LMFDB - Embedded newform 336.2.b.c.223.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [336,2,Mod(223,336)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(336, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0, 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("336.223");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 336 = 2^{4} \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	336.b (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:	$2.68297350792$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{-6}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 6 $ x^2 + 6
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			223.1
Root			$-2.44949i$ of defining polynomial
Character	$\chi$	$=$	336.223
Dual form			336.2.b.c.223.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} -2.44949i q^{5} +(-1.00000 - 2.44949i) q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -2.44949i q^{5} +(-1.00000 - 2.44949i) q^{7} +1.00000 q^{9} +2.44949i q^{11} -4.89898i q^{13} -2.44949i q^{15} -2.44949i q^{17} -2.00000 q^{19} +(-1.00000 - 2.44949i) q^{21} +7.34847i q^{23} -1.00000 q^{25} +1.00000 q^{27} +6.00000 q^{29} +8.00000 q^{31} +2.44949i q^{33} +(-6.00000 + 2.44949i) q^{35} +4.00000 q^{37} -4.89898i q^{39} +7.34847i q^{41} -4.89898i q^{43} -2.44949i q^{45} -12.0000 q^{47} +(-5.00000 + 4.89898i) q^{49} -2.44949i q^{51} -6.00000 q^{53} +6.00000 q^{55} -2.00000 q^{57} +12.0000 q^{59} +(-1.00000 - 2.44949i) q^{63} -12.0000 q^{65} +7.34847i q^{69} +12.2474i q^{71} +14.6969i q^{73} -1.00000 q^{75} +(6.00000 - 2.44949i) q^{77} -9.79796i q^{79} +1.00000 q^{81} -6.00000 q^{85} +6.00000 q^{87} +12.2474i q^{89} +(-12.0000 + 4.89898i) q^{91} +8.00000 q^{93} +4.89898i q^{95} -4.89898i q^{97} +2.44949i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} - 2 q^{7} + 2 q^{9}+O(q^{10}) $ 2 * q + 2 * q^3 - 2 * q^7 + 2 * q^9	$ 2 q + 2 q^{3} - 2 q^{7} + 2 q^{9} - 4 q^{19} - 2 q^{21} - 2 q^{25} + 2 q^{27} + 12 q^{29} + 16 q^{31} - 12 q^{35} + 8 q^{37} - 24 q^{47} - 10 q^{49} - 12 q^{53} + 12 q^{55} - 4 q^{57} + 24 q^{59} - 2 q^{63} - 24 q^{65} - 2 q^{75} + 12 q^{77} + 2 q^{81} - 12 q^{85} + 12 q^{87} - 24 q^{91} + 16 q^{93}+O(q^{100}) $ 2 * q + 2 * q^3 - 2 * q^7 + 2 * q^9 - 4 * q^19 - 2 * q^21 - 2 * q^25 + 2 * q^27 + 12 * q^29 + 16 * q^31 - 12 * q^35 + 8 * q^37 - 24 * q^47 - 10 * q^49 - 12 * q^53 + 12 * q^55 - 4 * q^57 + 24 * q^59 - 2 * q^63 - 24 * q^65 - 2 * q^75 + 12 * q^77 + 2 * q^81 - 12 * q^85 + 12 * q^87 - 24 * q^91 + 16 * q^93

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/336\mathbb{Z}\right)^\times$.

$n$	$85$	$113$	$127$	$241$
$\chi(n)$	$1$	$1$	$-1$	$-1$

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
$3$	1.00000			0.577350
$3$	1.00000			0.577350
$4$		0			0
$5$		−	2.44949i		−	1.09545i	−0.836660	−	0.547723i	$-0.815495\pi$
$5$		−	2.44949i		−	1.09545i	0.836660	−	0.547723i	$-0.184505\pi$
$6$		0			0
$7$	−1.00000	−	2.44949i	−0.377964	−	0.925820i
$7$	−1.00000	−	2.44949i	−0.377964	−	0.925820i
$8$		0			0
$9$	1.00000			0.333333
$10$		0			0
$11$			2.44949i			0.738549i	0.929320	+	0.369274i	$0.120394\pi$
$11$			2.44949i			0.738549i	−0.929320	+	0.369274i	$0.879606\pi$
$12$		0			0
$13$		−	4.89898i		−	1.35873i	−0.733799	−	0.679366i	$-0.762255\pi$
$13$		−	4.89898i		−	1.35873i	0.733799	−	0.679366i	$-0.237745\pi$
$14$		0			0
$15$		−	2.44949i		−	0.632456i
$16$		0			0
$17$		−	2.44949i		−	0.594089i	−0.954864	−	0.297044i	$-0.903999\pi$
$17$		−	2.44949i		−	0.594089i	0.954864	−	0.297044i	$-0.0960009\pi$
$18$		0			0
$19$	−2.00000			−0.458831			−0.229416	−	0.973329i	$-0.573682\pi$
$19$	−2.00000			−0.458831			−0.229416	+	0.973329i	$0.573682\pi$
$20$		0			0
$21$	−1.00000	−	2.44949i	−0.218218	−	0.534522i
$22$		0			0
$23$			7.34847i			1.53226i	0.642685	+	0.766131i	$0.277821\pi$
$23$			7.34847i			1.53226i	−0.642685	+	0.766131i	$0.722179\pi$
$24$		0			0
$25$	−1.00000			−0.200000
$26$		0			0
$27$	1.00000			0.192450
$28$		0			0
$29$	6.00000			1.11417			0.557086	−	0.830455i	$-0.311919\pi$
$29$	6.00000			1.11417			0.557086	+	0.830455i	$0.311919\pi$
$30$		0			0
$31$	8.00000			1.43684			0.718421	−	0.695608i	$-0.244865\pi$
$31$	8.00000			1.43684			0.718421	+	0.695608i	$0.244865\pi$
$32$		0			0
$33$			2.44949i			0.426401i
$34$		0			0
$35$	−6.00000	+	2.44949i	−1.01419	+	0.414039i
$36$		0			0
$37$	4.00000			0.657596			0.328798	−	0.944400i	$-0.393356\pi$
$37$	4.00000			0.657596			0.328798	+	0.944400i	$0.393356\pi$
$38$		0			0
$39$		−	4.89898i		−	0.784465i
$40$		0			0
$41$			7.34847i			1.14764i	0.818982	+	0.573819i	$0.194539\pi$
$41$			7.34847i			1.14764i	−0.818982	+	0.573819i	$0.805461\pi$
$42$		0			0
$43$		−	4.89898i		−	0.747087i	−0.927613	−	0.373544i	$-0.878143\pi$
$43$		−	4.89898i		−	0.747087i	0.927613	−	0.373544i	$-0.121857\pi$
$44$		0			0
$45$		−	2.44949i		−	0.365148i
$46$		0			0
$47$	−12.0000			−1.75038			−0.875190	−	0.483779i	$-0.839264\pi$
$47$	−12.0000			−1.75038			−0.875190	+	0.483779i	$0.839264\pi$
$48$		0			0
$49$	−5.00000	+	4.89898i	−0.714286	+	0.699854i
$50$		0			0
$51$		−	2.44949i		−	0.342997i
$52$		0			0
$53$	−6.00000			−0.824163			−0.412082	−	0.911147i	$-0.635198\pi$
$53$	−6.00000			−0.824163			−0.412082	+	0.911147i	$0.635198\pi$
$54$		0			0
$55$	6.00000			0.809040
$56$		0			0
$57$	−2.00000			−0.264906
$58$		0			0
$59$	12.0000			1.56227			0.781133	−	0.624364i	$-0.214642\pi$
$59$	12.0000			1.56227			0.781133	+	0.624364i	$0.214642\pi$
$60$		0			0
$61$		0			0		1.00000			$0$
$61$		0			0		−1.00000			$\pi$
$62$		0			0
$63$	−1.00000	−	2.44949i	−0.125988	−	0.308607i
$64$		0			0
$65$	−12.0000			−1.48842
$66$		0			0
$67$		0			0		1.00000			$0$
$67$		0			0		−1.00000			$\pi$
$68$		0			0
$69$			7.34847i			0.884652i
$70$		0			0
$71$			12.2474i			1.45350i	0.686900	+	0.726752i	$0.258971\pi$
$71$			12.2474i			1.45350i	−0.686900	+	0.726752i	$0.741029\pi$
$72$		0			0
$73$			14.6969i			1.72015i	0.510171	+	0.860073i	$0.329582\pi$
$73$			14.6969i			1.72015i	−0.510171	+	0.860073i	$0.670418\pi$
$74$		0			0
$75$	−1.00000			−0.115470
$76$		0			0
$77$	6.00000	−	2.44949i	0.683763	−	0.279145i
$78$		0			0
$79$		−	9.79796i		−	1.10236i	−0.834388	−	0.551178i	$-0.814178\pi$
$79$		−	9.79796i		−	1.10236i	0.834388	−	0.551178i	$-0.185822\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$		0			0			−	1.00000i	$-0.5\pi$
$83$		0			0				1.00000i	$0.5\pi$
$84$		0			0
$85$	−6.00000			−0.650791
$86$		0			0
$87$	6.00000			0.643268
$88$		0			0
$89$			12.2474i			1.29823i	0.760692	+	0.649113i	$0.224860\pi$
$89$			12.2474i			1.29823i	−0.760692	+	0.649113i	$0.775140\pi$
$90$		0			0
$91$	−12.0000	+	4.89898i	−1.25794	+	0.513553i
$92$		0			0
$93$	8.00000			0.829561
$94$		0			0
$95$			4.89898i			0.502625i
$96$		0			0
$97$		−	4.89898i		−	0.497416i	−0.968579	−	0.248708i	$-0.919994\pi$
$97$		−	4.89898i		−	0.497416i	0.968579	−	0.248708i	$-0.0800060\pi$
$98$		0			0
$99$			2.44949i			0.246183i
$100$		0			0
$101$			2.44949i			0.243733i	0.992546	+	0.121867i	$0.0388880\pi$
$101$			2.44949i			0.243733i	−0.992546	+	0.121867i	$0.961112\pi$
$102$		0			0
$103$	−16.0000			−1.57653			−0.788263	−	0.615338i	$-0.789020\pi$
$103$	−16.0000			−1.57653			−0.788263	+	0.615338i	$0.789020\pi$
$104$		0			0
$105$	−6.00000	+	2.44949i	−0.585540	+	0.239046i
$106$		0			0
$107$		−	12.2474i		−	1.18401i	−0.805936	−	0.592003i	$-0.798337\pi$
$107$		−	12.2474i		−	1.18401i	0.805936	−	0.592003i	$-0.201663\pi$
$108$		0			0
$109$	2.00000			0.191565			0.0957826	−	0.995402i	$-0.469465\pi$
$109$	2.00000			0.191565			0.0957826	+	0.995402i	$0.469465\pi$
$110$		0			0
$111$	4.00000			0.379663
$112$		0			0
$113$	6.00000			0.564433			0.282216	−	0.959351i	$-0.408930\pi$
$113$	6.00000			0.564433			0.282216	+	0.959351i	$0.408930\pi$
$114$		0			0
$115$	18.0000			1.67851
$116$		0			0
$117$		−	4.89898i		−	0.452911i
$118$		0			0
$119$	−6.00000	+	2.44949i	−0.550019	+	0.224544i
$120$		0			0
$121$	5.00000			0.454545
$122$		0			0
$123$			7.34847i			0.662589i
$124$		0			0
$125$		−	9.79796i		−	0.876356i
$126$		0			0
$127$			9.79796i			0.869428i	0.900568	+	0.434714i	$0.143151\pi$
$127$			9.79796i			0.869428i	−0.900568	+	0.434714i	$0.856849\pi$
$128$		0			0
$129$		−	4.89898i		−	0.431331i
$130$		0			0
$131$		0			0			−	1.00000i	$-0.5\pi$
$131$		0			0				1.00000i	$0.5\pi$
$132$		0			0
$133$	2.00000	+	4.89898i	0.173422	+	0.424795i
$134$		0			0
$135$		−	2.44949i		−	0.210819i
$136$		0			0
$137$	6.00000			0.512615			0.256307	−	0.966595i	$-0.417494\pi$
$137$	6.00000			0.512615			0.256307	+	0.966595i	$0.417494\pi$
$138$		0			0
$139$	−4.00000			−0.339276			−0.169638	−	0.985506i	$-0.554260\pi$
$139$	−4.00000			−0.339276			−0.169638	+	0.985506i	$0.554260\pi$
$140$		0			0
$141$	−12.0000			−1.01058
$142$		0			0
$143$	12.0000			1.00349
$144$		0			0
$145$		−	14.6969i		−	1.22051i
$146$		0			0
$147$	−5.00000	+	4.89898i	−0.412393	+	0.404061i
$148$		0			0
$149$	−18.0000			−1.47462			−0.737309	−	0.675556i	$-0.763904\pi$
$149$	−18.0000			−1.47462			−0.737309	+	0.675556i	$0.763904\pi$
$150$		0			0
$151$		−	14.6969i		−	1.19602i	−0.801489	−	0.598010i	$-0.795958\pi$
$151$		−	14.6969i		−	1.19602i	0.801489	−	0.598010i	$-0.204042\pi$
$152$		0			0
$153$		−	2.44949i		−	0.198030i
$154$		0			0
$155$		−	19.5959i		−	1.57398i
$156$		0			0
$157$		−	19.5959i		−	1.56392i	−0.623326	−	0.781962i	$-0.714219\pi$
$157$		−	19.5959i		−	1.56392i	0.623326	−	0.781962i	$-0.285781\pi$
$158$		0			0
$159$	−6.00000			−0.475831
$160$		0			0
$161$	18.0000	−	7.34847i	1.41860	−	0.579141i
$162$		0			0
$163$			19.5959i			1.53487i	0.641126	+	0.767435i	$0.278467\pi$
$163$			19.5959i			1.53487i	−0.641126	+	0.767435i	$0.721533\pi$
$164$		0			0
$165$	6.00000			0.467099
$166$		0			0
$167$	12.0000			0.928588			0.464294	−	0.885681i	$-0.346308\pi$
$167$	12.0000			0.928588			0.464294	+	0.885681i	$0.346308\pi$
$168$		0			0
$169$	−11.0000			−0.846154
$170$		0			0
$171$	−2.00000			−0.152944
$172$		0			0
$173$		−	7.34847i		−	0.558694i	−0.960190	−	0.279347i	$-0.909882\pi$
$173$		−	7.34847i		−	0.558694i	0.960190	−	0.279347i	$-0.0901179\pi$
$174$		0			0
$175$	1.00000	+	2.44949i	0.0755929	+	0.185164i
$176$		0			0
$177$	12.0000			0.901975
$178$		0			0
$179$		−	17.1464i		−	1.28158i	−0.767714	−	0.640792i	$-0.778606\pi$
$179$		−	17.1464i		−	1.28158i	0.767714	−	0.640792i	$-0.221394\pi$
$180$		0			0
$181$			4.89898i			0.364138i	0.983286	+	0.182069i	$0.0582795\pi$
$181$			4.89898i			0.364138i	−0.983286	+	0.182069i	$0.941721\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		−	9.79796i		−	0.720360i
$186$		0			0
$187$	6.00000			0.438763
$188$		0			0
$189$	−1.00000	−	2.44949i	−0.0727393	−	0.178174i
$190$		0			0
$191$		−	2.44949i		−	0.177239i	−0.996066	−	0.0886194i	$-0.971755\pi$
$191$		−	2.44949i		−	0.177239i	0.996066	−	0.0886194i	$-0.0282455\pi$
$192$		0			0
$193$	16.0000			1.15171			0.575853	−	0.817554i	$-0.304670\pi$
$193$	16.0000			1.15171			0.575853	+	0.817554i	$0.304670\pi$
$194$		0			0
$195$	−12.0000			−0.859338
$196$		0			0
$197$	18.0000			1.28245			0.641223	−	0.767354i	$-0.278427\pi$
$197$	18.0000			1.28245			0.641223	+	0.767354i	$0.278427\pi$
$198$		0			0
$199$	−2.00000			−0.141776			−0.0708881	−	0.997484i	$-0.522583\pi$
$199$	−2.00000			−0.141776			−0.0708881	+	0.997484i	$0.522583\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−6.00000	−	14.6969i	−0.421117	−	1.03152i
$204$		0			0
$205$	18.0000			1.25717
$206$		0			0
$207$			7.34847i			0.510754i
$208$		0			0
$209$		−	4.89898i		−	0.338869i
$210$		0			0
$211$			4.89898i			0.337260i	0.985679	+	0.168630i	$0.0539342\pi$
$211$			4.89898i			0.337260i	−0.985679	+	0.168630i	$0.946066\pi$
$212$		0			0
$213$			12.2474i			0.839181i
$214$		0			0
$215$	−12.0000			−0.818393
$216$		0			0
$217$	−8.00000	−	19.5959i	−0.543075	−	1.33026i
$218$		0			0
$219$			14.6969i			0.993127i
$220$		0			0
$221$	−12.0000			−0.807207
$222$		0			0
$223$	10.0000			0.669650			0.334825	−	0.942280i	$-0.391323\pi$
$223$	10.0000			0.669650			0.334825	+	0.942280i	$0.391323\pi$
$224$		0			0
$225$	−1.00000			−0.0666667
$226$		0			0
$227$	12.0000			0.796468			0.398234	−	0.917284i	$-0.369623\pi$
$227$	12.0000			0.796468			0.398234	+	0.917284i	$0.369623\pi$
$228$		0			0
$229$			24.4949i			1.61867i	0.587348	+	0.809334i	$0.300172\pi$
$229$			24.4949i			1.61867i	−0.587348	+	0.809334i	$0.699828\pi$
$230$		0			0
$231$	6.00000	−	2.44949i	0.394771	−	0.161165i
$232$		0			0
$233$	−30.0000			−1.96537			−0.982683	−	0.185296i	$-0.940675\pi$
$233$	−30.0000			−1.96537			−0.982683	+	0.185296i	$0.940675\pi$
$234$		0			0
$235$			29.3939i			1.91745i
$236$		0			0
$237$		−	9.79796i		−	0.636446i
$238$		0			0
$239$		−	12.2474i		−	0.792222i	−0.918203	−	0.396111i	$-0.870360\pi$
$239$		−	12.2474i		−	0.792222i	0.918203	−	0.396111i	$-0.129640\pi$
$240$		0			0
$241$		−	4.89898i		−	0.315571i	−0.987473	−	0.157786i	$-0.949565\pi$
$241$		−	4.89898i		−	0.315571i	0.987473	−	0.157786i	$-0.0504355\pi$
$242$		0			0
$243$	1.00000			0.0641500
$244$		0			0
$245$	12.0000	+	12.2474i	0.766652	+	0.782461i
$246$		0			0
$247$			9.79796i			0.623429i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0			−	1.00000i	$-0.5\pi$
$251$		0			0				1.00000i	$0.5\pi$
$252$		0			0
$253$	−18.0000			−1.13165
$254$		0			0
$255$	−6.00000			−0.375735
$256$		0			0
$257$		−	22.0454i		−	1.37515i	−0.726111	−	0.687577i	$-0.758674\pi$
$257$		−	22.0454i		−	1.37515i	0.726111	−	0.687577i	$-0.241326\pi$
$258$		0			0
$259$	−4.00000	−	9.79796i	−0.248548	−	0.608816i
$260$		0			0
$261$	6.00000			0.371391
$262$		0			0
$263$			2.44949i			0.151042i	0.997144	+	0.0755210i	$0.0240620\pi$
$263$			2.44949i			0.151042i	−0.997144	+	0.0755210i	$0.975938\pi$
$264$		0			0
$265$			14.6969i			0.902826i
$266$		0			0
$267$			12.2474i			0.749532i
$268$		0			0
$269$			22.0454i			1.34413i	0.740491	+	0.672066i	$0.234593\pi$
$269$			22.0454i			1.34413i	−0.740491	+	0.672066i	$0.765407\pi$
$270$		0			0
$271$	−16.0000			−0.971931			−0.485965	−	0.873978i	$-0.661532\pi$
$271$	−16.0000			−0.971931			−0.485965	+	0.873978i	$0.661532\pi$
$272$		0			0
$273$	−12.0000	+	4.89898i	−0.726273	+	0.296500i
$274$		0			0
$275$		−	2.44949i		−	0.147710i
$276$		0			0
$277$	−8.00000			−0.480673			−0.240337	−	0.970690i	$-0.577258\pi$
$277$	−8.00000			−0.480673			−0.240337	+	0.970690i	$0.577258\pi$
$278$		0			0
$279$	8.00000			0.478947
$280$		0			0
$281$	6.00000			0.357930			0.178965	−	0.983855i	$-0.442725\pi$
$281$	6.00000			0.357930			0.178965	+	0.983855i	$0.442725\pi$
$282$		0			0
$283$	−14.0000			−0.832214			−0.416107	−	0.909316i	$-0.636606\pi$
$283$	−14.0000			−0.832214			−0.416107	+	0.909316i	$0.636606\pi$
$284$		0			0
$285$			4.89898i			0.290191i
$286$		0			0
$287$	18.0000	−	7.34847i	1.06251	−	0.433766i
$288$		0			0
$289$	11.0000			0.647059
$290$		0			0
$291$		−	4.89898i		−	0.287183i
$292$		0			0
$293$			22.0454i			1.28791i	0.765065	+	0.643953i	$0.222707\pi$
$293$			22.0454i			1.28791i	−0.765065	+	0.643953i	$0.777293\pi$
$294$		0			0
$295$		−	29.3939i		−	1.71138i
$296$		0			0
$297$			2.44949i			0.142134i
$298$		0			0
$299$	36.0000			2.08193
$300$		0			0
$301$	−12.0000	+	4.89898i	−0.691669	+	0.282372i
$302$		0			0
$303$			2.44949i			0.140720i
$304$		0			0
$305$		0			0
$306$		0			0
$307$	−26.0000			−1.48390			−0.741949	−	0.670456i	$-0.766098\pi$
$307$	−26.0000			−1.48390			−0.741949	+	0.670456i	$0.766098\pi$
$308$		0			0
$309$	−16.0000			−0.910208
$310$		0			0
$311$	−24.0000			−1.36092			−0.680458	−	0.732787i	$-0.738219\pi$
$311$	−24.0000			−1.36092			−0.680458	+	0.732787i	$0.738219\pi$
$312$		0			0
$313$			9.79796i			0.553813i	0.960897	+	0.276907i	$0.0893093\pi$
$313$			9.79796i			0.553813i	−0.960897	+	0.276907i	$0.910691\pi$
$314$		0			0
$315$	−6.00000	+	2.44949i	−0.338062	+	0.138013i
$316$		0			0
$317$	−18.0000			−1.01098			−0.505490	−	0.862832i	$-0.668688\pi$
$317$	−18.0000			−1.01098			−0.505490	+	0.862832i	$0.668688\pi$
$318$		0			0
$319$			14.6969i			0.822871i
$320$		0			0
$321$		−	12.2474i		−	0.683586i
$322$		0			0
$323$			4.89898i			0.272587i
$324$		0			0
$325$			4.89898i			0.271746i
$326$		0			0
$327$	2.00000			0.110600
$328$		0			0
$329$	12.0000	+	29.3939i	0.661581	+	1.62054i
$330$		0			0
$331$		−	24.4949i		−	1.34636i	−0.739478	−	0.673181i	$-0.764928\pi$
$331$		−	24.4949i		−	1.34636i	0.739478	−	0.673181i	$-0.235072\pi$
$332$		0			0
$333$	4.00000			0.219199
$334$		0			0
$335$		0			0
$336$		0			0
$337$	14.0000			0.762629			0.381314	−	0.924445i	$-0.375472\pi$
$337$	14.0000			0.762629			0.381314	+	0.924445i	$0.375472\pi$
$338$		0			0
$339$	6.00000			0.325875
$340$		0			0
$341$			19.5959i			1.06118i
$342$		0			0
$343$	17.0000	+	7.34847i	0.917914	+	0.396780i
$344$		0			0
$345$	18.0000			0.969087
$346$		0			0
$347$		−	17.1464i		−	0.920468i	−0.887798	−	0.460234i	$-0.847765\pi$
$347$		−	17.1464i		−	0.920468i	0.887798	−	0.460234i	$-0.152235\pi$
$348$		0			0
$349$			19.5959i			1.04895i	0.851427	+	0.524473i	$0.175738\pi$
$349$			19.5959i			1.04895i	−0.851427	+	0.524473i	$0.824262\pi$
$350$		0			0
$351$		−	4.89898i		−	0.261488i
$352$		0			0
$353$			2.44949i			0.130373i	0.997873	+	0.0651866i	$0.0207643\pi$
$353$			2.44949i			0.130373i	−0.997873	+	0.0651866i	$0.979236\pi$
$354$		0			0
$355$	30.0000			1.59223
$356$		0			0
$357$	−6.00000	+	2.44949i	−0.317554	+	0.129641i
$358$		0			0
$359$			12.2474i			0.646396i	0.946331	+	0.323198i	$0.104758\pi$
$359$			12.2474i			0.646396i	−0.946331	+	0.323198i	$0.895242\pi$
$360$		0			0
$361$	−15.0000			−0.789474
$362$		0			0
$363$	5.00000			0.262432
$364$		0			0
$365$	36.0000			1.88433
$366$		0			0
$367$	−14.0000			−0.730794			−0.365397	−	0.930852i	$-0.619067\pi$
$367$	−14.0000			−0.730794			−0.365397	+	0.930852i	$0.619067\pi$
$368$		0			0
$369$			7.34847i			0.382546i
$370$		0			0
$371$	6.00000	+	14.6969i	0.311504	+	0.763027i
$372$		0			0
$373$	−10.0000			−0.517780			−0.258890	−	0.965907i	$-0.583357\pi$
$373$	−10.0000			−0.517780			−0.258890	+	0.965907i	$0.583357\pi$
$374$		0			0
$375$		−	9.79796i		−	0.505964i
$376$		0			0
$377$		−	29.3939i		−	1.51386i
$378$		0			0
$379$			34.2929i			1.76151i	0.473576	+	0.880753i	$0.342963\pi$
$379$			34.2929i			1.76151i	−0.473576	+	0.880753i	$0.657037\pi$
$380$		0			0
$381$			9.79796i			0.501965i
$382$		0			0
$383$	−12.0000			−0.613171			−0.306586	−	0.951843i	$-0.599187\pi$
$383$	−12.0000			−0.613171			−0.306586	+	0.951843i	$0.599187\pi$
$384$		0			0
$385$	−6.00000	−	14.6969i	−0.305788	−	0.749025i
$386$		0			0
$387$		−	4.89898i		−	0.249029i
$388$		0			0
$389$	−30.0000			−1.52106			−0.760530	−	0.649303i	$-0.775061\pi$
$389$	−30.0000			−1.52106			−0.760530	+	0.649303i	$0.775061\pi$
$390$		0			0
$391$	18.0000			0.910299
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−24.0000			−1.20757
$396$		0			0
$397$		−	9.79796i		−	0.491745i	−0.969302	−	0.245873i	$-0.920925\pi$
$397$		−	9.79796i		−	0.491745i	0.969302	−	0.245873i	$-0.0790745\pi$
$398$		0			0
$399$	2.00000	+	4.89898i	0.100125	+	0.245256i
$400$		0			0
$401$	−18.0000			−0.898877			−0.449439	−	0.893311i	$-0.648376\pi$
$401$	−18.0000			−0.898877			−0.449439	+	0.893311i	$0.648376\pi$
$402$		0			0
$403$		−	39.1918i		−	1.95228i
$404$		0			0
$405$		−	2.44949i		−	0.121716i
$406$		0			0
$407$			9.79796i			0.485667i
$408$		0			0
$409$		−	4.89898i		−	0.242239i	−0.992638	−	0.121119i	$-0.961352\pi$
$409$		−	4.89898i		−	0.242239i	0.992638	−	0.121119i	$-0.0386484\pi$
$410$		0			0
$411$	6.00000			0.295958
$412$		0			0
$413$	−12.0000	−	29.3939i	−0.590481	−	1.44638i
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−4.00000			−0.195881
$418$		0			0
$419$	−36.0000			−1.75872			−0.879358	−	0.476162i	$-0.842028\pi$
$419$	−36.0000			−1.75872			−0.879358	+	0.476162i	$0.842028\pi$
$420$		0			0
$421$	−8.00000			−0.389896			−0.194948	−	0.980814i	$-0.562454\pi$
$421$	−8.00000			−0.389896			−0.194948	+	0.980814i	$0.562454\pi$
$422$		0			0
$423$	−12.0000			−0.583460
$424$		0			0
$425$			2.44949i			0.118818i
$426$		0			0
$427$		0			0
$428$		0			0
$429$	12.0000			0.579365
$430$		0			0
$431$			17.1464i			0.825914i	0.910750	+	0.412957i	$0.135504\pi$
$431$			17.1464i			0.825914i	−0.910750	+	0.412957i	$0.864496\pi$
$432$		0			0
$433$			9.79796i			0.470860i	0.971891	+	0.235430i	$0.0756498\pi$
$433$			9.79796i			0.470860i	−0.971891	+	0.235430i	$0.924350\pi$
$434$		0			0
$435$		−	14.6969i		−	0.704664i
$436$		0			0
$437$		−	14.6969i		−	0.703050i
$438$		0			0
$439$	22.0000			1.05000			0.525001	−	0.851101i	$-0.324065\pi$
$439$	22.0000			1.05000			0.525001	+	0.851101i	$0.324065\pi$
$440$		0			0
$441$	−5.00000	+	4.89898i	−0.238095	+	0.233285i
$442$		0			0
$443$		−	2.44949i		−	0.116379i	−0.998306	−	0.0581894i	$-0.981467\pi$
$443$		−	2.44949i		−	0.116379i	0.998306	−	0.0581894i	$-0.0185327\pi$
$444$		0			0
$445$	30.0000			1.42214
$446$		0			0
$447$	−18.0000			−0.851371
$448$		0			0
$449$	6.00000			0.283158			0.141579	−	0.989927i	$-0.454782\pi$
$449$	6.00000			0.283158			0.141579	+	0.989927i	$0.454782\pi$
$450$		0			0
$451$	−18.0000			−0.847587
$452$		0			0
$453$		−	14.6969i		−	0.690522i
$454$		0			0
$455$	12.0000	+	29.3939i	0.562569	+	1.37801i
$456$		0			0
$457$	38.0000			1.77757			0.888783	−	0.458329i	$-0.151552\pi$
$457$	38.0000			1.77757			0.888783	+	0.458329i	$0.151552\pi$
$458$		0			0
$459$		−	2.44949i		−	0.114332i
$460$		0			0
$461$		−	2.44949i		−	0.114084i	−0.998372	−	0.0570421i	$-0.981833\pi$
$461$		−	2.44949i		−	0.114084i	0.998372	−	0.0570421i	$-0.0181669\pi$
$462$		0			0
$463$			9.79796i			0.455350i	0.973737	+	0.227675i	$0.0731123\pi$
$463$			9.79796i			0.455350i	−0.973737	+	0.227675i	$0.926888\pi$
$464$		0			0
$465$		−	19.5959i		−	0.908739i
$466$		0			0
$467$	24.0000			1.11059			0.555294	−	0.831654i	$-0.312606\pi$
$467$	24.0000			1.11059			0.555294	+	0.831654i	$0.312606\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		−	19.5959i		−	0.902932i
$472$		0			0
$473$	12.0000			0.551761
$474$		0			0
$475$	2.00000			0.0917663
$476$		0			0
$477$	−6.00000			−0.274721
$478$		0			0
$479$	−24.0000			−1.09659			−0.548294	−	0.836286i	$-0.684723\pi$
$479$	−24.0000			−1.09659			−0.548294	+	0.836286i	$0.684723\pi$
$480$		0			0
$481$		−	19.5959i		−	0.893497i
$482$		0			0
$483$	18.0000	−	7.34847i	0.819028	−	0.334367i
$484$		0			0
$485$	−12.0000			−0.544892
$486$		0			0
$487$			4.89898i			0.221994i	0.993821	+	0.110997i	$0.0354044\pi$
$487$			4.89898i			0.221994i	−0.993821	+	0.110997i	$0.964596\pi$
$488$		0			0
$489$			19.5959i			0.886158i
$490$		0			0
$491$			36.7423i			1.65816i	0.559131	+	0.829079i	$0.311135\pi$
$491$			36.7423i			1.65816i	−0.559131	+	0.829079i	$0.688865\pi$
$492$		0			0
$493$		−	14.6969i		−	0.661917i
$494$		0			0
$495$	6.00000			0.269680
$496$		0			0
$497$	30.0000	−	12.2474i	1.34568	−	0.549373i
$498$		0			0
$499$		−	14.6969i		−	0.657925i	−0.944343	−	0.328963i	$-0.893301\pi$
$499$		−	14.6969i		−	0.657925i	0.944343	−	0.328963i	$-0.106699\pi$
$500$		0			0
$501$	12.0000			0.536120
$502$		0			0
$503$	12.0000			0.535054			0.267527	−	0.963550i	$-0.413794\pi$
$503$	12.0000			0.535054			0.267527	+	0.963550i	$0.413794\pi$
$504$		0			0
$505$	6.00000			0.266996
$506$		0			0
$507$	−11.0000			−0.488527
$508$		0			0
$509$		−	17.1464i		−	0.760002i	−0.924986	−	0.380001i	$-0.875924\pi$
$509$		−	17.1464i		−	0.760002i	0.924986	−	0.380001i	$-0.124076\pi$
$510$		0			0
$511$	36.0000	−	14.6969i	1.59255	−	0.650154i
$512$		0			0
$513$	−2.00000			−0.0883022
$514$		0			0
$515$			39.1918i			1.72700i
$516$		0			0
$517$		−	29.3939i		−	1.29274i
$518$		0			0
$519$		−	7.34847i		−	0.322562i
$520$		0			0
$521$			2.44949i			0.107314i	0.998559	+	0.0536570i	$0.0170878\pi$
$521$			2.44949i			0.107314i	−0.998559	+	0.0536570i	$0.982912\pi$
$522$		0			0
$523$	20.0000			0.874539			0.437269	−	0.899331i	$-0.355946\pi$
$523$	20.0000			0.874539			0.437269	+	0.899331i	$0.355946\pi$
$524$		0			0
$525$	1.00000	+	2.44949i	0.0436436	+	0.106904i
$526$		0			0
$527$		−	19.5959i		−	0.853612i
$528$		0			0
$529$	−31.0000			−1.34783
$530$		0			0
$531$	12.0000			0.520756
$532$		0			0
$533$	36.0000			1.55933
$534$		0			0
$535$	−30.0000			−1.29701
$536$		0			0
$537$		−	17.1464i		−	0.739923i
$538$		0			0
$539$	−12.0000	−	12.2474i	−0.516877	−	0.527535i
$540$		0			0
$541$	16.0000			0.687894			0.343947	−	0.938989i	$-0.388236\pi$
$541$	16.0000			0.687894			0.343947	+	0.938989i	$0.388236\pi$
$542$		0			0
$543$			4.89898i			0.210235i
$544$		0			0
$545$		−	4.89898i		−	0.209849i
$546$		0			0
$547$		−	29.3939i		−	1.25679i	−0.777894	−	0.628396i	$-0.783712\pi$
$547$		−	29.3939i		−	1.25679i	0.777894	−	0.628396i	$-0.216288\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−12.0000			−0.511217
$552$		0			0
$553$	−24.0000	+	9.79796i	−1.02058	+	0.416652i
$554$		0			0
$555$		−	9.79796i		−	0.415900i
$556$		0			0
$557$	6.00000			0.254228			0.127114	−	0.991888i	$-0.459429\pi$
$557$	6.00000			0.254228			0.127114	+	0.991888i	$0.459429\pi$
$558$		0			0
$559$	−24.0000			−1.01509
$560$		0			0
$561$	6.00000			0.253320
$562$		0			0
$563$	−24.0000			−1.01148			−0.505740	−	0.862686i	$-0.668780\pi$
$563$	−24.0000			−1.01148			−0.505740	+	0.862686i	$0.668780\pi$
$564$		0			0
$565$		−	14.6969i		−	0.618305i
$566$		0			0
$567$	−1.00000	−	2.44949i	−0.0419961	−	0.102869i
$568$		0			0
$569$	−6.00000			−0.251533			−0.125767	−	0.992060i	$-0.540139\pi$
$569$	−6.00000			−0.251533			−0.125767	+	0.992060i	$0.540139\pi$
$570$		0			0
$571$		−	19.5959i		−	0.820064i	−0.912071	−	0.410032i	$-0.865518\pi$
$571$		−	19.5959i		−	0.820064i	0.912071	−	0.410032i	$-0.134482\pi$
$572$		0			0
$573$		−	2.44949i		−	0.102329i
$574$		0			0
$575$		−	7.34847i		−	0.306452i
$576$		0			0
$577$		0			0		1.00000			$0$
$577$		0			0		−1.00000			$\pi$
$578$		0			0
$579$	16.0000			0.664937
$580$		0			0
$581$		0			0
$582$		0			0
$583$		−	14.6969i		−	0.608685i
$584$		0			0
$585$	−12.0000			−0.496139
$586$		0			0
$587$		0			0			−	1.00000i	$-0.5\pi$
$587$		0			0				1.00000i	$0.5\pi$
$588$		0			0
$589$	−16.0000			−0.659269
$590$		0			0
$591$	18.0000			0.740421
$592$		0			0
$593$			46.5403i			1.91118i	0.294697	+	0.955591i	$0.404781\pi$
$593$			46.5403i			1.91118i	−0.294697	+	0.955591i	$0.595219\pi$
$594$		0			0
$595$	6.00000	+	14.6969i	0.245976	+	0.602516i
$596$		0			0
$597$	−2.00000			−0.0818546
$598$		0			0
$599$			2.44949i			0.100083i	0.998747	+	0.0500417i	$0.0159354\pi$
$599$			2.44949i			0.100083i	−0.998747	+	0.0500417i	$0.984065\pi$
$600$		0			0
$601$		−	19.5959i		−	0.799334i	−0.916660	−	0.399667i	$-0.869126\pi$
$601$		−	19.5959i		−	0.799334i	0.916660	−	0.399667i	$-0.130874\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$		−	12.2474i		−	0.497930i
$606$		0			0
$607$	−14.0000			−0.568242			−0.284121	−	0.958788i	$-0.591702\pi$
$607$	−14.0000			−0.568242			−0.284121	+	0.958788i	$0.591702\pi$
$608$		0			0
$609$	−6.00000	−	14.6969i	−0.243132	−	0.595550i
$610$		0			0
$611$			58.7878i			2.37830i
$612$		0			0
$613$	26.0000			1.05013			0.525065	−	0.851062i	$-0.324041\pi$
$613$	26.0000			1.05013			0.525065	+	0.851062i	$0.324041\pi$
$614$		0			0
$615$	18.0000			0.725830
$616$		0			0
$617$	−42.0000			−1.69086			−0.845428	−	0.534089i	$-0.820655\pi$
$617$	−42.0000			−1.69086			−0.845428	+	0.534089i	$0.820655\pi$
$618$		0			0
$619$	−4.00000			−0.160774			−0.0803868	−	0.996764i	$-0.525616\pi$
$619$	−4.00000			−0.160774			−0.0803868	+	0.996764i	$0.525616\pi$
$620$		0			0
$621$			7.34847i			0.294884i
$622$		0			0
$623$	30.0000	−	12.2474i	1.20192	−	0.490684i
$624$		0			0
$625$	−29.0000			−1.16000
$626$		0			0
$627$		−	4.89898i		−	0.195646i
$628$		0			0
$629$		−	9.79796i		−	0.390670i
$630$		0			0
$631$		−	19.5959i		−	0.780101i	−0.920793	−	0.390051i	$-0.872458\pi$
$631$		−	19.5959i		−	0.780101i	0.920793	−	0.390051i	$-0.127542\pi$
$632$		0			0
$633$			4.89898i			0.194717i
$634$		0			0
$635$	24.0000			0.952411
$636$		0			0
$637$	24.0000	+	24.4949i	0.950915	+	0.970523i
$638$		0			0
$639$			12.2474i			0.484502i
$640$		0			0
$641$	−6.00000			−0.236986			−0.118493	−	0.992955i	$-0.537806\pi$
$641$	−6.00000			−0.236986			−0.118493	+	0.992955i	$0.537806\pi$
$642$		0			0
$643$	−14.0000			−0.552106			−0.276053	−	0.961142i	$-0.589027\pi$
$643$	−14.0000			−0.552106			−0.276053	+	0.961142i	$0.589027\pi$
$644$		0			0
$645$	−12.0000			−0.472500
$646$		0			0
$647$	−24.0000			−0.943537			−0.471769	−	0.881722i	$-0.656384\pi$
$647$	−24.0000			−0.943537			−0.471769	+	0.881722i	$0.656384\pi$
$648$		0			0
$649$			29.3939i			1.15381i
$650$		0			0
$651$	−8.00000	−	19.5959i	−0.313545	−	0.768025i
$652$		0			0
$653$	−6.00000			−0.234798			−0.117399	−	0.993085i	$-0.537456\pi$
$653$	−6.00000			−0.234798			−0.117399	+	0.993085i	$0.537456\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$			14.6969i			0.573382i
$658$		0			0
$659$			7.34847i			0.286256i	0.989704	+	0.143128i	$0.0457160\pi$
$659$			7.34847i			0.286256i	−0.989704	+	0.143128i	$0.954284\pi$
$660$		0			0
$661$			29.3939i			1.14329i	0.820501	+	0.571645i	$0.193694\pi$
$661$			29.3939i			1.14329i	−0.820501	+	0.571645i	$0.806306\pi$
$662$		0			0
$663$	−12.0000			−0.466041
$664$		0			0
$665$	12.0000	−	4.89898i	0.465340	−	0.189974i
$666$		0			0
$667$			44.0908i			1.70720i
$668$		0			0
$669$	10.0000			0.386622
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−8.00000			−0.308377			−0.154189	−	0.988041i	$-0.549276\pi$
$673$	−8.00000			−0.308377			−0.154189	+	0.988041i	$0.549276\pi$
$674$		0			0
$675$	−1.00000			−0.0384900
$676$		0			0
$677$		−	12.2474i		−	0.470708i	−0.971910	−	0.235354i	$-0.924375\pi$
$677$		−	12.2474i		−	0.470708i	0.971910	−	0.235354i	$-0.0756249\pi$
$678$		0			0
$679$	−12.0000	+	4.89898i	−0.460518	+	0.188006i
$680$		0			0
$681$	12.0000			0.459841
$682$		0			0
$683$			7.34847i			0.281181i	0.990068	+	0.140591i	$0.0449002\pi$
$683$			7.34847i			0.281181i	−0.990068	+	0.140591i	$0.955100\pi$
$684$		0			0
$685$		−	14.6969i		−	0.561541i
$686$		0			0
$687$			24.4949i			0.934539i
$688$		0			0
$689$			29.3939i			1.11982i
$690$		0			0
$691$	20.0000			0.760836			0.380418	−	0.924815i	$-0.375780\pi$
$691$	20.0000			0.760836			0.380418	+	0.924815i	$0.375780\pi$
$692$		0			0
$693$	6.00000	−	2.44949i	0.227921	−	0.0930484i
$694$		0			0
$695$			9.79796i			0.371658i
$696$		0			0
$697$	18.0000			0.681799
$698$		0			0
$699$	−30.0000			−1.13470
$700$		0			0
$701$	6.00000			0.226617			0.113308	−	0.993560i	$-0.463855\pi$
$701$	6.00000			0.226617			0.113308	+	0.993560i	$0.463855\pi$
$702$		0			0
$703$	−8.00000			−0.301726
$704$		0			0
$705$			29.3939i			1.10704i
$706$		0			0
$707$	6.00000	−	2.44949i	0.225653	−	0.0921225i
$708$		0			0
$709$	−10.0000			−0.375558			−0.187779	−	0.982211i	$-0.560129\pi$
$709$	−10.0000			−0.375558			−0.187779	+	0.982211i	$0.560129\pi$
$710$		0			0
$711$		−	9.79796i		−	0.367452i
$712$		0			0
$713$			58.7878i			2.20162i
$714$		0			0
$715$		−	29.3939i		−	1.09927i
$716$		0			0
$717$		−	12.2474i		−	0.457389i
$718$		0			0
$719$	24.0000			0.895049			0.447524	−	0.894272i	$-0.352306\pi$
$719$	24.0000			0.895049			0.447524	+	0.894272i	$0.352306\pi$
$720$		0			0
$721$	16.0000	+	39.1918i	0.595871	+	1.45958i
$722$		0			0
$723$		−	4.89898i		−	0.182195i
$724$		0			0
$725$	−6.00000			−0.222834
$726$		0			0
$727$	32.0000			1.18681			0.593407	−	0.804902i	$-0.297782\pi$
$727$	32.0000			1.18681			0.593407	+	0.804902i	$0.297782\pi$
$728$		0			0
$729$	1.00000			0.0370370
$730$		0			0
$731$	−12.0000			−0.443836
$732$		0			0
$733$		−	4.89898i		−	0.180948i	−0.995899	−	0.0904740i	$-0.971162\pi$
$733$		−	4.89898i		−	0.180948i	0.995899	−	0.0904740i	$-0.0288382\pi$
$734$		0			0
$735$	12.0000	+	12.2474i	0.442627	+	0.451754i
$736$		0			0
$737$		0			0
$738$		0			0
$739$			9.79796i			0.360424i	0.983628	+	0.180212i	$0.0576783\pi$
$739$			9.79796i			0.360424i	−0.983628	+	0.180212i	$0.942322\pi$
$740$		0			0
$741$			9.79796i			0.359937i
$742$		0			0
$743$		−	12.2474i		−	0.449315i	−0.974438	−	0.224658i	$-0.927874\pi$
$743$		−	12.2474i		−	0.449315i	0.974438	−	0.224658i	$-0.0721264\pi$
$744$		0			0
$745$			44.0908i			1.61536i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−30.0000	+	12.2474i	−1.09618	+	0.447512i
$750$		0			0
$751$			34.2929i			1.25136i	0.780078	+	0.625682i	$0.215179\pi$
$751$			34.2929i			1.25136i	−0.780078	+	0.625682i	$0.784821\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−36.0000			−1.31017
$756$		0			0
$757$	−10.0000			−0.363456			−0.181728	−	0.983349i	$-0.558169\pi$
$757$	−10.0000			−0.363456			−0.181728	+	0.983349i	$0.558169\pi$
$758$		0			0
$759$	−18.0000			−0.653359
$760$		0			0
$761$		−	41.6413i		−	1.50950i	−0.656014	−	0.754748i	$-0.727759\pi$
$761$		−	41.6413i		−	1.50950i	0.656014	−	0.754748i	$-0.272241\pi$
$762$		0			0
$763$	−2.00000	−	4.89898i	−0.0724049	−	0.177355i
$764$		0			0
$765$	−6.00000			−0.216930
$766$		0			0
$767$		−	58.7878i		−	2.12270i
$768$		0			0
$769$			9.79796i			0.353323i	0.984272	+	0.176662i	$0.0565299\pi$
$769$			9.79796i			0.353323i	−0.984272	+	0.176662i	$0.943470\pi$
$770$		0			0
$771$		−	22.0454i		−	0.793946i
$772$		0			0
$773$		−	22.0454i		−	0.792918i	−0.918052	−	0.396459i	$-0.870239\pi$
$773$		−	22.0454i		−	0.792918i	0.918052	−	0.396459i	$-0.129761\pi$
$774$		0			0
$775$	−8.00000			−0.287368
$776$		0			0
$777$	−4.00000	−	9.79796i	−0.143499	−	0.351500i
$778$		0			0
$779$		−	14.6969i		−	0.526572i
$780$		0			0
$781$	−30.0000			−1.07348
$782$		0			0
$783$	6.00000			0.214423
$784$		0			0
$785$	−48.0000			−1.71319
$786$		0			0
$787$	20.0000			0.712923			0.356462	−	0.934310i	$-0.383983\pi$
$787$	20.0000			0.712923			0.356462	+	0.934310i	$0.383983\pi$
$788$		0			0
$789$			2.44949i			0.0872041i
$790$		0			0
$791$	−6.00000	−	14.6969i	−0.213335	−	0.522563i
$792$		0			0
$793$		0			0
$794$		0			0
$795$			14.6969i			0.521247i
$796$		0			0
$797$			36.7423i			1.30148i	0.759300	+	0.650740i	$0.225541\pi$
$797$			36.7423i			1.30148i	−0.759300	+	0.650740i	$0.774459\pi$
$798$		0			0
$799$			29.3939i			1.03988i
$800$		0			0
$801$			12.2474i			0.432742i
$802$		0			0
$803$	−36.0000			−1.27041
$804$		0			0
$805$	−18.0000	−	44.0908i	−0.634417	−	1.55400i
$806$		0			0
$807$			22.0454i			0.776035i
$808$		0			0
$809$	18.0000			0.632846			0.316423	−	0.948618i	$-0.397518\pi$
$809$	18.0000			0.632846			0.316423	+	0.948618i	$0.397518\pi$
$810$		0			0
$811$	20.0000			0.702295			0.351147	−	0.936320i	$-0.385792\pi$
$811$	20.0000			0.702295			0.351147	+	0.936320i	$0.385792\pi$
$812$		0			0
$813$	−16.0000			−0.561144
$814$		0			0
$815$	48.0000			1.68137
$816$		0			0
$817$			9.79796i			0.342787i
$818$		0			0
$819$	−12.0000	+	4.89898i	−0.419314	+	0.171184i
$820$		0			0
$821$	18.0000			0.628204			0.314102	−	0.949389i	$-0.398297\pi$
$821$	18.0000			0.628204			0.314102	+	0.949389i	$0.398297\pi$
$822$		0			0
$823$			19.5959i			0.683071i	0.939869	+	0.341535i	$0.110947\pi$
$823$			19.5959i			0.683071i	−0.939869	+	0.341535i	$0.889053\pi$
$824$		0			0
$825$		−	2.44949i		−	0.0852803i
$826$		0			0
$827$		−	46.5403i		−	1.61836i	−0.587557	−	0.809182i	$-0.699910\pi$
$827$		−	46.5403i		−	1.61836i	0.587557	−	0.809182i	$-0.300090\pi$
$828$		0			0
$829$		−	4.89898i		−	0.170149i	−0.996375	−	0.0850743i	$-0.972887\pi$
$829$		−	4.89898i		−	0.170149i	0.996375	−	0.0850743i	$-0.0271128\pi$
$830$		0			0
$831$	−8.00000			−0.277517
$832$		0			0
$833$	12.0000	+	12.2474i	0.415775	+	0.424349i
$834$		0			0
$835$		−	29.3939i		−	1.01722i
$836$		0			0
$837$	8.00000			0.276520
$838$		0			0
$839$	12.0000			0.414286			0.207143	−	0.978311i	$-0.433583\pi$
$839$	12.0000			0.414286			0.207143	+	0.978311i	$0.433583\pi$
$840$		0			0
$841$	7.00000			0.241379
$842$		0			0
$843$	6.00000			0.206651
$844$		0			0
$845$			26.9444i			0.926915i
$846$		0			0
$847$	−5.00000	−	12.2474i	−0.171802	−	0.420827i
$848$		0			0
$849$	−14.0000			−0.480479
$850$		0			0
$851$			29.3939i			1.00761i
$852$		0			0
$853$		−	29.3939i		−	1.00643i	−0.864162	−	0.503214i	$-0.832151\pi$
$853$		−	29.3939i		−	1.00643i	0.864162	−	0.503214i	$-0.167849\pi$
$854$		0			0
$855$			4.89898i			0.167542i
$856$		0			0
$857$		−	22.0454i		−	0.753057i	−0.926405	−	0.376528i	$-0.877118\pi$
$857$		−	22.0454i		−	0.753057i	0.926405	−	0.376528i	$-0.122882\pi$
$858$		0			0
$859$	22.0000			0.750630			0.375315	−	0.926897i	$-0.377534\pi$
$859$	22.0000			0.750630			0.375315	+	0.926897i	$0.377534\pi$
$860$		0			0
$861$	18.0000	−	7.34847i	0.613438	−	0.250435i
$862$		0			0
$863$		−	51.4393i		−	1.75101i	−0.483206	−	0.875507i	$-0.660528\pi$
$863$		−	51.4393i		−	1.75101i	0.483206	−	0.875507i	$-0.339472\pi$
$864$		0			0
$865$	−18.0000			−0.612018
$866$		0			0
$867$	11.0000			0.373580
$868$		0			0
$869$	24.0000			0.814144
$870$		0			0
$871$		0			0
$872$		0			0
$873$		−	4.89898i		−	0.165805i
$874$		0			0
$875$	−24.0000	+	9.79796i	−0.811348	+	0.331231i
$876$		0			0
$877$	−22.0000			−0.742887			−0.371444	−	0.928456i	$-0.621137\pi$
$877$	−22.0000			−0.742887			−0.371444	+	0.928456i	$0.621137\pi$
$878$		0			0
$879$			22.0454i			0.743573i
$880$		0			0
$881$			2.44949i			0.0825254i	0.999148	+	0.0412627i	$0.0131381\pi$
$881$			2.44949i			0.0825254i	−0.999148	+	0.0412627i	$0.986862\pi$
$882$		0			0
$883$			34.2929i			1.15405i	0.816728	+	0.577023i	$0.195786\pi$
$883$			34.2929i			1.15405i	−0.816728	+	0.577023i	$0.804214\pi$
$884$		0			0
$885$		−	29.3939i		−	0.988064i
$886$		0			0
$887$	12.0000			0.402921			0.201460	−	0.979497i	$-0.435431\pi$
$887$	12.0000			0.402921			0.201460	+	0.979497i	$0.435431\pi$
$888$		0			0
$889$	24.0000	−	9.79796i	0.804934	−	0.328613i
$890$		0			0
$891$			2.44949i			0.0820610i
$892$		0			0
$893$	24.0000			0.803129
$894$		0			0
$895$	−42.0000			−1.40391
$896$		0			0
$897$	36.0000			1.20201
$898$		0			0
$899$	48.0000			1.60089
$900$		0			0
$901$			14.6969i			0.489626i
$902$		0			0
$903$	−12.0000	+	4.89898i	−0.399335	+	0.163028i
$904$		0			0
$905$	12.0000			0.398893
$906$		0			0
$907$			24.4949i			0.813340i	0.913575	+	0.406670i	$0.133310\pi$
$907$			24.4949i			0.813340i	−0.913575	+	0.406670i	$0.866690\pi$
$908$		0			0
$909$			2.44949i			0.0812444i
$910$		0			0
$911$		−	7.34847i		−	0.243466i	−0.992563	−	0.121733i	$-0.961155\pi$
$911$		−	7.34847i		−	0.243466i	0.992563	−	0.121733i	$-0.0388451\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$		−	14.6969i		−	0.484807i	−0.970175	−	0.242404i	$-0.922064\pi$
$919$		−	14.6969i		−	0.484807i	0.970175	−	0.242404i	$-0.0779358\pi$
$920$		0			0
$921$	−26.0000			−0.856729
$922$		0			0
$923$	60.0000			1.97492
$924$		0			0
$925$	−4.00000			−0.131519
$926$		0			0
$927$	−16.0000			−0.525509
$928$		0			0
$929$		−	12.2474i		−	0.401826i	−0.979609	−	0.200913i	$-0.935609\pi$
$929$		−	12.2474i		−	0.401826i	0.979609	−	0.200913i	$-0.0643908\pi$
$930$		0			0
$931$	10.0000	−	9.79796i	0.327737	−	0.321115i
$932$		0			0
$933$	−24.0000			−0.785725
$934$		0			0
$935$		−	14.6969i		−	0.480641i
$936$		0			0
$937$		−	39.1918i		−	1.28034i	−0.768233	−	0.640171i	$-0.778864\pi$
$937$		−	39.1918i		−	1.28034i	0.768233	−	0.640171i	$-0.221136\pi$
$938$		0			0
$939$			9.79796i			0.319744i
$940$		0			0
$941$		−	12.2474i		−	0.399255i	−0.979872	−	0.199628i	$-0.936027\pi$
$941$		−	12.2474i		−	0.399255i	0.979872	−	0.199628i	$-0.0639733\pi$
$942$		0			0
$943$	−54.0000			−1.75848
$944$		0			0
$945$	−6.00000	+	2.44949i	−0.195180	+	0.0796819i
$946$		0			0
$947$		−	26.9444i		−	0.875575i	−0.899078	−	0.437787i	$-0.855762\pi$
$947$		−	26.9444i		−	0.875575i	0.899078	−	0.437787i	$-0.144238\pi$
$948$		0			0
$949$	72.0000			2.33722
$950$		0			0
$951$	−18.0000			−0.583690
$952$		0			0
$953$	−18.0000			−0.583077			−0.291539	−	0.956559i	$-0.594167\pi$
$953$	−18.0000			−0.583077			−0.291539	+	0.956559i	$0.594167\pi$
$954$		0			0
$955$	−6.00000			−0.194155
$956$		0			0
$957$			14.6969i			0.475085i
$958$		0			0
$959$	−6.00000	−	14.6969i	−0.193750	−	0.474589i
$960$		0			0
$961$	33.0000			1.06452
$962$		0			0
$963$		−	12.2474i		−	0.394669i
$964$		0			0
$965$		−	39.1918i		−	1.26163i
$966$		0			0
$967$			58.7878i			1.89049i	0.326366	+	0.945243i	$0.394176\pi$
$967$			58.7878i			1.89049i	−0.326366	+	0.945243i	$0.605824\pi$
$968$		0			0
$969$			4.89898i			0.157378i
$970$		0			0
$971$	−36.0000			−1.15529			−0.577647	−	0.816286i	$-0.696029\pi$
$971$	−36.0000			−1.15529			−0.577647	+	0.816286i	$0.696029\pi$
$972$		0			0
$973$	4.00000	+	9.79796i	0.128234	+	0.314108i
$974$		0			0
$975$			4.89898i			0.156893i
$976$		0			0
$977$	6.00000			0.191957			0.0959785	−	0.995383i	$-0.469402\pi$
$977$	6.00000			0.191957			0.0959785	+	0.995383i	$0.469402\pi$
$978$		0			0
$979$	−30.0000			−0.958804
$980$		0			0
$981$	2.00000			0.0638551
$982$		0			0
$983$	12.0000			0.382741			0.191370	−	0.981518i	$-0.438707\pi$
$983$	12.0000			0.382741			0.191370	+	0.981518i	$0.438707\pi$
$984$		0			0
$985$		−	44.0908i		−	1.40485i
$986$		0			0
$987$	12.0000	+	29.3939i	0.381964	+	0.935617i
$988$		0			0
$989$	36.0000			1.14473
$990$		0			0
$991$		−	14.6969i		−	0.466864i	−0.972373	−	0.233432i	$-0.925004\pi$
$991$		−	14.6969i		−	0.466864i	0.972373	−	0.233432i	$-0.0749956\pi$
$992$		0			0
$993$		−	24.4949i		−	0.777322i
$994$		0			0
$995$			4.89898i			0.155308i
$996$		0			0
$997$		0			0		1.00000			$0$
$997$		0			0		−1.00000			$\pi$
$998$		0			0
$999$	4.00000			0.126554

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	336.2.b.c.223.1	yes	2
3.2	odd	2		1008.2.b.d.559.2		2
4.3	odd	2		336.2.b.b.223.1	✓	2
7.2	even	3		2352.2.bl.m.31.1		4
7.3	odd	6		2352.2.bl.n.607.1		4
7.4	even	3		2352.2.bl.m.607.2		4
7.5	odd	6		2352.2.bl.n.31.2		4
7.6	odd	2		336.2.b.b.223.2	yes	2
8.3	odd	2		1344.2.b.d.895.2		2
8.5	even	2		1344.2.b.a.895.2		2
12.11	even	2		1008.2.b.e.559.2		2
21.20	even	2		1008.2.b.e.559.1		2
24.5	odd	2		4032.2.b.d.3583.1		2
24.11	even	2		4032.2.b.f.3583.1		2
28.3	even	6		2352.2.bl.m.607.1		4
28.11	odd	6		2352.2.bl.n.607.2		4
28.19	even	6		2352.2.bl.m.31.2		4
28.23	odd	6		2352.2.bl.n.31.1		4
28.27	even	2	inner	336.2.b.c.223.2	yes	2
56.13	odd	2		1344.2.b.d.895.1		2
56.27	even	2		1344.2.b.a.895.1		2
84.83	odd	2		1008.2.b.d.559.1		2
168.83	odd	2		4032.2.b.d.3583.2		2
168.125	even	2		4032.2.b.f.3583.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
336.2.b.b.223.1	✓	2	4.3	odd	2
336.2.b.b.223.2	yes	2	7.6	odd	2
336.2.b.c.223.1	yes	2	1.1	even	1	trivial
336.2.b.c.223.2	yes	2	28.27	even	2	inner
1008.2.b.d.559.1		2	84.83	odd	2
1008.2.b.d.559.2		2	3.2	odd	2
1008.2.b.e.559.1		2	21.20	even	2
1008.2.b.e.559.2		2	12.11	even	2
1344.2.b.a.895.1		2	56.27	even	2
1344.2.b.a.895.2		2	8.5	even	2
1344.2.b.d.895.1		2	56.13	odd	2
1344.2.b.d.895.2		2	8.3	odd	2
2352.2.bl.m.31.1		4	7.2	even	3
2352.2.bl.m.31.2		4	28.19	even	6
2352.2.bl.m.607.1		4	28.3	even	6
2352.2.bl.m.607.2		4	7.4	even	3
2352.2.bl.n.31.1		4	28.23	odd	6
2352.2.bl.n.31.2		4	7.5	odd	6
2352.2.bl.n.607.1		4	7.3	odd	6
2352.2.bl.n.607.2		4	28.11	odd	6
4032.2.b.d.3583.1		2	24.5	odd	2
4032.2.b.d.3583.2		2	168.83	odd	2
4032.2.b.f.3583.1		2	24.11	even	2
4032.2.b.f.3583.2		2	168.125	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:	\( N \)	\(=\)	\( 336 = 2^{4} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	336.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -2.44949i q^{5} +(-1.00000 - 2.44949i) q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -2.44949i q^{5} +(-1.00000 - 2.44949i) q^{7} +1.00000 q^{9} +2.44949i q^{11} -4.89898i q^{13} -2.44949i q^{15} -2.44949i q^{17} -2.00000 q^{19} +(-1.00000 - 2.44949i) q^{21} +7.34847i q^{23} -1.00000 q^{25} +1.00000 q^{27} +6.00000 q^{29} +8.00000 q^{31} +2.44949i q^{33} +(-6.00000 + 2.44949i) q^{35} +4.00000 q^{37} -4.89898i q^{39} +7.34847i q^{41} -4.89898i q^{43} -2.44949i q^{45} -12.0000 q^{47} +(-5.00000 + 4.89898i) q^{49} -2.44949i q^{51} -6.00000 q^{53} +6.00000 q^{55} -2.00000 q^{57} +12.0000 q^{59} +(-1.00000 - 2.44949i) q^{63} -12.0000 q^{65} +7.34847i q^{69} +12.2474i q^{71} +14.6969i q^{73} -1.00000 q^{75} +(6.00000 - 2.44949i) q^{77} -9.79796i q^{79} +1.00000 q^{81} -6.00000 q^{85} +6.00000 q^{87} +12.2474i q^{89} +(-12.0000 + 4.89898i) q^{91} +8.00000 q^{93} +4.89898i q^{95} -4.89898i q^{97} +2.44949i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} - 2 q^{7} + 2 q^{9}+O(q^{10}) \) 2 * q + 2 * q^3 - 2 * q^7 + 2 * q^9	\( 2 q + 2 q^{3} - 2 q^{7} + 2 q^{9} - 4 q^{19} - 2 q^{21} - 2 q^{25} + 2 q^{27} + 12 q^{29} + 16 q^{31} - 12 q^{35} + 8 q^{37} - 24 q^{47} - 10 q^{49} - 12 q^{53} + 12 q^{55} - 4 q^{57} + 24 q^{59} - 2 q^{63} - 24 q^{65} - 2 q^{75} + 12 q^{77} + 2 q^{81} - 12 q^{85} + 12 q^{87} - 24 q^{91} + 16 q^{93}+O(q^{100}) \) 2 * q + 2 * q^3 - 2 * q^7 + 2 * q^9 - 4 * q^19 - 2 * q^21 - 2 * q^25 + 2 * q^27 + 12 * q^29 + 16 * q^31 - 12 * q^35 + 8 * q^37 - 24 * q^47 - 10 * q^49 - 12 * q^53 + 12 * q^55 - 4 * q^57 + 24 * q^59 - 2 * q^63 - 24 * q^65 - 2 * q^75 + 12 * q^77 + 2 * q^81 - 12 * q^85 + 12 * q^87 - 24 * q^91 + 16 * q^93

Introduction

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