LMFDB - Embedded newform 6014.2.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6014,2,Mod(1,6014)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6014, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("6014.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6014 = 2 \cdot 31 \cdot 97 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6014.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:	$48.0220317756$
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	$\chi$	$=$	6014.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +3.00000 q^{5} -1.00000 q^{6} +1.00000 q^{8} -2.00000 q^{9} +O(q^{10})$

\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +3.00000 q^{5} -1.00000 q^{6} +1.00000 q^{8} -2.00000 q^{9} +3.00000 q^{10} +1.00000 q^{11} -1.00000 q^{12} +6.00000 q^{13} -3.00000 q^{15} +1.00000 q^{16} -1.00000 q^{17} -2.00000 q^{18} +7.00000 q^{19} +3.00000 q^{20} +1.00000 q^{22} +7.00000 q^{23} -1.00000 q^{24} +4.00000 q^{25} +6.00000 q^{26} +5.00000 q^{27} -3.00000 q^{30} +1.00000 q^{31} +1.00000 q^{32} -1.00000 q^{33} -1.00000 q^{34} -2.00000 q^{36} -4.00000 q^{37} +7.00000 q^{38} -6.00000 q^{39} +3.00000 q^{40} +2.00000 q^{41} -11.0000 q^{43} +1.00000 q^{44} -6.00000 q^{45} +7.00000 q^{46} -13.0000 q^{47} -1.00000 q^{48} -7.00000 q^{49} +4.00000 q^{50} +1.00000 q^{51} +6.00000 q^{52} -1.00000 q^{53} +5.00000 q^{54} +3.00000 q^{55} -7.00000 q^{57} +7.00000 q^{59} -3.00000 q^{60} +10.0000 q^{61} +1.00000 q^{62} +1.00000 q^{64} +18.0000 q^{65} -1.00000 q^{66} -4.00000 q^{67} -1.00000 q^{68} -7.00000 q^{69} +10.0000 q^{71} -2.00000 q^{72} -4.00000 q^{73} -4.00000 q^{74} -4.00000 q^{75} +7.00000 q^{76} -6.00000 q^{78} -12.0000 q^{79} +3.00000 q^{80} +1.00000 q^{81} +2.00000 q^{82} -2.00000 q^{83} -3.00000 q^{85} -11.0000 q^{86} +1.00000 q^{88} +6.00000 q^{89} -6.00000 q^{90} +7.00000 q^{92} -1.00000 q^{93} -13.0000 q^{94} +21.0000 q^{95} -1.00000 q^{96} +1.00000 q^{97} -7.00000 q^{98} -2.00000 q^{99} +O(q^{100})\)

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 $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	−1.00000	−0.577350	−0.288675	−	0.957427i	$-0.593215\pi$
$3$	−1.00000	−0.577350	−0.288675	+	0.957427i	$0.593215\pi$
$4$	1.00000	0.500000
$5$	3.00000	1.34164	0.670820	−	0.741620i	$-0.265942\pi$
$5$	3.00000	1.34164	0.670820	+	0.741620i	$0.265942\pi$
$6$	−1.00000	−0.408248
$7$	0	0		−	1.00000i	$-0.5\pi$
$7$	0	0			1.00000i	$0.5\pi$
$8$	1.00000	0.353553
$9$	−2.00000	−0.666667
$10$	3.00000	0.948683
$11$	1.00000	0.301511	0.150756	−	0.988571i	$-0.451829\pi$
$11$	1.00000	0.301511	0.150756	+	0.988571i	$0.451829\pi$
$12$	−1.00000	−0.288675
$13$	6.00000	1.66410	0.832050	−	0.554700i	$-0.187167\pi$
$13$	6.00000	1.66410	0.832050	+	0.554700i	$0.187167\pi$
$14$	0	0
$15$	−3.00000	−0.774597
$16$	1.00000	0.250000
$17$	−1.00000	−0.242536	−0.121268	−	0.992620i	$-0.538696\pi$
$17$	−1.00000	−0.242536	−0.121268	+	0.992620i	$0.538696\pi$
$18$	−2.00000	−0.471405
$19$	7.00000	1.60591	0.802955	−	0.596040i	$-0.203260\pi$
$19$	7.00000	1.60591	0.802955	+	0.596040i	$0.203260\pi$
$20$	3.00000	0.670820
$21$	0	0
$22$	1.00000	0.213201
$23$	7.00000	1.45960	0.729800	−	0.683660i	$-0.239613\pi$
$23$	7.00000	1.45960	0.729800	+	0.683660i	$0.239613\pi$
$24$	−1.00000	−0.204124
$25$	4.00000	0.800000
$26$	6.00000	1.17670
$27$	5.00000	0.962250
$28$	0	0
$29$	0	0		−	1.00000i	$-0.5\pi$
$29$	0	0			1.00000i	$0.5\pi$
$30$	−3.00000	−0.547723
$31$	1.00000	0.179605
$31$	1.00000	0.179605
$32$	1.00000	0.176777
$33$	−1.00000	−0.174078
$34$	−1.00000	−0.171499
$35$	0	0
$36$	−2.00000	−0.333333
$37$	−4.00000	−0.657596	−0.328798	−	0.944400i	$-0.606644\pi$
$37$	−4.00000	−0.657596	−0.328798	+	0.944400i	$0.606644\pi$
$38$	7.00000	1.13555
$39$	−6.00000	−0.960769
$40$	3.00000	0.474342
$41$	2.00000	0.312348	0.156174	−	0.987730i	$-0.450084\pi$
$41$	2.00000	0.312348	0.156174	+	0.987730i	$0.450084\pi$
$42$	0	0
$43$	−11.0000	−1.67748	−0.838742	−	0.544529i	$-0.816708\pi$
$43$	−11.0000	−1.67748	−0.838742	+	0.544529i	$0.816708\pi$
$44$	1.00000	0.150756
$45$	−6.00000	−0.894427
$46$	7.00000	1.03209
$47$	−13.0000	−1.89624	−0.948122	−	0.317905i	$-0.897021\pi$
$47$	−13.0000	−1.89624	−0.948122	+	0.317905i	$0.897021\pi$
$48$	−1.00000	−0.144338
$49$	−7.00000	−1.00000
$50$	4.00000	0.565685
$51$	1.00000	0.140028
$52$	6.00000	0.832050
$53$	−1.00000	−0.137361	−0.0686803	−	0.997639i	$-0.521879\pi$
$53$	−1.00000	−0.137361	−0.0686803	+	0.997639i	$0.521879\pi$
$54$	5.00000	0.680414
$55$	3.00000	0.404520
$56$	0	0
$57$	−7.00000	−0.927173
$58$	0	0
$59$	7.00000	0.911322	0.455661	−	0.890153i	$-0.349403\pi$
$59$	7.00000	0.911322	0.455661	+	0.890153i	$0.349403\pi$
$60$	−3.00000	−0.387298
$61$	10.0000	1.28037	0.640184	−	0.768221i	$-0.278858\pi$
$61$	10.0000	1.28037	0.640184	+	0.768221i	$0.278858\pi$
$62$	1.00000	0.127000
$63$	0	0
$64$	1.00000	0.125000
$65$	18.0000	2.23263
$66$	−1.00000	−0.123091
$67$	−4.00000	−0.488678	−0.244339	−	0.969690i	$-0.578571\pi$
$67$	−4.00000	−0.488678	−0.244339	+	0.969690i	$0.578571\pi$
$68$	−1.00000	−0.121268
$69$	−7.00000	−0.842701
$70$	0	0
$71$	10.0000	1.18678	0.593391	−	0.804914i	$-0.297789\pi$
$71$	10.0000	1.18678	0.593391	+	0.804914i	$0.297789\pi$
$72$	−2.00000	−0.235702
$73$	−4.00000	−0.468165	−0.234082	−	0.972217i	$-0.575209\pi$
$73$	−4.00000	−0.468165	−0.234082	+	0.972217i	$0.575209\pi$
$74$	−4.00000	−0.464991
$75$	−4.00000	−0.461880
$76$	7.00000	0.802955
$77$	0	0
$78$	−6.00000	−0.679366
$79$	−12.0000	−1.35011	−0.675053	−	0.737769i	$-0.735879\pi$
$79$	−12.0000	−1.35011	−0.675053	+	0.737769i	$0.735879\pi$
$80$	3.00000	0.335410
$81$	1.00000	0.111111
$82$	2.00000	0.220863
$83$	−2.00000	−0.219529	−0.109764	−	0.993958i	$-0.535010\pi$
$83$	−2.00000	−0.219529	−0.109764	+	0.993958i	$0.535010\pi$
$84$	0	0
$85$	−3.00000	−0.325396
$86$	−11.0000	−1.18616
$87$	0	0
$88$	1.00000	0.106600
$89$	6.00000	0.635999	0.317999	−	0.948091i	$-0.396989\pi$
$89$	6.00000	0.635999	0.317999	+	0.948091i	$0.396989\pi$
$90$	−6.00000	−0.632456
$91$	0	0
$92$	7.00000	0.729800
$93$	−1.00000	−0.103695
$94$	−13.0000	−1.34085
$95$	21.0000	2.15455
$96$	−1.00000	−0.102062
$97$	1.00000	0.101535
$97$	1.00000	0.101535
$98$	−7.00000	−0.707107
$99$	−2.00000	−0.201008
$100$	4.00000	0.400000
$101$	−16.0000	−1.59206	−0.796030	−	0.605257i	$-0.793070\pi$
$101$	−16.0000	−1.59206	−0.796030	+	0.605257i	$0.793070\pi$
$102$	1.00000	0.0990148
$103$	−5.00000	−0.492665	−0.246332	−	0.969185i	$-0.579225\pi$
$103$	−5.00000	−0.492665	−0.246332	+	0.969185i	$0.579225\pi$
$104$	6.00000	0.588348
$105$	0	0
$106$	−1.00000	−0.0971286
$107$	16.0000	1.54678	0.773389	−	0.633932i	$-0.218560\pi$
$107$	16.0000	1.54678	0.773389	+	0.633932i	$0.218560\pi$
$108$	5.00000	0.481125
$109$	18.0000	1.72409	0.862044	−	0.506834i	$-0.169184\pi$
$109$	18.0000	1.72409	0.862044	+	0.506834i	$0.169184\pi$
$110$	3.00000	0.286039
$111$	4.00000	0.379663
$112$	0	0
$113$	−15.0000	−1.41108	−0.705541	−	0.708669i	$-0.749296\pi$
$113$	−15.0000	−1.41108	−0.705541	+	0.708669i	$0.749296\pi$
$114$	−7.00000	−0.655610
$115$	21.0000	1.95826
$116$	0	0
$117$	−12.0000	−1.10940
$118$	7.00000	0.644402
$119$	0	0
$120$	−3.00000	−0.273861
$121$	−10.0000	−0.909091
$122$	10.0000	0.905357
$123$	−2.00000	−0.180334
$124$	1.00000	0.0898027
$125$	−3.00000	−0.268328
$126$	0	0
$127$	20.0000	1.77471	0.887357	−	0.461084i	$-0.152539\pi$
$127$	20.0000	1.77471	0.887357	+	0.461084i	$0.152539\pi$
$128$	1.00000	0.0883883
$129$	11.0000	0.968496
$130$	18.0000	1.57870
$131$	3.00000	0.262111	0.131056	−	0.991375i	$-0.458163\pi$
$131$	3.00000	0.262111	0.131056	+	0.991375i	$0.458163\pi$
$132$	−1.00000	−0.0870388
$133$	0	0
$134$	−4.00000	−0.345547
$135$	15.0000	1.29099
$136$	−1.00000	−0.0857493
$137$	11.0000	0.939793	0.469897	−	0.882721i	$-0.344291\pi$
$137$	11.0000	0.939793	0.469897	+	0.882721i	$0.344291\pi$
$138$	−7.00000	−0.595880
$139$	−2.00000	−0.169638	−0.0848189	−	0.996396i	$-0.527031\pi$
$139$	−2.00000	−0.169638	−0.0848189	+	0.996396i	$0.527031\pi$
$140$	0	0
$141$	13.0000	1.09480
$142$	10.0000	0.839181
$143$	6.00000	0.501745
$144$	−2.00000	−0.166667
$145$	0	0
$146$	−4.00000	−0.331042
$147$	7.00000	0.577350
$148$	−4.00000	−0.328798
$149$	6.00000	0.491539	0.245770	−	0.969328i	$-0.420959\pi$
$149$	6.00000	0.491539	0.245770	+	0.969328i	$0.420959\pi$
$150$	−4.00000	−0.326599
$151$	12.0000	0.976546	0.488273	−	0.872691i	$-0.337627\pi$
$151$	12.0000	0.976546	0.488273	+	0.872691i	$0.337627\pi$
$152$	7.00000	0.567775
$153$	2.00000	0.161690
$154$	0	0
$155$	3.00000	0.240966
$156$	−6.00000	−0.480384
$157$	11.0000	0.877896	0.438948	−	0.898513i	$-0.355351\pi$
$157$	11.0000	0.877896	0.438948	+	0.898513i	$0.355351\pi$
$158$	−12.0000	−0.954669
$159$	1.00000	0.0793052
$160$	3.00000	0.237171
$161$	0	0
$162$	1.00000	0.0785674
$163$	12.0000	0.939913	0.469956	−	0.882690i	$-0.344270\pi$
$163$	12.0000	0.939913	0.469956	+	0.882690i	$0.344270\pi$
$164$	2.00000	0.156174
$165$	−3.00000	−0.233550
$166$	−2.00000	−0.155230
$167$	2.00000	0.154765	0.0773823	−	0.997001i	$-0.475344\pi$
$167$	2.00000	0.154765	0.0773823	+	0.997001i	$0.475344\pi$
$168$	0	0
$169$	23.0000	1.76923
$170$	−3.00000	−0.230089
$171$	−14.0000	−1.07061
$172$	−11.0000	−0.838742
$173$	1.00000	0.0760286	0.0380143	−	0.999277i	$-0.487897\pi$
$173$	1.00000	0.0760286	0.0380143	+	0.999277i	$0.487897\pi$
$174$	0	0
$175$	0	0
$176$	1.00000	0.0753778
$177$	−7.00000	−0.526152
$178$	6.00000	0.449719
$179$	−24.0000	−1.79384	−0.896922	−	0.442189i	$-0.854202\pi$
$179$	−24.0000	−1.79384	−0.896922	+	0.442189i	$0.854202\pi$
$180$	−6.00000	−0.447214
$181$	14.0000	1.04061	0.520306	−	0.853980i	$-0.325818\pi$
$181$	14.0000	1.04061	0.520306	+	0.853980i	$0.325818\pi$
$182$	0	0
$183$	−10.0000	−0.739221
$184$	7.00000	0.516047
$185$	−12.0000	−0.882258
$186$	−1.00000	−0.0733236
$187$	−1.00000	−0.0731272
$188$	−13.0000	−0.948122
$189$	0	0
$190$	21.0000	1.52350
$191$	20.0000	1.44715	0.723575	−	0.690246i	$-0.242498\pi$
$191$	20.0000	1.44715	0.723575	+	0.690246i	$0.242498\pi$
$192$	−1.00000	−0.0721688
$193$	−2.00000	−0.143963	−0.0719816	−	0.997406i	$-0.522932\pi$
$193$	−2.00000	−0.143963	−0.0719816	+	0.997406i	$0.522932\pi$
$194$	1.00000	0.0717958
$195$	−18.0000	−1.28901
$196$	−7.00000	−0.500000
$197$	−6.00000	−0.427482	−0.213741	−	0.976890i	$-0.568565\pi$
$197$	−6.00000	−0.427482	−0.213741	+	0.976890i	$0.568565\pi$
$198$	−2.00000	−0.142134
$199$	1.00000	0.0708881	0.0354441	−	0.999372i	$-0.488715\pi$
$199$	1.00000	0.0708881	0.0354441	+	0.999372i	$0.488715\pi$
$200$	4.00000	0.282843
$201$	4.00000	0.282138
$202$	−16.0000	−1.12576
$203$	0	0
$204$	1.00000	0.0700140
$205$	6.00000	0.419058
$206$	−5.00000	−0.348367
$207$	−14.0000	−0.973067
$208$	6.00000	0.416025
$209$	7.00000	0.484200
$210$	0	0
$211$	−1.00000	−0.0688428	−0.0344214	−	0.999407i	$-0.510959\pi$
$211$	−1.00000	−0.0688428	−0.0344214	+	0.999407i	$0.510959\pi$
$212$	−1.00000	−0.0686803
$213$	−10.0000	−0.685189
$214$	16.0000	1.09374
$215$	−33.0000	−2.25058
$216$	5.00000	0.340207
$217$	0	0
$218$	18.0000	1.21911
$219$	4.00000	0.270295
$220$	3.00000	0.202260
$221$	−6.00000	−0.403604
$222$	4.00000	0.268462
$223$	−21.0000	−1.40626	−0.703132	−	0.711059i	$-0.748216\pi$
$223$	−21.0000	−1.40626	−0.703132	+	0.711059i	$0.748216\pi$
$224$	0	0
$225$	−8.00000	−0.533333
$226$	−15.0000	−0.997785
$227$	18.0000	1.19470	0.597351	−	0.801980i	$-0.296220\pi$
$227$	18.0000	1.19470	0.597351	+	0.801980i	$0.296220\pi$
$228$	−7.00000	−0.463586
$229$	6.00000	0.396491	0.198246	−	0.980152i	$-0.436476\pi$
$229$	6.00000	0.396491	0.198246	+	0.980152i	$0.436476\pi$
$230$	21.0000	1.38470
$231$	0	0
$232$	0	0
$233$	−12.0000	−0.786146	−0.393073	−	0.919507i	$-0.628588\pi$
$233$	−12.0000	−0.786146	−0.393073	+	0.919507i	$0.628588\pi$
$234$	−12.0000	−0.784465
$235$	−39.0000	−2.54408
$236$	7.00000	0.455661
$237$	12.0000	0.779484
$238$	0	0
$239$	−9.00000	−0.582162	−0.291081	−	0.956698i	$-0.594015\pi$
$239$	−9.00000	−0.582162	−0.291081	+	0.956698i	$0.594015\pi$
$240$	−3.00000	−0.193649
$241$	10.0000	0.644157	0.322078	−	0.946713i	$-0.395619\pi$
$241$	10.0000	0.644157	0.322078	+	0.946713i	$0.395619\pi$
$242$	−10.0000	−0.642824
$243$	−16.0000	−1.02640
$244$	10.0000	0.640184
$245$	−21.0000	−1.34164
$246$	−2.00000	−0.127515
$247$	42.0000	2.67240
$248$	1.00000	0.0635001
$249$	2.00000	0.126745
$250$	−3.00000	−0.189737
$251$	−28.0000	−1.76734	−0.883672	−	0.468106i	$-0.844936\pi$
$251$	−28.0000	−1.76734	−0.883672	+	0.468106i	$0.844936\pi$
$252$	0	0
$253$	7.00000	0.440086
$254$	20.0000	1.25491
$255$	3.00000	0.187867
$256$	1.00000	0.0625000
$257$	−14.0000	−0.873296	−0.436648	−	0.899632i	$-0.643834\pi$
$257$	−14.0000	−0.873296	−0.436648	+	0.899632i	$0.643834\pi$
$258$	11.0000	0.684830
$259$	0	0
$260$	18.0000	1.11631
$261$	0	0
$262$	3.00000	0.185341
$263$	29.0000	1.78822	0.894108	−	0.447851i	$-0.147810\pi$
$263$	29.0000	1.78822	0.894108	+	0.447851i	$0.147810\pi$
$264$	−1.00000	−0.0615457
$265$	−3.00000	−0.184289
$266$	0	0
$267$	−6.00000	−0.367194
$268$	−4.00000	−0.244339
$269$	−13.0000	−0.792624	−0.396312	−	0.918116i	$-0.629710\pi$
$269$	−13.0000	−0.792624	−0.396312	+	0.918116i	$0.629710\pi$
$270$	15.0000	0.912871
$271$	−28.0000	−1.70088	−0.850439	−	0.526073i	$-0.823664\pi$
$271$	−28.0000	−1.70088	−0.850439	+	0.526073i	$0.823664\pi$
$272$	−1.00000	−0.0606339
$273$	0	0
$274$	11.0000	0.664534
$275$	4.00000	0.241209
$276$	−7.00000	−0.421350
$277$	−2.00000	−0.120168	−0.0600842	−	0.998193i	$-0.519137\pi$
$277$	−2.00000	−0.120168	−0.0600842	+	0.998193i	$0.519137\pi$
$278$	−2.00000	−0.119952
$279$	−2.00000	−0.119737
$280$	0	0
$281$	16.0000	0.954480	0.477240	−	0.878773i	$-0.341637\pi$
$281$	16.0000	0.954480	0.477240	+	0.878773i	$0.341637\pi$
$282$	13.0000	0.774139
$283$	0	0		−	1.00000i	$-0.5\pi$
$283$	0	0			1.00000i	$0.5\pi$
$284$	10.0000	0.593391
$285$	−21.0000	−1.24393
$286$	6.00000	0.354787
$287$	0	0
$288$	−2.00000	−0.117851
$289$	−16.0000	−0.941176
$290$	0	0
$291$	−1.00000	−0.0586210
$292$	−4.00000	−0.234082
$293$	−16.0000	−0.934730	−0.467365	−	0.884064i	$-0.654797\pi$
$293$	−16.0000	−0.934730	−0.467365	+	0.884064i	$0.654797\pi$
$294$	7.00000	0.408248
$295$	21.0000	1.22267
$296$	−4.00000	−0.232495
$297$	5.00000	0.290129
$298$	6.00000	0.347571
$299$	42.0000	2.42892
$300$	−4.00000	−0.230940
$301$	0	0
$302$	12.0000	0.690522
$303$	16.0000	0.919176
$304$	7.00000	0.401478
$305$	30.0000	1.71780
$306$	2.00000	0.114332
$307$	22.0000	1.25561	0.627803	−	0.778372i	$-0.283954\pi$
$307$	22.0000	1.25561	0.627803	+	0.778372i	$0.283954\pi$
$308$	0	0
$309$	5.00000	0.284440
$310$	3.00000	0.170389
$311$	30.0000	1.70114	0.850572	−	0.525859i	$-0.176256\pi$
$311$	30.0000	1.70114	0.850572	+	0.525859i	$0.176256\pi$
$312$	−6.00000	−0.339683
$313$	−28.0000	−1.58265	−0.791327	−	0.611393i	$-0.790609\pi$
$313$	−28.0000	−1.58265	−0.791327	+	0.611393i	$0.790609\pi$
$314$	11.0000	0.620766
$315$	0	0
$316$	−12.0000	−0.675053
$317$	−15.0000	−0.842484	−0.421242	−	0.906948i	$-0.638406\pi$
$317$	−15.0000	−0.842484	−0.421242	+	0.906948i	$0.638406\pi$
$318$	1.00000	0.0560772
$319$	0	0
$320$	3.00000	0.167705
$321$	−16.0000	−0.893033
$322$	0	0
$323$	−7.00000	−0.389490
$324$	1.00000	0.0555556
$325$	24.0000	1.33128
$326$	12.0000	0.664619
$327$	−18.0000	−0.995402
$328$	2.00000	0.110432
$329$	0	0
$330$	−3.00000	−0.165145
$331$	−10.0000	−0.549650	−0.274825	−	0.961494i	$-0.588620\pi$
$331$	−10.0000	−0.549650	−0.274825	+	0.961494i	$0.588620\pi$
$332$	−2.00000	−0.109764
$333$	8.00000	0.438397
$334$	2.00000	0.109435
$335$	−12.0000	−0.655630
$336$	0	0
$337$	−9.00000	−0.490261	−0.245131	−	0.969490i	$-0.578831\pi$
$337$	−9.00000	−0.490261	−0.245131	+	0.969490i	$0.578831\pi$
$338$	23.0000	1.25104
$339$	15.0000	0.814688
$340$	−3.00000	−0.162698
$341$	1.00000	0.0541530
$342$	−14.0000	−0.757033
$343$	0	0
$344$	−11.0000	−0.593080
$345$	−21.0000	−1.13060
$346$	1.00000	0.0537603
$347$	−16.0000	−0.858925	−0.429463	−	0.903085i	$-0.641297\pi$
$347$	−16.0000	−0.858925	−0.429463	+	0.903085i	$0.641297\pi$
$348$	0	0
$349$	−26.0000	−1.39175	−0.695874	−	0.718164i	$-0.744983\pi$
$349$	−26.0000	−1.39175	−0.695874	+	0.718164i	$0.744983\pi$
$350$	0	0
$351$	30.0000	1.60128
$352$	1.00000	0.0533002
$353$	−14.0000	−0.745145	−0.372572	−	0.928003i	$-0.621524\pi$
$353$	−14.0000	−0.745145	−0.372572	+	0.928003i	$0.621524\pi$
$354$	−7.00000	−0.372046
$355$	30.0000	1.59223
$356$	6.00000	0.317999
$357$	0	0
$358$	−24.0000	−1.26844
$359$	20.0000	1.05556	0.527780	−	0.849381i	$-0.323025\pi$
$359$	20.0000	1.05556	0.527780	+	0.849381i	$0.323025\pi$
$360$	−6.00000	−0.316228
$361$	30.0000	1.57895
$362$	14.0000	0.735824
$363$	10.0000	0.524864
$364$	0	0
$365$	−12.0000	−0.628109
$366$	−10.0000	−0.522708
$367$	33.0000	1.72259	0.861293	−	0.508109i	$-0.169655\pi$
$367$	33.0000	1.72259	0.861293	+	0.508109i	$0.169655\pi$
$368$	7.00000	0.364900
$369$	−4.00000	−0.208232
$370$	−12.0000	−0.623850
$371$	0	0
$372$	−1.00000	−0.0518476
$373$	1.00000	0.0517780	0.0258890	−	0.999665i	$-0.491758\pi$
$373$	1.00000	0.0517780	0.0258890	+	0.999665i	$0.491758\pi$
$374$	−1.00000	−0.0517088
$375$	3.00000	0.154919
$376$	−13.0000	−0.670424
$377$	0	0
$378$	0	0
$379$	28.0000	1.43826	0.719132	−	0.694874i	$-0.244540\pi$
$379$	28.0000	1.43826	0.719132	+	0.694874i	$0.244540\pi$
$380$	21.0000	1.07728
$381$	−20.0000	−1.02463
$382$	20.0000	1.02329
$383$	−8.00000	−0.408781	−0.204390	−	0.978889i	$-0.565521\pi$
$383$	−8.00000	−0.408781	−0.204390	+	0.978889i	$0.565521\pi$
$384$	−1.00000	−0.0510310
$385$	0	0
$386$	−2.00000	−0.101797
$387$	22.0000	1.11832
$388$	1.00000	0.0507673
$389$	−26.0000	−1.31825	−0.659126	−	0.752032i	$-0.729074\pi$
$389$	−26.0000	−1.31825	−0.659126	+	0.752032i	$0.729074\pi$
$390$	−18.0000	−0.911465
$391$	−7.00000	−0.354005
$392$	−7.00000	−0.353553
$393$	−3.00000	−0.151330
$394$	−6.00000	−0.302276
$395$	−36.0000	−1.81136
$396$	−2.00000	−0.100504
$397$	−32.0000	−1.60603	−0.803017	−	0.595956i	$-0.796773\pi$
$397$	−32.0000	−1.60603	−0.803017	+	0.595956i	$0.796773\pi$
$398$	1.00000	0.0501255
$399$	0	0
$400$	4.00000	0.200000
$401$	6.00000	0.299626	0.149813	−	0.988714i	$-0.452133\pi$
$401$	6.00000	0.299626	0.149813	+	0.988714i	$0.452133\pi$
$402$	4.00000	0.199502
$403$	6.00000	0.298881
$404$	−16.0000	−0.796030
$405$	3.00000	0.149071
$406$	0	0
$407$	−4.00000	−0.198273
$408$	1.00000	0.0495074
$409$	9.00000	0.445021	0.222511	−	0.974930i	$-0.428575\pi$
$409$	9.00000	0.445021	0.222511	+	0.974930i	$0.428575\pi$
$410$	6.00000	0.296319
$411$	−11.0000	−0.542590
$412$	−5.00000	−0.246332
$413$	0	0
$414$	−14.0000	−0.688062
$415$	−6.00000	−0.294528
$416$	6.00000	0.294174
$417$	2.00000	0.0979404
$418$	7.00000	0.342381
$419$	30.0000	1.46560	0.732798	−	0.680446i	$-0.238214\pi$
$419$	30.0000	1.46560	0.732798	+	0.680446i	$0.238214\pi$
$420$	0	0
$421$	−26.0000	−1.26716	−0.633581	−	0.773676i	$-0.718416\pi$
$421$	−26.0000	−1.26716	−0.633581	+	0.773676i	$0.718416\pi$
$422$	−1.00000	−0.0486792
$423$	26.0000	1.26416
$424$	−1.00000	−0.0485643
$425$	−4.00000	−0.194029
$426$	−10.0000	−0.484502
$427$	0	0
$428$	16.0000	0.773389
$429$	−6.00000	−0.289683
$430$	−33.0000	−1.59140
$431$	19.0000	0.915198	0.457599	−	0.889159i	$-0.348710\pi$
$431$	19.0000	0.915198	0.457599	+	0.889159i	$0.348710\pi$
$432$	5.00000	0.240563
$433$	23.0000	1.10531	0.552655	−	0.833410i	$-0.313615\pi$
$433$	23.0000	1.10531	0.552655	+	0.833410i	$0.313615\pi$
$434$	0	0
$435$	0	0
$436$	18.0000	0.862044
$437$	49.0000	2.34399
$438$	4.00000	0.191127
$439$	20.0000	0.954548	0.477274	−	0.878755i	$-0.341625\pi$
$439$	20.0000	0.954548	0.477274	+	0.878755i	$0.341625\pi$
$440$	3.00000	0.143019
$441$	14.0000	0.666667
$442$	−6.00000	−0.285391
$443$	−4.00000	−0.190046	−0.0950229	−	0.995475i	$-0.530292\pi$
$443$	−4.00000	−0.190046	−0.0950229	+	0.995475i	$0.530292\pi$
$444$	4.00000	0.189832
$445$	18.0000	0.853282
$446$	−21.0000	−0.994379
$447$	−6.00000	−0.283790
$448$	0	0
$449$	6.00000	0.283158	0.141579	−	0.989927i	$-0.454782\pi$
$449$	6.00000	0.283158	0.141579	+	0.989927i	$0.454782\pi$
$450$	−8.00000	−0.377124
$451$	2.00000	0.0941763
$452$	−15.0000	−0.705541
$453$	−12.0000	−0.563809
$454$	18.0000	0.844782
$455$	0	0
$456$	−7.00000	−0.327805
$457$	−2.00000	−0.0935561	−0.0467780	−	0.998905i	$-0.514895\pi$
$457$	−2.00000	−0.0935561	−0.0467780	+	0.998905i	$0.514895\pi$
$458$	6.00000	0.280362
$459$	−5.00000	−0.233380
$460$	21.0000	0.979130
$461$	−18.0000	−0.838344	−0.419172	−	0.907907i	$-0.637680\pi$
$461$	−18.0000	−0.838344	−0.419172	+	0.907907i	$0.637680\pi$
$462$	0	0
$463$	10.0000	0.464739	0.232370	−	0.972628i	$-0.425352\pi$
$463$	10.0000	0.464739	0.232370	+	0.972628i	$0.425352\pi$
$464$	0	0
$465$	−3.00000	−0.139122
$466$	−12.0000	−0.555889
$467$	38.0000	1.75843	0.879215	−	0.476425i	$-0.158068\pi$
$467$	38.0000	1.75843	0.879215	+	0.476425i	$0.158068\pi$
$468$	−12.0000	−0.554700
$469$	0	0
$470$	−39.0000	−1.79894
$471$	−11.0000	−0.506853
$472$	7.00000	0.322201
$473$	−11.0000	−0.505781
$474$	12.0000	0.551178
$475$	28.0000	1.28473
$476$	0	0
$477$	2.00000	0.0915737
$478$	−9.00000	−0.411650
$479$	−29.0000	−1.32504	−0.662522	−	0.749043i	$-0.730514\pi$
$479$	−29.0000	−1.32504	−0.662522	+	0.749043i	$0.730514\pi$
$480$	−3.00000	−0.136931
$481$	−24.0000	−1.09431
$482$	10.0000	0.455488
$483$	0	0
$484$	−10.0000	−0.454545
$485$	3.00000	0.136223
$486$	−16.0000	−0.725775
$487$	2.00000	0.0906287	0.0453143	−	0.998973i	$-0.485571\pi$
$487$	2.00000	0.0906287	0.0453143	+	0.998973i	$0.485571\pi$
$488$	10.0000	0.452679
$489$	−12.0000	−0.542659
$490$	−21.0000	−0.948683
$491$	12.0000	0.541552	0.270776	−	0.962642i	$-0.412720\pi$
$491$	12.0000	0.541552	0.270776	+	0.962642i	$0.412720\pi$
$492$	−2.00000	−0.0901670
$493$	0	0
$494$	42.0000	1.88967
$495$	−6.00000	−0.269680
$496$	1.00000	0.0449013
$497$	0	0
$498$	2.00000	0.0896221
$499$	32.0000	1.43252	0.716258	−	0.697835i	$-0.245853\pi$
$499$	32.0000	1.43252	0.716258	+	0.697835i	$0.245853\pi$
$500$	−3.00000	−0.134164
$501$	−2.00000	−0.0893534
$502$	−28.0000	−1.24970
$503$	1.00000	0.0445878	0.0222939	−	0.999751i	$-0.492903\pi$
$503$	1.00000	0.0445878	0.0222939	+	0.999751i	$0.492903\pi$
$504$	0	0
$505$	−48.0000	−2.13597
$506$	7.00000	0.311188
$507$	−23.0000	−1.02147
$508$	20.0000	0.887357
$509$	−35.0000	−1.55135	−0.775674	−	0.631134i	$-0.782590\pi$
$509$	−35.0000	−1.55135	−0.775674	+	0.631134i	$0.782590\pi$
$510$	3.00000	0.132842
$511$	0	0
$512$	1.00000	0.0441942
$513$	35.0000	1.54529
$514$	−14.0000	−0.617514
$515$	−15.0000	−0.660979
$516$	11.0000	0.484248
$517$	−13.0000	−0.571739
$518$	0	0
$519$	−1.00000	−0.0438951
$520$	18.0000	0.789352
$521$	19.0000	0.832405	0.416203	−	0.909272i	$-0.363361\pi$
$521$	19.0000	0.832405	0.416203	+	0.909272i	$0.363361\pi$
$522$	0	0
$523$	−14.0000	−0.612177	−0.306089	−	0.952003i	$-0.599020\pi$
$523$	−14.0000	−0.612177	−0.306089	+	0.952003i	$0.599020\pi$
$524$	3.00000	0.131056
$525$	0	0
$526$	29.0000	1.26446
$527$	−1.00000	−0.0435607
$528$	−1.00000	−0.0435194
$529$	26.0000	1.13043
$530$	−3.00000	−0.130312
$531$	−14.0000	−0.607548
$532$	0	0
$533$	12.0000	0.519778
$534$	−6.00000	−0.259645
$535$	48.0000	2.07522
$536$	−4.00000	−0.172774
$537$	24.0000	1.03568
$538$	−13.0000	−0.560470
$539$	−7.00000	−0.301511
$540$	15.0000	0.645497
$541$	7.00000	0.300954	0.150477	−	0.988614i	$-0.451919\pi$
$541$	7.00000	0.300954	0.150477	+	0.988614i	$0.451919\pi$
$542$	−28.0000	−1.20270
$543$	−14.0000	−0.600798
$544$	−1.00000	−0.0428746
$545$	54.0000	2.31311
$546$	0	0
$547$	0	0		−	1.00000i	$-0.5\pi$
$547$	0	0			1.00000i	$0.5\pi$
$548$	11.0000	0.469897
$549$	−20.0000	−0.853579
$550$	4.00000	0.170561
$551$	0	0
$552$	−7.00000	−0.297940
$553$	0	0
$554$	−2.00000	−0.0849719
$555$	12.0000	0.509372
$556$	−2.00000	−0.0848189
$557$	−18.0000	−0.762684	−0.381342	−	0.924434i	$-0.624538\pi$
$557$	−18.0000	−0.762684	−0.381342	+	0.924434i	$0.624538\pi$
$558$	−2.00000	−0.0846668
$559$	−66.0000	−2.79150
$560$	0	0
$561$	1.00000	0.0422200
$562$	16.0000	0.674919
$563$	12.0000	0.505740	0.252870	−	0.967500i	$-0.418626\pi$
$563$	12.0000	0.505740	0.252870	+	0.967500i	$0.418626\pi$
$564$	13.0000	0.547399
$565$	−45.0000	−1.89316
$566$	0	0
$567$	0	0
$568$	10.0000	0.419591
$569$	−42.0000	−1.76073	−0.880366	−	0.474295i	$-0.842703\pi$
$569$	−42.0000	−1.76073	−0.880366	+	0.474295i	$0.842703\pi$
$570$	−21.0000	−0.879593
$571$	−33.0000	−1.38101	−0.690504	−	0.723329i	$-0.742611\pi$
$571$	−33.0000	−1.38101	−0.690504	+	0.723329i	$0.742611\pi$
$572$	6.00000	0.250873
$573$	−20.0000	−0.835512
$574$	0	0
$575$	28.0000	1.16768
$576$	−2.00000	−0.0833333
$577$	−30.0000	−1.24892	−0.624458	−	0.781058i	$-0.714680\pi$
$577$	−30.0000	−1.24892	−0.624458	+	0.781058i	$0.714680\pi$
$578$	−16.0000	−0.665512
$579$	2.00000	0.0831172
$580$	0	0
$581$	0	0
$582$	−1.00000	−0.0414513
$583$	−1.00000	−0.0414158
$584$	−4.00000	−0.165521
$585$	−36.0000	−1.48842
$586$	−16.0000	−0.660954
$587$	−14.0000	−0.577842	−0.288921	−	0.957353i	$-0.593296\pi$
$587$	−14.0000	−0.577842	−0.288921	+	0.957353i	$0.593296\pi$
$588$	7.00000	0.288675
$589$	7.00000	0.288430
$590$	21.0000	0.864556
$591$	6.00000	0.246807
$592$	−4.00000	−0.164399
$593$	6.00000	0.246390	0.123195	−	0.992382i	$-0.460686\pi$
$593$	6.00000	0.246390	0.123195	+	0.992382i	$0.460686\pi$
$594$	5.00000	0.205152
$595$	0	0
$596$	6.00000	0.245770
$597$	−1.00000	−0.0409273
$598$	42.0000	1.71751
$599$	0	0		−	1.00000i	$-0.5\pi$
$599$	0	0			1.00000i	$0.5\pi$
$600$	−4.00000	−0.163299
$601$	6.00000	0.244745	0.122373	−	0.992484i	$-0.460950\pi$
$601$	6.00000	0.244745	0.122373	+	0.992484i	$0.460950\pi$
$602$	0	0
$603$	8.00000	0.325785
$604$	12.0000	0.488273
$605$	−30.0000	−1.21967
$606$	16.0000	0.649956
$607$	−5.00000	−0.202944	−0.101472	−	0.994838i	$-0.532355\pi$
$607$	−5.00000	−0.202944	−0.101472	+	0.994838i	$0.532355\pi$
$608$	7.00000	0.283887
$609$	0	0
$610$	30.0000	1.21466
$611$	−78.0000	−3.15554
$612$	2.00000	0.0808452
$613$	−9.00000	−0.363507	−0.181753	−	0.983344i	$-0.558177\pi$
$613$	−9.00000	−0.363507	−0.181753	+	0.983344i	$0.558177\pi$
$614$	22.0000	0.887848
$615$	−6.00000	−0.241943
$616$	0	0
$617$	9.00000	0.362326	0.181163	−	0.983453i	$-0.442014\pi$
$617$	9.00000	0.362326	0.181163	+	0.983453i	$0.442014\pi$
$618$	5.00000	0.201129
$619$	−22.0000	−0.884255	−0.442127	−	0.896952i	$-0.645776\pi$
$619$	−22.0000	−0.884255	−0.442127	+	0.896952i	$0.645776\pi$
$620$	3.00000	0.120483
$621$	35.0000	1.40450
$622$	30.0000	1.20289
$623$	0	0
$624$	−6.00000	−0.240192
$625$	−29.0000	−1.16000
$626$	−28.0000	−1.11911
$627$	−7.00000	−0.279553
$628$	11.0000	0.438948
$629$	4.00000	0.159490
$630$	0	0
$631$	28.0000	1.11466	0.557331	−	0.830290i	$-0.311825\pi$
$631$	28.0000	1.11466	0.557331	+	0.830290i	$0.311825\pi$
$632$	−12.0000	−0.477334
$633$	1.00000	0.0397464
$634$	−15.0000	−0.595726
$635$	60.0000	2.38103
$636$	1.00000	0.0396526
$637$	−42.0000	−1.66410
$638$	0	0
$639$	−20.0000	−0.791188
$640$	3.00000	0.118585
$641$	18.0000	0.710957	0.355479	−	0.934684i	$-0.384318\pi$
$641$	18.0000	0.710957	0.355479	+	0.934684i	$0.384318\pi$
$642$	−16.0000	−0.631470
$643$	19.0000	0.749287	0.374643	−	0.927169i	$-0.377765\pi$
$643$	19.0000	0.749287	0.374643	+	0.927169i	$0.377765\pi$
$644$	0	0
$645$	33.0000	1.29937
$646$	−7.00000	−0.275411
$647$	18.0000	0.707653	0.353827	−	0.935311i	$-0.384880\pi$
$647$	18.0000	0.707653	0.353827	+	0.935311i	$0.384880\pi$
$648$	1.00000	0.0392837
$649$	7.00000	0.274774
$650$	24.0000	0.941357
$651$	0	0
$652$	12.0000	0.469956
$653$	−3.00000	−0.117399	−0.0586995	−	0.998276i	$-0.518695\pi$
$653$	−3.00000	−0.117399	−0.0586995	+	0.998276i	$0.518695\pi$
$654$	−18.0000	−0.703856
$655$	9.00000	0.351659
$656$	2.00000	0.0780869
$657$	8.00000	0.312110
$658$	0	0
$659$	−5.00000	−0.194772	−0.0973862	−	0.995247i	$-0.531048\pi$
$659$	−5.00000	−0.194772	−0.0973862	+	0.995247i	$0.531048\pi$
$660$	−3.00000	−0.116775
$661$	−20.0000	−0.777910	−0.388955	−	0.921257i	$-0.627164\pi$
$661$	−20.0000	−0.777910	−0.388955	+	0.921257i	$0.627164\pi$
$662$	−10.0000	−0.388661
$663$	6.00000	0.233021
$664$	−2.00000	−0.0776151
$665$	0	0
$666$	8.00000	0.309994
$667$	0	0
$668$	2.00000	0.0773823
$669$	21.0000	0.811907
$670$	−12.0000	−0.463600
$671$	10.0000	0.386046
$672$	0	0
$673$	8.00000	0.308377	0.154189	−	0.988041i	$-0.450724\pi$
$673$	8.00000	0.308377	0.154189	+	0.988041i	$0.450724\pi$
$674$	−9.00000	−0.346667
$675$	20.0000	0.769800
$676$	23.0000	0.884615
$677$	9.00000	0.345898	0.172949	−	0.984931i	$-0.444670\pi$
$677$	9.00000	0.345898	0.172949	+	0.984931i	$0.444670\pi$
$678$	15.0000	0.576072
$679$	0	0
$680$	−3.00000	−0.115045
$681$	−18.0000	−0.689761
$682$	1.00000	0.0382920
$683$	−30.0000	−1.14792	−0.573959	−	0.818884i	$-0.694593\pi$
$683$	−30.0000	−1.14792	−0.573959	+	0.818884i	$0.694593\pi$
$684$	−14.0000	−0.535303
$685$	33.0000	1.26087
$686$	0	0
$687$	−6.00000	−0.228914
$688$	−11.0000	−0.419371
$689$	−6.00000	−0.228582
$690$	−21.0000	−0.799456
$691$	−22.0000	−0.836919	−0.418460	−	0.908235i	$-0.637430\pi$
$691$	−22.0000	−0.836919	−0.418460	+	0.908235i	$0.637430\pi$
$692$	1.00000	0.0380143
$693$	0	0
$694$	−16.0000	−0.607352
$695$	−6.00000	−0.227593
$696$	0	0
$697$	−2.00000	−0.0757554
$698$	−26.0000	−0.984115
$699$	12.0000	0.453882
$700$	0	0
$701$	−18.0000	−0.679851	−0.339925	−	0.940452i	$-0.610402\pi$
$701$	−18.0000	−0.679851	−0.339925	+	0.940452i	$0.610402\pi$
$702$	30.0000	1.13228
$703$	−28.0000	−1.05604
$704$	1.00000	0.0376889
$705$	39.0000	1.46882
$706$	−14.0000	−0.526897
$707$	0	0
$708$	−7.00000	−0.263076
$709$	−38.0000	−1.42712	−0.713560	−	0.700594i	$-0.752918\pi$
$709$	−38.0000	−1.42712	−0.713560	+	0.700594i	$0.752918\pi$
$710$	30.0000	1.12588
$711$	24.0000	0.900070
$712$	6.00000	0.224860
$713$	7.00000	0.262152
$714$	0	0
$715$	18.0000	0.673162
$716$	−24.0000	−0.896922
$717$	9.00000	0.336111
$718$	20.0000	0.746393
$719$	−49.0000	−1.82739	−0.913696	−	0.406399i	$-0.866784\pi$
$719$	−49.0000	−1.82739	−0.913696	+	0.406399i	$0.866784\pi$
$720$	−6.00000	−0.223607
$721$	0	0
$722$	30.0000	1.11648
$723$	−10.0000	−0.371904
$724$	14.0000	0.520306
$725$	0	0
$726$	10.0000	0.371135
$727$	−15.0000	−0.556319	−0.278160	−	0.960535i	$-0.589724\pi$
$727$	−15.0000	−0.556319	−0.278160	+	0.960535i	$0.589724\pi$
$728$	0	0
$729$	13.0000	0.481481
$730$	−12.0000	−0.444140
$731$	11.0000	0.406850
$732$	−10.0000	−0.369611
$733$	22.0000	0.812589	0.406294	−	0.913742i	$-0.366821\pi$
$733$	22.0000	0.812589	0.406294	+	0.913742i	$0.366821\pi$
$734$	33.0000	1.21805
$735$	21.0000	0.774597
$736$	7.00000	0.258023
$737$	−4.00000	−0.147342
$738$	−4.00000	−0.147242
$739$	−20.0000	−0.735712	−0.367856	−	0.929883i	$-0.619908\pi$
$739$	−20.0000	−0.735712	−0.367856	+	0.929883i	$0.619908\pi$
$740$	−12.0000	−0.441129
$741$	−42.0000	−1.54291
$742$	0	0
$743$	46.0000	1.68758	0.843788	−	0.536676i	$-0.180320\pi$
$743$	46.0000	1.68758	0.843788	+	0.536676i	$0.180320\pi$
$744$	−1.00000	−0.0366618
$745$	18.0000	0.659469
$746$	1.00000	0.0366126
$747$	4.00000	0.146352
$748$	−1.00000	−0.0365636
$749$	0	0
$750$	3.00000	0.109545
$751$	39.0000	1.42313	0.711565	−	0.702620i	$-0.247987\pi$
$751$	39.0000	1.42313	0.711565	+	0.702620i	$0.247987\pi$
$752$	−13.0000	−0.474061
$753$	28.0000	1.02038
$754$	0	0
$755$	36.0000	1.31017
$756$	0	0
$757$	−14.0000	−0.508839	−0.254419	−	0.967094i	$-0.581884\pi$
$757$	−14.0000	−0.508839	−0.254419	+	0.967094i	$0.581884\pi$
$758$	28.0000	1.01701
$759$	−7.00000	−0.254084
$760$	21.0000	0.761750
$761$	3.00000	0.108750	0.0543750	−	0.998521i	$-0.482683\pi$
$761$	3.00000	0.108750	0.0543750	+	0.998521i	$0.482683\pi$
$762$	−20.0000	−0.724524
$763$	0	0
$764$	20.0000	0.723575
$765$	6.00000	0.216930
$766$	−8.00000	−0.289052
$767$	42.0000	1.51653
$768$	−1.00000	−0.0360844
$769$	14.0000	0.504853	0.252426	−	0.967616i	$-0.418771\pi$
$769$	14.0000	0.504853	0.252426	+	0.967616i	$0.418771\pi$
$770$	0	0
$771$	14.0000	0.504198
$772$	−2.00000	−0.0719816
$773$	−21.0000	−0.755318	−0.377659	−	0.925945i	$-0.623271\pi$
$773$	−21.0000	−0.755318	−0.377659	+	0.925945i	$0.623271\pi$
$774$	22.0000	0.790774
$775$	4.00000	0.143684
$776$	1.00000	0.0358979
$777$	0	0
$778$	−26.0000	−0.932145
$779$	14.0000	0.501602
$780$	−18.0000	−0.644503
$781$	10.0000	0.357828
$782$	−7.00000	−0.250319
$783$	0	0
$784$	−7.00000	−0.250000
$785$	33.0000	1.17782
$786$	−3.00000	−0.107006
$787$	−4.00000	−0.142585	−0.0712923	−	0.997455i	$-0.522712\pi$
$787$	−4.00000	−0.142585	−0.0712923	+	0.997455i	$0.522712\pi$
$788$	−6.00000	−0.213741
$789$	−29.0000	−1.03243
$790$	−36.0000	−1.28082
$791$	0	0
$792$	−2.00000	−0.0710669
$793$	60.0000	2.13066
$794$	−32.0000	−1.13564
$795$	3.00000	0.106399
$796$	1.00000	0.0354441
$797$	−26.0000	−0.920967	−0.460484	−	0.887668i	$-0.652324\pi$
$797$	−26.0000	−0.920967	−0.460484	+	0.887668i	$0.652324\pi$
$798$	0	0
$799$	13.0000	0.459907
$800$	4.00000	0.141421
$801$	−12.0000	−0.423999
$802$	6.00000	0.211867
$803$	−4.00000	−0.141157
$804$	4.00000	0.141069
$805$	0	0
$806$	6.00000	0.211341
$807$	13.0000	0.457622
$808$	−16.0000	−0.562878
$809$	−16.0000	−0.562530	−0.281265	−	0.959630i	$-0.590754\pi$
$809$	−16.0000	−0.562530	−0.281265	+	0.959630i	$0.590754\pi$
$810$	3.00000	0.105409
$811$	0	0		−	1.00000i	$-0.5\pi$
$811$	0	0			1.00000i	$0.5\pi$
$812$	0	0
$813$	28.0000	0.982003
$814$	−4.00000	−0.140200
$815$	36.0000	1.26102
$816$	1.00000	0.0350070
$817$	−77.0000	−2.69389
$818$	9.00000	0.314678
$819$	0	0
$820$	6.00000	0.209529
$821$	−36.0000	−1.25641	−0.628204	−	0.778048i	$-0.716210\pi$
$821$	−36.0000	−1.25641	−0.628204	+	0.778048i	$0.716210\pi$
$822$	−11.0000	−0.383669
$823$	−44.0000	−1.53374	−0.766872	−	0.641800i	$-0.778188\pi$
$823$	−44.0000	−1.53374	−0.766872	+	0.641800i	$0.778188\pi$
$824$	−5.00000	−0.174183
$825$	−4.00000	−0.139262
$826$	0	0
$827$	46.0000	1.59958	0.799788	−	0.600282i	$-0.204945\pi$
$827$	46.0000	1.59958	0.799788	+	0.600282i	$0.204945\pi$
$828$	−14.0000	−0.486534
$829$	7.00000	0.243120	0.121560	−	0.992584i	$-0.461210\pi$
$829$	7.00000	0.243120	0.121560	+	0.992584i	$0.461210\pi$
$830$	−6.00000	−0.208263
$831$	2.00000	0.0693792
$832$	6.00000	0.208013
$833$	7.00000	0.242536
$834$	2.00000	0.0692543
$835$	6.00000	0.207639
$836$	7.00000	0.242100
$837$	5.00000	0.172825
$838$	30.0000	1.03633
$839$	6.00000	0.207143	0.103572	−	0.994622i	$-0.466973\pi$
$839$	6.00000	0.207143	0.103572	+	0.994622i	$0.466973\pi$
$840$	0	0
$841$	−29.0000	−1.00000
$842$	−26.0000	−0.896019
$843$	−16.0000	−0.551069
$844$	−1.00000	−0.0344214
$845$	69.0000	2.37367
$846$	26.0000	0.893898
$847$	0	0
$848$	−1.00000	−0.0343401
$849$	0	0
$850$	−4.00000	−0.137199
$851$	−28.0000	−0.959828
$852$	−10.0000	−0.342594
$853$	26.0000	0.890223	0.445112	−	0.895475i	$-0.353164\pi$
$853$	26.0000	0.890223	0.445112	+	0.895475i	$0.353164\pi$
$854$	0	0
$855$	−42.0000	−1.43637
$856$	16.0000	0.546869
$857$	10.0000	0.341593	0.170797	−	0.985306i	$-0.445366\pi$
$857$	10.0000	0.341593	0.170797	+	0.985306i	$0.445366\pi$
$858$	−6.00000	−0.204837
$859$	−40.0000	−1.36478	−0.682391	−	0.730987i	$-0.739060\pi$
$859$	−40.0000	−1.36478	−0.682391	+	0.730987i	$0.739060\pi$
$860$	−33.0000	−1.12529
$861$	0	0
$862$	19.0000	0.647143
$863$	−16.0000	−0.544646	−0.272323	−	0.962206i	$-0.587792\pi$
$863$	−16.0000	−0.544646	−0.272323	+	0.962206i	$0.587792\pi$
$864$	5.00000	0.170103
$865$	3.00000	0.102003
$866$	23.0000	0.781572
$867$	16.0000	0.543388
$868$	0	0
$869$	−12.0000	−0.407072
$870$	0	0
$871$	−24.0000	−0.813209
$872$	18.0000	0.609557
$873$	−2.00000	−0.0676897
$874$	49.0000	1.65745
$875$	0	0
$876$	4.00000	0.135147
$877$	−6.00000	−0.202606	−0.101303	−	0.994856i	$-0.532301\pi$
$877$	−6.00000	−0.202606	−0.101303	+	0.994856i	$0.532301\pi$
$878$	20.0000	0.674967
$879$	16.0000	0.539667
$880$	3.00000	0.101130
$881$	2.00000	0.0673817	0.0336909	−	0.999432i	$-0.489274\pi$
$881$	2.00000	0.0673817	0.0336909	+	0.999432i	$0.489274\pi$
$882$	14.0000	0.471405
$883$	−16.0000	−0.538443	−0.269221	−	0.963078i	$-0.586766\pi$
$883$	−16.0000	−0.538443	−0.269221	+	0.963078i	$0.586766\pi$
$884$	−6.00000	−0.201802
$885$	−21.0000	−0.705907
$886$	−4.00000	−0.134383
$887$	−12.0000	−0.402921	−0.201460	−	0.979497i	$-0.564569\pi$
$887$	−12.0000	−0.402921	−0.201460	+	0.979497i	$0.564569\pi$
$888$	4.00000	0.134231
$889$	0	0
$890$	18.0000	0.603361
$891$	1.00000	0.0335013
$892$	−21.0000	−0.703132
$893$	−91.0000	−3.04520
$894$	−6.00000	−0.200670
$895$	−72.0000	−2.40669
$896$	0	0
$897$	−42.0000	−1.40234
$898$	6.00000	0.200223
$899$	0	0
$900$	−8.00000	−0.266667
$901$	1.00000	0.0333148
$902$	2.00000	0.0665927
$903$	0	0
$904$	−15.0000	−0.498893
$905$	42.0000	1.39613
$906$	−12.0000	−0.398673
$907$	19.0000	0.630885	0.315442	−	0.948945i	$-0.397847\pi$
$907$	19.0000	0.630885	0.315442	+	0.948945i	$0.397847\pi$
$908$	18.0000	0.597351
$909$	32.0000	1.06137
$910$	0	0
$911$	−27.0000	−0.894550	−0.447275	−	0.894397i	$-0.647605\pi$
$911$	−27.0000	−0.894550	−0.447275	+	0.894397i	$0.647605\pi$
$912$	−7.00000	−0.231793
$913$	−2.00000	−0.0661903
$914$	−2.00000	−0.0661541
$915$	−30.0000	−0.991769
$916$	6.00000	0.198246
$917$	0	0
$918$	−5.00000	−0.165025
$919$	−6.00000	−0.197922	−0.0989609	−	0.995091i	$-0.531552\pi$
$919$	−6.00000	−0.197922	−0.0989609	+	0.995091i	$0.531552\pi$
$920$	21.0000	0.692349
$921$	−22.0000	−0.724925
$922$	−18.0000	−0.592798
$923$	60.0000	1.97492
$924$	0	0
$925$	−16.0000	−0.526077
$926$	10.0000	0.328620
$927$	10.0000	0.328443
$928$	0	0
$929$	−58.0000	−1.90292	−0.951459	−	0.307775i	$-0.900416\pi$
$929$	−58.0000	−1.90292	−0.951459	+	0.307775i	$0.900416\pi$
$930$	−3.00000	−0.0983739
$931$	−49.0000	−1.60591
$932$	−12.0000	−0.393073
$933$	−30.0000	−0.982156
$934$	38.0000	1.24340
$935$	−3.00000	−0.0981105
$936$	−12.0000	−0.392232
$937$	−35.0000	−1.14340	−0.571700	−	0.820463i	$-0.693716\pi$
$937$	−35.0000	−1.14340	−0.571700	+	0.820463i	$0.693716\pi$
$938$	0	0
$939$	28.0000	0.913745
$940$	−39.0000	−1.27204
$941$	42.0000	1.36916	0.684580	−	0.728937i	$-0.259985\pi$
$941$	42.0000	1.36916	0.684580	+	0.728937i	$0.259985\pi$
$942$	−11.0000	−0.358399
$943$	14.0000	0.455903
$944$	7.00000	0.227831
$945$	0	0
$946$	−11.0000	−0.357641
$947$	18.0000	0.584921	0.292461	−	0.956278i	$-0.405526\pi$
$947$	18.0000	0.584921	0.292461	+	0.956278i	$0.405526\pi$
$948$	12.0000	0.389742
$949$	−24.0000	−0.779073
$950$	28.0000	0.908440
$951$	15.0000	0.486408
$952$	0	0
$953$	−6.00000	−0.194359	−0.0971795	−	0.995267i	$-0.530982\pi$
$953$	−6.00000	−0.194359	−0.0971795	+	0.995267i	$0.530982\pi$
$954$	2.00000	0.0647524
$955$	60.0000	1.94155
$956$	−9.00000	−0.291081
$957$	0	0
$958$	−29.0000	−0.936947
$959$	0	0
$960$	−3.00000	−0.0968246
$961$	1.00000	0.0322581
$962$	−24.0000	−0.773791
$963$	−32.0000	−1.03119
$964$	10.0000	0.322078
$965$	−6.00000	−0.193147
$966$	0	0
$967$	−54.0000	−1.73652	−0.868261	−	0.496107i	$-0.834762\pi$
$967$	−54.0000	−1.73652	−0.868261	+	0.496107i	$0.834762\pi$
$968$	−10.0000	−0.321412
$969$	7.00000	0.224872
$970$	3.00000	0.0963242
$971$	14.0000	0.449281	0.224641	−	0.974442i	$-0.427879\pi$
$971$	14.0000	0.449281	0.224641	+	0.974442i	$0.427879\pi$
$972$	−16.0000	−0.513200
$973$	0	0
$974$	2.00000	0.0640841
$975$	−24.0000	−0.768615
$976$	10.0000	0.320092
$977$	12.0000	0.383914	0.191957	−	0.981403i	$-0.438517\pi$
$977$	12.0000	0.383914	0.191957	+	0.981403i	$0.438517\pi$
$978$	−12.0000	−0.383718
$979$	6.00000	0.191761
$980$	−21.0000	−0.670820
$981$	−36.0000	−1.14939
$982$	12.0000	0.382935
$983$	21.0000	0.669796	0.334898	−	0.942254i	$-0.391298\pi$
$983$	21.0000	0.669796	0.334898	+	0.942254i	$0.391298\pi$
$984$	−2.00000	−0.0637577
$985$	−18.0000	−0.573528
$986$	0	0
$987$	0	0
$988$	42.0000	1.33620
$989$	−77.0000	−2.44846
$990$	−6.00000	−0.190693
$991$	11.0000	0.349427	0.174713	−	0.984619i	$-0.444100\pi$
$991$	11.0000	0.349427	0.174713	+	0.984619i	$0.444100\pi$
$992$	1.00000	0.0317500
$993$	10.0000	0.317340
$994$	0	0
$995$	3.00000	0.0951064
$996$	2.00000	0.0633724
$997$	−26.0000	−0.823428	−0.411714	−	0.911313i	$-0.635070\pi$
$997$	−26.0000	−0.823428	−0.411714	+	0.911313i	$0.635070\pi$
$998$	32.0000	1.01294
$999$	−20.0000	−0.632772

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6014.2.a.a.1.1	✓	1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6014.2.a.a.1.1	✓	1	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 6014 = 2 \cdot 31 \cdot 97 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6014.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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