LMFDB - Embedded newform 2151.4.a.b.1.7

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			1.7
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)

$f(q)$	$=$	$q-3.55218 q^{2} +4.61801 q^{4} -1.20455 q^{5} +9.27993 q^{7} +12.0135 q^{8} +O(q^{10})$	\(q-3.55218 q^{2} +4.61801 q^{4} -1.20455 q^{5} +9.27993 q^{7} +12.0135 q^{8} +4.27878 q^{10} -9.91778 q^{11} -27.5292 q^{13} -32.9640 q^{14} -79.6181 q^{16} +108.425 q^{17} +114.186 q^{19} -5.56262 q^{20} +35.2298 q^{22} -144.115 q^{23} -123.549 q^{25} +97.7889 q^{26} +42.8548 q^{28} +138.176 q^{29} -200.217 q^{31} +186.710 q^{32} -385.145 q^{34} -11.1781 q^{35} -22.0567 q^{37} -405.609 q^{38} -14.4708 q^{40} +333.414 q^{41} +54.7616 q^{43} -45.8004 q^{44} +511.921 q^{46} -69.6964 q^{47} -256.883 q^{49} +438.869 q^{50} -127.130 q^{52} +214.839 q^{53} +11.9464 q^{55} +111.484 q^{56} -490.826 q^{58} -586.914 q^{59} -629.512 q^{61} +711.207 q^{62} -26.2851 q^{64} +33.1603 q^{65} -621.632 q^{67} +500.707 q^{68} +39.7067 q^{70} +841.529 q^{71} -222.150 q^{73} +78.3495 q^{74} +527.311 q^{76} -92.0363 q^{77} -66.1137 q^{79} +95.9038 q^{80} -1184.35 q^{82} -569.154 q^{83} -130.603 q^{85} -194.523 q^{86} -119.147 q^{88} +779.215 q^{89} -255.469 q^{91} -665.522 q^{92} +247.574 q^{94} -137.542 q^{95} +349.756 q^{97} +912.496 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−3.55218	−1.25589	−0.627943	−	0.778259i	$-0.716103\pi$
$2$	−3.55218	−1.25589	−0.627943	+	0.778259i	$0.716103\pi$
$3$	0	0
$3$	0	0
$4$	4.61801	0.577251
$5$	−1.20455	−0.107738	−0.0538690	−	0.998548i	$-0.517155\pi$
$5$	−1.20455	−0.107738	−0.0538690	+	0.998548i	$0.517155\pi$
$6$	0	0
$7$	9.27993	0.501069	0.250534	−	0.968108i	$-0.419394\pi$
$7$	9.27993	0.501069	0.250534	+	0.968108i	$0.419394\pi$
$8$	12.0135	0.530925
$9$	0	0
$10$	4.27878	0.135307
$11$	−9.91778	−0.271848	−0.135924	−	0.990719i	$-0.543400\pi$
$11$	−9.91778	−0.271848	−0.135924	+	0.990719i	$0.543400\pi$
$12$	0	0
$13$	−27.5292	−0.587326	−0.293663	−	0.955909i	$-0.594874\pi$
$13$	−27.5292	−0.587326	−0.293663	+	0.955909i	$0.594874\pi$
$14$	−32.9640	−0.629286
$15$	0	0
$16$	−79.6181	−1.24403
$17$	108.425	1.54688	0.773438	−	0.633872i	$-0.218535\pi$
$17$	108.425	1.54688	0.773438	+	0.633872i	$0.218535\pi$
$18$	0	0
$19$	114.186	1.37874	0.689368	−	0.724411i	$-0.257888\pi$
$19$	114.186	1.37874	0.689368	+	0.724411i	$0.257888\pi$
$20$	−5.56262	−0.0621919
$21$	0	0
$22$	35.2298	0.341410
$23$	−144.115	−1.30652	−0.653260	−	0.757134i	$-0.726599\pi$
$23$	−144.115	−1.30652	−0.653260	+	0.757134i	$0.726599\pi$
$24$	0	0
$25$	−123.549	−0.988393
$26$	97.7889	0.737614
$27$	0	0
$28$	42.8548	0.289243
$29$	138.176	0.884780	0.442390	−	0.896823i	$-0.354131\pi$
$29$	138.176	0.884780	0.442390	+	0.896823i	$0.354131\pi$
$30$	0	0
$31$	−200.217	−1.16000	−0.580000	−	0.814616i	$-0.696947\pi$
$31$	−200.217	−1.16000	−0.580000	+	0.814616i	$0.696947\pi$
$32$	186.710	1.03144
$33$	0	0
$34$	−385.145	−1.94270
$35$	−11.1781	−0.0539842
$36$	0	0
$37$	−22.0567	−0.0980027	−0.0490014	−	0.998799i	$-0.515604\pi$
$37$	−22.0567	−0.0980027	−0.0490014	+	0.998799i	$0.515604\pi$
$38$	−405.609	−1.73154
$39$	0	0
$40$	−14.4708	−0.0572008
$41$	333.414	1.27001	0.635006	−	0.772507i	$-0.280998\pi$
$41$	333.414	1.27001	0.635006	+	0.772507i	$0.280998\pi$
$42$	0	0
$43$	54.7616	0.194211	0.0971054	−	0.995274i	$-0.469042\pi$
$43$	54.7616	0.194211	0.0971054	+	0.995274i	$0.469042\pi$
$44$	−45.8004	−0.156924
$45$	0	0
$46$	511.921	1.64084
$47$	−69.6964	−0.216303	−0.108152	−	0.994134i	$-0.534493\pi$
$47$	−69.6964	−0.216303	−0.108152	+	0.994134i	$0.534493\pi$
$48$	0	0
$49$	−256.883	−0.748930
$50$	438.869	1.24131
$51$	0	0
$52$	−127.130	−0.339034
$53$	214.839	0.556800	0.278400	−	0.960465i	$-0.410196\pi$
$53$	214.839	0.556800	0.278400	+	0.960465i	$0.410196\pi$
$54$	0	0
$55$	11.9464	0.0292883
$56$	111.484	0.266030
$57$	0	0
$58$	−490.826	−1.11118
$59$	−586.914	−1.29508	−0.647540	−	0.762031i	$-0.724202\pi$
$59$	−586.914	−1.29508	−0.647540	+	0.762031i	$0.724202\pi$
$60$	0	0
$61$	−629.512	−1.32132	−0.660661	−	0.750684i	$-0.729724\pi$
$61$	−629.512	−1.32132	−0.660661	+	0.750684i	$0.729724\pi$
$62$	711.207	1.45683
$63$	0	0
$64$	−26.2851	−0.0513381
$65$	33.1603	0.0632773
$66$	0	0
$67$	−621.632	−1.13350	−0.566749	−	0.823890i	$-0.691799\pi$
$67$	−621.632	−1.13350	−0.566749	+	0.823890i	$0.691799\pi$
$68$	500.707	0.892936
$69$	0	0
$70$	39.7067	0.0677980
$71$	841.529	1.40664	0.703318	−	0.710875i	$-0.251701\pi$
$71$	841.529	1.40664	0.703318	+	0.710875i	$0.251701\pi$
$72$	0	0
$73$	−222.150	−0.356174	−0.178087	−	0.984015i	$-0.556991\pi$
$73$	−222.150	−0.356174	−0.178087	+	0.984015i	$0.556991\pi$
$74$	78.3495	0.123080
$75$	0	0
$76$	527.311	0.795878
$77$	−92.0363	−0.136214
$78$	0	0
$79$	−66.1137	−0.0941566	−0.0470783	−	0.998891i	$-0.514991\pi$
$79$	−66.1137	−0.0941566	−0.0470783	+	0.998891i	$0.514991\pi$
$80$	95.9038	0.134030
$81$	0	0
$82$	−1184.35	−1.59499
$83$	−569.154	−0.752684	−0.376342	−	0.926481i	$-0.622818\pi$
$83$	−569.154	−0.752684	−0.376342	+	0.926481i	$0.622818\pi$
$84$	0	0
$85$	−130.603	−0.166657
$86$	−194.523	−0.243907
$87$	0	0
$88$	−119.147	−0.144331
$89$	779.215	0.928052	0.464026	−	0.885822i	$-0.346404\pi$
$89$	779.215	0.928052	0.464026	+	0.885822i	$0.346404\pi$
$90$	0	0
$91$	−255.469	−0.294291
$92$	−665.522	−0.754190
$93$	0	0
$94$	247.574	0.271653
$95$	−137.542	−0.148542
$96$	0	0
$97$	349.756	0.366106	0.183053	−	0.983103i	$-0.441402\pi$
$97$	349.756	0.366106	0.183053	+	0.983103i	$0.441402\pi$
$98$	912.496	0.940571
$99$	0	0
$100$	−570.551	−0.570551
$101$	−952.777	−0.938661	−0.469331	−	0.883022i	$-0.655505\pi$
$101$	−952.777	−0.938661	−0.469331	+	0.883022i	$0.655505\pi$
$102$	0	0
$103$	540.883	0.517425	0.258712	−	0.965954i	$-0.416702\pi$
$103$	540.883	0.517425	0.258712	+	0.965954i	$0.416702\pi$
$104$	−330.721	−0.311826
$105$	0	0
$106$	−763.148	−0.699278
$107$	1734.38	1.56700	0.783500	−	0.621392i	$-0.213433\pi$
$107$	1734.38	1.56700	0.783500	+	0.621392i	$0.213433\pi$
$108$	0	0
$109$	−1670.08	−1.46756	−0.733781	−	0.679386i	$-0.762246\pi$
$109$	−1670.08	−1.46756	−0.733781	+	0.679386i	$0.762246\pi$
$110$	−42.4360	−0.0367828
$111$	0	0
$112$	−738.850	−0.623346
$113$	−1031.54	−0.858757	−0.429379	−	0.903125i	$-0.641267\pi$
$113$	−1031.54	−0.858757	−0.429379	+	0.903125i	$0.641267\pi$
$114$	0	0
$115$	173.593	0.140762
$116$	638.097	0.510740
$117$	0	0
$118$	2084.83	1.62647
$119$	1006.17	0.775092
$120$	0	0
$121$	−1232.64	−0.926099
$122$	2236.14	1.65943
$123$	0	0
$124$	−924.603	−0.669611
$125$	299.389	0.214226
$126$	0	0
$127$	2118.93	1.48051	0.740255	−	0.672327i	$-0.234705\pi$
$127$	2118.93	1.48051	0.740255	+	0.672327i	$0.234705\pi$
$128$	−1400.31	−0.966964
$129$	0	0
$130$	−117.791	−0.0794691
$131$	1710.84	1.14104	0.570521	−	0.821283i	$-0.306741\pi$
$131$	1710.84	1.14104	0.570521	+	0.821283i	$0.306741\pi$
$132$	0	0
$133$	1059.64	0.690842
$134$	2208.15	1.42355
$135$	0	0
$136$	1302.56	0.821275
$137$	−106.979	−0.0667142	−0.0333571	−	0.999443i	$-0.510620\pi$
$137$	−106.979	−0.0667142	−0.0333571	+	0.999443i	$0.510620\pi$
$138$	0	0
$139$	2111.44	1.28842	0.644208	−	0.764850i	$-0.277187\pi$
$139$	2111.44	1.28842	0.644208	+	0.764850i	$0.277187\pi$
$140$	−51.6207	−0.0311624
$141$	0	0
$142$	−2989.27	−1.76658
$143$	273.029	0.159663
$144$	0	0
$145$	−166.439	−0.0953244
$146$	789.117	0.447314
$147$	0	0
$148$	−101.858	−0.0565722
$149$	−1296.43	−0.712806	−0.356403	−	0.934332i	$-0.615997\pi$
$149$	−1296.43	−0.712806	−0.356403	+	0.934332i	$0.615997\pi$
$150$	0	0
$151$	645.324	0.347786	0.173893	−	0.984765i	$-0.444365\pi$
$151$	645.324	0.347786	0.173893	+	0.984765i	$0.444365\pi$
$152$	1371.76	0.732005
$153$	0	0
$154$	326.930	0.171070
$155$	241.171	0.124976
$156$	0	0
$157$	2114.28	1.07476	0.537382	−	0.843339i	$-0.319413\pi$
$157$	2114.28	1.07476	0.537382	+	0.843339i	$0.319413\pi$
$158$	234.848	0.118250
$159$	0	0
$160$	−224.902	−0.111125
$161$	−1337.37	−0.654656
$162$	0	0
$163$	427.645	0.205495	0.102748	−	0.994707i	$-0.467237\pi$
$163$	427.645	0.205495	0.102748	+	0.994707i	$0.467237\pi$
$164$	1539.71	0.733116
$165$	0	0
$166$	2021.74	0.945286
$167$	−1840.32	−0.852746	−0.426373	−	0.904548i	$-0.640209\pi$
$167$	−1840.32	−0.852746	−0.426373	+	0.904548i	$0.640209\pi$
$168$	0	0
$169$	−1439.14	−0.655049
$170$	463.926	0.209303
$171$	0	0
$172$	252.890	0.112108
$173$	483.983	0.212697	0.106348	−	0.994329i	$-0.466084\pi$
$173$	483.983	0.212697	0.106348	+	0.994329i	$0.466084\pi$
$174$	0	0
$175$	−1146.53	−0.495253
$176$	789.634	0.338187
$177$	0	0
$178$	−2767.92	−1.16553
$179$	−3197.27	−1.33505	−0.667527	−	0.744585i	$-0.732647\pi$
$179$	−3197.27	−1.33505	−0.667527	+	0.744585i	$0.732647\pi$
$180$	0	0
$181$	2076.04	0.852548	0.426274	−	0.904594i	$-0.359826\pi$
$181$	2076.04	0.852548	0.426274	+	0.904594i	$0.359826\pi$
$182$	907.473	0.369596
$183$	0	0
$184$	−1731.31	−0.693663
$185$	26.5684	0.0105586
$186$	0	0
$187$	−1075.33	−0.420514
$188$	−321.859	−0.124861
$189$	0	0
$190$	488.575	0.186552
$191$	1976.64	0.748820	0.374410	−	0.927263i	$-0.377845\pi$
$191$	1976.64	0.748820	0.374410	+	0.927263i	$0.377845\pi$
$192$	0	0
$193$	82.8973	0.0309175	0.0154588	−	0.999881i	$-0.495079\pi$
$193$	82.8973	0.0309175	0.0154588	+	0.999881i	$0.495079\pi$
$194$	−1242.40	−0.459788
$195$	0	0
$196$	−1186.29	−0.432321
$197$	−669.025	−0.241960	−0.120980	−	0.992655i	$-0.538604\pi$
$197$	−669.025	−0.241960	−0.120980	+	0.992655i	$0.538604\pi$
$198$	0	0
$199$	−3167.26	−1.12825	−0.564123	−	0.825691i	$-0.690785\pi$
$199$	−3167.26	−1.12825	−0.564123	+	0.825691i	$0.690785\pi$
$200$	−1484.25	−0.524762
$201$	0	0
$202$	3384.44	1.17885
$203$	1282.26	0.443336
$204$	0	0
$205$	−401.613	−0.136829
$206$	−1921.32	−0.649827
$207$	0	0
$208$	2191.82	0.730652
$209$	−1132.47	−0.374806
$210$	0	0
$211$	5803.33	1.89345	0.946724	−	0.322047i	$-0.104371\pi$
$211$	5803.33	1.89345	0.946724	+	0.322047i	$0.104371\pi$
$212$	992.129	0.321414
$213$	0	0
$214$	−6160.84	−1.96797
$215$	−65.9630	−0.0209239
$216$	0	0
$217$	−1858.00	−0.581240
$218$	5932.41	1.84309
$219$	0	0
$220$	55.1688	0.0169067
$221$	−2984.85	−0.908520
$222$	0	0
$223$	−4046.52	−1.21514	−0.607568	−	0.794268i	$-0.707855\pi$
$223$	−4046.52	−1.21514	−0.607568	+	0.794268i	$0.707855\pi$
$224$	1732.66	0.516822
$225$	0	0
$226$	3664.24	1.07850
$227$	2076.71	0.607207	0.303603	−	0.952798i	$-0.401810\pi$
$227$	2076.71	0.607207	0.303603	+	0.952798i	$0.401810\pi$
$228$	0	0
$229$	3252.98	0.938703	0.469352	−	0.883011i	$-0.344488\pi$
$229$	3252.98	0.938703	0.469352	+	0.883011i	$0.344488\pi$
$230$	−616.634	−0.176781
$231$	0	0
$232$	1659.97	0.469751
$233$	−343.641	−0.0966208	−0.0483104	−	0.998832i	$-0.515384\pi$
$233$	−343.641	−0.0966208	−0.0483104	+	0.998832i	$0.515384\pi$
$234$	0	0
$235$	83.9526	0.0233041
$236$	−2710.38	−0.747587
$237$	0	0
$238$	−3574.12	−0.973427
$239$	−239.000	−0.0646846
$239$	−239.000	−0.0646846
$240$	0	0
$241$	6706.98	1.79268	0.896338	−	0.443372i	$-0.146218\pi$
$241$	6706.98	1.79268	0.896338	+	0.443372i	$0.146218\pi$
$242$	4378.56	1.16308
$243$	0	0
$244$	−2907.09	−0.762735
$245$	309.428	0.0806883
$246$	0	0
$247$	−3143.44	−0.809767
$248$	−2405.29	−0.615872
$249$	0	0
$250$	−1063.49	−0.269043
$251$	−4527.43	−1.13852	−0.569260	−	0.822157i	$-0.692770\pi$
$251$	−4527.43	−1.13852	−0.569260	+	0.822157i	$0.692770\pi$
$252$	0	0
$253$	1429.30	0.355174
$254$	−7526.83	−1.85935
$255$	0	0
$256$	5184.45	1.26574
$257$	−3437.20	−0.834267	−0.417134	−	0.908845i	$-0.636965\pi$
$257$	−3437.20	−0.834267	−0.417134	+	0.908845i	$0.636965\pi$
$258$	0	0
$259$	−204.685	−0.0491061
$260$	153.134	0.0365269
$261$	0	0
$262$	−6077.20	−1.43302
$263$	−2716.75	−0.636967	−0.318483	−	0.947928i	$-0.603173\pi$
$263$	−2716.75	−0.636967	−0.318483	+	0.947928i	$0.603173\pi$
$264$	0	0
$265$	−258.784	−0.0599886
$266$	−3764.02	−0.867619
$267$	0	0
$268$	−2870.70	−0.654314
$269$	−4788.87	−1.08544	−0.542718	−	0.839915i	$-0.682605\pi$
$269$	−4788.87	−1.08544	−0.542718	+	0.839915i	$0.682605\pi$
$270$	0	0
$271$	−1320.61	−0.296020	−0.148010	−	0.988986i	$-0.547287\pi$
$271$	−1320.61	−0.296020	−0.148010	+	0.988986i	$0.547287\pi$
$272$	−8632.58	−1.92436
$273$	0	0
$274$	380.010	0.0837855
$275$	1225.33	0.268692
$276$	0	0
$277$	−2197.76	−0.476716	−0.238358	−	0.971177i	$-0.576609\pi$
$277$	−2197.76	−0.476716	−0.238358	+	0.971177i	$0.576609\pi$
$278$	−7500.21	−1.61810
$279$	0	0
$280$	−134.288	−0.0286615
$281$	633.194	0.134424	0.0672120	−	0.997739i	$-0.478590\pi$
$281$	633.194	0.134424	0.0672120	+	0.997739i	$0.478590\pi$
$282$	0	0
$283$	−8058.94	−1.69277	−0.846386	−	0.532570i	$-0.821226\pi$
$283$	−8058.94	−1.69277	−0.846386	+	0.532570i	$0.821226\pi$
$284$	3886.19	0.811982
$285$	0	0
$286$	−969.848	−0.200519
$287$	3094.05	0.636363
$288$	0	0
$289$	6842.95	1.39283
$290$	591.223	0.119717
$291$	0	0
$292$	−1025.89	−0.205602
$293$	6554.25	1.30684	0.653419	−	0.756997i	$-0.273334\pi$
$293$	6554.25	1.30684	0.653419	+	0.756997i	$0.273334\pi$
$294$	0	0
$295$	706.967	0.139529
$296$	−264.977	−0.0520320
$297$	0	0
$298$	4605.17	0.895203
$299$	3967.36	0.767352
$300$	0	0
$301$	508.184	0.0973130
$302$	−2292.31	−0.436780
$303$	0	0
$304$	−9091.25	−1.71519
$305$	758.277	0.142357
$306$	0	0
$307$	−4634.60	−0.861597	−0.430799	−	0.902448i	$-0.641768\pi$
$307$	−4634.60	−0.861597	−0.430799	+	0.902448i	$0.641768\pi$
$308$	−425.024	−0.0786299
$309$	0	0
$310$	−856.683	−0.156956
$311$	10594.5	1.93170	0.965848	−	0.259110i	$-0.0834293\pi$
$311$	10594.5	1.93170	0.965848	+	0.259110i	$0.0834293\pi$
$312$	0	0
$313$	−3210.95	−0.579851	−0.289926	−	0.957049i	$-0.593631\pi$
$313$	−3210.95	−0.579851	−0.289926	+	0.957049i	$0.593631\pi$
$314$	−7510.32	−1.34978
$315$	0	0
$316$	−305.314	−0.0543520
$317$	3091.57	0.547759	0.273880	−	0.961764i	$-0.411693\pi$
$317$	3091.57	0.547759	0.273880	+	0.961764i	$0.411693\pi$
$318$	0	0
$319$	−1370.40	−0.240525
$320$	31.6617	0.00553106
$321$	0	0
$322$	4750.59	0.822174
$323$	12380.6	2.13274
$324$	0	0
$325$	3401.21	0.580508
$326$	−1519.07	−0.258079
$327$	0	0
$328$	4005.45	0.674280
$329$	−646.777	−0.108383
$330$	0	0
$331$	−2315.94	−0.384579	−0.192290	−	0.981338i	$-0.561591\pi$
$331$	−2315.94	−0.384579	−0.192290	+	0.981338i	$0.561591\pi$
$332$	−2628.36	−0.434488
$333$	0	0
$334$	6537.17	1.07095
$335$	748.786	0.122121
$336$	0	0
$337$	−4000.94	−0.646721	−0.323360	−	0.946276i	$-0.604813\pi$
$337$	−4000.94	−0.646721	−0.323360	+	0.946276i	$0.604813\pi$
$338$	5112.10	0.822667
$339$	0	0
$340$	−603.126	−0.0962032
$341$	1985.71	0.315343
$342$	0	0
$343$	−5566.87	−0.876334
$344$	657.876	0.103111
$345$	0	0
$346$	−1719.20	−0.267123
$347$	−7098.08	−1.09811	−0.549056	−	0.835785i	$-0.685013\pi$
$347$	−7098.08	−1.09811	−0.549056	+	0.835785i	$0.685013\pi$
$348$	0	0
$349$	741.183	0.113681	0.0568404	−	0.998383i	$-0.481897\pi$
$349$	741.183	0.113681	0.0568404	+	0.998383i	$0.481897\pi$
$350$	4072.67	0.621981
$351$	0	0
$352$	−1851.75	−0.280394
$353$	−5787.72	−0.872661	−0.436330	−	0.899787i	$-0.643722\pi$
$353$	−5787.72	−0.872661	−0.436330	+	0.899787i	$0.643722\pi$
$354$	0	0
$355$	−1013.66	−0.151548
$356$	3598.42	0.535719
$357$	0	0
$358$	11357.3	1.67668
$359$	−4048.29	−0.595155	−0.297577	−	0.954698i	$-0.596179\pi$
$359$	−4048.29	−0.595155	−0.297577	+	0.954698i	$0.596179\pi$
$360$	0	0
$361$	6179.38	0.900915
$362$	−7374.49	−1.07070
$363$	0	0
$364$	−1179.76	−0.169880
$365$	267.590	0.0383735
$366$	0	0
$367$	1693.86	0.240923	0.120461	−	0.992718i	$-0.461563\pi$
$367$	1693.86	0.240923	0.120461	+	0.992718i	$0.461563\pi$
$368$	11474.1	1.62535
$369$	0	0
$370$	−94.3757	−0.0132604
$371$	1993.69	0.278995
$372$	0	0
$373$	−9723.35	−1.34975	−0.674874	−	0.737933i	$-0.735802\pi$
$373$	−9723.35	−1.34975	−0.674874	+	0.737933i	$0.735802\pi$
$374$	3819.78	0.528119
$375$	0	0
$376$	−837.294	−0.114841
$377$	−3803.87	−0.519654
$378$	0	0
$379$	−1899.64	−0.257462	−0.128731	−	0.991680i	$-0.541090\pi$
$379$	−1899.64	−0.257462	−0.128731	+	0.991680i	$0.541090\pi$
$380$	−635.171	−0.0857463
$381$	0	0
$382$	−7021.39	−0.940433
$383$	−1398.21	−0.186541	−0.0932705	−	0.995641i	$-0.529732\pi$
$383$	−1398.21	−0.186541	−0.0932705	+	0.995641i	$0.529732\pi$
$384$	0	0
$385$	110.862	0.0146755
$386$	−294.467	−0.0388289
$387$	0	0
$388$	1615.18	0.211335
$389$	1717.79	0.223895	0.111948	−	0.993714i	$-0.464291\pi$
$389$	1717.79	0.223895	0.111948	+	0.993714i	$0.464291\pi$
$390$	0	0
$391$	−15625.6	−2.02102
$392$	−3086.05	−0.397625
$393$	0	0
$394$	2376.50	0.303874
$395$	79.6371	0.0101443
$396$	0	0
$397$	−4753.52	−0.600938	−0.300469	−	0.953792i	$-0.597143\pi$
$397$	−4753.52	−0.600938	−0.300469	+	0.953792i	$0.597143\pi$
$398$	11250.7	1.41695
$399$	0	0
$400$	9836.74	1.22959
$401$	−812.614	−0.101197	−0.0505985	−	0.998719i	$-0.516113\pi$
$401$	−812.614	−0.101197	−0.0505985	+	0.998719i	$0.516113\pi$
$402$	0	0
$403$	5511.81	0.681298
$404$	−4399.93	−0.541843
$405$	0	0
$406$	−4554.83	−0.556779
$407$	218.754	0.0266418
$408$	0	0
$409$	−9985.23	−1.20718	−0.603592	−	0.797294i	$-0.706264\pi$
$409$	−9985.23	−1.20718	−0.603592	+	0.797294i	$0.706264\pi$
$410$	1426.60	0.171841
$411$	0	0
$412$	2497.80	0.298684
$413$	−5446.52	−0.648925
$414$	0	0
$415$	685.573	0.0810927
$416$	−5139.99	−0.605790
$417$	0	0
$418$	4022.74	0.470714
$419$	−12660.8	−1.47618	−0.738089	−	0.674704i	$-0.764271\pi$
$419$	−12660.8	−1.47618	−0.738089	+	0.674704i	$0.764271\pi$
$420$	0	0
$421$	2483.44	0.287495	0.143748	−	0.989614i	$-0.454085\pi$
$421$	2483.44	0.287495	0.143748	+	0.989614i	$0.454085\pi$
$422$	−20614.5	−2.37796
$423$	0	0
$424$	2580.96	0.295619
$425$	−13395.8	−1.52892
$426$	0	0
$427$	−5841.82	−0.662074
$428$	8009.39	0.904552
$429$	0	0
$430$	234.313	0.0262780
$431$	−12824.3	−1.43324	−0.716620	−	0.697464i	$-0.754312\pi$
$431$	−12824.3	−1.43324	−0.716620	+	0.697464i	$0.754312\pi$
$432$	0	0
$433$	−16254.5	−1.80402	−0.902012	−	0.431711i	$-0.857910\pi$
$433$	−16254.5	−1.80402	−0.902012	+	0.431711i	$0.857910\pi$
$434$	6599.95	0.729971
$435$	0	0
$436$	−7712.42	−0.847152
$437$	−16455.8	−1.80135
$438$	0	0
$439$	4356.34	0.473615	0.236807	−	0.971557i	$-0.423899\pi$
$439$	4356.34	0.473615	0.236807	+	0.971557i	$0.423899\pi$
$440$	143.518	0.0155499
$441$	0	0
$442$	10602.7	1.14100
$443$	13850.7	1.48547	0.742737	−	0.669583i	$-0.233527\pi$
$443$	13850.7	1.48547	0.742737	+	0.669583i	$0.233527\pi$
$444$	0	0
$445$	−938.602	−0.0999865
$446$	14374.0	1.52607
$447$	0	0
$448$	−243.924	−0.0257239
$449$	−5633.45	−0.592113	−0.296057	−	0.955170i	$-0.595672\pi$
$449$	−5633.45	−0.592113	−0.296057	+	0.955170i	$0.595672\pi$
$450$	0	0
$451$	−3306.72	−0.345249
$452$	−4763.68	−0.495719
$453$	0	0
$454$	−7376.85	−0.762583
$455$	307.725	0.0317063
$456$	0	0
$457$	−19187.9	−1.96405	−0.982024	−	0.188754i	$-0.939555\pi$
$457$	−19187.9	−1.96405	−0.982024	+	0.188754i	$0.939555\pi$
$458$	−11555.2	−1.17891
$459$	0	0
$460$	801.654	0.0812550
$461$	3761.53	0.380025	0.190013	−	0.981782i	$-0.439147\pi$
$461$	3761.53	0.380025	0.190013	+	0.981782i	$0.439147\pi$
$462$	0	0
$463$	−11913.2	−1.19579	−0.597896	−	0.801573i	$-0.703997\pi$
$463$	−11913.2	−1.19579	−0.597896	+	0.801573i	$0.703997\pi$
$464$	−11001.3	−1.10069
$465$	0	0
$466$	1220.67	0.121345
$467$	−11139.4	−1.10379	−0.551894	−	0.833914i	$-0.686095\pi$
$467$	−11139.4	−1.10379	−0.551894	+	0.833914i	$0.686095\pi$
$468$	0	0
$469$	−5768.70	−0.567961
$470$	−298.215	−0.0292673
$471$	0	0
$472$	−7050.87	−0.687590
$473$	−543.114	−0.0527957
$474$	0	0
$475$	−14107.5	−1.36273
$476$	4646.53	0.447423
$477$	0	0
$478$	848.972	0.0812366
$479$	−6363.89	−0.607043	−0.303521	−	0.952825i	$-0.598162\pi$
$479$	−6363.89	−0.607043	−0.303521	+	0.952825i	$0.598162\pi$
$480$	0	0
$481$	607.204	0.0575595
$482$	−23824.4	−2.25140
$483$	0	0
$484$	−5692.33	−0.534592
$485$	−421.298	−0.0394436
$486$	0	0
$487$	8103.59	0.754022	0.377011	−	0.926209i	$-0.376952\pi$
$487$	8103.59	0.754022	0.377011	+	0.926209i	$0.376952\pi$
$488$	−7562.61	−0.701523
$489$	0	0
$490$	−1099.14	−0.101335
$491$	−12836.2	−1.17982	−0.589909	−	0.807470i	$-0.700836\pi$
$491$	−12836.2	−1.17982	−0.589909	+	0.807470i	$0.700836\pi$
$492$	0	0
$493$	14981.7	1.36864
$494$	11166.1	1.01698
$495$	0	0
$496$	15940.9	1.44308
$497$	7809.33	0.704822
$498$	0	0
$499$	−5775.82	−0.518159	−0.259080	−	0.965856i	$-0.583419\pi$
$499$	−5775.82	−0.518159	−0.259080	+	0.965856i	$0.583419\pi$
$500$	1382.58	0.123662
$501$	0	0
$502$	16082.3	1.42985
$503$	−17743.4	−1.57284	−0.786421	−	0.617691i	$-0.788068\pi$
$503$	−17743.4	−1.57284	−0.786421	+	0.617691i	$0.788068\pi$
$504$	0	0
$505$	1147.67	0.101130
$506$	−5077.12	−0.446058
$507$	0	0
$508$	9785.24	0.854626
$509$	13158.7	1.14587	0.572936	−	0.819600i	$-0.305804\pi$
$509$	13158.7	1.14587	0.572936	+	0.819600i	$0.305804\pi$
$510$	0	0
$511$	−2061.54	−0.178468
$512$	−7213.62	−0.622656
$513$	0	0
$514$	12209.6	1.04775
$515$	−651.519	−0.0557464
$516$	0	0
$517$	691.233	0.0588016
$518$	727.077	0.0616717
$519$	0	0
$520$	398.369	0.0335955
$521$	−20311.1	−1.70796	−0.853978	−	0.520309i	$-0.825817\pi$
$521$	−20311.1	−1.70796	−0.853978	+	0.520309i	$0.825817\pi$
$522$	0	0
$523$	1677.52	0.140254	0.0701269	−	0.997538i	$-0.477660\pi$
$523$	1677.52	0.140254	0.0701269	+	0.997538i	$0.477660\pi$
$524$	7900.66	0.658667
$525$	0	0
$526$	9650.41	0.799958
$527$	−21708.5	−1.79438
$528$	0	0
$529$	8601.99	0.706994
$530$	919.248	0.0753388
$531$	0	0
$532$	4893.40	0.398789
$533$	−9178.62	−0.745910
$534$	0	0
$535$	−2089.15	−0.168825
$536$	−7467.95	−0.601802
$537$	0	0
$538$	17010.9	1.36318
$539$	2547.71	0.203595
$540$	0	0
$541$	−18649.8	−1.48210	−0.741050	−	0.671449i	$-0.765672\pi$
$541$	−18649.8	−1.48210	−0.741050	+	0.671449i	$0.765672\pi$
$542$	4691.05	0.371767
$543$	0	0
$544$	20244.1	1.59551
$545$	2011.69	0.158112
$546$	0	0
$547$	−7956.40	−0.621921	−0.310961	−	0.950423i	$-0.600651\pi$
$547$	−7956.40	−0.621921	−0.310961	+	0.950423i	$0.600651\pi$
$548$	−494.031	−0.0385109
$549$	0	0
$550$	−4352.61	−0.337447
$551$	15777.7	1.21988
$552$	0	0
$553$	−613.530	−0.0471790
$554$	7806.84	0.598702
$555$	0	0
$556$	9750.64	0.743740
$557$	407.459	0.0309957	0.0154978	−	0.999880i	$-0.495067\pi$
$557$	407.459	0.0309957	0.0154978	+	0.999880i	$0.495067\pi$
$558$	0	0
$559$	−1507.54	−0.114065
$560$	889.980	0.0671581
$561$	0	0
$562$	−2249.22	−0.168821
$563$	−15181.3	−1.13644	−0.568222	−	0.822876i	$-0.692368\pi$
$563$	−15181.3	−1.13644	−0.568222	+	0.822876i	$0.692368\pi$
$564$	0	0
$565$	1242.55	0.0925209
$566$	28626.8	2.12593
$567$	0	0
$568$	10109.7	0.746818
$569$	18208.1	1.34152	0.670759	−	0.741675i	$-0.265968\pi$
$569$	18208.1	1.34152	0.670759	+	0.741675i	$0.265968\pi$
$570$	0	0
$571$	−12564.3	−0.920838	−0.460419	−	0.887702i	$-0.652301\pi$
$571$	−12564.3	−0.920838	−0.460419	+	0.887702i	$0.652301\pi$
$572$	1260.85	0.0921657
$573$	0	0
$574$	−10990.7	−0.799200
$575$	17805.2	1.29135
$576$	0	0
$577$	−3338.46	−0.240870	−0.120435	−	0.992721i	$-0.538429\pi$
$577$	−3338.46	−0.240870	−0.120435	+	0.992721i	$0.538429\pi$
$578$	−24307.4	−1.74923
$579$	0	0
$580$	−768.619	−0.0550261
$581$	−5281.71	−0.377147
$582$	0	0
$583$	−2130.73	−0.151365
$584$	−2668.79	−0.189101
$585$	0	0
$586$	−23281.9	−1.64124
$587$	−4420.26	−0.310807	−0.155403	−	0.987851i	$-0.549668\pi$
$587$	−4420.26	−0.310807	−0.155403	+	0.987851i	$0.549668\pi$
$588$	0	0
$589$	−22861.9	−1.59933
$590$	−2511.28	−0.175233
$591$	0	0
$592$	1756.11	0.121919
$593$	16260.3	1.12602	0.563011	−	0.826450i	$-0.309643\pi$
$593$	16260.3	1.12602	0.563011	+	0.826450i	$0.309643\pi$
$594$	0	0
$595$	−1211.99	−0.0835069
$596$	−5986.95	−0.411468
$597$	0	0
$598$	−14092.8	−0.963708
$599$	9718.13	0.662892	0.331446	−	0.943474i	$-0.392464\pi$
$599$	9718.13	0.662892	0.331446	+	0.943474i	$0.392464\pi$
$600$	0	0
$601$	15963.2	1.08345	0.541725	−	0.840556i	$-0.317771\pi$
$601$	15963.2	1.08345	0.541725	+	0.840556i	$0.317771\pi$
$602$	−1805.16	−0.122214
$603$	0	0
$604$	2980.11	0.200760
$605$	1484.77	0.0997761
$606$	0	0
$607$	1590.91	0.106380	0.0531902	−	0.998584i	$-0.483061\pi$
$607$	1590.91	0.106380	0.0531902	+	0.998584i	$0.483061\pi$
$608$	21319.7	1.42208
$609$	0	0
$610$	−2693.54	−0.178784
$611$	1918.69	0.127041
$612$	0	0
$613$	8009.91	0.527761	0.263880	−	0.964555i	$-0.414998\pi$
$613$	8009.91	0.527761	0.263880	+	0.964555i	$0.414998\pi$
$614$	16462.9	1.08207
$615$	0	0
$616$	−1105.67	−0.0723195
$617$	14776.6	0.964154	0.482077	−	0.876129i	$-0.339882\pi$
$617$	14776.6	0.964154	0.482077	+	0.876129i	$0.339882\pi$
$618$	0	0
$619$	−29551.6	−1.91886	−0.959432	−	0.281939i	$-0.909022\pi$
$619$	−29551.6	−1.91886	−0.959432	+	0.281939i	$0.909022\pi$
$620$	1113.73	0.0721426
$621$	0	0
$622$	−37633.5	−2.42599
$623$	7231.06	0.465018
$624$	0	0
$625$	15083.0	0.965312
$626$	11405.9	0.728227
$627$	0	0
$628$	9763.77	0.620409
$629$	−2391.50	−0.151598
$630$	0	0
$631$	15431.5	0.973561	0.486781	−	0.873524i	$-0.338171\pi$
$631$	15431.5	0.973561	0.486781	+	0.873524i	$0.338171\pi$
$632$	−794.254	−0.0499901
$633$	0	0
$634$	−10981.8	−0.687923
$635$	−2552.35	−0.159507
$636$	0	0
$637$	7071.79	0.439866
$638$	4867.90	0.302072
$639$	0	0
$640$	1686.75	0.104179
$641$	−27751.8	−1.71003	−0.855016	−	0.518602i	$-0.826453\pi$
$641$	−27751.8	−1.71003	−0.855016	+	0.518602i	$0.826453\pi$
$642$	0	0
$643$	−23403.5	−1.43537	−0.717687	−	0.696366i	$-0.754799\pi$
$643$	−23403.5	−1.43537	−0.717687	+	0.696366i	$0.754799\pi$
$644$	−6176.00	−0.377901
$645$	0	0
$646$	−43978.1	−2.67847
$647$	6547.18	0.397830	0.198915	−	0.980017i	$-0.436258\pi$
$647$	6547.18	0.397830	0.198915	+	0.980017i	$0.436258\pi$
$648$	0	0
$649$	5820.89	0.352064
$650$	−12081.7	−0.729052
$651$	0	0
$652$	1974.87	0.118622
$653$	15829.1	0.948607	0.474303	−	0.880361i	$-0.342700\pi$
$653$	15829.1	0.948607	0.474303	+	0.880361i	$0.342700\pi$
$654$	0	0
$655$	−2060.78	−0.122934
$656$	−26545.8	−1.57994
$657$	0	0
$658$	2297.47	0.136117
$659$	−2409.78	−0.142446	−0.0712230	−	0.997460i	$-0.522690\pi$
$659$	−2409.78	−0.142446	−0.0712230	+	0.997460i	$0.522690\pi$
$660$	0	0
$661$	22705.0	1.33604	0.668019	−	0.744144i	$-0.267143\pi$
$661$	22705.0	1.33604	0.668019	+	0.744144i	$0.267143\pi$
$662$	8226.66	0.482988
$663$	0	0
$664$	−6837.50	−0.399618
$665$	−1276.38	−0.0744300
$666$	0	0
$667$	−19913.1	−1.15598
$668$	−8498.63	−0.492248
$669$	0	0
$670$	−2659.82	−0.153370
$671$	6243.36	0.359198
$672$	0	0
$673$	−28529.7	−1.63409	−0.817044	−	0.576576i	$-0.804389\pi$
$673$	−28529.7	−1.63409	−0.817044	+	0.576576i	$0.804389\pi$
$674$	14212.1	0.812208
$675$	0	0
$676$	−6645.97	−0.378128
$677$	21641.9	1.22861	0.614304	−	0.789070i	$-0.289437\pi$
$677$	21641.9	1.22861	0.614304	+	0.789070i	$0.289437\pi$
$678$	0	0
$679$	3245.71	0.183444
$680$	−1568.99	−0.0884825
$681$	0	0
$682$	−7053.59	−0.396035
$683$	3423.99	0.191823	0.0959117	−	0.995390i	$-0.469423\pi$
$683$	3423.99	0.191823	0.0959117	+	0.995390i	$0.469423\pi$
$684$	0	0
$685$	128.862	0.00718766
$686$	19774.5	1.10058
$687$	0	0
$688$	−4360.01	−0.241605
$689$	−5914.35	−0.327023
$690$	0	0
$691$	3581.90	0.197195	0.0985975	−	0.995127i	$-0.468564\pi$
$691$	3581.90	0.197195	0.0985975	+	0.995127i	$0.468564\pi$
$692$	2235.04	0.122780
$693$	0	0
$694$	25213.7	1.37911
$695$	−2543.33	−0.138811
$696$	0	0
$697$	36150.3	1.96455
$698$	−2632.82	−0.142770
$699$	0	0
$700$	−5294.67	−0.285885
$701$	−25176.1	−1.35647	−0.678237	−	0.734843i	$-0.737256\pi$
$701$	−25176.1	−1.35647	−0.678237	+	0.734843i	$0.737256\pi$
$702$	0	0
$703$	−2518.56	−0.135120
$704$	260.690	0.0139561
$705$	0	0
$706$	20559.0	1.09596
$707$	−8841.70	−0.470334
$708$	0	0
$709$	−25615.8	−1.35687	−0.678436	−	0.734659i	$-0.737342\pi$
$709$	−25615.8	−1.35687	−0.678436	+	0.734659i	$0.737342\pi$
$710$	3600.72	0.190327
$711$	0	0
$712$	9361.06	0.492726
$713$	28854.1	1.51556
$714$	0	0
$715$	−328.876	−0.0172018
$716$	−14765.0	−0.770662
$717$	0	0
$718$	14380.3	0.747447
$719$	25820.6	1.33928	0.669642	−	0.742684i	$-0.266448\pi$
$719$	25820.6	1.33928	0.669642	+	0.742684i	$0.266448\pi$
$720$	0	0
$721$	5019.35	0.259266
$722$	−21950.3	−1.13145
$723$	0	0
$724$	9587.20	0.492134
$725$	−17071.5	−0.874509
$726$	0	0
$727$	28756.7	1.46703	0.733513	−	0.679676i	$-0.237879\pi$
$727$	28756.7	1.46703	0.733513	+	0.679676i	$0.237879\pi$
$728$	−3069.07	−0.156246
$729$	0	0
$730$	−950.530	−0.0481927
$731$	5937.52	0.300420
$732$	0	0
$733$	24819.8	1.25067	0.625334	−	0.780357i	$-0.284963\pi$
$733$	24819.8	1.25067	0.625334	+	0.780357i	$0.284963\pi$
$734$	−6016.89	−0.302571
$735$	0	0
$736$	−26907.7	−1.34760
$737$	6165.21	0.308139
$738$	0	0
$739$	−37701.1	−1.87667	−0.938335	−	0.345727i	$-0.887632\pi$
$739$	−37701.1	−1.87667	−0.938335	+	0.345727i	$0.887632\pi$
$740$	122.693	0.00609498
$741$	0	0
$742$	−7081.95	−0.350386
$743$	27382.2	1.35202	0.676012	−	0.736890i	$-0.263707\pi$
$743$	27382.2	1.35202	0.676012	+	0.736890i	$0.263707\pi$
$744$	0	0
$745$	1561.62	0.0767963
$746$	34539.1	1.69513
$747$	0	0
$748$	−4965.90	−0.242742
$749$	16094.9	0.785175
$750$	0	0
$751$	−11798.6	−0.573284	−0.286642	−	0.958038i	$-0.592539\pi$
$751$	−11798.6	−0.573284	−0.286642	+	0.958038i	$0.592539\pi$
$752$	5549.09	0.269088
$753$	0	0
$754$	13512.1	0.652626
$755$	−777.324	−0.0374698
$756$	0	0
$757$	−30011.5	−1.44093	−0.720466	−	0.693490i	$-0.756072\pi$
$757$	−30011.5	−1.44093	−0.720466	+	0.693490i	$0.756072\pi$
$758$	6747.87	0.323343
$759$	0	0
$760$	−1652.36	−0.0788648
$761$	16683.3	0.794704	0.397352	−	0.917666i	$-0.369929\pi$
$761$	16683.3	0.794704	0.397352	+	0.917666i	$0.369929\pi$
$762$	0	0
$763$	−15498.2	−0.735349
$764$	9128.14	0.432257
$765$	0	0
$766$	4966.70	0.234274
$767$	16157.3	0.760634
$768$	0	0
$769$	−16399.5	−0.769026	−0.384513	−	0.923120i	$-0.625631\pi$
$769$	−16399.5	−0.769026	−0.384513	+	0.923120i	$0.625631\pi$
$770$	−393.803	−0.0184307
$771$	0	0
$772$	382.821	0.0178472
$773$	−14247.9	−0.662949	−0.331475	−	0.943464i	$-0.607546\pi$
$773$	−14247.9	−0.662949	−0.331475	+	0.943464i	$0.607546\pi$
$774$	0	0
$775$	24736.6	1.14654
$776$	4201.77	0.194375
$777$	0	0
$778$	−6101.90	−0.281187
$779$	38071.1	1.75101
$780$	0	0
$781$	−8346.10	−0.382391
$782$	55505.0	2.53818
$783$	0	0
$784$	20452.5	0.931693
$785$	−2546.75	−0.115793
$786$	0	0
$787$	32858.0	1.48826	0.744131	−	0.668034i	$-0.232864\pi$
$787$	32858.0	1.48826	0.744131	+	0.668034i	$0.232864\pi$
$788$	−3089.57	−0.139672
$789$	0	0
$790$	−282.886	−0.0127400
$791$	−9572.66	−0.430297
$792$	0	0
$793$	17330.0	0.776047
$794$	16885.4	0.754710
$795$	0	0
$796$	−14626.4	−0.651281
$797$	−27443.2	−1.21968	−0.609841	−	0.792524i	$-0.708767\pi$
$797$	−27443.2	−1.21968	−0.609841	+	0.792524i	$0.708767\pi$
$798$	0	0
$799$	−7556.82	−0.334595
$800$	−23067.9	−1.01947
$801$	0	0
$802$	2886.56	0.127092
$803$	2203.23	0.0968250
$804$	0	0
$805$	1610.93	0.0705314
$806$	−19579.0	−0.855633
$807$	0	0
$808$	−11446.1	−0.498358
$809$	−609.012	−0.0264669	−0.0132334	−	0.999912i	$-0.504212\pi$
$809$	−609.012	−0.0264669	−0.0132334	+	0.999912i	$0.504212\pi$
$810$	0	0
$811$	45186.3	1.95648	0.978241	−	0.207474i	$-0.0665242\pi$
$811$	45186.3	1.95648	0.978241	+	0.207474i	$0.0665242\pi$
$812$	5921.50	0.255916
$813$	0	0
$814$	−777.053	−0.0334591
$815$	−515.119	−0.0221397
$816$	0	0
$817$	6252.99	0.267766
$818$	35469.4	1.51609
$819$	0	0
$820$	−1854.65	−0.0789845
$821$	21814.8	0.927335	0.463667	−	0.886009i	$-0.346533\pi$
$821$	21814.8	0.927335	0.463667	+	0.886009i	$0.346533\pi$
$822$	0	0
$823$	43074.1	1.82439	0.912193	−	0.409761i	$-0.134388\pi$
$823$	43074.1	1.82439	0.912193	+	0.409761i	$0.134388\pi$
$824$	6497.87	0.274714
$825$	0	0
$826$	19347.0	0.814976
$827$	21454.4	0.902106	0.451053	−	0.892497i	$-0.351049\pi$
$827$	21454.4	0.902106	0.451053	+	0.892497i	$0.351049\pi$
$828$	0	0
$829$	1189.99	0.0498552	0.0249276	−	0.999689i	$-0.492064\pi$
$829$	1189.99	0.0498552	0.0249276	+	0.999689i	$0.492064\pi$
$830$	−2435.28	−0.101843
$831$	0	0
$832$	723.608	0.0301522
$833$	−27852.5	−1.15850
$834$	0	0
$835$	2216.76	0.0918732
$836$	−5229.75	−0.216357
$837$	0	0
$838$	44973.3	1.85391
$839$	−45246.6	−1.86184	−0.930921	−	0.365221i	$-0.880993\pi$
$839$	−45246.6	−1.86184	−0.930921	+	0.365221i	$0.880993\pi$
$840$	0	0
$841$	−5296.44	−0.217165
$842$	−8821.65	−0.361062
$843$	0	0
$844$	26799.8	1.09299
$845$	1733.52	0.0705737
$846$	0	0
$847$	−11438.8	−0.464039
$848$	−17105.1	−0.692677
$849$	0	0
$850$	47584.3	1.92015
$851$	3178.69	0.128042
$852$	0	0
$853$	−25550.9	−1.02561	−0.512805	−	0.858505i	$-0.671394\pi$
$853$	−25550.9	−1.02561	−0.512805	+	0.858505i	$0.671394\pi$
$854$	20751.2	0.831490
$855$	0	0
$856$	20835.9	0.831959
$857$	−17765.4	−0.708114	−0.354057	−	0.935224i	$-0.615198\pi$
$857$	−17765.4	−0.708114	−0.354057	+	0.935224i	$0.615198\pi$
$858$	0	0
$859$	−30996.5	−1.23118	−0.615592	−	0.788065i	$-0.711083\pi$
$859$	−30996.5	−1.23118	−0.615592	+	0.788065i	$0.711083\pi$
$860$	−304.618	−0.0120783
$861$	0	0
$862$	45554.4	1.79999
$863$	40190.7	1.58529	0.792647	−	0.609681i	$-0.208703\pi$
$863$	40190.7	1.58529	0.792647	+	0.609681i	$0.208703\pi$
$864$	0	0
$865$	−582.981	−0.0229156
$866$	57739.0	2.26565
$867$	0	0
$868$	−8580.25	−0.335521
$869$	655.701	0.0255962
$870$	0	0
$871$	17113.0	0.665733
$872$	−20063.4	−0.779164
$873$	0	0
$874$	58454.1	2.26229
$875$	2778.31	0.107342
$876$	0	0
$877$	−16311.7	−0.628057	−0.314028	−	0.949414i	$-0.601679\pi$
$877$	−16311.7	−0.628057	−0.314028	+	0.949414i	$0.601679\pi$
$878$	−15474.5	−0.594806
$879$	0	0
$880$	−951.153	−0.0364356
$881$	−31308.6	−1.19729	−0.598645	−	0.801015i	$-0.704294\pi$
$881$	−31308.6	−1.19729	−0.598645	+	0.801015i	$0.704294\pi$
$882$	0	0
$883$	7151.98	0.272574	0.136287	−	0.990669i	$-0.456483\pi$
$883$	7151.98	0.272574	0.136287	+	0.990669i	$0.456483\pi$
$884$	−13784.1	−0.524444
$885$	0	0
$886$	−49200.1	−1.86559
$887$	30037.6	1.13705	0.568526	−	0.822665i	$-0.307514\pi$
$887$	30037.6	1.13705	0.568526	+	0.822665i	$0.307514\pi$
$888$	0	0
$889$	19663.5	0.741837
$890$	3334.09	0.125572
$891$	0	0
$892$	−18686.9	−0.701438
$893$	−7958.33	−0.298226
$894$	0	0
$895$	3851.26	0.143836
$896$	−12994.8	−0.484516
$897$	0	0
$898$	20011.0	0.743627
$899$	−27665.1	−1.02634
$900$	0	0
$901$	23293.9	0.861301
$902$	11746.1	0.433594
$903$	0	0
$904$	−12392.4	−0.455935
$905$	−2500.70	−0.0918519
$906$	0	0
$907$	14848.1	0.543576	0.271788	−	0.962357i	$-0.412385\pi$
$907$	14848.1	0.543576	0.271788	+	0.962357i	$0.412385\pi$
$908$	9590.26	0.350511
$909$	0	0
$910$	−1093.10	−0.0398195
$911$	29002.6	1.05478	0.527388	−	0.849625i	$-0.323172\pi$
$911$	29002.6	1.05478	0.527388	+	0.849625i	$0.323172\pi$
$912$	0	0
$913$	5644.74	0.204615
$914$	68158.8	2.46662
$915$	0	0
$916$	15022.3	0.541868
$917$	15876.4	0.571740
$918$	0	0
$919$	−16668.8	−0.598315	−0.299158	−	0.954204i	$-0.596706\pi$
$919$	−16668.8	−0.598315	−0.299158	+	0.954204i	$0.596706\pi$
$920$	2085.45	0.0747340
$921$	0	0
$922$	−13361.6	−0.477269
$923$	−23166.7	−0.826153
$924$	0	0
$925$	2725.09	0.0968651
$926$	42317.8	1.50178
$927$	0	0
$928$	25798.9	0.912596
$929$	3893.81	0.137515	0.0687577	−	0.997633i	$-0.478096\pi$
$929$	3893.81	0.137515	0.0687577	+	0.997633i	$0.478096\pi$
$930$	0	0
$931$	−29332.4	−1.03258
$932$	−1586.94	−0.0557745
$933$	0	0
$934$	39569.1	1.38623
$935$	1295.29	0.0453054
$936$	0	0
$937$	4657.69	0.162391	0.0811954	−	0.996698i	$-0.474126\pi$
$937$	4657.69	0.162391	0.0811954	+	0.996698i	$0.474126\pi$
$938$	20491.5	0.713295
$939$	0	0
$940$	387.694	0.0134523
$941$	10178.6	0.352616	0.176308	−	0.984335i	$-0.443585\pi$
$941$	10178.6	0.352616	0.176308	+	0.984335i	$0.443585\pi$
$942$	0	0
$943$	−48049.8	−1.65930
$944$	46729.0	1.61112
$945$	0	0
$946$	1929.24	0.0663055
$947$	−27536.1	−0.944880	−0.472440	−	0.881363i	$-0.656627\pi$
$947$	−27536.1	−0.944880	−0.472440	+	0.881363i	$0.656627\pi$
$948$	0	0
$949$	6115.62	0.209190
$950$	50112.6	1.71144
$951$	0	0
$952$	12087.6	0.411515
$953$	−40274.3	−1.36895	−0.684477	−	0.729034i	$-0.739969\pi$
$953$	−40274.3	−1.36895	−0.684477	+	0.729034i	$0.739969\pi$
$954$	0	0
$955$	−2380.96	−0.0806764
$956$	−1103.70	−0.0373393
$957$	0	0
$958$	22605.7	0.762377
$959$	−992.759	−0.0334284
$960$	0	0
$961$	10295.8	0.345600
$962$	−2156.90	−0.0722882
$963$	0	0
$964$	30972.9	1.03482
$965$	−99.8538	−0.00333099
$966$	0	0
$967$	−14039.2	−0.466877	−0.233438	−	0.972372i	$-0.574998\pi$
$967$	−14039.2	−0.466877	−0.233438	+	0.972372i	$0.574998\pi$
$968$	−14808.2	−0.491689
$969$	0	0
$970$	1496.53	0.0495367
$971$	−12723.2	−0.420503	−0.210251	−	0.977647i	$-0.567428\pi$
$971$	−12723.2	−0.420503	−0.210251	+	0.977647i	$0.567428\pi$
$972$	0	0
$973$	19594.0	0.645585
$974$	−28785.4	−0.946966
$975$	0	0
$976$	50120.5	1.64377
$977$	28114.9	0.920649	0.460324	−	0.887751i	$-0.347733\pi$
$977$	28114.9	0.920649	0.460324	+	0.887751i	$0.347733\pi$
$978$	0	0
$979$	−7728.08	−0.252289
$980$	1428.94	0.0465774
$981$	0	0
$982$	45596.6	1.48172
$983$	−42634.2	−1.38334	−0.691668	−	0.722216i	$-0.743124\pi$
$983$	−42634.2	−1.38334	−0.691668	+	0.722216i	$0.743124\pi$
$984$	0	0
$985$	805.873	0.0260683
$986$	−53217.7	−1.71886
$987$	0	0
$988$	−14516.5	−0.467439
$989$	−7891.94	−0.253740
$990$	0	0
$991$	14151.1	0.453607	0.226804	−	0.973941i	$-0.427172\pi$
$991$	14151.1	0.453607	0.226804	+	0.973941i	$0.427172\pi$
$992$	−37382.6	−1.19647
$993$	0	0
$994$	−27740.2	−0.885176
$995$	3815.11	0.121555
$996$	0	0
$997$	−15789.5	−0.501563	−0.250782	−	0.968044i	$-0.580688\pi$
$997$	−15789.5	−0.501563	−0.250782	+	0.968044i	$0.580688\pi$
$998$	20516.8	0.650749
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2151.4.a.b.1.7		28
3.2	odd	2		717.4.a.b.1.22	✓	28

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
717.4.a.b.1.22	✓	28	3.2	odd	2
2151.4.a.b.1.7		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)

\(f(q)\)	\(=\)	\(q-3.55218 q^{2} +4.61801 q^{4} -1.20455 q^{5} +9.27993 q^{7} +12.0135 q^{8} +O(q^{10})\)	\(q-3.55218 q^{2} +4.61801 q^{4} -1.20455 q^{5} +9.27993 q^{7} +12.0135 q^{8} +4.27878 q^{10} -9.91778 q^{11} -27.5292 q^{13} -32.9640 q^{14} -79.6181 q^{16} +108.425 q^{17} +114.186 q^{19} -5.56262 q^{20} +35.2298 q^{22} -144.115 q^{23} -123.549 q^{25} +97.7889 q^{26} +42.8548 q^{28} +138.176 q^{29} -200.217 q^{31} +186.710 q^{32} -385.145 q^{34} -11.1781 q^{35} -22.0567 q^{37} -405.609 q^{38} -14.4708 q^{40} +333.414 q^{41} +54.7616 q^{43} -45.8004 q^{44} +511.921 q^{46} -69.6964 q^{47} -256.883 q^{49} +438.869 q^{50} -127.130 q^{52} +214.839 q^{53} +11.9464 q^{55} +111.484 q^{56} -490.826 q^{58} -586.914 q^{59} -629.512 q^{61} +711.207 q^{62} -26.2851 q^{64} +33.1603 q^{65} -621.632 q^{67} +500.707 q^{68} +39.7067 q^{70} +841.529 q^{71} -222.150 q^{73} +78.3495 q^{74} +527.311 q^{76} -92.0363 q^{77} -66.1137 q^{79} +95.9038 q^{80} -1184.35 q^{82} -569.154 q^{83} -130.603 q^{85} -194.523 q^{86} -119.147 q^{88} +779.215 q^{89} -255.469 q^{91} -665.522 q^{92} +247.574 q^{94} -137.542 q^{95} +349.756 q^{97} +912.496 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q + 5 q^{2} + 103 q^{4} - 6 q^{5} - 68 q^{7} + 39 q^{8}+O(q^{10}) \) 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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