LMFDB - Embedded newform 1008.2.h.b.575.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1008,2,Mod(575,1008)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1008, 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("1008.575");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1008 = 2^{4} \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1008.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:	$8.04892052375$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\Q(\zeta_{24})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{4} + 1 $ x^8 - x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{8} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			575.5
Root			$0.258819 + 0.965926i$ of defining polynomial
Character	$\chi$	$=$	1008.575
Dual form			1008.2.h.b.575.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.03528i q^{5} -1.00000i q^{7} +O(q^{10})$	$q+1.03528i q^{5} -1.00000i q^{7} +0.378937 q^{11} +2.00000 q^{13} +3.86370i q^{17} -1.46410i q^{19} +5.27792 q^{23} +3.92820 q^{25} +3.48477i q^{29} -2.53590i q^{31} +1.03528 q^{35} +2.92820 q^{37} +8.76268i q^{41} -4.00000i q^{43} +8.48528 q^{47} -1.00000 q^{49} +7.07107i q^{53} +0.392305i q^{55} -2.82843 q^{59} +3.46410 q^{61} +2.07055i q^{65} +8.53590i q^{67} +10.1769 q^{71} +7.46410 q^{73} -0.378937i q^{77} -10.3923i q^{79} -17.5254 q^{83} -4.00000 q^{85} -1.03528i q^{89} -2.00000i q^{91} +1.51575 q^{95} +4.53590 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q+O(q^{10}) $ 8 * q	$ 8 q + 16 q^{13} - 24 q^{25} - 32 q^{37} - 8 q^{49} + 32 q^{73} - 32 q^{85} + 64 q^{97}+O(q^{100}) $ 8 * q + 16 * q^13 - 24 * q^25 - 32 * q^37 - 8 * q^49 + 32 * q^73 - 32 * q^85 + 64 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1008\mathbb{Z}\right)^\times$.

$n$	$127$	$577$	$757$	$785$
$\chi(n)$	$-1$	$1$	$1$	$-1$

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
$5$	1.03528i	0.462990i	0.972836	+	0.231495i	$0.0743616\pi$
$5$	1.03528i	0.462990i	−0.972836	+	0.231495i	$0.925638\pi$
$6$	0	0
$7$	− 1.00000i	− 0.377964i
$7$	− 1.00000i	− 0.377964i
$8$	0	0
$9$	0	0
$10$	0	0
$11$	0.378937	0.114254	0.0571270	−	0.998367i	$-0.481806\pi$
$11$	0.378937	0.114254	0.0571270	+	0.998367i	$0.481806\pi$
$12$	0	0
$13$	2.00000	0.554700	0.277350	−	0.960769i	$-0.410544\pi$
$13$	2.00000	0.554700	0.277350	+	0.960769i	$0.410544\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	3.86370i	0.937086i	0.883441	+	0.468543i	$0.155221\pi$
$17$	3.86370i	0.937086i	−0.883441	+	0.468543i	$0.844779\pi$
$18$	0	0
$19$	− 1.46410i	− 0.335888i	−0.985797	−	0.167944i	$-0.946287\pi$
$19$	− 1.46410i	− 0.335888i	0.985797	−	0.167944i	$-0.0537128\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	5.27792	1.10052	0.550261	−	0.834993i	$-0.314528\pi$
$23$	5.27792	1.10052	0.550261	+	0.834993i	$0.314528\pi$
$24$	0	0
$25$	3.92820	0.785641
$26$	0	0
$27$	0	0
$28$	0	0
$29$	3.48477i	0.647105i	0.946210	+	0.323552i	$0.104877\pi$
$29$	3.48477i	0.647105i	−0.946210	+	0.323552i	$0.895123\pi$
$30$	0	0
$31$	− 2.53590i	− 0.455461i	−0.973724	−	0.227730i	$-0.926870\pi$
$31$	− 2.53590i	− 0.455461i	0.973724	−	0.227730i	$-0.0731305\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	1.03528	0.174994
$36$	0	0
$37$	2.92820	0.481394	0.240697	−	0.970600i	$-0.422624\pi$
$37$	2.92820	0.481394	0.240697	+	0.970600i	$0.422624\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	8.76268i	1.36850i	0.729247	+	0.684251i	$0.239871\pi$
$41$	8.76268i	1.36850i	−0.729247	+	0.684251i	$0.760129\pi$
$42$	0	0
$43$	− 4.00000i	− 0.609994i	−0.952353	−	0.304997i	$-0.901344\pi$
$43$	− 4.00000i	− 0.609994i	0.952353	−	0.304997i	$-0.0986555\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	8.48528	1.23771	0.618853	−	0.785507i	$-0.287598\pi$
$47$	8.48528	1.23771	0.618853	+	0.785507i	$0.287598\pi$
$48$	0	0
$49$	−1.00000	−0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	7.07107i	0.971286i	0.874157	+	0.485643i	$0.161414\pi$
$53$	7.07107i	0.971286i	−0.874157	+	0.485643i	$0.838586\pi$
$54$	0	0
$55$	0.392305i	0.0528984i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−2.82843	−0.368230	−0.184115	−	0.982905i	$-0.558942\pi$
$59$	−2.82843	−0.368230	−0.184115	+	0.982905i	$0.558942\pi$
$60$	0	0
$61$	3.46410	0.443533	0.221766	−	0.975100i	$-0.428818\pi$
$61$	3.46410	0.443533	0.221766	+	0.975100i	$0.428818\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	2.07055i	0.256820i
$66$	0	0
$67$	8.53590i	1.04283i	0.853304	+	0.521413i	$0.174595\pi$
$67$	8.53590i	1.04283i	−0.853304	+	0.521413i	$0.825405\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	10.1769	1.20778	0.603888	−	0.797069i	$-0.293618\pi$
$71$	10.1769	1.20778	0.603888	+	0.797069i	$0.293618\pi$
$72$	0	0
$73$	7.46410	0.873607	0.436804	−	0.899557i	$-0.356111\pi$
$73$	7.46410	0.873607	0.436804	+	0.899557i	$0.356111\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 0.378937i	− 0.0431839i
$78$	0	0
$79$	− 10.3923i	− 1.16923i	−0.811312	−	0.584613i	$-0.801246\pi$
$79$	− 10.3923i	− 1.16923i	0.811312	−	0.584613i	$-0.198754\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−17.5254	−1.92366	−0.961829	−	0.273650i	$-0.911769\pi$
$83$	−17.5254	−1.92366	−0.961829	+	0.273650i	$0.911769\pi$
$84$	0	0
$85$	−4.00000	−0.433861
$86$	0	0
$87$	0	0
$88$	0	0
$89$	− 1.03528i	− 0.109739i	−0.998494	−	0.0548695i	$-0.982526\pi$
$89$	− 1.03528i	− 0.109739i	0.998494	−	0.0548695i	$-0.0174743\pi$
$90$	0	0
$91$	− 2.00000i	− 0.209657i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	1.51575	0.155513
$96$	0	0
$97$	4.53590	0.460551	0.230275	−	0.973126i	$-0.426037\pi$
$97$	4.53590	0.460551	0.230275	+	0.973126i	$0.426037\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	0.277401i	0.0276025i	0.999905	+	0.0138012i	$0.00439321\pi$
$101$	0.277401i	0.0276025i	−0.999905	+	0.0138012i	$0.995607\pi$
$102$	0	0
$103$	− 1.46410i	− 0.144262i	−0.997395	−	0.0721311i	$-0.977020\pi$
$103$	− 1.46410i	− 0.144262i	0.997395	−	0.0721311i	$-0.0229800\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	4.52004	0.436969	0.218484	−	0.975840i	$-0.429889\pi$
$107$	4.52004	0.436969	0.218484	+	0.975840i	$0.429889\pi$
$108$	0	0
$109$	8.00000	0.766261	0.383131	−	0.923694i	$-0.374846\pi$
$109$	8.00000	0.766261	0.383131	+	0.923694i	$0.374846\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 9.89949i	− 0.931266i	−0.884978	−	0.465633i	$-0.845827\pi$
$113$	− 9.89949i	− 0.931266i	0.884978	−	0.465633i	$-0.154173\pi$
$114$	0	0
$115$	5.46410i	0.509530i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	3.86370	0.354185
$120$	0	0
$121$	−10.8564	−0.986946
$122$	0	0
$123$	0	0
$124$	0	0
$125$	9.24316i	0.826733i
$126$	0	0
$127$	− 14.3923i	− 1.27711i	−0.769576	−	0.638555i	$-0.779532\pi$
$127$	− 14.3923i	− 1.27711i	0.769576	−	0.638555i	$-0.220468\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−1.51575	−0.132432	−0.0662158	−	0.997805i	$-0.521093\pi$
$131$	−1.51575	−0.132432	−0.0662158	+	0.997805i	$0.521093\pi$
$132$	0	0
$133$	−1.46410	−0.126954
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 6.31319i	− 0.539372i	−0.962948	−	0.269686i	$-0.913080\pi$
$137$	− 6.31319i	− 0.539372i	0.962948	−	0.269686i	$-0.0869200\pi$
$138$	0	0
$139$	21.8564i	1.85384i	0.375264	+	0.926918i	$0.377552\pi$
$139$	21.8564i	1.85384i	−0.375264	+	0.926918i	$0.622448\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	0.757875	0.0633767
$144$	0	0
$145$	−3.60770	−0.299603
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 16.8690i	− 1.38196i	−0.722872	−	0.690982i	$-0.757178\pi$
$149$	− 16.8690i	− 1.38196i	0.722872	−	0.690982i	$-0.242822\pi$
$150$	0	0
$151$	2.92820i	0.238294i	0.992877	+	0.119147i	$0.0380159\pi$
$151$	2.92820i	0.238294i	−0.992877	+	0.119147i	$0.961984\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	2.62536	0.210874
$156$	0	0
$157$	−17.3205	−1.38233	−0.691164	−	0.722698i	$-0.742902\pi$
$157$	−17.3205	−1.38233	−0.691164	+	0.722698i	$0.742902\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	− 5.27792i	− 0.415958i
$162$	0	0
$163$	0.535898i	0.0419748i	0.999780	+	0.0209874i	$0.00668099\pi$
$163$	0.535898i	0.0419748i	−0.999780	+	0.0209874i	$0.993319\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−22.4243	−1.73525	−0.867624	−	0.497221i	$-0.834354\pi$
$167$	−22.4243	−1.73525	−0.867624	+	0.497221i	$0.834354\pi$
$168$	0	0
$169$	−9.00000	−0.692308
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 11.5911i	− 0.881256i	−0.897690	−	0.440628i	$-0.854756\pi$
$173$	− 11.5911i	− 0.881256i	0.897690	−	0.440628i	$-0.145244\pi$
$174$	0	0
$175$	− 3.92820i	− 0.296944i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−26.3896	−1.97245	−0.986225	−	0.165409i	$-0.947106\pi$
$179$	−26.3896	−1.97245	−0.986225	+	0.165409i	$0.947106\pi$
$180$	0	0
$181$	12.9282	0.960946	0.480473	−	0.877010i	$-0.340465\pi$
$181$	12.9282	0.960946	0.480473	+	0.877010i	$0.340465\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	3.03150i	0.222880i
$186$	0	0
$187$	1.46410i	0.107066i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	20.7327	1.50017	0.750084	−	0.661343i	$-0.230013\pi$
$191$	20.7327	1.50017	0.750084	+	0.661343i	$0.230013\pi$
$192$	0	0
$193$	−13.8564	−0.997406	−0.498703	−	0.866773i	$-0.666190\pi$
$193$	−13.8564	−0.997406	−0.498703	+	0.866773i	$0.666190\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 15.3533i	− 1.09388i	−0.837173	−	0.546938i	$-0.815793\pi$
$197$	− 15.3533i	− 1.09388i	0.837173	−	0.546938i	$-0.184207\pi$
$198$	0	0
$199$	4.00000i	0.283552i	0.989899	+	0.141776i	$0.0452813\pi$
$199$	4.00000i	0.283552i	−0.989899	+	0.141776i	$0.954719\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	3.48477	0.244583
$204$	0	0
$205$	−9.07180	−0.633602
$206$	0	0
$207$	0	0
$208$	0	0
$209$	− 0.554803i	− 0.0383765i
$210$	0	0
$211$	9.07180i	0.624528i	0.949995	+	0.312264i	$0.101087\pi$
$211$	9.07180i	0.624528i	−0.949995	+	0.312264i	$0.898913\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	4.14110	0.282421
$216$	0	0
$217$	−2.53590	−0.172148
$218$	0	0
$219$	0	0
$220$	0	0
$221$	7.72741i	0.519802i
$222$	0	0
$223$	17.8564i	1.19575i	0.801588	+	0.597877i	$0.203989\pi$
$223$	17.8564i	1.19575i	−0.801588	+	0.597877i	$0.796011\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−23.9401	−1.58896	−0.794480	−	0.607290i	$-0.792256\pi$
$227$	−23.9401	−1.58896	−0.794480	+	0.607290i	$0.792256\pi$
$228$	0	0
$229$	−25.7128	−1.69915	−0.849575	−	0.527467i	$-0.823142\pi$
$229$	−25.7128	−1.69915	−0.849575	+	0.527467i	$0.823142\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 10.6574i	− 0.698188i	−0.937088	−	0.349094i	$-0.886489\pi$
$233$	− 10.6574i	− 0.698188i	0.937088	−	0.349094i	$-0.113511\pi$
$234$	0	0
$235$	8.78461i	0.573045i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	17.3495	1.12225	0.561123	−	0.827732i	$-0.310369\pi$
$239$	17.3495	1.12225	0.561123	+	0.827732i	$0.310369\pi$
$240$	0	0
$241$	14.3923	0.927090	0.463545	−	0.886073i	$-0.346577\pi$
$241$	14.3923	0.927090	0.463545	+	0.886073i	$0.346577\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	− 1.03528i	− 0.0661414i
$246$	0	0
$247$	− 2.92820i	− 0.186317i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−24.4949	−1.54610	−0.773052	−	0.634343i	$-0.781271\pi$
$251$	−24.4949	−1.54610	−0.773052	+	0.634343i	$0.781271\pi$
$252$	0	0
$253$	2.00000	0.125739
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 14.4195i	− 0.899466i	−0.893163	−	0.449733i	$-0.851519\pi$
$257$	− 14.4195i	− 0.899466i	0.893163	−	0.449733i	$-0.148481\pi$
$258$	0	0
$259$	− 2.92820i	− 0.181950i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−17.3495	−1.06982	−0.534908	−	0.844910i	$-0.679654\pi$
$263$	−17.3495	−1.06982	−0.534908	+	0.844910i	$0.679654\pi$
$264$	0	0
$265$	−7.32051	−0.449695
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 18.0058i	− 1.09784i	−0.835876	−	0.548918i	$-0.815040\pi$
$269$	− 18.0058i	− 1.09784i	0.835876	−	0.548918i	$-0.184960\pi$
$270$	0	0
$271$	− 31.3205i	− 1.90259i	−0.308289	−	0.951293i	$-0.599756\pi$
$271$	− 31.3205i	− 1.90259i	0.308289	−	0.951293i	$-0.400244\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	1.48854	0.0897625
$276$	0	0
$277$	0.143594	0.00862770	0.00431385	−	0.999991i	$-0.498627\pi$
$277$	0.143594	0.00862770	0.00431385	+	0.999991i	$0.498627\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	− 10.6574i	− 0.635765i	−0.948130	−	0.317883i	$-0.897028\pi$
$281$	− 10.6574i	− 0.635765i	0.948130	−	0.317883i	$-0.102972\pi$
$282$	0	0
$283$	14.5359i	0.864069i	0.901857	+	0.432035i	$0.142204\pi$
$283$	14.5359i	0.864069i	−0.901857	+	0.432035i	$0.857796\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	8.76268	0.517245
$288$	0	0
$289$	2.07180	0.121870
$290$	0	0
$291$	0	0
$292$	0	0
$293$	13.1069i	0.765711i	0.923808	+	0.382855i	$0.125059\pi$
$293$	13.1069i	0.765711i	−0.923808	+	0.382855i	$0.874941\pi$
$294$	0	0
$295$	− 2.92820i	− 0.170487i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	10.5558	0.610460
$300$	0	0
$301$	−4.00000	−0.230556
$302$	0	0
$303$	0	0
$304$	0	0
$305$	3.58630i	0.205351i
$306$	0	0
$307$	21.4641i	1.22502i	0.790462	+	0.612510i	$0.209840\pi$
$307$	21.4641i	1.22502i	−0.790462	+	0.612510i	$0.790160\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−1.31268	−0.0744351	−0.0372176	−	0.999307i	$-0.511849\pi$
$311$	−1.31268	−0.0744351	−0.0372176	+	0.999307i	$0.511849\pi$
$312$	0	0
$313$	11.0718	0.625815	0.312907	−	0.949784i	$-0.398697\pi$
$313$	11.0718	0.625815	0.312907	+	0.949784i	$0.398697\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 23.8386i	− 1.33891i	−0.742854	−	0.669453i	$-0.766528\pi$
$317$	− 23.8386i	− 1.33891i	0.742854	−	0.669453i	$-0.233472\pi$
$318$	0	0
$319$	1.32051i	0.0739343i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	5.65685	0.314756
$324$	0	0
$325$	7.85641	0.435795
$326$	0	0
$327$	0	0
$328$	0	0
$329$	− 8.48528i	− 0.467809i
$330$	0	0
$331$	− 28.7846i	− 1.58215i	−0.611722	−	0.791073i	$-0.709523\pi$
$331$	− 28.7846i	− 1.58215i	0.611722	−	0.791073i	$-0.290477\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−8.83701	−0.482818
$336$	0	0
$337$	−28.7846	−1.56800	−0.783999	−	0.620762i	$-0.786823\pi$
$337$	−28.7846	−1.56800	−0.783999	+	0.620762i	$0.786823\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	− 0.960947i	− 0.0520382i
$342$	0	0
$343$	1.00000i	0.0539949i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−10.9348	−0.587009	−0.293505	−	0.955958i	$-0.594822\pi$
$347$	−10.9348	−0.587009	−0.293505	+	0.955958i	$0.594822\pi$
$348$	0	0
$349$	21.3205	1.14126	0.570630	−	0.821207i	$-0.306699\pi$
$349$	21.3205	1.14126	0.570630	+	0.821207i	$0.306699\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	20.8343i	1.10890i	0.832218	+	0.554448i	$0.187071\pi$
$353$	20.8343i	1.10890i	−0.832218	+	0.554448i	$0.812929\pi$
$354$	0	0
$355$	10.5359i	0.559187i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	9.41902	0.497117	0.248558	−	0.968617i	$-0.420043\pi$
$359$	9.41902	0.497117	0.248558	+	0.968617i	$0.420043\pi$
$360$	0	0
$361$	16.8564	0.887179
$362$	0	0
$363$	0	0
$364$	0	0
$365$	7.72741i	0.404471i
$366$	0	0
$367$	2.92820i	0.152851i	0.997075	+	0.0764255i	$0.0243507\pi$
$367$	2.92820i	0.152851i	−0.997075	+	0.0764255i	$0.975649\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	7.07107	0.367112
$372$	0	0
$373$	−22.0000	−1.13912	−0.569558	−	0.821951i	$-0.692886\pi$
$373$	−22.0000	−1.13912	−0.569558	+	0.821951i	$0.692886\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	6.96953i	0.358949i
$378$	0	0
$379$	− 20.7846i	− 1.06763i	−0.845600	−	0.533817i	$-0.820757\pi$
$379$	− 20.7846i	− 1.06763i	0.845600	−	0.533817i	$-0.179243\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	13.9391	0.712253	0.356126	−	0.934438i	$-0.384097\pi$
$383$	13.9391	0.712253	0.356126	+	0.934438i	$0.384097\pi$
$384$	0	0
$385$	0.392305	0.0199937
$386$	0	0
$387$	0	0
$388$	0	0
$389$	23.2838i	1.18053i	0.807208	+	0.590267i	$0.200977\pi$
$389$	23.2838i	1.18053i	−0.807208	+	0.590267i	$0.799023\pi$
$390$	0	0
$391$	20.3923i	1.03128i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	10.7589	0.541339
$396$	0	0
$397$	−4.53590	−0.227650	−0.113825	−	0.993501i	$-0.536310\pi$
$397$	−4.53590	−0.227650	−0.113825	+	0.993501i	$0.536310\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	24.5964i	1.22829i	0.789194	+	0.614144i	$0.210498\pi$
$401$	24.5964i	1.22829i	−0.789194	+	0.614144i	$0.789502\pi$
$402$	0	0
$403$	− 5.07180i	− 0.252644i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	1.10961	0.0550011
$408$	0	0
$409$	3.46410	0.171289	0.0856444	−	0.996326i	$-0.472705\pi$
$409$	3.46410	0.171289	0.0856444	+	0.996326i	$0.472705\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	2.82843i	0.139178i
$414$	0	0
$415$	− 18.1436i	− 0.890634i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−26.0106	−1.27070	−0.635352	−	0.772223i	$-0.719145\pi$
$419$	−26.0106	−1.27070	−0.635352	+	0.772223i	$0.719145\pi$
$420$	0	0
$421$	29.7128	1.44811	0.724057	−	0.689740i	$-0.242275\pi$
$421$	29.7128	1.44811	0.724057	+	0.689740i	$0.242275\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	15.1774i	0.736213i
$426$	0	0
$427$	− 3.46410i	− 0.167640i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	6.79367	0.327239	0.163620	−	0.986523i	$-0.447683\pi$
$431$	6.79367	0.327239	0.163620	+	0.986523i	$0.447683\pi$
$432$	0	0
$433$	18.7846	0.902731	0.451365	−	0.892339i	$-0.350937\pi$
$433$	18.7846	0.902731	0.451365	+	0.892339i	$0.350937\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 7.72741i	− 0.369652i
$438$	0	0
$439$	13.8564i	0.661330i	0.943748	+	0.330665i	$0.107273\pi$
$439$	13.8564i	0.661330i	−0.943748	+	0.330665i	$0.892727\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	23.0064	1.09306	0.546532	−	0.837438i	$-0.315948\pi$
$443$	23.0064	1.09306	0.546532	+	0.837438i	$0.315948\pi$
$444$	0	0
$445$	1.07180	0.0508080
$446$	0	0
$447$	0	0
$448$	0	0
$449$	− 11.4152i	− 0.538719i	−0.963040	−	0.269359i	$-0.913188\pi$
$449$	− 11.4152i	− 0.538719i	0.963040	−	0.269359i	$-0.0868119\pi$
$450$	0	0
$451$	3.32051i	0.156357i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	2.07055	0.0970690
$456$	0	0
$457$	0	0		−	1.00000i	$-0.5\pi$
$457$	0	0			1.00000i	$0.5\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	37.8048i	1.76075i	0.474282	+	0.880373i	$0.342708\pi$
$461$	37.8048i	1.76075i	−0.474282	+	0.880373i	$0.657292\pi$
$462$	0	0
$463$	− 7.46410i	− 0.346886i	−0.984844	−	0.173443i	$-0.944511\pi$
$463$	− 7.46410i	− 0.346886i	0.984844	−	0.173443i	$-0.0554893\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−0.757875	−0.0350703	−0.0175351	−	0.999846i	$-0.505582\pi$
$467$	−0.757875	−0.0350703	−0.0175351	+	0.999846i	$0.505582\pi$
$468$	0	0
$469$	8.53590	0.394151
$470$	0	0
$471$	0	0
$472$	0	0
$473$	− 1.51575i	− 0.0696942i
$474$	0	0
$475$	− 5.75129i	− 0.263887i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	9.04008	0.413052	0.206526	−	0.978441i	$-0.433784\pi$
$479$	9.04008	0.413052	0.206526	+	0.978441i	$0.433784\pi$
$480$	0	0
$481$	5.85641	0.267029
$482$	0	0
$483$	0	0
$484$	0	0
$485$	4.69591i	0.213230i
$486$	0	0
$487$	10.9282i	0.495204i	0.968862	+	0.247602i	$0.0796426\pi$
$487$	10.9282i	0.495204i	−0.968862	+	0.247602i	$0.920357\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	39.2190	1.76993	0.884965	−	0.465657i	$-0.154182\pi$
$491$	39.2190	1.76993	0.884965	+	0.465657i	$0.154182\pi$
$492$	0	0
$493$	−13.4641	−0.606393
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 10.1769i	− 0.456496i
$498$	0	0
$499$	6.14359i	0.275025i	0.990500	+	0.137513i	$0.0439107\pi$
$499$	6.14359i	0.275025i	−0.990500	+	0.137513i	$0.956089\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−23.1822	−1.03364	−0.516822	−	0.856093i	$-0.672885\pi$
$503$	−23.1822	−1.03364	−0.516822	+	0.856093i	$0.672885\pi$
$504$	0	0
$505$	−0.287187	−0.0127797
$506$	0	0
$507$	0	0
$508$	0	0
$509$	8.76268i	0.388399i	0.980962	+	0.194200i	$0.0622109\pi$
$509$	8.76268i	0.388399i	−0.980962	+	0.194200i	$0.937789\pi$
$510$	0	0
$511$	− 7.46410i	− 0.330192i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	1.51575	0.0667919
$516$	0	0
$517$	3.21539	0.141413
$518$	0	0
$519$	0	0
$520$	0	0
$521$	40.4302i	1.77128i	0.464374	+	0.885639i	$0.346279\pi$
$521$	40.4302i	1.77128i	−0.464374	+	0.885639i	$0.653721\pi$
$522$	0	0
$523$	− 43.7128i	− 1.91143i	−0.294297	−	0.955714i	$-0.595086\pi$
$523$	− 43.7128i	− 1.91143i	0.294297	−	0.955714i	$-0.404914\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	9.79796	0.426806
$528$	0	0
$529$	4.85641	0.211148
$530$	0	0
$531$	0	0
$532$	0	0
$533$	17.5254i	0.759108i
$534$	0	0
$535$	4.67949i	0.202312i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−0.378937	−0.0163220
$540$	0	0
$541$	25.7128	1.10548	0.552740	−	0.833354i	$-0.313582\pi$
$541$	25.7128	1.10548	0.552740	+	0.833354i	$0.313582\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	8.28221i	0.354771i
$546$	0	0
$547$	− 11.4641i	− 0.490170i	−0.969502	−	0.245085i	$-0.921184\pi$
$547$	− 11.4641i	− 0.490170i	0.969502	−	0.245085i	$-0.0788158\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	5.10205	0.217355
$552$	0	0
$553$	−10.3923	−0.441926
$554$	0	0
$555$	0	0
$556$	0	0
$557$	2.72689i	0.115542i	0.998330	+	0.0577710i	$0.0183993\pi$
$557$	2.72689i	0.115542i	−0.998330	+	0.0577710i	$0.981601\pi$
$558$	0	0
$559$	− 8.00000i	− 0.338364i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	1.31268	0.0553228	0.0276614	−	0.999617i	$-0.491194\pi$
$563$	1.31268	0.0553228	0.0276614	+	0.999617i	$0.491194\pi$
$564$	0	0
$565$	10.2487	0.431167
$566$	0	0
$567$	0	0
$568$	0	0
$569$	0.656339i	0.0275152i	0.999905	+	0.0137576i	$0.00437931\pi$
$569$	0.656339i	0.0275152i	−0.999905	+	0.0137576i	$0.995621\pi$
$570$	0	0
$571$	− 2.39230i	− 0.100115i	−0.998746	−	0.0500574i	$-0.984060\pi$
$571$	− 2.39230i	− 0.100115i	0.998746	−	0.0500574i	$-0.0159404\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	20.7327	0.864615
$576$	0	0
$577$	31.0718	1.29354	0.646768	−	0.762687i	$-0.276120\pi$
$577$	31.0718	1.29354	0.646768	+	0.762687i	$0.276120\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	17.5254i	0.727075i
$582$	0	0
$583$	2.67949i	0.110973i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	19.7990	0.817192	0.408596	−	0.912715i	$-0.366019\pi$
$587$	19.7990	0.817192	0.408596	+	0.912715i	$0.366019\pi$
$588$	0	0
$589$	−3.71281	−0.152984
$590$	0	0
$591$	0	0
$592$	0	0
$593$	− 0.277401i	− 0.0113915i	−0.999984	−	0.00569576i	$-0.998187\pi$
$593$	− 0.277401i	− 0.0113915i	0.999984	−	0.00569576i	$-0.00181303\pi$
$594$	0	0
$595$	4.00000i	0.163984i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−18.4591	−0.754219	−0.377109	−	0.926169i	$-0.623082\pi$
$599$	−18.4591	−0.754219	−0.377109	+	0.926169i	$0.623082\pi$
$600$	0	0
$601$	8.92820	0.364189	0.182095	−	0.983281i	$-0.441712\pi$
$601$	8.92820	0.364189	0.182095	+	0.983281i	$0.441712\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	− 11.2394i	− 0.456946i
$606$	0	0
$607$	16.7846i	0.681266i	0.940196	+	0.340633i	$0.110641\pi$
$607$	16.7846i	0.681266i	−0.940196	+	0.340633i	$0.889359\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	16.9706	0.686555
$612$	0	0
$613$	30.7846	1.24338	0.621689	−	0.783264i	$-0.286447\pi$
$613$	30.7846	1.24338	0.621689	+	0.783264i	$0.286447\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 23.0807i	− 0.929193i	−0.885522	−	0.464597i	$-0.846199\pi$
$617$	− 23.0807i	− 0.929193i	0.885522	−	0.464597i	$-0.153801\pi$
$618$	0	0
$619$	− 41.8564i	− 1.68235i	−0.540762	−	0.841176i	$-0.681864\pi$
$619$	− 41.8564i	− 1.68235i	0.540762	−	0.841176i	$-0.318136\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−1.03528	−0.0414775
$624$	0	0
$625$	10.0718	0.402872
$626$	0	0
$627$	0	0
$628$	0	0
$629$	11.3137i	0.451107i
$630$	0	0
$631$	− 8.53590i	− 0.339809i	−0.985461	−	0.169904i	$-0.945654\pi$
$631$	− 8.53590i	− 0.339809i	0.985461	−	0.169904i	$-0.0543459\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	14.9000	0.591289
$636$	0	0
$637$	−2.00000	−0.0792429
$638$	0	0
$639$	0	0
$640$	0	0
$641$	14.5954i	0.576484i	0.957558	+	0.288242i	$0.0930707\pi$
$641$	14.5954i	0.576484i	−0.957558	+	0.288242i	$0.906929\pi$
$642$	0	0
$643$	10.2487i	0.404170i	0.979368	+	0.202085i	$0.0647717\pi$
$643$	10.2487i	0.404170i	−0.979368	+	0.202085i	$0.935228\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−39.3949	−1.54877	−0.774387	−	0.632712i	$-0.781942\pi$
$647$	−39.3949	−1.54877	−0.774387	+	0.632712i	$0.781942\pi$
$648$	0	0
$649$	−1.07180	−0.0420717
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 19.1427i	− 0.749110i	−0.927205	−	0.374555i	$-0.877796\pi$
$653$	− 19.1427i	− 0.749110i	0.927205	−	0.374555i	$-0.122204\pi$
$654$	0	0
$655$	− 1.56922i	− 0.0613145i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−18.8652	−0.734886	−0.367443	−	0.930046i	$-0.619767\pi$
$659$	−18.8652	−0.734886	−0.367443	+	0.930046i	$0.619767\pi$
$660$	0	0
$661$	−15.4641	−0.601484	−0.300742	−	0.953706i	$-0.597234\pi$
$661$	−15.4641	−0.601484	−0.300742	+	0.953706i	$0.597234\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	− 1.51575i	− 0.0587782i
$666$	0	0
$667$	18.3923i	0.712153i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	1.31268	0.0506754
$672$	0	0
$673$	−9.85641	−0.379937	−0.189968	−	0.981790i	$-0.560839\pi$
$673$	−9.85641	−0.379937	−0.189968	+	0.981790i	$0.560839\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 31.1870i	− 1.19861i	−0.800519	−	0.599307i	$-0.795443\pi$
$677$	− 31.1870i	− 1.19861i	0.800519	−	0.599307i	$-0.204557\pi$
$678$	0	0
$679$	− 4.53590i	− 0.174072i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	33.5622	1.28422	0.642111	−	0.766612i	$-0.278059\pi$
$683$	33.5622	1.28422	0.642111	+	0.766612i	$0.278059\pi$
$684$	0	0
$685$	6.53590	0.249724
$686$	0	0
$687$	0	0
$688$	0	0
$689$	14.1421i	0.538772i
$690$	0	0
$691$	1.85641i	0.0706210i	0.999376	+	0.0353105i	$0.0112420\pi$
$691$	1.85641i	0.0706210i	−0.999376	+	0.0353105i	$0.988758\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−22.6274	−0.858307
$696$	0	0
$697$	−33.8564	−1.28240
$698$	0	0
$699$	0	0
$700$	0	0
$701$	41.5670i	1.56996i	0.619519	+	0.784982i	$0.287328\pi$
$701$	41.5670i	1.56996i	−0.619519	+	0.784982i	$0.712672\pi$
$702$	0	0
$703$	− 4.28719i	− 0.161694i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0.277401	0.0104328
$708$	0	0
$709$	−13.8564	−0.520388	−0.260194	−	0.965556i	$-0.583787\pi$
$709$	−13.8564	−0.520388	−0.260194	+	0.965556i	$0.583787\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	− 13.3843i	− 0.501245i
$714$	0	0
$715$	0.784610i	0.0293427i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−1.51575	−0.0565279	−0.0282640	−	0.999600i	$-0.508998\pi$
$719$	−1.51575	−0.0565279	−0.0282640	+	0.999600i	$0.508998\pi$
$720$	0	0
$721$	−1.46410	−0.0545260
$722$	0	0
$723$	0	0
$724$	0	0
$725$	13.6889i	0.508392i
$726$	0	0
$727$	22.5359i	0.835810i	0.908491	+	0.417905i	$0.137236\pi$
$727$	22.5359i	0.835810i	−0.908491	+	0.417905i	$0.862764\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	15.4548	0.571617
$732$	0	0
$733$	4.92820	0.182027	0.0910137	−	0.995850i	$-0.470989\pi$
$733$	4.92820	0.182027	0.0910137	+	0.995850i	$0.470989\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	3.23457i	0.119147i
$738$	0	0
$739$	37.3205i	1.37286i	0.727197	+	0.686429i	$0.240823\pi$
$739$	37.3205i	1.37286i	−0.727197	+	0.686429i	$0.759177\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−31.2886	−1.14787	−0.573933	−	0.818902i	$-0.694583\pi$
$743$	−31.2886	−1.14787	−0.573933	+	0.818902i	$0.694583\pi$
$744$	0	0
$745$	17.4641	0.639835
$746$	0	0
$747$	0	0
$748$	0	0
$749$	− 4.52004i	− 0.165159i
$750$	0	0
$751$	− 5.85641i	− 0.213703i	−0.994275	−	0.106852i	$-0.965923\pi$
$751$	− 5.85641i	− 0.213703i	0.994275	−	0.106852i	$-0.0340770\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−3.03150	−0.110328
$756$	0	0
$757$	−10.1436	−0.368675	−0.184338	−	0.982863i	$-0.559014\pi$
$757$	−10.1436	−0.368675	−0.184338	+	0.982863i	$0.559014\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	31.9449i	1.15800i	0.815327	+	0.579001i	$0.196557\pi$
$761$	31.9449i	1.15800i	−0.815327	+	0.579001i	$0.803443\pi$
$762$	0	0
$763$	− 8.00000i	− 0.289619i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−5.65685	−0.204257
$768$	0	0
$769$	−34.0000	−1.22607	−0.613036	−	0.790055i	$-0.710052\pi$
$769$	−34.0000	−1.22607	−0.613036	+	0.790055i	$0.710052\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	− 36.8439i	− 1.32518i	−0.748981	−	0.662591i	$-0.769457\pi$
$773$	− 36.8439i	− 1.32518i	0.748981	−	0.662591i	$-0.230543\pi$
$774$	0	0
$775$	− 9.96152i	− 0.357829i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	12.8295	0.459663
$780$	0	0
$781$	3.85641	0.137993
$782$	0	0
$783$	0	0
$784$	0	0
$785$	− 17.9315i	− 0.640003i
$786$	0	0
$787$	20.7846i	0.740891i	0.928854	+	0.370446i	$0.120795\pi$
$787$	20.7846i	0.740891i	−0.928854	+	0.370446i	$0.879205\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−9.89949	−0.351986
$792$	0	0
$793$	6.92820	0.246028
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 22.9048i	− 0.811330i	−0.914022	−	0.405665i	$-0.867040\pi$
$797$	− 22.9048i	− 0.811330i	0.914022	−	0.405665i	$-0.132960\pi$
$798$	0	0
$799$	32.7846i	1.15984i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	2.82843	0.0998130
$804$	0	0
$805$	5.46410	0.192584
$806$	0	0
$807$	0	0
$808$	0	0
$809$	− 50.4040i	− 1.77211i	−0.463580	−	0.886055i	$-0.653435\pi$
$809$	− 50.4040i	− 1.77211i	0.463580	−	0.886055i	$-0.346565\pi$
$810$	0	0
$811$	− 49.8564i	− 1.75070i	−0.483494	−	0.875348i	$-0.660633\pi$
$811$	− 49.8564i	− 1.75070i	0.483494	−	0.875348i	$-0.339367\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−0.554803	−0.0194339
$816$	0	0
$817$	−5.85641	−0.204890
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 1.41421i	− 0.0493564i	−0.999695	−	0.0246782i	$-0.992144\pi$
$821$	− 1.41421i	− 0.0493564i	0.999695	−	0.0246782i	$-0.00785611\pi$
$822$	0	0
$823$	13.6077i	0.474334i	0.971469	+	0.237167i	$0.0762189\pi$
$823$	13.6077i	0.474334i	−0.971469	+	0.237167i	$0.923781\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−8.66115	−0.301178	−0.150589	−	0.988596i	$-0.548117\pi$
$827$	−8.66115	−0.301178	−0.150589	+	0.988596i	$0.548117\pi$
$828$	0	0
$829$	−40.9282	−1.42150	−0.710748	−	0.703447i	$-0.751643\pi$
$829$	−40.9282	−1.42150	−0.710748	+	0.703447i	$0.751643\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	− 3.86370i	− 0.133869i
$834$	0	0
$835$	− 23.2154i	− 0.803402i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−7.93048	−0.273791	−0.136895	−	0.990586i	$-0.543712\pi$
$839$	−7.93048	−0.273791	−0.136895	+	0.990586i	$0.543712\pi$
$840$	0	0
$841$	16.8564	0.581255
$842$	0	0
$843$	0	0
$844$	0	0
$845$	− 9.31749i	− 0.320531i
$846$	0	0
$847$	10.8564i	0.373031i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	15.4548	0.529784
$852$	0	0
$853$	−13.3205	−0.456086	−0.228043	−	0.973651i	$-0.573233\pi$
$853$	−13.3205	−0.456086	−0.228043	+	0.973651i	$0.573233\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	9.72363i	0.332153i	0.986113	+	0.166076i	$0.0531098\pi$
$857$	9.72363i	0.332153i	−0.986113	+	0.166076i	$0.946890\pi$
$858$	0	0
$859$	29.4641i	1.00530i	0.864489	+	0.502651i	$0.167642\pi$
$859$	29.4641i	1.00530i	−0.864489	+	0.502651i	$0.832358\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−11.6926	−0.398022	−0.199011	−	0.979997i	$-0.563773\pi$
$863$	−11.6926	−0.398022	−0.199011	+	0.979997i	$0.563773\pi$
$864$	0	0
$865$	12.0000	0.408012
$866$	0	0
$867$	0	0
$868$	0	0
$869$	− 3.93803i	− 0.133589i
$870$	0	0
$871$	17.0718i	0.578456i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	9.24316	0.312476
$876$	0	0
$877$	−46.7846	−1.57980	−0.789902	−	0.613233i	$-0.789869\pi$
$877$	−46.7846	−1.57980	−0.789902	+	0.613233i	$0.789869\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	− 18.7637i	− 0.632166i	−0.948732	−	0.316083i	$-0.897632\pi$
$881$	− 18.7637i	− 0.632166i	0.948732	−	0.316083i	$-0.102368\pi$
$882$	0	0
$883$	− 1.85641i	− 0.0624731i	−0.999512	−	0.0312365i	$-0.990055\pi$
$883$	− 1.85641i	− 0.0624731i	0.999512	−	0.0312365i	$-0.00994451\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−48.0833	−1.61448	−0.807239	−	0.590225i	$-0.799039\pi$
$887$	−48.0833	−1.61448	−0.807239	+	0.590225i	$0.799039\pi$
$888$	0	0
$889$	−14.3923	−0.482702
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 12.4233i	− 0.415730i
$894$	0	0
$895$	− 27.3205i	− 0.913224i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	8.83701	0.294731
$900$	0	0
$901$	−27.3205	−0.910178
$902$	0	0
$903$	0	0
$904$	0	0
$905$	13.3843i	0.444908i
$906$	0	0
$907$	− 47.7128i	− 1.58428i	−0.610341	−	0.792139i	$-0.708967\pi$
$907$	− 47.7128i	− 1.58428i	0.610341	−	0.792139i	$-0.291033\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−26.3896	−0.874326	−0.437163	−	0.899382i	$-0.644017\pi$
$911$	−26.3896	−0.874326	−0.437163	+	0.899382i	$0.644017\pi$
$912$	0	0
$913$	−6.64102	−0.219786
$914$	0	0
$915$	0	0
$916$	0	0
$917$	1.51575i	0.0500545i
$918$	0	0
$919$	34.9282i	1.15218i	0.817388	+	0.576088i	$0.195421\pi$
$919$	34.9282i	1.15218i	−0.817388	+	0.576088i	$0.804579\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	20.3538	0.669953
$924$	0	0
$925$	11.5026	0.378202
$926$	0	0
$927$	0	0
$928$	0	0
$929$	− 25.7332i	− 0.844280i	−0.906531	−	0.422140i	$-0.861279\pi$
$929$	− 25.7332i	− 0.844280i	0.906531	−	0.422140i	$-0.138721\pi$
$930$	0	0
$931$	1.46410i	0.0479840i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−1.51575	−0.0495703
$936$	0	0
$937$	−39.5692	−1.29267	−0.646335	−	0.763054i	$-0.723699\pi$
$937$	−39.5692	−1.29267	−0.646335	+	0.763054i	$0.723699\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	− 34.4216i	− 1.12211i	−0.827778	−	0.561056i	$-0.810395\pi$
$941$	− 34.4216i	− 1.12211i	0.827778	−	0.561056i	$-0.189605\pi$
$942$	0	0
$943$	46.2487i	1.50607i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−40.3286	−1.31050	−0.655252	−	0.755410i	$-0.727438\pi$
$947$	−40.3286	−1.31050	−0.655252	+	0.755410i	$0.727438\pi$
$948$	0	0
$949$	14.9282	0.484590
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 25.1512i	− 0.814728i	−0.913266	−	0.407364i	$-0.866448\pi$
$953$	− 25.1512i	− 0.814728i	0.913266	−	0.407364i	$-0.133552\pi$
$954$	0	0
$955$	21.4641i	0.694562i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−6.31319	−0.203864
$960$	0	0
$961$	24.5692	0.792555
$962$	0	0
$963$	0	0
$964$	0	0
$965$	− 14.3452i	− 0.461789i
$966$	0	0
$967$	16.5359i	0.531759i	0.964006	+	0.265879i	$0.0856623\pi$
$967$	16.5359i	0.531759i	−0.964006	+	0.265879i	$0.914338\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	26.2137	0.841238	0.420619	−	0.907237i	$-0.361813\pi$
$971$	26.2137	0.841238	0.420619	+	0.907237i	$0.361813\pi$
$972$	0	0
$973$	21.8564	0.700684
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 47.2239i	− 1.51082i	−0.655250	−	0.755412i	$-0.727437\pi$
$977$	− 47.2239i	− 1.51082i	0.655250	−	0.755412i	$-0.272563\pi$
$978$	0	0
$979$	− 0.392305i	− 0.0125381i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−12.2747	−0.391501	−0.195750	−	0.980654i	$-0.562714\pi$
$983$	−12.2747	−0.391501	−0.195750	+	0.980654i	$0.562714\pi$
$984$	0	0
$985$	15.8949	0.506453
$986$	0	0
$987$	0	0
$988$	0	0
$989$	− 21.1117i	− 0.671312i
$990$	0	0
$991$	14.6410i	0.465087i	0.972586	+	0.232544i	$0.0747048\pi$
$991$	14.6410i	0.465087i	−0.972586	+	0.232544i	$0.925295\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−4.14110	−0.131282
$996$	0	0
$997$	−35.4641	−1.12316	−0.561580	−	0.827423i	$-0.689806\pi$
$997$	−35.4641	−1.12316	−0.561580	+	0.827423i	$0.689806\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1008.2.h.b.575.5	yes	8
3.2	odd	2	inner	1008.2.h.b.575.3	✓	8
4.3	odd	2	inner	1008.2.h.b.575.6	yes	8
7.6	odd	2		7056.2.h.k.4607.4		8
8.3	odd	2		4032.2.h.f.575.4		8
8.5	even	2		4032.2.h.f.575.3		8
12.11	even	2	inner	1008.2.h.b.575.4	yes	8
21.20	even	2		7056.2.h.k.4607.5		8
24.5	odd	2		4032.2.h.f.575.5		8
24.11	even	2		4032.2.h.f.575.6		8
28.27	even	2		7056.2.h.k.4607.3		8
84.83	odd	2		7056.2.h.k.4607.6		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1008.2.h.b.575.3	✓	8	3.2	odd	2	inner
1008.2.h.b.575.4	yes	8	12.11	even	2	inner
1008.2.h.b.575.5	yes	8	1.1	even	1	trivial
1008.2.h.b.575.6	yes	8	4.3	odd	2	inner
4032.2.h.f.575.3		8	8.5	even	2
4032.2.h.f.575.4		8	8.3	odd	2
4032.2.h.f.575.5		8	24.5	odd	2
4032.2.h.f.575.6		8	24.11	even	2
7056.2.h.k.4607.3		8	28.27	even	2
7056.2.h.k.4607.4		8	7.6	odd	2
7056.2.h.k.4607.5		8	21.20	even	2
7056.2.h.k.4607.6		8	84.83	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1008 = 2^{4} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1008.h (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.03528i q^{5} -1.00000i q^{7} +O(q^{10})\)	\(q+1.03528i q^{5} -1.00000i q^{7} +0.378937 q^{11} +2.00000 q^{13} +3.86370i q^{17} -1.46410i q^{19} +5.27792 q^{23} +3.92820 q^{25} +3.48477i q^{29} -2.53590i q^{31} +1.03528 q^{35} +2.92820 q^{37} +8.76268i q^{41} -4.00000i q^{43} +8.48528 q^{47} -1.00000 q^{49} +7.07107i q^{53} +0.392305i q^{55} -2.82843 q^{59} +3.46410 q^{61} +2.07055i q^{65} +8.53590i q^{67} +10.1769 q^{71} +7.46410 q^{73} -0.378937i q^{77} -10.3923i q^{79} -17.5254 q^{83} -4.00000 q^{85} -1.03528i q^{89} -2.00000i q^{91} +1.51575 q^{95} +4.53590 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q+O(q^{10}) \) 8 * q	\( 8 q + 16 q^{13} - 24 q^{25} - 32 q^{37} - 8 q^{49} + 32 q^{73} - 32 q^{85} + 64 q^{97}+O(q^{100}) \) 8 * q + 16 * q^13 - 24 * q^25 - 32 * q^37 - 8 * q^49 + 32 * q^73 - 32 * q^85 + 64 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more