LMFDB - Embedded newform 5472.2.d.e.2015.3

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 = [8,0,0,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,0,0] 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:	$8$
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			2015.3
Root			$0.965926 + 0.258819i$ of defining polynomial
Character	$\chi$	$=$	5472.2015
Dual form			5472.2.d.e.2015.6

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-0.896575i q^{5} -4.46410i q^{7} -2.96713 q^{11} +5.46410 q^{13} +7.20977i q^{17} -1.00000i q^{19} +0.378937 q^{23} +4.19615 q^{25} +8.48528i q^{29} +7.46410i q^{31} -4.00240 q^{35} +9.46410 q^{37} -2.82843i q^{41} +2.26795i q^{43} +8.24504 q^{47} -12.9282 q^{49} +7.45001i q^{53} +2.66025i q^{55} -3.10583 q^{59} -7.19615 q^{61} -4.89898i q^{65} +4.92820i q^{67} +9.52056 q^{71} +7.92820 q^{73} +13.2456i q^{77} +11.4641i q^{79} +8.10634 q^{83} +6.46410 q^{85} +5.93426i q^{89} -24.3923i q^{91} -0.896575 q^{95} +10.9282 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 16 q^{13} - 8 q^{25} + 48 q^{37} - 48 q^{49} - 16 q^{61} + 8 q^{73} + 24 q^{85} + 32 q^{97}+O(q^{100}) $ 8 * q + 16 * q^13 - 8 * q^25 + 48 * q^37 - 48 * q^49 - 16 * q^61 + 8 * q^73 + 24 * q^85 + 32 * 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
$5$	− 0.896575i	− 0.400961i	−0.979698	−	0.200480i	$-0.935750\pi$
$5$	− 0.896575i	− 0.400961i	0.979698	−	0.200480i	$-0.0642503\pi$
$6$	0	0
$7$	− 4.46410i	− 1.68727i	−0.536916	−	0.843636i	$-0.680411\pi$
$7$	− 4.46410i	− 1.68727i	0.536916	−	0.843636i	$-0.319589\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−2.96713	−0.894623	−0.447311	−	0.894378i	$-0.647618\pi$
$11$	−2.96713	−0.894623	−0.447311	+	0.894378i	$0.647618\pi$
$12$	0	0
$13$	5.46410	1.51547	0.757735	−	0.652563i	$-0.226306\pi$
$13$	5.46410	1.51547	0.757735	+	0.652563i	$0.226306\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	7.20977i	1.74863i	0.485363	+	0.874313i	$0.338688\pi$
$17$	7.20977i	1.74863i	−0.485363	+	0.874313i	$0.661312\pi$
$18$	0	0
$19$	− 1.00000i	− 0.229416i
$19$	− 1.00000i	− 0.229416i
$20$	0	0
$21$	0	0
$22$	0	0
$23$	0.378937	0.0790139	0.0395070	−	0.999219i	$-0.487421\pi$
$23$	0.378937	0.0790139	0.0395070	+	0.999219i	$0.487421\pi$
$24$	0	0
$25$	4.19615	0.839230
$26$	0	0
$27$	0	0
$28$	0	0
$29$	8.48528i	1.57568i	0.615882	+	0.787839i	$0.288800\pi$
$29$	8.48528i	1.57568i	−0.615882	+	0.787839i	$0.711200\pi$
$30$	0	0
$31$	7.46410i	1.34059i	0.742094	+	0.670296i	$0.233833\pi$
$31$	7.46410i	1.34059i	−0.742094	+	0.670296i	$0.766167\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−4.00240	−0.676530
$36$	0	0
$37$	9.46410	1.55589	0.777944	−	0.628333i	$-0.216263\pi$
$37$	9.46410	1.55589	0.777944	+	0.628333i	$0.216263\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 2.82843i	− 0.441726i	−0.975305	−	0.220863i	$-0.929113\pi$
$41$	− 2.82843i	− 0.441726i	0.975305	−	0.220863i	$-0.0708874\pi$
$42$	0	0
$43$	2.26795i	0.345859i	0.984934	+	0.172930i	$0.0553233\pi$
$43$	2.26795i	0.345859i	−0.984934	+	0.172930i	$0.944677\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	8.24504	1.20266	0.601332	−	0.799000i	$-0.294637\pi$
$47$	8.24504	1.20266	0.601332	+	0.799000i	$0.294637\pi$
$48$	0	0
$49$	−12.9282	−1.84689
$50$	0	0
$51$	0	0
$52$	0	0
$53$	7.45001i	1.02334i	0.859183	+	0.511668i	$0.170972\pi$
$53$	7.45001i	1.02334i	−0.859183	+	0.511668i	$0.829028\pi$
$54$	0	0
$55$	2.66025i	0.358709i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−3.10583	−0.404344	−0.202172	−	0.979350i	$-0.564800\pi$
$59$	−3.10583	−0.404344	−0.202172	+	0.979350i	$0.564800\pi$
$60$	0	0
$61$	−7.19615	−0.921373	−0.460686	−	0.887563i	$-0.652397\pi$
$61$	−7.19615	−0.921373	−0.460686	+	0.887563i	$0.652397\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	− 4.89898i	− 0.607644i
$66$	0	0
$67$	4.92820i	0.602076i	0.953612	+	0.301038i	$0.0973331\pi$
$67$	4.92820i	0.602076i	−0.953612	+	0.301038i	$0.902667\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	9.52056	1.12988	0.564941	−	0.825131i	$-0.308899\pi$
$71$	9.52056	1.12988	0.564941	+	0.825131i	$0.308899\pi$
$72$	0	0
$73$	7.92820	0.927926	0.463963	−	0.885855i	$-0.346427\pi$
$73$	7.92820	0.927926	0.463963	+	0.885855i	$0.346427\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	13.2456i	1.50947i
$78$	0	0
$79$	11.4641i	1.28981i	0.764262	+	0.644906i	$0.223104\pi$
$79$	11.4641i	1.28981i	−0.764262	+	0.644906i	$0.776896\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	8.10634	0.889787	0.444893	−	0.895584i	$-0.353242\pi$
$83$	8.10634	0.889787	0.444893	+	0.895584i	$0.353242\pi$
$84$	0	0
$85$	6.46410	0.701130
$86$	0	0
$87$	0	0
$88$	0	0
$89$	5.93426i	0.629030i	0.949253	+	0.314515i	$0.101842\pi$
$89$	5.93426i	0.629030i	−0.949253	+	0.314515i	$0.898158\pi$
$90$	0	0
$91$	− 24.3923i	− 2.55701i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−0.896575	−0.0919867
$96$	0	0
$97$	10.9282	1.10959	0.554795	−	0.831987i	$-0.312797\pi$
$97$	10.9282	1.10959	0.554795	+	0.831987i	$0.312797\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	− 16.1112i	− 1.60312i	−0.597915	−	0.801560i	$-0.704004\pi$
$101$	− 16.1112i	− 1.60312i	0.597915	−	0.801560i	$-0.295996\pi$
$102$	0	0
$103$	− 18.3923i	− 1.81225i	−0.423013	−	0.906124i	$-0.639027\pi$
$103$	− 18.3923i	− 1.81225i	0.423013	−	0.906124i	$-0.360973\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−6.69213	−0.646953	−0.323476	−	0.946236i	$-0.604852\pi$
$107$	−6.69213	−0.646953	−0.323476	+	0.946236i	$0.604852\pi$
$108$	0	0
$109$	−6.39230	−0.612272	−0.306136	−	0.951988i	$-0.599036\pi$
$109$	−6.39230	−0.612272	−0.306136	+	0.951988i	$0.599036\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 1.51575i	− 0.142590i	−0.997455	−	0.0712949i	$-0.977287\pi$
$113$	− 1.51575i	− 0.142590i	0.997455	−	0.0712949i	$-0.0227131\pi$
$114$	0	0
$115$	− 0.339746i	− 0.0316815i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	32.1851	2.95041
$120$	0	0
$121$	−2.19615	−0.199650
$122$	0	0
$123$	0	0
$124$	0	0
$125$	− 8.24504i	− 0.737459i
$126$	0	0
$127$	− 10.9282i	− 0.969721i	−0.874591	−	0.484861i	$-0.838870\pi$
$127$	− 10.9282i	− 0.969721i	0.874591	−	0.484861i	$-0.161130\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−6.93237	−0.605684	−0.302842	−	0.953041i	$-0.597935\pi$
$131$	−6.93237	−0.605684	−0.302842	+	0.953041i	$0.597935\pi$
$132$	0	0
$133$	−4.46410	−0.387087
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 13.8004i	− 1.17904i	−0.807752	−	0.589522i	$-0.799316\pi$
$137$	− 13.8004i	− 1.17904i	0.807752	−	0.589522i	$-0.200684\pi$
$138$	0	0
$139$	4.07180i	0.345365i	0.984978	+	0.172683i	$0.0552435\pi$
$139$	4.07180i	0.345365i	−0.984978	+	0.172683i	$0.944756\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−16.2127	−1.35577
$144$	0	0
$145$	7.60770	0.631785
$146$	0	0
$147$	0	0
$148$	0	0
$149$	6.93237i	0.567922i	0.958836	+	0.283961i	$0.0916486\pi$
$149$	6.93237i	0.567922i	−0.958836	+	0.283961i	$0.908351\pi$
$150$	0	0
$151$	18.3923i	1.49674i	0.663279	+	0.748372i	$0.269164\pi$
$151$	18.3923i	1.49674i	−0.663279	+	0.748372i	$0.730836\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	6.69213	0.537525
$156$	0	0
$157$	4.00000	0.319235	0.159617	−	0.987179i	$-0.448974\pi$
$157$	4.00000	0.319235	0.159617	+	0.987179i	$0.448974\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	− 1.69161i	− 0.133318i
$162$	0	0
$163$	− 12.0000i	− 0.939913i	−0.882690	−	0.469956i	$-0.844270\pi$
$163$	− 12.0000i	− 0.939913i	0.882690	−	0.469956i	$-0.155730\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	2.82843	0.218870	0.109435	−	0.993994i	$-0.465096\pi$
$167$	2.82843	0.218870	0.109435	+	0.993994i	$0.465096\pi$
$168$	0	0
$169$	16.8564	1.29665
$170$	0	0
$171$	0	0
$172$	0	0
$173$	18.5606i	1.41114i	0.708640	+	0.705570i	$0.249309\pi$
$173$	18.5606i	1.41114i	−0.708640	+	0.705570i	$0.750691\pi$
$174$	0	0
$175$	− 18.7321i	− 1.41601i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	11.3137	0.845626	0.422813	−	0.906217i	$-0.361043\pi$
$179$	11.3137	0.845626	0.422813	+	0.906217i	$0.361043\pi$
$180$	0	0
$181$	−4.92820	−0.366310	−0.183155	−	0.983084i	$-0.558631\pi$
$181$	−4.92820	−0.366310	−0.183155	+	0.983084i	$0.558631\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	− 8.48528i	− 0.623850i
$186$	0	0
$187$	− 21.3923i	− 1.56436i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−20.5940	−1.49013	−0.745066	−	0.666991i	$-0.767582\pi$
$191$	−20.5940	−1.49013	−0.745066	+	0.666991i	$0.767582\pi$
$192$	0	0
$193$	8.53590	0.614427	0.307214	−	0.951641i	$-0.400603\pi$
$193$	8.53590	0.614427	0.307214	+	0.951641i	$0.400603\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 10.4543i	− 0.744838i	−0.928065	−	0.372419i	$-0.878528\pi$
$197$	− 10.4543i	− 0.744838i	0.928065	−	0.372419i	$-0.121472\pi$
$198$	0	0
$199$	− 20.2679i	− 1.43676i	−0.695653	−	0.718378i	$-0.744885\pi$
$199$	− 20.2679i	− 1.43676i	0.695653	−	0.718378i	$-0.255115\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	37.8792	2.65860
$204$	0	0
$205$	−2.53590	−0.177115
$206$	0	0
$207$	0	0
$208$	0	0
$209$	2.96713i	0.205241i
$210$	0	0
$211$	− 2.00000i	− 0.137686i	−0.997628	−	0.0688428i	$-0.978069\pi$
$211$	− 2.00000i	− 0.137686i	0.997628	−	0.0688428i	$-0.0219307\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	2.03339	0.138676
$216$	0	0
$217$	33.3205	2.26194
$218$	0	0
$219$	0	0
$220$	0	0
$221$	39.3949i	2.64999i
$222$	0	0
$223$	− 22.9282i	− 1.53539i	−0.640818	−	0.767693i	$-0.721405\pi$
$223$	− 22.9282i	− 1.53539i	0.640818	−	0.767693i	$-0.278595\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	22.6274	1.50183	0.750917	−	0.660396i	$-0.229612\pi$
$227$	22.6274	1.50183	0.750917	+	0.660396i	$0.229612\pi$
$228$	0	0
$229$	−27.9808	−1.84902	−0.924510	−	0.381157i	$-0.875526\pi$
$229$	−27.9808	−1.84902	−0.924510	+	0.381157i	$0.875526\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 19.5588i	− 1.28134i	−0.767818	−	0.640668i	$-0.778657\pi$
$233$	− 19.5588i	− 1.28134i	0.767818	−	0.640668i	$-0.221343\pi$
$234$	0	0
$235$	− 7.39230i	− 0.482221i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−3.62347	−0.234383	−0.117191	−	0.993109i	$-0.537389\pi$
$239$	−3.62347	−0.234383	−0.117191	+	0.993109i	$0.537389\pi$
$240$	0	0
$241$	12.0000	0.772988	0.386494	−	0.922292i	$-0.373686\pi$
$241$	12.0000	0.772988	0.386494	+	0.922292i	$0.373686\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	11.5911i	0.740529i
$246$	0	0
$247$	− 5.46410i	− 0.347672i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−19.9377	−1.25846	−0.629228	−	0.777221i	$-0.716629\pi$
$251$	−19.9377	−1.25846	−0.629228	+	0.777221i	$0.716629\pi$
$252$	0	0
$253$	−1.12436	−0.0706876
$254$	0	0
$255$	0	0
$256$	0	0
$257$	4.14110i	0.258315i	0.991624	+	0.129158i	$0.0412273\pi$
$257$	4.14110i	0.258315i	−0.991624	+	0.129158i	$0.958773\pi$
$258$	0	0
$259$	− 42.2487i	− 2.62521i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	25.1141	1.54860	0.774300	−	0.632819i	$-0.218102\pi$
$263$	25.1141	1.54860	0.774300	+	0.632819i	$0.218102\pi$
$264$	0	0
$265$	6.67949	0.410318
$266$	0	0
$267$	0	0
$268$	0	0
$269$	23.4596i	1.43036i	0.698941	+	0.715179i	$0.253655\pi$
$269$	23.4596i	1.43036i	−0.698941	+	0.715179i	$0.746345\pi$
$270$	0	0
$271$	− 0.535898i	− 0.0325535i	−0.999868	−	0.0162768i	$-0.994819\pi$
$271$	− 0.535898i	− 0.0325535i	0.999868	−	0.0162768i	$-0.00518128\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−12.4505	−0.750795
$276$	0	0
$277$	10.1244	0.608314	0.304157	−	0.952622i	$-0.401625\pi$
$277$	10.1244	0.608314	0.304157	+	0.952622i	$0.401625\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	− 1.79315i	− 0.106970i	−0.998569	−	0.0534852i	$-0.982967\pi$
$281$	− 1.79315i	− 0.106970i	0.998569	−	0.0534852i	$-0.0170330\pi$
$282$	0	0
$283$	15.5885i	0.926638i	0.886192	+	0.463319i	$0.153342\pi$
$283$	15.5885i	0.926638i	−0.886192	+	0.463319i	$0.846658\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−12.6264	−0.745312
$288$	0	0
$289$	−34.9808	−2.05769
$290$	0	0
$291$	0	0
$292$	0	0
$293$	29.5969i	1.72907i	0.502571	+	0.864536i	$0.332388\pi$
$293$	29.5969i	1.72907i	−0.502571	+	0.864536i	$0.667612\pi$
$294$	0	0
$295$	2.78461i	0.162126i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	2.07055	0.119743
$300$	0	0
$301$	10.1244	0.583558
$302$	0	0
$303$	0	0
$304$	0	0
$305$	6.45189i	0.369434i
$306$	0	0
$307$	22.7846i	1.30039i	0.759769	+	0.650193i	$0.225312\pi$
$307$	22.7846i	1.30039i	−0.759769	+	0.650193i	$0.774688\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−26.8057	−1.52001	−0.760006	−	0.649917i	$-0.774804\pi$
$311$	−26.8057	−1.52001	−0.760006	+	0.649917i	$0.774804\pi$
$312$	0	0
$313$	25.8564	1.46149	0.730745	−	0.682650i	$-0.239173\pi$
$313$	25.8564	1.46149	0.730745	+	0.682650i	$0.239173\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 1.79315i	− 0.100713i	−0.998731	−	0.0503567i	$-0.983964\pi$
$317$	− 1.79315i	− 0.100713i	0.998731	−	0.0503567i	$-0.0160358\pi$
$318$	0	0
$319$	− 25.1769i	− 1.40964i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	7.20977	0.401162
$324$	0	0
$325$	22.9282	1.27183
$326$	0	0
$327$	0	0
$328$	0	0
$329$	− 36.8067i	− 2.02922i
$330$	0	0
$331$	27.8564i	1.53113i	0.643361	+	0.765563i	$0.277540\pi$
$331$	27.8564i	1.53113i	−0.643361	+	0.765563i	$0.722460\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	4.41851	0.241409
$336$	0	0
$337$	−9.46410	−0.515542	−0.257771	−	0.966206i	$-0.582988\pi$
$337$	−9.46410	−0.515542	−0.257771	+	0.966206i	$0.582988\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	− 22.1469i	− 1.19932i
$342$	0	0
$343$	26.4641i	1.42893i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	10.5187	0.564672	0.282336	−	0.959316i	$-0.408891\pi$
$347$	10.5187	0.564672	0.282336	+	0.959316i	$0.408891\pi$
$348$	0	0
$349$	16.3205	0.873617	0.436808	−	0.899555i	$-0.356109\pi$
$349$	16.3205	0.873617	0.436808	+	0.899555i	$0.356109\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 13.2827i	− 0.706968i	−0.935441	−	0.353484i	$-0.884997\pi$
$353$	− 13.2827i	− 0.706968i	0.935441	−	0.353484i	$-0.115003\pi$
$354$	0	0
$355$	− 8.53590i	− 0.453038i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	7.69024	0.405875	0.202938	−	0.979192i	$-0.434951\pi$
$359$	7.69024	0.405875	0.202938	+	0.979192i	$0.434951\pi$
$360$	0	0
$361$	−1.00000	−0.0526316
$362$	0	0
$363$	0	0
$364$	0	0
$365$	− 7.10823i	− 0.372062i
$366$	0	0
$367$	24.2487i	1.26577i	0.774245	+	0.632886i	$0.218130\pi$
$367$	24.2487i	1.26577i	−0.774245	+	0.632886i	$0.781870\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	33.2576	1.72665
$372$	0	0
$373$	4.78461	0.247738	0.123869	−	0.992299i	$-0.460470\pi$
$373$	4.78461	0.247738	0.123869	+	0.992299i	$0.460470\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	46.3644i	2.38789i
$378$	0	0
$379$	− 13.6077i	− 0.698980i	−0.936940	−	0.349490i	$-0.886355\pi$
$379$	− 13.6077i	− 0.698980i	0.936940	−	0.349490i	$-0.113645\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−14.9743	−0.765153	−0.382577	−	0.923924i	$-0.624963\pi$
$383$	−14.9743	−0.765153	−0.382577	+	0.923924i	$0.624963\pi$
$384$	0	0
$385$	11.8756	0.605239
$386$	0	0
$387$	0	0
$388$	0	0
$389$	30.5951i	1.55123i	0.631206	+	0.775616i	$0.282560\pi$
$389$	30.5951i	1.55123i	−0.631206	+	0.775616i	$0.717440\pi$
$390$	0	0
$391$	2.73205i	0.138166i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	10.2784	0.517164
$396$	0	0
$397$	6.32051	0.317217	0.158609	−	0.987342i	$-0.449299\pi$
$397$	6.32051	0.317217	0.158609	+	0.987342i	$0.449299\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	20.6312i	1.03027i	0.857108	+	0.515136i	$0.172259\pi$
$401$	20.6312i	1.03027i	−0.857108	+	0.515136i	$0.827741\pi$
$402$	0	0
$403$	40.7846i	2.03163i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−28.0812	−1.39193
$408$	0	0
$409$	−33.3205	−1.64759	−0.823797	−	0.566886i	$-0.808148\pi$
$409$	−33.3205	−1.64759	−0.823797	+	0.566886i	$0.808148\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	13.8647i	0.682239i
$414$	0	0
$415$	− 7.26795i	− 0.356770i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−29.7728	−1.45450	−0.727249	−	0.686374i	$-0.759201\pi$
$419$	−29.7728	−1.45450	−0.727249	+	0.686374i	$0.759201\pi$
$420$	0	0
$421$	10.0000	0.487370	0.243685	−	0.969854i	$-0.421644\pi$
$421$	10.0000	0.487370	0.243685	+	0.969854i	$0.421644\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	30.2533i	1.46750i
$426$	0	0
$427$	32.1244i	1.55461i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	33.7381	1.62510	0.812552	−	0.582888i	$-0.198077\pi$
$431$	33.7381	1.62510	0.812552	+	0.582888i	$0.198077\pi$
$432$	0	0
$433$	20.3923	0.979992	0.489996	−	0.871725i	$-0.336998\pi$
$433$	20.3923	0.979992	0.489996	+	0.871725i	$0.336998\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 0.378937i	− 0.0181270i
$438$	0	0
$439$	− 9.85641i	− 0.470421i	−0.971945	−	0.235210i	$-0.924422\pi$
$439$	− 9.85641i	− 0.470421i	0.971945	−	0.235210i	$-0.0755779\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	3.82654	0.181804	0.0909022	−	0.995860i	$-0.471025\pi$
$443$	3.82654	0.181804	0.0909022	+	0.995860i	$0.471025\pi$
$444$	0	0
$445$	5.32051	0.252216
$446$	0	0
$447$	0	0
$448$	0	0
$449$	− 11.1106i	− 0.524343i	−0.965021	−	0.262172i	$-0.915561\pi$
$449$	− 11.1106i	− 0.524343i	0.965021	−	0.262172i	$-0.0844387\pi$
$450$	0	0
$451$	8.39230i	0.395178i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−21.8695	−1.02526
$456$	0	0
$457$	30.8564	1.44340	0.721701	−	0.692205i	$-0.243361\pi$
$457$	30.8564	1.44340	0.721701	+	0.692205i	$0.243361\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	− 14.1793i	− 0.660396i	−0.943912	−	0.330198i	$-0.892885\pi$
$461$	− 14.1793i	− 0.660396i	0.943912	−	0.330198i	$-0.107115\pi$
$462$	0	0
$463$	− 23.9808i	− 1.11448i	−0.830351	−	0.557240i	$-0.811860\pi$
$463$	− 23.9808i	− 1.11448i	0.830351	−	0.557240i	$-0.188140\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−38.2953	−1.77209	−0.886047	−	0.463596i	$-0.846559\pi$
$467$	−38.2953	−1.77209	−0.886047	+	0.463596i	$0.846559\pi$
$468$	0	0
$469$	22.0000	1.01587
$470$	0	0
$471$	0	0
$472$	0	0
$473$	− 6.72930i	− 0.309413i
$474$	0	0
$475$	− 4.19615i	− 0.192533i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	35.4297	1.61882	0.809411	−	0.587242i	$-0.199786\pi$
$479$	35.4297	1.61882	0.809411	+	0.587242i	$0.199786\pi$
$480$	0	0
$481$	51.7128	2.35790
$482$	0	0
$483$	0	0
$484$	0	0
$485$	− 9.79796i	− 0.444902i
$486$	0	0
$487$	19.8564i	0.899780i	0.893084	+	0.449890i	$0.148537\pi$
$487$	19.8564i	0.899780i	−0.893084	+	0.449890i	$0.851463\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−6.79367	−0.306594	−0.153297	−	0.988180i	$-0.548989\pi$
$491$	−6.79367	−0.306594	−0.153297	+	0.988180i	$0.548989\pi$
$492$	0	0
$493$	−61.1769	−2.75527
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 42.5007i	− 1.90642i
$498$	0	0
$499$	− 15.0000i	− 0.671492i	−0.941953	−	0.335746i	$-0.891012\pi$
$499$	− 15.0000i	− 0.671492i	0.941953	−	0.335746i	$-0.108988\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	9.41902	0.419973	0.209987	−	0.977704i	$-0.432658\pi$
$503$	9.41902	0.419973	0.209987	+	0.977704i	$0.432658\pi$
$504$	0	0
$505$	−14.4449	−0.642788
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 11.3137i	− 0.501471i	−0.968056	−	0.250736i	$-0.919328\pi$
$509$	− 11.3137i	− 0.501471i	0.968056	−	0.250736i	$-0.0806725\pi$
$510$	0	0
$511$	− 35.3923i	− 1.56566i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−16.4901	−0.726640
$516$	0	0
$517$	−24.4641	−1.07593
$518$	0	0
$519$	0	0
$520$	0	0
$521$	− 26.2880i	− 1.15170i	−0.817555	−	0.575850i	$-0.804671\pi$
$521$	− 26.2880i	− 1.15170i	0.817555	−	0.575850i	$-0.195329\pi$
$522$	0	0
$523$	− 11.4641i	− 0.501290i	−0.968079	−	0.250645i	$-0.919357\pi$
$523$	− 11.4641i	− 0.501290i	0.968079	−	0.250645i	$-0.0806427\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−53.8144	−2.34419
$528$	0	0
$529$	−22.8564	−0.993757
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 15.4548i	− 0.669422i
$534$	0	0
$535$	6.00000i	0.259403i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	38.3596	1.65227
$540$	0	0
$541$	−0.607695	−0.0261269	−0.0130634	−	0.999915i	$-0.504158\pi$
$541$	−0.607695	−0.0261269	−0.0130634	+	0.999915i	$0.504158\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	5.73118i	0.245497i
$546$	0	0
$547$	− 26.5359i	− 1.13459i	−0.823514	−	0.567297i	$-0.807989\pi$
$547$	− 26.5359i	− 1.13459i	0.823514	−	0.567297i	$-0.192011\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	8.48528	0.361485
$552$	0	0
$553$	51.1769	2.17626
$554$	0	0
$555$	0	0
$556$	0	0
$557$	38.9788i	1.65159i	0.563974	+	0.825793i	$0.309272\pi$
$557$	38.9788i	1.65159i	−0.563974	+	0.825793i	$0.690728\pi$
$558$	0	0
$559$	12.3923i	0.524139i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−4.34418	−0.183085	−0.0915426	−	0.995801i	$-0.529180\pi$
$563$	−4.34418	−0.183085	−0.0915426	+	0.995801i	$0.529180\pi$
$564$	0	0
$565$	−1.35898	−0.0571729
$566$	0	0
$567$	0	0
$568$	0	0
$569$	− 22.4243i	− 0.940077i	−0.882646	−	0.470039i	$-0.844240\pi$
$569$	− 22.4243i	− 0.940077i	0.882646	−	0.470039i	$-0.155760\pi$
$570$	0	0
$571$	40.2487i	1.68436i	0.539199	+	0.842178i	$0.318727\pi$
$571$	40.2487i	1.68436i	−0.539199	+	0.842178i	$0.681273\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	1.59008	0.0663109
$576$	0	0
$577$	17.1962	0.715885	0.357943	−	0.933744i	$-0.383478\pi$
$577$	17.1962	0.715885	0.357943	+	0.933744i	$0.383478\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	− 36.1875i	− 1.50131i
$582$	0	0
$583$	− 22.1051i	− 0.915500i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	11.2493	0.464310	0.232155	−	0.972679i	$-0.425422\pi$
$587$	11.2493	0.464310	0.232155	+	0.972679i	$0.425422\pi$
$588$	0	0
$589$	7.46410	0.307553
$590$	0	0
$591$	0	0
$592$	0	0
$593$	40.0512i	1.64471i	0.568976	+	0.822354i	$0.307340\pi$
$593$	40.0512i	1.64471i	−0.568976	+	0.822354i	$0.692660\pi$
$594$	0	0
$595$	− 28.8564i	− 1.18300i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	14.4195	0.589166	0.294583	−	0.955626i	$-0.404819\pi$
$599$	14.4195	0.589166	0.294583	+	0.955626i	$0.404819\pi$
$600$	0	0
$601$	−10.0000	−0.407909	−0.203954	−	0.978980i	$-0.565379\pi$
$601$	−10.0000	−0.407909	−0.203954	+	0.978980i	$0.565379\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	1.96902i	0.0800519i
$606$	0	0
$607$	29.4641i	1.19591i	0.801529	+	0.597955i	$0.204020\pi$
$607$	29.4641i	1.19591i	−0.801529	+	0.597955i	$0.795980\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	45.0518	1.82260
$612$	0	0
$613$	−20.2679	−0.818615	−0.409307	−	0.912397i	$-0.634230\pi$
$613$	−20.2679	−0.818615	−0.409307	+	0.912397i	$0.634230\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 19.4572i	− 0.783318i	−0.920110	−	0.391659i	$-0.871901\pi$
$617$	− 19.4572i	− 0.783318i	0.920110	−	0.391659i	$-0.128099\pi$
$618$	0	0
$619$	33.0718i	1.32927i	0.747169	+	0.664634i	$0.231412\pi$
$619$	33.0718i	1.32927i	−0.747169	+	0.664634i	$0.768588\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	26.4911	1.06134
$624$	0	0
$625$	13.5885	0.543538
$626$	0	0
$627$	0	0
$628$	0	0
$629$	68.2340i	2.72067i
$630$	0	0
$631$	7.24871i	0.288567i	0.989536	+	0.144283i	$0.0460877\pi$
$631$	7.24871i	0.288567i	−0.989536	+	0.144283i	$0.953912\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−9.79796	−0.388820
$636$	0	0
$637$	−70.6410	−2.79890
$638$	0	0
$639$	0	0
$640$	0	0
$641$	5.10205i	0.201519i	0.994911	+	0.100759i	$0.0321273\pi$
$641$	5.10205i	0.201519i	−0.994911	+	0.100759i	$0.967873\pi$
$642$	0	0
$643$	− 27.5885i	− 1.08798i	−0.839091	−	0.543991i	$-0.816912\pi$
$643$	− 27.5885i	− 1.08798i	0.839091	−	0.543991i	$-0.183088\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	16.8319	0.661729	0.330864	−	0.943678i	$-0.392660\pi$
$647$	16.8319	0.661729	0.330864	+	0.943678i	$0.392660\pi$
$648$	0	0
$649$	9.21539	0.361736
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 34.4316i	− 1.34741i	−0.739000	−	0.673705i	$-0.764702\pi$
$653$	− 34.4316i	− 1.34741i	0.739000	−	0.673705i	$-0.235298\pi$
$654$	0	0
$655$	6.21539i	0.242855i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−4.41851	−0.172121	−0.0860603	−	0.996290i	$-0.527428\pi$
$659$	−4.41851	−0.172121	−0.0860603	+	0.996290i	$0.527428\pi$
$660$	0	0
$661$	3.32051	0.129153	0.0645764	−	0.997913i	$-0.479430\pi$
$661$	3.32051	0.129153	0.0645764	+	0.997913i	$0.479430\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	4.00240i	0.155207i
$666$	0	0
$667$	3.21539i	0.124500i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	21.3519	0.824281
$672$	0	0
$673$	16.2487	0.626342	0.313171	−	0.949697i	$-0.398609\pi$
$673$	16.2487	0.626342	0.313171	+	0.949697i	$0.398609\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	5.45378i	0.209606i	0.994493	+	0.104803i	$0.0334212\pi$
$677$	5.45378i	0.209606i	−0.994493	+	0.104803i	$0.966579\pi$
$678$	0	0
$679$	− 48.7846i	− 1.87218i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	22.0726	0.844585	0.422293	−	0.906460i	$-0.361225\pi$
$683$	22.0726	0.844585	0.422293	+	0.906460i	$0.361225\pi$
$684$	0	0
$685$	−12.3731	−0.472751
$686$	0	0
$687$	0	0
$688$	0	0
$689$	40.7076i	1.55084i
$690$	0	0
$691$	− 35.8372i	− 1.36331i	−0.731674	−	0.681655i	$-0.761260\pi$
$691$	− 35.8372i	− 1.36331i	0.731674	−	0.681655i	$-0.238740\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	3.65067	0.138478
$696$	0	0
$697$	20.3923	0.772414
$698$	0	0
$699$	0	0
$700$	0	0
$701$	23.2838i	0.879415i	0.898141	+	0.439708i	$0.144918\pi$
$701$	23.2838i	0.879415i	−0.898141	+	0.439708i	$0.855082\pi$
$702$	0	0
$703$	− 9.46410i	− 0.356945i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−71.9218	−2.70490
$708$	0	0
$709$	−31.8564	−1.19639	−0.598196	−	0.801350i	$-0.704116\pi$
$709$	−31.8564	−1.19639	−0.598196	+	0.801350i	$0.704116\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	2.82843i	0.105925i
$714$	0	0
$715$	14.5359i	0.543612i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	25.9463	0.967633	0.483816	−	0.875170i	$-0.339250\pi$
$719$	25.9463	0.967633	0.483816	+	0.875170i	$0.339250\pi$
$720$	0	0
$721$	−82.1051	−3.05775
$722$	0	0
$723$	0	0
$724$	0	0
$725$	35.6055i	1.32236i
$726$	0	0
$727$	7.19615i	0.266891i	0.991056	+	0.133445i	$0.0426040\pi$
$727$	7.19615i	0.266891i	−0.991056	+	0.133445i	$0.957396\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−16.3514	−0.604778
$732$	0	0
$733$	−27.8564	−1.02890	−0.514450	−	0.857520i	$-0.672004\pi$
$733$	−27.8564	−1.02890	−0.514450	+	0.857520i	$0.672004\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 14.6226i	− 0.538631i
$738$	0	0
$739$	− 46.5692i	− 1.71308i	−0.516084	−	0.856538i	$-0.672611\pi$
$739$	− 46.5692i	− 1.71308i	0.516084	−	0.856538i	$-0.327389\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	50.7829	1.86305	0.931523	−	0.363683i	$-0.118481\pi$
$743$	50.7829	1.86305	0.931523	+	0.363683i	$0.118481\pi$
$744$	0	0
$745$	6.21539	0.227714
$746$	0	0
$747$	0	0
$748$	0	0
$749$	29.8744i	1.09159i
$750$	0	0
$751$	5.85641i	0.213703i	0.994275	+	0.106852i	$0.0340770\pi$
$751$	5.85641i	0.213703i	−0.994275	+	0.106852i	$0.965923\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	16.4901	0.600136
$756$	0	0
$757$	17.5885	0.639263	0.319632	−	0.947542i	$-0.396441\pi$
$757$	17.5885	0.639263	0.319632	+	0.947542i	$0.396441\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	− 46.3273i	− 1.67936i	−0.543080	−	0.839681i	$-0.682742\pi$
$761$	− 46.3273i	− 1.67936i	0.543080	−	0.839681i	$-0.317258\pi$
$762$	0	0
$763$	28.5359i	1.03307i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−16.9706	−0.612772
$768$	0	0
$769$	16.6603	0.600784	0.300392	−	0.953816i	$-0.402882\pi$
$769$	16.6603	0.600784	0.300392	+	0.953816i	$0.402882\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	14.6226i	0.525939i	0.964804	+	0.262969i	$0.0847018\pi$
$773$	14.6226i	0.525939i	−0.964804	+	0.262969i	$0.915298\pi$
$774$	0	0
$775$	31.3205i	1.12507i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−2.82843	−0.101339
$780$	0	0
$781$	−28.2487	−1.01082
$782$	0	0
$783$	0	0
$784$	0	0
$785$	− 3.58630i	− 0.128001i
$786$	0	0
$787$	− 44.3923i	− 1.58241i	−0.611548	−	0.791207i	$-0.709453\pi$
$787$	− 44.3923i	− 1.58241i	0.611548	−	0.791207i	$-0.290547\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−6.76646	−0.240588
$792$	0	0
$793$	−39.3205	−1.39631
$794$	0	0
$795$	0	0
$796$	0	0
$797$	16.2870i	0.576916i	0.957493	+	0.288458i	$0.0931425\pi$
$797$	16.2870i	0.576916i	−0.957493	+	0.288458i	$0.906857\pi$
$798$	0	0
$799$	59.4449i	2.10301i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−23.5240	−0.830144
$804$	0	0
$805$	−1.51666	−0.0534553
$806$	0	0
$807$	0	0
$808$	0	0
$809$	− 35.4940i	− 1.24790i	−0.781463	−	0.623952i	$-0.785526\pi$
$809$	− 35.4940i	− 1.24790i	0.781463	−	0.623952i	$-0.214474\pi$
$810$	0	0
$811$	− 18.6410i	− 0.654575i	−0.944925	−	0.327287i	$-0.893866\pi$
$811$	− 18.6410i	− 0.654575i	0.944925	−	0.327287i	$-0.106134\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−10.7589	−0.376868
$816$	0	0
$817$	2.26795	0.0793455
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 23.1178i	− 0.806818i	−0.915020	−	0.403409i	$-0.867825\pi$
$821$	− 23.1178i	− 0.806818i	0.915020	−	0.403409i	$-0.132175\pi$
$822$	0	0
$823$	14.4641i	0.504187i	0.967703	+	0.252093i	$0.0811190\pi$
$823$	14.4641i	0.504187i	−0.967703	+	0.252093i	$0.918881\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−32.7028	−1.13719	−0.568593	−	0.822619i	$-0.692512\pi$
$827$	−32.7028	−1.13719	−0.568593	+	0.822619i	$0.692512\pi$
$828$	0	0
$829$	38.5359	1.33841	0.669204	−	0.743079i	$-0.266635\pi$
$829$	38.5359	1.33841	0.669204	+	0.743079i	$0.266635\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	− 93.2094i	− 3.22951i
$834$	0	0
$835$	− 2.53590i	− 0.0877584i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	1.99622	0.0689173	0.0344586	−	0.999406i	$-0.489029\pi$
$839$	1.99622	0.0689173	0.0344586	+	0.999406i	$0.489029\pi$
$840$	0	0
$841$	−43.0000	−1.48276
$842$	0	0
$843$	0	0
$844$	0	0
$845$	− 15.1130i	− 0.519904i
$846$	0	0
$847$	9.80385i	0.336864i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	3.58630	0.122937
$852$	0	0
$853$	−39.8564	−1.36466	−0.682329	−	0.731046i	$-0.739033\pi$
$853$	−39.8564	−1.36466	−0.682329	+	0.731046i	$0.739033\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 41.7429i	− 1.42591i	−0.701210	−	0.712954i	$-0.747357\pi$
$857$	− 41.7429i	− 1.42591i	0.701210	−	0.712954i	$-0.252643\pi$
$858$	0	0
$859$	14.8564i	0.506894i	0.967349	+	0.253447i	$0.0815644\pi$
$859$	14.8564i	0.506894i	−0.967349	+	0.253447i	$0.918436\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	3.93803	0.134052	0.0670261	−	0.997751i	$-0.478649\pi$
$863$	3.93803	0.134052	0.0670261	+	0.997751i	$0.478649\pi$
$864$	0	0
$865$	16.6410	0.565812
$866$	0	0
$867$	0	0
$868$	0	0
$869$	− 34.0155i	− 1.15390i
$870$	0	0
$871$	26.9282i	0.912427i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−36.8067	−1.24429
$876$	0	0
$877$	−19.8564	−0.670503	−0.335252	−	0.942129i	$-0.608821\pi$
$877$	−19.8564	−0.670503	−0.335252	+	0.942129i	$0.608821\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	6.35036i	0.213949i	0.994262	+	0.106974i	$0.0341163\pi$
$881$	6.35036i	0.213949i	−0.994262	+	0.106974i	$0.965884\pi$
$882$	0	0
$883$	17.6410i	0.593667i	0.954929	+	0.296834i	$0.0959307\pi$
$883$	17.6410i	0.593667i	−0.954929	+	0.296834i	$0.904069\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	2.34795	0.0788366	0.0394183	−	0.999223i	$-0.487450\pi$
$887$	2.34795	0.0788366	0.0394183	+	0.999223i	$0.487450\pi$
$888$	0	0
$889$	−48.7846	−1.63618
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 8.24504i	− 0.275910i
$894$	0	0
$895$	− 10.1436i	− 0.339063i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−63.3350	−2.11234
$900$	0	0
$901$	−53.7128	−1.78943
$902$	0	0
$903$	0	0
$904$	0	0
$905$	4.41851i	0.146876i
$906$	0	0
$907$	20.9282i	0.694910i	0.937697	+	0.347455i	$0.112954\pi$
$907$	20.9282i	0.694910i	−0.937697	+	0.347455i	$0.887046\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−53.2596	−1.76457	−0.882285	−	0.470715i	$-0.843996\pi$
$911$	−53.2596	−1.76457	−0.882285	+	0.470715i	$0.843996\pi$
$912$	0	0
$913$	−24.0526	−0.796024
$914$	0	0
$915$	0	0
$916$	0	0
$917$	30.9468i	1.02195i
$918$	0	0
$919$	27.7128i	0.914161i	0.889425	+	0.457081i	$0.151105\pi$
$919$	27.7128i	0.914161i	−0.889425	+	0.457081i	$0.848895\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	52.0213	1.71230
$924$	0	0
$925$	39.7128	1.30575
$926$	0	0
$927$	0	0
$928$	0	0
$929$	− 41.5670i	− 1.36377i	−0.731460	−	0.681884i	$-0.761161\pi$
$929$	− 41.5670i	− 1.36377i	0.731460	−	0.681884i	$-0.238839\pi$
$930$	0	0
$931$	12.9282i	0.423705i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−19.1798	−0.627247
$936$	0	0
$937$	−29.1436	−0.952080	−0.476040	−	0.879424i	$-0.657928\pi$
$937$	−29.1436	−0.952080	−0.476040	+	0.879424i	$0.657928\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	− 43.8134i	− 1.42828i	−0.700005	−	0.714138i	$-0.746819\pi$
$941$	− 43.8134i	− 1.42828i	0.700005	−	0.714138i	$-0.253181\pi$
$942$	0	0
$943$	− 1.07180i	− 0.0349025i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	4.52004	0.146882	0.0734408	−	0.997300i	$-0.476602\pi$
$947$	4.52004	0.146882	0.0734408	+	0.997300i	$0.476602\pi$
$948$	0	0
$949$	43.3205	1.40624
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 23.1079i	− 0.748538i	−0.927320	−	0.374269i	$-0.877894\pi$
$953$	− 23.1079i	− 0.748538i	0.927320	−	0.374269i	$-0.122106\pi$
$954$	0	0
$955$	18.4641i	0.597484i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−61.6062	−1.98937
$960$	0	0
$961$	−24.7128	−0.797188
$962$	0	0
$963$	0	0
$964$	0	0
$965$	− 7.65308i	− 0.246361i
$966$	0	0
$967$	− 7.21539i	− 0.232031i	−0.993247	−	0.116016i	$-0.962988\pi$
$967$	− 7.21539i	− 0.232031i	0.993247	−	0.116016i	$-0.0370123\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−7.72741	−0.247984	−0.123992	−	0.992283i	$-0.539570\pi$
$971$	−7.72741	−0.247984	−0.123992	+	0.992283i	$0.539570\pi$
$972$	0	0
$973$	18.1769	0.582725
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 44.0165i	− 1.40821i	−0.710095	−	0.704106i	$-0.751348\pi$
$977$	− 44.0165i	− 1.40821i	0.710095	−	0.704106i	$-0.248652\pi$
$978$	0	0
$979$	− 17.6077i	− 0.562744i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−19.7990	−0.631490	−0.315745	−	0.948844i	$-0.602254\pi$
$983$	−19.7990	−0.631490	−0.315745	+	0.948844i	$0.602254\pi$
$984$	0	0
$985$	−9.37307	−0.298651
$986$	0	0
$987$	0	0
$988$	0	0
$989$	0.859411i	0.0273277i
$990$	0	0
$991$	− 10.9282i	− 0.347146i	−0.984821	−	0.173573i	$-0.944469\pi$
$991$	− 10.9282i	− 0.347146i	0.984821	−	0.173573i	$-0.0555312\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−18.1717	−0.576083
$996$	0	0
$997$	27.9808	0.886160	0.443080	−	0.896482i	$-0.353886\pi$
$997$	27.9808	0.886160	0.443080	+	0.896482i	$0.353886\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5472.2.d.e.2015.3	✓	8
3.2	odd	2	inner	5472.2.d.e.2015.5	yes	8
4.3	odd	2	inner	5472.2.d.e.2015.4	yes	8
12.11	even	2	inner	5472.2.d.e.2015.6	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5472.2.d.e.2015.3	✓	8	1.1	even	1	trivial
5472.2.d.e.2015.4	yes	8	4.3	odd	2	inner
5472.2.d.e.2015.5	yes	8	3.2	odd	2	inner
5472.2.d.e.2015.6	yes	8	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)

\(f(q)\)	\(=\)	\(q-0.896575i q^{5} -4.46410i q^{7} -2.96713 q^{11} +5.46410 q^{13} +7.20977i q^{17} -1.00000i q^{19} +0.378937 q^{23} +4.19615 q^{25} +8.48528i q^{29} +7.46410i q^{31} -4.00240 q^{35} +9.46410 q^{37} -2.82843i q^{41} +2.26795i q^{43} +8.24504 q^{47} -12.9282 q^{49} +7.45001i q^{53} +2.66025i q^{55} -3.10583 q^{59} -7.19615 q^{61} -4.89898i q^{65} +4.92820i q^{67} +9.52056 q^{71} +7.92820 q^{73} +13.2456i q^{77} +11.4641i q^{79} +8.10634 q^{83} +6.46410 q^{85} +5.93426i q^{89} -24.3923i q^{91} -0.896575 q^{95} +10.9282 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 16 q^{13} - 8 q^{25} + 48 q^{37} - 48 q^{49} - 16 q^{61} + 8 q^{73} + 24 q^{85} + 32 q^{97}+O(q^{100}) \) 8 * q + 16 * q^13 - 8 * q^25 + 48 * q^37 - 48 * q^49 - 16 * q^61 + 8 * q^73 + 24 * q^85 + 32 * q^97

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