LMFDB - Embedded newform 2151.4.a.b.1.15

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("2151.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

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

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$126.913108422$
Analytic rank:	$1$
Dimension:	$28$
Twist minimal:	no (minimal twist has level 717)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.15
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)

$f(q)$	$=$	$q+0.609904 q^{2} -7.62802 q^{4} +11.6127 q^{5} -26.3682 q^{7} -9.53159 q^{8} +O(q^{10})$	\(q+0.609904 q^{2} -7.62802 q^{4} +11.6127 q^{5} -26.3682 q^{7} -9.53159 q^{8} +7.08263 q^{10} +0.610836 q^{11} +42.5600 q^{13} -16.0821 q^{14} +55.2108 q^{16} -4.31924 q^{17} +84.0403 q^{19} -88.5818 q^{20} +0.372552 q^{22} -97.3553 q^{23} +9.85471 q^{25} +25.9575 q^{26} +201.137 q^{28} -80.7760 q^{29} -155.154 q^{31} +109.926 q^{32} -2.63432 q^{34} -306.206 q^{35} -181.347 q^{37} +51.2565 q^{38} -110.688 q^{40} +77.9913 q^{41} +481.313 q^{43} -4.65947 q^{44} -59.3774 q^{46} +215.363 q^{47} +352.281 q^{49} +6.01043 q^{50} -324.648 q^{52} +180.490 q^{53} +7.09346 q^{55} +251.331 q^{56} -49.2656 q^{58} +321.812 q^{59} +464.989 q^{61} -94.6294 q^{62} -374.642 q^{64} +494.236 q^{65} +392.572 q^{67} +32.9472 q^{68} -186.756 q^{70} +179.410 q^{71} -216.122 q^{73} -110.604 q^{74} -641.061 q^{76} -16.1067 q^{77} -49.0631 q^{79} +641.146 q^{80} +47.5672 q^{82} -1050.90 q^{83} -50.1580 q^{85} +293.555 q^{86} -5.82225 q^{88} -561.329 q^{89} -1122.23 q^{91} +742.628 q^{92} +131.351 q^{94} +975.934 q^{95} -750.775 q^{97} +214.858 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 28 q + 5 q^{2} + 103 q^{4} - 6 q^{5} - 68 q^{7} + 39 q^{8}+O(q^{10}) $ 28 * q + 5 * q^2 + 103 * q^4 - 6 * q^5 - 68 * q^7 + 39 * q^8	$ 28 q + 5 q^{2} + 103 q^{4} - 6 q^{5} - 68 q^{7} + 39 q^{8} - 88 q^{10} + 110 q^{11} - 82 q^{13} - 126 q^{14} + 271 q^{16} - 100 q^{17} - 292 q^{19} + 52 q^{20} - 351 q^{22} + 276 q^{23} + 386 q^{25} - 84 q^{26} - 1010 q^{28} + 38 q^{29} - 432 q^{31} + 452 q^{32} - 524 q^{34} + 166 q^{35} - 936 q^{37} + 41 q^{38} - 1183 q^{40} - 1054 q^{41} - 1804 q^{43} + 341 q^{44} - 888 q^{46} + 560 q^{47} + 1074 q^{49} + 1054 q^{50} - 632 q^{52} + 160 q^{53} - 842 q^{55} - 509 q^{56} - 1266 q^{58} - 846 q^{59} - 2220 q^{61} - 82 q^{62} - 1565 q^{64} - 296 q^{65} - 4752 q^{67} + 1719 q^{68} - 5601 q^{70} + 802 q^{71} - 2732 q^{73} + 4581 q^{74} - 5614 q^{76} + 1008 q^{77} - 3172 q^{79} + 732 q^{80} - 9709 q^{82} + 4780 q^{83} - 4624 q^{85} + 2009 q^{86} - 9331 q^{88} - 4372 q^{89} - 7398 q^{91} + 6138 q^{92} - 7068 q^{94} + 3160 q^{95} - 4846 q^{97} + 3772 q^{98}+O(q^{100}) $ 28 * q + 5 * q^2 + 103 * q^4 - 6 * q^5 - 68 * q^7 + 39 * q^8 - 88 * q^10 + 110 * q^11 - 82 * q^13 - 126 * q^14 + 271 * q^16 - 100 * q^17 - 292 * q^19 + 52 * q^20 - 351 * q^22 + 276 * q^23 + 386 * q^25 - 84 * q^26 - 1010 * q^28 + 38 * q^29 - 432 * q^31 + 452 * q^32 - 524 * q^34 + 166 * q^35 - 936 * q^37 + 41 * q^38 - 1183 * q^40 - 1054 * q^41 - 1804 * q^43 + 341 * q^44 - 888 * q^46 + 560 * q^47 + 1074 * q^49 + 1054 * q^50 - 632 * q^52 + 160 * q^53 - 842 * q^55 - 509 * q^56 - 1266 * q^58 - 846 * q^59 - 2220 * q^61 - 82 * q^62 - 1565 * q^64 - 296 * q^65 - 4752 * q^67 + 1719 * q^68 - 5601 * q^70 + 802 * q^71 - 2732 * q^73 + 4581 * q^74 - 5614 * q^76 + 1008 * q^77 - 3172 * q^79 + 732 * q^80 - 9709 * q^82 + 4780 * q^83 - 4624 * q^85 + 2009 * q^86 - 9331 * q^88 - 4372 * q^89 - 7398 * q^91 + 6138 * q^92 - 7068 * q^94 + 3160 * q^95 - 4846 * q^97 + 3772 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0.609904	0.215634	0.107817	−	0.994171i	$-0.465614\pi$
$2$	0.609904	0.215634	0.107817	+	0.994171i	$0.465614\pi$
$3$	0	0
$3$	0	0
$4$	−7.62802	−0.953502
$5$	11.6127	1.03867	0.519336	−	0.854570i	$-0.326179\pi$
$5$	11.6127	1.03867	0.519336	+	0.854570i	$0.326179\pi$
$6$	0	0
$7$	−26.3682	−1.42375	−0.711874	−	0.702307i	$-0.752153\pi$
$7$	−26.3682	−1.42375	−0.711874	+	0.702307i	$0.752153\pi$
$8$	−9.53159	−0.421241
$9$	0	0
$10$	7.08263	0.223973
$11$	0.610836	0.0167431	0.00837155	−	0.999965i	$-0.497335\pi$
$11$	0.610836	0.0167431	0.00837155	+	0.999965i	$0.497335\pi$
$12$	0	0
$13$	42.5600	0.908001	0.454001	−	0.891001i	$-0.349996\pi$
$13$	42.5600	0.908001	0.454001	+	0.891001i	$0.349996\pi$
$14$	−16.0821	−0.307008
$15$	0	0
$16$	55.2108	0.862668
$17$	−4.31924	−0.0616217	−0.0308108	−	0.999525i	$-0.509809\pi$
$17$	−4.31924	−0.0616217	−0.0308108	+	0.999525i	$0.509809\pi$
$18$	0	0
$19$	84.0403	1.01475	0.507373	−	0.861727i	$-0.330617\pi$
$19$	84.0403	1.01475	0.507373	+	0.861727i	$0.330617\pi$
$20$	−88.5818	−0.990375
$21$	0	0
$22$	0.372552	0.00361038
$23$	−97.3553	−0.882608	−0.441304	−	0.897358i	$-0.645484\pi$
$23$	−97.3553	−0.882608	−0.441304	+	0.897358i	$0.645484\pi$
$24$	0	0
$25$	9.85471	0.0788377
$26$	25.9575	0.195796
$27$	0	0
$28$	201.137	1.35755
$29$	−80.7760	−0.517232	−0.258616	−	0.965980i	$-0.583266\pi$
$29$	−80.7760	−0.517232	−0.258616	+	0.965980i	$0.583266\pi$
$30$	0	0
$31$	−155.154	−0.898921	−0.449461	−	0.893300i	$-0.648384\pi$
$31$	−155.154	−0.898921	−0.449461	+	0.893300i	$0.648384\pi$
$32$	109.926	0.607261
$33$	0	0
$34$	−2.63432	−0.0132877
$35$	−306.206	−1.47881
$36$	0	0
$37$	−181.347	−0.805765	−0.402883	−	0.915252i	$-0.631992\pi$
$37$	−181.347	−0.805765	−0.402883	+	0.915252i	$0.631992\pi$
$38$	51.2565	0.218813
$39$	0	0
$40$	−110.688	−0.437531
$41$	77.9913	0.297078	0.148539	−	0.988907i	$-0.452543\pi$
$41$	77.9913	0.297078	0.148539	+	0.988907i	$0.452543\pi$
$42$	0	0
$43$	481.313	1.70697	0.853484	−	0.521119i	$-0.174485\pi$
$43$	481.313	1.70697	0.853484	+	0.521119i	$0.174485\pi$
$44$	−4.65947	−0.0159646
$45$	0	0
$46$	−59.3774	−0.190320
$47$	215.363	0.668381	0.334191	−	0.942506i	$-0.391537\pi$
$47$	215.363	0.668381	0.334191	+	0.942506i	$0.391537\pi$
$48$	0	0
$49$	352.281	1.02706
$50$	6.01043	0.0170001
$51$	0	0
$52$	−324.648	−0.865781
$53$	180.490	0.467778	0.233889	−	0.972263i	$-0.424855\pi$
$53$	180.490	0.467778	0.233889	+	0.972263i	$0.424855\pi$
$54$	0	0
$55$	7.09346	0.0173906
$56$	251.331	0.599741
$57$	0	0
$58$	−49.2656	−0.111533
$59$	321.812	0.710108	0.355054	−	0.934846i	$-0.384462\pi$
$59$	321.812	0.710108	0.355054	+	0.934846i	$0.384462\pi$
$60$	0	0
$61$	464.989	0.975995	0.487998	−	0.872845i	$-0.337727\pi$
$61$	464.989	0.975995	0.487998	+	0.872845i	$0.337727\pi$
$62$	−94.6294	−0.193838
$63$	0	0
$64$	−374.642	−0.731722
$65$	494.236	0.943115
$66$	0	0
$67$	392.572	0.715826	0.357913	−	0.933755i	$-0.383488\pi$
$67$	392.572	0.715826	0.357913	+	0.933755i	$0.383488\pi$
$68$	32.9472	0.0587564
$69$	0	0
$70$	−186.756	−0.318881
$71$	179.410	0.299889	0.149944	−	0.988694i	$-0.452091\pi$
$71$	179.410	0.299889	0.149944	+	0.988694i	$0.452091\pi$
$72$	0	0
$73$	−216.122	−0.346510	−0.173255	−	0.984877i	$-0.555428\pi$
$73$	−216.122	−0.346510	−0.173255	+	0.984877i	$0.555428\pi$
$74$	−110.604	−0.173750
$75$	0	0
$76$	−641.061	−0.967562
$77$	−16.1067	−0.0238380
$78$	0	0
$79$	−49.0631	−0.0698737	−0.0349369	−	0.999390i	$-0.511123\pi$
$79$	−49.0631	−0.0698737	−0.0349369	+	0.999390i	$0.511123\pi$
$80$	641.146	0.896029
$81$	0	0
$82$	47.5672	0.0640600
$83$	−1050.90	−1.38978	−0.694888	−	0.719118i	$-0.744546\pi$
$83$	−1050.90	−1.38978	−0.694888	+	0.719118i	$0.744546\pi$
$84$	0	0
$85$	−50.1580	−0.0640047
$86$	293.555	0.368080
$87$	0	0
$88$	−5.82225	−0.00705288
$89$	−561.329	−0.668547	−0.334274	−	0.942476i	$-0.608491\pi$
$89$	−561.329	−0.668547	−0.334274	+	0.942476i	$0.608491\pi$
$90$	0	0
$91$	−1122.23	−1.29277
$92$	742.628	0.841569
$93$	0	0
$94$	131.351	0.144126
$95$	975.934	1.05399
$96$	0	0
$97$	−750.775	−0.785873	−0.392936	−	0.919566i	$-0.628541\pi$
$97$	−750.775	−0.785873	−0.392936	+	0.919566i	$0.628541\pi$
$98$	214.858	0.221469
$99$	0	0
$100$	−75.1719	−0.0751719
$101$	−1475.27	−1.45342	−0.726708	−	0.686947i	$-0.758950\pi$
$101$	−1475.27	−1.45342	−0.726708	+	0.686947i	$0.758950\pi$
$102$	0	0
$103$	−78.9844	−0.0755588	−0.0377794	−	0.999286i	$-0.512028\pi$
$103$	−78.9844	−0.0755588	−0.0377794	+	0.999286i	$0.512028\pi$
$104$	−405.665	−0.382487
$105$	0	0
$106$	110.082	0.100869
$107$	1089.62	0.984466	0.492233	−	0.870463i	$-0.336181\pi$
$107$	1089.62	0.984466	0.492233	+	0.870463i	$0.336181\pi$
$108$	0	0
$109$	848.618	0.745715	0.372857	−	0.927889i	$-0.378378\pi$
$109$	848.618	0.745715	0.372857	+	0.927889i	$0.378378\pi$
$110$	4.32633	0.00374999
$111$	0	0
$112$	−1455.81	−1.22822
$113$	−455.657	−0.379333	−0.189666	−	0.981849i	$-0.560741\pi$
$113$	−455.657	−0.379333	−0.189666	+	0.981849i	$0.560741\pi$
$114$	0	0
$115$	−1130.56	−0.916740
$116$	616.161	0.493182
$117$	0	0
$118$	196.275	0.153123
$119$	113.890	0.0877338
$120$	0	0
$121$	−1330.63	−0.999720
$122$	283.599	0.210458
$123$	0	0
$124$	1183.52	0.857123
$125$	−1337.15	−0.956785
$126$	0	0
$127$	−439.908	−0.307366	−0.153683	−	0.988120i	$-0.549113\pi$
$127$	−439.908	−0.307366	−0.153683	+	0.988120i	$0.549113\pi$
$128$	−1107.90	−0.765045
$129$	0	0
$130$	301.437	0.203367
$131$	536.678	0.357937	0.178969	−	0.983855i	$-0.442724\pi$
$131$	536.678	0.357937	0.178969	+	0.983855i	$0.442724\pi$
$132$	0	0
$133$	−2215.99	−1.44474
$134$	239.432	0.154356
$135$	0	0
$136$	41.1692	0.0259576
$137$	130.546	0.0814108	0.0407054	−	0.999171i	$-0.487039\pi$
$137$	130.546	0.0814108	0.0407054	+	0.999171i	$0.487039\pi$
$138$	0	0
$139$	−1806.60	−1.10240	−0.551201	−	0.834373i	$-0.685830\pi$
$139$	−1806.60	−1.10240	−0.551201	+	0.834373i	$0.685830\pi$
$140$	2335.74	1.41004
$141$	0	0
$142$	109.423	0.0646661
$143$	25.9972	0.0152028
$144$	0	0
$145$	−938.027	−0.537234
$146$	−131.814	−0.0747191
$147$	0	0
$148$	1383.32	0.768299
$149$	−2275.84	−1.25130	−0.625651	−	0.780103i	$-0.715167\pi$
$149$	−2275.84	−1.25130	−0.625651	+	0.780103i	$0.715167\pi$
$150$	0	0
$151$	−1331.62	−0.717652	−0.358826	−	0.933404i	$-0.616823\pi$
$151$	−1331.62	−0.717652	−0.358826	+	0.933404i	$0.616823\pi$
$152$	−801.038	−0.427452
$153$	0	0
$154$	−9.82352	−0.00514027
$155$	−1801.76	−0.933684
$156$	0	0
$157$	−1368.28	−0.695548	−0.347774	−	0.937578i	$-0.613062\pi$
$157$	−1368.28	−0.695548	−0.347774	+	0.937578i	$0.613062\pi$
$158$	−29.9238	−0.0150671
$159$	0	0
$160$	1276.54	0.630745
$161$	2567.08	1.25661
$162$	0	0
$163$	−326.170	−0.156734	−0.0783669	−	0.996925i	$-0.524971\pi$
$163$	−326.170	−0.156734	−0.0783669	+	0.996925i	$0.524971\pi$
$164$	−594.919	−0.283264
$165$	0	0
$166$	−640.950	−0.299683
$167$	−1569.33	−0.727175	−0.363587	−	0.931560i	$-0.618448\pi$
$167$	−1569.33	−0.727175	−0.363587	+	0.931560i	$0.618448\pi$
$168$	0	0
$169$	−385.647	−0.175533
$170$	−30.5916	−0.0138016
$171$	0	0
$172$	−3671.47	−1.62760
$173$	−593.745	−0.260934	−0.130467	−	0.991453i	$-0.541648\pi$
$173$	−593.745	−0.260934	−0.130467	+	0.991453i	$0.541648\pi$
$174$	0	0
$175$	−259.851	−0.112245
$176$	33.7248	0.0144437
$177$	0	0
$178$	−342.357	−0.144161
$179$	4326.77	1.80669	0.903346	−	0.428913i	$-0.141103\pi$
$179$	4326.77	1.80669	0.903346	+	0.428913i	$0.141103\pi$
$180$	0	0
$181$	2305.67	0.946847	0.473423	−	0.880835i	$-0.343018\pi$
$181$	2305.67	0.946847	0.473423	+	0.880835i	$0.343018\pi$
$182$	−684.453	−0.278764
$183$	0	0
$184$	927.952	0.371791
$185$	−2105.93	−0.836925
$186$	0	0
$187$	−2.63835	−0.00103174
$188$	−1642.79	−0.637303
$189$	0	0
$190$	595.226	0.227275
$191$	−1939.23	−0.734647	−0.367323	−	0.930093i	$-0.619726\pi$
$191$	−1939.23	−0.734647	−0.367323	+	0.930093i	$0.619726\pi$
$192$	0	0
$193$	−4774.70	−1.78078	−0.890389	−	0.455200i	$-0.849568\pi$
$193$	−4774.70	−1.78078	−0.890389	+	0.455200i	$0.849568\pi$
$194$	−457.901	−0.169461
$195$	0	0
$196$	−2687.21	−0.979303
$197$	1385.29	0.501004	0.250502	−	0.968116i	$-0.419404\pi$
$197$	1385.29	0.501004	0.250502	+	0.968116i	$0.419404\pi$
$198$	0	0
$199$	250.056	0.0890756	0.0445378	−	0.999008i	$-0.485818\pi$
$199$	250.056	0.0890756	0.0445378	+	0.999008i	$0.485818\pi$
$200$	−93.9311	−0.0332097
$201$	0	0
$202$	−899.774	−0.313405
$203$	2129.92	0.736408
$204$	0	0
$205$	905.689	0.308566
$206$	−48.1729	−0.0162930
$207$	0	0
$208$	2349.77	0.783304
$209$	51.3349	0.0169900
$210$	0	0
$211$	−1654.97	−0.539968	−0.269984	−	0.962865i	$-0.587018\pi$
$211$	−1654.97	−0.539968	−0.269984	+	0.962865i	$0.587018\pi$
$212$	−1376.78	−0.446027
$213$	0	0
$214$	664.566	0.212284
$215$	5589.35	1.77298
$216$	0	0
$217$	4091.14	1.27984
$218$	517.576	0.160801
$219$	0	0
$220$	−54.1090	−0.0165820
$221$	−183.827	−0.0559526
$222$	0	0
$223$	−3093.89	−0.929068	−0.464534	−	0.885555i	$-0.653778\pi$
$223$	−3093.89	−0.929068	−0.464534	+	0.885555i	$0.653778\pi$
$224$	−2898.55	−0.864587
$225$	0	0
$226$	−277.907	−0.0817969
$227$	3794.73	1.10954	0.554768	−	0.832005i	$-0.312807\pi$
$227$	3794.73	1.10954	0.554768	+	0.832005i	$0.312807\pi$
$228$	0	0
$229$	4550.67	1.31317	0.656587	−	0.754250i	$-0.272000\pi$
$229$	4550.67	1.31317	0.656587	+	0.754250i	$0.272000\pi$
$230$	−689.532	−0.197680
$231$	0	0
$232$	769.924	0.217879
$233$	−621.034	−0.174615	−0.0873076	−	0.996181i	$-0.527826\pi$
$233$	−621.034	−0.174615	−0.0873076	+	0.996181i	$0.527826\pi$
$234$	0	0
$235$	2500.94	0.694228
$236$	−2454.79	−0.677089
$237$	0	0
$238$	69.4623	0.0189184
$239$	−239.000	−0.0646846
$239$	−239.000	−0.0646846
$240$	0	0
$241$	−3875.21	−1.03579	−0.517893	−	0.855445i	$-0.673283\pi$
$241$	−3875.21	−1.03579	−0.517893	+	0.855445i	$0.673283\pi$
$242$	−811.555	−0.215573
$243$	0	0
$244$	−3546.94	−0.930614
$245$	4090.94	1.06678
$246$	0	0
$247$	3576.75	0.921390
$248$	1478.87	0.378662
$249$	0	0
$250$	−815.532	−0.206315
$251$	−2110.17	−0.530649	−0.265324	−	0.964159i	$-0.585479\pi$
$251$	−2110.17	−0.530649	−0.265324	+	0.964159i	$0.585479\pi$
$252$	0	0
$253$	−59.4682	−0.0147776
$254$	−268.302	−0.0662785
$255$	0	0
$256$	2321.42	0.566753
$257$	−3910.55	−0.949158	−0.474579	−	0.880213i	$-0.657400\pi$
$257$	−3910.55	−0.949158	−0.474579	+	0.880213i	$0.657400\pi$
$258$	0	0
$259$	4781.80	1.14721
$260$	−3770.04	−0.899262
$261$	0	0
$262$	327.322	0.0771834
$263$	1160.82	0.272164	0.136082	−	0.990698i	$-0.456549\pi$
$263$	1160.82	0.272164	0.136082	+	0.990698i	$0.456549\pi$
$264$	0	0
$265$	2095.98	0.485867
$266$	−1351.54	−0.311535
$267$	0	0
$268$	−2994.55	−0.682542
$269$	−66.5096	−0.0150750	−0.00753748	−	0.999972i	$-0.502399\pi$
$269$	−66.5096	−0.0150750	−0.00753748	+	0.999972i	$0.502399\pi$
$270$	0	0
$271$	−6223.00	−1.39491	−0.697455	−	0.716629i	$-0.745684\pi$
$271$	−6223.00	−1.39491	−0.697455	+	0.716629i	$0.745684\pi$
$272$	−238.468	−0.0531591
$273$	0	0
$274$	79.6205	0.0175549
$275$	6.01962	0.00131999
$276$	0	0
$277$	1122.29	0.243436	0.121718	−	0.992565i	$-0.461160\pi$
$277$	1122.29	0.243436	0.121718	+	0.992565i	$0.461160\pi$
$278$	−1101.85	−0.237715
$279$	0	0
$280$	2918.63	0.622934
$281$	−1756.34	−0.372863	−0.186432	−	0.982468i	$-0.559692\pi$
$281$	−1756.34	−0.372863	−0.186432	+	0.982468i	$0.559692\pi$
$282$	0	0
$283$	−5815.90	−1.22162	−0.610811	−	0.791776i	$-0.709157\pi$
$283$	−5815.90	−1.22162	−0.610811	+	0.791776i	$0.709157\pi$
$284$	−1368.55	−0.285944
$285$	0	0
$286$	15.8558	0.00327823
$287$	−2056.49	−0.422964
$288$	0	0
$289$	−4894.34	−0.996203
$290$	−572.107	−0.115846
$291$	0	0
$292$	1648.58	0.330398
$293$	−6447.65	−1.28558	−0.642791	−	0.766041i	$-0.722224\pi$
$293$	−6447.65	−1.28558	−0.642791	+	0.766041i	$0.722224\pi$
$294$	0	0
$295$	3737.11	0.737569
$296$	1728.53	0.339421
$297$	0	0
$298$	−1388.04	−0.269823
$299$	−4143.44	−0.801409
$300$	0	0
$301$	−12691.4	−2.43029
$302$	−812.159	−0.154750
$303$	0	0
$304$	4639.93	0.875389
$305$	5399.77	1.01374
$306$	0	0
$307$	−10357.7	−1.92555	−0.962774	−	0.270307i	$-0.912875\pi$
$307$	−10357.7	−1.92555	−0.962774	+	0.270307i	$0.912875\pi$
$308$	122.862	0.0227295
$309$	0	0
$310$	−1098.90	−0.201334
$311$	−8342.79	−1.52115	−0.760573	−	0.649252i	$-0.775082\pi$
$311$	−8342.79	−1.52115	−0.760573	+	0.649252i	$0.775082\pi$
$312$	0	0
$313$	−3580.58	−0.646602	−0.323301	−	0.946296i	$-0.604793\pi$
$313$	−3580.58	−0.646602	−0.323301	+	0.946296i	$0.604793\pi$
$314$	−834.523	−0.149984
$315$	0	0
$316$	374.254	0.0666247
$317$	4222.71	0.748174	0.374087	−	0.927394i	$-0.377956\pi$
$317$	4222.71	0.748174	0.374087	+	0.927394i	$0.377956\pi$
$318$	0	0
$319$	−49.3409	−0.00866006
$320$	−4350.60	−0.760019
$321$	0	0
$322$	1565.68	0.270968
$323$	−362.990	−0.0625303
$324$	0	0
$325$	419.416	0.0715847
$326$	−198.933	−0.0337971
$327$	0	0
$328$	−743.381	−0.125141
$329$	−5678.73	−0.951607
$330$	0	0
$331$	6719.91	1.11589	0.557945	−	0.829878i	$-0.311590\pi$
$331$	6719.91	1.11589	0.557945	+	0.829878i	$0.311590\pi$
$332$	8016.30	1.32515
$333$	0	0
$334$	−957.139	−0.156803
$335$	4558.82	0.743508
$336$	0	0
$337$	2129.31	0.344187	0.172093	−	0.985081i	$-0.444947\pi$
$337$	2129.31	0.344187	0.172093	+	0.985081i	$0.444947\pi$
$338$	−235.208	−0.0378509
$339$	0	0
$340$	382.606	0.0610286
$341$	−94.7740	−0.0150507
$342$	0	0
$343$	−244.735	−0.0385261
$344$	−4587.68	−0.719045
$345$	0	0
$346$	−362.128	−0.0562662
$347$	691.250	0.106940	0.0534701	−	0.998569i	$-0.482972\pi$
$347$	691.250	0.106940	0.0534701	+	0.998569i	$0.482972\pi$
$348$	0	0
$349$	1683.98	0.258285	0.129143	−	0.991626i	$-0.458778\pi$
$349$	1683.98	0.258285	0.129143	+	0.991626i	$0.458778\pi$
$350$	−158.484	−0.0242038
$351$	0	0
$352$	67.1468	0.0101674
$353$	−11113.1	−1.67561	−0.837803	−	0.545972i	$-0.816161\pi$
$353$	−11113.1	−1.67561	−0.837803	+	0.545972i	$0.816161\pi$
$354$	0	0
$355$	2083.44	0.311486
$356$	4281.82	0.637461
$357$	0	0
$358$	2638.91	0.389584
$359$	−8120.92	−1.19389	−0.596944	−	0.802283i	$-0.703619\pi$
$359$	−8120.92	−1.19389	−0.596944	+	0.802283i	$0.703619\pi$
$360$	0	0
$361$	203.766	0.0297079
$362$	1406.24	0.204172
$363$	0	0
$364$	8560.39	1.23265
$365$	−2509.76	−0.359909
$366$	0	0
$367$	1738.27	0.247240	0.123620	−	0.992330i	$-0.460550\pi$
$367$	1738.27	0.247240	0.123620	+	0.992330i	$0.460550\pi$
$368$	−5375.06	−0.761398
$369$	0	0
$370$	−1284.42	−0.180469
$371$	−4759.20	−0.665998
$372$	0	0
$373$	5972.57	0.829083	0.414542	−	0.910030i	$-0.363942\pi$
$373$	5972.57	0.829083	0.414542	+	0.910030i	$0.363942\pi$
$374$	−1.60914	−0.000222478 0
$375$	0	0
$376$	−2052.75	−0.281550
$377$	−3437.83	−0.469647
$378$	0	0
$379$	7219.74	0.978505	0.489252	−	0.872142i	$-0.337270\pi$
$379$	7219.74	0.978505	0.489252	+	0.872142i	$0.337270\pi$
$380$	−7444.44	−1.00498
$381$	0	0
$382$	−1182.74	−0.158415
$383$	13095.3	1.74709	0.873547	−	0.486739i	$-0.161814\pi$
$383$	13095.3	1.74709	0.873547	+	0.486739i	$0.161814\pi$
$384$	0	0
$385$	−187.042	−0.0247598
$386$	−2912.11	−0.383996
$387$	0	0
$388$	5726.92	0.749331
$389$	−6867.72	−0.895135	−0.447567	−	0.894250i	$-0.647710\pi$
$389$	−6867.72	−0.895135	−0.447567	+	0.894250i	$0.647710\pi$
$390$	0	0
$391$	420.501	0.0543878
$392$	−3357.80	−0.432640
$393$	0	0
$394$	844.894	0.108033
$395$	−569.754	−0.0725758
$396$	0	0
$397$	−10544.5	−1.33303	−0.666515	−	0.745492i	$-0.732215\pi$
$397$	−10544.5	−1.33303	−0.666515	+	0.745492i	$0.732215\pi$
$398$	152.511	0.0192077
$399$	0	0
$400$	544.086	0.0680108
$401$	4079.72	0.508058	0.254029	−	0.967197i	$-0.418244\pi$
$401$	4079.72	0.508058	0.254029	+	0.967197i	$0.418244\pi$
$402$	0	0
$403$	−6603.37	−0.816222
$404$	11253.4	1.38583
$405$	0	0
$406$	1299.05	0.158794
$407$	−110.774	−0.0134910
$408$	0	0
$409$	−13394.2	−1.61932	−0.809660	−	0.586899i	$-0.800348\pi$
$409$	−13394.2	−1.61932	−0.809660	+	0.586899i	$0.800348\pi$
$410$	552.384	0.0665373
$411$	0	0
$412$	602.494	0.0720455
$413$	−8485.60	−1.01102
$414$	0	0
$415$	−12203.8	−1.44352
$416$	4678.45	0.551394
$417$	0	0
$418$	31.3094	0.00366361
$419$	−252.526	−0.0294432	−0.0147216	−	0.999892i	$-0.504686\pi$
$419$	−252.526	−0.0294432	−0.0147216	+	0.999892i	$0.504686\pi$
$420$	0	0
$421$	−4157.51	−0.481294	−0.240647	−	0.970613i	$-0.577360\pi$
$421$	−4157.51	−0.481294	−0.240647	+	0.970613i	$0.577360\pi$
$422$	−1009.38	−0.116435
$423$	0	0
$424$	−1720.36	−0.197047
$425$	−42.5648	−0.00485811
$426$	0	0
$427$	−12260.9	−1.38957
$428$	−8311.67	−0.938691
$429$	0	0
$430$	3408.97	0.382314
$431$	5873.75	0.656447	0.328224	−	0.944600i	$-0.393550\pi$
$431$	5873.75	0.656447	0.328224	+	0.944600i	$0.393550\pi$
$432$	0	0
$433$	5516.18	0.612219	0.306109	−	0.951996i	$-0.400973\pi$
$433$	5516.18	0.612219	0.306109	+	0.951996i	$0.400973\pi$
$434$	2495.21	0.275976
$435$	0	0
$436$	−6473.28	−0.711041
$437$	−8181.77	−0.895622
$438$	0	0
$439$	7202.33	0.783026	0.391513	−	0.920173i	$-0.371952\pi$
$439$	7202.33	0.783026	0.391513	+	0.920173i	$0.371952\pi$
$440$	−67.6120	−0.00732562
$441$	0	0
$442$	−112.117	−0.0120653
$443$	−3887.48	−0.416929	−0.208465	−	0.978030i	$-0.566847\pi$
$443$	−3887.48	−0.416929	−0.208465	+	0.978030i	$0.566847\pi$
$444$	0	0
$445$	−6518.54	−0.694401
$446$	−1886.98	−0.200338
$447$	0	0
$448$	9878.63	1.04179
$449$	−9834.75	−1.03370	−0.516849	−	0.856076i	$-0.672895\pi$
$449$	−9834.75	−1.03370	−0.516849	+	0.856076i	$0.672895\pi$
$450$	0	0
$451$	47.6399	0.00497400
$452$	3475.76	0.361694
$453$	0	0
$454$	2314.42	0.239253
$455$	−13032.1	−1.34276
$456$	0	0
$457$	344.172	0.0352291	0.0176145	−	0.999845i	$-0.494393\pi$
$457$	344.172	0.0352291	0.0176145	+	0.999845i	$0.494393\pi$
$458$	2775.47	0.283165
$459$	0	0
$460$	8623.91	0.874113
$461$	−328.390	−0.0331771	−0.0165886	−	0.999862i	$-0.505281\pi$
$461$	−328.390	−0.0331771	−0.0165886	+	0.999862i	$0.505281\pi$
$462$	0	0
$463$	−6501.18	−0.652561	−0.326280	−	0.945273i	$-0.605795\pi$
$463$	−6501.18	−0.652561	−0.326280	+	0.945273i	$0.605795\pi$
$464$	−4459.70	−0.446200
$465$	0	0
$466$	−378.772	−0.0376529
$467$	940.207	0.0931640	0.0465820	−	0.998914i	$-0.485167\pi$
$467$	940.207	0.0931640	0.0465820	+	0.998914i	$0.485167\pi$
$468$	0	0
$469$	−10351.4	−1.01916
$470$	1525.34	0.149699
$471$	0	0
$472$	−3067.38	−0.299127
$473$	294.004	0.0285799
$474$	0	0
$475$	828.192	0.0800002
$476$	−868.758	−0.0836543
$477$	0	0
$478$	−145.767	−0.0139482
$479$	13933.4	1.32909	0.664543	−	0.747250i	$-0.268626\pi$
$479$	13933.4	1.32909	0.664543	+	0.747250i	$0.268626\pi$
$480$	0	0
$481$	−7718.14	−0.731636
$482$	−2363.51	−0.223350
$483$	0	0
$484$	10150.0	0.953235
$485$	−8718.52	−0.816263
$486$	0	0
$487$	−2384.25	−0.221850	−0.110925	−	0.993829i	$-0.535381\pi$
$487$	−2384.25	−0.221850	−0.110925	+	0.993829i	$0.535381\pi$
$488$	−4432.09	−0.411129
$489$	0	0
$490$	2495.08	0.230033
$491$	16076.6	1.47765	0.738824	−	0.673898i	$-0.235382\pi$
$491$	16076.6	1.47765	0.738824	+	0.673898i	$0.235382\pi$
$492$	0	0
$493$	348.891	0.0318727
$494$	2181.48	0.198683
$495$	0	0
$496$	−8566.20	−0.775471
$497$	−4730.73	−0.426966
$498$	0	0
$499$	−16647.1	−1.49344	−0.746719	−	0.665140i	$-0.768372\pi$
$499$	−16647.1	−1.49344	−0.746719	+	0.665140i	$0.768372\pi$
$500$	10199.8	0.912296
$501$	0	0
$502$	−1287.00	−0.114426
$503$	18757.4	1.66272	0.831362	−	0.555732i	$-0.187562\pi$
$503$	18757.4	1.66272	0.831362	+	0.555732i	$0.187562\pi$
$504$	0	0
$505$	−17131.9	−1.50962
$506$	−36.2699	−0.00318655
$507$	0	0
$508$	3355.62	0.293074
$509$	13290.0	1.15730	0.578652	−	0.815575i	$-0.303579\pi$
$509$	13290.0	1.15730	0.578652	+	0.815575i	$0.303579\pi$
$510$	0	0
$511$	5698.75	0.493342
$512$	10279.1	0.887256
$513$	0	0
$514$	−2385.06	−0.204671
$515$	−917.221	−0.0784808
$516$	0	0
$517$	131.552	0.0111908
$518$	2916.44	0.247376
$519$	0	0
$520$	−4710.86	−0.397279
$521$	14633.7	1.23055	0.615273	−	0.788314i	$-0.289046\pi$
$521$	14633.7	1.23055	0.615273	+	0.788314i	$0.289046\pi$
$522$	0	0
$523$	−2718.79	−0.227312	−0.113656	−	0.993520i	$-0.536256\pi$
$523$	−2718.79	−0.227312	−0.113656	+	0.993520i	$0.536256\pi$
$524$	−4093.79	−0.341294
$525$	0	0
$526$	707.988	0.0586877
$527$	670.149	0.0553930
$528$	0	0
$529$	−2688.94	−0.221003
$530$	1278.35	0.104769
$531$	0	0
$532$	16903.6	1.37756
$533$	3319.31	0.269747
$534$	0	0
$535$	12653.5	1.02254
$536$	−3741.84	−0.301535
$537$	0	0
$538$	−40.5645	−0.00325067
$539$	215.186	0.0171962
$540$	0	0
$541$	−10239.0	−0.813692	−0.406846	−	0.913497i	$-0.633372\pi$
$541$	−10239.0	−0.813692	−0.406846	+	0.913497i	$0.633372\pi$
$542$	−3795.43	−0.300789
$543$	0	0
$544$	−474.797	−0.0374205
$545$	9854.75	0.774552
$546$	0	0
$547$	15940.6	1.24602	0.623008	−	0.782216i	$-0.285910\pi$
$547$	15940.6	1.24602	0.623008	+	0.782216i	$0.285910\pi$
$548$	−995.806	−0.0776254
$549$	0	0
$550$	3.67139	0.000284634 0
$551$	−6788.43	−0.524859
$552$	0	0
$553$	1293.70	0.0994826
$554$	684.488	0.0524929
$555$	0	0
$556$	13780.8	1.05114
$557$	12629.3	0.960716	0.480358	−	0.877072i	$-0.340507\pi$
$557$	12629.3	0.960716	0.480358	+	0.877072i	$0.340507\pi$
$558$	0	0
$559$	20484.7	1.54993
$560$	−16905.9	−1.27572
$561$	0	0
$562$	−1071.20	−0.0804019
$563$	3779.83	0.282950	0.141475	−	0.989942i	$-0.454816\pi$
$563$	3779.83	0.282950	0.141475	+	0.989942i	$0.454816\pi$
$564$	0	0
$565$	−5291.40	−0.394002
$566$	−3547.14	−0.263423
$567$	0	0
$568$	−1710.07	−0.126325
$569$	−4697.19	−0.346075	−0.173037	−	0.984915i	$-0.555358\pi$
$569$	−4697.19	−0.346075	−0.173037	+	0.984915i	$0.555358\pi$
$570$	0	0
$571$	−8430.65	−0.617884	−0.308942	−	0.951081i	$-0.599975\pi$
$571$	−8430.65	−0.617884	−0.308942	+	0.951081i	$0.599975\pi$
$572$	−198.307	−0.0144959
$573$	0	0
$574$	−1254.26	−0.0912053
$575$	−959.408	−0.0695828
$576$	0	0
$577$	12071.8	0.870978	0.435489	−	0.900194i	$-0.356575\pi$
$577$	12071.8	0.870978	0.435489	+	0.900194i	$0.356575\pi$
$578$	−2985.08	−0.214815
$579$	0	0
$580$	7155.29	0.512254
$581$	27710.4	1.97869
$582$	0	0
$583$	110.250	0.00783205
$584$	2059.99	0.145964
$585$	0	0
$586$	−3932.45	−0.277215
$587$	−16790.6	−1.18062	−0.590310	−	0.807177i	$-0.700994\pi$
$587$	−16790.6	−1.18062	−0.590310	+	0.807177i	$0.700994\pi$
$588$	0	0
$589$	−13039.2	−0.912176
$590$	2279.28	0.159045
$591$	0	0
$592$	−10012.3	−0.695108
$593$	19380.7	1.34211	0.671053	−	0.741410i	$-0.265842\pi$
$593$	19380.7	1.34211	0.671053	+	0.741410i	$0.265842\pi$
$594$	0	0
$595$	1322.58	0.0911265
$596$	17360.1	1.19312
$597$	0	0
$598$	−2527.10	−0.172811
$599$	950.137	0.0648106	0.0324053	−	0.999475i	$-0.489683\pi$
$599$	950.137	0.0648106	0.0324053	+	0.999475i	$0.489683\pi$
$600$	0	0
$601$	12765.0	0.866384	0.433192	−	0.901302i	$-0.357387\pi$
$601$	12765.0	0.866384	0.433192	+	0.901302i	$0.357387\pi$
$602$	−7740.52	−0.524053
$603$	0	0
$604$	10157.6	0.684283
$605$	−15452.2	−1.03838
$606$	0	0
$607$	10819.3	0.723463	0.361731	−	0.932282i	$-0.382186\pi$
$607$	10819.3	0.723463	0.361731	+	0.932282i	$0.382186\pi$
$608$	9238.21	0.616216
$609$	0	0
$610$	3293.35	0.218596
$611$	9165.85	0.606891
$612$	0	0
$613$	−337.338	−0.0222267	−0.0111133	−	0.999938i	$-0.503538\pi$
$613$	−337.338	−0.0222267	−0.0111133	+	0.999938i	$0.503538\pi$
$614$	−6317.19	−0.415213
$615$	0	0
$616$	153.522	0.0100415
$617$	26564.0	1.73327	0.866635	−	0.498943i	$-0.166278\pi$
$617$	26564.0	1.73327	0.866635	+	0.498943i	$0.166278\pi$
$618$	0	0
$619$	1885.69	0.122443	0.0612216	−	0.998124i	$-0.480500\pi$
$619$	1885.69	0.122443	0.0612216	+	0.998124i	$0.480500\pi$
$620$	13743.9	0.890269
$621$	0	0
$622$	−5088.31	−0.328010
$623$	14801.2	0.951843
$624$	0	0
$625$	−16759.7	−1.07262
$626$	−2183.81	−0.139429
$627$	0	0
$628$	10437.3	0.663206
$629$	783.282	0.0496526
$630$	0	0
$631$	12553.5	0.791990	0.395995	−	0.918253i	$-0.370400\pi$
$631$	12553.5	0.791990	0.395995	+	0.918253i	$0.370400\pi$
$632$	467.649	0.0294337
$633$	0	0
$634$	2575.45	0.161332
$635$	−5108.51	−0.319252
$636$	0	0
$637$	14993.1	0.932572
$638$	−30.0932	−0.00186740
$639$	0	0
$640$	−12865.8	−0.794631
$641$	−18271.8	−1.12588	−0.562942	−	0.826497i	$-0.690330\pi$
$641$	−18271.8	−1.12588	−0.562942	+	0.826497i	$0.690330\pi$
$642$	0	0
$643$	−17090.5	−1.04819	−0.524094	−	0.851660i	$-0.675596\pi$
$643$	−17090.5	−1.04819	−0.524094	+	0.851660i	$0.675596\pi$
$644$	−19581.8	−1.19818
$645$	0	0
$646$	−221.389	−0.0134836
$647$	27559.8	1.67463	0.837317	−	0.546717i	$-0.184123\pi$
$647$	27559.8	1.67463	0.837317	+	0.546717i	$0.184123\pi$
$648$	0	0
$649$	196.575	0.0118894
$650$	255.804	0.0154361
$651$	0	0
$652$	2488.03	0.149446
$653$	−1182.84	−0.0708855	−0.0354428	−	0.999372i	$-0.511284\pi$
$653$	−1182.84	−0.0708855	−0.0354428	+	0.999372i	$0.511284\pi$
$654$	0	0
$655$	6232.28	0.371779
$656$	4305.96	0.256280
$657$	0	0
$658$	−3463.48	−0.205199
$659$	−18185.8	−1.07499	−0.537494	−	0.843268i	$-0.680629\pi$
$659$	−18185.8	−1.07499	−0.537494	+	0.843268i	$0.680629\pi$
$660$	0	0
$661$	4320.27	0.254219	0.127110	−	0.991889i	$-0.459430\pi$
$661$	4320.27	0.254219	0.127110	+	0.991889i	$0.459430\pi$
$662$	4098.50	0.240623
$663$	0	0
$664$	10016.8	0.585431
$665$	−25733.6	−1.50061
$666$	0	0
$667$	7863.97	0.456513
$668$	11970.8	0.693362
$669$	0	0
$670$	2780.45	0.160325
$671$	284.032	0.0163412
$672$	0	0
$673$	−31436.2	−1.80056	−0.900279	−	0.435314i	$-0.856637\pi$
$673$	−31436.2	−1.80056	−0.900279	+	0.435314i	$0.856637\pi$
$674$	1298.67	0.0742182
$675$	0	0
$676$	2941.72	0.167372
$677$	9087.92	0.515919	0.257960	−	0.966156i	$-0.416950\pi$
$677$	9087.92	0.515919	0.257960	+	0.966156i	$0.416950\pi$
$678$	0	0
$679$	19796.6	1.11889
$680$	478.086	0.0269614
$681$	0	0
$682$	−57.8031	−0.00324545
$683$	6304.85	0.353219	0.176609	−	0.984281i	$-0.443487\pi$
$683$	6304.85	0.353219	0.176609	+	0.984281i	$0.443487\pi$
$684$	0	0
$685$	1515.99	0.0845591
$686$	−149.265	−0.00830752
$687$	0	0
$688$	26573.7	1.47255
$689$	7681.66	0.424743
$690$	0	0
$691$	−32948.2	−1.81390	−0.906952	−	0.421234i	$-0.861597\pi$
$691$	−32948.2	−1.81390	−0.906952	+	0.421234i	$0.861597\pi$
$692$	4529.10	0.248801
$693$	0	0
$694$	421.597	0.0230599
$695$	−20979.5	−1.14503
$696$	0	0
$697$	−336.863	−0.0183064
$698$	1027.07	0.0556950
$699$	0	0
$700$	1982.15	0.107026
$701$	27336.7	1.47289	0.736444	−	0.676499i	$-0.236504\pi$
$701$	27336.7	1.47289	0.736444	+	0.676499i	$0.236504\pi$
$702$	0	0
$703$	−15240.5	−0.817646
$704$	−228.845	−0.0122513
$705$	0	0
$706$	−6777.91	−0.361317
$707$	38900.2	2.06930
$708$	0	0
$709$	−4032.45	−0.213599	−0.106800	−	0.994281i	$-0.534060\pi$
$709$	−4032.45	−0.213599	−0.106800	+	0.994281i	$0.534060\pi$
$710$	1270.70	0.0671668
$711$	0	0
$712$	5350.36	0.281620
$713$	15105.1	0.793395
$714$	0	0
$715$	301.898	0.0157907
$716$	−33004.7	−1.72268
$717$	0	0
$718$	−4952.99	−0.257443
$719$	11582.1	0.600751	0.300376	−	0.953821i	$-0.402888\pi$
$719$	11582.1	0.600751	0.300376	+	0.953821i	$0.402888\pi$
$720$	0	0
$721$	2082.67	0.107577
$722$	124.278	0.00640602
$723$	0	0
$724$	−17587.7	−0.902821
$725$	−796.024	−0.0407774
$726$	0	0
$727$	−9801.86	−0.500042	−0.250021	−	0.968240i	$-0.580438\pi$
$727$	−9801.86	−0.500042	−0.250021	+	0.968240i	$0.580438\pi$
$728$	10696.6	0.544566
$729$	0	0
$730$	−1530.71	−0.0776086
$731$	−2078.91	−0.105186
$732$	0	0
$733$	−12949.9	−0.652544	−0.326272	−	0.945276i	$-0.605793\pi$
$733$	−12949.9	−0.652544	−0.326272	+	0.945276i	$0.605793\pi$
$734$	1060.18	0.0533133
$735$	0	0
$736$	−10701.9	−0.535974
$737$	239.798	0.0119852
$738$	0	0
$739$	−14041.7	−0.698961	−0.349481	−	0.936944i	$-0.613642\pi$
$739$	−14041.7	−0.698961	−0.349481	+	0.936944i	$0.613642\pi$
$740$	16064.1	0.798010
$741$	0	0
$742$	−2902.65	−0.143612
$743$	20939.9	1.03393	0.516965	−	0.856006i	$-0.327062\pi$
$743$	20939.9	1.03393	0.516965	+	0.856006i	$0.327062\pi$
$744$	0	0
$745$	−26428.6	−1.29969
$746$	3642.70	0.178778
$747$	0	0
$748$	20.1254	0.000983764 0
$749$	−28731.4	−1.40163
$750$	0	0
$751$	20089.9	0.976155	0.488078	−	0.872800i	$-0.337698\pi$
$751$	20089.9	0.976155	0.488078	+	0.872800i	$0.337698\pi$
$752$	11890.4	0.576591
$753$	0	0
$754$	−2096.74	−0.101272
$755$	−15463.7	−0.745404
$756$	0	0
$757$	−7114.19	−0.341571	−0.170786	−	0.985308i	$-0.554631\pi$
$757$	−7114.19	−0.341571	−0.170786	+	0.985308i	$0.554631\pi$
$758$	4403.35	0.210999
$759$	0	0
$760$	−9302.21	−0.443982
$761$	−29975.9	−1.42789	−0.713945	−	0.700202i	$-0.753093\pi$
$761$	−29975.9	−1.42789	−0.713945	+	0.700202i	$0.753093\pi$
$762$	0	0
$763$	−22376.5	−1.06171
$764$	14792.5	0.700487
$765$	0	0
$766$	7986.86	0.376732
$767$	13696.3	0.644779
$768$	0	0
$769$	−12744.9	−0.597648	−0.298824	−	0.954308i	$-0.596594\pi$
$769$	−12744.9	−0.597648	−0.298824	+	0.954308i	$0.596594\pi$
$770$	−114.078	−0.00533905
$771$	0	0
$772$	36421.5	1.69798
$773$	−39410.4	−1.83376	−0.916879	−	0.399166i	$-0.869300\pi$
$773$	−39410.4	−1.83376	−0.916879	+	0.399166i	$0.869300\pi$
$774$	0	0
$775$	−1529.00	−0.0708689
$776$	7156.08	0.331042
$777$	0	0
$778$	−4188.65	−0.193021
$779$	6554.41	0.301458
$780$	0	0
$781$	109.590	0.00502106
$782$	256.465	0.0117278
$783$	0	0
$784$	19449.7	0.886012
$785$	−15889.5	−0.722445
$786$	0	0
$787$	−2001.93	−0.0906746	−0.0453373	−	0.998972i	$-0.514436\pi$
$787$	−2001.93	−0.0906746	−0.0453373	+	0.998972i	$0.514436\pi$
$788$	−10567.0	−0.477709
$789$	0	0
$790$	−347.496	−0.0156498
$791$	12014.8	0.540074
$792$	0	0
$793$	19789.9	0.886205
$794$	−6431.13	−0.287446
$795$	0	0
$796$	−1907.44	−0.0849337
$797$	11274.3	0.501073	0.250537	−	0.968107i	$-0.419393\pi$
$797$	11274.3	0.501073	0.250537	+	0.968107i	$0.419393\pi$
$798$	0	0
$799$	−930.204	−0.0411868
$800$	1083.29	0.0478751
$801$	0	0
$802$	2488.24	0.109555
$803$	−132.015	−0.00580164
$804$	0	0
$805$	29810.8	1.30521
$806$	−4027.43	−0.176005
$807$	0	0
$808$	14061.7	0.612238
$809$	−481.333	−0.0209181	−0.0104591	−	0.999945i	$-0.503329\pi$
$809$	−481.333	−0.0209181	−0.0104591	+	0.999945i	$0.503329\pi$
$810$	0	0
$811$	−4898.44	−0.212093	−0.106047	−	0.994361i	$-0.533819\pi$
$811$	−4898.44	−0.212093	−0.106047	+	0.994361i	$0.533819\pi$
$812$	−16247.0	−0.702167
$813$	0	0
$814$	−67.5612	−0.00290912
$815$	−3787.71	−0.162795
$816$	0	0
$817$	40449.7	1.73214
$818$	−8169.20	−0.349180
$819$	0	0
$820$	−6908.61	−0.294219
$821$	−32029.8	−1.36157	−0.680785	−	0.732483i	$-0.738361\pi$
$821$	−32029.8	−1.36157	−0.680785	+	0.732483i	$0.738361\pi$
$822$	0	0
$823$	−31092.5	−1.31691	−0.658455	−	0.752620i	$-0.728790\pi$
$823$	−31092.5	−1.31691	−0.658455	+	0.752620i	$0.728790\pi$
$824$	752.847	0.0318285
$825$	0	0
$826$	−5175.41	−0.218009
$827$	3565.93	0.149939	0.0749694	−	0.997186i	$-0.476114\pi$
$827$	3565.93	0.149939	0.0749694	+	0.997186i	$0.476114\pi$
$828$	0	0
$829$	10136.3	0.424666	0.212333	−	0.977197i	$-0.431894\pi$
$829$	10136.3	0.424666	0.212333	+	0.977197i	$0.431894\pi$
$830$	−7443.15	−0.311272
$831$	0	0
$832$	−15944.8	−0.664405
$833$	−1521.59	−0.0632891
$834$	0	0
$835$	−18224.1	−0.755295
$836$	−391.583	−0.0162000
$837$	0	0
$838$	−154.017	−0.00634895
$839$	−35920.1	−1.47807	−0.739035	−	0.673667i	$-0.764718\pi$
$839$	−35920.1	−1.47807	−0.739035	+	0.673667i	$0.764718\pi$
$840$	0	0
$841$	−17864.2	−0.732471
$842$	−2535.68	−0.103783
$843$	0	0
$844$	12624.2	0.514860
$845$	−4478.40	−0.182322
$846$	0	0
$847$	35086.2	1.42335
$848$	9965.00	0.403537
$849$	0	0
$850$	−25.9605	−0.00104757
$851$	17655.1	0.711175
$852$	0	0
$853$	1824.87	0.0732499	0.0366250	−	0.999329i	$-0.488339\pi$
$853$	1824.87	0.0732499	0.0366250	+	0.999329i	$0.488339\pi$
$854$	−7477.99	−0.299639
$855$	0	0
$856$	−10385.8	−0.414697
$857$	−34966.7	−1.39375	−0.696873	−	0.717195i	$-0.745426\pi$
$857$	−34966.7	−1.39375	−0.696873	+	0.717195i	$0.745426\pi$
$858$	0	0
$859$	16921.8	0.672135	0.336068	−	0.941838i	$-0.390903\pi$
$859$	16921.8	0.672135	0.336068	+	0.941838i	$0.390903\pi$
$860$	−42635.6	−1.69054
$861$	0	0
$862$	3582.43	0.141552
$863$	17454.9	0.688497	0.344248	−	0.938879i	$-0.388134\pi$
$863$	17454.9	0.688497	0.344248	+	0.938879i	$0.388134\pi$
$864$	0	0
$865$	−6894.98	−0.271025
$866$	3364.34	0.132015
$867$	0	0
$868$	−31207.3	−1.22033
$869$	−29.9695	−0.00116990
$870$	0	0
$871$	16707.9	0.649971
$872$	−8088.69	−0.314126
$873$	0	0
$874$	−4990.10	−0.193126
$875$	35258.2	1.36222
$876$	0	0
$877$	20777.4	0.800004	0.400002	−	0.916514i	$-0.369009\pi$
$877$	20777.4	0.800004	0.400002	+	0.916514i	$0.369009\pi$
$878$	4392.73	0.168847
$879$	0	0
$880$	391.635	0.0150023
$881$	−34676.1	−1.32607	−0.663034	−	0.748589i	$-0.730732\pi$
$881$	−34676.1	−1.32607	−0.663034	+	0.748589i	$0.730732\pi$
$882$	0	0
$883$	5916.62	0.225493	0.112746	−	0.993624i	$-0.464035\pi$
$883$	5916.62	0.225493	0.112746	+	0.993624i	$0.464035\pi$
$884$	1402.23	0.0533509
$885$	0	0
$886$	−2370.99	−0.0899040
$887$	−254.592	−0.00963739	−0.00481869	−	0.999988i	$-0.501534\pi$
$887$	−254.592	−0.00963739	−0.00481869	+	0.999988i	$0.501534\pi$
$888$	0	0
$889$	11599.6	0.437612
$890$	−3975.68	−0.149736
$891$	0	0
$892$	23600.2	0.885868
$893$	18099.2	0.678237
$894$	0	0
$895$	50245.4	1.87656
$896$	29213.4	1.08923
$897$	0	0
$898$	−5998.26	−0.222900
$899$	12532.8	0.464951
$900$	0	0
$901$	−779.579	−0.0288253
$902$	29.0558	0.00107256
$903$	0	0
$904$	4343.14	0.159790
$905$	26775.1	0.983463
$906$	0	0
$907$	13349.7	0.488719	0.244360	−	0.969685i	$-0.421422\pi$
$907$	13349.7	0.488719	0.244360	+	0.969685i	$0.421422\pi$
$908$	−28946.2	−1.05795
$909$	0	0
$910$	−7948.34	−0.289544
$911$	2273.47	0.0826822	0.0413411	−	0.999145i	$-0.486837\pi$
$911$	2273.47	0.0826822	0.0413411	+	0.999145i	$0.486837\pi$
$912$	0	0
$913$	−641.929	−0.0232692
$914$	209.912	0.00759658
$915$	0	0
$916$	−34712.6	−1.25211
$917$	−14151.2	−0.509613
$918$	0	0
$919$	−6511.37	−0.233722	−0.116861	−	0.993148i	$-0.537283\pi$
$919$	−6511.37	−0.233722	−0.116861	+	0.993148i	$0.537283\pi$
$920$	10776.0	0.386168
$921$	0	0
$922$	−200.287	−0.00715411
$923$	7635.70	0.272299
$924$	0	0
$925$	−1787.12	−0.0635246
$926$	−3965.10	−0.140714
$927$	0	0
$928$	−8879.38	−0.314095
$929$	−810.144	−0.0286114	−0.0143057	−	0.999898i	$-0.504554\pi$
$929$	−810.144	−0.0286114	−0.0143057	+	0.999898i	$0.504554\pi$
$930$	0	0
$931$	29605.8	1.04220
$932$	4737.26	0.166496
$933$	0	0
$934$	573.436	0.0200893
$935$	−30.6383	−0.00107164
$936$	0	0
$937$	−21391.8	−0.745825	−0.372913	−	0.927866i	$-0.621641\pi$
$937$	−21391.8	−0.745825	−0.372913	+	0.927866i	$0.621641\pi$
$938$	−6313.38	−0.219765
$939$	0	0
$940$	−19077.2	−0.661948
$941$	−12923.9	−0.447721	−0.223861	−	0.974621i	$-0.571866\pi$
$941$	−12923.9	−0.447721	−0.223861	+	0.974621i	$0.571866\pi$
$942$	0	0
$943$	−7592.87	−0.262203
$944$	17767.5	0.612588
$945$	0	0
$946$	179.314	0.00616280
$947$	−33461.9	−1.14822	−0.574110	−	0.818778i	$-0.694652\pi$
$947$	−33461.9	−1.14822	−0.574110	+	0.818778i	$0.694652\pi$
$948$	0	0
$949$	−9198.16	−0.314631
$950$	505.118	0.0172507
$951$	0	0
$952$	−1085.56	−0.0369571
$953$	−47235.7	−1.60558	−0.802788	−	0.596264i	$-0.796651\pi$
$953$	−47235.7	−1.60558	−0.802788	+	0.596264i	$0.796651\pi$
$954$	0	0
$955$	−22519.7	−0.763057
$956$	1823.10	0.0616769
$957$	0	0
$958$	8498.02	0.286596
$959$	−3442.26	−0.115909
$960$	0	0
$961$	−5718.10	−0.191940
$962$	−4707.33	−0.157765
$963$	0	0
$964$	29560.2	0.987624
$965$	−55447.1	−1.84964
$966$	0	0
$967$	−20651.7	−0.686778	−0.343389	−	0.939193i	$-0.611575\pi$
$967$	−20651.7	−0.686778	−0.343389	+	0.939193i	$0.611575\pi$
$968$	12683.0	0.421123
$969$	0	0
$970$	−5317.46	−0.176014
$971$	4803.07	0.158741	0.0793707	−	0.996845i	$-0.474709\pi$
$971$	4803.07	0.158741	0.0793707	+	0.996845i	$0.474709\pi$
$972$	0	0
$973$	47636.8	1.56954
$974$	−1454.17	−0.0478383
$975$	0	0
$976$	25672.4	0.841960
$977$	−14726.1	−0.482221	−0.241110	−	0.970498i	$-0.577512\pi$
$977$	−14726.1	−0.482221	−0.241110	+	0.970498i	$0.577512\pi$
$978$	0	0
$979$	−342.880	−0.0111936
$980$	−31205.7	−1.01717
$981$	0	0
$982$	9805.16	0.318631
$983$	21788.2	0.706953	0.353477	−	0.935443i	$-0.384999\pi$
$983$	21788.2	0.706953	0.353477	+	0.935443i	$0.384999\pi$
$984$	0	0
$985$	16087.0	0.520379
$986$	212.790	0.00687283
$987$	0	0
$988$	−27283.5	−0.878547
$989$	−46858.4	−1.50658
$990$	0	0
$991$	40218.9	1.28920	0.644600	−	0.764520i	$-0.277024\pi$
$991$	40218.9	1.28920	0.644600	+	0.764520i	$0.277024\pi$
$992$	−17055.5	−0.545880
$993$	0	0
$994$	−2885.29	−0.0920683
$995$	2903.83	0.0925202
$996$	0	0
$997$	56745.1	1.80254	0.901271	−	0.433256i	$-0.142635\pi$
$997$	56745.1	1.80254	0.901271	+	0.433256i	$0.142635\pi$
$998$	−10153.1	−0.322035
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2151.4.a.b.1.15		28
3.2	odd	2		717.4.a.b.1.14	✓	28

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
717.4.a.b.1.14	✓	28	3.2	odd	2
2151.4.a.b.1.15		28	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

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

\(f(q)\)	\(=\)	\(q+0.609904 q^{2} -7.62802 q^{4} +11.6127 q^{5} -26.3682 q^{7} -9.53159 q^{8} +O(q^{10})\)	\(q+0.609904 q^{2} -7.62802 q^{4} +11.6127 q^{5} -26.3682 q^{7} -9.53159 q^{8} +7.08263 q^{10} +0.610836 q^{11} +42.5600 q^{13} -16.0821 q^{14} +55.2108 q^{16} -4.31924 q^{17} +84.0403 q^{19} -88.5818 q^{20} +0.372552 q^{22} -97.3553 q^{23} +9.85471 q^{25} +25.9575 q^{26} +201.137 q^{28} -80.7760 q^{29} -155.154 q^{31} +109.926 q^{32} -2.63432 q^{34} -306.206 q^{35} -181.347 q^{37} +51.2565 q^{38} -110.688 q^{40} +77.9913 q^{41} +481.313 q^{43} -4.65947 q^{44} -59.3774 q^{46} +215.363 q^{47} +352.281 q^{49} +6.01043 q^{50} -324.648 q^{52} +180.490 q^{53} +7.09346 q^{55} +251.331 q^{56} -49.2656 q^{58} +321.812 q^{59} +464.989 q^{61} -94.6294 q^{62} -374.642 q^{64} +494.236 q^{65} +392.572 q^{67} +32.9472 q^{68} -186.756 q^{70} +179.410 q^{71} -216.122 q^{73} -110.604 q^{74} -641.061 q^{76} -16.1067 q^{77} -49.0631 q^{79} +641.146 q^{80} +47.5672 q^{82} -1050.90 q^{83} -50.1580 q^{85} +293.555 q^{86} -5.82225 q^{88} -561.329 q^{89} -1122.23 q^{91} +742.628 q^{92} +131.351 q^{94} +975.934 q^{95} -750.775 q^{97} +214.858 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q + 5 q^{2} + 103 q^{4} - 6 q^{5} - 68 q^{7} + 39 q^{8}+O(q^{10}) \) 28 * q + 5 * q^2 + 103 * q^4 - 6 * q^5 - 68 * q^7 + 39 * q^8	\( 28 q + 5 q^{2} + 103 q^{4} - 6 q^{5} - 68 q^{7} + 39 q^{8} - 88 q^{10} + 110 q^{11} - 82 q^{13} - 126 q^{14} + 271 q^{16} - 100 q^{17} - 292 q^{19} + 52 q^{20} - 351 q^{22} + 276 q^{23} + 386 q^{25} - 84 q^{26} - 1010 q^{28} + 38 q^{29} - 432 q^{31} + 452 q^{32} - 524 q^{34} + 166 q^{35} - 936 q^{37} + 41 q^{38} - 1183 q^{40} - 1054 q^{41} - 1804 q^{43} + 341 q^{44} - 888 q^{46} + 560 q^{47} + 1074 q^{49} + 1054 q^{50} - 632 q^{52} + 160 q^{53} - 842 q^{55} - 509 q^{56} - 1266 q^{58} - 846 q^{59} - 2220 q^{61} - 82 q^{62} - 1565 q^{64} - 296 q^{65} - 4752 q^{67} + 1719 q^{68} - 5601 q^{70} + 802 q^{71} - 2732 q^{73} + 4581 q^{74} - 5614 q^{76} + 1008 q^{77} - 3172 q^{79} + 732 q^{80} - 9709 q^{82} + 4780 q^{83} - 4624 q^{85} + 2009 q^{86} - 9331 q^{88} - 4372 q^{89} - 7398 q^{91} + 6138 q^{92} - 7068 q^{94} + 3160 q^{95} - 4846 q^{97} + 3772 q^{98}+O(q^{100}) \) 28 * q + 5 * q^2 + 103 * q^4 - 6 * q^5 - 68 * q^7 + 39 * q^8 - 88 * q^10 + 110 * q^11 - 82 * q^13 - 126 * q^14 + 271 * q^16 - 100 * q^17 - 292 * q^19 + 52 * q^20 - 351 * q^22 + 276 * q^23 + 386 * q^25 - 84 * q^26 - 1010 * q^28 + 38 * q^29 - 432 * q^31 + 452 * q^32 - 524 * q^34 + 166 * q^35 - 936 * q^37 + 41 * q^38 - 1183 * q^40 - 1054 * q^41 - 1804 * q^43 + 341 * q^44 - 888 * q^46 + 560 * q^47 + 1074 * q^49 + 1054 * q^50 - 632 * q^52 + 160 * q^53 - 842 * q^55 - 509 * q^56 - 1266 * q^58 - 846 * q^59 - 2220 * q^61 - 82 * q^62 - 1565 * q^64 - 296 * q^65 - 4752 * q^67 + 1719 * q^68 - 5601 * q^70 + 802 * q^71 - 2732 * q^73 + 4581 * q^74 - 5614 * q^76 + 1008 * q^77 - 3172 * q^79 + 732 * q^80 - 9709 * q^82 + 4780 * q^83 - 4624 * q^85 + 2009 * q^86 - 9331 * q^88 - 4372 * q^89 - 7398 * q^91 + 6138 * q^92 - 7068 * q^94 + 3160 * q^95 - 4846 * q^97 + 3772 * q^98

Introduction

L-functions

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