LMFDB - Embedded newform 8028.2.h.a.4013.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8028,2,Mod(4013,8028)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8028, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("8028.4013");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8028 = 2^{2} \cdot 3^{2} \cdot 223 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8028.h (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$64.1039027427$
Analytic rank:	$0$
Dimension:	$76$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			4013.4
Character	$\chi$	$=$	8028.4013
Dual form			8028.2.h.a.4013.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.84036 q^{5} +1.75909 q^{7} +O(q^{10})$	$q-3.84036 q^{5} +1.75909 q^{7} +5.98889 q^{11} -5.14669i q^{13} -7.41667i q^{17} +8.31871 q^{19} -2.63152 q^{23} +9.74838 q^{25} -5.44889i q^{29} +4.99898 q^{31} -6.75556 q^{35} +5.71440 q^{37} +2.69328i q^{41} -7.69616 q^{43} -11.4380i q^{47} -3.90559 q^{49} -3.06965i q^{53} -22.9995 q^{55} -1.03626 q^{59} +9.82914i q^{61} +19.7651i q^{65} +1.69857i q^{67} +7.83513 q^{71} +5.44707 q^{73} +10.5350 q^{77} -6.91980i q^{79} +8.11970i q^{83} +28.4827i q^{85} +9.06835i q^{89} -9.05351i q^{91} -31.9469 q^{95} -0.305441i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 76 q + 8 q^{7}+O(q^{10}) $ 76 * q + 8 * q^7	$ 76 q + 8 q^{7} + 16 q^{19} + 100 q^{25} - 8 q^{31} + 32 q^{37} - 24 q^{43} + 68 q^{49} + 24 q^{55} + 8 q^{73}+O(q^{100}) $ 76 * q + 8 * q^7 + 16 * q^19 + 100 * q^25 - 8 * q^31 + 32 * q^37 - 24 * q^43 + 68 * q^49 + 24 * q^55 + 8 * q^73

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/8028\mathbb{Z}\right)^\times$.

$n$	$893$	$2233$	$4015$
$\chi(n)$	$-1$	$-1$	$1$

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
$5$	−3.84036			−1.71746			−0.858731	−	0.512427i	$-0.828747\pi$
$5$	−3.84036			−1.71746			−0.858731	+	0.512427i	$0.828747\pi$
$6$		0			0
$7$	1.75909			0.664875			0.332438	−	0.943125i	$-0.392129\pi$
$7$	1.75909			0.664875			0.332438	+	0.943125i	$0.392129\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	5.98889			1.80572			0.902859	−	0.429936i	$-0.141464\pi$
$11$	5.98889			1.80572			0.902859	+	0.429936i	$0.141464\pi$
$12$		0			0
$13$		−	5.14669i		−	1.42743i	−0.700434	−	0.713717i	$-0.747010\pi$
$13$		−	5.14669i		−	1.42743i	0.700434	−	0.713717i	$-0.252990\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		−	7.41667i		−	1.79881i	−0.437119	−	0.899404i	$-0.644001\pi$
$17$		−	7.41667i		−	1.79881i	0.437119	−	0.899404i	$-0.355999\pi$
$18$		0			0
$19$	8.31871			1.90844			0.954222	−	0.299101i	$-0.0966867\pi$
$19$	8.31871			1.90844			0.954222	+	0.299101i	$0.0966867\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−2.63152			−0.548711			−0.274355	−	0.961628i	$-0.588464\pi$
$23$	−2.63152			−0.548711			−0.274355	+	0.961628i	$0.588464\pi$
$24$		0			0
$25$	9.74838			1.94968
$26$		0			0
$27$		0			0
$28$		0			0
$29$		−	5.44889i		−	1.01183i	−0.862582	−	0.505917i	$-0.831154\pi$
$29$		−	5.44889i		−	1.01183i	0.862582	−	0.505917i	$-0.168846\pi$
$30$		0			0
$31$	4.99898			0.897843			0.448922	−	0.893571i	$-0.351808\pi$
$31$	4.99898			0.897843			0.448922	+	0.893571i	$0.351808\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−6.75556			−1.14190
$36$		0			0
$37$	5.71440			0.939441			0.469721	−	0.882815i	$-0.344355\pi$
$37$	5.71440			0.939441			0.469721	+	0.882815i	$0.344355\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$			2.69328i			0.420620i	0.977635	+	0.210310i	$0.0674473\pi$
$41$			2.69328i			0.420620i	−0.977635	+	0.210310i	$0.932553\pi$
$42$		0			0
$43$	−7.69616			−1.17365			−0.586827	−	0.809713i	$-0.699623\pi$
$43$	−7.69616			−1.17365			−0.586827	+	0.809713i	$0.699623\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		−	11.4380i		−	1.66840i	−0.551459	−	0.834202i	$-0.685929\pi$
$47$		−	11.4380i		−	1.66840i	0.551459	−	0.834202i	$-0.314071\pi$
$48$		0			0
$49$	−3.90559			−0.557941
$50$		0			0
$51$		0			0
$52$		0			0
$53$		−	3.06965i		−	0.421649i	−0.977524	−	0.210825i	$-0.932385\pi$
$53$		−	3.06965i		−	0.421649i	0.977524	−	0.210825i	$-0.0676149\pi$
$54$		0			0
$55$	−22.9995			−3.10125
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−1.03626			−0.134910			−0.0674548	−	0.997722i	$-0.521488\pi$
$59$	−1.03626			−0.134910			−0.0674548	+	0.997722i	$0.521488\pi$
$60$		0			0
$61$			9.82914i			1.25849i	0.777206	+	0.629246i	$0.216636\pi$
$61$			9.82914i			1.25849i	−0.777206	+	0.629246i	$0.783364\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$			19.7651i			2.45156i
$66$		0			0
$67$			1.69857i			0.207514i	0.994603	+	0.103757i	$0.0330864\pi$
$67$			1.69857i			0.207514i	−0.994603	+	0.103757i	$0.966914\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	7.83513			0.929858			0.464929	−	0.885348i	$-0.346080\pi$
$71$	7.83513			0.929858			0.464929	+	0.885348i	$0.346080\pi$
$72$		0			0
$73$	5.44707			0.637531			0.318765	−	0.947834i	$-0.396732\pi$
$73$	5.44707			0.637531			0.318765	+	0.947834i	$0.396732\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	10.5350			1.20058
$78$		0			0
$79$		−	6.91980i		−	0.778538i	−0.921124	−	0.389269i	$-0.872728\pi$
$79$		−	6.91980i		−	0.778538i	0.921124	−	0.389269i	$-0.127272\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$			8.11970i			0.891253i	0.895219	+	0.445626i	$0.147019\pi$
$83$			8.11970i			0.891253i	−0.895219	+	0.445626i	$0.852981\pi$
$84$		0			0
$85$			28.4827i			3.08938i
$86$		0			0
$87$		0			0
$88$		0			0
$89$			9.06835i			0.961243i	0.876928	+	0.480621i	$0.159589\pi$
$89$			9.06835i			0.961243i	−0.876928	+	0.480621i	$0.840411\pi$
$90$		0			0
$91$		−	9.05351i		−	0.949066i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−31.9469			−3.27768
$96$		0			0
$97$		−	0.305441i		−	0.0310129i	−0.999880	−	0.0155064i	$-0.995064\pi$
$97$		−	0.305441i		−	0.0310129i	0.999880	−	0.0155064i	$-0.00493605\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$		−	0.669968i		−	0.0666643i	−0.999444	−	0.0333321i	$-0.989388\pi$
$101$		−	0.669968i		−	0.0666643i	0.999444	−	0.0333321i	$-0.0106119\pi$
$102$		0			0
$103$			7.99641i			0.787910i	0.919130	+	0.393955i	$0.128893\pi$
$103$			7.99641i			0.787910i	−0.919130	+	0.393955i	$0.871107\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−3.13989			−0.303545			−0.151772	−	0.988415i	$-0.548498\pi$
$107$	−3.13989			−0.303545			−0.151772	+	0.988415i	$0.548498\pi$
$108$		0			0
$109$	5.26082			0.503895			0.251948	−	0.967741i	$-0.418929\pi$
$109$	5.26082			0.503895			0.251948	+	0.967741i	$0.418929\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−4.35771			−0.409939			−0.204969	−	0.978768i	$-0.565709\pi$
$113$	−4.35771			−0.409939			−0.204969	+	0.978768i	$0.565709\pi$
$114$		0			0
$115$	10.1060			0.942390
$116$		0			0
$117$		0			0
$118$		0			0
$119$		−	13.0466i		−	1.19598i
$120$		0			0
$121$	24.8668			2.26062
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−18.2355			−1.63103
$126$		0			0
$127$	−2.65602			−0.235684			−0.117842	−	0.993032i	$-0.537598\pi$
$127$	−2.65602			−0.235684			−0.117842	+	0.993032i	$0.537598\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$			7.82739i			0.683883i	0.939721	+	0.341941i	$0.111084\pi$
$131$			7.82739i			0.683883i	−0.939721	+	0.341941i	$0.888916\pi$
$132$		0			0
$133$	14.6334			1.26888
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−9.24368			−0.789741			−0.394870	−	0.918737i	$-0.629210\pi$
$137$	−9.24368			−0.789741			−0.394870	+	0.918737i	$0.629210\pi$
$138$		0			0
$139$	−11.3748			−0.964795			−0.482398	−	0.875952i	$-0.660234\pi$
$139$	−11.3748			−0.964795			−0.482398	+	0.875952i	$0.660234\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		−	30.8230i		−	2.57755i
$144$		0			0
$145$			20.9257i			1.73779i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−15.4080			−1.26228			−0.631138	−	0.775671i	$-0.717412\pi$
$149$	−15.4080			−1.26228			−0.631138	+	0.775671i	$0.717412\pi$
$150$		0			0
$151$		−	18.8560i		−	1.53448i	−0.641361	−	0.767239i	$-0.721630\pi$
$151$		−	18.8560i		−	1.53448i	0.641361	−	0.767239i	$-0.278370\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−19.1979			−1.54201
$156$		0			0
$157$		−	15.0313i		−	1.19962i	−0.800141	−	0.599812i	$-0.795242\pi$
$157$		−	15.0313i		−	1.19962i	0.800141	−	0.599812i	$-0.204758\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−4.62910			−0.364824
$162$		0			0
$163$			13.3802i			1.04802i	0.851712	+	0.524010i	$0.175565\pi$
$163$			13.3802i			1.04802i	−0.851712	+	0.524010i	$0.824435\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−11.3366			−0.877252			−0.438626	−	0.898670i	$-0.644535\pi$
$167$	−11.3366			−0.877252			−0.438626	+	0.898670i	$0.644535\pi$
$168$		0			0
$169$	−13.4884			−1.03757
$170$		0			0
$171$		0			0
$172$		0			0
$173$	12.8260			0.975146			0.487573	−	0.873082i	$-0.337882\pi$
$173$	12.8260			0.975146			0.487573	+	0.873082i	$0.337882\pi$
$174$		0			0
$175$	17.1483			1.29629
$176$		0			0
$177$		0			0
$178$		0			0
$179$		−	20.2660i		−	1.51475i	−0.652979	−	0.757376i	$-0.726481\pi$
$179$		−	20.2660i		−	1.51475i	0.652979	−	0.757376i	$-0.273519\pi$
$180$		0			0
$181$	17.7440			1.31890			0.659452	−	0.751747i	$-0.270788\pi$
$181$	17.7440			1.31890			0.659452	+	0.751747i	$0.270788\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−21.9454			−1.61346
$186$		0			0
$187$		−	44.4177i		−	3.24814i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−0.399844			−0.0289317			−0.0144659	−	0.999895i	$-0.504605\pi$
$191$	−0.399844			−0.0289317			−0.0144659	+	0.999895i	$0.504605\pi$
$192$		0			0
$193$			9.86875i			0.710368i	0.934796	+	0.355184i	$0.115582\pi$
$193$			9.86875i			0.710368i	−0.934796	+	0.355184i	$0.884418\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			7.40836i			0.527824i	0.964547	+	0.263912i	$0.0850128\pi$
$197$			7.40836i			0.527824i	−0.964547	+	0.263912i	$0.914987\pi$
$198$		0			0
$199$	12.3243			0.873649			0.436825	−	0.899547i	$-0.356103\pi$
$199$	12.3243			0.873649			0.436825	+	0.899547i	$0.356103\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		−	9.58511i		−	0.672743i
$204$		0			0
$205$		−	10.3432i		−	0.722399i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	49.8198			3.44611
$210$		0			0
$211$	−7.20156			−0.495776			−0.247888	−	0.968789i	$-0.579737\pi$
$211$	−7.20156			−0.495776			−0.247888	+	0.968789i	$0.579737\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	29.5560			2.01570
$216$		0			0
$217$	8.79367			0.596953
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−38.1713			−2.56768
$222$		0			0
$223$	−12.1797	−	8.64024i	−0.815616	−	0.578593i
$223$	−12.1797	−	8.64024i	−0.815616	−	0.578593i
$224$		0			0
$225$		0			0
$226$		0			0
$227$		−	11.6670i		−	0.774367i	−0.922003	−	0.387183i	$-0.873448\pi$
$227$		−	11.6670i		−	0.774367i	0.922003	−	0.387183i	$-0.126552\pi$
$228$		0			0
$229$			6.50468i			0.429842i	0.976631	+	0.214921i	$0.0689493\pi$
$229$			6.50468i			0.429842i	−0.976631	+	0.214921i	$0.931051\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−22.5344			−1.47628			−0.738138	−	0.674650i	$-0.764295\pi$
$233$	−22.5344			−1.47628			−0.738138	+	0.674650i	$0.764295\pi$
$234$		0			0
$235$			43.9261i			2.86542i
$236$		0			0
$237$		0			0
$238$		0			0
$239$			22.3441i			1.44532i	0.691204	+	0.722659i	$0.257080\pi$
$239$			22.3441i			1.44532i	−0.691204	+	0.722659i	$0.742920\pi$
$240$		0			0
$241$	25.7219			1.65689			0.828446	−	0.560070i	$-0.189226\pi$
$241$	25.7219			1.65689			0.828446	+	0.560070i	$0.189226\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	14.9989			0.958243
$246$		0			0
$247$		−	42.8138i		−	2.72418i
$248$		0			0
$249$		0			0
$250$		0			0
$251$			19.9642i			1.26013i	0.776542	+	0.630065i	$0.216972\pi$
$251$			19.9642i			1.26013i	−0.776542	+	0.630065i	$0.783028\pi$
$252$		0			0
$253$	−15.7599			−0.990817
$254$		0			0
$255$		0			0
$256$		0			0
$257$			12.5567i			0.783265i	0.920122	+	0.391632i	$0.128089\pi$
$257$			12.5567i			0.783265i	−0.920122	+	0.391632i	$0.871911\pi$
$258$		0			0
$259$	10.0522			0.624611
$260$		0			0
$261$		0			0
$262$		0			0
$263$	24.3133			1.49922			0.749612	−	0.661877i	$-0.230240\pi$
$263$	24.3133			1.49922			0.749612	+	0.661877i	$0.230240\pi$
$264$		0			0
$265$			11.7886i			0.724167i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−5.07527			−0.309445			−0.154722	−	0.987958i	$-0.549448\pi$
$269$	−5.07527			−0.309445			−0.154722	+	0.987958i	$0.549448\pi$
$270$		0			0
$271$		−	0.814354i		−	0.0494685i	−0.999694	−	0.0247342i	$-0.992126\pi$
$271$		−	0.814354i		−	0.0494685i	0.999694	−	0.0247342i	$-0.00787395\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	58.3820			3.52057
$276$		0			0
$277$		−	26.1934i		−	1.57381i	−0.617074	−	0.786905i	$-0.711682\pi$
$277$		−	26.1934i		−	1.57381i	0.617074	−	0.786905i	$-0.288318\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	24.6745i		−	1.47196i	−0.677004	−	0.735979i	$-0.736722\pi$
$281$		−	24.6745i		−	1.47196i	0.677004	−	0.735979i	$-0.263278\pi$
$282$		0			0
$283$	14.0745			0.836642			0.418321	−	0.908299i	$-0.362619\pi$
$283$	14.0745			0.836642			0.418321	+	0.908299i	$0.362619\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$			4.73774i			0.279660i
$288$		0			0
$289$	−38.0071			−2.23571
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−14.5077			−0.847548			−0.423774	−	0.905768i	$-0.639295\pi$
$293$	−14.5077			−0.847548			−0.423774	+	0.905768i	$0.639295\pi$
$294$		0			0
$295$	3.97961			0.231702
$296$		0			0
$297$		0			0
$298$		0			0
$299$			13.5436i			0.783249i
$300$		0			0
$301$	−13.5383			−0.780333
$302$		0			0
$303$		0			0
$304$		0			0
$305$		−	37.7475i		−	2.16141i
$306$		0			0
$307$		−	34.6272i		−	1.97628i	−0.153553	−	0.988140i	$-0.549071\pi$
$307$		−	34.6272i		−	1.97628i	0.153553	−	0.988140i	$-0.450929\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	7.71486			0.437470			0.218735	−	0.975784i	$-0.429807\pi$
$311$	7.71486			0.437470			0.218735	+	0.975784i	$0.429807\pi$
$312$		0			0
$313$			28.4539i			1.60831i	0.594420	+	0.804155i	$0.297382\pi$
$313$			28.4539i			1.60831i	−0.594420	+	0.804155i	$0.702618\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	31.4378i		−	1.76572i	−0.469632	−	0.882862i	$-0.655613\pi$
$317$		−	31.4378i		−	1.76572i	0.469632	−	0.882862i	$-0.344387\pi$
$318$		0			0
$319$		−	32.6328i		−	1.82709i
$320$		0			0
$321$		0			0
$322$		0			0
$323$		−	61.6972i		−	3.43292i
$324$		0			0
$325$		−	50.1719i		−	2.78304i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		−	20.1205i		−	1.10928i
$330$		0			0
$331$		−	12.4935i		−	0.686703i	−0.939207	−	0.343351i	$-0.888438\pi$
$331$		−	12.4935i		−	0.686703i	0.939207	−	0.343351i	$-0.111562\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		−	6.52314i		−	0.356397i
$336$		0			0
$337$			34.8365i			1.89766i	0.315781	+	0.948832i	$0.397734\pi$
$337$			34.8365i			1.89766i	−0.315781	+	0.948832i	$0.602266\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	29.9383			1.62125
$342$		0			0
$343$	−19.1840			−1.03584
$344$		0			0
$345$		0			0
$346$		0			0
$347$		−	11.8567i		−	0.636501i	−0.948007	−	0.318251i	$-0.896905\pi$
$347$		−	11.8567i		−	0.636501i	0.948007	−	0.318251i	$-0.103095\pi$
$348$		0			0
$349$	29.7922			1.59474			0.797371	−	0.603489i	$-0.206223\pi$
$349$	29.7922			1.59474			0.797371	+	0.603489i	$0.206223\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$			19.2735i			1.02583i	0.858441	+	0.512913i	$0.171434\pi$
$353$			19.2735i			1.02583i	−0.858441	+	0.512913i	$0.828566\pi$
$354$		0			0
$355$	−30.0897			−1.59700
$356$		0			0
$357$		0			0
$358$		0			0
$359$			2.05711i			0.108570i	0.998525	+	0.0542850i	$0.0172879\pi$
$359$			2.05711i			0.108570i	−0.998525	+	0.0542850i	$0.982712\pi$
$360$		0			0
$361$	50.2009			2.64215
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−20.9187			−1.09494
$366$		0			0
$367$	−18.8162			−0.982199			−0.491100	−	0.871103i	$-0.663405\pi$
$367$	−18.8162			−0.982199			−0.491100	+	0.871103i	$0.663405\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		−	5.39981i		−	0.280344i
$372$		0			0
$373$		−	8.51663i		−	0.440974i	−0.975390	−	0.220487i	$-0.929235\pi$
$373$		−	8.51663i		−	0.440974i	0.975390	−	0.220487i	$-0.0707647\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−28.0438			−1.44433
$378$		0			0
$379$	−29.8622			−1.53392			−0.766959	−	0.641696i	$-0.778231\pi$
$379$	−29.8622			−1.53392			−0.766959	+	0.641696i	$0.778231\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	18.4880			0.944691			0.472346	−	0.881413i	$-0.343407\pi$
$383$	18.4880			0.944691			0.472346	+	0.881413i	$0.343407\pi$
$384$		0			0
$385$	−40.4583			−2.06195
$386$		0			0
$387$		0			0
$388$		0			0
$389$		−	17.9103i		−	0.908088i	−0.890979	−	0.454044i	$-0.849981\pi$
$389$		−	17.9103i		−	0.908088i	0.890979	−	0.454044i	$-0.150019\pi$
$390$		0			0
$391$			19.5172i			0.987025i
$392$		0			0
$393$		0			0
$394$		0			0
$395$			26.5745i			1.33711i
$396$		0			0
$397$			2.78904i			0.139978i	0.997548	+	0.0699891i	$0.0222964\pi$
$397$			2.78904i			0.139978i	−0.997548	+	0.0699891i	$0.977704\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		−	7.26702i		−	0.362898i	−0.983400	−	0.181449i	$-0.941921\pi$
$401$		−	7.26702i		−	0.362898i	0.983400	−	0.181449i	$-0.0580787\pi$
$402$		0			0
$403$		−	25.7282i		−	1.28161i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	34.2229			1.69637
$408$		0			0
$409$			21.1416i			1.04538i	0.852521	+	0.522692i	$0.175072\pi$
$409$			21.1416i			1.04538i	−0.852521	+	0.522692i	$0.824928\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−1.82288			−0.0896980
$414$		0			0
$415$		−	31.1826i		−	1.53069i
$416$		0			0
$417$		0			0
$418$		0			0
$419$			35.1158i			1.71552i	0.514051	+	0.857760i	$0.328144\pi$
$419$			35.1158i			1.71552i	−0.514051	+	0.857760i	$0.671856\pi$
$420$		0			0
$421$		−	25.0049i		−	1.21866i	−0.792916	−	0.609331i	$-0.791438\pi$
$421$		−	25.0049i		−	1.21866i	0.792916	−	0.609331i	$-0.208562\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		−	72.3006i		−	3.50709i
$426$		0			0
$427$			17.2904i			0.836740i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	5.81487			0.280093			0.140046	−	0.990145i	$-0.455275\pi$
$431$	5.81487			0.280093			0.140046	+	0.990145i	$0.455275\pi$
$432$		0			0
$433$	0.534136			0.0256689			0.0128345	−	0.999918i	$-0.495915\pi$
$433$	0.534136			0.0256689			0.0128345	+	0.999918i	$0.495915\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−21.8909			−1.04718
$438$		0			0
$439$			26.9522i			1.28636i	0.765717	+	0.643178i	$0.222384\pi$
$439$			26.9522i			1.28636i	−0.765717	+	0.643178i	$0.777616\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$			12.8170i			0.608955i	0.952520	+	0.304477i	$0.0984818\pi$
$443$			12.8170i			0.608955i	−0.952520	+	0.304477i	$0.901518\pi$
$444$		0			0
$445$		−	34.8257i		−	1.65090i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	35.0760			1.65534			0.827669	−	0.561217i	$-0.189667\pi$
$449$	35.0760			1.65534			0.827669	+	0.561217i	$0.189667\pi$
$450$		0			0
$451$			16.1298i			0.759521i
$452$		0			0
$453$		0			0
$454$		0			0
$455$			34.7688i			1.62998i
$456$		0			0
$457$		−	5.87919i		−	0.275017i	−0.990501	−	0.137508i	$-0.956091\pi$
$457$		−	5.87919i		−	0.275017i	0.990501	−	0.137508i	$-0.0439094\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		−	10.4554i		−	0.486958i	−0.969906	−	0.243479i	$-0.921711\pi$
$461$		−	10.4554i		−	0.486958i	0.969906	−	0.243479i	$-0.0782887\pi$
$462$		0			0
$463$	−16.1470			−0.750413			−0.375206	−	0.926941i	$-0.622428\pi$
$463$	−16.1470			−0.750413			−0.375206	+	0.926941i	$0.622428\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−33.4585			−1.54827			−0.774137	−	0.633018i	$-0.781816\pi$
$467$	−33.4585			−1.54827			−0.774137	+	0.633018i	$0.781816\pi$
$468$		0			0
$469$			2.98795i			0.137971i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−46.0915			−2.11929
$474$		0			0
$475$	81.0940			3.72085
$476$		0			0
$477$		0			0
$478$		0			0
$479$		−	35.2904i		−	1.61246i	−0.591602	−	0.806230i	$-0.701504\pi$
$479$		−	35.2904i		−	1.61246i	0.591602	−	0.806230i	$-0.298496\pi$
$480$		0			0
$481$		−	29.4102i		−	1.34099i
$482$		0			0
$483$		0			0
$484$		0			0
$485$			1.17301i			0.0532634i
$486$		0			0
$487$	−15.4665			−0.700852			−0.350426	−	0.936590i	$-0.613963\pi$
$487$	−15.4665			−0.700852			−0.350426	+	0.936590i	$0.613963\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−2.32103			−0.104747			−0.0523733	−	0.998628i	$-0.516679\pi$
$491$	−2.32103			−0.104747			−0.0523733	+	0.998628i	$0.516679\pi$
$492$		0			0
$493$	−40.4127			−1.82009
$494$		0			0
$495$		0			0
$496$		0			0
$497$	13.7827			0.618240
$498$		0			0
$499$	−4.10308			−0.183679			−0.0918396	−	0.995774i	$-0.529275\pi$
$499$	−4.10308			−0.183679			−0.0918396	+	0.995774i	$0.529275\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−6.19123			−0.276053			−0.138027	−	0.990429i	$-0.544076\pi$
$503$	−6.19123			−0.276053			−0.138027	+	0.990429i	$0.544076\pi$
$504$		0			0
$505$			2.57292i			0.114493i
$506$		0			0
$507$		0			0
$508$		0			0
$509$			21.3919i			0.948178i	0.880477	+	0.474089i	$0.157223\pi$
$509$			21.3919i			0.948178i	−0.880477	+	0.474089i	$0.842777\pi$
$510$		0			0
$511$	9.58190			0.423878
$512$		0			0
$513$		0			0
$514$		0			0
$515$		−	30.7091i		−	1.35321i
$516$		0			0
$517$		−	68.5010i		−	3.01267i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	21.3684			0.936166			0.468083	−	0.883684i	$-0.344945\pi$
$521$	21.3684			0.936166			0.468083	+	0.883684i	$0.344945\pi$
$522$		0			0
$523$		−	20.1591i		−	0.881498i	−0.897630	−	0.440749i	$-0.854713\pi$
$523$		−	20.1591i		−	0.881498i	0.897630	−	0.440749i	$-0.145287\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		−	37.0758i		−	1.61505i
$528$		0			0
$529$	−16.0751			−0.698917
$530$		0			0
$531$		0			0
$532$		0			0
$533$	13.8615			0.600408
$534$		0			0
$535$	12.0583			0.521327
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−23.3901			−1.00748
$540$		0			0
$541$		−	23.9544i		−	1.02988i	−0.857226	−	0.514941i	$-0.827814\pi$
$541$		−	23.9544i		−	1.02988i	0.857226	−	0.514941i	$-0.172186\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−20.2035			−0.865421
$546$		0			0
$547$	30.7380			1.31426			0.657132	−	0.753776i	$-0.271770\pi$
$547$	30.7380			1.31426			0.657132	+	0.753776i	$0.271770\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		−	45.3278i		−	1.93103i
$552$		0			0
$553$		−	12.1726i		−	0.517631i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−3.09972			−0.131339			−0.0656697	−	0.997841i	$-0.520918\pi$
$557$	−3.09972			−0.131339			−0.0656697	+	0.997841i	$0.520918\pi$
$558$		0			0
$559$			39.6097i			1.67531i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	2.04850			0.0863340			0.0431670	−	0.999068i	$-0.486255\pi$
$563$	2.04850			0.0863340			0.0431670	+	0.999068i	$0.486255\pi$
$564$		0			0
$565$	16.7352			0.704054
$566$		0			0
$567$		0			0
$568$		0			0
$569$	40.2770			1.68850			0.844250	−	0.535950i	$-0.180046\pi$
$569$	40.2770			1.68850			0.844250	+	0.535950i	$0.180046\pi$
$570$		0			0
$571$			29.3204i			1.22702i	0.789687	+	0.613509i	$0.210243\pi$
$571$			29.3204i			1.22702i	−0.789687	+	0.613509i	$0.789757\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−25.6531			−1.06981
$576$		0			0
$577$	0.782350			0.0325696			0.0162848	−	0.999867i	$-0.494816\pi$
$577$	0.782350			0.0325696			0.0162848	+	0.999867i	$0.494816\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$			14.2833i			0.592572i
$582$		0			0
$583$		−	18.3838i		−	0.761380i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−20.9552			−0.864912			−0.432456	−	0.901655i	$-0.642353\pi$
$587$	−20.9552			−0.864912			−0.432456	+	0.901655i	$0.642353\pi$
$588$		0			0
$589$	41.5851			1.71348
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−26.7414			−1.09814			−0.549069	−	0.835777i	$-0.685018\pi$
$593$	−26.7414			−1.09814			−0.549069	+	0.835777i	$0.685018\pi$
$594$		0			0
$595$			50.1038i			2.05405i
$596$		0			0
$597$		0			0
$598$		0			0
$599$			8.10910i			0.331329i	0.986182	+	0.165664i	$0.0529768\pi$
$599$			8.10910i			0.331329i	−0.986182	+	0.165664i	$0.947023\pi$
$600$		0			0
$601$			38.3430i			1.56404i	0.623251	+	0.782022i	$0.285812\pi$
$601$			38.3430i			1.56404i	−0.623251	+	0.782022i	$0.714188\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−95.4976			−3.88253
$606$		0			0
$607$			1.02394i			0.0415606i	0.999784	+	0.0207803i	$0.00661506\pi$
$607$			1.02394i			0.0415606i	−0.999784	+	0.0207803i	$0.993385\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−58.8678			−2.38154
$612$		0			0
$613$		−	16.8647i		−	0.681158i	−0.940216	−	0.340579i	$-0.889377\pi$
$613$		−	16.8647i		−	0.681158i	0.940216	−	0.340579i	$-0.110623\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			21.4891i			0.865118i	0.901606	+	0.432559i	$0.142389\pi$
$617$			21.4891i			0.865118i	−0.901606	+	0.432559i	$0.857611\pi$
$618$		0			0
$619$			34.4819i			1.38594i	0.720964	+	0.692972i	$0.243699\pi$
$619$			34.4819i			1.38594i	−0.720964	+	0.692972i	$0.756301\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$			15.9521i			0.639106i
$624$		0			0
$625$	21.2890			0.851561
$626$		0			0
$627$		0			0
$628$		0			0
$629$		−	42.3818i		−	1.68987i
$630$		0			0
$631$		−	12.7254i		−	0.506591i	−0.967389	−	0.253296i	$-0.918485\pi$
$631$		−	12.7254i		−	0.506591i	0.967389	−	0.253296i	$-0.0815145\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	10.2001			0.404778
$636$		0			0
$637$			20.1008i			0.796425i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−29.5625			−1.16765			−0.583824	−	0.811880i	$-0.698444\pi$
$641$	−29.5625			−1.16765			−0.583824	+	0.811880i	$0.698444\pi$
$642$		0			0
$643$	14.8720			0.586494			0.293247	−	0.956037i	$-0.405264\pi$
$643$	14.8720			0.586494			0.293247	+	0.956037i	$0.405264\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			7.74488i			0.304483i	0.988343	+	0.152241i	$0.0486491\pi$
$647$			7.74488i			0.304483i	−0.988343	+	0.152241i	$0.951351\pi$
$648$		0			0
$649$	−6.20605			−0.243609
$650$		0			0
$651$		0			0
$652$		0			0
$653$	16.7105			0.653934			0.326967	−	0.945036i	$-0.393973\pi$
$653$	16.7105			0.653934			0.326967	+	0.945036i	$0.393973\pi$
$654$		0			0
$655$		−	30.0600i		−	1.17454i
$656$		0			0
$657$		0			0
$658$		0			0
$659$			9.24886i			0.360285i	0.983641	+	0.180142i	$0.0576558\pi$
$659$			9.24886i			0.360285i	−0.983641	+	0.180142i	$0.942344\pi$
$660$		0			0
$661$			8.04614i			0.312958i	0.987681	+	0.156479i	$0.0500144\pi$
$661$			8.04614i			0.312958i	−0.987681	+	0.156479i	$0.949986\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−56.1975			−2.17925
$666$		0			0
$667$			14.3389i			0.555204i
$668$		0			0
$669$		0			0
$670$		0			0
$671$			58.8657i			2.27248i
$672$		0			0
$673$	−24.1004			−0.929003			−0.464502	−	0.885572i	$-0.653767\pi$
$673$	−24.1004			−0.929003			−0.464502	+	0.885572i	$0.653767\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		−	18.6147i		−	0.715420i	−0.933833	−	0.357710i	$-0.883558\pi$
$677$		−	18.6147i		−	0.715420i	0.933833	−	0.357710i	$-0.116442\pi$
$678$		0			0
$679$		−	0.537300i		−	0.0206197i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		−	9.45352i		−	0.361729i	−0.983508	−	0.180864i	$-0.942110\pi$
$683$		−	9.45352i		−	0.361729i	0.983508	−	0.180864i	$-0.0578895\pi$
$684$		0			0
$685$	35.4991			1.35635
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−15.7985			−0.601877
$690$		0			0
$691$		−	6.23717i		−	0.237273i	−0.992938	−	0.118637i	$-0.962148\pi$
$691$		−	6.23717i		−	0.237273i	0.992938	−	0.118637i	$-0.0378523\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	43.6832			1.65700
$696$		0			0
$697$	19.9752			0.756615
$698$		0			0
$699$		0			0
$700$		0			0
$701$			38.5793i			1.45712i	0.684981	+	0.728560i	$0.259810\pi$
$701$			38.5793i			1.45712i	−0.684981	+	0.728560i	$0.740190\pi$
$702$		0			0
$703$	47.5364			1.79287
$704$		0			0
$705$		0			0
$706$		0			0
$707$		−	1.17854i		−	0.0443234i
$708$		0			0
$709$		−	26.8480i		−	1.00830i	−0.863616	−	0.504149i	$-0.831806\pi$
$709$		−	26.8480i		−	1.00830i	0.863616	−	0.504149i	$-0.168194\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−13.1549			−0.492656
$714$		0			0
$715$			118.371i			4.42684i
$716$		0			0
$717$		0			0
$718$		0			0
$719$		−	17.1810i		−	0.640742i	−0.947292	−	0.320371i	$-0.896192\pi$
$719$		−	17.1810i		−	0.640742i	0.947292	−	0.320371i	$-0.103808\pi$
$720$		0			0
$721$			14.0664i			0.523862i
$722$		0			0
$723$		0			0
$724$		0			0
$725$		−	53.1179i		−	1.97275i
$726$		0			0
$727$	−40.7727			−1.51218			−0.756088	−	0.654470i	$-0.772892\pi$
$727$	−40.7727			−1.51218			−0.756088	+	0.654470i	$0.772892\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$			57.0799i			2.11118i
$732$		0			0
$733$	−12.0337			−0.444475			−0.222238	−	0.974993i	$-0.571336\pi$
$733$	−12.0337			−0.444475			−0.222238	+	0.974993i	$0.571336\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			10.1726i			0.374711i
$738$		0			0
$739$		−	44.1979i		−	1.62585i	−0.582370	−	0.812924i	$-0.697875\pi$
$739$		−	44.1979i		−	1.62585i	0.582370	−	0.812924i	$-0.302125\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	17.0993i		−	0.627314i	−0.949536	−	0.313657i	$-0.898446\pi$
$743$		−	17.0993i		−	0.627314i	0.949536	−	0.313657i	$-0.101554\pi$
$744$		0			0
$745$	59.1724			2.16791
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−5.52336			−0.201819
$750$		0			0
$751$	−7.52779			−0.274693			−0.137346	−	0.990523i	$-0.543857\pi$
$751$	−7.52779			−0.274693			−0.137346	+	0.990523i	$0.543857\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$			72.4138i			2.63541i
$756$		0			0
$757$		−	19.3568i		−	0.703536i	−0.936087	−	0.351768i	$-0.885581\pi$
$757$		−	19.3568i		−	0.703536i	0.936087	−	0.351768i	$-0.114419\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−3.85457			−0.139728			−0.0698640	−	0.997557i	$-0.522257\pi$
$761$	−3.85457			−0.139728			−0.0698640	+	0.997557i	$0.522257\pi$
$762$		0			0
$763$	9.25428			0.335027
$764$		0			0
$765$		0			0
$766$		0			0
$767$			5.33331i			0.192575i
$768$		0			0
$769$	42.7949			1.54322			0.771612	−	0.636094i	$-0.219451\pi$
$769$	42.7949			1.54322			0.771612	+	0.636094i	$0.219451\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	41.9392			1.50845			0.754224	−	0.656617i	$-0.228013\pi$
$773$	41.9392			1.50845			0.754224	+	0.656617i	$0.228013\pi$
$774$		0			0
$775$	48.7320			1.75050
$776$		0			0
$777$		0			0
$778$		0			0
$779$			22.4046i			0.802729i
$780$		0			0
$781$	46.9237			1.67906
$782$		0			0
$783$		0			0
$784$		0			0
$785$			57.7255i			2.06031i
$786$		0			0
$787$		−	7.73416i		−	0.275693i	−0.990454	−	0.137847i	$-0.955982\pi$
$787$		−	7.73416i		−	0.275693i	0.990454	−	0.137847i	$-0.0440181\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−7.66562			−0.272558
$792$		0			0
$793$	50.5875			1.79642
$794$		0			0
$795$		0			0
$796$		0			0
$797$		−	40.6414i		−	1.43959i	−0.694185	−	0.719796i	$-0.744235\pi$
$797$		−	40.6414i		−	1.43959i	0.694185	−	0.719796i	$-0.255765\pi$
$798$		0			0
$799$	−84.8320			−3.00114
$800$		0			0
$801$		0			0
$802$		0			0
$803$	32.6219			1.15120
$804$		0			0
$805$	17.7774			0.626571
$806$		0			0
$807$		0			0
$808$		0			0
$809$	45.1580			1.58767			0.793835	−	0.608133i	$-0.208081\pi$
$809$	45.1580			1.58767			0.793835	+	0.608133i	$0.208081\pi$
$810$		0			0
$811$			30.6080i			1.07479i	0.843329	+	0.537397i	$0.180592\pi$
$811$			30.6080i			1.07479i	−0.843329	+	0.537397i	$0.819408\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		−	51.3849i		−	1.79994i
$816$		0			0
$817$	−64.0221			−2.23985
$818$		0			0
$819$		0			0
$820$		0			0
$821$			47.9518i			1.67353i	0.547563	+	0.836765i	$0.315556\pi$
$821$			47.9518i			1.67353i	−0.547563	+	0.836765i	$0.684444\pi$
$822$		0			0
$823$		−	34.8971i		−	1.21644i	−0.793770	−	0.608218i	$-0.791885\pi$
$823$		−	34.8971i		−	1.21644i	0.793770	−	0.608218i	$-0.208115\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	47.2514			1.64309			0.821545	−	0.570143i	$-0.193112\pi$
$827$	47.2514			1.64309			0.821545	+	0.570143i	$0.193112\pi$
$828$		0			0
$829$			4.89282i			0.169935i	0.996384	+	0.0849674i	$0.0270786\pi$
$829$			4.89282i			0.169935i	−0.996384	+	0.0849674i	$0.972921\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$			28.9665i			1.00363i
$834$		0			0
$835$	43.5366			1.50665
$836$		0			0
$837$		0			0
$838$		0			0
$839$	8.46132			0.292117			0.146059	−	0.989276i	$-0.453341\pi$
$839$	8.46132			0.292117			0.146059	+	0.989276i	$0.453341\pi$
$840$		0			0
$841$	−0.690425			−0.0238078
$842$		0			0
$843$		0			0
$844$		0			0
$845$	51.8004			1.78199
$846$		0			0
$847$	43.7431			1.50303
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−15.0376			−0.515482
$852$		0			0
$853$			42.5391i			1.45651i	0.685306	+	0.728255i	$0.259668\pi$
$853$			42.5391i			1.45651i	−0.685306	+	0.728255i	$0.740332\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$			51.7346i			1.76722i	0.468223	+	0.883610i	$0.344894\pi$
$857$			51.7346i			1.76722i	−0.468223	+	0.883610i	$0.655106\pi$
$858$		0			0
$859$		−	43.9477i		−	1.49947i	−0.661736	−	0.749737i	$-0.730180\pi$
$859$		−	43.9477i		−	1.49947i	0.661736	−	0.749737i	$-0.269820\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−33.2934			−1.13332			−0.566661	−	0.823951i	$-0.691765\pi$
$863$	−33.2934			−1.13332			−0.566661	+	0.823951i	$0.691765\pi$
$864$		0			0
$865$	−49.2567			−1.67478
$866$		0			0
$867$		0			0
$868$		0			0
$869$		−	41.4419i		−	1.40582i
$870$		0			0
$871$	8.74203			0.296212
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−32.0780			−1.08443
$876$		0			0
$877$			21.7256i			0.733621i	0.930296	+	0.366811i	$0.119550\pi$
$877$			21.7256i			0.733621i	−0.930296	+	0.366811i	$0.880450\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	32.9824i		−	1.11121i	−0.831448	−	0.555603i	$-0.812487\pi$
$881$		−	32.9824i		−	1.11121i	0.831448	−	0.555603i	$-0.187513\pi$
$882$		0			0
$883$			45.4984i			1.53114i	0.643350	+	0.765572i	$0.277544\pi$
$883$			45.4984i			1.53114i	−0.643350	+	0.765572i	$0.722456\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		−	3.28662i		−	0.110354i	−0.998477	−	0.0551770i	$-0.982428\pi$
$887$		−	3.28662i		−	0.110354i	0.998477	−	0.0551770i	$-0.0175723\pi$
$888$		0			0
$889$	−4.67220			−0.156700
$890$		0			0
$891$		0			0
$892$		0			0
$893$		−	95.1494i		−	3.18405i
$894$		0			0
$895$			77.8288i			2.60153i
$896$		0			0
$897$		0			0
$898$		0			0
$899$		−	27.2389i		−	0.908468i
$900$		0			0
$901$	−22.7666			−0.758466
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−68.1435			−2.26517
$906$		0			0
$907$	−52.6858			−1.74940			−0.874702	−	0.484662i	$-0.838943\pi$
$907$	−52.6858			−1.74940			−0.874702	+	0.484662i	$0.838943\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			13.9976i			0.463760i	0.972744	+	0.231880i	$0.0744877\pi$
$911$			13.9976i			0.463760i	−0.972744	+	0.231880i	$0.925512\pi$
$912$		0			0
$913$			48.6280i			1.60935i
$914$		0			0
$915$		0			0
$916$		0			0
$917$			13.7691i			0.454696i
$918$		0			0
$919$			43.0324i			1.41951i	0.704450	+	0.709754i	$0.251194\pi$
$919$			43.0324i			1.41951i	−0.704450	+	0.709754i	$0.748806\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		−	40.3250i		−	1.32731i
$924$		0			0
$925$	55.7061			1.83161
$926$		0			0
$927$		0			0
$928$		0			0
$929$		−	5.80809i		−	0.190557i	−0.995451	−	0.0952786i	$-0.969626\pi$
$929$		−	5.80809i		−	0.190557i	0.995451	−	0.0952786i	$-0.0303742\pi$
$930$		0			0
$931$	−32.4895			−1.06480
$932$		0			0
$933$		0			0
$934$		0			0
$935$			170.580i			5.57856i
$936$		0			0
$937$		−	4.80069i		−	0.156832i	−0.996921	−	0.0784158i	$-0.975014\pi$
$937$		−	4.80069i		−	0.156832i	0.996921	−	0.0784158i	$-0.0249862\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$			6.34405i			0.206810i	0.994639	+	0.103405i	$0.0329738\pi$
$941$			6.34405i			0.206810i	−0.994639	+	0.103405i	$0.967026\pi$
$942$		0			0
$943$		−	7.08744i		−	0.230799i
$944$		0			0
$945$		0			0
$946$		0			0
$947$		−	10.0695i		−	0.327214i	−0.986526	−	0.163607i	$-0.947687\pi$
$947$		−	10.0695i		−	0.327214i	0.986526	−	0.163607i	$-0.0523129\pi$
$948$		0			0
$949$		−	28.0344i		−	0.910034i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	57.2952			1.85597			0.927986	−	0.372615i	$-0.121539\pi$
$953$	57.2952			1.85597			0.927986	+	0.372615i	$0.121539\pi$
$954$		0			0
$955$	1.53555			0.0496891
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−16.2605			−0.525079
$960$		0			0
$961$	−6.01021			−0.193878
$962$		0			0
$963$		0			0
$964$		0			0
$965$		−	37.8996i		−	1.22003i
$966$		0			0
$967$		−	33.1170i		−	1.06497i	−0.846439	−	0.532486i	$-0.821258\pi$
$967$		−	33.1170i		−	1.06497i	0.846439	−	0.532486i	$-0.178742\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−2.24321			−0.0719880			−0.0359940	−	0.999352i	$-0.511460\pi$
$971$	−2.24321			−0.0719880			−0.0359940	+	0.999352i	$0.511460\pi$
$972$		0			0
$973$	−20.0093			−0.641468
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−14.1526			−0.452783			−0.226392	−	0.974036i	$-0.572693\pi$
$977$	−14.1526			−0.452783			−0.226392	+	0.974036i	$0.572693\pi$
$978$		0			0
$979$			54.3093i			1.73573i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	7.01691			0.223805			0.111902	−	0.993719i	$-0.464306\pi$
$983$	7.01691			0.223805			0.111902	+	0.993719i	$0.464306\pi$
$984$		0			0
$985$		−	28.4508i		−	0.906517i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	20.2526			0.643996
$990$		0			0
$991$			36.6977i			1.16574i	0.812565	+	0.582871i	$0.198071\pi$
$991$			36.6977i			1.16574i	−0.812565	+	0.582871i	$0.801929\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−47.3299			−1.50046
$996$		0			0
$997$	2.38591			0.0755625			0.0377812	−	0.999286i	$-0.487971\pi$
$997$	2.38591			0.0755625			0.0377812	+	0.999286i	$0.487971\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8028.2.h.a.4013.4	yes	76
3.2	odd	2	inner	8028.2.h.a.4013.74	yes	76
223.222	odd	2	inner	8028.2.h.a.4013.73	yes	76
669.668	even	2	inner	8028.2.h.a.4013.3	✓	76

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8028.2.h.a.4013.3	✓	76	669.668	even	2	inner
8028.2.h.a.4013.4	yes	76	1.1	even	1	trivial
8028.2.h.a.4013.73	yes	76	223.222	odd	2	inner
8028.2.h.a.4013.74	yes	76	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:	\( N \)	\(=\)	\( 8028 = 2^{2} \cdot 3^{2} \cdot 223 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8028.h (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-3.84036 q^{5} +1.75909 q^{7} +O(q^{10})\)	\(q-3.84036 q^{5} +1.75909 q^{7} +5.98889 q^{11} -5.14669i q^{13} -7.41667i q^{17} +8.31871 q^{19} -2.63152 q^{23} +9.74838 q^{25} -5.44889i q^{29} +4.99898 q^{31} -6.75556 q^{35} +5.71440 q^{37} +2.69328i q^{41} -7.69616 q^{43} -11.4380i q^{47} -3.90559 q^{49} -3.06965i q^{53} -22.9995 q^{55} -1.03626 q^{59} +9.82914i q^{61} +19.7651i q^{65} +1.69857i q^{67} +7.83513 q^{71} +5.44707 q^{73} +10.5350 q^{77} -6.91980i q^{79} +8.11970i q^{83} +28.4827i q^{85} +9.06835i q^{89} -9.05351i q^{91} -31.9469 q^{95} -0.305441i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 76 q + 8 q^{7}+O(q^{10}) \) 76 * q + 8 * q^7	\( 76 q + 8 q^{7} + 16 q^{19} + 100 q^{25} - 8 q^{31} + 32 q^{37} - 24 q^{43} + 68 q^{49} + 24 q^{55} + 8 q^{73}+O(q^{100}) \) 76 * q + 8 * q^7 + 16 * q^19 + 100 * q^25 - 8 * q^31 + 32 * q^37 - 24 * q^43 + 68 * q^49 + 24 * q^55 + 8 * q^73

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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