LMFDB - Embedded newform 7056.2.h.g.4607.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7056,2,Mod(4607,7056)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7056, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 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("7056.4607");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7056 = 2^{4} \cdot 3^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7056.h (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:	$56.3424436662$
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:	$\Z[a_1, \ldots, a_{29}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			4607.4
Root			$-1.22474 - 0.707107i$ of defining polynomial
Character	$\chi$	$=$	7056.4607
Dual form			7056.2.h.g.4607.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.82843i q^{5} +O(q^{10})$	$q+2.82843i q^{5} +2.44949 q^{11} +2.82843i q^{17} -6.92820i q^{19} -2.44949 q^{23} -3.00000 q^{25} +4.24264i q^{29} -6.92820i q^{31} +4.00000 q^{37} -2.82843i q^{41} +6.92820i q^{43} +9.79796 q^{47} +4.24264i q^{53} +6.92820i q^{55} +9.79796 q^{59} +12.0000 q^{61} +10.3923i q^{67} +12.2474 q^{71} +12.0000 q^{73} -3.46410i q^{79} -9.79796 q^{83} -8.00000 q^{85} -2.82843i q^{89} +19.5959 q^{95} -12.0000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q+O(q^{10}) $ 4 * q	$ 4 q - 12 q^{25} + 16 q^{37} + 48 q^{61} + 48 q^{73} - 32 q^{85} - 48 q^{97}+O(q^{100}) $ 4 * q - 12 * q^25 + 16 * q^37 + 48 * q^61 + 48 * q^73 - 32 * q^85 - 48 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/7056\mathbb{Z}\right)^\times$.

$n$	$785$	$1765$	$4609$	$6175$
$\chi(n)$	$-1$	$1$	$1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	2.82843i	1.26491i	0.774597	+	0.632456i	$0.217953\pi$
$5$	2.82843i	1.26491i	−0.774597	+	0.632456i	$0.782047\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	0	0
$10$	0	0
$11$	2.44949	0.738549	0.369274	−	0.929320i	$-0.379606\pi$
$11$	2.44949	0.738549	0.369274	+	0.929320i	$0.379606\pi$
$12$	0	0
$13$	0	0		−	1.00000i	$-0.5\pi$
$13$	0	0			1.00000i	$0.5\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	2.82843i	0.685994i	0.939336	+	0.342997i	$0.111442\pi$
$17$	2.82843i	0.685994i	−0.939336	+	0.342997i	$0.888558\pi$
$18$	0	0
$19$	− 6.92820i	− 1.58944i	−0.606977	−	0.794719i	$-0.707618\pi$
$19$	− 6.92820i	− 1.58944i	0.606977	−	0.794719i	$-0.292382\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−2.44949	−0.510754	−0.255377	−	0.966842i	$-0.582200\pi$
$23$	−2.44949	−0.510754	−0.255377	+	0.966842i	$0.582200\pi$
$24$	0	0
$25$	−3.00000	−0.600000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	4.24264i	0.787839i	0.919145	+	0.393919i	$0.128881\pi$
$29$	4.24264i	0.787839i	−0.919145	+	0.393919i	$0.871119\pi$
$30$	0	0
$31$	− 6.92820i	− 1.24434i	−0.782881	−	0.622171i	$-0.786251\pi$
$31$	− 6.92820i	− 1.24434i	0.782881	−	0.622171i	$-0.213749\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	4.00000	0.657596	0.328798	−	0.944400i	$-0.393356\pi$
$37$	4.00000	0.657596	0.328798	+	0.944400i	$0.393356\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 2.82843i	− 0.441726i	−0.975305	−	0.220863i	$-0.929113\pi$
$41$	− 2.82843i	− 0.441726i	0.975305	−	0.220863i	$-0.0708874\pi$
$42$	0	0
$43$	6.92820i	1.05654i	0.849076	+	0.528271i	$0.177159\pi$
$43$	6.92820i	1.05654i	−0.849076	+	0.528271i	$0.822841\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	9.79796	1.42918	0.714590	−	0.699544i	$-0.246613\pi$
$47$	9.79796	1.42918	0.714590	+	0.699544i	$0.246613\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	0	0
$52$	0	0
$53$	4.24264i	0.582772i	0.956606	+	0.291386i	$0.0941163\pi$
$53$	4.24264i	0.582772i	−0.956606	+	0.291386i	$0.905884\pi$
$54$	0	0
$55$	6.92820i	0.934199i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	9.79796	1.27559	0.637793	−	0.770208i	$-0.279848\pi$
$59$	9.79796	1.27559	0.637793	+	0.770208i	$0.279848\pi$
$60$	0	0
$61$	12.0000	1.53644	0.768221	−	0.640184i	$-0.221142\pi$
$61$	12.0000	1.53644	0.768221	+	0.640184i	$0.221142\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0	0
$66$	0	0
$67$	10.3923i	1.26962i	0.772667	+	0.634811i	$0.218922\pi$
$67$	10.3923i	1.26962i	−0.772667	+	0.634811i	$0.781078\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	12.2474	1.45350	0.726752	−	0.686900i	$-0.241029\pi$
$71$	12.2474	1.45350	0.726752	+	0.686900i	$0.241029\pi$
$72$	0	0
$73$	12.0000	1.40449	0.702247	−	0.711934i	$-0.252180\pi$
$73$	12.0000	1.40449	0.702247	+	0.711934i	$0.252180\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	− 3.46410i	− 0.389742i	−0.980829	−	0.194871i	$-0.937571\pi$
$79$	− 3.46410i	− 0.389742i	0.980829	−	0.194871i	$-0.0624288\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−9.79796	−1.07547	−0.537733	−	0.843115i	$-0.680719\pi$
$83$	−9.79796	−1.07547	−0.537733	+	0.843115i	$0.680719\pi$
$84$	0	0
$85$	−8.00000	−0.867722
$86$	0	0
$87$	0	0
$88$	0	0
$89$	− 2.82843i	− 0.299813i	−0.988700	−	0.149906i	$-0.952103\pi$
$89$	− 2.82843i	− 0.299813i	0.988700	−	0.149906i	$-0.0478972\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	19.5959	2.01050
$96$	0	0
$97$	−12.0000	−1.21842	−0.609208	−	0.793011i	$-0.708512\pi$
$97$	−12.0000	−1.21842	−0.609208	+	0.793011i	$0.708512\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	− 2.82843i	− 0.281439i	−0.990050	−	0.140720i	$-0.955058\pi$
$101$	− 2.82843i	− 0.281439i	0.990050	−	0.140720i	$-0.0449416\pi$
$102$	0	0
$103$	− 6.92820i	− 0.682656i	−0.939944	−	0.341328i	$-0.889123\pi$
$103$	− 6.92820i	− 0.682656i	0.939944	−	0.341328i	$-0.110877\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−7.34847	−0.710403	−0.355202	−	0.934790i	$-0.615588\pi$
$107$	−7.34847	−0.710403	−0.355202	+	0.934790i	$0.615588\pi$
$108$	0	0
$109$	−4.00000	−0.383131	−0.191565	−	0.981480i	$-0.561356\pi$
$109$	−4.00000	−0.383131	−0.191565	+	0.981480i	$0.561356\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	4.24264i	0.399114i	0.979886	+	0.199557i	$0.0639503\pi$
$113$	4.24264i	0.399114i	−0.979886	+	0.199557i	$0.936050\pi$
$114$	0	0
$115$	− 6.92820i	− 0.646058i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−5.00000	−0.454545
$122$	0	0
$123$	0	0
$124$	0	0
$125$	5.65685i	0.505964i
$126$	0	0
$127$	3.46410i	0.307389i	0.988118	+	0.153695i	$0.0491172\pi$
$127$	3.46410i	0.307389i	−0.988118	+	0.153695i	$0.950883\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	0	0		−	1.00000i	$-0.5\pi$
$131$	0	0			1.00000i	$0.5\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	0	0
$137$	21.2132i	1.81237i	0.422885	+	0.906183i	$0.361017\pi$
$137$	21.2132i	1.81237i	−0.422885	+	0.906183i	$0.638983\pi$
$138$	0	0
$139$	0	0	1.00000			$0$
$139$	0	0	−1.00000			$\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−12.0000	−0.996546
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 21.2132i	− 1.73785i	−0.494941	−	0.868927i	$-0.664810\pi$
$149$	− 21.2132i	− 1.73785i	0.494941	−	0.868927i	$-0.335190\pi$
$150$	0	0
$151$	13.8564i	1.12762i	0.825905	+	0.563809i	$0.190665\pi$
$151$	13.8564i	1.12762i	−0.825905	+	0.563809i	$0.809335\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	19.5959	1.57398
$156$	0	0
$157$	−12.0000	−0.957704	−0.478852	−	0.877896i	$-0.658947\pi$
$157$	−12.0000	−0.957704	−0.478852	+	0.877896i	$0.658947\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	− 17.3205i	− 1.35665i	−0.734763	−	0.678323i	$-0.762707\pi$
$163$	− 17.3205i	− 1.35665i	0.734763	−	0.678323i	$-0.237293\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	9.79796	0.758189	0.379094	−	0.925358i	$-0.376236\pi$
$167$	9.79796	0.758189	0.379094	+	0.925358i	$0.376236\pi$
$168$	0	0
$169$	−13.0000	−1.00000
$170$	0	0
$171$	0	0
$172$	0	0
$173$	14.1421i	1.07521i	0.843198	+	0.537603i	$0.180670\pi$
$173$	14.1421i	1.07521i	−0.843198	+	0.537603i	$0.819330\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	22.0454	1.64775	0.823876	−	0.566771i	$-0.191807\pi$
$179$	22.0454	1.64775	0.823876	+	0.566771i	$0.191807\pi$
$180$	0	0
$181$	24.0000	1.78391	0.891953	−	0.452128i	$-0.149335\pi$
$181$	24.0000	1.78391	0.891953	+	0.452128i	$0.149335\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	11.3137i	0.831800i
$186$	0	0
$187$	6.92820i	0.506640i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−12.2474	−0.886194	−0.443097	−	0.896474i	$-0.646120\pi$
$191$	−12.2474	−0.886194	−0.443097	+	0.896474i	$0.646120\pi$
$192$	0	0
$193$	−16.0000	−1.15171	−0.575853	−	0.817554i	$-0.695330\pi$
$193$	−16.0000	−1.15171	−0.575853	+	0.817554i	$0.695330\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	12.7279i	0.906827i	0.891300	+	0.453413i	$0.149794\pi$
$197$	12.7279i	0.906827i	−0.891300	+	0.453413i	$0.850206\pi$
$198$	0	0
$199$	13.8564i	0.982255i	0.871088	+	0.491127i	$0.163415\pi$
$199$	13.8564i	0.982255i	−0.871088	+	0.491127i	$0.836585\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	0	0
$204$	0	0
$205$	8.00000	0.558744
$206$	0	0
$207$	0	0
$208$	0	0
$209$	− 16.9706i	− 1.17388i
$210$	0	0
$211$	20.7846i	1.43087i	0.698679	+	0.715436i	$0.253772\pi$
$211$	20.7846i	1.43087i	−0.698679	+	0.715436i	$0.746228\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−19.5959	−1.33643
$216$	0	0
$217$	0	0
$218$	0	0
$219$	0	0
$220$	0	0
$221$	0	0
$222$	0	0
$223$	− 13.8564i	− 0.927894i	−0.885863	−	0.463947i	$-0.846433\pi$
$223$	− 13.8564i	− 0.927894i	0.885863	−	0.463947i	$-0.153567\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−29.3939	−1.95094	−0.975470	−	0.220132i	$-0.929351\pi$
$227$	−29.3939	−1.95094	−0.975470	+	0.220132i	$0.929351\pi$
$228$	0	0
$229$	0	0		−	1.00000i	$-0.5\pi$
$229$	0	0			1.00000i	$0.5\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	12.7279i	0.833834i	0.908945	+	0.416917i	$0.136889\pi$
$233$	12.7279i	0.833834i	−0.908945	+	0.416917i	$0.863111\pi$
$234$	0	0
$235$	27.7128i	1.80778i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	22.0454	1.42600	0.712999	−	0.701165i	$-0.247336\pi$
$239$	22.0454	1.42600	0.712999	+	0.701165i	$0.247336\pi$
$240$	0	0
$241$	−12.0000	−0.772988	−0.386494	−	0.922292i	$-0.626314\pi$
$241$	−12.0000	−0.772988	−0.386494	+	0.922292i	$0.626314\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	0	0
$248$	0	0
$249$	0	0
$250$	0	0
$251$	19.5959	1.23688	0.618442	−	0.785831i	$-0.287764\pi$
$251$	19.5959	1.23688	0.618442	+	0.785831i	$0.287764\pi$
$252$	0	0
$253$	−6.00000	−0.377217
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 19.7990i	− 1.23503i	−0.786560	−	0.617514i	$-0.788140\pi$
$257$	− 19.7990i	− 1.23503i	0.786560	−	0.617514i	$-0.211860\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	0	0
$262$	0	0
$263$	26.9444	1.66146	0.830731	−	0.556674i	$-0.187923\pi$
$263$	26.9444	1.66146	0.830731	+	0.556674i	$0.187923\pi$
$264$	0	0
$265$	−12.0000	−0.737154
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 14.1421i	− 0.862261i	−0.902290	−	0.431131i	$-0.858115\pi$
$269$	− 14.1421i	− 0.862261i	0.902290	−	0.431131i	$-0.141885\pi$
$270$	0	0
$271$	20.7846i	1.26258i	0.775549	+	0.631288i	$0.217473\pi$
$271$	20.7846i	1.26258i	−0.775549	+	0.631288i	$0.782527\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−7.34847	−0.443129
$276$	0	0
$277$	−2.00000	−0.120168	−0.0600842	−	0.998193i	$-0.519137\pi$
$277$	−2.00000	−0.120168	−0.0600842	+	0.998193i	$0.519137\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	12.7279i	0.759284i	0.925133	+	0.379642i	$0.123953\pi$
$281$	12.7279i	0.759284i	−0.925133	+	0.379642i	$0.876047\pi$
$282$	0	0
$283$	20.7846i	1.23552i	0.786368	+	0.617758i	$0.211959\pi$
$283$	20.7846i	1.23552i	−0.786368	+	0.617758i	$0.788041\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	9.00000	0.529412
$290$	0	0
$291$	0	0
$292$	0	0
$293$	31.1127i	1.81762i	0.417207	+	0.908812i	$0.363009\pi$
$293$	31.1127i	1.81762i	−0.417207	+	0.908812i	$0.636991\pi$
$294$	0	0
$295$	27.7128i	1.61350i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	0	0
$300$	0	0
$301$	0	0
$302$	0	0
$303$	0	0
$304$	0	0
$305$	33.9411i	1.94346i
$306$	0	0
$307$	− 6.92820i	− 0.395413i	−0.980261	−	0.197707i	$-0.936651\pi$
$307$	− 6.92820i	− 0.395413i	0.980261	−	0.197707i	$-0.0633494\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	9.79796	0.555591	0.277796	−	0.960640i	$-0.410396\pi$
$311$	9.79796	0.555591	0.277796	+	0.960640i	$0.410396\pi$
$312$	0	0
$313$	−24.0000	−1.35656	−0.678280	−	0.734803i	$-0.737274\pi$
$313$	−24.0000	−1.35656	−0.678280	+	0.734803i	$0.737274\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 12.7279i	− 0.714871i	−0.933938	−	0.357436i	$-0.883651\pi$
$317$	− 12.7279i	− 0.714871i	0.933938	−	0.357436i	$-0.116349\pi$
$318$	0	0
$319$	10.3923i	0.581857i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	19.5959	1.09035
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0	0
$328$	0	0
$329$	0	0
$330$	0	0
$331$	6.92820i	0.380808i	0.981706	+	0.190404i	$0.0609799\pi$
$331$	6.92820i	0.380808i	−0.981706	+	0.190404i	$0.939020\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−29.3939	−1.60596
$336$	0	0
$337$	8.00000	0.435788	0.217894	−	0.975972i	$-0.430081\pi$
$337$	8.00000	0.435788	0.217894	+	0.975972i	$0.430081\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	− 16.9706i	− 0.919007i
$342$	0	0
$343$	0	0
$344$	0	0
$345$	0	0
$346$	0	0
$347$	31.8434	1.70944	0.854721	−	0.519088i	$-0.173728\pi$
$347$	31.8434	1.70944	0.854721	+	0.519088i	$0.173728\pi$
$348$	0	0
$349$	12.0000	0.642345	0.321173	−	0.947021i	$-0.395923\pi$
$349$	12.0000	0.642345	0.321173	+	0.947021i	$0.395923\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 31.1127i	− 1.65596i	−0.560756	−	0.827981i	$-0.689490\pi$
$353$	− 31.1127i	− 1.65596i	0.560756	−	0.827981i	$-0.310510\pi$
$354$	0	0
$355$	34.6410i	1.83855i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−22.0454	−1.16351	−0.581756	−	0.813363i	$-0.697634\pi$
$359$	−22.0454	−1.16351	−0.581756	+	0.813363i	$0.697634\pi$
$360$	0	0
$361$	−29.0000	−1.52632
$362$	0	0
$363$	0	0
$364$	0	0
$365$	33.9411i	1.77656i
$366$	0	0
$367$	13.8564i	0.723299i	0.932314	+	0.361649i	$0.117786\pi$
$367$	13.8564i	0.723299i	−0.932314	+	0.361649i	$0.882214\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	26.0000	1.34623	0.673114	−	0.739538i	$-0.264956\pi$
$373$	26.0000	1.34623	0.673114	+	0.739538i	$0.264956\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	0	0
$378$	0	0
$379$	− 6.92820i	− 0.355878i	−0.984042	−	0.177939i	$-0.943057\pi$
$379$	− 6.92820i	− 0.355878i	0.984042	−	0.177939i	$-0.0569430\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−19.5959	−1.00130	−0.500652	−	0.865648i	$-0.666906\pi$
$383$	−19.5959	−1.00130	−0.500652	+	0.865648i	$0.666906\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	− 4.24264i	− 0.215110i	−0.994199	−	0.107555i	$-0.965698\pi$
$389$	− 4.24264i	− 0.215110i	0.994199	−	0.107555i	$-0.0343022\pi$
$390$	0	0
$391$	− 6.92820i	− 0.350374i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	9.79796	0.492989
$396$	0	0
$397$	12.0000	0.602263	0.301131	−	0.953583i	$-0.402636\pi$
$397$	12.0000	0.602263	0.301131	+	0.953583i	$0.402636\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	4.24264i	0.211867i	0.994373	+	0.105934i	$0.0337831\pi$
$401$	4.24264i	0.211867i	−0.994373	+	0.105934i	$0.966217\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	0	0
$406$	0	0
$407$	9.79796	0.485667
$408$	0	0
$409$	12.0000	0.593362	0.296681	−	0.954977i	$-0.404120\pi$
$409$	12.0000	0.593362	0.296681	+	0.954977i	$0.404120\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	0	0
$414$	0	0
$415$	− 27.7128i	− 1.36037i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	19.5959	0.957323	0.478662	−	0.877999i	$-0.341122\pi$
$419$	19.5959	0.957323	0.478662	+	0.877999i	$0.341122\pi$
$420$	0	0
$421$	−14.0000	−0.682318	−0.341159	−	0.940006i	$-0.610819\pi$
$421$	−14.0000	−0.682318	−0.341159	+	0.940006i	$0.610819\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	− 8.48528i	− 0.411597i
$426$	0	0
$427$	0	0
$428$	0	0
$429$	0	0
$430$	0	0
$431$	17.1464	0.825914	0.412957	−	0.910750i	$-0.364496\pi$
$431$	17.1464	0.825914	0.412957	+	0.910750i	$0.364496\pi$
$432$	0	0
$433$	−24.0000	−1.15337	−0.576683	−	0.816968i	$-0.695653\pi$
$433$	−24.0000	−1.15337	−0.576683	+	0.816968i	$0.695653\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	16.9706i	0.811812i
$438$	0	0
$439$	0	0	1.00000			$0$
$439$	0	0	−1.00000			$\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−17.1464	−0.814651	−0.407326	−	0.913283i	$-0.633539\pi$
$443$	−17.1464	−0.814651	−0.407326	+	0.913283i	$0.633539\pi$
$444$	0	0
$445$	8.00000	0.379236
$446$	0	0
$447$	0	0
$448$	0	0
$449$	− 12.7279i	− 0.600668i	−0.953834	−	0.300334i	$-0.902902\pi$
$449$	− 12.7279i	− 0.600668i	0.953834	−	0.300334i	$-0.0970981\pi$
$450$	0	0
$451$	− 6.92820i	− 0.326236i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−32.0000	−1.49690	−0.748448	−	0.663193i	$-0.769201\pi$
$457$	−32.0000	−1.49690	−0.748448	+	0.663193i	$0.769201\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	− 19.7990i	− 0.922131i	−0.887366	−	0.461065i	$-0.847467\pi$
$461$	− 19.7990i	− 0.922131i	0.887366	−	0.461065i	$-0.152533\pi$
$462$	0	0
$463$	24.2487i	1.12693i	0.826139	+	0.563467i	$0.190533\pi$
$463$	24.2487i	1.12693i	−0.826139	+	0.563467i	$0.809467\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	0	0		−	1.00000i	$-0.5\pi$
$467$	0	0			1.00000i	$0.5\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0	0
$472$	0	0
$473$	16.9706i	0.780307i
$474$	0	0
$475$	20.7846i	0.953663i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−39.1918	−1.79072	−0.895360	−	0.445342i	$-0.853082\pi$
$479$	−39.1918	−1.79072	−0.895360	+	0.445342i	$0.853082\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	0	0
$484$	0	0
$485$	− 33.9411i	− 1.54119i
$486$	0	0
$487$	41.5692i	1.88368i	0.336060	+	0.941841i	$0.390905\pi$
$487$	41.5692i	1.88368i	−0.336060	+	0.941841i	$0.609095\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	17.1464	0.773807	0.386904	−	0.922120i	$-0.373545\pi$
$491$	17.1464	0.773807	0.386904	+	0.922120i	$0.373545\pi$
$492$	0	0
$493$	−12.0000	−0.540453
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	− 6.92820i	− 0.310149i	−0.987903	−	0.155074i	$-0.950438\pi$
$499$	− 6.92820i	− 0.310149i	0.987903	−	0.155074i	$-0.0495618\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	9.79796	0.436869	0.218435	−	0.975852i	$-0.429905\pi$
$503$	9.79796	0.436869	0.218435	+	0.975852i	$0.429905\pi$
$504$	0	0
$505$	8.00000	0.355995
$506$	0	0
$507$	0	0
$508$	0	0
$509$	19.7990i	0.877575i	0.898591	+	0.438787i	$0.144592\pi$
$509$	19.7990i	0.877575i	−0.898591	+	0.438787i	$0.855408\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	0	0
$514$	0	0
$515$	19.5959	0.863499
$516$	0	0
$517$	24.0000	1.05552
$518$	0	0
$519$	0	0
$520$	0	0
$521$	36.7696i	1.61090i	0.592661	+	0.805452i	$0.298077\pi$
$521$	36.7696i	1.61090i	−0.592661	+	0.805452i	$0.701923\pi$
$522$	0	0
$523$	− 27.7128i	− 1.21180i	−0.795542	−	0.605898i	$-0.792814\pi$
$523$	− 27.7128i	− 1.21180i	0.795542	−	0.605898i	$-0.207186\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	19.5959	0.853612
$528$	0	0
$529$	−17.0000	−0.739130
$530$	0	0
$531$	0	0
$532$	0	0
$533$	0	0
$534$	0	0
$535$	− 20.7846i	− 0.898597i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	0	0
$540$	0	0
$541$	22.0000	0.945854	0.472927	−	0.881102i	$-0.343197\pi$
$541$	22.0000	0.945854	0.472927	+	0.881102i	$0.343197\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	− 11.3137i	− 0.484626i
$546$	0	0
$547$	17.3205i	0.740571i	0.928918	+	0.370286i	$0.120740\pi$
$547$	17.3205i	0.740571i	−0.928918	+	0.370286i	$0.879260\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	29.3939	1.25222
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 4.24264i	− 0.179766i	−0.995952	−	0.0898832i	$-0.971351\pi$
$557$	− 4.24264i	− 0.179766i	0.995952	−	0.0898832i	$-0.0286494\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0	0
$562$	0	0
$563$	29.3939	1.23880	0.619402	−	0.785074i	$-0.287375\pi$
$563$	29.3939	1.23880	0.619402	+	0.785074i	$0.287375\pi$
$564$	0	0
$565$	−12.0000	−0.504844
$566$	0	0
$567$	0	0
$568$	0	0
$569$	29.6985i	1.24503i	0.782610	+	0.622513i	$0.213888\pi$
$569$	29.6985i	1.24503i	−0.782610	+	0.622513i	$0.786112\pi$
$570$	0	0
$571$	− 3.46410i	− 0.144968i	−0.997370	−	0.0724841i	$-0.976907\pi$
$571$	− 3.46410i	− 0.144968i	0.997370	−	0.0724841i	$-0.0230926\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	7.34847	0.306452
$576$	0	0
$577$	−24.0000	−0.999133	−0.499567	−	0.866276i	$-0.666507\pi$
$577$	−24.0000	−0.999133	−0.499567	+	0.866276i	$0.666507\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	10.3923i	0.430405i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−29.3939	−1.21322	−0.606608	−	0.795001i	$-0.707470\pi$
$587$	−29.3939	−1.21322	−0.606608	+	0.795001i	$0.707470\pi$
$588$	0	0
$589$	−48.0000	−1.97781
$590$	0	0
$591$	0	0
$592$	0	0
$593$	36.7696i	1.50994i	0.655757	+	0.754972i	$0.272350\pi$
$593$	36.7696i	1.50994i	−0.655757	+	0.754972i	$0.727650\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−12.2474	−0.500417	−0.250209	−	0.968192i	$-0.580499\pi$
$599$	−12.2474	−0.500417	−0.250209	+	0.968192i	$0.580499\pi$
$600$	0	0
$601$	24.0000	0.978980	0.489490	−	0.872009i	$-0.337183\pi$
$601$	24.0000	0.978980	0.489490	+	0.872009i	$0.337183\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	− 14.1421i	− 0.574960i
$606$	0	0
$607$	− 41.5692i	− 1.68724i	−0.536939	−	0.843621i	$-0.680419\pi$
$607$	− 41.5692i	− 1.68724i	0.536939	−	0.843621i	$-0.319581\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	0	0
$612$	0	0
$613$	10.0000	0.403896	0.201948	−	0.979396i	$-0.435273\pi$
$613$	10.0000	0.403896	0.201948	+	0.979396i	$0.435273\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 4.24264i	− 0.170802i	−0.996347	−	0.0854011i	$-0.972783\pi$
$617$	− 4.24264i	− 0.170802i	0.996347	−	0.0854011i	$-0.0272172\pi$
$618$	0	0
$619$	− 13.8564i	− 0.556936i	−0.960446	−	0.278468i	$-0.910173\pi$
$619$	− 13.8564i	− 0.556936i	0.960446	−	0.278468i	$-0.0898266\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−31.0000	−1.24000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	11.3137i	0.451107i
$630$	0	0
$631$	− 38.1051i	− 1.51694i	−0.651707	−	0.758470i	$-0.725947\pi$
$631$	− 38.1051i	− 1.51694i	0.651707	−	0.758470i	$-0.274053\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−9.79796	−0.388820
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	0	0
$641$	29.6985i	1.17302i	0.809942	+	0.586510i	$0.199498\pi$
$641$	29.6985i	1.17302i	−0.809942	+	0.586510i	$0.800502\pi$
$642$	0	0
$643$	− 6.92820i	− 0.273222i	−0.990625	−	0.136611i	$-0.956379\pi$
$643$	− 6.92820i	− 0.273222i	0.990625	−	0.136611i	$-0.0436210\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	29.3939	1.15559	0.577796	−	0.816181i	$-0.303913\pi$
$647$	29.3939	1.15559	0.577796	+	0.816181i	$0.303913\pi$
$648$	0	0
$649$	24.0000	0.942082
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 29.6985i	− 1.16219i	−0.813835	−	0.581096i	$-0.802624\pi$
$653$	− 29.6985i	− 1.16219i	0.813835	−	0.581096i	$-0.197376\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0	0
$658$	0	0
$659$	17.1464	0.667930	0.333965	−	0.942585i	$-0.391613\pi$
$659$	17.1464	0.667930	0.333965	+	0.942585i	$0.391613\pi$
$660$	0	0
$661$	36.0000	1.40024	0.700119	−	0.714026i	$-0.253130\pi$
$661$	36.0000	1.40024	0.700119	+	0.714026i	$0.253130\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 10.3923i	− 0.402392i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	29.3939	1.13474
$672$	0	0
$673$	−8.00000	−0.308377	−0.154189	−	0.988041i	$-0.549276\pi$
$673$	−8.00000	−0.308377	−0.154189	+	0.988041i	$0.549276\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	14.1421i	0.543526i	0.962364	+	0.271763i	$0.0876068\pi$
$677$	14.1421i	0.543526i	−0.962364	+	0.271763i	$0.912393\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−2.44949	−0.0937271	−0.0468636	−	0.998901i	$-0.514923\pi$
$683$	−2.44949	−0.0937271	−0.0468636	+	0.998901i	$0.514923\pi$
$684$	0	0
$685$	−60.0000	−2.29248
$686$	0	0
$687$	0	0
$688$	0	0
$689$	0	0
$690$	0	0
$691$	− 41.5692i	− 1.58137i	−0.612225	−	0.790684i	$-0.709725\pi$
$691$	− 41.5692i	− 1.58137i	0.612225	−	0.790684i	$-0.290275\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	8.00000	0.303022
$698$	0	0
$699$	0	0
$700$	0	0
$701$	21.2132i	0.801212i	0.916250	+	0.400606i	$0.131200\pi$
$701$	21.2132i	0.801212i	−0.916250	+	0.400606i	$0.868800\pi$
$702$	0	0
$703$	− 27.7128i	− 1.04521i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	20.0000	0.751116	0.375558	−	0.926799i	$-0.377451\pi$
$709$	20.0000	0.751116	0.375558	+	0.926799i	$0.377451\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	16.9706i	0.635553i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	39.1918	1.46161	0.730804	−	0.682587i	$-0.239145\pi$
$719$	39.1918	1.46161	0.730804	+	0.682587i	$0.239145\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	0	0
$724$	0	0
$725$	− 12.7279i	− 0.472703i
$726$	0	0
$727$	20.7846i	0.770859i	0.922737	+	0.385429i	$0.125947\pi$
$727$	20.7846i	0.770859i	−0.922737	+	0.385429i	$0.874053\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−19.5959	−0.724781
$732$	0	0
$733$	24.0000	0.886460	0.443230	−	0.896408i	$-0.353832\pi$
$733$	24.0000	0.886460	0.443230	+	0.896408i	$0.353832\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	25.4558i	0.937678i
$738$	0	0
$739$	− 10.3923i	− 0.382287i	−0.981562	−	0.191144i	$-0.938780\pi$
$739$	− 10.3923i	− 0.382287i	0.981562	−	0.191144i	$-0.0612196\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−2.44949	−0.0898631	−0.0449315	−	0.998990i	$-0.514307\pi$
$743$	−2.44949	−0.0898631	−0.0449315	+	0.998990i	$0.514307\pi$
$744$	0	0
$745$	60.0000	2.19823
$746$	0	0
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	0	0	1.00000			$0$
$751$	0	0	−1.00000			$\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−39.1918	−1.42634
$756$	0	0
$757$	44.0000	1.59921	0.799604	−	0.600528i	$-0.205043\pi$
$757$	44.0000	1.59921	0.799604	+	0.600528i	$0.205043\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	− 19.7990i	− 0.717713i	−0.933393	−	0.358856i	$-0.883167\pi$
$761$	− 19.7990i	− 0.717713i	0.933393	−	0.358856i	$-0.116833\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0	0
$769$	0	0		−	1.00000i	$-0.5\pi$
$769$	0	0			1.00000i	$0.5\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	14.1421i	0.508657i	0.967118	+	0.254329i	$0.0818545\pi$
$773$	14.1421i	0.508657i	−0.967118	+	0.254329i	$0.918146\pi$
$774$	0	0
$775$	20.7846i	0.746605i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−19.5959	−0.702097
$780$	0	0
$781$	30.0000	1.07348
$782$	0	0
$783$	0	0
$784$	0	0
$785$	− 33.9411i	− 1.21141i
$786$	0	0
$787$	0	0	1.00000			$0$
$787$	0	0	−1.00000			$\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	0	0
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 19.7990i	− 0.701316i	−0.936504	−	0.350658i	$-0.885958\pi$
$797$	− 19.7990i	− 0.701316i	0.936504	−	0.350658i	$-0.114042\pi$
$798$	0	0
$799$	27.7128i	0.980409i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	29.3939	1.03729
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	− 4.24264i	− 0.149163i	−0.997215	−	0.0745817i	$-0.976238\pi$
$809$	− 4.24264i	− 0.149163i	0.997215	−	0.0745817i	$-0.0237621\pi$
$810$	0	0
$811$	− 13.8564i	− 0.486564i	−0.969956	−	0.243282i	$-0.921776\pi$
$811$	− 13.8564i	− 0.486564i	0.969956	−	0.243282i	$-0.0782241\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	48.9898	1.71604
$816$	0	0
$817$	48.0000	1.67931
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 38.1838i	− 1.33262i	−0.745674	−	0.666311i	$-0.767872\pi$
$821$	− 38.1838i	− 1.33262i	0.745674	−	0.666311i	$-0.232128\pi$
$822$	0	0
$823$	− 17.3205i	− 0.603755i	−0.953347	−	0.301877i	$-0.902387\pi$
$823$	− 17.3205i	− 0.603755i	0.953347	−	0.301877i	$-0.0976134\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−31.8434	−1.10730	−0.553651	−	0.832749i	$-0.686766\pi$
$827$	−31.8434	−1.10730	−0.553651	+	0.832749i	$0.686766\pi$
$828$	0	0
$829$	−24.0000	−0.833554	−0.416777	−	0.909009i	$-0.636840\pi$
$829$	−24.0000	−0.833554	−0.416777	+	0.909009i	$0.636840\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	0	0
$834$	0	0
$835$	27.7128i	0.959041i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−19.5959	−0.676526	−0.338263	−	0.941052i	$-0.609839\pi$
$839$	−19.5959	−0.676526	−0.338263	+	0.941052i	$0.609839\pi$
$840$	0	0
$841$	11.0000	0.379310
$842$	0	0
$843$	0	0
$844$	0	0
$845$	− 36.7696i	− 1.26491i
$846$	0	0
$847$	0	0
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−9.79796	−0.335870
$852$	0	0
$853$	−12.0000	−0.410872	−0.205436	−	0.978671i	$-0.565861\pi$
$853$	−12.0000	−0.410872	−0.205436	+	0.978671i	$0.565861\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 19.7990i	− 0.676321i	−0.941089	−	0.338160i	$-0.890195\pi$
$857$	− 19.7990i	− 0.676321i	0.941089	−	0.338160i	$-0.109805\pi$
$858$	0	0
$859$	− 6.92820i	− 0.236387i	−0.992991	−	0.118194i	$-0.962290\pi$
$859$	− 6.92820i	− 0.236387i	0.992991	−	0.118194i	$-0.0377103\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	7.34847	0.250145	0.125072	−	0.992148i	$-0.460084\pi$
$863$	7.34847	0.250145	0.125072	+	0.992148i	$0.460084\pi$
$864$	0	0
$865$	−40.0000	−1.36004
$866$	0	0
$867$	0	0
$868$	0	0
$869$	− 8.48528i	− 0.287843i
$870$	0	0
$871$	0	0
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−2.00000	−0.0675352	−0.0337676	−	0.999430i	$-0.510751\pi$
$877$	−2.00000	−0.0675352	−0.0337676	+	0.999430i	$0.510751\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	2.82843i	0.0952921i	0.998864	+	0.0476461i	$0.0151720\pi$
$881$	2.82843i	0.0952921i	−0.998864	+	0.0476461i	$0.984828\pi$
$882$	0	0
$883$	− 48.4974i	− 1.63207i	−0.578004	−	0.816034i	$-0.696168\pi$
$883$	− 48.4974i	− 1.63207i	0.578004	−	0.816034i	$-0.303832\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	9.79796	0.328983	0.164492	−	0.986378i	$-0.447402\pi$
$887$	9.79796	0.328983	0.164492	+	0.986378i	$0.447402\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 67.8823i	− 2.27159i
$894$	0	0
$895$	62.3538i	2.08426i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	29.3939	0.980341
$900$	0	0
$901$	−12.0000	−0.399778
$902$	0	0
$903$	0	0
$904$	0	0
$905$	67.8823i	2.25648i
$906$	0	0
$907$	− 6.92820i	− 0.230047i	−0.993363	−	0.115024i	$-0.963306\pi$
$907$	− 6.92820i	− 0.230047i	0.993363	−	0.115024i	$-0.0366944\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−36.7423	−1.21733	−0.608664	−	0.793428i	$-0.708294\pi$
$911$	−36.7423	−1.21733	−0.608664	+	0.793428i	$0.708294\pi$
$912$	0	0
$913$	−24.0000	−0.794284
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	− 27.7128i	− 0.914161i	−0.889425	−	0.457081i	$-0.848895\pi$
$919$	− 27.7128i	− 0.914161i	0.889425	−	0.457081i	$-0.151105\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	0	0
$924$	0	0
$925$	−12.0000	−0.394558
$926$	0	0
$927$	0	0
$928$	0	0
$929$	14.1421i	0.463988i	0.972717	+	0.231994i	$0.0745250\pi$
$929$	14.1421i	0.463988i	−0.972717	+	0.231994i	$0.925475\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−19.5959	−0.640855
$936$	0	0
$937$	48.0000	1.56809	0.784046	−	0.620703i	$-0.213153\pi$
$937$	48.0000	1.56809	0.784046	+	0.620703i	$0.213153\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	2.82843i	0.0922041i	0.998937	+	0.0461020i	$0.0146799\pi$
$941$	2.82843i	0.0922041i	−0.998937	+	0.0461020i	$0.985320\pi$
$942$	0	0
$943$	6.92820i	0.225613i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	41.6413	1.35316	0.676581	−	0.736369i	$-0.263461\pi$
$947$	41.6413	1.35316	0.676581	+	0.736369i	$0.263461\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	0	0
$952$	0	0
$953$	12.7279i	0.412298i	0.978521	+	0.206149i	$0.0660931\pi$
$953$	12.7279i	0.412298i	−0.978521	+	0.206149i	$0.933907\pi$
$954$	0	0
$955$	− 34.6410i	− 1.12096i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−17.0000	−0.548387
$962$	0	0
$963$	0	0
$964$	0	0
$965$	− 45.2548i	− 1.45680i
$966$	0	0
$967$	10.3923i	0.334194i	0.985940	+	0.167097i	$0.0534393\pi$
$967$	10.3923i	0.334194i	−0.985940	+	0.167097i	$0.946561\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−9.79796	−0.314431	−0.157216	−	0.987564i	$-0.550252\pi$
$971$	−9.79796	−0.314431	−0.157216	+	0.987564i	$0.550252\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	46.6690i	1.49308i	0.665343	+	0.746538i	$0.268285\pi$
$977$	46.6690i	1.49308i	−0.665343	+	0.746538i	$0.731715\pi$
$978$	0	0
$979$	− 6.92820i	− 0.221426i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−48.9898	−1.56253	−0.781266	−	0.624198i	$-0.785426\pi$
$983$	−48.9898	−1.56253	−0.781266	+	0.624198i	$0.785426\pi$
$984$	0	0
$985$	−36.0000	−1.14706
$986$	0	0
$987$	0	0
$988$	0	0
$989$	− 16.9706i	− 0.539633i
$990$	0	0
$991$	− 13.8564i	− 0.440163i	−0.975481	−	0.220082i	$-0.929368\pi$
$991$	− 13.8564i	− 0.440163i	0.975481	−	0.220082i	$-0.0706324\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−39.1918	−1.24246
$996$	0	0
$997$	36.0000	1.14013	0.570066	−	0.821599i	$-0.306918\pi$
$997$	36.0000	1.14013	0.570066	+	0.821599i	$0.306918\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7056.2.h.g.4607.4	yes	4
3.2	odd	2	inner	7056.2.h.g.4607.1	yes	4
4.3	odd	2	inner	7056.2.h.g.4607.3	yes	4
7.6	odd	2		7056.2.h.f.4607.2	yes	4
12.11	even	2	inner	7056.2.h.g.4607.2	yes	4
21.20	even	2		7056.2.h.f.4607.3	yes	4
28.27	even	2		7056.2.h.f.4607.1	✓	4
84.83	odd	2		7056.2.h.f.4607.4	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
7056.2.h.f.4607.1	✓	4	28.27	even	2
7056.2.h.f.4607.2	yes	4	7.6	odd	2
7056.2.h.f.4607.3	yes	4	21.20	even	2
7056.2.h.f.4607.4	yes	4	84.83	odd	2
7056.2.h.g.4607.1	yes	4	3.2	odd	2	inner
7056.2.h.g.4607.2	yes	4	12.11	even	2	inner
7056.2.h.g.4607.3	yes	4	4.3	odd	2	inner
7056.2.h.g.4607.4	yes	4	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 7056 = 2^{4} \cdot 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7056.h (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+2.82843i q^{5} +O(q^{10})\)	\(q+2.82843i q^{5} +2.44949 q^{11} +2.82843i q^{17} -6.92820i q^{19} -2.44949 q^{23} -3.00000 q^{25} +4.24264i q^{29} -6.92820i q^{31} +4.00000 q^{37} -2.82843i q^{41} +6.92820i q^{43} +9.79796 q^{47} +4.24264i q^{53} +6.92820i q^{55} +9.79796 q^{59} +12.0000 q^{61} +10.3923i q^{67} +12.2474 q^{71} +12.0000 q^{73} -3.46410i q^{79} -9.79796 q^{83} -8.00000 q^{85} -2.82843i q^{89} +19.5959 q^{95} -12.0000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q+O(q^{10}) \) 4 * q	\( 4 q - 12 q^{25} + 16 q^{37} + 48 q^{61} + 48 q^{73} - 32 q^{85} - 48 q^{97}+O(q^{100}) \) 4 * q - 12 * q^25 + 16 * q^37 + 48 * q^61 + 48 * q^73 - 32 * q^85 - 48 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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