LMFDB - Embedded newform 2107.4.a.h.1.27

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2107,4,Mod(1,2107)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2107, 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("2107.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2107 = 7^{2} \cdot 43 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2107.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:	$124.317024382$
Analytic rank:	$1$
Dimension:	$30$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.27
Character	$\chi$	$=$	2107.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+4.05038 q^{2} -0.841255 q^{3} +8.40554 q^{4} +17.3009 q^{5} -3.40740 q^{6} +1.64258 q^{8} -26.2923 q^{9} +O(q^{10})$	\(q+4.05038 q^{2} -0.841255 q^{3} +8.40554 q^{4} +17.3009 q^{5} -3.40740 q^{6} +1.64258 q^{8} -26.2923 q^{9} +70.0752 q^{10} +64.6067 q^{11} -7.07120 q^{12} -30.9328 q^{13} -14.5545 q^{15} -60.5912 q^{16} -94.7937 q^{17} -106.494 q^{18} -72.9649 q^{19} +145.424 q^{20} +261.681 q^{22} -173.091 q^{23} -1.38183 q^{24} +174.322 q^{25} -125.289 q^{26} +44.8324 q^{27} -127.780 q^{29} -58.9511 q^{30} -316.510 q^{31} -258.558 q^{32} -54.3507 q^{33} -383.950 q^{34} -221.001 q^{36} -235.065 q^{37} -295.535 q^{38} +26.0223 q^{39} +28.4182 q^{40} +5.63902 q^{41} -43.0000 q^{43} +543.054 q^{44} -454.881 q^{45} -701.083 q^{46} +399.144 q^{47} +50.9727 q^{48} +706.069 q^{50} +79.7457 q^{51} -260.006 q^{52} -312.551 q^{53} +181.588 q^{54} +1117.76 q^{55} +61.3821 q^{57} -517.558 q^{58} +285.592 q^{59} -122.338 q^{60} +710.510 q^{61} -1281.99 q^{62} -562.527 q^{64} -535.165 q^{65} -220.141 q^{66} +371.855 q^{67} -796.792 q^{68} +145.614 q^{69} -186.290 q^{71} -43.1873 q^{72} +449.935 q^{73} -952.100 q^{74} -146.649 q^{75} -613.309 q^{76} +105.400 q^{78} -422.158 q^{79} -1048.28 q^{80} +672.176 q^{81} +22.8401 q^{82} -573.897 q^{83} -1640.02 q^{85} -174.166 q^{86} +107.496 q^{87} +106.122 q^{88} +579.477 q^{89} -1842.44 q^{90} -1454.92 q^{92} +266.266 q^{93} +1616.68 q^{94} -1262.36 q^{95} +217.513 q^{96} -583.089 q^{97} -1698.66 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 30 q - 8 q^{2} + 100 q^{4} - 90 q^{8} + 264 q^{9}+O(q^{10}) $ 30 * q - 8 * q^2 + 100 * q^4 - 90 * q^8 + 264 * q^9	$ 30 q - 8 q^{2} + 100 q^{4} - 90 q^{8} + 264 q^{9} - 60 q^{11} - 96 q^{15} + 372 q^{16} - 1008 q^{18} - 234 q^{22} - 214 q^{23} + 520 q^{25} - 870 q^{29} - 12 q^{30} - 1548 q^{32} + 1142 q^{36} - 1246 q^{37} - 416 q^{39} - 1290 q^{43} - 446 q^{44} - 660 q^{46} - 278 q^{50} - 3702 q^{51} - 2960 q^{53} + 620 q^{57} - 3634 q^{58} - 898 q^{60} + 2578 q^{64} - 4848 q^{65} + 928 q^{67} - 1708 q^{71} - 7900 q^{72} + 1714 q^{74} - 138 q^{78} - 3562 q^{79} + 2210 q^{81} - 948 q^{85} + 344 q^{86} + 2502 q^{88} - 3848 q^{92} - 11986 q^{93} - 2894 q^{95} - 2804 q^{99}+O(q^{100}) $ 30 * q - 8 * q^2 + 100 * q^4 - 90 * q^8 + 264 * q^9 - 60 * q^11 - 96 * q^15 + 372 * q^16 - 1008 * q^18 - 234 * q^22 - 214 * q^23 + 520 * q^25 - 870 * q^29 - 12 * q^30 - 1548 * q^32 + 1142 * q^36 - 1246 * q^37 - 416 * q^39 - 1290 * q^43 - 446 * q^44 - 660 * q^46 - 278 * q^50 - 3702 * q^51 - 2960 * q^53 + 620 * q^57 - 3634 * q^58 - 898 * q^60 + 2578 * q^64 - 4848 * q^65 + 928 * q^67 - 1708 * q^71 - 7900 * q^72 + 1714 * q^74 - 138 * q^78 - 3562 * q^79 + 2210 * q^81 - 948 * q^85 + 344 * q^86 + 2502 * q^88 - 3848 * q^92 - 11986 * q^93 - 2894 * q^95 - 2804 * q^99

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 $

$2$	4.05038	1.43202	0.716012	−	0.698088i	$-0.245966\pi$
$2$	4.05038	1.43202	0.716012	+	0.698088i	$0.245966\pi$
$3$	−0.841255	−0.161900	−0.0809498	−	0.996718i	$-0.525795\pi$
$3$	−0.841255	−0.161900	−0.0809498	+	0.996718i	$0.525795\pi$
$4$	8.40554	1.05069
$5$	17.3009	1.54744	0.773721	−	0.633527i	$-0.218393\pi$
$5$	17.3009	1.54744	0.773721	+	0.633527i	$0.218393\pi$
$6$	−3.40740	−0.231844
$7$	0	0
$7$	0	0
$8$	1.64258	0.0725927
$9$	−26.2923	−0.973789
$10$	70.0752	2.21597
$11$	64.6067	1.77088	0.885439	−	0.464756i	$-0.153858\pi$
$11$	64.6067	1.77088	0.885439	+	0.464756i	$0.153858\pi$
$12$	−7.07120	−0.170107
$13$	−30.9328	−0.659939	−0.329969	−	0.943992i	$-0.607038\pi$
$13$	−30.9328	−0.659939	−0.329969	+	0.943992i	$0.607038\pi$
$14$	0	0
$15$	−14.5545	−0.250530
$16$	−60.5912	−0.946738
$17$	−94.7937	−1.35240	−0.676201	−	0.736717i	$-0.736375\pi$
$17$	−94.7937	−1.35240	−0.676201	+	0.736717i	$0.736375\pi$
$18$	−106.494	−1.39449
$19$	−72.9649	−0.881015	−0.440508	−	0.897749i	$-0.645202\pi$
$19$	−72.9649	−0.881015	−0.440508	+	0.897749i	$0.645202\pi$
$20$	145.424	1.62588
$21$	0	0
$22$	261.681	2.53594
$23$	−173.091	−1.56921	−0.784607	−	0.619994i	$-0.787135\pi$
$23$	−173.091	−1.56921	−0.784607	+	0.619994i	$0.787135\pi$
$24$	−1.38183	−0.0117527
$25$	174.322	1.39458
$26$	−125.289	−0.945048
$27$	44.8324	0.319556
$28$	0	0
$29$	−127.780	−0.818214	−0.409107	−	0.912486i	$-0.634160\pi$
$29$	−127.780	−0.818214	−0.409107	+	0.912486i	$0.634160\pi$
$30$	−58.9511	−0.358765
$31$	−316.510	−1.83377	−0.916886	−	0.399150i	$-0.869305\pi$
$31$	−316.510	−1.83377	−0.916886	+	0.399150i	$0.869305\pi$
$32$	−258.558	−1.42834
$33$	−54.3507	−0.286705
$34$	−383.950	−1.93667
$35$	0	0
$36$	−221.001	−1.02315
$37$	−235.065	−1.04444	−0.522221	−	0.852810i	$-0.674896\pi$
$37$	−235.065	−1.04444	−0.522221	+	0.852810i	$0.674896\pi$
$38$	−295.535	−1.26164
$39$	26.0223	0.106844
$40$	28.4182	0.112333
$41$	5.63902	0.0214797	0.0107398	−	0.999942i	$-0.496581\pi$
$41$	5.63902	0.0214797	0.0107398	+	0.999942i	$0.496581\pi$
$42$	0	0
$43$	−43.0000	−0.152499
$43$	−43.0000	−0.152499
$44$	543.054	1.86065
$45$	−454.881	−1.50688
$46$	−701.083	−2.24715
$47$	399.144	1.23875	0.619374	−	0.785096i	$-0.287387\pi$
$47$	399.144	1.23875	0.619374	+	0.785096i	$0.287387\pi$
$48$	50.9727	0.153277
$49$	0	0
$50$	706.069	1.99706
$51$	79.7457	0.218954
$52$	−260.006	−0.693392
$53$	−312.551	−0.810041	−0.405020	−	0.914308i	$-0.632736\pi$
$53$	−312.551	−0.810041	−0.405020	+	0.914308i	$0.632736\pi$
$54$	181.588	0.457611
$55$	1117.76	2.74033
$56$	0	0
$57$	61.3821	0.142636
$58$	−517.558	−1.17170
$59$	285.592	0.630184	0.315092	−	0.949061i	$-0.397965\pi$
$59$	285.592	0.630184	0.315092	+	0.949061i	$0.397965\pi$
$60$	−122.338	−0.263230
$61$	710.510	1.49134	0.745668	−	0.666318i	$-0.232131\pi$
$61$	710.510	1.49134	0.745668	+	0.666318i	$0.232131\pi$
$62$	−1281.99	−2.62600
$63$	0	0
$64$	−562.527	−1.09868
$65$	−535.165	−1.02122
$66$	−220.141	−0.410568
$67$	371.855	0.678050	0.339025	−	0.940777i	$-0.389903\pi$
$67$	371.855	0.678050	0.339025	+	0.940777i	$0.389903\pi$
$68$	−796.792	−1.42096
$69$	145.614	0.254055
$70$	0	0
$71$	−186.290	−0.311388	−0.155694	−	0.987805i	$-0.549761\pi$
$71$	−186.290	−0.311388	−0.155694	+	0.987805i	$0.549761\pi$
$72$	−43.1873	−0.0706899
$73$	449.935	0.721382	0.360691	−	0.932685i	$-0.382541\pi$
$73$	449.935	0.721382	0.360691	+	0.932685i	$0.382541\pi$
$74$	−952.100	−1.49567
$75$	−146.649	−0.225781
$76$	−613.309	−0.925676
$77$	0	0
$78$	105.400	0.153003
$79$	−422.158	−0.601221	−0.300611	−	0.953747i	$-0.597190\pi$
$79$	−422.158	−0.601221	−0.300611	+	0.953747i	$0.597190\pi$
$80$	−1048.28	−1.46502
$81$	672.176	0.922053
$82$	22.8401	0.0307594
$83$	−573.897	−0.758956	−0.379478	−	0.925201i	$-0.623896\pi$
$83$	−573.897	−0.758956	−0.379478	+	0.925201i	$0.623896\pi$
$84$	0	0
$85$	−1640.02	−2.09276
$86$	−174.166	−0.218382
$87$	107.496	0.132469
$88$	106.122	0.128553
$89$	579.477	0.690162	0.345081	−	0.938573i	$-0.387851\pi$
$89$	579.477	0.690162	0.345081	+	0.938573i	$0.387851\pi$
$90$	−1842.44	−2.15789
$91$	0	0
$92$	−1454.92	−1.64876
$93$	266.266	0.296887
$94$	1616.68	1.77392
$95$	−1262.36	−1.36332
$96$	217.513	0.231248
$97$	−583.089	−0.610347	−0.305174	−	0.952297i	$-0.598714\pi$
$97$	−583.089	−0.610347	−0.305174	+	0.952297i	$0.598714\pi$
$98$	0	0
$99$	−1698.66	−1.72446
$100$	1465.27	1.46527
$101$	1758.08	1.73203	0.866017	−	0.500014i	$-0.166672\pi$
$101$	1758.08	1.73203	0.866017	+	0.500014i	$0.166672\pi$
$102$	323.000	0.313547
$103$	−1126.35	−1.07750	−0.538750	−	0.842466i	$-0.681103\pi$
$103$	−1126.35	−1.07750	−0.538750	+	0.842466i	$0.681103\pi$
$104$	−50.8097	−0.0479067
$105$	0	0
$106$	−1265.95	−1.16000
$107$	−1089.58	−0.984429	−0.492215	−	0.870474i	$-0.663812\pi$
$107$	−1089.58	−0.984429	−0.492215	+	0.870474i	$0.663812\pi$
$108$	376.841	0.335755
$109$	−237.661	−0.208842	−0.104421	−	0.994533i	$-0.533299\pi$
$109$	−237.661	−0.208842	−0.104421	+	0.994533i	$0.533299\pi$
$110$	4527.33	3.92422
$111$	197.749	0.169095
$112$	0	0
$113$	655.205	0.545456	0.272728	−	0.962091i	$-0.412074\pi$
$113$	655.205	0.545456	0.272728	+	0.962091i	$0.412074\pi$
$114$	248.621	0.204258
$115$	−2994.63	−2.42827
$116$	−1074.06	−0.859691
$117$	813.293	0.642641
$118$	1156.75	0.902438
$119$	0	0
$120$	−23.9070	−0.0181867
$121$	2843.03	2.13601
$122$	2877.83	2.13563
$123$	−4.74385	−0.00347755
$124$	−2660.44	−1.92673
$125$	853.314	0.610582
$126$	0	0
$127$	1402.64	0.980033	0.490017	−	0.871713i	$-0.336991\pi$
$127$	1402.64	0.980033	0.490017	+	0.871713i	$0.336991\pi$
$128$	−209.981	−0.144999
$129$	36.1740	0.0246895
$130$	−2167.62	−1.46241
$131$	2416.06	1.61139	0.805694	−	0.592333i	$-0.201793\pi$
$131$	2416.06	1.61139	0.805694	+	0.592333i	$0.201793\pi$
$132$	−456.847	−0.301238
$133$	0	0
$134$	1506.15	0.970983
$135$	775.642	0.494494
$136$	−155.707	−0.0981745
$137$	−2396.92	−1.49477	−0.747383	−	0.664393i	$-0.768690\pi$
$137$	−2396.92	−1.49477	−0.747383	+	0.664393i	$0.768690\pi$
$138$	589.789	0.363813
$139$	2024.41	1.23531	0.617657	−	0.786448i	$-0.288082\pi$
$139$	2024.41	1.23531	0.617657	+	0.786448i	$0.288082\pi$
$140$	0	0
$141$	−335.782	−0.200553
$142$	−754.545	−0.445915
$143$	−1998.46	−1.16867
$144$	1593.08	0.921923
$145$	−2210.72	−1.26614
$146$	1822.40	1.03304
$147$	0	0
$148$	−1975.84	−1.09739
$149$	−1373.83	−0.755358	−0.377679	−	0.925937i	$-0.623278\pi$
$149$	−1373.83	−0.755358	−0.377679	+	0.925937i	$0.623278\pi$
$150$	−593.984	−0.323324
$151$	−1325.59	−0.714405	−0.357203	−	0.934027i	$-0.616269\pi$
$151$	−1325.59	−0.714405	−0.357203	+	0.934027i	$0.616269\pi$
$152$	−119.851	−0.0639553
$153$	2492.34	1.31695
$154$	0	0
$155$	−5475.92	−2.83765
$156$	218.732	0.112260
$157$	1117.32	0.567972	0.283986	−	0.958828i	$-0.408343\pi$
$157$	1117.32	0.567972	0.283986	+	0.958828i	$0.408343\pi$
$158$	−1709.90	−0.860963
$159$	262.935	0.131145
$160$	−4473.29	−2.21028
$161$	0	0
$162$	2722.57	1.32040
$163$	88.6480	0.0425979	0.0212989	−	0.999773i	$-0.493220\pi$
$163$	88.6480	0.0425979	0.0212989	+	0.999773i	$0.493220\pi$
$164$	47.3990	0.0225685
$165$	−940.318	−0.443658
$166$	−2324.50	−1.08684
$167$	−1462.11	−0.677495	−0.338747	−	0.940877i	$-0.610003\pi$
$167$	−1462.11	−0.677495	−0.338747	+	0.940877i	$0.610003\pi$
$168$	0	0
$169$	−1240.16	−0.564481
$170$	−6642.69	−2.99689
$171$	1918.41	0.857923
$172$	−361.438	−0.160229
$173$	−374.257	−0.164475	−0.0822376	−	0.996613i	$-0.526207\pi$
$173$	−374.257	−0.164475	−0.0822376	+	0.996613i	$0.526207\pi$
$174$	435.398	0.189698
$175$	0	0
$176$	−3914.60	−1.67656
$177$	−240.255	−0.102027
$178$	2347.10	0.988328
$179$	−4305.70	−1.79789	−0.898947	−	0.438056i	$-0.855667\pi$
$179$	−4305.70	−1.79789	−0.898947	+	0.438056i	$0.855667\pi$
$180$	−3823.52	−1.58327
$181$	4089.09	1.67923	0.839613	−	0.543185i	$-0.182782\pi$
$181$	4089.09	1.67923	0.839613	+	0.543185i	$0.182782\pi$
$182$	0	0
$183$	−597.720	−0.241447
$184$	−284.316	−0.113913
$185$	−4066.83	−1.61621
$186$	1078.48	0.425149
$187$	−6124.31	−2.39494
$188$	3355.02	1.30154
$189$	0	0
$190$	−5113.03	−1.95231
$191$	4064.19	1.53966	0.769829	−	0.638250i	$-0.220341\pi$
$191$	4064.19	1.53966	0.769829	+	0.638250i	$0.220341\pi$
$192$	473.228	0.177877
$193$	997.396	0.371990	0.185995	−	0.982551i	$-0.440449\pi$
$193$	997.396	0.371990	0.185995	+	0.982551i	$0.440449\pi$
$194$	−2361.73	−0.874032
$195$	450.210	0.165335
$196$	0	0
$197$	102.888	0.0372104	0.0186052	−	0.999827i	$-0.494077\pi$
$197$	102.888	0.0372104	0.0186052	+	0.999827i	$0.494077\pi$
$198$	−6880.20	−2.46947
$199$	−1234.31	−0.439688	−0.219844	−	0.975535i	$-0.570555\pi$
$199$	−1234.31	−0.439688	−0.219844	+	0.975535i	$0.570555\pi$
$200$	286.338	0.101236
$201$	−312.825	−0.109776
$202$	7120.88	2.48031
$203$	0	0
$204$	670.305	0.230053
$205$	97.5602	0.0332385
$206$	−4562.14	−1.54301
$207$	4550.95	1.52808
$208$	1874.25	0.624789
$209$	−4714.02	−1.56017
$210$	0	0
$211$	984.762	0.321298	0.160649	−	0.987012i	$-0.448641\pi$
$211$	984.762	0.321298	0.160649	+	0.987012i	$0.448641\pi$
$212$	−2627.16	−0.851104
$213$	156.717	0.0504136
$214$	−4413.22	−1.40973
$215$	−743.940	−0.235983
$216$	73.6410	0.0231974
$217$	0	0
$218$	−962.615	−0.299067
$219$	−378.510	−0.116791
$220$	9395.34	2.87924
$221$	2932.23	0.892503
$222$	800.959	0.242148
$223$	1134.20	0.340590	0.170295	−	0.985393i	$-0.445528\pi$
$223$	1134.20	0.340590	0.170295	+	0.985393i	$0.445528\pi$
$224$	0	0
$225$	−4583.32	−1.35802
$226$	2653.83	0.781106
$227$	498.056	0.145626	0.0728131	−	0.997346i	$-0.476802\pi$
$227$	498.056	0.145626	0.0728131	+	0.997346i	$0.476802\pi$
$228$	515.950	0.149867
$229$	−2875.50	−0.829775	−0.414888	−	0.909873i	$-0.636179\pi$
$229$	−2875.50	−0.829775	−0.414888	+	0.909873i	$0.636179\pi$
$230$	−12129.4	−3.47734
$231$	0	0
$232$	−209.890	−0.0593963
$233$	2741.36	0.770782	0.385391	−	0.922753i	$-0.374067\pi$
$233$	2741.36	0.770782	0.385391	+	0.922753i	$0.374067\pi$
$234$	3294.14	0.920277
$235$	6905.56	1.91689
$236$	2400.55	0.662129
$237$	355.143	0.0973375
$238$	0	0
$239$	−3410.23	−0.922968	−0.461484	−	0.887149i	$-0.652683\pi$
$239$	−3410.23	−0.922968	−0.461484	+	0.887149i	$0.652683\pi$
$240$	881.874	0.237186
$241$	−3694.59	−0.987508	−0.493754	−	0.869602i	$-0.664376\pi$
$241$	−3694.59	−0.987508	−0.493754	+	0.869602i	$0.664376\pi$
$242$	11515.3	3.05882
$243$	−1775.95	−0.468836
$244$	5972.22	1.56694
$245$	0	0
$246$	−19.2144	−0.00497994
$247$	2257.00	0.581416
$248$	−519.895	−0.133118
$249$	482.794	0.122875
$250$	3456.24	0.874368
$251$	−190.016	−0.0477837	−0.0238918	−	0.999715i	$-0.507606\pi$
$251$	−190.016	−0.0477837	−0.0238918	+	0.999715i	$0.507606\pi$
$252$	0	0
$253$	−11182.8	−2.77889
$254$	5681.22	1.40343
$255$	1379.67	0.338818
$256$	3649.71	0.891043
$257$	−962.449	−0.233603	−0.116801	−	0.993155i	$-0.537264\pi$
$257$	−962.449	−0.233603	−0.116801	+	0.993155i	$0.537264\pi$
$258$	146.518	0.0353559
$259$	0	0
$260$	−4498.35	−1.07298
$261$	3359.64	0.796767
$262$	9785.93	2.30754
$263$	−6706.31	−1.57235	−0.786177	−	0.618002i	$-0.787943\pi$
$263$	−6706.31	−1.57235	−0.786177	+	0.618002i	$0.787943\pi$
$264$	−89.2757	−0.0208126
$265$	−5407.42	−1.25349
$266$	0	0
$267$	−487.488	−0.111737
$268$	3125.64	0.712421
$269$	−480.260	−0.108855	−0.0544274	−	0.998518i	$-0.517333\pi$
$269$	−480.260	−0.108855	−0.0544274	+	0.998518i	$0.517333\pi$
$270$	3141.64	0.708127
$271$	3108.15	0.696704	0.348352	−	0.937364i	$-0.386741\pi$
$271$	3108.15	0.696704	0.348352	+	0.937364i	$0.386741\pi$
$272$	5743.67	1.28037
$273$	0	0
$274$	−9708.44	−2.14054
$275$	11262.4	2.46962
$276$	1223.96	0.266934
$277$	−3180.23	−0.689825	−0.344913	−	0.938635i	$-0.612091\pi$
$277$	−3180.23	−0.689825	−0.344913	+	0.938635i	$0.612091\pi$
$278$	8199.64	1.76900
$279$	8321.78	1.78571
$280$	0	0
$281$	6323.08	1.34236	0.671181	−	0.741294i	$-0.265787\pi$
$281$	6323.08	1.34236	0.671181	+	0.741294i	$0.265787\pi$
$282$	−1360.04	−0.287196
$283$	245.304	0.0515258	0.0257629	−	0.999668i	$-0.491799\pi$
$283$	245.304	0.0515258	0.0257629	+	0.999668i	$0.491799\pi$
$284$	−1565.87	−0.327173
$285$	1061.97	0.220721
$286$	−8094.53	−1.67356
$287$	0	0
$288$	6798.08	1.39090
$289$	4072.84	0.828993
$290$	−8954.23	−1.81314
$291$	490.526	0.0988150
$292$	3781.94	0.757951
$293$	−5089.75	−1.01483	−0.507417	−	0.861701i	$-0.669400\pi$
$293$	−5089.75	−1.01483	−0.507417	+	0.861701i	$0.669400\pi$
$294$	0	0
$295$	4941.00	0.975173
$296$	−386.113	−0.0758189
$297$	2896.48	0.565894
$298$	−5564.52	−1.08169
$299$	5354.17	1.03558
$300$	−1232.67	−0.237227
$301$	0	0
$302$	−5369.15	−1.02305
$303$	−1478.99	−0.280416
$304$	4421.03	0.834091
$305$	12292.5	2.30776
$306$	10094.9	1.88591
$307$	−8217.56	−1.52769	−0.763845	−	0.645399i	$-0.776691\pi$
$307$	−8217.56	−1.52769	−0.763845	+	0.645399i	$0.776691\pi$
$308$	0	0
$309$	947.548	0.174447
$310$	−22179.5	−4.06359
$311$	−9724.13	−1.77301	−0.886503	−	0.462723i	$-0.846873\pi$
$311$	−9724.13	−1.77301	−0.886503	+	0.462723i	$0.846873\pi$
$312$	42.7439	0.00775608
$313$	8174.03	1.47611	0.738057	−	0.674738i	$-0.235744\pi$
$313$	8174.03	1.47611	0.738057	+	0.674738i	$0.235744\pi$
$314$	4525.55	0.813350
$315$	0	0
$316$	−3548.46	−0.631698
$317$	−9049.53	−1.60338	−0.801691	−	0.597738i	$-0.796066\pi$
$317$	−9049.53	−1.60338	−0.801691	+	0.597738i	$0.796066\pi$
$318$	1064.99	0.187803
$319$	−8255.46	−1.44896
$320$	−9732.23	−1.70015
$321$	916.617	0.159379
$322$	0	0
$323$	6916.61	1.19149
$324$	5650.00	0.968794
$325$	−5392.26	−0.920334
$326$	359.058	0.0610011
$327$	199.933	0.0338114
$328$	9.26256	0.00155927
$329$	0	0
$330$	−3808.64	−0.635329
$331$	−2043.02	−0.339258	−0.169629	−	0.985508i	$-0.554257\pi$
$331$	−2043.02	−0.339258	−0.169629	+	0.985508i	$0.554257\pi$
$332$	−4823.91	−0.797429
$333$	6180.39	1.01707
$334$	−5922.10	−0.970189
$335$	6433.43	1.04924
$336$	0	0
$337$	−1542.27	−0.249295	−0.124648	−	0.992201i	$-0.539780\pi$
$337$	−1542.27	−0.249295	−0.124648	+	0.992201i	$0.539780\pi$
$338$	−5023.13	−0.808350
$339$	−551.195	−0.0883091
$340$	−13785.2	−2.19885
$341$	−20448.7	−3.24739
$342$	7770.30	1.22857
$343$	0	0
$344$	−70.6311	−0.0110703
$345$	2519.25	0.393136
$346$	−1515.88	−0.235532
$347$	12551.6	1.94180	0.970902	−	0.239478i	$-0.0769764\pi$
$347$	12551.6	1.94180	0.970902	+	0.239478i	$0.0769764\pi$
$348$	903.560	0.139184
$349$	−12887.5	−1.97666	−0.988331	−	0.152325i	$-0.951324\pi$
$349$	−12887.5	−1.97666	−0.988331	+	0.152325i	$0.951324\pi$
$350$	0	0
$351$	−1386.79	−0.210887
$352$	−16704.6	−2.52942
$353$	9549.46	1.43985	0.719924	−	0.694052i	$-0.244176\pi$
$353$	9549.46	1.43985	0.719924	+	0.694052i	$0.244176\pi$
$354$	−973.124	−0.146104
$355$	−3222.99	−0.481855
$356$	4870.81	0.725148
$357$	0	0
$358$	−17439.7	−2.57463
$359$	−7088.89	−1.04217	−0.521083	−	0.853506i	$-0.674472\pi$
$359$	−7088.89	−1.04217	−0.521083	+	0.853506i	$0.674472\pi$
$360$	−747.180	−0.109388
$361$	−1535.12	−0.223812
$362$	16562.4	2.40469
$363$	−2391.71	−0.345819
$364$	0	0
$365$	7784.29	1.11630
$366$	−2420.99	−0.345758
$367$	915.833	0.130262	0.0651309	−	0.997877i	$-0.479253\pi$
$367$	915.833	0.130262	0.0651309	+	0.997877i	$0.479253\pi$
$368$	10487.8	1.48563
$369$	−148.263	−0.0209167
$370$	−16472.2	−2.31446
$371$	0	0
$372$	2238.11	0.311937
$373$	−12829.4	−1.78092	−0.890458	−	0.455066i	$-0.849616\pi$
$373$	−12829.4	−1.78092	−0.890458	+	0.455066i	$0.849616\pi$
$374$	−24805.7	−3.42961
$375$	−717.855	−0.0988530
$376$	655.628	0.0899240
$377$	3952.60	0.539971
$378$	0	0
$379$	−1552.45	−0.210407	−0.105203	−	0.994451i	$-0.533549\pi$
$379$	−1552.45	−0.210407	−0.105203	+	0.994451i	$0.533549\pi$
$380$	−10610.8	−1.43243
$381$	−1179.98	−0.158667
$382$	16461.5	2.20483
$383$	13050.3	1.74109	0.870546	−	0.492087i	$-0.163766\pi$
$383$	13050.3	1.74109	0.870546	+	0.492087i	$0.163766\pi$
$384$	176.647	0.0234753
$385$	0	0
$386$	4039.83	0.532699
$387$	1130.57	0.148501
$388$	−4901.17	−0.641287
$389$	−6213.90	−0.809915	−0.404958	−	0.914335i	$-0.632714\pi$
$389$	−6213.90	−0.809915	−0.404958	+	0.914335i	$0.632714\pi$
$390$	1823.52	0.236763
$391$	16407.9	2.12221
$392$	0	0
$393$	−2032.52	−0.260883
$394$	416.734	0.0532862
$395$	−7303.72	−0.930355
$396$	−14278.1	−1.81188
$397$	−10864.0	−1.37343	−0.686713	−	0.726929i	$-0.740947\pi$
$397$	−10864.0	−1.37343	−0.686713	+	0.726929i	$0.740947\pi$
$398$	−4999.42	−0.629643
$399$	0	0
$400$	−10562.4	−1.32030
$401$	11490.0	1.43089	0.715443	−	0.698671i	$-0.246225\pi$
$401$	11490.0	1.43089	0.715443	+	0.698671i	$0.246225\pi$
$402$	−1267.06	−0.157202
$403$	9790.53	1.21018
$404$	14777.6	1.81984
$405$	11629.3	1.42682
$406$	0	0
$407$	−15186.7	−1.84958
$408$	130.989	0.0158944
$409$	5558.73	0.672033	0.336017	−	0.941856i	$-0.390920\pi$
$409$	5558.73	0.672033	0.336017	+	0.941856i	$0.390920\pi$
$410$	395.155	0.0475984
$411$	2016.43	0.242002
$412$	−9467.58	−1.13212
$413$	0	0
$414$	18433.1	2.18825
$415$	−9928.94	−1.17444
$416$	7997.91	0.942619
$417$	−1703.05	−0.199997
$418$	−19093.6	−2.23420
$419$	−12023.4	−1.40186	−0.700930	−	0.713230i	$-0.747232\pi$
$419$	−12023.4	−1.40186	−0.700930	+	0.713230i	$0.747232\pi$
$420$	0	0
$421$	10834.8	1.25429	0.627147	−	0.778901i	$-0.284222\pi$
$421$	10834.8	1.25429	0.627147	+	0.778901i	$0.284222\pi$
$422$	3988.65	0.460106
$423$	−10494.4	−1.20628
$424$	−513.391	−0.0588030
$425$	−16524.6	−1.88603
$426$	634.765	0.0721935
$427$	0	0
$428$	−9158.53	−1.03433
$429$	1681.22	0.189207
$430$	−3013.23	−0.337933
$431$	−2137.09	−0.238840	−0.119420	−	0.992844i	$-0.538104\pi$
$431$	−2137.09	−0.238840	−0.119420	+	0.992844i	$0.538104\pi$
$432$	−2716.45	−0.302535
$433$	−7065.99	−0.784225	−0.392113	−	0.919917i	$-0.628256\pi$
$433$	−7065.99	−0.784225	−0.392113	+	0.919917i	$0.628256\pi$
$434$	0	0
$435$	1859.78	0.204987
$436$	−1997.67	−0.219429
$437$	12629.5	1.38250
$438$	−1533.11	−0.167248
$439$	−3650.27	−0.396852	−0.198426	−	0.980116i	$-0.563583\pi$
$439$	−3650.27	−0.396852	−0.198426	+	0.980116i	$0.563583\pi$
$440$	1836.01	0.198928
$441$	0	0
$442$	11876.6	1.27809
$443$	5859.41	0.628418	0.314209	−	0.949354i	$-0.398261\pi$
$443$	5859.41	0.628418	0.314209	+	0.949354i	$0.398261\pi$
$444$	1662.19	0.177667
$445$	10025.5	1.06798
$446$	4593.93	0.487733
$447$	1155.74	0.122292
$448$	0	0
$449$	1296.91	0.136314	0.0681568	−	0.997675i	$-0.478288\pi$
$449$	1296.91	0.136314	0.0681568	+	0.997675i	$0.478288\pi$
$450$	−18564.2	−1.94472
$451$	364.318	0.0380379
$452$	5507.35	0.573107
$453$	1115.16	0.115662
$454$	2017.32	0.208540
$455$	0	0
$456$	100.825	0.0103543
$457$	769.167	0.0787311	0.0393656	−	0.999225i	$-0.487466\pi$
$457$	769.167	0.0787311	0.0393656	+	0.999225i	$0.487466\pi$
$458$	−11646.9	−1.18826
$459$	−4249.83	−0.432168
$460$	−25171.5	−2.55136
$461$	−13267.7	−1.34043	−0.670214	−	0.742168i	$-0.733798\pi$
$461$	−13267.7	−1.34043	−0.670214	+	0.742168i	$0.733798\pi$
$462$	0	0
$463$	−19255.4	−1.93278	−0.966389	−	0.257082i	$-0.917239\pi$
$463$	−19255.4	−1.93278	−0.966389	+	0.257082i	$0.917239\pi$
$464$	7742.36	0.774634
$465$	4606.65	0.459415
$466$	11103.5	1.10378
$467$	3728.75	0.369478	0.184739	−	0.982788i	$-0.440856\pi$
$467$	3728.75	0.369478	0.184739	+	0.982788i	$0.440856\pi$
$468$	6836.16	0.675218
$469$	0	0
$470$	27970.1	2.74503
$471$	−939.949	−0.0919545
$472$	469.108	0.0457467
$473$	−2778.09	−0.270056
$474$	1438.46	0.139390
$475$	−12719.4	−1.22864
$476$	0	0
$477$	8217.68	0.788808
$478$	−13812.7	−1.32171
$479$	8466.64	0.807621	0.403811	−	0.914843i	$-0.367685\pi$
$479$	8466.64	0.807621	0.403811	+	0.914843i	$0.367685\pi$
$480$	3763.18	0.357843
$481$	7271.19	0.689268
$482$	−14964.5	−1.41413
$483$	0	0
$484$	23897.2	2.24429
$485$	−10088.0	−0.944477
$486$	−7193.25	−0.671384
$487$	−639.778	−0.0595300	−0.0297650	−	0.999557i	$-0.509476\pi$
$487$	−639.778	−0.0595300	−0.0297650	+	0.999557i	$0.509476\pi$
$488$	1167.07	0.108260
$489$	−74.5756	−0.00689658
$490$	0	0
$491$	−1759.02	−0.161677	−0.0808386	−	0.996727i	$-0.525760\pi$
$491$	−1759.02	−0.161677	−0.0808386	+	0.996727i	$0.525760\pi$
$492$	−39.8746	−0.00365384
$493$	12112.8	1.10655
$494$	9141.72	0.832602
$495$	−29388.4	−2.66850
$496$	19177.7	1.73610
$497$	0	0
$498$	1955.50	0.175960
$499$	−1207.88	−0.108361	−0.0541804	−	0.998531i	$-0.517255\pi$
$499$	−1207.88	−0.108361	−0.0541804	+	0.998531i	$0.517255\pi$
$500$	7172.57	0.641534
$501$	1230.01	0.109686
$502$	−769.636	−0.0684273
$503$	7241.51	0.641915	0.320957	−	0.947094i	$-0.395995\pi$
$503$	7241.51	0.641915	0.320957	+	0.947094i	$0.395995\pi$
$504$	0	0
$505$	30416.4	2.68022
$506$	−45294.6	−3.97943
$507$	1043.30	0.0913893
$508$	11789.9	1.02971
$509$	20876.5	1.81795	0.908973	−	0.416855i	$-0.136868\pi$
$509$	20876.5	1.81795	0.908973	+	0.416855i	$0.136868\pi$
$510$	5588.20	0.485195
$511$	0	0
$512$	16462.5	1.42099
$513$	−3271.19	−0.281533
$514$	−3898.28	−0.334525
$515$	−19486.9	−1.66737
$516$	304.062	0.0259410
$517$	25787.4	2.19367
$518$	0	0
$519$	314.845	0.0266285
$520$	−879.054	−0.0741328
$521$	4680.33	0.393568	0.196784	−	0.980447i	$-0.436950\pi$
$521$	4680.33	0.393568	0.196784	+	0.980447i	$0.436950\pi$
$522$	13607.8	1.14099
$523$	22978.4	1.92118	0.960589	−	0.277974i	$-0.0896630\pi$
$523$	22978.4	1.92118	0.960589	+	0.277974i	$0.0896630\pi$
$524$	20308.2	1.69307
$525$	0	0
$526$	−27163.1	−2.25165
$527$	30003.2	2.48000
$528$	3293.18	0.271434
$529$	17793.4	1.46243
$530$	−21902.1	−1.79503
$531$	−7508.86	−0.613666
$532$	0	0
$533$	−174.430	−0.0141753
$534$	−1974.51	−0.160010
$535$	−18850.8	−1.52335
$536$	610.803	0.0492214
$537$	3622.19	0.291079
$538$	−1945.23	−0.155883
$539$	0	0
$540$	6519.69	0.519561
$541$	−12685.1	−1.00809	−0.504045	−	0.863678i	$-0.668155\pi$
$541$	−12685.1	−1.00809	−0.504045	+	0.863678i	$0.668155\pi$
$542$	12589.2	0.997697
$543$	−3439.97	−0.271866
$544$	24509.7	1.93170
$545$	−4111.75	−0.323171
$546$	0	0
$547$	−12493.5	−0.976573	−0.488286	−	0.872683i	$-0.662378\pi$
$547$	−12493.5	−0.976573	−0.488286	+	0.872683i	$0.662378\pi$
$548$	−20147.4	−1.57054
$549$	−18680.9	−1.45225
$550$	45616.8	3.53656
$551$	9323.47	0.720859
$552$	239.182	0.0184425
$553$	0	0
$554$	−12881.1	−0.987846
$555$	3421.24	0.261664
$556$	17016.3	1.29793
$557$	154.876	0.0117815	0.00589076	−	0.999983i	$-0.498125\pi$
$557$	154.876	0.0117815	0.00589076	+	0.999983i	$0.498125\pi$
$558$	33706.3	2.55717
$559$	1330.11	0.100640
$560$	0	0
$561$	5152.11	0.387740
$562$	25610.9	1.92229
$563$	−23173.9	−1.73475	−0.867376	−	0.497654i	$-0.834195\pi$
$563$	−23173.9	−1.73475	−0.867376	+	0.497654i	$0.834195\pi$
$564$	−2822.43	−0.210719
$565$	11335.7	0.844061
$566$	993.573	0.0737862
$567$	0	0
$568$	−305.997	−0.0226045
$569$	10634.9	0.783550	0.391775	−	0.920061i	$-0.371861\pi$
$569$	10634.9	0.783550	0.391775	+	0.920061i	$0.371861\pi$
$570$	4301.36	0.316078
$571$	−2496.20	−0.182947	−0.0914736	−	0.995808i	$-0.529158\pi$
$571$	−2496.20	−0.182947	−0.0914736	+	0.995808i	$0.529158\pi$
$572$	−16798.2	−1.22791
$573$	−3419.02	−0.249270
$574$	0	0
$575$	−30173.5	−2.18839
$576$	14790.1	1.06989
$577$	−8480.62	−0.611877	−0.305938	−	0.952051i	$-0.598970\pi$
$577$	−8480.62	−0.611877	−0.305938	+	0.952051i	$0.598970\pi$
$578$	16496.5	1.18714
$579$	−839.065	−0.0602251
$580$	−18582.3	−1.33032
$581$	0	0
$582$	1986.82	0.141505
$583$	−20192.9	−1.43448
$584$	739.056	0.0523670
$585$	14070.7	0.994449
$586$	−20615.4	−1.45327
$587$	8946.85	0.629090	0.314545	−	0.949243i	$-0.398148\pi$
$587$	8946.85	0.629090	0.314545	+	0.949243i	$0.398148\pi$
$588$	0	0
$589$	23094.1	1.61558
$590$	20012.9	1.39647
$591$	−86.5549	−0.00602435
$592$	14242.8	0.988813
$593$	1302.90	0.0902255	0.0451128	−	0.998982i	$-0.485635\pi$
$593$	1302.90	0.0902255	0.0451128	+	0.998982i	$0.485635\pi$
$594$	11731.8	0.810374
$595$	0	0
$596$	−11547.8	−0.793649
$597$	1038.37	0.0711853
$598$	21686.4	1.48298
$599$	−21831.6	−1.48918	−0.744588	−	0.667524i	$-0.767354\pi$
$599$	−21831.6	−1.48918	−0.744588	+	0.667524i	$0.767354\pi$
$600$	−240.884	−0.0163901
$601$	−22578.1	−1.53241	−0.766207	−	0.642593i	$-0.777858\pi$
$601$	−22578.1	−1.53241	−0.766207	+	0.642593i	$0.777858\pi$
$602$	0	0
$603$	−9776.92	−0.660277
$604$	−11142.3	−0.750620
$605$	49187.0	3.30535
$606$	−5990.48	−0.401562
$607$	−25653.1	−1.71537	−0.857684	−	0.514177i	$-0.828097\pi$
$607$	−25653.1	−1.71537	−0.857684	+	0.514177i	$0.828097\pi$
$608$	18865.6	1.25839
$609$	0	0
$610$	49789.2	3.30476
$611$	−12346.6	−0.817497
$612$	20949.5	1.38371
$613$	14381.0	0.947540	0.473770	−	0.880649i	$-0.342893\pi$
$613$	14381.0	0.947540	0.473770	+	0.880649i	$0.342893\pi$
$614$	−33284.2	−2.18769
$615$	−82.0730	−0.00538131
$616$	0	0
$617$	−17361.8	−1.13283	−0.566417	−	0.824119i	$-0.691671\pi$
$617$	−17361.8	−1.13283	−0.566417	+	0.824119i	$0.691671\pi$
$618$	3837.92	0.249812
$619$	10690.1	0.694140	0.347070	−	0.937839i	$-0.387177\pi$
$619$	10690.1	0.694140	0.347070	+	0.937839i	$0.387177\pi$
$620$	−46028.0	−2.98150
$621$	−7760.08	−0.501451
$622$	−39386.4	−2.53899
$623$	0	0
$624$	−1576.73	−0.101153
$625$	−7027.11	−0.449735
$626$	33107.9	2.11383
$627$	3965.70	0.252591
$628$	9391.65	0.596764
$629$	22282.6	1.41251
$630$	0	0
$631$	−20644.3	−1.30243	−0.651217	−	0.758891i	$-0.725741\pi$
$631$	−20644.3	−1.30243	−0.651217	+	0.758891i	$0.725741\pi$
$632$	−693.430	−0.0436442
$633$	−828.436	−0.0520180
$634$	−36654.0	−2.29608
$635$	24267.0	1.51654
$636$	2210.11	0.137793
$637$	0	0
$638$	−33437.7	−2.07494
$639$	4897.99	0.303226
$640$	−3632.86	−0.224377
$641$	−9791.15	−0.603318	−0.301659	−	0.953416i	$-0.597540\pi$
$641$	−9791.15	−0.603318	−0.301659	+	0.953416i	$0.597540\pi$
$642$	3712.64	0.228234
$643$	15261.3	0.935998	0.467999	−	0.883729i	$-0.344975\pi$
$643$	15261.3	0.935998	0.467999	+	0.883729i	$0.344975\pi$
$644$	0	0
$645$	625.843	0.0382055
$646$	28014.9	1.70624
$647$	15280.9	0.928521	0.464260	−	0.885699i	$-0.346320\pi$
$647$	15280.9	0.928521	0.464260	+	0.885699i	$0.346320\pi$
$648$	1104.11	0.0669343
$649$	18451.1	1.11598
$650$	−21840.7	−1.31794
$651$	0	0
$652$	745.134	0.0447572
$653$	9070.69	0.543589	0.271795	−	0.962355i	$-0.412383\pi$
$653$	9070.69	0.543589	0.271795	+	0.962355i	$0.412383\pi$
$654$	809.805	0.0484188
$655$	41800.0	2.49353
$656$	−341.675	−0.0203356
$657$	−11829.8	−0.702474
$658$	0	0
$659$	14157.3	0.836858	0.418429	−	0.908250i	$-0.362581\pi$
$659$	14157.3	0.836858	0.418429	+	0.908250i	$0.362581\pi$
$660$	−7903.88	−0.466149
$661$	9472.21	0.557377	0.278689	−	0.960382i	$-0.410100\pi$
$661$	9472.21	0.557377	0.278689	+	0.960382i	$0.410100\pi$
$662$	−8274.98	−0.485825
$663$	−2466.75	−0.144496
$664$	−942.674	−0.0550947
$665$	0	0
$666$	25032.9	1.45646
$667$	22117.6	1.28395
$668$	−12289.8	−0.711839
$669$	−954.150	−0.0551414
$670$	26057.8	1.50254
$671$	45903.7	2.64097
$672$	0	0
$673$	−13128.6	−0.751964	−0.375982	−	0.926627i	$-0.622694\pi$
$673$	−13128.6	−0.751964	−0.375982	+	0.926627i	$0.622694\pi$
$674$	−6246.75	−0.356997
$675$	7815.27	0.445644
$676$	−10424.3	−0.593096
$677$	−9742.12	−0.553058	−0.276529	−	0.961006i	$-0.589184\pi$
$677$	−9742.12	−0.553058	−0.276529	+	0.961006i	$0.589184\pi$
$678$	−2232.55	−0.126461
$679$	0	0
$680$	−2693.87	−0.151919
$681$	−418.993	−0.0235768
$682$	−82824.9	−4.65033
$683$	1110.24	0.0621992	0.0310996	−	0.999516i	$-0.490099\pi$
$683$	1110.24	0.0621992	0.0310996	+	0.999516i	$0.490099\pi$
$684$	16125.3	0.901413
$685$	−41469.0	−2.31306
$686$	0	0
$687$	2419.03	0.134340
$688$	2605.42	0.144376
$689$	9668.06	0.534577
$690$	10203.9	0.562979
$691$	16348.1	0.900015	0.450007	−	0.893025i	$-0.351421\pi$
$691$	16348.1	0.900015	0.450007	+	0.893025i	$0.351421\pi$
$692$	−3145.83	−0.172813
$693$	0	0
$694$	50838.7	2.78071
$695$	35024.2	1.91158
$696$	176.571	0.00961624
$697$	−534.543	−0.0290492
$698$	−52199.4	−2.83063
$699$	−2306.18	−0.124789
$700$	0	0
$701$	2514.17	0.135462	0.0677309	−	0.997704i	$-0.478424\pi$
$701$	2514.17	0.135462	0.0677309	+	0.997704i	$0.478424\pi$
$702$	−5617.02	−0.301995
$703$	17151.5	0.920170
$704$	−36343.0	−1.94564
$705$	−5809.34	−0.310344
$706$	38678.9	2.06190
$707$	0	0
$708$	−2019.48	−0.107199
$709$	33318.6	1.76489	0.882444	−	0.470417i	$-0.155896\pi$
$709$	33318.6	1.76489	0.882444	+	0.470417i	$0.155896\pi$
$710$	−13054.3	−0.690028
$711$	11099.5	0.585462
$712$	951.839	0.0501007
$713$	54785.0	2.87758
$714$	0	0
$715$	−34575.3	−1.80845
$716$	−36191.7	−1.88903
$717$	2868.87	0.149428
$718$	−28712.7	−1.49241
$719$	25807.4	1.33860	0.669300	−	0.742993i	$-0.266594\pi$
$719$	25807.4	1.33860	0.669300	+	0.742993i	$0.266594\pi$
$720$	27561.8	1.42662
$721$	0	0
$722$	−6217.83	−0.320504
$723$	3108.09	0.159877
$724$	34371.0	1.76435
$725$	−22274.9	−1.14106
$726$	−9687.33	−0.495221
$727$	1433.87	0.0731490	0.0365745	−	0.999331i	$-0.488355\pi$
$727$	1433.87	0.0731490	0.0365745	+	0.999331i	$0.488355\pi$
$728$	0	0
$729$	−16654.7	−0.846148
$730$	31529.3	1.59856
$731$	4076.13	0.206239
$732$	−5024.16	−0.253686
$733$	−5606.65	−0.282519	−0.141260	−	0.989973i	$-0.545115\pi$
$733$	−5606.65	−0.282519	−0.141260	+	0.989973i	$0.545115\pi$
$734$	3709.47	0.186538
$735$	0	0
$736$	44754.0	2.24138
$737$	24024.3	1.20074
$738$	−600.519	−0.0299531
$739$	25503.3	1.26949	0.634745	−	0.772722i	$-0.281105\pi$
$739$	25503.3	1.26949	0.634745	+	0.772722i	$0.281105\pi$
$740$	−34183.9	−1.69814
$741$	−1898.72	−0.0941311
$742$	0	0
$743$	−15649.4	−0.772705	−0.386353	−	0.922351i	$-0.626265\pi$
$743$	−15649.4	−0.772705	−0.386353	+	0.922351i	$0.626265\pi$
$744$	437.364	0.0215518
$745$	−23768.5	−1.16887
$746$	−51963.9	−2.55031
$747$	15089.1	0.739063
$748$	−51478.1	−2.51635
$749$	0	0
$750$	−2907.58	−0.141560
$751$	9849.49	0.478579	0.239290	−	0.970948i	$-0.423085\pi$
$751$	9849.49	0.478579	0.239290	+	0.970948i	$0.423085\pi$
$752$	−24184.6	−1.17277
$753$	159.852	0.00773616
$754$	16009.5	0.773251
$755$	−22934.0	−1.10550
$756$	0	0
$757$	1284.51	0.0616728	0.0308364	−	0.999524i	$-0.490183\pi$
$757$	1284.51	0.0616728	0.0308364	+	0.999524i	$0.490183\pi$
$758$	−6288.02	−0.301308
$759$	9407.61	0.449901
$760$	−2073.53	−0.0989670
$761$	−21649.7	−1.03128	−0.515638	−	0.856806i	$-0.672445\pi$
$761$	−21649.7	−1.03128	−0.515638	+	0.856806i	$0.672445\pi$
$762$	−4779.36	−0.227215
$763$	0	0
$764$	34161.7	1.61771
$765$	43119.8	2.03791
$766$	52858.6	2.49329
$767$	−8834.13	−0.415883
$768$	−3070.34	−0.144260
$769$	8950.61	0.419723	0.209862	−	0.977731i	$-0.432699\pi$
$769$	8950.61	0.419723	0.209862	+	0.977731i	$0.432699\pi$
$770$	0	0
$771$	809.665	0.0378202
$772$	8383.65	0.390848
$773$	−19897.2	−0.925810	−0.462905	−	0.886408i	$-0.653193\pi$
$773$	−19897.2	−0.925810	−0.462905	+	0.886408i	$0.653193\pi$
$774$	4579.23	0.212657
$775$	−55174.7	−2.55733
$776$	−957.772	−0.0443067
$777$	0	0
$778$	−25168.6	−1.15982
$779$	−411.450	−0.0189239
$780$	3784.26	0.173716
$781$	−12035.6	−0.551430
$782$	66458.2	3.03905
$783$	−5728.70	−0.261465
$784$	0	0
$785$	19330.6	0.878904
$786$	−8232.46	−0.373591
$787$	−19833.0	−0.898310	−0.449155	−	0.893454i	$-0.648275\pi$
$787$	−19833.0	−0.898310	−0.449155	+	0.893454i	$0.648275\pi$
$788$	864.827	0.0390967
$789$	5641.72	0.254563
$790$	−29582.8	−1.33229
$791$	0	0
$792$	−2790.19	−0.125183
$793$	−21978.0	−0.984190
$794$	−44003.4	−1.96678
$795$	4549.02	0.202940
$796$	−10375.0	−0.461977
$797$	−4235.19	−0.188229	−0.0941143	−	0.995561i	$-0.530002\pi$
$797$	−4235.19	−0.188229	−0.0941143	+	0.995561i	$0.530002\pi$
$798$	0	0
$799$	−37836.3	−1.67529
$800$	−45072.3	−1.99193
$801$	−15235.8	−0.672072
$802$	46539.0	2.04906
$803$	29068.8	1.27748
$804$	−2629.46	−0.115341
$805$	0	0
$806$	39655.3	1.73300
$807$	404.021	0.0176236
$808$	2887.79	0.125733
$809$	40635.0	1.76595	0.882974	−	0.469423i	$-0.155538\pi$
$809$	40635.0	1.76595	0.882974	+	0.469423i	$0.155538\pi$
$810$	47102.9	2.04324
$811$	42890.6	1.85708	0.928541	−	0.371229i	$-0.121064\pi$
$811$	42890.6	1.85708	0.928541	+	0.371229i	$0.121064\pi$
$812$	0	0
$813$	−2614.75	−0.112796
$814$	−61512.0	−2.64864
$815$	1533.69	0.0659177
$816$	−4831.89	−0.207292
$817$	3137.49	0.134354
$818$	22515.0	0.962368
$819$	0	0
$820$	820.046	0.0349235
$821$	41795.4	1.77670	0.888349	−	0.459168i	$-0.151853\pi$
$821$	41795.4	1.77670	0.888349	+	0.459168i	$0.151853\pi$
$822$	8167.28	0.346553
$823$	−21149.8	−0.895790	−0.447895	−	0.894086i	$-0.647826\pi$
$823$	−21149.8	−0.895790	−0.447895	+	0.894086i	$0.647826\pi$
$824$	−1850.12	−0.0782186
$825$	−9474.52	−0.399831
$826$	0	0
$827$	−30183.5	−1.26915	−0.634573	−	0.772863i	$-0.718824\pi$
$827$	−30183.5	−1.26915	−0.634573	+	0.772863i	$0.718824\pi$
$828$	38253.2	1.60554
$829$	−15114.7	−0.633240	−0.316620	−	0.948552i	$-0.602548\pi$
$829$	−15114.7	−0.633240	−0.316620	+	0.948552i	$0.602548\pi$
$830$	−40215.9	−1.68183
$831$	2675.39	0.111682
$832$	17400.5	0.725064
$833$	0	0
$834$	−6897.99	−0.286400
$835$	−25295.9	−1.04838
$836$	−39623.9	−1.63926
$837$	−14189.9	−0.585992
$838$	−48699.1	−2.00750
$839$	−6629.48	−0.272795	−0.136398	−	0.990654i	$-0.543552\pi$
$839$	−6629.48	−0.272795	−0.136398	+	0.990654i	$0.543552\pi$
$840$	0	0
$841$	−8061.20	−0.330526
$842$	43885.2	1.79618
$843$	−5319.33	−0.217328
$844$	8277.45	0.337585
$845$	−21456.0	−0.873501
$846$	−42506.3	−1.72742
$847$	0	0
$848$	18937.8	0.766896
$849$	−206.363	−0.00834201
$850$	−66930.9	−2.70084
$851$	40687.5	1.63895
$852$	1317.29	0.0529692
$853$	−29206.5	−1.17235	−0.586174	−	0.810185i	$-0.699366\pi$
$853$	−29206.5	−1.17235	−0.586174	+	0.810185i	$0.699366\pi$
$854$	0	0
$855$	33190.3	1.32759
$856$	−1789.73	−0.0714623
$857$	−15223.8	−0.606811	−0.303405	−	0.952862i	$-0.598124\pi$
$857$	−15223.8	−0.606811	−0.303405	+	0.952862i	$0.598124\pi$
$858$	6809.56	0.270949
$859$	13261.7	0.526758	0.263379	−	0.964692i	$-0.415163\pi$
$859$	13261.7	0.526758	0.263379	+	0.964692i	$0.415163\pi$
$860$	−6253.21	−0.247945
$861$	0	0
$862$	−8656.02	−0.342025
$863$	11695.6	0.461323	0.230662	−	0.973034i	$-0.425911\pi$
$863$	11695.6	0.461323	0.230662	+	0.973034i	$0.425911\pi$
$864$	−11591.8	−0.456435
$865$	−6474.99	−0.254516
$866$	−28619.9	−1.12303
$867$	−3426.30	−0.134214
$868$	0	0
$869$	−27274.2	−1.06469
$870$	7532.79	0.293547
$871$	−11502.5	−0.447471
$872$	−390.378	−0.0151604
$873$	15330.7	0.594349
$874$	51154.4	1.97978
$875$	0	0
$876$	−3181.58	−0.122712
$877$	−14346.1	−0.552377	−0.276189	−	0.961103i	$-0.589071\pi$
$877$	−14346.1	−0.552377	−0.276189	+	0.961103i	$0.589071\pi$
$878$	−14785.0	−0.568301
$879$	4281.78	0.164301
$880$	−67726.2	−2.59437
$881$	−6815.99	−0.260655	−0.130327	−	0.991471i	$-0.541603\pi$
$881$	−6815.99	−0.260655	−0.130327	+	0.991471i	$0.541603\pi$
$882$	0	0
$883$	−89.5757	−0.00341389	−0.00170694	−	0.999999i	$-0.500543\pi$
$883$	−89.5757	−0.00341389	−0.00170694	+	0.999999i	$0.500543\pi$
$884$	24647.0	0.937746
$885$	−4156.64	−0.157880
$886$	23732.8	0.899909
$887$	34619.3	1.31049	0.655243	−	0.755418i	$-0.272566\pi$
$887$	34619.3	1.31049	0.655243	+	0.755418i	$0.272566\pi$
$888$	324.820	0.0122750
$889$	0	0
$890$	40606.9	1.52938
$891$	43427.1	1.63284
$892$	9533.55	0.357855
$893$	−29123.5	−1.09136
$894$	4681.18	0.175125
$895$	−74492.6	−2.78214
$896$	0	0
$897$	−4504.23	−0.167661
$898$	5252.96	0.195204
$899$	40443.8	1.50042
$900$	−38525.3	−1.42686
$901$	29627.8	1.09550
$902$	1475.63	0.0544711
$903$	0	0
$904$	1076.23	0.0395961
$905$	70745.1	2.59850
$906$	4516.82	0.165631
$907$	−20696.1	−0.757664	−0.378832	−	0.925465i	$-0.623674\pi$
$907$	−20696.1	−0.757664	−0.378832	+	0.925465i	$0.623674\pi$
$908$	4186.43	0.153008
$909$	−46223.9	−1.68663
$910$	0	0
$911$	34194.3	1.24359	0.621793	−	0.783182i	$-0.286405\pi$
$911$	34194.3	1.24359	0.621793	+	0.783182i	$0.286405\pi$
$912$	−3719.22	−0.135039
$913$	−37077.6	−1.34402
$914$	3115.41	0.112745
$915$	−10341.1	−0.373625
$916$	−24170.1	−0.871838
$917$	0	0
$918$	−17213.4	−0.618875
$919$	−11276.2	−0.404751	−0.202375	−	0.979308i	$-0.564866\pi$
$919$	−11276.2	−0.404751	−0.202375	+	0.979308i	$0.564866\pi$
$920$	−4918.93	−0.176274
$921$	6913.07	0.247333
$922$	−53739.1	−1.91952
$923$	5762.46	0.205497
$924$	0	0
$925$	−40976.9	−1.45655
$926$	−77991.8	−2.76779
$927$	29614.3	1.04926
$928$	33038.6	1.16869
$929$	10671.1	0.376865	0.188433	−	0.982086i	$-0.439659\pi$
$929$	10671.1	0.376865	0.188433	+	0.982086i	$0.439659\pi$
$930$	18658.6	0.657894
$931$	0	0
$932$	23042.6	0.809855
$933$	8180.48	0.287049
$934$	15102.8	0.529101
$935$	−105956.	−3.70603
$936$	1335.90	0.0466510
$937$	−24613.4	−0.858147	−0.429073	−	0.903270i	$-0.641160\pi$
$937$	−24613.4	−0.858147	−0.429073	+	0.903270i	$0.641160\pi$
$938$	0	0
$939$	−6876.45	−0.238982
$940$	58044.9	2.01406
$941$	−23993.5	−0.831206	−0.415603	−	0.909546i	$-0.636429\pi$
$941$	−23993.5	−0.831206	−0.415603	+	0.909546i	$0.636429\pi$
$942$	−3807.15	−0.131681
$943$	−976.062	−0.0337062
$944$	−17304.3	−0.596619
$945$	0	0
$946$	−11252.3	−0.386727
$947$	32980.7	1.13171	0.565856	−	0.824504i	$-0.308546\pi$
$947$	32980.7	1.13171	0.565856	+	0.824504i	$0.308546\pi$
$948$	2985.16	0.102272
$949$	−13917.7	−0.476068
$950$	−51518.3	−1.75945
$951$	7612.97	0.259587
$952$	0	0
$953$	30764.0	1.04569	0.522846	−	0.852427i	$-0.324870\pi$
$953$	30764.0	1.04569	0.522846	+	0.852427i	$0.324870\pi$
$954$	33284.7	1.12959
$955$	70314.3	2.38253
$956$	−28664.8	−0.969755
$957$	6944.95	0.234586
$958$	34293.1	1.15653
$959$	0	0
$960$	8187.29	0.275254
$961$	70387.7	2.36272
$962$	29451.1	0.987048
$963$	28647.6	0.958626
$964$	−31055.0	−1.03757
$965$	17255.9	0.575634
$966$	0	0
$967$	−18108.4	−0.602200	−0.301100	−	0.953593i	$-0.597354\pi$
$967$	−18108.4	−0.602200	−0.301100	+	0.953593i	$0.597354\pi$
$968$	4669.91	0.155059
$969$	−5818.64	−0.192901
$970$	−40860.1	−1.35251
$971$	20700.5	0.684151	0.342076	−	0.939672i	$-0.388870\pi$
$971$	20700.5	0.684151	0.342076	+	0.939672i	$0.388870\pi$
$972$	−14927.8	−0.492602
$973$	0	0
$974$	−2591.34	−0.0852484
$975$	4536.26	0.149002
$976$	−43050.7	−1.41190
$977$	−35494.5	−1.16230	−0.581151	−	0.813796i	$-0.697397\pi$
$977$	−35494.5	−1.16230	−0.581151	+	0.813796i	$0.697397\pi$
$978$	−302.059	−0.00987606
$979$	37438.1	1.22219
$980$	0	0
$981$	6248.64	0.203368
$982$	−7124.70	−0.231526
$983$	25466.3	0.826296	0.413148	−	0.910664i	$-0.364429\pi$
$983$	25466.3	0.826296	0.413148	+	0.910664i	$0.364429\pi$
$984$	−7.79218	−0.000252445 0
$985$	1780.05	0.0575809
$986$	49061.2	1.58461
$987$	0	0
$988$	18971.3	0.610889
$989$	7442.90	0.239303
$990$	−119034.	−3.82136
$991$	−18966.3	−0.607956	−0.303978	−	0.952679i	$-0.598315\pi$
$991$	−18966.3	−0.607956	−0.303978	+	0.952679i	$0.598315\pi$
$992$	81836.2	2.61926
$993$	1718.70	0.0549257
$994$	0	0
$995$	−21354.7	−0.680391
$996$	4058.14	0.129104
$997$	50589.7	1.60701	0.803507	−	0.595295i	$-0.202965\pi$
$997$	50589.7	1.60701	0.803507	+	0.595295i	$0.202965\pi$
$998$	−4892.36	−0.155175
$999$	−10538.5	−0.333758

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2107.4.a.h.1.27	✓	30
7.6	odd	2	inner	2107.4.a.h.1.28	yes	30

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2107.4.a.h.1.27	✓	30	1.1	even	1	trivial
2107.4.a.h.1.28	yes	30	7.6	odd	2	inner

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:	\( N \)	\(=\)	\( 2107 = 7^{2} \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2107.a (trivial)

\(f(q)\)	\(=\)	\(q+4.05038 q^{2} -0.841255 q^{3} +8.40554 q^{4} +17.3009 q^{5} -3.40740 q^{6} +1.64258 q^{8} -26.2923 q^{9} +O(q^{10})\)	\(q+4.05038 q^{2} -0.841255 q^{3} +8.40554 q^{4} +17.3009 q^{5} -3.40740 q^{6} +1.64258 q^{8} -26.2923 q^{9} +70.0752 q^{10} +64.6067 q^{11} -7.07120 q^{12} -30.9328 q^{13} -14.5545 q^{15} -60.5912 q^{16} -94.7937 q^{17} -106.494 q^{18} -72.9649 q^{19} +145.424 q^{20} +261.681 q^{22} -173.091 q^{23} -1.38183 q^{24} +174.322 q^{25} -125.289 q^{26} +44.8324 q^{27} -127.780 q^{29} -58.9511 q^{30} -316.510 q^{31} -258.558 q^{32} -54.3507 q^{33} -383.950 q^{34} -221.001 q^{36} -235.065 q^{37} -295.535 q^{38} +26.0223 q^{39} +28.4182 q^{40} +5.63902 q^{41} -43.0000 q^{43} +543.054 q^{44} -454.881 q^{45} -701.083 q^{46} +399.144 q^{47} +50.9727 q^{48} +706.069 q^{50} +79.7457 q^{51} -260.006 q^{52} -312.551 q^{53} +181.588 q^{54} +1117.76 q^{55} +61.3821 q^{57} -517.558 q^{58} +285.592 q^{59} -122.338 q^{60} +710.510 q^{61} -1281.99 q^{62} -562.527 q^{64} -535.165 q^{65} -220.141 q^{66} +371.855 q^{67} -796.792 q^{68} +145.614 q^{69} -186.290 q^{71} -43.1873 q^{72} +449.935 q^{73} -952.100 q^{74} -146.649 q^{75} -613.309 q^{76} +105.400 q^{78} -422.158 q^{79} -1048.28 q^{80} +672.176 q^{81} +22.8401 q^{82} -573.897 q^{83} -1640.02 q^{85} -174.166 q^{86} +107.496 q^{87} +106.122 q^{88} +579.477 q^{89} -1842.44 q^{90} -1454.92 q^{92} +266.266 q^{93} +1616.68 q^{94} -1262.36 q^{95} +217.513 q^{96} -583.089 q^{97} -1698.66 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 30 q - 8 q^{2} + 100 q^{4} - 90 q^{8} + 264 q^{9}+O(q^{10}) \) 30 * q - 8 * q^2 + 100 * q^4 - 90 * q^8 + 264 * q^9	\( 30 q - 8 q^{2} + 100 q^{4} - 90 q^{8} + 264 q^{9} - 60 q^{11} - 96 q^{15} + 372 q^{16} - 1008 q^{18} - 234 q^{22} - 214 q^{23} + 520 q^{25} - 870 q^{29} - 12 q^{30} - 1548 q^{32} + 1142 q^{36} - 1246 q^{37} - 416 q^{39} - 1290 q^{43} - 446 q^{44} - 660 q^{46} - 278 q^{50} - 3702 q^{51} - 2960 q^{53} + 620 q^{57} - 3634 q^{58} - 898 q^{60} + 2578 q^{64} - 4848 q^{65} + 928 q^{67} - 1708 q^{71} - 7900 q^{72} + 1714 q^{74} - 138 q^{78} - 3562 q^{79} + 2210 q^{81} - 948 q^{85} + 344 q^{86} + 2502 q^{88} - 3848 q^{92} - 11986 q^{93} - 2894 q^{95} - 2804 q^{99}+O(q^{100}) \) 30 * q - 8 * q^2 + 100 * q^4 - 90 * q^8 + 264 * q^9 - 60 * q^11 - 96 * q^15 + 372 * q^16 - 1008 * q^18 - 234 * q^22 - 214 * q^23 + 520 * q^25 - 870 * q^29 - 12 * q^30 - 1548 * q^32 + 1142 * q^36 - 1246 * q^37 - 416 * q^39 - 1290 * q^43 - 446 * q^44 - 660 * q^46 - 278 * q^50 - 3702 * q^51 - 2960 * q^53 + 620 * q^57 - 3634 * q^58 - 898 * q^60 + 2578 * q^64 - 4848 * q^65 + 928 * q^67 - 1708 * q^71 - 7900 * q^72 + 1714 * q^74 - 138 * q^78 - 3562 * q^79 + 2210 * q^81 - 948 * q^85 + 344 * q^86 + 2502 * q^88 - 3848 * q^92 - 11986 * q^93 - 2894 * q^95 - 2804 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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