LMFDB - Embedded newform 1150.4.a.w.1.6

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1150,4,Mod(1,1150)] 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("1150.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1150, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 1150 = 2 \cdot 5^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1150.a (trivial)

Newform invariants

comment:select newform

sage:traces = [6,-12,5] 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:	$67.8521965066$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} - 109x^{4} + 94x^{3} + 2808x^{2} + 81x - 9774 $ x^6 - x^5 - 109x^4 + 94x^3 + 2808x^2 + 81x - 9774
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$-8.71212$ of defining polynomial
Character	$\chi$	$=$	1150.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-2.00000 q^{2} +9.71212 q^{3} +4.00000 q^{4} -19.4242 q^{6} +33.1288 q^{7} -8.00000 q^{8} +67.3252 q^{9} +1.59705 q^{11} +38.8485 q^{12} -15.1238 q^{13} -66.2575 q^{14} +16.0000 q^{16} -63.9661 q^{17} -134.650 q^{18} -110.153 q^{19} +321.750 q^{21} -3.19409 q^{22} +23.0000 q^{23} -77.6969 q^{24} +30.2477 q^{26} +391.643 q^{27} +132.515 q^{28} +191.979 q^{29} +292.477 q^{31} -32.0000 q^{32} +15.5107 q^{33} +127.932 q^{34} +269.301 q^{36} -62.7459 q^{37} +220.306 q^{38} -146.885 q^{39} +296.314 q^{41} -643.501 q^{42} -90.3963 q^{43} +6.38819 q^{44} -46.0000 q^{46} +142.749 q^{47} +155.394 q^{48} +754.515 q^{49} -621.246 q^{51} -60.4954 q^{52} +589.530 q^{53} -783.286 q^{54} -265.030 q^{56} -1069.82 q^{57} -383.959 q^{58} -175.690 q^{59} -809.615 q^{61} -584.955 q^{62} +2230.40 q^{63} +64.0000 q^{64} -31.0214 q^{66} -431.173 q^{67} -255.864 q^{68} +223.379 q^{69} +453.657 q^{71} -538.602 q^{72} +111.788 q^{73} +125.492 q^{74} -440.612 q^{76} +52.9082 q^{77} +293.769 q^{78} -1175.57 q^{79} +1985.90 q^{81} -592.628 q^{82} -95.0748 q^{83} +1287.00 q^{84} +180.793 q^{86} +1864.52 q^{87} -12.7764 q^{88} +836.699 q^{89} -501.034 q^{91} +92.0000 q^{92} +2840.57 q^{93} -285.499 q^{94} -310.788 q^{96} +599.979 q^{97} -1509.03 q^{98} +107.522 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 12 q^{2} + 5 q^{3} + 24 q^{4} - 10 q^{6} + 42 q^{7} - 48 q^{8} + 61 q^{9} - 49 q^{11} + 20 q^{12} + 16 q^{13} - 84 q^{14} + 96 q^{16} + 175 q^{17} - 122 q^{18} - 229 q^{19} + 92 q^{21} + 98 q^{22}+ \cdots - 142 q^{99}+O(q^{100}) $ 6 * q - 12 * q^2 + 5 * q^3 + 24 * q^4 - 10 * q^6 + 42 * q^7 - 48 * q^8 + 61 * q^9 - 49 * q^11 + 20 * q^12 + 16 * q^13 - 84 * q^14 + 96 * q^16 + 175 * q^17 - 122 * q^18 - 229 * q^19 + 92 * q^21 + 98 * q^22 + 138 * q^23 - 40 * q^24 - 32 * q^26 + 344 * q^27 + 168 * q^28 - 182 * q^29 + 114 * q^31 - 192 * q^32 + 641 * q^33 - 350 * q^34 + 244 * q^36 + 64 * q^37 + 458 * q^38 - 143 * q^39 + 243 * q^41 - 184 * q^42 + 182 * q^43 - 196 * q^44 - 276 * q^46 + 498 * q^47 + 80 * q^48 + 648 * q^49 - 1031 * q^51 + 64 * q^52 + 1290 * q^53 - 688 * q^54 - 336 * q^56 - 353 * q^57 + 364 * q^58 - 559 * q^59 - 688 * q^61 - 228 * q^62 + 3656 * q^63 + 384 * q^64 - 1282 * q^66 - 2069 * q^67 + 700 * q^68 + 115 * q^69 + 584 * q^71 - 488 * q^72 + 2485 * q^73 - 128 * q^74 - 916 * q^76 - 810 * q^77 + 286 * q^78 - 1432 * q^79 + 34 * q^81 - 486 * q^82 + 2089 * q^83 + 368 * q^84 - 364 * q^86 + 527 * q^87 + 392 * q^88 - 591 * q^89 - 2864 * q^91 + 552 * q^92 + 4899 * q^93 - 996 * q^94 - 160 * q^96 - 968 * q^97 - 1296 * q^98 - 142 * 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 $

$2$	−2.00000	−0.707107
$2$	−2.00000	−0.707107
$3$	9.71212	1.86910	0.934549	−	0.355835i	$-0.115803\pi$
$3$	9.71212	1.86910	0.934549	+	0.355835i	$0.115803\pi$
$4$	4.00000	0.500000
$5$	0	0
$5$	0	0
$6$	−19.4242	−1.32165
$7$	33.1288	1.78878	0.894392	−	0.447283i	$-0.147608\pi$
$7$	33.1288	1.78878	0.894392	+	0.447283i	$0.147608\pi$
$8$	−8.00000	−0.353553
$9$	67.3252	2.49353
$10$	0	0
$11$	1.59705	0.0437753	0.0218876	−	0.999760i	$-0.493032\pi$
$11$	1.59705	0.0437753	0.0218876	+	0.999760i	$0.493032\pi$
$12$	38.8485	0.934549
$13$	−15.1238	−0.322661	−0.161331	−	0.986900i	$-0.551579\pi$
$13$	−15.1238	−0.322661	−0.161331	+	0.986900i	$0.551579\pi$
$14$	−66.2575	−1.26486
$15$	0	0
$16$	16.0000	0.250000
$17$	−63.9661	−0.912591	−0.456296	−	0.889828i	$-0.650824\pi$
$17$	−63.9661	−0.912591	−0.456296	+	0.889828i	$0.650824\pi$
$18$	−134.650	−1.76319
$19$	−110.153	−1.33004	−0.665022	−	0.746824i	$-0.731578\pi$
$19$	−110.153	−1.33004	−0.665022	+	0.746824i	$0.731578\pi$
$20$	0	0
$21$	321.750	3.34341
$22$	−3.19409	−0.0309538
$23$	23.0000	0.208514
$23$	23.0000	0.208514
$24$	−77.6969	−0.660826
$25$	0	0
$26$	30.2477	0.228156
$27$	391.643	2.79155
$28$	132.515	0.894392
$29$	191.979	1.22930	0.614649	−	0.788801i	$-0.289298\pi$
$29$	191.979	1.22930	0.614649	+	0.788801i	$0.289298\pi$
$30$	0	0
$31$	292.477	1.69453	0.847266	−	0.531169i	$-0.178247\pi$
$31$	292.477	1.69453	0.847266	+	0.531169i	$0.178247\pi$
$32$	−32.0000	−0.176777
$33$	15.5107	0.0818202
$34$	127.932	0.645299
$35$	0	0
$36$	269.301	1.24676
$37$	−62.7459	−0.278794	−0.139397	−	0.990237i	$-0.544516\pi$
$37$	−62.7459	−0.278794	−0.139397	+	0.990237i	$0.544516\pi$
$38$	220.306	0.940483
$39$	−146.885	−0.603086
$40$	0	0
$41$	296.314	1.12869	0.564347	−	0.825538i	$-0.309128\pi$
$41$	296.314	1.12869	0.564347	+	0.825538i	$0.309128\pi$
$42$	−643.501	−2.36415
$43$	−90.3963	−0.320589	−0.160294	−	0.987069i	$-0.551244\pi$
$43$	−90.3963	−0.320589	−0.160294	+	0.987069i	$0.551244\pi$
$44$	6.38819	0.0218876
$45$	0	0
$46$	−46.0000	−0.147442
$47$	142.749	0.443024	0.221512	−	0.975158i	$-0.428901\pi$
$47$	142.749	0.443024	0.221512	+	0.975158i	$0.428901\pi$
$48$	155.394	0.467274
$49$	754.515	2.19975
$50$	0	0
$51$	−621.246	−1.70572
$52$	−60.4954	−0.161331
$53$	589.530	1.52789	0.763945	−	0.645281i	$-0.223260\pi$
$53$	589.530	1.52789	0.763945	+	0.645281i	$0.223260\pi$
$54$	−783.286	−1.97392
$55$	0	0
$56$	−265.030	−0.632431
$57$	−1069.82	−2.48598
$58$	−383.959	−0.869245
$59$	−175.690	−0.387675	−0.193838	−	0.981034i	$-0.562093\pi$
$59$	−175.690	−0.387675	−0.193838	+	0.981034i	$0.562093\pi$
$60$	0	0
$61$	−809.615	−1.69935	−0.849677	−	0.527303i	$-0.823203\pi$
$61$	−809.615	−1.69935	−0.849677	+	0.527303i	$0.823203\pi$
$62$	−584.955	−1.19821
$63$	2230.40	4.46038
$64$	64.0000	0.125000
$65$	0	0
$66$	−31.0214	−0.0578556
$67$	−431.173	−0.786212	−0.393106	−	0.919493i	$-0.628599\pi$
$67$	−431.173	−0.786212	−0.393106	+	0.919493i	$0.628599\pi$
$68$	−255.864	−0.456296
$69$	223.379	0.389734
$70$	0	0
$71$	453.657	0.758298	0.379149	−	0.925336i	$-0.376217\pi$
$71$	453.657	0.758298	0.379149	+	0.925336i	$0.376217\pi$
$72$	−538.602	−0.881595
$73$	111.788	0.179230	0.0896149	−	0.995976i	$-0.471436\pi$
$73$	111.788	0.179230	0.0896149	+	0.995976i	$0.471436\pi$
$74$	125.492	0.197137
$75$	0	0
$76$	−440.612	−0.665022
$77$	52.9082	0.0783045
$78$	293.769	0.426446
$79$	−1175.57	−1.67420	−0.837102	−	0.547047i	$-0.815752\pi$
$79$	−1175.57	−1.67420	−0.837102	+	0.547047i	$0.815752\pi$
$80$	0	0
$81$	1985.90	2.72415
$82$	−592.628	−0.798107
$83$	−95.0748	−0.125733	−0.0628663	−	0.998022i	$-0.520024\pi$
$83$	−95.0748	−0.125733	−0.0628663	+	0.998022i	$0.520024\pi$
$84$	1287.00	1.67171
$85$	0	0
$86$	180.793	0.226690
$87$	1864.52	2.29768
$88$	−12.7764	−0.0154769
$89$	836.699	0.996516	0.498258	−	0.867029i	$-0.333973\pi$
$89$	836.699	0.996516	0.498258	+	0.867029i	$0.333973\pi$
$90$	0	0
$91$	−501.034	−0.577172
$92$	92.0000	0.104257
$93$	2840.57	3.16724
$94$	−285.499	−0.313265
$95$	0	0
$96$	−310.788	−0.330413
$97$	599.979	0.628027	0.314013	−	0.949419i	$-0.398326\pi$
$97$	599.979	0.628027	0.314013	+	0.949419i	$0.398326\pi$
$98$	−1509.03	−1.55546
$99$	107.522	0.109155
$100$	0	0
$101$	845.212	0.832691	0.416345	−	0.909207i	$-0.363311\pi$
$101$	845.212	0.832691	0.416345	+	0.909207i	$0.363311\pi$
$102$	1242.49	1.20613
$103$	1103.23	1.05538	0.527690	−	0.849437i	$-0.323058\pi$
$103$	1103.23	1.05538	0.527690	+	0.849437i	$0.323058\pi$
$104$	120.991	0.114078
$105$	0	0
$106$	−1179.06	−1.08038
$107$	−1102.86	−0.996426	−0.498213	−	0.867055i	$-0.666010\pi$
$107$	−1102.86	−0.996426	−0.498213	+	0.867055i	$0.666010\pi$
$108$	1566.57	1.39577
$109$	−1377.62	−1.21057	−0.605284	−	0.796010i	$-0.706940\pi$
$109$	−1377.62	−1.21057	−0.605284	+	0.796010i	$0.706940\pi$
$110$	0	0
$111$	−609.395	−0.521092
$112$	530.060	0.447196
$113$	989.295	0.823584	0.411792	−	0.911278i	$-0.364903\pi$
$113$	989.295	0.823584	0.411792	+	0.911278i	$0.364903\pi$
$114$	2139.64	1.75785
$115$	0	0
$116$	767.917	0.614649
$117$	−1018.22	−0.804565
$118$	351.379	0.274128
$119$	−2119.12	−1.63243
$120$	0	0
$121$	−1328.45	−0.998084
$122$	1619.23	1.20163
$123$	2877.83	2.10964
$124$	1169.91	0.847266
$125$	0	0
$126$	−4460.80	−3.15397
$127$	2088.22	1.45905	0.729525	−	0.683954i	$-0.239741\pi$
$127$	2088.22	1.45905	0.729525	+	0.683954i	$0.239741\pi$
$128$	−128.000	−0.0883883
$129$	−877.940	−0.599211
$130$	0	0
$131$	−2153.34	−1.43617	−0.718085	−	0.695955i	$-0.754981\pi$
$131$	−2153.34	−1.43617	−0.718085	+	0.695955i	$0.754981\pi$
$132$	62.0428	0.0409101
$133$	−3649.23	−2.37916
$134$	862.346	0.555936
$135$	0	0
$136$	511.729	0.322650
$137$	−1308.13	−0.815771	−0.407886	−	0.913033i	$-0.633734\pi$
$137$	−1308.13	−0.815771	−0.407886	+	0.913033i	$0.633734\pi$
$138$	−446.757	−0.275583
$139$	−2122.45	−1.29513	−0.647567	−	0.762008i	$-0.724214\pi$
$139$	−2122.45	−1.29513	−0.647567	+	0.762008i	$0.724214\pi$
$140$	0	0
$141$	1386.40	0.828055
$142$	−907.314	−0.536198
$143$	−24.1535	−0.0141246
$144$	1077.20	0.623381
$145$	0	0
$146$	−223.576	−0.126735
$147$	7327.93	4.11155
$148$	−250.984	−0.139397
$149$	−1631.01	−0.896765	−0.448382	−	0.893842i	$-0.648000\pi$
$149$	−1631.01	−0.896765	−0.448382	+	0.893842i	$0.648000\pi$
$150$	0	0
$151$	1130.37	0.609192	0.304596	−	0.952482i	$-0.401479\pi$
$151$	1130.37	0.609192	0.304596	+	0.952482i	$0.401479\pi$
$152$	881.224	0.470241
$153$	−4306.53	−2.27557
$154$	−105.816	−0.0553697
$155$	0	0
$156$	−587.538	−0.301543
$157$	411.116	0.208985	0.104493	−	0.994526i	$-0.466678\pi$
$157$	411.116	0.208985	0.104493	+	0.994526i	$0.466678\pi$
$158$	2351.14	1.18384
$159$	5725.58	2.85578
$160$	0	0
$161$	761.961	0.372987
$162$	−3971.80	−1.92626
$163$	−3150.89	−1.51409	−0.757046	−	0.653362i	$-0.773358\pi$
$163$	−3150.89	−1.51409	−0.757046	+	0.653362i	$0.773358\pi$
$164$	1185.26	0.564347
$165$	0	0
$166$	190.150	0.0889064
$167$	−3685.50	−1.70774	−0.853871	−	0.520485i	$-0.825751\pi$
$167$	−3685.50	−1.70774	−0.853871	+	0.520485i	$0.825751\pi$
$168$	−2574.00	−1.18208
$169$	−1968.27	−0.895890
$170$	0	0
$171$	−7416.07	−3.31650
$172$	−361.585	−0.160294
$173$	4516.07	1.98468	0.992342	−	0.123522i	$-0.0394190\pi$
$173$	4516.07	1.98468	0.992342	+	0.123522i	$0.0394190\pi$
$174$	−3729.05	−1.62470
$175$	0	0
$176$	25.5528	0.0109438
$177$	−1706.32	−0.724603
$178$	−1673.40	−0.704643
$179$	3047.24	1.27241	0.636206	−	0.771519i	$-0.280503\pi$
$179$	3047.24	1.27241	0.636206	+	0.771519i	$0.280503\pi$
$180$	0	0
$181$	−185.447	−0.0761558	−0.0380779	−	0.999275i	$-0.512124\pi$
$181$	−185.447	−0.0761558	−0.0380779	+	0.999275i	$0.512124\pi$
$182$	1002.07	0.408122
$183$	−7863.08	−3.17626
$184$	−184.000	−0.0737210
$185$	0	0
$186$	−5681.15	−2.23958
$187$	−102.157	−0.0399489
$188$	570.998	0.221512
$189$	12974.6	4.99347
$190$	0	0
$191$	3449.76	1.30689	0.653445	−	0.756974i	$-0.273323\pi$
$191$	3449.76	1.30689	0.653445	+	0.756974i	$0.273323\pi$
$192$	621.575	0.233637
$193$	1593.41	0.594279	0.297140	−	0.954834i	$-0.403967\pi$
$193$	1593.41	0.594279	0.297140	+	0.954834i	$0.403967\pi$
$194$	−1199.96	−0.444082
$195$	0	0
$196$	3018.06	1.09988
$197$	−91.9830	−0.0332666	−0.0166333	−	0.999862i	$-0.505295\pi$
$197$	−91.9830	−0.0332666	−0.0166333	+	0.999862i	$0.505295\pi$
$198$	−215.043	−0.0771841
$199$	−3252.36	−1.15856	−0.579281	−	0.815128i	$-0.696667\pi$
$199$	−3252.36	−1.15856	−0.579281	+	0.815128i	$0.696667\pi$
$200$	0	0
$201$	−4187.60	−1.46951
$202$	−1690.42	−0.588801
$203$	6360.03	2.19895
$204$	−2484.98	−0.852861
$205$	0	0
$206$	−2206.45	−0.746267
$207$	1548.48	0.519936
$208$	−241.981	−0.0806654
$209$	−175.919	−0.0582230
$210$	0	0
$211$	1591.56	0.519278	0.259639	−	0.965706i	$-0.416396\pi$
$211$	1591.56	0.519278	0.259639	+	0.965706i	$0.416396\pi$
$212$	2358.12	0.763945
$213$	4405.97	1.41733
$214$	2205.72	0.704580
$215$	0	0
$216$	−3133.14	−0.986960
$217$	9689.41	3.03115
$218$	2755.24	0.856000
$219$	1085.70	0.334998
$220$	0	0
$221$	967.413	0.294458
$222$	1218.79	0.368468
$223$	2299.64	0.690561	0.345280	−	0.938500i	$-0.387784\pi$
$223$	2299.64	0.690561	0.345280	+	0.938500i	$0.387784\pi$
$224$	−1060.12	−0.316215
$225$	0	0
$226$	−1978.59	−0.582362
$227$	3254.18	0.951488	0.475744	−	0.879584i	$-0.342179\pi$
$227$	3254.18	0.951488	0.475744	+	0.879584i	$0.342179\pi$
$228$	−4279.27	−1.24299
$229$	−4934.80	−1.42402	−0.712010	−	0.702169i	$-0.752215\pi$
$229$	−4934.80	−1.42402	−0.712010	+	0.702169i	$0.752215\pi$
$230$	0	0
$231$	513.850	0.146359
$232$	−1535.83	−0.434623
$233$	−1194.48	−0.335849	−0.167925	−	0.985800i	$-0.553706\pi$
$233$	−1194.48	−0.335849	−0.167925	+	0.985800i	$0.553706\pi$
$234$	2036.43	0.568913
$235$	0	0
$236$	−702.758	−0.193838
$237$	−11417.3	−3.12925
$238$	4238.23	1.15430
$239$	1552.61	0.420208	0.210104	−	0.977679i	$-0.432620\pi$
$239$	1552.61	0.420208	0.210104	+	0.977679i	$0.432620\pi$
$240$	0	0
$241$	−4178.20	−1.11677	−0.558384	−	0.829582i	$-0.688579\pi$
$241$	−4178.20	−1.11677	−0.558384	+	0.829582i	$0.688579\pi$
$242$	2656.90	0.705752
$243$	8712.95	2.30015
$244$	−3238.46	−0.849677
$245$	0	0
$246$	−5755.67	−1.49174
$247$	1665.94	0.429154
$248$	−2339.82	−0.599107
$249$	−923.377	−0.235007
$250$	0	0
$251$	672.494	0.169113	0.0845567	−	0.996419i	$-0.473053\pi$
$251$	672.494	0.169113	0.0845567	+	0.996419i	$0.473053\pi$
$252$	8921.60	2.23019
$253$	36.7321	0.00912777
$254$	−4176.43	−1.03170
$255$	0	0
$256$	256.000	0.0625000
$257$	−1986.84	−0.482239	−0.241119	−	0.970495i	$-0.577515\pi$
$257$	−1986.84	−0.482239	−0.241119	+	0.970495i	$0.577515\pi$
$258$	1755.88	0.423707
$259$	−2078.69	−0.498702
$260$	0	0
$261$	12925.0	3.06529
$262$	4306.68	1.01553
$263$	−1583.85	−0.371347	−0.185673	−	0.982612i	$-0.559447\pi$
$263$	−1583.85	−0.371347	−0.185673	+	0.982612i	$0.559447\pi$
$264$	−124.086	−0.0289278
$265$	0	0
$266$	7298.46	1.68232
$267$	8126.12	1.86259
$268$	−1724.69	−0.393106
$269$	−5309.66	−1.20348	−0.601739	−	0.798693i	$-0.705525\pi$
$269$	−5309.66	−1.20348	−0.601739	+	0.798693i	$0.705525\pi$
$270$	0	0
$271$	−1083.04	−0.242767	−0.121383	−	0.992606i	$-0.538733\pi$
$271$	−1083.04	−0.242767	−0.121383	+	0.992606i	$0.538733\pi$
$272$	−1023.46	−0.228148
$273$	−4866.10	−1.07879
$274$	2616.25	0.576837
$275$	0	0
$276$	893.515	0.194867
$277$	−8793.99	−1.90751	−0.953753	−	0.300590i	$-0.902816\pi$
$277$	−8793.99	−1.90751	−0.953753	+	0.300590i	$0.902816\pi$
$278$	4244.90	0.915799
$279$	19691.1	4.22536
$280$	0	0
$281$	6603.86	1.40197	0.700984	−	0.713177i	$-0.252744\pi$
$281$	6603.86	1.40197	0.700984	+	0.713177i	$0.252744\pi$
$282$	−2772.80	−0.585524
$283$	7387.79	1.55180	0.775898	−	0.630858i	$-0.217297\pi$
$283$	7387.79	1.55180	0.775898	+	0.630858i	$0.217297\pi$
$284$	1814.63	0.379149
$285$	0	0
$286$	48.3070	0.00998759
$287$	9816.51	2.01899
$288$	−2154.41	−0.440797
$289$	−821.341	−0.167177
$290$	0	0
$291$	5827.06	1.17384
$292$	447.151	0.0896149
$293$	4539.89	0.905198	0.452599	−	0.891714i	$-0.350497\pi$
$293$	4539.89	0.905198	0.452599	+	0.891714i	$0.350497\pi$
$294$	−14655.9	−2.90730
$295$	0	0
$296$	501.967	0.0985684
$297$	625.472	0.122201
$298$	3262.03	0.634108
$299$	−347.848	−0.0672796
$300$	0	0
$301$	−2994.72	−0.573464
$302$	−2260.73	−0.430764
$303$	8208.80	1.55638
$304$	−1762.45	−0.332511
$305$	0	0
$306$	8613.06	1.60907
$307$	−2906.64	−0.540360	−0.270180	−	0.962810i	$-0.587083\pi$
$307$	−2906.64	−0.540360	−0.270180	+	0.962810i	$0.587083\pi$
$308$	211.633	0.0391523
$309$	10714.7	1.97261
$310$	0	0
$311$	−8212.89	−1.49746	−0.748730	−	0.662875i	$-0.769336\pi$
$311$	−8212.89	−1.49746	−0.748730	+	0.662875i	$0.769336\pi$
$312$	1175.08	0.213223
$313$	−201.103	−0.0363163	−0.0181582	−	0.999835i	$-0.505780\pi$
$313$	−201.103	−0.0363163	−0.0181582	+	0.999835i	$0.505780\pi$
$314$	−822.233	−0.147775
$315$	0	0
$316$	−4702.29	−0.837102
$317$	−2545.20	−0.450954	−0.225477	−	0.974248i	$-0.572394\pi$
$317$	−2545.20	−0.450954	−0.225477	+	0.974248i	$0.572394\pi$
$318$	−11451.2	−2.01934
$319$	306.600	0.0538129
$320$	0	0
$321$	−10711.1	−1.86242
$322$	−1523.92	−0.263742
$323$	7046.05	1.21379
$324$	7943.61	1.36207
$325$	0	0
$326$	6301.79	1.07062
$327$	−13379.6	−2.26267
$328$	−2370.51	−0.399053
$329$	4729.11	0.792475
$330$	0	0
$331$	7330.61	1.21730	0.608651	−	0.793438i	$-0.291711\pi$
$331$	7330.61	1.21730	0.608651	+	0.793438i	$0.291711\pi$
$332$	−380.299	−0.0628663
$333$	−4224.38	−0.695179
$334$	7371.01	1.20756
$335$	0	0
$336$	5148.01	0.835853
$337$	7988.83	1.29133	0.645666	−	0.763620i	$-0.276580\pi$
$337$	7988.83	1.29133	0.645666	+	0.763620i	$0.276580\pi$
$338$	3936.54	0.633490
$339$	9608.15	1.53936
$340$	0	0
$341$	467.100	0.0741786
$342$	14832.1	2.34512
$343$	13633.0	2.14610
$344$	723.171	0.113345
$345$	0	0
$346$	−9032.14	−1.40338
$347$	−9300.59	−1.43885	−0.719427	−	0.694569i	$-0.755595\pi$
$347$	−9300.59	−1.43885	−0.719427	+	0.694569i	$0.755595\pi$
$348$	7458.10	1.14884
$349$	−627.258	−0.0962073	−0.0481037	−	0.998842i	$-0.515318\pi$
$349$	−627.258	−0.0962073	−0.0481037	+	0.998842i	$0.515318\pi$
$350$	0	0
$351$	−5923.15	−0.900724
$352$	−51.1055	−0.00773845
$353$	−10275.9	−1.54938	−0.774691	−	0.632339i	$-0.782095\pi$
$353$	−10275.9	−1.54938	−0.774691	+	0.632339i	$0.782095\pi$
$354$	3412.63	0.512371
$355$	0	0
$356$	3346.80	0.498258
$357$	−20581.1	−3.05117
$358$	−6094.49	−0.899731
$359$	9083.80	1.33544	0.667722	−	0.744411i	$-0.267270\pi$
$359$	9083.80	1.33544	0.667722	+	0.744411i	$0.267270\pi$
$360$	0	0
$361$	5274.67	0.769015
$362$	370.895	0.0538503
$363$	−12902.1	−1.86552
$364$	−2004.14	−0.288586
$365$	0	0
$366$	15726.2	2.24595
$367$	−8578.97	−1.22021	−0.610107	−	0.792319i	$-0.708873\pi$
$367$	−8578.97	−1.22021	−0.610107	+	0.792319i	$0.708873\pi$
$368$	368.000	0.0521286
$369$	19949.4	2.81443
$370$	0	0
$371$	19530.4	2.73307
$372$	11362.3	1.58362
$373$	−10462.6	−1.45236	−0.726181	−	0.687503i	$-0.758707\pi$
$373$	−10462.6	−1.45236	−0.726181	+	0.687503i	$0.758707\pi$
$374$	204.314	0.0282482
$375$	0	0
$376$	−1142.00	−0.156633
$377$	−2903.46	−0.396647
$378$	−25949.3	−3.53092
$379$	−7320.67	−0.992184	−0.496092	−	0.868270i	$-0.665232\pi$
$379$	−7320.67	−0.992184	−0.496092	+	0.868270i	$0.665232\pi$
$380$	0	0
$381$	20281.0	2.72711
$382$	−6899.52	−0.924111
$383$	2226.12	0.296995	0.148498	−	0.988913i	$-0.452556\pi$
$383$	2226.12	0.296995	0.148498	+	0.988913i	$0.452556\pi$
$384$	−1243.15	−0.165206
$385$	0	0
$386$	−3186.81	−0.420219
$387$	−6085.95	−0.799396
$388$	2399.92	0.314013
$389$	1358.18	0.177024	0.0885121	−	0.996075i	$-0.471789\pi$
$389$	1358.18	0.177024	0.0885121	+	0.996075i	$0.471789\pi$
$390$	0	0
$391$	−1471.22	−0.190288
$392$	−6036.12	−0.777729
$393$	−20913.5	−2.68434
$394$	183.966	0.0235230
$395$	0	0
$396$	430.086	0.0545774
$397$	9938.92	1.25647	0.628237	−	0.778022i	$-0.283777\pi$
$397$	9938.92	1.25647	0.628237	+	0.778022i	$0.283777\pi$
$398$	6504.73	0.819227
$399$	−35441.7	−4.44688
$400$	0	0
$401$	−13384.5	−1.66680	−0.833401	−	0.552669i	$-0.813609\pi$
$401$	−13384.5	−1.66680	−0.833401	+	0.552669i	$0.813609\pi$
$402$	8375.21	1.03910
$403$	−4423.38	−0.546760
$404$	3380.85	0.416345
$405$	0	0
$406$	−12720.1	−1.55489
$407$	−100.208	−0.0122043
$408$	4969.97	0.603064
$409$	2957.56	0.357560	0.178780	−	0.983889i	$-0.442785\pi$
$409$	2957.56	0.357560	0.178780	+	0.983889i	$0.442785\pi$
$410$	0	0
$411$	−12704.7	−1.52476
$412$	4412.91	0.527690
$413$	−5820.38	−0.693467
$414$	−3096.96	−0.367650
$415$	0	0
$416$	483.963	0.0570390
$417$	−20613.5	−2.42073
$418$	351.839	0.0411699
$419$	−13178.4	−1.53654	−0.768268	−	0.640128i	$-0.778881\pi$
$419$	−13178.4	−1.53654	−0.768268	+	0.640128i	$0.778881\pi$
$420$	0	0
$421$	−1367.55	−0.158314	−0.0791569	−	0.996862i	$-0.525223\pi$
$421$	−1367.55	−0.158314	−0.0791569	+	0.996862i	$0.525223\pi$
$422$	−3183.12	−0.367185
$423$	9610.63	1.10469
$424$	−4716.24	−0.540191
$425$	0	0
$426$	−8811.94	−1.00221
$427$	−26821.6	−3.03978
$428$	−4411.44	−0.498213
$429$	−234.581	−0.0264002
$430$	0	0
$431$	34.6193	0.00386903	0.00193451	−	0.999998i	$-0.499384\pi$
$431$	34.6193	0.00386903	0.00193451	+	0.999998i	$0.499384\pi$
$432$	6266.29	0.697886
$433$	3270.08	0.362933	0.181466	−	0.983397i	$-0.441916\pi$
$433$	3270.08	0.362933	0.181466	+	0.983397i	$0.441916\pi$
$434$	−19378.8	−2.14335
$435$	0	0
$436$	−5510.47	−0.605284
$437$	−2533.52	−0.277333
$438$	−2171.39	−0.236879
$439$	6565.41	0.713781	0.356891	−	0.934146i	$-0.383837\pi$
$439$	6565.41	0.713781	0.356891	+	0.934146i	$0.383837\pi$
$440$	0	0
$441$	50797.8	5.48514
$442$	−1934.83	−0.208213
$443$	8809.74	0.944838	0.472419	−	0.881374i	$-0.343381\pi$
$443$	8809.74	0.944838	0.472419	+	0.881374i	$0.343381\pi$
$444$	−2437.58	−0.260546
$445$	0	0
$446$	−4599.27	−0.488300
$447$	−15840.6	−1.67614
$448$	2120.24	0.223598
$449$	−8977.39	−0.943584	−0.471792	−	0.881710i	$-0.656393\pi$
$449$	−8977.39	−0.943584	−0.471792	+	0.881710i	$0.656393\pi$
$450$	0	0
$451$	473.227	0.0494089
$452$	3957.18	0.411792
$453$	10978.3	1.13864
$454$	−6508.37	−0.672803
$455$	0	0
$456$	8558.55	0.878927
$457$	−856.195	−0.0876392	−0.0438196	−	0.999039i	$-0.513953\pi$
$457$	−856.195	−0.0876392	−0.0438196	+	0.999039i	$0.513953\pi$
$458$	9869.60	1.00693
$459$	−25051.9	−2.54754
$460$	0	0
$461$	367.756	0.0371543	0.0185771	−	0.999827i	$-0.494086\pi$
$461$	367.756	0.0371543	0.0185771	+	0.999827i	$0.494086\pi$
$462$	−1027.70	−0.103491
$463$	7330.47	0.735801	0.367901	−	0.929865i	$-0.380077\pi$
$463$	7330.47	0.735801	0.367901	+	0.929865i	$0.380077\pi$
$464$	3071.67	0.307325
$465$	0	0
$466$	2388.96	0.237481
$467$	−11503.1	−1.13983	−0.569914	−	0.821705i	$-0.693023\pi$
$467$	−11503.1	−1.13983	−0.569914	+	0.821705i	$0.693023\pi$
$468$	−4072.86	−0.402282
$469$	−14284.2	−1.40636
$470$	0	0
$471$	3992.81	0.390614
$472$	1405.52	0.137064
$473$	−144.367	−0.0140339
$474$	22834.6	2.21271
$475$	0	0
$476$	−8476.47	−0.816215
$477$	39690.2	3.80983
$478$	−3105.21	−0.297132
$479$	8580.47	0.818479	0.409240	−	0.912427i	$-0.365794\pi$
$479$	8580.47	0.818479	0.409240	+	0.912427i	$0.365794\pi$
$480$	0	0
$481$	948.959	0.0899559
$482$	8356.39	0.789675
$483$	7400.26	0.697150
$484$	−5313.80	−0.499042
$485$	0	0
$486$	−17425.9	−1.62645
$487$	15826.8	1.47265	0.736325	−	0.676628i	$-0.236559\pi$
$487$	15826.8	1.47265	0.736325	+	0.676628i	$0.236559\pi$
$488$	6476.92	0.600813
$489$	−30601.8	−2.82999
$490$	0	0
$491$	−6219.11	−0.571618	−0.285809	−	0.958287i	$-0.592262\pi$
$491$	−6219.11	−0.571618	−0.285809	+	0.958287i	$0.592262\pi$
$492$	11511.3	1.05482
$493$	−12280.2	−1.12185
$494$	−3331.87	−0.303457
$495$	0	0
$496$	4679.64	0.423633
$497$	15029.1	1.35643
$498$	1846.75	0.166175
$499$	12820.1	1.15011	0.575056	−	0.818114i	$-0.304980\pi$
$499$	12820.1	1.15011	0.575056	+	0.818114i	$0.304980\pi$
$500$	0	0
$501$	−35794.0	−3.19194
$502$	−1344.99	−0.119581
$503$	−423.446	−0.0375358	−0.0187679	−	0.999824i	$-0.505974\pi$
$503$	−423.446	−0.0375358	−0.0187679	+	0.999824i	$0.505974\pi$
$504$	−17843.2	−1.57698
$505$	0	0
$506$	−73.4642	−0.00645431
$507$	−19116.1	−1.67451
$508$	8352.87	0.729525
$509$	−13039.2	−1.13546	−0.567731	−	0.823214i	$-0.692179\pi$
$509$	−13039.2	−1.13546	−0.567731	+	0.823214i	$0.692179\pi$
$510$	0	0
$511$	3703.39	0.320603
$512$	−512.000	−0.0441942
$513$	−43140.6	−3.71288
$514$	3973.67	0.340994
$515$	0	0
$516$	−3511.76	−0.299606
$517$	227.978	0.0193935
$518$	4157.39	0.352635
$519$	43860.6	3.70957
$520$	0	0
$521$	13783.7	1.15907	0.579535	−	0.814948i	$-0.303234\pi$
$521$	13783.7	1.15907	0.579535	+	0.814948i	$0.303234\pi$
$522$	−25850.1	−2.16749
$523$	4223.40	0.353109	0.176555	−	0.984291i	$-0.443505\pi$
$523$	4223.40	0.353109	0.176555	+	0.984291i	$0.443505\pi$
$524$	−8613.36	−0.718085
$525$	0	0
$526$	3167.69	0.262582
$527$	−18708.6	−1.54641
$528$	248.171	0.0204551
$529$	529.000	0.0434783
$530$	0	0
$531$	−11828.3	−0.966678
$532$	−14596.9	−1.18958
$533$	−4481.40	−0.364186
$534$	−16252.2	−1.31705
$535$	0	0
$536$	3449.39	0.277968
$537$	29595.2	2.37826
$538$	10619.3	0.850987
$539$	1205.00	0.0962947
$540$	0	0
$541$	−8572.42	−0.681252	−0.340626	−	0.940199i	$-0.610639\pi$
$541$	−8572.42	−0.681252	−0.340626	+	0.940199i	$0.610639\pi$
$542$	2166.07	0.171662
$543$	−1801.09	−0.142343
$544$	2046.91	0.161325
$545$	0	0
$546$	9732.20	0.762820
$547$	17318.1	1.35369	0.676845	−	0.736125i	$-0.263347\pi$
$547$	17318.1	1.35369	0.676845	+	0.736125i	$0.263347\pi$
$548$	−5232.50	−0.407886
$549$	−54507.5	−4.23738
$550$	0	0
$551$	−21147.1	−1.63502
$552$	−1787.03	−0.137792
$553$	−38945.2	−2.99479
$554$	17588.0	1.34881
$555$	0	0
$556$	−8489.79	−0.647567
$557$	−17447.9	−1.32727	−0.663635	−	0.748057i	$-0.730987\pi$
$557$	−17447.9	−1.32727	−0.663635	+	0.748057i	$0.730987\pi$
$558$	−39382.2	−2.98778
$559$	1367.14	0.103442
$560$	0	0
$561$	−992.159	−0.0746684
$562$	−13207.7	−0.991341
$563$	−22775.6	−1.70493	−0.852465	−	0.522784i	$-0.824893\pi$
$563$	−22775.6	−1.70493	−0.852465	+	0.522784i	$0.824893\pi$
$564$	5545.59	0.414028
$565$	0	0
$566$	−14775.6	−1.09729
$567$	65790.5	4.87291
$568$	−3629.26	−0.268099
$569$	−14993.4	−1.10467	−0.552333	−	0.833624i	$-0.686262\pi$
$569$	−14993.4	−1.10467	−0.552333	+	0.833624i	$0.686262\pi$
$570$	0	0
$571$	5161.10	0.378258	0.189129	−	0.981952i	$-0.439434\pi$
$571$	5161.10	0.378258	0.189129	+	0.981952i	$0.439434\pi$
$572$	−96.6140	−0.00706229
$573$	33504.5	2.44270
$574$	−19633.0	−1.42764
$575$	0	0
$576$	4308.81	0.311691
$577$	7474.63	0.539294	0.269647	−	0.962959i	$-0.413093\pi$
$577$	7474.63	0.539294	0.269647	+	0.962959i	$0.413093\pi$
$578$	1642.68	0.118212
$579$	15475.3	1.11077
$580$	0	0
$581$	−3149.71	−0.224909
$582$	−11654.1	−0.830033
$583$	941.507	0.0668838
$584$	−894.302	−0.0633673
$585$	0	0
$586$	−9079.78	−0.640072
$587$	10806.8	0.759869	0.379934	−	0.925013i	$-0.375947\pi$
$587$	10806.8	0.759869	0.379934	+	0.925013i	$0.375947\pi$
$588$	29311.7	2.05577
$589$	−32217.2	−2.25380
$590$	0	0
$591$	−893.349	−0.0621785
$592$	−1003.93	−0.0696984
$593$	2031.95	0.140712	0.0703559	−	0.997522i	$-0.477587\pi$
$593$	2031.95	0.140712	0.0703559	+	0.997522i	$0.477587\pi$
$594$	−1250.94	−0.0864089
$595$	0	0
$596$	−6524.06	−0.448382
$597$	−31587.3	−2.16547
$598$	695.697	0.0475738
$599$	−17544.8	−1.19677	−0.598383	−	0.801210i	$-0.704190\pi$
$599$	−17544.8	−1.19677	−0.598383	+	0.801210i	$0.704190\pi$
$600$	0	0
$601$	38.6142	0.00262081	0.00131041	−	0.999999i	$-0.499583\pi$
$601$	38.6142	0.00262081	0.00131041	+	0.999999i	$0.499583\pi$
$602$	5989.44	0.405500
$603$	−29028.8	−1.96044
$604$	4521.47	0.304596
$605$	0	0
$606$	−16417.6	−1.10053
$607$	−10452.9	−0.698961	−0.349480	−	0.936944i	$-0.613642\pi$
$607$	−10452.9	−0.698961	−0.349480	+	0.936944i	$0.613642\pi$
$608$	3524.89	0.235121
$609$	61769.4	4.11005
$610$	0	0
$611$	−2158.92	−0.142947
$612$	−17226.1	−1.13778
$613$	−27674.5	−1.82343	−0.911715	−	0.410823i	$-0.865241\pi$
$613$	−27674.5	−1.82343	−0.911715	+	0.410823i	$0.865241\pi$
$614$	5813.27	0.382092
$615$	0	0
$616$	−423.266	−0.0276848
$617$	24533.5	1.60078	0.800390	−	0.599480i	$-0.204626\pi$
$617$	24533.5	1.60078	0.800390	+	0.599480i	$0.204626\pi$
$618$	−21429.3	−1.39485
$619$	−1794.30	−0.116509	−0.0582545	−	0.998302i	$-0.518554\pi$
$619$	−1794.30	−0.116509	−0.0582545	+	0.998302i	$0.518554\pi$
$620$	0	0
$621$	9007.79	0.582077
$622$	16425.8	1.05886
$623$	27718.8	1.78255
$624$	−2350.15	−0.150771
$625$	0	0
$626$	402.206	0.0256795
$627$	−1708.55	−0.108824
$628$	1644.47	0.104493
$629$	4013.61	0.254425
$630$	0	0
$631$	−12771.3	−0.805735	−0.402868	−	0.915258i	$-0.631986\pi$
$631$	−12771.3	−0.805735	−0.402868	+	0.915258i	$0.631986\pi$
$632$	9404.57	0.591921
$633$	15457.4	0.970581
$634$	5090.39	0.318873
$635$	0	0
$636$	22902.3	1.42789
$637$	−11411.2	−0.709775
$638$	−613.200	−0.0380514
$639$	30542.5	1.89084
$640$	0	0
$641$	15626.1	0.962863	0.481432	−	0.876484i	$-0.340117\pi$
$641$	15626.1	0.962863	0.481432	+	0.876484i	$0.340117\pi$
$642$	21422.2	1.31693
$643$	12756.7	0.782384	0.391192	−	0.920309i	$-0.372063\pi$
$643$	12756.7	0.782384	0.391192	+	0.920309i	$0.372063\pi$
$644$	3047.85	0.186494
$645$	0	0
$646$	−14092.1	−0.858276
$647$	22.8807	0.00139031	0.000695157	−	1.00000i	$-0.499779\pi$
$647$	22.8807	0.00139031	0.000695157		1.00000i	$0.499779\pi$
$648$	−15887.2	−0.963131
$649$	−280.584	−0.0169706
$650$	0	0
$651$	94104.7	5.66552
$652$	−12603.6	−0.757046
$653$	5985.07	0.358674	0.179337	−	0.983788i	$-0.442605\pi$
$653$	5985.07	0.358674	0.179337	+	0.983788i	$0.442605\pi$
$654$	26759.2	1.59995
$655$	0	0
$656$	4741.02	0.282173
$657$	7526.13	0.446914
$658$	−9458.22	−0.560364
$659$	−12224.7	−0.722622	−0.361311	−	0.932445i	$-0.617671\pi$
$659$	−12224.7	−0.722622	−0.361311	+	0.932445i	$0.617671\pi$
$660$	0	0
$661$	−24940.4	−1.46758	−0.733790	−	0.679376i	$-0.762251\pi$
$661$	−24940.4	−1.46758	−0.733790	+	0.679376i	$0.762251\pi$
$662$	−14661.2	−0.860762
$663$	9395.62	0.550371
$664$	760.598	0.0444532
$665$	0	0
$666$	8448.76	0.491566
$667$	4415.52	0.256326
$668$	−14742.0	−0.853871
$669$	22334.3	1.29073
$670$	0	0
$671$	−1292.99	−0.0743897
$672$	−10296.0	−0.591038
$673$	15461.4	0.885576	0.442788	−	0.896626i	$-0.353989\pi$
$673$	15461.4	0.885576	0.442788	+	0.896626i	$0.353989\pi$
$674$	−15977.7	−0.913110
$675$	0	0
$676$	−7873.08	−0.447945
$677$	6904.24	0.391952	0.195976	−	0.980609i	$-0.437212\pi$
$677$	6904.24	0.391952	0.195976	+	0.980609i	$0.437212\pi$
$678$	−19216.3	−1.08849
$679$	19876.6	1.12341
$680$	0	0
$681$	31605.0	1.77842
$682$	−934.200	−0.0524522
$683$	9644.73	0.540330	0.270165	−	0.962814i	$-0.412922\pi$
$683$	9644.73	0.540330	0.270165	+	0.962814i	$0.412922\pi$
$684$	−29664.3	−1.65825
$685$	0	0
$686$	−27265.9	−1.51752
$687$	−47927.4	−2.66163
$688$	−1446.34	−0.0801472
$689$	−8915.96	−0.492991
$690$	0	0
$691$	−704.001	−0.0387575	−0.0193788	−	0.999812i	$-0.506169\pi$
$691$	−704.001	−0.0387575	−0.0193788	+	0.999812i	$0.506169\pi$
$692$	18064.3	0.992342
$693$	3562.05	0.195254
$694$	18601.2	1.01742
$695$	0	0
$696$	−14916.2	−0.812352
$697$	−18954.0	−1.03004
$698$	1254.52	0.0680289
$699$	−11600.9	−0.627735
$700$	0	0
$701$	2557.38	0.137790	0.0688952	−	0.997624i	$-0.478053\pi$
$701$	2557.38	0.137790	0.0688952	+	0.997624i	$0.478053\pi$
$702$	11846.3	0.636908
$703$	6911.64	0.370807
$704$	102.211	0.00547191
$705$	0	0
$706$	20551.8	1.09558
$707$	28000.8	1.48950
$708$	−6825.27	−0.362301
$709$	−17999.7	−0.953448	−0.476724	−	0.879053i	$-0.658176\pi$
$709$	−17999.7	−0.953448	−0.476724	+	0.879053i	$0.658176\pi$
$710$	0	0
$711$	−79145.6	−4.17467
$712$	−6693.59	−0.352322
$713$	6726.98	0.353334
$714$	41162.2	2.15750
$715$	0	0
$716$	12189.0	0.636206
$717$	15079.1	0.785410
$718$	−18167.6	−0.944301
$719$	−24565.1	−1.27416	−0.637082	−	0.770796i	$-0.719859\pi$
$719$	−24565.1	−1.27416	−0.637082	+	0.770796i	$0.719859\pi$
$720$	0	0
$721$	36548.5	1.88785
$722$	−10549.3	−0.543775
$723$	−40579.1	−2.08735
$724$	−741.790	−0.0380779
$725$	0	0
$726$	25804.1	1.31912
$727$	16301.8	0.831638	0.415819	−	0.909447i	$-0.363495\pi$
$727$	16301.8	0.831638	0.415819	+	0.909447i	$0.363495\pi$
$728$	4008.27	0.204061
$729$	31001.8	1.57506
$730$	0	0
$731$	5782.30	0.292566
$732$	−31452.3	−1.58813
$733$	6884.70	0.346920	0.173460	−	0.984841i	$-0.444505\pi$
$733$	6884.70	0.346920	0.173460	+	0.984841i	$0.444505\pi$
$734$	17157.9	0.862821
$735$	0	0
$736$	−736.000	−0.0368605
$737$	−688.604	−0.0344166
$738$	−39898.8	−1.99010
$739$	−32833.4	−1.63436	−0.817182	−	0.576380i	$-0.804465\pi$
$739$	−32833.4	−1.63436	−0.817182	+	0.576380i	$0.804465\pi$
$740$	0	0
$741$	16179.8	0.802130
$742$	−39060.8	−1.93257
$743$	32228.8	1.59133	0.795666	−	0.605736i	$-0.207121\pi$
$743$	32228.8	1.59133	0.795666	+	0.605736i	$0.207121\pi$
$744$	−22724.6	−1.11979
$745$	0	0
$746$	20925.1	1.02698
$747$	−6400.93	−0.313518
$748$	−408.627	−0.0199745
$749$	−36536.4	−1.78239
$750$	0	0
$751$	−10938.3	−0.531481	−0.265741	−	0.964045i	$-0.585616\pi$
$751$	−10938.3	−0.531481	−0.265741	+	0.964045i	$0.585616\pi$
$752$	2283.99	0.110756
$753$	6531.34	0.316090
$754$	5806.93	0.280472
$755$	0	0
$756$	51898.6	2.49674
$757$	−13881.7	−0.666496	−0.333248	−	0.942839i	$-0.608145\pi$
$757$	−13881.7	−0.666496	−0.333248	+	0.942839i	$0.608145\pi$
$758$	14641.3	0.701580
$759$	356.746	0.0170607
$760$	0	0
$761$	−32811.6	−1.56297	−0.781485	−	0.623924i	$-0.785538\pi$
$761$	−32811.6	−1.56297	−0.781485	+	0.623924i	$0.785538\pi$
$762$	−40562.0	−1.92836
$763$	−45638.8	−2.16544
$764$	13799.0	0.653445
$765$	0	0
$766$	−4452.23	−0.210007
$767$	2657.10	0.125088
$768$	2486.30	0.116819
$769$	9987.66	0.468354	0.234177	−	0.972194i	$-0.424760\pi$
$769$	9987.66	0.468354	0.234177	+	0.972194i	$0.424760\pi$
$770$	0	0
$771$	−19296.4	−0.901352
$772$	6373.62	0.297140
$773$	−3484.56	−0.162135	−0.0810677	−	0.996709i	$-0.525833\pi$
$773$	−3484.56	−0.162135	−0.0810677	+	0.996709i	$0.525833\pi$
$774$	12171.9	0.565258
$775$	0	0
$776$	−4799.83	−0.222041
$777$	−20188.5	−0.932122
$778$	−2716.36	−0.125175
$779$	−32639.8	−1.50121
$780$	0	0
$781$	724.512	0.0331947
$782$	2942.44	0.134554
$783$	75187.3	3.43164
$784$	12072.2	0.549938
$785$	0	0
$786$	41827.0	1.89812
$787$	12421.1	0.562597	0.281298	−	0.959620i	$-0.409235\pi$
$787$	12421.1	0.562597	0.281298	+	0.959620i	$0.409235\pi$
$788$	−367.932	−0.0166333
$789$	−15382.5	−0.694083
$790$	0	0
$791$	32774.1	1.47322
$792$	−860.172	−0.0385920
$793$	12244.5	0.548316
$794$	−19877.8	−0.888461
$795$	0	0
$796$	−13009.5	−0.579281
$797$	24083.7	1.07037	0.535187	−	0.844734i	$-0.320241\pi$
$797$	24083.7	1.07037	0.535187	+	0.844734i	$0.320241\pi$
$798$	70883.5	3.14442
$799$	−9131.12	−0.404300
$800$	0	0
$801$	56330.9	2.48484
$802$	26768.9	1.17861
$803$	178.530	0.00784583
$804$	−16750.4	−0.734753
$805$	0	0
$806$	8846.76	0.386618
$807$	−51568.0	−2.24942
$808$	−6761.70	−0.294401
$809$	3687.22	0.160242	0.0801209	−	0.996785i	$-0.474469\pi$
$809$	3687.22	0.160242	0.0801209	+	0.996785i	$0.474469\pi$
$810$	0	0
$811$	19516.5	0.845028	0.422514	−	0.906356i	$-0.361148\pi$
$811$	19516.5	0.845028	0.422514	+	0.906356i	$0.361148\pi$
$812$	25440.1	1.09948
$813$	−10518.6	−0.453755
$814$	200.416	0.00862971
$815$	0	0
$816$	−9939.93	−0.426431
$817$	9957.42	0.426397
$818$	−5915.12	−0.252833
$819$	−33732.2	−1.43919
$820$	0	0
$821$	−25386.0	−1.07914	−0.539571	−	0.841940i	$-0.681414\pi$
$821$	−25386.0	−1.07914	−0.539571	+	0.841940i	$0.681414\pi$
$822$	25409.3	1.07817
$823$	−34018.2	−1.44083	−0.720413	−	0.693545i	$-0.756048\pi$
$823$	−34018.2	−1.44083	−0.720413	+	0.693545i	$0.756048\pi$
$824$	−8825.82	−0.373133
$825$	0	0
$826$	11640.8	0.490355
$827$	23934.0	1.00637	0.503183	−	0.864180i	$-0.332162\pi$
$827$	23934.0	1.00637	0.503183	+	0.864180i	$0.332162\pi$
$828$	6193.92	0.259968
$829$	−26149.2	−1.09554	−0.547768	−	0.836630i	$-0.684522\pi$
$829$	−26149.2	−1.09554	−0.547768	+	0.836630i	$0.684522\pi$
$830$	0	0
$831$	−85408.2	−3.56532
$832$	−967.926	−0.0403327
$833$	−48263.3	−2.00747
$834$	41226.9	1.71172
$835$	0	0
$836$	−703.678	−0.0291115
$837$	114547.	4.73036
$838$	26356.9	1.08649
$839$	13611.0	0.560074	0.280037	−	0.959989i	$-0.409653\pi$
$839$	13611.0	0.560074	0.280037	+	0.959989i	$0.409653\pi$
$840$	0	0
$841$	12467.0	0.511175
$842$	2735.09	0.111945
$843$	64137.4	2.62042
$844$	6366.24	0.259639
$845$	0	0
$846$	−19221.3	−0.781135
$847$	−44009.9	−1.78536
$848$	9432.48	0.381973
$849$	71751.0	2.90046
$850$	0	0
$851$	−1443.16	−0.0581325
$852$	17623.9	0.708667
$853$	27108.9	1.08815	0.544073	−	0.839038i	$-0.316881\pi$
$853$	27108.9	1.08815	0.544073	+	0.839038i	$0.316881\pi$
$854$	53643.1	2.14945
$855$	0	0
$856$	8822.89	0.352290
$857$	11549.4	0.460349	0.230175	−	0.973149i	$-0.426070\pi$
$857$	11549.4	0.460349	0.230175	+	0.973149i	$0.426070\pi$
$858$	469.163	0.0186678
$859$	−15491.8	−0.615335	−0.307668	−	0.951494i	$-0.599549\pi$
$859$	−15491.8	−0.615335	−0.307668	+	0.951494i	$0.599549\pi$
$860$	0	0
$861$	95339.1	3.77369
$862$	−69.2386	−0.00273582
$863$	4709.17	0.185750	0.0928749	−	0.995678i	$-0.470394\pi$
$863$	4709.17	0.185750	0.0928749	+	0.995678i	$0.470394\pi$
$864$	−12532.6	−0.493480
$865$	0	0
$866$	−6540.15	−0.256632
$867$	−7976.96	−0.312470
$868$	38757.6	1.51558
$869$	−1877.44	−0.0732887
$870$	0	0
$871$	6520.99	0.253680
$872$	11020.9	0.428000
$873$	40393.7	1.56600
$874$	5067.04	0.196104
$875$	0	0
$876$	4342.78	0.167499
$877$	−22566.1	−0.868875	−0.434438	−	0.900702i	$-0.643053\pi$
$877$	−22566.1	−0.868875	−0.434438	+	0.900702i	$0.643053\pi$
$878$	−13130.8	−0.504720
$879$	44091.9	1.69190
$880$	0	0
$881$	15121.2	0.578261	0.289130	−	0.957290i	$-0.406634\pi$
$881$	15121.2	0.578261	0.289130	+	0.957290i	$0.406634\pi$
$882$	−101596.	−3.87858
$883$	48306.7	1.84105	0.920527	−	0.390678i	$-0.127760\pi$
$883$	48306.7	1.84105	0.920527	+	0.390678i	$0.127760\pi$
$884$	3869.65	0.147229
$885$	0	0
$886$	−17619.5	−0.668101
$887$	−15632.8	−0.591767	−0.295883	−	0.955224i	$-0.595614\pi$
$887$	−15632.8	−0.591767	−0.295883	+	0.955224i	$0.595614\pi$
$888$	4875.16	0.184234
$889$	69180.0	2.60993
$890$	0	0
$891$	3171.58	0.119250
$892$	9198.55	0.345280
$893$	−15724.3	−0.589241
$894$	31681.2	1.18521
$895$	0	0
$896$	−4240.48	−0.158108
$897$	−3378.34	−0.125752
$898$	17954.8	0.667215
$899$	56149.6	2.08309
$900$	0	0
$901$	−37709.9	−1.39434
$902$	−946.454	−0.0349373
$903$	−29085.1	−1.07186
$904$	−7914.36	−0.291181
$905$	0	0
$906$	−21956.5	−0.805139
$907$	15321.2	0.560897	0.280449	−	0.959869i	$-0.409517\pi$
$907$	15321.2	0.560897	0.280449	+	0.959869i	$0.409517\pi$
$908$	13016.7	0.475744
$909$	56904.1	2.07634
$910$	0	0
$911$	27566.7	1.00255	0.501275	−	0.865288i	$-0.332864\pi$
$911$	27566.7	1.00255	0.501275	+	0.865288i	$0.332864\pi$
$912$	−17117.1	−0.621495
$913$	−151.839	−0.00550398
$914$	1712.39	0.0619702
$915$	0	0
$916$	−19739.2	−0.712010
$917$	−71337.5	−2.56900
$918$	50103.7	1.80138
$919$	6372.54	0.228738	0.114369	−	0.993438i	$-0.463515\pi$
$919$	6372.54	0.228738	0.114369	+	0.993438i	$0.463515\pi$
$920$	0	0
$921$	−28229.6	−1.00999
$922$	−735.512	−0.0262720
$923$	−6861.04	−0.244674
$924$	2055.40	0.0731794
$925$	0	0
$926$	−14660.9	−0.520290
$927$	74275.0	2.63162
$928$	−6143.34	−0.217311
$929$	39918.6	1.40978	0.704890	−	0.709316i	$-0.250996\pi$
$929$	39918.6	1.40978	0.704890	+	0.709316i	$0.250996\pi$
$930$	0	0
$931$	−83112.0	−2.92576
$932$	−4777.91	−0.167925
$933$	−79764.5	−2.79890
$934$	23006.2	0.805980
$935$	0	0
$936$	8145.72	0.284457
$937$	−11758.4	−0.409957	−0.204979	−	0.978766i	$-0.565712\pi$
$937$	−11758.4	−0.409957	−0.204979	+	0.978766i	$0.565712\pi$
$938$	28568.5	0.994449
$939$	−1953.14	−0.0678788
$940$	0	0
$941$	−26312.9	−0.911559	−0.455779	−	0.890093i	$-0.650639\pi$
$941$	−26312.9	−0.911559	−0.455779	+	0.890093i	$0.650639\pi$
$942$	−7985.62	−0.276206
$943$	6815.22	0.235349
$944$	−2811.03	−0.0969188
$945$	0	0
$946$	288.734	0.00992343
$947$	−8785.42	−0.301466	−0.150733	−	0.988575i	$-0.548163\pi$
$947$	−8785.42	−0.301466	−0.150733	+	0.988575i	$0.548163\pi$
$948$	−45669.1	−1.56463
$949$	−1690.66	−0.0578305
$950$	0	0
$951$	−24719.2	−0.842877
$952$	16952.9	0.577151
$953$	11133.8	0.378447	0.189223	−	0.981934i	$-0.439403\pi$
$953$	11133.8	0.378447	0.189223	+	0.981934i	$0.439403\pi$
$954$	−79380.5	−2.69396
$955$	0	0
$956$	6210.42	0.210104
$957$	2977.73	0.100581
$958$	−17160.9	−0.578752
$959$	−43336.6	−1.45924
$960$	0	0
$961$	55752.0	1.87144
$962$	−1897.92	−0.0636084
$963$	−74250.3	−2.48461
$964$	−16712.8	−0.558384
$965$	0	0
$966$	−14800.5	−0.492959
$967$	−30546.1	−1.01582	−0.507910	−	0.861410i	$-0.669582\pi$
$967$	−30546.1	−1.01582	−0.507910	+	0.861410i	$0.669582\pi$
$968$	10627.6	0.352876
$969$	68432.1	2.26868
$970$	0	0
$971$	−26757.1	−0.884322	−0.442161	−	0.896936i	$-0.645788\pi$
$971$	−26757.1	−0.884322	−0.442161	+	0.896936i	$0.645788\pi$
$972$	34851.8	1.15007
$973$	−70314.1	−2.31672
$974$	−31653.6	−1.04132
$975$	0	0
$976$	−12953.8	−0.424839
$977$	−36550.8	−1.19689	−0.598447	−	0.801163i	$-0.704215\pi$
$977$	−36550.8	−1.19689	−0.598447	+	0.801163i	$0.704215\pi$
$978$	61203.7	2.00110
$979$	1336.25	0.0436228
$980$	0	0
$981$	−92748.4	−3.01858
$982$	12438.2	0.404195
$983$	7459.39	0.242032	0.121016	−	0.992651i	$-0.461385\pi$
$983$	7459.39	0.242032	0.121016	+	0.992651i	$0.461385\pi$
$984$	−23022.7	−0.745870
$985$	0	0
$986$	24560.3	0.793266
$987$	45929.7	1.48121
$988$	6663.74	0.214577
$989$	−2079.12	−0.0668474
$990$	0	0
$991$	−56932.3	−1.82494	−0.912470	−	0.409145i	$-0.865827\pi$
$991$	−56932.3	−1.82494	−0.912470	+	0.409145i	$0.865827\pi$
$992$	−9359.27	−0.299554
$993$	71195.7	2.27526
$994$	−30058.2	−0.959142
$995$	0	0
$996$	−3693.51	−0.117503
$997$	−34243.6	−1.08777	−0.543885	−	0.839160i	$-0.683047\pi$
$997$	−34243.6	−1.08777	−0.543885	+	0.839160i	$0.683047\pi$
$998$	−25640.2	−0.813252
$999$	−24574.0	−0.778265

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1150.4.a.w.1.6	✓	6
5.2	odd	4		1150.4.b.s.599.1		12
5.3	odd	4		1150.4.b.s.599.12		12
5.4	even	2		1150.4.a.x.1.1	yes	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1150.4.a.w.1.6	✓	6	1.1	even	1	trivial
1150.4.a.x.1.1	yes	6	5.4	even	2
1150.4.b.s.599.1		12	5.2	odd	4
1150.4.b.s.599.12		12	5.3	odd	4

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:	\( N \)	\(=\)	\( 1150 = 2 \cdot 5^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1150.a (trivial)

\(f(q)\)	\(=\)	\(q-2.00000 q^{2} +9.71212 q^{3} +4.00000 q^{4} -19.4242 q^{6} +33.1288 q^{7} -8.00000 q^{8} +67.3252 q^{9} +1.59705 q^{11} +38.8485 q^{12} -15.1238 q^{13} -66.2575 q^{14} +16.0000 q^{16} -63.9661 q^{17} -134.650 q^{18} -110.153 q^{19} +321.750 q^{21} -3.19409 q^{22} +23.0000 q^{23} -77.6969 q^{24} +30.2477 q^{26} +391.643 q^{27} +132.515 q^{28} +191.979 q^{29} +292.477 q^{31} -32.0000 q^{32} +15.5107 q^{33} +127.932 q^{34} +269.301 q^{36} -62.7459 q^{37} +220.306 q^{38} -146.885 q^{39} +296.314 q^{41} -643.501 q^{42} -90.3963 q^{43} +6.38819 q^{44} -46.0000 q^{46} +142.749 q^{47} +155.394 q^{48} +754.515 q^{49} -621.246 q^{51} -60.4954 q^{52} +589.530 q^{53} -783.286 q^{54} -265.030 q^{56} -1069.82 q^{57} -383.959 q^{58} -175.690 q^{59} -809.615 q^{61} -584.955 q^{62} +2230.40 q^{63} +64.0000 q^{64} -31.0214 q^{66} -431.173 q^{67} -255.864 q^{68} +223.379 q^{69} +453.657 q^{71} -538.602 q^{72} +111.788 q^{73} +125.492 q^{74} -440.612 q^{76} +52.9082 q^{77} +293.769 q^{78} -1175.57 q^{79} +1985.90 q^{81} -592.628 q^{82} -95.0748 q^{83} +1287.00 q^{84} +180.793 q^{86} +1864.52 q^{87} -12.7764 q^{88} +836.699 q^{89} -501.034 q^{91} +92.0000 q^{92} +2840.57 q^{93} -285.499 q^{94} -310.788 q^{96} +599.979 q^{97} -1509.03 q^{98} +107.522 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 12 q^{2} + 5 q^{3} + 24 q^{4} - 10 q^{6} + 42 q^{7} - 48 q^{8} + 61 q^{9} - 49 q^{11} + 20 q^{12} + 16 q^{13} - 84 q^{14} + 96 q^{16} + 175 q^{17} - 122 q^{18} - 229 q^{19} + 92 q^{21} + 98 q^{22}+ \cdots - 142 q^{99}+O(q^{100}) \) 6 * q - 12 * q^2 + 5 * q^3 + 24 * q^4 - 10 * q^6 + 42 * q^7 - 48 * q^8 + 61 * q^9 - 49 * q^11 + 20 * q^12 + 16 * q^13 - 84 * q^14 + 96 * q^16 + 175 * q^17 - 122 * q^18 - 229 * q^19 + 92 * q^21 + 98 * q^22 + 138 * q^23 - 40 * q^24 - 32 * q^26 + 344 * q^27 + 168 * q^28 - 182 * q^29 + 114 * q^31 - 192 * q^32 + 641 * q^33 - 350 * q^34 + 244 * q^36 + 64 * q^37 + 458 * q^38 - 143 * q^39 + 243 * q^41 - 184 * q^42 + 182 * q^43 - 196 * q^44 - 276 * q^46 + 498 * q^47 + 80 * q^48 + 648 * q^49 - 1031 * q^51 + 64 * q^52 + 1290 * q^53 - 688 * q^54 - 336 * q^56 - 353 * q^57 + 364 * q^58 - 559 * q^59 - 688 * q^61 - 228 * q^62 + 3656 * q^63 + 384 * q^64 - 1282 * q^66 - 2069 * q^67 + 700 * q^68 + 115 * q^69 + 584 * q^71 - 488 * q^72 + 2485 * q^73 - 128 * q^74 - 916 * q^76 - 810 * q^77 + 286 * q^78 - 1432 * q^79 + 34 * q^81 - 486 * q^82 + 2089 * q^83 + 368 * q^84 - 364 * q^86 + 527 * q^87 + 392 * q^88 - 591 * q^89 - 2864 * q^91 + 552 * q^92 + 4899 * q^93 - 996 * q^94 - 160 * q^96 - 968 * q^97 - 1296 * q^98 - 142 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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