LMFDB - Embedded newform 6048.2.h.a.2591.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6048,2,Mod(2591,6048)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6048, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0, 1, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("6048.2591");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6048 = 2^{5} \cdot 3^{3} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6048.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:	$48.2935231425$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\Q(\zeta_{24})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{4} + 1 $ x^8 - x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2591.7
Root			$0.965926 + 0.258819i$ of defining polynomial
Character	$\chi$	$=$	6048.2591
Dual form			6048.2.h.a.2591.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.66390i q^{5} +1.00000i q^{7} +O(q^{10})$	$q+2.66390i q^{5} +1.00000i q^{7} -0.164525 q^{11} +3.56048 q^{13} +3.41421i q^{17} -3.89898i q^{19} -5.29958 q^{23} -2.09638 q^{25} -7.84304i q^{29} +8.72741i q^{31} -2.66390 q^{35} +7.82843 q^{37} +1.62863i q^{41} +11.2173i q^{43} +0.585786 q^{47} -1.00000 q^{49} +14.2268i q^{53} -0.438278i q^{55} -6.98539 q^{59} +4.92820 q^{61} +9.48477i q^{65} +6.83183i q^{67} +10.2144 q^{71} +5.48993 q^{73} -0.164525i q^{77} -9.09513i q^{79} -1.95492 q^{83} -9.09513 q^{85} -6.05745i q^{89} +3.56048i q^{91} +10.3865 q^{95} +4.72865 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q+O(q^{10}) $ 8 * q	$ 8 q - 8 q^{11} - 8 q^{13} - 8 q^{23} - 8 q^{25} + 8 q^{35} + 40 q^{37} + 16 q^{47} - 8 q^{49} - 64 q^{59} - 16 q^{61} + 72 q^{71} + 24 q^{73} + 32 q^{83} + 8 q^{85} + 56 q^{95} + 48 q^{97}+O(q^{100}) $ 8 * q - 8 * q^11 - 8 * q^13 - 8 * q^23 - 8 * q^25 + 8 * q^35 + 40 * q^37 + 16 * q^47 - 8 * q^49 - 64 * q^59 - 16 * q^61 + 72 * q^71 + 24 * q^73 + 32 * q^83 + 8 * q^85 + 56 * q^95 + 48 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/6048\mathbb{Z}\right)^\times$.

$n$	$2593$	$3781$	$3809$	$4159$
$\chi(n)$	$1$	$1$	$-1$	$-1$

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
$5$	2.66390i	1.19133i	0.803232	+	0.595667i	$0.203112\pi$
$5$	2.66390i	1.19133i	−0.803232	+	0.595667i	$0.796888\pi$
$6$	0	0
$7$	1.00000i	0.377964i
$7$	1.00000i	0.377964i
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−0.164525	−0.0496061	−0.0248030	−	0.999692i	$-0.507896\pi$
$11$	−0.164525	−0.0496061	−0.0248030	+	0.999692i	$0.507896\pi$
$12$	0	0
$13$	3.56048	0.987499	0.493749	−	0.869604i	$-0.335626\pi$
$13$	3.56048	0.987499	0.493749	+	0.869604i	$0.335626\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	3.41421i	0.828068i	0.910261	+	0.414034i	$0.135881\pi$
$17$	3.41421i	0.828068i	−0.910261	+	0.414034i	$0.864119\pi$
$18$	0	0
$19$	− 3.89898i	− 0.894487i	−0.894412	−	0.447244i	$-0.852406\pi$
$19$	− 3.89898i	− 0.894487i	0.894412	−	0.447244i	$-0.147594\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−5.29958	−1.10504	−0.552519	−	0.833500i	$-0.686333\pi$
$23$	−5.29958	−1.10504	−0.552519	+	0.833500i	$0.686333\pi$
$24$	0	0
$25$	−2.09638	−0.419275
$26$	0	0
$27$	0	0
$28$	0	0
$29$	− 7.84304i	− 1.45642i	−0.685356	−	0.728208i	$-0.740354\pi$
$29$	− 7.84304i	− 1.45642i	0.685356	−	0.728208i	$-0.259646\pi$
$30$	0	0
$31$	8.72741i	1.56749i	0.621084	+	0.783744i	$0.286693\pi$
$31$	8.72741i	1.56749i	−0.621084	+	0.783744i	$0.713307\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−2.66390	−0.450282
$36$	0	0
$37$	7.82843	1.28699	0.643493	−	0.765452i	$-0.277485\pi$
$37$	7.82843	1.28699	0.643493	+	0.765452i	$0.277485\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	1.62863i	0.254349i	0.991880	+	0.127174i	$0.0405908\pi$
$41$	1.62863i	0.254349i	−0.991880	+	0.127174i	$0.959409\pi$
$42$	0	0
$43$	11.2173i	1.71063i	0.518111	+	0.855314i	$0.326636\pi$
$43$	11.2173i	1.71063i	−0.518111	+	0.855314i	$0.673364\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	0.585786	0.0854457	0.0427229	−	0.999087i	$-0.486397\pi$
$47$	0.585786	0.0854457	0.0427229	+	0.999087i	$0.486397\pi$
$48$	0	0
$49$	−1.00000	−0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	14.2268i	1.95420i	0.212783	+	0.977100i	$0.431747\pi$
$53$	14.2268i	1.95420i	−0.212783	+	0.977100i	$0.568253\pi$
$54$	0	0
$55$	− 0.438278i	− 0.0590973i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−6.98539	−0.909420	−0.454710	−	0.890640i	$-0.650257\pi$
$59$	−6.98539	−0.909420	−0.454710	+	0.890640i	$0.650257\pi$
$60$	0	0
$61$	4.92820	0.630992	0.315496	−	0.948927i	$-0.397829\pi$
$61$	4.92820	0.630992	0.315496	+	0.948927i	$0.397829\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	9.48477i	1.17644i
$66$	0	0
$67$	6.83183i	0.834640i	0.908759	+	0.417320i	$0.137031\pi$
$67$	6.83183i	0.834640i	−0.908759	+	0.417320i	$0.862969\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	10.2144	1.21223	0.606114	−	0.795378i	$-0.292728\pi$
$71$	10.2144	1.21223	0.606114	+	0.795378i	$0.292728\pi$
$72$	0	0
$73$	5.48993	0.642547	0.321274	−	0.946986i	$-0.395889\pi$
$73$	5.48993	0.642547	0.321274	+	0.946986i	$0.395889\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 0.164525i	− 0.0187493i
$78$	0	0
$79$	− 9.09513i	− 1.02328i	−0.859199	−	0.511641i	$-0.829038\pi$
$79$	− 9.09513i	− 1.02328i	0.859199	−	0.511641i	$-0.170962\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−1.95492	−0.214580	−0.107290	−	0.994228i	$-0.534217\pi$
$83$	−1.95492	−0.214580	−0.107290	+	0.994228i	$0.534217\pi$
$84$	0	0
$85$	−9.09513	−0.986506
$86$	0	0
$87$	0	0
$88$	0	0
$89$	− 6.05745i	− 0.642089i	−0.947064	−	0.321044i	$-0.895966\pi$
$89$	− 6.05745i	− 0.642089i	0.947064	−	0.321044i	$-0.104034\pi$
$90$	0	0
$91$	3.56048i	0.373240i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	10.3865	1.06563
$96$	0	0
$97$	4.72865	0.480122	0.240061	−	0.970758i	$-0.422833\pi$
$97$	4.72865	0.480122	0.240061	+	0.970758i	$0.422833\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	10.6563i	1.06035i	0.847890	+	0.530173i	$0.177873\pi$
$101$	10.6563i	1.06035i	−0.847890	+	0.530173i	$0.822127\pi$
$102$	0	0
$103$	− 6.58630i	− 0.648968i	−0.945891	−	0.324484i	$-0.894809\pi$
$103$	− 6.58630i	− 0.648968i	0.945891	−	0.324484i	$-0.105191\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−15.1416	−1.46380	−0.731898	−	0.681414i	$-0.761365\pi$
$107$	−15.1416	−1.46380	−0.731898	+	0.681414i	$0.761365\pi$
$108$	0	0
$109$	−15.1163	−1.44788	−0.723940	−	0.689863i	$-0.757671\pi$
$109$	−15.1163	−1.44788	−0.723940	+	0.689863i	$0.757671\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	5.69818i	0.536040i	0.963413	+	0.268020i	$0.0863693\pi$
$113$	5.69818i	0.536040i	−0.963413	+	0.268020i	$0.913631\pi$
$114$	0	0
$115$	− 14.1176i	− 1.31647i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−3.41421	−0.312980
$120$	0	0
$121$	−10.9729	−0.997539
$122$	0	0
$123$	0	0
$124$	0	0
$125$	7.73497i	0.691837i
$126$	0	0
$127$	− 5.63103i	− 0.499673i	−0.968288	−	0.249837i	$-0.919623\pi$
$127$	− 5.63103i	− 0.499673i	0.968288	−	0.249837i	$-0.0803769\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−6.71279	−0.586500	−0.293250	−	0.956036i	$-0.594737\pi$
$131$	−6.71279	−0.586500	−0.293250	+	0.956036i	$0.594737\pi$
$132$	0	0
$133$	3.89898	0.338084
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 8.91484i	− 0.761646i	−0.924648	−	0.380823i	$-0.875641\pi$
$137$	− 8.91484i	− 0.761646i	0.924648	−	0.380823i	$-0.124359\pi$
$138$	0	0
$139$	9.46410i	0.802735i	0.915917	+	0.401367i	$0.131465\pi$
$139$	9.46410i	0.802735i	−0.915917	+	0.401367i	$0.868535\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−0.585786	−0.0489859
$144$	0	0
$145$	20.8931	1.73508
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 15.0280i	− 1.23114i	−0.788082	−	0.615570i	$-0.788926\pi$
$149$	− 15.0280i	− 1.23114i	0.788082	−	0.615570i	$-0.211074\pi$
$150$	0	0
$151$	0	0	1.00000			$0$
$151$	0	0	−1.00000			$\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−23.2490	−1.86740
$156$	0	0
$157$	17.2173	1.37409	0.687046	−	0.726614i	$-0.258907\pi$
$157$	17.2173	1.37409	0.687046	+	0.726614i	$0.258907\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	− 5.29958i	− 0.417665i
$162$	0	0
$163$	17.3584i	1.35962i	0.733389	+	0.679809i	$0.237937\pi$
$163$	17.3584i	1.35962i	−0.733389	+	0.679809i	$0.762063\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	1.18670	0.0918296	0.0459148	−	0.998945i	$-0.485380\pi$
$167$	1.18670	0.0918296	0.0459148	+	0.998945i	$0.485380\pi$
$168$	0	0
$169$	−0.322997	−0.0248459
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 7.73497i	− 0.588079i	−0.955793	−	0.294039i	$-0.905000\pi$
$173$	− 7.73497i	− 0.588079i	0.955793	−	0.294039i	$-0.0949997\pi$
$174$	0	0
$175$	− 2.09638i	− 0.158471i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−11.8844	−0.888279	−0.444140	−	0.895958i	$-0.646491\pi$
$179$	−11.8844	−0.888279	−0.444140	+	0.895958i	$0.646491\pi$
$180$	0	0
$181$	0.141105	0.0104882	0.00524412	−	0.999986i	$-0.498331\pi$
$181$	0.141105	0.0104882	0.00524412	+	0.999986i	$0.498331\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	20.8542i	1.53323i
$186$	0	0
$187$	− 0.561722i	− 0.0410772i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−14.5841	−1.05527	−0.527633	−	0.849473i	$-0.676920\pi$
$191$	−14.5841	−1.05527	−0.527633	+	0.849473i	$0.676920\pi$
$192$	0	0
$193$	−14.1928	−1.02162	−0.510808	−	0.859695i	$-0.670654\pi$
$193$	−14.1928	−1.02162	−0.510808	+	0.859695i	$0.670654\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	24.8824i	1.77280i	0.462923	+	0.886399i	$0.346801\pi$
$197$	24.8824i	1.77280i	−0.462923	+	0.886399i	$0.653199\pi$
$198$	0	0
$199$	− 23.4181i	− 1.66007i	−0.557713	−	0.830034i	$-0.688321\pi$
$199$	− 23.4181i	− 1.66007i	0.557713	−	0.830034i	$-0.311679\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	7.84304	0.550473
$204$	0	0
$205$	−4.33850	−0.303014
$206$	0	0
$207$	0	0
$208$	0	0
$209$	0.641478i	0.0443720i
$210$	0	0
$211$	15.1210i	1.04097i	0.853871	+	0.520485i	$0.174249\pi$
$211$	15.1210i	1.04097i	−0.853871	+	0.520485i	$0.825751\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−29.8819	−2.03793
$216$	0	0
$217$	−8.72741	−0.592455
$218$	0	0
$219$	0	0
$220$	0	0
$221$	12.1562i	0.817717i
$222$	0	0
$223$	5.01654i	0.335932i	0.985793	+	0.167966i	$0.0537199\pi$
$223$	5.01654i	0.335932i	−0.985793	+	0.167966i	$0.946280\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−10.6556	−0.707238	−0.353619	−	0.935390i	$-0.615049\pi$
$227$	−10.6556	−0.707238	−0.353619	+	0.935390i	$0.615049\pi$
$228$	0	0
$229$	−18.9189	−1.25020	−0.625099	−	0.780546i	$-0.714941\pi$
$229$	−18.9189	−1.25020	−0.625099	+	0.780546i	$0.714941\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	8.89898i	0.582992i	0.956572	+	0.291496i	$0.0941529\pi$
$233$	8.89898i	0.582992i	−0.956572	+	0.291496i	$0.905847\pi$
$234$	0	0
$235$	1.56048i	0.101794i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−5.11439	−0.330822	−0.165411	−	0.986225i	$-0.552895\pi$
$239$	−5.11439	−0.330822	−0.165411	+	0.986225i	$0.552895\pi$
$240$	0	0
$241$	6.03262	0.388595	0.194298	−	0.980943i	$-0.437757\pi$
$241$	6.03262	0.388595	0.194298	+	0.980943i	$0.437757\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	− 2.66390i	− 0.170190i
$246$	0	0
$247$	− 13.8822i	− 0.883305i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	7.95867	0.502347	0.251173	−	0.967942i	$-0.419184\pi$
$251$	7.95867	0.502347	0.251173	+	0.967942i	$0.419184\pi$
$252$	0	0
$253$	0.871911	0.0548166
$254$	0	0
$255$	0	0
$256$	0	0
$257$	24.3565i	1.51932i	0.650321	+	0.759660i	$0.274635\pi$
$257$	24.3565i	1.51932i	−0.650321	+	0.759660i	$0.725365\pi$
$258$	0	0
$259$	7.82843i	0.486435i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−16.9106	−1.04275	−0.521377	−	0.853327i	$-0.674581\pi$
$263$	−16.9106	−1.04275	−0.521377	+	0.853327i	$0.674581\pi$
$264$	0	0
$265$	−37.8988	−2.32810
$266$	0	0
$267$	0	0
$268$	0	0
$269$	4.20512i	0.256391i	0.991749	+	0.128195i	$0.0409185\pi$
$269$	4.20512i	0.256391i	−0.991749	+	0.128195i	$0.959082\pi$
$270$	0	0
$271$	− 25.6544i	− 1.55839i	−0.626781	−	0.779196i	$-0.715628\pi$
$271$	− 25.6544i	− 1.55839i	0.626781	−	0.779196i	$-0.284372\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0.344906	0.0207986
$276$	0	0
$277$	13.0320	0.783019	0.391510	−	0.920174i	$-0.371953\pi$
$277$	13.0320	0.783019	0.391510	+	0.920174i	$0.371953\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	20.1252i	1.20057i	0.799786	+	0.600286i	$0.204947\pi$
$281$	20.1252i	1.20057i	−0.799786	+	0.600286i	$0.795053\pi$
$282$	0	0
$283$	− 3.70850i	− 0.220448i	−0.993907	−	0.110224i	$-0.964843\pi$
$283$	− 3.70850i	− 0.220448i	0.993907	−	0.110224i	$-0.0351568\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−1.62863	−0.0961348
$288$	0	0
$289$	5.34315	0.314303
$290$	0	0
$291$	0	0
$292$	0	0
$293$	32.3696i	1.89106i	0.325542	+	0.945528i	$0.394453\pi$
$293$	32.3696i	1.89106i	−0.325542	+	0.945528i	$0.605547\pi$
$294$	0	0
$295$	− 18.6084i	− 1.08342i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−18.8690	−1.09122
$300$	0	0
$301$	−11.2173	−0.646556
$302$	0	0
$303$	0	0
$304$	0	0
$305$	13.1283i	0.751722i
$306$	0	0
$307$	0.300573i	0.0171546i	0.999963	+	0.00857730i	$0.00273027\pi$
$307$	0.300573i	0.0171546i	−0.999963	+	0.00857730i	$0.997270\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	11.5140	0.652898	0.326449	−	0.945215i	$-0.394148\pi$
$311$	11.5140	0.652898	0.326449	+	0.945215i	$0.394148\pi$
$312$	0	0
$313$	−3.28788	−0.185842	−0.0929211	−	0.995673i	$-0.529620\pi$
$313$	−3.28788	−0.185842	−0.0929211	+	0.995673i	$0.529620\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 21.9267i	− 1.23153i	−0.787931	−	0.615764i	$-0.788848\pi$
$317$	− 21.9267i	− 1.23153i	0.787931	−	0.615764i	$-0.211152\pi$
$318$	0	0
$319$	1.29037i	0.0722470i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	13.3119	0.740697
$324$	0	0
$325$	−7.46410	−0.414034
$326$	0	0
$327$	0	0
$328$	0	0
$329$	0.585786i	0.0322955i
$330$	0	0
$331$	− 24.8673i	− 1.36683i	−0.730031	−	0.683414i	$-0.760494\pi$
$331$	− 24.8673i	− 1.36683i	0.730031	−	0.683414i	$-0.239506\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−18.1993	−0.994335
$336$	0	0
$337$	−2.80725	−0.152921	−0.0764603	−	0.997073i	$-0.524362\pi$
$337$	−2.80725	−0.152921	−0.0764603	+	0.997073i	$0.524362\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	− 1.43587i	− 0.0777569i
$342$	0	0
$343$	− 1.00000i	− 0.0539949i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	9.04765	0.485703	0.242852	−	0.970063i	$-0.421917\pi$
$347$	9.04765	0.485703	0.242852	+	0.970063i	$0.421917\pi$
$348$	0	0
$349$	−24.7169	−1.32306	−0.661532	−	0.749917i	$-0.730093\pi$
$349$	−24.7169	−1.32306	−0.661532	+	0.749917i	$0.730093\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 14.0161i	− 0.746003i	−0.927831	−	0.373001i	$-0.878329\pi$
$353$	− 14.0161i	− 0.746003i	0.927831	−	0.373001i	$-0.121671\pi$
$354$	0	0
$355$	27.2102i	1.44417i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	20.2870	1.07071	0.535353	−	0.844628i	$-0.320178\pi$
$359$	20.2870	1.07071	0.535353	+	0.844628i	$0.320178\pi$
$360$	0	0
$361$	3.79796	0.199893
$362$	0	0
$363$	0	0
$364$	0	0
$365$	14.6246i	0.765488i
$366$	0	0
$367$	− 28.3887i	− 1.48188i	−0.671573	−	0.740939i	$-0.734381\pi$
$367$	− 28.3887i	− 1.48188i	0.671573	−	0.740939i	$-0.265619\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−14.2268	−0.738618
$372$	0	0
$373$	29.6522	1.53533	0.767667	−	0.640849i	$-0.221418\pi$
$373$	29.6522	1.53533	0.767667	+	0.640849i	$0.221418\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 27.9250i	− 1.43821i
$378$	0	0
$379$	− 7.15888i	− 0.367727i	−0.982952	−	0.183864i	$-0.941140\pi$
$379$	− 7.15888i	− 0.367727i	0.982952	−	0.183864i	$-0.0588605\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−0.0413286	−0.00211179	−0.00105590	−	0.999999i	$-0.500336\pi$
$383$	−0.0413286	−0.00211179	−0.00105590	+	0.999999i	$0.500336\pi$
$384$	0	0
$385$	0.438278	0.0223367
$386$	0	0
$387$	0	0
$388$	0	0
$389$	− 16.2147i	− 0.822117i	−0.911609	−	0.411058i	$-0.865159\pi$
$389$	− 16.2147i	− 0.822117i	0.911609	−	0.411058i	$-0.134841\pi$
$390$	0	0
$391$	− 18.0939i	− 0.915047i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	24.2285	1.21907
$396$	0	0
$397$	−19.5536	−0.981365	−0.490682	−	0.871338i	$-0.663252\pi$
$397$	−19.5536	−0.981365	−0.490682	+	0.871338i	$0.663252\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	22.5964i	1.12841i	0.825634	+	0.564206i	$0.190818\pi$
$401$	22.5964i	1.12841i	−0.825634	+	0.564206i	$0.809182\pi$
$402$	0	0
$403$	31.0737i	1.54789i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−1.28797	−0.0638423
$408$	0	0
$409$	27.5298	1.36126	0.680630	−	0.732627i	$-0.261706\pi$
$409$	27.5298	1.36126	0.680630	+	0.732627i	$0.261706\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	− 6.98539i	− 0.343728i
$414$	0	0
$415$	− 5.20772i	− 0.255637i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−27.1691	−1.32730	−0.663648	−	0.748045i	$-0.730993\pi$
$419$	−27.1691	−1.32730	−0.663648	+	0.748045i	$0.730993\pi$
$420$	0	0
$421$	−25.6264	−1.24895	−0.624477	−	0.781043i	$-0.714688\pi$
$421$	−25.6264	−1.24895	−0.624477	+	0.781043i	$0.714688\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	− 7.15748i	− 0.347189i
$426$	0	0
$427$	4.92820i	0.238492i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−18.6123	−0.896521	−0.448260	−	0.893903i	$-0.647956\pi$
$431$	−18.6123	−0.896521	−0.448260	+	0.893903i	$0.647956\pi$
$432$	0	0
$433$	20.5592	0.988014	0.494007	−	0.869458i	$-0.335532\pi$
$433$	20.5592	0.988014	0.494007	+	0.869458i	$0.335532\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	20.6629i	0.988443i
$438$	0	0
$439$	23.6476i	1.12864i	0.825557	+	0.564318i	$0.190861\pi$
$439$	23.6476i	1.12864i	−0.825557	+	0.564318i	$0.809139\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	3.87880	0.184287	0.0921436	−	0.995746i	$-0.470628\pi$
$443$	3.87880	0.184287	0.0921436	+	0.995746i	$0.470628\pi$
$444$	0	0
$445$	16.1365	0.764942
$446$	0	0
$447$	0	0
$448$	0	0
$449$	22.6536i	1.06909i	0.845140	+	0.534545i	$0.179517\pi$
$449$	22.6536i	1.06909i	−0.845140	+	0.534545i	$0.820483\pi$
$450$	0	0
$451$	− 0.267949i	− 0.0126172i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−9.48477	−0.444653
$456$	0	0
$457$	3.57826	0.167384	0.0836919	−	0.996492i	$-0.473329\pi$
$457$	3.57826	0.167384	0.0836919	+	0.996492i	$0.473329\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	7.46383i	0.347625i	0.984779	+	0.173813i	$0.0556087\pi$
$461$	7.46383i	0.347625i	−0.984779	+	0.173813i	$0.944391\pi$
$462$	0	0
$463$	17.7060i	0.822868i	0.911440	+	0.411434i	$0.134972\pi$
$463$	17.7060i	0.822868i	−0.911440	+	0.411434i	$0.865028\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−3.59864	−0.166525	−0.0832626	−	0.996528i	$-0.526534\pi$
$467$	−3.59864	−0.166525	−0.0832626	+	0.996528i	$0.526534\pi$
$468$	0	0
$469$	−6.83183	−0.315464
$470$	0	0
$471$	0	0
$472$	0	0
$473$	− 1.84553i	− 0.0848575i
$474$	0	0
$475$	8.17373i	0.375036i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−10.7287	−0.490205	−0.245102	−	0.969497i	$-0.578822\pi$
$479$	−10.7287	−0.490205	−0.245102	+	0.969497i	$0.578822\pi$
$480$	0	0
$481$	27.8729	1.27090
$482$	0	0
$483$	0	0
$484$	0	0
$485$	12.5967i	0.571985i
$486$	0	0
$487$	8.41133i	0.381154i	0.981672	+	0.190577i	$0.0610358\pi$
$487$	8.41133i	0.381154i	−0.981672	+	0.190577i	$0.938964\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	41.1908	1.85892	0.929458	−	0.368929i	$-0.120275\pi$
$491$	41.1908	1.85892	0.929458	+	0.368929i	$0.120275\pi$
$492$	0	0
$493$	26.7778	1.20601
$494$	0	0
$495$	0	0
$496$	0	0
$497$	10.2144i	0.458179i
$498$	0	0
$499$	− 4.23997i	− 0.189807i	−0.995486	−	0.0949036i	$-0.969746\pi$
$499$	− 4.23997i	− 0.189807i	0.995486	−	0.0949036i	$-0.0302543\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−30.5551	−1.36238	−0.681192	−	0.732104i	$-0.738538\pi$
$503$	−30.5551	−1.36238	−0.681192	+	0.732104i	$0.738538\pi$
$504$	0	0
$505$	−28.3874	−1.26322
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 16.1136i	− 0.714225i	−0.934061	−	0.357112i	$-0.883761\pi$
$509$	− 16.1136i	− 0.714225i	0.934061	−	0.357112i	$-0.116239\pi$
$510$	0	0
$511$	5.48993i	0.242860i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	17.5453	0.773137
$516$	0	0
$517$	−0.0963763	−0.00423863
$518$	0	0
$519$	0	0
$520$	0	0
$521$	− 16.6291i	− 0.728536i	−0.931294	−	0.364268i	$-0.881319\pi$
$521$	− 16.6291i	− 0.728536i	0.931294	−	0.364268i	$-0.118681\pi$
$522$	0	0
$523$	17.7812i	0.777518i	0.921339	+	0.388759i	$0.127096\pi$
$523$	17.7812i	0.777518i	−0.921339	+	0.388759i	$0.872904\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−29.7972	−1.29799
$528$	0	0
$529$	5.08552	0.221109
$530$	0	0
$531$	0	0
$532$	0	0
$533$	5.79869i	0.251169i
$534$	0	0
$535$	− 40.3358i	− 1.74387i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	0.164525	0.00708658
$540$	0	0
$541$	45.3392	1.94928	0.974642	−	0.223769i	$-0.0718361\pi$
$541$	45.3392	1.94928	0.974642	+	0.223769i	$0.0718361\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	− 40.2684i	− 1.72491i
$546$	0	0
$547$	22.3004i	0.953495i	0.879040	+	0.476747i	$0.158184\pi$
$547$	22.3004i	0.953495i	−0.879040	+	0.476747i	$0.841816\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−30.5798	−1.30275
$552$	0	0
$553$	9.09513	0.386764
$554$	0	0
$555$	0	0
$556$	0	0
$557$	40.0247i	1.69590i	0.530074	+	0.847951i	$0.322164\pi$
$557$	40.0247i	1.69590i	−0.530074	+	0.847951i	$0.677836\pi$
$558$	0	0
$559$	39.9391i	1.68924i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	23.8844	1.00661	0.503303	−	0.864110i	$-0.332118\pi$
$563$	23.8844	1.00661	0.503303	+	0.864110i	$0.332118\pi$
$564$	0	0
$565$	−15.1794	−0.638602
$566$	0	0
$567$	0	0
$568$	0	0
$569$	− 22.4677i	− 0.941894i	−0.882162	−	0.470947i	$-0.843912\pi$
$569$	− 22.4677i	− 0.941894i	0.882162	−	0.470947i	$-0.156088\pi$
$570$	0	0
$571$	10.3948i	0.435009i	0.976059	+	0.217504i	$0.0697916\pi$
$571$	10.3948i	0.435009i	−0.976059	+	0.217504i	$0.930208\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	11.1099	0.463315
$576$	0	0
$577$	−8.27292	−0.344406	−0.172203	−	0.985061i	$-0.555089\pi$
$577$	−8.27292	−0.344406	−0.172203	+	0.985061i	$0.555089\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	− 1.95492i	− 0.0811037i
$582$	0	0
$583$	− 2.34066i	− 0.0969401i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−35.5531	−1.46743	−0.733717	−	0.679455i	$-0.762216\pi$
$587$	−35.5531	−1.46743	−0.733717	+	0.679455i	$0.762216\pi$
$588$	0	0
$589$	34.0280	1.40210
$590$	0	0
$591$	0	0
$592$	0	0
$593$	17.7443i	0.728669i	0.931268	+	0.364335i	$0.118704\pi$
$593$	17.7443i	0.728669i	−0.931268	+	0.364335i	$0.881296\pi$
$594$	0	0
$595$	− 9.09513i	− 0.372864i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	31.9951	1.30728	0.653641	−	0.756804i	$-0.273241\pi$
$599$	31.9951	1.30728	0.653641	+	0.756804i	$0.273241\pi$
$600$	0	0
$601$	8.89558	0.362858	0.181429	−	0.983404i	$-0.441928\pi$
$601$	8.89558	0.362858	0.181429	+	0.983404i	$0.441928\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	− 29.2308i	− 1.18840i
$606$	0	0
$607$	− 21.7060i	− 0.881020i	−0.897748	−	0.440510i	$-0.854798\pi$
$607$	− 21.7060i	− 0.881020i	0.897748	−	0.440510i	$-0.145202\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	2.08568	0.0843776
$612$	0	0
$613$	−38.4771	−1.55408	−0.777038	−	0.629454i	$-0.783279\pi$
$613$	−38.4771	−1.55408	−0.777038	+	0.629454i	$0.783279\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	21.7856i	0.877056i	0.898717	+	0.438528i	$0.144500\pi$
$617$	21.7856i	0.877056i	−0.898717	+	0.438528i	$0.855500\pi$
$618$	0	0
$619$	44.1549i	1.77474i	0.461061	+	0.887368i	$0.347469\pi$
$619$	44.1549i	1.77474i	−0.461061	+	0.887368i	$0.652531\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	6.05745	0.242687
$624$	0	0
$625$	−31.0871	−1.24348
$626$	0	0
$627$	0	0
$628$	0	0
$629$	26.7279i	1.06571i
$630$	0	0
$631$	30.9355i	1.23152i	0.787933	+	0.615760i	$0.211151\pi$
$631$	30.9355i	1.23152i	−0.787933	+	0.615760i	$0.788849\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	15.0005	0.595277
$636$	0	0
$637$	−3.56048	−0.141071
$638$	0	0
$639$	0	0
$640$	0	0
$641$	18.5427i	0.732393i	0.930538	+	0.366196i	$0.119340\pi$
$641$	18.5427i	0.732393i	−0.930538	+	0.366196i	$0.880660\pi$
$642$	0	0
$643$	− 10.5243i	− 0.415039i	−0.978231	−	0.207520i	$-0.933461\pi$
$643$	− 10.5243i	− 0.415039i	0.978231	−	0.207520i	$-0.0665391\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	19.6531	0.772643	0.386322	−	0.922364i	$-0.373745\pi$
$647$	19.6531	0.772643	0.386322	+	0.922364i	$0.373745\pi$
$648$	0	0
$649$	1.14927	0.0451127
$650$	0	0
$651$	0	0
$652$	0	0
$653$	23.0308i	0.901264i	0.892710	+	0.450632i	$0.148801\pi$
$653$	23.0308i	0.901264i	−0.892710	+	0.450632i	$0.851199\pi$
$654$	0	0
$655$	− 17.8822i	− 0.698717i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	28.4901	1.10981	0.554907	−	0.831912i	$-0.312754\pi$
$659$	28.4901	1.10981	0.554907	+	0.831912i	$0.312754\pi$
$660$	0	0
$661$	−3.94349	−0.153384	−0.0766921	−	0.997055i	$-0.524436\pi$
$661$	−3.94349	−0.153384	−0.0766921	+	0.997055i	$0.524436\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	10.3865i	0.402771i
$666$	0	0
$667$	41.5648i	1.60940i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−0.810811	−0.0313010
$672$	0	0
$673$	−10.0000	−0.385472	−0.192736	−	0.981251i	$-0.561736\pi$
$673$	−10.0000	−0.385472	−0.192736	+	0.981251i	$0.561736\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 39.1588i	− 1.50499i	−0.658595	−	0.752497i	$-0.728849\pi$
$677$	− 39.1588i	− 1.50499i	0.658595	−	0.752497i	$-0.271151\pi$
$678$	0	0
$679$	4.72865i	0.181469i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−3.23582	−0.123815	−0.0619077	−	0.998082i	$-0.519718\pi$
$683$	−3.23582	−0.123815	−0.0619077	+	0.998082i	$0.519718\pi$
$684$	0	0
$685$	23.7483	0.907374
$686$	0	0
$687$	0	0
$688$	0	0
$689$	50.6542i	1.92977i
$690$	0	0
$691$	7.22219i	0.274745i	0.990519	+	0.137373i	$0.0438657\pi$
$691$	7.22219i	0.274745i	−0.990519	+	0.137373i	$0.956134\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−25.2114	−0.956325
$696$	0	0
$697$	−5.56048	−0.210618
$698$	0	0
$699$	0	0
$700$	0	0
$701$	− 14.9289i	− 0.563858i	−0.959435	−	0.281929i	$-0.909026\pi$
$701$	− 14.9289i	− 0.563858i	0.959435	−	0.281929i	$-0.0909743\pi$
$702$	0	0
$703$	− 30.5229i	− 1.15119i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−10.6563	−0.400773
$708$	0	0
$709$	−16.3317	−0.613350	−0.306675	−	0.951814i	$-0.599216\pi$
$709$	−16.3317	−0.613350	−0.306675	+	0.951814i	$0.599216\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	− 46.2516i	− 1.73213i
$714$	0	0
$715$	− 1.56048i	− 0.0583586i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	46.0552	1.71757	0.858785	−	0.512336i	$-0.171220\pi$
$719$	46.0552	1.71757	0.858785	+	0.512336i	$0.171220\pi$
$720$	0	0
$721$	6.58630	0.245287
$722$	0	0
$723$	0	0
$724$	0	0
$725$	16.4420i	0.610639i
$726$	0	0
$727$	− 25.5984i	− 0.949392i	−0.880150	−	0.474696i	$-0.842558\pi$
$727$	− 25.5984i	− 0.949392i	0.880150	−	0.474696i	$-0.157442\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−38.2984	−1.41652
$732$	0	0
$733$	41.3584	1.52761	0.763804	−	0.645448i	$-0.223329\pi$
$733$	41.3584	1.52761	0.763804	+	0.645448i	$0.223329\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 1.12400i	− 0.0414032i
$738$	0	0
$739$	− 11.0109i	− 0.405041i	−0.979278	−	0.202521i	$-0.935087\pi$
$739$	− 11.0109i	− 0.405041i	0.979278	−	0.202521i	$-0.0649133\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	27.4506	1.00706	0.503532	−	0.863976i	$-0.332033\pi$
$743$	27.4506	1.00706	0.503532	+	0.863976i	$0.332033\pi$
$744$	0	0
$745$	40.0331	1.46670
$746$	0	0
$747$	0	0
$748$	0	0
$749$	− 15.1416i	− 0.553263i
$750$	0	0
$751$	− 26.8414i	− 0.979458i	−0.871875	−	0.489729i	$-0.837096\pi$
$751$	− 26.8414i	− 0.979458i	0.871875	−	0.489729i	$-0.162904\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−11.9975	−0.436057	−0.218028	−	0.975942i	$-0.569963\pi$
$757$	−11.9975	−0.436057	−0.218028	+	0.975942i	$0.569963\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	− 22.5682i	− 0.818096i	−0.912513	−	0.409048i	$-0.865861\pi$
$761$	− 22.5682i	− 0.818096i	0.912513	−	0.409048i	$-0.134139\pi$
$762$	0	0
$763$	− 15.1163i	− 0.547247i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−24.8713	−0.898051
$768$	0	0
$769$	0.549945	0.0198315	0.00991576	−	0.999951i	$-0.496844\pi$
$769$	0.549945	0.0198315	0.00991576	+	0.999951i	$0.496844\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	− 14.5062i	− 0.521752i	−0.965372	−	0.260876i	$-0.915989\pi$
$773$	− 14.5062i	− 0.521752i	0.965372	−	0.260876i	$-0.0840114\pi$
$774$	0	0
$775$	− 18.2959i	− 0.657209i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	6.34998	0.227512
$780$	0	0
$781$	−1.68052	−0.0601338
$782$	0	0
$783$	0	0
$784$	0	0
$785$	45.8653i	1.63700i
$786$	0	0
$787$	8.63671i	0.307865i	0.988081	+	0.153933i	$0.0491939\pi$
$787$	8.63671i	0.307865i	−0.988081	+	0.153933i	$0.950806\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−5.69818	−0.202604
$792$	0	0
$793$	17.5468	0.623104
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 51.2737i	− 1.81621i	−0.418745	−	0.908104i	$-0.637530\pi$
$797$	− 51.2737i	− 1.81621i	0.418745	−	0.908104i	$-0.362470\pi$
$798$	0	0
$799$	2.00000i	0.0707549i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−0.903228	−0.0318742
$804$	0	0
$805$	14.1176	0.497578
$806$	0	0
$807$	0	0
$808$	0	0
$809$	− 23.4293i	− 0.823732i	−0.911244	−	0.411866i	$-0.864877\pi$
$809$	− 23.4293i	− 0.823732i	0.911244	−	0.411866i	$-0.135123\pi$
$810$	0	0
$811$	51.9554i	1.82440i	0.409745	+	0.912200i	$0.365618\pi$
$811$	51.9554i	1.82440i	−0.409745	+	0.912200i	$0.634382\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−46.2412	−1.61976
$816$	0	0
$817$	43.7361	1.53013
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 8.94310i	− 0.312116i	−0.987748	−	0.156058i	$-0.950121\pi$
$821$	− 8.94310i	− 0.312116i	0.987748	−	0.156058i	$-0.0498787\pi$
$822$	0	0
$823$	− 3.07180i	− 0.107076i	−0.998566	−	0.0535381i	$-0.982950\pi$
$823$	− 3.07180i	− 0.107076i	0.998566	−	0.0535381i	$-0.0170498\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	4.73319	0.164589	0.0822946	−	0.996608i	$-0.473775\pi$
$827$	4.73319	0.164589	0.0822946	+	0.996608i	$0.473775\pi$
$828$	0	0
$829$	−12.7378	−0.442403	−0.221201	−	0.975228i	$-0.570998\pi$
$829$	−12.7378	−0.442403	−0.221201	+	0.975228i	$0.570998\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	− 3.41421i	− 0.118295i
$834$	0	0
$835$	3.16125i	0.109400i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−31.3976	−1.08397	−0.541983	−	0.840389i	$-0.682326\pi$
$839$	−31.3976	−1.08397	−0.541983	+	0.840389i	$0.682326\pi$
$840$	0	0
$841$	−32.5133	−1.12115
$842$	0	0
$843$	0	0
$844$	0	0
$845$	− 0.860432i	− 0.0295998i
$846$	0	0
$847$	− 10.9729i	− 0.377034i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−41.4874	−1.42217
$852$	0	0
$853$	−9.89670	−0.338857	−0.169428	−	0.985543i	$-0.554192\pi$
$853$	−9.89670	−0.338857	−0.169428	+	0.985543i	$0.554192\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 8.24092i	− 0.281504i	−0.990045	−	0.140752i	$-0.955048\pi$
$857$	− 8.24092i	− 0.281504i	0.990045	−	0.140752i	$-0.0449521\pi$
$858$	0	0
$859$	− 17.3021i	− 0.590342i	−0.955445	−	0.295171i	$-0.904623\pi$
$859$	− 17.3021i	− 0.590342i	0.955445	−	0.295171i	$-0.0953765\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	23.9146	0.814063	0.407032	−	0.913414i	$-0.366564\pi$
$863$	23.9146	0.814063	0.407032	+	0.913414i	$0.366564\pi$
$864$	0	0
$865$	20.6052	0.700598
$866$	0	0
$867$	0	0
$868$	0	0
$869$	1.49637i	0.0507610i
$870$	0	0
$871$	24.3246i	0.824207i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−7.73497	−0.261490
$876$	0	0
$877$	23.1862	0.782942	0.391471	−	0.920190i	$-0.371966\pi$
$877$	23.1862	0.782942	0.391471	+	0.920190i	$0.371966\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	− 35.6992i	− 1.20274i	−0.798972	−	0.601368i	$-0.794623\pi$
$881$	− 35.6992i	− 1.20274i	0.798972	−	0.601368i	$-0.205377\pi$
$882$	0	0
$883$	54.2704i	1.82635i	0.407573	+	0.913173i	$0.366375\pi$
$883$	54.2704i	1.82635i	−0.407573	+	0.913173i	$0.633625\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	20.8136	0.698852	0.349426	−	0.936964i	$-0.386377\pi$
$887$	20.8136	0.698852	0.349426	+	0.936964i	$0.386377\pi$
$888$	0	0
$889$	5.63103	0.188859
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 2.28397i	− 0.0764301i
$894$	0	0
$895$	− 31.6588i	− 1.05824i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	68.4494	2.28291
$900$	0	0
$901$	−48.5733	−1.61821
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0.375889i	0.0124950i
$906$	0	0
$907$	12.1386i	0.403056i	0.979483	+	0.201528i	$0.0645907\pi$
$907$	12.1386i	0.403056i	−0.979483	+	0.201528i	$0.935409\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	27.8093	0.921364	0.460682	−	0.887565i	$-0.347605\pi$
$911$	27.8093	0.921364	0.460682	+	0.887565i	$0.347605\pi$
$912$	0	0
$913$	0.321633	0.0106445
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 6.71279i	− 0.221676i
$918$	0	0
$919$	− 2.77101i	− 0.0914072i	−0.998955	−	0.0457036i	$-0.985447\pi$
$919$	− 2.77101i	− 0.0914072i	0.998955	−	0.0457036i	$-0.0145530\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	36.3682	1.19707
$924$	0	0
$925$	−16.4113	−0.539601
$926$	0	0
$927$	0	0
$928$	0	0
$929$	− 57.9933i	− 1.90270i	−0.308113	−	0.951350i	$-0.599697\pi$
$929$	− 57.9933i	− 1.90270i	0.308113	−	0.951350i	$-0.400303\pi$
$930$	0	0
$931$	3.89898i	0.127784i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	1.49637	0.0489366
$936$	0	0
$937$	−36.4813	−1.19179	−0.595897	−	0.803061i	$-0.703203\pi$
$937$	−36.4813	−1.19179	−0.595897	+	0.803061i	$0.703203\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	53.8744i	1.75625i	0.478427	+	0.878127i	$0.341207\pi$
$941$	53.8744i	1.75625i	−0.478427	+	0.878127i	$0.658793\pi$
$942$	0	0
$943$	− 8.63103i	− 0.281065i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−25.5198	−0.829282	−0.414641	−	0.909985i	$-0.636093\pi$
$947$	−25.5198	−0.829282	−0.414641	+	0.909985i	$0.636093\pi$
$948$	0	0
$949$	19.5468	0.634515
$950$	0	0
$951$	0	0
$952$	0	0
$953$	40.9794i	1.32745i	0.747975	+	0.663727i	$0.231026\pi$
$953$	40.9794i	1.32745i	−0.747975	+	0.663727i	$0.768974\pi$
$954$	0	0
$955$	− 38.8505i	− 1.25717i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	8.91484	0.287875
$960$	0	0
$961$	−45.1676	−1.45702
$962$	0	0
$963$	0	0
$964$	0	0
$965$	− 37.8081i	− 1.21709i
$966$	0	0
$967$	− 5.62129i	− 0.180769i	−0.995907	−	0.0903843i	$-0.971190\pi$
$967$	− 5.62129i	− 0.180769i	0.995907	−	0.0903843i	$-0.0288095\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−38.2654	−1.22799	−0.613997	−	0.789308i	$-0.710439\pi$
$971$	−38.2654	−1.22799	−0.613997	+	0.789308i	$0.710439\pi$
$972$	0	0
$973$	−9.46410	−0.303405
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 31.1822i	− 0.997608i	−0.866715	−	0.498804i	$-0.833773\pi$
$977$	− 31.1822i	− 0.997608i	0.866715	−	0.498804i	$-0.166227\pi$
$978$	0	0
$979$	0.996600i	0.0318515i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	57.5907	1.83686	0.918429	−	0.395585i	$-0.129458\pi$
$983$	57.5907	1.83686	0.918429	+	0.395585i	$0.129458\pi$
$984$	0	0
$985$	−66.2843	−2.11199
$986$	0	0
$987$	0	0
$988$	0	0
$989$	− 59.4471i	− 1.89031i
$990$	0	0
$991$	− 25.3278i	− 0.804563i	−0.915516	−	0.402281i	$-0.868217\pi$
$991$	− 25.3278i	− 0.804563i	0.915516	−	0.402281i	$-0.131783\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	62.3836	1.97769
$996$	0	0
$997$	−35.3697	−1.12017	−0.560084	−	0.828436i	$-0.689231\pi$
$997$	−35.3697	−1.12017	−0.560084	+	0.828436i	$0.689231\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6048.2.h.a.2591.7	yes	8
3.2	odd	2		6048.2.h.g.2591.2	yes	8
4.3	odd	2		6048.2.h.g.2591.7	yes	8
12.11	even	2	inner	6048.2.h.a.2591.2	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6048.2.h.a.2591.2	✓	8	12.11	even	2	inner
6048.2.h.a.2591.7	yes	8	1.1	even	1	trivial
6048.2.h.g.2591.2	yes	8	3.2	odd	2
6048.2.h.g.2591.7	yes	8	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 6048 = 2^{5} \cdot 3^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6048.h (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+2.66390i q^{5} +1.00000i q^{7} +O(q^{10})\)	\(q+2.66390i q^{5} +1.00000i q^{7} -0.164525 q^{11} +3.56048 q^{13} +3.41421i q^{17} -3.89898i q^{19} -5.29958 q^{23} -2.09638 q^{25} -7.84304i q^{29} +8.72741i q^{31} -2.66390 q^{35} +7.82843 q^{37} +1.62863i q^{41} +11.2173i q^{43} +0.585786 q^{47} -1.00000 q^{49} +14.2268i q^{53} -0.438278i q^{55} -6.98539 q^{59} +4.92820 q^{61} +9.48477i q^{65} +6.83183i q^{67} +10.2144 q^{71} +5.48993 q^{73} -0.164525i q^{77} -9.09513i q^{79} -1.95492 q^{83} -9.09513 q^{85} -6.05745i q^{89} +3.56048i q^{91} +10.3865 q^{95} +4.72865 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q+O(q^{10}) \) 8 * q	\( 8 q - 8 q^{11} - 8 q^{13} - 8 q^{23} - 8 q^{25} + 8 q^{35} + 40 q^{37} + 16 q^{47} - 8 q^{49} - 64 q^{59} - 16 q^{61} + 72 q^{71} + 24 q^{73} + 32 q^{83} + 8 q^{85} + 56 q^{95} + 48 q^{97}+O(q^{100}) \) 8 * q - 8 * q^11 - 8 * q^13 - 8 * q^23 - 8 * q^25 + 8 * q^35 + 40 * q^37 + 16 * q^47 - 8 * q^49 - 64 * q^59 - 16 * q^61 + 72 * q^71 + 24 * q^73 + 32 * q^83 + 8 * q^85 + 56 * q^95 + 48 * q^97

Introduction

L-functions

Modular forms

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