LMFDB - Embedded newform 69.2.c.a.68.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [69,2,Mod(68,69)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(69, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("69.68");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 69 = 3 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	69.c (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:	$0.550967773947$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.8869743.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 3x^{3} + 8 $ x^6 - 3*x^3 + 8
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2\cdot 3 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			68.3
Root			$1.33454 - 0.467979i$ of defining polynomial
Character	$\chi$	$=$	69.68
Dual form			69.2.c.a.68.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.935958i q^{2} +(-0.227452 + 1.71705i) q^{3} +1.12398 q^{4} +(1.60709 + 0.212885i) q^{6} -2.92392i q^{8} +(-2.89653 - 0.781094i) q^{9} +O(q^{10})$	\(q-0.935958i q^{2} +(-0.227452 + 1.71705i) q^{3} +1.12398 q^{4} +(1.60709 + 0.212885i) q^{6} -2.92392i q^{8} +(-2.89653 - 0.781094i) q^{9} +(-0.255652 + 1.92994i) q^{12} -4.88325 q^{13} -0.488695 q^{16} +(-0.731071 + 2.71103i) q^{18} +4.79583i q^{23} +(5.02051 + 0.665051i) q^{24} -5.00000 q^{25} +4.57052i q^{26} +(2.00000 - 4.79583i) q^{27} -8.43039i q^{29} +11.1312 q^{31} -5.39043i q^{32} +(-3.25565 - 0.877936i) q^{36} +(1.11071 - 8.38480i) q^{39} +12.1742i q^{41} +4.48870 q^{46} +6.55848i q^{47} +(0.111155 - 0.839115i) q^{48} +7.00000 q^{49} +4.67979i q^{50} -5.48870 q^{52} +(-4.48870 - 1.87192i) q^{54} -7.89049 q^{58} +9.59166i q^{59} -10.4184i q^{62} -6.02261 q^{64} +(-8.23469 - 1.09082i) q^{69} -14.0461i q^{71} +(-2.28385 + 8.46921i) q^{72} -7.61268 q^{73} +(1.13726 - 8.58526i) q^{75} +(-7.84782 - 1.03957i) q^{78} +(7.77979 + 4.52492i) q^{81} +11.3946 q^{82} +(14.4754 + 1.91751i) q^{87} +5.39043i q^{92} +(-2.53182 + 19.1129i) q^{93} +6.13846 q^{94} +(9.25565 + 1.22607i) q^{96} -6.55170i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 12 q^{4} + 3 q^{6}+O(q^{10}) $ 6 * q - 12 * q^4 + 3 * q^6	$ 6 q - 12 q^{4} + 3 q^{6} - 15 q^{12} + 24 q^{16} + 21 q^{18} - 6 q^{24} - 30 q^{25} + 12 q^{27} - 33 q^{36} - 24 q^{39} + 69 q^{48} + 42 q^{49} - 6 q^{52} + 30 q^{58} - 90 q^{64} - 42 q^{72} - 51 q^{78} + 66 q^{82} + 48 q^{87} - 6 q^{93} - 78 q^{94} + 69 q^{96}+O(q^{100}) $ 6 * q - 12 * q^4 + 3 * q^6 - 15 * q^12 + 24 * q^16 + 21 * q^18 - 6 * q^24 - 30 * q^25 + 12 * q^27 - 33 * q^36 - 24 * q^39 + 69 * q^48 + 42 * q^49 - 6 * q^52 + 30 * q^58 - 90 * q^64 - 42 * q^72 - 51 * q^78 + 66 * q^82 + 48 * q^87 - 6 * q^93 - 78 * q^94 + 69 * q^96

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/69\mathbb{Z}\right)^\times$.

$n$	$28$	$47$
$\chi(n)$	$-1$	$-1$

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 $

$2$		−	0.935958i		−	0.661822i	−0.943662	−	0.330911i	$-0.892644\pi$
$2$		−	0.935958i		−	0.661822i	0.943662	−	0.330911i	$-0.107356\pi$
$3$	−0.227452	+	1.71705i	−0.131319	+	0.991340i
$3$	−0.227452	+	1.71705i	−0.131319	+	0.991340i
$4$	1.12398			0.561992
$5$		0			0			−	1.00000i	$-0.5\pi$
$5$		0			0				1.00000i	$0.5\pi$
$6$	1.60709	+	0.212885i	0.656091	+	0.0869101i
$7$		0			0		1.00000			$0$
$7$		0			0		−1.00000			$\pi$
$8$		−	2.92392i		−	1.03376i
$9$	−2.89653	−	0.781094i	−0.965510	−	0.260365i
$10$		0			0
$11$		0			0			−	1.00000i	$-0.5\pi$
$11$		0			0				1.00000i	$0.5\pi$
$12$	−0.255652	+	1.92994i	−0.0738005	+	0.557125i
$13$	−4.88325			−1.35437			−0.677185	−	0.735812i	$-0.736801\pi$
$13$	−4.88325			−1.35437			−0.677185	+	0.735812i	$0.736801\pi$
$14$		0			0
$15$		0			0
$16$	−0.488695			−0.122174
$17$		0			0			−	1.00000i	$-0.5\pi$
$17$		0			0				1.00000i	$0.5\pi$
$18$	−0.731071	+	2.71103i	−0.172315	+	0.638996i
$19$		0			0		1.00000			$0$
$19$		0			0		−1.00000			$\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$			4.79583i			1.00000i
$23$			4.79583i			1.00000i
$24$	5.02051	+	0.665051i	1.02481	+	0.135753i
$25$	−5.00000			−1.00000
$26$			4.57052i			0.896353i
$27$	2.00000	−	4.79583i	0.384900	−	0.922958i
$28$		0			0
$29$		−	8.43039i		−	1.56548i	−0.622346	−	0.782742i	$-0.713820\pi$
$29$		−	8.43039i		−	1.56548i	0.622346	−	0.782742i	$-0.286180\pi$
$30$		0			0
$31$	11.1312			1.99923			0.999613	−	0.0278144i	$-0.00885474\pi$
$31$	11.1312			1.99923			0.999613	+	0.0278144i	$0.00885474\pi$
$32$		−	5.39043i		−	0.952903i
$33$		0			0
$34$		0			0
$35$		0			0
$36$	−3.25565	−	0.877936i	−0.542609	−	0.146323i
$37$		0			0		1.00000			$0$
$37$		0			0		−1.00000			$\pi$
$38$		0			0
$39$	1.11071	−	8.38480i	0.177855	−	1.34264i
$40$		0			0
$41$			12.1742i			1.90129i	0.310274	+	0.950647i	$0.399579\pi$
$41$			12.1742i			1.90129i	−0.310274	+	0.950647i	$0.600421\pi$
$42$		0			0
$43$		0			0		1.00000			$0$
$43$		0			0		−1.00000			$\pi$
$44$		0			0
$45$		0			0
$46$	4.48870			0.661822
$47$			6.55848i			0.956652i	0.878182	+	0.478326i	$0.158756\pi$
$47$			6.55848i			0.956652i	−0.878182	+	0.478326i	$0.841244\pi$
$48$	0.111155	−	0.839115i	0.0160438	−	0.121116i
$49$	7.00000			1.00000
$50$			4.67979i			0.661822i
$51$		0			0
$52$	−5.48870			−0.761145
$53$		0			0			−	1.00000i	$-0.5\pi$
$53$		0			0				1.00000i	$0.5\pi$
$54$	−4.48870	−	1.87192i	−0.610834	−	0.254735i
$55$		0			0
$56$		0			0
$57$		0			0
$58$	−7.89049			−1.03607
$59$			9.59166i			1.24873i	0.781133	+	0.624364i	$0.214642\pi$
$59$			9.59166i			1.24873i	−0.781133	+	0.624364i	$0.785358\pi$
$60$		0			0
$61$		0			0		1.00000			$0$
$61$		0			0		−1.00000			$\pi$
$62$		−	10.4184i		−	1.32313i
$63$		0			0
$64$	−6.02261			−0.752826
$65$		0			0
$66$		0			0
$67$		0			0		1.00000			$0$
$67$		0			0		−1.00000			$\pi$
$68$		0			0
$69$	−8.23469	−	1.09082i	−0.991340	−	0.131319i
$70$		0			0
$71$		−	14.0461i		−	1.66697i	−0.552542	−	0.833485i	$-0.686342\pi$
$71$		−	14.0461i		−	1.66697i	0.552542	−	0.833485i	$-0.313658\pi$
$72$	−2.28385	+	8.46921i	−0.269155	+	0.998106i
$73$	−7.61268			−0.890997			−0.445498	−	0.895283i	$-0.646973\pi$
$73$	−7.61268			−0.890997			−0.445498	+	0.895283i	$0.646973\pi$
$74$		0			0
$75$	1.13726	−	8.58526i	0.131319	−	0.991340i
$76$		0			0
$77$		0			0
$78$	−7.84782	−	1.03957i	−0.888590	−	0.117709i
$79$		0			0		1.00000			$0$
$79$		0			0		−1.00000			$\pi$
$80$		0			0
$81$	7.77979	+	4.52492i	0.864421	+	0.502769i
$82$	11.3946			1.25832
$83$		0			0			−	1.00000i	$-0.5\pi$
$83$		0			0				1.00000i	$0.5\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	14.4754	+	1.91751i	1.55193	+	0.205579i
$88$		0			0
$89$		0			0			−	1.00000i	$-0.5\pi$
$89$		0			0				1.00000i	$0.5\pi$
$90$		0			0
$91$		0			0
$92$			5.39043i			0.561992i
$93$	−2.53182	+	19.1129i	−0.262537	+	1.98191i
$94$	6.13846			0.633134
$95$		0			0
$96$	9.25565	+	1.22607i	0.944651	+	0.125135i
$97$		0			0		1.00000			$0$
$97$		0			0		−1.00000			$\pi$
$98$		−	6.55170i		−	0.661822i
$99$		0			0
$100$	−5.61992			−0.561992
$101$		−	19.1833i		−	1.90881i	−0.298511	−	0.954406i	$-0.596490\pi$
$101$		−	19.1833i		−	1.90881i	0.298511	−	0.954406i	$-0.403510\pi$
$102$		0			0
$103$		0			0		1.00000			$0$
$103$		0			0		−1.00000			$\pi$
$104$			14.2782i			1.40010i
$105$		0			0
$106$		0			0
$107$		0			0			−	1.00000i	$-0.5\pi$
$107$		0			0				1.00000i	$0.5\pi$
$108$	2.24797	−	5.39043i	0.216311	−	0.518695i
$109$		0			0		1.00000			$0$
$109$		0			0		−1.00000			$\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		0			0			−	1.00000i	$-0.5\pi$
$113$		0			0				1.00000i	$0.5\pi$
$114$		0			0
$115$		0			0
$116$		−	9.47562i		−	0.879789i
$117$	14.1445	+	3.81428i	1.30766	+	0.352630i
$118$	8.97739			0.826436
$119$		0			0
$120$		0			0
$121$	−11.0000			−1.00000
$122$		0			0
$123$	−20.9038	−	2.76905i	−1.88483	−	0.249677i
$124$	12.5113			1.12355
$125$		0			0
$126$		0			0
$127$	8.40180			0.745539			0.372769	−	0.927924i	$-0.378408\pi$
$127$	8.40180			0.745539			0.372769	+	0.927924i	$0.378408\pi$
$128$		−	5.14396i		−	0.454666i
$129$		0			0
$130$		0			0
$131$		−	2.81465i		−	0.245917i	−0.992412	−	0.122958i	$-0.960762\pi$
$131$		−	2.81465i		−	0.245917i	0.992412	−	0.122958i	$-0.0392382\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0			−	1.00000i	$-0.5\pi$
$137$		0			0				1.00000i	$0.5\pi$
$138$	−1.02096	+	7.70732i	−0.0869101	+	0.656091i
$139$	−20.8977			−1.77252			−0.886261	−	0.463186i	$-0.846706\pi$
$139$	−20.8977			−1.77252			−0.886261	+	0.463186i	$0.846706\pi$
$140$		0			0
$141$	−11.2612	−	1.49174i	−0.948368	−	0.125627i
$142$	−13.1466			−1.10324
$143$		0			0
$144$	1.41552	+	0.381717i	0.117960	+	0.0318097i
$145$		0			0
$146$			7.12515i			0.589681i
$147$	−1.59216	+	12.0194i	−0.131319	+	0.991340i
$148$		0			0
$149$		0			0			−	1.00000i	$-0.5\pi$
$149$		0			0				1.00000i	$0.5\pi$
$150$	−8.03544	−	1.06443i	−0.656091	−	0.0869101i
$151$	13.8606			1.12796			0.563982	−	0.825787i	$-0.309269\pi$
$151$	13.8606			1.12796			0.563982	+	0.825787i	$0.309269\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$	1.24842	−	9.42437i	0.0999532	−	0.754554i
$157$		0			0		1.00000			$0$
$157$		0			0		−1.00000			$\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$	4.23514	−	7.28155i	0.332744	−	0.572093i
$163$	−23.6272			−1.85062			−0.925311	−	0.379210i	$-0.876196\pi$
$163$	−23.6272			−1.85062			−0.925311	+	0.379210i	$0.876196\pi$
$164$			13.6836i			1.06851i
$165$		0			0
$166$		0			0
$167$			9.59166i			0.742225i	0.928588	+	0.371113i	$0.121024\pi$
$167$			9.59166i			0.742225i	−0.928588	+	0.371113i	$0.878976\pi$
$168$		0			0
$169$	10.8462			0.834321
$170$		0			0
$171$		0			0
$172$		0			0
$173$		−	19.1833i		−	1.45848i	−0.684257	−	0.729241i	$-0.739873\pi$
$173$		−	19.1833i		−	1.45848i	0.684257	−	0.729241i	$-0.260127\pi$
$174$	1.79471	−	13.5484i	0.136056	−	1.02710i
$175$		0			0
$176$		0			0
$177$	−16.4694	−	2.18164i	−1.23791	−	0.163982i
$178$		0			0
$179$			17.7900i			1.32968i	0.746984	+	0.664842i	$0.231501\pi$
$179$			17.7900i			1.32968i	−0.746984	+	0.664842i	$0.768499\pi$
$180$		0			0
$181$		0			0		1.00000			$0$
$181$		0			0		−1.00000			$\pi$
$182$		0			0
$183$		0			0
$184$	14.0226			1.03376
$185$		0			0
$186$	17.8888	+	2.36968i	1.31167	+	0.173753i
$187$		0			0
$188$			7.37162i			0.537631i
$189$		0			0
$190$		0			0
$191$		0			0			−	1.00000i	$-0.5\pi$
$191$		0			0				1.00000i	$0.5\pi$
$192$	1.36985	−	10.3411i	0.0988608	−	0.746307i
$193$	27.1457			1.95399			0.976995	−	0.213262i	$-0.0684089\pi$
$193$	27.1457			1.95399			0.976995	+	0.213262i	$0.0684089\pi$
$194$		0			0
$195$		0			0
$196$	7.86788			0.561992
$197$			0.942731i			0.0671668i	0.999436	+	0.0335834i	$0.0106919\pi$
$197$			0.942731i			0.0671668i	−0.999436	+	0.0335834i	$0.989308\pi$
$198$		0			0
$199$		0			0		1.00000			$0$
$199$		0			0		−1.00000			$\pi$
$200$			14.6196i			1.03376i
$201$		0			0
$202$	−17.9548			−1.26329
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$	3.74599	−	13.8913i	0.260365	−	0.965510i
$208$	2.38642			0.165469
$209$		0			0
$210$		0			0
$211$	−4.00000			−0.275371			−0.137686	−	0.990476i	$-0.543966\pi$
$211$	−4.00000			−0.275371			−0.137686	+	0.990476i	$0.543966\pi$
$212$		0			0
$213$	24.1179	+	3.19482i	1.65253	+	0.218906i
$214$		0			0
$215$		0			0
$216$	−14.0226	−	5.84783i	−0.954118	−	0.397895i
$217$		0			0
$218$		0			0
$219$	1.73152	−	13.0714i	0.117005	−	0.883281i
$220$		0			0
$221$		0			0
$222$		0			0
$223$	8.00000			0.535720			0.267860	−	0.963458i	$-0.413684\pi$
$223$	8.00000			0.535720			0.267860	+	0.963458i	$0.413684\pi$
$224$		0			0
$225$	14.4827	+	3.90547i	0.965510	+	0.260365i
$226$		0			0
$227$		0			0			−	1.00000i	$-0.5\pi$
$227$		0			0				1.00000i	$0.5\pi$
$228$		0			0
$229$		0			0		1.00000			$0$
$229$		0			0		−1.00000			$\pi$
$230$		0			0
$231$		0			0
$232$	−24.6498			−1.61834
$233$		−	29.0350i		−	1.90215i	−0.308965	−	0.951073i	$-0.599983\pi$
$233$		−	29.0350i		−	1.90215i	0.308965	−	0.951073i	$-0.400017\pi$
$234$	3.57000	−	13.2387i	0.233378	−	0.865438i
$235$		0			0
$236$			10.7809i			0.701775i
$237$		0			0
$238$		0			0
$239$			27.1631i			1.75703i	0.477711	+	0.878517i	$0.341467\pi$
$239$			27.1631i			1.75703i	−0.477711	+	0.878517i	$0.658533\pi$
$240$		0			0
$241$		0			0		1.00000			$0$
$241$		0			0		−1.00000			$\pi$
$242$			10.2955i			0.661822i
$243$	−9.53906	+	12.3291i	−0.611931	+	0.790911i
$244$		0			0
$245$		0			0
$246$	−2.59172	+	19.5650i	−0.165242	+	1.24742i
$247$		0			0
$248$		−	32.5468i		−	2.06672i
$249$		0			0
$250$		0			0
$251$		0			0			−	1.00000i	$-0.5\pi$
$251$		0			0				1.00000i	$0.5\pi$
$252$		0			0
$253$		0			0
$254$		−	7.86373i		−	0.493414i
$255$		0			0
$256$	−16.8597			−1.05373
$257$		−	19.6619i		−	1.22647i	−0.789899	−	0.613237i	$-0.789867\pi$
$257$		−	19.6619i		−	1.22647i	0.789899	−	0.613237i	$-0.210133\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	−6.58493	+	24.4189i	−0.407597	+	1.51149i
$262$	−2.63439			−0.162753
$263$		0			0			−	1.00000i	$-0.5\pi$
$263$		0			0				1.00000i	$0.5\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$			32.7788i			1.99856i	0.0379247	+	0.999281i	$0.487925\pi$
$269$			32.7788i			1.99856i	−0.0379247	+	0.999281i	$0.512075\pi$
$270$		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
$273$		0			0
$274$		0			0
$275$		0			0
$276$	−9.25565	−	1.22607i	−0.557125	−	0.0738005i
$277$	29.8751			1.79502			0.897511	−	0.440992i	$-0.145373\pi$
$277$	29.8751			1.79502			0.897511	+	0.440992i	$0.145373\pi$
$278$			19.5594i			1.17309i
$279$	−32.2419	−	8.69453i	−1.93027	−	0.520528i
$280$		0			0
$281$		0			0			−	1.00000i	$-0.5\pi$
$281$		0			0				1.00000i	$0.5\pi$
$282$	−1.39620	+	10.5400i	−0.0831428	+	0.627651i
$283$		0			0		1.00000			$0$
$283$		0			0		−1.00000			$\pi$
$284$		−	15.7876i		−	0.936823i
$285$		0			0
$286$		0			0
$287$		0			0
$288$	−4.21043	+	15.6136i	−0.248102	+	0.920038i
$289$	−17.0000			−1.00000
$290$		0			0
$291$		0			0
$292$	−8.55652			−0.500733
$293$		0			0			−	1.00000i	$-0.5\pi$
$293$		0			0				1.00000i	$0.5\pi$
$294$	11.2496	+	1.49020i	0.656091	+	0.0869101i
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		−	23.4193i		−	1.35437i
$300$	1.27826	−	9.64968i	0.0738005	−	0.557125i
$301$		0			0
$302$		−	12.9730i		−	0.746511i
$303$	32.9388	+	4.36329i	1.89228	+	0.250664i
$304$		0			0
$305$		0			0
$306$		0			0
$307$	20.0000			1.14146			0.570730	−	0.821138i	$-0.306660\pi$
$307$	20.0000			1.14146			0.570730	+	0.821138i	$0.306660\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		−	34.6508i		−	1.96486i	−0.186621	−	0.982432i	$-0.559754\pi$
$311$		−	34.6508i		−	1.96486i	0.186621	−	0.982432i	$-0.440246\pi$
$312$	−24.5164	−	3.24761i	−1.38797	−	0.183860i
$313$		0			0		1.00000			$0$
$313$		0			0		−1.00000			$\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	19.1833i		−	1.07744i	−0.842484	−	0.538721i	$-0.818908\pi$
$317$		−	19.1833i		−	1.07744i	0.842484	−	0.538721i	$-0.181092\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$	8.74435	+	5.08594i	0.485797	+	0.282552i
$325$	24.4163			1.35437
$326$			22.1140i			1.22478i
$327$		0			0
$328$	35.5964			1.96548
$329$		0			0
$330$		0			0
$331$	−18.1683			−0.998620			−0.499310	−	0.866423i	$-0.666413\pi$
$331$	−18.1683			−0.998620			−0.499310	+	0.866423i	$0.666413\pi$
$332$		0			0
$333$		0			0
$334$	8.97739			0.491221
$335$		0			0
$336$		0			0
$337$		0			0		1.00000			$0$
$337$		0			0		−1.00000			$\pi$
$338$		−	10.1516i		−	0.552172i
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$	−17.9548			−0.965255
$347$			9.59166i			0.514907i	0.966291	+	0.257454i	$0.0828835\pi$
$347$			9.59166i			0.514907i	−0.966291	+	0.257454i	$0.917117\pi$
$348$	16.2701	+	2.15525i	0.872170	+	0.115533i
$349$	−10.3421			−0.553600			−0.276800	−	0.960928i	$-0.589274\pi$
$349$	−10.3421			−0.553600			−0.276800	+	0.960928i	$0.589274\pi$
$350$		0			0
$351$	−9.76651	+	23.4193i	−0.521298	+	1.25003i
$352$		0			0
$353$			21.5473i			1.14685i	0.819258	+	0.573425i	$0.194386\pi$
$353$			21.5473i			1.14685i	−0.819258	+	0.573425i	$0.805614\pi$
$354$	−2.04193	+	15.4146i	−0.108527	+	0.819279i
$355$		0			0
$356$		0			0
$357$		0			0
$358$	16.6507			0.880015
$359$		0			0			−	1.00000i	$-0.5\pi$
$359$		0			0				1.00000i	$0.5\pi$
$360$		0			0
$361$	19.0000			1.00000
$362$		0			0
$363$	2.50197	−	18.8876i	0.131319	−	0.991340i
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0		1.00000			$0$
$367$		0			0		−1.00000			$\pi$
$368$		−	2.34370i		−	0.122174i
$369$	9.50921	−	35.2630i	0.495030	−	1.83572i
$370$		0			0
$371$		0			0
$372$	−2.84572	+	21.4826i	−0.147544	+	1.11382i
$373$		0			0		1.00000			$0$
$373$		0			0		−1.00000			$\pi$
$374$		0			0
$375$		0			0
$376$	19.1764			0.988949
$377$			41.1678i			2.12025i
$378$		0			0
$379$		0			0		1.00000			$0$
$379$		0			0		−1.00000			$\pi$
$380$		0			0
$381$	−1.91101	+	14.4263i	−0.0979038	+	0.739083i
$382$		0			0
$383$		0			0			−	1.00000i	$-0.5\pi$
$383$		0			0				1.00000i	$0.5\pi$
$384$	8.83244	+	1.17000i	0.450729	+	0.0597065i
$385$		0			0
$386$		−	25.4072i		−	1.29319i
$387$		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
$392$		−	20.4674i		−	1.03376i
$393$	4.83289	+	0.640197i	0.243787	+	0.0322937i
$394$	0.882357			0.0444525
$395$		0			0
$396$		0			0
$397$	−39.6416			−1.98956			−0.994778	−	0.102061i	$-0.967456\pi$
$397$	−39.6416			−1.98956			−0.994778	+	0.102061i	$0.967456\pi$
$398$		0			0
$399$		0			0
$400$	2.44348			0.122174
$401$		0			0			−	1.00000i	$-0.5\pi$
$401$		0			0				1.00000i	$0.5\pi$
$402$		0			0
$403$	−54.3566			−2.70769
$404$		−	21.5617i		−	1.07274i
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−36.9122			−1.82519			−0.912595	−	0.408864i	$-0.865925\pi$
$409$	−36.9122			−1.82519			−0.912595	+	0.408864i	$0.865925\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$	−13.0016	−	3.50609i	−0.638996	−	0.172315i
$415$		0			0
$416$			26.3229i			1.29058i
$417$	4.75323	−	35.8825i	0.232767	−	1.75717i
$418$		0			0
$419$		0			0			−	1.00000i	$-0.5\pi$
$419$		0			0				1.00000i	$0.5\pi$
$420$		0			0
$421$		0			0		1.00000			$0$
$421$		0			0		−1.00000			$\pi$
$422$			3.74383i			0.182247i
$423$	5.12279	−	18.9968i	0.249078	−	0.923658i
$424$		0			0
$425$		0			0
$426$	2.99022	−	22.5734i	0.144877	−	1.09368i
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0			−	1.00000i	$-0.5\pi$
$431$		0			0				1.00000i	$0.5\pi$
$432$	−0.977391	+	2.34370i	−0.0470247	+	0.112761i
$433$		0			0		1.00000			$0$
$433$		0			0		−1.00000			$\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$	−12.2342	−	1.62063i	−0.584575	−	0.0774366i
$439$	5.67237			0.270728			0.135364	−	0.990796i	$-0.456780\pi$
$439$	5.67237			0.270728			0.135364	+	0.990796i	$0.456780\pi$
$440$		0			0
$441$	−20.2757	−	5.46766i	−0.965510	−	0.260365i
$442$		0			0
$443$			38.3946i			1.82418i	0.409988	+	0.912091i	$0.365533\pi$
$443$			38.3946i			1.82418i	−0.409988	+	0.912091i	$0.634467\pi$
$444$		0			0
$445$		0			0
$446$		−	7.48766i		−	0.354551i
$447$		0			0
$448$		0			0
$449$			38.3667i			1.81063i	0.424736	+	0.905317i	$0.360367\pi$
$449$			38.3667i			1.81063i	−0.424736	+	0.905317i	$0.639633\pi$
$450$	3.65535	−	13.5552i	0.172315	−	0.638996i
$451$		0			0
$452$		0			0
$453$	−3.15263	+	23.7994i	−0.148124	+	1.11820i
$454$		0			0
$455$		0			0
$456$		0			0
$457$		0			0		1.00000			$0$
$457$		0			0		−1.00000			$\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$			23.4057i			1.09011i	0.838399	+	0.545056i	$0.183492\pi$
$461$			23.4057i			1.09011i	−0.838399	+	0.545056i	$0.816508\pi$
$462$		0			0
$463$	32.0000			1.48717			0.743583	−	0.668644i	$-0.233125\pi$
$463$	32.0000			1.48717			0.743583	+	0.668644i	$0.233125\pi$
$464$			4.11989i			0.191261i
$465$		0			0
$466$	−27.1755			−1.25888
$467$		0			0			−	1.00000i	$-0.5\pi$
$467$		0			0				1.00000i	$0.5\pi$
$468$	15.8982	+	4.28719i	0.734894	+	0.198175i
$469$		0			0
$470$		0			0
$471$		0			0
$472$	28.0452			1.29089
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$		0			0
$478$	25.4235			1.16284
$479$		0			0			−	1.00000i	$-0.5\pi$
$479$		0			0				1.00000i	$0.5\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$	−12.3638			−0.561992
$485$		0			0
$486$	11.5395	+	8.92815i	0.523443	+	0.404989i
$487$	16.5901			0.751768			0.375884	−	0.926667i	$-0.377339\pi$
$487$	16.5901			0.751768			0.375884	+	0.926667i	$0.377339\pi$
$488$		0			0
$489$	5.37404	−	40.5690i	0.243023	−	1.83460i
$490$		0			0
$491$		−	4.67301i		−	0.210890i	−0.994425	−	0.105445i	$-0.966373\pi$
$491$		−	4.67301i		−	0.210890i	0.994425	−	0.105445i	$-0.0336267\pi$
$492$	−23.4955	−	3.11237i	−1.05926	−	0.140316i
$493$		0			0
$494$		0			0
$495$		0			0
$496$	−5.43978			−0.244253
$497$		0			0
$498$		0			0
$499$	43.1602			1.93211			0.966057	−	0.258328i	$-0.0831715\pi$
$499$	43.1602			1.93211			0.966057	+	0.258328i	$0.0831715\pi$
$500$		0			0
$501$	−16.4694	−	2.18164i	−0.735798	−	0.0974686i
$502$		0			0
$503$		0			0			−	1.00000i	$-0.5\pi$
$503$		0			0				1.00000i	$0.5\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−2.46698	+	18.6234i	−0.109563	+	0.827096i
$508$	9.44348			0.418987
$509$		−	17.8035i		−	0.789127i	−0.918869	−	0.394564i	$-0.870896\pi$
$509$		−	17.8035i		−	0.789127i	0.918869	−	0.394564i	$-0.129104\pi$
$510$		0			0
$511$		0			0
$512$			5.49209i			0.242718i
$513$		0			0
$514$	−18.4027			−0.811708
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$	32.9388	+	4.36329i	1.44585	+	0.191527i
$520$		0			0
$521$		0			0			−	1.00000i	$-0.5\pi$
$521$		0			0				1.00000i	$0.5\pi$
$522$	22.8551	+	6.16321i	1.00034	+	0.269757i
$523$		0			0		1.00000			$0$
$523$		0			0		−1.00000			$\pi$
$524$		−	3.16362i		−	0.138203i
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−23.0000			−1.00000
$530$		0			0
$531$	7.49199	−	27.7826i	0.325125	−	1.20566i
$532$		0			0
$533$		−	59.4498i		−	2.57506i
$534$		0			0
$535$		0			0
$536$		0			0
$537$	−30.5463	−	4.04637i	−1.31817	−	0.174614i
$538$	30.6796			1.32269
$539$		0			0
$540$		0			0
$541$	0.575595			0.0247468			0.0123734	−	0.999923i	$-0.496061\pi$
$541$	0.575595			0.0247468			0.0123734	+	0.999923i	$0.496061\pi$
$542$			14.9753i			0.643245i
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	45.8896			1.96210			0.981049	−	0.193761i	$-0.0620688\pi$
$547$	45.8896			1.96210			0.981049	+	0.193761i	$0.0620688\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		0			0
$552$	−3.18947	+	24.0775i	−0.135753	+	1.02481i
$553$		0			0
$554$		−	27.9618i		−	1.18799i
$555$		0			0
$556$	−23.4887			−0.996143
$557$		0			0			−	1.00000i	$-0.5\pi$
$557$		0			0				1.00000i	$0.5\pi$
$558$	−8.13771	+	30.1771i	−0.344497	+	1.27750i
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0			−	1.00000i	$-0.5\pi$
$563$		0			0				1.00000i	$0.5\pi$
$564$	−12.6574	−	1.67669i	−0.532975	−	0.0706014i
$565$		0			0
$566$		0			0
$567$		0			0
$568$	−41.0697			−1.72325
$569$		0			0			−	1.00000i	$-0.5\pi$
$569$		0			0				1.00000i	$0.5\pi$
$570$		0			0
$571$		0			0		1.00000			$0$
$571$		0			0		−1.00000			$\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		−	23.9792i		−	1.00000i
$576$	17.4447	+	4.70422i	0.726861	+	0.196009i
$577$	32.6045			1.35734			0.678672	−	0.734441i	$-0.262556\pi$
$577$	32.6045			1.35734			0.678672	+	0.734441i	$0.262556\pi$
$578$			15.9113i			0.661822i
$579$	−6.17434	+	46.6106i	−0.256597	+	1.93707i
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$			22.2588i			0.921077i
$585$		0			0
$586$		0			0
$587$		−	25.2776i		−	1.04332i	−0.853154	−	0.521660i	$-0.825313\pi$
$587$		−	25.2776i		−	1.04332i	0.853154	−	0.521660i	$-0.174687\pi$
$588$	−1.78957	+	13.5096i	−0.0738005	+	0.557125i
$589$		0			0
$590$		0			0
$591$	−1.61872	−	0.214426i	−0.0665852	−	0.00882032i
$592$		0			0
$593$			38.3667i			1.57553i	0.615976	+	0.787765i	$0.288762\pi$
$593$			38.3667i			1.57553i	−0.615976	+	0.787765i	$0.711238\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$	−21.9194			−0.896353
$599$			9.59166i			0.391905i	0.980613	+	0.195952i	$0.0627798\pi$
$599$			9.59166i			0.391905i	−0.980613	+	0.195952i	$0.937220\pi$
$600$	−25.1026	−	3.32525i	−1.02481	−	0.135753i
$601$	−42.3711			−1.72835			−0.864176	−	0.503190i	$-0.832159\pi$
$601$	−42.3711			−1.72835			−0.864176	+	0.503190i	$0.832159\pi$
$602$		0			0
$603$		0			0
$604$	15.5791			0.633906
$605$		0			0
$606$	4.08385	−	30.8293i	0.165895	−	1.25235i
$607$	−40.0000			−1.62355			−0.811775	−	0.583970i	$-0.801498\pi$
$607$	−40.0000			−1.62355			−0.811775	+	0.583970i	$0.801498\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		−	32.0267i		−	1.29566i
$612$		0			0
$613$		0			0		1.00000			$0$
$613$		0			0		−1.00000			$\pi$
$614$		−	18.7192i		−	0.755444i
$615$		0			0
$616$		0			0
$617$		0			0			−	1.00000i	$-0.5\pi$
$617$		0			0				1.00000i	$0.5\pi$
$618$		0			0
$619$		0			0		1.00000			$0$
$619$		0			0		−1.00000			$\pi$
$620$		0			0
$621$	23.0000	+	9.59166i	0.922958	+	0.384900i
$622$	−32.4316			−1.30039
$623$		0			0
$624$	−0.542797	+	4.09761i	−0.0217293	+	0.164036i
$625$	25.0000			1.00000
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$		0			0		1.00000			$0$
$631$		0			0		−1.00000			$\pi$
$632$		0			0
$633$	0.909808	−	6.86821i	0.0361616	−	0.272987i
$634$	−17.9548			−0.713075
$635$		0			0
$636$		0			0
$637$	−34.1828			−1.35437
$638$		0			0
$639$	−10.9714	+	40.6851i	−0.434020	+	1.60948i
$640$		0			0
$641$		0			0			−	1.00000i	$-0.5\pi$
$641$		0			0				1.00000i	$0.5\pi$
$642$		0			0
$643$		0			0		1.00000			$0$
$643$		0			0		−1.00000			$\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			47.7677i			1.87794i	0.343996	+	0.938971i	$0.388219\pi$
$647$			47.7677i			1.87794i	−0.343996	+	0.938971i	$0.611781\pi$
$648$	13.2305	−	22.7474i	0.519743	−	0.893604i
$649$		0			0
$650$		−	22.8526i		−	0.896353i
$651$		0			0
$652$	−26.5565			−1.04003
$653$		−	49.6396i		−	1.94255i	−0.237962	−	0.971274i	$-0.576480\pi$
$653$		−	49.6396i		−	1.94255i	0.237962	−	0.971274i	$-0.423520\pi$
$654$		0			0
$655$		0			0
$656$		−	5.94949i		−	0.232288i
$657$	22.0504	+	5.94622i	0.860267	+	0.231984i
$658$		0			0
$659$		0			0			−	1.00000i	$-0.5\pi$
$659$		0			0				1.00000i	$0.5\pi$
$660$		0			0
$661$		0			0		1.00000			$0$
$661$		0			0		−1.00000			$\pi$
$662$			17.0048i			0.660909i
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	40.4307			1.56548
$668$			10.7809i			0.417124i
$669$	−1.81962	+	13.7364i	−0.0703504	+	0.531080i
$670$		0			0
$671$		0			0
$672$		0			0
$673$	21.6868			0.835966			0.417983	−	0.908455i	$-0.362737\pi$
$673$	21.6868			0.835966			0.417983	+	0.908455i	$0.362737\pi$
$674$		0			0
$675$	−10.0000	+	23.9792i	−0.384900	+	0.922958i
$676$	12.1909			0.468881
$677$		0			0			−	1.00000i	$-0.5\pi$
$677$		0			0				1.00000i	$0.5\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$		−	44.0239i		−	1.68453i	−0.539066	−	0.842263i	$-0.681223\pi$
$683$		−	44.0239i		−	1.68453i	0.539066	−	0.842263i	$-0.318777\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$		0			0
$689$		0			0
$690$		0			0
$691$	44.0000			1.67384			0.836919	−	0.547326i	$-0.184354\pi$
$691$	44.0000			1.67384			0.836919	+	0.547326i	$0.184354\pi$
$692$		−	21.5617i		−	0.819654i
$693$		0			0
$694$	8.97739			0.340777
$695$		0			0
$696$	5.60664	−	42.3249i	0.212519	−	1.60432i
$697$		0			0
$698$			9.67977i			0.366385i
$699$	49.8546	+	6.60407i	1.88567	+	0.249789i
$700$		0			0
$701$		0			0			−	1.00000i	$-0.5\pi$
$701$		0			0				1.00000i	$0.5\pi$
$702$	21.9194	+	9.14104i	0.827296	+	0.345006i
$703$		0			0
$704$		0			0
$705$		0			0
$706$	20.1674			0.759010
$707$		0			0
$708$	−18.5113	−	2.45213i	−0.695697	−	0.0921567i
$709$		0			0		1.00000			$0$
$709$		0			0		−1.00000			$\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$			53.3835i			1.99923i
$714$		0			0
$715$		0			0
$716$			19.9956i			0.747272i
$717$	−46.6404	−	6.17830i	−1.74182	−	0.230733i
$718$		0			0
$719$		−	47.9583i		−	1.78854i	−0.447524	−	0.894272i	$-0.647694\pi$
$719$		−	47.9583i		−	1.78854i	0.447524	−	0.894272i	$-0.352306\pi$
$720$		0			0
$721$		0			0
$722$		−	17.7832i		−	0.661822i
$723$		0			0
$724$		0			0
$725$			42.1520i			1.56548i
$726$	−17.6780	−	2.34174i	−0.656091	−	0.0869101i
$727$		0			0		1.00000			$0$
$727$		0			0		−1.00000			$\pi$
$728$		0			0
$729$	−19.0000	−	19.1833i	−0.703704	−	0.710494i
$730$		0			0
$731$		0			0
$732$		0			0
$733$		0			0		1.00000			$0$
$733$		0			0		−1.00000			$\pi$
$734$		0			0
$735$		0			0
$736$	25.8516			0.952903
$737$		0			0
$738$	−33.0047	−	8.90022i	−1.21492	−	0.327622i
$739$	−15.4389			−0.567928			−0.283964	−	0.958835i	$-0.591650\pi$
$739$	−15.4389			−0.567928			−0.283964	+	0.958835i	$0.591650\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0			−	1.00000i	$-0.5\pi$
$743$		0			0				1.00000i	$0.5\pi$
$744$	55.8845	+	7.40283i	2.04882	+	0.271401i
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$		0			0		1.00000			$0$
$751$		0			0		−1.00000			$\pi$
$752$		−	3.20510i		−	0.116878i
$753$		0			0
$754$	38.5313			1.40323
$755$		0			0
$756$		0			0
$757$		0			0		1.00000			$0$
$757$		0			0		−1.00000			$\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$			44.0103i			1.59537i	0.603072	+	0.797687i	$0.293943\pi$
$761$			44.0103i			1.59537i	−0.603072	+	0.797687i	$0.706057\pi$
$762$	13.5024	+	1.78862i	0.489141	+	0.0647949i
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$		−	46.8385i		−	1.69124i
$768$	3.83478	−	28.9491i	0.138376	−	1.04461i
$769$		0			0		1.00000			$0$
$769$		0			0		−1.00000			$\pi$
$770$		0			0
$771$	33.7605	+	4.47214i	1.21585	+	0.161060i
$772$	30.5113			1.09813
$773$		0			0			−	1.00000i	$-0.5\pi$
$773$		0			0				1.00000i	$0.5\pi$
$774$		0			0
$775$	−55.6561			−1.99923
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$	−40.4307	−	16.8608i	−1.44488	−	0.602555i
$784$	−3.42087			−0.122174
$785$		0			0
$786$	0.599197	−	4.52338i	0.0213727	−	0.161344i
$787$		0			0		1.00000			$0$
$787$		0			0		−1.00000			$\pi$
$788$			1.05961i			0.0377472i
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$			37.1029i			1.31673i
$795$		0			0
$796$		0			0
$797$		0			0			−	1.00000i	$-0.5\pi$
$797$		0			0				1.00000i	$0.5\pi$
$798$		0			0
$799$		0			0
$800$			26.9522i			0.952903i
$801$		0			0
$802$		0			0
$803$		0			0
$804$		0			0
$805$		0			0
$806$			50.8755i			1.79201i
$807$	−56.2830	−	7.45561i	−1.98125	−	0.262450i
$808$	−56.0904			−1.97325
$809$			38.3667i			1.34890i	0.738321	+	0.674450i	$0.235619\pi$
$809$			38.3667i			1.34890i	−0.738321	+	0.674450i	$0.764381\pi$
$810$		0			0
$811$	48.6190			1.70724			0.853622	−	0.520892i	$-0.174401\pi$
$811$	48.6190			1.70724			0.853622	+	0.520892i	$0.174401\pi$
$812$		0			0
$813$	3.63923	−	27.4728i	0.127633	−	0.963514i
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$			34.5483i			1.20795i
$819$		0			0
$820$		0			0
$821$		−	19.1833i		−	0.669503i	−0.942306	−	0.334751i	$-0.891348\pi$
$821$		−	19.1833i		−	0.669503i	0.942306	−	0.334751i	$-0.108652\pi$
$822$		0			0
$823$	−52.9267			−1.84491			−0.922454	−	0.386107i	$-0.873820\pi$
$823$	−52.9267			−1.84491			−0.922454	+	0.386107i	$0.873820\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0			−	1.00000i	$-0.5\pi$
$827$		0			0				1.00000i	$0.5\pi$
$828$	4.21043	−	15.6136i	0.146323	−	0.542609i
$829$	2.00000			0.0694629			0.0347314	−	0.999397i	$-0.488942\pi$
$829$	2.00000			0.0694629			0.0347314	+	0.999397i	$0.488942\pi$
$830$		0			0
$831$	−6.79516	+	51.2971i	−0.235721	+	1.77948i
$832$	29.4099			1.01961
$833$		0			0
$834$	−33.5845	−	4.44882i	−1.16294	−	0.154050i
$835$		0			0
$836$		0			0
$837$	22.2624	−	53.3835i	0.769503	−	1.84520i
$838$		0			0
$839$		0			0			−	1.00000i	$-0.5\pi$
$839$		0			0				1.00000i	$0.5\pi$
$840$		0			0
$841$	−42.0715			−1.45074
$842$		0			0
$843$		0			0
$844$	−4.49593			−0.154756
$845$		0			0
$846$	−17.7802	−	4.79471i	−0.611297	−	0.164846i
$847$		0			0
$848$		0			0
$849$		0			0
$850$		0			0
$851$		0			0
$852$	27.1082	+	3.59093i	0.928710	+	0.123023i
$853$	−10.0000			−0.342393			−0.171197	−	0.985237i	$-0.554763\pi$
$853$	−10.0000			−0.342393			−0.171197	+	0.985237i	$0.554763\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	38.4081i		−	1.31200i	−0.754762	−	0.655998i	$-0.772248\pi$
$857$		−	38.4081i		−	1.31200i	0.754762	−	0.655998i	$-0.227752\pi$
$858$		0			0
$859$	−29.0860			−0.992402			−0.496201	−	0.868208i	$-0.665272\pi$
$859$	−29.0860			−0.992402			−0.496201	+	0.868208i	$0.665272\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	12.1878i		−	0.414877i	−0.978248	−	0.207438i	$-0.933487\pi$
$863$		−	12.1878i		−	0.414877i	0.978248	−	0.207438i	$-0.0665126\pi$
$864$	−25.8516	−	10.7809i	−0.879490	−	0.366773i
$865$		0			0
$866$		0			0
$867$	3.86668	−	29.1899i	0.131319	−	0.991340i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$	1.94620	−	14.6920i	0.0657560	−	0.496396i
$877$	14.0000			0.472746			0.236373	−	0.971662i	$-0.424041\pi$
$877$	14.0000			0.472746			0.236373	+	0.971662i	$0.424041\pi$
$878$		−	5.30910i		−	0.179173i
$879$		0			0
$880$		0			0
$881$		0			0			−	1.00000i	$-0.5\pi$
$881$		0			0				1.00000i	$0.5\pi$
$882$	−5.11749	+	18.9772i	−0.172315	+	0.638996i
$883$	−52.0000			−1.74994			−0.874970	−	0.484178i	$-0.839119\pi$
$883$	−52.0000			−1.74994			−0.874970	+	0.484178i	$0.839119\pi$
$884$		0			0
$885$		0			0
$886$	35.9357			1.20728
$887$			29.0215i			0.974445i	0.873278	+	0.487223i	$0.161990\pi$
$887$			29.0215i			0.974445i	−0.873278	+	0.487223i	$0.838010\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$	8.99187			0.301070
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$	40.2121	+	5.32676i	1.34264	+	0.177855i
$898$	35.9096			1.19832
$899$		−	93.8406i		−	3.12976i
$900$	16.2783	+	4.38968i	0.542609	+	0.146323i
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$	22.2753	+	2.95073i	0.740046	+	0.0980314i
$907$		0			0		1.00000			$0$
$907$		0			0		−1.00000			$\pi$
$908$		0			0
$909$	−14.9840	+	55.5651i	−0.496987	+	1.84298i
$910$		0			0
$911$		0			0			−	1.00000i	$-0.5\pi$
$911$		0			0				1.00000i	$0.5\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$		0			0		1.00000			$0$
$919$		0			0		−1.00000			$\pi$
$920$		0			0
$921$	−4.54904	+	34.3410i	−0.149896	+	1.13158i
$922$	21.9068			0.721461
$923$			68.5909i			2.25770i
$924$		0			0
$925$		0			0
$926$		−	29.9506i		−	0.984239i
$927$		0			0
$928$	−45.4435			−1.49176
$929$		−	10.2888i		−	0.337563i	−0.985653	−	0.168782i	$-0.946017\pi$
$929$		−	10.2888i		−	0.337563i	0.985653	−	0.168782i	$-0.0539833\pi$
$930$		0			0
$931$		0			0
$932$		−	32.6349i		−	1.06899i
$933$	59.4971	+	7.88139i	1.94785	+	0.258025i
$934$		0			0
$935$		0			0
$936$	11.1526	−	41.3573i	0.364535	−	1.35181i
$937$		0			0		1.00000			$0$
$937$		0			0		−1.00000			$\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		0			0			−	1.00000i	$-0.5\pi$
$941$		0			0				1.00000i	$0.5\pi$
$942$		0			0
$943$	−58.3855			−1.90129
$944$		−	4.68740i		−	0.152562i
$945$		0			0
$946$		0			0
$947$			36.5362i			1.18727i	0.804735	+	0.593634i	$0.202307\pi$
$947$			36.5362i			1.18727i	−0.804735	+	0.593634i	$0.797693\pi$
$948$		0			0
$949$	37.1746			1.20674
$950$		0			0
$951$	32.9388	+	4.36329i	1.06811	+	0.141489i
$952$		0			0
$953$		0			0			−	1.00000i	$-0.5\pi$
$953$		0			0				1.00000i	$0.5\pi$
$954$		0			0
$955$		0			0
$956$			30.5309i			0.987439i
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	92.9041			2.99691
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$	2.94295			0.0946388			0.0473194	−	0.998880i	$-0.484932\pi$
$967$	2.94295			0.0946388			0.0473194	+	0.998880i	$0.484932\pi$
$968$			32.1631i			1.03376i
$969$		0			0
$970$		0			0
$971$		0			0			−	1.00000i	$-0.5\pi$
$971$		0			0				1.00000i	$0.5\pi$
$972$	−10.7217	+	13.8577i	−0.343900	+	0.444486i
$973$		0			0
$974$		−	15.5276i		−	0.497537i
$975$	−5.55353	+	41.9240i	−0.177855	+	1.34264i
$976$		0			0
$977$		0			0			−	1.00000i	$-0.5\pi$
$977$		0			0				1.00000i	$0.5\pi$
$978$	−37.9709	−	5.02988i	−1.21418	−	0.160838i
$979$		0			0
$980$		0			0
$981$		0			0
$982$	−4.37374			−0.139572
$983$		0			0			−	1.00000i	$-0.5\pi$
$983$		0			0				1.00000i	$0.5\pi$
$984$	−8.09648	+	61.1209i	−0.258106	+	1.94846i
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$	56.0000			1.77890			0.889449	−	0.457034i	$-0.151088\pi$
$991$	56.0000			1.77890			0.889449	+	0.457034i	$0.151088\pi$
$992$		−	60.0021i		−	1.90507i
$993$	4.13242	−	31.1959i	0.131138	−	0.989972i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	26.0000			0.823428			0.411714	−	0.911313i	$-0.364930\pi$
$997$	26.0000			0.823428			0.411714	+	0.911313i	$0.364930\pi$
$998$		−	40.3961i		−	1.27872i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	69.2.c.a.68.3	✓	6
3.2	odd	2	inner	69.2.c.a.68.4	yes	6
4.3	odd	2		1104.2.m.a.689.3		6
12.11	even	2		1104.2.m.a.689.4		6
23.22	odd	2	CM	69.2.c.a.68.3	✓	6
69.68	even	2	inner	69.2.c.a.68.4	yes	6
92.91	even	2		1104.2.m.a.689.3		6
276.275	odd	2		1104.2.m.a.689.4		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
69.2.c.a.68.3	✓	6	1.1	even	1	trivial
69.2.c.a.68.3	✓	6	23.22	odd	2	CM
69.2.c.a.68.4	yes	6	3.2	odd	2	inner
69.2.c.a.68.4	yes	6	69.68	even	2	inner
1104.2.m.a.689.3		6	4.3	odd	2
1104.2.m.a.689.3		6	92.91	even	2
1104.2.m.a.689.4		6	12.11	even	2
1104.2.m.a.689.4		6	276.275	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 69 = 3 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	69.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-0.935958i q^{2} +(-0.227452 + 1.71705i) q^{3} +1.12398 q^{4} +(1.60709 + 0.212885i) q^{6} -2.92392i q^{8} +(-2.89653 - 0.781094i) q^{9} +O(q^{10})\)	\(q-0.935958i q^{2} +(-0.227452 + 1.71705i) q^{3} +1.12398 q^{4} +(1.60709 + 0.212885i) q^{6} -2.92392i q^{8} +(-2.89653 - 0.781094i) q^{9} +(-0.255652 + 1.92994i) q^{12} -4.88325 q^{13} -0.488695 q^{16} +(-0.731071 + 2.71103i) q^{18} +4.79583i q^{23} +(5.02051 + 0.665051i) q^{24} -5.00000 q^{25} +4.57052i q^{26} +(2.00000 - 4.79583i) q^{27} -8.43039i q^{29} +11.1312 q^{31} -5.39043i q^{32} +(-3.25565 - 0.877936i) q^{36} +(1.11071 - 8.38480i) q^{39} +12.1742i q^{41} +4.48870 q^{46} +6.55848i q^{47} +(0.111155 - 0.839115i) q^{48} +7.00000 q^{49} +4.67979i q^{50} -5.48870 q^{52} +(-4.48870 - 1.87192i) q^{54} -7.89049 q^{58} +9.59166i q^{59} -10.4184i q^{62} -6.02261 q^{64} +(-8.23469 - 1.09082i) q^{69} -14.0461i q^{71} +(-2.28385 + 8.46921i) q^{72} -7.61268 q^{73} +(1.13726 - 8.58526i) q^{75} +(-7.84782 - 1.03957i) q^{78} +(7.77979 + 4.52492i) q^{81} +11.3946 q^{82} +(14.4754 + 1.91751i) q^{87} +5.39043i q^{92} +(-2.53182 + 19.1129i) q^{93} +6.13846 q^{94} +(9.25565 + 1.22607i) q^{96} -6.55170i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 12 q^{4} + 3 q^{6}+O(q^{10}) \) 6 * q - 12 * q^4 + 3 * q^6	\( 6 q - 12 q^{4} + 3 q^{6} - 15 q^{12} + 24 q^{16} + 21 q^{18} - 6 q^{24} - 30 q^{25} + 12 q^{27} - 33 q^{36} - 24 q^{39} + 69 q^{48} + 42 q^{49} - 6 q^{52} + 30 q^{58} - 90 q^{64} - 42 q^{72} - 51 q^{78} + 66 q^{82} + 48 q^{87} - 6 q^{93} - 78 q^{94} + 69 q^{96}+O(q^{100}) \) 6 * q - 12 * q^4 + 3 * q^6 - 15 * q^12 + 24 * q^16 + 21 * q^18 - 6 * q^24 - 30 * q^25 + 12 * q^27 - 33 * q^36 - 24 * q^39 + 69 * q^48 + 42 * q^49 - 6 * q^52 + 30 * q^58 - 90 * q^64 - 42 * q^72 - 51 * q^78 + 66 * q^82 + 48 * q^87 - 6 * q^93 - 78 * q^94 + 69 * q^96

Introduction

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