LMFDB - Embedded newform 1950.4.a.y.1.2

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			1.2
Root			$-5.17891$ 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)

$f(q)$	$=$	\(q+2.00000 q^{2} +3.00000 q^{3} +4.00000 q^{4} +6.00000 q^{6} +20.7156 q^{7} +8.00000 q^{8} +9.00000 q^{9} -21.4313 q^{11} +12.0000 q^{12} -13.0000 q^{13} +41.4313 q^{14} +16.0000 q^{16} -22.7156 q^{17} +18.0000 q^{18} +34.1469 q^{19} +62.1469 q^{21} -42.8625 q^{22} +143.578 q^{23} +24.0000 q^{24} -26.0000 q^{26} +27.0000 q^{27} +82.8625 q^{28} +82.7156 q^{29} -58.8625 q^{31} +32.0000 q^{32} -64.2938 q^{33} -45.4313 q^{34} +36.0000 q^{36} +101.706 q^{37} +68.2938 q^{38} -39.0000 q^{39} +267.431 q^{41} +124.294 q^{42} +205.431 q^{43} -85.7251 q^{44} +287.156 q^{46} -550.313 q^{47} +48.0000 q^{48} +86.1375 q^{49} -68.1469 q^{51} -52.0000 q^{52} -160.275 q^{53} +54.0000 q^{54} +165.725 q^{56} +102.441 q^{57} +165.431 q^{58} +813.431 q^{59} +632.863 q^{61} -117.725 q^{62} +186.441 q^{63} +64.0000 q^{64} -128.588 q^{66} +448.881 q^{67} -90.8625 q^{68} +430.735 q^{69} +823.744 q^{71} +72.0000 q^{72} -911.047 q^{73} +203.412 q^{74} +136.588 q^{76} -443.962 q^{77} -78.0000 q^{78} +0.256066 q^{79} +81.0000 q^{81} +534.863 q^{82} -129.394 q^{83} +248.588 q^{84} +410.863 q^{86} +248.147 q^{87} -171.450 q^{88} +1136.64 q^{89} -269.303 q^{91} +574.313 q^{92} -176.588 q^{93} -1100.63 q^{94} +96.0000 q^{96} +318.166 q^{97} +172.275 q^{98} -192.881 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$7$	20.7156	1.11854	0.559270	−	0.828986i	$-0.311082\pi$
$7$	20.7156	1.11854	0.559270	+	0.828986i	$0.311082\pi$
$8$	8.00000	0.353553
$9$	9.00000	0.333333
$10$	0	0
$11$	−21.4313	−0.587434	−0.293717	−	0.955892i	$-0.594892\pi$
$11$	−21.4313	−0.587434	−0.293717	+	0.955892i	$0.594892\pi$
$12$	12.0000	0.288675
$13$	−13.0000	−0.277350
$13$	−13.0000	−0.277350
$14$	41.4313	0.790927
$15$	0	0
$16$	16.0000	0.250000
$17$	−22.7156	−0.324079	−0.162040	−	0.986784i	$-0.551807\pi$
$17$	−22.7156	−0.324079	−0.162040	+	0.986784i	$0.551807\pi$
$18$	18.0000	0.235702
$19$	34.1469	0.412307	0.206154	−	0.978520i	$-0.433905\pi$
$19$	34.1469	0.412307	0.206154	+	0.978520i	$0.433905\pi$
$20$	0	0
$21$	62.1469	0.645789
$22$	−42.8625	−0.415378
$23$	143.578	1.30166	0.650829	−	0.759225i	$-0.274422\pi$
$23$	143.578	1.30166	0.650829	+	0.759225i	$0.274422\pi$
$24$	24.0000	0.204124
$25$	0	0
$26$	−26.0000	−0.196116
$27$	27.0000	0.192450
$28$	82.8625	0.559270
$29$	82.7156	0.529652	0.264826	−	0.964296i	$-0.414685\pi$
$29$	82.7156	0.529652	0.264826	+	0.964296i	$0.414685\pi$
$30$	0	0
$31$	−58.8625	−0.341033	−0.170517	−	0.985355i	$-0.554544\pi$
$31$	−58.8625	−0.341033	−0.170517	+	0.985355i	$0.554544\pi$
$32$	32.0000	0.176777
$33$	−64.2938	−0.339155
$34$	−45.4313	−0.229159
$35$	0	0
$36$	36.0000	0.166667
$37$	101.706	0.451903	0.225951	−	0.974139i	$-0.427451\pi$
$37$	101.706	0.451903	0.225951	+	0.974139i	$0.427451\pi$
$38$	68.2938	0.291545
$39$	−39.0000	−0.160128
$40$	0	0
$41$	267.431	1.01868	0.509338	−	0.860566i	$-0.329890\pi$
$41$	267.431	1.01868	0.509338	+	0.860566i	$0.329890\pi$
$42$	124.294	0.456642
$43$	205.431	0.728558	0.364279	−	0.931290i	$-0.381316\pi$
$43$	205.431	0.728558	0.364279	+	0.931290i	$0.381316\pi$
$44$	−85.7251	−0.293717
$45$	0	0
$46$	287.156	0.920411
$47$	−550.313	−1.70790	−0.853951	−	0.520354i	$-0.825800\pi$
$47$	−550.313	−1.70790	−0.853951	+	0.520354i	$0.825800\pi$
$48$	48.0000	0.144338
$49$	86.1375	0.251130
$50$	0	0
$51$	−68.1469	−0.187107
$52$	−52.0000	−0.138675
$53$	−160.275	−0.415386	−0.207693	−	0.978194i	$-0.566595\pi$
$53$	−160.275	−0.415386	−0.207693	+	0.978194i	$0.566595\pi$
$54$	54.0000	0.136083
$55$	0	0
$56$	165.725	0.395463
$57$	102.441	0.238046
$58$	165.431	0.374521
$59$	813.431	1.79491	0.897455	−	0.441105i	$-0.145413\pi$
$59$	813.431	1.79491	0.897455	+	0.441105i	$0.145413\pi$
$60$	0	0
$61$	632.863	1.32836	0.664178	−	0.747574i	$-0.268782\pi$
$61$	632.863	1.32836	0.664178	+	0.747574i	$0.268782\pi$
$62$	−117.725	−0.241147
$63$	186.441	0.372846
$64$	64.0000	0.125000
$65$	0	0
$66$	−128.588	−0.239819
$67$	448.881	0.818501	0.409251	−	0.912422i	$-0.365790\pi$
$67$	448.881	0.818501	0.409251	+	0.912422i	$0.365790\pi$
$68$	−90.8625	−0.162040
$69$	430.735	0.751512
$70$	0	0
$71$	823.744	1.37691	0.688454	−	0.725280i	$-0.258290\pi$
$71$	823.744	1.37691	0.688454	+	0.725280i	$0.258290\pi$
$72$	72.0000	0.117851
$73$	−911.047	−1.46069	−0.730343	−	0.683081i	$-0.760640\pi$
$73$	−911.047	−1.46069	−0.730343	+	0.683081i	$0.760640\pi$
$74$	203.412	0.319543
$75$	0	0
$76$	136.588	0.206154
$77$	−443.962	−0.657067
$78$	−78.0000	−0.113228
$79$	0.256066	0.000364679 0	0.000182339	−	1.00000i	$-0.499942\pi$
$79$	0.256066	0.000364679 0	0.000182339		1.00000i	$0.499942\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	534.863	0.720313
$83$	−129.394	−0.171118	−0.0855589	−	0.996333i	$-0.527268\pi$
$83$	−129.394	−0.171118	−0.0855589	+	0.996333i	$0.527268\pi$
$84$	248.588	0.322894
$85$	0	0
$86$	410.863	0.515168
$87$	248.147	0.305795
$88$	−171.450	−0.207689
$89$	1136.64	1.35375	0.676877	−	0.736096i	$-0.263333\pi$
$89$	1136.64	1.35375	0.676877	+	0.736096i	$0.263333\pi$
$90$	0	0
$91$	−269.303	−0.310227
$92$	574.313	0.650829
$93$	−176.588	−0.196895
$94$	−1100.63	−1.20767
$95$	0	0
$96$	96.0000	0.102062
$97$	318.166	0.333040	0.166520	−	0.986038i	$-0.446747\pi$
$97$	318.166	0.333040	0.166520	+	0.986038i	$0.446747\pi$
$98$	172.275	0.177575
$99$	−192.881	−0.195811
$100$	0	0
$101$	1364.15	1.34394	0.671969	−	0.740579i	$-0.265449\pi$
$101$	1364.15	1.34394	0.671969	+	0.740579i	$0.265449\pi$
$102$	−136.294	−0.132305
$103$	−39.7439	−0.0380203	−0.0190101	−	0.999819i	$-0.506051\pi$
$103$	−39.7439	−0.0380203	−0.0190101	+	0.999819i	$0.506051\pi$
$104$	−104.000	−0.0980581
$105$	0	0
$106$	−320.550	−0.293722
$107$	−257.213	−0.232390	−0.116195	−	0.993226i	$-0.537070\pi$
$107$	−257.213	−0.232390	−0.116195	+	0.993226i	$0.537070\pi$
$108$	108.000	0.0962250
$109$	−859.009	−0.754846	−0.377423	−	0.926041i	$-0.623190\pi$
$109$	−859.009	−0.754846	−0.377423	+	0.926041i	$0.623190\pi$
$110$	0	0
$111$	305.119	0.260906
$112$	331.450	0.279635
$113$	290.204	0.241593	0.120797	−	0.992677i	$-0.461455\pi$
$113$	290.204	0.241593	0.120797	+	0.992677i	$0.461455\pi$
$114$	204.881	0.168324
$115$	0	0
$116$	330.863	0.264826
$117$	−117.000	−0.0924500
$118$	1626.86	1.26919
$119$	−470.569	−0.362496
$120$	0	0
$121$	−871.701	−0.654922
$122$	1265.73	0.939290
$123$	802.294	0.588133
$124$	−235.450	−0.170517
$125$	0	0
$126$	372.881	0.263642
$127$	−1030.64	−0.720117	−0.360059	−	0.932930i	$-0.617243\pi$
$127$	−1030.64	−0.720117	−0.360059	+	0.932930i	$0.617243\pi$
$128$	128.000	0.0883883
$129$	616.294	0.420633
$130$	0	0
$131$	−1328.20	−0.885845	−0.442922	−	0.896560i	$-0.646058\pi$
$131$	−1328.20	−0.885845	−0.442922	+	0.896560i	$0.646058\pi$
$132$	−257.175	−0.169577
$133$	707.375	0.461182
$134$	897.763	0.578768
$135$	0	0
$136$	−181.725	−0.114579
$137$	−710.550	−0.443112	−0.221556	−	0.975148i	$-0.571114\pi$
$137$	−710.550	−0.443112	−0.221556	+	0.975148i	$0.571114\pi$
$138$	861.469	0.531399
$139$	−1089.98	−0.665115	−0.332558	−	0.943083i	$-0.607912\pi$
$139$	−1089.98	−0.665115	−0.332558	+	0.943083i	$0.607912\pi$
$140$	0	0
$141$	−1650.94	−0.986057
$142$	1647.49	0.973620
$143$	278.606	0.162925
$144$	144.000	0.0833333
$145$	0	0
$146$	−1822.09	−1.03286
$147$	258.412	0.144990
$148$	406.825	0.225951
$149$	−893.270	−0.491138	−0.245569	−	0.969379i	$-0.578975\pi$
$149$	−893.270	−0.491138	−0.245569	+	0.969379i	$0.578975\pi$
$150$	0	0
$151$	2068.70	1.11489	0.557445	−	0.830214i	$-0.311782\pi$
$151$	2068.70	1.11489	0.557445	+	0.830214i	$0.311782\pi$
$152$	273.175	0.145773
$153$	−204.441	−0.108026
$154$	−887.925	−0.464617
$155$	0	0
$156$	−156.000	−0.0800641
$157$	−1020.39	−0.518700	−0.259350	−	0.965783i	$-0.583508\pi$
$157$	−1020.39	−0.518700	−0.259350	+	0.965783i	$0.583508\pi$
$158$	0.512131	0.000257867 0
$159$	−480.825	−0.239823
$160$	0	0
$161$	2974.31	1.45595
$162$	162.000	0.0785674
$163$	−3731.64	−1.79316	−0.896578	−	0.442885i	$-0.853955\pi$
$163$	−3731.64	−1.79316	−0.896578	+	0.442885i	$0.853955\pi$
$164$	1069.73	0.509338
$165$	0	0
$166$	−258.787	−0.120999
$167$	1497.43	0.693861	0.346930	−	0.937891i	$-0.387224\pi$
$167$	1497.43	0.693861	0.346930	+	0.937891i	$0.387224\pi$
$168$	497.175	0.228321
$169$	169.000	0.0769231
$170$	0	0
$171$	307.322	0.137436
$172$	821.725	0.364279
$173$	−282.113	−0.123981	−0.0619904	−	0.998077i	$-0.519745\pi$
$173$	−282.113	−0.123981	−0.0619904	+	0.998077i	$0.519745\pi$
$174$	496.294	0.216230
$175$	0	0
$176$	−342.900	−0.146858
$177$	2440.29	1.03629
$178$	2273.29	0.957248
$179$	−2683.32	−1.12045	−0.560226	−	0.828340i	$-0.689286\pi$
$179$	−2683.32	−1.12045	−0.560226	+	0.828340i	$0.689286\pi$
$180$	0	0
$181$	1513.12	0.621377	0.310688	−	0.950512i	$-0.399440\pi$
$181$	1513.12	0.621377	0.310688	+	0.950512i	$0.399440\pi$
$182$	−538.606	−0.219364
$183$	1898.59	0.766927
$184$	1148.63	0.460205
$185$	0	0
$186$	−353.175	−0.139226
$187$	486.825	0.190375
$188$	−2201.25	−0.853951
$189$	559.322	0.215263
$190$	0	0
$191$	4930.35	1.86779	0.933894	−	0.357549i	$-0.116388\pi$
$191$	4930.35	1.86779	0.933894	+	0.357549i	$0.116388\pi$
$192$	192.000	0.0721688
$193$	4493.47	1.67589	0.837946	−	0.545753i	$-0.183756\pi$
$193$	4493.47	1.67589	0.837946	+	0.545753i	$0.183756\pi$
$194$	636.332	0.235495
$195$	0	0
$196$	344.550	0.125565
$197$	2923.91	1.05746	0.528732	−	0.848789i	$-0.322668\pi$
$197$	2923.91	1.05746	0.528732	+	0.848789i	$0.322668\pi$
$198$	−385.763	−0.138459
$199$	−3760.99	−1.33975	−0.669874	−	0.742475i	$-0.733652\pi$
$199$	−3760.99	−1.33975	−0.669874	+	0.742475i	$0.733652\pi$
$200$	0	0
$201$	1346.64	0.472562
$202$	2728.29	0.950307
$203$	1713.51	0.592436
$204$	−272.588	−0.0935537
$205$	0	0
$206$	−79.4879	−0.0268844
$207$	1292.20	0.433886
$208$	−208.000	−0.0693375
$209$	−731.811	−0.242203
$210$	0	0
$211$	−1166.35	−0.380544	−0.190272	−	0.981731i	$-0.560937\pi$
$211$	−1166.35	−0.380544	−0.190272	+	0.981731i	$0.560937\pi$
$212$	−641.100	−0.207693
$213$	2471.23	0.794958
$214$	−514.426	−0.164324
$215$	0	0
$216$	216.000	0.0680414
$217$	−1219.37	−0.381459
$218$	−1718.02	−0.533757
$219$	−2733.14	−0.843327
$220$	0	0
$221$	295.303	0.0898835
$222$	610.237	0.184488
$223$	961.085	0.288605	0.144303	−	0.989534i	$-0.453906\pi$
$223$	961.085	0.288605	0.144303	+	0.989534i	$0.453906\pi$
$224$	662.900	0.197732
$225$	0	0
$226$	580.407	0.170832
$227$	−2586.98	−0.756404	−0.378202	−	0.925723i	$-0.623457\pi$
$227$	−2586.98	−0.756404	−0.378202	+	0.925723i	$0.623457\pi$
$228$	409.763	0.119023
$229$	−3254.02	−0.939004	−0.469502	−	0.882931i	$-0.655567\pi$
$229$	−3254.02	−0.939004	−0.469502	+	0.882931i	$0.655567\pi$
$230$	0	0
$231$	−1331.89	−0.379358
$232$	661.725	0.187260
$233$	630.497	0.177276	0.0886379	−	0.996064i	$-0.471749\pi$
$233$	630.497	0.177276	0.0886379	+	0.996064i	$0.471749\pi$
$234$	−234.000	−0.0653720
$235$	0	0
$236$	3253.73	0.897455
$237$	0.768197	0.000210547 0
$238$	−941.137	−0.256323
$239$	−4066.43	−1.10057	−0.550283	−	0.834978i	$-0.685480\pi$
$239$	−4066.43	−1.10057	−0.550283	+	0.834978i	$0.685480\pi$
$240$	0	0
$241$	2960.82	0.791384	0.395692	−	0.918383i	$-0.370505\pi$
$241$	2960.82	0.791384	0.395692	+	0.918383i	$0.370505\pi$
$242$	−1743.40	−0.463100
$243$	243.000	0.0641500
$244$	2531.45	0.664178
$245$	0	0
$246$	1604.59	0.415873
$247$	−443.910	−0.114353
$248$	−470.900	−0.120573
$249$	−388.181	−0.0987950
$250$	0	0
$251$	4686.67	1.17856	0.589282	−	0.807927i	$-0.299411\pi$
$251$	4686.67	1.17856	0.589282	+	0.807927i	$0.299411\pi$
$252$	745.763	0.186423
$253$	−3077.06	−0.764637
$254$	−2061.29	−0.509200
$255$	0	0
$256$	256.000	0.0625000
$257$	2530.72	0.614248	0.307124	−	0.951670i	$-0.400633\pi$
$257$	2530.72	0.614248	0.307124	+	0.951670i	$0.400633\pi$
$258$	1232.59	0.297432
$259$	2106.91	0.505471
$260$	0	0
$261$	744.441	0.176551
$262$	−2656.41	−0.626387
$263$	1632.42	0.382736	0.191368	−	0.981518i	$-0.438708\pi$
$263$	1632.42	0.382736	0.191368	+	0.981518i	$0.438708\pi$
$264$	−514.350	−0.119909
$265$	0	0
$266$	1414.75	0.326105
$267$	3409.93	0.781590
$268$	1795.53	0.409251
$269$	5804.45	1.31563	0.657813	−	0.753181i	$-0.271482\pi$
$269$	5804.45	1.31563	0.657813	+	0.753181i	$0.271482\pi$
$270$	0	0
$271$	1247.56	0.279646	0.139823	−	0.990177i	$-0.455347\pi$
$271$	1247.56	0.279646	0.139823	+	0.990177i	$0.455347\pi$
$272$	−363.450	−0.0810199
$273$	−807.910	−0.179110
$274$	−1421.10	−0.313328
$275$	0	0
$276$	1722.94	0.375756
$277$	1991.88	0.432059	0.216029	−	0.976387i	$-0.430689\pi$
$277$	1991.88	0.432059	0.216029	+	0.976387i	$0.430689\pi$
$278$	−2179.96	−0.470307
$279$	−529.763	−0.113678
$280$	0	0
$281$	5127.39	1.08852	0.544261	−	0.838916i	$-0.316810\pi$
$281$	5127.39	1.08852	0.544261	+	0.838916i	$0.316810\pi$
$282$	−3301.88	−0.697248
$283$	5686.32	1.19441	0.597203	−	0.802090i	$-0.296279\pi$
$283$	5686.32	1.19441	0.597203	+	0.802090i	$0.296279\pi$
$284$	3294.98	0.688454
$285$	0	0
$286$	557.213	0.115205
$287$	5540.01	1.13943
$288$	288.000	0.0589256
$289$	−4397.00	−0.894973
$290$	0	0
$291$	954.497	0.192280
$292$	−3644.19	−0.730343
$293$	−3431.40	−0.684180	−0.342090	−	0.939667i	$-0.611135\pi$
$293$	−3431.40	−0.684180	−0.342090	+	0.939667i	$0.611135\pi$
$294$	516.825	0.102523
$295$	0	0
$296$	813.650	0.159772
$297$	−578.644	−0.113052
$298$	−1786.54	−0.347287
$299$	−1866.52	−0.361015
$300$	0	0
$301$	4255.64	0.814920
$302$	4137.40	0.788347
$303$	4092.44	0.775923
$304$	546.350	0.103077
$305$	0	0
$306$	−408.881	−0.0763863
$307$	1138.21	0.211599	0.105800	−	0.994387i	$-0.466260\pi$
$307$	1138.21	0.211599	0.105800	+	0.994387i	$0.466260\pi$
$308$	−1775.85	−0.328534
$309$	−119.232	−0.0219510
$310$	0	0
$311$	6008.81	1.09559	0.547795	−	0.836612i	$-0.315467\pi$
$311$	6008.81	1.09559	0.547795	+	0.836612i	$0.315467\pi$
$312$	−312.000	−0.0566139
$313$	−1106.77	−0.199867	−0.0999333	−	0.994994i	$-0.531863\pi$
$313$	−1106.77	−0.199867	−0.0999333	+	0.994994i	$0.531863\pi$
$314$	−2040.78	−0.366776
$315$	0	0
$316$	1.02426	0.000182339 0
$317$	−5147.36	−0.912002	−0.456001	−	0.889979i	$-0.650719\pi$
$317$	−5147.36	−0.912002	−0.456001	+	0.889979i	$0.650719\pi$
$318$	−961.650	−0.169581
$319$	−1772.70	−0.311135
$320$	0	0
$321$	−771.639	−0.134170
$322$	5948.63	1.02952
$323$	−775.668	−0.133620
$324$	324.000	0.0555556
$325$	0	0
$326$	−7463.28	−1.26795
$327$	−2577.03	−0.435810
$328$	2139.45	0.360157
$329$	−11400.1	−1.91035
$330$	0	0
$331$	2710.11	0.450033	0.225017	−	0.974355i	$-0.427756\pi$
$331$	2710.11	0.450033	0.225017	+	0.974355i	$0.427756\pi$
$332$	−517.574	−0.0855589
$333$	915.356	0.150634
$334$	2994.86	0.490633
$335$	0	0
$336$	994.350	0.161447
$337$	−5234.99	−0.846197	−0.423098	−	0.906084i	$-0.639058\pi$
$337$	−5234.99	−0.846197	−0.423098	+	0.906084i	$0.639058\pi$
$338$	338.000	0.0543928
$339$	870.611	0.139484
$340$	0	0
$341$	1261.50	0.200334
$342$	614.644	0.0971817
$343$	−5321.07	−0.837641
$344$	1643.45	0.257584
$345$	0	0
$346$	−564.226	−0.0876676
$347$	−2603.16	−0.402723	−0.201361	−	0.979517i	$-0.564537\pi$
$347$	−2603.16	−0.402723	−0.201361	+	0.979517i	$0.564537\pi$
$348$	992.588	0.152897
$349$	5726.03	0.878245	0.439122	−	0.898427i	$-0.355289\pi$
$349$	5726.03	0.878245	0.439122	+	0.898427i	$0.355289\pi$
$350$	0	0
$351$	−351.000	−0.0533761
$352$	−685.801	−0.103845
$353$	−1962.00	−0.295826	−0.147913	−	0.989000i	$-0.547256\pi$
$353$	−1962.00	−0.295826	−0.147913	+	0.989000i	$0.547256\pi$
$354$	4880.59	0.732769
$355$	0	0
$356$	4546.58	0.676877
$357$	−1411.71	−0.209287
$358$	−5366.64	−0.792279
$359$	−9472.48	−1.39259	−0.696293	−	0.717757i	$-0.745169\pi$
$359$	−9472.48	−1.39259	−0.696293	+	0.717757i	$0.745169\pi$
$360$	0	0
$361$	−5692.99	−0.830003
$362$	3026.24	0.439380
$363$	−2615.10	−0.378119
$364$	−1077.21	−0.155113
$365$	0	0
$366$	3797.18	0.542299
$367$	1417.36	0.201595	0.100798	−	0.994907i	$-0.467861\pi$
$367$	1417.36	0.201595	0.100798	+	0.994907i	$0.467861\pi$
$368$	2297.25	0.325414
$369$	2406.88	0.339559
$370$	0	0
$371$	−3320.20	−0.464625
$372$	−706.350	−0.0984477
$373$	−11107.4	−1.54187	−0.770935	−	0.636914i	$-0.780211\pi$
$373$	−11107.4	−1.54187	−0.770935	+	0.636914i	$0.780211\pi$
$374$	973.650	0.134616
$375$	0	0
$376$	−4402.50	−0.603834
$377$	−1075.30	−0.146899
$378$	1118.64	0.152214
$379$	6340.09	0.859284	0.429642	−	0.902999i	$-0.358640\pi$
$379$	6340.09	0.859284	0.429642	+	0.902999i	$0.358640\pi$
$380$	0	0
$381$	−3091.93	−0.415760
$382$	9860.70	1.32073
$383$	−6087.10	−0.812105	−0.406052	−	0.913850i	$-0.633095\pi$
$383$	−6087.10	−0.812105	−0.406052	+	0.913850i	$0.633095\pi$
$384$	384.000	0.0510310
$385$	0	0
$386$	8986.95	1.18504
$387$	1848.88	0.242853
$388$	1272.66	0.166520
$389$	−13403.3	−1.74698	−0.873488	−	0.486846i	$-0.838147\pi$
$389$	−13403.3	−1.74698	−0.873488	+	0.486846i	$0.838147\pi$
$390$	0	0
$391$	−3261.47	−0.421840
$392$	689.100	0.0887877
$393$	−3984.61	−0.511443
$394$	5847.83	0.747739
$395$	0	0
$396$	−771.526	−0.0979056
$397$	−3933.74	−0.497302	−0.248651	−	0.968593i	$-0.579987\pi$
$397$	−3933.74	−0.497302	−0.248651	+	0.968593i	$0.579987\pi$
$398$	−7521.99	−0.947345
$399$	2122.12	0.266263
$400$	0	0
$401$	8357.57	1.04079	0.520396	−	0.853925i	$-0.325785\pi$
$401$	8357.57	1.04079	0.520396	+	0.853925i	$0.325785\pi$
$402$	2693.29	0.334152
$403$	765.213	0.0945855
$404$	5456.59	0.671969
$405$	0	0
$406$	3427.01	0.418916
$407$	−2179.69	−0.265463
$408$	−545.175	−0.0661524
$409$	8925.71	1.07909	0.539545	−	0.841957i	$-0.318596\pi$
$409$	8925.71	1.07909	0.539545	+	0.841957i	$0.318596\pi$
$410$	0	0
$411$	−2131.65	−0.255831
$412$	−158.976	−0.0190101
$413$	16850.7	2.00768
$414$	2584.41	0.306804
$415$	0	0
$416$	−416.000	−0.0490290
$417$	−3269.94	−0.384004
$418$	−1463.62	−0.171263
$419$	−16604.5	−1.93599	−0.967997	−	0.250962i	$-0.919253\pi$
$419$	−16604.5	−1.93599	−0.967997	+	0.250962i	$0.919253\pi$
$420$	0	0
$421$	−13456.6	−1.55781	−0.778903	−	0.627145i	$-0.784223\pi$
$421$	−13456.6	−1.55781	−0.778903	+	0.627145i	$0.784223\pi$
$422$	−2332.70	−0.269086
$423$	−4952.81	−0.569300
$424$	−1282.20	−0.146861
$425$	0	0
$426$	4942.46	0.562120
$427$	13110.1	1.48582
$428$	−1028.85	−0.116195
$429$	835.819	0.0940647
$430$	0	0
$431$	−10889.5	−1.21700	−0.608501	−	0.793553i	$-0.708229\pi$
$431$	−10889.5	−1.21700	−0.608501	+	0.793553i	$0.708229\pi$
$432$	432.000	0.0481125
$433$	7812.57	0.867086	0.433543	−	0.901133i	$-0.357263\pi$
$433$	7812.57	0.867086	0.433543	+	0.901133i	$0.357263\pi$
$434$	−2438.75	−0.269732
$435$	0	0
$436$	−3436.04	−0.377423
$437$	4902.75	0.536683
$438$	−5466.28	−0.596322
$439$	−468.097	−0.0508908	−0.0254454	−	0.999676i	$-0.508100\pi$
$439$	−468.097	−0.0508908	−0.0254454	+	0.999676i	$0.508100\pi$
$440$	0	0
$441$	775.237	0.0837099
$442$	590.606	0.0635572
$443$	4542.09	0.487136	0.243568	−	0.969884i	$-0.421682\pi$
$443$	4542.09	0.487136	0.243568	+	0.969884i	$0.421682\pi$
$444$	1220.47	0.130453
$445$	0	0
$446$	1922.17	0.204075
$447$	−2679.81	−0.283558
$448$	1325.80	0.139817
$449$	7248.51	0.761867	0.380934	−	0.924602i	$-0.375603\pi$
$449$	7248.51	0.761867	0.380934	+	0.924602i	$0.375603\pi$
$450$	0	0
$451$	−5731.39	−0.598405
$452$	1160.81	0.120797
$453$	6206.10	0.643682
$454$	−5173.95	−0.534858
$455$	0	0
$456$	819.526	0.0841618
$457$	−573.209	−0.0586730	−0.0293365	−	0.999570i	$-0.509339\pi$
$457$	−573.209	−0.0586730	−0.0293365	+	0.999570i	$0.509339\pi$
$458$	−6508.05	−0.663976
$459$	−613.322	−0.0623691
$460$	0	0
$461$	−1061.96	−0.107290	−0.0536448	−	0.998560i	$-0.517084\pi$
$461$	−1061.96	−0.107290	−0.0536448	+	0.998560i	$0.517084\pi$
$462$	−2663.77	−0.268247
$463$	−8714.97	−0.874771	−0.437385	−	0.899274i	$-0.644095\pi$
$463$	−8714.97	−0.874771	−0.437385	+	0.899274i	$0.644095\pi$
$464$	1323.45	0.132413
$465$	0	0
$466$	1260.99	0.125353
$467$	5528.93	0.547855	0.273927	−	0.961750i	$-0.411677\pi$
$467$	5528.93	0.547855	0.273927	+	0.961750i	$0.411677\pi$
$468$	−468.000	−0.0462250
$469$	9298.86	0.915526
$470$	0	0
$471$	−3061.16	−0.299471
$472$	6507.45	0.634597
$473$	−4402.65	−0.427979
$474$	1.53639	0.000148880 0
$475$	0	0
$476$	−1882.27	−0.181248
$477$	−1442.47	−0.138462
$478$	−8132.85	−0.778218
$479$	6348.49	0.605574	0.302787	−	0.953058i	$-0.402083\pi$
$479$	6348.49	0.605574	0.302787	+	0.953058i	$0.402083\pi$
$480$	0	0
$481$	−1322.18	−0.125335
$482$	5921.65	0.559593
$483$	8922.94	0.840596
$484$	−3486.80	−0.327461
$485$	0	0
$486$	486.000	0.0453609
$487$	−149.198	−0.0138826	−0.00694129	−	0.999976i	$-0.502209\pi$
$487$	−149.198	−0.0138826	−0.00694129	+	0.999976i	$0.502209\pi$
$488$	5062.90	0.469645
$489$	−11194.9	−1.03528
$490$	0	0
$491$	−9556.38	−0.878357	−0.439178	−	0.898400i	$-0.644730\pi$
$491$	−9556.38	−0.878357	−0.439178	+	0.898400i	$0.644730\pi$
$492$	3209.18	0.294067
$493$	−1878.94	−0.171649
$494$	−887.819	−0.0808601
$495$	0	0
$496$	−941.801	−0.0852583
$497$	17064.4	1.54012
$498$	−776.361	−0.0698586
$499$	−9280.84	−0.832600	−0.416300	−	0.909227i	$-0.636673\pi$
$499$	−9280.84	−0.832600	−0.416300	+	0.909227i	$0.636673\pi$
$500$	0	0
$501$	4492.29	0.400601
$502$	9373.33	0.833371
$503$	6093.50	0.540151	0.270075	−	0.962839i	$-0.412951\pi$
$503$	6093.50	0.540151	0.270075	+	0.962839i	$0.412951\pi$
$504$	1491.53	0.131821
$505$	0	0
$506$	−6154.12	−0.540680
$507$	507.000	0.0444116
$508$	−4122.58	−0.360059
$509$	−353.925	−0.0308201	−0.0154100	−	0.999881i	$-0.504905\pi$
$509$	−353.925	−0.0308201	−0.0154100	+	0.999881i	$0.504905\pi$
$510$	0	0
$511$	−18872.9	−1.63383
$512$	512.000	0.0441942
$513$	921.966	0.0793486
$514$	5061.43	0.434339
$515$	0	0
$516$	2465.18	0.210316
$517$	11793.9	1.00328
$518$	4213.82	0.357422
$519$	−846.340	−0.0715803
$520$	0	0
$521$	−210.504	−0.0177012	−0.00885062	−	0.999961i	$-0.502817\pi$
$521$	−210.504	−0.0177012	−0.00885062	+	0.999961i	$0.502817\pi$
$522$	1488.88	0.124840
$523$	−8588.45	−0.718063	−0.359031	−	0.933325i	$-0.616893\pi$
$523$	−8588.45	−0.718063	−0.359031	+	0.933325i	$0.616893\pi$
$524$	−5312.81	−0.442922
$525$	0	0
$526$	3264.84	0.270635
$527$	1337.10	0.110522
$528$	−1028.70	−0.0847887
$529$	8447.69	0.694312
$530$	0	0
$531$	7320.88	0.598304
$532$	2829.50	0.230591
$533$	−3476.61	−0.282530
$534$	6819.87	0.552667
$535$	0	0
$536$	3591.05	0.289384
$537$	−8049.97	−0.646893
$538$	11608.9	0.930288
$539$	−1846.04	−0.147522
$540$	0	0
$541$	10116.0	0.803922	0.401961	−	0.915657i	$-0.368329\pi$
$541$	10116.0	0.803922	0.401961	+	0.915657i	$0.368329\pi$
$542$	2495.13	0.197740
$543$	4539.36	0.358752
$544$	−726.900	−0.0572897
$545$	0	0
$546$	−1615.82	−0.126650
$547$	−9779.04	−0.764390	−0.382195	−	0.924082i	$-0.624832\pi$
$547$	−9779.04	−0.764390	−0.382195	+	0.924082i	$0.624832\pi$
$548$	−2842.20	−0.221556
$549$	5695.76	0.442785
$550$	0	0
$551$	2824.48	0.218379
$552$	3445.88	0.265700
$553$	5.30456	0.000407908 0
$554$	3983.75	0.305512
$555$	0	0
$556$	−4359.92	−0.332558
$557$	−22023.3	−1.67532	−0.837662	−	0.546189i	$-0.816078\pi$
$557$	−22023.3	−1.67532	−0.837662	+	0.546189i	$0.816078\pi$
$558$	−1059.53	−0.0803822
$559$	−2670.61	−0.202066
$560$	0	0
$561$	1460.47	0.109913
$562$	10254.8	0.769701
$563$	14263.8	1.06776	0.533878	−	0.845561i	$-0.320734\pi$
$563$	14263.8	1.06776	0.533878	+	0.845561i	$0.320734\pi$
$564$	−6603.75	−0.493029
$565$	0	0
$566$	11372.6	0.844572
$567$	1677.97	0.124282
$568$	6589.95	0.486810
$569$	6525.44	0.480774	0.240387	−	0.970677i	$-0.422726\pi$
$569$	6525.44	0.480774	0.240387	+	0.970677i	$0.422726\pi$
$570$	0	0
$571$	14699.1	1.07730	0.538650	−	0.842530i	$-0.318935\pi$
$571$	14699.1	1.07730	0.538650	+	0.842530i	$0.318935\pi$
$572$	1114.43	0.0814624
$573$	14791.1	1.07837
$574$	11080.0	0.805698
$575$	0	0
$576$	576.000	0.0416667
$577$	10515.6	0.758698	0.379349	−	0.925254i	$-0.376148\pi$
$577$	10515.6	0.758698	0.379349	+	0.925254i	$0.376148\pi$
$578$	−8794.00	−0.632841
$579$	13480.4	0.967577
$580$	0	0
$581$	−2680.47	−0.191402
$582$	1908.99	0.135963
$583$	3434.89	0.244012
$584$	−7288.38	−0.516430
$585$	0	0
$586$	−6862.80	−0.483788
$587$	−21022.9	−1.47821	−0.739103	−	0.673592i	$-0.764750\pi$
$587$	−21022.9	−1.47821	−0.739103	+	0.673592i	$0.764750\pi$
$588$	1033.65	0.0724949
$589$	−2009.97	−0.140610
$590$	0	0
$591$	8771.74	0.610527
$592$	1627.30	0.112976
$593$	−14914.6	−1.03283	−0.516414	−	0.856339i	$-0.672734\pi$
$593$	−14914.6	−1.03283	−0.516414	+	0.856339i	$0.672734\pi$
$594$	−1157.29	−0.0799396
$595$	0	0
$596$	−3573.08	−0.245569
$597$	−11283.0	−0.773504
$598$	−3733.03	−0.255276
$599$	−19508.5	−1.33071	−0.665354	−	0.746528i	$-0.731720\pi$
$599$	−19508.5	−1.33071	−0.665354	+	0.746528i	$0.731720\pi$
$600$	0	0
$601$	−11353.1	−0.770552	−0.385276	−	0.922801i	$-0.625894\pi$
$601$	−11353.1	−0.770552	−0.385276	+	0.922801i	$0.625894\pi$
$602$	8511.28	0.576236
$603$	4039.93	0.272834
$604$	8274.80	0.557445
$605$	0	0
$606$	8184.88	0.548660
$607$	−22554.3	−1.50815	−0.754077	−	0.656786i	$-0.771915\pi$
$607$	−22554.3	−1.50815	−0.754077	+	0.656786i	$0.771915\pi$
$608$	1092.70	0.0728863
$609$	5140.52	0.342043
$610$	0	0
$611$	7154.06	0.473687
$612$	−817.763	−0.0540132
$613$	−17954.7	−1.18301	−0.591505	−	0.806302i	$-0.701466\pi$
$613$	−17954.7	−1.18301	−0.591505	+	0.806302i	$0.701466\pi$
$614$	2276.42	0.149623
$615$	0	0
$616$	−3551.70	−0.232308
$617$	17969.4	1.17248	0.586241	−	0.810137i	$-0.300607\pi$
$617$	17969.4	1.17248	0.586241	+	0.810137i	$0.300607\pi$
$618$	−238.464	−0.0155217
$619$	−7547.70	−0.490093	−0.245047	−	0.969511i	$-0.578803\pi$
$619$	−7547.70	−0.490093	−0.245047	+	0.969511i	$0.578803\pi$
$620$	0	0
$621$	3876.61	0.250504
$622$	12017.6	0.774699
$623$	23546.3	1.51423
$624$	−624.000	−0.0400320
$625$	0	0
$626$	−2213.54	−0.141327
$627$	−2195.43	−0.139836
$628$	−4081.55	−0.259350
$629$	−2310.32	−0.146452
$630$	0	0
$631$	−25226.5	−1.59152	−0.795761	−	0.605611i	$-0.792929\pi$
$631$	−25226.5	−1.59152	−0.795761	+	0.605611i	$0.792929\pi$
$632$	2.04852	0.000128933 0
$633$	−3499.05	−0.219707
$634$	−10294.7	−0.644883
$635$	0	0
$636$	−1923.30	−0.119912
$637$	−1119.79	−0.0696508
$638$	−3545.40	−0.220006
$639$	7413.70	0.458969
$640$	0	0
$641$	−738.189	−0.0454863	−0.0227431	−	0.999741i	$-0.507240\pi$
$641$	−738.189	−0.0454863	−0.0227431	+	0.999741i	$0.507240\pi$
$642$	−1543.28	−0.0948728
$643$	−1061.36	−0.0650945	−0.0325473	−	0.999470i	$-0.510362\pi$
$643$	−1061.36	−0.0650945	−0.0325473	+	0.999470i	$0.510362\pi$
$644$	11897.3	0.727977
$645$	0	0
$646$	−1551.34	−0.0944838
$647$	−19890.0	−1.20859	−0.604293	−	0.796762i	$-0.706544\pi$
$647$	−19890.0	−1.20859	−0.604293	+	0.796762i	$0.706544\pi$
$648$	648.000	0.0392837
$649$	−17432.9	−1.05439
$650$	0	0
$651$	−3658.12	−0.220235
$652$	−14926.6	−0.896578
$653$	18758.5	1.12416	0.562079	−	0.827083i	$-0.310002\pi$
$653$	18758.5	1.12416	0.562079	+	0.827083i	$0.310002\pi$
$654$	−5154.06	−0.308164
$655$	0	0
$656$	4278.90	0.254669
$657$	−8199.42	−0.486895
$658$	−22800.2	−1.35082
$659$	13840.7	0.818144	0.409072	−	0.912502i	$-0.365852\pi$
$659$	13840.7	0.818144	0.409072	+	0.912502i	$0.365852\pi$
$660$	0	0
$661$	13279.4	0.781403	0.390702	−	0.920517i	$-0.372232\pi$
$661$	13279.4	0.781403	0.390702	+	0.920517i	$0.372232\pi$
$662$	5420.22	0.318222
$663$	885.910	0.0518942
$664$	−1035.15	−0.0604993
$665$	0	0
$666$	1830.71	0.106514
$667$	11876.2	0.689425
$668$	5989.73	0.346930
$669$	2883.25	0.166626
$670$	0	0
$671$	−13563.0	−0.780321
$672$	1988.70	0.114160
$673$	3784.18	0.216745	0.108373	−	0.994110i	$-0.465436\pi$
$673$	3784.18	0.216745	0.108373	+	0.994110i	$0.465436\pi$
$674$	−10470.0	−0.598352
$675$	0	0
$676$	676.000	0.0384615
$677$	−6019.74	−0.341739	−0.170870	−	0.985294i	$-0.554658\pi$
$677$	−6019.74	−0.341739	−0.170870	+	0.985294i	$0.554658\pi$
$678$	1741.22	0.0986301
$679$	6591.01	0.372518
$680$	0	0
$681$	−7760.93	−0.436710
$682$	2523.00	0.141658
$683$	−17525.2	−0.981822	−0.490911	−	0.871210i	$-0.663336\pi$
$683$	−17525.2	−0.981822	−0.490911	+	0.871210i	$0.663336\pi$
$684$	1229.29	0.0687179
$685$	0	0
$686$	−10642.1	−0.592301
$687$	−9762.07	−0.542134
$688$	3286.90	0.182139
$689$	2083.57	0.115207
$690$	0	0
$691$	20112.7	1.10727	0.553635	−	0.832760i	$-0.313240\pi$
$691$	20112.7	1.10727	0.553635	+	0.832760i	$0.313240\pi$
$692$	−1128.45	−0.0619904
$693$	−3995.66	−0.219022
$694$	−5206.31	−0.284768
$695$	0	0
$696$	1985.18	0.108115
$697$	−6074.87	−0.330132
$698$	11452.1	0.621013
$699$	1891.49	0.102350
$700$	0	0
$701$	13700.4	0.738168	0.369084	−	0.929396i	$-0.379671\pi$
$701$	13700.4	0.738168	0.369084	+	0.929396i	$0.379671\pi$
$702$	−702.000	−0.0377426
$703$	3472.95	0.186323
$704$	−1371.60	−0.0734292
$705$	0	0
$706$	−3924.00	−0.209181
$707$	28259.2	1.50325
$708$	9761.18	0.518146
$709$	−19689.8	−1.04297	−0.521485	−	0.853260i	$-0.674622\pi$
$709$	−19689.8	−1.04297	−0.521485	+	0.853260i	$0.674622\pi$
$710$	0	0
$711$	2.30459	0.000121560 0
$712$	9093.15	0.478624
$713$	−8451.37	−0.443908
$714$	−2823.41	−0.147988
$715$	0	0
$716$	−10733.3	−0.560226
$717$	−12199.3	−0.635412
$718$	−18945.0	−0.984707
$719$	−22867.2	−1.18610	−0.593048	−	0.805167i	$-0.702076\pi$
$719$	−22867.2	−1.18610	−0.593048	+	0.805167i	$0.702076\pi$
$720$	0	0
$721$	−823.321	−0.0425271
$722$	−11386.0	−0.586901
$723$	8882.47	0.456906
$724$	6052.47	0.310688
$725$	0	0
$726$	−5230.20	−0.267371
$727$	−20365.9	−1.03897	−0.519484	−	0.854480i	$-0.673876\pi$
$727$	−20365.9	−1.03897	−0.519484	+	0.854480i	$0.673876\pi$
$728$	−2154.43	−0.109682
$729$	729.000	0.0370370
$730$	0	0
$731$	−4666.50	−0.236111
$732$	7594.35	0.383463
$733$	−28247.6	−1.42340	−0.711699	−	0.702485i	$-0.752074\pi$
$733$	−28247.6	−1.42340	−0.711699	+	0.702485i	$0.752074\pi$
$734$	2834.71	0.142549
$735$	0	0
$736$	4594.50	0.230103
$737$	−9620.10	−0.480815
$738$	4813.76	0.240104
$739$	34918.9	1.73817	0.869087	−	0.494659i	$-0.164707\pi$
$739$	34918.9	1.73817	0.869087	+	0.494659i	$0.164707\pi$
$740$	0	0
$741$	−1331.73	−0.0660220
$742$	−6640.39	−0.328540
$743$	−6021.06	−0.297297	−0.148648	−	0.988890i	$-0.547492\pi$
$743$	−6021.06	−0.297297	−0.148648	+	0.988890i	$0.547492\pi$
$744$	−1412.70	−0.0696131
$745$	0	0
$746$	−22214.7	−1.09027
$747$	−1164.54	−0.0570393
$748$	1947.30	0.0951876
$749$	−5328.33	−0.259937
$750$	0	0
$751$	−27511.2	−1.33675	−0.668374	−	0.743826i	$-0.733009\pi$
$751$	−27511.2	−1.33675	−0.668374	+	0.743826i	$0.733009\pi$
$752$	−8805.00	−0.426975
$753$	14060.0	0.680445
$754$	−2150.61	−0.103873
$755$	0	0
$756$	2237.29	0.107631
$757$	−10600.2	−0.508945	−0.254473	−	0.967080i	$-0.581902\pi$
$757$	−10600.2	−0.508945	−0.254473	+	0.967080i	$0.581902\pi$
$758$	12680.2	0.607605
$759$	−9231.19	−0.441464
$760$	0	0
$761$	29906.9	1.42460	0.712302	−	0.701873i	$-0.247653\pi$
$761$	29906.9	1.42460	0.712302	+	0.701873i	$0.247653\pi$
$762$	−6183.87	−0.293987
$763$	−17794.9	−0.844324
$764$	19721.4	0.933894
$765$	0	0
$766$	−12174.2	−0.574245
$767$	−10574.6	−0.497819
$768$	768.000	0.0360844
$769$	−28994.9	−1.35966	−0.679832	−	0.733367i	$-0.737947\pi$
$769$	−28994.9	−1.35966	−0.679832	+	0.733367i	$0.737947\pi$
$770$	0	0
$771$	7592.15	0.354636
$772$	17973.9	0.837946
$773$	−9556.54	−0.444663	−0.222332	−	0.974971i	$-0.571367\pi$
$773$	−9556.54	−0.444663	−0.222332	+	0.974971i	$0.571367\pi$
$774$	3697.76	0.171723
$775$	0	0
$776$	2545.33	0.117747
$777$	6320.73	0.291834
$778$	−26806.6	−1.23530
$779$	9131.95	0.420008
$780$	0	0
$781$	−17653.9	−0.808842
$782$	−6522.94	−0.298286
$783$	2233.32	0.101932
$784$	1378.20	0.0627824
$785$	0	0
$786$	−7969.22	−0.361645
$787$	−21610.5	−0.978819	−0.489409	−	0.872054i	$-0.662788\pi$
$787$	−21610.5	−0.978819	−0.489409	+	0.872054i	$0.662788\pi$
$788$	11695.7	0.528732
$789$	4897.27	0.220972
$790$	0	0
$791$	6011.75	0.270232
$792$	−1543.05	−0.0692297
$793$	−8227.21	−0.368420
$794$	−7867.49	−0.351646
$795$	0	0
$796$	−15044.0	−0.669874
$797$	22274.0	0.989946	0.494973	−	0.868908i	$-0.335178\pi$
$797$	22274.0	0.989946	0.494973	+	0.868908i	$0.335178\pi$
$798$	4244.25	0.188277
$799$	12500.7	0.553496
$800$	0	0
$801$	10229.8	0.451251
$802$	16715.1	0.735950
$803$	19524.9	0.858056
$804$	5386.58	0.236281
$805$	0	0
$806$	1530.43	0.0668821
$807$	17413.3	0.759577
$808$	10913.2	0.475154
$809$	35070.2	1.52411	0.762053	−	0.647515i	$-0.224192\pi$
$809$	35070.2	1.52411	0.762053	+	0.647515i	$0.224192\pi$
$810$	0	0
$811$	−34343.4	−1.48700	−0.743502	−	0.668733i	$-0.766837\pi$
$811$	−34343.4	−1.48700	−0.743502	+	0.668733i	$0.766837\pi$
$812$	6854.03	0.296218
$813$	3742.69	0.161454
$814$	−4359.39	−0.187711
$815$	0	0
$816$	−1090.35	−0.0467768
$817$	7014.84	0.300389
$818$	17851.4	0.763032
$819$	−2423.73	−0.103409
$820$	0	0
$821$	−26.6496	−0.00113286	−0.000566429	−	1.00000i	$-0.500180\pi$
$821$	−26.6496	−0.00113286	−0.000566429		1.00000i	$0.500180\pi$
$822$	−4263.30	−0.180900
$823$	−8777.13	−0.371752	−0.185876	−	0.982573i	$-0.559512\pi$
$823$	−8777.13	−0.371752	−0.185876	+	0.982573i	$0.559512\pi$
$824$	−317.951	−0.0134422
$825$	0	0
$826$	33701.5	1.41964
$827$	−20451.6	−0.859941	−0.429970	−	0.902843i	$-0.641476\pi$
$827$	−20451.6	−0.859941	−0.429970	+	0.902843i	$0.641476\pi$
$828$	5168.81	0.216943
$829$	17505.7	0.733411	0.366706	−	0.930337i	$-0.380486\pi$
$829$	17505.7	0.733411	0.366706	+	0.930337i	$0.380486\pi$
$830$	0	0
$831$	5975.63	0.249449
$832$	−832.000	−0.0346688
$833$	−1956.67	−0.0813860
$834$	−6539.89	−0.271532
$835$	0	0
$836$	−2927.25	−0.121102
$837$	−1589.29	−0.0656318
$838$	−33208.9	−1.36895
$839$	−17453.3	−0.718181	−0.359090	−	0.933303i	$-0.616913\pi$
$839$	−17453.3	−0.718181	−0.359090	+	0.933303i	$0.616913\pi$
$840$	0	0
$841$	−17547.1	−0.719469
$842$	−26913.3	−1.10153
$843$	15382.2	0.628458
$844$	−4665.40	−0.190272
$845$	0	0
$846$	−9905.63	−0.402556
$847$	−18057.8	−0.732556
$848$	−2564.40	−0.103846
$849$	17059.0	0.689590
$850$	0	0
$851$	14602.8	0.588222
$852$	9884.93	0.397479
$853$	−10442.5	−0.419160	−0.209580	−	0.977791i	$-0.567210\pi$
$853$	−10442.5	−0.419160	−0.209580	+	0.977791i	$0.567210\pi$
$854$	26220.3	1.05063
$855$	0	0
$856$	−2057.70	−0.0821622
$857$	−34377.8	−1.37027	−0.685137	−	0.728414i	$-0.740258\pi$
$857$	−34377.8	−1.37027	−0.685137	+	0.728414i	$0.740258\pi$
$858$	1671.64	0.0665138
$859$	8502.73	0.337729	0.168865	−	0.985639i	$-0.445990\pi$
$859$	8502.73	0.337729	0.168865	+	0.985639i	$0.445990\pi$
$860$	0	0
$861$	16620.0	0.657850
$862$	−21779.0	−0.860550
$863$	28799.8	1.13599	0.567994	−	0.823033i	$-0.307720\pi$
$863$	28799.8	1.13599	0.567994	+	0.823033i	$0.307720\pi$
$864$	864.000	0.0340207
$865$	0	0
$866$	15625.1	0.613122
$867$	−13191.0	−0.516713
$868$	−4877.50	−0.190729
$869$	−5.48781	−0.000214225 0
$870$	0	0
$871$	−5835.46	−0.227011
$872$	−6872.08	−0.266878
$873$	2863.49	0.111013
$874$	9805.50	0.379492
$875$	0	0
$876$	−10932.6	−0.421663
$877$	−9704.26	−0.373648	−0.186824	−	0.982393i	$-0.559819\pi$
$877$	−9704.26	−0.373648	−0.186824	+	0.982393i	$0.559819\pi$
$878$	−936.194	−0.0359852
$879$	−10294.2	−0.395011
$880$	0	0
$881$	−39564.0	−1.51299	−0.756495	−	0.654000i	$-0.773090\pi$
$881$	−39564.0	−1.51299	−0.756495	+	0.654000i	$0.773090\pi$
$882$	1550.47	0.0591918
$883$	−10752.5	−0.409798	−0.204899	−	0.978783i	$-0.565687\pi$
$883$	−10752.5	−0.409798	−0.204899	+	0.978783i	$0.565687\pi$
$884$	1181.21	0.0449417
$885$	0	0
$886$	9084.19	0.344457
$887$	−42492.1	−1.60850	−0.804252	−	0.594288i	$-0.797434\pi$
$887$	−42492.1	−1.60850	−0.804252	+	0.594288i	$0.797434\pi$
$888$	2440.95	0.0922442
$889$	−21350.4	−0.805479
$890$	0	0
$891$	−1735.93	−0.0652704
$892$	3844.34	0.144303
$893$	−18791.5	−0.704180
$894$	−5359.62	−0.200506
$895$	0	0
$896$	2651.60	0.0988658
$897$	−5599.55	−0.208432
$898$	14497.0	0.538721
$899$	−4868.85	−0.180629
$900$	0	0
$901$	3640.75	0.134618
$902$	−11462.8	−0.423136
$903$	12766.9	0.470494
$904$	2321.63	0.0854161
$905$	0	0
$906$	12412.2	0.455152
$907$	25613.0	0.937670	0.468835	−	0.883286i	$-0.344674\pi$
$907$	25613.0	0.937670	0.468835	+	0.883286i	$0.344674\pi$
$908$	−10347.9	−0.378202
$909$	12277.3	0.447979
$910$	0	0
$911$	−13608.6	−0.494922	−0.247461	−	0.968898i	$-0.579596\pi$
$911$	−13608.6	−0.494922	−0.247461	+	0.968898i	$0.579596\pi$
$912$	1639.05	0.0595114
$913$	2773.07	0.100520
$914$	−1146.42	−0.0414881
$915$	0	0
$916$	−13016.1	−0.469502
$917$	−27514.6	−0.990852
$918$	−1226.64	−0.0441016
$919$	28792.1	1.03348	0.516738	−	0.856143i	$-0.327146\pi$
$919$	28792.1	1.03348	0.516738	+	0.856143i	$0.327146\pi$
$920$	0	0
$921$	3414.62	0.122167
$922$	−2123.92	−0.0758652
$923$	−10708.7	−0.381885
$924$	−5327.55	−0.189679
$925$	0	0
$926$	−17429.9	−0.618556
$927$	−357.695	−0.0126734
$928$	2646.90	0.0936301
$929$	4623.87	0.163299	0.0816493	−	0.996661i	$-0.473981\pi$
$929$	4623.87	0.163299	0.0816493	+	0.996661i	$0.473981\pi$
$930$	0	0
$931$	2941.33	0.103543
$932$	2521.99	0.0886379
$933$	18026.4	0.632539
$934$	11057.9	0.387392
$935$	0	0
$936$	−936.000	−0.0326860
$937$	−35625.6	−1.24209	−0.621044	−	0.783776i	$-0.713291\pi$
$937$	−35625.6	−1.24209	−0.621044	+	0.783776i	$0.713291\pi$
$938$	18597.7	0.647374
$939$	−3320.30	−0.115393
$940$	0	0
$941$	35533.9	1.23100	0.615500	−	0.788137i	$-0.288954\pi$
$941$	35533.9	1.23100	0.615500	+	0.788137i	$0.288954\pi$
$942$	−6122.33	−0.211758
$943$	38397.3	1.32597
$944$	13014.9	0.448728
$945$	0	0
$946$	−8805.30	−0.302627
$947$	−28063.2	−0.962970	−0.481485	−	0.876454i	$-0.659902\pi$
$947$	−28063.2	−0.962970	−0.481485	+	0.876454i	$0.659902\pi$
$948$	3.07279	0.000105274 0
$949$	11843.6	0.405121
$950$	0	0
$951$	−15442.1	−0.526545
$952$	−3764.55	−0.128162
$953$	−10551.8	−0.358663	−0.179332	−	0.983789i	$-0.557393\pi$
$953$	−10551.8	−0.358663	−0.179332	+	0.983789i	$0.557393\pi$
$954$	−2884.95	−0.0979074
$955$	0	0
$956$	−16265.7	−0.550283
$957$	−5318.10	−0.179634
$958$	12697.0	0.428205
$959$	−14719.5	−0.495638
$960$	0	0
$961$	−26326.2	−0.883696
$962$	−2644.36	−0.0886254
$963$	−2314.92	−0.0774633
$964$	11843.3	0.395692
$965$	0	0
$966$	17845.9	0.594391
$967$	−5237.65	−0.174179	−0.0870897	−	0.996200i	$-0.527757\pi$
$967$	−5237.65	−0.174179	−0.0870897	+	0.996200i	$0.527757\pi$
$968$	−6973.61	−0.231550
$969$	−2327.01	−0.0771457
$970$	0	0
$971$	27828.6	0.919736	0.459868	−	0.887987i	$-0.347897\pi$
$971$	27828.6	0.919736	0.459868	+	0.887987i	$0.347897\pi$
$972$	972.000	0.0320750
$973$	−22579.6	−0.743957
$974$	−298.396	−0.00981646
$975$	0	0
$976$	10125.8	0.332089
$977$	23900.6	0.782648	0.391324	−	0.920253i	$-0.372017\pi$
$977$	23900.6	0.782648	0.391324	+	0.920253i	$0.372017\pi$
$978$	−22389.8	−0.732053
$979$	−24359.7	−0.795240
$980$	0	0
$981$	−7731.08	−0.251615
$982$	−19112.8	−0.621092
$983$	34690.4	1.12559	0.562794	−	0.826597i	$-0.309727\pi$
$983$	34690.4	1.12559	0.562794	+	0.826597i	$0.309727\pi$
$984$	6418.35	0.207937
$985$	0	0
$986$	−3757.88	−0.121374
$987$	−34200.2	−1.10294
$988$	−1775.64	−0.0571767
$989$	29495.4	0.948332
$990$	0	0
$991$	26562.5	0.851448	0.425724	−	0.904853i	$-0.360019\pi$
$991$	26562.5	0.851448	0.425724	+	0.904853i	$0.360019\pi$
$992$	−1883.60	−0.0602867
$993$	8130.33	0.259827
$994$	34128.8	1.08903
$995$	0	0
$996$	−1552.72	−0.0493975
$997$	27416.7	0.870908	0.435454	−	0.900211i	$-0.356588\pi$
$997$	27416.7	0.870908	0.435454	+	0.900211i	$0.356588\pi$
$998$	−18561.7	−0.588737
$999$	2746.07	0.0869687

Twists

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

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

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

\(f(q)\)	\(=\)	\(q+2.00000 q^{2} +3.00000 q^{3} +4.00000 q^{4} +6.00000 q^{6} +20.7156 q^{7} +8.00000 q^{8} +9.00000 q^{9} -21.4313 q^{11} +12.0000 q^{12} -13.0000 q^{13} +41.4313 q^{14} +16.0000 q^{16} -22.7156 q^{17} +18.0000 q^{18} +34.1469 q^{19} +62.1469 q^{21} -42.8625 q^{22} +143.578 q^{23} +24.0000 q^{24} -26.0000 q^{26} +27.0000 q^{27} +82.8625 q^{28} +82.7156 q^{29} -58.8625 q^{31} +32.0000 q^{32} -64.2938 q^{33} -45.4313 q^{34} +36.0000 q^{36} +101.706 q^{37} +68.2938 q^{38} -39.0000 q^{39} +267.431 q^{41} +124.294 q^{42} +205.431 q^{43} -85.7251 q^{44} +287.156 q^{46} -550.313 q^{47} +48.0000 q^{48} +86.1375 q^{49} -68.1469 q^{51} -52.0000 q^{52} -160.275 q^{53} +54.0000 q^{54} +165.725 q^{56} +102.441 q^{57} +165.431 q^{58} +813.431 q^{59} +632.863 q^{61} -117.725 q^{62} +186.441 q^{63} +64.0000 q^{64} -128.588 q^{66} +448.881 q^{67} -90.8625 q^{68} +430.735 q^{69} +823.744 q^{71} +72.0000 q^{72} -911.047 q^{73} +203.412 q^{74} +136.588 q^{76} -443.962 q^{77} -78.0000 q^{78} +0.256066 q^{79} +81.0000 q^{81} +534.863 q^{82} -129.394 q^{83} +248.588 q^{84} +410.863 q^{86} +248.147 q^{87} -171.450 q^{88} +1136.64 q^{89} -269.303 q^{91} +574.313 q^{92} -176.588 q^{93} -1100.63 q^{94} +96.0000 q^{96} +318.166 q^{97} +172.275 q^{98} -192.881 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 4 q^{2} + 6 q^{3} + 8 q^{4} + 12 q^{6} - 4 q^{7} + 16 q^{8} + 18 q^{9} + 48 q^{11} + 24 q^{12} - 26 q^{13} - 8 q^{14} + 32 q^{16} + 36 q^{18} - 68 q^{19} - 12 q^{21} + 96 q^{22} + 60 q^{23} + 48 q^{24}+ \cdots + 432 q^{99}+O(q^{100}) \) 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

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more