LMFDB - Embedded newform 2151.4.a.f.1.10

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:	$0$
Dimension:	$37$
Twist minimal:	no (minimal twist has level 239)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.10
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)

$f(q)$	$=$	$q-3.37030 q^{2} +3.35895 q^{4} +1.43026 q^{5} +27.2799 q^{7} +15.6418 q^{8} +O(q^{10})$	\(q-3.37030 q^{2} +3.35895 q^{4} +1.43026 q^{5} +27.2799 q^{7} +15.6418 q^{8} -4.82040 q^{10} -6.62635 q^{11} +84.5743 q^{13} -91.9416 q^{14} -79.5890 q^{16} +8.04823 q^{17} +27.6752 q^{19} +4.80416 q^{20} +22.3328 q^{22} -16.9673 q^{23} -122.954 q^{25} -285.041 q^{26} +91.6317 q^{28} +118.315 q^{29} +184.699 q^{31} +143.105 q^{32} -27.1250 q^{34} +39.0173 q^{35} +353.342 q^{37} -93.2738 q^{38} +22.3718 q^{40} +488.043 q^{41} -427.878 q^{43} -22.2575 q^{44} +57.1851 q^{46} +468.788 q^{47} +401.193 q^{49} +414.394 q^{50} +284.080 q^{52} +122.499 q^{53} -9.47739 q^{55} +426.706 q^{56} -398.759 q^{58} +210.060 q^{59} -452.647 q^{61} -622.491 q^{62} +154.405 q^{64} +120.963 q^{65} +379.919 q^{67} +27.0336 q^{68} -131.500 q^{70} -1145.80 q^{71} +713.382 q^{73} -1190.87 q^{74} +92.9594 q^{76} -180.766 q^{77} +85.2500 q^{79} -113.833 q^{80} -1644.85 q^{82} +400.621 q^{83} +11.5110 q^{85} +1442.08 q^{86} -103.648 q^{88} +1665.53 q^{89} +2307.18 q^{91} -56.9923 q^{92} -1579.96 q^{94} +39.5827 q^{95} -1183.67 q^{97} -1352.14 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 37 q - 4 q^{2} + 170 q^{4} - 43 q^{5} + 60 q^{7} - 27 q^{8}+O(q^{10}) $ 37 * q - 4 * q^2 + 170 * q^4 - 43 * q^5 + 60 * q^7 - 27 * q^8	$ 37 q - 4 q^{2} + 170 q^{4} - 43 q^{5} + 60 q^{7} - 27 q^{8} + 147 q^{10} - 55 q^{11} + 250 q^{13} - 169 q^{14} + 918 q^{16} - 189 q^{17} + 550 q^{19} - 486 q^{20} + 226 q^{22} - 74 q^{23} + 1604 q^{25} - 560 q^{26} + 829 q^{28} - 389 q^{29} + 1107 q^{31} - 125 q^{32} + 1423 q^{34} - 270 q^{35} + 1002 q^{37} - 1037 q^{38} + 1536 q^{40} - 1518 q^{41} + 1098 q^{43} - 1037 q^{44} + 1030 q^{46} - 1214 q^{47} + 4663 q^{49} - 929 q^{50} + 2895 q^{52} - 904 q^{53} + 1350 q^{55} - 2556 q^{56} + 1396 q^{58} - 1658 q^{59} + 2313 q^{61} + 4519 q^{62} + 3807 q^{64} + 56 q^{65} + 1535 q^{67} + 6526 q^{68} - 4099 q^{70} + 3255 q^{71} + 3154 q^{73} + 2629 q^{74} + 1981 q^{76} + 3734 q^{77} + 2260 q^{79} + 8242 q^{80} - 9898 q^{82} + 939 q^{83} + 1272 q^{85} + 3457 q^{86} - 1808 q^{88} - 1486 q^{89} + 174 q^{91} + 14076 q^{92} - 984 q^{94} + 1828 q^{95} + 6148 q^{97} + 6243 q^{98}+O(q^{100}) $ 37 * q - 4 * q^2 + 170 * q^4 - 43 * q^5 + 60 * q^7 - 27 * q^8 + 147 * q^10 - 55 * q^11 + 250 * q^13 - 169 * q^14 + 918 * q^16 - 189 * q^17 + 550 * q^19 - 486 * q^20 + 226 * q^22 - 74 * q^23 + 1604 * q^25 - 560 * q^26 + 829 * q^28 - 389 * q^29 + 1107 * q^31 - 125 * q^32 + 1423 * q^34 - 270 * q^35 + 1002 * q^37 - 1037 * q^38 + 1536 * q^40 - 1518 * q^41 + 1098 * q^43 - 1037 * q^44 + 1030 * q^46 - 1214 * q^47 + 4663 * q^49 - 929 * q^50 + 2895 * q^52 - 904 * q^53 + 1350 * q^55 - 2556 * q^56 + 1396 * q^58 - 1658 * q^59 + 2313 * q^61 + 4519 * q^62 + 3807 * q^64 + 56 * q^65 + 1535 * q^67 + 6526 * q^68 - 4099 * q^70 + 3255 * q^71 + 3154 * q^73 + 2629 * q^74 + 1981 * q^76 + 3734 * q^77 + 2260 * q^79 + 8242 * q^80 - 9898 * q^82 + 939 * q^83 + 1272 * q^85 + 3457 * q^86 - 1808 * q^88 - 1486 * q^89 + 174 * q^91 + 14076 * q^92 - 984 * q^94 + 1828 * q^95 + 6148 * q^97 + 6243 * 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 $

$2$	−3.37030	−1.19158	−0.595791	−	0.803139i	$-0.703161\pi$
$2$	−3.37030	−1.19158	−0.595791	+	0.803139i	$0.703161\pi$
$3$	0	0
$3$	0	0
$4$	3.35895	0.419868
$5$	1.43026	0.127926	0.0639631	−	0.997952i	$-0.479626\pi$
$5$	1.43026	0.127926	0.0639631	+	0.997952i	$0.479626\pi$
$6$	0	0
$7$	27.2799	1.47298	0.736488	−	0.676450i	$-0.236483\pi$
$7$	27.2799	1.47298	0.736488	+	0.676450i	$0.236483\pi$
$8$	15.6418	0.691275
$9$	0	0
$10$	−4.82040	−0.152435
$11$	−6.62635	−0.181629	−0.0908145	−	0.995868i	$-0.528947\pi$
$11$	−6.62635	−0.181629	−0.0908145	+	0.995868i	$0.528947\pi$
$12$	0	0
$13$	84.5743	1.80436	0.902180	−	0.431359i	$-0.141966\pi$
$13$	84.5743	1.80436	0.902180	+	0.431359i	$0.141966\pi$
$14$	−91.9416	−1.75517
$15$	0	0
$16$	−79.5890	−1.24358
$17$	8.04823	0.114822	0.0574112	−	0.998351i	$-0.481715\pi$
$17$	8.04823	0.114822	0.0574112	+	0.998351i	$0.481715\pi$
$18$	0	0
$19$	27.6752	0.334164	0.167082	−	0.985943i	$-0.446565\pi$
$19$	27.6752	0.334164	0.167082	+	0.985943i	$0.446565\pi$
$20$	4.80416	0.0537121
$21$	0	0
$22$	22.3328	0.216426
$23$	−16.9673	−0.153823	−0.0769116	−	0.997038i	$-0.524506\pi$
$23$	−16.9673	−0.153823	−0.0769116	+	0.997038i	$0.524506\pi$
$24$	0	0
$25$	−122.954	−0.983635
$26$	−285.041	−2.15004
$27$	0	0
$28$	91.6317	0.618456
$29$	118.315	0.757607	0.378804	−	0.925477i	$-0.376336\pi$
$29$	118.315	0.757607	0.378804	+	0.925477i	$0.376336\pi$
$30$	0	0
$31$	184.699	1.07009	0.535047	−	0.844823i	$-0.320294\pi$
$31$	184.699	1.07009	0.535047	+	0.844823i	$0.320294\pi$
$32$	143.105	0.790552
$33$	0	0
$34$	−27.1250	−0.136820
$35$	39.0173	0.188432
$36$	0	0
$37$	353.342	1.56998	0.784988	−	0.619510i	$-0.212669\pi$
$37$	353.342	1.56998	0.784988	+	0.619510i	$0.212669\pi$
$38$	−93.2738	−0.398184
$39$	0	0
$40$	22.3718	0.0884322
$41$	488.043	1.85901	0.929507	−	0.368805i	$-0.120233\pi$
$41$	488.043	1.85901	0.929507	+	0.368805i	$0.120233\pi$
$42$	0	0
$43$	−427.878	−1.51746	−0.758730	−	0.651405i	$-0.774180\pi$
$43$	−427.878	−1.51746	−0.758730	+	0.651405i	$0.774180\pi$
$44$	−22.2575	−0.0762602
$45$	0	0
$46$	57.1851	0.183293
$47$	468.788	1.45489	0.727444	−	0.686167i	$-0.240708\pi$
$47$	468.788	1.45489	0.727444	+	0.686167i	$0.240708\pi$
$48$	0	0
$49$	401.193	1.16966
$50$	414.394	1.17208
$51$	0	0
$52$	284.080	0.757594
$53$	122.499	0.317481	0.158741	−	0.987320i	$-0.449257\pi$
$53$	122.499	0.317481	0.158741	+	0.987320i	$0.449257\pi$
$54$	0	0
$55$	−9.47739	−0.0232351
$56$	426.706	1.01823
$57$	0	0
$58$	−398.759	−0.902752
$59$	210.060	0.463516	0.231758	−	0.972773i	$-0.425552\pi$
$59$	210.060	0.463516	0.231758	+	0.972773i	$0.425552\pi$
$60$	0	0
$61$	−452.647	−0.950091	−0.475045	−	0.879961i	$-0.657568\pi$
$61$	−452.647	−0.950091	−0.475045	+	0.879961i	$0.657568\pi$
$62$	−622.491	−1.27510
$63$	0	0
$64$	154.405	0.301571
$65$	120.963	0.230825
$66$	0	0
$67$	379.919	0.692754	0.346377	−	0.938095i	$-0.387412\pi$
$67$	379.919	0.692754	0.346377	+	0.938095i	$0.387412\pi$
$68$	27.0336	0.0482103
$69$	0	0
$70$	−131.500	−0.224533
$71$	−1145.80	−1.91523	−0.957617	−	0.288045i	$-0.906995\pi$
$71$	−1145.80	−1.91523	−0.957617	+	0.288045i	$0.906995\pi$
$72$	0	0
$73$	713.382	1.14377	0.571884	−	0.820335i	$-0.306213\pi$
$73$	713.382	1.14377	0.571884	+	0.820335i	$0.306213\pi$
$74$	−1190.87	−1.87076
$75$	0	0
$76$	92.9594	0.140305
$77$	−180.766	−0.267535
$78$	0	0
$79$	85.2500	0.121410	0.0607049	−	0.998156i	$-0.480665\pi$
$79$	85.2500	0.121410	0.0607049	+	0.998156i	$0.480665\pi$
$80$	−113.833	−0.159086
$81$	0	0
$82$	−1644.85	−2.21517
$83$	400.621	0.529806	0.264903	−	0.964275i	$-0.414660\pi$
$83$	400.621	0.529806	0.264903	+	0.964275i	$0.414660\pi$
$84$	0	0
$85$	11.5110	0.0146888
$86$	1442.08	1.80818
$87$	0	0
$88$	−103.648	−0.125556
$89$	1665.53	1.98366	0.991830	−	0.127569i	$-0.0407173\pi$
$89$	1665.53	1.98366	0.991830	+	0.127569i	$0.0407173\pi$
$90$	0	0
$91$	2307.18	2.65778
$92$	−56.9923	−0.0645855
$93$	0	0
$94$	−1579.96	−1.73362
$95$	39.5827	0.0427484
$96$	0	0
$97$	−1183.67	−1.23900	−0.619501	−	0.784996i	$-0.712665\pi$
$97$	−1183.67	−1.23900	−0.619501	+	0.784996i	$0.712665\pi$
$98$	−1352.14	−1.39375
$99$	0	0
$100$	−412.997	−0.412997
$101$	−739.422	−0.728468	−0.364234	−	0.931308i	$-0.618669\pi$
$101$	−739.422	−0.728468	−0.364234	+	0.931308i	$0.618669\pi$
$102$	0	0
$103$	1082.77	1.03581	0.517904	−	0.855439i	$-0.326712\pi$
$103$	1082.77	1.03581	0.517904	+	0.855439i	$0.326712\pi$
$104$	1322.89	1.24731
$105$	0	0
$106$	−412.858	−0.378305
$107$	−1008.05	−0.910764	−0.455382	−	0.890296i	$-0.650497\pi$
$107$	−1008.05	−0.910764	−0.455382	+	0.890296i	$0.650497\pi$
$108$	0	0
$109$	−917.247	−0.806022	−0.403011	−	0.915195i	$-0.632036\pi$
$109$	−917.247	−0.806022	−0.403011	+	0.915195i	$0.632036\pi$
$110$	31.9417	0.0276865
$111$	0	0
$112$	−2171.18	−1.83176
$113$	12.6055	0.0104941	0.00524703	−	0.999986i	$-0.498330\pi$
$113$	12.6055	0.0104941	0.00524703	+	0.999986i	$0.498330\pi$
$114$	0	0
$115$	−24.2677	−0.0196780
$116$	397.415	0.318095
$117$	0	0
$118$	−707.965	−0.552318
$119$	219.555	0.169131
$120$	0	0
$121$	−1287.09	−0.967011
$122$	1525.56	1.13211
$123$	0	0
$124$	620.393	0.449298
$125$	−354.639	−0.253759
$126$	0	0
$127$	−320.750	−0.224110	−0.112055	−	0.993702i	$-0.535743\pi$
$127$	−320.750	−0.224110	−0.112055	+	0.993702i	$0.535743\pi$
$128$	−1665.23	−1.14990
$129$	0	0
$130$	−407.682	−0.275047
$131$	−1962.29	−1.30875	−0.654374	−	0.756171i	$-0.727068\pi$
$131$	−1962.29	−1.30875	−0.654374	+	0.756171i	$0.727068\pi$
$132$	0	0
$133$	754.976	0.492216
$134$	−1280.44	−0.825473
$135$	0	0
$136$	125.888	0.0793739
$137$	2437.40	1.52001	0.760004	−	0.649918i	$-0.225197\pi$
$137$	2437.40	1.52001	0.760004	+	0.649918i	$0.225197\pi$
$138$	0	0
$139$	−1218.76	−0.743695	−0.371847	−	0.928294i	$-0.621275\pi$
$139$	−1218.76	−0.743695	−0.371847	+	0.928294i	$0.621275\pi$
$140$	131.057	0.0791167
$141$	0	0
$142$	3861.70	2.28216
$143$	−560.419	−0.327724
$144$	0	0
$145$	169.222	0.0969178
$146$	−2404.31	−1.36289
$147$	0	0
$148$	1186.86	0.659183
$149$	−2093.39	−1.15099	−0.575494	−	0.817806i	$-0.695190\pi$
$149$	−2093.39	−1.15099	−0.575494	+	0.817806i	$0.695190\pi$
$150$	0	0
$151$	−2170.25	−1.16962	−0.584811	−	0.811170i	$-0.698831\pi$
$151$	−2170.25	−1.16962	−0.584811	+	0.811170i	$0.698831\pi$
$152$	432.889	0.230999
$153$	0	0
$154$	609.237	0.318790
$155$	264.167	0.136893
$156$	0	0
$157$	2876.47	1.46221	0.731106	−	0.682264i	$-0.239005\pi$
$157$	2876.47	1.46221	0.731106	+	0.682264i	$0.239005\pi$
$158$	−287.318	−0.144670
$159$	0	0
$160$	204.677	0.101132
$161$	−462.867	−0.226578
$162$	0	0
$163$	1463.09	0.703056	0.351528	−	0.936177i	$-0.385662\pi$
$163$	1463.09	0.703056	0.351528	+	0.936177i	$0.385662\pi$
$164$	1639.31	0.780541
$165$	0	0
$166$	−1350.21	−0.631307
$167$	−2582.08	−1.19645	−0.598225	−	0.801328i	$-0.704127\pi$
$167$	−2582.08	−1.19645	−0.598225	+	0.801328i	$0.704127\pi$
$168$	0	0
$169$	4955.81	2.25572
$170$	−38.7957	−0.0175029
$171$	0	0
$172$	−1437.22	−0.637133
$173$	1463.21	0.643038	0.321519	−	0.946903i	$-0.395807\pi$
$173$	1463.21	0.643038	0.321519	+	0.946903i	$0.395807\pi$
$174$	0	0
$175$	−3354.18	−1.44887
$176$	527.385	0.225870
$177$	0	0
$178$	−5613.34	−2.36369
$179$	−1445.51	−0.603588	−0.301794	−	0.953373i	$-0.597586\pi$
$179$	−1445.51	−0.603588	−0.301794	+	0.953373i	$0.597586\pi$
$180$	0	0
$181$	664.382	0.272835	0.136418	−	0.990651i	$-0.456441\pi$
$181$	664.382	0.272835	0.136418	+	0.990651i	$0.456441\pi$
$182$	−7775.89	−3.16696
$183$	0	0
$184$	−265.399	−0.106334
$185$	505.371	0.200841
$186$	0	0
$187$	−53.3304	−0.0208551
$188$	1574.63	0.610861
$189$	0	0
$190$	−133.406	−0.0509382
$191$	−2439.75	−0.924263	−0.462132	−	0.886811i	$-0.652915\pi$
$191$	−2439.75	−0.924263	−0.462132	+	0.886811i	$0.652915\pi$
$192$	0	0
$193$	1259.42	0.469715	0.234857	−	0.972030i	$-0.424538\pi$
$193$	1259.42	0.469715	0.234857	+	0.972030i	$0.424538\pi$
$194$	3989.32	1.47637
$195$	0	0
$196$	1347.59	0.491103
$197$	1756.42	0.635226	0.317613	−	0.948220i	$-0.397119\pi$
$197$	1756.42	0.635226	0.317613	+	0.948220i	$0.397119\pi$
$198$	0	0
$199$	−2307.55	−0.822001	−0.411000	−	0.911635i	$-0.634821\pi$
$199$	−2307.55	−0.822001	−0.411000	+	0.911635i	$0.634821\pi$
$200$	−1923.22	−0.679962
$201$	0	0
$202$	2492.08	0.868029
$203$	3227.63	1.11594
$204$	0	0
$205$	698.028	0.237817
$206$	−3649.26	−1.23425
$207$	0	0
$208$	−6731.19	−2.24386
$209$	−183.385	−0.0606939
$210$	0	0
$211$	2321.08	0.757298	0.378649	−	0.925540i	$-0.376389\pi$
$211$	2321.08	0.757298	0.378649	+	0.925540i	$0.376389\pi$
$212$	411.467	0.133300
$213$	0	0
$214$	3397.43	1.08525
$215$	−611.976	−0.194123
$216$	0	0
$217$	5038.57	1.57622
$218$	3091.40	0.960441
$219$	0	0
$220$	−31.8340	−0.00975568
$221$	680.673	0.207181
$222$	0	0
$223$	4602.38	1.38206	0.691028	−	0.722828i	$-0.257158\pi$
$223$	4602.38	1.38206	0.691028	+	0.722828i	$0.257158\pi$
$224$	3903.89	1.16446
$225$	0	0
$226$	−42.4845	−0.0125045
$227$	−851.726	−0.249035	−0.124518	−	0.992217i	$-0.539738\pi$
$227$	−851.726	−0.249035	−0.124518	+	0.992217i	$0.539738\pi$
$228$	0	0
$229$	−3547.80	−1.02378	−0.511889	−	0.859052i	$-0.671054\pi$
$229$	−3547.80	−1.02378	−0.511889	+	0.859052i	$0.671054\pi$
$230$	81.7894	0.0234480
$231$	0	0
$232$	1850.66	0.523715
$233$	−3774.75	−1.06134	−0.530670	−	0.847578i	$-0.678060\pi$
$233$	−3774.75	−1.06134	−0.530670	+	0.847578i	$0.678060\pi$
$234$	0	0
$235$	670.488	0.186118
$236$	705.580	0.194616
$237$	0	0
$238$	−739.967	−0.201533
$239$	239.000	0.0646846
$239$	239.000	0.0646846
$240$	0	0
$241$	−1513.79	−0.404612	−0.202306	−	0.979322i	$-0.564844\pi$
$241$	−1513.79	−0.404612	−0.202306	+	0.979322i	$0.564844\pi$
$242$	4337.89	1.15227
$243$	0	0
$244$	−1520.42	−0.398913
$245$	573.810	0.149630
$246$	0	0
$247$	2340.61	0.602953
$248$	2889.01	0.739728
$249$	0	0
$250$	1195.24	0.302375
$251$	−1188.68	−0.298921	−0.149460	−	0.988768i	$-0.547754\pi$
$251$	−1188.68	−0.298921	−0.149460	+	0.988768i	$0.547754\pi$
$252$	0	0
$253$	112.431	0.0279388
$254$	1081.02	0.267045
$255$	0	0
$256$	4377.10	1.06863
$257$	−3820.94	−0.927407	−0.463703	−	0.885990i	$-0.653480\pi$
$257$	−3820.94	−0.927407	−0.463703	+	0.885990i	$0.653480\pi$
$258$	0	0
$259$	9639.15	2.31254
$260$	406.308	0.0969161
$261$	0	0
$262$	6613.51	1.55948
$263$	−1913.79	−0.448705	−0.224353	−	0.974508i	$-0.572027\pi$
$263$	−1913.79	−0.448705	−0.224353	+	0.974508i	$0.572027\pi$
$264$	0	0
$265$	175.205	0.0406142
$266$	−2544.50	−0.586516
$267$	0	0
$268$	1276.13	0.290865
$269$	−1299.54	−0.294551	−0.147275	−	0.989096i	$-0.547050\pi$
$269$	−1299.54	−0.294551	−0.147275	+	0.989096i	$0.547050\pi$
$270$	0	0
$271$	3050.88	0.683867	0.341933	−	0.939724i	$-0.388918\pi$
$271$	3050.88	0.683867	0.341933	+	0.939724i	$0.388918\pi$
$272$	−640.551	−0.142791
$273$	0	0
$274$	−8214.78	−1.81122
$275$	814.738	0.178657
$276$	0	0
$277$	9190.69	1.99356	0.996778	−	0.0802149i	$-0.0255607\pi$
$277$	9190.69	1.99356	0.996778	+	0.0802149i	$0.0255607\pi$
$278$	4107.58	0.886173
$279$	0	0
$280$	610.299	0.130258
$281$	−3502.70	−0.743607	−0.371803	−	0.928312i	$-0.621260\pi$
$281$	−3502.70	−0.743607	−0.371803	+	0.928312i	$0.621260\pi$
$282$	0	0
$283$	−2189.09	−0.459816	−0.229908	−	0.973212i	$-0.573843\pi$
$283$	−2189.09	−0.459816	−0.229908	+	0.973212i	$0.573843\pi$
$284$	−3848.68	−0.804146
$285$	0	0
$286$	1888.78	0.390510
$287$	13313.8	2.73828
$288$	0	0
$289$	−4848.23	−0.986816
$290$	−570.328	−0.115486
$291$	0	0
$292$	2396.21	0.480232
$293$	−6782.24	−1.35230	−0.676148	−	0.736766i	$-0.736352\pi$
$293$	−6782.24	−1.35230	−0.676148	+	0.736766i	$0.736352\pi$
$294$	0	0
$295$	300.440	0.0592959
$296$	5526.90	1.08529
$297$	0	0
$298$	7055.36	1.37150
$299$	−1435.00	−0.277552
$300$	0	0
$301$	−11672.5	−2.23518
$302$	7314.42	1.39370
$303$	0	0
$304$	−2202.64	−0.415560
$305$	−647.402	−0.121541
$306$	0	0
$307$	−2119.59	−0.394043	−0.197022	−	0.980399i	$-0.563127\pi$
$307$	−2119.59	−0.394043	−0.197022	+	0.980399i	$0.563127\pi$
$308$	−607.184	−0.112330
$309$	0	0
$310$	−890.323	−0.163119
$311$	484.675	0.0883710	0.0441855	−	0.999023i	$-0.485931\pi$
$311$	484.675	0.0883710	0.0441855	+	0.999023i	$0.485931\pi$
$312$	0	0
$313$	−1645.83	−0.297213	−0.148607	−	0.988896i	$-0.547479\pi$
$313$	−1645.83	−0.297213	−0.148607	+	0.988896i	$0.547479\pi$
$314$	−9694.57	−1.74235
$315$	0	0
$316$	286.350	0.0509761
$317$	−1689.22	−0.299294	−0.149647	−	0.988740i	$-0.547814\pi$
$317$	−1689.22	−0.299294	−0.149647	+	0.988740i	$0.547814\pi$
$318$	0	0
$319$	−783.999	−0.137603
$320$	220.838	0.0385789
$321$	0	0
$322$	1560.00	0.269986
$323$	222.736	0.0383696
$324$	0	0
$325$	−10398.8	−1.77483
$326$	−4931.06	−0.837749
$327$	0	0
$328$	7633.86	1.28509
$329$	12788.5	2.14302
$330$	0	0
$331$	−5387.08	−0.894563	−0.447282	−	0.894393i	$-0.647608\pi$
$331$	−5387.08	−0.894563	−0.447282	+	0.894393i	$0.647608\pi$
$332$	1345.66	0.222449
$333$	0	0
$334$	8702.39	1.42567
$335$	543.382	0.0886213
$336$	0	0
$337$	7237.63	1.16991	0.584954	−	0.811066i	$-0.301113\pi$
$337$	7237.63	1.16991	0.584954	+	0.811066i	$0.301113\pi$
$338$	−16702.6	−2.68787
$339$	0	0
$340$	38.6650	0.00616736
$341$	−1223.88	−0.194360
$342$	0	0
$343$	1587.50	0.249904
$344$	−6692.77	−1.04898
$345$	0	0
$346$	−4931.45	−0.766233
$347$	−5767.19	−0.892216	−0.446108	−	0.894979i	$-0.647190\pi$
$347$	−5767.19	−0.892216	−0.446108	+	0.894979i	$0.647190\pi$
$348$	0	0
$349$	−12511.8	−1.91903	−0.959516	−	0.281655i	$-0.909117\pi$
$349$	−12511.8	−1.91903	−0.959516	+	0.281655i	$0.909117\pi$
$350$	11304.6	1.72645
$351$	0	0
$352$	−948.264	−0.143587
$353$	5967.37	0.899749	0.449874	−	0.893092i	$-0.351469\pi$
$353$	5967.37	0.899749	0.449874	+	0.893092i	$0.351469\pi$
$354$	0	0
$355$	−1638.79	−0.245009
$356$	5594.42	0.832876
$357$	0	0
$358$	4871.80	0.719225
$359$	10202.4	1.49989	0.749946	−	0.661499i	$-0.230079\pi$
$359$	10202.4	1.49989	0.749946	+	0.661499i	$0.230079\pi$
$360$	0	0
$361$	−6093.08	−0.888334
$362$	−2239.17	−0.325105
$363$	0	0
$364$	7749.69	1.11592
$365$	1020.32	0.146318
$366$	0	0
$367$	9924.32	1.41157	0.705783	−	0.708428i	$-0.250595\pi$
$367$	9924.32	1.41157	0.705783	+	0.708428i	$0.250595\pi$
$368$	1350.41	0.191291
$369$	0	0
$370$	−1703.25	−0.239319
$371$	3341.76	0.467643
$372$	0	0
$373$	11508.9	1.59761	0.798807	−	0.601588i	$-0.205465\pi$
$373$	11508.9	1.59761	0.798807	+	0.601588i	$0.205465\pi$
$374$	179.740	0.0248506
$375$	0	0
$376$	7332.67	1.00573
$377$	10006.4	1.36700
$378$	0	0
$379$	−9943.30	−1.34763	−0.673817	−	0.738899i	$-0.735346\pi$
$379$	−9943.30	−1.34763	−0.673817	+	0.738899i	$0.735346\pi$
$380$	132.956	0.0179487
$381$	0	0
$382$	8222.71	1.10134
$383$	10306.8	1.37507	0.687535	−	0.726151i	$-0.258693\pi$
$383$	10306.8	1.37507	0.687535	+	0.726151i	$0.258693\pi$
$384$	0	0
$385$	−258.542	−0.0342248
$386$	−4244.62	−0.559704
$387$	0	0
$388$	−3975.87	−0.520217
$389$	12872.4	1.67778	0.838891	−	0.544299i	$-0.183204\pi$
$389$	12872.4	1.67778	0.838891	+	0.544299i	$0.183204\pi$
$390$	0	0
$391$	−136.557	−0.0176624
$392$	6275.37	0.808556
$393$	0	0
$394$	−5919.66	−0.756924
$395$	121.930	0.0155315
$396$	0	0
$397$	8982.83	1.13560	0.567802	−	0.823165i	$-0.307794\pi$
$397$	8982.83	1.13560	0.567802	+	0.823165i	$0.307794\pi$
$398$	7777.16	0.979481
$399$	0	0
$400$	9785.82	1.22323
$401$	4611.29	0.574257	0.287128	−	0.957892i	$-0.407299\pi$
$401$	4611.29	0.574257	0.287128	+	0.957892i	$0.407299\pi$
$402$	0	0
$403$	15620.8	1.93083
$404$	−2483.68	−0.305860
$405$	0	0
$406$	−10878.1	−1.32973
$407$	−2341.37	−0.285153
$408$	0	0
$409$	8936.58	1.08040	0.540202	−	0.841535i	$-0.318348\pi$
$409$	8936.58	1.08040	0.540202	+	0.841535i	$0.318348\pi$
$410$	−2352.57	−0.283378
$411$	0	0
$412$	3636.96	0.434903
$413$	5730.41	0.682749
$414$	0	0
$415$	572.992	0.0677760
$416$	12103.0	1.42644
$417$	0	0
$418$	618.064	0.0723218
$419$	−3088.91	−0.360151	−0.180075	−	0.983653i	$-0.557634\pi$
$419$	−3088.91	−0.360151	−0.180075	+	0.983653i	$0.557634\pi$
$420$	0	0
$421$	−6449.81	−0.746662	−0.373331	−	0.927698i	$-0.621784\pi$
$421$	−6449.81	−0.746662	−0.373331	+	0.927698i	$0.621784\pi$
$422$	−7822.75	−0.902383
$423$	0	0
$424$	1916.10	0.219467
$425$	−989.565	−0.112943
$426$	0	0
$427$	−12348.2	−1.39946
$428$	−3385.98	−0.382401
$429$	0	0
$430$	2062.55	0.231314
$431$	384.304	0.0429496	0.0214748	−	0.999769i	$-0.493164\pi$
$431$	384.304	0.0429496	0.0214748	+	0.999769i	$0.493164\pi$
$432$	0	0
$433$	−6910.86	−0.767009	−0.383505	−	0.923539i	$-0.625283\pi$
$433$	−6910.86	−0.767009	−0.383505	+	0.923539i	$0.625283\pi$
$434$	−16981.5	−1.87820
$435$	0	0
$436$	−3080.98	−0.338423
$437$	−469.574	−0.0514022
$438$	0	0
$439$	−8632.21	−0.938480	−0.469240	−	0.883071i	$-0.655472\pi$
$439$	−8632.21	−0.938480	−0.469240	+	0.883071i	$0.655472\pi$
$440$	−148.243	−0.0160618
$441$	0	0
$442$	−2294.08	−0.246873
$443$	12829.9	1.37600	0.687998	−	0.725712i	$-0.258490\pi$
$443$	12829.9	1.37600	0.687998	+	0.725712i	$0.258490\pi$
$444$	0	0
$445$	2382.14	0.253762
$446$	−15511.4	−1.64683
$447$	0	0
$448$	4212.14	0.444208
$449$	−2809.69	−0.295317	−0.147658	−	0.989038i	$-0.547174\pi$
$449$	−2809.69	−0.295317	−0.147658	+	0.989038i	$0.547174\pi$
$450$	0	0
$451$	−3233.94	−0.337651
$452$	42.3413	0.00440612
$453$	0	0
$454$	2870.57	0.296746
$455$	3299.86	0.340000
$456$	0	0
$457$	5096.12	0.521633	0.260817	−	0.965388i	$-0.416008\pi$
$457$	5096.12	0.521633	0.260817	+	0.965388i	$0.416008\pi$
$458$	11957.2	1.21992
$459$	0	0
$460$	−81.5138	−0.00826217
$461$	4066.14	0.410800	0.205400	−	0.978678i	$-0.434150\pi$
$461$	4066.14	0.410800	0.205400	+	0.978678i	$0.434150\pi$
$462$	0	0
$463$	4030.39	0.404553	0.202276	−	0.979328i	$-0.435166\pi$
$463$	4030.39	0.404553	0.202276	+	0.979328i	$0.435166\pi$
$464$	−9416.61	−0.942145
$465$	0	0
$466$	12722.1	1.26467
$467$	6881.13	0.681843	0.340921	−	0.940092i	$-0.389261\pi$
$467$	6881.13	0.681843	0.340921	+	0.940092i	$0.389261\pi$
$468$	0	0
$469$	10364.2	1.02041
$470$	−2259.75	−0.221775
$471$	0	0
$472$	3285.71	0.320417
$473$	2835.27	0.275615
$474$	0	0
$475$	−3402.78	−0.328696
$476$	737.473	0.0710126
$477$	0	0
$478$	−805.503	−0.0770770
$479$	8730.87	0.832826	0.416413	−	0.909176i	$-0.363287\pi$
$479$	8730.87	0.832826	0.416413	+	0.909176i	$0.363287\pi$
$480$	0	0
$481$	29883.7	2.83280
$482$	5101.92	0.482129
$483$	0	0
$484$	−4323.27	−0.406017
$485$	−1692.95	−0.158501
$486$	0	0
$487$	1144.00	0.106447	0.0532233	−	0.998583i	$-0.483050\pi$
$487$	1144.00	0.106447	0.0532233	+	0.998583i	$0.483050\pi$
$488$	−7080.20	−0.656774
$489$	0	0
$490$	−1933.91	−0.178297
$491$	9951.31	0.914656	0.457328	−	0.889298i	$-0.348806\pi$
$491$	9951.31	0.914656	0.457328	+	0.889298i	$0.348806\pi$
$492$	0	0
$493$	952.229	0.0869904
$494$	−7888.56	−0.718468
$495$	0	0
$496$	−14700.0	−1.33075
$497$	−31257.3	−2.82109
$498$	0	0
$499$	−8390.02	−0.752683	−0.376342	−	0.926481i	$-0.622818\pi$
$499$	−8390.02	−0.752683	−0.376342	+	0.926481i	$0.622818\pi$
$500$	−1191.21	−0.106545
$501$	0	0
$502$	4006.23	0.356188
$503$	−10551.0	−0.935277	−0.467639	−	0.883920i	$-0.654895\pi$
$503$	−10551.0	−0.935277	−0.467639	+	0.883920i	$0.654895\pi$
$504$	0	0
$505$	−1057.56	−0.0931901
$506$	−378.928	−0.0332913
$507$	0	0
$508$	−1077.38	−0.0940966
$509$	2287.01	0.199155	0.0995776	−	0.995030i	$-0.468251\pi$
$509$	2287.01	0.199155	0.0995776	+	0.995030i	$0.468251\pi$
$510$	0	0
$511$	19461.0	1.68474
$512$	−1430.30	−0.123459
$513$	0	0
$514$	12877.7	1.10508
$515$	1548.64	0.132507
$516$	0	0
$517$	−3106.35	−0.264250
$518$	−32486.9	−2.75558
$519$	0	0
$520$	1892.08	0.159563
$521$	−4208.33	−0.353878	−0.176939	−	0.984222i	$-0.556619\pi$
$521$	−4208.33	−0.353878	−0.176939	+	0.984222i	$0.556619\pi$
$522$	0	0
$523$	11032.8	0.922426	0.461213	−	0.887290i	$-0.347414\pi$
$523$	11032.8	0.922426	0.461213	+	0.887290i	$0.347414\pi$
$524$	−6591.22	−0.549501
$525$	0	0
$526$	6450.06	0.534669
$527$	1486.50	0.122871
$528$	0	0
$529$	−11879.1	−0.976338
$530$	−590.494	−0.0483952
$531$	0	0
$532$	2535.92	0.206666
$533$	41275.9	3.35433
$534$	0	0
$535$	−1441.77	−0.116511
$536$	5942.60	0.478883
$537$	0	0
$538$	4379.83	0.350981
$539$	−2658.45	−0.212444
$540$	0	0
$541$	13020.1	1.03471	0.517354	−	0.855772i	$-0.326917\pi$
$541$	13020.1	1.03471	0.517354	+	0.855772i	$0.326917\pi$
$542$	−10282.4	−0.814883
$543$	0	0
$544$	1151.74	0.0907731
$545$	−1311.90	−0.103111
$546$	0	0
$547$	−7969.25	−0.622926	−0.311463	−	0.950258i	$-0.600819\pi$
$547$	−7969.25	−0.622926	−0.311463	+	0.950258i	$0.600819\pi$
$548$	8187.10	0.638203
$549$	0	0
$550$	−2745.92	−0.212884
$551$	3274.40	0.253165
$552$	0	0
$553$	2325.61	0.178834
$554$	−30975.4	−2.37548
$555$	0	0
$556$	−4093.74	−0.312254
$557$	8985.15	0.683506	0.341753	−	0.939790i	$-0.388979\pi$
$557$	8985.15	0.683506	0.341753	+	0.939790i	$0.388979\pi$
$558$	0	0
$559$	−36187.5	−2.73805
$560$	−3105.35	−0.234330
$561$	0	0
$562$	11805.2	0.886069
$563$	12293.1	0.920237	0.460118	−	0.887858i	$-0.347807\pi$
$563$	12293.1	0.920237	0.460118	+	0.887858i	$0.347807\pi$
$564$	0	0
$565$	18.0292	0.00134246
$566$	7377.90	0.547908
$567$	0	0
$568$	−17922.3	−1.32395
$569$	−1743.02	−0.128420	−0.0642102	−	0.997936i	$-0.520453\pi$
$569$	−1743.02	−0.128420	−0.0642102	+	0.997936i	$0.520453\pi$
$570$	0	0
$571$	14716.1	1.07854	0.539272	−	0.842132i	$-0.318699\pi$
$571$	14716.1	1.07854	0.539272	+	0.842132i	$0.318699\pi$
$572$	−1882.42	−0.137601
$573$	0	0
$574$	−44871.5	−3.26289
$575$	2086.21	0.151306
$576$	0	0
$577$	5325.84	0.384259	0.192130	−	0.981370i	$-0.438461\pi$
$577$	5325.84	0.384259	0.192130	+	0.981370i	$0.438461\pi$
$578$	16340.0	1.17587
$579$	0	0
$580$	568.406	0.0406927
$581$	10928.9	0.780391
$582$	0	0
$583$	−811.720	−0.0576638
$584$	11158.5	0.790658
$585$	0	0
$586$	22858.2	1.61137
$587$	−21489.8	−1.51104	−0.755520	−	0.655125i	$-0.772616\pi$
$587$	−21489.8	−1.51104	−0.755520	+	0.655125i	$0.772616\pi$
$588$	0	0
$589$	5111.57	0.357587
$590$	−1012.57	−0.0706559
$591$	0	0
$592$	−28122.2	−1.95239
$593$	3166.61	0.219287	0.109644	−	0.993971i	$-0.465029\pi$
$593$	3166.61	0.219287	0.109644	+	0.993971i	$0.465029\pi$
$594$	0	0
$595$	314.020	0.0216363
$596$	−7031.58	−0.483263
$597$	0	0
$598$	4836.39	0.330727
$599$	22373.1	1.52611	0.763055	−	0.646333i	$-0.223698\pi$
$599$	22373.1	1.52611	0.763055	+	0.646333i	$0.223698\pi$
$600$	0	0
$601$	−2655.02	−0.180200	−0.0901002	−	0.995933i	$-0.528719\pi$
$601$	−2655.02	−0.180200	−0.0901002	+	0.995933i	$0.528719\pi$
$602$	39339.8	2.66341
$603$	0	0
$604$	−7289.77	−0.491087
$605$	−1840.87	−0.123706
$606$	0	0
$607$	−6835.91	−0.457102	−0.228551	−	0.973532i	$-0.573399\pi$
$607$	−6835.91	−0.457102	−0.228551	+	0.973532i	$0.573399\pi$
$608$	3960.46	0.264174
$609$	0	0
$610$	2181.94	0.144827
$611$	39647.4	2.62514
$612$	0	0
$613$	19181.6	1.26385	0.631923	−	0.775031i	$-0.282266\pi$
$613$	19181.6	1.26385	0.631923	+	0.775031i	$0.282266\pi$
$614$	7143.65	0.469535
$615$	0	0
$616$	−2827.50	−0.184940
$617$	−25690.4	−1.67627	−0.838134	−	0.545464i	$-0.816354\pi$
$617$	−25690.4	−1.67627	−0.838134	+	0.545464i	$0.816354\pi$
$618$	0	0
$619$	8513.41	0.552799	0.276400	−	0.961043i	$-0.410859\pi$
$619$	8513.41	0.552799	0.276400	+	0.961043i	$0.410859\pi$
$620$	887.323	0.0574770
$621$	0	0
$622$	−1633.50	−0.105301
$623$	45435.5	2.92188
$624$	0	0
$625$	14862.1	0.951172
$626$	5546.94	0.354154
$627$	0	0
$628$	9661.90	0.613936
$629$	2843.78	0.180269
$630$	0	0
$631$	4370.79	0.275751	0.137875	−	0.990450i	$-0.455973\pi$
$631$	4370.79	0.275751	0.137875	+	0.990450i	$0.455973\pi$
$632$	1333.46	0.0839275
$633$	0	0
$634$	5693.19	0.356633
$635$	−458.755	−0.0286695
$636$	0	0
$637$	33930.6	2.11049
$638$	2642.31	0.163966
$639$	0	0
$640$	−2381.71	−0.147102
$641$	429.911	0.0264906	0.0132453	−	0.999912i	$-0.495784\pi$
$641$	429.911	0.0264906	0.0132453	+	0.999912i	$0.495784\pi$
$642$	0	0
$643$	10486.6	0.643160	0.321580	−	0.946882i	$-0.395786\pi$
$643$	10486.6	0.643160	0.321580	+	0.946882i	$0.395786\pi$
$644$	−1554.75	−0.0951329
$645$	0	0
$646$	−750.689	−0.0457205
$647$	17231.1	1.04702	0.523512	−	0.852018i	$-0.324622\pi$
$647$	17231.1	1.04702	0.523512	+	0.852018i	$0.324622\pi$
$648$	0	0
$649$	−1391.93	−0.0841880
$650$	35047.0	2.11486
$651$	0	0
$652$	4914.44	0.295191
$653$	−14451.1	−0.866023	−0.433012	−	0.901388i	$-0.642549\pi$
$653$	−14451.1	−0.866023	−0.433012	+	0.901388i	$0.642549\pi$
$654$	0	0
$655$	−2806.58	−0.167423
$656$	−38842.9	−2.31183
$657$	0	0
$658$	−43101.1	−2.55358
$659$	22139.4	1.30870	0.654348	−	0.756194i	$-0.272943\pi$
$659$	22139.4	1.30870	0.654348	+	0.756194i	$0.272943\pi$
$660$	0	0
$661$	10161.7	0.597948	0.298974	−	0.954261i	$-0.403356\pi$
$661$	10161.7	0.597948	0.298974	+	0.954261i	$0.403356\pi$
$662$	18156.1	1.06595
$663$	0	0
$664$	6266.42	0.366241
$665$	1079.81	0.0629674
$666$	0	0
$667$	−2007.50	−0.116538
$668$	−8673.06	−0.502351
$669$	0	0
$670$	−1831.36	−0.105600
$671$	2999.40	0.172564
$672$	0	0
$673$	−13486.9	−0.772484	−0.386242	−	0.922397i	$-0.626227\pi$
$673$	−13486.9	−0.772484	−0.386242	+	0.922397i	$0.626227\pi$
$674$	−24393.0	−1.39404
$675$	0	0
$676$	16646.3	0.947104
$677$	−27398.0	−1.55538	−0.777689	−	0.628649i	$-0.783608\pi$
$677$	−27398.0	−1.55538	−0.777689	+	0.628649i	$0.783608\pi$
$678$	0	0
$679$	−32290.3	−1.82502
$680$	180.053	0.0101540
$681$	0	0
$682$	4124.84	0.231596
$683$	−28674.0	−1.60641	−0.803205	−	0.595702i	$-0.796874\pi$
$683$	−28674.0	−1.60641	−0.803205	+	0.595702i	$0.796874\pi$
$684$	0	0
$685$	3486.11	0.194449
$686$	−5350.37	−0.297781
$687$	0	0
$688$	34054.4	1.88708
$689$	10360.3	0.572851
$690$	0	0
$691$	−19357.7	−1.06570	−0.532852	−	0.846208i	$-0.678880\pi$
$691$	−19357.7	−1.06570	−0.532852	+	0.846208i	$0.678880\pi$
$692$	4914.83	0.269991
$693$	0	0
$694$	19437.2	1.06315
$695$	−1743.14	−0.0951380
$696$	0	0
$697$	3927.88	0.213457
$698$	42168.6	2.28668
$699$	0	0
$700$	−11266.5	−0.608335
$701$	−703.445	−0.0379012	−0.0189506	−	0.999820i	$-0.506033\pi$
$701$	−703.445	−0.0379012	−0.0189506	+	0.999820i	$0.506033\pi$
$702$	0	0
$703$	9778.82	0.524630
$704$	−1023.14	−0.0547741
$705$	0	0
$706$	−20111.9	−1.07212
$707$	−20171.4	−1.07302
$708$	0	0
$709$	−15148.8	−0.802433	−0.401217	−	0.915983i	$-0.631413\pi$
$709$	−15148.8	−0.802433	−0.401217	+	0.915983i	$0.631413\pi$
$710$	5523.23	0.291948
$711$	0	0
$712$	26051.8	1.37125
$713$	−3133.85	−0.164605
$714$	0	0
$715$	−801.543	−0.0419245
$716$	−4855.38	−0.253428
$717$	0	0
$718$	−34385.1	−1.78724
$719$	−19554.6	−1.01427	−0.507137	−	0.861866i	$-0.669296\pi$
$719$	−19554.6	−1.01427	−0.507137	+	0.861866i	$0.669296\pi$
$720$	0	0
$721$	29537.8	1.52572
$722$	20535.5	1.05852
$723$	0	0
$724$	2231.62	0.114555
$725$	−14547.4	−0.745209
$726$	0	0
$727$	15889.5	0.810606	0.405303	−	0.914182i	$-0.367166\pi$
$727$	15889.5	0.810606	0.405303	+	0.914182i	$0.367166\pi$
$728$	36088.3	1.83726
$729$	0	0
$730$	−3438.79	−0.174350
$731$	−3443.66	−0.174239
$732$	0	0
$733$	−6094.65	−0.307109	−0.153555	−	0.988140i	$-0.549072\pi$
$733$	−6094.65	−0.307109	−0.153555	+	0.988140i	$0.549072\pi$
$734$	−33448.0	−1.68200
$735$	0	0
$736$	−2428.11	−0.121605
$737$	−2517.47	−0.125824
$738$	0	0
$739$	−11499.0	−0.572393	−0.286196	−	0.958171i	$-0.592391\pi$
$739$	−11499.0	−0.572393	−0.286196	+	0.958171i	$0.592391\pi$
$740$	1697.51	0.0843268
$741$	0	0
$742$	−11262.7	−0.557235
$743$	6346.79	0.313380	0.156690	−	0.987648i	$-0.449918\pi$
$743$	6346.79	0.313380	0.156690	+	0.987648i	$0.449918\pi$
$744$	0	0
$745$	−2994.09	−0.147242
$746$	−38788.6	−1.90369
$747$	0	0
$748$	−179.134	−0.00875639
$749$	−27499.5	−1.34153
$750$	0	0
$751$	−6045.02	−0.293723	−0.146862	−	0.989157i	$-0.546917\pi$
$751$	−6045.02	−0.293723	−0.146862	+	0.989157i	$0.546917\pi$
$752$	−37310.4	−1.80927
$753$	0	0
$754$	−33724.7	−1.62889
$755$	−3104.03	−0.149625
$756$	0	0
$757$	33341.0	1.60079	0.800396	−	0.599472i	$-0.204623\pi$
$757$	33341.0	1.60079	0.800396	+	0.599472i	$0.204623\pi$
$758$	33511.9	1.60582
$759$	0	0
$760$	619.143	0.0295509
$761$	−26445.5	−1.25972	−0.629862	−	0.776707i	$-0.716888\pi$
$761$	−26445.5	−1.25972	−0.629862	+	0.776707i	$0.716888\pi$
$762$	0	0
$763$	−25022.4	−1.18725
$764$	−8195.00	−0.388069
$765$	0	0
$766$	−34737.0	−1.63851
$767$	17765.7	0.836350
$768$	0	0
$769$	−21472.2	−1.00690	−0.503451	−	0.864024i	$-0.667936\pi$
$769$	−21472.2	−1.00690	−0.503451	+	0.864024i	$0.667936\pi$
$770$	871.366	0.0407816
$771$	0	0
$772$	4230.32	0.197218
$773$	16975.8	0.789881	0.394940	−	0.918707i	$-0.370765\pi$
$773$	16975.8	0.789881	0.394940	+	0.918707i	$0.370765\pi$
$774$	0	0
$775$	−22709.5	−1.05258
$776$	−18514.6	−0.856490
$777$	0	0
$778$	−43383.9	−1.99922
$779$	13506.7	0.621216
$780$	0	0
$781$	7592.48	0.347862
$782$	460.238	0.0210462
$783$	0	0
$784$	−31930.6	−1.45456
$785$	4114.09	0.187055
$786$	0	0
$787$	−23552.1	−1.06676	−0.533381	−	0.845875i	$-0.679079\pi$
$787$	−23552.1	−1.06676	−0.533381	+	0.845875i	$0.679079\pi$
$788$	5899.71	0.266711
$789$	0	0
$790$	−410.940	−0.0185071
$791$	343.878	0.0154575
$792$	0	0
$793$	−38282.3	−1.71431
$794$	−30274.8	−1.35317
$795$	0	0
$796$	−7750.95	−0.345132
$797$	37695.4	1.67533	0.837666	−	0.546182i	$-0.183919\pi$
$797$	37695.4	1.67533	0.837666	+	0.546182i	$0.183919\pi$
$798$	0	0
$799$	3772.91	0.167054
$800$	−17595.4	−0.777614
$801$	0	0
$802$	−15541.5	−0.684274
$803$	−4727.12	−0.207741
$804$	0	0
$805$	−662.020	−0.0289853
$806$	−52646.7	−2.30075
$807$	0	0
$808$	−11565.9	−0.503571
$809$	7349.53	0.319402	0.159701	−	0.987165i	$-0.448947\pi$
$809$	7349.53	0.319402	0.159701	+	0.987165i	$0.448947\pi$
$810$	0	0
$811$	−3250.04	−0.140721	−0.0703604	−	0.997522i	$-0.522415\pi$
$811$	−3250.04	−0.140721	−0.0703604	+	0.997522i	$0.522415\pi$
$812$	10841.4	0.468547
$813$	0	0
$814$	7891.13	0.339784
$815$	2092.60	0.0899393
$816$	0	0
$817$	−11841.6	−0.507081
$818$	−30119.0	−1.28739
$819$	0	0
$820$	2344.64	0.0998516
$821$	−35780.6	−1.52101	−0.760506	−	0.649331i	$-0.775049\pi$
$821$	−35780.6	−1.52101	−0.760506	+	0.649331i	$0.775049\pi$
$822$	0	0
$823$	35589.3	1.50737	0.753684	−	0.657237i	$-0.228275\pi$
$823$	35589.3	1.50737	0.753684	+	0.657237i	$0.228275\pi$
$824$	16936.4	0.716028
$825$	0	0
$826$	−19313.2	−0.813551
$827$	−29543.8	−1.24225	−0.621123	−	0.783713i	$-0.713323\pi$
$827$	−29543.8	−1.24225	−0.621123	+	0.783713i	$0.713323\pi$
$828$	0	0
$829$	−32475.4	−1.36057	−0.680287	−	0.732946i	$-0.738145\pi$
$829$	−32475.4	−1.36057	−0.680287	+	0.732946i	$0.738145\pi$
$830$	−1931.16	−0.0807607
$831$	0	0
$832$	13058.7	0.544144
$833$	3228.89	0.134303
$834$	0	0
$835$	−3693.04	−0.153057
$836$	−615.982	−0.0254835
$837$	0	0
$838$	10410.6	0.429149
$839$	−5398.55	−0.222144	−0.111072	−	0.993812i	$-0.535428\pi$
$839$	−5398.55	−0.222144	−0.111072	+	0.993812i	$0.535428\pi$
$840$	0	0
$841$	−10390.5	−0.426031
$842$	21737.8	0.889709
$843$	0	0
$844$	7796.39	0.317965
$845$	7088.09	0.288565
$846$	0	0
$847$	−35111.7	−1.42438
$848$	−9749.57	−0.394813
$849$	0	0
$850$	3335.13	0.134581
$851$	−5995.28	−0.241499
$852$	0	0
$853$	24758.6	0.993807	0.496904	−	0.867806i	$-0.334470\pi$
$853$	24758.6	0.993807	0.496904	+	0.867806i	$0.334470\pi$
$854$	41617.1	1.66757
$855$	0	0
$856$	−15767.7	−0.629588
$857$	20312.0	0.809619	0.404809	−	0.914401i	$-0.367338\pi$
$857$	20312.0	0.809619	0.404809	+	0.914401i	$0.367338\pi$
$858$	0	0
$859$	−2687.82	−0.106760	−0.0533802	−	0.998574i	$-0.517000\pi$
$859$	−2687.82	−0.106760	−0.0533802	+	0.998574i	$0.517000\pi$
$860$	−2055.60	−0.0815061
$861$	0	0
$862$	−1295.22	−0.0511780
$863$	26875.9	1.06010	0.530050	−	0.847967i	$-0.322173\pi$
$863$	26875.9	1.06010	0.530050	+	0.847967i	$0.322173\pi$
$864$	0	0
$865$	2092.76	0.0822614
$866$	23291.7	0.913954
$867$	0	0
$868$	16924.3	0.661805
$869$	−564.896	−0.0220515
$870$	0	0
$871$	32131.4	1.24998
$872$	−14347.4	−0.557182
$873$	0	0
$874$	1582.61	0.0612500
$875$	−9674.51	−0.373781
$876$	0	0
$877$	6974.07	0.268526	0.134263	−	0.990946i	$-0.457133\pi$
$877$	6974.07	0.268526	0.134263	+	0.990946i	$0.457133\pi$
$878$	29093.2	1.11828
$879$	0	0
$880$	754.296	0.0288947
$881$	−36541.3	−1.39740	−0.698698	−	0.715417i	$-0.746237\pi$
$881$	−36541.3	−1.39740	−0.698698	+	0.715417i	$0.746237\pi$
$882$	0	0
$883$	−37573.5	−1.43199	−0.715996	−	0.698105i	$-0.754027\pi$
$883$	−37573.5	−1.43199	−0.715996	+	0.698105i	$0.754027\pi$
$884$	2286.34	0.0869888
$885$	0	0
$886$	−43240.6	−1.63961
$887$	12616.1	0.477574	0.238787	−	0.971072i	$-0.423250\pi$
$887$	12616.1	0.477574	0.238787	+	0.971072i	$0.423250\pi$
$888$	0	0
$889$	−8750.03	−0.330109
$890$	−8028.52	−0.302378
$891$	0	0
$892$	15459.2	0.580281
$893$	12973.8	0.486172
$894$	0	0
$895$	−2067.45	−0.0772148
$896$	−45427.4	−1.69377
$897$	0	0
$898$	9469.49	0.351894
$899$	21852.7	0.810710
$900$	0	0
$901$	985.899	0.0364540
$902$	10899.4	0.402339
$903$	0	0
$904$	197.173	0.00725428
$905$	950.238	0.0349028
$906$	0	0
$907$	40730.1	1.49109	0.745546	−	0.666454i	$-0.232189\pi$
$907$	40730.1	1.49109	0.745546	+	0.666454i	$0.232189\pi$
$908$	−2860.90	−0.104562
$909$	0	0
$910$	−11121.5	−0.405138
$911$	−20688.0	−0.752386	−0.376193	−	0.926541i	$-0.622767\pi$
$911$	−20688.0	−0.752386	−0.376193	+	0.926541i	$0.622767\pi$
$912$	0	0
$913$	−2654.65	−0.0962281
$914$	−17175.5	−0.621569
$915$	0	0
$916$	−11916.9	−0.429852
$917$	−53531.0	−1.92775
$918$	0	0
$919$	−34402.3	−1.23485	−0.617426	−	0.786629i	$-0.711824\pi$
$919$	−34402.3	−1.23485	−0.617426	+	0.786629i	$0.711824\pi$
$920$	−379.589	−0.0136029
$921$	0	0
$922$	−13704.1	−0.489502
$923$	−96905.3	−3.45577
$924$	0	0
$925$	−43445.0	−1.54428
$926$	−13583.6	−0.482058
$927$	0	0
$928$	16931.5	0.598928
$929$	−43106.7	−1.52237	−0.761187	−	0.648533i	$-0.775383\pi$
$929$	−43106.7	−1.52237	−0.761187	+	0.648533i	$0.775383\pi$
$930$	0	0
$931$	11103.1	0.390858
$932$	−12679.2	−0.445623
$933$	0	0
$934$	−23191.5	−0.812472
$935$	−76.2762	−0.00266791
$936$	0	0
$937$	6344.56	0.221204	0.110602	−	0.993865i	$-0.464722\pi$
$937$	6344.56	0.221204	0.110602	+	0.993865i	$0.464722\pi$
$938$	−34930.3	−1.21590
$939$	0	0
$940$	2252.13	0.0781452
$941$	13009.1	0.450675	0.225337	−	0.974281i	$-0.427652\pi$
$941$	13009.1	0.450675	0.225337	+	0.974281i	$0.427652\pi$
$942$	0	0
$943$	−8280.79	−0.285959
$944$	−16718.5	−0.576419
$945$	0	0
$946$	−9555.72	−0.328418
$947$	20028.5	0.687263	0.343632	−	0.939104i	$-0.388343\pi$
$947$	20028.5	0.687263	0.343632	+	0.939104i	$0.388343\pi$
$948$	0	0
$949$	60333.8	2.06377
$950$	11468.4	0.391668
$951$	0	0
$952$	3434.23	0.116916
$953$	−13132.9	−0.446397	−0.223198	−	0.974773i	$-0.571650\pi$
$953$	−13132.9	−0.446397	−0.223198	+	0.974773i	$0.571650\pi$
$954$	0	0
$955$	−3489.48	−0.118238
$956$	802.788	0.0271590
$957$	0	0
$958$	−29425.7	−0.992381
$959$	66492.1	2.23894
$960$	0	0
$961$	4322.65	0.145099
$962$	−100717.	−3.37552
$963$	0	0
$964$	−5084.73	−0.169884
$965$	1801.29	0.0600888
$966$	0	0
$967$	6491.96	0.215892	0.107946	−	0.994157i	$-0.465573\pi$
$967$	6491.96	0.215892	0.107946	+	0.994157i	$0.465573\pi$
$968$	−20132.4	−0.668470
$969$	0	0
$970$	5705.75	0.188867
$971$	−52373.6	−1.73095	−0.865473	−	0.500956i	$-0.832982\pi$
$971$	−52373.6	−1.73095	−0.865473	+	0.500956i	$0.832982\pi$
$972$	0	0
$973$	−33247.5	−1.09544
$974$	−3855.62	−0.126840
$975$	0	0
$976$	36025.8	1.18151
$977$	23121.4	0.757135	0.378567	−	0.925574i	$-0.376417\pi$
$977$	23121.4	0.757135	0.378567	+	0.925574i	$0.376417\pi$
$978$	0	0
$979$	−11036.4	−0.360290
$980$	1927.40	0.0628249
$981$	0	0
$982$	−33538.9	−1.08989
$983$	−14960.8	−0.485429	−0.242714	−	0.970098i	$-0.578038\pi$
$983$	−14960.8	−0.485429	−0.242714	+	0.970098i	$0.578038\pi$
$984$	0	0
$985$	2512.13	0.0812620
$986$	−3209.30	−0.103656
$987$	0	0
$988$	7861.98	0.253161
$989$	7259.95	0.233421
$990$	0	0
$991$	39192.0	1.25628	0.628140	−	0.778100i	$-0.283816\pi$
$991$	39192.0	1.25628	0.628140	+	0.778100i	$0.283816\pi$
$992$	26431.4	0.845964
$993$	0	0
$994$	105347.	3.36156
$995$	−3300.40	−0.105155
$996$	0	0
$997$	43875.3	1.39373	0.696864	−	0.717204i	$-0.254578\pi$
$997$	43875.3	1.39373	0.696864	+	0.717204i	$0.254578\pi$
$998$	28276.9	0.896884
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2151.4.a.f.1.10		37
3.2	odd	2		239.4.a.b.1.28	✓	37

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
239.4.a.b.1.28	✓	37	3.2	odd	2
2151.4.a.f.1.10		37	1.1	even	1	trivial

Overview	Random
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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)

\(f(q)\)	\(=\)	\(q-3.37030 q^{2} +3.35895 q^{4} +1.43026 q^{5} +27.2799 q^{7} +15.6418 q^{8} +O(q^{10})\)	\(q-3.37030 q^{2} +3.35895 q^{4} +1.43026 q^{5} +27.2799 q^{7} +15.6418 q^{8} -4.82040 q^{10} -6.62635 q^{11} +84.5743 q^{13} -91.9416 q^{14} -79.5890 q^{16} +8.04823 q^{17} +27.6752 q^{19} +4.80416 q^{20} +22.3328 q^{22} -16.9673 q^{23} -122.954 q^{25} -285.041 q^{26} +91.6317 q^{28} +118.315 q^{29} +184.699 q^{31} +143.105 q^{32} -27.1250 q^{34} +39.0173 q^{35} +353.342 q^{37} -93.2738 q^{38} +22.3718 q^{40} +488.043 q^{41} -427.878 q^{43} -22.2575 q^{44} +57.1851 q^{46} +468.788 q^{47} +401.193 q^{49} +414.394 q^{50} +284.080 q^{52} +122.499 q^{53} -9.47739 q^{55} +426.706 q^{56} -398.759 q^{58} +210.060 q^{59} -452.647 q^{61} -622.491 q^{62} +154.405 q^{64} +120.963 q^{65} +379.919 q^{67} +27.0336 q^{68} -131.500 q^{70} -1145.80 q^{71} +713.382 q^{73} -1190.87 q^{74} +92.9594 q^{76} -180.766 q^{77} +85.2500 q^{79} -113.833 q^{80} -1644.85 q^{82} +400.621 q^{83} +11.5110 q^{85} +1442.08 q^{86} -103.648 q^{88} +1665.53 q^{89} +2307.18 q^{91} -56.9923 q^{92} -1579.96 q^{94} +39.5827 q^{95} -1183.67 q^{97} -1352.14 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 37 q - 4 q^{2} + 170 q^{4} - 43 q^{5} + 60 q^{7} - 27 q^{8}+O(q^{10}) \) 37 * q - 4 * q^2 + 170 * q^4 - 43 * q^5 + 60 * q^7 - 27 * q^8	\( 37 q - 4 q^{2} + 170 q^{4} - 43 q^{5} + 60 q^{7} - 27 q^{8} + 147 q^{10} - 55 q^{11} + 250 q^{13} - 169 q^{14} + 918 q^{16} - 189 q^{17} + 550 q^{19} - 486 q^{20} + 226 q^{22} - 74 q^{23} + 1604 q^{25} - 560 q^{26} + 829 q^{28} - 389 q^{29} + 1107 q^{31} - 125 q^{32} + 1423 q^{34} - 270 q^{35} + 1002 q^{37} - 1037 q^{38} + 1536 q^{40} - 1518 q^{41} + 1098 q^{43} - 1037 q^{44} + 1030 q^{46} - 1214 q^{47} + 4663 q^{49} - 929 q^{50} + 2895 q^{52} - 904 q^{53} + 1350 q^{55} - 2556 q^{56} + 1396 q^{58} - 1658 q^{59} + 2313 q^{61} + 4519 q^{62} + 3807 q^{64} + 56 q^{65} + 1535 q^{67} + 6526 q^{68} - 4099 q^{70} + 3255 q^{71} + 3154 q^{73} + 2629 q^{74} + 1981 q^{76} + 3734 q^{77} + 2260 q^{79} + 8242 q^{80} - 9898 q^{82} + 939 q^{83} + 1272 q^{85} + 3457 q^{86} - 1808 q^{88} - 1486 q^{89} + 174 q^{91} + 14076 q^{92} - 984 q^{94} + 1828 q^{95} + 6148 q^{97} + 6243 q^{98}+O(q^{100}) \) 37 * q - 4 * q^2 + 170 * q^4 - 43 * q^5 + 60 * q^7 - 27 * q^8 + 147 * q^10 - 55 * q^11 + 250 * q^13 - 169 * q^14 + 918 * q^16 - 189 * q^17 + 550 * q^19 - 486 * q^20 + 226 * q^22 - 74 * q^23 + 1604 * q^25 - 560 * q^26 + 829 * q^28 - 389 * q^29 + 1107 * q^31 - 125 * q^32 + 1423 * q^34 - 270 * q^35 + 1002 * q^37 - 1037 * q^38 + 1536 * q^40 - 1518 * q^41 + 1098 * q^43 - 1037 * q^44 + 1030 * q^46 - 1214 * q^47 + 4663 * q^49 - 929 * q^50 + 2895 * q^52 - 904 * q^53 + 1350 * q^55 - 2556 * q^56 + 1396 * q^58 - 1658 * q^59 + 2313 * q^61 + 4519 * q^62 + 3807 * q^64 + 56 * q^65 + 1535 * q^67 + 6526 * q^68 - 4099 * q^70 + 3255 * q^71 + 3154 * q^73 + 2629 * q^74 + 1981 * q^76 + 3734 * q^77 + 2260 * q^79 + 8242 * q^80 - 9898 * q^82 + 939 * q^83 + 1272 * q^85 + 3457 * q^86 - 1808 * q^88 - 1486 * q^89 + 174 * q^91 + 14076 * q^92 - 984 * q^94 + 1828 * q^95 + 6148 * q^97 + 6243 * q^98

Introduction

L-functions

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