LMFDB - Embedded newform 3822.2.a.f.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:	$1$
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)

$f(q)$

$=$

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

\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} -3.00000 q^{11} -1.00000 q^{12} -1.00000 q^{13} -1.00000 q^{15} +1.00000 q^{16} -2.00000 q^{17} -1.00000 q^{18} +1.00000 q^{20} +3.00000 q^{22} +6.00000 q^{23} +1.00000 q^{24} -4.00000 q^{25} +1.00000 q^{26} -1.00000 q^{27} +9.00000 q^{29} +1.00000 q^{30} +5.00000 q^{31} -1.00000 q^{32} +3.00000 q^{33} +2.00000 q^{34} +1.00000 q^{36} -8.00000 q^{37} +1.00000 q^{39} -1.00000 q^{40} -4.00000 q^{41} -3.00000 q^{44} +1.00000 q^{45} -6.00000 q^{46} -1.00000 q^{48} +4.00000 q^{50} +2.00000 q^{51} -1.00000 q^{52} +1.00000 q^{53} +1.00000 q^{54} -3.00000 q^{55} -9.00000 q^{58} -7.00000 q^{59} -1.00000 q^{60} -4.00000 q^{61} -5.00000 q^{62} +1.00000 q^{64} -1.00000 q^{65} -3.00000 q^{66} -4.00000 q^{67} -2.00000 q^{68} -6.00000 q^{69} +6.00000 q^{71} -1.00000 q^{72} +6.00000 q^{73} +8.00000 q^{74} +4.00000 q^{75} -1.00000 q^{78} -13.0000 q^{79} +1.00000 q^{80} +1.00000 q^{81} +4.00000 q^{82} -3.00000 q^{83} -2.00000 q^{85} -9.00000 q^{87} +3.00000 q^{88} -8.00000 q^{89} -1.00000 q^{90} +6.00000 q^{92} -5.00000 q^{93} +1.00000 q^{96} -15.0000 q^{97} -3.00000 q^{99} +O(q^{100})\)

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	$-0.428217\pi$
$5$	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
$11$	−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
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	−1.00000	−0.258199
$16$	1.00000	0.250000
$17$	−2.00000	−0.485071	−0.242536	−	0.970143i	$-0.577979\pi$
$17$	−2.00000	−0.485071	−0.242536	+	0.970143i	$0.577979\pi$
$18$	−1.00000	−0.235702
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	1.00000	0.223607
$21$	0	0
$22$	3.00000	0.639602
$23$	6.00000	1.25109	0.625543	−	0.780189i	$-0.284877\pi$
$23$	6.00000	1.25109	0.625543	+	0.780189i	$0.284877\pi$
$24$	1.00000	0.204124
$25$	−4.00000	−0.800000
$26$	1.00000	0.196116
$27$	−1.00000	−0.192450
$28$	0	0
$29$	9.00000	1.67126	0.835629	−	0.549294i	$-0.185103\pi$
$29$	9.00000	1.67126	0.835629	+	0.549294i	$0.185103\pi$
$30$	1.00000	0.182574
$31$	5.00000	0.898027	0.449013	−	0.893525i	$-0.351776\pi$
$31$	5.00000	0.898027	0.449013	+	0.893525i	$0.351776\pi$
$32$	−1.00000	−0.176777
$33$	3.00000	0.522233
$34$	2.00000	0.342997
$35$	0	0
$36$	1.00000	0.166667
$37$	−8.00000	−1.31519	−0.657596	−	0.753371i	$-0.728427\pi$
$37$	−8.00000	−1.31519	−0.657596	+	0.753371i	$0.728427\pi$
$38$	0	0
$39$	1.00000	0.160128
$40$	−1.00000	−0.158114
$41$	−4.00000	−0.624695	−0.312348	−	0.949968i	$-0.601115\pi$
$41$	−4.00000	−0.624695	−0.312348	+	0.949968i	$0.601115\pi$
$42$	0	0
$43$	0	0		−	1.00000i	$-0.5\pi$
$43$	0	0			1.00000i	$0.5\pi$
$44$	−3.00000	−0.452267
$45$	1.00000	0.149071
$46$	−6.00000	−0.884652
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	−1.00000	−0.144338
$49$	0	0
$50$	4.00000	0.565685
$51$	2.00000	0.280056
$52$	−1.00000	−0.138675
$53$	1.00000	0.137361	0.0686803	−	0.997639i	$-0.478121\pi$
$53$	1.00000	0.137361	0.0686803	+	0.997639i	$0.478121\pi$
$54$	1.00000	0.136083
$55$	−3.00000	−0.404520
$56$	0	0
$57$	0	0
$58$	−9.00000	−1.18176
$59$	−7.00000	−0.911322	−0.455661	−	0.890153i	$-0.650597\pi$
$59$	−7.00000	−0.911322	−0.455661	+	0.890153i	$0.650597\pi$
$60$	−1.00000	−0.129099
$61$	−4.00000	−0.512148	−0.256074	−	0.966657i	$-0.582429\pi$
$61$	−4.00000	−0.512148	−0.256074	+	0.966657i	$0.582429\pi$
$62$	−5.00000	−0.635001
$63$	0	0
$64$	1.00000	0.125000
$65$	−1.00000	−0.124035
$66$	−3.00000	−0.369274
$67$	−4.00000	−0.488678	−0.244339	−	0.969690i	$-0.578571\pi$
$67$	−4.00000	−0.488678	−0.244339	+	0.969690i	$0.578571\pi$
$68$	−2.00000	−0.242536
$69$	−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
$73$	6.00000	0.702247	0.351123	−	0.936329i	$-0.385800\pi$
$73$	6.00000	0.702247	0.351123	+	0.936329i	$0.385800\pi$
$74$	8.00000	0.929981
$75$	4.00000	0.461880
$76$	0	0
$77$	0	0
$78$	−1.00000	−0.113228
$79$	−13.0000	−1.46261	−0.731307	−	0.682048i	$-0.761089\pi$
$79$	−13.0000	−1.46261	−0.731307	+	0.682048i	$0.761089\pi$
$80$	1.00000	0.111803
$81$	1.00000	0.111111
$82$	4.00000	0.441726
$83$	−3.00000	−0.329293	−0.164646	−	0.986353i	$-0.552648\pi$
$83$	−3.00000	−0.329293	−0.164646	+	0.986353i	$0.552648\pi$
$84$	0	0
$85$	−2.00000	−0.216930
$86$	0	0
$87$	−9.00000	−0.964901
$88$	3.00000	0.319801
$89$	−8.00000	−0.847998	−0.423999	−	0.905663i	$-0.639374\pi$
$89$	−8.00000	−0.847998	−0.423999	+	0.905663i	$0.639374\pi$
$90$	−1.00000	−0.105409
$91$	0	0
$92$	6.00000	0.625543
$93$	−5.00000	−0.518476
$94$	0	0
$95$	0	0
$96$	1.00000	0.102062
$97$	−15.0000	−1.52302	−0.761510	−	0.648154i	$-0.775541\pi$
$97$	−15.0000	−1.52302	−0.761510	+	0.648154i	$0.775541\pi$
$98$	0	0
$99$	−3.00000	−0.301511
$100$	−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	$0.531726\pi$
$102$	−2.00000	−0.198030
$103$	−16.0000	−1.57653	−0.788263	−	0.615338i	$-0.789020\pi$
$103$	−16.0000	−1.57653	−0.788263	+	0.615338i	$0.789020\pi$
$104$	1.00000	0.0980581
$105$	0	0
$106$	−1.00000	−0.0971286
$107$	17.0000	1.64345	0.821726	−	0.569883i	$-0.193011\pi$
$107$	17.0000	1.64345	0.821726	+	0.569883i	$0.193011\pi$
$108$	−1.00000	−0.0962250
$109$	16.0000	1.53252	0.766261	−	0.642529i	$-0.222115\pi$
$109$	16.0000	1.53252	0.766261	+	0.642529i	$0.222115\pi$
$110$	3.00000	0.286039
$111$	8.00000	0.759326
$112$	0	0
$113$	12.0000	1.12887	0.564433	−	0.825479i	$-0.309095\pi$
$113$	12.0000	1.12887	0.564433	+	0.825479i	$0.309095\pi$
$114$	0	0
$115$	6.00000	0.559503
$116$	9.00000	0.835629
$117$	−1.00000	−0.0924500
$118$	7.00000	0.644402
$119$	0	0
$120$	1.00000	0.0912871
$121$	−2.00000	−0.181818
$122$	4.00000	0.362143
$123$	4.00000	0.360668
$124$	5.00000	0.449013
$125$	−9.00000	−0.804984
$126$	0	0
$127$	5.00000	0.443678	0.221839	−	0.975083i	$-0.428794\pi$
$127$	5.00000	0.443678	0.221839	+	0.975083i	$0.428794\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	1.00000	0.0877058
$131$	15.0000	1.31056	0.655278	−	0.755388i	$-0.272551\pi$
$131$	15.0000	1.31056	0.655278	+	0.755388i	$0.272551\pi$
$132$	3.00000	0.261116
$133$	0	0
$134$	4.00000	0.345547
$135$	−1.00000	−0.0860663
$136$	2.00000	0.171499
$137$	−12.0000	−1.02523	−0.512615	−	0.858619i	$-0.671323\pi$
$137$	−12.0000	−1.02523	−0.512615	+	0.858619i	$0.671323\pi$
$138$	6.00000	0.510754
$139$	4.00000	0.339276	0.169638	−	0.985506i	$-0.445740\pi$
$139$	4.00000	0.339276	0.169638	+	0.985506i	$0.445740\pi$
$140$	0	0
$141$	0	0
$142$	−6.00000	−0.503509
$143$	3.00000	0.250873
$144$	1.00000	0.0833333
$145$	9.00000	0.747409
$146$	−6.00000	−0.496564
$147$	0	0
$148$	−8.00000	−0.657596
$149$	−6.00000	−0.491539	−0.245770	−	0.969328i	$-0.579041\pi$
$149$	−6.00000	−0.491539	−0.245770	+	0.969328i	$0.579041\pi$
$150$	−4.00000	−0.326599
$151$	−11.0000	−0.895167	−0.447584	−	0.894242i	$-0.647715\pi$
$151$	−11.0000	−0.895167	−0.447584	+	0.894242i	$0.647715\pi$
$152$	0	0
$153$	−2.00000	−0.161690
$154$	0	0
$155$	5.00000	0.401610
$156$	1.00000	0.0800641
$157$	−14.0000	−1.11732	−0.558661	−	0.829396i	$-0.688685\pi$
$157$	−14.0000	−1.11732	−0.558661	+	0.829396i	$0.688685\pi$
$158$	13.0000	1.03422
$159$	−1.00000	−0.0793052
$160$	−1.00000	−0.0790569
$161$	0	0
$162$	−1.00000	−0.0785674
$163$	−4.00000	−0.313304	−0.156652	−	0.987654i	$-0.550070\pi$
$163$	−4.00000	−0.313304	−0.156652	+	0.987654i	$0.550070\pi$
$164$	−4.00000	−0.312348
$165$	3.00000	0.233550
$166$	3.00000	0.232845
$167$	−4.00000	−0.309529	−0.154765	−	0.987951i	$-0.549462\pi$
$167$	−4.00000	−0.309529	−0.154765	+	0.987951i	$0.549462\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	2.00000	0.153393
$171$	0	0
$172$	0	0
$173$	−14.0000	−1.06440	−0.532200	−	0.846619i	$-0.678635\pi$
$173$	−14.0000	−1.06440	−0.532200	+	0.846619i	$0.678635\pi$
$174$	9.00000	0.682288
$175$	0	0
$176$	−3.00000	−0.226134
$177$	7.00000	0.526152
$178$	8.00000	0.599625
$179$	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
$181$	−8.00000	−0.594635	−0.297318	−	0.954779i	$-0.596092\pi$
$181$	−8.00000	−0.594635	−0.297318	+	0.954779i	$0.596092\pi$
$182$	0	0
$183$	4.00000	0.295689
$184$	−6.00000	−0.442326
$185$	−8.00000	−0.588172
$186$	5.00000	0.366618
$187$	6.00000	0.438763
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−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
$193$	19.0000	1.36765	0.683825	−	0.729646i	$-0.260315\pi$
$193$	19.0000	1.36765	0.683825	+	0.729646i	$0.260315\pi$
$194$	15.0000	1.07694
$195$	1.00000	0.0716115
$196$	0	0
$197$	−22.0000	−1.56744	−0.783718	−	0.621117i	$-0.786679\pi$
$197$	−22.0000	−1.56744	−0.783718	+	0.621117i	$0.786679\pi$
$198$	3.00000	0.213201
$199$	−24.0000	−1.70131	−0.850657	−	0.525720i	$-0.823796\pi$
$199$	−24.0000	−1.70131	−0.850657	+	0.525720i	$0.823796\pi$
$200$	4.00000	0.282843
$201$	4.00000	0.282138
$202$	2.00000	0.140720
$203$	0	0
$204$	2.00000	0.140028
$205$	−4.00000	−0.279372
$206$	16.0000	1.11477
$207$	6.00000	0.417029
$208$	−1.00000	−0.0693375
$209$	0	0
$210$	0	0
$211$	20.0000	1.37686	0.688428	−	0.725304i	$-0.258301\pi$
$211$	20.0000	1.37686	0.688428	+	0.725304i	$0.258301\pi$
$212$	1.00000	0.0686803
$213$	−6.00000	−0.411113
$214$	−17.0000	−1.16210
$215$	0	0
$216$	1.00000	0.0680414
$217$	0	0
$218$	−16.0000	−1.08366
$219$	−6.00000	−0.405442
$220$	−3.00000	−0.202260
$221$	2.00000	0.134535
$222$	−8.00000	−0.536925
$223$	−1.00000	−0.0669650	−0.0334825	−	0.999439i	$-0.510660\pi$
$223$	−1.00000	−0.0669650	−0.0334825	+	0.999439i	$0.510660\pi$
$224$	0	0
$225$	−4.00000	−0.266667
$226$	−12.0000	−0.798228
$227$	−13.0000	−0.862840	−0.431420	−	0.902151i	$-0.641987\pi$
$227$	−13.0000	−0.862840	−0.431420	+	0.902151i	$0.641987\pi$
$228$	0	0
$229$	−14.0000	−0.925146	−0.462573	−	0.886581i	$-0.653074\pi$
$229$	−14.0000	−0.925146	−0.462573	+	0.886581i	$0.653074\pi$
$230$	−6.00000	−0.395628
$231$	0	0
$232$	−9.00000	−0.590879
$233$	−14.0000	−0.917170	−0.458585	−	0.888650i	$-0.651644\pi$
$233$	−14.0000	−0.917170	−0.458585	+	0.888650i	$0.651644\pi$
$234$	1.00000	0.0653720
$235$	0	0
$236$	−7.00000	−0.455661
$237$	13.0000	0.844441
$238$	0	0
$239$	22.0000	1.42306	0.711531	−	0.702655i	$-0.248002\pi$
$239$	22.0000	1.42306	0.711531	+	0.702655i	$0.248002\pi$
$240$	−1.00000	−0.0645497
$241$	−27.0000	−1.73922	−0.869611	−	0.493737i	$-0.835631\pi$
$241$	−27.0000	−1.73922	−0.869611	+	0.493737i	$0.835631\pi$
$242$	2.00000	0.128565
$243$	−1.00000	−0.0641500
$244$	−4.00000	−0.256074
$245$	0	0
$246$	−4.00000	−0.255031
$247$	0	0
$248$	−5.00000	−0.317500
$249$	3.00000	0.190117
$250$	9.00000	0.569210
$251$	23.0000	1.45175	0.725874	−	0.687828i	$-0.241436\pi$
$251$	23.0000	1.45175	0.725874	+	0.687828i	$0.241436\pi$
$252$	0	0
$253$	−18.0000	−1.13165
$254$	−5.00000	−0.313728
$255$	2.00000	0.125245
$256$	1.00000	0.0625000
$257$	−26.0000	−1.62184	−0.810918	−	0.585160i	$-0.801032\pi$
$257$	−26.0000	−1.62184	−0.810918	+	0.585160i	$0.801032\pi$
$258$	0	0
$259$	0	0
$260$	−1.00000	−0.0620174
$261$	9.00000	0.557086
$262$	−15.0000	−0.926703
$263$	−26.0000	−1.60323	−0.801614	−	0.597841i	$-0.796025\pi$
$263$	−26.0000	−1.60323	−0.801614	+	0.597841i	$0.796025\pi$
$264$	−3.00000	−0.184637
$265$	1.00000	0.0614295
$266$	0	0
$267$	8.00000	0.489592
$268$	−4.00000	−0.244339
$269$	−3.00000	−0.182913	−0.0914566	−	0.995809i	$-0.529152\pi$
$269$	−3.00000	−0.182913	−0.0914566	+	0.995809i	$0.529152\pi$
$270$	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
$274$	12.0000	0.724947
$275$	12.0000	0.723627
$276$	−6.00000	−0.361158
$277$	−22.0000	−1.32185	−0.660926	−	0.750451i	$-0.729836\pi$
$277$	−22.0000	−1.32185	−0.660926	+	0.750451i	$0.729836\pi$
$278$	−4.00000	−0.239904
$279$	5.00000	0.299342
$280$	0	0
$281$	0	0		−	1.00000i	$-0.5\pi$
$281$	0	0			1.00000i	$0.5\pi$
$282$	0	0
$283$	14.0000	0.832214	0.416107	−	0.909316i	$-0.363394\pi$
$283$	14.0000	0.832214	0.416107	+	0.909316i	$0.363394\pi$
$284$	6.00000	0.356034
$285$	0	0
$286$	−3.00000	−0.177394
$287$	0	0
$288$	−1.00000	−0.0589256
$289$	−13.0000	−0.764706
$290$	−9.00000	−0.528498
$291$	15.0000	0.879316
$292$	6.00000	0.351123
$293$	15.0000	0.876309	0.438155	−	0.898900i	$-0.355632\pi$
$293$	15.0000	0.876309	0.438155	+	0.898900i	$0.355632\pi$
$294$	0	0
$295$	−7.00000	−0.407556
$296$	8.00000	0.464991
$297$	3.00000	0.174078
$298$	6.00000	0.347571
$299$	−6.00000	−0.346989
$300$	4.00000	0.230940
$301$	0	0
$302$	11.0000	0.632979
$303$	2.00000	0.114897
$304$	0	0
$305$	−4.00000	−0.229039
$306$	2.00000	0.114332
$307$	−18.0000	−1.02731	−0.513657	−	0.857996i	$-0.671710\pi$
$307$	−18.0000	−1.02731	−0.513657	+	0.857996i	$0.671710\pi$
$308$	0	0
$309$	16.0000	0.910208
$310$	−5.00000	−0.283981
$311$	2.00000	0.113410	0.0567048	−	0.998391i	$-0.481941\pi$
$311$	2.00000	0.113410	0.0567048	+	0.998391i	$0.481941\pi$
$312$	−1.00000	−0.0566139
$313$	13.0000	0.734803	0.367402	−	0.930062i	$-0.380247\pi$
$313$	13.0000	0.734803	0.367402	+	0.930062i	$0.380247\pi$
$314$	14.0000	0.790066
$315$	0	0
$316$	−13.0000	−0.731307
$317$	3.00000	0.168497	0.0842484	−	0.996445i	$-0.473151\pi$
$317$	3.00000	0.168497	0.0842484	+	0.996445i	$0.473151\pi$
$318$	1.00000	0.0560772
$319$	−27.0000	−1.51171
$320$	1.00000	0.0559017
$321$	−17.0000	−0.948847
$322$	0	0
$323$	0	0
$324$	1.00000	0.0555556
$325$	4.00000	0.221880
$326$	4.00000	0.221540
$327$	−16.0000	−0.884802
$328$	4.00000	0.220863
$329$	0	0
$330$	−3.00000	−0.165145
$331$	8.00000	0.439720	0.219860	−	0.975531i	$-0.429440\pi$
$331$	8.00000	0.439720	0.219860	+	0.975531i	$0.429440\pi$
$332$	−3.00000	−0.164646
$333$	−8.00000	−0.438397
$334$	4.00000	0.218870
$335$	−4.00000	−0.218543
$336$	0	0
$337$	1.00000	0.0544735	0.0272367	−	0.999629i	$-0.491329\pi$
$337$	1.00000	0.0544735	0.0272367	+	0.999629i	$0.491329\pi$
$338$	−1.00000	−0.0543928
$339$	−12.0000	−0.651751
$340$	−2.00000	−0.108465
$341$	−15.0000	−0.812296
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−6.00000	−0.323029
$346$	14.0000	0.752645
$347$	−28.0000	−1.50312	−0.751559	−	0.659665i	$-0.770698\pi$
$347$	−28.0000	−1.50312	−0.751559	+	0.659665i	$0.770698\pi$
$348$	−9.00000	−0.482451
$349$	−34.0000	−1.81998	−0.909989	−	0.414632i	$-0.863910\pi$
$349$	−34.0000	−1.81998	−0.909989	+	0.414632i	$0.863910\pi$
$350$	0	0
$351$	1.00000	0.0533761
$352$	3.00000	0.159901
$353$	6.00000	0.319348	0.159674	−	0.987170i	$-0.448956\pi$
$353$	6.00000	0.319348	0.159674	+	0.987170i	$0.448956\pi$
$354$	−7.00000	−0.372046
$355$	6.00000	0.318447
$356$	−8.00000	−0.423999
$357$	0	0
$358$	−12.0000	−0.634220
$359$	−2.00000	−0.105556	−0.0527780	−	0.998606i	$-0.516808\pi$
$359$	−2.00000	−0.105556	−0.0527780	+	0.998606i	$0.516808\pi$
$360$	−1.00000	−0.0527046
$361$	−19.0000	−1.00000
$362$	8.00000	0.420471
$363$	2.00000	0.104973
$364$	0	0
$365$	6.00000	0.314054
$366$	−4.00000	−0.209083
$367$	−3.00000	−0.156599	−0.0782994	−	0.996930i	$-0.524949\pi$
$367$	−3.00000	−0.156599	−0.0782994	+	0.996930i	$0.524949\pi$
$368$	6.00000	0.312772
$369$	−4.00000	−0.208232
$370$	8.00000	0.415900
$371$	0	0
$372$	−5.00000	−0.259238
$373$	−26.0000	−1.34623	−0.673114	−	0.739538i	$-0.735044\pi$
$373$	−26.0000	−1.34623	−0.673114	+	0.739538i	$0.735044\pi$
$374$	−6.00000	−0.310253
$375$	9.00000	0.464758
$376$	0	0
$377$	−9.00000	−0.463524
$378$	0	0
$379$	−24.0000	−1.23280	−0.616399	−	0.787434i	$-0.711409\pi$
$379$	−24.0000	−1.23280	−0.616399	+	0.787434i	$0.711409\pi$
$380$	0	0
$381$	−5.00000	−0.256158
$382$	6.00000	0.306987
$383$	−26.0000	−1.32854	−0.664269	−	0.747494i	$-0.731257\pi$
$383$	−26.0000	−1.32854	−0.664269	+	0.747494i	$0.731257\pi$
$384$	1.00000	0.0510310
$385$	0	0
$386$	−19.0000	−0.967075
$387$	0	0
$388$	−15.0000	−0.761510
$389$	34.0000	1.72387	0.861934	−	0.507020i	$-0.169253\pi$
$389$	34.0000	1.72387	0.861934	+	0.507020i	$0.169253\pi$
$390$	−1.00000	−0.0506370
$391$	−12.0000	−0.606866
$392$	0	0
$393$	−15.0000	−0.756650
$394$	22.0000	1.10834
$395$	−13.0000	−0.654101
$396$	−3.00000	−0.150756
$397$	34.0000	1.70641	0.853206	−	0.521575i	$-0.174655\pi$
$397$	34.0000	1.70641	0.853206	+	0.521575i	$0.174655\pi$
$398$	24.0000	1.20301
$399$	0	0
$400$	−4.00000	−0.200000
$401$	6.00000	0.299626	0.149813	−	0.988714i	$-0.452133\pi$
$401$	6.00000	0.299626	0.149813	+	0.988714i	$0.452133\pi$
$402$	−4.00000	−0.199502
$403$	−5.00000	−0.249068
$404$	−2.00000	−0.0995037
$405$	1.00000	0.0496904
$406$	0	0
$407$	24.0000	1.18964
$408$	−2.00000	−0.0990148
$409$	−11.0000	−0.543915	−0.271957	−	0.962309i	$-0.587671\pi$
$409$	−11.0000	−0.543915	−0.271957	+	0.962309i	$0.587671\pi$
$410$	4.00000	0.197546
$411$	12.0000	0.591916
$412$	−16.0000	−0.788263
$413$	0	0
$414$	−6.00000	−0.294884
$415$	−3.00000	−0.147264
$416$	1.00000	0.0490290
$417$	−4.00000	−0.195881
$418$	0	0
$419$	0	0		−	1.00000i	$-0.5\pi$
$419$	0	0			1.00000i	$0.5\pi$
$420$	0	0
$421$	−12.0000	−0.584844	−0.292422	−	0.956289i	$-0.594461\pi$
$421$	−12.0000	−0.584844	−0.292422	+	0.956289i	$0.594461\pi$
$422$	−20.0000	−0.973585
$423$	0	0
$424$	−1.00000	−0.0485643
$425$	8.00000	0.388057
$426$	6.00000	0.290701
$427$	0	0
$428$	17.0000	0.821726
$429$	−3.00000	−0.144841
$430$	0	0
$431$	26.0000	1.25238	0.626188	−	0.779672i	$-0.284614\pi$
$431$	26.0000	1.25238	0.626188	+	0.779672i	$0.284614\pi$
$432$	−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
$435$	−9.00000	−0.431517
$436$	16.0000	0.766261
$437$	0	0
$438$	6.00000	0.286691
$439$	−29.0000	−1.38409	−0.692047	−	0.721852i	$-0.743291\pi$
$439$	−29.0000	−1.38409	−0.692047	+	0.721852i	$0.743291\pi$
$440$	3.00000	0.143019
$441$	0	0
$442$	−2.00000	−0.0951303
$443$	39.0000	1.85295	0.926473	−	0.376361i	$-0.122825\pi$
$443$	39.0000	1.85295	0.926473	+	0.376361i	$0.122825\pi$
$444$	8.00000	0.379663
$445$	−8.00000	−0.379236
$446$	1.00000	0.0473514
$447$	6.00000	0.283790
$448$	0	0
$449$	8.00000	0.377543	0.188772	−	0.982021i	$-0.439549\pi$
$449$	8.00000	0.377543	0.188772	+	0.982021i	$0.439549\pi$
$450$	4.00000	0.188562
$451$	12.0000	0.565058
$452$	12.0000	0.564433
$453$	11.0000	0.516825
$454$	13.0000	0.610120
$455$	0	0
$456$	0	0
$457$	1.00000	0.0467780	0.0233890	−	0.999726i	$-0.492554\pi$
$457$	1.00000	0.0467780	0.0233890	+	0.999726i	$0.492554\pi$
$458$	14.0000	0.654177
$459$	2.00000	0.0933520
$460$	6.00000	0.279751
$461$	−26.0000	−1.21094	−0.605470	−	0.795868i	$-0.707015\pi$
$461$	−26.0000	−1.21094	−0.605470	+	0.795868i	$0.707015\pi$
$462$	0	0
$463$	−8.00000	−0.371792	−0.185896	−	0.982569i	$-0.559519\pi$
$463$	−8.00000	−0.371792	−0.185896	+	0.982569i	$0.559519\pi$
$464$	9.00000	0.417815
$465$	−5.00000	−0.231869
$466$	14.0000	0.648537
$467$	12.0000	0.555294	0.277647	−	0.960683i	$-0.410445\pi$
$467$	12.0000	0.555294	0.277647	+	0.960683i	$0.410445\pi$
$468$	−1.00000	−0.0462250
$469$	0	0
$470$	0	0
$471$	14.0000	0.645086
$472$	7.00000	0.322201
$473$	0	0
$474$	−13.0000	−0.597110
$475$	0	0
$476$	0	0
$477$	1.00000	0.0457869
$478$	−22.0000	−1.00626
$479$	30.0000	1.37073	0.685367	−	0.728197i	$-0.259642\pi$
$479$	30.0000	1.37073	0.685367	+	0.728197i	$0.259642\pi$
$480$	1.00000	0.0456435
$481$	8.00000	0.364769
$482$	27.0000	1.22982
$483$	0	0
$484$	−2.00000	−0.0909091
$485$	−15.0000	−0.681115
$486$	1.00000	0.0453609
$487$	−9.00000	−0.407829	−0.203914	−	0.978989i	$-0.565366\pi$
$487$	−9.00000	−0.407829	−0.203914	+	0.978989i	$0.565366\pi$
$488$	4.00000	0.181071
$489$	4.00000	0.180886
$490$	0	0
$491$	−29.0000	−1.30875	−0.654376	−	0.756169i	$-0.727069\pi$
$491$	−29.0000	−1.30875	−0.654376	+	0.756169i	$0.727069\pi$
$492$	4.00000	0.180334
$493$	−18.0000	−0.810679
$494$	0	0
$495$	−3.00000	−0.134840
$496$	5.00000	0.224507
$497$	0	0
$498$	−3.00000	−0.134433
$499$	16.0000	0.716258	0.358129	−	0.933672i	$-0.383415\pi$
$499$	16.0000	0.716258	0.358129	+	0.933672i	$0.383415\pi$
$500$	−9.00000	−0.402492
$501$	4.00000	0.178707
$502$	−23.0000	−1.02654
$503$	−26.0000	−1.15928	−0.579641	−	0.814872i	$-0.696807\pi$
$503$	−26.0000	−1.15928	−0.579641	+	0.814872i	$0.696807\pi$
$504$	0	0
$505$	−2.00000	−0.0889988
$506$	18.0000	0.800198
$507$	−1.00000	−0.0444116
$508$	5.00000	0.221839
$509$	−21.0000	−0.930809	−0.465404	−	0.885098i	$-0.654091\pi$
$509$	−21.0000	−0.930809	−0.465404	+	0.885098i	$0.654091\pi$
$510$	−2.00000	−0.0885615
$511$	0	0
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	26.0000	1.14681
$515$	−16.0000	−0.705044
$516$	0	0
$517$	0	0
$518$	0	0
$519$	14.0000	0.614532
$520$	1.00000	0.0438529
$521$	28.0000	1.22670	0.613351	−	0.789810i	$-0.289821\pi$
$521$	28.0000	1.22670	0.613351	+	0.789810i	$0.289821\pi$
$522$	−9.00000	−0.393919
$523$	−12.0000	−0.524723	−0.262362	−	0.964970i	$-0.584501\pi$
$523$	−12.0000	−0.524723	−0.262362	+	0.964970i	$0.584501\pi$
$524$	15.0000	0.655278
$525$	0	0
$526$	26.0000	1.13365
$527$	−10.0000	−0.435607
$528$	3.00000	0.130558
$529$	13.0000	0.565217
$530$	−1.00000	−0.0434372
$531$	−7.00000	−0.303774
$532$	0	0
$533$	4.00000	0.173259
$534$	−8.00000	−0.346194
$535$	17.0000	0.734974
$536$	4.00000	0.172774
$537$	−12.0000	−0.517838
$538$	3.00000	0.129339
$539$	0	0
$540$	−1.00000	−0.0430331
$541$	−32.0000	−1.37579	−0.687894	−	0.725811i	$-0.741464\pi$
$541$	−32.0000	−1.37579	−0.687894	+	0.725811i	$0.741464\pi$
$542$	−9.00000	−0.386583
$543$	8.00000	0.343313
$544$	2.00000	0.0857493
$545$	16.0000	0.685365
$546$	0	0
$547$	8.00000	0.342055	0.171028	−	0.985266i	$-0.445291\pi$
$547$	8.00000	0.342055	0.171028	+	0.985266i	$0.445291\pi$
$548$	−12.0000	−0.512615
$549$	−4.00000	−0.170716
$550$	−12.0000	−0.511682
$551$	0	0
$552$	6.00000	0.255377
$553$	0	0
$554$	22.0000	0.934690
$555$	8.00000	0.339581
$556$	4.00000	0.169638
$557$	17.0000	0.720313	0.360157	−	0.932892i	$-0.382723\pi$
$557$	17.0000	0.720313	0.360157	+	0.932892i	$0.382723\pi$
$558$	−5.00000	−0.211667
$559$	0	0
$560$	0	0
$561$	−6.00000	−0.253320
$562$	0	0
$563$	3.00000	0.126435	0.0632175	−	0.998000i	$-0.479864\pi$
$563$	3.00000	0.126435	0.0632175	+	0.998000i	$0.479864\pi$
$564$	0	0
$565$	12.0000	0.504844
$566$	−14.0000	−0.588464
$567$	0	0
$568$	−6.00000	−0.251754
$569$	0	0		−	1.00000i	$-0.5\pi$
$569$	0	0			1.00000i	$0.5\pi$
$570$	0	0
$571$	20.0000	0.836974	0.418487	−	0.908223i	$-0.362561\pi$
$571$	20.0000	0.836974	0.418487	+	0.908223i	$0.362561\pi$
$572$	3.00000	0.125436
$573$	6.00000	0.250654
$574$	0	0
$575$	−24.0000	−1.00087
$576$	1.00000	0.0416667
$577$	−3.00000	−0.124892	−0.0624458	−	0.998048i	$-0.519890\pi$
$577$	−3.00000	−0.124892	−0.0624458	+	0.998048i	$0.519890\pi$
$578$	13.0000	0.540729
$579$	−19.0000	−0.789613
$580$	9.00000	0.373705
$581$	0	0
$582$	−15.0000	−0.621770
$583$	−3.00000	−0.124247
$584$	−6.00000	−0.248282
$585$	−1.00000	−0.0413449
$586$	−15.0000	−0.619644
$587$	−9.00000	−0.371470	−0.185735	−	0.982600i	$-0.559467\pi$
$587$	−9.00000	−0.371470	−0.185735	+	0.982600i	$0.559467\pi$
$588$	0	0
$589$	0	0
$590$	7.00000	0.288185
$591$	22.0000	0.904959
$592$	−8.00000	−0.328798
$593$	−40.0000	−1.64260	−0.821302	−	0.570494i	$-0.806752\pi$
$593$	−40.0000	−1.64260	−0.821302	+	0.570494i	$0.806752\pi$
$594$	−3.00000	−0.123091
$595$	0	0
$596$	−6.00000	−0.245770
$597$	24.0000	0.982255
$598$	6.00000	0.245358
$599$	18.0000	0.735460	0.367730	−	0.929933i	$-0.380135\pi$
$599$	18.0000	0.735460	0.367730	+	0.929933i	$0.380135\pi$
$600$	−4.00000	−0.163299
$601$	19.0000	0.775026	0.387513	−	0.921864i	$-0.373334\pi$
$601$	19.0000	0.775026	0.387513	+	0.921864i	$0.373334\pi$
$602$	0	0
$603$	−4.00000	−0.162893
$604$	−11.0000	−0.447584
$605$	−2.00000	−0.0813116
$606$	−2.00000	−0.0812444
$607$	1.00000	0.0405887	0.0202944	−	0.999794i	$-0.493540\pi$
$607$	1.00000	0.0405887	0.0202944	+	0.999794i	$0.493540\pi$
$608$	0	0
$609$	0	0
$610$	4.00000	0.161955
$611$	0	0
$612$	−2.00000	−0.0808452
$613$	−28.0000	−1.13091	−0.565455	−	0.824779i	$-0.691299\pi$
$613$	−28.0000	−1.13091	−0.565455	+	0.824779i	$0.691299\pi$
$614$	18.0000	0.726421
$615$	4.00000	0.161296
$616$	0	0
$617$	4.00000	0.161034	0.0805170	−	0.996753i	$-0.474343\pi$
$617$	4.00000	0.161034	0.0805170	+	0.996753i	$0.474343\pi$
$618$	−16.0000	−0.643614
$619$	32.0000	1.28619	0.643094	−	0.765787i	$-0.277650\pi$
$619$	32.0000	1.28619	0.643094	+	0.765787i	$0.277650\pi$
$620$	5.00000	0.200805
$621$	−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
$627$	0	0
$628$	−14.0000	−0.558661
$629$	16.0000	0.637962
$630$	0	0
$631$	−13.0000	−0.517522	−0.258761	−	0.965941i	$-0.583314\pi$
$631$	−13.0000	−0.517522	−0.258761	+	0.965941i	$0.583314\pi$
$632$	13.0000	0.517112
$633$	−20.0000	−0.794929
$634$	−3.00000	−0.119145
$635$	5.00000	0.198419
$636$	−1.00000	−0.0396526
$637$	0	0
$638$	27.0000	1.06894
$639$	6.00000	0.237356
$640$	−1.00000	−0.0395285
$641$	12.0000	0.473972	0.236986	−	0.971513i	$-0.423841\pi$
$641$	12.0000	0.473972	0.236986	+	0.971513i	$0.423841\pi$
$642$	17.0000	0.670936
$643$	−32.0000	−1.26196	−0.630978	−	0.775800i	$-0.717346\pi$
$643$	−32.0000	−1.26196	−0.630978	+	0.775800i	$0.717346\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	36.0000	1.41531	0.707653	−	0.706560i	$-0.249754\pi$
$647$	36.0000	1.41531	0.707653	+	0.706560i	$0.249754\pi$
$648$	−1.00000	−0.0392837
$649$	21.0000	0.824322
$650$	−4.00000	−0.156893
$651$	0	0
$652$	−4.00000	−0.156652
$653$	39.0000	1.52619	0.763094	−	0.646288i	$-0.223679\pi$
$653$	39.0000	1.52619	0.763094	+	0.646288i	$0.223679\pi$
$654$	16.0000	0.625650
$655$	15.0000	0.586098
$656$	−4.00000	−0.156174
$657$	6.00000	0.234082
$658$	0	0
$659$	−12.0000	−0.467454	−0.233727	−	0.972302i	$-0.575092\pi$
$659$	−12.0000	−0.467454	−0.233727	+	0.972302i	$0.575092\pi$
$660$	3.00000	0.116775
$661$	26.0000	1.01128	0.505641	−	0.862744i	$-0.331256\pi$
$661$	26.0000	1.01128	0.505641	+	0.862744i	$0.331256\pi$
$662$	−8.00000	−0.310929
$663$	−2.00000	−0.0776736
$664$	3.00000	0.116423
$665$	0	0
$666$	8.00000	0.309994
$667$	54.0000	2.09089
$668$	−4.00000	−0.154765
$669$	1.00000	0.0386622
$670$	4.00000	0.154533
$671$	12.0000	0.463255
$672$	0	0
$673$	17.0000	0.655302	0.327651	−	0.944799i	$-0.393743\pi$
$673$	17.0000	0.655302	0.327651	+	0.944799i	$0.393743\pi$
$674$	−1.00000	−0.0385186
$675$	4.00000	0.153960
$676$	1.00000	0.0384615
$677$	31.0000	1.19143	0.595713	−	0.803197i	$-0.296869\pi$
$677$	31.0000	1.19143	0.595713	+	0.803197i	$0.296869\pi$
$678$	12.0000	0.460857
$679$	0	0
$680$	2.00000	0.0766965
$681$	13.0000	0.498161
$682$	15.0000	0.574380
$683$	−41.0000	−1.56882	−0.784411	−	0.620242i	$-0.787034\pi$
$683$	−41.0000	−1.56882	−0.784411	+	0.620242i	$0.787034\pi$
$684$	0	0
$685$	−12.0000	−0.458496
$686$	0	0
$687$	14.0000	0.534133
$688$	0	0
$689$	−1.00000	−0.0380970
$690$	6.00000	0.228416
$691$	8.00000	0.304334	0.152167	−	0.988355i	$-0.451375\pi$
$691$	8.00000	0.304334	0.152167	+	0.988355i	$0.451375\pi$
$692$	−14.0000	−0.532200
$693$	0	0
$694$	28.0000	1.06287
$695$	4.00000	0.151729
$696$	9.00000	0.341144
$697$	8.00000	0.303022
$698$	34.0000	1.28692
$699$	14.0000	0.529529
$700$	0	0
$701$	−39.0000	−1.47301	−0.736505	−	0.676432i	$-0.763525\pi$
$701$	−39.0000	−1.47301	−0.736505	+	0.676432i	$0.763525\pi$
$702$	−1.00000	−0.0377426
$703$	0	0
$704$	−3.00000	−0.113067
$705$	0	0
$706$	−6.00000	−0.225813
$707$	0	0
$708$	7.00000	0.263076
$709$	−26.0000	−0.976450	−0.488225	−	0.872718i	$-0.662356\pi$
$709$	−26.0000	−0.976450	−0.488225	+	0.872718i	$0.662356\pi$
$710$	−6.00000	−0.225176
$711$	−13.0000	−0.487538
$712$	8.00000	0.299813
$713$	30.0000	1.12351
$714$	0	0
$715$	3.00000	0.112194
$716$	12.0000	0.448461
$717$	−22.0000	−0.821605
$718$	2.00000	0.0746393
$719$	2.00000	0.0745874	0.0372937	−	0.999304i	$-0.488126\pi$
$719$	2.00000	0.0745874	0.0372937	+	0.999304i	$0.488126\pi$
$720$	1.00000	0.0372678
$721$	0	0
$722$	19.0000	0.707107
$723$	27.0000	1.00414
$724$	−8.00000	−0.297318
$725$	−36.0000	−1.33701
$726$	−2.00000	−0.0742270
$727$	−9.00000	−0.333792	−0.166896	−	0.985975i	$-0.553374\pi$
$727$	−9.00000	−0.333792	−0.166896	+	0.985975i	$0.553374\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	−6.00000	−0.222070
$731$	0	0
$732$	4.00000	0.147844
$733$	22.0000	0.812589	0.406294	−	0.913742i	$-0.366821\pi$
$733$	22.0000	0.812589	0.406294	+	0.913742i	$0.366821\pi$
$734$	3.00000	0.110732
$735$	0	0
$736$	−6.00000	−0.221163
$737$	12.0000	0.442026
$738$	4.00000	0.147242
$739$	−18.0000	−0.662141	−0.331070	−	0.943606i	$-0.607410\pi$
$739$	−18.0000	−0.662141	−0.331070	+	0.943606i	$0.607410\pi$
$740$	−8.00000	−0.294086
$741$	0	0
$742$	0	0
$743$	−20.0000	−0.733729	−0.366864	−	0.930274i	$-0.619569\pi$
$743$	−20.0000	−0.733729	−0.366864	+	0.930274i	$0.619569\pi$
$744$	5.00000	0.183309
$745$	−6.00000	−0.219823
$746$	26.0000	0.951928
$747$	−3.00000	−0.109764
$748$	6.00000	0.219382
$749$	0	0
$750$	−9.00000	−0.328634
$751$	13.0000	0.474377	0.237188	−	0.971464i	$-0.423774\pi$
$751$	13.0000	0.474377	0.237188	+	0.971464i	$0.423774\pi$
$752$	0	0
$753$	−23.0000	−0.838167
$754$	9.00000	0.327761
$755$	−11.0000	−0.400331
$756$	0	0
$757$	22.0000	0.799604	0.399802	−	0.916602i	$-0.369079\pi$
$757$	22.0000	0.799604	0.399802	+	0.916602i	$0.369079\pi$
$758$	24.0000	0.871719
$759$	18.0000	0.653359
$760$	0	0
$761$	−32.0000	−1.16000	−0.580000	−	0.814617i	$-0.696947\pi$
$761$	−32.0000	−1.16000	−0.580000	+	0.814617i	$0.696947\pi$
$762$	5.00000	0.181131
$763$	0	0
$764$	−6.00000	−0.217072
$765$	−2.00000	−0.0723102
$766$	26.0000	0.939418
$767$	7.00000	0.252755
$768$	−1.00000	−0.0360844
$769$	−13.0000	−0.468792	−0.234396	−	0.972141i	$-0.575311\pi$
$769$	−13.0000	−0.468792	−0.234396	+	0.972141i	$0.575311\pi$
$770$	0	0
$771$	26.0000	0.936367
$772$	19.0000	0.683825
$773$	18.0000	0.647415	0.323708	−	0.946157i	$-0.395071\pi$
$773$	18.0000	0.647415	0.323708	+	0.946157i	$0.395071\pi$
$774$	0	0
$775$	−20.0000	−0.718421
$776$	15.0000	0.538469
$777$	0	0
$778$	−34.0000	−1.21896
$779$	0	0
$780$	1.00000	0.0358057
$781$	−18.0000	−0.644091
$782$	12.0000	0.429119
$783$	−9.00000	−0.321634
$784$	0	0
$785$	−14.0000	−0.499681
$786$	15.0000	0.535032
$787$	28.0000	0.998092	0.499046	−	0.866575i	$-0.333684\pi$
$787$	28.0000	0.998092	0.499046	+	0.866575i	$0.333684\pi$
$788$	−22.0000	−0.783718
$789$	26.0000	0.925625
$790$	13.0000	0.462519
$791$	0	0
$792$	3.00000	0.106600
$793$	4.00000	0.142044
$794$	−34.0000	−1.20661
$795$	−1.00000	−0.0354663
$796$	−24.0000	−0.850657
$797$	7.00000	0.247953	0.123976	−	0.992285i	$-0.460435\pi$
$797$	7.00000	0.247953	0.123976	+	0.992285i	$0.460435\pi$
$798$	0	0
$799$	0	0
$800$	4.00000	0.141421
$801$	−8.00000	−0.282666
$802$	−6.00000	−0.211867
$803$	−18.0000	−0.635206
$804$	4.00000	0.141069
$805$	0	0
$806$	5.00000	0.176117
$807$	3.00000	0.105605
$808$	2.00000	0.0703598
$809$	4.00000	0.140633	0.0703163	−	0.997525i	$-0.477599\pi$
$809$	4.00000	0.140633	0.0703163	+	0.997525i	$0.477599\pi$
$810$	−1.00000	−0.0351364
$811$	36.0000	1.26413	0.632065	−	0.774915i	$-0.282207\pi$
$811$	36.0000	1.26413	0.632065	+	0.774915i	$0.282207\pi$
$812$	0	0
$813$	−9.00000	−0.315644
$814$	−24.0000	−0.841200
$815$	−4.00000	−0.140114
$816$	2.00000	0.0700140
$817$	0	0
$818$	11.0000	0.384606
$819$	0	0
$820$	−4.00000	−0.139686
$821$	3.00000	0.104701	0.0523504	−	0.998629i	$-0.483329\pi$
$821$	3.00000	0.104701	0.0523504	+	0.998629i	$0.483329\pi$
$822$	−12.0000	−0.418548
$823$	20.0000	0.697156	0.348578	−	0.937280i	$-0.386665\pi$
$823$	20.0000	0.697156	0.348578	+	0.937280i	$0.386665\pi$
$824$	16.0000	0.557386
$825$	−12.0000	−0.417786
$826$	0	0
$827$	−27.0000	−0.938882	−0.469441	−	0.882964i	$-0.655545\pi$
$827$	−27.0000	−0.938882	−0.469441	+	0.882964i	$0.655545\pi$
$828$	6.00000	0.208514
$829$	38.0000	1.31979	0.659897	−	0.751356i	$-0.270600\pi$
$829$	38.0000	1.31979	0.659897	+	0.751356i	$0.270600\pi$
$830$	3.00000	0.104132
$831$	22.0000	0.763172
$832$	−1.00000	−0.0346688
$833$	0	0
$834$	4.00000	0.138509
$835$	−4.00000	−0.138426
$836$	0	0
$837$	−5.00000	−0.172825
$838$	0	0
$839$	44.0000	1.51905	0.759524	−	0.650479i	$-0.225432\pi$
$839$	44.0000	1.51905	0.759524	+	0.650479i	$0.225432\pi$
$840$	0	0
$841$	52.0000	1.79310
$842$	12.0000	0.413547
$843$	0	0
$844$	20.0000	0.688428
$845$	1.00000	0.0344010
$846$	0	0
$847$	0	0
$848$	1.00000	0.0343401
$849$	−14.0000	−0.480479
$850$	−8.00000	−0.274398
$851$	−48.0000	−1.64542
$852$	−6.00000	−0.205557
$853$	58.0000	1.98588	0.992941	−	0.118609i	$-0.0378434\pi$
$853$	58.0000	1.98588	0.992941	+	0.118609i	$0.0378434\pi$
$854$	0	0
$855$	0	0
$856$	−17.0000	−0.581048
$857$	−50.0000	−1.70797	−0.853984	−	0.520300i	$-0.825820\pi$
$857$	−50.0000	−1.70797	−0.853984	+	0.520300i	$0.825820\pi$
$858$	3.00000	0.102418
$859$	40.0000	1.36478	0.682391	−	0.730987i	$-0.260940\pi$
$859$	40.0000	1.36478	0.682391	+	0.730987i	$0.260940\pi$
$860$	0	0
$861$	0	0
$862$	−26.0000	−0.885564
$863$	40.0000	1.36162	0.680808	−	0.732462i	$-0.261629\pi$
$863$	40.0000	1.36162	0.680808	+	0.732462i	$0.261629\pi$
$864$	1.00000	0.0340207
$865$	−14.0000	−0.476014
$866$	38.0000	1.29129
$867$	13.0000	0.441503
$868$	0	0
$869$	39.0000	1.32298
$870$	9.00000	0.305129
$871$	4.00000	0.135535
$872$	−16.0000	−0.541828
$873$	−15.0000	−0.507673
$874$	0	0
$875$	0	0
$876$	−6.00000	−0.202721
$877$	−18.0000	−0.607817	−0.303908	−	0.952701i	$-0.598292\pi$
$877$	−18.0000	−0.607817	−0.303908	+	0.952701i	$0.598292\pi$
$878$	29.0000	0.978703
$879$	−15.0000	−0.505937
$880$	−3.00000	−0.101130
$881$	50.0000	1.68454	0.842271	−	0.539054i	$-0.181218\pi$
$881$	50.0000	1.68454	0.842271	+	0.539054i	$0.181218\pi$
$882$	0	0
$883$	10.0000	0.336527	0.168263	−	0.985742i	$-0.446184\pi$
$883$	10.0000	0.336527	0.168263	+	0.985742i	$0.446184\pi$
$884$	2.00000	0.0672673
$885$	7.00000	0.235302
$886$	−39.0000	−1.31023
$887$	24.0000	0.805841	0.402921	−	0.915235i	$-0.367995\pi$
$887$	24.0000	0.805841	0.402921	+	0.915235i	$0.367995\pi$
$888$	−8.00000	−0.268462
$889$	0	0
$890$	8.00000	0.268161
$891$	−3.00000	−0.100504
$892$	−1.00000	−0.0334825
$893$	0	0
$894$	−6.00000	−0.200670
$895$	12.0000	0.401116
$896$	0	0
$897$	6.00000	0.200334
$898$	−8.00000	−0.266963
$899$	45.0000	1.50083
$900$	−4.00000	−0.133333
$901$	−2.00000	−0.0666297
$902$	−12.0000	−0.399556
$903$	0	0
$904$	−12.0000	−0.399114
$905$	−8.00000	−0.265929
$906$	−11.0000	−0.365451
$907$	−8.00000	−0.265636	−0.132818	−	0.991140i	$-0.542403\pi$
$907$	−8.00000	−0.265636	−0.132818	+	0.991140i	$0.542403\pi$
$908$	−13.0000	−0.431420
$909$	−2.00000	−0.0663358
$910$	0	0
$911$	−50.0000	−1.65657	−0.828287	−	0.560304i	$-0.810684\pi$
$911$	−50.0000	−1.65657	−0.828287	+	0.560304i	$0.810684\pi$
$912$	0	0
$913$	9.00000	0.297857
$914$	−1.00000	−0.0330771
$915$	4.00000	0.132236
$916$	−14.0000	−0.462573
$917$	0	0
$918$	−2.00000	−0.0660098
$919$	−56.0000	−1.84727	−0.923635	−	0.383274i	$-0.874797\pi$
$919$	−56.0000	−1.84727	−0.923635	+	0.383274i	$0.874797\pi$
$920$	−6.00000	−0.197814
$921$	18.0000	0.593120
$922$	26.0000	0.856264
$923$	−6.00000	−0.197492
$924$	0	0
$925$	32.0000	1.05215
$926$	8.00000	0.262896
$927$	−16.0000	−0.525509
$928$	−9.00000	−0.295439
$929$	28.0000	0.918650	0.459325	−	0.888268i	$-0.348091\pi$
$929$	28.0000	0.918650	0.459325	+	0.888268i	$0.348091\pi$
$930$	5.00000	0.163956
$931$	0	0
$932$	−14.0000	−0.458585
$933$	−2.00000	−0.0654771
$934$	−12.0000	−0.392652
$935$	6.00000	0.196221
$936$	1.00000	0.0326860
$937$	−37.0000	−1.20874	−0.604369	−	0.796705i	$-0.706575\pi$
$937$	−37.0000	−1.20874	−0.604369	+	0.796705i	$0.706575\pi$
$938$	0	0
$939$	−13.0000	−0.424239
$940$	0	0
$941$	−39.0000	−1.27136	−0.635682	−	0.771951i	$-0.719281\pi$
$941$	−39.0000	−1.27136	−0.635682	+	0.771951i	$0.719281\pi$
$942$	−14.0000	−0.456145
$943$	−24.0000	−0.781548
$944$	−7.00000	−0.227831
$945$	0	0
$946$	0	0
$947$	12.0000	0.389948	0.194974	−	0.980808i	$-0.437538\pi$
$947$	12.0000	0.389948	0.194974	+	0.980808i	$0.437538\pi$
$948$	13.0000	0.422220
$949$	−6.00000	−0.194768
$950$	0	0
$951$	−3.00000	−0.0972817
$952$	0	0
$953$	12.0000	0.388718	0.194359	−	0.980930i	$-0.437737\pi$
$953$	12.0000	0.388718	0.194359	+	0.980930i	$0.437737\pi$
$954$	−1.00000	−0.0323762
$955$	−6.00000	−0.194155
$956$	22.0000	0.711531
$957$	27.0000	0.872786
$958$	−30.0000	−0.969256
$959$	0	0
$960$	−1.00000	−0.0322749
$961$	−6.00000	−0.193548
$962$	−8.00000	−0.257930
$963$	17.0000	0.547817
$964$	−27.0000	−0.869611
$965$	19.0000	0.611632
$966$	0	0
$967$	21.0000	0.675314	0.337657	−	0.941269i	$-0.390366\pi$
$967$	21.0000	0.675314	0.337657	+	0.941269i	$0.390366\pi$
$968$	2.00000	0.0642824
$969$	0	0
$970$	15.0000	0.481621
$971$	13.0000	0.417190	0.208595	−	0.978002i	$-0.433111\pi$
$971$	13.0000	0.417190	0.208595	+	0.978002i	$0.433111\pi$
$972$	−1.00000	−0.0320750
$973$	0	0
$974$	9.00000	0.288379
$975$	−4.00000	−0.128103
$976$	−4.00000	−0.128037
$977$	−22.0000	−0.703842	−0.351921	−	0.936030i	$-0.614471\pi$
$977$	−22.0000	−0.703842	−0.351921	+	0.936030i	$0.614471\pi$
$978$	−4.00000	−0.127906
$979$	24.0000	0.767043
$980$	0	0
$981$	16.0000	0.510841
$982$	29.0000	0.925427
$983$	6.00000	0.191370	0.0956851	−	0.995412i	$-0.469496\pi$
$983$	6.00000	0.191370	0.0956851	+	0.995412i	$0.469496\pi$
$984$	−4.00000	−0.127515
$985$	−22.0000	−0.700978
$986$	18.0000	0.573237
$987$	0	0
$988$	0	0
$989$	0	0
$990$	3.00000	0.0953463
$991$	15.0000	0.476491	0.238245	−	0.971205i	$-0.423428\pi$
$991$	15.0000	0.476491	0.238245	+	0.971205i	$0.423428\pi$
$992$	−5.00000	−0.158750
$993$	−8.00000	−0.253872
$994$	0	0
$995$	−24.0000	−0.760851
$996$	3.00000	0.0950586
$997$	8.00000	0.253363	0.126681	−	0.991943i	$-0.459567\pi$
$997$	8.00000	0.253363	0.126681	+	0.991943i	$0.459567\pi$
$998$	−16.0000	−0.506471
$999$	8.00000	0.253109

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3822.2.a.f.1.1		1
7.2	even	3		546.2.i.f.235.1	yes	2
7.4	even	3		546.2.i.f.79.1	✓	2
7.6	odd	2		3822.2.a.l.1.1		1
21.2	odd	6		1638.2.j.e.235.1		2
21.11	odd	6		1638.2.j.e.1171.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
546.2.i.f.79.1	✓	2	7.4	even	3
546.2.i.f.235.1	yes	2	7.2	even	3
1638.2.j.e.235.1		2	21.2	odd	6
1638.2.j.e.1171.1		2	21.11	odd	6
3822.2.a.f.1.1		1	1.1	even	1	trivial
3822.2.a.l.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

Related objects

Downloads

Learn more