LMFDB - Embedded newform 4032.2.k.f.3905.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4032,2,Mod(3905,4032)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4032, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 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("4032.3905");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4032 = 2^{6} \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4032.k (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:	$32.1956820950$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\Q(\zeta_{16})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 1 $ x^8 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{9} $
Twist minimal:	no (minimal twist has level 504)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			3905.1
Root			$0.923880 + 0.382683i$ of defining polynomial
Character	$\chi$	$=$	4032.3905
Dual form			4032.2.k.f.3905.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.69552 q^{5} +(2.41421 - 1.08239i) q^{7} +O(q^{10})$	$q-3.69552 q^{5} +(2.41421 - 1.08239i) q^{7} +3.41421i q^{11} -5.22625i q^{13} -6.75699 q^{17} +2.16478i q^{19} -6.24264i q^{23} +8.65685 q^{25} -2.58579i q^{29} +10.4525i q^{31} +(-8.92177 + 4.00000i) q^{35} -4.00000 q^{37} +0.634051 q^{41} +6.48528 q^{43} +3.06147 q^{47} +(4.65685 - 5.22625i) q^{49} +2.58579i q^{53} -12.6173i q^{55} +19.3137i q^{65} -1.65685 q^{67} +7.41421i q^{71} -0.896683i q^{73} +(3.69552 + 8.24264i) q^{77} -13.6569 q^{79} +11.7206 q^{83} +24.9706 q^{85} -6.75699 q^{89} +(-5.65685 - 12.6173i) q^{91} -8.00000i q^{95} +9.55582i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 8 q^{7}+O(q^{10}) $ 8 * q + 8 * q^7	$ 8 q + 8 q^{7} + 24 q^{25} - 32 q^{37} - 16 q^{43} - 8 q^{49} + 32 q^{67} - 64 q^{79} + 64 q^{85}+O(q^{100}) $ 8 * q + 8 * q^7 + 24 * q^25 - 32 * q^37 - 16 * q^43 - 8 * q^49 + 32 * q^67 - 64 * q^79 + 64 * q^85

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4032\mathbb{Z}\right)^\times$.

$n$	$127$	$577$	$1793$	$3781$
$\chi(n)$	$1$	$-1$	$-1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	−3.69552			−1.65269			−0.826343	−	0.563167i	$-0.809583\pi$
$5$	−3.69552			−1.65269			−0.826343	+	0.563167i	$0.809583\pi$
$6$		0			0
$7$	2.41421	−	1.08239i	0.912487	−	0.409106i
$7$	2.41421	−	1.08239i	0.912487	−	0.409106i
$8$		0			0
$9$		0			0
$10$		0			0
$11$			3.41421i			1.02942i	0.857363	+	0.514712i	$0.172101\pi$
$11$			3.41421i			1.02942i	−0.857363	+	0.514712i	$0.827899\pi$
$12$		0			0
$13$		−	5.22625i		−	1.44950i	−0.689011	−	0.724751i	$-0.741955\pi$
$13$		−	5.22625i		−	1.44950i	0.689011	−	0.724751i	$-0.258045\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−6.75699			−1.63881			−0.819405	−	0.573215i	$-0.805696\pi$
$17$	−6.75699			−1.63881			−0.819405	+	0.573215i	$0.805696\pi$
$18$		0			0
$19$			2.16478i			0.496636i	0.968679	+	0.248318i	$0.0798777\pi$
$19$			2.16478i			0.496636i	−0.968679	+	0.248318i	$0.920122\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		−	6.24264i		−	1.30168i	−0.759215	−	0.650840i	$-0.774417\pi$
$23$		−	6.24264i		−	1.30168i	0.759215	−	0.650840i	$-0.225583\pi$
$24$		0			0
$25$	8.65685			1.73137
$26$		0			0
$27$		0			0
$28$		0			0
$29$		−	2.58579i		−	0.480168i	−0.970752	−	0.240084i	$-0.922825\pi$
$29$		−	2.58579i		−	0.480168i	0.970752	−	0.240084i	$-0.0771751\pi$
$30$		0			0
$31$			10.4525i			1.87733i	0.344837	+	0.938663i	$0.387934\pi$
$31$			10.4525i			1.87733i	−0.344837	+	0.938663i	$0.612066\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−8.92177	+	4.00000i	−1.50805	+	0.676123i
$36$		0			0
$37$	−4.00000			−0.657596			−0.328798	−	0.944400i	$-0.606644\pi$
$37$	−4.00000			−0.657596			−0.328798	+	0.944400i	$0.606644\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	0.634051			0.0990221			0.0495110	−	0.998774i	$-0.484234\pi$
$41$	0.634051			0.0990221			0.0495110	+	0.998774i	$0.484234\pi$
$42$		0			0
$43$	6.48528			0.988996			0.494498	−	0.869179i	$-0.335352\pi$
$43$	6.48528			0.988996			0.494498	+	0.869179i	$0.335352\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	3.06147			0.446561			0.223280	−	0.974754i	$-0.428323\pi$
$47$	3.06147			0.446561			0.223280	+	0.974754i	$0.428323\pi$
$48$		0			0
$49$	4.65685	−	5.22625i	0.665265	−	0.746607i
$50$		0			0
$51$		0			0
$52$		0			0
$53$			2.58579i			0.355185i	0.984104	+	0.177593i	$0.0568309\pi$
$53$			2.58579i			0.355185i	−0.984104	+	0.177593i	$0.943169\pi$
$54$		0			0
$55$		−	12.6173i		−	1.70131i
$56$		0			0
$57$		0			0
$58$		0			0
$59$		0			0			−	1.00000i	$-0.5\pi$
$59$		0			0				1.00000i	$0.5\pi$
$60$		0			0
$61$		0			0		1.00000			$0$
$61$		0			0		−1.00000			$\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$			19.3137i			2.39557i
$66$		0			0
$67$	−1.65685			−0.202417			−0.101208	−	0.994865i	$-0.532271\pi$
$67$	−1.65685			−0.202417			−0.101208	+	0.994865i	$0.532271\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			7.41421i			0.879905i	0.898021	+	0.439953i	$0.145005\pi$
$71$			7.41421i			0.879905i	−0.898021	+	0.439953i	$0.854995\pi$
$72$		0			0
$73$		−	0.896683i		−	0.104949i	−0.998622	−	0.0524744i	$-0.983289\pi$
$73$		−	0.896683i		−	0.104949i	0.998622	−	0.0524744i	$-0.0167108\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	3.69552	+	8.24264i	0.421143	+	0.939336i
$78$		0			0
$79$	−13.6569			−1.53652			−0.768258	−	0.640140i	$-0.778876\pi$
$79$	−13.6569			−1.53652			−0.768258	+	0.640140i	$0.778876\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	11.7206			1.28650			0.643252	−	0.765655i	$-0.277585\pi$
$83$	11.7206			1.28650			0.643252	+	0.765655i	$0.277585\pi$
$84$		0			0
$85$	24.9706			2.70844
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−6.75699			−0.716239			−0.358120	−	0.933676i	$-0.616582\pi$
$89$	−6.75699			−0.716239			−0.358120	+	0.933676i	$0.616582\pi$
$90$		0			0
$91$	−5.65685	−	12.6173i	−0.592999	−	1.32265i
$92$		0			0
$93$		0			0
$94$		0			0
$95$		−	8.00000i		−	0.820783i
$96$		0			0
$97$			9.55582i			0.970247i	0.874446	+	0.485123i	$0.161225\pi$
$97$			9.55582i			0.970247i	−0.874446	+	0.485123i	$0.838775\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−2.42742			−0.241537			−0.120768	−	0.992681i	$-0.538536\pi$
$101$	−2.42742			−0.241537			−0.120768	+	0.992681i	$0.538536\pi$
$102$		0			0
$103$		−	6.12293i		−	0.603311i	−0.953417	−	0.301655i	$-0.902461\pi$
$103$		−	6.12293i		−	0.603311i	0.953417	−	0.301655i	$-0.0975392\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			1.07107i			0.103544i	0.998659	+	0.0517720i	$0.0164869\pi$
$107$			1.07107i			0.103544i	−0.998659	+	0.0517720i	$0.983513\pi$
$108$		0			0
$109$	−1.65685			−0.158698			−0.0793489	−	0.996847i	$-0.525284\pi$
$109$	−1.65685			−0.158698			−0.0793489	+	0.996847i	$0.525284\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			11.7574i			1.10604i	0.833168	+	0.553020i	$0.186525\pi$
$113$			11.7574i			1.10604i	−0.833168	+	0.553020i	$0.813475\pi$
$114$		0			0
$115$			23.0698i			2.15127i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−16.3128	+	7.31371i	−1.49539	+	0.670447i
$120$		0			0
$121$	−0.656854			−0.0597140
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−13.5140			−1.20873
$126$		0			0
$127$	−16.9706			−1.50589			−0.752947	−	0.658081i	$-0.771368\pi$
$127$	−16.9706			−1.50589			−0.752947	+	0.658081i	$0.771368\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	3.06147			0.267482			0.133741	−	0.991016i	$-0.457301\pi$
$131$	3.06147			0.267482			0.133741	+	0.991016i	$0.457301\pi$
$132$		0			0
$133$	2.34315	+	5.22625i	0.203177	+	0.453174i
$134$		0			0
$135$		0			0
$136$		0			0
$137$			9.89949i			0.845771i	0.906183	+	0.422885i	$0.138983\pi$
$137$			9.89949i			0.845771i	−0.906183	+	0.422885i	$0.861017\pi$
$138$		0			0
$139$		−	4.32957i		−	0.367229i	−0.982998	−	0.183615i	$-0.941220\pi$
$139$		−	4.32957i		−	0.367229i	0.982998	−	0.183615i	$-0.0587798\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	17.8435			1.49215
$144$		0			0
$145$			9.55582i			0.793568i
$146$		0			0
$147$		0			0
$148$		0			0
$149$			7.75736i			0.635508i	0.948173	+	0.317754i	$0.102929\pi$
$149$			7.75736i			0.635508i	−0.948173	+	0.317754i	$0.897071\pi$
$150$		0			0
$151$	16.1421			1.31363			0.656814	−	0.754052i	$-0.271904\pi$
$151$	16.1421			1.31363			0.656814	+	0.754052i	$0.271904\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		−	38.6274i		−	3.10263i
$156$		0			0
$157$			19.1116i			1.52528i	0.646826	+	0.762638i	$0.276096\pi$
$157$			19.1116i			1.52528i	−0.646826	+	0.762638i	$0.723904\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−6.75699	−	15.0711i	−0.532525	−	1.18777i
$162$		0			0
$163$	17.6569			1.38299			0.691496	−	0.722380i	$-0.256952\pi$
$163$	17.6569			1.38299			0.691496	+	0.722380i	$0.256952\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	11.7206			0.906968			0.453484	−	0.891265i	$-0.350181\pi$
$167$	11.7206			0.906968			0.453484	+	0.891265i	$0.350181\pi$
$168$		0			0
$169$	−14.3137			−1.10105
$170$		0			0
$171$		0			0
$172$		0			0
$173$	12.3547			0.939307			0.469654	−	0.882851i	$-0.344379\pi$
$173$	12.3547			0.939307			0.469654	+	0.882851i	$0.344379\pi$
$174$		0			0
$175$	20.8995	−	9.37011i	1.57985	−	0.708314i
$176$		0			0
$177$		0			0
$178$		0			0
$179$			15.8995i			1.18838i	0.804323	+	0.594192i	$0.202528\pi$
$179$			15.8995i			1.18838i	−0.804323	+	0.594192i	$0.797472\pi$
$180$		0			0
$181$			24.3379i			1.80902i	0.426451	+	0.904511i	$0.359764\pi$
$181$			24.3379i			1.80902i	−0.426451	+	0.904511i	$0.640236\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	14.7821			1.08680
$186$		0			0
$187$		−	23.0698i		−	1.68703i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		−	2.72792i		−	0.197386i	−0.995118	−	0.0986928i	$-0.968534\pi$
$191$		−	2.72792i		−	0.197386i	0.995118	−	0.0986928i	$-0.0314661\pi$
$192$		0			0
$193$	5.65685			0.407189			0.203595	−	0.979055i	$-0.434738\pi$
$193$	5.65685			0.407189			0.203595	+	0.979055i	$0.434738\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			22.3848i			1.59485i	0.603419	+	0.797425i	$0.293805\pi$
$197$			22.3848i			1.59485i	−0.603419	+	0.797425i	$0.706195\pi$
$198$		0			0
$199$			23.0698i			1.63537i	0.575663	+	0.817687i	$0.304744\pi$
$199$			23.0698i			1.63537i	−0.575663	+	0.817687i	$0.695256\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−2.79884	−	6.24264i	−0.196440	−	0.438147i
$204$		0			0
$205$	−2.34315			−0.163652
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−7.39104			−0.511249
$210$		0			0
$211$	−19.1716			−1.31983			−0.659913	−	0.751342i	$-0.729407\pi$
$211$	−19.1716			−1.31983			−0.659913	+	0.751342i	$0.729407\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−23.9665			−1.63450
$216$		0			0
$217$	11.3137	+	25.2346i	0.768025	+	1.71303i
$218$		0			0
$219$		0			0
$220$		0			0
$221$			35.3137i			2.37546i
$222$		0			0
$223$			23.0698i			1.54487i	0.635095	+	0.772434i	$0.280961\pi$
$223$			23.0698i			1.54487i	−0.635095	+	0.772434i	$0.719039\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	14.7821			0.981121			0.490560	−	0.871407i	$-0.336792\pi$
$227$	14.7821			0.981121			0.490560	+	0.871407i	$0.336792\pi$
$228$		0			0
$229$			3.43289i			0.226851i	0.993546	+	0.113426i	$0.0361824\pi$
$229$			3.43289i			0.226851i	−0.993546	+	0.113426i	$0.963818\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			20.7279i			1.35793i	0.734170	+	0.678966i	$0.237571\pi$
$233$			20.7279i			1.35793i	−0.734170	+	0.678966i	$0.762429\pi$
$234$		0			0
$235$	−11.3137			−0.738025
$236$		0			0
$237$		0			0
$238$		0			0
$239$			2.92893i			0.189457i	0.995503	+	0.0947284i	$0.0301983\pi$
$239$			2.92893i			0.189457i	−0.995503	+	0.0947284i	$0.969802\pi$
$240$		0			0
$241$			11.3492i			0.731065i	0.930799	+	0.365533i	$0.119113\pi$
$241$			11.3492i			0.731065i	−0.930799	+	0.365533i	$0.880887\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−17.2095	+	19.3137i	−1.09947	+	1.23391i
$246$		0			0
$247$	11.3137			0.719874
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−5.59767			−0.353322			−0.176661	−	0.984272i	$-0.556530\pi$
$251$	−5.59767			−0.353322			−0.176661	+	0.984272i	$0.556530\pi$
$252$		0			0
$253$	21.3137			1.33998
$254$		0			0
$255$		0			0
$256$		0			0
$257$	22.8072			1.42267			0.711336	−	0.702852i	$-0.248091\pi$
$257$	22.8072			1.42267			0.711336	+	0.702852i	$0.248091\pi$
$258$		0			0
$259$	−9.65685	+	4.32957i	−0.600048	+	0.269026i
$260$		0			0
$261$		0			0
$262$		0			0
$263$			18.7279i			1.15481i	0.816457	+	0.577407i	$0.195935\pi$
$263$			18.7279i			1.15481i	−0.816457	+	0.577407i	$0.804065\pi$
$264$		0			0
$265$		−	9.55582i		−	0.587009i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−11.0866			−0.675959			−0.337980	−	0.941153i	$-0.609743\pi$
$269$	−11.0866			−0.675959			−0.337980	+	0.941153i	$0.609743\pi$
$270$		0			0
$271$		−	19.1116i		−	1.16095i	−0.814278	−	0.580475i	$-0.802867\pi$
$271$		−	19.1116i		−	1.16095i	0.814278	−	0.580475i	$-0.197133\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$			29.5563i			1.78231i
$276$		0			0
$277$	−19.6569			−1.18107			−0.590533	−	0.807014i	$-0.701082\pi$
$277$	−19.6569			−1.18107			−0.590533	+	0.807014i	$0.701082\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	9.89949i		−	0.590554i	−0.955412	−	0.295277i	$-0.904588\pi$
$281$		−	9.89949i		−	0.590554i	0.955412	−	0.295277i	$-0.0954120\pi$
$282$		0			0
$283$			14.4107i			0.856624i	0.903631	+	0.428312i	$0.140892\pi$
$283$			14.4107i			0.856624i	−0.903631	+	0.428312i	$0.859108\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	1.53073	−	0.686292i	0.0903564	−	0.0405105i
$288$		0			0
$289$	28.6569			1.68570
$290$		0			0
$291$		0			0
$292$		0			0
$293$	25.8686			1.51126			0.755631	−	0.654998i	$-0.227331\pi$
$293$	25.8686			1.51126			0.755631	+	0.654998i	$0.227331\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−32.6256			−1.88679
$300$		0			0
$301$	15.6569	−	7.01962i	0.902446	−	0.404604i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$			10.8239i			0.617754i	0.951102	+	0.308877i	$0.0999531\pi$
$307$			10.8239i			0.617754i	−0.951102	+	0.308877i	$0.900047\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−12.2459			−0.694400			−0.347200	−	0.937791i	$-0.612867\pi$
$311$	−12.2459			−0.694400			−0.347200	+	0.937791i	$0.612867\pi$
$312$		0			0
$313$		−	20.9050i		−	1.18162i	−0.806810	−	0.590810i	$-0.798808\pi$
$313$		−	20.9050i		−	1.18162i	0.806810	−	0.590810i	$-0.201192\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	22.3848i		−	1.25725i	−0.777707	−	0.628627i	$-0.783617\pi$
$317$		−	22.3848i		−	1.25725i	0.777707	−	0.628627i	$-0.216383\pi$
$318$		0			0
$319$	8.82843			0.494297
$320$		0			0
$321$		0			0
$322$		0			0
$323$		−	14.6274i		−	0.813891i
$324$		0			0
$325$		−	45.2429i		−	2.50962i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	7.39104	−	3.31371i	0.407481	−	0.182691i
$330$		0			0
$331$	−4.82843			−0.265394			−0.132697	−	0.991157i	$-0.542364\pi$
$331$	−4.82843			−0.265394			−0.132697	+	0.991157i	$0.542364\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	6.12293			0.334532
$336$		0			0
$337$	−8.00000			−0.435788			−0.217894	−	0.975972i	$-0.569919\pi$
$337$	−8.00000			−0.435788			−0.217894	+	0.975972i	$0.569919\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−35.6871			−1.93256
$342$		0			0
$343$	5.58579	−	17.6578i	0.301604	−	0.953433i
$344$		0			0
$345$		0			0
$346$		0			0
$347$			13.5563i			0.727743i	0.931449	+	0.363871i	$0.118545\pi$
$347$			13.5563i			0.727743i	−0.931449	+	0.363871i	$0.881455\pi$
$348$		0			0
$349$		−	19.1116i		−	1.02302i	−0.859277	−	0.511511i	$-0.829086\pi$
$349$		−	19.1116i		−	1.02302i	0.859277	−	0.511511i	$-0.170914\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−21.5391			−1.14641			−0.573204	−	0.819413i	$-0.694300\pi$
$353$	−21.5391			−1.14641			−0.573204	+	0.819413i	$0.694300\pi$
$354$		0			0
$355$		−	27.3994i		−	1.45421i
$356$		0			0
$357$		0			0
$358$		0			0
$359$		−	8.38478i		−	0.442532i	−0.975214	−	0.221266i	$-0.928981\pi$
$359$		−	8.38478i		−	0.442532i	0.975214	−	0.221266i	$-0.0710188\pi$
$360$		0			0
$361$	14.3137			0.753353
$362$		0			0
$363$		0			0
$364$		0			0
$365$			3.31371i			0.173447i
$366$		0			0
$367$		−	6.49435i		−	0.339002i	−0.985530	−	0.169501i	$-0.945784\pi$
$367$		−	6.49435i		−	0.339002i	0.985530	−	0.169501i	$-0.0542157\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	2.79884	+	6.24264i	0.145308	+	0.324102i
$372$		0			0
$373$	17.3137			0.896470			0.448235	−	0.893916i	$-0.352053\pi$
$373$	17.3137			0.896470			0.448235	+	0.893916i	$0.352053\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−13.5140			−0.696005
$378$		0			0
$379$	−11.1716			−0.573845			−0.286923	−	0.957954i	$-0.592632\pi$
$379$	−11.1716			−0.573845			−0.286923	+	0.957954i	$0.592632\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−5.59767			−0.286028			−0.143014	−	0.989721i	$-0.545679\pi$
$383$	−5.59767			−0.286028			−0.143014	+	0.989721i	$0.545679\pi$
$384$		0			0
$385$	−13.6569	−	30.4608i	−0.696018	−	1.55243i
$386$		0			0
$387$		0			0
$388$		0			0
$389$		−	24.7279i		−	1.25376i	−0.779118	−	0.626878i	$-0.784333\pi$
$389$		−	24.7279i		−	1.25376i	0.779118	−	0.626878i	$-0.215667\pi$
$390$		0			0
$391$			42.1814i			2.13321i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	50.4692			2.53938
$396$		0			0
$397$			20.9050i			1.04919i	0.851351	+	0.524596i	$0.175784\pi$
$397$			20.9050i			1.04919i	−0.851351	+	0.524596i	$0.824216\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		−	0.443651i		−	0.0221549i	−0.999939	−	0.0110774i	$-0.996474\pi$
$401$		−	0.443651i		−	0.0221549i	0.999939	−	0.0110774i	$-0.00352613\pi$
$402$		0			0
$403$	54.6274			2.72119
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	13.6569i		−	0.676945i
$408$		0			0
$409$		−	0.896683i		−	0.0443381i	−0.999754	−	0.0221691i	$-0.992943\pi$
$409$		−	0.896683i		−	0.0443381i	0.999754	−	0.0221691i	$-0.00705721\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$	−43.3137			−2.12619
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−20.9050			−1.02128			−0.510638	−	0.859796i	$-0.670591\pi$
$419$	−20.9050			−1.02128			−0.510638	+	0.859796i	$0.670591\pi$
$420$		0			0
$421$	−18.9706			−0.924569			−0.462284	−	0.886732i	$-0.652970\pi$
$421$	−18.9706			−0.924569			−0.462284	+	0.886732i	$0.652970\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−58.4942			−2.83739
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	1.75736i		−	0.0846490i	−0.999104	−	0.0423245i	$-0.986524\pi$
$431$		−	1.75736i		−	0.0846490i	0.999104	−	0.0423245i	$-0.0134763\pi$
$432$		0			0
$433$		−	22.6984i		−	1.09081i	−0.838171	−	0.545407i	$-0.816375\pi$
$433$		−	22.6984i		−	1.09081i	0.838171	−	0.545407i	$-0.183625\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	13.5140			0.646461
$438$		0			0
$439$		−	23.0698i		−	1.10106i	−0.834815	−	0.550531i	$-0.814425\pi$
$439$		−	23.0698i		−	1.10106i	0.834815	−	0.550531i	$-0.185575\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$		−	11.4142i		−	0.542306i	−0.962536	−	0.271153i	$-0.912595\pi$
$443$		−	11.4142i		−	0.542306i	0.962536	−	0.271153i	$-0.0874049\pi$
$444$		0			0
$445$	24.9706			1.18372
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	6.58579i		−	0.310802i	−0.987851	−	0.155401i	$-0.950333\pi$
$449$		−	6.58579i		−	0.310802i	0.987851	−	0.155401i	$-0.0496670\pi$
$450$		0			0
$451$			2.16478i			0.101936i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	20.9050	+	46.6274i	0.980042	+	2.18593i
$456$		0			0
$457$	11.3137			0.529233			0.264616	−	0.964354i	$-0.414755\pi$
$457$	11.3137			0.529233			0.264616	+	0.964354i	$0.414755\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−25.8686			−1.20482			−0.602411	−	0.798186i	$-0.705793\pi$
$461$	−25.8686			−1.20482			−0.602411	+	0.798186i	$0.705793\pi$
$462$		0			0
$463$	−18.3431			−0.852478			−0.426239	−	0.904611i	$-0.640162\pi$
$463$	−18.3431			−0.852478			−0.426239	+	0.904611i	$0.640162\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−30.0894			−1.39237			−0.696186	−	0.717862i	$-0.745121\pi$
$467$	−30.0894			−1.39237			−0.696186	+	0.717862i	$0.745121\pi$
$468$		0			0
$469$	−4.00000	+	1.79337i	−0.184703	+	0.0828100i
$470$		0			0
$471$		0			0
$472$		0			0
$473$			22.1421i			1.01810i
$474$		0			0
$475$			18.7402i			0.859860i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	27.0279			1.23494			0.617469	−	0.786595i	$-0.288158\pi$
$479$	27.0279			1.23494			0.617469	+	0.786595i	$0.288158\pi$
$480$		0			0
$481$			20.9050i			0.953186i
$482$		0			0
$483$		0			0
$484$		0			0
$485$		−	35.3137i		−	1.60351i
$486$		0			0
$487$	−11.4558			−0.519114			−0.259557	−	0.965728i	$-0.583577\pi$
$487$	−11.4558			−0.519114			−0.259557	+	0.965728i	$0.583577\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		−	38.5269i		−	1.73870i	−0.494201	−	0.869348i	$-0.664539\pi$
$491$		−	38.5269i		−	1.73870i	0.494201	−	0.869348i	$-0.335461\pi$
$492$		0			0
$493$			17.4721i			0.786905i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	8.02509	+	17.8995i	0.359974	+	0.802902i
$498$		0			0
$499$	12.8284			0.574279			0.287140	−	0.957889i	$-0.407296\pi$
$499$	12.8284			0.574279			0.287140	+	0.957889i	$0.407296\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−26.5027			−1.18170			−0.590848	−	0.806783i	$-0.701207\pi$
$503$	−26.5027			−1.18170			−0.590848	+	0.806783i	$0.701207\pi$
$504$		0			0
$505$	8.97056			0.399185
$506$		0			0
$507$		0			0
$508$		0			0
$509$	17.2095			0.762797			0.381399	−	0.924411i	$-0.375443\pi$
$509$	17.2095			0.762797			0.381399	+	0.924411i	$0.375443\pi$
$510$		0			0
$511$	−0.970563	−	2.16478i	−0.0429352	−	0.0957644i
$512$		0			0
$513$		0			0
$514$		0			0
$515$			22.6274i			0.997083i
$516$		0			0
$517$			10.4525i			0.459701i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	6.75699			0.296029			0.148014	−	0.988985i	$-0.452712\pi$
$521$	6.75699			0.296029			0.148014	+	0.988985i	$0.452712\pi$
$522$		0			0
$523$			20.9050i			0.914112i	0.889438	+	0.457056i	$0.151096\pi$
$523$			20.9050i			0.914112i	−0.889438	+	0.457056i	$0.848904\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		−	70.6274i		−	3.07658i
$528$		0			0
$529$	−15.9706			−0.694372
$530$		0			0
$531$		0			0
$532$		0			0
$533$		−	3.31371i		−	0.143533i
$534$		0			0
$535$		−	3.95815i		−	0.171126i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	17.8435	+	15.8995i	0.768576	+	0.684840i
$540$		0			0
$541$	−27.6569			−1.18906			−0.594531	−	0.804073i	$-0.702662\pi$
$541$	−27.6569			−1.18906			−0.594531	+	0.804073i	$0.702662\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	6.12293			0.262278
$546$		0			0
$547$	−17.6569			−0.754953			−0.377476	−	0.926019i	$-0.623208\pi$
$547$	−17.6569			−0.754953			−0.377476	+	0.926019i	$0.623208\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	5.59767			0.238469
$552$		0			0
$553$	−32.9706	+	14.7821i	−1.40205	+	0.628598i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		−	45.8995i		−	1.94482i	−0.233270	−	0.972412i	$-0.574943\pi$
$557$		−	45.8995i		−	1.94482i	0.233270	−	0.972412i	$-0.425057\pi$
$558$		0			0
$559$		−	33.8937i		−	1.43355i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−36.2123			−1.52617			−0.763084	−	0.646299i	$-0.776316\pi$
$563$	−36.2123			−1.52617			−0.763084	+	0.646299i	$0.776316\pi$
$564$		0			0
$565$		−	43.4495i		−	1.82794i
$566$		0			0
$567$		0			0
$568$		0			0
$569$		−	4.24264i		−	0.177861i	−0.996038	−	0.0889304i	$-0.971655\pi$
$569$		−	4.24264i		−	0.177861i	0.996038	−	0.0889304i	$-0.0283449\pi$
$570$		0			0
$571$	−12.9706			−0.542801			−0.271401	−	0.962466i	$-0.587487\pi$
$571$	−12.9706			−0.542801			−0.271401	+	0.962466i	$0.587487\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		−	54.0416i		−	2.25369i
$576$		0			0
$577$		−	10.4525i		−	0.435143i	−0.976044	−	0.217572i	$-0.930186\pi$
$577$		−	10.4525i		−	0.435143i	0.976044	−	0.217572i	$-0.0698136\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	28.2960	−	12.6863i	1.17392	−	0.526316i
$582$		0			0
$583$	−8.82843			−0.365636
$584$		0			0
$585$		0			0
$586$		0			0
$587$	32.6256			1.34660			0.673302	−	0.739368i	$-0.264876\pi$
$587$	32.6256			1.34660			0.673302	+	0.739368i	$0.264876\pi$
$588$		0			0
$589$	−22.6274			−0.932346
$590$		0			0
$591$		0			0
$592$		0			0
$593$	28.9301			1.18802			0.594008	−	0.804459i	$-0.297545\pi$
$593$	28.9301			1.18802			0.594008	+	0.804459i	$0.297545\pi$
$594$		0			0
$595$	60.2843	−	27.0279i	2.47141	−	1.10804i
$596$		0			0
$597$		0			0
$598$		0			0
$599$			31.2132i			1.27534i	0.770311	+	0.637668i	$0.220101\pi$
$599$			31.2132i			1.27534i	−0.770311	+	0.637668i	$0.779899\pi$
$600$		0			0
$601$			22.6984i			0.925886i	0.886388	+	0.462943i	$0.153207\pi$
$601$			22.6984i			0.925886i	−0.886388	+	0.462943i	$0.846793\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	2.42742			0.0986885
$606$		0			0
$607$		−	2.16478i		−	0.0878659i	−0.999034	−	0.0439329i	$-0.986011\pi$
$607$		−	2.16478i		−	0.0878659i	0.999034	−	0.0439329i	$-0.0139888\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		−	16.0000i		−	0.647291i
$612$		0			0
$613$	21.3137			0.860853			0.430426	−	0.902626i	$-0.358363\pi$
$613$	21.3137			0.860853			0.430426	+	0.902626i	$0.358363\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			3.75736i			0.151266i	0.997136	+	0.0756328i	$0.0240977\pi$
$617$			3.75736i			0.151266i	−0.997136	+	0.0756328i	$0.975902\pi$
$618$		0			0
$619$		−	16.5754i		−	0.666223i	−0.942887	−	0.333112i	$-0.891901\pi$
$619$		−	16.5754i		−	0.666223i	0.942887	−	0.333112i	$-0.108099\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−16.3128	+	7.31371i	−0.653559	+	0.293018i
$624$		0			0
$625$	6.65685			0.266274
$626$		0			0
$627$		0			0
$628$		0			0
$629$	27.0279			1.07767
$630$		0			0
$631$	39.5980			1.57637			0.788185	−	0.615438i	$-0.211021\pi$
$631$	39.5980			1.57637			0.788185	+	0.615438i	$0.211021\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	62.7150			2.48877
$636$		0			0
$637$	−27.3137	−	24.3379i	−1.08221	−	0.964302i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		−	34.8701i		−	1.37728i	−0.725101	−	0.688642i	$-0.758207\pi$
$641$		−	34.8701i		−	1.37728i	0.725101	−	0.688642i	$-0.241793\pi$
$642$		0			0
$643$			18.7402i			0.739042i	0.929222	+	0.369521i	$0.120478\pi$
$643$			18.7402i			0.739042i	−0.929222	+	0.369521i	$0.879522\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	50.4692			1.98415			0.992074	−	0.125658i	$-0.0401043\pi$
$647$	50.4692			1.98415			0.992074	+	0.125658i	$0.0401043\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$		0			0
$653$		−	1.21320i		−	0.0474763i	−0.999718	−	0.0237382i	$-0.992443\pi$
$653$		−	1.21320i		−	0.0474763i	0.999718	−	0.0237382i	$-0.00755680\pi$
$654$		0			0
$655$	−11.3137			−0.442063
$656$		0			0
$657$		0			0
$658$		0			0
$659$			44.5858i			1.73682i	0.495851	+	0.868408i	$0.334856\pi$
$659$			44.5858i			1.73682i	−0.495851	+	0.868408i	$0.665144\pi$
$660$		0			0
$661$			27.7708i			1.08016i	0.841614	+	0.540079i	$0.181606\pi$
$661$			27.7708i			1.08016i	−0.841614	+	0.540079i	$0.818394\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−8.65914	−	19.3137i	−0.335787	−	0.748953i
$666$		0			0
$667$	−16.1421			−0.625026
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−2.34315			−0.0903216			−0.0451608	−	0.998980i	$-0.514380\pi$
$673$	−2.34315			−0.0903216			−0.0451608	+	0.998980i	$0.514380\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	14.6733			0.563940			0.281970	−	0.959423i	$-0.409012\pi$
$677$	14.6733			0.563940			0.281970	+	0.959423i	$0.409012\pi$
$678$		0			0
$679$	10.3431	+	23.0698i	0.396934	+	0.885337i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		−	9.27208i		−	0.354786i	−0.984140	−	0.177393i	$-0.943234\pi$
$683$		−	9.27208i		−	0.354786i	0.984140	−	0.177393i	$-0.0567664\pi$
$684$		0			0
$685$		−	36.5838i		−	1.39779i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	13.5140			0.514841
$690$		0			0
$691$			12.9887i			0.494114i	0.969001	+	0.247057i	$0.0794634\pi$
$691$			12.9887i			0.494114i	−0.969001	+	0.247057i	$0.920537\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$			16.0000i			0.606915i
$696$		0			0
$697$	−4.28427			−0.162278
$698$		0			0
$699$		0			0
$700$		0			0
$701$			43.0711i			1.62677i	0.581725	+	0.813386i	$0.302378\pi$
$701$			43.0711i			1.62677i	−0.581725	+	0.813386i	$0.697622\pi$
$702$		0			0
$703$		−	8.65914i		−	0.326586i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−5.86030	+	2.62742i	−0.220399	+	0.0988142i
$708$		0			0
$709$	−32.2843			−1.21246			−0.606231	−	0.795289i	$-0.707319\pi$
$709$	−32.2843			−1.21246			−0.606231	+	0.795289i	$0.707319\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	65.2512			2.44368
$714$		0			0
$715$	−65.9411			−2.46606
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−8.65914			−0.322931			−0.161466	−	0.986878i	$-0.551622\pi$
$719$	−8.65914			−0.322931			−0.161466	+	0.986878i	$0.551622\pi$
$720$		0			0
$721$	−6.62742	−	14.7821i	−0.246818	−	0.550513i
$722$		0			0
$723$		0			0
$724$		0			0
$725$		−	22.3848i		−	0.831350i
$726$		0			0
$727$		−	14.7821i		−	0.548237i	−0.961696	−	0.274118i	$-0.911614\pi$
$727$		−	14.7821i		−	0.548237i	0.961696	−	0.274118i	$-0.0883860\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−43.8210			−1.62078
$732$		0			0
$733$			15.6788i			0.579108i	0.957162	+	0.289554i	$0.0935070\pi$
$733$			15.6788i			0.579108i	−0.957162	+	0.289554i	$0.906493\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	5.65685i		−	0.208373i
$738$		0			0
$739$	16.2843			0.599027			0.299513	−	0.954092i	$-0.403176\pi$
$739$	16.2843			0.599027			0.299513	+	0.954092i	$0.403176\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	11.6985i		−	0.429176i	−0.976705	−	0.214588i	$-0.931159\pi$
$743$		−	11.6985i		−	0.429176i	0.976705	−	0.214588i	$-0.0688408\pi$
$744$		0			0
$745$		−	28.6675i		−	1.05029i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	1.15932	+	2.58579i	0.0423605	+	0.0944826i
$750$		0			0
$751$	30.4853			1.11242			0.556212	−	0.831041i	$-0.312254\pi$
$751$	30.4853			1.11242			0.556212	+	0.831041i	$0.312254\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−59.6536			−2.17102
$756$		0			0
$757$	−51.5980			−1.87536			−0.937680	−	0.347499i	$-0.887031\pi$
$757$	−51.5980			−1.87536			−0.937680	+	0.347499i	$0.887031\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	33.7849			1.22470			0.612351	−	0.790586i	$-0.290224\pi$
$761$	33.7849			1.22470			0.612351	+	0.790586i	$0.290224\pi$
$762$		0			0
$763$	−4.00000	+	1.79337i	−0.144810	+	0.0649242i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$			22.6984i			0.818524i	0.912417	+	0.409262i	$0.134214\pi$
$769$			22.6984i			0.818524i	−0.912417	+	0.409262i	$0.865786\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−2.42742			−0.0873081			−0.0436541	−	0.999047i	$-0.513900\pi$
$773$	−2.42742			−0.0873081			−0.0436541	+	0.999047i	$0.513900\pi$
$774$		0			0
$775$			90.4858i			3.25035i
$776$		0			0
$777$		0			0
$778$		0			0
$779$			1.37258i			0.0491779i
$780$		0			0
$781$	−25.3137			−0.905796
$782$		0			0
$783$		0			0
$784$		0			0
$785$		−	70.6274i		−	2.52080i
$786$		0			0
$787$		−	33.1509i		−	1.18170i	−0.806781	−	0.590851i	$-0.798792\pi$
$787$		−	33.1509i		−	1.18170i	0.806781	−	0.590851i	$-0.201208\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	12.7261	+	28.3848i	0.452487	+	1.00925i
$792$		0			0
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−12.3547			−0.437624			−0.218812	−	0.975767i	$-0.570218\pi$
$797$	−12.3547			−0.437624			−0.218812	+	0.975767i	$0.570218\pi$
$798$		0			0
$799$	−20.6863			−0.731828
$800$		0			0
$801$		0			0
$802$		0			0
$803$	3.06147			0.108037
$804$		0			0
$805$	24.9706	+	55.6954i	0.880097	+	1.96301i
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	7.07107i		−	0.248606i	−0.992244	−	0.124303i	$-0.960331\pi$
$809$		−	7.07107i		−	0.248606i	0.992244	−	0.124303i	$-0.0396694\pi$
$810$		0			0
$811$		−	29.5641i		−	1.03814i	−0.854732	−	0.519069i	$-0.826279\pi$
$811$		−	29.5641i		−	1.03814i	0.854732	−	0.519069i	$-0.173721\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−65.2512			−2.28565
$816$		0			0
$817$			14.0392i			0.491171i
$818$		0			0
$819$		0			0
$820$		0			0
$821$			31.3553i			1.09431i	0.837032	+	0.547154i	$0.184289\pi$
$821$			31.3553i			1.09431i	−0.837032	+	0.547154i	$0.815711\pi$
$822$		0			0
$823$	10.3431			0.360539			0.180270	−	0.983617i	$-0.442303\pi$
$823$	10.3431			0.360539			0.180270	+	0.983617i	$0.442303\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	27.0122i		−	0.939306i	−0.882851	−	0.469653i	$-0.844379\pi$
$827$		−	27.0122i		−	0.939306i	0.882851	−	0.469653i	$-0.155621\pi$
$828$		0			0
$829$			26.1313i			0.907576i	0.891110	+	0.453788i	$0.149928\pi$
$829$			26.1313i			0.907576i	−0.891110	+	0.453788i	$0.850072\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−31.4663	+	35.3137i	−1.09024	+	1.22355i
$834$		0			0
$835$	−43.3137			−1.49893
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−21.4303			−0.739855			−0.369928	−	0.929061i	$-0.620618\pi$
$839$	−21.4303			−0.739855			−0.369928	+	0.929061i	$0.620618\pi$
$840$		0			0
$841$	22.3137			0.769438
$842$		0			0
$843$		0			0
$844$		0			0
$845$	52.8966			1.81970
$846$		0			0
$847$	−1.58579	+	0.710974i	−0.0544883	+	0.0244294i
$848$		0			0
$849$		0			0
$850$		0			0
$851$			24.9706i			0.855980i
$852$		0			0
$853$			12.2459i			0.419291i	0.977778	+	0.209645i	$0.0672309\pi$
$853$			12.2459i			0.419291i	−0.977778	+	0.209645i	$0.932769\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	6.75699			0.230814			0.115407	−	0.993318i	$-0.463183\pi$
$857$	6.75699			0.230814			0.115407	+	0.993318i	$0.463183\pi$
$858$		0			0
$859$			18.7402i			0.639408i	0.947517	+	0.319704i	$0.103583\pi$
$859$			18.7402i			0.639408i	−0.947517	+	0.319704i	$0.896417\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			23.4142i			0.797029i	0.917162	+	0.398515i	$0.130474\pi$
$863$			23.4142i			0.797029i	−0.917162	+	0.398515i	$0.869526\pi$
$864$		0			0
$865$	−45.6569			−1.55238
$866$		0			0
$867$		0			0
$868$		0			0
$869$		−	46.6274i		−	1.58173i
$870$		0			0
$871$			8.65914i			0.293404i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−32.6256	+	14.6274i	−1.10295	+	0.494497i
$876$		0			0
$877$	14.0000			0.472746			0.236373	−	0.971662i	$-0.424041\pi$
$877$	14.0000			0.472746			0.236373	+	0.971662i	$0.424041\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−31.4663			−1.06013			−0.530063	−	0.847958i	$-0.677832\pi$
$881$	−31.4663			−1.06013			−0.530063	+	0.847958i	$0.677832\pi$
$882$		0			0
$883$	−24.1421			−0.812448			−0.406224	−	0.913774i	$-0.633155\pi$
$883$	−24.1421			−0.812448			−0.406224	+	0.913774i	$0.633155\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	23.9665			0.804715			0.402358	−	0.915483i	$-0.368191\pi$
$887$	23.9665			0.804715			0.402358	+	0.915483i	$0.368191\pi$
$888$		0			0
$889$	−40.9706	+	18.3688i	−1.37411	+	0.616070i
$890$		0			0
$891$		0			0
$892$		0			0
$893$			6.62742i			0.221778i
$894$		0			0
$895$		−	58.7569i		−	1.96403i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	27.0279			0.901432
$900$		0			0
$901$		−	17.4721i		−	0.582081i
$902$		0			0
$903$		0			0
$904$		0			0
$905$		−	89.9411i		−	2.98974i
$906$		0			0
$907$	49.7990			1.65355			0.826774	−	0.562534i	$-0.190173\pi$
$907$	49.7990			1.65355			0.826774	+	0.562534i	$0.190173\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	56.1838i		−	1.86145i	−0.365718	−	0.930726i	$-0.619177\pi$
$911$		−	56.1838i		−	1.86145i	0.365718	−	0.930726i	$-0.380823\pi$
$912$		0			0
$913$			40.0166i			1.32436i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	7.39104	−	3.31371i	0.244074	−	0.109428i
$918$		0			0
$919$	1.79899			0.0593432			0.0296716	−	0.999560i	$-0.490554\pi$
$919$	1.79899			0.0593432			0.0296716	+	0.999560i	$0.490554\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	38.7485			1.27542
$924$		0			0
$925$	−34.6274			−1.13854
$926$		0			0
$927$		0			0
$928$		0			0
$929$	55.9580			1.83592			0.917962	−	0.396669i	$-0.129834\pi$
$929$	55.9580			1.83592			0.917962	+	0.396669i	$0.129834\pi$
$930$		0			0
$931$	11.3137	+	10.0811i	0.370792	+	0.330394i
$932$		0			0
$933$		0			0
$934$		0			0
$935$			85.2548i			2.78813i
$936$		0			0
$937$			50.4692i			1.64876i	0.566040	+	0.824378i	$0.308475\pi$
$937$			50.4692i			1.64876i	−0.566040	+	0.824378i	$0.691525\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	12.3547			0.402750			0.201375	−	0.979514i	$-0.435459\pi$
$941$	12.3547			0.402750			0.201375	+	0.979514i	$0.435459\pi$
$942$		0			0
$943$		−	3.95815i		−	0.128895i
$944$		0			0
$945$		0			0
$946$		0			0
$947$			27.4142i			0.890842i	0.895321	+	0.445421i	$0.146946\pi$
$947$			27.4142i			0.890842i	−0.895321	+	0.445421i	$0.853054\pi$
$948$		0			0
$949$	−4.68629			−0.152123
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	10.7868i		−	0.349419i	−0.984620	−	0.174709i	$-0.944101\pi$
$953$		−	10.7868i		−	0.349419i	0.984620	−	0.174709i	$-0.0558986\pi$
$954$		0			0
$955$			10.0811i			0.326216i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	10.7151	+	23.8995i	0.346010	+	0.771755i
$960$		0			0
$961$	−78.2548			−2.52435
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−20.9050			−0.672956
$966$		0			0
$967$	−18.3431			−0.589876			−0.294938	−	0.955516i	$-0.595299\pi$
$967$	−18.3431			−0.589876			−0.294938	+	0.955516i	$0.595299\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	27.0279			0.867368			0.433684	−	0.901065i	$-0.357213\pi$
$971$	27.0279			0.867368			0.433684	+	0.901065i	$0.357213\pi$
$972$		0			0
$973$	−4.68629	−	10.4525i	−0.150236	−	0.335092i
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	1.89949i		−	0.0607702i	−0.999538	−	0.0303851i	$-0.990327\pi$
$977$		−	1.89949i		−	0.0607702i	0.999538	−	0.0303851i	$-0.00967337\pi$
$978$		0			0
$979$		−	23.0698i		−	0.737314i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−59.6536			−1.90265			−0.951326	−	0.308185i	$-0.900278\pi$
$983$	−59.6536			−1.90265			−0.951326	+	0.308185i	$0.900278\pi$
$984$		0			0
$985$		−	82.7233i		−	2.63578i
$986$		0			0
$987$		0			0
$988$		0			0
$989$		−	40.4853i		−	1.28736i
$990$		0			0
$991$	56.1421			1.78341			0.891707	−	0.452613i	$-0.149508\pi$
$991$	56.1421			1.78341			0.891707	+	0.452613i	$0.149508\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		−	85.2548i		−	2.70276i
$996$		0			0
$997$		−	46.8824i		−	1.48478i	−0.669967	−	0.742391i	$-0.733692\pi$
$997$		−	46.8824i		−	1.48478i	0.669967	−	0.742391i	$-0.266308\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4032.2.k.f.3905.1		8
3.2	odd	2	inner	4032.2.k.f.3905.7		8
4.3	odd	2		4032.2.k.e.3905.2		8
7.6	odd	2	inner	4032.2.k.f.3905.8		8
8.3	odd	2		1008.2.k.c.881.8		8
8.5	even	2		504.2.k.a.377.7	yes	8
12.11	even	2		4032.2.k.e.3905.8		8
21.20	even	2	inner	4032.2.k.f.3905.2		8
24.5	odd	2		504.2.k.a.377.1	✓	8
24.11	even	2		1008.2.k.c.881.2		8
28.27	even	2		4032.2.k.e.3905.7		8
56.5	odd	6		3528.2.bl.b.521.8		16
56.13	odd	2		504.2.k.a.377.2	yes	8
56.27	even	2		1008.2.k.c.881.1		8
56.37	even	6		3528.2.bl.b.521.2		16
56.45	odd	6		3528.2.bl.b.1097.7		16
56.53	even	6		3528.2.bl.b.1097.1		16
84.83	odd	2		4032.2.k.e.3905.1		8
168.5	even	6		3528.2.bl.b.521.1		16
168.53	odd	6		3528.2.bl.b.1097.8		16
168.83	odd	2		1008.2.k.c.881.7		8
168.101	even	6		3528.2.bl.b.1097.2		16
168.125	even	2		504.2.k.a.377.8	yes	8
168.149	odd	6		3528.2.bl.b.521.7		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
504.2.k.a.377.1	✓	8	24.5	odd	2
504.2.k.a.377.2	yes	8	56.13	odd	2
504.2.k.a.377.7	yes	8	8.5	even	2
504.2.k.a.377.8	yes	8	168.125	even	2
1008.2.k.c.881.1		8	56.27	even	2
1008.2.k.c.881.2		8	24.11	even	2
1008.2.k.c.881.7		8	168.83	odd	2
1008.2.k.c.881.8		8	8.3	odd	2
3528.2.bl.b.521.1		16	168.5	even	6
3528.2.bl.b.521.2		16	56.37	even	6
3528.2.bl.b.521.7		16	168.149	odd	6
3528.2.bl.b.521.8		16	56.5	odd	6
3528.2.bl.b.1097.1		16	56.53	even	6
3528.2.bl.b.1097.2		16	168.101	even	6
3528.2.bl.b.1097.7		16	56.45	odd	6
3528.2.bl.b.1097.8		16	168.53	odd	6
4032.2.k.e.3905.1		8	84.83	odd	2
4032.2.k.e.3905.2		8	4.3	odd	2
4032.2.k.e.3905.7		8	28.27	even	2
4032.2.k.e.3905.8		8	12.11	even	2
4032.2.k.f.3905.1		8	1.1	even	1	trivial
4032.2.k.f.3905.2		8	21.20	even	2	inner
4032.2.k.f.3905.7		8	3.2	odd	2	inner
4032.2.k.f.3905.8		8	7.6	odd	2	inner

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:	\( N \)	\(=\)	\( 4032 = 2^{6} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4032.k (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-3.69552 q^{5} +(2.41421 - 1.08239i) q^{7} +O(q^{10})\)	\(q-3.69552 q^{5} +(2.41421 - 1.08239i) q^{7} +3.41421i q^{11} -5.22625i q^{13} -6.75699 q^{17} +2.16478i q^{19} -6.24264i q^{23} +8.65685 q^{25} -2.58579i q^{29} +10.4525i q^{31} +(-8.92177 + 4.00000i) q^{35} -4.00000 q^{37} +0.634051 q^{41} +6.48528 q^{43} +3.06147 q^{47} +(4.65685 - 5.22625i) q^{49} +2.58579i q^{53} -12.6173i q^{55} +19.3137i q^{65} -1.65685 q^{67} +7.41421i q^{71} -0.896683i q^{73} +(3.69552 + 8.24264i) q^{77} -13.6569 q^{79} +11.7206 q^{83} +24.9706 q^{85} -6.75699 q^{89} +(-5.65685 - 12.6173i) q^{91} -8.00000i q^{95} +9.55582i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{7}+O(q^{10}) \) 8 * q + 8 * q^7	\( 8 q + 8 q^{7} + 24 q^{25} - 32 q^{37} - 16 q^{43} - 8 q^{49} + 32 q^{67} - 64 q^{79} + 64 q^{85}+O(q^{100}) \) 8 * q + 8 * q^7 + 24 * q^25 - 32 * q^37 - 16 * q^43 - 8 * q^49 + 32 * q^67 - 64 * q^79 + 64 * q^85

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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