LMFDB - Embedded newform 6762.2.a.bh.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6762,2,Mod(1,6762)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6762, 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("6762.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6762 = 2 \cdot 3 \cdot 7^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6762.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:	$53.9948418468$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 966)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	6762.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

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

\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} -3.00000 q^{11} +1.00000 q^{12} -2.00000 q^{13} -1.00000 q^{15} +1.00000 q^{16} +2.00000 q^{17} +1.00000 q^{18} +2.00000 q^{19} -1.00000 q^{20} -3.00000 q^{22} +1.00000 q^{23} +1.00000 q^{24} -4.00000 q^{25} -2.00000 q^{26} +1.00000 q^{27} -7.00000 q^{29} -1.00000 q^{30} -7.00000 q^{31} +1.00000 q^{32} -3.00000 q^{33} +2.00000 q^{34} +1.00000 q^{36} -2.00000 q^{37} +2.00000 q^{38} -2.00000 q^{39} -1.00000 q^{40} +2.00000 q^{41} -6.00000 q^{43} -3.00000 q^{44} -1.00000 q^{45} +1.00000 q^{46} -6.00000 q^{47} +1.00000 q^{48} -4.00000 q^{50} +2.00000 q^{51} -2.00000 q^{52} -9.00000 q^{53} +1.00000 q^{54} +3.00000 q^{55} +2.00000 q^{57} -7.00000 q^{58} +9.00000 q^{59} -1.00000 q^{60} -6.00000 q^{61} -7.00000 q^{62} +1.00000 q^{64} +2.00000 q^{65} -3.00000 q^{66} +10.0000 q^{67} +2.00000 q^{68} +1.00000 q^{69} -8.00000 q^{71} +1.00000 q^{72} +10.0000 q^{73} -2.00000 q^{74} -4.00000 q^{75} +2.00000 q^{76} -2.00000 q^{78} -15.0000 q^{79} -1.00000 q^{80} +1.00000 q^{81} +2.00000 q^{82} -1.00000 q^{83} -2.00000 q^{85} -6.00000 q^{86} -7.00000 q^{87} -3.00000 q^{88} -1.00000 q^{90} +1.00000 q^{92} -7.00000 q^{93} -6.00000 q^{94} -2.00000 q^{95} +1.00000 q^{96} -5.00000 q^{97} -3.00000 q^{99} +O(q^{100})\)

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
$3$	1.00000	0.577350
$4$	1.00000	0.500000
$5$	−1.00000	−0.447214	−0.223607	−	0.974679i	$-0.571783\pi$
$5$	−1.00000	−0.447214	−0.223607	+	0.974679i	$0.571783\pi$
$6$	1.00000	0.408248
$7$	0	0
$7$	0	0
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	−1.00000	−0.316228
$11$	−3.00000	−0.904534	−0.452267	−	0.891883i	$-0.649385\pi$
$11$	−3.00000	−0.904534	−0.452267	+	0.891883i	$0.649385\pi$
$12$	1.00000	0.288675
$13$	−2.00000	−0.554700	−0.277350	−	0.960769i	$-0.589456\pi$
$13$	−2.00000	−0.554700	−0.277350	+	0.960769i	$0.589456\pi$
$14$	0	0
$15$	−1.00000	−0.258199
$16$	1.00000	0.250000
$17$	2.00000	0.485071	0.242536	−	0.970143i	$-0.422021\pi$
$17$	2.00000	0.485071	0.242536	+	0.970143i	$0.422021\pi$
$18$	1.00000	0.235702
$19$	2.00000	0.458831	0.229416	−	0.973329i	$-0.426318\pi$
$19$	2.00000	0.458831	0.229416	+	0.973329i	$0.426318\pi$
$20$	−1.00000	−0.223607
$21$	0	0
$22$	−3.00000	−0.639602
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	1.00000	0.204124
$25$	−4.00000	−0.800000
$26$	−2.00000	−0.392232
$27$	1.00000	0.192450
$28$	0	0
$29$	−7.00000	−1.29987	−0.649934	−	0.759991i	$-0.725203\pi$
$29$	−7.00000	−1.29987	−0.649934	+	0.759991i	$0.725203\pi$
$30$	−1.00000	−0.182574
$31$	−7.00000	−1.25724	−0.628619	−	0.777714i	$-0.716379\pi$
$31$	−7.00000	−1.25724	−0.628619	+	0.777714i	$0.716379\pi$
$32$	1.00000	0.176777
$33$	−3.00000	−0.522233
$34$	2.00000	0.342997
$35$	0	0
$36$	1.00000	0.166667
$37$	−2.00000	−0.328798	−0.164399	−	0.986394i	$-0.552568\pi$
$37$	−2.00000	−0.328798	−0.164399	+	0.986394i	$0.552568\pi$
$38$	2.00000	0.324443
$39$	−2.00000	−0.320256
$40$	−1.00000	−0.158114
$41$	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
$43$	−6.00000	−0.914991	−0.457496	−	0.889212i	$-0.651253\pi$
$43$	−6.00000	−0.914991	−0.457496	+	0.889212i	$0.651253\pi$
$44$	−3.00000	−0.452267
$45$	−1.00000	−0.149071
$46$	1.00000	0.147442
$47$	−6.00000	−0.875190	−0.437595	−	0.899172i	$-0.644170\pi$
$47$	−6.00000	−0.875190	−0.437595	+	0.899172i	$0.644170\pi$
$48$	1.00000	0.144338
$49$	0	0
$50$	−4.00000	−0.565685
$51$	2.00000	0.280056
$52$	−2.00000	−0.277350
$53$	−9.00000	−1.23625	−0.618123	−	0.786082i	$-0.712106\pi$
$53$	−9.00000	−1.23625	−0.618123	+	0.786082i	$0.712106\pi$
$54$	1.00000	0.136083
$55$	3.00000	0.404520
$56$	0	0
$57$	2.00000	0.264906
$58$	−7.00000	−0.919145
$59$	9.00000	1.17170	0.585850	−	0.810419i	$-0.300761\pi$
$59$	9.00000	1.17170	0.585850	+	0.810419i	$0.300761\pi$
$60$	−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	$0.625492\pi$
$62$	−7.00000	−0.889001
$63$	0	0
$64$	1.00000	0.125000
$65$	2.00000	0.248069
$66$	−3.00000	−0.369274
$67$	10.0000	1.22169	0.610847	−	0.791748i	$-0.290829\pi$
$67$	10.0000	1.22169	0.610847	+	0.791748i	$0.290829\pi$
$68$	2.00000	0.242536
$69$	1.00000	0.120386
$70$	0	0
$71$	−8.00000	−0.949425	−0.474713	−	0.880141i	$-0.657448\pi$
$71$	−8.00000	−0.949425	−0.474713	+	0.880141i	$0.657448\pi$
$72$	1.00000	0.117851
$73$	10.0000	1.17041	0.585206	−	0.810885i	$-0.301014\pi$
$73$	10.0000	1.17041	0.585206	+	0.810885i	$0.301014\pi$
$74$	−2.00000	−0.232495
$75$	−4.00000	−0.461880
$76$	2.00000	0.229416
$77$	0	0
$78$	−2.00000	−0.226455
$79$	−15.0000	−1.68763	−0.843816	−	0.536633i	$-0.819696\pi$
$79$	−15.0000	−1.68763	−0.843816	+	0.536633i	$0.819696\pi$
$80$	−1.00000	−0.111803
$81$	1.00000	0.111111
$82$	2.00000	0.220863
$83$	−1.00000	−0.109764	−0.0548821	−	0.998493i	$-0.517478\pi$
$83$	−1.00000	−0.109764	−0.0548821	+	0.998493i	$0.517478\pi$
$84$	0	0
$85$	−2.00000	−0.216930
$86$	−6.00000	−0.646997
$87$	−7.00000	−0.750479
$88$	−3.00000	−0.319801
$89$	0	0		−	1.00000i	$-0.5\pi$
$89$	0	0			1.00000i	$0.5\pi$
$90$	−1.00000	−0.105409
$91$	0	0
$92$	1.00000	0.104257
$93$	−7.00000	−0.725866
$94$	−6.00000	−0.618853
$95$	−2.00000	−0.205196
$96$	1.00000	0.102062
$97$	−5.00000	−0.507673	−0.253837	−	0.967247i	$-0.581693\pi$
$97$	−5.00000	−0.507673	−0.253837	+	0.967247i	$0.581693\pi$
$98$	0	0
$99$	−3.00000	−0.301511
$100$	−4.00000	−0.400000
$101$	10.0000	0.995037	0.497519	−	0.867453i	$-0.334245\pi$
$101$	10.0000	0.995037	0.497519	+	0.867453i	$0.334245\pi$
$102$	2.00000	0.198030
$103$	8.00000	0.788263	0.394132	−	0.919054i	$-0.371045\pi$
$103$	8.00000	0.788263	0.394132	+	0.919054i	$0.371045\pi$
$104$	−2.00000	−0.196116
$105$	0	0
$106$	−9.00000	−0.874157
$107$	3.00000	0.290021	0.145010	−	0.989430i	$-0.453678\pi$
$107$	3.00000	0.290021	0.145010	+	0.989430i	$0.453678\pi$
$108$	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	$0.341031\pi$
$110$	3.00000	0.286039
$111$	−2.00000	−0.189832
$112$	0	0
$113$	10.0000	0.940721	0.470360	−	0.882474i	$-0.344124\pi$
$113$	10.0000	0.940721	0.470360	+	0.882474i	$0.344124\pi$
$114$	2.00000	0.187317
$115$	−1.00000	−0.0932505
$116$	−7.00000	−0.649934
$117$	−2.00000	−0.184900
$118$	9.00000	0.828517
$119$	0	0
$120$	−1.00000	−0.0912871
$121$	−2.00000	−0.181818
$122$	−6.00000	−0.543214
$123$	2.00000	0.180334
$124$	−7.00000	−0.628619
$125$	9.00000	0.804984
$126$	0	0
$127$	−7.00000	−0.621150	−0.310575	−	0.950549i	$-0.600522\pi$
$127$	−7.00000	−0.621150	−0.310575	+	0.950549i	$0.600522\pi$
$128$	1.00000	0.0883883
$129$	−6.00000	−0.528271
$130$	2.00000	0.175412
$131$	−7.00000	−0.611593	−0.305796	−	0.952097i	$-0.598923\pi$
$131$	−7.00000	−0.611593	−0.305796	+	0.952097i	$0.598923\pi$
$132$	−3.00000	−0.261116
$133$	0	0
$134$	10.0000	0.863868
$135$	−1.00000	−0.0860663
$136$	2.00000	0.171499
$137$	−6.00000	−0.512615	−0.256307	−	0.966595i	$-0.582506\pi$
$137$	−6.00000	−0.512615	−0.256307	+	0.966595i	$0.582506\pi$
$138$	1.00000	0.0851257
$139$	−14.0000	−1.18746	−0.593732	−	0.804663i	$-0.702346\pi$
$139$	−14.0000	−1.18746	−0.593732	+	0.804663i	$0.702346\pi$
$140$	0	0
$141$	−6.00000	−0.505291
$142$	−8.00000	−0.671345
$143$	6.00000	0.501745
$144$	1.00000	0.0833333
$145$	7.00000	0.581318
$146$	10.0000	0.827606
$147$	0	0
$148$	−2.00000	−0.164399
$149$	−22.0000	−1.80231	−0.901155	−	0.433497i	$-0.857280\pi$
$149$	−22.0000	−1.80231	−0.901155	+	0.433497i	$0.857280\pi$
$150$	−4.00000	−0.326599
$151$	−5.00000	−0.406894	−0.203447	−	0.979086i	$-0.565214\pi$
$151$	−5.00000	−0.406894	−0.203447	+	0.979086i	$0.565214\pi$
$152$	2.00000	0.162221
$153$	2.00000	0.161690
$154$	0	0
$155$	7.00000	0.562254
$156$	−2.00000	−0.160128
$157$	−6.00000	−0.478852	−0.239426	−	0.970915i	$-0.576959\pi$
$157$	−6.00000	−0.478852	−0.239426	+	0.970915i	$0.576959\pi$
$158$	−15.0000	−1.19334
$159$	−9.00000	−0.713746
$160$	−1.00000	−0.0790569
$161$	0	0
$162$	1.00000	0.0785674
$163$	2.00000	0.156652	0.0783260	−	0.996928i	$-0.475042\pi$
$163$	2.00000	0.156652	0.0783260	+	0.996928i	$0.475042\pi$
$164$	2.00000	0.156174
$165$	3.00000	0.233550
$166$	−1.00000	−0.0776151
$167$	−6.00000	−0.464294	−0.232147	−	0.972681i	$-0.574575\pi$
$167$	−6.00000	−0.464294	−0.232147	+	0.972681i	$0.574575\pi$
$168$	0	0
$169$	−9.00000	−0.692308
$170$	−2.00000	−0.153393
$171$	2.00000	0.152944
$172$	−6.00000	−0.457496
$173$	6.00000	0.456172	0.228086	−	0.973641i	$-0.426753\pi$
$173$	6.00000	0.456172	0.228086	+	0.973641i	$0.426753\pi$
$174$	−7.00000	−0.530669
$175$	0	0
$176$	−3.00000	−0.226134
$177$	9.00000	0.676481
$178$	0	0
$179$	−12.0000	−0.896922	−0.448461	−	0.893802i	$-0.648028\pi$
$179$	−12.0000	−0.896922	−0.448461	+	0.893802i	$0.648028\pi$
$180$	−1.00000	−0.0745356
$181$	−16.0000	−1.18927	−0.594635	−	0.803996i	$-0.702704\pi$
$181$	−16.0000	−1.18927	−0.594635	+	0.803996i	$0.702704\pi$
$182$	0	0
$183$	−6.00000	−0.443533
$184$	1.00000	0.0737210
$185$	2.00000	0.147043
$186$	−7.00000	−0.513265
$187$	−6.00000	−0.438763
$188$	−6.00000	−0.437595
$189$	0	0
$190$	−2.00000	−0.145095
$191$	10.0000	0.723575	0.361787	−	0.932261i	$-0.382167\pi$
$191$	10.0000	0.723575	0.361787	+	0.932261i	$0.382167\pi$
$192$	1.00000	0.0721688
$193$	−23.0000	−1.65558	−0.827788	−	0.561041i	$-0.810401\pi$
$193$	−23.0000	−1.65558	−0.827788	+	0.561041i	$0.810401\pi$
$194$	−5.00000	−0.358979
$195$	2.00000	0.143223
$196$	0	0
$197$	14.0000	0.997459	0.498729	−	0.866758i	$-0.333800\pi$
$197$	14.0000	0.997459	0.498729	+	0.866758i	$0.333800\pi$
$198$	−3.00000	−0.213201
$199$	4.00000	0.283552	0.141776	−	0.989899i	$-0.454719\pi$
$199$	4.00000	0.283552	0.141776	+	0.989899i	$0.454719\pi$
$200$	−4.00000	−0.282843
$201$	10.0000	0.705346
$202$	10.0000	0.703598
$203$	0	0
$204$	2.00000	0.140028
$205$	−2.00000	−0.139686
$206$	8.00000	0.557386
$207$	1.00000	0.0695048
$208$	−2.00000	−0.138675
$209$	−6.00000	−0.415029
$210$	0	0
$211$	6.00000	0.413057	0.206529	−	0.978441i	$-0.433783\pi$
$211$	6.00000	0.413057	0.206529	+	0.978441i	$0.433783\pi$
$212$	−9.00000	−0.618123
$213$	−8.00000	−0.548151
$214$	3.00000	0.205076
$215$	6.00000	0.409197
$216$	1.00000	0.0680414
$217$	0	0
$218$	10.0000	0.677285
$219$	10.0000	0.675737
$220$	3.00000	0.202260
$221$	−4.00000	−0.269069
$222$	−2.00000	−0.134231
$223$	−13.0000	−0.870544	−0.435272	−	0.900299i	$-0.643348\pi$
$223$	−13.0000	−0.870544	−0.435272	+	0.900299i	$0.643348\pi$
$224$	0	0
$225$	−4.00000	−0.266667
$226$	10.0000	0.665190
$227$	9.00000	0.597351	0.298675	−	0.954355i	$-0.403455\pi$
$227$	9.00000	0.597351	0.298675	+	0.954355i	$0.403455\pi$
$228$	2.00000	0.132453
$229$	−22.0000	−1.45380	−0.726900	−	0.686743i	$-0.759040\pi$
$229$	−22.0000	−1.45380	−0.726900	+	0.686743i	$0.759040\pi$
$230$	−1.00000	−0.0659380
$231$	0	0
$232$	−7.00000	−0.459573
$233$	−6.00000	−0.393073	−0.196537	−	0.980497i	$-0.562969\pi$
$233$	−6.00000	−0.393073	−0.196537	+	0.980497i	$0.562969\pi$
$234$	−2.00000	−0.130744
$235$	6.00000	0.391397
$236$	9.00000	0.585850
$237$	−15.0000	−0.974355
$238$	0	0
$239$	−12.0000	−0.776215	−0.388108	−	0.921614i	$-0.626871\pi$
$239$	−12.0000	−0.776215	−0.388108	+	0.921614i	$0.626871\pi$
$240$	−1.00000	−0.0645497
$241$	27.0000	1.73922	0.869611	−	0.493737i	$-0.164369\pi$
$241$	27.0000	1.73922	0.869611	+	0.493737i	$0.164369\pi$
$242$	−2.00000	−0.128565
$243$	1.00000	0.0641500
$244$	−6.00000	−0.384111
$245$	0	0
$246$	2.00000	0.127515
$247$	−4.00000	−0.254514
$248$	−7.00000	−0.444500
$249$	−1.00000	−0.0633724
$250$	9.00000	0.569210
$251$	3.00000	0.189358	0.0946792	−	0.995508i	$-0.469817\pi$
$251$	3.00000	0.189358	0.0946792	+	0.995508i	$0.469817\pi$
$252$	0	0
$253$	−3.00000	−0.188608
$254$	−7.00000	−0.439219
$255$	−2.00000	−0.125245
$256$	1.00000	0.0625000
$257$	4.00000	0.249513	0.124757	−	0.992187i	$-0.460185\pi$
$257$	4.00000	0.249513	0.124757	+	0.992187i	$0.460185\pi$
$258$	−6.00000	−0.373544
$259$	0	0
$260$	2.00000	0.124035
$261$	−7.00000	−0.433289
$262$	−7.00000	−0.432461
$263$	−2.00000	−0.123325	−0.0616626	−	0.998097i	$-0.519640\pi$
$263$	−2.00000	−0.123325	−0.0616626	+	0.998097i	$0.519640\pi$
$264$	−3.00000	−0.184637
$265$	9.00000	0.552866
$266$	0	0
$267$	0	0
$268$	10.0000	0.610847
$269$	27.0000	1.64622	0.823110	−	0.567883i	$-0.192237\pi$
$269$	27.0000	1.64622	0.823110	+	0.567883i	$0.192237\pi$
$270$	−1.00000	−0.0608581
$271$	9.00000	0.546711	0.273356	−	0.961913i	$-0.411866\pi$
$271$	9.00000	0.546711	0.273356	+	0.961913i	$0.411866\pi$
$272$	2.00000	0.121268
$273$	0	0
$274$	−6.00000	−0.362473
$275$	12.0000	0.723627
$276$	1.00000	0.0601929
$277$	−8.00000	−0.480673	−0.240337	−	0.970690i	$-0.577258\pi$
$277$	−8.00000	−0.480673	−0.240337	+	0.970690i	$0.577258\pi$
$278$	−14.0000	−0.839664
$279$	−7.00000	−0.419079
$280$	0	0
$281$	−4.00000	−0.238620	−0.119310	−	0.992857i	$-0.538068\pi$
$281$	−4.00000	−0.238620	−0.119310	+	0.992857i	$0.538068\pi$
$282$	−6.00000	−0.357295
$283$	−6.00000	−0.356663	−0.178331	−	0.983970i	$-0.557070\pi$
$283$	−6.00000	−0.356663	−0.178331	+	0.983970i	$0.557070\pi$
$284$	−8.00000	−0.474713
$285$	−2.00000	−0.118470
$286$	6.00000	0.354787
$287$	0	0
$288$	1.00000	0.0589256
$289$	−13.0000	−0.764706
$290$	7.00000	0.411054
$291$	−5.00000	−0.293105
$292$	10.0000	0.585206
$293$	−19.0000	−1.10999	−0.554996	−	0.831853i	$-0.687280\pi$
$293$	−19.0000	−1.10999	−0.554996	+	0.831853i	$0.687280\pi$
$294$	0	0
$295$	−9.00000	−0.524000
$296$	−2.00000	−0.116248
$297$	−3.00000	−0.174078
$298$	−22.0000	−1.27443
$299$	−2.00000	−0.115663
$300$	−4.00000	−0.230940
$301$	0	0
$302$	−5.00000	−0.287718
$303$	10.0000	0.574485
$304$	2.00000	0.114708
$305$	6.00000	0.343559
$306$	2.00000	0.114332
$307$	2.00000	0.114146	0.0570730	−	0.998370i	$-0.481823\pi$
$307$	2.00000	0.114146	0.0570730	+	0.998370i	$0.481823\pi$
$308$	0	0
$309$	8.00000	0.455104
$310$	7.00000	0.397573
$311$	2.00000	0.113410	0.0567048	−	0.998391i	$-0.481941\pi$
$311$	2.00000	0.113410	0.0567048	+	0.998391i	$0.481941\pi$
$312$	−2.00000	−0.113228
$313$	25.0000	1.41308	0.706542	−	0.707671i	$-0.250254\pi$
$313$	25.0000	1.41308	0.706542	+	0.707671i	$0.250254\pi$
$314$	−6.00000	−0.338600
$315$	0	0
$316$	−15.0000	−0.843816
$317$	−31.0000	−1.74113	−0.870567	−	0.492050i	$-0.836248\pi$
$317$	−31.0000	−1.74113	−0.870567	+	0.492050i	$0.836248\pi$
$318$	−9.00000	−0.504695
$319$	21.0000	1.17577
$320$	−1.00000	−0.0559017
$321$	3.00000	0.167444
$322$	0	0
$323$	4.00000	0.222566
$324$	1.00000	0.0555556
$325$	8.00000	0.443760
$326$	2.00000	0.110770
$327$	10.0000	0.553001
$328$	2.00000	0.110432
$329$	0	0
$330$	3.00000	0.165145
$331$	10.0000	0.549650	0.274825	−	0.961494i	$-0.411380\pi$
$331$	10.0000	0.549650	0.274825	+	0.961494i	$0.411380\pi$
$332$	−1.00000	−0.0548821
$333$	−2.00000	−0.109599
$334$	−6.00000	−0.328305
$335$	−10.0000	−0.546358
$336$	0	0
$337$	−9.00000	−0.490261	−0.245131	−	0.969490i	$-0.578831\pi$
$337$	−9.00000	−0.490261	−0.245131	+	0.969490i	$0.578831\pi$
$338$	−9.00000	−0.489535
$339$	10.0000	0.543125
$340$	−2.00000	−0.108465
$341$	21.0000	1.13721
$342$	2.00000	0.108148
$343$	0	0
$344$	−6.00000	−0.323498
$345$	−1.00000	−0.0538382
$346$	6.00000	0.322562
$347$	36.0000	1.93258	0.966291	−	0.257454i	$-0.0828835\pi$
$347$	36.0000	1.93258	0.966291	+	0.257454i	$0.0828835\pi$
$348$	−7.00000	−0.375239
$349$	−16.0000	−0.856460	−0.428230	−	0.903670i	$-0.640863\pi$
$349$	−16.0000	−0.856460	−0.428230	+	0.903670i	$0.640863\pi$
$350$	0	0
$351$	−2.00000	−0.106752
$352$	−3.00000	−0.159901
$353$	26.0000	1.38384	0.691920	−	0.721974i	$-0.256765\pi$
$353$	26.0000	1.38384	0.691920	+	0.721974i	$0.256765\pi$
$354$	9.00000	0.478345
$355$	8.00000	0.424596
$356$	0	0
$357$	0	0
$358$	−12.0000	−0.634220
$359$	30.0000	1.58334	0.791670	−	0.610949i	$-0.209212\pi$
$359$	30.0000	1.58334	0.791670	+	0.610949i	$0.209212\pi$
$360$	−1.00000	−0.0527046
$361$	−15.0000	−0.789474
$362$	−16.0000	−0.840941
$363$	−2.00000	−0.104973
$364$	0	0
$365$	−10.0000	−0.523424
$366$	−6.00000	−0.313625
$367$	−7.00000	−0.365397	−0.182699	−	0.983169i	$-0.558483\pi$
$367$	−7.00000	−0.365397	−0.182699	+	0.983169i	$0.558483\pi$
$368$	1.00000	0.0521286
$369$	2.00000	0.104116
$370$	2.00000	0.103975
$371$	0	0
$372$	−7.00000	−0.362933
$373$	32.0000	1.65690	0.828449	−	0.560065i	$-0.189224\pi$
$373$	32.0000	1.65690	0.828449	+	0.560065i	$0.189224\pi$
$374$	−6.00000	−0.310253
$375$	9.00000	0.464758
$376$	−6.00000	−0.309426
$377$	14.0000	0.721037
$378$	0	0
$379$	−4.00000	−0.205466	−0.102733	−	0.994709i	$-0.532759\pi$
$379$	−4.00000	−0.205466	−0.102733	+	0.994709i	$0.532759\pi$
$380$	−2.00000	−0.102598
$381$	−7.00000	−0.358621
$382$	10.0000	0.511645
$383$	−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
$386$	−23.0000	−1.17067
$387$	−6.00000	−0.304997
$388$	−5.00000	−0.253837
$389$	38.0000	1.92668	0.963338	−	0.268290i	$-0.0864585\pi$
$389$	38.0000	1.92668	0.963338	+	0.268290i	$0.0864585\pi$
$390$	2.00000	0.101274
$391$	2.00000	0.101144
$392$	0	0
$393$	−7.00000	−0.353103
$394$	14.0000	0.705310
$395$	15.0000	0.754732
$396$	−3.00000	−0.150756
$397$	38.0000	1.90717	0.953583	−	0.301131i	$-0.0973643\pi$
$397$	38.0000	1.90717	0.953583	+	0.301131i	$0.0973643\pi$
$398$	4.00000	0.200502
$399$	0	0
$400$	−4.00000	−0.200000
$401$	4.00000	0.199750	0.0998752	−	0.995000i	$-0.468156\pi$
$401$	4.00000	0.199750	0.0998752	+	0.995000i	$0.468156\pi$
$402$	10.0000	0.498755
$403$	14.0000	0.697390
$404$	10.0000	0.497519
$405$	−1.00000	−0.0496904
$406$	0	0
$407$	6.00000	0.297409
$408$	2.00000	0.0990148
$409$	29.0000	1.43396	0.716979	−	0.697095i	$-0.245524\pi$
$409$	29.0000	1.43396	0.716979	+	0.697095i	$0.245524\pi$
$410$	−2.00000	−0.0987730
$411$	−6.00000	−0.295958
$412$	8.00000	0.394132
$413$	0	0
$414$	1.00000	0.0491473
$415$	1.00000	0.0490881
$416$	−2.00000	−0.0980581
$417$	−14.0000	−0.685583
$418$	−6.00000	−0.293470
$419$	36.0000	1.75872	0.879358	−	0.476162i	$-0.157972\pi$
$419$	36.0000	1.75872	0.879358	+	0.476162i	$0.157972\pi$
$420$	0	0
$421$	−18.0000	−0.877266	−0.438633	−	0.898666i	$-0.644537\pi$
$421$	−18.0000	−0.877266	−0.438633	+	0.898666i	$0.644537\pi$
$422$	6.00000	0.292075
$423$	−6.00000	−0.291730
$424$	−9.00000	−0.437079
$425$	−8.00000	−0.388057
$426$	−8.00000	−0.387601
$427$	0	0
$428$	3.00000	0.145010
$429$	6.00000	0.289683
$430$	6.00000	0.289346
$431$	0	0		−	1.00000i	$-0.5\pi$
$431$	0	0			1.00000i	$0.5\pi$
$432$	1.00000	0.0481125
$433$	−38.0000	−1.82616	−0.913082	−	0.407777i	$-0.866304\pi$
$433$	−38.0000	−1.82616	−0.913082	+	0.407777i	$0.866304\pi$
$434$	0	0
$435$	7.00000	0.335624
$436$	10.0000	0.478913
$437$	2.00000	0.0956730
$438$	10.0000	0.477818
$439$	−35.0000	−1.67046	−0.835229	−	0.549902i	$-0.814665\pi$
$439$	−35.0000	−1.67046	−0.835229	+	0.549902i	$0.814665\pi$
$440$	3.00000	0.143019
$441$	0	0
$442$	−4.00000	−0.190261
$443$	−29.0000	−1.37783	−0.688916	−	0.724841i	$-0.741913\pi$
$443$	−29.0000	−1.37783	−0.688916	+	0.724841i	$0.741913\pi$
$444$	−2.00000	−0.0949158
$445$	0	0
$446$	−13.0000	−0.615568
$447$	−22.0000	−1.04056
$448$	0	0
$449$	−30.0000	−1.41579	−0.707894	−	0.706319i	$-0.750354\pi$
$449$	−30.0000	−1.41579	−0.707894	+	0.706319i	$0.750354\pi$
$450$	−4.00000	−0.188562
$451$	−6.00000	−0.282529
$452$	10.0000	0.470360
$453$	−5.00000	−0.234920
$454$	9.00000	0.422391
$455$	0	0
$456$	2.00000	0.0936586
$457$	1.00000	0.0467780	0.0233890	−	0.999726i	$-0.492554\pi$
$457$	1.00000	0.0467780	0.0233890	+	0.999726i	$0.492554\pi$
$458$	−22.0000	−1.02799
$459$	2.00000	0.0933520
$460$	−1.00000	−0.0466252
$461$	−2.00000	−0.0931493	−0.0465746	−	0.998915i	$-0.514831\pi$
$461$	−2.00000	−0.0931493	−0.0465746	+	0.998915i	$0.514831\pi$
$462$	0	0
$463$	−8.00000	−0.371792	−0.185896	−	0.982569i	$-0.559519\pi$
$463$	−8.00000	−0.371792	−0.185896	+	0.982569i	$0.559519\pi$
$464$	−7.00000	−0.324967
$465$	7.00000	0.324617
$466$	−6.00000	−0.277945
$467$	−4.00000	−0.185098	−0.0925490	−	0.995708i	$-0.529501\pi$
$467$	−4.00000	−0.185098	−0.0925490	+	0.995708i	$0.529501\pi$
$468$	−2.00000	−0.0924500
$469$	0	0
$470$	6.00000	0.276759
$471$	−6.00000	−0.276465
$472$	9.00000	0.414259
$473$	18.0000	0.827641
$474$	−15.0000	−0.688973
$475$	−8.00000	−0.367065
$476$	0	0
$477$	−9.00000	−0.412082
$478$	−12.0000	−0.548867
$479$	−28.0000	−1.27935	−0.639676	−	0.768644i	$-0.720932\pi$
$479$	−28.0000	−1.27935	−0.639676	+	0.768644i	$0.720932\pi$
$480$	−1.00000	−0.0456435
$481$	4.00000	0.182384
$482$	27.0000	1.22982
$483$	0	0
$484$	−2.00000	−0.0909091
$485$	5.00000	0.227038
$486$	1.00000	0.0453609
$487$	−15.0000	−0.679715	−0.339857	−	0.940477i	$-0.610379\pi$
$487$	−15.0000	−0.679715	−0.339857	+	0.940477i	$0.610379\pi$
$488$	−6.00000	−0.271607
$489$	2.00000	0.0904431
$490$	0	0
$491$	19.0000	0.857458	0.428729	−	0.903433i	$-0.358962\pi$
$491$	19.0000	0.857458	0.428729	+	0.903433i	$0.358962\pi$
$492$	2.00000	0.0901670
$493$	−14.0000	−0.630528
$494$	−4.00000	−0.179969
$495$	3.00000	0.134840
$496$	−7.00000	−0.314309
$497$	0	0
$498$	−1.00000	−0.0448111
$499$	6.00000	0.268597	0.134298	−	0.990941i	$-0.457122\pi$
$499$	6.00000	0.268597	0.134298	+	0.990941i	$0.457122\pi$
$500$	9.00000	0.402492
$501$	−6.00000	−0.268060
$502$	3.00000	0.133897
$503$	28.0000	1.24846	0.624229	−	0.781241i	$-0.285413\pi$
$503$	28.0000	1.24846	0.624229	+	0.781241i	$0.285413\pi$
$504$	0	0
$505$	−10.0000	−0.444994
$506$	−3.00000	−0.133366
$507$	−9.00000	−0.399704
$508$	−7.00000	−0.310575
$509$	−17.0000	−0.753512	−0.376756	−	0.926313i	$-0.622960\pi$
$509$	−17.0000	−0.753512	−0.376756	+	0.926313i	$0.622960\pi$
$510$	−2.00000	−0.0885615
$511$	0	0
$512$	1.00000	0.0441942
$513$	2.00000	0.0883022
$514$	4.00000	0.176432
$515$	−8.00000	−0.352522
$516$	−6.00000	−0.264135
$517$	18.0000	0.791639
$518$	0	0
$519$	6.00000	0.263371
$520$	2.00000	0.0877058
$521$	18.0000	0.788594	0.394297	−	0.918983i	$-0.370988\pi$
$521$	18.0000	0.788594	0.394297	+	0.918983i	$0.370988\pi$
$522$	−7.00000	−0.306382
$523$	12.0000	0.524723	0.262362	−	0.964970i	$-0.415499\pi$
$523$	12.0000	0.524723	0.262362	+	0.964970i	$0.415499\pi$
$524$	−7.00000	−0.305796
$525$	0	0
$526$	−2.00000	−0.0872041
$527$	−14.0000	−0.609850
$528$	−3.00000	−0.130558
$529$	1.00000	0.0434783
$530$	9.00000	0.390935
$531$	9.00000	0.390567
$532$	0	0
$533$	−4.00000	−0.173259
$534$	0	0
$535$	−3.00000	−0.129701
$536$	10.0000	0.431934
$537$	−12.0000	−0.517838
$538$	27.0000	1.16405
$539$	0	0
$540$	−1.00000	−0.0430331
$541$	−8.00000	−0.343947	−0.171973	−	0.985102i	$-0.555014\pi$
$541$	−8.00000	−0.343947	−0.171973	+	0.985102i	$0.555014\pi$
$542$	9.00000	0.386583
$543$	−16.0000	−0.686626
$544$	2.00000	0.0857493
$545$	−10.0000	−0.428353
$546$	0	0
$547$	−38.0000	−1.62476	−0.812381	−	0.583127i	$-0.801829\pi$
$547$	−38.0000	−1.62476	−0.812381	+	0.583127i	$0.801829\pi$
$548$	−6.00000	−0.256307
$549$	−6.00000	−0.256074
$550$	12.0000	0.511682
$551$	−14.0000	−0.596420
$552$	1.00000	0.0425628
$553$	0	0
$554$	−8.00000	−0.339887
$555$	2.00000	0.0848953
$556$	−14.0000	−0.593732
$557$	33.0000	1.39825	0.699127	−	0.714997i	$-0.253572\pi$
$557$	33.0000	1.39825	0.699127	+	0.714997i	$0.253572\pi$
$558$	−7.00000	−0.296334
$559$	12.0000	0.507546
$560$	0	0
$561$	−6.00000	−0.253320
$562$	−4.00000	−0.168730
$563$	3.00000	0.126435	0.0632175	−	0.998000i	$-0.479864\pi$
$563$	3.00000	0.126435	0.0632175	+	0.998000i	$0.479864\pi$
$564$	−6.00000	−0.252646
$565$	−10.0000	−0.420703
$566$	−6.00000	−0.252199
$567$	0	0
$568$	−8.00000	−0.335673
$569$	−36.0000	−1.50920	−0.754599	−	0.656186i	$-0.772169\pi$
$569$	−36.0000	−1.50920	−0.754599	+	0.656186i	$0.772169\pi$
$570$	−2.00000	−0.0837708
$571$	20.0000	0.836974	0.418487	−	0.908223i	$-0.362561\pi$
$571$	20.0000	0.836974	0.418487	+	0.908223i	$0.362561\pi$
$572$	6.00000	0.250873
$573$	10.0000	0.417756
$574$	0	0
$575$	−4.00000	−0.166812
$576$	1.00000	0.0416667
$577$	33.0000	1.37381	0.686904	−	0.726748i	$-0.258969\pi$
$577$	33.0000	1.37381	0.686904	+	0.726748i	$0.258969\pi$
$578$	−13.0000	−0.540729
$579$	−23.0000	−0.955847
$580$	7.00000	0.290659
$581$	0	0
$582$	−5.00000	−0.207257
$583$	27.0000	1.11823
$584$	10.0000	0.413803
$585$	2.00000	0.0826898
$586$	−19.0000	−0.784883
$587$	3.00000	0.123823	0.0619116	−	0.998082i	$-0.480280\pi$
$587$	3.00000	0.123823	0.0619116	+	0.998082i	$0.480280\pi$
$588$	0	0
$589$	−14.0000	−0.576860
$590$	−9.00000	−0.370524
$591$	14.0000	0.575883
$592$	−2.00000	−0.0821995
$593$	12.0000	0.492781	0.246390	−	0.969171i	$-0.420755\pi$
$593$	12.0000	0.492781	0.246390	+	0.969171i	$0.420755\pi$
$594$	−3.00000	−0.123091
$595$	0	0
$596$	−22.0000	−0.901155
$597$	4.00000	0.163709
$598$	−2.00000	−0.0817861
$599$	24.0000	0.980613	0.490307	−	0.871550i	$-0.336885\pi$
$599$	24.0000	0.980613	0.490307	+	0.871550i	$0.336885\pi$
$600$	−4.00000	−0.163299
$601$	−23.0000	−0.938190	−0.469095	−	0.883148i	$-0.655420\pi$
$601$	−23.0000	−0.938190	−0.469095	+	0.883148i	$0.655420\pi$
$602$	0	0
$603$	10.0000	0.407231
$604$	−5.00000	−0.203447
$605$	2.00000	0.0813116
$606$	10.0000	0.406222
$607$	−21.0000	−0.852364	−0.426182	−	0.904638i	$-0.640142\pi$
$607$	−21.0000	−0.852364	−0.426182	+	0.904638i	$0.640142\pi$
$608$	2.00000	0.0811107
$609$	0	0
$610$	6.00000	0.242933
$611$	12.0000	0.485468
$612$	2.00000	0.0808452
$613$	36.0000	1.45403	0.727013	−	0.686624i	$-0.240908\pi$
$613$	36.0000	1.45403	0.727013	+	0.686624i	$0.240908\pi$
$614$	2.00000	0.0807134
$615$	−2.00000	−0.0806478
$616$	0	0
$617$	−6.00000	−0.241551	−0.120775	−	0.992680i	$-0.538538\pi$
$617$	−6.00000	−0.241551	−0.120775	+	0.992680i	$0.538538\pi$
$618$	8.00000	0.321807
$619$	16.0000	0.643094	0.321547	−	0.946894i	$-0.395797\pi$
$619$	16.0000	0.643094	0.321547	+	0.946894i	$0.395797\pi$
$620$	7.00000	0.281127
$621$	1.00000	0.0401286
$622$	2.00000	0.0801927
$623$	0	0
$624$	−2.00000	−0.0800641
$625$	11.0000	0.440000
$626$	25.0000	0.999201
$627$	−6.00000	−0.239617
$628$	−6.00000	−0.239426
$629$	−4.00000	−0.159490
$630$	0	0
$631$	31.0000	1.23409	0.617045	−	0.786928i	$-0.288330\pi$
$631$	31.0000	1.23409	0.617045	+	0.786928i	$0.288330\pi$
$632$	−15.0000	−0.596668
$633$	6.00000	0.238479
$634$	−31.0000	−1.23117
$635$	7.00000	0.277787
$636$	−9.00000	−0.356873
$637$	0	0
$638$	21.0000	0.831398
$639$	−8.00000	−0.316475
$640$	−1.00000	−0.0395285
$641$	−12.0000	−0.473972	−0.236986	−	0.971513i	$-0.576159\pi$
$641$	−12.0000	−0.473972	−0.236986	+	0.971513i	$0.576159\pi$
$642$	3.00000	0.118401
$643$	−4.00000	−0.157745	−0.0788723	−	0.996885i	$-0.525132\pi$
$643$	−4.00000	−0.157745	−0.0788723	+	0.996885i	$0.525132\pi$
$644$	0	0
$645$	6.00000	0.236250
$646$	4.00000	0.157378
$647$	−38.0000	−1.49393	−0.746967	−	0.664861i	$-0.768491\pi$
$647$	−38.0000	−1.49393	−0.746967	+	0.664861i	$0.768491\pi$
$648$	1.00000	0.0392837
$649$	−27.0000	−1.05984
$650$	8.00000	0.313786
$651$	0	0
$652$	2.00000	0.0783260
$653$	11.0000	0.430463	0.215232	−	0.976563i	$-0.430949\pi$
$653$	11.0000	0.430463	0.215232	+	0.976563i	$0.430949\pi$
$654$	10.0000	0.391031
$655$	7.00000	0.273513
$656$	2.00000	0.0780869
$657$	10.0000	0.390137
$658$	0	0
$659$	28.0000	1.09073	0.545363	−	0.838200i	$-0.316392\pi$
$659$	28.0000	1.09073	0.545363	+	0.838200i	$0.316392\pi$
$660$	3.00000	0.116775
$661$	−38.0000	−1.47803	−0.739014	−	0.673690i	$-0.764708\pi$
$661$	−38.0000	−1.47803	−0.739014	+	0.673690i	$0.764708\pi$
$662$	10.0000	0.388661
$663$	−4.00000	−0.155347
$664$	−1.00000	−0.0388075
$665$	0	0
$666$	−2.00000	−0.0774984
$667$	−7.00000	−0.271041
$668$	−6.00000	−0.232147
$669$	−13.0000	−0.502609
$670$	−10.0000	−0.386334
$671$	18.0000	0.694882
$672$	0	0
$673$	25.0000	0.963679	0.481840	−	0.876259i	$-0.339969\pi$
$673$	25.0000	0.963679	0.481840	+	0.876259i	$0.339969\pi$
$674$	−9.00000	−0.346667
$675$	−4.00000	−0.153960
$676$	−9.00000	−0.346154
$677$	35.0000	1.34516	0.672580	−	0.740025i	$-0.265186\pi$
$677$	35.0000	1.34516	0.672580	+	0.740025i	$0.265186\pi$
$678$	10.0000	0.384048
$679$	0	0
$680$	−2.00000	−0.0766965
$681$	9.00000	0.344881
$682$	21.0000	0.804132
$683$	−23.0000	−0.880071	−0.440035	−	0.897980i	$-0.645034\pi$
$683$	−23.0000	−0.880071	−0.440035	+	0.897980i	$0.645034\pi$
$684$	2.00000	0.0764719
$685$	6.00000	0.229248
$686$	0	0
$687$	−22.0000	−0.839352
$688$	−6.00000	−0.228748
$689$	18.0000	0.685745
$690$	−1.00000	−0.0380693
$691$	22.0000	0.836919	0.418460	−	0.908235i	$-0.362570\pi$
$691$	22.0000	0.836919	0.418460	+	0.908235i	$0.362570\pi$
$692$	6.00000	0.228086
$693$	0	0
$694$	36.0000	1.36654
$695$	14.0000	0.531050
$696$	−7.00000	−0.265334
$697$	4.00000	0.151511
$698$	−16.0000	−0.605609
$699$	−6.00000	−0.226941
$700$	0	0
$701$	−1.00000	−0.0377695	−0.0188847	−	0.999822i	$-0.506012\pi$
$701$	−1.00000	−0.0377695	−0.0188847	+	0.999822i	$0.506012\pi$
$702$	−2.00000	−0.0754851
$703$	−4.00000	−0.150863
$704$	−3.00000	−0.113067
$705$	6.00000	0.225973
$706$	26.0000	0.978523
$707$	0	0
$708$	9.00000	0.338241
$709$	−28.0000	−1.05156	−0.525781	−	0.850620i	$-0.676227\pi$
$709$	−28.0000	−1.05156	−0.525781	+	0.850620i	$0.676227\pi$
$710$	8.00000	0.300235
$711$	−15.0000	−0.562544
$712$	0	0
$713$	−7.00000	−0.262152
$714$	0	0
$715$	−6.00000	−0.224387
$716$	−12.0000	−0.448461
$717$	−12.0000	−0.448148
$718$	30.0000	1.11959
$719$	30.0000	1.11881	0.559406	−	0.828894i	$-0.311029\pi$
$719$	30.0000	1.11881	0.559406	+	0.828894i	$0.311029\pi$
$720$	−1.00000	−0.0372678
$721$	0	0
$722$	−15.0000	−0.558242
$723$	27.0000	1.00414
$724$	−16.0000	−0.594635
$725$	28.0000	1.03989
$726$	−2.00000	−0.0742270
$727$	−5.00000	−0.185440	−0.0927199	−	0.995692i	$-0.529556\pi$
$727$	−5.00000	−0.185440	−0.0927199	+	0.995692i	$0.529556\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	−10.0000	−0.370117
$731$	−12.0000	−0.443836
$732$	−6.00000	−0.221766
$733$	28.0000	1.03420	0.517102	−	0.855924i	$-0.327011\pi$
$733$	28.0000	1.03420	0.517102	+	0.855924i	$0.327011\pi$
$734$	−7.00000	−0.258375
$735$	0	0
$736$	1.00000	0.0368605
$737$	−30.0000	−1.10506
$738$	2.00000	0.0736210
$739$	−14.0000	−0.514998	−0.257499	−	0.966279i	$-0.582898\pi$
$739$	−14.0000	−0.514998	−0.257499	+	0.966279i	$0.582898\pi$
$740$	2.00000	0.0735215
$741$	−4.00000	−0.146944
$742$	0	0
$743$	48.0000	1.76095	0.880475	−	0.474093i	$-0.157224\pi$
$743$	48.0000	1.76095	0.880475	+	0.474093i	$0.157224\pi$
$744$	−7.00000	−0.256632
$745$	22.0000	0.806018
$746$	32.0000	1.17160
$747$	−1.00000	−0.0365881
$748$	−6.00000	−0.219382
$749$	0	0
$750$	9.00000	0.328634
$751$	11.0000	0.401396	0.200698	−	0.979653i	$-0.435679\pi$
$751$	11.0000	0.401396	0.200698	+	0.979653i	$0.435679\pi$
$752$	−6.00000	−0.218797
$753$	3.00000	0.109326
$754$	14.0000	0.509850
$755$	5.00000	0.181969
$756$	0	0
$757$	8.00000	0.290765	0.145382	−	0.989376i	$-0.453559\pi$
$757$	8.00000	0.290765	0.145382	+	0.989376i	$0.453559\pi$
$758$	−4.00000	−0.145287
$759$	−3.00000	−0.108893
$760$	−2.00000	−0.0725476
$761$	−18.0000	−0.652499	−0.326250	−	0.945284i	$-0.605785\pi$
$761$	−18.0000	−0.652499	−0.326250	+	0.945284i	$0.605785\pi$
$762$	−7.00000	−0.253583
$763$	0	0
$764$	10.0000	0.361787
$765$	−2.00000	−0.0723102
$766$	−8.00000	−0.289052
$767$	−18.0000	−0.649942
$768$	1.00000	0.0360844
$769$	−15.0000	−0.540914	−0.270457	−	0.962732i	$-0.587175\pi$
$769$	−15.0000	−0.540914	−0.270457	+	0.962732i	$0.587175\pi$
$770$	0	0
$771$	4.00000	0.144056
$772$	−23.0000	−0.827788
$773$	14.0000	0.503545	0.251773	−	0.967786i	$-0.418987\pi$
$773$	14.0000	0.503545	0.251773	+	0.967786i	$0.418987\pi$
$774$	−6.00000	−0.215666
$775$	28.0000	1.00579
$776$	−5.00000	−0.179490
$777$	0	0
$778$	38.0000	1.36237
$779$	4.00000	0.143315
$780$	2.00000	0.0716115
$781$	24.0000	0.858788
$782$	2.00000	0.0715199
$783$	−7.00000	−0.250160
$784$	0	0
$785$	6.00000	0.214149
$786$	−7.00000	−0.249682
$787$	14.0000	0.499046	0.249523	−	0.968369i	$-0.419726\pi$
$787$	14.0000	0.499046	0.249523	+	0.968369i	$0.419726\pi$
$788$	14.0000	0.498729
$789$	−2.00000	−0.0712019
$790$	15.0000	0.533676
$791$	0	0
$792$	−3.00000	−0.106600
$793$	12.0000	0.426132
$794$	38.0000	1.34857
$795$	9.00000	0.319197
$796$	4.00000	0.141776
$797$	15.0000	0.531327	0.265664	−	0.964066i	$-0.414409\pi$
$797$	15.0000	0.531327	0.265664	+	0.964066i	$0.414409\pi$
$798$	0	0
$799$	−12.0000	−0.424529
$800$	−4.00000	−0.141421
$801$	0	0
$802$	4.00000	0.141245
$803$	−30.0000	−1.05868
$804$	10.0000	0.352673
$805$	0	0
$806$	14.0000	0.493129
$807$	27.0000	0.950445
$808$	10.0000	0.351799
$809$	−34.0000	−1.19538	−0.597688	−	0.801729i	$-0.703914\pi$
$809$	−34.0000	−1.19538	−0.597688	+	0.801729i	$0.703914\pi$
$810$	−1.00000	−0.0351364
$811$	−46.0000	−1.61528	−0.807639	−	0.589677i	$-0.799255\pi$
$811$	−46.0000	−1.61528	−0.807639	+	0.589677i	$0.799255\pi$
$812$	0	0
$813$	9.00000	0.315644
$814$	6.00000	0.210300
$815$	−2.00000	−0.0700569
$816$	2.00000	0.0700140
$817$	−12.0000	−0.419827
$818$	29.0000	1.01396
$819$	0	0
$820$	−2.00000	−0.0698430
$821$	17.0000	0.593304	0.296652	−	0.954986i	$-0.404130\pi$
$821$	17.0000	0.593304	0.296652	+	0.954986i	$0.404130\pi$
$822$	−6.00000	−0.209274
$823$	16.0000	0.557725	0.278862	−	0.960331i	$-0.410043\pi$
$823$	16.0000	0.557725	0.278862	+	0.960331i	$0.410043\pi$
$824$	8.00000	0.278693
$825$	12.0000	0.417786
$826$	0	0
$827$	5.00000	0.173867	0.0869335	−	0.996214i	$-0.472293\pi$
$827$	5.00000	0.173867	0.0869335	+	0.996214i	$0.472293\pi$
$828$	1.00000	0.0347524
$829$	20.0000	0.694629	0.347314	−	0.937749i	$-0.387094\pi$
$829$	20.0000	0.694629	0.347314	+	0.937749i	$0.387094\pi$
$830$	1.00000	0.0347105
$831$	−8.00000	−0.277517
$832$	−2.00000	−0.0693375
$833$	0	0
$834$	−14.0000	−0.484780
$835$	6.00000	0.207639
$836$	−6.00000	−0.207514
$837$	−7.00000	−0.241955
$838$	36.0000	1.24360
$839$	26.0000	0.897620	0.448810	−	0.893627i	$-0.351848\pi$
$839$	26.0000	0.897620	0.448810	+	0.893627i	$0.351848\pi$
$840$	0	0
$841$	20.0000	0.689655
$842$	−18.0000	−0.620321
$843$	−4.00000	−0.137767
$844$	6.00000	0.206529
$845$	9.00000	0.309609
$846$	−6.00000	−0.206284
$847$	0	0
$848$	−9.00000	−0.309061
$849$	−6.00000	−0.205919
$850$	−8.00000	−0.274398
$851$	−2.00000	−0.0685591
$852$	−8.00000	−0.274075
$853$	−54.0000	−1.84892	−0.924462	−	0.381273i	$-0.875486\pi$
$853$	−54.0000	−1.84892	−0.924462	+	0.381273i	$0.875486\pi$
$854$	0	0
$855$	−2.00000	−0.0683986
$856$	3.00000	0.102538
$857$	−16.0000	−0.546550	−0.273275	−	0.961936i	$-0.588107\pi$
$857$	−16.0000	−0.546550	−0.273275	+	0.961936i	$0.588107\pi$
$858$	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	$0.739060\pi$
$860$	6.00000	0.204598
$861$	0	0
$862$	0	0
$863$	0	0		−	1.00000i	$-0.5\pi$
$863$	0	0			1.00000i	$0.5\pi$
$864$	1.00000	0.0340207
$865$	−6.00000	−0.204006
$866$	−38.0000	−1.29129
$867$	−13.0000	−0.441503
$868$	0	0
$869$	45.0000	1.52652
$870$	7.00000	0.237322
$871$	−20.0000	−0.677674
$872$	10.0000	0.338643
$873$	−5.00000	−0.169224
$874$	2.00000	0.0676510
$875$	0	0
$876$	10.0000	0.337869
$877$	50.0000	1.68838	0.844190	−	0.536044i	$-0.180082\pi$
$877$	50.0000	1.68838	0.844190	+	0.536044i	$0.180082\pi$
$878$	−35.0000	−1.18119
$879$	−19.0000	−0.640854
$880$	3.00000	0.101130
$881$	18.0000	0.606435	0.303218	−	0.952921i	$-0.401939\pi$
$881$	18.0000	0.606435	0.303218	+	0.952921i	$0.401939\pi$
$882$	0	0
$883$	2.00000	0.0673054	0.0336527	−	0.999434i	$-0.489286\pi$
$883$	2.00000	0.0673054	0.0336527	+	0.999434i	$0.489286\pi$
$884$	−4.00000	−0.134535
$885$	−9.00000	−0.302532
$886$	−29.0000	−0.974274
$887$	36.0000	1.20876	0.604381	−	0.796696i	$-0.293421\pi$
$887$	36.0000	1.20876	0.604381	+	0.796696i	$0.293421\pi$
$888$	−2.00000	−0.0671156
$889$	0	0
$890$	0	0
$891$	−3.00000	−0.100504
$892$	−13.0000	−0.435272
$893$	−12.0000	−0.401565
$894$	−22.0000	−0.735790
$895$	12.0000	0.401116
$896$	0	0
$897$	−2.00000	−0.0667781
$898$	−30.0000	−1.00111
$899$	49.0000	1.63424
$900$	−4.00000	−0.133333
$901$	−18.0000	−0.599667
$902$	−6.00000	−0.199778
$903$	0	0
$904$	10.0000	0.332595
$905$	16.0000	0.531858
$906$	−5.00000	−0.166114
$907$	46.0000	1.52740	0.763702	−	0.645568i	$-0.223379\pi$
$907$	46.0000	1.52740	0.763702	+	0.645568i	$0.223379\pi$
$908$	9.00000	0.298675
$909$	10.0000	0.331679
$910$	0	0
$911$	22.0000	0.728893	0.364446	−	0.931224i	$-0.381258\pi$
$911$	22.0000	0.728893	0.364446	+	0.931224i	$0.381258\pi$
$912$	2.00000	0.0662266
$913$	3.00000	0.0992855
$914$	1.00000	0.0330771
$915$	6.00000	0.198354
$916$	−22.0000	−0.726900
$917$	0	0
$918$	2.00000	0.0660098
$919$	32.0000	1.05558	0.527791	−	0.849374i	$-0.323020\pi$
$919$	32.0000	1.05558	0.527791	+	0.849374i	$0.323020\pi$
$920$	−1.00000	−0.0329690
$921$	2.00000	0.0659022
$922$	−2.00000	−0.0658665
$923$	16.0000	0.526646
$924$	0	0
$925$	8.00000	0.263038
$926$	−8.00000	−0.262896
$927$	8.00000	0.262754
$928$	−7.00000	−0.229786
$929$	−32.0000	−1.04989	−0.524943	−	0.851137i	$-0.675913\pi$
$929$	−32.0000	−1.04989	−0.524943	+	0.851137i	$0.675913\pi$
$930$	7.00000	0.229539
$931$	0	0
$932$	−6.00000	−0.196537
$933$	2.00000	0.0654771
$934$	−4.00000	−0.130884
$935$	6.00000	0.196221
$936$	−2.00000	−0.0653720
$937$	−29.0000	−0.947389	−0.473694	−	0.880689i	$-0.657080\pi$
$937$	−29.0000	−0.947389	−0.473694	+	0.880689i	$0.657080\pi$
$938$	0	0
$939$	25.0000	0.815844
$940$	6.00000	0.195698
$941$	15.0000	0.488986	0.244493	−	0.969651i	$-0.421378\pi$
$941$	15.0000	0.488986	0.244493	+	0.969651i	$0.421378\pi$
$942$	−6.00000	−0.195491
$943$	2.00000	0.0651290
$944$	9.00000	0.292925
$945$	0	0
$946$	18.0000	0.585230
$947$	−24.0000	−0.779895	−0.389948	−	0.920837i	$-0.627507\pi$
$947$	−24.0000	−0.779895	−0.389948	+	0.920837i	$0.627507\pi$
$948$	−15.0000	−0.487177
$949$	−20.0000	−0.649227
$950$	−8.00000	−0.259554
$951$	−31.0000	−1.00524
$952$	0	0
$953$	−26.0000	−0.842223	−0.421111	−	0.907009i	$-0.638360\pi$
$953$	−26.0000	−0.842223	−0.421111	+	0.907009i	$0.638360\pi$
$954$	−9.00000	−0.291386
$955$	−10.0000	−0.323592
$956$	−12.0000	−0.388108
$957$	21.0000	0.678834
$958$	−28.0000	−0.904639
$959$	0	0
$960$	−1.00000	−0.0322749
$961$	18.0000	0.580645
$962$	4.00000	0.128965
$963$	3.00000	0.0966736
$964$	27.0000	0.869611
$965$	23.0000	0.740396
$966$	0	0
$967$	−13.0000	−0.418052	−0.209026	−	0.977910i	$-0.567029\pi$
$967$	−13.0000	−0.418052	−0.209026	+	0.977910i	$0.567029\pi$
$968$	−2.00000	−0.0642824
$969$	4.00000	0.128499
$970$	5.00000	0.160540
$971$	17.0000	0.545556	0.272778	−	0.962077i	$-0.412058\pi$
$971$	17.0000	0.545556	0.272778	+	0.962077i	$0.412058\pi$
$972$	1.00000	0.0320750
$973$	0	0
$974$	−15.0000	−0.480631
$975$	8.00000	0.256205
$976$	−6.00000	−0.192055
$977$	30.0000	0.959785	0.479893	−	0.877327i	$-0.340676\pi$
$977$	30.0000	0.959785	0.479893	+	0.877327i	$0.340676\pi$
$978$	2.00000	0.0639529
$979$	0	0
$980$	0	0
$981$	10.0000	0.319275
$982$	19.0000	0.606314
$983$	50.0000	1.59475	0.797376	−	0.603483i	$-0.206221\pi$
$983$	50.0000	1.59475	0.797376	+	0.603483i	$0.206221\pi$
$984$	2.00000	0.0637577
$985$	−14.0000	−0.446077
$986$	−14.0000	−0.445851
$987$	0	0
$988$	−4.00000	−0.127257
$989$	−6.00000	−0.190789
$990$	3.00000	0.0953463
$991$	3.00000	0.0952981	0.0476491	−	0.998864i	$-0.484827\pi$
$991$	3.00000	0.0952981	0.0476491	+	0.998864i	$0.484827\pi$
$992$	−7.00000	−0.222250
$993$	10.0000	0.317340
$994$	0	0
$995$	−4.00000	−0.126809
$996$	−1.00000	−0.0316862
$997$	−36.0000	−1.14013	−0.570066	−	0.821599i	$-0.693082\pi$
$997$	−36.0000	−1.14013	−0.570066	+	0.821599i	$0.693082\pi$
$998$	6.00000	0.189927
$999$	−2.00000	−0.0632772

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6762.2.a.bh.1.1		1
7.3	odd	6		966.2.i.c.415.1	yes	2
7.5	odd	6		966.2.i.c.277.1	✓	2
7.6	odd	2		6762.2.a.bb.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
966.2.i.c.277.1	✓	2	7.5	odd	6
966.2.i.c.415.1	yes	2	7.3	odd	6
6762.2.a.bb.1.1		1	7.6	odd	2
6762.2.a.bh.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:	\( N \)	\(=\)	\( 6762 = 2 \cdot 3 \cdot 7^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6762.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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