LMFDB - Embedded newform 162.4.a.e.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [162,4,Mod(1,162)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("162.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(162, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 162 = 2 \cdot 3^{4} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	162.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,-4,0,8,-12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	yes
Analytic conductor:	$9.55830942093$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{12})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 3 $ x^2 - 3
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 3 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.73205$ of defining polynomial
Character	$\chi$	$=$	162.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-2.00000 q^{2} +4.00000 q^{4} -11.1962 q^{5} +18.3923 q^{7} -8.00000 q^{8} +22.3923 q^{10} +23.5692 q^{11} -67.7461 q^{13} -36.7846 q^{14} +16.0000 q^{16} -117.158 q^{17} +110.315 q^{19} -44.7846 q^{20} -47.1384 q^{22} -69.2154 q^{23} +0.353829 q^{25} +135.492 q^{26} +73.5692 q^{28} -198.373 q^{29} -311.061 q^{31} -32.0000 q^{32} +234.315 q^{34} -205.923 q^{35} -206.608 q^{37} -220.631 q^{38} +89.5692 q^{40} +132.631 q^{41} -335.177 q^{43} +94.2769 q^{44} +138.431 q^{46} +379.061 q^{47} -4.72312 q^{49} -0.707658 q^{50} -270.985 q^{52} -190.908 q^{53} -263.885 q^{55} -147.138 q^{56} +396.746 q^{58} +337.723 q^{59} +277.469 q^{61} +622.123 q^{62} +64.0000 q^{64} +758.496 q^{65} +665.069 q^{67} -468.631 q^{68} +411.846 q^{70} +528.431 q^{71} -73.8306 q^{73} +413.215 q^{74} +441.261 q^{76} +433.492 q^{77} -479.808 q^{79} -179.138 q^{80} -265.261 q^{82} +179.769 q^{83} +1311.72 q^{85} +670.354 q^{86} -188.554 q^{88} -846.458 q^{89} -1246.01 q^{91} -276.862 q^{92} -758.123 q^{94} -1235.11 q^{95} -672.985 q^{97} +9.44624 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 4 q^{2} + 8 q^{4} - 12 q^{5} + 16 q^{7} - 16 q^{8} + 24 q^{10} - 36 q^{11} + 10 q^{13} - 32 q^{14} + 32 q^{16} - 120 q^{17} - 8 q^{19} - 48 q^{20} + 72 q^{22} - 180 q^{23} - 124 q^{25} - 20 q^{26}+ \cdots + 684 q^{98}+O(q^{100}) $ 2 * q - 4 * q^2 + 8 * q^4 - 12 * q^5 + 16 * q^7 - 16 * q^8 + 24 * q^10 - 36 * q^11 + 10 * q^13 - 32 * q^14 + 32 * q^16 - 120 * q^17 - 8 * q^19 - 48 * q^20 + 72 * q^22 - 180 * q^23 - 124 * q^25 - 20 * q^26 + 64 * q^28 - 324 * q^29 - 248 * q^31 - 64 * q^32 + 240 * q^34 - 204 * q^35 - 434 * q^37 + 16 * q^38 + 96 * q^40 - 192 * q^41 - 608 * q^43 - 144 * q^44 + 360 * q^46 + 384 * q^47 - 342 * q^49 + 248 * q^50 + 40 * q^52 + 408 * q^53 - 216 * q^55 - 128 * q^56 + 648 * q^58 + 1008 * q^59 + 742 * q^61 + 496 * q^62 + 128 * q^64 + 696 * q^65 - 104 * q^67 - 480 * q^68 + 408 * q^70 + 1140 * q^71 + 850 * q^73 + 868 * q^74 - 32 * q^76 + 576 * q^77 - 440 * q^79 - 192 * q^80 + 384 * q^82 - 264 * q^83 + 1314 * q^85 + 1216 * q^86 + 288 * q^88 - 768 * q^89 - 1432 * q^91 - 720 * q^92 - 768 * q^94 - 1140 * q^95 - 764 * q^97 + 684 * q^98

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 $

$2$	−2.00000	−0.707107
$2$	−2.00000	−0.707107
$3$	0	0
$3$	0	0
$4$	4.00000	0.500000
$5$	−11.1962	−1.00141	−0.500707	−	0.865617i	$-0.666927\pi$
$5$	−11.1962	−1.00141	−0.500707	+	0.865617i	$0.666927\pi$
$6$	0	0
$7$	18.3923	0.993091	0.496546	−	0.868011i	$-0.334602\pi$
$7$	18.3923	0.993091	0.496546	+	0.868011i	$0.334602\pi$
$8$	−8.00000	−0.353553
$9$	0	0
$10$	22.3923	0.708107
$11$	23.5692	0.646035	0.323018	−	0.946393i	$-0.395303\pi$
$11$	23.5692	0.646035	0.323018	+	0.946393i	$0.395303\pi$
$12$	0	0
$13$	−67.7461	−1.44534	−0.722669	−	0.691194i	$-0.757085\pi$
$13$	−67.7461	−1.44534	−0.722669	+	0.691194i	$0.757085\pi$
$14$	−36.7846	−0.702221
$15$	0	0
$16$	16.0000	0.250000
$17$	−117.158	−1.67147	−0.835733	−	0.549137i	$-0.814957\pi$
$17$	−117.158	−1.67147	−0.835733	+	0.549137i	$0.814957\pi$
$18$	0	0
$19$	110.315	1.33200	0.666002	−	0.745950i	$-0.268004\pi$
$19$	110.315	1.33200	0.666002	+	0.745950i	$0.268004\pi$
$20$	−44.7846	−0.500707
$21$	0	0
$22$	−47.1384	−0.456816
$23$	−69.2154	−0.627496	−0.313748	−	0.949506i	$-0.601585\pi$
$23$	−69.2154	−0.627496	−0.313748	+	0.949506i	$0.601585\pi$
$24$	0	0
$25$	0.353829	0.00283063
$26$	135.492	1.02201
$27$	0	0
$28$	73.5692	0.496546
$29$	−198.373	−1.27024	−0.635120	−	0.772414i	$-0.719049\pi$
$29$	−198.373	−1.27024	−0.635120	+	0.772414i	$0.719049\pi$
$30$	0	0
$31$	−311.061	−1.80220	−0.901101	−	0.433608i	$-0.857240\pi$
$31$	−311.061	−1.80220	−0.901101	+	0.433608i	$0.857240\pi$
$32$	−32.0000	−0.176777
$33$	0	0
$34$	234.315	1.18190
$35$	−205.923	−0.994496
$36$	0	0
$37$	−206.608	−0.918003	−0.459001	−	0.888436i	$-0.651793\pi$
$37$	−206.608	−0.918003	−0.459001	+	0.888436i	$0.651793\pi$
$38$	−220.631	−0.941869
$39$	0	0
$40$	89.5692	0.354053
$41$	132.631	0.505206	0.252603	−	0.967570i	$-0.418713\pi$
$41$	132.631	0.505206	0.252603	+	0.967570i	$0.418713\pi$
$42$	0	0
$43$	−335.177	−1.18870	−0.594349	−	0.804207i	$-0.702590\pi$
$43$	−335.177	−1.18870	−0.594349	+	0.804207i	$0.702590\pi$
$44$	94.2769	0.323018
$45$	0	0
$46$	138.431	0.443707
$47$	379.061	1.17642	0.588211	−	0.808708i	$-0.299833\pi$
$47$	379.061	1.17642	0.588211	+	0.808708i	$0.299833\pi$
$48$	0	0
$49$	−4.72312	−0.0137700
$50$	−0.707658	−0.00200156
$51$	0	0
$52$	−270.985	−0.722669
$53$	−190.908	−0.494777	−0.247388	−	0.968916i	$-0.579572\pi$
$53$	−190.908	−0.494777	−0.247388	+	0.968916i	$0.579572\pi$
$54$	0	0
$55$	−263.885	−0.646949
$56$	−147.138	−0.351111
$57$	0	0
$58$	396.746	0.898195
$59$	337.723	0.745217	0.372609	−	0.927989i	$-0.378463\pi$
$59$	337.723	0.745217	0.372609	+	0.927989i	$0.378463\pi$
$60$	0	0
$61$	277.469	0.582398	0.291199	−	0.956662i	$-0.405946\pi$
$61$	277.469	0.582398	0.291199	+	0.956662i	$0.405946\pi$
$62$	622.123	1.27435
$63$	0	0
$64$	64.0000	0.125000
$65$	758.496	1.44738
$66$	0	0
$67$	665.069	1.21270	0.606352	−	0.795197i	$-0.292632\pi$
$67$	665.069	1.21270	0.606352	+	0.795197i	$0.292632\pi$
$68$	−468.631	−0.835733
$69$	0	0
$70$	411.846	0.703215
$71$	528.431	0.883284	0.441642	−	0.897191i	$-0.354396\pi$
$71$	528.431	0.883284	0.441642	+	0.897191i	$0.354396\pi$
$72$	0	0
$73$	−73.8306	−0.118373	−0.0591865	−	0.998247i	$-0.518851\pi$
$73$	−73.8306	−0.118373	−0.0591865	+	0.998247i	$0.518851\pi$
$74$	413.215	0.649126
$75$	0	0
$76$	441.261	0.666002
$77$	433.492	0.641572
$78$	0	0
$79$	−479.808	−0.683324	−0.341662	−	0.939823i	$-0.610990\pi$
$79$	−479.808	−0.683324	−0.341662	+	0.939823i	$0.610990\pi$
$80$	−179.138	−0.250354
$81$	0	0
$82$	−265.261	−0.357234
$83$	179.769	0.237738	0.118869	−	0.992910i	$-0.462073\pi$
$83$	179.769	0.237738	0.118869	+	0.992910i	$0.462073\pi$
$84$	0	0
$85$	1311.72	1.67383
$86$	670.354	0.840536
$87$	0	0
$88$	−188.554	−0.228408
$89$	−846.458	−1.00814	−0.504069	−	0.863663i	$-0.668164\pi$
$89$	−846.458	−1.00814	−0.504069	+	0.863663i	$0.668164\pi$
$90$	0	0
$91$	−1246.01	−1.43535
$92$	−276.862	−0.313748
$93$	0	0
$94$	−758.123	−0.831855
$95$	−1235.11	−1.33389
$96$	0	0
$97$	−672.985	−0.704446	−0.352223	−	0.935916i	$-0.614574\pi$
$97$	−672.985	−0.704446	−0.352223	+	0.935916i	$0.614574\pi$
$98$	9.44624	0.00973689
$99$	0	0
$100$	1.41532	0.00141532
$101$	−206.523	−0.203463	−0.101732	−	0.994812i	$-0.532438\pi$
$101$	−206.523	−0.203463	−0.101732	+	0.994812i	$0.532438\pi$
$102$	0	0
$103$	−1371.77	−1.31228	−0.656138	−	0.754641i	$-0.727811\pi$
$103$	−1371.77	−1.31228	−0.656138	+	0.754641i	$0.727811\pi$
$104$	541.969	0.511004
$105$	0	0
$106$	381.815	0.349860
$107$	1267.00	1.14472	0.572362	−	0.820001i	$-0.306027\pi$
$107$	1267.00	1.14472	0.572362	+	0.820001i	$0.306027\pi$
$108$	0	0
$109$	1725.13	1.51594	0.757970	−	0.652289i	$-0.226191\pi$
$109$	1725.13	1.51594	0.757970	+	0.652289i	$0.226191\pi$
$110$	527.769	0.457462
$111$	0	0
$112$	294.277	0.248273
$113$	1740.20	1.44871	0.724353	−	0.689429i	$-0.242138\pi$
$113$	1740.20	1.44871	0.724353	+	0.689429i	$0.242138\pi$
$114$	0	0
$115$	774.946	0.628383
$116$	−793.492	−0.635120
$117$	0	0
$118$	−675.446	−0.526948
$119$	−2154.80	−1.65992
$120$	0	0
$121$	−775.492	−0.582639
$122$	−554.939	−0.411818
$123$	0	0
$124$	−1244.25	−0.901101
$125$	1395.56	0.998580
$126$	0	0
$127$	492.131	0.343855	0.171927	−	0.985110i	$-0.445001\pi$
$127$	492.131	0.343855	0.171927	+	0.985110i	$0.445001\pi$
$128$	−128.000	−0.0883883
$129$	0	0
$130$	−1516.99	−1.02345
$131$	−1919.28	−1.28006	−0.640030	−	0.768350i	$-0.721078\pi$
$131$	−1919.28	−1.28006	−0.640030	+	0.768350i	$0.721078\pi$
$132$	0	0
$133$	2028.95	1.32280
$134$	−1330.14	−0.857511
$135$	0	0
$136$	937.261	0.590952
$137$	−2622.86	−1.63566	−0.817832	−	0.575458i	$-0.804824\pi$
$137$	−2622.86	−1.63566	−0.817832	+	0.575458i	$0.804824\pi$
$138$	0	0
$139$	−628.569	−0.383558	−0.191779	−	0.981438i	$-0.561426\pi$
$139$	−628.569	−0.383558	−0.191779	+	0.981438i	$0.561426\pi$
$140$	−823.692	−0.497248
$141$	0	0
$142$	−1056.86	−0.624576
$143$	−1596.72	−0.933739
$144$	0	0
$145$	2221.02	1.27204
$146$	147.661	0.0837023
$147$	0	0
$148$	−826.431	−0.459001
$149$	−568.312	−0.312469	−0.156235	−	0.987720i	$-0.549936\pi$
$149$	−568.312	−0.312469	−0.156235	+	0.987720i	$0.549936\pi$
$150$	0	0
$151$	357.430	0.192631	0.0963155	−	0.995351i	$-0.469294\pi$
$151$	357.430	0.192631	0.0963155	+	0.995351i	$0.469294\pi$
$152$	−882.523	−0.470935
$153$	0	0
$154$	−866.985	−0.453660
$155$	3482.69	1.80475
$156$	0	0
$157$	−727.253	−0.369689	−0.184844	−	0.982768i	$-0.559178\pi$
$157$	−727.253	−0.369689	−0.184844	+	0.982768i	$0.559178\pi$
$158$	959.615	0.483183
$159$	0	0
$160$	358.277	0.177027
$161$	−1273.03	−0.623161
$162$	0	0
$163$	−396.554	−0.190555	−0.0952776	−	0.995451i	$-0.530374\pi$
$163$	−396.554	−0.190555	−0.0952776	+	0.995451i	$0.530374\pi$
$164$	530.523	0.252603
$165$	0	0
$166$	−359.538	−0.168106
$167$	−3178.52	−1.47282	−0.736411	−	0.676534i	$-0.763481\pi$
$167$	−3178.52	−1.47282	−0.736411	+	0.676534i	$0.763481\pi$
$168$	0	0
$169$	2392.54	1.08900
$170$	−2623.43	−1.18358
$171$	0	0
$172$	−1340.71	−0.594349
$173$	−2152.65	−0.946029	−0.473014	−	0.881055i	$-0.656834\pi$
$173$	−2152.65	−0.946029	−0.473014	+	0.881055i	$0.656834\pi$
$174$	0	0
$175$	6.50773	0.00281108
$176$	377.108	0.161509
$177$	0	0
$178$	1692.92	0.712862
$179$	4490.29	1.87497	0.937487	−	0.348022i	$-0.113146\pi$
$179$	4490.29	1.87497	0.937487	+	0.348022i	$0.113146\pi$
$180$	0	0
$181$	1407.32	0.577931	0.288966	−	0.957340i	$-0.406689\pi$
$181$	1407.32	0.577931	0.288966	+	0.957340i	$0.406689\pi$
$182$	2492.02	1.01495
$183$	0	0
$184$	553.723	0.221853
$185$	2313.21	0.919301
$186$	0	0
$187$	−2761.31	−1.07983
$188$	1516.25	0.588211
$189$	0	0
$190$	2470.22	0.943201
$191$	−772.277	−0.292565	−0.146283	−	0.989243i	$-0.546731\pi$
$191$	−772.277	−0.292565	−0.146283	+	0.989243i	$0.546731\pi$
$192$	0	0
$193$	3652.68	1.36231	0.681154	−	0.732140i	$-0.261478\pi$
$193$	3652.68	1.36231	0.681154	+	0.732140i	$0.261478\pi$
$194$	1345.97	0.498118
$195$	0	0
$196$	−18.8925	−0.00688502
$197$	−2647.40	−0.957460	−0.478730	−	0.877962i	$-0.658903\pi$
$197$	−2647.40	−0.957460	−0.478730	+	0.877962i	$0.658903\pi$
$198$	0	0
$199$	1470.22	0.523723	0.261861	−	0.965106i	$-0.415664\pi$
$199$	1470.22	0.523723	0.261861	+	0.965106i	$0.415664\pi$
$200$	−2.83063	−0.00100078
$201$	0	0
$202$	413.046	0.143870
$203$	−3648.54	−1.26146
$204$	0	0
$205$	−1484.95	−0.505920
$206$	2743.54	0.927919
$207$	0	0
$208$	−1083.94	−0.361335
$209$	2600.05	0.860522
$210$	0	0
$211$	1536.01	0.501152	0.250576	−	0.968097i	$-0.419380\pi$
$211$	1536.01	0.501152	0.250576	+	0.968097i	$0.419380\pi$
$212$	−763.630	−0.247388
$213$	0	0
$214$	−2534.00	−0.809442
$215$	3752.69	1.19038
$216$	0	0
$217$	−5721.14	−1.78975
$218$	−3450.26	−1.07193
$219$	0	0
$220$	−1055.54	−0.323474
$221$	7936.98	2.41583
$222$	0	0
$223$	−1657.53	−0.497742	−0.248871	−	0.968537i	$-0.580060\pi$
$223$	−1657.53	−0.497742	−0.248871	+	0.968537i	$0.580060\pi$
$224$	−588.554	−0.175555
$225$	0	0
$226$	−3480.39	−1.02439
$227$	1514.26	0.442753	0.221377	−	0.975188i	$-0.428945\pi$
$227$	1514.26	0.442753	0.221377	+	0.975188i	$0.428945\pi$
$228$	0	0
$229$	4299.04	1.24056	0.620280	−	0.784380i	$-0.287019\pi$
$229$	4299.04	1.24056	0.620280	+	0.784380i	$0.287019\pi$
$230$	−1549.89	−0.444334
$231$	0	0
$232$	1586.98	0.449098
$233$	−1336.78	−0.375860	−0.187930	−	0.982182i	$-0.560178\pi$
$233$	−1336.78	−0.375860	−0.187930	+	0.982182i	$0.560178\pi$
$234$	0	0
$235$	−4244.03	−1.17809
$236$	1350.89	0.372609
$237$	0	0
$238$	4309.60	1.17374
$239$	6878.63	1.86168	0.930840	−	0.365427i	$-0.119077\pi$
$239$	6878.63	1.86168	0.930840	+	0.365427i	$0.119077\pi$
$240$	0	0
$241$	1531.29	0.409291	0.204646	−	0.978836i	$-0.434396\pi$
$241$	1531.29	0.409291	0.204646	+	0.978836i	$0.434396\pi$
$242$	1550.98	0.411988
$243$	0	0
$244$	1109.88	0.291199
$245$	52.8808	0.0137895
$246$	0	0
$247$	−7473.44	−1.92520
$248$	2488.49	0.637175
$249$	0	0
$250$	−2791.12	−0.706102
$251$	−1181.60	−0.297139	−0.148570	−	0.988902i	$-0.547467\pi$
$251$	−1181.60	−0.297139	−0.148570	+	0.988902i	$0.547467\pi$
$252$	0	0
$253$	−1631.35	−0.405384
$254$	−984.261	−0.243142
$255$	0	0
$256$	256.000	0.0625000
$257$	−4381.35	−1.06343	−0.531714	−	0.846924i	$-0.678452\pi$
$257$	−4381.35	−1.06343	−0.531714	+	0.846924i	$0.678452\pi$
$258$	0	0
$259$	−3799.99	−0.911660
$260$	3033.98	0.723691
$261$	0	0
$262$	3838.55	0.905140
$263$	2800.08	0.656502	0.328251	−	0.944590i	$-0.393541\pi$
$263$	2800.08	0.656502	0.328251	+	0.944590i	$0.393541\pi$
$264$	0	0
$265$	2137.43	0.495477
$266$	−4057.91	−0.935362
$267$	0	0
$268$	2660.28	0.606352
$269$	−2803.73	−0.635489	−0.317745	−	0.948176i	$-0.602925\pi$
$269$	−2803.73	−0.635489	−0.317745	+	0.948176i	$0.602925\pi$
$270$	0	0
$271$	−6332.36	−1.41942	−0.709711	−	0.704493i	$-0.751175\pi$
$271$	−6332.36	−1.41942	−0.709711	+	0.704493i	$0.751175\pi$
$272$	−1874.52	−0.417866
$273$	0	0
$274$	5245.71	1.15659
$275$	8.33948	0.00182869
$276$	0	0
$277$	927.661	0.201219	0.100610	−	0.994926i	$-0.467921\pi$
$277$	927.661	0.201219	0.100610	+	0.994926i	$0.467921\pi$
$278$	1257.14	0.271216
$279$	0	0
$280$	1647.38	0.351607
$281$	−1141.53	−0.242341	−0.121170	−	0.992632i	$-0.538665\pi$
$281$	−1141.53	−0.242341	−0.121170	+	0.992632i	$0.538665\pi$
$282$	0	0
$283$	3611.05	0.758496	0.379248	−	0.925295i	$-0.376183\pi$
$283$	3611.05	0.758496	0.379248	+	0.925295i	$0.376183\pi$
$284$	2113.72	0.441642
$285$	0	0
$286$	3193.45	0.660253
$287$	2439.38	0.501715
$288$	0	0
$289$	8812.92	1.79380
$290$	−4442.03	−0.899465
$291$	0	0
$292$	−295.323	−0.0591865
$293$	−2324.21	−0.463419	−0.231710	−	0.972785i	$-0.574432\pi$
$293$	−2324.21	−0.463419	−0.231710	+	0.972785i	$0.574432\pi$
$294$	0	0
$295$	−3781.20	−0.746271
$296$	1652.86	0.324563
$297$	0	0
$298$	1136.62	0.220949
$299$	4689.08	0.906944
$300$	0	0
$301$	−6164.68	−1.18049
$302$	−714.861	−0.136211
$303$	0	0
$304$	1765.05	0.333001
$305$	−3106.59	−0.583222
$306$	0	0
$307$	6968.51	1.29548	0.647742	−	0.761860i	$-0.275713\pi$
$307$	6968.51	1.29548	0.647742	+	0.761860i	$0.275713\pi$
$308$	1733.97	0.320786
$309$	0	0
$310$	−6965.38	−1.27615
$311$	−6340.31	−1.15603	−0.578016	−	0.816026i	$-0.696173\pi$
$311$	−6340.31	−1.15603	−0.578016	+	0.816026i	$0.696173\pi$
$312$	0	0
$313$	2388.83	0.431389	0.215694	−	0.976461i	$-0.430799\pi$
$313$	2388.83	0.431389	0.215694	+	0.976461i	$0.430799\pi$
$314$	1454.51	0.261409
$315$	0	0
$316$	−1919.23	−0.341662
$317$	5861.26	1.03849	0.519245	−	0.854625i	$-0.326213\pi$
$317$	5861.26	1.03849	0.519245	+	0.854625i	$0.326213\pi$
$318$	0	0
$319$	−4675.50	−0.820620
$320$	−716.554	−0.125177
$321$	0	0
$322$	2546.06	0.440641
$323$	−12924.3	−2.22640
$324$	0	0
$325$	−23.9706	−0.00409122
$326$	793.108	0.134743
$327$	0	0
$328$	−1061.05	−0.178617
$329$	6971.81	1.16829
$330$	0	0
$331$	4964.96	0.824468	0.412234	−	0.911078i	$-0.364749\pi$
$331$	4964.96	0.824468	0.412234	+	0.911078i	$0.364749\pi$
$332$	719.077	0.118869
$333$	0	0
$334$	6357.04	1.04144
$335$	−7446.21	−1.21442
$336$	0	0
$337$	−3097.66	−0.500713	−0.250357	−	0.968154i	$-0.580548\pi$
$337$	−3097.66	−0.500713	−0.250357	+	0.968154i	$0.580548\pi$
$338$	−4785.08	−0.770041
$339$	0	0
$340$	5246.86	0.836915
$341$	−7331.48	−1.16429
$342$	0	0
$343$	−6395.43	−1.00677
$344$	2681.42	0.420268
$345$	0	0
$346$	4305.30	0.668943
$347$	−8044.20	−1.24448	−0.622241	−	0.782826i	$-0.713778\pi$
$347$	−8044.20	−1.24448	−0.622241	+	0.782826i	$0.713778\pi$
$348$	0	0
$349$	−1144.71	−0.175572	−0.0877862	−	0.996139i	$-0.527979\pi$
$349$	−1144.71	−0.175572	−0.0877862	+	0.996139i	$0.527979\pi$
$350$	−13.0155	−0.00198773
$351$	0	0
$352$	−754.215	−0.114204
$353$	−7801.26	−1.17626	−0.588129	−	0.808767i	$-0.700135\pi$
$353$	−7801.26	−1.17626	−0.588129	+	0.808767i	$0.700135\pi$
$354$	0	0
$355$	−5916.39	−0.884534
$356$	−3385.83	−0.504069
$357$	0	0
$358$	−8980.58	−1.32581
$359$	−341.307	−0.0501769	−0.0250885	−	0.999685i	$-0.507987\pi$
$359$	−341.307	−0.0501769	−0.0250885	+	0.999685i	$0.507987\pi$
$360$	0	0
$361$	5310.48	0.774235
$362$	−2814.65	−0.408659
$363$	0	0
$364$	−4984.03	−0.717676
$365$	826.619	0.118540
$366$	0	0
$367$	119.368	0.0169781	0.00848903	−	0.999964i	$-0.497298\pi$
$367$	119.368	0.0169781	0.00848903	+	0.999964i	$0.497298\pi$
$368$	−1107.45	−0.156874
$369$	0	0
$370$	−4626.42	−0.650044
$371$	−3511.23	−0.491359
$372$	0	0
$373$	−4374.43	−0.607237	−0.303619	−	0.952794i	$-0.598195\pi$
$373$	−4374.43	−0.607237	−0.303619	+	0.952794i	$0.598195\pi$
$374$	5522.63	0.763552
$375$	0	0
$376$	−3032.49	−0.415928
$377$	13439.0	1.83593
$378$	0	0
$379$	8949.46	1.21294	0.606468	−	0.795108i	$-0.292586\pi$
$379$	8949.46	1.21294	0.606468	+	0.795108i	$0.292586\pi$
$380$	−4940.43	−0.666944
$381$	0	0
$382$	1544.55	0.206875
$383$	−206.463	−0.0275451	−0.0137726	−	0.999905i	$-0.504384\pi$
$383$	−206.463	−0.0275451	−0.0137726	+	0.999905i	$0.504384\pi$
$384$	0	0
$385$	−4853.45	−0.642479
$386$	−7305.35	−0.963297
$387$	0	0
$388$	−2691.94	−0.352223
$389$	2028.94	0.264451	0.132225	−	0.991220i	$-0.457788\pi$
$389$	2028.94	0.264451	0.132225	+	0.991220i	$0.457788\pi$
$390$	0	0
$391$	8109.11	1.04884
$392$	37.7850	0.00486844
$393$	0	0
$394$	5294.81	0.677027
$395$	5372.00	0.684290
$396$	0	0
$397$	−6646.07	−0.840193	−0.420097	−	0.907479i	$-0.638004\pi$
$397$	−6646.07	−0.840193	−0.420097	+	0.907479i	$0.638004\pi$
$398$	−2940.43	−0.370328
$399$	0	0
$400$	5.66127	0.000707658 0
$401$	1.63448	0.000203546 0	0.000101773	−	1.00000i	$-0.499968\pi$
$401$	1.63448	0.000203546 0	0.000101773		1.00000i	$0.499968\pi$
$402$	0	0
$403$	21073.2	2.60479
$404$	−826.091	−0.101732
$405$	0	0
$406$	7297.08	0.891990
$407$	−4869.58	−0.593062
$408$	0	0
$409$	−6203.35	−0.749966	−0.374983	−	0.927032i	$-0.622351\pi$
$409$	−6203.35	−0.749966	−0.374983	+	0.927032i	$0.622351\pi$
$410$	2969.91	0.357740
$411$	0	0
$412$	−5487.08	−0.656138
$413$	6211.51	0.740068
$414$	0	0
$415$	−2012.72	−0.238074
$416$	2167.88	0.255502
$417$	0	0
$418$	−5200.09	−0.608481
$419$	9750.37	1.13684	0.568421	−	0.822738i	$-0.307555\pi$
$419$	9750.37	1.13684	0.568421	+	0.822738i	$0.307555\pi$
$420$	0	0
$421$	−10061.4	−1.16475	−0.582377	−	0.812919i	$-0.697877\pi$
$421$	−10061.4	−1.16475	−0.582377	+	0.812919i	$0.697877\pi$
$422$	−3072.01	−0.354368
$423$	0	0
$424$	1527.26	0.174930
$425$	−41.4538	−0.00473130
$426$	0	0
$427$	5103.30	0.578375
$428$	5068.00	0.572362
$429$	0	0
$430$	−7505.38	−0.841725
$431$	6763.21	0.755853	0.377926	−	0.925836i	$-0.376637\pi$
$431$	6763.21	0.755853	0.377926	+	0.925836i	$0.376637\pi$
$432$	0	0
$433$	−10601.4	−1.17660	−0.588302	−	0.808641i	$-0.700204\pi$
$433$	−10601.4	−1.17660	−0.588302	+	0.808641i	$0.700204\pi$
$434$	11442.3	1.26555
$435$	0	0
$436$	6900.52	0.757970
$437$	−7635.52	−0.835827
$438$	0	0
$439$	−12568.9	−1.36647	−0.683237	−	0.730197i	$-0.739428\pi$
$439$	−12568.9	−1.36647	−0.683237	+	0.730197i	$0.739428\pi$
$440$	2111.08	0.228731
$441$	0	0
$442$	−15874.0	−1.70825
$443$	10255.4	1.09989	0.549944	−	0.835202i	$-0.314649\pi$
$443$	10255.4	1.09989	0.549944	+	0.835202i	$0.314649\pi$
$444$	0	0
$445$	9477.07	1.00956
$446$	3315.06	0.351957
$447$	0	0
$448$	1177.11	0.124136
$449$	4080.23	0.428860	0.214430	−	0.976739i	$-0.431211\pi$
$449$	4080.23	0.428860	0.214430	+	0.976739i	$0.431211\pi$
$450$	0	0
$451$	3126.00	0.326381
$452$	6960.78	0.724353
$453$	0	0
$454$	−3028.52	−0.313074
$455$	13950.5	1.43738
$456$	0	0
$457$	−2183.20	−0.223470	−0.111735	−	0.993738i	$-0.535641\pi$
$457$	−2183.20	−0.223470	−0.111735	+	0.993738i	$0.535641\pi$
$458$	−8598.08	−0.877209
$459$	0	0
$460$	3099.78	0.314192
$461$	−3250.26	−0.328372	−0.164186	−	0.986429i	$-0.552500\pi$
$461$	−3250.26	−0.328372	−0.164186	+	0.986429i	$0.552500\pi$
$462$	0	0
$463$	18991.1	1.90625	0.953124	−	0.302580i	$-0.0978479\pi$
$463$	18991.1	1.90625	0.953124	+	0.302580i	$0.0978479\pi$
$464$	−3173.97	−0.317560
$465$	0	0
$466$	2673.56	0.265773
$467$	−6906.52	−0.684359	−0.342180	−	0.939635i	$-0.611165\pi$
$467$	−6906.52	−0.684359	−0.342180	+	0.939635i	$0.611165\pi$
$468$	0	0
$469$	12232.2	1.20432
$470$	8488.06	0.833032
$471$	0	0
$472$	−2701.78	−0.263474
$473$	−7899.86	−0.767941
$474$	0	0
$475$	39.0328	0.00377041
$476$	−8619.20	−0.829959
$477$	0	0
$478$	−13757.3	−1.31641
$479$	−7380.27	−0.703994	−0.351997	−	0.936001i	$-0.614497\pi$
$479$	−7380.27	−0.703994	−0.351997	+	0.936001i	$0.614497\pi$
$480$	0	0
$481$	13996.9	1.32682
$482$	−3062.58	−0.289413
$483$	0	0
$484$	−3101.97	−0.291319
$485$	7534.84	0.705442
$486$	0	0
$487$	−8756.51	−0.814774	−0.407387	−	0.913256i	$-0.633560\pi$
$487$	−8756.51	−0.814774	−0.407387	+	0.913256i	$0.633560\pi$
$488$	−2219.75	−0.205909
$489$	0	0
$490$	−105.762	−0.00975066
$491$	−11837.9	−1.08806	−0.544031	−	0.839065i	$-0.683103\pi$
$491$	−11837.9	−1.08806	−0.544031	+	0.839065i	$0.683103\pi$
$492$	0	0
$493$	23240.9	2.12316
$494$	14946.9	1.36132
$495$	0	0
$496$	−4976.98	−0.450551
$497$	9719.06	0.877182
$498$	0	0
$499$	8578.38	0.769581	0.384791	−	0.923004i	$-0.374274\pi$
$499$	8578.38	0.769581	0.384791	+	0.923004i	$0.374274\pi$
$500$	5582.23	0.499290
$501$	0	0
$502$	2363.20	0.210109
$503$	3611.17	0.320107	0.160054	−	0.987108i	$-0.448833\pi$
$503$	3611.17	0.320107	0.160054	+	0.987108i	$0.448833\pi$
$504$	0	0
$505$	2312.26	0.203751
$506$	3262.71	0.286650
$507$	0	0
$508$	1968.52	0.171927
$509$	−9059.74	−0.788931	−0.394465	−	0.918911i	$-0.629070\pi$
$509$	−9059.74	−0.788931	−0.394465	+	0.918911i	$0.629070\pi$
$510$	0	0
$511$	−1357.92	−0.117555
$512$	−512.000	−0.0441942
$513$	0	0
$514$	8762.70	0.751957
$515$	15358.5	1.31413
$516$	0	0
$517$	8934.18	0.760010
$518$	7599.98	0.644641
$519$	0	0
$520$	−6067.97	−0.511727
$521$	12834.1	1.07922	0.539609	−	0.841916i	$-0.318572\pi$
$521$	12834.1	1.07922	0.539609	+	0.841916i	$0.318572\pi$
$522$	0	0
$523$	7061.21	0.590373	0.295187	−	0.955440i	$-0.404618\pi$
$523$	7061.21	0.590373	0.295187	+	0.955440i	$0.404618\pi$
$524$	−7677.11	−0.640030
$525$	0	0
$526$	−5600.15	−0.464217
$527$	36443.2	3.01232
$528$	0	0
$529$	−7376.23	−0.606249
$530$	−4274.86	−0.350355
$531$	0	0
$532$	8115.81	0.661401
$533$	−8985.22	−0.730193
$534$	0	0
$535$	−14185.5	−1.14634
$536$	−5320.55	−0.428755
$537$	0	0
$538$	5607.47	0.449359
$539$	−111.320	−0.00889593
$540$	0	0
$541$	−21645.6	−1.72018	−0.860091	−	0.510141i	$-0.829593\pi$
$541$	−21645.6	−1.72018	−0.860091	+	0.510141i	$0.829593\pi$
$542$	12664.7	1.00368
$543$	0	0
$544$	3749.05	0.295476
$545$	−19314.8	−1.51808
$546$	0	0
$547$	8507.05	0.664964	0.332482	−	0.943110i	$-0.392114\pi$
$547$	8507.05	0.664964	0.332482	+	0.943110i	$0.392114\pi$
$548$	−10491.4	−0.817832
$549$	0	0
$550$	−16.6790	−0.00129308
$551$	−21883.6	−1.69196
$552$	0	0
$553$	−8824.77	−0.678603
$554$	−1855.32	−0.142284
$555$	0	0
$556$	−2514.28	−0.191779
$557$	636.486	0.0484179	0.0242090	−	0.999707i	$-0.492293\pi$
$557$	636.486	0.0484179	0.0242090	+	0.999707i	$0.492293\pi$
$558$	0	0
$559$	22706.9	1.71807
$560$	−3294.77	−0.248624
$561$	0	0
$562$	2283.05	0.171361
$563$	−13562.6	−1.01527	−0.507635	−	0.861572i	$-0.669480\pi$
$563$	−13562.6	−1.01527	−0.507635	+	0.861572i	$0.669480\pi$
$564$	0	0
$565$	−19483.5	−1.45076
$566$	−7222.09	−0.536338
$567$	0	0
$568$	−4227.45	−0.312288
$569$	21116.7	1.55582	0.777909	−	0.628377i	$-0.216281\pi$
$569$	21116.7	1.55582	0.777909	+	0.628377i	$0.216281\pi$
$570$	0	0
$571$	−14527.1	−1.06469	−0.532347	−	0.846526i	$-0.678690\pi$
$571$	−14527.1	−1.06469	−0.532347	+	0.846526i	$0.678690\pi$
$572$	−6386.89	−0.466870
$573$	0	0
$574$	−4878.77	−0.354766
$575$	−24.4904	−0.00177621
$576$	0	0
$577$	−14590.9	−1.05273	−0.526366	−	0.850258i	$-0.676446\pi$
$577$	−14590.9	−1.05273	−0.526366	+	0.850258i	$0.676446\pi$
$578$	−17625.8	−1.26841
$579$	0	0
$580$	8884.06	0.636018
$581$	3306.37	0.236095
$582$	0	0
$583$	−4499.54	−0.319643
$584$	590.645	0.0418511
$585$	0	0
$586$	4648.42	0.327687
$587$	−18341.5	−1.28967	−0.644833	−	0.764324i	$-0.723073\pi$
$587$	−18341.5	−1.28967	−0.644833	+	0.764324i	$0.723073\pi$
$588$	0	0
$589$	−34314.9	−2.40054
$590$	7562.40	0.527693
$591$	0	0
$592$	−3305.72	−0.229501
$593$	−28380.4	−1.96534	−0.982668	−	0.185376i	$-0.940650\pi$
$593$	−28380.4	−1.96534	−0.982668	+	0.185376i	$0.940650\pi$
$594$	0	0
$595$	24125.5	1.66227
$596$	−2273.25	−0.156235
$597$	0	0
$598$	−9378.15	−0.641306
$599$	23881.1	1.62897	0.814487	−	0.580181i	$-0.197018\pi$
$599$	23881.1	1.62897	0.814487	+	0.580181i	$0.197018\pi$
$600$	0	0
$601$	16620.8	1.12808	0.564039	−	0.825748i	$-0.309247\pi$
$601$	16620.8	1.12808	0.564039	+	0.825748i	$0.309247\pi$
$602$	12329.4	0.834729
$603$	0	0
$604$	1429.72	0.0963155
$605$	8682.53	0.583463
$606$	0	0
$607$	−1570.78	−0.105034	−0.0525172	−	0.998620i	$-0.516724\pi$
$607$	−1570.78	−0.105034	−0.0525172	+	0.998620i	$0.516724\pi$
$608$	−3530.09	−0.235467
$609$	0	0
$610$	6213.18	0.412400
$611$	−25680.0	−1.70033
$612$	0	0
$613$	−5766.40	−0.379939	−0.189969	−	0.981790i	$-0.560839\pi$
$613$	−5766.40	−0.379939	−0.189969	+	0.981790i	$0.560839\pi$
$614$	−13937.0	−0.916046
$615$	0	0
$616$	−3467.94	−0.226830
$617$	−6962.57	−0.454299	−0.227149	−	0.973860i	$-0.572941\pi$
$617$	−6962.57	−0.454299	−0.227149	+	0.973860i	$0.572941\pi$
$618$	0	0
$619$	2660.42	0.172748	0.0863741	−	0.996263i	$-0.472472\pi$
$619$	2660.42	0.172748	0.0863741	+	0.996263i	$0.472472\pi$
$620$	13930.8	0.902376
$621$	0	0
$622$	12680.6	0.817438
$623$	−15568.3	−1.00117
$624$	0	0
$625$	−15669.1	−1.00282
$626$	−4777.66	−0.305038
$627$	0	0
$628$	−2909.01	−0.184844
$629$	24205.7	1.53441
$630$	0	0
$631$	−20432.5	−1.28908	−0.644538	−	0.764573i	$-0.722950\pi$
$631$	−20432.5	−1.28908	−0.644538	+	0.764573i	$0.722950\pi$
$632$	3838.46	0.241591
$633$	0	0
$634$	−11722.5	−0.734323
$635$	−5509.97	−0.344341
$636$	0	0
$637$	319.973	0.0199024
$638$	9351.00	0.580266
$639$	0	0
$640$	1433.11	0.0885134
$641$	−26068.3	−1.60630	−0.803149	−	0.595779i	$-0.796844\pi$
$641$	−26068.3	−1.60630	−0.803149	+	0.595779i	$0.796844\pi$
$642$	0	0
$643$	21170.4	1.29841	0.649207	−	0.760612i	$-0.275101\pi$
$643$	21170.4	1.29841	0.649207	+	0.760612i	$0.275101\pi$
$644$	−5092.12	−0.311580
$645$	0	0
$646$	25848.6	1.57430
$647$	−8291.45	−0.503818	−0.251909	−	0.967751i	$-0.581058\pi$
$647$	−8291.45	−0.503818	−0.251909	+	0.967751i	$0.581058\pi$
$648$	0	0
$649$	7959.87	0.481436
$650$	47.9411	0.00289293
$651$	0	0
$652$	−1586.22	−0.0952776
$653$	−24685.0	−1.47932	−0.739662	−	0.672979i	$-0.765015\pi$
$653$	−24685.0	−1.47932	−0.739662	+	0.672979i	$0.765015\pi$
$654$	0	0
$655$	21488.5	1.28187
$656$	2122.09	0.126301
$657$	0	0
$658$	−13943.6	−0.826108
$659$	29954.4	1.77065	0.885325	−	0.464973i	$-0.153936\pi$
$659$	29954.4	1.77065	0.885325	+	0.464973i	$0.153936\pi$
$660$	0	0
$661$	−4126.76	−0.242833	−0.121416	−	0.992602i	$-0.538744\pi$
$661$	−4126.76	−0.242833	−0.121416	+	0.992602i	$0.538744\pi$
$662$	−9929.92	−0.582987
$663$	0	0
$664$	−1438.15	−0.0840530
$665$	−22716.5	−1.32467
$666$	0	0
$667$	13730.5	0.797070
$668$	−12714.1	−0.736411
$669$	0	0
$670$	14892.4	0.858723
$671$	6539.73	0.376250
$672$	0	0
$673$	−26329.7	−1.50808	−0.754039	−	0.656830i	$-0.771897\pi$
$673$	−26329.7	−1.50808	−0.754039	+	0.656830i	$0.771897\pi$
$674$	6195.32	0.354058
$675$	0	0
$676$	9570.15	0.544501
$677$	−6603.86	−0.374900	−0.187450	−	0.982274i	$-0.560022\pi$
$677$	−6603.86	−0.374900	−0.187450	+	0.982274i	$0.560022\pi$
$678$	0	0
$679$	−12377.7	−0.699579
$680$	−10493.7	−0.591788
$681$	0	0
$682$	14663.0	0.823275
$683$	−12706.4	−0.711854	−0.355927	−	0.934514i	$-0.615835\pi$
$683$	−12706.4	−0.711854	−0.355927	+	0.934514i	$0.615835\pi$
$684$	0	0
$685$	29365.9	1.63798
$686$	12790.9	0.711891
$687$	0	0
$688$	−5362.83	−0.297174
$689$	12933.3	0.715120
$690$	0	0
$691$	−15730.2	−0.865999	−0.432999	−	0.901394i	$-0.642545\pi$
$691$	−15730.2	−0.865999	−0.432999	+	0.901394i	$0.642545\pi$
$692$	−8610.60	−0.473014
$693$	0	0
$694$	16088.4	0.879982
$695$	7037.55	0.384100
$696$	0	0
$697$	−15538.7	−0.844434
$698$	2289.41	0.124148
$699$	0	0
$700$	26.0309	0.00140554
$701$	652.959	0.0351811	0.0175905	−	0.999845i	$-0.494400\pi$
$701$	652.959	0.0351811	0.0175905	+	0.999845i	$0.494400\pi$
$702$	0	0
$703$	−22792.0	−1.22278
$704$	1508.43	0.0807544
$705$	0	0
$706$	15602.5	0.831740
$707$	−3798.43	−0.202058
$708$	0	0
$709$	−24884.4	−1.31813	−0.659065	−	0.752086i	$-0.729048\pi$
$709$	−24884.4	−1.31813	−0.659065	+	0.752086i	$0.729048\pi$
$710$	11832.8	0.625460
$711$	0	0
$712$	6771.66	0.356431
$713$	21530.2	1.13088
$714$	0	0
$715$	17877.2	0.935060
$716$	17961.2	0.937487
$717$	0	0
$718$	682.615	0.0354804
$719$	−21323.8	−1.10604	−0.553020	−	0.833168i	$-0.686525\pi$
$719$	−21323.8	−1.10604	−0.553020	+	0.833168i	$0.686525\pi$
$720$	0	0
$721$	−25230.0	−1.30321
$722$	−10621.0	−0.547467
$723$	0	0
$724$	5629.29	0.288966
$725$	−70.1902	−0.00359558
$726$	0	0
$727$	8032.01	0.409753	0.204877	−	0.978788i	$-0.434321\pi$
$727$	8032.01	0.409753	0.204877	+	0.978788i	$0.434321\pi$
$728$	9968.06	0.507474
$729$	0	0
$730$	−1653.24	−0.0838207
$731$	39268.5	1.98687
$732$	0	0
$733$	28457.4	1.43397	0.716984	−	0.697090i	$-0.245522\pi$
$733$	28457.4	1.43397	0.716984	+	0.697090i	$0.245522\pi$
$734$	−238.736	−0.0120053
$735$	0	0
$736$	2214.89	0.110927
$737$	15675.2	0.783449
$738$	0	0
$739$	−11006.3	−0.547868	−0.273934	−	0.961748i	$-0.588325\pi$
$739$	−11006.3	−0.547868	−0.273934	+	0.961748i	$0.588325\pi$
$740$	9252.84	0.459650
$741$	0	0
$742$	7022.46	0.347443
$743$	−4652.91	−0.229742	−0.114871	−	0.993380i	$-0.536646\pi$
$743$	−4652.91	−0.229742	−0.114871	+	0.993380i	$0.536646\pi$
$744$	0	0
$745$	6362.91	0.312911
$746$	8748.86	0.429381
$747$	0	0
$748$	−11045.3	−0.539913
$749$	23303.0	1.13682
$750$	0	0
$751$	−17357.7	−0.843400	−0.421700	−	0.906735i	$-0.638566\pi$
$751$	−17357.7	−0.843400	−0.421700	+	0.906735i	$0.638566\pi$
$752$	6064.98	0.294105
$753$	0	0
$754$	−26878.0	−1.29820
$755$	−4001.85	−0.192903
$756$	0	0
$757$	119.139	0.00572019	0.00286010	−	0.999996i	$-0.499090\pi$
$757$	119.139	0.00572019	0.00286010	+	0.999996i	$0.499090\pi$
$758$	−17898.9	−0.857675
$759$	0	0
$760$	9880.86	0.471601
$761$	8843.02	0.421234	0.210617	−	0.977569i	$-0.432453\pi$
$761$	8843.02	0.421234	0.210617	+	0.977569i	$0.432453\pi$
$762$	0	0
$763$	31729.1	1.50547
$764$	−3089.11	−0.146283
$765$	0	0
$766$	412.926	0.0194773
$767$	−22879.4	−1.07709
$768$	0	0
$769$	2693.10	0.126288	0.0631442	−	0.998004i	$-0.479887\pi$
$769$	2693.10	0.126288	0.0631442	+	0.998004i	$0.479887\pi$
$770$	9706.89	0.454301
$771$	0	0
$772$	14610.7	0.681154
$773$	18116.5	0.842958	0.421479	−	0.906838i	$-0.361511\pi$
$773$	18116.5	0.842958	0.421479	+	0.906838i	$0.361511\pi$
$774$	0	0
$775$	−110.063	−0.00510137
$776$	5383.88	0.249059
$777$	0	0
$778$	−4057.88	−0.186995
$779$	14631.2	0.672936
$780$	0	0
$781$	12454.7	0.570633
$782$	−16218.2	−0.741640
$783$	0	0
$784$	−75.5700	−0.00344251
$785$	8142.44	0.370212
$786$	0	0
$787$	31747.5	1.43796	0.718980	−	0.695031i	$-0.244609\pi$
$787$	31747.5	1.43796	0.718980	+	0.695031i	$0.244609\pi$
$788$	−10589.6	−0.478730
$789$	0	0
$790$	−10744.0	−0.483866
$791$	32006.2	1.43870
$792$	0	0
$793$	−18797.5	−0.841763
$794$	13292.1	0.594106
$795$	0	0
$796$	5880.86	0.261861
$797$	28265.8	1.25624	0.628122	−	0.778115i	$-0.283824\pi$
$797$	28265.8	1.25624	0.628122	+	0.778115i	$0.283824\pi$
$798$	0	0
$799$	−44410.0	−1.96635
$800$	−11.3225	−0.000500390 0
$801$	0	0
$802$	−3.26896	−0.000143929 0
$803$	−1740.13	−0.0764731
$804$	0	0
$805$	14253.0	0.624042
$806$	−42146.4	−1.84187
$807$	0	0
$808$	1652.18	0.0719351
$809$	42553.4	1.84932	0.924659	−	0.380795i	$-0.124350\pi$
$809$	42553.4	1.84932	0.924659	+	0.380795i	$0.124350\pi$
$810$	0	0
$811$	−6900.03	−0.298758	−0.149379	−	0.988780i	$-0.547727\pi$
$811$	−6900.03	−0.298758	−0.149379	+	0.988780i	$0.547727\pi$
$812$	−14594.2	−0.630732
$813$	0	0
$814$	9739.16	0.419358
$815$	4439.88	0.190825
$816$	0	0
$817$	−36975.2	−1.58335
$818$	12406.7	0.530306
$819$	0	0
$820$	−5939.81	−0.252960
$821$	−22359.2	−0.950476	−0.475238	−	0.879857i	$-0.657638\pi$
$821$	−22359.2	−0.950476	−0.475238	+	0.879857i	$0.657638\pi$
$822$	0	0
$823$	791.570	0.0335266	0.0167633	−	0.999859i	$-0.494664\pi$
$823$	791.570	0.0335266	0.0167633	+	0.999859i	$0.494664\pi$
$824$	10974.2	0.463960
$825$	0	0
$826$	−12423.0	−0.523307
$827$	−23005.9	−0.967343	−0.483671	−	0.875250i	$-0.660697\pi$
$827$	−23005.9	−0.967343	−0.483671	+	0.875250i	$0.660697\pi$
$828$	0	0
$829$	−15420.2	−0.646040	−0.323020	−	0.946392i	$-0.604698\pi$
$829$	−15420.2	−0.646040	−0.323020	+	0.946392i	$0.604698\pi$
$830$	4025.45	0.168344
$831$	0	0
$832$	−4335.75	−0.180667
$833$	553.350	0.0230161
$834$	0	0
$835$	35587.2	1.47491
$836$	10400.2	0.430261
$837$	0	0
$838$	−19500.7	−0.803869
$839$	−46290.1	−1.90478	−0.952391	−	0.304878i	$-0.901384\pi$
$839$	−46290.1	−1.90478	−0.952391	+	0.304878i	$0.901384\pi$
$840$	0	0
$841$	14962.9	0.613509
$842$	20122.8	0.823605
$843$	0	0
$844$	6144.03	0.250576
$845$	−26787.2	−1.09054
$846$	0	0
$847$	−14263.1	−0.578613
$848$	−3054.52	−0.123694
$849$	0	0
$850$	82.9076	0.00334554
$851$	14300.4	0.576043
$852$	0	0
$853$	27902.4	1.12000	0.559999	−	0.828493i	$-0.310801\pi$
$853$	27902.4	1.12000	0.559999	+	0.828493i	$0.310801\pi$
$854$	−10206.6	−0.408973
$855$	0	0
$856$	−10136.0	−0.404721
$857$	−6552.93	−0.261195	−0.130597	−	0.991435i	$-0.541690\pi$
$857$	−6552.93	−0.261195	−0.130597	+	0.991435i	$0.541690\pi$
$858$	0	0
$859$	34857.7	1.38455	0.692275	−	0.721634i	$-0.256608\pi$
$859$	34857.7	1.38455	0.692275	+	0.721634i	$0.256608\pi$
$860$	15010.8	0.595189
$861$	0	0
$862$	−13526.4	−0.534469
$863$	10885.2	0.429359	0.214680	−	0.976685i	$-0.431129\pi$
$863$	10885.2	0.429359	0.214680	+	0.976685i	$0.431129\pi$
$864$	0	0
$865$	24101.4	0.947367
$866$	21202.8	0.831985
$867$	0	0
$868$	−22884.6	−0.894876
$869$	−11308.7	−0.441451
$870$	0	0
$871$	−45055.9	−1.75277
$872$	−13801.0	−0.535966
$873$	0	0
$874$	15271.0	0.591019
$875$	25667.5	0.991681
$876$	0	0
$877$	−27913.8	−1.07478	−0.537390	−	0.843334i	$-0.680590\pi$
$877$	−27913.8	−1.07478	−0.537390	+	0.843334i	$0.680590\pi$
$878$	25137.8	0.966243
$879$	0	0
$880$	−4222.15	−0.161737
$881$	−10694.5	−0.408975	−0.204488	−	0.978869i	$-0.565553\pi$
$881$	−10694.5	−0.408975	−0.204488	+	0.978869i	$0.565553\pi$
$882$	0	0
$883$	3265.74	0.124463	0.0622315	−	0.998062i	$-0.480178\pi$
$883$	3265.74	0.124463	0.0622315	+	0.998062i	$0.480178\pi$
$884$	31747.9	1.20792
$885$	0	0
$886$	−20510.9	−0.777738
$887$	−9392.88	−0.355560	−0.177780	−	0.984070i	$-0.556892\pi$
$887$	−9392.88	−0.355560	−0.177780	+	0.984070i	$0.556892\pi$
$888$	0	0
$889$	9051.41	0.341479
$890$	−18954.1	−0.713870
$891$	0	0
$892$	−6630.12	−0.248871
$893$	41816.3	1.56700
$894$	0	0
$895$	−50274.0	−1.87762
$896$	−2354.22	−0.0877777
$897$	0	0
$898$	−8160.46	−0.303250
$899$	61706.2	2.28923
$900$	0	0
$901$	22366.3	0.827002
$902$	−6252.00	−0.230786
$903$	0	0
$904$	−13921.6	−0.512195
$905$	−15756.6	−0.578748
$906$	0	0
$907$	13538.5	0.495631	0.247816	−	0.968807i	$-0.420287\pi$
$907$	13538.5	0.495631	0.247816	+	0.968807i	$0.420287\pi$
$908$	6057.04	0.221377
$909$	0	0
$910$	−27901.0	−1.01638
$911$	−37806.4	−1.37495	−0.687476	−	0.726207i	$-0.741282\pi$
$911$	−37806.4	−1.37495	−0.687476	+	0.726207i	$0.741282\pi$
$912$	0	0
$913$	4237.02	0.153587
$914$	4366.40	0.158017
$915$	0	0
$916$	17196.2	0.620280
$917$	−35299.9	−1.27122
$918$	0	0
$919$	30674.2	1.10103	0.550515	−	0.834825i	$-0.314431\pi$
$919$	30674.2	1.10103	0.550515	+	0.834825i	$0.314431\pi$
$920$	−6199.57	−0.222167
$921$	0	0
$922$	6500.52	0.232194
$923$	−35799.1	−1.27664
$924$	0	0
$925$	−73.1038	−0.00259853
$926$	−37982.3	−1.34792
$927$	0	0
$928$	6347.94	0.224549
$929$	−28117.4	−0.993004	−0.496502	−	0.868035i	$-0.665383\pi$
$929$	−28117.4	−0.993004	−0.496502	+	0.868035i	$0.665383\pi$
$930$	0	0
$931$	−521.033	−0.0183417
$932$	−5347.12	−0.187930
$933$	0	0
$934$	13813.0	0.483915
$935$	30916.1	1.08135
$936$	0	0
$937$	31859.0	1.11077	0.555384	−	0.831594i	$-0.312571\pi$
$937$	31859.0	1.11077	0.555384	+	0.831594i	$0.312571\pi$
$938$	−24464.3	−0.851586
$939$	0	0
$940$	−16976.1	−0.589043
$941$	2263.14	0.0784018	0.0392009	−	0.999231i	$-0.487519\pi$
$941$	2263.14	0.0784018	0.0392009	+	0.999231i	$0.487519\pi$
$942$	0	0
$943$	−9180.09	−0.317015
$944$	5403.57	0.186304
$945$	0	0
$946$	15799.7	0.543016
$947$	6985.33	0.239697	0.119848	−	0.992792i	$-0.461759\pi$
$947$	6985.33	0.239697	0.119848	+	0.992792i	$0.461759\pi$
$948$	0	0
$949$	5001.74	0.171089
$950$	−78.0656	−0.00266609
$951$	0	0
$952$	17238.4	0.586869
$953$	−26436.8	−0.898608	−0.449304	−	0.893379i	$-0.648328\pi$
$953$	−26436.8	−0.898608	−0.449304	+	0.893379i	$0.648328\pi$
$954$	0	0
$955$	8646.53	0.292979
$956$	27514.5	0.930840
$957$	0	0
$958$	14760.5	0.497799
$959$	−48240.4	−1.62436
$960$	0	0
$961$	66968.2	2.24794
$962$	−27993.7	−0.938206
$963$	0	0
$964$	6125.17	0.204646
$965$	−40895.9	−1.36423
$966$	0	0
$967$	−100.156	−0.00333072	−0.00166536	−	0.999999i	$-0.500530\pi$
$967$	−100.156	−0.00333072	−0.00166536	+	0.999999i	$0.500530\pi$
$968$	6203.94	0.205994
$969$	0	0
$970$	−15069.7	−0.498823
$971$	−678.145	−0.0224127	−0.0112063	−	0.999937i	$-0.503567\pi$
$971$	−678.145	−0.0224127	−0.0112063	+	0.999937i	$0.503567\pi$
$972$	0	0
$973$	−11560.8	−0.380908
$974$	17513.0	0.576133
$975$	0	0
$976$	4439.51	0.145600
$977$	9748.25	0.319216	0.159608	−	0.987180i	$-0.448977\pi$
$977$	9748.25	0.319216	0.159608	+	0.987180i	$0.448977\pi$
$978$	0	0
$979$	−19950.3	−0.651293
$980$	211.523	0.00689476
$981$	0	0
$982$	23675.9	0.769376
$983$	3515.69	0.114072	0.0570361	−	0.998372i	$-0.481835\pi$
$983$	3515.69	0.114072	0.0570361	+	0.998372i	$0.481835\pi$
$984$	0	0
$985$	29640.7	0.958815
$986$	−46481.9	−1.50130
$987$	0	0
$988$	−29893.8	−0.962598
$989$	23199.4	0.745903
$990$	0	0
$991$	−612.517	−0.0196339	−0.00981697	−	0.999952i	$-0.503125\pi$
$991$	−612.517	−0.0196339	−0.00981697	+	0.999952i	$0.503125\pi$
$992$	9953.97	0.318587
$993$	0	0
$994$	−19438.1	−0.620261
$995$	−16460.8	−0.524463
$996$	0	0
$997$	22956.4	0.729223	0.364611	−	0.931160i	$-0.381202\pi$
$997$	22956.4	0.729223	0.364611	+	0.931160i	$0.381202\pi$
$998$	−17156.8	−0.544176
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	162.4.a.e.1.1	✓	2
3.2	odd	2		162.4.a.h.1.2	yes	2
4.3	odd	2		1296.4.a.j.1.1		2
9.2	odd	6		162.4.c.i.109.1		4
9.4	even	3		162.4.c.j.55.2		4
9.5	odd	6		162.4.c.i.55.1		4
9.7	even	3		162.4.c.j.109.2		4
12.11	even	2		1296.4.a.s.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
162.4.a.e.1.1	✓	2	1.1	even	1	trivial
162.4.a.h.1.2	yes	2	3.2	odd	2
162.4.c.i.55.1		4	9.5	odd	6
162.4.c.i.109.1		4	9.2	odd	6
162.4.c.j.55.2		4	9.4	even	3
162.4.c.j.109.2		4	9.7	even	3
1296.4.a.j.1.1		2	4.3	odd	2
1296.4.a.s.1.2		2	12.11	even	2

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 162 = 2 \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	162.a (trivial)

\(f(q)\)	\(=\)	\(q-2.00000 q^{2} +4.00000 q^{4} -11.1962 q^{5} +18.3923 q^{7} -8.00000 q^{8} +22.3923 q^{10} +23.5692 q^{11} -67.7461 q^{13} -36.7846 q^{14} +16.0000 q^{16} -117.158 q^{17} +110.315 q^{19} -44.7846 q^{20} -47.1384 q^{22} -69.2154 q^{23} +0.353829 q^{25} +135.492 q^{26} +73.5692 q^{28} -198.373 q^{29} -311.061 q^{31} -32.0000 q^{32} +234.315 q^{34} -205.923 q^{35} -206.608 q^{37} -220.631 q^{38} +89.5692 q^{40} +132.631 q^{41} -335.177 q^{43} +94.2769 q^{44} +138.431 q^{46} +379.061 q^{47} -4.72312 q^{49} -0.707658 q^{50} -270.985 q^{52} -190.908 q^{53} -263.885 q^{55} -147.138 q^{56} +396.746 q^{58} +337.723 q^{59} +277.469 q^{61} +622.123 q^{62} +64.0000 q^{64} +758.496 q^{65} +665.069 q^{67} -468.631 q^{68} +411.846 q^{70} +528.431 q^{71} -73.8306 q^{73} +413.215 q^{74} +441.261 q^{76} +433.492 q^{77} -479.808 q^{79} -179.138 q^{80} -265.261 q^{82} +179.769 q^{83} +1311.72 q^{85} +670.354 q^{86} -188.554 q^{88} -846.458 q^{89} -1246.01 q^{91} -276.862 q^{92} -758.123 q^{94} -1235.11 q^{95} -672.985 q^{97} +9.44624 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 4 q^{2} + 8 q^{4} - 12 q^{5} + 16 q^{7} - 16 q^{8} + 24 q^{10} - 36 q^{11} + 10 q^{13} - 32 q^{14} + 32 q^{16} - 120 q^{17} - 8 q^{19} - 48 q^{20} + 72 q^{22} - 180 q^{23} - 124 q^{25} - 20 q^{26}+ \cdots + 684 q^{98}+O(q^{100}) \) 2 * q - 4 * q^2 + 8 * q^4 - 12 * q^5 + 16 * q^7 - 16 * q^8 + 24 * q^10 - 36 * q^11 + 10 * q^13 - 32 * q^14 + 32 * q^16 - 120 * q^17 - 8 * q^19 - 48 * q^20 + 72 * q^22 - 180 * q^23 - 124 * q^25 - 20 * q^26 + 64 * q^28 - 324 * q^29 - 248 * q^31 - 64 * q^32 + 240 * q^34 - 204 * q^35 - 434 * q^37 + 16 * q^38 + 96 * q^40 - 192 * q^41 - 608 * q^43 - 144 * q^44 + 360 * q^46 + 384 * q^47 - 342 * q^49 + 248 * q^50 + 40 * q^52 + 408 * q^53 - 216 * q^55 - 128 * q^56 + 648 * q^58 + 1008 * q^59 + 742 * q^61 + 496 * q^62 + 128 * q^64 + 696 * q^65 - 104 * q^67 - 480 * q^68 + 408 * q^70 + 1140 * q^71 + 850 * q^73 + 868 * q^74 - 32 * q^76 + 576 * q^77 - 440 * q^79 - 192 * q^80 + 384 * q^82 - 264 * q^83 + 1314 * q^85 + 1216 * q^86 + 288 * q^88 - 768 * q^89 - 1432 * q^91 - 720 * q^92 - 768 * q^94 - 1140 * q^95 - 764 * q^97 + 684 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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