LMFDB - Embedded newform 1792.2.f.g.1791.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1792,2,Mod(1791,1792)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1792, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1792.1791");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1792 = 2^{8} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1792.f (of order $2$, degree $1$, not 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:	$14.3091920422$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{2}, \sqrt{-3})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 2x^{2} + 4 $ x^4 + 2*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 896)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1791.1
Root			$-0.707107 + 1.22474i$ of defining polynomial
Character	$\chi$	$=$	1792.1791
Dual form			1792.2.f.g.1791.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.41421 q^{3} -2.44949i q^{5} +(2.00000 - 1.73205i) q^{7} -1.00000 q^{9} +O(q^{10})$	$q-1.41421 q^{3} -2.44949i q^{5} +(2.00000 - 1.73205i) q^{7} -1.00000 q^{9} +4.89898i q^{11} -2.44949i q^{13} +3.46410i q^{15} +6.92820i q^{17} -4.24264 q^{19} +(-2.82843 + 2.44949i) q^{21} +6.92820i q^{23} -1.00000 q^{25} +5.65685 q^{27} -5.65685 q^{29} +4.00000 q^{31} -6.92820i q^{33} +(-4.24264 - 4.89898i) q^{35} +3.46410i q^{39} +4.89898i q^{43} +2.44949i q^{45} +12.0000 q^{47} +(1.00000 - 6.92820i) q^{49} -9.79796i q^{51} -5.65685 q^{53} +12.0000 q^{55} +6.00000 q^{57} +9.89949 q^{59} -7.34847i q^{61} +(-2.00000 + 1.73205i) q^{63} -6.00000 q^{65} +4.89898i q^{67} -9.79796i q^{69} +3.46410i q^{71} -6.92820i q^{73} +1.41421 q^{75} +(8.48528 + 9.79796i) q^{77} +10.3923i q^{79} -5.00000 q^{81} -1.41421 q^{83} +16.9706 q^{85} +8.00000 q^{87} +6.92820i q^{89} +(-4.24264 - 4.89898i) q^{91} -5.65685 q^{93} +10.3923i q^{95} +6.92820i q^{97} -4.89898i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 8 q^{7} - 4 q^{9}+O(q^{10}) $ 4 * q + 8 * q^7 - 4 * q^9	$ 4 q + 8 q^{7} - 4 q^{9} - 4 q^{25} + 16 q^{31} + 48 q^{47} + 4 q^{49} + 48 q^{55} + 24 q^{57} - 8 q^{63} - 24 q^{65} - 20 q^{81} + 32 q^{87}+O(q^{100}) $ 4 * q + 8 * q^7 - 4 * q^9 - 4 * q^25 + 16 * q^31 + 48 * q^47 + 4 * q^49 + 48 * q^55 + 24 * q^57 - 8 * q^63 - 24 * q^65 - 20 * q^81 + 32 * q^87

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1792\mathbb{Z}\right)^\times$.

$n$	$1023$	$1025$	$1541$
$\chi(n)$	$-1$	$-1$	$1$

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
$3$	−1.41421			−0.816497			−0.408248	−	0.912871i	$-0.633860\pi$
$3$	−1.41421			−0.816497			−0.408248	+	0.912871i	$0.633860\pi$
$4$		0			0
$5$		−	2.44949i		−	1.09545i	−0.836660	−	0.547723i	$-0.815495\pi$
$5$		−	2.44949i		−	1.09545i	0.836660	−	0.547723i	$-0.184505\pi$
$6$		0			0
$7$	2.00000	−	1.73205i	0.755929	−	0.654654i
$7$	2.00000	−	1.73205i	0.755929	−	0.654654i
$8$		0			0
$9$	−1.00000			−0.333333
$10$		0			0
$11$			4.89898i			1.47710i	0.674200	+	0.738549i	$0.264489\pi$
$11$			4.89898i			1.47710i	−0.674200	+	0.738549i	$0.735511\pi$
$12$		0			0
$13$		−	2.44949i		−	0.679366i	−0.940540	−	0.339683i	$-0.889680\pi$
$13$		−	2.44949i		−	0.679366i	0.940540	−	0.339683i	$-0.110320\pi$
$14$		0			0
$15$			3.46410i			0.894427i
$16$		0			0
$17$			6.92820i			1.68034i	0.542326	+	0.840168i	$0.317544\pi$
$17$			6.92820i			1.68034i	−0.542326	+	0.840168i	$0.682456\pi$
$18$		0			0
$19$	−4.24264			−0.973329			−0.486664	−	0.873589i	$-0.661786\pi$
$19$	−4.24264			−0.973329			−0.486664	+	0.873589i	$0.661786\pi$
$20$		0			0
$21$	−2.82843	+	2.44949i	−0.617213	+	0.534522i
$22$		0			0
$23$			6.92820i			1.44463i	0.691564	+	0.722315i	$0.256922\pi$
$23$			6.92820i			1.44463i	−0.691564	+	0.722315i	$0.743078\pi$
$24$		0			0
$25$	−1.00000			−0.200000
$26$		0			0
$27$	5.65685			1.08866
$28$		0			0
$29$	−5.65685			−1.05045			−0.525226	−	0.850963i	$-0.676019\pi$
$29$	−5.65685			−1.05045			−0.525226	+	0.850963i	$0.676019\pi$
$30$		0			0
$31$	4.00000			0.718421			0.359211	−	0.933257i	$-0.383046\pi$
$31$	4.00000			0.718421			0.359211	+	0.933257i	$0.383046\pi$
$32$		0			0
$33$		−	6.92820i		−	1.20605i
$34$		0			0
$35$	−4.24264	−	4.89898i	−0.717137	−	0.828079i
$36$		0			0
$37$		0			0			−	1.00000i	$-0.5\pi$
$37$		0			0				1.00000i	$0.5\pi$
$38$		0			0
$39$			3.46410i			0.554700i
$40$		0			0
$41$		0			0		1.00000			$0$
$41$		0			0		−1.00000			$\pi$
$42$		0			0
$43$			4.89898i			0.747087i	0.927613	+	0.373544i	$0.121857\pi$
$43$			4.89898i			0.747087i	−0.927613	+	0.373544i	$0.878143\pi$
$44$		0			0
$45$			2.44949i			0.365148i
$46$		0			0
$47$	12.0000			1.75038			0.875190	−	0.483779i	$-0.160736\pi$
$47$	12.0000			1.75038			0.875190	+	0.483779i	$0.160736\pi$
$48$		0			0
$49$	1.00000	−	6.92820i	0.142857	−	0.989743i
$50$		0			0
$51$		−	9.79796i		−	1.37199i
$52$		0			0
$53$	−5.65685			−0.777029			−0.388514	−	0.921443i	$-0.627012\pi$
$53$	−5.65685			−0.777029			−0.388514	+	0.921443i	$0.627012\pi$
$54$		0			0
$55$	12.0000			1.61808
$56$		0			0
$57$	6.00000			0.794719
$58$		0			0
$59$	9.89949			1.28880			0.644402	−	0.764687i	$-0.277106\pi$
$59$	9.89949			1.28880			0.644402	+	0.764687i	$0.277106\pi$
$60$		0			0
$61$		−	7.34847i		−	0.940875i	−0.882433	−	0.470438i	$-0.844096\pi$
$61$		−	7.34847i		−	0.940875i	0.882433	−	0.470438i	$-0.155904\pi$
$62$		0			0
$63$	−2.00000	+	1.73205i	−0.251976	+	0.218218i
$64$		0			0
$65$	−6.00000			−0.744208
$66$		0			0
$67$			4.89898i			0.598506i	0.954174	+	0.299253i	$0.0967374\pi$
$67$			4.89898i			0.598506i	−0.954174	+	0.299253i	$0.903263\pi$
$68$		0			0
$69$		−	9.79796i		−	1.17954i
$70$		0			0
$71$			3.46410i			0.411113i	0.978645	+	0.205557i	$0.0659005\pi$
$71$			3.46410i			0.411113i	−0.978645	+	0.205557i	$0.934100\pi$
$72$		0			0
$73$		−	6.92820i		−	0.810885i	−0.914121	−	0.405442i	$-0.867117\pi$
$73$		−	6.92820i		−	0.810885i	0.914121	−	0.405442i	$-0.132883\pi$
$74$		0			0
$75$	1.41421			0.163299
$76$		0			0
$77$	8.48528	+	9.79796i	0.966988	+	1.11658i
$78$		0			0
$79$			10.3923i			1.16923i	0.811312	+	0.584613i	$0.198754\pi$
$79$			10.3923i			1.16923i	−0.811312	+	0.584613i	$0.801246\pi$
$80$		0			0
$81$	−5.00000			−0.555556
$82$		0			0
$83$	−1.41421			−0.155230			−0.0776151	−	0.996983i	$-0.524731\pi$
$83$	−1.41421			−0.155230			−0.0776151	+	0.996983i	$0.524731\pi$
$84$		0			0
$85$	16.9706			1.84072
$86$		0			0
$87$	8.00000			0.857690
$88$		0			0
$89$			6.92820i			0.734388i	0.930144	+	0.367194i	$0.119682\pi$
$89$			6.92820i			0.734388i	−0.930144	+	0.367194i	$0.880318\pi$
$90$		0			0
$91$	−4.24264	−	4.89898i	−0.444750	−	0.513553i
$92$		0			0
$93$	−5.65685			−0.586588
$94$		0			0
$95$			10.3923i			1.06623i
$96$		0			0
$97$			6.92820i			0.703452i	0.936103	+	0.351726i	$0.114405\pi$
$97$			6.92820i			0.703452i	−0.936103	+	0.351726i	$0.885595\pi$
$98$		0			0
$99$		−	4.89898i		−	0.492366i
$100$		0			0
$101$			12.2474i			1.21867i	0.792914	+	0.609333i	$0.208563\pi$
$101$			12.2474i			1.21867i	−0.792914	+	0.609333i	$0.791437\pi$
$102$		0			0
$103$	−8.00000			−0.788263			−0.394132	−	0.919054i	$-0.628955\pi$
$103$	−8.00000			−0.788263			−0.394132	+	0.919054i	$0.628955\pi$
$104$		0			0
$105$	6.00000	+	6.92820i	0.585540	+	0.676123i
$106$		0			0
$107$		−	4.89898i		−	0.473602i	−0.971558	−	0.236801i	$-0.923901\pi$
$107$		−	4.89898i		−	0.473602i	0.971558	−	0.236801i	$-0.0760990\pi$
$108$		0			0
$109$	16.9706			1.62549			0.812743	−	0.582623i	$-0.197974\pi$
$109$	16.9706			1.62549			0.812743	+	0.582623i	$0.197974\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	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
$115$	16.9706			1.58251
$116$		0			0
$117$			2.44949i			0.226455i
$118$		0			0
$119$	12.0000	+	13.8564i	1.10004	+	1.27021i
$120$		0			0
$121$	−13.0000			−1.18182
$122$		0			0
$123$		0			0
$124$		0			0
$125$		−	9.79796i		−	0.876356i
$126$		0			0
$127$		−	6.92820i		−	0.614779i	−0.951584	−	0.307389i	$-0.900545\pi$
$127$		−	6.92820i		−	0.614779i	0.951584	−	0.307389i	$-0.0994554\pi$
$128$		0			0
$129$		−	6.92820i		−	0.609994i
$130$		0			0
$131$	−1.41421			−0.123560			−0.0617802	−	0.998090i	$-0.519678\pi$
$131$	−1.41421			−0.123560			−0.0617802	+	0.998090i	$0.519678\pi$
$132$		0			0
$133$	−8.48528	+	7.34847i	−0.735767	+	0.637193i
$134$		0			0
$135$		−	13.8564i		−	1.19257i
$136$		0			0
$137$	−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
$139$	−12.7279			−1.07957			−0.539784	−	0.841803i	$-0.681494\pi$
$139$	−12.7279			−1.07957			−0.539784	+	0.841803i	$0.681494\pi$
$140$		0			0
$141$	−16.9706			−1.42918
$142$		0			0
$143$	12.0000			1.00349
$144$		0			0
$145$			13.8564i			1.15071i
$146$		0			0
$147$	−1.41421	+	9.79796i	−0.116642	+	0.808122i
$148$		0			0
$149$	11.3137			0.926855			0.463428	−	0.886135i	$-0.346619\pi$
$149$	11.3137			0.926855			0.463428	+	0.886135i	$0.346619\pi$
$150$		0			0
$151$			6.92820i			0.563809i	0.959442	+	0.281905i	$0.0909662\pi$
$151$			6.92820i			0.563809i	−0.959442	+	0.281905i	$0.909034\pi$
$152$		0			0
$153$		−	6.92820i		−	0.560112i
$154$		0			0
$155$		−	9.79796i		−	0.786991i
$156$		0			0
$157$			17.1464i			1.36843i	0.729279	+	0.684217i	$0.239856\pi$
$157$			17.1464i			1.36843i	−0.729279	+	0.684217i	$0.760144\pi$
$158$		0			0
$159$	8.00000			0.634441
$160$		0			0
$161$	12.0000	+	13.8564i	0.945732	+	1.09204i
$162$		0			0
$163$			24.4949i			1.91859i	0.282409	+	0.959294i	$0.408867\pi$
$163$			24.4949i			1.91859i	−0.282409	+	0.959294i	$0.591133\pi$
$164$		0			0
$165$	−16.9706			−1.32116
$166$		0			0
$167$		0			0			−	1.00000i	$-0.5\pi$
$167$		0			0				1.00000i	$0.5\pi$
$168$		0			0
$169$	7.00000			0.538462
$170$		0			0
$171$	4.24264			0.324443
$172$		0			0
$173$			17.1464i			1.30362i	0.758383	+	0.651809i	$0.225990\pi$
$173$			17.1464i			1.30362i	−0.758383	+	0.651809i	$0.774010\pi$
$174$		0			0
$175$	−2.00000	+	1.73205i	−0.151186	+	0.130931i
$176$		0			0
$177$	−14.0000			−1.05230
$178$		0			0
$179$		−	4.89898i		−	0.366167i	−0.983097	−	0.183083i	$-0.941392\pi$
$179$		−	4.89898i		−	0.366167i	0.983097	−	0.183083i	$-0.0586079\pi$
$180$		0			0
$181$		−	2.44949i		−	0.182069i	−0.995848	−	0.0910346i	$-0.970983\pi$
$181$		−	2.44949i		−	0.182069i	0.995848	−	0.0910346i	$-0.0290174\pi$
$182$		0			0
$183$			10.3923i			0.768221i
$184$		0			0
$185$		0			0
$186$		0			0
$187$	−33.9411			−2.48202
$188$		0			0
$189$	11.3137	−	9.79796i	0.822951	−	0.712697i
$190$		0			0
$191$		−	10.3923i		−	0.751961i	−0.926628	−	0.375980i	$-0.877306\pi$
$191$		−	10.3923i		−	0.751961i	0.926628	−	0.375980i	$-0.122694\pi$
$192$		0			0
$193$	4.00000			0.287926			0.143963	−	0.989583i	$-0.454015\pi$
$193$	4.00000			0.287926			0.143963	+	0.989583i	$0.454015\pi$
$194$		0			0
$195$	8.48528			0.607644
$196$		0			0
$197$	22.6274			1.61214			0.806068	−	0.591822i	$-0.201591\pi$
$197$	22.6274			1.61214			0.806068	+	0.591822i	$0.201591\pi$
$198$		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
$201$		−	6.92820i		−	0.488678i
$202$		0			0
$203$	−11.3137	+	9.79796i	−0.794067	+	0.687682i
$204$		0			0
$205$		0			0
$206$		0			0
$207$		−	6.92820i		−	0.481543i
$208$		0			0
$209$		−	20.7846i		−	1.43770i
$210$		0			0
$211$			24.4949i			1.68630i	0.537680	+	0.843149i	$0.319301\pi$
$211$			24.4949i			1.68630i	−0.537680	+	0.843149i	$0.680699\pi$
$212$		0			0
$213$		−	4.89898i		−	0.335673i
$214$		0			0
$215$	12.0000			0.818393
$216$		0			0
$217$	8.00000	−	6.92820i	0.543075	−	0.470317i
$218$		0			0
$219$			9.79796i			0.662085i
$220$		0			0
$221$	16.9706			1.14156
$222$		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
$227$	18.3848			1.22024			0.610120	−	0.792309i	$-0.291121\pi$
$227$	18.3848			1.22024			0.610120	+	0.792309i	$0.291121\pi$
$228$		0			0
$229$		−	7.34847i		−	0.485601i	−0.970076	−	0.242800i	$-0.921934\pi$
$229$		−	7.34847i		−	0.485601i	0.970076	−	0.242800i	$-0.0780660\pi$
$230$		0			0
$231$	−12.0000	−	13.8564i	−0.789542	−	0.911685i
$232$		0			0
$233$	−18.0000			−1.17922			−0.589610	−	0.807688i	$-0.700718\pi$
$233$	−18.0000			−1.17922			−0.589610	+	0.807688i	$0.700718\pi$
$234$		0			0
$235$		−	29.3939i		−	1.91745i
$236$		0			0
$237$		−	14.6969i		−	0.954669i
$238$		0			0
$239$			20.7846i			1.34444i	0.740349	+	0.672222i	$0.234660\pi$
$239$			20.7846i			1.34444i	−0.740349	+	0.672222i	$0.765340\pi$
$240$		0			0
$241$		−	20.7846i		−	1.33885i	−0.742878	−	0.669427i	$-0.766540\pi$
$241$		−	20.7846i		−	1.33885i	0.742878	−	0.669427i	$-0.233460\pi$
$242$		0			0
$243$	−9.89949			−0.635053
$244$		0			0
$245$	−16.9706	−	2.44949i	−1.08421	−	0.156492i
$246$		0			0
$247$			10.3923i			0.661247i
$248$		0			0
$249$	2.00000			0.126745
$250$		0			0
$251$	7.07107			0.446322			0.223161	−	0.974782i	$-0.428362\pi$
$251$	7.07107			0.446322			0.223161	+	0.974782i	$0.428362\pi$
$252$		0			0
$253$	−33.9411			−2.13386
$254$		0			0
$255$	−24.0000			−1.50294
$256$		0			0
$257$		0			0		1.00000			$0$
$257$		0			0		−1.00000			$\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	5.65685			0.350150
$262$		0			0
$263$		−	24.2487i		−	1.49524i	−0.664127	−	0.747620i	$-0.731197\pi$
$263$		−	24.2487i		−	1.49524i	0.664127	−	0.747620i	$-0.268803\pi$
$264$		0			0
$265$			13.8564i			0.851192i
$266$		0			0
$267$		−	9.79796i		−	0.599625i
$268$		0			0
$269$		−	22.0454i		−	1.34413i	−0.740491	−	0.672066i	$-0.765407\pi$
$269$		−	22.0454i		−	1.34413i	0.740491	−	0.672066i	$-0.234593\pi$
$270$		0			0
$271$	−16.0000			−0.971931			−0.485965	−	0.873978i	$-0.661532\pi$
$271$	−16.0000			−0.971931			−0.485965	+	0.873978i	$0.661532\pi$
$272$		0			0
$273$	6.00000	+	6.92820i	0.363137	+	0.419314i
$274$		0			0
$275$		−	4.89898i		−	0.295420i
$276$		0			0
$277$	−16.9706			−1.01966			−0.509831	−	0.860274i	$-0.670292\pi$
$277$	−16.9706			−1.01966			−0.509831	+	0.860274i	$0.670292\pi$
$278$		0			0
$279$	−4.00000			−0.239474
$280$		0			0
$281$	18.0000			1.07379			0.536895	−	0.843649i	$-0.319597\pi$
$281$	18.0000			1.07379			0.536895	+	0.843649i	$0.319597\pi$
$282$		0			0
$283$	−4.24264			−0.252199			−0.126099	−	0.992018i	$-0.540246\pi$
$283$	−4.24264			−0.252199			−0.126099	+	0.992018i	$0.540246\pi$
$284$		0			0
$285$		−	14.6969i		−	0.870572i
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−31.0000			−1.82353
$290$		0			0
$291$		−	9.79796i		−	0.574367i
$292$		0			0
$293$		−	7.34847i		−	0.429302i	−0.976691	−	0.214651i	$-0.931139\pi$
$293$		−	7.34847i		−	0.429302i	0.976691	−	0.214651i	$-0.0688614\pi$
$294$		0			0
$295$		−	24.2487i		−	1.41181i
$296$		0			0
$297$			27.7128i			1.60806i
$298$		0			0
$299$	16.9706			0.981433
$300$		0			0
$301$	8.48528	+	9.79796i	0.489083	+	0.564745i
$302$		0			0
$303$		−	17.3205i		−	0.995037i
$304$		0			0
$305$	−18.0000			−1.03068
$306$		0			0
$307$	21.2132			1.21070			0.605351	−	0.795959i	$-0.293033\pi$
$307$	21.2132			1.21070			0.605351	+	0.795959i	$0.293033\pi$
$308$		0			0
$309$	11.3137			0.643614
$310$		0			0
$311$	12.0000			0.680458			0.340229	−	0.940343i	$-0.389495\pi$
$311$	12.0000			0.680458			0.340229	+	0.940343i	$0.389495\pi$
$312$		0			0
$313$		−	13.8564i		−	0.783210i	−0.920133	−	0.391605i	$-0.871920\pi$
$313$		−	13.8564i		−	0.783210i	0.920133	−	0.391605i	$-0.128080\pi$
$314$		0			0
$315$	4.24264	+	4.89898i	0.239046	+	0.276026i
$316$		0			0
$317$	−11.3137			−0.635441			−0.317721	−	0.948184i	$-0.602917\pi$
$317$	−11.3137			−0.635441			−0.317721	+	0.948184i	$0.602917\pi$
$318$		0			0
$319$		−	27.7128i		−	1.55162i
$320$		0			0
$321$			6.92820i			0.386695i
$322$		0			0
$323$		−	29.3939i		−	1.63552i
$324$		0			0
$325$			2.44949i			0.135873i
$326$		0			0
$327$	−24.0000			−1.32720
$328$		0			0
$329$	24.0000	−	20.7846i	1.32316	−	1.14589i
$330$		0			0
$331$		−	4.89898i		−	0.269272i	−0.990895	−	0.134636i	$-0.957013\pi$
$331$		−	4.89898i		−	0.269272i	0.990895	−	0.134636i	$-0.0429866\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	12.0000			0.655630
$336$		0			0
$337$	28.0000			1.52526			0.762629	−	0.646837i	$-0.223908\pi$
$337$	28.0000			1.52526			0.762629	+	0.646837i	$0.223908\pi$
$338$		0			0
$339$	−16.9706			−0.921714
$340$		0			0
$341$			19.5959i			1.06118i
$342$		0			0
$343$	−10.0000	−	15.5885i	−0.539949	−	0.841698i
$344$		0			0
$345$	−24.0000			−1.29212
$346$		0			0
$347$			4.89898i			0.262991i	0.991317	+	0.131495i	$0.0419779\pi$
$347$			4.89898i			0.262991i	−0.991317	+	0.131495i	$0.958022\pi$
$348$		0			0
$349$			31.8434i			1.70454i	0.523105	+	0.852268i	$0.324774\pi$
$349$			31.8434i			1.70454i	−0.523105	+	0.852268i	$0.675226\pi$
$350$		0			0
$351$		−	13.8564i		−	0.739600i
$352$		0			0
$353$			13.8564i			0.737502i	0.929528	+	0.368751i	$0.120215\pi$
$353$			13.8564i			0.737502i	−0.929528	+	0.368751i	$0.879785\pi$
$354$		0			0
$355$	8.48528			0.450352
$356$		0			0
$357$	−16.9706	−	19.5959i	−0.898177	−	1.03713i
$358$		0			0
$359$			6.92820i			0.365657i	0.983145	+	0.182828i	$0.0585252\pi$
$359$			6.92820i			0.365657i	−0.983145	+	0.182828i	$0.941475\pi$
$360$		0			0
$361$	−1.00000			−0.0526316
$362$		0			0
$363$	18.3848			0.964951
$364$		0			0
$365$	−16.9706			−0.888280
$366$		0			0
$367$	−8.00000			−0.417597			−0.208798	−	0.977959i	$-0.566955\pi$
$367$	−8.00000			−0.417597			−0.208798	+	0.977959i	$0.566955\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−11.3137	+	9.79796i	−0.587378	+	0.508685i
$372$		0			0
$373$	−16.9706			−0.878702			−0.439351	−	0.898315i	$-0.644792\pi$
$373$	−16.9706			−0.878702			−0.439351	+	0.898315i	$0.644792\pi$
$374$		0			0
$375$			13.8564i			0.715542i
$376$		0			0
$377$			13.8564i			0.713641i
$378$		0			0
$379$			14.6969i			0.754931i	0.926024	+	0.377466i	$0.123204\pi$
$379$			14.6969i			0.754931i	−0.926024	+	0.377466i	$0.876796\pi$
$380$		0			0
$381$			9.79796i			0.501965i
$382$		0			0
$383$	−36.0000			−1.83951			−0.919757	−	0.392488i	$-0.871614\pi$
$383$	−36.0000			−1.83951			−0.919757	+	0.392488i	$0.871614\pi$
$384$		0			0
$385$	24.0000	−	20.7846i	1.22315	−	1.05928i
$386$		0			0
$387$		−	4.89898i		−	0.249029i
$388$		0			0
$389$	−11.3137			−0.573628			−0.286814	−	0.957986i	$-0.592596\pi$
$389$	−11.3137			−0.573628			−0.286814	+	0.957986i	$0.592596\pi$
$390$		0			0
$391$	−48.0000			−2.42746
$392$		0			0
$393$	2.00000			0.100887
$394$		0			0
$395$	25.4558			1.28082
$396$		0			0
$397$		−	7.34847i		−	0.368809i	−0.982850	−	0.184405i	$-0.940964\pi$
$397$		−	7.34847i		−	0.368809i	0.982850	−	0.184405i	$-0.0590357\pi$
$398$		0			0
$399$	12.0000	−	10.3923i	0.600751	−	0.520266i
$400$		0			0
$401$	−12.0000			−0.599251			−0.299626	−	0.954057i	$-0.596862\pi$
$401$	−12.0000			−0.599251			−0.299626	+	0.954057i	$0.596862\pi$
$402$		0			0
$403$		−	9.79796i		−	0.488071i
$404$		0			0
$405$			12.2474i			0.608581i
$406$		0			0
$407$		0			0
$408$		0			0
$409$			13.8564i			0.685155i	0.939490	+	0.342578i	$0.111300\pi$
$409$			13.8564i			0.685155i	−0.939490	+	0.342578i	$0.888700\pi$
$410$		0			0
$411$	8.48528			0.418548
$412$		0			0
$413$	19.7990	−	17.1464i	0.974245	−	0.843721i
$414$		0			0
$415$			3.46410i			0.170046i
$416$		0			0
$417$	18.0000			0.881464
$418$		0			0
$419$	−35.3553			−1.72722			−0.863611	−	0.504159i	$-0.831802\pi$
$419$	−35.3553			−1.72722			−0.863611	+	0.504159i	$0.831802\pi$
$420$		0			0
$421$	33.9411			1.65419			0.827095	−	0.562063i	$-0.189992\pi$
$421$	33.9411			1.65419			0.827095	+	0.562063i	$0.189992\pi$
$422$		0			0
$423$	−12.0000			−0.583460
$424$		0			0
$425$		−	6.92820i		−	0.336067i
$426$		0			0
$427$	−12.7279	−	14.6969i	−0.615947	−	0.711235i
$428$		0			0
$429$	−16.9706			−0.819346
$430$		0			0
$431$			20.7846i			1.00116i	0.865690	+	0.500580i	$0.166880\pi$
$431$			20.7846i			1.00116i	−0.865690	+	0.500580i	$0.833120\pi$
$432$		0			0
$433$			34.6410i			1.66474i	0.554220	+	0.832370i	$0.313017\pi$
$433$			34.6410i			1.66474i	−0.554220	+	0.832370i	$0.686983\pi$
$434$		0			0
$435$		−	19.5959i		−	0.939552i
$436$		0			0
$437$		−	29.3939i		−	1.40610i
$438$		0			0
$439$	28.0000			1.33637			0.668184	−	0.743996i	$-0.267072\pi$
$439$	28.0000			1.33637			0.668184	+	0.743996i	$0.267072\pi$
$440$		0			0
$441$	−1.00000	+	6.92820i	−0.0476190	+	0.329914i
$442$		0			0
$443$			14.6969i			0.698273i	0.937072	+	0.349136i	$0.113525\pi$
$443$			14.6969i			0.698273i	−0.937072	+	0.349136i	$0.886475\pi$
$444$		0			0
$445$	16.9706			0.804482
$446$		0			0
$447$	−16.0000			−0.756774
$448$		0			0
$449$	−6.00000			−0.283158			−0.141579	−	0.989927i	$-0.545218\pi$
$449$	−6.00000			−0.283158			−0.141579	+	0.989927i	$0.545218\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$		−	9.79796i		−	0.460348i
$454$		0			0
$455$	−12.0000	+	10.3923i	−0.562569	+	0.487199i
$456$		0			0
$457$	−28.0000			−1.30978			−0.654892	−	0.755722i	$-0.727286\pi$
$457$	−28.0000			−1.30978			−0.654892	+	0.755722i	$0.727286\pi$
$458$		0			0
$459$			39.1918i			1.82932i
$460$		0			0
$461$			12.2474i			0.570421i	0.958465	+	0.285210i	$0.0920634\pi$
$461$			12.2474i			0.570421i	−0.958465	+	0.285210i	$0.907937\pi$
$462$		0			0
$463$		−	38.1051i		−	1.77090i	−0.464739	−	0.885448i	$-0.653852\pi$
$463$		−	38.1051i		−	1.77090i	0.464739	−	0.885448i	$-0.346148\pi$
$464$		0			0
$465$			13.8564i			0.642575i
$466$		0			0
$467$	−15.5563			−0.719862			−0.359931	−	0.932979i	$-0.617200\pi$
$467$	−15.5563			−0.719862			−0.359931	+	0.932979i	$0.617200\pi$
$468$		0			0
$469$	8.48528	+	9.79796i	0.391814	+	0.452428i
$470$		0			0
$471$		−	24.2487i		−	1.11732i
$472$		0			0
$473$	−24.0000			−1.10352
$474$		0			0
$475$	4.24264			0.194666
$476$		0			0
$477$	5.65685			0.259010
$478$		0			0
$479$	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
$481$		0			0
$482$		0			0
$483$	−16.9706	−	19.5959i	−0.772187	−	0.891645i
$484$		0			0
$485$	16.9706			0.770594
$486$		0			0
$487$		−	20.7846i		−	0.941841i	−0.882176	−	0.470920i	$-0.843922\pi$
$487$		−	20.7846i		−	0.941841i	0.882176	−	0.470920i	$-0.156078\pi$
$488$		0			0
$489$		−	34.6410i		−	1.56652i
$490$		0			0
$491$			24.4949i			1.10544i	0.833367	+	0.552720i	$0.186410\pi$
$491$			24.4949i			1.10544i	−0.833367	+	0.552720i	$0.813590\pi$
$492$		0			0
$493$		−	39.1918i		−	1.76511i
$494$		0			0
$495$	−12.0000			−0.539360
$496$		0			0
$497$	6.00000	+	6.92820i	0.269137	+	0.310772i
$498$		0			0
$499$		−	14.6969i		−	0.657925i	−0.944343	−	0.328963i	$-0.893301\pi$
$499$		−	14.6969i		−	0.657925i	0.944343	−	0.328963i	$-0.106699\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	12.0000			0.535054			0.267527	−	0.963550i	$-0.413794\pi$
$503$	12.0000			0.535054			0.267527	+	0.963550i	$0.413794\pi$
$504$		0			0
$505$	30.0000			1.33498
$506$		0			0
$507$	−9.89949			−0.439652
$508$		0			0
$509$			12.2474i			0.542859i	0.962458	+	0.271429i	$0.0874963\pi$
$509$			12.2474i			0.542859i	−0.962458	+	0.271429i	$0.912504\pi$
$510$		0			0
$511$	−12.0000	−	13.8564i	−0.530849	−	0.612971i
$512$		0			0
$513$	−24.0000			−1.05963
$514$		0			0
$515$			19.5959i			0.863499i
$516$		0			0
$517$			58.7878i			2.58548i
$518$		0			0
$519$		−	24.2487i		−	1.06440i
$520$		0			0
$521$		−	27.7128i		−	1.21412i	−0.794656	−	0.607060i	$-0.792349\pi$
$521$		−	27.7128i		−	1.21412i	0.794656	−	0.607060i	$-0.207651\pi$
$522$		0			0
$523$	21.2132			0.927589			0.463794	−	0.885943i	$-0.346488\pi$
$523$	21.2132			0.927589			0.463794	+	0.885943i	$0.346488\pi$
$524$		0			0
$525$	2.82843	−	2.44949i	0.123443	−	0.106904i
$526$		0			0
$527$			27.7128i			1.20719i
$528$		0			0
$529$	−25.0000			−1.08696
$530$		0			0
$531$	−9.89949			−0.429601
$532$		0			0
$533$		0			0
$534$		0			0
$535$	−12.0000			−0.518805
$536$		0			0
$537$			6.92820i			0.298974i
$538$		0			0
$539$	33.9411	+	4.89898i	1.46195	+	0.211014i
$540$		0			0
$541$	33.9411			1.45924			0.729621	−	0.683851i	$-0.239696\pi$
$541$	33.9411			1.45924			0.729621	+	0.683851i	$0.239696\pi$
$542$		0			0
$543$			3.46410i			0.148659i
$544$		0			0
$545$		−	41.5692i		−	1.78063i
$546$		0			0
$547$			44.0908i			1.88519i	0.333942	+	0.942594i	$0.391621\pi$
$547$			44.0908i			1.88519i	−0.333942	+	0.942594i	$0.608379\pi$
$548$		0			0
$549$			7.34847i			0.313625i
$550$		0			0
$551$	24.0000			1.02243
$552$		0			0
$553$	18.0000	+	20.7846i	0.765438	+	0.883852i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	5.65685			0.239689			0.119844	−	0.992793i	$-0.461760\pi$
$557$	5.65685			0.239689			0.119844	+	0.992793i	$0.461760\pi$
$558$		0			0
$559$	12.0000			0.507546
$560$		0			0
$561$	48.0000			2.02656
$562$		0			0
$563$	−26.8701			−1.13244			−0.566219	−	0.824255i	$-0.691594\pi$
$563$	−26.8701			−1.13244			−0.566219	+	0.824255i	$0.691594\pi$
$564$		0			0
$565$		−	29.3939i		−	1.23661i
$566$		0			0
$567$	−10.0000	+	8.66025i	−0.419961	+	0.363696i
$568$		0			0
$569$	−12.0000			−0.503066			−0.251533	−	0.967849i	$-0.580935\pi$
$569$	−12.0000			−0.503066			−0.251533	+	0.967849i	$0.580935\pi$
$570$		0			0
$571$			4.89898i			0.205016i	0.994732	+	0.102508i	$0.0326867\pi$
$571$			4.89898i			0.205016i	−0.994732	+	0.102508i	$0.967313\pi$
$572$		0			0
$573$			14.6969i			0.613973i
$574$		0			0
$575$		−	6.92820i		−	0.288926i
$576$		0			0
$577$		0			0		1.00000			$0$
$577$		0			0		−1.00000			$\pi$
$578$		0			0
$579$	−5.65685			−0.235091
$580$		0			0
$581$	−2.82843	+	2.44949i	−0.117343	+	0.101622i
$582$		0			0
$583$		−	27.7128i		−	1.14775i
$584$		0			0
$585$	6.00000			0.248069
$586$		0			0
$587$	−35.3553			−1.45927			−0.729636	−	0.683836i	$-0.760310\pi$
$587$	−35.3553			−1.45927			−0.729636	+	0.683836i	$0.760310\pi$
$588$		0			0
$589$	−16.9706			−0.699260
$590$		0			0
$591$	−32.0000			−1.31630
$592$		0			0
$593$		−	41.5692i		−	1.70704i	−0.521057	−	0.853522i	$-0.674462\pi$
$593$		−	41.5692i		−	1.70704i	0.521057	−	0.853522i	$-0.325538\pi$
$594$		0			0
$595$	33.9411	−	29.3939i	1.39145	−	1.20503i
$596$		0			0
$597$	22.6274			0.926079
$598$		0			0
$599$			24.2487i			0.990775i	0.868672	+	0.495388i	$0.164974\pi$
$599$			24.2487i			0.990775i	−0.868672	+	0.495388i	$0.835026\pi$
$600$		0			0
$601$		−	6.92820i		−	0.282607i	−0.989966	−	0.141304i	$-0.954871\pi$
$601$		−	6.92820i		−	0.282607i	0.989966	−	0.141304i	$-0.0451294\pi$
$602$		0			0
$603$		−	4.89898i		−	0.199502i
$604$		0			0
$605$			31.8434i			1.29462i
$606$		0			0
$607$	−8.00000			−0.324710			−0.162355	−	0.986732i	$-0.551909\pi$
$607$	−8.00000			−0.324710			−0.162355	+	0.986732i	$0.551909\pi$
$608$		0			0
$609$	16.0000	−	13.8564i	0.648353	−	0.561490i
$610$		0			0
$611$		−	29.3939i		−	1.18915i
$612$		0			0
$613$		0			0			−	1.00000i	$-0.5\pi$
$613$		0			0				1.00000i	$0.5\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	12.0000			0.483102			0.241551	−	0.970388i	$-0.422344\pi$
$617$	12.0000			0.483102			0.241551	+	0.970388i	$0.422344\pi$
$618$		0			0
$619$	−4.24264			−0.170526			−0.0852631	−	0.996358i	$-0.527173\pi$
$619$	−4.24264			−0.170526			−0.0852631	+	0.996358i	$0.527173\pi$
$620$		0			0
$621$			39.1918i			1.57271i
$622$		0			0
$623$	12.0000	+	13.8564i	0.480770	+	0.555145i
$624$		0			0
$625$	−29.0000			−1.16000
$626$		0			0
$627$			29.3939i			1.17388i
$628$		0			0
$629$		0			0
$630$		0			0
$631$			24.2487i			0.965326i	0.875806	+	0.482663i	$0.160330\pi$
$631$			24.2487i			0.965326i	−0.875806	+	0.482663i	$0.839670\pi$
$632$		0			0
$633$		−	34.6410i		−	1.37686i
$634$		0			0
$635$	−16.9706			−0.673456
$636$		0			0
$637$	−16.9706	−	2.44949i	−0.672398	−	0.0970523i
$638$		0			0
$639$		−	3.46410i		−	0.137038i
$640$		0			0
$641$	−12.0000			−0.473972			−0.236986	−	0.971513i	$-0.576159\pi$
$641$	−12.0000			−0.473972			−0.236986	+	0.971513i	$0.576159\pi$
$642$		0			0
$643$	−4.24264			−0.167313			−0.0836567	−	0.996495i	$-0.526660\pi$
$643$	−4.24264			−0.167313			−0.0836567	+	0.996495i	$0.526660\pi$
$644$		0			0
$645$	−16.9706			−0.668215
$646$		0			0
$647$		0			0			−	1.00000i	$-0.5\pi$
$647$		0			0				1.00000i	$0.5\pi$
$648$		0			0
$649$			48.4974i			1.90369i
$650$		0			0
$651$	−11.3137	+	9.79796i	−0.443419	+	0.384012i
$652$		0			0
$653$	−11.3137			−0.442740			−0.221370	−	0.975190i	$-0.571053\pi$
$653$	−11.3137			−0.442740			−0.221370	+	0.975190i	$0.571053\pi$
$654$		0			0
$655$			3.46410i			0.135354i
$656$		0			0
$657$			6.92820i			0.270295i
$658$		0			0
$659$		−	4.89898i		−	0.190837i	−0.995437	−	0.0954186i	$-0.969581\pi$
$659$		−	4.89898i		−	0.190837i	0.995437	−	0.0954186i	$-0.0304190\pi$
$660$		0			0
$661$		−	7.34847i		−	0.285822i	−0.989736	−	0.142911i	$-0.954354\pi$
$661$		−	7.34847i		−	0.285822i	0.989736	−	0.142911i	$-0.0456463\pi$
$662$		0			0
$663$	−24.0000			−0.932083
$664$		0			0
$665$	18.0000	+	20.7846i	0.698010	+	0.805993i
$666$		0			0
$667$		−	39.1918i		−	1.51751i
$668$		0			0
$669$	11.3137			0.437413
$670$		0			0
$671$	36.0000			1.38976
$672$		0			0
$673$	−10.0000			−0.385472			−0.192736	−	0.981251i	$-0.561736\pi$
$673$	−10.0000			−0.385472			−0.192736	+	0.981251i	$0.561736\pi$
$674$		0			0
$675$	−5.65685			−0.217732
$676$		0			0
$677$			36.7423i			1.41212i	0.708150	+	0.706062i	$0.249530\pi$
$677$			36.7423i			1.41212i	−0.708150	+	0.706062i	$0.750470\pi$
$678$		0			0
$679$	12.0000	+	13.8564i	0.460518	+	0.531760i
$680$		0			0
$681$	−26.0000			−0.996322
$682$		0			0
$683$			14.6969i			0.562363i	0.959655	+	0.281181i	$0.0907262\pi$
$683$			14.6969i			0.562363i	−0.959655	+	0.281181i	$0.909274\pi$
$684$		0			0
$685$			14.6969i			0.561541i
$686$		0			0
$687$			10.3923i			0.396491i
$688$		0			0
$689$			13.8564i			0.527887i
$690$		0			0
$691$	21.2132			0.806988			0.403494	−	0.914982i	$-0.367796\pi$
$691$	21.2132			0.806988			0.403494	+	0.914982i	$0.367796\pi$
$692$		0			0
$693$	−8.48528	−	9.79796i	−0.322329	−	0.372194i
$694$		0			0
$695$			31.1769i			1.18261i
$696$		0			0
$697$		0			0
$698$		0			0
$699$	25.4558			0.962828
$700$		0			0
$701$	22.6274			0.854626			0.427313	−	0.904104i	$-0.359460\pi$
$701$	22.6274			0.854626			0.427313	+	0.904104i	$0.359460\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$			41.5692i			1.56559i
$706$		0			0
$707$	21.2132	+	24.4949i	0.797805	+	0.921225i
$708$		0			0
$709$	−16.9706			−0.637343			−0.318671	−	0.947865i	$-0.603237\pi$
$709$	−16.9706			−0.637343			−0.318671	+	0.947865i	$0.603237\pi$
$710$		0			0
$711$		−	10.3923i		−	0.389742i
$712$		0			0
$713$			27.7128i			1.03785i
$714$		0			0
$715$		−	29.3939i		−	1.09927i
$716$		0			0
$717$		−	29.3939i		−	1.09773i
$718$		0			0
$719$	36.0000			1.34257			0.671287	−	0.741198i	$-0.265742\pi$
$719$	36.0000			1.34257			0.671287	+	0.741198i	$0.265742\pi$
$720$		0			0
$721$	−16.0000	+	13.8564i	−0.595871	+	0.516040i
$722$		0			0
$723$			29.3939i			1.09317i
$724$		0			0
$725$	5.65685			0.210090
$726$		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
$729$	29.0000			1.07407
$730$		0			0
$731$	−33.9411			−1.25536
$732$		0			0
$733$		−	46.5403i		−	1.71901i	−0.511131	−	0.859503i	$-0.670773\pi$
$733$		−	46.5403i		−	1.71901i	0.511131	−	0.859503i	$-0.329227\pi$
$734$		0			0
$735$	24.0000	+	3.46410i	0.885253	+	0.127775i
$736$		0			0
$737$	−24.0000			−0.884051
$738$		0			0
$739$		−	14.6969i		−	0.540636i	−0.962771	−	0.270318i	$-0.912871\pi$
$739$		−	14.6969i		−	0.540636i	0.962771	−	0.270318i	$-0.0871288\pi$
$740$		0			0
$741$		−	14.6969i		−	0.539906i
$742$		0			0
$743$			34.6410i			1.27086i	0.772160	+	0.635428i	$0.219176\pi$
$743$			34.6410i			1.27086i	−0.772160	+	0.635428i	$0.780824\pi$
$744$		0			0
$745$		−	27.7128i		−	1.01532i
$746$		0			0
$747$	1.41421			0.0517434
$748$		0			0
$749$	−8.48528	−	9.79796i	−0.310045	−	0.358010i
$750$		0			0
$751$		−	34.6410i		−	1.26407i	−0.774940	−	0.632034i	$-0.782220\pi$
$751$		−	34.6410i		−	1.26407i	0.774940	−	0.632034i	$-0.217780\pi$
$752$		0			0
$753$	−10.0000			−0.364420
$754$		0			0
$755$	16.9706			0.617622
$756$		0			0
$757$		0			0			−	1.00000i	$-0.5\pi$
$757$		0			0				1.00000i	$0.5\pi$
$758$		0			0
$759$	48.0000			1.74229
$760$		0			0
$761$			13.8564i			0.502294i	0.967949	+	0.251147i	$0.0808078\pi$
$761$			13.8564i			0.502294i	−0.967949	+	0.251147i	$0.919192\pi$
$762$		0			0
$763$	33.9411	−	29.3939i	1.22875	−	1.06413i
$764$		0			0
$765$	−16.9706			−0.613572
$766$		0			0
$767$		−	24.2487i		−	0.875570i
$768$		0			0
$769$			6.92820i			0.249837i	0.992167	+	0.124919i	$0.0398670\pi$
$769$			6.92820i			0.249837i	−0.992167	+	0.124919i	$0.960133\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		−	22.0454i		−	0.792918i	−0.918052	−	0.396459i	$-0.870239\pi$
$773$		−	22.0454i		−	0.792918i	0.918052	−	0.396459i	$-0.129761\pi$
$774$		0			0
$775$	−4.00000			−0.143684
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$	−16.9706			−0.607254
$782$		0			0
$783$	−32.0000			−1.14359
$784$		0			0
$785$	42.0000			1.49904
$786$		0			0
$787$	29.6985			1.05864			0.529318	−	0.848423i	$-0.322448\pi$
$787$	29.6985			1.05864			0.529318	+	0.848423i	$0.322448\pi$
$788$		0			0
$789$			34.2929i			1.22086i
$790$		0			0
$791$	24.0000	−	20.7846i	0.853342	−	0.739016i
$792$		0			0
$793$	−18.0000			−0.639199
$794$		0			0
$795$		−	19.5959i		−	0.694996i
$796$		0			0
$797$		−	2.44949i		−	0.0867654i	−0.999059	−	0.0433827i	$-0.986187\pi$
$797$		−	2.44949i		−	0.0867654i	0.999059	−	0.0433827i	$-0.0138135\pi$
$798$		0			0
$799$			83.1384i			2.94123i
$800$		0			0
$801$		−	6.92820i		−	0.244796i
$802$		0			0
$803$	33.9411			1.19776
$804$		0			0
$805$	33.9411	−	29.3939i	1.19627	−	1.03600i
$806$		0			0
$807$			31.1769i			1.09748i
$808$		0			0
$809$	−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
$811$	−38.1838			−1.34081			−0.670407	−	0.741994i	$-0.733880\pi$
$811$	−38.1838			−1.34081			−0.670407	+	0.741994i	$0.733880\pi$
$812$		0			0
$813$	22.6274			0.793578
$814$		0			0
$815$	60.0000			2.10171
$816$		0			0
$817$		−	20.7846i		−	0.727161i
$818$		0			0
$819$	4.24264	+	4.89898i	0.148250	+	0.171184i
$820$		0			0
$821$	−5.65685			−0.197426			−0.0987128	−	0.995116i	$-0.531473\pi$
$821$	−5.65685			−0.197426			−0.0987128	+	0.995116i	$0.531473\pi$
$822$		0			0
$823$		−	24.2487i		−	0.845257i	−0.906303	−	0.422628i	$-0.861108\pi$
$823$		−	24.2487i		−	0.845257i	0.906303	−	0.422628i	$-0.138892\pi$
$824$		0			0
$825$			6.92820i			0.241209i
$826$		0			0
$827$		−	14.6969i		−	0.511063i	−0.966801	−	0.255531i	$-0.917750\pi$
$827$		−	14.6969i		−	0.511063i	0.966801	−	0.255531i	$-0.0822504\pi$
$828$		0			0
$829$		−	7.34847i		−	0.255223i	−0.991824	−	0.127611i	$-0.959269\pi$
$829$		−	7.34847i		−	0.255223i	0.991824	−	0.127611i	$-0.0407310\pi$
$830$		0			0
$831$	24.0000			0.832551
$832$		0			0
$833$	48.0000	+	6.92820i	1.66310	+	0.240048i
$834$		0			0
$835$		0			0
$836$		0			0
$837$	22.6274			0.782118
$838$		0			0
$839$	48.0000			1.65714			0.828572	−	0.559883i	$-0.189154\pi$
$839$	48.0000			1.65714			0.828572	+	0.559883i	$0.189154\pi$
$840$		0			0
$841$	3.00000			0.103448
$842$		0			0
$843$	−25.4558			−0.876746
$844$		0			0
$845$		−	17.1464i		−	0.589855i
$846$		0			0
$847$	−26.0000	+	22.5167i	−0.893371	+	0.773682i
$848$		0			0
$849$	6.00000			0.205919
$850$		0			0
$851$		0			0
$852$		0			0
$853$		−	26.9444i		−	0.922558i	−0.887255	−	0.461279i	$-0.847391\pi$
$853$		−	26.9444i		−	0.922558i	0.887255	−	0.461279i	$-0.152609\pi$
$854$		0			0
$855$		−	10.3923i		−	0.355409i
$856$		0			0
$857$		−	13.8564i		−	0.473326i	−0.971592	−	0.236663i	$-0.923946\pi$
$857$		−	13.8564i		−	0.473326i	0.971592	−	0.236663i	$-0.0760537\pi$
$858$		0			0
$859$	−4.24264			−0.144757			−0.0723785	−	0.997377i	$-0.523059\pi$
$859$	−4.24264			−0.144757			−0.0723785	+	0.997377i	$0.523059\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			17.3205i			0.589597i	0.955559	+	0.294798i	$0.0952525\pi$
$863$			17.3205i			0.589597i	−0.955559	+	0.294798i	$0.904747\pi$
$864$		0			0
$865$	42.0000			1.42804
$866$		0			0
$867$	43.8406			1.48891
$868$		0			0
$869$	−50.9117			−1.72706
$870$		0			0
$871$	12.0000			0.406604
$872$		0			0
$873$		−	6.92820i		−	0.234484i
$874$		0			0
$875$	−16.9706	−	19.5959i	−0.573710	−	0.662463i
$876$		0			0
$877$	33.9411			1.14611			0.573055	−	0.819517i	$-0.305758\pi$
$877$	33.9411			1.14611			0.573055	+	0.819517i	$0.305758\pi$
$878$		0			0
$879$			10.3923i			0.350524i
$880$		0			0
$881$			13.8564i			0.466834i	0.972377	+	0.233417i	$0.0749907\pi$
$881$			13.8564i			0.466834i	−0.972377	+	0.233417i	$0.925009\pi$
$882$		0			0
$883$		−	24.4949i		−	0.824319i	−0.911112	−	0.412159i	$-0.864775\pi$
$883$		−	24.4949i		−	0.824319i	0.911112	−	0.412159i	$-0.135225\pi$
$884$		0			0
$885$			34.2929i			1.15274i
$886$		0			0
$887$	24.0000			0.805841			0.402921	−	0.915235i	$-0.367995\pi$
$887$	24.0000			0.805841			0.402921	+	0.915235i	$0.367995\pi$
$888$		0			0
$889$	−12.0000	−	13.8564i	−0.402467	−	0.464729i
$890$		0			0
$891$		−	24.4949i		−	0.820610i
$892$		0			0
$893$	−50.9117			−1.70369
$894$		0			0
$895$	−12.0000			−0.401116
$896$		0			0
$897$	−24.0000			−0.801337
$898$		0			0
$899$	−22.6274			−0.754667
$900$		0			0
$901$		−	39.1918i		−	1.30567i
$902$		0			0
$903$	−12.0000	−	13.8564i	−0.399335	−	0.461112i
$904$		0			0
$905$	−6.00000			−0.199447
$906$		0			0
$907$		−	44.0908i		−	1.46401i	−0.681298	−	0.732006i	$-0.738584\pi$
$907$		−	44.0908i		−	1.46401i	0.681298	−	0.732006i	$-0.261416\pi$
$908$		0			0
$909$		−	12.2474i		−	0.406222i
$910$		0			0
$911$			20.7846i			0.688625i	0.938855	+	0.344312i	$0.111888\pi$
$911$			20.7846i			0.688625i	−0.938855	+	0.344312i	$0.888112\pi$
$912$		0			0
$913$		−	6.92820i		−	0.229290i
$914$		0			0
$915$	25.4558			0.841544
$916$		0			0
$917$	−2.82843	+	2.44949i	−0.0934029	+	0.0808893i
$918$		0			0
$919$			3.46410i			0.114270i	0.998366	+	0.0571351i	$0.0181966\pi$
$919$			3.46410i			0.114270i	−0.998366	+	0.0571351i	$0.981803\pi$
$920$		0			0
$921$	−30.0000			−0.988534
$922$		0			0
$923$	8.48528			0.279296
$924$		0			0
$925$		0			0
$926$		0			0
$927$	8.00000			0.262754
$928$		0			0
$929$			20.7846i			0.681921i	0.940078	+	0.340960i	$0.110752\pi$
$929$			20.7846i			0.681921i	−0.940078	+	0.340960i	$0.889248\pi$
$930$		0			0
$931$	−4.24264	+	29.3939i	−0.139047	+	0.963345i
$932$		0			0
$933$	−16.9706			−0.555591
$934$		0			0
$935$			83.1384i			2.71892i
$936$		0			0
$937$		−	34.6410i		−	1.13167i	−0.824518	−	0.565836i	$-0.808553\pi$
$937$		−	34.6410i		−	1.13167i	0.824518	−	0.565836i	$-0.191447\pi$
$938$		0			0
$939$			19.5959i			0.639489i
$940$		0			0
$941$			17.1464i			0.558958i	0.960152	+	0.279479i	$0.0901617\pi$
$941$			17.1464i			0.558958i	−0.960152	+	0.279479i	$0.909838\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$	−24.0000	−	27.7128i	−0.780720	−	0.901498i
$946$		0			0
$947$			53.8888i			1.75115i	0.483082	+	0.875575i	$0.339517\pi$
$947$			53.8888i			1.75115i	−0.483082	+	0.875575i	$0.660483\pi$
$948$		0			0
$949$	−16.9706			−0.550888
$950$		0			0
$951$	16.0000			0.518836
$952$		0			0
$953$	6.00000			0.194359			0.0971795	−	0.995267i	$-0.469018\pi$
$953$	6.00000			0.194359			0.0971795	+	0.995267i	$0.469018\pi$
$954$		0			0
$955$	−25.4558			−0.823732
$956$		0			0
$957$			39.1918i			1.26689i
$958$		0			0
$959$	−12.0000	+	10.3923i	−0.387500	+	0.335585i
$960$		0			0
$961$	−15.0000			−0.483871
$962$		0			0
$963$			4.89898i			0.157867i
$964$		0			0
$965$		−	9.79796i		−	0.315407i
$966$		0			0
$967$			6.92820i			0.222796i	0.993776	+	0.111398i	$0.0355328\pi$
$967$			6.92820i			0.222796i	−0.993776	+	0.111398i	$0.964467\pi$
$968$		0			0
$969$			41.5692i			1.33540i
$970$		0			0
$971$	32.5269			1.04384			0.521919	−	0.852995i	$-0.325216\pi$
$971$	32.5269			1.04384			0.521919	+	0.852995i	$0.325216\pi$
$972$		0			0
$973$	−25.4558	+	22.0454i	−0.816077	+	0.706743i
$974$		0			0
$975$		−	3.46410i		−	0.110940i
$976$		0			0
$977$	−54.0000			−1.72761			−0.863807	−	0.503824i	$-0.831926\pi$
$977$	−54.0000			−1.72761			−0.863807	+	0.503824i	$0.831926\pi$
$978$		0			0
$979$	−33.9411			−1.08476
$980$		0			0
$981$	−16.9706			−0.541828
$982$		0			0
$983$	−48.0000			−1.53096			−0.765481	−	0.643458i	$-0.777499\pi$
$983$	−48.0000			−1.53096			−0.765481	+	0.643458i	$0.777499\pi$
$984$		0			0
$985$		−	55.4256i		−	1.76601i
$986$		0			0
$987$	−33.9411	+	29.3939i	−1.08036	+	0.935617i
$988$		0			0
$989$	−33.9411			−1.07927
$990$		0			0
$991$			38.1051i			1.21045i	0.796055	+	0.605224i	$0.206917\pi$
$991$			38.1051i			1.21045i	−0.796055	+	0.605224i	$0.793083\pi$
$992$		0			0
$993$			6.92820i			0.219860i
$994$		0			0
$995$			39.1918i			1.24246i
$996$		0			0
$997$		−	2.44949i		−	0.0775761i	−0.999247	−	0.0387881i	$-0.987650\pi$
$997$		−	2.44949i		−	0.0775761i	0.999247	−	0.0387881i	$-0.0123497\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1792.2.f.g.1791.1		4
4.3	odd	2		1792.2.f.c.1791.3		4
7.6	odd	2		1792.2.f.c.1791.4		4
8.3	odd	2		1792.2.f.c.1791.2		4
8.5	even	2	inner	1792.2.f.g.1791.4		4
16.3	odd	4		896.2.e.d.447.3	yes	4
16.5	even	4		896.2.e.c.447.4	yes	4
16.11	odd	4		896.2.e.d.447.2	yes	4
16.13	even	4		896.2.e.c.447.1	✓	4
28.27	even	2	inner	1792.2.f.g.1791.2		4
56.13	odd	2		1792.2.f.c.1791.1		4
56.27	even	2	inner	1792.2.f.g.1791.3		4
112.13	odd	4		896.2.e.d.447.4	yes	4
112.27	even	4		896.2.e.c.447.3	yes	4
112.69	odd	4		896.2.e.d.447.1	yes	4
112.83	even	4		896.2.e.c.447.2	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
896.2.e.c.447.1	✓	4	16.13	even	4
896.2.e.c.447.2	yes	4	112.83	even	4
896.2.e.c.447.3	yes	4	112.27	even	4
896.2.e.c.447.4	yes	4	16.5	even	4
896.2.e.d.447.1	yes	4	112.69	odd	4
896.2.e.d.447.2	yes	4	16.11	odd	4
896.2.e.d.447.3	yes	4	16.3	odd	4
896.2.e.d.447.4	yes	4	112.13	odd	4
1792.2.f.c.1791.1		4	56.13	odd	2
1792.2.f.c.1791.2		4	8.3	odd	2
1792.2.f.c.1791.3		4	4.3	odd	2
1792.2.f.c.1791.4		4	7.6	odd	2
1792.2.f.g.1791.1		4	1.1	even	1	trivial
1792.2.f.g.1791.2		4	28.27	even	2	inner
1792.2.f.g.1791.3		4	56.27	even	2	inner
1792.2.f.g.1791.4		4	8.5	even	2	inner

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:	\( N \)	\(=\)	\( 1792 = 2^{8} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1792.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.41421 q^{3} -2.44949i q^{5} +(2.00000 - 1.73205i) q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q-1.41421 q^{3} -2.44949i q^{5} +(2.00000 - 1.73205i) q^{7} -1.00000 q^{9} +4.89898i q^{11} -2.44949i q^{13} +3.46410i q^{15} +6.92820i q^{17} -4.24264 q^{19} +(-2.82843 + 2.44949i) q^{21} +6.92820i q^{23} -1.00000 q^{25} +5.65685 q^{27} -5.65685 q^{29} +4.00000 q^{31} -6.92820i q^{33} +(-4.24264 - 4.89898i) q^{35} +3.46410i q^{39} +4.89898i q^{43} +2.44949i q^{45} +12.0000 q^{47} +(1.00000 - 6.92820i) q^{49} -9.79796i q^{51} -5.65685 q^{53} +12.0000 q^{55} +6.00000 q^{57} +9.89949 q^{59} -7.34847i q^{61} +(-2.00000 + 1.73205i) q^{63} -6.00000 q^{65} +4.89898i q^{67} -9.79796i q^{69} +3.46410i q^{71} -6.92820i q^{73} +1.41421 q^{75} +(8.48528 + 9.79796i) q^{77} +10.3923i q^{79} -5.00000 q^{81} -1.41421 q^{83} +16.9706 q^{85} +8.00000 q^{87} +6.92820i q^{89} +(-4.24264 - 4.89898i) q^{91} -5.65685 q^{93} +10.3923i q^{95} +6.92820i q^{97} -4.89898i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 8 q^{7} - 4 q^{9}+O(q^{10}) \) 4 * q + 8 * q^7 - 4 * q^9	\( 4 q + 8 q^{7} - 4 q^{9} - 4 q^{25} + 16 q^{31} + 48 q^{47} + 4 q^{49} + 48 q^{55} + 24 q^{57} - 8 q^{63} - 24 q^{65} - 20 q^{81} + 32 q^{87}+O(q^{100}) \) 4 * q + 8 * q^7 - 4 * q^9 - 4 * q^25 + 16 * q^31 + 48 * q^47 + 4 * q^49 + 48 * q^55 + 24 * q^57 - 8 * q^63 - 24 * q^65 - 20 * q^81 + 32 * q^87

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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