LMFDB - Embedded newform 392.2.b.b.197.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [392,2,Mod(197,392)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(392, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("392.197");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 392 = 2^{3} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	392.b (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:	$3.13013575923$
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			197.2
Root			$1.41421i$ of defining polynomial
Character	$\chi$	$=$	392.197
Dual form			392.2.b.b.197.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.41421i q^{2} -1.41421i q^{3} -2.00000 q^{4} +1.41421i q^{5} +2.00000 q^{6} -2.82843i q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+1.41421i q^{2} -1.41421i q^{3} -2.00000 q^{4} +1.41421i q^{5} +2.00000 q^{6} -2.82843i q^{8} +1.00000 q^{9} -2.00000 q^{10} +2.82843i q^{11} +2.82843i q^{12} +4.24264i q^{13} +2.00000 q^{15} +4.00000 q^{16} +6.00000 q^{17} +1.41421i q^{18} +4.24264i q^{19} -2.82843i q^{20} -4.00000 q^{22} -6.00000 q^{23} -4.00000 q^{24} +3.00000 q^{25} -6.00000 q^{26} -5.65685i q^{27} +2.82843i q^{29} +2.82843i q^{30} +4.00000 q^{31} +5.65685i q^{32} +4.00000 q^{33} +8.48528i q^{34} -2.00000 q^{36} +8.48528i q^{37} -6.00000 q^{38} +6.00000 q^{39} +4.00000 q^{40} -6.00000 q^{41} -8.48528i q^{43} -5.65685i q^{44} +1.41421i q^{45} -8.48528i q^{46} -5.65685i q^{48} +4.24264i q^{50} -8.48528i q^{51} -8.48528i q^{52} -5.65685i q^{53} +8.00000 q^{54} -4.00000 q^{55} +6.00000 q^{57} -4.00000 q^{58} +1.41421i q^{59} -4.00000 q^{60} -12.7279i q^{61} +5.65685i q^{62} -8.00000 q^{64} -6.00000 q^{65} +5.65685i q^{66} -12.0000 q^{68} +8.48528i q^{69} -2.82843i q^{72} -2.00000 q^{73} -12.0000 q^{74} -4.24264i q^{75} -8.48528i q^{76} +8.48528i q^{78} +8.00000 q^{79} +5.65685i q^{80} -5.00000 q^{81} -8.48528i q^{82} -15.5563i q^{83} +8.48528i q^{85} +12.0000 q^{86} +4.00000 q^{87} +8.00000 q^{88} -6.00000 q^{89} -2.00000 q^{90} +12.0000 q^{92} -5.65685i q^{93} -6.00000 q^{95} +8.00000 q^{96} +10.0000 q^{97} +2.82843i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 4 q^{4} + 4 q^{6} + 2 q^{9}+O(q^{10}) $ 2 * q - 4 * q^4 + 4 * q^6 + 2 * q^9	$ 2 q - 4 q^{4} + 4 q^{6} + 2 q^{9} - 4 q^{10} + 4 q^{15} + 8 q^{16} + 12 q^{17} - 8 q^{22} - 12 q^{23} - 8 q^{24} + 6 q^{25} - 12 q^{26} + 8 q^{31} + 8 q^{33} - 4 q^{36} - 12 q^{38} + 12 q^{39} + 8 q^{40} - 12 q^{41} + 16 q^{54} - 8 q^{55} + 12 q^{57} - 8 q^{58} - 8 q^{60} - 16 q^{64} - 12 q^{65} - 24 q^{68} - 4 q^{73} - 24 q^{74} + 16 q^{79} - 10 q^{81} + 24 q^{86} + 8 q^{87} + 16 q^{88} - 12 q^{89} - 4 q^{90} + 24 q^{92} - 12 q^{95} + 16 q^{96} + 20 q^{97}+O(q^{100}) $ 2 * q - 4 * q^4 + 4 * q^6 + 2 * q^9 - 4 * q^10 + 4 * q^15 + 8 * q^16 + 12 * q^17 - 8 * q^22 - 12 * q^23 - 8 * q^24 + 6 * q^25 - 12 * q^26 + 8 * q^31 + 8 * q^33 - 4 * q^36 - 12 * q^38 + 12 * q^39 + 8 * q^40 - 12 * q^41 + 16 * q^54 - 8 * q^55 + 12 * q^57 - 8 * q^58 - 8 * q^60 - 16 * q^64 - 12 * q^65 - 24 * q^68 - 4 * q^73 - 24 * q^74 + 16 * q^79 - 10 * q^81 + 24 * q^86 + 8 * q^87 + 16 * q^88 - 12 * q^89 - 4 * q^90 + 24 * q^92 - 12 * q^95 + 16 * q^96 + 20 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/392\mathbb{Z}\right)^\times$.

$n$	$197$	$295$	$297$
$\chi(n)$	$-1$	$1$	$1$

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
$3$	− 1.41421i	− 0.816497i	−0.912871	−	0.408248i	$-0.866140\pi$
$3$	− 1.41421i	− 0.816497i	0.912871	−	0.408248i	$-0.133860\pi$
$4$	−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	$0.897584\pi$
$6$	2.00000	0.816497
$7$	0	0
$7$	0	0
$8$	− 2.82843i	− 1.00000i
$9$	1.00000	0.333333
$10$	−2.00000	−0.632456
$11$	2.82843i	0.852803i	0.904534	+	0.426401i	$0.140219\pi$
$11$	2.82843i	0.852803i	−0.904534	+	0.426401i	$0.859781\pi$
$12$	2.82843i	0.816497i
$13$	4.24264i	1.17670i	0.808608	+	0.588348i	$0.200222\pi$
$13$	4.24264i	1.17670i	−0.808608	+	0.588348i	$0.799778\pi$
$14$	0	0
$15$	2.00000	0.516398
$16$	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	$0.240633\pi$
$18$	1.41421i	0.333333i
$19$	4.24264i	0.973329i	0.873589	+	0.486664i	$0.161786\pi$
$19$	4.24264i	0.973329i	−0.873589	+	0.486664i	$0.838214\pi$
$20$	− 2.82843i	− 0.632456i
$21$	0	0
$22$	−4.00000	−0.852803
$23$	−6.00000	−1.25109	−0.625543	−	0.780189i	$-0.715123\pi$
$23$	−6.00000	−1.25109	−0.625543	+	0.780189i	$0.715123\pi$
$24$	−4.00000	−0.816497
$25$	3.00000	0.600000
$26$	−6.00000	−1.17670
$27$	− 5.65685i	− 1.08866i
$28$	0	0
$29$	2.82843i	0.525226i	0.964901	+	0.262613i	$0.0845842\pi$
$29$	2.82843i	0.525226i	−0.964901	+	0.262613i	$0.915416\pi$
$30$	2.82843i	0.516398i
$31$	4.00000	0.718421	0.359211	−	0.933257i	$-0.383046\pi$
$31$	4.00000	0.718421	0.359211	+	0.933257i	$0.383046\pi$
$32$	5.65685i	1.00000i
$33$	4.00000	0.696311
$34$	8.48528i	1.45521i
$35$	0	0
$36$	−2.00000	−0.333333
$37$	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
$39$	6.00000	0.960769
$40$	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
$45$	1.41421i	0.210819i
$46$	− 8.48528i	− 1.25109i
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	− 5.65685i	− 0.816497i
$49$	0	0
$50$	4.24264i	0.600000i
$51$	− 8.48528i	− 1.18818i
$52$	− 8.48528i	− 1.17670i
$53$	− 5.65685i	− 0.777029i	−0.921443	−	0.388514i	$-0.872988\pi$
$53$	− 5.65685i	− 0.777029i	0.921443	−	0.388514i	$-0.127012\pi$
$54$	8.00000	1.08866
$55$	−4.00000	−0.539360
$56$	0	0
$57$	6.00000	0.794719
$58$	−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	$0.970656\pi$
$60$	−4.00000	−0.516398
$61$	− 12.7279i	− 1.62964i	−0.579712	−	0.814822i	$-0.696835\pi$
$61$	− 12.7279i	− 1.62964i	0.579712	−	0.814822i	$-0.303165\pi$
$62$	5.65685i	0.718421i
$63$	0	0
$64$	−8.00000	−1.00000
$65$	−6.00000	−0.744208
$66$	5.65685i	0.696311i
$67$	0	0	1.00000			$0$
$67$	0	0	−1.00000			$\pi$
$68$	−12.0000	−1.45521
$69$	8.48528i	1.02151i
$70$	0	0
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	− 2.82843i	− 0.333333i
$73$	−2.00000	−0.234082	−0.117041	−	0.993127i	$-0.537341\pi$
$73$	−2.00000	−0.234082	−0.117041	+	0.993127i	$0.537341\pi$
$74$	−12.0000	−1.39497
$75$	− 4.24264i	− 0.489898i
$76$	− 8.48528i	− 0.973329i
$77$	0	0
$78$	8.48528i	0.960769i
$79$	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
$81$	−5.00000	−0.555556
$82$	− 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
$87$	4.00000	0.428845
$88$	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	$0.603011\pi$
$90$	−2.00000	−0.210819
$91$	0	0
$92$	12.0000	1.25109
$93$	− 5.65685i	− 0.586588i
$94$	0	0
$95$	−6.00000	−0.615587
$96$	8.00000	0.816497
$97$	10.0000	1.01535	0.507673	−	0.861550i	$-0.330506\pi$
$97$	10.0000	1.01535	0.507673	+	0.861550i	$0.330506\pi$
$98$	0	0
$99$	2.82843i	0.284268i
$100$	−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	$0.836076\pi$
$102$	12.0000	1.18818
$103$	4.00000	0.394132	0.197066	−	0.980390i	$-0.436859\pi$
$103$	4.00000	0.394132	0.197066	+	0.980390i	$0.436859\pi$
$104$	12.0000	1.17670
$105$	0	0
$106$	8.00000	0.777029
$107$	− 5.65685i	− 0.546869i	−0.961891	−	0.273434i	$-0.911840\pi$
$107$	− 5.65685i	− 0.546869i	0.961891	−	0.273434i	$-0.0881596\pi$
$108$	11.3137i	1.08866i
$109$	− 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
$111$	12.0000	1.13899
$112$	0	0
$113$	−12.0000	−1.12887	−0.564433	−	0.825479i	$-0.690905\pi$
$113$	−12.0000	−1.12887	−0.564433	+	0.825479i	$0.690905\pi$
$114$	8.48528i	0.794719i
$115$	− 8.48528i	− 0.791257i
$116$	− 5.65685i	− 0.525226i
$117$	4.24264i	0.392232i
$118$	−2.00000	−0.184115
$119$	0	0
$120$	− 5.65685i	− 0.516398i
$121$	3.00000	0.272727
$122$	18.0000	1.62964
$123$	8.48528i	0.765092i
$124$	−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
$129$	−12.0000	−1.05654
$130$	− 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	$0.980322\pi$
$132$	−8.00000	−0.696311
$133$	0	0
$134$	0	0
$135$	8.00000	0.688530
$136$	− 16.9706i	− 1.45521i
$137$	6.00000	0.512615	0.256307	−	0.966595i	$-0.417494\pi$
$137$	6.00000	0.512615	0.256307	+	0.966595i	$0.417494\pi$
$138$	−12.0000	−1.02151
$139$	4.24264i	0.359856i	0.983680	+	0.179928i	$0.0575865\pi$
$139$	4.24264i	0.359856i	−0.983680	+	0.179928i	$0.942414\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−12.0000	−1.00349
$144$	4.00000	0.333333
$145$	−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	$0.153381\pi$
$149$	11.3137i	0.926855i	−0.886135	+	0.463428i	$0.846619\pi$
$150$	6.00000	0.489898
$151$	−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
$153$	6.00000	0.485071
$154$	0	0
$155$	5.65685i	0.454369i
$156$	−12.0000	−0.960769
$157$	− 12.7279i	− 1.01580i	−0.861416	−	0.507899i	$-0.830422\pi$
$157$	− 12.7279i	− 1.01580i	0.861416	−	0.507899i	$-0.169578\pi$
$158$	11.3137i	0.900070i
$159$	−8.00000	−0.634441
$160$	−8.00000	−0.632456
$161$	0	0
$162$	− 7.07107i	− 0.555556i
$163$	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
$165$	5.65685i	0.440386i
$166$	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
$171$	4.24264i	0.324443i
$172$	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	$0.877189\pi$
$174$	5.65685i	0.428845i
$175$	0	0
$176$	11.3137i	0.852803i
$177$	2.00000	0.150329
$178$	− 8.48528i	− 0.635999i
$179$	− 5.65685i	− 0.422813i	−0.977398	−	0.211407i	$-0.932196\pi$
$179$	− 5.65685i	− 0.422813i	0.977398	−	0.211407i	$-0.0678044\pi$
$180$	− 2.82843i	− 0.210819i
$181$	− 12.7279i	− 0.946059i	−0.881047	−	0.473029i	$-0.843160\pi$
$181$	− 12.7279i	− 0.946059i	0.881047	−	0.473029i	$-0.156840\pi$
$182$	0	0
$183$	−18.0000	−1.33060
$184$	16.9706i	1.25109i
$185$	−12.0000	−0.882258
$186$	8.00000	0.586588
$187$	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	$0.5\pi$
$192$	11.3137i	0.816497i
$193$	−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
$195$	8.48528i	0.607644i
$196$	0	0
$197$	− 5.65685i	− 0.403034i	−0.979485	−	0.201517i	$-0.935413\pi$
$197$	− 5.65685i	− 0.403034i	0.979485	−	0.201517i	$-0.0645872\pi$
$198$	−4.00000	−0.284268
$199$	−20.0000	−1.41776	−0.708881	−	0.705328i	$-0.750800\pi$
$199$	−20.0000	−1.41776	−0.708881	+	0.705328i	$0.750800\pi$
$200$	− 8.48528i	− 0.600000i
$201$	0	0
$202$	−14.0000	−0.985037
$203$	0	0
$204$	16.9706i	1.18818i
$205$	− 8.48528i	− 0.592638i
$206$	5.65685i	0.394132i
$207$	−6.00000	−0.417029
$208$	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
$216$	−16.0000	−1.08866
$217$	0	0
$218$	12.0000	0.812743
$219$	2.82843i	0.191127i
$220$	8.00000	0.539360
$221$	25.4558i	1.71235i
$222$	16.9706i	1.13899i
$223$	28.0000	1.87502	0.937509	−	0.347960i	$-0.113126\pi$
$223$	28.0000	1.87502	0.937509	+	0.347960i	$0.113126\pi$
$224$	0	0
$225$	3.00000	0.200000
$226$	− 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	$0.893448\pi$
$228$	−12.0000	−0.794719
$229$	− 4.24264i	− 0.280362i	−0.990126	−	0.140181i	$-0.955232\pi$
$229$	− 4.24264i	− 0.280362i	0.990126	−	0.140181i	$-0.0447684\pi$
$230$	12.0000	0.791257
$231$	0	0
$232$	8.00000	0.525226
$233$	6.00000	0.393073	0.196537	−	0.980497i	$-0.437031\pi$
$233$	6.00000	0.393073	0.196537	+	0.980497i	$0.437031\pi$
$234$	−6.00000	−0.392232
$235$	0	0
$236$	− 2.82843i	− 0.184115i
$237$	− 11.3137i	− 0.734904i
$238$	0	0
$239$	6.00000	0.388108	0.194054	−	0.980991i	$-0.437836\pi$
$239$	6.00000	0.388108	0.194054	+	0.980991i	$0.437836\pi$
$240$	8.00000	0.516398
$241$	10.0000	0.644157	0.322078	−	0.946713i	$-0.395619\pi$
$241$	10.0000	0.644157	0.322078	+	0.946713i	$0.395619\pi$
$242$	4.24264i	0.272727i
$243$	− 9.89949i	− 0.635053i
$244$	25.4558i	1.62964i
$245$	0	0
$246$	−12.0000	−0.765092
$247$	−18.0000	−1.14531
$248$	− 11.3137i	− 0.718421i
$249$	−22.0000	−1.39419
$250$	−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
$255$	12.0000	0.751469
$256$	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	$0.440080\pi$
$258$	− 16.9706i	− 1.05654i
$259$	0	0
$260$	12.0000	0.744208
$261$	2.82843i	0.175075i
$262$	−2.00000	−0.123560
$263$	24.0000	1.47990	0.739952	−	0.672660i	$-0.234848\pi$
$263$	24.0000	1.47990	0.739952	+	0.672660i	$0.234848\pi$
$264$	− 11.3137i	− 0.696311i
$265$	8.00000	0.491436
$266$	0	0
$267$	8.48528i	0.519291i
$268$	0	0
$269$	− 7.07107i	− 0.431131i	−0.976489	−	0.215565i	$-0.930841\pi$
$269$	− 7.07107i	− 0.431131i	0.976489	−	0.215565i	$-0.0691594\pi$
$270$	11.3137i	0.688530i
$271$	−20.0000	−1.21491	−0.607457	−	0.794353i	$-0.707810\pi$
$271$	−20.0000	−1.21491	−0.607457	+	0.794353i	$0.707810\pi$
$272$	24.0000	1.45521
$273$	0	0
$274$	8.48528i	0.512615i
$275$	8.48528i	0.511682i
$276$	− 16.9706i	− 1.02151i
$277$	− 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
$279$	4.00000	0.239474
$280$	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
$283$	− 12.7279i	− 0.756596i	−0.925684	−	0.378298i	$-0.876509\pi$
$283$	− 12.7279i	− 0.756596i	0.925684	−	0.378298i	$-0.123491\pi$
$284$	0	0
$285$	8.48528i	0.502625i
$286$	− 16.9706i	− 1.00349i
$287$	0	0
$288$	5.65685i	0.333333i
$289$	19.0000	1.11765
$290$	− 5.65685i	− 0.332182i
$291$	− 14.1421i	− 0.829027i
$292$	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
$297$	16.0000	0.928414
$298$	−16.0000	−0.926855
$299$	− 25.4558i	− 1.47215i
$300$	8.48528i	0.489898i
$301$	0	0
$302$	− 14.1421i	− 0.813788i
$303$	14.0000	0.804279
$304$	16.9706i	0.973329i
$305$	18.0000	1.03068
$306$	8.48528i	0.485071i
$307$	12.7279i	0.726421i	0.931707	+	0.363210i	$0.118319\pi$
$307$	12.7279i	0.726421i	−0.931707	+	0.363210i	$0.881681\pi$
$308$	0	0
$309$	− 5.65685i	− 0.321807i
$310$	−8.00000	−0.454369
$311$	0	0		−	1.00000i	$-0.5\pi$
$311$	0	0			1.00000i	$0.5\pi$
$312$	− 16.9706i	− 0.960769i
$313$	10.0000	0.565233	0.282617	−	0.959233i	$-0.408798\pi$
$313$	10.0000	0.565233	0.282617	+	0.959233i	$0.408798\pi$
$314$	18.0000	1.01580
$315$	0	0
$316$	−16.0000	−0.900070
$317$	− 22.6274i	− 1.27088i	−0.772149	−	0.635441i	$-0.780818\pi$
$317$	− 22.6274i	− 1.27088i	0.772149	−	0.635441i	$-0.219182\pi$
$318$	− 11.3137i	− 0.634441i
$319$	−8.00000	−0.447914
$320$	− 11.3137i	− 0.632456i
$321$	−8.00000	−0.446516
$322$	0	0
$323$	25.4558i	1.41640i
$324$	10.0000	0.555556
$325$	12.7279i	0.706018i
$326$	−12.0000	−0.664619
$327$	−12.0000	−0.663602
$328$	16.9706i	0.937043i
$329$	0	0
$330$	−8.00000	−0.440386
$331$	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
$333$	8.48528i	0.464991i
$334$	− 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
$339$	16.9706i	0.921714i
$340$	− 16.9706i	− 0.920358i
$341$	11.3137i	0.612672i
$342$	−6.00000	−0.324443
$343$	0	0
$344$	−24.0000	−1.29399
$345$	−12.0000	−0.646058
$346$	−14.0000	−0.752645
$347$	− 14.1421i	− 0.759190i	−0.925153	−	0.379595i	$-0.876063\pi$
$347$	− 14.1421i	− 0.759190i	0.925153	−	0.379595i	$-0.123937\pi$
$348$	−8.00000	−0.428845
$349$	− 4.24264i	− 0.227103i	−0.993532	−	0.113552i	$-0.963777\pi$
$349$	− 4.24264i	− 0.227103i	0.993532	−	0.113552i	$-0.0362227\pi$
$350$	0	0
$351$	24.0000	1.28103
$352$	−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	$0.551044\pi$
$354$	2.82843i	0.150329i
$355$	0	0
$356$	12.0000	0.635999
$357$	0	0
$358$	8.00000	0.422813
$359$	−30.0000	−1.58334	−0.791670	−	0.610949i	$-0.790788\pi$
$359$	−30.0000	−1.58334	−0.791670	+	0.610949i	$0.790788\pi$
$360$	4.00000	0.210819
$361$	1.00000	0.0526316
$362$	18.0000	0.946059
$363$	− 4.24264i	− 0.222681i
$364$	0	0
$365$	− 2.82843i	− 0.148047i
$366$	− 25.4558i	− 1.33060i
$367$	28.0000	1.46159	0.730794	−	0.682598i	$-0.239150\pi$
$367$	28.0000	1.46159	0.730794	+	0.682598i	$0.239150\pi$
$368$	−24.0000	−1.25109
$369$	−6.00000	−0.312348
$370$	− 16.9706i	− 0.882258i
$371$	0	0
$372$	11.3137i	0.586588i
$373$	− 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
$375$	16.0000	0.826236
$376$	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
$381$	− 2.82843i	− 0.144905i
$382$	0	0
$383$	24.0000	1.22634	0.613171	−	0.789950i	$-0.289894\pi$
$383$	24.0000	1.22634	0.613171	+	0.789950i	$0.289894\pi$
$384$	−16.0000	−0.816497
$385$	0	0
$386$	− 5.65685i	− 0.287926i
$387$	− 8.48528i	− 0.431331i
$388$	−20.0000	−1.01535
$389$	2.82843i	0.143407i	0.997426	+	0.0717035i	$0.0228435\pi$
$389$	2.82843i	0.143407i	−0.997426	+	0.0717035i	$0.977156\pi$
$390$	−12.0000	−0.607644
$391$	−36.0000	−1.82060
$392$	0	0
$393$	2.00000	0.100887
$394$	8.00000	0.403034
$395$	11.3137i	0.569254i
$396$	− 5.65685i	− 0.284268i
$397$	− 21.2132i	− 1.06466i	−0.846537	−	0.532330i	$-0.821317\pi$
$397$	− 21.2132i	− 1.06466i	0.846537	−	0.532330i	$-0.178683\pi$
$398$	− 28.2843i	− 1.41776i
$399$	0	0
$400$	12.0000	0.600000
$401$	−24.0000	−1.19850	−0.599251	−	0.800561i	$-0.704535\pi$
$401$	−24.0000	−1.19850	−0.599251	+	0.800561i	$0.704535\pi$
$402$	0	0
$403$	16.9706i	0.845364i
$404$	− 19.7990i	− 0.985037i
$405$	− 7.07107i	− 0.351364i
$406$	0	0
$407$	−24.0000	−1.18964
$408$	−24.0000	−1.18818
$409$	22.0000	1.08783	0.543915	−	0.839140i	$-0.316941\pi$
$409$	22.0000	1.08783	0.543915	+	0.839140i	$0.316941\pi$
$410$	12.0000	0.592638
$411$	− 8.48528i	− 0.418548i
$412$	−8.00000	−0.394132
$413$	0	0
$414$	− 8.48528i	− 0.417029i
$415$	22.0000	1.07994
$416$	−24.0000	−1.17670
$417$	6.00000	0.293821
$418$	− 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
$427$	0	0
$428$	11.3137i	0.546869i
$429$	16.9706i	0.819346i
$430$	16.9706i	0.818393i
$431$	−6.00000	−0.289010	−0.144505	−	0.989504i	$-0.546159\pi$
$431$	−6.00000	−0.289010	−0.144505	+	0.989504i	$0.546159\pi$
$432$	− 22.6274i	− 1.08866i
$433$	−2.00000	−0.0961139	−0.0480569	−	0.998845i	$-0.515303\pi$
$433$	−2.00000	−0.0961139	−0.0480569	+	0.998845i	$0.515303\pi$
$434$	0	0
$435$	5.65685i	0.271225i
$436$	16.9706i	0.812743i
$437$	− 25.4558i	− 1.21772i
$438$	−4.00000	−0.191127
$439$	28.0000	1.33637	0.668184	−	0.743996i	$-0.267072\pi$
$439$	28.0000	1.33637	0.668184	+	0.743996i	$0.267072\pi$
$440$	11.3137i	0.539360i
$441$	0	0
$442$	−36.0000	−1.71235
$443$	− 22.6274i	− 1.07506i	−0.843244	−	0.537531i	$-0.819357\pi$
$443$	− 22.6274i	− 1.07506i	0.843244	−	0.537531i	$-0.180643\pi$
$444$	−24.0000	−1.13899
$445$	− 8.48528i	− 0.402241i
$446$	39.5980i	1.87502i
$447$	16.0000	0.756774
$448$	0	0
$449$	−18.0000	−0.849473	−0.424736	−	0.905317i	$-0.639633\pi$
$449$	−18.0000	−0.849473	−0.424736	+	0.905317i	$0.639633\pi$
$450$	4.24264i	0.200000i
$451$	− 16.9706i	− 0.799113i
$452$	24.0000	1.12887
$453$	14.1421i	0.664455i
$454$	−14.0000	−0.657053
$455$	0	0
$456$	− 16.9706i	− 0.794719i
$457$	−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
$459$	− 33.9411i	− 1.58424i
$460$	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
$465$	8.00000	0.370991
$466$	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	$-0.0523123\pi$
$468$	− 8.48528i	− 0.392232i
$469$	0	0
$470$	0	0
$471$	−18.0000	−0.829396
$472$	4.00000	0.184115
$473$	24.0000	1.10352
$474$	16.0000	0.734904
$475$	12.7279i	0.583997i
$476$	0	0
$477$	− 5.65685i	− 0.259010i
$478$	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	$0.411605\pi$
$480$	11.3137i	0.516398i
$481$	−36.0000	−1.64146
$482$	14.1421i	0.644157i
$483$	0	0
$484$	−6.00000	−0.272727
$485$	14.1421i	0.642161i
$486$	14.0000	0.635053
$487$	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
$489$	12.0000	0.542659
$490$	0	0
$491$	− 39.5980i	− 1.78703i	−0.449032	−	0.893516i	$-0.648231\pi$
$491$	− 39.5980i	− 1.78703i	0.449032	−	0.893516i	$-0.351769\pi$
$492$	− 16.9706i	− 0.765092i
$493$	16.9706i	0.764316i
$494$	− 25.4558i	− 1.14531i
$495$	−4.00000	−0.179787
$496$	16.0000	0.718421
$497$	0	0
$498$	− 31.1127i	− 1.39419i
$499$	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
$501$	33.9411i	1.51638i
$502$	−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
$507$	7.07107i	0.314037i
$508$	−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	$0.990022\pi$
$510$	16.9706i	0.751469i
$511$	0	0
$512$	22.6274i	1.00000i
$513$	24.0000	1.05963
$514$	8.48528i	0.374270i
$515$	5.65685i	0.249271i
$516$	24.0000	1.05654
$517$	0	0
$518$	0	0
$519$	14.0000	0.614532
$520$	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	$0.458042\pi$
$522$	−4.00000	−0.175075
$523$	29.6985i	1.29862i	0.760522	+	0.649312i	$0.224943\pi$
$523$	29.6985i	1.29862i	−0.760522	+	0.649312i	$0.775057\pi$
$524$	− 2.82843i	− 0.123560i
$525$	0	0
$526$	33.9411i	1.47990i
$527$	24.0000	1.04546
$528$	16.0000	0.696311
$529$	13.0000	0.565217
$530$	11.3137i	0.491436i
$531$	1.41421i	0.0613716i
$532$	0	0
$533$	− 25.4558i	− 1.10262i
$534$	−12.0000	−0.519291
$535$	8.00000	0.345870
$536$	0	0
$537$	−8.00000	−0.345225
$538$	10.0000	0.431131
$539$	0	0
$540$	−16.0000	−0.688530
$541$	− 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
$543$	−18.0000	−0.772454
$544$	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
$549$	− 12.7279i	− 0.543214i
$550$	−12.0000	−0.511682
$551$	−12.0000	−0.511217
$552$	24.0000	1.02151
$553$	0	0
$554$	24.0000	1.01966
$555$	16.9706i	0.720360i
$556$	− 8.48528i	− 0.359856i
$557$	− 5.65685i	− 0.239689i	−0.992793	−	0.119844i	$-0.961760\pi$
$557$	− 5.65685i	− 0.239689i	0.992793	−	0.119844i	$-0.0382395\pi$
$558$	5.65685i	0.239474i
$559$	36.0000	1.52264
$560$	0	0
$561$	24.0000	1.01328
$562$	− 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	$0.5\pi$
$570$	−12.0000	−0.502625
$571$	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
$574$	0	0
$575$	−18.0000	−0.750652
$576$	−8.00000	−0.333333
$577$	−38.0000	−1.58196	−0.790980	−	0.611842i	$-0.790429\pi$
$577$	−38.0000	−1.58196	−0.790980	+	0.611842i	$0.790429\pi$
$578$	26.8701i	1.11765i
$579$	5.65685i	0.235091i
$580$	8.00000	0.332182
$581$	0	0
$582$	20.0000	0.829027
$583$	16.0000	0.662652
$584$	5.65685i	0.234082i
$585$	−6.00000	−0.248069
$586$	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
$591$	−8.00000	−0.329076
$592$	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	$0.288762\pi$
$594$	22.6274i	0.928414i
$595$	0	0
$596$	− 22.6274i	− 0.926855i
$597$	28.2843i	1.15760i
$598$	36.0000	1.47215
$599$	0	0		−	1.00000i	$-0.5\pi$
$599$	0	0			1.00000i	$0.5\pi$
$600$	−12.0000	−0.489898
$601$	10.0000	0.407909	0.203954	−	0.978980i	$-0.434621\pi$
$601$	10.0000	0.407909	0.203954	+	0.978980i	$0.434621\pi$
$602$	0	0
$603$	0	0
$604$	20.0000	0.813788
$605$	4.24264i	0.172488i
$606$	19.7990i	0.804279i
$607$	−32.0000	−1.29884	−0.649420	−	0.760430i	$-0.724988\pi$
$607$	−32.0000	−1.29884	−0.649420	+	0.760430i	$0.724988\pi$
$608$	−24.0000	−0.973329
$609$	0	0
$610$	25.4558i	1.03068i
$611$	0	0
$612$	−12.0000	−0.485071
$613$	− 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
$615$	−12.0000	−0.483887
$616$	0	0
$617$	24.0000	0.966204	0.483102	−	0.875564i	$-0.339510\pi$
$617$	24.0000	0.966204	0.483102	+	0.875564i	$0.339510\pi$
$618$	8.00000	0.321807
$619$	4.24264i	0.170526i	0.996358	+	0.0852631i	$0.0271731\pi$
$619$	4.24264i	0.170526i	−0.996358	+	0.0852631i	$0.972827\pi$
$620$	− 11.3137i	− 0.454369i
$621$	33.9411i	1.36201i
$622$	0	0
$623$	0	0
$624$	24.0000	0.960769
$625$	−1.00000	−0.0400000
$626$	14.1421i	0.565233i
$627$	16.9706i	0.677739i
$628$	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
$633$	−24.0000	−0.953914
$634$	32.0000	1.27088
$635$	2.82843i	0.112243i
$636$	16.0000	0.634441
$637$	0	0
$638$	− 11.3137i	− 0.447914i
$639$	0	0
$640$	16.0000	0.632456
$641$	−48.0000	−1.89589	−0.947943	−	0.318440i	$-0.896841\pi$
$641$	−48.0000	−1.89589	−0.947943	+	0.318440i	$0.896841\pi$
$642$	− 11.3137i	− 0.446516i
$643$	− 21.2132i	− 0.836567i	−0.908317	−	0.418284i	$-0.862632\pi$
$643$	− 21.2132i	− 0.836567i	0.908317	−	0.418284i	$-0.137368\pi$
$644$	0	0
$645$	− 16.9706i	− 0.668215i
$646$	−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	$0.575799\pi$
$648$	14.1421i	0.555556i
$649$	−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	$0.255607\pi$
$653$	36.7696i	1.43890i	−0.694542	+	0.719452i	$0.744393\pi$
$654$	− 16.9706i	− 0.663602i
$655$	−2.00000	−0.0781465
$656$	−24.0000	−0.937043
$657$	−2.00000	−0.0780274
$658$	0	0
$659$	2.82843i	0.110180i	0.998481	+	0.0550899i	$0.0175446\pi$
$659$	2.82843i	0.110180i	−0.998481	+	0.0550899i	$0.982455\pi$
$660$	− 11.3137i	− 0.440386i
$661$	38.1838i	1.48518i	0.669748	+	0.742588i	$0.266402\pi$
$661$	38.1838i	1.48518i	−0.669748	+	0.742588i	$0.733598\pi$
$662$	−36.0000	−1.39918
$663$	36.0000	1.39812
$664$	−44.0000	−1.70753
$665$	0	0
$666$	−12.0000	−0.464991
$667$	− 16.9706i	− 0.657103i
$668$	48.0000	1.85718
$669$	− 39.5980i	− 1.53095i
$670$	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
$675$	− 16.9706i	− 0.653197i
$676$	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	$0.939075\pi$
$678$	−24.0000	−0.921714
$679$	0	0
$680$	24.0000	0.920358
$681$	14.0000	0.536481
$682$	−16.0000	−0.612672
$683$	− 5.65685i	− 0.216454i	−0.994126	−	0.108227i	$-0.965483\pi$
$683$	− 5.65685i	− 0.216454i	0.994126	−	0.108227i	$-0.0345173\pi$
$684$	− 8.48528i	− 0.324443i
$685$	8.48528i	0.324206i
$686$	0	0
$687$	−6.00000	−0.228914
$688$	− 33.9411i	− 1.29399i
$689$	24.0000	0.914327
$690$	− 16.9706i	− 0.646058i
$691$	12.7279i	0.484193i	0.970252	+	0.242096i	$0.0778351\pi$
$691$	12.7279i	0.484193i	−0.970252	+	0.242096i	$0.922165\pi$
$692$	− 19.7990i	− 0.752645i
$693$	0	0
$694$	20.0000	0.759190
$695$	−6.00000	−0.227593
$696$	− 11.3137i	− 0.428845i
$697$	−36.0000	−1.36360
$698$	6.00000	0.227103
$699$	− 8.48528i	− 0.320943i
$700$	0	0
$701$	19.7990i	0.747798i	0.927470	+	0.373899i	$0.121979\pi$
$701$	19.7990i	0.747798i	−0.927470	+	0.373899i	$0.878021\pi$
$702$	33.9411i	1.28103i
$703$	−36.0000	−1.35777
$704$	− 22.6274i	− 0.852803i
$705$	0	0
$706$	− 8.48528i	− 0.319348i
$707$	0	0
$708$	−4.00000	−0.150329
$709$	− 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
$711$	8.00000	0.300023
$712$	16.9706i	0.635999i
$713$	−24.0000	−0.898807
$714$	0	0
$715$	− 16.9706i	− 0.634663i
$716$	11.3137i	0.422813i
$717$	− 8.48528i	− 0.316889i
$718$	− 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	$0.647694\pi$
$720$	5.65685i	0.210819i
$721$	0	0
$722$	1.41421i	0.0526316i
$723$	− 14.1421i	− 0.525952i
$724$	25.4558i	0.946059i
$725$	8.48528i	0.315135i
$726$	6.00000	0.222681
$727$	−8.00000	−0.296704	−0.148352	−	0.988935i	$-0.547397\pi$
$727$	−8.00000	−0.296704	−0.148352	+	0.988935i	$0.547397\pi$
$728$	0	0
$729$	−29.0000	−1.07407
$730$	4.00000	0.148047
$731$	− 50.9117i	− 1.88304i
$732$	36.0000	1.33060
$733$	− 29.6985i	− 1.09694i	−0.836171	−	0.548469i	$-0.815211\pi$
$733$	− 29.6985i	− 1.09694i	0.836171	−	0.548469i	$-0.184789\pi$
$734$	39.5980i	1.46159i
$735$	0	0
$736$	− 33.9411i	− 1.25109i
$737$	0	0
$738$	− 8.48528i	− 0.312348i
$739$	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
$741$	25.4558i	0.935144i
$742$	0	0
$743$	−30.0000	−1.10059	−0.550297	−	0.834969i	$-0.685485\pi$
$743$	−30.0000	−1.10059	−0.550297	+	0.834969i	$0.685485\pi$
$744$	−16.0000	−0.586588
$745$	−16.0000	−0.586195
$746$	48.0000	1.75740
$747$	− 15.5563i	− 0.569177i
$748$	− 33.9411i	− 1.24101i
$749$	0	0
$750$	22.6274i	0.826236i
$751$	−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
$753$	26.0000	0.947493
$754$	− 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
$759$	−24.0000	−0.871145
$760$	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	$0.682996\pi$
$762$	4.00000	0.144905
$763$	0	0
$764$	0	0
$765$	8.48528i	0.306786i
$766$	33.9411i	1.22634i
$767$	−6.00000	−0.216647
$768$	− 22.6274i	− 0.816497i
$769$	−14.0000	−0.504853	−0.252426	−	0.967616i	$-0.581229\pi$
$769$	−14.0000	−0.504853	−0.252426	+	0.967616i	$0.581229\pi$
$770$	0	0
$771$	− 8.48528i	− 0.305590i
$772$	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	$-0.198888\pi$
$774$	12.0000	0.431331
$775$	12.0000	0.431053
$776$	− 28.2843i	− 1.01535i
$777$	0	0
$778$	−4.00000	−0.143407
$779$	− 25.4558i	− 0.912050i
$780$	− 16.9706i	− 0.607644i
$781$	0	0
$782$	− 50.9117i	− 1.82060i
$783$	16.0000	0.571793
$784$	0	0
$785$	18.0000	0.642448
$786$	2.82843i	0.100887i
$787$	38.1838i	1.36110i	0.732700	+	0.680552i	$0.238260\pi$
$787$	38.1838i	1.36110i	−0.732700	+	0.680552i	$0.761740\pi$
$788$	11.3137i	0.403034i
$789$	− 33.9411i	− 1.20834i
$790$	−16.0000	−0.569254
$791$	0	0
$792$	8.00000	0.284268
$793$	54.0000	1.91760
$794$	30.0000	1.06466
$795$	− 11.3137i	− 0.401256i
$796$	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
$801$	−6.00000	−0.212000
$802$	− 33.9411i	− 1.19850i
$803$	− 5.65685i	− 0.199626i
$804$	0	0
$805$	0	0
$806$	−24.0000	−0.845364
$807$	−10.0000	−0.352017
$808$	28.0000	0.985037
$809$	12.0000	0.421898	0.210949	−	0.977497i	$-0.432345\pi$
$809$	12.0000	0.421898	0.210949	+	0.977497i	$0.432345\pi$
$810$	10.0000	0.351364
$811$	− 12.7279i	− 0.446938i	−0.974711	−	0.223469i	$-0.928262\pi$
$811$	− 12.7279i	− 0.446938i	0.974711	−	0.223469i	$-0.0717381\pi$
$812$	0	0
$813$	28.2843i	0.991973i
$814$	− 33.9411i	− 1.18964i
$815$	−12.0000	−0.420342
$816$	− 33.9411i	− 1.18818i
$817$	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	$-0.870796\pi$
$821$	− 22.6274i	− 0.789702i	0.918745	−	0.394851i	$-0.129204\pi$
$822$	12.0000	0.418548
$823$	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
$825$	12.0000	0.417786
$826$	0	0
$827$	45.2548i	1.57366i	0.617167	+	0.786832i	$0.288280\pi$
$827$	45.2548i	1.57366i	−0.617167	+	0.786832i	$0.711720\pi$
$828$	12.0000	0.417029
$829$	21.2132i	0.736765i	0.929674	+	0.368383i	$0.120088\pi$
$829$	21.2132i	0.736765i	−0.929674	+	0.368383i	$0.879912\pi$
$830$	31.1127i	1.07994i
$831$	−24.0000	−0.832551
$832$	− 33.9411i	− 1.17670i
$833$	0	0
$834$	8.48528i	0.293821i
$835$	− 33.9411i	− 1.17458i
$836$	24.0000	0.830057
$837$	− 22.6274i	− 0.782118i
$838$	−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
$843$	8.48528i	0.292249i
$844$	33.9411i	1.16830i
$845$	− 7.07107i	− 0.243252i
$846$	0	0
$847$	0	0
$848$	− 22.6274i	− 0.777029i
$849$	−18.0000	−0.617758
$850$	25.4558i	0.873128i
$851$	− 50.9117i	− 1.74523i
$852$	0	0
$853$	− 4.24264i	− 0.145265i	−0.997359	−	0.0726326i	$-0.976860\pi$
$853$	− 4.24264i	− 0.145265i	0.997359	−	0.0726326i	$-0.0231401\pi$
$854$	0	0
$855$	−6.00000	−0.205196
$856$	−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	$0.754643\pi$
$858$	−24.0000	−0.819346
$859$	46.6690i	1.59233i	0.605081	+	0.796164i	$0.293141\pi$
$859$	46.6690i	1.59233i	−0.605081	+	0.796164i	$0.706859\pi$
$860$	−24.0000	−0.818393
$861$	0	0
$862$	− 8.48528i	− 0.289010i
$863$	24.0000	0.816970	0.408485	−	0.912765i	$-0.366057\pi$
$863$	24.0000	0.816970	0.408485	+	0.912765i	$0.366057\pi$
$864$	32.0000	1.08866
$865$	−14.0000	−0.476014
$866$	− 2.82843i	− 0.0961139i
$867$	− 26.8701i	− 0.912555i
$868$	0	0
$869$	22.6274i	0.767583i
$870$	−8.00000	−0.271225
$871$	0	0
$872$	−24.0000	−0.812743
$873$	10.0000	0.338449
$874$	36.0000	1.21772
$875$	0	0
$876$	− 5.65685i	− 0.191127i
$877$	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
$879$	−34.0000	−1.14679
$880$	−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
$885$	2.82843i	0.0950765i
$886$	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	$0.798286\pi$
$888$	− 33.9411i	− 1.13899i
$889$	0	0
$890$	12.0000	0.402241
$891$	− 14.1421i	− 0.473779i
$892$	−56.0000	−1.87502
$893$	0	0
$894$	22.6274i	0.756774i
$895$	8.00000	0.267411
$896$	0	0
$897$	−36.0000	−1.20201
$898$	− 25.4558i	− 0.849473i
$899$	11.3137i	0.377333i
$900$	−6.00000	−0.200000
$901$	− 33.9411i	− 1.13074i
$902$	24.0000	0.799113
$903$	0	0
$904$	33.9411i	1.12887i
$905$	18.0000	0.598340
$906$	−20.0000	−0.664455
$907$	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
$909$	9.89949i	0.328346i
$910$	0	0
$911$	30.0000	0.993944	0.496972	−	0.867766i	$-0.334445\pi$
$911$	30.0000	0.993944	0.496972	+	0.867766i	$0.334445\pi$
$912$	24.0000	0.794719
$913$	44.0000	1.45619
$914$	− 39.5980i	− 1.30978i
$915$	− 25.4558i	− 0.841544i
$916$	8.48528i	0.280362i
$917$	0	0
$918$	48.0000	1.58424
$919$	−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
$921$	18.0000	0.593120
$922$	22.0000	0.724531
$923$	0	0
$924$	0	0
$925$	25.4558i	0.836983i
$926$	45.2548i	1.48717i
$927$	4.00000	0.131377
$928$	−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	$0.595413\pi$
$930$	11.3137i	0.370991i
$931$	0	0
$932$	−12.0000	−0.393073
$933$	0	0
$934$	10.0000	0.327210
$935$	−24.0000	−0.784884
$936$	12.0000	0.392232
$937$	−2.00000	−0.0653372	−0.0326686	−	0.999466i	$-0.510401\pi$
$937$	−2.00000	−0.0653372	−0.0326686	+	0.999466i	$0.510401\pi$
$938$	0	0
$939$	− 14.1421i	− 0.461511i
$940$	0	0
$941$	52.3259i	1.70578i	0.522094	+	0.852888i	$0.325151\pi$
$941$	52.3259i	1.70578i	−0.522094	+	0.852888i	$0.674849\pi$
$942$	− 25.4558i	− 0.829396i
$943$	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	$0.0146333\pi$
$947$	2.82843i	0.0919115i	−0.998943	+	0.0459558i	$0.985367\pi$
$948$	22.6274i	0.734904i
$949$	− 8.48528i	− 0.275444i
$950$	−18.0000	−0.583997
$951$	−32.0000	−1.03767
$952$	0	0
$953$	18.0000	0.583077	0.291539	−	0.956559i	$-0.405833\pi$
$953$	18.0000	0.583077	0.291539	+	0.956559i	$0.405833\pi$
$954$	8.00000	0.259010
$955$	0	0
$956$	−12.0000	−0.388108
$957$	11.3137i	0.365720i
$958$	16.9706i	0.548294i
$959$	0	0
$960$	−16.0000	−0.516398
$961$	−15.0000	−0.483871
$962$	− 50.9117i	− 1.64146i
$963$	− 5.65685i	− 0.182290i
$964$	−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
$969$	36.0000	1.15649
$970$	−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	$-0.174784\pi$
$972$	19.7990i	0.635053i
$973$	0	0
$974$	2.82843i	0.0906287i
$975$	18.0000	0.576461
$976$	− 50.9117i	− 1.62964i
$977$	−18.0000	−0.575871	−0.287936	−	0.957650i	$-0.592969\pi$
$977$	−18.0000	−0.575871	−0.287936	+	0.957650i	$0.592969\pi$
$978$	16.9706i	0.542659i
$979$	− 16.9706i	− 0.542382i
$980$	0	0
$981$	− 8.48528i	− 0.270914i
$982$	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	$0.438707\pi$
$984$	24.0000	0.765092
$985$	8.00000	0.254901
$986$	−24.0000	−0.764316
$987$	0	0
$988$	36.0000	1.14531
$989$	50.9117i	1.61890i
$990$	− 5.65685i	− 0.179787i
$991$	−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
$993$	36.0000	1.14243
$994$	0	0
$995$	− 28.2843i	− 0.896672i
$996$	44.0000	1.39419
$997$	− 21.2132i	− 0.671829i	−0.941893	−	0.335914i	$-0.890955\pi$
$997$	− 21.2132i	− 0.671829i	0.941893	−	0.335914i	$-0.109045\pi$
$998$	−24.0000	−0.759707
$999$	48.0000	1.51865

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	392.2.b.b.197.2		2
4.3	odd	2		1568.2.b.a.785.2		2
7.2	even	3		392.2.p.b.165.2		4
7.3	odd	6		392.2.p.a.373.1		4
7.4	even	3		392.2.p.b.373.1		4
7.5	odd	6		392.2.p.a.165.2		4
7.6	odd	2		56.2.b.a.29.2	yes	2
8.3	odd	2		1568.2.b.a.785.1		2
8.5	even	2	inner	392.2.b.b.197.1		2
21.20	even	2		504.2.c.a.253.1		2
28.3	even	6		1568.2.t.c.177.1		4
28.11	odd	6		1568.2.t.b.177.2		4
28.19	even	6		1568.2.t.c.753.2		4
28.23	odd	6		1568.2.t.b.753.1		4
28.27	even	2		224.2.b.a.113.1		2
56.3	even	6		1568.2.t.c.177.2		4
56.5	odd	6		392.2.p.a.165.1		4
56.11	odd	6		1568.2.t.b.177.1		4
56.13	odd	2		56.2.b.a.29.1	✓	2
56.19	even	6		1568.2.t.c.753.1		4
56.27	even	2		224.2.b.a.113.2		2
56.37	even	6		392.2.p.b.165.1		4
56.45	odd	6		392.2.p.a.373.2		4
56.51	odd	6		1568.2.t.b.753.2		4
56.53	even	6		392.2.p.b.373.2		4
84.83	odd	2		2016.2.c.a.1009.2		2
112.13	odd	4		1792.2.a.n.1.1		2
112.27	even	4		1792.2.a.p.1.1		2
112.69	odd	4		1792.2.a.n.1.2		2
112.83	even	4		1792.2.a.p.1.2		2
168.83	odd	2		2016.2.c.a.1009.1		2
168.125	even	2		504.2.c.a.253.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
56.2.b.a.29.1	✓	2	56.13	odd	2
56.2.b.a.29.2	yes	2	7.6	odd	2
224.2.b.a.113.1		2	28.27	even	2
224.2.b.a.113.2		2	56.27	even	2
392.2.b.b.197.1		2	8.5	even	2	inner
392.2.b.b.197.2		2	1.1	even	1	trivial
392.2.p.a.165.1		4	56.5	odd	6
392.2.p.a.165.2		4	7.5	odd	6
392.2.p.a.373.1		4	7.3	odd	6
392.2.p.a.373.2		4	56.45	odd	6
392.2.p.b.165.1		4	56.37	even	6
392.2.p.b.165.2		4	7.2	even	3
392.2.p.b.373.1		4	7.4	even	3
392.2.p.b.373.2		4	56.53	even	6
504.2.c.a.253.1		2	21.20	even	2
504.2.c.a.253.2		2	168.125	even	2
1568.2.b.a.785.1		2	8.3	odd	2
1568.2.b.a.785.2		2	4.3	odd	2
1568.2.t.b.177.1		4	56.11	odd	6
1568.2.t.b.177.2		4	28.11	odd	6
1568.2.t.b.753.1		4	28.23	odd	6
1568.2.t.b.753.2		4	56.51	odd	6
1568.2.t.c.177.1		4	28.3	even	6
1568.2.t.c.177.2		4	56.3	even	6
1568.2.t.c.753.1		4	56.19	even	6
1568.2.t.c.753.2		4	28.19	even	6
1792.2.a.n.1.1		2	112.13	odd	4
1792.2.a.n.1.2		2	112.69	odd	4
1792.2.a.p.1.1		2	112.27	even	4
1792.2.a.p.1.2		2	112.83	even	4
2016.2.c.a.1009.1		2	168.83	odd	2
2016.2.c.a.1009.2		2	84.83	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:	\( N \)	\(=\)	\( 392 = 2^{3} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	392.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.41421i q^{2} -1.41421i q^{3} -2.00000 q^{4} +1.41421i q^{5} +2.00000 q^{6} -2.82843i q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+1.41421i q^{2} -1.41421i q^{3} -2.00000 q^{4} +1.41421i q^{5} +2.00000 q^{6} -2.82843i q^{8} +1.00000 q^{9} -2.00000 q^{10} +2.82843i q^{11} +2.82843i q^{12} +4.24264i q^{13} +2.00000 q^{15} +4.00000 q^{16} +6.00000 q^{17} +1.41421i q^{18} +4.24264i q^{19} -2.82843i q^{20} -4.00000 q^{22} -6.00000 q^{23} -4.00000 q^{24} +3.00000 q^{25} -6.00000 q^{26} -5.65685i q^{27} +2.82843i q^{29} +2.82843i q^{30} +4.00000 q^{31} +5.65685i q^{32} +4.00000 q^{33} +8.48528i q^{34} -2.00000 q^{36} +8.48528i q^{37} -6.00000 q^{38} +6.00000 q^{39} +4.00000 q^{40} -6.00000 q^{41} -8.48528i q^{43} -5.65685i q^{44} +1.41421i q^{45} -8.48528i q^{46} -5.65685i q^{48} +4.24264i q^{50} -8.48528i q^{51} -8.48528i q^{52} -5.65685i q^{53} +8.00000 q^{54} -4.00000 q^{55} +6.00000 q^{57} -4.00000 q^{58} +1.41421i q^{59} -4.00000 q^{60} -12.7279i q^{61} +5.65685i q^{62} -8.00000 q^{64} -6.00000 q^{65} +5.65685i q^{66} -12.0000 q^{68} +8.48528i q^{69} -2.82843i q^{72} -2.00000 q^{73} -12.0000 q^{74} -4.24264i q^{75} -8.48528i q^{76} +8.48528i q^{78} +8.00000 q^{79} +5.65685i q^{80} -5.00000 q^{81} -8.48528i q^{82} -15.5563i q^{83} +8.48528i q^{85} +12.0000 q^{86} +4.00000 q^{87} +8.00000 q^{88} -6.00000 q^{89} -2.00000 q^{90} +12.0000 q^{92} -5.65685i q^{93} -6.00000 q^{95} +8.00000 q^{96} +10.0000 q^{97} +2.82843i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 4 q^{4} + 4 q^{6} + 2 q^{9}+O(q^{10}) \) 2 * q - 4 * q^4 + 4 * q^6 + 2 * q^9	\( 2 q - 4 q^{4} + 4 q^{6} + 2 q^{9} - 4 q^{10} + 4 q^{15} + 8 q^{16} + 12 q^{17} - 8 q^{22} - 12 q^{23} - 8 q^{24} + 6 q^{25} - 12 q^{26} + 8 q^{31} + 8 q^{33} - 4 q^{36} - 12 q^{38} + 12 q^{39} + 8 q^{40} - 12 q^{41} + 16 q^{54} - 8 q^{55} + 12 q^{57} - 8 q^{58} - 8 q^{60} - 16 q^{64} - 12 q^{65} - 24 q^{68} - 4 q^{73} - 24 q^{74} + 16 q^{79} - 10 q^{81} + 24 q^{86} + 8 q^{87} + 16 q^{88} - 12 q^{89} - 4 q^{90} + 24 q^{92} - 12 q^{95} + 16 q^{96} + 20 q^{97}+O(q^{100}) \) 2 * q - 4 * q^4 + 4 * q^6 + 2 * q^9 - 4 * q^10 + 4 * q^15 + 8 * q^16 + 12 * q^17 - 8 * q^22 - 12 * q^23 - 8 * q^24 + 6 * q^25 - 12 * q^26 + 8 * q^31 + 8 * q^33 - 4 * q^36 - 12 * q^38 + 12 * q^39 + 8 * q^40 - 12 * q^41 + 16 * q^54 - 8 * q^55 + 12 * q^57 - 8 * q^58 - 8 * q^60 - 16 * q^64 - 12 * q^65 - 24 * q^68 - 4 * q^73 - 24 * q^74 + 16 * q^79 - 10 * q^81 + 24 * q^86 + 8 * q^87 + 16 * q^88 - 12 * q^89 - 4 * q^90 + 24 * q^92 - 12 * q^95 + 16 * q^96 + 20 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more