LMFDB - Embedded newform 5472.2.k.d.2431.18

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5472,2,Mod(2431,5472)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5472, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 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("5472.2431");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$43.6941399860$
Analytic rank:	$0$
Dimension:	$40$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2431.18
Character	$\chi$	$=$	5472.2431
Dual form			5472.2.k.d.2431.19

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.27046 q^{5} +2.45392i q^{7} +O(q^{10})$	$q-1.27046 q^{5} +2.45392i q^{7} -4.60851i q^{11} -5.38751i q^{13} -2.30591 q^{17} +(-0.0756671 - 4.35824i) q^{19} +3.46935i q^{23} -3.38594 q^{25} +4.46101i q^{29} -0.666790 q^{31} -3.11759i q^{35} -4.23961i q^{37} +7.19331i q^{41} -1.91031i q^{43} +1.97513i q^{47} +0.978285 q^{49} +7.96209i q^{53} +5.85491i q^{55} -0.672646 q^{59} +6.33054 q^{61} +6.84460i q^{65} -3.25638 q^{67} -5.48237 q^{71} -5.78693 q^{73} +11.3089 q^{77} +6.85186 q^{79} +7.51209i q^{83} +2.92955 q^{85} -1.80707i q^{89} +13.2205 q^{91} +(0.0961316 + 5.53695i) q^{95} +6.69843i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 40 q+O(q^{10}) $ 40 * q	$ 40 q + 40 q^{25} - 24 q^{49} + 48 q^{61} + 16 q^{73} - 16 q^{85}+O(q^{100}) $ 40 * q + 40 * q^25 - 24 * q^49 + 48 * q^61 + 16 * q^73 - 16 * q^85

Display 100 coefficients 10 coefficients

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
$5$	−1.27046			−0.568165			−0.284082	−	0.958800i	$-0.591689\pi$
$5$	−1.27046			−0.568165			−0.284082	+	0.958800i	$0.591689\pi$
$6$		0			0
$7$			2.45392i			0.927494i	0.885968	+	0.463747i	$0.153495\pi$
$7$			2.45392i			0.927494i	−0.885968	+	0.463747i	$0.846505\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$		−	4.60851i		−	1.38952i	−0.719242	−	0.694759i	$-0.755511\pi$
$11$		−	4.60851i		−	1.38952i	0.719242	−	0.694759i	$-0.244489\pi$
$12$		0			0
$13$		−	5.38751i		−	1.49423i	−0.664696	−	0.747114i	$-0.731439\pi$
$13$		−	5.38751i		−	1.49423i	0.664696	−	0.747114i	$-0.268561\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−2.30591			−0.559264			−0.279632	−	0.960107i	$-0.590213\pi$
$17$	−2.30591			−0.559264			−0.279632	+	0.960107i	$0.590213\pi$
$18$		0			0
$19$	−0.0756671	−	4.35824i	−0.0173592	−	0.999849i
$19$	−0.0756671	−	4.35824i	−0.0173592	−	0.999849i
$20$		0			0
$21$		0			0
$22$		0			0
$23$			3.46935i			0.723410i	0.932293	+	0.361705i	$0.117805\pi$
$23$			3.46935i			0.723410i	−0.932293	+	0.361705i	$0.882195\pi$
$24$		0			0
$25$	−3.38594			−0.677189
$26$		0			0
$27$		0			0
$28$		0			0
$29$			4.46101i			0.828390i	0.910188	+	0.414195i	$0.135937\pi$
$29$			4.46101i			0.828390i	−0.910188	+	0.414195i	$0.864063\pi$
$30$		0			0
$31$	−0.666790			−0.119759			−0.0598795	−	0.998206i	$-0.519072\pi$
$31$	−0.666790			−0.119759			−0.0598795	+	0.998206i	$0.519072\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		−	3.11759i		−	0.526970i
$36$		0			0
$37$		−	4.23961i		−	0.696988i	−0.937311	−	0.348494i	$-0.886693\pi$
$37$		−	4.23961i		−	0.696988i	0.937311	−	0.348494i	$-0.113307\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$			7.19331i			1.12341i	0.827339	+	0.561703i	$0.189854\pi$
$41$			7.19331i			1.12341i	−0.827339	+	0.561703i	$0.810146\pi$
$42$		0			0
$43$		−	1.91031i		−	0.291320i	−0.989335	−	0.145660i	$-0.953470\pi$
$43$		−	1.91031i		−	0.291320i	0.989335	−	0.145660i	$-0.0465305\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$			1.97513i			0.288103i	0.989570	+	0.144051i	$0.0460131\pi$
$47$			1.97513i			0.288103i	−0.989570	+	0.144051i	$0.953987\pi$
$48$		0			0
$49$	0.978285			0.139755
$50$		0			0
$51$		0			0
$52$		0			0
$53$			7.96209i			1.09368i	0.837238	+	0.546838i	$0.184169\pi$
$53$			7.96209i			1.09368i	−0.837238	+	0.546838i	$0.815831\pi$
$54$		0			0
$55$			5.85491i			0.789476i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−0.672646			−0.0875711			−0.0437855	−	0.999041i	$-0.513942\pi$
$59$	−0.672646			−0.0875711			−0.0437855	+	0.999041i	$0.513942\pi$
$60$		0			0
$61$	6.33054			0.810543			0.405271	−	0.914196i	$-0.367177\pi$
$61$	6.33054			0.810543			0.405271	+	0.914196i	$0.367177\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$			6.84460i			0.848968i
$66$		0			0
$67$	−3.25638			−0.397830			−0.198915	−	0.980017i	$-0.563742\pi$
$67$	−3.25638			−0.397830			−0.198915	+	0.980017i	$0.563742\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−5.48237			−0.650638			−0.325319	−	0.945604i	$-0.605472\pi$
$71$	−5.48237			−0.650638			−0.325319	+	0.945604i	$0.605472\pi$
$72$		0			0
$73$	−5.78693			−0.677309			−0.338655	−	0.940911i	$-0.609972\pi$
$73$	−5.78693			−0.677309			−0.338655	+	0.940911i	$0.609972\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	11.3089			1.28877
$78$		0			0
$79$	6.85186			0.770895			0.385447	−	0.922730i	$-0.374047\pi$
$79$	6.85186			0.770895			0.385447	+	0.922730i	$0.374047\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$			7.51209i			0.824559i	0.911058	+	0.412279i	$0.135267\pi$
$83$			7.51209i			0.824559i	−0.911058	+	0.412279i	$0.864733\pi$
$84$		0			0
$85$	2.92955			0.317754
$86$		0			0
$87$		0			0
$88$		0			0
$89$		−	1.80707i		−	0.191549i	−0.995403	−	0.0957744i	$-0.969467\pi$
$89$		−	1.80707i		−	0.191549i	0.995403	−	0.0957744i	$-0.0305327\pi$
$90$		0			0
$91$	13.2205			1.38589
$92$		0			0
$93$		0			0
$94$		0			0
$95$	0.0961316	+	5.53695i	0.00986290	+	0.568079i
$96$		0			0
$97$			6.69843i			0.680123i	0.940403	+	0.340061i	$0.110448\pi$
$97$			6.69843i			0.680123i	−0.940403	+	0.340061i	$0.889552\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−5.19974			−0.517393			−0.258697	−	0.965959i	$-0.583293\pi$
$101$	−5.19974			−0.517393			−0.258697	+	0.965959i	$0.583293\pi$
$102$		0			0
$103$	−9.29735			−0.916095			−0.458048	−	0.888928i	$-0.651451\pi$
$103$	−9.29735			−0.916095			−0.458048	+	0.888928i	$0.651451\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−18.4989			−1.78836			−0.894179	−	0.447709i	$-0.852240\pi$
$107$	−18.4989			−1.78836			−0.894179	+	0.447709i	$0.852240\pi$
$108$		0			0
$109$			15.5030i			1.48492i	0.669892	+	0.742459i	$0.266340\pi$
$109$			15.5030i			1.48492i	−0.669892	+	0.742459i	$0.733660\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			5.03753i			0.473891i	0.971523	+	0.236945i	$0.0761463\pi$
$113$			5.03753i			0.473891i	−0.971523	+	0.236945i	$0.923854\pi$
$114$		0			0
$115$		−	4.40766i		−	0.411016i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		−	5.65850i		−	0.518714i
$120$		0			0
$121$	−10.2384			−0.930762
$122$		0			0
$123$		0			0
$124$		0			0
$125$	10.6540			0.952920
$126$		0			0
$127$	14.8495			1.31768			0.658841	−	0.752282i	$-0.271047\pi$
$127$	14.8495			1.31768			0.658841	+	0.752282i	$0.271047\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$			4.39449i			0.383949i	0.981400	+	0.191974i	$0.0614890\pi$
$131$			4.39449i			0.383949i	−0.981400	+	0.191974i	$0.938511\pi$
$132$		0			0
$133$	10.6948	−	0.185681i	0.927354	−	0.0161006i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−2.96085			−0.252962			−0.126481	−	0.991969i	$-0.540368\pi$
$137$	−2.96085			−0.252962			−0.126481	+	0.991969i	$0.540368\pi$
$138$		0			0
$139$			6.31797i			0.535883i	0.963435	+	0.267942i	$0.0863434\pi$
$139$			6.31797i			0.535883i	−0.963435	+	0.267942i	$0.913657\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−24.8284			−2.07626
$144$		0			0
$145$		−	5.66752i		−	0.470662i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	9.35592			0.766467			0.383234	−	0.923651i	$-0.374810\pi$
$149$	9.35592			0.766467			0.383234	+	0.923651i	$0.374810\pi$
$150$		0			0
$151$	1.62737			0.132433			0.0662166	−	0.997805i	$-0.478907\pi$
$151$	1.62737			0.132433			0.0662166	+	0.997805i	$0.478907\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	0.847127			0.0680429
$156$		0			0
$157$	8.17288			0.652267			0.326133	−	0.945324i	$-0.394254\pi$
$157$	8.17288			0.652267			0.326133	+	0.945324i	$0.394254\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−8.51351			−0.670958
$162$		0			0
$163$		−	4.30883i		−	0.337493i	−0.985659	−	0.168747i	$-0.946028\pi$
$163$		−	4.30883i		−	0.337493i	0.985659	−	0.168747i	$-0.0539720\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−13.7070			−1.06068			−0.530339	−	0.847786i	$-0.677935\pi$
$167$	−13.7070			−1.06068			−0.530339	+	0.847786i	$0.677935\pi$
$168$		0			0
$169$	−16.0253			−1.23272
$170$		0			0
$171$		0			0
$172$		0			0
$173$			7.96209i			0.605346i	0.953094	+	0.302673i	$0.0978791\pi$
$173$			7.96209i			0.605346i	−0.953094	+	0.302673i	$0.902121\pi$
$174$		0			0
$175$		−	8.30883i		−	0.628088i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−19.3638			−1.44732			−0.723660	−	0.690156i	$-0.757542\pi$
$179$	−19.3638			−1.44732			−0.723660	+	0.690156i	$0.757542\pi$
$180$		0			0
$181$			13.9837i			1.03940i	0.854348	+	0.519701i	$0.173957\pi$
$181$			13.9837i			1.03940i	−0.854348	+	0.519701i	$0.826043\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$			5.38624i			0.396004i
$186$		0			0
$187$			10.6268i			0.777108i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		−	20.4158i		−	1.47723i	−0.674125	−	0.738617i	$-0.735479\pi$
$191$		−	20.4158i		−	1.47723i	0.674125	−	0.738617i	$-0.264521\pi$
$192$		0			0
$193$		−	8.33468i		−	0.599943i	−0.953948	−	0.299972i	$-0.903023\pi$
$193$		−	8.33468i		−	0.599943i	0.953948	−	0.299972i	$-0.0969773\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−25.7047			−1.83138			−0.915691	−	0.401882i	$-0.868356\pi$
$197$	−25.7047			−1.83138			−0.915691	+	0.401882i	$0.868356\pi$
$198$		0			0
$199$		−	2.45392i		−	0.173954i	−0.996210	−	0.0869768i	$-0.972279\pi$
$199$		−	2.45392i		−	0.173954i	0.996210	−	0.0869768i	$-0.0277206\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−10.9470			−0.768326
$204$		0			0
$205$		−	9.13878i		−	0.638280i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−20.0850	+	0.348713i	−1.38931	+	0.0241210i
$210$		0			0
$211$	−25.4393			−1.75132			−0.875658	−	0.482932i	$-0.839572\pi$
$211$	−25.4393			−1.75132			−0.875658	+	0.482932i	$0.839572\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$			2.42696i			0.165518i
$216$		0			0
$217$		−	1.63625i		−	0.111076i
$218$		0			0
$219$		0			0
$220$		0			0
$221$			12.4231i			0.835668i
$222$		0			0
$223$	−20.7319			−1.38831			−0.694156	−	0.719824i	$-0.744222\pi$
$223$	−20.7319			−1.38831			−0.694156	+	0.719824i	$0.744222\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−11.1392			−0.739336			−0.369668	−	0.929164i	$-0.620529\pi$
$227$	−11.1392			−0.739336			−0.369668	+	0.929164i	$0.620529\pi$
$228$		0			0
$229$	−0.114236			−0.00754894			−0.00377447	−	0.999993i	$-0.501201\pi$
$229$	−0.114236			−0.00754894			−0.00377447	+	0.999993i	$0.501201\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	22.8528			1.49714			0.748569	−	0.663057i	$-0.230741\pi$
$233$	22.8528			1.49714			0.748569	+	0.663057i	$0.230741\pi$
$234$		0			0
$235$		−	2.50932i		−	0.163690i
$236$		0			0
$237$		0			0
$238$		0			0
$239$		−	8.39525i		−	0.543044i	−0.962432	−	0.271522i	$-0.912473\pi$
$239$		−	8.39525i		−	0.543044i	0.962432	−	0.271522i	$-0.0875270\pi$
$240$		0			0
$241$		−	9.74412i		−	0.627674i	−0.949477	−	0.313837i	$-0.898385\pi$
$241$		−	9.74412i		−	0.627674i	0.949477	−	0.313837i	$-0.101615\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−1.24287			−0.0794039
$246$		0			0
$247$	−23.4801	+	0.407657i	−1.49400	+	0.0259386i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		−	11.9753i		−	0.755871i	−0.925832	−	0.377936i	$-0.876634\pi$
$251$		−	11.9753i		−	0.755871i	0.925832	−	0.377936i	$-0.123366\pi$
$252$		0			0
$253$	15.9885			1.00519
$254$		0			0
$255$		0			0
$256$		0			0
$257$			6.41754i			0.400315i	0.979764	+	0.200157i	$0.0641454\pi$
$257$			6.41754i			0.400315i	−0.979764	+	0.200157i	$0.935855\pi$
$258$		0			0
$259$	10.4037			0.646452
$260$		0			0
$261$		0			0
$262$		0			0
$263$		−	15.3564i		−	0.946919i	−0.880816	−	0.473459i	$-0.843005\pi$
$263$		−	15.3564i		−	0.946919i	0.880816	−	0.473459i	$-0.156995\pi$
$264$		0			0
$265$		−	10.1155i		−	0.621389i
$266$		0			0
$267$		0			0
$268$		0			0
$269$			13.4267i			0.818639i	0.912391	+	0.409320i	$0.134234\pi$
$269$			13.4267i			0.818639i	−0.912391	+	0.409320i	$0.865766\pi$
$270$		0			0
$271$			0.419630i			0.0254907i	0.999919	+	0.0127453i	$0.00405708\pi$
$271$			0.419630i			0.0254907i	−0.999919	+	0.0127453i	$0.995943\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$			15.6042i			0.940966i
$276$		0			0
$277$	3.77496			0.226815			0.113408	−	0.993549i	$-0.463823\pi$
$277$	3.77496			0.226815			0.113408	+	0.993549i	$0.463823\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			26.7316i			1.59467i	0.603534	+	0.797337i	$0.293759\pi$
$281$			26.7316i			1.59467i	−0.603534	+	0.797337i	$0.706241\pi$
$282$		0			0
$283$			3.22580i			0.191754i	0.995393	+	0.0958771i	$0.0305656\pi$
$283$			3.22580i			0.191754i	−0.995393	+	0.0958771i	$0.969434\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−17.6518			−1.04195
$288$		0			0
$289$	−11.6828			−0.687223
$290$		0			0
$291$		0			0
$292$		0			0
$293$			30.5733i			1.78611i	0.449946	+	0.893056i	$0.351443\pi$
$293$			30.5733i			1.78611i	−0.449946	+	0.893056i	$0.648557\pi$
$294$		0			0
$295$	0.854567			0.0497548
$296$		0			0
$297$		0			0
$298$		0			0
$299$	18.6912			1.08094
$300$		0			0
$301$	4.68774			0.270197
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−8.04267			−0.460522
$306$		0			0
$307$	−21.0669			−1.20235			−0.601174	−	0.799118i	$-0.705300\pi$
$307$	−21.0669			−1.20235			−0.601174	+	0.799118i	$0.705300\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$			24.1520i			1.36954i	0.728761	+	0.684768i	$0.240096\pi$
$311$			24.1520i			1.36954i	−0.728761	+	0.684768i	$0.759904\pi$
$312$		0			0
$313$	−19.4834			−1.10127			−0.550634	−	0.834747i	$-0.685614\pi$
$313$	−19.4834			−1.10127			−0.550634	+	0.834747i	$0.685614\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	15.0773i		−	0.846825i	−0.905937	−	0.423413i	$-0.860832\pi$
$317$		−	15.0773i		−	0.846825i	0.905937	−	0.423413i	$-0.139168\pi$
$318$		0			0
$319$	20.5586			1.15106
$320$		0			0
$321$		0			0
$322$		0			0
$323$	0.174481	+	10.0497i	0.00970839	+	0.559180i
$324$		0			0
$325$			18.2418i			1.01187i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−4.84682			−0.267214
$330$		0			0
$331$	−14.8568			−0.816602			−0.408301	−	0.912847i	$-0.633879\pi$
$331$	−14.8568			−0.816602			−0.408301	+	0.912847i	$0.633879\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	4.13708			0.226033
$336$		0			0
$337$			36.0682i			1.96476i	0.186899	+	0.982379i	$0.440156\pi$
$337$			36.0682i			1.96476i	−0.186899	+	0.982379i	$0.559844\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$			3.07291i			0.166407i
$342$		0			0
$343$			19.5781i			1.05712i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		−	5.76188i		−	0.309314i	−0.987968	−	0.154657i	$-0.950573\pi$
$347$		−	5.76188i		−	0.309314i	0.987968	−	0.154657i	$-0.0494272\pi$
$348$		0			0
$349$	32.1782			1.72246			0.861230	−	0.508216i	$-0.169695\pi$
$349$	32.1782			1.72246			0.861230	+	0.508216i	$0.169695\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	26.5195			1.41149			0.705746	−	0.708465i	$-0.250612\pi$
$353$	26.5195			1.41149			0.705746	+	0.708465i	$0.250612\pi$
$354$		0			0
$355$	6.96511			0.369670
$356$		0			0
$357$		0			0
$358$		0			0
$359$			10.9847i			0.579752i	0.957064	+	0.289876i	$0.0936141\pi$
$359$			10.9847i			0.579752i	−0.957064	+	0.289876i	$0.906386\pi$
$360$		0			0
$361$	−18.9885	+	0.659551i	−0.999397	+	0.0347132i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	7.35204			0.384823
$366$		0			0
$367$		−	25.8526i		−	1.34949i	−0.738049	−	0.674747i	$-0.764253\pi$
$367$		−	25.8526i		−	1.34949i	0.738049	−	0.674747i	$-0.235747\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−19.5383			−1.01438
$372$		0			0
$373$		−	30.0478i		−	1.55581i	−0.628379	−	0.777907i	$-0.716281\pi$
$373$		−	30.0478i		−	1.55581i	0.628379	−	0.777907i	$-0.283719\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	24.0338			1.23780
$378$		0			0
$379$	29.3964			1.50999			0.754996	−	0.655730i	$-0.227639\pi$
$379$	29.3964			1.50999			0.754996	+	0.655730i	$0.227639\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	18.1930			0.929620			0.464810	−	0.885410i	$-0.346123\pi$
$383$	18.1930			0.929620			0.464810	+	0.885410i	$0.346123\pi$
$384$		0			0
$385$	−14.3675			−0.732234
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−12.6070			−0.639198			−0.319599	−	0.947553i	$-0.603548\pi$
$389$	−12.6070			−0.639198			−0.319599	+	0.947553i	$0.603548\pi$
$390$		0			0
$391$		−	8.00000i		−	0.404577i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−8.70499			−0.437995
$396$		0			0
$397$	22.3625			1.12234			0.561171	−	0.827700i	$-0.310351\pi$
$397$	22.3625			1.12234			0.561171	+	0.827700i	$0.310351\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			25.2733i			1.26209i	0.775748	+	0.631043i	$0.217373\pi$
$401$			25.2733i			1.26209i	−0.775748	+	0.631043i	$0.782627\pi$
$402$		0			0
$403$			3.59234i			0.178947i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−19.5383			−0.968478
$408$		0			0
$409$			30.0293i			1.48485i	0.669928	+	0.742426i	$0.266325\pi$
$409$			30.0293i			1.48485i	−0.669928	+	0.742426i	$0.733675\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		−	1.65062i		−	0.0812217i
$414$		0			0
$415$		−	9.54377i		−	0.468485i
$416$		0			0
$417$		0			0
$418$		0			0
$419$		−	6.80197i		−	0.332298i	−0.986101	−	0.166149i	$-0.946867\pi$
$419$		−	6.80197i		−	0.332298i	0.986101	−	0.166149i	$-0.0531333\pi$
$420$		0			0
$421$		−	6.86145i		−	0.334407i	−0.985922	−	0.167203i	$-0.946526\pi$
$421$		−	6.86145i		−	0.334407i	0.985922	−	0.167203i	$-0.0534736\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	7.80767			0.378727
$426$		0			0
$427$			15.5346i			0.751773i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	2.74223			0.132089			0.0660443	−	0.997817i	$-0.478962\pi$
$431$	2.74223			0.132089			0.0660443	+	0.997817i	$0.478962\pi$
$432$		0			0
$433$		−	12.1845i		−	0.585548i	−0.956182	−	0.292774i	$-0.905422\pi$
$433$		−	12.1845i		−	0.585548i	0.956182	−	0.292774i	$-0.0945784\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	15.1203	−	0.262516i	0.723301	−	0.0125578i
$438$		0			0
$439$	12.1409			0.579452			0.289726	−	0.957110i	$-0.406436\pi$
$439$	12.1409			0.579452			0.289726	+	0.957110i	$0.406436\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$			39.0200i			1.85389i	0.375192	+	0.926947i	$0.377577\pi$
$443$			39.0200i			1.85389i	−0.375192	+	0.926947i	$0.622423\pi$
$444$		0			0
$445$			2.29580i			0.108831i
$446$		0			0
$447$		0			0
$448$		0			0
$449$			26.6533i			1.25785i	0.777468	+	0.628923i	$0.216504\pi$
$449$			26.6533i			1.25785i	−0.777468	+	0.628923i	$0.783496\pi$
$450$		0			0
$451$	33.1505			1.56099
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−16.7961			−0.787412
$456$		0			0
$457$	−17.8119			−0.833205			−0.416602	−	0.909089i	$-0.636779\pi$
$457$	−17.8119			−0.833205			−0.416602	+	0.909089i	$0.636779\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−34.2369			−1.59457			−0.797285	−	0.603604i	$-0.793731\pi$
$461$	−34.2369			−1.59457			−0.797285	+	0.603604i	$0.793731\pi$
$462$		0			0
$463$			4.45887i			0.207221i	0.994618	+	0.103611i	$0.0330396\pi$
$463$			4.45887i			0.207221i	−0.994618	+	0.103611i	$0.966960\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		−	24.9576i		−	1.15490i	−0.816426	−	0.577450i	$-0.804048\pi$
$467$		−	24.9576i		−	1.15490i	0.816426	−	0.577450i	$-0.195952\pi$
$468$		0			0
$469$		−	7.99088i		−	0.368985i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−8.80369			−0.404794
$474$		0			0
$475$	0.256204	+	14.7568i	0.0117555	+	0.677087i
$476$		0			0
$477$		0			0
$478$		0			0
$479$			15.9179i			0.727308i	0.931534	+	0.363654i	$0.118471\pi$
$479$			15.9179i			0.727308i	−0.931534	+	0.363654i	$0.881529\pi$
$480$		0			0
$481$	−22.8410			−1.04146
$482$		0			0
$483$		0			0
$484$		0			0
$485$		−	8.51006i		−	0.386422i
$486$		0			0
$487$	4.03337			0.182769			0.0913847	−	0.995816i	$-0.470871\pi$
$487$	4.03337			0.182769			0.0913847	+	0.995816i	$0.470871\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		−	32.1588i		−	1.45131i	−0.688060	−	0.725654i	$-0.741537\pi$
$491$		−	32.1588i		−	1.45131i	0.688060	−	0.725654i	$-0.258463\pi$
$492$		0			0
$493$		−	10.2867i		−	0.463289i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		−	13.4533i		−	0.603463i
$498$		0			0
$499$		−	8.72068i		−	0.390391i	−0.980764	−	0.195196i	$-0.937466\pi$
$499$		−	8.72068i		−	0.390391i	0.980764	−	0.195196i	$-0.0625342\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$			17.5476i			0.782408i	0.920304	+	0.391204i	$0.127941\pi$
$503$			17.5476i			0.782408i	−0.920304	+	0.391204i	$0.872059\pi$
$504$		0			0
$505$	6.60603			0.293965
$506$		0			0
$507$		0			0
$508$		0			0
$509$			33.0781i			1.46616i	0.680143	+	0.733079i	$0.261918\pi$
$509$			33.0781i			1.46616i	−0.680143	+	0.733079i	$0.738082\pi$
$510$		0			0
$511$		−	14.2007i		−	0.628200i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	11.8119			0.520493
$516$		0			0
$517$	9.10243			0.400324
$518$		0			0
$519$		0			0
$520$		0			0
$521$			19.0329i			0.833847i	0.908942	+	0.416923i	$0.136892\pi$
$521$			19.0329i			0.833847i	−0.908942	+	0.416923i	$0.863108\pi$
$522$		0			0
$523$	−38.3296			−1.67604			−0.838019	−	0.545642i	$-0.816286\pi$
$523$	−38.3296			−1.67604			−0.838019	+	0.545642i	$0.816286\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	1.53756			0.0669770
$528$		0			0
$529$	10.9636			0.476678
$530$		0			0
$531$		0			0
$532$		0			0
$533$	38.7541			1.67862
$534$		0			0
$535$	23.5021			1.01608
$536$		0			0
$537$		0			0
$538$		0			0
$539$		−	4.50844i		−	0.194192i
$540$		0			0
$541$	37.5358			1.61379			0.806894	−	0.590696i	$-0.201147\pi$
$541$	37.5358			1.61379			0.806894	+	0.590696i	$0.201147\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		−	19.6959i		−	0.843678i
$546$		0			0
$547$	30.1514			1.28918			0.644591	−	0.764528i	$-0.277028\pi$
$547$	30.1514			1.28918			0.644591	+	0.764528i	$0.277028\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	19.4422	−	0.337552i	0.828265	−	0.0143802i
$552$		0			0
$553$			16.8139i			0.715000i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−16.6936			−0.707330			−0.353665	−	0.935372i	$-0.615065\pi$
$557$	−16.6936			−0.707330			−0.353665	+	0.935372i	$0.615065\pi$
$558$		0			0
$559$	−10.2918			−0.435298
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−0.690428			−0.0290981			−0.0145490	−	0.999894i	$-0.504631\pi$
$563$	−0.690428			−0.0290981			−0.0145490	+	0.999894i	$0.504631\pi$
$564$		0			0
$565$		−	6.39996i		−	0.269248i
$566$		0			0
$567$		0			0
$568$		0			0
$569$		−	32.3538i		−	1.35634i	−0.734905	−	0.678170i	$-0.762773\pi$
$569$		−	32.3538i		−	1.35634i	0.734905	−	0.678170i	$-0.237227\pi$
$570$		0			0
$571$		−	8.01197i		−	0.335291i	−0.985847	−	0.167645i	$-0.946384\pi$
$571$		−	8.01197i		−	0.335291i	0.985847	−	0.167645i	$-0.0536163\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		−	11.7470i		−	0.489885i
$576$		0			0
$577$	−34.5978			−1.44033			−0.720163	−	0.693805i	$-0.755933\pi$
$577$	−34.5978			−1.44033			−0.720163	+	0.693805i	$0.755933\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−18.4340			−0.764773
$582$		0			0
$583$	36.6934			1.51968
$584$		0			0
$585$		0			0
$586$		0			0
$587$		−	23.5387i		−	0.971544i	−0.874085	−	0.485772i	$-0.838538\pi$
$587$		−	23.5387i		−	0.971544i	0.874085	−	0.485772i	$-0.161462\pi$
$588$		0			0
$589$	0.0504541	+	2.90603i	0.00207892	+	0.119741i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−4.37170			−0.179524			−0.0897622	−	0.995963i	$-0.528611\pi$
$593$	−4.37170			−0.179524			−0.0897622	+	0.995963i	$0.528611\pi$
$594$		0			0
$595$			7.18888i			0.294715i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−6.02360			−0.246118			−0.123059	−	0.992399i	$-0.539270\pi$
$599$	−6.02360			−0.246118			−0.123059	+	0.992399i	$0.539270\pi$
$600$		0			0
$601$		−	38.9693i		−	1.58959i	−0.606877	−	0.794796i	$-0.707578\pi$
$601$		−	38.9693i		−	1.58959i	0.606877	−	0.794796i	$-0.292422\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	13.0074			0.528826
$606$		0			0
$607$	−23.0261			−0.934600			−0.467300	−	0.884099i	$-0.654773\pi$
$607$	−23.0261			−0.934600			−0.467300	+	0.884099i	$0.654773\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	10.6411			0.430491
$612$		0			0
$613$	−2.39792			−0.0968509			−0.0484254	−	0.998827i	$-0.515420\pi$
$613$	−2.39792			−0.0968509			−0.0484254	+	0.998827i	$0.515420\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−31.3663			−1.26276			−0.631380	−	0.775473i	$-0.717511\pi$
$617$	−31.3663			−1.26276			−0.631380	+	0.775473i	$0.717511\pi$
$618$		0			0
$619$			33.3341i			1.33981i	0.742446	+	0.669906i	$0.233666\pi$
$619$			33.3341i			1.33981i	−0.742446	+	0.669906i	$0.766334\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	4.43440			0.177660
$624$		0			0
$625$	3.39432			0.135773
$626$		0			0
$627$		0			0
$628$		0			0
$629$			9.77615i			0.389801i
$630$		0			0
$631$			28.5045i			1.13475i	0.823460	+	0.567374i	$0.192041\pi$
$631$			28.5045i			1.13475i	−0.823460	+	0.567374i	$0.807959\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−18.8657			−0.748661
$636$		0			0
$637$		−	5.27053i		−	0.208826i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		−	38.8841i		−	1.53583i	−0.640552	−	0.767915i	$-0.721294\pi$
$641$		−	38.8841i		−	1.53583i	0.640552	−	0.767915i	$-0.278706\pi$
$642$		0			0
$643$		−	29.7376i		−	1.17274i	−0.810045	−	0.586368i	$-0.800557\pi$
$643$		−	29.7376i		−	1.17274i	0.810045	−	0.586368i	$-0.199443\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			16.0309i			0.630239i	0.949052	+	0.315119i	$0.102045\pi$
$647$			16.0309i			0.630239i	−0.949052	+	0.315119i	$0.897955\pi$
$648$		0			0
$649$			3.09990i			0.121682i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−32.8718			−1.28637			−0.643186	−	0.765710i	$-0.722388\pi$
$653$	−32.8718			−1.28637			−0.643186	+	0.765710i	$0.722388\pi$
$654$		0			0
$655$		−	5.58301i		−	0.218146i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	27.2217			1.06041			0.530203	−	0.847871i	$-0.322116\pi$
$659$	27.2217			1.06041			0.530203	+	0.847871i	$0.322116\pi$
$660$		0			0
$661$			29.3882i			1.14307i	0.820578	+	0.571534i	$0.193652\pi$
$661$			29.3882i			1.14307i	−0.820578	+	0.571534i	$0.806348\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−13.5872	+	0.235899i	−0.526890	+	0.00914778i
$666$		0			0
$667$	−15.4768			−0.599265
$668$		0			0
$669$		0			0
$670$		0			0
$671$		−	29.1744i		−	1.12626i
$672$		0			0
$673$		−	5.37933i		−	0.207358i	−0.994611	−	0.103679i	$-0.966939\pi$
$673$		−	5.37933i		−	0.207358i	0.994611	−	0.103679i	$-0.0330615\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		−	9.76891i		−	0.375450i	−0.982222	−	0.187725i	$-0.939889\pi$
$677$		−	9.76891i		−	0.375450i	0.982222	−	0.187725i	$-0.0601113\pi$
$678$		0			0
$679$	−16.4374			−0.630810
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−11.2959			−0.432227			−0.216113	−	0.976368i	$-0.569338\pi$
$683$	−11.2959			−0.432227			−0.216113	+	0.976368i	$0.569338\pi$
$684$		0			0
$685$	3.76163			0.143724
$686$		0			0
$687$		0			0
$688$		0			0
$689$	42.8959			1.63420
$690$		0			0
$691$		−	25.4168i		−	0.966901i	−0.875372	−	0.483450i	$-0.839383\pi$
$691$		−	25.4168i		−	0.966901i	0.875372	−	0.483450i	$-0.160617\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		−	8.02670i		−	0.304470i
$696$		0			0
$697$		−	16.5871i		−	0.628281i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−5.38467			−0.203376			−0.101688	−	0.994816i	$-0.532424\pi$
$701$	−5.38467			−0.203376			−0.101688	+	0.994816i	$0.532424\pi$
$702$		0			0
$703$	−18.4773	+	0.320799i	−0.696883	+	0.0120992i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		−	12.7597i		−	0.479879i
$708$		0			0
$709$	34.6931			1.30293			0.651463	−	0.758681i	$-0.274156\pi$
$709$	34.6931			1.30293			0.651463	+	0.758681i	$0.274156\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		−	2.31333i		−	0.0866349i
$714$		0			0
$715$	31.5434			1.17966
$716$		0			0
$717$		0			0
$718$		0			0
$719$			3.34252i			0.124655i	0.998056	+	0.0623275i	$0.0198523\pi$
$719$			3.34252i			0.124655i	−0.998056	+	0.0623275i	$0.980148\pi$
$720$		0			0
$721$		−	22.8149i		−	0.849673i
$722$		0			0
$723$		0			0
$724$		0			0
$725$		−	15.1047i		−	0.560976i
$726$		0			0
$727$			23.6836i			0.878375i	0.898395	+	0.439187i	$0.144734\pi$
$727$			23.6836i			0.878375i	−0.898395	+	0.439187i	$0.855266\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$			4.40500i			0.162925i
$732$		0			0
$733$	−11.3477			−0.419136			−0.209568	−	0.977794i	$-0.567206\pi$
$733$	−11.3477			−0.419136			−0.209568	+	0.977794i	$0.567206\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			15.0070i			0.552792i
$738$		0			0
$739$		−	29.0415i		−	1.06831i	−0.845387	−	0.534154i	$-0.820630\pi$
$739$		−	29.0415i		−	1.06831i	0.845387	−	0.534154i	$-0.179370\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	15.2585			0.559781			0.279891	−	0.960032i	$-0.409702\pi$
$743$	15.2585			0.559781			0.279891	+	0.960032i	$0.409702\pi$
$744$		0			0
$745$	−11.8863			−0.435480
$746$		0			0
$747$		0			0
$748$		0			0
$749$		−	45.3948i		−	1.65869i
$750$		0			0
$751$	−20.4437			−0.746003			−0.373001	−	0.927831i	$-0.621671\pi$
$751$	−20.4437			−0.746003			−0.373001	+	0.927831i	$0.621671\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−2.06750			−0.0752439
$756$		0			0
$757$	−9.92991			−0.360909			−0.180454	−	0.983583i	$-0.557757\pi$
$757$	−9.92991			−0.360909			−0.180454	+	0.983583i	$0.557757\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−25.5787			−0.927226			−0.463613	−	0.886038i	$-0.653447\pi$
$761$	−25.5787			−0.927226			−0.463613	+	0.886038i	$0.653447\pi$
$762$		0			0
$763$	−38.0431			−1.37725
$764$		0			0
$765$		0			0
$766$		0			0
$767$			3.62389i			0.130851i
$768$		0			0
$769$	−36.1768			−1.30457			−0.652285	−	0.757974i	$-0.726189\pi$
$769$	−36.1768			−1.30457			−0.652285	+	0.757974i	$0.726189\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		−	13.9243i		−	0.500821i	−0.968140	−	0.250411i	$-0.919434\pi$
$773$		−	13.9243i		−	0.500821i	0.968140	−	0.250411i	$-0.0805656\pi$
$774$		0			0
$775$	2.25771			0.0810995
$776$		0			0
$777$		0			0
$778$		0			0
$779$	31.3502	−	0.544297i	1.12324	−	0.0195015i
$780$		0			0
$781$			25.2656i			0.904074i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−10.3833			−0.370595
$786$		0			0
$787$	−6.36142			−0.226760			−0.113380	−	0.993552i	$-0.536168\pi$
$787$	−6.36142			−0.226760			−0.113380	+	0.993552i	$0.536168\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−12.3617			−0.439531
$792$		0			0
$793$		−	34.1059i		−	1.21114i
$794$		0			0
$795$		0			0
$796$		0			0
$797$		−	48.1621i		−	1.70599i	−0.521920	−	0.852995i	$-0.674784\pi$
$797$		−	48.1621i		−	1.70599i	0.521920	−	0.852995i	$-0.325216\pi$
$798$		0			0
$799$		−	4.55447i		−	0.161126i
$800$		0			0
$801$		0			0
$802$		0			0
$803$			26.6691i			0.941134i
$804$		0			0
$805$	10.8160			0.381215
$806$		0			0
$807$		0			0
$808$		0			0
$809$	11.4744			0.403417			0.201708	−	0.979446i	$-0.435351\pi$
$809$	11.4744			0.403417			0.201708	+	0.979446i	$0.435351\pi$
$810$		0			0
$811$	−37.3279			−1.31076			−0.655380	−	0.755299i	$-0.727492\pi$
$811$	−37.3279			−1.31076			−0.655380	+	0.755299i	$0.727492\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$			5.47417i			0.191752i
$816$		0			0
$817$	−8.32559	+	0.144548i	−0.291276	+	0.00505708i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	39.9825			1.39540			0.697700	−	0.716390i	$-0.254207\pi$
$821$	39.9825			1.39540			0.697700	+	0.716390i	$0.254207\pi$
$822$		0			0
$823$			29.2330i			1.01900i	0.860471	+	0.509499i	$0.170169\pi$
$823$			29.2330i			1.01900i	−0.860471	+	0.509499i	$0.829831\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−41.9912			−1.46018			−0.730089	−	0.683352i	$-0.760522\pi$
$827$	−41.9912			−1.46018			−0.730089	+	0.683352i	$0.760522\pi$
$828$		0			0
$829$		−	6.20190i		−	0.215401i	−0.994183	−	0.107700i	$-0.965651\pi$
$829$		−	6.20190i		−	0.215401i	0.994183	−	0.107700i	$-0.0343487\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−2.25583			−0.0781600
$834$		0			0
$835$	17.4141			0.602640
$836$		0			0
$837$		0			0
$838$		0			0
$839$	43.1621			1.49012			0.745060	−	0.666998i	$-0.232421\pi$
$839$	43.1621			1.49012			0.745060	+	0.666998i	$0.232421\pi$
$840$		0			0
$841$	9.09935			0.313771
$842$		0			0
$843$		0			0
$844$		0			0
$845$	20.3594			0.700386
$846$		0			0
$847$		−	25.1241i		−	0.863276i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	14.7087			0.504208
$852$		0			0
$853$	19.3461			0.662398			0.331199	−	0.943561i	$-0.392547\pi$
$853$	19.3461			0.662398			0.331199	+	0.943561i	$0.392547\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	28.2678i		−	0.965611i	−0.875728	−	0.482805i	$-0.839618\pi$
$857$		−	28.2678i		−	0.965611i	0.875728	−	0.482805i	$-0.160382\pi$
$858$		0			0
$859$			1.99717i			0.0681425i	0.999419	+	0.0340713i	$0.0108473\pi$
$859$			1.99717i			0.0681425i	−0.999419	+	0.0340713i	$0.989153\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−40.2297			−1.36943			−0.684717	−	0.728809i	$-0.740074\pi$
$863$	−40.2297			−1.36943			−0.684717	+	0.728809i	$0.740074\pi$
$864$		0			0
$865$		−	10.1155i		−	0.343937i
$866$		0			0
$867$		0			0
$868$		0			0
$869$		−	31.5769i		−	1.07117i
$870$		0			0
$871$			17.5438i			0.594448i
$872$		0			0
$873$		0			0
$874$		0			0
$875$			26.1440i			0.883827i
$876$		0			0
$877$		−	20.9090i		−	0.706046i	−0.935615	−	0.353023i	$-0.885154\pi$
$877$		−	20.9090i		−	0.706046i	0.935615	−	0.353023i	$-0.114846\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	22.7977			0.768072			0.384036	−	0.923318i	$-0.374534\pi$
$881$	22.7977			0.768072			0.384036	+	0.923318i	$0.374534\pi$
$882$		0			0
$883$		−	50.6844i		−	1.70567i	−0.522183	−	0.852834i	$-0.674882\pi$
$883$		−	50.6844i		−	1.70567i	0.522183	−	0.852834i	$-0.325118\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	2.39327			0.0803581			0.0401790	−	0.999192i	$-0.487207\pi$
$887$	2.39327			0.0803581			0.0401790	+	0.999192i	$0.487207\pi$
$888$		0			0
$889$			36.4395i			1.22214i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	8.60811	−	0.149453i	0.288059	−	0.00500124i
$894$		0			0
$895$	24.6009			0.822317
$896$		0			0
$897$		0			0
$898$		0			0
$899$		−	2.97456i		−	0.0992072i
$900$		0			0
$901$		−	18.3598i		−	0.611654i
$902$		0			0
$903$		0			0
$904$		0			0
$905$		−	17.7657i		−	0.590552i
$906$		0			0
$907$	25.3541			0.841870			0.420935	−	0.907091i	$-0.361702\pi$
$907$	25.3541			0.841870			0.420935	+	0.907091i	$0.361702\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−3.43807			−0.113908			−0.0569541	−	0.998377i	$-0.518139\pi$
$911$	−3.43807			−0.113908			−0.0569541	+	0.998377i	$0.518139\pi$
$912$		0			0
$913$	34.6195			1.14574
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−10.7837			−0.356110
$918$		0			0
$919$			36.8423i			1.21532i	0.794198	+	0.607658i	$0.207891\pi$
$919$			36.8423i			1.21532i	−0.794198	+	0.607658i	$0.792109\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$			29.5364i			0.972201i
$924$		0			0
$925$			14.3551i			0.471993i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	35.2384			1.15614			0.578068	−	0.815989i	$-0.303807\pi$
$929$	35.2384			1.15614			0.578068	+	0.815989i	$0.303807\pi$
$930$		0			0
$931$	−0.0740240	−	4.26360i	−0.00242604	−	0.139734i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		−	13.5009i		−	0.441526i
$936$		0			0
$937$	−31.9499			−1.04376			−0.521879	−	0.853019i	$-0.674769\pi$
$937$	−31.9499			−1.04376			−0.521879	+	0.853019i	$0.674769\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$			12.2575i			0.399582i	0.979839	+	0.199791i	$0.0640263\pi$
$941$			12.2575i			0.399582i	−0.979839	+	0.199791i	$0.935974\pi$
$942$		0			0
$943$	−24.9561			−0.812683
$944$		0			0
$945$		0			0
$946$		0			0
$947$			38.6087i			1.25461i	0.778772	+	0.627307i	$0.215843\pi$
$947$			38.6087i			1.25461i	−0.778772	+	0.627307i	$0.784157\pi$
$948$		0			0
$949$			31.1772i			1.01205i
$950$		0			0
$951$		0			0
$952$		0			0
$953$			29.5848i			0.958347i	0.877720	+	0.479173i	$0.159063\pi$
$953$			29.5848i			0.958347i	−0.877720	+	0.479173i	$0.840937\pi$
$954$		0			0
$955$			25.9373i			0.839313i
$956$		0			0
$957$		0			0
$958$		0			0
$959$		−	7.26568i		−	0.234621i
$960$		0			0
$961$	−30.5554			−0.985658
$962$		0			0
$963$		0			0
$964$		0			0
$965$			10.5888i			0.340867i
$966$		0			0
$967$			22.2213i			0.714587i	0.933992	+	0.357294i	$0.116300\pi$
$967$			22.2213i			0.714587i	−0.933992	+	0.357294i	$0.883700\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	30.8520			0.990088			0.495044	−	0.868868i	$-0.335152\pi$
$971$	30.8520			0.990088			0.495044	+	0.868868i	$0.335152\pi$
$972$		0			0
$973$	−15.5038			−0.497028
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	25.3525i		−	0.811097i	−0.914073	−	0.405549i	$-0.867080\pi$
$977$		−	25.3525i		−	0.811097i	0.914073	−	0.405549i	$-0.132920\pi$
$978$		0			0
$979$	−8.32789			−0.266161
$980$		0			0
$981$		0			0
$982$		0			0
$983$	37.1901			1.18618			0.593090	−	0.805136i	$-0.297908\pi$
$983$	37.1901			1.18618			0.593090	+	0.805136i	$0.297908\pi$
$984$		0			0
$985$	32.6567			1.04053
$986$		0			0
$987$		0			0
$988$		0			0
$989$	6.62754			0.210743
$990$		0			0
$991$	−21.8833			−0.695147			−0.347573	−	0.937653i	$-0.612994\pi$
$991$	−21.8833			−0.695147			−0.347573	+	0.937653i	$0.612994\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$			3.11759i			0.0988344i
$996$		0			0
$997$	35.5057			1.12448			0.562238	−	0.826976i	$-0.309941\pi$
$997$	35.5057			1.12448			0.562238	+	0.826976i	$0.309941\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5472.2.k.d.2431.18	yes	40
3.2	odd	2	inner	5472.2.k.d.2431.24	yes	40
4.3	odd	2	inner	5472.2.k.d.2431.17	✓	40
12.11	even	2	inner	5472.2.k.d.2431.22	yes	40
19.18	odd	2	inner	5472.2.k.d.2431.20	yes	40
57.56	even	2	inner	5472.2.k.d.2431.23	yes	40
76.75	even	2	inner	5472.2.k.d.2431.19	yes	40
228.227	odd	2	inner	5472.2.k.d.2431.21	yes	40

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5472.2.k.d.2431.17	✓	40	4.3	odd	2	inner
5472.2.k.d.2431.18	yes	40	1.1	even	1	trivial
5472.2.k.d.2431.19	yes	40	76.75	even	2	inner
5472.2.k.d.2431.20	yes	40	19.18	odd	2	inner
5472.2.k.d.2431.21	yes	40	228.227	odd	2	inner
5472.2.k.d.2431.22	yes	40	12.11	even	2	inner
5472.2.k.d.2431.23	yes	40	57.56	even	2	inner
5472.2.k.d.2431.24	yes	40	3.2	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-1.27046 q^{5} +2.45392i q^{7} +O(q^{10})\)	\(q-1.27046 q^{5} +2.45392i q^{7} -4.60851i q^{11} -5.38751i q^{13} -2.30591 q^{17} +(-0.0756671 - 4.35824i) q^{19} +3.46935i q^{23} -3.38594 q^{25} +4.46101i q^{29} -0.666790 q^{31} -3.11759i q^{35} -4.23961i q^{37} +7.19331i q^{41} -1.91031i q^{43} +1.97513i q^{47} +0.978285 q^{49} +7.96209i q^{53} +5.85491i q^{55} -0.672646 q^{59} +6.33054 q^{61} +6.84460i q^{65} -3.25638 q^{67} -5.48237 q^{71} -5.78693 q^{73} +11.3089 q^{77} +6.85186 q^{79} +7.51209i q^{83} +2.92955 q^{85} -1.80707i q^{89} +13.2205 q^{91} +(0.0961316 + 5.53695i) q^{95} +6.69843i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q+O(q^{10}) \) 40 * q	\( 40 q + 40 q^{25} - 24 q^{49} + 48 q^{61} + 16 q^{73} - 16 q^{85}+O(q^{100}) \) 40 * q + 40 * q^25 - 24 * q^49 + 48 * q^61 + 16 * q^73 - 16 * q^85

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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