0.125118
$726$	−258.697	−0.0132247
$727$	−11312.0	−0.577081	−0.288541	−	0.957468i	$-0.593170\pi$
$727$	−11312.0	−0.577081	−0.288541	+	0.957468i	$0.593170\pi$
$728$	0	0
$729$	12206.1	0.620132
$730$	−5487.79	−0.278236
$731$	−3234.83	−0.163672
$732$	−9233.97	−0.466253
$733$	−19038.0	−0.959326	−0.479663	−	0.877453i	$-0.659241\pi$
$733$	−19038.0	−0.959326	−0.479663	+	0.877453i	$0.659241\pi$
$734$	5744.07	0.288852
$735$	16836.1	0.844912
$736$	22802.4	1.14200
$737$	5199.85	0.259890
$738$	703.316	0.0350805
$739$	6575.85	0.327329	0.163665	−	0.986516i	$-0.447668\pi$
$739$	6575.85	0.327329	0.163665	+	0.986516i	$0.447668\pi$
$740$	−19559.7	−0.971658
$741$	0	0
$742$	4169.31	0.206280
$743$	13415.8	0.662420	0.331210	−	0.943557i	$-0.392543\pi$
$743$	13415.8	0.662420	0.331210	+	0.943557i	$0.392543\pi$
$744$	5602.72	0.276083
$745$	−17801.7	−0.875440
$746$	−9196.49	−0.451350
$747$	18183.3	0.890617
$748$	−3161.93	−0.154561
$749$	37637.4	1.83610
$750$	−3133.24	−0.152546
$751$	−21737.5	−1.05621	−0.528104	−	0.849180i	$-0.677097\pi$
$751$	−21737.5	−1.05621	−0.528104	+	0.849180i	$0.677097\pi$
$752$	3312.37	0.160625
$753$	4539.55	0.219695
$754$	0	0
$755$	31435.2	1.51529
$756$	30760.5	1.47982
$757$	22482.0	1.07942	0.539712	−	0.841850i	$-0.318533\pi$
$757$	22482.0	1.07942	0.539712	+	0.841850i	$0.318533\pi$
$758$	7390.26	0.354124
$759$	6809.97	0.325674
$760$	−7685.98	−0.366842
$761$	−8880.80	−0.423034	−0.211517	−	0.977374i	$-0.567840\pi$
$761$	−8880.80	−0.423034	−0.211517	+	0.977374i	$0.567840\pi$
$762$	34.0080	0.00161677
$763$	−32293.9	−1.53226
$764$	−34135.9	−1.61648
$765$	−6669.89	−0.315229
$766$	8121.45	0.383081
$767$	0	0
$768$	−6475.98	−0.304273
$769$	−9358.13	−0.438833	−0.219417	−	0.975631i	$-0.570415\pi$
$769$	−9358.13	−0.438833	−0.219417	+	0.975631i	$0.570415\pi$
$770$	−2232.04	−0.104464
$771$	−22615.7	−1.05640
$772$	−9103.66	−0.424414
$773$	−19918.0	−0.926780	−0.463390	−	0.886155i	$-0.653367\pi$
$773$	−19918.0	−0.926780	−0.463390	+	0.886155i	$0.653367\pi$
$774$	−942.648	−0.0437762
$775$	−2403.29	−0.111392
$776$	−18169.0	−0.840503
$777$	−22924.6	−1.05845
$778$	−4524.78	−0.208511
$779$	4479.65	0.206034
$780$	0	0
$781$	−3792.89	−0.173778
$782$	−4871.11	−0.222750
$783$	24030.1	1.09676
$784$	26666.3	1.21476
$785$	23968.8	1.08979
$786$	2904.53	0.131808
$787$	−2511.97	−0.113777	−0.0568883	−	0.998381i	$-0.518118\pi$
$787$	−2511.97	−0.113777	−0.0568883	+	0.998381i	$0.518118\pi$
$788$	−2450.66	−0.110788
$789$	−20956.7	−0.945599
$790$	−2856.40	−0.128641
$791$	−10781.6	−0.484641
$792$	−1896.73	−0.0850977
$793$	0	0
$794$	−1879.09	−0.0839880
$795$	−7314.39	−0.326308
$796$	−32880.2	−1.46408
$797$	12645.8	0.562028	0.281014	−	0.959704i	$-0.409329\pi$
$797$	12645.8	0.562028	0.281014	+	0.959704i	$0.409329\pi$
$798$	−4376.06	−0.194124
$799$	−2351.26	−0.104107
$800$	−1689.41	−0.0746622
$801$	−11040.3	−0.487004
$802$	9214.85	0.405720
$803$	−8625.77	−0.379075
$804$	−11485.2	−0.503795
$805$	58756.6	2.57254
$806$	0	0
$807$	18859.3	0.822649
$808$	−517.239	−0.0225203
$809$	−14134.1	−0.614250	−0.307125	−	0.951669i	$-0.599367\pi$
$809$	−14134.1	−0.614250	−0.307125	+	0.951669i	$0.599367\pi$
$810$	9.14794	0.000396822 0
$811$	−250.820	−0.0108600	−0.00543002	−	0.999985i	$-0.501728\pi$
$811$	−250.820	−0.0108600	−0.00543002	+	0.999985i	$0.501728\pi$
$812$	37512.5	1.62122
$813$	23222.9	1.00180
$814$	1799.21	0.0774722
$815$	−7187.86	−0.308932
$816$	6551.29	0.281055
$817$	−6004.04	−0.257105
$818$	5924.33	0.253227
$819$	0	0
$820$	−5046.73	−0.214926
$821$	−13243.8	−0.562986	−0.281493	−	0.959563i	$-0.590830\pi$
$821$	−13243.8	−0.562986	−0.281493	+	0.959563i	$0.590830\pi$
$822$	1496.65	0.0635056
$823$	24277.3	1.02825	0.514127	−	0.857714i	$-0.328116\pi$
$823$	24277.3	1.02825	0.514127	+	0.857714i	$0.328116\pi$
$824$	−14361.3	−0.607161
$825$	−504.545	−0.0212921
$826$	12541.0	0.528279
$827$	34821.6	1.46417	0.732083	−	0.681215i	$-0.238548\pi$
$827$	34821.6	1.46417	0.732083	+	0.681215i	$0.238548\pi$
$828$	24255.2	1.01803
$829$	6701.26	0.280753	0.140377	−	0.990098i	$-0.455169\pi$
$829$	6701.26	0.280753	0.140377	+	0.990098i	$0.455169\pi$
$830$	7635.76	0.319326
$831$	−17919.5	−0.748039
$832$	0	0
$833$	−18928.9	−0.787332
$834$	4709.70	0.195544
$835$	−1723.67	−0.0714371
$836$	−5868.73	−0.242792
$837$	23644.7	0.976440
$838$	139.010	0.00573035
$839$	37306.6	1.53512	0.767562	−	0.640975i	$-0.221470\pi$
$839$	37306.6	1.53512	0.767562	+	0.640975i	$0.221470\pi$
$840$	10148.6	0.416856
$841$	4915.77	0.201557
$842$	4491.02	0.183813
$843$	267.010	0.0109090
$844$	−30079.9	−1.22677
$845$	0	0
$846$	−685.172	−0.0278448
$847$	−3508.35	−0.142324
$848$	−11585.1	−0.469143
$849$	−14308.8	−0.578419
$850$	360.896	0.0145631
$851$	−47362.8	−1.90784
$852$	8377.55	0.336867
$853$	36525.1	1.46611	0.733057	−	0.680167i	$-0.238093\pi$
$853$	36525.1	1.46611	0.733057	+	0.680167i	$0.238093\pi$
$854$	7328.60	0.293653
$855$	−12379.7	−0.495178
$856$	13430.9	0.536282
$857$	−5408.05	−0.215561	−0.107780	−	0.994175i	$-0.534374\pi$
$857$	−5408.05	−0.215561	−0.107780	+	0.994175i	$0.534374\pi$
$858$	0	0
$859$	38851.4	1.54318	0.771591	−	0.636119i	$-0.219461\pi$
$859$	38851.4	1.54318	0.771591	+	0.636119i	$0.219461\pi$
$860$	6764.08	0.268201
$861$	−5914.93	−0.234123
$862$	452.443	0.0178773
$863$	35630.7	1.40543	0.702713	−	0.711474i	$-0.251972\pi$
$863$	35630.7	1.40543	0.702713	+	0.711474i	$0.251972\pi$
$864$	16621.3	0.654476
$865$	−22114.8	−0.869277
$866$	−7503.65	−0.294439
$867$	11143.8	0.436520
$868$	36910.9	1.44336
$869$	−4489.72	−0.175263
$870$	3851.34	0.150083
$871$	0	0
$872$	−11524.0	−0.447537
$873$	−29264.6	−1.13454
$874$	−9041.07	−0.349907
$875$	−42491.8	−1.64170
$876$	19052.2	0.734834
$877$	−23572.2	−0.907612	−0.453806	−	0.891100i	$-0.649934\pi$
$877$	−23572.2	−0.907612	−0.453806	+	0.891100i	$0.649934\pi$
$878$	3145.94	0.120923
$879$	−2408.39	−0.0924154
$880$	6202.07	0.237582
$881$	1507.11	0.0576343	0.0288171	−	0.999585i	$-0.490826\pi$
$881$	1507.11	0.0576343	0.0288171	+	0.999585i	$0.490826\pi$
$882$	−5515.99	−0.210582
$883$	12632.0	0.481428	0.240714	−	0.970596i	$-0.422618\pi$
$883$	12632.0	0.481428	0.240714	+	0.970596i	$0.422618\pi$
$884$	0	0
$885$	−22001.3	−0.835666
$886$	−445.570	−0.0168953
$887$	3320.10	0.125680	0.0628399	−	0.998024i	$-0.479984\pi$
$887$	3320.10	0.125680	0.0628399	+	0.998024i	$0.479984\pi$
$888$	−8180.60	−0.309147
$889$	461.204	0.0173997
$890$	−4636.19	−0.174613
$891$	14.3788	0.000540639 0
$892$	13844.3	0.519664
$893$	−4364.09	−0.163537
$894$	−3616.84	−0.135308
$895$	1563.61	0.0583974
$896$	34212.4	1.27562
$897$	0	0
$898$	−1083.71	−0.0402715
$899$	28834.8	1.06974
$900$	−1797.05	−0.0665573
$901$	8223.58	0.304070
$902$	464.228	0.0171365
$903$	7927.73	0.292157
$904$	−3847.41	−0.141552
$905$	−25873.1	−0.950332
$906$	6386.82	0.234203
$907$	20570.1	0.753054	0.376527	−	0.926406i	$-0.377118\pi$
$907$	20570.1	0.753054	0.376527	+	0.926406i	$0.377118\pi$
$908$	26860.4	0.981712
$909$	−833.110	−0.0303988
$910$	0	0
$911$	51427.6	1.87033	0.935165	−	0.354211i	$-0.115251\pi$
$911$	51427.6	1.87033	0.935165	+	0.354211i	$0.115251\pi$
$912$	12159.6	0.441496
$913$	12002.0	0.435057
$914$	−7661.90	−0.277279
$915$	−12856.9	−0.464519
$916$	−3973.34	−0.143322
$917$	39390.1	1.41851
$918$	−3550.67	−0.127658
$919$	−9430.87	−0.338515	−0.169258	−	0.985572i	$-0.554137\pi$
$919$	−9430.87	−0.338515	−0.169258	+	0.985572i	$0.554137\pi$
$920$	20967.2	0.751379
$921$	20092.4	0.718857
$922$	−5496.55	−0.196333
$923$	0	0
$924$	7749.07	0.275894
$925$	3509.07	0.124732
$926$	5634.36	0.199953
$927$	−23131.6	−0.819571
$928$	20269.7	0.717011
$929$	22666.2	0.800487	0.400244	−	0.916409i	$-0.368926\pi$
$929$	22666.2	0.800487	0.400244	+	0.916409i	$0.368926\pi$
$930$	3789.57	0.133618
$931$	−35133.2	−1.23678
$932$	−342.575	−0.0120401
$933$	−14949.2	−0.524560
$934$	4587.13	0.160702
$935$	−4402.50	−0.153986
$936$	0	0
$937$	39917.7	1.39173	0.695867	−	0.718171i	$-0.255020\pi$
$937$	39917.7	1.39173	0.695867	+	0.718171i	$0.255020\pi$
$938$	9115.30	0.317297
$939$	8036.93	0.279313
$940$	4916.53	0.170595
$941$	−3426.69	−0.118711	−0.0593554	−	0.998237i	$-0.518905\pi$
$941$	−3426.69	−0.118711	−0.0593554	+	0.998237i	$0.518905\pi$
$942$	4869.83	0.168437
$943$	−12220.4	−0.422005
$944$	−34847.2	−1.20146
$945$	42829.2	1.47432
$946$	−622.200	−0.0213842
$947$	−27293.3	−0.936549	−0.468275	−	0.883583i	$-0.655124\pi$
$947$	−27293.3	−0.936549	−0.468275	+	0.883583i	$0.655124\pi$
$948$	9916.67	0.339745
$949$	0	0
$950$	669.845	0.0228765
$951$	29763.4	1.01487
$952$	−11410.1	−0.388447
$953$	−33407.3	−1.13554	−0.567770	−	0.823187i	$-0.692194\pi$
$953$	−33407.3	−1.13554	−0.567770	+	0.823187i	$0.692194\pi$
$954$	2396.40	0.0813273
$955$	−47528.9	−1.61047
$956$	−16087.5	−0.544256
$957$	6053.57	0.204477
$958$	−6445.87	−0.217387
$959$	20297.0	0.683445
$960$	−11836.6	−0.397943
$961$	−1418.61	−0.0476189
$962$	0	0
$963$	21632.9	0.723895
$964$	38438.9	1.28427
$965$	−12675.4	−0.422836
$966$	11937.8	0.397612
$967$	17254.6	0.573807	0.286904	−	0.957959i	$-0.407374\pi$
$967$	17254.6	0.573807	0.286904	+	0.957959i	$0.407374\pi$
$968$	−1251.95	−0.0415694
$969$	−8631.39	−0.286151
$970$	−12289.2	−0.406785
$971$	10977.0	0.362789	0.181394	−	0.983410i	$-0.441939\pi$
$971$	10977.0	0.362789	0.181394	+	0.983410i	$0.441939\pi$
$972$	28612.6	0.944188
$973$	63871.2	2.10444
$974$	2711.70	0.0892080
$975$	0	0
$976$	−20363.7	−0.667853
$977$	24694.6	0.808648	0.404324	−	0.914616i	$-0.367507\pi$
$977$	24694.6	0.808648	0.404324	+	0.914616i	$0.367507\pi$
$978$	−1460.39	−0.0477485
$979$	−7287.22	−0.237896
$980$	39580.6	1.29016
$981$	−18561.6	−0.604104
$982$	−13618.3	−0.442543
$983$	−4763.40	−0.154556	−0.0772782	−	0.997010i	$-0.524623\pi$
$983$	−4763.40	−0.154556	−0.0772782	+	0.997010i	$0.524623\pi$
$984$	−2110.73	−0.0683819
$985$	−3412.17	−0.110376
$986$	−4330.06	−0.139855
$987$	5762.34	0.185833
$988$	0	0
$989$	16378.9	0.526611
$990$	−1282.91	−0.0411855
$991$	−27692.2	−0.887660	−0.443830	−	0.896111i	$-0.646381\pi$
$991$	−27692.2	−0.887660	−0.443830	+	0.896111i	$0.646381\pi$
$992$	19944.6	0.638350
$993$	−17808.3	−0.569112
$994$	−6648.90	−0.212163
$995$	−45780.5	−1.45863
$996$	−26509.4	−0.843355
$997$	−3336.44	−0.105984	−0.0529920	−	0.998595i	$-0.516876\pi$
$997$	−3336.44	−0.105984	−0.0529920	+	0.998595i	$0.516876\pi$
$998$	7823.59	0.248148
$999$	−34523.9	−1.09338

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1859.4.a.l.1.19		36
13.6	odd	12		143.4.j.a.23.18	✓	72
13.11	odd	12		143.4.j.a.56.18	yes	72
13.12	even	2		1859.4.a.m.1.18		36

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
143.4.j.a.23.18	✓	72	13.6	odd	12
143.4.j.a.56.18	yes	72	13.11	odd	12
1859.4.a.l.1.19		36	1.1	even	1	trivial
1859.4.a.m.1.18		36	13.12	even	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:	\( N \)	\(=\)	\( 1859 = 11 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1859.a (trivial)

\(f(q)\)	\(=\)	\(q+0.665052 q^{2} -3.21477 q^{3} -7.55771 q^{4} -10.5229 q^{5} -2.13799 q^{6} -28.9946 q^{7} -10.3467 q^{8} -16.6653 q^{9} -6.99830 q^{10} -11.0000 q^{11} +24.2963 q^{12} -19.2829 q^{14} +33.8288 q^{15} +53.5806 q^{16} -38.0338 q^{17} -11.0833 q^{18} -70.5930 q^{19} +79.5293 q^{20} +93.2109 q^{21} -7.31557 q^{22} +192.576 q^{23} +33.2622 q^{24} -14.2678 q^{25} +140.374 q^{27} +219.133 q^{28} +171.186 q^{29} +22.4979 q^{30} +168.441 q^{31} +118.407 q^{32} +35.3625 q^{33} -25.2944 q^{34} +305.108 q^{35} +125.951 q^{36} -245.943 q^{37} -46.9480 q^{38} +108.877 q^{40} -63.4575 q^{41} +61.9901 q^{42} +85.0515 q^{43} +83.1348 q^{44} +175.367 q^{45} +128.073 q^{46} +61.8204 q^{47} -172.249 q^{48} +497.686 q^{49} -9.48883 q^{50} +122.270 q^{51} -216.218 q^{53} +93.3558 q^{54} +115.752 q^{55} +299.998 q^{56} +226.940 q^{57} +113.848 q^{58} -650.371 q^{59} -255.668 q^{60} -380.057 q^{61} +112.022 q^{62} +483.202 q^{63} -349.898 q^{64} +23.5179 q^{66} -472.714 q^{67} +287.448 q^{68} -619.088 q^{69} +202.913 q^{70} +344.808 q^{71} +172.430 q^{72} +784.161 q^{73} -163.565 q^{74} +45.8677 q^{75} +533.521 q^{76} +318.940 q^{77} +408.156 q^{79} -563.825 q^{80} -1.30717 q^{81} -42.2025 q^{82} -1091.09 q^{83} -704.461 q^{84} +400.227 q^{85} +56.5636 q^{86} -550.325 q^{87} +113.813 q^{88} +662.474 q^{89} +116.628 q^{90} -1455.43 q^{92} -541.499 q^{93} +41.1138 q^{94} +742.845 q^{95} -380.652 q^{96} +1756.02 q^{97} +330.987 q^{98} +183.318 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 36 q - 4 q^{2} + 12 q^{3} + 152 q^{4} - 40 q^{5} - 98 q^{6} - 56 q^{7} - 84 q^{8} + 360 q^{9} - 56 q^{10} - 396 q^{11} + 66 q^{12} + 164 q^{14} - 120 q^{15} + 644 q^{16} + 138 q^{17} + 28 q^{18} - 498 q^{19}+ \cdots - 3960 q^{99}+O(q^{100}) \) 36 * q - 4 * q^2 + 12 * q^3 + 152 * q^4 - 40 * q^5 - 98 * q^6 - 56 * q^7 - 84 * q^8 + 360 * q^9 - 56 * q^10 - 396 * q^11 + 66 * q^12 + 164 * q^14 - 120 * q^15 + 644 * q^16 + 138 * q^17 + 28 * q^18 - 498 * q^19 - 320 * q^20 - 404 * q^21 + 44 * q^22 + 46 * q^23 - 1340 * q^24 + 818 * q^25 + 24 * q^27 - 1382 * q^28 + 262 * q^29 + 104 * q^30 - 660 * q^31 - 106 * q^32 - 132 * q^33 + 640 * q^34 + 68 * q^35 + 1960 * q^36 - 1372 * q^37 + 224 * q^38 + 108 * q^40 - 1216 * q^41 + 82 * q^42 + 436 * q^43 - 1672 * q^44 - 820 * q^45 - 2512 * q^46 + 140 * q^47 - 1944 * q^48 + 1192 * q^49 - 3484 * q^50 + 712 * q^51 + 146 * q^53 - 3078 * q^54 + 440 * q^55 - 102 * q^56 - 3656 * q^57 + 324 * q^58 - 404 * q^59 + 2944 * q^60 - 590 * q^61 + 1776 * q^62 - 4364 * q^63 + 2514 * q^64 + 1078 * q^66 - 2236 * q^67 - 1530 * q^68 + 12 * q^69 - 4852 * q^70 - 292 * q^71 - 2990 * q^72 - 1936 * q^73 + 1568 * q^74 + 4688 * q^75 - 7398 * q^76 + 616 * q^77 - 1464 * q^79 + 808 * q^80 + 3780 * q^81 - 4072 * q^82 - 4422 * q^83 + 2678 * q^84 - 3700 * q^85 + 2588 * q^86 - 1832 * q^87 + 924 * q^88 - 6856 * q^89 - 3422 * q^90 - 668 * q^92 - 840 * q^93 - 4166 * q^94 - 256 * q^95 - 18290 * q^96 - 2776 * q^97 - 4608 * q^98 - 3960 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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