LMFDB - Embedded newform 3840.2.f.m.769.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3840,2,Mod(769,3840)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3840, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("3840.769");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3840 = 2^{8} \cdot 3 \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3840.f (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:	$30.6625543762$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	12.0.180227832610816.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} + x^{10} - 8x^{6} + 16x^{2} + 64 $ x^12 + x^10 - 8x^6 + 16x^2 + 64
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{12} $
Twist minimal:	no (minimal twist has level 120)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			769.7
Root			$-0.450129 - 1.34067i$ of defining polynomial
Character	$\chi$	$=$	3840.769
Dual form			3840.2.f.m.769.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{3} +(-2.22158 - 0.254102i) q^{5} +2.64265i q^{7} -1.00000 q^{9} -1.51363 q^{11} -3.87086i q^{13} +(0.254102 - 2.22158i) q^{15} +3.31415i q^{17} +7.08582 q^{19} -2.64265 q^{21} -4.82778i q^{23} +(4.87086 + 1.12902i) q^{25} -1.00000i q^{27} +2.18513 q^{29} +7.36266 q^{31} -1.51363i q^{33} +(0.671502 - 5.87086i) q^{35} -7.87086i q^{37} +3.87086 q^{39} -8.72532 q^{41} -1.01641i q^{43} +(2.22158 + 0.254102i) q^{45} +7.08582i q^{47} +0.0164068 q^{49} -3.31415 q^{51} +4.50820i q^{53} +(3.36266 + 0.384617i) q^{55} +7.08582i q^{57} +6.79893 q^{59} -3.60104 q^{61} -2.64265i q^{63} +(-0.983593 + 8.59945i) q^{65} -1.01641i q^{67} +4.82778 q^{69} -6.72532 q^{71} +15.5146i q^{73} +(-1.12902 + 4.87086i) q^{75} -4.00000i q^{77} -7.36266 q^{79} +1.00000 q^{81} +7.74173i q^{83} +(0.842131 - 7.36266i) q^{85} +2.18513i q^{87} +14.7581 q^{89} +10.2293 q^{91} +7.36266i q^{93} +(-15.7417 - 1.80052i) q^{95} +11.1444i q^{97} +1.51363 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 12 q^{9} - 4 q^{25} + 32 q^{31} - 16 q^{39} + 8 q^{41} - 12 q^{49} - 16 q^{55} - 24 q^{65} + 32 q^{71} - 32 q^{79} + 12 q^{81} + 40 q^{89} - 64 q^{95}+O(q^{100}) $ 12 * q - 12 * q^9 - 4 * q^25 + 32 * q^31 - 16 * q^39 + 8 * q^41 - 12 * q^49 - 16 * q^55 - 24 * q^65 + 32 * q^71 - 32 * q^79 + 12 * q^81 + 40 * q^89 - 64 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3840\mathbb{Z}\right)^\times$.

$n$	$511$	$1537$	$2561$	$2821$
$\chi(n)$	$1$	$-1$	$1$	$1$

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
$5$	−2.22158	−	0.254102i	−0.993522	−	0.113638i
$5$	−2.22158	−	0.254102i	−0.993522	−	0.113638i
$6$		0			0
$7$			2.64265i			0.998827i	0.866364	+	0.499414i	$0.166451\pi$
$7$			2.64265i			0.998827i	−0.866364	+	0.499414i	$0.833549\pi$
$8$		0			0
$9$	−1.00000			−0.333333
$10$		0			0
$11$	−1.51363			−0.456377			−0.228189	−	0.973617i	$-0.573280\pi$
$11$	−1.51363			−0.456377			−0.228189	+	0.973617i	$0.573280\pi$
$12$		0			0
$13$		−	3.87086i		−	1.07358i	−0.843714	−	0.536792i	$-0.819636\pi$
$13$		−	3.87086i		−	1.07358i	0.843714	−	0.536792i	$-0.180364\pi$
$14$		0			0
$15$	0.254102	−	2.22158i	0.0656088	−	0.573610i
$16$		0			0
$17$			3.31415i			0.803800i	0.915684	+	0.401900i	$0.131650\pi$
$17$			3.31415i			0.803800i	−0.915684	+	0.401900i	$0.868350\pi$
$18$		0			0
$19$	7.08582			1.62560			0.812799	−	0.582545i	$-0.197943\pi$
$19$	7.08582			1.62560			0.812799	+	0.582545i	$0.197943\pi$
$20$		0			0
$21$	−2.64265			−0.576673
$22$		0			0
$23$		−	4.82778i		−	1.00666i	−0.864094	−	0.503331i	$-0.832108\pi$
$23$		−	4.82778i		−	1.00666i	0.864094	−	0.503331i	$-0.167892\pi$
$24$		0			0
$25$	4.87086	+	1.12902i	0.974173	+	0.225803i
$26$		0			0
$27$		−	1.00000i		−	0.192450i
$28$		0			0
$29$	2.18513			0.405769			0.202885	−	0.979203i	$-0.434968\pi$
$29$	2.18513			0.405769			0.202885	+	0.979203i	$0.434968\pi$
$30$		0			0
$31$	7.36266			1.32237			0.661187	−	0.750222i	$-0.270053\pi$
$31$	7.36266			1.32237			0.661187	+	0.750222i	$0.270053\pi$
$32$		0			0
$33$		−	1.51363i		−	0.263490i
$34$		0			0
$35$	0.671502	−	5.87086i	0.113504	−	0.992357i
$36$		0			0
$37$		−	7.87086i		−	1.29396i	−0.762506	−	0.646981i	$-0.776031\pi$
$37$		−	7.87086i		−	1.29396i	0.762506	−	0.646981i	$-0.223969\pi$
$38$		0			0
$39$	3.87086			0.619834
$40$		0			0
$41$	−8.72532			−1.36267			−0.681333	−	0.731973i	$-0.738600\pi$
$41$	−8.72532			−1.36267			−0.681333	+	0.731973i	$0.738600\pi$
$42$		0			0
$43$		−	1.01641i		−	0.155001i	−0.996992	−	0.0775003i	$-0.975306\pi$
$43$		−	1.01641i		−	0.155001i	0.996992	−	0.0775003i	$-0.0246939\pi$
$44$		0			0
$45$	2.22158	+	0.254102i	0.331174	+	0.0378792i
$46$		0			0
$47$			7.08582i			1.03357i	0.856114	+	0.516786i	$0.172872\pi$
$47$			7.08582i			1.03357i	−0.856114	+	0.516786i	$0.827128\pi$
$48$		0			0
$49$	0.0164068			0.00234382
$50$		0			0
$51$	−3.31415			−0.464074
$52$		0			0
$53$			4.50820i			0.619249i	0.950859	+	0.309625i	$0.100203\pi$
$53$			4.50820i			0.619249i	−0.950859	+	0.309625i	$0.899797\pi$
$54$		0			0
$55$	3.36266	+	0.384617i	0.453421	+	0.0518617i
$56$		0			0
$57$			7.08582i			0.938539i
$58$		0			0
$59$	6.79893			0.885145			0.442573	−	0.896733i	$-0.354066\pi$
$59$	6.79893			0.885145			0.442573	+	0.896733i	$0.354066\pi$
$60$		0			0
$61$	−3.60104			−0.461065			−0.230533	−	0.973065i	$-0.574047\pi$
$61$	−3.60104			−0.461065			−0.230533	+	0.973065i	$0.574047\pi$
$62$		0			0
$63$		−	2.64265i		−	0.332942i
$64$		0			0
$65$	−0.983593	+	8.59945i	−0.122000	+	1.06663i
$66$		0			0
$67$		−	1.01641i		−	0.124174i	−0.998071	−	0.0620869i	$-0.980224\pi$
$67$		−	1.01641i		−	0.124174i	0.998071	−	0.0620869i	$-0.0197756\pi$
$68$		0			0
$69$	4.82778			0.581197
$70$		0			0
$71$	−6.72532			−0.798149			−0.399074	−	0.916919i	$-0.630669\pi$
$71$	−6.72532			−0.798149			−0.399074	+	0.916919i	$0.630669\pi$
$72$		0			0
$73$			15.5146i			1.81585i	0.419132	+	0.907925i	$0.362334\pi$
$73$			15.5146i			1.81585i	−0.419132	+	0.907925i	$0.637666\pi$
$74$		0			0
$75$	−1.12902	+	4.87086i	−0.130368	+	0.562439i
$76$		0			0
$77$		−	4.00000i		−	0.455842i
$78$		0			0
$79$	−7.36266			−0.828364			−0.414182	−	0.910194i	$-0.635932\pi$
$79$	−7.36266			−0.828364			−0.414182	+	0.910194i	$0.635932\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$			7.74173i			0.849765i	0.905248	+	0.424883i	$0.139685\pi$
$83$			7.74173i			0.849765i	−0.905248	+	0.424883i	$0.860315\pi$
$84$		0			0
$85$	0.842131	−	7.36266i	0.0913420	−	0.798593i
$86$		0			0
$87$			2.18513i			0.234271i
$88$		0			0
$89$	14.7581			1.56436			0.782180	−	0.623053i	$-0.214108\pi$
$89$	14.7581			1.56436			0.782180	+	0.623053i	$0.214108\pi$
$90$		0			0
$91$	10.2293			1.07233
$92$		0			0
$93$			7.36266i			0.763472i
$94$		0			0
$95$	−15.7417	−	1.80052i	−1.61507	−	0.184729i
$96$		0			0
$97$			11.1444i			1.13154i	0.824563	+	0.565769i	$0.191421\pi$
$97$			11.1444i			1.13154i	−0.824563	+	0.565769i	$0.808579\pi$
$98$		0			0
$99$	1.51363			0.152126
$100$		0			0
$101$	13.3295			1.32633			0.663167	−	0.748471i	$-0.269212\pi$
$101$	13.3295			1.32633			0.663167	+	0.748471i	$0.269212\pi$
$102$		0			0
$103$			0.958386i			0.0944326i	0.998885	+	0.0472163i	$0.0150350\pi$
$103$			0.958386i			0.0944326i	−0.998885	+	0.0472163i	$0.984965\pi$
$104$		0			0
$105$	5.87086	+	0.671502i	0.572938	+	0.0655318i
$106$		0			0
$107$			4.00000i			0.386695i	0.981130	+	0.193347i	$0.0619344\pi$
$107$			4.00000i			0.386695i	−0.981130	+	0.193347i	$0.938066\pi$
$108$		0			0
$109$	−0.769233			−0.0736792			−0.0368396	−	0.999321i	$-0.511729\pi$
$109$	−0.769233			−0.0736792			−0.0368396	+	0.999321i	$0.511729\pi$
$110$		0			0
$111$	7.87086			0.747069
$112$		0			0
$113$			14.4585i			1.36014i	0.733146	+	0.680071i	$0.238051\pi$
$113$			14.4585i			1.36014i	−0.733146	+	0.680071i	$0.761949\pi$
$114$		0			0
$115$	−1.22675	+	10.7253i	−0.114395	+	1.00014i
$116$		0			0
$117$			3.87086i			0.357862i
$118$		0			0
$119$	−8.75814			−0.802857
$120$		0			0
$121$	−8.70892			−0.791720
$122$		0			0
$123$		−	8.72532i		−	0.786736i
$124$		0			0
$125$	−10.5341	−	3.74590i	−0.942203	−	0.335043i
$126$		0			0
$127$		−	11.5290i		−	1.02303i	−0.859274	−	0.511516i	$-0.829084\pi$
$127$		−	11.5290i		−	1.02303i	0.859274	−	0.511516i	$-0.170916\pi$
$128$		0			0
$129$	1.01641			0.0894896
$130$		0			0
$131$	−7.37270			−0.644156			−0.322078	−	0.946713i	$-0.604381\pi$
$131$	−7.37270			−0.644156			−0.322078	+	0.946713i	$0.604381\pi$
$132$		0			0
$133$			18.7253i			1.62369i
$134$		0			0
$135$	−0.254102	+	2.22158i	−0.0218696	+	0.191203i
$136$		0			0
$137$			3.88792i			0.332167i	0.986112	+	0.166084i	$0.0531122\pi$
$137$			3.88792i			0.332167i	−0.986112	+	0.166084i	$0.946888\pi$
$138$		0			0
$139$	14.6291			1.24083			0.620414	−	0.784275i	$-0.286965\pi$
$139$	14.6291			1.24083			0.620414	+	0.784275i	$0.286965\pi$
$140$		0			0
$141$	−7.08582			−0.596733
$142$		0			0
$143$			5.85907i			0.489960i
$144$		0			0
$145$	−4.85446	−	0.555246i	−0.403141	−	0.0461107i
$146$		0			0
$147$			0.0164068i			0.00135321i
$148$		0			0
$149$	−11.0715			−0.907010			−0.453505	−	0.891254i	$-0.649827\pi$
$149$	−11.0715			−0.907010			−0.453505	+	0.891254i	$0.649827\pi$
$150$		0			0
$151$	0.637339			0.0518659			0.0259329	−	0.999664i	$-0.491744\pi$
$151$	0.637339			0.0518659			0.0259329	+	0.999664i	$0.491744\pi$
$152$		0			0
$153$		−	3.31415i		−	0.267933i
$154$		0			0
$155$	−16.3568	−	1.87086i	−1.31381	−	0.150271i
$156$		0			0
$157$		−	0.129135i		−	0.0103061i	−0.999987	−	0.00515306i	$-0.998360\pi$
$157$		−	0.129135i		−	0.0103061i	0.999987	−	0.00515306i	$-0.00164028\pi$
$158$		0			0
$159$	−4.50820			−0.357524
$160$		0			0
$161$	12.7581			1.00548
$162$		0			0
$163$		−	19.4835i		−	1.52606i	−0.646362	−	0.763031i	$-0.723710\pi$
$163$		−	19.4835i		−	1.52606i	0.646362	−	0.763031i	$-0.276290\pi$
$164$		0			0
$165$	−0.384617	+	3.36266i	−0.0299424	+	0.261783i
$166$		0			0
$167$			1.80052i			0.139328i	0.997571	+	0.0696641i	$0.0221928\pi$
$167$			1.80052i			0.139328i	−0.997571	+	0.0696641i	$0.977807\pi$
$168$		0			0
$169$	−1.98359			−0.152584
$170$		0			0
$171$	−7.08582			−0.541866
$172$		0			0
$173$			23.2335i			1.76641i	0.468985	+	0.883206i	$0.344620\pi$
$173$			23.2335i			1.76641i	−0.468985	+	0.883206i	$0.655380\pi$
$174$		0			0
$175$	−2.98359	+	12.8720i	−0.225538	+	0.973031i
$176$		0			0
$177$			6.79893i			0.511039i
$178$		0			0
$179$	−2.85664			−0.213515			−0.106757	−	0.994285i	$-0.534047\pi$
$179$	−2.85664			−0.213515			−0.106757	+	0.994285i	$0.534047\pi$
$180$		0			0
$181$	−5.28530			−0.392853			−0.196427	−	0.980519i	$-0.562934\pi$
$181$	−5.28530			−0.392853			−0.196427	+	0.980519i	$0.562934\pi$
$182$		0			0
$183$		−	3.60104i		−	0.266196i
$184$		0			0
$185$	−2.00000	+	17.4858i	−0.147043	+	1.28558i
$186$		0			0
$187$		−	5.01641i		−	0.366836i
$188$		0			0
$189$	2.64265			0.192224
$190$		0			0
$191$	5.96719			0.431770			0.215885	−	0.976419i	$-0.430736\pi$
$191$	5.96719			0.431770			0.215885	+	0.976419i	$0.430736\pi$
$192$		0			0
$193$			14.9409i			1.07547i	0.843115	+	0.537733i	$0.180719\pi$
$193$			14.9409i			1.07547i	−0.843115	+	0.537733i	$0.819281\pi$
$194$		0			0
$195$	−8.59945	−	0.983593i	−0.615819	−	0.0704366i
$196$		0			0
$197$			3.23353i			0.230379i	0.993344	+	0.115190i	$0.0367476\pi$
$197$			3.23353i			0.230379i	−0.993344	+	0.115190i	$0.963252\pi$
$198$		0			0
$199$	8.12080			0.575668			0.287834	−	0.957680i	$-0.407065\pi$
$199$	8.12080			0.575668			0.287834	+	0.957680i	$0.407065\pi$
$200$		0			0
$201$	1.01641			0.0716918
$202$		0			0
$203$			5.77454i			0.405293i
$204$		0			0
$205$	19.3840	+	2.21712i	1.35384	+	0.154850i
$206$		0			0
$207$			4.82778i			0.335554i
$208$		0			0
$209$	−10.7253			−0.741886
$210$		0			0
$211$	13.7141			0.944119			0.472059	−	0.881567i	$-0.343511\pi$
$211$	13.7141			0.944119			0.472059	+	0.881567i	$0.343511\pi$
$212$		0			0
$213$		−	6.72532i		−	0.460812i
$214$		0			0
$215$	−0.258271	+	2.25803i	−0.0176139	+	0.153997i
$216$		0			0
$217$			19.4569i			1.32082i
$218$		0			0
$219$	−15.5146			−1.04838
$220$		0			0
$221$	12.8286			0.862947
$222$		0			0
$223$		−	9.84472i		−	0.659251i	−0.944112	−	0.329626i	$-0.893077\pi$
$223$		−	9.84472i		−	0.659251i	0.944112	−	0.329626i	$-0.106923\pi$
$224$		0			0
$225$	−4.87086	−	1.12902i	−0.324724	−	0.0752677i
$226$		0			0
$227$		−	5.70892i		−	0.378914i	−0.981889	−	0.189457i	$-0.939327\pi$
$227$		−	5.70892i		−	0.378914i	0.981889	−	0.189457i	$-0.0606728\pi$
$228$		0			0
$229$	−0.769233			−0.0508324			−0.0254162	−	0.999677i	$-0.508091\pi$
$229$	−0.769233			−0.0508324			−0.0254162	+	0.999677i	$0.508091\pi$
$230$		0			0
$231$	4.00000			0.263181
$232$		0			0
$233$			18.4008i			1.20548i	0.797939	+	0.602739i	$0.205924\pi$
$233$			18.4008i			1.20548i	−0.797939	+	0.602739i	$0.794076\pi$
$234$		0			0
$235$	1.80052	−	15.7417i	0.117453	−	1.02688i
$236$		0			0
$237$		−	7.36266i		−	0.478256i
$238$		0			0
$239$	10.0328			0.648969			0.324484	−	0.945891i	$-0.394809\pi$
$239$	10.0328			0.648969			0.324484	+	0.945891i	$0.394809\pi$
$240$		0			0
$241$	10.7581			0.692992			0.346496	−	0.938051i	$-0.387371\pi$
$241$	10.7581			0.692992			0.346496	+	0.938051i	$0.387371\pi$
$242$		0			0
$243$			1.00000i			0.0641500i
$244$		0			0
$245$	−0.0364490	−	0.00416898i	−0.00232864	−	0.000266347i
$246$		0			0
$247$		−	27.4282i		−	1.74522i
$248$		0			0
$249$	−7.74173			−0.490612
$250$		0			0
$251$	−12.6580			−0.798966			−0.399483	−	0.916741i	$-0.630810\pi$
$251$	−12.6580			−0.798966			−0.399483	+	0.916741i	$0.630810\pi$
$252$		0			0
$253$			7.30749i			0.459418i
$254$		0			0
$255$	7.36266	+	0.842131i	0.461068	+	0.0527363i
$256$		0			0
$257$		−	13.3110i		−	0.830316i	−0.909749	−	0.415158i	$-0.863726\pi$
$257$		−	13.3110i		−	0.830316i	0.909749	−	0.415158i	$-0.136274\pi$
$258$		0			0
$259$	20.7999			1.29244
$260$		0			0
$261$	−2.18513			−0.135256
$262$		0			0
$263$			18.4256i			1.13617i	0.822969	+	0.568087i	$0.192316\pi$
$263$			18.4256i			1.13617i	−0.822969	+	0.568087i	$0.807684\pi$
$264$		0			0
$265$	1.14554	−	10.0153i	0.0703701	−	0.615238i
$266$		0			0
$267$			14.7581i			0.903183i
$268$		0			0
$269$	3.86940			0.235921			0.117961	−	0.993018i	$-0.462364\pi$
$269$	3.86940			0.235921			0.117961	+	0.993018i	$0.462364\pi$
$270$		0			0
$271$	17.3955			1.05670			0.528350	−	0.849027i	$-0.322811\pi$
$271$	17.3955			1.05670			0.528350	+	0.849027i	$0.322811\pi$
$272$		0			0
$273$			10.2293i			0.619108i
$274$		0			0
$275$	−7.37270	−	1.70892i	−0.444591	−	0.103052i
$276$		0			0
$277$			0.887271i			0.0533110i	0.999645	+	0.0266555i	$0.00848571\pi$
$277$			0.887271i			0.0533110i	−0.999645	+	0.0266555i	$0.991514\pi$
$278$		0			0
$279$	−7.36266			−0.440791
$280$		0			0
$281$	13.4835			0.804356			0.402178	−	0.915562i	$-0.368253\pi$
$281$	13.4835			0.804356			0.402178	+	0.915562i	$0.368253\pi$
$282$		0			0
$283$			28.4342i			1.69024i	0.534577	+	0.845120i	$0.320471\pi$
$283$			28.4342i			1.69024i	−0.534577	+	0.845120i	$0.679529\pi$
$284$		0			0
$285$	1.80052	−	15.7417i	0.106653	−	0.932460i
$286$		0			0
$287$		−	23.0580i		−	1.36107i
$288$		0			0
$289$	6.01641			0.353906
$290$		0			0
$291$	−11.1444			−0.653294
$292$		0			0
$293$		−	7.99166i		−	0.466878i	−0.972371	−	0.233439i	$-0.925002\pi$
$293$		−	7.99166i		−	0.466878i	0.972371	−	0.233439i	$-0.0749979\pi$
$294$		0			0
$295$	−15.1044	−	1.72762i	−0.879412	−	0.100586i
$296$		0			0
$297$			1.51363i			0.0878299i
$298$		0			0
$299$	−18.6877			−1.08074
$300$		0			0
$301$	2.68601			0.154819
$302$		0			0
$303$			13.3295i			0.765760i
$304$		0			0
$305$	8.00000	+	0.915029i	0.458079	+	0.0523944i
$306$		0			0
$307$			17.4506i			0.995961i	0.867188	+	0.497980i	$0.165925\pi$
$307$			17.4506i			0.995961i	−0.867188	+	0.497980i	$0.834075\pi$
$308$		0			0
$309$	−0.958386			−0.0545207
$310$		0			0
$311$	21.4506			1.21635			0.608177	−	0.793801i	$-0.291901\pi$
$311$	21.4506			1.21635			0.608177	+	0.793801i	$0.291901\pi$
$312$		0			0
$313$		−	7.73879i		−	0.437422i	−0.975790	−	0.218711i	$-0.929815\pi$
$313$		−	7.73879i		−	0.437422i	0.975790	−	0.218711i	$-0.0701853\pi$
$314$		0			0
$315$	−0.671502	+	5.87086i	−0.0378348	+	0.330786i
$316$		0			0
$317$		−	11.2335i		−	0.630938i	−0.948936	−	0.315469i	$-0.897838\pi$
$317$		−	11.2335i		−	0.630938i	0.948936	−	0.315469i	$-0.102162\pi$
$318$		0			0
$319$	−3.30749			−0.185184
$320$		0			0
$321$	−4.00000			−0.223258
$322$		0			0
$323$			23.4835i			1.30665i
$324$		0			0
$325$	4.37027	−	18.8545i	0.242419	−	1.04586i
$326$		0			0
$327$		−	0.769233i		−	0.0425387i
$328$		0			0
$329$	−18.7253			−1.03236
$330$		0			0
$331$	8.00084			0.439766			0.219883	−	0.975526i	$-0.429432\pi$
$331$	8.00084			0.439766			0.219883	+	0.975526i	$0.429432\pi$
$332$		0			0
$333$			7.87086i			0.431321i
$334$		0			0
$335$	−0.258271	+	2.25803i	−0.0141108	+	0.123369i
$336$		0			0
$337$			21.5692i			1.17495i	0.809243	+	0.587474i	$0.199877\pi$
$337$			21.5692i			1.17495i	−0.809243	+	0.587474i	$0.800123\pi$
$338$		0			0
$339$	−14.4585			−0.785279
$340$		0			0
$341$	−11.1444			−0.603501
$342$		0			0
$343$			18.5419i			1.00117i
$344$		0			0
$345$	−10.7253	−	1.22675i	−0.577432	−	0.0660459i
$346$		0			0
$347$		−	21.7089i		−	1.16540i	−0.812689	−	0.582698i	$-0.801997\pi$
$347$		−	21.7089i		−	1.16540i	0.812689	−	0.582698i	$-0.198003\pi$
$348$		0			0
$349$	−24.7422			−1.32442			−0.662211	−	0.749318i	$-0.730382\pi$
$349$	−24.7422			−1.32442			−0.662211	+	0.749318i	$0.730382\pi$
$350$		0			0
$351$	−3.87086			−0.206611
$352$		0			0
$353$			3.31415i			0.176394i	0.996103	+	0.0881972i	$0.0281106\pi$
$353$			3.31415i			0.176394i	−0.996103	+	0.0881972i	$0.971889\pi$
$354$		0			0
$355$	14.9409	+	1.70892i	0.792979	+	0.0906998i
$356$		0			0
$357$		−	8.75814i		−	0.463530i
$358$		0			0
$359$	16.7581			0.884461			0.442230	−	0.896902i	$-0.354187\pi$
$359$	16.7581			0.884461			0.442230	+	0.896902i	$0.354187\pi$
$360$		0			0
$361$	31.2088			1.64257
$362$		0			0
$363$		−	8.70892i		−	0.457100i
$364$		0			0
$365$	3.94229	−	34.4671i	0.206349	−	1.80409i
$366$		0			0
$367$			28.5324i			1.48938i	0.667411	+	0.744690i	$0.267403\pi$
$367$			28.5324i			1.48938i	−0.667411	+	0.744690i	$0.732597\pi$
$368$		0			0
$369$	8.72532			0.454222
$370$		0			0
$371$	−11.9136			−0.618523
$372$		0			0
$373$			37.5798i			1.94581i	0.231211	+	0.972904i	$0.425731\pi$
$373$			37.5798i			1.94581i	−0.231211	+	0.972904i	$0.574269\pi$
$374$		0			0
$375$	3.74590	−	10.5341i	0.193437	−	0.543981i
$376$		0			0
$377$		−	8.45836i		−	0.435628i
$378$		0			0
$379$	6.74456			0.346445			0.173222	−	0.984883i	$-0.444582\pi$
$379$	6.74456			0.346445			0.173222	+	0.984883i	$0.444582\pi$
$380$		0			0
$381$	11.5290			0.590648
$382$		0			0
$383$		−	21.8312i		−	1.11552i	−0.830001	−	0.557762i	$-0.811660\pi$
$383$		−	21.8312i		−	1.11552i	0.830001	−	0.557762i	$-0.188340\pi$
$384$		0			0
$385$	−1.01641	+	8.88633i	−0.0518009	+	0.452889i
$386$		0			0
$387$			1.01641i			0.0516669i
$388$		0			0
$389$	8.81344			0.446859			0.223429	−	0.974720i	$-0.428275\pi$
$389$	8.81344			0.446859			0.223429	+	0.974720i	$0.428275\pi$
$390$		0			0
$391$	16.0000			0.809155
$392$		0			0
$393$		−	7.37270i		−	0.371904i
$394$		0			0
$395$	16.3568	+	1.87086i	0.822998	+	0.0941334i
$396$		0			0
$397$			0.821644i			0.0412372i	0.999787	+	0.0206186i	$0.00656356\pi$
$397$			0.821644i			0.0412372i	−0.999787	+	0.0206186i	$0.993436\pi$
$398$		0			0
$399$	−18.7253			−0.937439
$400$		0			0
$401$	−12.7253			−0.635472			−0.317736	−	0.948179i	$-0.602923\pi$
$401$	−12.7253			−0.635472			−0.317736	+	0.948179i	$0.602923\pi$
$402$		0			0
$403$		−	28.4999i		−	1.41968i
$404$		0			0
$405$	−2.22158	−	0.254102i	−0.110391	−	0.0126264i
$406$		0			0
$407$			11.9136i			0.590535i
$408$		0			0
$409$	2.25827			0.111664			0.0558321	−	0.998440i	$-0.482219\pi$
$409$	2.25827			0.111664			0.0558321	+	0.998440i	$0.482219\pi$
$410$		0			0
$411$	−3.88792			−0.191777
$412$		0			0
$413$			17.9672i			0.884107i
$414$		0			0
$415$	1.96719	−	17.1989i	0.0965654	−	0.844261i
$416$		0			0
$417$			14.6291i			0.716392i
$418$		0			0
$419$	−33.4579			−1.63453			−0.817263	−	0.576264i	$-0.804510\pi$
$419$	−33.4579			−1.63453			−0.817263	+	0.576264i	$0.804510\pi$
$420$		0			0
$421$	11.3398			0.552669			0.276335	−	0.961061i	$-0.410880\pi$
$421$	11.3398			0.552669			0.276335	+	0.961061i	$0.410880\pi$
$422$		0			0
$423$		−	7.08582i		−	0.344524i
$424$		0			0
$425$	−3.74173	+	16.1428i	−0.181501	+	0.783040i
$426$		0			0
$427$		−	9.51627i		−	0.460525i
$428$		0			0
$429$	−5.85907			−0.282878
$430$		0			0
$431$	−10.6597			−0.513459			−0.256730	−	0.966483i	$-0.582645\pi$
$431$	−10.6597			−0.513459			−0.256730	+	0.966483i	$0.582645\pi$
$432$		0			0
$433$		−	26.5132i		−	1.27414i	−0.770805	−	0.637072i	$-0.780146\pi$
$433$		−	26.5132i		−	1.27414i	0.770805	−	0.637072i	$-0.219854\pi$
$434$		0			0
$435$	0.555246	−	4.85446i	0.0266220	−	0.232753i
$436$		0			0
$437$		−	34.2088i		−	1.63643i
$438$		0			0
$439$	−32.8789			−1.56923			−0.784613	−	0.619986i	$-0.787138\pi$
$439$	−32.8789			−1.56923			−0.784613	+	0.619986i	$0.787138\pi$
$440$		0			0
$441$	−0.0164068			−0.000781274
$442$		0			0
$443$		−	5.70892i		−	0.271239i	−0.990761	−	0.135619i	$-0.956698\pi$
$443$		−	5.70892i		−	0.271239i	0.990761	−	0.135619i	$-0.0433024\pi$
$444$		0			0
$445$	−32.7864	−	3.75007i	−1.55423	−	0.177770i
$446$		0			0
$447$		−	11.0715i		−	0.523662i
$448$		0			0
$449$	2.00000			0.0943858			0.0471929	−	0.998886i	$-0.484972\pi$
$449$	2.00000			0.0943858			0.0471929	+	0.998886i	$0.484972\pi$
$450$		0			0
$451$	13.2069			0.621890
$452$		0			0
$453$			0.637339i			0.0299448i
$454$		0			0
$455$	−22.7253	−	2.59929i	−1.06538	−	0.121857i
$456$		0			0
$457$			3.94229i			0.184413i	0.995740	+	0.0922064i	$0.0293920\pi$
$457$			3.94229i			0.184413i	−0.995740	+	0.0922064i	$0.970608\pi$
$458$		0			0
$459$	3.31415			0.154691
$460$		0			0
$461$	−33.8969			−1.57874			−0.789369	−	0.613920i	$-0.789592\pi$
$461$	−33.8969			−1.57874			−0.789369	+	0.613920i	$0.789592\pi$
$462$		0			0
$463$			22.8688i			1.06280i	0.847120	+	0.531402i	$0.178335\pi$
$463$			22.8688i			1.06280i	−0.847120	+	0.531402i	$0.821665\pi$
$464$		0			0
$465$	1.87086	−	16.3568i	0.0867593	−	0.758527i
$466$		0			0
$467$		−	15.7417i		−	0.728440i	−0.931313	−	0.364220i	$-0.881336\pi$
$467$		−	15.7417i		−	0.728440i	0.931313	−	0.364220i	$-0.118664\pi$
$468$		0			0
$469$	2.68601			0.124028
$470$		0			0
$471$	0.129135			0.00595024
$472$		0			0
$473$			1.53847i			0.0707388i
$474$		0			0
$475$	34.5140	+	8.00000i	1.58361	+	0.367065i
$476$		0			0
$477$		−	4.50820i		−	0.206416i
$478$		0			0
$479$	20.6925			0.945465			0.472732	−	0.881206i	$-0.343268\pi$
$479$	20.6925			0.945465			0.472732	+	0.881206i	$0.343268\pi$
$480$		0			0
$481$	−30.4671			−1.38918
$482$		0			0
$483$			12.7581i			0.580515i
$484$		0			0
$485$	2.83180	−	24.7581i	0.128586	−	1.12421i
$486$		0			0
$487$			30.8401i			1.39750i	0.715366	+	0.698750i	$0.246260\pi$
$487$			30.8401i			1.39750i	−0.715366	+	0.698750i	$0.753740\pi$
$488$		0			0
$489$	19.4835			0.881072
$490$		0			0
$491$	−10.9737			−0.495238			−0.247619	−	0.968858i	$-0.579648\pi$
$491$	−10.9737			−0.495238			−0.247619	+	0.968858i	$0.579648\pi$
$492$		0			0
$493$			7.24186i			0.326157i
$494$		0			0
$495$	−3.36266	−	0.384617i	−0.151140	−	0.0172872i
$496$		0			0
$497$		−	17.7727i		−	0.797213i
$498$		0			0
$499$	3.71729			0.166409			0.0832044	−	0.996533i	$-0.473485\pi$
$499$	3.71729			0.166409			0.0832044	+	0.996533i	$0.473485\pi$
$500$		0			0
$501$	−1.80052			−0.0804412
$502$		0			0
$503$		−	39.9451i		−	1.78107i	−0.454919	−	0.890533i	$-0.650332\pi$
$503$		−	39.9451i		−	1.78107i	0.454919	−	0.890533i	$-0.349668\pi$
$504$		0			0
$505$	−29.6126	−	3.38705i	−1.31774	−	0.150722i
$506$		0			0
$507$		−	1.98359i		−	0.0880945i
$508$		0			0
$509$	0.0728979			0.00323114			0.00161557	−	0.999999i	$-0.499486\pi$
$509$	0.0728979			0.00323114			0.00161557	+	0.999999i	$0.499486\pi$
$510$		0			0
$511$	−40.9997			−1.81372
$512$		0			0
$513$		−	7.08582i		−	0.312846i
$514$		0			0
$515$	0.243528	−	2.12914i	0.0107311	−	0.0938209i
$516$		0			0
$517$		−	10.7253i		−	0.471699i
$518$		0			0
$519$	−23.2335			−1.01984
$520$		0			0
$521$	11.9672			0.524292			0.262146	−	0.965028i	$-0.415570\pi$
$521$	11.9672			0.524292			0.262146	+	0.965028i	$0.415570\pi$
$522$		0			0
$523$		−	16.0656i		−	0.702501i	−0.936282	−	0.351250i	$-0.885757\pi$
$523$		−	16.0656i		−	0.702501i	0.936282	−	0.351250i	$-0.114243\pi$
$524$		0			0
$525$	−12.8720	−	2.98359i	−0.561779	−	0.130215i
$526$		0			0
$527$			24.4010i			1.06292i
$528$		0			0
$529$	−0.307491			−0.0133692
$530$		0			0
$531$	−6.79893			−0.295048
$532$		0			0
$533$			33.7745i			1.46294i
$534$		0			0
$535$	1.01641	−	8.88633i	0.0439431	−	0.384190i
$536$		0			0
$537$		−	2.85664i		−	0.123273i
$538$		0			0
$539$	−0.0248338			−0.00106967
$540$		0			0
$541$	15.8559			0.681698			0.340849	−	0.940118i	$-0.389285\pi$
$541$	15.8559			0.681698			0.340849	+	0.940118i	$0.389285\pi$
$542$		0			0
$543$		−	5.28530i		−	0.226814i
$544$		0			0
$545$	1.70892	+	0.195463i	0.0732019	+	0.00837274i
$546$		0			0
$547$			4.95078i			0.211680i	0.994383	+	0.105840i	$0.0337531\pi$
$547$			4.95078i			0.211680i	−0.994383	+	0.105840i	$0.966247\pi$
$548$		0			0
$549$	3.60104			0.153688
$550$		0			0
$551$	15.4835			0.659618
$552$		0			0
$553$		−	19.4569i		−	0.827393i
$554$		0			0
$555$	−17.4858	−	2.00000i	−0.742230	−	0.0848953i
$556$		0			0
$557$			1.26634i			0.0536565i	0.999640	+	0.0268283i	$0.00854073\pi$
$557$			1.26634i			0.0536565i	−0.999640	+	0.0268283i	$0.991459\pi$
$558$		0			0
$559$	−3.93437			−0.166406
$560$		0			0
$561$	5.01641			0.211793
$562$		0			0
$563$		−	5.70892i		−	0.240602i	−0.992737	−	0.120301i	$-0.961614\pi$
$563$		−	5.70892i		−	0.240602i	0.992737	−	0.120301i	$-0.0383860\pi$
$564$		0			0
$565$	3.67393	−	32.1208i	0.154564	−	1.35133i
$566$		0			0
$567$			2.64265i			0.110981i
$568$		0			0
$569$	−2.75814			−0.115627			−0.0578135	−	0.998327i	$-0.518413\pi$
$569$	−2.75814			−0.115627			−0.0578135	+	0.998327i	$0.518413\pi$
$570$		0			0
$571$	25.7735			1.07859			0.539294	−	0.842118i	$-0.318691\pi$
$571$	25.7735			1.07859			0.539294	+	0.842118i	$0.318691\pi$
$572$		0			0
$573$			5.96719i			0.249283i
$574$		0			0
$575$	5.45065	−	23.5155i	0.227308	−	0.980663i
$576$		0			0
$577$		−	32.7135i		−	1.36188i	−0.732338	−	0.680941i	$-0.761571\pi$
$577$		−	32.7135i		−	1.36188i	0.732338	−	0.680941i	$-0.238429\pi$
$578$		0			0
$579$	−14.9409			−0.620921
$580$		0			0
$581$	−20.4587			−0.848769
$582$		0			0
$583$		−	6.82376i		−	0.282611i
$584$		0			0
$585$	0.983593	−	8.59945i	0.0406666	−	0.355543i
$586$		0			0
$587$			43.4835i			1.79475i	0.441264	+	0.897377i	$0.354530\pi$
$587$			43.4835i			1.79475i	−0.441264	+	0.897377i	$0.645470\pi$
$588$		0			0
$589$	52.1705			2.14965
$590$		0			0
$591$	−3.23353			−0.133009
$592$		0			0
$593$		−	7.83021i		−	0.321548i	−0.986991	−	0.160774i	$-0.948601\pi$
$593$		−	7.83021i		−	0.321548i	0.986991	−	0.160774i	$-0.0513991\pi$
$594$		0			0
$595$	19.4569	+	2.22546i	0.797656	+	0.0912348i
$596$		0			0
$597$			8.12080i			0.332362i
$598$		0			0
$599$	32.7581			1.33846			0.669231	−	0.743055i	$-0.266624\pi$
$599$	32.7581			1.33846			0.669231	+	0.743055i	$0.266624\pi$
$600$		0			0
$601$	−17.8074			−0.726377			−0.363189	−	0.931716i	$-0.618312\pi$
$601$	−17.8074			−0.726377			−0.363189	+	0.931716i	$0.618312\pi$
$602$		0			0
$603$			1.01641i			0.0413913i
$604$		0			0
$605$	19.3476	+	2.21295i	0.786591	+	0.0899692i
$606$		0			0
$607$			3.41188i			0.138484i	0.997600	+	0.0692420i	$0.0220581\pi$
$607$			3.41188i			0.138484i	−0.997600	+	0.0692420i	$0.977942\pi$
$608$		0			0
$609$	−5.77454			−0.233996
$610$		0			0
$611$	27.4282			1.10963
$612$		0			0
$613$			36.6290i			1.47943i	0.672920	+	0.739716i	$0.265040\pi$
$613$			36.6290i			1.47943i	−0.672920	+	0.739716i	$0.734960\pi$
$614$		0			0
$615$	−2.21712	+	19.3840i	−0.0894029	+	0.781640i
$616$		0			0
$617$		−	40.3979i		−	1.62636i	−0.582012	−	0.813180i	$-0.697734\pi$
$617$		−	40.3979i		−	1.62636i	0.582012	−	0.813180i	$-0.302266\pi$
$618$		0			0
$619$	24.5172			0.985430			0.492715	−	0.870191i	$-0.336004\pi$
$619$	24.5172			0.985430			0.492715	+	0.870191i	$0.336004\pi$
$620$		0			0
$621$	−4.82778			−0.193732
$622$		0			0
$623$			39.0006i			1.56252i
$624$		0			0
$625$	22.4506	+	10.9986i	0.898026	+	0.439943i
$626$		0			0
$627$		−	10.7253i		−	0.428328i
$628$		0			0
$629$	26.0852			1.04009
$630$		0			0
$631$	18.7805			0.747640			0.373820	−	0.927501i	$-0.378048\pi$
$631$	18.7805			0.747640			0.373820	+	0.927501i	$0.378048\pi$
$632$		0			0
$633$			13.7141i			0.545087i
$634$		0			0
$635$	−2.92953	+	25.6126i	−0.116255	+	1.01640i
$636$		0			0
$637$		−	0.0635083i		−	0.00251629i
$638$		0			0
$639$	6.72532			0.266050
$640$		0			0
$641$	−15.5163			−0.612856			−0.306428	−	0.951894i	$-0.599134\pi$
$641$	−15.5163			−0.612856			−0.306428	+	0.951894i	$0.599134\pi$
$642$		0			0
$643$			17.4506i			0.688186i	0.938936	+	0.344093i	$0.111814\pi$
$643$			17.4506i			0.688186i	−0.938936	+	0.344093i	$0.888186\pi$
$644$		0			0
$645$	−2.25803	−	0.258271i	−0.0889099	−	0.0101694i
$646$		0			0
$647$		−	13.1403i		−	0.516600i	−0.966065	−	0.258300i	$-0.916838\pi$
$647$		−	13.1403i		−	0.516600i	0.966065	−	0.258300i	$-0.0831624\pi$
$648$		0			0
$649$	−10.2911			−0.403960
$650$		0			0
$651$	−19.4569			−0.762577
$652$		0			0
$653$			14.7993i			0.579141i	0.957157	+	0.289570i	$0.0935124\pi$
$653$			14.7993i			0.579141i	−0.957157	+	0.289570i	$0.906488\pi$
$654$		0			0
$655$	16.3791	+	1.87342i	0.639983	+	0.0732004i
$656$		0			0
$657$		−	15.5146i		−	0.605284i
$658$		0			0
$659$	7.99614			0.311485			0.155743	−	0.987798i	$-0.450223\pi$
$659$	7.99614			0.311485			0.155743	+	0.987798i	$0.450223\pi$
$660$		0			0
$661$	0.915029			0.0355905			0.0177953	−	0.999842i	$-0.494335\pi$
$661$	0.915029			0.0355905			0.0177953	+	0.999842i	$0.494335\pi$
$662$		0			0
$663$			12.8286i			0.498223i
$664$		0			0
$665$	4.75814	−	41.5999i	0.184513	−	1.61317i
$666$		0			0
$667$		−	10.5494i		−	0.408473i
$668$		0			0
$669$	9.84472			0.380619
$670$		0			0
$671$	5.45065			0.210420
$672$		0			0
$673$		−	34.3978i		−	1.32594i	−0.748647	−	0.662969i	$-0.769296\pi$
$673$		−	34.3978i		−	1.32594i	0.748647	−	0.662969i	$-0.230704\pi$
$674$		0			0
$675$	1.12902	−	4.87086i	0.0434559	−	0.187480i
$676$		0			0
$677$			40.1676i			1.54377i	0.635764	+	0.771884i	$0.280685\pi$
$677$			40.1676i			1.54377i	−0.635764	+	0.771884i	$0.719315\pi$
$678$		0			0
$679$	−29.4506			−1.13021
$680$		0			0
$681$	5.70892			0.218766
$682$		0			0
$683$		−	33.2580i		−	1.27258i	−0.771449	−	0.636291i	$-0.780468\pi$
$683$		−	33.2580i		−	1.27258i	0.771449	−	0.636291i	$-0.219532\pi$
$684$		0			0
$685$	0.987927	−	8.63734i	0.0377468	−	0.330016i
$686$		0			0
$687$		−	0.769233i		−	0.0293481i
$688$		0			0
$689$	17.4506			0.664817
$690$		0			0
$691$	50.2241			1.91062			0.955308	−	0.295611i	$-0.0955233\pi$
$691$	50.2241			1.91062			0.955308	+	0.295611i	$0.0955233\pi$
$692$		0			0
$693$			4.00000i			0.151947i
$694$		0			0
$695$	−32.4999	−	3.71729i	−1.23279	−	0.141005i
$696$		0			0
$697$		−	28.9170i		−	1.09531i
$698$		0			0
$699$	−18.4008			−0.695983
$700$		0			0
$701$	23.7543			0.897188			0.448594	−	0.893736i	$-0.351925\pi$
$701$	23.7543			0.897188			0.448594	+	0.893736i	$0.351925\pi$
$702$		0			0
$703$		−	55.7715i		−	2.10346i
$704$		0			0
$705$	15.7417	+	1.80052i	0.592868	+	0.0678114i
$706$		0			0
$707$			35.2252i			1.32478i
$708$		0			0
$709$	−36.3146			−1.36382			−0.681911	−	0.731435i	$-0.738851\pi$
$709$	−36.3146			−1.36382			−0.681911	+	0.731435i	$0.738851\pi$
$710$		0			0
$711$	7.36266			0.276121
$712$		0			0
$713$		−	35.5453i		−	1.33118i
$714$		0			0
$715$	1.48880	−	13.0164i	0.0556779	−	0.486786i
$716$		0			0
$717$			10.0328i			0.374682i
$718$		0			0
$719$	30.7253			1.14586			0.572931	−	0.819604i	$-0.305806\pi$
$719$	30.7253			1.14586			0.572931	+	0.819604i	$0.305806\pi$
$720$		0			0
$721$	−2.53268			−0.0943219
$722$		0			0
$723$			10.7581i			0.400099i
$724$		0			0
$725$	10.6435	+	2.46705i	0.395289	+	0.0916240i
$726$		0			0
$727$			5.47445i			0.203036i	0.994834	+	0.101518i	$0.0323700\pi$
$727$			5.47445i			0.203036i	−0.994834	+	0.101518i	$0.967630\pi$
$728$		0			0
$729$	−1.00000			−0.0370370
$730$		0			0
$731$	3.36852			0.124589
$732$		0			0
$733$			17.1455i			0.633285i	0.948545	+	0.316643i	$0.102556\pi$
$733$			17.1455i			0.633285i	−0.948545	+	0.316643i	$0.897444\pi$
$734$		0			0
$735$	0.00416898	−	0.0364490i	0.000153775	−	0.00134444i
$736$		0			0
$737$			1.53847i			0.0566701i
$738$		0			0
$739$	11.6019			0.426782			0.213391	−	0.976967i	$-0.431549\pi$
$739$	11.6019			0.426782			0.213391	+	0.976967i	$0.431549\pi$
$740$		0			0
$741$	27.4282			1.00760
$742$		0			0
$743$			23.6613i			0.868048i	0.900901	+	0.434024i	$0.142907\pi$
$743$			23.6613i			0.868048i	−0.900901	+	0.434024i	$0.857093\pi$
$744$		0			0
$745$	24.5962	+	2.81328i	0.901135	+	0.103071i
$746$		0			0
$747$		−	7.74173i		−	0.283255i
$748$		0			0
$749$	−10.5706			−0.386241
$750$		0			0
$751$	11.4283			0.417024			0.208512	−	0.978020i	$-0.433138\pi$
$751$	11.4283			0.417024			0.208512	+	0.978020i	$0.433138\pi$
$752$		0			0
$753$		−	12.6580i		−	0.461283i
$754$		0			0
$755$	−1.41590	−	0.161949i	−0.0515299	−	0.00589392i
$756$		0			0
$757$		−	19.1784i		−	0.697049i	−0.937300	−	0.348525i	$-0.886683\pi$
$757$		−	19.1784i		−	0.697049i	0.937300	−	0.348525i	$-0.113317\pi$
$758$		0			0
$759$	−7.30749			−0.265245
$760$		0			0
$761$	−4.03281			−0.146189			−0.0730947	−	0.997325i	$-0.523288\pi$
$761$	−4.03281			−0.146189			−0.0730947	+	0.997325i	$0.523288\pi$
$762$		0			0
$763$		−	2.03281i		−	0.0735928i
$764$		0			0
$765$	−0.842131	+	7.36266i	−0.0304473	+	0.266198i
$766$		0			0
$767$		−	26.3177i		−	0.950279i
$768$		0			0
$769$	2.95078			0.106408			0.0532039	−	0.998584i	$-0.483057\pi$
$769$	2.95078			0.106408			0.0532039	+	0.998584i	$0.483057\pi$
$770$		0			0
$771$	13.3110			0.479383
$772$		0			0
$773$			45.2663i			1.62812i	0.580783	+	0.814059i	$0.302747\pi$
$773$			45.2663i			1.62812i	−0.580783	+	0.814059i	$0.697253\pi$
$774$		0			0
$775$	35.8625	+	8.31256i	1.28822	+	0.298596i
$776$		0			0
$777$			20.7999i			0.746193i
$778$		0			0
$779$	−61.8260			−2.21515
$780$		0			0
$781$	10.1797			0.364257
$782$		0			0
$783$		−	2.18513i		−	0.0780903i
$784$		0			0
$785$	−0.0328135	+	0.286885i	−0.00117116	+	0.0102394i
$786$		0			0
$787$		−	52.9997i		−	1.88924i	−0.328171	−	0.944618i	$-0.606432\pi$
$787$		−	52.9997i		−	1.88924i	0.328171	−	0.944618i	$-0.393568\pi$
$788$		0			0
$789$	−18.4256			−0.655970
$790$		0			0
$791$	−38.2088			−1.35855
$792$		0			0
$793$			13.9391i			0.494993i
$794$		0			0
$795$	10.0153	+	1.14554i	0.355208	+	0.0406282i
$796$		0			0
$797$		−	16.5738i		−	0.587075i	−0.955948	−	0.293538i	$-0.905167\pi$
$797$		−	16.5738i		−	0.587075i	0.955948	−	0.293538i	$-0.0948326\pi$
$798$		0			0
$799$	−23.4835			−0.830785
$800$		0			0
$801$	−14.7581			−0.521453
$802$		0			0
$803$		−	23.4835i		−	0.828713i
$804$		0			0
$805$	−28.3433	−	3.24186i	−0.998969	−	0.114261i
$806$		0			0
$807$			3.86940i			0.136209i
$808$		0			0
$809$	−37.5491			−1.32016			−0.660078	−	0.751197i	$-0.729477\pi$
$809$	−37.5491			−1.32016			−0.660078	+	0.751197i	$0.729477\pi$
$810$		0			0
$811$	32.1102			1.12754			0.563771	−	0.825931i	$-0.309350\pi$
$811$	32.1102			1.12754			0.563771	+	0.825931i	$0.309350\pi$
$812$		0			0
$813$			17.3955i			0.610086i
$814$		0			0
$815$	−4.95078	+	43.2841i	−0.173418	+	1.51618i
$816$		0			0
$817$		−	7.20207i		−	0.251969i
$818$		0			0
$819$	−10.2293			−0.357442
$820$		0			0
$821$	−29.3809			−1.02540			−0.512699	−	0.858568i	$-0.671354\pi$
$821$	−29.3809			−1.02540			−0.512699	+	0.858568i	$0.671354\pi$
$822$		0			0
$823$		−	28.3866i		−	0.989495i	−0.869037	−	0.494748i	$-0.835260\pi$
$823$		−	28.3866i		−	0.989495i	0.869037	−	0.494748i	$-0.164740\pi$
$824$		0			0
$825$	1.70892	−	7.37270i	0.0594968	−	0.256684i
$826$		0			0
$827$			1.45065i			0.0504439i	0.999682	+	0.0252219i	$0.00802924\pi$
$827$			1.45065i			0.0504439i	−0.999682	+	0.0252219i	$0.991971\pi$
$828$		0			0
$829$	−37.4621			−1.30111			−0.650556	−	0.759458i	$-0.725464\pi$
$829$	−37.4621			−1.30111			−0.650556	+	0.759458i	$0.725464\pi$
$830$		0			0
$831$	−0.887271			−0.0307791
$832$		0			0
$833$			0.0543744i			0.00188396i
$834$		0			0
$835$	0.457515	−	4.00000i	0.0158329	−	0.138426i
$836$		0			0
$837$		−	7.36266i		−	0.254491i
$838$		0			0
$839$	48.7581			1.68332			0.841659	−	0.540010i	$-0.181579\pi$
$839$	48.7581			1.68332			0.841659	+	0.540010i	$0.181579\pi$
$840$		0			0
$841$	−24.2252			−0.835351
$842$		0			0
$843$			13.4835i			0.464395i
$844$		0			0
$845$	4.40672	+	0.504034i	0.151596	+	0.0173393i
$846$		0			0
$847$		−	23.0146i		−	0.790791i
$848$		0			0
$849$	−28.4342			−0.975861
$850$		0			0
$851$	−37.9988			−1.30258
$852$		0			0
$853$		−	4.37073i		−	0.149651i	−0.997197	−	0.0748255i	$-0.976160\pi$
$853$		−	4.37073i		−	0.149651i	0.997197	−	0.0748255i	$-0.0238400\pi$
$854$		0			0
$855$	15.7417	+	1.80052i	0.538356	+	0.0615764i
$856$		0			0
$857$		−	20.5130i		−	0.700712i	−0.936617	−	0.350356i	$-0.886061\pi$
$857$		−	20.5130i		−	0.700712i	0.936617	−	0.350356i	$-0.113939\pi$
$858$		0			0
$859$	10.1131			0.345054			0.172527	−	0.985005i	$-0.444807\pi$
$859$	10.1131			0.345054			0.172527	+	0.985005i	$0.444807\pi$
$860$		0			0
$861$	23.0580			0.785813
$862$		0			0
$863$			13.2861i			0.452266i	0.974096	+	0.226133i	$0.0726083\pi$
$863$			13.2861i			0.452266i	−0.974096	+	0.226133i	$0.927392\pi$
$864$		0			0
$865$	5.90368	−	51.6152i	0.200731	−	1.75497i
$866$		0			0
$867$			6.01641i			0.204328i
$868$		0			0
$869$	11.1444			0.378047
$870$		0			0
$871$	−3.93437			−0.133311
$872$		0			0
$873$		−	11.1444i		−	0.377180i
$874$		0			0
$875$	9.89909	−	27.8381i	0.334650	−	0.941098i
$876$		0			0
$877$		−	33.6454i		−	1.13612i	−0.822986	−	0.568062i	$-0.807693\pi$
$877$		−	33.6454i		−	1.13612i	0.822986	−	0.568062i	$-0.192307\pi$
$878$		0			0
$879$	7.99166			0.269552
$880$		0			0
$881$	−32.7909			−1.10476			−0.552378	−	0.833594i	$-0.686279\pi$
$881$	−32.7909			−1.10476			−0.552378	+	0.833594i	$0.686279\pi$
$882$		0			0
$883$			33.4506i			1.12570i	0.826558	+	0.562852i	$0.190296\pi$
$883$			33.4506i			1.12570i	−0.826558	+	0.562852i	$0.809704\pi$
$884$		0			0
$885$	1.72762	−	15.1044i	0.0580733	−	0.507729i
$886$		0			0
$887$		−	34.8924i		−	1.17157i	−0.810466	−	0.585785i	$-0.800786\pi$
$887$		−	34.8924i		−	1.17157i	0.810466	−	0.585785i	$-0.199214\pi$
$888$		0			0
$889$	30.4671			1.02183
$890$		0			0
$891$	−1.51363			−0.0507086
$892$		0			0
$893$			50.2088i			1.68017i
$894$		0			0
$895$	6.34625	+	0.725876i	0.212132	+	0.0242634i
$896$		0			0
$897$		−	18.6877i		−	0.623964i
$898$		0			0
$899$	16.0884			0.536578
$900$		0			0
$901$	−14.9409			−0.497752
$902$		0			0
$903$			2.68601i			0.0893847i
$904$		0			0
$905$	11.7417	+	1.34300i	0.390308	+	0.0446429i
$906$		0			0
$907$			30.9836i			1.02879i	0.857552	+	0.514397i	$0.171984\pi$
$907$			30.9836i			1.02879i	−0.857552	+	0.514397i	$0.828016\pi$
$908$		0			0
$909$	−13.3295			−0.442112
$910$		0			0
$911$	−16.0000			−0.530104			−0.265052	−	0.964234i	$-0.585389\pi$
$911$	−16.0000			−0.530104			−0.265052	+	0.964234i	$0.585389\pi$
$912$		0			0
$913$		−	11.7181i		−	0.387814i
$914$		0			0
$915$	−0.915029	+	8.00000i	−0.0302499	+	0.264472i
$916$		0			0
$917$		−	19.4835i		−	0.643400i
$918$		0			0
$919$	15.6043			0.514737			0.257368	−	0.966313i	$-0.417145\pi$
$919$	15.6043			0.514737			0.257368	+	0.966313i	$0.417145\pi$
$920$		0			0
$921$	−17.4506			−0.575018
$922$		0			0
$923$			26.0328i			0.856880i
$924$		0			0
$925$	8.88633	−	38.3379i	0.292181	−	1.26054i
$926$		0			0
$927$		−	0.958386i		−	0.0314775i
$928$		0			0
$929$	38.9341			1.27739			0.638693	−	0.769461i	$-0.279475\pi$
$929$	38.9341			1.27739			0.638693	+	0.769461i	$0.279475\pi$
$930$		0			0
$931$	0.116255			0.00381011
$932$		0			0
$933$			21.4506i			0.702263i
$934$		0			0
$935$	−1.27468	+	11.1444i	−0.0416864	+	0.364460i
$936$		0			0
$937$		−	19.6027i		−	0.640393i	−0.947351	−	0.320197i	$-0.896251\pi$
$937$		−	19.6027i		−	0.640393i	0.947351	−	0.320197i	$-0.103749\pi$
$938$		0			0
$939$	7.73879			0.252546
$940$		0			0
$941$	−25.0476			−0.816530			−0.408265	−	0.912864i	$-0.633866\pi$
$941$	−25.0476			−0.816530			−0.408265	+	0.912864i	$0.633866\pi$
$942$		0			0
$943$			42.1240i			1.37175i
$944$		0			0
$945$	−5.87086	−	0.671502i	−0.190979	−	0.0218439i
$946$		0			0
$947$		−	7.93437i		−	0.257832i	−0.991655	−	0.128916i	$-0.958850\pi$
$947$		−	7.93437i		−	0.257832i	0.991655	−	0.128916i	$-0.0411498\pi$
$948$		0			0
$949$	60.0550			1.94947
$950$		0			0
$951$	11.2335			0.364272
$952$		0			0
$953$		−	11.4809i		−	0.371903i	−0.982559	−	0.185952i	$-0.940463\pi$
$953$		−	11.4809i		−	0.371903i	0.982559	−	0.185952i	$-0.0595368\pi$
$954$		0			0
$955$	−13.2566	−	1.51627i	−0.428974	−	0.0490654i
$956$		0			0
$957$		−	3.30749i		−	0.106916i
$958$		0			0
$959$	−10.2744			−0.331778
$960$		0			0
$961$	23.2088			0.748670
$962$		0			0
$963$		−	4.00000i		−	0.128898i
$964$		0			0
$965$	3.79650	−	33.1924i	0.122214	−	1.06850i
$966$		0			0
$967$			15.8993i			0.511285i	0.966771	+	0.255643i	$0.0822871\pi$
$967$			15.8993i			0.511285i	−0.966771	+	0.255643i	$0.917713\pi$
$968$		0			0
$969$	−23.4835			−0.754397
$970$		0			0
$971$	40.6600			1.30484			0.652421	−	0.757857i	$-0.273754\pi$
$971$	40.6600			1.30484			0.652421	+	0.757857i	$0.273754\pi$
$972$		0			0
$973$			38.6597i			1.23937i
$974$		0			0
$975$	18.8545	+	4.37027i	0.603826	+	0.139961i
$976$		0			0
$977$		−	26.5676i		−	0.849972i	−0.905200	−	0.424986i	$-0.860279\pi$
$977$		−	26.5676i		−	0.849972i	0.905200	−	0.424986i	$-0.139721\pi$
$978$		0			0
$979$	−22.3384			−0.713938
$980$		0			0
$981$	0.769233			0.0245597
$982$		0			0
$983$		−	9.88057i		−	0.315141i	−0.987508	−	0.157571i	$-0.949634\pi$
$983$		−	9.88057i		−	0.315141i	0.987508	−	0.157571i	$-0.0503662\pi$
$984$		0			0
$985$	0.821644	−	7.18355i	0.0261798	−	0.228887i
$986$		0			0
$987$		−	18.7253i		−	0.596034i
$988$		0			0
$989$	−4.90699			−0.156033
$990$		0			0
$991$	53.0549			1.68534			0.842672	−	0.538427i	$-0.180981\pi$
$991$	53.0549			1.68534			0.842672	+	0.538427i	$0.180981\pi$
$992$		0			0
$993$			8.00084i			0.253899i
$994$		0			0
$995$	−18.0410	−	2.06351i	−0.571939	−	0.0654176i
$996$		0			0
$997$		−	32.3051i		−	1.02311i	−0.859250	−	0.511556i	$-0.829069\pi$
$997$		−	32.3051i		−	1.02311i	0.859250	−	0.511556i	$-0.170931\pi$
$998$		0			0
$999$	−7.87086			−0.249023

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3840.2.f.m.769.7		12
4.3	odd	2		3840.2.f.l.769.1		12
5.4	even	2	inner	3840.2.f.m.769.1		12
8.3	odd	2		3840.2.f.l.769.12		12
8.5	even	2	inner	3840.2.f.m.769.6		12
16.3	odd	4		120.2.d.a.109.1	✓	6
16.5	even	4		480.2.d.b.49.3		6
16.11	odd	4		120.2.d.b.109.5	yes	6
16.13	even	4		480.2.d.a.49.4		6
20.19	odd	2		3840.2.f.l.769.7		12
40.19	odd	2		3840.2.f.l.769.6		12
40.29	even	2	inner	3840.2.f.m.769.12		12
48.5	odd	4		1440.2.d.f.1009.4		6
48.11	even	4		360.2.d.e.109.2		6
48.29	odd	4		1440.2.d.e.1009.3		6
48.35	even	4		360.2.d.f.109.6		6
80.3	even	4		600.2.k.f.301.6		12
80.13	odd	4		2400.2.k.f.1201.2		12
80.19	odd	4		120.2.d.b.109.6	yes	6
80.27	even	4		600.2.k.f.301.8		12
80.29	even	4		480.2.d.b.49.4		6
80.37	odd	4		2400.2.k.f.1201.5		12
80.43	even	4		600.2.k.f.301.5		12
80.53	odd	4		2400.2.k.f.1201.8		12
80.59	odd	4		120.2.d.a.109.2	yes	6
80.67	even	4		600.2.k.f.301.7		12
80.69	even	4		480.2.d.a.49.3		6
80.77	odd	4		2400.2.k.f.1201.11		12
240.29	odd	4		1440.2.d.f.1009.3		6
240.53	even	4		7200.2.k.u.3601.4		12
240.59	even	4		360.2.d.f.109.5		6
240.77	even	4		7200.2.k.u.3601.9		12
240.83	odd	4		1800.2.k.u.901.7		12
240.107	odd	4		1800.2.k.u.901.5		12
240.149	odd	4		1440.2.d.e.1009.4		6
240.173	even	4		7200.2.k.u.3601.3		12
240.179	even	4		360.2.d.e.109.1		6
240.197	even	4		7200.2.k.u.3601.10		12
240.203	odd	4		1800.2.k.u.901.8		12
240.227	odd	4		1800.2.k.u.901.6		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
120.2.d.a.109.1	✓	6	16.3	odd	4
120.2.d.a.109.2	yes	6	80.59	odd	4
120.2.d.b.109.5	yes	6	16.11	odd	4
120.2.d.b.109.6	yes	6	80.19	odd	4
360.2.d.e.109.1		6	240.179	even	4
360.2.d.e.109.2		6	48.11	even	4
360.2.d.f.109.5		6	240.59	even	4
360.2.d.f.109.6		6	48.35	even	4
480.2.d.a.49.3		6	80.69	even	4
480.2.d.a.49.4		6	16.13	even	4
480.2.d.b.49.3		6	16.5	even	4
480.2.d.b.49.4		6	80.29	even	4
600.2.k.f.301.5		12	80.43	even	4
600.2.k.f.301.6		12	80.3	even	4
600.2.k.f.301.7		12	80.67	even	4
600.2.k.f.301.8		12	80.27	even	4
1440.2.d.e.1009.3		6	48.29	odd	4
1440.2.d.e.1009.4		6	240.149	odd	4
1440.2.d.f.1009.3		6	240.29	odd	4
1440.2.d.f.1009.4		6	48.5	odd	4
1800.2.k.u.901.5		12	240.107	odd	4
1800.2.k.u.901.6		12	240.227	odd	4
1800.2.k.u.901.7		12	240.83	odd	4
1800.2.k.u.901.8		12	240.203	odd	4
2400.2.k.f.1201.2		12	80.13	odd	4
2400.2.k.f.1201.5		12	80.37	odd	4
2400.2.k.f.1201.8		12	80.53	odd	4
2400.2.k.f.1201.11		12	80.77	odd	4
3840.2.f.l.769.1		12	4.3	odd	2
3840.2.f.l.769.6		12	40.19	odd	2
3840.2.f.l.769.7		12	20.19	odd	2
3840.2.f.l.769.12		12	8.3	odd	2
3840.2.f.m.769.1		12	5.4	even	2	inner
3840.2.f.m.769.6		12	8.5	even	2	inner
3840.2.f.m.769.7		12	1.1	even	1	trivial
3840.2.f.m.769.12		12	40.29	even	2	inner
7200.2.k.u.3601.3		12	240.173	even	4
7200.2.k.u.3601.4		12	240.53	even	4
7200.2.k.u.3601.9		12	240.77	even	4
7200.2.k.u.3601.10		12	240.197	even	4

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:	\( N \)	\(=\)	\( 3840 = 2^{8} \cdot 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3840.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{3} +(-2.22158 - 0.254102i) q^{5} +2.64265i q^{7} -1.00000 q^{9} -1.51363 q^{11} -3.87086i q^{13} +(0.254102 - 2.22158i) q^{15} +3.31415i q^{17} +7.08582 q^{19} -2.64265 q^{21} -4.82778i q^{23} +(4.87086 + 1.12902i) q^{25} -1.00000i q^{27} +2.18513 q^{29} +7.36266 q^{31} -1.51363i q^{33} +(0.671502 - 5.87086i) q^{35} -7.87086i q^{37} +3.87086 q^{39} -8.72532 q^{41} -1.01641i q^{43} +(2.22158 + 0.254102i) q^{45} +7.08582i q^{47} +0.0164068 q^{49} -3.31415 q^{51} +4.50820i q^{53} +(3.36266 + 0.384617i) q^{55} +7.08582i q^{57} +6.79893 q^{59} -3.60104 q^{61} -2.64265i q^{63} +(-0.983593 + 8.59945i) q^{65} -1.01641i q^{67} +4.82778 q^{69} -6.72532 q^{71} +15.5146i q^{73} +(-1.12902 + 4.87086i) q^{75} -4.00000i q^{77} -7.36266 q^{79} +1.00000 q^{81} +7.74173i q^{83} +(0.842131 - 7.36266i) q^{85} +2.18513i q^{87} +14.7581 q^{89} +10.2293 q^{91} +7.36266i q^{93} +(-15.7417 - 1.80052i) q^{95} +11.1444i q^{97} +1.51363 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 12 q^{9} - 4 q^{25} + 32 q^{31} - 16 q^{39} + 8 q^{41} - 12 q^{49} - 16 q^{55} - 24 q^{65} + 32 q^{71} - 32 q^{79} + 12 q^{81} + 40 q^{89} - 64 q^{95}+O(q^{100}) \) 12 * q - 12 * q^9 - 4 * q^25 + 32 * q^31 - 16 * q^39 + 8 * q^41 - 12 * q^49 - 16 * q^55 - 24 * q^65 + 32 * q^71 - 32 * q^79 + 12 * q^81 + 40 * q^89 - 64 * q^95

Introduction

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