LMFDB - Embedded newform 4550.2.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4550,2,Mod(1,4550)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4550, 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("4550.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4550 = 2 \cdot 5^{2} \cdot 7 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4550.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:	$36.3319329197$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 182)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	4550.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-1.00000 q^{2} -3.00000 q^{3} +1.00000 q^{4} +3.00000 q^{6} +1.00000 q^{7} -1.00000 q^{8} +6.00000 q^{9} +O(q^{10})$

\(q-1.00000 q^{2} -3.00000 q^{3} +1.00000 q^{4} +3.00000 q^{6} +1.00000 q^{7} -1.00000 q^{8} +6.00000 q^{9} +1.00000 q^{11} -3.00000 q^{12} +1.00000 q^{13} -1.00000 q^{14} +1.00000 q^{16} -6.00000 q^{18} -6.00000 q^{19} -3.00000 q^{21} -1.00000 q^{22} +7.00000 q^{23} +3.00000 q^{24} -1.00000 q^{26} -9.00000 q^{27} +1.00000 q^{28} -4.00000 q^{29} +7.00000 q^{31} -1.00000 q^{32} -3.00000 q^{33} +6.00000 q^{36} -9.00000 q^{37} +6.00000 q^{38} -3.00000 q^{39} -3.00000 q^{41} +3.00000 q^{42} -4.00000 q^{43} +1.00000 q^{44} -7.00000 q^{46} -7.00000 q^{47} -3.00000 q^{48} +1.00000 q^{49} +1.00000 q^{52} +9.00000 q^{54} -1.00000 q^{56} +18.0000 q^{57} +4.00000 q^{58} -10.0000 q^{59} +1.00000 q^{61} -7.00000 q^{62} +6.00000 q^{63} +1.00000 q^{64} +3.00000 q^{66} -1.00000 q^{67} -21.0000 q^{69} +16.0000 q^{71} -6.00000 q^{72} -5.00000 q^{73} +9.00000 q^{74} -6.00000 q^{76} +1.00000 q^{77} +3.00000 q^{78} +11.0000 q^{79} +9.00000 q^{81} +3.00000 q^{82} -3.00000 q^{84} +4.00000 q^{86} +12.0000 q^{87} -1.00000 q^{88} -6.00000 q^{89} +1.00000 q^{91} +7.00000 q^{92} -21.0000 q^{93} +7.00000 q^{94} +3.00000 q^{96} +1.00000 q^{97} -1.00000 q^{98} +6.00000 q^{99} +O(q^{100})\)

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
$2$	−1.00000	−0.707107
$3$	−3.00000	−1.73205	−0.866025	−	0.500000i	$-0.833333\pi$
$3$	−3.00000	−1.73205	−0.866025	+	0.500000i	$0.833333\pi$
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	3.00000	1.22474
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	−1.00000	−0.353553
$9$	6.00000	2.00000
$10$	0	0
$11$	1.00000	0.301511	0.150756	−	0.988571i	$-0.451829\pi$
$11$	1.00000	0.301511	0.150756	+	0.988571i	$0.451829\pi$
$12$	−3.00000	−0.866025
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	−1.00000	−0.267261
$15$	0	0
$16$	1.00000	0.250000
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	−6.00000	−1.41421
$19$	−6.00000	−1.37649	−0.688247	−	0.725476i	$-0.741620\pi$
$19$	−6.00000	−1.37649	−0.688247	+	0.725476i	$0.741620\pi$
$20$	0	0
$21$	−3.00000	−0.654654
$22$	−1.00000	−0.213201
$23$	7.00000	1.45960	0.729800	−	0.683660i	$-0.239613\pi$
$23$	7.00000	1.45960	0.729800	+	0.683660i	$0.239613\pi$
$24$	3.00000	0.612372
$25$	0	0
$26$	−1.00000	−0.196116
$27$	−9.00000	−1.73205
$28$	1.00000	0.188982
$29$	−4.00000	−0.742781	−0.371391	−	0.928477i	$-0.621119\pi$
$29$	−4.00000	−0.742781	−0.371391	+	0.928477i	$0.621119\pi$
$30$	0	0
$31$	7.00000	1.25724	0.628619	−	0.777714i	$-0.283621\pi$
$31$	7.00000	1.25724	0.628619	+	0.777714i	$0.283621\pi$
$32$	−1.00000	−0.176777
$33$	−3.00000	−0.522233
$34$	0	0
$35$	0	0
$36$	6.00000	1.00000
$37$	−9.00000	−1.47959	−0.739795	−	0.672832i	$-0.765078\pi$
$37$	−9.00000	−1.47959	−0.739795	+	0.672832i	$0.765078\pi$
$38$	6.00000	0.973329
$39$	−3.00000	−0.480384
$40$	0	0
$41$	−3.00000	−0.468521	−0.234261	−	0.972174i	$-0.575267\pi$
$41$	−3.00000	−0.468521	−0.234261	+	0.972174i	$0.575267\pi$
$42$	3.00000	0.462910
$43$	−4.00000	−0.609994	−0.304997	−	0.952353i	$-0.598656\pi$
$43$	−4.00000	−0.609994	−0.304997	+	0.952353i	$0.598656\pi$
$44$	1.00000	0.150756
$45$	0	0
$46$	−7.00000	−1.03209
$47$	−7.00000	−1.02105	−0.510527	−	0.859861i	$-0.670550\pi$
$47$	−7.00000	−1.02105	−0.510527	+	0.859861i	$0.670550\pi$
$48$	−3.00000	−0.433013
$49$	1.00000	0.142857
$50$	0	0
$51$	0	0
$52$	1.00000	0.138675
$53$	0	0		−	1.00000i	$-0.5\pi$
$53$	0	0			1.00000i	$0.5\pi$
$54$	9.00000	1.22474
$55$	0	0
$56$	−1.00000	−0.133631
$57$	18.0000	2.38416
$58$	4.00000	0.525226
$59$	−10.0000	−1.30189	−0.650945	−	0.759125i	$-0.725627\pi$
$59$	−10.0000	−1.30189	−0.650945	+	0.759125i	$0.725627\pi$
$60$	0	0
$61$	1.00000	0.128037	0.0640184	−	0.997949i	$-0.479608\pi$
$61$	1.00000	0.128037	0.0640184	+	0.997949i	$0.479608\pi$
$62$	−7.00000	−0.889001
$63$	6.00000	0.755929
$64$	1.00000	0.125000
$65$	0	0
$66$	3.00000	0.369274
$67$	−1.00000	−0.122169	−0.0610847	−	0.998133i	$-0.519456\pi$
$67$	−1.00000	−0.122169	−0.0610847	+	0.998133i	$0.519456\pi$
$68$	0	0
$69$	−21.0000	−2.52810
$70$	0	0
$71$	16.0000	1.89885	0.949425	−	0.313993i	$-0.101667\pi$
$71$	16.0000	1.89885	0.949425	+	0.313993i	$0.101667\pi$
$72$	−6.00000	−0.707107
$73$	−5.00000	−0.585206	−0.292603	−	0.956234i	$-0.594521\pi$
$73$	−5.00000	−0.585206	−0.292603	+	0.956234i	$0.594521\pi$
$74$	9.00000	1.04623
$75$	0	0
$76$	−6.00000	−0.688247
$77$	1.00000	0.113961
$78$	3.00000	0.339683
$79$	11.0000	1.23760	0.618798	−	0.785550i	$-0.287620\pi$
$79$	11.0000	1.23760	0.618798	+	0.785550i	$0.287620\pi$
$80$	0	0
$81$	9.00000	1.00000
$82$	3.00000	0.331295
$83$	0	0		−	1.00000i	$-0.5\pi$
$83$	0	0			1.00000i	$0.5\pi$
$84$	−3.00000	−0.327327
$85$	0	0
$86$	4.00000	0.431331
$87$	12.0000	1.28654
$88$	−1.00000	−0.106600
$89$	−6.00000	−0.635999	−0.317999	−	0.948091i	$-0.603011\pi$
$89$	−6.00000	−0.635999	−0.317999	+	0.948091i	$0.603011\pi$
$90$	0	0
$91$	1.00000	0.104828
$92$	7.00000	0.729800
$93$	−21.0000	−2.17760
$94$	7.00000	0.721995
$95$	0	0
$96$	3.00000	0.306186
$97$	1.00000	0.101535	0.0507673	−	0.998711i	$-0.483833\pi$
$97$	1.00000	0.101535	0.0507673	+	0.998711i	$0.483833\pi$
$98$	−1.00000	−0.101015
$99$	6.00000	0.603023
$100$	0	0
$101$	−5.00000	−0.497519	−0.248759	−	0.968565i	$-0.580023\pi$
$101$	−5.00000	−0.497519	−0.248759	+	0.968565i	$0.580023\pi$
$102$	0	0
$103$	14.0000	1.37946	0.689730	−	0.724066i	$-0.257729\pi$
$103$	14.0000	1.37946	0.689730	+	0.724066i	$0.257729\pi$
$104$	−1.00000	−0.0980581
$105$	0	0
$106$	0	0
$107$	4.00000	0.386695	0.193347	−	0.981130i	$-0.438066\pi$
$107$	4.00000	0.386695	0.193347	+	0.981130i	$0.438066\pi$
$108$	−9.00000	−0.866025
$109$	−14.0000	−1.34096	−0.670478	−	0.741929i	$-0.733911\pi$
$109$	−14.0000	−1.34096	−0.670478	+	0.741929i	$0.733911\pi$
$110$	0	0
$111$	27.0000	2.56273
$112$	1.00000	0.0944911
$113$	7.00000	0.658505	0.329252	−	0.944242i	$-0.393203\pi$
$113$	7.00000	0.658505	0.329252	+	0.944242i	$0.393203\pi$
$114$	−18.0000	−1.68585
$115$	0	0
$116$	−4.00000	−0.371391
$117$	6.00000	0.554700
$118$	10.0000	0.920575
$119$	0	0
$120$	0	0
$121$	−10.0000	−0.909091
$122$	−1.00000	−0.0905357
$123$	9.00000	0.811503
$124$	7.00000	0.628619
$125$	0	0
$126$	−6.00000	−0.534522
$127$	11.0000	0.976092	0.488046	−	0.872818i	$-0.337710\pi$
$127$	11.0000	0.976092	0.488046	+	0.872818i	$0.337710\pi$
$128$	−1.00000	−0.0883883
$129$	12.0000	1.05654
$130$	0	0
$131$	8.00000	0.698963	0.349482	−	0.936943i	$-0.386358\pi$
$131$	8.00000	0.698963	0.349482	+	0.936943i	$0.386358\pi$
$132$	−3.00000	−0.261116
$133$	−6.00000	−0.520266
$134$	1.00000	0.0863868
$135$	0	0
$136$	0	0
$137$	6.00000	0.512615	0.256307	−	0.966595i	$-0.417494\pi$
$137$	6.00000	0.512615	0.256307	+	0.966595i	$0.417494\pi$
$138$	21.0000	1.78764
$139$	4.00000	0.339276	0.169638	−	0.985506i	$-0.445740\pi$
$139$	4.00000	0.339276	0.169638	+	0.985506i	$0.445740\pi$
$140$	0	0
$141$	21.0000	1.76852
$142$	−16.0000	−1.34269
$143$	1.00000	0.0836242
$144$	6.00000	0.500000
$145$	0	0
$146$	5.00000	0.413803
$147$	−3.00000	−0.247436
$148$	−9.00000	−0.739795
$149$	9.00000	0.737309	0.368654	−	0.929567i	$-0.379819\pi$
$149$	9.00000	0.737309	0.368654	+	0.929567i	$0.379819\pi$
$150$	0	0
$151$	0	0		−	1.00000i	$-0.5\pi$
$151$	0	0			1.00000i	$0.5\pi$
$152$	6.00000	0.486664
$153$	0	0
$154$	−1.00000	−0.0805823
$155$	0	0
$156$	−3.00000	−0.240192
$157$	−5.00000	−0.399043	−0.199522	−	0.979893i	$-0.563939\pi$
$157$	−5.00000	−0.399043	−0.199522	+	0.979893i	$0.563939\pi$
$158$	−11.0000	−0.875113
$159$	0	0
$160$	0	0
$161$	7.00000	0.551677
$162$	−9.00000	−0.707107
$163$	4.00000	0.313304	0.156652	−	0.987654i	$-0.449930\pi$
$163$	4.00000	0.313304	0.156652	+	0.987654i	$0.449930\pi$
$164$	−3.00000	−0.234261
$165$	0	0
$166$	0	0
$167$	0	0		−	1.00000i	$-0.5\pi$
$167$	0	0			1.00000i	$0.5\pi$
$168$	3.00000	0.231455
$169$	1.00000	0.0769231
$170$	0	0
$171$	−36.0000	−2.75299
$172$	−4.00000	−0.304997
$173$	−2.00000	−0.152057	−0.0760286	−	0.997106i	$-0.524224\pi$
$173$	−2.00000	−0.152057	−0.0760286	+	0.997106i	$0.524224\pi$
$174$	−12.0000	−0.909718
$175$	0	0
$176$	1.00000	0.0753778
$177$	30.0000	2.25494
$178$	6.00000	0.449719
$179$	−6.00000	−0.448461	−0.224231	−	0.974536i	$-0.571987\pi$
$179$	−6.00000	−0.448461	−0.224231	+	0.974536i	$0.571987\pi$
$180$	0	0
$181$	15.0000	1.11494	0.557471	−	0.830197i	$-0.311772\pi$
$181$	15.0000	1.11494	0.557471	+	0.830197i	$0.311772\pi$
$182$	−1.00000	−0.0741249
$183$	−3.00000	−0.221766
$184$	−7.00000	−0.516047
$185$	0	0
$186$	21.0000	1.53979
$187$	0	0
$188$	−7.00000	−0.510527
$189$	−9.00000	−0.654654
$190$	0	0
$191$	−8.00000	−0.578860	−0.289430	−	0.957199i	$-0.593466\pi$
$191$	−8.00000	−0.578860	−0.289430	+	0.957199i	$0.593466\pi$
$192$	−3.00000	−0.216506
$193$	−20.0000	−1.43963	−0.719816	−	0.694165i	$-0.755774\pi$
$193$	−20.0000	−1.43963	−0.719816	+	0.694165i	$0.755774\pi$
$194$	−1.00000	−0.0717958
$195$	0	0
$196$	1.00000	0.0714286
$197$	27.0000	1.92367	0.961835	−	0.273629i	$-0.0882242\pi$
$197$	27.0000	1.92367	0.961835	+	0.273629i	$0.0882242\pi$
$198$	−6.00000	−0.426401
$199$	−4.00000	−0.283552	−0.141776	−	0.989899i	$-0.545281\pi$
$199$	−4.00000	−0.283552	−0.141776	+	0.989899i	$0.545281\pi$
$200$	0	0
$201$	3.00000	0.211604
$202$	5.00000	0.351799
$203$	−4.00000	−0.280745
$204$	0	0
$205$	0	0
$206$	−14.0000	−0.975426
$207$	42.0000	2.91920
$208$	1.00000	0.0693375
$209$	−6.00000	−0.415029
$210$	0	0
$211$	−10.0000	−0.688428	−0.344214	−	0.938891i	$-0.611855\pi$
$211$	−10.0000	−0.688428	−0.344214	+	0.938891i	$0.611855\pi$
$212$	0	0
$213$	−48.0000	−3.28891
$214$	−4.00000	−0.273434
$215$	0	0
$216$	9.00000	0.612372
$217$	7.00000	0.475191
$218$	14.0000	0.948200
$219$	15.0000	1.01361
$220$	0	0
$221$	0	0
$222$	−27.0000	−1.81212
$223$	−21.0000	−1.40626	−0.703132	−	0.711059i	$-0.748216\pi$
$223$	−21.0000	−1.40626	−0.703132	+	0.711059i	$0.748216\pi$
$224$	−1.00000	−0.0668153
$225$	0	0
$226$	−7.00000	−0.465633
$227$	−24.0000	−1.59294	−0.796468	−	0.604681i	$-0.793301\pi$
$227$	−24.0000	−1.59294	−0.796468	+	0.604681i	$0.793301\pi$
$228$	18.0000	1.19208
$229$	−24.0000	−1.58596	−0.792982	−	0.609245i	$-0.791473\pi$
$229$	−24.0000	−1.58596	−0.792982	+	0.609245i	$0.791473\pi$
$230$	0	0
$231$	−3.00000	−0.197386
$232$	4.00000	0.262613
$233$	5.00000	0.327561	0.163780	−	0.986497i	$-0.447631\pi$
$233$	5.00000	0.327561	0.163780	+	0.986497i	$0.447631\pi$
$234$	−6.00000	−0.392232
$235$	0	0
$236$	−10.0000	−0.650945
$237$	−33.0000	−2.14358
$238$	0	0
$239$	6.00000	0.388108	0.194054	−	0.980991i	$-0.437836\pi$
$239$	6.00000	0.388108	0.194054	+	0.980991i	$0.437836\pi$
$240$	0	0
$241$	−18.0000	−1.15948	−0.579741	−	0.814801i	$-0.696846\pi$
$241$	−18.0000	−1.15948	−0.579741	+	0.814801i	$0.696846\pi$
$242$	10.0000	0.642824
$243$	0	0
$244$	1.00000	0.0640184
$245$	0	0
$246$	−9.00000	−0.573819
$247$	−6.00000	−0.381771
$248$	−7.00000	−0.444500
$249$	0	0
$250$	0	0
$251$	3.00000	0.189358	0.0946792	−	0.995508i	$-0.469817\pi$
$251$	3.00000	0.189358	0.0946792	+	0.995508i	$0.469817\pi$
$252$	6.00000	0.377964
$253$	7.00000	0.440086
$254$	−11.0000	−0.690201
$255$	0	0
$256$	1.00000	0.0625000
$257$	−24.0000	−1.49708	−0.748539	−	0.663090i	$-0.769245\pi$
$257$	−24.0000	−1.49708	−0.748539	+	0.663090i	$0.769245\pi$
$258$	−12.0000	−0.747087
$259$	−9.00000	−0.559233
$260$	0	0
$261$	−24.0000	−1.48556
$262$	−8.00000	−0.494242
$263$	−24.0000	−1.47990	−0.739952	−	0.672660i	$-0.765152\pi$
$263$	−24.0000	−1.47990	−0.739952	+	0.672660i	$0.765152\pi$
$264$	3.00000	0.184637
$265$	0	0
$266$	6.00000	0.367884
$267$	18.0000	1.10158
$268$	−1.00000	−0.0610847
$269$	9.00000	0.548740	0.274370	−	0.961624i	$-0.411531\pi$
$269$	9.00000	0.548740	0.274370	+	0.961624i	$0.411531\pi$
$270$	0	0
$271$	−23.0000	−1.39715	−0.698575	−	0.715537i	$-0.746182\pi$
$271$	−23.0000	−1.39715	−0.698575	+	0.715537i	$0.746182\pi$
$272$	0	0
$273$	−3.00000	−0.181568
$274$	−6.00000	−0.362473
$275$	0	0
$276$	−21.0000	−1.26405
$277$	18.0000	1.08152	0.540758	−	0.841178i	$-0.318138\pi$
$277$	18.0000	1.08152	0.540758	+	0.841178i	$0.318138\pi$
$278$	−4.00000	−0.239904
$279$	42.0000	2.51447
$280$	0	0
$281$	−8.00000	−0.477240	−0.238620	−	0.971113i	$-0.576695\pi$
$281$	−8.00000	−0.477240	−0.238620	+	0.971113i	$0.576695\pi$
$282$	−21.0000	−1.25053
$283$	−19.0000	−1.12943	−0.564716	−	0.825285i	$-0.691014\pi$
$283$	−19.0000	−1.12943	−0.564716	+	0.825285i	$0.691014\pi$
$284$	16.0000	0.949425
$285$	0	0
$286$	−1.00000	−0.0591312
$287$	−3.00000	−0.177084
$288$	−6.00000	−0.353553
$289$	−17.0000	−1.00000
$290$	0	0
$291$	−3.00000	−0.175863
$292$	−5.00000	−0.292603
$293$	26.0000	1.51894	0.759468	−	0.650545i	$-0.225459\pi$
$293$	26.0000	1.51894	0.759468	+	0.650545i	$0.225459\pi$
$294$	3.00000	0.174964
$295$	0	0
$296$	9.00000	0.523114
$297$	−9.00000	−0.522233
$298$	−9.00000	−0.521356
$299$	7.00000	0.404820
$300$	0	0
$301$	−4.00000	−0.230556
$302$	0	0
$303$	15.0000	0.861727
$304$	−6.00000	−0.344124
$305$	0	0
$306$	0	0
$307$	4.00000	0.228292	0.114146	−	0.993464i	$-0.463587\pi$
$307$	4.00000	0.228292	0.114146	+	0.993464i	$0.463587\pi$
$308$	1.00000	0.0569803
$309$	−42.0000	−2.38930
$310$	0	0
$311$	30.0000	1.70114	0.850572	−	0.525859i	$-0.176256\pi$
$311$	30.0000	1.70114	0.850572	+	0.525859i	$0.176256\pi$
$312$	3.00000	0.169842
$313$	−14.0000	−0.791327	−0.395663	−	0.918396i	$-0.629485\pi$
$313$	−14.0000	−0.791327	−0.395663	+	0.918396i	$0.629485\pi$
$314$	5.00000	0.282166
$315$	0	0
$316$	11.0000	0.618798
$317$	−21.0000	−1.17948	−0.589739	−	0.807594i	$-0.700769\pi$
$317$	−21.0000	−1.17948	−0.589739	+	0.807594i	$0.700769\pi$
$318$	0	0
$319$	−4.00000	−0.223957
$320$	0	0
$321$	−12.0000	−0.669775
$322$	−7.00000	−0.390095
$323$	0	0
$324$	9.00000	0.500000
$325$	0	0
$326$	−4.00000	−0.221540
$327$	42.0000	2.32261
$328$	3.00000	0.165647
$329$	−7.00000	−0.385922
$330$	0	0
$331$	−7.00000	−0.384755	−0.192377	−	0.981321i	$-0.561620\pi$
$331$	−7.00000	−0.384755	−0.192377	+	0.981321i	$0.561620\pi$
$332$	0	0
$333$	−54.0000	−2.95918
$334$	0	0
$335$	0	0
$336$	−3.00000	−0.163663
$337$	−17.0000	−0.926049	−0.463025	−	0.886345i	$-0.653236\pi$
$337$	−17.0000	−0.926049	−0.463025	+	0.886345i	$0.653236\pi$
$338$	−1.00000	−0.0543928
$339$	−21.0000	−1.14056
$340$	0	0
$341$	7.00000	0.379071
$342$	36.0000	1.94666
$343$	1.00000	0.0539949
$344$	4.00000	0.215666
$345$	0	0
$346$	2.00000	0.107521
$347$	24.0000	1.28839	0.644194	−	0.764862i	$-0.277193\pi$
$347$	24.0000	1.28839	0.644194	+	0.764862i	$0.277193\pi$
$348$	12.0000	0.643268
$349$	−26.0000	−1.39175	−0.695874	−	0.718164i	$-0.744983\pi$
$349$	−26.0000	−1.39175	−0.695874	+	0.718164i	$0.744983\pi$
$350$	0	0
$351$	−9.00000	−0.480384
$352$	−1.00000	−0.0533002
$353$	−25.0000	−1.33062	−0.665308	−	0.746569i	$-0.731700\pi$
$353$	−25.0000	−1.33062	−0.665308	+	0.746569i	$0.731700\pi$
$354$	−30.0000	−1.59448
$355$	0	0
$356$	−6.00000	−0.317999
$357$	0	0
$358$	6.00000	0.317110
$359$	12.0000	0.633336	0.316668	−	0.948536i	$-0.397436\pi$
$359$	12.0000	0.633336	0.316668	+	0.948536i	$0.397436\pi$
$360$	0	0
$361$	17.0000	0.894737
$362$	−15.0000	−0.788382
$363$	30.0000	1.57459
$364$	1.00000	0.0524142
$365$	0	0
$366$	3.00000	0.156813
$367$	−32.0000	−1.67039	−0.835193	−	0.549957i	$-0.814644\pi$
$367$	−32.0000	−1.67039	−0.835193	+	0.549957i	$0.814644\pi$
$368$	7.00000	0.364900
$369$	−18.0000	−0.937043
$370$	0	0
$371$	0	0
$372$	−21.0000	−1.08880
$373$	16.0000	0.828449	0.414224	−	0.910175i	$-0.364053\pi$
$373$	16.0000	0.828449	0.414224	+	0.910175i	$0.364053\pi$
$374$	0	0
$375$	0	0
$376$	7.00000	0.360997
$377$	−4.00000	−0.206010
$378$	9.00000	0.462910
$379$	−8.00000	−0.410932	−0.205466	−	0.978664i	$-0.565871\pi$
$379$	−8.00000	−0.410932	−0.205466	+	0.978664i	$0.565871\pi$
$380$	0	0
$381$	−33.0000	−1.69064
$382$	8.00000	0.409316
$383$	15.0000	0.766464	0.383232	−	0.923652i	$-0.374811\pi$
$383$	15.0000	0.766464	0.383232	+	0.923652i	$0.374811\pi$
$384$	3.00000	0.153093
$385$	0	0
$386$	20.0000	1.01797
$387$	−24.0000	−1.21999
$388$	1.00000	0.0507673
$389$	−36.0000	−1.82527	−0.912636	−	0.408773i	$-0.865957\pi$
$389$	−36.0000	−1.82527	−0.912636	+	0.408773i	$0.865957\pi$
$390$	0	0
$391$	0	0
$392$	−1.00000	−0.0505076
$393$	−24.0000	−1.21064
$394$	−27.0000	−1.36024
$395$	0	0
$396$	6.00000	0.301511
$397$	28.0000	1.40528	0.702640	−	0.711546i	$-0.252005\pi$
$397$	28.0000	1.40528	0.702640	+	0.711546i	$0.252005\pi$
$398$	4.00000	0.200502
$399$	18.0000	0.901127
$400$	0	0
$401$	12.0000	0.599251	0.299626	−	0.954057i	$-0.403138\pi$
$401$	12.0000	0.599251	0.299626	+	0.954057i	$0.403138\pi$
$402$	−3.00000	−0.149626
$403$	7.00000	0.348695
$404$	−5.00000	−0.248759
$405$	0	0
$406$	4.00000	0.198517
$407$	−9.00000	−0.446113
$408$	0	0
$409$	−26.0000	−1.28562	−0.642809	−	0.766027i	$-0.722231\pi$
$409$	−26.0000	−1.28562	−0.642809	+	0.766027i	$0.722231\pi$
$410$	0	0
$411$	−18.0000	−0.887875
$412$	14.0000	0.689730
$413$	−10.0000	−0.492068
$414$	−42.0000	−2.06419
$415$	0	0
$416$	−1.00000	−0.0490290
$417$	−12.0000	−0.587643
$418$	6.00000	0.293470
$419$	15.0000	0.732798	0.366399	−	0.930458i	$-0.380591\pi$
$419$	15.0000	0.732798	0.366399	+	0.930458i	$0.380591\pi$
$420$	0	0
$421$	19.0000	0.926003	0.463002	−	0.886357i	$-0.346772\pi$
$421$	19.0000	0.926003	0.463002	+	0.886357i	$0.346772\pi$
$422$	10.0000	0.486792
$423$	−42.0000	−2.04211
$424$	0	0
$425$	0	0
$426$	48.0000	2.32561
$427$	1.00000	0.0483934
$428$	4.00000	0.193347
$429$	−3.00000	−0.144841
$430$	0	0
$431$	18.0000	0.867029	0.433515	−	0.901146i	$-0.357273\pi$
$431$	18.0000	0.867029	0.433515	+	0.901146i	$0.357273\pi$
$432$	−9.00000	−0.433013
$433$	−34.0000	−1.63394	−0.816968	−	0.576683i	$-0.804347\pi$
$433$	−34.0000	−1.63394	−0.816968	+	0.576683i	$0.804347\pi$
$434$	−7.00000	−0.336011
$435$	0	0
$436$	−14.0000	−0.670478
$437$	−42.0000	−2.00913
$438$	−15.0000	−0.716728
$439$	2.00000	0.0954548	0.0477274	−	0.998860i	$-0.484802\pi$
$439$	2.00000	0.0954548	0.0477274	+	0.998860i	$0.484802\pi$
$440$	0	0
$441$	6.00000	0.285714
$442$	0	0
$443$	−12.0000	−0.570137	−0.285069	−	0.958507i	$-0.592016\pi$
$443$	−12.0000	−0.570137	−0.285069	+	0.958507i	$0.592016\pi$
$444$	27.0000	1.28136
$445$	0	0
$446$	21.0000	0.994379
$447$	−27.0000	−1.27706
$448$	1.00000	0.0472456
$449$	−2.00000	−0.0943858	−0.0471929	−	0.998886i	$-0.515028\pi$
$449$	−2.00000	−0.0943858	−0.0471929	+	0.998886i	$0.515028\pi$
$450$	0	0
$451$	−3.00000	−0.141264
$452$	7.00000	0.329252
$453$	0	0
$454$	24.0000	1.12638
$455$	0	0
$456$	−18.0000	−0.842927
$457$	26.0000	1.21623	0.608114	−	0.793849i	$-0.291926\pi$
$457$	26.0000	1.21623	0.608114	+	0.793849i	$0.291926\pi$
$458$	24.0000	1.12145
$459$	0	0
$460$	0	0
$461$	12.0000	0.558896	0.279448	−	0.960161i	$-0.409849\pi$
$461$	12.0000	0.558896	0.279448	+	0.960161i	$0.409849\pi$
$462$	3.00000	0.139573
$463$	−16.0000	−0.743583	−0.371792	−	0.928316i	$-0.621256\pi$
$463$	−16.0000	−0.743583	−0.371792	+	0.928316i	$0.621256\pi$
$464$	−4.00000	−0.185695
$465$	0	0
$466$	−5.00000	−0.231621
$467$	4.00000	0.185098	0.0925490	−	0.995708i	$-0.470499\pi$
$467$	4.00000	0.185098	0.0925490	+	0.995708i	$0.470499\pi$
$468$	6.00000	0.277350
$469$	−1.00000	−0.0461757
$470$	0	0
$471$	15.0000	0.691164
$472$	10.0000	0.460287
$473$	−4.00000	−0.183920
$474$	33.0000	1.51574
$475$	0	0
$476$	0	0
$477$	0	0
$478$	−6.00000	−0.274434
$479$	24.0000	1.09659	0.548294	−	0.836286i	$-0.315277\pi$
$479$	24.0000	1.09659	0.548294	+	0.836286i	$0.315277\pi$
$480$	0	0
$481$	−9.00000	−0.410365
$482$	18.0000	0.819878
$483$	−21.0000	−0.955533
$484$	−10.0000	−0.454545
$485$	0	0
$486$	0	0
$487$	−38.0000	−1.72194	−0.860972	−	0.508652i	$-0.830144\pi$
$487$	−38.0000	−1.72194	−0.860972	+	0.508652i	$0.830144\pi$
$488$	−1.00000	−0.0452679
$489$	−12.0000	−0.542659
$490$	0	0
$491$	16.0000	0.722070	0.361035	−	0.932552i	$-0.382424\pi$
$491$	16.0000	0.722070	0.361035	+	0.932552i	$0.382424\pi$
$492$	9.00000	0.405751
$493$	0	0
$494$	6.00000	0.269953
$495$	0	0
$496$	7.00000	0.314309
$497$	16.0000	0.717698
$498$	0	0
$499$	−5.00000	−0.223831	−0.111915	−	0.993718i	$-0.535699\pi$
$499$	−5.00000	−0.223831	−0.111915	+	0.993718i	$0.535699\pi$
$500$	0	0
$501$	0	0
$502$	−3.00000	−0.133897
$503$	20.0000	0.891756	0.445878	−	0.895094i	$-0.352892\pi$
$503$	20.0000	0.891756	0.445878	+	0.895094i	$0.352892\pi$
$504$	−6.00000	−0.267261
$505$	0	0
$506$	−7.00000	−0.311188
$507$	−3.00000	−0.133235
$508$	11.0000	0.488046
$509$	10.0000	0.443242	0.221621	−	0.975133i	$-0.428865\pi$
$509$	10.0000	0.443242	0.221621	+	0.975133i	$0.428865\pi$
$510$	0	0
$511$	−5.00000	−0.221187
$512$	−1.00000	−0.0441942
$513$	54.0000	2.38416
$514$	24.0000	1.05859
$515$	0	0
$516$	12.0000	0.528271
$517$	−7.00000	−0.307860
$518$	9.00000	0.395437
$519$	6.00000	0.263371
$520$	0	0
$521$	−18.0000	−0.788594	−0.394297	−	0.918983i	$-0.629012\pi$
$521$	−18.0000	−0.788594	−0.394297	+	0.918983i	$0.629012\pi$
$522$	24.0000	1.05045
$523$	−27.0000	−1.18063	−0.590314	−	0.807174i	$-0.700996\pi$
$523$	−27.0000	−1.18063	−0.590314	+	0.807174i	$0.700996\pi$
$524$	8.00000	0.349482
$525$	0	0
$526$	24.0000	1.04645
$527$	0	0
$528$	−3.00000	−0.130558
$529$	26.0000	1.13043
$530$	0	0
$531$	−60.0000	−2.60378
$532$	−6.00000	−0.260133
$533$	−3.00000	−0.129944
$534$	−18.0000	−0.778936
$535$	0	0
$536$	1.00000	0.0431934
$537$	18.0000	0.776757
$538$	−9.00000	−0.388018
$539$	1.00000	0.0430730
$540$	0	0
$541$	−10.0000	−0.429934	−0.214967	−	0.976621i	$-0.568964\pi$
$541$	−10.0000	−0.429934	−0.214967	+	0.976621i	$0.568964\pi$
$542$	23.0000	0.987935
$543$	−45.0000	−1.93113
$544$	0	0
$545$	0	0
$546$	3.00000	0.128388
$547$	−28.0000	−1.19719	−0.598597	−	0.801050i	$-0.704275\pi$
$547$	−28.0000	−1.19719	−0.598597	+	0.801050i	$0.704275\pi$
$548$	6.00000	0.256307
$549$	6.00000	0.256074
$550$	0	0
$551$	24.0000	1.02243
$552$	21.0000	0.893819
$553$	11.0000	0.467768
$554$	−18.0000	−0.764747
$555$	0	0
$556$	4.00000	0.169638
$557$	−9.00000	−0.381342	−0.190671	−	0.981654i	$-0.561066\pi$
$557$	−9.00000	−0.381342	−0.190671	+	0.981654i	$0.561066\pi$
$558$	−42.0000	−1.77800
$559$	−4.00000	−0.169182
$560$	0	0
$561$	0	0
$562$	8.00000	0.337460
$563$	−11.0000	−0.463595	−0.231797	−	0.972764i	$-0.574461\pi$
$563$	−11.0000	−0.463595	−0.231797	+	0.972764i	$0.574461\pi$
$564$	21.0000	0.884260
$565$	0	0
$566$	19.0000	0.798630
$567$	9.00000	0.377964
$568$	−16.0000	−0.671345
$569$	−35.0000	−1.46728	−0.733638	−	0.679540i	$-0.762179\pi$
$569$	−35.0000	−1.46728	−0.733638	+	0.679540i	$0.762179\pi$
$570$	0	0
$571$	6.00000	0.251092	0.125546	−	0.992088i	$-0.459932\pi$
$571$	6.00000	0.251092	0.125546	+	0.992088i	$0.459932\pi$
$572$	1.00000	0.0418121
$573$	24.0000	1.00261
$574$	3.00000	0.125218
$575$	0	0
$576$	6.00000	0.250000
$577$	−30.0000	−1.24892	−0.624458	−	0.781058i	$-0.714680\pi$
$577$	−30.0000	−1.24892	−0.624458	+	0.781058i	$0.714680\pi$
$578$	17.0000	0.707107
$579$	60.0000	2.49351
$580$	0	0
$581$	0	0
$582$	3.00000	0.124354
$583$	0	0
$584$	5.00000	0.206901
$585$	0	0
$586$	−26.0000	−1.07405
$587$	−14.0000	−0.577842	−0.288921	−	0.957353i	$-0.593296\pi$
$587$	−14.0000	−0.577842	−0.288921	+	0.957353i	$0.593296\pi$
$588$	−3.00000	−0.123718
$589$	−42.0000	−1.73058
$590$	0	0
$591$	−81.0000	−3.33189
$592$	−9.00000	−0.369898
$593$	−14.0000	−0.574911	−0.287456	−	0.957794i	$-0.592809\pi$
$593$	−14.0000	−0.574911	−0.287456	+	0.957794i	$0.592809\pi$
$594$	9.00000	0.369274
$595$	0	0
$596$	9.00000	0.368654
$597$	12.0000	0.491127
$598$	−7.00000	−0.286251
$599$	−29.0000	−1.18491	−0.592454	−	0.805604i	$-0.701841\pi$
$599$	−29.0000	−1.18491	−0.592454	+	0.805604i	$0.701841\pi$
$600$	0	0
$601$	−26.0000	−1.06056	−0.530281	−	0.847822i	$-0.677914\pi$
$601$	−26.0000	−1.06056	−0.530281	+	0.847822i	$0.677914\pi$
$602$	4.00000	0.163028
$603$	−6.00000	−0.244339
$604$	0	0
$605$	0	0
$606$	−15.0000	−0.609333
$607$	−6.00000	−0.243532	−0.121766	−	0.992559i	$-0.538856\pi$
$607$	−6.00000	−0.243532	−0.121766	+	0.992559i	$0.538856\pi$
$608$	6.00000	0.243332
$609$	12.0000	0.486265
$610$	0	0
$611$	−7.00000	−0.283190
$612$	0	0
$613$	49.0000	1.97909	0.989546	−	0.144220i	$-0.0460672\pi$
$613$	49.0000	1.97909	0.989546	+	0.144220i	$0.0460672\pi$
$614$	−4.00000	−0.161427
$615$	0	0
$616$	−1.00000	−0.0402911
$617$	−22.0000	−0.885687	−0.442843	−	0.896599i	$-0.646030\pi$
$617$	−22.0000	−0.885687	−0.442843	+	0.896599i	$0.646030\pi$
$618$	42.0000	1.68949
$619$	22.0000	0.884255	0.442127	−	0.896952i	$-0.354224\pi$
$619$	22.0000	0.884255	0.442127	+	0.896952i	$0.354224\pi$
$620$	0	0
$621$	−63.0000	−2.52810
$622$	−30.0000	−1.20289
$623$	−6.00000	−0.240385
$624$	−3.00000	−0.120096
$625$	0	0
$626$	14.0000	0.559553
$627$	18.0000	0.718851
$628$	−5.00000	−0.199522
$629$	0	0
$630$	0	0
$631$	−2.00000	−0.0796187	−0.0398094	−	0.999207i	$-0.512675\pi$
$631$	−2.00000	−0.0796187	−0.0398094	+	0.999207i	$0.512675\pi$
$632$	−11.0000	−0.437557
$633$	30.0000	1.19239
$634$	21.0000	0.834017
$635$	0	0
$636$	0	0
$637$	1.00000	0.0396214
$638$	4.00000	0.158362
$639$	96.0000	3.79770
$640$	0	0
$641$	7.00000	0.276483	0.138242	−	0.990399i	$-0.455855\pi$
$641$	7.00000	0.276483	0.138242	+	0.990399i	$0.455855\pi$
$642$	12.0000	0.473602
$643$	16.0000	0.630978	0.315489	−	0.948929i	$-0.397831\pi$
$643$	16.0000	0.630978	0.315489	+	0.948929i	$0.397831\pi$
$644$	7.00000	0.275839
$645$	0	0
$646$	0	0
$647$	8.00000	0.314512	0.157256	−	0.987558i	$-0.449735\pi$
$647$	8.00000	0.314512	0.157256	+	0.987558i	$0.449735\pi$
$648$	−9.00000	−0.353553
$649$	−10.0000	−0.392534
$650$	0	0
$651$	−21.0000	−0.823055
$652$	4.00000	0.156652
$653$	36.0000	1.40879	0.704394	−	0.709809i	$-0.251219\pi$
$653$	36.0000	1.40879	0.704394	+	0.709809i	$0.251219\pi$
$654$	−42.0000	−1.64233
$655$	0	0
$656$	−3.00000	−0.117130
$657$	−30.0000	−1.17041
$658$	7.00000	0.272888
$659$	−18.0000	−0.701180	−0.350590	−	0.936529i	$-0.614019\pi$
$659$	−18.0000	−0.701180	−0.350590	+	0.936529i	$0.614019\pi$
$660$	0	0
$661$	−4.00000	−0.155582	−0.0777910	−	0.996970i	$-0.524787\pi$
$661$	−4.00000	−0.155582	−0.0777910	+	0.996970i	$0.524787\pi$
$662$	7.00000	0.272063
$663$	0	0
$664$	0	0
$665$	0	0
$666$	54.0000	2.09246
$667$	−28.0000	−1.08416
$668$	0	0
$669$	63.0000	2.43572
$670$	0	0
$671$	1.00000	0.0386046
$672$	3.00000	0.115728
$673$	−7.00000	−0.269830	−0.134915	−	0.990857i	$-0.543076\pi$
$673$	−7.00000	−0.269830	−0.134915	+	0.990857i	$0.543076\pi$
$674$	17.0000	0.654816
$675$	0	0
$676$	1.00000	0.0384615
$677$	−39.0000	−1.49889	−0.749446	−	0.662066i	$-0.769680\pi$
$677$	−39.0000	−1.49889	−0.749446	+	0.662066i	$0.769680\pi$
$678$	21.0000	0.806500
$679$	1.00000	0.0383765
$680$	0	0
$681$	72.0000	2.75905
$682$	−7.00000	−0.268044
$683$	−1.00000	−0.0382639	−0.0191320	−	0.999817i	$-0.506090\pi$
$683$	−1.00000	−0.0382639	−0.0191320	+	0.999817i	$0.506090\pi$
$684$	−36.0000	−1.37649
$685$	0	0
$686$	−1.00000	−0.0381802
$687$	72.0000	2.74697
$688$	−4.00000	−0.152499
$689$	0	0
$690$	0	0
$691$	−32.0000	−1.21734	−0.608669	−	0.793424i	$-0.708296\pi$
$691$	−32.0000	−1.21734	−0.608669	+	0.793424i	$0.708296\pi$
$692$	−2.00000	−0.0760286
$693$	6.00000	0.227921
$694$	−24.0000	−0.911028
$695$	0	0
$696$	−12.0000	−0.454859
$697$	0	0
$698$	26.0000	0.984115
$699$	−15.0000	−0.567352
$700$	0	0
$701$	0	0		−	1.00000i	$-0.5\pi$
$701$	0	0			1.00000i	$0.5\pi$
$702$	9.00000	0.339683
$703$	54.0000	2.03665
$704$	1.00000	0.0376889
$705$	0	0
$706$	25.0000	0.940887
$707$	−5.00000	−0.188044
$708$	30.0000	1.12747
$709$	−13.0000	−0.488225	−0.244113	−	0.969747i	$-0.578497\pi$
$709$	−13.0000	−0.488225	−0.244113	+	0.969747i	$0.578497\pi$
$710$	0	0
$711$	66.0000	2.47519
$712$	6.00000	0.224860
$713$	49.0000	1.83506
$714$	0	0
$715$	0	0
$716$	−6.00000	−0.224231
$717$	−18.0000	−0.672222
$718$	−12.0000	−0.447836
$719$	2.00000	0.0745874	0.0372937	−	0.999304i	$-0.488126\pi$
$719$	2.00000	0.0745874	0.0372937	+	0.999304i	$0.488126\pi$
$720$	0	0
$721$	14.0000	0.521387
$722$	−17.0000	−0.632674
$723$	54.0000	2.00828
$724$	15.0000	0.557471
$725$	0	0
$726$	−30.0000	−1.11340
$727$	26.0000	0.964287	0.482143	−	0.876092i	$-0.339858\pi$
$727$	26.0000	0.964287	0.482143	+	0.876092i	$0.339858\pi$
$728$	−1.00000	−0.0370625
$729$	−27.0000	−1.00000
$730$	0	0
$731$	0	0
$732$	−3.00000	−0.110883
$733$	4.00000	0.147743	0.0738717	−	0.997268i	$-0.476464\pi$
$733$	4.00000	0.147743	0.0738717	+	0.997268i	$0.476464\pi$
$734$	32.0000	1.18114
$735$	0	0
$736$	−7.00000	−0.258023
$737$	−1.00000	−0.0368355
$738$	18.0000	0.662589
$739$	24.0000	0.882854	0.441427	−	0.897297i	$-0.354472\pi$
$739$	24.0000	0.882854	0.441427	+	0.897297i	$0.354472\pi$
$740$	0	0
$741$	18.0000	0.661247
$742$	0	0
$743$	−16.0000	−0.586983	−0.293492	−	0.955962i	$-0.594817\pi$
$743$	−16.0000	−0.586983	−0.293492	+	0.955962i	$0.594817\pi$
$744$	21.0000	0.769897
$745$	0	0
$746$	−16.0000	−0.585802
$747$	0	0
$748$	0	0
$749$	4.00000	0.146157
$750$	0	0
$751$	−27.0000	−0.985244	−0.492622	−	0.870243i	$-0.663961\pi$
$751$	−27.0000	−0.985244	−0.492622	+	0.870243i	$0.663961\pi$
$752$	−7.00000	−0.255264
$753$	−9.00000	−0.327978
$754$	4.00000	0.145671
$755$	0	0
$756$	−9.00000	−0.327327
$757$	−54.0000	−1.96266	−0.981332	−	0.192323i	$-0.938398\pi$
$757$	−54.0000	−1.96266	−0.981332	+	0.192323i	$0.938398\pi$
$758$	8.00000	0.290573
$759$	−21.0000	−0.762252
$760$	0	0
$761$	45.0000	1.63125	0.815624	−	0.578582i	$-0.196394\pi$
$761$	45.0000	1.63125	0.815624	+	0.578582i	$0.196394\pi$
$762$	33.0000	1.19546
$763$	−14.0000	−0.506834
$764$	−8.00000	−0.289430
$765$	0	0
$766$	−15.0000	−0.541972
$767$	−10.0000	−0.361079
$768$	−3.00000	−0.108253
$769$	21.0000	0.757279	0.378640	−	0.925544i	$-0.376392\pi$
$769$	21.0000	0.757279	0.378640	+	0.925544i	$0.376392\pi$
$770$	0	0
$771$	72.0000	2.59302
$772$	−20.0000	−0.719816
$773$	−14.0000	−0.503545	−0.251773	−	0.967786i	$-0.581013\pi$
$773$	−14.0000	−0.503545	−0.251773	+	0.967786i	$0.581013\pi$
$774$	24.0000	0.862662
$775$	0	0
$776$	−1.00000	−0.0358979
$777$	27.0000	0.968620
$778$	36.0000	1.29066
$779$	18.0000	0.644917
$780$	0	0
$781$	16.0000	0.572525
$782$	0	0
$783$	36.0000	1.28654
$784$	1.00000	0.0357143
$785$	0	0
$786$	24.0000	0.856052
$787$	−32.0000	−1.14068	−0.570338	−	0.821410i	$-0.693188\pi$
$787$	−32.0000	−1.14068	−0.570338	+	0.821410i	$0.693188\pi$
$788$	27.0000	0.961835
$789$	72.0000	2.56327
$790$	0	0
$791$	7.00000	0.248891
$792$	−6.00000	−0.213201
$793$	1.00000	0.0355110
$794$	−28.0000	−0.993683
$795$	0	0
$796$	−4.00000	−0.141776
$797$	−15.0000	−0.531327	−0.265664	−	0.964066i	$-0.585591\pi$
$797$	−15.0000	−0.531327	−0.265664	+	0.964066i	$0.585591\pi$
$798$	−18.0000	−0.637193
$799$	0	0
$800$	0	0
$801$	−36.0000	−1.27200
$802$	−12.0000	−0.423735
$803$	−5.00000	−0.176446
$804$	3.00000	0.105802
$805$	0	0
$806$	−7.00000	−0.246564
$807$	−27.0000	−0.950445
$808$	5.00000	0.175899
$809$	6.00000	0.210949	0.105474	−	0.994422i	$-0.466364\pi$
$809$	6.00000	0.210949	0.105474	+	0.994422i	$0.466364\pi$
$810$	0	0
$811$	−28.0000	−0.983213	−0.491606	−	0.870817i	$-0.663590\pi$
$811$	−28.0000	−0.983213	−0.491606	+	0.870817i	$0.663590\pi$
$812$	−4.00000	−0.140372
$813$	69.0000	2.41994
$814$	9.00000	0.315450
$815$	0	0
$816$	0	0
$817$	24.0000	0.839654
$818$	26.0000	0.909069
$819$	6.00000	0.209657
$820$	0	0
$821$	−46.0000	−1.60541	−0.802706	−	0.596376i	$-0.796607\pi$
$821$	−46.0000	−1.60541	−0.802706	+	0.596376i	$0.796607\pi$
$822$	18.0000	0.627822
$823$	3.00000	0.104573	0.0522867	−	0.998632i	$-0.483349\pi$
$823$	3.00000	0.104573	0.0522867	+	0.998632i	$0.483349\pi$
$824$	−14.0000	−0.487713
$825$	0	0
$826$	10.0000	0.347945
$827$	12.0000	0.417281	0.208640	−	0.977992i	$-0.433096\pi$
$827$	12.0000	0.417281	0.208640	+	0.977992i	$0.433096\pi$
$828$	42.0000	1.45960
$829$	−14.0000	−0.486240	−0.243120	−	0.969996i	$-0.578171\pi$
$829$	−14.0000	−0.486240	−0.243120	+	0.969996i	$0.578171\pi$
$830$	0	0
$831$	−54.0000	−1.87324
$832$	1.00000	0.0346688
$833$	0	0
$834$	12.0000	0.415526
$835$	0	0
$836$	−6.00000	−0.207514
$837$	−63.0000	−2.17760
$838$	−15.0000	−0.518166
$839$	17.0000	0.586905	0.293453	−	0.955974i	$-0.405196\pi$
$839$	17.0000	0.586905	0.293453	+	0.955974i	$0.405196\pi$
$840$	0	0
$841$	−13.0000	−0.448276
$842$	−19.0000	−0.654783
$843$	24.0000	0.826604
$844$	−10.0000	−0.344214
$845$	0	0
$846$	42.0000	1.44399
$847$	−10.0000	−0.343604
$848$	0	0
$849$	57.0000	1.95623
$850$	0	0
$851$	−63.0000	−2.15961
$852$	−48.0000	−1.64445
$853$	−16.0000	−0.547830	−0.273915	−	0.961754i	$-0.588319\pi$
$853$	−16.0000	−0.547830	−0.273915	+	0.961754i	$0.588319\pi$
$854$	−1.00000	−0.0342193
$855$	0	0
$856$	−4.00000	−0.136717
$857$	−22.0000	−0.751506	−0.375753	−	0.926720i	$-0.622616\pi$
$857$	−22.0000	−0.751506	−0.375753	+	0.926720i	$0.622616\pi$
$858$	3.00000	0.102418
$859$	5.00000	0.170598	0.0852989	−	0.996355i	$-0.472815\pi$
$859$	5.00000	0.170598	0.0852989	+	0.996355i	$0.472815\pi$
$860$	0	0
$861$	9.00000	0.306719
$862$	−18.0000	−0.613082
$863$	−28.0000	−0.953131	−0.476566	−	0.879139i	$-0.658119\pi$
$863$	−28.0000	−0.953131	−0.476566	+	0.879139i	$0.658119\pi$
$864$	9.00000	0.306186
$865$	0	0
$866$	34.0000	1.15537
$867$	51.0000	1.73205
$868$	7.00000	0.237595
$869$	11.0000	0.373149
$870$	0	0
$871$	−1.00000	−0.0338837
$872$	14.0000	0.474100
$873$	6.00000	0.203069
$874$	42.0000	1.42067
$875$	0	0
$876$	15.0000	0.506803
$877$	45.0000	1.51954	0.759771	−	0.650191i	$-0.225311\pi$
$877$	45.0000	1.51954	0.759771	+	0.650191i	$0.225311\pi$
$878$	−2.00000	−0.0674967
$879$	−78.0000	−2.63087
$880$	0	0
$881$	18.0000	0.606435	0.303218	−	0.952921i	$-0.401939\pi$
$881$	18.0000	0.606435	0.303218	+	0.952921i	$0.401939\pi$
$882$	−6.00000	−0.202031
$883$	14.0000	0.471138	0.235569	−	0.971858i	$-0.424305\pi$
$883$	14.0000	0.471138	0.235569	+	0.971858i	$0.424305\pi$
$884$	0	0
$885$	0	0
$886$	12.0000	0.403148
$887$	38.0000	1.27592	0.637958	−	0.770072i	$-0.279780\pi$
$887$	38.0000	1.27592	0.637958	+	0.770072i	$0.279780\pi$
$888$	−27.0000	−0.906061
$889$	11.0000	0.368928
$890$	0	0
$891$	9.00000	0.301511
$892$	−21.0000	−0.703132
$893$	42.0000	1.40548
$894$	27.0000	0.903015
$895$	0	0
$896$	−1.00000	−0.0334077
$897$	−21.0000	−0.701170
$898$	2.00000	0.0667409
$899$	−28.0000	−0.933852
$900$	0	0
$901$	0	0
$902$	3.00000	0.0998891
$903$	12.0000	0.399335
$904$	−7.00000	−0.232817
$905$	0	0
$906$	0	0
$907$	30.0000	0.996134	0.498067	−	0.867139i	$-0.334043\pi$
$907$	30.0000	0.996134	0.498067	+	0.867139i	$0.334043\pi$
$908$	−24.0000	−0.796468
$909$	−30.0000	−0.995037
$910$	0	0
$911$	36.0000	1.19273	0.596367	−	0.802712i	$-0.296610\pi$
$911$	36.0000	1.19273	0.596367	+	0.802712i	$0.296610\pi$
$912$	18.0000	0.596040
$913$	0	0
$914$	−26.0000	−0.860004
$915$	0	0
$916$	−24.0000	−0.792982
$917$	8.00000	0.264183
$918$	0	0
$919$	27.0000	0.890648	0.445324	−	0.895370i	$-0.353089\pi$
$919$	27.0000	0.890648	0.445324	+	0.895370i	$0.353089\pi$
$920$	0	0
$921$	−12.0000	−0.395413
$922$	−12.0000	−0.395199
$923$	16.0000	0.526646
$924$	−3.00000	−0.0986928
$925$	0	0
$926$	16.0000	0.525793
$927$	84.0000	2.75892
$928$	4.00000	0.131306
$929$	−15.0000	−0.492134	−0.246067	−	0.969253i	$-0.579138\pi$
$929$	−15.0000	−0.492134	−0.246067	+	0.969253i	$0.579138\pi$
$930$	0	0
$931$	−6.00000	−0.196642
$932$	5.00000	0.163780
$933$	−90.0000	−2.94647
$934$	−4.00000	−0.130884
$935$	0	0
$936$	−6.00000	−0.196116
$937$	36.0000	1.17607	0.588034	−	0.808836i	$-0.299902\pi$
$937$	36.0000	1.17607	0.588034	+	0.808836i	$0.299902\pi$
$938$	1.00000	0.0326512
$939$	42.0000	1.37062
$940$	0	0
$941$	−50.0000	−1.62995	−0.814977	−	0.579494i	$-0.803250\pi$
$941$	−50.0000	−1.62995	−0.814977	+	0.579494i	$0.803250\pi$
$942$	−15.0000	−0.488726
$943$	−21.0000	−0.683854
$944$	−10.0000	−0.325472
$945$	0	0
$946$	4.00000	0.130051
$947$	−24.0000	−0.779895	−0.389948	−	0.920837i	$-0.627507\pi$
$947$	−24.0000	−0.779895	−0.389948	+	0.920837i	$0.627507\pi$
$948$	−33.0000	−1.07179
$949$	−5.00000	−0.162307
$950$	0	0
$951$	63.0000	2.04291
$952$	0	0
$953$	34.0000	1.10137	0.550684	−	0.834714i	$-0.314367\pi$
$953$	34.0000	1.10137	0.550684	+	0.834714i	$0.314367\pi$
$954$	0	0
$955$	0	0
$956$	6.00000	0.194054
$957$	12.0000	0.387905
$958$	−24.0000	−0.775405
$959$	6.00000	0.193750
$960$	0	0
$961$	18.0000	0.580645
$962$	9.00000	0.290172
$963$	24.0000	0.773389
$964$	−18.0000	−0.579741
$965$	0	0
$966$	21.0000	0.675664
$967$	22.0000	0.707472	0.353736	−	0.935345i	$-0.384911\pi$
$967$	22.0000	0.707472	0.353736	+	0.935345i	$0.384911\pi$
$968$	10.0000	0.321412
$969$	0	0
$970$	0	0
$971$	−15.0000	−0.481373	−0.240686	−	0.970603i	$-0.577373\pi$
$971$	−15.0000	−0.481373	−0.240686	+	0.970603i	$0.577373\pi$
$972$	0	0
$973$	4.00000	0.128234
$974$	38.0000	1.21760
$975$	0	0
$976$	1.00000	0.0320092
$977$	−42.0000	−1.34370	−0.671850	−	0.740688i	$-0.734500\pi$
$977$	−42.0000	−1.34370	−0.671850	+	0.740688i	$0.734500\pi$
$978$	12.0000	0.383718
$979$	−6.00000	−0.191761
$980$	0	0
$981$	−84.0000	−2.68191
$982$	−16.0000	−0.510581
$983$	40.0000	1.27580	0.637901	−	0.770118i	$-0.279803\pi$
$983$	40.0000	1.27580	0.637901	+	0.770118i	$0.279803\pi$
$984$	−9.00000	−0.286910
$985$	0	0
$986$	0	0
$987$	21.0000	0.668437
$988$	−6.00000	−0.190885
$989$	−28.0000	−0.890348
$990$	0	0
$991$	−57.0000	−1.81066	−0.905332	−	0.424704i	$-0.860378\pi$
$991$	−57.0000	−1.81066	−0.905332	+	0.424704i	$0.860378\pi$
$992$	−7.00000	−0.222250
$993$	21.0000	0.666415
$994$	−16.0000	−0.507489
$995$	0	0
$996$	0	0
$997$	−5.00000	−0.158352	−0.0791758	−	0.996861i	$-0.525229\pi$
$997$	−5.00000	−0.158352	−0.0791758	+	0.996861i	$0.525229\pi$
$998$	5.00000	0.158272
$999$	81.0000	2.56273

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4550.2.a.a.1.1		1
5.4	even	2		182.2.a.e.1.1	✓	1
15.14	odd	2		1638.2.a.j.1.1		1
20.19	odd	2		1456.2.a.a.1.1		1
35.4	even	6		1274.2.f.b.79.1		2
35.9	even	6		1274.2.f.b.1145.1		2
35.19	odd	6		1274.2.f.k.1145.1		2
35.24	odd	6		1274.2.f.k.79.1		2
35.34	odd	2		1274.2.a.h.1.1		1
40.19	odd	2		5824.2.a.bf.1.1		1
40.29	even	2		5824.2.a.b.1.1		1
65.34	odd	4		2366.2.d.j.337.2		2
65.44	odd	4		2366.2.d.j.337.1		2
65.64	even	2		2366.2.a.h.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
182.2.a.e.1.1	✓	1	5.4	even	2
1274.2.a.h.1.1		1	35.34	odd	2
1274.2.f.b.79.1		2	35.4	even	6
1274.2.f.b.1145.1		2	35.9	even	6
1274.2.f.k.79.1		2	35.24	odd	6
1274.2.f.k.1145.1		2	35.19	odd	6
1456.2.a.a.1.1		1	20.19	odd	2
1638.2.a.j.1.1		1	15.14	odd	2
2366.2.a.h.1.1		1	65.64	even	2
2366.2.d.j.337.1		2	65.44	odd	4
2366.2.d.j.337.2		2	65.34	odd	4
4550.2.a.a.1.1		1	1.1	even	1	trivial
5824.2.a.b.1.1		1	40.29	even	2
5824.2.a.bf.1.1		1	40.19	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:	\( N \)	\(=\)	\( 4550 = 2 \cdot 5^{2} \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4550.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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