LMFDB - Embedded newform 690.4.a.g.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [690,4,Mod(1,690)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(690, 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("690.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 690 = 2 \cdot 3 \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	690.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:	$40.7113179040$
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	$\chi$	$=$	690.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

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

\(q+2.00000 q^{2} +3.00000 q^{3} +4.00000 q^{4} -5.00000 q^{5} +6.00000 q^{6} -5.00000 q^{7} +8.00000 q^{8} +9.00000 q^{9} -10.0000 q^{10} -58.0000 q^{11} +12.0000 q^{12} +10.0000 q^{13} -10.0000 q^{14} -15.0000 q^{15} +16.0000 q^{16} +27.0000 q^{17} +18.0000 q^{18} -154.000 q^{19} -20.0000 q^{20} -15.0000 q^{21} -116.000 q^{22} +23.0000 q^{23} +24.0000 q^{24} +25.0000 q^{25} +20.0000 q^{26} +27.0000 q^{27} -20.0000 q^{28} -205.000 q^{29} -30.0000 q^{30} -103.000 q^{31} +32.0000 q^{32} -174.000 q^{33} +54.0000 q^{34} +25.0000 q^{35} +36.0000 q^{36} +143.000 q^{37} -308.000 q^{38} +30.0000 q^{39} -40.0000 q^{40} -447.000 q^{41} -30.0000 q^{42} +264.000 q^{43} -232.000 q^{44} -45.0000 q^{45} +46.0000 q^{46} +128.000 q^{47} +48.0000 q^{48} -318.000 q^{49} +50.0000 q^{50} +81.0000 q^{51} +40.0000 q^{52} -521.000 q^{53} +54.0000 q^{54} +290.000 q^{55} -40.0000 q^{56} -462.000 q^{57} -410.000 q^{58} +565.000 q^{59} -60.0000 q^{60} -492.000 q^{61} -206.000 q^{62} -45.0000 q^{63} +64.0000 q^{64} -50.0000 q^{65} -348.000 q^{66} +371.000 q^{67} +108.000 q^{68} +69.0000 q^{69} +50.0000 q^{70} -65.0000 q^{71} +72.0000 q^{72} +530.000 q^{73} +286.000 q^{74} +75.0000 q^{75} -616.000 q^{76} +290.000 q^{77} +60.0000 q^{78} -740.000 q^{79} -80.0000 q^{80} +81.0000 q^{81} -894.000 q^{82} -457.000 q^{83} -60.0000 q^{84} -135.000 q^{85} +528.000 q^{86} -615.000 q^{87} -464.000 q^{88} -144.000 q^{89} -90.0000 q^{90} -50.0000 q^{91} +92.0000 q^{92} -309.000 q^{93} +256.000 q^{94} +770.000 q^{95} +96.0000 q^{96} -38.0000 q^{97} -636.000 q^{98} -522.000 q^{99} +O(q^{100})\)

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
$7$	−5.00000	−0.269975	−0.134987	−	0.990847i	$-0.543099\pi$
$7$	−5.00000	−0.269975	−0.134987	+	0.990847i	$0.543099\pi$
$8$	8.00000	0.353553
$9$	9.00000	0.333333
$10$	−10.0000	−0.316228
$11$	−58.0000	−1.58979	−0.794894	−	0.606749i	$-0.792473\pi$
$11$	−58.0000	−1.58979	−0.794894	+	0.606749i	$0.792473\pi$
$12$	12.0000	0.288675
$13$	10.0000	0.213346	0.106673	−	0.994294i	$-0.465980\pi$
$13$	10.0000	0.213346	0.106673	+	0.994294i	$0.465980\pi$
$14$	−10.0000	−0.190901
$15$	−15.0000	−0.258199
$16$	16.0000	0.250000
$17$	27.0000	0.385204	0.192602	−	0.981277i	$-0.438307\pi$
$17$	27.0000	0.385204	0.192602	+	0.981277i	$0.438307\pi$
$18$	18.0000	0.235702
$19$	−154.000	−1.85947	−0.929737	−	0.368223i	$-0.879966\pi$
$19$	−154.000	−1.85947	−0.929737	+	0.368223i	$0.879966\pi$
$20$	−20.0000	−0.223607
$21$	−15.0000	−0.155870
$22$	−116.000	−1.12415
$23$	23.0000	0.208514
$23$	23.0000	0.208514
$24$	24.0000	0.204124
$25$	25.0000	0.200000
$26$	20.0000	0.150859
$27$	27.0000	0.192450
$28$	−20.0000	−0.134987
$29$	−205.000	−1.31267	−0.656337	−	0.754468i	$-0.727895\pi$
$29$	−205.000	−1.31267	−0.656337	+	0.754468i	$0.727895\pi$
$30$	−30.0000	−0.182574
$31$	−103.000	−0.596753	−0.298377	−	0.954448i	$-0.596445\pi$
$31$	−103.000	−0.596753	−0.298377	+	0.954448i	$0.596445\pi$
$32$	32.0000	0.176777
$33$	−174.000	−0.917864
$34$	54.0000	0.272380
$35$	25.0000	0.120736
$36$	36.0000	0.166667
$37$	143.000	0.635380	0.317690	−	0.948195i	$-0.397093\pi$
$37$	143.000	0.635380	0.317690	+	0.948195i	$0.397093\pi$
$38$	−308.000	−1.31485
$39$	30.0000	0.123176
$40$	−40.0000	−0.158114
$41$	−447.000	−1.70267	−0.851337	−	0.524618i	$-0.824208\pi$
$41$	−447.000	−1.70267	−0.851337	+	0.524618i	$0.824208\pi$
$42$	−30.0000	−0.110217
$43$	264.000	0.936270	0.468135	−	0.883657i	$-0.344926\pi$
$43$	264.000	0.936270	0.468135	+	0.883657i	$0.344926\pi$
$44$	−232.000	−0.794894
$45$	−45.0000	−0.149071
$46$	46.0000	0.147442
$47$	128.000	0.397249	0.198625	−	0.980076i	$-0.436353\pi$
$47$	128.000	0.397249	0.198625	+	0.980076i	$0.436353\pi$
$48$	48.0000	0.144338
$49$	−318.000	−0.927114
$50$	50.0000	0.141421
$51$	81.0000	0.222397
$52$	40.0000	0.106673
$53$	−521.000	−1.35028	−0.675140	−	0.737690i	$-0.735917\pi$
$53$	−521.000	−1.35028	−0.675140	+	0.737690i	$0.735917\pi$
$54$	54.0000	0.136083
$55$	290.000	0.710974
$56$	−40.0000	−0.0954504
$57$	−462.000	−1.07357
$58$	−410.000	−0.928201
$59$	565.000	1.24672	0.623362	−	0.781933i	$-0.285766\pi$
$59$	565.000	1.24672	0.623362	+	0.781933i	$0.285766\pi$
$60$	−60.0000	−0.129099
$61$	−492.000	−1.03269	−0.516345	−	0.856380i	$-0.672708\pi$
$61$	−492.000	−1.03269	−0.516345	+	0.856380i	$0.672708\pi$
$62$	−206.000	−0.421968
$63$	−45.0000	−0.0899915
$64$	64.0000	0.125000
$65$	−50.0000	−0.0954113
$66$	−348.000	−0.649028
$67$	371.000	0.676491	0.338245	−	0.941058i	$-0.390167\pi$
$67$	371.000	0.676491	0.338245	+	0.941058i	$0.390167\pi$
$68$	108.000	0.192602
$69$	69.0000	0.120386
$70$	50.0000	0.0853735
$71$	−65.0000	−0.108649	−0.0543245	−	0.998523i	$-0.517301\pi$
$71$	−65.0000	−0.108649	−0.0543245	+	0.998523i	$0.517301\pi$
$72$	72.0000	0.117851
$73$	530.000	0.849751	0.424875	−	0.905252i	$-0.360318\pi$
$73$	530.000	0.849751	0.424875	+	0.905252i	$0.360318\pi$
$74$	286.000	0.449281
$75$	75.0000	0.115470
$76$	−616.000	−0.929737
$77$	290.000	0.429202
$78$	60.0000	0.0870982
$79$	−740.000	−1.05388	−0.526940	−	0.849903i	$-0.676661\pi$
$79$	−740.000	−1.05388	−0.526940	+	0.849903i	$0.676661\pi$
$80$	−80.0000	−0.111803
$81$	81.0000	0.111111
$82$	−894.000	−1.20397
$83$	−457.000	−0.604365	−0.302182	−	0.953250i	$-0.597715\pi$
$83$	−457.000	−0.604365	−0.302182	+	0.953250i	$0.597715\pi$
$84$	−60.0000	−0.0779350
$85$	−135.000	−0.172268
$86$	528.000	0.662043
$87$	−615.000	−0.757873
$88$	−464.000	−0.562075
$89$	−144.000	−0.171505	−0.0857526	−	0.996316i	$-0.527329\pi$
$89$	−144.000	−0.171505	−0.0857526	+	0.996316i	$0.527329\pi$
$90$	−90.0000	−0.105409
$91$	−50.0000	−0.0575981
$92$	92.0000	0.104257
$93$	−309.000	−0.344536
$94$	256.000	0.280898
$95$	770.000	0.831582
$96$	96.0000	0.102062
$97$	−38.0000	−0.0397764	−0.0198882	−	0.999802i	$-0.506331\pi$
$97$	−38.0000	−0.0397764	−0.0198882	+	0.999802i	$0.506331\pi$
$98$	−636.000	−0.655568
$99$	−522.000	−0.529929
$100$	100.000	0.100000
$101$	−425.000	−0.418704	−0.209352	−	0.977840i	$-0.567135\pi$
$101$	−425.000	−0.418704	−0.209352	+	0.977840i	$0.567135\pi$
$102$	162.000	0.157259
$103$	−8.00000	−0.00765304	−0.00382652	−	0.999993i	$-0.501218\pi$
$103$	−8.00000	−0.00765304	−0.00382652	+	0.999993i	$0.501218\pi$
$104$	80.0000	0.0754293
$105$	75.0000	0.0697071
$106$	−1042.00	−0.954792
$107$	−649.000	−0.586366	−0.293183	−	0.956056i	$-0.594715\pi$
$107$	−649.000	−0.586366	−0.293183	+	0.956056i	$0.594715\pi$
$108$	108.000	0.0962250
$109$	1378.00	1.21090	0.605452	−	0.795882i	$-0.292992\pi$
$109$	1378.00	1.21090	0.605452	+	0.795882i	$0.292992\pi$
$110$	580.000	0.502735
$111$	429.000	0.366837
$112$	−80.0000	−0.0674937
$113$	−985.000	−0.820009	−0.410004	−	0.912084i	$-0.634473\pi$
$113$	−985.000	−0.820009	−0.410004	+	0.912084i	$0.634473\pi$
$114$	−924.000	−0.759127
$115$	−115.000	−0.0932505
$116$	−820.000	−0.656337
$117$	90.0000	0.0711154
$118$	1130.00	0.881567
$119$	−135.000	−0.103995
$120$	−120.000	−0.0912871
$121$	2033.00	1.52742
$122$	−984.000	−0.730223
$123$	−1341.00	−0.983040
$124$	−412.000	−0.298377
$125$	−125.000	−0.0894427
$126$	−90.0000	−0.0636336
$127$	1112.00	0.776961	0.388480	−	0.921457i	$-0.373000\pi$
$127$	1112.00	0.776961	0.388480	+	0.921457i	$0.373000\pi$
$128$	128.000	0.0883883
$129$	792.000	0.540556
$130$	−100.000	−0.0674660
$131$	2956.00	1.97150	0.985752	−	0.168208i	$-0.0537979\pi$
$131$	2956.00	1.97150	0.985752	+	0.168208i	$0.0537979\pi$
$132$	−696.000	−0.458932
$133$	770.000	0.502011
$134$	742.000	0.478351
$135$	−135.000	−0.0860663
$136$	216.000	0.136190
$137$	2094.00	1.30586	0.652929	−	0.757419i	$-0.273540\pi$
$137$	2094.00	1.30586	0.652929	+	0.757419i	$0.273540\pi$
$138$	138.000	0.0851257
$139$	−917.000	−0.559561	−0.279780	−	0.960064i	$-0.590262\pi$
$139$	−917.000	−0.559561	−0.279780	+	0.960064i	$0.590262\pi$
$140$	100.000	0.0603682
$141$	384.000	0.229352
$142$	−130.000	−0.0768265
$143$	−580.000	−0.339175
$144$	144.000	0.0833333
$145$	1025.00	0.587046
$146$	1060.00	0.600865
$147$	−954.000	−0.535269
$148$	572.000	0.317690
$149$	−3100.00	−1.70444	−0.852221	−	0.523182i	$-0.824745\pi$
$149$	−3100.00	−1.70444	−0.852221	+	0.523182i	$0.824745\pi$
$150$	150.000	0.0816497
$151$	1580.00	0.851514	0.425757	−	0.904838i	$-0.360008\pi$
$151$	1580.00	0.851514	0.425757	+	0.904838i	$0.360008\pi$
$152$	−1232.00	−0.657424
$153$	243.000	0.128401
$154$	580.000	0.303492
$155$	515.000	0.266876
$156$	120.000	0.0615878
$157$	−2867.00	−1.45740	−0.728699	−	0.684834i	$-0.759875\pi$
$157$	−2867.00	−1.45740	−0.728699	+	0.684834i	$0.759875\pi$
$158$	−1480.00	−0.745206
$159$	−1563.00	−0.779585
$160$	−160.000	−0.0790569
$161$	−115.000	−0.0562936
$162$	162.000	0.0785674
$163$	1786.00	0.858223	0.429111	−	0.903252i	$-0.358827\pi$
$163$	1786.00	0.858223	0.429111	+	0.903252i	$0.358827\pi$
$164$	−1788.00	−0.851337
$165$	870.000	0.410481
$166$	−914.000	−0.427350
$167$	−464.000	−0.215002	−0.107501	−	0.994205i	$-0.534285\pi$
$167$	−464.000	−0.215002	−0.107501	+	0.994205i	$0.534285\pi$
$168$	−120.000	−0.0551083
$169$	−2097.00	−0.954483
$170$	−270.000	−0.121812
$171$	−1386.00	−0.619825
$172$	1056.00	0.468135
$173$	1228.00	0.539671	0.269836	−	0.962906i	$-0.413031\pi$
$173$	1228.00	0.539671	0.269836	+	0.962906i	$0.413031\pi$
$174$	−1230.00	−0.535897
$175$	−125.000	−0.0539949
$176$	−928.000	−0.397447
$177$	1695.00	0.719797
$178$	−288.000	−0.121273
$179$	−2176.00	−0.908614	−0.454307	−	0.890845i	$-0.650113\pi$
$179$	−2176.00	−0.908614	−0.454307	+	0.890845i	$0.650113\pi$
$180$	−180.000	−0.0745356
$181$	−1964.00	−0.806536	−0.403268	−	0.915082i	$-0.632126\pi$
$181$	−1964.00	−0.806536	−0.403268	+	0.915082i	$0.632126\pi$
$182$	−100.000	−0.0407280
$183$	−1476.00	−0.596224
$184$	184.000	0.0737210
$185$	−715.000	−0.284151
$186$	−618.000	−0.243623
$187$	−1566.00	−0.612392
$188$	512.000	0.198625
$189$	−135.000	−0.0519566
$190$	1540.00	0.588018
$191$	−688.000	−0.260638	−0.130319	−	0.991472i	$-0.541600\pi$
$191$	−688.000	−0.260638	−0.130319	+	0.991472i	$0.541600\pi$
$192$	192.000	0.0721688
$193$	3188.00	1.18900	0.594501	−	0.804095i	$-0.297350\pi$
$193$	3188.00	1.18900	0.594501	+	0.804095i	$0.297350\pi$
$194$	−76.0000	−0.0281262
$195$	−150.000	−0.0550858
$196$	−1272.00	−0.463557
$197$	2078.00	0.751530	0.375765	−	0.926715i	$-0.377380\pi$
$197$	2078.00	0.751530	0.375765	+	0.926715i	$0.377380\pi$
$198$	−1044.00	−0.374716
$199$	1786.00	0.636212	0.318106	−	0.948055i	$-0.396953\pi$
$199$	1786.00	0.636212	0.318106	+	0.948055i	$0.396953\pi$
$200$	200.000	0.0707107
$201$	1113.00	0.390572
$202$	−850.000	−0.296068
$203$	1025.00	0.354389
$204$	324.000	0.111199
$205$	2235.00	0.761459
$206$	−16.0000	−0.00541152
$207$	207.000	0.0695048
$208$	160.000	0.0533366
$209$	8932.00	2.95617
$210$	150.000	0.0492904
$211$	−2495.00	−0.814042	−0.407021	−	0.913419i	$-0.633432\pi$
$211$	−2495.00	−0.814042	−0.407021	+	0.913419i	$0.633432\pi$
$212$	−2084.00	−0.675140
$213$	−195.000	−0.0627285
$214$	−1298.00	−0.414624
$215$	−1320.00	−0.418713
$216$	216.000	0.0680414
$217$	515.000	0.161108
$218$	2756.00	0.856238
$219$	1590.00	0.490604
$220$	1160.00	0.355487
$221$	270.000	0.0821817
$222$	858.000	0.259393
$223$	2798.00	0.840215	0.420107	−	0.907474i	$-0.361992\pi$
$223$	2798.00	0.840215	0.420107	+	0.907474i	$0.361992\pi$
$224$	−160.000	−0.0477252
$225$	225.000	0.0666667
$226$	−1970.00	−0.579834
$227$	−6364.00	−1.86076	−0.930382	−	0.366591i	$-0.880525\pi$
$227$	−6364.00	−1.86076	−0.930382	+	0.366591i	$0.880525\pi$
$228$	−1848.00	−0.536784
$229$	−3916.00	−1.13003	−0.565014	−	0.825081i	$-0.691129\pi$
$229$	−3916.00	−1.13003	−0.565014	+	0.825081i	$0.691129\pi$
$230$	−230.000	−0.0659380
$231$	870.000	0.247800
$232$	−1640.00	−0.464100
$233$	2676.00	0.752406	0.376203	−	0.926537i	$-0.377230\pi$
$233$	2676.00	0.752406	0.376203	+	0.926537i	$0.377230\pi$
$234$	180.000	0.0502862
$235$	−640.000	−0.177655
$236$	2260.00	0.623362
$237$	−2220.00	−0.608458
$238$	−270.000	−0.0735357
$239$	4121.00	1.11534	0.557668	−	0.830064i	$-0.311696\pi$
$239$	4121.00	1.11534	0.557668	+	0.830064i	$0.311696\pi$
$240$	−240.000	−0.0645497
$241$	1042.00	0.278511	0.139255	−	0.990256i	$-0.455529\pi$
$241$	1042.00	0.278511	0.139255	+	0.990256i	$0.455529\pi$
$242$	4066.00	1.08005
$243$	243.000	0.0641500
$244$	−1968.00	−0.516345
$245$	1590.00	0.414618
$246$	−2682.00	−0.695114
$247$	−1540.00	−0.396712
$248$	−824.000	−0.210984
$249$	−1371.00	−0.348930
$250$	−250.000	−0.0632456
$251$	2902.00	0.729771	0.364886	−	0.931052i	$-0.381108\pi$
$251$	2902.00	0.729771	0.364886	+	0.931052i	$0.381108\pi$
$252$	−180.000	−0.0449958
$253$	−1334.00	−0.331494
$254$	2224.00	0.549394
$255$	−405.000	−0.0994592
$256$	256.000	0.0625000
$257$	−3048.00	−0.739802	−0.369901	−	0.929071i	$-0.620608\pi$
$257$	−3048.00	−0.739802	−0.369901	+	0.929071i	$0.620608\pi$
$258$	1584.00	0.382231
$259$	−715.000	−0.171536
$260$	−200.000	−0.0477057
$261$	−1845.00	−0.437558
$262$	5912.00	1.39406
$263$	2233.00	0.523546	0.261773	−	0.965129i	$-0.415693\pi$
$263$	2233.00	0.523546	0.261773	+	0.965129i	$0.415693\pi$
$264$	−1392.00	−0.324514
$265$	2605.00	0.603864
$266$	1540.00	0.354975
$267$	−432.000	−0.0990186
$268$	1484.00	0.338245
$269$	−1503.00	−0.340667	−0.170334	−	0.985386i	$-0.554485\pi$
$269$	−1503.00	−0.340667	−0.170334	+	0.985386i	$0.554485\pi$
$270$	−270.000	−0.0608581
$271$	7687.00	1.72307	0.861535	−	0.507698i	$-0.169503\pi$
$271$	7687.00	1.72307	0.861535	+	0.507698i	$0.169503\pi$
$272$	432.000	0.0963009
$273$	−150.000	−0.0332543
$274$	4188.00	0.923381
$275$	−1450.00	−0.317957
$276$	276.000	0.0601929
$277$	6438.00	1.39647	0.698235	−	0.715869i	$-0.253969\pi$
$277$	6438.00	1.39647	0.698235	+	0.715869i	$0.253969\pi$
$278$	−1834.00	−0.395669
$279$	−927.000	−0.198918
$280$	200.000	0.0426867
$281$	−8878.00	−1.88476	−0.942379	−	0.334547i	$-0.891417\pi$
$281$	−8878.00	−1.88476	−0.942379	+	0.334547i	$0.891417\pi$
$282$	768.000	0.162176
$283$	4445.00	0.933667	0.466834	−	0.884345i	$-0.345395\pi$
$283$	4445.00	0.933667	0.466834	+	0.884345i	$0.345395\pi$
$284$	−260.000	−0.0543245
$285$	2310.00	0.480114
$286$	−1160.00	−0.239833
$287$	2235.00	0.459679
$288$	288.000	0.0589256
$289$	−4184.00	−0.851618
$290$	2050.00	0.415104
$291$	−114.000	−0.0229649
$292$	2120.00	0.424875
$293$	6461.00	1.28824	0.644122	−	0.764923i	$-0.277223\pi$
$293$	6461.00	1.28824	0.644122	+	0.764923i	$0.277223\pi$
$294$	−1908.00	−0.378493
$295$	−2825.00	−0.557552
$296$	1144.00	0.224641
$297$	−1566.00	−0.305955
$298$	−6200.00	−1.20522
$299$	230.000	0.0444858
$300$	300.000	0.0577350
$301$	−1320.00	−0.252769
$302$	3160.00	0.602111
$303$	−1275.00	−0.241739
$304$	−2464.00	−0.464869
$305$	2460.00	0.461833
$306$	486.000	0.0907934
$307$	5914.00	1.09945	0.549723	−	0.835347i	$-0.314733\pi$
$307$	5914.00	1.09945	0.549723	+	0.835347i	$0.314733\pi$
$308$	1160.00	0.214601
$309$	−24.0000	−0.00441849
$310$	1030.00	0.188710
$311$	−6992.00	−1.27486	−0.637428	−	0.770510i	$-0.720001\pi$
$311$	−6992.00	−1.27486	−0.637428	+	0.770510i	$0.720001\pi$
$312$	240.000	0.0435491
$313$	7805.00	1.40947	0.704736	−	0.709470i	$-0.251065\pi$
$313$	7805.00	1.40947	0.704736	+	0.709470i	$0.251065\pi$
$314$	−5734.00	−1.03054
$315$	225.000	0.0402454
$316$	−2960.00	−0.526940
$317$	−10272.0	−1.81998	−0.909989	−	0.414632i	$-0.863910\pi$
$317$	−10272.0	−1.81998	−0.909989	+	0.414632i	$0.863910\pi$
$318$	−3126.00	−0.551250
$319$	11890.0	2.08687
$320$	−320.000	−0.0559017
$321$	−1947.00	−0.338539
$322$	−230.000	−0.0398056
$323$	−4158.00	−0.716276
$324$	324.000	0.0555556
$325$	250.000	0.0426692
$326$	3572.00	0.606855
$327$	4134.00	0.699115
$328$	−3576.00	−0.601986
$329$	−640.000	−0.107247
$330$	1740.00	0.290254
$331$	5469.00	0.908167	0.454084	−	0.890959i	$-0.349967\pi$
$331$	5469.00	0.908167	0.454084	+	0.890959i	$0.349967\pi$
$332$	−1828.00	−0.302182
$333$	1287.00	0.211793
$334$	−928.000	−0.152030
$335$	−1855.00	−0.302536
$336$	−240.000	−0.0389675
$337$	6226.00	1.00639	0.503193	−	0.864174i	$-0.332159\pi$
$337$	6226.00	1.00639	0.503193	+	0.864174i	$0.332159\pi$
$338$	−4194.00	−0.674922
$339$	−2955.00	−0.473432
$340$	−540.000	−0.0861342
$341$	5974.00	0.948710
$342$	−2772.00	−0.438282
$343$	3305.00	0.520272
$344$	2112.00	0.331022
$345$	−345.000	−0.0538382
$346$	2456.00	0.381605
$347$	−9528.00	−1.47403	−0.737017	−	0.675874i	$-0.763766\pi$
$347$	−9528.00	−1.47403	−0.737017	+	0.675874i	$0.763766\pi$
$348$	−2460.00	−0.378936
$349$	5079.00	0.779005	0.389502	−	0.921025i	$-0.372647\pi$
$349$	5079.00	0.779005	0.389502	+	0.921025i	$0.372647\pi$
$350$	−250.000	−0.0381802
$351$	270.000	0.0410585
$352$	−1856.00	−0.281037
$353$	−2368.00	−0.357042	−0.178521	−	0.983936i	$-0.557131\pi$
$353$	−2368.00	−0.357042	−0.178521	+	0.983936i	$0.557131\pi$
$354$	3390.00	0.508973
$355$	325.000	0.0485893
$356$	−576.000	−0.0857526
$357$	−405.000	−0.0600417
$358$	−4352.00	−0.642487
$359$	5552.00	0.816221	0.408111	−	0.912933i	$-0.366188\pi$
$359$	5552.00	0.816221	0.408111	+	0.912933i	$0.366188\pi$
$360$	−360.000	−0.0527046
$361$	16857.0	2.45765
$362$	−3928.00	−0.570307
$363$	6099.00	0.881858
$364$	−200.000	−0.0287990
$365$	−2650.00	−0.380020
$366$	−2952.00	−0.421594
$367$	−63.0000	−0.00896069	−0.00448035	−	0.999990i	$-0.501426\pi$
$367$	−63.0000	−0.00896069	−0.00448035	+	0.999990i	$0.501426\pi$
$368$	368.000	0.0521286
$369$	−4023.00	−0.567558
$370$	−1430.00	−0.200925
$371$	2605.00	0.364541
$372$	−1236.00	−0.172268
$373$	462.000	0.0641326	0.0320663	−	0.999486i	$-0.489791\pi$
$373$	462.000	0.0641326	0.0320663	+	0.999486i	$0.489791\pi$
$374$	−3132.00	−0.433026
$375$	−375.000	−0.0516398
$376$	1024.00	0.140449
$377$	−2050.00	−0.280054
$378$	−270.000	−0.0367389
$379$	−2512.00	−0.340456	−0.170228	−	0.985405i	$-0.554450\pi$
$379$	−2512.00	−0.340456	−0.170228	+	0.985405i	$0.554450\pi$
$380$	3080.00	0.415791
$381$	3336.00	0.448579
$382$	−1376.00	−0.184299
$383$	4473.00	0.596761	0.298381	−	0.954447i	$-0.403553\pi$
$383$	4473.00	0.596761	0.298381	+	0.954447i	$0.403553\pi$
$384$	384.000	0.0510310
$385$	−1450.00	−0.191945
$386$	6376.00	0.840751
$387$	2376.00	0.312090
$388$	−152.000	−0.0198882
$389$	−2462.00	−0.320896	−0.160448	−	0.987044i	$-0.551294\pi$
$389$	−2462.00	−0.320896	−0.160448	+	0.987044i	$0.551294\pi$
$390$	−300.000	−0.0389515
$391$	621.000	0.0803205
$392$	−2544.00	−0.327784
$393$	8868.00	1.13825
$394$	4156.00	0.531412
$395$	3700.00	0.471309
$396$	−2088.00	−0.264965
$397$	−7264.00	−0.918312	−0.459156	−	0.888356i	$-0.651848\pi$
$397$	−7264.00	−0.918312	−0.459156	+	0.888356i	$0.651848\pi$
$398$	3572.00	0.449870
$399$	2310.00	0.289836
$400$	400.000	0.0500000
$401$	−1630.00	−0.202988	−0.101494	−	0.994836i	$-0.532362\pi$
$401$	−1630.00	−0.202988	−0.101494	+	0.994836i	$0.532362\pi$
$402$	2226.00	0.276176
$403$	−1030.00	−0.127315
$404$	−1700.00	−0.209352
$405$	−405.000	−0.0496904
$406$	2050.00	0.250591
$407$	−8294.00	−1.01012
$408$	648.000	0.0786294
$409$	−5467.00	−0.660943	−0.330472	−	0.943816i	$-0.607208\pi$
$409$	−5467.00	−0.660943	−0.330472	+	0.943816i	$0.607208\pi$
$410$	4470.00	0.538433
$411$	6282.00	0.753937
$412$	−32.0000	−0.00382652
$413$	−2825.00	−0.336584
$414$	414.000	0.0491473
$415$	2285.00	0.270280
$416$	320.000	0.0377146
$417$	−2751.00	−0.323062
$418$	17864.0	2.09033
$419$	−8574.00	−0.999683	−0.499842	−	0.866117i	$-0.666608\pi$
$419$	−8574.00	−0.999683	−0.499842	+	0.866117i	$0.666608\pi$
$420$	300.000	0.0348536
$421$	7182.00	0.831423	0.415712	−	0.909496i	$-0.363533\pi$
$421$	7182.00	0.831423	0.415712	+	0.909496i	$0.363533\pi$
$422$	−4990.00	−0.575615
$423$	1152.00	0.132416
$424$	−4168.00	−0.477396
$425$	675.000	0.0770407
$426$	−390.000	−0.0443558
$427$	2460.00	0.278800
$428$	−2596.00	−0.293183
$429$	−1740.00	−0.195823
$430$	−2640.00	−0.296075
$431$	4040.00	0.451508	0.225754	−	0.974184i	$-0.427515\pi$
$431$	4040.00	0.451508	0.225754	+	0.974184i	$0.427515\pi$
$432$	432.000	0.0481125
$433$	7871.00	0.873571	0.436785	−	0.899566i	$-0.356117\pi$
$433$	7871.00	0.873571	0.436785	+	0.899566i	$0.356117\pi$
$434$	1030.00	0.113921
$435$	3075.00	0.338931
$436$	5512.00	0.605452
$437$	−3542.00	−0.387727
$438$	3180.00	0.346909
$439$	−9608.00	−1.04457	−0.522283	−	0.852772i	$-0.674920\pi$
$439$	−9608.00	−1.04457	−0.522283	+	0.852772i	$0.674920\pi$
$440$	2320.00	0.251367
$441$	−2862.00	−0.309038
$442$	540.000	0.0581113
$443$	−8332.00	−0.893601	−0.446801	−	0.894634i	$-0.647437\pi$
$443$	−8332.00	−0.893601	−0.446801	+	0.894634i	$0.647437\pi$
$444$	1716.00	0.183418
$445$	720.000	0.0766995
$446$	5596.00	0.594122
$447$	−9300.00	−0.984060
$448$	−320.000	−0.0337468
$449$	−13293.0	−1.39718	−0.698592	−	0.715520i	$-0.746190\pi$
$449$	−13293.0	−1.39718	−0.698592	+	0.715520i	$0.746190\pi$
$450$	450.000	0.0471405
$451$	25926.0	2.70689
$452$	−3940.00	−0.410004
$453$	4740.00	0.491622
$454$	−12728.0	−1.31576
$455$	250.000	0.0257586
$456$	−3696.00	−0.379564
$457$	−14951.0	−1.53037	−0.765184	−	0.643812i	$-0.777352\pi$
$457$	−14951.0	−1.53037	−0.765184	+	0.643812i	$0.777352\pi$
$458$	−7832.00	−0.799051
$459$	729.000	0.0741325
$460$	−460.000	−0.0466252
$461$	−9498.00	−0.959579	−0.479790	−	0.877384i	$-0.659287\pi$
$461$	−9498.00	−0.959579	−0.479790	+	0.877384i	$0.659287\pi$
$462$	1740.00	0.175221
$463$	4622.00	0.463936	0.231968	−	0.972723i	$-0.425483\pi$
$463$	4622.00	0.463936	0.231968	+	0.972723i	$0.425483\pi$
$464$	−3280.00	−0.328168
$465$	1545.00	0.154081
$466$	5352.00	0.532031
$467$	8607.00	0.852858	0.426429	−	0.904521i	$-0.359771\pi$
$467$	8607.00	0.852858	0.426429	+	0.904521i	$0.359771\pi$
$468$	360.000	0.0355577
$469$	−1855.00	−0.182635
$470$	−1280.00	−0.125621
$471$	−8601.00	−0.841429
$472$	4520.00	0.440784
$473$	−15312.0	−1.48847
$474$	−4440.00	−0.430245
$475$	−3850.00	−0.371895
$476$	−540.000	−0.0519976
$477$	−4689.00	−0.450093
$478$	8242.00	0.788662
$479$	−5340.00	−0.509375	−0.254688	−	0.967023i	$-0.581973\pi$
$479$	−5340.00	−0.509375	−0.254688	+	0.967023i	$0.581973\pi$
$480$	−480.000	−0.0456435
$481$	1430.00	0.135556
$482$	2084.00	0.196937
$483$	−345.000	−0.0325011
$484$	8132.00	0.763711
$485$	190.000	0.0177886
$486$	486.000	0.0453609
$487$	−13936.0	−1.29672	−0.648358	−	0.761336i	$-0.724544\pi$
$487$	−13936.0	−1.29672	−0.648358	+	0.761336i	$0.724544\pi$
$488$	−3936.00	−0.365111
$489$	5358.00	0.495495
$490$	3180.00	0.293179
$491$	13383.0	1.23007	0.615037	−	0.788498i	$-0.289141\pi$
$491$	13383.0	1.23007	0.615037	+	0.788498i	$0.289141\pi$
$492$	−5364.00	−0.491520
$493$	−5535.00	−0.505647
$494$	−3080.00	−0.280518
$495$	2610.00	0.236991
$496$	−1648.00	−0.149188
$497$	325.000	0.0293325
$498$	−2742.00	−0.246731
$499$	2711.00	0.243208	0.121604	−	0.992579i	$-0.461196\pi$
$499$	2711.00	0.243208	0.121604	+	0.992579i	$0.461196\pi$
$500$	−500.000	−0.0447214
$501$	−1392.00	−0.124132
$502$	5804.00	0.516026
$503$	−387.000	−0.0343051	−0.0171526	−	0.999853i	$-0.505460\pi$
$503$	−387.000	−0.0343051	−0.0171526	+	0.999853i	$0.505460\pi$
$504$	−360.000	−0.0318168
$505$	2125.00	0.187250
$506$	−2668.00	−0.234401
$507$	−6291.00	−0.551071
$508$	4448.00	0.388480
$509$	6450.00	0.561672	0.280836	−	0.959756i	$-0.409388\pi$
$509$	6450.00	0.561672	0.280836	+	0.959756i	$0.409388\pi$
$510$	−810.000	−0.0703282
$511$	−2650.00	−0.229411
$512$	512.000	0.0441942
$513$	−4158.00	−0.357856
$514$	−6096.00	−0.523119
$515$	40.0000	0.00342254
$516$	3168.00	0.270278
$517$	−7424.00	−0.631542
$518$	−1430.00	−0.121295
$519$	3684.00	0.311579
$520$	−400.000	−0.0337330
$521$	−10614.0	−0.892529	−0.446265	−	0.894901i	$-0.647246\pi$
$521$	−10614.0	−0.892529	−0.446265	+	0.894901i	$0.647246\pi$
$522$	−3690.00	−0.309400
$523$	−13748.0	−1.14944	−0.574721	−	0.818349i	$-0.694889\pi$
$523$	−13748.0	−1.14944	−0.574721	+	0.818349i	$0.694889\pi$
$524$	11824.0	0.985752
$525$	−375.000	−0.0311740
$526$	4466.00	0.370203
$527$	−2781.00	−0.229871
$528$	−2784.00	−0.229466
$529$	529.000	0.0434783
$530$	5210.00	0.426996
$531$	5085.00	0.415575
$532$	3080.00	0.251006
$533$	−4470.00	−0.363259
$534$	−864.000	−0.0700167
$535$	3245.00	0.262231
$536$	2968.00	0.239176
$537$	−6528.00	−0.524588
$538$	−3006.00	−0.240888
$539$	18444.0	1.47391
$540$	−540.000	−0.0430331
$541$	16126.0	1.28154	0.640768	−	0.767735i	$-0.278616\pi$
$541$	16126.0	1.28154	0.640768	+	0.767735i	$0.278616\pi$
$542$	15374.0	1.21839
$543$	−5892.00	−0.465654
$544$	864.000	0.0680950
$545$	−6890.00	−0.541532
$546$	−300.000	−0.0235143
$547$	−9680.00	−0.756649	−0.378324	−	0.925673i	$-0.623500\pi$
$547$	−9680.00	−0.756649	−0.378324	+	0.925673i	$0.623500\pi$
$548$	8376.00	0.652929
$549$	−4428.00	−0.344230
$550$	−2900.00	−0.224830
$551$	31570.0	2.44088
$552$	552.000	0.0425628
$553$	3700.00	0.284521
$554$	12876.0	0.987453
$555$	−2145.00	−0.164054
$556$	−3668.00	−0.279780
$557$	2815.00	0.214139	0.107069	−	0.994252i	$-0.465853\pi$
$557$	2815.00	0.214139	0.107069	+	0.994252i	$0.465853\pi$
$558$	−1854.00	−0.140656
$559$	2640.00	0.199750
$560$	400.000	0.0301841
$561$	−4698.00	−0.353565
$562$	−17756.0	−1.33273
$563$	−5583.00	−0.417931	−0.208966	−	0.977923i	$-0.567010\pi$
$563$	−5583.00	−0.417931	−0.208966	+	0.977923i	$0.567010\pi$
$564$	1536.00	0.114676
$565$	4925.00	0.366719
$566$	8890.00	0.660202
$567$	−405.000	−0.0299972
$568$	−520.000	−0.0384132
$569$	−13310.0	−0.980640	−0.490320	−	0.871542i	$-0.663120\pi$
$569$	−13310.0	−0.980640	−0.490320	+	0.871542i	$0.663120\pi$
$570$	4620.00	0.339492
$571$	−15756.0	−1.15476	−0.577380	−	0.816475i	$-0.695925\pi$
$571$	−15756.0	−1.15476	−0.577380	+	0.816475i	$0.695925\pi$
$572$	−2320.00	−0.169588
$573$	−2064.00	−0.150480
$574$	4470.00	0.325042
$575$	575.000	0.0417029
$576$	576.000	0.0416667
$577$	−16504.0	−1.19076	−0.595382	−	0.803443i	$-0.702999\pi$
$577$	−16504.0	−1.19076	−0.595382	+	0.803443i	$0.702999\pi$
$578$	−8368.00	−0.602185
$579$	9564.00	0.686470
$580$	4100.00	0.293523
$581$	2285.00	0.163163
$582$	−228.000	−0.0162387
$583$	30218.0	2.14666
$584$	4240.00	0.300432
$585$	−450.000	−0.0318038
$586$	12922.0	0.910926
$587$	−7506.00	−0.527778	−0.263889	−	0.964553i	$-0.585005\pi$
$587$	−7506.00	−0.527778	−0.263889	+	0.964553i	$0.585005\pi$
$588$	−3816.00	−0.267635
$589$	15862.0	1.10965
$590$	−5650.00	−0.394249
$591$	6234.00	0.433896
$592$	2288.00	0.158845
$593$	4944.00	0.342371	0.171185	−	0.985239i	$-0.445240\pi$
$593$	4944.00	0.342371	0.171185	+	0.985239i	$0.445240\pi$
$594$	−3132.00	−0.216343
$595$	675.000	0.0465081
$596$	−12400.0	−0.852221
$597$	5358.00	0.367317
$598$	460.000	0.0314562
$599$	−16936.0	−1.15524	−0.577618	−	0.816307i	$-0.696018\pi$
$599$	−16936.0	−1.15524	−0.577618	+	0.816307i	$0.696018\pi$
$600$	600.000	0.0408248
$601$	−7733.00	−0.524851	−0.262426	−	0.964952i	$-0.584522\pi$
$601$	−7733.00	−0.524851	−0.262426	+	0.964952i	$0.584522\pi$
$602$	−2640.00	−0.178735
$603$	3339.00	0.225497
$604$	6320.00	0.425757
$605$	−10165.0	−0.683084
$606$	−2550.00	−0.170935
$607$	9156.00	0.612241	0.306121	−	0.951993i	$-0.400969\pi$
$607$	9156.00	0.612241	0.306121	+	0.951993i	$0.400969\pi$
$608$	−4928.00	−0.328712
$609$	3075.00	0.204606
$610$	4920.00	0.326566
$611$	1280.00	0.0847516
$612$	972.000	0.0642006
$613$	−23930.0	−1.57671	−0.788355	−	0.615220i	$-0.789067\pi$
$613$	−23930.0	−1.57671	−0.788355	+	0.615220i	$0.789067\pi$
$614$	11828.0	0.777425
$615$	6705.00	0.439629
$616$	2320.00	0.151746
$617$	−27699.0	−1.80733	−0.903663	−	0.428245i	$-0.859132\pi$
$617$	−27699.0	−1.80733	−0.903663	+	0.428245i	$0.859132\pi$
$618$	−48.0000	−0.00312434
$619$	−17480.0	−1.13503	−0.567513	−	0.823365i	$-0.692094\pi$
$619$	−17480.0	−1.13503	−0.567513	+	0.823365i	$0.692094\pi$
$620$	2060.00	0.133438
$621$	621.000	0.0401286
$622$	−13984.0	−0.901459
$623$	720.000	0.0463021
$624$	480.000	0.0307939
$625$	625.000	0.0400000
$626$	15610.0	0.996647
$627$	26796.0	1.70675
$628$	−11468.0	−0.728699
$629$	3861.00	0.244751
$630$	450.000	0.0284578
$631$	14164.0	0.893597	0.446799	−	0.894635i	$-0.352564\pi$
$631$	14164.0	0.893597	0.446799	+	0.894635i	$0.352564\pi$
$632$	−5920.00	−0.372603
$633$	−7485.00	−0.469987
$634$	−20544.0	−1.28692
$635$	−5560.00	−0.347468
$636$	−6252.00	−0.389792
$637$	−3180.00	−0.197796
$638$	23780.0	1.47564
$639$	−585.000	−0.0362163
$640$	−640.000	−0.0395285
$641$	−25506.0	−1.57165	−0.785824	−	0.618450i	$-0.787761\pi$
$641$	−25506.0	−1.57165	−0.785824	+	0.618450i	$0.787761\pi$
$642$	−3894.00	−0.239383
$643$	−13817.0	−0.847417	−0.423709	−	0.905799i	$-0.639272\pi$
$643$	−13817.0	−0.847417	−0.423709	+	0.905799i	$0.639272\pi$
$644$	−460.000	−0.0281468
$645$	−3960.00	−0.241744
$646$	−8316.00	−0.506484
$647$	−18562.0	−1.12789	−0.563947	−	0.825811i	$-0.690718\pi$
$647$	−18562.0	−1.12789	−0.563947	+	0.825811i	$0.690718\pi$
$648$	648.000	0.0392837
$649$	−32770.0	−1.98203
$650$	500.000	0.0301717
$651$	1545.00	0.0930159
$652$	7144.00	0.429111
$653$	−29022.0	−1.73923	−0.869616	−	0.493729i	$-0.835634\pi$
$653$	−29022.0	−1.73923	−0.869616	+	0.493729i	$0.835634\pi$
$654$	8268.00	0.494349
$655$	−14780.0	−0.881683
$656$	−7152.00	−0.425669
$657$	4770.00	0.283250
$658$	−1280.00	−0.0758353
$659$	−28038.0	−1.65737	−0.828684	−	0.559717i	$-0.810910\pi$
$659$	−28038.0	−1.65737	−0.828684	+	0.559717i	$0.810910\pi$
$660$	3480.00	0.205241
$661$	−8950.00	−0.526648	−0.263324	−	0.964707i	$-0.584819\pi$
$661$	−8950.00	−0.526648	−0.263324	+	0.964707i	$0.584819\pi$
$662$	10938.0	0.642171
$663$	810.000	0.0474477
$664$	−3656.00	−0.213675
$665$	−3850.00	−0.224506
$666$	2574.00	0.149760
$667$	−4715.00	−0.273711
$668$	−1856.00	−0.107501
$669$	8394.00	0.485098
$670$	−3710.00	−0.213925
$671$	28536.0	1.64176
$672$	−480.000	−0.0275542
$673$	−1420.00	−0.0813328	−0.0406664	−	0.999173i	$-0.512948\pi$
$673$	−1420.00	−0.0813328	−0.0406664	+	0.999173i	$0.512948\pi$
$674$	12452.0	0.711622
$675$	675.000	0.0384900
$676$	−8388.00	−0.477242
$677$	−3287.00	−0.186602	−0.0933011	−	0.995638i	$-0.529742\pi$
$677$	−3287.00	−0.186602	−0.0933011	+	0.995638i	$0.529742\pi$
$678$	−5910.00	−0.334767
$679$	190.000	0.0107386
$680$	−1080.00	−0.0609060
$681$	−19092.0	−1.07431
$682$	11948.0	0.670840
$683$	−14520.0	−0.813459	−0.406729	−	0.913549i	$-0.633331\pi$
$683$	−14520.0	−0.813459	−0.406729	+	0.913549i	$0.633331\pi$
$684$	−5544.00	−0.309912
$685$	−10470.0	−0.583997
$686$	6610.00	0.367888
$687$	−11748.0	−0.652422
$688$	4224.00	0.234068
$689$	−5210.00	−0.288077
$690$	−690.000	−0.0380693
$691$	−21460.0	−1.18144	−0.590721	−	0.806876i	$-0.701157\pi$
$691$	−21460.0	−1.18144	−0.590721	+	0.806876i	$0.701157\pi$
$692$	4912.00	0.269836
$693$	2610.00	0.143067
$694$	−19056.0	−1.04230
$695$	4585.00	0.250243
$696$	−4920.00	−0.267948
$697$	−12069.0	−0.655877
$698$	10158.0	0.550839
$699$	8028.00	0.434402
$700$	−500.000	−0.0269975
$701$	13512.0	0.728019	0.364009	−	0.931395i	$-0.381408\pi$
$701$	13512.0	0.728019	0.364009	+	0.931395i	$0.381408\pi$
$702$	540.000	0.0290327
$703$	−22022.0	−1.18147
$704$	−3712.00	−0.198723
$705$	−1920.00	−0.102569
$706$	−4736.00	−0.252467
$707$	2125.00	0.113039
$708$	6780.00	0.359898
$709$	19528.0	1.03440	0.517200	−	0.855865i	$-0.326974\pi$
$709$	19528.0	1.03440	0.517200	+	0.855865i	$0.326974\pi$
$710$	650.000	0.0343578
$711$	−6660.00	−0.351293
$712$	−1152.00	−0.0606363
$713$	−2369.00	−0.124432
$714$	−810.000	−0.0424559
$715$	2900.00	0.151684
$716$	−8704.00	−0.454307
$717$	12363.0	0.643940
$718$	11104.0	0.577155
$719$	−12503.0	−0.648516	−0.324258	−	0.945969i	$-0.605115\pi$
$719$	−12503.0	−0.648516	−0.324258	+	0.945969i	$0.605115\pi$
$720$	−720.000	−0.0372678
$721$	40.0000	0.00206613
$722$	33714.0	1.73782
$723$	3126.00	0.160798
$724$	−7856.00	−0.403268
$725$	−5125.00	−0.262535
$726$	12198.0	0.623568
$727$	13409.0	0.684061	0.342030	−	0.939689i	$-0.388885\pi$
$727$	13409.0	0.684061	0.342030	+	0.939689i	$0.388885\pi$
$728$	−400.000	−0.0203640
$729$	729.000	0.0370370
$730$	−5300.00	−0.268715
$731$	7128.00	0.360655
$732$	−5904.00	−0.298112
$733$	30095.0	1.51649	0.758243	−	0.651972i	$-0.226058\pi$
$733$	30095.0	1.51649	0.758243	+	0.651972i	$0.226058\pi$
$734$	−126.000	−0.00633616
$735$	4770.00	0.239380
$736$	736.000	0.0368605
$737$	−21518.0	−1.07548
$738$	−8046.00	−0.401324
$739$	−7831.00	−0.389808	−0.194904	−	0.980822i	$-0.562440\pi$
$739$	−7831.00	−0.389808	−0.194904	+	0.980822i	$0.562440\pi$
$740$	−2860.00	−0.142075
$741$	−4620.00	−0.229042
$742$	5210.00	0.257770
$743$	20408.0	1.00767	0.503834	−	0.863801i	$-0.331922\pi$
$743$	20408.0	1.00767	0.503834	+	0.863801i	$0.331922\pi$
$744$	−2472.00	−0.121812
$745$	15500.0	0.762250
$746$	924.000	0.0453486
$747$	−4113.00	−0.201455
$748$	−6264.00	−0.306196
$749$	3245.00	0.158304
$750$	−750.000	−0.0365148
$751$	−3118.00	−0.151501	−0.0757506	−	0.997127i	$-0.524135\pi$
$751$	−3118.00	−0.151501	−0.0757506	+	0.997127i	$0.524135\pi$
$752$	2048.00	0.0993123
$753$	8706.00	0.421334
$754$	−4100.00	−0.198028
$755$	−7900.00	−0.380809
$756$	−540.000	−0.0259783
$757$	−17639.0	−0.846896	−0.423448	−	0.905920i	$-0.639180\pi$
$757$	−17639.0	−0.846896	−0.423448	+	0.905920i	$0.639180\pi$
$758$	−5024.00	−0.240739
$759$	−4002.00	−0.191388
$760$	6160.00	0.294009
$761$	−2499.00	−0.119039	−0.0595195	−	0.998227i	$-0.518957\pi$
$761$	−2499.00	−0.119039	−0.0595195	+	0.998227i	$0.518957\pi$
$762$	6672.00	0.317193
$763$	−6890.00	−0.326913
$764$	−2752.00	−0.130319
$765$	−1215.00	−0.0574228
$766$	8946.00	0.421974
$767$	5650.00	0.265984
$768$	768.000	0.0360844
$769$	11552.0	0.541711	0.270856	−	0.962620i	$-0.412693\pi$
$769$	11552.0	0.541711	0.270856	+	0.962620i	$0.412693\pi$
$770$	−2900.00	−0.135726
$771$	−9144.00	−0.427125
$772$	12752.0	0.594501
$773$	−41226.0	−1.91824	−0.959118	−	0.283007i	$-0.908668\pi$
$773$	−41226.0	−1.91824	−0.959118	+	0.283007i	$0.908668\pi$
$774$	4752.00	0.220681
$775$	−2575.00	−0.119351
$776$	−304.000	−0.0140631
$777$	−2145.00	−0.0990366
$778$	−4924.00	−0.226907
$779$	68838.0	3.16608
$780$	−600.000	−0.0275429
$781$	3770.00	0.172729
$782$	1242.00	0.0567952
$783$	−5535.00	−0.252624
$784$	−5088.00	−0.231778
$785$	14335.0	0.651768
$786$	17736.0	0.804863
$787$	−4489.00	−0.203323	−0.101662	−	0.994819i	$-0.532416\pi$
$787$	−4489.00	−0.203323	−0.101662	+	0.994819i	$0.532416\pi$
$788$	8312.00	0.375765
$789$	6699.00	0.302270
$790$	7400.00	0.333266
$791$	4925.00	0.221382
$792$	−4176.00	−0.187358
$793$	−4920.00	−0.220321
$794$	−14528.0	−0.649344
$795$	7815.00	0.348641
$796$	7144.00	0.318106
$797$	−34671.0	−1.54092	−0.770458	−	0.637491i	$-0.779972\pi$
$797$	−34671.0	−1.54092	−0.770458	+	0.637491i	$0.779972\pi$
$798$	4620.00	0.204945
$799$	3456.00	0.153022
$800$	800.000	0.0353553
$801$	−1296.00	−0.0571684
$802$	−3260.00	−0.143534
$803$	−30740.0	−1.35092
$804$	4452.00	0.195286
$805$	575.000	0.0251753
$806$	−2060.00	−0.0900253
$807$	−4509.00	−0.196684
$808$	−3400.00	−0.148034
$809$	28911.0	1.25644	0.628218	−	0.778037i	$-0.283785\pi$
$809$	28911.0	1.25644	0.628218	+	0.778037i	$0.283785\pi$
$810$	−810.000	−0.0351364
$811$	−24495.0	−1.06059	−0.530293	−	0.847814i	$-0.677918\pi$
$811$	−24495.0	−1.06059	−0.530293	+	0.847814i	$0.677918\pi$
$812$	4100.00	0.177194
$813$	23061.0	0.994815
$814$	−16588.0	−0.714262
$815$	−8930.00	−0.383809
$816$	1296.00	0.0555994
$817$	−40656.0	−1.74097
$818$	−10934.0	−0.467357
$819$	−450.000	−0.0191994
$820$	8940.00	0.380730
$821$	25314.0	1.07608	0.538042	−	0.842918i	$-0.319164\pi$
$821$	25314.0	1.07608	0.538042	+	0.842918i	$0.319164\pi$
$822$	12564.0	0.533114
$823$	36060.0	1.52731	0.763653	−	0.645627i	$-0.223404\pi$
$823$	36060.0	1.52731	0.763653	+	0.645627i	$0.223404\pi$
$824$	−64.0000	−0.00270576
$825$	−4350.00	−0.183573
$826$	−5650.00	−0.238001
$827$	−16539.0	−0.695426	−0.347713	−	0.937601i	$-0.613042\pi$
$827$	−16539.0	−0.695426	−0.347713	+	0.937601i	$0.613042\pi$
$828$	828.000	0.0347524
$829$	−37799.0	−1.58361	−0.791806	−	0.610773i	$-0.790859\pi$
$829$	−37799.0	−1.58361	−0.791806	+	0.610773i	$0.790859\pi$
$830$	4570.00	0.191117
$831$	19314.0	0.806252
$832$	640.000	0.0266683
$833$	−8586.00	−0.357128
$834$	−5502.00	−0.228440
$835$	2320.00	0.0961520
$836$	35728.0	1.47808
$837$	−2781.00	−0.114845
$838$	−17148.0	−0.706883
$839$	78.0000	0.00320961	0.00160480	−	0.999999i	$-0.499489\pi$
$839$	78.0000	0.00320961	0.00160480	+	0.999999i	$0.499489\pi$
$840$	600.000	0.0246452
$841$	17636.0	0.723113
$842$	14364.0	0.587905
$843$	−26634.0	−1.08817
$844$	−9980.00	−0.407021
$845$	10485.0	0.426858
$846$	2304.00	0.0936326
$847$	−10165.0	−0.412365
$848$	−8336.00	−0.337570
$849$	13335.0	0.539053
$850$	1350.00	0.0544760
$851$	3289.00	0.132486
$852$	−780.000	−0.0313643
$853$	17102.0	0.686473	0.343236	−	0.939249i	$-0.388477\pi$
$853$	17102.0	0.686473	0.343236	+	0.939249i	$0.388477\pi$
$854$	4920.00	0.197142
$855$	6930.00	0.277194
$856$	−5192.00	−0.207312
$857$	23886.0	0.952077	0.476039	−	0.879424i	$-0.342072\pi$
$857$	23886.0	0.952077	0.476039	+	0.879424i	$0.342072\pi$
$858$	−3480.00	−0.138468
$859$	−7675.00	−0.304852	−0.152426	−	0.988315i	$-0.548709\pi$
$859$	−7675.00	−0.304852	−0.152426	+	0.988315i	$0.548709\pi$
$860$	−5280.00	−0.209356
$861$	6705.00	0.265396
$862$	8080.00	0.319264
$863$	31554.0	1.24462	0.622312	−	0.782769i	$-0.286193\pi$
$863$	31554.0	1.24462	0.622312	+	0.782769i	$0.286193\pi$
$864$	864.000	0.0340207
$865$	−6140.00	−0.241348
$866$	15742.0	0.617708
$867$	−12552.0	−0.491682
$868$	2060.00	0.0805541
$869$	42920.0	1.67544
$870$	6150.00	0.239660
$871$	3710.00	0.144327
$872$	11024.0	0.428119
$873$	−342.000	−0.0132588
$874$	−7084.00	−0.274165
$875$	625.000	0.0241473
$876$	6360.00	0.245302
$877$	26534.0	1.02165	0.510826	−	0.859684i	$-0.329339\pi$
$877$	26534.0	1.02165	0.510826	+	0.859684i	$0.329339\pi$
$878$	−19216.0	−0.738620
$879$	19383.0	0.743768
$880$	4640.00	0.177744
$881$	−34524.0	−1.32025	−0.660127	−	0.751154i	$-0.729497\pi$
$881$	−34524.0	−1.32025	−0.660127	+	0.751154i	$0.729497\pi$
$882$	−5724.00	−0.218523
$883$	−13876.0	−0.528839	−0.264419	−	0.964408i	$-0.585180\pi$
$883$	−13876.0	−0.528839	−0.264419	+	0.964408i	$0.585180\pi$
$884$	1080.00	0.0410909
$885$	−8475.00	−0.321903
$886$	−16664.0	−0.631871
$887$	1494.00	0.0565542	0.0282771	−	0.999600i	$-0.490998\pi$
$887$	1494.00	0.0565542	0.0282771	+	0.999600i	$0.490998\pi$
$888$	3432.00	0.129696
$889$	−5560.00	−0.209760
$890$	1440.00	0.0542347
$891$	−4698.00	−0.176643
$892$	11192.0	0.420107
$893$	−19712.0	−0.738675
$894$	−18600.0	−0.695836
$895$	10880.0	0.406344
$896$	−640.000	−0.0238626
$897$	690.000	0.0256839
$898$	−26586.0	−0.987958
$899$	21115.0	0.783342
$900$	900.000	0.0333333
$901$	−14067.0	−0.520133
$902$	51852.0	1.91406
$903$	−3960.00	−0.145936
$904$	−7880.00	−0.289917
$905$	9820.00	0.360694
$906$	9480.00	0.347629
$907$	−1709.00	−0.0625650	−0.0312825	−	0.999511i	$-0.509959\pi$
$907$	−1709.00	−0.0625650	−0.0312825	+	0.999511i	$0.509959\pi$
$908$	−25456.0	−0.930382
$909$	−3825.00	−0.139568
$910$	500.000	0.0182141
$911$	53100.0	1.93115	0.965577	−	0.260117i	$-0.0837611\pi$
$911$	53100.0	1.93115	0.965577	+	0.260117i	$0.0837611\pi$
$912$	−7392.00	−0.268392
$913$	26506.0	0.960811
$914$	−29902.0	−1.08213
$915$	7380.00	0.266640
$916$	−15664.0	−0.565014
$917$	−14780.0	−0.532256
$918$	1458.00	0.0524196
$919$	33272.0	1.19428	0.597139	−	0.802138i	$-0.296304\pi$
$919$	33272.0	1.19428	0.597139	+	0.802138i	$0.296304\pi$
$920$	−920.000	−0.0329690
$921$	17742.0	0.634765
$922$	−18996.0	−0.678525
$923$	−650.000	−0.0231799
$924$	3480.00	0.123900
$925$	3575.00	0.127076
$926$	9244.00	0.328053
$927$	−72.0000	−0.00255101
$928$	−6560.00	−0.232050
$929$	20883.0	0.737512	0.368756	−	0.929526i	$-0.379784\pi$
$929$	20883.0	0.737512	0.368756	+	0.929526i	$0.379784\pi$
$930$	3090.00	0.108952
$931$	48972.0	1.72394
$932$	10704.0	0.376203
$933$	−20976.0	−0.736038
$934$	17214.0	0.603061
$935$	7830.00	0.273870
$936$	720.000	0.0251431
$937$	5370.00	0.187225	0.0936127	−	0.995609i	$-0.470158\pi$
$937$	5370.00	0.187225	0.0936127	+	0.995609i	$0.470158\pi$
$938$	−3710.00	−0.129143
$939$	23415.0	0.813759
$940$	−2560.00	−0.0888277
$941$	30212.0	1.04663	0.523317	−	0.852138i	$-0.324694\pi$
$941$	30212.0	1.04663	0.523317	+	0.852138i	$0.324694\pi$
$942$	−17202.0	−0.594980
$943$	−10281.0	−0.355032
$944$	9040.00	0.311681
$945$	675.000	0.0232357
$946$	−30624.0	−1.05251
$947$	26764.0	0.918388	0.459194	−	0.888336i	$-0.348138\pi$
$947$	26764.0	0.918388	0.459194	+	0.888336i	$0.348138\pi$
$948$	−8880.00	−0.304229
$949$	5300.00	0.181291
$950$	−7700.00	−0.262969
$951$	−30816.0	−1.05076
$952$	−1080.00	−0.0367679
$953$	−48646.0	−1.65351	−0.826757	−	0.562559i	$-0.809817\pi$
$953$	−48646.0	−1.65351	−0.826757	+	0.562559i	$0.809817\pi$
$954$	−9378.00	−0.318264
$955$	3440.00	0.116561
$956$	16484.0	0.557668
$957$	35670.0	1.20486
$958$	−10680.0	−0.360183
$959$	−10470.0	−0.352548
$960$	−960.000	−0.0322749
$961$	−19182.0	−0.643886
$962$	2860.00	0.0958525
$963$	−5841.00	−0.195455
$964$	4168.00	0.139255
$965$	−15940.0	−0.531738
$966$	−690.000	−0.0229818
$967$	26164.0	0.870091	0.435045	−	0.900409i	$-0.356732\pi$
$967$	26164.0	0.870091	0.435045	+	0.900409i	$0.356732\pi$
$968$	16264.0	0.540026
$969$	−12474.0	−0.413542
$970$	380.000	0.0125784
$971$	45876.0	1.51620	0.758100	−	0.652138i	$-0.226128\pi$
$971$	45876.0	1.51620	0.758100	+	0.652138i	$0.226128\pi$
$972$	972.000	0.0320750
$973$	4585.00	0.151067
$974$	−27872.0	−0.916916
$975$	750.000	0.0246351
$976$	−7872.00	−0.258173
$977$	43561.0	1.42645	0.713224	−	0.700936i	$-0.247234\pi$
$977$	43561.0	1.42645	0.713224	+	0.700936i	$0.247234\pi$
$978$	10716.0	0.350368
$979$	8352.00	0.272657
$980$	6360.00	0.207309
$981$	12402.0	0.403634
$982$	26766.0	0.869794
$983$	−3573.00	−0.115932	−0.0579659	−	0.998319i	$-0.518461\pi$
$983$	−3573.00	−0.115932	−0.0579659	+	0.998319i	$0.518461\pi$
$984$	−10728.0	−0.347557
$985$	−10390.0	−0.336094
$986$	−11070.0	−0.357546
$987$	−1920.00	−0.0619192
$988$	−6160.00	−0.198356
$989$	6072.00	0.195226
$990$	5220.00	0.167578
$991$	3589.00	0.115044	0.0575219	−	0.998344i	$-0.481680\pi$
$991$	3589.00	0.115044	0.0575219	+	0.998344i	$0.481680\pi$
$992$	−3296.00	−0.105492
$993$	16407.0	0.524331
$994$	650.000	0.0207412
$995$	−8930.00	−0.284523
$996$	−5484.00	−0.174465
$997$	−41432.0	−1.31611	−0.658056	−	0.752969i	$-0.728621\pi$
$997$	−41432.0	−1.31611	−0.658056	+	0.752969i	$0.728621\pi$
$998$	5422.00	0.171974
$999$	3861.00	0.122279

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	690.4.a.g.1.1	✓	1
3.2	odd	2		2070.4.a.f.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
690.4.a.g.1.1	✓	1	1.1	even	1	trivial
2070.4.a.f.1.1		1	3.2	odd	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:	\( N \)	\(=\)	\( 690 = 2 \cdot 3 \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	690.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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