LMFDB - Embedded newform 4056.2.c.m.337.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4056,2,Mod(337,4056)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4056, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("4056.337");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4056 = 2^{3} \cdot 3 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4056.c (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:	$32.3873230598$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.44836416.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} + 12x^{4} + 36x^{2} + 1 $ x^6 + 12x^4 + 36x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 312)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			337.1
Root			$-2.36147i$ of defining polynomial
Character	$\chi$	$=$	4056.337
Dual form			4056.2.c.m.337.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} -3.93800i q^{5} -1.78493i q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} -3.93800i q^{5} -1.78493i q^{7} +1.00000 q^{9} -4.72294i q^{11} +3.93800i q^{15} -0.784934 q^{17} +5.15307i q^{19} +1.78493i q^{21} -4.72294 q^{23} -10.5079 q^{25} -1.00000 q^{27} -7.93800 q^{29} -2.93800i q^{31} +4.72294i q^{33} -7.02908 q^{35} -1.21507i q^{37} -8.66094i q^{41} +6.21507 q^{43} -3.93800i q^{45} -0.722938i q^{47} +3.81401 q^{49} +0.784934 q^{51} -13.8140 q^{53} -18.5989 q^{55} -5.15307i q^{57} -0.430132i q^{59} +4.15307 q^{61} -1.78493i q^{63} +9.78493i q^{67} +4.72294 q^{69} +8.72294i q^{71} -4.87601i q^{73} +10.5079 q^{75} -8.43013 q^{77} +16.3839 q^{79} +1.00000 q^{81} +6.59894i q^{83} +3.09107i q^{85} +7.93800 q^{87} -15.8760i q^{89} +2.93800i q^{93} +20.2928 q^{95} +3.06200i q^{97} -4.72294i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 6 q^{3} + 6 q^{9}+O(q^{10}) $ 6 * q - 6 * q^3 + 6 * q^9	$ 6 q - 6 q^{3} + 6 q^{9} - 30 q^{25} - 6 q^{27} - 24 q^{29} + 24 q^{35} + 42 q^{43} - 48 q^{49} - 12 q^{53} - 36 q^{55} + 6 q^{61} + 30 q^{75} - 60 q^{77} + 18 q^{79} + 6 q^{81} + 24 q^{87} + 84 q^{95}+O(q^{100}) $ 6 * q - 6 * q^3 + 6 * q^9 - 30 * q^25 - 6 * q^27 - 24 * q^29 + 24 * q^35 + 42 * q^43 - 48 * q^49 - 12 * q^53 - 36 * q^55 + 6 * q^61 + 30 * q^75 - 60 * q^77 + 18 * q^79 + 6 * q^81 + 24 * q^87 + 84 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4056\mathbb{Z}\right)^\times$.

$n$	$1015$	$2029$	$2705$	$3889$
$\chi(n)$	$1$	$1$	$1$	$-1$

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	− 3.93800i	− 1.76113i	−0.473927	−	0.880564i	$-0.657164\pi$
$5$	− 3.93800i	− 1.76113i	0.473927	−	0.880564i	$-0.342836\pi$
$6$	0	0
$7$	− 1.78493i	− 0.674642i	−0.941390	−	0.337321i	$-0.890479\pi$
$7$	− 1.78493i	− 0.674642i	0.941390	−	0.337321i	$-0.109521\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	− 4.72294i	− 1.42402i	−0.702170	−	0.712010i	$-0.747785\pi$
$11$	− 4.72294i	− 1.42402i	0.702170	−	0.712010i	$-0.252215\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	0	0
$15$	3.93800i	1.01679i
$16$	0	0
$17$	−0.784934	−0.190374	−0.0951872	−	0.995459i	$-0.530345\pi$
$17$	−0.784934	−0.190374	−0.0951872	+	0.995459i	$0.530345\pi$
$18$	0	0
$19$	5.15307i	1.18220i	0.806600	+	0.591098i	$0.201305\pi$
$19$	5.15307i	1.18220i	−0.806600	+	0.591098i	$0.798695\pi$
$20$	0	0
$21$	1.78493i	0.389505i
$22$	0	0
$23$	−4.72294	−0.984801	−0.492400	−	0.870369i	$-0.663880\pi$
$23$	−4.72294	−0.984801	−0.492400	+	0.870369i	$0.663880\pi$
$24$	0	0
$25$	−10.5079	−2.10157
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−7.93800	−1.47405	−0.737025	−	0.675865i	$-0.763770\pi$
$29$	−7.93800	−1.47405	−0.737025	+	0.675865i	$0.763770\pi$
$30$	0	0
$31$	− 2.93800i	− 0.527681i	−0.964566	−	0.263841i	$-0.915011\pi$
$31$	− 2.93800i	− 0.527681i	0.964566	−	0.263841i	$-0.0849893\pi$
$32$	0	0
$33$	4.72294i	0.822158i
$34$	0	0
$35$	−7.02908	−1.18813
$36$	0	0
$37$	− 1.21507i	− 0.199756i	−0.995000	−	0.0998778i	$-0.968155\pi$
$37$	− 1.21507i	− 0.199756i	0.995000	−	0.0998778i	$-0.0318452\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 8.66094i	− 1.35261i	−0.736621	−	0.676306i	$-0.763580\pi$
$41$	− 8.66094i	− 1.35261i	0.736621	−	0.676306i	$-0.236420\pi$
$42$	0	0
$43$	6.21507	0.947789	0.473894	−	0.880582i	$-0.342848\pi$
$43$	6.21507	0.947789	0.473894	+	0.880582i	$0.342848\pi$
$44$	0	0
$45$	− 3.93800i	− 0.587043i
$46$	0	0
$47$	− 0.722938i	− 0.105451i	−0.998609	−	0.0527256i	$-0.983209\pi$
$47$	− 0.722938i	− 0.105451i	0.998609	−	0.0527256i	$-0.0167909\pi$
$48$	0	0
$49$	3.81401	0.544859
$50$	0	0
$51$	0.784934	0.109913
$52$	0	0
$53$	−13.8140	−1.89750	−0.948750	−	0.316027i	$-0.897651\pi$
$53$	−13.8140	−1.89750	−0.948750	+	0.316027i	$0.897651\pi$
$54$	0	0
$55$	−18.5989	−2.50788
$56$	0	0
$57$	− 5.15307i	− 0.682541i
$58$	0	0
$59$	− 0.430132i	− 0.0559984i	−0.999608	−	0.0279992i	$-0.991086\pi$
$59$	− 0.430132i	− 0.0559984i	0.999608	−	0.0279992i	$-0.00891359\pi$
$60$	0	0
$61$	4.15307	0.531746	0.265873	−	0.964008i	$-0.414340\pi$
$61$	4.15307	0.531746	0.265873	+	0.964008i	$0.414340\pi$
$62$	0	0
$63$	− 1.78493i	− 0.224881i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	9.78493i	1.19542i	0.801712	+	0.597710i	$0.203923\pi$
$67$	9.78493i	1.19542i	−0.801712	+	0.597710i	$0.796077\pi$
$68$	0	0
$69$	4.72294	0.568575
$70$	0	0
$71$	8.72294i	1.03522i	0.855616	+	0.517611i	$0.173179\pi$
$71$	8.72294i	1.03522i	−0.855616	+	0.517611i	$0.826821\pi$
$72$	0	0
$73$	− 4.87601i	− 0.570693i	−0.958424	−	0.285347i	$-0.907891\pi$
$73$	− 4.87601i	− 0.570693i	0.958424	−	0.285347i	$-0.0921088\pi$
$74$	0	0
$75$	10.5079	1.21334
$76$	0	0
$77$	−8.43013	−0.960703
$78$	0	0
$79$	16.3839	1.84333	0.921665	−	0.387986i	$-0.126829\pi$
$79$	16.3839	1.84333	0.921665	+	0.387986i	$0.126829\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	6.59894i	0.724328i	0.932114	+	0.362164i	$0.117962\pi$
$83$	6.59894i	0.724328i	−0.932114	+	0.362164i	$0.882038\pi$
$84$	0	0
$85$	3.09107i	0.335274i
$86$	0	0
$87$	7.93800	0.851043
$88$	0	0
$89$	− 15.8760i	− 1.68285i	−0.540371	−	0.841427i	$-0.681716\pi$
$89$	− 15.8760i	− 1.68285i	0.540371	−	0.841427i	$-0.318284\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	2.93800i	0.304657i
$94$	0	0
$95$	20.2928	2.08200
$96$	0	0
$97$	3.06200i	0.310899i	0.987844	+	0.155449i	$0.0496825\pi$
$97$	3.06200i	0.310899i	−0.987844	+	0.155449i	$0.950317\pi$
$98$	0	0
$99$	− 4.72294i	− 0.474673i
$100$	0	0
$101$	1.63186	0.162377	0.0811883	−	0.996699i	$-0.474128\pi$
$101$	1.63186	0.162377	0.0811883	+	0.996699i	$0.474128\pi$
$102$	0	0
$103$	7.66094	0.754855	0.377427	−	0.926039i	$-0.376809\pi$
$103$	7.66094	0.754855	0.377427	+	0.926039i	$0.376809\pi$
$104$	0	0
$105$	7.02908	0.685968
$106$	0	0
$107$	0.722938	0.0698890	0.0349445	−	0.999389i	$-0.488875\pi$
$107$	0.722938	0.0698890	0.0349445	+	0.999389i	$0.488875\pi$
$108$	0	0
$109$	5.36814i	0.514174i	0.966388	+	0.257087i	$0.0827627\pi$
$109$	5.36814i	0.514174i	−0.966388	+	0.257087i	$0.917237\pi$
$110$	0	0
$111$	1.21507i	0.115329i
$112$	0	0
$113$	13.2151	1.24317	0.621584	−	0.783347i	$-0.286489\pi$
$113$	13.2151	1.24317	0.621584	+	0.783347i	$0.286489\pi$
$114$	0	0
$115$	18.5989i	1.73436i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	1.40106i	0.128435i
$120$	0	0
$121$	−11.3061	−1.02783
$122$	0	0
$123$	8.66094i	0.780931i
$124$	0	0
$125$	21.6900i	1.94001i
$126$	0	0
$127$	−16.3839	−1.45383	−0.726917	−	0.686725i	$-0.759048\pi$
$127$	−16.3839	−1.45383	−0.726917	+	0.686725i	$0.759048\pi$
$128$	0	0
$129$	−6.21507	−0.547206
$130$	0	0
$131$	−21.0157	−1.83615	−0.918077	−	0.396402i	$-0.870259\pi$
$131$	−21.0157	−1.83615	−0.918077	+	0.396402i	$0.870259\pi$
$132$	0	0
$133$	9.19789	0.797558
$134$	0	0
$135$	3.93800i	0.338929i
$136$	0	0
$137$	− 12.5369i	− 1.07110i	−0.844502	−	0.535552i	$-0.820104\pi$
$137$	− 12.5369i	− 1.07110i	0.844502	−	0.535552i	$-0.179896\pi$
$138$	0	0
$139$	8.81401	0.747595	0.373797	−	0.927510i	$-0.378056\pi$
$139$	8.81401	0.747595	0.373797	+	0.927510i	$0.378056\pi$
$140$	0	0
$141$	0.722938i	0.0608823i
$142$	0	0
$143$	0	0
$144$	0	0
$145$	31.2599i	2.59599i
$146$	0	0
$147$	−3.81401	−0.314574
$148$	0	0
$149$	3.63186i	0.297534i	0.988872	+	0.148767i	$0.0475304\pi$
$149$	3.63186i	0.297534i	−0.988872	+	0.148767i	$0.952470\pi$
$150$	0	0
$151$	14.5989i	1.18805i	0.804448	+	0.594023i	$0.202461\pi$
$151$	14.5989i	1.18805i	−0.804448	+	0.594023i	$0.797539\pi$
$152$	0	0
$153$	−0.784934	−0.0634582
$154$	0	0
$155$	−11.5699	−0.929314
$156$	0	0
$157$	−15.5989	−1.24493	−0.622466	−	0.782647i	$-0.713869\pi$
$157$	−15.5989	−1.24493	−0.622466	+	0.782647i	$0.713869\pi$
$158$	0	0
$159$	13.8140	1.09552
$160$	0	0
$161$	8.43013i	0.664387i
$162$	0	0
$163$	− 10.9380i	− 0.856731i	−0.903606	−	0.428365i	$-0.859090\pi$
$163$	− 10.9380i	− 0.856731i	0.903606	−	0.428365i	$-0.140910\pi$
$164$	0	0
$165$	18.5989	1.44793
$166$	0	0
$167$	20.5856i	1.59296i	0.604663	+	0.796481i	$0.293308\pi$
$167$	20.5856i	1.59296i	−0.604663	+	0.796481i	$0.706692\pi$
$168$	0	0
$169$	0	0
$170$	0	0
$171$	5.15307i	0.394065i
$172$	0	0
$173$	0.306139	0.0232753	0.0116377	−	0.999932i	$-0.496296\pi$
$173$	0.306139	0.0232753	0.0116377	+	0.999932i	$0.496296\pi$
$174$	0	0
$175$	18.7559i	1.41781i
$176$	0	0
$177$	0.430132i	0.0323307i
$178$	0	0
$179$	2.84693	0.212790	0.106395	−	0.994324i	$-0.466069\pi$
$179$	2.84693	0.212790	0.106395	+	0.994324i	$0.466069\pi$
$180$	0	0
$181$	−13.2151	−0.982268	−0.491134	−	0.871084i	$-0.663417\pi$
$181$	−13.2151	−0.982268	−0.491134	+	0.871084i	$0.663417\pi$
$182$	0	0
$183$	−4.15307	−0.307004
$184$	0	0
$185$	−4.78493	−0.351795
$186$	0	0
$187$	3.70719i	0.271097i
$188$	0	0
$189$	1.78493i	0.129835i
$190$	0	0
$191$	−3.56987	−0.258307	−0.129153	−	0.991625i	$-0.541226\pi$
$191$	−3.56987	−0.258307	−0.129153	+	0.991625i	$0.541226\pi$
$192$	0	0
$193$	− 12.4459i	− 0.895874i	−0.894065	−	0.447937i	$-0.852159\pi$
$193$	− 12.4459i	− 0.895874i	0.894065	−	0.447937i	$-0.147841\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 8.73627i	− 0.622434i	−0.950339	−	0.311217i	$-0.899263\pi$
$197$	− 8.73627i	− 0.622434i	0.950339	−	0.311217i	$-0.100737\pi$
$198$	0	0
$199$	16.0911	1.14067	0.570333	−	0.821414i	$-0.306814\pi$
$199$	16.0911	1.14067	0.570333	+	0.821414i	$0.306814\pi$
$200$	0	0
$201$	− 9.78493i	− 0.690176i
$202$	0	0
$203$	14.1688i	0.994456i
$204$	0	0
$205$	−34.1068	−2.38212
$206$	0	0
$207$	−4.72294	−0.328267
$208$	0	0
$209$	24.3376	1.68347
$210$	0	0
$211$	4.38388	0.301799	0.150899	−	0.988549i	$-0.451783\pi$
$211$	4.38388	0.301799	0.150899	+	0.988549i	$0.451783\pi$
$212$	0	0
$213$	− 8.72294i	− 0.597686i
$214$	0	0
$215$	− 24.4750i	− 1.66918i
$216$	0	0
$217$	−5.24414	−0.355996
$218$	0	0
$219$	4.87601i	0.329490i
$220$	0	0
$221$	0	0
$222$	0	0
$223$	16.0000i	1.07144i	0.844396	+	0.535720i	$0.179960\pi$
$223$	16.0000i	1.07144i	−0.844396	+	0.535720i	$0.820040\pi$
$224$	0	0
$225$	−10.5079	−0.700525
$226$	0	0
$227$	− 17.7387i	− 1.17736i	−0.808367	−	0.588679i	$-0.799648\pi$
$227$	− 17.7387i	− 1.17736i	0.808367	−	0.588679i	$-0.200352\pi$
$228$	0	0
$229$	− 11.4459i	− 0.756365i	−0.925731	−	0.378182i	$-0.876549\pi$
$229$	− 11.4459i	− 0.756365i	0.925731	−	0.378182i	$-0.123451\pi$
$230$	0	0
$231$	8.43013	0.554662
$232$	0	0
$233$	9.75201	0.638876	0.319438	−	0.947607i	$-0.396506\pi$
$233$	9.75201	0.638876	0.319438	+	0.947607i	$0.396506\pi$
$234$	0	0
$235$	−2.84693	−0.185713
$236$	0	0
$237$	−16.3839	−1.06425
$238$	0	0
$239$	6.59894i	0.426850i	0.976959	+	0.213425i	$0.0684619\pi$
$239$	6.59894i	0.426850i	−0.976959	+	0.213425i	$0.931538\pi$
$240$	0	0
$241$	− 23.8140i	− 1.53400i	−0.641650	−	0.766998i	$-0.721750\pi$
$241$	− 23.8140i	− 1.53400i	0.641650	−	0.766998i	$-0.278250\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	− 15.0196i	− 0.959566i
$246$	0	0
$247$	0	0
$248$	0	0
$249$	− 6.59894i	− 0.418191i
$250$	0	0
$251$	−9.44588	−0.596218	−0.298109	−	0.954532i	$-0.596356\pi$
$251$	−9.44588	−0.596218	−0.298109	+	0.954532i	$0.596356\pi$
$252$	0	0
$253$	22.3061i	1.40237i
$254$	0	0
$255$	− 3.09107i	− 0.193570i
$256$	0	0
$257$	12.6609	0.789768	0.394884	−	0.918731i	$-0.370785\pi$
$257$	12.6609	0.789768	0.394884	+	0.918731i	$0.370785\pi$
$258$	0	0
$259$	−2.16881	−0.134763
$260$	0	0
$261$	−7.93800	−0.491350
$262$	0	0
$263$	−0.292806	−0.0180552	−0.00902758	−	0.999959i	$-0.502874\pi$
$263$	−0.292806	−0.0180552	−0.00902758	+	0.999959i	$0.502874\pi$
$264$	0	0
$265$	54.3996i	3.34174i
$266$	0	0
$267$	15.8760i	0.971596i
$268$	0	0
$269$	5.56987	0.339601	0.169800	−	0.985478i	$-0.445688\pi$
$269$	5.56987	0.339601	0.169800	+	0.985478i	$0.445688\pi$
$270$	0	0
$271$	− 11.9537i	− 0.726138i	−0.931762	−	0.363069i	$-0.881729\pi$
$271$	− 11.9537i	− 0.726138i	0.931762	−	0.363069i	$-0.118271\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	49.6280i	2.99268i
$276$	0	0
$277$	−18.6609	−1.12123	−0.560614	−	0.828078i	$-0.689435\pi$
$277$	−18.6609	−1.12123	−0.560614	+	0.828078i	$0.689435\pi$
$278$	0	0
$279$	− 2.93800i	− 0.175894i
$280$	0	0
$281$	− 4.96708i	− 0.296311i	−0.988964	−	0.148156i	$-0.952666\pi$
$281$	− 4.96708i	− 0.296311i	0.988964	−	0.148156i	$-0.0473336\pi$
$282$	0	0
$283$	10.8007	0.642034	0.321017	−	0.947073i	$-0.395975\pi$
$283$	10.8007	0.642034	0.321017	+	0.947073i	$0.395975\pi$
$284$	0	0
$285$	−20.2928	−1.20204
$286$	0	0
$287$	−15.4592	−0.912528
$288$	0	0
$289$	−16.3839	−0.963758
$290$	0	0
$291$	− 3.06200i	− 0.179497i
$292$	0	0
$293$	26.2441i	1.53320i	0.642125	+	0.766600i	$0.278053\pi$
$293$	26.2441i	1.53320i	−0.642125	+	0.766600i	$0.721947\pi$
$294$	0	0
$295$	−1.69386	−0.0986204
$296$	0	0
$297$	4.72294i	0.274053i
$298$	0	0
$299$	0	0
$300$	0	0
$301$	− 11.0935i	− 0.639418i
$302$	0	0
$303$	−1.63186	−0.0937482
$304$	0	0
$305$	− 16.3548i	− 0.936473i
$306$	0	0
$307$	− 21.5369i	− 1.22918i	−0.788847	−	0.614589i	$-0.789322\pi$
$307$	− 21.5369i	− 1.22918i	0.788847	−	0.614589i	$-0.210678\pi$
$308$	0	0
$309$	−7.66094	−0.435816
$310$	0	0
$311$	−21.1531	−1.19948	−0.599740	−	0.800195i	$-0.704729\pi$
$311$	−21.1531	−1.19948	−0.599740	+	0.800195i	$0.704729\pi$
$312$	0	0
$313$	9.79827	0.553831	0.276915	−	0.960894i	$-0.410688\pi$
$313$	9.79827	0.553831	0.276915	+	0.960894i	$0.410688\pi$
$314$	0	0
$315$	−7.02908	−0.396044
$316$	0	0
$317$	22.5236i	1.26505i	0.774539	+	0.632526i	$0.217982\pi$
$317$	22.5236i	1.26505i	−0.774539	+	0.632526i	$0.782018\pi$
$318$	0	0
$319$	37.4907i	2.09908i
$320$	0	0
$321$	−0.722938	−0.0403504
$322$	0	0
$323$	− 4.04482i	− 0.225060i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 5.36814i	− 0.296859i
$328$	0	0
$329$	−1.29040	−0.0711418
$330$	0	0
$331$	21.0620i	1.15767i	0.815444	+	0.578836i	$0.196493\pi$
$331$	21.0620i	1.15767i	−0.815444	+	0.578836i	$0.803507\pi$
$332$	0	0
$333$	− 1.21507i	− 0.0665852i
$334$	0	0
$335$	38.5331	2.10529
$336$	0	0
$337$	2.01574	0.109805	0.0549023	−	0.998492i	$-0.482515\pi$
$337$	2.01574	0.109805	0.0549023	+	0.998492i	$0.482515\pi$
$338$	0	0
$339$	−13.2151	−0.717744
$340$	0	0
$341$	−13.8760	−0.751428
$342$	0	0
$343$	− 19.3023i	− 1.04223i
$344$	0	0
$345$	− 18.5989i	− 1.00133i
$346$	0	0
$347$	22.5989	1.21317	0.606587	−	0.795017i	$-0.292538\pi$
$347$	22.5989	1.21317	0.606587	+	0.795017i	$0.292538\pi$
$348$	0	0
$349$	4.25989i	0.228026i	0.993479	+	0.114013i	$0.0363706\pi$
$349$	4.25989i	0.228026i	−0.993479	+	0.114013i	$0.963629\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	3.52120i	0.187415i	0.995600	+	0.0937074i	$0.0298718\pi$
$353$	3.52120i	0.187415i	−0.995600	+	0.0937074i	$0.970128\pi$
$354$	0	0
$355$	34.3510	1.82316
$356$	0	0
$357$	− 1.40106i	− 0.0741517i
$358$	0	0
$359$	− 9.15307i	− 0.483081i	−0.970391	−	0.241540i	$-0.922347\pi$
$359$	− 9.15307i	− 0.483081i	0.970391	−	0.241540i	$-0.0776526\pi$
$360$	0	0
$361$	−7.55412	−0.397586
$362$	0	0
$363$	11.3061	0.593418
$364$	0	0
$365$	−19.2017	−1.00506
$366$	0	0
$367$	2.21507	0.115626	0.0578128	−	0.998327i	$-0.481587\pi$
$367$	2.21507	0.115626	0.0578128	+	0.998327i	$0.481587\pi$
$368$	0	0
$369$	− 8.66094i	− 0.450871i
$370$	0	0
$371$	24.6571i	1.28013i
$372$	0	0
$373$	−28.4592	−1.47356	−0.736781	−	0.676131i	$-0.763655\pi$
$373$	−28.4592	−1.47356	−0.736781	+	0.676131i	$0.763655\pi$
$374$	0	0
$375$	− 21.6900i	− 1.12007i
$376$	0	0
$377$	0	0
$378$	0	0
$379$	− 34.0119i	− 1.74707i	−0.486758	−	0.873537i	$-0.661821\pi$
$379$	− 34.0119i	− 1.74707i	0.486758	−	0.873537i	$-0.338179\pi$
$380$	0	0
$381$	16.3839	0.839372
$382$	0	0
$383$	26.8918i	1.37410i	0.726608	+	0.687052i	$0.241096\pi$
$383$	26.8918i	1.37410i	−0.726608	+	0.687052i	$0.758904\pi$
$384$	0	0
$385$	33.1979i	1.69192i
$386$	0	0
$387$	6.21507	0.315930
$388$	0	0
$389$	−34.2756	−1.73784	−0.868922	−	0.494950i	$-0.835187\pi$
$389$	−34.2756	−1.73784	−0.868922	+	0.494950i	$0.835187\pi$
$390$	0	0
$391$	3.70719	0.187481
$392$	0	0
$393$	21.0157	1.06010
$394$	0	0
$395$	− 64.5198i	− 3.24634i
$396$	0	0
$397$	27.3996i	1.37515i	0.726115	+	0.687574i	$0.241324\pi$
$397$	27.3996i	1.37515i	−0.726115	+	0.687574i	$0.758676\pi$
$398$	0	0
$399$	−9.19789	−0.460470
$400$	0	0
$401$	− 17.3972i	− 0.868775i	−0.900726	−	0.434388i	$-0.856965\pi$
$401$	− 17.3972i	− 0.868775i	0.900726	−	0.434388i	$-0.143035\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	− 3.93800i	− 0.195681i
$406$	0	0
$407$	−5.73868	−0.284456
$408$	0	0
$409$	35.5198i	1.75634i	0.478349	+	0.878170i	$0.341235\pi$
$409$	35.5198i	1.75634i	−0.478349	+	0.878170i	$0.658765\pi$
$410$	0	0
$411$	12.5369i	0.618402i
$412$	0	0
$413$	−0.767757	−0.0377789
$414$	0	0
$415$	25.9867	1.27564
$416$	0	0
$417$	−8.81401	−0.431624
$418$	0	0
$419$	7.13974	0.348799	0.174399	−	0.984675i	$-0.444202\pi$
$419$	7.13974	0.348799	0.174399	+	0.984675i	$0.444202\pi$
$420$	0	0
$421$	18.0291i	0.878683i	0.898320	+	0.439342i	$0.144788\pi$
$421$	18.0291i	0.878683i	−0.898320	+	0.439342i	$0.855212\pi$
$422$	0	0
$423$	− 0.722938i	− 0.0351504i
$424$	0	0
$425$	8.24799	0.400086
$426$	0	0
$427$	− 7.41296i	− 0.358738i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	− 5.40106i	− 0.260160i	−0.991504	−	0.130080i	$-0.958477\pi$
$431$	− 5.40106i	− 0.260160i	0.991504	−	0.130080i	$-0.0415234\pi$
$432$	0	0
$433$	−33.8918	−1.62873	−0.814367	−	0.580351i	$-0.802916\pi$
$433$	−33.8918	−1.62873	−0.814367	+	0.580351i	$0.802916\pi$
$434$	0	0
$435$	− 31.2599i	− 1.49880i
$436$	0	0
$437$	− 24.3376i	− 1.16423i
$438$	0	0
$439$	−6.64520	−0.317158	−0.158579	−	0.987346i	$-0.550691\pi$
$439$	−6.64520	−0.317158	−0.158579	+	0.987346i	$0.550691\pi$
$440$	0	0
$441$	3.81401	0.181620
$442$	0	0
$443$	9.44588	0.448787	0.224394	−	0.974499i	$-0.427960\pi$
$443$	9.44588	0.448787	0.224394	+	0.974499i	$0.427960\pi$
$444$	0	0
$445$	−62.5198	−2.96372
$446$	0	0
$447$	− 3.63186i	− 0.171781i
$448$	0	0
$449$	9.75201i	0.460226i	0.973164	+	0.230113i	$0.0739096\pi$
$449$	9.75201i	0.460226i	−0.973164	+	0.230113i	$0.926090\pi$
$450$	0	0
$451$	−40.9051	−1.92615
$452$	0	0
$453$	− 14.5989i	− 0.685918i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	17.4616i	0.816820i	0.912799	+	0.408410i	$0.133917\pi$
$457$	17.4616i	0.816820i	−0.912799	+	0.408410i	$0.866083\pi$
$458$	0	0
$459$	0.784934	0.0366376
$460$	0	0
$461$	0.244142i	0.0113708i	0.999984	+	0.00568542i	$0.00180974\pi$
$461$	0.244142i	0.0113708i	−0.999984	+	0.00568542i	$0.998190\pi$
$462$	0	0
$463$	0.0910730i	0.00423252i	0.999998	+	0.00211626i	$0.000673627\pi$
$463$	0.0910730i	0.00423252i	−0.999998	+	0.00211626i	$0.999326\pi$
$464$	0	0
$465$	11.5699	0.536540
$466$	0	0
$467$	14.1688	0.655654	0.327827	−	0.944738i	$-0.393684\pi$
$467$	14.1688	0.655654	0.327827	+	0.944738i	$0.393684\pi$
$468$	0	0
$469$	17.4655	0.806480
$470$	0	0
$471$	15.5989	0.718761
$472$	0	0
$473$	− 29.3534i	− 1.34967i
$474$	0	0
$475$	− 54.1478i	− 2.48447i
$476$	0	0
$477$	−13.8140	−0.632500
$478$	0	0
$479$	− 1.69386i	− 0.0773945i	−0.999251	−	0.0386972i	$-0.987679\pi$
$479$	− 1.69386i	− 0.0773945i	0.999251	−	0.0386972i	$-0.0123208\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	− 8.43013i	− 0.383584i
$484$	0	0
$485$	12.0582	0.547533
$486$	0	0
$487$	18.8469i	0.854036i	0.904243	+	0.427018i	$0.140436\pi$
$487$	18.8469i	0.854036i	−0.904243	+	0.427018i	$0.859564\pi$
$488$	0	0
$489$	10.9380i	0.494634i
$490$	0	0
$491$	−21.9208	−0.989273	−0.494637	−	0.869100i	$-0.664699\pi$
$491$	−21.9208	−0.989273	−0.494637	+	0.869100i	$0.664699\pi$
$492$	0	0
$493$	6.23081	0.280622
$494$	0	0
$495$	−18.5989	−0.835960
$496$	0	0
$497$	15.5699	0.698404
$498$	0	0
$499$	13.4459i	0.601920i	0.953637	+	0.300960i	$0.0973071\pi$
$499$	13.4459i	0.601920i	−0.953637	+	0.300960i	$0.902693\pi$
$500$	0	0
$501$	− 20.5856i	− 0.919697i
$502$	0	0
$503$	9.92083	0.442348	0.221174	−	0.975234i	$-0.429011\pi$
$503$	9.92083	0.442348	0.221174	+	0.975234i	$0.429011\pi$
$504$	0	0
$505$	− 6.42629i	− 0.285966i
$506$	0	0
$507$	0	0
$508$	0	0
$509$	20.3996i	0.904197i	0.891968	+	0.452099i	$0.149325\pi$
$509$	20.3996i	0.904197i	−0.891968	+	0.452099i	$0.850675\pi$
$510$	0	0
$511$	−8.70335	−0.385014
$512$	0	0
$513$	− 5.15307i	− 0.227514i
$514$	0	0
$515$	− 30.1688i	− 1.32940i
$516$	0	0
$517$	−3.41439	−0.150165
$518$	0	0
$519$	−0.306139	−0.0134380
$520$	0	0
$521$	−1.49454	−0.0654769	−0.0327385	−	0.999464i	$-0.510423\pi$
$521$	−1.49454	−0.0654769	−0.0327385	+	0.999464i	$0.510423\pi$
$522$	0	0
$523$	−32.0448	−1.40122	−0.700611	−	0.713543i	$-0.747089\pi$
$523$	−32.0448	−1.40122	−0.700611	+	0.713543i	$0.747089\pi$
$524$	0	0
$525$	− 18.7559i	− 0.818573i
$526$	0	0
$527$	2.30614i	0.100457i
$528$	0	0
$529$	−0.693861	−0.0301679
$530$	0	0
$531$	− 0.430132i	− 0.0186661i
$532$	0	0
$533$	0	0
$534$	0	0
$535$	− 2.84693i	− 0.123084i
$536$	0	0
$537$	−2.84693	−0.122854
$538$	0	0
$539$	− 18.0133i	− 0.775889i
$540$	0	0
$541$	− 1.44347i	− 0.0620594i	−0.999518	−	0.0310297i	$-0.990121\pi$
$541$	− 1.44347i	− 0.0620594i	0.999518	−	0.0310297i	$-0.00987865\pi$
$542$	0	0
$543$	13.2151	0.567113
$544$	0	0
$545$	21.1397	0.905527
$546$	0	0
$547$	−18.3972	−0.786608	−0.393304	−	0.919408i	$-0.628668\pi$
$547$	−18.3972	−0.786608	−0.393304	+	0.919408i	$0.628668\pi$
$548$	0	0
$549$	4.15307	0.177249
$550$	0	0
$551$	− 40.9051i	− 1.74262i
$552$	0	0
$553$	− 29.2441i	− 1.24359i
$554$	0	0
$555$	4.78493	0.203109
$556$	0	0
$557$	16.0935i	0.681903i	0.940081	+	0.340951i	$0.110749\pi$
$557$	16.0935i	0.681903i	−0.940081	+	0.340951i	$0.889251\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	− 3.70719i	− 0.156518i
$562$	0	0
$563$	−13.0424	−0.549672	−0.274836	−	0.961491i	$-0.588624\pi$
$563$	−13.0424	−0.549672	−0.274836	+	0.961491i	$0.588624\pi$
$564$	0	0
$565$	− 52.0410i	− 2.18938i
$566$	0	0
$567$	− 1.78493i	− 0.0749602i
$568$	0	0
$569$	−6.58561	−0.276083	−0.138042	−	0.990426i	$-0.544081\pi$
$569$	−6.58561	−0.276083	−0.138042	+	0.990426i	$0.544081\pi$
$570$	0	0
$571$	26.8469	1.12351	0.561755	−	0.827304i	$-0.310127\pi$
$571$	26.8469	1.12351	0.561755	+	0.827304i	$0.310127\pi$
$572$	0	0
$573$	3.56987	0.149133
$574$	0	0
$575$	49.6280	2.06963
$576$	0	0
$577$	− 22.5818i	− 0.940091i	−0.882642	−	0.470046i	$-0.844237\pi$
$577$	− 22.5818i	− 0.940091i	0.882642	−	0.470046i	$-0.155763\pi$
$578$	0	0
$579$	12.4459i	0.517233i
$580$	0	0
$581$	11.7787	0.488662
$582$	0	0
$583$	65.2427i	2.70208i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 5.87601i	− 0.242529i	−0.992620	−	0.121264i	$-0.961305\pi$
$587$	− 5.87601i	− 0.242529i	0.992620	−	0.121264i	$-0.0386949\pi$
$588$	0	0
$589$	15.1397	0.623822
$590$	0	0
$591$	8.73627i	0.359362i
$592$	0	0
$593$	12.4788i	0.512443i	0.966618	+	0.256221i	$0.0824776\pi$
$593$	12.4788i	0.512443i	−0.966618	+	0.256221i	$0.917522\pi$
$594$	0	0
$595$	5.51736	0.226190
$596$	0	0
$597$	−16.0911	−0.658564
$598$	0	0
$599$	41.6280	1.70087	0.850437	−	0.526076i	$-0.176337\pi$
$599$	41.6280	1.70087	0.850437	+	0.526076i	$0.176337\pi$
$600$	0	0
$601$	29.2599	1.19354	0.596768	−	0.802414i	$-0.296451\pi$
$601$	29.2599	1.19354	0.596768	+	0.802414i	$0.296451\pi$
$602$	0	0
$603$	9.78493i	0.398473i
$604$	0	0
$605$	44.5236i	1.81014i
$606$	0	0
$607$	−18.0582	−0.732958	−0.366479	−	0.930426i	$-0.619437\pi$
$607$	−18.0582	−0.732958	−0.366479	+	0.930426i	$0.619437\pi$
$608$	0	0
$609$	− 14.1688i	− 0.574149i
$610$	0	0
$611$	0	0
$612$	0	0
$613$	− 17.9051i	− 0.723180i	−0.932337	−	0.361590i	$-0.882234\pi$
$613$	− 17.9051i	− 0.723180i	0.932337	−	0.361590i	$-0.117766\pi$
$614$	0	0
$615$	34.1068	1.37532
$616$	0	0
$617$	− 23.3972i	− 0.941936i	−0.882150	−	0.470968i	$-0.843905\pi$
$617$	− 23.3972i	− 0.941936i	0.882150	−	0.470968i	$-0.156095\pi$
$618$	0	0
$619$	− 43.9537i	− 1.76665i	−0.468761	−	0.883325i	$-0.655299\pi$
$619$	− 43.9537i	− 1.76665i	0.468761	−	0.883325i	$-0.344701\pi$
$620$	0	0
$621$	4.72294	0.189525
$622$	0	0
$623$	−28.3376	−1.13532
$624$	0	0
$625$	32.8760	1.31504
$626$	0	0
$627$	−24.3376	−0.971951
$628$	0	0
$629$	0.953747i	0.0380284i
$630$	0	0
$631$	9.67427i	0.385127i	0.981285	+	0.192563i	$0.0616801\pi$
$631$	9.67427i	0.385127i	−0.981285	+	0.192563i	$0.938320\pi$
$632$	0	0
$633$	−4.38388	−0.174244
$634$	0	0
$635$	64.5198i	2.56039i
$636$	0	0
$637$	0	0
$638$	0	0
$639$	8.72294i	0.345074i
$640$	0	0
$641$	−14.4130	−0.569277	−0.284639	−	0.958635i	$-0.591874\pi$
$641$	−14.4130	−0.569277	−0.284639	+	0.958635i	$0.591874\pi$
$642$	0	0
$643$	18.6900i	0.737062i	0.929615	+	0.368531i	$0.120139\pi$
$643$	18.6900i	0.737062i	−0.929615	+	0.368531i	$0.879861\pi$
$644$	0	0
$645$	24.4750i	0.963700i
$646$	0	0
$647$	−22.7363	−0.893855	−0.446928	−	0.894570i	$-0.647482\pi$
$647$	−22.7363	−0.893855	−0.446928	+	0.894570i	$0.647482\pi$
$648$	0	0
$649$	−2.03149	−0.0797428
$650$	0	0
$651$	5.24414	0.205534
$652$	0	0
$653$	20.2136	0.791021	0.395510	−	0.918462i	$-0.370568\pi$
$653$	20.2136	0.791021	0.395510	+	0.918462i	$0.370568\pi$
$654$	0	0
$655$	82.7601i	3.23370i
$656$	0	0
$657$	− 4.87601i	− 0.190231i
$658$	0	0
$659$	19.7520	0.769429	0.384715	−	0.923036i	$-0.374300\pi$
$659$	19.7520	0.769429	0.384715	+	0.923036i	$0.374300\pi$
$660$	0	0
$661$	10.7653i	0.418723i	0.977838	+	0.209362i	$0.0671386\pi$
$661$	10.7653i	0.418723i	−0.977838	+	0.209362i	$0.932861\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	− 36.2213i	− 1.40460i
$666$	0	0
$667$	37.4907	1.45165
$668$	0	0
$669$	− 16.0000i	− 0.618596i
$670$	0	0
$671$	− 19.6147i	− 0.757217i
$672$	0	0
$673$	−46.5040	−1.79260	−0.896299	−	0.443450i	$-0.853754\pi$
$673$	−46.5040	−1.79260	−0.896299	+	0.443450i	$0.853754\pi$
$674$	0	0
$675$	10.5079	0.404448
$676$	0	0
$677$	2.18215	0.0838667	0.0419333	−	0.999120i	$-0.486648\pi$
$677$	2.18215	0.0838667	0.0419333	+	0.999120i	$0.486648\pi$
$678$	0	0
$679$	5.46546	0.209745
$680$	0	0
$681$	17.7387i	0.679748i
$682$	0	0
$683$	5.87601i	0.224839i	0.993661	+	0.112420i	$0.0358601\pi$
$683$	5.87601i	0.224839i	−0.993661	+	0.112420i	$0.964140\pi$
$684$	0	0
$685$	−49.3705	−1.88635
$686$	0	0
$687$	11.4459i	0.436687i
$688$	0	0
$689$	0	0
$690$	0	0
$691$	− 11.4788i	− 0.436674i	−0.975873	−	0.218337i	$-0.929937\pi$
$691$	− 11.4788i	− 0.436674i	0.975873	−	0.218337i	$-0.0700632\pi$
$692$	0	0
$693$	−8.43013	−0.320234
$694$	0	0
$695$	− 34.7096i	− 1.31661i
$696$	0	0
$697$	6.79827i	0.257503i
$698$	0	0
$699$	−9.75201	−0.368855
$700$	0	0
$701$	26.4301	0.998252	0.499126	−	0.866529i	$-0.333654\pi$
$701$	26.4301	0.998252	0.499126	+	0.866529i	$0.333654\pi$
$702$	0	0
$703$	6.26132	0.236150
$704$	0	0
$705$	2.84693	0.107222
$706$	0	0
$707$	− 2.91277i	− 0.109546i
$708$	0	0
$709$	− 52.0606i	− 1.95518i	−0.210528	−	0.977588i	$-0.567518\pi$
$709$	− 52.0606i	− 1.95518i	0.210528	−	0.977588i	$-0.432482\pi$
$710$	0	0
$711$	16.3839	0.614443
$712$	0	0
$713$	13.8760i	0.519661i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 6.59894i	− 0.246442i
$718$	0	0
$719$	−34.4616	−1.28520	−0.642601	−	0.766201i	$-0.722145\pi$
$719$	−34.4616	−1.28520	−0.642601	+	0.766201i	$0.722145\pi$
$720$	0	0
$721$	− 13.6743i	− 0.509257i
$722$	0	0
$723$	23.8140i	0.885653i
$724$	0	0
$725$	83.4115	3.09783
$726$	0	0
$727$	29.1335	1.08050	0.540251	−	0.841504i	$-0.318329\pi$
$727$	29.1335	1.08050	0.540251	+	0.841504i	$0.318329\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−4.87842	−0.180435
$732$	0	0
$733$	− 37.7229i	− 1.39333i	−0.717397	−	0.696664i	$-0.754667\pi$
$733$	− 37.7229i	− 1.39333i	0.717397	−	0.696664i	$-0.245333\pi$
$734$	0	0
$735$	15.0196i	0.554006i
$736$	0	0
$737$	46.2136	1.70230
$738$	0	0
$739$	− 27.7520i	− 1.02087i	−0.859915	−	0.510437i	$-0.829484\pi$
$739$	− 27.7520i	− 1.02087i	0.859915	−	0.510437i	$-0.170516\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 24.1821i	− 0.887157i	−0.896236	−	0.443578i	$-0.853709\pi$
$743$	− 24.1821i	− 0.887157i	0.896236	−	0.443578i	$-0.146291\pi$
$744$	0	0
$745$	14.3023	0.523996
$746$	0	0
$747$	6.59894i	0.241443i
$748$	0	0
$749$	− 1.29040i	− 0.0471500i
$750$	0	0
$751$	−43.7968	−1.59817	−0.799085	−	0.601219i	$-0.794682\pi$
$751$	−43.7968	−1.59817	−0.799085	+	0.601219i	$0.794682\pi$
$752$	0	0
$753$	9.44588	0.344227
$754$	0	0
$755$	57.4907	2.09230
$756$	0	0
$757$	16.8918	0.613941	0.306971	−	0.951719i	$-0.400685\pi$
$757$	16.8918	0.613941	0.306971	+	0.951719i	$0.400685\pi$
$758$	0	0
$759$	− 22.3061i	− 0.809662i
$760$	0	0
$761$	0.123993i	0.00449474i	0.999997	+	0.00224737i	$0.000715361\pi$
$761$	0.123993i	0.00449474i	−0.999997	+	0.00224737i	$0.999285\pi$
$762$	0	0
$763$	9.58177	0.346883
$764$	0	0
$765$	3.09107i	0.111758i
$766$	0	0
$767$	0	0
$768$	0	0
$769$	− 43.8102i	− 1.57984i	−0.613213	−	0.789918i	$-0.710123\pi$
$769$	− 43.8102i	− 1.57984i	0.613213	−	0.789918i	$-0.289877\pi$
$770$	0	0
$771$	−12.6609	−0.455973
$772$	0	0
$773$	− 4.55412i	− 0.163800i	−0.996641	−	0.0819002i	$-0.973901\pi$
$773$	− 4.55412i	− 0.163800i	0.996641	−	0.0819002i	$-0.0260989\pi$
$774$	0	0
$775$	30.8722i	1.10896i
$776$	0	0
$777$	2.16881	0.0778057
$778$	0	0
$779$	44.6304	1.59905
$780$	0	0
$781$	41.1979	1.47418
$782$	0	0
$783$	7.93800	0.283681
$784$	0	0
$785$	61.4287i	2.19248i
$786$	0	0
$787$	− 0.0462534i	− 0.00164875i	−1.00000		0.000824377i	$-0.999738\pi$
$787$	− 0.0462534i	− 0.00164875i	1.00000		0.000824377i	$-0.000262407\pi$
$788$	0	0
$789$	0.292806	0.0104242
$790$	0	0
$791$	− 23.5880i	− 0.838693i
$792$	0	0
$793$	0	0
$794$	0	0
$795$	− 54.3996i	− 1.92936i
$796$	0	0
$797$	−18.5198	−0.656004	−0.328002	−	0.944677i	$-0.606375\pi$
$797$	−18.5198	−0.656004	−0.328002	+	0.944677i	$0.606375\pi$
$798$	0	0
$799$	0.567458i	0.0200752i
$800$	0	0
$801$	− 15.8760i	− 0.560951i
$802$	0	0
$803$	−23.0291	−0.812678
$804$	0	0
$805$	33.1979	1.17007
$806$	0	0
$807$	−5.56987	−0.196069
$808$	0	0
$809$	43.7034	1.53653	0.768264	−	0.640133i	$-0.221121\pi$
$809$	43.7034	1.53653	0.768264	+	0.640133i	$0.221121\pi$
$810$	0	0
$811$	12.2017i	0.428461i	0.976783	+	0.214230i	$0.0687243\pi$
$811$	12.2017i	0.428461i	−0.976783	+	0.214230i	$0.931276\pi$
$812$	0	0
$813$	11.9537i	0.419236i
$814$	0	0
$815$	−43.0739	−1.50881
$816$	0	0
$817$	32.0267i	1.12047i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 38.5856i	− 1.34665i	−0.739348	−	0.673324i	$-0.764866\pi$
$821$	− 38.5856i	− 1.34665i	0.739348	−	0.673324i	$-0.235134\pi$
$822$	0	0
$823$	−48.3376	−1.68494	−0.842472	−	0.538740i	$-0.818900\pi$
$823$	−48.3376	−1.68494	−0.842472	+	0.538740i	$0.818900\pi$
$824$	0	0
$825$	− 49.6280i	− 1.72783i
$826$	0	0
$827$	1.19789i	0.0416547i	0.999783	+	0.0208273i	$0.00663003\pi$
$827$	1.19789i	0.0416547i	−0.999783	+	0.0208273i	$0.993370\pi$
$828$	0	0
$829$	18.3352	0.636808	0.318404	−	0.947955i	$-0.396853\pi$
$829$	18.3352	0.636808	0.318404	+	0.947955i	$0.396853\pi$
$830$	0	0
$831$	18.6609	0.647341
$832$	0	0
$833$	−2.99375	−0.103727
$834$	0	0
$835$	81.0662	2.80541
$836$	0	0
$837$	2.93800i	0.101552i
$838$	0	0
$839$	− 46.4883i	− 1.60495i	−0.596683	−	0.802477i	$-0.703515\pi$
$839$	− 46.4883i	− 1.60495i	0.596683	−	0.802477i	$-0.296485\pi$
$840$	0	0
$841$	34.0119	1.17282
$842$	0	0
$843$	4.96708i	0.171075i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	20.1807i	0.693417i
$848$	0	0
$849$	−10.8007	−0.370678
$850$	0	0
$851$	5.73868i	0.196719i
$852$	0	0
$853$	10.7653i	0.368598i	0.982870	+	0.184299i	$0.0590016\pi$
$853$	10.7653i	0.368598i	−0.982870	+	0.184299i	$0.940998\pi$
$854$	0	0
$855$	20.2928	0.693999
$856$	0	0
$857$	23.3972	0.799234	0.399617	−	0.916682i	$-0.369143\pi$
$857$	23.3972	0.799234	0.399617	+	0.916682i	$0.369143\pi$
$858$	0	0
$859$	0.339059	0.0115685	0.00578427	−	0.999983i	$-0.498159\pi$
$859$	0.339059	0.0115685	0.00578427	+	0.999983i	$0.498159\pi$
$860$	0	0
$861$	15.4592	0.526848
$862$	0	0
$863$	13.9208i	0.473870i	0.971525	+	0.236935i	$0.0761429\pi$
$863$	13.9208i	0.473870i	−0.971525	+	0.236935i	$0.923857\pi$
$864$	0	0
$865$	− 1.20558i	− 0.0409908i
$866$	0	0
$867$	16.3839	0.556426
$868$	0	0
$869$	− 77.3800i	− 2.62494i
$870$	0	0
$871$	0	0
$872$	0	0
$873$	3.06200i	0.103633i
$874$	0	0
$875$	38.7153	1.30881
$876$	0	0
$877$	− 7.37055i	− 0.248886i	−0.992227	−	0.124443i	$-0.960286\pi$
$877$	− 7.37055i	− 0.248886i	0.992227	−	0.124443i	$-0.0397143\pi$
$878$	0	0
$879$	− 26.2441i	− 0.885193i
$880$	0	0
$881$	30.9671	1.04331	0.521654	−	0.853157i	$-0.325315\pi$
$881$	30.9671	1.04331	0.521654	+	0.853157i	$0.325315\pi$
$882$	0	0
$883$	−24.5660	−0.826713	−0.413356	−	0.910569i	$-0.635644\pi$
$883$	−24.5660	−0.826713	−0.413356	+	0.910569i	$0.635644\pi$
$884$	0	0
$885$	1.69386	0.0569385
$886$	0	0
$887$	−20.1821	−0.677650	−0.338825	−	0.940849i	$-0.610029\pi$
$887$	−20.1821	−0.677650	−0.338825	+	0.940849i	$0.610029\pi$
$888$	0	0
$889$	29.2441i	0.980817i
$890$	0	0
$891$	− 4.72294i	− 0.158224i
$892$	0	0
$893$	3.72535	0.124664
$894$	0	0
$895$	− 11.2112i	− 0.374750i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	23.3219i	0.777828i
$900$	0	0
$901$	10.8431	0.361236
$902$	0	0
$903$	11.0935i	0.369168i
$904$	0	0
$905$	52.0410i	1.72990i
$906$	0	0
$907$	−47.2294	−1.56823	−0.784113	−	0.620618i	$-0.786882\pi$
$907$	−47.2294	−1.56823	−0.784113	+	0.620618i	$0.786882\pi$
$908$	0	0
$909$	1.63186	0.0541255
$910$	0	0
$911$	−48.3376	−1.60150	−0.800748	−	0.599001i	$-0.795565\pi$
$911$	−48.3376	−1.60150	−0.800748	+	0.599001i	$0.795565\pi$
$912$	0	0
$913$	31.1664	1.03146
$914$	0	0
$915$	16.3548i	0.540673i
$916$	0	0
$917$	37.5117i	1.23875i
$918$	0	0
$919$	−10.0582	−0.331788	−0.165894	−	0.986144i	$-0.553051\pi$
$919$	−10.0582	−0.331788	−0.165894	+	0.986144i	$0.553051\pi$
$920$	0	0
$921$	21.5369i	0.709667i
$922$	0	0
$923$	0	0
$924$	0	0
$925$	12.7678i	0.419801i
$926$	0	0
$927$	7.66094	0.251618
$928$	0	0
$929$	− 12.0487i	− 0.395304i	−0.980272	−	0.197652i	$-0.936668\pi$
$929$	− 12.0487i	− 0.395304i	0.980272	−	0.197652i	$-0.0633316\pi$
$930$	0	0
$931$	19.6539i	0.644129i
$932$	0	0
$933$	21.1531	0.692520
$934$	0	0
$935$	14.5989	0.477437
$936$	0	0
$937$	−22.8298	−0.745816	−0.372908	−	0.927868i	$-0.621639\pi$
$937$	−22.8298	−0.745816	−0.372908	+	0.927868i	$0.621639\pi$
$938$	0	0
$939$	−9.79827	−0.319754
$940$	0	0
$941$	− 39.8102i	− 1.29777i	−0.760885	−	0.648887i	$-0.775235\pi$
$941$	− 39.8102i	− 1.29777i	0.760885	−	0.648887i	$-0.224765\pi$
$942$	0	0
$943$	40.9051i	1.33205i
$944$	0	0
$945$	7.02908	0.228656
$946$	0	0
$947$	30.3061i	0.984817i	0.870364	+	0.492409i	$0.163883\pi$
$947$	30.3061i	0.984817i	−0.870364	+	0.492409i	$0.836117\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	− 22.5236i	− 0.730378i
$952$	0	0
$953$	−10.0267	−0.324796	−0.162398	−	0.986725i	$-0.551923\pi$
$953$	−10.0267	−0.324796	−0.162398	+	0.986725i	$0.551923\pi$
$954$	0	0
$955$	14.0582i	0.454911i
$956$	0	0
$957$	− 37.4907i	− 1.21190i
$958$	0	0
$959$	−22.3776	−0.722611
$960$	0	0
$961$	22.3681	0.721553
$962$	0	0
$963$	0.722938	0.0232963
$964$	0	0
$965$	−49.0119	−1.57775
$966$	0	0
$967$	− 43.7702i	− 1.40755i	−0.710421	−	0.703777i	$-0.751495\pi$
$967$	− 43.7702i	− 1.40755i	0.710421	−	0.703777i	$-0.248505\pi$
$968$	0	0
$969$	4.04482i	0.129938i
$970$	0	0
$971$	−48.9232	−1.57002	−0.785011	−	0.619482i	$-0.787343\pi$
$971$	−48.9232	−1.57002	−0.785011	+	0.619482i	$0.787343\pi$
$972$	0	0
$973$	− 15.7324i	− 0.504358i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 27.5794i	− 0.882342i	−0.897423	−	0.441171i	$-0.854563\pi$
$977$	− 27.5794i	− 0.882342i	0.897423	−	0.441171i	$-0.145437\pi$
$978$	0	0
$979$	−74.9814	−2.39642
$980$	0	0
$981$	5.36814i	0.171391i
$982$	0	0
$983$	34.7096i	1.10706i	0.832828	+	0.553532i	$0.186720\pi$
$983$	34.7096i	1.10706i	−0.832828	+	0.553532i	$0.813280\pi$
$984$	0	0
$985$	−34.4035	−1.09619
$986$	0	0
$987$	1.29040	0.0410738
$988$	0	0
$989$	−29.3534	−0.933383
$990$	0	0
$991$	−32.6571	−1.03739	−0.518693	−	0.854960i	$-0.673581\pi$
$991$	−32.6571	−1.03739	−0.518693	+	0.854960i	$0.673581\pi$
$992$	0	0
$993$	− 21.0620i	− 0.668382i
$994$	0	0
$995$	− 63.3667i	− 2.00886i
$996$	0	0
$997$	5.59894	0.177320	0.0886602	−	0.996062i	$-0.471741\pi$
$997$	5.59894	0.177320	0.0886602	+	0.996062i	$0.471741\pi$
$998$	0	0
$999$	1.21507i	0.0384430i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4056.2.c.m.337.1		6
13.2	odd	12		312.2.q.e.217.1	✓	6
13.5	odd	4		4056.2.a.z.1.1		3
13.6	odd	12		312.2.q.e.289.1	yes	6
13.8	odd	4		4056.2.a.y.1.3		3
13.12	even	2	inner	4056.2.c.m.337.6		6
39.2	even	12		936.2.t.h.217.3		6
39.32	even	12		936.2.t.h.289.3		6
52.15	even	12		624.2.q.j.529.1		6
52.19	even	12		624.2.q.j.289.1		6
52.31	even	4		8112.2.a.ck.1.1		3
52.47	even	4		8112.2.a.cl.1.3		3
156.71	odd	12		1872.2.t.u.289.3		6
156.119	odd	12		1872.2.t.u.1153.3		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
312.2.q.e.217.1	✓	6	13.2	odd	12
312.2.q.e.289.1	yes	6	13.6	odd	12
624.2.q.j.289.1		6	52.19	even	12
624.2.q.j.529.1		6	52.15	even	12
936.2.t.h.217.3		6	39.2	even	12
936.2.t.h.289.3		6	39.32	even	12
1872.2.t.u.289.3		6	156.71	odd	12
1872.2.t.u.1153.3		6	156.119	odd	12
4056.2.a.y.1.3		3	13.8	odd	4
4056.2.a.z.1.1		3	13.5	odd	4
4056.2.c.m.337.1		6	1.1	even	1	trivial
4056.2.c.m.337.6		6	13.12	even	2	inner
8112.2.a.ck.1.1		3	52.31	even	4
8112.2.a.cl.1.3		3	52.47	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 4056 = 2^{3} \cdot 3 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4056.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -3.93800i q^{5} -1.78493i q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} -3.93800i q^{5} -1.78493i q^{7} +1.00000 q^{9} -4.72294i q^{11} +3.93800i q^{15} -0.784934 q^{17} +5.15307i q^{19} +1.78493i q^{21} -4.72294 q^{23} -10.5079 q^{25} -1.00000 q^{27} -7.93800 q^{29} -2.93800i q^{31} +4.72294i q^{33} -7.02908 q^{35} -1.21507i q^{37} -8.66094i q^{41} +6.21507 q^{43} -3.93800i q^{45} -0.722938i q^{47} +3.81401 q^{49} +0.784934 q^{51} -13.8140 q^{53} -18.5989 q^{55} -5.15307i q^{57} -0.430132i q^{59} +4.15307 q^{61} -1.78493i q^{63} +9.78493i q^{67} +4.72294 q^{69} +8.72294i q^{71} -4.87601i q^{73} +10.5079 q^{75} -8.43013 q^{77} +16.3839 q^{79} +1.00000 q^{81} +6.59894i q^{83} +3.09107i q^{85} +7.93800 q^{87} -15.8760i q^{89} +2.93800i q^{93} +20.2928 q^{95} +3.06200i q^{97} -4.72294i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 6 q^{3} + 6 q^{9}+O(q^{10}) \) 6 * q - 6 * q^3 + 6 * q^9	\( 6 q - 6 q^{3} + 6 q^{9} - 30 q^{25} - 6 q^{27} - 24 q^{29} + 24 q^{35} + 42 q^{43} - 48 q^{49} - 12 q^{53} - 36 q^{55} + 6 q^{61} + 30 q^{75} - 60 q^{77} + 18 q^{79} + 6 q^{81} + 24 q^{87} + 84 q^{95}+O(q^{100}) \) 6 * q - 6 * q^3 + 6 * q^9 - 30 * q^25 - 6 * q^27 - 24 * q^29 + 24 * q^35 + 42 * q^43 - 48 * q^49 - 12 * q^53 - 36 * q^55 + 6 * q^61 + 30 * q^75 - 60 * q^77 + 18 * q^79 + 6 * q^81 + 24 * q^87 + 84 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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