LMFDB - Embedded newform 2151.4.a.g.1.53

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:	$59$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.53
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)

$f(q)$	$=$	$q+4.45221 q^{2} +11.8222 q^{4} -12.0796 q^{5} -15.4984 q^{7} +17.0170 q^{8} +O(q^{10})$	\(q+4.45221 q^{2} +11.8222 q^{4} -12.0796 q^{5} -15.4984 q^{7} +17.0170 q^{8} -53.7808 q^{10} +9.43863 q^{11} +58.4141 q^{13} -69.0020 q^{14} -18.8139 q^{16} +38.5888 q^{17} +155.155 q^{19} -142.807 q^{20} +42.0227 q^{22} +9.96595 q^{23} +20.9160 q^{25} +260.072 q^{26} -183.224 q^{28} -268.647 q^{29} +126.828 q^{31} -219.900 q^{32} +171.805 q^{34} +187.214 q^{35} -392.339 q^{37} +690.781 q^{38} -205.558 q^{40} -336.132 q^{41} -79.3956 q^{43} +111.585 q^{44} +44.3705 q^{46} +43.3712 q^{47} -102.800 q^{49} +93.1225 q^{50} +690.581 q^{52} +23.7667 q^{53} -114.015 q^{55} -263.736 q^{56} -1196.07 q^{58} -17.5671 q^{59} +679.230 q^{61} +564.665 q^{62} -828.527 q^{64} -705.617 q^{65} -890.593 q^{67} +456.203 q^{68} +833.515 q^{70} +237.928 q^{71} +638.534 q^{73} -1746.78 q^{74} +1834.26 q^{76} -146.283 q^{77} -713.642 q^{79} +227.264 q^{80} -1496.53 q^{82} -532.550 q^{83} -466.136 q^{85} -353.486 q^{86} +160.617 q^{88} -1585.59 q^{89} -905.324 q^{91} +117.819 q^{92} +193.098 q^{94} -1874.20 q^{95} +1522.63 q^{97} -457.688 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 59 q - 8 q^{2} + 238 q^{4} - 80 q^{5} - 10 q^{7} - 96 q^{8}+O(q^{10}) $ 59 * q - 8 * q^2 + 238 * q^4 - 80 * q^5 - 10 * q^7 - 96 * q^8	$ 59 q - 8 q^{2} + 238 q^{4} - 80 q^{5} - 10 q^{7} - 96 q^{8} - 36 q^{10} - 132 q^{11} + 104 q^{13} - 280 q^{14} + 822 q^{16} - 408 q^{17} + 20 q^{19} - 800 q^{20} - 2 q^{22} - 276 q^{23} + 1477 q^{25} - 780 q^{26} + 224 q^{28} - 696 q^{29} - 380 q^{31} - 896 q^{32} - 72 q^{34} - 700 q^{35} + 224 q^{37} - 988 q^{38} - 258 q^{40} - 2706 q^{41} - 156 q^{43} - 1584 q^{44} + 428 q^{46} - 1316 q^{47} + 2135 q^{49} - 1400 q^{50} + 1092 q^{52} - 1484 q^{53} - 992 q^{55} - 3360 q^{56} - 120 q^{58} - 3186 q^{59} - 254 q^{61} - 1240 q^{62} + 3054 q^{64} - 5120 q^{65} + 288 q^{67} - 9420 q^{68} + 1108 q^{70} - 4468 q^{71} - 1770 q^{73} - 6214 q^{74} + 720 q^{76} - 6352 q^{77} - 746 q^{79} - 7040 q^{80} + 276 q^{82} - 5484 q^{83} + 588 q^{85} - 10152 q^{86} + 1186 q^{88} - 11570 q^{89} + 1768 q^{91} - 15366 q^{92} - 2142 q^{94} - 5736 q^{95} + 2390 q^{97} - 6912 q^{98}+O(q^{100}) $ 59 * q - 8 * q^2 + 238 * q^4 - 80 * q^5 - 10 * q^7 - 96 * q^8 - 36 * q^10 - 132 * q^11 + 104 * q^13 - 280 * q^14 + 822 * q^16 - 408 * q^17 + 20 * q^19 - 800 * q^20 - 2 * q^22 - 276 * q^23 + 1477 * q^25 - 780 * q^26 + 224 * q^28 - 696 * q^29 - 380 * q^31 - 896 * q^32 - 72 * q^34 - 700 * q^35 + 224 * q^37 - 988 * q^38 - 258 * q^40 - 2706 * q^41 - 156 * q^43 - 1584 * q^44 + 428 * q^46 - 1316 * q^47 + 2135 * q^49 - 1400 * q^50 + 1092 * q^52 - 1484 * q^53 - 992 * q^55 - 3360 * q^56 - 120 * q^58 - 3186 * q^59 - 254 * q^61 - 1240 * q^62 + 3054 * q^64 - 5120 * q^65 + 288 * q^67 - 9420 * q^68 + 1108 * q^70 - 4468 * q^71 - 1770 * q^73 - 6214 * q^74 + 720 * q^76 - 6352 * q^77 - 746 * q^79 - 7040 * q^80 + 276 * q^82 - 5484 * q^83 + 588 * q^85 - 10152 * q^86 + 1186 * q^88 - 11570 * q^89 + 1768 * q^91 - 15366 * q^92 - 2142 * q^94 - 5736 * q^95 + 2390 * q^97 - 6912 * 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 $

$2$	4.45221	1.57409	0.787047	−	0.616894i	$-0.211609\pi$
$2$	4.45221	1.57409	0.787047	+	0.616894i	$0.211609\pi$
$3$	0	0
$3$	0	0
$4$	11.8222	1.47777
$5$	−12.0796	−1.08043	−0.540215	−	0.841527i	$-0.681657\pi$
$5$	−12.0796	−1.08043	−0.540215	+	0.841527i	$0.681657\pi$
$6$	0	0
$7$	−15.4984	−0.836834	−0.418417	−	0.908255i	$-0.637415\pi$
$7$	−15.4984	−0.836834	−0.418417	+	0.908255i	$0.637415\pi$
$8$	17.0170	0.752053
$9$	0	0
$10$	−53.7808	−1.70070
$11$	9.43863	0.258714	0.129357	−	0.991598i	$-0.458709\pi$
$11$	9.43863	0.258714	0.129357	+	0.991598i	$0.458709\pi$
$12$	0	0
$13$	58.4141	1.24624	0.623122	−	0.782125i	$-0.285864\pi$
$13$	58.4141	1.24624	0.623122	+	0.782125i	$0.285864\pi$
$14$	−69.0020	−1.31725
$15$	0	0
$16$	−18.8139	−0.293968
$17$	38.5888	0.550539	0.275269	−	0.961367i	$-0.411233\pi$
$17$	38.5888	0.550539	0.275269	+	0.961367i	$0.411233\pi$
$18$	0	0
$19$	155.155	1.87342	0.936709	−	0.350108i	$-0.113855\pi$
$19$	155.155	1.87342	0.936709	+	0.350108i	$0.113855\pi$
$20$	−142.807	−1.59663
$21$	0	0
$22$	42.0227	0.407240
$23$	9.96595	0.0903497	0.0451749	−	0.998979i	$-0.485616\pi$
$23$	9.96595	0.0903497	0.0451749	+	0.998979i	$0.485616\pi$
$24$	0	0
$25$	20.9160	0.167328
$26$	260.072	1.96170
$27$	0	0
$28$	−183.224	−1.23665
$29$	−268.647	−1.72023	−0.860113	−	0.510104i	$-0.829607\pi$
$29$	−268.647	−1.72023	−0.860113	+	0.510104i	$0.829607\pi$
$30$	0	0
$31$	126.828	0.734807	0.367403	−	0.930062i	$-0.380247\pi$
$31$	126.828	0.734807	0.367403	+	0.930062i	$0.380247\pi$
$32$	−219.900	−1.21479
$33$	0	0
$34$	171.805	0.866599
$35$	187.214	0.904140
$36$	0	0
$37$	−392.339	−1.74325	−0.871624	−	0.490175i	$-0.836933\pi$
$37$	−392.339	−1.74325	−0.871624	+	0.490175i	$0.836933\pi$
$38$	690.781	2.94893
$39$	0	0
$40$	−205.558	−0.812540
$41$	−336.132	−1.28037	−0.640183	−	0.768222i	$-0.721142\pi$
$41$	−336.132	−1.28037	−0.640183	+	0.768222i	$0.721142\pi$
$42$	0	0
$43$	−79.3956	−0.281575	−0.140787	−	0.990040i	$-0.544963\pi$
$43$	−79.3956	−0.281575	−0.140787	+	0.990040i	$0.544963\pi$
$44$	111.585	0.382319
$45$	0	0
$46$	44.3705	0.142219
$47$	43.3712	0.134603	0.0673015	−	0.997733i	$-0.478561\pi$
$47$	43.3712	0.134603	0.0673015	+	0.997733i	$0.478561\pi$
$48$	0	0
$49$	−102.800	−0.299709
$50$	93.1225	0.263390
$51$	0	0
$52$	690.581	1.84166
$53$	23.7667	0.0615963	0.0307982	−	0.999526i	$-0.490195\pi$
$53$	23.7667	0.0615963	0.0307982	+	0.999526i	$0.490195\pi$
$54$	0	0
$55$	−114.015	−0.279522
$56$	−263.736	−0.629343
$57$	0	0
$58$	−1196.07	−2.70779
$59$	−17.5671	−0.0387634	−0.0193817	−	0.999812i	$-0.506170\pi$
$59$	−17.5671	−0.0387634	−0.0193817	+	0.999812i	$0.506170\pi$
$60$	0	0
$61$	679.230	1.42568	0.712840	−	0.701327i	$-0.247409\pi$
$61$	679.230	1.42568	0.712840	+	0.701327i	$0.247409\pi$
$62$	564.665	1.15665
$63$	0	0
$64$	−828.527	−1.61822
$65$	−705.617	−1.34648
$66$	0	0
$67$	−890.593	−1.62393	−0.811964	−	0.583707i	$-0.801602\pi$
$67$	−890.593	−1.62393	−0.811964	+	0.583707i	$0.801602\pi$
$68$	456.203	0.813569
$69$	0	0
$70$	833.515	1.42320
$71$	237.928	0.397703	0.198851	−	0.980030i	$-0.436279\pi$
$71$	237.928	0.397703	0.198851	+	0.980030i	$0.436279\pi$
$72$	0	0
$73$	638.534	1.02376	0.511882	−	0.859056i	$-0.328949\pi$
$73$	638.534	1.02376	0.511882	+	0.859056i	$0.328949\pi$
$74$	−1746.78	−2.74403
$75$	0	0
$76$	1834.26	2.76848
$77$	−146.283	−0.216501
$78$	0	0
$79$	−713.642	−1.01634	−0.508171	−	0.861256i	$-0.669678\pi$
$79$	−713.642	−1.01634	−0.508171	+	0.861256i	$0.669678\pi$
$80$	227.264	0.317611
$81$	0	0
$82$	−1496.53	−2.01542
$83$	−532.550	−0.704277	−0.352139	−	0.935948i	$-0.614545\pi$
$83$	−532.550	−0.704277	−0.352139	+	0.935948i	$0.614545\pi$
$84$	0	0
$85$	−466.136	−0.594818
$86$	−353.486	−0.443225
$87$	0	0
$88$	160.617	0.194567
$89$	−1585.59	−1.88845	−0.944224	−	0.329303i	$-0.893186\pi$
$89$	−1585.59	−1.88845	−0.944224	+	0.329303i	$0.893186\pi$
$90$	0	0
$91$	−905.324	−1.04290
$92$	117.819	0.133516
$93$	0	0
$94$	193.098	0.211878
$95$	−1874.20	−2.02410
$96$	0	0
$97$	1522.63	1.59381	0.796907	−	0.604102i	$-0.206468\pi$
$97$	1522.63	1.59381	0.796907	+	0.604102i	$0.206468\pi$
$98$	−457.688	−0.471770
$99$	0	0
$100$	247.272	0.247272
$101$	−8.88424	−0.00875262	−0.00437631	−	0.999990i	$-0.501393\pi$
$101$	−8.88424	−0.00875262	−0.00437631	+	0.999990i	$0.501393\pi$
$102$	0	0
$103$	−1479.46	−1.41530	−0.707650	−	0.706563i	$-0.750245\pi$
$103$	−1479.46	−1.41530	−0.707650	+	0.706563i	$0.750245\pi$
$104$	994.034	0.937241
$105$	0	0
$106$	105.814	0.0969584
$107$	−1015.63	−0.917612	−0.458806	−	0.888536i	$-0.651723\pi$
$107$	−1015.63	−0.917612	−0.458806	+	0.888536i	$0.651723\pi$
$108$	0	0
$109$	390.042	0.342745	0.171373	−	0.985206i	$-0.445180\pi$
$109$	390.042	0.342745	0.171373	+	0.985206i	$0.445180\pi$
$110$	−507.616	−0.439994
$111$	0	0
$112$	291.585	0.246002
$113$	−1722.71	−1.43415	−0.717075	−	0.696996i	$-0.754520\pi$
$113$	−1722.71	−1.43415	−0.717075	+	0.696996i	$0.754520\pi$
$114$	0	0
$115$	−120.384	−0.0976165
$116$	−3175.99	−2.54210
$117$	0	0
$118$	−78.2123	−0.0610171
$119$	−598.064	−0.460710
$120$	0	0
$121$	−1241.91	−0.933067
$122$	3024.07	2.24415
$123$	0	0
$124$	1499.38	1.08587
$125$	1257.29	0.899643
$126$	0	0
$127$	−2181.69	−1.52436	−0.762181	−	0.647364i	$-0.775871\pi$
$127$	−2181.69	−1.52436	−0.762181	+	0.647364i	$0.775871\pi$
$128$	−1929.58	−1.33244
$129$	0	0
$130$	−3141.56	−2.11948
$131$	−1785.44	−1.19080	−0.595400	−	0.803430i	$-0.703006\pi$
$131$	−1785.44	−1.19080	−0.595400	+	0.803430i	$0.703006\pi$
$132$	0	0
$133$	−2404.65	−1.56774
$134$	−3965.10	−2.55621
$135$	0	0
$136$	656.666	0.414034
$137$	133.999	0.0835641	0.0417821	−	0.999127i	$-0.486696\pi$
$137$	133.999	0.0835641	0.0417821	+	0.999127i	$0.486696\pi$
$138$	0	0
$139$	2290.40	1.39762	0.698810	−	0.715307i	$-0.253713\pi$
$139$	2290.40	1.39762	0.698810	+	0.715307i	$0.253713\pi$
$140$	2213.27	1.33611
$141$	0	0
$142$	1059.31	0.626021
$143$	551.349	0.322420
$144$	0	0
$145$	3245.14	1.85858
$146$	2842.89	1.61150
$147$	0	0
$148$	−4638.29	−2.57612
$149$	2306.25	1.26802	0.634010	−	0.773325i	$-0.281408\pi$
$149$	2306.25	1.26802	0.634010	+	0.773325i	$0.281408\pi$
$150$	0	0
$151$	−551.248	−0.297086	−0.148543	−	0.988906i	$-0.547458\pi$
$151$	−551.248	−0.297086	−0.148543	+	0.988906i	$0.547458\pi$
$152$	2640.27	1.40891
$153$	0	0
$154$	−651.284	−0.340792
$155$	−1532.03	−0.793907
$156$	0	0
$157$	742.891	0.377638	0.188819	−	0.982012i	$-0.439534\pi$
$157$	742.891	0.377638	0.188819	+	0.982012i	$0.439534\pi$
$158$	−3177.28	−1.59982
$159$	0	0
$160$	2656.29	1.31249
$161$	−154.456	−0.0756077
$162$	0	0
$163$	−1676.37	−0.805541	−0.402771	−	0.915301i	$-0.631953\pi$
$163$	−1676.37	−0.805541	−0.402771	+	0.915301i	$0.631953\pi$
$164$	−3973.81	−1.89209
$165$	0	0
$166$	−2371.02	−1.10860
$167$	1282.24	0.594150	0.297075	−	0.954854i	$-0.403989\pi$
$167$	1282.24	0.594150	0.297075	+	0.954854i	$0.403989\pi$
$168$	0	0
$169$	1215.21	0.553122
$170$	−2075.34	−0.936300
$171$	0	0
$172$	−938.627	−0.416102
$173$	399.531	0.175583	0.0877913	−	0.996139i	$-0.472019\pi$
$173$	399.531	0.175583	0.0877913	+	0.996139i	$0.472019\pi$
$174$	0	0
$175$	−324.164	−0.140026
$176$	−177.578	−0.0760535
$177$	0	0
$178$	−7059.36	−2.97259
$179$	−3206.34	−1.33884	−0.669422	−	0.742882i	$-0.733458\pi$
$179$	−3206.34	−1.33884	−0.669422	+	0.742882i	$0.733458\pi$
$180$	0	0
$181$	−3402.47	−1.39726	−0.698628	−	0.715485i	$-0.746206\pi$
$181$	−3402.47	−1.39726	−0.698628	+	0.715485i	$0.746206\pi$
$182$	−4030.69	−1.64162
$183$	0	0
$184$	169.591	0.0679478
$185$	4739.29	1.88346
$186$	0	0
$187$	364.225	0.142432
$188$	512.741	0.198912
$189$	0	0
$190$	−8344.34	−3.18612
$191$	1298.73	0.492003	0.246001	−	0.969269i	$-0.420883\pi$
$191$	1298.73	0.492003	0.246001	+	0.969269i	$0.420883\pi$
$192$	0	0
$193$	2380.31	0.887762	0.443881	−	0.896086i	$-0.353601\pi$
$193$	2380.31	0.887762	0.443881	+	0.896086i	$0.353601\pi$
$194$	6779.08	2.50881
$195$	0	0
$196$	−1215.32	−0.442901
$197$	4261.27	1.54113	0.770566	−	0.637360i	$-0.219974\pi$
$197$	4261.27	1.54113	0.770566	+	0.637360i	$0.219974\pi$
$198$	0	0
$199$	373.752	0.133138	0.0665692	−	0.997782i	$-0.478795\pi$
$199$	373.752	0.133138	0.0665692	+	0.997782i	$0.478795\pi$
$200$	355.928	0.125840
$201$	0	0
$202$	−39.5545	−0.0137774
$203$	4163.60	1.43954
$204$	0	0
$205$	4060.33	1.38335
$206$	−6586.88	−2.22782
$207$	0	0
$208$	−1099.00	−0.366355
$209$	1464.45	0.484679
$210$	0	0
$211$	−1747.25	−0.570074	−0.285037	−	0.958516i	$-0.592006\pi$
$211$	−1747.25	−0.570074	−0.285037	+	0.958516i	$0.592006\pi$
$212$	280.973	0.0910252
$213$	0	0
$214$	−4521.79	−1.44441
$215$	959.065	0.304222
$216$	0	0
$217$	−1965.63	−0.614911
$218$	1736.55	0.539513
$219$	0	0
$220$	−1347.90	−0.413069
$221$	2254.13	0.686105
$222$	0	0
$223$	−2024.31	−0.607882	−0.303941	−	0.952691i	$-0.598303\pi$
$223$	−2024.31	−0.607882	−0.303941	+	0.952691i	$0.598303\pi$
$224$	3408.09	1.01657
$225$	0	0
$226$	−7669.87	−2.25749
$227$	−4270.42	−1.24862	−0.624312	−	0.781175i	$-0.714621\pi$
$227$	−4270.42	−1.24862	−0.624312	+	0.781175i	$0.714621\pi$
$228$	0	0
$229$	−4802.09	−1.38572	−0.692862	−	0.721070i	$-0.743651\pi$
$229$	−4802.09	−1.38572	−0.692862	+	0.721070i	$0.743651\pi$
$230$	−535.976	−0.153657
$231$	0	0
$232$	−4571.57	−1.29370
$233$	3112.20	0.875051	0.437526	−	0.899206i	$-0.355855\pi$
$233$	3112.20	0.875051	0.437526	+	0.899206i	$0.355855\pi$
$234$	0	0
$235$	−523.905	−0.145429
$236$	−207.681	−0.0572833
$237$	0	0
$238$	−2662.71	−0.725200
$239$	239.000	0.0646846
$239$	239.000	0.0646846
$240$	0	0
$241$	4310.85	1.15223	0.576113	−	0.817370i	$-0.304569\pi$
$241$	4310.85	1.15223	0.576113	+	0.817370i	$0.304569\pi$
$242$	−5529.25	−1.46873
$243$	0	0
$244$	8029.96	2.10682
$245$	1241.78	0.323815
$246$	0	0
$247$	9063.23	2.33473
$248$	2158.24	0.552613
$249$	0	0
$250$	5597.71	1.41612
$251$	6020.03	1.51387	0.756935	−	0.653491i	$-0.226696\pi$
$251$	6020.03	1.51387	0.756935	+	0.653491i	$0.226696\pi$
$252$	0	0
$253$	94.0648	0.0233747
$254$	−9713.35	−2.39949
$255$	0	0
$256$	−1962.67	−0.479167
$257$	−1622.09	−0.393710	−0.196855	−	0.980433i	$-0.563073\pi$
$257$	−1622.09	−0.393710	−0.196855	+	0.980433i	$0.563073\pi$
$258$	0	0
$259$	6080.62	1.45881
$260$	−8341.92	−1.98978
$261$	0	0
$262$	−7949.16	−1.87443
$263$	−5730.73	−1.34362	−0.671809	−	0.740724i	$-0.734483\pi$
$263$	−5730.73	−1.34362	−0.671809	+	0.740724i	$0.734483\pi$
$264$	0	0
$265$	−287.091	−0.0665505
$266$	−10706.0	−2.46777
$267$	0	0
$268$	−10528.7	−2.39979
$269$	−6414.10	−1.45381	−0.726904	−	0.686739i	$-0.759041\pi$
$269$	−6414.10	−1.45381	−0.726904	+	0.686739i	$0.759041\pi$
$270$	0	0
$271$	992.636	0.222503	0.111252	−	0.993792i	$-0.464514\pi$
$271$	992.636	0.222503	0.111252	+	0.993792i	$0.464514\pi$
$272$	−726.007	−0.161841
$273$	0	0
$274$	596.590	0.131538
$275$	197.419	0.0432901
$276$	0	0
$277$	6606.52	1.43302	0.716511	−	0.697575i	$-0.245738\pi$
$277$	6606.52	1.43302	0.716511	+	0.697575i	$0.245738\pi$
$278$	10197.3	2.19998
$279$	0	0
$280$	3185.82	0.679961
$281$	6616.73	1.40470	0.702351	−	0.711831i	$-0.252134\pi$
$281$	6616.73	1.40470	0.702351	+	0.711831i	$0.252134\pi$
$282$	0	0
$283$	−4172.85	−0.876502	−0.438251	−	0.898853i	$-0.644402\pi$
$283$	−4172.85	−0.876502	−0.438251	+	0.898853i	$0.644402\pi$
$284$	2812.83	0.587713
$285$	0	0
$286$	2454.72	0.507520
$287$	5209.51	1.07145
$288$	0	0
$289$	−3423.90	−0.696907
$290$	14448.1	2.92558
$291$	0	0
$292$	7548.85	1.51289
$293$	2788.75	0.556043	0.278021	−	0.960575i	$-0.410321\pi$
$293$	2788.75	0.556043	0.278021	+	0.960575i	$0.410321\pi$
$294$	0	0
$295$	212.203	0.0418811
$296$	−6676.44	−1.31101
$297$	0	0
$298$	10267.9	1.99598
$299$	582.152	0.112598
$300$	0	0
$301$	1230.50	0.235631
$302$	−2454.27	−0.467641
$303$	0	0
$304$	−2919.07	−0.550724
$305$	−8204.80	−1.54035
$306$	0	0
$307$	467.489	0.0869088	0.0434544	−	0.999055i	$-0.486164\pi$
$307$	467.489	0.0869088	0.0434544	+	0.999055i	$0.486164\pi$
$308$	−1729.38	−0.319938
$309$	0	0
$310$	−6820.91	−1.24968
$311$	−9268.32	−1.68990	−0.844949	−	0.534847i	$-0.820369\pi$
$311$	−9268.32	−1.68990	−0.844949	+	0.534847i	$0.820369\pi$
$312$	0	0
$313$	5146.60	0.929403	0.464702	−	0.885467i	$-0.346162\pi$
$313$	5146.60	0.929403	0.464702	+	0.885467i	$0.346162\pi$
$314$	3307.50	0.594437
$315$	0	0
$316$	−8436.78	−1.50192
$317$	3101.92	0.549594	0.274797	−	0.961502i	$-0.411389\pi$
$317$	3101.92	0.549594	0.274797	+	0.961502i	$0.411389\pi$
$318$	0	0
$319$	−2535.66	−0.445046
$320$	10008.3	1.74837
$321$	0	0
$322$	−687.670	−0.119014
$323$	5987.24	1.03139
$324$	0	0
$325$	1221.79	0.208532
$326$	−7463.54	−1.26800
$327$	0	0
$328$	−5719.97	−0.962904
$329$	−672.183	−0.112640
$330$	0	0
$331$	8290.74	1.37674	0.688369	−	0.725361i	$-0.258327\pi$
$331$	8290.74	1.37674	0.688369	+	0.725361i	$0.258327\pi$
$332$	−6295.89	−1.04076
$333$	0	0
$334$	5708.82	0.935247
$335$	10758.0	1.75454
$336$	0	0
$337$	6843.83	1.10625	0.553126	−	0.833097i	$-0.313435\pi$
$337$	6843.83	1.10625	0.553126	+	0.833097i	$0.313435\pi$
$338$	5410.36	0.870666
$339$	0	0
$340$	−5510.73	−0.879004
$341$	1197.08	0.190105
$342$	0	0
$343$	6909.18	1.08764
$344$	−1351.08	−0.211759
$345$	0	0
$346$	1778.79	0.276383
$347$	−8518.74	−1.31790	−0.658948	−	0.752189i	$-0.728998\pi$
$347$	−8518.74	−1.31790	−0.658948	+	0.752189i	$0.728998\pi$
$348$	0	0
$349$	−5096.98	−0.781762	−0.390881	−	0.920441i	$-0.627830\pi$
$349$	−5096.98	−0.781762	−0.390881	+	0.920441i	$0.627830\pi$
$350$	−1443.25	−0.220414
$351$	0	0
$352$	−2075.55	−0.314282
$353$	−6701.17	−1.01039	−0.505194	−	0.863006i	$-0.668579\pi$
$353$	−6701.17	−1.01039	−0.505194	+	0.863006i	$0.668579\pi$
$354$	0	0
$355$	−2874.07	−0.429690
$356$	−18745.1	−2.79069
$357$	0	0
$358$	−14275.3	−2.10746
$359$	−6336.49	−0.931551	−0.465776	−	0.884903i	$-0.654225\pi$
$359$	−6336.49	−0.931551	−0.465776	+	0.884903i	$0.654225\pi$
$360$	0	0
$361$	17214.0	2.50970
$362$	−15148.5	−2.19941
$363$	0	0
$364$	−10702.9	−1.54116
$365$	−7713.22	−1.10610
$366$	0	0
$367$	4634.23	0.659142	0.329571	−	0.944131i	$-0.393096\pi$
$367$	4634.23	0.659142	0.329571	+	0.944131i	$0.393096\pi$
$368$	−187.499	−0.0265599
$369$	0	0
$370$	21100.3	2.96474
$371$	−368.345	−0.0515459
$372$	0	0
$373$	−12169.6	−1.68933	−0.844665	−	0.535296i	$-0.820200\pi$
$373$	−12169.6	−1.68933	−0.844665	+	0.535296i	$0.820200\pi$
$374$	1621.61	0.224201
$375$	0	0
$376$	738.048	0.101229
$377$	−15692.8	−2.14382
$378$	0	0
$379$	6615.75	0.896645	0.448322	−	0.893872i	$-0.352022\pi$
$379$	6615.75	0.896645	0.448322	+	0.893872i	$0.352022\pi$
$380$	−22157.1	−2.99115
$381$	0	0
$382$	5782.20	0.774458
$383$	14005.4	1.86852	0.934260	−	0.356594i	$-0.116062\pi$
$383$	14005.4	1.86852	0.934260	+	0.356594i	$0.116062\pi$
$384$	0	0
$385$	1767.04	0.233914
$386$	10597.6	1.39742
$387$	0	0
$388$	18000.8	2.35529
$389$	−14343.8	−1.86956	−0.934779	−	0.355229i	$-0.884403\pi$
$389$	−14343.8	−1.86956	−0.934779	+	0.355229i	$0.884403\pi$
$390$	0	0
$391$	384.574	0.0497410
$392$	−1749.35	−0.225397
$393$	0	0
$394$	18972.1	2.42589
$395$	8620.48	1.09809
$396$	0	0
$397$	8729.84	1.10362	0.551811	−	0.833969i	$-0.313937\pi$
$397$	8729.84	1.10362	0.551811	+	0.833969i	$0.313937\pi$
$398$	1664.02	0.209572
$399$	0	0
$400$	−393.513	−0.0491891
$401$	−3137.18	−0.390681	−0.195341	−	0.980735i	$-0.562581\pi$
$401$	−3137.18	−0.390681	−0.195341	+	0.980735i	$0.562581\pi$
$402$	0	0
$403$	7408.55	0.915748
$404$	−105.031	−0.0129343
$405$	0	0
$406$	18537.2	2.26597
$407$	−3703.14	−0.451002
$408$	0	0
$409$	−10168.2	−1.22931	−0.614653	−	0.788798i	$-0.710704\pi$
$409$	−10168.2	−1.22931	−0.614653	+	0.788798i	$0.710704\pi$
$410$	18077.4	2.17752
$411$	0	0
$412$	−17490.5	−2.09149
$413$	272.261	0.0324385
$414$	0	0
$415$	6432.98	0.760922
$416$	−12845.2	−1.51392
$417$	0	0
$418$	6520.03	0.762930
$419$	7237.46	0.843850	0.421925	−	0.906631i	$-0.361355\pi$
$419$	7237.46	0.843850	0.421925	+	0.906631i	$0.361355\pi$
$420$	0	0
$421$	5373.23	0.622032	0.311016	−	0.950405i	$-0.399331\pi$
$421$	5373.23	0.622032	0.311016	+	0.950405i	$0.399331\pi$
$422$	−7779.12	−0.897350
$423$	0	0
$424$	404.438	0.0463237
$425$	807.124	0.0921207
$426$	0	0
$427$	−10527.0	−1.19306
$428$	−12006.9	−1.35602
$429$	0	0
$430$	4269.95	0.478873
$431$	−1674.28	−0.187117	−0.0935585	−	0.995614i	$-0.529824\pi$
$431$	−1674.28	−0.187117	−0.0935585	+	0.995614i	$0.529824\pi$
$432$	0	0
$433$	−11105.6	−1.23257	−0.616284	−	0.787524i	$-0.711363\pi$
$433$	−11105.6	−1.23257	−0.616284	+	0.787524i	$0.711363\pi$
$434$	−8751.39	−0.967927
$435$	0	0
$436$	4611.13	0.506498
$437$	1546.26	0.169263
$438$	0	0
$439$	16347.4	1.77726	0.888630	−	0.458626i	$-0.151658\pi$
$439$	16347.4	1.77726	0.888630	+	0.458626i	$0.151658\pi$
$440$	−1940.19	−0.210215
$441$	0	0
$442$	10035.9	1.07999
$443$	8720.88	0.935309	0.467654	−	0.883911i	$-0.345099\pi$
$443$	8720.88	0.935309	0.467654	+	0.883911i	$0.345099\pi$
$444$	0	0
$445$	19153.2	2.04034
$446$	−9012.64	−0.956863
$447$	0	0
$448$	12840.8	1.35418
$449$	6224.16	0.654201	0.327101	−	0.944990i	$-0.393928\pi$
$449$	6224.16	0.654201	0.327101	+	0.944990i	$0.393928\pi$
$450$	0	0
$451$	−3172.63	−0.331249
$452$	−20366.2	−2.11934
$453$	0	0
$454$	−19012.8	−1.96545
$455$	10935.9	1.12678
$456$	0	0
$457$	11792.8	1.20709	0.603547	−	0.797328i	$-0.293754\pi$
$457$	11792.8	1.20709	0.603547	+	0.797328i	$0.293754\pi$
$458$	−21379.9	−2.18126
$459$	0	0
$460$	−1423.20	−0.144255
$461$	−7664.01	−0.774292	−0.387146	−	0.922018i	$-0.626539\pi$
$461$	−7664.01	−0.774292	−0.387146	+	0.922018i	$0.626539\pi$
$462$	0	0
$463$	−2555.99	−0.256559	−0.128279	−	0.991738i	$-0.540945\pi$
$463$	−2555.99	−0.256559	−0.128279	+	0.991738i	$0.540945\pi$
$464$	5054.31	0.505691
$465$	0	0
$466$	13856.2	1.37741
$467$	3113.80	0.308543	0.154272	−	0.988028i	$-0.450697\pi$
$467$	3113.80	0.308543	0.154272	+	0.988028i	$0.450697\pi$
$468$	0	0
$469$	13802.7	1.35896
$470$	−2332.54	−0.228919
$471$	0	0
$472$	−298.939	−0.0291521
$473$	−749.385	−0.0728473
$474$	0	0
$475$	3245.22	0.313476
$476$	−7070.40	−0.680822
$477$	0	0
$478$	1064.08	0.101820
$479$	5888.63	0.561708	0.280854	−	0.959750i	$-0.409382\pi$
$479$	5888.63	0.561708	0.280854	+	0.959750i	$0.409382\pi$
$480$	0	0
$481$	−22918.2	−2.17251
$482$	19192.8	1.81371
$483$	0	0
$484$	−14682.1	−1.37886
$485$	−18392.8	−1.72200
$486$	0	0
$487$	8277.47	0.770201	0.385100	−	0.922875i	$-0.374167\pi$
$487$	8277.47	0.770201	0.385100	+	0.922875i	$0.374167\pi$
$488$	11558.5	1.07219
$489$	0	0
$490$	5528.67	0.509714
$491$	−13500.7	−1.24089	−0.620447	−	0.784248i	$-0.713049\pi$
$491$	−13500.7	−1.24089	−0.620447	+	0.784248i	$0.713049\pi$
$492$	0	0
$493$	−10366.8	−0.947051
$494$	40351.4	3.67509
$495$	0	0
$496$	−2386.14	−0.216009
$497$	−3687.51	−0.332811
$498$	0	0
$499$	−18025.8	−1.61713	−0.808563	−	0.588410i	$-0.799754\pi$
$499$	−18025.8	−1.61713	−0.808563	+	0.588410i	$0.799754\pi$
$500$	14863.9	1.32947
$501$	0	0
$502$	26802.4	2.38297
$503$	−10382.8	−0.920375	−0.460187	−	0.887822i	$-0.652218\pi$
$503$	−10382.8	−0.920375	−0.460187	+	0.887822i	$0.652218\pi$
$504$	0	0
$505$	107.318	0.00945659
$506$	418.796	0.0367940
$507$	0	0
$508$	−25792.3	−2.25265
$509$	2917.87	0.254091	0.127045	−	0.991897i	$-0.459451\pi$
$509$	2917.87	0.254091	0.127045	+	0.991897i	$0.459451\pi$
$510$	0	0
$511$	−9896.24	−0.856720
$512$	6698.43	0.578187
$513$	0	0
$514$	−7221.89	−0.619736
$515$	17871.3	1.52913
$516$	0	0
$517$	409.365	0.0348237
$518$	27072.2	2.29630
$519$	0	0
$520$	−12007.5	−1.01262
$521$	5398.65	0.453972	0.226986	−	0.973898i	$-0.427113\pi$
$521$	5398.65	0.453972	0.226986	+	0.973898i	$0.427113\pi$
$522$	0	0
$523$	−4608.30	−0.385290	−0.192645	−	0.981269i	$-0.561707\pi$
$523$	−4608.30	−0.385290	−0.192645	+	0.981269i	$0.561707\pi$
$524$	−21107.8	−1.75973
$525$	0	0
$526$	−25514.4	−2.11498
$527$	4894.15	0.404540
$528$	0	0
$529$	−12067.7	−0.991837
$530$	−1278.19	−0.104757
$531$	0	0
$532$	−28428.1	−2.31676
$533$	−19634.9	−1.59565
$534$	0	0
$535$	12268.4	0.991415
$536$	−15155.2	−1.22128
$537$	0	0
$538$	−28556.9	−2.28843
$539$	−970.293	−0.0775389
$540$	0	0
$541$	−3575.27	−0.284127	−0.142064	−	0.989858i	$-0.545374\pi$
$541$	−3575.27	−0.284127	−0.142064	+	0.989858i	$0.545374\pi$
$542$	4419.42	0.350240
$543$	0	0
$544$	−8485.67	−0.668787
$545$	−4711.54	−0.370312
$546$	0	0
$547$	4167.64	0.325768	0.162884	−	0.986645i	$-0.447920\pi$
$547$	4167.64	0.325768	0.162884	+	0.986645i	$0.447920\pi$
$548$	1584.15	0.123488
$549$	0	0
$550$	878.948	0.0681427
$551$	−41681.9	−3.22270
$552$	0	0
$553$	11060.3	0.850509
$554$	29413.6	2.25571
$555$	0	0
$556$	27077.5	2.06536
$557$	−8423.86	−0.640809	−0.320404	−	0.947281i	$-0.603819\pi$
$557$	−8423.86	−0.640809	−0.320404	+	0.947281i	$0.603819\pi$
$558$	0	0
$559$	−4637.82	−0.350911
$560$	−3522.23	−0.265788
$561$	0	0
$562$	29459.1	2.21113
$563$	−23644.9	−1.77000	−0.885002	−	0.465587i	$-0.845843\pi$
$563$	−23644.9	−1.77000	−0.885002	+	0.465587i	$0.845843\pi$
$564$	0	0
$565$	20809.6	1.54950
$566$	−18578.4	−1.37970
$567$	0	0
$568$	4048.83	0.299094
$569$	−14266.1	−1.05108	−0.525540	−	0.850769i	$-0.676137\pi$
$569$	−14266.1	−1.05108	−0.525540	+	0.850769i	$0.676137\pi$
$570$	0	0
$571$	7683.48	0.563124	0.281562	−	0.959543i	$-0.409148\pi$
$571$	7683.48	0.563124	0.281562	+	0.959543i	$0.409148\pi$
$572$	6518.13	0.476463
$573$	0	0
$574$	23193.8	1.68657
$575$	208.448	0.0151181
$576$	0	0
$577$	11833.0	0.853749	0.426874	−	0.904311i	$-0.359615\pi$
$577$	11833.0	0.853749	0.426874	+	0.904311i	$0.359615\pi$
$578$	−15243.9	−1.09700
$579$	0	0
$580$	38364.6	2.74656
$581$	8253.67	0.589363
$582$	0	0
$583$	224.325	0.0159358
$584$	10865.9	0.769925
$585$	0	0
$586$	12416.1	0.875263
$587$	986.552	0.0693686	0.0346843	−	0.999398i	$-0.488957\pi$
$587$	986.552	0.0693686	0.0346843	+	0.999398i	$0.488957\pi$
$588$	0	0
$589$	19678.0	1.37660
$590$	944.770	0.0659247
$591$	0	0
$592$	7381.44	0.512459
$593$	−6807.18	−0.471395	−0.235698	−	0.971826i	$-0.575737\pi$
$593$	−6807.18	−0.471395	−0.235698	+	0.971826i	$0.575737\pi$
$594$	0	0
$595$	7224.36	0.497764
$596$	27264.8	1.87384
$597$	0	0
$598$	2591.86	0.177239
$599$	54.1935	0.00369664	0.00184832	−	0.999998i	$-0.499412\pi$
$599$	54.1935	0.00369664	0.00184832	+	0.999998i	$0.499412\pi$
$600$	0	0
$601$	44.6639	0.00303141	0.00151570	−	0.999999i	$-0.499518\pi$
$601$	44.6639	0.00303141	0.00151570	+	0.999999i	$0.499518\pi$
$602$	5478.45	0.370906
$603$	0	0
$604$	−6516.94	−0.439024
$605$	15001.8	1.00811
$606$	0	0
$607$	1299.31	0.0868818	0.0434409	−	0.999056i	$-0.486168\pi$
$607$	1299.31	0.0868818	0.0434409	+	0.999056i	$0.486168\pi$
$608$	−34118.5	−2.27580
$609$	0	0
$610$	−36529.5	−2.42465
$611$	2533.49	0.167748
$612$	0	0
$613$	3353.42	0.220951	0.110476	−	0.993879i	$-0.464763\pi$
$613$	3353.42	0.220951	0.110476	+	0.993879i	$0.464763\pi$
$614$	2081.36	0.136802
$615$	0	0
$616$	−2489.31	−0.162820
$617$	−24209.6	−1.57964	−0.789822	−	0.613336i	$-0.789827\pi$
$617$	−24209.6	−1.57964	−0.789822	+	0.613336i	$0.789827\pi$
$618$	0	0
$619$	53.0613	0.00344542	0.00172271	−	0.999999i	$-0.499452\pi$
$619$	53.0613	0.00344542	0.00172271	+	0.999999i	$0.499452\pi$
$620$	−18111.9	−1.17321
$621$	0	0
$622$	−41264.5	−2.66006
$623$	24574.0	1.58032
$624$	0	0
$625$	−17802.0	−1.13933
$626$	22913.8	1.46297
$627$	0	0
$628$	8782.57	0.558061
$629$	−15139.9	−0.959726
$630$	0	0
$631$	899.565	0.0567530	0.0283765	−	0.999597i	$-0.490966\pi$
$631$	899.565	0.0567530	0.0283765	+	0.999597i	$0.490966\pi$
$632$	−12144.1	−0.764342
$633$	0	0
$634$	13810.4	0.865112
$635$	26353.9	1.64697
$636$	0	0
$637$	−6004.98	−0.373510
$638$	−11289.3	−0.700544
$639$	0	0
$640$	23308.5	1.43961
$641$	1537.41	0.0947331	0.0473665	−	0.998878i	$-0.484917\pi$
$641$	1537.41	0.0947331	0.0473665	+	0.998878i	$0.484917\pi$
$642$	0	0
$643$	25594.7	1.56976	0.784881	−	0.619647i	$-0.212724\pi$
$643$	25594.7	1.56976	0.784881	+	0.619647i	$0.212724\pi$
$644$	−1826.00	−0.111731
$645$	0	0
$646$	26656.4	1.62350
$647$	−15216.0	−0.924580	−0.462290	−	0.886729i	$-0.652972\pi$
$647$	−15216.0	−0.924580	−0.462290	+	0.886729i	$0.652972\pi$
$648$	0	0
$649$	−165.809	−0.0100286
$650$	5439.67	0.328248
$651$	0	0
$652$	−19818.3	−1.19040
$653$	−29819.4	−1.78702	−0.893510	−	0.449044i	$-0.851765\pi$
$653$	−29819.4	−1.78702	−0.893510	+	0.449044i	$0.851765\pi$
$654$	0	0
$655$	21567.4	1.28658
$656$	6323.97	0.376387
$657$	0	0
$658$	−2992.70	−0.177306
$659$	−28251.3	−1.66998	−0.834988	−	0.550269i	$-0.814525\pi$
$659$	−28251.3	−1.66998	−0.834988	+	0.550269i	$0.814525\pi$
$660$	0	0
$661$	14182.3	0.834534	0.417267	−	0.908784i	$-0.362988\pi$
$661$	14182.3	0.834534	0.417267	+	0.908784i	$0.362988\pi$
$662$	36912.1	2.16711
$663$	0	0
$664$	−9062.42	−0.529654
$665$	29047.1	1.69383
$666$	0	0
$667$	−2677.32	−0.155422
$668$	15158.9	0.878016
$669$	0	0
$670$	47896.8	2.76181
$671$	6410.99	0.368843
$672$	0	0
$673$	−22102.0	−1.26593	−0.632963	−	0.774182i	$-0.718161\pi$
$673$	−22102.0	−1.26593	−0.632963	+	0.774182i	$0.718161\pi$
$674$	30470.2	1.74134
$675$	0	0
$676$	14366.4	0.817387
$677$	−11041.8	−0.626839	−0.313419	−	0.949615i	$-0.601475\pi$
$677$	−11041.8	−0.626839	−0.313419	+	0.949615i	$0.601475\pi$
$678$	0	0
$679$	−23598.4	−1.33376
$680$	−7932.25	−0.447335
$681$	0	0
$682$	5329.66	0.299242
$683$	24708.8	1.38427	0.692135	−	0.721768i	$-0.256670\pi$
$683$	24708.8	1.38427	0.692135	+	0.721768i	$0.256670\pi$
$684$	0	0
$685$	−1618.65	−0.0902851
$686$	30761.1	1.71205
$687$	0	0
$688$	1493.74	0.0827739
$689$	1388.31	0.0767640
$690$	0	0
$691$	−11954.9	−0.658156	−0.329078	−	0.944303i	$-0.606738\pi$
$691$	−11954.9	−0.658156	−0.329078	+	0.944303i	$0.606738\pi$
$692$	4723.32	0.259470
$693$	0	0
$694$	−37927.2	−2.07449
$695$	−27667.0	−1.51003
$696$	0	0
$697$	−12970.9	−0.704892
$698$	−22692.8	−1.23057
$699$	0	0
$700$	−3832.32	−0.206926
$701$	4351.75	0.234470	0.117235	−	0.993104i	$-0.462597\pi$
$701$	4351.75	0.234470	0.117235	+	0.993104i	$0.462597\pi$
$702$	0	0
$703$	−60873.3	−3.26583
$704$	−7820.16	−0.418655
$705$	0	0
$706$	−29835.0	−1.59045
$707$	137.691	0.00732449
$708$	0	0
$709$	29815.0	1.57931	0.789653	−	0.613554i	$-0.210261\pi$
$709$	29815.0	1.57931	0.789653	+	0.613554i	$0.210261\pi$
$710$	−12796.0	−0.676372
$711$	0	0
$712$	−26982.0	−1.42021
$713$	1263.96	0.0663896
$714$	0	0
$715$	−6660.06	−0.348353
$716$	−37905.8	−1.97850
$717$	0	0
$718$	−28211.3	−1.46635
$719$	29875.8	1.54962	0.774811	−	0.632193i	$-0.217845\pi$
$719$	29875.8	1.54962	0.774811	+	0.632193i	$0.217845\pi$
$720$	0	0
$721$	22929.3	1.18437
$722$	76640.3	3.95050
$723$	0	0
$724$	−40224.5	−2.06482
$725$	−5619.03	−0.287842
$726$	0	0
$727$	21050.1	1.07387	0.536936	−	0.843623i	$-0.319582\pi$
$727$	21050.1	1.07387	0.536936	+	0.843623i	$0.319582\pi$
$728$	−15405.9	−0.784315
$729$	0	0
$730$	−34340.8	−1.74111
$731$	−3063.78	−0.155018
$732$	0	0
$733$	18976.2	0.956212	0.478106	−	0.878302i	$-0.341324\pi$
$733$	18976.2	0.956212	0.478106	+	0.878302i	$0.341324\pi$
$734$	20632.6	1.03755
$735$	0	0
$736$	−2191.51	−0.109756
$737$	−8405.97	−0.420133
$738$	0	0
$739$	8704.26	0.433277	0.216638	−	0.976252i	$-0.430491\pi$
$739$	8704.26	0.433277	0.216638	+	0.976252i	$0.430491\pi$
$740$	56028.6	2.78331
$741$	0	0
$742$	−1639.95	−0.0811380
$743$	40169.6	1.98342	0.991709	−	0.128502i	$-0.0410170\pi$
$743$	40169.6	1.98342	0.991709	+	0.128502i	$0.0410170\pi$
$744$	0	0
$745$	−27858.5	−1.37001
$746$	−54181.7	−2.65916
$747$	0	0
$748$	4305.93	0.210482
$749$	15740.6	0.767889
$750$	0	0
$751$	13712.0	0.666255	0.333127	−	0.942882i	$-0.391896\pi$
$751$	13712.0	0.666255	0.333127	+	0.942882i	$0.391896\pi$
$752$	−815.983	−0.0395689
$753$	0	0
$754$	−69867.6	−3.37457
$755$	6658.84	0.320980
$756$	0	0
$757$	−16061.4	−0.771153	−0.385576	−	0.922676i	$-0.625997\pi$
$757$	−16061.4	−0.771153	−0.385576	+	0.922676i	$0.625997\pi$
$758$	29454.7	1.41140
$759$	0	0
$760$	−31893.3	−1.52223
$761$	−29960.0	−1.42713	−0.713567	−	0.700587i	$-0.752922\pi$
$761$	−29960.0	−1.42713	−0.713567	+	0.700587i	$0.752922\pi$
$762$	0	0
$763$	−6045.01	−0.286821
$764$	15353.7	0.727066
$765$	0	0
$766$	62355.0	2.94122
$767$	−1026.17	−0.0483086
$768$	0	0
$769$	9759.16	0.457639	0.228819	−	0.973469i	$-0.426513\pi$
$769$	9759.16	0.457639	0.228819	+	0.973469i	$0.426513\pi$
$770$	7867.23	0.368202
$771$	0	0
$772$	28140.3	1.31191
$773$	25860.4	1.20328	0.601638	−	0.798769i	$-0.294515\pi$
$773$	25860.4	1.20328	0.601638	+	0.798769i	$0.294515\pi$
$774$	0	0
$775$	2652.74	0.122954
$776$	25910.7	1.19863
$777$	0	0
$778$	−63861.5	−2.94286
$779$	−52152.5	−2.39866
$780$	0	0
$781$	2245.72	0.102891
$782$	1712.20	0.0782970
$783$	0	0
$784$	1934.08	0.0881048
$785$	−8973.80	−0.408011
$786$	0	0
$787$	28918.2	1.30981	0.654906	−	0.755710i	$-0.272708\pi$
$787$	28918.2	1.30981	0.654906	+	0.755710i	$0.272708\pi$
$788$	50377.4	2.27744
$789$	0	0
$790$	38380.2	1.72849
$791$	26699.2	1.20015
$792$	0	0
$793$	39676.6	1.77674
$794$	38867.1	1.73720
$795$	0	0
$796$	4418.55	0.196748
$797$	17215.9	0.765142	0.382571	−	0.923926i	$-0.375039\pi$
$797$	17215.9	0.765142	0.382571	+	0.923926i	$0.375039\pi$
$798$	0	0
$799$	1673.64	0.0741042
$800$	−4599.43	−0.203268
$801$	0	0
$802$	−13967.4	−0.614969
$803$	6026.88	0.264862
$804$	0	0
$805$	1865.76	0.0816888
$806$	32984.4	1.44147
$807$	0	0
$808$	−151.183	−0.00658243
$809$	8937.15	0.388397	0.194199	−	0.980962i	$-0.437789\pi$
$809$	8937.15	0.388397	0.194199	+	0.980962i	$0.437789\pi$
$810$	0	0
$811$	−31148.1	−1.34865	−0.674326	−	0.738434i	$-0.735566\pi$
$811$	−31148.1	−1.34865	−0.674326	+	0.738434i	$0.735566\pi$
$812$	49222.7	2.12731
$813$	0	0
$814$	−16487.2	−0.709920
$815$	20249.8	0.870331
$816$	0	0
$817$	−12318.6	−0.527507
$818$	−45271.0	−1.93504
$819$	0	0
$820$	48001.9	2.04427
$821$	26976.7	1.14676	0.573382	−	0.819288i	$-0.305631\pi$
$821$	26976.7	1.14676	0.573382	+	0.819288i	$0.305631\pi$
$822$	0	0
$823$	−42010.5	−1.77934	−0.889668	−	0.456608i	$-0.849064\pi$
$823$	−42010.5	−1.77934	−0.889668	+	0.456608i	$0.849064\pi$
$824$	−25176.1	−1.06438
$825$	0	0
$826$	1212.16	0.0510612
$827$	−19180.8	−0.806508	−0.403254	−	0.915088i	$-0.632121\pi$
$827$	−19180.8	−0.806508	−0.403254	+	0.915088i	$0.632121\pi$
$828$	0	0
$829$	29952.7	1.25489	0.627443	−	0.778663i	$-0.284102\pi$
$829$	29952.7	1.25489	0.627443	+	0.778663i	$0.284102\pi$
$830$	28641.0	1.19776
$831$	0	0
$832$	−48397.7	−2.01669
$833$	−3966.94	−0.165001
$834$	0	0
$835$	−15489.0	−0.641937
$836$	17312.9	0.716244
$837$	0	0
$838$	32222.7	1.32830
$839$	3668.38	0.150950	0.0754748	−	0.997148i	$-0.475953\pi$
$839$	3668.38	0.150950	0.0754748	+	0.997148i	$0.475953\pi$
$840$	0	0
$841$	47782.3	1.95918
$842$	23922.7	0.979136
$843$	0	0
$844$	−20656.3	−0.842438
$845$	−14679.2	−0.597609
$846$	0	0
$847$	19247.6	0.780822
$848$	−447.145	−0.0181073
$849$	0	0
$850$	3593.49	0.145007
$851$	−3910.03	−0.157502
$852$	0	0
$853$	44013.3	1.76669	0.883345	−	0.468724i	$-0.155286\pi$
$853$	44013.3	1.76669	0.883345	+	0.468724i	$0.155286\pi$
$854$	−46868.2	−1.87798
$855$	0	0
$856$	−17283.0	−0.690093
$857$	−12232.4	−0.487574	−0.243787	−	0.969829i	$-0.578390\pi$
$857$	−12232.4	−0.487574	−0.243787	+	0.969829i	$0.578390\pi$
$858$	0	0
$859$	45812.7	1.81969	0.909843	−	0.414954i	$-0.136202\pi$
$859$	45812.7	1.81969	0.909843	+	0.414954i	$0.136202\pi$
$860$	11338.2	0.449569
$861$	0	0
$862$	−7454.26	−0.294540
$863$	−33.9685	−0.00133986	−0.000669932	−	1.00000i	$-0.500213\pi$
$863$	−33.9685	−0.00133986	−0.000669932		1.00000i	$0.500213\pi$
$864$	0	0
$865$	−4826.16	−0.189705
$866$	−49444.5	−1.94018
$867$	0	0
$868$	−23238.0	−0.908696
$869$	−6735.80	−0.262942
$870$	0	0
$871$	−52023.2	−2.02381
$872$	6637.35	0.257762
$873$	0	0
$874$	6884.29	0.266435
$875$	−19486.0	−0.752852
$876$	0	0
$877$	−16690.0	−0.642624	−0.321312	−	0.946973i	$-0.604124\pi$
$877$	−16690.0	−0.642624	−0.321312	+	0.946973i	$0.604124\pi$
$878$	72781.8	2.79757
$879$	0	0
$880$	2145.06	0.0821705
$881$	−7575.04	−0.289682	−0.144841	−	0.989455i	$-0.546267\pi$
$881$	−7575.04	−0.289682	−0.144841	+	0.989455i	$0.546267\pi$
$882$	0	0
$883$	−19841.4	−0.756191	−0.378096	−	0.925767i	$-0.623421\pi$
$883$	−19841.4	−0.756191	−0.378096	+	0.925767i	$0.623421\pi$
$884$	26648.7	1.01391
$885$	0	0
$886$	38827.2	1.47226
$887$	6016.47	0.227749	0.113874	−	0.993495i	$-0.463674\pi$
$887$	6016.47	0.227749	0.113874	+	0.993495i	$0.463674\pi$
$888$	0	0
$889$	33812.7	1.27564
$890$	85274.1	3.21168
$891$	0	0
$892$	−23931.7	−0.898309
$893$	6729.25	0.252168
$894$	0	0
$895$	38731.2	1.44653
$896$	29905.3	1.11503
$897$	0	0
$898$	27711.3	1.02977
$899$	−34072.0	−1.26403
$900$	0	0
$901$	917.128	0.0339112
$902$	−14125.2	−0.521416
$903$	0	0
$904$	−29315.4	−1.07856
$905$	41100.3	1.50964
$906$	0	0
$907$	−33324.2	−1.21997	−0.609985	−	0.792413i	$-0.708825\pi$
$907$	−33324.2	−1.21997	−0.609985	+	0.792413i	$0.708825\pi$
$908$	−50485.5	−1.84518
$909$	0	0
$910$	48689.0	1.77365
$911$	38706.7	1.40770	0.703848	−	0.710351i	$-0.251464\pi$
$911$	38706.7	1.40770	0.703848	+	0.710351i	$0.251464\pi$
$912$	0	0
$913$	−5026.54	−0.182206
$914$	52503.8	1.90008
$915$	0	0
$916$	−56771.0	−2.04778
$917$	27671.5	0.996501
$918$	0	0
$919$	32552.7	1.16846	0.584230	−	0.811588i	$-0.301397\pi$
$919$	32552.7	1.16846	0.584230	+	0.811588i	$0.301397\pi$
$920$	−2048.58	−0.0734128
$921$	0	0
$922$	−34121.8	−1.21881
$923$	13898.4	0.495635
$924$	0	0
$925$	−8206.18	−0.291695
$926$	−11379.8	−0.403848
$927$	0	0
$928$	59075.4	2.08970
$929$	−15272.2	−0.539358	−0.269679	−	0.962950i	$-0.586918\pi$
$929$	−15272.2	−0.539358	−0.269679	+	0.962950i	$0.586918\pi$
$930$	0	0
$931$	−15949.9	−0.561480
$932$	36792.9	1.29312
$933$	0	0
$934$	13863.3	0.485676
$935$	−4399.69	−0.153888
$936$	0	0
$937$	−31426.2	−1.09568	−0.547838	−	0.836584i	$-0.684549\pi$
$937$	−31426.2	−1.09568	−0.547838	+	0.836584i	$0.684549\pi$
$938$	61452.7	2.13913
$939$	0	0
$940$	−6193.69	−0.214911
$941$	−28418.9	−0.984518	−0.492259	−	0.870449i	$-0.663829\pi$
$941$	−28418.9	−0.984518	−0.492259	+	0.870449i	$0.663829\pi$
$942$	0	0
$943$	−3349.88	−0.115681
$944$	330.506	0.0113952
$945$	0	0
$946$	−3336.42	−0.114668
$947$	−35394.9	−1.21455	−0.607275	−	0.794492i	$-0.707737\pi$
$947$	−35394.9	−1.21455	−0.607275	+	0.794492i	$0.707737\pi$
$948$	0	0
$949$	37299.4	1.27586
$950$	14448.4	0.493440
$951$	0	0
$952$	−10177.3	−0.346478
$953$	−20490.3	−0.696481	−0.348241	−	0.937405i	$-0.613221\pi$
$953$	−20490.3	−0.696481	−0.348241	+	0.937405i	$0.613221\pi$
$954$	0	0
$955$	−15688.1	−0.531574
$956$	2825.49	0.0955889
$957$	0	0
$958$	26217.4	0.884181
$959$	−2076.76	−0.0699293
$960$	0	0
$961$	−13705.6	−0.460059
$962$	−102036.	−3.41973
$963$	0	0
$964$	50963.5	1.70272
$965$	−28753.1	−0.959165
$966$	0	0
$967$	−45429.2	−1.51076	−0.755380	−	0.655287i	$-0.772548\pi$
$967$	−45429.2	−1.51076	−0.755380	+	0.655287i	$0.772548\pi$
$968$	−21133.6	−0.701716
$969$	0	0
$970$	−81888.4	−2.71060
$971$	32589.7	1.07709	0.538544	−	0.842597i	$-0.318974\pi$
$971$	32589.7	1.07709	0.538544	+	0.842597i	$0.318974\pi$
$972$	0	0
$973$	−35497.5	−1.16958
$974$	36853.0	1.21237
$975$	0	0
$976$	−12779.0	−0.419104
$977$	−35416.5	−1.15975	−0.579874	−	0.814706i	$-0.696898\pi$
$977$	−35416.5	−1.15975	−0.579874	+	0.814706i	$0.696898\pi$
$978$	0	0
$979$	−14965.8	−0.488568
$980$	14680.5	0.478523
$981$	0	0
$982$	−60108.0	−1.95328
$983$	10203.0	0.331054	0.165527	−	0.986205i	$-0.447068\pi$
$983$	10203.0	0.331054	0.165527	+	0.986205i	$0.447068\pi$
$984$	0	0
$985$	−51474.3	−1.66509
$986$	−46155.0	−1.49075
$987$	0	0
$988$	107147.	3.45020
$989$	−791.252	−0.0254402
$990$	0	0
$991$	33015.0	1.05828	0.529141	−	0.848534i	$-0.322514\pi$
$991$	33015.0	1.05828	0.529141	+	0.848534i	$0.322514\pi$
$992$	−27889.5	−0.892632
$993$	0	0
$994$	−16417.5	−0.523876
$995$	−4514.76	−0.143847
$996$	0	0
$997$	15027.1	0.477346	0.238673	−	0.971100i	$-0.423288\pi$
$997$	15027.1	0.477346	0.238673	+	0.971100i	$0.423288\pi$
$998$	−80254.6	−2.54551
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2151.4.a.g.1.53	✓	59
3.2	odd	2		2151.4.a.h.1.7	yes	59

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2151.4.a.g.1.53	✓	59	1.1	even	1	trivial
2151.4.a.h.1.7	yes	59	3.2	odd	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 $\F_{q}$

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

\(f(q)\)	\(=\)	\(q+4.45221 q^{2} +11.8222 q^{4} -12.0796 q^{5} -15.4984 q^{7} +17.0170 q^{8} +O(q^{10})\)	\(q+4.45221 q^{2} +11.8222 q^{4} -12.0796 q^{5} -15.4984 q^{7} +17.0170 q^{8} -53.7808 q^{10} +9.43863 q^{11} +58.4141 q^{13} -69.0020 q^{14} -18.8139 q^{16} +38.5888 q^{17} +155.155 q^{19} -142.807 q^{20} +42.0227 q^{22} +9.96595 q^{23} +20.9160 q^{25} +260.072 q^{26} -183.224 q^{28} -268.647 q^{29} +126.828 q^{31} -219.900 q^{32} +171.805 q^{34} +187.214 q^{35} -392.339 q^{37} +690.781 q^{38} -205.558 q^{40} -336.132 q^{41} -79.3956 q^{43} +111.585 q^{44} +44.3705 q^{46} +43.3712 q^{47} -102.800 q^{49} +93.1225 q^{50} +690.581 q^{52} +23.7667 q^{53} -114.015 q^{55} -263.736 q^{56} -1196.07 q^{58} -17.5671 q^{59} +679.230 q^{61} +564.665 q^{62} -828.527 q^{64} -705.617 q^{65} -890.593 q^{67} +456.203 q^{68} +833.515 q^{70} +237.928 q^{71} +638.534 q^{73} -1746.78 q^{74} +1834.26 q^{76} -146.283 q^{77} -713.642 q^{79} +227.264 q^{80} -1496.53 q^{82} -532.550 q^{83} -466.136 q^{85} -353.486 q^{86} +160.617 q^{88} -1585.59 q^{89} -905.324 q^{91} +117.819 q^{92} +193.098 q^{94} -1874.20 q^{95} +1522.63 q^{97} -457.688 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 59 q - 8 q^{2} + 238 q^{4} - 80 q^{5} - 10 q^{7} - 96 q^{8}+O(q^{10}) \) 59 * q - 8 * q^2 + 238 * q^4 - 80 * q^5 - 10 * q^7 - 96 * q^8	\( 59 q - 8 q^{2} + 238 q^{4} - 80 q^{5} - 10 q^{7} - 96 q^{8} - 36 q^{10} - 132 q^{11} + 104 q^{13} - 280 q^{14} + 822 q^{16} - 408 q^{17} + 20 q^{19} - 800 q^{20} - 2 q^{22} - 276 q^{23} + 1477 q^{25} - 780 q^{26} + 224 q^{28} - 696 q^{29} - 380 q^{31} - 896 q^{32} - 72 q^{34} - 700 q^{35} + 224 q^{37} - 988 q^{38} - 258 q^{40} - 2706 q^{41} - 156 q^{43} - 1584 q^{44} + 428 q^{46} - 1316 q^{47} + 2135 q^{49} - 1400 q^{50} + 1092 q^{52} - 1484 q^{53} - 992 q^{55} - 3360 q^{56} - 120 q^{58} - 3186 q^{59} - 254 q^{61} - 1240 q^{62} + 3054 q^{64} - 5120 q^{65} + 288 q^{67} - 9420 q^{68} + 1108 q^{70} - 4468 q^{71} - 1770 q^{73} - 6214 q^{74} + 720 q^{76} - 6352 q^{77} - 746 q^{79} - 7040 q^{80} + 276 q^{82} - 5484 q^{83} + 588 q^{85} - 10152 q^{86} + 1186 q^{88} - 11570 q^{89} + 1768 q^{91} - 15366 q^{92} - 2142 q^{94} - 5736 q^{95} + 2390 q^{97} - 6912 q^{98}+O(q^{100}) \) 59 * q - 8 * q^2 + 238 * q^4 - 80 * q^5 - 10 * q^7 - 96 * q^8 - 36 * q^10 - 132 * q^11 + 104 * q^13 - 280 * q^14 + 822 * q^16 - 408 * q^17 + 20 * q^19 - 800 * q^20 - 2 * q^22 - 276 * q^23 + 1477 * q^25 - 780 * q^26 + 224 * q^28 - 696 * q^29 - 380 * q^31 - 896 * q^32 - 72 * q^34 - 700 * q^35 + 224 * q^37 - 988 * q^38 - 258 * q^40 - 2706 * q^41 - 156 * q^43 - 1584 * q^44 + 428 * q^46 - 1316 * q^47 + 2135 * q^49 - 1400 * q^50 + 1092 * q^52 - 1484 * q^53 - 992 * q^55 - 3360 * q^56 - 120 * q^58 - 3186 * q^59 - 254 * q^61 - 1240 * q^62 + 3054 * q^64 - 5120 * q^65 + 288 * q^67 - 9420 * q^68 + 1108 * q^70 - 4468 * q^71 - 1770 * q^73 - 6214 * q^74 + 720 * q^76 - 6352 * q^77 - 746 * q^79 - 7040 * q^80 + 276 * q^82 - 5484 * q^83 + 588 * q^85 - 10152 * q^86 + 1186 * q^88 - 11570 * q^89 + 1768 * q^91 - 15366 * q^92 - 2142 * q^94 - 5736 * q^95 + 2390 * q^97 - 6912 * q^98

Introduction

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