LMFDB - Embedded newform 1950.4.a.y.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1950,4,Mod(1,1950)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1950, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1950.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1950 = 2 \cdot 3 \cdot 5^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1950.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$115.053724511$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{129}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 32 $ x^2 - x - 32
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 390)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$6.17891$ of defining polynomial
Character	$\chi$	$=$	1950.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+2.00000 q^{2} +3.00000 q^{3} +4.00000 q^{4} +6.00000 q^{6} -24.7156 q^{7} +8.00000 q^{8} +9.00000 q^{9} +69.4313 q^{11} +12.0000 q^{12} -13.0000 q^{13} -49.4313 q^{14} +16.0000 q^{16} +22.7156 q^{17} +18.0000 q^{18} -102.147 q^{19} -74.1469 q^{21} +138.863 q^{22} -83.5782 q^{23} +24.0000 q^{24} -26.0000 q^{26} +27.0000 q^{27} -98.8625 q^{28} +37.2844 q^{29} +122.863 q^{31} +32.0000 q^{32} +208.294 q^{33} +45.4313 q^{34} +36.0000 q^{36} +374.294 q^{37} -204.294 q^{38} -39.0000 q^{39} +176.569 q^{41} -148.294 q^{42} +114.569 q^{43} +277.725 q^{44} -167.156 q^{46} +358.313 q^{47} +48.0000 q^{48} +267.863 q^{49} +68.1469 q^{51} -52.0000 q^{52} -523.725 q^{53} +54.0000 q^{54} -197.725 q^{56} -306.441 q^{57} +74.5687 q^{58} +722.569 q^{59} +451.137 q^{61} +245.725 q^{62} -222.441 q^{63} +64.0000 q^{64} +416.588 q^{66} -368.881 q^{67} +90.8625 q^{68} -250.735 q^{69} -175.744 q^{71} +72.0000 q^{72} +679.047 q^{73} +748.588 q^{74} -408.588 q^{76} -1716.04 q^{77} -78.0000 q^{78} +999.744 q^{79} +81.0000 q^{81} +353.137 q^{82} -1310.61 q^{83} -296.588 q^{84} +229.137 q^{86} +111.853 q^{87} +555.450 q^{88} -1316.64 q^{89} +321.303 q^{91} -334.313 q^{92} +368.588 q^{93} +716.625 q^{94} +96.0000 q^{96} -454.166 q^{97} +535.725 q^{98} +624.881 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 4 q^{2} + 6 q^{3} + 8 q^{4} + 12 q^{6} - 4 q^{7} + 16 q^{8} + 18 q^{9} + 48 q^{11} + 24 q^{12} - 26 q^{13} - 8 q^{14} + 32 q^{16} + 36 q^{18} - 68 q^{19} - 12 q^{21} + 96 q^{22} + 60 q^{23} + 48 q^{24}+ \cdots + 432 q^{99}+O(q^{100}) $ 2 * q + 4 * q^2 + 6 * q^3 + 8 * q^4 + 12 * q^6 - 4 * q^7 + 16 * q^8 + 18 * q^9 + 48 * q^11 + 24 * q^12 - 26 * q^13 - 8 * q^14 + 32 * q^16 + 36 * q^18 - 68 * q^19 - 12 * q^21 + 96 * q^22 + 60 * q^23 + 48 * q^24 - 52 * q^26 + 54 * q^27 - 16 * q^28 + 120 * q^29 + 64 * q^31 + 64 * q^32 + 144 * q^33 + 72 * q^36 + 476 * q^37 - 136 * q^38 - 78 * q^39 + 444 * q^41 - 24 * q^42 + 320 * q^43 + 192 * q^44 + 120 * q^46 - 192 * q^47 + 96 * q^48 + 354 * q^49 - 104 * q^52 - 684 * q^53 + 108 * q^54 - 32 * q^56 - 204 * q^57 + 240 * q^58 + 1536 * q^59 + 1084 * q^61 + 128 * q^62 - 36 * q^63 + 128 * q^64 + 288 * q^66 + 80 * q^67 + 180 * q^69 + 648 * q^71 + 144 * q^72 - 232 * q^73 + 952 * q^74 - 272 * q^76 - 2160 * q^77 - 156 * q^78 + 1000 * q^79 + 162 * q^81 + 888 * q^82 - 1440 * q^83 - 48 * q^84 + 640 * q^86 + 360 * q^87 + 384 * q^88 - 180 * q^89 + 52 * q^91 + 240 * q^92 + 192 * q^93 - 384 * q^94 + 192 * q^96 - 136 * q^97 + 708 * q^98 + 432 * 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
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	4.00000	0.500000
$5$	0	0
$5$	0	0
$6$	6.00000	0.408248
$7$	−24.7156	−1.33452	−0.667259	−	0.744825i	$-0.732533\pi$
$7$	−24.7156	−1.33452	−0.667259	+	0.744825i	$0.732533\pi$
$8$	8.00000	0.353553
$9$	9.00000	0.333333
$10$	0	0
$11$	69.4313	1.90312	0.951560	−	0.307464i	$-0.0994803\pi$
$11$	69.4313	1.90312	0.951560	+	0.307464i	$0.0994803\pi$
$12$	12.0000	0.288675
$13$	−13.0000	−0.277350
$13$	−13.0000	−0.277350
$14$	−49.4313	−0.943647
$15$	0	0
$16$	16.0000	0.250000
$17$	22.7156	0.324079	0.162040	−	0.986784i	$-0.448193\pi$
$17$	22.7156	0.324079	0.162040	+	0.986784i	$0.448193\pi$
$18$	18.0000	0.235702
$19$	−102.147	−1.23337	−0.616687	−	0.787208i	$-0.711526\pi$
$19$	−102.147	−1.23337	−0.616687	+	0.787208i	$0.711526\pi$
$20$	0	0
$21$	−74.1469	−0.770485
$22$	138.863	1.34571
$23$	−83.5782	−0.757707	−0.378853	−	0.925457i	$-0.623682\pi$
$23$	−83.5782	−0.757707	−0.378853	+	0.925457i	$0.623682\pi$
$24$	24.0000	0.204124
$25$	0	0
$26$	−26.0000	−0.196116
$27$	27.0000	0.192450
$28$	−98.8625	−0.667259
$29$	37.2844	0.238743	0.119371	−	0.992850i	$-0.461912\pi$
$29$	37.2844	0.238743	0.119371	+	0.992850i	$0.461912\pi$
$30$	0	0
$31$	122.863	0.711831	0.355916	−	0.934518i	$-0.384169\pi$
$31$	122.863	0.711831	0.355916	+	0.934518i	$0.384169\pi$
$32$	32.0000	0.176777
$33$	208.294	1.09877
$34$	45.4313	0.229159
$35$	0	0
$36$	36.0000	0.166667
$37$	374.294	1.66307	0.831534	−	0.555474i	$-0.187463\pi$
$37$	374.294	1.66307	0.831534	+	0.555474i	$0.187463\pi$
$38$	−204.294	−0.872127
$39$	−39.0000	−0.160128
$40$	0	0
$41$	176.569	0.672571	0.336285	−	0.941760i	$-0.390829\pi$
$41$	176.569	0.672571	0.336285	+	0.941760i	$0.390829\pi$
$42$	−148.294	−0.544815
$43$	114.569	0.406316	0.203158	−	0.979146i	$-0.434880\pi$
$43$	114.569	0.406316	0.203158	+	0.979146i	$0.434880\pi$
$44$	277.725	0.951560
$45$	0	0
$46$	−167.156	−0.535779
$47$	358.313	1.11203	0.556014	−	0.831173i	$-0.312330\pi$
$47$	358.313	1.11203	0.556014	+	0.831173i	$0.312330\pi$
$48$	48.0000	0.144338
$49$	267.863	0.780940
$50$	0	0
$51$	68.1469	0.187107
$52$	−52.0000	−0.138675
$53$	−523.725	−1.35734	−0.678671	−	0.734442i	$-0.737444\pi$
$53$	−523.725	−1.35734	−0.678671	+	0.734442i	$0.737444\pi$
$54$	54.0000	0.136083
$55$	0	0
$56$	−197.725	−0.471824
$57$	−306.441	−0.712089
$58$	74.5687	0.168816
$59$	722.569	1.59441	0.797207	−	0.603706i	$-0.206310\pi$
$59$	722.569	1.59441	0.797207	+	0.603706i	$0.206310\pi$
$60$	0	0
$61$	451.137	0.946922	0.473461	−	0.880815i	$-0.343005\pi$
$61$	451.137	0.946922	0.473461	+	0.880815i	$0.343005\pi$
$62$	245.725	0.503341
$63$	−222.441	−0.444840
$64$	64.0000	0.125000
$65$	0	0
$66$	416.588	0.776945
$67$	−368.881	−0.672627	−0.336314	−	0.941750i	$-0.609180\pi$
$67$	−368.881	−0.672627	−0.336314	+	0.941750i	$0.609180\pi$
$68$	90.8625	0.162040
$69$	−250.735	−0.437462
$70$	0	0
$71$	−175.744	−0.293760	−0.146880	−	0.989154i	$-0.546923\pi$
$71$	−175.744	−0.293760	−0.146880	+	0.989154i	$0.546923\pi$
$72$	72.0000	0.117851
$73$	679.047	1.08872	0.544359	−	0.838852i	$-0.316773\pi$
$73$	679.047	1.08872	0.544359	+	0.838852i	$0.316773\pi$
$74$	748.588	1.17597
$75$	0	0
$76$	−408.588	−0.616687
$77$	−1716.04	−2.53975
$78$	−78.0000	−0.113228
$79$	999.744	1.42380	0.711899	−	0.702282i	$-0.247836\pi$
$79$	999.744	1.42380	0.711899	+	0.702282i	$0.247836\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	353.137	0.475579
$83$	−1310.61	−1.73323	−0.866613	−	0.498981i	$-0.833708\pi$
$83$	−1310.61	−1.73323	−0.866613	+	0.498981i	$0.833708\pi$
$84$	−296.588	−0.385242
$85$	0	0
$86$	229.137	0.287308
$87$	111.853	0.137838
$88$	555.450	0.672854
$89$	−1316.64	−1.56813	−0.784067	−	0.620676i	$-0.786858\pi$
$89$	−1316.64	−1.56813	−0.784067	+	0.620676i	$0.786858\pi$
$90$	0	0
$91$	321.303	0.370129
$92$	−334.313	−0.378853
$93$	368.588	0.410976
$94$	716.625	0.786322
$95$	0	0
$96$	96.0000	0.102062
$97$	−454.166	−0.475397	−0.237699	−	0.971339i	$-0.576393\pi$
$97$	−454.166	−0.475397	−0.237699	+	0.971339i	$0.576393\pi$
$98$	535.725	0.552208
$99$	624.881	0.634373
$100$	0	0
$101$	1227.85	1.20966	0.604831	−	0.796354i	$-0.293240\pi$
$101$	1227.85	1.20966	0.604831	+	0.796354i	$0.293240\pi$
$102$	136.294	0.132305
$103$	959.744	0.918120	0.459060	−	0.888405i	$-0.348186\pi$
$103$	959.744	0.918120	0.459060	+	0.888405i	$0.348186\pi$
$104$	−104.000	−0.0980581
$105$	0	0
$106$	−1047.45	−0.959786
$107$	2105.21	1.90204	0.951022	−	0.309125i	$-0.100036\pi$
$107$	2105.21	1.90204	0.951022	+	0.309125i	$0.100036\pi$
$108$	108.000	0.0962250
$109$	−540.991	−0.475390	−0.237695	−	0.971340i	$-0.576392\pi$
$109$	−540.991	−0.475390	−0.237695	+	0.971340i	$0.576392\pi$
$110$	0	0
$111$	1122.88	0.960173
$112$	−395.450	−0.333630
$113$	−1754.20	−1.46037	−0.730184	−	0.683251i	$-0.760566\pi$
$113$	−1754.20	−1.46037	−0.730184	+	0.683251i	$0.760566\pi$
$114$	−612.881	−0.503523
$115$	0	0
$116$	149.137	0.119371
$117$	−117.000	−0.0924500
$118$	1445.14	1.12742
$119$	−561.431	−0.432490
$120$	0	0
$121$	3489.70	2.62186
$122$	902.275	0.669575
$123$	529.706	0.388309
$124$	491.450	0.355916
$125$	0	0
$126$	−444.881	−0.314549
$127$	1422.64	0.994010	0.497005	−	0.867748i	$-0.334433\pi$
$127$	1422.64	0.994010	0.497005	+	0.867748i	$0.334433\pi$
$128$	128.000	0.0883883
$129$	343.706	0.234586
$130$	0	0
$131$	716.204	0.477672	0.238836	−	0.971060i	$-0.423234\pi$
$131$	716.204	0.477672	0.238836	+	0.971060i	$0.423234\pi$
$132$	833.175	0.549383
$133$	2524.63	1.64596
$134$	−737.763	−0.475619
$135$	0	0
$136$	181.725	0.114579
$137$	−1437.45	−0.896421	−0.448210	−	0.893928i	$-0.647938\pi$
$137$	−1437.45	−0.896421	−0.448210	+	0.893928i	$0.647938\pi$
$138$	−501.469	−0.309332
$139$	−1726.02	−1.05323	−0.526615	−	0.850104i	$-0.676539\pi$
$139$	−1726.02	−1.05323	−0.526615	+	0.850104i	$0.676539\pi$
$140$	0	0
$141$	1074.94	0.642029
$142$	−351.488	−0.207720
$143$	−902.606	−0.527830
$144$	144.000	0.0833333
$145$	0	0
$146$	1358.09	0.769840
$147$	803.588	0.450876
$148$	1497.18	0.831534
$149$	3377.27	1.85689	0.928445	−	0.371469i	$-0.121146\pi$
$149$	3377.27	1.85689	0.928445	+	0.371469i	$0.121146\pi$
$150$	0	0
$151$	−2292.70	−1.23561	−0.617806	−	0.786331i	$-0.711978\pi$
$151$	−2292.70	−1.23561	−0.617806	+	0.786331i	$0.711978\pi$
$152$	−817.175	−0.436064
$153$	204.441	0.108026
$154$	−3432.08	−1.79587
$155$	0	0
$156$	−156.000	−0.0800641
$157$	2432.39	1.23647	0.618235	−	0.785994i	$-0.287848\pi$
$157$	2432.39	1.23647	0.618235	+	0.785994i	$0.287848\pi$
$158$	1999.49	1.00678
$159$	−1571.18	−0.783662
$160$	0	0
$161$	2065.69	1.01117
$162$	162.000	0.0785674
$163$	3355.64	1.61248	0.806239	−	0.591590i	$-0.201500\pi$
$163$	3355.64	1.61248	0.806239	+	0.591590i	$0.201500\pi$
$164$	706.275	0.336285
$165$	0	0
$166$	−2621.21	−1.22558
$167$	1406.57	0.651758	0.325879	−	0.945412i	$-0.394340\pi$
$167$	1406.57	0.651758	0.325879	+	0.945412i	$0.394340\pi$
$168$	−593.175	−0.272408
$169$	169.000	0.0769231
$170$	0	0
$171$	−919.322	−0.411125
$172$	458.275	0.203158
$173$	3534.11	1.55314	0.776571	−	0.630029i	$-0.216957\pi$
$173$	3534.11	1.55314	0.776571	+	0.630029i	$0.216957\pi$
$174$	223.706	0.0974662
$175$	0	0
$176$	1110.90	0.475780
$177$	2167.71	0.920535
$178$	−2633.29	−1.10884
$179$	−1456.68	−0.608253	−0.304126	−	0.952632i	$-0.598364\pi$
$179$	−1456.68	−0.608253	−0.304126	+	0.952632i	$0.598364\pi$
$180$	0	0
$181$	2330.88	0.957199	0.478600	−	0.878033i	$-0.341145\pi$
$181$	2330.88	0.957199	0.478600	+	0.878033i	$0.341145\pi$
$182$	642.606	0.261721
$183$	1353.41	0.546706
$184$	−668.625	−0.267890
$185$	0	0
$186$	737.175	0.290604
$187$	1577.18	0.616762
$188$	1433.25	0.556014
$189$	−667.322	−0.256828
$190$	0	0
$191$	2749.65	1.04166	0.520832	−	0.853659i	$-0.325622\pi$
$191$	2749.65	1.04166	0.520832	+	0.853659i	$0.325622\pi$
$192$	192.000	0.0721688
$193$	−1821.47	−0.679339	−0.339670	−	0.940545i	$-0.610315\pi$
$193$	−1821.47	−0.679339	−0.339670	+	0.940545i	$0.610315\pi$
$194$	−908.332	−0.336157
$195$	0	0
$196$	1071.45	0.390470
$197$	−3799.91	−1.37428	−0.687139	−	0.726526i	$-0.741134\pi$
$197$	−3799.91	−1.37428	−0.687139	+	0.726526i	$0.741134\pi$
$198$	1249.76	0.448570
$199$	872.995	0.310980	0.155490	−	0.987837i	$-0.450304\pi$
$199$	872.995	0.310980	0.155490	+	0.987837i	$0.450304\pi$
$200$	0	0
$201$	−1106.64	−0.388342
$202$	2455.71	0.855361
$203$	−921.507	−0.318606
$204$	272.588	0.0935537
$205$	0	0
$206$	1919.49	0.649209
$207$	−752.204	−0.252569
$208$	−208.000	−0.0693375
$209$	−7092.19	−2.34726
$210$	0	0
$211$	1014.35	0.330951	0.165476	−	0.986214i	$-0.447084\pi$
$211$	1014.35	0.330951	0.165476	+	0.986214i	$0.447084\pi$
$212$	−2094.90	−0.678671
$213$	−527.232	−0.169602
$214$	4210.43	1.34495
$215$	0	0
$216$	216.000	0.0680414
$217$	−3036.63	−0.949952
$218$	−1081.98	−0.336151
$219$	2037.14	0.628572
$220$	0	0
$221$	−295.303	−0.0898835
$222$	2245.76	0.678945
$223$	−1901.08	−0.570879	−0.285440	−	0.958397i	$-0.592140\pi$
$223$	−1901.08	−0.570879	−0.285440	+	0.958397i	$0.592140\pi$
$224$	−790.900	−0.235912
$225$	0	0
$226$	−3508.41	−1.03264
$227$	1410.98	0.412554	0.206277	−	0.978494i	$-0.433865\pi$
$227$	1410.98	0.412554	0.206277	+	0.978494i	$0.433865\pi$
$228$	−1225.76	−0.356044
$229$	2334.02	0.673522	0.336761	−	0.941590i	$-0.390669\pi$
$229$	2334.02	0.673522	0.336761	+	0.941590i	$0.390669\pi$
$230$	0	0
$231$	−5148.11	−1.46632
$232$	298.275	0.0844082
$233$	−1686.50	−0.474189	−0.237095	−	0.971487i	$-0.576195\pi$
$233$	−1686.50	−0.474189	−0.237095	+	0.971487i	$0.576195\pi$
$234$	−234.000	−0.0653720
$235$	0	0
$236$	2890.27	0.797207
$237$	2999.23	0.822030
$238$	−1122.86	−0.305817
$239$	658.426	0.178201	0.0891005	−	0.996023i	$-0.471601\pi$
$239$	658.426	0.178201	0.0891005	+	0.996023i	$0.471601\pi$
$240$	0	0
$241$	4051.18	1.08282	0.541409	−	0.840759i	$-0.317891\pi$
$241$	4051.18	1.08282	0.541409	+	0.840759i	$0.317891\pi$
$242$	6979.40	1.85394
$243$	243.000	0.0641500
$244$	1804.55	0.473461
$245$	0	0
$246$	1059.41	0.274576
$247$	1327.91	0.342076
$248$	982.900	0.251670
$249$	−3931.82	−1.00068
$250$	0	0
$251$	−3354.67	−0.843604	−0.421802	−	0.906688i	$-0.638602\pi$
$251$	−3354.67	−0.843604	−0.421802	+	0.906688i	$0.638602\pi$
$252$	−889.763	−0.222420
$253$	−5802.94	−1.44201
$254$	2845.29	0.702871
$255$	0	0
$256$	256.000	0.0625000
$257$	2485.28	0.603221	0.301610	−	0.953431i	$-0.402476\pi$
$257$	2485.28	0.603221	0.301610	+	0.953431i	$0.402476\pi$
$258$	687.412	0.165878
$259$	−9250.91	−2.21940
$260$	0	0
$261$	335.559	0.0795808
$262$	1432.41	0.337765
$263$	1859.58	0.435994	0.217997	−	0.975949i	$-0.430048\pi$
$263$	1859.58	0.435994	0.217997	+	0.975949i	$0.430048\pi$
$264$	1666.35	0.388473
$265$	0	0
$266$	5049.25	1.16387
$267$	−3949.93	−0.905363
$268$	−1475.53	−0.336314
$269$	−4508.45	−1.02188	−0.510939	−	0.859617i	$-0.670702\pi$
$269$	−4508.45	−1.02188	−0.510939	+	0.859617i	$0.670702\pi$
$270$	0	0
$271$	−3295.56	−0.738713	−0.369356	−	0.929288i	$-0.620422\pi$
$271$	−3295.56	−0.738713	−0.369356	+	0.929288i	$0.620422\pi$
$272$	363.450	0.0810199
$273$	963.910	0.213694
$274$	−2874.90	−0.633865
$275$	0	0
$276$	−1002.94	−0.218731
$277$	−3459.88	−0.750483	−0.375242	−	0.926927i	$-0.622440\pi$
$277$	−3459.88	−0.750483	−0.375242	+	0.926927i	$0.622440\pi$
$278$	−3452.04	−0.744746
$279$	1105.76	0.237277
$280$	0	0
$281$	6308.61	1.33929	0.669644	−	0.742682i	$-0.266447\pi$
$281$	6308.61	1.33929	0.669644	+	0.742682i	$0.266447\pi$
$282$	2149.88	0.453983
$283$	−5126.32	−1.07678	−0.538389	−	0.842696i	$-0.680967\pi$
$283$	−5126.32	−1.07678	−0.538389	+	0.842696i	$0.680967\pi$
$284$	−702.976	−0.146880
$285$	0	0
$286$	−1805.21	−0.373232
$287$	−4364.01	−0.897558
$288$	288.000	0.0589256
$289$	−4397.00	−0.894973
$290$	0	0
$291$	−1362.50	−0.274471
$292$	2716.19	0.544359
$293$	5291.40	1.05504	0.527520	−	0.849542i	$-0.323122\pi$
$293$	5291.40	1.05504	0.527520	+	0.849542i	$0.323122\pi$
$294$	1607.18	0.318818
$295$	0	0
$296$	2994.35	0.587983
$297$	1874.64	0.366256
$298$	6754.54	1.31302
$299$	1086.52	0.210150
$300$	0	0
$301$	−2831.64	−0.542236
$302$	−4585.40	−0.873709
$303$	3683.56	0.698399
$304$	−1634.35	−0.308344
$305$	0	0
$306$	408.881	0.0763863
$307$	−5858.21	−1.08907	−0.544537	−	0.838737i	$-0.683295\pi$
$307$	−5858.21	−1.08907	−0.544537	+	0.838737i	$0.683295\pi$
$308$	−6864.15	−1.26987
$309$	2879.23	0.530077
$310$	0	0
$311$	−2168.81	−0.395441	−0.197721	−	0.980258i	$-0.563354\pi$
$311$	−2168.81	−0.395441	−0.197721	+	0.980258i	$0.563354\pi$
$312$	−312.000	−0.0566139
$313$	−4105.23	−0.741346	−0.370673	−	0.928763i	$-0.620873\pi$
$313$	−4105.23	−0.741346	−0.370673	+	0.928763i	$0.620873\pi$
$314$	4864.78	0.874316
$315$	0	0
$316$	3998.98	0.711899
$317$	2303.36	0.408107	0.204053	−	0.978960i	$-0.434588\pi$
$317$	2303.36	0.408107	0.204053	+	0.978960i	$0.434588\pi$
$318$	−3142.35	−0.554133
$319$	2588.70	0.454356
$320$	0	0
$321$	6315.64	1.09815
$322$	4131.37	0.715008
$323$	−2320.33	−0.399711
$324$	324.000	0.0555556
$325$	0	0
$326$	6711.28	1.14019
$327$	−1622.97	−0.274466
$328$	1412.55	0.237790
$329$	−8855.92	−1.48402
$330$	0	0
$331$	3845.89	0.638638	0.319319	−	0.947647i	$-0.396546\pi$
$331$	3845.89	0.638638	0.319319	+	0.947647i	$0.396546\pi$
$332$	−5242.43	−0.866613
$333$	3368.64	0.554356
$334$	2813.14	0.460862
$335$	0	0
$336$	−1186.35	−0.192621
$337$	−601.005	−0.0971479	−0.0485740	−	0.998820i	$-0.515468\pi$
$337$	−601.005	−0.0971479	−0.0485740	+	0.998820i	$0.515468\pi$
$338$	338.000	0.0543928
$339$	−5262.61	−0.843144
$340$	0	0
$341$	8530.50	1.35470
$342$	−1838.64	−0.290709
$343$	1857.07	0.292339
$344$	916.550	0.143654
$345$	0	0
$346$	7068.23	1.09824
$347$	−2148.84	−0.332438	−0.166219	−	0.986089i	$-0.553156\pi$
$347$	−2148.84	−0.332438	−0.166219	+	0.986089i	$0.553156\pi$
$348$	447.412	0.0689190
$349$	−9766.03	−1.49789	−0.748945	−	0.662632i	$-0.769439\pi$
$349$	−9766.03	−1.49789	−0.748945	+	0.662632i	$0.769439\pi$
$350$	0	0
$351$	−351.000	−0.0533761
$352$	2221.80	0.336427
$353$	−1962.00	−0.295826	−0.147913	−	0.989000i	$-0.547256\pi$
$353$	−1962.00	−0.295826	−0.147913	+	0.989000i	$0.547256\pi$
$354$	4335.41	0.650917
$355$	0	0
$356$	−5266.58	−0.784067
$357$	−1684.29	−0.249698
$358$	−2913.36	−0.430100
$359$	−2839.52	−0.417448	−0.208724	−	0.977975i	$-0.566931\pi$
$359$	−2839.52	−0.417448	−0.208724	+	0.977975i	$0.566931\pi$
$360$	0	0
$361$	3574.99	0.521211
$362$	4661.76	0.676842
$363$	10469.1	1.51373
$364$	1285.21	0.185064
$365$	0	0
$366$	2706.82	0.386579
$367$	3870.64	0.550534	0.275267	−	0.961368i	$-0.411234\pi$
$367$	3870.64	0.550534	0.275267	+	0.961368i	$0.411234\pi$
$368$	−1337.25	−0.189427
$369$	1589.12	0.224190
$370$	0	0
$371$	12944.2	1.81140
$372$	1474.35	0.205488
$373$	−3656.64	−0.507596	−0.253798	−	0.967257i	$-0.581680\pi$
$373$	−3656.64	−0.507596	−0.253798	+	0.967257i	$0.581680\pi$
$374$	3154.35	0.436117
$375$	0	0
$376$	2866.50	0.393161
$377$	−484.697	−0.0662153
$378$	−1334.64	−0.181605
$379$	8111.91	1.09942	0.549711	−	0.835355i	$-0.314738\pi$
$379$	8111.91	1.09942	0.549711	+	0.835355i	$0.314738\pi$
$380$	0	0
$381$	4267.93	0.573892
$382$	5499.30	0.736567
$383$	11631.1	1.55175	0.775876	−	0.630885i	$-0.217308\pi$
$383$	11631.1	1.55175	0.775876	+	0.630885i	$0.217308\pi$
$384$	384.000	0.0510310
$385$	0	0
$386$	−3642.95	−0.480365
$387$	1031.12	0.135439
$388$	−1816.66	−0.237699
$389$	5723.28	0.745969	0.372984	−	0.927838i	$-0.378334\pi$
$389$	5723.28	0.745969	0.372984	+	0.927838i	$0.378334\pi$
$390$	0	0
$391$	−1898.53	−0.245557
$392$	2142.90	0.276104
$393$	2148.61	0.275784
$394$	−7599.83	−0.971761
$395$	0	0
$396$	2499.53	0.317187
$397$	−2934.26	−0.370947	−0.185474	−	0.982649i	$-0.559382\pi$
$397$	−2934.26	−0.370947	−0.185474	+	0.982649i	$0.559382\pi$
$398$	1745.99	0.219896
$399$	7573.88	0.950296
$400$	0	0
$401$	−6089.57	−0.758351	−0.379175	−	0.925325i	$-0.623792\pi$
$401$	−6089.57	−0.758351	−0.379175	+	0.925325i	$0.623792\pi$
$402$	−2213.29	−0.274599
$403$	−1597.21	−0.197426
$404$	4911.41	0.604831
$405$	0	0
$406$	−1843.01	−0.225289
$407$	25987.7	3.16502
$408$	545.175	0.0661524
$409$	9198.29	1.11204	0.556022	−	0.831167i	$-0.312327\pi$
$409$	9198.29	1.11204	0.556022	+	0.831167i	$0.312327\pi$
$410$	0	0
$411$	−4312.35	−0.517549
$412$	3838.98	0.459060
$413$	−17858.7	−2.12778
$414$	−1504.41	−0.178593
$415$	0	0
$416$	−416.000	−0.0490290
$417$	−5178.06	−0.608083
$418$	−14184.4	−1.65976
$419$	−5655.53	−0.659405	−0.329703	−	0.944085i	$-0.606948\pi$
$419$	−5655.53	−0.659405	−0.329703	+	0.944085i	$0.606948\pi$
$420$	0	0
$421$	−6687.37	−0.774163	−0.387081	−	0.922046i	$-0.626517\pi$
$421$	−6687.37	−0.774163	−0.387081	+	0.922046i	$0.626517\pi$
$422$	2028.70	0.234018
$423$	3224.81	0.370676
$424$	−4189.80	−0.479893
$425$	0	0
$426$	−1054.46	−0.119927
$427$	−11150.1	−1.26368
$428$	8420.85	0.951022
$429$	−2707.82	−0.304743
$430$	0	0
$431$	377.477	0.0421866	0.0210933	−	0.999778i	$-0.493285\pi$
$431$	377.477	0.0421866	0.0210933	+	0.999778i	$0.493285\pi$
$432$	432.000	0.0481125
$433$	7903.43	0.877170	0.438585	−	0.898690i	$-0.355480\pi$
$433$	7903.43	0.877170	0.438585	+	0.898690i	$0.355480\pi$
$434$	−6073.25	−0.671717
$435$	0	0
$436$	−2163.96	−0.237695
$437$	8537.25	0.934536
$438$	4074.28	0.444468
$439$	−16459.9	−1.78950	−0.894748	−	0.446572i	$-0.852645\pi$
$439$	−16459.9	−1.78950	−0.894748	+	0.446572i	$0.852645\pi$
$440$	0	0
$441$	2410.76	0.260313
$442$	−590.606	−0.0635572
$443$	1361.91	0.146063	0.0730317	−	0.997330i	$-0.476733\pi$
$443$	1361.91	0.146063	0.0730317	+	0.997330i	$0.476733\pi$
$444$	4491.53	0.480086
$445$	0	0
$446$	−3802.17	−0.403673
$447$	10131.8	1.07208
$448$	−1581.80	−0.166815
$449$	−9924.51	−1.04313	−0.521566	−	0.853211i	$-0.674652\pi$
$449$	−9924.51	−1.04313	−0.521566	+	0.853211i	$0.674652\pi$
$450$	0	0
$451$	12259.4	1.27998
$452$	−7016.81	−0.730184
$453$	−6878.10	−0.713381
$454$	2821.95	0.291720
$455$	0	0
$456$	−2451.53	−0.251761
$457$	−3162.79	−0.323740	−0.161870	−	0.986812i	$-0.551753\pi$
$457$	−3162.79	−0.323740	−0.161870	+	0.986812i	$0.551753\pi$
$458$	4668.05	0.476252
$459$	613.322	0.0623691
$460$	0	0
$461$	−2334.04	−0.235807	−0.117903	−	0.993025i	$-0.537617\pi$
$461$	−2334.04	−0.235807	−0.117903	+	0.993025i	$0.537617\pi$
$462$	−10296.2	−1.03685
$463$	9502.97	0.953867	0.476934	−	0.878939i	$-0.341748\pi$
$463$	9502.97	0.953867	0.476934	+	0.878939i	$0.341748\pi$
$464$	596.550	0.0596856
$465$	0	0
$466$	−3372.99	−0.335302
$467$	−6464.93	−0.640602	−0.320301	−	0.947316i	$-0.603784\pi$
$467$	−6464.93	−0.640602	−0.320301	+	0.947316i	$0.603784\pi$
$468$	−468.000	−0.0462250
$469$	9117.14	0.897634
$470$	0	0
$471$	7297.16	0.713876
$472$	5780.55	0.563711
$473$	7954.65	0.773267
$474$	5998.46	0.581263
$475$	0	0
$476$	−2245.73	−0.216245
$477$	−4713.53	−0.452448
$478$	1316.85	0.126007
$479$	−10188.5	−0.971866	−0.485933	−	0.873996i	$-0.661520\pi$
$479$	−10188.5	−0.971866	−0.485933	+	0.873996i	$0.661520\pi$
$480$	0	0
$481$	−4865.82	−0.461252
$482$	8102.35	0.765668
$483$	6197.06	0.583801
$484$	13958.8	1.31093
$485$	0	0
$486$	486.000	0.0453609
$487$	6529.20	0.607528	0.303764	−	0.952747i	$-0.401757\pi$
$487$	6529.20	0.607528	0.303764	+	0.952747i	$0.401757\pi$
$488$	3609.10	0.334787
$489$	10066.9	0.930965
$490$	0	0
$491$	−20959.6	−1.92647	−0.963233	−	0.268668i	$-0.913416\pi$
$491$	−20959.6	−1.92647	−0.963233	+	0.268668i	$0.913416\pi$
$492$	2118.82	0.194154
$493$	846.938	0.0773715
$494$	2655.82	0.241885
$495$	0	0
$496$	1965.80	0.177958
$497$	4343.62	0.392028
$498$	−7863.64	−0.707587
$499$	4484.84	0.402342	0.201171	−	0.979556i	$-0.435525\pi$
$499$	4484.84	0.402342	0.201171	+	0.979556i	$0.435525\pi$
$500$	0	0
$501$	4219.71	0.376293
$502$	−6709.33	−0.596518
$503$	−10761.5	−0.953939	−0.476970	−	0.878920i	$-0.658265\pi$
$503$	−10761.5	−0.953939	−0.476970	+	0.878920i	$0.658265\pi$
$504$	−1779.53	−0.157275
$505$	0	0
$506$	−11605.9	−1.01965
$507$	507.000	0.0444116
$508$	5690.58	0.497005
$509$	−2898.08	−0.252367	−0.126184	−	0.992007i	$-0.540273\pi$
$509$	−2898.08	−0.252367	−0.126184	+	0.992007i	$0.540273\pi$
$510$	0	0
$511$	−16783.1	−1.45292
$512$	512.000	0.0441942
$513$	−2757.97	−0.237363
$514$	4970.57	0.426542
$515$	0	0
$516$	1374.82	0.117293
$517$	24878.1	2.11632
$518$	−18501.8	−1.56935
$519$	10602.3	0.896707
$520$	0	0
$521$	−12113.5	−1.01862	−0.509311	−	0.860583i	$-0.670100\pi$
$521$	−12113.5	−1.01862	−0.509311	+	0.860583i	$0.670100\pi$
$522$	671.119	0.0562722
$523$	6676.45	0.558204	0.279102	−	0.960261i	$-0.409963\pi$
$523$	6676.45	0.558204	0.279102	+	0.960261i	$0.409963\pi$
$524$	2864.81	0.238836
$525$	0	0
$526$	3719.16	0.308295
$527$	2790.90	0.230690
$528$	3332.70	0.274692
$529$	−5181.69	−0.425881
$530$	0	0
$531$	6503.12	0.531471
$532$	10098.5	0.822980
$533$	−2295.39	−0.186538
$534$	−7899.87	−0.640188
$535$	0	0
$536$	−2951.05	−0.237810
$537$	−4370.03	−0.351175
$538$	−9016.90	−0.722576
$539$	18598.0	1.48622
$540$	0	0
$541$	−24548.0	−1.95084	−0.975418	−	0.220363i	$-0.929276\pi$
$541$	−24548.0	−1.95084	−0.975418	+	0.220363i	$0.929276\pi$
$542$	−6591.13	−0.522349
$543$	6992.64	0.552639
$544$	726.900	0.0572897
$545$	0	0
$546$	1927.82	0.151104
$547$	−23045.0	−1.80134	−0.900669	−	0.434507i	$-0.856923\pi$
$547$	−23045.0	−1.80134	−0.900669	+	0.434507i	$0.856923\pi$
$548$	−5749.80	−0.448210
$549$	4060.24	0.315641
$550$	0	0
$551$	−3808.48	−0.294459
$552$	−2005.88	−0.154666
$553$	−24709.3	−1.90008
$554$	−6919.75	−0.530672
$555$	0	0
$556$	−6904.08	−0.526615
$557$	1419.27	0.107965	0.0539823	−	0.998542i	$-0.482809\pi$
$557$	1419.27	0.107965	0.0539823	+	0.998542i	$0.482809\pi$
$558$	2211.53	0.167780
$559$	−1489.39	−0.112692
$560$	0	0
$561$	4731.53	0.356088
$562$	12617.2	0.947020
$563$	11992.2	0.897712	0.448856	−	0.893604i	$-0.351832\pi$
$563$	11992.2	0.897712	0.448856	+	0.893604i	$0.351832\pi$
$564$	4299.75	0.321015
$565$	0	0
$566$	−10252.6	−0.761397
$567$	−2001.97	−0.148280
$568$	−1405.95	−0.103860
$569$	−22641.4	−1.66815	−0.834076	−	0.551650i	$-0.813999\pi$
$569$	−22641.4	−1.66815	−0.834076	+	0.551650i	$0.813999\pi$
$570$	0	0
$571$	−3019.10	−0.221270	−0.110635	−	0.993861i	$-0.535288\pi$
$571$	−3019.10	−0.221270	−0.110635	+	0.993861i	$0.535288\pi$
$572$	−3610.43	−0.263915
$573$	8248.95	0.601405
$574$	−8728.02	−0.634670
$575$	0	0
$576$	576.000	0.0416667
$577$	10924.4	0.788198	0.394099	−	0.919068i	$-0.371057\pi$
$577$	10924.4	0.788198	0.394099	+	0.919068i	$0.371057\pi$
$578$	−8794.00	−0.632841
$579$	−5464.42	−0.392217
$580$	0	0
$581$	32392.5	2.31302
$582$	−2724.99	−0.194080
$583$	−36362.9	−2.58319
$584$	5432.38	0.384920
$585$	0	0
$586$	10582.8	0.746027
$587$	−1033.12	−0.0726431	−0.0363215	−	0.999340i	$-0.511564\pi$
$587$	−1033.12	−0.0726431	−0.0363215	+	0.999340i	$0.511564\pi$
$588$	3214.35	0.225438
$589$	−12550.0	−0.877954
$590$	0	0
$591$	−11399.7	−0.793439
$592$	5988.70	0.415767
$593$	−5737.44	−0.397316	−0.198658	−	0.980069i	$-0.563658\pi$
$593$	−5737.44	−0.397316	−0.198658	+	0.980069i	$0.563658\pi$
$594$	3749.29	0.258982
$595$	0	0
$596$	13509.1	0.928445
$597$	2618.98	0.179544
$598$	2173.03	0.148598
$599$	−22779.5	−1.55383	−0.776917	−	0.629603i	$-0.783217\pi$
$599$	−22779.5	−1.55383	−0.776917	+	0.629603i	$0.783217\pi$
$600$	0	0
$601$	−23346.9	−1.58459	−0.792297	−	0.610136i	$-0.791115\pi$
$601$	−23346.9	−1.58459	−0.792297	+	0.610136i	$0.791115\pi$
$602$	−5663.28	−0.383419
$603$	−3319.93	−0.224209
$604$	−9170.80	−0.617806
$605$	0	0
$606$	7367.12	0.493843
$607$	5522.26	0.369261	0.184631	−	0.982808i	$-0.440891\pi$
$607$	5522.26	0.369261	0.184631	+	0.982808i	$0.440891\pi$
$608$	−3268.70	−0.218032
$609$	−2764.52	−0.183947
$610$	0	0
$611$	−4658.06	−0.308421
$612$	817.763	0.0540132
$613$	−22225.3	−1.46439	−0.732194	−	0.681096i	$-0.761504\pi$
$613$	−22225.3	−1.46439	−0.732194	+	0.681096i	$0.761504\pi$
$614$	−11716.4	−0.770091
$615$	0	0
$616$	−13728.3	−0.897937
$617$	−19829.4	−1.29384	−0.646922	−	0.762556i	$-0.723944\pi$
$617$	−19829.4	−1.29384	−0.646922	+	0.762556i	$0.723944\pi$
$618$	5758.46	0.374821
$619$	6399.70	0.415550	0.207775	−	0.978177i	$-0.433378\pi$
$619$	6399.70	0.415550	0.207775	+	0.978177i	$0.433378\pi$
$620$	0	0
$621$	−2256.61	−0.145821
$622$	−4337.63	−0.279619
$623$	32541.7	2.09271
$624$	−624.000	−0.0400320
$625$	0	0
$626$	−8210.46	−0.524211
$627$	−21276.6	−1.35519
$628$	9729.55	0.618235
$629$	8502.32	0.538966
$630$	0	0
$631$	10482.5	0.661333	0.330667	−	0.943748i	$-0.392726\pi$
$631$	10482.5	0.661333	0.330667	+	0.943748i	$0.392726\pi$
$632$	7997.95	0.503388
$633$	3043.05	0.191075
$634$	4606.73	0.288575
$635$	0	0
$636$	−6284.70	−0.391831
$637$	−3482.21	−0.216594
$638$	5177.40	0.321278
$639$	−1581.70	−0.0979200
$640$	0	0
$641$	5622.19	0.346432	0.173216	−	0.984884i	$-0.444584\pi$
$641$	5622.19	0.346432	0.173216	+	0.984884i	$0.444584\pi$
$642$	12631.3	0.776506
$643$	−3514.64	−0.215558	−0.107779	−	0.994175i	$-0.534374\pi$
$643$	−3514.64	−0.215558	−0.107779	+	0.994175i	$0.534374\pi$
$644$	8262.75	0.505587
$645$	0	0
$646$	−4640.66	−0.282638
$647$	22134.0	1.34494	0.672470	−	0.740125i	$-0.265234\pi$
$647$	22134.0	1.34494	0.672470	+	0.740125i	$0.265234\pi$
$648$	648.000	0.0392837
$649$	50168.9	3.03436
$650$	0	0
$651$	−9109.88	−0.548455
$652$	13422.6	0.806239
$653$	−6410.46	−0.384166	−0.192083	−	0.981379i	$-0.561524\pi$
$653$	−6410.46	−0.384166	−0.192083	+	0.981379i	$0.561524\pi$
$654$	−3245.94	−0.194077
$655$	0	0
$656$	2825.10	0.168143
$657$	6111.42	0.362906
$658$	−17711.8	−1.04936
$659$	−14644.7	−0.865670	−0.432835	−	0.901473i	$-0.642487\pi$
$659$	−14644.7	−0.865670	−0.432835	+	0.901473i	$0.642487\pi$
$660$	0	0
$661$	20048.6	1.17973	0.589865	−	0.807502i	$-0.299181\pi$
$661$	20048.6	1.17973	0.589865	+	0.807502i	$0.299181\pi$
$662$	7691.78	0.451585
$663$	−885.910	−0.0518942
$664$	−10484.9	−0.612788
$665$	0	0
$666$	6737.29	0.391989
$667$	−3116.16	−0.180897
$668$	5626.27	0.325879
$669$	−5703.25	−0.329597
$670$	0	0
$671$	31323.0	1.80211
$672$	−2372.70	−0.136204
$673$	26499.8	1.51782	0.758910	−	0.651196i	$-0.225732\pi$
$673$	26499.8	1.51782	0.758910	+	0.651196i	$0.225732\pi$
$674$	−1202.01	−0.0686940
$675$	0	0
$676$	676.000	0.0384615
$677$	−34096.3	−1.93564	−0.967818	−	0.251651i	$-0.919027\pi$
$677$	−34096.3	−1.93564	−0.967818	+	0.251651i	$0.919027\pi$
$678$	−10525.2	−0.596193
$679$	11225.0	0.634427
$680$	0	0
$681$	4232.93	0.238188
$682$	17061.0	0.957917
$683$	14549.2	0.815097	0.407548	−	0.913184i	$-0.366384\pi$
$683$	14549.2	0.815097	0.407548	+	0.913184i	$0.366384\pi$
$684$	−3677.29	−0.205562
$685$	0	0
$686$	3714.14	0.206715
$687$	7002.07	0.388858
$688$	1833.10	0.101579
$689$	6808.43	0.376459
$690$	0	0
$691$	−8372.70	−0.460944	−0.230472	−	0.973079i	$-0.574027\pi$
$691$	−8372.70	−0.460944	−0.230472	+	0.973079i	$0.574027\pi$
$692$	14136.5	0.776571
$693$	−15444.3	−0.846583
$694$	−4297.69	−0.235069
$695$	0	0
$696$	894.825	0.0487331
$697$	4010.87	0.217966
$698$	−19532.1	−1.05917
$699$	−5059.49	−0.273773
$700$	0	0
$701$	5931.63	0.319593	0.159796	−	0.987150i	$-0.448916\pi$
$701$	5931.63	0.319593	0.159796	+	0.987150i	$0.448916\pi$
$702$	−702.000	−0.0377426
$703$	−38233.0	−2.05119
$704$	4443.60	0.237890
$705$	0	0
$706$	−3924.00	−0.209181
$707$	−30347.2	−1.61432
$708$	8670.82	0.460268
$709$	16609.8	0.879822	0.439911	−	0.898041i	$-0.355010\pi$
$709$	16609.8	0.879822	0.439911	+	0.898041i	$0.355010\pi$
$710$	0	0
$711$	8997.70	0.474599
$712$	−10533.2	−0.554419
$713$	−10268.6	−0.539359
$714$	−3368.59	−0.176563
$715$	0	0
$716$	−5826.71	−0.304126
$717$	1975.28	0.102884
$718$	−5679.04	−0.295181
$719$	18475.2	0.958289	0.479144	−	0.877736i	$-0.340947\pi$
$719$	18475.2	0.958289	0.479144	+	0.877736i	$0.340947\pi$
$720$	0	0
$721$	−23720.7	−1.22525
$722$	7149.98	0.368552
$723$	12153.5	0.625165
$724$	9323.53	0.478600
$725$	0	0
$726$	20938.2	1.07037
$727$	4893.89	0.249662	0.124831	−	0.992178i	$-0.460161\pi$
$727$	4893.89	0.249662	0.124831	+	0.992178i	$0.460161\pi$
$728$	2570.43	0.130860
$729$	729.000	0.0370370
$730$	0	0
$731$	2602.50	0.131679
$732$	5413.65	0.273353
$733$	36355.6	1.83196	0.915979	−	0.401225i	$-0.131416\pi$
$733$	36355.6	1.83196	0.915979	+	0.401225i	$0.131416\pi$
$734$	7741.29	0.389286
$735$	0	0
$736$	−2674.50	−0.133945
$737$	−25611.9	−1.28009
$738$	3178.24	0.158526
$739$	−17826.9	−0.887377	−0.443688	−	0.896181i	$-0.646330\pi$
$739$	−17826.9	−0.887377	−0.443688	+	0.896181i	$0.646330\pi$
$740$	0	0
$741$	3983.73	0.197498
$742$	25888.4	1.28085
$743$	39501.1	1.95041	0.975204	−	0.221307i	$-0.0710322\pi$
$743$	39501.1	1.95041	0.975204	+	0.221307i	$0.0710322\pi$
$744$	2948.70	0.145302
$745$	0	0
$746$	−7313.27	−0.358925
$747$	−11795.5	−0.577742
$748$	6308.70	0.308381
$749$	−52031.7	−2.53831
$750$	0	0
$751$	−35688.8	−1.73409	−0.867046	−	0.498228i	$-0.833984\pi$
$751$	−35688.8	−1.73409	−0.867046	+	0.498228i	$0.833984\pi$
$752$	5733.00	0.278007
$753$	−10064.0	−0.487055
$754$	−969.394	−0.0468213
$755$	0	0
$756$	−2669.29	−0.128414
$757$	6300.22	0.302490	0.151245	−	0.988496i	$-0.451672\pi$
$757$	6300.22	0.302490	0.151245	+	0.988496i	$0.451672\pi$
$758$	16223.8	0.777409
$759$	−17408.8	−0.832543
$760$	0	0
$761$	−9254.88	−0.440853	−0.220426	−	0.975404i	$-0.570745\pi$
$761$	−9254.88	−0.440853	−0.220426	+	0.975404i	$0.570745\pi$
$762$	8535.87	0.405803
$763$	13370.9	0.634417
$764$	10998.6	0.520832
$765$	0	0
$766$	23262.2	1.09725
$767$	−9393.39	−0.442211
$768$	768.000	0.0360844
$769$	20070.9	0.941189	0.470595	−	0.882350i	$-0.344039\pi$
$769$	20070.9	0.941189	0.470595	+	0.882350i	$0.344039\pi$
$770$	0	0
$771$	7455.85	0.348270
$772$	−7285.89	−0.339670
$773$	−1015.46	−0.0472491	−0.0236246	−	0.999721i	$-0.507521\pi$
$773$	−1015.46	−0.0472491	−0.0236246	+	0.999721i	$0.507521\pi$
$774$	2062.24	0.0957695
$775$	0	0
$776$	−3633.33	−0.168078
$777$	−27752.7	−1.28137
$778$	11446.6	0.527480
$779$	−18035.9	−0.829531
$780$	0	0
$781$	−12202.1	−0.559061
$782$	−3797.06	−0.173635
$783$	1006.68	0.0459460
$784$	4285.80	0.195235
$785$	0	0
$786$	4297.22	0.195009
$787$	4194.48	0.189984	0.0949918	−	0.995478i	$-0.469718\pi$
$787$	4194.48	0.189984	0.0949918	+	0.995478i	$0.469718\pi$
$788$	−15199.7	−0.687139
$789$	5578.73	0.251721
$790$	0	0
$791$	43356.3	1.94889
$792$	4999.05	0.224285
$793$	−5864.79	−0.262629
$794$	−5868.51	−0.262299
$795$	0	0
$796$	3491.98	0.155490
$797$	−17342.0	−0.770748	−0.385374	−	0.922760i	$-0.625928\pi$
$797$	−17342.0	−0.770748	−0.385374	+	0.922760i	$0.625928\pi$
$798$	15147.8	0.671961
$799$	8139.30	0.360385
$800$	0	0
$801$	−11849.8	−0.522712
$802$	−12179.1	−0.536235
$803$	47147.1	2.07196
$804$	−4426.58	−0.194171
$805$	0	0
$806$	−3194.43	−0.139602
$807$	−13525.3	−0.589981
$808$	9822.82	0.427680
$809$	−28170.2	−1.22424	−0.612120	−	0.790765i	$-0.709683\pi$
$809$	−28170.2	−1.22424	−0.612120	+	0.790765i	$0.709683\pi$
$810$	0	0
$811$	−860.578	−0.0372614	−0.0186307	−	0.999826i	$-0.505931\pi$
$811$	−860.578	−0.0372614	−0.0186307	+	0.999826i	$0.505931\pi$
$812$	−3686.03	−0.159303
$813$	−9886.69	−0.426496
$814$	51975.4	2.23801
$815$	0	0
$816$	1090.35	0.0467768
$817$	−11702.8	−0.501139
$818$	18396.6	0.786334
$819$	2891.73	0.123376
$820$	0	0
$821$	31502.6	1.33916	0.669580	−	0.742740i	$-0.266474\pi$
$821$	31502.6	1.33916	0.669580	+	0.742740i	$0.266474\pi$
$822$	−8624.70	−0.365962
$823$	−18862.9	−0.798929	−0.399464	−	0.916749i	$-0.630804\pi$
$823$	−18862.9	−0.798929	−0.399464	+	0.916749i	$0.630804\pi$
$824$	7677.95	0.324605
$825$	0	0
$826$	−35717.5	−1.50456
$827$	−44348.4	−1.86475	−0.932374	−	0.361496i	$-0.882266\pi$
$827$	−44348.4	−1.86475	−0.932374	+	0.361496i	$0.882266\pi$
$828$	−3008.81	−0.126284
$829$	−10661.7	−0.446678	−0.223339	−	0.974741i	$-0.571696\pi$
$829$	−10661.7	−0.446678	−0.223339	+	0.974741i	$0.571696\pi$
$830$	0	0
$831$	−10379.6	−0.433292
$832$	−832.000	−0.0346688
$833$	6084.67	0.253087
$834$	−10356.1	−0.429979
$835$	0	0
$836$	−28368.8	−1.17363
$837$	3317.29	0.136992
$838$	−11311.1	−0.466270
$839$	44333.3	1.82426	0.912130	−	0.409901i	$-0.134437\pi$
$839$	44333.3	1.82426	0.912130	+	0.409901i	$0.134437\pi$
$840$	0	0
$841$	−22998.9	−0.943002
$842$	−13374.7	−0.547416
$843$	18925.8	0.773238
$844$	4057.40	0.165476
$845$	0	0
$846$	6449.63	0.262107
$847$	−86250.2	−3.49893
$848$	−8379.60	−0.339336
$849$	−15379.0	−0.621678
$850$	0	0
$851$	−31282.8	−1.26012
$852$	−2108.93	−0.0848012
$853$	34534.5	1.38621	0.693105	−	0.720836i	$-0.256242\pi$
$853$	34534.5	1.38621	0.693105	+	0.720836i	$0.256242\pi$
$854$	−22300.3	−0.893560
$855$	0	0
$856$	16841.7	0.672474
$857$	−35150.2	−1.40106	−0.700529	−	0.713624i	$-0.747053\pi$
$857$	−35150.2	−1.40106	−0.700529	+	0.713624i	$0.747053\pi$
$858$	−5415.64	−0.215486
$859$	−6398.73	−0.254158	−0.127079	−	0.991893i	$-0.540560\pi$
$859$	−6398.73	−0.254158	−0.127079	+	0.991893i	$0.540560\pi$
$860$	0	0
$861$	−13092.0	−0.518206
$862$	754.954	0.0298304
$863$	5448.16	0.214899	0.107449	−	0.994211i	$-0.465732\pi$
$863$	5448.16	0.214899	0.107449	+	0.994211i	$0.465732\pi$
$864$	864.000	0.0340207
$865$	0	0
$866$	15806.9	0.620253
$867$	−13191.0	−0.516713
$868$	−12146.5	−0.474976
$869$	69413.5	2.70966
$870$	0	0
$871$	4795.46	0.186553
$872$	−4327.92	−0.168076
$873$	−4087.49	−0.158466
$874$	17074.5	0.660817
$875$	0	0
$876$	8148.57	0.314286
$877$	−29875.7	−1.15032	−0.575161	−	0.818040i	$-0.695061\pi$
$877$	−29875.7	−1.15032	−0.575161	+	0.818040i	$0.695061\pi$
$878$	−32919.8	−1.26536
$879$	15874.2	0.609128
$880$	0	0
$881$	16680.0	0.637868	0.318934	−	0.947777i	$-0.396675\pi$
$881$	16680.0	0.637868	0.318934	+	0.947777i	$0.396675\pi$
$882$	4821.53	0.184069
$883$	−31287.5	−1.19242	−0.596210	−	0.802828i	$-0.703327\pi$
$883$	−31287.5	−1.19242	−0.596210	+	0.802828i	$0.703327\pi$
$884$	−1181.21	−0.0449417
$885$	0	0
$886$	2723.81	0.103282
$887$	2712.06	0.102663	0.0513314	−	0.998682i	$-0.483654\pi$
$887$	2712.06	0.102663	0.0513314	+	0.998682i	$0.483654\pi$
$888$	8983.05	0.339472
$889$	−35161.6	−1.32652
$890$	0	0
$891$	5623.93	0.211458
$892$	−7604.34	−0.285440
$893$	−36600.5	−1.37155
$894$	20263.6	0.758072
$895$	0	0
$896$	−3163.60	−0.117956
$897$	3259.55	0.121330
$898$	−19849.0	−0.737606
$899$	4580.85	0.169944
$900$	0	0
$901$	−11896.7	−0.439887
$902$	24518.8	0.905084
$903$	−8494.92	−0.313060
$904$	−14033.6	−0.516318
$905$	0	0
$906$	−13756.2	−0.504436
$907$	−38445.0	−1.40744	−0.703719	−	0.710478i	$-0.748479\pi$
$907$	−38445.0	−1.40744	−0.703719	+	0.710478i	$0.748479\pi$
$908$	5643.90	0.206277
$909$	11050.7	0.403221
$910$	0	0
$911$	−21695.4	−0.789023	−0.394512	−	0.918891i	$-0.629086\pi$
$911$	−21695.4	−0.789023	−0.394512	+	0.918891i	$0.629086\pi$
$912$	−4903.05	−0.178022
$913$	−90997.1	−3.29854
$914$	−6325.58	−0.228919
$915$	0	0
$916$	9336.09	0.336761
$917$	−17701.4	−0.637462
$918$	1226.64	0.0441016
$919$	43511.9	1.56183	0.780916	−	0.624636i	$-0.214753\pi$
$919$	43511.9	1.56183	0.780916	+	0.624636i	$0.214753\pi$
$920$	0	0
$921$	−17574.6	−0.628777
$922$	−4668.08	−0.166741
$923$	2284.67	0.0814744
$924$	−20592.5	−0.733162
$925$	0	0
$926$	19005.9	0.674486
$927$	8637.70	0.306040
$928$	1193.10	0.0422041
$929$	47420.1	1.67471	0.837354	−	0.546661i	$-0.184101\pi$
$929$	47420.1	1.67471	0.837354	+	0.546661i	$0.184101\pi$
$930$	0	0
$931$	−27361.3	−0.963192
$932$	−6745.99	−0.237095
$933$	−6506.44	−0.228308
$934$	−12929.9	−0.452974
$935$	0	0
$936$	−936.000	−0.0326860
$937$	45605.6	1.59004	0.795021	−	0.606583i	$-0.207460\pi$
$937$	45605.6	1.59004	0.795021	+	0.606583i	$0.207460\pi$
$938$	18234.3	0.634723
$939$	−12315.7	−0.428017
$940$	0	0
$941$	−27433.9	−0.950392	−0.475196	−	0.879880i	$-0.657623\pi$
$941$	−27433.9	−0.950392	−0.475196	+	0.879880i	$0.657623\pi$
$942$	14594.3	0.504787
$943$	−14757.3	−0.509611
$944$	11561.1	0.398604
$945$	0	0
$946$	15909.3	0.546782
$947$	13279.2	0.455667	0.227834	−	0.973700i	$-0.426836\pi$
$947$	13279.2	0.455667	0.227834	+	0.973700i	$0.426836\pi$
$948$	11996.9	0.411015
$949$	−8827.61	−0.301956
$950$	0	0
$951$	6910.09	0.235620
$952$	−4491.45	−0.152908
$953$	44919.8	1.52686	0.763428	−	0.645892i	$-0.223515\pi$
$953$	44919.8	1.52686	0.763428	+	0.645892i	$0.223515\pi$
$954$	−9427.05	−0.319929
$955$	0	0
$956$	2633.70	0.0891005
$957$	7766.10	0.262322
$958$	−20377.0	−0.687213
$959$	35527.5	1.19629
$960$	0	0
$961$	−14695.8	−0.493297
$962$	−9731.64	−0.326154
$963$	18946.9	0.634014
$964$	16204.7	0.541409
$965$	0	0
$966$	12394.1	0.412810
$967$	16705.7	0.555551	0.277775	−	0.960646i	$-0.410403\pi$
$967$	16705.7	0.555551	0.277775	+	0.960646i	$0.410403\pi$
$968$	27917.6	0.926969
$969$	−6960.99	−0.230773
$970$	0	0
$971$	−8016.63	−0.264950	−0.132475	−	0.991186i	$-0.542292\pi$
$971$	−8016.63	−0.264950	−0.132475	+	0.991186i	$0.542292\pi$
$972$	972.000	0.0320750
$973$	42659.6	1.40556
$974$	13058.4	0.429587
$975$	0	0
$976$	7218.20	0.236730
$977$	23991.4	0.785623	0.392812	−	0.919619i	$-0.371502\pi$
$977$	23991.4	0.785623	0.392812	+	0.919619i	$0.371502\pi$
$978$	20133.8	0.658291
$979$	−91416.3	−2.98435
$980$	0	0
$981$	−4868.92	−0.158463
$982$	−41919.2	−1.36222
$983$	1525.59	0.0495002	0.0247501	−	0.999694i	$-0.492121\pi$
$983$	1525.59	0.0495002	0.0247501	+	0.999694i	$0.492121\pi$
$984$	4237.65	0.137288
$985$	0	0
$986$	1693.88	0.0547099
$987$	−26567.8	−0.856800
$988$	5311.64	0.171038
$989$	−9575.44	−0.307868
$990$	0	0
$991$	−9146.49	−0.293186	−0.146593	−	0.989197i	$-0.546831\pi$
$991$	−9146.49	−0.293186	−0.146593	+	0.989197i	$0.546831\pi$
$992$	3931.60	0.125835
$993$	11537.7	0.368718
$994$	8687.25	0.277206
$995$	0	0
$996$	−15727.3	−0.500339
$997$	13787.3	0.437962	0.218981	−	0.975729i	$-0.429727\pi$
$997$	13787.3	0.437962	0.218981	+	0.975729i	$0.429727\pi$
$998$	8969.67	0.284499
$999$	10105.9	0.320058

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1950.4.a.y.1.1		2
5.4	even	2		390.4.a.m.1.2	✓	2
15.14	odd	2		1170.4.a.x.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
390.4.a.m.1.2	✓	2	5.4	even	2
1170.4.a.x.1.2		2	15.14	odd	2
1950.4.a.y.1.1		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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 1950 = 2 \cdot 3 \cdot 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1950.a (trivial)

\(f(q)\)	\(=\)	\(q+2.00000 q^{2} +3.00000 q^{3} +4.00000 q^{4} +6.00000 q^{6} -24.7156 q^{7} +8.00000 q^{8} +9.00000 q^{9} +69.4313 q^{11} +12.0000 q^{12} -13.0000 q^{13} -49.4313 q^{14} +16.0000 q^{16} +22.7156 q^{17} +18.0000 q^{18} -102.147 q^{19} -74.1469 q^{21} +138.863 q^{22} -83.5782 q^{23} +24.0000 q^{24} -26.0000 q^{26} +27.0000 q^{27} -98.8625 q^{28} +37.2844 q^{29} +122.863 q^{31} +32.0000 q^{32} +208.294 q^{33} +45.4313 q^{34} +36.0000 q^{36} +374.294 q^{37} -204.294 q^{38} -39.0000 q^{39} +176.569 q^{41} -148.294 q^{42} +114.569 q^{43} +277.725 q^{44} -167.156 q^{46} +358.313 q^{47} +48.0000 q^{48} +267.863 q^{49} +68.1469 q^{51} -52.0000 q^{52} -523.725 q^{53} +54.0000 q^{54} -197.725 q^{56} -306.441 q^{57} +74.5687 q^{58} +722.569 q^{59} +451.137 q^{61} +245.725 q^{62} -222.441 q^{63} +64.0000 q^{64} +416.588 q^{66} -368.881 q^{67} +90.8625 q^{68} -250.735 q^{69} -175.744 q^{71} +72.0000 q^{72} +679.047 q^{73} +748.588 q^{74} -408.588 q^{76} -1716.04 q^{77} -78.0000 q^{78} +999.744 q^{79} +81.0000 q^{81} +353.137 q^{82} -1310.61 q^{83} -296.588 q^{84} +229.137 q^{86} +111.853 q^{87} +555.450 q^{88} -1316.64 q^{89} +321.303 q^{91} -334.313 q^{92} +368.588 q^{93} +716.625 q^{94} +96.0000 q^{96} -454.166 q^{97} +535.725 q^{98} +624.881 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 4 q^{2} + 6 q^{3} + 8 q^{4} + 12 q^{6} - 4 q^{7} + 16 q^{8} + 18 q^{9} + 48 q^{11} + 24 q^{12} - 26 q^{13} - 8 q^{14} + 32 q^{16} + 36 q^{18} - 68 q^{19} - 12 q^{21} + 96 q^{22} + 60 q^{23} + 48 q^{24}+ \cdots + 432 q^{99}+O(q^{100}) \) 2 * q + 4 * q^2 + 6 * q^3 + 8 * q^4 + 12 * q^6 - 4 * q^7 + 16 * q^8 + 18 * q^9 + 48 * q^11 + 24 * q^12 - 26 * q^13 - 8 * q^14 + 32 * q^16 + 36 * q^18 - 68 * q^19 - 12 * q^21 + 96 * q^22 + 60 * q^23 + 48 * q^24 - 52 * q^26 + 54 * q^27 - 16 * q^28 + 120 * q^29 + 64 * q^31 + 64 * q^32 + 144 * q^33 + 72 * q^36 + 476 * q^37 - 136 * q^38 - 78 * q^39 + 444 * q^41 - 24 * q^42 + 320 * q^43 + 192 * q^44 + 120 * q^46 - 192 * q^47 + 96 * q^48 + 354 * q^49 - 104 * q^52 - 684 * q^53 + 108 * q^54 - 32 * q^56 - 204 * q^57 + 240 * q^58 + 1536 * q^59 + 1084 * q^61 + 128 * q^62 - 36 * q^63 + 128 * q^64 + 288 * q^66 + 80 * q^67 + 180 * q^69 + 648 * q^71 + 144 * q^72 - 232 * q^73 + 952 * q^74 - 272 * q^76 - 2160 * q^77 - 156 * q^78 + 1000 * q^79 + 162 * q^81 + 888 * q^82 - 1440 * q^83 - 48 * q^84 + 640 * q^86 + 360 * q^87 + 384 * q^88 - 180 * q^89 + 52 * q^91 + 240 * q^92 + 192 * q^93 - 384 * q^94 + 192 * q^96 - 136 * q^97 + 708 * q^98 + 432 * q^99

Introduction

L-functions

Modular forms

Varieties

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Representations

Groups

Database

Properties

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