LMFDB - Embedded newform 672.2.p.a.559.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [672,2,Mod(559,672)]

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([1, 1, 0, 1]))

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

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

chi := DirichletCharacter("672.559");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 672 = 2^{5} \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	672.p (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:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} + x^{14} - 4x^{12} - 4x^{10} + 16x^{8} - 16x^{6} - 64x^{4} + 64x^{2} + 256 $ x^16 + x^14 - 4x^12 - 4x^10 + 16x^8 - 16x^6 - 64x^4 + 64x^2 + 256
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{17} $
Twist minimal:	no (minimal twist has level 168)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			559.8
Root			$1.40199 + 0.185533i$ of defining polynomial
Character	$\chi$	$=$	672.559
Dual form			672.2.p.a.559.16

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{3} +3.84444 q^{5} +(-1.62140 - 2.09071i) q^{7} -1.00000 q^{9} +O(q^{10})$	$q-1.00000i q^{3} +3.84444 q^{5} +(-1.62140 - 2.09071i) q^{7} -1.00000 q^{9} +4.54637 q^{11} -1.81625 q^{13} -3.84444i q^{15} -3.49124i q^{17} -1.68700i q^{19} +(-2.09071 + 1.62140i) q^{21} +5.00632i q^{23} +9.77975 q^{25} +1.00000i q^{27} +1.81625i q^{29} -5.34329 q^{31} -4.54637i q^{33} +(-6.23338 - 8.03761i) q^{35} +1.42654i q^{37} +1.81625i q^{39} -8.97551i q^{41} +8.03761 q^{43} -3.84444 q^{45} -4.83580 q^{47} +(-1.74213 + 6.77975i) q^{49} -3.49124 q^{51} -5.87263i q^{53} +17.4783 q^{55} -1.68700 q^{57} +8.46675i q^{59} +3.01955 q^{61} +(1.62140 + 2.09071i) q^{63} -6.98249 q^{65} +4.42914 q^{67} +5.00632 q^{69} +1.47928i q^{71} -6.98249i q^{73} -9.77975i q^{75} +(-7.37148 - 9.50514i) q^{77} -2.97813i q^{79} +1.00000 q^{81} -10.5770i q^{83} -13.4219i q^{85} +1.81625 q^{87} +15.9580i q^{89} +(2.94487 + 3.79726i) q^{91} +5.34329i q^{93} -6.48559i q^{95} +11.6085i q^{97} -4.54637 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 16 q^{9}+O(q^{10}) $ 16 * q - 16 * q^9	$ 16 q - 16 q^{9} + 8 q^{11} + 16 q^{25} - 24 q^{35} + 8 q^{43} - 8 q^{49} - 16 q^{57} + 40 q^{67} + 16 q^{81} + 56 q^{91} - 8 q^{99}+O(q^{100}) $ 16 * q - 16 * q^9 + 8 * q^11 + 16 * q^25 - 24 * q^35 + 8 * q^43 - 8 * q^49 - 16 * q^57 + 40 * q^67 + 16 * q^81 + 56 * q^91 - 8 * q^99

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
$3$		−	1.00000i		−	0.577350i
$3$		−	1.00000i		−	0.577350i
$4$		0			0
$5$	3.84444			1.71929			0.859644	−	0.510894i	$-0.170686\pi$
$5$	3.84444			1.71929			0.859644	+	0.510894i	$0.170686\pi$
$6$		0			0
$7$	−1.62140	−	2.09071i	−0.612831	−	0.790214i
$7$	−1.62140	−	2.09071i	−0.612831	−	0.790214i
$8$		0			0
$9$	−1.00000			−0.333333
$10$		0			0
$11$	4.54637			1.37078			0.685391	−	0.728175i	$-0.259631\pi$
$11$	4.54637			1.37078			0.685391	+	0.728175i	$0.259631\pi$
$12$		0			0
$13$	−1.81625			−0.503738			−0.251869	−	0.967761i	$-0.581045\pi$
$13$	−1.81625			−0.503738			−0.251869	+	0.967761i	$0.581045\pi$
$14$		0			0
$15$		−	3.84444i		−	0.992631i
$16$		0			0
$17$		−	3.49124i		−	0.846751i	−0.905954	−	0.423375i	$-0.860845\pi$
$17$		−	3.49124i		−	0.846751i	0.905954	−	0.423375i	$-0.139155\pi$
$18$		0			0
$19$		−	1.68700i		−	0.387025i	−0.981098	−	0.193513i	$-0.938012\pi$
$19$		−	1.68700i		−	0.387025i	0.981098	−	0.193513i	$-0.0619881\pi$
$20$		0			0
$21$	−2.09071	+	1.62140i	−0.456230	+	0.353818i
$22$		0			0
$23$			5.00632i			1.04389i	0.852979	+	0.521945i	$0.174793\pi$
$23$			5.00632i			1.04389i	−0.852979	+	0.521945i	$0.825207\pi$
$24$		0			0
$25$	9.77975			1.95595
$26$		0			0
$27$			1.00000i			0.192450i
$28$		0			0
$29$			1.81625i			0.337270i	0.985679	+	0.168635i	$0.0539359\pi$
$29$			1.81625i			0.337270i	−0.985679	+	0.168635i	$0.946064\pi$
$30$		0			0
$31$	−5.34329			−0.959683			−0.479842	−	0.877355i	$-0.659306\pi$
$31$	−5.34329			−0.959683			−0.479842	+	0.877355i	$0.659306\pi$
$32$		0			0
$33$		−	4.54637i		−	0.791422i
$34$		0			0
$35$	−6.23338	−	8.03761i	−1.05363	−	1.35860i
$36$		0			0
$37$			1.42654i			0.234522i	0.993101	+	0.117261i	$0.0374114\pi$
$37$			1.42654i			0.234522i	−0.993101	+	0.117261i	$0.962589\pi$
$38$		0			0
$39$			1.81625i			0.290833i
$40$		0			0
$41$		−	8.97551i		−	1.40174i	−0.713290	−	0.700869i	$-0.752796\pi$
$41$		−	8.97551i		−	1.40174i	0.713290	−	0.700869i	$-0.247204\pi$
$42$		0			0
$43$	8.03761			1.22572			0.612862	−	0.790190i	$-0.290018\pi$
$43$	8.03761			1.22572			0.612862	+	0.790190i	$0.290018\pi$
$44$		0			0
$45$	−3.84444			−0.573096
$46$		0			0
$47$	−4.83580			−0.705374			−0.352687	−	0.935741i	$-0.614732\pi$
$47$	−4.83580			−0.705374			−0.352687	+	0.935741i	$0.614732\pi$
$48$		0			0
$49$	−1.74213	+	6.77975i	−0.248876	+	0.968535i
$50$		0			0
$51$	−3.49124			−0.488872
$52$		0			0
$53$		−	5.87263i		−	0.806668i	−0.915053	−	0.403334i	$-0.867851\pi$
$53$		−	5.87263i		−	0.806668i	0.915053	−	0.403334i	$-0.132149\pi$
$54$		0			0
$55$	17.4783			2.35677
$56$		0			0
$57$	−1.68700			−0.223449
$58$		0			0
$59$			8.46675i			1.10228i	0.834414	+	0.551139i	$0.185806\pi$
$59$			8.46675i			1.10228i	−0.834414	+	0.551139i	$0.814194\pi$
$60$		0			0
$61$	3.01955			0.386613			0.193307	−	0.981138i	$-0.438079\pi$
$61$	3.01955			0.386613			0.193307	+	0.981138i	$0.438079\pi$
$62$		0			0
$63$	1.62140	+	2.09071i	0.204277	+	0.263405i
$64$		0			0
$65$	−6.98249			−0.866071
$66$		0			0
$67$	4.42914			0.541105			0.270553	−	0.962705i	$-0.412794\pi$
$67$	4.42914			0.541105			0.270553	+	0.962705i	$0.412794\pi$
$68$		0			0
$69$	5.00632			0.602690
$70$		0			0
$71$			1.47928i			0.175558i	0.996140	+	0.0877791i	$0.0279769\pi$
$71$			1.47928i			0.175558i	−0.996140	+	0.0877791i	$0.972023\pi$
$72$		0			0
$73$		−	6.98249i		−	0.817238i	−0.912705	−	0.408619i	$-0.866010\pi$
$73$		−	6.98249i		−	0.817238i	0.912705	−	0.408619i	$-0.133990\pi$
$74$		0			0
$75$		−	9.77975i		−	1.12927i
$76$		0			0
$77$	−7.37148	−	9.50514i	−0.840058	−	1.08321i
$78$		0			0
$79$		−	2.97813i		−	0.335065i	−0.985867	−	0.167533i	$-0.946420\pi$
$79$		−	2.97813i		−	0.335065i	0.985867	−	0.167533i	$-0.0535800\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$		−	10.5770i		−	1.16098i	−0.814268	−	0.580489i	$-0.802862\pi$
$83$		−	10.5770i		−	1.16098i	0.814268	−	0.580489i	$-0.197138\pi$
$84$		0			0
$85$		−	13.4219i		−	1.45581i
$86$		0			0
$87$	1.81625			0.194723
$88$		0			0
$89$			15.9580i			1.69154i	0.533544	+	0.845772i	$0.320860\pi$
$89$			15.9580i			1.69154i	−0.533544	+	0.845772i	$0.679140\pi$
$90$		0			0
$91$	2.94487	+	3.79726i	0.308707	+	0.398061i
$92$		0			0
$93$			5.34329i			0.554073i
$94$		0			0
$95$		−	6.48559i		−	0.665408i
$96$		0			0
$97$			11.6085i			1.17866i	0.807892	+	0.589331i	$0.200609\pi$
$97$			11.6085i			1.17866i	−0.807892	+	0.589331i	$0.799391\pi$
$98$		0			0
$99$	−4.54637			−0.456928
$100$		0			0
$101$	−11.5333			−1.14761			−0.573805	−	0.818992i	$-0.694533\pi$
$101$	−11.5333			−1.14761			−0.573805	+	0.818992i	$0.694533\pi$
$102$		0			0
$103$	−1.14230			−0.112555			−0.0562773	−	0.998415i	$-0.517923\pi$
$103$	−1.14230			−0.112555			−0.0562773	+	0.998415i	$0.517923\pi$
$104$		0			0
$105$	−8.03761	+	6.23338i	−0.784391	+	0.608315i
$106$		0			0
$107$	−0.0796192			−0.00769707			−0.00384854	−	0.999993i	$-0.501225\pi$
$107$	−0.0796192			−0.00769707			−0.00384854	+	0.999993i	$0.501225\pi$
$108$		0			0
$109$			20.1081i			1.92601i	0.269489	+	0.963004i	$0.413145\pi$
$109$			20.1081i			1.92601i	−0.269489	+	0.963004i	$0.586855\pi$
$110$		0			0
$111$	1.42654			0.135401
$112$		0			0
$113$	4.98249			0.468713			0.234356	−	0.972151i	$-0.424702\pi$
$113$	4.98249			0.468713			0.234356	+	0.972151i	$0.424702\pi$
$114$		0			0
$115$			19.2465i			1.79475i
$116$		0			0
$117$	1.81625			0.167913
$118$		0			0
$119$	−7.29918	+	5.66070i	−0.669114	+	0.518915i
$120$		0			0
$121$	9.66949			0.879045
$122$		0			0
$123$	−8.97551			−0.809294
$124$		0			0
$125$	18.3755			1.64355
$126$		0			0
$127$			5.38471i			0.477816i	0.971042	+	0.238908i	$0.0767894\pi$
$127$			5.38471i			0.477816i	−0.971042	+	0.238908i	$0.923211\pi$
$128$		0			0
$129$		−	8.03761i		−	0.707673i
$130$		0			0
$131$			13.0927i			1.14392i	0.820282	+	0.571959i	$0.193816\pi$
$131$			13.0927i			1.14392i	−0.820282	+	0.571959i	$0.806184\pi$
$132$		0			0
$133$	−3.52704	+	2.73531i	−0.305833	+	0.237181i
$134$		0			0
$135$			3.84444i			0.330877i
$136$		0			0
$137$	−19.5595			−1.67108			−0.835540	−	0.549429i	$-0.814845\pi$
$137$	−19.5595			−1.67108			−0.835540	+	0.549429i	$0.814845\pi$
$138$		0			0
$139$			14.5910i			1.23759i	0.785553	+	0.618795i	$0.212379\pi$
$139$			14.5910i			1.23759i	−0.785553	+	0.618795i	$0.787621\pi$
$140$		0			0
$141$			4.83580i			0.407248i
$142$		0			0
$143$	−8.25737			−0.690516
$144$		0			0
$145$			6.98249i			0.579864i
$146$		0			0
$147$	6.77975	+	1.74213i	0.559184	+	0.143689i
$148$		0			0
$149$			18.3973i			1.50717i	0.657352	+	0.753584i	$0.271676\pi$
$149$			18.3973i			1.50717i	−0.657352	+	0.753584i	$0.728324\pi$
$150$		0			0
$151$			9.88759i			0.804641i	0.915499	+	0.402320i	$0.131796\pi$
$151$			9.88759i			0.804641i	−0.915499	+	0.402320i	$0.868204\pi$
$152$		0			0
$153$			3.49124i			0.282250i
$154$		0			0
$155$	−20.5420			−1.64997
$156$		0			0
$157$	−12.3582			−0.986294			−0.493147	−	0.869946i	$-0.664154\pi$
$157$	−12.3582			−0.986294			−0.493147	+	0.869946i	$0.664154\pi$
$158$		0			0
$159$	−5.87263			−0.465730
$160$		0			0
$161$	10.4668	−	8.11723i	0.824896	−	0.639728i
$162$		0			0
$163$	3.57086			0.279692			0.139846	−	0.990173i	$-0.455339\pi$
$163$	3.57086			0.279692			0.139846	+	0.990173i	$0.455339\pi$
$164$		0			0
$165$		−	17.4783i		−	1.36068i
$166$		0			0
$167$	19.5788			1.51505			0.757525	−	0.652806i	$-0.226408\pi$
$167$	19.5788			1.51505			0.757525	+	0.652806i	$0.226408\pi$
$168$		0			0
$169$	−9.70122			−0.746248
$170$		0			0
$171$			1.68700i			0.129008i
$172$		0			0
$173$	5.49424			0.417719			0.208860	−	0.977946i	$-0.433025\pi$
$173$	5.49424			0.417719			0.208860	+	0.977946i	$0.433025\pi$
$174$		0			0
$175$	−15.8569	−	20.4466i	−1.19867	−	1.54562i
$176$		0			0
$177$	8.46675			0.636400
$178$		0			0
$179$	−10.0306			−0.749725			−0.374862	−	0.927080i	$-0.622310\pi$
$179$	−10.0306			−0.749725			−0.374862	+	0.927080i	$0.622310\pi$
$180$		0			0
$181$	24.2481			1.80235			0.901174	−	0.433458i	$-0.142707\pi$
$181$	24.2481			1.80235			0.901174	+	0.433458i	$0.142707\pi$
$182$		0			0
$183$		−	3.01955i		−	0.223211i
$184$		0			0
$185$			5.48426i			0.403211i
$186$		0			0
$187$		−	15.8725i		−	1.16071i
$188$		0			0
$189$	2.09071	−	1.62140i	0.152077	−	0.117939i
$190$		0			0
$191$		−	8.49422i		−	0.614620i	−0.951609	−	0.307310i	$-0.900571\pi$
$191$		−	8.49422i		−	0.614620i	0.951609	−	0.307310i	$-0.0994288\pi$
$192$		0			0
$193$	−11.2955			−0.813067			−0.406533	−	0.913636i	$-0.633262\pi$
$193$	−11.2955			−0.813067			−0.406533	+	0.913636i	$0.633262\pi$
$194$		0			0
$195$			6.98249i			0.500026i
$196$		0			0
$197$			2.38473i			0.169905i	0.996385	+	0.0849526i	$0.0270739\pi$
$197$			2.38473i			0.169905i	−0.996385	+	0.0849526i	$0.972926\pi$
$198$		0			0
$199$	−2.46338			−0.174624			−0.0873120	−	0.996181i	$-0.527828\pi$
$199$	−2.46338			−0.174624			−0.0873120	+	0.996181i	$0.527828\pi$
$200$		0			0
$201$		−	4.42914i		−	0.312407i
$202$		0			0
$203$	3.79726	−	2.94487i	0.266515	−	0.206690i
$204$		0			0
$205$		−	34.5058i		−	2.40999i
$206$		0			0
$207$		−	5.00632i		−	0.347963i
$208$		0			0
$209$		−	7.66975i		−	0.530528i
$210$		0			0
$211$	0.0376150			0.00258952			0.00129476	−	0.999999i	$-0.499588\pi$
$211$	0.0376150			0.00258952			0.00129476	+	0.999999i	$0.499588\pi$
$212$		0			0
$213$	1.47928			0.101359
$214$		0			0
$215$	30.9002			2.10737
$216$		0			0
$217$	8.66361	+	11.1713i	0.588124	+	0.758355i
$218$		0			0
$219$	−6.98249			−0.471833
$220$		0			0
$221$			6.34099i			0.426541i
$222$		0			0
$223$	−12.9144			−0.864812			−0.432406	−	0.901679i	$-0.642335\pi$
$223$	−12.9144			−0.864812			−0.432406	+	0.901679i	$0.642335\pi$
$224$		0			0
$225$	−9.77975			−0.651983
$226$		0			0
$227$			1.48426i			0.0985141i	0.998786	+	0.0492571i	$0.0156854\pi$
$227$			1.48426i			0.0985141i	−0.998786	+	0.0492571i	$0.984315\pi$
$228$		0			0
$229$	4.66934			0.308559			0.154279	−	0.988027i	$-0.450694\pi$
$229$	4.66934			0.308559			0.154279	+	0.988027i	$0.450694\pi$
$230$		0			0
$231$	−9.50514	+	7.37148i	−0.625392	+	0.485008i
$232$		0			0
$233$	−8.35650			−0.547452			−0.273726	−	0.961808i	$-0.588256\pi$
$233$	−8.35650			−0.547452			−0.273726	+	0.961808i	$0.588256\pi$
$234$		0			0
$235$	−18.5910			−1.21274
$236$		0			0
$237$	−2.97813			−0.193450
$238$		0			0
$239$		−	19.8547i		−	1.28430i	−0.766580	−	0.642148i	$-0.778043\pi$
$239$		−	19.8547i		−	1.28430i	0.766580	−	0.642148i	$-0.221957\pi$
$240$		0			0
$241$			6.57701i			0.423662i	0.977306	+	0.211831i	$0.0679427\pi$
$241$			6.57701i			0.423662i	−0.977306	+	0.211831i	$0.932057\pi$
$242$		0			0
$243$		−	1.00000i		−	0.0641500i
$244$		0			0
$245$	−6.69753	+	26.0644i	−0.427889	+	1.66519i
$246$		0			0
$247$			3.06403i			0.194960i
$248$		0			0
$249$	−10.5770			−0.670291
$250$		0			0
$251$		−	4.85827i		−	0.306652i	−0.988176	−	0.153326i	$-0.951002\pi$
$251$		−	4.85827i		−	0.306652i	0.988176	−	0.153326i	$-0.0489984\pi$
$252$		0			0
$253$			22.7606i			1.43094i
$254$		0			0
$255$	−13.4219			−0.840511
$256$		0			0
$257$			9.20998i			0.574503i	0.957855	+	0.287251i	$0.0927415\pi$
$257$			9.20998i			0.574503i	−0.957855	+	0.287251i	$0.907259\pi$
$258$		0			0
$259$	2.98249	−	2.31300i	0.185323	−	0.143723i
$260$		0			0
$261$		−	1.81625i		−	0.112423i
$262$		0			0
$263$			3.90849i			0.241008i	0.992713	+	0.120504i	$0.0384511\pi$
$263$			3.90849i			0.241008i	−0.992713	+	0.120504i	$0.961549\pi$
$264$		0			0
$265$		−	22.5770i		−	1.38689i
$266$		0			0
$267$	15.9580			0.976613
$268$		0			0
$269$	−9.24872			−0.563905			−0.281952	−	0.959428i	$-0.590982\pi$
$269$	−9.24872			−0.563905			−0.281952	+	0.959428i	$0.590982\pi$
$270$		0			0
$271$	16.5201			1.00352			0.501762	−	0.865006i	$-0.332685\pi$
$271$	16.5201			1.00352			0.501762	+	0.865006i	$0.332685\pi$
$272$		0			0
$273$	3.79726	−	2.94487i	0.229821	−	0.178232i
$274$		0			0
$275$	44.4624			2.68118
$276$		0			0
$277$		−	3.26465i		−	0.196154i	−0.995179	−	0.0980769i	$-0.968731\pi$
$277$		−	3.26465i		−	0.196154i	0.995179	−	0.0980769i	$-0.0312691\pi$
$278$		0			0
$279$	5.34329			0.319894
$280$		0			0
$281$	19.1680			1.14347			0.571733	−	0.820440i	$-0.306271\pi$
$281$	19.1680			1.14347			0.571733	+	0.820440i	$0.306271\pi$
$282$		0			0
$283$		−	20.2780i		−	1.20540i	−0.797968	−	0.602700i	$-0.794092\pi$
$283$		−	20.2780i		−	1.20540i	0.797968	−	0.602700i	$-0.205908\pi$
$284$		0			0
$285$	−6.48559			−0.384173
$286$		0			0
$287$	−18.7652	+	14.5529i	−1.10767	+	0.859029i
$288$		0			0
$289$	4.81122			0.283013
$290$		0			0
$291$	11.6085			0.680501
$292$		0			0
$293$	−5.07037			−0.296214			−0.148107	−	0.988971i	$-0.547318\pi$
$293$	−5.07037			−0.296214			−0.148107	+	0.988971i	$0.547318\pi$
$294$		0			0
$295$			32.5500i			1.89513i
$296$		0			0
$297$			4.54637i			0.263807i
$298$		0			0
$299$		−	9.09274i		−	0.525847i
$300$		0			0
$301$	−13.0322	−	16.8043i	−0.751162	−	0.968585i
$302$		0			0
$303$			11.5333i			0.662573i
$304$		0			0
$305$	11.6085			0.664700
$306$		0			0
$307$		−	15.2465i		−	0.870164i	−0.900391	−	0.435082i	$-0.856719\pi$
$307$		−	15.2465i		−	0.870164i	0.900391	−	0.435082i	$-0.143281\pi$
$308$		0			0
$309$			1.14230i			0.0649834i
$310$		0			0
$311$	−8.91481			−0.505513			−0.252756	−	0.967530i	$-0.581337\pi$
$311$	−8.91481			−0.505513			−0.252756	+	0.967530i	$0.581337\pi$
$312$		0			0
$313$		−	24.9335i		−	1.40932i	−0.709543	−	0.704662i	$-0.751098\pi$
$313$		−	24.9335i		−	1.40932i	0.709543	−	0.704662i	$-0.248902\pi$
$314$		0			0
$315$	6.23338	+	8.03761i	0.351211	+	0.452868i
$316$		0			0
$317$		−	16.1127i		−	0.904980i	−0.891769	−	0.452490i	$-0.850536\pi$
$317$		−	16.1127i		−	0.904980i	0.891769	−	0.452490i	$-0.149464\pi$
$318$		0			0
$319$			8.25737i			0.462324i
$320$		0			0
$321$			0.0796192i			0.00444391i
$322$		0			0
$323$	−5.88974			−0.327714
$324$		0			0
$325$	−17.7625			−0.985287
$326$		0			0
$327$	20.1081			1.11198
$328$		0			0
$329$	7.84076	+	10.1103i	0.432275	+	0.557396i
$330$		0			0
$331$	−22.3802			−1.23012			−0.615062	−	0.788479i	$-0.710869\pi$
$331$	−22.3802			−1.23012			−0.615062	+	0.788479i	$0.710869\pi$
$332$		0			0
$333$		−	1.42654i		−	0.0781741i
$334$		0			0
$335$	17.0276			0.930315
$336$		0			0
$337$	2.22051			0.120959			0.0604795	−	0.998169i	$-0.480737\pi$
$337$	2.22051			0.120959			0.0604795	+	0.998169i	$0.480737\pi$
$338$		0			0
$339$		−	4.98249i		−	0.270612i
$340$		0			0
$341$	−24.2926			−1.31552
$342$		0			0
$343$	16.9992	−	7.35038i	0.917869	−	0.396883i
$344$		0			0
$345$	19.2465			1.03620
$346$		0			0
$347$	17.4186			0.935080			0.467540	−	0.883972i	$-0.345140\pi$
$347$	17.4186			0.935080			0.467540	+	0.883972i	$0.345140\pi$
$348$		0			0
$349$	29.0839			1.55683			0.778413	−	0.627753i	$-0.216025\pi$
$349$	29.0839			1.55683			0.778413	+	0.627753i	$0.216025\pi$
$350$		0			0
$351$		−	1.81625i		−	0.0969445i
$352$		0			0
$353$		−	7.02449i		−	0.373876i	−0.982372	−	0.186938i	$-0.940144\pi$
$353$		−	7.02449i		−	0.373876i	0.982372	−	0.186938i	$-0.0598563\pi$
$354$		0			0
$355$			5.68700i			0.301835i
$356$		0			0
$357$	5.66070	+	7.29918i	0.299596	+	0.386313i
$358$		0			0
$359$		−	11.1509i		−	0.588521i	−0.955725	−	0.294260i	$-0.904927\pi$
$359$		−	11.1509i		−	0.588521i	0.955725	−	0.294260i	$-0.0950733\pi$
$360$		0			0
$361$	16.1540			0.850211
$362$		0			0
$363$		−	9.66949i		−	0.507517i
$364$		0			0
$365$		−	26.8438i		−	1.40507i
$366$		0			0
$367$	−6.42880			−0.335581			−0.167790	−	0.985823i	$-0.553663\pi$
$367$	−6.42880			−0.335581			−0.167790	+	0.985823i	$0.553663\pi$
$368$		0			0
$369$			8.97551i			0.467246i
$370$		0			0
$371$	−12.2780	+	9.52188i	−0.637440	+	0.494351i
$372$		0			0
$373$			1.75527i			0.0908842i	0.998967	+	0.0454421i	$0.0144697\pi$
$373$			1.75527i			0.0908842i	−0.998967	+	0.0454421i	$0.985530\pi$
$374$		0			0
$375$		−	18.3755i		−	0.948905i
$376$		0			0
$377$		−	3.29878i		−	0.169896i
$378$		0			0
$379$	19.0061			0.976280			0.488140	−	0.872765i	$-0.337676\pi$
$379$	19.0061			0.976280			0.488140	+	0.872765i	$0.337676\pi$
$380$		0			0
$381$	5.38471			0.275867
$382$		0			0
$383$	−28.4936			−1.45595			−0.727977	−	0.685602i	$-0.759539\pi$
$383$	−28.4936			−1.45595			−0.727977	+	0.685602i	$0.759539\pi$
$384$		0			0
$385$	−28.3392	−	36.5420i	−1.44430	−	1.86235i
$386$		0			0
$387$	−8.03761			−0.408575
$388$		0			0
$389$		−	29.3858i		−	1.48992i	−0.667110	−	0.744960i	$-0.732469\pi$
$389$		−	29.3858i		−	1.48992i	0.667110	−	0.744960i	$-0.267531\pi$
$390$		0			0
$391$	17.4783			0.883914
$392$		0			0
$393$	13.0927			0.660442
$394$		0			0
$395$		−	11.4492i		−	0.576074i
$396$		0			0
$397$	−15.5442			−0.780143			−0.390071	−	0.920785i	$-0.627550\pi$
$397$	−15.5442			−0.780143			−0.390071	+	0.920785i	$0.627550\pi$
$398$		0			0
$399$	2.73531	+	3.52704i	0.136937	+	0.176573i
$400$		0			0
$401$	−15.5595			−0.777004			−0.388502	−	0.921448i	$-0.627007\pi$
$401$	−15.5595			−0.777004			−0.388502	+	0.921448i	$0.627007\pi$
$402$		0			0
$403$	9.70478			0.483429
$404$		0			0
$405$	3.84444			0.191032
$406$		0			0
$407$			6.48559i			0.321479i
$408$		0			0
$409$			18.3565i			0.907670i	0.891086	+	0.453835i	$0.149945\pi$
$409$			18.3565i			0.907670i	−0.891086	+	0.453835i	$0.850055\pi$
$410$		0			0
$411$			19.5595i			0.964799i
$412$		0			0
$413$	17.7015	−	13.7280i	0.871035	−	0.675510i
$414$		0			0
$415$		−	40.6627i		−	1.99605i
$416$		0			0
$417$	14.5910			0.714523
$418$		0			0
$419$			21.1680i			1.03412i	0.855948	+	0.517062i	$0.172974\pi$
$419$			21.1680i			1.03412i	−0.855948	+	0.517062i	$0.827026\pi$
$420$		0			0
$421$			2.20597i			0.107512i	0.998554	+	0.0537561i	$0.0171193\pi$
$421$			2.20597i			0.107512i	−0.998554	+	0.0537561i	$0.982881\pi$
$422$		0			0
$423$	4.83580			0.235125
$424$		0			0
$425$		−	34.1435i		−	1.65620i
$426$		0			0
$427$	−4.89589	−	6.31300i	−0.236929	−	0.305507i
$428$		0			0
$429$			8.25737i			0.398669i
$430$		0			0
$431$		−	17.4255i		−	0.839358i	−0.907673	−	0.419679i	$-0.862143\pi$
$431$		−	17.4255i		−	0.839358i	0.907673	−	0.419679i	$-0.137857\pi$
$432$		0			0
$433$		−	2.18549i		−	0.105028i	−0.998620	−	0.0525139i	$-0.983277\pi$
$433$		−	2.18549i		−	0.105028i	0.998620	−	0.0525139i	$-0.0167234\pi$
$434$		0			0
$435$	6.98249			0.334785
$436$		0			0
$437$	8.44568			0.404012
$438$		0			0
$439$	−27.5128			−1.31311			−0.656556	−	0.754277i	$-0.727987\pi$
$439$	−27.5128			−1.31311			−0.656556	+	0.754277i	$0.727987\pi$
$440$		0			0
$441$	1.74213	−	6.77975i	0.0829587	−	0.322845i
$442$		0			0
$443$	−3.37840			−0.160513			−0.0802563	−	0.996774i	$-0.525574\pi$
$443$	−3.37840			−0.160513			−0.0802563	+	0.996774i	$0.525574\pi$
$444$		0			0
$445$			61.3496i			2.90825i
$446$		0			0
$447$	18.3973			0.870163
$448$		0			0
$449$	1.76553			0.0833206			0.0416603	−	0.999132i	$-0.486735\pi$
$449$	1.76553			0.0833206			0.0416603	+	0.999132i	$0.486735\pi$
$450$		0			0
$451$		−	40.8060i		−	1.92148i
$452$		0			0
$453$	9.88759			0.464560
$454$		0			0
$455$	11.3214	+	14.5984i	0.530755	+	0.684381i
$456$		0			0
$457$	−24.4178			−1.14222			−0.571108	−	0.820875i	$-0.693486\pi$
$457$	−24.4178			−1.14222			−0.571108	+	0.820875i	$0.693486\pi$
$458$		0			0
$459$	3.49124			0.162957
$460$		0			0
$461$	7.77884			0.362297			0.181148	−	0.983456i	$-0.442019\pi$
$461$	7.77884			0.362297			0.181148	+	0.983456i	$0.442019\pi$
$462$		0			0
$463$		−	16.5615i		−	0.769678i	−0.922984	−	0.384839i	$-0.874257\pi$
$463$		−	16.5615i		−	0.769678i	0.922984	−	0.384839i	$-0.125743\pi$
$464$		0			0
$465$			20.5420i			0.952612i
$466$		0			0
$467$		−	14.1103i		−	0.652945i	−0.945207	−	0.326472i	$-0.894140\pi$
$467$		−	14.1103i		−	0.652945i	0.945207	−	0.326472i	$-0.105860\pi$
$468$		0			0
$469$	−7.18140	−	9.26004i	−0.331606	−	0.427589i
$470$		0			0
$471$			12.3582i			0.569437i
$472$		0			0
$473$	36.5420			1.68020
$474$		0			0
$475$		−	16.4985i		−	0.757002i
$476$		0			0
$477$			5.87263i			0.268889i
$478$		0			0
$479$	−8.74757			−0.399687			−0.199843	−	0.979828i	$-0.564043\pi$
$479$	−8.74757			−0.399687			−0.199843	+	0.979828i	$0.564043\pi$
$480$		0			0
$481$		−	2.59097i		−	0.118138i
$482$		0			0
$483$	−8.11723	−	10.4668i	−0.369347	−	0.476254i
$484$		0			0
$485$			44.6281i			2.02646i
$486$		0			0
$487$			31.0252i			1.40589i	0.711246	+	0.702943i	$0.248131\pi$
$487$			31.0252i			1.40589i	−0.711246	+	0.702943i	$0.751869\pi$
$488$		0			0
$489$		−	3.57086i		−	0.161480i
$490$		0			0
$491$	22.0446			0.994859			0.497429	−	0.867505i	$-0.334277\pi$
$491$	22.0446			0.994859			0.497429	+	0.867505i	$0.334277\pi$
$492$		0			0
$493$	6.34099			0.285584
$494$		0			0
$495$	−17.4783			−0.785590
$496$		0			0
$497$	3.09274	−	2.39850i	0.138728	−	0.107587i
$498$		0			0
$499$	−40.6409			−1.81934			−0.909668	−	0.415337i	$-0.863664\pi$
$499$	−40.6409			−1.81934			−0.909668	+	0.415337i	$0.863664\pi$
$500$		0			0
$501$		−	19.5788i		−	0.874715i
$502$		0			0
$503$	−33.1848			−1.47964			−0.739818	−	0.672807i	$-0.765088\pi$
$503$	−33.1848			−1.47964			−0.739818	+	0.672807i	$0.765088\pi$
$504$		0			0
$505$	−44.3392			−1.97307
$506$		0			0
$507$			9.70122i			0.430846i
$508$		0			0
$509$	35.5240			1.57457			0.787287	−	0.616586i	$-0.211485\pi$
$509$	35.5240			1.57457			0.787287	+	0.616586i	$0.211485\pi$
$510$		0			0
$511$	−14.5984	+	11.3214i	−0.645793	+	0.500829i
$512$		0			0
$513$	1.68700			0.0744831
$514$		0			0
$515$	−4.39152			−0.193514
$516$		0			0
$517$	−21.9853			−0.966914
$518$		0			0
$519$		−	5.49424i		−	0.241170i
$520$		0			0
$521$			26.4737i			1.15984i	0.814675	+	0.579918i	$0.196915\pi$
$521$			26.4737i			1.15984i	−0.814675	+	0.579918i	$0.803085\pi$
$522$		0			0
$523$			41.7100i			1.82385i	0.410358	+	0.911924i	$0.365404\pi$
$523$			41.7100i			1.82385i	−0.410358	+	0.911924i	$0.634596\pi$
$524$		0			0
$525$	−20.4466	+	15.8569i	−0.892363	+	0.692051i
$526$		0			0
$527$			18.6547i			0.812613i
$528$		0			0
$529$	−2.06320			−0.0897043
$530$		0			0
$531$		−	8.46675i		−	0.367426i
$532$		0			0
$533$			16.3018i			0.706110i
$534$		0			0
$535$	−0.306091			−0.0132335
$536$		0			0
$537$			10.0306i			0.432854i
$538$		0			0
$539$	−7.92038	+	30.8232i	−0.341155	+	1.32765i
$540$		0			0
$541$		−	10.3187i		−	0.443637i	−0.975088	−	0.221818i	$-0.928801\pi$
$541$		−	10.3187i		−	0.443637i	0.975088	−	0.221818i	$-0.0711992\pi$
$542$		0			0
$543$		−	24.2481i		−	1.04059i
$544$		0			0
$545$			77.3045i			3.31136i
$546$		0			0
$547$	−19.6461			−0.840006			−0.420003	−	0.907523i	$-0.637971\pi$
$547$	−19.6461			−0.840006			−0.420003	+	0.907523i	$0.637971\pi$
$548$		0			0
$549$	−3.01955			−0.128871
$550$		0			0
$551$	3.06403			0.130532
$552$		0			0
$553$	−6.22640	+	4.82873i	−0.264773	+	0.205339i
$554$		0			0
$555$	5.48426			0.232794
$556$		0			0
$557$		−	30.0989i		−	1.27533i	−0.770314	−	0.637665i	$-0.779900\pi$
$557$		−	30.0989i		−	1.27533i	0.770314	−	0.637665i	$-0.220100\pi$
$558$		0			0
$559$	−14.5984			−0.617445
$560$		0			0
$561$	−15.8725			−0.670137
$562$		0			0
$563$		−	6.35650i		−	0.267894i	−0.990988	−	0.133947i	$-0.957235\pi$
$563$		−	6.35650i		−	0.267894i	0.990988	−	0.133947i	$-0.0427653\pi$
$564$		0			0
$565$	19.1549			0.805852
$566$		0			0
$567$	−1.62140	−	2.09071i	−0.0680923	−	0.0878015i
$568$		0			0
$569$	−37.7589			−1.58294			−0.791469	−	0.611210i	$-0.790683\pi$
$569$	−37.7589			−1.58294			−0.791469	+	0.611210i	$0.790683\pi$
$570$		0			0
$571$	29.0324			1.21497			0.607484	−	0.794332i	$-0.292179\pi$
$571$	29.0324			1.21497			0.607484	+	0.794332i	$0.292179\pi$
$572$		0			0
$573$	−8.49422			−0.354851
$574$		0			0
$575$			48.9605i			2.04179i
$576$		0			0
$577$		−	38.4060i		−	1.59886i	−0.600758	−	0.799431i	$-0.705134\pi$
$577$		−	38.4060i		−	1.59886i	0.600758	−	0.799431i	$-0.294866\pi$
$578$		0			0
$579$			11.2955i			0.469424i
$580$		0			0
$581$	−22.1134	+	17.1495i	−0.917420	+	0.711483i
$582$		0			0
$583$		−	26.6992i		−	1.10577i
$584$		0			0
$585$	6.98249			0.288690
$586$		0			0
$587$			0.157054i			0.00648231i	0.999995	+	0.00324116i	$0.00103169\pi$
$587$			0.157054i			0.00648231i	−0.999995	+	0.00324116i	$0.998968\pi$
$588$		0			0
$589$			9.01416i			0.371422i
$590$		0			0
$591$	2.38473			0.0980948
$592$		0			0
$593$		−	37.5927i		−	1.54375i	−0.635776	−	0.771874i	$-0.719320\pi$
$593$		−	37.5927i		−	1.54375i	0.635776	−	0.771874i	$-0.280680\pi$
$594$		0			0
$595$	−28.0613	+	21.7622i	−1.15040	+	0.892165i
$596$		0			0
$597$			2.46338i			0.100819i
$598$		0			0
$599$		−	18.7569i		−	0.766387i	−0.923668	−	0.383194i	$-0.874824\pi$
$599$		−	18.7569i		−	0.766387i	0.923668	−	0.383194i	$-0.125176\pi$
$600$		0			0
$601$		−	2.44051i		−	0.0995503i	−0.998760	−	0.0497751i	$-0.984150\pi$
$601$		−	2.44051i		−	0.0995503i	0.998760	−	0.0497751i	$-0.0158505\pi$
$602$		0			0
$603$	−4.42914			−0.180368
$604$		0			0
$605$	37.1738			1.51133
$606$		0			0
$607$	40.4839			1.64319			0.821596	−	0.570070i	$-0.193084\pi$
$607$	40.4839			1.64319			0.821596	+	0.570070i	$0.193084\pi$
$608$		0			0
$609$	−2.94487	−	3.79726i	−0.119332	−	0.153873i
$610$		0			0
$611$	8.78304			0.355324
$612$		0			0
$613$		−	8.57378i		−	0.346292i	−0.984896	−	0.173146i	$-0.944607\pi$
$613$		−	8.57378i		−	0.346292i	0.984896	−	0.173146i	$-0.0553932\pi$
$614$		0			0
$615$	−34.5058			−1.39141
$616$		0			0
$617$	31.3390			1.26166			0.630830	−	0.775921i	$-0.282715\pi$
$617$	31.3390			1.26166			0.630830	+	0.775921i	$0.282715\pi$
$618$		0			0
$619$			41.5595i			1.67042i	0.549933	+	0.835209i	$0.314653\pi$
$619$			41.5595i			1.67042i	−0.549933	+	0.835209i	$0.685347\pi$
$620$		0			0
$621$	−5.00632			−0.200897
$622$		0			0
$623$	33.3635	−	25.8743i	1.33668	−	1.03663i
$624$		0			0
$625$	21.7447			0.869789
$626$		0			0
$627$	−7.66975			−0.306300
$628$		0			0
$629$	4.98041			0.198582
$630$		0			0
$631$		−	48.0528i		−	1.91295i	−0.291814	−	0.956475i	$-0.594259\pi$
$631$		−	48.0528i		−	1.91295i	0.291814	−	0.956475i	$-0.405741\pi$
$632$		0			0
$633$		−	0.0376150i		−	0.00149506i
$634$		0			0
$635$			20.7012i			0.821503i
$636$		0			0
$637$	3.16416	−	12.3137i	0.125368	−	0.487888i
$638$		0			0
$639$		−	1.47928i		−	0.0585194i
$640$		0			0
$641$	−26.9335			−1.06381			−0.531905	−	0.846804i	$-0.678524\pi$
$641$	−26.9335			−1.06381			−0.531905	+	0.846804i	$0.678524\pi$
$642$		0			0
$643$		−	20.2780i		−	0.799685i	−0.916584	−	0.399843i	$-0.869065\pi$
$643$		−	20.2780i		−	0.799685i	0.916584	−	0.399843i	$-0.130935\pi$
$644$		0			0
$645$		−	30.9002i		−	1.21669i
$646$		0			0
$647$	16.3928			0.644466			0.322233	−	0.946660i	$-0.395567\pi$
$647$	16.3928			0.644466			0.322233	+	0.946660i	$0.395567\pi$
$648$		0			0
$649$			38.4930i			1.51098i
$650$		0			0
$651$	11.1713	−	8.66361i	0.437837	−	0.339553i
$652$		0			0
$653$		−	27.4567i		−	1.07447i	−0.843434	−	0.537233i	$-0.819470\pi$
$653$		−	27.4567i		−	1.07447i	0.843434	−	0.537233i	$-0.180530\pi$
$654$		0			0
$655$			50.3343i			1.96672i
$656$		0			0
$657$			6.98249i			0.272413i
$658$		0			0
$659$	−34.1059			−1.32858			−0.664288	−	0.747477i	$-0.731265\pi$
$659$	−34.1059			−1.32858			−0.664288	+	0.747477i	$0.731265\pi$
$660$		0			0
$661$	42.0551			1.63575			0.817877	−	0.575393i	$-0.195151\pi$
$661$	42.0551			1.63575			0.817877	+	0.575393i	$0.195151\pi$
$662$		0			0
$663$	6.34099			0.246263
$664$		0			0
$665$	−13.5595	+	10.5157i	−0.525815	+	0.407783i
$666$		0			0
$667$	−9.09274			−0.352072
$668$		0			0
$669$			12.9144i			0.499300i
$670$		0			0
$671$	13.7280			0.529963
$672$		0			0
$673$	43.0405			1.65909			0.829544	−	0.558441i	$-0.188600\pi$
$673$	43.0405			1.65909			0.829544	+	0.558441i	$0.188600\pi$
$674$		0			0
$675$			9.77975i			0.376423i
$676$		0			0
$677$	−5.91811			−0.227451			−0.113726	−	0.993512i	$-0.536278\pi$
$677$	−5.91811			−0.227451			−0.113726	+	0.993512i	$0.536278\pi$
$678$		0			0
$679$	24.2700	−	18.8220i	0.931395	−	0.722321i
$680$		0			0
$681$	1.48426			0.0568772
$682$		0			0
$683$	1.50261			0.0574958			0.0287479	−	0.999587i	$-0.490848\pi$
$683$	1.50261			0.0574958			0.0287479	+	0.999587i	$0.490848\pi$
$684$		0			0
$685$	−75.1954			−2.87307
$686$		0			0
$687$		−	4.66934i		−	0.178146i
$688$		0			0
$689$			10.6662i			0.406350i
$690$		0			0
$691$		−	8.62599i		−	0.328148i	−0.986448	−	0.164074i	$-0.947536\pi$
$691$		−	8.62599i		−	0.328148i	0.986448	−	0.164074i	$-0.0524636\pi$
$692$		0			0
$693$	7.37148	+	9.50514i	0.280019	+	0.361070i
$694$		0			0
$695$			56.0941i			2.12777i
$696$		0			0
$697$	−31.3357			−1.18692
$698$		0			0
$699$			8.35650i			0.316072i
$700$		0			0
$701$		−	10.4065i		−	0.393050i	−0.980499	−	0.196525i	$-0.937034\pi$
$701$		−	10.4065i		−	0.393050i	0.980499	−	0.196525i	$-0.0629656\pi$
$702$		0			0
$703$	2.40659			0.0907661
$704$		0			0
$705$			18.5910i			0.700176i
$706$		0			0
$707$	18.7001	+	24.1128i	0.703291	+	0.906857i
$708$		0			0
$709$		−	34.8737i		−	1.30971i	−0.755755	−	0.654855i	$-0.772730\pi$
$709$		−	34.8737i		−	1.30971i	0.755755	−	0.654855i	$-0.227270\pi$
$710$		0			0
$711$			2.97813i			0.111688i
$712$		0			0
$713$		−	26.7502i		−	1.00180i
$714$		0			0
$715$	−31.7450			−1.18719
$716$		0			0
$717$	−19.8547			−0.741489
$718$		0			0
$719$	−8.74757			−0.326229			−0.163115	−	0.986607i	$-0.552154\pi$
$719$	−8.74757			−0.326229			−0.163115	+	0.986607i	$0.552154\pi$
$720$		0			0
$721$	1.85213	+	2.38823i	0.0689769	+	0.0889422i
$722$		0			0
$723$	6.57701			0.244602
$724$		0			0
$725$			17.7625i			0.659683i
$726$		0			0
$727$	23.7851			0.882140			0.441070	−	0.897473i	$-0.354599\pi$
$727$	23.7851			0.882140			0.441070	+	0.897473i	$0.354599\pi$
$728$		0			0
$729$	−1.00000			−0.0370370
$730$		0			0
$731$		−	28.0613i		−	1.03788i
$732$		0			0
$733$	4.79132			0.176971			0.0884857	−	0.996077i	$-0.471797\pi$
$733$	4.79132			0.176971			0.0884857	+	0.996077i	$0.471797\pi$
$734$		0			0
$735$	26.0644	+	6.69753i	0.961398	+	0.247042i
$736$		0			0
$737$	20.1365			0.741738
$738$		0			0
$739$	−13.7563			−0.506035			−0.253018	−	0.967462i	$-0.581423\pi$
$739$	−13.7563			−0.506035			−0.253018	+	0.967462i	$0.581423\pi$
$740$		0			0
$741$	3.06403			0.112560
$742$		0			0
$743$		−	11.5310i		−	0.423033i	−0.977374	−	0.211517i	$-0.932160\pi$
$743$		−	11.5310i		−	0.423033i	0.977374	−	0.211517i	$-0.0678402\pi$
$744$		0			0
$745$			70.7275i			2.59125i
$746$		0			0
$747$			10.5770i			0.386992i
$748$		0			0
$749$	0.129094	+	0.166461i	0.00471701	+	0.00608234i
$750$		0			0
$751$			40.8541i			1.49079i	0.666625	+	0.745393i	$0.267738\pi$
$751$			40.8541i			1.49079i	−0.666625	+	0.745393i	$0.732262\pi$
$752$		0			0
$753$	−4.85827			−0.177045
$754$		0			0
$755$			38.0123i			1.38341i
$756$		0			0
$757$			27.8196i			1.01112i	0.862791	+	0.505561i	$0.168714\pi$
$757$			27.8196i			1.01112i	−0.862791	+	0.505561i	$0.831286\pi$
$758$		0			0
$759$	22.7606			0.826156
$760$		0			0
$761$		−	26.4737i		−	0.959672i	−0.877358	−	0.479836i	$-0.840696\pi$
$761$		−	26.4737i		−	0.959672i	0.877358	−	0.479836i	$-0.159304\pi$
$762$		0			0
$763$	42.0402	−	32.6033i	1.52196	−	1.18032i
$764$		0			0
$765$			13.4219i			0.485269i
$766$		0			0
$767$		−	15.3778i		−	0.555259i
$768$		0			0
$769$			31.5945i			1.13933i	0.821878	+	0.569664i	$0.192927\pi$
$769$			31.5945i			1.13933i	−0.821878	+	0.569664i	$0.807073\pi$
$770$		0			0
$771$	9.20998			0.331689
$772$		0			0
$773$	−33.5413			−1.20640			−0.603199	−	0.797591i	$-0.706107\pi$
$773$	−33.5413			−1.20640			−0.603199	+	0.797591i	$0.706107\pi$
$774$		0			0
$775$	−52.2560			−1.87709
$776$		0			0
$777$	−2.31300	−	2.98249i	−0.0829782	−	0.106996i
$778$		0			0
$779$	−15.1417			−0.542509
$780$		0			0
$781$			6.72535i			0.240652i
$782$		0			0
$783$	−1.81625			−0.0649076
$784$		0			0
$785$	−47.5105			−1.69572
$786$		0			0
$787$			1.03147i			0.0367679i	0.999831	+	0.0183840i	$0.00585213\pi$
$787$			1.03147i			0.0367679i	−0.999831	+	0.0183840i	$0.994148\pi$
$788$		0			0
$789$	3.90849			0.139146
$790$		0			0
$791$	−8.07860	−	10.4169i	−0.287242	−	0.370384i
$792$		0			0
$793$	−5.48426			−0.194752
$794$		0			0
$795$	−22.5770			−0.800724
$796$		0			0
$797$	−22.2199			−0.787070			−0.393535	−	0.919310i	$-0.628748\pi$
$797$	−22.2199			−0.787070			−0.393535	+	0.919310i	$0.628748\pi$
$798$		0			0
$799$			16.8830i			0.597276i
$800$		0			0
$801$		−	15.9580i		−	0.563848i
$802$		0			0
$803$		−	31.7450i		−	1.12026i
$804$		0			0
$805$	40.2388	−	31.2062i	1.41823	−	1.09988i
$806$		0			0
$807$			9.24872i			0.325570i
$808$		0			0
$809$	−52.4790			−1.84506			−0.922532	−	0.385920i	$-0.873884\pi$
$809$	−52.4790			−1.84506			−0.922532	+	0.385920i	$0.873884\pi$
$810$		0			0
$811$		−	4.40548i		−	0.154697i	−0.997004	−	0.0773487i	$-0.975355\pi$
$811$		−	4.40548i		−	0.154697i	0.997004	−	0.0773487i	$-0.0246455\pi$
$812$		0			0
$813$		−	16.5201i		−	0.579384i
$814$		0			0
$815$	13.7280			0.480870
$816$		0			0
$817$		−	13.5595i		−	0.474387i
$818$		0			0
$819$	−2.94487	−	3.79726i	−0.102902	−	0.132687i
$820$		0			0
$821$			21.2067i			0.740119i	0.929008	+	0.370060i	$0.120663\pi$
$821$			21.2067i			0.740119i	−0.929008	+	0.370060i	$0.879337\pi$
$822$		0			0
$823$			29.3609i			1.02346i	0.859148	+	0.511728i	$0.170994\pi$
$823$			29.3609i			1.02346i	−0.859148	+	0.511728i	$0.829006\pi$
$824$		0			0
$825$		−	44.4624i		−	1.54798i
$826$		0			0
$827$	28.2161			0.981171			0.490585	−	0.871393i	$-0.336783\pi$
$827$	28.2161			0.981171			0.490585	+	0.871393i	$0.336783\pi$
$828$		0			0
$829$	9.77173			0.339386			0.169693	−	0.985497i	$-0.445722\pi$
$829$	9.77173			0.339386			0.169693	+	0.985497i	$0.445722\pi$
$830$		0			0
$831$	−3.26465			−0.113249
$832$		0			0
$833$	23.6698	+	6.08221i	0.820108	+	0.210736i
$834$		0			0
$835$	75.2694			2.60481
$836$		0			0
$837$		−	5.34329i		−	0.184691i
$838$		0			0
$839$	−35.0785			−1.21104			−0.605522	−	0.795828i	$-0.707036\pi$
$839$	−35.0785			−1.21104			−0.605522	+	0.795828i	$0.707036\pi$
$840$		0			0
$841$	25.7012			0.886249
$842$		0			0
$843$		−	19.1680i		−	0.660180i
$844$		0			0
$845$	−37.2958			−1.28301
$846$		0			0
$847$	−15.6781	−	20.2161i	−0.538706	−	0.694633i
$848$		0			0
$849$	−20.2780			−0.695938
$850$		0			0
$851$	−7.14173			−0.244815
$852$		0			0
$853$	−18.3536			−0.628416			−0.314208	−	0.949354i	$-0.601739\pi$
$853$	−18.3536			−0.628416			−0.314208	+	0.949354i	$0.601739\pi$
$854$		0			0
$855$			6.48559i			0.221803i
$856$		0			0
$857$		−	7.62775i		−	0.260559i	−0.991477	−	0.130279i	$-0.958413\pi$
$857$		−	7.62775i		−	0.260559i	0.991477	−	0.130279i	$-0.0415874\pi$
$858$		0			0
$859$			13.9075i			0.474518i	0.971446	+	0.237259i	$0.0762491\pi$
$859$			13.9075i			0.474518i	−0.971446	+	0.237259i	$0.923751\pi$
$860$		0			0
$861$	14.5529	+	18.7652i	0.495961	+	0.639516i
$862$		0			0
$863$		−	13.3074i		−	0.452989i	−0.974012	−	0.226494i	$-0.927274\pi$
$863$		−	13.3074i		−	0.452989i	0.974012	−	0.226494i	$-0.0727265\pi$
$864$		0			0
$865$	21.1223			0.718179
$866$		0			0
$867$		−	4.81122i		−	0.163398i
$868$		0			0
$869$		−	13.5397i		−	0.459302i
$870$		0			0
$871$	−8.04444			−0.272575
$872$		0			0
$873$		−	11.6085i		−	0.392887i
$874$		0			0
$875$	−29.7940	−	38.4178i	−1.00722	−	1.29876i
$876$		0			0
$877$			25.0885i			0.847179i	0.905854	+	0.423589i	$0.139230\pi$
$877$			25.0885i			0.847179i	−0.905854	+	0.423589i	$0.860770\pi$
$878$		0			0
$879$			5.07037i			0.171019i
$880$		0			0
$881$			5.29180i			0.178285i	0.996019	+	0.0891426i	$0.0284127\pi$
$881$			5.29180i			0.178285i	−0.996019	+	0.0891426i	$0.971587\pi$
$882$		0			0
$883$	21.5219			0.724269			0.362134	−	0.932126i	$-0.382048\pi$
$883$	21.5219			0.724269			0.362134	+	0.932126i	$0.382048\pi$
$884$		0			0
$885$	32.5500			1.09415
$886$		0			0
$887$	−14.7656			−0.495780			−0.247890	−	0.968788i	$-0.579737\pi$
$887$	−14.7656			−0.495780			−0.247890	+	0.968788i	$0.579737\pi$
$888$		0			0
$889$	11.2579	−	8.73076i	0.377577	−	0.292820i
$890$		0			0
$891$	4.54637			0.152309
$892$		0			0
$893$			8.15802i			0.272998i
$894$		0			0
$895$	−38.5622			−1.28899
$896$		0			0
$897$	−9.09274			−0.303598
$898$		0			0
$899$		−	9.70478i		−	0.323672i
$900$		0			0
$901$	−20.5028			−0.683047
$902$		0			0
$903$	−16.8043	+	13.0322i	−0.559213	+	0.433684i
$904$		0			0
$905$	93.2205			3.09875
$906$		0			0
$907$	−46.8609			−1.55599			−0.777995	−	0.628271i	$-0.783763\pi$
$907$	−46.8609			−1.55599			−0.777995	+	0.628271i	$0.783763\pi$
$908$		0			0
$909$	11.5333			0.382536
$910$		0			0
$911$		−	54.6440i		−	1.81044i	−0.424945	−	0.905219i	$-0.639707\pi$
$911$		−	54.6440i		−	1.81044i	0.424945	−	0.905219i	$-0.360293\pi$
$912$		0			0
$913$		−	48.0870i		−	1.59145i
$914$		0			0
$915$		−	11.6085i		−	0.383764i
$916$		0			0
$917$	27.3731	−	21.2286i	0.903940	−	0.701029i
$918$		0			0
$919$		−	28.6186i		−	0.944041i	−0.881588	−	0.472020i	$-0.843525\pi$
$919$		−	28.6186i		−	0.944041i	0.881588	−	0.472020i	$-0.156475\pi$
$920$		0			0
$921$	−15.2465			−0.502389
$922$		0			0
$923$		−	2.68675i		−	0.0884353i
$924$		0			0
$925$			13.9512i			0.458714i
$926$		0			0
$927$	1.14230			0.0375182
$928$		0			0
$929$			29.6885i			0.974048i	0.873389	+	0.487024i	$0.161918\pi$
$929$			29.6885i			0.974048i	−0.873389	+	0.487024i	$0.838082\pi$
$930$		0			0
$931$	11.4375	+	2.93899i	0.374848	+	0.0963214i
$932$		0			0
$933$			8.91481i			0.291858i
$934$		0			0
$935$		−	61.0209i		−	1.99560i
$936$		0			0
$937$		−	32.4055i		−	1.05864i	−0.848422	−	0.529320i	$-0.822447\pi$
$937$		−	32.4055i		−	1.05864i	0.848422	−	0.529320i	$-0.177553\pi$
$938$		0			0
$939$	−24.9335			−0.813674
$940$		0			0
$941$	10.5656			0.344429			0.172214	−	0.985060i	$-0.444908\pi$
$941$	10.5656			0.344429			0.172214	+	0.985060i	$0.444908\pi$
$942$		0			0
$943$	44.9342			1.46326
$944$		0			0
$945$	8.03761	−	6.23338i	0.261464	−	0.202772i
$946$		0			0
$947$	3.44186			0.111845			0.0559226	−	0.998435i	$-0.482190\pi$
$947$	3.44186			0.111845			0.0559226	+	0.998435i	$0.482190\pi$
$948$		0			0
$949$			12.6820i			0.411674i
$950$		0			0
$951$	−16.1127			−0.522491
$952$		0			0
$953$	−38.5559			−1.24895			−0.624475	−	0.781045i	$-0.714687\pi$
$953$	−38.5559			−1.24895			−0.624475	+	0.781045i	$0.714687\pi$
$954$		0			0
$955$		−	32.6555i		−	1.05671i
$956$		0			0
$957$	8.25737			0.266923
$958$		0			0
$959$	31.7137	+	40.8932i	1.02409	+	1.32051i
$960$		0			0
$961$	−2.44924			−0.0790077
$962$		0			0
$963$	0.0796192			0.00256569
$964$		0			0
$965$	−43.4248			−1.39790
$966$		0			0
$967$			10.0322i			0.322614i	0.986904	+	0.161307i	$0.0515709\pi$
$967$			10.0322i			0.322614i	−0.986904	+	0.161307i	$0.948429\pi$
$968$		0			0
$969$			5.88974i			0.189206i
$970$		0			0
$971$		−	28.4807i		−	0.913989i	−0.889469	−	0.456995i	$-0.848926\pi$
$971$		−	28.4807i		−	0.913989i	0.889469	−	0.456995i	$-0.151074\pi$
$972$		0			0
$973$	30.5055	−	23.6578i	0.977960	−	0.758433i
$974$		0			0
$975$			17.7625i			0.568856i
$976$		0			0
$977$	8.98249			0.287375			0.143688	−	0.989623i	$-0.454104\pi$
$977$	8.98249			0.287375			0.143688	+	0.989623i	$0.454104\pi$
$978$		0			0
$979$			72.5510i			2.31874i
$980$		0			0
$981$		−	20.1081i		−	0.642002i
$982$		0			0
$983$	11.1768			0.356484			0.178242	−	0.983987i	$-0.442959\pi$
$983$	11.1768			0.356484			0.178242	+	0.983987i	$0.442959\pi$
$984$		0			0
$985$			9.16797i			0.292116i
$986$		0			0
$987$	10.1103	−	7.84076i	0.321813	−	0.249574i
$988$		0			0
$989$			40.2388i			1.27952i
$990$		0			0
$991$			21.5500i			0.684559i	0.939598	+	0.342279i	$0.111199\pi$
$991$			21.5500i			0.684559i	−0.939598	+	0.342279i	$0.888801\pi$
$992$		0			0
$993$			22.3802i			0.710213i
$994$		0			0
$995$	−9.47031			−0.300229
$996$		0			0
$997$	38.2116			1.21017			0.605087	−	0.796159i	$-0.293138\pi$
$997$	38.2116			1.21017			0.605087	+	0.796159i	$0.293138\pi$
$998$		0			0
$999$	−1.42654			−0.0451338

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	672.2.p.a.559.8		16
3.2	odd	2		2016.2.p.g.559.1		16
4.3	odd	2		168.2.p.a.139.8	yes	16
7.6	odd	2	inner	672.2.p.a.559.9		16
8.3	odd	2	inner	672.2.p.a.559.1		16
8.5	even	2		168.2.p.a.139.6	yes	16
12.11	even	2		504.2.p.g.307.9		16
21.20	even	2		2016.2.p.g.559.15		16
24.5	odd	2		504.2.p.g.307.12		16
24.11	even	2		2016.2.p.g.559.16		16
28.27	even	2		168.2.p.a.139.7	yes	16
56.13	odd	2		168.2.p.a.139.5	✓	16
56.27	even	2	inner	672.2.p.a.559.16		16
84.83	odd	2		504.2.p.g.307.10		16
168.83	odd	2		2016.2.p.g.559.2		16
168.125	even	2		504.2.p.g.307.11		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
168.2.p.a.139.5	✓	16	56.13	odd	2
168.2.p.a.139.6	yes	16	8.5	even	2
168.2.p.a.139.7	yes	16	28.27	even	2
168.2.p.a.139.8	yes	16	4.3	odd	2
504.2.p.g.307.9		16	12.11	even	2
504.2.p.g.307.10		16	84.83	odd	2
504.2.p.g.307.11		16	168.125	even	2
504.2.p.g.307.12		16	24.5	odd	2
672.2.p.a.559.1		16	8.3	odd	2	inner
672.2.p.a.559.8		16	1.1	even	1	trivial
672.2.p.a.559.9		16	7.6	odd	2	inner
672.2.p.a.559.16		16	56.27	even	2	inner
2016.2.p.g.559.1		16	3.2	odd	2
2016.2.p.g.559.2		16	168.83	odd	2
2016.2.p.g.559.15		16	21.20	even	2
2016.2.p.g.559.16		16	24.11	even	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:	\( N \)	\(=\)	\( 672 = 2^{5} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	672.p (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{3} +3.84444 q^{5} +(-1.62140 - 2.09071i) q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q-1.00000i q^{3} +3.84444 q^{5} +(-1.62140 - 2.09071i) q^{7} -1.00000 q^{9} +4.54637 q^{11} -1.81625 q^{13} -3.84444i q^{15} -3.49124i q^{17} -1.68700i q^{19} +(-2.09071 + 1.62140i) q^{21} +5.00632i q^{23} +9.77975 q^{25} +1.00000i q^{27} +1.81625i q^{29} -5.34329 q^{31} -4.54637i q^{33} +(-6.23338 - 8.03761i) q^{35} +1.42654i q^{37} +1.81625i q^{39} -8.97551i q^{41} +8.03761 q^{43} -3.84444 q^{45} -4.83580 q^{47} +(-1.74213 + 6.77975i) q^{49} -3.49124 q^{51} -5.87263i q^{53} +17.4783 q^{55} -1.68700 q^{57} +8.46675i q^{59} +3.01955 q^{61} +(1.62140 + 2.09071i) q^{63} -6.98249 q^{65} +4.42914 q^{67} +5.00632 q^{69} +1.47928i q^{71} -6.98249i q^{73} -9.77975i q^{75} +(-7.37148 - 9.50514i) q^{77} -2.97813i q^{79} +1.00000 q^{81} -10.5770i q^{83} -13.4219i q^{85} +1.81625 q^{87} +15.9580i q^{89} +(2.94487 + 3.79726i) q^{91} +5.34329i q^{93} -6.48559i q^{95} +11.6085i q^{97} -4.54637 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 16 q^{9}+O(q^{10}) \) 16 * q - 16 * q^9	\( 16 q - 16 q^{9} + 8 q^{11} + 16 q^{25} - 24 q^{35} + 8 q^{43} - 8 q^{49} - 16 q^{57} + 40 q^{67} + 16 q^{81} + 56 q^{91} - 8 q^{99}+O(q^{100}) \) 16 * q - 16 * q^9 + 8 * q^11 + 16 * q^25 - 24 * q^35 + 8 * q^43 - 8 * q^49 - 16 * q^57 + 40 * q^67 + 16 * q^81 + 56 * q^91 - 8 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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