LMFDB - Embedded newform 639.2.a.k.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [639,2,Mod(1,639)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(639, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("639.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 639 = 3^{2} \cdot 71 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	639.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:	$5.10244068916$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.8112.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 5x^{2} + 3 $ x^4 - 5*x^2 + 3
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$2.07431$ of defining polynomial
Character	$\chi$	$=$	639.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.07431 q^{2} +2.30278 q^{4} -0.0743133 q^{5} +1.39959 q^{7} +0.628052 q^{8} +O(q^{10})$	\(q+2.07431 q^{2} +2.30278 q^{4} -0.0743133 q^{5} +1.39959 q^{7} +0.628052 q^{8} -0.154149 q^{10} +6.37709 q^{11} -2.33042 q^{13} +2.90319 q^{14} -3.30278 q^{16} -0.377089 q^{17} +7.75418 q^{19} -0.171127 q^{20} +13.2281 q^{22} +5.47390 q^{23} -4.99448 q^{25} -4.83401 q^{26} +3.22294 q^{28} -8.56402 q^{29} -3.25610 q^{31} -8.10709 q^{32} -0.782201 q^{34} -0.104008 q^{35} -7.65222 q^{37} +16.0846 q^{38} -0.0466726 q^{40} +9.41539 q^{41} -0.589748 q^{43} +14.6850 q^{44} +11.3546 q^{46} -7.65736 q^{47} -5.04115 q^{49} -10.3601 q^{50} -5.36643 q^{52} +3.19016 q^{53} -0.473902 q^{55} +0.879014 q^{56} -17.7645 q^{58} +5.43956 q^{59} -11.4099 q^{61} -6.75418 q^{62} -10.2111 q^{64} +0.173181 q^{65} -8.47904 q^{67} -0.868351 q^{68} -0.215745 q^{70} +1.00000 q^{71} +9.40473 q^{73} -15.8731 q^{74} +17.8561 q^{76} +8.92530 q^{77} -9.88700 q^{79} +0.245440 q^{80} +19.5305 q^{82} +1.08498 q^{83} +0.0280227 q^{85} -1.22332 q^{86} +4.00514 q^{88} +5.17832 q^{89} -3.26163 q^{91} +12.6052 q^{92} -15.8838 q^{94} -0.576238 q^{95} -7.48110 q^{97} -10.4569 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{4} + 8 q^{5} + 2 q^{7}+O(q^{10}) $ 4 * q + 2 * q^4 + 8 * q^5 + 2 * q^7	$ 4 q + 2 q^{4} + 8 q^{5} + 2 q^{7} - 10 q^{10} + 10 q^{11} + 4 q^{13} + 8 q^{14} - 6 q^{16} + 14 q^{17} + 4 q^{20} + 10 q^{22} + 10 q^{23} + 6 q^{25} - 6 q^{26} - 12 q^{28} - 4 q^{29} - 8 q^{31} - 10 q^{34} - 4 q^{35} - 14 q^{37} + 20 q^{38} + 2 q^{40} + 24 q^{41} + 2 q^{43} + 18 q^{44} + 18 q^{46} + 4 q^{47} + 8 q^{49} - 40 q^{50} + 2 q^{52} + 12 q^{53} + 10 q^{55} + 14 q^{56} - 4 q^{58} - 6 q^{59} - 6 q^{61} + 4 q^{62} - 12 q^{64} + 14 q^{65} - 4 q^{67} - 6 q^{68} + 24 q^{70} + 4 q^{71} + 16 q^{73} - 4 q^{74} + 26 q^{76} - 2 q^{79} - 12 q^{80} - 16 q^{82} + 4 q^{83} + 38 q^{85} - 24 q^{86} - 2 q^{88} + 16 q^{89} - 34 q^{91} - 8 q^{92} - 12 q^{94} - 20 q^{95} - 18 q^{97} - 44 q^{98}+O(q^{100}) $ 4 * q + 2 * q^4 + 8 * q^5 + 2 * q^7 - 10 * q^10 + 10 * q^11 + 4 * q^13 + 8 * q^14 - 6 * q^16 + 14 * q^17 + 4 * q^20 + 10 * q^22 + 10 * q^23 + 6 * q^25 - 6 * q^26 - 12 * q^28 - 4 * q^29 - 8 * q^31 - 10 * q^34 - 4 * q^35 - 14 * q^37 + 20 * q^38 + 2 * q^40 + 24 * q^41 + 2 * q^43 + 18 * q^44 + 18 * q^46 + 4 * q^47 + 8 * q^49 - 40 * q^50 + 2 * q^52 + 12 * q^53 + 10 * q^55 + 14 * q^56 - 4 * q^58 - 6 * q^59 - 6 * q^61 + 4 * q^62 - 12 * q^64 + 14 * q^65 - 4 * q^67 - 6 * q^68 + 24 * q^70 + 4 * q^71 + 16 * q^73 - 4 * q^74 + 26 * q^76 - 2 * q^79 - 12 * q^80 - 16 * q^82 + 4 * q^83 + 38 * q^85 - 24 * q^86 - 2 * q^88 + 16 * q^89 - 34 * q^91 - 8 * q^92 - 12 * q^94 - 20 * q^95 - 18 * q^97 - 44 * 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 $

$2$	2.07431	1.46676	0.733380	−	0.679818i	$-0.237941\pi$
$2$	2.07431	1.46676	0.733380	+	0.679818i	$0.237941\pi$
$3$	0	0
$3$	0	0
$4$	2.30278	1.15139
$5$	−0.0743133	−0.0332339	−0.0166170	−	0.999862i	$-0.505290\pi$
$5$	−0.0743133	−0.0332339	−0.0166170	+	0.999862i	$0.505290\pi$
$6$	0	0
$7$	1.39959	0.528995	0.264497	−	0.964386i	$-0.414794\pi$
$7$	1.39959	0.528995	0.264497	+	0.964386i	$0.414794\pi$
$8$	0.628052	0.222050
$9$	0	0
$10$	−0.154149	−0.0487462
$11$	6.37709	1.92276	0.961382	−	0.275216i	$-0.0887495\pi$
$11$	6.37709	1.92276	0.961382	+	0.275216i	$0.0887495\pi$
$12$	0	0
$13$	−2.33042	−0.646341	−0.323171	−	0.946341i	$-0.604749\pi$
$13$	−2.33042	−0.646341	−0.323171	+	0.946341i	$0.604749\pi$
$14$	2.90319	0.775909
$15$	0	0
$16$	−3.30278	−0.825694
$17$	−0.377089	−0.0914575	−0.0457287	−	0.998954i	$-0.514561\pi$
$17$	−0.377089	−0.0914575	−0.0457287	+	0.998954i	$0.514561\pi$
$18$	0	0
$19$	7.75418	1.77893	0.889465	−	0.457003i	$-0.151077\pi$
$19$	7.75418	1.77893	0.889465	+	0.457003i	$0.151077\pi$
$20$	−0.171127	−0.0382651
$21$	0	0
$22$	13.2281	2.82024
$23$	5.47390	1.14139	0.570694	−	0.821163i	$-0.306674\pi$
$23$	5.47390	1.14139	0.570694	+	0.821163i	$0.306674\pi$
$24$	0	0
$25$	−4.99448	−0.998896
$26$	−4.83401	−0.948028
$27$	0	0
$28$	3.22294	0.609078
$29$	−8.56402	−1.59030	−0.795149	−	0.606414i	$-0.792608\pi$
$29$	−8.56402	−1.59030	−0.795149	+	0.606414i	$0.792608\pi$
$30$	0	0
$31$	−3.25610	−0.584813	−0.292407	−	0.956294i	$-0.594456\pi$
$31$	−3.25610	−0.584813	−0.292407	+	0.956294i	$0.594456\pi$
$32$	−8.10709	−1.43315
$33$	0	0
$34$	−0.782201	−0.134146
$35$	−0.104008	−0.0175806
$36$	0	0
$37$	−7.65222	−1.25802	−0.629009	−	0.777398i	$-0.716539\pi$
$37$	−7.65222	−1.25802	−0.629009	+	0.777398i	$0.716539\pi$
$38$	16.0846	2.60927
$39$	0	0
$40$	−0.0466726	−0.00737958
$41$	9.41539	1.47044	0.735219	−	0.677830i	$-0.237079\pi$
$41$	9.41539	1.47044	0.735219	+	0.677830i	$0.237079\pi$
$42$	0	0
$43$	−0.589748	−0.0899357	−0.0449679	−	0.998988i	$-0.514319\pi$
$43$	−0.589748	−0.0899357	−0.0449679	+	0.998988i	$0.514319\pi$
$44$	14.6850	2.21385
$45$	0	0
$46$	11.3546	1.67414
$47$	−7.65736	−1.11694	−0.558471	−	0.829524i	$-0.688612\pi$
$47$	−7.65736	−1.11694	−0.558471	+	0.829524i	$0.688612\pi$
$48$	0	0
$49$	−5.04115	−0.720164
$50$	−10.3601	−1.46514
$51$	0	0
$52$	−5.36643	−0.744189
$53$	3.19016	0.438202	0.219101	−	0.975702i	$-0.429688\pi$
$53$	3.19016	0.438202	0.219101	+	0.975702i	$0.429688\pi$
$54$	0	0
$55$	−0.473902	−0.0639010
$56$	0.879014	0.117463
$57$	0	0
$58$	−17.7645	−2.33259
$59$	5.43956	0.708171	0.354086	−	0.935213i	$-0.384792\pi$
$59$	5.43956	0.708171	0.354086	+	0.935213i	$0.384792\pi$
$60$	0	0
$61$	−11.4099	−1.46088	−0.730442	−	0.682975i	$-0.760686\pi$
$61$	−11.4099	−1.46088	−0.730442	+	0.682975i	$0.760686\pi$
$62$	−6.75418	−0.857781
$63$	0	0
$64$	−10.2111	−1.27639
$65$	0.173181	0.0214804
$66$	0	0
$67$	−8.47904	−1.03588	−0.517940	−	0.855417i	$-0.673301\pi$
$67$	−8.47904	−1.03588	−0.517940	+	0.855417i	$0.673301\pi$
$68$	−0.868351	−0.105303
$69$	0	0
$70$	−0.215745	−0.0257865
$71$	1.00000	0.118678
$71$	1.00000	0.118678
$72$	0	0
$73$	9.40473	1.10074	0.550370	−	0.834921i	$-0.314487\pi$
$73$	9.40473	1.10074	0.550370	+	0.834921i	$0.314487\pi$
$74$	−15.8731	−1.84521
$75$	0	0
$76$	17.8561	2.04824
$77$	8.92530	1.01713
$78$	0	0
$79$	−9.88700	−1.11237	−0.556187	−	0.831057i	$-0.687736\pi$
$79$	−9.88700	−1.11237	−0.556187	+	0.831057i	$0.687736\pi$
$80$	0.245440	0.0274410
$81$	0	0
$82$	19.5305	2.15678
$83$	1.08498	0.119092	0.0595458	−	0.998226i	$-0.481035\pi$
$83$	1.08498	0.119092	0.0595458	+	0.998226i	$0.481035\pi$
$84$	0	0
$85$	0.0280227	0.00303949
$86$	−1.22332	−0.131914
$87$	0	0
$88$	4.00514	0.426949
$89$	5.17832	0.548901	0.274450	−	0.961601i	$-0.411504\pi$
$89$	5.17832	0.548901	0.274450	+	0.961601i	$0.411504\pi$
$90$	0	0
$91$	−3.26163	−0.341911
$92$	12.6052	1.31418
$93$	0	0
$94$	−15.8838	−1.63829
$95$	−0.576238	−0.0591208
$96$	0	0
$97$	−7.48110	−0.759590	−0.379795	−	0.925071i	$-0.624006\pi$
$97$	−7.48110	−0.759590	−0.379795	+	0.925071i	$0.624006\pi$
$98$	−10.4569	−1.05631
$99$	0	0
$100$	−11.5012	−1.15012
$101$	13.0739	1.30090	0.650452	−	0.759547i	$-0.274579\pi$
$101$	13.0739	1.30090	0.650452	+	0.759547i	$0.274579\pi$
$102$	0	0
$103$	2.55336	0.251590	0.125795	−	0.992056i	$-0.459852\pi$
$103$	2.55336	0.251590	0.125795	+	0.992056i	$0.459852\pi$
$104$	−1.46362	−0.143520
$105$	0	0
$106$	6.61739	0.642738
$107$	−10.2557	−0.991458	−0.495729	−	0.868477i	$-0.665099\pi$
$107$	−10.2557	−0.991458	−0.495729	+	0.868477i	$0.665099\pi$
$108$	0	0
$109$	−7.34225	−0.703260	−0.351630	−	0.936139i	$-0.614373\pi$
$109$	−7.34225	−0.703260	−0.351630	+	0.936139i	$0.614373\pi$
$110$	−0.983022	−0.0937275
$111$	0	0
$112$	−4.62253	−0.436788
$113$	−2.59527	−0.244142	−0.122071	−	0.992521i	$-0.538954\pi$
$113$	−2.59527	−0.244142	−0.122071	+	0.992521i	$0.538954\pi$
$114$	0	0
$115$	−0.406784	−0.0379328
$116$	−19.7210	−1.83105
$117$	0	0
$118$	11.2834	1.03872
$119$	−0.527770	−0.0483806
$120$	0	0
$121$	29.6673	2.69702
$122$	−23.6676	−2.14277
$123$	0	0
$124$	−7.49807	−0.673347
$125$	0.742723	0.0664311
$126$	0	0
$127$	−17.5798	−1.55996	−0.779978	−	0.625806i	$-0.784770\pi$
$127$	−17.5798	−1.55996	−0.779978	+	0.625806i	$0.784770\pi$
$128$	−4.96684	−0.439010
$129$	0	0
$130$	0.359231	0.0315067
$131$	1.20082	0.104916	0.0524581	−	0.998623i	$-0.483294\pi$
$131$	1.20082	0.104916	0.0524581	+	0.998623i	$0.483294\pi$
$132$	0	0
$133$	10.8527	0.941045
$134$	−17.5882	−1.51939
$135$	0	0
$136$	−0.236831	−0.0203081
$137$	0.0450005	0.00384465	0.00192233	−	0.999998i	$-0.499388\pi$
$137$	0.0450005	0.00384465	0.00192233	+	0.999998i	$0.499388\pi$
$138$	0	0
$139$	16.2906	1.38175	0.690873	−	0.722976i	$-0.257226\pi$
$139$	16.2906	1.38175	0.690873	+	0.722976i	$0.257226\pi$
$140$	−0.239507	−0.0202421
$141$	0	0
$142$	2.07431	0.174073
$143$	−14.8613	−1.24276
$144$	0	0
$145$	0.636420	0.0528518
$146$	19.5084	1.61452
$147$	0	0
$148$	−17.6214	−1.44847
$149$	3.32528	0.272417	0.136209	−	0.990680i	$-0.456508\pi$
$149$	3.32528	0.272417	0.136209	+	0.990680i	$0.456508\pi$
$150$	0	0
$151$	2.50360	0.203740	0.101870	−	0.994798i	$-0.467517\pi$
$151$	2.50360	0.203740	0.101870	+	0.994798i	$0.467517\pi$
$152$	4.87002	0.395011
$153$	0	0
$154$	18.5139	1.49189
$155$	0.241972	0.0194356
$156$	0	0
$157$	21.6783	1.73012	0.865059	−	0.501671i	$-0.167281\pi$
$157$	21.6783	1.73012	0.865059	+	0.501671i	$0.167281\pi$
$158$	−20.5087	−1.63159
$159$	0	0
$160$	0.602465	0.0476290
$161$	7.66121	0.603788
$162$	0	0
$163$	15.6416	1.22514	0.612571	−	0.790416i	$-0.290135\pi$
$163$	15.6416	1.22514	0.612571	+	0.790416i	$0.290135\pi$
$164$	21.6815	1.69304
$165$	0	0
$166$	2.25058	0.174679
$167$	−20.7214	−1.60347	−0.801735	−	0.597680i	$-0.796089\pi$
$167$	−20.7214	−1.60347	−0.801735	+	0.597680i	$0.796089\pi$
$168$	0	0
$169$	−7.56916	−0.582243
$170$	0.0581279	0.00445821
$171$	0	0
$172$	−1.35806	−0.103551
$173$	−18.9163	−1.43818	−0.719090	−	0.694917i	$-0.755441\pi$
$173$	−18.9163	−1.43818	−0.719090	+	0.694917i	$0.755441\pi$
$174$	0	0
$175$	−6.99022	−0.528411
$176$	−21.0621	−1.58762
$177$	0	0
$178$	10.7415	0.805107
$179$	14.0724	1.05182	0.525909	−	0.850541i	$-0.323725\pi$
$179$	14.0724	1.05182	0.525909	+	0.850541i	$0.323725\pi$
$180$	0	0
$181$	8.21060	0.610289	0.305145	−	0.952306i	$-0.401295\pi$
$181$	8.21060	0.610289	0.305145	+	0.952306i	$0.401295\pi$
$182$	−6.76563	−0.501502
$183$	0	0
$184$	3.43789	0.253445
$185$	0.568662	0.0418089
$186$	0	0
$187$	−2.40473	−0.175851
$188$	−17.6332	−1.28603
$189$	0	0
$190$	−1.19530	−0.0867161
$191$	0.551444	0.0399011	0.0199506	−	0.999801i	$-0.493649\pi$
$191$	0.551444	0.0399011	0.0199506	+	0.999801i	$0.493649\pi$
$192$	0	0
$193$	−10.1751	−0.732419	−0.366210	−	0.930532i	$-0.619345\pi$
$193$	−10.1751	−0.732419	−0.366210	+	0.930532i	$0.619345\pi$
$194$	−15.5181	−1.11414
$195$	0	0
$196$	−11.6086	−0.829188
$197$	15.7262	1.12044	0.560221	−	0.828343i	$-0.310716\pi$
$197$	15.7262	1.12044	0.560221	+	0.828343i	$0.310716\pi$
$198$	0	0
$199$	19.9272	1.41260	0.706301	−	0.707911i	$-0.250362\pi$
$199$	19.9272	1.41260	0.706301	+	0.707911i	$0.250362\pi$
$200$	−3.13679	−0.221804
$201$	0	0
$202$	27.1194	1.90812
$203$	−11.9861	−0.841260
$204$	0	0
$205$	−0.699689	−0.0488684
$206$	5.29646	0.369022
$207$	0	0
$208$	7.69684	0.533680
$209$	49.4491	3.42046
$210$	0	0
$211$	−10.0553	−0.692234	−0.346117	−	0.938191i	$-0.612500\pi$
$211$	−10.0553	−0.692234	−0.346117	+	0.938191i	$0.612500\pi$
$212$	7.34622	0.504540
$213$	0	0
$214$	−21.2736	−1.45423
$215$	0.0438261	0.00298892
$216$	0	0
$217$	−4.55721	−0.309363
$218$	−15.2301	−1.03151
$219$	0	0
$220$	−1.09129	−0.0735748
$221$	0.878774	0.0591128
$222$	0	0
$223$	16.1661	1.08256	0.541281	−	0.840842i	$-0.317939\pi$
$223$	16.1661	1.08256	0.541281	+	0.840842i	$0.317939\pi$
$224$	−11.3466	−0.758127
$225$	0	0
$226$	−5.38340	−0.358099
$227$	0.113791	0.00755260	0.00377630	−	0.999993i	$-0.498798\pi$
$227$	0.113791	0.00755260	0.00377630	+	0.999993i	$0.498798\pi$
$228$	0	0
$229$	−9.31858	−0.615789	−0.307895	−	0.951420i	$-0.599624\pi$
$229$	−9.31858	−0.615789	−0.307895	+	0.951420i	$0.599624\pi$
$230$	−0.843797	−0.0556383
$231$	0	0
$232$	−5.37865	−0.353125
$233$	11.9585	0.783425	0.391713	−	0.920088i	$-0.371883\pi$
$233$	11.9585	0.783425	0.391713	+	0.920088i	$0.371883\pi$
$234$	0	0
$235$	0.569044	0.0371203
$236$	12.5261	0.815379
$237$	0	0
$238$	−1.09476	−0.0709627
$239$	23.1415	1.49690	0.748451	−	0.663190i	$-0.230798\pi$
$239$	23.1415	1.49690	0.748451	+	0.663190i	$0.230798\pi$
$240$	0	0
$241$	−20.8807	−1.34504	−0.672522	−	0.740077i	$-0.734789\pi$
$241$	−20.8807	−1.34504	−0.672522	+	0.740077i	$0.734789\pi$
$242$	61.5392	3.95589
$243$	0	0
$244$	−26.2744	−1.68204
$245$	0.374624	0.0239339
$246$	0	0
$247$	−18.0705	−1.14980
$248$	−2.04500	−0.129858
$249$	0	0
$250$	1.54064	0.0974386
$251$	−8.81175	−0.556193	−0.278096	−	0.960553i	$-0.589703\pi$
$251$	−8.81175	−0.556193	−0.278096	+	0.960553i	$0.589703\pi$
$252$	0	0
$253$	34.9076	2.19462
$254$	−36.4661	−2.28808
$255$	0	0
$256$	10.1194	0.632464
$257$	−18.3289	−1.14332	−0.571661	−	0.820490i	$-0.693701\pi$
$257$	−18.3289	−1.14332	−0.571661	+	0.820490i	$0.693701\pi$
$258$	0	0
$259$	−10.7100	−0.665485
$260$	0.398797	0.0247323
$261$	0	0
$262$	2.49088	0.153887
$263$	20.7004	1.27644	0.638221	−	0.769853i	$-0.279670\pi$
$263$	20.7004	1.27644	0.638221	+	0.769853i	$0.279670\pi$
$264$	0	0
$265$	−0.237071	−0.0145632
$266$	22.5118	1.38029
$267$	0	0
$268$	−19.5253	−1.19270
$269$	9.10748	0.555293	0.277646	−	0.960683i	$-0.410446\pi$
$269$	9.10748	0.555293	0.277646	+	0.960683i	$0.410446\pi$
$270$	0	0
$271$	9.01982	0.547915	0.273958	−	0.961742i	$-0.411667\pi$
$271$	9.01982	0.547915	0.273958	+	0.961742i	$0.411667\pi$
$272$	1.24544	0.0755159
$273$	0	0
$274$	0.0933452	0.00563919
$275$	−31.8502	−1.92064
$276$	0	0
$277$	−6.43727	−0.386778	−0.193389	−	0.981122i	$-0.561948\pi$
$277$	−6.43727	−0.386778	−0.193389	+	0.981122i	$0.561948\pi$
$278$	33.7917	2.02669
$279$	0	0
$280$	−0.0653224	−0.00390376
$281$	−1.08665	−0.0648240	−0.0324120	−	0.999475i	$-0.510319\pi$
$281$	−1.08665	−0.0648240	−0.0324120	+	0.999475i	$0.510319\pi$
$282$	0	0
$283$	4.70918	0.279932	0.139966	−	0.990156i	$-0.455301\pi$
$283$	4.70918	0.279932	0.139966	+	0.990156i	$0.455301\pi$
$284$	2.30278	0.136645
$285$	0	0
$286$	−30.8269	−1.82283
$287$	13.1777	0.777854
$288$	0	0
$289$	−16.8578	−0.991636
$290$	1.32014	0.0775210
$291$	0	0
$292$	21.6570	1.26738
$293$	−9.79762	−0.572383	−0.286192	−	0.958172i	$-0.592389\pi$
$293$	−9.79762	−0.572383	−0.286192	+	0.958172i	$0.592389\pi$
$294$	0	0
$295$	−0.404232	−0.0235353
$296$	−4.80599	−0.279343
$297$	0	0
$298$	6.89766	0.399571
$299$	−12.7565	−0.737726
$300$	0	0
$301$	−0.825405	−0.0475755
$302$	5.19325	0.298838
$303$	0	0
$304$	−25.6103	−1.46885
$305$	0.847905	0.0485509
$306$	0	0
$307$	24.8961	1.42090	0.710448	−	0.703750i	$-0.248492\pi$
$307$	24.8961	1.42090	0.710448	+	0.703750i	$0.248492\pi$
$308$	20.5530	1.17111
$309$	0	0
$310$	0.501925	0.0285074
$311$	1.61621	0.0916471	0.0458235	−	0.998950i	$-0.485409\pi$
$311$	1.61621	0.0916471	0.0458235	+	0.998950i	$0.485409\pi$
$312$	0	0
$313$	29.1551	1.64794	0.823971	−	0.566633i	$-0.191754\pi$
$313$	29.1551	1.64794	0.823971	+	0.566633i	$0.191754\pi$
$314$	44.9676	2.53767
$315$	0	0
$316$	−22.7675	−1.28077
$317$	−20.1799	−1.13341	−0.566707	−	0.823920i	$-0.691783\pi$
$317$	−20.1799	−1.13341	−0.566707	+	0.823920i	$0.691783\pi$
$318$	0	0
$319$	−54.6135	−3.05777
$320$	0.758821	0.0424194
$321$	0	0
$322$	15.8918	0.885613
$323$	−2.92401	−0.162697
$324$	0	0
$325$	11.6392	0.645627
$326$	32.4455	1.79699
$327$	0	0
$328$	5.91335	0.326510
$329$	−10.7172	−0.590856
$330$	0	0
$331$	16.1309	0.886634	0.443317	−	0.896365i	$-0.353802\pi$
$331$	16.1309	0.886634	0.443317	+	0.896365i	$0.353802\pi$
$332$	2.49846	0.137121
$333$	0	0
$334$	−42.9827	−2.35191
$335$	0.630106	0.0344263
$336$	0	0
$337$	7.06762	0.384998	0.192499	−	0.981297i	$-0.438341\pi$
$337$	7.06762	0.384998	0.192499	+	0.981297i	$0.438341\pi$
$338$	−15.7008	−0.854011
$339$	0	0
$340$	0.0645300	0.00349963
$341$	−20.7645	−1.12446
$342$	0	0
$343$	−16.8527	−0.909958
$344$	−0.370392	−0.0199702
$345$	0	0
$346$	−39.2384	−2.10947
$347$	−18.4696	−0.991498	−0.495749	−	0.868466i	$-0.665106\pi$
$347$	−18.4696	−0.991498	−0.495749	+	0.868466i	$0.665106\pi$
$348$	0	0
$349$	−3.51426	−0.188114	−0.0940570	−	0.995567i	$-0.529984\pi$
$349$	−3.51426	−0.188114	−0.0940570	+	0.995567i	$0.529984\pi$
$350$	−14.4999	−0.775052
$351$	0	0
$352$	−51.6997	−2.75560
$353$	13.5009	0.718582	0.359291	−	0.933226i	$-0.383019\pi$
$353$	13.5009	0.718582	0.359291	+	0.933226i	$0.383019\pi$
$354$	0	0
$355$	−0.0743133	−0.00394414
$356$	11.9245	0.631998
$357$	0	0
$358$	29.1905	1.54277
$359$	−31.6345	−1.66960	−0.834802	−	0.550550i	$-0.814418\pi$
$359$	−31.6345	−1.66960	−0.834802	+	0.550550i	$0.814418\pi$
$360$	0	0
$361$	41.1273	2.16459
$362$	17.0314	0.895149
$363$	0	0
$364$	−7.51079	−0.393672
$365$	−0.698896	−0.0365819
$366$	0	0
$367$	2.08139	0.108648	0.0543239	−	0.998523i	$-0.482700\pi$
$367$	2.08139	0.108648	0.0543239	+	0.998523i	$0.482700\pi$
$368$	−18.0791	−0.942437
$369$	0	0
$370$	1.17958	0.0613236
$371$	4.46491	0.231807
$372$	0	0
$373$	6.89085	0.356795	0.178397	−	0.983959i	$-0.442909\pi$
$373$	6.89085	0.356795	0.178397	+	0.983959i	$0.442909\pi$
$374$	−4.98816	−0.257932
$375$	0	0
$376$	−4.80922	−0.248017
$377$	19.9577	1.02788
$378$	0	0
$379$	17.9961	0.924400	0.462200	−	0.886776i	$-0.347060\pi$
$379$	17.9961	0.924400	0.462200	+	0.886776i	$0.347060\pi$
$380$	−1.32695	−0.0680710
$381$	0	0
$382$	1.14387	0.0585254
$383$	5.73682	0.293138	0.146569	−	0.989200i	$-0.453177\pi$
$383$	5.73682	0.293138	0.146569	+	0.989200i	$0.453177\pi$
$384$	0	0
$385$	−0.663269	−0.0338033
$386$	−21.1063	−1.07428
$387$	0	0
$388$	−17.2273	−0.874583
$389$	12.6711	0.642451	0.321225	−	0.947003i	$-0.395905\pi$
$389$	12.6711	0.642451	0.321225	+	0.947003i	$0.395905\pi$
$390$	0	0
$391$	−2.06415	−0.104388
$392$	−3.16610	−0.159912
$393$	0	0
$394$	32.6210	1.64342
$395$	0.734736	0.0369686
$396$	0	0
$397$	2.56211	0.128588	0.0642942	−	0.997931i	$-0.479520\pi$
$397$	2.56211	0.128588	0.0642942	+	0.997931i	$0.479520\pi$
$398$	41.3353	2.07195
$399$	0	0
$400$	16.4956	0.824782
$401$	32.0897	1.60248	0.801242	−	0.598340i	$-0.204173\pi$
$401$	32.0897	1.60248	0.801242	+	0.598340i	$0.204173\pi$
$402$	0	0
$403$	7.58808	0.377989
$404$	30.1063	1.49785
$405$	0	0
$406$	−24.8629	−1.23393
$407$	−48.7989	−2.41887
$408$	0	0
$409$	−24.4301	−1.20799	−0.603995	−	0.796988i	$-0.706425\pi$
$409$	−24.4301	−1.20799	−0.603995	+	0.796988i	$0.706425\pi$
$410$	−1.45137	−0.0716782
$411$	0	0
$412$	5.87981	0.289677
$413$	7.61316	0.374619
$414$	0	0
$415$	−0.0806282	−0.00395788
$416$	18.8929	0.926301
$417$	0	0
$418$	102.573	5.01700
$419$	−19.3460	−0.945113	−0.472556	−	0.881300i	$-0.656669\pi$
$419$	−19.3460	−0.945113	−0.472556	+	0.881300i	$0.656669\pi$
$420$	0	0
$421$	−0.0882047	−0.00429883	−0.00214942	−	0.999998i	$-0.500684\pi$
$421$	−0.0882047	−0.00429883	−0.00214942	+	0.999998i	$0.500684\pi$
$422$	−20.8578	−1.01534
$423$	0	0
$424$	2.00358	0.0973026
$425$	1.88336	0.0913565
$426$	0	0
$427$	−15.9691	−0.772800
$428$	−23.6166	−1.14155
$429$	0	0
$430$	0.0909091	0.00438403
$431$	21.3822	1.02995	0.514973	−	0.857206i	$-0.327802\pi$
$431$	21.3822	1.02995	0.514973	+	0.857206i	$0.327802\pi$
$432$	0	0
$433$	−31.5968	−1.51845	−0.759223	−	0.650831i	$-0.774421\pi$
$433$	−31.5968	−1.51845	−0.759223	+	0.650831i	$0.774421\pi$
$434$	−9.45307	−0.453762
$435$	0	0
$436$	−16.9076	−0.809725
$437$	42.4456	2.03045
$438$	0	0
$439$	36.0196	1.71912	0.859560	−	0.511035i	$-0.170738\pi$
$439$	36.0196	1.71912	0.859560	+	0.511035i	$0.170738\pi$
$440$	−0.297635	−0.0141892
$441$	0	0
$442$	1.82285	0.0867043
$443$	−27.8670	−1.32400	−0.662000	−	0.749504i	$-0.730292\pi$
$443$	−27.8670	−1.32400	−0.662000	+	0.749504i	$0.730292\pi$
$444$	0	0
$445$	−0.384818	−0.0182421
$446$	33.5336	1.58786
$447$	0	0
$448$	−14.2913	−0.675203
$449$	22.6879	1.07071	0.535353	−	0.844628i	$-0.320179\pi$
$449$	22.6879	1.07071	0.535353	+	0.844628i	$0.320179\pi$
$450$	0	0
$451$	60.0428	2.82730
$452$	−5.97633	−0.281103
$453$	0	0
$454$	0.236039	0.0110779
$455$	0.242382	0.0113630
$456$	0	0
$457$	−9.28257	−0.434220	−0.217110	−	0.976147i	$-0.569663\pi$
$457$	−9.28257	−0.434220	−0.217110	+	0.976147i	$0.569663\pi$
$458$	−19.3297	−0.903215
$459$	0	0
$460$	−0.936732	−0.0436753
$461$	−17.1447	−0.798506	−0.399253	−	0.916841i	$-0.630731\pi$
$461$	−17.1447	−0.798506	−0.399253	+	0.916841i	$0.630731\pi$
$462$	0	0
$463$	20.7037	0.962180	0.481090	−	0.876671i	$-0.340241\pi$
$463$	20.7037	0.962180	0.481090	+	0.876671i	$0.340241\pi$
$464$	28.2850	1.31310
$465$	0	0
$466$	24.8056	1.14910
$467$	22.1557	1.02524	0.512621	−	0.858615i	$-0.328674\pi$
$467$	22.1557	1.02524	0.512621	+	0.858615i	$0.328674\pi$
$468$	0	0
$469$	−11.8672	−0.547975
$470$	1.18038	0.0544467
$471$	0	0
$472$	3.41633	0.157249
$473$	−3.76087	−0.172925
$474$	0	0
$475$	−38.7281	−1.77697
$476$	−1.21533	−0.0557048
$477$	0	0
$478$	48.0028	2.19560
$479$	15.1800	0.693592	0.346796	−	0.937941i	$-0.387270\pi$
$479$	15.1800	0.693592	0.346796	+	0.937941i	$0.387270\pi$
$480$	0	0
$481$	17.8329	0.813109
$482$	−43.3131	−1.97286
$483$	0	0
$484$	68.3171	3.10532
$485$	0.555945	0.0252442
$486$	0	0
$487$	−14.3420	−0.649899	−0.324949	−	0.945731i	$-0.605347\pi$
$487$	−14.3420	−0.649899	−0.324949	+	0.945731i	$0.605347\pi$
$488$	−7.16599	−0.324389
$489$	0	0
$490$	0.777089	0.0351053
$491$	−16.2633	−0.733952	−0.366976	−	0.930230i	$-0.619607\pi$
$491$	−16.2633	−0.733952	−0.366976	+	0.930230i	$0.619607\pi$
$492$	0	0
$493$	3.22940	0.145445
$494$	−37.4838	−1.68648
$495$	0	0
$496$	10.7542	0.482877
$497$	1.39959	0.0627802
$498$	0	0
$499$	−23.4973	−1.05188	−0.525942	−	0.850520i	$-0.676287\pi$
$499$	−23.4973	−1.05188	−0.525942	+	0.850520i	$0.676287\pi$
$500$	1.71032	0.0764880
$501$	0	0
$502$	−18.2783	−0.815802
$503$	34.2808	1.52851	0.764253	−	0.644917i	$-0.223108\pi$
$503$	34.2808	1.52851	0.764253	+	0.644917i	$0.223108\pi$
$504$	0	0
$505$	−0.971567	−0.0432342
$506$	72.4092	3.21898
$507$	0	0
$508$	−40.4824	−1.79612
$509$	−19.1867	−0.850434	−0.425217	−	0.905091i	$-0.639802\pi$
$509$	−19.1867	−0.850434	−0.425217	+	0.905091i	$0.639802\pi$
$510$	0	0
$511$	13.1628	0.582286
$512$	30.9245	1.36668
$513$	0	0
$514$	−38.0198	−1.67698
$515$	−0.189748	−0.00836131
$516$	0	0
$517$	−48.8317	−2.14762
$518$	−22.2158	−0.976108
$519$	0	0
$520$	0.108767	0.00476973
$521$	−10.9131	−0.478111	−0.239055	−	0.971006i	$-0.576838\pi$
$521$	−10.9131	−0.478111	−0.239055	+	0.971006i	$0.576838\pi$
$522$	0	0
$523$	−2.25146	−0.0984495	−0.0492247	−	0.998788i	$-0.515675\pi$
$523$	−2.25146	−0.0984495	−0.0492247	+	0.998788i	$0.515675\pi$
$524$	2.76522	0.120799
$525$	0	0
$526$	42.9392	1.87224
$527$	1.22784	0.0534856
$528$	0	0
$529$	6.96361	0.302766
$530$	−0.491760	−0.0213607
$531$	0	0
$532$	24.9912	1.08351
$533$	−21.9418	−0.950404
$534$	0	0
$535$	0.762136	0.0329500
$536$	−5.32528	−0.230017
$537$	0	0
$538$	18.8918	0.814482
$539$	−32.1479	−1.38471
$540$	0	0
$541$	−18.1743	−0.781375	−0.390688	−	0.920523i	$-0.627763\pi$
$541$	−18.1743	−0.781375	−0.390688	+	0.920523i	$0.627763\pi$
$542$	18.7099	0.803661
$543$	0	0
$544$	3.05710	0.131072
$545$	0.545627	0.0233721
$546$	0	0
$547$	19.8973	0.850746	0.425373	−	0.905018i	$-0.360143\pi$
$547$	19.8973	0.850746	0.425373	+	0.905018i	$0.360143\pi$
$548$	0.103626	0.00442669
$549$	0	0
$550$	−66.0674	−2.81712
$551$	−66.4069	−2.82903
$552$	0	0
$553$	−13.8377	−0.588441
$554$	−13.3529	−0.567311
$555$	0	0
$556$	37.5135	1.59093
$557$	−31.2012	−1.32204	−0.661019	−	0.750369i	$-0.729876\pi$
$557$	−31.2012	−1.32204	−0.661019	+	0.750369i	$0.729876\pi$
$558$	0	0
$559$	1.37436	0.0581292
$560$	0.343515	0.0145162
$561$	0	0
$562$	−2.25405	−0.0950813
$563$	−36.0582	−1.51967	−0.759837	−	0.650114i	$-0.774721\pi$
$563$	−36.0582	−1.51967	−0.759837	+	0.650114i	$0.774721\pi$
$564$	0	0
$565$	0.192863	0.00811381
$566$	9.76831	0.410593
$567$	0	0
$568$	0.628052	0.0263525
$569$	−4.38942	−0.184014	−0.0920071	−	0.995758i	$-0.529328\pi$
$569$	−4.38942	−0.184014	−0.0920071	+	0.995758i	$0.529328\pi$
$570$	0	0
$571$	10.5455	0.441315	0.220657	−	0.975351i	$-0.429180\pi$
$571$	10.5455	0.441315	0.220657	+	0.975351i	$0.429180\pi$
$572$	−34.2222	−1.43090
$573$	0	0
$574$	27.3346	1.14093
$575$	−27.3393	−1.14013
$576$	0	0
$577$	−15.6134	−0.649995	−0.324998	−	0.945715i	$-0.605364\pi$
$577$	−15.6134	−0.649995	−0.324998	+	0.945715i	$0.605364\pi$
$578$	−34.9684	−1.45449
$579$	0	0
$580$	1.46553	0.0608530
$581$	1.51852	0.0629989
$582$	0	0
$583$	20.3439	0.842559
$584$	5.90665	0.244419
$585$	0	0
$586$	−20.3233	−0.839549
$587$	−28.8095	−1.18909	−0.594547	−	0.804061i	$-0.702669\pi$
$587$	−28.8095	−1.18909	−0.594547	+	0.804061i	$0.702669\pi$
$588$	0	0
$589$	−25.2484	−1.04034
$590$	−0.838504	−0.0345207
$591$	0	0
$592$	25.2736	1.03874
$593$	19.9428	0.818952	0.409476	−	0.912321i	$-0.365712\pi$
$593$	19.9428	0.818952	0.409476	+	0.912321i	$0.365712\pi$
$594$	0	0
$595$	0.0392203	0.00160788
$596$	7.65736	0.313658
$597$	0	0
$598$	−26.4609	−1.08207
$599$	3.91743	0.160062	0.0800310	−	0.996792i	$-0.474498\pi$
$599$	3.91743	0.160062	0.0800310	+	0.996792i	$0.474498\pi$
$600$	0	0
$601$	−6.26999	−0.255758	−0.127879	−	0.991790i	$-0.540817\pi$
$601$	−6.26999	−0.255758	−0.127879	+	0.991790i	$0.540817\pi$
$602$	−1.71215	−0.0697819
$603$	0	0
$604$	5.76522	0.234584
$605$	−2.20467	−0.0896327
$606$	0	0
$607$	35.0082	1.42094	0.710470	−	0.703727i	$-0.248482\pi$
$607$	35.0082	1.42094	0.710470	+	0.703727i	$0.248482\pi$
$608$	−62.8639	−2.54947
$609$	0	0
$610$	1.75882	0.0712126
$611$	17.8448	0.721925
$612$	0	0
$613$	−27.9265	−1.12794	−0.563969	−	0.825796i	$-0.690726\pi$
$613$	−27.9265	−1.12794	−0.563969	+	0.825796i	$0.690726\pi$
$614$	51.6423	2.08411
$615$	0	0
$616$	5.60555	0.225854
$617$	40.4595	1.62884	0.814419	−	0.580277i	$-0.197056\pi$
$617$	40.4595	1.62884	0.814419	+	0.580277i	$0.197056\pi$
$618$	0	0
$619$	−19.7172	−0.792500	−0.396250	−	0.918143i	$-0.629689\pi$
$619$	−19.7172	−0.792500	−0.396250	+	0.918143i	$0.629689\pi$
$620$	0.557207	0.0223780
$621$	0	0
$622$	3.35253	0.134424
$623$	7.24752	0.290366
$624$	0	0
$625$	24.9172	0.996688
$626$	60.4767	2.41714
$627$	0	0
$628$	49.9203	1.99204
$629$	2.88557	0.115055
$630$	0	0
$631$	−2.78758	−0.110972	−0.0554859	−	0.998459i	$-0.517671\pi$
$631$	−2.78758	−0.110972	−0.0554859	+	0.998459i	$0.517671\pi$
$632$	−6.20955	−0.247002
$633$	0	0
$634$	−41.8593	−1.66245
$635$	1.30641	0.0518435
$636$	0	0
$637$	11.7480	0.465472
$638$	−113.286	−4.48502
$639$	0	0
$640$	0.369102	0.0145900
$641$	−20.4112	−0.806192	−0.403096	−	0.915158i	$-0.632066\pi$
$641$	−20.4112	−0.806192	−0.403096	+	0.915158i	$0.632066\pi$
$642$	0	0
$643$	−40.7646	−1.60760	−0.803800	−	0.594900i	$-0.797192\pi$
$643$	−40.7646	−1.60760	−0.803800	+	0.594900i	$0.797192\pi$
$644$	17.6421	0.695194
$645$	0	0
$646$	−6.06532	−0.238637
$647$	44.9608	1.76759	0.883796	−	0.467873i	$-0.154980\pi$
$647$	44.9608	1.76759	0.883796	+	0.467873i	$0.154980\pi$
$648$	0	0
$649$	34.6886	1.36165
$650$	24.1434	0.946981
$651$	0	0
$652$	36.0190	1.41061
$653$	−10.6603	−0.417171	−0.208586	−	0.978004i	$-0.566886\pi$
$653$	−10.6603	−0.417171	−0.208586	+	0.978004i	$0.566886\pi$
$654$	0	0
$655$	−0.0892370	−0.00348678
$656$	−31.0969	−1.21413
$657$	0	0
$658$	−22.2308	−0.866645
$659$	−13.6558	−0.531955	−0.265977	−	0.963979i	$-0.585695\pi$
$659$	−13.6558	−0.531955	−0.265977	+	0.963979i	$0.585695\pi$
$660$	0	0
$661$	−19.0000	−0.739013	−0.369507	−	0.929228i	$-0.620473\pi$
$661$	−19.0000	−0.739013	−0.369507	+	0.929228i	$0.620473\pi$
$662$	33.4605	1.30048
$663$	0	0
$664$	0.681421	0.0264443
$665$	−0.806497	−0.0312746
$666$	0	0
$667$	−46.8786	−1.81515
$668$	−47.7167	−1.84622
$669$	0	0
$670$	1.30704	0.0504952
$671$	−72.7618	−2.80894
$672$	0	0
$673$	−19.7087	−0.759714	−0.379857	−	0.925045i	$-0.624027\pi$
$673$	−19.7087	−0.759714	−0.379857	+	0.925045i	$0.624027\pi$
$674$	14.6605	0.564699
$675$	0	0
$676$	−17.4301	−0.670388
$677$	31.4390	1.20830	0.604150	−	0.796870i	$-0.293513\pi$
$677$	31.4390	1.20830	0.604150	+	0.796870i	$0.293513\pi$
$678$	0	0
$679$	−10.4705	−0.401819
$680$	0.0175997	0.000674918 0
$681$	0	0
$682$	−43.0720	−1.64931
$683$	−5.39074	−0.206271	−0.103135	−	0.994667i	$-0.532888\pi$
$683$	−5.39074	−0.206271	−0.103135	+	0.994667i	$0.532888\pi$
$684$	0	0
$685$	−0.00334414	−0.000127773 0
$686$	−34.9577	−1.33469
$687$	0	0
$688$	1.94780	0.0742594
$689$	−7.43440	−0.283228
$690$	0	0
$691$	−40.1669	−1.52802	−0.764010	−	0.645205i	$-0.776772\pi$
$691$	−40.1669	−1.52802	−0.764010	+	0.645205i	$0.776772\pi$
$692$	−43.5600	−1.65590
$693$	0	0
$694$	−38.3116	−1.45429
$695$	−1.21060	−0.0459209
$696$	0	0
$697$	−3.55044	−0.134483
$698$	−7.28968	−0.275918
$699$	0	0
$700$	−16.0969	−0.608406
$701$	−8.56290	−0.323416	−0.161708	−	0.986839i	$-0.551700\pi$
$701$	−8.56290	−0.323416	−0.161708	+	0.986839i	$0.551700\pi$
$702$	0	0
$703$	−59.3367	−2.23793
$704$	−65.1171	−2.45419
$705$	0	0
$706$	28.0051	1.05399
$707$	18.2981	0.688172
$708$	0	0
$709$	4.33400	0.162767	0.0813834	−	0.996683i	$-0.474066\pi$
$709$	4.33400	0.162767	0.0813834	+	0.996683i	$0.474066\pi$
$710$	−0.154149	−0.00578511
$711$	0	0
$712$	3.25225	0.121883
$713$	−17.8236	−0.667499
$714$	0	0
$715$	1.10439	0.0413018
$716$	32.4055	1.21105
$717$	0	0
$718$	−65.6198	−2.44891
$719$	20.8072	0.775976	0.387988	−	0.921664i	$-0.373170\pi$
$719$	20.8072	0.775976	0.387988	+	0.921664i	$0.373170\pi$
$720$	0	0
$721$	3.57365	0.133090
$722$	85.3109	3.17494
$723$	0	0
$724$	18.9072	0.702680
$725$	42.7728	1.58854
$726$	0	0
$727$	9.29496	0.344731	0.172365	−	0.985033i	$-0.444859\pi$
$727$	9.29496	0.344731	0.172365	+	0.985033i	$0.444859\pi$
$728$	−2.04847	−0.0759213
$729$	0	0
$730$	−1.44973	−0.0536569
$731$	0.222387	0.00822530
$732$	0	0
$733$	26.2296	0.968813	0.484406	−	0.874843i	$-0.339036\pi$
$733$	26.2296	0.968813	0.484406	+	0.874843i	$0.339036\pi$
$734$	4.31746	0.159360
$735$	0	0
$736$	−44.3774	−1.63577
$737$	−54.0716	−1.99175
$738$	0	0
$739$	−4.50668	−0.165781	−0.0828905	−	0.996559i	$-0.526415\pi$
$739$	−4.50668	−0.165781	−0.0828905	+	0.996559i	$0.526415\pi$
$740$	1.30950	0.0481382
$741$	0	0
$742$	9.26163	0.340005
$743$	23.3083	0.855098	0.427549	−	0.903992i	$-0.359377\pi$
$743$	23.3083	0.855098	0.427549	+	0.903992i	$0.359377\pi$
$744$	0	0
$745$	−0.247112	−0.00905349
$746$	14.2938	0.523333
$747$	0	0
$748$	−5.53755	−0.202473
$749$	−14.3538	−0.524476
$750$	0	0
$751$	37.3898	1.36437	0.682187	−	0.731178i	$-0.261029\pi$
$751$	37.3898	1.36437	0.682187	+	0.731178i	$0.261029\pi$
$752$	25.2906	0.922252
$753$	0	0
$754$	41.3986	1.50765
$755$	−0.186051	−0.00677107
$756$	0	0
$757$	4.32986	0.157372	0.0786858	−	0.996899i	$-0.474928\pi$
$757$	4.32986	0.157372	0.0786858	+	0.996899i	$0.474928\pi$
$758$	37.3297	1.35587
$759$	0	0
$760$	−0.361907	−0.0131278
$761$	33.1837	1.20291	0.601454	−	0.798907i	$-0.294588\pi$
$761$	33.1837	1.20291	0.601454	+	0.798907i	$0.294588\pi$
$762$	0	0
$763$	−10.2761	−0.372021
$764$	1.26985	0.0459416
$765$	0	0
$766$	11.9000	0.429963
$767$	−12.6765	−0.457720
$768$	0	0
$769$	−19.1229	−0.689589	−0.344795	−	0.938678i	$-0.612051\pi$
$769$	−19.1229	−0.689589	−0.344795	+	0.938678i	$0.612051\pi$
$770$	−1.37583	−0.0495814
$771$	0	0
$772$	−23.4310	−0.843299
$773$	−37.0413	−1.33228	−0.666141	−	0.745826i	$-0.732055\pi$
$773$	−37.0413	−1.33228	−0.666141	+	0.745826i	$0.732055\pi$
$774$	0	0
$775$	16.2625	0.584167
$776$	−4.69851	−0.168667
$777$	0	0
$778$	26.2839	0.942322
$779$	73.0086	2.61581
$780$	0	0
$781$	6.37709	0.228190
$782$	−4.28169	−0.153113
$783$	0	0
$784$	16.6498	0.594635
$785$	−1.61099	−0.0574986
$786$	0	0
$787$	−43.2532	−1.54181	−0.770904	−	0.636951i	$-0.780195\pi$
$787$	−43.2532	−1.54181	−0.770904	+	0.636951i	$0.780195\pi$
$788$	36.2138	1.29006
$789$	0	0
$790$	1.52407	0.0542240
$791$	−3.63231	−0.129150
$792$	0	0
$793$	26.5897	0.944230
$794$	5.31461	0.188609
$795$	0	0
$796$	45.8879	1.62645
$797$	37.5369	1.32963	0.664813	−	0.747010i	$-0.268511\pi$
$797$	37.5369	1.32963	0.664813	+	0.747010i	$0.268511\pi$
$798$	0	0
$799$	2.88751	0.102153
$800$	40.4907	1.43156
$801$	0	0
$802$	66.5642	2.35046
$803$	59.9748	2.11646
$804$	0	0
$805$	−0.569330	−0.0200662
$806$	15.7400	0.554419
$807$	0	0
$808$	8.21110	0.288866
$809$	−40.0546	−1.40825	−0.704123	−	0.710078i	$-0.748660\pi$
$809$	−40.0546	−1.40825	−0.704123	+	0.710078i	$0.748660\pi$
$810$	0	0
$811$	21.1469	0.742569	0.371285	−	0.928519i	$-0.378917\pi$
$811$	21.1469	0.742569	0.371285	+	0.928519i	$0.378917\pi$
$812$	−27.6013	−0.968616
$813$	0	0
$814$	−101.224	−3.54791
$815$	−1.16238	−0.0407163
$816$	0	0
$817$	−4.57301	−0.159989
$818$	−50.6756	−1.77183
$819$	0	0
$820$	−1.61123	−0.0562665
$821$	6.90474	0.240977	0.120489	−	0.992715i	$-0.461554\pi$
$821$	6.90474	0.240977	0.120489	+	0.992715i	$0.461554\pi$
$822$	0	0
$823$	−31.8285	−1.10947	−0.554736	−	0.832027i	$-0.687181\pi$
$823$	−31.8285	−1.10947	−0.554736	+	0.832027i	$0.687181\pi$
$824$	1.60364	0.0558654
$825$	0	0
$826$	15.7921	0.549476
$827$	−41.3427	−1.43763	−0.718814	−	0.695203i	$-0.755315\pi$
$827$	−41.3427	−1.43763	−0.718814	+	0.695203i	$0.755315\pi$
$828$	0	0
$829$	5.94780	0.206576	0.103288	−	0.994652i	$-0.467064\pi$
$829$	5.94780	0.206576	0.103288	+	0.994652i	$0.467064\pi$
$830$	−0.167248	−0.00580526
$831$	0	0
$832$	23.7961	0.824982
$833$	1.90096	0.0658644
$834$	0	0
$835$	1.53988	0.0532896
$836$	113.870	3.93828
$837$	0	0
$838$	−40.1296	−1.38625
$839$	−21.1517	−0.730238	−0.365119	−	0.930961i	$-0.618972\pi$
$839$	−21.1517	−0.730238	−0.365119	+	0.930961i	$0.618972\pi$
$840$	0	0
$841$	44.3424	1.52905
$842$	−0.182964	−0.00630536
$843$	0	0
$844$	−23.1551	−0.797030
$845$	0.562489	0.0193502
$846$	0	0
$847$	41.5220	1.42671
$848$	−10.5364	−0.361821
$849$	0	0
$850$	3.90668	0.133998
$851$	−41.8875	−1.43589
$852$	0	0
$853$	7.67355	0.262737	0.131369	−	0.991334i	$-0.458063\pi$
$853$	7.67355	0.262737	0.131369	+	0.991334i	$0.458063\pi$
$854$	−33.1250	−1.13351
$855$	0	0
$856$	−6.44112	−0.220153
$857$	44.0666	1.50529	0.752643	−	0.658428i	$-0.228778\pi$
$857$	44.0666	1.50529	0.752643	+	0.658428i	$0.228778\pi$
$858$	0	0
$859$	23.5158	0.802350	0.401175	−	0.916001i	$-0.368602\pi$
$859$	23.5158	0.802350	0.401175	+	0.916001i	$0.368602\pi$
$860$	0.100922	0.00344140
$861$	0	0
$862$	44.3534	1.51068
$863$	−15.5167	−0.528194	−0.264097	−	0.964496i	$-0.585074\pi$
$863$	−15.5167	−0.528194	−0.264097	+	0.964496i	$0.585074\pi$
$864$	0	0
$865$	1.40573	0.0477964
$866$	−65.5417	−2.22720
$867$	0	0
$868$	−10.4942	−0.356197
$869$	−63.0503	−2.13883
$870$	0	0
$871$	19.7597	0.669532
$872$	−4.61131	−0.156159
$873$	0	0
$874$	88.0455	2.97818
$875$	1.03951	0.0351417
$876$	0	0
$877$	−18.3731	−0.620415	−0.310208	−	0.950669i	$-0.600399\pi$
$877$	−18.3731	−0.620415	−0.310208	+	0.950669i	$0.600399\pi$
$878$	74.7158	2.52154
$879$	0	0
$880$	1.56519	0.0527627
$881$	46.4405	1.56462	0.782310	−	0.622890i	$-0.214041\pi$
$881$	46.4405	1.56462	0.782310	+	0.622890i	$0.214041\pi$
$882$	0	0
$883$	−57.3538	−1.93011	−0.965055	−	0.262048i	$-0.915602\pi$
$883$	−57.3538	−1.93011	−0.965055	+	0.262048i	$0.915602\pi$
$884$	2.02362	0.0680617
$885$	0	0
$886$	−57.8048	−1.94199
$887$	3.40091	0.114191	0.0570956	−	0.998369i	$-0.481816\pi$
$887$	3.40091	0.114191	0.0570956	+	0.998369i	$0.481816\pi$
$888$	0	0
$889$	−24.6045	−0.825209
$890$	−0.798233	−0.0267568
$891$	0	0
$892$	37.2269	1.24645
$893$	−59.3766	−1.98696
$894$	0	0
$895$	−1.04576	−0.0349561
$896$	−6.95153	−0.232234
$897$	0	0
$898$	47.0617	1.57047
$899$	27.8853	0.930028
$900$	0	0
$901$	−1.20297	−0.0400769
$902$	124.548	4.14698
$903$	0	0
$904$	−1.62996	−0.0542118
$905$	−0.610157	−0.0202823
$906$	0	0
$907$	−34.1546	−1.13408	−0.567042	−	0.823689i	$-0.691912\pi$
$907$	−34.1546	−1.13408	−0.567042	+	0.823689i	$0.691912\pi$
$908$	0.262036	0.00869597
$909$	0	0
$910$	0.502776	0.0166669
$911$	−38.4177	−1.27284	−0.636418	−	0.771344i	$-0.719585\pi$
$911$	−38.4177	−1.27284	−0.636418	+	0.771344i	$0.719585\pi$
$912$	0	0
$913$	6.91899	0.228985
$914$	−19.2550	−0.636897
$915$	0	0
$916$	−21.4586	−0.709012
$917$	1.68066	0.0555002
$918$	0	0
$919$	−37.2512	−1.22880	−0.614402	−	0.788993i	$-0.710603\pi$
$919$	−37.2512	−1.22880	−0.614402	+	0.788993i	$0.710603\pi$
$920$	−0.255481	−0.00842296
$921$	0	0
$922$	−35.5634	−1.17122
$923$	−2.33042	−0.0767066
$924$	0	0
$925$	38.2189	1.25663
$926$	42.9459	1.41129
$927$	0	0
$928$	69.4293	2.27913
$929$	33.3171	1.09310	0.546549	−	0.837427i	$-0.315941\pi$
$929$	33.3171	1.09310	0.546549	+	0.837427i	$0.315941\pi$
$930$	0	0
$931$	−39.0900	−1.28112
$932$	27.5377	0.902026
$933$	0	0
$934$	45.9578	1.50379
$935$	0.178703	0.00584423
$936$	0	0
$937$	−6.51990	−0.212996	−0.106498	−	0.994313i	$-0.533964\pi$
$937$	−6.51990	−0.212996	−0.106498	+	0.994313i	$0.533964\pi$
$938$	−24.6162	−0.803749
$939$	0	0
$940$	1.31038	0.0427399
$941$	39.9406	1.30203	0.651013	−	0.759066i	$-0.274344\pi$
$941$	39.9406	1.30203	0.651013	+	0.759066i	$0.274344\pi$
$942$	0	0
$943$	51.5389	1.67834
$944$	−17.9657	−0.584733
$945$	0	0
$946$	−7.80123	−0.253640
$947$	12.1311	0.394209	0.197104	−	0.980383i	$-0.436846\pi$
$947$	12.1311	0.394209	0.197104	+	0.980383i	$0.436846\pi$
$948$	0	0
$949$	−21.9169	−0.711454
$950$	−80.3341	−2.60638
$951$	0	0
$952$	−0.331466	−0.0107429
$953$	19.3506	0.626828	0.313414	−	0.949617i	$-0.398527\pi$
$953$	19.3506	0.626828	0.313414	+	0.949617i	$0.398527\pi$
$954$	0	0
$955$	−0.0409796	−0.00132607
$956$	53.2898	1.72352
$957$	0	0
$958$	31.4881	1.01733
$959$	0.0629822	0.00203380
$960$	0	0
$961$	−20.3978	−0.657993
$962$	36.9910	1.19264
$963$	0	0
$964$	−48.0835	−1.54867
$965$	0.756145	0.0243412
$966$	0	0
$967$	9.26356	0.297896	0.148948	−	0.988845i	$-0.452411\pi$
$967$	9.26356	0.297896	0.148948	+	0.988845i	$0.452411\pi$
$968$	18.6326	0.598873
$969$	0	0
$970$	1.15320	0.0370271
$971$	56.9143	1.82647	0.913233	−	0.407438i	$-0.133578\pi$
$971$	56.9143	1.82647	0.913233	+	0.407438i	$0.133578\pi$
$972$	0	0
$973$	22.8001	0.730937
$974$	−29.7498	−0.953246
$975$	0	0
$976$	37.6842	1.20624
$977$	22.7260	0.727069	0.363534	−	0.931581i	$-0.381570\pi$
$977$	22.7260	0.727069	0.363534	+	0.931581i	$0.381570\pi$
$978$	0	0
$979$	33.0226	1.05541
$980$	0.862676	0.0275572
$981$	0	0
$982$	−33.7352	−1.07653
$983$	2.94023	0.0937787	0.0468894	−	0.998900i	$-0.485069\pi$
$983$	2.94023	0.0937787	0.0468894	+	0.998900i	$0.485069\pi$
$984$	0	0
$985$	−1.16866	−0.0372367
$986$	6.69878	0.213333
$987$	0	0
$988$	−41.6122	−1.32386
$989$	−3.22822	−0.102652
$990$	0	0
$991$	−17.5577	−0.557739	−0.278869	−	0.960329i	$-0.589960\pi$
$991$	−17.5577	−0.557739	−0.278869	+	0.960329i	$0.589960\pi$
$992$	26.3975	0.838123
$993$	0	0
$994$	2.90319	0.0920835
$995$	−1.48086	−0.0469463
$996$	0	0
$997$	−3.01965	−0.0956334	−0.0478167	−	0.998856i	$-0.515226\pi$
$997$	−3.01965	−0.0956334	−0.0478167	+	0.998856i	$0.515226\pi$
$998$	−48.7408	−1.54286
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	639.2.a.k.1.4	yes	4
3.2	odd	2		639.2.a.j.1.1	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
639.2.a.j.1.1	✓	4	3.2	odd	2
639.2.a.k.1.4	yes	4	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:	\( N \)	\(=\)	\( 639 = 3^{2} \cdot 71 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	639.a (trivial)

\(f(q)\)	\(=\)	\(q+2.07431 q^{2} +2.30278 q^{4} -0.0743133 q^{5} +1.39959 q^{7} +0.628052 q^{8} +O(q^{10})\)	\(q+2.07431 q^{2} +2.30278 q^{4} -0.0743133 q^{5} +1.39959 q^{7} +0.628052 q^{8} -0.154149 q^{10} +6.37709 q^{11} -2.33042 q^{13} +2.90319 q^{14} -3.30278 q^{16} -0.377089 q^{17} +7.75418 q^{19} -0.171127 q^{20} +13.2281 q^{22} +5.47390 q^{23} -4.99448 q^{25} -4.83401 q^{26} +3.22294 q^{28} -8.56402 q^{29} -3.25610 q^{31} -8.10709 q^{32} -0.782201 q^{34} -0.104008 q^{35} -7.65222 q^{37} +16.0846 q^{38} -0.0466726 q^{40} +9.41539 q^{41} -0.589748 q^{43} +14.6850 q^{44} +11.3546 q^{46} -7.65736 q^{47} -5.04115 q^{49} -10.3601 q^{50} -5.36643 q^{52} +3.19016 q^{53} -0.473902 q^{55} +0.879014 q^{56} -17.7645 q^{58} +5.43956 q^{59} -11.4099 q^{61} -6.75418 q^{62} -10.2111 q^{64} +0.173181 q^{65} -8.47904 q^{67} -0.868351 q^{68} -0.215745 q^{70} +1.00000 q^{71} +9.40473 q^{73} -15.8731 q^{74} +17.8561 q^{76} +8.92530 q^{77} -9.88700 q^{79} +0.245440 q^{80} +19.5305 q^{82} +1.08498 q^{83} +0.0280227 q^{85} -1.22332 q^{86} +4.00514 q^{88} +5.17832 q^{89} -3.26163 q^{91} +12.6052 q^{92} -15.8838 q^{94} -0.576238 q^{95} -7.48110 q^{97} -10.4569 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{4} + 8 q^{5} + 2 q^{7}+O(q^{10}) \) 4 * q + 2 * q^4 + 8 * q^5 + 2 * q^7	\( 4 q + 2 q^{4} + 8 q^{5} + 2 q^{7} - 10 q^{10} + 10 q^{11} + 4 q^{13} + 8 q^{14} - 6 q^{16} + 14 q^{17} + 4 q^{20} + 10 q^{22} + 10 q^{23} + 6 q^{25} - 6 q^{26} - 12 q^{28} - 4 q^{29} - 8 q^{31} - 10 q^{34} - 4 q^{35} - 14 q^{37} + 20 q^{38} + 2 q^{40} + 24 q^{41} + 2 q^{43} + 18 q^{44} + 18 q^{46} + 4 q^{47} + 8 q^{49} - 40 q^{50} + 2 q^{52} + 12 q^{53} + 10 q^{55} + 14 q^{56} - 4 q^{58} - 6 q^{59} - 6 q^{61} + 4 q^{62} - 12 q^{64} + 14 q^{65} - 4 q^{67} - 6 q^{68} + 24 q^{70} + 4 q^{71} + 16 q^{73} - 4 q^{74} + 26 q^{76} - 2 q^{79} - 12 q^{80} - 16 q^{82} + 4 q^{83} + 38 q^{85} - 24 q^{86} - 2 q^{88} + 16 q^{89} - 34 q^{91} - 8 q^{92} - 12 q^{94} - 20 q^{95} - 18 q^{97} - 44 q^{98}+O(q^{100}) \) 4 * q + 2 * q^4 + 8 * q^5 + 2 * q^7 - 10 * q^10 + 10 * q^11 + 4 * q^13 + 8 * q^14 - 6 * q^16 + 14 * q^17 + 4 * q^20 + 10 * q^22 + 10 * q^23 + 6 * q^25 - 6 * q^26 - 12 * q^28 - 4 * q^29 - 8 * q^31 - 10 * q^34 - 4 * q^35 - 14 * q^37 + 20 * q^38 + 2 * q^40 + 24 * q^41 + 2 * q^43 + 18 * q^44 + 18 * q^46 + 4 * q^47 + 8 * q^49 - 40 * q^50 + 2 * q^52 + 12 * q^53 + 10 * q^55 + 14 * q^56 - 4 * q^58 - 6 * q^59 - 6 * q^61 + 4 * q^62 - 12 * q^64 + 14 * q^65 - 4 * q^67 - 6 * q^68 + 24 * q^70 + 4 * q^71 + 16 * q^73 - 4 * q^74 + 26 * q^76 - 2 * q^79 - 12 * q^80 - 16 * q^82 + 4 * q^83 + 38 * q^85 - 24 * q^86 - 2 * q^88 + 16 * q^89 - 34 * q^91 - 8 * q^92 - 12 * q^94 - 20 * q^95 - 18 * q^97 - 44 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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