LMFDB - Embedded newform 4032.2.k.f.3905.4

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			3905.4
Root			$0.382683 + 0.923880i$ of defining polynomial
Character	$\chi$	$=$	4032.3905
Dual form			4032.2.k.f.3905.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.53073 q^{5} +(-0.414214 + 2.61313i) q^{7} +O(q^{10})$	$q-1.53073 q^{5} +(-0.414214 + 2.61313i) q^{7} -0.585786i q^{11} -2.16478i q^{13} +5.86030 q^{17} -5.22625i q^{19} -2.24264i q^{23} -2.65685 q^{25} +5.41421i q^{29} +4.32957i q^{31} +(0.634051 - 4.00000i) q^{35} -4.00000 q^{37} +8.92177 q^{41} -10.4853 q^{43} -7.39104 q^{47} +(-6.65685 - 2.16478i) q^{49} -5.41421i q^{53} +0.896683i q^{55} +3.31371i q^{65} +9.65685 q^{67} -4.58579i q^{71} -12.6173i q^{73} +(1.53073 + 0.242641i) q^{77} -2.34315 q^{79} +13.5140 q^{83} -8.97056 q^{85} +5.86030 q^{89} +(5.65685 + 0.896683i) q^{91} +8.00000i q^{95} -8.28772i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	−1.53073			−0.684565			−0.342282	−	0.939597i	$-0.611200\pi$
$5$	−1.53073			−0.684565			−0.342282	+	0.939597i	$0.611200\pi$
$6$		0			0
$7$	−0.414214	+	2.61313i	−0.156558	+	0.987669i
$7$	−0.414214	+	2.61313i	−0.156558	+	0.987669i
$8$		0			0
$9$		0			0
$10$		0			0
$11$		−	0.585786i		−	0.176621i	−0.996093	−	0.0883106i	$-0.971853\pi$
$11$		−	0.585786i		−	0.176621i	0.996093	−	0.0883106i	$-0.0281468\pi$
$12$		0			0
$13$		−	2.16478i		−	0.600403i	−0.953876	−	0.300202i	$-0.902946\pi$
$13$		−	2.16478i		−	0.600403i	0.953876	−	0.300202i	$-0.0970540\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	5.86030			1.42133			0.710666	−	0.703529i	$-0.248394\pi$
$17$	5.86030			1.42133			0.710666	+	0.703529i	$0.248394\pi$
$18$		0			0
$19$		−	5.22625i		−	1.19898i	−0.800381	−	0.599492i	$-0.795369\pi$
$19$		−	5.22625i		−	1.19898i	0.800381	−	0.599492i	$-0.204631\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		−	2.24264i		−	0.467623i	−0.972282	−	0.233811i	$-0.924880\pi$
$23$		−	2.24264i		−	0.467623i	0.972282	−	0.233811i	$-0.0751198\pi$
$24$		0			0
$25$	−2.65685			−0.531371
$26$		0			0
$27$		0			0
$28$		0			0
$29$			5.41421i			1.00539i	0.864463	+	0.502697i	$0.167659\pi$
$29$			5.41421i			1.00539i	−0.864463	+	0.502697i	$0.832341\pi$
$30$		0			0
$31$			4.32957i			0.777614i	0.921319	+	0.388807i	$0.127113\pi$
$31$			4.32957i			0.777614i	−0.921319	+	0.388807i	$0.872887\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	0.634051	−	4.00000i	0.107174	−	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
$41$	8.92177			1.39335			0.696673	−	0.717389i	$-0.254663\pi$
$41$	8.92177			1.39335			0.696673	+	0.717389i	$0.254663\pi$
$42$		0			0
$43$	−10.4853			−1.59899			−0.799495	−	0.600672i	$-0.794900\pi$
$43$	−10.4853			−1.59899			−0.799495	+	0.600672i	$0.794900\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−7.39104			−1.07809			−0.539047	−	0.842276i	$-0.681215\pi$
$47$	−7.39104			−1.07809			−0.539047	+	0.842276i	$0.681215\pi$
$48$		0			0
$49$	−6.65685	−	2.16478i	−0.950979	−	0.309255i
$50$		0			0
$51$		0			0
$52$		0			0
$53$		−	5.41421i		−	0.743699i	−0.928293	−	0.371850i	$-0.878724\pi$
$53$		−	5.41421i		−	0.743699i	0.928293	−	0.371850i	$-0.121276\pi$
$54$		0			0
$55$			0.896683i			0.120909i
$56$		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
$65$			3.31371i			0.411015i
$66$		0			0
$67$	9.65685			1.17977			0.589886	−	0.807486i	$-0.299173\pi$
$67$	9.65685			1.17977			0.589886	+	0.807486i	$0.299173\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	4.58579i		−	0.544233i	−0.962264	−	0.272116i	$-0.912276\pi$
$71$		−	4.58579i		−	0.544233i	0.962264	−	0.272116i	$-0.0877236\pi$
$72$		0			0
$73$		−	12.6173i		−	1.47674i	−0.674395	−	0.738371i	$-0.735595\pi$
$73$		−	12.6173i		−	1.47674i	0.674395	−	0.738371i	$-0.264405\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	1.53073	+	0.242641i	0.174443	+	0.0276515i
$78$		0			0
$79$	−2.34315			−0.263624			−0.131812	−	0.991275i	$-0.542080\pi$
$79$	−2.34315			−0.263624			−0.131812	+	0.991275i	$0.542080\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	13.5140			1.48335			0.741676	−	0.670759i	$-0.234031\pi$
$83$	13.5140			1.48335			0.741676	+	0.670759i	$0.234031\pi$
$84$		0			0
$85$	−8.97056			−0.972994
$86$		0			0
$87$		0			0
$88$		0			0
$89$	5.86030			0.621191			0.310595	−	0.950542i	$-0.399472\pi$
$89$	5.86030			0.621191			0.310595	+	0.950542i	$0.399472\pi$
$90$		0			0
$91$	5.65685	+	0.896683i	0.592999	+	0.0939979i
$92$		0			0
$93$		0			0
$94$		0			0
$95$			8.00000i			0.820783i
$96$		0			0
$97$		−	8.28772i		−	0.841490i	−0.907179	−	0.420745i	$-0.861769\pi$
$97$		−	8.28772i		−	0.841490i	0.907179	−	0.420745i	$-0.138231\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	16.3128			1.62318			0.811592	−	0.584224i	$-0.198601\pi$
$101$	16.3128			1.62318			0.811592	+	0.584224i	$0.198601\pi$
$102$		0			0
$103$		−	14.7821i		−	1.45652i	−0.685300	−	0.728260i	$-0.740329\pi$
$103$		−	14.7821i		−	1.45652i	0.685300	−	0.728260i	$-0.259671\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			13.0711i			1.26363i	0.775120	+	0.631814i	$0.217689\pi$
$107$			13.0711i			1.26363i	−0.775120	+	0.631814i	$0.782311\pi$
$108$		0			0
$109$	9.65685			0.924959			0.462479	−	0.886630i	$-0.346960\pi$
$109$	9.65685			0.924959			0.462479	+	0.886630i	$0.346960\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		−	20.2426i		−	1.90427i	−0.305682	−	0.952134i	$-0.598884\pi$
$113$		−	20.2426i		−	1.90427i	0.305682	−	0.952134i	$-0.401116\pi$
$114$		0			0
$115$			3.43289i			0.320118i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−2.42742	+	15.3137i	−0.222521	+	1.40381i
$120$		0			0
$121$	10.6569			0.968805
$122$		0			0
$123$		0			0
$124$		0			0
$125$	11.7206			1.04832
$126$		0			0
$127$	16.9706			1.50589			0.752947	−	0.658081i	$-0.228632\pi$
$127$	16.9706			1.50589			0.752947	+	0.658081i	$0.228632\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−7.39104			−0.645758			−0.322879	−	0.946440i	$-0.604651\pi$
$131$	−7.39104			−0.645758			−0.322879	+	0.946440i	$0.604651\pi$
$132$		0			0
$133$	13.6569	+	2.16478i	1.18420	+	0.187711i
$134$		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
$139$			10.4525i			0.886570i	0.896381	+	0.443285i	$0.146187\pi$
$139$			10.4525i			0.886570i	−0.896381	+	0.443285i	$0.853813\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−1.26810			−0.106044
$144$		0			0
$145$		−	8.28772i		−	0.688258i
$146$		0			0
$147$		0			0
$148$		0			0
$149$		−	16.2426i		−	1.33065i	−0.746554	−	0.665324i	$-0.768293\pi$
$149$		−	16.2426i		−	1.33065i	0.746554	−	0.665324i	$-0.231707\pi$
$150$		0			0
$151$	−12.1421			−0.988113			−0.494056	−	0.869430i	$-0.664486\pi$
$151$	−12.1421			−0.988113			−0.494056	+	0.869430i	$0.664486\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		−	6.62742i		−	0.532327i
$156$		0			0
$157$		−	16.5754i		−	1.32286i	−0.750005	−	0.661432i	$-0.769949\pi$
$157$		−	16.5754i		−	1.32286i	0.750005	−	0.661432i	$-0.230051\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	5.86030	+	0.928932i	0.461857	+	0.0732101i
$162$		0			0
$163$	6.34315			0.496834			0.248417	−	0.968653i	$-0.420090\pi$
$163$	6.34315			0.496834			0.248417	+	0.968653i	$0.420090\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	13.5140			1.04574			0.522871	−	0.852412i	$-0.324861\pi$
$167$	13.5140			1.04574			0.522871	+	0.852412i	$0.324861\pi$
$168$		0			0
$169$	8.31371			0.639516
$170$		0			0
$171$		0			0
$172$		0			0
$173$	22.4357			1.70576			0.852879	−	0.522109i	$-0.174855\pi$
$173$	22.4357			1.70576			0.852879	+	0.522109i	$0.174855\pi$
$174$		0			0
$175$	1.10051	−	6.94269i	0.0831904	−	0.524818i
$176$		0			0
$177$		0			0
$178$		0			0
$179$			3.89949i			0.291462i	0.989324	+	0.145731i	$0.0465534\pi$
$179$			3.89949i			0.291462i	−0.989324	+	0.145731i	$0.953447\pi$
$180$		0			0
$181$		−	14.4107i		−	1.07114i	−0.844492	−	0.535568i	$-0.820098\pi$
$181$		−	14.4107i		−	1.07114i	0.844492	−	0.535568i	$-0.179902\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	6.12293			0.450167
$186$		0			0
$187$		−	3.43289i		−	0.251037i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		−	22.7279i		−	1.64453i	−0.569101	−	0.822267i	$-0.692709\pi$
$191$		−	22.7279i		−	1.64453i	0.569101	−	0.822267i	$-0.307291\pi$
$192$		0			0
$193$	−5.65685			−0.407189			−0.203595	−	0.979055i	$-0.565262\pi$
$193$	−5.65685			−0.407189			−0.203595	+	0.979055i	$0.565262\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			14.3848i			1.02487i	0.858725	+	0.512436i	$0.171257\pi$
$197$			14.3848i			1.02487i	−0.858725	+	0.512436i	$0.828743\pi$
$198$		0			0
$199$			3.43289i			0.243351i	0.992570	+	0.121675i	$0.0388267\pi$
$199$			3.43289i			0.243351i	−0.992570	+	0.121675i	$0.961173\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−14.1480	−	2.24264i	−0.992996	−	0.157403i
$204$		0			0
$205$	−13.6569			−0.953836
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−3.06147			−0.211766
$210$		0			0
$211$	−24.8284			−1.70926			−0.854630	−	0.519238i	$-0.826216\pi$
$211$	−24.8284			−1.70926			−0.854630	+	0.519238i	$0.826216\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	16.0502			1.09461
$216$		0			0
$217$	−11.3137	−	1.79337i	−0.768025	−	0.121742i
$218$		0			0
$219$		0			0
$220$		0			0
$221$		−	12.6863i		−	0.853372i
$222$		0			0
$223$			3.43289i			0.229883i	0.993372	+	0.114942i	$0.0366681\pi$
$223$			3.43289i			0.229883i	−0.993372	+	0.114942i	$0.963332\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	6.12293			0.406393			0.203197	−	0.979138i	$-0.434867\pi$
$227$	6.12293			0.406393			0.203197	+	0.979138i	$0.434867\pi$
$228$		0			0
$229$		−	23.0698i		−	1.52449i	−0.647286	−	0.762247i	$-0.724096\pi$
$229$		−	23.0698i		−	1.52449i	0.647286	−	0.762247i	$-0.275904\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			4.72792i			0.309736i	0.987935	+	0.154868i	$0.0494953\pi$
$233$			4.72792i			0.309736i	−0.987935	+	0.154868i	$0.950505\pi$
$234$		0			0
$235$	11.3137			0.738025
$236$		0			0
$237$		0			0
$238$		0			0
$239$		−	17.0711i		−	1.10424i	−0.833766	−	0.552118i	$-0.813820\pi$
$239$		−	17.0711i		−	1.10424i	0.833766	−	0.552118i	$-0.186180\pi$
$240$		0			0
$241$			16.9469i			1.09164i	0.837901	+	0.545822i	$0.183782\pi$
$241$			16.9469i			1.09164i	−0.837901	+	0.545822i	$0.816218\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	10.1899	+	3.31371i	0.651007	+	0.211705i
$246$		0			0
$247$	−11.3137			−0.719874
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−28.2960			−1.78603			−0.893015	−	0.450026i	$-0.851415\pi$
$251$	−28.2960			−1.78603			−0.893015	+	0.450026i	$0.851415\pi$
$252$		0			0
$253$	−1.31371			−0.0825921
$254$		0			0
$255$		0			0
$256$		0			0
$257$	18.1062			1.12943			0.564716	−	0.825285i	$-0.308986\pi$
$257$	18.1062			1.12943			0.564716	+	0.825285i	$0.308986\pi$
$258$		0			0
$259$	1.65685	−	10.4525i	0.102952	−	0.649487i
$260$		0			0
$261$		0			0
$262$		0			0
$263$			6.72792i			0.414861i	0.978250	+	0.207431i	$0.0665102\pi$
$263$			6.72792i			0.414861i	−0.978250	+	0.207431i	$0.933490\pi$
$264$		0			0
$265$			8.28772i			0.509111i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−4.59220			−0.279991			−0.139996	−	0.990152i	$-0.544709\pi$
$269$	−4.59220			−0.279991			−0.139996	+	0.990152i	$0.544709\pi$
$270$		0			0
$271$			16.5754i			1.00689i	0.864028	+	0.503443i	$0.167934\pi$
$271$			16.5754i			1.00689i	−0.864028	+	0.503443i	$0.832066\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$			1.55635i			0.0938514i
$276$		0			0
$277$	−8.34315			−0.501291			−0.250646	−	0.968079i	$-0.580643\pi$
$277$	−8.34315			−0.501291			−0.250646	+	0.968079i	$0.580643\pi$
$278$		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
$283$			24.3379i			1.44674i	0.690462	+	0.723369i	$0.257407\pi$
$283$			24.3379i			1.44674i	−0.690462	+	0.723369i	$0.742593\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−3.69552	+	23.3137i	−0.218140	+	1.37616i
$288$		0			0
$289$	17.3431			1.02019
$290$		0			0
$291$		0			0
$292$		0			0
$293$	10.7151			0.625985			0.312992	−	0.949756i	$-0.398669\pi$
$293$	10.7151			0.625985			0.312992	+	0.949756i	$0.398669\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−4.85483			−0.280762
$300$		0			0
$301$	4.34315	−	27.3994i	0.250335	−	1.57927i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$		−	26.1313i		−	1.49139i	−0.666288	−	0.745695i	$-0.732118\pi$
$307$		−	26.1313i		−	1.49139i	0.666288	−	0.745695i	$-0.267882\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	29.5641			1.67643			0.838214	−	0.545341i	$-0.183600\pi$
$311$	29.5641			1.67643			0.838214	+	0.545341i	$0.183600\pi$
$312$		0			0
$313$		−	8.65914i		−	0.489443i	−0.969593	−	0.244722i	$-0.921303\pi$
$313$		−	8.65914i		−	0.489443i	0.969593	−	0.244722i	$-0.0786966\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	14.3848i		−	0.807930i	−0.914775	−	0.403965i	$-0.867632\pi$
$317$		−	14.3848i		−	0.807930i	0.914775	−	0.403965i	$-0.132368\pi$
$318$		0			0
$319$	3.17157			0.177574
$320$		0			0
$321$		0			0
$322$		0			0
$323$		−	30.6274i		−	1.70416i
$324$		0			0
$325$			5.75152i			0.319037i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	3.06147	−	19.3137i	0.168784	−	1.06480i
$330$		0			0
$331$	0.828427			0.0455345			0.0227672	−	0.999741i	$-0.492752\pi$
$331$	0.828427			0.0455345			0.0227672	+	0.999741i	$0.492752\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−14.7821			−0.807631
$336$		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
$341$	2.53620			0.137343
$342$		0			0
$343$	8.41421	−	16.4985i	0.454325	−	0.890836i
$344$		0			0
$345$		0			0
$346$		0			0
$347$			17.5563i			0.942474i	0.882007	+	0.471237i	$0.156192\pi$
$347$			17.5563i			0.942474i	−0.882007	+	0.471237i	$0.843808\pi$
$348$		0			0
$349$			16.5754i			0.887263i	0.896209	+	0.443631i	$0.146310\pi$
$349$			16.5754i			0.887263i	−0.896209	+	0.443631i	$0.853690\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−0.262632			−0.0139785			−0.00698926	−	0.999976i	$-0.502225\pi$
$353$	−0.262632			−0.0139785			−0.00698926	+	0.999976i	$0.502225\pi$
$354$		0			0
$355$			7.01962i			0.372563i
$356$		0			0
$357$		0			0
$358$		0			0
$359$		−	28.3848i		−	1.49809i	−0.662518	−	0.749046i	$-0.730512\pi$
$359$		−	28.3848i		−	1.49809i	0.662518	−	0.749046i	$-0.269488\pi$
$360$		0			0
$361$	−8.31371			−0.437564
$362$		0			0
$363$		0			0
$364$		0			0
$365$			19.3137i			1.01093i
$366$		0			0
$367$			15.6788i			0.818424i	0.912439	+	0.409212i	$0.134196\pi$
$367$			15.6788i			0.818424i	−0.912439	+	0.409212i	$0.865804\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	14.1480	+	2.24264i	0.734529	+	0.116432i
$372$		0			0
$373$	−5.31371			−0.275133			−0.137567	−	0.990493i	$-0.543928\pi$
$373$	−5.31371			−0.275133			−0.137567	+	0.990493i	$0.543928\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	11.7206			0.603642
$378$		0			0
$379$	−16.8284			−0.864418			−0.432209	−	0.901773i	$-0.642266\pi$
$379$	−16.8284			−0.864418			−0.432209	+	0.901773i	$0.642266\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−28.2960			−1.44586			−0.722930	−	0.690921i	$-0.757205\pi$
$383$	−28.2960			−1.44586			−0.722930	+	0.690921i	$0.757205\pi$
$384$		0			0
$385$	−2.34315	−	0.371418i	−0.119418	−	0.0189292i
$386$		0			0
$387$		0			0
$388$		0			0
$389$		−	0.727922i		−	0.0369071i	−0.999830	−	0.0184536i	$-0.994126\pi$
$389$		−	0.727922i		−	0.0369071i	0.999830	−	0.0184536i	$-0.00587428\pi$
$390$		0			0
$391$		−	13.1426i		−	0.664647i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	3.58673			0.180468
$396$		0			0
$397$			8.65914i			0.434590i	0.976106	+	0.217295i	$0.0697233\pi$
$397$			8.65914i			0.434590i	−0.976106	+	0.217295i	$0.930277\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			31.5563i			1.57585i	0.615772	+	0.787924i	$0.288844\pi$
$401$			31.5563i			1.57585i	−0.615772	+	0.787924i	$0.711156\pi$
$402$		0			0
$403$	9.37258			0.466882
$404$		0			0
$405$		0			0
$406$		0			0
$407$			2.34315i			0.116145i
$408$		0			0
$409$		−	12.6173i		−	0.623885i	−0.950101	−	0.311942i	$-0.899020\pi$
$409$		−	12.6173i		−	0.623885i	0.950101	−	0.311942i	$-0.100980\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$	−20.6863			−1.01545
$416$		0			0
$417$		0			0
$418$		0			0
$419$	8.65914			0.423027			0.211513	−	0.977375i	$-0.432161\pi$
$419$	8.65914			0.423027			0.211513	+	0.977375i	$0.432161\pi$
$420$		0			0
$421$	14.9706			0.729621			0.364810	−	0.931082i	$-0.381134\pi$
$421$	14.9706			0.729621			0.364810	+	0.931082i	$0.381134\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−15.5700			−0.755254
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$			10.2426i			0.493371i	0.969096	+	0.246685i	$0.0793414\pi$
$431$			10.2426i			0.493371i	−0.969096	+	0.246685i	$0.920659\pi$
$432$		0			0
$433$		−	33.8937i		−	1.62883i	−0.580284	−	0.814414i	$-0.697059\pi$
$433$		−	33.8937i		−	1.62883i	0.580284	−	0.814414i	$-0.302941\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−11.7206			−0.560673
$438$		0			0
$439$		−	3.43289i		−	0.163843i	−0.996639	−	0.0819213i	$-0.973894\pi$
$439$		−	3.43289i		−	0.163843i	0.996639	−	0.0819213i	$-0.0261056\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$			8.58579i			0.407923i	0.978979	+	0.203962i	$0.0653817\pi$
$443$			8.58579i			0.407923i	−0.978979	+	0.203962i	$0.934618\pi$
$444$		0			0
$445$	−8.97056			−0.425245
$446$		0			0
$447$		0			0
$448$		0			0
$449$			9.41421i			0.444284i	0.975014	+	0.222142i	$0.0713049\pi$
$449$			9.41421i			0.444284i	−0.975014	+	0.222142i	$0.928695\pi$
$450$		0			0
$451$		−	5.22625i		−	0.246095i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−8.65914	−	1.37258i	−0.405947	−	0.0643477i
$456$		0			0
$457$	−11.3137			−0.529233			−0.264616	−	0.964354i	$-0.585245\pi$
$457$	−11.3137			−0.529233			−0.264616	+	0.964354i	$0.585245\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−10.7151			−0.499054			−0.249527	−	0.968368i	$-0.580275\pi$
$461$	−10.7151			−0.499054			−0.249527	+	0.968368i	$0.580275\pi$
$462$		0			0
$463$	−29.6569			−1.37827			−0.689135	−	0.724633i	$-0.742010\pi$
$463$	−29.6569			−1.37827			−0.689135	+	0.724633i	$0.742010\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	30.8322			1.42675			0.713373	−	0.700784i	$-0.247167\pi$
$467$	30.8322			1.42675			0.713373	+	0.700784i	$0.247167\pi$
$468$		0			0
$469$	−4.00000	+	25.2346i	−0.184703	+	1.16522i
$470$		0			0
$471$		0			0
$472$		0			0
$473$			6.14214i			0.282416i
$474$		0			0
$475$			13.8854i			0.637105i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−23.4412			−1.07106			−0.535528	−	0.844517i	$-0.679887\pi$
$479$	−23.4412			−1.07106			−0.535528	+	0.844517i	$0.679887\pi$
$480$		0			0
$481$			8.65914i			0.394823i
$482$		0			0
$483$		0			0
$484$		0			0
$485$			12.6863i			0.576055i
$486$		0			0
$487$	39.4558			1.78791			0.893957	−	0.448152i	$-0.147918\pi$
$487$	39.4558			1.78791			0.893957	+	0.448152i	$0.147918\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		−	26.5269i		−	1.19714i	−0.801069	−	0.598571i	$-0.795735\pi$
$491$		−	26.5269i		−	1.19714i	0.801069	−	0.598571i	$-0.204265\pi$
$492$		0			0
$493$			31.7289i			1.42900i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	11.9832	+	1.89949i	0.537522	+	0.0852040i
$498$		0			0
$499$	7.17157			0.321044			0.160522	−	0.987032i	$-0.448682\pi$
$499$	7.17157			0.321044			0.160522	+	0.987032i	$0.448682\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−19.6369			−0.875566			−0.437783	−	0.899081i	$-0.644236\pi$
$503$	−19.6369			−0.875566			−0.437783	+	0.899081i	$0.644236\pi$
$504$		0			0
$505$	−24.9706			−1.11118
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−10.1899			−0.451658			−0.225829	−	0.974167i	$-0.572509\pi$
$509$	−10.1899			−0.451658			−0.225829	+	0.974167i	$0.572509\pi$
$510$		0			0
$511$	32.9706	+	5.22625i	1.45853	+	0.231196i
$512$		0			0
$513$		0			0
$514$		0			0
$515$			22.6274i			0.997083i
$516$		0			0
$517$			4.32957i			0.190414i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−5.86030			−0.256745			−0.128372	−	0.991726i	$-0.540975\pi$
$521$	−5.86030			−0.256745			−0.128372	+	0.991726i	$0.540975\pi$
$522$		0			0
$523$			8.65914i			0.378638i	0.981916	+	0.189319i	$0.0606280\pi$
$523$			8.65914i			0.378638i	−0.981916	+	0.189319i	$0.939372\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$			25.3726i			1.10525i
$528$		0			0
$529$	17.9706			0.781329
$530$		0			0
$531$		0			0
$532$		0			0
$533$		−	19.3137i		−	0.836570i
$534$		0			0
$535$		−	20.0083i		−	0.865035i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−1.26810	+	3.89949i	−0.0546210	+	0.167963i
$540$		0			0
$541$	−16.3431			−0.702647			−0.351323	−	0.936254i	$-0.614268\pi$
$541$	−16.3431			−0.702647			−0.351323	+	0.936254i	$0.614268\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−14.7821			−0.633194
$546$		0			0
$547$	−6.34315			−0.271213			−0.135607	−	0.990763i	$-0.543298\pi$
$547$	−6.34315			−0.271213			−0.135607	+	0.990763i	$0.543298\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	28.2960			1.20545
$552$		0			0
$553$	0.970563	−	6.12293i	0.0412725	−	0.260374i
$554$		0			0
$555$		0			0
$556$		0			0
$557$			26.1005i			1.10591i	0.833210	+	0.552957i	$0.186501\pi$
$557$			26.1005i			1.10591i	−0.833210	+	0.552957i	$0.813499\pi$
$558$		0			0
$559$			22.6984i			0.960039i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	45.6143			1.92241			0.961207	−	0.275827i	$-0.0889518\pi$
$563$	45.6143			1.92241			0.961207	+	0.275827i	$0.0889518\pi$
$564$		0			0
$565$			30.9861i			1.30359i
$566$		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
$571$	20.9706			0.877591			0.438795	−	0.898587i	$-0.355405\pi$
$571$	20.9706			0.877591			0.438795	+	0.898587i	$0.355405\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$			5.95837i			0.248481i
$576$		0			0
$577$		−	4.32957i		−	0.180242i	−0.995931	−	0.0901212i	$-0.971275\pi$
$577$		−	4.32957i		−	0.180242i	0.995931	−	0.0901212i	$-0.0287254\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−5.59767	+	35.3137i	−0.232230	+	1.46506i
$582$		0			0
$583$	−3.17157			−0.131353
$584$		0			0
$585$		0			0
$586$		0			0
$587$	4.85483			0.200380			0.100190	−	0.994968i	$-0.468055\pi$
$587$	4.85483			0.200380			0.100190	+	0.994968i	$0.468055\pi$
$588$		0			0
$589$	22.6274			0.932346
$590$		0			0
$591$		0			0
$592$		0			0
$593$	3.32410			0.136504			0.0682522	−	0.997668i	$-0.478258\pi$
$593$	3.32410			0.136504			0.0682522	+	0.997668i	$0.478258\pi$
$594$		0			0
$595$	3.71573	−	23.4412i	0.152330	−	0.960996i
$596$		0			0
$597$		0			0
$598$		0			0
$599$			11.2132i			0.458159i	0.973408	+	0.229080i	$0.0735716\pi$
$599$			11.2132i			0.458159i	−0.973408	+	0.229080i	$0.926428\pi$
$600$		0			0
$601$			33.8937i			1.38255i	0.722590	+	0.691277i	$0.242951\pi$
$601$			33.8937i			1.38255i	−0.722590	+	0.691277i	$0.757049\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−16.3128			−0.663210
$606$		0			0
$607$			5.22625i			0.212127i	0.994359	+	0.106064i	$0.0338247\pi$
$607$			5.22625i			0.212127i	−0.994359	+	0.106064i	$0.966175\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$			16.0000i			0.647291i
$612$		0			0
$613$	−1.31371			−0.0530602			−0.0265301	−	0.999648i	$-0.508446\pi$
$613$	−1.31371			−0.0530602			−0.0265301	+	0.999648i	$0.508446\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	12.2426i		−	0.492870i	−0.969159	−	0.246435i	$-0.920741\pi$
$617$		−	12.2426i		−	0.492870i	0.969159	−	0.246435i	$-0.0792592\pi$
$618$		0			0
$619$		−	19.1116i		−	0.768162i	−0.923299	−	0.384081i	$-0.874518\pi$
$619$		−	19.1116i		−	0.768162i	0.923299	−	0.384081i	$-0.125482\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−2.42742	+	15.3137i	−0.0972524	+	0.613531i
$624$		0			0
$625$	−4.65685			−0.186274
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−23.4412			−0.934662
$630$		0			0
$631$	−39.5980			−1.57637			−0.788185	−	0.615438i	$-0.788979\pi$
$631$	−39.5980			−1.57637			−0.788185	+	0.615438i	$0.788979\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−25.9774			−1.03088
$636$		0			0
$637$	−4.68629	+	14.4107i	−0.185678	+	0.570971i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		−	18.8701i		−	0.745322i	−0.927967	−	0.372661i	$-0.878445\pi$
$641$		−	18.8701i		−	0.745322i	0.927967	−	0.372661i	$-0.121555\pi$
$642$		0			0
$643$			13.8854i			0.547586i	0.961789	+	0.273793i	$0.0882784\pi$
$643$			13.8854i			0.547586i	−0.961789	+	0.273793i	$0.911722\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	3.58673			0.141009			0.0705045	−	0.997511i	$-0.477539\pi$
$647$	3.58673			0.141009			0.0705045	+	0.997511i	$0.477539\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$		0			0
$653$		−	41.2132i		−	1.61280i	−0.591372	−	0.806399i	$-0.701414\pi$
$653$		−	41.2132i		−	1.61280i	0.591372	−	0.806399i	$-0.298586\pi$
$654$		0			0
$655$	11.3137			0.442063
$656$		0			0
$657$		0			0
$658$		0			0
$659$		−	47.4142i		−	1.84700i	−0.383604	−	0.923498i	$-0.625317\pi$
$659$		−	47.4142i		−	1.84700i	0.383604	−	0.923498i	$-0.374683\pi$
$660$		0			0
$661$		−	37.4804i		−	1.45782i	−0.684609	−	0.728910i	$-0.740027\pi$
$661$		−	37.4804i		−	1.45782i	0.684609	−	0.728910i	$-0.259973\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−20.9050	−	3.31371i	−0.810661	−	0.128500i
$666$		0			0
$667$	12.1421			0.470145
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−13.6569			−0.526433			−0.263217	−	0.964737i	$-0.584783\pi$
$673$	−13.6569			−0.526433			−0.263217	+	0.964737i	$0.584783\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−45.8770			−1.76319			−0.881597	−	0.472002i	$-0.843531\pi$
$677$	−45.8770			−1.76319			−0.881597	+	0.472002i	$0.843531\pi$
$678$		0			0
$679$	21.6569	+	3.43289i	0.831114	+	0.131742i
$680$		0			0
$681$		0			0
$682$		0			0
$683$			34.7279i			1.32883i	0.747365	+	0.664414i	$0.231319\pi$
$683$			34.7279i			1.32883i	−0.747365	+	0.664414i	$0.768681\pi$
$684$		0			0
$685$		−	15.1535i		−	0.578985i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−11.7206			−0.446519
$690$		0			0
$691$		−	31.3575i		−	1.19290i	−0.802652	−	0.596448i	$-0.796578\pi$
$691$		−	31.3575i		−	1.19290i	0.802652	−	0.596448i	$-0.203422\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		−	16.0000i		−	0.606915i
$696$		0			0
$697$	52.2843			1.98041
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	28.9289i		−	1.09263i	−0.837580	−	0.546315i	$-0.816030\pi$
$701$		−	28.9289i		−	1.09263i	0.837580	−	0.546315i	$-0.183970\pi$
$702$		0			0
$703$			20.9050i			0.788447i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−6.75699	+	42.6274i	−0.254123	+	1.60317i
$708$		0			0
$709$	24.2843			0.912015			0.456007	−	0.889976i	$-0.349279\pi$
$709$	24.2843			0.912015			0.456007	+	0.889976i	$0.349279\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	9.70967			0.363630
$714$		0			0
$715$	1.94113			0.0725940
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−20.9050			−0.779625			−0.389813	−	0.920894i	$-0.627460\pi$
$719$	−20.9050			−0.779625			−0.389813	+	0.920894i	$0.627460\pi$
$720$		0			0
$721$	38.6274	+	6.12293i	1.43856	+	0.228030i
$722$		0			0
$723$		0			0
$724$		0			0
$725$		−	14.3848i		−	0.534237i
$726$		0			0
$727$			6.12293i			0.227087i	0.993533	+	0.113544i	$0.0362201\pi$
$727$			6.12293i			0.227087i	−0.993533	+	0.113544i	$0.963780\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−61.4469			−2.27270
$732$		0			0
$733$			6.49435i			0.239874i	0.992781	+	0.119937i	$0.0382693\pi$
$733$			6.49435i			0.239874i	−0.992781	+	0.119937i	$0.961731\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	5.65685i		−	0.208373i
$738$		0			0
$739$	−40.2843			−1.48188			−0.740940	−	0.671571i	$-0.765620\pi$
$739$	−40.2843			−1.48188			−0.740940	+	0.671571i	$0.765620\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	47.6985i		−	1.74989i	−0.484224	−	0.874944i	$-0.660898\pi$
$743$		−	47.6985i		−	1.74989i	0.484224	−	0.874944i	$-0.339102\pi$
$744$		0			0
$745$			24.8632i			0.910916i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−34.1563	−	5.41421i	−1.24805	−	0.197831i
$750$		0			0
$751$	13.5147			0.493159			0.246580	−	0.969123i	$-0.420693\pi$
$751$	13.5147			0.493159			0.246580	+	0.969123i	$0.420693\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	18.5864			0.676428
$756$		0			0
$757$	27.5980			1.00307			0.501533	−	0.865139i	$-0.332770\pi$
$757$	27.5980			1.00307			0.501533	+	0.865139i	$0.332770\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−29.3015			−1.06218			−0.531090	−	0.847316i	$-0.678217\pi$
$761$	−29.3015			−1.06218			−0.531090	+	0.847316i	$0.678217\pi$
$762$		0			0
$763$	−4.00000	+	25.2346i	−0.144810	+	0.913553i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$			33.8937i			1.22224i	0.791539	+	0.611119i	$0.209280\pi$
$769$			33.8937i			1.22224i	−0.791539	+	0.611119i	$0.790720\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	16.3128			0.586731			0.293365	−	0.956000i	$-0.405225\pi$
$773$	16.3128			0.586731			0.293365	+	0.956000i	$0.405225\pi$
$774$		0			0
$775$		−	11.5030i		−	0.413201i
$776$		0			0
$777$		0			0
$778$		0			0
$779$		−	46.6274i		−	1.67060i
$780$		0			0
$781$	−2.68629			−0.0961231
$782$		0			0
$783$		0			0
$784$		0			0
$785$			25.3726i			0.905586i
$786$		0			0
$787$		−	38.2233i		−	1.36251i	−0.732045	−	0.681256i	$-0.761434\pi$
$787$		−	38.2233i		−	1.36251i	0.732045	−	0.681256i	$-0.238566\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	52.8966	+	8.38478i	1.88079	+	0.298128i
$792$		0			0
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−22.4357			−0.794715			−0.397357	−	0.917664i	$-0.630073\pi$
$797$	−22.4357			−0.794715			−0.397357	+	0.917664i	$0.630073\pi$
$798$		0			0
$799$	−43.3137			−1.53233
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−7.39104			−0.260824
$804$		0			0
$805$	−8.97056	−	1.42195i	−0.316171	−	0.0501171i
$806$		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
$811$			12.2459i			0.430011i	0.976613	+	0.215005i	$0.0689769\pi$
$811$			12.2459i			0.430011i	−0.976613	+	0.215005i	$0.931023\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−9.70967			−0.340115
$816$		0			0
$817$			54.7987i			1.91716i
$818$		0			0
$819$		0			0
$820$		0			0
$821$			39.3553i			1.37351i	0.726889	+	0.686755i	$0.240966\pi$
$821$			39.3553i			1.37351i	−0.726889	+	0.686755i	$0.759034\pi$
$822$		0			0
$823$	21.6569			0.754910			0.377455	−	0.926028i	$-0.376799\pi$
$823$	21.6569			0.754910			0.377455	+	0.926028i	$0.376799\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	55.0122i		−	1.91296i	−0.291796	−	0.956481i	$-0.594253\pi$
$827$		−	55.0122i		−	1.91296i	0.291796	−	0.956481i	$-0.405747\pi$
$828$		0			0
$829$			10.8239i			0.375930i	0.982176	+	0.187965i	$0.0601892\pi$
$829$			10.8239i			0.375930i	−0.982176	+	0.187965i	$0.939811\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−39.0112	−	12.6863i	−1.35166	−	0.439554i
$834$		0			0
$835$	−20.6863			−0.715879
$836$		0			0
$837$		0			0
$838$		0			0
$839$	51.7373			1.78617			0.893084	−	0.449890i	$-0.148537\pi$
$839$	51.7373			1.78617			0.893084	+	0.449890i	$0.148537\pi$
$840$		0			0
$841$	−0.313708			−0.0108175
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−12.7261			−0.437790
$846$		0			0
$847$	−4.41421	+	27.8477i	−0.151674	+	0.956858i
$848$		0			0
$849$		0			0
$850$		0			0
$851$			8.97056i			0.307507i
$852$		0			0
$853$			29.5641i			1.01226i	0.862458	+	0.506129i	$0.168924\pi$
$853$			29.5641i			1.01226i	−0.862458	+	0.506129i	$0.831076\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−5.86030			−0.200184			−0.100092	−	0.994978i	$-0.531914\pi$
$857$	−5.86030			−0.200184			−0.100092	+	0.994978i	$0.531914\pi$
$858$		0			0
$859$			13.8854i			0.473763i	0.971539	+	0.236882i	$0.0761254\pi$
$859$			13.8854i			0.473763i	−0.971539	+	0.236882i	$0.923875\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	20.5858i		−	0.700748i	−0.936610	−	0.350374i	$-0.886054\pi$
$863$		−	20.5858i		−	0.700748i	0.936610	−	0.350374i	$-0.113946\pi$
$864$		0			0
$865$	−34.3431			−1.16770
$866$		0			0
$867$		0			0
$868$		0			0
$869$			1.37258i			0.0465617i
$870$		0			0
$871$		−	20.9050i		−	0.708339i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−4.85483	+	30.6274i	−0.164123	+	1.03540i
$876$		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
$881$	−39.0112			−1.31432			−0.657160	−	0.753751i	$-0.728242\pi$
$881$	−39.0112			−1.31432			−0.657160	+	0.753751i	$0.728242\pi$
$882$		0			0
$883$	4.14214			0.139394			0.0696970	−	0.997568i	$-0.477797\pi$
$883$	4.14214			0.139394			0.0696970	+	0.997568i	$0.477797\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−16.0502			−0.538912			−0.269456	−	0.963013i	$-0.586844\pi$
$887$	−16.0502			−0.538912			−0.269456	+	0.963013i	$0.586844\pi$
$888$		0			0
$889$	−7.02944	+	44.3462i	−0.235760	+	1.48732i
$890$		0			0
$891$		0			0
$892$		0			0
$893$			38.6274i			1.29262i
$894$		0			0
$895$		−	5.96909i		−	0.199525i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−23.4412			−0.781808
$900$		0			0
$901$		−	31.7289i		−	1.05704i
$902$		0			0
$903$		0			0
$904$		0			0
$905$			22.0589i			0.733262i
$906$		0			0
$907$	10.2010			0.338719			0.169359	−	0.985554i	$-0.445830\pi$
$907$	10.2010			0.338719			0.169359	+	0.985554i	$0.445830\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	20.1838i		−	0.668718i	−0.942446	−	0.334359i	$-0.891480\pi$
$911$		−	20.1838i		−	0.668718i	0.942446	−	0.334359i	$-0.108520\pi$
$912$		0			0
$913$		−	7.91630i		−	0.261991i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	3.06147	−	19.3137i	0.101099	−	0.637795i
$918$		0			0
$919$	−37.7990			−1.24687			−0.623437	−	0.781874i	$-0.714264\pi$
$919$	−37.7990			−1.24687			−0.623437	+	0.781874i	$0.714264\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−9.92724			−0.326759
$924$		0			0
$925$	10.6274			0.349427
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−20.1171			−0.660021			−0.330010	−	0.943977i	$-0.607052\pi$
$929$	−20.1171			−0.660021			−0.330010	+	0.943977i	$0.607052\pi$
$930$		0			0
$931$	−11.3137	+	34.7904i	−0.370792	+	1.14021i
$932$		0			0
$933$		0			0
$934$		0			0
$935$			5.25483i			0.171851i
$936$		0			0
$937$		−	3.58673i		−	0.117173i	−0.998282	−	0.0585867i	$-0.981341\pi$
$937$		−	3.58673i		−	0.117173i	0.998282	−	0.0585867i	$-0.0186594\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	22.4357			0.731384			0.365692	−	0.930736i	$-0.380832\pi$
$941$	22.4357			0.731384			0.365692	+	0.930736i	$0.380832\pi$
$942$		0			0
$943$		−	20.0083i		−	0.651561i
$944$		0			0
$945$		0			0
$946$		0			0
$947$		−	24.5858i		−	0.798931i	−0.916748	−	0.399465i	$-0.869196\pi$
$947$		−	24.5858i		−	0.798931i	0.916748	−	0.399465i	$-0.130804\pi$
$948$		0			0
$949$	−27.3137			−0.886640
$950$		0			0
$951$		0			0
$952$		0			0
$953$			53.2132i			1.72374i	0.507125	+	0.861872i	$0.330708\pi$
$953$			53.2132i			1.72374i	−0.507125	+	0.861872i	$0.669292\pi$
$954$		0			0
$955$			34.7904i			1.12579i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−25.8686	−	4.10051i	−0.835342	−	0.132412i
$960$		0			0
$961$	12.2548			0.395317
$962$		0			0
$963$		0			0
$964$		0			0
$965$	8.65914			0.278748
$966$		0			0
$967$	−29.6569			−0.953700			−0.476850	−	0.878985i	$-0.658222\pi$
$967$	−29.6569			−0.953700			−0.476850	+	0.878985i	$0.658222\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−23.4412			−0.752264			−0.376132	−	0.926566i	$-0.622746\pi$
$971$	−23.4412			−0.752264			−0.376132	+	0.926566i	$0.622746\pi$
$972$		0			0
$973$	−27.3137	−	4.32957i	−0.875637	−	0.138800i
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	17.8995i		−	0.572656i	−0.958132	−	0.286328i	$-0.907565\pi$
$977$		−	17.8995i		−	0.572656i	0.958132	−	0.286328i	$-0.0924346\pi$
$978$		0			0
$979$		−	3.43289i		−	0.109716i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	18.5864			0.592813			0.296407	−	0.955062i	$-0.404212\pi$
$983$	18.5864			0.592813			0.296407	+	0.955062i	$0.404212\pi$
$984$		0			0
$985$		−	22.0193i		−	0.701592i
$986$		0			0
$987$		0			0
$988$		0			0
$989$			23.5147i			0.747725i
$990$		0			0
$991$	27.8579			0.884934			0.442467	−	0.896785i	$-0.354103\pi$
$991$	27.8579			0.884934			0.442467	+	0.896785i	$0.354103\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		−	5.25483i		−	0.166589i
$996$		0			0
$997$			54.0559i			1.71197i	0.517003	+	0.855984i	$0.327048\pi$
$997$			54.0559i			1.71197i	−0.517003	+	0.855984i	$0.672952\pi$
$998$		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.3905.4		8
3.2	odd	2	inner	4032.2.k.f.3905.6		8
4.3	odd	2		4032.2.k.e.3905.3		8
7.6	odd	2	inner	4032.2.k.f.3905.5		8
8.3	odd	2		1008.2.k.c.881.5		8
8.5	even	2		504.2.k.a.377.6	yes	8
12.11	even	2		4032.2.k.e.3905.5		8
21.20	even	2	inner	4032.2.k.f.3905.3		8
24.5	odd	2		504.2.k.a.377.4	yes	8
24.11	even	2		1008.2.k.c.881.3		8
28.27	even	2		4032.2.k.e.3905.6		8
56.5	odd	6		3528.2.bl.b.521.5		16
56.13	odd	2		504.2.k.a.377.3	✓	8
56.27	even	2		1008.2.k.c.881.4		8
56.37	even	6		3528.2.bl.b.521.3		16
56.45	odd	6		3528.2.bl.b.1097.6		16
56.53	even	6		3528.2.bl.b.1097.4		16
84.83	odd	2		4032.2.k.e.3905.4		8
168.5	even	6		3528.2.bl.b.521.4		16
168.53	odd	6		3528.2.bl.b.1097.5		16
168.83	odd	2		1008.2.k.c.881.6		8
168.101	even	6		3528.2.bl.b.1097.3		16
168.125	even	2		504.2.k.a.377.5	yes	8
168.149	odd	6		3528.2.bl.b.521.6		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
504.2.k.a.377.3	✓	8	56.13	odd	2
504.2.k.a.377.4	yes	8	24.5	odd	2
504.2.k.a.377.5	yes	8	168.125	even	2
504.2.k.a.377.6	yes	8	8.5	even	2
1008.2.k.c.881.3		8	24.11	even	2
1008.2.k.c.881.4		8	56.27	even	2
1008.2.k.c.881.5		8	8.3	odd	2
1008.2.k.c.881.6		8	168.83	odd	2
3528.2.bl.b.521.3		16	56.37	even	6
3528.2.bl.b.521.4		16	168.5	even	6
3528.2.bl.b.521.5		16	56.5	odd	6
3528.2.bl.b.521.6		16	168.149	odd	6
3528.2.bl.b.1097.3		16	168.101	even	6
3528.2.bl.b.1097.4		16	56.53	even	6
3528.2.bl.b.1097.5		16	168.53	odd	6
3528.2.bl.b.1097.6		16	56.45	odd	6
4032.2.k.e.3905.3		8	4.3	odd	2
4032.2.k.e.3905.4		8	84.83	odd	2
4032.2.k.e.3905.5		8	12.11	even	2
4032.2.k.e.3905.6		8	28.27	even	2
4032.2.k.f.3905.3		8	21.20	even	2	inner
4032.2.k.f.3905.4		8	1.1	even	1	trivial
4032.2.k.f.3905.5		8	7.6	odd	2	inner
4032.2.k.f.3905.6		8	3.2	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)

\(f(q)\)	\(=\)	\(q-1.53073 q^{5} +(-0.414214 + 2.61313i) q^{7} +O(q^{10})\)	\(q-1.53073 q^{5} +(-0.414214 + 2.61313i) q^{7} -0.585786i q^{11} -2.16478i q^{13} +5.86030 q^{17} -5.22625i q^{19} -2.24264i q^{23} -2.65685 q^{25} +5.41421i q^{29} +4.32957i q^{31} +(0.634051 - 4.00000i) q^{35} -4.00000 q^{37} +8.92177 q^{41} -10.4853 q^{43} -7.39104 q^{47} +(-6.65685 - 2.16478i) q^{49} -5.41421i q^{53} +0.896683i q^{55} +3.31371i q^{65} +9.65685 q^{67} -4.58579i q^{71} -12.6173i q^{73} +(1.53073 + 0.242641i) q^{77} -2.34315 q^{79} +13.5140 q^{83} -8.97056 q^{85} +5.86030 q^{89} +(5.65685 + 0.896683i) q^{91} +8.00000i q^{95} -8.28772i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{7}+O(q^{10}) \) 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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