LMFDB - Embedded newform 1911.4.a.w.1.8

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1911,4,Mod(1,1911)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1911, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 4, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1911.1"); S:= CuspForms(chi, 4); N := Newforms(S);

Level:	$ N $	$=$	$ 1911 = 3 \cdot 7^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1911.a (trivial)

Newform invariants

comment:select newform

sage:traces = [11,-1,33,31,-17] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$112.752650021$
Analytic rank:	$1$
Dimension:	$11$
Coefficient field:	$\mathbb{Q}[x]/(x^{11} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{11} - x^{10} - 59 x^{9} + 36 x^{8} + 1220 x^{7} - 339 x^{6} - 10807 x^{5} + 58 x^{4} + 40509 x^{3} + \cdots - 27208 $ x^11 - x^10 - 59x^9 + 36x^8 + 1220x^7 - 339x^6 - 10807x^5 + 58x^4 + 40509x^3 + 10990x^2 - 51360*x - 27208
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 273)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Root			$-1.96658$ of defining polynomial
Character	$\chi$	$=$	1911.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.96658 q^{2} +3.00000 q^{3} -4.13255 q^{4} -10.3068 q^{5} +5.89975 q^{6} -23.8597 q^{8} +9.00000 q^{9} -20.2692 q^{10} +65.9110 q^{11} -12.3977 q^{12} +13.0000 q^{13} -30.9205 q^{15} -13.8616 q^{16} -108.032 q^{17} +17.6992 q^{18} -11.0814 q^{19} +42.5935 q^{20} +129.619 q^{22} +68.6342 q^{23} -71.5790 q^{24} -18.7694 q^{25} +25.5656 q^{26} +27.0000 q^{27} -20.9819 q^{29} -60.8077 q^{30} +80.2525 q^{31} +163.617 q^{32} +197.733 q^{33} -212.455 q^{34} -37.1930 q^{36} +155.964 q^{37} -21.7924 q^{38} +39.0000 q^{39} +245.917 q^{40} +216.970 q^{41} -194.723 q^{43} -272.380 q^{44} -92.7614 q^{45} +134.975 q^{46} +253.170 q^{47} -41.5849 q^{48} -36.9115 q^{50} -324.097 q^{51} -53.7232 q^{52} -707.877 q^{53} +53.0977 q^{54} -679.333 q^{55} -33.2441 q^{57} -41.2626 q^{58} -768.060 q^{59} +127.780 q^{60} -392.039 q^{61} +157.823 q^{62} +432.660 q^{64} -133.989 q^{65} +388.858 q^{66} +830.212 q^{67} +446.449 q^{68} +205.903 q^{69} +647.743 q^{71} -214.737 q^{72} +127.715 q^{73} +306.716 q^{74} -56.3081 q^{75} +45.7943 q^{76} +76.6967 q^{78} -1185.68 q^{79} +142.869 q^{80} +81.0000 q^{81} +426.690 q^{82} +785.987 q^{83} +1113.47 q^{85} -382.939 q^{86} -62.9457 q^{87} -1572.61 q^{88} -1529.76 q^{89} -182.423 q^{90} -283.634 q^{92} +240.757 q^{93} +497.880 q^{94} +114.214 q^{95} +490.852 q^{96} -631.573 q^{97} +593.199 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 11 q - q^{2} + 33 q^{3} + 31 q^{4} - 17 q^{5} - 3 q^{6} - 54 q^{8} + 99 q^{9} - 75 q^{10} + 7 q^{11} + 93 q^{12} + 143 q^{13} - 51 q^{15} + 23 q^{16} + 20 q^{17} - 9 q^{18} - 242 q^{19} - 254 q^{20} - 290 q^{22}+ \cdots + 63 q^{99}+O(q^{100}) $ 11 * q - q^2 + 33 * q^3 + 31 * q^4 - 17 * q^5 - 3 * q^6 - 54 * q^8 + 99 * q^9 - 75 * q^10 + 7 * q^11 + 93 * q^12 + 143 * q^13 - 51 * q^15 + 23 * q^16 + 20 * q^17 - 9 * q^18 - 242 * q^19 - 254 * q^20 - 290 * q^22 - 122 * q^23 - 162 * q^24 + 208 * q^25 - 13 * q^26 + 297 * q^27 - 177 * q^29 - 225 * q^30 - 493 * q^31 - 448 * q^32 + 21 * q^33 - 855 * q^34 + 279 * q^36 - 660 * q^37 - 458 * q^38 + 429 * q^39 + 530 * q^40 - 854 * q^41 - 1018 * q^43 + 1122 * q^44 - 153 * q^45 - 757 * q^46 - 414 * q^47 + 69 * q^48 + 851 * q^50 + 60 * q^51 + 403 * q^52 - 961 * q^53 - 27 * q^54 - 1003 * q^55 - 726 * q^57 + 801 * q^58 - 2053 * q^59 - 762 * q^60 - 916 * q^61 + 653 * q^62 + 1366 * q^64 - 221 * q^65 - 870 * q^66 - 2788 * q^67 + 2458 * q^68 - 366 * q^69 + 886 * q^71 - 486 * q^72 - 1916 * q^73 - 1557 * q^74 + 624 * q^75 - 2292 * q^76 - 39 * q^78 + 651 * q^79 - 4438 * q^80 + 891 * q^81 + 2148 * q^82 - 185 * q^83 - 2052 * q^85 + 4025 * q^86 - 531 * q^87 - 4503 * q^88 - 1490 * q^89 - 675 * q^90 + 586 * q^92 - 1479 * q^93 - 1796 * q^94 - 2496 * q^95 - 1344 * q^96 - 2705 * q^97 + 63 * 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 $

$2$	1.96658	0.695292	0.347646	−	0.937626i	$-0.386981\pi$
$2$	1.96658	0.695292	0.347646	+	0.937626i	$0.386981\pi$
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	−4.13255	−0.516569
$5$	−10.3068	−0.921870	−0.460935	−	0.887434i	$-0.652486\pi$
$5$	−10.3068	−0.921870	−0.460935	+	0.887434i	$0.652486\pi$
$6$	5.89975	0.401427
$7$	0	0
$7$	0	0
$8$	−23.8597	−1.05446
$9$	9.00000	0.333333
$10$	−20.2692	−0.640969
$11$	65.9110	1.80663	0.903314	−	0.428980i	$-0.141127\pi$
$11$	65.9110	1.80663	0.903314	+	0.428980i	$0.141127\pi$
$12$	−12.3977	−0.298241
$13$	13.0000	0.277350
$13$	13.0000	0.277350
$14$	0	0
$15$	−30.9205	−0.532242
$16$	−13.8616	−0.216588
$17$	−108.032	−1.54128	−0.770638	−	0.637273i	$-0.780062\pi$
$17$	−108.032	−1.54128	−0.770638	+	0.637273i	$0.780062\pi$
$18$	17.6992	0.231764
$19$	−11.0814	−0.133802	−0.0669011	−	0.997760i	$-0.521311\pi$
$19$	−11.0814	−0.133802	−0.0669011	+	0.997760i	$0.521311\pi$
$20$	42.5935	0.476210
$21$	0	0
$22$	129.619	1.25613
$23$	68.6342	0.622227	0.311114	−	0.950373i	$-0.399298\pi$
$23$	68.6342	0.622227	0.311114	+	0.950373i	$0.399298\pi$
$24$	−71.5790	−0.608792
$25$	−18.7694	−0.150155
$26$	25.5656	0.192839
$27$	27.0000	0.192450
$28$	0	0
$29$	−20.9819	−0.134353	−0.0671765	−	0.997741i	$-0.521399\pi$
$29$	−20.9819	−0.134353	−0.0671765	+	0.997741i	$0.521399\pi$
$30$	−60.8077	−0.370064
$31$	80.2525	0.464960	0.232480	−	0.972601i	$-0.425316\pi$
$31$	80.2525	0.464960	0.232480	+	0.972601i	$0.425316\pi$
$32$	163.617	0.903867
$33$	197.733	1.04306
$34$	−212.455	−1.07164
$35$	0	0
$36$	−37.1930	−0.172190
$37$	155.964	0.692980	0.346490	−	0.938054i	$-0.387373\pi$
$37$	155.964	0.692980	0.346490	+	0.938054i	$0.387373\pi$
$38$	−21.7924	−0.0930316
$39$	39.0000	0.160128
$40$	245.917	0.972074
$41$	216.970	0.826465	0.413232	−	0.910626i	$-0.364400\pi$
$41$	216.970	0.826465	0.413232	+	0.910626i	$0.364400\pi$
$42$	0	0
$43$	−194.723	−0.690582	−0.345291	−	0.938496i	$-0.612220\pi$
$43$	−194.723	−0.690582	−0.345291	+	0.938496i	$0.612220\pi$
$44$	−272.380	−0.933248
$45$	−92.7614	−0.307290
$46$	134.975	0.432630
$47$	253.170	0.785715	0.392858	−	0.919599i	$-0.371486\pi$
$47$	253.170	0.785715	0.392858	+	0.919599i	$0.371486\pi$
$48$	−41.5849	−0.125047
$49$	0	0
$50$	−36.9115	−0.104402
$51$	−324.097	−0.889856
$52$	−53.7232	−0.143270
$53$	−707.877	−1.83461	−0.917305	−	0.398185i	$-0.869640\pi$
$53$	−707.877	−1.83461	−0.917305	+	0.398185i	$0.869640\pi$
$54$	53.0977	0.133809
$55$	−679.333	−1.66548
$56$	0	0
$57$	−33.2441	−0.0772507
$58$	−41.2626	−0.0934146
$59$	−768.060	−1.69479	−0.847397	−	0.530960i	$-0.821832\pi$
$59$	−768.060	−1.69479	−0.847397	+	0.530960i	$0.821832\pi$
$60$	127.780	0.274940
$61$	−392.039	−0.822875	−0.411438	−	0.911438i	$-0.634973\pi$
$61$	−392.039	−0.822875	−0.411438	+	0.911438i	$0.634973\pi$
$62$	157.823	0.323283
$63$	0	0
$64$	432.660	0.845039
$65$	−133.989	−0.255681
$66$	388.858	0.725229
$67$	830.212	1.51383	0.756914	−	0.653514i	$-0.226706\pi$
$67$	830.212	1.51383	0.756914	+	0.653514i	$0.226706\pi$
$68$	446.449	0.796175
$69$	205.903	0.359243
$70$	0	0
$71$	647.743	1.08272	0.541359	−	0.840792i	$-0.317910\pi$
$71$	647.743	1.08272	0.541359	+	0.840792i	$0.317910\pi$
$72$	−214.737	−0.351486
$73$	127.715	0.204767	0.102383	−	0.994745i	$-0.467353\pi$
$73$	127.715	0.204767	0.102383	+	0.994745i	$0.467353\pi$
$74$	306.716	0.481824
$75$	−56.3081	−0.0866920
$76$	45.7943	0.0691180
$77$	0	0
$78$	76.6967	0.111336
$79$	−1185.68	−1.68860	−0.844298	−	0.535873i	$-0.819982\pi$
$79$	−1185.68	−1.68860	−0.844298	+	0.535873i	$0.819982\pi$
$80$	142.869	0.199666
$81$	81.0000	0.111111
$82$	426.690	0.574634
$83$	785.987	1.03944	0.519719	−	0.854337i	$-0.326037\pi$
$83$	785.987	1.03944	0.519719	+	0.854337i	$0.326037\pi$
$84$	0	0
$85$	1113.47	1.42086
$86$	−382.939	−0.480156
$87$	−62.9457	−0.0775688
$88$	−1572.61	−1.90501
$89$	−1529.76	−1.82196	−0.910980	−	0.412451i	$-0.864673\pi$
$89$	−1529.76	−1.82196	−0.910980	+	0.412451i	$0.864673\pi$
$90$	−182.423	−0.213656
$91$	0	0
$92$	−283.634	−0.321423
$93$	240.757	0.268445
$94$	497.880	0.546302
$95$	114.214	0.123348
$96$	490.852	0.521848
$97$	−631.573	−0.661098	−0.330549	−	0.943789i	$-0.607234\pi$
$97$	−631.573	−0.661098	−0.330549	+	0.943789i	$0.607234\pi$
$98$	0	0
$99$	593.199	0.602209
$100$	77.5654	0.0775654
$101$	−47.6116	−0.0469062	−0.0234531	−	0.999725i	$-0.507466\pi$
$101$	−47.6116	−0.0469062	−0.0234531	+	0.999725i	$0.507466\pi$
$102$	−637.364	−0.618710
$103$	−1167.47	−1.11684	−0.558421	−	0.829558i	$-0.688593\pi$
$103$	−1167.47	−1.11684	−0.558421	+	0.829558i	$0.688593\pi$
$104$	−310.176	−0.292454
$105$	0	0
$106$	−1392.10	−1.27559
$107$	529.441	0.478346	0.239173	−	0.970977i	$-0.423124\pi$
$107$	529.441	0.478346	0.239173	+	0.970977i	$0.423124\pi$
$108$	−111.579	−0.0994137
$109$	−20.0691	−0.0176355	−0.00881775	−	0.999961i	$-0.502807\pi$
$109$	−20.0691	−0.0176355	−0.00881775	+	0.999961i	$0.502807\pi$
$110$	−1335.96	−1.15799
$111$	467.891	0.400092
$112$	0	0
$113$	−1555.90	−1.29528	−0.647640	−	0.761947i	$-0.724244\pi$
$113$	−1555.90	−1.29528	−0.647640	+	0.761947i	$0.724244\pi$
$114$	−65.3773	−0.0537118
$115$	−707.401	−0.573613
$116$	86.7087	0.0694026
$117$	117.000	0.0924500
$118$	−1510.45	−1.17838
$119$	0	0
$120$	737.752	0.561227
$121$	3013.26	2.26390
$122$	−770.976	−0.572139
$123$	650.911	0.477160
$124$	−331.647	−0.240184
$125$	1481.81	1.06029
$126$	0	0
$127$	−1331.46	−0.930297	−0.465148	−	0.885233i	$-0.653999\pi$
$127$	−1331.46	−0.930297	−0.465148	+	0.885233i	$0.653999\pi$
$128$	−458.077	−0.316318
$129$	−584.170	−0.398708
$130$	−263.500	−0.177773
$131$	−2157.40	−1.43888	−0.719440	−	0.694555i	$-0.755601\pi$
$131$	−2157.40	−1.43888	−0.719440	+	0.694555i	$0.755601\pi$
$132$	−817.141	−0.538811
$133$	0	0
$134$	1632.68	1.05255
$135$	−278.284	−0.177414
$136$	2577.62	1.62521
$137$	−1348.91	−0.841206	−0.420603	−	0.907245i	$-0.638181\pi$
$137$	−1348.91	−0.841206	−0.420603	+	0.907245i	$0.638181\pi$
$138$	404.925	0.249779
$139$	−2064.71	−1.25990	−0.629952	−	0.776634i	$-0.716926\pi$
$139$	−2064.71	−1.25990	−0.629952	+	0.776634i	$0.716926\pi$
$140$	0	0
$141$	759.510	0.453633
$142$	1273.84	0.752805
$143$	856.843	0.501068
$144$	−124.755	−0.0721959
$145$	216.257	0.123856
$146$	251.163	0.142373
$147$	0	0
$148$	−644.528	−0.357972
$149$	−1706.69	−0.938374	−0.469187	−	0.883099i	$-0.655453\pi$
$149$	−1706.69	−0.938374	−0.469187	+	0.883099i	$0.655453\pi$
$150$	−110.735	−0.0602763
$151$	316.050	0.170330	0.0851650	−	0.996367i	$-0.472858\pi$
$151$	316.050	0.170330	0.0851650	+	0.996367i	$0.472858\pi$
$152$	264.398	0.141089
$153$	−972.291	−0.513759
$154$	0	0
$155$	−827.148	−0.428633
$156$	−161.169	−0.0827172
$157$	−3585.62	−1.82270	−0.911348	−	0.411637i	$-0.864957\pi$
$157$	−3585.62	−1.82270	−0.911348	+	0.411637i	$0.864957\pi$
$158$	−2331.73	−1.17407
$159$	−2123.63	−1.05921
$160$	−1686.38	−0.833248
$161$	0	0
$162$	159.293	0.0772547
$163$	−2178.29	−1.04673	−0.523365	−	0.852109i	$-0.675323\pi$
$163$	−2178.29	−1.04673	−0.523365	+	0.852109i	$0.675323\pi$
$164$	−896.640	−0.426926
$165$	−2038.00	−0.961563
$166$	1545.71	0.722713
$167$	−2597.93	−1.20379	−0.601897	−	0.798574i	$-0.705588\pi$
$167$	−2597.93	−1.20379	−0.601897	+	0.798574i	$0.705588\pi$
$168$	0	0
$169$	169.000	0.0769231
$170$	2189.73	0.987911
$171$	−99.7324	−0.0446007
$172$	804.704	0.356733
$173$	−3184.79	−1.39963	−0.699813	−	0.714326i	$-0.746733\pi$
$173$	−3184.79	−1.39963	−0.699813	+	0.714326i	$0.746733\pi$
$174$	−123.788	−0.0539330
$175$	0	0
$176$	−913.633	−0.391294
$177$	−2304.18	−0.978490
$178$	−3008.40	−1.26679
$179$	2586.34	1.07996	0.539978	−	0.841679i	$-0.318432\pi$
$179$	2586.34	1.07996	0.539978	+	0.841679i	$0.318432\pi$
$180$	383.341	0.158737
$181$	320.751	0.131720	0.0658598	−	0.997829i	$-0.479021\pi$
$181$	320.751	0.131720	0.0658598	+	0.997829i	$0.479021\pi$
$182$	0	0
$183$	−1176.12	−0.475087
$184$	−1637.59	−0.656113
$185$	−1607.49	−0.638838
$186$	473.469	0.186648
$187$	−7120.52	−2.78451
$188$	−1046.24	−0.405876
$189$	0	0
$190$	224.611	0.0857631
$191$	−1054.33	−0.399416	−0.199708	−	0.979855i	$-0.563999\pi$
$191$	−1054.33	−0.399416	−0.199708	+	0.979855i	$0.563999\pi$
$192$	1297.98	0.487884
$193$	3259.93	1.21583	0.607915	−	0.794002i	$-0.292006\pi$
$193$	3259.93	1.21583	0.607915	+	0.794002i	$0.292006\pi$
$194$	−1242.04	−0.459656
$195$	−401.966	−0.147617
$196$	0	0
$197$	2770.81	1.00209	0.501047	−	0.865420i	$-0.332949\pi$
$197$	2770.81	1.00209	0.501047	+	0.865420i	$0.332949\pi$
$198$	1166.57	0.418711
$199$	1328.01	0.473068	0.236534	−	0.971623i	$-0.423989\pi$
$199$	1328.01	0.473068	0.236534	+	0.971623i	$0.423989\pi$
$200$	447.831	0.158332
$201$	2490.64	0.874009
$202$	−93.6321	−0.0326135
$203$	0	0
$204$	1339.35	0.459672
$205$	−2236.27	−0.761893
$206$	−2295.94	−0.776531
$207$	617.708	0.207409
$208$	−180.201	−0.0600706
$209$	−730.384	−0.241731
$210$	0	0
$211$	−200.402	−0.0653850	−0.0326925	−	0.999465i	$-0.510408\pi$
$211$	−200.402	−0.0653850	−0.0326925	+	0.999465i	$0.510408\pi$
$212$	2925.34	0.947702
$213$	1943.23	0.625107
$214$	1041.19	0.332590
$215$	2006.98	0.636627
$216$	−644.211	−0.202931
$217$	0	0
$218$	−39.4675	−0.0122618
$219$	383.146	0.118222
$220$	2807.38	0.860333
$221$	−1404.42	−0.427473
$222$	920.147	0.278181
$223$	1054.09	0.316533	0.158267	−	0.987396i	$-0.449409\pi$
$223$	1054.09	0.316533	0.158267	+	0.987396i	$0.449409\pi$
$224$	0	0
$225$	−168.924	−0.0500517
$226$	−3059.80	−0.900598
$227$	1417.97	0.414600	0.207300	−	0.978277i	$-0.433532\pi$
$227$	1417.97	0.414600	0.207300	+	0.978277i	$0.433532\pi$
$228$	137.383	0.0399053
$229$	−3808.57	−1.09903	−0.549514	−	0.835484i	$-0.685187\pi$
$229$	−3808.57	−1.09903	−0.549514	+	0.835484i	$0.685187\pi$
$230$	−1391.16	−0.398829
$231$	0	0
$232$	500.621	0.141670
$233$	−3189.67	−0.896835	−0.448417	−	0.893824i	$-0.648012\pi$
$233$	−3189.67	−0.896835	−0.448417	+	0.893824i	$0.648012\pi$
$234$	230.090	0.0642798
$235$	−2609.38	−0.724328
$236$	3174.05	0.875478
$237$	−3557.03	−0.974912
$238$	0	0
$239$	236.764	0.0640794	0.0320397	−	0.999487i	$-0.489800\pi$
$239$	236.764	0.0640794	0.0320397	+	0.999487i	$0.489800\pi$
$240$	428.608	0.115277
$241$	−6110.84	−1.63334	−0.816668	−	0.577108i	$-0.804181\pi$
$241$	−6110.84	−1.63334	−0.816668	+	0.577108i	$0.804181\pi$
$242$	5925.82	1.57407
$243$	243.000	0.0641500
$244$	1620.12	0.425072
$245$	0	0
$246$	1280.07	0.331765
$247$	−144.058	−0.0371101
$248$	−1914.80	−0.490281
$249$	2357.96	0.600119
$250$	2914.09	0.737214
$251$	3253.27	0.818106	0.409053	−	0.912511i	$-0.365859\pi$
$251$	3253.27	0.818106	0.409053	+	0.912511i	$0.365859\pi$
$252$	0	0
$253$	4523.75	1.12413
$254$	−2618.42	−0.646828
$255$	3340.41	0.820332
$256$	−4362.13	−1.06497
$257$	4998.99	1.21334	0.606670	−	0.794954i	$-0.292505\pi$
$257$	4998.99	1.21334	0.606670	+	0.794954i	$0.292505\pi$
$258$	−1148.82	−0.277218
$259$	0	0
$260$	553.715	0.132077
$261$	−188.837	−0.0447844
$262$	−4242.71	−1.00044
$263$	−4207.34	−0.986448	−0.493224	−	0.869902i	$-0.664182\pi$
$263$	−4207.34	−0.986448	−0.493224	+	0.869902i	$0.664182\pi$
$264$	−4717.84	−1.09986
$265$	7295.96	1.69127
$266$	0	0
$267$	−4589.28	−1.05191
$268$	−3430.89	−0.781997
$269$	638.057	0.144621	0.0723104	−	0.997382i	$-0.476963\pi$
$269$	638.057	0.144621	0.0723104	+	0.997382i	$0.476963\pi$
$270$	−547.269	−0.123355
$271$	−3925.27	−0.879864	−0.439932	−	0.898031i	$-0.644997\pi$
$271$	−3925.27	−0.879864	−0.439932	+	0.898031i	$0.644997\pi$
$272$	1497.50	0.333822
$273$	0	0
$274$	−2652.75	−0.584884
$275$	−1237.11	−0.271274
$276$	−850.903	−0.185574
$277$	7696.13	1.66937	0.834685	−	0.550727i	$-0.185650\pi$
$277$	7696.13	1.66937	0.834685	+	0.550727i	$0.185650\pi$
$278$	−4060.43	−0.876002
$279$	722.272	0.154987
$280$	0	0
$281$	8259.88	1.75353	0.876767	−	0.480916i	$-0.159696\pi$
$281$	8259.88	1.75353	0.876767	+	0.480916i	$0.159696\pi$
$282$	1493.64	0.315407
$283$	−1700.91	−0.357275	−0.178637	−	0.983915i	$-0.557169\pi$
$283$	−1700.91	−0.357275	−0.178637	+	0.983915i	$0.557169\pi$
$284$	−2676.83	−0.559298
$285$	342.641	0.0712152
$286$	1685.05	0.348389
$287$	0	0
$288$	1472.56	0.301289
$289$	6757.99	1.37553
$290$	425.287	0.0861162
$291$	−1894.72	−0.381685
$292$	−527.791	−0.105776
$293$	5896.29	1.17565	0.587824	−	0.808989i	$-0.299985\pi$
$293$	5896.29	1.17565	0.587824	+	0.808989i	$0.299985\pi$
$294$	0	0
$295$	7916.26	1.56238
$296$	−3721.24	−0.730719
$297$	1779.60	0.347686
$298$	−3356.35	−0.652444
$299$	892.245	0.172575
$300$	232.696	0.0447824
$301$	0	0
$302$	621.539	0.118429
$303$	−142.835	−0.0270813
$304$	153.606	0.0289799
$305$	4040.67	0.758584
$306$	−1912.09	−0.357212
$307$	−804.548	−0.149570	−0.0747850	−	0.997200i	$-0.523827\pi$
$307$	−804.548	−0.149570	−0.0747850	+	0.997200i	$0.523827\pi$
$308$	0	0
$309$	−3502.42	−0.644809
$310$	−1626.66	−0.298025
$311$	−1310.68	−0.238977	−0.119489	−	0.992836i	$-0.538125\pi$
$311$	−1310.68	−0.238977	−0.119489	+	0.992836i	$0.538125\pi$
$312$	−930.527	−0.168848
$313$	−8819.69	−1.59271	−0.796355	−	0.604830i	$-0.793241\pi$
$313$	−8819.69	−1.59271	−0.796355	+	0.604830i	$0.793241\pi$
$314$	−7051.41	−1.26731
$315$	0	0
$316$	4899.87	0.872276
$317$	2952.84	0.523180	0.261590	−	0.965179i	$-0.415753\pi$
$317$	2952.84	0.523180	0.261590	+	0.965179i	$0.415753\pi$
$318$	−4176.29	−0.736462
$319$	−1382.94	−0.242726
$320$	−4459.35	−0.779017
$321$	1588.32	0.276173
$322$	0	0
$323$	1197.15	0.206226
$324$	−334.737	−0.0573965
$325$	−244.002	−0.0416455
$326$	−4283.79	−0.727783
$327$	−60.2073	−0.0101819
$328$	−5176.84	−0.871473
$329$	0	0
$330$	−4007.89	−0.668568
$331$	−1925.35	−0.319719	−0.159859	−	0.987140i	$-0.551104\pi$
$331$	−1925.35	−0.319719	−0.159859	+	0.987140i	$0.551104\pi$
$332$	−3248.13	−0.536941
$333$	1403.67	0.230993
$334$	−5109.04	−0.836989
$335$	−8556.85	−1.39555
$336$	0	0
$337$	7061.36	1.14141	0.570707	−	0.821154i	$-0.306669\pi$
$337$	7061.36	1.14141	0.570707	+	0.821154i	$0.306669\pi$
$338$	332.353	0.0534840
$339$	−4667.69	−0.747830
$340$	−4601.47	−0.733970
$341$	5289.52	0.840010
$342$	−196.132	−0.0310105
$343$	0	0
$344$	4646.03	0.728190
$345$	−2122.20	−0.331176
$346$	−6263.16	−0.973148
$347$	−3784.91	−0.585547	−0.292773	−	0.956182i	$-0.594578\pi$
$347$	−3784.91	−0.585547	−0.292773	+	0.956182i	$0.594578\pi$
$348$	260.126	0.0400696
$349$	6267.40	0.961279	0.480639	−	0.876918i	$-0.340405\pi$
$349$	6267.40	0.961279	0.480639	+	0.876918i	$0.340405\pi$
$350$	0	0
$351$	351.000	0.0533761
$352$	10784.2	1.63295
$353$	−7092.57	−1.06940	−0.534702	−	0.845041i	$-0.679576\pi$
$353$	−7092.57	−1.06940	−0.534702	+	0.845041i	$0.679576\pi$
$354$	−4531.36	−0.680336
$355$	−6676.17	−0.998125
$356$	6321.82	0.941167
$357$	0	0
$358$	5086.26	0.750886
$359$	−1502.90	−0.220947	−0.110473	−	0.993879i	$-0.535237\pi$
$359$	−1502.90	−0.220947	−0.110473	+	0.993879i	$0.535237\pi$
$360$	2213.26	0.324025
$361$	−6736.20	−0.982097
$362$	630.784	0.0915836
$363$	9039.77	1.30707
$364$	0	0
$365$	−1316.34	−0.188768
$366$	−2312.93	−0.330324
$367$	474.710	0.0675195	0.0337598	−	0.999430i	$-0.489252\pi$
$367$	474.710	0.0675195	0.0337598	+	0.999430i	$0.489252\pi$
$368$	−951.382	−0.134767
$369$	1952.73	0.275488
$370$	−3161.26	−0.444179
$371$	0	0
$372$	−994.942	−0.138670
$373$	8739.85	1.21322	0.606612	−	0.794998i	$-0.292528\pi$
$373$	8739.85	1.21322	0.606612	+	0.794998i	$0.292528\pi$
$374$	−14003.1	−1.93605
$375$	4445.42	0.612161
$376$	−6040.55	−0.828504
$377$	−272.765	−0.0372628
$378$	0	0
$379$	5951.16	0.806572	0.403286	−	0.915074i	$-0.367868\pi$
$379$	5951.16	0.806572	0.403286	+	0.915074i	$0.367868\pi$
$380$	−471.994	−0.0637179
$381$	−3994.37	−0.537107
$382$	−2073.43	−0.277711
$383$	−6526.43	−0.870718	−0.435359	−	0.900257i	$-0.643379\pi$
$383$	−6526.43	−0.870718	−0.435359	+	0.900257i	$0.643379\pi$
$384$	−1374.23	−0.182626
$385$	0	0
$386$	6410.93	0.845357
$387$	−1752.51	−0.230194
$388$	2610.01	0.341503
$389$	−4854.73	−0.632762	−0.316381	−	0.948632i	$-0.602468\pi$
$389$	−4854.73	−0.632762	−0.316381	+	0.948632i	$0.602468\pi$
$390$	−790.500	−0.102637
$391$	−7414.72	−0.959024
$392$	0	0
$393$	−6472.21	−0.830738
$394$	5449.04	0.696747
$395$	12220.6	1.55667
$396$	−2451.42	−0.311083
$397$	4128.64	0.521941	0.260971	−	0.965347i	$-0.415957\pi$
$397$	4128.64	0.521941	0.260971	+	0.965347i	$0.415957\pi$
$398$	2611.65	0.328920
$399$	0	0
$400$	260.174	0.0325217
$401$	4604.18	0.573371	0.286686	−	0.958025i	$-0.407447\pi$
$401$	4604.18	0.573371	0.286686	+	0.958025i	$0.407447\pi$
$402$	4898.04	0.607692
$403$	1043.28	0.128957
$404$	196.757	0.0242303
$405$	−834.853	−0.102430
$406$	0	0
$407$	10279.7	1.25196
$408$	7732.85	0.938316
$409$	10392.6	1.25643	0.628217	−	0.778038i	$-0.283785\pi$
$409$	10392.6	1.25643	0.628217	+	0.778038i	$0.283785\pi$
$410$	−4397.82	−0.529738
$411$	−4046.73	−0.485671
$412$	4824.65	0.576926
$413$	0	0
$414$	1214.77	0.144210
$415$	−8101.03	−0.958227
$416$	2127.03	0.250687
$417$	−6194.14	−0.727406
$418$	−1436.36	−0.168074
$419$	−3735.98	−0.435596	−0.217798	−	0.975994i	$-0.569887\pi$
$419$	−3735.98	−0.435596	−0.217798	+	0.975994i	$0.569887\pi$
$420$	0	0
$421$	4603.90	0.532970	0.266485	−	0.963839i	$-0.414138\pi$
$421$	4603.90	0.532970	0.266485	+	0.963839i	$0.414138\pi$
$422$	−394.107	−0.0454617
$423$	2278.53	0.261905
$424$	16889.7	1.93452
$425$	2027.70	0.231430
$426$	3821.52	0.434632
$427$	0	0
$428$	−2187.94	−0.247098
$429$	2570.53	0.289292
$430$	3946.89	0.442642
$431$	−7231.42	−0.808179	−0.404089	−	0.914720i	$-0.632412\pi$
$431$	−7231.42	−0.808179	−0.404089	+	0.914720i	$0.632412\pi$
$432$	−374.264	−0.0416823
$433$	5871.38	0.651641	0.325820	−	0.945432i	$-0.394360\pi$
$433$	5871.38	0.651641	0.325820	+	0.945432i	$0.394360\pi$
$434$	0	0
$435$	648.770	0.0715084
$436$	82.9365	0.00910995
$437$	−760.562	−0.0832554
$438$	753.489	0.0821989
$439$	7552.55	0.821102	0.410551	−	0.911838i	$-0.365336\pi$
$439$	7552.55	0.821102	0.410551	+	0.911838i	$0.365336\pi$
$440$	16208.7	1.75618
$441$	0	0
$442$	−2761.91	−0.297219
$443$	−2640.78	−0.283221	−0.141611	−	0.989922i	$-0.545228\pi$
$443$	−2640.78	−0.283221	−0.141611	+	0.989922i	$0.545228\pi$
$444$	−1933.58	−0.206675
$445$	15767.0	1.67961
$446$	2072.95	0.220083
$447$	−5120.08	−0.541770
$448$	0	0
$449$	8934.03	0.939027	0.469513	−	0.882925i	$-0.344429\pi$
$449$	8934.03	0.939027	0.469513	+	0.882925i	$0.344429\pi$
$450$	−332.204	−0.0348005
$451$	14300.7	1.49311
$452$	6429.83	0.669101
$453$	948.151	0.0983400
$454$	2788.56	0.288268
$455$	0	0
$456$	793.194	0.0814577
$457$	−9931.16	−1.01654	−0.508271	−	0.861197i	$-0.669715\pi$
$457$	−9931.16	−1.01654	−0.508271	+	0.861197i	$0.669715\pi$
$458$	−7489.87	−0.764146
$459$	−2916.87	−0.296619
$460$	2923.37	0.296311
$461$	−6986.75	−0.705869	−0.352934	−	0.935648i	$-0.614816\pi$
$461$	−6986.75	−0.705869	−0.352934	+	0.935648i	$0.614816\pi$
$462$	0	0
$463$	−6918.40	−0.694439	−0.347220	−	0.937784i	$-0.612874\pi$
$463$	−6918.40	−0.694439	−0.347220	+	0.937784i	$0.612874\pi$
$464$	290.843	0.0290992
$465$	−2481.44	−0.247471
$466$	−6272.76	−0.623562
$467$	−8224.96	−0.815001	−0.407501	−	0.913205i	$-0.633600\pi$
$467$	−8224.96	−0.815001	−0.407501	+	0.913205i	$0.633600\pi$
$468$	−483.508	−0.0477568
$469$	0	0
$470$	−5131.56	−0.503619
$471$	−10756.8	−1.05233
$472$	18325.6	1.78709
$473$	−12834.4	−1.24762
$474$	−6995.20	−0.677849
$475$	207.990	0.0200911
$476$	0	0
$477$	−6370.89	−0.611537
$478$	465.615	0.0445539
$479$	4476.57	0.427014	0.213507	−	0.976941i	$-0.431511\pi$
$479$	4476.57	0.427014	0.213507	+	0.976941i	$0.431511\pi$
$480$	−5059.13	−0.481076
$481$	2027.53	0.192198
$482$	−12017.5	−1.13565
$483$	0	0
$484$	−12452.4	−1.16946
$485$	6509.51	0.609447
$486$	477.880	0.0446030
$487$	20683.2	1.92453	0.962263	−	0.272121i	$-0.0877250\pi$
$487$	20683.2	1.92453	0.962263	+	0.272121i	$0.0877250\pi$
$488$	9353.91	0.867688
$489$	−6534.87	−0.604329
$490$	0	0
$491$	10117.4	0.929922	0.464961	−	0.885331i	$-0.346068\pi$
$491$	10117.4	0.929922	0.464961	+	0.885331i	$0.346068\pi$
$492$	−2689.92	−0.246486
$493$	2266.72	0.207075
$494$	−283.302	−0.0258023
$495$	−6114.00	−0.555159
$496$	−1112.43	−0.100705
$497$	0	0
$498$	4637.13	0.417258
$499$	10268.1	0.921165	0.460583	−	0.887617i	$-0.347641\pi$
$499$	10268.1	0.921165	0.460583	+	0.887617i	$0.347641\pi$
$500$	−6123.64	−0.547715
$501$	−7793.78	−0.695011
$502$	6397.82	0.568822
$503$	2506.71	0.222204	0.111102	−	0.993809i	$-0.464562\pi$
$503$	2506.71	0.222204	0.111102	+	0.993809i	$0.464562\pi$
$504$	0	0
$505$	490.724	0.0432415
$506$	8896.33	0.781601
$507$	507.000	0.0444116
$508$	5502.31	0.480562
$509$	14404.1	1.25433	0.627163	−	0.778888i	$-0.284216\pi$
$509$	14404.1	1.25433	0.627163	+	0.778888i	$0.284216\pi$
$510$	6569.20	0.570370
$511$	0	0
$512$	−4913.87	−0.424149
$513$	−299.197	−0.0257502
$514$	9830.92	0.843625
$515$	12033.0	1.02958
$516$	2414.11	0.205960
$517$	16686.7	1.41950
$518$	0	0
$519$	−9554.37	−0.808074
$520$	3196.93	0.269605
$521$	9809.30	0.824862	0.412431	−	0.910989i	$-0.364680\pi$
$521$	9809.30	0.824862	0.412431	+	0.910989i	$0.364680\pi$
$522$	−371.364	−0.0311382
$523$	1928.74	0.161258	0.0806292	−	0.996744i	$-0.474307\pi$
$523$	1928.74	0.161258	0.0806292	+	0.996744i	$0.474307\pi$
$524$	8915.58	0.743281
$525$	0	0
$526$	−8274.09	−0.685869
$527$	−8669.86	−0.716632
$528$	−2740.90	−0.225913
$529$	−7456.34	−0.612833
$530$	14348.1	1.17593
$531$	−6912.54	−0.564931
$532$	0	0
$533$	2820.61	0.229220
$534$	−9025.21	−0.731384
$535$	−5456.86	−0.440973
$536$	−19808.6	−1.59627
$537$	7759.03	0.623513
$538$	1254.79	0.100554
$539$	0	0
$540$	1150.02	0.0916466
$541$	−19621.1	−1.55929	−0.779645	−	0.626221i	$-0.784601\pi$
$541$	−19621.1	−1.55929	−0.779645	+	0.626221i	$0.784601\pi$
$542$	−7719.37	−0.611762
$543$	962.254	0.0760484
$544$	−17676.0	−1.39311
$545$	206.849	0.0162577
$546$	0	0
$547$	−18394.7	−1.43784	−0.718921	−	0.695092i	$-0.755364\pi$
$547$	−18394.7	−1.43784	−0.718921	+	0.695092i	$0.755364\pi$
$548$	5574.44	0.434541
$549$	−3528.35	−0.274292
$550$	−2432.87	−0.188615
$551$	232.508	0.0179767
$552$	−4912.77	−0.378807
$553$	0	0
$554$	15135.1	1.16070
$555$	−4822.47	−0.368833
$556$	8532.54	0.650828
$557$	2658.08	0.202202	0.101101	−	0.994876i	$-0.467763\pi$
$557$	2658.08	0.202202	0.101101	+	0.994876i	$0.467763\pi$
$558$	1420.41	0.107761
$559$	−2531.40	−0.191533
$560$	0	0
$561$	−21361.6	−1.60764
$562$	16243.7	1.21922
$563$	−18361.4	−1.37450	−0.687249	−	0.726422i	$-0.741182\pi$
$563$	−18361.4	−1.37450	−0.687249	+	0.726422i	$0.741182\pi$
$564$	−3138.71	−0.234333
$565$	16036.4	1.19408
$566$	−3344.99	−0.248410
$567$	0	0
$568$	−15454.9	−1.14168
$569$	−22360.5	−1.64745	−0.823725	−	0.566989i	$-0.808108\pi$
$569$	−22360.5	−1.64745	−0.823725	+	0.566989i	$0.808108\pi$
$570$	673.833	0.0495153
$571$	−26613.2	−1.95049	−0.975245	−	0.221127i	$-0.929026\pi$
$571$	−26613.2	−1.95049	−0.975245	+	0.221127i	$0.929026\pi$
$572$	−3540.95	−0.258836
$573$	−3162.99	−0.230603
$574$	0	0
$575$	−1288.22	−0.0934305
$576$	3893.94	0.281680
$577$	−2670.66	−0.192688	−0.0963440	−	0.995348i	$-0.530715\pi$
$577$	−2670.66	−0.192688	−0.0963440	+	0.995348i	$0.530715\pi$
$578$	13290.1	0.956396
$579$	9779.80	0.701960
$580$	−893.692	−0.0639802
$581$	0	0
$582$	−3726.12	−0.265383
$583$	−46656.8	−3.31446
$584$	−3047.25	−0.215918
$585$	−1205.90	−0.0852269
$586$	11595.5	0.817419
$587$	16289.7	1.14540	0.572698	−	0.819767i	$-0.305897\pi$
$587$	16289.7	1.14540	0.572698	+	0.819767i	$0.305897\pi$
$588$	0	0
$589$	−889.308	−0.0622127
$590$	15568.0	1.08631
$591$	8312.44	0.578559
$592$	−2161.91	−0.150091
$593$	4984.73	0.345191	0.172596	−	0.984993i	$-0.444785\pi$
$593$	4984.73	0.345191	0.172596	+	0.984993i	$0.444785\pi$
$594$	3499.72	0.241743
$595$	0	0
$596$	7052.99	0.484735
$597$	3984.04	0.273126
$598$	1754.67	0.119990
$599$	−8261.79	−0.563552	−0.281776	−	0.959480i	$-0.590924\pi$
$599$	−8261.79	−0.563552	−0.281776	+	0.959480i	$0.590924\pi$
$600$	1343.49	0.0914131
$601$	−7867.97	−0.534012	−0.267006	−	0.963695i	$-0.586034\pi$
$601$	−7867.97	−0.534012	−0.267006	+	0.963695i	$0.586034\pi$
$602$	0	0
$603$	7471.91	0.504610
$604$	−1306.09	−0.0879871
$605$	−31057.1	−2.08703
$606$	−280.896	−0.0188294
$607$	15106.2	1.01012	0.505059	−	0.863085i	$-0.331471\pi$
$607$	15106.2	1.01012	0.505059	+	0.863085i	$0.331471\pi$
$608$	−1813.10	−0.120939
$609$	0	0
$610$	7946.32	0.527438
$611$	3291.21	0.217918
$612$	4018.04	0.265392
$613$	−144.119	−0.00949580	−0.00474790	−	0.999989i	$-0.501511\pi$
$613$	−144.119	−0.00949580	−0.00474790	+	0.999989i	$0.501511\pi$
$614$	−1582.21	−0.103995
$615$	−6708.82	−0.439879
$616$	0	0
$617$	−17656.7	−1.15208	−0.576038	−	0.817423i	$-0.695402\pi$
$617$	−17656.7	−1.15208	−0.576038	+	0.817423i	$0.695402\pi$
$618$	−6887.81	−0.448331
$619$	−15069.2	−0.978487	−0.489244	−	0.872147i	$-0.662727\pi$
$619$	−15069.2	−0.978487	−0.489244	+	0.872147i	$0.662727\pi$
$620$	3418.23	0.221418
$621$	1853.12	0.119748
$622$	−2577.56	−0.166159
$623$	0	0
$624$	−540.603	−0.0346818
$625$	−12926.5	−0.827299
$626$	−17344.6	−1.10740
$627$	−2191.15	−0.139563
$628$	14817.7	0.941548
$629$	−16849.1	−1.06807
$630$	0	0
$631$	−2524.21	−0.159251	−0.0796253	−	0.996825i	$-0.525372\pi$
$631$	−2524.21	−0.159251	−0.0796253	+	0.996825i	$0.525372\pi$
$632$	28289.9	1.78056
$633$	−601.205	−0.0377500
$634$	5807.01	0.363763
$635$	13723.1	0.857613
$636$	8776.01	0.547156
$637$	0	0
$638$	−2719.66	−0.168765
$639$	5829.69	0.360906
$640$	4721.32	0.291604
$641$	18446.5	1.13665	0.568325	−	0.822804i	$-0.307592\pi$
$641$	18446.5	1.13665	0.568325	+	0.822804i	$0.307592\pi$
$642$	3123.57	0.192021
$643$	9541.38	0.585187	0.292594	−	0.956237i	$-0.405482\pi$
$643$	9541.38	0.585187	0.292594	+	0.956237i	$0.405482\pi$
$644$	0	0
$645$	6020.94	0.367557
$646$	2354.29	0.143387
$647$	15243.0	0.926219	0.463109	−	0.886301i	$-0.346734\pi$
$647$	15243.0	0.926219	0.463109	+	0.886301i	$0.346734\pi$
$648$	−1932.63	−0.117162
$649$	−50623.6	−3.06186
$650$	−479.850	−0.0289558
$651$	0	0
$652$	9001.89	0.540708
$653$	9063.17	0.543138	0.271569	−	0.962419i	$-0.412457\pi$
$653$	9063.17	0.543138	0.271569	+	0.962419i	$0.412457\pi$
$654$	−118.403	−0.00707937
$655$	22236.0	1.32646
$656$	−3007.56	−0.179002
$657$	1149.44	0.0682555
$658$	0	0
$659$	18173.0	1.07423	0.537116	−	0.843508i	$-0.319514\pi$
$659$	18173.0	1.07423	0.537116	+	0.843508i	$0.319514\pi$
$660$	8422.13	0.496714
$661$	−1041.37	−0.0612779	−0.0306390	−	0.999531i	$-0.509754\pi$
$661$	−1041.37	−0.0612779	−0.0306390	+	0.999531i	$0.509754\pi$
$662$	−3786.37	−0.222298
$663$	−4213.26	−0.246802
$664$	−18753.4	−1.09604
$665$	0	0
$666$	2760.44	0.160608
$667$	−1440.08	−0.0835981
$668$	10736.1	0.621843
$669$	3162.26	0.182751
$670$	−16827.8	−0.970318
$671$	−25839.6	−1.48663
$672$	0	0
$673$	15334.5	0.878310	0.439155	−	0.898411i	$-0.355278\pi$
$673$	15334.5	0.878310	0.439155	+	0.898411i	$0.355278\pi$
$674$	13886.7	0.793617
$675$	−506.773	−0.0288973
$676$	−698.401	−0.0397361
$677$	−2380.19	−0.135123	−0.0675613	−	0.997715i	$-0.521522\pi$
$677$	−2380.19	−0.135123	−0.0675613	+	0.997715i	$0.521522\pi$
$678$	−9179.41	−0.519960
$679$	0	0
$680$	−26567.0	−1.49823
$681$	4253.92	0.239369
$682$	10402.3	0.584052
$683$	−1206.19	−0.0675749	−0.0337874	−	0.999429i	$-0.510757\pi$
$683$	−1206.19	−0.0675749	−0.0337874	+	0.999429i	$0.510757\pi$
$684$	412.149	0.0230393
$685$	13903.0	0.775483
$686$	0	0
$687$	−11425.7	−0.634524
$688$	2699.18	0.149572
$689$	−9202.40	−0.508829
$690$	−4173.49	−0.230264
$691$	−18505.6	−1.01879	−0.509395	−	0.860533i	$-0.670131\pi$
$691$	−18505.6	−1.01879	−0.509395	+	0.860533i	$0.670131\pi$
$692$	13161.3	0.723003
$693$	0	0
$694$	−7443.35	−0.407126
$695$	21280.6	1.16147
$696$	1501.86	0.0817931
$697$	−23439.8	−1.27381
$698$	12325.4	0.668370
$699$	−9569.02	−0.517788
$700$	0	0
$701$	−10639.5	−0.573249	−0.286625	−	0.958043i	$-0.592533\pi$
$701$	−10639.5	−0.573249	−0.286625	+	0.958043i	$0.592533\pi$
$702$	690.271	0.0371119
$703$	−1728.29	−0.0927223
$704$	28517.0	1.52667
$705$	−7828.13	−0.418191
$706$	−13948.1	−0.743548
$707$	0	0
$708$	9522.14	0.505457
$709$	7984.88	0.422960	0.211480	−	0.977382i	$-0.432172\pi$
$709$	7984.88	0.422960	0.211480	+	0.977382i	$0.432172\pi$
$710$	−13129.3	−0.693989
$711$	−10671.1	−0.562866
$712$	36499.6	1.92118
$713$	5508.07	0.289311
$714$	0	0
$715$	−8831.33	−0.461920
$716$	−10688.2	−0.557872
$717$	710.291	0.0369962
$718$	−2955.57	−0.153622
$719$	−17213.1	−0.892823	−0.446412	−	0.894828i	$-0.647298\pi$
$719$	−17213.1	−0.892823	−0.446412	+	0.894828i	$0.647298\pi$
$720$	1285.82	0.0665553
$721$	0	0
$722$	−13247.3	−0.682844
$723$	−18332.5	−0.943007
$724$	−1325.52	−0.0680423
$725$	393.817	0.0201738
$726$	17777.5	0.908793
$727$	−10806.2	−0.551278	−0.275639	−	0.961261i	$-0.588889\pi$
$727$	−10806.2	−0.551278	−0.275639	+	0.961261i	$0.588889\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	−2588.69	−0.131249
$731$	21036.4	1.06438
$732$	4860.36	0.245415
$733$	−26918.9	−1.35644	−0.678222	−	0.734857i	$-0.737249\pi$
$733$	−26918.9	−1.35644	−0.678222	+	0.734857i	$0.737249\pi$
$734$	933.557	0.0469458
$735$	0	0
$736$	11229.8	0.562410
$737$	54720.1	2.73492
$738$	3840.21	0.191545
$739$	−28907.8	−1.43896	−0.719480	−	0.694513i	$-0.755620\pi$
$739$	−28907.8	−1.43896	−0.719480	+	0.694513i	$0.755620\pi$
$740$	6643.03	0.330004
$741$	−432.174	−0.0214255
$742$	0	0
$743$	19192.8	0.947666	0.473833	−	0.880615i	$-0.342870\pi$
$743$	19192.8	0.947666	0.473833	+	0.880615i	$0.342870\pi$
$744$	−5744.39	−0.283064
$745$	17590.6	0.865059
$746$	17187.6	0.843545
$747$	7073.88	0.346479
$748$	29425.9	1.43839
$749$	0	0
$750$	8742.28	0.425631
$751$	−35433.6	−1.72169	−0.860845	−	0.508867i	$-0.830065\pi$
$751$	−35433.6	−1.72169	−0.860845	+	0.508867i	$0.830065\pi$
$752$	−3509.34	−0.170176
$753$	9759.81	0.472334
$754$	−536.414	−0.0259086
$755$	−3257.48	−0.157022
$756$	0	0
$757$	−16832.2	−0.808162	−0.404081	−	0.914723i	$-0.632409\pi$
$757$	−16832.2	−0.808162	−0.404081	+	0.914723i	$0.632409\pi$
$758$	11703.5	0.560803
$759$	13571.2	0.649019
$760$	−2725.10	−0.130066
$761$	10219.5	0.486802	0.243401	−	0.969926i	$-0.421737\pi$
$761$	10219.5	0.486802	0.243401	+	0.969926i	$0.421737\pi$
$762$	−7855.26	−0.373446
$763$	0	0
$764$	4357.07	0.206326
$765$	10021.2	0.473619
$766$	−12834.8	−0.605403
$767$	−9984.78	−0.470051
$768$	−13086.4	−0.614862
$769$	−30338.0	−1.42265	−0.711324	−	0.702865i	$-0.751904\pi$
$769$	−30338.0	−1.42265	−0.711324	+	0.702865i	$0.751904\pi$
$770$	0	0
$771$	14997.0	0.700522
$772$	−13471.8	−0.628060
$773$	−21757.6	−1.01237	−0.506187	−	0.862424i	$-0.668945\pi$
$773$	−21757.6	−1.01237	−0.506187	+	0.862424i	$0.668945\pi$
$774$	−3446.46	−0.160052
$775$	−1506.29	−0.0698161
$776$	15069.1	0.697100
$777$	0	0
$778$	−9547.22	−0.439954
$779$	−2404.33	−0.110583
$780$	1661.15	0.0762546
$781$	42693.4	1.95607
$782$	−14581.7	−0.666802
$783$	−566.511	−0.0258563
$784$	0	0
$785$	36956.3	1.68029
$786$	−12728.1	−0.577605
$787$	−7178.12	−0.325124	−0.162562	−	0.986698i	$-0.551976\pi$
$787$	−7178.12	−0.325124	−0.162562	+	0.986698i	$0.551976\pi$
$788$	−11450.5	−0.517650
$789$	−12622.0	−0.569526
$790$	24032.8	1.08234
$791$	0	0
$792$	−14153.5	−0.635005
$793$	−5096.50	−0.228225
$794$	8119.32	0.362902
$795$	21887.9	0.976457
$796$	−5488.09	−0.244372
$797$	17696.9	0.786521	0.393261	−	0.919427i	$-0.371347\pi$
$797$	17696.9	0.786521	0.393261	+	0.919427i	$0.371347\pi$
$798$	0	0
$799$	−27350.5	−1.21100
$800$	−3070.99	−0.135720
$801$	−13767.9	−0.607320
$802$	9054.50	0.398660
$803$	8417.85	0.369937
$804$	−10292.7	−0.451486
$805$	0	0
$806$	2051.70	0.0896626
$807$	1914.17	0.0834969
$808$	1136.00	0.0494607
$809$	45279.3	1.96778	0.983891	−	0.178769i	$-0.0572114\pi$
$809$	45279.3	1.96778	0.983891	+	0.178769i	$0.0572114\pi$
$810$	−1641.81	−0.0712188
$811$	−28130.9	−1.21801	−0.609007	−	0.793165i	$-0.708432\pi$
$811$	−28130.9	−1.21801	−0.609007	+	0.793165i	$0.708432\pi$
$812$	0	0
$813$	−11775.8	−0.507990
$814$	20215.9	0.870476
$815$	22451.3	0.964949
$816$	4492.51	0.192732
$817$	2157.80	0.0924014
$818$	20437.9	0.873588
$819$	0	0
$820$	9241.52	0.393570
$821$	1213.20	0.0515725	0.0257862	−	0.999667i	$-0.491791\pi$
$821$	1213.20	0.0515725	0.0257862	+	0.999667i	$0.491791\pi$
$822$	−7958.24	−0.337683
$823$	−35692.8	−1.51175	−0.755876	−	0.654715i	$-0.772789\pi$
$823$	−35692.8	−1.51175	−0.755876	+	0.654715i	$0.772789\pi$
$824$	27855.6	1.17766
$825$	−3711.32	−0.156620
$826$	0	0
$827$	−43406.1	−1.82513	−0.912563	−	0.408937i	$-0.865900\pi$
$827$	−43406.1	−1.82513	−0.912563	+	0.408937i	$0.865900\pi$
$828$	−2552.71	−0.107141
$829$	−2918.73	−0.122282	−0.0611409	−	0.998129i	$-0.519474\pi$
$829$	−2918.73	−0.122282	−0.0611409	+	0.998129i	$0.519474\pi$
$830$	−15931.4	−0.666247
$831$	23088.4	0.963811
$832$	5624.58	0.234372
$833$	0	0
$834$	−12181.3	−0.505760
$835$	26776.4	1.10974
$836$	3018.35	0.124871
$837$	2166.82	0.0894816
$838$	−7347.12	−0.302866
$839$	34816.9	1.43267	0.716336	−	0.697756i	$-0.245818\pi$
$839$	34816.9	1.43267	0.716336	+	0.697756i	$0.245818\pi$
$840$	0	0
$841$	−23948.8	−0.981949
$842$	9053.96	0.370570
$843$	24779.6	1.01240
$844$	828.171	0.0337758
$845$	−1741.85	−0.0709131
$846$	4480.92	0.182101
$847$	0	0
$848$	9812.32	0.397354
$849$	−5102.74	−0.206273
$850$	3987.64	0.160912
$851$	10704.4	0.431191
$852$	−8030.49	−0.322911
$853$	19224.6	0.771674	0.385837	−	0.922567i	$-0.373913\pi$
$853$	19224.6	0.771674	0.385837	+	0.922567i	$0.373913\pi$
$854$	0	0
$855$	1027.92	0.0411161
$856$	−12632.3	−0.504396
$857$	41170.2	1.64101	0.820507	−	0.571637i	$-0.193691\pi$
$857$	41170.2	1.64101	0.820507	+	0.571637i	$0.193691\pi$
$858$	5055.16	0.201142
$859$	−9066.63	−0.360127	−0.180064	−	0.983655i	$-0.557630\pi$
$859$	−9066.63	−0.360127	−0.180064	+	0.983655i	$0.557630\pi$
$860$	−8293.94	−0.328862
$861$	0	0
$862$	−14221.2	−0.561920
$863$	36433.1	1.43708	0.718538	−	0.695488i	$-0.244812\pi$
$863$	36433.1	1.43708	0.718538	+	0.695488i	$0.244812\pi$
$864$	4417.67	0.173949
$865$	32825.1	1.29027
$866$	11546.6	0.453081
$867$	20274.0	0.794164
$868$	0	0
$869$	−78149.2	−3.05067
$870$	1275.86	0.0497192
$871$	10792.8	0.419861
$872$	478.842	0.0185959
$873$	−5684.16	−0.220366
$874$	−1495.71	−0.0578868
$875$	0	0
$876$	−1583.37	−0.0610698
$877$	5148.99	0.198254	0.0991272	−	0.995075i	$-0.468395\pi$
$877$	5148.99	0.198254	0.0991272	+	0.995075i	$0.468395\pi$
$878$	14852.7	0.570906
$879$	17688.9	0.678761
$880$	9416.65	0.360722
$881$	−35179.3	−1.34531	−0.672657	−	0.739955i	$-0.734847\pi$
$881$	−35179.3	−1.34531	−0.672657	+	0.739955i	$0.734847\pi$
$882$	0	0
$883$	14167.5	0.539949	0.269974	−	0.962867i	$-0.412985\pi$
$883$	14167.5	0.539949	0.269974	+	0.962867i	$0.412985\pi$
$884$	5803.84	0.220819
$885$	23748.8	0.902041
$886$	−5193.30	−0.196922
$887$	35159.3	1.33093	0.665464	−	0.746430i	$-0.268234\pi$
$887$	35159.3	1.33093	0.665464	+	0.746430i	$0.268234\pi$
$888$	−11163.7	−0.421881
$889$	0	0
$890$	31007.1	1.16782
$891$	5338.79	0.200736
$892$	−4356.07	−0.163511
$893$	−2805.47	−0.105130
$894$	−10069.1	−0.376689
$895$	−26657.0	−0.995580
$896$	0	0
$897$	2676.74	0.0996361
$898$	17569.5	0.652898
$899$	−1683.85	−0.0624688
$900$	698.088	0.0258551
$901$	76473.6	2.82764
$902$	28123.6	1.03815
$903$	0	0
$904$	37123.2	1.36582
$905$	−3305.93	−0.121428
$906$	1864.62	0.0683751
$907$	14634.5	0.535757	0.267879	−	0.963453i	$-0.413677\pi$
$907$	14634.5	0.535757	0.267879	+	0.963453i	$0.413677\pi$
$908$	−5859.84	−0.214169
$909$	−428.504	−0.0156354
$910$	0	0
$911$	38854.2	1.41306	0.706530	−	0.707683i	$-0.250259\pi$
$911$	38854.2	1.41306	0.706530	+	0.707683i	$0.250259\pi$
$912$	460.817	0.0167316
$913$	51805.2	1.87788
$914$	−19530.4	−0.706794
$915$	12122.0	0.437969
$916$	15739.1	0.567724
$917$	0	0
$918$	−5736.27	−0.206237
$919$	−38399.2	−1.37832	−0.689158	−	0.724611i	$-0.742020\pi$
$919$	−38399.2	−1.37832	−0.689158	+	0.724611i	$0.742020\pi$
$920$	16878.4	0.604851
$921$	−2413.64	−0.0863542
$922$	−13740.0	−0.490785
$923$	8420.66	0.300292
$924$	0	0
$925$	−2927.34	−0.104054
$926$	−13605.6	−0.482838
$927$	−10507.3	−0.372281
$928$	−3433.00	−0.121437
$929$	−23361.9	−0.825058	−0.412529	−	0.910944i	$-0.635354\pi$
$929$	−23361.9	−0.825058	−0.412529	+	0.910944i	$0.635354\pi$
$930$	−4879.97	−0.172065
$931$	0	0
$932$	13181.5	0.463277
$933$	−3932.04	−0.137973
$934$	−16175.1	−0.566664
$935$	73389.9	2.56696
$936$	−2791.58	−0.0974847
$937$	21769.0	0.758978	0.379489	−	0.925196i	$-0.376100\pi$
$937$	21769.0	0.758978	0.379489	+	0.925196i	$0.376100\pi$
$938$	0	0
$939$	−26459.1	−0.919551
$940$	10783.4	0.374165
$941$	−42983.4	−1.48907	−0.744536	−	0.667582i	$-0.767329\pi$
$941$	−42983.4	−1.48907	−0.744536	+	0.667582i	$0.767329\pi$
$942$	−21154.2	−0.731679
$943$	14891.6	0.514249
$944$	10646.5	0.367072
$945$	0	0
$946$	−25239.9	−0.867463
$947$	17612.7	0.604367	0.302183	−	0.953250i	$-0.402285\pi$
$947$	17612.7	0.604367	0.302183	+	0.953250i	$0.402285\pi$
$948$	14699.6	0.503609
$949$	1660.30	0.0567920
$950$	409.030	0.0139692
$951$	8858.53	0.302058
$952$	0	0
$953$	15162.6	0.515389	0.257694	−	0.966227i	$-0.417037\pi$
$953$	15162.6	0.515389	0.257694	+	0.966227i	$0.417037\pi$
$954$	−12528.9	−0.425197
$955$	10866.8	0.368210
$956$	−978.438	−0.0331014
$957$	−4148.81	−0.140138
$958$	8803.56	0.296900
$959$	0	0
$960$	−13378.1	−0.449765
$961$	−23350.5	−0.783812
$962$	3987.30	0.133634
$963$	4764.97	0.159449
$964$	25253.4	0.843730
$965$	−33599.6	−1.12084
$966$	0	0
$967$	−25202.6	−0.838118	−0.419059	−	0.907959i	$-0.637640\pi$
$967$	−25202.6	−0.838118	−0.419059	+	0.907959i	$0.637640\pi$
$968$	−71895.3	−2.38719
$969$	3591.44	0.119065
$970$	12801.5	0.423743
$971$	−2022.40	−0.0668403	−0.0334202	−	0.999441i	$-0.510640\pi$
$971$	−2022.40	−0.0668403	−0.0334202	+	0.999441i	$0.510640\pi$
$972$	−1004.21	−0.0331379
$973$	0	0
$974$	40675.2	1.33811
$975$	−732.005	−0.0240440
$976$	5434.29	0.178225
$977$	50725.2	1.66105	0.830524	−	0.556983i	$-0.188041\pi$
$977$	50725.2	1.66105	0.830524	+	0.556983i	$0.188041\pi$
$978$	−12851.4	−0.420185
$979$	−100828.	−3.29160
$980$	0	0
$981$	−180.622	−0.00587850
$982$	19896.7	0.646567
$983$	−30468.7	−0.988608	−0.494304	−	0.869289i	$-0.664577\pi$
$983$	−30468.7	−0.988608	−0.494304	+	0.869289i	$0.664577\pi$
$984$	−15530.5	−0.503145
$985$	−28558.3	−0.923800
$986$	4457.70	0.143978
$987$	0	0
$988$	595.326	0.0191699
$989$	−13364.7	−0.429699
$990$	−12023.7	−0.385998
$991$	−573.809	−0.0183932	−0.00919660	−	0.999958i	$-0.502927\pi$
$991$	−573.809	−0.0183932	−0.00919660	+	0.999958i	$0.502927\pi$
$992$	13130.7	0.420262
$993$	−5776.06	−0.184590
$994$	0	0
$995$	−13687.6	−0.436107
$996$	−9744.40	−0.310003
$997$	−6382.35	−0.202739	−0.101370	−	0.994849i	$-0.532322\pi$
$997$	−6382.35	−0.202739	−0.101370	+	0.994849i	$0.532322\pi$
$998$	20193.0	0.640479
$999$	4211.02	0.133364

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1911.4.a.w.1.8		11
7.3	odd	6		273.4.i.d.79.4	✓	22
7.5	odd	6		273.4.i.d.235.4	yes	22
7.6	odd	2		1911.4.a.v.1.8		11

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
273.4.i.d.79.4	✓	22	7.3	odd	6
273.4.i.d.235.4	yes	22	7.5	odd	6
1911.4.a.v.1.8		11	7.6	odd	2
1911.4.a.w.1.8		11	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1911 = 3 \cdot 7^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1911.a (trivial)

\(f(q)\)	\(=\)	\(q+1.96658 q^{2} +3.00000 q^{3} -4.13255 q^{4} -10.3068 q^{5} +5.89975 q^{6} -23.8597 q^{8} +9.00000 q^{9} -20.2692 q^{10} +65.9110 q^{11} -12.3977 q^{12} +13.0000 q^{13} -30.9205 q^{15} -13.8616 q^{16} -108.032 q^{17} +17.6992 q^{18} -11.0814 q^{19} +42.5935 q^{20} +129.619 q^{22} +68.6342 q^{23} -71.5790 q^{24} -18.7694 q^{25} +25.5656 q^{26} +27.0000 q^{27} -20.9819 q^{29} -60.8077 q^{30} +80.2525 q^{31} +163.617 q^{32} +197.733 q^{33} -212.455 q^{34} -37.1930 q^{36} +155.964 q^{37} -21.7924 q^{38} +39.0000 q^{39} +245.917 q^{40} +216.970 q^{41} -194.723 q^{43} -272.380 q^{44} -92.7614 q^{45} +134.975 q^{46} +253.170 q^{47} -41.5849 q^{48} -36.9115 q^{50} -324.097 q^{51} -53.7232 q^{52} -707.877 q^{53} +53.0977 q^{54} -679.333 q^{55} -33.2441 q^{57} -41.2626 q^{58} -768.060 q^{59} +127.780 q^{60} -392.039 q^{61} +157.823 q^{62} +432.660 q^{64} -133.989 q^{65} +388.858 q^{66} +830.212 q^{67} +446.449 q^{68} +205.903 q^{69} +647.743 q^{71} -214.737 q^{72} +127.715 q^{73} +306.716 q^{74} -56.3081 q^{75} +45.7943 q^{76} +76.6967 q^{78} -1185.68 q^{79} +142.869 q^{80} +81.0000 q^{81} +426.690 q^{82} +785.987 q^{83} +1113.47 q^{85} -382.939 q^{86} -62.9457 q^{87} -1572.61 q^{88} -1529.76 q^{89} -182.423 q^{90} -283.634 q^{92} +240.757 q^{93} +497.880 q^{94} +114.214 q^{95} +490.852 q^{96} -631.573 q^{97} +593.199 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 11 q - q^{2} + 33 q^{3} + 31 q^{4} - 17 q^{5} - 3 q^{6} - 54 q^{8} + 99 q^{9} - 75 q^{10} + 7 q^{11} + 93 q^{12} + 143 q^{13} - 51 q^{15} + 23 q^{16} + 20 q^{17} - 9 q^{18} - 242 q^{19} - 254 q^{20} - 290 q^{22}+ \cdots + 63 q^{99}+O(q^{100}) \) 11 * q - q^2 + 33 * q^3 + 31 * q^4 - 17 * q^5 - 3 * q^6 - 54 * q^8 + 99 * q^9 - 75 * q^10 + 7 * q^11 + 93 * q^12 + 143 * q^13 - 51 * q^15 + 23 * q^16 + 20 * q^17 - 9 * q^18 - 242 * q^19 - 254 * q^20 - 290 * q^22 - 122 * q^23 - 162 * q^24 + 208 * q^25 - 13 * q^26 + 297 * q^27 - 177 * q^29 - 225 * q^30 - 493 * q^31 - 448 * q^32 + 21 * q^33 - 855 * q^34 + 279 * q^36 - 660 * q^37 - 458 * q^38 + 429 * q^39 + 530 * q^40 - 854 * q^41 - 1018 * q^43 + 1122 * q^44 - 153 * q^45 - 757 * q^46 - 414 * q^47 + 69 * q^48 + 851 * q^50 + 60 * q^51 + 403 * q^52 - 961 * q^53 - 27 * q^54 - 1003 * q^55 - 726 * q^57 + 801 * q^58 - 2053 * q^59 - 762 * q^60 - 916 * q^61 + 653 * q^62 + 1366 * q^64 - 221 * q^65 - 870 * q^66 - 2788 * q^67 + 2458 * q^68 - 366 * q^69 + 886 * q^71 - 486 * q^72 - 1916 * q^73 - 1557 * q^74 + 624 * q^75 - 2292 * q^76 - 39 * q^78 + 651 * q^79 - 4438 * q^80 + 891 * q^81 + 2148 * q^82 - 185 * q^83 - 2052 * q^85 + 4025 * q^86 - 531 * q^87 - 4503 * q^88 - 1490 * q^89 - 675 * q^90 + 586 * q^92 - 1479 * q^93 - 1796 * q^94 - 2496 * q^95 - 1344 * q^96 - 2705 * q^97 + 63 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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