LMFDB - Embedded newform 4033.2.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4033,2,Mod(1,4033)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4033, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("4033.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4033 = 37 \cdot 109 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4033.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:	$32.2036671352$
Analytic rank:	$1$
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	$\chi$	$=$	4033.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-1.00000 q^{2} -1.00000 q^{4} +2.00000 q^{5} +3.00000 q^{8} -3.00000 q^{9} +O(q^{10})$

\(q-1.00000 q^{2} -1.00000 q^{4} +2.00000 q^{5} +3.00000 q^{8} -3.00000 q^{9} -2.00000 q^{10} +2.00000 q^{11} -2.00000 q^{13} -1.00000 q^{16} +6.00000 q^{17} +3.00000 q^{18} -4.00000 q^{19} -2.00000 q^{20} -2.00000 q^{22} -4.00000 q^{23} -1.00000 q^{25} +2.00000 q^{26} -6.00000 q^{29} +10.0000 q^{31} -5.00000 q^{32} -6.00000 q^{34} +3.00000 q^{36} -1.00000 q^{37} +4.00000 q^{38} +6.00000 q^{40} +10.0000 q^{41} -10.0000 q^{43} -2.00000 q^{44} -6.00000 q^{45} +4.00000 q^{46} +2.00000 q^{47} -7.00000 q^{49} +1.00000 q^{50} +2.00000 q^{52} +2.00000 q^{53} +4.00000 q^{55} +6.00000 q^{58} -12.0000 q^{59} -14.0000 q^{61} -10.0000 q^{62} +7.00000 q^{64} -4.00000 q^{65} +2.00000 q^{67} -6.00000 q^{68} +12.0000 q^{71} -9.00000 q^{72} -2.00000 q^{73} +1.00000 q^{74} +4.00000 q^{76} -8.00000 q^{79} -2.00000 q^{80} +9.00000 q^{81} -10.0000 q^{82} +16.0000 q^{83} +12.0000 q^{85} +10.0000 q^{86} +6.00000 q^{88} -6.00000 q^{89} +6.00000 q^{90} +4.00000 q^{92} -2.00000 q^{94} -8.00000 q^{95} +2.00000 q^{97} +7.00000 q^{98} -6.00000 q^{99} +O(q^{100})\)

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	$0.615027\pi$
$3$	0	0		−	1.00000i	$-0.5\pi$
$3$	0	0			1.00000i	$0.5\pi$
$4$	−1.00000	−0.500000
$5$	2.00000	0.894427	0.447214	−	0.894427i	$-0.352416\pi$
$5$	2.00000	0.894427	0.447214	+	0.894427i	$0.352416\pi$
$6$	0	0
$7$	0	0		−	1.00000i	$-0.5\pi$
$7$	0	0			1.00000i	$0.5\pi$
$8$	3.00000	1.06066
$9$	−3.00000	−1.00000
$10$	−2.00000	−0.632456
$11$	2.00000	0.603023	0.301511	−	0.953463i	$-0.402509\pi$
$11$	2.00000	0.603023	0.301511	+	0.953463i	$0.402509\pi$
$12$	0	0
$13$	−2.00000	−0.554700	−0.277350	−	0.960769i	$-0.589456\pi$
$13$	−2.00000	−0.554700	−0.277350	+	0.960769i	$0.589456\pi$
$14$	0	0
$15$	0	0
$16$	−1.00000	−0.250000
$17$	6.00000	1.45521	0.727607	−	0.685994i	$-0.240633\pi$
$17$	6.00000	1.45521	0.727607	+	0.685994i	$0.240633\pi$
$18$	3.00000	0.707107
$19$	−4.00000	−0.917663	−0.458831	−	0.888523i	$-0.651732\pi$
$19$	−4.00000	−0.917663	−0.458831	+	0.888523i	$0.651732\pi$
$20$	−2.00000	−0.447214
$21$	0	0
$22$	−2.00000	−0.426401
$23$	−4.00000	−0.834058	−0.417029	−	0.908893i	$-0.636929\pi$
$23$	−4.00000	−0.834058	−0.417029	+	0.908893i	$0.636929\pi$
$24$	0	0
$25$	−1.00000	−0.200000
$26$	2.00000	0.392232
$27$	0	0
$28$	0	0
$29$	−6.00000	−1.11417	−0.557086	−	0.830455i	$-0.688081\pi$
$29$	−6.00000	−1.11417	−0.557086	+	0.830455i	$0.688081\pi$
$30$	0	0
$31$	10.0000	1.79605	0.898027	−	0.439941i	$-0.145001\pi$
$31$	10.0000	1.79605	0.898027	+	0.439941i	$0.145001\pi$
$32$	−5.00000	−0.883883
$33$	0	0
$34$	−6.00000	−1.02899
$35$	0	0
$36$	3.00000	0.500000
$37$	−1.00000	−0.164399
$37$	−1.00000	−0.164399
$38$	4.00000	0.648886
$39$	0	0
$40$	6.00000	0.948683
$41$	10.0000	1.56174	0.780869	−	0.624695i	$-0.214777\pi$
$41$	10.0000	1.56174	0.780869	+	0.624695i	$0.214777\pi$
$42$	0	0
$43$	−10.0000	−1.52499	−0.762493	−	0.646997i	$-0.776025\pi$
$43$	−10.0000	−1.52499	−0.762493	+	0.646997i	$0.776025\pi$
$44$	−2.00000	−0.301511
$45$	−6.00000	−0.894427
$46$	4.00000	0.589768
$47$	2.00000	0.291730	0.145865	−	0.989305i	$-0.453403\pi$
$47$	2.00000	0.291730	0.145865	+	0.989305i	$0.453403\pi$
$48$	0	0
$49$	−7.00000	−1.00000
$50$	1.00000	0.141421
$51$	0	0
$52$	2.00000	0.277350
$53$	2.00000	0.274721	0.137361	−	0.990521i	$-0.456138\pi$
$53$	2.00000	0.274721	0.137361	+	0.990521i	$0.456138\pi$
$54$	0	0
$55$	4.00000	0.539360
$56$	0	0
$57$	0	0
$58$	6.00000	0.787839
$59$	−12.0000	−1.56227	−0.781133	−	0.624364i	$-0.785358\pi$
$59$	−12.0000	−1.56227	−0.781133	+	0.624364i	$0.785358\pi$
$60$	0	0
$61$	−14.0000	−1.79252	−0.896258	−	0.443533i	$-0.853725\pi$
$61$	−14.0000	−1.79252	−0.896258	+	0.443533i	$0.853725\pi$
$62$	−10.0000	−1.27000
$63$	0	0
$64$	7.00000	0.875000
$65$	−4.00000	−0.496139
$66$	0	0
$67$	2.00000	0.244339	0.122169	−	0.992509i	$-0.461015\pi$
$67$	2.00000	0.244339	0.122169	+	0.992509i	$0.461015\pi$
$68$	−6.00000	−0.727607
$69$	0	0
$70$	0	0
$71$	12.0000	1.42414	0.712069	−	0.702109i	$-0.247758\pi$
$71$	12.0000	1.42414	0.712069	+	0.702109i	$0.247758\pi$
$72$	−9.00000	−1.06066
$73$	−2.00000	−0.234082	−0.117041	−	0.993127i	$-0.537341\pi$
$73$	−2.00000	−0.234082	−0.117041	+	0.993127i	$0.537341\pi$
$74$	1.00000	0.116248
$75$	0	0
$76$	4.00000	0.458831
$77$	0	0
$78$	0	0
$79$	−8.00000	−0.900070	−0.450035	−	0.893011i	$-0.648589\pi$
$79$	−8.00000	−0.900070	−0.450035	+	0.893011i	$0.648589\pi$
$80$	−2.00000	−0.223607
$81$	9.00000	1.00000
$82$	−10.0000	−1.10432
$83$	16.0000	1.75623	0.878114	−	0.478451i	$-0.158802\pi$
$83$	16.0000	1.75623	0.878114	+	0.478451i	$0.158802\pi$
$84$	0	0
$85$	12.0000	1.30158
$86$	10.0000	1.07833
$87$	0	0
$88$	6.00000	0.639602
$89$	−6.00000	−0.635999	−0.317999	−	0.948091i	$-0.603011\pi$
$89$	−6.00000	−0.635999	−0.317999	+	0.948091i	$0.603011\pi$
$90$	6.00000	0.632456
$91$	0	0
$92$	4.00000	0.417029
$93$	0	0
$94$	−2.00000	−0.206284
$95$	−8.00000	−0.820783
$96$	0	0
$97$	2.00000	0.203069	0.101535	−	0.994832i	$-0.467625\pi$
$97$	2.00000	0.203069	0.101535	+	0.994832i	$0.467625\pi$
$98$	7.00000	0.707107
$99$	−6.00000	−0.603023
$100$	1.00000	0.100000
$101$	2.00000	0.199007	0.0995037	−	0.995037i	$-0.468274\pi$
$101$	2.00000	0.199007	0.0995037	+	0.995037i	$0.468274\pi$
$102$	0	0
$103$	−20.0000	−1.97066	−0.985329	−	0.170664i	$-0.945409\pi$
$103$	−20.0000	−1.97066	−0.985329	+	0.170664i	$0.945409\pi$
$104$	−6.00000	−0.588348
$105$	0	0
$106$	−2.00000	−0.194257
$107$	6.00000	0.580042	0.290021	−	0.957020i	$-0.406338\pi$
$107$	6.00000	0.580042	0.290021	+	0.957020i	$0.406338\pi$
$108$	0	0
$109$	−1.00000	−0.0957826
$109$	−1.00000	−0.0957826
$110$	−4.00000	−0.381385
$111$	0	0
$112$	0	0
$113$	18.0000	1.69330	0.846649	−	0.532152i	$-0.178617\pi$
$113$	18.0000	1.69330	0.846649	+	0.532152i	$0.178617\pi$
$114$	0	0
$115$	−8.00000	−0.746004
$116$	6.00000	0.557086
$117$	6.00000	0.554700
$118$	12.0000	1.10469
$119$	0	0
$120$	0	0
$121$	−7.00000	−0.636364
$122$	14.0000	1.26750
$123$	0	0
$124$	−10.0000	−0.898027
$125$	−12.0000	−1.07331
$126$	0	0
$127$	−14.0000	−1.24230	−0.621150	−	0.783692i	$-0.713334\pi$
$127$	−14.0000	−1.24230	−0.621150	+	0.783692i	$0.713334\pi$
$128$	3.00000	0.265165
$129$	0	0
$130$	4.00000	0.350823
$131$	−2.00000	−0.174741	−0.0873704	−	0.996176i	$-0.527846\pi$
$131$	−2.00000	−0.174741	−0.0873704	+	0.996176i	$0.527846\pi$
$132$	0	0
$133$	0	0
$134$	−2.00000	−0.172774
$135$	0	0
$136$	18.0000	1.54349
$137$	−18.0000	−1.53784	−0.768922	−	0.639343i	$-0.779207\pi$
$137$	−18.0000	−1.53784	−0.768922	+	0.639343i	$0.779207\pi$
$138$	0	0
$139$	−2.00000	−0.169638	−0.0848189	−	0.996396i	$-0.527031\pi$
$139$	−2.00000	−0.169638	−0.0848189	+	0.996396i	$0.527031\pi$
$140$	0	0
$141$	0	0
$142$	−12.0000	−1.00702
$143$	−4.00000	−0.334497
$144$	3.00000	0.250000
$145$	−12.0000	−0.996546
$146$	2.00000	0.165521
$147$	0	0
$148$	1.00000	0.0821995
$149$	10.0000	0.819232	0.409616	−	0.912258i	$-0.365663\pi$
$149$	10.0000	0.819232	0.409616	+	0.912258i	$0.365663\pi$
$150$	0	0
$151$	6.00000	0.488273	0.244137	−	0.969741i	$-0.421495\pi$
$151$	6.00000	0.488273	0.244137	+	0.969741i	$0.421495\pi$
$152$	−12.0000	−0.973329
$153$	−18.0000	−1.45521
$154$	0	0
$155$	20.0000	1.60644
$156$	0	0
$157$	−18.0000	−1.43656	−0.718278	−	0.695756i	$-0.755069\pi$
$157$	−18.0000	−1.43656	−0.718278	+	0.695756i	$0.755069\pi$
$158$	8.00000	0.636446
$159$	0	0
$160$	−10.0000	−0.790569
$161$	0	0
$162$	−9.00000	−0.707107
$163$	−20.0000	−1.56652	−0.783260	−	0.621694i	$-0.786445\pi$
$163$	−20.0000	−1.56652	−0.783260	+	0.621694i	$0.786445\pi$
$164$	−10.0000	−0.780869
$165$	0	0
$166$	−16.0000	−1.24184
$167$	8.00000	0.619059	0.309529	−	0.950890i	$-0.399829\pi$
$167$	8.00000	0.619059	0.309529	+	0.950890i	$0.399829\pi$
$168$	0	0
$169$	−9.00000	−0.692308
$170$	−12.0000	−0.920358
$171$	12.0000	0.917663
$172$	10.0000	0.762493
$173$	18.0000	1.36851	0.684257	−	0.729241i	$-0.260127\pi$
$173$	18.0000	1.36851	0.684257	+	0.729241i	$0.260127\pi$
$174$	0	0
$175$	0	0
$176$	−2.00000	−0.150756
$177$	0	0
$178$	6.00000	0.449719
$179$	20.0000	1.49487	0.747435	−	0.664335i	$-0.231285\pi$
$179$	20.0000	1.49487	0.747435	+	0.664335i	$0.231285\pi$
$180$	6.00000	0.447214
$181$	18.0000	1.33793	0.668965	−	0.743294i	$-0.266738\pi$
$181$	18.0000	1.33793	0.668965	+	0.743294i	$0.266738\pi$
$182$	0	0
$183$	0	0
$184$	−12.0000	−0.884652
$185$	−2.00000	−0.147043
$186$	0	0
$187$	12.0000	0.877527
$188$	−2.00000	−0.145865
$189$	0	0
$190$	8.00000	0.580381
$191$	−10.0000	−0.723575	−0.361787	−	0.932261i	$-0.617833\pi$
$191$	−10.0000	−0.723575	−0.361787	+	0.932261i	$0.617833\pi$
$192$	0	0
$193$	−6.00000	−0.431889	−0.215945	−	0.976406i	$-0.569283\pi$
$193$	−6.00000	−0.431889	−0.215945	+	0.976406i	$0.569283\pi$
$194$	−2.00000	−0.143592
$195$	0	0
$196$	7.00000	0.500000
$197$	14.0000	0.997459	0.498729	−	0.866758i	$-0.333800\pi$
$197$	14.0000	0.997459	0.498729	+	0.866758i	$0.333800\pi$
$198$	6.00000	0.426401
$199$	−8.00000	−0.567105	−0.283552	−	0.958957i	$-0.591513\pi$
$199$	−8.00000	−0.567105	−0.283552	+	0.958957i	$0.591513\pi$
$200$	−3.00000	−0.212132
$201$	0	0
$202$	−2.00000	−0.140720
$203$	0	0
$204$	0	0
$205$	20.0000	1.39686
$206$	20.0000	1.39347
$207$	12.0000	0.834058
$208$	2.00000	0.138675
$209$	−8.00000	−0.553372
$210$	0	0
$211$	−16.0000	−1.10149	−0.550743	−	0.834675i	$-0.685655\pi$
$211$	−16.0000	−1.10149	−0.550743	+	0.834675i	$0.685655\pi$
$212$	−2.00000	−0.137361
$213$	0	0
$214$	−6.00000	−0.410152
$215$	−20.0000	−1.36399
$216$	0	0
$217$	0	0
$218$	1.00000	0.0677285
$219$	0	0
$220$	−4.00000	−0.269680
$221$	−12.0000	−0.807207
$222$	0	0
$223$	16.0000	1.07144	0.535720	−	0.844396i	$-0.320040\pi$
$223$	16.0000	1.07144	0.535720	+	0.844396i	$0.320040\pi$
$224$	0	0
$225$	3.00000	0.200000
$226$	−18.0000	−1.19734
$227$	−6.00000	−0.398234	−0.199117	−	0.979976i	$-0.563807\pi$
$227$	−6.00000	−0.398234	−0.199117	+	0.979976i	$0.563807\pi$
$228$	0	0
$229$	2.00000	0.132164	0.0660819	−	0.997814i	$-0.478950\pi$
$229$	2.00000	0.132164	0.0660819	+	0.997814i	$0.478950\pi$
$230$	8.00000	0.527504
$231$	0	0
$232$	−18.0000	−1.18176
$233$	−6.00000	−0.393073	−0.196537	−	0.980497i	$-0.562969\pi$
$233$	−6.00000	−0.393073	−0.196537	+	0.980497i	$0.562969\pi$
$234$	−6.00000	−0.392232
$235$	4.00000	0.260931
$236$	12.0000	0.781133
$237$	0	0
$238$	0	0
$239$	−22.0000	−1.42306	−0.711531	−	0.702655i	$-0.751998\pi$
$239$	−22.0000	−1.42306	−0.711531	+	0.702655i	$0.751998\pi$
$240$	0	0
$241$	−26.0000	−1.67481	−0.837404	−	0.546585i	$-0.815928\pi$
$241$	−26.0000	−1.67481	−0.837404	+	0.546585i	$0.815928\pi$
$242$	7.00000	0.449977
$243$	0	0
$244$	14.0000	0.896258
$245$	−14.0000	−0.894427
$246$	0	0
$247$	8.00000	0.509028
$248$	30.0000	1.90500
$249$	0	0
$250$	12.0000	0.758947
$251$	−24.0000	−1.51487	−0.757433	−	0.652913i	$-0.773547\pi$
$251$	−24.0000	−1.51487	−0.757433	+	0.652913i	$0.773547\pi$
$252$	0	0
$253$	−8.00000	−0.502956
$254$	14.0000	0.878438
$255$	0	0
$256$	−17.0000	−1.06250
$257$	14.0000	0.873296	0.436648	−	0.899632i	$-0.356166\pi$
$257$	14.0000	0.873296	0.436648	+	0.899632i	$0.356166\pi$
$258$	0	0
$259$	0	0
$260$	4.00000	0.248069
$261$	18.0000	1.11417
$262$	2.00000	0.123560
$263$	−28.0000	−1.72655	−0.863277	−	0.504730i	$-0.831592\pi$
$263$	−28.0000	−1.72655	−0.863277	+	0.504730i	$0.831592\pi$
$264$	0	0
$265$	4.00000	0.245718
$266$	0	0
$267$	0	0
$268$	−2.00000	−0.122169
$269$	−6.00000	−0.365826	−0.182913	−	0.983129i	$-0.558553\pi$
$269$	−6.00000	−0.365826	−0.182913	+	0.983129i	$0.558553\pi$
$270$	0	0
$271$	−14.0000	−0.850439	−0.425220	−	0.905090i	$-0.639803\pi$
$271$	−14.0000	−0.850439	−0.425220	+	0.905090i	$0.639803\pi$
$272$	−6.00000	−0.363803
$273$	0	0
$274$	18.0000	1.08742
$275$	−2.00000	−0.120605
$276$	0	0
$277$	−6.00000	−0.360505	−0.180253	−	0.983620i	$-0.557691\pi$
$277$	−6.00000	−0.360505	−0.180253	+	0.983620i	$0.557691\pi$
$278$	2.00000	0.119952
$279$	−30.0000	−1.79605
$280$	0	0
$281$	−6.00000	−0.357930	−0.178965	−	0.983855i	$-0.557275\pi$
$281$	−6.00000	−0.357930	−0.178965	+	0.983855i	$0.557275\pi$
$282$	0	0
$283$	24.0000	1.42665	0.713326	−	0.700832i	$-0.247188\pi$
$283$	24.0000	1.42665	0.713326	+	0.700832i	$0.247188\pi$
$284$	−12.0000	−0.712069
$285$	0	0
$286$	4.00000	0.236525
$287$	0	0
$288$	15.0000	0.883883
$289$	19.0000	1.11765
$290$	12.0000	0.704664
$291$	0	0
$292$	2.00000	0.117041
$293$	−6.00000	−0.350524	−0.175262	−	0.984522i	$-0.556077\pi$
$293$	−6.00000	−0.350524	−0.175262	+	0.984522i	$0.556077\pi$
$294$	0	0
$295$	−24.0000	−1.39733
$296$	−3.00000	−0.174371
$297$	0	0
$298$	−10.0000	−0.579284
$299$	8.00000	0.462652
$300$	0	0
$301$	0	0
$302$	−6.00000	−0.345261
$303$	0	0
$304$	4.00000	0.229416
$305$	−28.0000	−1.60328
$306$	18.0000	1.02899
$307$	−8.00000	−0.456584	−0.228292	−	0.973593i	$-0.573314\pi$
$307$	−8.00000	−0.456584	−0.228292	+	0.973593i	$0.573314\pi$
$308$	0	0
$309$	0	0
$310$	−20.0000	−1.13592
$311$	10.0000	0.567048	0.283524	−	0.958965i	$-0.408496\pi$
$311$	10.0000	0.567048	0.283524	+	0.958965i	$0.408496\pi$
$312$	0	0
$313$	−6.00000	−0.339140	−0.169570	−	0.985518i	$-0.554238\pi$
$313$	−6.00000	−0.339140	−0.169570	+	0.985518i	$0.554238\pi$
$314$	18.0000	1.01580
$315$	0	0
$316$	8.00000	0.450035
$317$	−22.0000	−1.23564	−0.617822	−	0.786318i	$-0.711985\pi$
$317$	−22.0000	−1.23564	−0.617822	+	0.786318i	$0.711985\pi$
$318$	0	0
$319$	−12.0000	−0.671871
$320$	14.0000	0.782624
$321$	0	0
$322$	0	0
$323$	−24.0000	−1.33540
$324$	−9.00000	−0.500000
$325$	2.00000	0.110940
$326$	20.0000	1.10770
$327$	0	0
$328$	30.0000	1.65647
$329$	0	0
$330$	0	0
$331$	−26.0000	−1.42909	−0.714545	−	0.699590i	$-0.753366\pi$
$331$	−26.0000	−1.42909	−0.714545	+	0.699590i	$0.753366\pi$
$332$	−16.0000	−0.878114
$333$	3.00000	0.164399
$334$	−8.00000	−0.437741
$335$	4.00000	0.218543
$336$	0	0
$337$	−14.0000	−0.762629	−0.381314	−	0.924445i	$-0.624528\pi$
$337$	−14.0000	−0.762629	−0.381314	+	0.924445i	$0.624528\pi$
$338$	9.00000	0.489535
$339$	0	0
$340$	−12.0000	−0.650791
$341$	20.0000	1.08306
$342$	−12.0000	−0.648886
$343$	0	0
$344$	−30.0000	−1.61749
$345$	0	0
$346$	−18.0000	−0.967686
$347$	2.00000	0.107366	0.0536828	−	0.998558i	$-0.482904\pi$
$347$	2.00000	0.107366	0.0536828	+	0.998558i	$0.482904\pi$
$348$	0	0
$349$	14.0000	0.749403	0.374701	−	0.927146i	$-0.377745\pi$
$349$	14.0000	0.749403	0.374701	+	0.927146i	$0.377745\pi$
$350$	0	0
$351$	0	0
$352$	−10.0000	−0.533002
$353$	−6.00000	−0.319348	−0.159674	−	0.987170i	$-0.551044\pi$
$353$	−6.00000	−0.319348	−0.159674	+	0.987170i	$0.551044\pi$
$354$	0	0
$355$	24.0000	1.27379
$356$	6.00000	0.317999
$357$	0	0
$358$	−20.0000	−1.05703
$359$	−22.0000	−1.16112	−0.580558	−	0.814219i	$-0.697165\pi$
$359$	−22.0000	−1.16112	−0.580558	+	0.814219i	$0.697165\pi$
$360$	−18.0000	−0.948683
$361$	−3.00000	−0.157895
$362$	−18.0000	−0.946059
$363$	0	0
$364$	0	0
$365$	−4.00000	−0.209370
$366$	0	0
$367$	−34.0000	−1.77479	−0.887393	−	0.461014i	$-0.847486\pi$
$367$	−34.0000	−1.77479	−0.887393	+	0.461014i	$0.847486\pi$
$368$	4.00000	0.208514
$369$	−30.0000	−1.56174
$370$	2.00000	0.103975
$371$	0	0
$372$	0	0
$373$	10.0000	0.517780	0.258890	−	0.965907i	$-0.416643\pi$
$373$	10.0000	0.517780	0.258890	+	0.965907i	$0.416643\pi$
$374$	−12.0000	−0.620505
$375$	0	0
$376$	6.00000	0.309426
$377$	12.0000	0.618031
$378$	0	0
$379$	26.0000	1.33553	0.667765	−	0.744372i	$-0.267251\pi$
$379$	26.0000	1.33553	0.667765	+	0.744372i	$0.267251\pi$
$380$	8.00000	0.410391
$381$	0	0
$382$	10.0000	0.511645
$383$	−32.0000	−1.63512	−0.817562	−	0.575841i	$-0.804675\pi$
$383$	−32.0000	−1.63512	−0.817562	+	0.575841i	$0.804675\pi$
$384$	0	0
$385$	0	0
$386$	6.00000	0.305392
$387$	30.0000	1.52499
$388$	−2.00000	−0.101535
$389$	−10.0000	−0.507020	−0.253510	−	0.967333i	$-0.581585\pi$
$389$	−10.0000	−0.507020	−0.253510	+	0.967333i	$0.581585\pi$
$390$	0	0
$391$	−24.0000	−1.21373
$392$	−21.0000	−1.06066
$393$	0	0
$394$	−14.0000	−0.705310
$395$	−16.0000	−0.805047
$396$	6.00000	0.301511
$397$	18.0000	0.903394	0.451697	−	0.892171i	$-0.350819\pi$
$397$	18.0000	0.903394	0.451697	+	0.892171i	$0.350819\pi$
$398$	8.00000	0.401004
$399$	0	0
$400$	1.00000	0.0500000
$401$	2.00000	0.0998752	0.0499376	−	0.998752i	$-0.484098\pi$
$401$	2.00000	0.0998752	0.0499376	+	0.998752i	$0.484098\pi$
$402$	0	0
$403$	−20.0000	−0.996271
$404$	−2.00000	−0.0995037
$405$	18.0000	0.894427
$406$	0	0
$407$	−2.00000	−0.0991363
$408$	0	0
$409$	−14.0000	−0.692255	−0.346128	−	0.938187i	$-0.612504\pi$
$409$	−14.0000	−0.692255	−0.346128	+	0.938187i	$0.612504\pi$
$410$	−20.0000	−0.987730
$411$	0	0
$412$	20.0000	0.985329
$413$	0	0
$414$	−12.0000	−0.589768
$415$	32.0000	1.57082
$416$	10.0000	0.490290
$417$	0	0
$418$	8.00000	0.391293
$419$	30.0000	1.46560	0.732798	−	0.680446i	$-0.238214\pi$
$419$	30.0000	1.46560	0.732798	+	0.680446i	$0.238214\pi$
$420$	0	0
$421$	−38.0000	−1.85201	−0.926003	−	0.377515i	$-0.876779\pi$
$421$	−38.0000	−1.85201	−0.926003	+	0.377515i	$0.876779\pi$
$422$	16.0000	0.778868
$423$	−6.00000	−0.291730
$424$	6.00000	0.291386
$425$	−6.00000	−0.291043
$426$	0	0
$427$	0	0
$428$	−6.00000	−0.290021
$429$	0	0
$430$	20.0000	0.964486
$431$	−14.0000	−0.674356	−0.337178	−	0.941441i	$-0.609472\pi$
$431$	−14.0000	−0.674356	−0.337178	+	0.941441i	$0.609472\pi$
$432$	0	0
$433$	34.0000	1.63394	0.816968	−	0.576683i	$-0.195653\pi$
$433$	34.0000	1.63394	0.816968	+	0.576683i	$0.195653\pi$
$434$	0	0
$435$	0	0
$436$	1.00000	0.0478913
$437$	16.0000	0.765384
$438$	0	0
$439$	−26.0000	−1.24091	−0.620456	−	0.784241i	$-0.713053\pi$
$439$	−26.0000	−1.24091	−0.620456	+	0.784241i	$0.713053\pi$
$440$	12.0000	0.572078
$441$	21.0000	1.00000
$442$	12.0000	0.570782
$443$	−24.0000	−1.14027	−0.570137	−	0.821549i	$-0.693110\pi$
$443$	−24.0000	−1.14027	−0.570137	+	0.821549i	$0.693110\pi$
$444$	0	0
$445$	−12.0000	−0.568855
$446$	−16.0000	−0.757622
$447$	0	0
$448$	0	0
$449$	14.0000	0.660701	0.330350	−	0.943858i	$-0.392833\pi$
$449$	14.0000	0.660701	0.330350	+	0.943858i	$0.392833\pi$
$450$	−3.00000	−0.141421
$451$	20.0000	0.941763
$452$	−18.0000	−0.846649
$453$	0	0
$454$	6.00000	0.281594
$455$	0	0
$456$	0	0
$457$	10.0000	0.467780	0.233890	−	0.972263i	$-0.424854\pi$
$457$	10.0000	0.467780	0.233890	+	0.972263i	$0.424854\pi$
$458$	−2.00000	−0.0934539
$459$	0	0
$460$	8.00000	0.373002
$461$	34.0000	1.58354	0.791769	−	0.610821i	$-0.209160\pi$
$461$	34.0000	1.58354	0.791769	+	0.610821i	$0.209160\pi$
$462$	0	0
$463$	−14.0000	−0.650635	−0.325318	−	0.945605i	$-0.605471\pi$
$463$	−14.0000	−0.650635	−0.325318	+	0.945605i	$0.605471\pi$
$464$	6.00000	0.278543
$465$	0	0
$466$	6.00000	0.277945
$467$	−18.0000	−0.832941	−0.416470	−	0.909149i	$-0.636733\pi$
$467$	−18.0000	−0.832941	−0.416470	+	0.909149i	$0.636733\pi$
$468$	−6.00000	−0.277350
$469$	0	0
$470$	−4.00000	−0.184506
$471$	0	0
$472$	−36.0000	−1.65703
$473$	−20.0000	−0.919601
$474$	0	0
$475$	4.00000	0.183533
$476$	0	0
$477$	−6.00000	−0.274721
$478$	22.0000	1.00626
$479$	−10.0000	−0.456912	−0.228456	−	0.973554i	$-0.573368\pi$
$479$	−10.0000	−0.456912	−0.228456	+	0.973554i	$0.573368\pi$
$480$	0	0
$481$	2.00000	0.0911922
$482$	26.0000	1.18427
$483$	0	0
$484$	7.00000	0.318182
$485$	4.00000	0.181631
$486$	0	0
$487$	28.0000	1.26880	0.634401	−	0.773004i	$-0.281247\pi$
$487$	28.0000	1.26880	0.634401	+	0.773004i	$0.281247\pi$
$488$	−42.0000	−1.90125
$489$	0	0
$490$	14.0000	0.632456
$491$	−10.0000	−0.451294	−0.225647	−	0.974209i	$-0.572450\pi$
$491$	−10.0000	−0.451294	−0.225647	+	0.974209i	$0.572450\pi$
$492$	0	0
$493$	−36.0000	−1.62136
$494$	−8.00000	−0.359937
$495$	−12.0000	−0.539360
$496$	−10.0000	−0.449013
$497$	0	0
$498$	0	0
$499$	−10.0000	−0.447661	−0.223831	−	0.974628i	$-0.571856\pi$
$499$	−10.0000	−0.447661	−0.223831	+	0.974628i	$0.571856\pi$
$500$	12.0000	0.536656
$501$	0	0
$502$	24.0000	1.07117
$503$	24.0000	1.07011	0.535054	−	0.844818i	$-0.320291\pi$
$503$	24.0000	1.07011	0.535054	+	0.844818i	$0.320291\pi$
$504$	0	0
$505$	4.00000	0.177998
$506$	8.00000	0.355643
$507$	0	0
$508$	14.0000	0.621150
$509$	18.0000	0.797836	0.398918	−	0.916987i	$-0.369386\pi$
$509$	18.0000	0.797836	0.398918	+	0.916987i	$0.369386\pi$
$510$	0	0
$511$	0	0
$512$	11.0000	0.486136
$513$	0	0
$514$	−14.0000	−0.617514
$515$	−40.0000	−1.76261
$516$	0	0
$517$	4.00000	0.175920
$518$	0	0
$519$	0	0
$520$	−12.0000	−0.526235
$521$	−30.0000	−1.31432	−0.657162	−	0.753749i	$-0.728243\pi$
$521$	−30.0000	−1.31432	−0.657162	+	0.753749i	$0.728243\pi$
$522$	−18.0000	−0.787839
$523$	14.0000	0.612177	0.306089	−	0.952003i	$-0.400980\pi$
$523$	14.0000	0.612177	0.306089	+	0.952003i	$0.400980\pi$
$524$	2.00000	0.0873704
$525$	0	0
$526$	28.0000	1.22086
$527$	60.0000	2.61364
$528$	0	0
$529$	−7.00000	−0.304348
$530$	−4.00000	−0.173749
$531$	36.0000	1.56227
$532$	0	0
$533$	−20.0000	−0.866296
$534$	0	0
$535$	12.0000	0.518805
$536$	6.00000	0.259161
$537$	0	0
$538$	6.00000	0.258678
$539$	−14.0000	−0.603023
$540$	0	0
$541$	−38.0000	−1.63375	−0.816874	−	0.576816i	$-0.804295\pi$
$541$	−38.0000	−1.63375	−0.816874	+	0.576816i	$0.804295\pi$
$542$	14.0000	0.601351
$543$	0	0
$544$	−30.0000	−1.28624
$545$	−2.00000	−0.0856706
$546$	0	0
$547$	−12.0000	−0.513083	−0.256541	−	0.966533i	$-0.582583\pi$
$547$	−12.0000	−0.513083	−0.256541	+	0.966533i	$0.582583\pi$
$548$	18.0000	0.768922
$549$	42.0000	1.79252
$550$	2.00000	0.0852803
$551$	24.0000	1.02243
$552$	0	0
$553$	0	0
$554$	6.00000	0.254916
$555$	0	0
$556$	2.00000	0.0848189
$557$	−38.0000	−1.61011	−0.805056	−	0.593199i	$-0.797865\pi$
$557$	−38.0000	−1.61011	−0.805056	+	0.593199i	$0.797865\pi$
$558$	30.0000	1.27000
$559$	20.0000	0.845910
$560$	0	0
$561$	0	0
$562$	6.00000	0.253095
$563$	24.0000	1.01148	0.505740	−	0.862686i	$-0.331220\pi$
$563$	24.0000	1.01148	0.505740	+	0.862686i	$0.331220\pi$
$564$	0	0
$565$	36.0000	1.51453
$566$	−24.0000	−1.00880
$567$	0	0
$568$	36.0000	1.51053
$569$	−6.00000	−0.251533	−0.125767	−	0.992060i	$-0.540139\pi$
$569$	−6.00000	−0.251533	−0.125767	+	0.992060i	$0.540139\pi$
$570$	0	0
$571$	12.0000	0.502184	0.251092	−	0.967963i	$-0.419210\pi$
$571$	12.0000	0.502184	0.251092	+	0.967963i	$0.419210\pi$
$572$	4.00000	0.167248
$573$	0	0
$574$	0	0
$575$	4.00000	0.166812
$576$	−21.0000	−0.875000
$577$	38.0000	1.58196	0.790980	−	0.611842i	$-0.209571\pi$
$577$	38.0000	1.58196	0.790980	+	0.611842i	$0.209571\pi$
$578$	−19.0000	−0.790296
$579$	0	0
$580$	12.0000	0.498273
$581$	0	0
$582$	0	0
$583$	4.00000	0.165663
$584$	−6.00000	−0.248282
$585$	12.0000	0.496139
$586$	6.00000	0.247858
$587$	36.0000	1.48588	0.742940	−	0.669359i	$-0.233431\pi$
$587$	36.0000	1.48588	0.742940	+	0.669359i	$0.233431\pi$
$588$	0	0
$589$	−40.0000	−1.64817
$590$	24.0000	0.988064
$591$	0	0
$592$	1.00000	0.0410997
$593$	−34.0000	−1.39621	−0.698106	−	0.715994i	$-0.745974\pi$
$593$	−34.0000	−1.39621	−0.698106	+	0.715994i	$0.745974\pi$
$594$	0	0
$595$	0	0
$596$	−10.0000	−0.409616
$597$	0	0
$598$	−8.00000	−0.327144
$599$	18.0000	0.735460	0.367730	−	0.929933i	$-0.380135\pi$
$599$	18.0000	0.735460	0.367730	+	0.929933i	$0.380135\pi$
$600$	0	0
$601$	−22.0000	−0.897399	−0.448699	−	0.893683i	$-0.648113\pi$
$601$	−22.0000	−0.897399	−0.448699	+	0.893683i	$0.648113\pi$
$602$	0	0
$603$	−6.00000	−0.244339
$604$	−6.00000	−0.244137
$605$	−14.0000	−0.569181
$606$	0	0
$607$	12.0000	0.487065	0.243532	−	0.969893i	$-0.421694\pi$
$607$	12.0000	0.487065	0.243532	+	0.969893i	$0.421694\pi$
$608$	20.0000	0.811107
$609$	0	0
$610$	28.0000	1.13369
$611$	−4.00000	−0.161823
$612$	18.0000	0.727607
$613$	26.0000	1.05013	0.525065	−	0.851062i	$-0.324041\pi$
$613$	26.0000	1.05013	0.525065	+	0.851062i	$0.324041\pi$
$614$	8.00000	0.322854
$615$	0	0
$616$	0	0
$617$	−38.0000	−1.52982	−0.764911	−	0.644136i	$-0.777217\pi$
$617$	−38.0000	−1.52982	−0.764911	+	0.644136i	$0.777217\pi$
$618$	0	0
$619$	4.00000	0.160774	0.0803868	−	0.996764i	$-0.474384\pi$
$619$	4.00000	0.160774	0.0803868	+	0.996764i	$0.474384\pi$
$620$	−20.0000	−0.803219
$621$	0	0
$622$	−10.0000	−0.400963
$623$	0	0
$624$	0	0
$625$	−19.0000	−0.760000
$626$	6.00000	0.239808
$627$	0	0
$628$	18.0000	0.718278
$629$	−6.00000	−0.239236
$630$	0	0
$631$	−36.0000	−1.43314	−0.716569	−	0.697517i	$-0.754288\pi$
$631$	−36.0000	−1.43314	−0.716569	+	0.697517i	$0.754288\pi$
$632$	−24.0000	−0.954669
$633$	0	0
$634$	22.0000	0.873732
$635$	−28.0000	−1.11115
$636$	0	0
$637$	14.0000	0.554700
$638$	12.0000	0.475085
$639$	−36.0000	−1.42414
$640$	6.00000	0.237171
$641$	−30.0000	−1.18493	−0.592464	−	0.805597i	$-0.701845\pi$
$641$	−30.0000	−1.18493	−0.592464	+	0.805597i	$0.701845\pi$
$642$	0	0
$643$	−44.0000	−1.73519	−0.867595	−	0.497271i	$-0.834335\pi$
$643$	−44.0000	−1.73519	−0.867595	+	0.497271i	$0.834335\pi$
$644$	0	0
$645$	0	0
$646$	24.0000	0.944267
$647$	18.0000	0.707653	0.353827	−	0.935311i	$-0.384880\pi$
$647$	18.0000	0.707653	0.353827	+	0.935311i	$0.384880\pi$
$648$	27.0000	1.06066
$649$	−24.0000	−0.942082
$650$	−2.00000	−0.0784465
$651$	0	0
$652$	20.0000	0.783260
$653$	18.0000	0.704394	0.352197	−	0.935926i	$-0.385435\pi$
$653$	18.0000	0.704394	0.352197	+	0.935926i	$0.385435\pi$
$654$	0	0
$655$	−4.00000	−0.156293
$656$	−10.0000	−0.390434
$657$	6.00000	0.234082
$658$	0	0
$659$	24.0000	0.934907	0.467454	−	0.884018i	$-0.345171\pi$
$659$	24.0000	0.934907	0.467454	+	0.884018i	$0.345171\pi$
$660$	0	0
$661$	42.0000	1.63361	0.816805	−	0.576913i	$-0.195743\pi$
$661$	42.0000	1.63361	0.816805	+	0.576913i	$0.195743\pi$
$662$	26.0000	1.01052
$663$	0	0
$664$	48.0000	1.86276
$665$	0	0
$666$	−3.00000	−0.116248
$667$	24.0000	0.929284
$668$	−8.00000	−0.309529
$669$	0	0
$670$	−4.00000	−0.154533
$671$	−28.0000	−1.08093
$672$	0	0
$673$	26.0000	1.00223	0.501113	−	0.865382i	$-0.332924\pi$
$673$	26.0000	1.00223	0.501113	+	0.865382i	$0.332924\pi$
$674$	14.0000	0.539260
$675$	0	0
$676$	9.00000	0.346154
$677$	18.0000	0.691796	0.345898	−	0.938272i	$-0.387574\pi$
$677$	18.0000	0.691796	0.345898	+	0.938272i	$0.387574\pi$
$678$	0	0
$679$	0	0
$680$	36.0000	1.38054
$681$	0	0
$682$	−20.0000	−0.765840
$683$	6.00000	0.229584	0.114792	−	0.993390i	$-0.463380\pi$
$683$	6.00000	0.229584	0.114792	+	0.993390i	$0.463380\pi$
$684$	−12.0000	−0.458831
$685$	−36.0000	−1.37549
$686$	0	0
$687$	0	0
$688$	10.0000	0.381246
$689$	−4.00000	−0.152388
$690$	0	0
$691$	46.0000	1.74992	0.874961	−	0.484193i	$-0.160887\pi$
$691$	46.0000	1.74992	0.874961	+	0.484193i	$0.160887\pi$
$692$	−18.0000	−0.684257
$693$	0	0
$694$	−2.00000	−0.0759190
$695$	−4.00000	−0.151729
$696$	0	0
$697$	60.0000	2.27266
$698$	−14.0000	−0.529908
$699$	0	0
$700$	0	0
$701$	18.0000	0.679851	0.339925	−	0.940452i	$-0.389598\pi$
$701$	18.0000	0.679851	0.339925	+	0.940452i	$0.389598\pi$
$702$	0	0
$703$	4.00000	0.150863
$704$	14.0000	0.527645
$705$	0	0
$706$	6.00000	0.225813
$707$	0	0
$708$	0	0
$709$	22.0000	0.826227	0.413114	−	0.910679i	$-0.364441\pi$
$709$	22.0000	0.826227	0.413114	+	0.910679i	$0.364441\pi$
$710$	−24.0000	−0.900704
$711$	24.0000	0.900070
$712$	−18.0000	−0.674579
$713$	−40.0000	−1.49801
$714$	0	0
$715$	−8.00000	−0.299183
$716$	−20.0000	−0.747435
$717$	0	0
$718$	22.0000	0.821033
$719$	14.0000	0.522112	0.261056	−	0.965324i	$-0.415929\pi$
$719$	14.0000	0.522112	0.261056	+	0.965324i	$0.415929\pi$
$720$	6.00000	0.223607
$721$	0	0
$722$	3.00000	0.111648
$723$	0	0
$724$	−18.0000	−0.668965
$725$	6.00000	0.222834
$726$	0	0
$727$	6.00000	0.222528	0.111264	−	0.993791i	$-0.464510\pi$
$727$	6.00000	0.222528	0.111264	+	0.993791i	$0.464510\pi$
$728$	0	0
$729$	−27.0000	−1.00000
$730$	4.00000	0.148047
$731$	−60.0000	−2.21918
$732$	0	0
$733$	−14.0000	−0.517102	−0.258551	−	0.965998i	$-0.583245\pi$
$733$	−14.0000	−0.517102	−0.258551	+	0.965998i	$0.583245\pi$
$734$	34.0000	1.25496
$735$	0	0
$736$	20.0000	0.737210
$737$	4.00000	0.147342
$738$	30.0000	1.10432
$739$	6.00000	0.220714	0.110357	−	0.993892i	$-0.464801\pi$
$739$	6.00000	0.220714	0.110357	+	0.993892i	$0.464801\pi$
$740$	2.00000	0.0735215
$741$	0	0
$742$	0	0
$743$	−4.00000	−0.146746	−0.0733729	−	0.997305i	$-0.523376\pi$
$743$	−4.00000	−0.146746	−0.0733729	+	0.997305i	$0.523376\pi$
$744$	0	0
$745$	20.0000	0.732743
$746$	−10.0000	−0.366126
$747$	−48.0000	−1.75623
$748$	−12.0000	−0.438763
$749$	0	0
$750$	0	0
$751$	−52.0000	−1.89751	−0.948753	−	0.316017i	$-0.897654\pi$
$751$	−52.0000	−1.89751	−0.948753	+	0.316017i	$0.897654\pi$
$752$	−2.00000	−0.0729325
$753$	0	0
$754$	−12.0000	−0.437014
$755$	12.0000	0.436725
$756$	0	0
$757$	−22.0000	−0.799604	−0.399802	−	0.916602i	$-0.630921\pi$
$757$	−22.0000	−0.799604	−0.399802	+	0.916602i	$0.630921\pi$
$758$	−26.0000	−0.944363
$759$	0	0
$760$	−24.0000	−0.870572
$761$	−22.0000	−0.797499	−0.398750	−	0.917060i	$-0.630556\pi$
$761$	−22.0000	−0.797499	−0.398750	+	0.917060i	$0.630556\pi$
$762$	0	0
$763$	0	0
$764$	10.0000	0.361787
$765$	−36.0000	−1.30158
$766$	32.0000	1.15621
$767$	24.0000	0.866590
$768$	0	0
$769$	46.0000	1.65880	0.829401	−	0.558653i	$-0.188682\pi$
$769$	46.0000	1.65880	0.829401	+	0.558653i	$0.188682\pi$
$770$	0	0
$771$	0	0
$772$	6.00000	0.215945
$773$	18.0000	0.647415	0.323708	−	0.946157i	$-0.395071\pi$
$773$	18.0000	0.647415	0.323708	+	0.946157i	$0.395071\pi$
$774$	−30.0000	−1.07833
$775$	−10.0000	−0.359211
$776$	6.00000	0.215387
$777$	0	0
$778$	10.0000	0.358517
$779$	−40.0000	−1.43315
$780$	0	0
$781$	24.0000	0.858788
$782$	24.0000	0.858238
$783$	0	0
$784$	7.00000	0.250000
$785$	−36.0000	−1.28490
$786$	0	0
$787$	−18.0000	−0.641631	−0.320815	−	0.947142i	$-0.603957\pi$
$787$	−18.0000	−0.641631	−0.320815	+	0.947142i	$0.603957\pi$
$788$	−14.0000	−0.498729
$789$	0	0
$790$	16.0000	0.569254
$791$	0	0
$792$	−18.0000	−0.639602
$793$	28.0000	0.994309
$794$	−18.0000	−0.638796
$795$	0	0
$796$	8.00000	0.283552
$797$	−22.0000	−0.779280	−0.389640	−	0.920967i	$-0.627401\pi$
$797$	−22.0000	−0.779280	−0.389640	+	0.920967i	$0.627401\pi$
$798$	0	0
$799$	12.0000	0.424529
$800$	5.00000	0.176777
$801$	18.0000	0.635999
$802$	−2.00000	−0.0706225
$803$	−4.00000	−0.141157
$804$	0	0
$805$	0	0
$806$	20.0000	0.704470
$807$	0	0
$808$	6.00000	0.211079
$809$	18.0000	0.632846	0.316423	−	0.948618i	$-0.397518\pi$
$809$	18.0000	0.632846	0.316423	+	0.948618i	$0.397518\pi$
$810$	−18.0000	−0.632456
$811$	28.0000	0.983213	0.491606	−	0.870817i	$-0.336410\pi$
$811$	28.0000	0.983213	0.491606	+	0.870817i	$0.336410\pi$
$812$	0	0
$813$	0	0
$814$	2.00000	0.0701000
$815$	−40.0000	−1.40114
$816$	0	0
$817$	40.0000	1.39942
$818$	14.0000	0.489499
$819$	0	0
$820$	−20.0000	−0.698430
$821$	18.0000	0.628204	0.314102	−	0.949389i	$-0.398297\pi$
$821$	18.0000	0.628204	0.314102	+	0.949389i	$0.398297\pi$
$822$	0	0
$823$	12.0000	0.418294	0.209147	−	0.977884i	$-0.432931\pi$
$823$	12.0000	0.418294	0.209147	+	0.977884i	$0.432931\pi$
$824$	−60.0000	−2.09020
$825$	0	0
$826$	0	0
$827$	6.00000	0.208640	0.104320	−	0.994544i	$-0.466733\pi$
$827$	6.00000	0.208640	0.104320	+	0.994544i	$0.466733\pi$
$828$	−12.0000	−0.417029
$829$	2.00000	0.0694629	0.0347314	−	0.999397i	$-0.488942\pi$
$829$	2.00000	0.0694629	0.0347314	+	0.999397i	$0.488942\pi$
$830$	−32.0000	−1.11074
$831$	0	0
$832$	−14.0000	−0.485363
$833$	−42.0000	−1.45521
$834$	0	0
$835$	16.0000	0.553703
$836$	8.00000	0.276686
$837$	0	0
$838$	−30.0000	−1.03633
$839$	30.0000	1.03572	0.517858	−	0.855467i	$-0.326730\pi$
$839$	30.0000	1.03572	0.517858	+	0.855467i	$0.326730\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	38.0000	1.30957
$843$	0	0
$844$	16.0000	0.550743
$845$	−18.0000	−0.619219
$846$	6.00000	0.206284
$847$	0	0
$848$	−2.00000	−0.0686803
$849$	0	0
$850$	6.00000	0.205798
$851$	4.00000	0.137118
$852$	0	0
$853$	38.0000	1.30110	0.650548	−	0.759465i	$-0.274539\pi$
$853$	38.0000	1.30110	0.650548	+	0.759465i	$0.274539\pi$
$854$	0	0
$855$	24.0000	0.820783
$856$	18.0000	0.615227
$857$	42.0000	1.43469	0.717346	−	0.696717i	$-0.245357\pi$
$857$	42.0000	1.43469	0.717346	+	0.696717i	$0.245357\pi$
$858$	0	0
$859$	16.0000	0.545913	0.272956	−	0.962026i	$-0.411998\pi$
$859$	16.0000	0.545913	0.272956	+	0.962026i	$0.411998\pi$
$860$	20.0000	0.681994
$861$	0	0
$862$	14.0000	0.476842
$863$	12.0000	0.408485	0.204242	−	0.978920i	$-0.434527\pi$
$863$	12.0000	0.408485	0.204242	+	0.978920i	$0.434527\pi$
$864$	0	0
$865$	36.0000	1.22404
$866$	−34.0000	−1.15537
$867$	0	0
$868$	0	0
$869$	−16.0000	−0.542763
$870$	0	0
$871$	−4.00000	−0.135535
$872$	−3.00000	−0.101593
$873$	−6.00000	−0.203069
$874$	−16.0000	−0.541208
$875$	0	0
$876$	0	0
$877$	46.0000	1.55331	0.776655	−	0.629926i	$-0.216915\pi$
$877$	46.0000	1.55331	0.776655	+	0.629926i	$0.216915\pi$
$878$	26.0000	0.877457
$879$	0	0
$880$	−4.00000	−0.134840
$881$	22.0000	0.741199	0.370599	−	0.928793i	$-0.379152\pi$
$881$	22.0000	0.741199	0.370599	+	0.928793i	$0.379152\pi$
$882$	−21.0000	−0.707107
$883$	−8.00000	−0.269221	−0.134611	−	0.990899i	$-0.542978\pi$
$883$	−8.00000	−0.269221	−0.134611	+	0.990899i	$0.542978\pi$
$884$	12.0000	0.403604
$885$	0	0
$886$	24.0000	0.806296
$887$	12.0000	0.402921	0.201460	−	0.979497i	$-0.435431\pi$
$887$	12.0000	0.402921	0.201460	+	0.979497i	$0.435431\pi$
$888$	0	0
$889$	0	0
$890$	12.0000	0.402241
$891$	18.0000	0.603023
$892$	−16.0000	−0.535720
$893$	−8.00000	−0.267710
$894$	0	0
$895$	40.0000	1.33705
$896$	0	0
$897$	0	0
$898$	−14.0000	−0.467186
$899$	−60.0000	−2.00111
$900$	−3.00000	−0.100000
$901$	12.0000	0.399778
$902$	−20.0000	−0.665927
$903$	0	0
$904$	54.0000	1.79601
$905$	36.0000	1.19668
$906$	0	0
$907$	18.0000	0.597680	0.298840	−	0.954303i	$-0.403400\pi$
$907$	18.0000	0.597680	0.298840	+	0.954303i	$0.403400\pi$
$908$	6.00000	0.199117
$909$	−6.00000	−0.199007
$910$	0	0
$911$	16.0000	0.530104	0.265052	−	0.964234i	$-0.414611\pi$
$911$	16.0000	0.530104	0.265052	+	0.964234i	$0.414611\pi$
$912$	0	0
$913$	32.0000	1.05905
$914$	−10.0000	−0.330771
$915$	0	0
$916$	−2.00000	−0.0660819
$917$	0	0
$918$	0	0
$919$	−16.0000	−0.527791	−0.263896	−	0.964551i	$-0.585007\pi$
$919$	−16.0000	−0.527791	−0.263896	+	0.964551i	$0.585007\pi$
$920$	−24.0000	−0.791257
$921$	0	0
$922$	−34.0000	−1.11973
$923$	−24.0000	−0.789970
$924$	0	0
$925$	1.00000	0.0328798
$926$	14.0000	0.460069
$927$	60.0000	1.97066
$928$	30.0000	0.984798
$929$	50.0000	1.64045	0.820223	−	0.572043i	$-0.193849\pi$
$929$	50.0000	1.64045	0.820223	+	0.572043i	$0.193849\pi$
$930$	0	0
$931$	28.0000	0.917663
$932$	6.00000	0.196537
$933$	0	0
$934$	18.0000	0.588978
$935$	24.0000	0.784884
$936$	18.0000	0.588348
$937$	2.00000	0.0653372	0.0326686	−	0.999466i	$-0.489599\pi$
$937$	2.00000	0.0653372	0.0326686	+	0.999466i	$0.489599\pi$
$938$	0	0
$939$	0	0
$940$	−4.00000	−0.130466
$941$	18.0000	0.586783	0.293392	−	0.955992i	$-0.405216\pi$
$941$	18.0000	0.586783	0.293392	+	0.955992i	$0.405216\pi$
$942$	0	0
$943$	−40.0000	−1.30258
$944$	12.0000	0.390567
$945$	0	0
$946$	20.0000	0.650256
$947$	18.0000	0.584921	0.292461	−	0.956278i	$-0.405526\pi$
$947$	18.0000	0.584921	0.292461	+	0.956278i	$0.405526\pi$
$948$	0	0
$949$	4.00000	0.129845
$950$	−4.00000	−0.129777
$951$	0	0
$952$	0	0
$953$	−18.0000	−0.583077	−0.291539	−	0.956559i	$-0.594167\pi$
$953$	−18.0000	−0.583077	−0.291539	+	0.956559i	$0.594167\pi$
$954$	6.00000	0.194257
$955$	−20.0000	−0.647185
$956$	22.0000	0.711531
$957$	0	0
$958$	10.0000	0.323085
$959$	0	0
$960$	0	0
$961$	69.0000	2.22581
$962$	−2.00000	−0.0644826
$963$	−18.0000	−0.580042
$964$	26.0000	0.837404
$965$	−12.0000	−0.386294
$966$	0	0
$967$	−8.00000	−0.257263	−0.128631	−	0.991692i	$-0.541058\pi$
$967$	−8.00000	−0.257263	−0.128631	+	0.991692i	$0.541058\pi$
$968$	−21.0000	−0.674966
$969$	0	0
$970$	−4.00000	−0.128432
$971$	14.0000	0.449281	0.224641	−	0.974442i	$-0.427879\pi$
$971$	14.0000	0.449281	0.224641	+	0.974442i	$0.427879\pi$
$972$	0	0
$973$	0	0
$974$	−28.0000	−0.897178
$975$	0	0
$976$	14.0000	0.448129
$977$	−30.0000	−0.959785	−0.479893	−	0.877327i	$-0.659324\pi$
$977$	−30.0000	−0.959785	−0.479893	+	0.877327i	$0.659324\pi$
$978$	0	0
$979$	−12.0000	−0.383522
$980$	14.0000	0.447214
$981$	3.00000	0.0957826
$982$	10.0000	0.319113
$983$	−18.0000	−0.574111	−0.287055	−	0.957914i	$-0.592676\pi$
$983$	−18.0000	−0.574111	−0.287055	+	0.957914i	$0.592676\pi$
$984$	0	0
$985$	28.0000	0.892154
$986$	36.0000	1.14647
$987$	0	0
$988$	−8.00000	−0.254514
$989$	40.0000	1.27193
$990$	12.0000	0.381385
$991$	−32.0000	−1.01651	−0.508257	−	0.861206i	$-0.669710\pi$
$991$	−32.0000	−1.01651	−0.508257	+	0.861206i	$0.669710\pi$
$992$	−50.0000	−1.58750
$993$	0	0
$994$	0	0
$995$	−16.0000	−0.507234
$996$	0	0
$997$	18.0000	0.570066	0.285033	−	0.958518i	$-0.407995\pi$
$997$	18.0000	0.570066	0.285033	+	0.958518i	$0.407995\pi$
$998$	10.0000	0.316544
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4033.2.a.a.1.1	✓	1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4033.2.a.a.1.1	✓	1	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 4033 = 37 \cdot 109 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4033.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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