LMFDB - Embedded newform 2151.4.a.d.1.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2151,4,Mod(1,2151)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2151, 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("2151.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2151 = 3^{2} \cdot 239 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2151.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:	$126.913108422$
Analytic rank:	$1$
Dimension:	$32$
Twist minimal:	no (minimal twist has level 717)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Character	$\chi$	$=$	2151.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.87098 q^{2} +6.98450 q^{4} +17.8390 q^{5} +4.77620 q^{7} +3.93099 q^{8} +O(q^{10})$	\(q-3.87098 q^{2} +6.98450 q^{4} +17.8390 q^{5} +4.77620 q^{7} +3.93099 q^{8} -69.0545 q^{10} -1.33177 q^{11} +66.8431 q^{13} -18.4886 q^{14} -71.0928 q^{16} -126.834 q^{17} +41.7374 q^{19} +124.597 q^{20} +5.15526 q^{22} +43.9321 q^{23} +193.230 q^{25} -258.749 q^{26} +33.3593 q^{28} -191.949 q^{29} +83.8113 q^{31} +243.751 q^{32} +490.973 q^{34} +85.2026 q^{35} -4.03299 q^{37} -161.565 q^{38} +70.1251 q^{40} -55.8613 q^{41} -79.8065 q^{43} -9.30175 q^{44} -170.060 q^{46} -590.519 q^{47} -320.188 q^{49} -747.991 q^{50} +466.866 q^{52} -253.603 q^{53} -23.7575 q^{55} +18.7752 q^{56} +743.032 q^{58} -429.806 q^{59} -305.167 q^{61} -324.432 q^{62} -374.813 q^{64} +1192.42 q^{65} -304.421 q^{67} -885.874 q^{68} -329.818 q^{70} -421.142 q^{71} -385.054 q^{73} +15.6116 q^{74} +291.515 q^{76} -6.36080 q^{77} -112.266 q^{79} -1268.22 q^{80} +216.238 q^{82} -110.186 q^{83} -2262.60 q^{85} +308.929 q^{86} -5.23518 q^{88} -809.268 q^{89} +319.256 q^{91} +306.843 q^{92} +2285.89 q^{94} +744.554 q^{95} +80.5607 q^{97} +1239.44 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q - 11 q^{2} + 147 q^{4} - 66 q^{5} + 58 q^{7} - 153 q^{8}+O(q^{10}) $ 32 * q - 11 * q^2 + 147 * q^4 - 66 * q^5 + 58 * q^7 - 153 * q^8	$ 32 q - 11 q^{2} + 147 q^{4} - 66 q^{5} + 58 q^{7} - 153 q^{8} + 52 q^{10} - 270 q^{11} + 48 q^{13} - 184 q^{14} + 775 q^{16} - 384 q^{17} + 216 q^{19} - 534 q^{20} + 437 q^{22} - 712 q^{23} + 1190 q^{25} - 436 q^{26} + 598 q^{28} - 562 q^{29} + 384 q^{31} - 1770 q^{32} + 452 q^{34} - 1026 q^{35} + 770 q^{37} - 733 q^{38} + 877 q^{40} - 1648 q^{41} + 1592 q^{43} - 1595 q^{44} + 532 q^{46} - 1540 q^{47} + 2134 q^{49} - 1646 q^{50} - 144 q^{52} - 1708 q^{53} + 1282 q^{55} - 2155 q^{56} + 1086 q^{58} - 2396 q^{59} + 364 q^{61} - 2180 q^{62} + 1663 q^{64} - 1520 q^{65} + 2728 q^{67} - 1545 q^{68} - 4609 q^{70} - 3322 q^{71} - 188 q^{73} - 1111 q^{74} - 3134 q^{76} - 556 q^{77} - 462 q^{79} - 6076 q^{80} - 7965 q^{82} - 4604 q^{83} - 852 q^{85} - 549 q^{86} - 1127 q^{88} - 6742 q^{89} + 1390 q^{91} - 1802 q^{92} - 2796 q^{94} - 448 q^{95} - 1322 q^{97} - 1000 q^{98}+O(q^{100}) $ 32 * q - 11 * q^2 + 147 * q^4 - 66 * q^5 + 58 * q^7 - 153 * q^8 + 52 * q^10 - 270 * q^11 + 48 * q^13 - 184 * q^14 + 775 * q^16 - 384 * q^17 + 216 * q^19 - 534 * q^20 + 437 * q^22 - 712 * q^23 + 1190 * q^25 - 436 * q^26 + 598 * q^28 - 562 * q^29 + 384 * q^31 - 1770 * q^32 + 452 * q^34 - 1026 * q^35 + 770 * q^37 - 733 * q^38 + 877 * q^40 - 1648 * q^41 + 1592 * q^43 - 1595 * q^44 + 532 * q^46 - 1540 * q^47 + 2134 * q^49 - 1646 * q^50 - 144 * q^52 - 1708 * q^53 + 1282 * q^55 - 2155 * q^56 + 1086 * q^58 - 2396 * q^59 + 364 * q^61 - 2180 * q^62 + 1663 * q^64 - 1520 * q^65 + 2728 * q^67 - 1545 * q^68 - 4609 * q^70 - 3322 * q^71 - 188 * q^73 - 1111 * q^74 - 3134 * q^76 - 556 * q^77 - 462 * q^79 - 6076 * q^80 - 7965 * q^82 - 4604 * q^83 - 852 * q^85 - 549 * q^86 - 1127 * q^88 - 6742 * q^89 + 1390 * q^91 - 1802 * q^92 - 2796 * q^94 - 448 * q^95 - 1322 * q^97 - 1000 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−3.87098	−1.36860	−0.684299	−	0.729201i	$-0.739892\pi$
$2$	−3.87098	−1.36860	−0.684299	+	0.729201i	$0.739892\pi$
$3$	0	0
$3$	0	0
$4$	6.98450	0.873062
$5$	17.8390	1.59557	0.797785	−	0.602942i	$-0.206005\pi$
$5$	17.8390	1.59557	0.797785	+	0.602942i	$0.206005\pi$
$6$	0	0
$7$	4.77620	0.257890	0.128945	−	0.991652i	$-0.458841\pi$
$7$	4.77620	0.257890	0.128945	+	0.991652i	$0.458841\pi$
$8$	3.93099	0.173727
$9$	0	0
$10$	−69.0545	−2.18369
$11$	−1.33177	−0.0365040	−0.0182520	−	0.999833i	$-0.505810\pi$
$11$	−1.33177	−0.0365040	−0.0182520	+	0.999833i	$0.505810\pi$
$12$	0	0
$13$	66.8431	1.42607	0.713037	−	0.701127i	$-0.247319\pi$
$13$	66.8431	1.42607	0.713037	+	0.701127i	$0.247319\pi$
$14$	−18.4886	−0.352948
$15$	0	0
$16$	−71.0928	−1.11082
$17$	−126.834	−1.80952	−0.904760	−	0.425921i	$-0.859950\pi$
$17$	−126.834	−1.80952	−0.904760	+	0.425921i	$0.859950\pi$
$18$	0	0
$19$	41.7374	0.503959	0.251979	−	0.967733i	$-0.418918\pi$
$19$	41.7374	0.503959	0.251979	+	0.967733i	$0.418918\pi$
$20$	124.597	1.39303
$21$	0	0
$22$	5.15526	0.0499593
$23$	43.9321	0.398281	0.199141	−	0.979971i	$-0.436185\pi$
$23$	43.9321	0.398281	0.199141	+	0.979971i	$0.436185\pi$
$24$	0	0
$25$	193.230	1.54584
$26$	−258.749	−1.95172
$27$	0	0
$28$	33.3593	0.225154
$29$	−191.949	−1.22911	−0.614553	−	0.788875i	$-0.710664\pi$
$29$	−191.949	−1.22911	−0.614553	+	0.788875i	$0.710664\pi$
$30$	0	0
$31$	83.8113	0.485579	0.242789	−	0.970079i	$-0.421938\pi$
$31$	83.8113	0.485579	0.242789	+	0.970079i	$0.421938\pi$
$32$	243.751	1.34655
$33$	0	0
$34$	490.973	2.47651
$35$	85.2026	0.411482
$36$	0	0
$37$	−4.03299	−0.0179195	−0.00895973	−	0.999960i	$-0.502852\pi$
$37$	−4.03299	−0.0179195	−0.00895973	+	0.999960i	$0.502852\pi$
$38$	−161.565	−0.689717
$39$	0	0
$40$	70.1251	0.277194
$41$	−55.8613	−0.212782	−0.106391	−	0.994324i	$-0.533930\pi$
$41$	−55.8613	−0.212782	−0.106391	+	0.994324i	$0.533930\pi$
$42$	0	0
$43$	−79.8065	−0.283032	−0.141516	−	0.989936i	$-0.545198\pi$
$43$	−79.8065	−0.283032	−0.141516	+	0.989936i	$0.545198\pi$
$44$	−9.30175	−0.0318703
$45$	0	0
$46$	−170.060	−0.545087
$47$	−590.519	−1.83268	−0.916340	−	0.400400i	$-0.868871\pi$
$47$	−590.519	−1.83268	−0.916340	+	0.400400i	$0.868871\pi$
$48$	0	0
$49$	−320.188	−0.933493
$50$	−747.991	−2.11564
$51$	0	0
$52$	466.866	1.24505
$53$	−253.603	−0.657265	−0.328632	−	0.944458i	$-0.606588\pi$
$53$	−253.603	−0.657265	−0.328632	+	0.944458i	$0.606588\pi$
$54$	0	0
$55$	−23.7575	−0.0582447
$56$	18.7752	0.0448025
$57$	0	0
$58$	743.032	1.68215
$59$	−429.806	−0.948405	−0.474203	−	0.880416i	$-0.657264\pi$
$59$	−429.806	−0.948405	−0.474203	+	0.880416i	$0.657264\pi$
$60$	0	0
$61$	−305.167	−0.640535	−0.320268	−	0.947327i	$-0.603773\pi$
$61$	−305.167	−0.640535	−0.320268	+	0.947327i	$0.603773\pi$
$62$	−324.432	−0.664563
$63$	0	0
$64$	−374.813	−0.732056
$65$	1192.42	2.27540
$66$	0	0
$67$	−304.421	−0.555089	−0.277545	−	0.960713i	$-0.589521\pi$
$67$	−304.421	−0.555089	−0.277545	+	0.960713i	$0.589521\pi$
$68$	−885.874	−1.57982
$69$	0	0
$70$	−329.818	−0.563154
$71$	−421.142	−0.703948	−0.351974	−	0.936010i	$-0.614489\pi$
$71$	−421.142	−0.703948	−0.351974	+	0.936010i	$0.614489\pi$
$72$	0	0
$73$	−385.054	−0.617358	−0.308679	−	0.951166i	$-0.599887\pi$
$73$	−385.054	−0.617358	−0.308679	+	0.951166i	$0.599887\pi$
$74$	15.6116	0.0245245
$75$	0	0
$76$	291.515	0.439987
$77$	−6.36080	−0.00941403
$78$	0	0
$79$	−112.266	−0.159885	−0.0799427	−	0.996799i	$-0.525474\pi$
$79$	−112.266	−0.159885	−0.0799427	+	0.996799i	$0.525474\pi$
$80$	−1268.22	−1.77240
$81$	0	0
$82$	216.238	0.291213
$83$	−110.186	−0.145716	−0.0728582	−	0.997342i	$-0.523212\pi$
$83$	−110.186	−0.145716	−0.0728582	+	0.997342i	$0.523212\pi$
$84$	0	0
$85$	−2262.60	−2.88722
$86$	308.929	0.387357
$87$	0	0
$88$	−5.23518	−0.00634173
$89$	−809.268	−0.963846	−0.481923	−	0.876214i	$-0.660061\pi$
$89$	−809.268	−0.963846	−0.481923	+	0.876214i	$0.660061\pi$
$90$	0	0
$91$	319.256	0.367770
$92$	306.843	0.347724
$93$	0	0
$94$	2285.89	2.50820
$95$	744.554	0.804101
$96$	0	0
$97$	80.5607	0.0843268	0.0421634	−	0.999111i	$-0.486575\pi$
$97$	80.5607	0.0843268	0.0421634	+	0.999111i	$0.486575\pi$
$98$	1239.44	1.27758
$99$	0	0
$100$	1349.62	1.34962
$101$	978.223	0.963731	0.481865	−	0.876245i	$-0.339959\pi$
$101$	978.223	0.963731	0.481865	+	0.876245i	$0.339959\pi$
$102$	0	0
$103$	−769.032	−0.735679	−0.367840	−	0.929889i	$-0.619902\pi$
$103$	−769.032	−0.735679	−0.367840	+	0.929889i	$0.619902\pi$
$104$	262.760	0.247748
$105$	0	0
$106$	981.692	0.899531
$107$	−1772.18	−1.60115	−0.800574	−	0.599234i	$-0.795472\pi$
$107$	−1772.18	−1.60115	−0.800574	+	0.599234i	$0.795472\pi$
$108$	0	0
$109$	−848.889	−0.745952	−0.372976	−	0.927841i	$-0.621663\pi$
$109$	−848.889	−0.745952	−0.372976	+	0.927841i	$0.621663\pi$
$110$	91.9647	0.0797136
$111$	0	0
$112$	−339.553	−0.286471
$113$	−826.155	−0.687771	−0.343885	−	0.939012i	$-0.611743\pi$
$113$	−826.155	−0.687771	−0.343885	+	0.939012i	$0.611743\pi$
$114$	0	0
$115$	783.705	0.635486
$116$	−1340.67	−1.07309
$117$	0	0
$118$	1663.77	1.29799
$119$	−605.786	−0.466658
$120$	0	0
$121$	−1329.23	−0.998667
$122$	1181.30	0.876635
$123$	0	0
$124$	585.380	0.423941
$125$	1217.16	0.870929
$126$	0	0
$127$	−1228.92	−0.858656	−0.429328	−	0.903149i	$-0.641250\pi$
$127$	−1228.92	−0.858656	−0.429328	+	0.903149i	$0.641250\pi$
$128$	−499.114	−0.344655
$129$	0	0
$130$	−4615.82	−3.11411
$131$	699.708	0.466670	0.233335	−	0.972396i	$-0.425036\pi$
$131$	699.708	0.466670	0.233335	+	0.972396i	$0.425036\pi$
$132$	0	0
$133$	199.346	0.129966
$134$	1178.41	0.759694
$135$	0	0
$136$	−498.585	−0.314363
$137$	1306.42	0.814707	0.407353	−	0.913271i	$-0.366452\pi$
$137$	1306.42	0.814707	0.407353	+	0.913271i	$0.366452\pi$
$138$	0	0
$139$	−738.032	−0.450353	−0.225177	−	0.974318i	$-0.572296\pi$
$139$	−738.032	−0.450353	−0.225177	+	0.974318i	$0.572296\pi$
$140$	595.097	0.359249
$141$	0	0
$142$	1630.23	0.963422
$143$	−89.0197	−0.0520574
$144$	0	0
$145$	−3424.19	−1.96113
$146$	1490.54	0.844915
$147$	0	0
$148$	−28.1684	−0.0156448
$149$	143.257	0.0787655	0.0393827	−	0.999224i	$-0.487461\pi$
$149$	143.257	0.0787655	0.0393827	+	0.999224i	$0.487461\pi$
$150$	0	0
$151$	−1713.17	−0.923285	−0.461643	−	0.887066i	$-0.652740\pi$
$151$	−1713.17	−0.923285	−0.461643	+	0.887066i	$0.652740\pi$
$152$	164.070	0.0875513
$153$	0	0
$154$	24.6225	0.0128840
$155$	1495.11	0.774775
$156$	0	0
$157$	2126.20	1.08082	0.540412	−	0.841401i	$-0.318268\pi$
$157$	2126.20	1.08082	0.540412	+	0.841401i	$0.318268\pi$
$158$	434.581	0.218819
$159$	0	0
$160$	4348.27	2.14851
$161$	209.828	0.102713
$162$	0	0
$163$	−496.628	−0.238644	−0.119322	−	0.992856i	$-0.538072\pi$
$163$	−496.628	−0.238644	−0.119322	+	0.992856i	$0.538072\pi$
$164$	−390.163	−0.185772
$165$	0	0
$166$	426.527	0.199427
$167$	11.7491	0.00544413	0.00272207	−	0.999996i	$-0.499134\pi$
$167$	11.7491	0.00544413	0.00272207	+	0.999996i	$0.499134\pi$
$168$	0	0
$169$	2271.01	1.03369
$170$	8758.48	3.95144
$171$	0	0
$172$	−557.408	−0.247105
$173$	3868.34	1.70002	0.850012	−	0.526763i	$-0.176595\pi$
$173$	3868.34	1.70002	0.850012	+	0.526763i	$0.176595\pi$
$174$	0	0
$175$	922.906	0.398658
$176$	94.6793	0.0405495
$177$	0	0
$178$	3132.66	1.31912
$179$	641.199	0.267740	0.133870	−	0.990999i	$-0.457260\pi$
$179$	641.199	0.267740	0.133870	+	0.990999i	$0.457260\pi$
$180$	0	0
$181$	2977.91	1.22291	0.611454	−	0.791280i	$-0.290585\pi$
$181$	2977.91	1.22291	0.611454	+	0.791280i	$0.290585\pi$
$182$	−1235.83	−0.503330
$183$	0	0
$184$	172.697	0.0691922
$185$	−71.9446	−0.0285917
$186$	0	0
$187$	168.914	0.0660547
$188$	−4124.48	−1.60004
$189$	0	0
$190$	−2882.16	−1.10049
$191$	2188.88	0.829223	0.414611	−	0.909999i	$-0.363917\pi$
$191$	2188.88	0.829223	0.414611	+	0.909999i	$0.363917\pi$
$192$	0	0
$193$	180.655	0.0673772	0.0336886	−	0.999432i	$-0.489275\pi$
$193$	180.655	0.0673772	0.0336886	+	0.999432i	$0.489275\pi$
$194$	−311.849	−0.115410
$195$	0	0
$196$	−2236.35	−0.814997
$197$	−3320.18	−1.20078	−0.600389	−	0.799708i	$-0.704987\pi$
$197$	−3320.18	−1.20078	−0.600389	+	0.799708i	$0.704987\pi$
$198$	0	0
$199$	3069.97	1.09359	0.546795	−	0.837267i	$-0.315848\pi$
$199$	3069.97	1.09359	0.546795	+	0.837267i	$0.315848\pi$
$200$	759.587	0.268555
$201$	0	0
$202$	−3786.68	−1.31896
$203$	−916.788	−0.316975
$204$	0	0
$205$	−996.509	−0.339508
$206$	2976.91	1.00685
$207$	0	0
$208$	−4752.07	−1.58412
$209$	−55.5847	−0.0183965
$210$	0	0
$211$	2836.05	0.925316	0.462658	−	0.886537i	$-0.346896\pi$
$211$	2836.05	0.925316	0.462658	+	0.886537i	$0.346896\pi$
$212$	−1771.29	−0.573833
$213$	0	0
$214$	6860.07	2.19133
$215$	−1423.67	−0.451597
$216$	0	0
$217$	400.299	0.125226
$218$	3286.03	1.02091
$219$	0	0
$220$	−165.934	−0.0508512
$221$	−8478.01	−2.58051
$222$	0	0
$223$	4310.08	1.29428	0.647140	−	0.762372i	$-0.275965\pi$
$223$	4310.08	1.29428	0.647140	+	0.762372i	$0.275965\pi$
$224$	1164.20	0.347261
$225$	0	0
$226$	3198.03	0.941282
$227$	1698.27	0.496556	0.248278	−	0.968689i	$-0.420135\pi$
$227$	1698.27	0.496556	0.248278	+	0.968689i	$0.420135\pi$
$228$	0	0
$229$	5190.84	1.49791	0.748953	−	0.662623i	$-0.230557\pi$
$229$	5190.84	1.49791	0.748953	+	0.662623i	$0.230557\pi$
$230$	−3033.71	−0.869725
$231$	0	0
$232$	−754.552	−0.213529
$233$	1916.41	0.538834	0.269417	−	0.963024i	$-0.413169\pi$
$233$	1916.41	0.538834	0.269417	+	0.963024i	$0.413169\pi$
$234$	0	0
$235$	−10534.3	−2.92417
$236$	−3001.98	−0.828017
$237$	0	0
$238$	2344.98	0.638667
$239$	−239.000	−0.0646846
$239$	−239.000	−0.0646846
$240$	0	0
$241$	1899.75	0.507775	0.253887	−	0.967234i	$-0.418291\pi$
$241$	1899.75	0.507775	0.253887	+	0.967234i	$0.418291\pi$
$242$	5145.41	1.36677
$243$	0	0
$244$	−2131.44	−0.559227
$245$	−5711.84	−1.48945
$246$	0	0
$247$	2789.86	0.718682
$248$	329.462	0.0843582
$249$	0	0
$250$	−4711.61	−1.19195
$251$	−6874.34	−1.72870	−0.864352	−	0.502888i	$-0.832271\pi$
$251$	−6874.34	−1.72870	−0.864352	+	0.502888i	$0.832271\pi$
$252$	0	0
$253$	−58.5074	−0.0145389
$254$	4757.14	1.17516
$255$	0	0
$256$	4930.56	1.20375
$257$	−4517.11	−1.09638	−0.548190	−	0.836354i	$-0.684683\pi$
$257$	−4517.11	−1.09638	−0.548190	+	0.836354i	$0.684683\pi$
$258$	0	0
$259$	−19.2624	−0.00462125
$260$	8328.42	1.98656
$261$	0	0
$262$	−2708.56	−0.638684
$263$	3317.95	0.777921	0.388961	−	0.921254i	$-0.372834\pi$
$263$	3317.95	0.777921	0.388961	+	0.921254i	$0.372834\pi$
$264$	0	0
$265$	−4524.02	−1.04871
$266$	−771.665	−0.177871
$267$	0	0
$268$	−2126.23	−0.484627
$269$	1753.30	0.397401	0.198700	−	0.980060i	$-0.436328\pi$
$269$	1753.30	0.397401	0.198700	+	0.980060i	$0.436328\pi$
$270$	0	0
$271$	−4781.53	−1.07180	−0.535898	−	0.844282i	$-0.680027\pi$
$271$	−4781.53	−1.07180	−0.535898	+	0.844282i	$0.680027\pi$
$272$	9017.01	2.01006
$273$	0	0
$274$	−5057.12	−1.11501
$275$	−257.338	−0.0564294
$276$	0	0
$277$	−7661.62	−1.66188	−0.830942	−	0.556359i	$-0.812198\pi$
$277$	−7661.62	−1.66188	−0.830942	+	0.556359i	$0.812198\pi$
$278$	2856.91	0.616353
$279$	0	0
$280$	334.931	0.0714855
$281$	3290.89	0.698640	0.349320	−	0.937004i	$-0.386413\pi$
$281$	3290.89	0.698640	0.349320	+	0.937004i	$0.386413\pi$
$282$	0	0
$283$	2898.72	0.608873	0.304436	−	0.952533i	$-0.401532\pi$
$283$	2898.72	0.608873	0.304436	+	0.952533i	$0.401532\pi$
$284$	−2941.46	−0.614590
$285$	0	0
$286$	344.594	0.0712456
$287$	−266.804	−0.0548744
$288$	0	0
$289$	11174.0	2.27436
$290$	13255.0	2.68399
$291$	0	0
$292$	−2689.41	−0.538992
$293$	7669.52	1.52921	0.764605	−	0.644500i	$-0.222934\pi$
$293$	7669.52	1.52921	0.764605	+	0.644500i	$0.222934\pi$
$294$	0	0
$295$	−7667.31	−1.51325
$296$	−15.8537	−0.00311309
$297$	0	0
$298$	−554.545	−0.107798
$299$	2936.56	0.567978
$300$	0	0
$301$	−381.171	−0.0729912
$302$	6631.66	1.26361
$303$	0	0
$304$	−2967.23	−0.559810
$305$	−5443.88	−1.02202
$306$	0	0
$307$	−5688.02	−1.05743	−0.528717	−	0.848798i	$-0.677327\pi$
$307$	−5688.02	−1.05743	−0.528717	+	0.848798i	$0.677327\pi$
$308$	−44.4270	−0.00821903
$309$	0	0
$310$	−5787.54	−1.06036
$311$	−818.692	−0.149273	−0.0746363	−	0.997211i	$-0.523780\pi$
$311$	−818.692	−0.149273	−0.0746363	+	0.997211i	$0.523780\pi$
$312$	0	0
$313$	−4592.60	−0.829358	−0.414679	−	0.909968i	$-0.636106\pi$
$313$	−4592.60	−0.829358	−0.414679	+	0.909968i	$0.636106\pi$
$314$	−8230.48	−1.47921
$315$	0	0
$316$	−784.124	−0.139590
$317$	3123.28	0.553379	0.276689	−	0.960959i	$-0.410763\pi$
$317$	3123.28	0.553379	0.276689	+	0.960959i	$0.410763\pi$
$318$	0	0
$319$	255.632	0.0448673
$320$	−6686.29	−1.16805
$321$	0	0
$322$	−812.241	−0.140573
$323$	−5293.74	−0.911924
$324$	0	0
$325$	12916.1	2.20448
$326$	1922.44	0.326607
$327$	0	0
$328$	−219.590	−0.0369660
$329$	−2820.43	−0.472631
$330$	0	0
$331$	3969.00	0.659081	0.329541	−	0.944141i	$-0.393106\pi$
$331$	3969.00	0.659081	0.329541	+	0.944141i	$0.393106\pi$
$332$	−769.593	−0.127220
$333$	0	0
$334$	−45.4804	−0.00745083
$335$	−5430.57	−0.885683
$336$	0	0
$337$	6792.75	1.09800	0.548998	−	0.835824i	$-0.315009\pi$
$337$	6792.75	1.09800	0.548998	+	0.835824i	$0.315009\pi$
$338$	−8791.02	−1.41470
$339$	0	0
$340$	−15803.1	−2.52072
$341$	−111.617	−0.0177256
$342$	0	0
$343$	−3167.52	−0.498629
$344$	−313.719	−0.0491703
$345$	0	0
$346$	−14974.3	−2.32665
$347$	−9171.16	−1.41883	−0.709415	−	0.704791i	$-0.751041\pi$
$347$	−9171.16	−1.41883	−0.709415	+	0.704791i	$0.751041\pi$
$348$	0	0
$349$	−509.248	−0.0781072	−0.0390536	−	0.999237i	$-0.512434\pi$
$349$	−509.248	−0.0781072	−0.0390536	+	0.999237i	$0.512434\pi$
$350$	−3572.55	−0.545602
$351$	0	0
$352$	−324.620	−0.0491543
$353$	−12698.0	−1.91457	−0.957287	−	0.289139i	$-0.906631\pi$
$353$	−12698.0	−1.91457	−0.957287	+	0.289139i	$0.906631\pi$
$354$	0	0
$355$	−7512.75	−1.12320
$356$	−5652.33	−0.841497
$357$	0	0
$358$	−2482.07	−0.366428
$359$	−893.956	−0.131424	−0.0657120	−	0.997839i	$-0.520932\pi$
$359$	−893.956	−0.131424	−0.0657120	+	0.997839i	$0.520932\pi$
$360$	0	0
$361$	−5116.99	−0.746025
$362$	−11527.4	−1.67367
$363$	0	0
$364$	2229.84	0.321086
$365$	−6868.98	−0.985038
$366$	0	0
$367$	−10694.3	−1.52109	−0.760544	−	0.649287i	$-0.775068\pi$
$367$	−10694.3	−1.52109	−0.760544	+	0.649287i	$0.775068\pi$
$368$	−3123.25	−0.442421
$369$	0	0
$370$	278.496	0.0391306
$371$	−1211.26	−0.169502
$372$	0	0
$373$	1613.30	0.223951	0.111976	−	0.993711i	$-0.464282\pi$
$373$	1613.30	0.223951	0.111976	+	0.993711i	$0.464282\pi$
$374$	−653.864	−0.0904024
$375$	0	0
$376$	−2321.33	−0.318386
$377$	−12830.5	−1.75280
$378$	0	0
$379$	7387.35	1.00122	0.500611	−	0.865673i	$-0.333109\pi$
$379$	7387.35	1.00122	0.500611	+	0.865673i	$0.333109\pi$
$380$	5200.34	0.702030
$381$	0	0
$382$	−8473.10	−1.13487
$383$	7872.20	1.05026	0.525131	−	0.851021i	$-0.324016\pi$
$383$	7872.20	1.05026	0.525131	+	0.851021i	$0.324016\pi$
$384$	0	0
$385$	−113.470	−0.0150207
$386$	−699.310	−0.0922123
$387$	0	0
$388$	562.676	0.0736226
$389$	2665.26	0.347388	0.173694	−	0.984800i	$-0.444430\pi$
$389$	2665.26	0.347388	0.173694	+	0.984800i	$0.444430\pi$
$390$	0	0
$391$	−5572.10	−0.720698
$392$	−1258.66	−0.162173
$393$	0	0
$394$	12852.4	1.64338
$395$	−2002.72	−0.255108
$396$	0	0
$397$	−4945.06	−0.625152	−0.312576	−	0.949893i	$-0.601192\pi$
$397$	−4945.06	−0.625152	−0.312576	+	0.949893i	$0.601192\pi$
$398$	−11883.8	−1.49669
$399$	0	0
$400$	−13737.3	−1.71716
$401$	3479.84	0.433354	0.216677	−	0.976243i	$-0.430478\pi$
$401$	3479.84	0.433354	0.216677	+	0.976243i	$0.430478\pi$
$402$	0	0
$403$	5602.21	0.692471
$404$	6832.39	0.841397
$405$	0	0
$406$	3548.87	0.433811
$407$	5.37102	0.000654132 0
$408$	0	0
$409$	−14116.6	−1.70665	−0.853324	−	0.521381i	$-0.825417\pi$
$409$	−14116.6	−1.70665	−0.853324	+	0.521381i	$0.825417\pi$
$410$	3857.47	0.464651
$411$	0	0
$412$	−5371.30	−0.642294
$413$	−2052.84	−0.244585
$414$	0	0
$415$	−1965.61	−0.232501
$416$	16293.1	1.92027
$417$	0	0
$418$	215.167	0.0251774
$419$	−3491.29	−0.407066	−0.203533	−	0.979068i	$-0.565242\pi$
$419$	−3491.29	−0.407066	−0.203533	+	0.979068i	$0.565242\pi$
$420$	0	0
$421$	−9501.05	−1.09989	−0.549944	−	0.835202i	$-0.685351\pi$
$421$	−9501.05	−1.09989	−0.549944	+	0.835202i	$0.685351\pi$
$422$	−10978.3	−1.26639
$423$	0	0
$424$	−996.911	−0.114185
$425$	−24508.2	−2.79723
$426$	0	0
$427$	−1457.54	−0.165188
$428$	−12377.8	−1.39790
$429$	0	0
$430$	5511.00	0.618055
$431$	3433.18	0.383690	0.191845	−	0.981425i	$-0.438553\pi$
$431$	3433.18	0.383690	0.191845	+	0.981425i	$0.438553\pi$
$432$	0	0
$433$	13492.9	1.49752	0.748759	−	0.662842i	$-0.230650\pi$
$433$	13492.9	1.49752	0.748759	+	0.662842i	$0.230650\pi$
$434$	−1549.55	−0.171384
$435$	0	0
$436$	−5929.06	−0.651263
$437$	1833.61	0.200717
$438$	0	0
$439$	−2345.78	−0.255029	−0.127515	−	0.991837i	$-0.540700\pi$
$439$	−2345.78	−0.255029	−0.127515	+	0.991837i	$0.540700\pi$
$440$	−93.3905	−0.0101187
$441$	0	0
$442$	32818.2	3.53168
$443$	9894.05	1.06113	0.530565	−	0.847644i	$-0.321980\pi$
$443$	9894.05	1.06113	0.530565	+	0.847644i	$0.321980\pi$
$444$	0	0
$445$	−14436.5	−1.53788
$446$	−16684.2	−1.77135
$447$	0	0
$448$	−1790.18	−0.188790
$449$	4872.44	0.512126	0.256063	−	0.966660i	$-0.417575\pi$
$449$	4872.44	0.512126	0.256063	+	0.966660i	$0.417575\pi$
$450$	0	0
$451$	74.3944	0.00776739
$452$	−5770.27	−0.600467
$453$	0	0
$454$	−6573.97	−0.679585
$455$	5695.21	0.586803
$456$	0	0
$457$	−17229.6	−1.76360	−0.881801	−	0.471623i	$-0.843669\pi$
$457$	−17229.6	−1.76360	−0.881801	+	0.471623i	$0.843669\pi$
$458$	−20093.7	−2.05003
$459$	0	0
$460$	5473.78	0.554818
$461$	−6338.48	−0.640374	−0.320187	−	0.947354i	$-0.603746\pi$
$461$	−6338.48	−0.640374	−0.320187	+	0.947354i	$0.603746\pi$
$462$	0	0
$463$	−9341.33	−0.937642	−0.468821	−	0.883293i	$-0.655321\pi$
$463$	−9341.33	−0.937642	−0.468821	+	0.883293i	$0.655321\pi$
$464$	13646.2	1.36532
$465$	0	0
$466$	−7418.40	−0.737448
$467$	273.650	0.0271157	0.0135578	−	0.999908i	$-0.495684\pi$
$467$	273.650	0.0271157	0.0135578	+	0.999908i	$0.495684\pi$
$468$	0	0
$469$	−1453.98	−0.143152
$470$	40778.0	4.00201
$471$	0	0
$472$	−1689.56	−0.164764
$473$	106.284	0.0103318
$474$	0	0
$475$	8064.93	0.779041
$476$	−4231.11	−0.407421
$477$	0	0
$478$	925.165	0.0885273
$479$	19220.8	1.83345	0.916723	−	0.399524i	$-0.130825\pi$
$479$	19220.8	1.83345	0.916723	+	0.399524i	$0.130825\pi$
$480$	0	0
$481$	−269.578	−0.0255545
$482$	−7353.90	−0.694940
$483$	0	0
$484$	−9283.98	−0.871899
$485$	1437.12	0.134549
$486$	0	0
$487$	19110.0	1.77815	0.889074	−	0.457763i	$-0.151349\pi$
$487$	19110.0	1.77815	0.889074	+	0.457763i	$0.151349\pi$
$488$	−1199.61	−0.111278
$489$	0	0
$490$	22110.4	2.03846
$491$	3486.04	0.320413	0.160206	−	0.987084i	$-0.448784\pi$
$491$	3486.04	0.320413	0.160206	+	0.987084i	$0.448784\pi$
$492$	0	0
$493$	24345.8	2.22409
$494$	−10799.5	−0.983588
$495$	0	0
$496$	−5958.38	−0.539393
$497$	−2011.45	−0.181541
$498$	0	0
$499$	1788.37	0.160438	0.0802188	−	0.996777i	$-0.474438\pi$
$499$	1788.37	0.160438	0.0802188	+	0.996777i	$0.474438\pi$
$500$	8501.25	0.760375
$501$	0	0
$502$	26610.4	2.36590
$503$	15616.4	1.38429	0.692146	−	0.721758i	$-0.256665\pi$
$503$	15616.4	1.38429	0.692146	+	0.721758i	$0.256665\pi$
$504$	0	0
$505$	17450.5	1.53770
$506$	226.481	0.0198979
$507$	0	0
$508$	−8583.41	−0.749660
$509$	15019.8	1.30794	0.653968	−	0.756522i	$-0.273103\pi$
$509$	15019.8	1.30794	0.653968	+	0.756522i	$0.273103\pi$
$510$	0	0
$511$	−1839.09	−0.159211
$512$	−15093.2	−1.30280
$513$	0	0
$514$	17485.7	1.50051
$515$	−13718.8	−1.17383
$516$	0	0
$517$	786.435	0.0669002
$518$	74.5642	0.00632464
$519$	0	0
$520$	4687.38	0.395298
$521$	−10960.6	−0.921677	−0.460838	−	0.887484i	$-0.652451\pi$
$521$	−10960.6	−0.921677	−0.460838	+	0.887484i	$0.652451\pi$
$522$	0	0
$523$	−7932.97	−0.663259	−0.331629	−	0.943410i	$-0.607598\pi$
$523$	−7932.97	−0.663259	−0.331629	+	0.943410i	$0.607598\pi$
$524$	4887.11	0.407432
$525$	0	0
$526$	−12843.7	−1.06466
$527$	−10630.1	−0.878665
$528$	0	0
$529$	−10237.0	−0.841372
$530$	17512.4	1.43526
$531$	0	0
$532$	1392.33	0.113468
$533$	−3733.94	−0.303443
$534$	0	0
$535$	−31613.9	−2.55474
$536$	−1196.68	−0.0964340
$537$	0	0
$538$	−6787.00	−0.543882
$539$	426.417	0.0340762
$540$	0	0
$541$	−17581.1	−1.39717	−0.698587	−	0.715525i	$-0.746188\pi$
$541$	−17581.1	−1.39717	−0.698587	+	0.715525i	$0.746188\pi$
$542$	18509.2	1.46686
$543$	0	0
$544$	−30916.0	−2.43660
$545$	−15143.3	−1.19022
$546$	0	0
$547$	12792.7	0.999958	0.499979	−	0.866037i	$-0.333341\pi$
$547$	12792.7	0.999958	0.499979	+	0.866037i	$0.333341\pi$
$548$	9124.67	0.711290
$549$	0	0
$550$	996.152	0.0772292
$551$	−8011.47	−0.619419
$552$	0	0
$553$	−536.206	−0.0412329
$554$	29658.0	2.27445
$555$	0	0
$556$	−5154.78	−0.393186
$557$	−7066.04	−0.537518	−0.268759	−	0.963207i	$-0.586614\pi$
$557$	−7066.04	−0.537518	−0.268759	+	0.963207i	$0.586614\pi$
$558$	0	0
$559$	−5334.52	−0.403624
$560$	−6057.29	−0.457084
$561$	0	0
$562$	−12739.0	−0.956158
$563$	19917.9	1.49101	0.745504	−	0.666501i	$-0.232209\pi$
$563$	19917.9	1.49101	0.745504	+	0.666501i	$0.232209\pi$
$564$	0	0
$565$	−14737.8	−1.09739
$566$	−11220.9	−0.833303
$567$	0	0
$568$	−1655.51	−0.122295
$569$	−12727.1	−0.937694	−0.468847	−	0.883279i	$-0.655330\pi$
$569$	−12727.1	−0.937694	−0.468847	+	0.883279i	$0.655330\pi$
$570$	0	0
$571$	20573.2	1.50782	0.753908	−	0.656980i	$-0.228167\pi$
$571$	20573.2	1.50782	0.753908	+	0.656980i	$0.228167\pi$
$572$	−621.758	−0.0454493
$573$	0	0
$574$	1032.79	0.0751010
$575$	8489.01	0.615680
$576$	0	0
$577$	−7967.03	−0.574821	−0.287411	−	0.957807i	$-0.592794\pi$
$577$	−7967.03	−0.574821	−0.287411	+	0.957807i	$0.592794\pi$
$578$	−43254.2	−3.11269
$579$	0	0
$580$	−23916.2	−1.71218
$581$	−526.269	−0.0375789
$582$	0	0
$583$	337.741	0.0239928
$584$	−1513.64	−0.107252
$585$	0	0
$586$	−29688.6	−2.09287
$587$	24928.1	1.75280	0.876400	−	0.481584i	$-0.159939\pi$
$587$	24928.1	1.75280	0.876400	+	0.481584i	$0.159939\pi$
$588$	0	0
$589$	3498.07	0.244712
$590$	29680.0	2.07103
$591$	0	0
$592$	286.717	0.0199054
$593$	−9501.79	−0.657996	−0.328998	−	0.944331i	$-0.606711\pi$
$593$	−9501.79	−0.657996	−0.328998	+	0.944331i	$0.606711\pi$
$594$	0	0
$595$	−10806.6	−0.744585
$596$	1000.58	0.0687672
$597$	0	0
$598$	−11367.4	−0.777334
$599$	−23187.5	−1.58166	−0.790832	−	0.612033i	$-0.790352\pi$
$599$	−23187.5	−1.58166	−0.790832	+	0.612033i	$0.790352\pi$
$600$	0	0
$601$	−3900.87	−0.264759	−0.132379	−	0.991199i	$-0.542262\pi$
$601$	−3900.87	−0.264759	−0.132379	+	0.991199i	$0.542262\pi$
$602$	1475.51	0.0998957
$603$	0	0
$604$	−11965.7	−0.806085
$605$	−23712.1	−1.59344
$606$	0	0
$607$	−4596.63	−0.307367	−0.153683	−	0.988120i	$-0.549114\pi$
$607$	−4596.63	−0.307367	−0.153683	+	0.988120i	$0.549114\pi$
$608$	10173.5	0.678604
$609$	0	0
$610$	21073.1	1.39873
$611$	−39472.1	−2.61354
$612$	0	0
$613$	10103.4	0.665694	0.332847	−	0.942981i	$-0.391991\pi$
$613$	10103.4	0.665694	0.332847	+	0.942981i	$0.391991\pi$
$614$	22018.2	1.44720
$615$	0	0
$616$	−25.0043	−0.00163547
$617$	−8299.40	−0.541526	−0.270763	−	0.962646i	$-0.587276\pi$
$617$	−8299.40	−0.541526	−0.270763	+	0.962646i	$0.587276\pi$
$618$	0	0
$619$	19970.7	1.29676	0.648378	−	0.761319i	$-0.275448\pi$
$619$	19970.7	1.29676	0.648378	+	0.761319i	$0.275448\pi$
$620$	10442.6	0.676427
$621$	0	0
$622$	3169.14	0.204294
$623$	−3865.22	−0.248567
$624$	0	0
$625$	−2440.85	−0.156214
$626$	17777.9	1.13506
$627$	0	0
$628$	14850.4	0.943626
$629$	511.522	0.0324256
$630$	0	0
$631$	18487.8	1.16638	0.583192	−	0.812334i	$-0.301803\pi$
$631$	18487.8	1.16638	0.583192	+	0.812334i	$0.301803\pi$
$632$	−441.318	−0.0277764
$633$	0	0
$634$	−12090.2	−0.757353
$635$	−21922.8	−1.37005
$636$	0	0
$637$	−21402.4	−1.33123
$638$	−989.549	−0.0614053
$639$	0	0
$640$	−8903.69	−0.549921
$641$	−22217.1	−1.36899	−0.684495	−	0.729017i	$-0.739977\pi$
$641$	−22217.1	−1.36899	−0.684495	+	0.729017i	$0.739977\pi$
$642$	0	0
$643$	1943.35	0.119188	0.0595942	−	0.998223i	$-0.481019\pi$
$643$	1943.35	0.119188	0.0595942	+	0.998223i	$0.481019\pi$
$644$	1465.54	0.0896747
$645$	0	0
$646$	20492.0	1.24806
$647$	14471.7	0.879353	0.439677	−	0.898156i	$-0.355093\pi$
$647$	14471.7	0.879353	0.439677	+	0.898156i	$0.355093\pi$
$648$	0	0
$649$	572.402	0.0346206
$650$	−49998.1	−3.01705
$651$	0	0
$652$	−3468.70	−0.208351
$653$	−24864.8	−1.49010	−0.745050	−	0.667009i	$-0.767574\pi$
$653$	−24864.8	−1.49010	−0.745050	+	0.667009i	$0.767574\pi$
$654$	0	0
$655$	12482.1	0.744605
$656$	3971.33	0.236363
$657$	0	0
$658$	10917.8	0.646842
$659$	6228.59	0.368181	0.184091	−	0.982909i	$-0.441066\pi$
$659$	6228.59	0.368181	0.184091	+	0.982909i	$0.441066\pi$
$660$	0	0
$661$	2212.84	0.130211	0.0651054	−	0.997878i	$-0.479262\pi$
$661$	2212.84	0.130211	0.0651054	+	0.997878i	$0.479262\pi$
$662$	−15363.9	−0.902018
$663$	0	0
$664$	−433.140	−0.0253149
$665$	3556.14	0.207370
$666$	0	0
$667$	−8432.73	−0.489530
$668$	82.0613	0.00475307
$669$	0	0
$670$	21021.6	1.21214
$671$	406.412	0.0233821
$672$	0	0
$673$	13960.8	0.799628	0.399814	−	0.916596i	$-0.369075\pi$
$673$	13960.8	0.799628	0.399814	+	0.916596i	$0.369075\pi$
$674$	−26294.6	−1.50272
$675$	0	0
$676$	15861.8	0.902471
$677$	27543.2	1.56362	0.781810	−	0.623516i	$-0.214297\pi$
$677$	27543.2	1.56362	0.781810	+	0.623516i	$0.214297\pi$
$678$	0	0
$679$	384.774	0.0217471
$680$	−8894.27	−0.501588
$681$	0	0
$682$	432.069	0.0242592
$683$	−15179.6	−0.850409	−0.425205	−	0.905097i	$-0.639798\pi$
$683$	−15179.6	−0.850409	−0.425205	+	0.905097i	$0.639798\pi$
$684$	0	0
$685$	23305.2	1.29992
$686$	12261.4	0.682423
$687$	0	0
$688$	5673.67	0.314399
$689$	−16951.6	−0.937307
$690$	0	0
$691$	−14456.5	−0.795879	−0.397939	−	0.917412i	$-0.630275\pi$
$691$	−14456.5	−0.795879	−0.397939	+	0.917412i	$0.630275\pi$
$692$	27018.4	1.48423
$693$	0	0
$694$	35501.4	1.94181
$695$	−13165.8	−0.718570
$696$	0	0
$697$	7085.13	0.385033
$698$	1971.29	0.106897
$699$	0	0
$700$	6446.03	0.348053
$701$	27756.7	1.49551	0.747757	−	0.663972i	$-0.231131\pi$
$701$	27756.7	1.49551	0.747757	+	0.663972i	$0.231131\pi$
$702$	0	0
$703$	−168.327	−0.00903067
$704$	499.165	0.0267230
$705$	0	0
$706$	49153.6	2.62028
$707$	4672.18	0.248537
$708$	0	0
$709$	−23563.0	−1.24813	−0.624066	−	0.781372i	$-0.714520\pi$
$709$	−23563.0	−1.24813	−0.624066	+	0.781372i	$0.714520\pi$
$710$	29081.7	1.53721
$711$	0	0
$712$	−3181.23	−0.167446
$713$	3682.00	0.193397
$714$	0	0
$715$	−1588.02	−0.0830612
$716$	4478.45	0.233754
$717$	0	0
$718$	3460.49	0.179867
$719$	−18023.3	−0.934850	−0.467425	−	0.884033i	$-0.654818\pi$
$719$	−18023.3	−0.934850	−0.467425	+	0.884033i	$0.654818\pi$
$720$	0	0
$721$	−3673.05	−0.189725
$722$	19807.8	1.02101
$723$	0	0
$724$	20799.2	1.06768
$725$	−37090.4	−1.90001
$726$	0	0
$727$	25082.9	1.27961	0.639803	−	0.768539i	$-0.279016\pi$
$727$	25082.9	1.27961	0.639803	+	0.768539i	$0.279016\pi$
$728$	1254.99	0.0638917
$729$	0	0
$730$	26589.7	1.34812
$731$	10122.2	0.512152
$732$	0	0
$733$	−26538.5	−1.33727	−0.668637	−	0.743589i	$-0.733122\pi$
$733$	−26538.5	−1.33727	−0.668637	+	0.743589i	$0.733122\pi$
$734$	41397.5	2.08176
$735$	0	0
$736$	10708.5	0.536304
$737$	405.419	0.0202630
$738$	0	0
$739$	23939.6	1.19165	0.595826	−	0.803113i	$-0.296825\pi$
$739$	23939.6	1.19165	0.595826	+	0.803113i	$0.296825\pi$
$740$	−502.497	−0.0249624
$741$	0	0
$742$	4688.75	0.231980
$743$	−13408.1	−0.662041	−0.331021	−	0.943624i	$-0.607393\pi$
$743$	−13408.1	−0.662041	−0.331021	+	0.943624i	$0.607393\pi$
$744$	0	0
$745$	2555.56	0.125676
$746$	−6245.07	−0.306499
$747$	0	0
$748$	1179.78	0.0576699
$749$	−8464.27	−0.412921
$750$	0	0
$751$	−4508.43	−0.219061	−0.109530	−	0.993983i	$-0.534935\pi$
$751$	−4508.43	−0.219061	−0.109530	+	0.993983i	$0.534935\pi$
$752$	41981.6	2.03579
$753$	0	0
$754$	49666.6	2.39887
$755$	−30561.3	−1.47317
$756$	0	0
$757$	−6228.74	−0.299059	−0.149529	−	0.988757i	$-0.547776\pi$
$757$	−6228.74	−0.299059	−0.149529	+	0.988757i	$0.547776\pi$
$758$	−28596.3	−1.37027
$759$	0	0
$760$	2926.84	0.139694
$761$	36780.8	1.75204	0.876020	−	0.482276i	$-0.160190\pi$
$761$	36780.8	1.75204	0.876020	+	0.482276i	$0.160190\pi$
$762$	0	0
$763$	−4054.46	−0.192374
$764$	15288.2	0.723963
$765$	0	0
$766$	−30473.1	−1.43739
$767$	−28729.6	−1.35250
$768$	0	0
$769$	18159.5	0.851557	0.425778	−	0.904827i	$-0.360000\pi$
$769$	18159.5	0.851557	0.425778	+	0.904827i	$0.360000\pi$
$770$	439.241	0.0205574
$771$	0	0
$772$	1261.78	0.0588245
$773$	−21832.7	−1.01587	−0.507935	−	0.861395i	$-0.669591\pi$
$773$	−21832.7	−1.01587	−0.507935	+	0.861395i	$0.669591\pi$
$774$	0	0
$775$	16194.9	0.750628
$776$	316.684	0.0146499
$777$	0	0
$778$	−10317.2	−0.475435
$779$	−2331.50	−0.107233
$780$	0	0
$781$	560.864	0.0256969
$782$	21569.5	0.986347
$783$	0	0
$784$	22763.1	1.03695
$785$	37929.3	1.72453
$786$	0	0
$787$	−27987.5	−1.26766	−0.633828	−	0.773474i	$-0.718517\pi$
$787$	−27987.5	−1.26766	−0.633828	+	0.773474i	$0.718517\pi$
$788$	−23189.8	−1.04835
$789$	0	0
$790$	7752.49	0.349141
$791$	−3945.88	−0.177369
$792$	0	0
$793$	−20398.3	−0.913450
$794$	19142.2	0.855582
$795$	0	0
$796$	21442.2	0.954772
$797$	−32065.9	−1.42514	−0.712568	−	0.701603i	$-0.752468\pi$
$797$	−32065.9	−1.42514	−0.712568	+	0.701603i	$0.752468\pi$
$798$	0	0
$799$	74898.0	3.31627
$800$	47100.0	2.08155
$801$	0	0
$802$	−13470.4	−0.593088
$803$	512.803	0.0225360
$804$	0	0
$805$	3743.13	0.163886
$806$	−21686.0	−0.947715
$807$	0	0
$808$	3845.39	0.167426
$809$	−24475.8	−1.06369	−0.531843	−	0.846843i	$-0.678500\pi$
$809$	−24475.8	−1.06369	−0.531843	+	0.846843i	$0.678500\pi$
$810$	0	0
$811$	33867.5	1.46640	0.733199	−	0.680014i	$-0.238026\pi$
$811$	33867.5	1.46640	0.733199	+	0.680014i	$0.238026\pi$
$812$	−6403.30	−0.276739
$813$	0	0
$814$	−20.7911	−0.000895244 0
$815$	−8859.35	−0.380772
$816$	0	0
$817$	−3330.92	−0.142637
$818$	54645.0	2.33572
$819$	0	0
$820$	−6960.12	−0.296412
$821$	−42226.7	−1.79503	−0.897517	−	0.440981i	$-0.854631\pi$
$821$	−42226.7	−1.79503	−0.897517	+	0.440981i	$0.854631\pi$
$822$	0	0
$823$	1467.15	0.0621407	0.0310703	−	0.999517i	$-0.490108\pi$
$823$	1467.15	0.0621407	0.0310703	+	0.999517i	$0.490108\pi$
$824$	−3023.06	−0.127807
$825$	0	0
$826$	7946.49	0.334738
$827$	29308.2	1.23234	0.616170	−	0.787613i	$-0.288683\pi$
$827$	29308.2	1.23234	0.616170	+	0.787613i	$0.288683\pi$
$828$	0	0
$829$	−38303.7	−1.60475	−0.802377	−	0.596818i	$-0.796432\pi$
$829$	−38303.7	−1.60475	−0.802377	+	0.596818i	$0.796432\pi$
$830$	7608.82	0.318200
$831$	0	0
$832$	−25053.7	−1.04397
$833$	40610.8	1.68917
$834$	0	0
$835$	209.592	0.00868649
$836$	−388.231	−0.0160613
$837$	0	0
$838$	13514.7	0.557109
$839$	−5749.25	−0.236575	−0.118287	−	0.992979i	$-0.537740\pi$
$839$	−5749.25	−0.236575	−0.118287	+	0.992979i	$0.537740\pi$
$840$	0	0
$841$	12455.6	0.510704
$842$	36778.4	1.50531
$843$	0	0
$844$	19808.4	0.807858
$845$	40512.5	1.64932
$846$	0	0
$847$	−6348.64	−0.257547
$848$	18029.3	0.730106
$849$	0	0
$850$	94870.9	3.82829
$851$	−177.178	−0.00713699
$852$	0	0
$853$	7317.44	0.293722	0.146861	−	0.989157i	$-0.453083\pi$
$853$	7317.44	0.293722	0.146861	+	0.989157i	$0.453083\pi$
$854$	5642.10	0.226076
$855$	0	0
$856$	−6966.42	−0.278163
$857$	6624.19	0.264035	0.132018	−	0.991247i	$-0.457854\pi$
$857$	6624.19	0.264035	0.132018	+	0.991247i	$0.457854\pi$
$858$	0	0
$859$	−31859.5	−1.26546	−0.632731	−	0.774372i	$-0.718066\pi$
$859$	−31859.5	−1.26546	−0.632731	+	0.774372i	$0.718066\pi$
$860$	−9943.61	−0.394272
$861$	0	0
$862$	−13289.8	−0.525117
$863$	41325.2	1.63004	0.815021	−	0.579431i	$-0.196725\pi$
$863$	41325.2	1.63004	0.815021	+	0.579431i	$0.196725\pi$
$864$	0	0
$865$	69007.3	2.71251
$866$	−52230.6	−2.04950
$867$	0	0
$868$	2795.89	0.109330
$869$	149.513	0.00583646
$870$	0	0
$871$	−20348.5	−0.791598
$872$	−3336.98	−0.129592
$873$	0	0
$874$	−7097.87	−0.274702
$875$	5813.40	0.224604
$876$	0	0
$877$	−33204.6	−1.27850	−0.639248	−	0.769001i	$-0.720754\pi$
$877$	−33204.6	−1.27850	−0.639248	+	0.769001i	$0.720754\pi$
$878$	9080.45	0.349032
$879$	0	0
$880$	1688.98	0.0646996
$881$	28247.2	1.08022	0.540108	−	0.841596i	$-0.318383\pi$
$881$	28247.2	1.08022	0.540108	+	0.841596i	$0.318383\pi$
$882$	0	0
$883$	27760.6	1.05800	0.529002	−	0.848620i	$-0.322566\pi$
$883$	27760.6	1.05800	0.529002	+	0.848620i	$0.322566\pi$
$884$	−59214.6	−2.25294
$885$	0	0
$886$	−38299.7	−1.45226
$887$	41637.2	1.57615	0.788073	−	0.615582i	$-0.211079\pi$
$887$	41637.2	1.57615	0.788073	+	0.615582i	$0.211079\pi$
$888$	0	0
$889$	−5869.58	−0.221439
$890$	55883.6	2.10474
$891$	0	0
$892$	30103.7	1.12999
$893$	−24646.7	−0.923596
$894$	0	0
$895$	11438.3	0.427198
$896$	−2383.86	−0.0888831
$897$	0	0
$898$	−18861.1	−0.700895
$899$	−16087.5	−0.596828
$900$	0	0
$901$	32165.5	1.18933
$902$	−287.979	−0.0106304
$903$	0	0
$904$	−3247.61	−0.119484
$905$	53123.0	1.95124
$906$	0	0
$907$	−5064.02	−0.185389	−0.0926947	−	0.995695i	$-0.529548\pi$
$907$	−5064.02	−0.185389	−0.0926947	+	0.995695i	$0.529548\pi$
$908$	11861.6	0.433524
$909$	0	0
$910$	−22046.0	−0.803098
$911$	18104.4	0.658427	0.329213	−	0.944256i	$-0.393216\pi$
$911$	18104.4	0.658427	0.329213	+	0.944256i	$0.393216\pi$
$912$	0	0
$913$	146.742	0.00531923
$914$	66695.4	2.41366
$915$	0	0
$916$	36255.4	1.30777
$917$	3341.94	0.120350
$918$	0	0
$919$	18026.5	0.647051	0.323525	−	0.946220i	$-0.395132\pi$
$919$	18026.5	0.647051	0.323525	+	0.946220i	$0.395132\pi$
$920$	3080.74	0.110401
$921$	0	0
$922$	24536.1	0.876415
$923$	−28150.4	−1.00388
$924$	0	0
$925$	−779.296	−0.0277007
$926$	36160.1	1.28326
$927$	0	0
$928$	−46787.8	−1.65505
$929$	51008.8	1.80145	0.900724	−	0.434391i	$-0.143037\pi$
$929$	51008.8	1.80145	0.900724	+	0.434391i	$0.143037\pi$
$930$	0	0
$931$	−13363.8	−0.470442
$932$	13385.2	0.470436
$933$	0	0
$934$	−1059.30	−0.0371105
$935$	3013.26	0.105395
$936$	0	0
$937$	−20078.9	−0.700053	−0.350026	−	0.936740i	$-0.613827\pi$
$937$	−20078.9	−0.700053	−0.350026	+	0.936740i	$0.613827\pi$
$938$	5628.31	0.195918
$939$	0	0
$940$	−73576.6	−2.55298
$941$	−12075.6	−0.418335	−0.209168	−	0.977880i	$-0.567075\pi$
$941$	−12075.6	−0.418335	−0.209168	+	0.977880i	$0.567075\pi$
$942$	0	0
$943$	−2454.10	−0.0847471
$944$	30556.1	1.05351
$945$	0	0
$946$	−411.423	−0.0141401
$947$	−30441.4	−1.04458	−0.522288	−	0.852769i	$-0.674921\pi$
$947$	−30441.4	−1.04458	−0.522288	+	0.852769i	$0.674921\pi$
$948$	0	0
$949$	−25738.2	−0.880398
$950$	−31219.2	−1.06619
$951$	0	0
$952$	−2381.34	−0.0810711
$953$	5133.90	0.174505	0.0872526	−	0.996186i	$-0.472191\pi$
$953$	5133.90	0.174505	0.0872526	+	0.996186i	$0.472191\pi$
$954$	0	0
$955$	39047.4	1.32308
$956$	−1669.29	−0.0564737
$957$	0	0
$958$	−74403.3	−2.50925
$959$	6239.71	0.210105
$960$	0	0
$961$	−22766.7	−0.764213
$962$	1043.53	0.0349738
$963$	0	0
$964$	13268.8	0.443319
$965$	3222.70	0.107505
$966$	0	0
$967$	−46.5820	−0.00154910	−0.000774549	−	1.00000i	$-0.500247\pi$
$967$	−46.5820	−0.00154910	−0.000774549		1.00000i	$0.500247\pi$
$968$	−5225.18	−0.173496
$969$	0	0
$970$	−5563.08	−0.184144
$971$	−15582.0	−0.514983	−0.257492	−	0.966281i	$-0.582896\pi$
$971$	−15582.0	−0.514983	−0.257492	+	0.966281i	$0.582896\pi$
$972$	0	0
$973$	−3524.99	−0.116142
$974$	−73974.6	−2.43357
$975$	0	0
$976$	21695.2	0.711522
$977$	−16308.1	−0.534026	−0.267013	−	0.963693i	$-0.586037\pi$
$977$	−16308.1	−0.534026	−0.267013	+	0.963693i	$0.586037\pi$
$978$	0	0
$979$	1077.76	0.0351842
$980$	−39894.3	−1.30038
$981$	0	0
$982$	−13494.4	−0.438516
$983$	39876.8	1.29387	0.646934	−	0.762546i	$-0.276051\pi$
$983$	39876.8	1.29387	0.646934	+	0.762546i	$0.276051\pi$
$984$	0	0
$985$	−59228.7	−1.91592
$986$	−94242.0	−3.04389
$987$	0	0
$988$	19485.8	0.627454
$989$	−3506.06	−0.112726
$990$	0	0
$991$	−39627.0	−1.27023	−0.635113	−	0.772419i	$-0.719047\pi$
$991$	−39627.0	−1.27023	−0.635113	+	0.772419i	$0.719047\pi$
$992$	20429.1	0.653854
$993$	0	0
$994$	7786.30	0.248457
$995$	54765.2	1.74490
$996$	0	0
$997$	−36562.6	−1.16143	−0.580717	−	0.814106i	$-0.697228\pi$
$997$	−36562.6	−1.16143	−0.580717	+	0.814106i	$0.697228\pi$
$998$	−6922.74	−0.219575
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2151.4.a.d.1.8		32
3.2	odd	2		717.4.a.d.1.25	✓	32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
717.4.a.d.1.25	✓	32	3.2	odd	2
2151.4.a.d.1.8		32	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 2151 = 3^{2} \cdot 239 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2151.a (trivial)

\(f(q)\)	\(=\)	\(q-3.87098 q^{2} +6.98450 q^{4} +17.8390 q^{5} +4.77620 q^{7} +3.93099 q^{8} +O(q^{10})\)	\(q-3.87098 q^{2} +6.98450 q^{4} +17.8390 q^{5} +4.77620 q^{7} +3.93099 q^{8} -69.0545 q^{10} -1.33177 q^{11} +66.8431 q^{13} -18.4886 q^{14} -71.0928 q^{16} -126.834 q^{17} +41.7374 q^{19} +124.597 q^{20} +5.15526 q^{22} +43.9321 q^{23} +193.230 q^{25} -258.749 q^{26} +33.3593 q^{28} -191.949 q^{29} +83.8113 q^{31} +243.751 q^{32} +490.973 q^{34} +85.2026 q^{35} -4.03299 q^{37} -161.565 q^{38} +70.1251 q^{40} -55.8613 q^{41} -79.8065 q^{43} -9.30175 q^{44} -170.060 q^{46} -590.519 q^{47} -320.188 q^{49} -747.991 q^{50} +466.866 q^{52} -253.603 q^{53} -23.7575 q^{55} +18.7752 q^{56} +743.032 q^{58} -429.806 q^{59} -305.167 q^{61} -324.432 q^{62} -374.813 q^{64} +1192.42 q^{65} -304.421 q^{67} -885.874 q^{68} -329.818 q^{70} -421.142 q^{71} -385.054 q^{73} +15.6116 q^{74} +291.515 q^{76} -6.36080 q^{77} -112.266 q^{79} -1268.22 q^{80} +216.238 q^{82} -110.186 q^{83} -2262.60 q^{85} +308.929 q^{86} -5.23518 q^{88} -809.268 q^{89} +319.256 q^{91} +306.843 q^{92} +2285.89 q^{94} +744.554 q^{95} +80.5607 q^{97} +1239.44 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 11 q^{2} + 147 q^{4} - 66 q^{5} + 58 q^{7} - 153 q^{8}+O(q^{10}) \) 32 * q - 11 * q^2 + 147 * q^4 - 66 * q^5 + 58 * q^7 - 153 * q^8	\( 32 q - 11 q^{2} + 147 q^{4} - 66 q^{5} + 58 q^{7} - 153 q^{8} + 52 q^{10} - 270 q^{11} + 48 q^{13} - 184 q^{14} + 775 q^{16} - 384 q^{17} + 216 q^{19} - 534 q^{20} + 437 q^{22} - 712 q^{23} + 1190 q^{25} - 436 q^{26} + 598 q^{28} - 562 q^{29} + 384 q^{31} - 1770 q^{32} + 452 q^{34} - 1026 q^{35} + 770 q^{37} - 733 q^{38} + 877 q^{40} - 1648 q^{41} + 1592 q^{43} - 1595 q^{44} + 532 q^{46} - 1540 q^{47} + 2134 q^{49} - 1646 q^{50} - 144 q^{52} - 1708 q^{53} + 1282 q^{55} - 2155 q^{56} + 1086 q^{58} - 2396 q^{59} + 364 q^{61} - 2180 q^{62} + 1663 q^{64} - 1520 q^{65} + 2728 q^{67} - 1545 q^{68} - 4609 q^{70} - 3322 q^{71} - 188 q^{73} - 1111 q^{74} - 3134 q^{76} - 556 q^{77} - 462 q^{79} - 6076 q^{80} - 7965 q^{82} - 4604 q^{83} - 852 q^{85} - 549 q^{86} - 1127 q^{88} - 6742 q^{89} + 1390 q^{91} - 1802 q^{92} - 2796 q^{94} - 448 q^{95} - 1322 q^{97} - 1000 q^{98}+O(q^{100}) \) 32 * q - 11 * q^2 + 147 * q^4 - 66 * q^5 + 58 * q^7 - 153 * q^8 + 52 * q^10 - 270 * q^11 + 48 * q^13 - 184 * q^14 + 775 * q^16 - 384 * q^17 + 216 * q^19 - 534 * q^20 + 437 * q^22 - 712 * q^23 + 1190 * q^25 - 436 * q^26 + 598 * q^28 - 562 * q^29 + 384 * q^31 - 1770 * q^32 + 452 * q^34 - 1026 * q^35 + 770 * q^37 - 733 * q^38 + 877 * q^40 - 1648 * q^41 + 1592 * q^43 - 1595 * q^44 + 532 * q^46 - 1540 * q^47 + 2134 * q^49 - 1646 * q^50 - 144 * q^52 - 1708 * q^53 + 1282 * q^55 - 2155 * q^56 + 1086 * q^58 - 2396 * q^59 + 364 * q^61 - 2180 * q^62 + 1663 * q^64 - 1520 * q^65 + 2728 * q^67 - 1545 * q^68 - 4609 * q^70 - 3322 * q^71 - 188 * q^73 - 1111 * q^74 - 3134 * q^76 - 556 * q^77 - 462 * q^79 - 6076 * q^80 - 7965 * q^82 - 4604 * q^83 - 852 * q^85 - 549 * q^86 - 1127 * q^88 - 6742 * q^89 + 1390 * q^91 - 1802 * q^92 - 2796 * q^94 - 448 * q^95 - 1322 * q^97 - 1000 * q^98

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