LMFDB - Embedded newform 504.2.c.a.253.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [504,2,Mod(253,504)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(504, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 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("504.253");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 504 = 2^{3} \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	504.c (of order $2$, degree $1$, minimal)

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:	no
Analytic conductor:	$4.02446026187$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{-2}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 2 $ x^2 + 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 56)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			253.1
Root			$-1.41421i$ of defining polynomial
Character	$\chi$	$=$	504.253
Dual form			504.2.c.a.253.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.41421i q^{2} -2.00000 q^{4} +1.41421i q^{5} +1.00000 q^{7} +2.82843i q^{8} +O(q^{10})$	$q-1.41421i q^{2} -2.00000 q^{4} +1.41421i q^{5} +1.00000 q^{7} +2.82843i q^{8} +2.00000 q^{10} -2.82843i q^{11} -4.24264i q^{13} -1.41421i q^{14} +4.00000 q^{16} +6.00000 q^{17} -4.24264i q^{19} -2.82843i q^{20} -4.00000 q^{22} +6.00000 q^{23} +3.00000 q^{25} -6.00000 q^{26} -2.00000 q^{28} -2.82843i q^{29} -4.00000 q^{31} -5.65685i q^{32} -8.48528i q^{34} +1.41421i q^{35} +8.48528i q^{37} -6.00000 q^{38} -4.00000 q^{40} -6.00000 q^{41} -8.48528i q^{43} +5.65685i q^{44} -8.48528i q^{46} +1.00000 q^{49} -4.24264i q^{50} +8.48528i q^{52} +5.65685i q^{53} +4.00000 q^{55} +2.82843i q^{56} -4.00000 q^{58} +1.41421i q^{59} +12.7279i q^{61} +5.65685i q^{62} -8.00000 q^{64} +6.00000 q^{65} -12.0000 q^{68} +2.00000 q^{70} +2.00000 q^{73} +12.0000 q^{74} +8.48528i q^{76} -2.82843i q^{77} +8.00000 q^{79} +5.65685i q^{80} +8.48528i q^{82} -15.5563i q^{83} +8.48528i q^{85} -12.0000 q^{86} +8.00000 q^{88} -6.00000 q^{89} -4.24264i q^{91} -12.0000 q^{92} +6.00000 q^{95} -10.0000 q^{97} -1.41421i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 4 q^{4} + 2 q^{7}+O(q^{10}) $ 2 * q - 4 * q^4 + 2 * q^7	$ 2 q - 4 q^{4} + 2 q^{7} + 4 q^{10} + 8 q^{16} + 12 q^{17} - 8 q^{22} + 12 q^{23} + 6 q^{25} - 12 q^{26} - 4 q^{28} - 8 q^{31} - 12 q^{38} - 8 q^{40} - 12 q^{41} + 2 q^{49} + 8 q^{55} - 8 q^{58} - 16 q^{64} + 12 q^{65} - 24 q^{68} + 4 q^{70} + 4 q^{73} + 24 q^{74} + 16 q^{79} - 24 q^{86} + 16 q^{88} - 12 q^{89} - 24 q^{92} + 12 q^{95} - 20 q^{97}+O(q^{100}) $ 2 * q - 4 * q^4 + 2 * q^7 + 4 * q^10 + 8 * q^16 + 12 * q^17 - 8 * q^22 + 12 * q^23 + 6 * q^25 - 12 * q^26 - 4 * q^28 - 8 * q^31 - 12 * q^38 - 8 * q^40 - 12 * q^41 + 2 * q^49 + 8 * q^55 - 8 * q^58 - 16 * q^64 + 12 * q^65 - 24 * q^68 + 4 * q^70 + 4 * q^73 + 24 * q^74 + 16 * q^79 - 24 * q^86 + 16 * q^88 - 12 * q^89 - 24 * q^92 + 12 * q^95 - 20 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/504\mathbb{Z}\right)^\times$.

$n$	$73$	$127$	$253$	$281$
$\chi(n)$	$1$	$1$	$-1$	$1$

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 $

$2$	− 1.41421i	− 1.00000i
$2$	− 1.41421i	− 1.00000i
$3$	0	0
$3$	0	0
$4$	−2.00000	−1.00000
$5$	1.41421i	0.632456i	0.948683	+	0.316228i	$0.102416\pi$
$5$	1.41421i	0.632456i	−0.948683	+	0.316228i	$0.897584\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	2.82843i	1.00000i
$9$	0	0
$10$	2.00000	0.632456
$11$	− 2.82843i	− 0.852803i	−0.904534	−	0.426401i	$-0.859781\pi$
$11$	− 2.82843i	− 0.852803i	0.904534	−	0.426401i	$-0.140219\pi$
$12$	0	0
$13$	− 4.24264i	− 1.17670i	−0.808608	−	0.588348i	$-0.799778\pi$
$13$	− 4.24264i	− 1.17670i	0.808608	−	0.588348i	$-0.200222\pi$
$14$	− 1.41421i	− 0.377964i
$15$	0	0
$16$	4.00000	1.00000
$17$	6.00000	1.45521	0.727607	−	0.685994i	$-0.240633\pi$
$17$	6.00000	1.45521	0.727607	+	0.685994i	$0.240633\pi$
$18$	0	0
$19$	− 4.24264i	− 0.973329i	−0.873589	−	0.486664i	$-0.838214\pi$
$19$	− 4.24264i	− 0.973329i	0.873589	−	0.486664i	$-0.161786\pi$
$20$	− 2.82843i	− 0.632456i
$21$	0	0
$22$	−4.00000	−0.852803
$23$	6.00000	1.25109	0.625543	−	0.780189i	$-0.284877\pi$
$23$	6.00000	1.25109	0.625543	+	0.780189i	$0.284877\pi$
$24$	0	0
$25$	3.00000	0.600000
$26$	−6.00000	−1.17670
$27$	0	0
$28$	−2.00000	−0.377964
$29$	− 2.82843i	− 0.525226i	−0.964901	−	0.262613i	$-0.915416\pi$
$29$	− 2.82843i	− 0.525226i	0.964901	−	0.262613i	$-0.0845842\pi$
$30$	0	0
$31$	−4.00000	−0.718421	−0.359211	−	0.933257i	$-0.616954\pi$
$31$	−4.00000	−0.718421	−0.359211	+	0.933257i	$0.616954\pi$
$32$	− 5.65685i	− 1.00000i
$33$	0	0
$34$	− 8.48528i	− 1.45521i
$35$	1.41421i	0.239046i
$36$	0	0
$37$	8.48528i	1.39497i	0.716599	+	0.697486i	$0.245698\pi$
$37$	8.48528i	1.39497i	−0.716599	+	0.697486i	$0.754302\pi$
$38$	−6.00000	−0.973329
$39$	0	0
$40$	−4.00000	−0.632456
$41$	−6.00000	−0.937043	−0.468521	−	0.883452i	$-0.655213\pi$
$41$	−6.00000	−0.937043	−0.468521	+	0.883452i	$0.655213\pi$
$42$	0	0
$43$	− 8.48528i	− 1.29399i	−0.762493	−	0.646997i	$-0.776025\pi$
$43$	− 8.48528i	− 1.29399i	0.762493	−	0.646997i	$-0.223975\pi$
$44$	5.65685i	0.852803i
$45$	0	0
$46$	− 8.48528i	− 1.25109i
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	− 4.24264i	− 0.600000i
$51$	0	0
$52$	8.48528i	1.17670i
$53$	5.65685i	0.777029i	0.921443	+	0.388514i	$0.127012\pi$
$53$	5.65685i	0.777029i	−0.921443	+	0.388514i	$0.872988\pi$
$54$	0	0
$55$	4.00000	0.539360
$56$	2.82843i	0.377964i
$57$	0	0
$58$	−4.00000	−0.525226
$59$	1.41421i	0.184115i	0.995754	+	0.0920575i	$0.0293443\pi$
$59$	1.41421i	0.184115i	−0.995754	+	0.0920575i	$0.970656\pi$
$60$	0	0
$61$	12.7279i	1.62964i	0.579712	+	0.814822i	$0.303165\pi$
$61$	12.7279i	1.62964i	−0.579712	+	0.814822i	$0.696835\pi$
$62$	5.65685i	0.718421i
$63$	0	0
$64$	−8.00000	−1.00000
$65$	6.00000	0.744208
$66$	0	0
$67$	0	0	1.00000			$0$
$67$	0	0	−1.00000			$\pi$
$68$	−12.0000	−1.45521
$69$	0	0
$70$	2.00000	0.239046
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	0	0
$73$	2.00000	0.234082	0.117041	−	0.993127i	$-0.462659\pi$
$73$	2.00000	0.234082	0.117041	+	0.993127i	$0.462659\pi$
$74$	12.0000	1.39497
$75$	0	0
$76$	8.48528i	0.973329i
$77$	− 2.82843i	− 0.322329i
$78$	0	0
$79$	8.00000	0.900070	0.450035	−	0.893011i	$-0.351411\pi$
$79$	8.00000	0.900070	0.450035	+	0.893011i	$0.351411\pi$
$80$	5.65685i	0.632456i
$81$	0	0
$82$	8.48528i	0.937043i
$83$	− 15.5563i	− 1.70753i	−0.520658	−	0.853766i	$-0.674313\pi$
$83$	− 15.5563i	− 1.70753i	0.520658	−	0.853766i	$-0.325687\pi$
$84$	0	0
$85$	8.48528i	0.920358i
$86$	−12.0000	−1.29399
$87$	0	0
$88$	8.00000	0.852803
$89$	−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
$91$	− 4.24264i	− 0.444750i
$92$	−12.0000	−1.25109
$93$	0	0
$94$	0	0
$95$	6.00000	0.615587
$96$	0	0
$97$	−10.0000	−1.01535	−0.507673	−	0.861550i	$-0.669494\pi$
$97$	−10.0000	−1.01535	−0.507673	+	0.861550i	$0.669494\pi$
$98$	− 1.41421i	− 0.142857i
$99$	0	0
$100$	−6.00000	−0.600000
$101$	9.89949i	0.985037i	0.870302	+	0.492518i	$0.163924\pi$
$101$	9.89949i	0.985037i	−0.870302	+	0.492518i	$0.836076\pi$
$102$	0	0
$103$	−4.00000	−0.394132	−0.197066	−	0.980390i	$-0.563141\pi$
$103$	−4.00000	−0.394132	−0.197066	+	0.980390i	$0.563141\pi$
$104$	12.0000	1.17670
$105$	0	0
$106$	8.00000	0.777029
$107$	5.65685i	0.546869i	0.961891	+	0.273434i	$0.0881596\pi$
$107$	5.65685i	0.546869i	−0.961891	+	0.273434i	$0.911840\pi$
$108$	0	0
$109$	− 8.48528i	− 0.812743i	−0.913708	−	0.406371i	$-0.866794\pi$
$109$	− 8.48528i	− 0.812743i	0.913708	−	0.406371i	$-0.133206\pi$
$110$	− 5.65685i	− 0.539360i
$111$	0	0
$112$	4.00000	0.377964
$113$	12.0000	1.12887	0.564433	−	0.825479i	$-0.309095\pi$
$113$	12.0000	1.12887	0.564433	+	0.825479i	$0.309095\pi$
$114$	0	0
$115$	8.48528i	0.791257i
$116$	5.65685i	0.525226i
$117$	0	0
$118$	2.00000	0.184115
$119$	6.00000	0.550019
$120$	0	0
$121$	3.00000	0.272727
$122$	18.0000	1.62964
$123$	0	0
$124$	8.00000	0.718421
$125$	11.3137i	1.01193i
$126$	0	0
$127$	2.00000	0.177471	0.0887357	−	0.996055i	$-0.471717\pi$
$127$	2.00000	0.177471	0.0887357	+	0.996055i	$0.471717\pi$
$128$	11.3137i	1.00000i
$129$	0	0
$130$	− 8.48528i	− 0.744208i
$131$	1.41421i	0.123560i	0.998090	+	0.0617802i	$0.0196778\pi$
$131$	1.41421i	0.123560i	−0.998090	+	0.0617802i	$0.980322\pi$
$132$	0	0
$133$	− 4.24264i	− 0.367884i
$134$	0	0
$135$	0	0
$136$	16.9706i	1.45521i
$137$	−6.00000	−0.512615	−0.256307	−	0.966595i	$-0.582506\pi$
$137$	−6.00000	−0.512615	−0.256307	+	0.966595i	$0.582506\pi$
$138$	0	0
$139$	− 4.24264i	− 0.359856i	−0.983680	−	0.179928i	$-0.942414\pi$
$139$	− 4.24264i	− 0.359856i	0.983680	−	0.179928i	$-0.0575865\pi$
$140$	− 2.82843i	− 0.239046i
$141$	0	0
$142$	0	0
$143$	−12.0000	−1.00349
$144$	0	0
$145$	4.00000	0.332182
$146$	− 2.82843i	− 0.234082i
$147$	0	0
$148$	− 16.9706i	− 1.39497i
$149$	− 11.3137i	− 0.926855i	−0.886135	−	0.463428i	$-0.846619\pi$
$149$	− 11.3137i	− 0.926855i	0.886135	−	0.463428i	$-0.153381\pi$
$150$	0	0
$151$	−10.0000	−0.813788	−0.406894	−	0.913475i	$-0.633388\pi$
$151$	−10.0000	−0.813788	−0.406894	+	0.913475i	$0.633388\pi$
$152$	12.0000	0.973329
$153$	0	0
$154$	−4.00000	−0.322329
$155$	− 5.65685i	− 0.454369i
$156$	0	0
$157$	12.7279i	1.01580i	0.861416	+	0.507899i	$0.169578\pi$
$157$	12.7279i	1.01580i	−0.861416	+	0.507899i	$0.830422\pi$
$158$	− 11.3137i	− 0.900070i
$159$	0	0
$160$	8.00000	0.632456
$161$	6.00000	0.472866
$162$	0	0
$163$	8.48528i	0.664619i	0.943170	+	0.332309i	$0.107828\pi$
$163$	8.48528i	0.664619i	−0.943170	+	0.332309i	$0.892172\pi$
$164$	12.0000	0.937043
$165$	0	0
$166$	−22.0000	−1.70753
$167$	−24.0000	−1.85718	−0.928588	−	0.371113i	$-0.878976\pi$
$167$	−24.0000	−1.85718	−0.928588	+	0.371113i	$0.878976\pi$
$168$	0	0
$169$	−5.00000	−0.384615
$170$	12.0000	0.920358
$171$	0	0
$172$	16.9706i	1.29399i
$173$	9.89949i	0.752645i	0.926489	+	0.376322i	$0.122811\pi$
$173$	9.89949i	0.752645i	−0.926489	+	0.376322i	$0.877189\pi$
$174$	0	0
$175$	3.00000	0.226779
$176$	− 11.3137i	− 0.852803i
$177$	0	0
$178$	8.48528i	0.635999i
$179$	5.65685i	0.422813i	0.977398	+	0.211407i	$0.0678044\pi$
$179$	5.65685i	0.422813i	−0.977398	+	0.211407i	$0.932196\pi$
$180$	0	0
$181$	12.7279i	0.946059i	0.881047	+	0.473029i	$0.156840\pi$
$181$	12.7279i	0.946059i	−0.881047	+	0.473029i	$0.843160\pi$
$182$	−6.00000	−0.444750
$183$	0	0
$184$	16.9706i	1.25109i
$185$	−12.0000	−0.882258
$186$	0	0
$187$	− 16.9706i	− 1.24101i
$188$	0	0
$189$	0	0
$190$	− 8.48528i	− 0.615587i
$191$	0	0		−	1.00000i	$-0.5\pi$
$191$	0	0			1.00000i	$0.5\pi$
$192$	0	0
$193$	−4.00000	−0.287926	−0.143963	−	0.989583i	$-0.545985\pi$
$193$	−4.00000	−0.287926	−0.143963	+	0.989583i	$0.545985\pi$
$194$	14.1421i	1.01535i
$195$	0	0
$196$	−2.00000	−0.142857
$197$	5.65685i	0.403034i	0.979485	+	0.201517i	$0.0645872\pi$
$197$	5.65685i	0.403034i	−0.979485	+	0.201517i	$0.935413\pi$
$198$	0	0
$199$	20.0000	1.41776	0.708881	−	0.705328i	$-0.249200\pi$
$199$	20.0000	1.41776	0.708881	+	0.705328i	$0.249200\pi$
$200$	8.48528i	0.600000i
$201$	0	0
$202$	14.0000	0.985037
$203$	− 2.82843i	− 0.198517i
$204$	0	0
$205$	− 8.48528i	− 0.592638i
$206$	5.65685i	0.394132i
$207$	0	0
$208$	− 16.9706i	− 1.17670i
$209$	−12.0000	−0.830057
$210$	0	0
$211$	− 16.9706i	− 1.16830i	−0.811645	−	0.584151i	$-0.801428\pi$
$211$	− 16.9706i	− 1.16830i	0.811645	−	0.584151i	$-0.198572\pi$
$212$	− 11.3137i	− 0.777029i
$213$	0	0
$214$	8.00000	0.546869
$215$	12.0000	0.818393
$216$	0	0
$217$	−4.00000	−0.271538
$218$	−12.0000	−0.812743
$219$	0	0
$220$	−8.00000	−0.539360
$221$	− 25.4558i	− 1.71235i
$222$	0	0
$223$	−28.0000	−1.87502	−0.937509	−	0.347960i	$-0.886874\pi$
$223$	−28.0000	−1.87502	−0.937509	+	0.347960i	$0.886874\pi$
$224$	− 5.65685i	− 0.377964i
$225$	0	0
$226$	− 16.9706i	− 1.12887i
$227$	9.89949i	0.657053i	0.944495	+	0.328526i	$0.106552\pi$
$227$	9.89949i	0.657053i	−0.944495	+	0.328526i	$0.893448\pi$
$228$	0	0
$229$	4.24264i	0.280362i	0.990126	+	0.140181i	$0.0447684\pi$
$229$	4.24264i	0.280362i	−0.990126	+	0.140181i	$0.955232\pi$
$230$	12.0000	0.791257
$231$	0	0
$232$	8.00000	0.525226
$233$	−6.00000	−0.393073	−0.196537	−	0.980497i	$-0.562969\pi$
$233$	−6.00000	−0.393073	−0.196537	+	0.980497i	$0.562969\pi$
$234$	0	0
$235$	0	0
$236$	− 2.82843i	− 0.184115i
$237$	0	0
$238$	− 8.48528i	− 0.550019i
$239$	−6.00000	−0.388108	−0.194054	−	0.980991i	$-0.562164\pi$
$239$	−6.00000	−0.388108	−0.194054	+	0.980991i	$0.562164\pi$
$240$	0	0
$241$	−10.0000	−0.644157	−0.322078	−	0.946713i	$-0.604381\pi$
$241$	−10.0000	−0.644157	−0.322078	+	0.946713i	$0.604381\pi$
$242$	− 4.24264i	− 0.272727i
$243$	0	0
$244$	− 25.4558i	− 1.62964i
$245$	1.41421i	0.0903508i
$246$	0	0
$247$	−18.0000	−1.14531
$248$	− 11.3137i	− 0.718421i
$249$	0	0
$250$	16.0000	1.01193
$251$	18.3848i	1.16044i	0.814461	+	0.580218i	$0.197033\pi$
$251$	18.3848i	1.16044i	−0.814461	+	0.580218i	$0.802967\pi$
$252$	0	0
$253$	− 16.9706i	− 1.06693i
$254$	− 2.82843i	− 0.177471i
$255$	0	0
$256$	16.0000	1.00000
$257$	6.00000	0.374270	0.187135	−	0.982334i	$-0.440080\pi$
$257$	6.00000	0.374270	0.187135	+	0.982334i	$0.440080\pi$
$258$	0	0
$259$	8.48528i	0.527250i
$260$	−12.0000	−0.744208
$261$	0	0
$262$	2.00000	0.123560
$263$	−24.0000	−1.47990	−0.739952	−	0.672660i	$-0.765152\pi$
$263$	−24.0000	−1.47990	−0.739952	+	0.672660i	$0.765152\pi$
$264$	0	0
$265$	−8.00000	−0.491436
$266$	−6.00000	−0.367884
$267$	0	0
$268$	0	0
$269$	− 7.07107i	− 0.431131i	−0.976489	−	0.215565i	$-0.930841\pi$
$269$	− 7.07107i	− 0.431131i	0.976489	−	0.215565i	$-0.0691594\pi$
$270$	0	0
$271$	20.0000	1.21491	0.607457	−	0.794353i	$-0.292190\pi$
$271$	20.0000	1.21491	0.607457	+	0.794353i	$0.292190\pi$
$272$	24.0000	1.45521
$273$	0	0
$274$	8.48528i	0.512615i
$275$	− 8.48528i	− 0.511682i
$276$	0	0
$277$	− 16.9706i	− 1.01966i	−0.860274	−	0.509831i	$-0.829708\pi$
$277$	− 16.9706i	− 1.01966i	0.860274	−	0.509831i	$-0.170292\pi$
$278$	−6.00000	−0.359856
$279$	0	0
$280$	−4.00000	−0.239046
$281$	6.00000	0.357930	0.178965	−	0.983855i	$-0.442725\pi$
$281$	6.00000	0.357930	0.178965	+	0.983855i	$0.442725\pi$
$282$	0	0
$283$	12.7279i	0.756596i	0.925684	+	0.378298i	$0.123491\pi$
$283$	12.7279i	0.756596i	−0.925684	+	0.378298i	$0.876509\pi$
$284$	0	0
$285$	0	0
$286$	16.9706i	1.00349i
$287$	−6.00000	−0.354169
$288$	0	0
$289$	19.0000	1.11765
$290$	− 5.65685i	− 0.332182i
$291$	0	0
$292$	−4.00000	−0.234082
$293$	− 24.0416i	− 1.40453i	−0.711917	−	0.702264i	$-0.752173\pi$
$293$	− 24.0416i	− 1.40453i	0.711917	−	0.702264i	$-0.247827\pi$
$294$	0	0
$295$	−2.00000	−0.116445
$296$	−24.0000	−1.39497
$297$	0	0
$298$	−16.0000	−0.926855
$299$	− 25.4558i	− 1.47215i
$300$	0	0
$301$	− 8.48528i	− 0.489083i
$302$	14.1421i	0.813788i
$303$	0	0
$304$	− 16.9706i	− 0.973329i
$305$	−18.0000	−1.03068
$306$	0	0
$307$	− 12.7279i	− 0.726421i	−0.931707	−	0.363210i	$-0.881681\pi$
$307$	− 12.7279i	− 0.726421i	0.931707	−	0.363210i	$-0.118319\pi$
$308$	5.65685i	0.322329i
$309$	0	0
$310$	−8.00000	−0.454369
$311$	0	0		−	1.00000i	$-0.5\pi$
$311$	0	0			1.00000i	$0.5\pi$
$312$	0	0
$313$	−10.0000	−0.565233	−0.282617	−	0.959233i	$-0.591202\pi$
$313$	−10.0000	−0.565233	−0.282617	+	0.959233i	$0.591202\pi$
$314$	18.0000	1.01580
$315$	0	0
$316$	−16.0000	−0.900070
$317$	22.6274i	1.27088i	0.772149	+	0.635441i	$0.219182\pi$
$317$	22.6274i	1.27088i	−0.772149	+	0.635441i	$0.780818\pi$
$318$	0	0
$319$	−8.00000	−0.447914
$320$	− 11.3137i	− 0.632456i
$321$	0	0
$322$	− 8.48528i	− 0.472866i
$323$	− 25.4558i	− 1.41640i
$324$	0	0
$325$	− 12.7279i	− 0.706018i
$326$	12.0000	0.664619
$327$	0	0
$328$	− 16.9706i	− 0.937043i
$329$	0	0
$330$	0	0
$331$	25.4558i	1.39918i	0.714545	+	0.699590i	$0.246634\pi$
$331$	25.4558i	1.39918i	−0.714545	+	0.699590i	$0.753366\pi$
$332$	31.1127i	1.70753i
$333$	0	0
$334$	33.9411i	1.85718i
$335$	0	0
$336$	0	0
$337$	32.0000	1.74315	0.871576	−	0.490261i	$-0.163099\pi$
$337$	32.0000	1.74315	0.871576	+	0.490261i	$0.163099\pi$
$338$	7.07107i	0.384615i
$339$	0	0
$340$	− 16.9706i	− 0.920358i
$341$	11.3137i	0.612672i
$342$	0	0
$343$	1.00000	0.0539949
$344$	24.0000	1.29399
$345$	0	0
$346$	14.0000	0.752645
$347$	14.1421i	0.759190i	0.925153	+	0.379595i	$0.123937\pi$
$347$	14.1421i	0.759190i	−0.925153	+	0.379595i	$0.876063\pi$
$348$	0	0
$349$	4.24264i	0.227103i	0.993532	+	0.113552i	$0.0362227\pi$
$349$	4.24264i	0.227103i	−0.993532	+	0.113552i	$0.963777\pi$
$350$	− 4.24264i	− 0.226779i
$351$	0	0
$352$	−16.0000	−0.852803
$353$	−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
$355$	0	0
$356$	12.0000	0.635999
$357$	0	0
$358$	8.00000	0.422813
$359$	30.0000	1.58334	0.791670	−	0.610949i	$-0.209212\pi$
$359$	30.0000	1.58334	0.791670	+	0.610949i	$0.209212\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	18.0000	0.946059
$363$	0	0
$364$	8.48528i	0.444750i
$365$	2.82843i	0.148047i
$366$	0	0
$367$	−28.0000	−1.46159	−0.730794	−	0.682598i	$-0.760850\pi$
$367$	−28.0000	−1.46159	−0.730794	+	0.682598i	$0.760850\pi$
$368$	24.0000	1.25109
$369$	0	0
$370$	16.9706i	0.882258i
$371$	5.65685i	0.293689i
$372$	0	0
$373$	− 33.9411i	− 1.75740i	−0.477370	−	0.878702i	$-0.658410\pi$
$373$	− 33.9411i	− 1.75740i	0.477370	−	0.878702i	$-0.341590\pi$
$374$	−24.0000	−1.24101
$375$	0	0
$376$	0	0
$377$	−12.0000	−0.618031
$378$	0	0
$379$	25.4558i	1.30758i	0.756677	+	0.653789i	$0.226822\pi$
$379$	25.4558i	1.30758i	−0.756677	+	0.653789i	$0.773178\pi$
$380$	−12.0000	−0.615587
$381$	0	0
$382$	0	0
$383$	24.0000	1.22634	0.613171	−	0.789950i	$-0.289894\pi$
$383$	24.0000	1.22634	0.613171	+	0.789950i	$0.289894\pi$
$384$	0	0
$385$	4.00000	0.203859
$386$	5.65685i	0.287926i
$387$	0	0
$388$	20.0000	1.01535
$389$	− 2.82843i	− 0.143407i	−0.997426	−	0.0717035i	$-0.977156\pi$
$389$	− 2.82843i	− 0.143407i	0.997426	−	0.0717035i	$-0.0228435\pi$
$390$	0	0
$391$	36.0000	1.82060
$392$	2.82843i	0.142857i
$393$	0	0
$394$	8.00000	0.403034
$395$	11.3137i	0.569254i
$396$	0	0
$397$	21.2132i	1.06466i	0.846537	+	0.532330i	$0.178683\pi$
$397$	21.2132i	1.06466i	−0.846537	+	0.532330i	$0.821317\pi$
$398$	− 28.2843i	− 1.41776i
$399$	0	0
$400$	12.0000	0.600000
$401$	24.0000	1.19850	0.599251	−	0.800561i	$-0.295465\pi$
$401$	24.0000	1.19850	0.599251	+	0.800561i	$0.295465\pi$
$402$	0	0
$403$	16.9706i	0.845364i
$404$	− 19.7990i	− 0.985037i
$405$	0	0
$406$	−4.00000	−0.198517
$407$	24.0000	1.18964
$408$	0	0
$409$	−22.0000	−1.08783	−0.543915	−	0.839140i	$-0.683059\pi$
$409$	−22.0000	−1.08783	−0.543915	+	0.839140i	$0.683059\pi$
$410$	−12.0000	−0.592638
$411$	0	0
$412$	8.00000	0.394132
$413$	1.41421i	0.0695889i
$414$	0	0
$415$	22.0000	1.07994
$416$	−24.0000	−1.17670
$417$	0	0
$418$	16.9706i	0.830057i
$419$	26.8701i	1.31269i	0.754462	+	0.656344i	$0.227898\pi$
$419$	26.8701i	1.31269i	−0.754462	+	0.656344i	$0.772102\pi$
$420$	0	0
$421$	16.9706i	0.827095i	0.910483	+	0.413547i	$0.135710\pi$
$421$	16.9706i	0.827095i	−0.910483	+	0.413547i	$0.864290\pi$
$422$	−24.0000	−1.16830
$423$	0	0
$424$	−16.0000	−0.777029
$425$	18.0000	0.873128
$426$	0	0
$427$	12.7279i	0.615947i
$428$	− 11.3137i	− 0.546869i
$429$	0	0
$430$	− 16.9706i	− 0.818393i
$431$	6.00000	0.289010	0.144505	−	0.989504i	$-0.453841\pi$
$431$	6.00000	0.289010	0.144505	+	0.989504i	$0.453841\pi$
$432$	0	0
$433$	2.00000	0.0961139	0.0480569	−	0.998845i	$-0.484697\pi$
$433$	2.00000	0.0961139	0.0480569	+	0.998845i	$0.484697\pi$
$434$	5.65685i	0.271538i
$435$	0	0
$436$	16.9706i	0.812743i
$437$	− 25.4558i	− 1.21772i
$438$	0	0
$439$	−28.0000	−1.33637	−0.668184	−	0.743996i	$-0.732928\pi$
$439$	−28.0000	−1.33637	−0.668184	+	0.743996i	$0.732928\pi$
$440$	11.3137i	0.539360i
$441$	0	0
$442$	−36.0000	−1.71235
$443$	22.6274i	1.07506i	0.843244	+	0.537531i	$0.180643\pi$
$443$	22.6274i	1.07506i	−0.843244	+	0.537531i	$0.819357\pi$
$444$	0	0
$445$	− 8.48528i	− 0.402241i
$446$	39.5980i	1.87502i
$447$	0	0
$448$	−8.00000	−0.377964
$449$	18.0000	0.849473	0.424736	−	0.905317i	$-0.360367\pi$
$449$	18.0000	0.849473	0.424736	+	0.905317i	$0.360367\pi$
$450$	0	0
$451$	16.9706i	0.799113i
$452$	−24.0000	−1.12887
$453$	0	0
$454$	14.0000	0.657053
$455$	6.00000	0.281284
$456$	0	0
$457$	−28.0000	−1.30978	−0.654892	−	0.755722i	$-0.727286\pi$
$457$	−28.0000	−1.30978	−0.654892	+	0.755722i	$0.727286\pi$
$458$	6.00000	0.280362
$459$	0	0
$460$	− 16.9706i	− 0.791257i
$461$	− 15.5563i	− 0.724531i	−0.932075	−	0.362266i	$-0.882003\pi$
$461$	− 15.5563i	− 0.724531i	0.932075	−	0.362266i	$-0.117997\pi$
$462$	0	0
$463$	32.0000	1.48717	0.743583	−	0.668644i	$-0.233125\pi$
$463$	32.0000	1.48717	0.743583	+	0.668644i	$0.233125\pi$
$464$	− 11.3137i	− 0.525226i
$465$	0	0
$466$	8.48528i	0.393073i
$467$	− 7.07107i	− 0.327210i	−0.986526	−	0.163605i	$-0.947688\pi$
$467$	− 7.07107i	− 0.327210i	0.986526	−	0.163605i	$-0.0523123\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0	0
$472$	−4.00000	−0.184115
$473$	−24.0000	−1.10352
$474$	0	0
$475$	− 12.7279i	− 0.583997i
$476$	−12.0000	−0.550019
$477$	0	0
$478$	8.48528i	0.388108i
$479$	12.0000	0.548294	0.274147	−	0.961688i	$-0.411605\pi$
$479$	12.0000	0.548294	0.274147	+	0.961688i	$0.411605\pi$
$480$	0	0
$481$	36.0000	1.64146
$482$	14.1421i	0.644157i
$483$	0	0
$484$	−6.00000	−0.272727
$485$	− 14.1421i	− 0.642161i
$486$	0	0
$487$	2.00000	0.0906287	0.0453143	−	0.998973i	$-0.485571\pi$
$487$	2.00000	0.0906287	0.0453143	+	0.998973i	$0.485571\pi$
$488$	−36.0000	−1.62964
$489$	0	0
$490$	2.00000	0.0903508
$491$	39.5980i	1.78703i	0.449032	+	0.893516i	$0.351769\pi$
$491$	39.5980i	1.78703i	−0.449032	+	0.893516i	$0.648231\pi$
$492$	0	0
$493$	− 16.9706i	− 0.764316i
$494$	25.4558i	1.14531i
$495$	0	0
$496$	−16.0000	−0.718421
$497$	0	0
$498$	0	0
$499$	16.9706i	0.759707i	0.925047	+	0.379853i	$0.124026\pi$
$499$	16.9706i	0.759707i	−0.925047	+	0.379853i	$0.875974\pi$
$500$	− 22.6274i	− 1.01193i
$501$	0	0
$502$	26.0000	1.16044
$503$	−36.0000	−1.60516	−0.802580	−	0.596544i	$-0.796540\pi$
$503$	−36.0000	−1.60516	−0.802580	+	0.596544i	$0.796540\pi$
$504$	0	0
$505$	−14.0000	−0.622992
$506$	−24.0000	−1.06693
$507$	0	0
$508$	−4.00000	−0.177471
$509$	1.41421i	0.0626839i	0.999509	+	0.0313420i	$0.00997809\pi$
$509$	1.41421i	0.0626839i	−0.999509	+	0.0313420i	$0.990022\pi$
$510$	0	0
$511$	2.00000	0.0884748
$512$	− 22.6274i	− 1.00000i
$513$	0	0
$514$	− 8.48528i	− 0.374270i
$515$	− 5.65685i	− 0.249271i
$516$	0	0
$517$	0	0
$518$	12.0000	0.527250
$519$	0	0
$520$	16.9706i	0.744208i
$521$	6.00000	0.262865	0.131432	−	0.991325i	$-0.458042\pi$
$521$	6.00000	0.262865	0.131432	+	0.991325i	$0.458042\pi$
$522$	0	0
$523$	− 29.6985i	− 1.29862i	−0.760522	−	0.649312i	$-0.775057\pi$
$523$	− 29.6985i	− 1.29862i	0.760522	−	0.649312i	$-0.224943\pi$
$524$	− 2.82843i	− 0.123560i
$525$	0	0
$526$	33.9411i	1.47990i
$527$	−24.0000	−1.04546
$528$	0	0
$529$	13.0000	0.565217
$530$	11.3137i	0.491436i
$531$	0	0
$532$	8.48528i	0.367884i
$533$	25.4558i	1.10262i
$534$	0	0
$535$	−8.00000	−0.345870
$536$	0	0
$537$	0	0
$538$	−10.0000	−0.431131
$539$	− 2.82843i	− 0.121829i
$540$	0	0
$541$	− 16.9706i	− 0.729621i	−0.931082	−	0.364811i	$-0.881134\pi$
$541$	− 16.9706i	− 0.729621i	0.931082	−	0.364811i	$-0.118866\pi$
$542$	− 28.2843i	− 1.21491i
$543$	0	0
$544$	− 33.9411i	− 1.45521i
$545$	12.0000	0.514024
$546$	0	0
$547$	− 8.48528i	− 0.362804i	−0.983409	−	0.181402i	$-0.941936\pi$
$547$	− 8.48528i	− 0.362804i	0.983409	−	0.181402i	$-0.0580636\pi$
$548$	12.0000	0.512615
$549$	0	0
$550$	−12.0000	−0.511682
$551$	−12.0000	−0.511217
$552$	0	0
$553$	8.00000	0.340195
$554$	−24.0000	−1.01966
$555$	0	0
$556$	8.48528i	0.359856i
$557$	5.65685i	0.239689i	0.992793	+	0.119844i	$0.0382395\pi$
$557$	5.65685i	0.239689i	−0.992793	+	0.119844i	$0.961760\pi$
$558$	0	0
$559$	−36.0000	−1.52264
$560$	5.65685i	0.239046i
$561$	0	0
$562$	− 8.48528i	− 0.357930i
$563$	1.41421i	0.0596020i	0.999556	+	0.0298010i	$0.00948736\pi$
$563$	1.41421i	0.0596020i	−0.999556	+	0.0298010i	$0.990513\pi$
$564$	0	0
$565$	16.9706i	0.713957i
$566$	18.0000	0.756596
$567$	0	0
$568$	0	0
$569$	0	0		−	1.00000i	$-0.5\pi$
$569$	0	0			1.00000i	$0.5\pi$
$570$	0	0
$571$	25.4558i	1.06529i	0.846338	+	0.532647i	$0.178803\pi$
$571$	25.4558i	1.06529i	−0.846338	+	0.532647i	$0.821197\pi$
$572$	24.0000	1.00349
$573$	0	0
$574$	8.48528i	0.354169i
$575$	18.0000	0.750652
$576$	0	0
$577$	38.0000	1.58196	0.790980	−	0.611842i	$-0.209571\pi$
$577$	38.0000	1.58196	0.790980	+	0.611842i	$0.209571\pi$
$578$	− 26.8701i	− 1.11765i
$579$	0	0
$580$	−8.00000	−0.332182
$581$	− 15.5563i	− 0.645386i
$582$	0	0
$583$	16.0000	0.662652
$584$	5.65685i	0.234082i
$585$	0	0
$586$	−34.0000	−1.40453
$587$	− 41.0122i	− 1.69275i	−0.532584	−	0.846377i	$-0.678779\pi$
$587$	− 41.0122i	− 1.69275i	0.532584	−	0.846377i	$-0.321221\pi$
$588$	0	0
$589$	16.9706i	0.699260i
$590$	2.82843i	0.116445i
$591$	0	0
$592$	33.9411i	1.39497i
$593$	30.0000	1.23195	0.615976	−	0.787765i	$-0.288762\pi$
$593$	30.0000	1.23195	0.615976	+	0.787765i	$0.288762\pi$
$594$	0	0
$595$	8.48528i	0.347863i
$596$	22.6274i	0.926855i
$597$	0	0
$598$	−36.0000	−1.47215
$599$	0	0		−	1.00000i	$-0.5\pi$
$599$	0	0			1.00000i	$0.5\pi$
$600$	0	0
$601$	−10.0000	−0.407909	−0.203954	−	0.978980i	$-0.565379\pi$
$601$	−10.0000	−0.407909	−0.203954	+	0.978980i	$0.565379\pi$
$602$	−12.0000	−0.489083
$603$	0	0
$604$	20.0000	0.813788
$605$	4.24264i	0.172488i
$606$	0	0
$607$	32.0000	1.29884	0.649420	−	0.760430i	$-0.275012\pi$
$607$	32.0000	1.29884	0.649420	+	0.760430i	$0.275012\pi$
$608$	−24.0000	−0.973329
$609$	0	0
$610$	25.4558i	1.03068i
$611$	0	0
$612$	0	0
$613$	− 8.48528i	− 0.342717i	−0.985209	−	0.171359i	$-0.945184\pi$
$613$	− 8.48528i	− 0.342717i	0.985209	−	0.171359i	$-0.0548157\pi$
$614$	−18.0000	−0.726421
$615$	0	0
$616$	8.00000	0.322329
$617$	−24.0000	−0.966204	−0.483102	−	0.875564i	$-0.660490\pi$
$617$	−24.0000	−0.966204	−0.483102	+	0.875564i	$0.660490\pi$
$618$	0	0
$619$	− 4.24264i	− 0.170526i	−0.996358	−	0.0852631i	$-0.972827\pi$
$619$	− 4.24264i	− 0.170526i	0.996358	−	0.0852631i	$-0.0271731\pi$
$620$	11.3137i	0.454369i
$621$	0	0
$622$	0	0
$623$	−6.00000	−0.240385
$624$	0	0
$625$	−1.00000	−0.0400000
$626$	14.1421i	0.565233i
$627$	0	0
$628$	− 25.4558i	− 1.01580i
$629$	50.9117i	2.02998i
$630$	0	0
$631$	−16.0000	−0.636950	−0.318475	−	0.947931i	$-0.603171\pi$
$631$	−16.0000	−0.636950	−0.318475	+	0.947931i	$0.603171\pi$
$632$	22.6274i	0.900070i
$633$	0	0
$634$	32.0000	1.27088
$635$	2.82843i	0.112243i
$636$	0	0
$637$	− 4.24264i	− 0.168100i
$638$	11.3137i	0.447914i
$639$	0	0
$640$	−16.0000	−0.632456
$641$	48.0000	1.89589	0.947943	−	0.318440i	$-0.103159\pi$
$641$	48.0000	1.89589	0.947943	+	0.318440i	$0.103159\pi$
$642$	0	0
$643$	21.2132i	0.836567i	0.908317	+	0.418284i	$0.137368\pi$
$643$	21.2132i	0.836567i	−0.908317	+	0.418284i	$0.862632\pi$
$644$	−12.0000	−0.472866
$645$	0	0
$646$	−36.0000	−1.41640
$647$	−12.0000	−0.471769	−0.235884	−	0.971781i	$-0.575799\pi$
$647$	−12.0000	−0.471769	−0.235884	+	0.971781i	$0.575799\pi$
$648$	0	0
$649$	4.00000	0.157014
$650$	−18.0000	−0.706018
$651$	0	0
$652$	− 16.9706i	− 0.664619i
$653$	− 36.7696i	− 1.43890i	−0.694542	−	0.719452i	$-0.744393\pi$
$653$	− 36.7696i	− 1.43890i	0.694542	−	0.719452i	$-0.255607\pi$
$654$	0	0
$655$	−2.00000	−0.0781465
$656$	−24.0000	−0.937043
$657$	0	0
$658$	0	0
$659$	− 2.82843i	− 0.110180i	−0.998481	−	0.0550899i	$-0.982455\pi$
$659$	− 2.82843i	− 0.110180i	0.998481	−	0.0550899i	$-0.0175446\pi$
$660$	0	0
$661$	− 38.1838i	− 1.48518i	−0.669748	−	0.742588i	$-0.733598\pi$
$661$	− 38.1838i	− 1.48518i	0.669748	−	0.742588i	$-0.266402\pi$
$662$	36.0000	1.39918
$663$	0	0
$664$	44.0000	1.70753
$665$	6.00000	0.232670
$666$	0	0
$667$	− 16.9706i	− 0.657103i
$668$	48.0000	1.85718
$669$	0	0
$670$	0	0
$671$	36.0000	1.38976
$672$	0	0
$673$	−46.0000	−1.77317	−0.886585	−	0.462566i	$-0.846929\pi$
$673$	−46.0000	−1.77317	−0.886585	+	0.462566i	$0.846929\pi$
$674$	− 45.2548i	− 1.74315i
$675$	0	0
$676$	10.0000	0.384615
$677$	9.89949i	0.380468i	0.981739	+	0.190234i	$0.0609248\pi$
$677$	9.89949i	0.380468i	−0.981739	+	0.190234i	$0.939075\pi$
$678$	0	0
$679$	−10.0000	−0.383765
$680$	−24.0000	−0.920358
$681$	0	0
$682$	16.0000	0.612672
$683$	5.65685i	0.216454i	0.994126	+	0.108227i	$0.0345173\pi$
$683$	5.65685i	0.216454i	−0.994126	+	0.108227i	$0.965483\pi$
$684$	0	0
$685$	− 8.48528i	− 0.324206i
$686$	− 1.41421i	− 0.0539949i
$687$	0	0
$688$	− 33.9411i	− 1.29399i
$689$	24.0000	0.914327
$690$	0	0
$691$	− 12.7279i	− 0.484193i	−0.970252	−	0.242096i	$-0.922165\pi$
$691$	− 12.7279i	− 0.484193i	0.970252	−	0.242096i	$-0.0778351\pi$
$692$	− 19.7990i	− 0.752645i
$693$	0	0
$694$	20.0000	0.759190
$695$	6.00000	0.227593
$696$	0	0
$697$	−36.0000	−1.36360
$698$	6.00000	0.227103
$699$	0	0
$700$	−6.00000	−0.226779
$701$	− 19.7990i	− 0.747798i	−0.927470	−	0.373899i	$-0.878021\pi$
$701$	− 19.7990i	− 0.747798i	0.927470	−	0.373899i	$-0.121979\pi$
$702$	0	0
$703$	36.0000	1.35777
$704$	22.6274i	0.852803i
$705$	0	0
$706$	8.48528i	0.319348i
$707$	9.89949i	0.372309i
$708$	0	0
$709$	− 25.4558i	− 0.956014i	−0.878356	−	0.478007i	$-0.841359\pi$
$709$	− 25.4558i	− 0.956014i	0.878356	−	0.478007i	$-0.158641\pi$
$710$	0	0
$711$	0	0
$712$	− 16.9706i	− 0.635999i
$713$	−24.0000	−0.898807
$714$	0	0
$715$	− 16.9706i	− 0.634663i
$716$	− 11.3137i	− 0.422813i
$717$	0	0
$718$	− 42.4264i	− 1.58334i
$719$	−24.0000	−0.895049	−0.447524	−	0.894272i	$-0.647694\pi$
$719$	−24.0000	−0.895049	−0.447524	+	0.894272i	$0.647694\pi$
$720$	0	0
$721$	−4.00000	−0.148968
$722$	− 1.41421i	− 0.0526316i
$723$	0	0
$724$	− 25.4558i	− 0.946059i
$725$	− 8.48528i	− 0.315135i
$726$	0	0
$727$	8.00000	0.296704	0.148352	−	0.988935i	$-0.452603\pi$
$727$	8.00000	0.296704	0.148352	+	0.988935i	$0.452603\pi$
$728$	12.0000	0.444750
$729$	0	0
$730$	4.00000	0.148047
$731$	− 50.9117i	− 1.88304i
$732$	0	0
$733$	29.6985i	1.09694i	0.836171	+	0.548469i	$0.184789\pi$
$733$	29.6985i	1.09694i	−0.836171	+	0.548469i	$0.815211\pi$
$734$	39.5980i	1.46159i
$735$	0	0
$736$	− 33.9411i	− 1.25109i
$737$	0	0
$738$	0	0
$739$	42.4264i	1.56068i	0.625355	+	0.780340i	$0.284954\pi$
$739$	42.4264i	1.56068i	−0.625355	+	0.780340i	$0.715046\pi$
$740$	24.0000	0.882258
$741$	0	0
$742$	8.00000	0.293689
$743$	30.0000	1.10059	0.550297	−	0.834969i	$-0.314515\pi$
$743$	30.0000	1.10059	0.550297	+	0.834969i	$0.314515\pi$
$744$	0	0
$745$	16.0000	0.586195
$746$	−48.0000	−1.75740
$747$	0	0
$748$	33.9411i	1.24101i
$749$	5.65685i	0.206697i
$750$	0	0
$751$	−22.0000	−0.802791	−0.401396	−	0.915905i	$-0.631475\pi$
$751$	−22.0000	−0.802791	−0.401396	+	0.915905i	$0.631475\pi$
$752$	0	0
$753$	0	0
$754$	16.9706i	0.618031i
$755$	− 14.1421i	− 0.514685i
$756$	0	0
$757$	25.4558i	0.925208i	0.886565	+	0.462604i	$0.153085\pi$
$757$	25.4558i	0.925208i	−0.886565	+	0.462604i	$0.846915\pi$
$758$	36.0000	1.30758
$759$	0	0
$760$	16.9706i	0.615587i
$761$	−30.0000	−1.08750	−0.543750	−	0.839248i	$-0.682996\pi$
$761$	−30.0000	−1.08750	−0.543750	+	0.839248i	$0.682996\pi$
$762$	0	0
$763$	− 8.48528i	− 0.307188i
$764$	0	0
$765$	0	0
$766$	− 33.9411i	− 1.22634i
$767$	6.00000	0.216647
$768$	0	0
$769$	14.0000	0.504853	0.252426	−	0.967616i	$-0.418771\pi$
$769$	14.0000	0.504853	0.252426	+	0.967616i	$0.418771\pi$
$770$	− 5.65685i	− 0.203859i
$771$	0	0
$772$	8.00000	0.287926
$773$	− 32.5269i	− 1.16991i	−0.811065	−	0.584956i	$-0.801112\pi$
$773$	− 32.5269i	− 1.16991i	0.811065	−	0.584956i	$-0.198888\pi$
$774$	0	0
$775$	−12.0000	−0.431053
$776$	− 28.2843i	− 1.01535i
$777$	0	0
$778$	−4.00000	−0.143407
$779$	25.4558i	0.912050i
$780$	0	0
$781$	0	0
$782$	− 50.9117i	− 1.82060i
$783$	0	0
$784$	4.00000	0.142857
$785$	−18.0000	−0.642448
$786$	0	0
$787$	− 38.1838i	− 1.36110i	−0.732700	−	0.680552i	$-0.761740\pi$
$787$	− 38.1838i	− 1.36110i	0.732700	−	0.680552i	$-0.238260\pi$
$788$	− 11.3137i	− 0.403034i
$789$	0	0
$790$	16.0000	0.569254
$791$	12.0000	0.426671
$792$	0	0
$793$	54.0000	1.91760
$794$	30.0000	1.06466
$795$	0	0
$796$	−40.0000	−1.41776
$797$	26.8701i	0.951786i	0.879503	+	0.475893i	$0.157875\pi$
$797$	26.8701i	0.951786i	−0.879503	+	0.475893i	$0.842125\pi$
$798$	0	0
$799$	0	0
$800$	− 16.9706i	− 0.600000i
$801$	0	0
$802$	− 33.9411i	− 1.19850i
$803$	− 5.65685i	− 0.199626i
$804$	0	0
$805$	8.48528i	0.299067i
$806$	24.0000	0.845364
$807$	0	0
$808$	−28.0000	−0.985037
$809$	−12.0000	−0.421898	−0.210949	−	0.977497i	$-0.567655\pi$
$809$	−12.0000	−0.421898	−0.210949	+	0.977497i	$0.567655\pi$
$810$	0	0
$811$	12.7279i	0.446938i	0.974711	+	0.223469i	$0.0717381\pi$
$811$	12.7279i	0.446938i	−0.974711	+	0.223469i	$0.928262\pi$
$812$	5.65685i	0.198517i
$813$	0	0
$814$	− 33.9411i	− 1.18964i
$815$	−12.0000	−0.420342
$816$	0	0
$817$	−36.0000	−1.25948
$818$	31.1127i	1.08783i
$819$	0	0
$820$	16.9706i	0.592638i
$821$	22.6274i	0.789702i	0.918745	+	0.394851i	$0.129204\pi$
$821$	22.6274i	0.789702i	−0.918745	+	0.394851i	$0.870796\pi$
$822$	0	0
$823$	32.0000	1.11545	0.557725	−	0.830026i	$-0.311674\pi$
$823$	32.0000	1.11545	0.557725	+	0.830026i	$0.311674\pi$
$824$	− 11.3137i	− 0.394132i
$825$	0	0
$826$	2.00000	0.0695889
$827$	− 45.2548i	− 1.57366i	−0.617167	−	0.786832i	$-0.711720\pi$
$827$	− 45.2548i	− 1.57366i	0.617167	−	0.786832i	$-0.288280\pi$
$828$	0	0
$829$	− 21.2132i	− 0.736765i	−0.929674	−	0.368383i	$-0.879912\pi$
$829$	− 21.2132i	− 0.736765i	0.929674	−	0.368383i	$-0.120088\pi$
$830$	− 31.1127i	− 1.07994i
$831$	0	0
$832$	33.9411i	1.17670i
$833$	6.00000	0.207888
$834$	0	0
$835$	− 33.9411i	− 1.17458i
$836$	24.0000	0.830057
$837$	0	0
$838$	38.0000	1.31269
$839$	−12.0000	−0.414286	−0.207143	−	0.978311i	$-0.566417\pi$
$839$	−12.0000	−0.414286	−0.207143	+	0.978311i	$0.566417\pi$
$840$	0	0
$841$	21.0000	0.724138
$842$	24.0000	0.827095
$843$	0	0
$844$	33.9411i	1.16830i
$845$	− 7.07107i	− 0.243252i
$846$	0	0
$847$	3.00000	0.103081
$848$	22.6274i	0.777029i
$849$	0	0
$850$	− 25.4558i	− 0.873128i
$851$	50.9117i	1.74523i
$852$	0	0
$853$	4.24264i	0.145265i	0.997359	+	0.0726326i	$0.0231401\pi$
$853$	4.24264i	0.145265i	−0.997359	+	0.0726326i	$0.976860\pi$
$854$	18.0000	0.615947
$855$	0	0
$856$	−16.0000	−0.546869
$857$	−42.0000	−1.43469	−0.717346	−	0.696717i	$-0.754643\pi$
$857$	−42.0000	−1.43469	−0.717346	+	0.696717i	$0.754643\pi$
$858$	0	0
$859$	− 46.6690i	− 1.59233i	−0.605081	−	0.796164i	$-0.706859\pi$
$859$	− 46.6690i	− 1.59233i	0.605081	−	0.796164i	$-0.293141\pi$
$860$	−24.0000	−0.818393
$861$	0	0
$862$	− 8.48528i	− 0.289010i
$863$	−24.0000	−0.816970	−0.408485	−	0.912765i	$-0.633943\pi$
$863$	−24.0000	−0.816970	−0.408485	+	0.912765i	$0.633943\pi$
$864$	0	0
$865$	−14.0000	−0.476014
$866$	− 2.82843i	− 0.0961139i
$867$	0	0
$868$	8.00000	0.271538
$869$	− 22.6274i	− 0.767583i
$870$	0	0
$871$	0	0
$872$	24.0000	0.812743
$873$	0	0
$874$	−36.0000	−1.21772
$875$	11.3137i	0.382473i
$876$	0	0
$877$	42.4264i	1.43264i	0.697773	+	0.716319i	$0.254174\pi$
$877$	42.4264i	1.43264i	−0.697773	+	0.716319i	$0.745826\pi$
$878$	39.5980i	1.33637i
$879$	0	0
$880$	16.0000	0.539360
$881$	18.0000	0.606435	0.303218	−	0.952921i	$-0.401939\pi$
$881$	18.0000	0.606435	0.303218	+	0.952921i	$0.401939\pi$
$882$	0	0
$883$	− 50.9117i	− 1.71331i	−0.515886	−	0.856657i	$-0.672537\pi$
$883$	− 50.9117i	− 1.71331i	0.515886	−	0.856657i	$-0.327463\pi$
$884$	50.9117i	1.71235i
$885$	0	0
$886$	32.0000	1.07506
$887$	−48.0000	−1.61168	−0.805841	−	0.592132i	$-0.798286\pi$
$887$	−48.0000	−1.61168	−0.805841	+	0.592132i	$0.798286\pi$
$888$	0	0
$889$	2.00000	0.0670778
$890$	−12.0000	−0.402241
$891$	0	0
$892$	56.0000	1.87502
$893$	0	0
$894$	0	0
$895$	−8.00000	−0.267411
$896$	11.3137i	0.377964i
$897$	0	0
$898$	− 25.4558i	− 0.849473i
$899$	11.3137i	0.377333i
$900$	0	0
$901$	33.9411i	1.13074i
$902$	24.0000	0.799113
$903$	0	0
$904$	33.9411i	1.12887i
$905$	−18.0000	−0.598340
$906$	0	0
$907$	33.9411i	1.12700i	0.826117	+	0.563498i	$0.190545\pi$
$907$	33.9411i	1.12700i	−0.826117	+	0.563498i	$0.809455\pi$
$908$	− 19.7990i	− 0.657053i
$909$	0	0
$910$	− 8.48528i	− 0.281284i
$911$	−30.0000	−0.993944	−0.496972	−	0.867766i	$-0.665555\pi$
$911$	−30.0000	−0.993944	−0.496972	+	0.867766i	$0.665555\pi$
$912$	0	0
$913$	−44.0000	−1.45619
$914$	39.5980i	1.30978i
$915$	0	0
$916$	− 8.48528i	− 0.280362i
$917$	1.41421i	0.0467014i
$918$	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
$922$	−22.0000	−0.724531
$923$	0	0
$924$	0	0
$925$	25.4558i	0.836983i
$926$	− 45.2548i	− 1.48717i
$927$	0	0
$928$	−16.0000	−0.525226
$929$	−18.0000	−0.590561	−0.295280	−	0.955411i	$-0.595413\pi$
$929$	−18.0000	−0.590561	−0.295280	+	0.955411i	$0.595413\pi$
$930$	0	0
$931$	− 4.24264i	− 0.139047i
$932$	12.0000	0.393073
$933$	0	0
$934$	−10.0000	−0.327210
$935$	24.0000	0.784884
$936$	0	0
$937$	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
$940$	0	0
$941$	52.3259i	1.70578i	0.522094	+	0.852888i	$0.325151\pi$
$941$	52.3259i	1.70578i	−0.522094	+	0.852888i	$0.674849\pi$
$942$	0	0
$943$	−36.0000	−1.17232
$944$	5.65685i	0.184115i
$945$	0	0
$946$	33.9411i	1.10352i
$947$	− 2.82843i	− 0.0919115i	−0.998943	−	0.0459558i	$-0.985367\pi$
$947$	− 2.82843i	− 0.0919115i	0.998943	−	0.0459558i	$-0.0146333\pi$
$948$	0	0
$949$	− 8.48528i	− 0.275444i
$950$	−18.0000	−0.583997
$951$	0	0
$952$	16.9706i	0.550019i
$953$	−18.0000	−0.583077	−0.291539	−	0.956559i	$-0.594167\pi$
$953$	−18.0000	−0.583077	−0.291539	+	0.956559i	$0.594167\pi$
$954$	0	0
$955$	0	0
$956$	12.0000	0.388108
$957$	0	0
$958$	− 16.9706i	− 0.548294i
$959$	−6.00000	−0.193750
$960$	0	0
$961$	−15.0000	−0.483871
$962$	− 50.9117i	− 1.64146i
$963$	0	0
$964$	20.0000	0.644157
$965$	− 5.65685i	− 0.182101i
$966$	0	0
$967$	−22.0000	−0.707472	−0.353736	−	0.935345i	$-0.615089\pi$
$967$	−22.0000	−0.707472	−0.353736	+	0.935345i	$0.615089\pi$
$968$	8.48528i	0.272727i
$969$	0	0
$970$	−20.0000	−0.642161
$971$	− 32.5269i	− 1.04384i	−0.852995	−	0.521919i	$-0.825216\pi$
$971$	− 32.5269i	− 1.04384i	0.852995	−	0.521919i	$-0.174784\pi$
$972$	0	0
$973$	− 4.24264i	− 0.136013i
$974$	− 2.82843i	− 0.0906287i
$975$	0	0
$976$	50.9117i	1.62964i
$977$	18.0000	0.575871	0.287936	−	0.957650i	$-0.407031\pi$
$977$	18.0000	0.575871	0.287936	+	0.957650i	$0.407031\pi$
$978$	0	0
$979$	16.9706i	0.542382i
$980$	− 2.82843i	− 0.0903508i
$981$	0	0
$982$	56.0000	1.78703
$983$	12.0000	0.382741	0.191370	−	0.981518i	$-0.438707\pi$
$983$	12.0000	0.382741	0.191370	+	0.981518i	$0.438707\pi$
$984$	0	0
$985$	−8.00000	−0.254901
$986$	−24.0000	−0.764316
$987$	0	0
$988$	36.0000	1.14531
$989$	− 50.9117i	− 1.61890i
$990$	0	0
$991$	−16.0000	−0.508257	−0.254128	−	0.967170i	$-0.581789\pi$
$991$	−16.0000	−0.508257	−0.254128	+	0.967170i	$0.581789\pi$
$992$	22.6274i	0.718421i
$993$	0	0
$994$	0	0
$995$	28.2843i	0.896672i
$996$	0	0
$997$	21.2132i	0.671829i	0.941893	+	0.335914i	$0.109045\pi$
$997$	21.2132i	0.671829i	−0.941893	+	0.335914i	$0.890955\pi$
$998$	24.0000	0.759707
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	504.2.c.a.253.1		2
3.2	odd	2		56.2.b.a.29.2	yes	2
4.3	odd	2		2016.2.c.a.1009.2		2
8.3	odd	2		2016.2.c.a.1009.1		2
8.5	even	2	inner	504.2.c.a.253.2		2
12.11	even	2		224.2.b.a.113.1		2
21.2	odd	6		392.2.p.a.165.2		4
21.5	even	6		392.2.p.b.165.2		4
21.11	odd	6		392.2.p.a.373.1		4
21.17	even	6		392.2.p.b.373.1		4
21.20	even	2		392.2.b.b.197.2		2
24.5	odd	2		56.2.b.a.29.1	✓	2
24.11	even	2		224.2.b.a.113.2		2
48.5	odd	4		1792.2.a.n.1.2		2
48.11	even	4		1792.2.a.p.1.1		2
48.29	odd	4		1792.2.a.n.1.1		2
48.35	even	4		1792.2.a.p.1.2		2
84.11	even	6		1568.2.t.c.177.1		4
84.23	even	6		1568.2.t.c.753.2		4
84.47	odd	6		1568.2.t.b.753.1		4
84.59	odd	6		1568.2.t.b.177.2		4
84.83	odd	2		1568.2.b.a.785.2		2
168.5	even	6		392.2.p.b.165.1		4
168.11	even	6		1568.2.t.c.177.2		4
168.53	odd	6		392.2.p.a.373.2		4
168.59	odd	6		1568.2.t.b.177.1		4
168.83	odd	2		1568.2.b.a.785.1		2
168.101	even	6		392.2.p.b.373.2		4
168.107	even	6		1568.2.t.c.753.1		4
168.125	even	2		392.2.b.b.197.1		2
168.131	odd	6		1568.2.t.b.753.2		4
168.149	odd	6		392.2.p.a.165.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
56.2.b.a.29.1	✓	2	24.5	odd	2
56.2.b.a.29.2	yes	2	3.2	odd	2
224.2.b.a.113.1		2	12.11	even	2
224.2.b.a.113.2		2	24.11	even	2
392.2.b.b.197.1		2	168.125	even	2
392.2.b.b.197.2		2	21.20	even	2
392.2.p.a.165.1		4	168.149	odd	6
392.2.p.a.165.2		4	21.2	odd	6
392.2.p.a.373.1		4	21.11	odd	6
392.2.p.a.373.2		4	168.53	odd	6
392.2.p.b.165.1		4	168.5	even	6
392.2.p.b.165.2		4	21.5	even	6
392.2.p.b.373.1		4	21.17	even	6
392.2.p.b.373.2		4	168.101	even	6
504.2.c.a.253.1		2	1.1	even	1	trivial
504.2.c.a.253.2		2	8.5	even	2	inner
1568.2.b.a.785.1		2	168.83	odd	2
1568.2.b.a.785.2		2	84.83	odd	2
1568.2.t.b.177.1		4	168.59	odd	6
1568.2.t.b.177.2		4	84.59	odd	6
1568.2.t.b.753.1		4	84.47	odd	6
1568.2.t.b.753.2		4	168.131	odd	6
1568.2.t.c.177.1		4	84.11	even	6
1568.2.t.c.177.2		4	168.11	even	6
1568.2.t.c.753.1		4	168.107	even	6
1568.2.t.c.753.2		4	84.23	even	6
1792.2.a.n.1.1		2	48.29	odd	4
1792.2.a.n.1.2		2	48.5	odd	4
1792.2.a.p.1.1		2	48.11	even	4
1792.2.a.p.1.2		2	48.35	even	4
2016.2.c.a.1009.1		2	8.3	odd	2
2016.2.c.a.1009.2		2	4.3	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:	\( N \)	\(=\)	\( 504 = 2^{3} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	504.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.41421i q^{2} -2.00000 q^{4} +1.41421i q^{5} +1.00000 q^{7} +2.82843i q^{8} +O(q^{10})\)	\(q-1.41421i q^{2} -2.00000 q^{4} +1.41421i q^{5} +1.00000 q^{7} +2.82843i q^{8} +2.00000 q^{10} -2.82843i q^{11} -4.24264i q^{13} -1.41421i q^{14} +4.00000 q^{16} +6.00000 q^{17} -4.24264i q^{19} -2.82843i q^{20} -4.00000 q^{22} +6.00000 q^{23} +3.00000 q^{25} -6.00000 q^{26} -2.00000 q^{28} -2.82843i q^{29} -4.00000 q^{31} -5.65685i q^{32} -8.48528i q^{34} +1.41421i q^{35} +8.48528i q^{37} -6.00000 q^{38} -4.00000 q^{40} -6.00000 q^{41} -8.48528i q^{43} +5.65685i q^{44} -8.48528i q^{46} +1.00000 q^{49} -4.24264i q^{50} +8.48528i q^{52} +5.65685i q^{53} +4.00000 q^{55} +2.82843i q^{56} -4.00000 q^{58} +1.41421i q^{59} +12.7279i q^{61} +5.65685i q^{62} -8.00000 q^{64} +6.00000 q^{65} -12.0000 q^{68} +2.00000 q^{70} +2.00000 q^{73} +12.0000 q^{74} +8.48528i q^{76} -2.82843i q^{77} +8.00000 q^{79} +5.65685i q^{80} +8.48528i q^{82} -15.5563i q^{83} +8.48528i q^{85} -12.0000 q^{86} +8.00000 q^{88} -6.00000 q^{89} -4.24264i q^{91} -12.0000 q^{92} +6.00000 q^{95} -10.0000 q^{97} -1.41421i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 4 q^{4} + 2 q^{7}+O(q^{10}) \) 2 * q - 4 * q^4 + 2 * q^7	\( 2 q - 4 q^{4} + 2 q^{7} + 4 q^{10} + 8 q^{16} + 12 q^{17} - 8 q^{22} + 12 q^{23} + 6 q^{25} - 12 q^{26} - 4 q^{28} - 8 q^{31} - 12 q^{38} - 8 q^{40} - 12 q^{41} + 2 q^{49} + 8 q^{55} - 8 q^{58} - 16 q^{64} + 12 q^{65} - 24 q^{68} + 4 q^{70} + 4 q^{73} + 24 q^{74} + 16 q^{79} - 24 q^{86} + 16 q^{88} - 12 q^{89} - 24 q^{92} + 12 q^{95} - 20 q^{97}+O(q^{100}) \) 2 * q - 4 * q^4 + 2 * q^7 + 4 * q^10 + 8 * q^16 + 12 * q^17 - 8 * q^22 + 12 * q^23 + 6 * q^25 - 12 * q^26 - 4 * q^28 - 8 * q^31 - 12 * q^38 - 8 * q^40 - 12 * q^41 + 2 * q^49 + 8 * q^55 - 8 * q^58 - 16 * q^64 + 12 * q^65 - 24 * q^68 + 4 * q^70 + 4 * q^73 + 24 * q^74 + 16 * q^79 - 24 * q^86 + 16 * q^88 - 12 * q^89 - 24 * q^92 + 12 * q^95 - 20 * q^97

Introduction

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