LMFDB - Embedded newform 2151.4.a.f.1.20

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:	$0$
Dimension:	$37$
Twist minimal:	no (minimal twist has level 239)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.20
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)

$f(q)$	$=$	$q-0.123948 q^{2} -7.98464 q^{4} -3.85596 q^{5} -18.4037 q^{7} +1.98126 q^{8} +O(q^{10})$	\(q-0.123948 q^{2} -7.98464 q^{4} -3.85596 q^{5} -18.4037 q^{7} +1.98126 q^{8} +0.477937 q^{10} -29.4804 q^{11} +31.2243 q^{13} +2.28109 q^{14} +63.6315 q^{16} -113.373 q^{17} +89.3172 q^{19} +30.7884 q^{20} +3.65403 q^{22} -129.390 q^{23} -110.132 q^{25} -3.87018 q^{26} +146.947 q^{28} -62.2031 q^{29} -250.107 q^{31} -23.7370 q^{32} +14.0523 q^{34} +70.9638 q^{35} -98.9201 q^{37} -11.0707 q^{38} -7.63965 q^{40} +169.570 q^{41} -524.710 q^{43} +235.390 q^{44} +16.0376 q^{46} +554.220 q^{47} -4.30529 q^{49} +13.6505 q^{50} -249.315 q^{52} -702.263 q^{53} +113.675 q^{55} -36.4624 q^{56} +7.70992 q^{58} -629.087 q^{59} -264.362 q^{61} +31.0001 q^{62} -506.110 q^{64} -120.400 q^{65} -93.7760 q^{67} +905.240 q^{68} -8.79579 q^{70} -745.038 q^{71} +142.534 q^{73} +12.2609 q^{74} -713.165 q^{76} +542.548 q^{77} -662.180 q^{79} -245.361 q^{80} -21.0178 q^{82} +217.606 q^{83} +437.161 q^{85} +65.0365 q^{86} -58.4083 q^{88} -995.759 q^{89} -574.641 q^{91} +1033.13 q^{92} -68.6943 q^{94} -344.404 q^{95} +1515.36 q^{97} +0.533630 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 37 q - 4 q^{2} + 170 q^{4} - 43 q^{5} + 60 q^{7} - 27 q^{8}+O(q^{10}) $ 37 * q - 4 * q^2 + 170 * q^4 - 43 * q^5 + 60 * q^7 - 27 * q^8	$ 37 q - 4 q^{2} + 170 q^{4} - 43 q^{5} + 60 q^{7} - 27 q^{8} + 147 q^{10} - 55 q^{11} + 250 q^{13} - 169 q^{14} + 918 q^{16} - 189 q^{17} + 550 q^{19} - 486 q^{20} + 226 q^{22} - 74 q^{23} + 1604 q^{25} - 560 q^{26} + 829 q^{28} - 389 q^{29} + 1107 q^{31} - 125 q^{32} + 1423 q^{34} - 270 q^{35} + 1002 q^{37} - 1037 q^{38} + 1536 q^{40} - 1518 q^{41} + 1098 q^{43} - 1037 q^{44} + 1030 q^{46} - 1214 q^{47} + 4663 q^{49} - 929 q^{50} + 2895 q^{52} - 904 q^{53} + 1350 q^{55} - 2556 q^{56} + 1396 q^{58} - 1658 q^{59} + 2313 q^{61} + 4519 q^{62} + 3807 q^{64} + 56 q^{65} + 1535 q^{67} + 6526 q^{68} - 4099 q^{70} + 3255 q^{71} + 3154 q^{73} + 2629 q^{74} + 1981 q^{76} + 3734 q^{77} + 2260 q^{79} + 8242 q^{80} - 9898 q^{82} + 939 q^{83} + 1272 q^{85} + 3457 q^{86} - 1808 q^{88} - 1486 q^{89} + 174 q^{91} + 14076 q^{92} - 984 q^{94} + 1828 q^{95} + 6148 q^{97} + 6243 q^{98}+O(q^{100}) $ 37 * q - 4 * q^2 + 170 * q^4 - 43 * q^5 + 60 * q^7 - 27 * q^8 + 147 * q^10 - 55 * q^11 + 250 * q^13 - 169 * q^14 + 918 * q^16 - 189 * q^17 + 550 * q^19 - 486 * q^20 + 226 * q^22 - 74 * q^23 + 1604 * q^25 - 560 * q^26 + 829 * q^28 - 389 * q^29 + 1107 * q^31 - 125 * q^32 + 1423 * q^34 - 270 * q^35 + 1002 * q^37 - 1037 * q^38 + 1536 * q^40 - 1518 * q^41 + 1098 * q^43 - 1037 * q^44 + 1030 * q^46 - 1214 * q^47 + 4663 * q^49 - 929 * q^50 + 2895 * q^52 - 904 * q^53 + 1350 * q^55 - 2556 * q^56 + 1396 * q^58 - 1658 * q^59 + 2313 * q^61 + 4519 * q^62 + 3807 * q^64 + 56 * q^65 + 1535 * q^67 + 6526 * q^68 - 4099 * q^70 + 3255 * q^71 + 3154 * q^73 + 2629 * q^74 + 1981 * q^76 + 3734 * q^77 + 2260 * q^79 + 8242 * q^80 - 9898 * q^82 + 939 * q^83 + 1272 * q^85 + 3457 * q^86 - 1808 * q^88 - 1486 * q^89 + 174 * q^91 + 14076 * q^92 - 984 * q^94 + 1828 * q^95 + 6148 * q^97 + 6243 * 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 $

$2$	−0.123948	−0.0438221	−0.0219110	−	0.999760i	$-0.506975\pi$
$2$	−0.123948	−0.0438221	−0.0219110	+	0.999760i	$0.506975\pi$
$3$	0	0
$3$	0	0
$4$	−7.98464	−0.998080
$5$	−3.85596	−0.344888	−0.172444	−	0.985019i	$-0.555166\pi$
$5$	−3.85596	−0.344888	−0.172444	+	0.985019i	$0.555166\pi$
$6$	0	0
$7$	−18.4037	−0.993704	−0.496852	−	0.867835i	$-0.665511\pi$
$7$	−18.4037	−0.993704	−0.496852	+	0.867835i	$0.665511\pi$
$8$	1.98126	0.0875600
$9$	0	0
$10$	0.477937	0.0151137
$11$	−29.4804	−0.808062	−0.404031	−	0.914745i	$-0.632391\pi$
$11$	−29.4804	−0.808062	−0.404031	+	0.914745i	$0.632391\pi$
$12$	0	0
$13$	31.2243	0.666159	0.333079	−	0.942899i	$-0.391912\pi$
$13$	31.2243	0.666159	0.333079	+	0.942899i	$0.391912\pi$
$14$	2.28109	0.0435462
$15$	0	0
$16$	63.6315	0.994243
$17$	−113.373	−1.61747	−0.808733	−	0.588176i	$-0.799846\pi$
$17$	−113.373	−1.61747	−0.808733	+	0.588176i	$0.799846\pi$
$18$	0	0
$19$	89.3172	1.07846	0.539231	−	0.842158i	$-0.318715\pi$
$19$	89.3172	1.07846	0.539231	+	0.842158i	$0.318715\pi$
$20$	30.7884	0.344225
$21$	0	0
$22$	3.65403	0.0354110
$23$	−129.390	−1.17303	−0.586515	−	0.809938i	$-0.699501\pi$
$23$	−129.390	−1.17303	−0.586515	+	0.809938i	$0.699501\pi$
$24$	0	0
$25$	−110.132	−0.881053
$26$	−3.87018	−0.0291925
$27$	0	0
$28$	146.947	0.991796
$29$	−62.2031	−0.398304	−0.199152	−	0.979969i	$-0.563819\pi$
$29$	−62.2031	−0.398304	−0.199152	+	0.979969i	$0.563819\pi$
$30$	0	0
$31$	−250.107	−1.44905	−0.724523	−	0.689250i	$-0.757940\pi$
$31$	−250.107	−1.44905	−0.724523	+	0.689250i	$0.757940\pi$
$32$	−23.7370	−0.131130
$33$	0	0
$34$	14.0523	0.0708808
$35$	70.9638	0.342716
$36$	0	0
$37$	−98.9201	−0.439523	−0.219762	−	0.975554i	$-0.570528\pi$
$37$	−98.9201	−0.439523	−0.219762	+	0.975554i	$0.570528\pi$
$38$	−11.0707	−0.0472604
$39$	0	0
$40$	−7.63965	−0.0301984
$41$	169.570	0.645911	0.322955	−	0.946414i	$-0.395324\pi$
$41$	169.570	0.645911	0.322955	+	0.946414i	$0.395324\pi$
$42$	0	0
$43$	−524.710	−1.86087	−0.930436	−	0.366454i	$-0.880572\pi$
$43$	−524.710	−1.86087	−0.930436	+	0.366454i	$0.880572\pi$
$44$	235.390	0.806510
$45$	0	0
$46$	16.0376	0.0514046
$47$	554.220	1.72003	0.860014	−	0.510270i	$-0.170454\pi$
$47$	554.220	1.72003	0.860014	+	0.510270i	$0.170454\pi$
$48$	0	0
$49$	−4.30529	−0.0125519
$50$	13.6505	0.0386096
$51$	0	0
$52$	−249.315	−0.664879
$53$	−702.263	−1.82006	−0.910031	−	0.414540i	$-0.863943\pi$
$53$	−702.263	−1.82006	−0.910031	+	0.414540i	$0.863943\pi$
$54$	0	0
$55$	113.675	0.278690
$56$	−36.4624	−0.0870088
$57$	0	0
$58$	7.70992	0.0174545
$59$	−629.087	−1.38814	−0.694069	−	0.719909i	$-0.744184\pi$
$59$	−629.087	−1.38814	−0.694069	+	0.719909i	$0.744184\pi$
$60$	0	0
$61$	−264.362	−0.554886	−0.277443	−	0.960742i	$-0.589487\pi$
$61$	−264.362	−0.554886	−0.277443	+	0.960742i	$0.589487\pi$
$62$	31.0001	0.0635003
$63$	0	0
$64$	−506.110	−0.988496
$65$	−120.400	−0.229750
$66$	0	0
$67$	−93.7760	−0.170993	−0.0854967	−	0.996338i	$-0.527248\pi$
$67$	−93.7760	−0.170993	−0.0854967	+	0.996338i	$0.527248\pi$
$68$	905.240	1.61436
$69$	0	0
$70$	−8.79579	−0.0150185
$71$	−745.038	−1.24535	−0.622675	−	0.782481i	$-0.713954\pi$
$71$	−745.038	−1.24535	−0.622675	+	0.782481i	$0.713954\pi$
$72$	0	0
$73$	142.534	0.228525	0.114262	−	0.993451i	$-0.463550\pi$
$73$	142.534	0.228525	0.114262	+	0.993451i	$0.463550\pi$
$74$	12.2609	0.0192608
$75$	0	0
$76$	−713.165	−1.07639
$77$	542.548	0.802974
$78$	0	0
$79$	−662.180	−0.943052	−0.471526	−	0.881852i	$-0.656297\pi$
$79$	−662.180	−0.943052	−0.471526	+	0.881852i	$0.656297\pi$
$80$	−245.361	−0.342902
$81$	0	0
$82$	−21.0178	−0.0283052
$83$	217.606	0.287775	0.143887	−	0.989594i	$-0.454040\pi$
$83$	217.606	0.287775	0.143887	+	0.989594i	$0.454040\pi$
$84$	0	0
$85$	437.161	0.557844
$86$	65.0365	0.0815473
$87$	0	0
$88$	−58.4083	−0.0707539
$89$	−995.759	−1.18596	−0.592979	−	0.805218i	$-0.702048\pi$
$89$	−995.759	−1.18596	−0.592979	+	0.805218i	$0.702048\pi$
$90$	0	0
$91$	−574.641	−0.661965
$92$	1033.13	1.17078
$93$	0	0
$94$	−68.6943	−0.0753753
$95$	−344.404	−0.371948
$96$	0	0
$97$	1515.36	1.58620	0.793101	−	0.609090i	$-0.208465\pi$
$97$	1515.36	1.58620	0.793101	+	0.609090i	$0.208465\pi$
$98$	0.533630	0.000550049 0
$99$	0	0
$100$	879.361	0.879361
$101$	421.891	0.415640	0.207820	−	0.978167i	$-0.433363\pi$
$101$	421.891	0.415640	0.207820	+	0.978167i	$0.433363\pi$
$102$	0	0
$103$	−999.584	−0.956232	−0.478116	−	0.878297i	$-0.658680\pi$
$103$	−999.584	−0.956232	−0.478116	+	0.878297i	$0.658680\pi$
$104$	61.8634	0.0583289
$105$	0	0
$106$	87.0439	0.0797589
$107$	1010.67	0.913135	0.456568	−	0.889689i	$-0.349079\pi$
$107$	1010.67	0.913135	0.456568	+	0.889689i	$0.349079\pi$
$108$	0	0
$109$	−479.724	−0.421553	−0.210776	−	0.977534i	$-0.567599\pi$
$109$	−479.724	−0.421553	−0.210776	+	0.977534i	$0.567599\pi$
$110$	−14.0898	−0.0122128
$111$	0	0
$112$	−1171.05	−0.987983
$113$	1352.00	1.12553	0.562767	−	0.826615i	$-0.309737\pi$
$113$	1352.00	1.12553	0.562767	+	0.826615i	$0.309737\pi$
$114$	0	0
$115$	498.923	0.404564
$116$	496.669	0.397539
$117$	0	0
$118$	77.9738	0.0608311
$119$	2086.47	1.60728
$120$	0	0
$121$	−461.905	−0.347036
$122$	32.7670	0.0243163
$123$	0	0
$124$	1997.01	1.44626
$125$	906.658	0.648752
$126$	0	0
$127$	−1394.77	−0.974532	−0.487266	−	0.873254i	$-0.662006\pi$
$127$	−1394.77	−0.974532	−0.487266	+	0.873254i	$0.662006\pi$
$128$	252.627	0.174448
$129$	0	0
$130$	14.9232	0.0100681
$131$	−961.698	−0.641404	−0.320702	−	0.947180i	$-0.603919\pi$
$131$	−961.698	−0.641404	−0.320702	+	0.947180i	$0.603919\pi$
$132$	0	0
$133$	−1643.76	−1.07167
$134$	11.6233	0.00749329
$135$	0	0
$136$	−224.621	−0.141625
$137$	−1011.04	−0.630503	−0.315251	−	0.949008i	$-0.602089\pi$
$137$	−1011.04	−0.630503	−0.315251	+	0.949008i	$0.602089\pi$
$138$	0	0
$139$	3102.70	1.89329	0.946645	−	0.322277i	$-0.104448\pi$
$139$	3102.70	1.89329	0.946645	+	0.322277i	$0.104448\pi$
$140$	−566.620	−0.342058
$141$	0	0
$142$	92.3457	0.0545738
$143$	−920.505	−0.538297
$144$	0	0
$145$	239.853	0.137370
$146$	−17.6667	−0.0100144
$147$	0	0
$148$	789.841	0.438679
$149$	271.639	0.149352	0.0746762	−	0.997208i	$-0.476208\pi$
$149$	271.639	0.149352	0.0746762	+	0.997208i	$0.476208\pi$
$150$	0	0
$151$	356.605	0.192186	0.0960930	−	0.995372i	$-0.469365\pi$
$151$	356.605	0.192186	0.0960930	+	0.995372i	$0.469365\pi$
$152$	176.960	0.0944301
$153$	0	0
$154$	−67.2475	−0.0351880
$155$	964.401	0.499758
$156$	0	0
$157$	−1869.29	−0.950227	−0.475114	−	0.879924i	$-0.657593\pi$
$157$	−1869.29	−0.950227	−0.475114	+	0.879924i	$0.657593\pi$
$158$	82.0756	0.0413265
$159$	0	0
$160$	91.5291	0.0452250
$161$	2381.25	1.16565
$162$	0	0
$163$	1741.42	0.836799	0.418400	−	0.908263i	$-0.362591\pi$
$163$	1741.42	0.836799	0.418400	+	0.908263i	$0.362591\pi$
$164$	−1353.95	−0.644670
$165$	0	0
$166$	−26.9717	−0.0126109
$167$	−2356.55	−1.09195	−0.545974	−	0.837802i	$-0.683840\pi$
$167$	−2356.55	−1.09195	−0.545974	+	0.837802i	$0.683840\pi$
$168$	0	0
$169$	−1222.04	−0.556233
$170$	−54.1850	−0.0244459
$171$	0	0
$172$	4189.62	1.85730
$173$	2015.70	0.885844	0.442922	−	0.896560i	$-0.353942\pi$
$173$	2015.70	0.885844	0.442922	+	0.896560i	$0.353942\pi$
$174$	0	0
$175$	2026.82	0.875506
$176$	−1875.88	−0.803409
$177$	0	0
$178$	123.422	0.0519711
$179$	3372.85	1.40837	0.704186	−	0.710015i	$-0.251312\pi$
$179$	3372.85	1.40837	0.704186	+	0.710015i	$0.251312\pi$
$180$	0	0
$181$	−1441.64	−0.592023	−0.296011	−	0.955184i	$-0.595657\pi$
$181$	−1441.64	−0.592023	−0.296011	+	0.955184i	$0.595657\pi$
$182$	71.2254	0.0290087
$183$	0	0
$184$	−256.355	−0.102711
$185$	381.432	0.151586
$186$	0	0
$187$	3342.28	1.30701
$188$	−4425.25	−1.71673
$189$	0	0
$190$	42.6880	0.0162995
$191$	3371.21	1.27713	0.638566	−	0.769567i	$-0.279528\pi$
$191$	3371.21	1.27713	0.638566	+	0.769567i	$0.279528\pi$
$192$	0	0
$193$	2619.50	0.976971	0.488486	−	0.872572i	$-0.337549\pi$
$193$	2619.50	0.976971	0.488486	+	0.872572i	$0.337549\pi$
$194$	−187.825	−0.0695107
$195$	0	0
$196$	34.3762	0.0125278
$197$	−1916.66	−0.693179	−0.346590	−	0.938017i	$-0.612660\pi$
$197$	−1916.66	−0.693179	−0.346590	+	0.938017i	$0.612660\pi$
$198$	0	0
$199$	−1731.72	−0.616876	−0.308438	−	0.951244i	$-0.599806\pi$
$199$	−1731.72	−0.616876	−0.308438	+	0.951244i	$0.599806\pi$
$200$	−218.199	−0.0771450
$201$	0	0
$202$	−52.2923	−0.0182142
$203$	1144.76	0.395797
$204$	0	0
$205$	−653.854	−0.222767
$206$	123.896	0.0419041
$207$	0	0
$208$	1986.85	0.662323
$209$	−2633.11	−0.871463
$210$	0	0
$211$	3238.72	1.05669	0.528347	−	0.849029i	$-0.322812\pi$
$211$	3238.72	1.05669	0.528347	+	0.849029i	$0.322812\pi$
$212$	5607.32	1.81657
$213$	0	0
$214$	−125.270	−0.0400155
$215$	2023.26	0.641792
$216$	0	0
$217$	4602.88	1.43992
$218$	59.4606	0.0184733
$219$	0	0
$220$	−907.656	−0.278155
$221$	−3539.98	−1.07749
$222$	0	0
$223$	−2864.63	−0.860224	−0.430112	−	0.902776i	$-0.641526\pi$
$223$	−2864.63	−0.860224	−0.430112	+	0.902776i	$0.641526\pi$
$224$	436.848	0.130304
$225$	0	0
$226$	−167.577	−0.0493233
$227$	−4189.32	−1.22491	−0.612456	−	0.790505i	$-0.709818\pi$
$227$	−4189.32	−1.22491	−0.612456	+	0.790505i	$0.709818\pi$
$228$	0	0
$229$	57.2610	0.0165236	0.00826182	−	0.999966i	$-0.497370\pi$
$229$	57.2610	0.0165236	0.00826182	+	0.999966i	$0.497370\pi$
$230$	−61.8403	−0.0177288
$231$	0	0
$232$	−123.240	−0.0348755
$233$	−99.8782	−0.0280826	−0.0140413	−	0.999901i	$-0.504470\pi$
$233$	−99.8782	−0.0280826	−0.0140413	+	0.999901i	$0.504470\pi$
$234$	0	0
$235$	−2137.05	−0.593217
$236$	5023.03	1.38547
$237$	0	0
$238$	−258.613	−0.0704345
$239$	239.000	0.0646846
$239$	239.000	0.0646846
$240$	0	0
$241$	−3859.85	−1.03168	−0.515839	−	0.856685i	$-0.672520\pi$
$241$	−3859.85	−1.03168	−0.515839	+	0.856685i	$0.672520\pi$
$242$	57.2520	0.0152079
$243$	0	0
$244$	2110.83	0.553820
$245$	16.6010	0.00432898
$246$	0	0
$247$	2788.87	0.718426
$248$	−495.525	−0.126879
$249$	0	0
$250$	−112.378	−0.0284297
$251$	1123.54	0.282539	0.141270	−	0.989971i	$-0.454882\pi$
$251$	1123.54	0.282539	0.141270	+	0.989971i	$0.454882\pi$
$252$	0	0
$253$	3814.47	0.947881
$254$	172.878	0.0427060
$255$	0	0
$256$	4017.57	0.980852
$257$	1665.41	0.404223	0.202111	−	0.979363i	$-0.435220\pi$
$257$	1665.41	0.404223	0.202111	+	0.979363i	$0.435220\pi$
$258$	0	0
$259$	1820.49	0.436756
$260$	961.348	0.229309
$261$	0	0
$262$	119.200	0.0281077
$263$	3029.19	0.710219	0.355109	−	0.934825i	$-0.384444\pi$
$263$	3029.19	0.710219	0.355109	+	0.934825i	$0.384444\pi$
$264$	0	0
$265$	2707.90	0.627717
$266$	203.740	0.0469629
$267$	0	0
$268$	748.767	0.170665
$269$	−204.886	−0.0464391	−0.0232196	−	0.999730i	$-0.507392\pi$
$269$	−204.886	−0.0464391	−0.0232196	+	0.999730i	$0.507392\pi$
$270$	0	0
$271$	−4713.22	−1.05649	−0.528243	−	0.849093i	$-0.677149\pi$
$271$	−4713.22	−1.05649	−0.528243	+	0.849093i	$0.677149\pi$
$272$	−7214.08	−1.60815
$273$	0	0
$274$	125.316	0.0276299
$275$	3246.72	0.711945
$276$	0	0
$277$	3822.72	0.829189	0.414594	−	0.910006i	$-0.363923\pi$
$277$	3822.72	0.829189	0.414594	+	0.910006i	$0.363923\pi$
$278$	−384.572	−0.0829680
$279$	0	0
$280$	140.598	0.0300082
$281$	7450.89	1.58179	0.790895	−	0.611952i	$-0.209615\pi$
$281$	7450.89	1.58179	0.790895	+	0.611952i	$0.209615\pi$
$282$	0	0
$283$	−5470.82	−1.14914	−0.574570	−	0.818456i	$-0.694831\pi$
$283$	−5470.82	−1.14914	−0.574570	+	0.818456i	$0.694831\pi$
$284$	5948.86	1.24296
$285$	0	0
$286$	114.094	0.0235893
$287$	−3120.70	−0.641844
$288$	0	0
$289$	7940.38	1.61620
$290$	−29.7292	−0.00601985
$291$	0	0
$292$	−1138.08	−0.228086
$293$	5663.51	1.12923	0.564617	−	0.825353i	$-0.309024\pi$
$293$	5663.51	1.12923	0.564617	+	0.825353i	$0.309024\pi$
$294$	0	0
$295$	2425.73	0.478751
$296$	−195.986	−0.0384847
$297$	0	0
$298$	−33.6690	−0.00654493
$299$	−4040.11	−0.781424
$300$	0	0
$301$	9656.58	1.84916
$302$	−44.2003	−0.00842199
$303$	0	0
$304$	5683.39	1.07225
$305$	1019.37	0.191373
$306$	0	0
$307$	−6774.96	−1.25950	−0.629752	−	0.776796i	$-0.716843\pi$
$307$	−6774.96	−1.25950	−0.629752	+	0.776796i	$0.716843\pi$
$308$	−4332.04	−0.801432
$309$	0	0
$310$	−119.535	−0.0219005
$311$	1478.85	0.269639	0.134819	−	0.990870i	$-0.456955\pi$
$311$	1478.85	0.269639	0.134819	+	0.990870i	$0.456955\pi$
$312$	0	0
$313$	−4317.67	−0.779709	−0.389855	−	0.920876i	$-0.627475\pi$
$313$	−4317.67	−0.779709	−0.389855	+	0.920876i	$0.627475\pi$
$314$	231.694	0.0416410
$315$	0	0
$316$	5287.27	0.941241
$317$	−4738.07	−0.839485	−0.419742	−	0.907643i	$-0.637880\pi$
$317$	−4738.07	−0.839485	−0.419742	+	0.907643i	$0.637880\pi$
$318$	0	0
$319$	1833.77	0.321854
$320$	1951.54	0.340920
$321$	0	0
$322$	−295.150	−0.0510810
$323$	−10126.1	−1.74438
$324$	0	0
$325$	−3438.78	−0.586921
$326$	−215.844	−0.0366703
$327$	0	0
$328$	335.961	0.0565560
$329$	−10199.7	−1.70920
$330$	0	0
$331$	6147.35	1.02081	0.510406	−	0.859933i	$-0.329495\pi$
$331$	6147.35	1.02081	0.510406	+	0.859933i	$0.329495\pi$
$332$	−1737.50	−0.287222
$333$	0	0
$334$	292.089	0.0478515
$335$	361.596	0.0589735
$336$	0	0
$337$	−4082.42	−0.659892	−0.329946	−	0.944000i	$-0.607031\pi$
$337$	−4082.42	−0.659892	−0.329946	+	0.944000i	$0.607031\pi$
$338$	151.469	0.0243753
$339$	0	0
$340$	−3490.57	−0.556773
$341$	7373.24	1.17092
$342$	0	0
$343$	6391.69	1.00618
$344$	−1039.59	−0.162938
$345$	0	0
$346$	−249.841	−0.0388195
$347$	−4349.07	−0.672825	−0.336413	−	0.941715i	$-0.609214\pi$
$347$	−4349.07	−0.672825	−0.336413	+	0.941715i	$0.609214\pi$
$348$	0	0
$349$	5489.52	0.841970	0.420985	−	0.907068i	$-0.361684\pi$
$349$	5489.52	0.841970	0.420985	+	0.907068i	$0.361684\pi$
$350$	−251.220	−0.0383665
$351$	0	0
$352$	699.778	0.105961
$353$	−2794.52	−0.421352	−0.210676	−	0.977556i	$-0.567566\pi$
$353$	−2794.52	−0.421352	−0.210676	+	0.977556i	$0.567566\pi$
$354$	0	0
$355$	2872.84	0.429505
$356$	7950.77	1.18368
$357$	0	0
$358$	−418.057	−0.0617178
$359$	9694.78	1.42527	0.712634	−	0.701536i	$-0.247502\pi$
$359$	9694.78	1.42527	0.712634	+	0.701536i	$0.247502\pi$
$360$	0	0
$361$	1118.56	0.163079
$362$	178.688	0.0259437
$363$	0	0
$364$	4588.30	0.660693
$365$	−549.604	−0.0788153
$366$	0	0
$367$	3653.01	0.519579	0.259789	−	0.965665i	$-0.416347\pi$
$367$	3653.01	0.519579	0.259789	+	0.965665i	$0.416347\pi$
$368$	−8233.29	−1.16628
$369$	0	0
$370$	−47.2776	−0.00664282
$371$	12924.2	1.80860
$372$	0	0
$373$	−5128.98	−0.711980	−0.355990	−	0.934490i	$-0.615856\pi$
$373$	−5128.98	−0.711980	−0.355990	+	0.934490i	$0.615856\pi$
$374$	−414.267	−0.0572760
$375$	0	0
$376$	1098.05	0.150606
$377$	−1942.25	−0.265334
$378$	0	0
$379$	−3027.74	−0.410355	−0.205178	−	0.978725i	$-0.565777\pi$
$379$	−3027.74	−0.410355	−0.205178	+	0.978725i	$0.565777\pi$
$380$	2749.94	0.371234
$381$	0	0
$382$	−417.853	−0.0559666
$383$	−2688.58	−0.358695	−0.179347	−	0.983786i	$-0.557399\pi$
$383$	−2688.58	−0.358695	−0.179347	+	0.983786i	$0.557399\pi$
$384$	0	0
$385$	−2092.04	−0.276936
$386$	−324.680	−0.0428129
$387$	0	0
$388$	−12099.6	−1.58316
$389$	−9538.97	−1.24330	−0.621652	−	0.783294i	$-0.713538\pi$
$389$	−9538.97	−1.24330	−0.621652	+	0.783294i	$0.713538\pi$
$390$	0	0
$391$	14669.3	1.89734
$392$	−8.52989	−0.00109904
$393$	0	0
$394$	237.565	0.0303766
$395$	2553.34	0.325247
$396$	0	0
$397$	8824.59	1.11560	0.557800	−	0.829975i	$-0.311646\pi$
$397$	8824.59	1.11560	0.557800	+	0.829975i	$0.311646\pi$
$398$	214.642	0.0270328
$399$	0	0
$400$	−7007.84	−0.875980
$401$	3196.44	0.398061	0.199031	−	0.979993i	$-0.436221\pi$
$401$	3196.44	0.398061	0.199031	+	0.979993i	$0.436221\pi$
$402$	0	0
$403$	−7809.40	−0.965295
$404$	−3368.64	−0.414842
$405$	0	0
$406$	−141.891	−0.0173446
$407$	2916.21	0.355162
$408$	0	0
$409$	−9582.89	−1.15854	−0.579271	−	0.815135i	$-0.696663\pi$
$409$	−9582.89	−1.15854	−0.579271	+	0.815135i	$0.696663\pi$
$410$	81.0436	0.00976210
$411$	0	0
$412$	7981.31	0.954396
$413$	11577.5	1.37940
$414$	0	0
$415$	−839.078	−0.0992500
$416$	−741.172	−0.0873533
$417$	0	0
$418$	326.367	0.0381894
$419$	−3798.09	−0.442838	−0.221419	−	0.975179i	$-0.571069\pi$
$419$	−3798.09	−0.442838	−0.221419	+	0.975179i	$0.571069\pi$
$420$	0	0
$421$	13023.8	1.50769	0.753847	−	0.657049i	$-0.228196\pi$
$421$	13023.8	1.50769	0.753847	+	0.657049i	$0.228196\pi$
$422$	−401.431	−0.0463065
$423$	0	0
$424$	−1391.36	−0.159365
$425$	12485.9	1.42507
$426$	0	0
$427$	4865.22	0.551392
$428$	−8069.86	−0.911382
$429$	0	0
$430$	−250.778	−0.0281247
$431$	−7332.10	−0.819431	−0.409716	−	0.912213i	$-0.634372\pi$
$431$	−7332.10	−0.819431	−0.409716	+	0.912213i	$0.634372\pi$
$432$	0	0
$433$	17749.2	1.96991	0.984955	−	0.172811i	$-0.0552849\pi$
$433$	17749.2	1.96991	0.984955	+	0.172811i	$0.0552849\pi$
$434$	−570.515	−0.0631005
$435$	0	0
$436$	3830.42	0.420743
$437$	−11556.8	−1.26507
$438$	0	0
$439$	−308.137	−0.0335002	−0.0167501	−	0.999860i	$-0.505332\pi$
$439$	−308.137	−0.0335002	−0.0167501	+	0.999860i	$0.505332\pi$
$440$	225.220	0.0244021
$441$	0	0
$442$	438.773	0.0472178
$443$	−6784.38	−0.727620	−0.363810	−	0.931473i	$-0.618524\pi$
$443$	−6784.38	−0.727620	−0.363810	+	0.931473i	$0.618524\pi$
$444$	0	0
$445$	3839.61	0.409022
$446$	355.064	0.0376968
$447$	0	0
$448$	9314.28	0.982273
$449$	−10502.1	−1.10384	−0.551919	−	0.833898i	$-0.686104\pi$
$449$	−10502.1	−1.10384	−0.551919	+	0.833898i	$0.686104\pi$
$450$	0	0
$451$	−4998.99	−0.521936
$452$	−10795.2	−1.12337
$453$	0	0
$454$	519.256	0.0536782
$455$	2215.79	0.228303
$456$	0	0
$457$	6714.75	0.687315	0.343657	−	0.939095i	$-0.388334\pi$
$457$	6714.75	0.687315	0.343657	+	0.939095i	$0.388334\pi$
$458$	−7.09736	−0.000724100 0
$459$	0	0
$460$	−3983.72	−0.403787
$461$	15510.0	1.56697	0.783485	−	0.621410i	$-0.213440\pi$
$461$	15510.0	1.56697	0.783485	+	0.621410i	$0.213440\pi$
$462$	0	0
$463$	−9791.96	−0.982874	−0.491437	−	0.870913i	$-0.663528\pi$
$463$	−9791.96	−0.982874	−0.491437	+	0.870913i	$0.663528\pi$
$464$	−3958.08	−0.396011
$465$	0	0
$466$	12.3797	0.00123064
$467$	−916.878	−0.0908523	−0.0454262	−	0.998968i	$-0.514465\pi$
$467$	−916.878	−0.0908523	−0.0454262	+	0.998968i	$0.514465\pi$
$468$	0	0
$469$	1725.82	0.169917
$470$	264.882	0.0259960
$471$	0	0
$472$	−1246.38	−0.121545
$473$	15468.7	1.50370
$474$	0	0
$475$	−9836.64	−0.950181
$476$	−16659.7	−1.60420
$477$	0	0
$478$	−29.6235	−0.00283462
$479$	217.192	0.0207177	0.0103588	−	0.999946i	$-0.496703\pi$
$479$	217.192	0.0207177	0.0103588	+	0.999946i	$0.496703\pi$
$480$	0	0
$481$	−3088.71	−0.292792
$482$	478.419	0.0452103
$483$	0	0
$484$	3688.15	0.346370
$485$	−5843.17	−0.547061
$486$	0	0
$487$	−17769.0	−1.65337	−0.826685	−	0.562665i	$-0.809776\pi$
$487$	−17769.0	−1.65337	−0.826685	+	0.562665i	$0.809776\pi$
$488$	−523.768	−0.0485858
$489$	0	0
$490$	−2.05766	−0.000189705 0
$491$	12873.3	1.18322	0.591611	−	0.806224i	$-0.298492\pi$
$491$	12873.3	1.18322	0.591611	+	0.806224i	$0.298492\pi$
$492$	0	0
$493$	7052.13	0.644244
$494$	−345.673	−0.0314829
$495$	0	0
$496$	−15914.7	−1.44070
$497$	13711.4	1.23751
$498$	0	0
$499$	9873.65	0.885782	0.442891	−	0.896576i	$-0.353953\pi$
$499$	9873.65	0.885782	0.442891	+	0.896576i	$0.353953\pi$
$500$	−7239.34	−0.647506
$501$	0	0
$502$	−139.260	−0.0123815
$503$	18092.2	1.60376	0.801881	−	0.597484i	$-0.203833\pi$
$503$	18092.2	1.60376	0.801881	+	0.597484i	$0.203833\pi$
$504$	0	0
$505$	−1626.79	−0.143349
$506$	−472.795	−0.0415381
$507$	0	0
$508$	11136.7	0.972660
$509$	−17328.6	−1.50899	−0.754495	−	0.656306i	$-0.772118\pi$
$509$	−17328.6	−1.50899	−0.754495	+	0.656306i	$0.772118\pi$
$510$	0	0
$511$	−2623.14	−0.227086
$512$	−2518.99	−0.217431
$513$	0	0
$514$	−206.423	−0.0177139
$515$	3854.36	0.329793
$516$	0	0
$517$	−16338.6	−1.38989
$518$	−225.646	−0.0191396
$519$	0	0
$520$	−238.543	−0.0201169
$521$	−11507.5	−0.967664	−0.483832	−	0.875161i	$-0.660755\pi$
$521$	−11507.5	−0.967664	−0.483832	+	0.875161i	$0.660755\pi$
$522$	0	0
$523$	−7887.71	−0.659475	−0.329738	−	0.944073i	$-0.606960\pi$
$523$	−7887.71	−0.659475	−0.329738	+	0.944073i	$0.606960\pi$
$524$	7678.81	0.640172
$525$	0	0
$526$	−375.460	−0.0311233
$527$	28355.3	2.34378
$528$	0	0
$529$	4574.79	0.376000
$530$	−335.638	−0.0275079
$531$	0	0
$532$	13124.9	1.06961
$533$	5294.69	0.430279
$534$	0	0
$535$	−3897.12	−0.314929
$536$	−185.794	−0.0149722
$537$	0	0
$538$	25.3951	0.00203506
$539$	126.922	0.0101427
$540$	0	0
$541$	−848.660	−0.0674432	−0.0337216	−	0.999431i	$-0.510736\pi$
$541$	−848.660	−0.0674432	−0.0337216	+	0.999431i	$0.510736\pi$
$542$	584.193	0.0462975
$543$	0	0
$544$	2691.13	0.212098
$545$	1849.80	0.145388
$546$	0	0
$547$	7067.03	0.552403	0.276202	−	0.961100i	$-0.410924\pi$
$547$	7067.03	0.552403	0.276202	+	0.961100i	$0.410924\pi$
$548$	8072.78	0.629292
$549$	0	0
$550$	−402.424	−0.0311989
$551$	−5555.80	−0.429556
$552$	0	0
$553$	12186.5	0.937114
$554$	−473.818	−0.0363368
$555$	0	0
$556$	−24773.9	−1.88965
$557$	−5405.30	−0.411185	−0.205592	−	0.978638i	$-0.565912\pi$
$557$	−5405.30	−0.411185	−0.205592	+	0.978638i	$0.565912\pi$
$558$	0	0
$559$	−16383.7	−1.23964
$560$	4515.53	0.340743
$561$	0	0
$562$	−923.520	−0.0693173
$563$	−16064.5	−1.20255	−0.601277	−	0.799040i	$-0.705341\pi$
$563$	−16064.5	−1.20255	−0.601277	+	0.799040i	$0.705341\pi$
$564$	0	0
$565$	−5213.26	−0.388183
$566$	678.095	0.0503577
$567$	0	0
$568$	−1476.11	−0.109043
$569$	8075.71	0.594993	0.297497	−	0.954723i	$-0.403848\pi$
$569$	8075.71	0.594993	0.297497	+	0.954723i	$0.403848\pi$
$570$	0	0
$571$	17124.1	1.25503	0.627516	−	0.778604i	$-0.284072\pi$
$571$	17124.1	1.25503	0.627516	+	0.778604i	$0.284072\pi$
$572$	7349.90	0.537264
$573$	0	0
$574$	386.804	0.0281270
$575$	14249.9	1.03350
$576$	0	0
$577$	−25536.6	−1.84246	−0.921232	−	0.389014i	$-0.872816\pi$
$577$	−25536.6	−1.84246	−0.921232	+	0.389014i	$0.872816\pi$
$578$	−984.191	−0.0708252
$579$	0	0
$580$	−1915.14	−0.137106
$581$	−4004.74	−0.285963
$582$	0	0
$583$	20703.0	1.47072
$584$	282.396	0.0200096
$585$	0	0
$586$	−701.979	−0.0494854
$587$	9279.24	0.652462	0.326231	−	0.945290i	$-0.394221\pi$
$587$	9279.24	0.652462	0.326231	+	0.945290i	$0.394221\pi$
$588$	0	0
$589$	−22338.8	−1.56274
$590$	−300.664	−0.0209799
$591$	0	0
$592$	−6294.44	−0.436993
$593$	−6917.49	−0.479034	−0.239517	−	0.970892i	$-0.576989\pi$
$593$	−6917.49	−0.479034	−0.239517	+	0.970892i	$0.576989\pi$
$594$	0	0
$595$	−8045.36	−0.554332
$596$	−2168.94	−0.149066
$597$	0	0
$598$	500.762	0.0342436
$599$	−9024.07	−0.615548	−0.307774	−	0.951459i	$-0.599584\pi$
$599$	−9024.07	−0.615548	−0.307774	+	0.951459i	$0.599584\pi$
$600$	0	0
$601$	−20203.4	−1.37124	−0.685619	−	0.727960i	$-0.740468\pi$
$601$	−20203.4	−1.37124	−0.685619	+	0.727960i	$0.740468\pi$
$602$	−1196.91	−0.0810339
$603$	0	0
$604$	−2847.36	−0.191817
$605$	1781.09	0.119688
$606$	0	0
$607$	−3979.55	−0.266104	−0.133052	−	0.991109i	$-0.542478\pi$
$607$	−3979.55	−0.266104	−0.133052	+	0.991109i	$0.542478\pi$
$608$	−2120.12	−0.141418
$609$	0	0
$610$	−126.348	−0.00838637
$611$	17305.1	1.14581
$612$	0	0
$613$	26185.7	1.72533	0.862666	−	0.505774i	$-0.168793\pi$
$613$	26185.7	1.72533	0.862666	+	0.505774i	$0.168793\pi$
$614$	839.741	0.0551941
$615$	0	0
$616$	1074.93	0.0703085
$617$	−487.468	−0.0318067	−0.0159034	−	0.999874i	$-0.505062\pi$
$617$	−487.468	−0.0318067	−0.0159034	+	0.999874i	$0.505062\pi$
$618$	0	0
$619$	−10192.4	−0.661819	−0.330910	−	0.943662i	$-0.607356\pi$
$619$	−10192.4	−0.661819	−0.330910	+	0.943662i	$0.607356\pi$
$620$	−7700.39	−0.498799
$621$	0	0
$622$	−183.299	−0.0118161
$623$	18325.6	1.17849
$624$	0	0
$625$	10270.4	0.657306
$626$	535.165	0.0341685
$627$	0	0
$628$	14925.6	0.948403
$629$	11214.8	0.710914
$630$	0	0
$631$	−13447.9	−0.848420	−0.424210	−	0.905564i	$-0.639448\pi$
$631$	−13447.9	−0.848420	−0.424210	+	0.905564i	$0.639448\pi$
$632$	−1311.95	−0.0825736
$633$	0	0
$634$	587.273	0.0367880
$635$	5378.17	0.336104
$636$	0	0
$637$	−134.430	−0.00836154
$638$	−227.292	−0.0141043
$639$	0	0
$640$	−974.121	−0.0601649
$641$	15922.1	0.981101	0.490551	−	0.871413i	$-0.336796\pi$
$641$	15922.1	0.981101	0.490551	+	0.871413i	$0.336796\pi$
$642$	0	0
$643$	−2737.00	−0.167865	−0.0839323	−	0.996471i	$-0.526748\pi$
$643$	−2737.00	−0.167865	−0.0839323	+	0.996471i	$0.526748\pi$
$644$	−19013.4	−1.16341
$645$	0	0
$646$	1255.11	0.0764422
$647$	−2405.17	−0.146147	−0.0730734	−	0.997327i	$-0.523281\pi$
$647$	−2405.17	−0.146147	−0.0730734	+	0.997327i	$0.523281\pi$
$648$	0	0
$649$	18545.7	1.12170
$650$	426.229	0.0257201
$651$	0	0
$652$	−13904.6	−0.835192
$653$	−12292.3	−0.736652	−0.368326	−	0.929697i	$-0.620069\pi$
$653$	−12292.3	−0.736652	−0.368326	+	0.929697i	$0.620069\pi$
$654$	0	0
$655$	3708.27	0.221212
$656$	10790.0	0.642192
$657$	0	0
$658$	1264.23	0.0749007
$659$	−14323.9	−0.846708	−0.423354	−	0.905964i	$-0.639147\pi$
$659$	−14323.9	−0.846708	−0.423354	+	0.905964i	$0.639147\pi$
$660$	0	0
$661$	7334.75	0.431602	0.215801	−	0.976437i	$-0.430764\pi$
$661$	7334.75	0.431602	0.215801	+	0.976437i	$0.430764\pi$
$662$	−761.949	−0.0447341
$663$	0	0
$664$	431.133	0.0251976
$665$	6338.29	0.369606
$666$	0	0
$667$	8048.46	0.467223
$668$	18816.2	1.08985
$669$	0	0
$670$	−44.8190	−0.00258434
$671$	7793.49	0.448382
$672$	0	0
$673$	30070.5	1.72234	0.861168	−	0.508321i	$-0.169734\pi$
$673$	30070.5	1.72234	0.861168	+	0.508321i	$0.169734\pi$
$674$	506.007	0.0289179
$675$	0	0
$676$	9757.57	0.555165
$677$	16180.2	0.918548	0.459274	−	0.888295i	$-0.348110\pi$
$677$	16180.2	0.918548	0.459274	+	0.888295i	$0.348110\pi$
$678$	0	0
$679$	−27888.2	−1.57622
$680$	866.128	0.0488448
$681$	0	0
$682$	−913.896	−0.0513121
$683$	−12222.7	−0.684754	−0.342377	−	0.939563i	$-0.611232\pi$
$683$	−12222.7	−0.684754	−0.342377	+	0.939563i	$0.611232\pi$
$684$	0	0
$685$	3898.53	0.217453
$686$	−792.234	−0.0440928
$687$	0	0
$688$	−33388.1	−1.85016
$689$	−21927.7	−1.21245
$690$	0	0
$691$	33041.6	1.81905	0.909524	−	0.415651i	$-0.136446\pi$
$691$	33041.6	1.81905	0.909524	+	0.415651i	$0.136446\pi$
$692$	−16094.6	−0.884143
$693$	0	0
$694$	539.057	0.0294846
$695$	−11963.9	−0.652972
$696$	0	0
$697$	−19224.6	−1.04474
$698$	−680.413	−0.0368969
$699$	0	0
$700$	−16183.5	−0.873824
$701$	−15129.2	−0.815151	−0.407576	−	0.913171i	$-0.633626\pi$
$701$	−15129.2	−0.815151	−0.407576	+	0.913171i	$0.633626\pi$
$702$	0	0
$703$	−8835.27	−0.474009
$704$	14920.3	0.798766
$705$	0	0
$706$	346.374	0.0184645
$707$	−7764.33	−0.413024
$708$	0	0
$709$	30432.8	1.61203	0.806015	−	0.591895i	$-0.201620\pi$
$709$	30432.8	1.61203	0.806015	+	0.591895i	$0.201620\pi$
$710$	−356.081	−0.0188218
$711$	0	0
$712$	−1972.85	−0.103842
$713$	32361.3	1.69978
$714$	0	0
$715$	3549.43	0.185652
$716$	−26931.0	−1.40567
$717$	0	0
$718$	−1201.65	−0.0624582
$719$	−8363.30	−0.433795	−0.216897	−	0.976194i	$-0.569594\pi$
$719$	−8363.30	−0.433795	−0.216897	+	0.976194i	$0.569594\pi$
$720$	0	0
$721$	18396.0	0.950212
$722$	−138.643	−0.00714647
$723$	0	0
$724$	11511.0	0.590886
$725$	6850.52	0.350927
$726$	0	0
$727$	−15349.9	−0.783074	−0.391537	−	0.920162i	$-0.628056\pi$
$727$	−15349.9	−0.783074	−0.391537	+	0.920162i	$0.628056\pi$
$728$	−1138.51	−0.0579616
$729$	0	0
$730$	68.1221	0.00345385
$731$	59487.8	3.00990
$732$	0	0
$733$	11618.6	0.585459	0.292730	−	0.956195i	$-0.405436\pi$
$733$	11618.6	0.585459	0.292730	+	0.956195i	$0.405436\pi$
$734$	−452.781	−0.0227690
$735$	0	0
$736$	3071.34	0.153819
$737$	2764.55	0.138173
$738$	0	0
$739$	−35664.5	−1.77529	−0.887646	−	0.460527i	$-0.847660\pi$
$739$	−35664.5	−1.77529	−0.887646	+	0.460527i	$0.847660\pi$
$740$	−3045.60	−0.151295
$741$	0	0
$742$	−1601.93	−0.0792568
$743$	−11132.8	−0.549695	−0.274848	−	0.961488i	$-0.588627\pi$
$743$	−11132.8	−0.549695	−0.274848	+	0.961488i	$0.588627\pi$
$744$	0	0
$745$	−1047.43	−0.0515098
$746$	635.724	0.0312004
$747$	0	0
$748$	−26686.9	−1.30450
$749$	−18600.1	−0.907386
$750$	0	0
$751$	−33616.2	−1.63339	−0.816694	−	0.577071i	$-0.804195\pi$
$751$	−33616.2	−1.63339	−0.816694	+	0.577071i	$0.804195\pi$
$752$	35265.9	1.71013
$753$	0	0
$754$	240.737	0.0116275
$755$	−1375.05	−0.0662826
$756$	0	0
$757$	40188.5	1.92956	0.964779	−	0.263061i	$-0.0847320\pi$
$757$	40188.5	1.92956	0.964779	+	0.263061i	$0.0847320\pi$
$758$	375.281	0.0179826
$759$	0	0
$760$	−682.352	−0.0325678
$761$	−16205.6	−0.771949	−0.385975	−	0.922509i	$-0.626135\pi$
$761$	−16205.6	−0.771949	−0.385975	+	0.922509i	$0.626135\pi$
$762$	0	0
$763$	8828.68	0.418899
$764$	−26917.9	−1.27468
$765$	0	0
$766$	333.243	0.0157187
$767$	−19642.8	−0.924720
$768$	0	0
$769$	−12697.9	−0.595445	−0.297722	−	0.954652i	$-0.596227\pi$
$769$	−12697.9	−0.595445	−0.297722	+	0.954652i	$0.596227\pi$
$770$	259.304	0.0121359
$771$	0	0
$772$	−20915.7	−0.975095
$773$	−12373.2	−0.575722	−0.287861	−	0.957672i	$-0.592944\pi$
$773$	−12373.2	−0.575722	−0.287861	+	0.957672i	$0.592944\pi$
$774$	0	0
$775$	27544.6	1.27669
$776$	3002.32	0.138888
$777$	0	0
$778$	1182.33	0.0544841
$779$	15145.5	0.696590
$780$	0	0
$781$	21964.0	1.00632
$782$	−1818.23	−0.0831453
$783$	0	0
$784$	−273.952	−0.0124796
$785$	7207.91	0.327722
$786$	0	0
$787$	−34183.8	−1.54831	−0.774155	−	0.632996i	$-0.781825\pi$
$787$	−34183.8	−1.54831	−0.774155	+	0.632996i	$0.781825\pi$
$788$	15303.8	0.691848
$789$	0	0
$790$	−316.480	−0.0142530
$791$	−24881.7	−1.11845
$792$	0	0
$793$	−8254.50	−0.369642
$794$	−1093.79	−0.0488879
$795$	0	0
$796$	13827.1	0.615691
$797$	−34195.2	−1.51977	−0.759884	−	0.650058i	$-0.774745\pi$
$797$	−34195.2	−1.51977	−0.759884	+	0.650058i	$0.774745\pi$
$798$	0	0
$799$	−62833.5	−2.78209
$800$	2614.20	0.115532
$801$	0	0
$802$	−396.191	−0.0174439
$803$	−4201.95	−0.184662
$804$	0	0
$805$	−9182.01	−0.402017
$806$	967.956	0.0423012
$807$	0	0
$808$	835.874	0.0363935
$809$	19873.6	0.863684	0.431842	−	0.901949i	$-0.357864\pi$
$809$	19873.6	0.863684	0.431842	+	0.901949i	$0.357864\pi$
$810$	0	0
$811$	3153.23	0.136529	0.0682645	−	0.997667i	$-0.478254\pi$
$811$	3153.23	0.136529	0.0682645	+	0.997667i	$0.478254\pi$
$812$	−9140.53	−0.395036
$813$	0	0
$814$	−361.457	−0.0155639
$815$	−6714.83	−0.288602
$816$	0	0
$817$	−46865.6	−2.00688
$818$	1187.78	0.0507697
$819$	0	0
$820$	5220.79	0.222339
$821$	−19481.1	−0.828132	−0.414066	−	0.910247i	$-0.635892\pi$
$821$	−19481.1	−0.828132	−0.414066	+	0.910247i	$0.635892\pi$
$822$	0	0
$823$	9851.17	0.417242	0.208621	−	0.977997i	$-0.433102\pi$
$823$	9851.17	0.417242	0.208621	+	0.977997i	$0.433102\pi$
$824$	−1980.43	−0.0837277
$825$	0	0
$826$	−1435.00	−0.0604481
$827$	−23836.1	−1.00225	−0.501126	−	0.865375i	$-0.667081\pi$
$827$	−23836.1	−1.00225	−0.501126	+	0.865375i	$0.667081\pi$
$828$	0	0
$829$	24098.4	1.00962	0.504808	−	0.863232i	$-0.331563\pi$
$829$	24098.4	1.00962	0.504808	+	0.863232i	$0.331563\pi$
$830$	104.002	0.00434934
$831$	0	0
$832$	−15802.9	−0.658495
$833$	488.103	0.0203022
$834$	0	0
$835$	9086.77	0.376600
$836$	21024.4	0.869790
$837$	0	0
$838$	470.764	0.0194061
$839$	−796.151	−0.0327606	−0.0163803	−	0.999866i	$-0.505214\pi$
$839$	−796.151	−0.0327606	−0.0163803	+	0.999866i	$0.505214\pi$
$840$	0	0
$841$	−20519.8	−0.841354
$842$	−1614.26	−0.0660703
$843$	0	0
$844$	−25860.0	−1.05466
$845$	4712.15	0.191838
$846$	0	0
$847$	8500.75	0.344851
$848$	−44686.1	−1.80958
$849$	0	0
$850$	−1547.60	−0.0624497
$851$	12799.3	0.515574
$852$	0	0
$853$	−24114.1	−0.967938	−0.483969	−	0.875085i	$-0.660805\pi$
$853$	−24114.1	−0.967938	−0.483969	+	0.875085i	$0.660805\pi$
$854$	−603.032	−0.0241632
$855$	0	0
$856$	2002.40	0.0799541
$857$	26574.9	1.05925	0.529627	−	0.848231i	$-0.322332\pi$
$857$	26574.9	1.05925	0.529627	+	0.848231i	$0.322332\pi$
$858$	0	0
$859$	21880.3	0.869089	0.434544	−	0.900650i	$-0.356909\pi$
$859$	21880.3	0.869089	0.434544	+	0.900650i	$0.356909\pi$
$860$	−16155.0	−0.640559
$861$	0	0
$862$	908.796	0.0359092
$863$	−240.927	−0.00950318	−0.00475159	−	0.999989i	$-0.501512\pi$
$863$	−240.927	−0.00950318	−0.00475159	+	0.999989i	$0.501512\pi$
$864$	0	0
$865$	−7772.47	−0.305517
$866$	−2199.97	−0.0863256
$867$	0	0
$868$	−36752.3	−1.43716
$869$	19521.3	0.762044
$870$	0	0
$871$	−2928.09	−0.113909
$872$	−950.457	−0.0369112
$873$	0	0
$874$	1432.43	0.0554379
$875$	−16685.8	−0.644667
$876$	0	0
$877$	25933.7	0.998538	0.499269	−	0.866447i	$-0.333602\pi$
$877$	25933.7	0.998538	0.499269	+	0.866447i	$0.333602\pi$
$878$	38.1928	0.00146805
$879$	0	0
$880$	7233.33	0.277086
$881$	−49883.2	−1.90761	−0.953806	−	0.300423i	$-0.902872\pi$
$881$	−49883.2	−1.90761	−0.953806	+	0.300423i	$0.902872\pi$
$882$	0	0
$883$	−29662.7	−1.13050	−0.565249	−	0.824920i	$-0.691220\pi$
$883$	−29662.7	−1.13050	−0.565249	+	0.824920i	$0.691220\pi$
$884$	28265.5	1.07542
$885$	0	0
$886$	840.908	0.0318858
$887$	16649.3	0.630247	0.315124	−	0.949051i	$-0.397954\pi$
$887$	16649.3	0.630247	0.315124	+	0.949051i	$0.397954\pi$
$888$	0	0
$889$	25668.8	0.968397
$890$	−475.910	−0.0179242
$891$	0	0
$892$	22873.0	0.858572
$893$	49501.4	1.85498
$894$	0	0
$895$	−13005.6	−0.485730
$896$	−4649.27	−0.173350
$897$	0	0
$898$	1301.70	0.0483724
$899$	15557.4	0.577161
$900$	0	0
$901$	79617.5	2.94389
$902$	619.612	0.0228723
$903$	0	0
$904$	2678.66	0.0985518
$905$	5558.90	0.204181
$906$	0	0
$907$	10159.2	0.371921	0.185960	−	0.982557i	$-0.440460\pi$
$907$	10159.2	0.371921	0.185960	+	0.982557i	$0.440460\pi$
$908$	33450.2	1.22256
$909$	0	0
$910$	−274.642	−0.0100047
$911$	15435.2	0.561352	0.280676	−	0.959803i	$-0.409441\pi$
$911$	15435.2	0.561352	0.280676	+	0.959803i	$0.409441\pi$
$912$	0	0
$913$	−6415.10	−0.232540
$914$	−832.277	−0.0301196
$915$	0	0
$916$	−457.208	−0.0164919
$917$	17698.8	0.637366
$918$	0	0
$919$	49224.1	1.76687	0.883436	−	0.468553i	$-0.155224\pi$
$919$	49224.1	1.76687	0.883436	+	0.468553i	$0.155224\pi$
$920$	988.495	0.0354236
$921$	0	0
$922$	−1922.43	−0.0686679
$923$	−23263.3	−0.829600
$924$	0	0
$925$	10894.2	0.387243
$926$	1213.69	0.0430716
$927$	0	0
$928$	1476.52	0.0522296
$929$	−24472.3	−0.864272	−0.432136	−	0.901808i	$-0.642240\pi$
$929$	−24472.3	−0.864272	−0.432136	+	0.901808i	$0.642240\pi$
$930$	0	0
$931$	−384.536	−0.0135367
$932$	797.491	0.0280286
$933$	0	0
$934$	113.645	0.00398134
$935$	−12887.7	−0.450772
$936$	0	0
$937$	51903.3	1.80961	0.904807	−	0.425822i	$-0.140015\pi$
$937$	51903.3	1.80961	0.904807	+	0.425822i	$0.140015\pi$
$938$	−213.911	−0.00744611
$939$	0	0
$940$	17063.6	0.592077
$941$	24412.6	0.845725	0.422862	−	0.906194i	$-0.361025\pi$
$941$	24412.6	0.845725	0.422862	+	0.906194i	$0.361025\pi$
$942$	0	0
$943$	−21940.6	−0.757673
$944$	−40029.8	−1.38015
$945$	0	0
$946$	−1917.30	−0.0658953
$947$	−55016.2	−1.88784	−0.943921	−	0.330171i	$-0.892894\pi$
$947$	−55016.2	−1.88784	−0.943921	+	0.330171i	$0.892894\pi$
$948$	0	0
$949$	4450.51	0.152234
$950$	1219.23	0.0416389
$951$	0	0
$952$	4133.84	0.140734
$953$	35943.3	1.22174	0.610869	−	0.791732i	$-0.290820\pi$
$953$	35943.3	1.22174	0.610869	+	0.791732i	$0.290820\pi$
$954$	0	0
$955$	−12999.2	−0.440467
$956$	−1908.33	−0.0645604
$957$	0	0
$958$	−26.9205	−0.000907892 0
$959$	18606.8	0.626533
$960$	0	0
$961$	32762.3	1.09974
$962$	382.838	0.0128308
$963$	0	0
$964$	30819.5	1.02970
$965$	−10100.7	−0.336945
$966$	0	0
$967$	12954.4	0.430803	0.215401	−	0.976526i	$-0.430894\pi$
$967$	12954.4	0.430803	0.215401	+	0.976526i	$0.430894\pi$
$968$	−915.153	−0.0303865
$969$	0	0
$970$	724.247	0.0239734
$971$	−23718.3	−0.783890	−0.391945	−	0.919989i	$-0.628198\pi$
$971$	−23718.3	−0.783890	−0.391945	+	0.919989i	$0.628198\pi$
$972$	0	0
$973$	−57101.0	−1.88137
$974$	2202.43	0.0724542
$975$	0	0
$976$	−16821.7	−0.551691
$977$	−49573.5	−1.62333	−0.811667	−	0.584121i	$-0.801439\pi$
$977$	−49573.5	−1.62333	−0.811667	+	0.584121i	$0.801439\pi$
$978$	0	0
$979$	29355.4	0.958327
$980$	−132.553	−0.00432067
$981$	0	0
$982$	−1595.61	−0.0518512
$983$	42146.9	1.36752	0.683762	−	0.729705i	$-0.260343\pi$
$983$	42146.9	1.36752	0.683762	+	0.729705i	$0.260343\pi$
$984$	0	0
$985$	7390.56	0.239069
$986$	−874.095	−0.0282321
$987$	0	0
$988$	−22268.1	−0.717047
$989$	67892.2	2.18286
$990$	0	0
$991$	−35061.5	−1.12388	−0.561940	−	0.827178i	$-0.689945\pi$
$991$	−35061.5	−1.12388	−0.561940	+	0.827178i	$0.689945\pi$
$992$	5936.79	0.190013
$993$	0	0
$994$	−1699.50	−0.0542302
$995$	6677.44	0.212753
$996$	0	0
$997$	−19977.3	−0.634591	−0.317295	−	0.948327i	$-0.602775\pi$
$997$	−19977.3	−0.634591	−0.317295	+	0.948327i	$0.602775\pi$
$998$	−1223.81	−0.0388168
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2151.4.a.f.1.20		37
3.2	odd	2		239.4.a.b.1.18	✓	37

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
239.4.a.b.1.18	✓	37	3.2	odd	2
2151.4.a.f.1.20		37	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)

\(f(q)\)	\(=\)	\(q-0.123948 q^{2} -7.98464 q^{4} -3.85596 q^{5} -18.4037 q^{7} +1.98126 q^{8} +O(q^{10})\)	\(q-0.123948 q^{2} -7.98464 q^{4} -3.85596 q^{5} -18.4037 q^{7} +1.98126 q^{8} +0.477937 q^{10} -29.4804 q^{11} +31.2243 q^{13} +2.28109 q^{14} +63.6315 q^{16} -113.373 q^{17} +89.3172 q^{19} +30.7884 q^{20} +3.65403 q^{22} -129.390 q^{23} -110.132 q^{25} -3.87018 q^{26} +146.947 q^{28} -62.2031 q^{29} -250.107 q^{31} -23.7370 q^{32} +14.0523 q^{34} +70.9638 q^{35} -98.9201 q^{37} -11.0707 q^{38} -7.63965 q^{40} +169.570 q^{41} -524.710 q^{43} +235.390 q^{44} +16.0376 q^{46} +554.220 q^{47} -4.30529 q^{49} +13.6505 q^{50} -249.315 q^{52} -702.263 q^{53} +113.675 q^{55} -36.4624 q^{56} +7.70992 q^{58} -629.087 q^{59} -264.362 q^{61} +31.0001 q^{62} -506.110 q^{64} -120.400 q^{65} -93.7760 q^{67} +905.240 q^{68} -8.79579 q^{70} -745.038 q^{71} +142.534 q^{73} +12.2609 q^{74} -713.165 q^{76} +542.548 q^{77} -662.180 q^{79} -245.361 q^{80} -21.0178 q^{82} +217.606 q^{83} +437.161 q^{85} +65.0365 q^{86} -58.4083 q^{88} -995.759 q^{89} -574.641 q^{91} +1033.13 q^{92} -68.6943 q^{94} -344.404 q^{95} +1515.36 q^{97} +0.533630 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 37 q - 4 q^{2} + 170 q^{4} - 43 q^{5} + 60 q^{7} - 27 q^{8}+O(q^{10}) \) 37 * q - 4 * q^2 + 170 * q^4 - 43 * q^5 + 60 * q^7 - 27 * q^8	\( 37 q - 4 q^{2} + 170 q^{4} - 43 q^{5} + 60 q^{7} - 27 q^{8} + 147 q^{10} - 55 q^{11} + 250 q^{13} - 169 q^{14} + 918 q^{16} - 189 q^{17} + 550 q^{19} - 486 q^{20} + 226 q^{22} - 74 q^{23} + 1604 q^{25} - 560 q^{26} + 829 q^{28} - 389 q^{29} + 1107 q^{31} - 125 q^{32} + 1423 q^{34} - 270 q^{35} + 1002 q^{37} - 1037 q^{38} + 1536 q^{40} - 1518 q^{41} + 1098 q^{43} - 1037 q^{44} + 1030 q^{46} - 1214 q^{47} + 4663 q^{49} - 929 q^{50} + 2895 q^{52} - 904 q^{53} + 1350 q^{55} - 2556 q^{56} + 1396 q^{58} - 1658 q^{59} + 2313 q^{61} + 4519 q^{62} + 3807 q^{64} + 56 q^{65} + 1535 q^{67} + 6526 q^{68} - 4099 q^{70} + 3255 q^{71} + 3154 q^{73} + 2629 q^{74} + 1981 q^{76} + 3734 q^{77} + 2260 q^{79} + 8242 q^{80} - 9898 q^{82} + 939 q^{83} + 1272 q^{85} + 3457 q^{86} - 1808 q^{88} - 1486 q^{89} + 174 q^{91} + 14076 q^{92} - 984 q^{94} + 1828 q^{95} + 6148 q^{97} + 6243 q^{98}+O(q^{100}) \) 37 * q - 4 * q^2 + 170 * q^4 - 43 * q^5 + 60 * q^7 - 27 * q^8 + 147 * q^10 - 55 * q^11 + 250 * q^13 - 169 * q^14 + 918 * q^16 - 189 * q^17 + 550 * q^19 - 486 * q^20 + 226 * q^22 - 74 * q^23 + 1604 * q^25 - 560 * q^26 + 829 * q^28 - 389 * q^29 + 1107 * q^31 - 125 * q^32 + 1423 * q^34 - 270 * q^35 + 1002 * q^37 - 1037 * q^38 + 1536 * q^40 - 1518 * q^41 + 1098 * q^43 - 1037 * q^44 + 1030 * q^46 - 1214 * q^47 + 4663 * q^49 - 929 * q^50 + 2895 * q^52 - 904 * q^53 + 1350 * q^55 - 2556 * q^56 + 1396 * q^58 - 1658 * q^59 + 2313 * q^61 + 4519 * q^62 + 3807 * q^64 + 56 * q^65 + 1535 * q^67 + 6526 * q^68 - 4099 * q^70 + 3255 * q^71 + 3154 * q^73 + 2629 * q^74 + 1981 * q^76 + 3734 * q^77 + 2260 * q^79 + 8242 * q^80 - 9898 * q^82 + 939 * q^83 + 1272 * q^85 + 3457 * q^86 - 1808 * q^88 - 1486 * q^89 + 174 * q^91 + 14076 * q^92 - 984 * q^94 + 1828 * q^95 + 6148 * q^97 + 6243 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more