LMFDB - Embedded newform 2151.4.a.b.1.2

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:	$28$
Twist minimal:	no (minimal twist has level 717)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
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)

$f(q)$	$=$	$q-4.86739 q^{2} +15.6914 q^{4} +12.8877 q^{5} +25.7021 q^{7} -37.4372 q^{8} +O(q^{10})$	\(q-4.86739 q^{2} +15.6914 q^{4} +12.8877 q^{5} +25.7021 q^{7} -37.4372 q^{8} -62.7292 q^{10} +53.1026 q^{11} -81.7543 q^{13} -125.102 q^{14} +56.6898 q^{16} -36.6803 q^{17} +7.96807 q^{19} +202.226 q^{20} -258.471 q^{22} +53.4635 q^{23} +41.0919 q^{25} +397.930 q^{26} +403.302 q^{28} -104.112 q^{29} +18.8604 q^{31} +23.5667 q^{32} +178.537 q^{34} +331.240 q^{35} -258.224 q^{37} -38.7837 q^{38} -482.478 q^{40} +136.879 q^{41} -430.798 q^{43} +833.256 q^{44} -260.228 q^{46} -11.3640 q^{47} +317.596 q^{49} -200.010 q^{50} -1282.84 q^{52} +307.139 q^{53} +684.368 q^{55} -962.213 q^{56} +506.752 q^{58} -706.161 q^{59} -507.467 q^{61} -91.8008 q^{62} -568.226 q^{64} -1053.62 q^{65} -552.177 q^{67} -575.566 q^{68} -1612.27 q^{70} -986.525 q^{71} +144.680 q^{73} +1256.88 q^{74} +125.031 q^{76} +1364.85 q^{77} -507.444 q^{79} +730.599 q^{80} -666.242 q^{82} +502.367 q^{83} -472.723 q^{85} +2096.86 q^{86} -1988.01 q^{88} -1430.11 q^{89} -2101.26 q^{91} +838.920 q^{92} +55.3129 q^{94} +102.690 q^{95} -1299.67 q^{97} -1545.86 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 28 q + 5 q^{2} + 103 q^{4} - 6 q^{5} - 68 q^{7} + 39 q^{8}+O(q^{10}) $ 28 * q + 5 * q^2 + 103 * q^4 - 6 * q^5 - 68 * q^7 + 39 * q^8	$ 28 q + 5 q^{2} + 103 q^{4} - 6 q^{5} - 68 q^{7} + 39 q^{8} - 88 q^{10} + 110 q^{11} - 82 q^{13} - 126 q^{14} + 271 q^{16} - 100 q^{17} - 292 q^{19} + 52 q^{20} - 351 q^{22} + 276 q^{23} + 386 q^{25} - 84 q^{26} - 1010 q^{28} + 38 q^{29} - 432 q^{31} + 452 q^{32} - 524 q^{34} + 166 q^{35} - 936 q^{37} + 41 q^{38} - 1183 q^{40} - 1054 q^{41} - 1804 q^{43} + 341 q^{44} - 888 q^{46} + 560 q^{47} + 1074 q^{49} + 1054 q^{50} - 632 q^{52} + 160 q^{53} - 842 q^{55} - 509 q^{56} - 1266 q^{58} - 846 q^{59} - 2220 q^{61} - 82 q^{62} - 1565 q^{64} - 296 q^{65} - 4752 q^{67} + 1719 q^{68} - 5601 q^{70} + 802 q^{71} - 2732 q^{73} + 4581 q^{74} - 5614 q^{76} + 1008 q^{77} - 3172 q^{79} + 732 q^{80} - 9709 q^{82} + 4780 q^{83} - 4624 q^{85} + 2009 q^{86} - 9331 q^{88} - 4372 q^{89} - 7398 q^{91} + 6138 q^{92} - 7068 q^{94} + 3160 q^{95} - 4846 q^{97} + 3772 q^{98}+O(q^{100}) $ 28 * q + 5 * q^2 + 103 * q^4 - 6 * q^5 - 68 * q^7 + 39 * q^8 - 88 * q^10 + 110 * q^11 - 82 * q^13 - 126 * q^14 + 271 * q^16 - 100 * q^17 - 292 * q^19 + 52 * q^20 - 351 * q^22 + 276 * q^23 + 386 * q^25 - 84 * q^26 - 1010 * q^28 + 38 * q^29 - 432 * q^31 + 452 * q^32 - 524 * q^34 + 166 * q^35 - 936 * q^37 + 41 * q^38 - 1183 * q^40 - 1054 * q^41 - 1804 * q^43 + 341 * q^44 - 888 * q^46 + 560 * q^47 + 1074 * q^49 + 1054 * q^50 - 632 * q^52 + 160 * q^53 - 842 * q^55 - 509 * q^56 - 1266 * q^58 - 846 * q^59 - 2220 * q^61 - 82 * q^62 - 1565 * q^64 - 296 * q^65 - 4752 * q^67 + 1719 * q^68 - 5601 * q^70 + 802 * q^71 - 2732 * q^73 + 4581 * q^74 - 5614 * q^76 + 1008 * q^77 - 3172 * q^79 + 732 * q^80 - 9709 * q^82 + 4780 * q^83 - 4624 * q^85 + 2009 * q^86 - 9331 * q^88 - 4372 * q^89 - 7398 * q^91 + 6138 * q^92 - 7068 * q^94 + 3160 * q^95 - 4846 * q^97 + 3772 * 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 $

$2$	−4.86739	−1.72088	−0.860440	−	0.509551i	$-0.829811\pi$
$2$	−4.86739	−1.72088	−0.860440	+	0.509551i	$0.829811\pi$
$3$	0	0
$3$	0	0
$4$	15.6914	1.96143
$5$	12.8877	1.15271	0.576354	−	0.817200i	$-0.304475\pi$
$5$	12.8877	1.15271	0.576354	+	0.817200i	$0.304475\pi$
$6$	0	0
$7$	25.7021	1.38778	0.693891	−	0.720080i	$-0.255895\pi$
$7$	25.7021	1.38778	0.693891	+	0.720080i	$0.255895\pi$
$8$	−37.4372	−1.65451
$9$	0	0
$10$	−62.7292	−1.98367
$11$	53.1026	1.45555	0.727774	−	0.685817i	$-0.240555\pi$
$11$	53.1026	1.45555	0.727774	+	0.685817i	$0.240555\pi$
$12$	0	0
$13$	−81.7543	−1.74420	−0.872099	−	0.489329i	$-0.837242\pi$
$13$	−81.7543	−1.74420	−0.872099	+	0.489329i	$0.837242\pi$
$14$	−125.102	−2.38821
$15$	0	0
$16$	56.6898	0.885777
$17$	−36.6803	−0.523310	−0.261655	−	0.965161i	$-0.584268\pi$
$17$	−36.6803	−0.523310	−0.261655	+	0.965161i	$0.584268\pi$
$18$	0	0
$19$	7.96807	0.0962106	0.0481053	−	0.998842i	$-0.484682\pi$
$19$	7.96807	0.0962106	0.0481053	+	0.998842i	$0.484682\pi$
$20$	202.226	2.26096
$21$	0	0
$22$	−258.471	−2.50482
$23$	53.4635	0.484692	0.242346	−	0.970190i	$-0.422083\pi$
$23$	53.4635	0.484692	0.242346	+	0.970190i	$0.422083\pi$
$24$	0	0
$25$	41.0919	0.328735
$26$	397.930	3.00156
$27$	0	0
$28$	403.302	2.72204
$29$	−104.112	−0.666657	−0.333329	−	0.942811i	$-0.608172\pi$
$29$	−104.112	−0.666657	−0.333329	+	0.942811i	$0.608172\pi$
$30$	0	0
$31$	18.8604	0.109272	0.0546359	−	0.998506i	$-0.482600\pi$
$31$	18.8604	0.109272	0.0546359	+	0.998506i	$0.482600\pi$
$32$	23.5667	0.130189
$33$	0	0
$34$	178.537	0.900554
$35$	331.240	1.59971
$36$	0	0
$37$	−258.224	−1.14734	−0.573672	−	0.819085i	$-0.694482\pi$
$37$	−258.224	−1.14734	−0.573672	+	0.819085i	$0.694482\pi$
$38$	−38.7837	−0.165567
$39$	0	0
$40$	−482.478	−1.90716
$41$	136.879	0.521388	0.260694	−	0.965422i	$-0.416049\pi$
$41$	136.879	0.521388	0.260694	+	0.965422i	$0.416049\pi$
$42$	0	0
$43$	−430.798	−1.52782	−0.763908	−	0.645325i	$-0.776722\pi$
$43$	−430.798	−1.52782	−0.763908	+	0.645325i	$0.776722\pi$
$44$	833.256	2.85496
$45$	0	0
$46$	−260.228	−0.834097
$47$	−11.3640	−0.0352682	−0.0176341	−	0.999845i	$-0.505613\pi$
$47$	−11.3640	−0.0352682	−0.0176341	+	0.999845i	$0.505613\pi$
$48$	0	0
$49$	317.596	0.925937
$50$	−200.010	−0.565714
$51$	0	0
$52$	−1282.84	−3.42112
$53$	307.139	0.796015	0.398008	−	0.917382i	$-0.369702\pi$
$53$	307.139	0.796015	0.398008	+	0.917382i	$0.369702\pi$
$54$	0	0
$55$	684.368	1.67782
$56$	−962.213	−2.29609
$57$	0	0
$58$	506.752	1.14724
$59$	−706.161	−1.55821	−0.779105	−	0.626894i	$-0.784326\pi$
$59$	−706.161	−1.55821	−0.779105	+	0.626894i	$0.784326\pi$
$60$	0	0
$61$	−507.467	−1.06516	−0.532578	−	0.846381i	$-0.678777\pi$
$61$	−507.467	−1.06516	−0.532578	+	0.846381i	$0.678777\pi$
$62$	−91.8008	−0.188044
$63$	0	0
$64$	−568.226	−1.10982
$65$	−1053.62	−2.01055
$66$	0	0
$67$	−552.177	−1.00685	−0.503426	−	0.864038i	$-0.667928\pi$
$67$	−552.177	−1.00685	−0.503426	+	0.864038i	$0.667928\pi$
$68$	−575.566	−1.02644
$69$	0	0
$70$	−1612.27	−2.75290
$71$	−986.525	−1.64900	−0.824500	−	0.565862i	$-0.808544\pi$
$71$	−986.525	−1.64900	−0.824500	+	0.565862i	$0.808544\pi$
$72$	0	0
$73$	144.680	0.231966	0.115983	−	0.993251i	$-0.462998\pi$
$73$	144.680	0.231966	0.115983	+	0.993251i	$0.462998\pi$
$74$	1256.88	1.97444
$75$	0	0
$76$	125.031	0.188710
$77$	1364.85	2.01998
$78$	0	0
$79$	−507.444	−0.722683	−0.361341	−	0.932434i	$-0.617681\pi$
$79$	−507.444	−0.722683	−0.361341	+	0.932434i	$0.617681\pi$
$80$	730.599	1.02104
$81$	0	0
$82$	−666.242	−0.897246
$83$	502.367	0.664361	0.332180	−	0.943216i	$-0.392216\pi$
$83$	502.367	0.664361	0.332180	+	0.943216i	$0.392216\pi$
$84$	0	0
$85$	−472.723	−0.603223
$86$	2096.86	2.62919
$87$	0	0
$88$	−1988.01	−2.40821
$89$	−1430.11	−1.70327	−0.851635	−	0.524135i	$-0.824389\pi$
$89$	−1430.11	−1.70327	−0.851635	+	0.524135i	$0.824389\pi$
$90$	0	0
$91$	−2101.26	−2.42057
$92$	838.920	0.950689
$93$	0	0
$94$	55.3129	0.0606924
$95$	102.690	0.110903
$96$	0	0
$97$	−1299.67	−1.36043	−0.680215	−	0.733012i	$-0.738114\pi$
$97$	−1299.67	−1.36043	−0.680215	+	0.733012i	$0.738114\pi$
$98$	−1545.86	−1.59343
$99$	0	0
$100$	644.791	0.644791
$101$	−753.294	−0.742134	−0.371067	−	0.928606i	$-0.621008\pi$
$101$	−753.294	−0.742134	−0.371067	+	0.928606i	$0.621008\pi$
$102$	0	0
$103$	295.212	0.282409	0.141204	−	0.989980i	$-0.454903\pi$
$103$	295.212	0.282409	0.141204	+	0.989980i	$0.454903\pi$
$104$	3060.65	2.88579
$105$	0	0
$106$	−1494.97	−1.36985
$107$	176.010	0.159024	0.0795120	−	0.996834i	$-0.474664\pi$
$107$	176.010	0.159024	0.0795120	+	0.996834i	$0.474664\pi$
$108$	0	0
$109$	−616.259	−0.541532	−0.270766	−	0.962645i	$-0.587277\pi$
$109$	−616.259	−0.541532	−0.270766	+	0.962645i	$0.587277\pi$
$110$	−3331.08	−2.88733
$111$	0	0
$112$	1457.04	1.22927
$113$	1678.42	1.39728	0.698639	−	0.715475i	$-0.253790\pi$
$113$	1678.42	1.39728	0.698639	+	0.715475i	$0.253790\pi$
$114$	0	0
$115$	689.020	0.558708
$116$	−1633.66	−1.30760
$117$	0	0
$118$	3437.16	2.68149
$119$	−942.758	−0.726240
$120$	0	0
$121$	1488.88	1.11862
$122$	2470.04	1.83301
$123$	0	0
$124$	295.947	0.214329
$125$	−1081.38	−0.773772
$126$	0	0
$127$	165.496	0.115633	0.0578164	−	0.998327i	$-0.481586\pi$
$127$	165.496	0.115633	0.0578164	+	0.998327i	$0.481586\pi$
$128$	2577.24	1.77967
$129$	0	0
$130$	5128.39	3.45992
$131$	−2867.31	−1.91235	−0.956177	−	0.292789i	$-0.905417\pi$
$131$	−2867.31	−1.91235	−0.956177	+	0.292789i	$0.905417\pi$
$132$	0	0
$133$	204.796	0.133519
$134$	2687.66	1.73267
$135$	0	0
$136$	1373.21	0.865819
$137$	−933.866	−0.582377	−0.291188	−	0.956666i	$-0.594051\pi$
$137$	−933.866	−0.582377	−0.291188	+	0.956666i	$0.594051\pi$
$138$	0	0
$139$	−1152.86	−0.703486	−0.351743	−	0.936097i	$-0.614411\pi$
$139$	−1152.86	−0.703486	−0.351743	+	0.936097i	$0.614411\pi$
$140$	5197.63	3.13771
$141$	0	0
$142$	4801.80	2.83773
$143$	−4341.37	−2.53876
$144$	0	0
$145$	−1341.76	−0.768461
$146$	−704.214	−0.399186
$147$	0	0
$148$	−4051.91	−2.25044
$149$	1155.04	0.635064	0.317532	−	0.948248i	$-0.397146\pi$
$149$	1155.04	0.635064	0.317532	+	0.948248i	$0.397146\pi$
$150$	0	0
$151$	−418.063	−0.225308	−0.112654	−	0.993634i	$-0.535935\pi$
$151$	−418.063	−0.225308	−0.112654	+	0.993634i	$0.535935\pi$
$152$	−298.302	−0.159181
$153$	0	0
$154$	−6643.23	−3.47615
$155$	243.066	0.125958
$156$	0	0
$157$	786.803	0.399960	0.199980	−	0.979800i	$-0.435912\pi$
$157$	786.803	0.399960	0.199980	+	0.979800i	$0.435912\pi$
$158$	2469.93	1.24365
$159$	0	0
$160$	303.720	0.150070
$161$	1374.12	0.672646
$162$	0	0
$163$	3169.51	1.52304	0.761518	−	0.648143i	$-0.224454\pi$
$163$	3169.51	1.52304	0.761518	+	0.648143i	$0.224454\pi$
$164$	2147.83	1.02267
$165$	0	0
$166$	−2445.21	−1.14329
$167$	−3646.16	−1.68951	−0.844757	−	0.535151i	$-0.820255\pi$
$167$	−3646.16	−1.68951	−0.844757	+	0.535151i	$0.820255\pi$
$168$	0	0
$169$	4486.77	2.04223
$170$	2300.92	1.03808
$171$	0	0
$172$	−6759.84	−2.99670
$173$	−2371.65	−1.04227	−0.521137	−	0.853473i	$-0.674492\pi$
$173$	−2371.65	−1.04227	−0.521137	+	0.853473i	$0.674492\pi$
$174$	0	0
$175$	1056.15	0.456213
$176$	3010.37	1.28929
$177$	0	0
$178$	6960.88	2.93113
$179$	−547.509	−0.228618	−0.114309	−	0.993445i	$-0.536465\pi$
$179$	−547.509	−0.228618	−0.114309	+	0.993445i	$0.536465\pi$
$180$	0	0
$181$	−4221.82	−1.73373	−0.866866	−	0.498541i	$-0.833869\pi$
$181$	−4221.82	−1.73373	−0.866866	+	0.498541i	$0.833869\pi$
$182$	10227.6	4.16550
$183$	0	0
$184$	−2001.52	−0.801926
$185$	−3327.90	−1.32255
$186$	0	0
$187$	−1947.82	−0.761703
$188$	−178.317	−0.0691762
$189$	0	0
$190$	−499.831	−0.190850
$191$	−668.362	−0.253199	−0.126599	−	0.991954i	$-0.540406\pi$
$191$	−668.362	−0.253199	−0.126599	+	0.991954i	$0.540406\pi$
$192$	0	0
$193$	713.345	0.266050	0.133025	−	0.991113i	$-0.457531\pi$
$193$	713.345	0.266050	0.133025	+	0.991113i	$0.457531\pi$
$194$	6326.01	2.34114
$195$	0	0
$196$	4983.54	1.81616
$197$	4019.98	1.45387	0.726933	−	0.686708i	$-0.240945\pi$
$197$	4019.98	1.45387	0.726933	+	0.686708i	$0.240945\pi$
$198$	0	0
$199$	5076.26	1.80827	0.904136	−	0.427244i	$-0.140515\pi$
$199$	5076.26	1.80827	0.904136	+	0.427244i	$0.140515\pi$
$200$	−1538.37	−0.543894
$201$	0	0
$202$	3666.57	1.27712
$203$	−2675.89	−0.925175
$204$	0	0
$205$	1764.05	0.601007
$206$	−1436.91	−0.485992
$207$	0	0
$208$	−4634.63	−1.54497
$209$	423.125	0.140039
$210$	0	0
$211$	3896.66	1.27136	0.635681	−	0.771952i	$-0.280719\pi$
$211$	3896.66	1.27136	0.635681	+	0.771952i	$0.280719\pi$
$212$	4819.46	1.56133
$213$	0	0
$214$	−856.710	−0.273661
$215$	−5551.98	−1.76113
$216$	0	0
$217$	484.751	0.151645
$218$	2999.57	0.931911
$219$	0	0
$220$	10738.7	3.29093
$221$	2998.77	0.912756
$222$	0	0
$223$	−3340.77	−1.00320	−0.501601	−	0.865099i	$-0.667256\pi$
$223$	−3340.77	−1.00320	−0.501601	+	0.865099i	$0.667256\pi$
$224$	605.713	0.180674
$225$	0	0
$226$	−8169.51	−2.40455
$227$	5858.55	1.71298	0.856488	−	0.516166i	$-0.172641\pi$
$227$	5858.55	1.71298	0.856488	+	0.516166i	$0.172641\pi$
$228$	0	0
$229$	3218.01	0.928613	0.464306	−	0.885675i	$-0.346304\pi$
$229$	3218.01	0.928613	0.464306	+	0.885675i	$0.346304\pi$
$230$	−3353.73	−0.961470
$231$	0	0
$232$	3897.65	1.10299
$233$	−88.1635	−0.0247888	−0.0123944	−	0.999923i	$-0.503945\pi$
$233$	−88.1635	−0.0247888	−0.0123944	+	0.999923i	$0.503945\pi$
$234$	0	0
$235$	−146.455	−0.0406540
$236$	−11080.7	−3.05632
$237$	0	0
$238$	4588.77	1.24977
$239$	−239.000	−0.0646846
$239$	−239.000	−0.0646846
$240$	0	0
$241$	2475.88	0.661766	0.330883	−	0.943672i	$-0.392653\pi$
$241$	2475.88	0.661766	0.330883	+	0.943672i	$0.392653\pi$
$242$	−7246.97	−1.92501
$243$	0	0
$244$	−7962.89	−2.08923
$245$	4093.08	1.06733
$246$	0	0
$247$	−651.424	−0.167810
$248$	−706.080	−0.180791
$249$	0	0
$250$	5263.49	1.33157
$251$	−4657.91	−1.17133	−0.585667	−	0.810552i	$-0.699167\pi$
$251$	−4657.91	−1.17133	−0.585667	+	0.810552i	$0.699167\pi$
$252$	0	0
$253$	2839.05	0.705493
$254$	−805.531	−0.198990
$255$	0	0
$256$	−7998.62	−1.95279
$257$	5892.92	1.43031	0.715156	−	0.698965i	$-0.246356\pi$
$257$	5892.92	1.43031	0.715156	+	0.698965i	$0.246356\pi$
$258$	0	0
$259$	−6636.89	−1.59226
$260$	−16532.9	−3.94355
$261$	0	0
$262$	13956.3	3.29093
$263$	4062.87	0.952575	0.476288	−	0.879289i	$-0.341982\pi$
$263$	4062.87	0.952575	0.476288	+	0.879289i	$0.341982\pi$
$264$	0	0
$265$	3958.31	0.917573
$266$	−996.821	−0.229771
$267$	0	0
$268$	−8664.45	−1.97487
$269$	2492.51	0.564949	0.282474	−	0.959275i	$-0.408845\pi$
$269$	2492.51	0.564949	0.282474	+	0.959275i	$0.408845\pi$
$270$	0	0
$271$	−5129.98	−1.14990	−0.574952	−	0.818187i	$-0.694979\pi$
$271$	−5129.98	−1.14990	−0.574952	+	0.818187i	$0.694979\pi$
$272$	−2079.39	−0.463536
$273$	0	0
$274$	4545.49	1.00220
$275$	2182.09	0.478490
$276$	0	0
$277$	7795.14	1.69085	0.845423	−	0.534097i	$-0.179348\pi$
$277$	7795.14	1.69085	0.845423	+	0.534097i	$0.179348\pi$
$278$	5611.43	1.21062
$279$	0	0
$280$	−12400.7	−2.64672
$281$	1814.02	0.385109	0.192555	−	0.981286i	$-0.438323\pi$
$281$	1814.02	0.385109	0.192555	+	0.981286i	$0.438323\pi$
$282$	0	0
$283$	−281.625	−0.0591550	−0.0295775	−	0.999562i	$-0.509416\pi$
$283$	−281.625	−0.0591550	−0.0295775	+	0.999562i	$0.509416\pi$
$284$	−15480.0	−3.23440
$285$	0	0
$286$	21131.1	4.36891
$287$	3518.07	0.723572
$288$	0	0
$289$	−3567.56	−0.726147
$290$	6530.85	1.32243
$291$	0	0
$292$	2270.24	0.454986
$293$	625.851	0.124787	0.0623936	−	0.998052i	$-0.480127\pi$
$293$	625.851	0.124787	0.0623936	+	0.998052i	$0.480127\pi$
$294$	0	0
$295$	−9100.77	−1.79616
$296$	9667.18	1.89829
$297$	0	0
$298$	−5622.03	−1.09287
$299$	−4370.88	−0.845399
$300$	0	0
$301$	−11072.4	−2.12027
$302$	2034.88	0.387728
$303$	0	0
$304$	451.708	0.0852212
$305$	−6540.07	−1.22781
$306$	0	0
$307$	3498.07	0.650311	0.325155	−	0.945661i	$-0.394583\pi$
$307$	3498.07	0.650311	0.325155	+	0.945661i	$0.394583\pi$
$308$	21416.4	3.96205
$309$	0	0
$310$	−1183.10	−0.216759
$311$	43.1419	0.00786609	0.00393304	−	0.999992i	$-0.498748\pi$
$311$	43.1419	0.00786609	0.00393304	+	0.999992i	$0.498748\pi$
$312$	0	0
$313$	−11030.1	−1.99188	−0.995942	−	0.0900000i	$-0.971313\pi$
$313$	−11030.1	−1.99188	−0.995942	+	0.0900000i	$0.971313\pi$
$314$	−3829.67	−0.688283
$315$	0	0
$316$	−7962.53	−1.41749
$317$	−1750.06	−0.310073	−0.155036	−	0.987909i	$-0.549550\pi$
$317$	−1750.06	−0.310073	−0.155036	+	0.987909i	$0.549550\pi$
$318$	0	0
$319$	−5528.60	−0.970352
$320$	−7323.11	−1.27929
$321$	0	0
$322$	−6688.39	−1.15754
$323$	−292.271	−0.0503480
$324$	0	0
$325$	−3359.44	−0.573379
$326$	−15427.2	−2.62096
$327$	0	0
$328$	−5124.36	−0.862639
$329$	−292.078	−0.0489446
$330$	0	0
$331$	2650.32	0.440105	0.220052	−	0.975488i	$-0.429377\pi$
$331$	2650.32	0.440105	0.220052	+	0.975488i	$0.429377\pi$
$332$	7882.86	1.30310
$333$	0	0
$334$	17747.3	2.90745
$335$	−7116.27	−1.16061
$336$	0	0
$337$	−4402.78	−0.711677	−0.355838	−	0.934548i	$-0.615805\pi$
$337$	−4402.78	−0.711677	−0.355838	+	0.934548i	$0.615805\pi$
$338$	−21838.9	−3.51443
$339$	0	0
$340$	−7417.70	−1.18318
$341$	1001.54	0.159050
$342$	0	0
$343$	−652.927	−0.102783
$344$	16127.9	2.52778
$345$	0	0
$346$	11543.8	1.79363
$347$	9750.61	1.50847	0.754237	−	0.656602i	$-0.228007\pi$
$347$	9750.61	1.50847	0.754237	+	0.656602i	$0.228007\pi$
$348$	0	0
$349$	3046.79	0.467309	0.233655	−	0.972320i	$-0.424932\pi$
$349$	3046.79	0.467309	0.233655	+	0.972320i	$0.424932\pi$
$350$	−5140.67	−0.785087
$351$	0	0
$352$	1251.45	0.189496
$353$	5209.64	0.785499	0.392750	−	0.919645i	$-0.371524\pi$
$353$	5209.64	0.785499	0.392750	+	0.919645i	$0.371524\pi$
$354$	0	0
$355$	−12714.0	−1.90082
$356$	−22440.4	−3.34085
$357$	0	0
$358$	2664.93	0.393425
$359$	5418.81	0.796641	0.398320	−	0.917246i	$-0.369593\pi$
$359$	5418.81	0.796641	0.398320	+	0.917246i	$0.369593\pi$
$360$	0	0
$361$	−6795.51	−0.990744
$362$	20549.2	2.98355
$363$	0	0
$364$	−32971.7	−4.74777
$365$	1864.59	0.267389
$366$	0	0
$367$	−7155.67	−1.01777	−0.508887	−	0.860833i	$-0.669943\pi$
$367$	−7155.67	−1.01777	−0.508887	+	0.860833i	$0.669943\pi$
$368$	3030.83	0.429329
$369$	0	0
$370$	16198.2	2.27596
$371$	7894.11	1.10470
$372$	0	0
$373$	12097.4	1.67930	0.839650	−	0.543128i	$-0.182760\pi$
$373$	12097.4	1.67930	0.839650	+	0.543128i	$0.182760\pi$
$374$	9480.77	1.31080
$375$	0	0
$376$	425.436	0.0583515
$377$	8511.59	1.16278
$378$	0	0
$379$	−3039.72	−0.411978	−0.205989	−	0.978554i	$-0.566041\pi$
$379$	−3039.72	−0.411978	−0.205989	+	0.978554i	$0.566041\pi$
$380$	1611.35	0.217528
$381$	0	0
$382$	3253.18	0.435725
$383$	−10410.1	−1.38886	−0.694430	−	0.719560i	$-0.744343\pi$
$383$	−10410.1	−1.38886	−0.694430	+	0.719560i	$0.744343\pi$
$384$	0	0
$385$	17589.7	2.32845
$386$	−3472.12	−0.457841
$387$	0	0
$388$	−20393.7	−2.66839
$389$	12584.0	1.64020	0.820098	−	0.572223i	$-0.193919\pi$
$389$	12584.0	1.64020	0.820098	+	0.572223i	$0.193919\pi$
$390$	0	0
$391$	−1961.06	−0.253644
$392$	−11889.9	−1.53197
$393$	0	0
$394$	−19566.8	−2.50193
$395$	−6539.77	−0.833042
$396$	0	0
$397$	−5015.82	−0.634098	−0.317049	−	0.948409i	$-0.602692\pi$
$397$	−5015.82	−0.634098	−0.317049	+	0.948409i	$0.602692\pi$
$398$	−24708.1	−3.11182
$399$	0	0
$400$	2329.49	0.291186
$401$	10100.5	1.25784	0.628922	−	0.777469i	$-0.283497\pi$
$401$	10100.5	1.25784	0.628922	+	0.777469i	$0.283497\pi$
$402$	0	0
$403$	−1541.92	−0.190592
$404$	−11820.3	−1.45564
$405$	0	0
$406$	13024.6	1.59212
$407$	−13712.4	−1.67002
$408$	0	0
$409$	−3088.69	−0.373413	−0.186706	−	0.982416i	$-0.559781\pi$
$409$	−3088.69	−0.373413	−0.186706	+	0.982416i	$0.559781\pi$
$410$	−8586.31	−1.03426
$411$	0	0
$412$	4632.30	0.553925
$413$	−18149.8	−2.16245
$414$	0	0
$415$	6474.34	0.765814
$416$	−1926.68	−0.227075
$417$	0	0
$418$	−2059.51	−0.240991
$419$	−10633.0	−1.23976	−0.619879	−	0.784698i	$-0.712818\pi$
$419$	−10633.0	−1.23976	−0.619879	+	0.784698i	$0.712818\pi$
$420$	0	0
$421$	−883.014	−0.102222	−0.0511110	−	0.998693i	$-0.516276\pi$
$421$	−883.014	−0.102222	−0.0511110	+	0.998693i	$0.516276\pi$
$422$	−18966.6	−2.18786
$423$	0	0
$424$	−11498.4	−1.31701
$425$	−1507.26	−0.172030
$426$	0	0
$427$	−13043.0	−1.47820
$428$	2761.86	0.311914
$429$	0	0
$430$	27023.6	3.03069
$431$	1832.00	0.204743	0.102372	−	0.994746i	$-0.467357\pi$
$431$	1832.00	0.204743	0.102372	+	0.994746i	$0.467357\pi$
$432$	0	0
$433$	11930.2	1.32409	0.662044	−	0.749465i	$-0.269689\pi$
$433$	11930.2	1.32409	0.662044	+	0.749465i	$0.269689\pi$
$434$	−2359.47	−0.260964
$435$	0	0
$436$	−9670.00	−1.06218
$437$	426.001	0.0466325
$438$	0	0
$439$	−13299.9	−1.44594	−0.722972	−	0.690877i	$-0.757224\pi$
$439$	−13299.9	−1.44594	−0.722972	+	0.690877i	$0.757224\pi$
$440$	−25620.8	−2.77597
$441$	0	0
$442$	−14596.2	−1.57074
$443$	−8646.60	−0.927341	−0.463671	−	0.886008i	$-0.653468\pi$
$443$	−8646.60	−0.927341	−0.463671	+	0.886008i	$0.653468\pi$
$444$	0	0
$445$	−18430.7	−1.96337
$446$	16260.8	1.72639
$447$	0	0
$448$	−14604.6	−1.54018
$449$	−3986.26	−0.418983	−0.209492	−	0.977810i	$-0.567181\pi$
$449$	−3986.26	−0.418983	−0.209492	+	0.977810i	$0.567181\pi$
$450$	0	0
$451$	7268.62	0.758905
$452$	26336.8	2.74066
$453$	0	0
$454$	−28515.8	−2.94783
$455$	−27080.3	−2.79020
$456$	0	0
$457$	−2207.34	−0.225941	−0.112970	−	0.993598i	$-0.536037\pi$
$457$	−2207.34	−0.225941	−0.112970	+	0.993598i	$0.536037\pi$
$458$	−15663.3	−1.59803
$459$	0	0
$460$	10811.7	1.09587
$461$	−762.547	−0.0770398	−0.0385199	−	0.999258i	$-0.512264\pi$
$461$	−762.547	−0.0770398	−0.0385199	+	0.999258i	$0.512264\pi$
$462$	0	0
$463$	−10833.6	−1.08743	−0.543717	−	0.839268i	$-0.682984\pi$
$463$	−10833.6	−1.08743	−0.543717	+	0.839268i	$0.682984\pi$
$464$	−5902.07	−0.590510
$465$	0	0
$466$	429.126	0.0426585
$467$	837.565	0.0829933	0.0414967	−	0.999139i	$-0.486787\pi$
$467$	837.565	0.0829933	0.0414967	+	0.999139i	$0.486787\pi$
$468$	0	0
$469$	−14192.1	−1.39729
$470$	712.854	0.0699606
$471$	0	0
$472$	26436.7	2.57807
$473$	−22876.5	−2.22381
$474$	0	0
$475$	327.423	0.0316278
$476$	−14793.2	−1.42447
$477$	0	0
$478$	1163.31	0.111315
$479$	12686.8	1.21017	0.605086	−	0.796160i	$-0.293139\pi$
$479$	12686.8	1.21017	0.605086	+	0.796160i	$0.293139\pi$
$480$	0	0
$481$	21110.9	2.00120
$482$	−12051.1	−1.13882
$483$	0	0
$484$	23362.7	2.19410
$485$	−16749.7	−1.56818
$486$	0	0
$487$	17683.6	1.64542	0.822710	−	0.568461i	$-0.192461\pi$
$487$	17683.6	1.64542	0.822710	+	0.568461i	$0.192461\pi$
$488$	18998.2	1.76231
$489$	0	0
$490$	−19922.6	−1.83676
$491$	18906.9	1.73779	0.868896	−	0.494995i	$-0.164830\pi$
$491$	18906.9	1.73779	0.868896	+	0.494995i	$0.164830\pi$
$492$	0	0
$493$	3818.85	0.348868
$494$	3170.73	0.288781
$495$	0	0
$496$	1069.19	0.0967905
$497$	−25355.7	−2.28845
$498$	0	0
$499$	−4634.53	−0.415772	−0.207886	−	0.978153i	$-0.566658\pi$
$499$	−4634.53	−0.415772	−0.207886	+	0.978153i	$0.566658\pi$
$500$	−16968.4	−1.51770
$501$	0	0
$502$	22671.9	2.01573
$503$	1706.15	0.151239	0.0756196	−	0.997137i	$-0.475907\pi$
$503$	1706.15	0.151239	0.0756196	+	0.997137i	$0.475907\pi$
$504$	0	0
$505$	−9708.20	−0.855464
$506$	−13818.8	−1.21407
$507$	0	0
$508$	2596.86	0.226806
$509$	−2176.83	−0.189560	−0.0947801	−	0.995498i	$-0.530215\pi$
$509$	−2176.83	−0.189560	−0.0947801	+	0.995498i	$0.530215\pi$
$510$	0	0
$511$	3718.58	0.321918
$512$	18314.4	1.58084
$513$	0	0
$514$	−28683.1	−2.46140
$515$	3804.59	0.325535
$516$	0	0
$517$	−603.457	−0.0513346
$518$	32304.3	2.74010
$519$	0	0
$520$	39444.7	3.32647
$521$	4692.38	0.394581	0.197291	−	0.980345i	$-0.436786\pi$
$521$	4692.38	0.394581	0.197291	+	0.980345i	$0.436786\pi$
$522$	0	0
$523$	8435.46	0.705271	0.352636	−	0.935761i	$-0.385286\pi$
$523$	8435.46	0.705271	0.352636	+	0.935761i	$0.385286\pi$
$524$	−44992.3	−3.75095
$525$	0	0
$526$	−19775.6	−1.63927
$527$	−691.804	−0.0571830
$528$	0	0
$529$	−9308.65	−0.765074
$530$	−19266.6	−1.57903
$531$	0	0
$532$	3213.54	0.261889
$533$	−11190.4	−0.909403
$534$	0	0
$535$	2268.36	0.183308
$536$	20672.0	1.66584
$537$	0	0
$538$	−12132.0	−0.972210
$539$	16865.2	1.34775
$540$	0	0
$541$	4631.19	0.368042	0.184021	−	0.982922i	$-0.441089\pi$
$541$	4631.19	0.368042	0.184021	+	0.982922i	$0.441089\pi$
$542$	24969.6	1.97885
$543$	0	0
$544$	−864.433	−0.0681291
$545$	−7942.14	−0.624228
$546$	0	0
$547$	−15790.1	−1.23425	−0.617127	−	0.786864i	$-0.711703\pi$
$547$	−15790.1	−1.23425	−0.617127	+	0.786864i	$0.711703\pi$
$548$	−14653.7	−1.14229
$549$	0	0
$550$	−10621.1	−0.823424
$551$	−829.570	−0.0641395
$552$	0	0
$553$	−13042.4	−1.00293
$554$	−37942.0	−2.90975
$555$	0	0
$556$	−18090.1	−1.37984
$557$	9551.34	0.726577	0.363288	−	0.931677i	$-0.381654\pi$
$557$	9551.34	0.726577	0.363288	+	0.931677i	$0.381654\pi$
$558$	0	0
$559$	35219.6	2.66481
$560$	18777.9	1.41698
$561$	0	0
$562$	−8829.56	−0.662727
$563$	−11113.4	−0.831923	−0.415962	−	0.909382i	$-0.636555\pi$
$563$	−11113.4	−0.831923	−0.415962	+	0.909382i	$0.636555\pi$
$564$	0	0
$565$	21630.9	1.61065
$566$	1370.78	0.101799
$567$	0	0
$568$	36932.7	2.72828
$569$	14314.8	1.05467	0.527336	−	0.849657i	$-0.323191\pi$
$569$	14314.8	1.05467	0.527336	+	0.849657i	$0.323191\pi$
$570$	0	0
$571$	−176.918	−0.0129663	−0.00648317	−	0.999979i	$-0.502064\pi$
$571$	−176.918	−0.0129663	−0.00648317	+	0.999979i	$0.502064\pi$
$572$	−68122.3	−4.97961
$573$	0	0
$574$	−17123.8	−1.24518
$575$	2196.92	0.159335
$576$	0	0
$577$	20581.3	1.48494	0.742471	−	0.669878i	$-0.233654\pi$
$577$	20581.3	1.48494	0.742471	+	0.669878i	$0.233654\pi$
$578$	17364.7	1.24961
$579$	0	0
$580$	−21054.1	−1.50728
$581$	12911.9	0.921988
$582$	0	0
$583$	16309.9	1.15864
$584$	−5416.42	−0.383790
$585$	0	0
$586$	−3046.26	−0.214744
$587$	22471.3	1.58005	0.790026	−	0.613073i	$-0.210067\pi$
$587$	22471.3	1.58005	0.790026	+	0.613073i	$0.210067\pi$
$588$	0	0
$589$	150.281	0.0105131
$590$	44296.9	3.09098
$591$	0	0
$592$	−14638.7	−1.01629
$593$	7523.03	0.520967	0.260484	−	0.965478i	$-0.416118\pi$
$593$	7523.03	0.520967	0.260484	+	0.965478i	$0.416118\pi$
$594$	0	0
$595$	−12150.0	−0.837142
$596$	18124.2	1.24563
$597$	0	0
$598$	21274.7	1.45483
$599$	15953.6	1.08822	0.544112	−	0.839013i	$-0.316867\pi$
$599$	15953.6	1.08822	0.544112	+	0.839013i	$0.316867\pi$
$600$	0	0
$601$	953.599	0.0647223	0.0323612	−	0.999476i	$-0.489697\pi$
$601$	953.599	0.0647223	0.0323612	+	0.999476i	$0.489697\pi$
$602$	53893.6	3.64874
$603$	0	0
$604$	−6560.02	−0.441926
$605$	19188.2	1.28944
$606$	0	0
$607$	−27189.4	−1.81809	−0.909046	−	0.416696i	$-0.863188\pi$
$607$	−27189.4	−1.81809	−0.909046	+	0.416696i	$0.863188\pi$
$608$	187.781	0.0125255
$609$	0	0
$610$	31833.0	2.11292
$611$	929.055	0.0615148
$612$	0	0
$613$	26386.6	1.73857	0.869286	−	0.494309i	$-0.164579\pi$
$613$	26386.6	1.73857	0.869286	+	0.494309i	$0.164579\pi$
$614$	−17026.5	−1.11911
$615$	0	0
$616$	−51096.0	−3.34207
$617$	−8097.91	−0.528379	−0.264189	−	0.964471i	$-0.585104\pi$
$617$	−8097.91	−0.528379	−0.264189	+	0.964471i	$0.585104\pi$
$618$	0	0
$619$	−3529.62	−0.229188	−0.114594	−	0.993412i	$-0.536557\pi$
$619$	−3529.62	−0.229188	−0.114594	+	0.993412i	$0.536557\pi$
$620$	3814.06	0.247059
$621$	0	0
$622$	−209.988	−0.0135366
$623$	−36756.7	−2.36377
$624$	0	0
$625$	−19072.9	−1.22067
$626$	53687.9	3.42779
$627$	0	0
$628$	12346.1	0.784494
$629$	9471.72	0.600417
$630$	0	0
$631$	−7380.71	−0.465644	−0.232822	−	0.972519i	$-0.574796\pi$
$631$	−7380.71	−0.465644	−0.232822	+	0.972519i	$0.574796\pi$
$632$	18997.3	1.19568
$633$	0	0
$634$	8518.21	0.533598
$635$	2132.85	0.133291
$636$	0	0
$637$	−25964.9	−1.61502
$638$	26909.8	1.66986
$639$	0	0
$640$	33214.6	2.05144
$641$	−9645.99	−0.594374	−0.297187	−	0.954819i	$-0.596049\pi$
$641$	−9645.99	−0.594374	−0.297187	+	0.954819i	$0.596049\pi$
$642$	0	0
$643$	−27664.9	−1.69673	−0.848364	−	0.529413i	$-0.822412\pi$
$643$	−27664.9	−1.69673	−0.848364	+	0.529413i	$0.822412\pi$
$644$	21562.0	1.31935
$645$	0	0
$646$	1422.60	0.0866428
$647$	15623.4	0.949334	0.474667	−	0.880165i	$-0.342568\pi$
$647$	15623.4	0.949334	0.474667	+	0.880165i	$0.342568\pi$
$648$	0	0
$649$	−37499.0	−2.26805
$650$	16351.7	0.986717
$651$	0	0
$652$	49734.1	2.98733
$653$	156.806	0.00939706	0.00469853	−	0.999989i	$-0.498504\pi$
$653$	156.806	0.00939706	0.00469853	+	0.999989i	$0.498504\pi$
$654$	0	0
$655$	−36953.0	−2.20439
$656$	7759.63	0.461833
$657$	0	0
$658$	1421.66	0.0842278
$659$	−6405.08	−0.378614	−0.189307	−	0.981918i	$-0.560624\pi$
$659$	−6405.08	−0.378614	−0.189307	+	0.981918i	$0.560624\pi$
$660$	0	0
$661$	32471.6	1.91074	0.955369	−	0.295416i	$-0.0954582\pi$
$661$	32471.6	1.91074	0.955369	+	0.295416i	$0.0954582\pi$
$662$	−12900.1	−0.757368
$663$	0	0
$664$	−18807.2	−1.09919
$665$	2639.34	0.153909
$666$	0	0
$667$	−5566.18	−0.323124
$668$	−57213.6	−3.31386
$669$	0	0
$670$	34637.6	1.99727
$671$	−26947.8	−1.55039
$672$	0	0
$673$	−3967.84	−0.227264	−0.113632	−	0.993523i	$-0.536249\pi$
$673$	−3967.84	−0.227264	−0.113632	+	0.993523i	$0.536249\pi$
$674$	21430.1	1.22471
$675$	0	0
$676$	70403.9	4.00569
$677$	18433.7	1.04648	0.523238	−	0.852187i	$-0.324724\pi$
$677$	18433.7	1.04648	0.523238	+	0.852187i	$0.324724\pi$
$678$	0	0
$679$	−33404.3	−1.88798
$680$	17697.4	0.998037
$681$	0	0
$682$	−4874.86	−0.273707
$683$	10157.3	0.569043	0.284522	−	0.958670i	$-0.408165\pi$
$683$	10157.3	0.569043	0.284522	+	0.958670i	$0.408165\pi$
$684$	0	0
$685$	−12035.4	−0.671310
$686$	3178.05	0.176878
$687$	0	0
$688$	−24421.8	−1.35330
$689$	−25110.0	−1.38841
$690$	0	0
$691$	−201.380	−0.0110866	−0.00554331	−	0.999985i	$-0.501764\pi$
$691$	−201.380	−0.0110866	−0.00554331	+	0.999985i	$0.501764\pi$
$692$	−37214.7	−2.04435
$693$	0	0
$694$	−47460.0	−2.59590
$695$	−14857.7	−0.810914
$696$	0	0
$697$	−5020.75	−0.272847
$698$	−14829.9	−0.804183
$699$	0	0
$700$	16572.5	0.894829
$701$	2141.29	0.115372	0.0576859	−	0.998335i	$-0.481628\pi$
$701$	2141.29	0.115372	0.0576859	+	0.998335i	$0.481628\pi$
$702$	0	0
$703$	−2057.55	−0.110387
$704$	−30174.3	−1.61539
$705$	0	0
$706$	−25357.3	−1.35175
$707$	−19361.2	−1.02992
$708$	0	0
$709$	19332.5	1.02405	0.512023	−	0.858972i	$-0.328896\pi$
$709$	19332.5	1.02405	0.512023	+	0.858972i	$0.328896\pi$
$710$	61884.0	3.27108
$711$	0	0
$712$	53539.2	2.81807
$713$	1008.34	0.0529632
$714$	0	0
$715$	−55950.1	−2.92645
$716$	−8591.20	−0.448419
$717$	0	0
$718$	−26375.5	−1.37092
$719$	15337.0	0.795513	0.397756	−	0.917491i	$-0.369789\pi$
$719$	15337.0	0.795513	0.397756	+	0.917491i	$0.369789\pi$
$720$	0	0
$721$	7587.56	0.391922
$722$	33076.4	1.70495
$723$	0	0
$724$	−66246.5	−3.40059
$725$	−4278.15	−0.219154
$726$	0	0
$727$	20925.8	1.06753	0.533766	−	0.845632i	$-0.320776\pi$
$727$	20925.8	1.06753	0.533766	+	0.845632i	$0.320776\pi$
$728$	78665.1	4.00484
$729$	0	0
$730$	−9075.68	−0.460145
$731$	15801.8	0.799521
$732$	0	0
$733$	−21879.2	−1.10249	−0.551247	−	0.834342i	$-0.685848\pi$
$733$	−21879.2	−1.10249	−0.551247	+	0.834342i	$0.685848\pi$
$734$	34829.4	1.75147
$735$	0	0
$736$	1259.96	0.0631015
$737$	−29322.0	−1.46552
$738$	0	0
$739$	27062.8	1.34712	0.673560	−	0.739132i	$-0.264764\pi$
$739$	27062.8	1.34712	0.673560	+	0.739132i	$0.264764\pi$
$740$	−52219.6	−2.59410
$741$	0	0
$742$	−38423.7	−1.90105
$743$	−22905.7	−1.13100	−0.565498	−	0.824750i	$-0.691316\pi$
$743$	−22905.7	−1.13100	−0.565498	+	0.824750i	$0.691316\pi$
$744$	0	0
$745$	14885.8	0.732043
$746$	−58882.6	−2.88987
$747$	0	0
$748$	−30564.0	−1.49403
$749$	4523.83	0.220690
$750$	0	0
$751$	−9357.07	−0.454653	−0.227326	−	0.973819i	$-0.572998\pi$
$751$	−9357.07	−0.454653	−0.227326	+	0.973819i	$0.572998\pi$
$752$	−644.221	−0.0312398
$753$	0	0
$754$	−41429.2	−2.00101
$755$	−5387.86	−0.259714
$756$	0	0
$757$	6910.90	0.331811	0.165906	−	0.986142i	$-0.446945\pi$
$757$	6910.90	0.331811	0.165906	+	0.986142i	$0.446945\pi$
$758$	14795.5	0.708965
$759$	0	0
$760$	−3844.42	−0.183489
$761$	−28790.3	−1.37142	−0.685708	−	0.727877i	$-0.740507\pi$
$761$	−28790.3	−1.37142	−0.685708	+	0.727877i	$0.740507\pi$
$762$	0	0
$763$	−15839.1	−0.751527
$764$	−10487.6	−0.496632
$765$	0	0
$766$	50670.2	2.39006
$767$	57731.7	2.71783
$768$	0	0
$769$	1494.77	0.0700945	0.0350472	−	0.999386i	$-0.488842\pi$
$769$	1494.77	0.0700945	0.0350472	+	0.999386i	$0.488842\pi$
$770$	−85615.8	−4.00698
$771$	0	0
$772$	11193.4	0.521839
$773$	18782.4	0.873943	0.436971	−	0.899475i	$-0.356051\pi$
$773$	18782.4	0.873943	0.436971	+	0.899475i	$0.356051\pi$
$774$	0	0
$775$	775.009	0.0359215
$776$	48656.1	2.25084
$777$	0	0
$778$	−61251.4	−2.82258
$779$	1090.66	0.0501630
$780$	0	0
$781$	−52387.0	−2.40020
$782$	9545.21	0.436491
$783$	0	0
$784$	18004.5	0.820174
$785$	10140.1	0.461037
$786$	0	0
$787$	−3311.15	−0.149974	−0.0749872	−	0.997184i	$-0.523892\pi$
$787$	−3311.15	−0.149974	−0.0749872	+	0.997184i	$0.523892\pi$
$788$	63079.3	2.85166
$789$	0	0
$790$	31831.6	1.43357
$791$	43138.8	1.93911
$792$	0	0
$793$	41487.6	1.85784
$794$	24413.9	1.09121
$795$	0	0
$796$	79653.8	3.54680
$797$	18230.4	0.810231	0.405115	−	0.914266i	$-0.367231\pi$
$797$	18230.4	0.810231	0.405115	+	0.914266i	$0.367231\pi$
$798$	0	0
$799$	416.834	0.0184562
$800$	968.401	0.0427977
$801$	0	0
$802$	−49163.1	−2.16460
$803$	7682.89	0.337638
$804$	0	0
$805$	17709.2	0.775365
$806$	7505.11	0.327985
$807$	0	0
$808$	28201.2	1.22787
$809$	−30301.8	−1.31688	−0.658439	−	0.752634i	$-0.728783\pi$
$809$	−30301.8	−1.31688	−0.658439	+	0.752634i	$0.728783\pi$
$810$	0	0
$811$	−20841.4	−0.902395	−0.451197	−	0.892424i	$-0.649003\pi$
$811$	−20841.4	−0.902395	−0.451197	+	0.892424i	$0.649003\pi$
$812$	−41988.5	−1.81467
$813$	0	0
$814$	66743.3	2.87390
$815$	40847.5	1.75562
$816$	0	0
$817$	−3432.63	−0.146992
$818$	15033.8	0.642599
$819$	0	0
$820$	27680.5	1.17883
$821$	−1849.97	−0.0786410	−0.0393205	−	0.999227i	$-0.512519\pi$
$821$	−1849.97	−0.0786410	−0.0393205	+	0.999227i	$0.512519\pi$
$822$	0	0
$823$	13233.2	0.560485	0.280243	−	0.959929i	$-0.409585\pi$
$823$	13233.2	0.560485	0.280243	+	0.959929i	$0.409585\pi$
$824$	−11051.9	−0.467247
$825$	0	0
$826$	88342.1	3.72132
$827$	−4575.83	−0.192403	−0.0962015	−	0.995362i	$-0.530669\pi$
$827$	−4575.83	−0.192403	−0.0962015	+	0.995362i	$0.530669\pi$
$828$	0	0
$829$	25481.5	1.06756	0.533781	−	0.845623i	$-0.320771\pi$
$829$	25481.5	1.06756	0.533781	+	0.845623i	$0.320771\pi$
$830$	−31513.1	−1.31787
$831$	0	0
$832$	46455.0	1.93574
$833$	−11649.5	−0.484552
$834$	0	0
$835$	−46990.6	−1.94752
$836$	6639.44	0.274677
$837$	0	0
$838$	51755.1	2.13347
$839$	12457.9	0.512626	0.256313	−	0.966594i	$-0.417492\pi$
$839$	12457.9	0.512626	0.256313	+	0.966594i	$0.417492\pi$
$840$	0	0
$841$	−13549.7	−0.555568
$842$	4297.97	0.175912
$843$	0	0
$844$	61144.3	2.49369
$845$	57824.0	2.35409
$846$	0	0
$847$	38267.4	1.55240
$848$	17411.7	0.705093
$849$	0	0
$850$	7336.42	0.296044
$851$	−13805.6	−0.556109
$852$	0	0
$853$	−4017.73	−0.161271	−0.0806357	−	0.996744i	$-0.525695\pi$
$853$	−4017.73	−0.161271	−0.0806357	+	0.996744i	$0.525695\pi$
$854$	63485.1	2.54381
$855$	0	0
$856$	−6589.33	−0.263106
$857$	5839.09	0.232742	0.116371	−	0.993206i	$-0.462874\pi$
$857$	5839.09	0.232742	0.116371	+	0.993206i	$0.462874\pi$
$858$	0	0
$859$	38284.1	1.52065	0.760323	−	0.649545i	$-0.225041\pi$
$859$	38284.1	1.52065	0.760323	+	0.649545i	$0.225041\pi$
$860$	−87118.6	−3.45432
$861$	0	0
$862$	−8917.05	−0.352338
$863$	−35942.8	−1.41774	−0.708869	−	0.705340i	$-0.750794\pi$
$863$	−35942.8	−1.41774	−0.708869	+	0.705340i	$0.750794\pi$
$864$	0	0
$865$	−30565.1	−1.20144
$866$	−58069.0	−2.27860
$867$	0	0
$868$	7606.44	0.297442
$869$	−26946.6	−1.05190
$870$	0	0
$871$	45142.9	1.75615
$872$	23071.0	0.895967
$873$	0	0
$874$	−2073.51	−0.0802490
$875$	−27793.7	−1.07383
$876$	0	0
$877$	37783.3	1.45479	0.727396	−	0.686218i	$-0.240730\pi$
$877$	37783.3	1.45479	0.727396	+	0.686218i	$0.240730\pi$
$878$	64735.7	2.48830
$879$	0	0
$880$	38796.7	1.48618
$881$	−46281.4	−1.76987	−0.884937	−	0.465710i	$-0.845799\pi$
$881$	−46281.4	−1.76987	−0.884937	+	0.465710i	$0.845799\pi$
$882$	0	0
$883$	8154.17	0.310770	0.155385	−	0.987854i	$-0.450338\pi$
$883$	8154.17	0.310770	0.155385	+	0.987854i	$0.450338\pi$
$884$	47055.0	1.79031
$885$	0	0
$886$	42086.3	1.59584
$887$	−21950.9	−0.830934	−0.415467	−	0.909608i	$-0.636382\pi$
$887$	−21950.9	−0.830934	−0.415467	+	0.909608i	$0.636382\pi$
$888$	0	0
$889$	4253.58	0.160473
$890$	89709.5	3.37873
$891$	0	0
$892$	−52421.4	−1.96771
$893$	−90.5490	−0.00339318
$894$	0	0
$895$	−7056.11	−0.263530
$896$	66240.5	2.46980
$897$	0	0
$898$	19402.7	0.721020
$899$	−1963.59	−0.0728469
$900$	0	0
$901$	−11265.9	−0.416563
$902$	−35379.2	−1.30598
$903$	0	0
$904$	−62835.3	−2.31180
$905$	−54409.4	−1.99849
$906$	0	0
$907$	−35884.5	−1.31370	−0.656850	−	0.754021i	$-0.728112\pi$
$907$	−35884.5	−1.31370	−0.656850	+	0.754021i	$0.728112\pi$
$908$	91929.1	3.35988
$909$	0	0
$910$	131810.	4.80161
$911$	3990.76	0.145137	0.0725685	−	0.997363i	$-0.476880\pi$
$911$	3990.76	0.145137	0.0725685	+	0.997363i	$0.476880\pi$
$912$	0	0
$913$	26677.0	0.967009
$914$	10744.0	0.388817
$915$	0	0
$916$	50495.3	1.82141
$917$	−73695.9	−2.65393
$918$	0	0
$919$	−16836.4	−0.604334	−0.302167	−	0.953255i	$-0.597710\pi$
$919$	−16836.4	−0.604334	−0.302167	+	0.953255i	$0.597710\pi$
$920$	−25795.0	−0.924386
$921$	0	0
$922$	3711.61	0.132576
$923$	80652.7	2.87618
$924$	0	0
$925$	−10610.9	−0.377173
$926$	52731.5	1.87135
$927$	0	0
$928$	−2453.57	−0.0867914
$929$	12626.5	0.445922	0.222961	−	0.974827i	$-0.428428\pi$
$929$	12626.5	0.445922	0.222961	+	0.974827i	$0.428428\pi$
$930$	0	0
$931$	2530.63	0.0890849
$932$	−1383.41	−0.0486214
$933$	0	0
$934$	−4076.75	−0.142822
$935$	−25102.8	−0.878021
$936$	0	0
$937$	21058.2	0.734196	0.367098	−	0.930182i	$-0.380351\pi$
$937$	21058.2	0.734196	0.367098	+	0.930182i	$0.380351\pi$
$938$	69078.4	2.40457
$939$	0	0
$940$	−2298.09	−0.0797399
$941$	23153.1	0.802094	0.401047	−	0.916057i	$-0.368646\pi$
$941$	23153.1	0.802094	0.401047	+	0.916057i	$0.368646\pi$
$942$	0	0
$943$	7318.03	0.252712
$944$	−40032.1	−1.38023
$945$	0	0
$946$	111349.	3.82691
$947$	−6504.63	−0.223202	−0.111601	−	0.993753i	$-0.535598\pi$
$947$	−6504.63	−0.223202	−0.111601	+	0.993753i	$0.535598\pi$
$948$	0	0
$949$	−11828.2	−0.404595
$950$	−1593.70	−0.0544277
$951$	0	0
$952$	35294.2	1.20157
$953$	−15332.1	−0.521148	−0.260574	−	0.965454i	$-0.583912\pi$
$953$	−15332.1	−0.521148	−0.260574	+	0.965454i	$0.583912\pi$
$954$	0	0
$955$	−8613.63	−0.291864
$956$	−3750.25	−0.126874
$957$	0	0
$958$	−61751.3	−2.08256
$959$	−24002.3	−0.808211
$960$	0	0
$961$	−29435.3	−0.988060
$962$	−102755.	−3.44382
$963$	0	0
$964$	38850.2	1.29801
$965$	9193.35	0.306678
$966$	0	0
$967$	−18370.9	−0.610927	−0.305464	−	0.952204i	$-0.598811\pi$
$967$	−18370.9	−0.610927	−0.305464	+	0.952204i	$0.598811\pi$
$968$	−55739.7	−1.85076
$969$	0	0
$970$	81527.5	2.69865
$971$	−41221.7	−1.36238	−0.681188	−	0.732109i	$-0.738536\pi$
$971$	−41221.7	−1.36238	−0.681188	+	0.732109i	$0.738536\pi$
$972$	0	0
$973$	−29631.0	−0.976285
$974$	−86072.8	−2.83157
$975$	0	0
$976$	−28768.2	−0.943491
$977$	−11406.9	−0.373531	−0.186765	−	0.982405i	$-0.559800\pi$
$977$	−11406.9	−0.373531	−0.186765	+	0.982405i	$0.559800\pi$
$978$	0	0
$979$	−75942.4	−2.47919
$980$	64226.2	2.09350
$981$	0	0
$982$	−92027.1	−2.99053
$983$	56720.7	1.84040	0.920198	−	0.391454i	$-0.128028\pi$
$983$	56720.7	1.84040	0.920198	+	0.391454i	$0.128028\pi$
$984$	0	0
$985$	51808.1	1.67588
$986$	−18587.8	−0.600361
$987$	0	0
$988$	−10221.8	−0.329148
$989$	−23032.0	−0.740520
$990$	0	0
$991$	−53968.0	−1.72992	−0.864960	−	0.501842i	$-0.832656\pi$
$991$	−53968.0	−1.72992	−0.864960	+	0.501842i	$0.832656\pi$
$992$	444.477	0.0142260
$993$	0	0
$994$	123416.	3.93815
$995$	65421.1	2.08441
$996$	0	0
$997$	−6934.07	−0.220265	−0.110133	−	0.993917i	$-0.535128\pi$
$997$	−6934.07	−0.220265	−0.110133	+	0.993917i	$0.535128\pi$
$998$	22558.0	0.715493
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2151.4.a.b.1.2		28
3.2	odd	2		717.4.a.b.1.27	✓	28

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
717.4.a.b.1.27	✓	28	3.2	odd	2
2151.4.a.b.1.2		28	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)

\(f(q)\)	\(=\)	\(q-4.86739 q^{2} +15.6914 q^{4} +12.8877 q^{5} +25.7021 q^{7} -37.4372 q^{8} +O(q^{10})\)	\(q-4.86739 q^{2} +15.6914 q^{4} +12.8877 q^{5} +25.7021 q^{7} -37.4372 q^{8} -62.7292 q^{10} +53.1026 q^{11} -81.7543 q^{13} -125.102 q^{14} +56.6898 q^{16} -36.6803 q^{17} +7.96807 q^{19} +202.226 q^{20} -258.471 q^{22} +53.4635 q^{23} +41.0919 q^{25} +397.930 q^{26} +403.302 q^{28} -104.112 q^{29} +18.8604 q^{31} +23.5667 q^{32} +178.537 q^{34} +331.240 q^{35} -258.224 q^{37} -38.7837 q^{38} -482.478 q^{40} +136.879 q^{41} -430.798 q^{43} +833.256 q^{44} -260.228 q^{46} -11.3640 q^{47} +317.596 q^{49} -200.010 q^{50} -1282.84 q^{52} +307.139 q^{53} +684.368 q^{55} -962.213 q^{56} +506.752 q^{58} -706.161 q^{59} -507.467 q^{61} -91.8008 q^{62} -568.226 q^{64} -1053.62 q^{65} -552.177 q^{67} -575.566 q^{68} -1612.27 q^{70} -986.525 q^{71} +144.680 q^{73} +1256.88 q^{74} +125.031 q^{76} +1364.85 q^{77} -507.444 q^{79} +730.599 q^{80} -666.242 q^{82} +502.367 q^{83} -472.723 q^{85} +2096.86 q^{86} -1988.01 q^{88} -1430.11 q^{89} -2101.26 q^{91} +838.920 q^{92} +55.3129 q^{94} +102.690 q^{95} -1299.67 q^{97} -1545.86 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q + 5 q^{2} + 103 q^{4} - 6 q^{5} - 68 q^{7} + 39 q^{8}+O(q^{10}) \) 28 * q + 5 * q^2 + 103 * q^4 - 6 * q^5 - 68 * q^7 + 39 * q^8	\( 28 q + 5 q^{2} + 103 q^{4} - 6 q^{5} - 68 q^{7} + 39 q^{8} - 88 q^{10} + 110 q^{11} - 82 q^{13} - 126 q^{14} + 271 q^{16} - 100 q^{17} - 292 q^{19} + 52 q^{20} - 351 q^{22} + 276 q^{23} + 386 q^{25} - 84 q^{26} - 1010 q^{28} + 38 q^{29} - 432 q^{31} + 452 q^{32} - 524 q^{34} + 166 q^{35} - 936 q^{37} + 41 q^{38} - 1183 q^{40} - 1054 q^{41} - 1804 q^{43} + 341 q^{44} - 888 q^{46} + 560 q^{47} + 1074 q^{49} + 1054 q^{50} - 632 q^{52} + 160 q^{53} - 842 q^{55} - 509 q^{56} - 1266 q^{58} - 846 q^{59} - 2220 q^{61} - 82 q^{62} - 1565 q^{64} - 296 q^{65} - 4752 q^{67} + 1719 q^{68} - 5601 q^{70} + 802 q^{71} - 2732 q^{73} + 4581 q^{74} - 5614 q^{76} + 1008 q^{77} - 3172 q^{79} + 732 q^{80} - 9709 q^{82} + 4780 q^{83} - 4624 q^{85} + 2009 q^{86} - 9331 q^{88} - 4372 q^{89} - 7398 q^{91} + 6138 q^{92} - 7068 q^{94} + 3160 q^{95} - 4846 q^{97} + 3772 q^{98}+O(q^{100}) \) 28 * q + 5 * q^2 + 103 * q^4 - 6 * q^5 - 68 * q^7 + 39 * q^8 - 88 * q^10 + 110 * q^11 - 82 * q^13 - 126 * q^14 + 271 * q^16 - 100 * q^17 - 292 * q^19 + 52 * q^20 - 351 * q^22 + 276 * q^23 + 386 * q^25 - 84 * q^26 - 1010 * q^28 + 38 * q^29 - 432 * q^31 + 452 * q^32 - 524 * q^34 + 166 * q^35 - 936 * q^37 + 41 * q^38 - 1183 * q^40 - 1054 * q^41 - 1804 * q^43 + 341 * q^44 - 888 * q^46 + 560 * q^47 + 1074 * q^49 + 1054 * q^50 - 632 * q^52 + 160 * q^53 - 842 * q^55 - 509 * q^56 - 1266 * q^58 - 846 * q^59 - 2220 * q^61 - 82 * q^62 - 1565 * q^64 - 296 * q^65 - 4752 * q^67 + 1719 * q^68 - 5601 * q^70 + 802 * q^71 - 2732 * q^73 + 4581 * q^74 - 5614 * q^76 + 1008 * q^77 - 3172 * q^79 + 732 * q^80 - 9709 * q^82 + 4780 * q^83 - 4624 * q^85 + 2009 * q^86 - 9331 * q^88 - 4372 * q^89 - 7398 * q^91 + 6138 * q^92 - 7068 * q^94 + 3160 * q^95 - 4846 * q^97 + 3772 * q^98

Introduction

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