LMFDB - Embedded newform 5472.2.d.g.2015.5

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [5472,2,Mod(2015,5472)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(5472, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 1, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("5472.2015"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 5472 = 2^{5} \cdot 3^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5472.d (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [16,0,0,0,0,0,0,0,0,0,0,0,-16,0,0,0,0,0,0,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(23)] == traces)

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

Self dual:	no
Analytic conductor:	$43.6941399860$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 4x^{14} + 5x^{12} + 4x^{10} - 20x^{8} + 16x^{6} + 80x^{4} - 256x^{2} + 256 $ x^16 - 4x^14 + 5x^12 + 4x^10 - 20x^8 + 16x^6 + 80x^4 - 256*x^2 + 256
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{14} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2015.5
Root			$1.33768 + 0.458926i$ of defining polynomial
Character	$\chi$	$=$	5472.2015
Dual form			5472.2.d.g.2015.12

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-0.436116i q^{5} -3.45559i q^{7} -0.436116 q^{11} -5.08157 q^{13} -3.60784i q^{17} -1.00000i q^{19} +3.12943 q^{23} +4.80980 q^{25} +4.35800i q^{29} +3.08157i q^{31} -1.50704 q^{35} -5.08157 q^{37} -9.77387i q^{41} -6.23549i q^{43} +5.13741 q^{47} -4.94108 q^{49} -6.07321i q^{53} +0.190197i q^{55} -5.23023 q^{59} -1.92765 q^{61} +2.21615i q^{65} -2.91117i q^{67} -1.11321 q^{71} -5.94108 q^{73} +1.50704i q^{77} +13.2447i q^{79} -5.95786 q^{83} -1.57344 q^{85} -13.7155i q^{89} +17.5598i q^{91} -0.436116 q^{95} -9.88215 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 16 q^{13} - 48 q^{25} - 16 q^{37} + 8 q^{49} - 80 q^{61} - 8 q^{73} - 152 q^{85} + 16 q^{97}+O(q^{100}) $ 16 * q - 16 * q^13 - 48 * q^25 - 16 * q^37 + 8 * q^49 - 80 * q^61 - 8 * q^73 - 152 * q^85 + 16 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/5472\mathbb{Z}\right)^\times$.

$n$	$1217$	$2053$	$3745$	$4447$
$\chi(n)$	$-1$	$1$	$1$	$-1$

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
$5$	− 0.436116i	− 0.195037i	−0.995234	−	0.0975186i	$-0.968909\pi$
$5$	− 0.436116i	− 0.195037i	0.995234	−	0.0975186i	$-0.0310905\pi$
$6$	0	0
$7$	− 3.45559i	− 1.30609i	−0.757320	−	0.653044i	$-0.773491\pi$
$7$	− 3.45559i	− 1.30609i	0.757320	−	0.653044i	$-0.226509\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−0.436116	−0.131494	−0.0657470	−	0.997836i	$-0.520943\pi$
$11$	−0.436116	−0.131494	−0.0657470	+	0.997836i	$0.520943\pi$
$12$	0	0
$13$	−5.08157	−1.40937	−0.704687	−	0.709518i	$-0.748912\pi$
$13$	−5.08157	−1.40937	−0.704687	+	0.709518i	$0.748912\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 3.60784i	− 0.875029i	−0.899211	−	0.437514i	$-0.855859\pi$
$17$	− 3.60784i	− 0.875029i	0.899211	−	0.437514i	$-0.144141\pi$
$18$	0	0
$19$	− 1.00000i	− 0.229416i
$19$	− 1.00000i	− 0.229416i
$20$	0	0
$21$	0	0
$22$	0	0
$23$	3.12943	0.652531	0.326266	−	0.945278i	$-0.394210\pi$
$23$	3.12943	0.652531	0.326266	+	0.945278i	$0.394210\pi$
$24$	0	0
$25$	4.80980	0.961961
$26$	0	0
$27$	0	0
$28$	0	0
$29$	4.35800i	0.809260i	0.914481	+	0.404630i	$0.132600\pi$
$29$	4.35800i	0.809260i	−0.914481	+	0.404630i	$0.867400\pi$
$30$	0	0
$31$	3.08157i	0.553466i	0.960947	+	0.276733i	$0.0892518\pi$
$31$	3.08157i	0.553466i	−0.960947	+	0.276733i	$0.910748\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−1.50704	−0.254736
$36$	0	0
$37$	−5.08157	−0.835405	−0.417702	−	0.908584i	$-0.637165\pi$
$37$	−5.08157	−0.835405	−0.417702	+	0.908584i	$0.637165\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 9.77387i	− 1.52642i	−0.646149	−	0.763211i	$-0.723622\pi$
$41$	− 9.77387i	− 1.52642i	0.646149	−	0.763211i	$-0.276378\pi$
$42$	0	0
$43$	− 6.23549i	− 0.950903i	−0.879742	−	0.475451i	$-0.842285\pi$
$43$	− 6.23549i	− 0.950903i	0.879742	−	0.475451i	$-0.157715\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	5.13741	0.749368	0.374684	−	0.927153i	$-0.377751\pi$
$47$	5.13741	0.749368	0.374684	+	0.927153i	$0.377751\pi$
$48$	0	0
$49$	−4.94108	−0.705868
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 6.07321i	− 0.834220i	−0.908856	−	0.417110i	$-0.863043\pi$
$53$	− 6.07321i	− 0.834220i	0.908856	−	0.417110i	$-0.136957\pi$
$54$	0	0
$55$	0.190197i	0.0256462i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−5.23023	−0.680918	−0.340459	−	0.940259i	$-0.610582\pi$
$59$	−5.23023	−0.680918	−0.340459	+	0.940259i	$0.610582\pi$
$60$	0	0
$61$	−1.92765	−0.246811	−0.123405	−	0.992356i	$-0.539382\pi$
$61$	−1.92765	−0.246811	−0.123405	+	0.992356i	$0.539382\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	2.21615i	0.274880i
$66$	0	0
$67$	− 2.91117i	− 0.355656i	−0.984062	−	0.177828i	$-0.943093\pi$
$67$	− 2.91117i	− 0.355656i	0.984062	−	0.177828i	$-0.0569071\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−1.11321	−0.132114	−0.0660569	−	0.997816i	$-0.521042\pi$
$71$	−1.11321	−0.132114	−0.0660569	+	0.997816i	$0.521042\pi$
$72$	0	0
$73$	−5.94108	−0.695350	−0.347675	−	0.937615i	$-0.613029\pi$
$73$	−5.94108	−0.695350	−0.347675	+	0.937615i	$0.613029\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	1.50704i	0.171743i
$78$	0	0
$79$	13.2447i	1.49015i	0.666983	+	0.745073i	$0.267585\pi$
$79$	13.2447i	1.49015i	−0.666983	+	0.745073i	$0.732415\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−5.95786	−0.653960	−0.326980	−	0.945031i	$-0.606031\pi$
$83$	−5.95786	−0.653960	−0.326980	+	0.945031i	$0.606031\pi$
$84$	0	0
$85$	−1.57344	−0.170663
$86$	0	0
$87$	0	0
$88$	0	0
$89$	− 13.7155i	− 1.45384i	−0.686722	−	0.726921i	$-0.740951\pi$
$89$	− 13.7155i	− 1.45384i	0.686722	−	0.726921i	$-0.259049\pi$
$90$	0	0
$91$	17.5598i	1.84077i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−0.436116	−0.0447446
$96$	0	0
$97$	−9.88215	−1.00338	−0.501690	−	0.865047i	$-0.667288\pi$
$97$	−9.88215	−1.00338	−0.501690	+	0.865047i	$0.667288\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	9.88923i	0.984015i	0.870591	+	0.492008i	$0.163737\pi$
$101$	9.88923i	0.984015i	−0.870591	+	0.492008i	$0.836263\pi$
$102$	0	0
$103$	12.0525i	1.18757i	0.804623	+	0.593786i	$0.202368\pi$
$103$	12.0525i	1.18757i	−0.804623	+	0.593786i	$0.797632\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−12.2603	−1.18524	−0.592622	−	0.805481i	$-0.701907\pi$
$107$	−12.2603	−1.18524	−0.592622	+	0.805481i	$0.701907\pi$
$108$	0	0
$109$	−2.17040	−0.207886	−0.103943	−	0.994583i	$-0.533146\pi$
$109$	−2.17040	−0.207886	−0.103943	+	0.994583i	$0.533146\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	3.43043i	0.322708i	0.986897	+	0.161354i	$0.0515861\pi$
$113$	3.43043i	0.322708i	−0.986897	+	0.161354i	$0.948414\pi$
$114$	0	0
$115$	− 1.36480i	− 0.127268i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−12.4672	−1.14287
$120$	0	0
$121$	−10.8098	−0.982709
$122$	0	0
$123$	0	0
$124$	0	0
$125$	− 4.27821i	− 0.382655i
$126$	0	0
$127$	10.2553i	0.910009i	0.890489	+	0.455005i	$0.150362\pi$
$127$	10.2553i	0.910009i	−0.890489	+	0.455005i	$0.849638\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	0.352784	0.0308229	0.0154114	−	0.999881i	$-0.495094\pi$
$131$	0.352784	0.0308229	0.0154114	+	0.999881i	$0.495094\pi$
$132$	0	0
$133$	−3.45559	−0.299637
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 0.862741i	− 0.0737090i	−0.999321	−	0.0368545i	$-0.988266\pi$
$137$	− 0.862741i	− 0.0737090i	0.999321	−	0.0368545i	$-0.0117338\pi$
$138$	0	0
$139$	17.6601i	1.49791i	0.662621	+	0.748955i	$0.269444\pi$
$139$	17.6601i	1.49791i	−0.662621	+	0.748955i	$0.730556\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	2.21615	0.185324
$144$	0	0
$145$	1.90059	0.157836
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 11.2106i	− 0.918410i	−0.888330	−	0.459205i	$-0.848134\pi$
$149$	− 11.2106i	− 0.918410i	0.888330	−	0.459205i	$-0.151866\pi$
$150$	0	0
$151$	2.11059i	0.171757i	0.996306	+	0.0858787i	$0.0273698\pi$
$151$	2.11059i	0.171757i	−0.996306	+	0.0858787i	$0.972630\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	1.34392	0.107946
$156$	0	0
$157$	−9.19216	−0.733614	−0.366807	−	0.930297i	$-0.619549\pi$
$157$	−9.19216	−0.733614	−0.366807	+	0.930297i	$0.619549\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	− 10.8140i	− 0.852264i
$162$	0	0
$163$	− 11.1341i	− 0.872091i	−0.899925	−	0.436046i	$-0.856379\pi$
$163$	− 11.1341i	− 0.872091i	0.899925	−	0.436046i	$-0.143621\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−11.3880	−0.881232	−0.440616	−	0.897696i	$-0.645240\pi$
$167$	−11.3880	−0.881232	−0.440616	+	0.897696i	$0.645240\pi$
$168$	0	0
$169$	12.8223	0.986334
$170$	0	0
$171$	0	0
$172$	0	0
$173$	16.3030i	1.23949i	0.784803	+	0.619746i	$0.212764\pi$
$173$	16.3030i	1.23949i	−0.784803	+	0.619746i	$0.787236\pi$
$174$	0	0
$175$	− 16.6207i	− 1.25641i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−25.2307	−1.88583	−0.942915	−	0.333034i	$-0.891928\pi$
$179$	−25.2307	−1.88583	−0.942915	+	0.333034i	$0.891928\pi$
$180$	0	0
$181$	4.25529	0.316293	0.158146	−	0.987416i	$-0.449448\pi$
$181$	4.25529	0.316293	0.158146	+	0.987416i	$0.449448\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	2.21615i	0.162935i
$186$	0	0
$187$	1.57344i	0.115061i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−12.3531	−0.893837	−0.446919	−	0.894575i	$-0.647479\pi$
$191$	−12.3531	−0.893837	−0.446919	+	0.894575i	$0.647479\pi$
$192$	0	0
$193$	20.7520	1.49376	0.746879	−	0.664960i	$-0.231551\pi$
$193$	20.7520	1.49376	0.746879	+	0.664960i	$0.231551\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 17.0013i	− 1.21130i	−0.795733	−	0.605648i	$-0.792914\pi$
$197$	− 17.0013i	− 1.21130i	0.795733	−	0.605648i	$-0.207086\pi$
$198$	0	0
$199$	13.7914i	0.977644i	0.872384	+	0.488822i	$0.162573\pi$
$199$	13.7914i	0.977644i	−0.872384	+	0.488822i	$0.837427\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	15.0594	1.05696
$204$	0	0
$205$	−4.26254	−0.297709
$206$	0	0
$207$	0	0
$208$	0	0
$209$	0.436116i	0.0301668i
$210$	0	0
$211$	11.1324i	0.766383i	0.923669	+	0.383191i	$0.125175\pi$
$211$	11.1324i	0.766383i	−0.923669	+	0.383191i	$0.874825\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−2.71940	−0.185461
$216$	0	0
$217$	10.6486	0.722876
$218$	0	0
$219$	0	0
$220$	0	0
$221$	18.3335i	1.23324i
$222$	0	0
$223$	− 4.75922i	− 0.318701i	−0.987222	−	0.159350i	$-0.949060\pi$
$223$	− 4.75922i	− 0.318701i	0.987222	−	0.159350i	$-0.0509400\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	3.43043	0.227686	0.113843	−	0.993499i	$-0.463684\pi$
$227$	3.43043	0.227686	0.113843	+	0.993499i	$0.463684\pi$
$228$	0	0
$229$	1.42765	0.0943415	0.0471707	−	0.998887i	$-0.484980\pi$
$229$	1.42765	0.0943415	0.0471707	+	0.998887i	$0.484980\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 14.9215i	− 0.977543i	−0.872412	−	0.488771i	$-0.837445\pi$
$233$	− 14.9215i	− 0.977543i	0.872412	−	0.488771i	$-0.162555\pi$
$234$	0	0
$235$	− 2.24051i	− 0.146154i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	24.3139	1.57273	0.786366	−	0.617760i	$-0.211960\pi$
$239$	24.3139	1.57273	0.786366	+	0.617760i	$0.211960\pi$
$240$	0	0
$241$	−27.9257	−1.79885	−0.899425	−	0.437074i	$-0.856015\pi$
$241$	−27.9257	−1.79885	−0.899425	+	0.437074i	$0.856015\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	2.15488i	0.137670i
$246$	0	0
$247$	5.08157i	0.323332i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	26.1417	1.65005	0.825023	−	0.565099i	$-0.191162\pi$
$251$	26.1417	1.65005	0.825023	+	0.565099i	$0.191162\pi$
$252$	0	0
$253$	−1.36480	−0.0858040
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 8.93089i	− 0.557094i	−0.960423	−	0.278547i	$-0.910147\pi$
$257$	− 8.93089i	− 0.557094i	0.960423	−	0.278547i	$-0.0898527\pi$
$258$	0	0
$259$	17.5598i	1.09111i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−8.83682	−0.544902	−0.272451	−	0.962170i	$-0.587834\pi$
$263$	−8.83682	−0.544902	−0.272451	+	0.962170i	$0.587834\pi$
$264$	0	0
$265$	−2.64863	−0.162704
$266$	0	0
$267$	0	0
$268$	0	0
$269$	20.0487i	1.22239i	0.791480	+	0.611195i	$0.209311\pi$
$269$	20.0487i	1.22239i	−0.791480	+	0.611195i	$0.790689\pi$
$270$	0	0
$271$	15.3386i	0.931755i	0.884849	+	0.465877i	$0.154261\pi$
$271$	15.3386i	0.931755i	−0.884849	+	0.465877i	$0.845739\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−2.09763	−0.126492
$276$	0	0
$277$	−1.22098	−0.0733617	−0.0366808	−	0.999327i	$-0.511678\pi$
$277$	−1.22098	−0.0733617	−0.0366808	+	0.999327i	$0.511678\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	− 3.04016i	− 0.181361i	−0.995880	−	0.0906803i	$-0.971096\pi$
$281$	− 3.04016i	− 0.181361i	0.995880	−	0.0906803i	$-0.0289041\pi$
$282$	0	0
$283$	30.8657i	1.83477i	0.397997	+	0.917387i	$0.369706\pi$
$283$	30.8657i	1.83477i	−0.397997	+	0.917387i	$0.630294\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−33.7745	−1.99364
$288$	0	0
$289$	3.98352	0.234325
$290$	0	0
$291$	0	0
$292$	0	0
$293$	14.1421i	0.826192i	0.910687	+	0.413096i	$0.135553\pi$
$293$	14.1421i	0.826192i	−0.910687	+	0.413096i	$0.864447\pi$
$294$	0	0
$295$	2.28099i	0.132804i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−15.9024	−0.919661
$300$	0	0
$301$	−21.5473	−1.24196
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0.840681i	0.0481372i
$306$	0	0
$307$	− 11.7625i	− 0.671324i	−0.941983	−	0.335662i	$-0.891040\pi$
$307$	− 11.7625i	− 0.671324i	0.941983	−	0.335662i	$-0.108960\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	16.4511	0.932857	0.466429	−	0.884559i	$-0.345540\pi$
$311$	16.4511	0.932857	0.466429	+	0.884559i	$0.345540\pi$
$312$	0	0
$313$	−8.32628	−0.470629	−0.235314	−	0.971919i	$-0.575612\pi$
$313$	−8.32628	−0.470629	−0.235314	+	0.971919i	$0.575612\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 27.0787i	− 1.52089i	−0.649401	−	0.760446i	$-0.724980\pi$
$317$	− 27.0787i	− 1.52089i	0.649401	−	0.760446i	$-0.275020\pi$
$318$	0	0
$319$	− 1.90059i	− 0.106413i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−3.60784	−0.200745
$324$	0	0
$325$	−24.4413	−1.35576
$326$	0	0
$327$	0	0
$328$	0	0
$329$	− 17.7527i	− 0.978741i
$330$	0	0
$331$	21.0743i	1.15835i	0.815204	+	0.579174i	$0.196625\pi$
$331$	21.0743i	1.15835i	−0.815204	+	0.579174i	$0.803375\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−1.26961	−0.0693662
$336$	0	0
$337$	−7.33686	−0.399664	−0.199832	−	0.979830i	$-0.564040\pi$
$337$	−7.33686	−0.399664	−0.199832	+	0.979830i	$0.564040\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	− 1.34392i	− 0.0727775i
$342$	0	0
$343$	− 7.11479i	− 0.384163i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	15.7645	0.846284	0.423142	−	0.906063i	$-0.360927\pi$
$347$	15.7645	0.846284	0.423142	+	0.906063i	$0.360927\pi$
$348$	0	0
$349$	−33.3377	−1.78453	−0.892264	−	0.451514i	$-0.850884\pi$
$349$	−33.3377	−1.78453	−0.892264	+	0.451514i	$0.850884\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 0.151704i	− 0.00807438i	−0.999992	−	0.00403719i	$-0.998715\pi$
$353$	− 0.151704i	− 0.00807438i	0.999992	−	0.00403719i	$-0.00128508\pi$
$354$	0	0
$355$	0.485489i	0.0257671i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−7.49098	−0.395359	−0.197679	−	0.980267i	$-0.563340\pi$
$359$	−7.49098	−0.395359	−0.197679	+	0.980267i	$0.563340\pi$
$360$	0	0
$361$	−1.00000	−0.0526316
$362$	0	0
$363$	0	0
$364$	0	0
$365$	2.59100i	0.135619i
$366$	0	0
$367$	− 6.89273i	− 0.359798i	−0.983685	−	0.179899i	$-0.942423\pi$
$367$	− 6.89273i	− 0.359798i	0.983685	−	0.179899i	$-0.0575770\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−20.9865	−1.08957
$372$	0	0
$373$	−24.6994	−1.27889	−0.639444	−	0.768838i	$-0.720835\pi$
$373$	−24.6994	−1.27889	−0.639444	+	0.768838i	$0.720835\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 22.1455i	− 1.14055i
$378$	0	0
$379$	− 12.1302i	− 0.623085i	−0.950232	−	0.311543i	$-0.899154\pi$
$379$	− 12.1302i	− 0.623085i	0.950232	−	0.311543i	$-0.100846\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−20.4943	−1.04721	−0.523605	−	0.851961i	$-0.675413\pi$
$383$	−20.4943	−1.04721	−0.523605	+	0.851961i	$0.675413\pi$
$384$	0	0
$385$	0.657243	0.0334962
$386$	0	0
$387$	0	0
$388$	0	0
$389$	− 11.4548i	− 0.580780i	−0.956908	−	0.290390i	$-0.906215\pi$
$389$	− 11.4548i	− 0.580780i	0.956908	−	0.290390i	$-0.0937850\pi$
$390$	0	0
$391$	− 11.2905i	− 0.570984i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	5.77623	0.290634
$396$	0	0
$397$	−25.1003	−1.25975	−0.629873	−	0.776698i	$-0.716893\pi$
$397$	−25.1003	−1.25975	−0.629873	+	0.776698i	$0.716893\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	− 20.1308i	− 1.00528i	−0.864495	−	0.502641i	$-0.832362\pi$
$401$	− 20.1308i	− 1.00528i	0.864495	−	0.502641i	$-0.167638\pi$
$402$	0	0
$403$	− 15.6592i	− 0.780041i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	2.21615	0.109851
$408$	0	0
$409$	10.6715	0.527674	0.263837	−	0.964567i	$-0.415012\pi$
$409$	10.6715	0.527674	0.263837	+	0.964567i	$0.415012\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	18.0735i	0.889339i
$414$	0	0
$415$	2.59832i	0.127546i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	19.4775	0.951536	0.475768	−	0.879571i	$-0.342170\pi$
$419$	19.4775	0.951536	0.475768	+	0.879571i	$0.342170\pi$
$420$	0	0
$421$	2.53804	0.123696	0.0618482	−	0.998086i	$-0.480301\pi$
$421$	2.53804	0.123696	0.0618482	+	0.998086i	$0.480301\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	− 17.3530i	− 0.841743i
$426$	0	0
$427$	6.66117i	0.322357i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−19.8370	−0.955512	−0.477756	−	0.878492i	$-0.658550\pi$
$431$	−19.8370	−0.955512	−0.477756	+	0.878492i	$0.658550\pi$
$432$	0	0
$433$	−16.9380	−0.813989	−0.406995	−	0.913431i	$-0.633423\pi$
$433$	−16.9380	−0.813989	−0.406995	+	0.913431i	$0.633423\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 3.12943i	− 0.149701i
$438$	0	0
$439$	21.9514i	1.04768i	0.851816	+	0.523841i	$0.175501\pi$
$439$	21.9514i	1.04768i	−0.851816	+	0.523841i	$0.824499\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	17.2649	0.820278	0.410139	−	0.912023i	$-0.365480\pi$
$443$	17.2649	0.820278	0.410139	+	0.912023i	$0.365480\pi$
$444$	0	0
$445$	−5.98156	−0.283553
$446$	0	0
$447$	0	0
$448$	0	0
$449$	− 30.0185i	− 1.41666i	−0.705882	−	0.708330i	$-0.749449\pi$
$449$	− 30.0185i	− 1.41666i	0.705882	−	0.708330i	$-0.250551\pi$
$450$	0	0
$451$	4.26254i	0.200715i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	7.65811	0.359018
$456$	0	0
$457$	32.4969	1.52014	0.760071	−	0.649840i	$-0.225164\pi$
$457$	32.4969	1.52014	0.760071	+	0.649840i	$0.225164\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	− 16.3855i	− 0.763150i	−0.924338	−	0.381575i	$-0.875382\pi$
$461$	− 16.3855i	− 0.763150i	0.924338	−	0.381575i	$-0.124618\pi$
$462$	0	0
$463$	− 37.4110i	− 1.73863i	−0.494254	−	0.869317i	$-0.664559\pi$
$463$	− 37.4110i	− 1.73863i	0.494254	−	0.869317i	$-0.335441\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−7.44716	−0.344613	−0.172307	−	0.985043i	$-0.555122\pi$
$467$	−7.44716	−0.344613	−0.172307	+	0.985043i	$0.555122\pi$
$468$	0	0
$469$	−10.0598	−0.464519
$470$	0	0
$471$	0	0
$472$	0	0
$473$	2.71940i	0.125038i
$474$	0	0
$475$	− 4.80980i	− 0.220689i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−32.6463	−1.49165	−0.745824	−	0.666144i	$-0.767944\pi$
$479$	−32.6463	−1.49165	−0.745824	+	0.666144i	$0.767944\pi$
$480$	0	0
$481$	25.8223	1.17740
$482$	0	0
$483$	0	0
$484$	0	0
$485$	4.30977i	0.195696i
$486$	0	0
$487$	− 11.1229i	− 0.504028i	−0.967724	−	0.252014i	$-0.918907\pi$
$487$	− 11.1229i	− 0.504028i	0.967724	−	0.252014i	$-0.0810929\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−6.47778	−0.292338	−0.146169	−	0.989260i	$-0.546694\pi$
$491$	−6.47778	−0.292338	−0.146169	+	0.989260i	$0.546694\pi$
$492$	0	0
$493$	15.7229	0.708125
$494$	0	0
$495$	0	0
$496$	0	0
$497$	3.84679i	0.172552i
$498$	0	0
$499$	10.3254i	0.462228i	0.972927	+	0.231114i	$0.0742371\pi$
$499$	10.3254i	0.462228i	−0.972927	+	0.231114i	$0.925763\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	24.6729	1.10011	0.550055	−	0.835129i	$-0.314607\pi$
$503$	24.6729	1.10011	0.550055	+	0.835129i	$0.314607\pi$
$504$	0	0
$505$	4.31285	0.191919
$506$	0	0
$507$	0	0
$508$	0	0
$509$	19.8733i	0.880869i	0.897785	+	0.440434i	$0.145176\pi$
$509$	19.8733i	0.880869i	−0.897785	+	0.440434i	$0.854824\pi$
$510$	0	0
$511$	20.5299i	0.908189i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	5.25631	0.231621
$516$	0	0
$517$	−2.24051	−0.0985373
$518$	0	0
$519$	0	0
$520$	0	0
$521$	− 42.6658i	− 1.86922i	−0.355669	−	0.934612i	$-0.615747\pi$
$521$	− 42.6658i	− 1.86922i	0.355669	−	0.934612i	$-0.384253\pi$
$522$	0	0
$523$	1.76037i	0.0769758i	0.999259	+	0.0384879i	$0.0122541\pi$
$523$	1.76037i	0.0769758i	−0.999259	+	0.0384879i	$0.987746\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	11.1178	0.484299
$528$	0	0
$529$	−13.2067	−0.574203
$530$	0	0
$531$	0	0
$532$	0	0
$533$	49.6666i	2.15130i
$534$	0	0
$535$	5.34690i	0.231166i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	2.15488	0.0928174
$540$	0	0
$541$	−16.8109	−0.722756	−0.361378	−	0.932419i	$-0.617694\pi$
$541$	−16.8109	−0.722756	−0.361378	+	0.932419i	$0.617694\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0.946545i	0.0405455i
$546$	0	0
$547$	3.10727i	0.132857i	0.997791	+	0.0664287i	$0.0211605\pi$
$547$	3.10727i	0.132857i	−0.997791	+	0.0664287i	$0.978840\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	4.35800	0.185657
$552$	0	0
$553$	45.7682	1.94626
$554$	0	0
$555$	0	0
$556$	0	0
$557$	4.71980i	0.199984i	0.994988	+	0.0999922i	$0.0318818\pi$
$557$	4.71980i	0.199984i	−0.994988	+	0.0999922i	$0.968118\pi$
$558$	0	0
$559$	31.6861i	1.34018i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	2.61353	0.110147	0.0550736	−	0.998482i	$-0.482461\pi$
$563$	2.61353	0.110147	0.0550736	+	0.998482i	$0.482461\pi$
$564$	0	0
$565$	1.49607	0.0629401
$566$	0	0
$567$	0	0
$568$	0	0
$569$	26.5675i	1.11377i	0.830590	+	0.556884i	$0.188003\pi$
$569$	26.5675i	1.11377i	−0.830590	+	0.556884i	$0.811997\pi$
$570$	0	0
$571$	− 34.6570i	− 1.45035i	−0.688564	−	0.725176i	$-0.741758\pi$
$571$	− 34.6570i	− 1.45035i	0.688564	−	0.725176i	$-0.258242\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	15.0519	0.627709
$576$	0	0
$577$	−36.3388	−1.51280	−0.756402	−	0.654107i	$-0.773045\pi$
$577$	−36.3388	−1.51280	−0.756402	+	0.654107i	$0.773045\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	20.5879i	0.854130i
$582$	0	0
$583$	2.64863i	0.109695i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−27.8276	−1.14857	−0.574284	−	0.818656i	$-0.694720\pi$
$587$	−27.8276	−1.14857	−0.574284	+	0.818656i	$0.694720\pi$
$588$	0	0
$589$	3.08157	0.126974
$590$	0	0
$591$	0	0
$592$	0	0
$593$	6.53062i	0.268180i	0.990969	+	0.134090i	$0.0428112\pi$
$593$	6.53062i	0.268180i	−0.990969	+	0.134090i	$0.957189\pi$
$594$	0	0
$595$	5.43714i	0.222901i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	40.4599	1.65315	0.826574	−	0.562828i	$-0.190287\pi$
$599$	40.4599	1.65315	0.826574	+	0.562828i	$0.190287\pi$
$600$	0	0
$601$	−15.0761	−0.614966	−0.307483	−	0.951554i	$-0.599487\pi$
$601$	−15.0761	−0.614966	−0.307483	+	0.951554i	$0.599487\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	4.71433i	0.191665i
$606$	0	0
$607$	− 34.8441i	− 1.41428i	−0.707074	−	0.707139i	$-0.749985\pi$
$607$	− 34.8441i	− 1.41428i	0.707074	−	0.707139i	$-0.250015\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−26.1061	−1.05614
$612$	0	0
$613$	26.6198	1.07516	0.537582	−	0.843212i	$-0.319338\pi$
$613$	26.6198	1.07516	0.537582	+	0.843212i	$0.319338\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 12.5857i	− 0.506682i	−0.967377	−	0.253341i	$-0.918471\pi$
$617$	− 12.5857i	− 0.506682i	0.967377	−	0.253341i	$-0.0815294\pi$
$618$	0	0
$619$	− 31.1196i	− 1.25080i	−0.780303	−	0.625401i	$-0.784935\pi$
$619$	− 31.1196i	− 1.25080i	0.780303	−	0.625401i	$-0.215065\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−47.3951	−1.89885
$624$	0	0
$625$	22.1832	0.887329
$626$	0	0
$627$	0	0
$628$	0	0
$629$	18.3335i	0.731003i
$630$	0	0
$631$	− 16.6477i	− 0.662736i	−0.943502	−	0.331368i	$-0.892490\pi$
$631$	− 16.6477i	− 0.662736i	0.943502	−	0.331368i	$-0.107510\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	4.47250	0.177486
$636$	0	0
$637$	25.1084	0.994832
$638$	0	0
$639$	0	0
$640$	0	0
$641$	− 20.3086i	− 0.802143i	−0.916047	−	0.401072i	$-0.868638\pi$
$641$	− 20.3086i	− 0.802143i	0.916047	−	0.401072i	$-0.131362\pi$
$642$	0	0
$643$	24.8696i	0.980761i	0.871508	+	0.490381i	$0.163142\pi$
$643$	24.8696i	0.980761i	−0.871508	+	0.490381i	$0.836858\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	38.0835	1.49722	0.748608	−	0.663013i	$-0.230723\pi$
$647$	38.0835	1.49722	0.748608	+	0.663013i	$0.230723\pi$
$648$	0	0
$649$	2.28099	0.0895366
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 18.7593i	− 0.734109i	−0.930199	−	0.367054i	$-0.880366\pi$
$653$	− 18.7593i	− 0.734109i	0.930199	−	0.367054i	$-0.119634\pi$
$654$	0	0
$655$	− 0.153855i	− 0.00601160i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−31.6491	−1.23287	−0.616437	−	0.787404i	$-0.711425\pi$
$659$	−31.6491	−1.23287	−0.616437	+	0.787404i	$0.711425\pi$
$660$	0	0
$661$	−23.0090	−0.894947	−0.447473	−	0.894297i	$-0.647676\pi$
$661$	−23.0090	−0.894947	−0.447473	+	0.894297i	$0.647676\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	1.50704i	0.0584404i
$666$	0	0
$667$	13.6380i	0.528067i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	0.840681	0.0324541
$672$	0	0
$673$	40.4112	1.55774	0.778868	−	0.627188i	$-0.215794\pi$
$673$	40.4112	1.55774	0.778868	+	0.627188i	$0.215794\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 30.3235i	− 1.16543i	−0.812678	−	0.582713i	$-0.801991\pi$
$677$	− 30.3235i	− 1.16543i	0.812678	−	0.582713i	$-0.198009\pi$
$678$	0	0
$679$	34.1486i	1.31050i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	29.1431	1.11513	0.557564	−	0.830134i	$-0.311736\pi$
$683$	29.1431	1.11513	0.557564	+	0.830134i	$0.311736\pi$
$684$	0	0
$685$	−0.376255	−0.0143760
$686$	0	0
$687$	0	0
$688$	0	0
$689$	30.8615i	1.17573i
$690$	0	0
$691$	− 12.1488i	− 0.462163i	−0.972934	−	0.231082i	$-0.925774\pi$
$691$	− 12.1488i	− 0.462163i	0.972934	−	0.231082i	$-0.0742265\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	7.70185	0.292148
$696$	0	0
$697$	−35.2625	−1.33566
$698$	0	0
$699$	0	0
$700$	0	0
$701$	18.8565i	0.712199i	0.934448	+	0.356099i	$0.115894\pi$
$701$	18.8565i	0.712199i	−0.934448	+	0.356099i	$0.884106\pi$
$702$	0	0
$703$	5.08157i	0.191655i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	34.1731	1.28521
$708$	0	0
$709$	5.88039	0.220843	0.110421	−	0.993885i	$-0.464780\pi$
$709$	5.88039	0.220843	0.110421	+	0.993885i	$0.464780\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	9.64356i	0.361154i
$714$	0	0
$715$	− 0.966501i	− 0.0361451i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−25.9370	−0.967288	−0.483644	−	0.875265i	$-0.660687\pi$
$719$	−25.9370	−0.967288	−0.483644	+	0.875265i	$0.660687\pi$
$720$	0	0
$721$	41.6486	1.55108
$722$	0	0
$723$	0	0
$724$	0	0
$725$	20.9611i	0.778476i
$726$	0	0
$727$	− 2.45843i	− 0.0911781i	−0.998960	−	0.0455891i	$-0.985484\pi$
$727$	− 2.45843i	− 0.0911781i	0.998960	−	0.0455891i	$-0.0145165\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−22.4966	−0.832067
$732$	0	0
$733$	32.4525	1.19866	0.599331	−	0.800502i	$-0.295433\pi$
$733$	32.4525	1.19866	0.599331	+	0.800502i	$0.295433\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	1.26961i	0.0467667i
$738$	0	0
$739$	10.6311i	0.391070i	0.980697	+	0.195535i	$0.0626443\pi$
$739$	10.6311i	0.391070i	−0.980697	+	0.195535i	$0.937356\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−17.8223	−0.653835	−0.326918	−	0.945053i	$-0.606010\pi$
$743$	−17.8223	−0.653835	−0.326918	+	0.945053i	$0.606010\pi$
$744$	0	0
$745$	−4.88913	−0.179124
$746$	0	0
$747$	0	0
$748$	0	0
$749$	42.3664i	1.54803i
$750$	0	0
$751$	− 16.1972i	− 0.591046i	−0.955336	−	0.295523i	$-0.904506\pi$
$751$	− 16.1972i	− 0.591046i	0.955336	−	0.295523i	$-0.0954939\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0.920463	0.0334991
$756$	0	0
$757$	36.1194	1.31278	0.656391	−	0.754421i	$-0.272082\pi$
$757$	36.1194	1.31278	0.656391	+	0.754421i	$0.272082\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	− 6.83613i	− 0.247810i	−0.992294	−	0.123905i	$-0.960458\pi$
$761$	− 6.83613i	− 0.247810i	0.992294	−	0.123905i	$-0.0395417\pi$
$762$	0	0
$763$	7.49999i	0.271518i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	26.5778	0.959668
$768$	0	0
$769$	20.1031	0.724937	0.362468	−	0.931996i	$-0.381934\pi$
$769$	20.1031	0.724937	0.362468	+	0.931996i	$0.381934\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	− 1.79730i	− 0.0646445i	−0.999477	−	0.0323223i	$-0.989710\pi$
$773$	− 1.79730i	− 0.0646445i	0.999477	−	0.0323223i	$-0.0102903\pi$
$774$	0	0
$775$	14.8217i	0.532413i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−9.77387	−0.350185
$780$	0	0
$781$	0.485489	0.0173722
$782$	0	0
$783$	0	0
$784$	0	0
$785$	4.00885i	0.143082i
$786$	0	0
$787$	− 32.1319i	− 1.14538i	−0.819772	−	0.572690i	$-0.805900\pi$
$787$	− 32.1319i	− 1.14538i	0.819772	−	0.572690i	$-0.194100\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	11.8542	0.421485
$792$	0	0
$793$	9.79550	0.347848
$794$	0	0
$795$	0	0
$796$	0	0
$797$	1.17171i	0.0415040i	0.999785	+	0.0207520i	$0.00660603\pi$
$797$	1.17171i	0.0415040i	−0.999785	+	0.0207520i	$0.993394\pi$
$798$	0	0
$799$	− 18.5349i	− 0.655718i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	2.59100	0.0914344
$804$	0	0
$805$	−4.71617	−0.166223
$806$	0	0
$807$	0	0
$808$	0	0
$809$	47.6176i	1.67415i	0.547092	+	0.837073i	$0.315735\pi$
$809$	47.6176i	1.67415i	−0.547092	+	0.837073i	$0.684265\pi$
$810$	0	0
$811$	4.20666i	0.147716i	0.997269	+	0.0738580i	$0.0235312\pi$
$811$	4.20666i	0.147716i	−0.997269	+	0.0738580i	$0.976469\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−4.85577	−0.170090
$816$	0	0
$817$	−6.23549	−0.218152
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 9.89470i	− 0.345327i	−0.984981	−	0.172664i	$-0.944763\pi$
$821$	− 9.89470i	− 0.345327i	0.984981	−	0.172664i	$-0.0552374\pi$
$822$	0	0
$823$	− 21.0567i	− 0.733992i	−0.930222	−	0.366996i	$-0.880386\pi$
$823$	− 21.0567i	− 0.733992i	0.930222	−	0.366996i	$-0.119614\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−13.5006	−0.469462	−0.234731	−	0.972060i	$-0.575421\pi$
$827$	−13.5006	−0.469462	−0.234731	+	0.972060i	$0.575421\pi$
$828$	0	0
$829$	−17.5275	−0.608754	−0.304377	−	0.952552i	$-0.598448\pi$
$829$	−17.5275	−0.608754	−0.304377	+	0.952552i	$0.598448\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	17.8266i	0.617655i
$834$	0	0
$835$	4.96650i	0.171873i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	54.9687	1.89773	0.948864	−	0.315684i	$-0.102234\pi$
$839$	54.9687	1.89773	0.948864	+	0.315684i	$0.102234\pi$
$840$	0	0
$841$	10.0079	0.345099
$842$	0	0
$843$	0	0
$844$	0	0
$845$	− 5.59203i	− 0.192372i
$846$	0	0
$847$	37.3542i	1.28351i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−15.9024	−0.545128
$852$	0	0
$853$	27.7063	0.948644	0.474322	−	0.880351i	$-0.342693\pi$
$853$	27.7063	0.948644	0.474322	+	0.880351i	$0.342693\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 24.8041i	− 0.847290i	−0.905828	−	0.423645i	$-0.860750\pi$
$857$	− 24.8041i	− 0.847290i	0.905828	−	0.423645i	$-0.139250\pi$
$858$	0	0
$859$	7.94108i	0.270946i	0.990781	+	0.135473i	$0.0432554\pi$
$859$	7.94108i	0.270946i	−0.990781	+	0.135473i	$0.956745\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	7.57354	0.257806	0.128903	−	0.991657i	$-0.458854\pi$
$863$	7.57354	0.257806	0.128903	+	0.991657i	$0.458854\pi$
$864$	0	0
$865$	7.10999	0.241747
$866$	0	0
$867$	0	0
$868$	0	0
$869$	− 5.77623i	− 0.195945i
$870$	0	0
$871$	14.7933i	0.501253i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−14.7837	−0.499782
$876$	0	0
$877$	−36.9906	−1.24908	−0.624541	−	0.780992i	$-0.714714\pi$
$877$	−36.9906	−1.24908	−0.624541	+	0.780992i	$0.714714\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	32.2046i	1.08500i	0.840056	+	0.542500i	$0.182522\pi$
$881$	32.2046i	1.08500i	−0.840056	+	0.542500i	$0.817478\pi$
$882$	0	0
$883$	− 53.2383i	− 1.79161i	−0.444445	−	0.895806i	$-0.646599\pi$
$883$	− 53.2383i	− 1.79161i	0.444445	−	0.895806i	$-0.353401\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	11.0798	0.372024	0.186012	−	0.982547i	$-0.440444\pi$
$887$	11.0798	0.372024	0.186012	+	0.982547i	$0.440444\pi$
$888$	0	0
$889$	35.4380	1.18855
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 5.13741i	− 0.171917i
$894$	0	0
$895$	11.0035i	0.367807i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−13.4295	−0.447898
$900$	0	0
$901$	−21.9112	−0.729967
$902$	0	0
$903$	0	0
$904$	0	0
$905$	− 1.85580i	− 0.0616889i
$906$	0	0
$907$	− 21.6218i	− 0.717939i	−0.933349	−	0.358970i	$-0.883128\pi$
$907$	− 21.6218i	− 0.717939i	0.933349	−	0.358970i	$-0.116872\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−16.0004	−0.530118	−0.265059	−	0.964232i	$-0.585391\pi$
$911$	−16.0004	−0.530118	−0.265059	+	0.964232i	$0.585391\pi$
$912$	0	0
$913$	2.59832	0.0859918
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 1.21907i	− 0.0402574i
$918$	0	0
$919$	41.7933i	1.37863i	0.724460	+	0.689317i	$0.242089\pi$
$919$	41.7933i	1.37863i	−0.724460	+	0.689317i	$0.757911\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	5.65685	0.186198
$924$	0	0
$925$	−24.4413	−0.803626
$926$	0	0
$927$	0	0
$928$	0	0
$929$	− 43.9401i	− 1.44163i	−0.693128	−	0.720815i	$-0.743768\pi$
$929$	− 43.9401i	− 1.44163i	0.693128	−	0.720815i	$-0.256232\pi$
$930$	0	0
$931$	4.94108i	0.161937i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0.686201	0.0224412
$936$	0	0
$937$	−42.8830	−1.40093	−0.700464	−	0.713688i	$-0.747023\pi$
$937$	−42.8830	−1.40093	−0.700464	+	0.713688i	$0.747023\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	19.4545i	0.634197i	0.948393	+	0.317098i	$0.102709\pi$
$941$	19.4545i	0.634197i	−0.948393	+	0.317098i	$0.897291\pi$
$942$	0	0
$943$	− 30.5867i	− 0.996039i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	29.6867	0.964687	0.482344	−	0.875982i	$-0.339786\pi$
$947$	29.6867	0.964687	0.482344	+	0.875982i	$0.339786\pi$
$948$	0	0
$949$	30.1900	0.980008
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 9.21444i	− 0.298485i	−0.988801	−	0.149242i	$-0.952316\pi$
$953$	− 9.21444i	− 0.298485i	0.988801	−	0.149242i	$-0.0476835\pi$
$954$	0	0
$955$	5.38738i	0.174331i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−2.98128	−0.0962704
$960$	0	0
$961$	21.5039	0.693675
$962$	0	0
$963$	0	0
$964$	0	0
$965$	− 9.05027i	− 0.291338i
$966$	0	0
$967$	20.1051i	0.646536i	0.946307	+	0.323268i	$0.104782\pi$
$967$	20.1051i	0.646536i	−0.946307	+	0.323268i	$0.895218\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	26.5398	0.851703	0.425851	−	0.904793i	$-0.359975\pi$
$971$	26.5398	0.851703	0.425851	+	0.904793i	$0.359975\pi$
$972$	0	0
$973$	61.0260	1.95640
$974$	0	0
$975$	0	0
$976$	0	0
$977$	20.4666i	0.654784i	0.944889	+	0.327392i	$0.106170\pi$
$977$	20.4666i	0.654784i	−0.944889	+	0.327392i	$0.893830\pi$
$978$	0	0
$979$	5.98156i	0.191171i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	16.8039	0.535961	0.267981	−	0.963424i	$-0.413644\pi$
$983$	16.8039	0.535961	0.267981	+	0.963424i	$0.413644\pi$
$984$	0	0
$985$	−7.41456	−0.236248
$986$	0	0
$987$	0	0
$988$	0	0
$989$	− 19.5135i	− 0.620494i
$990$	0	0
$991$	− 46.6498i	− 1.48188i	−0.671572	−	0.740939i	$-0.734381\pi$
$991$	− 46.6498i	− 1.48188i	0.671572	−	0.740939i	$-0.265619\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	6.01464	0.190677
$996$	0	0
$997$	−40.6814	−1.28839	−0.644196	−	0.764860i	$-0.722808\pi$
$997$	−40.6814	−1.28839	−0.644196	+	0.764860i	$0.722808\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5472.2.d.g.2015.5	✓	16
3.2	odd	2	inner	5472.2.d.g.2015.11	yes	16
4.3	odd	2	inner	5472.2.d.g.2015.6	yes	16
12.11	even	2	inner	5472.2.d.g.2015.12	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5472.2.d.g.2015.5	✓	16	1.1	even	1	trivial
5472.2.d.g.2015.6	yes	16	4.3	odd	2	inner
5472.2.d.g.2015.11	yes	16	3.2	odd	2	inner
5472.2.d.g.2015.12	yes	16	12.11	even	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 5472 = 2^{5} \cdot 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5472.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-0.436116i q^{5} -3.45559i q^{7} -0.436116 q^{11} -5.08157 q^{13} -3.60784i q^{17} -1.00000i q^{19} +3.12943 q^{23} +4.80980 q^{25} +4.35800i q^{29} +3.08157i q^{31} -1.50704 q^{35} -5.08157 q^{37} -9.77387i q^{41} -6.23549i q^{43} +5.13741 q^{47} -4.94108 q^{49} -6.07321i q^{53} +0.190197i q^{55} -5.23023 q^{59} -1.92765 q^{61} +2.21615i q^{65} -2.91117i q^{67} -1.11321 q^{71} -5.94108 q^{73} +1.50704i q^{77} +13.2447i q^{79} -5.95786 q^{83} -1.57344 q^{85} -13.7155i q^{89} +17.5598i q^{91} -0.436116 q^{95} -9.88215 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 16 q^{13} - 48 q^{25} - 16 q^{37} + 8 q^{49} - 80 q^{61} - 8 q^{73} - 152 q^{85} + 16 q^{97}+O(q^{100}) \) 16 * q - 16 * q^13 - 48 * q^25 - 16 * q^37 + 8 * q^49 - 80 * q^61 - 8 * q^73 - 152 * q^85 + 16 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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