LMFDB - Embedded newform 390.4.a.c.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [390,4,Mod(1,390)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(390, 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("390.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 390 = 2 \cdot 3 \cdot 5 \cdot 13 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	390.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:	$23.0107449022$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	390.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-2.00000 q^{2} +3.00000 q^{3} +4.00000 q^{4} -5.00000 q^{5} -6.00000 q^{6} -15.0000 q^{7} -8.00000 q^{8} +9.00000 q^{9} +O(q^{10})$

\(q-2.00000 q^{2} +3.00000 q^{3} +4.00000 q^{4} -5.00000 q^{5} -6.00000 q^{6} -15.0000 q^{7} -8.00000 q^{8} +9.00000 q^{9} +10.0000 q^{10} +39.0000 q^{11} +12.0000 q^{12} -13.0000 q^{13} +30.0000 q^{14} -15.0000 q^{15} +16.0000 q^{16} -15.0000 q^{17} -18.0000 q^{18} +54.0000 q^{19} -20.0000 q^{20} -45.0000 q^{21} -78.0000 q^{22} -143.000 q^{23} -24.0000 q^{24} +25.0000 q^{25} +26.0000 q^{26} +27.0000 q^{27} -60.0000 q^{28} -122.000 q^{29} +30.0000 q^{30} -246.000 q^{31} -32.0000 q^{32} +117.000 q^{33} +30.0000 q^{34} +75.0000 q^{35} +36.0000 q^{36} -225.000 q^{37} -108.000 q^{38} -39.0000 q^{39} +40.0000 q^{40} +469.000 q^{41} +90.0000 q^{42} -484.000 q^{43} +156.000 q^{44} -45.0000 q^{45} +286.000 q^{46} +234.000 q^{47} +48.0000 q^{48} -118.000 q^{49} -50.0000 q^{50} -45.0000 q^{51} -52.0000 q^{52} +33.0000 q^{53} -54.0000 q^{54} -195.000 q^{55} +120.000 q^{56} +162.000 q^{57} +244.000 q^{58} -60.0000 q^{60} -831.000 q^{61} +492.000 q^{62} -135.000 q^{63} +64.0000 q^{64} +65.0000 q^{65} -234.000 q^{66} +772.000 q^{67} -60.0000 q^{68} -429.000 q^{69} -150.000 q^{70} -793.000 q^{71} -72.0000 q^{72} -998.000 q^{73} +450.000 q^{74} +75.0000 q^{75} +216.000 q^{76} -585.000 q^{77} +78.0000 q^{78} -681.000 q^{79} -80.0000 q^{80} +81.0000 q^{81} -938.000 q^{82} -772.000 q^{83} -180.000 q^{84} +75.0000 q^{85} +968.000 q^{86} -366.000 q^{87} -312.000 q^{88} -465.000 q^{89} +90.0000 q^{90} +195.000 q^{91} -572.000 q^{92} -738.000 q^{93} -468.000 q^{94} -270.000 q^{95} -96.0000 q^{96} -79.0000 q^{97} +236.000 q^{98} +351.000 q^{99} +O(q^{100})\)

Display 100 coefficients 10 coefficients

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
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	−6.00000	−0.408248
$7$	−15.0000	−0.809924	−0.404962	−	0.914334i	$-0.632715\pi$
$7$	−15.0000	−0.809924	−0.404962	+	0.914334i	$0.632715\pi$
$8$	−8.00000	−0.353553
$9$	9.00000	0.333333
$10$	10.0000	0.316228
$11$	39.0000	1.06899	0.534497	−	0.845170i	$-0.320501\pi$
$11$	39.0000	1.06899	0.534497	+	0.845170i	$0.320501\pi$
$12$	12.0000	0.288675
$13$	−13.0000	−0.277350
$13$	−13.0000	−0.277350
$14$	30.0000	0.572703
$15$	−15.0000	−0.258199
$16$	16.0000	0.250000
$17$	−15.0000	−0.214002	−0.107001	−	0.994259i	$-0.534125\pi$
$17$	−15.0000	−0.214002	−0.107001	+	0.994259i	$0.534125\pi$
$18$	−18.0000	−0.235702
$19$	54.0000	0.652024	0.326012	−	0.945366i	$-0.394295\pi$
$19$	54.0000	0.652024	0.326012	+	0.945366i	$0.394295\pi$
$20$	−20.0000	−0.223607
$21$	−45.0000	−0.467610
$22$	−78.0000	−0.755893
$23$	−143.000	−1.29642	−0.648208	−	0.761463i	$-0.724481\pi$
$23$	−143.000	−1.29642	−0.648208	+	0.761463i	$0.724481\pi$
$24$	−24.0000	−0.204124
$25$	25.0000	0.200000
$26$	26.0000	0.196116
$27$	27.0000	0.192450
$28$	−60.0000	−0.404962
$29$	−122.000	−0.781201	−0.390601	−	0.920560i	$-0.627733\pi$
$29$	−122.000	−0.781201	−0.390601	+	0.920560i	$0.627733\pi$
$30$	30.0000	0.182574
$31$	−246.000	−1.42525	−0.712627	−	0.701543i	$-0.752495\pi$
$31$	−246.000	−1.42525	−0.712627	+	0.701543i	$0.752495\pi$
$32$	−32.0000	−0.176777
$33$	117.000	0.617184
$34$	30.0000	0.151322
$35$	75.0000	0.362209
$36$	36.0000	0.166667
$37$	−225.000	−0.999724	−0.499862	−	0.866105i	$-0.666616\pi$
$37$	−225.000	−0.999724	−0.499862	+	0.866105i	$0.666616\pi$
$38$	−108.000	−0.461050
$39$	−39.0000	−0.160128
$40$	40.0000	0.158114
$41$	469.000	1.78648	0.893238	−	0.449585i	$-0.148428\pi$
$41$	469.000	1.78648	0.893238	+	0.449585i	$0.148428\pi$
$42$	90.0000	0.330650
$43$	−484.000	−1.71650	−0.858248	−	0.513236i	$-0.828447\pi$
$43$	−484.000	−1.71650	−0.858248	+	0.513236i	$0.828447\pi$
$44$	156.000	0.534497
$45$	−45.0000	−0.149071
$46$	286.000	0.916704
$47$	234.000	0.726221	0.363111	−	0.931746i	$-0.381715\pi$
$47$	234.000	0.726221	0.363111	+	0.931746i	$0.381715\pi$
$48$	48.0000	0.144338
$49$	−118.000	−0.344023
$50$	−50.0000	−0.141421
$51$	−45.0000	−0.123554
$52$	−52.0000	−0.138675
$53$	33.0000	0.0855264	0.0427632	−	0.999085i	$-0.486384\pi$
$53$	33.0000	0.0855264	0.0427632	+	0.999085i	$0.486384\pi$
$54$	−54.0000	−0.136083
$55$	−195.000	−0.478069
$56$	120.000	0.286351
$57$	162.000	0.376446
$58$	244.000	0.552393
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	−60.0000	−0.129099
$61$	−831.000	−1.74424	−0.872120	−	0.489292i	$-0.837255\pi$
$61$	−831.000	−1.74424	−0.872120	+	0.489292i	$0.837255\pi$
$62$	492.000	1.00781
$63$	−135.000	−0.269975
$64$	64.0000	0.125000
$65$	65.0000	0.124035
$66$	−234.000	−0.436415
$67$	772.000	1.40768	0.703842	−	0.710357i	$-0.251466\pi$
$67$	772.000	1.40768	0.703842	+	0.710357i	$0.251466\pi$
$68$	−60.0000	−0.107001
$69$	−429.000	−0.748486
$70$	−150.000	−0.256120
$71$	−793.000	−1.32552	−0.662759	−	0.748833i	$-0.730615\pi$
$71$	−793.000	−1.32552	−0.662759	+	0.748833i	$0.730615\pi$
$72$	−72.0000	−0.117851
$73$	−998.000	−1.60010	−0.800048	−	0.599935i	$-0.795193\pi$
$73$	−998.000	−1.60010	−0.800048	+	0.599935i	$0.795193\pi$
$74$	450.000	0.706911
$75$	75.0000	0.115470
$76$	216.000	0.326012
$77$	−585.000	−0.865804
$78$	78.0000	0.113228
$79$	−681.000	−0.969854	−0.484927	−	0.874555i	$-0.661154\pi$
$79$	−681.000	−0.969854	−0.484927	+	0.874555i	$0.661154\pi$
$80$	−80.0000	−0.111803
$81$	81.0000	0.111111
$82$	−938.000	−1.26323
$83$	−772.000	−1.02094	−0.510470	−	0.859896i	$-0.670529\pi$
$83$	−772.000	−1.02094	−0.510470	+	0.859896i	$0.670529\pi$
$84$	−180.000	−0.233805
$85$	75.0000	0.0957046
$86$	968.000	1.21375
$87$	−366.000	−0.451027
$88$	−312.000	−0.377947
$89$	−465.000	−0.553819	−0.276910	−	0.960896i	$-0.589310\pi$
$89$	−465.000	−0.553819	−0.276910	+	0.960896i	$0.589310\pi$
$90$	90.0000	0.105409
$91$	195.000	0.224632
$92$	−572.000	−0.648208
$93$	−738.000	−0.822871
$94$	−468.000	−0.513516
$95$	−270.000	−0.291594
$96$	−96.0000	−0.102062
$97$	−79.0000	−0.0826931	−0.0413466	−	0.999145i	$-0.513165\pi$
$97$	−79.0000	−0.0826931	−0.0413466	+	0.999145i	$0.513165\pi$
$98$	236.000	0.243261
$99$	351.000	0.356332
$100$	100.000	0.100000
$101$	648.000	0.638400	0.319200	−	0.947687i	$-0.396586\pi$
$101$	648.000	0.638400	0.319200	+	0.947687i	$0.396586\pi$
$102$	90.0000	0.0873660
$103$	−1140.00	−1.09056	−0.545279	−	0.838254i	$-0.683576\pi$
$103$	−1140.00	−1.09056	−0.545279	+	0.838254i	$0.683576\pi$
$104$	104.000	0.0980581
$105$	225.000	0.209121
$106$	−66.0000	−0.0604763
$107$	913.000	0.824888	0.412444	−	0.910983i	$-0.364675\pi$
$107$	913.000	0.824888	0.412444	+	0.910983i	$0.364675\pi$
$108$	108.000	0.0962250
$109$	1276.00	1.12127	0.560636	−	0.828062i	$-0.310557\pi$
$109$	1276.00	1.12127	0.560636	+	0.828062i	$0.310557\pi$
$110$	390.000	0.338046
$111$	−675.000	−0.577191
$112$	−240.000	−0.202481
$113$	−214.000	−0.178154	−0.0890771	−	0.996025i	$-0.528392\pi$
$113$	−214.000	−0.178154	−0.0890771	+	0.996025i	$0.528392\pi$
$114$	−324.000	−0.266188
$115$	715.000	0.579775
$116$	−488.000	−0.390601
$117$	−117.000	−0.0924500
$118$	0	0
$119$	225.000	0.173325
$120$	120.000	0.0912871
$121$	190.000	0.142750
$122$	1662.00	1.23336
$123$	1407.00	1.03142
$124$	−984.000	−0.712627
$125$	−125.000	−0.0894427
$126$	270.000	0.190901
$127$	−1074.00	−0.750410	−0.375205	−	0.926942i	$-0.622428\pi$
$127$	−1074.00	−0.750410	−0.375205	+	0.926942i	$0.622428\pi$
$128$	−128.000	−0.0883883
$129$	−1452.00	−0.991019
$130$	−130.000	−0.0877058
$131$	1482.00	0.988419	0.494210	−	0.869343i	$-0.335457\pi$
$131$	1482.00	0.988419	0.494210	+	0.869343i	$0.335457\pi$
$132$	468.000	0.308592
$133$	−810.000	−0.528090
$134$	−1544.00	−0.995383
$135$	−135.000	−0.0860663
$136$	120.000	0.0756611
$137$	666.000	0.415330	0.207665	−	0.978200i	$-0.433414\pi$
$137$	666.000	0.415330	0.207665	+	0.978200i	$0.433414\pi$
$138$	858.000	0.529259
$139$	−1583.00	−0.965959	−0.482980	−	0.875632i	$-0.660445\pi$
$139$	−1583.00	−0.965959	−0.482980	+	0.875632i	$0.660445\pi$
$140$	300.000	0.181104
$141$	702.000	0.419284
$142$	1586.00	0.937283
$143$	−507.000	−0.296486
$144$	144.000	0.0833333
$145$	610.000	0.349364
$146$	1996.00	1.13144
$147$	−354.000	−0.198622
$148$	−900.000	−0.499862
$149$	2135.00	1.17387	0.586933	−	0.809636i	$-0.300335\pi$
$149$	2135.00	1.17387	0.586933	+	0.809636i	$0.300335\pi$
$150$	−150.000	−0.0816497
$151$	1832.00	0.987325	0.493662	−	0.869654i	$-0.335658\pi$
$151$	1832.00	0.987325	0.493662	+	0.869654i	$0.335658\pi$
$152$	−432.000	−0.230525
$153$	−135.000	−0.0713340
$154$	1170.00	0.612216
$155$	1230.00	0.637393
$156$	−156.000	−0.0800641
$157$	−1766.00	−0.897721	−0.448860	−	0.893602i	$-0.648170\pi$
$157$	−1766.00	−0.897721	−0.448860	+	0.893602i	$0.648170\pi$
$158$	1362.00	0.685791
$159$	99.0000	0.0493787
$160$	160.000	0.0790569
$161$	2145.00	1.05000
$162$	−162.000	−0.0785674
$163$	3989.00	1.91683	0.958413	−	0.285385i	$-0.0921216\pi$
$163$	3989.00	1.91683	0.958413	+	0.285385i	$0.0921216\pi$
$164$	1876.00	0.893238
$165$	−585.000	−0.276013
$166$	1544.00	0.721914
$167$	−2496.00	−1.15656	−0.578282	−	0.815837i	$-0.696277\pi$
$167$	−2496.00	−1.15656	−0.578282	+	0.815837i	$0.696277\pi$
$168$	360.000	0.165325
$169$	169.000	0.0769231
$170$	−150.000	−0.0676734
$171$	486.000	0.217341
$172$	−1936.00	−0.858248
$173$	566.000	0.248741	0.124370	−	0.992236i	$-0.460309\pi$
$173$	566.000	0.248741	0.124370	+	0.992236i	$0.460309\pi$
$174$	732.000	0.318924
$175$	−375.000	−0.161985
$176$	624.000	0.267249
$177$	0	0
$178$	930.000	0.391609
$179$	−3478.00	−1.45228	−0.726139	−	0.687547i	$-0.758687\pi$
$179$	−3478.00	−1.45228	−0.726139	+	0.687547i	$0.758687\pi$
$180$	−180.000	−0.0745356
$181$	2573.00	1.05663	0.528314	−	0.849049i	$-0.322824\pi$
$181$	2573.00	1.05663	0.528314	+	0.849049i	$0.322824\pi$
$182$	−390.000	−0.158839
$183$	−2493.00	−1.00704
$184$	1144.00	0.458352
$185$	1125.00	0.447090
$186$	1476.00	0.581858
$187$	−585.000	−0.228767
$188$	936.000	0.363111
$189$	−405.000	−0.155870
$190$	540.000	0.206188
$191$	2592.00	0.981940	0.490970	−	0.871176i	$-0.336642\pi$
$191$	2592.00	0.981940	0.490970	+	0.871176i	$0.336642\pi$
$192$	192.000	0.0721688
$193$	−2725.00	−1.01632	−0.508160	−	0.861263i	$-0.669674\pi$
$193$	−2725.00	−1.01632	−0.508160	+	0.861263i	$0.669674\pi$
$194$	158.000	0.0584729
$195$	195.000	0.0716115
$196$	−472.000	−0.172012
$197$	4564.00	1.65062	0.825308	−	0.564682i	$-0.191001\pi$
$197$	4564.00	1.65062	0.825308	+	0.564682i	$0.191001\pi$
$198$	−702.000	−0.251964
$199$	4244.00	1.51180	0.755902	−	0.654684i	$-0.227198\pi$
$199$	4244.00	1.51180	0.755902	+	0.654684i	$0.227198\pi$
$200$	−200.000	−0.0707107
$201$	2316.00	0.812727
$202$	−1296.00	−0.451417
$203$	1830.00	0.632713
$204$	−180.000	−0.0617771
$205$	−2345.00	−0.798936
$206$	2280.00	0.771141
$207$	−1287.00	−0.432139
$208$	−208.000	−0.0693375
$209$	2106.00	0.697010
$210$	−450.000	−0.147871
$211$	2244.00	0.732148	0.366074	−	0.930586i	$-0.380702\pi$
$211$	2244.00	0.732148	0.366074	+	0.930586i	$0.380702\pi$
$212$	132.000	0.0427632
$213$	−2379.00	−0.765288
$214$	−1826.00	−0.583284
$215$	2420.00	0.767640
$216$	−216.000	−0.0680414
$217$	3690.00	1.15435
$218$	−2552.00	−0.792859
$219$	−2994.00	−0.923816
$220$	−780.000	−0.239034
$221$	195.000	0.0593535
$222$	1350.00	0.408135
$223$	−272.000	−0.0816792	−0.0408396	−	0.999166i	$-0.513003\pi$
$223$	−272.000	−0.0816792	−0.0408396	+	0.999166i	$0.513003\pi$
$224$	480.000	0.143176
$225$	225.000	0.0666667
$226$	428.000	0.125974
$227$	−1190.00	−0.347943	−0.173972	−	0.984751i	$-0.555660\pi$
$227$	−1190.00	−0.347943	−0.173972	+	0.984751i	$0.555660\pi$
$228$	648.000	0.188223
$229$	50.0000	0.0144284	0.00721418	−	0.999974i	$-0.497704\pi$
$229$	50.0000	0.0144284	0.00721418	+	0.999974i	$0.497704\pi$
$230$	−1430.00	−0.409963
$231$	−1755.00	−0.499872
$232$	976.000	0.276196
$233$	−3141.00	−0.883149	−0.441575	−	0.897225i	$-0.645580\pi$
$233$	−3141.00	−0.883149	−0.441575	+	0.897225i	$0.645580\pi$
$234$	234.000	0.0653720
$235$	−1170.00	−0.324776
$236$	0	0
$237$	−2043.00	−0.559946
$238$	−450.000	−0.122560
$239$	−2857.00	−0.773238	−0.386619	−	0.922239i	$-0.626357\pi$
$239$	−2857.00	−0.773238	−0.386619	+	0.922239i	$0.626357\pi$
$240$	−240.000	−0.0645497
$241$	−662.000	−0.176943	−0.0884713	−	0.996079i	$-0.528198\pi$
$241$	−662.000	−0.176943	−0.0884713	+	0.996079i	$0.528198\pi$
$242$	−380.000	−0.100939
$243$	243.000	0.0641500
$244$	−3324.00	−0.872120
$245$	590.000	0.153852
$246$	−2814.00	−0.729326
$247$	−702.000	−0.180839
$248$	1968.00	0.503904
$249$	−2316.00	−0.589440
$250$	250.000	0.0632456
$251$	−512.000	−0.128754	−0.0643768	−	0.997926i	$-0.520506\pi$
$251$	−512.000	−0.128754	−0.0643768	+	0.997926i	$0.520506\pi$
$252$	−540.000	−0.134987
$253$	−5577.00	−1.38586
$254$	2148.00	0.530620
$255$	225.000	0.0552551
$256$	256.000	0.0625000
$257$	−102.000	−0.0247571	−0.0123786	−	0.999923i	$-0.503940\pi$
$257$	−102.000	−0.0247571	−0.0123786	+	0.999923i	$0.503940\pi$
$258$	2904.00	0.700756
$259$	3375.00	0.809700
$260$	260.000	0.0620174
$261$	−1098.00	−0.260400
$262$	−2964.00	−0.698918
$263$	6448.00	1.51179	0.755895	−	0.654693i	$-0.227202\pi$
$263$	6448.00	1.51179	0.755895	+	0.654693i	$0.227202\pi$
$264$	−936.000	−0.218208
$265$	−165.000	−0.0382486
$266$	1620.00	0.373416
$267$	−1395.00	−0.319748
$268$	3088.00	0.703842
$269$	3976.00	0.901193	0.450597	−	0.892728i	$-0.351211\pi$
$269$	3976.00	0.901193	0.450597	+	0.892728i	$0.351211\pi$
$270$	270.000	0.0608581
$271$	7494.00	1.67981	0.839904	−	0.542735i	$-0.182611\pi$
$271$	7494.00	1.67981	0.839904	+	0.542735i	$0.182611\pi$
$272$	−240.000	−0.0535005
$273$	585.000	0.129692
$274$	−1332.00	−0.293683
$275$	975.000	0.213799
$276$	−1716.00	−0.374243
$277$	−5062.00	−1.09800	−0.549000	−	0.835822i	$-0.684991\pi$
$277$	−5062.00	−1.09800	−0.549000	+	0.835822i	$0.684991\pi$
$278$	3166.00	0.683036
$279$	−2214.00	−0.475085
$280$	−600.000	−0.128060
$281$	−3990.00	−0.847059	−0.423529	−	0.905882i	$-0.639209\pi$
$281$	−3990.00	−0.847059	−0.423529	+	0.905882i	$0.639209\pi$
$282$	−1404.00	−0.296479
$283$	−6468.00	−1.35860	−0.679298	−	0.733863i	$-0.737716\pi$
$283$	−6468.00	−1.35860	−0.679298	+	0.733863i	$0.737716\pi$
$284$	−3172.00	−0.662759
$285$	−810.000	−0.168352
$286$	1014.00	0.209647
$287$	−7035.00	−1.44691
$288$	−288.000	−0.0589256
$289$	−4688.00	−0.954203
$290$	−1220.00	−0.247037
$291$	−237.000	−0.0477429
$292$	−3992.00	−0.800048
$293$	−4748.00	−0.946693	−0.473347	−	0.880876i	$-0.656954\pi$
$293$	−4748.00	−0.946693	−0.473347	+	0.880876i	$0.656954\pi$
$294$	708.000	0.140447
$295$	0	0
$296$	1800.00	0.353456
$297$	1053.00	0.205728
$298$	−4270.00	−0.830049
$299$	1859.00	0.359561
$300$	300.000	0.0577350
$301$	7260.00	1.39023
$302$	−3664.00	−0.698144
$303$	1944.00	0.368580
$304$	864.000	0.163006
$305$	4155.00	0.780048
$306$	270.000	0.0504408
$307$	2981.00	0.554185	0.277092	−	0.960843i	$-0.410629\pi$
$307$	2981.00	0.554185	0.277092	+	0.960843i	$0.410629\pi$
$308$	−2340.00	−0.432902
$309$	−3420.00	−0.629634
$310$	−2460.00	−0.450705
$311$	2080.00	0.379248	0.189624	−	0.981857i	$-0.439273\pi$
$311$	2080.00	0.379248	0.189624	+	0.981857i	$0.439273\pi$
$312$	312.000	0.0566139
$313$	5618.00	1.01453	0.507265	−	0.861790i	$-0.330656\pi$
$313$	5618.00	1.01453	0.507265	+	0.861790i	$0.330656\pi$
$314$	3532.00	0.634784
$315$	675.000	0.120736
$316$	−2724.00	−0.484927
$317$	−8516.00	−1.50885	−0.754426	−	0.656385i	$-0.772085\pi$
$317$	−8516.00	−1.50885	−0.754426	+	0.656385i	$0.772085\pi$
$318$	−198.000	−0.0349160
$319$	−4758.00	−0.835100
$320$	−320.000	−0.0559017
$321$	2739.00	0.476249
$322$	−4290.00	−0.742461
$323$	−810.000	−0.139534
$324$	324.000	0.0555556
$325$	−325.000	−0.0554700
$326$	−7978.00	−1.35540
$327$	3828.00	0.647367
$328$	−3752.00	−0.631614
$329$	−3510.00	−0.588184
$330$	1170.00	0.195171
$331$	11324.0	1.88043	0.940217	−	0.340577i	$-0.110623\pi$
$331$	11324.0	1.88043	0.940217	+	0.340577i	$0.110623\pi$
$332$	−3088.00	−0.510470
$333$	−2025.00	−0.333241
$334$	4992.00	0.817815
$335$	−3860.00	−0.629535
$336$	−720.000	−0.116902
$337$	−1716.00	−0.277378	−0.138689	−	0.990336i	$-0.544289\pi$
$337$	−1716.00	−0.277378	−0.138689	+	0.990336i	$0.544289\pi$
$338$	−338.000	−0.0543928
$339$	−642.000	−0.102857
$340$	300.000	0.0478523
$341$	−9594.00	−1.52359
$342$	−972.000	−0.153683
$343$	6915.00	1.08856
$344$	3872.00	0.606873
$345$	2145.00	0.334733
$346$	−1132.00	−0.175886
$347$	8181.00	1.26565	0.632823	−	0.774297i	$-0.281896\pi$
$347$	8181.00	1.26565	0.632823	+	0.774297i	$0.281896\pi$
$348$	−1464.00	−0.225513
$349$	−11108.0	−1.70372	−0.851859	−	0.523771i	$-0.824525\pi$
$349$	−11108.0	−1.70372	−0.851859	+	0.523771i	$0.824525\pi$
$350$	750.000	0.114541
$351$	−351.000	−0.0533761
$352$	−1248.00	−0.188973
$353$	−4330.00	−0.652869	−0.326434	−	0.945220i	$-0.605847\pi$
$353$	−4330.00	−0.652869	−0.326434	+	0.945220i	$0.605847\pi$
$354$	0	0
$355$	3965.00	0.592790
$356$	−1860.00	−0.276910
$357$	675.000	0.100069
$358$	6956.00	1.02692
$359$	−1128.00	−0.165832	−0.0829158	−	0.996557i	$-0.526423\pi$
$359$	−1128.00	−0.165832	−0.0829158	+	0.996557i	$0.526423\pi$
$360$	360.000	0.0527046
$361$	−3943.00	−0.574865
$362$	−5146.00	−0.747148
$363$	570.000	0.0824166
$364$	780.000	0.112316
$365$	4990.00	0.715585
$366$	4986.00	0.712083
$367$	−1072.00	−0.152474	−0.0762370	−	0.997090i	$-0.524291\pi$
$367$	−1072.00	−0.152474	−0.0762370	+	0.997090i	$0.524291\pi$
$368$	−2288.00	−0.324104
$369$	4221.00	0.595492
$370$	−2250.00	−0.316140
$371$	−495.000	−0.0692699
$372$	−2952.00	−0.411436
$373$	10660.0	1.47977	0.739885	−	0.672734i	$-0.234880\pi$
$373$	10660.0	1.47977	0.739885	+	0.672734i	$0.234880\pi$
$374$	1170.00	0.161763
$375$	−375.000	−0.0516398
$376$	−1872.00	−0.256758
$377$	1586.00	0.216666
$378$	810.000	0.110217
$379$	6258.00	0.848158	0.424079	−	0.905625i	$-0.360598\pi$
$379$	6258.00	0.848158	0.424079	+	0.905625i	$0.360598\pi$
$380$	−1080.00	−0.145797
$381$	−3222.00	−0.433250
$382$	−5184.00	−0.694336
$383$	−3250.00	−0.433596	−0.216798	−	0.976216i	$-0.569561\pi$
$383$	−3250.00	−0.433596	−0.216798	+	0.976216i	$0.569561\pi$
$384$	−384.000	−0.0510310
$385$	2925.00	0.387199
$386$	5450.00	0.718647
$387$	−4356.00	−0.572165
$388$	−316.000	−0.0413466
$389$	−6156.00	−0.802369	−0.401185	−	0.915997i	$-0.631401\pi$
$389$	−6156.00	−0.802369	−0.401185	+	0.915997i	$0.631401\pi$
$390$	−390.000	−0.0506370
$391$	2145.00	0.277436
$392$	944.000	0.121631
$393$	4446.00	0.570664
$394$	−9128.00	−1.16716
$395$	3405.00	0.433732
$396$	1404.00	0.178166
$397$	6151.00	0.777607	0.388803	−	0.921321i	$-0.372889\pi$
$397$	6151.00	0.777607	0.388803	+	0.921321i	$0.372889\pi$
$398$	−8488.00	−1.06901
$399$	−2430.00	−0.304893
$400$	400.000	0.0500000
$401$	−8646.00	−1.07671	−0.538355	−	0.842718i	$-0.680954\pi$
$401$	−8646.00	−1.07671	−0.538355	+	0.842718i	$0.680954\pi$
$402$	−4632.00	−0.574684
$403$	3198.00	0.395295
$404$	2592.00	0.319200
$405$	−405.000	−0.0496904
$406$	−3660.00	−0.447396
$407$	−8775.00	−1.06870
$408$	360.000	0.0436830
$409$	−910.000	−0.110016	−0.0550081	−	0.998486i	$-0.517518\pi$
$409$	−910.000	−0.110016	−0.0550081	+	0.998486i	$0.517518\pi$
$410$	4690.00	0.564933
$411$	1998.00	0.239791
$412$	−4560.00	−0.545279
$413$	0	0
$414$	2574.00	0.305568
$415$	3860.00	0.456578
$416$	416.000	0.0490290
$417$	−4749.00	−0.557697
$418$	−4212.00	−0.492860
$419$	8502.00	0.991288	0.495644	−	0.868526i	$-0.334932\pi$
$419$	8502.00	0.991288	0.495644	+	0.868526i	$0.334932\pi$
$420$	900.000	0.104561
$421$	−3508.00	−0.406103	−0.203052	−	0.979168i	$-0.565086\pi$
$421$	−3508.00	−0.406103	−0.203052	+	0.979168i	$0.565086\pi$
$422$	−4488.00	−0.517707
$423$	2106.00	0.242074
$424$	−264.000	−0.0302381
$425$	−375.000	−0.0428004
$426$	4758.00	0.541141
$427$	12465.0	1.41270
$428$	3652.00	0.412444
$429$	−1521.00	−0.171176
$430$	−4840.00	−0.542804
$431$	8112.00	0.906592	0.453296	−	0.891360i	$-0.350248\pi$
$431$	8112.00	0.906592	0.453296	+	0.891360i	$0.350248\pi$
$432$	432.000	0.0481125
$433$	728.000	0.0807978	0.0403989	−	0.999184i	$-0.487137\pi$
$433$	728.000	0.0807978	0.0403989	+	0.999184i	$0.487137\pi$
$434$	−7380.00	−0.816247
$435$	1830.00	0.201705
$436$	5104.00	0.560636
$437$	−7722.00	−0.845294
$438$	5988.00	0.653237
$439$	−531.000	−0.0577295	−0.0288647	−	0.999583i	$-0.509189\pi$
$439$	−531.000	−0.0577295	−0.0288647	+	0.999583i	$0.509189\pi$
$440$	1560.00	0.169023
$441$	−1062.00	−0.114674
$442$	−390.000	−0.0419692
$443$	13441.0	1.44154	0.720769	−	0.693176i	$-0.243789\pi$
$443$	13441.0	1.44154	0.720769	+	0.693176i	$0.243789\pi$
$444$	−2700.00	−0.288595
$445$	2325.00	0.247675
$446$	544.000	0.0577559
$447$	6405.00	0.677732
$448$	−960.000	−0.101240
$449$	6029.00	0.633688	0.316844	−	0.948478i	$-0.397377\pi$
$449$	6029.00	0.633688	0.316844	+	0.948478i	$0.397377\pi$
$450$	−450.000	−0.0471405
$451$	18291.0	1.90973
$452$	−856.000	−0.0890771
$453$	5496.00	0.570032
$454$	2380.00	0.246033
$455$	−975.000	−0.100459
$456$	−1296.00	−0.133094
$457$	−12051.0	−1.23353	−0.616764	−	0.787148i	$-0.711557\pi$
$457$	−12051.0	−1.23353	−0.616764	+	0.787148i	$0.711557\pi$
$458$	−100.000	−0.0102024
$459$	−405.000	−0.0411847
$460$	2860.00	0.289887
$461$	−10099.0	−1.02030	−0.510149	−	0.860086i	$-0.670410\pi$
$461$	−10099.0	−1.02030	−0.510149	+	0.860086i	$0.670410\pi$
$462$	3510.00	0.353463
$463$	−12447.0	−1.24938	−0.624688	−	0.780874i	$-0.714774\pi$
$463$	−12447.0	−1.24938	−0.624688	+	0.780874i	$0.714774\pi$
$464$	−1952.00	−0.195300
$465$	3690.00	0.367999
$466$	6282.00	0.624481
$467$	−2135.00	−0.211555	−0.105777	−	0.994390i	$-0.533733\pi$
$467$	−2135.00	−0.211555	−0.105777	+	0.994390i	$0.533733\pi$
$468$	−468.000	−0.0462250
$469$	−11580.0	−1.14012
$470$	2340.00	0.229651
$471$	−5298.00	−0.518299
$472$	0	0
$473$	−18876.0	−1.83492
$474$	4086.00	0.395941
$475$	1350.00	0.130405
$476$	900.000	0.0866627
$477$	297.000	0.0285088
$478$	5714.00	0.546762
$479$	6405.00	0.610964	0.305482	−	0.952198i	$-0.401182\pi$
$479$	6405.00	0.610964	0.305482	+	0.952198i	$0.401182\pi$
$480$	480.000	0.0456435
$481$	2925.00	0.277273
$482$	1324.00	0.125117
$483$	6435.00	0.606217
$484$	760.000	0.0713749
$485$	395.000	0.0369815
$486$	−486.000	−0.0453609
$487$	7619.00	0.708932	0.354466	−	0.935069i	$-0.384663\pi$
$487$	7619.00	0.708932	0.354466	+	0.935069i	$0.384663\pi$
$488$	6648.00	0.616682
$489$	11967.0	1.10668
$490$	−1180.00	−0.108790
$491$	−9692.00	−0.890822	−0.445411	−	0.895326i	$-0.646943\pi$
$491$	−9692.00	−0.890822	−0.445411	+	0.895326i	$0.646943\pi$
$492$	5628.00	0.515711
$493$	1830.00	0.167179
$494$	1404.00	0.127872
$495$	−1755.00	−0.159356
$496$	−3936.00	−0.356314
$497$	11895.0	1.07357
$498$	4632.00	0.416797
$499$	−12994.0	−1.16571	−0.582857	−	0.812575i	$-0.698065\pi$
$499$	−12994.0	−1.16571	−0.582857	+	0.812575i	$0.698065\pi$
$500$	−500.000	−0.0447214
$501$	−7488.00	−0.667743
$502$	1024.00	0.0910425
$503$	12228.0	1.08394	0.541968	−	0.840399i	$-0.317679\pi$
$503$	12228.0	1.08394	0.541968	+	0.840399i	$0.317679\pi$
$504$	1080.00	0.0954504
$505$	−3240.00	−0.285501
$506$	11154.0	0.979952
$507$	507.000	0.0444116
$508$	−4296.00	−0.375205
$509$	7931.00	0.690639	0.345320	−	0.938485i	$-0.387771\pi$
$509$	7931.00	0.690639	0.345320	+	0.938485i	$0.387771\pi$
$510$	−450.000	−0.0390712
$511$	14970.0	1.29596
$512$	−512.000	−0.0441942
$513$	1458.00	0.125482
$514$	204.000	0.0175059
$515$	5700.00	0.487713
$516$	−5808.00	−0.495510
$517$	9126.00	0.776327
$518$	−6750.00	−0.572544
$519$	1698.00	0.143611
$520$	−520.000	−0.0438529
$521$	−7210.00	−0.606288	−0.303144	−	0.952945i	$-0.598036\pi$
$521$	−7210.00	−0.606288	−0.303144	+	0.952945i	$0.598036\pi$
$522$	2196.00	0.184131
$523$	−19700.0	−1.64708	−0.823538	−	0.567261i	$-0.808003\pi$
$523$	−19700.0	−1.64708	−0.823538	+	0.567261i	$0.808003\pi$
$524$	5928.00	0.494210
$525$	−1125.00	−0.0935220
$526$	−12896.0	−1.06900
$527$	3690.00	0.305007
$528$	1872.00	0.154296
$529$	8282.00	0.680694
$530$	330.000	0.0270458
$531$	0	0
$532$	−3240.00	−0.264045
$533$	−6097.00	−0.495479
$534$	2790.00	0.226096
$535$	−4565.00	−0.368901
$536$	−6176.00	−0.497691
$537$	−10434.0	−0.838474
$538$	−7952.00	−0.637240
$539$	−4602.00	−0.367759
$540$	−540.000	−0.0430331
$541$	7694.00	0.611443	0.305722	−	0.952121i	$-0.401102\pi$
$541$	7694.00	0.611443	0.305722	+	0.952121i	$0.401102\pi$
$542$	−14988.0	−1.18780
$543$	7719.00	0.610044
$544$	480.000	0.0378306
$545$	−6380.00	−0.501448
$546$	−1170.00	−0.0917058
$547$	76.0000	0.00594063	0.00297032	−	0.999996i	$-0.499055\pi$
$547$	76.0000	0.00594063	0.00297032	+	0.999996i	$0.499055\pi$
$548$	2664.00	0.207665
$549$	−7479.00	−0.581413
$550$	−1950.00	−0.151179
$551$	−6588.00	−0.509362
$552$	3432.00	0.264630
$553$	10215.0	0.785508
$554$	10124.0	0.776404
$555$	3375.00	0.258128
$556$	−6332.00	−0.482980
$557$	1914.00	0.145599	0.0727996	−	0.997347i	$-0.476807\pi$
$557$	1914.00	0.145599	0.0727996	+	0.997347i	$0.476807\pi$
$558$	4428.00	0.335936
$559$	6292.00	0.476070
$560$	1200.00	0.0905522
$561$	−1755.00	−0.132079
$562$	7980.00	0.598961
$563$	−21563.0	−1.61416	−0.807080	−	0.590442i	$-0.798953\pi$
$563$	−21563.0	−1.61416	−0.807080	+	0.590442i	$0.798953\pi$
$564$	2808.00	0.209642
$565$	1070.00	0.0796730
$566$	12936.0	0.960673
$567$	−1215.00	−0.0899915
$568$	6344.00	0.468641
$569$	−22752.0	−1.67630	−0.838149	−	0.545441i	$-0.816362\pi$
$569$	−22752.0	−1.67630	−0.838149	+	0.545441i	$0.816362\pi$
$570$	1620.00	0.119043
$571$	−25201.0	−1.84699	−0.923493	−	0.383615i	$-0.874679\pi$
$571$	−25201.0	−1.84699	−0.923493	+	0.383615i	$0.874679\pi$
$572$	−2028.00	−0.148243
$573$	7776.00	0.566923
$574$	14070.0	1.02312
$575$	−3575.00	−0.259283
$576$	576.000	0.0416667
$577$	8629.00	0.622582	0.311291	−	0.950315i	$-0.399239\pi$
$577$	8629.00	0.622582	0.311291	+	0.950315i	$0.399239\pi$
$578$	9376.00	0.674724
$579$	−8175.00	−0.586773
$580$	2440.00	0.174682
$581$	11580.0	0.826884
$582$	474.000	0.0337593
$583$	1287.00	0.0914273
$584$	7984.00	0.565720
$585$	585.000	0.0413449
$586$	9496.00	0.669413
$587$	7074.00	0.497402	0.248701	−	0.968580i	$-0.419996\pi$
$587$	7074.00	0.497402	0.248701	+	0.968580i	$0.419996\pi$
$588$	−1416.00	−0.0993110
$589$	−13284.0	−0.929300
$590$	0	0
$591$	13692.0	0.952984
$592$	−3600.00	−0.249931
$593$	3672.00	0.254285	0.127142	−	0.991884i	$-0.459419\pi$
$593$	3672.00	0.254285	0.127142	+	0.991884i	$0.459419\pi$
$594$	−2106.00	−0.145472
$595$	−1125.00	−0.0775135
$596$	8540.00	0.586933
$597$	12732.0	0.872841
$598$	−3718.00	−0.254248
$599$	−24516.0	−1.67228	−0.836141	−	0.548515i	$-0.815193\pi$
$599$	−24516.0	−1.67228	−0.836141	+	0.548515i	$0.815193\pi$
$600$	−600.000	−0.0408248
$601$	−18045.0	−1.22474	−0.612372	−	0.790570i	$-0.709784\pi$
$601$	−18045.0	−1.22474	−0.612372	+	0.790570i	$0.709784\pi$
$602$	−14520.0	−0.983042
$603$	6948.00	0.469228
$604$	7328.00	0.493662
$605$	−950.000	−0.0638397
$606$	−3888.00	−0.260626
$607$	25236.0	1.68748	0.843738	−	0.536756i	$-0.180350\pi$
$607$	25236.0	1.68748	0.843738	+	0.536756i	$0.180350\pi$
$608$	−1728.00	−0.115263
$609$	5490.00	0.365297
$610$	−8310.00	−0.551577
$611$	−3042.00	−0.201418
$612$	−540.000	−0.0356670
$613$	−26191.0	−1.72568	−0.862842	−	0.505473i	$-0.831318\pi$
$613$	−26191.0	−1.72568	−0.862842	+	0.505473i	$0.831318\pi$
$614$	−5962.00	−0.391868
$615$	−7035.00	−0.461266
$616$	4680.00	0.306108
$617$	18666.0	1.21793	0.608967	−	0.793196i	$-0.291584\pi$
$617$	18666.0	1.21793	0.608967	+	0.793196i	$0.291584\pi$
$618$	6840.00	0.445219
$619$	3082.00	0.200123	0.100061	−	0.994981i	$-0.468096\pi$
$619$	3082.00	0.200123	0.100061	+	0.994981i	$0.468096\pi$
$620$	4920.00	0.318697
$621$	−3861.00	−0.249495
$622$	−4160.00	−0.268168
$623$	6975.00	0.448551
$624$	−624.000	−0.0400320
$625$	625.000	0.0400000
$626$	−11236.0	−0.717382
$627$	6318.00	0.402419
$628$	−7064.00	−0.448860
$629$	3375.00	0.213943
$630$	−1350.00	−0.0853735
$631$	7452.00	0.470142	0.235071	−	0.971978i	$-0.424468\pi$
$631$	7452.00	0.470142	0.235071	+	0.971978i	$0.424468\pi$
$632$	5448.00	0.342895
$633$	6732.00	0.422706
$634$	17032.0	1.06692
$635$	5370.00	0.335594
$636$	396.000	0.0246893
$637$	1534.00	0.0954149
$638$	9516.00	0.590505
$639$	−7137.00	−0.441839
$640$	640.000	0.0395285
$641$	4768.00	0.293798	0.146899	−	0.989151i	$-0.453071\pi$
$641$	4768.00	0.293798	0.146899	+	0.989151i	$0.453071\pi$
$642$	−5478.00	−0.336759
$643$	10753.0	0.659498	0.329749	−	0.944069i	$-0.393036\pi$
$643$	10753.0	0.659498	0.329749	+	0.944069i	$0.393036\pi$
$644$	8580.00	0.524999
$645$	7260.00	0.443197
$646$	1620.00	0.0986657
$647$	5427.00	0.329764	0.164882	−	0.986313i	$-0.447276\pi$
$647$	5427.00	0.329764	0.164882	+	0.986313i	$0.447276\pi$
$648$	−648.000	−0.0392837
$649$	0	0
$650$	650.000	0.0392232
$651$	11070.0	0.666463
$652$	15956.0	0.958413
$653$	−5718.00	−0.342669	−0.171334	−	0.985213i	$-0.554808\pi$
$653$	−5718.00	−0.342669	−0.171334	+	0.985213i	$0.554808\pi$
$654$	−7656.00	−0.457757
$655$	−7410.00	−0.442035
$656$	7504.00	0.446619
$657$	−8982.00	−0.533366
$658$	7020.00	0.415909
$659$	−17248.0	−1.01955	−0.509777	−	0.860306i	$-0.670272\pi$
$659$	−17248.0	−1.01955	−0.509777	+	0.860306i	$0.670272\pi$
$660$	−2340.00	−0.138007
$661$	14532.0	0.855112	0.427556	−	0.903989i	$-0.359375\pi$
$661$	14532.0	0.855112	0.427556	+	0.903989i	$0.359375\pi$
$662$	−22648.0	−1.32967
$663$	585.000	0.0342677
$664$	6176.00	0.360957
$665$	4050.00	0.236169
$666$	4050.00	0.235637
$667$	17446.0	1.01276
$668$	−9984.00	−0.578282
$669$	−816.000	−0.0471575
$670$	7720.00	0.445149
$671$	−32409.0	−1.86458
$672$	1440.00	0.0826625
$673$	3198.00	0.183171	0.0915853	−	0.995797i	$-0.470807\pi$
$673$	3198.00	0.183171	0.0915853	+	0.995797i	$0.470807\pi$
$674$	3432.00	0.196136
$675$	675.000	0.0384900
$676$	676.000	0.0384615
$677$	−3711.00	−0.210673	−0.105336	−	0.994437i	$-0.533592\pi$
$677$	−3711.00	−0.210673	−0.105336	+	0.994437i	$0.533592\pi$
$678$	1284.00	0.0727312
$679$	1185.00	0.0669751
$680$	−600.000	−0.0338367
$681$	−3570.00	−0.200885
$682$	19188.0	1.07734
$683$	−4764.00	−0.266895	−0.133448	−	0.991056i	$-0.542605\pi$
$683$	−4764.00	−0.266895	−0.133448	+	0.991056i	$0.542605\pi$
$684$	1944.00	0.108671
$685$	−3330.00	−0.185741
$686$	−13830.0	−0.769726
$687$	150.000	0.00833021
$688$	−7744.00	−0.429124
$689$	−429.000	−0.0237208
$690$	−4290.00	−0.236692
$691$	23462.0	1.29166	0.645830	−	0.763482i	$-0.276512\pi$
$691$	23462.0	1.29166	0.645830	+	0.763482i	$0.276512\pi$
$692$	2264.00	0.124370
$693$	−5265.00	−0.288601
$694$	−16362.0	−0.894947
$695$	7915.00	0.431990
$696$	2928.00	0.159462
$697$	−7035.00	−0.382309
$698$	22216.0	1.20471
$699$	−9423.00	−0.509886
$700$	−1500.00	−0.0809924
$701$	−4908.00	−0.264440	−0.132220	−	0.991220i	$-0.542211\pi$
$701$	−4908.00	−0.264440	−0.132220	+	0.991220i	$0.542211\pi$
$702$	702.000	0.0377426
$703$	−12150.0	−0.651843
$704$	2496.00	0.133624
$705$	−3510.00	−0.187510
$706$	8660.00	0.461648
$707$	−9720.00	−0.517055
$708$	0	0
$709$	−1996.00	−0.105728	−0.0528641	−	0.998602i	$-0.516835\pi$
$709$	−1996.00	−0.105728	−0.0528641	+	0.998602i	$0.516835\pi$
$710$	−7930.00	−0.419166
$711$	−6129.00	−0.323285
$712$	3720.00	0.195805
$713$	35178.0	1.84772
$714$	−1350.00	−0.0707598
$715$	2535.00	0.132592
$716$	−13912.0	−0.726139
$717$	−8571.00	−0.446429
$718$	2256.00	0.117261
$719$	10296.0	0.534042	0.267021	−	0.963691i	$-0.413961\pi$
$719$	10296.0	0.534042	0.267021	+	0.963691i	$0.413961\pi$
$720$	−720.000	−0.0372678
$721$	17100.0	0.883269
$722$	7886.00	0.406491
$723$	−1986.00	−0.102158
$724$	10292.0	0.528314
$725$	−3050.00	−0.156240
$726$	−1140.00	−0.0582774
$727$	3954.00	0.201714	0.100857	−	0.994901i	$-0.467842\pi$
$727$	3954.00	0.201714	0.100857	+	0.994901i	$0.467842\pi$
$728$	−1560.00	−0.0794196
$729$	729.000	0.0370370
$730$	−9980.00	−0.505995
$731$	7260.00	0.367334
$732$	−9972.00	−0.503519
$733$	−25837.0	−1.30193	−0.650963	−	0.759110i	$-0.725635\pi$
$733$	−25837.0	−1.30193	−0.650963	+	0.759110i	$0.725635\pi$
$734$	2144.00	0.107815
$735$	1770.00	0.0888264
$736$	4576.00	0.229176
$737$	30108.0	1.50481
$738$	−8442.00	−0.421076
$739$	−1378.00	−0.0685934	−0.0342967	−	0.999412i	$-0.510919\pi$
$739$	−1378.00	−0.0685934	−0.0342967	+	0.999412i	$0.510919\pi$
$740$	4500.00	0.223545
$741$	−2106.00	−0.104407
$742$	990.000	0.0489812
$743$	7674.00	0.378912	0.189456	−	0.981889i	$-0.439328\pi$
$743$	7674.00	0.378912	0.189456	+	0.981889i	$0.439328\pi$
$744$	5904.00	0.290929
$745$	−10675.0	−0.524969
$746$	−21320.0	−1.04635
$747$	−6948.00	−0.340313
$748$	−2340.00	−0.114384
$749$	−13695.0	−0.668097
$750$	750.000	0.0365148
$751$	11937.0	0.580010	0.290005	−	0.957025i	$-0.406343\pi$
$751$	11937.0	0.580010	0.290005	+	0.957025i	$0.406343\pi$
$752$	3744.00	0.181555
$753$	−1536.00	−0.0743359
$754$	−3172.00	−0.153206
$755$	−9160.00	−0.441545
$756$	−1620.00	−0.0779350
$757$	3620.00	0.173806	0.0869030	−	0.996217i	$-0.472303\pi$
$757$	3620.00	0.173806	0.0869030	+	0.996217i	$0.472303\pi$
$758$	−12516.0	−0.599738
$759$	−16731.0	−0.800128
$760$	2160.00	0.103094
$761$	−36170.0	−1.72295	−0.861473	−	0.507804i	$-0.830457\pi$
$761$	−36170.0	−1.72295	−0.861473	+	0.507804i	$0.830457\pi$
$762$	6444.00	0.306354
$763$	−19140.0	−0.908145
$764$	10368.0	0.490970
$765$	675.000	0.0319015
$766$	6500.00	0.306599
$767$	0	0
$768$	768.000	0.0360844
$769$	−17936.0	−0.841078	−0.420539	−	0.907275i	$-0.638159\pi$
$769$	−17936.0	−0.841078	−0.420539	+	0.907275i	$0.638159\pi$
$770$	−5850.00	−0.273791
$771$	−306.000	−0.0142935
$772$	−10900.0	−0.508160
$773$	21364.0	0.994062	0.497031	−	0.867733i	$-0.334424\pi$
$773$	21364.0	0.994062	0.497031	+	0.867733i	$0.334424\pi$
$774$	8712.00	0.404582
$775$	−6150.00	−0.285051
$776$	632.000	0.0292364
$777$	10125.0	0.467481
$778$	12312.0	0.567361
$779$	25326.0	1.16482
$780$	780.000	0.0358057
$781$	−30927.0	−1.41697
$782$	−4290.00	−0.196177
$783$	−3294.00	−0.150342
$784$	−1888.00	−0.0860058
$785$	8830.00	0.401473
$786$	−8892.00	−0.403521
$787$	32004.0	1.44958	0.724790	−	0.688970i	$-0.241937\pi$
$787$	32004.0	1.44958	0.724790	+	0.688970i	$0.241937\pi$
$788$	18256.0	0.825308
$789$	19344.0	0.872832
$790$	−6810.00	−0.306695
$791$	3210.00	0.144291
$792$	−2808.00	−0.125982
$793$	10803.0	0.483765
$794$	−12302.0	−0.549851
$795$	−495.000	−0.0220828
$796$	16976.0	0.755902
$797$	−9909.00	−0.440395	−0.220197	−	0.975455i	$-0.570670\pi$
$797$	−9909.00	−0.440395	−0.220197	+	0.975455i	$0.570670\pi$
$798$	4860.00	0.215592
$799$	−3510.00	−0.155413
$800$	−800.000	−0.0353553
$801$	−4185.00	−0.184606
$802$	17292.0	0.761349
$803$	−38922.0	−1.71050
$804$	9264.00	0.406363
$805$	−10725.0	−0.469573
$806$	−6396.00	−0.279515
$807$	11928.0	0.520304
$808$	−5184.00	−0.225709
$809$	−20546.0	−0.892903	−0.446452	−	0.894808i	$-0.647313\pi$
$809$	−20546.0	−0.892903	−0.446452	+	0.894808i	$0.647313\pi$
$810$	810.000	0.0351364
$811$	−19468.0	−0.842927	−0.421464	−	0.906845i	$-0.638483\pi$
$811$	−19468.0	−0.842927	−0.421464	+	0.906845i	$0.638483\pi$
$812$	7320.00	0.316357
$813$	22482.0	0.969838
$814$	17550.0	0.755684
$815$	−19945.0	−0.857231
$816$	−720.000	−0.0308885
$817$	−26136.0	−1.11920
$818$	1820.00	0.0777932
$819$	1755.00	0.0748775
$820$	−9380.00	−0.399468
$821$	−38951.0	−1.65578	−0.827892	−	0.560887i	$-0.810460\pi$
$821$	−38951.0	−1.65578	−0.827892	+	0.560887i	$0.810460\pi$
$822$	−3996.00	−0.169558
$823$	−39908.0	−1.69029	−0.845143	−	0.534540i	$-0.820485\pi$
$823$	−39908.0	−1.69029	−0.845143	+	0.534540i	$0.820485\pi$
$824$	9120.00	0.385571
$825$	2925.00	0.123437
$826$	0	0
$827$	−3570.00	−0.150110	−0.0750551	−	0.997179i	$-0.523913\pi$
$827$	−3570.00	−0.150110	−0.0750551	+	0.997179i	$0.523913\pi$
$828$	−5148.00	−0.216069
$829$	−22266.0	−0.932847	−0.466423	−	0.884562i	$-0.654458\pi$
$829$	−22266.0	−0.932847	−0.466423	+	0.884562i	$0.654458\pi$
$830$	−7720.00	−0.322850
$831$	−15186.0	−0.633931
$832$	−832.000	−0.0346688
$833$	1770.00	0.0736217
$834$	9498.00	0.394351
$835$	12480.0	0.517231
$836$	8424.00	0.348505
$837$	−6642.00	−0.274290
$838$	−17004.0	−0.700947
$839$	36711.0	1.51061	0.755307	−	0.655372i	$-0.227488\pi$
$839$	36711.0	1.51061	0.755307	+	0.655372i	$0.227488\pi$
$840$	−1800.00	−0.0739356
$841$	−9505.00	−0.389725
$842$	7016.00	0.287158
$843$	−11970.0	−0.489049
$844$	8976.00	0.366074
$845$	−845.000	−0.0344010
$846$	−4212.00	−0.171172
$847$	−2850.00	−0.115616
$848$	528.000	0.0213816
$849$	−19404.0	−0.784386
$850$	750.000	0.0302645
$851$	32175.0	1.29606
$852$	−9516.00	−0.382644
$853$	−33167.0	−1.33132	−0.665660	−	0.746255i	$-0.731850\pi$
$853$	−33167.0	−1.33132	−0.665660	+	0.746255i	$0.731850\pi$
$854$	−24930.0	−0.998931
$855$	−2430.00	−0.0971979
$856$	−7304.00	−0.291642
$857$	10379.0	0.413699	0.206849	−	0.978373i	$-0.433679\pi$
$857$	10379.0	0.413699	0.206849	+	0.978373i	$0.433679\pi$
$858$	3042.00	0.121040
$859$	5053.00	0.200706	0.100353	−	0.994952i	$-0.468003\pi$
$859$	5053.00	0.200706	0.100353	+	0.994952i	$0.468003\pi$
$860$	9680.00	0.383820
$861$	−21105.0	−0.835373
$862$	−16224.0	−0.641058
$863$	−18054.0	−0.712127	−0.356063	−	0.934462i	$-0.615881\pi$
$863$	−18054.0	−0.712127	−0.356063	+	0.934462i	$0.615881\pi$
$864$	−864.000	−0.0340207
$865$	−2830.00	−0.111240
$866$	−1456.00	−0.0571327
$867$	−14064.0	−0.550909
$868$	14760.0	0.577174
$869$	−26559.0	−1.03677
$870$	−3660.00	−0.142627
$871$	−10036.0	−0.390421
$872$	−10208.0	−0.396429
$873$	−711.000	−0.0275644
$874$	15444.0	0.597713
$875$	1875.00	0.0724418
$876$	−11976.0	−0.461908
$877$	32434.0	1.24882	0.624412	−	0.781095i	$-0.285339\pi$
$877$	32434.0	1.24882	0.624412	+	0.781095i	$0.285339\pi$
$878$	1062.00	0.0408209
$879$	−14244.0	−0.546574
$880$	−3120.00	−0.119517
$881$	−35614.0	−1.36194	−0.680968	−	0.732313i	$-0.738441\pi$
$881$	−35614.0	−1.36194	−0.680968	+	0.732313i	$0.738441\pi$
$882$	2124.00	0.0810871
$883$	11548.0	0.440115	0.220057	−	0.975487i	$-0.429376\pi$
$883$	11548.0	0.440115	0.220057	+	0.975487i	$0.429376\pi$
$884$	780.000	0.0296767
$885$	0	0
$886$	−26882.0	−1.01932
$887$	27807.0	1.05261	0.526306	−	0.850295i	$-0.323576\pi$
$887$	27807.0	1.05261	0.526306	+	0.850295i	$0.323576\pi$
$888$	5400.00	0.204068
$889$	16110.0	0.607775
$890$	−4650.00	−0.175133
$891$	3159.00	0.118777
$892$	−1088.00	−0.0408396
$893$	12636.0	0.473514
$894$	−12810.0	−0.479229
$895$	17390.0	0.649479
$896$	1920.00	0.0715878
$897$	5577.00	0.207593
$898$	−12058.0	−0.448085
$899$	30012.0	1.11341
$900$	900.000	0.0333333
$901$	−495.000	−0.0183028
$902$	−36582.0	−1.35039
$903$	21780.0	0.802650
$904$	1712.00	0.0629870
$905$	−12865.0	−0.472538
$906$	−10992.0	−0.403074
$907$	−19190.0	−0.702529	−0.351264	−	0.936276i	$-0.614248\pi$
$907$	−19190.0	−0.702529	−0.351264	+	0.936276i	$0.614248\pi$
$908$	−4760.00	−0.173972
$909$	5832.00	0.212800
$910$	1950.00	0.0710350
$911$	20024.0	0.728238	0.364119	−	0.931352i	$-0.381370\pi$
$911$	20024.0	0.728238	0.364119	+	0.931352i	$0.381370\pi$
$912$	2592.00	0.0941115
$913$	−30108.0	−1.09138
$914$	24102.0	0.872236
$915$	12465.0	0.450361
$916$	200.000	0.00721418
$917$	−22230.0	−0.800544
$918$	810.000	0.0291220
$919$	42119.0	1.51184	0.755918	−	0.654666i	$-0.227191\pi$
$919$	42119.0	1.51184	0.755918	+	0.654666i	$0.227191\pi$
$920$	−5720.00	−0.204981
$921$	8943.00	0.319959
$922$	20198.0	0.721460
$923$	10309.0	0.367633
$924$	−7020.00	−0.249936
$925$	−5625.00	−0.199945
$926$	24894.0	0.883442
$927$	−10260.0	−0.363520
$928$	3904.00	0.138098
$929$	34999.0	1.23604	0.618019	−	0.786163i	$-0.287935\pi$
$929$	34999.0	1.23604	0.618019	+	0.786163i	$0.287935\pi$
$930$	−7380.00	−0.260215
$931$	−6372.00	−0.224311
$932$	−12564.0	−0.441575
$933$	6240.00	0.218959
$934$	4270.00	0.149592
$935$	2925.00	0.102308
$936$	936.000	0.0326860
$937$	16718.0	0.582874	0.291437	−	0.956590i	$-0.405867\pi$
$937$	16718.0	0.582874	0.291437	+	0.956590i	$0.405867\pi$
$938$	23160.0	0.806184
$939$	16854.0	0.585740
$940$	−4680.00	−0.162388
$941$	−30889.0	−1.07009	−0.535044	−	0.844824i	$-0.679705\pi$
$941$	−30889.0	−1.07009	−0.535044	+	0.844824i	$0.679705\pi$
$942$	10596.0	0.366493
$943$	−67067.0	−2.31601
$944$	0	0
$945$	2025.00	0.0697071
$946$	37752.0	1.29749
$947$	−4908.00	−0.168415	−0.0842073	−	0.996448i	$-0.526836\pi$
$947$	−4908.00	−0.168415	−0.0842073	+	0.996448i	$0.526836\pi$
$948$	−8172.00	−0.279973
$949$	12974.0	0.443787
$950$	−2700.00	−0.0922101
$951$	−25548.0	−0.871136
$952$	−1800.00	−0.0612798
$953$	11633.0	0.395414	0.197707	−	0.980261i	$-0.436650\pi$
$953$	11633.0	0.395414	0.197707	+	0.980261i	$0.436650\pi$
$954$	−594.000	−0.0201588
$955$	−12960.0	−0.439137
$956$	−11428.0	−0.386619
$957$	−14274.0	−0.482145
$958$	−12810.0	−0.432017
$959$	−9990.00	−0.336386
$960$	−960.000	−0.0322749
$961$	30725.0	1.03135
$962$	−5850.00	−0.196062
$963$	8217.00	0.274963
$964$	−2648.00	−0.0884713
$965$	13625.0	0.454512
$966$	−12870.0	−0.428660
$967$	7936.00	0.263914	0.131957	−	0.991255i	$-0.457874\pi$
$967$	7936.00	0.263914	0.131957	+	0.991255i	$0.457874\pi$
$968$	−1520.00	−0.0504697
$969$	−2430.00	−0.0805602
$970$	−790.000	−0.0261499
$971$	−6960.00	−0.230028	−0.115014	−	0.993364i	$-0.536691\pi$
$971$	−6960.00	−0.230028	−0.115014	+	0.993364i	$0.536691\pi$
$972$	972.000	0.0320750
$973$	23745.0	0.782353
$974$	−15238.0	−0.501291
$975$	−975.000	−0.0320256
$976$	−13296.0	−0.436060
$977$	35120.0	1.15004	0.575020	−	0.818140i	$-0.304994\pi$
$977$	35120.0	1.15004	0.575020	+	0.818140i	$0.304994\pi$
$978$	−23934.0	−0.782541
$979$	−18135.0	−0.592030
$980$	2360.00	0.0769260
$981$	11484.0	0.373757
$982$	19384.0	0.629907
$983$	7244.00	0.235043	0.117522	−	0.993070i	$-0.462505\pi$
$983$	7244.00	0.235043	0.117522	+	0.993070i	$0.462505\pi$
$984$	−11256.0	−0.364663
$985$	−22820.0	−0.738178
$986$	−3660.00	−0.118213
$987$	−10530.0	−0.339588
$988$	−2808.00	−0.0904194
$989$	69212.0	2.22529
$990$	3510.00	0.112682
$991$	16813.0	0.538933	0.269466	−	0.963010i	$-0.413153\pi$
$991$	16813.0	0.538933	0.269466	+	0.963010i	$0.413153\pi$
$992$	7872.00	0.251952
$993$	33972.0	1.08567
$994$	−23790.0	−0.759128
$995$	−21220.0	−0.676100
$996$	−9264.00	−0.294720
$997$	34112.0	1.08359	0.541794	−	0.840511i	$-0.317745\pi$
$997$	34112.0	1.08359	0.541794	+	0.840511i	$0.317745\pi$
$998$	25988.0	0.824284
$999$	−6075.00	−0.192397

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	390.4.a.c.1.1	✓	1
3.2	odd	2		1170.4.a.m.1.1		1
5.4	even	2		1950.4.a.n.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
390.4.a.c.1.1	✓	1	1.1	even	1	trivial
1170.4.a.m.1.1		1	3.2	odd	2
1950.4.a.n.1.1		1	5.4	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 390 = 2 \cdot 3 \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	390.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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