LMFDB - Embedded newform 3822.2.a.v.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3822,2,Mod(1,3822)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3822, 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("3822.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3822 = 2 \cdot 3 \cdot 7^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3822.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:	$30.5188236525$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 546)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	3822.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

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

\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} -1.00000 q^{11} -1.00000 q^{12} +1.00000 q^{13} -1.00000 q^{15} +1.00000 q^{16} +6.00000 q^{17} +1.00000 q^{18} +4.00000 q^{19} +1.00000 q^{20} -1.00000 q^{22} -6.00000 q^{23} -1.00000 q^{24} -4.00000 q^{25} +1.00000 q^{26} -1.00000 q^{27} +3.00000 q^{29} -1.00000 q^{30} +11.0000 q^{31} +1.00000 q^{32} +1.00000 q^{33} +6.00000 q^{34} +1.00000 q^{36} +4.00000 q^{37} +4.00000 q^{38} -1.00000 q^{39} +1.00000 q^{40} -12.0000 q^{41} -8.00000 q^{43} -1.00000 q^{44} +1.00000 q^{45} -6.00000 q^{46} +8.00000 q^{47} -1.00000 q^{48} -4.00000 q^{50} -6.00000 q^{51} +1.00000 q^{52} -5.00000 q^{53} -1.00000 q^{54} -1.00000 q^{55} -4.00000 q^{57} +3.00000 q^{58} +5.00000 q^{59} -1.00000 q^{60} -12.0000 q^{61} +11.0000 q^{62} +1.00000 q^{64} +1.00000 q^{65} +1.00000 q^{66} +16.0000 q^{67} +6.00000 q^{68} +6.00000 q^{69} +6.00000 q^{71} +1.00000 q^{72} +10.0000 q^{73} +4.00000 q^{74} +4.00000 q^{75} +4.00000 q^{76} -1.00000 q^{78} +7.00000 q^{79} +1.00000 q^{80} +1.00000 q^{81} -12.0000 q^{82} +17.0000 q^{83} +6.00000 q^{85} -8.00000 q^{86} -3.00000 q^{87} -1.00000 q^{88} +12.0000 q^{89} +1.00000 q^{90} -6.00000 q^{92} -11.0000 q^{93} +8.00000 q^{94} +4.00000 q^{95} -1.00000 q^{96} -13.0000 q^{97} -1.00000 q^{99} +O(q^{100})\)

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	$-0.428217\pi$
$5$	1.00000	0.447214	0.223607	+	0.974679i	$0.428217\pi$
$6$	−1.00000	−0.408248
$7$	0	0
$7$	0	0
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	1.00000	0.316228
$11$	−1.00000	−0.301511	−0.150756	−	0.988571i	$-0.548171\pi$
$11$	−1.00000	−0.301511	−0.150756	+	0.988571i	$0.548171\pi$
$12$	−1.00000	−0.288675
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	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
$19$	4.00000	0.917663	0.458831	−	0.888523i	$-0.348268\pi$
$19$	4.00000	0.917663	0.458831	+	0.888523i	$0.348268\pi$
$20$	1.00000	0.223607
$21$	0	0
$22$	−1.00000	−0.213201
$23$	−6.00000	−1.25109	−0.625543	−	0.780189i	$-0.715123\pi$
$23$	−6.00000	−1.25109	−0.625543	+	0.780189i	$0.715123\pi$
$24$	−1.00000	−0.204124
$25$	−4.00000	−0.800000
$26$	1.00000	0.196116
$27$	−1.00000	−0.192450
$28$	0	0
$29$	3.00000	0.557086	0.278543	−	0.960424i	$-0.410149\pi$
$29$	3.00000	0.557086	0.278543	+	0.960424i	$0.410149\pi$
$30$	−1.00000	−0.182574
$31$	11.0000	1.97566	0.987829	−	0.155543i	$-0.0497126\pi$
$31$	11.0000	1.97566	0.987829	+	0.155543i	$0.0497126\pi$
$32$	1.00000	0.176777
$33$	1.00000	0.174078
$34$	6.00000	1.02899
$35$	0	0
$36$	1.00000	0.166667
$37$	4.00000	0.657596	0.328798	−	0.944400i	$-0.393356\pi$
$37$	4.00000	0.657596	0.328798	+	0.944400i	$0.393356\pi$
$38$	4.00000	0.648886
$39$	−1.00000	−0.160128
$40$	1.00000	0.158114
$41$	−12.0000	−1.87409	−0.937043	−	0.349215i	$-0.886448\pi$
$41$	−12.0000	−1.87409	−0.937043	+	0.349215i	$0.886448\pi$
$42$	0	0
$43$	−8.00000	−1.21999	−0.609994	−	0.792406i	$-0.708828\pi$
$43$	−8.00000	−1.21999	−0.609994	+	0.792406i	$0.708828\pi$
$44$	−1.00000	−0.150756
$45$	1.00000	0.149071
$46$	−6.00000	−0.884652
$47$	8.00000	1.16692	0.583460	−	0.812142i	$-0.301699\pi$
$47$	8.00000	1.16692	0.583460	+	0.812142i	$0.301699\pi$
$48$	−1.00000	−0.144338
$49$	0	0
$50$	−4.00000	−0.565685
$51$	−6.00000	−0.840168
$52$	1.00000	0.138675
$53$	−5.00000	−0.686803	−0.343401	−	0.939189i	$-0.611579\pi$
$53$	−5.00000	−0.686803	−0.343401	+	0.939189i	$0.611579\pi$
$54$	−1.00000	−0.136083
$55$	−1.00000	−0.134840
$56$	0	0
$57$	−4.00000	−0.529813
$58$	3.00000	0.393919
$59$	5.00000	0.650945	0.325472	−	0.945552i	$-0.394477\pi$
$59$	5.00000	0.650945	0.325472	+	0.945552i	$0.394477\pi$
$60$	−1.00000	−0.129099
$61$	−12.0000	−1.53644	−0.768221	−	0.640184i	$-0.778858\pi$
$61$	−12.0000	−1.53644	−0.768221	+	0.640184i	$0.778858\pi$
$62$	11.0000	1.39700
$63$	0	0
$64$	1.00000	0.125000
$65$	1.00000	0.124035
$66$	1.00000	0.123091
$67$	16.0000	1.95471	0.977356	−	0.211604i	$-0.0678686\pi$
$67$	16.0000	1.95471	0.977356	+	0.211604i	$0.0678686\pi$
$68$	6.00000	0.727607
$69$	6.00000	0.722315
$70$	0	0
$71$	6.00000	0.712069	0.356034	−	0.934473i	$-0.384129\pi$
$71$	6.00000	0.712069	0.356034	+	0.934473i	$0.384129\pi$
$72$	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	$0.301014\pi$
$74$	4.00000	0.464991
$75$	4.00000	0.461880
$76$	4.00000	0.458831
$77$	0	0
$78$	−1.00000	−0.113228
$79$	7.00000	0.787562	0.393781	−	0.919204i	$-0.371167\pi$
$79$	7.00000	0.787562	0.393781	+	0.919204i	$0.371167\pi$
$80$	1.00000	0.111803
$81$	1.00000	0.111111
$82$	−12.0000	−1.32518
$83$	17.0000	1.86599	0.932996	−	0.359886i	$-0.117184\pi$
$83$	17.0000	1.86599	0.932996	+	0.359886i	$0.117184\pi$
$84$	0	0
$85$	6.00000	0.650791
$86$	−8.00000	−0.862662
$87$	−3.00000	−0.321634
$88$	−1.00000	−0.106600
$89$	12.0000	1.27200	0.635999	−	0.771690i	$-0.280588\pi$
$89$	12.0000	1.27200	0.635999	+	0.771690i	$0.280588\pi$
$90$	1.00000	0.105409
$91$	0	0
$92$	−6.00000	−0.625543
$93$	−11.0000	−1.14065
$94$	8.00000	0.825137
$95$	4.00000	0.410391
$96$	−1.00000	−0.102062
$97$	−13.0000	−1.31995	−0.659975	−	0.751288i	$-0.729433\pi$
$97$	−13.0000	−1.31995	−0.659975	+	0.751288i	$0.729433\pi$
$98$	0	0
$99$	−1.00000	−0.100504
$100$	−4.00000	−0.400000
$101$	−10.0000	−0.995037	−0.497519	−	0.867453i	$-0.665755\pi$
$101$	−10.0000	−0.995037	−0.497519	+	0.867453i	$0.665755\pi$
$102$	−6.00000	−0.594089
$103$	0	0		−	1.00000i	$-0.5\pi$
$103$	0	0			1.00000i	$0.5\pi$
$104$	1.00000	0.0980581
$105$	0	0
$106$	−5.00000	−0.485643
$107$	7.00000	0.676716	0.338358	−	0.941018i	$-0.390129\pi$
$107$	7.00000	0.676716	0.338358	+	0.941018i	$0.390129\pi$
$108$	−1.00000	−0.0962250
$109$	12.0000	1.14939	0.574696	−	0.818367i	$-0.305120\pi$
$109$	12.0000	1.14939	0.574696	+	0.818367i	$0.305120\pi$
$110$	−1.00000	−0.0953463
$111$	−4.00000	−0.379663
$112$	0	0
$113$	−4.00000	−0.376288	−0.188144	−	0.982141i	$-0.560247\pi$
$113$	−4.00000	−0.376288	−0.188144	+	0.982141i	$0.560247\pi$
$114$	−4.00000	−0.374634
$115$	−6.00000	−0.559503
$116$	3.00000	0.278543
$117$	1.00000	0.0924500
$118$	5.00000	0.460287
$119$	0	0
$120$	−1.00000	−0.0912871
$121$	−10.0000	−0.909091
$122$	−12.0000	−1.08643
$123$	12.0000	1.08200
$124$	11.0000	0.987829
$125$	−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
$129$	8.00000	0.704361
$130$	1.00000	0.0877058
$131$	−1.00000	−0.0873704	−0.0436852	−	0.999045i	$-0.513910\pi$
$131$	−1.00000	−0.0873704	−0.0436852	+	0.999045i	$0.513910\pi$
$132$	1.00000	0.0870388
$133$	0	0
$134$	16.0000	1.38219
$135$	−1.00000	−0.0860663
$136$	6.00000	0.514496
$137$	−8.00000	−0.683486	−0.341743	−	0.939793i	$-0.611017\pi$
$137$	−8.00000	−0.683486	−0.341743	+	0.939793i	$0.611017\pi$
$138$	6.00000	0.510754
$139$	0	0		−	1.00000i	$-0.5\pi$
$139$	0	0			1.00000i	$0.5\pi$
$140$	0	0
$141$	−8.00000	−0.673722
$142$	6.00000	0.503509
$143$	−1.00000	−0.0836242
$144$	1.00000	0.0833333
$145$	3.00000	0.249136
$146$	10.0000	0.827606
$147$	0	0
$148$	4.00000	0.328798
$149$	−10.0000	−0.819232	−0.409616	−	0.912258i	$-0.634337\pi$
$149$	−10.0000	−0.819232	−0.409616	+	0.912258i	$0.634337\pi$
$150$	4.00000	0.326599
$151$	13.0000	1.05792	0.528962	−	0.848645i	$-0.322581\pi$
$151$	13.0000	1.05792	0.528962	+	0.848645i	$0.322581\pi$
$152$	4.00000	0.324443
$153$	6.00000	0.485071
$154$	0	0
$155$	11.0000	0.883541
$156$	−1.00000	−0.0800641
$157$	18.0000	1.43656	0.718278	−	0.695756i	$-0.244931\pi$
$157$	18.0000	1.43656	0.718278	+	0.695756i	$0.244931\pi$
$158$	7.00000	0.556890
$159$	5.00000	0.396526
$160$	1.00000	0.0790569
$161$	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	$0.601436\pi$
$164$	−12.0000	−0.937043
$165$	1.00000	0.0778499
$166$	17.0000	1.31946
$167$	24.0000	1.85718	0.928588	−	0.371113i	$-0.121024\pi$
$167$	24.0000	1.85718	0.928588	+	0.371113i	$0.121024\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	6.00000	0.460179
$171$	4.00000	0.305888
$172$	−8.00000	−0.609994
$173$	−6.00000	−0.456172	−0.228086	−	0.973641i	$-0.573247\pi$
$173$	−6.00000	−0.456172	−0.228086	+	0.973641i	$0.573247\pi$
$174$	−3.00000	−0.227429
$175$	0	0
$176$	−1.00000	−0.0753778
$177$	−5.00000	−0.375823
$178$	12.0000	0.899438
$179$	12.0000	0.896922	0.448461	−	0.893802i	$-0.351972\pi$
$179$	12.0000	0.896922	0.448461	+	0.893802i	$0.351972\pi$
$180$	1.00000	0.0745356
$181$	16.0000	1.18927	0.594635	−	0.803996i	$-0.297296\pi$
$181$	16.0000	1.18927	0.594635	+	0.803996i	$0.297296\pi$
$182$	0	0
$183$	12.0000	0.887066
$184$	−6.00000	−0.442326
$185$	4.00000	0.294086
$186$	−11.0000	−0.806559
$187$	−6.00000	−0.438763
$188$	8.00000	0.583460
$189$	0	0
$190$	4.00000	0.290191
$191$	−6.00000	−0.434145	−0.217072	−	0.976156i	$-0.569651\pi$
$191$	−6.00000	−0.434145	−0.217072	+	0.976156i	$0.569651\pi$
$192$	−1.00000	−0.0721688
$193$	15.0000	1.07972	0.539862	−	0.841754i	$-0.318476\pi$
$193$	15.0000	1.07972	0.539862	+	0.841754i	$0.318476\pi$
$194$	−13.0000	−0.933346
$195$	−1.00000	−0.0716115
$196$	0	0
$197$	−18.0000	−1.28245	−0.641223	−	0.767354i	$-0.721573\pi$
$197$	−18.0000	−1.28245	−0.641223	+	0.767354i	$0.721573\pi$
$198$	−1.00000	−0.0710669
$199$	−8.00000	−0.567105	−0.283552	−	0.958957i	$-0.591513\pi$
$199$	−8.00000	−0.567105	−0.283552	+	0.958957i	$0.591513\pi$
$200$	−4.00000	−0.282843
$201$	−16.0000	−1.12855
$202$	−10.0000	−0.703598
$203$	0	0
$204$	−6.00000	−0.420084
$205$	−12.0000	−0.838116
$206$	0	0
$207$	−6.00000	−0.417029
$208$	1.00000	0.0693375
$209$	−4.00000	−0.276686
$210$	0	0
$211$	12.0000	0.826114	0.413057	−	0.910705i	$-0.364461\pi$
$211$	12.0000	0.826114	0.413057	+	0.910705i	$0.364461\pi$
$212$	−5.00000	−0.343401
$213$	−6.00000	−0.411113
$214$	7.00000	0.478510
$215$	−8.00000	−0.545595
$216$	−1.00000	−0.0680414
$217$	0	0
$218$	12.0000	0.812743
$219$	−10.0000	−0.675737
$220$	−1.00000	−0.0674200
$221$	6.00000	0.403604
$222$	−4.00000	−0.268462
$223$	−7.00000	−0.468755	−0.234377	−	0.972146i	$-0.575305\pi$
$223$	−7.00000	−0.468755	−0.234377	+	0.972146i	$0.575305\pi$
$224$	0	0
$225$	−4.00000	−0.266667
$226$	−4.00000	−0.266076
$227$	15.0000	0.995585	0.497792	−	0.867296i	$-0.334144\pi$
$227$	15.0000	0.995585	0.497792	+	0.867296i	$0.334144\pi$
$228$	−4.00000	−0.264906
$229$	−6.00000	−0.396491	−0.198246	−	0.980152i	$-0.563524\pi$
$229$	−6.00000	−0.396491	−0.198246	+	0.980152i	$0.563524\pi$
$230$	−6.00000	−0.395628
$231$	0	0
$232$	3.00000	0.196960
$233$	6.00000	0.393073	0.196537	−	0.980497i	$-0.437031\pi$
$233$	6.00000	0.393073	0.196537	+	0.980497i	$0.437031\pi$
$234$	1.00000	0.0653720
$235$	8.00000	0.521862
$236$	5.00000	0.325472
$237$	−7.00000	−0.454699
$238$	0	0
$239$	−6.00000	−0.388108	−0.194054	−	0.980991i	$-0.562164\pi$
$239$	−6.00000	−0.388108	−0.194054	+	0.980991i	$0.562164\pi$
$240$	−1.00000	−0.0645497
$241$	15.0000	0.966235	0.483117	−	0.875556i	$-0.339504\pi$
$241$	15.0000	0.966235	0.483117	+	0.875556i	$0.339504\pi$
$242$	−10.0000	−0.642824
$243$	−1.00000	−0.0641500
$244$	−12.0000	−0.768221
$245$	0	0
$246$	12.0000	0.765092
$247$	4.00000	0.254514
$248$	11.0000	0.698501
$249$	−17.0000	−1.07733
$250$	−9.00000	−0.569210
$251$	−17.0000	−1.07303	−0.536515	−	0.843891i	$-0.680260\pi$
$251$	−17.0000	−1.07303	−0.536515	+	0.843891i	$0.680260\pi$
$252$	0	0
$253$	6.00000	0.377217
$254$	−7.00000	−0.439219
$255$	−6.00000	−0.375735
$256$	1.00000	0.0625000
$257$	6.00000	0.374270	0.187135	−	0.982334i	$-0.440080\pi$
$257$	6.00000	0.374270	0.187135	+	0.982334i	$0.440080\pi$
$258$	8.00000	0.498058
$259$	0	0
$260$	1.00000	0.0620174
$261$	3.00000	0.185695
$262$	−1.00000	−0.0617802
$263$	−18.0000	−1.10993	−0.554964	−	0.831875i	$-0.687268\pi$
$263$	−18.0000	−1.10993	−0.554964	+	0.831875i	$0.687268\pi$
$264$	1.00000	0.0615457
$265$	−5.00000	−0.307148
$266$	0	0
$267$	−12.0000	−0.734388
$268$	16.0000	0.977356
$269$	9.00000	0.548740	0.274370	−	0.961624i	$-0.411531\pi$
$269$	9.00000	0.548740	0.274370	+	0.961624i	$0.411531\pi$
$270$	−1.00000	−0.0608581
$271$	7.00000	0.425220	0.212610	−	0.977137i	$-0.431804\pi$
$271$	7.00000	0.425220	0.212610	+	0.977137i	$0.431804\pi$
$272$	6.00000	0.363803
$273$	0	0
$274$	−8.00000	−0.483298
$275$	4.00000	0.241209
$276$	6.00000	0.361158
$277$	−10.0000	−0.600842	−0.300421	−	0.953807i	$-0.597127\pi$
$277$	−10.0000	−0.600842	−0.300421	+	0.953807i	$0.597127\pi$
$278$	0	0
$279$	11.0000	0.658553
$280$	0	0
$281$	4.00000	0.238620	0.119310	−	0.992857i	$-0.461932\pi$
$281$	4.00000	0.238620	0.119310	+	0.992857i	$0.461932\pi$
$282$	−8.00000	−0.476393
$283$	−10.0000	−0.594438	−0.297219	−	0.954809i	$-0.596059\pi$
$283$	−10.0000	−0.594438	−0.297219	+	0.954809i	$0.596059\pi$
$284$	6.00000	0.356034
$285$	−4.00000	−0.236940
$286$	−1.00000	−0.0591312
$287$	0	0
$288$	1.00000	0.0589256
$289$	19.0000	1.11765
$290$	3.00000	0.176166
$291$	13.0000	0.762073
$292$	10.0000	0.585206
$293$	7.00000	0.408944	0.204472	−	0.978872i	$-0.434452\pi$
$293$	7.00000	0.408944	0.204472	+	0.978872i	$0.434452\pi$
$294$	0	0
$295$	5.00000	0.291111
$296$	4.00000	0.232495
$297$	1.00000	0.0580259
$298$	−10.0000	−0.579284
$299$	−6.00000	−0.346989
$300$	4.00000	0.230940
$301$	0	0
$302$	13.0000	0.748066
$303$	10.0000	0.574485
$304$	4.00000	0.229416
$305$	−12.0000	−0.687118
$306$	6.00000	0.342997
$307$	−6.00000	−0.342438	−0.171219	−	0.985233i	$-0.554771\pi$
$307$	−6.00000	−0.342438	−0.171219	+	0.985233i	$0.554771\pi$
$308$	0	0
$309$	0	0
$310$	11.0000	0.624758
$311$	−2.00000	−0.113410	−0.0567048	−	0.998391i	$-0.518059\pi$
$311$	−2.00000	−0.113410	−0.0567048	+	0.998391i	$0.518059\pi$
$312$	−1.00000	−0.0566139
$313$	−13.0000	−0.734803	−0.367402	−	0.930062i	$-0.619753\pi$
$313$	−13.0000	−0.734803	−0.367402	+	0.930062i	$0.619753\pi$
$314$	18.0000	1.01580
$315$	0	0
$316$	7.00000	0.393781
$317$	−3.00000	−0.168497	−0.0842484	−	0.996445i	$-0.526849\pi$
$317$	−3.00000	−0.168497	−0.0842484	+	0.996445i	$0.526849\pi$
$318$	5.00000	0.280386
$319$	−3.00000	−0.167968
$320$	1.00000	0.0559017
$321$	−7.00000	−0.390702
$322$	0	0
$323$	24.0000	1.33540
$324$	1.00000	0.0555556
$325$	−4.00000	−0.221880
$326$	−8.00000	−0.443079
$327$	−12.0000	−0.663602
$328$	−12.0000	−0.662589
$329$	0	0
$330$	1.00000	0.0550482
$331$	−8.00000	−0.439720	−0.219860	−	0.975531i	$-0.570560\pi$
$331$	−8.00000	−0.439720	−0.219860	+	0.975531i	$0.570560\pi$
$332$	17.0000	0.932996
$333$	4.00000	0.219199
$334$	24.0000	1.31322
$335$	16.0000	0.874173
$336$	0	0
$337$	25.0000	1.36184	0.680918	−	0.732359i	$-0.261581\pi$
$337$	25.0000	1.36184	0.680918	+	0.732359i	$0.261581\pi$
$338$	1.00000	0.0543928
$339$	4.00000	0.217250
$340$	6.00000	0.325396
$341$	−11.0000	−0.595683
$342$	4.00000	0.216295
$343$	0	0
$344$	−8.00000	−0.431331
$345$	6.00000	0.323029
$346$	−6.00000	−0.322562
$347$	20.0000	1.07366	0.536828	−	0.843692i	$-0.319622\pi$
$347$	20.0000	1.07366	0.536828	+	0.843692i	$0.319622\pi$
$348$	−3.00000	−0.160817
$349$	−14.0000	−0.749403	−0.374701	−	0.927146i	$-0.622255\pi$
$349$	−14.0000	−0.749403	−0.374701	+	0.927146i	$0.622255\pi$
$350$	0	0
$351$	−1.00000	−0.0533761
$352$	−1.00000	−0.0533002
$353$	−30.0000	−1.59674	−0.798369	−	0.602168i	$-0.794304\pi$
$353$	−30.0000	−1.59674	−0.798369	+	0.602168i	$0.794304\pi$
$354$	−5.00000	−0.265747
$355$	6.00000	0.318447
$356$	12.0000	0.635999
$357$	0	0
$358$	12.0000	0.634220
$359$	−22.0000	−1.16112	−0.580558	−	0.814219i	$-0.697165\pi$
$359$	−22.0000	−1.16112	−0.580558	+	0.814219i	$0.697165\pi$
$360$	1.00000	0.0527046
$361$	−3.00000	−0.157895
$362$	16.0000	0.840941
$363$	10.0000	0.524864
$364$	0	0
$365$	10.0000	0.523424
$366$	12.0000	0.627250
$367$	7.00000	0.365397	0.182699	−	0.983169i	$-0.441517\pi$
$367$	7.00000	0.365397	0.182699	+	0.983169i	$0.441517\pi$
$368$	−6.00000	−0.312772
$369$	−12.0000	−0.624695
$370$	4.00000	0.207950
$371$	0	0
$372$	−11.0000	−0.570323
$373$	26.0000	1.34623	0.673114	−	0.739538i	$-0.264956\pi$
$373$	26.0000	1.34623	0.673114	+	0.739538i	$0.264956\pi$
$374$	−6.00000	−0.310253
$375$	9.00000	0.464758
$376$	8.00000	0.412568
$377$	3.00000	0.154508
$378$	0	0
$379$	16.0000	0.821865	0.410932	−	0.911666i	$-0.365203\pi$
$379$	16.0000	0.821865	0.410932	+	0.911666i	$0.365203\pi$
$380$	4.00000	0.205196
$381$	7.00000	0.358621
$382$	−6.00000	−0.306987
$383$	14.0000	0.715367	0.357683	−	0.933843i	$-0.383567\pi$
$383$	14.0000	0.715367	0.357683	+	0.933843i	$0.383567\pi$
$384$	−1.00000	−0.0510310
$385$	0	0
$386$	15.0000	0.763480
$387$	−8.00000	−0.406663
$388$	−13.0000	−0.659975
$389$	22.0000	1.11544	0.557722	−	0.830028i	$-0.311675\pi$
$389$	22.0000	1.11544	0.557722	+	0.830028i	$0.311675\pi$
$390$	−1.00000	−0.0506370
$391$	−36.0000	−1.82060
$392$	0	0
$393$	1.00000	0.0504433
$394$	−18.0000	−0.906827
$395$	7.00000	0.352208
$396$	−1.00000	−0.0502519
$397$	18.0000	0.903394	0.451697	−	0.892171i	$-0.350819\pi$
$397$	18.0000	0.903394	0.451697	+	0.892171i	$0.350819\pi$
$398$	−8.00000	−0.401004
$399$	0	0
$400$	−4.00000	−0.200000
$401$	−18.0000	−0.898877	−0.449439	−	0.893311i	$-0.648376\pi$
$401$	−18.0000	−0.898877	−0.449439	+	0.893311i	$0.648376\pi$
$402$	−16.0000	−0.798007
$403$	11.0000	0.547949
$404$	−10.0000	−0.497519
$405$	1.00000	0.0496904
$406$	0	0
$407$	−4.00000	−0.198273
$408$	−6.00000	−0.297044
$409$	−25.0000	−1.23617	−0.618085	−	0.786111i	$-0.712091\pi$
$409$	−25.0000	−1.23617	−0.618085	+	0.786111i	$0.712091\pi$
$410$	−12.0000	−0.592638
$411$	8.00000	0.394611
$412$	0	0
$413$	0	0
$414$	−6.00000	−0.294884
$415$	17.0000	0.834497
$416$	1.00000	0.0490290
$417$	0	0
$418$	−4.00000	−0.195646
$419$	−16.0000	−0.781651	−0.390826	−	0.920465i	$-0.627810\pi$
$419$	−16.0000	−0.781651	−0.390826	+	0.920465i	$0.627810\pi$
$420$	0	0
$421$	−36.0000	−1.75453	−0.877266	−	0.480004i	$-0.840635\pi$
$421$	−36.0000	−1.75453	−0.877266	+	0.480004i	$0.840635\pi$
$422$	12.0000	0.584151
$423$	8.00000	0.388973
$424$	−5.00000	−0.242821
$425$	−24.0000	−1.16417
$426$	−6.00000	−0.290701
$427$	0	0
$428$	7.00000	0.338358
$429$	1.00000	0.0482805
$430$	−8.00000	−0.385794
$431$	−38.0000	−1.83040	−0.915198	−	0.403005i	$-0.867966\pi$
$431$	−38.0000	−1.83040	−0.915198	+	0.403005i	$0.867966\pi$
$432$	−1.00000	−0.0481125
$433$	14.0000	0.672797	0.336399	−	0.941720i	$-0.390791\pi$
$433$	14.0000	0.672797	0.336399	+	0.941720i	$0.390791\pi$
$434$	0	0
$435$	−3.00000	−0.143839
$436$	12.0000	0.574696
$437$	−24.0000	−1.14808
$438$	−10.0000	−0.477818
$439$	17.0000	0.811366	0.405683	−	0.914014i	$-0.367034\pi$
$439$	17.0000	0.811366	0.405683	+	0.914014i	$0.367034\pi$
$440$	−1.00000	−0.0476731
$441$	0	0
$442$	6.00000	0.285391
$443$	−39.0000	−1.85295	−0.926473	−	0.376361i	$-0.877175\pi$
$443$	−39.0000	−1.85295	−0.926473	+	0.376361i	$0.877175\pi$
$444$	−4.00000	−0.189832
$445$	12.0000	0.568855
$446$	−7.00000	−0.331460
$447$	10.0000	0.472984
$448$	0	0
$449$	20.0000	0.943858	0.471929	−	0.881636i	$-0.343558\pi$
$449$	20.0000	0.943858	0.471929	+	0.881636i	$0.343558\pi$
$450$	−4.00000	−0.188562
$451$	12.0000	0.565058
$452$	−4.00000	−0.188144
$453$	−13.0000	−0.610793
$454$	15.0000	0.703985
$455$	0	0
$456$	−4.00000	−0.187317
$457$	−11.0000	−0.514558	−0.257279	−	0.966337i	$-0.582826\pi$
$457$	−11.0000	−0.514558	−0.257279	+	0.966337i	$0.582826\pi$
$458$	−6.00000	−0.280362
$459$	−6.00000	−0.280056
$460$	−6.00000	−0.279751
$461$	−34.0000	−1.58354	−0.791769	−	0.610821i	$-0.790840\pi$
$461$	−34.0000	−1.58354	−0.791769	+	0.610821i	$0.790840\pi$
$462$	0	0
$463$	−24.0000	−1.11537	−0.557687	−	0.830051i	$-0.688311\pi$
$463$	−24.0000	−1.11537	−0.557687	+	0.830051i	$0.688311\pi$
$464$	3.00000	0.139272
$465$	−11.0000	−0.510113
$466$	6.00000	0.277945
$467$	−12.0000	−0.555294	−0.277647	−	0.960683i	$-0.589555\pi$
$467$	−12.0000	−0.555294	−0.277647	+	0.960683i	$0.589555\pi$
$468$	1.00000	0.0462250
$469$	0	0
$470$	8.00000	0.369012
$471$	−18.0000	−0.829396
$472$	5.00000	0.230144
$473$	8.00000	0.367840
$474$	−7.00000	−0.321521
$475$	−16.0000	−0.734130
$476$	0	0
$477$	−5.00000	−0.228934
$478$	−6.00000	−0.274434
$479$	−14.0000	−0.639676	−0.319838	−	0.947472i	$-0.603629\pi$
$479$	−14.0000	−0.639676	−0.319838	+	0.947472i	$0.603629\pi$
$480$	−1.00000	−0.0456435
$481$	4.00000	0.182384
$482$	15.0000	0.683231
$483$	0	0
$484$	−10.0000	−0.454545
$485$	−13.0000	−0.590300
$486$	−1.00000	−0.0453609
$487$	23.0000	1.04223	0.521115	−	0.853487i	$-0.325516\pi$
$487$	23.0000	1.04223	0.521115	+	0.853487i	$0.325516\pi$
$488$	−12.0000	−0.543214
$489$	8.00000	0.361773
$490$	0	0
$491$	−35.0000	−1.57953	−0.789764	−	0.613411i	$-0.789797\pi$
$491$	−35.0000	−1.57953	−0.789764	+	0.613411i	$0.789797\pi$
$492$	12.0000	0.541002
$493$	18.0000	0.810679
$494$	4.00000	0.179969
$495$	−1.00000	−0.0449467
$496$	11.0000	0.493915
$497$	0	0
$498$	−17.0000	−0.761788
$499$	−28.0000	−1.25345	−0.626726	−	0.779240i	$-0.715605\pi$
$499$	−28.0000	−1.25345	−0.626726	+	0.779240i	$0.715605\pi$
$500$	−9.00000	−0.402492
$501$	−24.0000	−1.07224
$502$	−17.0000	−0.758747
$503$	6.00000	0.267527	0.133763	−	0.991013i	$-0.457294\pi$
$503$	6.00000	0.267527	0.133763	+	0.991013i	$0.457294\pi$
$504$	0	0
$505$	−10.0000	−0.444994
$506$	6.00000	0.266733
$507$	−1.00000	−0.0444116
$508$	−7.00000	−0.310575
$509$	−13.0000	−0.576215	−0.288107	−	0.957598i	$-0.593026\pi$
$509$	−13.0000	−0.576215	−0.288107	+	0.957598i	$0.593026\pi$
$510$	−6.00000	−0.265684
$511$	0	0
$512$	1.00000	0.0441942
$513$	−4.00000	−0.176604
$514$	6.00000	0.264649
$515$	0	0
$516$	8.00000	0.352180
$517$	−8.00000	−0.351840
$518$	0	0
$519$	6.00000	0.263371
$520$	1.00000	0.0438529
$521$	−28.0000	−1.22670	−0.613351	−	0.789810i	$-0.710179\pi$
$521$	−28.0000	−1.22670	−0.613351	+	0.789810i	$0.710179\pi$
$522$	3.00000	0.131306
$523$	−16.0000	−0.699631	−0.349816	−	0.936819i	$-0.613756\pi$
$523$	−16.0000	−0.699631	−0.349816	+	0.936819i	$0.613756\pi$
$524$	−1.00000	−0.0436852
$525$	0	0
$526$	−18.0000	−0.784837
$527$	66.0000	2.87501
$528$	1.00000	0.0435194
$529$	13.0000	0.565217
$530$	−5.00000	−0.217186
$531$	5.00000	0.216982
$532$	0	0
$533$	−12.0000	−0.519778
$534$	−12.0000	−0.519291
$535$	7.00000	0.302636
$536$	16.0000	0.691095
$537$	−12.0000	−0.517838
$538$	9.00000	0.388018
$539$	0	0
$540$	−1.00000	−0.0430331
$541$	32.0000	1.37579	0.687894	−	0.725811i	$-0.258536\pi$
$541$	32.0000	1.37579	0.687894	+	0.725811i	$0.258536\pi$
$542$	7.00000	0.300676
$543$	−16.0000	−0.686626
$544$	6.00000	0.257248
$545$	12.0000	0.514024
$546$	0	0
$547$	36.0000	1.53925	0.769624	−	0.638497i	$-0.220443\pi$
$547$	36.0000	1.53925	0.769624	+	0.638497i	$0.220443\pi$
$548$	−8.00000	−0.341743
$549$	−12.0000	−0.512148
$550$	4.00000	0.170561
$551$	12.0000	0.511217
$552$	6.00000	0.255377
$553$	0	0
$554$	−10.0000	−0.424859
$555$	−4.00000	−0.169791
$556$	0	0
$557$	−17.0000	−0.720313	−0.360157	−	0.932892i	$-0.617277\pi$
$557$	−17.0000	−0.720313	−0.360157	+	0.932892i	$0.617277\pi$
$558$	11.0000	0.465667
$559$	−8.00000	−0.338364
$560$	0	0
$561$	6.00000	0.253320
$562$	4.00000	0.168730
$563$	−21.0000	−0.885044	−0.442522	−	0.896758i	$-0.645916\pi$
$563$	−21.0000	−0.885044	−0.442522	+	0.896758i	$0.645916\pi$
$564$	−8.00000	−0.336861
$565$	−4.00000	−0.168281
$566$	−10.0000	−0.420331
$567$	0	0
$568$	6.00000	0.251754
$569$	−4.00000	−0.167689	−0.0838444	−	0.996479i	$-0.526720\pi$
$569$	−4.00000	−0.167689	−0.0838444	+	0.996479i	$0.526720\pi$
$570$	−4.00000	−0.167542
$571$	28.0000	1.17176	0.585882	−	0.810397i	$-0.300748\pi$
$571$	28.0000	1.17176	0.585882	+	0.810397i	$0.300748\pi$
$572$	−1.00000	−0.0418121
$573$	6.00000	0.250654
$574$	0	0
$575$	24.0000	1.00087
$576$	1.00000	0.0416667
$577$	−1.00000	−0.0416305	−0.0208153	−	0.999783i	$-0.506626\pi$
$577$	−1.00000	−0.0416305	−0.0208153	+	0.999783i	$0.506626\pi$
$578$	19.0000	0.790296
$579$	−15.0000	−0.623379
$580$	3.00000	0.124568
$581$	0	0
$582$	13.0000	0.538867
$583$	5.00000	0.207079
$584$	10.0000	0.413803
$585$	1.00000	0.0413449
$586$	7.00000	0.289167
$587$	−5.00000	−0.206372	−0.103186	−	0.994662i	$-0.532904\pi$
$587$	−5.00000	−0.206372	−0.103186	+	0.994662i	$0.532904\pi$
$588$	0	0
$589$	44.0000	1.81299
$590$	5.00000	0.205847
$591$	18.0000	0.740421
$592$	4.00000	0.164399
$593$	36.0000	1.47834	0.739171	−	0.673517i	$-0.235217\pi$
$593$	36.0000	1.47834	0.739171	+	0.673517i	$0.235217\pi$
$594$	1.00000	0.0410305
$595$	0	0
$596$	−10.0000	−0.409616
$597$	8.00000	0.327418
$598$	−6.00000	−0.245358
$599$	−10.0000	−0.408589	−0.204294	−	0.978909i	$-0.565490\pi$
$599$	−10.0000	−0.408589	−0.204294	+	0.978909i	$0.565490\pi$
$600$	4.00000	0.163299
$601$	−11.0000	−0.448699	−0.224350	−	0.974509i	$-0.572026\pi$
$601$	−11.0000	−0.448699	−0.224350	+	0.974509i	$0.572026\pi$
$602$	0	0
$603$	16.0000	0.651570
$604$	13.0000	0.528962
$605$	−10.0000	−0.406558
$606$	10.0000	0.406222
$607$	−29.0000	−1.17707	−0.588537	−	0.808470i	$-0.700296\pi$
$607$	−29.0000	−1.17707	−0.588537	+	0.808470i	$0.700296\pi$
$608$	4.00000	0.162221
$609$	0	0
$610$	−12.0000	−0.485866
$611$	8.00000	0.323645
$612$	6.00000	0.242536
$613$	0	0		−	1.00000i	$-0.5\pi$
$613$	0	0			1.00000i	$0.5\pi$
$614$	−6.00000	−0.242140
$615$	12.0000	0.483887
$616$	0	0
$617$	−36.0000	−1.44931	−0.724653	−	0.689114i	$-0.758000\pi$
$617$	−36.0000	−1.44931	−0.724653	+	0.689114i	$0.758000\pi$
$618$	0	0
$619$	8.00000	0.321547	0.160774	−	0.986991i	$-0.448601\pi$
$619$	8.00000	0.321547	0.160774	+	0.986991i	$0.448601\pi$
$620$	11.0000	0.441771
$621$	6.00000	0.240772
$622$	−2.00000	−0.0801927
$623$	0	0
$624$	−1.00000	−0.0400320
$625$	11.0000	0.440000
$626$	−13.0000	−0.519584
$627$	4.00000	0.159745
$628$	18.0000	0.718278
$629$	24.0000	0.956943
$630$	0	0
$631$	−29.0000	−1.15447	−0.577236	−	0.816577i	$-0.695869\pi$
$631$	−29.0000	−1.15447	−0.577236	+	0.816577i	$0.695869\pi$
$632$	7.00000	0.278445
$633$	−12.0000	−0.476957
$634$	−3.00000	−0.119145
$635$	−7.00000	−0.277787
$636$	5.00000	0.198263
$637$	0	0
$638$	−3.00000	−0.118771
$639$	6.00000	0.237356
$640$	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	$0.576159\pi$
$642$	−7.00000	−0.276268
$643$	−20.0000	−0.788723	−0.394362	−	0.918955i	$-0.629034\pi$
$643$	−20.0000	−0.788723	−0.394362	+	0.918955i	$0.629034\pi$
$644$	0	0
$645$	8.00000	0.315000
$646$	24.0000	0.944267
$647$	4.00000	0.157256	0.0786281	−	0.996904i	$-0.474946\pi$
$647$	4.00000	0.157256	0.0786281	+	0.996904i	$0.474946\pi$
$648$	1.00000	0.0392837
$649$	−5.00000	−0.196267
$650$	−4.00000	−0.156893
$651$	0	0
$652$	−8.00000	−0.313304
$653$	−11.0000	−0.430463	−0.215232	−	0.976563i	$-0.569051\pi$
$653$	−11.0000	−0.430463	−0.215232	+	0.976563i	$0.569051\pi$
$654$	−12.0000	−0.469237
$655$	−1.00000	−0.0390732
$656$	−12.0000	−0.468521
$657$	10.0000	0.390137
$658$	0	0
$659$	−12.0000	−0.467454	−0.233727	−	0.972302i	$-0.575092\pi$
$659$	−12.0000	−0.467454	−0.233727	+	0.972302i	$0.575092\pi$
$660$	1.00000	0.0389249
$661$	−22.0000	−0.855701	−0.427850	−	0.903850i	$-0.640729\pi$
$661$	−22.0000	−0.855701	−0.427850	+	0.903850i	$0.640729\pi$
$662$	−8.00000	−0.310929
$663$	−6.00000	−0.233021
$664$	17.0000	0.659728
$665$	0	0
$666$	4.00000	0.154997
$667$	−18.0000	−0.696963
$668$	24.0000	0.928588
$669$	7.00000	0.270636
$670$	16.0000	0.618134
$671$	12.0000	0.463255
$672$	0	0
$673$	−39.0000	−1.50334	−0.751670	−	0.659540i	$-0.770751\pi$
$673$	−39.0000	−1.50334	−0.751670	+	0.659540i	$0.770751\pi$
$674$	25.0000	0.962964
$675$	4.00000	0.153960
$676$	1.00000	0.0384615
$677$	35.0000	1.34516	0.672580	−	0.740025i	$-0.265186\pi$
$677$	35.0000	1.34516	0.672580	+	0.740025i	$0.265186\pi$
$678$	4.00000	0.153619
$679$	0	0
$680$	6.00000	0.230089
$681$	−15.0000	−0.574801
$682$	−11.0000	−0.421212
$683$	−27.0000	−1.03313	−0.516563	−	0.856249i	$-0.672789\pi$
$683$	−27.0000	−1.03313	−0.516563	+	0.856249i	$0.672789\pi$
$684$	4.00000	0.152944
$685$	−8.00000	−0.305664
$686$	0	0
$687$	6.00000	0.228914
$688$	−8.00000	−0.304997
$689$	−5.00000	−0.190485
$690$	6.00000	0.228416
$691$	40.0000	1.52167	0.760836	−	0.648944i	$-0.224789\pi$
$691$	40.0000	1.52167	0.760836	+	0.648944i	$0.224789\pi$
$692$	−6.00000	−0.228086
$693$	0	0
$694$	20.0000	0.759190
$695$	0	0
$696$	−3.00000	−0.113715
$697$	−72.0000	−2.72719
$698$	−14.0000	−0.529908
$699$	−6.00000	−0.226941
$700$	0	0
$701$	−13.0000	−0.491003	−0.245502	−	0.969396i	$-0.578953\pi$
$701$	−13.0000	−0.491003	−0.245502	+	0.969396i	$0.578953\pi$
$702$	−1.00000	−0.0377426
$703$	16.0000	0.603451
$704$	−1.00000	−0.0376889
$705$	−8.00000	−0.301297
$706$	−30.0000	−1.12906
$707$	0	0
$708$	−5.00000	−0.187912
$709$	6.00000	0.225335	0.112667	−	0.993633i	$-0.464061\pi$
$709$	6.00000	0.225335	0.112667	+	0.993633i	$0.464061\pi$
$710$	6.00000	0.225176
$711$	7.00000	0.262521
$712$	12.0000	0.449719
$713$	−66.0000	−2.47172
$714$	0	0
$715$	−1.00000	−0.0373979
$716$	12.0000	0.448461
$717$	6.00000	0.224074
$718$	−22.0000	−0.821033
$719$	−46.0000	−1.71551	−0.857755	−	0.514058i	$-0.828142\pi$
$719$	−46.0000	−1.71551	−0.857755	+	0.514058i	$0.828142\pi$
$720$	1.00000	0.0372678
$721$	0	0
$722$	−3.00000	−0.111648
$723$	−15.0000	−0.557856
$724$	16.0000	0.594635
$725$	−12.0000	−0.445669
$726$	10.0000	0.371135
$727$	−19.0000	−0.704671	−0.352335	−	0.935874i	$-0.614612\pi$
$727$	−19.0000	−0.704671	−0.352335	+	0.935874i	$0.614612\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	10.0000	0.370117
$731$	−48.0000	−1.77534
$732$	12.0000	0.443533
$733$	−2.00000	−0.0738717	−0.0369358	−	0.999318i	$-0.511760\pi$
$733$	−2.00000	−0.0738717	−0.0369358	+	0.999318i	$0.511760\pi$
$734$	7.00000	0.258375
$735$	0	0
$736$	−6.00000	−0.221163
$737$	−16.0000	−0.589368
$738$	−12.0000	−0.441726
$739$	−26.0000	−0.956425	−0.478213	−	0.878244i	$-0.658715\pi$
$739$	−26.0000	−0.956425	−0.478213	+	0.878244i	$0.658715\pi$
$740$	4.00000	0.147043
$741$	−4.00000	−0.146944
$742$	0	0
$743$	−32.0000	−1.17397	−0.586983	−	0.809599i	$-0.699684\pi$
$743$	−32.0000	−1.17397	−0.586983	+	0.809599i	$0.699684\pi$
$744$	−11.0000	−0.403280
$745$	−10.0000	−0.366372
$746$	26.0000	0.951928
$747$	17.0000	0.621997
$748$	−6.00000	−0.219382
$749$	0	0
$750$	9.00000	0.328634
$751$	25.0000	0.912263	0.456131	−	0.889912i	$-0.349235\pi$
$751$	25.0000	0.912263	0.456131	+	0.889912i	$0.349235\pi$
$752$	8.00000	0.291730
$753$	17.0000	0.619514
$754$	3.00000	0.109254
$755$	13.0000	0.473118
$756$	0	0
$757$	−38.0000	−1.38113	−0.690567	−	0.723269i	$-0.742639\pi$
$757$	−38.0000	−1.38113	−0.690567	+	0.723269i	$0.742639\pi$
$758$	16.0000	0.581146
$759$	−6.00000	−0.217786
$760$	4.00000	0.145095
$761$	28.0000	1.01500	0.507500	−	0.861652i	$-0.330570\pi$
$761$	28.0000	1.01500	0.507500	+	0.861652i	$0.330570\pi$
$762$	7.00000	0.253583
$763$	0	0
$764$	−6.00000	−0.217072
$765$	6.00000	0.216930
$766$	14.0000	0.505841
$767$	5.00000	0.180540
$768$	−1.00000	−0.0360844
$769$	−31.0000	−1.11789	−0.558944	−	0.829205i	$-0.688793\pi$
$769$	−31.0000	−1.11789	−0.558944	+	0.829205i	$0.688793\pi$
$770$	0	0
$771$	−6.00000	−0.216085
$772$	15.0000	0.539862
$773$	42.0000	1.51064	0.755318	−	0.655359i	$-0.227483\pi$
$773$	42.0000	1.51064	0.755318	+	0.655359i	$0.227483\pi$
$774$	−8.00000	−0.287554
$775$	−44.0000	−1.58053
$776$	−13.0000	−0.466673
$777$	0	0
$778$	22.0000	0.788738
$779$	−48.0000	−1.71978
$780$	−1.00000	−0.0358057
$781$	−6.00000	−0.214697
$782$	−36.0000	−1.28736
$783$	−3.00000	−0.107211
$784$	0	0
$785$	18.0000	0.642448
$786$	1.00000	0.0356688
$787$	12.0000	0.427754	0.213877	−	0.976861i	$-0.431391\pi$
$787$	12.0000	0.427754	0.213877	+	0.976861i	$0.431391\pi$
$788$	−18.0000	−0.641223
$789$	18.0000	0.640817
$790$	7.00000	0.249049
$791$	0	0
$792$	−1.00000	−0.0355335
$793$	−12.0000	−0.426132
$794$	18.0000	0.638796
$795$	5.00000	0.177332
$796$	−8.00000	−0.283552
$797$	3.00000	0.106265	0.0531327	−	0.998587i	$-0.483079\pi$
$797$	3.00000	0.106265	0.0531327	+	0.998587i	$0.483079\pi$
$798$	0	0
$799$	48.0000	1.69812
$800$	−4.00000	−0.141421
$801$	12.0000	0.423999
$802$	−18.0000	−0.635602
$803$	−10.0000	−0.352892
$804$	−16.0000	−0.564276
$805$	0	0
$806$	11.0000	0.387458
$807$	−9.00000	−0.316815
$808$	−10.0000	−0.351799
$809$	12.0000	0.421898	0.210949	−	0.977497i	$-0.432345\pi$
$809$	12.0000	0.421898	0.210949	+	0.977497i	$0.432345\pi$
$810$	1.00000	0.0351364
$811$	40.0000	1.40459	0.702295	−	0.711886i	$-0.252159\pi$
$811$	40.0000	1.40459	0.702295	+	0.711886i	$0.252159\pi$
$812$	0	0
$813$	−7.00000	−0.245501
$814$	−4.00000	−0.140200
$815$	−8.00000	−0.280228
$816$	−6.00000	−0.210042
$817$	−32.0000	−1.11954
$818$	−25.0000	−0.874105
$819$	0	0
$820$	−12.0000	−0.419058
$821$	−11.0000	−0.383903	−0.191951	−	0.981404i	$-0.561482\pi$
$821$	−11.0000	−0.383903	−0.191951	+	0.981404i	$0.561482\pi$
$822$	8.00000	0.279032
$823$	44.0000	1.53374	0.766872	−	0.641800i	$-0.221812\pi$
$823$	44.0000	1.53374	0.766872	+	0.641800i	$0.221812\pi$
$824$	0	0
$825$	−4.00000	−0.139262
$826$	0	0
$827$	−25.0000	−0.869335	−0.434668	−	0.900591i	$-0.643134\pi$
$827$	−25.0000	−0.869335	−0.434668	+	0.900591i	$0.643134\pi$
$828$	−6.00000	−0.208514
$829$	2.00000	0.0694629	0.0347314	−	0.999397i	$-0.488942\pi$
$829$	2.00000	0.0694629	0.0347314	+	0.999397i	$0.488942\pi$
$830$	17.0000	0.590079
$831$	10.0000	0.346896
$832$	1.00000	0.0346688
$833$	0	0
$834$	0	0
$835$	24.0000	0.830554
$836$	−4.00000	−0.138343
$837$	−11.0000	−0.380216
$838$	−16.0000	−0.552711
$839$	16.0000	0.552381	0.276191	−	0.961103i	$-0.410928\pi$
$839$	16.0000	0.552381	0.276191	+	0.961103i	$0.410928\pi$
$840$	0	0
$841$	−20.0000	−0.689655
$842$	−36.0000	−1.24064
$843$	−4.00000	−0.137767
$844$	12.0000	0.413057
$845$	1.00000	0.0344010
$846$	8.00000	0.275046
$847$	0	0
$848$	−5.00000	−0.171701
$849$	10.0000	0.343199
$850$	−24.0000	−0.823193
$851$	−24.0000	−0.822709
$852$	−6.00000	−0.205557
$853$	−26.0000	−0.890223	−0.445112	−	0.895475i	$-0.646836\pi$
$853$	−26.0000	−0.890223	−0.445112	+	0.895475i	$0.646836\pi$
$854$	0	0
$855$	4.00000	0.136797
$856$	7.00000	0.239255
$857$	−54.0000	−1.84460	−0.922302	−	0.386469i	$-0.873695\pi$
$857$	−54.0000	−1.84460	−0.922302	+	0.386469i	$0.873695\pi$
$858$	1.00000	0.0341394
$859$	8.00000	0.272956	0.136478	−	0.990643i	$-0.456422\pi$
$859$	8.00000	0.272956	0.136478	+	0.990643i	$0.456422\pi$
$860$	−8.00000	−0.272798
$861$	0	0
$862$	−38.0000	−1.29429
$863$	−4.00000	−0.136162	−0.0680808	−	0.997680i	$-0.521688\pi$
$863$	−4.00000	−0.136162	−0.0680808	+	0.997680i	$0.521688\pi$
$864$	−1.00000	−0.0340207
$865$	−6.00000	−0.204006
$866$	14.0000	0.475739
$867$	−19.0000	−0.645274
$868$	0	0
$869$	−7.00000	−0.237459
$870$	−3.00000	−0.101710
$871$	16.0000	0.542139
$872$	12.0000	0.406371
$873$	−13.0000	−0.439983
$874$	−24.0000	−0.811812
$875$	0	0
$876$	−10.0000	−0.337869
$877$	18.0000	0.607817	0.303908	−	0.952701i	$-0.401708\pi$
$877$	18.0000	0.607817	0.303908	+	0.952701i	$0.401708\pi$
$878$	17.0000	0.573722
$879$	−7.00000	−0.236104
$880$	−1.00000	−0.0337100
$881$	−30.0000	−1.01073	−0.505363	−	0.862907i	$-0.668641\pi$
$881$	−30.0000	−1.01073	−0.505363	+	0.862907i	$0.668641\pi$
$882$	0	0
$883$	30.0000	1.00958	0.504790	−	0.863242i	$-0.331570\pi$
$883$	30.0000	1.00958	0.504790	+	0.863242i	$0.331570\pi$
$884$	6.00000	0.201802
$885$	−5.00000	−0.168073
$886$	−39.0000	−1.31023
$887$	−36.0000	−1.20876	−0.604381	−	0.796696i	$-0.706579\pi$
$887$	−36.0000	−1.20876	−0.604381	+	0.796696i	$0.706579\pi$
$888$	−4.00000	−0.134231
$889$	0	0
$890$	12.0000	0.402241
$891$	−1.00000	−0.0335013
$892$	−7.00000	−0.234377
$893$	32.0000	1.07084
$894$	10.0000	0.334450
$895$	12.0000	0.401116
$896$	0	0
$897$	6.00000	0.200334
$898$	20.0000	0.667409
$899$	33.0000	1.10061
$900$	−4.00000	−0.133333
$901$	−30.0000	−0.999445
$902$	12.0000	0.399556
$903$	0	0
$904$	−4.00000	−0.133038
$905$	16.0000	0.531858
$906$	−13.0000	−0.431896
$907$	−36.0000	−1.19536	−0.597680	−	0.801735i	$-0.703911\pi$
$907$	−36.0000	−1.19536	−0.597680	+	0.801735i	$0.703911\pi$
$908$	15.0000	0.497792
$909$	−10.0000	−0.331679
$910$	0	0
$911$	−10.0000	−0.331315	−0.165657	−	0.986183i	$-0.552975\pi$
$911$	−10.0000	−0.331315	−0.165657	+	0.986183i	$0.552975\pi$
$912$	−4.00000	−0.132453
$913$	−17.0000	−0.562618
$914$	−11.0000	−0.363848
$915$	12.0000	0.396708
$916$	−6.00000	−0.198246
$917$	0	0
$918$	−6.00000	−0.198030
$919$	32.0000	1.05558	0.527791	−	0.849374i	$-0.323020\pi$
$919$	32.0000	1.05558	0.527791	+	0.849374i	$0.323020\pi$
$920$	−6.00000	−0.197814
$921$	6.00000	0.197707
$922$	−34.0000	−1.11973
$923$	6.00000	0.197492
$924$	0	0
$925$	−16.0000	−0.526077
$926$	−24.0000	−0.788689
$927$	0	0
$928$	3.00000	0.0984798
$929$	20.0000	0.656179	0.328089	−	0.944647i	$-0.393595\pi$
$929$	20.0000	0.656179	0.328089	+	0.944647i	$0.393595\pi$
$930$	−11.0000	−0.360704
$931$	0	0
$932$	6.00000	0.196537
$933$	2.00000	0.0654771
$934$	−12.0000	−0.392652
$935$	−6.00000	−0.196221
$936$	1.00000	0.0326860
$937$	13.0000	0.424691	0.212346	−	0.977195i	$-0.431890\pi$
$937$	13.0000	0.424691	0.212346	+	0.977195i	$0.431890\pi$
$938$	0	0
$939$	13.0000	0.424239
$940$	8.00000	0.260931
$941$	−15.0000	−0.488986	−0.244493	−	0.969651i	$-0.578622\pi$
$941$	−15.0000	−0.488986	−0.244493	+	0.969651i	$0.578622\pi$
$942$	−18.0000	−0.586472
$943$	72.0000	2.34464
$944$	5.00000	0.162736
$945$	0	0
$946$	8.00000	0.260102
$947$	36.0000	1.16984	0.584921	−	0.811090i	$-0.301125\pi$
$947$	36.0000	1.16984	0.584921	+	0.811090i	$0.301125\pi$
$948$	−7.00000	−0.227349
$949$	10.0000	0.324614
$950$	−16.0000	−0.519109
$951$	3.00000	0.0972817
$952$	0	0
$953$	−44.0000	−1.42530	−0.712650	−	0.701520i	$-0.752505\pi$
$953$	−44.0000	−1.42530	−0.712650	+	0.701520i	$0.752505\pi$
$954$	−5.00000	−0.161881
$955$	−6.00000	−0.194155
$956$	−6.00000	−0.194054
$957$	3.00000	0.0969762
$958$	−14.0000	−0.452319
$959$	0	0
$960$	−1.00000	−0.0322749
$961$	90.0000	2.90323
$962$	4.00000	0.128965
$963$	7.00000	0.225572
$964$	15.0000	0.483117
$965$	15.0000	0.482867
$966$	0	0
$967$	−11.0000	−0.353736	−0.176868	−	0.984235i	$-0.556597\pi$
$967$	−11.0000	−0.353736	−0.176868	+	0.984235i	$0.556597\pi$
$968$	−10.0000	−0.321412
$969$	−24.0000	−0.770991
$970$	−13.0000	−0.417405
$971$	29.0000	0.930654	0.465327	−	0.885139i	$-0.345937\pi$
$971$	29.0000	0.930654	0.465327	+	0.885139i	$0.345937\pi$
$972$	−1.00000	−0.0320750
$973$	0	0
$974$	23.0000	0.736968
$975$	4.00000	0.128103
$976$	−12.0000	−0.384111
$977$	42.0000	1.34370	0.671850	−	0.740688i	$-0.265500\pi$
$977$	42.0000	1.34370	0.671850	+	0.740688i	$0.265500\pi$
$978$	8.00000	0.255812
$979$	−12.0000	−0.383522
$980$	0	0
$981$	12.0000	0.383131
$982$	−35.0000	−1.11689
$983$	26.0000	0.829271	0.414636	−	0.909988i	$-0.363909\pi$
$983$	26.0000	0.829271	0.414636	+	0.909988i	$0.363909\pi$
$984$	12.0000	0.382546
$985$	−18.0000	−0.573528
$986$	18.0000	0.573237
$987$	0	0
$988$	4.00000	0.127257
$989$	48.0000	1.52631
$990$	−1.00000	−0.0317821
$991$	−5.00000	−0.158830	−0.0794151	−	0.996842i	$-0.525305\pi$
$991$	−5.00000	−0.158830	−0.0794151	+	0.996842i	$0.525305\pi$
$992$	11.0000	0.349250
$993$	8.00000	0.253872
$994$	0	0
$995$	−8.00000	−0.253617
$996$	−17.0000	−0.538666
$997$	28.0000	0.886769	0.443384	−	0.896332i	$-0.353778\pi$
$997$	28.0000	0.886769	0.443384	+	0.896332i	$0.353778\pi$
$998$	−28.0000	−0.886325
$999$	−4.00000	−0.126554

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3822.2.a.v.1.1		1
7.3	odd	6		546.2.i.b.79.1	✓	2
7.5	odd	6		546.2.i.b.235.1	yes	2
7.6	odd	2		3822.2.a.be.1.1		1
21.5	even	6		1638.2.j.h.235.1		2
21.17	even	6		1638.2.j.h.1171.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
546.2.i.b.79.1	✓	2	7.3	odd	6
546.2.i.b.235.1	yes	2	7.5	odd	6
1638.2.j.h.235.1		2	21.5	even	6
1638.2.j.h.1171.1		2	21.17	even	6
3822.2.a.v.1.1		1	1.1	even	1	trivial
3822.2.a.be.1.1		1	7.6	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 3822 = 2 \cdot 3 \cdot 7^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3822.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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