LMFDB - Embedded newform 6027.2.a.f.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6027,2,Mod(1,6027)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6027, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("6027.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6027 = 3 \cdot 7^{2} \cdot 41 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6027.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:	$48.1258372982$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 861)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	6027.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

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

\(q+1.00000 q^{2} -1.00000 q^{3} -1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} -3.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} -6.00000 q^{11} +1.00000 q^{12} -3.00000 q^{13} -1.00000 q^{15} -1.00000 q^{16} +6.00000 q^{17} +1.00000 q^{18} -6.00000 q^{19} -1.00000 q^{20} -6.00000 q^{22} -9.00000 q^{23} +3.00000 q^{24} -4.00000 q^{25} -3.00000 q^{26} -1.00000 q^{27} +5.00000 q^{29} -1.00000 q^{30} +8.00000 q^{31} +5.00000 q^{32} +6.00000 q^{33} +6.00000 q^{34} -1.00000 q^{36} -7.00000 q^{37} -6.00000 q^{38} +3.00000 q^{39} -3.00000 q^{40} -1.00000 q^{41} +2.00000 q^{43} +6.00000 q^{44} +1.00000 q^{45} -9.00000 q^{46} -3.00000 q^{47} +1.00000 q^{48} -4.00000 q^{50} -6.00000 q^{51} +3.00000 q^{52} +9.00000 q^{53} -1.00000 q^{54} -6.00000 q^{55} +6.00000 q^{57} +5.00000 q^{58} +12.0000 q^{59} +1.00000 q^{60} -6.00000 q^{61} +8.00000 q^{62} +7.00000 q^{64} -3.00000 q^{65} +6.00000 q^{66} +13.0000 q^{67} -6.00000 q^{68} +9.00000 q^{69} -6.00000 q^{71} -3.00000 q^{72} -4.00000 q^{73} -7.00000 q^{74} +4.00000 q^{75} +6.00000 q^{76} +3.00000 q^{78} -11.0000 q^{79} -1.00000 q^{80} +1.00000 q^{81} -1.00000 q^{82} -2.00000 q^{83} +6.00000 q^{85} +2.00000 q^{86} -5.00000 q^{87} +18.0000 q^{88} +4.00000 q^{89} +1.00000 q^{90} +9.00000 q^{92} -8.00000 q^{93} -3.00000 q^{94} -6.00000 q^{95} -5.00000 q^{96} +9.00000 q^{97} -6.00000 q^{99} +O(q^{100})\)

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	0.353553	−	0.935414i	$-0.384973\pi$
$2$	1.00000	0.707107	0.353553	+	0.935414i	$0.384973\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	−1.00000	−0.500000
$5$	1.00000	0.447214	0.223607	−	0.974679i	$-0.428217\pi$
$5$	1.00000	0.447214	0.223607	+	0.974679i	$0.428217\pi$
$6$	−1.00000	−0.408248
$7$	0	0
$7$	0	0
$8$	−3.00000	−1.06066
$9$	1.00000	0.333333
$10$	1.00000	0.316228
$11$	−6.00000	−1.80907	−0.904534	−	0.426401i	$-0.859781\pi$
$11$	−6.00000	−1.80907	−0.904534	+	0.426401i	$0.859781\pi$
$12$	1.00000	0.288675
$13$	−3.00000	−0.832050	−0.416025	−	0.909353i	$-0.636577\pi$
$13$	−3.00000	−0.832050	−0.416025	+	0.909353i	$0.636577\pi$
$14$	0	0
$15$	−1.00000	−0.258199
$16$	−1.00000	−0.250000
$17$	6.00000	1.45521	0.727607	−	0.685994i	$-0.240633\pi$
$17$	6.00000	1.45521	0.727607	+	0.685994i	$0.240633\pi$
$18$	1.00000	0.235702
$19$	−6.00000	−1.37649	−0.688247	−	0.725476i	$-0.741620\pi$
$19$	−6.00000	−1.37649	−0.688247	+	0.725476i	$0.741620\pi$
$20$	−1.00000	−0.223607
$21$	0	0
$22$	−6.00000	−1.27920
$23$	−9.00000	−1.87663	−0.938315	−	0.345782i	$-0.887614\pi$
$23$	−9.00000	−1.87663	−0.938315	+	0.345782i	$0.887614\pi$
$24$	3.00000	0.612372
$25$	−4.00000	−0.800000
$26$	−3.00000	−0.588348
$27$	−1.00000	−0.192450
$28$	0	0
$29$	5.00000	0.928477	0.464238	−	0.885710i	$-0.346328\pi$
$29$	5.00000	0.928477	0.464238	+	0.885710i	$0.346328\pi$
$30$	−1.00000	−0.182574
$31$	8.00000	1.43684	0.718421	−	0.695608i	$-0.244865\pi$
$31$	8.00000	1.43684	0.718421	+	0.695608i	$0.244865\pi$
$32$	5.00000	0.883883
$33$	6.00000	1.04447
$34$	6.00000	1.02899
$35$	0	0
$36$	−1.00000	−0.166667
$37$	−7.00000	−1.15079	−0.575396	−	0.817875i	$-0.695152\pi$
$37$	−7.00000	−1.15079	−0.575396	+	0.817875i	$0.695152\pi$
$38$	−6.00000	−0.973329
$39$	3.00000	0.480384
$40$	−3.00000	−0.474342
$41$	−1.00000	−0.156174
$41$	−1.00000	−0.156174
$42$	0	0
$43$	2.00000	0.304997	0.152499	−	0.988304i	$-0.451268\pi$
$43$	2.00000	0.304997	0.152499	+	0.988304i	$0.451268\pi$
$44$	6.00000	0.904534
$45$	1.00000	0.149071
$46$	−9.00000	−1.32698
$47$	−3.00000	−0.437595	−0.218797	−	0.975770i	$-0.570213\pi$
$47$	−3.00000	−0.437595	−0.218797	+	0.975770i	$0.570213\pi$
$48$	1.00000	0.144338
$49$	0	0
$50$	−4.00000	−0.565685
$51$	−6.00000	−0.840168
$52$	3.00000	0.416025
$53$	9.00000	1.23625	0.618123	−	0.786082i	$-0.287894\pi$
$53$	9.00000	1.23625	0.618123	+	0.786082i	$0.287894\pi$
$54$	−1.00000	−0.136083
$55$	−6.00000	−0.809040
$56$	0	0
$57$	6.00000	0.794719
$58$	5.00000	0.656532
$59$	12.0000	1.56227	0.781133	−	0.624364i	$-0.214642\pi$
$59$	12.0000	1.56227	0.781133	+	0.624364i	$0.214642\pi$
$60$	1.00000	0.129099
$61$	−6.00000	−0.768221	−0.384111	−	0.923287i	$-0.625492\pi$
$61$	−6.00000	−0.768221	−0.384111	+	0.923287i	$0.625492\pi$
$62$	8.00000	1.01600
$63$	0	0
$64$	7.00000	0.875000
$65$	−3.00000	−0.372104
$66$	6.00000	0.738549
$67$	13.0000	1.58820	0.794101	−	0.607785i	$-0.207942\pi$
$67$	13.0000	1.58820	0.794101	+	0.607785i	$0.207942\pi$
$68$	−6.00000	−0.727607
$69$	9.00000	1.08347
$70$	0	0
$71$	−6.00000	−0.712069	−0.356034	−	0.934473i	$-0.615871\pi$
$71$	−6.00000	−0.712069	−0.356034	+	0.934473i	$0.615871\pi$
$72$	−3.00000	−0.353553
$73$	−4.00000	−0.468165	−0.234082	−	0.972217i	$-0.575209\pi$
$73$	−4.00000	−0.468165	−0.234082	+	0.972217i	$0.575209\pi$
$74$	−7.00000	−0.813733
$75$	4.00000	0.461880
$76$	6.00000	0.688247
$77$	0	0
$78$	3.00000	0.339683
$79$	−11.0000	−1.23760	−0.618798	−	0.785550i	$-0.712380\pi$
$79$	−11.0000	−1.23760	−0.618798	+	0.785550i	$0.712380\pi$
$80$	−1.00000	−0.111803
$81$	1.00000	0.111111
$82$	−1.00000	−0.110432
$83$	−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
$85$	6.00000	0.650791
$86$	2.00000	0.215666
$87$	−5.00000	−0.536056
$88$	18.0000	1.91881
$89$	4.00000	0.423999	0.212000	−	0.977270i	$-0.432002\pi$
$89$	4.00000	0.423999	0.212000	+	0.977270i	$0.432002\pi$
$90$	1.00000	0.105409
$91$	0	0
$92$	9.00000	0.938315
$93$	−8.00000	−0.829561
$94$	−3.00000	−0.309426
$95$	−6.00000	−0.615587
$96$	−5.00000	−0.510310
$97$	9.00000	0.913812	0.456906	−	0.889515i	$-0.348958\pi$
$97$	9.00000	0.913812	0.456906	+	0.889515i	$0.348958\pi$
$98$	0	0
$99$	−6.00000	−0.603023
$100$	4.00000	0.400000
$101$	−2.00000	−0.199007	−0.0995037	−	0.995037i	$-0.531726\pi$
$101$	−2.00000	−0.199007	−0.0995037	+	0.995037i	$0.531726\pi$
$102$	−6.00000	−0.594089
$103$	1.00000	0.0985329	0.0492665	−	0.998786i	$-0.484312\pi$
$103$	1.00000	0.0985329	0.0492665	+	0.998786i	$0.484312\pi$
$104$	9.00000	0.882523
$105$	0	0
$106$	9.00000	0.874157
$107$	15.0000	1.45010	0.725052	−	0.688694i	$-0.241816\pi$
$107$	15.0000	1.45010	0.725052	+	0.688694i	$0.241816\pi$
$108$	1.00000	0.0962250
$109$	10.0000	0.957826	0.478913	−	0.877862i	$-0.341031\pi$
$109$	10.0000	0.957826	0.478913	+	0.877862i	$0.341031\pi$
$110$	−6.00000	−0.572078
$111$	7.00000	0.664411
$112$	0	0
$113$	−8.00000	−0.752577	−0.376288	−	0.926503i	$-0.622800\pi$
$113$	−8.00000	−0.752577	−0.376288	+	0.926503i	$0.622800\pi$
$114$	6.00000	0.561951
$115$	−9.00000	−0.839254
$116$	−5.00000	−0.464238
$117$	−3.00000	−0.277350
$118$	12.0000	1.10469
$119$	0	0
$120$	3.00000	0.273861
$121$	25.0000	2.27273
$122$	−6.00000	−0.543214
$123$	1.00000	0.0901670
$124$	−8.00000	−0.718421
$125$	−9.00000	−0.804984
$126$	0	0
$127$	−4.00000	−0.354943	−0.177471	−	0.984126i	$-0.556792\pi$
$127$	−4.00000	−0.354943	−0.177471	+	0.984126i	$0.556792\pi$
$128$	−3.00000	−0.265165
$129$	−2.00000	−0.176090
$130$	−3.00000	−0.263117
$131$	10.0000	0.873704	0.436852	−	0.899533i	$-0.356093\pi$
$131$	10.0000	0.873704	0.436852	+	0.899533i	$0.356093\pi$
$132$	−6.00000	−0.522233
$133$	0	0
$134$	13.0000	1.12303
$135$	−1.00000	−0.0860663
$136$	−18.0000	−1.54349
$137$	−11.0000	−0.939793	−0.469897	−	0.882721i	$-0.655709\pi$
$137$	−11.0000	−0.939793	−0.469897	+	0.882721i	$0.655709\pi$
$138$	9.00000	0.766131
$139$	3.00000	0.254457	0.127228	−	0.991873i	$-0.459392\pi$
$139$	3.00000	0.254457	0.127228	+	0.991873i	$0.459392\pi$
$140$	0	0
$141$	3.00000	0.252646
$142$	−6.00000	−0.503509
$143$	18.0000	1.50524
$144$	−1.00000	−0.0833333
$145$	5.00000	0.415227
$146$	−4.00000	−0.331042
$147$	0	0
$148$	7.00000	0.575396
$149$	10.0000	0.819232	0.409616	−	0.912258i	$-0.365663\pi$
$149$	10.0000	0.819232	0.409616	+	0.912258i	$0.365663\pi$
$150$	4.00000	0.326599
$151$	0	0		−	1.00000i	$-0.5\pi$
$151$	0	0			1.00000i	$0.5\pi$
$152$	18.0000	1.45999
$153$	6.00000	0.485071
$154$	0	0
$155$	8.00000	0.642575
$156$	−3.00000	−0.240192
$157$	14.0000	1.11732	0.558661	−	0.829396i	$-0.311315\pi$
$157$	14.0000	1.11732	0.558661	+	0.829396i	$0.311315\pi$
$158$	−11.0000	−0.875113
$159$	−9.00000	−0.713746
$160$	5.00000	0.395285
$161$	0	0
$162$	1.00000	0.0785674
$163$	−8.00000	−0.626608	−0.313304	−	0.949653i	$-0.601436\pi$
$163$	−8.00000	−0.626608	−0.313304	+	0.949653i	$0.601436\pi$
$164$	1.00000	0.0780869
$165$	6.00000	0.467099
$166$	−2.00000	−0.155230
$167$	−17.0000	−1.31550	−0.657750	−	0.753237i	$-0.728492\pi$
$167$	−17.0000	−1.31550	−0.657750	+	0.753237i	$0.728492\pi$
$168$	0	0
$169$	−4.00000	−0.307692
$170$	6.00000	0.460179
$171$	−6.00000	−0.458831
$172$	−2.00000	−0.152499
$173$	13.0000	0.988372	0.494186	−	0.869356i	$-0.335466\pi$
$173$	13.0000	0.988372	0.494186	+	0.869356i	$0.335466\pi$
$174$	−5.00000	−0.379049
$175$	0	0
$176$	6.00000	0.452267
$177$	−12.0000	−0.901975
$178$	4.00000	0.299813
$179$	−26.0000	−1.94333	−0.971666	−	0.236360i	$-0.924046\pi$
$179$	−26.0000	−1.94333	−0.971666	+	0.236360i	$0.924046\pi$
$180$	−1.00000	−0.0745356
$181$	−6.00000	−0.445976	−0.222988	−	0.974821i	$-0.571581\pi$
$181$	−6.00000	−0.445976	−0.222988	+	0.974821i	$0.571581\pi$
$182$	0	0
$183$	6.00000	0.443533
$184$	27.0000	1.99047
$185$	−7.00000	−0.514650
$186$	−8.00000	−0.586588
$187$	−36.0000	−2.63258
$188$	3.00000	0.218797
$189$	0	0
$190$	−6.00000	−0.435286
$191$	10.0000	0.723575	0.361787	−	0.932261i	$-0.382167\pi$
$191$	10.0000	0.723575	0.361787	+	0.932261i	$0.382167\pi$
$192$	−7.00000	−0.505181
$193$	14.0000	1.00774	0.503871	−	0.863779i	$-0.331909\pi$
$193$	14.0000	1.00774	0.503871	+	0.863779i	$0.331909\pi$
$194$	9.00000	0.646162
$195$	3.00000	0.214834
$196$	0	0
$197$	6.00000	0.427482	0.213741	−	0.976890i	$-0.431435\pi$
$197$	6.00000	0.427482	0.213741	+	0.976890i	$0.431435\pi$
$198$	−6.00000	−0.426401
$199$	14.0000	0.992434	0.496217	−	0.868199i	$-0.334722\pi$
$199$	14.0000	0.992434	0.496217	+	0.868199i	$0.334722\pi$
$200$	12.0000	0.848528
$201$	−13.0000	−0.916949
$202$	−2.00000	−0.140720
$203$	0	0
$204$	6.00000	0.420084
$205$	−1.00000	−0.0698430
$206$	1.00000	0.0696733
$207$	−9.00000	−0.625543
$208$	3.00000	0.208013
$209$	36.0000	2.49017
$210$	0	0
$211$	−17.0000	−1.17033	−0.585164	−	0.810915i	$-0.698970\pi$
$211$	−17.0000	−1.17033	−0.585164	+	0.810915i	$0.698970\pi$
$212$	−9.00000	−0.618123
$213$	6.00000	0.411113
$214$	15.0000	1.02538
$215$	2.00000	0.136399
$216$	3.00000	0.204124
$217$	0	0
$218$	10.0000	0.677285
$219$	4.00000	0.270295
$220$	6.00000	0.404520
$221$	−18.0000	−1.21081
$222$	7.00000	0.469809
$223$	19.0000	1.27233	0.636167	−	0.771551i	$-0.280519\pi$
$223$	19.0000	1.27233	0.636167	+	0.771551i	$0.280519\pi$
$224$	0	0
$225$	−4.00000	−0.266667
$226$	−8.00000	−0.532152
$227$	7.00000	0.464606	0.232303	−	0.972643i	$-0.425374\pi$
$227$	7.00000	0.464606	0.232303	+	0.972643i	$0.425374\pi$
$228$	−6.00000	−0.397360
$229$	−1.00000	−0.0660819	−0.0330409	−	0.999454i	$-0.510519\pi$
$229$	−1.00000	−0.0660819	−0.0330409	+	0.999454i	$0.510519\pi$
$230$	−9.00000	−0.593442
$231$	0	0
$232$	−15.0000	−0.984798
$233$	−18.0000	−1.17922	−0.589610	−	0.807688i	$-0.700718\pi$
$233$	−18.0000	−1.17922	−0.589610	+	0.807688i	$0.700718\pi$
$234$	−3.00000	−0.196116
$235$	−3.00000	−0.195698
$236$	−12.0000	−0.781133
$237$	11.0000	0.714527
$238$	0	0
$239$	8.00000	0.517477	0.258738	−	0.965947i	$-0.416693\pi$
$239$	8.00000	0.517477	0.258738	+	0.965947i	$0.416693\pi$
$240$	1.00000	0.0645497
$241$	14.0000	0.901819	0.450910	−	0.892570i	$-0.351100\pi$
$241$	14.0000	0.901819	0.450910	+	0.892570i	$0.351100\pi$
$242$	25.0000	1.60706
$243$	−1.00000	−0.0641500
$244$	6.00000	0.384111
$245$	0	0
$246$	1.00000	0.0637577
$247$	18.0000	1.14531
$248$	−24.0000	−1.52400
$249$	2.00000	0.126745
$250$	−9.00000	−0.569210
$251$	−30.0000	−1.89358	−0.946792	−	0.321847i	$-0.895696\pi$
$251$	−30.0000	−1.89358	−0.946792	+	0.321847i	$0.895696\pi$
$252$	0	0
$253$	54.0000	3.39495
$254$	−4.00000	−0.250982
$255$	−6.00000	−0.375735
$256$	−17.0000	−1.06250
$257$	12.0000	0.748539	0.374270	−	0.927320i	$-0.377893\pi$
$257$	12.0000	0.748539	0.374270	+	0.927320i	$0.377893\pi$
$258$	−2.00000	−0.124515
$259$	0	0
$260$	3.00000	0.186052
$261$	5.00000	0.309492
$262$	10.0000	0.617802
$263$	6.00000	0.369976	0.184988	−	0.982741i	$-0.440775\pi$
$263$	6.00000	0.369976	0.184988	+	0.982741i	$0.440775\pi$
$264$	−18.0000	−1.10782
$265$	9.00000	0.552866
$266$	0	0
$267$	−4.00000	−0.244796
$268$	−13.0000	−0.794101
$269$	21.0000	1.28039	0.640196	−	0.768211i	$-0.278853\pi$
$269$	21.0000	1.28039	0.640196	+	0.768211i	$0.278853\pi$
$270$	−1.00000	−0.0608581
$271$	−9.00000	−0.546711	−0.273356	−	0.961913i	$-0.588134\pi$
$271$	−9.00000	−0.546711	−0.273356	+	0.961913i	$0.588134\pi$
$272$	−6.00000	−0.363803
$273$	0	0
$274$	−11.0000	−0.664534
$275$	24.0000	1.44725
$276$	−9.00000	−0.541736
$277$	29.0000	1.74244	0.871221	−	0.490892i	$-0.163329\pi$
$277$	29.0000	1.74244	0.871221	+	0.490892i	$0.163329\pi$
$278$	3.00000	0.179928
$279$	8.00000	0.478947
$280$	0	0
$281$	29.0000	1.72999	0.864997	−	0.501776i	$-0.167320\pi$
$281$	29.0000	1.72999	0.864997	+	0.501776i	$0.167320\pi$
$282$	3.00000	0.178647
$283$	4.00000	0.237775	0.118888	−	0.992908i	$-0.462067\pi$
$283$	4.00000	0.237775	0.118888	+	0.992908i	$0.462067\pi$
$284$	6.00000	0.356034
$285$	6.00000	0.355409
$286$	18.0000	1.06436
$287$	0	0
$288$	5.00000	0.294628
$289$	19.0000	1.11765
$290$	5.00000	0.293610
$291$	−9.00000	−0.527589
$292$	4.00000	0.234082
$293$	22.0000	1.28525	0.642627	−	0.766179i	$-0.277845\pi$
$293$	22.0000	1.28525	0.642627	+	0.766179i	$0.277845\pi$
$294$	0	0
$295$	12.0000	0.698667
$296$	21.0000	1.22060
$297$	6.00000	0.348155
$298$	10.0000	0.579284
$299$	27.0000	1.56145
$300$	−4.00000	−0.230940
$301$	0	0
$302$	0	0
$303$	2.00000	0.114897
$304$	6.00000	0.344124
$305$	−6.00000	−0.343559
$306$	6.00000	0.342997
$307$	−7.00000	−0.399511	−0.199756	−	0.979846i	$-0.564015\pi$
$307$	−7.00000	−0.399511	−0.199756	+	0.979846i	$0.564015\pi$
$308$	0	0
$309$	−1.00000	−0.0568880
$310$	8.00000	0.454369
$311$	11.0000	0.623753	0.311876	−	0.950123i	$-0.399043\pi$
$311$	11.0000	0.623753	0.311876	+	0.950123i	$0.399043\pi$
$312$	−9.00000	−0.509525
$313$	3.00000	0.169570	0.0847850	−	0.996399i	$-0.472980\pi$
$313$	3.00000	0.169570	0.0847850	+	0.996399i	$0.472980\pi$
$314$	14.0000	0.790066
$315$	0	0
$316$	11.0000	0.618798
$317$	−3.00000	−0.168497	−0.0842484	−	0.996445i	$-0.526849\pi$
$317$	−3.00000	−0.168497	−0.0842484	+	0.996445i	$0.526849\pi$
$318$	−9.00000	−0.504695
$319$	−30.0000	−1.67968
$320$	7.00000	0.391312
$321$	−15.0000	−0.837218
$322$	0	0
$323$	−36.0000	−2.00309
$324$	−1.00000	−0.0555556
$325$	12.0000	0.665640
$326$	−8.00000	−0.443079
$327$	−10.0000	−0.553001
$328$	3.00000	0.165647
$329$	0	0
$330$	6.00000	0.330289
$331$	1.00000	0.0549650	0.0274825	−	0.999622i	$-0.491251\pi$
$331$	1.00000	0.0549650	0.0274825	+	0.999622i	$0.491251\pi$
$332$	2.00000	0.109764
$333$	−7.00000	−0.383598
$334$	−17.0000	−0.930199
$335$	13.0000	0.710266
$336$	0	0
$337$	−13.0000	−0.708155	−0.354078	−	0.935216i	$-0.615205\pi$
$337$	−13.0000	−0.708155	−0.354078	+	0.935216i	$0.615205\pi$
$338$	−4.00000	−0.217571
$339$	8.00000	0.434500
$340$	−6.00000	−0.325396
$341$	−48.0000	−2.59935
$342$	−6.00000	−0.324443
$343$	0	0
$344$	−6.00000	−0.323498
$345$	9.00000	0.484544
$346$	13.0000	0.698884
$347$	−2.00000	−0.107366	−0.0536828	−	0.998558i	$-0.517096\pi$
$347$	−2.00000	−0.107366	−0.0536828	+	0.998558i	$0.517096\pi$
$348$	5.00000	0.268028
$349$	20.0000	1.07058	0.535288	−	0.844670i	$-0.320203\pi$
$349$	20.0000	1.07058	0.535288	+	0.844670i	$0.320203\pi$
$350$	0	0
$351$	3.00000	0.160128
$352$	−30.0000	−1.59901
$353$	−2.00000	−0.106449	−0.0532246	−	0.998583i	$-0.516950\pi$
$353$	−2.00000	−0.106449	−0.0532246	+	0.998583i	$0.516950\pi$
$354$	−12.0000	−0.637793
$355$	−6.00000	−0.318447
$356$	−4.00000	−0.212000
$357$	0	0
$358$	−26.0000	−1.37414
$359$	5.00000	0.263890	0.131945	−	0.991257i	$-0.457878\pi$
$359$	5.00000	0.263890	0.131945	+	0.991257i	$0.457878\pi$
$360$	−3.00000	−0.158114
$361$	17.0000	0.894737
$362$	−6.00000	−0.315353
$363$	−25.0000	−1.31216
$364$	0	0
$365$	−4.00000	−0.209370
$366$	6.00000	0.313625
$367$	1.00000	0.0521996	0.0260998	−	0.999659i	$-0.491691\pi$
$367$	1.00000	0.0521996	0.0260998	+	0.999659i	$0.491691\pi$
$368$	9.00000	0.469157
$369$	−1.00000	−0.0520579
$370$	−7.00000	−0.363913
$371$	0	0
$372$	8.00000	0.414781
$373$	−7.00000	−0.362446	−0.181223	−	0.983442i	$-0.558006\pi$
$373$	−7.00000	−0.362446	−0.181223	+	0.983442i	$0.558006\pi$
$374$	−36.0000	−1.86152
$375$	9.00000	0.464758
$376$	9.00000	0.464140
$377$	−15.0000	−0.772539
$378$	0	0
$379$	10.0000	0.513665	0.256833	−	0.966456i	$-0.417321\pi$
$379$	10.0000	0.513665	0.256833	+	0.966456i	$0.417321\pi$
$380$	6.00000	0.307794
$381$	4.00000	0.204926
$382$	10.0000	0.511645
$383$	15.0000	0.766464	0.383232	−	0.923652i	$-0.374811\pi$
$383$	15.0000	0.766464	0.383232	+	0.923652i	$0.374811\pi$
$384$	3.00000	0.153093
$385$	0	0
$386$	14.0000	0.712581
$387$	2.00000	0.101666
$388$	−9.00000	−0.456906
$389$	−18.0000	−0.912636	−0.456318	−	0.889817i	$-0.650832\pi$
$389$	−18.0000	−0.912636	−0.456318	+	0.889817i	$0.650832\pi$
$390$	3.00000	0.151911
$391$	−54.0000	−2.73090
$392$	0	0
$393$	−10.0000	−0.504433
$394$	6.00000	0.302276
$395$	−11.0000	−0.553470
$396$	6.00000	0.301511
$397$	−14.0000	−0.702640	−0.351320	−	0.936255i	$-0.614267\pi$
$397$	−14.0000	−0.702640	−0.351320	+	0.936255i	$0.614267\pi$
$398$	14.0000	0.701757
$399$	0	0
$400$	4.00000	0.200000
$401$	−32.0000	−1.59800	−0.799002	−	0.601329i	$-0.794638\pi$
$401$	−32.0000	−1.59800	−0.799002	+	0.601329i	$0.794638\pi$
$402$	−13.0000	−0.648381
$403$	−24.0000	−1.19553
$404$	2.00000	0.0995037
$405$	1.00000	0.0496904
$406$	0	0
$407$	42.0000	2.08186
$408$	18.0000	0.891133
$409$	−4.00000	−0.197787	−0.0988936	−	0.995098i	$-0.531530\pi$
$409$	−4.00000	−0.197787	−0.0988936	+	0.995098i	$0.531530\pi$
$410$	−1.00000	−0.0493865
$411$	11.0000	0.542590
$412$	−1.00000	−0.0492665
$413$	0	0
$414$	−9.00000	−0.442326
$415$	−2.00000	−0.0981761
$416$	−15.0000	−0.735436
$417$	−3.00000	−0.146911
$418$	36.0000	1.76082
$419$	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
$421$	−22.0000	−1.07221	−0.536107	−	0.844150i	$-0.680106\pi$
$421$	−22.0000	−1.07221	−0.536107	+	0.844150i	$0.680106\pi$
$422$	−17.0000	−0.827547
$423$	−3.00000	−0.145865
$424$	−27.0000	−1.31124
$425$	−24.0000	−1.16417
$426$	6.00000	0.290701
$427$	0	0
$428$	−15.0000	−0.725052
$429$	−18.0000	−0.869048
$430$	2.00000	0.0964486
$431$	0	0		−	1.00000i	$-0.5\pi$
$431$	0	0			1.00000i	$0.5\pi$
$432$	1.00000	0.0481125
$433$	−6.00000	−0.288342	−0.144171	−	0.989553i	$-0.546051\pi$
$433$	−6.00000	−0.288342	−0.144171	+	0.989553i	$0.546051\pi$
$434$	0	0
$435$	−5.00000	−0.239732
$436$	−10.0000	−0.478913
$437$	54.0000	2.58317
$438$	4.00000	0.191127
$439$	28.0000	1.33637	0.668184	−	0.743996i	$-0.267072\pi$
$439$	28.0000	1.33637	0.668184	+	0.743996i	$0.267072\pi$
$440$	18.0000	0.858116
$441$	0	0
$442$	−18.0000	−0.856173
$443$	0	0		−	1.00000i	$-0.5\pi$
$443$	0	0			1.00000i	$0.5\pi$
$444$	−7.00000	−0.332205
$445$	4.00000	0.189618
$446$	19.0000	0.899676
$447$	−10.0000	−0.472984
$448$	0	0
$449$	16.0000	0.755087	0.377543	−	0.925992i	$-0.376769\pi$
$449$	16.0000	0.755087	0.377543	+	0.925992i	$0.376769\pi$
$450$	−4.00000	−0.188562
$451$	6.00000	0.282529
$452$	8.00000	0.376288
$453$	0	0
$454$	7.00000	0.328526
$455$	0	0
$456$	−18.0000	−0.842927
$457$	28.0000	1.30978	0.654892	−	0.755722i	$-0.272714\pi$
$457$	28.0000	1.30978	0.654892	+	0.755722i	$0.272714\pi$
$458$	−1.00000	−0.0467269
$459$	−6.00000	−0.280056
$460$	9.00000	0.419627
$461$	5.00000	0.232873	0.116437	−	0.993198i	$-0.462853\pi$
$461$	5.00000	0.232873	0.116437	+	0.993198i	$0.462853\pi$
$462$	0	0
$463$	−11.0000	−0.511213	−0.255607	−	0.966781i	$-0.582275\pi$
$463$	−11.0000	−0.511213	−0.255607	+	0.966781i	$0.582275\pi$
$464$	−5.00000	−0.232119
$465$	−8.00000	−0.370991
$466$	−18.0000	−0.833834
$467$	18.0000	0.832941	0.416470	−	0.909149i	$-0.363267\pi$
$467$	18.0000	0.832941	0.416470	+	0.909149i	$0.363267\pi$
$468$	3.00000	0.138675
$469$	0	0
$470$	−3.00000	−0.138380
$471$	−14.0000	−0.645086
$472$	−36.0000	−1.65703
$473$	−12.0000	−0.551761
$474$	11.0000	0.505247
$475$	24.0000	1.10120
$476$	0	0
$477$	9.00000	0.412082
$478$	8.00000	0.365911
$479$	−24.0000	−1.09659	−0.548294	−	0.836286i	$-0.684723\pi$
$479$	−24.0000	−1.09659	−0.548294	+	0.836286i	$0.684723\pi$
$480$	−5.00000	−0.228218
$481$	21.0000	0.957518
$482$	14.0000	0.637683
$483$	0	0
$484$	−25.0000	−1.13636
$485$	9.00000	0.408669
$486$	−1.00000	−0.0453609
$487$	−28.0000	−1.26880	−0.634401	−	0.773004i	$-0.718753\pi$
$487$	−28.0000	−1.26880	−0.634401	+	0.773004i	$0.718753\pi$
$488$	18.0000	0.814822
$489$	8.00000	0.361773
$490$	0	0
$491$	−8.00000	−0.361035	−0.180517	−	0.983572i	$-0.557777\pi$
$491$	−8.00000	−0.361035	−0.180517	+	0.983572i	$0.557777\pi$
$492$	−1.00000	−0.0450835
$493$	30.0000	1.35113
$494$	18.0000	0.809858
$495$	−6.00000	−0.269680
$496$	−8.00000	−0.359211
$497$	0	0
$498$	2.00000	0.0896221
$499$	−16.0000	−0.716258	−0.358129	−	0.933672i	$-0.616585\pi$
$499$	−16.0000	−0.716258	−0.358129	+	0.933672i	$0.616585\pi$
$500$	9.00000	0.402492
$501$	17.0000	0.759504
$502$	−30.0000	−1.33897
$503$	−11.0000	−0.490466	−0.245233	−	0.969464i	$-0.578864\pi$
$503$	−11.0000	−0.490466	−0.245233	+	0.969464i	$0.578864\pi$
$504$	0	0
$505$	−2.00000	−0.0889988
$506$	54.0000	2.40059
$507$	4.00000	0.177646
$508$	4.00000	0.177471
$509$	36.0000	1.59567	0.797836	−	0.602875i	$-0.205978\pi$
$509$	36.0000	1.59567	0.797836	+	0.602875i	$0.205978\pi$
$510$	−6.00000	−0.265684
$511$	0	0
$512$	−11.0000	−0.486136
$513$	6.00000	0.264906
$514$	12.0000	0.529297
$515$	1.00000	0.0440653
$516$	2.00000	0.0880451
$517$	18.0000	0.791639
$518$	0	0
$519$	−13.0000	−0.570637
$520$	9.00000	0.394676
$521$	−34.0000	−1.48957	−0.744784	−	0.667306i	$-0.767447\pi$
$521$	−34.0000	−1.48957	−0.744784	+	0.667306i	$0.767447\pi$
$522$	5.00000	0.218844
$523$	−12.0000	−0.524723	−0.262362	−	0.964970i	$-0.584501\pi$
$523$	−12.0000	−0.524723	−0.262362	+	0.964970i	$0.584501\pi$
$524$	−10.0000	−0.436852
$525$	0	0
$526$	6.00000	0.261612
$527$	48.0000	2.09091
$528$	−6.00000	−0.261116
$529$	58.0000	2.52174
$530$	9.00000	0.390935
$531$	12.0000	0.520756
$532$	0	0
$533$	3.00000	0.129944
$534$	−4.00000	−0.173097
$535$	15.0000	0.648507
$536$	−39.0000	−1.68454
$537$	26.0000	1.12198
$538$	21.0000	0.905374
$539$	0	0
$540$	1.00000	0.0430331
$541$	39.0000	1.67674	0.838370	−	0.545101i	$-0.183509\pi$
$541$	39.0000	1.67674	0.838370	+	0.545101i	$0.183509\pi$
$542$	−9.00000	−0.386583
$543$	6.00000	0.257485
$544$	30.0000	1.28624
$545$	10.0000	0.428353
$546$	0	0
$547$	−43.0000	−1.83855	−0.919274	−	0.393619i	$-0.871223\pi$
$547$	−43.0000	−1.83855	−0.919274	+	0.393619i	$0.871223\pi$
$548$	11.0000	0.469897
$549$	−6.00000	−0.256074
$550$	24.0000	1.02336
$551$	−30.0000	−1.27804
$552$	−27.0000	−1.14920
$553$	0	0
$554$	29.0000	1.23209
$555$	7.00000	0.297133
$556$	−3.00000	−0.127228
$557$	30.0000	1.27114	0.635570	−	0.772043i	$-0.280765\pi$
$557$	30.0000	1.27114	0.635570	+	0.772043i	$0.280765\pi$
$558$	8.00000	0.338667
$559$	−6.00000	−0.253773
$560$	0	0
$561$	36.0000	1.51992
$562$	29.0000	1.22329
$563$	21.0000	0.885044	0.442522	−	0.896758i	$-0.354084\pi$
$563$	21.0000	0.885044	0.442522	+	0.896758i	$0.354084\pi$
$564$	−3.00000	−0.126323
$565$	−8.00000	−0.336563
$566$	4.00000	0.168133
$567$	0	0
$568$	18.0000	0.755263
$569$	8.00000	0.335377	0.167689	−	0.985840i	$-0.446370\pi$
$569$	8.00000	0.335377	0.167689	+	0.985840i	$0.446370\pi$
$570$	6.00000	0.251312
$571$	−28.0000	−1.17176	−0.585882	−	0.810397i	$-0.699252\pi$
$571$	−28.0000	−1.17176	−0.585882	+	0.810397i	$0.699252\pi$
$572$	−18.0000	−0.752618
$573$	−10.0000	−0.417756
$574$	0	0
$575$	36.0000	1.50130
$576$	7.00000	0.291667
$577$	−18.0000	−0.749350	−0.374675	−	0.927156i	$-0.622246\pi$
$577$	−18.0000	−0.749350	−0.374675	+	0.927156i	$0.622246\pi$
$578$	19.0000	0.790296
$579$	−14.0000	−0.581820
$580$	−5.00000	−0.207614
$581$	0	0
$582$	−9.00000	−0.373062
$583$	−54.0000	−2.23645
$584$	12.0000	0.496564
$585$	−3.00000	−0.124035
$586$	22.0000	0.908812
$587$	20.0000	0.825488	0.412744	−	0.910847i	$-0.364570\pi$
$587$	20.0000	0.825488	0.412744	+	0.910847i	$0.364570\pi$
$588$	0	0
$589$	−48.0000	−1.97781
$590$	12.0000	0.494032
$591$	−6.00000	−0.246807
$592$	7.00000	0.287698
$593$	−4.00000	−0.164260	−0.0821302	−	0.996622i	$-0.526172\pi$
$593$	−4.00000	−0.164260	−0.0821302	+	0.996622i	$0.526172\pi$
$594$	6.00000	0.246183
$595$	0	0
$596$	−10.0000	−0.409616
$597$	−14.0000	−0.572982
$598$	27.0000	1.10411
$599$	−31.0000	−1.26663	−0.633313	−	0.773896i	$-0.718305\pi$
$599$	−31.0000	−1.26663	−0.633313	+	0.773896i	$0.718305\pi$
$600$	−12.0000	−0.489898
$601$	35.0000	1.42768	0.713840	−	0.700309i	$-0.246954\pi$
$601$	35.0000	1.42768	0.713840	+	0.700309i	$0.246954\pi$
$602$	0	0
$603$	13.0000	0.529401
$604$	0	0
$605$	25.0000	1.01639
$606$	2.00000	0.0812444
$607$	−40.0000	−1.62355	−0.811775	−	0.583970i	$-0.801498\pi$
$607$	−40.0000	−1.62355	−0.811775	+	0.583970i	$0.801498\pi$
$608$	−30.0000	−1.21666
$609$	0	0
$610$	−6.00000	−0.242933
$611$	9.00000	0.364101
$612$	−6.00000	−0.242536
$613$	19.0000	0.767403	0.383701	−	0.923457i	$-0.374649\pi$
$613$	19.0000	0.767403	0.383701	+	0.923457i	$0.374649\pi$
$614$	−7.00000	−0.282497
$615$	1.00000	0.0403239
$616$	0	0
$617$	−20.0000	−0.805170	−0.402585	−	0.915383i	$-0.631888\pi$
$617$	−20.0000	−0.805170	−0.402585	+	0.915383i	$0.631888\pi$
$618$	−1.00000	−0.0402259
$619$	−35.0000	−1.40677	−0.703384	−	0.710810i	$-0.748329\pi$
$619$	−35.0000	−1.40677	−0.703384	+	0.710810i	$0.748329\pi$
$620$	−8.00000	−0.321288
$621$	9.00000	0.361158
$622$	11.0000	0.441060
$623$	0	0
$624$	−3.00000	−0.120096
$625$	11.0000	0.440000
$626$	3.00000	0.119904
$627$	−36.0000	−1.43770
$628$	−14.0000	−0.558661
$629$	−42.0000	−1.67465
$630$	0	0
$631$	24.0000	0.955425	0.477712	−	0.878516i	$-0.341466\pi$
$631$	24.0000	0.955425	0.477712	+	0.878516i	$0.341466\pi$
$632$	33.0000	1.31267
$633$	17.0000	0.675689
$634$	−3.00000	−0.119145
$635$	−4.00000	−0.158735
$636$	9.00000	0.356873
$637$	0	0
$638$	−30.0000	−1.18771
$639$	−6.00000	−0.237356
$640$	−3.00000	−0.118585
$641$	−2.00000	−0.0789953	−0.0394976	−	0.999220i	$-0.512576\pi$
$641$	−2.00000	−0.0789953	−0.0394976	+	0.999220i	$0.512576\pi$
$642$	−15.0000	−0.592003
$643$	28.0000	1.10421	0.552106	−	0.833774i	$-0.313824\pi$
$643$	28.0000	1.10421	0.552106	+	0.833774i	$0.313824\pi$
$644$	0	0
$645$	−2.00000	−0.0787499
$646$	−36.0000	−1.41640
$647$	18.0000	0.707653	0.353827	−	0.935311i	$-0.384880\pi$
$647$	18.0000	0.707653	0.353827	+	0.935311i	$0.384880\pi$
$648$	−3.00000	−0.117851
$649$	−72.0000	−2.82625
$650$	12.0000	0.470679
$651$	0	0
$652$	8.00000	0.313304
$653$	34.0000	1.33052	0.665261	−	0.746611i	$-0.268320\pi$
$653$	34.0000	1.33052	0.665261	+	0.746611i	$0.268320\pi$
$654$	−10.0000	−0.391031
$655$	10.0000	0.390732
$656$	1.00000	0.0390434
$657$	−4.00000	−0.156055
$658$	0	0
$659$	14.0000	0.545363	0.272681	−	0.962104i	$-0.412090\pi$
$659$	14.0000	0.545363	0.272681	+	0.962104i	$0.412090\pi$
$660$	−6.00000	−0.233550
$661$	8.00000	0.311164	0.155582	−	0.987823i	$-0.450275\pi$
$661$	8.00000	0.311164	0.155582	+	0.987823i	$0.450275\pi$
$662$	1.00000	0.0388661
$663$	18.0000	0.699062
$664$	6.00000	0.232845
$665$	0	0
$666$	−7.00000	−0.271244
$667$	−45.0000	−1.74241
$668$	17.0000	0.657750
$669$	−19.0000	−0.734582
$670$	13.0000	0.502234
$671$	36.0000	1.38976
$672$	0	0
$673$	22.0000	0.848038	0.424019	−	0.905653i	$-0.360619\pi$
$673$	22.0000	0.848038	0.424019	+	0.905653i	$0.360619\pi$
$674$	−13.0000	−0.500741
$675$	4.00000	0.153960
$676$	4.00000	0.153846
$677$	−30.0000	−1.15299	−0.576497	−	0.817099i	$-0.695581\pi$
$677$	−30.0000	−1.15299	−0.576497	+	0.817099i	$0.695581\pi$
$678$	8.00000	0.307238
$679$	0	0
$680$	−18.0000	−0.690268
$681$	−7.00000	−0.268241
$682$	−48.0000	−1.83801
$683$	−4.00000	−0.153056	−0.0765279	−	0.997067i	$-0.524383\pi$
$683$	−4.00000	−0.153056	−0.0765279	+	0.997067i	$0.524383\pi$
$684$	6.00000	0.229416
$685$	−11.0000	−0.420288
$686$	0	0
$687$	1.00000	0.0381524
$688$	−2.00000	−0.0762493
$689$	−27.0000	−1.02862
$690$	9.00000	0.342624
$691$	40.0000	1.52167	0.760836	−	0.648944i	$-0.224789\pi$
$691$	40.0000	1.52167	0.760836	+	0.648944i	$0.224789\pi$
$692$	−13.0000	−0.494186
$693$	0	0
$694$	−2.00000	−0.0759190
$695$	3.00000	0.113796
$696$	15.0000	0.568574
$697$	−6.00000	−0.227266
$698$	20.0000	0.757011
$699$	18.0000	0.680823
$700$	0	0
$701$	0	0		−	1.00000i	$-0.5\pi$
$701$	0	0			1.00000i	$0.5\pi$
$702$	3.00000	0.113228
$703$	42.0000	1.58406
$704$	−42.0000	−1.58293
$705$	3.00000	0.112987
$706$	−2.00000	−0.0752710
$707$	0	0
$708$	12.0000	0.450988
$709$	22.0000	0.826227	0.413114	−	0.910679i	$-0.364441\pi$
$709$	22.0000	0.826227	0.413114	+	0.910679i	$0.364441\pi$
$710$	−6.00000	−0.225176
$711$	−11.0000	−0.412532
$712$	−12.0000	−0.449719
$713$	−72.0000	−2.69642
$714$	0	0
$715$	18.0000	0.673162
$716$	26.0000	0.971666
$717$	−8.00000	−0.298765
$718$	5.00000	0.186598
$719$	−28.0000	−1.04422	−0.522112	−	0.852877i	$-0.674856\pi$
$719$	−28.0000	−1.04422	−0.522112	+	0.852877i	$0.674856\pi$
$720$	−1.00000	−0.0372678
$721$	0	0
$722$	17.0000	0.632674
$723$	−14.0000	−0.520666
$724$	6.00000	0.222988
$725$	−20.0000	−0.742781
$726$	−25.0000	−0.927837
$727$	24.0000	0.890111	0.445055	−	0.895503i	$-0.353184\pi$
$727$	24.0000	0.890111	0.445055	+	0.895503i	$0.353184\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	−4.00000	−0.148047
$731$	12.0000	0.443836
$732$	−6.00000	−0.221766
$733$	−36.0000	−1.32969	−0.664845	−	0.746981i	$-0.731502\pi$
$733$	−36.0000	−1.32969	−0.664845	+	0.746981i	$0.731502\pi$
$734$	1.00000	0.0369107
$735$	0	0
$736$	−45.0000	−1.65872
$737$	−78.0000	−2.87317
$738$	−1.00000	−0.0368105
$739$	18.0000	0.662141	0.331070	−	0.943606i	$-0.392590\pi$
$739$	18.0000	0.662141	0.331070	+	0.943606i	$0.392590\pi$
$740$	7.00000	0.257325
$741$	−18.0000	−0.661247
$742$	0	0
$743$	−48.0000	−1.76095	−0.880475	−	0.474093i	$-0.842776\pi$
$743$	−48.0000	−1.76095	−0.880475	+	0.474093i	$0.842776\pi$
$744$	24.0000	0.879883
$745$	10.0000	0.366372
$746$	−7.00000	−0.256288
$747$	−2.00000	−0.0731762
$748$	36.0000	1.31629
$749$	0	0
$750$	9.00000	0.328634
$751$	−17.0000	−0.620339	−0.310169	−	0.950681i	$-0.600386\pi$
$751$	−17.0000	−0.620339	−0.310169	+	0.950681i	$0.600386\pi$
$752$	3.00000	0.109399
$753$	30.0000	1.09326
$754$	−15.0000	−0.546268
$755$	0	0
$756$	0	0
$757$	50.0000	1.81728	0.908640	−	0.417579i	$-0.137121\pi$
$757$	50.0000	1.81728	0.908640	+	0.417579i	$0.137121\pi$
$758$	10.0000	0.363216
$759$	−54.0000	−1.96008
$760$	18.0000	0.652929
$761$	−45.0000	−1.63125	−0.815624	−	0.578582i	$-0.803606\pi$
$761$	−45.0000	−1.63125	−0.815624	+	0.578582i	$0.803606\pi$
$762$	4.00000	0.144905
$763$	0	0
$764$	−10.0000	−0.361787
$765$	6.00000	0.216930
$766$	15.0000	0.541972
$767$	−36.0000	−1.29988
$768$	17.0000	0.613435
$769$	44.0000	1.58668	0.793340	−	0.608778i	$-0.208340\pi$
$769$	44.0000	1.58668	0.793340	+	0.608778i	$0.208340\pi$
$770$	0	0
$771$	−12.0000	−0.432169
$772$	−14.0000	−0.503871
$773$	−36.0000	−1.29483	−0.647415	−	0.762138i	$-0.724150\pi$
$773$	−36.0000	−1.29483	−0.647415	+	0.762138i	$0.724150\pi$
$774$	2.00000	0.0718885
$775$	−32.0000	−1.14947
$776$	−27.0000	−0.969244
$777$	0	0
$778$	−18.0000	−0.645331
$779$	6.00000	0.214972
$780$	−3.00000	−0.107417
$781$	36.0000	1.28818
$782$	−54.0000	−1.93104
$783$	−5.00000	−0.178685
$784$	0	0
$785$	14.0000	0.499681
$786$	−10.0000	−0.356688
$787$	−7.00000	−0.249523	−0.124762	−	0.992187i	$-0.539817\pi$
$787$	−7.00000	−0.249523	−0.124762	+	0.992187i	$0.539817\pi$
$788$	−6.00000	−0.213741
$789$	−6.00000	−0.213606
$790$	−11.0000	−0.391362
$791$	0	0
$792$	18.0000	0.639602
$793$	18.0000	0.639199
$794$	−14.0000	−0.496841
$795$	−9.00000	−0.319197
$796$	−14.0000	−0.496217
$797$	−30.0000	−1.06265	−0.531327	−	0.847167i	$-0.678307\pi$
$797$	−30.0000	−1.06265	−0.531327	+	0.847167i	$0.678307\pi$
$798$	0	0
$799$	−18.0000	−0.636794
$800$	−20.0000	−0.707107
$801$	4.00000	0.141333
$802$	−32.0000	−1.12996
$803$	24.0000	0.846942
$804$	13.0000	0.458475
$805$	0	0
$806$	−24.0000	−0.845364
$807$	−21.0000	−0.739235
$808$	6.00000	0.211079
$809$	−3.00000	−0.105474	−0.0527372	−	0.998608i	$-0.516795\pi$
$809$	−3.00000	−0.105474	−0.0527372	+	0.998608i	$0.516795\pi$
$810$	1.00000	0.0351364
$811$	47.0000	1.65039	0.825197	−	0.564846i	$-0.191064\pi$
$811$	47.0000	1.65039	0.825197	+	0.564846i	$0.191064\pi$
$812$	0	0
$813$	9.00000	0.315644
$814$	42.0000	1.47210
$815$	−8.00000	−0.280228
$816$	6.00000	0.210042
$817$	−12.0000	−0.419827
$818$	−4.00000	−0.139857
$819$	0	0
$820$	1.00000	0.0349215
$821$	6.00000	0.209401	0.104701	−	0.994504i	$-0.466612\pi$
$821$	6.00000	0.209401	0.104701	+	0.994504i	$0.466612\pi$
$822$	11.0000	0.383669
$823$	40.0000	1.39431	0.697156	−	0.716919i	$-0.254448\pi$
$823$	40.0000	1.39431	0.697156	+	0.716919i	$0.254448\pi$
$824$	−3.00000	−0.104510
$825$	−24.0000	−0.835573
$826$	0	0
$827$	−36.0000	−1.25184	−0.625921	−	0.779886i	$-0.715277\pi$
$827$	−36.0000	−1.25184	−0.625921	+	0.779886i	$0.715277\pi$
$828$	9.00000	0.312772
$829$	−54.0000	−1.87550	−0.937749	−	0.347314i	$-0.887094\pi$
$829$	−54.0000	−1.87550	−0.937749	+	0.347314i	$0.887094\pi$
$830$	−2.00000	−0.0694210
$831$	−29.0000	−1.00600
$832$	−21.0000	−0.728044
$833$	0	0
$834$	−3.00000	−0.103882
$835$	−17.0000	−0.588309
$836$	−36.0000	−1.24509
$837$	−8.00000	−0.276520
$838$	30.0000	1.03633
$839$	−51.0000	−1.76072	−0.880358	−	0.474310i	$-0.842698\pi$
$839$	−51.0000	−1.76072	−0.880358	+	0.474310i	$0.842698\pi$
$840$	0	0
$841$	−4.00000	−0.137931
$842$	−22.0000	−0.758170
$843$	−29.0000	−0.998813
$844$	17.0000	0.585164
$845$	−4.00000	−0.137604
$846$	−3.00000	−0.103142
$847$	0	0
$848$	−9.00000	−0.309061
$849$	−4.00000	−0.137280
$850$	−24.0000	−0.823193
$851$	63.0000	2.15961
$852$	−6.00000	−0.205557
$853$	−18.0000	−0.616308	−0.308154	−	0.951336i	$-0.599711\pi$
$853$	−18.0000	−0.616308	−0.308154	+	0.951336i	$0.599711\pi$
$854$	0	0
$855$	−6.00000	−0.205196
$856$	−45.0000	−1.53807
$857$	7.00000	0.239115	0.119558	−	0.992827i	$-0.461852\pi$
$857$	7.00000	0.239115	0.119558	+	0.992827i	$0.461852\pi$
$858$	−18.0000	−0.614510
$859$	13.0000	0.443554	0.221777	−	0.975097i	$-0.428814\pi$
$859$	13.0000	0.443554	0.221777	+	0.975097i	$0.428814\pi$
$860$	−2.00000	−0.0681994
$861$	0	0
$862$	0	0
$863$	−23.0000	−0.782929	−0.391465	−	0.920193i	$-0.628031\pi$
$863$	−23.0000	−0.782929	−0.391465	+	0.920193i	$0.628031\pi$
$864$	−5.00000	−0.170103
$865$	13.0000	0.442013
$866$	−6.00000	−0.203888
$867$	−19.0000	−0.645274
$868$	0	0
$869$	66.0000	2.23890
$870$	−5.00000	−0.169516
$871$	−39.0000	−1.32146
$872$	−30.0000	−1.01593
$873$	9.00000	0.304604
$874$	54.0000	1.82658
$875$	0	0
$876$	−4.00000	−0.135147
$877$	2.00000	0.0675352	0.0337676	−	0.999430i	$-0.489249\pi$
$877$	2.00000	0.0675352	0.0337676	+	0.999430i	$0.489249\pi$
$878$	28.0000	0.944954
$879$	−22.0000	−0.742042
$880$	6.00000	0.202260
$881$	−51.0000	−1.71823	−0.859117	−	0.511780i	$-0.828986\pi$
$881$	−51.0000	−1.71823	−0.859117	+	0.511780i	$0.828986\pi$
$882$	0	0
$883$	7.00000	0.235569	0.117784	−	0.993039i	$-0.462421\pi$
$883$	7.00000	0.235569	0.117784	+	0.993039i	$0.462421\pi$
$884$	18.0000	0.605406
$885$	−12.0000	−0.403376
$886$	0	0
$887$	−57.0000	−1.91387	−0.956936	−	0.290298i	$-0.906246\pi$
$887$	−57.0000	−1.91387	−0.956936	+	0.290298i	$0.906246\pi$
$888$	−21.0000	−0.704714
$889$	0	0
$890$	4.00000	0.134080
$891$	−6.00000	−0.201008
$892$	−19.0000	−0.636167
$893$	18.0000	0.602347
$894$	−10.0000	−0.334450
$895$	−26.0000	−0.869084
$896$	0	0
$897$	−27.0000	−0.901504
$898$	16.0000	0.533927
$899$	40.0000	1.33407
$900$	4.00000	0.133333
$901$	54.0000	1.79900
$902$	6.00000	0.199778
$903$	0	0
$904$	24.0000	0.798228
$905$	−6.00000	−0.199447
$906$	0	0
$907$	6.00000	0.199227	0.0996134	−	0.995026i	$-0.468239\pi$
$907$	6.00000	0.199227	0.0996134	+	0.995026i	$0.468239\pi$
$908$	−7.00000	−0.232303
$909$	−2.00000	−0.0663358
$910$	0	0
$911$	24.0000	0.795155	0.397578	−	0.917568i	$-0.369851\pi$
$911$	24.0000	0.795155	0.397578	+	0.917568i	$0.369851\pi$
$912$	−6.00000	−0.198680
$913$	12.0000	0.397142
$914$	28.0000	0.926158
$915$	6.00000	0.198354
$916$	1.00000	0.0330409
$917$	0	0
$918$	−6.00000	−0.198030
$919$	−11.0000	−0.362857	−0.181428	−	0.983404i	$-0.558072\pi$
$919$	−11.0000	−0.362857	−0.181428	+	0.983404i	$0.558072\pi$
$920$	27.0000	0.890164
$921$	7.00000	0.230658
$922$	5.00000	0.164666
$923$	18.0000	0.592477
$924$	0	0
$925$	28.0000	0.920634
$926$	−11.0000	−0.361482
$927$	1.00000	0.0328443
$928$	25.0000	0.820665
$929$	48.0000	1.57483	0.787414	−	0.616424i	$-0.211419\pi$
$929$	48.0000	1.57483	0.787414	+	0.616424i	$0.211419\pi$
$930$	−8.00000	−0.262330
$931$	0	0
$932$	18.0000	0.589610
$933$	−11.0000	−0.360124
$934$	18.0000	0.588978
$935$	−36.0000	−1.17733
$936$	9.00000	0.294174
$937$	38.0000	1.24141	0.620703	−	0.784046i	$-0.286847\pi$
$937$	38.0000	1.24141	0.620703	+	0.784046i	$0.286847\pi$
$938$	0	0
$939$	−3.00000	−0.0979013
$940$	3.00000	0.0978492
$941$	−45.0000	−1.46696	−0.733479	−	0.679712i	$-0.762105\pi$
$941$	−45.0000	−1.46696	−0.733479	+	0.679712i	$0.762105\pi$
$942$	−14.0000	−0.456145
$943$	9.00000	0.293080
$944$	−12.0000	−0.390567
$945$	0	0
$946$	−12.0000	−0.390154
$947$	33.0000	1.07236	0.536178	−	0.844105i	$-0.319868\pi$
$947$	33.0000	1.07236	0.536178	+	0.844105i	$0.319868\pi$
$948$	−11.0000	−0.357263
$949$	12.0000	0.389536
$950$	24.0000	0.778663
$951$	3.00000	0.0972817
$952$	0	0
$953$	2.00000	0.0647864	0.0323932	−	0.999475i	$-0.489687\pi$
$953$	2.00000	0.0647864	0.0323932	+	0.999475i	$0.489687\pi$
$954$	9.00000	0.291386
$955$	10.0000	0.323592
$956$	−8.00000	−0.258738
$957$	30.0000	0.969762
$958$	−24.0000	−0.775405
$959$	0	0
$960$	−7.00000	−0.225924
$961$	33.0000	1.06452
$962$	21.0000	0.677067
$963$	15.0000	0.483368
$964$	−14.0000	−0.450910
$965$	14.0000	0.450676
$966$	0	0
$967$	39.0000	1.25416	0.627078	−	0.778957i	$-0.284251\pi$
$967$	39.0000	1.25416	0.627078	+	0.778957i	$0.284251\pi$
$968$	−75.0000	−2.41059
$969$	36.0000	1.15649
$970$	9.00000	0.288973
$971$	21.0000	0.673922	0.336961	−	0.941519i	$-0.390601\pi$
$971$	21.0000	0.673922	0.336961	+	0.941519i	$0.390601\pi$
$972$	1.00000	0.0320750
$973$	0	0
$974$	−28.0000	−0.897178
$975$	−12.0000	−0.384308
$976$	6.00000	0.192055
$977$	26.0000	0.831814	0.415907	−	0.909407i	$-0.363464\pi$
$977$	26.0000	0.831814	0.415907	+	0.909407i	$0.363464\pi$
$978$	8.00000	0.255812
$979$	−24.0000	−0.767043
$980$	0	0
$981$	10.0000	0.319275
$982$	−8.00000	−0.255290
$983$	−56.0000	−1.78612	−0.893061	−	0.449935i	$-0.851447\pi$
$983$	−56.0000	−1.78612	−0.893061	+	0.449935i	$0.851447\pi$
$984$	−3.00000	−0.0956365
$985$	6.00000	0.191176
$986$	30.0000	0.955395
$987$	0	0
$988$	−18.0000	−0.572656
$989$	−18.0000	−0.572367
$990$	−6.00000	−0.190693
$991$	23.0000	0.730619	0.365310	−	0.930886i	$-0.380963\pi$
$991$	23.0000	0.730619	0.365310	+	0.930886i	$0.380963\pi$
$992$	40.0000	1.27000
$993$	−1.00000	−0.0317340
$994$	0	0
$995$	14.0000	0.443830
$996$	−2.00000	−0.0633724
$997$	−11.0000	−0.348373	−0.174187	−	0.984713i	$-0.555730\pi$
$997$	−11.0000	−0.348373	−0.174187	+	0.984713i	$0.555730\pi$
$998$	−16.0000	−0.506471
$999$	7.00000	0.221470

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6027.2.a.f.1.1		1
7.6	odd	2		861.2.a.d.1.1	✓	1
21.20	even	2		2583.2.a.a.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
861.2.a.d.1.1	✓	1	7.6	odd	2
2583.2.a.a.1.1		1	21.20	even	2
6027.2.a.f.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:	\( N \)	\(=\)	\( 6027 = 3 \cdot 7^{2} \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6027.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more