LMFDB - Embedded newform 2401.4.a.c.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2401,4,Mod(1,2401)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2401.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2401, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 2401 = 7^{4} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2401.a (trivial)

Newform invariants

comment:select newform

sage:traces = [39,1,-1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

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

Self dual:	yes
Analytic conductor:	$141.663585924$
Analytic rank:	$1$
Dimension:	$39$
Twist minimal:	no (minimal twist has level 49)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Character	$\chi$	$=$	2401.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-5.14785 q^{2} -6.96013 q^{3} +18.5003 q^{4} -3.24701 q^{5} +35.8297 q^{6} -54.0542 q^{8} +21.4435 q^{9} +16.7151 q^{10} +58.8488 q^{11} -128.765 q^{12} -47.9754 q^{13} +22.5996 q^{15} +130.260 q^{16} +84.9507 q^{17} -110.388 q^{18} -54.8763 q^{19} -60.0707 q^{20} -302.944 q^{22} +150.540 q^{23} +376.225 q^{24} -114.457 q^{25} +246.970 q^{26} +38.6742 q^{27} +44.2844 q^{29} -116.339 q^{30} -95.7503 q^{31} -238.126 q^{32} -409.595 q^{33} -437.313 q^{34} +396.712 q^{36} +209.605 q^{37} +282.495 q^{38} +333.915 q^{39} +175.514 q^{40} -354.943 q^{41} +221.030 q^{43} +1088.72 q^{44} -69.6271 q^{45} -774.956 q^{46} +235.194 q^{47} -906.628 q^{48} +589.207 q^{50} -591.268 q^{51} -887.561 q^{52} -508.784 q^{53} -199.089 q^{54} -191.082 q^{55} +381.947 q^{57} -227.969 q^{58} -439.523 q^{59} +418.100 q^{60} +51.4628 q^{61} +492.908 q^{62} +183.755 q^{64} +155.776 q^{65} +2108.53 q^{66} -263.337 q^{67} +1571.62 q^{68} -1047.78 q^{69} +203.727 q^{71} -1159.11 q^{72} +679.617 q^{73} -1079.02 q^{74} +796.636 q^{75} -1015.23 q^{76} -1718.94 q^{78} -189.020 q^{79} -422.955 q^{80} -848.151 q^{81} +1827.19 q^{82} -520.046 q^{83} -275.835 q^{85} -1137.83 q^{86} -308.225 q^{87} -3181.02 q^{88} -1494.87 q^{89} +358.430 q^{90} +2785.04 q^{92} +666.435 q^{93} -1210.74 q^{94} +178.184 q^{95} +1657.39 q^{96} -712.036 q^{97} +1261.92 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 39 q + q^{2} - q^{3} + 145 q^{4} - 27 q^{5} - 41 q^{6} - 12 q^{8} + 312 q^{9} - 78 q^{10} + q^{11} + 91 q^{12} - 77 q^{13} - 161 q^{15} + 461 q^{16} - 211 q^{17} + 8 q^{18} - 314 q^{19} - 476 q^{20}+ \cdots + 5384 q^{99}+O(q^{100}) $ 39 * q + q^2 - q^3 + 145 * q^4 - 27 * q^5 - 41 * q^6 - 12 * q^8 + 312 * q^9 - 78 * q^10 + q^11 + 91 * q^12 - 77 * q^13 - 161 * q^15 + 461 * q^16 - 211 * q^17 + 8 * q^18 - 314 * q^19 - 476 * q^20 - 61 * q^22 - 69 * q^23 - 330 * q^24 + 606 * q^25 - 504 * q^26 + 50 * q^27 + 57 * q^29 + 42 * q^30 - 638 * q^31 - 1600 * q^32 - 1574 * q^33 - 1343 * q^34 + 782 * q^36 + 71 * q^37 - 1359 * q^38 - 84 * q^39 + 155 * q^40 - 1393 * q^41 - 125 * q^43 + 52 * q^44 - 1129 * q^45 - 1454 * q^46 - 1483 * q^47 + 974 * q^48 + 3074 * q^50 - 2044 * q^51 - 3899 * q^52 + 2213 * q^53 - 1142 * q^54 - 1604 * q^55 + 98 * q^57 + 2403 * q^58 - 2073 * q^59 - 1519 * q^60 - 2575 * q^61 - 1742 * q^62 + 1358 * q^64 + 1876 * q^65 + 48 * q^66 + 176 * q^67 - 3038 * q^68 + 638 * q^69 - 1259 * q^71 - 3799 * q^72 + 307 * q^73 + 3845 * q^74 - 131 * q^75 - 1974 * q^76 - 6041 * q^78 + 22 * q^79 + 804 * q^80 + 795 * q^81 - 8043 * q^82 - 6349 * q^83 + 3094 * q^85 + 1745 * q^86 - 9508 * q^87 + 1299 * q^88 - 2253 * q^89 - 11156 * q^90 + 1284 * q^92 - 3430 * q^93 - 2738 * q^94 - 3290 * q^95 - 3031 * q^96 + 770 * q^97 + 5384 * q^99

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 $

$2$	−5.14785	−1.82004	−0.910020	−	0.414565i	$-0.863934\pi$
$2$	−5.14785	−1.82004	−0.910020	+	0.414565i	$0.863934\pi$
$3$	−6.96013	−1.33948	−0.669739	−	0.742596i	$-0.733594\pi$
$3$	−6.96013	−1.33948	−0.669739	+	0.742596i	$0.733594\pi$
$4$	18.5003	2.31254
$5$	−3.24701	−0.290421	−0.145210	−	0.989401i	$-0.546386\pi$
$5$	−3.24701	−0.290421	−0.145210	+	0.989401i	$0.546386\pi$
$6$	35.8297	2.43790
$7$	0	0
$7$	0	0
$8$	−54.0542	−2.38888
$9$	21.4435	0.794203
$10$	16.7151	0.528578
$11$	58.8488	1.61305	0.806526	−	0.591199i	$-0.201345\pi$
$11$	58.8488	1.61305	0.806526	+	0.591199i	$0.201345\pi$
$12$	−128.765	−3.09760
$13$	−47.9754	−1.02354	−0.511768	−	0.859123i	$-0.671009\pi$
$13$	−47.9754	−1.02354	−0.511768	+	0.859123i	$0.671009\pi$
$14$	0	0
$15$	22.5996	0.389013
$16$	130.260	2.03531
$17$	84.9507	1.21197	0.605987	−	0.795474i	$-0.292778\pi$
$17$	84.9507	1.21197	0.605987	+	0.795474i	$0.292778\pi$
$18$	−110.388	−1.44548
$19$	−54.8763	−0.662605	−0.331302	−	0.943525i	$-0.607488\pi$
$19$	−54.8763	−0.662605	−0.331302	+	0.943525i	$0.607488\pi$
$20$	−60.0707	−0.671611
$21$	0	0
$22$	−302.944	−2.93582
$23$	150.540	1.36477	0.682385	−	0.730993i	$-0.260943\pi$
$23$	150.540	1.36477	0.682385	+	0.730993i	$0.260943\pi$
$24$	376.225	3.19986
$25$	−114.457	−0.915656
$26$	246.970	1.86288
$27$	38.6742	0.275661
$28$	0	0
$29$	44.2844	0.283566	0.141783	−	0.989898i	$-0.454717\pi$
$29$	44.2844	0.283566	0.141783	+	0.989898i	$0.454717\pi$
$30$	−116.339	−0.708018
$31$	−95.7503	−0.554750	−0.277375	−	0.960762i	$-0.589464\pi$
$31$	−95.7503	−0.554750	−0.277375	+	0.960762i	$0.589464\pi$
$32$	−238.126	−1.31547
$33$	−409.595	−2.16065
$34$	−437.313	−2.20584
$35$	0	0
$36$	396.712	1.83663
$37$	209.605	0.931321	0.465661	−	0.884963i	$-0.345817\pi$
$37$	209.605	0.931321	0.465661	+	0.884963i	$0.345817\pi$
$38$	282.495	1.20597
$39$	333.915	1.37101
$40$	175.514	0.693781
$41$	−354.943	−1.35202	−0.676010	−	0.736893i	$-0.736292\pi$
$41$	−354.943	−1.35202	−0.676010	+	0.736893i	$0.736292\pi$
$42$	0	0
$43$	221.030	0.783877	0.391939	−	0.919991i	$-0.371805\pi$
$43$	221.030	0.783877	0.391939	+	0.919991i	$0.371805\pi$
$44$	1088.72	3.73025
$45$	−69.6271	−0.230653
$46$	−774.956	−2.48394
$47$	235.194	0.729928	0.364964	−	0.931022i	$-0.381081\pi$
$47$	235.194	0.729928	0.364964	+	0.931022i	$0.381081\pi$
$48$	−906.628	−2.72626
$49$	0	0
$50$	589.207	1.66653
$51$	−591.268	−1.62341
$52$	−887.561	−2.36697
$53$	−508.784	−1.31862	−0.659310	−	0.751871i	$-0.729151\pi$
$53$	−508.784	−1.31862	−0.659310	+	0.751871i	$0.729151\pi$
$54$	−199.089	−0.501714
$55$	−191.082	−0.468464
$56$	0	0
$57$	381.947	0.887545
$58$	−227.969	−0.516101
$59$	−439.523	−0.969847	−0.484924	−	0.874556i	$-0.661153\pi$
$59$	−439.523	−0.969847	−0.484924	+	0.874556i	$0.661153\pi$
$60$	418.100	0.899609
$61$	51.4628	0.108019	0.0540093	−	0.998540i	$-0.482800\pi$
$61$	51.4628	0.108019	0.0540093	+	0.998540i	$0.482800\pi$
$62$	492.908	1.00967
$63$	0	0
$64$	183.755	0.358896
$65$	155.776	0.297257
$66$	2108.53	3.93246
$67$	−263.337	−0.480175	−0.240087	−	0.970751i	$-0.577176\pi$
$67$	−263.337	−0.480175	−0.240087	+	0.970751i	$0.577176\pi$
$68$	1571.62	2.80274
$69$	−1047.78	−1.82808
$70$	0	0
$71$	203.727	0.340534	0.170267	−	0.985398i	$-0.445537\pi$
$71$	203.727	0.340534	0.170267	+	0.985398i	$0.445537\pi$
$72$	−1159.11	−1.89726
$73$	679.617	1.08963	0.544816	−	0.838556i	$-0.316600\pi$
$73$	679.617	1.08963	0.544816	+	0.838556i	$0.316600\pi$
$74$	−1079.02	−1.69504
$75$	796.636	1.22650
$76$	−1015.23	−1.53230
$77$	0	0
$78$	−1718.94	−2.49528
$79$	−189.020	−0.269196	−0.134598	−	0.990900i	$-0.542974\pi$
$79$	−189.020	−0.269196	−0.134598	+	0.990900i	$0.542974\pi$
$80$	−422.955	−0.591098
$81$	−848.151	−1.16344
$82$	1827.19	2.46073
$83$	−520.046	−0.687740	−0.343870	−	0.939017i	$-0.611738\pi$
$83$	−520.046	−0.687740	−0.343870	+	0.939017i	$0.611738\pi$
$84$	0	0
$85$	−275.835	−0.351983
$86$	−1137.83	−1.42669
$87$	−308.225	−0.379830
$88$	−3181.02	−3.85339
$89$	−1494.87	−1.78041	−0.890203	−	0.455565i	$-0.849437\pi$
$89$	−1494.87	−1.78041	−0.890203	+	0.455565i	$0.849437\pi$
$90$	358.430	0.419798
$91$	0	0
$92$	2785.04	3.15609
$93$	666.435	0.743076
$94$	−1210.74	−1.32850
$95$	178.184	0.192434
$96$	1657.39	1.76205
$97$	−712.036	−0.745322	−0.372661	−	0.927967i	$-0.621555\pi$
$97$	−712.036	−0.745322	−0.372661	+	0.927967i	$0.621555\pi$
$98$	0	0
$99$	1261.92	1.28109
$100$	−2117.49	−2.11749
$101$	−51.9916	−0.0512214	−0.0256107	−	0.999672i	$-0.508153\pi$
$101$	−51.9916	−0.0512214	−0.0256107	+	0.999672i	$0.508153\pi$
$102$	3043.76	2.95468
$103$	187.146	0.179030	0.0895149	−	0.995985i	$-0.471468\pi$
$103$	187.146	0.179030	0.0895149	+	0.995985i	$0.471468\pi$
$104$	2593.27	2.44511
$105$	0	0
$106$	2619.14	2.39994
$107$	626.000	0.565586	0.282793	−	0.959181i	$-0.408739\pi$
$107$	626.000	0.565586	0.282793	+	0.959181i	$0.408739\pi$
$108$	715.486	0.637478
$109$	−444.254	−0.390384	−0.195192	−	0.980765i	$-0.562533\pi$
$109$	−444.254	−0.390384	−0.195192	+	0.980765i	$0.562533\pi$
$110$	983.662	0.852623
$111$	−1458.88	−1.24748
$112$	0	0
$113$	−2043.53	−1.70123	−0.850616	−	0.525787i	$-0.823771\pi$
$113$	−2043.53	−1.70123	−0.850616	+	0.525787i	$0.823771\pi$
$114$	−1966.20	−1.61537
$115$	−488.803	−0.396358
$116$	819.276	0.655758
$117$	−1028.76	−0.812896
$118$	2262.60	1.76516
$119$	0	0
$120$	−1221.60	−0.929305
$121$	2132.18	1.60194
$122$	−264.923	−0.196598
$123$	2470.45	1.81100
$124$	−1771.41	−1.28288
$125$	777.518	0.556347
$126$	0	0
$127$	−36.3506	−0.0253984	−0.0126992	−	0.999919i	$-0.504042\pi$
$127$	−36.3506	−0.0253984	−0.0126992	+	0.999919i	$0.504042\pi$
$128$	959.065	0.662267
$129$	−1538.40	−1.04999
$130$	−801.913	−0.541019
$131$	−327.576	−0.218477	−0.109238	−	0.994016i	$-0.534841\pi$
$131$	−327.576	−0.218477	−0.109238	+	0.994016i	$0.534841\pi$
$132$	−7577.66	−4.99659
$133$	0	0
$134$	1355.62	0.873937
$135$	−125.575	−0.0800577
$136$	−4591.94	−2.89526
$137$	384.045	0.239498	0.119749	−	0.992804i	$-0.461791\pi$
$137$	384.045	0.239498	0.119749	+	0.992804i	$0.461791\pi$
$138$	5393.80	3.32718
$139$	215.972	0.131788	0.0658938	−	0.997827i	$-0.479010\pi$
$139$	215.972	0.131788	0.0658938	+	0.997827i	$0.479010\pi$
$140$	0	0
$141$	−1636.98	−0.977723
$142$	−1048.75	−0.619785
$143$	−2823.29	−1.65102
$144$	2793.23	1.61645
$145$	−143.792	−0.0823534
$146$	−3498.56	−1.98317
$147$	0	0
$148$	3877.77	2.15372
$149$	1874.22	1.03048	0.515242	−	0.857045i	$-0.327702\pi$
$149$	1874.22	1.03048	0.515242	+	0.857045i	$0.327702\pi$
$150$	−4100.96	−2.23228
$151$	−227.877	−0.122811	−0.0614053	−	0.998113i	$-0.519558\pi$
$151$	−227.877	−0.122811	−0.0614053	+	0.998113i	$0.519558\pi$
$152$	2966.30	1.58288
$153$	1821.64	0.962554
$154$	0	0
$155$	310.902	0.161111
$156$	6177.55	3.17051
$157$	−1353.42	−0.687992	−0.343996	−	0.938971i	$-0.611781\pi$
$157$	−1353.42	−0.687992	−0.343996	+	0.938971i	$0.611781\pi$
$158$	973.049	0.489947
$159$	3541.21	1.76626
$160$	773.196	0.382041
$161$	0	0
$162$	4366.15	2.11752
$163$	2541.27	1.22115	0.610575	−	0.791958i	$-0.290938\pi$
$163$	2541.27	1.22115	0.610575	+	0.791958i	$0.290938\pi$
$164$	−6566.57	−3.12660
$165$	1329.96	0.627497
$166$	2677.12	1.25171
$167$	−187.723	−0.0869847	−0.0434923	−	0.999054i	$-0.513848\pi$
$167$	−187.723	−0.0869847	−0.0434923	+	0.999054i	$0.513848\pi$
$168$	0	0
$169$	104.638	0.0476276
$170$	1419.96	0.640623
$171$	−1176.74	−0.526243
$172$	4089.13	1.81275
$173$	2854.37	1.25441	0.627206	−	0.778853i	$-0.284198\pi$
$173$	2854.37	1.25441	0.627206	+	0.778853i	$0.284198\pi$
$174$	1586.70	0.691306
$175$	0	0
$176$	7665.65	3.28307
$177$	3059.14	1.29909
$178$	7695.37	3.24041
$179$	355.983	0.148645	0.0743225	−	0.997234i	$-0.476321\pi$
$179$	355.983	0.148645	0.0743225	+	0.997234i	$0.476321\pi$
$180$	−1288.13	−0.533395
$181$	−266.530	−0.109453	−0.0547266	−	0.998501i	$-0.517429\pi$
$181$	−266.530	−0.109453	−0.0547266	+	0.998501i	$0.517429\pi$
$182$	0	0
$183$	−358.188	−0.144689
$184$	−8137.31	−3.26027
$185$	−680.589	−0.270475
$186$	−3430.71	−1.35243
$187$	4999.24	1.95498
$188$	4351.18	1.68799
$189$	0	0
$190$	−917.263	−0.350238
$191$	4485.01	1.69908	0.849540	−	0.527525i	$-0.176880\pi$
$191$	4485.01	1.69908	0.849540	+	0.527525i	$0.176880\pi$
$192$	−1278.96	−0.480733
$193$	3487.25	1.30061	0.650304	−	0.759674i	$-0.274641\pi$
$193$	3487.25	1.30061	0.650304	+	0.759674i	$0.274641\pi$
$194$	3665.45	1.35652
$195$	−1084.22	−0.398169
$196$	0	0
$197$	−2415.21	−0.873486	−0.436743	−	0.899586i	$-0.643868\pi$
$197$	−2415.21	−0.873486	−0.436743	+	0.899586i	$0.643868\pi$
$198$	−6496.18	−2.33163
$199$	5226.68	1.86186	0.930928	−	0.365203i	$-0.119000\pi$
$199$	5226.68	1.86186	0.930928	+	0.365203i	$0.119000\pi$
$200$	6186.88	2.18739
$201$	1832.86	0.643184
$202$	267.645	0.0932249
$203$	0	0
$204$	−10938.7	−3.75422
$205$	1152.50	0.392655
$206$	−963.400	−0.325841
$207$	3228.10	1.08390
$208$	−6249.28	−2.08322
$209$	−3229.40	−1.06882
$210$	0	0
$211$	−2239.69	−0.730743	−0.365372	−	0.930862i	$-0.619058\pi$
$211$	−2239.69	−0.730743	−0.365372	+	0.930862i	$0.619058\pi$
$212$	−9412.68	−3.04937
$213$	−1417.97	−0.456138
$214$	−3222.56	−1.02939
$215$	−717.685	−0.227654
$216$	−2090.50	−0.658521
$217$	0	0
$218$	2286.95	0.710514
$219$	−4730.22	−1.45954
$220$	−3535.09	−1.08334
$221$	−4075.54	−1.24050
$222$	7510.10	2.27047
$223$	1381.27	0.414784	0.207392	−	0.978258i	$-0.433502\pi$
$223$	1381.27	0.414784	0.207392	+	0.978258i	$0.433502\pi$
$224$	0	0
$225$	−2454.35	−0.727216
$226$	10519.8	3.09631
$227$	−295.331	−0.0863515	−0.0431757	−	0.999067i	$-0.513748\pi$
$227$	−295.331	−0.0863515	−0.0431757	+	0.999067i	$0.513748\pi$
$228$	7066.14	2.05249
$229$	−4468.09	−1.28934	−0.644672	−	0.764459i	$-0.723006\pi$
$229$	−4468.09	−1.28934	−0.644672	+	0.764459i	$0.723006\pi$
$230$	2516.29	0.721387
$231$	0	0
$232$	−2393.76	−0.677405
$233$	−2322.69	−0.653065	−0.326533	−	0.945186i	$-0.605880\pi$
$233$	−2322.69	−0.653065	−0.326533	+	0.945186i	$0.605880\pi$
$234$	5295.90	1.47950
$235$	−763.677	−0.211986
$236$	−8131.33	−2.24281
$237$	1315.61	0.360582
$238$	0	0
$239$	2.49690	0.000675777 0	0.000337889	−	1.00000i	$-0.499892\pi$
$239$	2.49690	0.000675777 0	0.000337889		1.00000i	$0.499892\pi$
$240$	2943.83	0.791763
$241$	5914.89	1.58096	0.790480	−	0.612487i	$-0.209831\pi$
$241$	5914.89	1.58096	0.790480	+	0.612487i	$0.209831\pi$
$242$	−10976.1	−2.91559
$243$	4859.04	1.28275
$244$	952.080	0.249798
$245$	0	0
$246$	−12717.5	−3.29609
$247$	2632.71	0.678200
$248$	5175.70	1.32523
$249$	3619.59	0.921213
$250$	−4002.55	−1.01257
$251$	596.482	0.149998	0.0749992	−	0.997184i	$-0.476105\pi$
$251$	596.482	0.149998	0.0749992	+	0.997184i	$0.476105\pi$
$252$	0	0
$253$	8859.08	2.20144
$254$	187.127	0.0462261
$255$	1919.85	0.471473
$256$	−6407.16	−1.56425
$257$	1023.45	0.248409	0.124205	−	0.992257i	$-0.460362\pi$
$257$	1023.45	0.248409	0.124205	+	0.992257i	$0.460362\pi$
$258$	7919.44	1.91102
$259$	0	0
$260$	2881.92	0.687419
$261$	949.611	0.225209
$262$	1686.31	0.397636
$263$	876.156	0.205422	0.102711	−	0.994711i	$-0.467248\pi$
$263$	876.156	0.205422	0.102711	+	0.994711i	$0.467248\pi$
$264$	22140.3	5.16153
$265$	1652.02	0.382955
$266$	0	0
$267$	10404.5	2.38481
$268$	−4871.82	−1.11043
$269$	−1408.69	−0.319291	−0.159645	−	0.987174i	$-0.551035\pi$
$269$	−1408.69	−0.319291	−0.159645	+	0.987174i	$0.551035\pi$
$270$	646.442	0.145708
$271$	1230.79	0.275885	0.137943	−	0.990440i	$-0.455951\pi$
$271$	1230.79	0.275885	0.137943	+	0.990440i	$0.455951\pi$
$272$	11065.7	2.46675
$273$	0	0
$274$	−1977.01	−0.435896
$275$	−6735.65	−1.47700
$276$	−19384.2	−4.22751
$277$	6840.25	1.48372	0.741860	−	0.670555i	$-0.233944\pi$
$277$	6840.25	1.48372	0.741860	+	0.670555i	$0.233944\pi$
$278$	−1111.79	−0.239859
$279$	−2053.22	−0.440584
$280$	0	0
$281$	2831.45	0.601104	0.300552	−	0.953765i	$-0.402829\pi$
$281$	2831.45	0.601104	0.300552	+	0.953765i	$0.402829\pi$
$282$	8426.95	1.77949
$283$	7110.88	1.49363	0.746816	−	0.665031i	$-0.231582\pi$
$283$	7110.88	1.49363	0.746816	+	0.665031i	$0.231582\pi$
$284$	3769.02	0.787500
$285$	−1240.18	−0.257762
$286$	14533.9	3.00492
$287$	0	0
$288$	−5106.24	−1.04475
$289$	2303.62	0.468883
$290$	740.217	0.149886
$291$	4955.86	0.998343
$292$	12573.1	2.51982
$293$	−6463.86	−1.28881	−0.644407	−	0.764682i	$-0.722896\pi$
$293$	−6463.86	−1.28881	−0.644407	+	0.764682i	$0.722896\pi$
$294$	0	0
$295$	1427.13	0.281664
$296$	−11330.0	−2.22482
$297$	2275.93	0.444655
$298$	−9648.20	−1.87552
$299$	−7222.20	−1.39689
$300$	14738.0	2.83634
$301$	0	0
$302$	1173.08	0.223520
$303$	361.869	0.0686099
$304$	−7148.20	−1.34861
$305$	−167.100	−0.0313709
$306$	−9377.52	−1.75189
$307$	6566.61	1.22077	0.610384	−	0.792105i	$-0.291015\pi$
$307$	6566.61	1.22077	0.610384	+	0.792105i	$0.291015\pi$
$308$	0	0
$309$	−1302.56	−0.239806
$310$	−1600.47	−0.293228
$311$	−7227.56	−1.31781	−0.658903	−	0.752228i	$-0.728979\pi$
$311$	−7227.56	−1.31781	−0.658903	+	0.752228i	$0.728979\pi$
$312$	−18049.5	−3.27517
$313$	−8437.42	−1.52368	−0.761839	−	0.647767i	$-0.775703\pi$
$313$	−8437.42	−1.52368	−0.761839	+	0.647767i	$0.775703\pi$
$314$	6967.20	1.25217
$315$	0	0
$316$	−3496.94	−0.622527
$317$	7128.11	1.26295	0.631474	−	0.775397i	$-0.282450\pi$
$317$	7128.11	1.26295	0.631474	+	0.775397i	$0.282450\pi$
$318$	−18229.6	−3.21467
$319$	2606.08	0.457406
$320$	−596.652	−0.104231
$321$	−4357.05	−0.757591
$322$	0	0
$323$	−4661.78	−0.803060
$324$	−15691.1	−2.69052
$325$	5491.12	0.937207
$326$	−13082.1	−2.22254
$327$	3092.07	0.522911
$328$	19186.2	3.22981
$329$	0	0
$330$	−6846.42	−1.14207
$331$	−3626.78	−0.602254	−0.301127	−	0.953584i	$-0.597363\pi$
$331$	−3626.78	−0.602254	−0.301127	+	0.953584i	$0.597363\pi$
$332$	−9621.03	−1.59043
$333$	4494.66	0.739658
$334$	966.370	0.158316
$335$	855.056	0.139453
$336$	0	0
$337$	−1275.65	−0.206199	−0.103099	−	0.994671i	$-0.532876\pi$
$337$	−1275.65	−0.206199	−0.103099	+	0.994671i	$0.532876\pi$
$338$	−538.659	−0.0866840
$339$	14223.3	2.27876
$340$	−5103.05	−0.813976
$341$	−5634.78	−0.894841
$342$	6057.67	0.957782
$343$	0	0
$344$	−11947.6	−1.87259
$345$	3402.14	0.530913
$346$	−14693.8	−2.28308
$347$	−6043.73	−0.934998	−0.467499	−	0.883994i	$-0.654845\pi$
$347$	−6043.73	−0.934998	−0.467499	+	0.883994i	$0.654845\pi$
$348$	−5702.27	−0.878374
$349$	−2389.00	−0.366419	−0.183210	−	0.983074i	$-0.558649\pi$
$349$	−2389.00	−0.366419	−0.183210	+	0.983074i	$0.558649\pi$
$350$	0	0
$351$	−1855.41	−0.282149
$352$	−14013.4	−2.12192
$353$	2524.06	0.380573	0.190287	−	0.981729i	$-0.439058\pi$
$353$	2524.06	0.380573	0.190287	+	0.981729i	$0.439058\pi$
$354$	−15748.0	−2.36439
$355$	−661.502	−0.0988982
$356$	−27655.7	−4.11727
$357$	0	0
$358$	−1832.55	−0.270540
$359$	4138.63	0.608437	0.304218	−	0.952602i	$-0.401605\pi$
$359$	4138.63	0.608437	0.304218	+	0.952602i	$0.401605\pi$
$360$	3763.64	0.551003
$361$	−3847.59	−0.560955
$362$	1372.06	0.199209
$363$	−14840.2	−2.14576
$364$	0	0
$365$	−2206.72	−0.316452
$366$	1843.90	0.263339
$367$	7614.50	1.08303	0.541517	−	0.840690i	$-0.317850\pi$
$367$	7614.50	1.08303	0.541517	+	0.840690i	$0.317850\pi$
$368$	19609.3	2.77774
$369$	−7611.21	−1.07378
$370$	3503.57	0.492276
$371$	0	0
$372$	12329.3	1.71840
$373$	−7415.95	−1.02945	−0.514723	−	0.857357i	$-0.672105\pi$
$373$	−7415.95	−1.02945	−0.514723	+	0.857357i	$0.672105\pi$
$374$	−25735.3	−3.55814
$375$	−5411.63	−0.745214
$376$	−12713.2	−1.74371
$377$	−2124.56	−0.290240
$378$	0	0
$379$	73.8067	0.0100032	0.00500158	−	0.999987i	$-0.498408\pi$
$379$	73.8067	0.0100032	0.00500158	+	0.999987i	$0.498408\pi$
$380$	3296.46	0.445013
$381$	253.005	0.0340206
$382$	−23088.2	−3.09239
$383$	8475.40	1.13074	0.565369	−	0.824838i	$-0.308734\pi$
$383$	8475.40	1.13074	0.565369	+	0.824838i	$0.308734\pi$
$384$	−6675.22	−0.887093
$385$	0	0
$386$	−17951.8	−2.36716
$387$	4739.65	0.622558
$388$	−13172.9	−1.72359
$389$	−7897.26	−1.02932	−0.514662	−	0.857393i	$-0.672082\pi$
$389$	−7897.26	−1.02932	−0.514662	+	0.857393i	$0.672082\pi$
$390$	5581.42	0.724683
$391$	12788.5	1.65407
$392$	0	0
$393$	2279.97	0.292645
$394$	12433.1	1.58978
$395$	613.750	0.0781801
$396$	23346.0	2.96258
$397$	10827.9	1.36886	0.684431	−	0.729077i	$-0.260051\pi$
$397$	10827.9	1.36886	0.684431	+	0.729077i	$0.260051\pi$
$398$	−26906.1	−3.38865
$399$	0	0
$400$	−14909.2	−1.86365
$401$	−2773.97	−0.345450	−0.172725	−	0.984970i	$-0.555257\pi$
$401$	−2773.97	−0.345450	−0.172725	+	0.984970i	$0.555257\pi$
$402$	−9435.29	−1.17062
$403$	4593.66	0.567807
$404$	−961.863	−0.118452
$405$	2753.95	0.337889
$406$	0	0
$407$	12335.0	1.50227
$408$	31960.5	3.87814
$409$	−14660.0	−1.77235	−0.886175	−	0.463350i	$-0.846647\pi$
$409$	−14660.0	−1.77235	−0.886175	+	0.463350i	$0.846647\pi$
$410$	−5932.91	−0.714647
$411$	−2673.01	−0.320802
$412$	3462.27	0.414014
$413$	0	0
$414$	−16617.7	−1.97275
$415$	1688.59	0.199734
$416$	11424.2	1.34643
$417$	−1503.19	−0.176527
$418$	16624.5	1.94529
$419$	−7935.55	−0.925243	−0.462622	−	0.886556i	$-0.653091\pi$
$419$	−7935.55	−0.925243	−0.462622	+	0.886556i	$0.653091\pi$
$420$	0	0
$421$	−12583.7	−1.45675	−0.728376	−	0.685177i	$-0.759725\pi$
$421$	−12583.7	−1.45675	−0.728376	+	0.685177i	$0.759725\pi$
$422$	11529.6	1.32998
$423$	5043.38	0.579711
$424$	27501.9	3.15003
$425$	−9723.20	−1.10975
$426$	7299.47	0.830189
$427$	0	0
$428$	11581.2	1.30794
$429$	19650.5	2.21150
$430$	3694.53	0.414340
$431$	−5401.64	−0.603684	−0.301842	−	0.953358i	$-0.597602\pi$
$431$	−5401.64	−0.603684	−0.301842	+	0.953358i	$0.597602\pi$
$432$	5037.70	0.561057
$433$	−5989.02	−0.664697	−0.332349	−	0.943157i	$-0.607841\pi$
$433$	−5989.02	−0.664697	−0.332349	+	0.943157i	$0.607841\pi$
$434$	0	0
$435$	1000.81	0.110311
$436$	−8218.86	−0.902780
$437$	−8261.07	−0.904303
$438$	24350.5	2.65642
$439$	−13191.9	−1.43421	−0.717103	−	0.696967i	$-0.754532\pi$
$439$	−13191.9	−1.43421	−0.717103	+	0.696967i	$0.754532\pi$
$440$	10328.8	1.11910
$441$	0	0
$442$	20980.3	2.25776
$443$	−15538.1	−1.66645	−0.833225	−	0.552934i	$-0.813508\pi$
$443$	−15538.1	−1.66645	−0.833225	+	0.552934i	$0.813508\pi$
$444$	−26989.8	−2.88486
$445$	4853.86	0.517067
$446$	−7110.58	−0.754923
$447$	−13044.8	−1.38031
$448$	0	0
$449$	7092.05	0.745422	0.372711	−	0.927948i	$-0.378428\pi$
$449$	7092.05	0.745422	0.372711	+	0.927948i	$0.378428\pi$
$450$	12634.6	1.32356
$451$	−20888.0	−2.18088
$452$	−37806.0	−3.93417
$453$	1586.06	0.164502
$454$	1520.32	0.157163
$455$	0	0
$456$	−20645.8	−2.12024
$457$	8539.94	0.874139	0.437069	−	0.899428i	$-0.356016\pi$
$457$	8539.94	0.874139	0.437069	+	0.899428i	$0.356016\pi$
$458$	23001.1	2.34666
$459$	3285.40	0.334094
$460$	−9043.03	−0.916595
$461$	6167.30	0.623080	0.311540	−	0.950233i	$-0.399155\pi$
$461$	6167.30	0.623080	0.311540	+	0.950233i	$0.399155\pi$
$462$	0	0
$463$	−8115.61	−0.814610	−0.407305	−	0.913292i	$-0.633531\pi$
$463$	−8115.61	−0.814610	−0.407305	+	0.913292i	$0.633531\pi$
$464$	5768.49	0.577145
$465$	−2163.92	−0.215805
$466$	11956.8	1.18860
$467$	−10883.1	−1.07840	−0.539199	−	0.842178i	$-0.681273\pi$
$467$	−10883.1	−1.07840	−0.539199	+	0.842178i	$0.681273\pi$
$468$	−19032.4	−1.87986
$469$	0	0
$470$	3931.29	0.385824
$471$	9419.99	0.921550
$472$	23758.1	2.31685
$473$	13007.3	1.26443
$474$	−6772.55	−0.656273
$475$	6280.98	0.606718
$476$	0	0
$477$	−10910.1	−1.04725
$478$	−12.8536	−0.00122994
$479$	−14758.0	−1.40775	−0.703873	−	0.710326i	$-0.748548\pi$
$479$	−14758.0	−1.40775	−0.703873	+	0.710326i	$0.748548\pi$
$480$	−5381.55	−0.511735
$481$	−10055.9	−0.953241
$482$	−30449.0	−2.87741
$483$	0	0
$484$	39446.0	3.70454
$485$	2311.98	0.216457
$486$	−25013.6	−2.33465
$487$	−2369.67	−0.220493	−0.110246	−	0.993904i	$-0.535164\pi$
$487$	−2369.67	−0.220493	−0.110246	+	0.993904i	$0.535164\pi$
$488$	−2781.78	−0.258044
$489$	−17687.6	−1.63570
$490$	0	0
$491$	17347.1	1.59443	0.797214	−	0.603696i	$-0.206306\pi$
$491$	17347.1	1.59443	0.797214	+	0.603696i	$0.206306\pi$
$492$	45704.2	4.18802
$493$	3761.99	0.343674
$494$	−13552.8	−1.23435
$495$	−4097.47	−0.372055
$496$	−12472.4	−1.12909
$497$	0	0
$498$	−18633.1	−1.67664
$499$	−14617.0	−1.31132	−0.655659	−	0.755058i	$-0.727609\pi$
$499$	−14617.0	−1.31132	−0.655659	+	0.755058i	$0.727609\pi$
$500$	14384.4	1.28658
$501$	1306.58	0.116514
$502$	−3070.60	−0.273003
$503$	−13355.9	−1.18392	−0.591958	−	0.805969i	$-0.701645\pi$
$503$	−13355.9	−1.18392	−0.591958	+	0.805969i	$0.701645\pi$
$504$	0	0
$505$	168.817	0.0148758
$506$	−45605.2	−4.00672
$507$	−728.293	−0.0637961
$508$	−672.499	−0.0587349
$509$	1359.50	0.118386	0.0591932	−	0.998247i	$-0.481147\pi$
$509$	1359.50	0.118386	0.0591932	+	0.998247i	$0.481147\pi$
$510$	−9883.10	−0.858100
$511$	0	0
$512$	25310.6	2.18473
$513$	−2122.30	−0.182654
$514$	−5268.57	−0.452114
$515$	−607.665	−0.0519940
$516$	−28460.9	−2.42814
$517$	13840.9	1.17741
$518$	0	0
$519$	−19866.8	−1.68026
$520$	−8420.37	−0.710111
$521$	10272.6	0.863817	0.431909	−	0.901917i	$-0.357840\pi$
$521$	10272.6	0.863817	0.431909	+	0.901917i	$0.357840\pi$
$522$	−4888.45	−0.409888
$523$	17081.2	1.42812	0.714060	−	0.700084i	$-0.246854\pi$
$523$	17081.2	1.42812	0.714060	+	0.700084i	$0.246854\pi$
$524$	−6060.27	−0.505237
$525$	0	0
$526$	−4510.32	−0.373877
$527$	−8134.05	−0.672343
$528$	−53353.9	−4.39760
$529$	10495.2	0.862597
$530$	−8504.37	−0.696993
$531$	−9424.90	−0.770255
$532$	0	0
$533$	17028.5	1.38384
$534$	−53560.8	−4.34046
$535$	−2032.63	−0.164258
$536$	14234.5	1.14708
$537$	−2477.69	−0.199107
$538$	7251.71	0.581122
$539$	0	0
$540$	−2323.19	−0.185137
$541$	10729.6	0.852683	0.426341	−	0.904562i	$-0.359802\pi$
$541$	10729.6	0.852683	0.426341	+	0.904562i	$0.359802\pi$
$542$	−6335.90	−0.502122
$543$	1855.08	0.146610
$544$	−20229.0	−1.59432
$545$	1442.50	0.113376
$546$	0	0
$547$	20050.0	1.56724	0.783618	−	0.621243i	$-0.213372\pi$
$547$	20050.0	1.56724	0.783618	+	0.621243i	$0.213372\pi$
$548$	7104.97	0.553849
$549$	1103.54	0.0857887
$550$	34674.1	2.68820
$551$	−2430.16	−0.187892
$552$	56636.8	4.36707
$553$	0	0
$554$	−35212.5	−2.70043
$555$	4736.99	0.362296
$556$	3995.55	0.304765
$557$	12051.4	0.916759	0.458380	−	0.888757i	$-0.348430\pi$
$557$	12051.4	0.916759	0.458380	+	0.888757i	$0.348430\pi$
$558$	10569.7	0.801880
$559$	−10604.0	−0.802327
$560$	0	0
$561$	−34795.4	−2.61865
$562$	−14575.9	−1.09403
$563$	−13594.0	−1.01762	−0.508810	−	0.860879i	$-0.669914\pi$
$563$	−13594.0	−1.01762	−0.508810	+	0.860879i	$0.669914\pi$
$564$	−30284.8	−2.26103
$565$	6635.36	0.494074
$566$	−36605.7	−2.71847
$567$	0	0
$568$	−11012.3	−0.813495
$569$	−10657.3	−0.785197	−0.392598	−	0.919710i	$-0.628424\pi$
$569$	−10657.3	−0.785197	−0.392598	+	0.919710i	$0.628424\pi$
$570$	6384.27	0.469136
$571$	−12337.4	−0.904209	−0.452105	−	0.891965i	$-0.649327\pi$
$571$	−12337.4	−0.904209	−0.452105	+	0.891965i	$0.649327\pi$
$572$	−52231.9	−3.81805
$573$	−31216.3	−2.27588
$574$	0	0
$575$	−17230.3	−1.24966
$576$	3940.34	0.285036
$577$	14808.4	1.06843	0.534214	−	0.845349i	$-0.320608\pi$
$577$	14808.4	1.06843	0.534214	+	0.845349i	$0.320608\pi$
$578$	−11858.7	−0.853385
$579$	−24271.7	−1.74214
$580$	−2660.19	−0.190446
$581$	0	0
$582$	−25512.0	−1.81702
$583$	−29941.3	−2.12700
$584$	−36736.2	−2.60300
$585$	3340.39	0.236082
$586$	33275.0	2.34569
$587$	−9084.07	−0.638739	−0.319369	−	0.947630i	$-0.603471\pi$
$587$	−9084.07	−0.638739	−0.319369	+	0.947630i	$0.603471\pi$
$588$	0	0
$589$	5254.42	0.367580
$590$	−7346.66	−0.512640
$591$	16810.2	1.17002
$592$	27303.2	1.89553
$593$	−27554.3	−1.90813	−0.954065	−	0.299600i	$-0.903147\pi$
$593$	−27554.3	−1.90813	−0.954065	+	0.299600i	$0.903147\pi$
$594$	−11716.1	−0.809290
$595$	0	0
$596$	34673.7	2.38304
$597$	−36378.4	−2.49392
$598$	37178.8	2.54240
$599$	15408.8	1.05106	0.525530	−	0.850775i	$-0.323867\pi$
$599$	15408.8	1.05106	0.525530	+	0.850775i	$0.323867\pi$
$600$	−43061.5	−2.92997
$601$	−26852.7	−1.82254	−0.911270	−	0.411810i	$-0.864897\pi$
$601$	−26852.7	−1.82254	−0.911270	+	0.411810i	$0.864897\pi$
$602$	0	0
$603$	−5646.86	−0.381356
$604$	−4215.81	−0.284005
$605$	−6923.19	−0.465236
$606$	−1862.85	−0.124873
$607$	12580.2	0.841213	0.420607	−	0.907243i	$-0.361817\pi$
$607$	12580.2	0.841213	0.420607	+	0.907243i	$0.361817\pi$
$608$	13067.5	0.871638
$609$	0	0
$610$	860.206	0.0570963
$611$	−11283.5	−0.747108
$612$	33700.9	2.22595
$613$	11022.4	0.726246	0.363123	−	0.931741i	$-0.381710\pi$
$613$	11022.4	0.726246	0.363123	+	0.931741i	$0.381710\pi$
$614$	−33803.9	−2.22185
$615$	−8021.57	−0.525953
$616$	0	0
$617$	18154.9	1.18459	0.592293	−	0.805723i	$-0.298223\pi$
$617$	18154.9	1.18459	0.592293	+	0.805723i	$0.298223\pi$
$618$	6705.40	0.436457
$619$	6517.56	0.423203	0.211602	−	0.977356i	$-0.432132\pi$
$619$	6517.56	0.423203	0.211602	+	0.977356i	$0.432132\pi$
$620$	5751.79	0.372576
$621$	5822.00	0.376214
$622$	37206.4	2.39846
$623$	0	0
$624$	43495.8	2.79043
$625$	11782.5	0.754081
$626$	43434.5	2.77315
$627$	22477.1	1.43166
$628$	−25038.7	−1.59101
$629$	17806.1	1.12874
$630$	0	0
$631$	−928.861	−0.0586012	−0.0293006	−	0.999571i	$-0.509328\pi$
$631$	−928.861	−0.0586012	−0.0293006	+	0.999571i	$0.509328\pi$
$632$	10217.4	0.643077
$633$	15588.6	0.978815
$634$	−36694.5	−2.29862
$635$	118.031	0.00737622
$636$	65513.5	4.08456
$637$	0	0
$638$	−13415.7	−0.832497
$639$	4368.61	0.270453
$640$	−3114.09	−0.192336
$641$	26729.8	1.64706	0.823528	−	0.567275i	$-0.192002\pi$
$641$	26729.8	1.64706	0.823528	+	0.567275i	$0.192002\pi$
$642$	22429.4	1.37885
$643$	−5311.51	−0.325763	−0.162881	−	0.986646i	$-0.552079\pi$
$643$	−5311.51	−0.325763	−0.162881	+	0.986646i	$0.552079\pi$
$644$	0	0
$645$	4995.18	0.304938
$646$	23998.1	1.46160
$647$	−10529.7	−0.639822	−0.319911	−	0.947448i	$-0.603653\pi$
$647$	−10529.7	−0.639822	−0.319911	+	0.947448i	$0.603653\pi$
$648$	45846.1	2.77933
$649$	−25865.4	−1.56441
$650$	−28267.4	−1.70575
$651$	0	0
$652$	47014.4	2.82396
$653$	247.029	0.0148039	0.00740197	−	0.999973i	$-0.497644\pi$
$653$	247.029	0.0148039	0.00740197	+	0.999973i	$0.497644\pi$
$654$	−15917.5	−0.951718
$655$	1063.64	0.0634503
$656$	−46234.9	−2.75178
$657$	14573.3	0.865389
$658$	0	0
$659$	−1307.57	−0.0772925	−0.0386463	−	0.999253i	$-0.512305\pi$
$659$	−1307.57	−0.0772925	−0.0386463	+	0.999253i	$0.512305\pi$
$660$	24604.7	1.45112
$661$	32906.1	1.93631	0.968153	−	0.250361i	$-0.0805491\pi$
$661$	32906.1	1.93631	0.968153	+	0.250361i	$0.0805491\pi$
$662$	18670.1	1.09613
$663$	28366.3	1.66162
$664$	28110.7	1.64293
$665$	0	0
$666$	−23137.9	−1.34621
$667$	6666.56	0.387002
$668$	−3472.94	−0.201156
$669$	−9613.84	−0.555594
$670$	−4401.70	−0.253810
$671$	3028.52	0.174240
$672$	0	0
$673$	−16392.4	−0.938900	−0.469450	−	0.882959i	$-0.655548\pi$
$673$	−16392.4	−0.938900	−0.469450	+	0.882959i	$0.655548\pi$
$674$	6566.85	0.375290
$675$	−4426.53	−0.252411
$676$	1935.84	0.110141
$677$	−5112.42	−0.290231	−0.145115	−	0.989415i	$-0.546355\pi$
$677$	−5112.42	−0.290231	−0.145115	+	0.989415i	$0.546355\pi$
$678$	−73219.2	−4.14744
$679$	0	0
$680$	14910.1	0.840845
$681$	2055.54	0.115666
$682$	29007.0	1.62865
$683$	−9655.89	−0.540955	−0.270477	−	0.962726i	$-0.587182\pi$
$683$	−9655.89	−0.540955	−0.270477	+	0.962726i	$0.587182\pi$
$684$	−21770.1	−1.21696
$685$	−1247.00	−0.0695552
$686$	0	0
$687$	31098.5	1.72705
$688$	28791.4	1.59544
$689$	24409.1	1.34966
$690$	−17513.7	−0.966282
$691$	13905.7	0.765552	0.382776	−	0.923841i	$-0.374968\pi$
$691$	13905.7	0.765552	0.382776	+	0.923841i	$0.374968\pi$
$692$	52806.8	2.90088
$693$	0	0
$694$	31112.2	1.70173
$695$	−701.261	−0.0382739
$696$	16660.9	0.907369
$697$	−30152.7	−1.63861
$698$	12298.2	0.666898
$699$	16166.2	0.874767
$700$	0	0
$701$	28770.3	1.55013	0.775064	−	0.631883i	$-0.217718\pi$
$701$	28770.3	1.55013	0.775064	+	0.631883i	$0.217718\pi$
$702$	9551.36	0.513523
$703$	−11502.4	−0.617098
$704$	10813.7	0.578917
$705$	5315.30	0.283951
$706$	−12993.5	−0.692658
$707$	0	0
$708$	56595.1	3.00420
$709$	−2627.73	−0.139191	−0.0695956	−	0.997575i	$-0.522171\pi$
$709$	−2627.73	−0.139191	−0.0695956	+	0.997575i	$0.522171\pi$
$710$	3405.31	0.179999
$711$	−4053.26	−0.213796
$712$	80804.1	4.25318
$713$	−14414.2	−0.757106
$714$	0	0
$715$	9167.24	0.479490
$716$	6585.82	0.343748
$717$	−17.3787	−0.000905189 0
$718$	−21305.1	−1.10738
$719$	−1386.68	−0.0719258	−0.0359629	−	0.999353i	$-0.511450\pi$
$719$	−1386.68	−0.0719258	−0.0359629	+	0.999353i	$0.511450\pi$
$720$	−9069.63	−0.469452
$721$	0	0
$722$	19806.8	1.02096
$723$	−41168.4	−2.11766
$724$	−4930.90	−0.253115
$725$	−5068.65	−0.259648
$726$	76395.3	3.90536
$727$	−28039.0	−1.43041	−0.715205	−	0.698915i	$-0.753667\pi$
$727$	−28039.0	−1.43041	−0.715205	+	0.698915i	$0.753667\pi$
$728$	0	0
$729$	−10919.5	−0.554769
$730$	11359.9	0.575955
$731$	18776.6	0.950040
$732$	−6626.61	−0.334599
$733$	23145.9	1.16632	0.583161	−	0.812356i	$-0.301816\pi$
$733$	23145.9	1.16632	0.583161	+	0.812356i	$0.301816\pi$
$734$	−39198.3	−1.97117
$735$	0	0
$736$	−35847.4	−1.79532
$737$	−15497.0	−0.774547
$738$	39181.4	1.95432
$739$	−10745.8	−0.534902	−0.267451	−	0.963572i	$-0.586181\pi$
$739$	−10745.8	−0.534902	−0.267451	+	0.963572i	$0.586181\pi$
$740$	−12591.1	−0.625486
$741$	−18324.0	−0.908435
$742$	0	0
$743$	−25041.1	−1.23643	−0.618216	−	0.786008i	$-0.712144\pi$
$743$	−25041.1	−1.23643	−0.618216	+	0.786008i	$0.712144\pi$
$744$	−36023.6	−1.77512
$745$	−6085.60	−0.299274
$746$	38176.2	1.87363
$747$	−11151.6	−0.546205
$748$	92487.7	4.52097
$749$	0	0
$750$	27858.3	1.35632
$751$	32944.8	1.60076	0.800381	−	0.599492i	$-0.204631\pi$
$751$	32944.8	1.60076	0.800381	+	0.599492i	$0.204631\pi$
$752$	30636.4	1.48563
$753$	−4151.59	−0.200920
$754$	10936.9	0.528248
$755$	739.919	0.0356668
$756$	0	0
$757$	−23803.0	−1.14285	−0.571424	−	0.820655i	$-0.693609\pi$
$757$	−23803.0	−1.14285	−0.571424	+	0.820655i	$0.693609\pi$
$758$	−379.946	−0.0182061
$759$	−61660.4	−2.94879
$760$	−9631.58	−0.459703
$761$	−34675.2	−1.65174	−0.825871	−	0.563858i	$-0.809316\pi$
$761$	−34675.2	−1.65174	−0.825871	+	0.563858i	$0.809316\pi$
$762$	−1302.43	−0.0619188
$763$	0	0
$764$	82974.3	3.92919
$765$	−5914.87	−0.279546
$766$	−43630.1	−2.05799
$767$	21086.3	0.992674
$768$	44594.7	2.09528
$769$	−28935.5	−1.35688	−0.678441	−	0.734655i	$-0.737344\pi$
$769$	−28935.5	−1.35688	−0.678441	+	0.734655i	$0.737344\pi$
$770$	0	0
$771$	−7123.36	−0.332739
$772$	64515.3	3.00771
$773$	16964.0	0.789331	0.394665	−	0.918825i	$-0.370861\pi$
$773$	16964.0	0.789331	0.394665	+	0.918825i	$0.370861\pi$
$774$	−24399.0	−1.13308
$775$	10959.3	0.507960
$776$	38488.5	1.78049
$777$	0	0
$778$	40653.9	1.87341
$779$	19478.0	0.895854
$780$	−20058.5	−0.920783
$781$	11989.1	0.549299
$782$	−65833.0	−3.01047
$783$	1712.66	0.0781680
$784$	0	0
$785$	4394.56	0.199807
$786$	−11737.0	−0.532626
$787$	−17051.3	−0.772318	−0.386159	−	0.922432i	$-0.626198\pi$
$787$	−17051.3	−0.772318	−0.386159	+	0.922432i	$0.626198\pi$
$788$	−44682.3	−2.01997
$789$	−6098.16	−0.275159
$790$	−3159.49	−0.142291
$791$	0	0
$792$	−68212.2	−3.06037
$793$	−2468.95	−0.110561
$794$	−55740.6	−2.49138
$795$	−11498.3	−0.512960
$796$	96695.4	4.30562
$797$	2866.63	0.127404	0.0637021	−	0.997969i	$-0.479709\pi$
$797$	2866.63	0.127404	0.0637021	+	0.997969i	$0.479709\pi$
$798$	0	0
$799$	19979.9	0.884654
$800$	27255.2	1.20452
$801$	−32055.2	−1.41400
$802$	14280.0	0.628732
$803$	39994.6	1.75763
$804$	33908.6	1.48739
$805$	0	0
$806$	−23647.4	−1.03343
$807$	9804.66	0.427683
$808$	2810.37	0.122362
$809$	5452.85	0.236974	0.118487	−	0.992956i	$-0.462196\pi$
$809$	5452.85	0.236974	0.118487	+	0.992956i	$0.462196\pi$
$810$	−14176.9	−0.614971
$811$	3643.68	0.157764	0.0788822	−	0.996884i	$-0.474865\pi$
$811$	3643.68	0.157764	0.0788822	+	0.996884i	$0.474865\pi$
$812$	0	0
$813$	−8566.44	−0.369543
$814$	−63498.7	−2.73419
$815$	−8251.51	−0.354648
$816$	−77018.7	−3.30416
$817$	−12129.3	−0.519401
$818$	75467.6	3.22575
$819$	0	0
$820$	21321.7	0.908031
$821$	21790.0	0.926281	0.463141	−	0.886285i	$-0.346723\pi$
$821$	21790.0	0.926281	0.463141	+	0.886285i	$0.346723\pi$
$822$	13760.2	0.583873
$823$	−5238.96	−0.221894	−0.110947	−	0.993826i	$-0.535388\pi$
$823$	−5238.96	−0.221894	−0.110947	+	0.993826i	$0.535388\pi$
$824$	−10116.0	−0.427681
$825$	46881.0	1.97841
$826$	0	0
$827$	−8639.97	−0.363291	−0.181645	−	0.983364i	$-0.558142\pi$
$827$	−8639.97	−0.363291	−0.181645	+	0.983364i	$0.558142\pi$
$828$	59720.9	2.50658
$829$	32197.9	1.34895	0.674474	−	0.738298i	$-0.264370\pi$
$829$	32197.9	1.34895	0.674474	+	0.738298i	$0.264370\pi$
$830$	−8692.61	−0.363524
$831$	−47609.0	−1.98741
$832$	−8815.70	−0.367343
$833$	0	0
$834$	7738.21	0.321286
$835$	609.538	0.0252622
$836$	−59745.1	−2.47168
$837$	−3703.06	−0.152923
$838$	40851.0	1.68398
$839$	40283.6	1.65762	0.828810	−	0.559530i	$-0.189018\pi$
$839$	40283.6	1.65762	0.828810	+	0.559530i	$0.189018\pi$
$840$	0	0
$841$	−22427.9	−0.919591
$842$	64779.1	2.65135
$843$	−19707.3	−0.805166
$844$	−41435.1	−1.68988
$845$	−339.759	−0.0138320
$846$	−25962.6	−1.05510
$847$	0	0
$848$	−66274.3	−2.68381
$849$	−49492.7	−2.00069
$850$	50053.6	2.01979
$851$	31553.9	1.27104
$852$	−26232.9	−1.05484
$853$	−44150.6	−1.77220	−0.886100	−	0.463493i	$-0.846596\pi$
$853$	−44150.6	−1.77220	−0.886100	+	0.463493i	$0.846596\pi$
$854$	0	0
$855$	3820.88	0.152832
$856$	−33838.0	−1.35112
$857$	−32361.4	−1.28990	−0.644951	−	0.764224i	$-0.723122\pi$
$857$	−32361.4	−1.28990	−0.644951	+	0.764224i	$0.723122\pi$
$858$	−101158.	−4.02502
$859$	−44224.6	−1.75660	−0.878302	−	0.478107i	$-0.841323\pi$
$859$	−44224.6	−1.75660	−0.878302	+	0.478107i	$0.841323\pi$
$860$	−13277.4	−0.526461
$861$	0	0
$862$	27806.8	1.09873
$863$	6285.13	0.247912	0.123956	−	0.992288i	$-0.460442\pi$
$863$	6285.13	0.247912	0.123956	+	0.992288i	$0.460442\pi$
$864$	−9209.32	−0.362624
$865$	−9268.14	−0.364308
$866$	30830.6	1.20978
$867$	−16033.5	−0.628058
$868$	0	0
$869$	−11123.6	−0.434227
$870$	−5152.01	−0.200770
$871$	12633.7	0.491477
$872$	24013.8	0.932581
$873$	−15268.5	−0.591937
$874$	42526.7	1.64587
$875$	0	0
$876$	−87510.8	−3.37525
$877$	10959.1	0.421963	0.210981	−	0.977490i	$-0.432334\pi$
$877$	10959.1	0.421963	0.210981	+	0.977490i	$0.432334\pi$
$878$	67910.1	2.61031
$879$	44989.3	1.72634
$880$	−24890.4	−0.953472
$881$	9900.29	0.378603	0.189301	−	0.981919i	$-0.439378\pi$
$881$	9900.29	0.378603	0.189301	+	0.981919i	$0.439378\pi$
$882$	0	0
$883$	−48125.0	−1.83413	−0.917065	−	0.398738i	$-0.869448\pi$
$883$	−48125.0	−1.83413	−0.917065	+	0.398738i	$0.869448\pi$
$884$	−75399.0	−2.86871
$885$	−9933.04	−0.377283
$886$	79987.8	3.03300
$887$	334.271	0.0126536	0.00632680	−	0.999980i	$-0.497986\pi$
$887$	334.271	0.0126536	0.00632680	+	0.999980i	$0.497986\pi$
$888$	78858.6	2.98009
$889$	0	0
$890$	−24986.9	−0.941083
$891$	−49912.6	−1.87670
$892$	25554.0	0.959206
$893$	−12906.6	−0.483654
$894$	67152.7	2.51222
$895$	−1155.88	−0.0431696
$896$	0	0
$897$	50267.5	1.87111
$898$	−36508.8	−1.35670
$899$	−4240.24	−0.157308
$900$	−45406.4	−1.68172
$901$	−43221.6	−1.59813
$902$	107528.	3.96928
$903$	0	0
$904$	110461.	4.06404
$905$	865.424	0.0317875
$906$	−8164.78	−0.299400
$907$	41015.9	1.50155	0.750777	−	0.660556i	$-0.229679\pi$
$907$	41015.9	1.50155	0.750777	+	0.660556i	$0.229679\pi$
$908$	−5463.72	−0.199692
$909$	−1114.88	−0.0406802
$910$	0	0
$911$	−40481.5	−1.47224	−0.736121	−	0.676849i	$-0.763345\pi$
$911$	−40481.5	−1.47224	−0.736121	+	0.676849i	$0.763345\pi$
$912$	49752.4	1.80643
$913$	−30604.0	−1.10936
$914$	−43962.3	−1.59097
$915$	1163.04	0.0420206
$916$	−82661.3	−2.98167
$917$	0	0
$918$	−16912.7	−0.608065
$919$	4752.86	0.170601	0.0853005	−	0.996355i	$-0.472815\pi$
$919$	4752.86	0.170601	0.0853005	+	0.996355i	$0.472815\pi$
$920$	26421.9	0.946852
$921$	−45704.5	−1.63519
$922$	−31748.3	−1.13403
$923$	−9773.87	−0.348549
$924$	0	0
$925$	−23990.8	−0.852770
$926$	41777.9	1.48262
$927$	4013.06	0.142186
$928$	−10545.3	−0.373023
$929$	32762.8	1.15707	0.578533	−	0.815659i	$-0.303626\pi$
$929$	32762.8	1.15707	0.578533	+	0.815659i	$0.303626\pi$
$930$	11139.5	0.392773
$931$	0	0
$932$	−42970.5	−1.51024
$933$	50304.8	1.76517
$934$	56024.8	1.96273
$935$	−16232.6	−0.567766
$936$	55608.8	1.94191
$937$	−37520.9	−1.30817	−0.654085	−	0.756421i	$-0.726946\pi$
$937$	−37520.9	−1.30817	−0.654085	+	0.756421i	$0.726946\pi$
$938$	0	0
$939$	58725.6	2.04093
$940$	−14128.3	−0.490228
$941$	−8686.80	−0.300937	−0.150468	−	0.988615i	$-0.548078\pi$
$941$	−8686.80	−0.300937	−0.150468	+	0.988615i	$0.548078\pi$
$942$	−48492.7	−1.67726
$943$	−53433.0	−1.84520
$944$	−57252.3	−1.97394
$945$	0	0
$946$	−66959.8	−2.30132
$947$	16321.2	0.560049	0.280024	−	0.959993i	$-0.409657\pi$
$947$	16321.2	0.560049	0.280024	+	0.959993i	$0.409657\pi$
$948$	24339.2	0.833861
$949$	−32604.9	−1.11528
$950$	−32333.5	−1.10425
$951$	−49612.6	−1.69169
$952$	0	0
$953$	16610.3	0.564596	0.282298	−	0.959327i	$-0.408903\pi$
$953$	16610.3	0.564596	0.282298	+	0.959327i	$0.408903\pi$
$954$	56163.5	1.90604
$955$	−14562.9	−0.493448
$956$	46.1934	0.00156276
$957$	−18138.7	−0.612685
$958$	75972.0	2.56215
$959$	0	0
$960$	4152.78	0.139615
$961$	−20622.9	−0.692252
$962$	51766.2	1.73494
$963$	13423.6	0.449190
$964$	109428.	3.65604
$965$	−11323.1	−0.377724
$966$	0	0
$967$	−7530.63	−0.250433	−0.125216	−	0.992129i	$-0.539963\pi$
$967$	−7530.63	−0.250433	−0.125216	+	0.992129i	$0.539963\pi$
$968$	−115253.	−3.82683
$969$	32446.6	1.07568
$970$	−11901.7	−0.393961
$971$	3882.76	0.128325	0.0641625	−	0.997939i	$-0.479562\pi$
$971$	3882.76	0.128325	0.0641625	+	0.997939i	$0.479562\pi$
$972$	89894.0	2.96641
$973$	0	0
$974$	12198.7	0.401305
$975$	−38218.9	−1.25537
$976$	6703.56	0.219852
$977$	−8185.31	−0.268036	−0.134018	−	0.990979i	$-0.542788\pi$
$977$	−8185.31	−0.268036	−0.134018	+	0.990979i	$0.542788\pi$
$978$	91052.9	2.97705
$979$	−87971.3	−2.87189
$980$	0	0
$981$	−9526.36	−0.310044
$982$	−89300.3	−2.90192
$983$	−29363.9	−0.952761	−0.476381	−	0.879239i	$-0.658052\pi$
$983$	−29363.9	−0.952761	−0.476381	+	0.879239i	$0.658052\pi$
$984$	−133538.	−4.32627
$985$	7842.21	0.253679
$986$	−19366.1	−0.625501
$987$	0	0
$988$	48706.1	1.56837
$989$	33273.8	1.06981
$990$	21093.1	0.677155
$991$	24197.0	0.775623	0.387811	−	0.921739i	$-0.373231\pi$
$991$	24197.0	0.775623	0.387811	+	0.921739i	$0.373231\pi$
$992$	22800.6	0.729758
$993$	25242.9	0.806706
$994$	0	0
$995$	−16971.0	−0.540722
$996$	66963.6	2.13035
$997$	−37381.7	−1.18745	−0.593726	−	0.804667i	$-0.702344\pi$
$997$	−37381.7	−1.18745	−0.593726	+	0.804667i	$0.702344\pi$
$998$	75246.2	2.38665
$999$	8106.31	0.256729

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2401.4.a.c.1.3		39
7.6	odd	2		2401.4.a.d.1.3		39
49.13	odd	14		49.4.e.a.22.13	✓	78
49.34	odd	14		49.4.e.a.29.13	yes	78

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
49.4.e.a.22.13	✓	78	49.13	odd	14
49.4.e.a.29.13	yes	78	49.34	odd	14
2401.4.a.c.1.3		39	1.1	even	1	trivial
2401.4.a.d.1.3		39	7.6	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 2401 = 7^{4} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2401.a (trivial)

\(f(q)\)	\(=\)	\(q-5.14785 q^{2} -6.96013 q^{3} +18.5003 q^{4} -3.24701 q^{5} +35.8297 q^{6} -54.0542 q^{8} +21.4435 q^{9} +16.7151 q^{10} +58.8488 q^{11} -128.765 q^{12} -47.9754 q^{13} +22.5996 q^{15} +130.260 q^{16} +84.9507 q^{17} -110.388 q^{18} -54.8763 q^{19} -60.0707 q^{20} -302.944 q^{22} +150.540 q^{23} +376.225 q^{24} -114.457 q^{25} +246.970 q^{26} +38.6742 q^{27} +44.2844 q^{29} -116.339 q^{30} -95.7503 q^{31} -238.126 q^{32} -409.595 q^{33} -437.313 q^{34} +396.712 q^{36} +209.605 q^{37} +282.495 q^{38} +333.915 q^{39} +175.514 q^{40} -354.943 q^{41} +221.030 q^{43} +1088.72 q^{44} -69.6271 q^{45} -774.956 q^{46} +235.194 q^{47} -906.628 q^{48} +589.207 q^{50} -591.268 q^{51} -887.561 q^{52} -508.784 q^{53} -199.089 q^{54} -191.082 q^{55} +381.947 q^{57} -227.969 q^{58} -439.523 q^{59} +418.100 q^{60} +51.4628 q^{61} +492.908 q^{62} +183.755 q^{64} +155.776 q^{65} +2108.53 q^{66} -263.337 q^{67} +1571.62 q^{68} -1047.78 q^{69} +203.727 q^{71} -1159.11 q^{72} +679.617 q^{73} -1079.02 q^{74} +796.636 q^{75} -1015.23 q^{76} -1718.94 q^{78} -189.020 q^{79} -422.955 q^{80} -848.151 q^{81} +1827.19 q^{82} -520.046 q^{83} -275.835 q^{85} -1137.83 q^{86} -308.225 q^{87} -3181.02 q^{88} -1494.87 q^{89} +358.430 q^{90} +2785.04 q^{92} +666.435 q^{93} -1210.74 q^{94} +178.184 q^{95} +1657.39 q^{96} -712.036 q^{97} +1261.92 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 39 q + q^{2} - q^{3} + 145 q^{4} - 27 q^{5} - 41 q^{6} - 12 q^{8} + 312 q^{9} - 78 q^{10} + q^{11} + 91 q^{12} - 77 q^{13} - 161 q^{15} + 461 q^{16} - 211 q^{17} + 8 q^{18} - 314 q^{19} - 476 q^{20}+ \cdots + 5384 q^{99}+O(q^{100}) \) 39 * q + q^2 - q^3 + 145 * q^4 - 27 * q^5 - 41 * q^6 - 12 * q^8 + 312 * q^9 - 78 * q^10 + q^11 + 91 * q^12 - 77 * q^13 - 161 * q^15 + 461 * q^16 - 211 * q^17 + 8 * q^18 - 314 * q^19 - 476 * q^20 - 61 * q^22 - 69 * q^23 - 330 * q^24 + 606 * q^25 - 504 * q^26 + 50 * q^27 + 57 * q^29 + 42 * q^30 - 638 * q^31 - 1600 * q^32 - 1574 * q^33 - 1343 * q^34 + 782 * q^36 + 71 * q^37 - 1359 * q^38 - 84 * q^39 + 155 * q^40 - 1393 * q^41 - 125 * q^43 + 52 * q^44 - 1129 * q^45 - 1454 * q^46 - 1483 * q^47 + 974 * q^48 + 3074 * q^50 - 2044 * q^51 - 3899 * q^52 + 2213 * q^53 - 1142 * q^54 - 1604 * q^55 + 98 * q^57 + 2403 * q^58 - 2073 * q^59 - 1519 * q^60 - 2575 * q^61 - 1742 * q^62 + 1358 * q^64 + 1876 * q^65 + 48 * q^66 + 176 * q^67 - 3038 * q^68 + 638 * q^69 - 1259 * q^71 - 3799 * q^72 + 307 * q^73 + 3845 * q^74 - 131 * q^75 - 1974 * q^76 - 6041 * q^78 + 22 * q^79 + 804 * q^80 + 795 * q^81 - 8043 * q^82 - 6349 * q^83 + 3094 * q^85 + 1745 * q^86 - 9508 * q^87 + 1299 * q^88 - 2253 * q^89 - 11156 * q^90 + 1284 * q^92 - 3430 * q^93 - 2738 * q^94 - 3290 * q^95 - 3031 * q^96 + 770 * q^97 + 5384 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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