LMFDB - Embedded newform 672.2.i.e.209.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [672,2,Mod(209,672)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(672, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("672.209");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 672 = 2^{5} \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	672.i (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:	$5.36594701583$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.10070523904.11
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 10x^{4} + 81 $ x^8 - 10*x^4 + 81
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{3}\cdot 3 $
Twist minimal:	no (minimal twist has level 168)
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			209.6
Root			$0.420861 + 1.68014i$ of defining polynomial
Character	$\chi$	$=$	672.209
Dual form			672.2.i.e.209.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.420861 + 1.68014i) q^{3} -2.16991i q^{5} -2.64575 q^{7} +(-2.64575 + 1.41421i) q^{9} +O(q^{10})$	$q+(0.420861 + 1.68014i) q^{3} -2.16991i q^{5} -2.64575 q^{7} +(-2.64575 + 1.41421i) q^{9} -5.59388 q^{13} +(3.64575 - 0.913230i) q^{15} -8.66259 q^{19} +(-1.11349 - 4.44524i) q^{21} +7.48331i q^{23} +0.291503 q^{25} +(-3.48957 - 3.85005i) q^{27} +5.74103i q^{35} +(-2.35425 - 9.39851i) q^{39} +(3.06871 + 5.74103i) q^{45} +7.00000 q^{49} +(-3.64575 - 14.5544i) q^{57} -14.4207i q^{59} -0.543544 q^{61} +(7.00000 - 3.74166i) q^{63} +12.1382i q^{65} +(-12.5730 + 3.14944i) q^{69} -5.65685i q^{71} +(0.122682 + 0.489766i) q^{75} -5.29150 q^{79} +(5.00000 - 7.48331i) q^{81} -1.40122i q^{83} +14.8000 q^{91} +18.7970i q^{95} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q+O(q^{10}) $ 8 * q	$ 8 q + 8 q^{15} - 40 q^{25} - 40 q^{39} + 56 q^{49} - 8 q^{57} + 56 q^{63} + 40 q^{81}+O(q^{100}) $ 8 * q + 8 * q^15 - 40 * q^25 - 40 * q^39 + 56 * q^49 - 8 * q^57 + 56 * q^63 + 40 * q^81

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/672\mathbb{Z}\right)^\times$.

$n$	$127$	$421$	$449$	$577$
$\chi(n)$	$1$	$-1$	$-1$	$-1$

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
$3$	0.420861	+	1.68014i	0.242984	+	0.970030i
$3$	0.420861	+	1.68014i	0.242984	+	0.970030i
$4$		0			0
$5$		−	2.16991i		−	0.970412i	−0.874400	−	0.485206i	$-0.838745\pi$
$5$		−	2.16991i		−	0.970412i	0.874400	−	0.485206i	$-0.161255\pi$
$6$		0			0
$7$	−2.64575			−1.00000
$7$	−2.64575			−1.00000
$8$		0			0
$9$	−2.64575	+	1.41421i	−0.881917	+	0.471405i
$10$		0			0
$11$		0			0			−	1.00000i	$-0.5\pi$
$11$		0			0				1.00000i	$0.5\pi$
$12$		0			0
$13$	−5.59388			−1.55146			−0.775732	−	0.631063i	$-0.782619\pi$
$13$	−5.59388			−1.55146			−0.775732	+	0.631063i	$0.782619\pi$
$14$		0			0
$15$	3.64575	−	0.913230i	0.941329	−	0.235795i
$16$		0			0
$17$		0			0			−	1.00000i	$-0.5\pi$
$17$		0			0				1.00000i	$0.5\pi$
$18$		0			0
$19$	−8.66259			−1.98734			−0.993668	−	0.112360i	$-0.964159\pi$
$19$	−8.66259			−1.98734			−0.993668	+	0.112360i	$0.964159\pi$
$20$		0			0
$21$	−1.11349	−	4.44524i	−0.242984	−	0.970030i
$22$		0			0
$23$			7.48331i			1.56038i	0.625543	+	0.780189i	$0.284877\pi$
$23$			7.48331i			1.56038i	−0.625543	+	0.780189i	$0.715123\pi$
$24$		0			0
$25$	0.291503			0.0583005
$26$		0			0
$27$	−3.48957	−	3.85005i	−0.671569	−	0.740942i
$28$		0			0
$29$		0			0			−	1.00000i	$-0.5\pi$
$29$		0			0				1.00000i	$0.5\pi$
$30$		0			0
$31$		0			0		1.00000			$0$
$31$		0			0		−1.00000			$\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$			5.74103i			0.970412i
$36$		0			0
$37$		0			0		1.00000			$0$
$37$		0			0		−1.00000			$\pi$
$38$		0			0
$39$	−2.35425	−	9.39851i	−0.376981	−	1.50497i
$40$		0			0
$41$		0			0			−	1.00000i	$-0.5\pi$
$41$		0			0				1.00000i	$0.5\pi$
$42$		0			0
$43$		0			0		1.00000			$0$
$43$		0			0		−1.00000			$\pi$
$44$		0			0
$45$	3.06871	+	5.74103i	0.457457	+	0.855823i
$46$		0			0
$47$		0			0			−	1.00000i	$-0.5\pi$
$47$		0			0				1.00000i	$0.5\pi$
$48$		0			0
$49$	7.00000			1.00000
$50$		0			0
$51$		0			0
$52$		0			0
$53$		0			0			−	1.00000i	$-0.5\pi$
$53$		0			0				1.00000i	$0.5\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	−3.64575	−	14.5544i	−0.482892	−	1.92778i
$58$		0			0
$59$		−	14.4207i		−	1.87741i	−0.344721	−	0.938705i	$-0.612026\pi$
$59$		−	14.4207i		−	1.87741i	0.344721	−	0.938705i	$-0.387974\pi$
$60$		0			0
$61$	−0.543544			−0.0695936			−0.0347968	−	0.999394i	$-0.511078\pi$
$61$	−0.543544			−0.0695936			−0.0347968	+	0.999394i	$0.511078\pi$
$62$		0			0
$63$	7.00000	−	3.74166i	0.881917	−	0.471405i
$64$		0			0
$65$			12.1382i			1.50556i
$66$		0			0
$67$		0			0		1.00000			$0$
$67$		0			0		−1.00000			$\pi$
$68$		0			0
$69$	−12.5730	+	3.14944i	−1.51361	+	0.379148i
$70$		0			0
$71$		−	5.65685i		−	0.671345i	−0.941979	−	0.335673i	$-0.891036\pi$
$71$		−	5.65685i		−	0.671345i	0.941979	−	0.335673i	$-0.108964\pi$
$72$		0			0
$73$		0			0		1.00000			$0$
$73$		0			0		−1.00000			$\pi$
$74$		0			0
$75$	0.122682	+	0.489766i	0.0141661	+	0.0565533i
$76$		0			0
$77$		0			0
$78$		0			0
$79$	−5.29150			−0.595341			−0.297670	−	0.954669i	$-0.596210\pi$
$79$	−5.29150			−0.595341			−0.297670	+	0.954669i	$0.596210\pi$
$80$		0			0
$81$	5.00000	−	7.48331i	0.555556	−	0.831479i
$82$		0			0
$83$		−	1.40122i		−	0.153804i	−0.997039	−	0.0769020i	$-0.975497\pi$
$83$		−	1.40122i		−	0.153804i	0.997039	−	0.0769020i	$-0.0245028\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0			−	1.00000i	$-0.5\pi$
$89$		0			0				1.00000i	$0.5\pi$
$90$		0			0
$91$	14.8000			1.55146
$92$		0			0
$93$		0			0
$94$		0			0
$95$			18.7970i			1.92853i
$96$		0			0
$97$		0			0		1.00000			$0$
$97$		0			0		−1.00000			$\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$			17.9918i			1.79025i	0.445815	+	0.895125i	$0.352914\pi$
$101$			17.9918i			1.79025i	−0.445815	+	0.895125i	$0.647086\pi$
$102$		0			0
$103$		0			0		1.00000			$0$
$103$		0			0		−1.00000			$\pi$
$104$		0			0
$105$	−9.64575	+	2.41618i	−0.941329	+	0.235795i
$106$		0			0
$107$		0			0			−	1.00000i	$-0.5\pi$
$107$		0			0				1.00000i	$0.5\pi$
$108$		0			0
$109$		0			0		1.00000			$0$
$109$		0			0		−1.00000			$\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			14.1421i			1.33038i	0.746674	+	0.665190i	$0.231650\pi$
$113$			14.1421i			1.33038i	−0.746674	+	0.665190i	$0.768350\pi$
$114$		0			0
$115$	16.2381			1.51421
$116$		0			0
$117$	14.8000	−	7.91094i	1.36826	−	0.731367i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−11.0000			−1.00000
$122$		0			0
$123$		0			0
$124$		0			0
$125$		−	11.4821i		−	1.02699i
$126$		0			0
$127$	−2.00000			−0.177471			−0.0887357	−	0.996055i	$-0.528283\pi$
$127$	−2.00000			−0.177471			−0.0887357	+	0.996055i	$0.528283\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$			18.7605i			1.63911i	0.573000	+	0.819555i	$0.305779\pi$
$131$			18.7605i			1.63911i	−0.573000	+	0.819555i	$0.694221\pi$
$132$		0			0
$133$	22.9191			1.98734
$134$		0			0
$135$	−8.35425	+	7.57205i	−0.719019	+	0.651698i
$136$		0			0
$137$			14.9666i			1.27869i	0.768922	+	0.639343i	$0.220793\pi$
$137$			14.9666i			1.27869i	−0.768922	+	0.639343i	$0.779207\pi$
$138$		0			0
$139$	−13.7129			−1.16312			−0.581558	−	0.813505i	$-0.697557\pi$
$139$	−13.7129			−1.16312			−0.581558	+	0.813505i	$0.697557\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$		0			0
$146$		0			0
$147$	2.94603	+	11.7610i	0.242984	+	0.970030i
$148$		0			0
$149$		0			0			−	1.00000i	$-0.5\pi$
$149$		0			0				1.00000i	$0.5\pi$
$150$		0			0
$151$	10.0000			0.813788			0.406894	−	0.913475i	$-0.366612\pi$
$151$	10.0000			0.813788			0.406894	+	0.913475i	$0.366612\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	17.8687			1.42608			0.713040	−	0.701123i	$-0.247318\pi$
$157$	17.8687			1.42608			0.713040	+	0.701123i	$0.247318\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		−	19.7990i		−	1.56038i
$162$		0			0
$163$		0			0		1.00000			$0$
$163$		0			0		−1.00000			$\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0			−	1.00000i	$-0.5\pi$
$167$		0			0				1.00000i	$0.5\pi$
$168$		0			0
$169$	18.2915			1.40704
$170$		0			0
$171$	22.9191	−	12.2508i	1.75266	−	0.936839i
$172$		0			0
$173$		−	9.31216i		−	0.707991i	−0.935247	−	0.353995i	$-0.884823\pi$
$173$		−	9.31216i		−	0.707991i	0.935247	−	0.353995i	$-0.115177\pi$
$174$		0			0
$175$	−0.771243			−0.0583005
$176$		0			0
$177$	24.2288	−	6.06910i	1.82114	−	0.456182i
$178$		0			0
$179$		0			0			−	1.00000i	$-0.5\pi$
$179$		0			0				1.00000i	$0.5\pi$
$180$		0			0
$181$	−10.6442			−0.791178			−0.395589	−	0.918428i	$-0.629460\pi$
$181$	−10.6442			−0.791178			−0.395589	+	0.918428i	$0.629460\pi$
$182$		0			0
$183$	−0.228757	−	0.913230i	−0.0169102	−	0.0675079i
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$	9.23254	+	10.1863i	0.671569	+	0.740942i
$190$		0			0
$191$		−	22.6274i		−	1.63726i	−0.574320	−	0.818631i	$-0.694733\pi$
$191$		−	22.6274i		−	1.63726i	0.574320	−	0.818631i	$-0.305267\pi$
$192$		0			0
$193$	−26.4575			−1.90445			−0.952227	−	0.305392i	$-0.901213\pi$
$193$	−26.4575			−1.90445			−0.952227	+	0.305392i	$0.901213\pi$
$194$		0			0
$195$	−20.3939	+	5.10850i	−1.46044	+	0.365827i
$196$		0			0
$197$		0			0			−	1.00000i	$-0.5\pi$
$197$		0			0				1.00000i	$0.5\pi$
$198$		0			0
$199$		0			0		1.00000			$0$
$199$		0			0		−1.00000			$\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$	−10.5830	−	19.7990i	−0.735570	−	1.37612i
$208$		0			0
$209$		0			0
$210$		0			0
$211$		0			0		1.00000			$0$
$211$		0			0		−1.00000			$\pi$
$212$		0			0
$213$	9.50432	−	2.38075i	0.651225	−	0.163126i
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0		1.00000			$0$
$223$		0			0		−1.00000			$\pi$
$224$		0			0
$225$	−0.771243	+	0.412247i	−0.0514162	+	0.0274831i
$226$		0			0
$227$			25.9027i			1.71922i	0.510947	+	0.859612i	$0.329295\pi$
$227$			25.9027i			1.71922i	−0.510947	+	0.859612i	$0.670705\pi$
$228$		0			0
$229$	−29.0565			−1.92011			−0.960053	−	0.279817i	$-0.909726\pi$
$229$	−29.0565			−1.92011			−0.960053	+	0.279817i	$0.909726\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	29.9333i		−	1.96099i	−0.196537	−	0.980497i	$-0.562969\pi$
$233$		−	29.9333i		−	1.96099i	0.196537	−	0.980497i	$-0.437031\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	−2.22699	−	8.89047i	−0.144659	−	0.577498i
$238$		0			0
$239$			7.48331i			0.484055i	0.970269	+	0.242028i	$0.0778125\pi$
$239$			7.48331i			0.484055i	−0.970269	+	0.242028i	$0.922188\pi$
$240$		0			0
$241$		0			0		1.00000			$0$
$241$		0			0		−1.00000			$\pi$
$242$		0			0
$243$	14.6773	+	5.25127i	0.941551	+	0.336869i
$244$		0			0
$245$		−	15.1894i		−	0.970412i
$246$		0			0
$247$	48.4575			3.08328
$248$		0			0
$249$	2.35425	−	0.589720i	0.149194	−	0.0373720i
$250$		0			0
$251$		−	21.5629i		−	1.36104i	−0.732730	−	0.680520i	$-0.761754\pi$
$251$		−	21.5629i		−	1.36104i	0.732730	−	0.680520i	$-0.238246\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$		0			0
$257$		0			0			−	1.00000i	$-0.5\pi$
$257$		0			0				1.00000i	$0.5\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$			28.2843i			1.74408i	0.489432	+	0.872041i	$0.337204\pi$
$263$			28.2843i			1.74408i	−0.489432	+	0.872041i	$0.662796\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$			4.97235i			0.303169i	0.988444	+	0.151585i	$0.0484376\pi$
$269$			4.97235i			0.303169i	−0.988444	+	0.151585i	$0.951562\pi$
$270$		0			0
$271$		0			0		1.00000			$0$
$271$		0			0		−1.00000			$\pi$
$272$		0			0
$273$	6.22876	+	24.8661i	0.376981	+	1.50497i
$274$		0			0
$275$		0			0
$276$		0			0
$277$		0			0		1.00000			$0$
$277$		0			0		−1.00000			$\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			14.9666i			0.892834i	0.894825	+	0.446417i	$0.147300\pi$
$281$			14.9666i			0.892834i	−0.894825	+	0.446417i	$0.852700\pi$
$282$		0			0
$283$	9.74968			0.579558			0.289779	−	0.957094i	$-0.406418\pi$
$283$	9.74968			0.579558			0.289779	+	0.957094i	$0.406418\pi$
$284$		0			0
$285$	−31.5817	+	7.91094i	−1.87074	+	0.468604i
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−17.0000			−1.00000
$290$		0			0
$291$		0			0
$292$		0			0
$293$			31.0112i			1.81170i	0.423603	+	0.905848i	$0.360765\pi$
$293$			31.0112i			1.81170i	−0.423603	+	0.905848i	$0.639235\pi$
$294$		0			0
$295$	−31.2915			−1.82186
$296$		0			0
$297$		0			0
$298$		0			0
$299$		−	41.8608i		−	2.42087i
$300$		0			0
$301$		0			0
$302$		0			0
$303$	−30.2288	+	7.57205i	−1.73660	+	0.435003i
$304$		0			0
$305$			1.17944i			0.0675345i
$306$		0			0
$307$	−32.1252			−1.83348			−0.916741	−	0.399482i	$-0.869190\pi$
$307$	−32.1252			−1.83348			−0.916741	+	0.399482i	$0.869190\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0			−	1.00000i	$-0.5\pi$
$311$		0			0				1.00000i	$0.5\pi$
$312$		0			0
$313$		0			0		1.00000			$0$
$313$		0			0		−1.00000			$\pi$
$314$		0			0
$315$	−8.11905	−	15.1894i	−0.457457	−	0.855823i
$316$		0			0
$317$		0			0			−	1.00000i	$-0.5\pi$
$317$		0			0				1.00000i	$0.5\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$	−1.63063			−0.0904511
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$		0			0		1.00000			$0$
$331$		0			0		−1.00000			$\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−26.4575			−1.44123			−0.720616	−	0.693334i	$-0.756141\pi$
$337$	−26.4575			−1.44123			−0.720616	+	0.693334i	$0.756141\pi$
$338$		0			0
$339$	−23.7608	+	5.95188i	−1.29051	+	0.323262i
$340$		0			0
$341$		0			0
$342$		0			0
$343$	−18.5203			−1.00000
$344$		0			0
$345$	6.83399	+	27.2823i	0.367930	+	1.46883i
$346$		0			0
$347$		0			0			−	1.00000i	$-0.5\pi$
$347$		0			0				1.00000i	$0.5\pi$
$348$		0			0
$349$	−24.0062			−1.28502			−0.642510	−	0.766277i	$-0.722107\pi$
$349$	−24.0062			−1.28502			−0.642510	+	0.766277i	$0.722107\pi$
$350$		0			0
$351$	19.5203	+	21.5367i	1.04191	+	1.14954i
$352$		0			0
$353$		0			0			−	1.00000i	$-0.5\pi$
$353$		0			0				1.00000i	$0.5\pi$
$354$		0			0
$355$	−12.2748			−0.651481
$356$		0			0
$357$		0			0
$358$		0			0
$359$		−	37.4166i		−	1.97477i	−0.158334	−	0.987386i	$-0.550612\pi$
$359$		−	37.4166i		−	1.97477i	0.158334	−	0.987386i	$-0.449388\pi$
$360$		0			0
$361$	56.0405			2.94950
$362$		0			0
$363$	−4.62948	−	18.4816i	−0.242984	−	0.970030i
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0		1.00000			$0$
$367$		0			0		−1.00000			$\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$		0			0		1.00000			$0$
$373$		0			0		−1.00000			$\pi$
$374$		0			0
$375$	19.2915	−	4.83236i	0.996209	−	0.249542i
$376$		0			0
$377$		0			0
$378$		0			0
$379$		0			0		1.00000			$0$
$379$		0			0		−1.00000			$\pi$
$380$		0			0
$381$	−0.841723	−	3.36028i	−0.0431228	−	0.172153i
$382$		0			0
$383$		0			0			−	1.00000i	$-0.5\pi$
$383$		0			0				1.00000i	$0.5\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$		0			0			−	1.00000i	$-0.5\pi$
$389$		0			0				1.00000i	$0.5\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$	−31.5203	+	7.89556i	−1.58999	+	0.398278i
$394$		0			0
$395$			11.4821i			0.577726i
$396$		0			0
$397$	12.8184			0.643337			0.321668	−	0.946852i	$-0.395756\pi$
$397$	12.8184			0.643337			0.321668	+	0.946852i	$0.395756\pi$
$398$		0			0
$399$	9.64575	+	38.5073i	0.482892	+	1.92778i
$400$		0			0
$401$		−	36.7696i		−	1.83618i	−0.396368	−	0.918092i	$-0.629729\pi$
$401$		−	36.7696i		−	1.83618i	0.396368	−	0.918092i	$-0.370271\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$	−16.2381	−	10.8495i	−0.806878	−	0.539118i
$406$		0			0
$407$		0			0
$408$		0			0
$409$		0			0		1.00000			$0$
$409$		0			0		−1.00000			$\pi$
$410$		0			0
$411$	−25.1461	+	6.29888i	−1.24036	+	0.310701i
$412$		0			0
$413$			38.1535i			1.87741i
$414$		0			0
$415$	−3.04052			−0.149253
$416$		0			0
$417$	−5.77124	−	23.0397i	−0.282619	−	1.12826i
$418$		0			0
$419$			12.8833i			0.629390i	0.949193	+	0.314695i	$0.101902\pi$
$419$			12.8833i			0.629390i	−0.949193	+	0.314695i	$0.898098\pi$
$420$		0			0
$421$		0			0		1.00000			$0$
$421$		0			0		−1.00000			$\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	1.43808			0.0695936
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	37.4166i		−	1.80229i	−0.433515	−	0.901146i	$-0.642727\pi$
$431$		−	37.4166i		−	1.80229i	0.433515	−	0.901146i	$-0.357273\pi$
$432$		0			0
$433$		0			0		1.00000			$0$
$433$		0			0		−1.00000			$\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		−	64.8249i		−	3.10100i
$438$		0			0
$439$		0			0		1.00000			$0$
$439$		0			0		−1.00000			$\pi$
$440$		0			0
$441$	−18.5203	+	9.89949i	−0.881917	+	0.471405i
$442$		0			0
$443$		0			0			−	1.00000i	$-0.5\pi$
$443$		0			0				1.00000i	$0.5\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	29.9333i		−	1.41264i	−0.707894	−	0.706319i	$-0.750354\pi$
$449$		−	29.9333i		−	1.41264i	0.707894	−	0.706319i	$-0.249646\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$	4.20861	+	16.8014i	0.197738	+	0.789399i
$454$		0			0
$455$		−	32.1147i		−	1.50556i
$456$		0			0
$457$	5.29150			0.247526			0.123763	−	0.992312i	$-0.460504\pi$
$457$	5.29150			0.247526			0.123763	+	0.992312i	$0.460504\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		−	42.4933i		−	1.97911i	−0.144154	−	0.989555i	$-0.546046\pi$
$461$		−	42.4933i		−	1.97911i	0.144154	−	0.989555i	$-0.453954\pi$
$462$		0			0
$463$	26.4575			1.22958			0.614792	−	0.788689i	$-0.289240\pi$
$463$	26.4575			1.22958			0.614792	+	0.788689i	$0.289240\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$			11.6182i			0.537627i	0.963192	+	0.268814i	$0.0866316\pi$
$467$			11.6182i			0.537627i	−0.963192	+	0.268814i	$0.913368\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	7.52026	+	30.0220i	0.346515	+	1.38334i
$472$		0			0
$473$		0			0
$474$		0			0
$475$	−2.52517			−0.115863
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0			−	1.00000i	$-0.5\pi$
$479$		0			0				1.00000i	$0.5\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$	33.2651	−	8.33263i	1.51361	−	0.379148i
$484$		0			0
$485$		0			0
$486$		0			0
$487$	−38.0000			−1.72194			−0.860972	−	0.508652i	$-0.830144\pi$
$487$	−38.0000			−1.72194			−0.860972	+	0.508652i	$0.830144\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		0			0			−	1.00000i	$-0.5\pi$
$491$		0			0				1.00000i	$0.5\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$			14.9666i			0.671345i
$498$		0			0
$499$		0			0		1.00000			$0$
$499$		0			0		−1.00000			$\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0			−	1.00000i	$-0.5\pi$
$503$		0			0				1.00000i	$0.5\pi$
$504$		0			0
$505$	39.0405			1.73728
$506$		0			0
$507$	7.69819	+	30.7323i	0.341889	+	1.36487i
$508$		0			0
$509$			25.1340i			1.11405i	0.830497	+	0.557024i	$0.188057\pi$
$509$			25.1340i			1.11405i	−0.830497	+	0.557024i	$0.811943\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$	30.2288	+	33.3514i	1.33463	+	1.47250i
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$	15.6458	−	3.91913i	0.686772	−	0.172031i
$520$		0			0
$521$		0			0			−	1.00000i	$-0.5\pi$
$521$		0			0				1.00000i	$0.5\pi$
$522$		0			0
$523$	38.2626			1.67311			0.836554	−	0.547885i	$-0.184567\pi$
$523$	38.2626			1.67311			0.836554	+	0.547885i	$0.184567\pi$
$524$		0			0
$525$	−0.324587	−	1.29580i	−0.0141661	−	0.0565533i
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−33.0000			−1.43478
$530$		0			0
$531$	20.3939	+	38.1535i	0.885020	+	1.65572i
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$		0			0		1.00000			$0$
$541$		0			0		−1.00000			$\pi$
$542$		0			0
$543$	−4.47974	−	17.8838i	−0.192244	−	0.767467i
$544$		0			0
$545$		0			0
$546$		0			0
$547$		0			0		1.00000			$0$
$547$		0			0		−1.00000			$\pi$
$548$		0			0
$549$	1.43808	−	0.768687i	0.0613758	−	0.0328068i
$550$		0			0
$551$		0			0
$552$		0			0
$553$	14.0000			0.595341
$554$		0			0
$555$		0			0
$556$		0			0
$557$		0			0			−	1.00000i	$-0.5\pi$
$557$		0			0				1.00000i	$0.5\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$		−	8.54348i		−	0.360065i	−0.983661	−	0.180032i	$-0.942380\pi$
$563$		−	8.54348i		−	0.360065i	0.983661	−	0.180032i	$-0.0576202\pi$
$564$		0			0
$565$	30.6871			1.29102
$566$		0			0
$567$	−13.2288	+	19.7990i	−0.555556	+	0.831479i
$568$		0			0
$569$		−	2.82843i		−	0.118574i	−0.998241	−	0.0592869i	$-0.981117\pi$
$569$		−	2.82843i		−	0.118574i	0.998241	−	0.0592869i	$-0.0188827\pi$
$570$		0			0
$571$		0			0		1.00000			$0$
$571$		0			0		−1.00000			$\pi$
$572$		0			0
$573$	38.0173	−	9.52301i	1.58819	−	0.397829i
$574$		0			0
$575$			2.18141i			0.0909709i
$576$		0			0
$577$		0			0		1.00000			$0$
$577$		0			0		−1.00000			$\pi$
$578$		0			0
$579$	−11.1349	−	44.4524i	−0.462753	−	1.84738i
$580$		0			0
$581$			3.70728i			0.153804i
$582$		0			0
$583$		0			0
$584$		0			0
$585$	−17.1660	−	32.1147i	−0.709727	−	1.32778i
$586$		0			0
$587$			31.7799i			1.31170i	0.754892	+	0.655849i	$0.227689\pi$
$587$			31.7799i			1.31170i	−0.754892	+	0.655849i	$0.772311\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$		0			0			−	1.00000i	$-0.5\pi$
$593$		0			0				1.00000i	$0.5\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$			11.3137i			0.462266i	0.972922	+	0.231133i	$0.0742432\pi$
$599$			11.3137i			0.462266i	−0.972922	+	0.231133i	$0.925757\pi$
$600$		0			0
$601$		0			0		1.00000			$0$
$601$		0			0		−1.00000			$\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$			23.8690i			0.970412i
$606$		0			0
$607$		0			0		1.00000			$0$
$607$		0			0		−1.00000			$\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$		0			0		1.00000			$0$
$613$		0			0		−1.00000			$\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			14.1421i			0.569341i	0.958625	+	0.284670i	$0.0918842\pi$
$617$			14.1421i			0.569341i	−0.958625	+	0.284670i	$0.908116\pi$
$618$		0			0
$619$	−42.2259			−1.69720			−0.848601	−	0.529034i	$-0.822554\pi$
$619$	−42.2259			−1.69720			−0.848601	+	0.529034i	$0.822554\pi$
$620$		0			0
$621$	28.8111	−	26.1136i	1.15615	−	1.04790i
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−23.4575			−0.938301
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$	−37.0405			−1.47456			−0.737280	−	0.675587i	$-0.763890\pi$
$631$	−37.0405			−1.47456			−0.737280	+	0.675587i	$0.763890\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$			4.33981i			0.172220i
$636$		0			0
$637$	−39.1572			−1.55146
$638$		0			0
$639$	8.00000	+	14.9666i	0.316475	+	0.592071i
$640$		0			0
$641$			48.0833i			1.89917i	0.313503	+	0.949587i	$0.398498\pi$
$641$			48.0833i			1.89917i	−0.313503	+	0.949587i	$0.601502\pi$
$642$		0			0
$643$	48.3633			1.90726			0.953631	−	0.300978i	$-0.0973129\pi$
$643$	48.3633			1.90726			0.953631	+	0.300978i	$0.0973129\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0			−	1.00000i	$-0.5\pi$
$647$		0			0				1.00000i	$0.5\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$		0			0
$653$		0			0			−	1.00000i	$-0.5\pi$
$653$		0			0				1.00000i	$0.5\pi$
$654$		0			0
$655$	40.7085			1.59061
$656$		0			0
$657$		0			0
$658$		0			0
$659$		0			0			−	1.00000i	$-0.5\pi$
$659$		0			0				1.00000i	$0.5\pi$
$660$		0			0
$661$	41.3313			1.60760			0.803801	−	0.594898i	$-0.202807\pi$
$661$	41.3313			1.60760			0.803801	+	0.594898i	$0.202807\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		−	49.7322i		−	1.92853i
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	26.0000			1.00223			0.501113	−	0.865382i	$-0.332924\pi$
$673$	26.0000			1.00223			0.501113	+	0.865382i	$0.332924\pi$
$674$		0			0
$675$	−1.01722	−	1.12230i	−0.0391528	−	0.0431973i
$676$		0			0
$677$			51.1729i			1.96674i	0.181625	+	0.983368i	$0.441864\pi$
$677$			51.1729i			1.96674i	−0.181625	+	0.983368i	$0.558136\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$	−43.5203	+	10.9015i	−1.66770	+	0.417745i
$682$		0			0
$683$		0			0			−	1.00000i	$-0.5\pi$
$683$		0			0				1.00000i	$0.5\pi$
$684$		0			0
$685$	32.4762			1.24085
$686$		0			0
$687$	−12.2288	−	48.8190i	−0.466556	−	1.86256i
$688$		0			0
$689$		0			0
$690$		0			0
$691$	4.69934			0.178771			0.0893857	−	0.995997i	$-0.471510\pi$
$691$	4.69934			0.178771			0.0893857	+	0.995997i	$0.471510\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$			29.7558i			1.12870i
$696$		0			0
$697$		0			0
$698$		0			0
$699$	50.2921	−	12.5978i	1.90222	−	0.476491i
$700$		0			0
$701$		0			0			−	1.00000i	$-0.5\pi$
$701$		0			0				1.00000i	$0.5\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$		0			0
$707$		−	47.6018i		−	1.79025i
$708$		0			0
$709$		0			0		1.00000			$0$
$709$		0			0		−1.00000			$\pi$
$710$		0			0
$711$	14.0000	−	7.48331i	0.525041	−	0.280646i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$	−12.5730	+	3.14944i	−0.469548	+	0.117618i
$718$		0			0
$719$		0			0			−	1.00000i	$-0.5\pi$
$719$		0			0				1.00000i	$0.5\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$		0			0
$724$		0			0
$725$		0			0
$726$		0			0
$727$		0			0		1.00000			$0$
$727$		0			0		−1.00000			$\pi$
$728$		0			0
$729$	−2.64575	+	26.8701i	−0.0979908	+	0.995187i
$730$		0			0
$731$		0			0
$732$		0			0
$733$	33.0197			1.21961			0.609806	−	0.792551i	$-0.291247\pi$
$733$	33.0197			1.21961			0.609806	+	0.792551i	$0.291247\pi$
$734$		0			0
$735$	25.5203	−	6.39261i	0.941329	−	0.235795i
$736$		0			0
$737$		0			0
$738$		0			0
$739$		0			0		1.00000			$0$
$739$		0			0		−1.00000			$\pi$
$740$		0			0
$741$	20.3939	+	81.4155i	0.749189	+	2.99087i
$742$		0			0
$743$			7.48331i			0.274536i	0.990534	+	0.137268i	$0.0438322\pi$
$743$			7.48331i			0.274536i	−0.990534	+	0.137268i	$0.956168\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	1.98162	+	3.70728i	0.0725039	+	0.135642i
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−50.0000			−1.82453			−0.912263	−	0.409605i	$-0.865667\pi$
$751$	−50.0000			−1.82453			−0.912263	+	0.409605i	$0.865667\pi$
$752$		0			0
$753$	36.2288	−	9.07500i	1.32025	−	0.330711i
$754$		0			0
$755$		−	21.6991i		−	0.789710i
$756$		0			0
$757$		0			0		1.00000			$0$
$757$		0			0		−1.00000			$\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0			−	1.00000i	$-0.5\pi$
$761$		0			0				1.00000i	$0.5\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$			80.6675i			2.91273i
$768$		0			0
$769$		0			0		1.00000			$0$
$769$		0			0		−1.00000			$\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		−	16.4544i		−	0.591824i	−0.955215	−	0.295912i	$-0.904376\pi$
$773$		−	16.4544i		−	0.591824i	0.955215	−	0.295912i	$-0.0956236\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$		0			0
$785$		−	38.7735i		−	1.38389i
$786$		0			0
$787$	33.2123			1.18389			0.591945	−	0.805978i	$-0.298360\pi$
$787$	33.2123			1.18389			0.591945	+	0.805978i	$0.298360\pi$
$788$		0			0
$789$	−47.5216	+	11.9038i	−1.69181	+	0.423785i
$790$		0			0
$791$		−	37.4166i		−	1.33038i
$792$		0			0
$793$	3.04052			0.107972
$794$		0			0
$795$		0			0
$796$		0			0
$797$		−	8.04710i		−	0.285043i	−0.989792	−	0.142521i	$-0.954479\pi$
$797$		−	8.04710i		−	0.285043i	0.989792	−	0.142521i	$-0.0455210\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$		0			0
$803$		0			0
$804$		0			0
$805$	−42.9620			−1.51421
$806$		0			0
$807$	−8.35425	+	2.09267i	−0.294083	+	0.0736655i
$808$		0			0
$809$			31.1127i			1.09386i	0.837177	+	0.546932i	$0.184204\pi$
$809$			31.1127i			1.09386i	−0.837177	+	0.546932i	$0.815796\pi$
$810$		0			0
$811$	29.9510			1.05172			0.525861	−	0.850570i	$-0.323743\pi$
$811$	29.9510			1.05172			0.525861	+	0.850570i	$0.323743\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$		0			0
$819$	−39.1572	+	20.9304i	−1.36826	+	0.731367i
$820$		0			0
$821$		0			0			−	1.00000i	$-0.5\pi$
$821$		0			0				1.00000i	$0.5\pi$
$822$		0			0
$823$	26.4575			0.922251			0.461125	−	0.887335i	$-0.347446\pi$
$823$	26.4575			0.922251			0.461125	+	0.887335i	$0.347446\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0			−	1.00000i	$-0.5\pi$
$827$		0			0				1.00000i	$0.5\pi$
$828$		0			0
$829$	−57.5694			−1.99947			−0.999735	−	0.0230361i	$-0.992667\pi$
$829$	−57.5694			−1.99947			−0.999735	+	0.0230361i	$0.992667\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0			−	1.00000i	$-0.5\pi$
$839$		0			0				1.00000i	$0.5\pi$
$840$		0			0
$841$	−29.0000			−1.00000
$842$		0			0
$843$	−25.1461	+	6.29888i	−0.866076	+	0.216945i
$844$		0			0
$845$		−	39.6909i		−	1.36541i
$846$		0			0
$847$	29.1033			1.00000
$848$		0			0
$849$	4.10326	+	16.3808i	0.140824	+	0.562189i
$850$		0			0
$851$		0			0
$852$		0			0
$853$	7.76806			0.265973			0.132987	−	0.991118i	$-0.457543\pi$
$853$	7.76806			0.265973			0.132987	+	0.991118i	$0.457543\pi$
$854$		0			0
$855$	−26.5830	−	49.7322i	−0.909120	−	1.70081i
$856$		0			0
$857$		0			0			−	1.00000i	$-0.5\pi$
$857$		0			0				1.00000i	$0.5\pi$
$858$		0			0
$859$	−22.0245			−0.751467			−0.375734	−	0.926728i	$-0.622609\pi$
$859$	−22.0245			−0.751467			−0.375734	+	0.926728i	$0.622609\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	56.5685i		−	1.92562i	−0.270187	−	0.962808i	$-0.587086\pi$
$863$		−	56.5685i		−	1.92562i	0.270187	−	0.962808i	$-0.412914\pi$
$864$		0			0
$865$	−20.2065			−0.687043
$866$		0			0
$867$	−7.15464	−	28.5624i	−0.242984	−	0.970030i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$		0			0
$874$		0			0
$875$			30.3787i			1.02699i
$876$		0			0
$877$		0			0		1.00000			$0$
$877$		0			0		−1.00000			$\pi$
$878$		0			0
$879$	−52.1033	+	13.0514i	−1.75740	+	0.440214i
$880$		0			0
$881$		0			0			−	1.00000i	$-0.5\pi$
$881$		0			0				1.00000i	$0.5\pi$
$882$		0			0
$883$		0			0		1.00000			$0$
$883$		0			0		−1.00000			$\pi$
$884$		0			0
$885$	−13.1694	−	52.5742i	−0.442684	−	1.76726i
$886$		0			0
$887$		0			0			−	1.00000i	$-0.5\pi$
$887$		0			0				1.00000i	$0.5\pi$
$888$		0			0
$889$	5.29150			0.177471
$890$		0			0
$891$		0			0
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$	70.3320	−	17.6176i	2.34832	−	0.588234i
$898$		0			0
$899$		0			0
$900$		0			0
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$			23.0970i			0.767769i
$906$		0			0
$907$		0			0		1.00000			$0$
$907$		0			0		−1.00000			$\pi$
$908$		0			0
$909$	−25.4442	−	47.6018i	−0.843932	−	1.57885i
$910$		0			0
$911$			52.3832i			1.73553i	0.496972	+	0.867766i	$0.334445\pi$
$911$			52.3832i			1.73553i	−0.496972	+	0.867766i	$0.665555\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$	−1.98162	+	0.496381i	−0.0655105	+	0.0164098i
$916$		0			0
$917$		−	49.6356i		−	1.63911i
$918$		0			0
$919$	58.2065			1.92006			0.960028	−	0.279904i	$-0.0903025\pi$
$919$	58.2065			1.92006			0.960028	+	0.279904i	$0.0903025\pi$
$920$		0			0
$921$	−13.5203	−	53.9749i	−0.445508	−	1.77853i
$922$		0			0
$923$			31.6438i			1.04157i
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$		0			0			−	1.00000i	$-0.5\pi$
$929$		0			0				1.00000i	$0.5\pi$
$930$		0			0
$931$	−60.6382			−1.98734
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$		0			0		1.00000			$0$
$937$		0			0		−1.00000			$\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$			58.3152i			1.90102i	0.310693	+	0.950510i	$0.399439\pi$
$941$			58.3152i			1.90102i	−0.310693	+	0.950510i	$0.600561\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$	22.1033	−	20.0338i	0.719019	−	0.651698i
$946$		0			0
$947$		0			0			−	1.00000i	$-0.5\pi$
$947$		0			0				1.00000i	$0.5\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	29.9333i		−	0.969633i	−0.874616	−	0.484817i	$-0.838886\pi$
$953$		−	29.9333i		−	0.969633i	0.874616	−	0.484817i	$-0.161114\pi$
$954$		0			0
$955$	−49.0994			−1.58882
$956$		0			0
$957$		0			0
$958$		0			0
$959$		−	39.5980i		−	1.27869i
$960$		0			0
$961$	31.0000			1.00000
$962$		0			0
$963$		0			0
$964$		0			0
$965$			57.4103i			1.84810i
$966$		0			0
$967$	58.0000			1.86515			0.932577	−	0.360971i	$-0.117555\pi$
$967$	58.0000			1.86515			0.932577	+	0.360971i	$0.117555\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		−	0.136153i		−	0.00436936i	−0.999998	−	0.00218468i	$-0.999305\pi$
$971$		−	0.136153i		−	0.00436936i	0.999998	−	0.00218468i	$-0.000695406\pi$
$972$		0			0
$973$	36.2810			1.16312
$974$		0			0
$975$	−0.686270	−	2.73969i	−0.0219782	−	0.0877403i
$976$		0			0
$977$			59.8665i			1.91530i	0.287936	+	0.957650i	$0.407031\pi$
$977$			59.8665i			1.91530i	−0.287936	+	0.957650i	$0.592969\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$		0			0
$982$		0			0
$983$		0			0			−	1.00000i	$-0.5\pi$
$983$		0			0				1.00000i	$0.5\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$	−37.0405			−1.17663			−0.588315	−	0.808632i	$-0.700209\pi$
$991$	−37.0405			−1.17663			−0.588315	+	0.808632i	$0.700209\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−47.4688			−1.50335			−0.751675	−	0.659533i	$-0.770754\pi$
$997$	−47.4688			−1.50335			−0.751675	+	0.659533i	$0.770754\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	672.2.i.e.209.6		8
3.2	odd	2	inner	672.2.i.e.209.5		8
4.3	odd	2		168.2.i.d.125.2	✓	8
7.6	odd	2	inner	672.2.i.e.209.3		8
8.3	odd	2		168.2.i.d.125.3	yes	8
8.5	even	2	inner	672.2.i.e.209.3		8
12.11	even	2		168.2.i.d.125.6	yes	8
21.20	even	2	inner	672.2.i.e.209.4		8
24.5	odd	2	inner	672.2.i.e.209.4		8
24.11	even	2		168.2.i.d.125.7	yes	8
28.27	even	2		168.2.i.d.125.3	yes	8
56.13	odd	2	CM	672.2.i.e.209.6		8
56.27	even	2		168.2.i.d.125.2	✓	8
84.83	odd	2		168.2.i.d.125.7	yes	8
168.83	odd	2		168.2.i.d.125.6	yes	8
168.125	even	2	inner	672.2.i.e.209.5		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
168.2.i.d.125.2	✓	8	4.3	odd	2
168.2.i.d.125.2	✓	8	56.27	even	2
168.2.i.d.125.3	yes	8	8.3	odd	2
168.2.i.d.125.3	yes	8	28.27	even	2
168.2.i.d.125.6	yes	8	12.11	even	2
168.2.i.d.125.6	yes	8	168.83	odd	2
168.2.i.d.125.7	yes	8	24.11	even	2
168.2.i.d.125.7	yes	8	84.83	odd	2
672.2.i.e.209.3		8	7.6	odd	2	inner
672.2.i.e.209.3		8	8.5	even	2	inner
672.2.i.e.209.4		8	21.20	even	2	inner
672.2.i.e.209.4		8	24.5	odd	2	inner
672.2.i.e.209.5		8	3.2	odd	2	inner
672.2.i.e.209.5		8	168.125	even	2	inner
672.2.i.e.209.6		8	1.1	even	1	trivial
672.2.i.e.209.6		8	56.13	odd	2	CM

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:	\( N \)	\(=\)	\( 672 = 2^{5} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	672.i (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+(0.420861 + 1.68014i) q^{3} -2.16991i q^{5} -2.64575 q^{7} +(-2.64575 + 1.41421i) q^{9} +O(q^{10})\)	\(q+(0.420861 + 1.68014i) q^{3} -2.16991i q^{5} -2.64575 q^{7} +(-2.64575 + 1.41421i) q^{9} -5.59388 q^{13} +(3.64575 - 0.913230i) q^{15} -8.66259 q^{19} +(-1.11349 - 4.44524i) q^{21} +7.48331i q^{23} +0.291503 q^{25} +(-3.48957 - 3.85005i) q^{27} +5.74103i q^{35} +(-2.35425 - 9.39851i) q^{39} +(3.06871 + 5.74103i) q^{45} +7.00000 q^{49} +(-3.64575 - 14.5544i) q^{57} -14.4207i q^{59} -0.543544 q^{61} +(7.00000 - 3.74166i) q^{63} +12.1382i q^{65} +(-12.5730 + 3.14944i) q^{69} -5.65685i q^{71} +(0.122682 + 0.489766i) q^{75} -5.29150 q^{79} +(5.00000 - 7.48331i) q^{81} -1.40122i q^{83} +14.8000 q^{91} +18.7970i q^{95} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q+O(q^{10}) \) 8 * q	\( 8 q + 8 q^{15} - 40 q^{25} - 40 q^{39} + 56 q^{49} - 8 q^{57} + 56 q^{63} + 40 q^{81}+O(q^{100}) \) 8 * q + 8 * q^15 - 40 * q^25 - 40 * q^39 + 56 * q^49 - 8 * q^57 + 56 * q^63 + 40 * q^81

Introduction

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