LMFDB - Embedded newform 3822.2.a.w.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} +1.00000 q^{10} +5.00000 q^{11} -1.00000 q^{12} +1.00000 q^{13} -1.00000 q^{15} +1.00000 q^{16} +3.00000 q^{17} +1.00000 q^{18} +1.00000 q^{19} +1.00000 q^{20} +5.00000 q^{22} +3.00000 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} -4.00000 q^{31} +1.00000 q^{32} -5.00000 q^{33} +3.00000 q^{34} +1.00000 q^{36} -11.0000 q^{37} +1.00000 q^{38} -1.00000 q^{39} +1.00000 q^{40} -5.00000 q^{43} +5.00000 q^{44} +1.00000 q^{45} +3.00000 q^{46} +8.00000 q^{47} -1.00000 q^{48} -4.00000 q^{50} -3.00000 q^{51} +1.00000 q^{52} -2.00000 q^{53} -1.00000 q^{54} +5.00000 q^{55} -1.00000 q^{57} +9.00000 q^{58} -4.00000 q^{59} -1.00000 q^{60} +15.0000 q^{61} -4.00000 q^{62} +1.00000 q^{64} +1.00000 q^{65} -5.00000 q^{66} -2.00000 q^{67} +3.00000 q^{68} -3.00000 q^{69} -12.0000 q^{71} +1.00000 q^{72} -11.0000 q^{73} -11.0000 q^{74} +4.00000 q^{75} +1.00000 q^{76} -1.00000 q^{78} +10.0000 q^{79} +1.00000 q^{80} +1.00000 q^{81} +14.0000 q^{83} +3.00000 q^{85} -5.00000 q^{86} -9.00000 q^{87} +5.00000 q^{88} -6.00000 q^{89} +1.00000 q^{90} +3.00000 q^{92} +4.00000 q^{93} +8.00000 q^{94} +1.00000 q^{95} -1.00000 q^{96} +14.0000 q^{97} +5.00000 q^{99} +O(q^{100})

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
$11$	5.00000	1.50756	0.753778	−	0.657129i	$-0.228229\pi$
$11$	5.00000	1.50756	0.753778	+	0.657129i	$0.228229\pi$
$12$	−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
$17$	3.00000	0.727607	0.363803	−	0.931476i	$-0.381478\pi$
$17$	3.00000	0.727607	0.363803	+	0.931476i	$0.381478\pi$
$18$	1.00000	0.235702
$19$	1.00000	0.229416	0.114708	−	0.993399i	$-0.463407\pi$
$19$	1.00000	0.229416	0.114708	+	0.993399i	$0.463407\pi$
$20$	1.00000	0.223607
$21$	0	0
$22$	5.00000	1.06600
$23$	3.00000	0.625543	0.312772	−	0.949828i	$-0.398743\pi$
$23$	3.00000	0.625543	0.312772	+	0.949828i	$0.398743\pi$
$24$	−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
$31$	−4.00000	−0.718421	−0.359211	−	0.933257i	$-0.616954\pi$
$31$	−4.00000	−0.718421	−0.359211	+	0.933257i	$0.616954\pi$
$32$	1.00000	0.176777
$33$	−5.00000	−0.870388
$34$	3.00000	0.514496
$35$	0	0
$36$	1.00000	0.166667
$37$	−11.0000	−1.80839	−0.904194	−	0.427121i	$-0.859528\pi$
$37$	−11.0000	−1.80839	−0.904194	+	0.427121i	$0.859528\pi$
$38$	1.00000	0.162221
$39$	−1.00000	−0.160128
$40$	1.00000	0.158114
$41$	0	0		−	1.00000i	$-0.5\pi$
$41$	0	0			1.00000i	$0.5\pi$
$42$	0	0
$43$	−5.00000	−0.762493	−0.381246	−	0.924473i	$-0.624505\pi$
$43$	−5.00000	−0.762493	−0.381246	+	0.924473i	$0.624505\pi$
$44$	5.00000	0.753778
$45$	1.00000	0.149071
$46$	3.00000	0.442326
$47$	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
$51$	−3.00000	−0.420084
$52$	1.00000	0.138675
$53$	−2.00000	−0.274721	−0.137361	−	0.990521i	$-0.543862\pi$
$53$	−2.00000	−0.274721	−0.137361	+	0.990521i	$0.543862\pi$
$54$	−1.00000	−0.136083
$55$	5.00000	0.674200
$56$	0	0
$57$	−1.00000	−0.132453
$58$	9.00000	1.18176
$59$	−4.00000	−0.520756	−0.260378	−	0.965507i	$-0.583847\pi$
$59$	−4.00000	−0.520756	−0.260378	+	0.965507i	$0.583847\pi$
$60$	−1.00000	−0.129099
$61$	15.0000	1.92055	0.960277	−	0.279050i	$-0.0900195\pi$
$61$	15.0000	1.92055	0.960277	+	0.279050i	$0.0900195\pi$
$62$	−4.00000	−0.508001
$63$	0	0
$64$	1.00000	0.125000
$65$	1.00000	0.124035
$66$	−5.00000	−0.615457
$67$	−2.00000	−0.244339	−0.122169	−	0.992509i	$-0.538985\pi$
$67$	−2.00000	−0.244339	−0.122169	+	0.992509i	$0.538985\pi$
$68$	3.00000	0.363803
$69$	−3.00000	−0.361158
$70$	0	0
$71$	−12.0000	−1.42414	−0.712069	−	0.702109i	$-0.752242\pi$
$71$	−12.0000	−1.42414	−0.712069	+	0.702109i	$0.752242\pi$
$72$	1.00000	0.117851
$73$	−11.0000	−1.28745	−0.643726	−	0.765256i	$-0.722612\pi$
$73$	−11.0000	−1.28745	−0.643726	+	0.765256i	$0.722612\pi$
$74$	−11.0000	−1.27872
$75$	4.00000	0.461880
$76$	1.00000	0.114708
$77$	0	0
$78$	−1.00000	−0.113228
$79$	10.0000	1.12509	0.562544	−	0.826767i	$-0.309823\pi$
$79$	10.0000	1.12509	0.562544	+	0.826767i	$0.309823\pi$
$80$	1.00000	0.111803
$81$	1.00000	0.111111
$82$	0	0
$83$	14.0000	1.53670	0.768350	−	0.640030i	$-0.221078\pi$
$83$	14.0000	1.53670	0.768350	+	0.640030i	$0.221078\pi$
$84$	0	0
$85$	3.00000	0.325396
$86$	−5.00000	−0.539164
$87$	−9.00000	−0.964901
$88$	5.00000	0.533002
$89$	−6.00000	−0.635999	−0.317999	−	0.948091i	$-0.603011\pi$
$89$	−6.00000	−0.635999	−0.317999	+	0.948091i	$0.603011\pi$
$90$	1.00000	0.105409
$91$	0	0
$92$	3.00000	0.312772
$93$	4.00000	0.414781
$94$	8.00000	0.825137
$95$	1.00000	0.102598
$96$	−1.00000	−0.102062
$97$	14.0000	1.42148	0.710742	−	0.703452i	$-0.248359\pi$
$97$	14.0000	1.42148	0.710742	+	0.703452i	$0.248359\pi$
$98$	0	0
$99$	5.00000	0.502519
$100$	−4.00000	−0.400000
$101$	−4.00000	−0.398015	−0.199007	−	0.979998i	$-0.563772\pi$
$101$	−4.00000	−0.398015	−0.199007	+	0.979998i	$0.563772\pi$
$102$	−3.00000	−0.297044
$103$	15.0000	1.47799	0.738997	−	0.673709i	$-0.235300\pi$
$103$	15.0000	1.47799	0.738997	+	0.673709i	$0.235300\pi$
$104$	1.00000	0.0980581
$105$	0	0
$106$	−2.00000	−0.194257
$107$	10.0000	0.966736	0.483368	−	0.875417i	$-0.339413\pi$
$107$	10.0000	0.966736	0.483368	+	0.875417i	$0.339413\pi$
$108$	−1.00000	−0.0962250
$109$	−9.00000	−0.862044	−0.431022	−	0.902342i	$-0.641847\pi$
$109$	−9.00000	−0.862044	−0.431022	+	0.902342i	$0.641847\pi$
$110$	5.00000	0.476731
$111$	11.0000	1.04407
$112$	0	0
$113$	2.00000	0.188144	0.0940721	−	0.995565i	$-0.470012\pi$
$113$	2.00000	0.188144	0.0940721	+	0.995565i	$0.470012\pi$
$114$	−1.00000	−0.0936586
$115$	3.00000	0.279751
$116$	9.00000	0.835629
$117$	1.00000	0.0924500
$118$	−4.00000	−0.368230
$119$	0	0
$120$	−1.00000	−0.0912871
$121$	14.0000	1.27273
$122$	15.0000	1.35804
$123$	0	0
$124$	−4.00000	−0.359211
$125$	−9.00000	−0.804984
$126$	0	0
$127$	2.00000	0.177471	0.0887357	−	0.996055i	$-0.471717\pi$
$127$	2.00000	0.177471	0.0887357	+	0.996055i	$0.471717\pi$
$128$	1.00000	0.0883883
$129$	5.00000	0.440225
$130$	1.00000	0.0877058
$131$	−13.0000	−1.13582	−0.567908	−	0.823092i	$-0.692247\pi$
$131$	−13.0000	−1.13582	−0.567908	+	0.823092i	$0.692247\pi$
$132$	−5.00000	−0.435194
$133$	0	0
$134$	−2.00000	−0.172774
$135$	−1.00000	−0.0860663
$136$	3.00000	0.257248
$137$	19.0000	1.62328	0.811640	−	0.584158i	$-0.198575\pi$
$137$	19.0000	1.62328	0.811640	+	0.584158i	$0.198575\pi$
$138$	−3.00000	−0.255377
$139$	0	0		−	1.00000i	$-0.5\pi$
$139$	0	0			1.00000i	$0.5\pi$
$140$	0	0
$141$	−8.00000	−0.673722
$142$	−12.0000	−1.00702
$143$	5.00000	0.418121
$144$	1.00000	0.0833333
$145$	9.00000	0.747409
$146$	−11.0000	−0.910366
$147$	0	0
$148$	−11.0000	−0.904194
$149$	−4.00000	−0.327693	−0.163846	−	0.986486i	$-0.552390\pi$
$149$	−4.00000	−0.327693	−0.163846	+	0.986486i	$0.552390\pi$
$150$	4.00000	0.326599
$151$	−23.0000	−1.87171	−0.935857	−	0.352381i	$-0.885372\pi$
$151$	−23.0000	−1.87171	−0.935857	+	0.352381i	$0.885372\pi$
$152$	1.00000	0.0811107
$153$	3.00000	0.242536
$154$	0	0
$155$	−4.00000	−0.321288
$156$	−1.00000	−0.0800641
$157$	3.00000	0.239426	0.119713	−	0.992809i	$-0.461803\pi$
$157$	3.00000	0.239426	0.119713	+	0.992809i	$0.461803\pi$
$158$	10.0000	0.795557
$159$	2.00000	0.158610
$160$	1.00000	0.0790569
$161$	0	0
$162$	1.00000	0.0785674
$163$	10.0000	0.783260	0.391630	−	0.920123i	$-0.371911\pi$
$163$	10.0000	0.783260	0.391630	+	0.920123i	$0.371911\pi$
$164$	0	0
$165$	−5.00000	−0.389249
$166$	14.0000	1.08661
$167$	21.0000	1.62503	0.812514	−	0.582941i	$-0.198098\pi$
$167$	21.0000	1.62503	0.812514	+	0.582941i	$0.198098\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	3.00000	0.230089
$171$	1.00000	0.0764719
$172$	−5.00000	−0.381246
$173$	−6.00000	−0.456172	−0.228086	−	0.973641i	$-0.573247\pi$
$173$	−6.00000	−0.456172	−0.228086	+	0.973641i	$0.573247\pi$
$174$	−9.00000	−0.682288
$175$	0	0
$176$	5.00000	0.376889
$177$	4.00000	0.300658
$178$	−6.00000	−0.449719
$179$	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
$181$	−14.0000	−1.04061	−0.520306	−	0.853980i	$-0.674182\pi$
$181$	−14.0000	−1.04061	−0.520306	+	0.853980i	$0.674182\pi$
$182$	0	0
$183$	−15.0000	−1.10883
$184$	3.00000	0.221163
$185$	−11.0000	−0.808736
$186$	4.00000	0.293294
$187$	15.0000	1.09691
$188$	8.00000	0.583460
$189$	0	0
$190$	1.00000	0.0725476
$191$	3.00000	0.217072	0.108536	−	0.994092i	$-0.465384\pi$
$191$	3.00000	0.217072	0.108536	+	0.994092i	$0.465384\pi$
$192$	−1.00000	−0.0721688
$193$	12.0000	0.863779	0.431889	−	0.901927i	$-0.357847\pi$
$193$	12.0000	0.863779	0.431889	+	0.901927i	$0.357847\pi$
$194$	14.0000	1.00514
$195$	−1.00000	−0.0716115
$196$	0	0
$197$	−12.0000	−0.854965	−0.427482	−	0.904024i	$-0.640599\pi$
$197$	−12.0000	−0.854965	−0.427482	+	0.904024i	$0.640599\pi$
$198$	5.00000	0.355335
$199$	−11.0000	−0.779769	−0.389885	−	0.920864i	$-0.627485\pi$
$199$	−11.0000	−0.779769	−0.389885	+	0.920864i	$0.627485\pi$
$200$	−4.00000	−0.282843
$201$	2.00000	0.141069
$202$	−4.00000	−0.281439
$203$	0	0
$204$	−3.00000	−0.210042
$205$	0	0
$206$	15.0000	1.04510
$207$	3.00000	0.208514
$208$	1.00000	0.0693375
$209$	5.00000	0.345857
$210$	0	0
$211$	9.00000	0.619586	0.309793	−	0.950804i	$-0.399740\pi$
$211$	9.00000	0.619586	0.309793	+	0.950804i	$0.399740\pi$
$212$	−2.00000	−0.137361
$213$	12.0000	0.822226
$214$	10.0000	0.683586
$215$	−5.00000	−0.340997
$216$	−1.00000	−0.0680414
$217$	0	0
$218$	−9.00000	−0.609557
$219$	11.0000	0.743311
$220$	5.00000	0.337100
$221$	3.00000	0.201802
$222$	11.0000	0.738272
$223$	14.0000	0.937509	0.468755	−	0.883328i	$-0.344703\pi$
$223$	14.0000	0.937509	0.468755	+	0.883328i	$0.344703\pi$
$224$	0	0
$225$	−4.00000	−0.266667
$226$	2.00000	0.133038
$227$	−18.0000	−1.19470	−0.597351	−	0.801980i	$-0.703780\pi$
$227$	−18.0000	−1.19470	−0.597351	+	0.801980i	$0.703780\pi$
$228$	−1.00000	−0.0662266
$229$	−6.00000	−0.396491	−0.198246	−	0.980152i	$-0.563524\pi$
$229$	−6.00000	−0.396491	−0.198246	+	0.980152i	$0.563524\pi$
$230$	3.00000	0.197814
$231$	0	0
$232$	9.00000	0.590879
$233$	24.0000	1.57229	0.786146	−	0.618041i	$-0.212073\pi$
$233$	24.0000	1.57229	0.786146	+	0.618041i	$0.212073\pi$
$234$	1.00000	0.0653720
$235$	8.00000	0.521862
$236$	−4.00000	−0.260378
$237$	−10.0000	−0.649570
$238$	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
$241$	−18.0000	−1.15948	−0.579741	−	0.814801i	$-0.696846\pi$
$241$	−18.0000	−1.15948	−0.579741	+	0.814801i	$0.696846\pi$
$242$	14.0000	0.899954
$243$	−1.00000	−0.0641500
$244$	15.0000	0.960277
$245$	0	0
$246$	0	0
$247$	1.00000	0.0636285
$248$	−4.00000	−0.254000
$249$	−14.0000	−0.887214
$250$	−9.00000	−0.569210
$251$	7.00000	0.441836	0.220918	−	0.975292i	$-0.429095\pi$
$251$	7.00000	0.441836	0.220918	+	0.975292i	$0.429095\pi$
$252$	0	0
$253$	15.0000	0.943042
$254$	2.00000	0.125491
$255$	−3.00000	−0.187867
$256$	1.00000	0.0625000
$257$	−6.00000	−0.374270	−0.187135	−	0.982334i	$-0.559920\pi$
$257$	−6.00000	−0.374270	−0.187135	+	0.982334i	$0.559920\pi$
$258$	5.00000	0.311286
$259$	0	0
$260$	1.00000	0.0620174
$261$	9.00000	0.557086
$262$	−13.0000	−0.803143
$263$	12.0000	0.739952	0.369976	−	0.929041i	$-0.379366\pi$
$263$	12.0000	0.739952	0.369976	+	0.929041i	$0.379366\pi$
$264$	−5.00000	−0.307729
$265$	−2.00000	−0.122859
$266$	0	0
$267$	6.00000	0.367194
$268$	−2.00000	−0.122169
$269$	−18.0000	−1.09748	−0.548740	−	0.835993i	$-0.684892\pi$
$269$	−18.0000	−1.09748	−0.548740	+	0.835993i	$0.684892\pi$
$270$	−1.00000	−0.0608581
$271$	22.0000	1.33640	0.668202	−	0.743980i	$-0.267064\pi$
$271$	22.0000	1.33640	0.668202	+	0.743980i	$0.267064\pi$
$272$	3.00000	0.181902
$273$	0	0
$274$	19.0000	1.14783
$275$	−20.0000	−1.20605
$276$	−3.00000	−0.180579
$277$	26.0000	1.56219	0.781094	−	0.624413i	$-0.214662\pi$
$277$	26.0000	1.56219	0.781094	+	0.624413i	$0.214662\pi$
$278$	0	0
$279$	−4.00000	−0.239474
$280$	0	0
$281$	−26.0000	−1.55103	−0.775515	−	0.631329i	$-0.782510\pi$
$281$	−26.0000	−1.55103	−0.775515	+	0.631329i	$0.782510\pi$
$282$	−8.00000	−0.476393
$283$	−4.00000	−0.237775	−0.118888	−	0.992908i	$-0.537933\pi$
$283$	−4.00000	−0.237775	−0.118888	+	0.992908i	$0.537933\pi$
$284$	−12.0000	−0.712069
$285$	−1.00000	−0.0592349
$286$	5.00000	0.295656
$287$	0	0
$288$	1.00000	0.0589256
$289$	−8.00000	−0.470588
$290$	9.00000	0.528498
$291$	−14.0000	−0.820695
$292$	−11.0000	−0.643726
$293$	−14.0000	−0.817889	−0.408944	−	0.912559i	$-0.634103\pi$
$293$	−14.0000	−0.817889	−0.408944	+	0.912559i	$0.634103\pi$
$294$	0	0
$295$	−4.00000	−0.232889
$296$	−11.0000	−0.639362
$297$	−5.00000	−0.290129
$298$	−4.00000	−0.231714
$299$	3.00000	0.173494
$300$	4.00000	0.230940
$301$	0	0
$302$	−23.0000	−1.32350
$303$	4.00000	0.229794
$304$	1.00000	0.0573539
$305$	15.0000	0.858898
$306$	3.00000	0.171499
$307$	0	0		−	1.00000i	$-0.5\pi$
$307$	0	0			1.00000i	$0.5\pi$
$308$	0	0
$309$	−15.0000	−0.853320
$310$	−4.00000	−0.227185
$311$	10.0000	0.567048	0.283524	−	0.958965i	$-0.408496\pi$
$311$	10.0000	0.567048	0.283524	+	0.958965i	$0.408496\pi$
$312$	−1.00000	−0.0566139
$313$	8.00000	0.452187	0.226093	−	0.974106i	$-0.427405\pi$
$313$	8.00000	0.452187	0.226093	+	0.974106i	$0.427405\pi$
$314$	3.00000	0.169300
$315$	0	0
$316$	10.0000	0.562544
$317$	24.0000	1.34797	0.673987	−	0.738743i	$-0.264580\pi$
$317$	24.0000	1.34797	0.673987	+	0.738743i	$0.264580\pi$
$318$	2.00000	0.112154
$319$	45.0000	2.51952
$320$	1.00000	0.0559017
$321$	−10.0000	−0.558146
$322$	0	0
$323$	3.00000	0.166924
$324$	1.00000	0.0555556
$325$	−4.00000	−0.221880
$326$	10.0000	0.553849
$327$	9.00000	0.497701
$328$	0	0
$329$	0	0
$330$	−5.00000	−0.275241
$331$	10.0000	0.549650	0.274825	−	0.961494i	$-0.411380\pi$
$331$	10.0000	0.549650	0.274825	+	0.961494i	$0.411380\pi$
$332$	14.0000	0.768350
$333$	−11.0000	−0.602796
$334$	21.0000	1.14907
$335$	−2.00000	−0.109272
$336$	0	0
$337$	−5.00000	−0.272367	−0.136184	−	0.990684i	$-0.543484\pi$
$337$	−5.00000	−0.272367	−0.136184	+	0.990684i	$0.543484\pi$
$338$	1.00000	0.0543928
$339$	−2.00000	−0.108625
$340$	3.00000	0.162698
$341$	−20.0000	−1.08306
$342$	1.00000	0.0540738
$343$	0	0
$344$	−5.00000	−0.269582
$345$	−3.00000	−0.161515
$346$	−6.00000	−0.322562
$347$	−16.0000	−0.858925	−0.429463	−	0.903085i	$-0.641297\pi$
$347$	−16.0000	−0.858925	−0.429463	+	0.903085i	$0.641297\pi$
$348$	−9.00000	−0.482451
$349$	−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
$352$	5.00000	0.266501
$353$	−18.0000	−0.958043	−0.479022	−	0.877803i	$-0.659008\pi$
$353$	−18.0000	−0.958043	−0.479022	+	0.877803i	$0.659008\pi$
$354$	4.00000	0.212598
$355$	−12.0000	−0.636894
$356$	−6.00000	−0.317999
$357$	0	0
$358$	12.0000	0.634220
$359$	−4.00000	−0.211112	−0.105556	−	0.994413i	$-0.533662\pi$
$359$	−4.00000	−0.211112	−0.105556	+	0.994413i	$0.533662\pi$
$360$	1.00000	0.0527046
$361$	−18.0000	−0.947368
$362$	−14.0000	−0.735824
$363$	−14.0000	−0.734809
$364$	0	0
$365$	−11.0000	−0.575766
$366$	−15.0000	−0.784063
$367$	−32.0000	−1.67039	−0.835193	−	0.549957i	$-0.814644\pi$
$367$	−32.0000	−1.67039	−0.835193	+	0.549957i	$0.814644\pi$
$368$	3.00000	0.156386
$369$	0	0
$370$	−11.0000	−0.571863
$371$	0	0
$372$	4.00000	0.207390
$373$	−4.00000	−0.207112	−0.103556	−	0.994624i	$-0.533022\pi$
$373$	−4.00000	−0.207112	−0.103556	+	0.994624i	$0.533022\pi$
$374$	15.0000	0.775632
$375$	9.00000	0.464758
$376$	8.00000	0.412568
$377$	9.00000	0.463524
$378$	0	0
$379$	16.0000	0.821865	0.410932	−	0.911666i	$-0.365203\pi$
$379$	16.0000	0.821865	0.410932	+	0.911666i	$0.365203\pi$
$380$	1.00000	0.0512989
$381$	−2.00000	−0.102463
$382$	3.00000	0.153493
$383$	29.0000	1.48183	0.740915	−	0.671598i	$-0.234392\pi$
$383$	29.0000	1.48183	0.740915	+	0.671598i	$0.234392\pi$
$384$	−1.00000	−0.0510310
$385$	0	0
$386$	12.0000	0.610784
$387$	−5.00000	−0.254164
$388$	14.0000	0.710742
$389$	−2.00000	−0.101404	−0.0507020	−	0.998714i	$-0.516146\pi$
$389$	−2.00000	−0.101404	−0.0507020	+	0.998714i	$0.516146\pi$
$390$	−1.00000	−0.0506370
$391$	9.00000	0.455150
$392$	0	0
$393$	13.0000	0.655763
$394$	−12.0000	−0.604551
$395$	10.0000	0.503155
$396$	5.00000	0.251259
$397$	−6.00000	−0.301131	−0.150566	−	0.988600i	$-0.548110\pi$
$397$	−6.00000	−0.301131	−0.150566	+	0.988600i	$0.548110\pi$
$398$	−11.0000	−0.551380
$399$	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	$0.648376\pi$
$402$	2.00000	0.0997509
$403$	−4.00000	−0.199254
$404$	−4.00000	−0.199007
$405$	1.00000	0.0496904
$406$	0	0
$407$	−55.0000	−2.72625
$408$	−3.00000	−0.148522
$409$	−25.0000	−1.23617	−0.618085	−	0.786111i	$-0.712091\pi$
$409$	−25.0000	−1.23617	−0.618085	+	0.786111i	$0.712091\pi$
$410$	0	0
$411$	−19.0000	−0.937201
$412$	15.0000	0.738997
$413$	0	0
$414$	3.00000	0.147442
$415$	14.0000	0.687233
$416$	1.00000	0.0490290
$417$	0	0
$418$	5.00000	0.244558
$419$	−7.00000	−0.341972	−0.170986	−	0.985273i	$-0.554695\pi$
$419$	−7.00000	−0.341972	−0.170986	+	0.985273i	$0.554695\pi$
$420$	0	0
$421$	30.0000	1.46211	0.731055	−	0.682318i	$-0.239028\pi$
$421$	30.0000	1.46211	0.731055	+	0.682318i	$0.239028\pi$
$422$	9.00000	0.438113
$423$	8.00000	0.388973
$424$	−2.00000	−0.0971286
$425$	−12.0000	−0.582086
$426$	12.0000	0.581402
$427$	0	0
$428$	10.0000	0.483368
$429$	−5.00000	−0.241402
$430$	−5.00000	−0.241121
$431$	−2.00000	−0.0963366	−0.0481683	−	0.998839i	$-0.515338\pi$
$431$	−2.00000	−0.0963366	−0.0481683	+	0.998839i	$0.515338\pi$
$432$	−1.00000	−0.0481125
$433$	−28.0000	−1.34559	−0.672797	−	0.739827i	$-0.734907\pi$
$433$	−28.0000	−1.34559	−0.672797	+	0.739827i	$0.734907\pi$
$434$	0	0
$435$	−9.00000	−0.431517
$436$	−9.00000	−0.431022
$437$	3.00000	0.143509
$438$	11.0000	0.525600
$439$	−13.0000	−0.620456	−0.310228	−	0.950662i	$-0.600405\pi$
$439$	−13.0000	−0.620456	−0.310228	+	0.950662i	$0.600405\pi$
$440$	5.00000	0.238366
$441$	0	0
$442$	3.00000	0.142695
$443$	−18.0000	−0.855206	−0.427603	−	0.903967i	$-0.640642\pi$
$443$	−18.0000	−0.855206	−0.427603	+	0.903967i	$0.640642\pi$
$444$	11.0000	0.522037
$445$	−6.00000	−0.284427
$446$	14.0000	0.662919
$447$	4.00000	0.189194
$448$	0	0
$449$	23.0000	1.08544	0.542719	−	0.839915i	$-0.317395\pi$
$449$	23.0000	1.08544	0.542719	+	0.839915i	$0.317395\pi$
$450$	−4.00000	−0.188562
$451$	0	0
$452$	2.00000	0.0940721
$453$	23.0000	1.08063
$454$	−18.0000	−0.844782
$455$	0	0
$456$	−1.00000	−0.0468293
$457$	−32.0000	−1.49690	−0.748448	−	0.663193i	$-0.769201\pi$
$457$	−32.0000	−1.49690	−0.748448	+	0.663193i	$0.769201\pi$
$458$	−6.00000	−0.280362
$459$	−3.00000	−0.140028
$460$	3.00000	0.139876
$461$	35.0000	1.63011	0.815056	−	0.579382i	$-0.196706\pi$
$461$	35.0000	1.63011	0.815056	+	0.579382i	$0.196706\pi$
$462$	0	0
$463$	9.00000	0.418265	0.209133	−	0.977887i	$-0.432936\pi$
$463$	9.00000	0.418265	0.209133	+	0.977887i	$0.432936\pi$
$464$	9.00000	0.417815
$465$	4.00000	0.185496
$466$	24.0000	1.11178
$467$	15.0000	0.694117	0.347059	−	0.937843i	$-0.387180\pi$
$467$	15.0000	0.694117	0.347059	+	0.937843i	$0.387180\pi$
$468$	1.00000	0.0462250
$469$	0	0
$470$	8.00000	0.369012
$471$	−3.00000	−0.138233
$472$	−4.00000	−0.184115
$473$	−25.0000	−1.14950
$474$	−10.0000	−0.459315
$475$	−4.00000	−0.183533
$476$	0	0
$477$	−2.00000	−0.0915737
$478$	−12.0000	−0.548867
$479$	31.0000	1.41643	0.708213	−	0.705999i	$-0.249502\pi$
$479$	31.0000	1.41643	0.708213	+	0.705999i	$0.249502\pi$
$480$	−1.00000	−0.0456435
$481$	−11.0000	−0.501557
$482$	−18.0000	−0.819878
$483$	0	0
$484$	14.0000	0.636364
$485$	14.0000	0.635707
$486$	−1.00000	−0.0453609
$487$	−16.0000	−0.725029	−0.362515	−	0.931978i	$-0.618082\pi$
$487$	−16.0000	−0.725029	−0.362515	+	0.931978i	$0.618082\pi$
$488$	15.0000	0.679018
$489$	−10.0000	−0.452216
$490$	0	0
$491$	−26.0000	−1.17336	−0.586682	−	0.809818i	$-0.699566\pi$
$491$	−26.0000	−1.17336	−0.586682	+	0.809818i	$0.699566\pi$
$492$	0	0
$493$	27.0000	1.21602
$494$	1.00000	0.0449921
$495$	5.00000	0.224733
$496$	−4.00000	−0.179605
$497$	0	0
$498$	−14.0000	−0.627355
$499$	−4.00000	−0.179065	−0.0895323	−	0.995984i	$-0.528537\pi$
$499$	−4.00000	−0.179065	−0.0895323	+	0.995984i	$0.528537\pi$
$500$	−9.00000	−0.402492
$501$	−21.0000	−0.938211
$502$	7.00000	0.312425
$503$	−42.0000	−1.87269	−0.936344	−	0.351085i	$-0.885813\pi$
$503$	−42.0000	−1.87269	−0.936344	+	0.351085i	$0.885813\pi$
$504$	0	0
$505$	−4.00000	−0.177998
$506$	15.0000	0.666831
$507$	−1.00000	−0.0444116
$508$	2.00000	0.0887357
$509$	−13.0000	−0.576215	−0.288107	−	0.957598i	$-0.593026\pi$
$509$	−13.0000	−0.576215	−0.288107	+	0.957598i	$0.593026\pi$
$510$	−3.00000	−0.132842
$511$	0	0
$512$	1.00000	0.0441942
$513$	−1.00000	−0.0441511
$514$	−6.00000	−0.264649
$515$	15.0000	0.660979
$516$	5.00000	0.220113
$517$	40.0000	1.75920
$518$	0	0
$519$	6.00000	0.263371
$520$	1.00000	0.0438529
$521$	17.0000	0.744784	0.372392	−	0.928076i	$-0.378538\pi$
$521$	17.0000	0.744784	0.372392	+	0.928076i	$0.378538\pi$
$522$	9.00000	0.393919
$523$	8.00000	0.349816	0.174908	−	0.984585i	$-0.444037\pi$
$523$	8.00000	0.349816	0.174908	+	0.984585i	$0.444037\pi$
$524$	−13.0000	−0.567908
$525$	0	0
$526$	12.0000	0.523225
$527$	−12.0000	−0.522728
$528$	−5.00000	−0.217597
$529$	−14.0000	−0.608696
$530$	−2.00000	−0.0868744
$531$	−4.00000	−0.173585
$532$	0	0
$533$	0	0
$534$	6.00000	0.259645
$535$	10.0000	0.432338
$536$	−2.00000	−0.0863868
$537$	−12.0000	−0.517838
$538$	−18.0000	−0.776035
$539$	0	0
$540$	−1.00000	−0.0430331
$541$	−25.0000	−1.07483	−0.537417	−	0.843317i	$-0.680600\pi$
$541$	−25.0000	−1.07483	−0.537417	+	0.843317i	$0.680600\pi$
$542$	22.0000	0.944981
$543$	14.0000	0.600798
$544$	3.00000	0.128624
$545$	−9.00000	−0.385518
$546$	0	0
$547$	−12.0000	−0.513083	−0.256541	−	0.966533i	$-0.582583\pi$
$547$	−12.0000	−0.513083	−0.256541	+	0.966533i	$0.582583\pi$
$548$	19.0000	0.811640
$549$	15.0000	0.640184
$550$	−20.0000	−0.852803
$551$	9.00000	0.383413
$552$	−3.00000	−0.127688
$553$	0	0
$554$	26.0000	1.10463
$555$	11.0000	0.466924
$556$	0	0
$557$	−44.0000	−1.86434	−0.932170	−	0.362021i	$-0.882087\pi$
$557$	−44.0000	−1.86434	−0.932170	+	0.362021i	$0.882087\pi$
$558$	−4.00000	−0.169334
$559$	−5.00000	−0.211477
$560$	0	0
$561$	−15.0000	−0.633300
$562$	−26.0000	−1.09674
$563$	3.00000	0.126435	0.0632175	−	0.998000i	$-0.479864\pi$
$563$	3.00000	0.126435	0.0632175	+	0.998000i	$0.479864\pi$
$564$	−8.00000	−0.336861
$565$	2.00000	0.0841406
$566$	−4.00000	−0.168133
$567$	0	0
$568$	−12.0000	−0.503509
$569$	−46.0000	−1.92842	−0.964210	−	0.265139i	$-0.914582\pi$
$569$	−46.0000	−1.92842	−0.964210	+	0.265139i	$0.914582\pi$
$570$	−1.00000	−0.0418854
$571$	40.0000	1.67395	0.836974	−	0.547243i	$-0.184323\pi$
$571$	40.0000	1.67395	0.836974	+	0.547243i	$0.184323\pi$
$572$	5.00000	0.209061
$573$	−3.00000	−0.125327
$574$	0	0
$575$	−12.0000	−0.500435
$576$	1.00000	0.0416667
$577$	38.0000	1.58196	0.790980	−	0.611842i	$-0.209571\pi$
$577$	38.0000	1.58196	0.790980	+	0.611842i	$0.209571\pi$
$578$	−8.00000	−0.332756
$579$	−12.0000	−0.498703
$580$	9.00000	0.373705
$581$	0	0
$582$	−14.0000	−0.580319
$583$	−10.0000	−0.414158
$584$	−11.0000	−0.455183
$585$	1.00000	0.0413449
$586$	−14.0000	−0.578335
$587$	−14.0000	−0.577842	−0.288921	−	0.957353i	$-0.593296\pi$
$587$	−14.0000	−0.577842	−0.288921	+	0.957353i	$0.593296\pi$
$588$	0	0
$589$	−4.00000	−0.164817
$590$	−4.00000	−0.164677
$591$	12.0000	0.493614
$592$	−11.0000	−0.452097
$593$	−6.00000	−0.246390	−0.123195	−	0.992382i	$-0.539314\pi$
$593$	−6.00000	−0.246390	−0.123195	+	0.992382i	$0.539314\pi$
$594$	−5.00000	−0.205152
$595$	0	0
$596$	−4.00000	−0.163846
$597$	11.0000	0.450200
$598$	3.00000	0.122679
$599$	5.00000	0.204294	0.102147	−	0.994769i	$-0.467429\pi$
$599$	5.00000	0.204294	0.102147	+	0.994769i	$0.467429\pi$
$600$	4.00000	0.163299
$601$	28.0000	1.14214	0.571072	−	0.820900i	$-0.306528\pi$
$601$	28.0000	1.14214	0.571072	+	0.820900i	$0.306528\pi$
$602$	0	0
$603$	−2.00000	−0.0814463
$604$	−23.0000	−0.935857
$605$	14.0000	0.569181
$606$	4.00000	0.162489
$607$	43.0000	1.74532	0.872658	−	0.488332i	$-0.162394\pi$
$607$	43.0000	1.74532	0.872658	+	0.488332i	$0.162394\pi$
$608$	1.00000	0.0405554
$609$	0	0
$610$	15.0000	0.607332
$611$	8.00000	0.323645
$612$	3.00000	0.121268
$613$	−9.00000	−0.363507	−0.181753	−	0.983344i	$-0.558177\pi$
$613$	−9.00000	−0.363507	−0.181753	+	0.983344i	$0.558177\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	9.00000	0.362326	0.181163	−	0.983453i	$-0.442014\pi$
$617$	9.00000	0.362326	0.181163	+	0.983453i	$0.442014\pi$
$618$	−15.0000	−0.603388
$619$	17.0000	0.683288	0.341644	−	0.939829i	$-0.389016\pi$
$619$	17.0000	0.683288	0.341644	+	0.939829i	$0.389016\pi$
$620$	−4.00000	−0.160644
$621$	−3.00000	−0.120386
$622$	10.0000	0.400963
$623$	0	0
$624$	−1.00000	−0.0400320
$625$	11.0000	0.440000
$626$	8.00000	0.319744
$627$	−5.00000	−0.199681
$628$	3.00000	0.119713
$629$	−33.0000	−1.31580
$630$	0	0
$631$	37.0000	1.47295	0.736473	−	0.676467i	$-0.236490\pi$
$631$	37.0000	1.47295	0.736473	+	0.676467i	$0.236490\pi$
$632$	10.0000	0.397779
$633$	−9.00000	−0.357718
$634$	24.0000	0.953162
$635$	2.00000	0.0793676
$636$	2.00000	0.0793052
$637$	0	0
$638$	45.0000	1.78157
$639$	−12.0000	−0.474713
$640$	1.00000	0.0395285
$641$	−30.0000	−1.18493	−0.592464	−	0.805597i	$-0.701845\pi$
$641$	−30.0000	−1.18493	−0.592464	+	0.805597i	$0.701845\pi$
$642$	−10.0000	−0.394669
$643$	7.00000	0.276053	0.138027	−	0.990429i	$-0.455924\pi$
$643$	7.00000	0.276053	0.138027	+	0.990429i	$0.455924\pi$
$644$	0	0
$645$	5.00000	0.196875
$646$	3.00000	0.118033
$647$	−32.0000	−1.25805	−0.629025	−	0.777385i	$-0.716546\pi$
$647$	−32.0000	−1.25805	−0.629025	+	0.777385i	$0.716546\pi$
$648$	1.00000	0.0392837
$649$	−20.0000	−0.785069
$650$	−4.00000	−0.156893
$651$	0	0
$652$	10.0000	0.391630
$653$	31.0000	1.21312	0.606562	−	0.795036i	$-0.292548\pi$
$653$	31.0000	1.21312	0.606562	+	0.795036i	$0.292548\pi$
$654$	9.00000	0.351928
$655$	−13.0000	−0.507952
$656$	0	0
$657$	−11.0000	−0.429151
$658$	0	0
$659$	30.0000	1.16863	0.584317	−	0.811525i	$-0.301362\pi$
$659$	30.0000	1.16863	0.584317	+	0.811525i	$0.301362\pi$
$660$	−5.00000	−0.194625
$661$	−46.0000	−1.78919	−0.894596	−	0.446875i	$-0.852537\pi$
$661$	−46.0000	−1.78919	−0.894596	+	0.446875i	$0.852537\pi$
$662$	10.0000	0.388661
$663$	−3.00000	−0.116510
$664$	14.0000	0.543305
$665$	0	0
$666$	−11.0000	−0.426241
$667$	27.0000	1.04544
$668$	21.0000	0.812514
$669$	−14.0000	−0.541271
$670$	−2.00000	−0.0772667
$671$	75.0000	2.89534
$672$	0	0
$673$	−33.0000	−1.27206	−0.636028	−	0.771666i	$-0.719424\pi$
$673$	−33.0000	−1.27206	−0.636028	+	0.771666i	$0.719424\pi$
$674$	−5.00000	−0.192593
$675$	4.00000	0.153960
$676$	1.00000	0.0384615
$677$	−34.0000	−1.30673	−0.653363	−	0.757045i	$-0.726642\pi$
$677$	−34.0000	−1.30673	−0.653363	+	0.757045i	$0.726642\pi$
$678$	−2.00000	−0.0768095
$679$	0	0
$680$	3.00000	0.115045
$681$	18.0000	0.689761
$682$	−20.0000	−0.765840
$683$	−9.00000	−0.344375	−0.172188	−	0.985064i	$-0.555084\pi$
$683$	−9.00000	−0.344375	−0.172188	+	0.985064i	$0.555084\pi$
$684$	1.00000	0.0382360
$685$	19.0000	0.725953
$686$	0	0
$687$	6.00000	0.228914
$688$	−5.00000	−0.190623
$689$	−2.00000	−0.0761939
$690$	−3.00000	−0.114208
$691$	−20.0000	−0.760836	−0.380418	−	0.924815i	$-0.624220\pi$
$691$	−20.0000	−0.760836	−0.380418	+	0.924815i	$0.624220\pi$
$692$	−6.00000	−0.228086
$693$	0	0
$694$	−16.0000	−0.607352
$695$	0	0
$696$	−9.00000	−0.341144
$697$	0	0
$698$	−14.0000	−0.529908
$699$	−24.0000	−0.907763
$700$	0	0
$701$	2.00000	0.0755390	0.0377695	−	0.999286i	$-0.487975\pi$
$701$	2.00000	0.0755390	0.0377695	+	0.999286i	$0.487975\pi$
$702$	−1.00000	−0.0377426
$703$	−11.0000	−0.414873
$704$	5.00000	0.188445
$705$	−8.00000	−0.301297
$706$	−18.0000	−0.677439
$707$	0	0
$708$	4.00000	0.150329
$709$	−18.0000	−0.676004	−0.338002	−	0.941145i	$-0.609751\pi$
$709$	−18.0000	−0.676004	−0.338002	+	0.941145i	$0.609751\pi$
$710$	−12.0000	−0.450352
$711$	10.0000	0.375029
$712$	−6.00000	−0.224860
$713$	−12.0000	−0.449404
$714$	0	0
$715$	5.00000	0.186989
$716$	12.0000	0.448461
$717$	12.0000	0.448148
$718$	−4.00000	−0.149279
$719$	−34.0000	−1.26799	−0.633993	−	0.773339i	$-0.718585\pi$
$719$	−34.0000	−1.26799	−0.633993	+	0.773339i	$0.718585\pi$
$720$	1.00000	0.0372678
$721$	0	0
$722$	−18.0000	−0.669891
$723$	18.0000	0.669427
$724$	−14.0000	−0.520306
$725$	−36.0000	−1.33701
$726$	−14.0000	−0.519589
$727$	35.0000	1.29808	0.649039	−	0.760755i	$-0.275171\pi$
$727$	35.0000	1.29808	0.649039	+	0.760755i	$0.275171\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	−11.0000	−0.407128
$731$	−15.0000	−0.554795
$732$	−15.0000	−0.554416
$733$	−20.0000	−0.738717	−0.369358	−	0.929287i	$-0.620423\pi$
$733$	−20.0000	−0.738717	−0.369358	+	0.929287i	$0.620423\pi$
$734$	−32.0000	−1.18114
$735$	0	0
$736$	3.00000	0.110581
$737$	−10.0000	−0.368355
$738$	0	0
$739$	−2.00000	−0.0735712	−0.0367856	−	0.999323i	$-0.511712\pi$
$739$	−2.00000	−0.0735712	−0.0367856	+	0.999323i	$0.511712\pi$
$740$	−11.0000	−0.404368
$741$	−1.00000	−0.0367359
$742$	0	0
$743$	−26.0000	−0.953847	−0.476924	−	0.878945i	$-0.658248\pi$
$743$	−26.0000	−0.953847	−0.476924	+	0.878945i	$0.658248\pi$
$744$	4.00000	0.146647
$745$	−4.00000	−0.146549
$746$	−4.00000	−0.146450
$747$	14.0000	0.512233
$748$	15.0000	0.548454
$749$	0	0
$750$	9.00000	0.328634
$751$	−32.0000	−1.16770	−0.583848	−	0.811863i	$-0.698454\pi$
$751$	−32.0000	−1.16770	−0.583848	+	0.811863i	$0.698454\pi$
$752$	8.00000	0.291730
$753$	−7.00000	−0.255094
$754$	9.00000	0.327761
$755$	−23.0000	−0.837056
$756$	0	0
$757$	−26.0000	−0.944986	−0.472493	−	0.881334i	$-0.656646\pi$
$757$	−26.0000	−0.944986	−0.472493	+	0.881334i	$0.656646\pi$
$758$	16.0000	0.581146
$759$	−15.0000	−0.544466
$760$	1.00000	0.0362738
$761$	−20.0000	−0.724999	−0.362500	−	0.931984i	$-0.618077\pi$
$761$	−20.0000	−0.724999	−0.362500	+	0.931984i	$0.618077\pi$
$762$	−2.00000	−0.0724524
$763$	0	0
$764$	3.00000	0.108536
$765$	3.00000	0.108465
$766$	29.0000	1.04781
$767$	−4.00000	−0.144432
$768$	−1.00000	−0.0360844
$769$	−49.0000	−1.76699	−0.883493	−	0.468445i	$-0.844814\pi$
$769$	−49.0000	−1.76699	−0.883493	+	0.468445i	$0.844814\pi$
$770$	0	0
$771$	6.00000	0.216085
$772$	12.0000	0.431889
$773$	3.00000	0.107903	0.0539513	−	0.998544i	$-0.482818\pi$
$773$	3.00000	0.107903	0.0539513	+	0.998544i	$0.482818\pi$
$774$	−5.00000	−0.179721
$775$	16.0000	0.574737
$776$	14.0000	0.502571
$777$	0	0
$778$	−2.00000	−0.0717035
$779$	0	0
$780$	−1.00000	−0.0358057
$781$	−60.0000	−2.14697
$782$	9.00000	0.321839
$783$	−9.00000	−0.321634
$784$	0	0
$785$	3.00000	0.107075
$786$	13.0000	0.463695
$787$	−39.0000	−1.39020	−0.695100	−	0.718913i	$-0.744640\pi$
$787$	−39.0000	−1.39020	−0.695100	+	0.718913i	$0.744640\pi$
$788$	−12.0000	−0.427482
$789$	−12.0000	−0.427211
$790$	10.0000	0.355784
$791$	0	0
$792$	5.00000	0.177667
$793$	15.0000	0.532666
$794$	−6.00000	−0.212932
$795$	2.00000	0.0709327
$796$	−11.0000	−0.389885
$797$	0	0		−	1.00000i	$-0.5\pi$
$797$	0	0			1.00000i	$0.5\pi$
$798$	0	0
$799$	24.0000	0.849059
$800$	−4.00000	−0.141421
$801$	−6.00000	−0.212000
$802$	−18.0000	−0.635602
$803$	−55.0000	−1.94091
$804$	2.00000	0.0705346
$805$	0	0
$806$	−4.00000	−0.140894
$807$	18.0000	0.633630
$808$	−4.00000	−0.140720
$809$	54.0000	1.89854	0.949269	−	0.314464i	$-0.101825\pi$
$809$	54.0000	1.89854	0.949269	+	0.314464i	$0.101825\pi$
$810$	1.00000	0.0351364
$811$	−35.0000	−1.22902	−0.614508	−	0.788911i	$-0.710645\pi$
$811$	−35.0000	−1.22902	−0.614508	+	0.788911i	$0.710645\pi$
$812$	0	0
$813$	−22.0000	−0.771574
$814$	−55.0000	−1.92775
$815$	10.0000	0.350285
$816$	−3.00000	−0.105021
$817$	−5.00000	−0.174928
$818$	−25.0000	−0.874105
$819$	0	0
$820$	0	0
$821$	−32.0000	−1.11681	−0.558404	−	0.829569i	$-0.688586\pi$
$821$	−32.0000	−1.11681	−0.558404	+	0.829569i	$0.688586\pi$
$822$	−19.0000	−0.662701
$823$	−16.0000	−0.557725	−0.278862	−	0.960331i	$-0.589957\pi$
$823$	−16.0000	−0.557725	−0.278862	+	0.960331i	$0.589957\pi$
$824$	15.0000	0.522550
$825$	20.0000	0.696311
$826$	0	0
$827$	−19.0000	−0.660695	−0.330347	−	0.943859i	$-0.607166\pi$
$827$	−19.0000	−0.660695	−0.330347	+	0.943859i	$0.607166\pi$
$828$	3.00000	0.104257
$829$	−25.0000	−0.868286	−0.434143	−	0.900844i	$-0.642949\pi$
$829$	−25.0000	−0.868286	−0.434143	+	0.900844i	$0.642949\pi$
$830$	14.0000	0.485947
$831$	−26.0000	−0.901930
$832$	1.00000	0.0346688
$833$	0	0
$834$	0	0
$835$	21.0000	0.726735
$836$	5.00000	0.172929
$837$	4.00000	0.138260
$838$	−7.00000	−0.241811
$839$	28.0000	0.966667	0.483334	−	0.875436i	$-0.339426\pi$
$839$	28.0000	0.966667	0.483334	+	0.875436i	$0.339426\pi$
$840$	0	0
$841$	52.0000	1.79310
$842$	30.0000	1.03387
$843$	26.0000	0.895488
$844$	9.00000	0.309793
$845$	1.00000	0.0344010
$846$	8.00000	0.275046
$847$	0	0
$848$	−2.00000	−0.0686803
$849$	4.00000	0.137280
$850$	−12.0000	−0.411597
$851$	−33.0000	−1.13123
$852$	12.0000	0.411113
$853$	28.0000	0.958702	0.479351	−	0.877623i	$-0.340872\pi$
$853$	28.0000	0.958702	0.479351	+	0.877623i	$0.340872\pi$
$854$	0	0
$855$	1.00000	0.0341993
$856$	10.0000	0.341793
$857$	−18.0000	−0.614868	−0.307434	−	0.951569i	$-0.599470\pi$
$857$	−18.0000	−0.614868	−0.307434	+	0.951569i	$0.599470\pi$
$858$	−5.00000	−0.170697
$859$	8.00000	0.272956	0.136478	−	0.990643i	$-0.456422\pi$
$859$	8.00000	0.272956	0.136478	+	0.990643i	$0.456422\pi$
$860$	−5.00000	−0.170499
$861$	0	0
$862$	−2.00000	−0.0681203
$863$	38.0000	1.29354	0.646768	−	0.762687i	$-0.276120\pi$
$863$	38.0000	1.29354	0.646768	+	0.762687i	$0.276120\pi$
$864$	−1.00000	−0.0340207
$865$	−6.00000	−0.204006
$866$	−28.0000	−0.951479
$867$	8.00000	0.271694
$868$	0	0
$869$	50.0000	1.69613
$870$	−9.00000	−0.305129
$871$	−2.00000	−0.0677674
$872$	−9.00000	−0.304778
$873$	14.0000	0.473828
$874$	3.00000	0.101477
$875$	0	0
$876$	11.0000	0.371656
$877$	−18.0000	−0.607817	−0.303908	−	0.952701i	$-0.598292\pi$
$877$	−18.0000	−0.607817	−0.303908	+	0.952701i	$0.598292\pi$
$878$	−13.0000	−0.438729
$879$	14.0000	0.472208
$880$	5.00000	0.168550
$881$	−21.0000	−0.707508	−0.353754	−	0.935339i	$-0.615095\pi$
$881$	−21.0000	−0.707508	−0.353754	+	0.935339i	$0.615095\pi$
$882$	0	0
$883$	51.0000	1.71629	0.858143	−	0.513410i	$-0.171618\pi$
$883$	51.0000	1.71629	0.858143	+	0.513410i	$0.171618\pi$
$884$	3.00000	0.100901
$885$	4.00000	0.134459
$886$	−18.0000	−0.604722
$887$	−48.0000	−1.61168	−0.805841	−	0.592132i	$-0.798286\pi$
$887$	−48.0000	−1.61168	−0.805841	+	0.592132i	$0.798286\pi$
$888$	11.0000	0.369136
$889$	0	0
$890$	−6.00000	−0.201120
$891$	5.00000	0.167506
$892$	14.0000	0.468755
$893$	8.00000	0.267710
$894$	4.00000	0.133780
$895$	12.0000	0.401116
$896$	0	0
$897$	−3.00000	−0.100167
$898$	23.0000	0.767520
$899$	−36.0000	−1.20067
$900$	−4.00000	−0.133333
$901$	−6.00000	−0.199889
$902$	0	0
$903$	0	0
$904$	2.00000	0.0665190
$905$	−14.0000	−0.465376
$906$	23.0000	0.764124
$907$	12.0000	0.398453	0.199227	−	0.979953i	$-0.436157\pi$
$907$	12.0000	0.398453	0.199227	+	0.979953i	$0.436157\pi$
$908$	−18.0000	−0.597351
$909$	−4.00000	−0.132672
$910$	0	0
$911$	−19.0000	−0.629498	−0.314749	−	0.949175i	$-0.601920\pi$
$911$	−19.0000	−0.629498	−0.314749	+	0.949175i	$0.601920\pi$
$912$	−1.00000	−0.0331133
$913$	70.0000	2.31666
$914$	−32.0000	−1.05847
$915$	−15.0000	−0.495885
$916$	−6.00000	−0.198246
$917$	0	0
$918$	−3.00000	−0.0990148
$919$	38.0000	1.25350	0.626752	−	0.779219i	$-0.284384\pi$
$919$	38.0000	1.25350	0.626752	+	0.779219i	$0.284384\pi$
$920$	3.00000	0.0989071
$921$	0	0
$922$	35.0000	1.15266
$923$	−12.0000	−0.394985
$924$	0	0
$925$	44.0000	1.44671
$926$	9.00000	0.295758
$927$	15.0000	0.492665
$928$	9.00000	0.295439
$929$	8.00000	0.262471	0.131236	−	0.991351i	$-0.458106\pi$
$929$	8.00000	0.262471	0.131236	+	0.991351i	$0.458106\pi$
$930$	4.00000	0.131165
$931$	0	0
$932$	24.0000	0.786146
$933$	−10.0000	−0.327385
$934$	15.0000	0.490815
$935$	15.0000	0.490552
$936$	1.00000	0.0326860
$937$	−14.0000	−0.457360	−0.228680	−	0.973502i	$-0.573441\pi$
$937$	−14.0000	−0.457360	−0.228680	+	0.973502i	$0.573441\pi$
$938$	0	0
$939$	−8.00000	−0.261070
$940$	8.00000	0.260931
$941$	−18.0000	−0.586783	−0.293392	−	0.955992i	$-0.594784\pi$
$941$	−18.0000	−0.586783	−0.293392	+	0.955992i	$0.594784\pi$
$942$	−3.00000	−0.0977453
$943$	0	0
$944$	−4.00000	−0.130189
$945$	0	0
$946$	−25.0000	−0.812820
$947$	−39.0000	−1.26733	−0.633665	−	0.773608i	$-0.718450\pi$
$947$	−39.0000	−1.26733	−0.633665	+	0.773608i	$0.718450\pi$
$948$	−10.0000	−0.324785
$949$	−11.0000	−0.357075
$950$	−4.00000	−0.129777
$951$	−24.0000	−0.778253
$952$	0	0
$953$	16.0000	0.518291	0.259145	−	0.965838i	$-0.416559\pi$
$953$	16.0000	0.518291	0.259145	+	0.965838i	$0.416559\pi$
$954$	−2.00000	−0.0647524
$955$	3.00000	0.0970777
$956$	−12.0000	−0.388108
$957$	−45.0000	−1.45464
$958$	31.0000	1.00156
$959$	0	0
$960$	−1.00000	−0.0322749
$961$	−15.0000	−0.483871
$962$	−11.0000	−0.354654
$963$	10.0000	0.322245
$964$	−18.0000	−0.579741
$965$	12.0000	0.386294
$966$	0	0
$967$	−5.00000	−0.160789	−0.0803946	−	0.996763i	$-0.525618\pi$
$967$	−5.00000	−0.160789	−0.0803946	+	0.996763i	$0.525618\pi$
$968$	14.0000	0.449977
$969$	−3.00000	−0.0963739
$970$	14.0000	0.449513
$971$	8.00000	0.256732	0.128366	−	0.991727i	$-0.459027\pi$
$971$	8.00000	0.256732	0.128366	+	0.991727i	$0.459027\pi$
$972$	−1.00000	−0.0320750
$973$	0	0
$974$	−16.0000	−0.512673
$975$	4.00000	0.128103
$976$	15.0000	0.480138
$977$	−51.0000	−1.63163	−0.815817	−	0.578310i	$-0.803713\pi$
$977$	−51.0000	−1.63163	−0.815817	+	0.578310i	$0.803713\pi$
$978$	−10.0000	−0.319765
$979$	−30.0000	−0.958804
$980$	0	0
$981$	−9.00000	−0.287348
$982$	−26.0000	−0.829693
$983$	−25.0000	−0.797376	−0.398688	−	0.917087i	$-0.630534\pi$
$983$	−25.0000	−0.797376	−0.398688	+	0.917087i	$0.630534\pi$
$984$	0	0
$985$	−12.0000	−0.382352
$986$	27.0000	0.859855
$987$	0	0
$988$	1.00000	0.0318142
$989$	−15.0000	−0.476972
$990$	5.00000	0.158910
$991$	−2.00000	−0.0635321	−0.0317660	−	0.999495i	$-0.510113\pi$
$991$	−2.00000	−0.0635321	−0.0317660	+	0.999495i	$0.510113\pi$
$992$	−4.00000	−0.127000
$993$	−10.0000	−0.317340
$994$	0	0
$995$	−11.0000	−0.348723
$996$	−14.0000	−0.443607
$997$	10.0000	0.316703	0.158352	−	0.987383i	$-0.449382\pi$
$997$	10.0000	0.316703	0.158352	+	0.987383i	$0.449382\pi$
$998$	−4.00000	−0.126618
$999$	11.0000	0.348025

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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3822.2.a.w.1.1		1
7.6	odd	2		546.2.a.f.1.1	✓	1
21.20	even	2		1638.2.a.h.1.1		1
28.27	even	2		4368.2.a.e.1.1		1
91.90	odd	2		7098.2.a.n.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
546.2.a.f.1.1	✓	1	7.6	odd	2
1638.2.a.h.1.1		1	21.20	even	2
3822.2.a.w.1.1		1	1.1	even	1	trivial
4368.2.a.e.1.1		1	28.27	even	2
7098.2.a.n.1.1		1	91.90	odd	2

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Classical	Maass
Hilbert	Bianchi

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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	3822.2.a.w.1.1

Level	$3822$
Weight	$2$
Character	3822.1
Self dual	yes
Analytic conductor	$30.519$
Analytic rank	$0$
Dimension	$1$
CM	no
Inner twists	$1$