LMFDB - Embedded newform 1925.4.a.l.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1925,4,Mod(1,1925)] 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("1925.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1925, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 1925 = 5^{2} \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1925.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,2,4,2,0,4,-14,18,-46,0,-22,4,80] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

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

Self dual:	yes
Analytic conductor:	$113.578676761$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 77)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.41421$ of defining polynomial
Character	$\chi$	$=$	1925.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+3.82843 q^{2} +2.00000 q^{3} +6.65685 q^{4} +7.65685 q^{6} -7.00000 q^{7} -5.14214 q^{8} -23.0000 q^{9} -11.0000 q^{11} +13.3137 q^{12} +45.6569 q^{13} -26.7990 q^{14} -72.9411 q^{16} +63.6812 q^{17} -88.0538 q^{18} -39.5147 q^{19} -14.0000 q^{21} -42.1127 q^{22} +78.9117 q^{23} -10.2843 q^{24} +174.794 q^{26} -100.000 q^{27} -46.5980 q^{28} +256.558 q^{29} -170.818 q^{31} -238.113 q^{32} -22.0000 q^{33} +243.799 q^{34} -153.108 q^{36} +223.941 q^{37} -151.279 q^{38} +91.3137 q^{39} +307.730 q^{41} -53.5980 q^{42} +316.000 q^{43} -73.2254 q^{44} +302.108 q^{46} +576.357 q^{47} -145.882 q^{48} +49.0000 q^{49} +127.362 q^{51} +303.931 q^{52} -173.951 q^{53} -382.843 q^{54} +35.9949 q^{56} -79.0294 q^{57} +982.215 q^{58} -82.6375 q^{59} -63.1472 q^{61} -653.966 q^{62} +161.000 q^{63} -328.068 q^{64} -84.2254 q^{66} +349.726 q^{67} +423.917 q^{68} +157.823 q^{69} -119.050 q^{71} +118.269 q^{72} -573.318 q^{73} +857.342 q^{74} -263.044 q^{76} +77.0000 q^{77} +349.588 q^{78} +568.362 q^{79} +421.000 q^{81} +1178.12 q^{82} -510.447 q^{83} -93.1960 q^{84} +1209.78 q^{86} +513.117 q^{87} +56.5635 q^{88} +1090.10 q^{89} -319.598 q^{91} +525.304 q^{92} -341.637 q^{93} +2206.54 q^{94} -476.225 q^{96} -396.530 q^{97} +187.593 q^{98} +253.000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} + 4 q^{3} + 2 q^{4} + 4 q^{6} - 14 q^{7} + 18 q^{8} - 46 q^{9} - 22 q^{11} + 4 q^{12} + 80 q^{13} - 14 q^{14} - 78 q^{16} - 48 q^{17} - 46 q^{18} - 96 q^{19} - 28 q^{21} - 22 q^{22} + 56 q^{23}+ \cdots + 506 q^{99}+O(q^{100}) $ 2 * q + 2 * q^2 + 4 * q^3 + 2 * q^4 + 4 * q^6 - 14 * q^7 + 18 * q^8 - 46 * q^9 - 22 * q^11 + 4 * q^12 + 80 * q^13 - 14 * q^14 - 78 * q^16 - 48 * q^17 - 46 * q^18 - 96 * q^19 - 28 * q^21 - 22 * q^22 + 56 * q^23 + 36 * q^24 + 112 * q^26 - 200 * q^27 - 14 * q^28 + 4 * q^29 + 60 * q^31 - 414 * q^32 - 44 * q^33 + 448 * q^34 - 46 * q^36 + 380 * q^37 - 48 * q^38 + 160 * q^39 + 112 * q^41 - 28 * q^42 + 632 * q^43 - 22 * q^44 + 344 * q^46 + 604 * q^47 - 156 * q^48 + 98 * q^49 - 96 * q^51 + 144 * q^52 - 676 * q^53 - 200 * q^54 - 126 * q^56 - 192 * q^57 + 1444 * q^58 - 516 * q^59 - 296 * q^61 - 1076 * q^62 + 322 * q^63 + 34 * q^64 - 44 * q^66 + 1152 * q^67 + 944 * q^68 + 112 * q^69 - 1064 * q^71 - 414 * q^72 - 168 * q^73 + 572 * q^74 + 154 * q^77 + 224 * q^78 - 368 * q^79 + 842 * q^81 + 1536 * q^82 - 1728 * q^83 - 28 * q^84 + 632 * q^86 + 8 * q^87 - 198 * q^88 + 1524 * q^89 - 560 * q^91 + 632 * q^92 + 120 * q^93 + 2156 * q^94 - 828 * q^96 - 1404 * q^97 + 98 * q^98 + 506 * 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 $

$2$	3.82843	1.35355	0.676777	−	0.736188i	$-0.263376\pi$
$2$	3.82843	1.35355	0.676777	+	0.736188i	$0.263376\pi$
$3$	2.00000	0.384900	0.192450	−	0.981307i	$-0.438357\pi$
$3$	2.00000	0.384900	0.192450	+	0.981307i	$0.438357\pi$
$4$	6.65685	0.832107
$5$	0	0
$5$	0	0
$6$	7.65685	0.520983
$7$	−7.00000	−0.377964
$7$	−7.00000	−0.377964
$8$	−5.14214	−0.227252
$9$	−23.0000	−0.851852
$10$	0	0
$11$	−11.0000	−0.301511
$11$	−11.0000	−0.301511
$12$	13.3137	0.320278
$13$	45.6569	0.974072	0.487036	−	0.873382i	$-0.338078\pi$
$13$	45.6569	0.974072	0.487036	+	0.873382i	$0.338078\pi$
$14$	−26.7990	−0.511595
$15$	0	0
$16$	−72.9411	−1.13971
$17$	63.6812	0.908528	0.454264	−	0.890867i	$-0.349902\pi$
$17$	63.6812	0.908528	0.454264	+	0.890867i	$0.349902\pi$
$18$	−88.0538	−1.15303
$19$	−39.5147	−0.477121	−0.238560	−	0.971128i	$-0.576676\pi$
$19$	−39.5147	−0.477121	−0.238560	+	0.971128i	$0.576676\pi$
$20$	0	0
$21$	−14.0000	−0.145479
$22$	−42.1127	−0.408112
$23$	78.9117	0.715401	0.357701	−	0.933836i	$-0.383561\pi$
$23$	78.9117	0.715401	0.357701	+	0.933836i	$0.383561\pi$
$24$	−10.2843	−0.0874695
$25$	0	0
$26$	174.794	1.31846
$27$	−100.000	−0.712778
$28$	−46.5980	−0.314507
$29$	256.558	1.64282	0.821409	−	0.570340i	$-0.193189\pi$
$29$	256.558	1.64282	0.821409	+	0.570340i	$0.193189\pi$
$30$	0	0
$31$	−170.818	−0.989673	−0.494837	−	0.868986i	$-0.664772\pi$
$31$	−170.818	−0.989673	−0.494837	+	0.868986i	$0.664772\pi$
$32$	−238.113	−1.31540
$33$	−22.0000	−0.116052
$34$	243.799	1.22974
$35$	0	0
$36$	−153.108	−0.708832
$37$	223.941	0.995019	0.497509	−	0.867459i	$-0.334248\pi$
$37$	223.941	0.995019	0.497509	+	0.867459i	$0.334248\pi$
$38$	−151.279	−0.645809
$39$	91.3137	0.374920
$40$	0	0
$41$	307.730	1.17218	0.586090	−	0.810246i	$-0.300667\pi$
$41$	307.730	1.17218	0.586090	+	0.810246i	$0.300667\pi$
$42$	−53.5980	−0.196913
$43$	316.000	1.12069	0.560344	−	0.828260i	$-0.310669\pi$
$43$	316.000	1.12069	0.560344	+	0.828260i	$0.310669\pi$
$44$	−73.2254	−0.250890
$45$	0	0
$46$	302.108	0.968334
$47$	576.357	1.78873	0.894366	−	0.447337i	$-0.147627\pi$
$47$	576.357	1.78873	0.894366	+	0.447337i	$0.147627\pi$
$48$	−145.882	−0.438673
$49$	49.0000	0.142857
$50$	0	0
$51$	127.362	0.349692
$52$	303.931	0.810532
$53$	−173.951	−0.450831	−0.225415	−	0.974263i	$-0.572374\pi$
$53$	−173.951	−0.450831	−0.225415	+	0.974263i	$0.572374\pi$
$54$	−382.843	−0.964783
$55$	0	0
$56$	35.9949	0.0858933
$57$	−79.0294	−0.183644
$58$	982.215	2.22364
$59$	−82.6375	−0.182347	−0.0911736	−	0.995835i	$-0.529062\pi$
$59$	−82.6375	−0.182347	−0.0911736	+	0.995835i	$0.529062\pi$
$60$	0	0
$61$	−63.1472	−0.132544	−0.0662719	−	0.997802i	$-0.521110\pi$
$61$	−63.1472	−0.132544	−0.0662719	+	0.997802i	$0.521110\pi$
$62$	−653.966	−1.33958
$63$	161.000	0.321970
$64$	−328.068	−0.640758
$65$	0	0
$66$	−84.2254	−0.157082
$67$	349.726	0.637699	0.318849	−	0.947805i	$-0.396704\pi$
$67$	349.726	0.637699	0.318849	+	0.947805i	$0.396704\pi$
$68$	423.917	0.755992
$69$	157.823	0.275358
$70$	0	0
$71$	−119.050	−0.198994	−0.0994971	−	0.995038i	$-0.531723\pi$
$71$	−119.050	−0.198994	−0.0994971	+	0.995038i	$0.531723\pi$
$72$	118.269	0.193585
$73$	−573.318	−0.919203	−0.459601	−	0.888125i	$-0.652008\pi$
$73$	−573.318	−0.919203	−0.459601	+	0.888125i	$0.652008\pi$
$74$	857.342	1.34681
$75$	0	0
$76$	−263.044	−0.397016
$77$	77.0000	0.113961
$78$	349.588	0.507475
$79$	568.362	0.809439	0.404719	−	0.914441i	$-0.367369\pi$
$79$	568.362	0.809439	0.404719	+	0.914441i	$0.367369\pi$
$80$	0	0
$81$	421.000	0.577503
$82$	1178.12	1.58661
$83$	−510.447	−0.675046	−0.337523	−	0.941317i	$-0.609589\pi$
$83$	−510.447	−0.675046	−0.337523	+	0.941317i	$0.609589\pi$
$84$	−93.1960	−0.121054
$85$	0	0
$86$	1209.78	1.51691
$87$	513.117	0.632321
$88$	56.5635	0.0685192
$89$	1090.10	1.29832	0.649158	−	0.760654i	$-0.275121\pi$
$89$	1090.10	1.29832	0.649158	+	0.760654i	$0.275121\pi$
$90$	0	0
$91$	−319.598	−0.368165
$92$	525.304	0.595290
$93$	−341.637	−0.380925
$94$	2206.54	2.42114
$95$	0	0
$96$	−476.225	−0.506297
$97$	−396.530	−0.415067	−0.207534	−	0.978228i	$-0.566544\pi$
$97$	−396.530	−0.415067	−0.207534	+	0.978228i	$0.566544\pi$
$98$	187.593	0.193365
$99$	253.000	0.256843
$100$	0	0
$101$	825.844	0.813609	0.406804	−	0.913515i	$-0.366643\pi$
$101$	825.844	0.813609	0.406804	+	0.913515i	$0.366643\pi$
$102$	487.598	0.473327
$103$	−1246.19	−1.19214	−0.596071	−	0.802932i	$-0.703272\pi$
$103$	−1246.19	−1.19214	−0.596071	+	0.802932i	$0.703272\pi$
$104$	−234.774	−0.221360
$105$	0	0
$106$	−665.960	−0.610224
$107$	1368.24	1.23620	0.618099	−	0.786100i	$-0.287903\pi$
$107$	1368.24	1.23620	0.618099	+	0.786100i	$0.287903\pi$
$108$	−665.685	−0.593107
$109$	−1312.80	−1.15361	−0.576806	−	0.816881i	$-0.695701\pi$
$109$	−1312.80	−1.15361	−0.576806	+	0.816881i	$0.695701\pi$
$110$	0	0
$111$	447.882	0.382983
$112$	510.588	0.430768
$113$	312.607	0.260244	0.130122	−	0.991498i	$-0.458463\pi$
$113$	312.607	0.260244	0.130122	+	0.991498i	$0.458463\pi$
$114$	−302.558	−0.248572
$115$	0	0
$116$	1707.87	1.36700
$117$	−1050.11	−0.829765
$118$	−316.372	−0.246817
$119$	−445.769	−0.343391
$120$	0	0
$121$	121.000	0.0909091
$122$	−241.754	−0.179405
$123$	615.460	0.451172
$124$	−1137.11	−0.823514
$125$	0	0
$126$	616.377	0.435803
$127$	1592.65	1.11280	0.556398	−	0.830916i	$-0.312183\pi$
$127$	1592.65	1.11280	0.556398	+	0.830916i	$0.312183\pi$
$128$	648.917	0.448099
$129$	632.000	0.431353
$130$	0	0
$131$	1291.43	0.861321	0.430661	−	0.902514i	$-0.358281\pi$
$131$	1291.43	0.861321	0.430661	+	0.902514i	$0.358281\pi$
$132$	−146.451	−0.0965675
$133$	276.603	0.180335
$134$	1338.90	0.863159
$135$	0	0
$136$	−327.458	−0.206465
$137$	2107.76	1.31444	0.657220	−	0.753699i	$-0.271732\pi$
$137$	2107.76	1.31444	0.657220	+	0.753699i	$0.271732\pi$
$138$	604.215	0.372712
$139$	2318.30	1.41464	0.707321	−	0.706892i	$-0.249903\pi$
$139$	2318.30	1.41464	0.707321	+	0.706892i	$0.249903\pi$
$140$	0	0
$141$	1152.71	0.688483
$142$	−455.773	−0.269349
$143$	−502.225	−0.293694
$144$	1677.65	0.970860
$145$	0	0
$146$	−2194.91	−1.24419
$147$	98.0000	0.0549857
$148$	1490.74	0.827962
$149$	602.061	0.331025	0.165513	−	0.986208i	$-0.447072\pi$
$149$	602.061	0.331025	0.165513	+	0.986208i	$0.447072\pi$
$150$	0	0
$151$	−1928.20	−1.03917	−0.519584	−	0.854420i	$-0.673913\pi$
$151$	−1928.20	−1.03917	−0.519584	+	0.854420i	$0.673913\pi$
$152$	203.190	0.108427
$153$	−1464.67	−0.773931
$154$	294.789	0.154252
$155$	0	0
$156$	607.862	0.311974
$157$	−2476.45	−1.25887	−0.629434	−	0.777054i	$-0.716713\pi$
$157$	−2476.45	−1.25887	−0.629434	+	0.777054i	$0.716713\pi$
$158$	2175.93	1.09562
$159$	−347.902	−0.173525
$160$	0	0
$161$	−552.382	−0.270396
$162$	1611.77	0.781682
$163$	1063.06	0.510829	0.255415	−	0.966832i	$-0.417788\pi$
$163$	1063.06	0.510829	0.255415	+	0.966832i	$0.417788\pi$
$164$	2048.51	0.975378
$165$	0	0
$166$	−1954.21	−0.913710
$167$	3720.65	1.72403	0.862015	−	0.506883i	$-0.169203\pi$
$167$	3720.65	1.72403	0.862015	+	0.506883i	$0.169203\pi$
$168$	71.9899	0.0330604
$169$	−112.452	−0.0511842
$170$	0	0
$171$	908.839	0.406436
$172$	2103.57	0.932531
$173$	909.421	0.399665	0.199832	−	0.979830i	$-0.435960\pi$
$173$	909.421	0.399665	0.199832	+	0.979830i	$0.435960\pi$
$174$	1964.43	0.855880
$175$	0	0
$176$	802.352	0.343634
$177$	−165.275	−0.0701855
$178$	4173.36	1.75734
$179$	1349.80	0.563626	0.281813	−	0.959469i	$-0.409064\pi$
$179$	1349.80	0.563626	0.281813	+	0.959469i	$0.409064\pi$
$180$	0	0
$181$	−4559.47	−1.87239	−0.936196	−	0.351479i	$-0.885679\pi$
$181$	−4559.47	−1.87239	−0.936196	+	0.351479i	$0.885679\pi$
$182$	−1223.56	−0.498330
$183$	−126.294	−0.0510161
$184$	−405.775	−0.162577
$185$	0	0
$186$	−1307.93	−0.515603
$187$	−700.494	−0.273931
$188$	3836.73	1.48842
$189$	700.000	0.269405
$190$	0	0
$191$	−2238.60	−0.848061	−0.424031	−	0.905648i	$-0.639385\pi$
$191$	−2238.60	−0.848061	−0.424031	+	0.905648i	$0.639385\pi$
$192$	−656.136	−0.246628
$193$	−2966.36	−1.10634	−0.553169	−	0.833069i	$-0.686582\pi$
$193$	−2966.36	−1.10634	−0.553169	+	0.833069i	$0.686582\pi$
$194$	−1518.09	−0.561815
$195$	0	0
$196$	326.186	0.118872
$197$	−3982.18	−1.44020	−0.720099	−	0.693872i	$-0.755903\pi$
$197$	−3982.18	−1.44020	−0.720099	+	0.693872i	$0.755903\pi$
$198$	968.592	0.347651
$199$	1895.03	0.675052	0.337526	−	0.941316i	$-0.390410\pi$
$199$	1895.03	0.675052	0.337526	+	0.941316i	$0.390410\pi$
$200$	0	0
$201$	699.452	0.245450
$202$	3161.68	1.10126
$203$	−1795.91	−0.620927
$204$	847.833	0.290981
$205$	0	0
$206$	−4770.94	−1.61363
$207$	−1814.97	−0.609416
$208$	−3330.26	−1.11015
$209$	434.662	0.143857
$210$	0	0
$211$	−1506.79	−0.491621	−0.245810	−	0.969318i	$-0.579054\pi$
$211$	−1506.79	−0.491621	−0.245810	+	0.969318i	$0.579054\pi$
$212$	−1157.97	−0.375139
$213$	−238.099	−0.0765929
$214$	5238.22	1.67326
$215$	0	0
$216$	514.214	0.161981
$217$	1195.73	0.374061
$218$	−5025.97	−1.56147
$219$	−1146.64	−0.353801
$220$	0	0
$221$	2907.49	0.884971
$222$	1714.68	0.518388
$223$	−4289.51	−1.28810	−0.644051	−	0.764983i	$-0.722748\pi$
$223$	−4289.51	−1.28810	−0.644051	+	0.764983i	$0.722748\pi$
$224$	1666.79	0.497174
$225$	0	0
$226$	1196.79	0.352255
$227$	2848.72	0.832934	0.416467	−	0.909151i	$-0.363268\pi$
$227$	2848.72	0.832934	0.416467	+	0.909151i	$0.363268\pi$
$228$	−526.087	−0.152811
$229$	3273.70	0.944681	0.472341	−	0.881416i	$-0.343409\pi$
$229$	3273.70	0.944681	0.472341	+	0.881416i	$0.343409\pi$
$230$	0	0
$231$	154.000	0.0438634
$232$	−1319.26	−0.373334
$233$	−4773.81	−1.34224	−0.671122	−	0.741347i	$-0.734187\pi$
$233$	−4773.81	−1.34224	−0.671122	+	0.741347i	$0.734187\pi$
$234$	−4020.26	−1.12313
$235$	0	0
$236$	−550.106	−0.151732
$237$	1136.72	0.311553
$238$	−1706.59	−0.464798
$239$	1976.50	0.534934	0.267467	−	0.963567i	$-0.413813\pi$
$239$	1976.50	0.534934	0.267467	+	0.963567i	$0.413813\pi$
$240$	0	0
$241$	−2236.73	−0.597845	−0.298923	−	0.954277i	$-0.596627\pi$
$241$	−2236.73	−0.597845	−0.298923	+	0.954277i	$0.596627\pi$
$242$	463.240	0.123050
$243$	3542.00	0.935059
$244$	−420.362	−0.110291
$245$	0	0
$246$	2356.24	0.610685
$247$	−1804.12	−0.464750
$248$	878.371	0.224906
$249$	−1020.89	−0.259825
$250$	0	0
$251$	−6364.56	−1.60051	−0.800254	−	0.599662i	$-0.795302\pi$
$251$	−6364.56	−1.60051	−0.800254	+	0.599662i	$0.795302\pi$
$252$	1071.75	0.267913
$253$	−868.029	−0.215702
$254$	6097.36	1.50623
$255$	0	0
$256$	5108.88	1.24728
$257$	−7947.46	−1.92898	−0.964492	−	0.264110i	$-0.914922\pi$
$257$	−7947.46	−1.92898	−0.964492	+	0.264110i	$0.914922\pi$
$258$	2419.57	0.583859
$259$	−1567.59	−0.376082
$260$	0	0
$261$	−5900.84	−1.39944
$262$	4944.16	1.16584
$263$	7330.82	1.71877	0.859387	−	0.511325i	$-0.170845\pi$
$263$	7330.82	1.71877	0.859387	+	0.511325i	$0.170845\pi$
$264$	113.127	0.0263730
$265$	0	0
$266$	1058.95	0.244093
$267$	2180.20	0.499722
$268$	2328.07	0.530633
$269$	−3504.02	−0.794216	−0.397108	−	0.917772i	$-0.629986\pi$
$269$	−3504.02	−0.794216	−0.397108	+	0.917772i	$0.629986\pi$
$270$	0	0
$271$	36.4996	0.00818152	0.00409076	−	0.999992i	$-0.498698\pi$
$271$	36.4996	0.00818152	0.00409076	+	0.999992i	$0.498698\pi$
$272$	−4644.98	−1.03545
$273$	−639.196	−0.141707
$274$	8069.42	1.77917
$275$	0	0
$276$	1050.61	0.229127
$277$	4965.14	1.07699	0.538495	−	0.842629i	$-0.318993\pi$
$277$	4965.14	1.07699	0.538495	+	0.842629i	$0.318993\pi$
$278$	8875.43	1.91479
$279$	3928.82	0.843055
$280$	0	0
$281$	−4560.42	−0.968156	−0.484078	−	0.875025i	$-0.660845\pi$
$281$	−4560.42	−0.968156	−0.484078	+	0.875025i	$0.660845\pi$
$282$	4413.08	0.931899
$283$	917.744	0.192771	0.0963855	−	0.995344i	$-0.469272\pi$
$283$	917.744	0.192771	0.0963855	+	0.995344i	$0.469272\pi$
$284$	−792.496	−0.165584
$285$	0	0
$286$	−1922.73	−0.397530
$287$	−2154.11	−0.443042
$288$	5476.59	1.12053
$289$	−857.700	−0.174578
$290$	0	0
$291$	−793.060	−0.159759
$292$	−3816.49	−0.764875
$293$	3983.97	0.794355	0.397177	−	0.917742i	$-0.369990\pi$
$293$	3983.97	0.794355	0.397177	+	0.917742i	$0.369990\pi$
$294$	375.186	0.0744261
$295$	0	0
$296$	−1151.54	−0.226120
$297$	1100.00	0.214911
$298$	2304.95	0.448060
$299$	3602.86	0.696852
$300$	0	0
$301$	−2212.00	−0.423580
$302$	−7381.95	−1.40657
$303$	1651.69	0.313158
$304$	2882.25	0.543777
$305$	0	0
$306$	−5607.38	−1.04756
$307$	7992.52	1.48585	0.742927	−	0.669372i	$-0.233437\pi$
$307$	7992.52	1.48585	0.742927	+	0.669372i	$0.233437\pi$
$308$	512.578	0.0948274
$309$	−2492.38	−0.458856
$310$	0	0
$311$	7217.65	1.31600	0.657999	−	0.753019i	$-0.271403\pi$
$311$	7217.65	1.31600	0.657999	+	0.753019i	$0.271403\pi$
$312$	−469.547	−0.0852016
$313$	8415.62	1.51974	0.759871	−	0.650074i	$-0.225262\pi$
$313$	8415.62	1.51974	0.759871	+	0.650074i	$0.225262\pi$
$314$	−9480.91	−1.70395
$315$	0	0
$316$	3783.50	0.673540
$317$	−632.365	−0.112041	−0.0560207	−	0.998430i	$-0.517841\pi$
$317$	−632.365	−0.112041	−0.0560207	+	0.998430i	$0.517841\pi$
$318$	−1331.92	−0.234875
$319$	−2822.14	−0.495328
$320$	0	0
$321$	2736.49	0.475813
$322$	−2114.75	−0.365996
$323$	−2516.35	−0.433478
$324$	2802.54	0.480545
$325$	0	0
$326$	4069.84	0.691434
$327$	−2625.60	−0.444025
$328$	−1582.39	−0.266381
$329$	−4034.50	−0.676077
$330$	0	0
$331$	425.759	0.0707004	0.0353502	−	0.999375i	$-0.488745\pi$
$331$	425.759	0.0707004	0.0353502	+	0.999375i	$0.488745\pi$
$332$	−3397.97	−0.561710
$333$	−5150.65	−0.847609
$334$	14244.3	2.33357
$335$	0	0
$336$	1021.18	0.165803
$337$	−1369.56	−0.221379	−0.110689	−	0.993855i	$-0.535306\pi$
$337$	−1369.56	−0.221379	−0.110689	+	0.993855i	$0.535306\pi$
$338$	−430.513	−0.0692805
$339$	625.214	0.100168
$340$	0	0
$341$	1879.00	0.298398
$342$	3479.42	0.550133
$343$	−343.000	−0.0539949
$344$	−1624.91	−0.254679
$345$	0	0
$346$	3481.65	0.540968
$347$	5531.76	0.855794	0.427897	−	0.903827i	$-0.359255\pi$
$347$	5531.76	0.855794	0.427897	+	0.903827i	$0.359255\pi$
$348$	3415.74	0.526158
$349$	10171.7	1.56011	0.780055	−	0.625711i	$-0.215191\pi$
$349$	10171.7	1.56011	0.780055	+	0.625711i	$0.215191\pi$
$350$	0	0
$351$	−4565.69	−0.694297
$352$	2619.24	0.396608
$353$	−9626.38	−1.45145	−0.725723	−	0.687987i	$-0.758495\pi$
$353$	−9626.38	−1.45145	−0.725723	+	0.687987i	$0.758495\pi$
$354$	−632.743	−0.0949998
$355$	0	0
$356$	7256.62	1.08034
$357$	−891.537	−0.132171
$358$	5167.62	0.762898
$359$	8491.00	1.24829	0.624147	−	0.781307i	$-0.285447\pi$
$359$	8491.00	1.24829	0.624147	+	0.781307i	$0.285447\pi$
$360$	0	0
$361$	−5297.59	−0.772356
$362$	−17455.6	−2.53438
$363$	242.000	0.0349909
$364$	−2127.52	−0.306352
$365$	0	0
$366$	−483.509	−0.0690530
$367$	−2948.02	−0.419307	−0.209653	−	0.977776i	$-0.567234\pi$
$367$	−2948.02	−0.419307	−0.209653	+	0.977776i	$0.567234\pi$
$368$	−5755.91	−0.815346
$369$	−7077.79	−0.998523
$370$	0	0
$371$	1217.66	0.170398
$372$	−2274.23	−0.316971
$373$	8295.57	1.15155	0.575775	−	0.817608i	$-0.304700\pi$
$373$	8295.57	1.15155	0.575775	+	0.817608i	$0.304700\pi$
$374$	−2681.79	−0.370781
$375$	0	0
$376$	−2963.71	−0.406494
$377$	11713.7	1.60022
$378$	2679.90	0.364654
$379$	−12137.3	−1.64499	−0.822495	−	0.568773i	$-0.807418\pi$
$379$	−12137.3	−1.64499	−0.822495	+	0.568773i	$0.807418\pi$
$380$	0	0
$381$	3185.31	0.428316
$382$	−8570.33	−1.14790
$383$	9053.44	1.20786	0.603929	−	0.797038i	$-0.293601\pi$
$383$	9053.44	1.20786	0.603929	+	0.797038i	$0.293601\pi$
$384$	1297.83	0.172473
$385$	0	0
$386$	−11356.5	−1.49749
$387$	−7268.00	−0.954659
$388$	−2639.64	−0.345380
$389$	−1130.94	−0.147406	−0.0737030	−	0.997280i	$-0.523482\pi$
$389$	−1130.94	−0.147406	−0.0737030	+	0.997280i	$0.523482\pi$
$390$	0	0
$391$	5025.19	0.649962
$392$	−251.965	−0.0324646
$393$	2582.87	0.331523
$394$	−15245.5	−1.94938
$395$	0	0
$396$	1684.18	0.213721
$397$	13113.4	1.65779	0.828896	−	0.559403i	$-0.188970\pi$
$397$	13113.4	1.65779	0.828896	+	0.559403i	$0.188970\pi$
$398$	7255.00	0.913719
$399$	553.206	0.0694109
$400$	0	0
$401$	12805.1	1.59465	0.797327	−	0.603548i	$-0.206247\pi$
$401$	12805.1	1.59465	0.797327	+	0.603548i	$0.206247\pi$
$402$	2677.80	0.332230
$403$	−7799.03	−0.964013
$404$	5497.52	0.677010
$405$	0	0
$406$	−6875.51	−0.840457
$407$	−2463.35	−0.300009
$408$	−654.915	−0.0794685
$409$	9184.62	1.11039	0.555196	−	0.831720i	$-0.312643\pi$
$409$	9184.62	1.11039	0.555196	+	0.831720i	$0.312643\pi$
$410$	0	0
$411$	4215.53	0.505928
$412$	−8295.70	−0.991990
$413$	578.463	0.0689208
$414$	−6948.48	−0.824877
$415$	0	0
$416$	−10871.5	−1.28129
$417$	4636.59	0.544496
$418$	1664.07	0.194719
$419$	−6249.25	−0.728629	−0.364315	−	0.931276i	$-0.618697\pi$
$419$	−6249.25	−0.728629	−0.364315	+	0.931276i	$0.618697\pi$
$420$	0	0
$421$	4703.21	0.544467	0.272233	−	0.962231i	$-0.412238\pi$
$421$	4703.21	0.544467	0.272233	+	0.962231i	$0.412238\pi$
$422$	−5768.65	−0.665435
$423$	−13256.2	−1.52373
$424$	894.481	0.102452
$425$	0	0
$426$	−911.546	−0.103673
$427$	442.030	0.0500968
$428$	9108.20	1.02865
$429$	−1004.45	−0.113043
$430$	0	0
$431$	7554.41	0.844276	0.422138	−	0.906532i	$-0.361280\pi$
$431$	7554.41	0.844276	0.422138	+	0.906532i	$0.361280\pi$
$432$	7294.11	0.812357
$433$	−2631.98	−0.292113	−0.146057	−	0.989276i	$-0.546658\pi$
$433$	−2631.98	−0.292113	−0.146057	+	0.989276i	$0.546658\pi$
$434$	4577.76	0.506312
$435$	0	0
$436$	−8739.13	−0.959928
$437$	−3118.17	−0.341333
$438$	−4389.81	−0.478889
$439$	12551.2	1.36455	0.682274	−	0.731097i	$-0.260991\pi$
$439$	12551.2	1.36455	0.682274	+	0.731097i	$0.260991\pi$
$440$	0	0
$441$	−1127.00	−0.121693
$442$	11131.1	1.19786
$443$	−11226.2	−1.20400	−0.602000	−	0.798496i	$-0.705629\pi$
$443$	−11226.2	−1.20400	−0.602000	+	0.798496i	$0.705629\pi$
$444$	2981.49	0.318683
$445$	0	0
$446$	−16422.1	−1.74351
$447$	1204.12	0.127412
$448$	2296.48	0.242184
$449$	−11177.4	−1.17482	−0.587410	−	0.809289i	$-0.699853\pi$
$449$	−11177.4	−1.17482	−0.587410	+	0.809289i	$0.699853\pi$
$450$	0	0
$451$	−3385.03	−0.353425
$452$	2080.98	0.216551
$453$	−3856.39	−0.399976
$454$	10906.1	1.12742
$455$	0	0
$456$	406.380	0.0417335
$457$	−7040.35	−0.720643	−0.360321	−	0.932828i	$-0.617333\pi$
$457$	−7040.35	−0.720643	−0.360321	+	0.932828i	$0.617333\pi$
$458$	12533.1	1.27868
$459$	−6368.12	−0.647579
$460$	0	0
$461$	−1483.35	−0.149862	−0.0749312	−	0.997189i	$-0.523874\pi$
$461$	−1483.35	−0.149862	−0.0749312	+	0.997189i	$0.523874\pi$
$462$	589.578	0.0593715
$463$	3106.19	0.311786	0.155893	−	0.987774i	$-0.450174\pi$
$463$	3106.19	0.311786	0.155893	+	0.987774i	$0.450174\pi$
$464$	−18713.7	−1.87233
$465$	0	0
$466$	−18276.2	−1.81680
$467$	9605.04	0.951752	0.475876	−	0.879512i	$-0.342131\pi$
$467$	9605.04	0.951752	0.475876	+	0.879512i	$0.342131\pi$
$468$	−6990.41	−0.690453
$469$	−2448.08	−0.241027
$470$	0	0
$471$	−4952.90	−0.484539
$472$	424.933	0.0414389
$473$	−3476.00	−0.337900
$474$	4351.86	0.421704
$475$	0	0
$476$	−2967.42	−0.285738
$477$	4000.88	0.384041
$478$	7566.88	0.724061
$479$	9118.53	0.869805	0.434902	−	0.900478i	$-0.356783\pi$
$479$	9118.53	0.869805	0.434902	+	0.900478i	$0.356783\pi$
$480$	0	0
$481$	10224.4	0.969220
$482$	−8563.17	−0.809215
$483$	−1104.76	−0.104076
$484$	805.479	0.0756461
$485$	0	0
$486$	13560.3	1.26565
$487$	−14045.1	−1.30687	−0.653434	−	0.756983i	$-0.726672\pi$
$487$	−14045.1	−1.30687	−0.653434	+	0.756983i	$0.726672\pi$
$488$	324.711	0.0301209
$489$	2126.12	0.196618
$490$	0	0
$491$	−1034.26	−0.0950620	−0.0475310	−	0.998870i	$-0.515135\pi$
$491$	−1034.26	−0.0950620	−0.0475310	+	0.998870i	$0.515135\pi$
$492$	4097.03	0.375423
$493$	16338.0	1.49255
$494$	−6906.93	−0.629064
$495$	0	0
$496$	12459.7	1.12794
$497$	833.347	0.0752128
$498$	−3908.42	−0.351687
$499$	−14332.3	−1.28577	−0.642886	−	0.765962i	$-0.722263\pi$
$499$	−14332.3	−1.28577	−0.642886	+	0.765962i	$0.722263\pi$
$500$	0	0
$501$	7441.31	0.663579
$502$	−24366.2	−2.16637
$503$	5007.72	0.443903	0.221951	−	0.975058i	$-0.428757\pi$
$503$	5007.72	0.443903	0.221951	+	0.975058i	$0.428757\pi$
$504$	−827.884	−0.0731684
$505$	0	0
$506$	−3323.18	−0.291964
$507$	−224.903	−0.0197008
$508$	10602.1	0.925966
$509$	−9234.40	−0.804140	−0.402070	−	0.915609i	$-0.631709\pi$
$509$	−9234.40	−0.804140	−0.402070	+	0.915609i	$0.631709\pi$
$510$	0	0
$511$	4013.23	0.347426
$512$	14367.6	1.24017
$513$	3951.47	0.340081
$514$	−30426.3	−2.61098
$515$	0	0
$516$	4207.13	0.358932
$517$	−6339.93	−0.539323
$518$	−6001.40	−0.509047
$519$	1818.84	0.153831
$520$	0	0
$521$	−5141.26	−0.432328	−0.216164	−	0.976357i	$-0.569355\pi$
$521$	−5141.26	−0.432328	−0.216164	+	0.976357i	$0.569355\pi$
$522$	−22591.0	−1.89421
$523$	19766.3	1.65262	0.826308	−	0.563218i	$-0.190437\pi$
$523$	19766.3	1.65262	0.826308	+	0.563218i	$0.190437\pi$
$524$	8596.89	0.716711
$525$	0	0
$526$	28065.5	2.32645
$527$	−10877.9	−0.899146
$528$	1604.70	0.132265
$529$	−5939.95	−0.488201
$530$	0	0
$531$	1900.66	0.155333
$532$	1841.31	0.150058
$533$	14050.0	1.14179
$534$	8346.72	0.676400
$535$	0	0
$536$	−1798.34	−0.144919
$537$	2699.61	0.216940
$538$	−13414.9	−1.07501
$539$	−539.000	−0.0430730
$540$	0	0
$541$	−16805.9	−1.33556	−0.667782	−	0.744356i	$-0.732756\pi$
$541$	−16805.9	−1.33556	−0.667782	+	0.744356i	$0.732756\pi$
$542$	139.736	0.0110741
$543$	−9118.95	−0.720684
$544$	−15163.3	−1.19508
$545$	0	0
$546$	−2447.12	−0.191807
$547$	−16284.0	−1.27286	−0.636428	−	0.771336i	$-0.719589\pi$
$547$	−16284.0	−1.27286	−0.636428	+	0.771336i	$0.719589\pi$
$548$	14031.1	1.09375
$549$	1452.39	0.112908
$550$	0	0
$551$	−10137.8	−0.783823
$552$	−811.549	−0.0625758
$553$	−3978.53	−0.305939
$554$	19008.7	1.45776
$555$	0	0
$556$	15432.6	1.17713
$557$	13972.6	1.06291	0.531454	−	0.847087i	$-0.321646\pi$
$557$	13972.6	1.06291	0.531454	+	0.847087i	$0.321646\pi$
$558$	15041.2	1.14112
$559$	14427.6	1.09163
$560$	0	0
$561$	−1400.99	−0.105436
$562$	−17459.2	−1.31045
$563$	−6396.33	−0.478816	−0.239408	−	0.970919i	$-0.576953\pi$
$563$	−6396.33	−0.478816	−0.239408	+	0.970919i	$0.576953\pi$
$564$	7673.45	0.572891
$565$	0	0
$566$	3513.51	0.260926
$567$	−2947.00	−0.218276
$568$	612.169	0.0452219
$569$	−15320.3	−1.12876	−0.564378	−	0.825517i	$-0.690884\pi$
$569$	−15320.3	−1.12876	−0.564378	+	0.825517i	$0.690884\pi$
$570$	0	0
$571$	−14347.1	−1.05150	−0.525750	−	0.850639i	$-0.676215\pi$
$571$	−14347.1	−1.05150	−0.525750	+	0.850639i	$0.676215\pi$
$572$	−3343.24	−0.244385
$573$	−4477.21	−0.326419
$574$	−8246.85	−0.599681
$575$	0	0
$576$	7545.57	0.545831
$577$	19882.3	1.43451	0.717256	−	0.696810i	$-0.245398\pi$
$577$	19882.3	1.43451	0.717256	+	0.696810i	$0.245398\pi$
$578$	−3283.64	−0.236300
$579$	−5932.72	−0.425829
$580$	0	0
$581$	3573.13	0.255143
$582$	−3036.17	−0.216243
$583$	1913.46	0.135931
$584$	2948.08	0.208891
$585$	0	0
$586$	15252.3	1.07520
$587$	−17868.5	−1.25641	−0.628204	−	0.778048i	$-0.716210\pi$
$587$	−17868.5	−1.25641	−0.628204	+	0.778048i	$0.716210\pi$
$588$	652.372	0.0457540
$589$	6749.84	0.472194
$590$	0	0
$591$	−7964.37	−0.554332
$592$	−16334.5	−1.13403
$593$	−24594.0	−1.70312	−0.851562	−	0.524253i	$-0.824344\pi$
$593$	−24594.0	−1.70312	−0.851562	+	0.524253i	$0.824344\pi$
$594$	4211.27	0.290893
$595$	0	0
$596$	4007.83	0.275448
$597$	3790.07	0.259828
$598$	13793.3	0.943226
$599$	13645.6	0.930789	0.465394	−	0.885103i	$-0.345913\pi$
$599$	13645.6	0.930789	0.465394	+	0.885103i	$0.345913\pi$
$600$	0	0
$601$	−23143.9	−1.57081	−0.785407	−	0.618980i	$-0.787546\pi$
$601$	−23143.9	−1.57081	−0.785407	+	0.618980i	$0.787546\pi$
$602$	−8468.48	−0.573338
$603$	−8043.69	−0.543225
$604$	−12835.7	−0.864698
$605$	0	0
$606$	6323.36	0.423876
$607$	11968.7	0.800321	0.400160	−	0.916445i	$-0.368954\pi$
$607$	11968.7	0.800321	0.400160	+	0.916445i	$0.368954\pi$
$608$	9408.96	0.627605
$609$	−3591.82	−0.238995
$610$	0	0
$611$	26314.7	1.74235
$612$	−9750.08	−0.643993
$613$	−7079.21	−0.466438	−0.233219	−	0.972424i	$-0.574926\pi$
$613$	−7079.21	−0.466438	−0.233219	+	0.972424i	$0.574926\pi$
$614$	30598.8	2.01118
$615$	0	0
$616$	−395.944	−0.0258978
$617$	24592.0	1.60460	0.802300	−	0.596921i	$-0.203610\pi$
$617$	24592.0	1.60460	0.802300	+	0.596921i	$0.203610\pi$
$618$	−9541.89	−0.621086
$619$	−5460.23	−0.354548	−0.177274	−	0.984162i	$-0.556728\pi$
$619$	−5460.23	−0.354548	−0.177274	+	0.984162i	$0.556728\pi$
$620$	0	0
$621$	−7891.17	−0.509922
$622$	27632.3	1.78127
$623$	−7630.68	−0.490717
$624$	−6660.52	−0.427299
$625$	0	0
$626$	32218.6	2.05705
$627$	869.324	0.0553707
$628$	−16485.4	−1.04751
$629$	14260.8	0.904002
$630$	0	0
$631$	−23358.4	−1.47367	−0.736833	−	0.676075i	$-0.763680\pi$
$631$	−23358.4	−1.47367	−0.736833	+	0.676075i	$0.763680\pi$
$632$	−2922.59	−0.183947
$633$	−3013.59	−0.189225
$634$	−2420.96	−0.151654
$635$	0	0
$636$	−2315.94	−0.144391
$637$	2237.19	0.139153
$638$	−10804.4	−0.670453
$639$	2738.14	0.169514
$640$	0	0
$641$	9328.68	0.574822	0.287411	−	0.957807i	$-0.407205\pi$
$641$	9328.68	0.574822	0.287411	+	0.957807i	$0.407205\pi$
$642$	10476.4	0.644038
$643$	267.859	0.0164282	0.00821409	−	0.999966i	$-0.497385\pi$
$643$	267.859	0.0164282	0.00821409	+	0.999966i	$0.497385\pi$
$644$	−3677.13	−0.224998
$645$	0	0
$646$	−9633.65	−0.586735
$647$	−11518.0	−0.699877	−0.349939	−	0.936773i	$-0.613798\pi$
$647$	−11518.0	−0.699877	−0.349939	+	0.936773i	$0.613798\pi$
$648$	−2164.84	−0.131239
$649$	909.013	0.0549798
$650$	0	0
$651$	2391.46	0.143976
$652$	7076.62	0.425064
$653$	12565.0	0.752994	0.376497	−	0.926418i	$-0.377129\pi$
$653$	12565.0	0.752994	0.376497	+	0.926418i	$0.377129\pi$
$654$	−10051.9	−0.601012
$655$	0	0
$656$	−22446.2	−1.33594
$657$	13186.3	0.783024
$658$	−15445.8	−0.915106
$659$	−4020.26	−0.237643	−0.118822	−	0.992916i	$-0.537912\pi$
$659$	−4020.26	−0.237643	−0.118822	+	0.992916i	$0.537912\pi$
$660$	0	0
$661$	−2080.20	−0.122406	−0.0612031	−	0.998125i	$-0.519494\pi$
$661$	−2080.20	−0.122406	−0.0612031	+	0.998125i	$0.519494\pi$
$662$	1629.99	0.0956968
$663$	5814.97	0.340626
$664$	2624.79	0.153406
$665$	0	0
$666$	−19718.9	−1.14728
$667$	20245.5	1.17527
$668$	24767.9	1.43458
$669$	−8579.02	−0.495791
$670$	0	0
$671$	694.619	0.0399634
$672$	3333.58	0.191362
$673$	17535.3	1.00436	0.502182	−	0.864762i	$-0.332531\pi$
$673$	17535.3	1.00436	0.502182	+	0.864762i	$0.332531\pi$
$674$	−5243.26	−0.299648
$675$	0	0
$676$	−748.574	−0.0425907
$677$	−12560.8	−0.713074	−0.356537	−	0.934281i	$-0.616043\pi$
$677$	−12560.8	−0.713074	−0.356537	+	0.934281i	$0.616043\pi$
$678$	2393.59	0.135583
$679$	2775.71	0.156881
$680$	0	0
$681$	5697.43	0.320596
$682$	7193.62	0.403897
$683$	4961.51	0.277961	0.138980	−	0.990295i	$-0.455618\pi$
$683$	4961.51	0.277961	0.138980	+	0.990295i	$0.455618\pi$
$684$	6050.01	0.338198
$685$	0	0
$686$	−1313.15	−0.0730850
$687$	6547.40	0.363608
$688$	−23049.4	−1.27725
$689$	−7942.07	−0.439142
$690$	0	0
$691$	−15429.4	−0.849441	−0.424720	−	0.905325i	$-0.639628\pi$
$691$	−15429.4	−0.849441	−0.424720	+	0.905325i	$0.639628\pi$
$692$	6053.89	0.332564
$693$	−1771.00	−0.0970775
$694$	21178.0	1.15836
$695$	0	0
$696$	−2638.52	−0.143696
$697$	19596.6	1.06496
$698$	38941.6	2.11169
$699$	−9547.62	−0.516630
$700$	0	0
$701$	20446.3	1.10163	0.550817	−	0.834626i	$-0.314316\pi$
$701$	20446.3	1.10163	0.550817	+	0.834626i	$0.314316\pi$
$702$	−17479.4	−0.939768
$703$	−8848.97	−0.474744
$704$	3608.75	0.193196
$705$	0	0
$706$	−36853.9	−1.96461
$707$	−5780.91	−0.307515
$708$	−1100.21	−0.0584018
$709$	−28090.1	−1.48794	−0.743968	−	0.668216i	$-0.767058\pi$
$709$	−28090.1	−1.48794	−0.743968	+	0.668216i	$0.767058\pi$
$710$	0	0
$711$	−13072.3	−0.689522
$712$	−5605.43	−0.295045
$713$	−13479.6	−0.708013
$714$	−3413.19	−0.178901
$715$	0	0
$716$	8985.44	0.468997
$717$	3953.00	0.205896
$718$	32507.2	1.68963
$719$	24923.7	1.29277	0.646383	−	0.763013i	$-0.276281\pi$
$719$	24923.7	1.29277	0.646383	+	0.763013i	$0.276281\pi$
$720$	0	0
$721$	8723.32	0.450587
$722$	−20281.4	−1.04542
$723$	−4473.47	−0.230111
$724$	−30351.7	−1.55803
$725$	0	0
$726$	926.479	0.0473621
$727$	24025.9	1.22568	0.612842	−	0.790205i	$-0.290026\pi$
$727$	24025.9	1.22568	0.612842	+	0.790205i	$0.290026\pi$
$728$	1643.42	0.0836663
$729$	−4283.00	−0.217599
$730$	0	0
$731$	20123.3	1.01818
$732$	−840.723	−0.0424509
$733$	−31004.0	−1.56229	−0.781145	−	0.624350i	$-0.785364\pi$
$733$	−31004.0	−1.56229	−0.781145	+	0.624350i	$0.785364\pi$
$734$	−11286.3	−0.567554
$735$	0	0
$736$	−18789.9	−0.941038
$737$	−3846.98	−0.192273
$738$	−27096.8	−1.35155
$739$	−5542.14	−0.275874	−0.137937	−	0.990441i	$-0.544047\pi$
$739$	−5542.14	−0.275874	−0.137937	+	0.990441i	$0.544047\pi$
$740$	0	0
$741$	−3608.24	−0.178882
$742$	4661.72	0.230643
$743$	−35227.3	−1.73939	−0.869694	−	0.493591i	$-0.835684\pi$
$743$	−35227.3	−1.73939	−0.869694	+	0.493591i	$0.835684\pi$
$744$	1756.74	0.0865662
$745$	0	0
$746$	31759.0	1.55868
$747$	11740.3	0.575039
$748$	−4663.08	−0.227940
$749$	−9577.71	−0.467239
$750$	0	0
$751$	19148.6	0.930415	0.465207	−	0.885202i	$-0.345980\pi$
$751$	19148.6	0.930415	0.465207	+	0.885202i	$0.345980\pi$
$752$	−42040.2	−2.03863
$753$	−12729.1	−0.616036
$754$	44844.9	2.16599
$755$	0	0
$756$	4659.80	0.224174
$757$	−12899.6	−0.619345	−0.309673	−	0.950843i	$-0.600219\pi$
$757$	−12899.6	−0.619345	−0.309673	+	0.950843i	$0.600219\pi$
$758$	−46466.7	−2.22658
$759$	−1736.06	−0.0830236
$760$	0	0
$761$	4568.70	0.217628	0.108814	−	0.994062i	$-0.465295\pi$
$761$	4568.70	0.217628	0.108814	+	0.994062i	$0.465295\pi$
$762$	12194.7	0.579748
$763$	9189.62	0.436024
$764$	−14902.1	−0.705677
$765$	0	0
$766$	34660.4	1.63490
$767$	−3772.97	−0.177619
$768$	10217.8	0.480080
$769$	6294.43	0.295166	0.147583	−	0.989050i	$-0.452851\pi$
$769$	6294.43	0.295166	0.147583	+	0.989050i	$0.452851\pi$
$770$	0	0
$771$	−15894.9	−0.742467
$772$	−19746.6	−0.920591
$773$	4218.70	0.196295	0.0981475	−	0.995172i	$-0.468708\pi$
$773$	4218.70	0.196295	0.0981475	+	0.995172i	$0.468708\pi$
$774$	−27825.0	−1.29218
$775$	0	0
$776$	2039.01	0.0943250
$777$	−3135.18	−0.144754
$778$	−4329.72	−0.199522
$779$	−12159.9	−0.559271
$780$	0	0
$781$	1309.55	0.0599990
$782$	19238.6	0.879758
$783$	−25655.8	−1.17096
$784$	−3574.12	−0.162815
$785$	0	0
$786$	9888.32	0.448734
$787$	−10453.6	−0.473482	−0.236741	−	0.971573i	$-0.576079\pi$
$787$	−10453.6	−0.473482	−0.236741	+	0.971573i	$0.576079\pi$
$788$	−26508.8	−1.19840
$789$	14661.6	0.661557
$790$	0	0
$791$	−2188.25	−0.0983631
$792$	−1300.96	−0.0583682
$793$	−2883.10	−0.129107
$794$	50203.7	2.24391
$795$	0	0
$796$	12615.0	0.561715
$797$	−2131.75	−0.0947436	−0.0473718	−	0.998877i	$-0.515085\pi$
$797$	−2131.75	−0.0947436	−0.0473718	+	0.998877i	$0.515085\pi$
$798$	2117.91	0.0939513
$799$	36703.2	1.62511
$800$	0	0
$801$	−25072.2	−1.10597
$802$	49023.4	2.15845
$803$	6306.50	0.277150
$804$	4656.15	0.204241
$805$	0	0
$806$	−29858.0	−1.30484
$807$	−7008.05	−0.305694
$808$	−4246.60	−0.184895
$809$	−35641.7	−1.54894	−0.774472	−	0.632608i	$-0.781984\pi$
$809$	−35641.7	−1.54894	−0.774472	+	0.632608i	$0.781984\pi$
$810$	0	0
$811$	−8034.96	−0.347898	−0.173949	−	0.984755i	$-0.555653\pi$
$811$	−8034.96	−0.347898	−0.173949	+	0.984755i	$0.555653\pi$
$812$	−11955.1	−0.516677
$813$	72.9991	0.00314907
$814$	−9430.77	−0.406079
$815$	0	0
$816$	−9289.96	−0.398546
$817$	−12486.7	−0.534703
$818$	35162.6	1.50297
$819$	7350.75	0.313622
$820$	0	0
$821$	37824.5	1.60790	0.803950	−	0.594697i	$-0.202728\pi$
$821$	37824.5	1.60790	0.803950	+	0.594697i	$0.202728\pi$
$822$	16138.8	0.684801
$823$	−19245.1	−0.815118	−0.407559	−	0.913179i	$-0.633620\pi$
$823$	−19245.1	−0.815118	−0.407559	+	0.913179i	$0.633620\pi$
$824$	6408.07	0.270917
$825$	0	0
$826$	2214.60	0.0932880
$827$	31294.1	1.31584	0.657921	−	0.753087i	$-0.271436\pi$
$827$	31294.1	1.31584	0.657921	+	0.753087i	$0.271436\pi$
$828$	−12082.0	−0.507099
$829$	−13121.7	−0.549740	−0.274870	−	0.961481i	$-0.588635\pi$
$829$	−13121.7	−0.549740	−0.274870	+	0.961481i	$0.588635\pi$
$830$	0	0
$831$	9930.27	0.414533
$832$	−14978.6	−0.624144
$833$	3120.38	0.129790
$834$	17750.9	0.737005
$835$	0	0
$836$	2893.48	0.119705
$837$	17081.8	0.705418
$838$	−23924.8	−0.986239
$839$	−5784.11	−0.238009	−0.119005	−	0.992894i	$-0.537970\pi$
$839$	−5784.11	−0.238009	−0.119005	+	0.992894i	$0.537970\pi$
$840$	0	0
$841$	41433.2	1.69885
$842$	18005.9	0.736965
$843$	−9120.84	−0.372643
$844$	−10030.5	−0.409081
$845$	0	0
$846$	−50750.5	−2.06246
$847$	−847.000	−0.0343604
$848$	12688.2	0.513814
$849$	1835.49	0.0741976
$850$	0	0
$851$	17671.6	0.711837
$852$	−1584.99	−0.0637335
$853$	−39103.1	−1.56959	−0.784797	−	0.619753i	$-0.787233\pi$
$853$	−39103.1	−1.56959	−0.784797	+	0.619753i	$0.787233\pi$
$854$	1692.28	0.0678087
$855$	0	0
$856$	−7035.70	−0.280929
$857$	−5793.14	−0.230910	−0.115455	−	0.993313i	$-0.536833\pi$
$857$	−5793.14	−0.230910	−0.115455	+	0.993313i	$0.536833\pi$
$858$	−3845.47	−0.153009
$859$	16365.8	0.650049	0.325025	−	0.945706i	$-0.394627\pi$
$859$	16365.8	0.650049	0.325025	+	0.945706i	$0.394627\pi$
$860$	0	0
$861$	−4308.22	−0.170527
$862$	28921.5	1.14277
$863$	20999.3	0.828303	0.414151	−	0.910208i	$-0.364078\pi$
$863$	20999.3	0.828303	0.414151	+	0.910208i	$0.364078\pi$
$864$	23811.3	0.937588
$865$	0	0
$866$	−10076.3	−0.395391
$867$	−1715.40	−0.0671949
$868$	7959.79	0.311259
$869$	−6251.98	−0.244055
$870$	0	0
$871$	15967.4	0.621164
$872$	6750.61	0.262161
$873$	9120.19	0.353576
$874$	−11937.7	−0.462012
$875$	0	0
$876$	−7632.99	−0.294400
$877$	−22042.8	−0.848727	−0.424364	−	0.905492i	$-0.639502\pi$
$877$	−22042.8	−0.848727	−0.424364	+	0.905492i	$0.639502\pi$
$878$	48051.4	1.84699
$879$	7967.94	0.305747
$880$	0	0
$881$	−7746.91	−0.296254	−0.148127	−	0.988968i	$-0.547324\pi$
$881$	−7746.91	−0.296254	−0.148127	+	0.988968i	$0.547324\pi$
$882$	−4314.64	−0.164718
$883$	12901.9	0.491715	0.245857	−	0.969306i	$-0.420930\pi$
$883$	12901.9	0.491715	0.245857	+	0.969306i	$0.420930\pi$
$884$	19354.7	0.736390
$885$	0	0
$886$	−42978.6	−1.62968
$887$	22230.9	0.841532	0.420766	−	0.907169i	$-0.361761\pi$
$887$	22230.9	0.841532	0.420766	+	0.907169i	$0.361761\pi$
$888$	−2303.07	−0.0870338
$889$	−11148.6	−0.420598
$890$	0	0
$891$	−4631.00	−0.174124
$892$	−28554.6	−1.07184
$893$	−22774.6	−0.853441
$894$	4609.89	0.172458
$895$	0	0
$896$	−4542.42	−0.169366
$897$	7205.72	0.268218
$898$	−42791.9	−1.59018
$899$	−43824.9	−1.62585
$900$	0	0
$901$	−11077.4	−0.409592
$902$	−12959.3	−0.478380
$903$	−4424.00	−0.163036
$904$	−1607.47	−0.0591412
$905$	0	0
$906$	−14763.9	−0.541389
$907$	35025.3	1.28225	0.641123	−	0.767438i	$-0.278469\pi$
$907$	35025.3	1.28225	0.641123	+	0.767438i	$0.278469\pi$
$908$	18963.5	0.693090
$909$	−18994.4	−0.693074
$910$	0	0
$911$	9605.98	0.349353	0.174676	−	0.984626i	$-0.444112\pi$
$911$	9605.98	0.349353	0.174676	+	0.984626i	$0.444112\pi$
$912$	5764.50	0.209300
$913$	5614.91	0.203534
$914$	−26953.5	−0.975428
$915$	0	0
$916$	21792.5	0.786076
$917$	−9040.04	−0.325549
$918$	−24379.9	−0.876532
$919$	−791.576	−0.0284132	−0.0142066	−	0.999899i	$-0.504522\pi$
$919$	−791.576	−0.0284132	−0.0142066	+	0.999899i	$0.504522\pi$
$920$	0	0
$921$	15985.0	0.571906
$922$	−5678.90	−0.202847
$923$	−5435.43	−0.193835
$924$	1025.16	0.0364991
$925$	0	0
$926$	11891.8	0.422019
$927$	28662.4	1.01553
$928$	−61089.8	−2.16096
$929$	25050.6	0.884695	0.442348	−	0.896844i	$-0.354146\pi$
$929$	25050.6	0.884695	0.442348	+	0.896844i	$0.354146\pi$
$930$	0	0
$931$	−1936.22	−0.0681601
$932$	−31778.6	−1.11689
$933$	14435.3	0.506528
$934$	36772.2	1.28825
$935$	0	0
$936$	5399.80	0.188566
$937$	−14769.7	−0.514948	−0.257474	−	0.966285i	$-0.582890\pi$
$937$	−14769.7	−0.514948	−0.257474	+	0.966285i	$0.582890\pi$
$938$	−9372.30	−0.326244
$939$	16831.2	0.584949
$940$	0	0
$941$	678.921	0.0235199	0.0117599	−	0.999931i	$-0.496257\pi$
$941$	678.921	0.0235199	0.0117599	+	0.999931i	$0.496257\pi$
$942$	−18961.8	−0.655849
$943$	24283.5	0.838578
$944$	6027.67	0.207822
$945$	0	0
$946$	−13307.6	−0.457366
$947$	21354.7	0.732772	0.366386	−	0.930463i	$-0.380595\pi$
$947$	21354.7	0.732772	0.366386	+	0.930463i	$0.380595\pi$
$948$	7567.00	0.259246
$949$	−26175.9	−0.895369
$950$	0	0
$951$	−1264.73	−0.0431248
$952$	2292.20	0.0780365
$953$	3942.06	0.133994	0.0669968	−	0.997753i	$-0.478658\pi$
$953$	3942.06	0.133994	0.0669968	+	0.997753i	$0.478658\pi$
$954$	15317.1	0.519820
$955$	0	0
$956$	13157.3	0.445122
$957$	−5644.29	−0.190652
$958$	34909.6	1.17733
$959$	−14754.3	−0.496812
$960$	0	0
$961$	−612.100	−0.0205465
$962$	39143.6	1.31189
$963$	−31469.6	−1.05306
$964$	−14889.6	−0.497471
$965$	0	0
$966$	−4229.51	−0.140872
$967$	27117.9	0.901813	0.450907	−	0.892571i	$-0.351101\pi$
$967$	27117.9	0.901813	0.450907	+	0.892571i	$0.351101\pi$
$968$	−622.198	−0.0206593
$969$	−5032.69	−0.166846
$970$	0	0
$971$	−28536.2	−0.943120	−0.471560	−	0.881834i	$-0.656309\pi$
$971$	−28536.2	−0.943120	−0.471560	+	0.881834i	$0.656309\pi$
$972$	23578.6	0.778069
$973$	−16228.1	−0.534685
$974$	−53770.7	−1.76892
$975$	0	0
$976$	4606.03	0.151061
$977$	−24692.4	−0.808577	−0.404288	−	0.914632i	$-0.632481\pi$
$977$	−24692.4	−0.808577	−0.404288	+	0.914632i	$0.632481\pi$
$978$	8139.68	0.266133
$979$	−11991.1	−0.391457
$980$	0	0
$981$	30194.5	0.982706
$982$	−3959.58	−0.128672
$983$	−26814.7	−0.870047	−0.435024	−	0.900419i	$-0.643260\pi$
$983$	−26814.7	−0.870047	−0.435024	+	0.900419i	$0.643260\pi$
$984$	−3164.78	−0.102530
$985$	0	0
$986$	62548.7	2.02024
$987$	−8069.00	−0.260222
$988$	−12009.7	−0.386722
$989$	24936.1	0.801741
$990$	0	0
$991$	46160.9	1.47967	0.739833	−	0.672790i	$-0.234904\pi$
$991$	46160.9	1.47967	0.739833	+	0.672790i	$0.234904\pi$
$992$	40674.0	1.30182
$993$	851.519	0.0272126
$994$	3190.41	0.101804
$995$	0	0
$996$	−6795.94	−0.216202
$997$	43200.4	1.37229	0.686143	−	0.727467i	$-0.259302\pi$
$997$	43200.4	1.37229	0.686143	+	0.727467i	$0.259302\pi$
$998$	−54870.0	−1.74036
$999$	−22394.1	−0.709228

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1925.4.a.l.1.2		2
5.4	even	2		77.4.a.b.1.1	✓	2
15.14	odd	2		693.4.a.i.1.2		2
20.19	odd	2		1232.4.a.m.1.2		2
35.34	odd	2		539.4.a.d.1.1		2
55.54	odd	2		847.4.a.c.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
77.4.a.b.1.1	✓	2	5.4	even	2
539.4.a.d.1.1		2	35.34	odd	2
693.4.a.i.1.2		2	15.14	odd	2
847.4.a.c.1.2		2	55.54	odd	2
1232.4.a.m.1.2		2	20.19	odd	2
1925.4.a.l.1.2		2	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1925 = 5^{2} \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1925.a (trivial)

\(f(q)\)	\(=\)	\(q+3.82843 q^{2} +2.00000 q^{3} +6.65685 q^{4} +7.65685 q^{6} -7.00000 q^{7} -5.14214 q^{8} -23.0000 q^{9} -11.0000 q^{11} +13.3137 q^{12} +45.6569 q^{13} -26.7990 q^{14} -72.9411 q^{16} +63.6812 q^{17} -88.0538 q^{18} -39.5147 q^{19} -14.0000 q^{21} -42.1127 q^{22} +78.9117 q^{23} -10.2843 q^{24} +174.794 q^{26} -100.000 q^{27} -46.5980 q^{28} +256.558 q^{29} -170.818 q^{31} -238.113 q^{32} -22.0000 q^{33} +243.799 q^{34} -153.108 q^{36} +223.941 q^{37} -151.279 q^{38} +91.3137 q^{39} +307.730 q^{41} -53.5980 q^{42} +316.000 q^{43} -73.2254 q^{44} +302.108 q^{46} +576.357 q^{47} -145.882 q^{48} +49.0000 q^{49} +127.362 q^{51} +303.931 q^{52} -173.951 q^{53} -382.843 q^{54} +35.9949 q^{56} -79.0294 q^{57} +982.215 q^{58} -82.6375 q^{59} -63.1472 q^{61} -653.966 q^{62} +161.000 q^{63} -328.068 q^{64} -84.2254 q^{66} +349.726 q^{67} +423.917 q^{68} +157.823 q^{69} -119.050 q^{71} +118.269 q^{72} -573.318 q^{73} +857.342 q^{74} -263.044 q^{76} +77.0000 q^{77} +349.588 q^{78} +568.362 q^{79} +421.000 q^{81} +1178.12 q^{82} -510.447 q^{83} -93.1960 q^{84} +1209.78 q^{86} +513.117 q^{87} +56.5635 q^{88} +1090.10 q^{89} -319.598 q^{91} +525.304 q^{92} -341.637 q^{93} +2206.54 q^{94} -476.225 q^{96} -396.530 q^{97} +187.593 q^{98} +253.000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + 4 q^{3} + 2 q^{4} + 4 q^{6} - 14 q^{7} + 18 q^{8} - 46 q^{9} - 22 q^{11} + 4 q^{12} + 80 q^{13} - 14 q^{14} - 78 q^{16} - 48 q^{17} - 46 q^{18} - 96 q^{19} - 28 q^{21} - 22 q^{22} + 56 q^{23}+ \cdots + 506 q^{99}+O(q^{100}) \) 2 * q + 2 * q^2 + 4 * q^3 + 2 * q^4 + 4 * q^6 - 14 * q^7 + 18 * q^8 - 46 * q^9 - 22 * q^11 + 4 * q^12 + 80 * q^13 - 14 * q^14 - 78 * q^16 - 48 * q^17 - 46 * q^18 - 96 * q^19 - 28 * q^21 - 22 * q^22 + 56 * q^23 + 36 * q^24 + 112 * q^26 - 200 * q^27 - 14 * q^28 + 4 * q^29 + 60 * q^31 - 414 * q^32 - 44 * q^33 + 448 * q^34 - 46 * q^36 + 380 * q^37 - 48 * q^38 + 160 * q^39 + 112 * q^41 - 28 * q^42 + 632 * q^43 - 22 * q^44 + 344 * q^46 + 604 * q^47 - 156 * q^48 + 98 * q^49 - 96 * q^51 + 144 * q^52 - 676 * q^53 - 200 * q^54 - 126 * q^56 - 192 * q^57 + 1444 * q^58 - 516 * q^59 - 296 * q^61 - 1076 * q^62 + 322 * q^63 + 34 * q^64 - 44 * q^66 + 1152 * q^67 + 944 * q^68 + 112 * q^69 - 1064 * q^71 - 414 * q^72 - 168 * q^73 + 572 * q^74 + 154 * q^77 + 224 * q^78 - 368 * q^79 + 842 * q^81 + 1536 * q^82 - 1728 * q^83 - 28 * q^84 + 632 * q^86 + 8 * q^87 - 198 * q^88 + 1524 * q^89 - 560 * q^91 + 632 * q^92 + 120 * q^93 + 2156 * q^94 - 828 * q^96 - 1404 * q^97 + 98 * q^98 + 506 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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