LMFDB - Embedded newform 2151.4.a.g.1.9

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:	$59$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.9
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)

$f(q)$	$=$	$q-4.52883 q^{2} +12.5103 q^{4} -9.53238 q^{5} -26.9020 q^{7} -20.4262 q^{8} +O(q^{10})$	\(q-4.52883 q^{2} +12.5103 q^{4} -9.53238 q^{5} -26.9020 q^{7} -20.4262 q^{8} +43.1705 q^{10} -71.2126 q^{11} -45.0662 q^{13} +121.834 q^{14} -7.57549 q^{16} -62.9039 q^{17} +125.071 q^{19} -119.252 q^{20} +322.509 q^{22} -81.9950 q^{23} -34.1338 q^{25} +204.097 q^{26} -336.551 q^{28} -42.5674 q^{29} +25.8084 q^{31} +197.717 q^{32} +284.881 q^{34} +256.440 q^{35} +145.133 q^{37} -566.424 q^{38} +194.710 q^{40} +338.284 q^{41} +265.559 q^{43} -890.888 q^{44} +371.341 q^{46} +225.799 q^{47} +380.716 q^{49} +154.586 q^{50} -563.790 q^{52} +10.7010 q^{53} +678.825 q^{55} +549.504 q^{56} +192.780 q^{58} -174.154 q^{59} -152.670 q^{61} -116.882 q^{62} -834.824 q^{64} +429.588 q^{65} +524.704 q^{67} -786.944 q^{68} -1161.37 q^{70} -368.349 q^{71} +582.804 q^{73} -657.282 q^{74} +1564.67 q^{76} +1915.76 q^{77} +747.284 q^{79} +72.2124 q^{80} -1532.03 q^{82} -321.115 q^{83} +599.624 q^{85} -1202.67 q^{86} +1454.60 q^{88} -819.245 q^{89} +1212.37 q^{91} -1025.78 q^{92} -1022.60 q^{94} -1192.22 q^{95} +1211.04 q^{97} -1724.20 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 59 q - 8 q^{2} + 238 q^{4} - 80 q^{5} - 10 q^{7} - 96 q^{8}+O(q^{10}) $ 59 * q - 8 * q^2 + 238 * q^4 - 80 * q^5 - 10 * q^7 - 96 * q^8	$ 59 q - 8 q^{2} + 238 q^{4} - 80 q^{5} - 10 q^{7} - 96 q^{8} - 36 q^{10} - 132 q^{11} + 104 q^{13} - 280 q^{14} + 822 q^{16} - 408 q^{17} + 20 q^{19} - 800 q^{20} - 2 q^{22} - 276 q^{23} + 1477 q^{25} - 780 q^{26} + 224 q^{28} - 696 q^{29} - 380 q^{31} - 896 q^{32} - 72 q^{34} - 700 q^{35} + 224 q^{37} - 988 q^{38} - 258 q^{40} - 2706 q^{41} - 156 q^{43} - 1584 q^{44} + 428 q^{46} - 1316 q^{47} + 2135 q^{49} - 1400 q^{50} + 1092 q^{52} - 1484 q^{53} - 992 q^{55} - 3360 q^{56} - 120 q^{58} - 3186 q^{59} - 254 q^{61} - 1240 q^{62} + 3054 q^{64} - 5120 q^{65} + 288 q^{67} - 9420 q^{68} + 1108 q^{70} - 4468 q^{71} - 1770 q^{73} - 6214 q^{74} + 720 q^{76} - 6352 q^{77} - 746 q^{79} - 7040 q^{80} + 276 q^{82} - 5484 q^{83} + 588 q^{85} - 10152 q^{86} + 1186 q^{88} - 11570 q^{89} + 1768 q^{91} - 15366 q^{92} - 2142 q^{94} - 5736 q^{95} + 2390 q^{97} - 6912 q^{98}+O(q^{100}) $ 59 * q - 8 * q^2 + 238 * q^4 - 80 * q^5 - 10 * q^7 - 96 * q^8 - 36 * q^10 - 132 * q^11 + 104 * q^13 - 280 * q^14 + 822 * q^16 - 408 * q^17 + 20 * q^19 - 800 * q^20 - 2 * q^22 - 276 * q^23 + 1477 * q^25 - 780 * q^26 + 224 * q^28 - 696 * q^29 - 380 * q^31 - 896 * q^32 - 72 * q^34 - 700 * q^35 + 224 * q^37 - 988 * q^38 - 258 * q^40 - 2706 * q^41 - 156 * q^43 - 1584 * q^44 + 428 * q^46 - 1316 * q^47 + 2135 * q^49 - 1400 * q^50 + 1092 * q^52 - 1484 * q^53 - 992 * q^55 - 3360 * q^56 - 120 * q^58 - 3186 * q^59 - 254 * q^61 - 1240 * q^62 + 3054 * q^64 - 5120 * q^65 + 288 * q^67 - 9420 * q^68 + 1108 * q^70 - 4468 * q^71 - 1770 * q^73 - 6214 * q^74 + 720 * q^76 - 6352 * q^77 - 746 * q^79 - 7040 * q^80 + 276 * q^82 - 5484 * q^83 + 588 * q^85 - 10152 * q^86 + 1186 * q^88 - 11570 * q^89 + 1768 * q^91 - 15366 * q^92 - 2142 * q^94 - 5736 * q^95 + 2390 * q^97 - 6912 * 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 $

$2$	−4.52883	−1.60118	−0.800591	−	0.599211i	$-0.795481\pi$
$2$	−4.52883	−1.60118	−0.800591	+	0.599211i	$0.795481\pi$
$3$	0	0
$3$	0	0
$4$	12.5103	1.56378
$5$	−9.53238	−0.852602	−0.426301	−	0.904581i	$-0.640183\pi$
$5$	−9.53238	−0.852602	−0.426301	+	0.904581i	$0.640183\pi$
$6$	0	0
$7$	−26.9020	−1.45257	−0.726285	−	0.687394i	$-0.758755\pi$
$7$	−26.9020	−1.45257	−0.726285	+	0.687394i	$0.758755\pi$
$8$	−20.4262	−0.902718
$9$	0	0
$10$	43.1705	1.36517
$11$	−71.2126	−1.95195	−0.975973	−	0.217893i	$-0.930082\pi$
$11$	−71.2126	−1.95195	−0.975973	+	0.217893i	$0.930082\pi$
$12$	0	0
$13$	−45.0662	−0.961470	−0.480735	−	0.876866i	$-0.659630\pi$
$13$	−45.0662	−0.961470	−0.480735	+	0.876866i	$0.659630\pi$
$14$	121.834	2.32583
$15$	0	0
$16$	−7.57549	−0.118367
$17$	−62.9039	−0.897437	−0.448719	−	0.893673i	$-0.648119\pi$
$17$	−62.9039	−0.897437	−0.448719	+	0.893673i	$0.648119\pi$
$18$	0	0
$19$	125.071	1.51017	0.755085	−	0.655627i	$-0.227596\pi$
$19$	125.071	1.51017	0.755085	+	0.655627i	$0.227596\pi$
$20$	−119.252	−1.33328
$21$	0	0
$22$	322.509	3.12542
$23$	−81.9950	−0.743354	−0.371677	−	0.928362i	$-0.621217\pi$
$23$	−81.9950	−0.743354	−0.371677	+	0.928362i	$0.621217\pi$
$24$	0	0
$25$	−34.1338	−0.273071
$26$	204.097	1.53949
$27$	0	0
$28$	−336.551	−2.27150
$29$	−42.5674	−0.272571	−0.136286	−	0.990670i	$-0.543516\pi$
$29$	−42.5674	−0.272571	−0.136286	+	0.990670i	$0.543516\pi$
$30$	0	0
$31$	25.8084	0.149527	0.0747634	−	0.997201i	$-0.476180\pi$
$31$	25.8084	0.149527	0.0747634	+	0.997201i	$0.476180\pi$
$32$	197.717	1.09225
$33$	0	0
$34$	284.881	1.43696
$35$	256.440	1.23846
$36$	0	0
$37$	145.133	0.644858	0.322429	−	0.946594i	$-0.395501\pi$
$37$	145.133	0.644858	0.322429	+	0.946594i	$0.395501\pi$
$38$	−566.424	−2.41805
$39$	0	0
$40$	194.710	0.769659
$41$	338.284	1.28856	0.644281	−	0.764789i	$-0.277157\pi$
$41$	338.284	1.28856	0.644281	+	0.764789i	$0.277157\pi$
$42$	0	0
$43$	265.559	0.941799	0.470899	−	0.882187i	$-0.343930\pi$
$43$	265.559	0.941799	0.470899	+	0.882187i	$0.343930\pi$
$44$	−890.888	−3.05242
$45$	0	0
$46$	371.341	1.19024
$47$	225.799	0.700769	0.350385	−	0.936606i	$-0.386051\pi$
$47$	225.799	0.700769	0.350385	+	0.936606i	$0.386051\pi$
$48$	0	0
$49$	380.716	1.10996
$50$	154.586	0.437236
$51$	0	0
$52$	−563.790	−1.50353
$53$	10.7010	0.0277339	0.0138669	−	0.999904i	$-0.495586\pi$
$53$	10.7010	0.0277339	0.0138669	+	0.999904i	$0.495586\pi$
$54$	0	0
$55$	678.825	1.66423
$56$	549.504	1.31126
$57$	0	0
$58$	192.780	0.436436
$59$	−174.154	−0.384287	−0.192143	−	0.981367i	$-0.561544\pi$
$59$	−174.154	−0.384287	−0.192143	+	0.981367i	$0.561544\pi$
$60$	0	0
$61$	−152.670	−0.320448	−0.160224	−	0.987081i	$-0.551222\pi$
$61$	−152.670	−0.320448	−0.160224	+	0.987081i	$0.551222\pi$
$62$	−116.882	−0.239420
$63$	0	0
$64$	−834.824	−1.63052
$65$	429.588	0.819751
$66$	0	0
$67$	524.704	0.956759	0.478379	−	0.878153i	$-0.341224\pi$
$67$	524.704	0.956759	0.478379	+	0.878153i	$0.341224\pi$
$68$	−786.944	−1.40340
$69$	0	0
$70$	−1161.37	−1.98300
$71$	−368.349	−0.615704	−0.307852	−	0.951434i	$-0.599610\pi$
$71$	−368.349	−0.615704	−0.307852	+	0.951434i	$0.599610\pi$
$72$	0	0
$73$	582.804	0.934412	0.467206	−	0.884149i	$-0.345261\pi$
$73$	582.804	0.934412	0.467206	+	0.884149i	$0.345261\pi$
$74$	−657.282	−1.03253
$75$	0	0
$76$	1564.67	2.36158
$77$	1915.76	2.83534
$78$	0	0
$79$	747.284	1.06425	0.532127	−	0.846665i	$-0.321393\pi$
$79$	747.284	1.06425	0.532127	+	0.846665i	$0.321393\pi$
$80$	72.2124	0.100920
$81$	0	0
$82$	−1532.03	−2.06322
$83$	−321.115	−0.424662	−0.212331	−	0.977198i	$-0.568105\pi$
$83$	−321.115	−0.424662	−0.212331	+	0.977198i	$0.568105\pi$
$84$	0	0
$85$	599.624	0.765157
$86$	−1202.67	−1.50799
$87$	0	0
$88$	1454.60	1.76206
$89$	−819.245	−0.975728	−0.487864	−	0.872920i	$-0.662224\pi$
$89$	−819.245	−0.975728	−0.487864	+	0.872920i	$0.662224\pi$
$90$	0	0
$91$	1212.37	1.39660
$92$	−1025.78	−1.16244
$93$	0	0
$94$	−1022.60	−1.12206
$95$	−1192.22	−1.28757
$96$	0	0
$97$	1211.04	1.26766	0.633828	−	0.773474i	$-0.281483\pi$
$97$	1211.04	1.26766	0.633828	+	0.773474i	$0.281483\pi$
$98$	−1724.20	−1.77725
$99$	0	0
$100$	−427.023	−0.427023
$101$	−1065.63	−1.04984	−0.524920	−	0.851151i	$-0.675905\pi$
$101$	−1065.63	−1.04984	−0.524920	+	0.851151i	$0.675905\pi$
$102$	0	0
$103$	1062.23	1.01616	0.508080	−	0.861310i	$-0.330356\pi$
$103$	1062.23	1.01616	0.508080	+	0.861310i	$0.330356\pi$
$104$	920.530	0.867936
$105$	0	0
$106$	−48.4629	−0.0444069
$107$	−657.809	−0.594325	−0.297162	−	0.954827i	$-0.596040\pi$
$107$	−657.809	−0.594325	−0.297162	+	0.954827i	$0.596040\pi$
$108$	0	0
$109$	−2011.87	−1.76791	−0.883955	−	0.467571i	$-0.845129\pi$
$109$	−2011.87	−1.76791	−0.883955	+	0.467571i	$0.845129\pi$
$110$	−3074.28	−2.66474
$111$	0	0
$112$	203.796	0.171936
$113$	−856.049	−0.712658	−0.356329	−	0.934361i	$-0.615972\pi$
$113$	−856.049	−0.712658	−0.356329	+	0.934361i	$0.615972\pi$
$114$	0	0
$115$	781.607	0.633785
$116$	−532.529	−0.426242
$117$	0	0
$118$	788.713	0.615313
$119$	1692.24	1.30359
$120$	0	0
$121$	3740.23	2.81009
$122$	691.414	0.513096
$123$	0	0
$124$	322.870	0.233827
$125$	1516.92	1.08542
$126$	0	0
$127$	−1178.42	−0.823366	−0.411683	−	0.911327i	$-0.635059\pi$
$127$	−1178.42	−0.823366	−0.411683	+	0.911327i	$0.635059\pi$
$128$	2199.03	1.51851
$129$	0	0
$130$	−1945.53	−1.31257
$131$	−1952.18	−1.30200	−0.651002	−	0.759076i	$-0.725651\pi$
$131$	−1952.18	−1.30200	−0.651002	+	0.759076i	$0.725651\pi$
$132$	0	0
$133$	−3364.65	−2.19363
$134$	−2376.29	−1.53194
$135$	0	0
$136$	1284.89	0.810133
$137$	−367.661	−0.229280	−0.114640	−	0.993407i	$-0.536572\pi$
$137$	−367.661	−0.229280	−0.114640	+	0.993407i	$0.536572\pi$
$138$	0	0
$139$	955.431	0.583011	0.291506	−	0.956569i	$-0.405844\pi$
$139$	955.431	0.583011	0.291506	+	0.956569i	$0.405844\pi$
$140$	3208.13	1.93669
$141$	0	0
$142$	1668.19	0.985854
$143$	3209.28	1.87674
$144$	0	0
$145$	405.768	0.232395
$146$	−2639.42	−1.49616
$147$	0	0
$148$	1815.65	1.00842
$149$	932.177	0.512529	0.256265	−	0.966607i	$-0.417508\pi$
$149$	932.177	0.512529	0.256265	+	0.966607i	$0.417508\pi$
$150$	0	0
$151$	−2352.28	−1.26772	−0.633861	−	0.773447i	$-0.718531\pi$
$151$	−2352.28	−1.26772	−0.633861	+	0.773447i	$0.718531\pi$
$152$	−2554.72	−1.36326
$153$	0	0
$154$	−8676.14	−4.53989
$155$	−246.016	−0.127487
$156$	0	0
$157$	2610.49	1.32700	0.663501	−	0.748175i	$-0.269070\pi$
$157$	2610.49	1.32700	0.663501	+	0.748175i	$0.269070\pi$
$158$	−3384.32	−1.70406
$159$	0	0
$160$	−1884.72	−0.931250
$161$	2205.83	1.07977
$162$	0	0
$163$	−2714.83	−1.30455	−0.652276	−	0.757982i	$-0.726186\pi$
$163$	−2714.83	−1.30455	−0.652276	+	0.757982i	$0.726186\pi$
$164$	4232.02	2.01503
$165$	0	0
$166$	1454.27	0.679961
$167$	1895.49	0.878306	0.439153	−	0.898412i	$-0.355279\pi$
$167$	1895.49	0.878306	0.439153	+	0.898412i	$0.355279\pi$
$168$	0	0
$169$	−166.040	−0.0755758
$170$	−2715.59	−1.22515
$171$	0	0
$172$	3322.21	1.47277
$173$	3244.24	1.42575	0.712876	−	0.701290i	$-0.247392\pi$
$173$	3244.24	1.42575	0.712876	+	0.701290i	$0.247392\pi$
$174$	0	0
$175$	918.267	0.396654
$176$	539.470	0.231046
$177$	0	0
$178$	3710.22	1.56232
$179$	−2294.90	−0.958262	−0.479131	−	0.877743i	$-0.659048\pi$
$179$	−2294.90	−0.958262	−0.479131	+	0.877743i	$0.659048\pi$
$180$	0	0
$181$	2792.91	1.14693	0.573467	−	0.819229i	$-0.305598\pi$
$181$	2792.91	1.14693	0.573467	+	0.819229i	$0.305598\pi$
$182$	−5490.61	−2.23621
$183$	0	0
$184$	1674.84	0.671039
$185$	−1383.46	−0.549807
$186$	0	0
$187$	4479.55	1.75175
$188$	2824.80	1.09585
$189$	0	0
$190$	5399.37	2.06164
$191$	−1823.07	−0.690644	−0.345322	−	0.938484i	$-0.612230\pi$
$191$	−1823.07	−0.690644	−0.345322	+	0.938484i	$0.612230\pi$
$192$	0	0
$193$	−1992.45	−0.743109	−0.371555	−	0.928411i	$-0.621175\pi$
$193$	−1992.45	−0.743109	−0.371555	+	0.928411i	$0.621175\pi$
$194$	−5484.60	−2.02975
$195$	0	0
$196$	4762.86	1.73574
$197$	2700.68	0.976727	0.488364	−	0.872640i	$-0.337594\pi$
$197$	2700.68	0.976727	0.488364	+	0.872640i	$0.337594\pi$
$198$	0	0
$199$	−4233.60	−1.50810	−0.754050	−	0.656817i	$-0.771903\pi$
$199$	−4233.60	−1.50810	−0.754050	+	0.656817i	$0.771903\pi$
$200$	697.223	0.246506
$201$	0	0
$202$	4826.04	1.68099
$203$	1145.15	0.395929
$204$	0	0
$205$	−3224.65	−1.09863
$206$	−4810.64	−1.62706
$207$	0	0
$208$	341.398	0.113806
$209$	−8906.61	−2.94777
$210$	0	0
$211$	3542.53	1.15582	0.577910	−	0.816100i	$-0.303868\pi$
$211$	3542.53	1.15582	0.577910	+	0.816100i	$0.303868\pi$
$212$	133.872	0.0433697
$213$	0	0
$214$	2979.10	0.951622
$215$	−2531.41	−0.802979
$216$	0	0
$217$	−694.298	−0.217198
$218$	9111.41	2.83075
$219$	0	0
$220$	8492.28	2.60250
$221$	2834.84	0.862859
$222$	0	0
$223$	1866.72	0.560560	0.280280	−	0.959918i	$-0.409573\pi$
$223$	1866.72	0.560560	0.280280	+	0.959918i	$0.409573\pi$
$224$	−5318.99	−1.58656
$225$	0	0
$226$	3876.90	1.14109
$227$	483.243	0.141295	0.0706475	−	0.997501i	$-0.477493\pi$
$227$	483.243	0.141295	0.0706475	+	0.997501i	$0.477493\pi$
$228$	0	0
$229$	3808.71	1.09907	0.549534	−	0.835472i	$-0.314805\pi$
$229$	3808.71	1.09907	0.549534	+	0.835472i	$0.314805\pi$
$230$	−3539.76	−1.01480
$231$	0	0
$232$	869.489	0.246055
$233$	−659.925	−0.185550	−0.0927750	−	0.995687i	$-0.529574\pi$
$233$	−659.925	−0.185550	−0.0927750	+	0.995687i	$0.529574\pi$
$234$	0	0
$235$	−2152.40	−0.597477
$236$	−2178.71	−0.600941
$237$	0	0
$238$	−7663.86	−2.08729
$239$	239.000	0.0646846
$239$	239.000	0.0646846
$240$	0	0
$241$	−1978.87	−0.528921	−0.264461	−	0.964396i	$-0.585194\pi$
$241$	−1978.87	−0.528921	−0.264461	+	0.964396i	$0.585194\pi$
$242$	−16938.9	−4.49947
$243$	0	0
$244$	−1909.94	−0.501111
$245$	−3629.13	−0.946353
$246$	0	0
$247$	−5636.46	−1.45198
$248$	−527.168	−0.134981
$249$	0	0
$250$	−6869.88	−1.73796
$251$	5508.66	1.38527	0.692636	−	0.721287i	$-0.256449\pi$
$251$	5508.66	1.38527	0.692636	+	0.721287i	$0.256449\pi$
$252$	0	0
$253$	5839.08	1.45099
$254$	5336.84	1.31836
$255$	0	0
$256$	−3280.44	−0.800889
$257$	4252.19	1.03208	0.516040	−	0.856564i	$-0.327406\pi$
$257$	4252.19	1.03208	0.516040	+	0.856564i	$0.327406\pi$
$258$	0	0
$259$	−3904.37	−0.936701
$260$	5374.25	1.28191
$261$	0	0
$262$	8841.07	2.08474
$263$	6326.04	1.48319	0.741597	−	0.670846i	$-0.234069\pi$
$263$	6326.04	1.48319	0.741597	+	0.670846i	$0.234069\pi$
$264$	0	0
$265$	−102.006	−0.0236459
$266$	15237.9	3.51239
$267$	0	0
$268$	6564.19	1.49616
$269$	6040.41	1.36911	0.684554	−	0.728962i	$-0.259997\pi$
$269$	6040.41	1.36911	0.684554	+	0.728962i	$0.259997\pi$
$270$	0	0
$271$	3322.50	0.744750	0.372375	−	0.928082i	$-0.378544\pi$
$271$	3322.50	0.744750	0.372375	+	0.928082i	$0.378544\pi$
$272$	476.528	0.106227
$273$	0	0
$274$	1665.07	0.367120
$275$	2430.76	0.533019
$276$	0	0
$277$	−1283.25	−0.278350	−0.139175	−	0.990268i	$-0.544445\pi$
$277$	−1283.25	−0.278350	−0.139175	+	0.990268i	$0.544445\pi$
$278$	−4326.98	−0.933507
$279$	0	0
$280$	−5238.08	−1.11798
$281$	2698.12	0.572798	0.286399	−	0.958110i	$-0.407542\pi$
$281$	2698.12	0.572798	0.286399	+	0.958110i	$0.407542\pi$
$282$	0	0
$283$	1175.60	0.246933	0.123466	−	0.992349i	$-0.460599\pi$
$283$	1175.60	0.246933	0.123466	+	0.992349i	$0.460599\pi$
$284$	−4608.14	−0.962827
$285$	0	0
$286$	−14534.3	−3.00500
$287$	−9100.50	−1.87173
$288$	0	0
$289$	−956.099	−0.194606
$290$	−1837.65	−0.372106
$291$	0	0
$292$	7291.03	1.46122
$293$	5415.07	1.07970	0.539850	−	0.841761i	$-0.318481\pi$
$293$	5415.07	1.07970	0.539850	+	0.841761i	$0.318481\pi$
$294$	0	0
$295$	1660.10	0.327643
$296$	−2964.51	−0.582125
$297$	0	0
$298$	−4221.67	−0.820653
$299$	3695.20	0.714712
$300$	0	0
$301$	−7144.06	−1.36803
$302$	10653.1	2.02985
$303$	0	0
$304$	−947.473	−0.178754
$305$	1455.30	0.273215
$306$	0	0
$307$	−4624.20	−0.859665	−0.429832	−	0.902909i	$-0.641427\pi$
$307$	−4624.20	−0.859665	−0.429832	+	0.902909i	$0.641427\pi$
$308$	23966.6	4.43385
$309$	0	0
$310$	1114.16	0.204129
$311$	4917.21	0.896558	0.448279	−	0.893894i	$-0.352037\pi$
$311$	4917.21	0.896558	0.448279	+	0.893894i	$0.352037\pi$
$312$	0	0
$313$	2750.52	0.496705	0.248352	−	0.968670i	$-0.420111\pi$
$313$	2750.52	0.496705	0.248352	+	0.968670i	$0.420111\pi$
$314$	−11822.4	−2.12477
$315$	0	0
$316$	9348.72	1.66426
$317$	6068.15	1.07515	0.537573	−	0.843217i	$-0.319341\pi$
$317$	6068.15	1.07515	0.537573	+	0.843217i	$0.319341\pi$
$318$	0	0
$319$	3031.33	0.532044
$320$	7957.86	1.39018
$321$	0	0
$322$	−9989.81	−1.72891
$323$	−7867.44	−1.35528
$324$	0	0
$325$	1538.28	0.262549
$326$	12295.0	2.08882
$327$	0	0
$328$	−6909.84	−1.16321
$329$	−6074.44	−1.01792
$330$	0	0
$331$	−11524.4	−1.91370	−0.956852	−	0.290577i	$-0.906153\pi$
$331$	−11524.4	−1.91370	−0.956852	+	0.290577i	$0.906153\pi$
$332$	−4017.23	−0.664079
$333$	0	0
$334$	−8584.33	−1.40633
$335$	−5001.68	−0.815734
$336$	0	0
$337$	−5396.43	−0.872291	−0.436146	−	0.899876i	$-0.643657\pi$
$337$	−5396.43	−0.872291	−0.436146	+	0.899876i	$0.643657\pi$
$338$	751.966	0.121011
$339$	0	0
$340$	7501.45	1.19654
$341$	−1837.88	−0.291868
$342$	0	0
$343$	−1014.64	−0.159724
$344$	−5424.35	−0.850178
$345$	0	0
$346$	−14692.6	−2.28289
$347$	−7811.17	−1.20843	−0.604215	−	0.796821i	$-0.706513\pi$
$347$	−7811.17	−1.20843	−0.604215	+	0.796821i	$0.706513\pi$
$348$	0	0
$349$	−7056.03	−1.08224	−0.541118	−	0.840947i	$-0.681999\pi$
$349$	−7056.03	−1.08224	−0.541118	+	0.840947i	$0.681999\pi$
$350$	−4158.67	−0.635115
$351$	0	0
$352$	−14080.0	−2.13200
$353$	−1348.27	−0.203290	−0.101645	−	0.994821i	$-0.532411\pi$
$353$	−1348.27	−0.203290	−0.101645	+	0.994821i	$0.532411\pi$
$354$	0	0
$355$	3511.24	0.524950
$356$	−10249.0	−1.52583
$357$	0	0
$358$	10393.2	1.53435
$359$	298.701	0.0439131	0.0219566	−	0.999759i	$-0.493010\pi$
$359$	298.701	0.0439131	0.0219566	+	0.999759i	$0.493010\pi$
$360$	0	0
$361$	8783.71	1.28061
$362$	−12648.6	−1.83645
$363$	0	0
$364$	15167.1	2.18398
$365$	−5555.51	−0.796681
$366$	0	0
$367$	1914.25	0.272270	0.136135	−	0.990690i	$-0.456532\pi$
$367$	1914.25	0.272270	0.136135	+	0.990690i	$0.456532\pi$
$368$	621.153	0.0879886
$369$	0	0
$370$	6265.46	0.880340
$371$	−287.878	−0.0402854
$372$	0	0
$373$	−4951.25	−0.687309	−0.343654	−	0.939096i	$-0.611665\pi$
$373$	−4951.25	−0.687309	−0.343654	+	0.939096i	$0.611665\pi$
$374$	−20287.1	−2.80487
$375$	0	0
$376$	−4612.21	−0.632597
$377$	1918.35	0.262069
$378$	0	0
$379$	7439.14	1.00824	0.504120	−	0.863634i	$-0.331817\pi$
$379$	7439.14	1.00824	0.504120	+	0.863634i	$0.331817\pi$
$380$	−14915.0	−2.01348
$381$	0	0
$382$	8256.38	1.10585
$383$	−1708.43	−0.227928	−0.113964	−	0.993485i	$-0.536355\pi$
$383$	−1708.43	−0.227928	−0.113964	+	0.993485i	$0.536355\pi$
$384$	0	0
$385$	−18261.7	−2.41741
$386$	9023.48	1.18985
$387$	0	0
$388$	15150.5	1.98234
$389$	−2335.00	−0.304343	−0.152171	−	0.988354i	$-0.548627\pi$
$389$	−2335.00	−0.304343	−0.152171	+	0.988354i	$0.548627\pi$
$390$	0	0
$391$	5157.81	0.667114
$392$	−7776.58	−1.00198
$393$	0	0
$394$	−12230.9	−1.56392
$395$	−7123.39	−0.907384
$396$	0	0
$397$	−7413.88	−0.937259	−0.468630	−	0.883395i	$-0.655252\pi$
$397$	−7413.88	−0.937259	−0.468630	+	0.883395i	$0.655252\pi$
$398$	19173.2	2.41474
$399$	0	0
$400$	258.580	0.0323226
$401$	−14407.7	−1.79423	−0.897113	−	0.441802i	$-0.854339\pi$
$401$	−14407.7	−1.79423	−0.897113	+	0.441802i	$0.854339\pi$
$402$	0	0
$403$	−1163.09	−0.143766
$404$	−13331.3	−1.64172
$405$	0	0
$406$	−5186.17	−0.633954
$407$	−10335.3	−1.25873
$408$	0	0
$409$	−12167.4	−1.47100	−0.735502	−	0.677523i	$-0.763053\pi$
$409$	−12167.4	−1.47100	−0.735502	+	0.677523i	$0.763053\pi$
$410$	14603.9	1.75911
$411$	0	0
$412$	13288.7	1.58905
$413$	4685.08	0.558203
$414$	0	0
$415$	3060.99	0.362067
$416$	−8910.37	−1.05016
$417$	0	0
$418$	40336.5	4.71991
$419$	−2369.22	−0.276239	−0.138120	−	0.990416i	$-0.544106\pi$
$419$	−2369.22	−0.276239	−0.138120	+	0.990416i	$0.544106\pi$
$420$	0	0
$421$	−3383.60	−0.391702	−0.195851	−	0.980634i	$-0.562747\pi$
$421$	−3383.60	−0.391702	−0.195851	+	0.980634i	$0.562747\pi$
$422$	−16043.5	−1.85068
$423$	0	0
$424$	−218.580	−0.0250359
$425$	2147.15	0.245064
$426$	0	0
$427$	4107.11	0.465473
$428$	−8229.36	−0.929395
$429$	0	0
$430$	11464.3	1.28572
$431$	−8341.46	−0.932237	−0.466119	−	0.884722i	$-0.654348\pi$
$431$	−8341.46	−0.932237	−0.466119	+	0.884722i	$0.654348\pi$
$432$	0	0
$433$	−4630.35	−0.513904	−0.256952	−	0.966424i	$-0.582718\pi$
$433$	−4630.35	−0.513904	−0.256952	+	0.966424i	$0.582718\pi$
$434$	3144.35	0.347774
$435$	0	0
$436$	−25169.0	−2.76463
$437$	−10255.2	−1.12259
$438$	0	0
$439$	−751.168	−0.0816658	−0.0408329	−	0.999166i	$-0.513001\pi$
$439$	−751.168	−0.0816658	−0.0408329	+	0.999166i	$0.513001\pi$
$440$	−13865.8	−1.50233
$441$	0	0
$442$	−12838.5	−1.38159
$443$	−2712.80	−0.290946	−0.145473	−	0.989362i	$-0.546470\pi$
$443$	−2712.80	−0.290946	−0.145473	+	0.989362i	$0.546470\pi$
$444$	0	0
$445$	7809.35	0.831907
$446$	−8454.05	−0.897558
$447$	0	0
$448$	22458.4	2.36844
$449$	881.973	0.0927013	0.0463506	−	0.998925i	$-0.485241\pi$
$449$	881.973	0.0927013	0.0463506	+	0.998925i	$0.485241\pi$
$450$	0	0
$451$	−24090.0	−2.51520
$452$	−10709.4	−1.11444
$453$	0	0
$454$	−2188.52	−0.226239
$455$	−11556.8	−1.19075
$456$	0	0
$457$	−16597.6	−1.69892	−0.849458	−	0.527656i	$-0.823071\pi$
$457$	−16597.6	−1.69892	−0.849458	+	0.527656i	$0.823071\pi$
$458$	−17249.0	−1.75981
$459$	0	0
$460$	9778.11	0.991101
$461$	−17653.8	−1.78355	−0.891776	−	0.452476i	$-0.850541\pi$
$461$	−17653.8	−1.78355	−0.891776	+	0.452476i	$0.850541\pi$
$462$	0	0
$463$	9223.29	0.925794	0.462897	−	0.886412i	$-0.346810\pi$
$463$	9223.29	0.925794	0.462897	+	0.886412i	$0.346810\pi$
$464$	322.469	0.0322635
$465$	0	0
$466$	2988.69	0.297099
$467$	15059.6	1.49224	0.746119	−	0.665813i	$-0.231915\pi$
$467$	15059.6	1.49224	0.746119	+	0.665813i	$0.231915\pi$
$468$	0	0
$469$	−14115.6	−1.38976
$470$	9747.84	0.956669
$471$	0	0
$472$	3557.30	0.346902
$473$	−18911.1	−1.83834
$474$	0	0
$475$	−4269.14	−0.412383
$476$	21170.4	2.03853
$477$	0	0
$478$	−1082.39	−0.103572
$479$	−2074.84	−0.197916	−0.0989581	−	0.995092i	$-0.531551\pi$
$479$	−2074.84	−0.197916	−0.0989581	+	0.995092i	$0.531551\pi$
$480$	0	0
$481$	−6540.59	−0.620011
$482$	8961.94	0.846899
$483$	0	0
$484$	46791.3	4.39437
$485$	−11544.1	−1.08081
$486$	0	0
$487$	2461.08	0.228998	0.114499	−	0.993423i	$-0.463474\pi$
$487$	2461.08	0.228998	0.114499	+	0.993423i	$0.463474\pi$
$488$	3118.46	0.289274
$489$	0	0
$490$	16435.7	1.51528
$491$	14648.7	1.34641	0.673205	−	0.739456i	$-0.264917\pi$
$491$	14648.7	1.34641	0.673205	+	0.739456i	$0.264917\pi$
$492$	0	0
$493$	2677.66	0.244616
$494$	25526.6	2.32489
$495$	0	0
$496$	−195.512	−0.0176991
$497$	9909.32	0.894353
$498$	0	0
$499$	4452.45	0.399437	0.199719	−	0.979853i	$-0.435997\pi$
$499$	4452.45	0.399437	0.199719	+	0.979853i	$0.435997\pi$
$500$	18977.1	1.69736
$501$	0	0
$502$	−24947.8	−2.21807
$503$	13281.3	1.17731	0.588654	−	0.808385i	$-0.299658\pi$
$503$	13281.3	1.17731	0.588654	+	0.808385i	$0.299658\pi$
$504$	0	0
$505$	10158.0	0.895096
$506$	−26444.2	−2.32329
$507$	0	0
$508$	−14742.3	−1.28757
$509$	−8587.44	−0.747802	−0.373901	−	0.927469i	$-0.621980\pi$
$509$	−8587.44	−0.747802	−0.373901	+	0.927469i	$0.621980\pi$
$510$	0	0
$511$	−15678.6	−1.35730
$512$	−2735.71	−0.236138
$513$	0	0
$514$	−19257.4	−1.65255
$515$	−10125.6	−0.866379
$516$	0	0
$517$	−16079.7	−1.36786
$518$	17682.2	1.49983
$519$	0	0
$520$	−8774.83	−0.740004
$521$	12785.8	1.07515	0.537577	−	0.843214i	$-0.319340\pi$
$521$	12785.8	1.07515	0.537577	+	0.843214i	$0.319340\pi$
$522$	0	0
$523$	11050.2	0.923884	0.461942	−	0.886910i	$-0.347153\pi$
$523$	11050.2	0.923884	0.461942	+	0.886910i	$0.347153\pi$
$524$	−24422.2	−2.03605
$525$	0	0
$526$	−28649.5	−2.37486
$527$	−1623.45	−0.134191
$528$	0	0
$529$	−5443.82	−0.447425
$530$	461.967	0.0378614
$531$	0	0
$532$	−42092.7	−3.43035
$533$	−15245.1	−1.23891
$534$	0	0
$535$	6270.48	0.506722
$536$	−10717.7	−0.863683
$537$	0	0
$538$	−27356.0	−2.19219
$539$	−27111.8	−2.16658
$540$	0	0
$541$	15026.0	1.19412	0.597061	−	0.802196i	$-0.296335\pi$
$541$	15026.0	1.19412	0.597061	+	0.802196i	$0.296335\pi$
$542$	−15047.0	−1.19248
$543$	0	0
$544$	−12437.2	−0.980222
$545$	19177.9	1.50732
$546$	0	0
$547$	−16029.5	−1.25297	−0.626483	−	0.779435i	$-0.715506\pi$
$547$	−16029.5	−1.25297	−0.626483	+	0.779435i	$0.715506\pi$
$548$	−4599.54	−0.358545
$549$	0	0
$550$	−11008.5	−0.853460
$551$	−5323.94	−0.411629
$552$	0	0
$553$	−20103.4	−1.54590
$554$	5811.61	0.445689
$555$	0	0
$556$	11952.7	0.911703
$557$	−10069.8	−0.766013	−0.383006	−	0.923746i	$-0.625111\pi$
$557$	−10069.8	−0.766013	−0.383006	+	0.923746i	$0.625111\pi$
$558$	0	0
$559$	−11967.7	−0.905511
$560$	−1942.66	−0.146593
$561$	0	0
$562$	−12219.3	−0.917153
$563$	−6732.03	−0.503945	−0.251973	−	0.967734i	$-0.581079\pi$
$563$	−6732.03	−0.503945	−0.251973	+	0.967734i	$0.581079\pi$
$564$	0	0
$565$	8160.18	0.607613
$566$	−5324.07	−0.395384
$567$	0	0
$568$	7523.96	0.555807
$569$	−14241.5	−1.04927	−0.524634	−	0.851328i	$-0.675798\pi$
$569$	−14241.5	−1.04927	−0.524634	+	0.851328i	$0.675798\pi$
$570$	0	0
$571$	−5698.78	−0.417665	−0.208832	−	0.977951i	$-0.566966\pi$
$571$	−5698.78	−0.417665	−0.208832	+	0.977951i	$0.566966\pi$
$572$	40148.9	2.93481
$573$	0	0
$574$	41214.6	2.99697
$575$	2798.80	0.202988
$576$	0	0
$577$	14087.1	1.01639	0.508194	−	0.861243i	$-0.330313\pi$
$577$	14087.1	1.01639	0.508194	+	0.861243i	$0.330313\pi$
$578$	4330.01	0.311599
$579$	0	0
$580$	5076.27	0.363415
$581$	8638.62	0.616851
$582$	0	0
$583$	−762.045	−0.0541350
$584$	−11904.5	−0.843510
$585$	0	0
$586$	−24523.9	−1.72879
$587$	−18371.7	−1.29179	−0.645894	−	0.763427i	$-0.723515\pi$
$587$	−18371.7	−1.29179	−0.645894	+	0.763427i	$0.723515\pi$
$588$	0	0
$589$	3227.88	0.225811
$590$	−7518.31	−0.524617
$591$	0	0
$592$	−1099.45	−0.0763299
$593$	−7010.72	−0.485490	−0.242745	−	0.970090i	$-0.578048\pi$
$593$	−7010.72	−0.485490	−0.242745	+	0.970090i	$0.578048\pi$
$594$	0	0
$595$	−16131.1	−1.11144
$596$	11661.8	0.801485
$597$	0	0
$598$	−16734.9	−1.14438
$599$	27511.6	1.87662	0.938310	−	0.345796i	$-0.112391\pi$
$599$	27511.6	1.87662	0.938310	+	0.345796i	$0.112391\pi$
$600$	0	0
$601$	16592.6	1.12617	0.563084	−	0.826400i	$-0.309615\pi$
$601$	16592.6	1.12617	0.563084	+	0.826400i	$0.309615\pi$
$602$	32354.2	2.19046
$603$	0	0
$604$	−29427.7	−1.98244
$605$	−35653.3	−2.39589
$606$	0	0
$607$	28624.3	1.91404	0.957020	−	0.290021i	$-0.0936622\pi$
$607$	28624.3	1.91404	0.957020	+	0.290021i	$0.0936622\pi$
$608$	24728.7	1.64947
$609$	0	0
$610$	−6590.82	−0.437466
$611$	−10175.9	−0.673768
$612$	0	0
$613$	8715.44	0.574247	0.287123	−	0.957894i	$-0.407301\pi$
$613$	8715.44	0.574247	0.287123	+	0.957894i	$0.407301\pi$
$614$	20942.2	1.37648
$615$	0	0
$616$	−39131.6	−2.55951
$617$	−13069.4	−0.852765	−0.426383	−	0.904543i	$-0.640212\pi$
$617$	−13069.4	−0.852765	−0.426383	+	0.904543i	$0.640212\pi$
$618$	0	0
$619$	21034.0	1.36580	0.682899	−	0.730513i	$-0.260719\pi$
$619$	21034.0	1.36580	0.682899	+	0.730513i	$0.260719\pi$
$620$	−3077.72	−0.199362
$621$	0	0
$622$	−22269.2	−1.43555
$623$	22039.3	1.41731
$624$	0	0
$625$	−10193.2	−0.652362
$626$	−12456.6	−0.795315
$627$	0	0
$628$	32657.8	2.07514
$629$	−9129.44	−0.578719
$630$	0	0
$631$	−17009.2	−1.07310	−0.536549	−	0.843869i	$-0.680272\pi$
$631$	−17009.2	−1.07310	−0.536549	+	0.843869i	$0.680272\pi$
$632$	−15264.2	−0.960721
$633$	0	0
$634$	−27481.6	−1.72150
$635$	11233.1	0.702003
$636$	0	0
$637$	−17157.4	−1.06719
$638$	−13728.4	−0.851900
$639$	0	0
$640$	−20962.0	−1.29468
$641$	27883.5	1.71815	0.859073	−	0.511853i	$-0.171041\pi$
$641$	27883.5	1.71815	0.859073	+	0.511853i	$0.171041\pi$
$642$	0	0
$643$	−19707.0	−1.20866	−0.604330	−	0.796734i	$-0.706559\pi$
$643$	−19707.0	−1.20866	−0.604330	+	0.796734i	$0.706559\pi$
$644$	27595.5	1.68853
$645$	0	0
$646$	35630.3	2.17005
$647$	22680.2	1.37813	0.689065	−	0.724700i	$-0.258022\pi$
$647$	22680.2	1.37813	0.689065	+	0.724700i	$0.258022\pi$
$648$	0	0
$649$	12401.9	0.750106
$650$	−6966.60	−0.420389
$651$	0	0
$652$	−33963.2	−2.04004
$653$	26254.3	1.57337	0.786685	−	0.617354i	$-0.211795\pi$
$653$	26254.3	1.57337	0.786685	+	0.617354i	$0.211795\pi$
$654$	0	0
$655$	18608.9	1.11009
$656$	−2562.66	−0.152523
$657$	0	0
$658$	27510.1	1.62987
$659$	−1949.72	−0.115251	−0.0576255	−	0.998338i	$-0.518353\pi$
$659$	−1949.72	−0.115251	−0.0576255	+	0.998338i	$0.518353\pi$
$660$	0	0
$661$	23777.0	1.39912	0.699561	−	0.714573i	$-0.253379\pi$
$661$	23777.0	1.39912	0.699561	+	0.714573i	$0.253379\pi$
$662$	52191.8	3.06419
$663$	0	0
$664$	6559.15	0.383350
$665$	32073.1	1.87029
$666$	0	0
$667$	3490.31	0.202617
$668$	23713.0	1.37348
$669$	0	0
$670$	22651.7	1.30614
$671$	10872.0	0.625497
$672$	0	0
$673$	27619.5	1.58195	0.790975	−	0.611848i	$-0.209574\pi$
$673$	27619.5	1.58195	0.790975	+	0.611848i	$0.209574\pi$
$674$	24439.5	1.39670
$675$	0	0
$676$	−2077.20	−0.118184
$677$	−13410.8	−0.761328	−0.380664	−	0.924713i	$-0.624305\pi$
$677$	−13410.8	−0.761328	−0.380664	+	0.924713i	$0.624305\pi$
$678$	0	0
$679$	−32579.4	−1.84136
$680$	−12248.0	−0.690721
$681$	0	0
$682$	8323.46	0.467334
$683$	−3714.81	−0.208116	−0.104058	−	0.994571i	$-0.533183\pi$
$683$	−3714.81	−0.208116	−0.104058	+	0.994571i	$0.533183\pi$
$684$	0	0
$685$	3504.68	0.195485
$686$	4595.13	0.255748
$687$	0	0
$688$	−2011.74	−0.111478
$689$	−482.253	−0.0266653
$690$	0	0
$691$	−20704.3	−1.13984	−0.569920	−	0.821701i	$-0.693026\pi$
$691$	−20704.3	−1.13984	−0.569920	+	0.821701i	$0.693026\pi$
$692$	40586.3	2.22957
$693$	0	0
$694$	35375.4	1.93492
$695$	−9107.52	−0.497076
$696$	0	0
$697$	−21279.4	−1.15640
$698$	31955.5	1.73286
$699$	0	0
$700$	11487.8	0.620281
$701$	12253.0	0.660186	0.330093	−	0.943948i	$-0.392920\pi$
$701$	12253.0	0.660186	0.330093	+	0.943948i	$0.392920\pi$
$702$	0	0
$703$	18151.9	0.973844
$704$	59450.0	3.18268
$705$	0	0
$706$	6106.10	0.325505
$707$	28667.5	1.52497
$708$	0	0
$709$	4597.24	0.243516	0.121758	−	0.992560i	$-0.461147\pi$
$709$	4597.24	0.243516	0.121758	+	0.992560i	$0.461147\pi$
$710$	−15901.8	−0.840541
$711$	0	0
$712$	16734.0	0.880807
$713$	−2116.16	−0.111151
$714$	0	0
$715$	−30592.0	−1.60011
$716$	−28709.8	−1.49851
$717$	0	0
$718$	−1352.76	−0.0703129
$719$	440.245	0.0228350	0.0114175	−	0.999935i	$-0.496366\pi$
$719$	440.245	0.0228350	0.0114175	+	0.999935i	$0.496366\pi$
$720$	0	0
$721$	−28576.0	−1.47604
$722$	−39779.9	−2.05049
$723$	0	0
$724$	34940.0	1.79356
$725$	1452.99	0.0744312
$726$	0	0
$727$	−19917.9	−1.01611	−0.508056	−	0.861324i	$-0.669636\pi$
$727$	−19917.9	−1.01611	−0.508056	+	0.861324i	$0.669636\pi$
$728$	−24764.1	−1.26074
$729$	0	0
$730$	25159.9	1.27563
$731$	−16704.7	−0.845205
$732$	0	0
$733$	−22373.8	−1.12742	−0.563708	−	0.825974i	$-0.690626\pi$
$733$	−22373.8	−1.12742	−0.563708	+	0.825974i	$0.690626\pi$
$734$	−8669.32	−0.435954
$735$	0	0
$736$	−16211.8	−0.811925
$737$	−37365.5	−1.86754
$738$	0	0
$739$	27750.1	1.38133	0.690665	−	0.723175i	$-0.257318\pi$
$739$	27750.1	1.38133	0.690665	+	0.723175i	$0.257318\pi$
$740$	−17307.5	−0.859778
$741$	0	0
$742$	1303.75	0.0645042
$743$	1598.81	0.0789430	0.0394715	−	0.999221i	$-0.487433\pi$
$743$	1598.81	0.0789430	0.0394715	+	0.999221i	$0.487433\pi$
$744$	0	0
$745$	−8885.86	−0.436983
$746$	22423.4	1.10051
$747$	0	0
$748$	56040.3	2.73935
$749$	17696.4	0.863299
$750$	0	0
$751$	13317.9	0.647106	0.323553	−	0.946210i	$-0.395123\pi$
$751$	13317.9	0.647106	0.323553	+	0.946210i	$0.395123\pi$
$752$	−1710.54	−0.0829480
$753$	0	0
$754$	−8687.87	−0.419620
$755$	22422.8	1.08086
$756$	0	0
$757$	−8542.95	−0.410170	−0.205085	−	0.978744i	$-0.565747\pi$
$757$	−8542.95	−0.410170	−0.205085	+	0.978744i	$0.565747\pi$
$758$	−33690.6	−1.61438
$759$	0	0
$760$	24352.5	1.16231
$761$	35185.7	1.67606	0.838030	−	0.545624i	$-0.183707\pi$
$761$	35185.7	1.67606	0.838030	+	0.545624i	$0.183707\pi$
$762$	0	0
$763$	54123.3	2.56801
$764$	−22807.1	−1.08002
$765$	0	0
$766$	7737.16	0.364954
$767$	7848.45	0.369480
$768$	0	0
$769$	−10784.9	−0.505738	−0.252869	−	0.967501i	$-0.581374\pi$
$769$	−10784.9	−0.505738	−0.252869	+	0.967501i	$0.581374\pi$
$770$	82704.2	3.87072
$771$	0	0
$772$	−24926.1	−1.16206
$773$	−28226.1	−1.31335	−0.656676	−	0.754172i	$-0.728038\pi$
$773$	−28226.1	−1.31335	−0.656676	+	0.754172i	$0.728038\pi$
$774$	0	0
$775$	−880.940	−0.0408314
$776$	−24737.0	−1.14434
$777$	0	0
$778$	10574.8	0.487308
$779$	42309.4	1.94595
$780$	0	0
$781$	26231.1	1.20182
$782$	−23358.8	−1.06817
$783$	0	0
$784$	−2884.11	−0.131383
$785$	−24884.1	−1.13140
$786$	0	0
$787$	25186.3	1.14078	0.570389	−	0.821374i	$-0.306792\pi$
$787$	25186.3	1.14078	0.570389	+	0.821374i	$0.306792\pi$
$788$	33786.2	1.52739
$789$	0	0
$790$	32260.6	1.45289
$791$	23029.4	1.03519
$792$	0	0
$793$	6880.24	0.308101
$794$	33576.2	1.50072
$795$	0	0
$796$	−52963.4	−2.35834
$797$	−35516.3	−1.57848	−0.789241	−	0.614083i	$-0.789526\pi$
$797$	−35516.3	−1.57848	−0.789241	+	0.614083i	$0.789526\pi$
$798$	0	0
$799$	−14203.6	−0.628897
$800$	−6748.85	−0.298260
$801$	0	0
$802$	65249.8	2.87288
$803$	−41503.0	−1.82392
$804$	0	0
$805$	−21026.8	−0.920617
$806$	5267.42	0.230195
$807$	0	0
$808$	21766.7	0.947710
$809$	−1221.43	−0.0530818	−0.0265409	−	0.999648i	$-0.508449\pi$
$809$	−1221.43	−0.0530818	−0.0265409	+	0.999648i	$0.508449\pi$
$810$	0	0
$811$	−25894.8	−1.12119	−0.560597	−	0.828089i	$-0.689428\pi$
$811$	−25894.8	−1.12119	−0.560597	+	0.828089i	$0.689428\pi$
$812$	14326.1	0.619147
$813$	0	0
$814$	46806.8	2.01545
$815$	25878.8	1.11226
$816$	0	0
$817$	33213.7	1.42228
$818$	55104.1	2.35534
$819$	0	0
$820$	−40341.2	−1.71802
$821$	14905.0	0.633602	0.316801	−	0.948492i	$-0.397391\pi$
$821$	14905.0	0.633602	0.316801	+	0.948492i	$0.397391\pi$
$822$	0	0
$823$	−18664.2	−0.790513	−0.395257	−	0.918571i	$-0.629344\pi$
$823$	−18664.2	−0.790513	−0.395257	+	0.918571i	$0.629344\pi$
$824$	−21697.2	−0.917305
$825$	0	0
$826$	−21217.9	−0.893785
$827$	40017.8	1.68265	0.841327	−	0.540527i	$-0.181775\pi$
$827$	40017.8	1.68265	0.841327	+	0.540527i	$0.181775\pi$
$828$	0	0
$829$	−3833.23	−0.160596	−0.0802978	−	0.996771i	$-0.525587\pi$
$829$	−3833.23	−0.160596	−0.0802978	+	0.996771i	$0.525587\pi$
$830$	−13862.7	−0.579736
$831$	0	0
$832$	37622.3	1.56769
$833$	−23948.5	−0.996120
$834$	0	0
$835$	−18068.5	−0.748845
$836$	−111424.	−4.60967
$837$	0	0
$838$	10729.8	0.442309
$839$	10518.1	0.432805	0.216403	−	0.976304i	$-0.430568\pi$
$839$	10518.1	0.432805	0.216403	+	0.976304i	$0.430568\pi$
$840$	0	0
$841$	−22577.0	−0.925705
$842$	15323.7	0.627185
$843$	0	0
$844$	44318.0	1.80745
$845$	1582.76	0.0644360
$846$	0	0
$847$	−100620.	−4.08185
$848$	−81.0653	−0.00328278
$849$	0	0
$850$	−9724.07	−0.392392
$851$	−11900.2	−0.479357
$852$	0	0
$853$	−11718.0	−0.470359	−0.235179	−	0.971952i	$-0.575568\pi$
$853$	−11718.0	−0.470359	−0.235179	+	0.971952i	$0.575568\pi$
$854$	−18600.4	−0.745308
$855$	0	0
$856$	13436.5	0.536508
$857$	−32239.1	−1.28503	−0.642513	−	0.766275i	$-0.722108\pi$
$857$	−32239.1	−1.28503	−0.642513	+	0.766275i	$0.722108\pi$
$858$	0	0
$859$	18098.0	0.718854	0.359427	−	0.933173i	$-0.382972\pi$
$859$	18098.0	0.718854	0.359427	+	0.933173i	$0.382972\pi$
$860$	−31668.5	−1.25568
$861$	0	0
$862$	37777.0	1.49268
$863$	38172.0	1.50567	0.752833	−	0.658211i	$-0.228687\pi$
$863$	38172.0	1.50567	0.752833	+	0.658211i	$0.228687\pi$
$864$	0	0
$865$	−30925.3	−1.21560
$866$	20970.0	0.822853
$867$	0	0
$868$	−8685.84	−0.339651
$869$	−53216.0	−2.07736
$870$	0	0
$871$	−23646.4	−0.919895
$872$	41094.8	1.59592
$873$	0	0
$874$	46443.9	1.79747
$875$	−40808.2	−1.57665
$876$	0	0
$877$	32257.1	1.24201	0.621006	−	0.783806i	$-0.286724\pi$
$877$	32257.1	1.24201	0.621006	+	0.783806i	$0.286724\pi$
$878$	3401.91	0.130762
$879$	0	0
$880$	−5142.43	−0.196990
$881$	−52107.0	−1.99266	−0.996329	−	0.0856123i	$-0.972715\pi$
$881$	−52107.0	−1.99266	−0.996329	+	0.0856123i	$0.972715\pi$
$882$	0	0
$883$	−1320.63	−0.0503315	−0.0251657	−	0.999683i	$-0.508011\pi$
$883$	−1320.63	−0.0503315	−0.0251657	+	0.999683i	$0.508011\pi$
$884$	35464.6	1.34932
$885$	0	0
$886$	12285.8	0.465858
$887$	47738.6	1.80711	0.903554	−	0.428474i	$-0.140949\pi$
$887$	47738.6	1.80711	0.903554	+	0.428474i	$0.140949\pi$
$888$	0	0
$889$	31701.7	1.19600
$890$	−35367.2	−1.33203
$891$	0	0
$892$	23353.1	0.876593
$893$	28240.9	1.05828
$894$	0	0
$895$	21875.9	0.817016
$896$	−59158.3	−2.20574
$897$	0	0
$898$	−3994.30	−0.148432
$899$	−1098.60	−0.0407567
$900$	0	0
$901$	−673.134	−0.0248894
$902$	109100.	4.02729
$903$	0	0
$904$	17485.8	0.643329
$905$	−26623.0	−0.977878
$906$	0	0
$907$	−43507.5	−1.59277	−0.796385	−	0.604789i	$-0.793257\pi$
$907$	−43507.5	−1.59277	−0.796385	+	0.604789i	$0.793257\pi$
$908$	6045.50	0.220955
$909$	0	0
$910$	52338.5	1.90660
$911$	6606.55	0.240269	0.120134	−	0.992758i	$-0.461667\pi$
$911$	6606.55	0.240269	0.120134	+	0.992758i	$0.461667\pi$
$912$	0	0
$913$	22867.4	0.828917
$914$	75167.8	2.72027
$915$	0	0
$916$	47647.9	1.71870
$917$	52517.4	1.89125
$918$	0	0
$919$	−5725.64	−0.205518	−0.102759	−	0.994706i	$-0.532767\pi$
$919$	−5725.64	−0.205518	−0.102759	+	0.994706i	$0.532767\pi$
$920$	−15965.2	−0.572129
$921$	0	0
$922$	79950.8	2.85579
$923$	16600.1	0.591981
$924$	0	0
$925$	−4953.95	−0.176092
$926$	−41770.7	−1.48236
$927$	0	0
$928$	−8416.32	−0.297715
$929$	−46480.5	−1.64153	−0.820763	−	0.571270i	$-0.806451\pi$
$929$	−46480.5	−1.64153	−0.820763	+	0.571270i	$0.806451\pi$
$930$	0	0
$931$	47616.5	1.67623
$932$	−8255.84	−0.290160
$933$	0	0
$934$	−68202.3	−2.38934
$935$	−42700.7	−1.49354
$936$	0	0
$937$	−45125.9	−1.57332	−0.786660	−	0.617387i	$-0.788191\pi$
$937$	−45125.9	−1.57332	−0.786660	+	0.617387i	$0.788191\pi$
$938$	63927.0	2.22526
$939$	0	0
$940$	−26927.1	−0.934324
$941$	11663.7	0.404065	0.202033	−	0.979379i	$-0.435245\pi$
$941$	11663.7	0.404065	0.202033	+	0.979379i	$0.435245\pi$
$942$	0	0
$943$	−27737.6	−0.957857
$944$	1319.30	0.0454869
$945$	0	0
$946$	85645.2	2.94352
$947$	27850.8	0.955681	0.477841	−	0.878447i	$-0.341420\pi$
$947$	27850.8	0.955681	0.477841	+	0.878447i	$0.341420\pi$
$948$	0	0
$949$	−26264.7	−0.898409
$950$	19334.2	0.660300
$951$	0	0
$952$	−34566.0	−1.17677
$953$	47479.3	1.61386	0.806929	−	0.590649i	$-0.201128\pi$
$953$	47479.3	1.61386	0.806929	+	0.590649i	$0.201128\pi$
$954$	0	0
$955$	17378.2	0.588844
$956$	2989.95	0.101153
$957$	0	0
$958$	9396.59	0.316900
$959$	9890.81	0.333046
$960$	0	0
$961$	−29124.9	−0.977642
$962$	29621.2	0.992750
$963$	0	0
$964$	−24756.1	−0.827118
$965$	18992.8	0.633576
$966$	0	0
$967$	−41490.9	−1.37979	−0.689895	−	0.723909i	$-0.742344\pi$
$967$	−41490.9	−1.37979	−0.689895	+	0.723909i	$0.742344\pi$
$968$	−76398.6	−2.53672
$969$	0	0
$970$	52281.3	1.73057
$971$	4741.55	0.156708	0.0783541	−	0.996926i	$-0.475034\pi$
$971$	4741.55	0.156708	0.0783541	+	0.996926i	$0.475034\pi$
$972$	0	0
$973$	−25703.0	−0.846865
$974$	−11145.8	−0.366668
$975$	0	0
$976$	1156.55	0.0379305
$977$	−23262.0	−0.761738	−0.380869	−	0.924629i	$-0.624375\pi$
$977$	−23262.0	−0.761738	−0.380869	+	0.924629i	$0.624375\pi$
$978$	0	0
$979$	58340.5	1.90457
$980$	−45401.4	−1.47989
$981$	0	0
$982$	−66341.5	−2.15585
$983$	46417.0	1.50607	0.753037	−	0.657978i	$-0.228588\pi$
$983$	46417.0	1.50607	0.753037	+	0.657978i	$0.228588\pi$
$984$	0	0
$985$	−25743.9	−0.832759
$986$	−12126.6	−0.391674
$987$	0	0
$988$	−70513.6	−2.27058
$989$	−21774.5	−0.700090
$990$	0	0
$991$	16178.6	0.518596	0.259298	−	0.965797i	$-0.416509\pi$
$991$	16178.6	0.518596	0.259298	+	0.965797i	$0.416509\pi$
$992$	5102.78	0.163320
$993$	0	0
$994$	−44877.6	−1.43202
$995$	40356.3	1.28581
$996$	0	0
$997$	9939.95	0.315749	0.157874	−	0.987459i	$-0.449536\pi$
$997$	9939.95	0.315749	0.157874	+	0.987459i	$0.449536\pi$
$998$	−20164.4	−0.639572
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2151.4.a.g.1.9	✓	59
3.2	odd	2		2151.4.a.h.1.51	yes	59

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2151.4.a.g.1.9	✓	59	1.1	even	1	trivial
2151.4.a.h.1.51	yes	59	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-4.52883 q^{2} +12.5103 q^{4} -9.53238 q^{5} -26.9020 q^{7} -20.4262 q^{8} +O(q^{10})\)	\(q-4.52883 q^{2} +12.5103 q^{4} -9.53238 q^{5} -26.9020 q^{7} -20.4262 q^{8} +43.1705 q^{10} -71.2126 q^{11} -45.0662 q^{13} +121.834 q^{14} -7.57549 q^{16} -62.9039 q^{17} +125.071 q^{19} -119.252 q^{20} +322.509 q^{22} -81.9950 q^{23} -34.1338 q^{25} +204.097 q^{26} -336.551 q^{28} -42.5674 q^{29} +25.8084 q^{31} +197.717 q^{32} +284.881 q^{34} +256.440 q^{35} +145.133 q^{37} -566.424 q^{38} +194.710 q^{40} +338.284 q^{41} +265.559 q^{43} -890.888 q^{44} +371.341 q^{46} +225.799 q^{47} +380.716 q^{49} +154.586 q^{50} -563.790 q^{52} +10.7010 q^{53} +678.825 q^{55} +549.504 q^{56} +192.780 q^{58} -174.154 q^{59} -152.670 q^{61} -116.882 q^{62} -834.824 q^{64} +429.588 q^{65} +524.704 q^{67} -786.944 q^{68} -1161.37 q^{70} -368.349 q^{71} +582.804 q^{73} -657.282 q^{74} +1564.67 q^{76} +1915.76 q^{77} +747.284 q^{79} +72.2124 q^{80} -1532.03 q^{82} -321.115 q^{83} +599.624 q^{85} -1202.67 q^{86} +1454.60 q^{88} -819.245 q^{89} +1212.37 q^{91} -1025.78 q^{92} -1022.60 q^{94} -1192.22 q^{95} +1211.04 q^{97} -1724.20 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 59 q - 8 q^{2} + 238 q^{4} - 80 q^{5} - 10 q^{7} - 96 q^{8}+O(q^{10}) \) 59 * q - 8 * q^2 + 238 * q^4 - 80 * q^5 - 10 * q^7 - 96 * q^8	\( 59 q - 8 q^{2} + 238 q^{4} - 80 q^{5} - 10 q^{7} - 96 q^{8} - 36 q^{10} - 132 q^{11} + 104 q^{13} - 280 q^{14} + 822 q^{16} - 408 q^{17} + 20 q^{19} - 800 q^{20} - 2 q^{22} - 276 q^{23} + 1477 q^{25} - 780 q^{26} + 224 q^{28} - 696 q^{29} - 380 q^{31} - 896 q^{32} - 72 q^{34} - 700 q^{35} + 224 q^{37} - 988 q^{38} - 258 q^{40} - 2706 q^{41} - 156 q^{43} - 1584 q^{44} + 428 q^{46} - 1316 q^{47} + 2135 q^{49} - 1400 q^{50} + 1092 q^{52} - 1484 q^{53} - 992 q^{55} - 3360 q^{56} - 120 q^{58} - 3186 q^{59} - 254 q^{61} - 1240 q^{62} + 3054 q^{64} - 5120 q^{65} + 288 q^{67} - 9420 q^{68} + 1108 q^{70} - 4468 q^{71} - 1770 q^{73} - 6214 q^{74} + 720 q^{76} - 6352 q^{77} - 746 q^{79} - 7040 q^{80} + 276 q^{82} - 5484 q^{83} + 588 q^{85} - 10152 q^{86} + 1186 q^{88} - 11570 q^{89} + 1768 q^{91} - 15366 q^{92} - 2142 q^{94} - 5736 q^{95} + 2390 q^{97} - 6912 q^{98}+O(q^{100}) \) 59 * q - 8 * q^2 + 238 * q^4 - 80 * q^5 - 10 * q^7 - 96 * q^8 - 36 * q^10 - 132 * q^11 + 104 * q^13 - 280 * q^14 + 822 * q^16 - 408 * q^17 + 20 * q^19 - 800 * q^20 - 2 * q^22 - 276 * q^23 + 1477 * q^25 - 780 * q^26 + 224 * q^28 - 696 * q^29 - 380 * q^31 - 896 * q^32 - 72 * q^34 - 700 * q^35 + 224 * q^37 - 988 * q^38 - 258 * q^40 - 2706 * q^41 - 156 * q^43 - 1584 * q^44 + 428 * q^46 - 1316 * q^47 + 2135 * q^49 - 1400 * q^50 + 1092 * q^52 - 1484 * q^53 - 992 * q^55 - 3360 * q^56 - 120 * q^58 - 3186 * q^59 - 254 * q^61 - 1240 * q^62 + 3054 * q^64 - 5120 * q^65 + 288 * q^67 - 9420 * q^68 + 1108 * q^70 - 4468 * q^71 - 1770 * q^73 - 6214 * q^74 + 720 * q^76 - 6352 * q^77 - 746 * q^79 - 7040 * q^80 + 276 * q^82 - 5484 * q^83 + 588 * q^85 - 10152 * q^86 + 1186 * q^88 - 11570 * q^89 + 1768 * q^91 - 15366 * q^92 - 2142 * q^94 - 5736 * q^95 + 2390 * q^97 - 6912 * q^98

Introduction

L-functions

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