LMFDB - Embedded newform 896.2.b.a.449.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [896,2,Mod(449,896)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(896, 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("896.449");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 896 = 2^{7} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	896.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:	$7.15459602111$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			449.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	896.449
Dual form			896.2.b.a.449.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.00000i q^{3} +2.00000i q^{5} -1.00000 q^{7} -1.00000 q^{9} +O(q^{10})$	$q-2.00000i q^{3} +2.00000i q^{5} -1.00000 q^{7} -1.00000 q^{9} +2.00000i q^{13} +4.00000 q^{15} +6.00000 q^{17} -6.00000i q^{19} +2.00000i q^{21} +4.00000 q^{23} +1.00000 q^{25} -4.00000i q^{27} +8.00000 q^{31} -2.00000i q^{35} -8.00000i q^{37} +4.00000 q^{39} -6.00000 q^{41} +8.00000i q^{43} -2.00000i q^{45} +1.00000 q^{49} -12.0000i q^{51} +4.00000i q^{53} -12.0000 q^{57} +6.00000i q^{59} -2.00000i q^{61} +1.00000 q^{63} -4.00000 q^{65} -4.00000i q^{67} -8.00000i q^{69} +16.0000 q^{71} +6.00000 q^{73} -2.00000i q^{75} -16.0000 q^{79} -11.0000 q^{81} -14.0000i q^{83} +12.0000i q^{85} +14.0000 q^{89} -2.00000i q^{91} -16.0000i q^{93} +12.0000 q^{95} -2.00000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{7} - 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^7 - 2 * q^9	$ 2 q - 2 q^{7} - 2 q^{9} + 8 q^{15} + 12 q^{17} + 8 q^{23} + 2 q^{25} + 16 q^{31} + 8 q^{39} - 12 q^{41} + 2 q^{49} - 24 q^{57} + 2 q^{63} - 8 q^{65} + 32 q^{71} + 12 q^{73} - 32 q^{79} - 22 q^{81} + 28 q^{89} + 24 q^{95} - 4 q^{97}+O(q^{100}) $ 2 * q - 2 * q^7 - 2 * q^9 + 8 * q^15 + 12 * q^17 + 8 * q^23 + 2 * q^25 + 16 * q^31 + 8 * q^39 - 12 * q^41 + 2 * q^49 - 24 * q^57 + 2 * q^63 - 8 * q^65 + 32 * q^71 + 12 * q^73 - 32 * q^79 - 22 * q^81 + 28 * q^89 + 24 * q^95 - 4 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/896\mathbb{Z}\right)^\times$.

$n$	$127$	$129$	$645$
$\chi(n)$	$1$	$1$	$-1$

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 $

$2$	0	0
$2$	0	0
$3$	− 2.00000i	− 1.15470i	−0.816497	−	0.577350i	$-0.804087\pi$
$3$	− 2.00000i	− 1.15470i	0.816497	−	0.577350i	$-0.195913\pi$
$4$	0	0
$5$	2.00000i	0.894427i	0.894427	+	0.447214i	$0.147584\pi$
$5$	2.00000i	0.894427i	−0.894427	+	0.447214i	$0.852416\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	−1.00000	−0.333333
$10$	0	0
$11$	0	0	1.00000			$0$
$11$	0	0	−1.00000			$\pi$
$12$	0	0
$13$	2.00000i	0.554700i	0.960769	+	0.277350i	$0.0894562\pi$
$13$	2.00000i	0.554700i	−0.960769	+	0.277350i	$0.910544\pi$
$14$	0	0
$15$	4.00000	1.03280
$16$	0	0
$17$	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
$19$	− 6.00000i	− 1.37649i	−0.725476	−	0.688247i	$-0.758380\pi$
$19$	− 6.00000i	− 1.37649i	0.725476	−	0.688247i	$-0.241620\pi$
$20$	0	0
$21$	2.00000i	0.436436i
$22$	0	0
$23$	4.00000	0.834058	0.417029	−	0.908893i	$-0.363071\pi$
$23$	4.00000	0.834058	0.417029	+	0.908893i	$0.363071\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	− 4.00000i	− 0.769800i
$28$	0	0
$29$	0	0	1.00000			$0$
$29$	0	0	−1.00000			$\pi$
$30$	0	0
$31$	8.00000	1.43684	0.718421	−	0.695608i	$-0.244865\pi$
$31$	8.00000	1.43684	0.718421	+	0.695608i	$0.244865\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	− 2.00000i	− 0.338062i
$36$	0	0
$37$	− 8.00000i	− 1.31519i	−0.753371	−	0.657596i	$-0.771573\pi$
$37$	− 8.00000i	− 1.31519i	0.753371	−	0.657596i	$-0.228427\pi$
$38$	0	0
$39$	4.00000	0.640513
$40$	0	0
$41$	−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
$43$	8.00000i	1.21999i	0.792406	+	0.609994i	$0.208828\pi$
$43$	8.00000i	1.21999i	−0.792406	+	0.609994i	$0.791172\pi$
$44$	0	0
$45$	− 2.00000i	− 0.298142i
$46$	0	0
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	− 12.0000i	− 1.68034i
$52$	0	0
$53$	4.00000i	0.549442i	0.961524	+	0.274721i	$0.0885855\pi$
$53$	4.00000i	0.549442i	−0.961524	+	0.274721i	$0.911414\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−12.0000	−1.58944
$58$	0	0
$59$	6.00000i	0.781133i	0.920575	+	0.390567i	$0.127721\pi$
$59$	6.00000i	0.781133i	−0.920575	+	0.390567i	$0.872279\pi$
$60$	0	0
$61$	− 2.00000i	− 0.256074i	−0.991769	−	0.128037i	$-0.959132\pi$
$61$	− 2.00000i	− 0.256074i	0.991769	−	0.128037i	$-0.0408676\pi$
$62$	0	0
$63$	1.00000	0.125988
$64$	0	0
$65$	−4.00000	−0.496139
$66$	0	0
$67$	− 4.00000i	− 0.488678i	−0.969690	−	0.244339i	$-0.921429\pi$
$67$	− 4.00000i	− 0.488678i	0.969690	−	0.244339i	$-0.0785709\pi$
$68$	0	0
$69$	− 8.00000i	− 0.963087i
$70$	0	0
$71$	16.0000	1.89885	0.949425	−	0.313993i	$-0.101667\pi$
$71$	16.0000	1.89885	0.949425	+	0.313993i	$0.101667\pi$
$72$	0	0
$73$	6.00000	0.702247	0.351123	−	0.936329i	$-0.385800\pi$
$73$	6.00000	0.702247	0.351123	+	0.936329i	$0.385800\pi$
$74$	0	0
$75$	− 2.00000i	− 0.230940i
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−16.0000	−1.80014	−0.900070	−	0.435745i	$-0.856485\pi$
$79$	−16.0000	−1.80014	−0.900070	+	0.435745i	$0.856485\pi$
$80$	0	0
$81$	−11.0000	−1.22222
$82$	0	0
$83$	− 14.0000i	− 1.53670i	−0.640030	−	0.768350i	$-0.721078\pi$
$83$	− 14.0000i	− 1.53670i	0.640030	−	0.768350i	$-0.278922\pi$
$84$	0	0
$85$	12.0000i	1.30158i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	14.0000	1.48400	0.741999	−	0.670402i	$-0.233878\pi$
$89$	14.0000	1.48400	0.741999	+	0.670402i	$0.233878\pi$
$90$	0	0
$91$	− 2.00000i	− 0.209657i
$92$	0	0
$93$	− 16.0000i	− 1.65912i
$94$	0	0
$95$	12.0000	1.23117
$96$	0	0
$97$	−2.00000	−0.203069	−0.101535	−	0.994832i	$-0.532375\pi$
$97$	−2.00000	−0.203069	−0.101535	+	0.994832i	$0.532375\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	− 14.0000i	− 1.39305i	−0.717532	−	0.696526i	$-0.754728\pi$
$101$	− 14.0000i	− 1.39305i	0.717532	−	0.696526i	$-0.245272\pi$
$102$	0	0
$103$	−16.0000	−1.57653	−0.788263	−	0.615338i	$-0.789020\pi$
$103$	−16.0000	−1.57653	−0.788263	+	0.615338i	$0.789020\pi$
$104$	0	0
$105$	−4.00000	−0.390360
$106$	0	0
$107$	4.00000i	0.386695i	0.981130	+	0.193347i	$0.0619344\pi$
$107$	4.00000i	0.386695i	−0.981130	+	0.193347i	$0.938066\pi$
$108$	0	0
$109$	16.0000i	1.53252i	0.642529	+	0.766261i	$0.277885\pi$
$109$	16.0000i	1.53252i	−0.642529	+	0.766261i	$0.722115\pi$
$110$	0	0
$111$	−16.0000	−1.51865
$112$	0	0
$113$	−10.0000	−0.940721	−0.470360	−	0.882474i	$-0.655876\pi$
$113$	−10.0000	−0.940721	−0.470360	+	0.882474i	$0.655876\pi$
$114$	0	0
$115$	8.00000i	0.746004i
$116$	0	0
$117$	− 2.00000i	− 0.184900i
$118$	0	0
$119$	−6.00000	−0.550019
$120$	0	0
$121$	11.0000	1.00000
$122$	0	0
$123$	12.0000i	1.08200i
$124$	0	0
$125$	12.0000i	1.07331i
$126$	0	0
$127$	4.00000	0.354943	0.177471	−	0.984126i	$-0.443208\pi$
$127$	4.00000	0.354943	0.177471	+	0.984126i	$0.443208\pi$
$128$	0	0
$129$	16.0000	1.40872
$130$	0	0
$131$	− 6.00000i	− 0.524222i	−0.965038	−	0.262111i	$-0.915581\pi$
$131$	− 6.00000i	− 0.524222i	0.965038	−	0.262111i	$-0.0844187\pi$
$132$	0	0
$133$	6.00000i	0.520266i
$134$	0	0
$135$	8.00000	0.688530
$136$	0	0
$137$	10.0000	0.854358	0.427179	−	0.904167i	$-0.359507\pi$
$137$	10.0000	0.854358	0.427179	+	0.904167i	$0.359507\pi$
$138$	0	0
$139$	10.0000i	0.848189i	0.905618	+	0.424094i	$0.139408\pi$
$139$	10.0000i	0.848189i	−0.905618	+	0.424094i	$0.860592\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	0	0
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 2.00000i	− 0.164957i
$148$	0	0
$149$	20.0000i	1.63846i	0.573462	+	0.819232i	$0.305600\pi$
$149$	20.0000i	1.63846i	−0.573462	+	0.819232i	$0.694400\pi$
$150$	0	0
$151$	−20.0000	−1.62758	−0.813788	−	0.581161i	$-0.802599\pi$
$151$	−20.0000	−1.62758	−0.813788	+	0.581161i	$0.802599\pi$
$152$	0	0
$153$	−6.00000	−0.485071
$154$	0	0
$155$	16.0000i	1.28515i
$156$	0	0
$157$	− 6.00000i	− 0.478852i	−0.970915	−	0.239426i	$-0.923041\pi$
$157$	− 6.00000i	− 0.478852i	0.970915	−	0.239426i	$-0.0769593\pi$
$158$	0	0
$159$	8.00000	0.634441
$160$	0	0
$161$	−4.00000	−0.315244
$162$	0	0
$163$	16.0000i	1.25322i	0.779334	+	0.626608i	$0.215557\pi$
$163$	16.0000i	1.25322i	−0.779334	+	0.626608i	$0.784443\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	8.00000	0.619059	0.309529	−	0.950890i	$-0.399829\pi$
$167$	8.00000	0.619059	0.309529	+	0.950890i	$0.399829\pi$
$168$	0	0
$169$	9.00000	0.692308
$170$	0	0
$171$	6.00000i	0.458831i
$172$	0	0
$173$	− 6.00000i	− 0.456172i	−0.973641	−	0.228086i	$-0.926753\pi$
$173$	− 6.00000i	− 0.456172i	0.973641	−	0.228086i	$-0.0732467\pi$
$174$	0	0
$175$	−1.00000	−0.0755929
$176$	0	0
$177$	12.0000	0.901975
$178$	0	0
$179$	4.00000i	0.298974i	0.988764	+	0.149487i	$0.0477622\pi$
$179$	4.00000i	0.298974i	−0.988764	+	0.149487i	$0.952238\pi$
$180$	0	0
$181$	6.00000i	0.445976i	0.974821	+	0.222988i	$0.0715812\pi$
$181$	6.00000i	0.445976i	−0.974821	+	0.222988i	$0.928419\pi$
$182$	0	0
$183$	−4.00000	−0.295689
$184$	0	0
$185$	16.0000	1.17634
$186$	0	0
$187$	0	0
$188$	0	0
$189$	4.00000i	0.290957i
$190$	0	0
$191$	−8.00000	−0.578860	−0.289430	−	0.957199i	$-0.593466\pi$
$191$	−8.00000	−0.578860	−0.289430	+	0.957199i	$0.593466\pi$
$192$	0	0
$193$	−22.0000	−1.58359	−0.791797	−	0.610784i	$-0.790854\pi$
$193$	−22.0000	−1.58359	−0.791797	+	0.610784i	$0.790854\pi$
$194$	0	0
$195$	8.00000i	0.572892i
$196$	0	0
$197$	12.0000i	0.854965i	0.904024	+	0.427482i	$0.140599\pi$
$197$	12.0000i	0.854965i	−0.904024	+	0.427482i	$0.859401\pi$
$198$	0	0
$199$	−16.0000	−1.13421	−0.567105	−	0.823646i	$-0.691937\pi$
$199$	−16.0000	−1.13421	−0.567105	+	0.823646i	$0.691937\pi$
$200$	0	0
$201$	−8.00000	−0.564276
$202$	0	0
$203$	0	0
$204$	0	0
$205$	− 12.0000i	− 0.838116i
$206$	0	0
$207$	−4.00000	−0.278019
$208$	0	0
$209$	0	0
$210$	0	0
$211$	28.0000i	1.92760i	0.266627	+	0.963800i	$0.414091\pi$
$211$	28.0000i	1.92760i	−0.266627	+	0.963800i	$0.585909\pi$
$212$	0	0
$213$	− 32.0000i	− 2.19260i
$214$	0	0
$215$	−16.0000	−1.09119
$216$	0	0
$217$	−8.00000	−0.543075
$218$	0	0
$219$	− 12.0000i	− 0.810885i
$220$	0	0
$221$	12.0000i	0.807207i
$222$	0	0
$223$	−8.00000	−0.535720	−0.267860	−	0.963458i	$-0.586316\pi$
$223$	−8.00000	−0.535720	−0.267860	+	0.963458i	$0.586316\pi$
$224$	0	0
$225$	−1.00000	−0.0666667
$226$	0	0
$227$	− 14.0000i	− 0.929213i	−0.885517	−	0.464606i	$-0.846196\pi$
$227$	− 14.0000i	− 0.929213i	0.885517	−	0.464606i	$-0.153804\pi$
$228$	0	0
$229$	− 2.00000i	− 0.132164i	−0.997814	−	0.0660819i	$-0.978950\pi$
$229$	− 2.00000i	− 0.132164i	0.997814	−	0.0660819i	$-0.0210498\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	6.00000	0.393073	0.196537	−	0.980497i	$-0.437031\pi$
$233$	6.00000	0.393073	0.196537	+	0.980497i	$0.437031\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	32.0000i	2.07862i
$238$	0	0
$239$	−20.0000	−1.29369	−0.646846	−	0.762620i	$-0.723912\pi$
$239$	−20.0000	−1.29369	−0.646846	+	0.762620i	$0.723912\pi$
$240$	0	0
$241$	14.0000	0.901819	0.450910	−	0.892570i	$-0.351100\pi$
$241$	14.0000	0.901819	0.450910	+	0.892570i	$0.351100\pi$
$242$	0	0
$243$	10.0000i	0.641500i
$244$	0	0
$245$	2.00000i	0.127775i
$246$	0	0
$247$	12.0000	0.763542
$248$	0	0
$249$	−28.0000	−1.77443
$250$	0	0
$251$	2.00000i	0.126239i	0.998006	+	0.0631194i	$0.0201049\pi$
$251$	2.00000i	0.126239i	−0.998006	+	0.0631194i	$0.979895\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	24.0000	1.50294
$256$	0	0
$257$	2.00000	0.124757	0.0623783	−	0.998053i	$-0.480131\pi$
$257$	2.00000	0.124757	0.0623783	+	0.998053i	$0.480131\pi$
$258$	0	0
$259$	8.00000i	0.497096i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	0	0		−	1.00000i	$-0.5\pi$
$263$	0	0			1.00000i	$0.5\pi$
$264$	0	0
$265$	−8.00000	−0.491436
$266$	0	0
$267$	− 28.0000i	− 1.71357i
$268$	0	0
$269$	− 30.0000i	− 1.82913i	−0.404436	−	0.914566i	$-0.632532\pi$
$269$	− 30.0000i	− 1.82913i	0.404436	−	0.914566i	$-0.367468\pi$
$270$	0	0
$271$	−8.00000	−0.485965	−0.242983	−	0.970031i	$-0.578126\pi$
$271$	−8.00000	−0.485965	−0.242983	+	0.970031i	$0.578126\pi$
$272$	0	0
$273$	−4.00000	−0.242091
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 20.0000i	− 1.20168i	−0.799368	−	0.600842i	$-0.794832\pi$
$277$	− 20.0000i	− 1.20168i	0.799368	−	0.600842i	$-0.205168\pi$
$278$	0	0
$279$	−8.00000	−0.478947
$280$	0	0
$281$	−26.0000	−1.55103	−0.775515	−	0.631329i	$-0.782510\pi$
$281$	−26.0000	−1.55103	−0.775515	+	0.631329i	$0.782510\pi$
$282$	0	0
$283$	− 2.00000i	− 0.118888i	−0.998232	−	0.0594438i	$-0.981067\pi$
$283$	− 2.00000i	− 0.118888i	0.998232	−	0.0594438i	$-0.0189327\pi$
$284$	0	0
$285$	− 24.0000i	− 1.42164i
$286$	0	0
$287$	6.00000	0.354169
$288$	0	0
$289$	19.0000	1.11765
$290$	0	0
$291$	4.00000i	0.234484i
$292$	0	0
$293$	− 30.0000i	− 1.75262i	−0.481749	−	0.876309i	$-0.659998\pi$
$293$	− 30.0000i	− 1.75262i	0.481749	−	0.876309i	$-0.340002\pi$
$294$	0	0
$295$	−12.0000	−0.698667
$296$	0	0
$297$	0	0
$298$	0	0
$299$	8.00000i	0.462652i
$300$	0	0
$301$	− 8.00000i	− 0.461112i
$302$	0	0
$303$	−28.0000	−1.60856
$304$	0	0
$305$	4.00000	0.229039
$306$	0	0
$307$	10.0000i	0.570730i	0.958419	+	0.285365i	$0.0921148\pi$
$307$	10.0000i	0.570730i	−0.958419	+	0.285365i	$0.907885\pi$
$308$	0	0
$309$	32.0000i	1.82042i
$310$	0	0
$311$	−8.00000	−0.453638	−0.226819	−	0.973937i	$-0.572833\pi$
$311$	−8.00000	−0.453638	−0.226819	+	0.973937i	$0.572833\pi$
$312$	0	0
$313$	−6.00000	−0.339140	−0.169570	−	0.985518i	$-0.554238\pi$
$313$	−6.00000	−0.339140	−0.169570	+	0.985518i	$0.554238\pi$
$314$	0	0
$315$	2.00000i	0.112687i
$316$	0	0
$317$	− 12.0000i	− 0.673987i	−0.941507	−	0.336994i	$-0.890590\pi$
$317$	− 12.0000i	− 0.673987i	0.941507	−	0.336994i	$-0.109410\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	8.00000	0.446516
$322$	0	0
$323$	− 36.0000i	− 2.00309i
$324$	0	0
$325$	2.00000i	0.110940i
$326$	0	0
$327$	32.0000	1.76960
$328$	0	0
$329$	0	0
$330$	0	0
$331$	8.00000i	0.439720i	0.975531	+	0.219860i	$0.0705600\pi$
$331$	8.00000i	0.439720i	−0.975531	+	0.219860i	$0.929440\pi$
$332$	0	0
$333$	8.00000i	0.438397i
$334$	0	0
$335$	8.00000	0.437087
$336$	0	0
$337$	−18.0000	−0.980522	−0.490261	−	0.871576i	$-0.663099\pi$
$337$	−18.0000	−0.980522	−0.490261	+	0.871576i	$0.663099\pi$
$338$	0	0
$339$	20.0000i	1.08625i
$340$	0	0
$341$	0	0
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	16.0000	0.861411
$346$	0	0
$347$	− 24.0000i	− 1.28839i	−0.764862	−	0.644194i	$-0.777193\pi$
$347$	− 24.0000i	− 1.28839i	0.764862	−	0.644194i	$-0.222807\pi$
$348$	0	0
$349$	34.0000i	1.81998i	0.414632	+	0.909989i	$0.363910\pi$
$349$	34.0000i	1.81998i	−0.414632	+	0.909989i	$0.636090\pi$
$350$	0	0
$351$	8.00000	0.427008
$352$	0	0
$353$	−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
$355$	32.0000i	1.69838i
$356$	0	0
$357$	12.0000i	0.635107i
$358$	0	0
$359$	20.0000	1.05556	0.527780	−	0.849381i	$-0.323025\pi$
$359$	20.0000	1.05556	0.527780	+	0.849381i	$0.323025\pi$
$360$	0	0
$361$	−17.0000	−0.894737
$362$	0	0
$363$	− 22.0000i	− 1.15470i
$364$	0	0
$365$	12.0000i	0.628109i
$366$	0	0
$367$	−24.0000	−1.25279	−0.626395	−	0.779506i	$-0.715470\pi$
$367$	−24.0000	−1.25279	−0.626395	+	0.779506i	$0.715470\pi$
$368$	0	0
$369$	6.00000	0.312348
$370$	0	0
$371$	− 4.00000i	− 0.207670i
$372$	0	0
$373$	4.00000i	0.207112i	0.994624	+	0.103556i	$0.0330221\pi$
$373$	4.00000i	0.207112i	−0.994624	+	0.103556i	$0.966978\pi$
$374$	0	0
$375$	24.0000	1.23935
$376$	0	0
$377$	0	0
$378$	0	0
$379$	− 16.0000i	− 0.821865i	−0.911666	−	0.410932i	$-0.865203\pi$
$379$	− 16.0000i	− 0.821865i	0.911666	−	0.410932i	$-0.134797\pi$
$380$	0	0
$381$	− 8.00000i	− 0.409852i
$382$	0	0
$383$	32.0000	1.63512	0.817562	−	0.575841i	$-0.195325\pi$
$383$	32.0000	1.63512	0.817562	+	0.575841i	$0.195325\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 8.00000i	− 0.406663i
$388$	0	0
$389$	24.0000i	1.21685i	0.793612	+	0.608424i	$0.208198\pi$
$389$	24.0000i	1.21685i	−0.793612	+	0.608424i	$0.791802\pi$
$390$	0	0
$391$	24.0000	1.21373
$392$	0	0
$393$	−12.0000	−0.605320
$394$	0	0
$395$	− 32.0000i	− 1.61009i
$396$	0	0
$397$	− 22.0000i	− 1.10415i	−0.833795	−	0.552074i	$-0.813837\pi$
$397$	− 22.0000i	− 1.10415i	0.833795	−	0.552074i	$-0.186163\pi$
$398$	0	0
$399$	12.0000	0.600751
$400$	0	0
$401$	18.0000	0.898877	0.449439	−	0.893311i	$-0.351624\pi$
$401$	18.0000	0.898877	0.449439	+	0.893311i	$0.351624\pi$
$402$	0	0
$403$	16.0000i	0.797017i
$404$	0	0
$405$	− 22.0000i	− 1.09319i
$406$	0	0
$407$	0	0
$408$	0	0
$409$	34.0000	1.68119	0.840596	−	0.541663i	$-0.182205\pi$
$409$	34.0000	1.68119	0.840596	+	0.541663i	$0.182205\pi$
$410$	0	0
$411$	− 20.0000i	− 0.986527i
$412$	0	0
$413$	− 6.00000i	− 0.295241i
$414$	0	0
$415$	28.0000	1.37447
$416$	0	0
$417$	20.0000	0.979404
$418$	0	0
$419$	22.0000i	1.07477i	0.843337	+	0.537385i	$0.180588\pi$
$419$	22.0000i	1.07477i	−0.843337	+	0.537385i	$0.819412\pi$
$420$	0	0
$421$	12.0000i	0.584844i	0.956289	+	0.292422i	$0.0944612\pi$
$421$	12.0000i	0.584844i	−0.956289	+	0.292422i	$0.905539\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	6.00000	0.291043
$426$	0	0
$427$	2.00000i	0.0967868i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−12.0000	−0.578020	−0.289010	−	0.957326i	$-0.593326\pi$
$431$	−12.0000	−0.578020	−0.289010	+	0.957326i	$0.593326\pi$
$432$	0	0
$433$	−26.0000	−1.24948	−0.624740	−	0.780833i	$-0.714795\pi$
$433$	−26.0000	−1.24948	−0.624740	+	0.780833i	$0.714795\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 24.0000i	− 1.14808i
$438$	0	0
$439$	0	0		−	1.00000i	$-0.5\pi$
$439$	0	0			1.00000i	$0.5\pi$
$440$	0	0
$441$	−1.00000	−0.0476190
$442$	0	0
$443$	12.0000i	0.570137i	0.958507	+	0.285069i	$0.0920164\pi$
$443$	12.0000i	0.570137i	−0.958507	+	0.285069i	$0.907984\pi$
$444$	0	0
$445$	28.0000i	1.32733i
$446$	0	0
$447$	40.0000	1.89194
$448$	0	0
$449$	−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
$451$	0	0
$452$	0	0
$453$	40.0000i	1.87936i
$454$	0	0
$455$	4.00000	0.187523
$456$	0	0
$457$	−2.00000	−0.0935561	−0.0467780	−	0.998905i	$-0.514895\pi$
$457$	−2.00000	−0.0935561	−0.0467780	+	0.998905i	$0.514895\pi$
$458$	0	0
$459$	− 24.0000i	− 1.12022i
$460$	0	0
$461$	18.0000i	0.838344i	0.907907	+	0.419172i	$0.137680\pi$
$461$	18.0000i	0.838344i	−0.907907	+	0.419172i	$0.862320\pi$
$462$	0	0
$463$	−24.0000	−1.11537	−0.557687	−	0.830051i	$-0.688311\pi$
$463$	−24.0000	−1.11537	−0.557687	+	0.830051i	$0.688311\pi$
$464$	0	0
$465$	32.0000	1.48396
$466$	0	0
$467$	− 38.0000i	− 1.75843i	−0.476425	−	0.879215i	$-0.658068\pi$
$467$	− 38.0000i	− 1.75843i	0.476425	−	0.879215i	$-0.341932\pi$
$468$	0	0
$469$	4.00000i	0.184703i
$470$	0	0
$471$	−12.0000	−0.552931
$472$	0	0
$473$	0	0
$474$	0	0
$475$	− 6.00000i	− 0.275299i
$476$	0	0
$477$	− 4.00000i	− 0.183147i
$478$	0	0
$479$	−8.00000	−0.365529	−0.182765	−	0.983157i	$-0.558505\pi$
$479$	−8.00000	−0.365529	−0.182765	+	0.983157i	$0.558505\pi$
$480$	0	0
$481$	16.0000	0.729537
$482$	0	0
$483$	8.00000i	0.364013i
$484$	0	0
$485$	− 4.00000i	− 0.181631i
$486$	0	0
$487$	−12.0000	−0.543772	−0.271886	−	0.962329i	$-0.587647\pi$
$487$	−12.0000	−0.543772	−0.271886	+	0.962329i	$0.587647\pi$
$488$	0	0
$489$	32.0000	1.44709
$490$	0	0
$491$	− 20.0000i	− 0.902587i	−0.892375	−	0.451294i	$-0.850963\pi$
$491$	− 20.0000i	− 0.902587i	0.892375	−	0.451294i	$-0.149037\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−16.0000	−0.717698
$498$	0	0
$499$	28.0000i	1.25345i	0.779240	+	0.626726i	$0.215605\pi$
$499$	28.0000i	1.25345i	−0.779240	+	0.626726i	$0.784395\pi$
$500$	0	0
$501$	− 16.0000i	− 0.714827i
$502$	0	0
$503$	16.0000	0.713405	0.356702	−	0.934218i	$-0.383901\pi$
$503$	16.0000	0.713405	0.356702	+	0.934218i	$0.383901\pi$
$504$	0	0
$505$	28.0000	1.24598
$506$	0	0
$507$	− 18.0000i	− 0.799408i
$508$	0	0
$509$	26.0000i	1.15243i	0.817298	+	0.576215i	$0.195471\pi$
$509$	26.0000i	1.15243i	−0.817298	+	0.576215i	$0.804529\pi$
$510$	0	0
$511$	−6.00000	−0.265424
$512$	0	0
$513$	−24.0000	−1.05963
$514$	0	0
$515$	− 32.0000i	− 1.41009i
$516$	0	0
$517$	0	0
$518$	0	0
$519$	−12.0000	−0.526742
$520$	0	0
$521$	−14.0000	−0.613351	−0.306676	−	0.951814i	$-0.599217\pi$
$521$	−14.0000	−0.613351	−0.306676	+	0.951814i	$0.599217\pi$
$522$	0	0
$523$	14.0000i	0.612177i	0.952003	+	0.306089i	$0.0990204\pi$
$523$	14.0000i	0.612177i	−0.952003	+	0.306089i	$0.900980\pi$
$524$	0	0
$525$	2.00000i	0.0872872i
$526$	0	0
$527$	48.0000	2.09091
$528$	0	0
$529$	−7.00000	−0.304348
$530$	0	0
$531$	− 6.00000i	− 0.260378i
$532$	0	0
$533$	− 12.0000i	− 0.519778i
$534$	0	0
$535$	−8.00000	−0.345870
$536$	0	0
$537$	8.00000	0.345225
$538$	0	0
$539$	0	0
$540$	0	0
$541$	− 36.0000i	− 1.54776i	−0.633332	−	0.773880i	$-0.718313\pi$
$541$	− 36.0000i	− 1.54776i	0.633332	−	0.773880i	$-0.281687\pi$
$542$	0	0
$543$	12.0000	0.514969
$544$	0	0
$545$	−32.0000	−1.37073
$546$	0	0
$547$	− 40.0000i	− 1.71028i	−0.518400	−	0.855138i	$-0.673472\pi$
$547$	− 40.0000i	− 1.71028i	0.518400	−	0.855138i	$-0.326528\pi$
$548$	0	0
$549$	2.00000i	0.0853579i
$550$	0	0
$551$	0	0
$552$	0	0
$553$	16.0000	0.680389
$554$	0	0
$555$	− 32.0000i	− 1.35832i
$556$	0	0
$557$	4.00000i	0.169485i	0.996403	+	0.0847427i	$0.0270068\pi$
$557$	4.00000i	0.169485i	−0.996403	+	0.0847427i	$0.972993\pi$
$558$	0	0
$559$	−16.0000	−0.676728
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 26.0000i	− 1.09577i	−0.836554	−	0.547885i	$-0.815433\pi$
$563$	− 26.0000i	− 1.09577i	0.836554	−	0.547885i	$-0.184567\pi$
$564$	0	0
$565$	− 20.0000i	− 0.841406i
$566$	0	0
$567$	11.0000	0.461957
$568$	0	0
$569$	−10.0000	−0.419222	−0.209611	−	0.977785i	$-0.567220\pi$
$569$	−10.0000	−0.419222	−0.209611	+	0.977785i	$0.567220\pi$
$570$	0	0
$571$	0	0	1.00000			$0$
$571$	0	0	−1.00000			$\pi$
$572$	0	0
$573$	16.0000i	0.668410i
$574$	0	0
$575$	4.00000	0.166812
$576$	0	0
$577$	−22.0000	−0.915872	−0.457936	−	0.888985i	$-0.651411\pi$
$577$	−22.0000	−0.915872	−0.457936	+	0.888985i	$0.651411\pi$
$578$	0	0
$579$	44.0000i	1.82858i
$580$	0	0
$581$	14.0000i	0.580818i
$582$	0	0
$583$	0	0
$584$	0	0
$585$	4.00000	0.165380
$586$	0	0
$587$	26.0000i	1.07313i	0.843857	+	0.536567i	$0.180279\pi$
$587$	26.0000i	1.07313i	−0.843857	+	0.536567i	$0.819721\pi$
$588$	0	0
$589$	− 48.0000i	− 1.97781i
$590$	0	0
$591$	24.0000	0.987228
$592$	0	0
$593$	−14.0000	−0.574911	−0.287456	−	0.957794i	$-0.592809\pi$
$593$	−14.0000	−0.574911	−0.287456	+	0.957794i	$0.592809\pi$
$594$	0	0
$595$	− 12.0000i	− 0.491952i
$596$	0	0
$597$	32.0000i	1.30967i
$598$	0	0
$599$	−24.0000	−0.980613	−0.490307	−	0.871550i	$-0.663115\pi$
$599$	−24.0000	−0.980613	−0.490307	+	0.871550i	$0.663115\pi$
$600$	0	0
$601$	−34.0000	−1.38689	−0.693444	−	0.720510i	$-0.743908\pi$
$601$	−34.0000	−1.38689	−0.693444	+	0.720510i	$0.743908\pi$
$602$	0	0
$603$	4.00000i	0.162893i
$604$	0	0
$605$	22.0000i	0.894427i
$606$	0	0
$607$	0	0		−	1.00000i	$-0.5\pi$
$607$	0	0			1.00000i	$0.5\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	0	0
$612$	0	0
$613$	− 16.0000i	− 0.646234i	−0.946359	−	0.323117i	$-0.895269\pi$
$613$	− 16.0000i	− 0.646234i	0.946359	−	0.323117i	$-0.104731\pi$
$614$	0	0
$615$	−24.0000	−0.967773
$616$	0	0
$617$	−6.00000	−0.241551	−0.120775	−	0.992680i	$-0.538538\pi$
$617$	−6.00000	−0.241551	−0.120775	+	0.992680i	$0.538538\pi$
$618$	0	0
$619$	− 10.0000i	− 0.401934i	−0.979598	−	0.200967i	$-0.935592\pi$
$619$	− 10.0000i	− 0.401934i	0.979598	−	0.200967i	$-0.0644084\pi$
$620$	0	0
$621$	− 16.0000i	− 0.642058i
$622$	0	0
$623$	−14.0000	−0.560898
$624$	0	0
$625$	−19.0000	−0.760000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 48.0000i	− 1.91389i
$630$	0	0
$631$	−8.00000	−0.318475	−0.159237	−	0.987240i	$-0.550904\pi$
$631$	−8.00000	−0.318475	−0.159237	+	0.987240i	$0.550904\pi$
$632$	0	0
$633$	56.0000	2.22580
$634$	0	0
$635$	8.00000i	0.317470i
$636$	0	0
$637$	2.00000i	0.0792429i
$638$	0	0
$639$	−16.0000	−0.632950
$640$	0	0
$641$	18.0000	0.710957	0.355479	−	0.934684i	$-0.384318\pi$
$641$	18.0000	0.710957	0.355479	+	0.934684i	$0.384318\pi$
$642$	0	0
$643$	22.0000i	0.867595i	0.901010	+	0.433798i	$0.142827\pi$
$643$	22.0000i	0.867595i	−0.901010	+	0.433798i	$0.857173\pi$
$644$	0	0
$645$	32.0000i	1.26000i
$646$	0	0
$647$	48.0000	1.88707	0.943537	−	0.331266i	$-0.107476\pi$
$647$	48.0000	1.88707	0.943537	+	0.331266i	$0.107476\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	16.0000i	0.627089i
$652$	0	0
$653$	− 40.0000i	− 1.56532i	−0.622449	−	0.782660i	$-0.713862\pi$
$653$	− 40.0000i	− 1.56532i	0.622449	−	0.782660i	$-0.286138\pi$
$654$	0	0
$655$	12.0000	0.468879
$656$	0	0
$657$	−6.00000	−0.234082
$658$	0	0
$659$	40.0000i	1.55818i	0.626913	+	0.779089i	$0.284318\pi$
$659$	40.0000i	1.55818i	−0.626913	+	0.779089i	$0.715682\pi$
$660$	0	0
$661$	− 22.0000i	− 0.855701i	−0.903850	−	0.427850i	$-0.859271\pi$
$661$	− 22.0000i	− 0.855701i	0.903850	−	0.427850i	$-0.140729\pi$
$662$	0	0
$663$	24.0000	0.932083
$664$	0	0
$665$	−12.0000	−0.465340
$666$	0	0
$667$	0	0
$668$	0	0
$669$	16.0000i	0.618596i
$670$	0	0
$671$	0	0
$672$	0	0
$673$	−18.0000	−0.693849	−0.346925	−	0.937893i	$-0.612774\pi$
$673$	−18.0000	−0.693849	−0.346925	+	0.937893i	$0.612774\pi$
$674$	0	0
$675$	− 4.00000i	− 0.153960i
$676$	0	0
$677$	6.00000i	0.230599i	0.993331	+	0.115299i	$0.0367827\pi$
$677$	6.00000i	0.230599i	−0.993331	+	0.115299i	$0.963217\pi$
$678$	0	0
$679$	2.00000	0.0767530
$680$	0	0
$681$	−28.0000	−1.07296
$682$	0	0
$683$	44.0000i	1.68361i	0.539779	+	0.841807i	$0.318508\pi$
$683$	44.0000i	1.68361i	−0.539779	+	0.841807i	$0.681492\pi$
$684$	0	0
$685$	20.0000i	0.764161i
$686$	0	0
$687$	−4.00000	−0.152610
$688$	0	0
$689$	−8.00000	−0.304776
$690$	0	0
$691$	− 22.0000i	− 0.836919i	−0.908235	−	0.418460i	$-0.862570\pi$
$691$	− 22.0000i	− 0.836919i	0.908235	−	0.418460i	$-0.137430\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−20.0000	−0.758643
$696$	0	0
$697$	−36.0000	−1.36360
$698$	0	0
$699$	− 12.0000i	− 0.453882i
$700$	0	0
$701$	− 24.0000i	− 0.906467i	−0.891392	−	0.453234i	$-0.850270\pi$
$701$	− 24.0000i	− 0.906467i	0.891392	−	0.453234i	$-0.149730\pi$
$702$	0	0
$703$	−48.0000	−1.81035
$704$	0	0
$705$	0	0
$706$	0	0
$707$	14.0000i	0.526524i
$708$	0	0
$709$	− 8.00000i	− 0.300446i	−0.988652	−	0.150223i	$-0.952001\pi$
$709$	− 8.00000i	− 0.300446i	0.988652	−	0.150223i	$-0.0479992\pi$
$710$	0	0
$711$	16.0000	0.600047
$712$	0	0
$713$	32.0000	1.19841
$714$	0	0
$715$	0	0
$716$	0	0
$717$	40.0000i	1.49383i
$718$	0	0
$719$	−16.0000	−0.596699	−0.298350	−	0.954457i	$-0.596436\pi$
$719$	−16.0000	−0.596699	−0.298350	+	0.954457i	$0.596436\pi$
$720$	0	0
$721$	16.0000	0.595871
$722$	0	0
$723$	− 28.0000i	− 1.04133i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−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
$729$	−13.0000	−0.481481
$730$	0	0
$731$	48.0000i	1.77534i
$732$	0	0
$733$	− 10.0000i	− 0.369358i	−0.982799	−	0.184679i	$-0.940875\pi$
$733$	− 10.0000i	− 0.369358i	0.982799	−	0.184679i	$-0.0591246\pi$
$734$	0	0
$735$	4.00000	0.147542
$736$	0	0
$737$	0	0
$738$	0	0
$739$	32.0000i	1.17714i	0.808447	+	0.588570i	$0.200309\pi$
$739$	32.0000i	1.17714i	−0.808447	+	0.588570i	$0.799691\pi$
$740$	0	0
$741$	− 24.0000i	− 0.881662i
$742$	0	0
$743$	−12.0000	−0.440237	−0.220119	−	0.975473i	$-0.570644\pi$
$743$	−12.0000	−0.440237	−0.220119	+	0.975473i	$0.570644\pi$
$744$	0	0
$745$	−40.0000	−1.46549
$746$	0	0
$747$	14.0000i	0.512233i
$748$	0	0
$749$	− 4.00000i	− 0.146157i
$750$	0	0
$751$	4.00000	0.145962	0.0729810	−	0.997333i	$-0.476749\pi$
$751$	4.00000	0.145962	0.0729810	+	0.997333i	$0.476749\pi$
$752$	0	0
$753$	4.00000	0.145768
$754$	0	0
$755$	− 40.0000i	− 1.45575i
$756$	0	0
$757$	16.0000i	0.581530i	0.956795	+	0.290765i	$0.0939098\pi$
$757$	16.0000i	0.581530i	−0.956795	+	0.290765i	$0.906090\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−6.00000	−0.217500	−0.108750	−	0.994069i	$-0.534685\pi$
$761$	−6.00000	−0.217500	−0.108750	+	0.994069i	$0.534685\pi$
$762$	0	0
$763$	− 16.0000i	− 0.579239i
$764$	0	0
$765$	− 12.0000i	− 0.433861i
$766$	0	0
$767$	−12.0000	−0.433295
$768$	0	0
$769$	30.0000	1.08183	0.540914	−	0.841078i	$-0.318079\pi$
$769$	30.0000	1.08183	0.540914	+	0.841078i	$0.318079\pi$
$770$	0	0
$771$	− 4.00000i	− 0.144056i
$772$	0	0
$773$	18.0000i	0.647415i	0.946157	+	0.323708i	$0.104929\pi$
$773$	18.0000i	0.647415i	−0.946157	+	0.323708i	$0.895071\pi$
$774$	0	0
$775$	8.00000	0.287368
$776$	0	0
$777$	16.0000	0.573997
$778$	0	0
$779$	36.0000i	1.28983i
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	0	0
$785$	12.0000	0.428298
$786$	0	0
$787$	10.0000i	0.356462i	0.983989	+	0.178231i	$0.0570374\pi$
$787$	10.0000i	0.356462i	−0.983989	+	0.178231i	$0.942963\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	10.0000	0.355559
$792$	0	0
$793$	4.00000	0.142044
$794$	0	0
$795$	16.0000i	0.567462i
$796$	0	0
$797$	− 30.0000i	− 1.06265i	−0.847167	−	0.531327i	$-0.821693\pi$
$797$	− 30.0000i	− 1.06265i	0.847167	−	0.531327i	$-0.178307\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	−14.0000	−0.494666
$802$	0	0
$803$	0	0
$804$	0	0
$805$	− 8.00000i	− 0.281963i
$806$	0	0
$807$	−60.0000	−2.11210
$808$	0	0
$809$	14.0000	0.492214	0.246107	−	0.969243i	$-0.420849\pi$
$809$	14.0000	0.492214	0.246107	+	0.969243i	$0.420849\pi$
$810$	0	0
$811$	30.0000i	1.05344i	0.850038	+	0.526721i	$0.176579\pi$
$811$	30.0000i	1.05344i	−0.850038	+	0.526721i	$0.823421\pi$
$812$	0	0
$813$	16.0000i	0.561144i
$814$	0	0
$815$	−32.0000	−1.12091
$816$	0	0
$817$	48.0000	1.67931
$818$	0	0
$819$	2.00000i	0.0698857i
$820$	0	0
$821$	28.0000i	0.977207i	0.872506	+	0.488603i	$0.162493\pi$
$821$	28.0000i	0.977207i	−0.872506	+	0.488603i	$0.837507\pi$
$822$	0	0
$823$	16.0000	0.557725	0.278862	−	0.960331i	$-0.410043\pi$
$823$	16.0000	0.557725	0.278862	+	0.960331i	$0.410043\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	36.0000i	1.25184i	0.779886	+	0.625921i	$0.215277\pi$
$827$	36.0000i	1.25184i	−0.779886	+	0.625921i	$0.784723\pi$
$828$	0	0
$829$	30.0000i	1.04194i	0.853574	+	0.520972i	$0.174430\pi$
$829$	30.0000i	1.04194i	−0.853574	+	0.520972i	$0.825570\pi$
$830$	0	0
$831$	−40.0000	−1.38758
$832$	0	0
$833$	6.00000	0.207888
$834$	0	0
$835$	16.0000i	0.553703i
$836$	0	0
$837$	− 32.0000i	− 1.10608i
$838$	0	0
$839$	−16.0000	−0.552381	−0.276191	−	0.961103i	$-0.589072\pi$
$839$	−16.0000	−0.552381	−0.276191	+	0.961103i	$0.589072\pi$
$840$	0	0
$841$	29.0000	1.00000
$842$	0	0
$843$	52.0000i	1.79098i
$844$	0	0
$845$	18.0000i	0.619219i
$846$	0	0
$847$	−11.0000	−0.377964
$848$	0	0
$849$	−4.00000	−0.137280
$850$	0	0
$851$	− 32.0000i	− 1.09695i
$852$	0	0
$853$	30.0000i	1.02718i	0.858036	+	0.513590i	$0.171685\pi$
$853$	30.0000i	1.02718i	−0.858036	+	0.513590i	$0.828315\pi$
$854$	0	0
$855$	−12.0000	−0.410391
$856$	0	0
$857$	2.00000	0.0683187	0.0341593	−	0.999416i	$-0.489125\pi$
$857$	2.00000	0.0683187	0.0341593	+	0.999416i	$0.489125\pi$
$858$	0	0
$859$	− 22.0000i	− 0.750630i	−0.926897	−	0.375315i	$-0.877534\pi$
$859$	− 22.0000i	− 0.750630i	0.926897	−	0.375315i	$-0.122466\pi$
$860$	0	0
$861$	− 12.0000i	− 0.408959i
$862$	0	0
$863$	24.0000	0.816970	0.408485	−	0.912765i	$-0.366057\pi$
$863$	24.0000	0.816970	0.408485	+	0.912765i	$0.366057\pi$
$864$	0	0
$865$	12.0000	0.408012
$866$	0	0
$867$	− 38.0000i	− 1.29055i
$868$	0	0
$869$	0	0
$870$	0	0
$871$	8.00000	0.271070
$872$	0	0
$873$	2.00000	0.0676897
$874$	0	0
$875$	− 12.0000i	− 0.405674i
$876$	0	0
$877$	8.00000i	0.270141i	0.990836	+	0.135070i	$0.0431261\pi$
$877$	8.00000i	0.270141i	−0.990836	+	0.135070i	$0.956874\pi$
$878$	0	0
$879$	−60.0000	−2.02375
$880$	0	0
$881$	26.0000	0.875962	0.437981	−	0.898984i	$-0.355694\pi$
$881$	26.0000	0.875962	0.437981	+	0.898984i	$0.355694\pi$
$882$	0	0
$883$	36.0000i	1.21150i	0.795656	+	0.605748i	$0.207126\pi$
$883$	36.0000i	1.21150i	−0.795656	+	0.605748i	$0.792874\pi$
$884$	0	0
$885$	24.0000i	0.806751i
$886$	0	0
$887$	−8.00000	−0.268614	−0.134307	−	0.990940i	$-0.542881\pi$
$887$	−8.00000	−0.268614	−0.134307	+	0.990940i	$0.542881\pi$
$888$	0	0
$889$	−4.00000	−0.134156
$890$	0	0
$891$	0	0
$892$	0	0
$893$	0	0
$894$	0	0
$895$	−8.00000	−0.267411
$896$	0	0
$897$	16.0000	0.534224
$898$	0	0
$899$	0	0
$900$	0	0
$901$	24.0000i	0.799556i
$902$	0	0
$903$	−16.0000	−0.532447
$904$	0	0
$905$	−12.0000	−0.398893
$906$	0	0
$907$	− 28.0000i	− 0.929725i	−0.885383	−	0.464862i	$-0.846104\pi$
$907$	− 28.0000i	− 0.929725i	0.885383	−	0.464862i	$-0.153896\pi$
$908$	0	0
$909$	14.0000i	0.464351i
$910$	0	0
$911$	60.0000	1.98789	0.993944	−	0.109885i	$-0.0350482\pi$
$911$	60.0000	1.98789	0.993944	+	0.109885i	$0.0350482\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	− 8.00000i	− 0.264472i
$916$	0	0
$917$	6.00000i	0.198137i
$918$	0	0
$919$	−32.0000	−1.05558	−0.527791	−	0.849374i	$-0.676980\pi$
$919$	−32.0000	−1.05558	−0.527791	+	0.849374i	$0.676980\pi$
$920$	0	0
$921$	20.0000	0.659022
$922$	0	0
$923$	32.0000i	1.05329i
$924$	0	0
$925$	− 8.00000i	− 0.263038i
$926$	0	0
$927$	16.0000	0.525509
$928$	0	0
$929$	−10.0000	−0.328089	−0.164045	−	0.986453i	$-0.552454\pi$
$929$	−10.0000	−0.328089	−0.164045	+	0.986453i	$0.552454\pi$
$930$	0	0
$931$	− 6.00000i	− 0.196642i
$932$	0	0
$933$	16.0000i	0.523816i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−26.0000	−0.849383	−0.424691	−	0.905338i	$-0.639617\pi$
$937$	−26.0000	−0.849383	−0.424691	+	0.905338i	$0.639617\pi$
$938$	0	0
$939$	12.0000i	0.391605i
$940$	0	0
$941$	6.00000i	0.195594i	0.995206	+	0.0977972i	$0.0311797\pi$
$941$	6.00000i	0.195594i	−0.995206	+	0.0977972i	$0.968820\pi$
$942$	0	0
$943$	−24.0000	−0.781548
$944$	0	0
$945$	−8.00000	−0.260240
$946$	0	0
$947$	− 24.0000i	− 0.779895i	−0.920837	−	0.389948i	$-0.872493\pi$
$947$	− 24.0000i	− 0.779895i	0.920837	−	0.389948i	$-0.127507\pi$
$948$	0	0
$949$	12.0000i	0.389536i
$950$	0	0
$951$	−24.0000	−0.778253
$952$	0	0
$953$	−22.0000	−0.712650	−0.356325	−	0.934362i	$-0.615970\pi$
$953$	−22.0000	−0.712650	−0.356325	+	0.934362i	$0.615970\pi$
$954$	0	0
$955$	− 16.0000i	− 0.517748i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−10.0000	−0.322917
$960$	0	0
$961$	33.0000	1.06452
$962$	0	0
$963$	− 4.00000i	− 0.128898i
$964$	0	0
$965$	− 44.0000i	− 1.41641i
$966$	0	0
$967$	52.0000	1.67221	0.836104	−	0.548572i	$-0.184828\pi$
$967$	52.0000	1.67221	0.836104	+	0.548572i	$0.184828\pi$
$968$	0	0
$969$	−72.0000	−2.31297
$970$	0	0
$971$	6.00000i	0.192549i	0.995355	+	0.0962746i	$0.0306927\pi$
$971$	6.00000i	0.192549i	−0.995355	+	0.0962746i	$0.969307\pi$
$972$	0	0
$973$	− 10.0000i	− 0.320585i
$974$	0	0
$975$	4.00000	0.128103
$976$	0	0
$977$	14.0000	0.447900	0.223950	−	0.974601i	$-0.428105\pi$
$977$	14.0000	0.447900	0.223950	+	0.974601i	$0.428105\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	− 16.0000i	− 0.510841i
$982$	0	0
$983$	−32.0000	−1.02064	−0.510321	−	0.859984i	$-0.670473\pi$
$983$	−32.0000	−1.02064	−0.510321	+	0.859984i	$0.670473\pi$
$984$	0	0
$985$	−24.0000	−0.764704
$986$	0	0
$987$	0	0
$988$	0	0
$989$	32.0000i	1.01754i
$990$	0	0
$991$	−16.0000	−0.508257	−0.254128	−	0.967170i	$-0.581789\pi$
$991$	−16.0000	−0.508257	−0.254128	+	0.967170i	$0.581789\pi$
$992$	0	0
$993$	16.0000	0.507745
$994$	0	0
$995$	− 32.0000i	− 1.01447i
$996$	0	0
$997$	2.00000i	0.0633406i	0.999498	+	0.0316703i	$0.0100827\pi$
$997$	2.00000i	0.0633406i	−0.999498	+	0.0316703i	$0.989917\pi$
$998$	0	0
$999$	−32.0000	−1.01244

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	896.2.b.a.449.1	✓	2
4.3	odd	2		896.2.b.c.449.2	yes	2
8.3	odd	2		896.2.b.c.449.1	yes	2
8.5	even	2	inner	896.2.b.a.449.2	yes	2
16.3	odd	4		1792.2.a.c.1.1		1
16.5	even	4		1792.2.a.b.1.1		1
16.11	odd	4		1792.2.a.f.1.1		1
16.13	even	4		1792.2.a.g.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
896.2.b.a.449.1	✓	2	1.1	even	1	trivial
896.2.b.a.449.2	yes	2	8.5	even	2	inner
896.2.b.c.449.1	yes	2	8.3	odd	2
896.2.b.c.449.2	yes	2	4.3	odd	2
1792.2.a.b.1.1		1	16.5	even	4
1792.2.a.c.1.1		1	16.3	odd	4
1792.2.a.f.1.1		1	16.11	odd	4
1792.2.a.g.1.1		1	16.13	even	4

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:	\( N \)	\(=\)	\( 896 = 2^{7} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	896.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-2.00000i q^{3} +2.00000i q^{5} -1.00000 q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q-2.00000i q^{3} +2.00000i q^{5} -1.00000 q^{7} -1.00000 q^{9} +2.00000i q^{13} +4.00000 q^{15} +6.00000 q^{17} -6.00000i q^{19} +2.00000i q^{21} +4.00000 q^{23} +1.00000 q^{25} -4.00000i q^{27} +8.00000 q^{31} -2.00000i q^{35} -8.00000i q^{37} +4.00000 q^{39} -6.00000 q^{41} +8.00000i q^{43} -2.00000i q^{45} +1.00000 q^{49} -12.0000i q^{51} +4.00000i q^{53} -12.0000 q^{57} +6.00000i q^{59} -2.00000i q^{61} +1.00000 q^{63} -4.00000 q^{65} -4.00000i q^{67} -8.00000i q^{69} +16.0000 q^{71} +6.00000 q^{73} -2.00000i q^{75} -16.0000 q^{79} -11.0000 q^{81} -14.0000i q^{83} +12.0000i q^{85} +14.0000 q^{89} -2.00000i q^{91} -16.0000i q^{93} +12.0000 q^{95} -2.00000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{7} - 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^7 - 2 * q^9	\( 2 q - 2 q^{7} - 2 q^{9} + 8 q^{15} + 12 q^{17} + 8 q^{23} + 2 q^{25} + 16 q^{31} + 8 q^{39} - 12 q^{41} + 2 q^{49} - 24 q^{57} + 2 q^{63} - 8 q^{65} + 32 q^{71} + 12 q^{73} - 32 q^{79} - 22 q^{81} + 28 q^{89} + 24 q^{95} - 4 q^{97}+O(q^{100}) \) 2 * q - 2 * q^7 - 2 * q^9 + 8 * q^15 + 12 * q^17 + 8 * q^23 + 2 * q^25 + 16 * q^31 + 8 * q^39 - 12 * q^41 + 2 * q^49 - 24 * q^57 + 2 * q^63 - 8 * q^65 + 32 * q^71 + 12 * q^73 - 32 * q^79 - 22 * q^81 + 28 * q^89 + 24 * q^95 - 4 * q^97

Introduction

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