LMFDB - Embedded newform 2001.2.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2001,2,Mod(1,2001)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2001, 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("2001.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2001 = 3 \cdot 23 \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2001.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:	$15.9780654445$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	2001.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

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

\(q-1.00000 q^{2} -1.00000 q^{3} -1.00000 q^{4} +3.00000 q^{5} +1.00000 q^{6} +3.00000 q^{8} +1.00000 q^{9} -3.00000 q^{10} +3.00000 q^{11} +1.00000 q^{12} +3.00000 q^{13} -3.00000 q^{15} -1.00000 q^{16} -1.00000 q^{18} +8.00000 q^{19} -3.00000 q^{20} -3.00000 q^{22} -1.00000 q^{23} -3.00000 q^{24} +4.00000 q^{25} -3.00000 q^{26} -1.00000 q^{27} +1.00000 q^{29} +3.00000 q^{30} +7.00000 q^{31} -5.00000 q^{32} -3.00000 q^{33} -1.00000 q^{36} -5.00000 q^{37} -8.00000 q^{38} -3.00000 q^{39} +9.00000 q^{40} +3.00000 q^{41} +2.00000 q^{43} -3.00000 q^{44} +3.00000 q^{45} +1.00000 q^{46} -6.00000 q^{47} +1.00000 q^{48} -7.00000 q^{49} -4.00000 q^{50} -3.00000 q^{52} -2.00000 q^{53} +1.00000 q^{54} +9.00000 q^{55} -8.00000 q^{57} -1.00000 q^{58} +1.00000 q^{59} +3.00000 q^{60} +7.00000 q^{61} -7.00000 q^{62} +7.00000 q^{64} +9.00000 q^{65} +3.00000 q^{66} -9.00000 q^{67} +1.00000 q^{69} +3.00000 q^{71} +3.00000 q^{72} -12.0000 q^{73} +5.00000 q^{74} -4.00000 q^{75} -8.00000 q^{76} +3.00000 q^{78} -14.0000 q^{79} -3.00000 q^{80} +1.00000 q^{81} -3.00000 q^{82} +14.0000 q^{83} -2.00000 q^{86} -1.00000 q^{87} +9.00000 q^{88} -10.0000 q^{89} -3.00000 q^{90} +1.00000 q^{92} -7.00000 q^{93} +6.00000 q^{94} +24.0000 q^{95} +5.00000 q^{96} +14.0000 q^{97} +7.00000 q^{98} +3.00000 q^{99} +O(q^{100})\)

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	−0.353553	−	0.935414i	$-0.615027\pi$
$2$	−1.00000	−0.707107	−0.353553	+	0.935414i	$0.615027\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	−1.00000	−0.500000
$5$	3.00000	1.34164	0.670820	−	0.741620i	$-0.265942\pi$
$5$	3.00000	1.34164	0.670820	+	0.741620i	$0.265942\pi$
$6$	1.00000	0.408248
$7$	0	0		−	1.00000i	$-0.5\pi$
$7$	0	0			1.00000i	$0.5\pi$
$8$	3.00000	1.06066
$9$	1.00000	0.333333
$10$	−3.00000	−0.948683
$11$	3.00000	0.904534	0.452267	−	0.891883i	$-0.350615\pi$
$11$	3.00000	0.904534	0.452267	+	0.891883i	$0.350615\pi$
$12$	1.00000	0.288675
$13$	3.00000	0.832050	0.416025	−	0.909353i	$-0.363423\pi$
$13$	3.00000	0.832050	0.416025	+	0.909353i	$0.363423\pi$
$14$	0	0
$15$	−3.00000	−0.774597
$16$	−1.00000	−0.250000
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	−1.00000	−0.235702
$19$	8.00000	1.83533	0.917663	−	0.397360i	$-0.130073\pi$
$19$	8.00000	1.83533	0.917663	+	0.397360i	$0.130073\pi$
$20$	−3.00000	−0.670820
$21$	0	0
$22$	−3.00000	−0.639602
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	−3.00000	−0.612372
$25$	4.00000	0.800000
$26$	−3.00000	−0.588348
$27$	−1.00000	−0.192450
$28$	0	0
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	3.00000	0.547723
$31$	7.00000	1.25724	0.628619	−	0.777714i	$-0.283621\pi$
$31$	7.00000	1.25724	0.628619	+	0.777714i	$0.283621\pi$
$32$	−5.00000	−0.883883
$33$	−3.00000	−0.522233
$34$	0	0
$35$	0	0
$36$	−1.00000	−0.166667
$37$	−5.00000	−0.821995	−0.410997	−	0.911636i	$-0.634819\pi$
$37$	−5.00000	−0.821995	−0.410997	+	0.911636i	$0.634819\pi$
$38$	−8.00000	−1.29777
$39$	−3.00000	−0.480384
$40$	9.00000	1.42302
$41$	3.00000	0.468521	0.234261	−	0.972174i	$-0.424733\pi$
$41$	3.00000	0.468521	0.234261	+	0.972174i	$0.424733\pi$
$42$	0	0
$43$	2.00000	0.304997	0.152499	−	0.988304i	$-0.451268\pi$
$43$	2.00000	0.304997	0.152499	+	0.988304i	$0.451268\pi$
$44$	−3.00000	−0.452267
$45$	3.00000	0.447214
$46$	1.00000	0.147442
$47$	−6.00000	−0.875190	−0.437595	−	0.899172i	$-0.644170\pi$
$47$	−6.00000	−0.875190	−0.437595	+	0.899172i	$0.644170\pi$
$48$	1.00000	0.144338
$49$	−7.00000	−1.00000
$50$	−4.00000	−0.565685
$51$	0	0
$52$	−3.00000	−0.416025
$53$	−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
$55$	9.00000	1.21356
$56$	0	0
$57$	−8.00000	−1.05963
$58$	−1.00000	−0.131306
$59$	1.00000	0.130189	0.0650945	−	0.997879i	$-0.479265\pi$
$59$	1.00000	0.130189	0.0650945	+	0.997879i	$0.479265\pi$
$60$	3.00000	0.387298
$61$	7.00000	0.896258	0.448129	−	0.893969i	$-0.352090\pi$
$61$	7.00000	0.896258	0.448129	+	0.893969i	$0.352090\pi$
$62$	−7.00000	−0.889001
$63$	0	0
$64$	7.00000	0.875000
$65$	9.00000	1.11631
$66$	3.00000	0.369274
$67$	−9.00000	−1.09952	−0.549762	−	0.835321i	$-0.685282\pi$
$67$	−9.00000	−1.09952	−0.549762	+	0.835321i	$0.685282\pi$
$68$	0	0
$69$	1.00000	0.120386
$70$	0	0
$71$	3.00000	0.356034	0.178017	−	0.984027i	$-0.443032\pi$
$71$	3.00000	0.356034	0.178017	+	0.984027i	$0.443032\pi$
$72$	3.00000	0.353553
$73$	−12.0000	−1.40449	−0.702247	−	0.711934i	$-0.747820\pi$
$73$	−12.0000	−1.40449	−0.702247	+	0.711934i	$0.747820\pi$
$74$	5.00000	0.581238
$75$	−4.00000	−0.461880
$76$	−8.00000	−0.917663
$77$	0	0
$78$	3.00000	0.339683
$79$	−14.0000	−1.57512	−0.787562	−	0.616236i	$-0.788657\pi$
$79$	−14.0000	−1.57512	−0.787562	+	0.616236i	$0.788657\pi$
$80$	−3.00000	−0.335410
$81$	1.00000	0.111111
$82$	−3.00000	−0.331295
$83$	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
$85$	0	0
$86$	−2.00000	−0.215666
$87$	−1.00000	−0.107211
$88$	9.00000	0.959403
$89$	−10.0000	−1.06000	−0.529999	−	0.847998i	$-0.677808\pi$
$89$	−10.0000	−1.06000	−0.529999	+	0.847998i	$0.677808\pi$
$90$	−3.00000	−0.316228
$91$	0	0
$92$	1.00000	0.104257
$93$	−7.00000	−0.725866
$94$	6.00000	0.618853
$95$	24.0000	2.46235
$96$	5.00000	0.510310
$97$	14.0000	1.42148	0.710742	−	0.703452i	$-0.248359\pi$
$97$	14.0000	1.42148	0.710742	+	0.703452i	$0.248359\pi$
$98$	7.00000	0.707107
$99$	3.00000	0.301511
$100$	−4.00000	−0.400000
$101$	5.00000	0.497519	0.248759	−	0.968565i	$-0.419977\pi$
$101$	5.00000	0.497519	0.248759	+	0.968565i	$0.419977\pi$
$102$	0	0
$103$	1.00000	0.0985329	0.0492665	−	0.998786i	$-0.484312\pi$
$103$	1.00000	0.0985329	0.0492665	+	0.998786i	$0.484312\pi$
$104$	9.00000	0.882523
$105$	0	0
$106$	2.00000	0.194257
$107$	8.00000	0.773389	0.386695	−	0.922208i	$-0.373617\pi$
$107$	8.00000	0.773389	0.386695	+	0.922208i	$0.373617\pi$
$108$	1.00000	0.0962250
$109$	2.00000	0.191565	0.0957826	−	0.995402i	$-0.469465\pi$
$109$	2.00000	0.191565	0.0957826	+	0.995402i	$0.469465\pi$
$110$	−9.00000	−0.858116
$111$	5.00000	0.474579
$112$	0	0
$113$	−6.00000	−0.564433	−0.282216	−	0.959351i	$-0.591070\pi$
$113$	−6.00000	−0.564433	−0.282216	+	0.959351i	$0.591070\pi$
$114$	8.00000	0.749269
$115$	−3.00000	−0.279751
$116$	−1.00000	−0.0928477
$117$	3.00000	0.277350
$118$	−1.00000	−0.0920575
$119$	0	0
$120$	−9.00000	−0.821584
$121$	−2.00000	−0.181818
$122$	−7.00000	−0.633750
$123$	−3.00000	−0.270501
$124$	−7.00000	−0.628619
$125$	−3.00000	−0.268328
$126$	0	0
$127$	−5.00000	−0.443678	−0.221839	−	0.975083i	$-0.571206\pi$
$127$	−5.00000	−0.443678	−0.221839	+	0.975083i	$0.571206\pi$
$128$	3.00000	0.265165
$129$	−2.00000	−0.176090
$130$	−9.00000	−0.789352
$131$	12.0000	1.04844	0.524222	−	0.851581i	$-0.324356\pi$
$131$	12.0000	1.04844	0.524222	+	0.851581i	$0.324356\pi$
$132$	3.00000	0.261116
$133$	0	0
$134$	9.00000	0.777482
$135$	−3.00000	−0.258199
$136$	0	0
$137$	8.00000	0.683486	0.341743	−	0.939793i	$-0.388983\pi$
$137$	8.00000	0.683486	0.341743	+	0.939793i	$0.388983\pi$
$138$	−1.00000	−0.0851257
$139$	18.0000	1.52674	0.763370	−	0.645961i	$-0.223543\pi$
$139$	18.0000	1.52674	0.763370	+	0.645961i	$0.223543\pi$
$140$	0	0
$141$	6.00000	0.505291
$142$	−3.00000	−0.251754
$143$	9.00000	0.752618
$144$	−1.00000	−0.0833333
$145$	3.00000	0.249136
$146$	12.0000	0.993127
$147$	7.00000	0.577350
$148$	5.00000	0.410997
$149$	−3.00000	−0.245770	−0.122885	−	0.992421i	$-0.539215\pi$
$149$	−3.00000	−0.245770	−0.122885	+	0.992421i	$0.539215\pi$
$150$	4.00000	0.326599
$151$	−8.00000	−0.651031	−0.325515	−	0.945537i	$-0.605538\pi$
$151$	−8.00000	−0.651031	−0.325515	+	0.945537i	$0.605538\pi$
$152$	24.0000	1.94666
$153$	0	0
$154$	0	0
$155$	21.0000	1.68676
$156$	3.00000	0.240192
$157$	2.00000	0.159617	0.0798087	−	0.996810i	$-0.474569\pi$
$157$	2.00000	0.159617	0.0798087	+	0.996810i	$0.474569\pi$
$158$	14.0000	1.11378
$159$	2.00000	0.158610
$160$	−15.0000	−1.18585
$161$	0	0
$162$	−1.00000	−0.0785674
$163$	−17.0000	−1.33154	−0.665771	−	0.746156i	$-0.731897\pi$
$163$	−17.0000	−1.33154	−0.665771	+	0.746156i	$0.731897\pi$
$164$	−3.00000	−0.234261
$165$	−9.00000	−0.700649
$166$	−14.0000	−1.08661
$167$	−3.00000	−0.232147	−0.116073	−	0.993241i	$-0.537031\pi$
$167$	−3.00000	−0.232147	−0.116073	+	0.993241i	$0.537031\pi$
$168$	0	0
$169$	−4.00000	−0.307692
$170$	0	0
$171$	8.00000	0.611775
$172$	−2.00000	−0.152499
$173$	18.0000	1.36851	0.684257	−	0.729241i	$-0.260127\pi$
$173$	18.0000	1.36851	0.684257	+	0.729241i	$0.260127\pi$
$174$	1.00000	0.0758098
$175$	0	0
$176$	−3.00000	−0.226134
$177$	−1.00000	−0.0751646
$178$	10.0000	0.749532
$179$	20.0000	1.49487	0.747435	−	0.664335i	$-0.231285\pi$
$179$	20.0000	1.49487	0.747435	+	0.664335i	$0.231285\pi$
$180$	−3.00000	−0.223607
$181$	10.0000	0.743294	0.371647	−	0.928374i	$-0.378793\pi$
$181$	10.0000	0.743294	0.371647	+	0.928374i	$0.378793\pi$
$182$	0	0
$183$	−7.00000	−0.517455
$184$	−3.00000	−0.221163
$185$	−15.0000	−1.10282
$186$	7.00000	0.513265
$187$	0	0
$188$	6.00000	0.437595
$189$	0	0
$190$	−24.0000	−1.74114
$191$	3.00000	0.217072	0.108536	−	0.994092i	$-0.465384\pi$
$191$	3.00000	0.217072	0.108536	+	0.994092i	$0.465384\pi$
$192$	−7.00000	−0.505181
$193$	4.00000	0.287926	0.143963	−	0.989583i	$-0.454015\pi$
$193$	4.00000	0.287926	0.143963	+	0.989583i	$0.454015\pi$
$194$	−14.0000	−1.00514
$195$	−9.00000	−0.644503
$196$	7.00000	0.500000
$197$	10.0000	0.712470	0.356235	−	0.934396i	$-0.384060\pi$
$197$	10.0000	0.712470	0.356235	+	0.934396i	$0.384060\pi$
$198$	−3.00000	−0.213201
$199$	21.0000	1.48865	0.744325	−	0.667817i	$-0.232771\pi$
$199$	21.0000	1.48865	0.744325	+	0.667817i	$0.232771\pi$
$200$	12.0000	0.848528
$201$	9.00000	0.634811
$202$	−5.00000	−0.351799
$203$	0	0
$204$	0	0
$205$	9.00000	0.628587
$206$	−1.00000	−0.0696733
$207$	−1.00000	−0.0695048
$208$	−3.00000	−0.208013
$209$	24.0000	1.66011
$210$	0	0
$211$	−3.00000	−0.206529	−0.103264	−	0.994654i	$-0.532929\pi$
$211$	−3.00000	−0.206529	−0.103264	+	0.994654i	$0.532929\pi$
$212$	2.00000	0.137361
$213$	−3.00000	−0.205557
$214$	−8.00000	−0.546869
$215$	6.00000	0.409197
$216$	−3.00000	−0.204124
$217$	0	0
$218$	−2.00000	−0.135457
$219$	12.0000	0.810885
$220$	−9.00000	−0.606780
$221$	0	0
$222$	−5.00000	−0.335578
$223$	8.00000	0.535720	0.267860	−	0.963458i	$-0.413684\pi$
$223$	8.00000	0.535720	0.267860	+	0.963458i	$0.413684\pi$
$224$	0	0
$225$	4.00000	0.266667
$226$	6.00000	0.399114
$227$	6.00000	0.398234	0.199117	−	0.979976i	$-0.436193\pi$
$227$	6.00000	0.398234	0.199117	+	0.979976i	$0.436193\pi$
$228$	8.00000	0.529813
$229$	−21.0000	−1.38772	−0.693860	−	0.720110i	$-0.744091\pi$
$229$	−21.0000	−1.38772	−0.693860	+	0.720110i	$0.744091\pi$
$230$	3.00000	0.197814
$231$	0	0
$232$	3.00000	0.196960
$233$	−28.0000	−1.83434	−0.917170	−	0.398495i	$-0.869533\pi$
$233$	−28.0000	−1.83434	−0.917170	+	0.398495i	$0.869533\pi$
$234$	−3.00000	−0.196116
$235$	−18.0000	−1.17419
$236$	−1.00000	−0.0650945
$237$	14.0000	0.909398
$238$	0	0
$239$	19.0000	1.22901	0.614504	−	0.788914i	$-0.289356\pi$
$239$	19.0000	1.22901	0.614504	+	0.788914i	$0.289356\pi$
$240$	3.00000	0.193649
$241$	12.0000	0.772988	0.386494	−	0.922292i	$-0.373686\pi$
$241$	12.0000	0.772988	0.386494	+	0.922292i	$0.373686\pi$
$242$	2.00000	0.128565
$243$	−1.00000	−0.0641500
$244$	−7.00000	−0.448129
$245$	−21.0000	−1.34164
$246$	3.00000	0.191273
$247$	24.0000	1.52708
$248$	21.0000	1.33350
$249$	−14.0000	−0.887214
$250$	3.00000	0.189737
$251$	−9.00000	−0.568075	−0.284037	−	0.958813i	$-0.591674\pi$
$251$	−9.00000	−0.568075	−0.284037	+	0.958813i	$0.591674\pi$
$252$	0	0
$253$	−3.00000	−0.188608
$254$	5.00000	0.313728
$255$	0	0
$256$	−17.0000	−1.06250
$257$	−6.00000	−0.374270	−0.187135	−	0.982334i	$-0.559920\pi$
$257$	−6.00000	−0.374270	−0.187135	+	0.982334i	$0.559920\pi$
$258$	2.00000	0.124515
$259$	0	0
$260$	−9.00000	−0.558156
$261$	1.00000	0.0618984
$262$	−12.0000	−0.741362
$263$	16.0000	0.986602	0.493301	−	0.869859i	$-0.335790\pi$
$263$	16.0000	0.986602	0.493301	+	0.869859i	$0.335790\pi$
$264$	−9.00000	−0.553912
$265$	−6.00000	−0.368577
$266$	0	0
$267$	10.0000	0.611990
$268$	9.00000	0.549762
$269$	−21.0000	−1.28039	−0.640196	−	0.768211i	$-0.721147\pi$
$269$	−21.0000	−1.28039	−0.640196	+	0.768211i	$0.721147\pi$
$270$	3.00000	0.182574
$271$	−15.0000	−0.911185	−0.455593	−	0.890188i	$-0.650573\pi$
$271$	−15.0000	−0.911185	−0.455593	+	0.890188i	$0.650573\pi$
$272$	0	0
$273$	0	0
$274$	−8.00000	−0.483298
$275$	12.0000	0.723627
$276$	−1.00000	−0.0601929
$277$	−17.0000	−1.02143	−0.510716	−	0.859750i	$-0.670619\pi$
$277$	−17.0000	−1.02143	−0.510716	+	0.859750i	$0.670619\pi$
$278$	−18.0000	−1.07957
$279$	7.00000	0.419079
$280$	0	0
$281$	−18.0000	−1.07379	−0.536895	−	0.843649i	$-0.680403\pi$
$281$	−18.0000	−1.07379	−0.536895	+	0.843649i	$0.680403\pi$
$282$	−6.00000	−0.357295
$283$	23.0000	1.36721	0.683604	−	0.729853i	$-0.260412\pi$
$283$	23.0000	1.36721	0.683604	+	0.729853i	$0.260412\pi$
$284$	−3.00000	−0.178017
$285$	−24.0000	−1.42164
$286$	−9.00000	−0.532181
$287$	0	0
$288$	−5.00000	−0.294628
$289$	−17.0000	−1.00000
$290$	−3.00000	−0.176166
$291$	−14.0000	−0.820695
$292$	12.0000	0.702247
$293$	20.0000	1.16841	0.584206	−	0.811605i	$-0.301406\pi$
$293$	20.0000	1.16841	0.584206	+	0.811605i	$0.301406\pi$
$294$	−7.00000	−0.408248
$295$	3.00000	0.174667
$296$	−15.0000	−0.871857
$297$	−3.00000	−0.174078
$298$	3.00000	0.173785
$299$	−3.00000	−0.173494
$300$	4.00000	0.230940
$301$	0	0
$302$	8.00000	0.460348
$303$	−5.00000	−0.287242
$304$	−8.00000	−0.458831
$305$	21.0000	1.20246
$306$	0	0
$307$	29.0000	1.65512	0.827559	−	0.561379i	$-0.189729\pi$
$307$	29.0000	1.65512	0.827559	+	0.561379i	$0.189729\pi$
$308$	0	0
$309$	−1.00000	−0.0568880
$310$	−21.0000	−1.19272
$311$	6.00000	0.340229	0.170114	−	0.985424i	$-0.445586\pi$
$311$	6.00000	0.340229	0.170114	+	0.985424i	$0.445586\pi$
$312$	−9.00000	−0.509525
$313$	−26.0000	−1.46961	−0.734803	−	0.678280i	$-0.762726\pi$
$313$	−26.0000	−1.46961	−0.734803	+	0.678280i	$0.762726\pi$
$314$	−2.00000	−0.112867
$315$	0	0
$316$	14.0000	0.787562
$317$	−7.00000	−0.393159	−0.196580	−	0.980488i	$-0.562983\pi$
$317$	−7.00000	−0.393159	−0.196580	+	0.980488i	$0.562983\pi$
$318$	−2.00000	−0.112154
$319$	3.00000	0.167968
$320$	21.0000	1.17394
$321$	−8.00000	−0.446516
$322$	0	0
$323$	0	0
$324$	−1.00000	−0.0555556
$325$	12.0000	0.665640
$326$	17.0000	0.941543
$327$	−2.00000	−0.110600
$328$	9.00000	0.496942
$329$	0	0
$330$	9.00000	0.495434
$331$	−28.0000	−1.53902	−0.769510	−	0.638635i	$-0.779499\pi$
$331$	−28.0000	−1.53902	−0.769510	+	0.638635i	$0.779499\pi$
$332$	−14.0000	−0.768350
$333$	−5.00000	−0.273998
$334$	3.00000	0.164153
$335$	−27.0000	−1.47517
$336$	0	0
$337$	−29.0000	−1.57973	−0.789865	−	0.613280i	$-0.789850\pi$
$337$	−29.0000	−1.57973	−0.789865	+	0.613280i	$0.789850\pi$
$338$	4.00000	0.217571
$339$	6.00000	0.325875
$340$	0	0
$341$	21.0000	1.13721
$342$	−8.00000	−0.432590
$343$	0	0
$344$	6.00000	0.323498
$345$	3.00000	0.161515
$346$	−18.0000	−0.967686
$347$	−12.0000	−0.644194	−0.322097	−	0.946707i	$-0.604388\pi$
$347$	−12.0000	−0.644194	−0.322097	+	0.946707i	$0.604388\pi$
$348$	1.00000	0.0536056
$349$	25.0000	1.33822	0.669110	−	0.743164i	$-0.266676\pi$
$349$	25.0000	1.33822	0.669110	+	0.743164i	$0.266676\pi$
$350$	0	0
$351$	−3.00000	−0.160128
$352$	−15.0000	−0.799503
$353$	2.00000	0.106449	0.0532246	−	0.998583i	$-0.483050\pi$
$353$	2.00000	0.106449	0.0532246	+	0.998583i	$0.483050\pi$
$354$	1.00000	0.0531494
$355$	9.00000	0.477670
$356$	10.0000	0.529999
$357$	0	0
$358$	−20.0000	−1.05703
$359$	−24.0000	−1.26667	−0.633336	−	0.773877i	$-0.718315\pi$
$359$	−24.0000	−1.26667	−0.633336	+	0.773877i	$0.718315\pi$
$360$	9.00000	0.474342
$361$	45.0000	2.36842
$362$	−10.0000	−0.525588
$363$	2.00000	0.104973
$364$	0	0
$365$	−36.0000	−1.88433
$366$	7.00000	0.365896
$367$	−34.0000	−1.77479	−0.887393	−	0.461014i	$-0.847486\pi$
$367$	−34.0000	−1.77479	−0.887393	+	0.461014i	$0.847486\pi$
$368$	1.00000	0.0521286
$369$	3.00000	0.156174
$370$	15.0000	0.779813
$371$	0	0
$372$	7.00000	0.362933
$373$	6.00000	0.310668	0.155334	−	0.987862i	$-0.450355\pi$
$373$	6.00000	0.310668	0.155334	+	0.987862i	$0.450355\pi$
$374$	0	0
$375$	3.00000	0.154919
$376$	−18.0000	−0.928279
$377$	3.00000	0.154508
$378$	0	0
$379$	−14.0000	−0.719132	−0.359566	−	0.933120i	$-0.617075\pi$
$379$	−14.0000	−0.719132	−0.359566	+	0.933120i	$0.617075\pi$
$380$	−24.0000	−1.23117
$381$	5.00000	0.256158
$382$	−3.00000	−0.153493
$383$	18.0000	0.919757	0.459879	−	0.887982i	$-0.347893\pi$
$383$	18.0000	0.919757	0.459879	+	0.887982i	$0.347893\pi$
$384$	−3.00000	−0.153093
$385$	0	0
$386$	−4.00000	−0.203595
$387$	2.00000	0.101666
$388$	−14.0000	−0.710742
$389$	20.0000	1.01404	0.507020	−	0.861934i	$-0.330747\pi$
$389$	20.0000	1.01404	0.507020	+	0.861934i	$0.330747\pi$
$390$	9.00000	0.455733
$391$	0	0
$392$	−21.0000	−1.06066
$393$	−12.0000	−0.605320
$394$	−10.0000	−0.503793
$395$	−42.0000	−2.11325
$396$	−3.00000	−0.150756
$397$	−2.00000	−0.100377	−0.0501886	−	0.998740i	$-0.515982\pi$
$397$	−2.00000	−0.100377	−0.0501886	+	0.998740i	$0.515982\pi$
$398$	−21.0000	−1.05263
$399$	0	0
$400$	−4.00000	−0.200000
$401$	−27.0000	−1.34832	−0.674158	−	0.738587i	$-0.735493\pi$
$401$	−27.0000	−1.34832	−0.674158	+	0.738587i	$0.735493\pi$
$402$	−9.00000	−0.448879
$403$	21.0000	1.04608
$404$	−5.00000	−0.248759
$405$	3.00000	0.149071
$406$	0	0
$407$	−15.0000	−0.743522
$408$	0	0
$409$	−38.0000	−1.87898	−0.939490	−	0.342578i	$-0.888700\pi$
$409$	−38.0000	−1.87898	−0.939490	+	0.342578i	$0.888700\pi$
$410$	−9.00000	−0.444478
$411$	−8.00000	−0.394611
$412$	−1.00000	−0.0492665
$413$	0	0
$414$	1.00000	0.0491473
$415$	42.0000	2.06170
$416$	−15.0000	−0.735436
$417$	−18.0000	−0.881464
$418$	−24.0000	−1.17388
$419$	6.00000	0.293119	0.146560	−	0.989202i	$-0.453180\pi$
$419$	6.00000	0.293119	0.146560	+	0.989202i	$0.453180\pi$
$420$	0	0
$421$	9.00000	0.438633	0.219317	−	0.975654i	$-0.429617\pi$
$421$	9.00000	0.438633	0.219317	+	0.975654i	$0.429617\pi$
$422$	3.00000	0.146038
$423$	−6.00000	−0.291730
$424$	−6.00000	−0.291386
$425$	0	0
$426$	3.00000	0.145350
$427$	0	0
$428$	−8.00000	−0.386695
$429$	−9.00000	−0.434524
$430$	−6.00000	−0.289346
$431$	30.0000	1.44505	0.722525	−	0.691345i	$-0.242982\pi$
$431$	30.0000	1.44505	0.722525	+	0.691345i	$0.242982\pi$
$432$	1.00000	0.0481125
$433$	−14.0000	−0.672797	−0.336399	−	0.941720i	$-0.609209\pi$
$433$	−14.0000	−0.672797	−0.336399	+	0.941720i	$0.609209\pi$
$434$	0	0
$435$	−3.00000	−0.143839
$436$	−2.00000	−0.0957826
$437$	−8.00000	−0.382692
$438$	−12.0000	−0.573382
$439$	38.0000	1.81364	0.906821	−	0.421517i	$-0.138502\pi$
$439$	38.0000	1.81364	0.906821	+	0.421517i	$0.138502\pi$
$440$	27.0000	1.28717
$441$	−7.00000	−0.333333
$442$	0	0
$443$	30.0000	1.42534	0.712672	−	0.701498i	$-0.247485\pi$
$443$	30.0000	1.42534	0.712672	+	0.701498i	$0.247485\pi$
$444$	−5.00000	−0.237289
$445$	−30.0000	−1.42214
$446$	−8.00000	−0.378811
$447$	3.00000	0.141895
$448$	0	0
$449$	7.00000	0.330350	0.165175	−	0.986264i	$-0.447181\pi$
$449$	7.00000	0.330350	0.165175	+	0.986264i	$0.447181\pi$
$450$	−4.00000	−0.188562
$451$	9.00000	0.423793
$452$	6.00000	0.282216
$453$	8.00000	0.375873
$454$	−6.00000	−0.281594
$455$	0	0
$456$	−24.0000	−1.12390
$457$	−16.0000	−0.748448	−0.374224	−	0.927338i	$-0.622091\pi$
$457$	−16.0000	−0.748448	−0.374224	+	0.927338i	$0.622091\pi$
$458$	21.0000	0.981266
$459$	0	0
$460$	3.00000	0.139876
$461$	−3.00000	−0.139724	−0.0698620	−	0.997557i	$-0.522256\pi$
$461$	−3.00000	−0.139724	−0.0698620	+	0.997557i	$0.522256\pi$
$462$	0	0
$463$	4.00000	0.185896	0.0929479	−	0.995671i	$-0.470371\pi$
$463$	4.00000	0.185896	0.0929479	+	0.995671i	$0.470371\pi$
$464$	−1.00000	−0.0464238
$465$	−21.0000	−0.973852
$466$	28.0000	1.29707
$467$	−25.0000	−1.15686	−0.578431	−	0.815731i	$-0.696335\pi$
$467$	−25.0000	−1.15686	−0.578431	+	0.815731i	$0.696335\pi$
$468$	−3.00000	−0.138675
$469$	0	0
$470$	18.0000	0.830278
$471$	−2.00000	−0.0921551
$472$	3.00000	0.138086
$473$	6.00000	0.275880
$474$	−14.0000	−0.643041
$475$	32.0000	1.46826
$476$	0	0
$477$	−2.00000	−0.0915737
$478$	−19.0000	−0.869040
$479$	−23.0000	−1.05090	−0.525448	−	0.850825i	$-0.676102\pi$
$479$	−23.0000	−1.05090	−0.525448	+	0.850825i	$0.676102\pi$
$480$	15.0000	0.684653
$481$	−15.0000	−0.683941
$482$	−12.0000	−0.546585
$483$	0	0
$484$	2.00000	0.0909091
$485$	42.0000	1.90712
$486$	1.00000	0.0453609
$487$	−40.0000	−1.81257	−0.906287	−	0.422664i	$-0.861095\pi$
$487$	−40.0000	−1.81257	−0.906287	+	0.422664i	$0.861095\pi$
$488$	21.0000	0.950625
$489$	17.0000	0.768767
$490$	21.0000	0.948683
$491$	30.0000	1.35388	0.676941	−	0.736038i	$-0.263305\pi$
$491$	30.0000	1.35388	0.676941	+	0.736038i	$0.263305\pi$
$492$	3.00000	0.135250
$493$	0	0
$494$	−24.0000	−1.07981
$495$	9.00000	0.404520
$496$	−7.00000	−0.314309
$497$	0	0
$498$	14.0000	0.627355
$499$	14.0000	0.626726	0.313363	−	0.949633i	$-0.398544\pi$
$499$	14.0000	0.626726	0.313363	+	0.949633i	$0.398544\pi$
$500$	3.00000	0.134164
$501$	3.00000	0.134030
$502$	9.00000	0.401690
$503$	−8.00000	−0.356702	−0.178351	−	0.983967i	$-0.557076\pi$
$503$	−8.00000	−0.356702	−0.178351	+	0.983967i	$0.557076\pi$
$504$	0	0
$505$	15.0000	0.667491
$506$	3.00000	0.133366
$507$	4.00000	0.177646
$508$	5.00000	0.221839
$509$	38.0000	1.68432	0.842160	−	0.539227i	$-0.181284\pi$
$509$	38.0000	1.68432	0.842160	+	0.539227i	$0.181284\pi$
$510$	0	0
$511$	0	0
$512$	11.0000	0.486136
$513$	−8.00000	−0.353209
$514$	6.00000	0.264649
$515$	3.00000	0.132196
$516$	2.00000	0.0880451
$517$	−18.0000	−0.791639
$518$	0	0
$519$	−18.0000	−0.790112
$520$	27.0000	1.18403
$521$	−2.00000	−0.0876216	−0.0438108	−	0.999040i	$-0.513950\pi$
$521$	−2.00000	−0.0876216	−0.0438108	+	0.999040i	$0.513950\pi$
$522$	−1.00000	−0.0437688
$523$	11.0000	0.480996	0.240498	−	0.970650i	$-0.422689\pi$
$523$	11.0000	0.480996	0.240498	+	0.970650i	$0.422689\pi$
$524$	−12.0000	−0.524222
$525$	0	0
$526$	−16.0000	−0.697633
$527$	0	0
$528$	3.00000	0.130558
$529$	1.00000	0.0434783
$530$	6.00000	0.260623
$531$	1.00000	0.0433963
$532$	0	0
$533$	9.00000	0.389833
$534$	−10.0000	−0.432742
$535$	24.0000	1.03761
$536$	−27.0000	−1.16622
$537$	−20.0000	−0.863064
$538$	21.0000	0.905374
$539$	−21.0000	−0.904534
$540$	3.00000	0.129099
$541$	−2.00000	−0.0859867	−0.0429934	−	0.999075i	$-0.513689\pi$
$541$	−2.00000	−0.0859867	−0.0429934	+	0.999075i	$0.513689\pi$
$542$	15.0000	0.644305
$543$	−10.0000	−0.429141
$544$	0	0
$545$	6.00000	0.257012
$546$	0	0
$547$	−28.0000	−1.19719	−0.598597	−	0.801050i	$-0.704275\pi$
$547$	−28.0000	−1.19719	−0.598597	+	0.801050i	$0.704275\pi$
$548$	−8.00000	−0.341743
$549$	7.00000	0.298753
$550$	−12.0000	−0.511682
$551$	8.00000	0.340811
$552$	3.00000	0.127688
$553$	0	0
$554$	17.0000	0.722261
$555$	15.0000	0.636715
$556$	−18.0000	−0.763370
$557$	−27.0000	−1.14403	−0.572013	−	0.820244i	$-0.693837\pi$
$557$	−27.0000	−1.14403	−0.572013	+	0.820244i	$0.693837\pi$
$558$	−7.00000	−0.296334
$559$	6.00000	0.253773
$560$	0	0
$561$	0	0
$562$	18.0000	0.759284
$563$	−11.0000	−0.463595	−0.231797	−	0.972764i	$-0.574461\pi$
$563$	−11.0000	−0.463595	−0.231797	+	0.972764i	$0.574461\pi$
$564$	−6.00000	−0.252646
$565$	−18.0000	−0.757266
$566$	−23.0000	−0.966762
$567$	0	0
$568$	9.00000	0.377632
$569$	28.0000	1.17382	0.586911	−	0.809652i	$-0.300344\pi$
$569$	28.0000	1.17382	0.586911	+	0.809652i	$0.300344\pi$
$570$	24.0000	1.00525
$571$	−31.0000	−1.29731	−0.648655	−	0.761083i	$-0.724668\pi$
$571$	−31.0000	−1.29731	−0.648655	+	0.761083i	$0.724668\pi$
$572$	−9.00000	−0.376309
$573$	−3.00000	−0.125327
$574$	0	0
$575$	−4.00000	−0.166812
$576$	7.00000	0.291667
$577$	−28.0000	−1.16566	−0.582828	−	0.812596i	$-0.698054\pi$
$577$	−28.0000	−1.16566	−0.582828	+	0.812596i	$0.698054\pi$
$578$	17.0000	0.707107
$579$	−4.00000	−0.166234
$580$	−3.00000	−0.124568
$581$	0	0
$582$	14.0000	0.580319
$583$	−6.00000	−0.248495
$584$	−36.0000	−1.48969
$585$	9.00000	0.372104
$586$	−20.0000	−0.826192
$587$	40.0000	1.65098	0.825488	−	0.564419i	$-0.190900\pi$
$587$	40.0000	1.65098	0.825488	+	0.564419i	$0.190900\pi$
$588$	−7.00000	−0.288675
$589$	56.0000	2.30744
$590$	−3.00000	−0.123508
$591$	−10.0000	−0.411345
$592$	5.00000	0.205499
$593$	−16.0000	−0.657041	−0.328521	−	0.944497i	$-0.606550\pi$
$593$	−16.0000	−0.657041	−0.328521	+	0.944497i	$0.606550\pi$
$594$	3.00000	0.123091
$595$	0	0
$596$	3.00000	0.122885
$597$	−21.0000	−0.859473
$598$	3.00000	0.122679
$599$	−12.0000	−0.490307	−0.245153	−	0.969484i	$-0.578838\pi$
$599$	−12.0000	−0.490307	−0.245153	+	0.969484i	$0.578838\pi$
$600$	−12.0000	−0.489898
$601$	6.00000	0.244745	0.122373	−	0.992484i	$-0.460950\pi$
$601$	6.00000	0.244745	0.122373	+	0.992484i	$0.460950\pi$
$602$	0	0
$603$	−9.00000	−0.366508
$604$	8.00000	0.325515
$605$	−6.00000	−0.243935
$606$	5.00000	0.203111
$607$	16.0000	0.649420	0.324710	−	0.945814i	$-0.394733\pi$
$607$	16.0000	0.649420	0.324710	+	0.945814i	$0.394733\pi$
$608$	−40.0000	−1.62221
$609$	0	0
$610$	−21.0000	−0.850265
$611$	−18.0000	−0.728202
$612$	0	0
$613$	38.0000	1.53481	0.767403	−	0.641165i	$-0.221549\pi$
$613$	38.0000	1.53481	0.767403	+	0.641165i	$0.221549\pi$
$614$	−29.0000	−1.17034
$615$	−9.00000	−0.362915
$616$	0	0
$617$	−14.0000	−0.563619	−0.281809	−	0.959470i	$-0.590935\pi$
$617$	−14.0000	−0.563619	−0.281809	+	0.959470i	$0.590935\pi$
$618$	1.00000	0.0402259
$619$	14.0000	0.562708	0.281354	−	0.959604i	$-0.409217\pi$
$619$	14.0000	0.562708	0.281354	+	0.959604i	$0.409217\pi$
$620$	−21.0000	−0.843380
$621$	1.00000	0.0401286
$622$	−6.00000	−0.240578
$623$	0	0
$624$	3.00000	0.120096
$625$	−29.0000	−1.16000
$626$	26.0000	1.03917
$627$	−24.0000	−0.958468
$628$	−2.00000	−0.0798087
$629$	0	0
$630$	0	0
$631$	23.0000	0.915616	0.457808	−	0.889051i	$-0.348635\pi$
$631$	23.0000	0.915616	0.457808	+	0.889051i	$0.348635\pi$
$632$	−42.0000	−1.67067
$633$	3.00000	0.119239
$634$	7.00000	0.278006
$635$	−15.0000	−0.595257
$636$	−2.00000	−0.0793052
$637$	−21.0000	−0.832050
$638$	−3.00000	−0.118771
$639$	3.00000	0.118678
$640$	9.00000	0.355756
$641$	12.0000	0.473972	0.236986	−	0.971513i	$-0.423841\pi$
$641$	12.0000	0.473972	0.236986	+	0.971513i	$0.423841\pi$
$642$	8.00000	0.315735
$643$	4.00000	0.157745	0.0788723	−	0.996885i	$-0.474868\pi$
$643$	4.00000	0.157745	0.0788723	+	0.996885i	$0.474868\pi$
$644$	0	0
$645$	−6.00000	−0.236250
$646$	0	0
$647$	−17.0000	−0.668339	−0.334169	−	0.942513i	$-0.608456\pi$
$647$	−17.0000	−0.668339	−0.334169	+	0.942513i	$0.608456\pi$
$648$	3.00000	0.117851
$649$	3.00000	0.117760
$650$	−12.0000	−0.470679
$651$	0	0
$652$	17.0000	0.665771
$653$	23.0000	0.900060	0.450030	−	0.893014i	$-0.351413\pi$
$653$	23.0000	0.900060	0.450030	+	0.893014i	$0.351413\pi$
$654$	2.00000	0.0782062
$655$	36.0000	1.40664
$656$	−3.00000	−0.117130
$657$	−12.0000	−0.468165
$658$	0	0
$659$	28.0000	1.09073	0.545363	−	0.838200i	$-0.316392\pi$
$659$	28.0000	1.09073	0.545363	+	0.838200i	$0.316392\pi$
$660$	9.00000	0.350325
$661$	−44.0000	−1.71140	−0.855701	−	0.517471i	$-0.826874\pi$
$661$	−44.0000	−1.71140	−0.855701	+	0.517471i	$0.826874\pi$
$662$	28.0000	1.08825
$663$	0	0
$664$	42.0000	1.62992
$665$	0	0
$666$	5.00000	0.193746
$667$	−1.00000	−0.0387202
$668$	3.00000	0.116073
$669$	−8.00000	−0.309298
$670$	27.0000	1.04310
$671$	21.0000	0.810696
$672$	0	0
$673$	−6.00000	−0.231283	−0.115642	−	0.993291i	$-0.536892\pi$
$673$	−6.00000	−0.231283	−0.115642	+	0.993291i	$0.536892\pi$
$674$	29.0000	1.11704
$675$	−4.00000	−0.153960
$676$	4.00000	0.153846
$677$	4.00000	0.153732	0.0768662	−	0.997041i	$-0.475509\pi$
$677$	4.00000	0.153732	0.0768662	+	0.997041i	$0.475509\pi$
$678$	−6.00000	−0.230429
$679$	0	0
$680$	0	0
$681$	−6.00000	−0.229920
$682$	−21.0000	−0.804132
$683$	9.00000	0.344375	0.172188	−	0.985064i	$-0.444916\pi$
$683$	9.00000	0.344375	0.172188	+	0.985064i	$0.444916\pi$
$684$	−8.00000	−0.305888
$685$	24.0000	0.916993
$686$	0	0
$687$	21.0000	0.801200
$688$	−2.00000	−0.0762493
$689$	−6.00000	−0.228582
$690$	−3.00000	−0.114208
$691$	34.0000	1.29342	0.646710	−	0.762736i	$-0.276144\pi$
$691$	34.0000	1.29342	0.646710	+	0.762736i	$0.276144\pi$
$692$	−18.0000	−0.684257
$693$	0	0
$694$	12.0000	0.455514
$695$	54.0000	2.04834
$696$	−3.00000	−0.113715
$697$	0	0
$698$	−25.0000	−0.946264
$699$	28.0000	1.05906
$700$	0	0
$701$	−11.0000	−0.415464	−0.207732	−	0.978186i	$-0.566608\pi$
$701$	−11.0000	−0.415464	−0.207732	+	0.978186i	$0.566608\pi$
$702$	3.00000	0.113228
$703$	−40.0000	−1.50863
$704$	21.0000	0.791467
$705$	18.0000	0.677919
$706$	−2.00000	−0.0752710
$707$	0	0
$708$	1.00000	0.0375823
$709$	−28.0000	−1.05156	−0.525781	−	0.850620i	$-0.676227\pi$
$709$	−28.0000	−1.05156	−0.525781	+	0.850620i	$0.676227\pi$
$710$	−9.00000	−0.337764
$711$	−14.0000	−0.525041
$712$	−30.0000	−1.12430
$713$	−7.00000	−0.262152
$714$	0	0
$715$	27.0000	1.00974
$716$	−20.0000	−0.747435
$717$	−19.0000	−0.709568
$718$	24.0000	0.895672
$719$	−15.0000	−0.559406	−0.279703	−	0.960087i	$-0.590236\pi$
$719$	−15.0000	−0.559406	−0.279703	+	0.960087i	$0.590236\pi$
$720$	−3.00000	−0.111803
$721$	0	0
$722$	−45.0000	−1.67473
$723$	−12.0000	−0.446285
$724$	−10.0000	−0.371647
$725$	4.00000	0.148556
$726$	−2.00000	−0.0742270
$727$	−34.0000	−1.26099	−0.630495	−	0.776193i	$-0.717148\pi$
$727$	−34.0000	−1.26099	−0.630495	+	0.776193i	$0.717148\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	36.0000	1.33242
$731$	0	0
$732$	7.00000	0.258727
$733$	−25.0000	−0.923396	−0.461698	−	0.887037i	$-0.652760\pi$
$733$	−25.0000	−0.923396	−0.461698	+	0.887037i	$0.652760\pi$
$734$	34.0000	1.25496
$735$	21.0000	0.774597
$736$	5.00000	0.184302
$737$	−27.0000	−0.994558
$738$	−3.00000	−0.110432
$739$	−15.0000	−0.551784	−0.275892	−	0.961189i	$-0.588973\pi$
$739$	−15.0000	−0.551784	−0.275892	+	0.961189i	$0.588973\pi$
$740$	15.0000	0.551411
$741$	−24.0000	−0.881662
$742$	0	0
$743$	−9.00000	−0.330178	−0.165089	−	0.986279i	$-0.552791\pi$
$743$	−9.00000	−0.330178	−0.165089	+	0.986279i	$0.552791\pi$
$744$	−21.0000	−0.769897
$745$	−9.00000	−0.329734
$746$	−6.00000	−0.219676
$747$	14.0000	0.512233
$748$	0	0
$749$	0	0
$750$	−3.00000	−0.109545
$751$	−12.0000	−0.437886	−0.218943	−	0.975738i	$-0.570261\pi$
$751$	−12.0000	−0.437886	−0.218943	+	0.975738i	$0.570261\pi$
$752$	6.00000	0.218797
$753$	9.00000	0.327978
$754$	−3.00000	−0.109254
$755$	−24.0000	−0.873449
$756$	0	0
$757$	−5.00000	−0.181728	−0.0908640	−	0.995863i	$-0.528963\pi$
$757$	−5.00000	−0.181728	−0.0908640	+	0.995863i	$0.528963\pi$
$758$	14.0000	0.508503
$759$	3.00000	0.108893
$760$	72.0000	2.61171
$761$	50.0000	1.81250	0.906249	−	0.422744i	$-0.138933\pi$
$761$	50.0000	1.81250	0.906249	+	0.422744i	$0.138933\pi$
$762$	−5.00000	−0.181131
$763$	0	0
$764$	−3.00000	−0.108536
$765$	0	0
$766$	−18.0000	−0.650366
$767$	3.00000	0.108324
$768$	17.0000	0.613435
$769$	27.0000	0.973645	0.486822	−	0.873501i	$-0.338156\pi$
$769$	27.0000	0.973645	0.486822	+	0.873501i	$0.338156\pi$
$770$	0	0
$771$	6.00000	0.216085
$772$	−4.00000	−0.143963
$773$	20.0000	0.719350	0.359675	−	0.933078i	$-0.382888\pi$
$773$	20.0000	0.719350	0.359675	+	0.933078i	$0.382888\pi$
$774$	−2.00000	−0.0718885
$775$	28.0000	1.00579
$776$	42.0000	1.50771
$777$	0	0
$778$	−20.0000	−0.717035
$779$	24.0000	0.859889
$780$	9.00000	0.322252
$781$	9.00000	0.322045
$782$	0	0
$783$	−1.00000	−0.0357371
$784$	7.00000	0.250000
$785$	6.00000	0.214149
$786$	12.0000	0.428026
$787$	17.0000	0.605985	0.302992	−	0.952993i	$-0.402014\pi$
$787$	17.0000	0.605985	0.302992	+	0.952993i	$0.402014\pi$
$788$	−10.0000	−0.356235
$789$	−16.0000	−0.569615
$790$	42.0000	1.49429
$791$	0	0
$792$	9.00000	0.319801
$793$	21.0000	0.745732
$794$	2.00000	0.0709773
$795$	6.00000	0.212798
$796$	−21.0000	−0.744325
$797$	−18.0000	−0.637593	−0.318796	−	0.947823i	$-0.603279\pi$
$797$	−18.0000	−0.637593	−0.318796	+	0.947823i	$0.603279\pi$
$798$	0	0
$799$	0	0
$800$	−20.0000	−0.707107
$801$	−10.0000	−0.353333
$802$	27.0000	0.953403
$803$	−36.0000	−1.27041
$804$	−9.00000	−0.317406
$805$	0	0
$806$	−21.0000	−0.739693
$807$	21.0000	0.739235
$808$	15.0000	0.527698
$809$	−31.0000	−1.08990	−0.544951	−	0.838468i	$-0.683452\pi$
$809$	−31.0000	−1.08990	−0.544951	+	0.838468i	$0.683452\pi$
$810$	−3.00000	−0.105409
$811$	12.0000	0.421377	0.210688	−	0.977553i	$-0.432429\pi$
$811$	12.0000	0.421377	0.210688	+	0.977553i	$0.432429\pi$
$812$	0	0
$813$	15.0000	0.526073
$814$	15.0000	0.525750
$815$	−51.0000	−1.78645
$816$	0	0
$817$	16.0000	0.559769
$818$	38.0000	1.32864
$819$	0	0
$820$	−9.00000	−0.314294
$821$	24.0000	0.837606	0.418803	−	0.908077i	$-0.362450\pi$
$821$	24.0000	0.837606	0.418803	+	0.908077i	$0.362450\pi$
$822$	8.00000	0.279032
$823$	−9.00000	−0.313720	−0.156860	−	0.987621i	$-0.550137\pi$
$823$	−9.00000	−0.313720	−0.156860	+	0.987621i	$0.550137\pi$
$824$	3.00000	0.104510
$825$	−12.0000	−0.417786
$826$	0	0
$827$	−35.0000	−1.21707	−0.608535	−	0.793527i	$-0.708242\pi$
$827$	−35.0000	−1.21707	−0.608535	+	0.793527i	$0.708242\pi$
$828$	1.00000	0.0347524
$829$	24.0000	0.833554	0.416777	−	0.909009i	$-0.363160\pi$
$829$	24.0000	0.833554	0.416777	+	0.909009i	$0.363160\pi$
$830$	−42.0000	−1.45784
$831$	17.0000	0.589723
$832$	21.0000	0.728044
$833$	0	0
$834$	18.0000	0.623289
$835$	−9.00000	−0.311458
$836$	−24.0000	−0.830057
$837$	−7.00000	−0.241955
$838$	−6.00000	−0.207267
$839$	33.0000	1.13929	0.569643	−	0.821892i	$-0.307081\pi$
$839$	33.0000	1.13929	0.569643	+	0.821892i	$0.307081\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	−9.00000	−0.310160
$843$	18.0000	0.619953
$844$	3.00000	0.103264
$845$	−12.0000	−0.412813
$846$	6.00000	0.206284
$847$	0	0
$848$	2.00000	0.0686803
$849$	−23.0000	−0.789358
$850$	0	0
$851$	5.00000	0.171398
$852$	3.00000	0.102778
$853$	40.0000	1.36957	0.684787	−	0.728743i	$-0.259895\pi$
$853$	40.0000	1.36957	0.684787	+	0.728743i	$0.259895\pi$
$854$	0	0
$855$	24.0000	0.820783
$856$	24.0000	0.820303
$857$	54.0000	1.84460	0.922302	−	0.386469i	$-0.126305\pi$
$857$	54.0000	1.84460	0.922302	+	0.386469i	$0.126305\pi$
$858$	9.00000	0.307255
$859$	20.0000	0.682391	0.341196	−	0.939992i	$-0.389168\pi$
$859$	20.0000	0.682391	0.341196	+	0.939992i	$0.389168\pi$
$860$	−6.00000	−0.204598
$861$	0	0
$862$	−30.0000	−1.02180
$863$	21.0000	0.714848	0.357424	−	0.933942i	$-0.383655\pi$
$863$	21.0000	0.714848	0.357424	+	0.933942i	$0.383655\pi$
$864$	5.00000	0.170103
$865$	54.0000	1.83606
$866$	14.0000	0.475739
$867$	17.0000	0.577350
$868$	0	0
$869$	−42.0000	−1.42475
$870$	3.00000	0.101710
$871$	−27.0000	−0.914860
$872$	6.00000	0.203186
$873$	14.0000	0.473828
$874$	8.00000	0.270604
$875$	0	0
$876$	−12.0000	−0.405442
$877$	−58.0000	−1.95852	−0.979260	−	0.202606i	$-0.935059\pi$
$877$	−58.0000	−1.95852	−0.979260	+	0.202606i	$0.935059\pi$
$878$	−38.0000	−1.28244
$879$	−20.0000	−0.674583
$880$	−9.00000	−0.303390
$881$	−54.0000	−1.81931	−0.909653	−	0.415369i	$-0.863653\pi$
$881$	−54.0000	−1.81931	−0.909653	+	0.415369i	$0.863653\pi$
$882$	7.00000	0.235702
$883$	−8.00000	−0.269221	−0.134611	−	0.990899i	$-0.542978\pi$
$883$	−8.00000	−0.269221	−0.134611	+	0.990899i	$0.542978\pi$
$884$	0	0
$885$	−3.00000	−0.100844
$886$	−30.0000	−1.00787
$887$	−8.00000	−0.268614	−0.134307	−	0.990940i	$-0.542881\pi$
$887$	−8.00000	−0.268614	−0.134307	+	0.990940i	$0.542881\pi$
$888$	15.0000	0.503367
$889$	0	0
$890$	30.0000	1.00560
$891$	3.00000	0.100504
$892$	−8.00000	−0.267860
$893$	−48.0000	−1.60626
$894$	−3.00000	−0.100335
$895$	60.0000	2.00558
$896$	0	0
$897$	3.00000	0.100167
$898$	−7.00000	−0.233593
$899$	7.00000	0.233463
$900$	−4.00000	−0.133333
$901$	0	0
$902$	−9.00000	−0.299667
$903$	0	0
$904$	−18.0000	−0.598671
$905$	30.0000	0.997234
$906$	−8.00000	−0.265782
$907$	10.0000	0.332045	0.166022	−	0.986122i	$-0.446908\pi$
$907$	10.0000	0.332045	0.166022	+	0.986122i	$0.446908\pi$
$908$	−6.00000	−0.199117
$909$	5.00000	0.165840
$910$	0	0
$911$	−9.00000	−0.298183	−0.149092	−	0.988823i	$-0.547635\pi$
$911$	−9.00000	−0.298183	−0.149092	+	0.988823i	$0.547635\pi$
$912$	8.00000	0.264906
$913$	42.0000	1.39000
$914$	16.0000	0.529233
$915$	−21.0000	−0.694239
$916$	21.0000	0.693860
$917$	0	0
$918$	0	0
$919$	−55.0000	−1.81428	−0.907141	−	0.420826i	$-0.861740\pi$
$919$	−55.0000	−1.81428	−0.907141	+	0.420826i	$0.861740\pi$
$920$	−9.00000	−0.296721
$921$	−29.0000	−0.955582
$922$	3.00000	0.0987997
$923$	9.00000	0.296239
$924$	0	0
$925$	−20.0000	−0.657596
$926$	−4.00000	−0.131448
$927$	1.00000	0.0328443
$928$	−5.00000	−0.164133
$929$	−38.0000	−1.24674	−0.623370	−	0.781927i	$-0.714237\pi$
$929$	−38.0000	−1.24674	−0.623370	+	0.781927i	$0.714237\pi$
$930$	21.0000	0.688617
$931$	−56.0000	−1.83533
$932$	28.0000	0.917170
$933$	−6.00000	−0.196431
$934$	25.0000	0.818025
$935$	0	0
$936$	9.00000	0.294174
$937$	−2.00000	−0.0653372	−0.0326686	−	0.999466i	$-0.510401\pi$
$937$	−2.00000	−0.0653372	−0.0326686	+	0.999466i	$0.510401\pi$
$938$	0	0
$939$	26.0000	0.848478
$940$	18.0000	0.587095
$941$	11.0000	0.358590	0.179295	−	0.983795i	$-0.442618\pi$
$941$	11.0000	0.358590	0.179295	+	0.983795i	$0.442618\pi$
$942$	2.00000	0.0651635
$943$	−3.00000	−0.0976934
$944$	−1.00000	−0.0325472
$945$	0	0
$946$	−6.00000	−0.195077
$947$	52.0000	1.68977	0.844886	−	0.534946i	$-0.179668\pi$
$947$	52.0000	1.68977	0.844886	+	0.534946i	$0.179668\pi$
$948$	−14.0000	−0.454699
$949$	−36.0000	−1.16861
$950$	−32.0000	−1.03822
$951$	7.00000	0.226991
$952$	0	0
$953$	−6.00000	−0.194359	−0.0971795	−	0.995267i	$-0.530982\pi$
$953$	−6.00000	−0.194359	−0.0971795	+	0.995267i	$0.530982\pi$
$954$	2.00000	0.0647524
$955$	9.00000	0.291233
$956$	−19.0000	−0.614504
$957$	−3.00000	−0.0969762
$958$	23.0000	0.743096
$959$	0	0
$960$	−21.0000	−0.677772
$961$	18.0000	0.580645
$962$	15.0000	0.483619
$963$	8.00000	0.257796
$964$	−12.0000	−0.386494
$965$	12.0000	0.386294
$966$	0	0
$967$	−32.0000	−1.02905	−0.514525	−	0.857475i	$-0.672032\pi$
$967$	−32.0000	−1.02905	−0.514525	+	0.857475i	$0.672032\pi$
$968$	−6.00000	−0.192847
$969$	0	0
$970$	−42.0000	−1.34854
$971$	1.00000	0.0320915	0.0160458	−	0.999871i	$-0.494892\pi$
$971$	1.00000	0.0320915	0.0160458	+	0.999871i	$0.494892\pi$
$972$	1.00000	0.0320750
$973$	0	0
$974$	40.0000	1.28168
$975$	−12.0000	−0.384308
$976$	−7.00000	−0.224065
$977$	−7.00000	−0.223950	−0.111975	−	0.993711i	$-0.535718\pi$
$977$	−7.00000	−0.223950	−0.111975	+	0.993711i	$0.535718\pi$
$978$	−17.0000	−0.543600
$979$	−30.0000	−0.958804
$980$	21.0000	0.670820
$981$	2.00000	0.0638551
$982$	−30.0000	−0.957338
$983$	−49.0000	−1.56286	−0.781429	−	0.623995i	$-0.785509\pi$
$983$	−49.0000	−1.56286	−0.781429	+	0.623995i	$0.785509\pi$
$984$	−9.00000	−0.286910
$985$	30.0000	0.955879
$986$	0	0
$987$	0	0
$988$	−24.0000	−0.763542
$989$	−2.00000	−0.0635963
$990$	−9.00000	−0.286039
$991$	20.0000	0.635321	0.317660	−	0.948205i	$-0.397103\pi$
$991$	20.0000	0.635321	0.317660	+	0.948205i	$0.397103\pi$
$992$	−35.0000	−1.11125
$993$	28.0000	0.888553
$994$	0	0
$995$	63.0000	1.99723
$996$	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	$0.449382\pi$
$998$	−14.0000	−0.443162
$999$	5.00000	0.158193

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2001.2.a.a.1.1	✓	1
3.2	odd	2		6003.2.a.c.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2001.2.a.a.1.1	✓	1	1.1	even	1	trivial
6003.2.a.c.1.1		1	3.2	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:	\( N \)	\(=\)	\( 2001 = 3 \cdot 23 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2001.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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