LMFDB - Embedded newform 2151.4.a.a.1.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2151, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("2151.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

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

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$126.913108422$
Analytic rank:	$1$
Dimension:	$22$
Twist minimal:	no (minimal twist has level 239)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

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

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.21535 q^{2} -3.09221 q^{4} -10.2488 q^{5} -27.8405 q^{7} +24.5732 q^{8} +O(q^{10})$	\(q-2.21535 q^{2} -3.09221 q^{4} -10.2488 q^{5} -27.8405 q^{7} +24.5732 q^{8} +22.7048 q^{10} -21.8780 q^{11} -22.8516 q^{13} +61.6766 q^{14} -29.7005 q^{16} +90.1399 q^{17} -24.5640 q^{19} +31.6916 q^{20} +48.4674 q^{22} -70.4601 q^{23} -19.9613 q^{25} +50.6243 q^{26} +86.0889 q^{28} -198.363 q^{29} -94.0164 q^{31} -130.788 q^{32} -199.692 q^{34} +285.333 q^{35} +102.654 q^{37} +54.4180 q^{38} -251.846 q^{40} +143.322 q^{41} -82.0409 q^{43} +67.6514 q^{44} +156.094 q^{46} +576.167 q^{47} +432.095 q^{49} +44.2214 q^{50} +70.6620 q^{52} +151.529 q^{53} +224.224 q^{55} -684.130 q^{56} +439.444 q^{58} -137.115 q^{59} -79.8377 q^{61} +208.280 q^{62} +527.346 q^{64} +234.202 q^{65} +642.365 q^{67} -278.732 q^{68} -632.114 q^{70} +1046.49 q^{71} +670.788 q^{73} -227.414 q^{74} +75.9572 q^{76} +609.095 q^{77} +304.508 q^{79} +304.396 q^{80} -317.509 q^{82} -670.634 q^{83} -923.829 q^{85} +181.750 q^{86} -537.611 q^{88} +871.291 q^{89} +636.201 q^{91} +217.878 q^{92} -1276.41 q^{94} +251.753 q^{95} -404.956 q^{97} -957.244 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 22 q + 4 q^{2} + 50 q^{4} + 37 q^{5} - 52 q^{7} + 69 q^{8}+O(q^{10}) $ 22 * q + 4 * q^2 + 50 * q^4 + 37 * q^5 - 52 * q^7 + 69 * q^8	$ 22 q + 4 q^{2} + 50 q^{4} + 37 q^{5} - 52 q^{7} + 69 q^{8} - 93 q^{10} + 77 q^{11} - 218 q^{13} + 111 q^{14} - 42 q^{16} + 219 q^{17} - 476 q^{19} + 314 q^{20} - 390 q^{22} + 202 q^{23} - 271 q^{25} + 220 q^{26} - 515 q^{28} + 307 q^{29} - 1001 q^{31} + 771 q^{32} - 1297 q^{34} + 430 q^{35} - 922 q^{37} - 49 q^{38} - 1344 q^{40} + 1188 q^{41} - 192 q^{43} + 547 q^{44} - 1178 q^{46} + 102 q^{47} - 1952 q^{49} + 471 q^{50} - 1785 q^{52} + 580 q^{53} - 1730 q^{55} + 804 q^{56} - 1156 q^{58} + 1528 q^{59} - 1631 q^{61} - 2206 q^{62} + 327 q^{64} - 44 q^{65} - 689 q^{67} - 2522 q^{68} + 1175 q^{70} - 341 q^{71} - 2260 q^{73} - 4027 q^{74} - 1855 q^{76} - 1578 q^{77} + 396 q^{79} - 6183 q^{80} + 4936 q^{82} - 1065 q^{83} + 144 q^{85} - 2915 q^{86} + 1068 q^{88} + 1984 q^{89} - 2186 q^{91} - 6720 q^{92} + 174 q^{94} - 2804 q^{95} - 4946 q^{97} - 7149 q^{98}+O(q^{100}) $ 22 * q + 4 * q^2 + 50 * q^4 + 37 * q^5 - 52 * q^7 + 69 * q^8 - 93 * q^10 + 77 * q^11 - 218 * q^13 + 111 * q^14 - 42 * q^16 + 219 * q^17 - 476 * q^19 + 314 * q^20 - 390 * q^22 + 202 * q^23 - 271 * q^25 + 220 * q^26 - 515 * q^28 + 307 * q^29 - 1001 * q^31 + 771 * q^32 - 1297 * q^34 + 430 * q^35 - 922 * q^37 - 49 * q^38 - 1344 * q^40 + 1188 * q^41 - 192 * q^43 + 547 * q^44 - 1178 * q^46 + 102 * q^47 - 1952 * q^49 + 471 * q^50 - 1785 * q^52 + 580 * q^53 - 1730 * q^55 + 804 * q^56 - 1156 * q^58 + 1528 * q^59 - 1631 * q^61 - 2206 * q^62 + 327 * q^64 - 44 * q^65 - 689 * q^67 - 2522 * q^68 + 1175 * q^70 - 341 * q^71 - 2260 * q^73 - 4027 * q^74 - 1855 * q^76 - 1578 * q^77 + 396 * q^79 - 6183 * q^80 + 4936 * q^82 - 1065 * q^83 + 144 * q^85 - 2915 * q^86 + 1068 * q^88 + 1984 * q^89 - 2186 * q^91 - 6720 * q^92 + 174 * q^94 - 2804 * q^95 - 4946 * q^97 - 7149 * 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 $

$2$	−2.21535	−0.783245	−0.391623	−	0.920126i	$-0.628086\pi$
$2$	−2.21535	−0.783245	−0.391623	+	0.920126i	$0.628086\pi$
$3$	0	0
$3$	0	0
$4$	−3.09221	−0.386527
$5$	−10.2488	−0.916684	−0.458342	−	0.888776i	$-0.651556\pi$
$5$	−10.2488	−0.916684	−0.458342	+	0.888776i	$0.651556\pi$
$6$	0	0
$7$	−27.8405	−1.50325	−0.751624	−	0.659592i	$-0.770729\pi$
$7$	−27.8405	−1.50325	−0.751624	+	0.659592i	$0.770729\pi$
$8$	24.5732	1.08599
$9$	0	0
$10$	22.7048	0.717989
$11$	−21.8780	−0.599678	−0.299839	−	0.953990i	$-0.596933\pi$
$11$	−21.8780	−0.599678	−0.299839	+	0.953990i	$0.596933\pi$
$12$	0	0
$13$	−22.8516	−0.487530	−0.243765	−	0.969834i	$-0.578383\pi$
$13$	−22.8516	−0.487530	−0.243765	+	0.969834i	$0.578383\pi$
$14$	61.6766	1.17741
$15$	0	0
$16$	−29.7005	−0.464071
$17$	90.1399	1.28601	0.643004	−	0.765863i	$-0.277688\pi$
$17$	90.1399	1.28601	0.643004	+	0.765863i	$0.277688\pi$
$18$	0	0
$19$	−24.5640	−0.296599	−0.148299	−	0.988943i	$-0.547380\pi$
$19$	−24.5640	−0.296599	−0.148299	+	0.988943i	$0.547380\pi$
$20$	31.6916	0.354323
$21$	0	0
$22$	48.4674	0.469695
$23$	−70.4601	−0.638780	−0.319390	−	0.947623i	$-0.603478\pi$
$23$	−70.4601	−0.638780	−0.319390	+	0.947623i	$0.603478\pi$
$24$	0	0
$25$	−19.9613	−0.159691
$26$	50.6243	0.381856
$27$	0	0
$28$	86.0889	0.581045
$29$	−198.363	−1.27017	−0.635087	−	0.772441i	$-0.719036\pi$
$29$	−198.363	−1.27017	−0.635087	+	0.772441i	$0.719036\pi$
$30$	0	0
$31$	−94.0164	−0.544705	−0.272352	−	0.962198i	$-0.587802\pi$
$31$	−94.0164	−0.544705	−0.272352	+	0.962198i	$0.587802\pi$
$32$	−130.788	−0.722509
$33$	0	0
$34$	−199.692	−1.00726
$35$	285.333	1.37800
$36$	0	0
$37$	102.654	0.456113	0.228056	−	0.973648i	$-0.426763\pi$
$37$	102.654	0.456113	0.228056	+	0.973648i	$0.426763\pi$
$38$	54.4180	0.232309
$39$	0	0
$40$	−251.846	−0.995510
$41$	143.322	0.545931	0.272966	−	0.962024i	$-0.411995\pi$
$41$	143.322	0.545931	0.272966	+	0.962024i	$0.411995\pi$
$42$	0	0
$43$	−82.0409	−0.290956	−0.145478	−	0.989361i	$-0.546472\pi$
$43$	−82.0409	−0.290956	−0.145478	+	0.989361i	$0.546472\pi$
$44$	67.6514	0.231791
$45$	0	0
$46$	156.094	0.500322
$47$	576.167	1.78814	0.894070	−	0.447928i	$-0.147838\pi$
$47$	576.167	1.78814	0.894070	+	0.447928i	$0.147838\pi$
$48$	0	0
$49$	432.095	1.25975
$50$	44.2214	0.125077
$51$	0	0
$52$	70.6620	0.188443
$53$	151.529	0.392719	0.196359	−	0.980532i	$-0.437088\pi$
$53$	151.529	0.392719	0.196359	+	0.980532i	$0.437088\pi$
$54$	0	0
$55$	224.224	0.549715
$56$	−684.130	−1.63251
$57$	0	0
$58$	439.444	0.994858
$59$	−137.115	−0.302557	−0.151278	−	0.988491i	$-0.548339\pi$
$59$	−137.115	−0.302557	−0.151278	+	0.988491i	$0.548339\pi$
$60$	0	0
$61$	−79.8377	−0.167577	−0.0837883	−	0.996484i	$-0.526702\pi$
$61$	−79.8377	−0.167577	−0.0837883	+	0.996484i	$0.526702\pi$
$62$	208.280	0.426637
$63$	0	0
$64$	527.346	1.02997
$65$	234.202	0.446911
$66$	0	0
$67$	642.365	1.17130	0.585652	−	0.810563i	$-0.300839\pi$
$67$	642.365	1.17130	0.585652	+	0.810563i	$0.300839\pi$
$68$	−278.732	−0.497076
$69$	0	0
$70$	−632.114	−1.07931
$71$	1046.49	1.74923	0.874617	−	0.484814i	$-0.161113\pi$
$71$	1046.49	1.74923	0.874617	+	0.484814i	$0.161113\pi$
$72$	0	0
$73$	670.788	1.07548	0.537738	−	0.843112i	$-0.319279\pi$
$73$	670.788	1.07548	0.537738	+	0.843112i	$0.319279\pi$
$74$	−227.414	−0.357248
$75$	0	0
$76$	75.9572	0.114643
$77$	609.095	0.901465
$78$	0	0
$79$	304.508	0.433669	0.216835	−	0.976208i	$-0.430427\pi$
$79$	304.508	0.433669	0.216835	+	0.976208i	$0.430427\pi$
$80$	304.396	0.425406
$81$	0	0
$82$	−317.509	−0.427598
$83$	−670.634	−0.886888	−0.443444	−	0.896302i	$-0.646243\pi$
$83$	−670.634	−0.886888	−0.443444	+	0.896302i	$0.646243\pi$
$84$	0	0
$85$	−923.829	−1.17886
$86$	181.750	0.227890
$87$	0	0
$88$	−537.611	−0.651245
$89$	871.291	1.03772	0.518858	−	0.854861i	$-0.326357\pi$
$89$	871.291	1.03772	0.518858	+	0.854861i	$0.326357\pi$
$90$	0	0
$91$	636.201	0.732878
$92$	217.878	0.246905
$93$	0	0
$94$	−1276.41	−1.40055
$95$	251.753	0.271887
$96$	0	0
$97$	−404.956	−0.423887	−0.211944	−	0.977282i	$-0.567979\pi$
$97$	−404.956	−0.423887	−0.211944	+	0.977282i	$0.567979\pi$
$98$	−957.244	−0.986696
$99$	0	0
$100$	61.7247	0.0617247
$101$	1588.14	1.56462	0.782308	−	0.622892i	$-0.214043\pi$
$101$	1588.14	1.56462	0.782308	+	0.622892i	$0.214043\pi$
$102$	0	0
$103$	1693.09	1.61967	0.809833	−	0.586661i	$-0.199558\pi$
$103$	1693.09	1.61967	0.809833	+	0.586661i	$0.199558\pi$
$104$	−561.536	−0.529453
$105$	0	0
$106$	−335.690	−0.307595
$107$	−1855.27	−1.67622	−0.838112	−	0.545498i	$-0.816340\pi$
$107$	−1855.27	−1.67622	−0.838112	+	0.545498i	$0.816340\pi$
$108$	0	0
$109$	217.152	0.190820	0.0954100	−	0.995438i	$-0.469584\pi$
$109$	217.152	0.190820	0.0954100	+	0.995438i	$0.469584\pi$
$110$	−496.735	−0.430562
$111$	0	0
$112$	826.878	0.697613
$113$	−1726.29	−1.43713	−0.718566	−	0.695459i	$-0.755201\pi$
$113$	−1726.29	−1.43713	−0.718566	+	0.695459i	$0.755201\pi$
$114$	0	0
$115$	722.134	0.585559
$116$	613.380	0.490956
$117$	0	0
$118$	303.758	0.236976
$119$	−2509.54	−1.93319
$120$	0	0
$121$	−852.354	−0.640386
$122$	176.869	0.131254
$123$	0	0
$124$	290.719	0.210543
$125$	1485.69	1.06307
$126$	0	0
$127$	−753.989	−0.526816	−0.263408	−	0.964684i	$-0.584847\pi$
$127$	−753.989	−0.526816	−0.263408	+	0.964684i	$0.584847\pi$
$128$	−121.952	−0.0842122
$129$	0	0
$130$	−518.841	−0.350041
$131$	1714.18	1.14327	0.571637	−	0.820506i	$-0.306308\pi$
$131$	1714.18	1.14327	0.571637	+	0.820506i	$0.306308\pi$
$132$	0	0
$133$	683.876	0.445861
$134$	−1423.06	−0.917419
$135$	0	0
$136$	2215.02	1.39659
$137$	989.369	0.616989	0.308495	−	0.951226i	$-0.400175\pi$
$137$	989.369	0.616989	0.308495	+	0.951226i	$0.400175\pi$
$138$	0	0
$139$	1256.97	0.767016	0.383508	−	0.923538i	$-0.374716\pi$
$139$	1256.97	0.767016	0.383508	+	0.923538i	$0.374716\pi$
$140$	−882.311	−0.532635
$141$	0	0
$142$	−2318.35	−1.37008
$143$	499.947	0.292361
$144$	0	0
$145$	2032.99	1.16435
$146$	−1486.03	−0.842362
$147$	0	0
$148$	−317.427	−0.176300
$149$	1429.90	0.786186	0.393093	−	0.919499i	$-0.371405\pi$
$149$	1429.90	0.786186	0.393093	+	0.919499i	$0.371405\pi$
$150$	0	0
$151$	−1265.93	−0.682252	−0.341126	−	0.940018i	$-0.610808\pi$
$151$	−1265.93	−0.682252	−0.341126	+	0.940018i	$0.610808\pi$
$152$	−603.616	−0.322103
$153$	0	0
$154$	−1349.36	−0.706068
$155$	963.559	0.499322
$156$	0	0
$157$	−1872.91	−0.952064	−0.476032	−	0.879428i	$-0.657925\pi$
$157$	−1872.91	−0.952064	−0.476032	+	0.879428i	$0.657925\pi$
$158$	−674.594	−0.339670
$159$	0	0
$160$	1340.43	0.662313
$161$	1961.65	0.960245
$162$	0	0
$163$	−2462.31	−1.18321	−0.591605	−	0.806228i	$-0.701505\pi$
$163$	−2462.31	−1.18321	−0.591605	+	0.806228i	$0.701505\pi$
$164$	−443.183	−0.211017
$165$	0	0
$166$	1485.69	0.694651
$167$	−1483.24	−0.687286	−0.343643	−	0.939100i	$-0.611661\pi$
$167$	−1483.24	−0.687286	−0.343643	+	0.939100i	$0.611661\pi$
$168$	0	0
$169$	−1674.80	−0.762314
$170$	2046.61	0.923339
$171$	0	0
$172$	253.688	0.112462
$173$	962.469	0.422978	0.211489	−	0.977380i	$-0.432169\pi$
$173$	962.469	0.422978	0.211489	+	0.977380i	$0.432169\pi$
$174$	0	0
$175$	555.734	0.240055
$176$	649.787	0.278293
$177$	0	0
$178$	−1930.22	−0.812786
$179$	2077.35	0.867420	0.433710	−	0.901053i	$-0.357204\pi$
$179$	2077.35	0.867420	0.433710	+	0.901053i	$0.357204\pi$
$180$	0	0
$181$	−1788.97	−0.734657	−0.367328	−	0.930091i	$-0.619727\pi$
$181$	−1788.97	−0.734657	−0.367328	+	0.930091i	$0.619727\pi$
$182$	−1409.41	−0.574024
$183$	0	0
$184$	−1731.43	−0.693709
$185$	−1052.08	−0.418111
$186$	0	0
$187$	−1972.08	−0.771191
$188$	−1781.63	−0.691163
$189$	0	0
$190$	−557.721	−0.212954
$191$	−1804.64	−0.683662	−0.341831	−	0.939761i	$-0.611047\pi$
$191$	−1804.64	−0.683662	−0.341831	+	0.939761i	$0.611047\pi$
$192$	0	0
$193$	−5279.30	−1.96898	−0.984488	−	0.175455i	$-0.943860\pi$
$193$	−5279.30	−1.96898	−0.984488	+	0.175455i	$0.943860\pi$
$194$	897.120	0.332008
$195$	0	0
$196$	−1336.13	−0.486928
$197$	−1242.17	−0.449242	−0.224621	−	0.974446i	$-0.572114\pi$
$197$	−1242.17	−0.449242	−0.224621	+	0.974446i	$0.572114\pi$
$198$	0	0
$199$	1397.03	0.497653	0.248827	−	0.968548i	$-0.419955\pi$
$199$	1397.03	0.497653	0.248827	+	0.968548i	$0.419955\pi$
$200$	−490.513	−0.173423
$201$	0	0
$202$	−3518.30	−1.22548
$203$	5522.53	1.90939
$204$	0	0
$205$	−1468.89	−0.500446
$206$	−3750.80	−1.26860
$207$	0	0
$208$	678.704	0.226248
$209$	537.411	0.177864
$210$	0	0
$211$	976.458	0.318588	0.159294	−	0.987231i	$-0.449078\pi$
$211$	976.458	0.318588	0.159294	+	0.987231i	$0.449078\pi$
$212$	−468.560	−0.151796
$213$	0	0
$214$	4110.08	1.31289
$215$	840.824	0.266715
$216$	0	0
$217$	2617.47	0.818826
$218$	−481.068	−0.149459
$219$	0	0
$220$	−693.348	−0.212480
$221$	−2059.84	−0.626967
$222$	0	0
$223$	2090.55	0.627774	0.313887	−	0.949460i	$-0.398369\pi$
$223$	2090.55	0.627774	0.313887	+	0.949460i	$0.398369\pi$
$224$	3641.21	1.08611
$225$	0	0
$226$	3824.35	1.12563
$227$	2254.95	0.659322	0.329661	−	0.944099i	$-0.393066\pi$
$227$	2254.95	0.659322	0.329661	+	0.944099i	$0.393066\pi$
$228$	0	0
$229$	−3683.18	−1.06285	−0.531423	−	0.847107i	$-0.678342\pi$
$229$	−3683.18	−1.06285	−0.531423	+	0.847107i	$0.678342\pi$
$230$	−1599.78	−0.458637
$231$	0	0
$232$	−4874.40	−1.37940
$233$	1506.90	0.423692	0.211846	−	0.977303i	$-0.432052\pi$
$233$	1506.90	0.423692	0.211846	+	0.977303i	$0.432052\pi$
$234$	0	0
$235$	−5905.04	−1.63916
$236$	423.988	0.116946
$237$	0	0
$238$	5559.52	1.51416
$239$	−239.000	−0.0646846
$239$	−239.000	−0.0646846
$240$	0	0
$241$	−4767.66	−1.27432	−0.637162	−	0.770730i	$-0.719892\pi$
$241$	−4767.66	−1.27432	−0.637162	+	0.770730i	$0.719892\pi$
$242$	1888.26	0.501580
$243$	0	0
$244$	246.875	0.0647728
$245$	−4428.48	−1.15480
$246$	0	0
$247$	561.327	0.144601
$248$	−2310.28	−0.591544
$249$	0	0
$250$	−3291.32	−0.832645
$251$	3267.74	0.821744	0.410872	−	0.911693i	$-0.365224\pi$
$251$	3267.74	0.821744	0.410872	+	0.911693i	$0.365224\pi$
$252$	0	0
$253$	1541.52	0.383062
$254$	1670.35	0.412627
$255$	0	0
$256$	−3948.60	−0.964014
$257$	−6625.79	−1.60819	−0.804096	−	0.594500i	$-0.797350\pi$
$257$	−6625.79	−1.60819	−0.804096	+	0.594500i	$0.797350\pi$
$258$	0	0
$259$	−2857.94	−0.685650
$260$	−724.203	−0.172743
$261$	0	0
$262$	−3797.52	−0.895465
$263$	675.912	0.158474	0.0792368	−	0.996856i	$-0.474752\pi$
$263$	675.912	0.158474	0.0792368	+	0.996856i	$0.474752\pi$
$264$	0	0
$265$	−1553.00	−0.359999
$266$	−1515.03	−0.349219
$267$	0	0
$268$	−1986.33	−0.452740
$269$	5162.52	1.17013	0.585064	−	0.810987i	$-0.301069\pi$
$269$	5162.52	1.17013	0.585064	+	0.810987i	$0.301069\pi$
$270$	0	0
$271$	3529.42	0.791133	0.395566	−	0.918437i	$-0.370548\pi$
$271$	3529.42	0.791133	0.395566	+	0.918437i	$0.370548\pi$
$272$	−2677.20	−0.596798
$273$	0	0
$274$	−2191.80	−0.483254
$275$	436.713	0.0957630
$276$	0	0
$277$	3766.22	0.816932	0.408466	−	0.912774i	$-0.366064\pi$
$277$	3766.22	0.816932	0.408466	+	0.912774i	$0.366064\pi$
$278$	−2784.64	−0.600762
$279$	0	0
$280$	7011.54	1.49650
$281$	1547.46	0.328518	0.164259	−	0.986417i	$-0.447477\pi$
$281$	1547.46	0.328518	0.164259	+	0.986417i	$0.447477\pi$
$282$	0	0
$283$	253.562	0.0532605	0.0266302	−	0.999645i	$-0.491522\pi$
$283$	253.562	0.0532605	0.0266302	+	0.999645i	$0.491522\pi$
$284$	−3235.97	−0.676125
$285$	0	0
$286$	−1107.56	−0.228990
$287$	−3990.17	−0.820670
$288$	0	0
$289$	3212.20	0.653816
$290$	−4503.79	−0.911971
$291$	0	0
$292$	−2074.22	−0.415700
$293$	2643.22	0.527026	0.263513	−	0.964656i	$-0.415119\pi$
$293$	2643.22	0.527026	0.263513	+	0.964656i	$0.415119\pi$
$294$	0	0
$295$	1405.27	0.277349
$296$	2522.53	0.495334
$297$	0	0
$298$	−3167.73	−0.615777
$299$	1610.12	0.311424
$300$	0	0
$301$	2284.06	0.437380
$302$	2804.48	0.534371
$303$	0	0
$304$	729.564	0.137643
$305$	818.244	0.153615
$306$	0	0
$307$	149.370	0.0277687	0.0138843	−	0.999904i	$-0.495580\pi$
$307$	149.370	0.0277687	0.0138843	+	0.999904i	$0.495580\pi$
$308$	−1883.45	−0.348440
$309$	0	0
$310$	−2134.62	−0.391092
$311$	219.799	0.0400760	0.0200380	−	0.999799i	$-0.493621\pi$
$311$	219.799	0.0400760	0.0200380	+	0.999799i	$0.493621\pi$
$312$	0	0
$313$	−7154.65	−1.29203	−0.646014	−	0.763326i	$-0.723565\pi$
$313$	−7154.65	−1.29203	−0.646014	+	0.763326i	$0.723565\pi$
$314$	4149.15	0.745700
$315$	0	0
$316$	−941.605	−0.167625
$317$	408.441	0.0723670	0.0361835	−	0.999345i	$-0.488480\pi$
$317$	408.441	0.0723670	0.0361835	+	0.999345i	$0.488480\pi$
$318$	0	0
$319$	4339.78	0.761696
$320$	−5404.68	−0.944160
$321$	0	0
$322$	−4345.74	−0.752107
$323$	−2214.20	−0.381428
$324$	0	0
$325$	456.148	0.0778540
$326$	5454.89	0.926744
$327$	0	0
$328$	3521.88	0.592876
$329$	−16040.8	−2.68802
$330$	0	0
$331$	11292.7	1.87524	0.937618	−	0.347668i	$-0.113026\pi$
$331$	11292.7	1.87524	0.937618	+	0.347668i	$0.113026\pi$
$332$	2073.74	0.342806
$333$	0	0
$334$	3285.91	0.538314
$335$	−6583.49	−1.07372
$336$	0	0
$337$	9597.55	1.55137	0.775685	−	0.631120i	$-0.217405\pi$
$337$	9597.55	1.55137	0.775685	+	0.631120i	$0.217405\pi$
$338$	3710.28	0.597079
$339$	0	0
$340$	2856.68	0.455662
$341$	2056.89	0.326647
$342$	0	0
$343$	−2480.47	−0.390474
$344$	−2016.01	−0.315976
$345$	0	0
$346$	−2132.21	−0.331296
$347$	12403.1	1.91883	0.959417	−	0.281992i	$-0.0909952\pi$
$347$	12403.1	1.91883	0.959417	+	0.281992i	$0.0909952\pi$
$348$	0	0
$349$	5008.19	0.768144	0.384072	−	0.923303i	$-0.374521\pi$
$349$	5008.19	0.768144	0.384072	+	0.923303i	$0.374521\pi$
$350$	−1231.15	−0.188022
$351$	0	0
$352$	2861.38	0.433273
$353$	−4673.62	−0.704679	−0.352340	−	0.935872i	$-0.614614\pi$
$353$	−4673.62	−0.704679	−0.352340	+	0.935872i	$0.614614\pi$
$354$	0	0
$355$	−10725.3	−1.60349
$356$	−2694.22	−0.401105
$357$	0	0
$358$	−4602.06	−0.679403
$359$	−12144.2	−1.78537	−0.892685	−	0.450681i	$-0.851181\pi$
$359$	−12144.2	−1.78537	−0.892685	+	0.450681i	$0.851181\pi$
$360$	0	0
$361$	−6255.61	−0.912029
$362$	3963.19	0.575416
$363$	0	0
$364$	−1967.27	−0.283277
$365$	−6874.79	−0.985872
$366$	0	0
$367$	−3110.84	−0.442464	−0.221232	−	0.975221i	$-0.571008\pi$
$367$	−3110.84	−0.442464	−0.221232	+	0.975221i	$0.571008\pi$
$368$	2092.70	0.296439
$369$	0	0
$370$	2330.73	0.327484
$371$	−4218.65	−0.590354
$372$	0	0
$373$	4149.75	0.576048	0.288024	−	0.957623i	$-0.407002\pi$
$373$	4149.75	0.576048	0.288024	+	0.957623i	$0.407002\pi$
$374$	4368.85	0.604032
$375$	0	0
$376$	14158.2	1.94190
$377$	4532.91	0.619248
$378$	0	0
$379$	−5874.93	−0.796240	−0.398120	−	0.917333i	$-0.630337\pi$
$379$	−5874.93	−0.796240	−0.398120	+	0.917333i	$0.630337\pi$
$380$	−778.473	−0.105092
$381$	0	0
$382$	3997.92	0.535475
$383$	−6039.02	−0.805690	−0.402845	−	0.915268i	$-0.631979\pi$
$383$	−6039.02	−0.805690	−0.402845	+	0.915268i	$0.631979\pi$
$384$	0	0
$385$	−6242.51	−0.826358
$386$	11695.5	1.54219
$387$	0	0
$388$	1252.21	0.163844
$389$	−6971.45	−0.908655	−0.454327	−	0.890835i	$-0.650120\pi$
$389$	−6971.45	−0.908655	−0.454327	+	0.890835i	$0.650120\pi$
$390$	0	0
$391$	−6351.26	−0.821476
$392$	10618.0	1.36808
$393$	0	0
$394$	2751.84	0.351867
$395$	−3120.86	−0.397538
$396$	0	0
$397$	−8002.21	−1.01164	−0.505818	−	0.862640i	$-0.668809\pi$
$397$	−8002.21	−1.01164	−0.505818	+	0.862640i	$0.668809\pi$
$398$	−3094.92	−0.389785
$399$	0	0
$400$	592.862	0.0741077
$401$	−9911.36	−1.23429	−0.617144	−	0.786850i	$-0.711710\pi$
$401$	−9911.36	−1.23429	−0.617144	+	0.786850i	$0.711710\pi$
$402$	0	0
$403$	2148.42	0.265560
$404$	−4910.88	−0.604765
$405$	0	0
$406$	−12234.3	−1.49552
$407$	−2245.86	−0.273521
$408$	0	0
$409$	7310.87	0.883861	0.441930	−	0.897049i	$-0.354294\pi$
$409$	7310.87	0.883861	0.441930	+	0.897049i	$0.354294\pi$
$410$	3254.10	0.391972
$411$	0	0
$412$	−5235.41	−0.626044
$413$	3817.35	0.454818
$414$	0	0
$415$	6873.22	0.812996
$416$	2988.72	0.352245
$417$	0	0
$418$	−1190.56	−0.139311
$419$	13335.6	1.55487	0.777433	−	0.628966i	$-0.216521\pi$
$419$	13335.6	1.55487	0.777433	+	0.628966i	$0.216521\pi$
$420$	0	0
$421$	−12588.5	−1.45730	−0.728651	−	0.684885i	$-0.759852\pi$
$421$	−12588.5	−1.45730	−0.728651	+	0.684885i	$0.759852\pi$
$422$	−2163.20	−0.249533
$423$	0	0
$424$	3723.54	0.426489
$425$	−1799.31	−0.205363
$426$	0	0
$427$	2222.73	0.251909
$428$	5736.90	0.647905
$429$	0	0
$430$	−1862.72	−0.208903
$431$	482.193	0.0538896	0.0269448	−	0.999637i	$-0.491422\pi$
$431$	482.193	0.0538896	0.0269448	+	0.999637i	$0.491422\pi$
$432$	0	0
$433$	−1924.02	−0.213539	−0.106770	−	0.994284i	$-0.534051\pi$
$433$	−1924.02	−0.213539	−0.106770	+	0.994284i	$0.534051\pi$
$434$	−5798.61	−0.641342
$435$	0	0
$436$	−671.480	−0.0737570
$437$	1730.78	0.189461
$438$	0	0
$439$	6788.41	0.738026	0.369013	−	0.929424i	$-0.379696\pi$
$439$	6788.41	0.738026	0.369013	+	0.929424i	$0.379696\pi$
$440$	5509.89	0.596986
$441$	0	0
$442$	4563.27	0.491069
$443$	−4844.77	−0.519598	−0.259799	−	0.965663i	$-0.583656\pi$
$443$	−4844.77	−0.519598	−0.259799	+	0.965663i	$0.583656\pi$
$444$	0	0
$445$	−8929.72	−0.951257
$446$	−4631.31	−0.491701
$447$	0	0
$448$	−14681.6	−1.54830
$449$	8980.25	0.943885	0.471942	−	0.881629i	$-0.343553\pi$
$449$	8980.25	0.943885	0.471942	+	0.881629i	$0.343553\pi$
$450$	0	0
$451$	−3135.60	−0.327383
$452$	5338.06	0.555490
$453$	0	0
$454$	−4995.50	−0.516411
$455$	−6520.32	−0.671818
$456$	0	0
$457$	−13376.3	−1.36919	−0.684594	−	0.728925i	$-0.740020\pi$
$457$	−13376.3	−1.36919	−0.684594	+	0.728925i	$0.740020\pi$
$458$	8159.55	0.832469
$459$	0	0
$460$	−2232.99	−0.226334
$461$	10422.0	1.05293	0.526464	−	0.850198i	$-0.323518\pi$
$461$	10422.0	1.05293	0.526464	+	0.850198i	$0.323518\pi$
$462$	0	0
$463$	−2268.51	−0.227704	−0.113852	−	0.993498i	$-0.536319\pi$
$463$	−2268.51	−0.227704	−0.113852	+	0.993498i	$0.536319\pi$
$464$	5891.48	0.589451
$465$	0	0
$466$	−3338.31	−0.331855
$467$	−3737.92	−0.370386	−0.185193	−	0.982702i	$-0.559291\pi$
$467$	−3737.92	−0.370386	−0.185193	+	0.982702i	$0.559291\pi$
$468$	0	0
$469$	−17883.8	−1.76076
$470$	13081.7	1.28386
$471$	0	0
$472$	−3369.35	−0.328574
$473$	1794.89	0.174480
$474$	0	0
$475$	490.330	0.0473640
$476$	7760.04	0.747229
$477$	0	0
$478$	529.469	0.0506639
$479$	−7571.40	−0.722225	−0.361113	−	0.932522i	$-0.617603\pi$
$479$	−7571.40	−0.722225	−0.361113	+	0.932522i	$0.617603\pi$
$480$	0	0
$481$	−2345.80	−0.222369
$482$	10562.1	0.998108
$483$	0	0
$484$	2635.66	0.247526
$485$	4150.33	0.388570
$486$	0	0
$487$	8567.85	0.797220	0.398610	−	0.917120i	$-0.369493\pi$
$487$	8567.85	0.797220	0.398610	+	0.917120i	$0.369493\pi$
$488$	−1961.87	−0.181987
$489$	0	0
$490$	9810.64	0.904489
$491$	−20123.0	−1.84957	−0.924785	−	0.380491i	$-0.875755\pi$
$491$	−20123.0	−1.84957	−0.924785	+	0.380491i	$0.875755\pi$
$492$	0	0
$493$	−17880.4	−1.63345
$494$	−1243.54	−0.113258
$495$	0	0
$496$	2792.34	0.252781
$497$	−29134.9	−2.62953
$498$	0	0
$499$	5518.82	0.495103	0.247551	−	0.968875i	$-0.420374\pi$
$499$	5518.82	0.495103	0.247551	+	0.968875i	$0.420374\pi$
$500$	−4594.05	−0.410905
$501$	0	0
$502$	−7239.19	−0.643627
$503$	−2987.11	−0.264788	−0.132394	−	0.991197i	$-0.542266\pi$
$503$	−2987.11	−0.264788	−0.132394	+	0.991197i	$0.542266\pi$
$504$	0	0
$505$	−16276.6	−1.43426
$506$	−3415.02	−0.300032
$507$	0	0
$508$	2331.49	0.203629
$509$	−3096.36	−0.269634	−0.134817	−	0.990871i	$-0.543045\pi$
$509$	−3096.36	−0.269634	−0.134817	+	0.990871i	$0.543045\pi$
$510$	0	0
$511$	−18675.1	−1.61671
$512$	9723.16	0.839272
$513$	0	0
$514$	14678.5	1.25961
$515$	−17352.2	−1.48472
$516$	0	0
$517$	−12605.4	−1.07231
$518$	6331.34	0.537033
$519$	0	0
$520$	5755.09	0.485341
$521$	−16573.7	−1.39368	−0.696838	−	0.717229i	$-0.745410\pi$
$521$	−16573.7	−1.39368	−0.696838	+	0.717229i	$0.745410\pi$
$522$	0	0
$523$	5974.41	0.499508	0.249754	−	0.968309i	$-0.419650\pi$
$523$	5974.41	0.499508	0.249754	+	0.968309i	$0.419650\pi$
$524$	−5300.62	−0.441906
$525$	0	0
$526$	−1497.38	−0.124124
$527$	−8474.63	−0.700494
$528$	0	0
$529$	−7202.38	−0.591960
$530$	3440.43	0.281968
$531$	0	0
$532$	−2114.69	−0.172337
$533$	−3275.14	−0.266158
$534$	0	0
$535$	19014.4	1.53657
$536$	15784.9	1.27203
$537$	0	0
$538$	−11436.8	−0.916498
$539$	−9453.38	−0.755447
$540$	0	0
$541$	3100.93	0.246432	0.123216	−	0.992380i	$-0.460679\pi$
$541$	3100.93	0.246432	0.123216	+	0.992380i	$0.460679\pi$
$542$	−7818.91	−0.619651
$543$	0	0
$544$	−11789.2	−0.929153
$545$	−2225.55	−0.174922
$546$	0	0
$547$	−656.275	−0.0512985	−0.0256493	−	0.999671i	$-0.508165\pi$
$547$	−656.275	−0.0512985	−0.0256493	+	0.999671i	$0.508165\pi$
$548$	−3059.34	−0.238483
$549$	0	0
$550$	−967.474	−0.0750059
$551$	4872.59	0.376732
$552$	0	0
$553$	−8477.68	−0.651913
$554$	−8343.50	−0.639858
$555$	0	0
$556$	−3886.83	−0.296472
$557$	−4263.76	−0.324347	−0.162173	−	0.986762i	$-0.551850\pi$
$557$	−4263.76	−0.324347	−0.162173	+	0.986762i	$0.551850\pi$
$558$	0	0
$559$	1874.77	0.141850
$560$	−8474.54	−0.639491
$561$	0	0
$562$	−3428.17	−0.257310
$563$	−15339.8	−1.14831	−0.574154	−	0.818747i	$-0.694669\pi$
$563$	−15339.8	−1.14831	−0.574154	+	0.818747i	$0.694669\pi$
$564$	0	0
$565$	17692.5	1.31740
$566$	−561.730	−0.0417160
$567$	0	0
$568$	25715.6	1.89965
$569$	−18370.8	−1.35350	−0.676752	−	0.736211i	$-0.736613\pi$
$569$	−18370.8	−1.35350	−0.676752	+	0.736211i	$0.736613\pi$
$570$	0	0
$571$	−17329.6	−1.27009	−0.635046	−	0.772475i	$-0.719019\pi$
$571$	−17329.6	−1.27009	−0.635046	+	0.772475i	$0.719019\pi$
$572$	−1545.94	−0.113005
$573$	0	0
$574$	8839.63	0.642786
$575$	1406.48	0.102007
$576$	0	0
$577$	−12392.1	−0.894091	−0.447046	−	0.894511i	$-0.647524\pi$
$577$	−12392.1	−0.894091	−0.447046	+	0.894511i	$0.647524\pi$
$578$	−7116.15	−0.512098
$579$	0	0
$580$	−6286.43	−0.450052
$581$	18670.8	1.33321
$582$	0	0
$583$	−3315.15	−0.235505
$584$	16483.4	1.16796
$585$	0	0
$586$	−5855.67	−0.412791
$587$	15717.8	1.10518	0.552591	−	0.833452i	$-0.313639\pi$
$587$	15717.8	1.10518	0.552591	+	0.833452i	$0.313639\pi$
$588$	0	0
$589$	2309.42	0.161559
$590$	−3113.17	−0.217232
$591$	0	0
$592$	−3048.87	−0.211669
$593$	28306.2	1.96020	0.980099	−	0.198509i	$-0.0636097\pi$
$593$	28306.2	1.96020	0.980099	+	0.198509i	$0.0636097\pi$
$594$	0	0
$595$	25719.9	1.77212
$596$	−4421.55	−0.303882
$597$	0	0
$598$	−3566.99	−0.243922
$599$	−11458.3	−0.781592	−0.390796	−	0.920477i	$-0.627800\pi$
$599$	−11458.3	−0.781592	−0.390796	+	0.920477i	$0.627800\pi$
$600$	0	0
$601$	−3300.56	−0.224014	−0.112007	−	0.993707i	$-0.535728\pi$
$601$	−3300.56	−0.224014	−0.112007	+	0.993707i	$0.535728\pi$
$602$	−5060.01	−0.342576
$603$	0	0
$604$	3914.53	0.263708
$605$	8735.64	0.587032
$606$	0	0
$607$	27868.9	1.86353	0.931765	−	0.363061i	$-0.118268\pi$
$607$	27868.9	1.86353	0.931765	+	0.363061i	$0.118268\pi$
$608$	3212.68	0.214295
$609$	0	0
$610$	−1812.70	−0.120318
$611$	−13166.3	−0.871772
$612$	0	0
$613$	25440.4	1.67623	0.838114	−	0.545495i	$-0.183658\pi$
$613$	25440.4	1.67623	0.838114	+	0.545495i	$0.183658\pi$
$614$	−330.907	−0.0217497
$615$	0	0
$616$	14967.4	0.978982
$617$	−13460.8	−0.878299	−0.439149	−	0.898414i	$-0.644720\pi$
$617$	−13460.8	−0.878299	−0.439149	+	0.898414i	$0.644720\pi$
$618$	0	0
$619$	11014.7	0.715213	0.357607	−	0.933872i	$-0.383593\pi$
$619$	11014.7	0.715213	0.357607	+	0.933872i	$0.383593\pi$
$620$	−2979.53	−0.193001
$621$	0	0
$622$	−486.932	−0.0313894
$623$	−24257.2	−1.55994
$624$	0	0
$625$	−12731.4	−0.814808
$626$	15850.1	1.01197
$627$	0	0
$628$	5791.42	0.367998
$629$	9253.20	0.586565
$630$	0	0
$631$	−20608.3	−1.30016	−0.650082	−	0.759864i	$-0.725266\pi$
$631$	−20608.3	−1.30016	−0.650082	+	0.759864i	$0.725266\pi$
$632$	7482.74	0.470961
$633$	0	0
$634$	−904.842	−0.0566812
$635$	7727.51	0.482924
$636$	0	0
$637$	−9874.07	−0.614168
$638$	−9614.14	−0.596595
$639$	0	0
$640$	1249.87	0.0771960
$641$	14510.0	0.894088	0.447044	−	0.894512i	$-0.352477\pi$
$641$	14510.0	0.894088	0.447044	+	0.894512i	$0.352477\pi$
$642$	0	0
$643$	28587.0	1.75328	0.876642	−	0.481144i	$-0.159779\pi$
$643$	28587.0	1.75328	0.876642	+	0.481144i	$0.159779\pi$
$644$	−6065.83	−0.371160
$645$	0	0
$646$	4905.23	0.298752
$647$	7488.06	0.455001	0.227501	−	0.973778i	$-0.426945\pi$
$647$	7488.06	0.455001	0.227501	+	0.973778i	$0.426945\pi$
$648$	0	0
$649$	2999.80	0.181437
$650$	−1010.53	−0.0609788
$651$	0	0
$652$	7613.99	0.457342
$653$	14144.1	0.847630	0.423815	−	0.905749i	$-0.360691\pi$
$653$	14144.1	0.847630	0.423815	+	0.905749i	$0.360691\pi$
$654$	0	0
$655$	−17568.4	−1.04802
$656$	−4256.75	−0.253351
$657$	0	0
$658$	35536.0	2.10538
$659$	17924.3	1.05953	0.529766	−	0.848144i	$-0.322280\pi$
$659$	17924.3	1.05953	0.529766	+	0.848144i	$0.322280\pi$
$660$	0	0
$661$	11051.4	0.650301	0.325151	−	0.945662i	$-0.394585\pi$
$661$	11051.4	0.650301	0.325151	+	0.945662i	$0.394585\pi$
$662$	−25017.3	−1.46877
$663$	0	0
$664$	−16479.6	−0.963152
$665$	−7008.93	−0.408714
$666$	0	0
$667$	13976.7	0.811362
$668$	4586.50	0.265654
$669$	0	0
$670$	14584.8	0.840983
$671$	1746.69	0.100492
$672$	0	0
$673$	−15162.6	−0.868463	−0.434232	−	0.900801i	$-0.642980\pi$
$673$	−15162.6	−0.868463	−0.434232	+	0.900801i	$0.642980\pi$
$674$	−21262.0	−1.21510
$675$	0	0
$676$	5178.85	0.294655
$677$	20148.4	1.14382	0.571909	−	0.820317i	$-0.306203\pi$
$677$	20148.4	1.14382	0.571909	+	0.820317i	$0.306203\pi$
$678$	0	0
$679$	11274.2	0.637207
$680$	−22701.4	−1.28023
$681$	0	0
$682$	−4556.73	−0.255845
$683$	−2717.56	−0.152247	−0.0761234	−	0.997098i	$-0.524254\pi$
$683$	−2717.56	−0.152247	−0.0761234	+	0.997098i	$0.524254\pi$
$684$	0	0
$685$	−10139.9	−0.565584
$686$	5495.11	0.305837
$687$	0	0
$688$	2436.66	0.135024
$689$	−3462.68	−0.191462
$690$	0	0
$691$	−19272.2	−1.06100	−0.530498	−	0.847686i	$-0.677995\pi$
$691$	−19272.2	−1.06100	−0.530498	+	0.847686i	$0.677995\pi$
$692$	−2976.16	−0.163492
$693$	0	0
$694$	−27477.3	−1.50292
$695$	−12882.5	−0.703111
$696$	0	0
$697$	12919.1	0.702072
$698$	−11094.9	−0.601645
$699$	0	0
$700$	−1718.45	−0.0927875
$701$	−20644.5	−1.11232	−0.556158	−	0.831077i	$-0.687725\pi$
$701$	−20644.5	−1.11232	−0.556158	+	0.831077i	$0.687725\pi$
$702$	0	0
$703$	−2521.59	−0.135282
$704$	−11537.3	−0.617652
$705$	0	0
$706$	10353.7	0.551937
$707$	−44214.7	−2.35200
$708$	0	0
$709$	21394.8	1.13328	0.566641	−	0.823965i	$-0.308243\pi$
$709$	21394.8	1.13328	0.566641	+	0.823965i	$0.308243\pi$
$710$	23760.4	1.25593
$711$	0	0
$712$	21410.4	1.12695
$713$	6624.40	0.347947
$714$	0	0
$715$	−5123.87	−0.268003
$716$	−6423.60	−0.335281
$717$	0	0
$718$	26903.7	1.39838
$719$	33578.5	1.74168	0.870838	−	0.491569i	$-0.163577\pi$
$719$	33578.5	1.74168	0.870838	+	0.491569i	$0.163577\pi$
$720$	0	0
$721$	−47136.6	−2.43476
$722$	13858.4	0.714343
$723$	0	0
$724$	5531.86	0.283964
$725$	3959.59	0.202835
$726$	0	0
$727$	1964.38	0.100213	0.0501065	−	0.998744i	$-0.484044\pi$
$727$	1964.38	0.100213	0.0501065	+	0.998744i	$0.484044\pi$
$728$	15633.5	0.795899
$729$	0	0
$730$	15230.1	0.772179
$731$	−7395.16	−0.374172
$732$	0	0
$733$	3877.80	0.195402	0.0977011	−	0.995216i	$-0.468851\pi$
$733$	3877.80	0.195402	0.0977011	+	0.995216i	$0.468851\pi$
$734$	6891.60	0.346558
$735$	0	0
$736$	9215.34	0.461525
$737$	−14053.6	−0.702405
$738$	0	0
$739$	−32891.8	−1.63727	−0.818636	−	0.574313i	$-0.805269\pi$
$739$	−32891.8	−1.63727	−0.818636	+	0.574313i	$0.805269\pi$
$740$	3253.26	0.161611
$741$	0	0
$742$	9345.79	0.462392
$743$	13930.1	0.687814	0.343907	−	0.939004i	$-0.388250\pi$
$743$	13930.1	0.687814	0.343907	+	0.939004i	$0.388250\pi$
$744$	0	0
$745$	−14654.8	−0.720684
$746$	−9193.17	−0.451187
$747$	0	0
$748$	6098.08	0.298086
$749$	51651.8	2.51978
$750$	0	0
$751$	5671.20	0.275559	0.137780	−	0.990463i	$-0.456003\pi$
$751$	5671.20	0.275559	0.137780	+	0.990463i	$0.456003\pi$
$752$	−17112.5	−0.829823
$753$	0	0
$754$	−10042.0	−0.485023
$755$	12974.3	0.625409
$756$	0	0
$757$	−4922.18	−0.236327	−0.118164	−	0.992994i	$-0.537701\pi$
$757$	−4922.18	−0.236327	−0.118164	+	0.992994i	$0.537701\pi$
$758$	13015.0	0.623651
$759$	0	0
$760$	6186.36	0.295267
$761$	2738.12	0.130430	0.0652148	−	0.997871i	$-0.479227\pi$
$761$	2738.12	0.130430	0.0652148	+	0.997871i	$0.479227\pi$
$762$	0	0
$763$	−6045.62	−0.286850
$764$	5580.34	0.264254
$765$	0	0
$766$	13378.5	0.631053
$767$	3133.29	0.147505
$768$	0	0
$769$	18076.4	0.847659	0.423830	−	0.905742i	$-0.360686\pi$
$769$	18076.4	0.847659	0.423830	+	0.905742i	$0.360686\pi$
$770$	13829.4	0.647241
$771$	0	0
$772$	16324.7	0.761061
$773$	−39916.7	−1.85731	−0.928657	−	0.370939i	$-0.879036\pi$
$773$	−39916.7	−1.85731	−0.928657	+	0.370939i	$0.879036\pi$
$774$	0	0
$775$	1876.69	0.0869842
$776$	−9951.05	−0.460337
$777$	0	0
$778$	15444.2	0.711700
$779$	−3520.57	−0.161922
$780$	0	0
$781$	−22895.1	−1.04898
$782$	14070.3	0.643417
$783$	0	0
$784$	−12833.5	−0.584615
$785$	19195.1	0.872742
$786$	0	0
$787$	−27502.9	−1.24571	−0.622853	−	0.782339i	$-0.714027\pi$
$787$	−27502.9	−1.24571	−0.622853	+	0.782339i	$0.714027\pi$
$788$	3841.05	0.173644
$789$	0	0
$790$	6913.80	0.311370
$791$	48060.9	2.16036
$792$	0	0
$793$	1824.42	0.0816986
$794$	17727.7	0.792359
$795$	0	0
$796$	−4319.92	−0.192356
$797$	22667.6	1.00744	0.503719	−	0.863867i	$-0.331965\pi$
$797$	22667.6	1.00744	0.503719	+	0.863867i	$0.331965\pi$
$798$	0	0
$799$	51935.6	2.29956
$800$	2610.71	0.115378
$801$	0	0
$802$	21957.2	0.966751
$803$	−14675.5	−0.644939
$804$	0	0
$805$	−20104.6	−0.880241
$806$	−4759.52	−0.207999
$807$	0	0
$808$	39025.7	1.69916
$809$	−3808.73	−0.165522	−0.0827612	−	0.996569i	$-0.526374\pi$
$809$	−3808.73	−0.165522	−0.0827612	+	0.996569i	$0.526374\pi$
$810$	0	0
$811$	−8292.93	−0.359068	−0.179534	−	0.983752i	$-0.557459\pi$
$811$	−8292.93	−0.359068	−0.179534	+	0.983752i	$0.557459\pi$
$812$	−17076.8	−0.738029
$813$	0	0
$814$	4975.36	0.214234
$815$	25235.8	1.08463
$816$	0	0
$817$	2015.26	0.0862973
$818$	−16196.1	−0.692280
$819$	0	0
$820$	4542.11	0.193436
$821$	−15574.1	−0.662048	−0.331024	−	0.943622i	$-0.607394\pi$
$821$	−15574.1	−0.662048	−0.331024	+	0.943622i	$0.607394\pi$
$822$	0	0
$823$	21639.9	0.916550	0.458275	−	0.888810i	$-0.348467\pi$
$823$	21639.9	0.916550	0.458275	+	0.888810i	$0.348467\pi$
$824$	41604.7	1.75894
$825$	0	0
$826$	−8456.78	−0.356234
$827$	−40211.4	−1.69080	−0.845398	−	0.534137i	$-0.820637\pi$
$827$	−40211.4	−1.69080	−0.845398	+	0.534137i	$0.820637\pi$
$828$	0	0
$829$	−5442.06	−0.227998	−0.113999	−	0.993481i	$-0.536366\pi$
$829$	−5442.06	−0.227998	−0.113999	+	0.993481i	$0.536366\pi$
$830$	−15226.6	−0.636775
$831$	0	0
$832$	−12050.7	−0.502143
$833$	38949.0	1.62005
$834$	0	0
$835$	15201.5	0.630024
$836$	−1661.79	−0.0687490
$837$	0	0
$838$	−29543.2	−1.21784
$839$	43164.2	1.77615	0.888076	−	0.459696i	$-0.152042\pi$
$839$	43164.2	1.77615	0.888076	+	0.459696i	$0.152042\pi$
$840$	0	0
$841$	14958.8	0.613343
$842$	27887.9	1.14143
$843$	0	0
$844$	−3019.42	−0.123143
$845$	17164.8	0.698801
$846$	0	0
$847$	23730.0	0.962659
$848$	−4500.49	−0.182249
$849$	0	0
$850$	3986.11	0.160850
$851$	−7232.99	−0.291356
$852$	0	0
$853$	45061.8	1.80878	0.904388	−	0.426711i	$-0.140328\pi$
$853$	45061.8	1.80878	0.904388	+	0.426711i	$0.140328\pi$
$854$	−4924.12	−0.197307
$855$	0	0
$856$	−45589.9	−1.82036
$857$	−45245.9	−1.80347	−0.901733	−	0.432292i	$-0.857705\pi$
$857$	−45245.9	−1.80347	−0.901733	+	0.432292i	$0.857705\pi$
$858$	0	0
$859$	−6435.73	−0.255628	−0.127814	−	0.991798i	$-0.540796\pi$
$859$	−6435.73	−0.255628	−0.127814	+	0.991798i	$0.540796\pi$
$860$	−2600.01	−0.103092
$861$	0	0
$862$	−1068.23	−0.0422088
$863$	−32153.6	−1.26828	−0.634138	−	0.773220i	$-0.718645\pi$
$863$	−32153.6	−1.26828	−0.634138	+	0.773220i	$0.718645\pi$
$864$	0	0
$865$	−9864.19	−0.387737
$866$	4262.39	0.167254
$867$	0	0
$868$	−8093.77	−0.316498
$869$	−6662.03	−0.260062
$870$	0	0
$871$	−14679.1	−0.571046
$872$	5336.11	0.207229
$873$	0	0
$874$	−3834.29	−0.148395
$875$	−41362.3	−1.59806
$876$	0	0
$877$	−13952.8	−0.537230	−0.268615	−	0.963248i	$-0.586566\pi$
$877$	−13952.8	−0.537230	−0.268615	+	0.963248i	$0.586566\pi$
$878$	−15038.7	−0.578055
$879$	0	0
$880$	−6659.57	−0.255107
$881$	49200.9	1.88152	0.940761	−	0.339071i	$-0.110113\pi$
$881$	49200.9	1.88152	0.940761	+	0.339071i	$0.110113\pi$
$882$	0	0
$883$	−10202.3	−0.388827	−0.194414	−	0.980920i	$-0.562280\pi$
$883$	−10202.3	−0.388827	−0.194414	+	0.980920i	$0.562280\pi$
$884$	6369.46	0.242340
$885$	0	0
$886$	10732.9	0.406972
$887$	27655.0	1.04686	0.523429	−	0.852069i	$-0.324653\pi$
$887$	27655.0	1.04686	0.523429	+	0.852069i	$0.324653\pi$
$888$	0	0
$889$	20991.5	0.791935
$890$	19782.5	0.745068
$891$	0	0
$892$	−6464.42	−0.242651
$893$	−14153.0	−0.530360
$894$	0	0
$895$	−21290.4	−0.795150
$896$	3395.22	0.126592
$897$	0	0
$898$	−19894.4	−0.739294
$899$	18649.4	0.691870
$900$	0	0
$901$	13658.8	0.505039
$902$	6946.46	0.256421
$903$	0	0
$904$	−42420.5	−1.56071
$905$	18334.8	0.673448
$906$	0	0
$907$	9311.73	0.340894	0.170447	−	0.985367i	$-0.445479\pi$
$907$	9311.73	0.340894	0.170447	+	0.985367i	$0.445479\pi$
$908$	−6972.77	−0.254845
$909$	0	0
$910$	14444.8	0.526198
$911$	−22161.9	−0.805988	−0.402994	−	0.915203i	$-0.632030\pi$
$911$	−22161.9	−0.805988	−0.402994	+	0.915203i	$0.632030\pi$
$912$	0	0
$913$	14672.1	0.531847
$914$	29633.3	1.07241
$915$	0	0
$916$	11389.2	0.410818
$917$	−47723.8	−1.71862
$918$	0	0
$919$	25869.8	0.928583	0.464291	−	0.885683i	$-0.346309\pi$
$919$	25869.8	0.928583	0.464291	+	0.885683i	$0.346309\pi$
$920$	17745.1	0.635912
$921$	0	0
$922$	−23088.3	−0.824701
$923$	−23914.0	−0.852804
$924$	0	0
$925$	−2049.11	−0.0728369
$926$	5025.56	0.178348
$927$	0	0
$928$	25943.5	0.917713
$929$	24629.8	0.869837	0.434919	−	0.900470i	$-0.356777\pi$
$929$	24629.8	0.869837	0.434919	+	0.900470i	$0.356777\pi$
$930$	0	0
$931$	−10614.0	−0.373641
$932$	−4659.65	−0.163768
$933$	0	0
$934$	8280.80	0.290103
$935$	20211.5	0.706938
$936$	0	0
$937$	−4766.88	−0.166198	−0.0830989	−	0.996541i	$-0.526482\pi$
$937$	−4766.88	−0.166198	−0.0830989	+	0.996541i	$0.526482\pi$
$938$	39618.9	1.37911
$939$	0	0
$940$	18259.6	0.633578
$941$	−6261.98	−0.216934	−0.108467	−	0.994100i	$-0.534594\pi$
$941$	−6261.98	−0.216934	−0.108467	+	0.994100i	$0.534594\pi$
$942$	0	0
$943$	−10098.5	−0.348730
$944$	4072.38	0.140408
$945$	0	0
$946$	−3976.31	−0.136661
$947$	52991.6	1.81837	0.909184	−	0.416394i	$-0.136706\pi$
$947$	52991.6	1.81837	0.909184	+	0.416394i	$0.136706\pi$
$948$	0	0
$949$	−15328.6	−0.524327
$950$	−1086.25	−0.0370976
$951$	0	0
$952$	−61667.4	−2.09942
$953$	−37225.5	−1.26532	−0.632662	−	0.774428i	$-0.718038\pi$
$953$	−37225.5	−1.26532	−0.632662	+	0.774428i	$0.718038\pi$
$954$	0	0
$955$	18495.5	0.626702
$956$	739.039	0.0250023
$957$	0	0
$958$	16773.3	0.565680
$959$	−27544.6	−0.927487
$960$	0	0
$961$	−20951.9	−0.703297
$962$	5196.78	0.174169
$963$	0	0
$964$	14742.6	0.492560
$965$	54106.7	1.80493
$966$	0	0
$967$	−13362.6	−0.444376	−0.222188	−	0.975004i	$-0.571320\pi$
$967$	−13362.6	−0.444376	−0.222188	+	0.975004i	$0.571320\pi$
$968$	−20945.0	−0.695453
$969$	0	0
$970$	−9194.44	−0.304346
$971$	−6037.96	−0.199554	−0.0997772	−	0.995010i	$-0.531813\pi$
$971$	−6037.96	−0.199554	−0.0997772	+	0.995010i	$0.531813\pi$
$972$	0	0
$973$	−34994.8	−1.15301
$974$	−18980.8	−0.624419
$975$	0	0
$976$	2371.22	0.0777674
$977$	17246.7	0.564759	0.282379	−	0.959303i	$-0.408876\pi$
$977$	17246.7	0.564759	0.282379	+	0.959303i	$0.408876\pi$
$978$	0	0
$979$	−19062.1	−0.622295
$980$	13693.8	0.446359
$981$	0	0
$982$	44579.6	1.44867
$983$	−14783.0	−0.479657	−0.239829	−	0.970815i	$-0.577091\pi$
$983$	−14783.0	−0.479657	−0.239829	+	0.970815i	$0.577091\pi$
$984$	0	0
$985$	12730.8	0.411813
$986$	39611.4	1.27940
$987$	0	0
$988$	−1735.74	−0.0558920
$989$	5780.61	0.185857
$990$	0	0
$991$	−34717.6	−1.11286	−0.556428	−	0.830896i	$-0.687828\pi$
$991$	−34717.6	−1.11286	−0.556428	+	0.830896i	$0.687828\pi$
$992$	12296.2	0.393554
$993$	0	0
$994$	64544.0	2.05957
$995$	−14318.0	−0.456191
$996$	0	0
$997$	−6648.86	−0.211205	−0.105602	−	0.994408i	$-0.533677\pi$
$997$	−6648.86	−0.211205	−0.105602	+	0.994408i	$0.533677\pi$
$998$	−12226.1	−0.387787
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2151.4.a.a.1.8		22
3.2	odd	2		239.4.a.a.1.15	✓	22

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
239.4.a.a.1.15	✓	22	3.2	odd	2
2151.4.a.a.1.8		22	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)

\(f(q)\)	\(=\)	\(q-2.21535 q^{2} -3.09221 q^{4} -10.2488 q^{5} -27.8405 q^{7} +24.5732 q^{8} +O(q^{10})\)	\(q-2.21535 q^{2} -3.09221 q^{4} -10.2488 q^{5} -27.8405 q^{7} +24.5732 q^{8} +22.7048 q^{10} -21.8780 q^{11} -22.8516 q^{13} +61.6766 q^{14} -29.7005 q^{16} +90.1399 q^{17} -24.5640 q^{19} +31.6916 q^{20} +48.4674 q^{22} -70.4601 q^{23} -19.9613 q^{25} +50.6243 q^{26} +86.0889 q^{28} -198.363 q^{29} -94.0164 q^{31} -130.788 q^{32} -199.692 q^{34} +285.333 q^{35} +102.654 q^{37} +54.4180 q^{38} -251.846 q^{40} +143.322 q^{41} -82.0409 q^{43} +67.6514 q^{44} +156.094 q^{46} +576.167 q^{47} +432.095 q^{49} +44.2214 q^{50} +70.6620 q^{52} +151.529 q^{53} +224.224 q^{55} -684.130 q^{56} +439.444 q^{58} -137.115 q^{59} -79.8377 q^{61} +208.280 q^{62} +527.346 q^{64} +234.202 q^{65} +642.365 q^{67} -278.732 q^{68} -632.114 q^{70} +1046.49 q^{71} +670.788 q^{73} -227.414 q^{74} +75.9572 q^{76} +609.095 q^{77} +304.508 q^{79} +304.396 q^{80} -317.509 q^{82} -670.634 q^{83} -923.829 q^{85} +181.750 q^{86} -537.611 q^{88} +871.291 q^{89} +636.201 q^{91} +217.878 q^{92} -1276.41 q^{94} +251.753 q^{95} -404.956 q^{97} -957.244 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 22 q + 4 q^{2} + 50 q^{4} + 37 q^{5} - 52 q^{7} + 69 q^{8}+O(q^{10}) \) 22 * q + 4 * q^2 + 50 * q^4 + 37 * q^5 - 52 * q^7 + 69 * q^8	\( 22 q + 4 q^{2} + 50 q^{4} + 37 q^{5} - 52 q^{7} + 69 q^{8} - 93 q^{10} + 77 q^{11} - 218 q^{13} + 111 q^{14} - 42 q^{16} + 219 q^{17} - 476 q^{19} + 314 q^{20} - 390 q^{22} + 202 q^{23} - 271 q^{25} + 220 q^{26} - 515 q^{28} + 307 q^{29} - 1001 q^{31} + 771 q^{32} - 1297 q^{34} + 430 q^{35} - 922 q^{37} - 49 q^{38} - 1344 q^{40} + 1188 q^{41} - 192 q^{43} + 547 q^{44} - 1178 q^{46} + 102 q^{47} - 1952 q^{49} + 471 q^{50} - 1785 q^{52} + 580 q^{53} - 1730 q^{55} + 804 q^{56} - 1156 q^{58} + 1528 q^{59} - 1631 q^{61} - 2206 q^{62} + 327 q^{64} - 44 q^{65} - 689 q^{67} - 2522 q^{68} + 1175 q^{70} - 341 q^{71} - 2260 q^{73} - 4027 q^{74} - 1855 q^{76} - 1578 q^{77} + 396 q^{79} - 6183 q^{80} + 4936 q^{82} - 1065 q^{83} + 144 q^{85} - 2915 q^{86} + 1068 q^{88} + 1984 q^{89} - 2186 q^{91} - 6720 q^{92} + 174 q^{94} - 2804 q^{95} - 4946 q^{97} - 7149 q^{98}+O(q^{100}) \) 22 * q + 4 * q^2 + 50 * q^4 + 37 * q^5 - 52 * q^7 + 69 * q^8 - 93 * q^10 + 77 * q^11 - 218 * q^13 + 111 * q^14 - 42 * q^16 + 219 * q^17 - 476 * q^19 + 314 * q^20 - 390 * q^22 + 202 * q^23 - 271 * q^25 + 220 * q^26 - 515 * q^28 + 307 * q^29 - 1001 * q^31 + 771 * q^32 - 1297 * q^34 + 430 * q^35 - 922 * q^37 - 49 * q^38 - 1344 * q^40 + 1188 * q^41 - 192 * q^43 + 547 * q^44 - 1178 * q^46 + 102 * q^47 - 1952 * q^49 + 471 * q^50 - 1785 * q^52 + 580 * q^53 - 1730 * q^55 + 804 * q^56 - 1156 * q^58 + 1528 * q^59 - 1631 * q^61 - 2206 * q^62 + 327 * q^64 - 44 * q^65 - 689 * q^67 - 2522 * q^68 + 1175 * q^70 - 341 * q^71 - 2260 * q^73 - 4027 * q^74 - 1855 * q^76 - 1578 * q^77 + 396 * q^79 - 6183 * q^80 + 4936 * q^82 - 1065 * q^83 + 144 * q^85 - 2915 * q^86 + 1068 * q^88 + 1984 * q^89 - 2186 * q^91 - 6720 * q^92 + 174 * q^94 - 2804 * q^95 - 4946 * q^97 - 7149 * q^98

Introduction

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