LMFDB - Embedded newform 390.4.a.i.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:	$0$
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)

$f(q)$

$=$

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

\(q+2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} -5.00000 q^{5} -6.00000 q^{6} +8.00000 q^{7} +8.00000 q^{8} +9.00000 q^{9} -10.0000 q^{10} +12.0000 q^{11} -12.0000 q^{12} +13.0000 q^{13} +16.0000 q^{14} +15.0000 q^{15} +16.0000 q^{16} -42.0000 q^{17} +18.0000 q^{18} -52.0000 q^{19} -20.0000 q^{20} -24.0000 q^{21} +24.0000 q^{22} +132.000 q^{23} -24.0000 q^{24} +25.0000 q^{25} +26.0000 q^{26} -27.0000 q^{27} +32.0000 q^{28} +282.000 q^{29} +30.0000 q^{30} +116.000 q^{31} +32.0000 q^{32} -36.0000 q^{33} -84.0000 q^{34} -40.0000 q^{35} +36.0000 q^{36} +398.000 q^{37} -104.000 q^{38} -39.0000 q^{39} -40.0000 q^{40} +174.000 q^{41} -48.0000 q^{42} -76.0000 q^{43} +48.0000 q^{44} -45.0000 q^{45} +264.000 q^{46} +456.000 q^{47} -48.0000 q^{48} -279.000 q^{49} +50.0000 q^{50} +126.000 q^{51} +52.0000 q^{52} +150.000 q^{53} -54.0000 q^{54} -60.0000 q^{55} +64.0000 q^{56} +156.000 q^{57} +564.000 q^{58} -156.000 q^{59} +60.0000 q^{60} +230.000 q^{61} +232.000 q^{62} +72.0000 q^{63} +64.0000 q^{64} -65.0000 q^{65} -72.0000 q^{66} -592.000 q^{67} -168.000 q^{68} -396.000 q^{69} -80.0000 q^{70} +408.000 q^{71} +72.0000 q^{72} -730.000 q^{73} +796.000 q^{74} -75.0000 q^{75} -208.000 q^{76} +96.0000 q^{77} -78.0000 q^{78} +728.000 q^{79} -80.0000 q^{80} +81.0000 q^{81} +348.000 q^{82} +36.0000 q^{83} -96.0000 q^{84} +210.000 q^{85} -152.000 q^{86} -846.000 q^{87} +96.0000 q^{88} -1482.00 q^{89} -90.0000 q^{90} +104.000 q^{91} +528.000 q^{92} -348.000 q^{93} +912.000 q^{94} +260.000 q^{95} -96.0000 q^{96} +1742.00 q^{97} -558.000 q^{98} +108.000 q^{99} +O(q^{100})\)

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
$7$	8.00000	0.431959	0.215980	−	0.976398i	$-0.430705\pi$
$7$	8.00000	0.431959	0.215980	+	0.976398i	$0.430705\pi$
$8$	8.00000	0.353553
$9$	9.00000	0.333333
$10$	−10.0000	−0.316228
$11$	12.0000	0.328921	0.164461	−	0.986384i	$-0.447412\pi$
$11$	12.0000	0.328921	0.164461	+	0.986384i	$0.447412\pi$
$12$	−12.0000	−0.288675
$13$	13.0000	0.277350
$13$	13.0000	0.277350
$14$	16.0000	0.305441
$15$	15.0000	0.258199
$16$	16.0000	0.250000
$17$	−42.0000	−0.599206	−0.299603	−	0.954064i	$-0.596854\pi$
$17$	−42.0000	−0.599206	−0.299603	+	0.954064i	$0.596854\pi$
$18$	18.0000	0.235702
$19$	−52.0000	−0.627875	−0.313937	−	0.949444i	$-0.601648\pi$
$19$	−52.0000	−0.627875	−0.313937	+	0.949444i	$0.601648\pi$
$20$	−20.0000	−0.223607
$21$	−24.0000	−0.249392
$22$	24.0000	0.232583
$23$	132.000	1.19669	0.598346	−	0.801238i	$-0.295825\pi$
$23$	132.000	1.19669	0.598346	+	0.801238i	$0.295825\pi$
$24$	−24.0000	−0.204124
$25$	25.0000	0.200000
$26$	26.0000	0.196116
$27$	−27.0000	−0.192450
$28$	32.0000	0.215980
$29$	282.000	1.80573	0.902864	−	0.429927i	$-0.141461\pi$
$29$	282.000	1.80573	0.902864	+	0.429927i	$0.141461\pi$
$30$	30.0000	0.182574
$31$	116.000	0.672071	0.336036	−	0.941849i	$-0.390914\pi$
$31$	116.000	0.672071	0.336036	+	0.941849i	$0.390914\pi$
$32$	32.0000	0.176777
$33$	−36.0000	−0.189903
$34$	−84.0000	−0.423702
$35$	−40.0000	−0.193178
$36$	36.0000	0.166667
$37$	398.000	1.76840	0.884200	−	0.467109i	$-0.154704\pi$
$37$	398.000	1.76840	0.884200	+	0.467109i	$0.154704\pi$
$38$	−104.000	−0.443974
$39$	−39.0000	−0.160128
$40$	−40.0000	−0.158114
$41$	174.000	0.662786	0.331393	−	0.943493i	$-0.392481\pi$
$41$	174.000	0.662786	0.331393	+	0.943493i	$0.392481\pi$
$42$	−48.0000	−0.176347
$43$	−76.0000	−0.269532	−0.134766	−	0.990877i	$-0.543028\pi$
$43$	−76.0000	−0.269532	−0.134766	+	0.990877i	$0.543028\pi$
$44$	48.0000	0.164461
$45$	−45.0000	−0.149071
$46$	264.000	0.846189
$47$	456.000	1.41520	0.707600	−	0.706613i	$-0.249778\pi$
$47$	456.000	1.41520	0.707600	+	0.706613i	$0.249778\pi$
$48$	−48.0000	−0.144338
$49$	−279.000	−0.813411
$50$	50.0000	0.141421
$51$	126.000	0.345952
$52$	52.0000	0.138675
$53$	150.000	0.388756	0.194378	−	0.980927i	$-0.437731\pi$
$53$	150.000	0.388756	0.194378	+	0.980927i	$0.437731\pi$
$54$	−54.0000	−0.136083
$55$	−60.0000	−0.147098
$56$	64.0000	0.152721
$57$	156.000	0.362504
$58$	564.000	1.27684
$59$	−156.000	−0.344228	−0.172114	−	0.985077i	$-0.555060\pi$
$59$	−156.000	−0.344228	−0.172114	+	0.985077i	$0.555060\pi$
$60$	60.0000	0.129099
$61$	230.000	0.482762	0.241381	−	0.970430i	$-0.422400\pi$
$61$	230.000	0.482762	0.241381	+	0.970430i	$0.422400\pi$
$62$	232.000	0.475226
$63$	72.0000	0.143986
$64$	64.0000	0.125000
$65$	−65.0000	−0.124035
$66$	−72.0000	−0.134282
$67$	−592.000	−1.07947	−0.539734	−	0.841836i	$-0.681475\pi$
$67$	−592.000	−1.07947	−0.539734	+	0.841836i	$0.681475\pi$
$68$	−168.000	−0.299603
$69$	−396.000	−0.690910
$70$	−80.0000	−0.136598
$71$	408.000	0.681982	0.340991	−	0.940067i	$-0.389238\pi$
$71$	408.000	0.681982	0.340991	+	0.940067i	$0.389238\pi$
$72$	72.0000	0.117851
$73$	−730.000	−1.17041	−0.585206	−	0.810885i	$-0.698986\pi$
$73$	−730.000	−1.17041	−0.585206	+	0.810885i	$0.698986\pi$
$74$	796.000	1.25045
$75$	−75.0000	−0.115470
$76$	−208.000	−0.313937
$77$	96.0000	0.142081
$78$	−78.0000	−0.113228
$79$	728.000	1.03679	0.518395	−	0.855141i	$-0.326530\pi$
$79$	728.000	1.03679	0.518395	+	0.855141i	$0.326530\pi$
$80$	−80.0000	−0.111803
$81$	81.0000	0.111111
$82$	348.000	0.468661
$83$	36.0000	0.0476086	0.0238043	−	0.999717i	$-0.492422\pi$
$83$	36.0000	0.0476086	0.0238043	+	0.999717i	$0.492422\pi$
$84$	−96.0000	−0.124696
$85$	210.000	0.267973
$86$	−152.000	−0.190588
$87$	−846.000	−1.04254
$88$	96.0000	0.116291
$89$	−1482.00	−1.76508	−0.882538	−	0.470242i	$-0.844167\pi$
$89$	−1482.00	−1.76508	−0.882538	+	0.470242i	$0.844167\pi$
$90$	−90.0000	−0.105409
$91$	104.000	0.119804
$92$	528.000	0.598346
$93$	−348.000	−0.388021
$94$	912.000	1.00070
$95$	260.000	0.280794
$96$	−96.0000	−0.102062
$97$	1742.00	1.82344	0.911718	−	0.410816i	$-0.134756\pi$
$97$	1742.00	1.82344	0.911718	+	0.410816i	$0.134756\pi$
$98$	−558.000	−0.575168
$99$	108.000	0.109640
$100$	100.000	0.100000
$101$	−534.000	−0.526089	−0.263044	−	0.964784i	$-0.584727\pi$
$101$	−534.000	−0.526089	−0.263044	+	0.964784i	$0.584727\pi$
$102$	252.000	0.244625
$103$	248.000	0.237244	0.118622	−	0.992939i	$-0.462152\pi$
$103$	248.000	0.237244	0.118622	+	0.992939i	$0.462152\pi$
$104$	104.000	0.0980581
$105$	120.000	0.111531
$106$	300.000	0.274892
$107$	1044.00	0.943246	0.471623	−	0.881800i	$-0.343668\pi$
$107$	1044.00	0.943246	0.471623	+	0.881800i	$0.343668\pi$
$108$	−108.000	−0.0962250
$109$	−1342.00	−1.17927	−0.589634	−	0.807670i	$-0.700728\pi$
$109$	−1342.00	−1.17927	−0.589634	+	0.807670i	$0.700728\pi$
$110$	−120.000	−0.104014
$111$	−1194.00	−1.02099
$112$	128.000	0.107990
$113$	−186.000	−0.154844	−0.0774222	−	0.996998i	$-0.524669\pi$
$113$	−186.000	−0.154844	−0.0774222	+	0.996998i	$0.524669\pi$
$114$	312.000	0.256329
$115$	−660.000	−0.535177
$116$	1128.00	0.902864
$117$	117.000	0.0924500
$118$	−312.000	−0.243406
$119$	−336.000	−0.258833
$120$	120.000	0.0912871
$121$	−1187.00	−0.891811
$122$	460.000	0.341364
$123$	−522.000	−0.382660
$124$	464.000	0.336036
$125$	−125.000	−0.0894427
$126$	144.000	0.101814
$127$	−1096.00	−0.765782	−0.382891	−	0.923794i	$-0.625071\pi$
$127$	−1096.00	−0.765782	−0.382891	+	0.923794i	$0.625071\pi$
$128$	128.000	0.0883883
$129$	228.000	0.155615
$130$	−130.000	−0.0877058
$131$	−1200.00	−0.800340	−0.400170	−	0.916441i	$-0.631049\pi$
$131$	−1200.00	−0.800340	−0.400170	+	0.916441i	$0.631049\pi$
$132$	−144.000	−0.0949514
$133$	−416.000	−0.271216
$134$	−1184.00	−0.763299
$135$	135.000	0.0860663
$136$	−336.000	−0.211851
$137$	534.000	0.333012	0.166506	−	0.986040i	$-0.446751\pi$
$137$	534.000	0.333012	0.166506	+	0.986040i	$0.446751\pi$
$138$	−792.000	−0.488547
$139$	−1756.00	−1.07153	−0.535763	−	0.844369i	$-0.679976\pi$
$139$	−1756.00	−1.07153	−0.535763	+	0.844369i	$0.679976\pi$
$140$	−160.000	−0.0965891
$141$	−1368.00	−0.817067
$142$	816.000	0.482234
$143$	156.000	0.0912264
$144$	144.000	0.0833333
$145$	−1410.00	−0.807546
$146$	−1460.00	−0.827606
$147$	837.000	0.469623
$148$	1592.00	0.884200
$149$	690.000	0.379376	0.189688	−	0.981844i	$-0.439252\pi$
$149$	690.000	0.379376	0.189688	+	0.981844i	$0.439252\pi$
$150$	−150.000	−0.0816497
$151$	308.000	0.165991	0.0829956	−	0.996550i	$-0.473551\pi$
$151$	308.000	0.165991	0.0829956	+	0.996550i	$0.473551\pi$
$152$	−416.000	−0.221987
$153$	−378.000	−0.199735
$154$	192.000	0.100466
$155$	−580.000	−0.300559
$156$	−156.000	−0.0800641
$157$	−490.000	−0.249084	−0.124542	−	0.992214i	$-0.539746\pi$
$157$	−490.000	−0.249084	−0.124542	+	0.992214i	$0.539746\pi$
$158$	1456.00	0.733121
$159$	−450.000	−0.224449
$160$	−160.000	−0.0790569
$161$	1056.00	0.516922
$162$	162.000	0.0785674
$163$	−952.000	−0.457463	−0.228731	−	0.973490i	$-0.573458\pi$
$163$	−952.000	−0.457463	−0.228731	+	0.973490i	$0.573458\pi$
$164$	696.000	0.331393
$165$	180.000	0.0849272
$166$	72.0000	0.0336644
$167$	3480.00	1.61252	0.806259	−	0.591563i	$-0.201489\pi$
$167$	3480.00	1.61252	0.806259	+	0.591563i	$0.201489\pi$
$168$	−192.000	−0.0881733
$169$	169.000	0.0769231
$170$	420.000	0.189485
$171$	−468.000	−0.209292
$172$	−304.000	−0.134766
$173$	−1530.00	−0.672392	−0.336196	−	0.941792i	$-0.609140\pi$
$173$	−1530.00	−0.672392	−0.336196	+	0.941792i	$0.609140\pi$
$174$	−1692.00	−0.737185
$175$	200.000	0.0863919
$176$	192.000	0.0822304
$177$	468.000	0.198740
$178$	−2964.00	−1.24810
$179$	2280.00	0.952040	0.476020	−	0.879434i	$-0.342079\pi$
$179$	2280.00	0.952040	0.476020	+	0.879434i	$0.342079\pi$
$180$	−180.000	−0.0745356
$181$	−2554.00	−1.04882	−0.524412	−	0.851464i	$-0.675715\pi$
$181$	−2554.00	−1.04882	−0.524412	+	0.851464i	$0.675715\pi$
$182$	208.000	0.0847142
$183$	−690.000	−0.278723
$184$	1056.00	0.423094
$185$	−1990.00	−0.790852
$186$	−696.000	−0.274372
$187$	−504.000	−0.197092
$188$	1824.00	0.707600
$189$	−216.000	−0.0831306
$190$	520.000	0.198551
$191$	360.000	0.136381	0.0681903	−	0.997672i	$-0.478278\pi$
$191$	360.000	0.136381	0.0681903	+	0.997672i	$0.478278\pi$
$192$	−192.000	−0.0721688
$193$	−2242.00	−0.836180	−0.418090	−	0.908406i	$-0.637300\pi$
$193$	−2242.00	−0.836180	−0.418090	+	0.908406i	$0.637300\pi$
$194$	3484.00	1.28936
$195$	195.000	0.0716115
$196$	−1116.00	−0.406706
$197$	−318.000	−0.115008	−0.0575040	−	0.998345i	$-0.518314\pi$
$197$	−318.000	−0.115008	−0.0575040	+	0.998345i	$0.518314\pi$
$198$	216.000	0.0775275
$199$	4376.00	1.55883	0.779413	−	0.626510i	$-0.215517\pi$
$199$	4376.00	1.55883	0.779413	+	0.626510i	$0.215517\pi$
$200$	200.000	0.0707107
$201$	1776.00	0.623231
$202$	−1068.00	−0.372001
$203$	2256.00	0.780001
$204$	504.000	0.172976
$205$	−870.000	−0.296407
$206$	496.000	0.167757
$207$	1188.00	0.398897
$208$	208.000	0.0693375
$209$	−624.000	−0.206521
$210$	240.000	0.0788646
$211$	1532.00	0.499845	0.249922	−	0.968266i	$-0.419595\pi$
$211$	1532.00	0.499845	0.249922	+	0.968266i	$0.419595\pi$
$212$	600.000	0.194378
$213$	−1224.00	−0.393742
$214$	2088.00	0.666975
$215$	380.000	0.120539
$216$	−216.000	−0.0680414
$217$	928.000	0.290308
$218$	−2684.00	−0.833869
$219$	2190.00	0.675737
$220$	−240.000	−0.0735491
$221$	−546.000	−0.166190
$222$	−2388.00	−0.721946
$223$	−5848.00	−1.75610	−0.878052	−	0.478566i	$-0.841157\pi$
$223$	−5848.00	−1.75610	−0.878052	+	0.478566i	$0.841157\pi$
$224$	256.000	0.0763604
$225$	225.000	0.0666667
$226$	−372.000	−0.109491
$227$	4884.00	1.42803	0.714014	−	0.700131i	$-0.246875\pi$
$227$	4884.00	1.42803	0.714014	+	0.700131i	$0.246875\pi$
$228$	624.000	0.181252
$229$	−5254.00	−1.51613	−0.758066	−	0.652178i	$-0.773855\pi$
$229$	−5254.00	−1.51613	−0.758066	+	0.652178i	$0.773855\pi$
$230$	−1320.00	−0.378427
$231$	−288.000	−0.0820303
$232$	2256.00	0.638421
$233$	2358.00	0.662994	0.331497	−	0.943456i	$-0.392446\pi$
$233$	2358.00	0.662994	0.331497	+	0.943456i	$0.392446\pi$
$234$	234.000	0.0653720
$235$	−2280.00	−0.632897
$236$	−624.000	−0.172114
$237$	−2184.00	−0.598591
$238$	−672.000	−0.183022
$239$	−2328.00	−0.630066	−0.315033	−	0.949081i	$-0.602016\pi$
$239$	−2328.00	−0.630066	−0.315033	+	0.949081i	$0.602016\pi$
$240$	240.000	0.0645497
$241$	−3670.00	−0.980936	−0.490468	−	0.871459i	$-0.663174\pi$
$241$	−3670.00	−0.980936	−0.490468	+	0.871459i	$0.663174\pi$
$242$	−2374.00	−0.630605
$243$	−243.000	−0.0641500
$244$	920.000	0.241381
$245$	1395.00	0.363768
$246$	−1044.00	−0.270581
$247$	−676.000	−0.174141
$248$	928.000	0.237613
$249$	−108.000	−0.0274868
$250$	−250.000	−0.0632456
$251$	−6312.00	−1.58729	−0.793645	−	0.608381i	$-0.791819\pi$
$251$	−6312.00	−1.58729	−0.793645	+	0.608381i	$0.791819\pi$
$252$	288.000	0.0719932
$253$	1584.00	0.393617
$254$	−2192.00	−0.541489
$255$	−630.000	−0.154714
$256$	256.000	0.0625000
$257$	6006.00	1.45776	0.728879	−	0.684642i	$-0.240042\pi$
$257$	6006.00	1.45776	0.728879	+	0.684642i	$0.240042\pi$
$258$	456.000	0.110036
$259$	3184.00	0.763877
$260$	−260.000	−0.0620174
$261$	2538.00	0.601909
$262$	−2400.00	−0.565926
$263$	5220.00	1.22387	0.611937	−	0.790906i	$-0.290390\pi$
$263$	5220.00	1.22387	0.611937	+	0.790906i	$0.290390\pi$
$264$	−288.000	−0.0671408
$265$	−750.000	−0.173857
$266$	−832.000	−0.191779
$267$	4446.00	1.01907
$268$	−2368.00	−0.539734
$269$	−126.000	−0.0285589	−0.0142795	−	0.999898i	$-0.504545\pi$
$269$	−126.000	−0.0285589	−0.0142795	+	0.999898i	$0.504545\pi$
$270$	270.000	0.0608581
$271$	−3244.00	−0.727155	−0.363577	−	0.931564i	$-0.618445\pi$
$271$	−3244.00	−0.727155	−0.363577	+	0.931564i	$0.618445\pi$
$272$	−672.000	−0.149801
$273$	−312.000	−0.0691689
$274$	1068.00	0.235475
$275$	300.000	0.0657843
$276$	−1584.00	−0.345455
$277$	−7666.00	−1.66284	−0.831418	−	0.555648i	$-0.812470\pi$
$277$	−7666.00	−1.66284	−0.831418	+	0.555648i	$0.812470\pi$
$278$	−3512.00	−0.757683
$279$	1044.00	0.224024
$280$	−320.000	−0.0682988
$281$	−1458.00	−0.309527	−0.154763	−	0.987952i	$-0.549462\pi$
$281$	−1458.00	−0.309527	−0.154763	+	0.987952i	$0.549462\pi$
$282$	−2736.00	−0.577753
$283$	188.000	0.0394892	0.0197446	−	0.999805i	$-0.493715\pi$
$283$	188.000	0.0394892	0.0197446	+	0.999805i	$0.493715\pi$
$284$	1632.00	0.340991
$285$	−780.000	−0.162117
$286$	312.000	0.0645068
$287$	1392.00	0.286297
$288$	288.000	0.0589256
$289$	−3149.00	−0.640953
$290$	−2820.00	−0.571021
$291$	−5226.00	−1.05276
$292$	−2920.00	−0.585206
$293$	3738.00	0.745312	0.372656	−	0.927970i	$-0.378447\pi$
$293$	3738.00	0.745312	0.372656	+	0.927970i	$0.378447\pi$
$294$	1674.00	0.332074
$295$	780.000	0.153944
$296$	3184.00	0.625224
$297$	−324.000	−0.0633010
$298$	1380.00	0.268259
$299$	1716.00	0.331902
$300$	−300.000	−0.0577350
$301$	−608.000	−0.116427
$302$	616.000	0.117374
$303$	1602.00	0.303738
$304$	−832.000	−0.156969
$305$	−1150.00	−0.215898
$306$	−756.000	−0.141234
$307$	8216.00	1.52740	0.763700	−	0.645571i	$-0.223381\pi$
$307$	8216.00	1.52740	0.763700	+	0.645571i	$0.223381\pi$
$308$	384.000	0.0710404
$309$	−744.000	−0.136973
$310$	−1160.00	−0.212528
$311$	6288.00	1.14649	0.573247	−	0.819382i	$-0.305683\pi$
$311$	6288.00	1.14649	0.573247	+	0.819382i	$0.305683\pi$
$312$	−312.000	−0.0566139
$313$	−2446.00	−0.441713	−0.220856	−	0.975306i	$-0.570885\pi$
$313$	−2446.00	−0.441713	−0.220856	+	0.975306i	$0.570885\pi$
$314$	−980.000	−0.176129
$315$	−360.000	−0.0643927
$316$	2912.00	0.518395
$317$	−5430.00	−0.962079	−0.481040	−	0.876699i	$-0.659741\pi$
$317$	−5430.00	−0.962079	−0.481040	+	0.876699i	$0.659741\pi$
$318$	−900.000	−0.158709
$319$	3384.00	0.593942
$320$	−320.000	−0.0559017
$321$	−3132.00	−0.544583
$322$	2112.00	0.365519
$323$	2184.00	0.376226
$324$	324.000	0.0555556
$325$	325.000	0.0554700
$326$	−1904.00	−0.323475
$327$	4026.00	0.680851
$328$	1392.00	0.234330
$329$	3648.00	0.611309
$330$	360.000	0.0600526
$331$	1628.00	0.270341	0.135171	−	0.990822i	$-0.456842\pi$
$331$	1628.00	0.270341	0.135171	+	0.990822i	$0.456842\pi$
$332$	144.000	0.0238043
$333$	3582.00	0.589467
$334$	6960.00	1.14022
$335$	2960.00	0.482752
$336$	−384.000	−0.0623480
$337$	5546.00	0.896468	0.448234	−	0.893916i	$-0.352053\pi$
$337$	5546.00	0.896468	0.448234	+	0.893916i	$0.352053\pi$
$338$	338.000	0.0543928
$339$	558.000	0.0893994
$340$	840.000	0.133986
$341$	1392.00	0.221059
$342$	−936.000	−0.147991
$343$	−4976.00	−0.783320
$344$	−608.000	−0.0952941
$345$	1980.00	0.308984
$346$	−3060.00	−0.475453
$347$	2772.00	0.428844	0.214422	−	0.976741i	$-0.431213\pi$
$347$	2772.00	0.428844	0.214422	+	0.976741i	$0.431213\pi$
$348$	−3384.00	−0.521269
$349$	10538.0	1.61629	0.808146	−	0.588982i	$-0.200471\pi$
$349$	10538.0	1.61629	0.808146	+	0.588982i	$0.200471\pi$
$350$	400.000	0.0610883
$351$	−351.000	−0.0533761
$352$	384.000	0.0581456
$353$	942.000	0.142033	0.0710164	−	0.997475i	$-0.477376\pi$
$353$	942.000	0.142033	0.0710164	+	0.997475i	$0.477376\pi$
$354$	936.000	0.140531
$355$	−2040.00	−0.304991
$356$	−5928.00	−0.882538
$357$	1008.00	0.149437
$358$	4560.00	0.673194
$359$	6000.00	0.882083	0.441042	−	0.897487i	$-0.354609\pi$
$359$	6000.00	0.882083	0.441042	+	0.897487i	$0.354609\pi$
$360$	−360.000	−0.0527046
$361$	−4155.00	−0.605773
$362$	−5108.00	−0.741631
$363$	3561.00	0.514887
$364$	416.000	0.0599020
$365$	3650.00	0.523424
$366$	−1380.00	−0.197087
$367$	8264.00	1.17541	0.587707	−	0.809073i	$-0.300031\pi$
$367$	8264.00	1.17541	0.587707	+	0.809073i	$0.300031\pi$
$368$	2112.00	0.299173
$369$	1566.00	0.220929
$370$	−3980.00	−0.559217
$371$	1200.00	0.167927
$372$	−1392.00	−0.194010
$373$	2654.00	0.368415	0.184208	−	0.982887i	$-0.441028\pi$
$373$	2654.00	0.368415	0.184208	+	0.982887i	$0.441028\pi$
$374$	−1008.00	−0.139365
$375$	375.000	0.0516398
$376$	3648.00	0.500349
$377$	3666.00	0.500819
$378$	−432.000	−0.0587822
$379$	−3076.00	−0.416896	−0.208448	−	0.978033i	$-0.566841\pi$
$379$	−3076.00	−0.416896	−0.208448	+	0.978033i	$0.566841\pi$
$380$	1040.00	0.140397
$381$	3288.00	0.442124
$382$	720.000	0.0964356
$383$	12864.0	1.71624	0.858120	−	0.513450i	$-0.171633\pi$
$383$	12864.0	1.71624	0.858120	+	0.513450i	$0.171633\pi$
$384$	−384.000	−0.0510310
$385$	−480.000	−0.0635404
$386$	−4484.00	−0.591268
$387$	−684.000	−0.0898441
$388$	6968.00	0.911718
$389$	−10974.0	−1.43034	−0.715172	−	0.698948i	$-0.753652\pi$
$389$	−10974.0	−1.43034	−0.715172	+	0.698948i	$0.753652\pi$
$390$	390.000	0.0506370
$391$	−5544.00	−0.717064
$392$	−2232.00	−0.287584
$393$	3600.00	0.462076
$394$	−636.000	−0.0813229
$395$	−3640.00	−0.463667
$396$	432.000	0.0548202
$397$	8582.00	1.08493	0.542466	−	0.840078i	$-0.317491\pi$
$397$	8582.00	1.08493	0.542466	+	0.840078i	$0.317491\pi$
$398$	8752.00	1.10226
$399$	1248.00	0.156587
$400$	400.000	0.0500000
$401$	2142.00	0.266749	0.133375	−	0.991066i	$-0.457419\pi$
$401$	2142.00	0.266749	0.133375	+	0.991066i	$0.457419\pi$
$402$	3552.00	0.440691
$403$	1508.00	0.186399
$404$	−2136.00	−0.263044
$405$	−405.000	−0.0496904
$406$	4512.00	0.551544
$407$	4776.00	0.581665
$408$	1008.00	0.122312
$409$	6122.00	0.740131	0.370065	−	0.929006i	$-0.379335\pi$
$409$	6122.00	0.740131	0.370065	+	0.929006i	$0.379335\pi$
$410$	−1740.00	−0.209591
$411$	−1602.00	−0.192265
$412$	992.000	0.118622
$413$	−1248.00	−0.148693
$414$	2376.00	0.282063
$415$	−180.000	−0.0212912
$416$	416.000	0.0490290
$417$	5268.00	0.618645
$418$	−1248.00	−0.146033
$419$	−13056.0	−1.52226	−0.761130	−	0.648599i	$-0.775355\pi$
$419$	−13056.0	−1.52226	−0.761130	+	0.648599i	$0.775355\pi$
$420$	480.000	0.0557657
$421$	−7582.00	−0.877729	−0.438865	−	0.898553i	$-0.644619\pi$
$421$	−7582.00	−0.877729	−0.438865	+	0.898553i	$0.644619\pi$
$422$	3064.00	0.353444
$423$	4104.00	0.471734
$424$	1200.00	0.137446
$425$	−1050.00	−0.119841
$426$	−2448.00	−0.278418
$427$	1840.00	0.208534
$428$	4176.00	0.471623
$429$	−468.000	−0.0526696
$430$	760.000	0.0852336
$431$	−3888.00	−0.434521	−0.217260	−	0.976114i	$-0.569712\pi$
$431$	−3888.00	−0.434521	−0.217260	+	0.976114i	$0.569712\pi$
$432$	−432.000	−0.0481125
$433$	−2710.00	−0.300772	−0.150386	−	0.988627i	$-0.548052\pi$
$433$	−2710.00	−0.300772	−0.150386	+	0.988627i	$0.548052\pi$
$434$	1856.00	0.205278
$435$	4230.00	0.466237
$436$	−5368.00	−0.589634
$437$	−6864.00	−0.751372
$438$	4380.00	0.477818
$439$	6008.00	0.653180	0.326590	−	0.945166i	$-0.394100\pi$
$439$	6008.00	0.653180	0.326590	+	0.945166i	$0.394100\pi$
$440$	−480.000	−0.0520071
$441$	−2511.00	−0.271137
$442$	−1092.00	−0.117514
$443$	13884.0	1.48905	0.744525	−	0.667595i	$-0.232676\pi$
$443$	13884.0	1.48905	0.744525	+	0.667595i	$0.232676\pi$
$444$	−4776.00	−0.510493
$445$	7410.00	0.789366
$446$	−11696.0	−1.24175
$447$	−2070.00	−0.219033
$448$	512.000	0.0539949
$449$	−8706.00	−0.915059	−0.457530	−	0.889194i	$-0.651266\pi$
$449$	−8706.00	−0.915059	−0.457530	+	0.889194i	$0.651266\pi$
$450$	450.000	0.0471405
$451$	2088.00	0.218005
$452$	−744.000	−0.0774222
$453$	−924.000	−0.0958351
$454$	9768.00	1.00977
$455$	−520.000	−0.0535780
$456$	1248.00	0.128164
$457$	5078.00	0.519779	0.259889	−	0.965638i	$-0.416314\pi$
$457$	5078.00	0.519779	0.259889	+	0.965638i	$0.416314\pi$
$458$	−10508.0	−1.07207
$459$	1134.00	0.115317
$460$	−2640.00	−0.267588
$461$	−14934.0	−1.50878	−0.754388	−	0.656429i	$-0.772066\pi$
$461$	−14934.0	−1.50878	−0.754388	+	0.656429i	$0.772066\pi$
$462$	−576.000	−0.0580042
$463$	−3448.00	−0.346095	−0.173048	−	0.984913i	$-0.555361\pi$
$463$	−3448.00	−0.346095	−0.173048	+	0.984913i	$0.555361\pi$
$464$	4512.00	0.451432
$465$	1740.00	0.173528
$466$	4716.00	0.468808
$467$	−10332.0	−1.02379	−0.511893	−	0.859049i	$-0.671056\pi$
$467$	−10332.0	−1.02379	−0.511893	+	0.859049i	$0.671056\pi$
$468$	468.000	0.0462250
$469$	−4736.00	−0.466286
$470$	−4560.00	−0.447526
$471$	1470.00	0.143809
$472$	−1248.00	−0.121703
$473$	−912.000	−0.0886550
$474$	−4368.00	−0.423268
$475$	−1300.00	−0.125575
$476$	−1344.00	−0.129416
$477$	1350.00	0.129585
$478$	−4656.00	−0.445524
$479$	−2328.00	−0.222065	−0.111032	−	0.993817i	$-0.535416\pi$
$479$	−2328.00	−0.222065	−0.111032	+	0.993817i	$0.535416\pi$
$480$	480.000	0.0456435
$481$	5174.00	0.490466
$482$	−7340.00	−0.693626
$483$	−3168.00	−0.298445
$484$	−4748.00	−0.445905
$485$	−8710.00	−0.815465
$486$	−486.000	−0.0453609
$487$	13160.0	1.22451	0.612255	−	0.790660i	$-0.290263\pi$
$487$	13160.0	1.22451	0.612255	+	0.790660i	$0.290263\pi$
$488$	1840.00	0.170682
$489$	2856.00	0.264116
$490$	2790.00	0.257223
$491$	8688.00	0.798542	0.399271	−	0.916833i	$-0.369263\pi$
$491$	8688.00	0.798542	0.399271	+	0.916833i	$0.369263\pi$
$492$	−2088.00	−0.191330
$493$	−11844.0	−1.08200
$494$	−1352.00	−0.123136
$495$	−540.000	−0.0490327
$496$	1856.00	0.168018
$497$	3264.00	0.294588
$498$	−216.000	−0.0194361
$499$	−15460.0	−1.38694	−0.693472	−	0.720484i	$-0.743920\pi$
$499$	−15460.0	−1.38694	−0.693472	+	0.720484i	$0.743920\pi$
$500$	−500.000	−0.0447214
$501$	−10440.0	−0.930988
$502$	−12624.0	−1.12238
$503$	−7116.00	−0.630789	−0.315394	−	0.948961i	$-0.602137\pi$
$503$	−7116.00	−0.630789	−0.315394	+	0.948961i	$0.602137\pi$
$504$	576.000	0.0509069
$505$	2670.00	0.235274
$506$	3168.00	0.278330
$507$	−507.000	−0.0444116
$508$	−4384.00	−0.382891
$509$	−14694.0	−1.27957	−0.639784	−	0.768555i	$-0.720976\pi$
$509$	−14694.0	−1.27957	−0.639784	+	0.768555i	$0.720976\pi$
$510$	−1260.00	−0.109399
$511$	−5840.00	−0.505570
$512$	512.000	0.0441942
$513$	1404.00	0.120835
$514$	12012.0	1.03079
$515$	−1240.00	−0.106099
$516$	912.000	0.0778073
$517$	5472.00	0.465490
$518$	6368.00	0.540143
$519$	4590.00	0.388205
$520$	−520.000	−0.0438529
$521$	−13782.0	−1.15893	−0.579463	−	0.814999i	$-0.696738\pi$
$521$	−13782.0	−1.15893	−0.579463	+	0.814999i	$0.696738\pi$
$522$	5076.00	0.425614
$523$	−5596.00	−0.467870	−0.233935	−	0.972252i	$-0.575160\pi$
$523$	−5596.00	−0.467870	−0.233935	+	0.972252i	$0.575160\pi$
$524$	−4800.00	−0.400170
$525$	−600.000	−0.0498784
$526$	10440.0	0.865410
$527$	−4872.00	−0.402709
$528$	−576.000	−0.0474757
$529$	5257.00	0.432070
$530$	−1500.00	−0.122936
$531$	−1404.00	−0.114743
$532$	−1664.00	−0.135608
$533$	2262.00	0.183824
$534$	8892.00	0.720589
$535$	−5220.00	−0.421832
$536$	−4736.00	−0.381649
$537$	−6840.00	−0.549661
$538$	−252.000	−0.0201942
$539$	−3348.00	−0.267548
$540$	540.000	0.0430331
$541$	7322.00	0.581881	0.290940	−	0.956741i	$-0.406032\pi$
$541$	7322.00	0.581881	0.290940	+	0.956741i	$0.406032\pi$
$542$	−6488.00	−0.514176
$543$	7662.00	0.605539
$544$	−1344.00	−0.105926
$545$	6710.00	0.527385
$546$	−624.000	−0.0489098
$547$	−15652.0	−1.22346	−0.611729	−	0.791068i	$-0.709526\pi$
$547$	−15652.0	−1.22346	−0.611729	+	0.791068i	$0.709526\pi$
$548$	2136.00	0.166506
$549$	2070.00	0.160921
$550$	600.000	0.0465165
$551$	−14664.0	−1.13377
$552$	−3168.00	−0.244274
$553$	5824.00	0.447851
$554$	−15332.0	−1.17580
$555$	5970.00	0.456599
$556$	−7024.00	−0.535763
$557$	13458.0	1.02376	0.511879	−	0.859057i	$-0.328949\pi$
$557$	13458.0	1.02376	0.511879	+	0.859057i	$0.328949\pi$
$558$	2088.00	0.158409
$559$	−988.000	−0.0747548
$560$	−640.000	−0.0482945
$561$	1512.00	0.113791
$562$	−2916.00	−0.218868
$563$	−18108.0	−1.35553	−0.677763	−	0.735280i	$-0.737050\pi$
$563$	−18108.0	−1.35553	−0.677763	+	0.735280i	$0.737050\pi$
$564$	−5472.00	−0.408533
$565$	930.000	0.0692485
$566$	376.000	0.0279231
$567$	648.000	0.0479955
$568$	3264.00	0.241117
$569$	−20118.0	−1.48223	−0.741116	−	0.671377i	$-0.765703\pi$
$569$	−20118.0	−1.48223	−0.741116	+	0.671377i	$0.765703\pi$
$570$	−1560.00	−0.114634
$571$	−4156.00	−0.304594	−0.152297	−	0.988335i	$-0.548667\pi$
$571$	−4156.00	−0.304594	−0.152297	+	0.988335i	$0.548667\pi$
$572$	624.000	0.0456132
$573$	−1080.00	−0.0787393
$574$	2784.00	0.202442
$575$	3300.00	0.239338
$576$	576.000	0.0416667
$577$	−7378.00	−0.532323	−0.266161	−	0.963929i	$-0.585755\pi$
$577$	−7378.00	−0.532323	−0.266161	+	0.963929i	$0.585755\pi$
$578$	−6298.00	−0.453222
$579$	6726.00	0.482769
$580$	−5640.00	−0.403773
$581$	288.000	0.0205650
$582$	−10452.0	−0.744415
$583$	1800.00	0.127870
$584$	−5840.00	−0.413803
$585$	−585.000	−0.0413449
$586$	7476.00	0.527015
$587$	6372.00	0.448042	0.224021	−	0.974584i	$-0.428082\pi$
$587$	6372.00	0.448042	0.224021	+	0.974584i	$0.428082\pi$
$588$	3348.00	0.234812
$589$	−6032.00	−0.421977
$590$	1560.00	0.108855
$591$	954.000	0.0663999
$592$	6368.00	0.442100
$593$	2382.00	0.164953	0.0824764	−	0.996593i	$-0.473717\pi$
$593$	2382.00	0.164953	0.0824764	+	0.996593i	$0.473717\pi$
$594$	−648.000	−0.0447605
$595$	1680.00	0.115753
$596$	2760.00	0.189688
$597$	−13128.0	−0.899989
$598$	3432.00	0.234690
$599$	10536.0	0.718680	0.359340	−	0.933207i	$-0.383002\pi$
$599$	10536.0	0.718680	0.359340	+	0.933207i	$0.383002\pi$
$600$	−600.000	−0.0408248
$601$	13610.0	0.923733	0.461866	−	0.886949i	$-0.347180\pi$
$601$	13610.0	0.923733	0.461866	+	0.886949i	$0.347180\pi$
$602$	−1216.00	−0.0823263
$603$	−5328.00	−0.359822
$604$	1232.00	0.0829956
$605$	5935.00	0.398830
$606$	3204.00	0.214775
$607$	13736.0	0.918496	0.459248	−	0.888308i	$-0.348119\pi$
$607$	13736.0	0.918496	0.459248	+	0.888308i	$0.348119\pi$
$608$	−1664.00	−0.110994
$609$	−6768.00	−0.450334
$610$	−2300.00	−0.152663
$611$	5928.00	0.392506
$612$	−1512.00	−0.0998676
$613$	−10882.0	−0.716998	−0.358499	−	0.933530i	$-0.616711\pi$
$613$	−10882.0	−0.716998	−0.358499	+	0.933530i	$0.616711\pi$
$614$	16432.0	1.08004
$615$	2610.00	0.171131
$616$	768.000	0.0502331
$617$	4902.00	0.319849	0.159925	−	0.987129i	$-0.448875\pi$
$617$	4902.00	0.319849	0.159925	+	0.987129i	$0.448875\pi$
$618$	−1488.00	−0.0968546
$619$	−20692.0	−1.34359	−0.671795	−	0.740737i	$-0.734476\pi$
$619$	−20692.0	−1.34359	−0.671795	+	0.740737i	$0.734476\pi$
$620$	−2320.00	−0.150280
$621$	−3564.00	−0.230303
$622$	12576.0	0.810694
$623$	−11856.0	−0.762441
$624$	−624.000	−0.0400320
$625$	625.000	0.0400000
$626$	−4892.00	−0.312338
$627$	1872.00	0.119235
$628$	−1960.00	−0.124542
$629$	−16716.0	−1.05964
$630$	−720.000	−0.0455325
$631$	14348.0	0.905206	0.452603	−	0.891712i	$-0.350496\pi$
$631$	14348.0	0.905206	0.452603	+	0.891712i	$0.350496\pi$
$632$	5824.00	0.366561
$633$	−4596.00	−0.288585
$634$	−10860.0	−0.680293
$635$	5480.00	0.342468
$636$	−1800.00	−0.112224
$637$	−3627.00	−0.225600
$638$	6768.00	0.419981
$639$	3672.00	0.227327
$640$	−640.000	−0.0395285
$641$	−1182.00	−0.0728334	−0.0364167	−	0.999337i	$-0.511594\pi$
$641$	−1182.00	−0.0728334	−0.0364167	+	0.999337i	$0.511594\pi$
$642$	−6264.00	−0.385078
$643$	−26080.0	−1.59953	−0.799763	−	0.600316i	$-0.795041\pi$
$643$	−26080.0	−1.59953	−0.799763	+	0.600316i	$0.795041\pi$
$644$	4224.00	0.258461
$645$	−1140.00	−0.0695930
$646$	4368.00	0.266032
$647$	−21852.0	−1.32781	−0.663903	−	0.747819i	$-0.731101\pi$
$647$	−21852.0	−1.32781	−0.663903	+	0.747819i	$0.731101\pi$
$648$	648.000	0.0392837
$649$	−1872.00	−0.113224
$650$	650.000	0.0392232
$651$	−2784.00	−0.167609
$652$	−3808.00	−0.228731
$653$	−12018.0	−0.720215	−0.360108	−	0.932911i	$-0.617260\pi$
$653$	−12018.0	−0.720215	−0.360108	+	0.932911i	$0.617260\pi$
$654$	8052.00	0.481434
$655$	6000.00	0.357923
$656$	2784.00	0.165697
$657$	−6570.00	−0.390137
$658$	7296.00	0.432261
$659$	23400.0	1.38321	0.691604	−	0.722277i	$-0.256904\pi$
$659$	23400.0	1.38321	0.691604	+	0.722277i	$0.256904\pi$
$660$	720.000	0.0424636
$661$	11354.0	0.668108	0.334054	−	0.942554i	$-0.391583\pi$
$661$	11354.0	0.668108	0.334054	+	0.942554i	$0.391583\pi$
$662$	3256.00	0.191160
$663$	1638.00	0.0959497
$664$	288.000	0.0168322
$665$	2080.00	0.121292
$666$	7164.00	0.416816
$667$	37224.0	2.16090
$668$	13920.0	0.806259
$669$	17544.0	1.01389
$670$	5920.00	0.341358
$671$	2760.00	0.158791
$672$	−768.000	−0.0440867
$673$	−23374.0	−1.33878	−0.669392	−	0.742909i	$-0.733445\pi$
$673$	−23374.0	−1.33878	−0.669392	+	0.742909i	$0.733445\pi$
$674$	11092.0	0.633899
$675$	−675.000	−0.0384900
$676$	676.000	0.0384615
$677$	−22866.0	−1.29810	−0.649049	−	0.760747i	$-0.724833\pi$
$677$	−22866.0	−1.29810	−0.649049	+	0.760747i	$0.724833\pi$
$678$	1116.00	0.0632149
$679$	13936.0	0.787650
$680$	1680.00	0.0947427
$681$	−14652.0	−0.824473
$682$	2784.00	0.156312
$683$	−13164.0	−0.737491	−0.368746	−	0.929530i	$-0.620213\pi$
$683$	−13164.0	−0.737491	−0.368746	+	0.929530i	$0.620213\pi$
$684$	−1872.00	−0.104646
$685$	−2670.00	−0.148928
$686$	−9952.00	−0.553891
$687$	15762.0	0.875339
$688$	−1216.00	−0.0673831
$689$	1950.00	0.107822
$690$	3960.00	0.218485
$691$	17012.0	0.936566	0.468283	−	0.883579i	$-0.344873\pi$
$691$	17012.0	0.936566	0.468283	+	0.883579i	$0.344873\pi$
$692$	−6120.00	−0.336196
$693$	864.000	0.0473602
$694$	5544.00	0.303238
$695$	8780.00	0.479201
$696$	−6768.00	−0.368592
$697$	−7308.00	−0.397145
$698$	21076.0	1.14289
$699$	−7074.00	−0.382780
$700$	800.000	0.0431959
$701$	−17910.0	−0.964981	−0.482490	−	0.875901i	$-0.660268\pi$
$701$	−17910.0	−0.964981	−0.482490	+	0.875901i	$0.660268\pi$
$702$	−702.000	−0.0377426
$703$	−20696.0	−1.11033
$704$	768.000	0.0411152
$705$	6840.00	0.365403
$706$	1884.00	0.100432
$707$	−4272.00	−0.227249
$708$	1872.00	0.0993702
$709$	−10918.0	−0.578327	−0.289164	−	0.957280i	$-0.593377\pi$
$709$	−10918.0	−0.578327	−0.289164	+	0.957280i	$0.593377\pi$
$710$	−4080.00	−0.215662
$711$	6552.00	0.345597
$712$	−11856.0	−0.624048
$713$	15312.0	0.804262
$714$	2016.00	0.105668
$715$	−780.000	−0.0407977
$716$	9120.00	0.476020
$717$	6984.00	0.363769
$718$	12000.0	0.623727
$719$	−26064.0	−1.35191	−0.675955	−	0.736943i	$-0.736269\pi$
$719$	−26064.0	−1.35191	−0.675955	+	0.736943i	$0.736269\pi$
$720$	−720.000	−0.0372678
$721$	1984.00	0.102480
$722$	−8310.00	−0.428347
$723$	11010.0	0.566343
$724$	−10216.0	−0.524412
$725$	7050.00	0.361145
$726$	7122.00	0.364080
$727$	1688.00	0.0861134	0.0430567	−	0.999073i	$-0.486290\pi$
$727$	1688.00	0.0861134	0.0430567	+	0.999073i	$0.486290\pi$
$728$	832.000	0.0423571
$729$	729.000	0.0370370
$730$	7300.00	0.370117
$731$	3192.00	0.161505
$732$	−2760.00	−0.139361
$733$	−20626.0	−1.03934	−0.519672	−	0.854366i	$-0.673946\pi$
$733$	−20626.0	−1.03934	−0.519672	+	0.854366i	$0.673946\pi$
$734$	16528.0	0.831144
$735$	−4185.00	−0.210022
$736$	4224.00	0.211547
$737$	−7104.00	−0.355060
$738$	3132.00	0.156220
$739$	16292.0	0.810976	0.405488	−	0.914100i	$-0.367102\pi$
$739$	16292.0	0.810976	0.405488	+	0.914100i	$0.367102\pi$
$740$	−7960.00	−0.395426
$741$	2028.00	0.100540
$742$	2400.00	0.118742
$743$	24048.0	1.18740	0.593698	−	0.804688i	$-0.297667\pi$
$743$	24048.0	1.18740	0.593698	+	0.804688i	$0.297667\pi$
$744$	−2784.00	−0.137186
$745$	−3450.00	−0.169662
$746$	5308.00	0.260509
$747$	324.000	0.0158695
$748$	−2016.00	−0.0985458
$749$	8352.00	0.407444
$750$	750.000	0.0365148
$751$	15248.0	0.740889	0.370444	−	0.928855i	$-0.379205\pi$
$751$	15248.0	0.740889	0.370444	+	0.928855i	$0.379205\pi$
$752$	7296.00	0.353800
$753$	18936.0	0.916423
$754$	7332.00	0.354132
$755$	−1540.00	−0.0742336
$756$	−864.000	−0.0415653
$757$	4286.00	0.205782	0.102891	−	0.994693i	$-0.467191\pi$
$757$	4286.00	0.205782	0.102891	+	0.994693i	$0.467191\pi$
$758$	−6152.00	−0.294790
$759$	−4752.00	−0.227255
$760$	2080.00	0.0992757
$761$	21294.0	1.01433	0.507166	−	0.861848i	$-0.330693\pi$
$761$	21294.0	1.01433	0.507166	+	0.861848i	$0.330693\pi$
$762$	6576.00	0.312629
$763$	−10736.0	−0.509396
$764$	1440.00	0.0681903
$765$	1890.00	0.0893243
$766$	25728.0	1.21356
$767$	−2028.00	−0.0954718
$768$	−768.000	−0.0360844
$769$	−7198.00	−0.337538	−0.168769	−	0.985656i	$-0.553979\pi$
$769$	−7198.00	−0.337538	−0.168769	+	0.985656i	$0.553979\pi$
$770$	−960.000	−0.0449299
$771$	−18018.0	−0.841637
$772$	−8968.00	−0.418090
$773$	11322.0	0.526810	0.263405	−	0.964685i	$-0.415154\pi$
$773$	11322.0	0.526810	0.263405	+	0.964685i	$0.415154\pi$
$774$	−1368.00	−0.0635294
$775$	2900.00	0.134414
$776$	13936.0	0.644682
$777$	−9552.00	−0.441025
$778$	−21948.0	−1.01141
$779$	−9048.00	−0.416147
$780$	780.000	0.0358057
$781$	4896.00	0.224318
$782$	−11088.0	−0.507041
$783$	−7614.00	−0.347512
$784$	−4464.00	−0.203353
$785$	2450.00	0.111394
$786$	7200.00	0.326737
$787$	13424.0	0.608023	0.304011	−	0.952668i	$-0.401674\pi$
$787$	13424.0	0.608023	0.304011	+	0.952668i	$0.401674\pi$
$788$	−1272.00	−0.0575040
$789$	−15660.0	−0.706604
$790$	−7280.00	−0.327862
$791$	−1488.00	−0.0668865
$792$	864.000	0.0387638
$793$	2990.00	0.133894
$794$	17164.0	0.767163
$795$	2250.00	0.100376
$796$	17504.0	0.779413
$797$	−23202.0	−1.03119	−0.515594	−	0.856833i	$-0.672429\pi$
$797$	−23202.0	−1.03119	−0.515594	+	0.856833i	$0.672429\pi$
$798$	2496.00	0.110724
$799$	−19152.0	−0.847996
$800$	800.000	0.0353553
$801$	−13338.0	−0.588358
$802$	4284.00	0.188620
$803$	−8760.00	−0.384973
$804$	7104.00	0.311615
$805$	−5280.00	−0.231175
$806$	3016.00	0.131804
$807$	378.000	0.0164885
$808$	−4272.00	−0.186001
$809$	−38814.0	−1.68681	−0.843404	−	0.537280i	$-0.819452\pi$
$809$	−38814.0	−1.68681	−0.843404	+	0.537280i	$0.819452\pi$
$810$	−810.000	−0.0351364
$811$	−9292.00	−0.402326	−0.201163	−	0.979558i	$-0.564472\pi$
$811$	−9292.00	−0.402326	−0.201163	+	0.979558i	$0.564472\pi$
$812$	9024.00	0.390000
$813$	9732.00	0.419823
$814$	9552.00	0.411299
$815$	4760.00	0.204583
$816$	2016.00	0.0864879
$817$	3952.00	0.169233
$818$	12244.0	0.523351
$819$	936.000	0.0399347
$820$	−3480.00	−0.148204
$821$	40338.0	1.71475	0.857373	−	0.514696i	$-0.172095\pi$
$821$	40338.0	1.71475	0.857373	+	0.514696i	$0.172095\pi$
$822$	−3204.00	−0.135952
$823$	−33352.0	−1.41261	−0.706305	−	0.707908i	$-0.749639\pi$
$823$	−33352.0	−1.41261	−0.706305	+	0.707908i	$0.749639\pi$
$824$	1984.00	0.0838785
$825$	−900.000	−0.0379806
$826$	−2496.00	−0.105142
$827$	35076.0	1.47486	0.737432	−	0.675422i	$-0.236038\pi$
$827$	35076.0	1.47486	0.737432	+	0.675422i	$0.236038\pi$
$828$	4752.00	0.199449
$829$	32678.0	1.36906	0.684532	−	0.728983i	$-0.260007\pi$
$829$	32678.0	1.36906	0.684532	+	0.728983i	$0.260007\pi$
$830$	−360.000	−0.0150552
$831$	22998.0	0.960038
$832$	832.000	0.0346688
$833$	11718.0	0.487401
$834$	10536.0	0.437448
$835$	−17400.0	−0.721140
$836$	−2496.00	−0.103261
$837$	−3132.00	−0.129340
$838$	−26112.0	−1.07640
$839$	11424.0	0.470084	0.235042	−	0.971985i	$-0.424477\pi$
$839$	11424.0	0.470084	0.235042	+	0.971985i	$0.424477\pi$
$840$	960.000	0.0394323
$841$	55135.0	2.26065
$842$	−15164.0	−0.620648
$843$	4374.00	0.178705
$844$	6128.00	0.249922
$845$	−845.000	−0.0344010
$846$	8208.00	0.333566
$847$	−9496.00	−0.385226
$848$	2400.00	0.0971891
$849$	−564.000	−0.0227991
$850$	−2100.00	−0.0847405
$851$	52536.0	2.11623
$852$	−4896.00	−0.196871
$853$	15374.0	0.617111	0.308556	−	0.951206i	$-0.400154\pi$
$853$	15374.0	0.617111	0.308556	+	0.951206i	$0.400154\pi$
$854$	3680.00	0.147456
$855$	2340.00	0.0935980
$856$	8352.00	0.333488
$857$	−17010.0	−0.678005	−0.339003	−	0.940785i	$-0.610090\pi$
$857$	−17010.0	−0.678005	−0.339003	+	0.940785i	$0.610090\pi$
$858$	−936.000	−0.0372430
$859$	−19516.0	−0.775177	−0.387589	−	0.921832i	$-0.626692\pi$
$859$	−19516.0	−0.775177	−0.387589	+	0.921832i	$0.626692\pi$
$860$	1520.00	0.0602693
$861$	−4176.00	−0.165293
$862$	−7776.00	−0.307252
$863$	−672.000	−0.0265065	−0.0132533	−	0.999912i	$-0.504219\pi$
$863$	−672.000	−0.0265065	−0.0132533	+	0.999912i	$0.504219\pi$
$864$	−864.000	−0.0340207
$865$	7650.00	0.300703
$866$	−5420.00	−0.212678
$867$	9447.00	0.370054
$868$	3712.00	0.145154
$869$	8736.00	0.341022
$870$	8460.00	0.329679
$871$	−7696.00	−0.299390
$872$	−10736.0	−0.416934
$873$	15678.0	0.607812
$874$	−13728.0	−0.531300
$875$	−1000.00	−0.0386356
$876$	8760.00	0.337869
$877$	35390.0	1.36264	0.681320	−	0.731986i	$-0.261406\pi$
$877$	35390.0	1.36264	0.681320	+	0.731986i	$0.261406\pi$
$878$	12016.0	0.461868
$879$	−11214.0	−0.430306
$880$	−960.000	−0.0367745
$881$	−10206.0	−0.390294	−0.195147	−	0.980774i	$-0.562518\pi$
$881$	−10206.0	−0.390294	−0.195147	+	0.980774i	$0.562518\pi$
$882$	−5022.00	−0.191723
$883$	21884.0	0.834038	0.417019	−	0.908898i	$-0.363075\pi$
$883$	21884.0	0.834038	0.417019	+	0.908898i	$0.363075\pi$
$884$	−2184.00	−0.0830949
$885$	−2340.00	−0.0888794
$886$	27768.0	1.05292
$887$	9588.00	0.362946	0.181473	−	0.983396i	$-0.441913\pi$
$887$	9588.00	0.362946	0.181473	+	0.983396i	$0.441913\pi$
$888$	−9552.00	−0.360973
$889$	−8768.00	−0.330787
$890$	14820.0	0.558166
$891$	972.000	0.0365468
$892$	−23392.0	−0.878052
$893$	−23712.0	−0.888569
$894$	−4140.00	−0.154880
$895$	−11400.0	−0.425765
$896$	1024.00	0.0381802
$897$	−5148.00	−0.191624
$898$	−17412.0	−0.647045
$899$	32712.0	1.21358
$900$	900.000	0.0333333
$901$	−6300.00	−0.232945
$902$	4176.00	0.154153
$903$	1824.00	0.0672192
$904$	−1488.00	−0.0547457
$905$	12770.0	0.469049
$906$	−1848.00	−0.0677657
$907$	−27124.0	−0.992985	−0.496493	−	0.868041i	$-0.665379\pi$
$907$	−27124.0	−0.992985	−0.496493	+	0.868041i	$0.665379\pi$
$908$	19536.0	0.714014
$909$	−4806.00	−0.175363
$910$	−1040.00	−0.0378853
$911$	3168.00	0.115215	0.0576073	−	0.998339i	$-0.481653\pi$
$911$	3168.00	0.115215	0.0576073	+	0.998339i	$0.481653\pi$
$912$	2496.00	0.0906259
$913$	432.000	0.0156595
$914$	10156.0	0.367539
$915$	3450.00	0.124649
$916$	−21016.0	−0.758066
$917$	−9600.00	−0.345714
$918$	2268.00	0.0815416
$919$	19784.0	0.710135	0.355067	−	0.934841i	$-0.384458\pi$
$919$	19784.0	0.710135	0.355067	+	0.934841i	$0.384458\pi$
$920$	−5280.00	−0.189214
$921$	−24648.0	−0.881845
$922$	−29868.0	−1.06687
$923$	5304.00	0.189148
$924$	−1152.00	−0.0410152
$925$	9950.00	0.353680
$926$	−6896.00	−0.244726
$927$	2232.00	0.0790814
$928$	9024.00	0.319210
$929$	32454.0	1.14616	0.573079	−	0.819500i	$-0.305749\pi$
$929$	32454.0	1.14616	0.573079	+	0.819500i	$0.305749\pi$
$930$	3480.00	0.122703
$931$	14508.0	0.510720
$932$	9432.00	0.331497
$933$	−18864.0	−0.661929
$934$	−20664.0	−0.723926
$935$	2520.00	0.0881420
$936$	936.000	0.0326860
$937$	−34198.0	−1.19232	−0.596158	−	0.802867i	$-0.703307\pi$
$937$	−34198.0	−1.19232	−0.596158	+	0.802867i	$0.703307\pi$
$938$	−9472.00	−0.329714
$939$	7338.00	0.255023
$940$	−9120.00	−0.316449
$941$	6090.00	0.210976	0.105488	−	0.994421i	$-0.466360\pi$
$941$	6090.00	0.210976	0.105488	+	0.994421i	$0.466360\pi$
$942$	2940.00	0.101688
$943$	22968.0	0.793151
$944$	−2496.00	−0.0860571
$945$	1080.00	0.0371771
$946$	−1824.00	−0.0626885
$947$	12324.0	0.422889	0.211445	−	0.977390i	$-0.432183\pi$
$947$	12324.0	0.422889	0.211445	+	0.977390i	$0.432183\pi$
$948$	−8736.00	−0.299295
$949$	−9490.00	−0.324614
$950$	−2600.00	−0.0887949
$951$	16290.0	0.555457
$952$	−2688.00	−0.0915111
$953$	−5034.00	−0.171109	−0.0855547	−	0.996333i	$-0.527266\pi$
$953$	−5034.00	−0.171109	−0.0855547	+	0.996333i	$0.527266\pi$
$954$	2700.00	0.0916307
$955$	−1800.00	−0.0609912
$956$	−9312.00	−0.315033
$957$	−10152.0	−0.342913
$958$	−4656.00	−0.157024
$959$	4272.00	0.143848
$960$	960.000	0.0322749
$961$	−16335.0	−0.548320
$962$	10348.0	0.346812
$963$	9396.00	0.314415
$964$	−14680.0	−0.490468
$965$	11210.0	0.373951
$966$	−6336.00	−0.211033
$967$	−21256.0	−0.706874	−0.353437	−	0.935458i	$-0.614987\pi$
$967$	−21256.0	−0.706874	−0.353437	+	0.935458i	$0.614987\pi$
$968$	−9496.00	−0.315303
$969$	−6552.00	−0.217214
$970$	−17420.0	−0.576621
$971$	11832.0	0.391047	0.195524	−	0.980699i	$-0.437359\pi$
$971$	11832.0	0.391047	0.195524	+	0.980699i	$0.437359\pi$
$972$	−972.000	−0.0320750
$973$	−14048.0	−0.462855
$974$	26320.0	0.865860
$975$	−975.000	−0.0320256
$976$	3680.00	0.120691
$977$	−34386.0	−1.12600	−0.563002	−	0.826456i	$-0.690354\pi$
$977$	−34386.0	−1.12600	−0.563002	+	0.826456i	$0.690354\pi$
$978$	5712.00	0.186758
$979$	−17784.0	−0.580571
$980$	5580.00	0.181884
$981$	−12078.0	−0.393090
$982$	17376.0	0.564654
$983$	10752.0	0.348866	0.174433	−	0.984669i	$-0.444191\pi$
$983$	10752.0	0.348866	0.174433	+	0.984669i	$0.444191\pi$
$984$	−4176.00	−0.135291
$985$	1590.00	0.0514331
$986$	−23688.0	−0.765091
$987$	−10944.0	−0.352940
$988$	−2704.00	−0.0870705
$989$	−10032.0	−0.322547
$990$	−1080.00	−0.0346714
$991$	32672.0	1.04729	0.523643	−	0.851938i	$-0.324573\pi$
$991$	32672.0	1.04729	0.523643	+	0.851938i	$0.324573\pi$
$992$	3712.00	0.118807
$993$	−4884.00	−0.156082
$994$	6528.00	0.208305
$995$	−21880.0	−0.697128
$996$	−432.000	−0.0137434
$997$	−34.0000	−0.00108003	−0.000540015	−	1.00000i	$-0.500172\pi$
$997$	−34.0000	−0.00108003	−0.000540015		1.00000i	$0.500172\pi$
$998$	−30920.0	−0.980717
$999$	−10746.0	−0.340329

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
390.4.a.i.1.1	✓	1	1.1	even	1	trivial
1170.4.a.g.1.1		1	3.2	odd	2
1950.4.a.e.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

Related objects

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