LMFDB - Embedded newform 2548.2.i.m.1745.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2548,2,Mod(165,2548)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2548, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 4, 4]))

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

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

chi := DirichletCharacter("2548.165");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2548 = 2^{2} \cdot 7^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2548.i (of order $3$, degree $2$, not minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$20.3458824350$
Analytic rank:	$0$
Dimension:	$12$
Relative dimension:	$6$ over $\Q(\zeta_{3})$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} + 9x^{10} + 66x^{8} + 127x^{6} + 189x^{4} + 60x^{2} + 16 $ x^12 + 9x^10 + 66x^8 + 127x^6 + 189x^4 + 60*x^2 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{9}]$
Coefficient ring index:	$ 3^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			1745.3
Root			$-0.286958 + 0.497025i$ of defining polynomial
Character	$\chi$	$=$	2548.1745
Dual form			2548.2.i.m.165.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-0.286958 + 0.497025i) q^{3} +(-1.45546 + 2.52093i) q^{5} +(1.33531 + 2.31283i) q^{9} +(-2.73673 + 4.74016i) q^{11} +(3.60003 - 0.199516i) q^{13} +(-0.835311 - 1.44680i) q^{15} -6.05222 q^{17} +(-0.286958 - 0.497025i) q^{19} +5.34124 q^{23} +(-1.73673 - 3.00811i) q^{25} -3.25446 q^{27} +(4.13815 + 7.16749i) q^{29} +(4.53815 + 7.86030i) q^{31} +(-1.57065 - 2.72045i) q^{33} -2.80284 q^{37} +(-0.933890 + 1.84656i) q^{39} +(-5.22725 - 9.05387i) q^{41} +(-3.63815 + 6.30146i) q^{43} -7.77397 q^{45} +(1.28369 - 2.22342i) q^{47} +(1.73673 - 3.00811i) q^{51} +(3.17062 + 5.49168i) q^{53} +(-7.96641 - 13.7982i) q^{55} +0.329378 q^{57} -4.17598 q^{59} +(-4.30981 - 7.46480i) q^{61} +(-4.73673 + 9.36581i) q^{65} +(7.24266 - 12.5447i) q^{67} +(-1.53271 + 2.65473i) q^{69} +(6.14408 - 10.6419i) q^{71} +(1.51204 + 2.61892i) q^{73} +1.99347 q^{75} +(1.67062 - 2.89360i) q^{79} +(-3.07204 + 5.32093i) q^{81} +13.5240 q^{83} +(8.80877 - 15.2572i) q^{85} -4.74989 q^{87} -14.9740 q^{89} -5.20902 q^{93} +1.67062 q^{95} +(4.13804 - 7.16729i) q^{97} -14.6175 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 6 q^{11} + 6 q^{15} + 6 q^{25} + 12 q^{29} - 12 q^{37} - 6 q^{39} - 6 q^{43} - 6 q^{51} + 6 q^{53} + 36 q^{57} - 30 q^{65} + 12 q^{67} - 12 q^{71} - 12 q^{79} + 6 q^{81} + 36 q^{85} + 12 q^{93} - 12 q^{95}+ \cdots - 36 q^{99}+O(q^{100}) $ 12 * q - 6 * q^11 + 6 * q^15 + 6 * q^25 + 12 * q^29 - 12 * q^37 - 6 * q^39 - 6 * q^43 - 6 * q^51 + 6 * q^53 + 36 * q^57 - 30 * q^65 + 12 * q^67 - 12 * q^71 - 12 * q^79 + 6 * q^81 + 36 * q^85 + 12 * q^93 - 12 * q^95 - 36 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2548\mathbb{Z}\right)^\times$.

$n$	$197$	$885$	$1275$
$\chi(n)$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{1}{3}\right)$	$1$

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
$3$	−0.286958	+	0.497025i	−0.165675	+	0.286958i	−0.936895	−	0.349611i	$-0.886314\pi$
$3$	−0.286958	+	0.497025i	−0.165675	+	0.286958i	0.771220	+	0.636569i	$0.219647\pi$
$4$		0			0
$5$	−1.45546	+	2.52093i	−0.650902	+	1.12739i	0.332003	+	0.943278i	$0.392276\pi$
$5$	−1.45546	+	2.52093i	−0.650902	+	1.12739i	−0.982904	+	0.184116i	$0.941058\pi$
$6$		0			0
$7$		0			0
$7$		0			0
$8$		0			0
$9$	1.33531	+	2.31283i	0.445104	+	0.770942i
$10$		0			0
$11$	−2.73673	+	4.74016i	−0.825155	+	1.42921i	0.0766449	+	0.997058i	$0.475579\pi$
$11$	−2.73673	+	4.74016i	−0.825155	+	1.42921i	−0.901800	+	0.432153i	$0.857754\pi$
$12$		0			0
$13$	3.60003	−	0.199516i	0.998468	−	0.0553358i
$13$	3.60003	−	0.199516i	0.998468	−	0.0553358i
$14$		0			0
$15$	−0.835311	−	1.44680i	−0.215676	−	0.373562i
$16$		0			0
$17$	−6.05222			−1.46788			−0.733940	−	0.679214i	$-0.762321\pi$
$17$	−6.05222			−1.46788			−0.733940	+	0.679214i	$0.762321\pi$
$18$		0			0
$19$	−0.286958	−	0.497025i	−0.0658326	−	0.114025i	0.831230	−	0.555928i	$-0.187637\pi$
$19$	−0.286958	−	0.497025i	−0.0658326	−	0.114025i	−0.897063	+	0.441903i	$0.854304\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	5.34124			1.11373			0.556863	−	0.830604i	$-0.312005\pi$
$23$	5.34124			1.11373			0.556863	+	0.830604i	$0.312005\pi$
$24$		0			0
$25$	−1.73673	−	3.00811i	−0.347346	−	0.601621i
$26$		0			0
$27$	−3.25446			−0.626320
$28$		0			0
$29$	4.13815	+	7.16749i	0.768435	+	1.33097i	0.938411	+	0.345521i	$0.112298\pi$
$29$	4.13815	+	7.16749i	0.768435	+	1.33097i	−0.169976	+	0.985448i	$0.554369\pi$
$30$		0			0
$31$	4.53815	+	7.86030i	0.815076	+	1.41175i	0.909274	+	0.416199i	$0.136638\pi$
$31$	4.53815	+	7.86030i	0.815076	+	1.41175i	−0.0941981	+	0.995553i	$0.530029\pi$
$32$		0			0
$33$	−1.57065	−	2.72045i	−0.273415	−	0.473569i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−2.80284			−0.460784			−0.230392	−	0.973098i	$-0.574001\pi$
$37$	−2.80284			−0.460784			−0.230392	+	0.973098i	$0.574001\pi$
$38$		0			0
$39$	−0.933890	+	1.84656i	−0.149542	+	0.295686i
$40$		0			0
$41$	−5.22725	−	9.05387i	−0.816360	−	1.41398i	−0.908347	−	0.418217i	$-0.862655\pi$
$41$	−5.22725	−	9.05387i	−0.816360	−	1.41398i	0.0919873	−	0.995760i	$-0.470678\pi$
$42$		0			0
$43$	−3.63815	+	6.30146i	−0.554813	+	0.960964i	0.443105	+	0.896470i	$0.353877\pi$
$43$	−3.63815	+	6.30146i	−0.554813	+	0.960964i	−0.997918	+	0.0644945i	$0.979457\pi$
$44$		0			0
$45$	−7.77397			−1.15887
$46$		0			0
$47$	1.28369	−	2.22342i	0.187246	−	0.324320i	−0.757085	−	0.653316i	$-0.773377\pi$
$47$	1.28369	−	2.22342i	0.187246	−	0.324320i	0.944331	+	0.328997i	$0.106711\pi$
$48$		0			0
$49$		0			0
$50$		0			0
$51$	1.73673	−	3.00811i	0.243191	−	0.421219i
$52$		0			0
$53$	3.17062	+	5.49168i	0.435518	+	0.754340i	0.997338	−	0.0729200i	$-0.0232318\pi$
$53$	3.17062	+	5.49168i	0.435518	+	0.754340i	−0.561819	+	0.827260i	$0.689898\pi$
$54$		0			0
$55$	−7.96641	−	13.7982i	−1.07419	−	1.86055i
$56$		0			0
$57$	0.329378			0.0436272
$58$		0			0
$59$	−4.17598			−0.543666			−0.271833	−	0.962344i	$-0.587630\pi$
$59$	−4.17598			−0.543666			−0.271833	+	0.962344i	$0.587630\pi$
$60$		0			0
$61$	−4.30981	−	7.46480i	−0.551814	−	0.955770i	−0.998144	−	0.0609014i	$-0.980602\pi$
$61$	−4.30981	−	7.46480i	−0.551814	−	0.955770i	0.446330	−	0.894869i	$-0.352731\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−4.73673	+	9.36581i	−0.587519	+	1.16169i
$66$		0			0
$67$	7.24266	−	12.5447i	0.884832	−	1.53257i	0.0389263	−	0.999242i	$-0.487606\pi$
$67$	7.24266	−	12.5447i	0.884832	−	1.53257i	0.845906	−	0.533332i	$-0.179060\pi$
$68$		0			0
$69$	−1.53271	+	2.65473i	−0.184517	+	0.319592i
$70$		0			0
$71$	6.14408	−	10.6419i	0.729169	−	1.26296i	−0.228066	−	0.973646i	$-0.573240\pi$
$71$	6.14408	−	10.6419i	0.729169	−	1.26296i	0.957235	−	0.289311i	$-0.0934263\pi$
$72$		0			0
$73$	1.51204	+	2.61892i	0.176971	+	0.306522i	0.940841	−	0.338847i	$-0.110037\pi$
$73$	1.51204	+	2.61892i	0.176971	+	0.306522i	−0.763871	+	0.645369i	$0.776704\pi$
$74$		0			0
$75$	1.99347			0.230186
$76$		0			0
$77$		0			0
$78$		0			0
$79$	1.67062	−	2.89360i	0.187960	−	0.325556i	−0.756610	−	0.653866i	$-0.773146\pi$
$79$	1.67062	−	2.89360i	0.187960	−	0.325556i	0.944570	+	0.328311i	$0.106479\pi$
$80$		0			0
$81$	−3.07204	+	5.32093i	−0.341338	+	0.591215i
$82$		0			0
$83$	13.5240			1.48445			0.742226	−	0.670149i	$-0.233770\pi$
$83$	13.5240			1.48445			0.742226	+	0.670149i	$0.233770\pi$
$84$		0			0
$85$	8.80877	−	15.2572i	0.955446	−	1.65488i
$86$		0			0
$87$	−4.74989			−0.509242
$88$		0			0
$89$	−14.9740			−1.58724			−0.793622	−	0.608412i	$-0.791807\pi$
$89$	−14.9740			−1.58724			−0.793622	+	0.608412i	$0.791807\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$	−5.20902			−0.540151
$94$		0			0
$95$	1.67062			0.171402
$96$		0			0
$97$	4.13804	−	7.16729i	0.420154	−	0.727728i	−0.575800	−	0.817591i	$-0.695309\pi$
$97$	4.13804	−	7.16729i	0.420154	−	0.727728i	0.995954	+	0.0898622i	$0.0286427\pi$
$98$		0			0
$99$	−14.6175			−1.46912
$100$		0			0
$101$	−8.42513	+	14.5928i	−0.838332	+	1.45203i	0.0529563	+	0.998597i	$0.483136\pi$
$101$	−8.42513	+	14.5928i	−0.838332	+	1.45203i	−0.891288	+	0.453437i	$0.850198\pi$
$102$		0			0
$103$	3.00544	−	5.20557i	0.296135	−	0.512920i	−0.679114	−	0.734033i	$-0.737636\pi$
$103$	3.00544	−	5.20557i	0.296135	−	0.512920i	0.975248	+	0.221113i	$0.0709689\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−14.4204			−1.39407			−0.697036	−	0.717036i	$-0.745498\pi$
$107$	−14.4204			−1.39407			−0.697036	+	0.717036i	$0.745498\pi$
$108$		0			0
$109$	−2.23080	−	3.86386i	−0.213672	−	0.370090i	0.739189	−	0.673498i	$-0.235209\pi$
$109$	−2.23080	−	3.86386i	−0.213672	−	0.370090i	−0.952861	+	0.303408i	$0.901876\pi$
$110$		0			0
$111$	0.804296	−	1.39308i	0.0763404	−	0.132225i
$112$		0			0
$113$	−3.17062	+	5.49168i	−0.298267	+	0.516614i	−0.975740	−	0.218934i	$-0.929742\pi$
$113$	−3.17062	+	5.49168i	−0.298267	+	0.516614i	0.677473	+	0.735548i	$0.263075\pi$
$114$		0			0
$115$	−7.77397	+	13.4649i	−0.724926	+	1.25561i
$116$		0			0
$117$	5.26860	+	8.05982i	0.487082	+	0.745131i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−9.47939	−	16.4188i	−0.861763	−	1.49262i
$122$		0			0
$123$	6.00000			0.541002
$124$		0			0
$125$	−4.44363			−0.397450
$126$		0			0
$127$	2.70309	+	4.68189i	0.239861	+	0.415451i	0.960674	−	0.277678i	$-0.0895649\pi$
$127$	2.70309	+	4.68189i	0.239861	+	0.415451i	−0.720814	+	0.693129i	$0.756232\pi$
$128$		0			0
$129$	−2.08799	−	3.61650i	−0.183837	−	0.318415i
$130$		0			0
$131$	−7.16211	+	12.4051i	−0.625757	+	1.08384i	0.362637	+	0.931930i	$0.381876\pi$
$131$	−7.16211	+	12.4051i	−0.625757	+	1.08384i	−0.988394	+	0.151912i	$0.951457\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$	4.73673	−	8.20426i	0.407673	−	0.706110i
$136$		0			0
$137$	0.276303			0.0236062			0.0118031	−	0.999930i	$-0.496243\pi$
$137$	0.276303			0.0236062			0.0118031	+	0.999930i	$0.496243\pi$
$138$		0			0
$139$	3.29240	−	5.70260i	0.279257	−	0.483688i	−0.691943	−	0.721952i	$-0.743245\pi$
$139$	3.29240	−	5.70260i	0.279257	−	0.483688i	0.971200	+	0.238264i	$0.0765784\pi$
$140$		0			0
$141$	0.736731	+	1.27606i	0.0620440	+	0.107463i
$142$		0			0
$143$	−8.90657	+	17.6107i	−0.744805	+	1.47268i
$144$		0			0
$145$	−24.0917			−2.00070
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−0.203092	−	0.351765i	−0.0166379	−	0.0288177i	0.857587	−	0.514340i	$-0.171963\pi$
$149$	−0.203092	−	0.351765i	−0.0166379	−	0.0288177i	−0.874224	+	0.485522i	$0.838630\pi$
$150$		0			0
$151$	−7.24266	−	12.5447i	−0.589400	−	1.02087i	−0.994311	−	0.106514i	$-0.966031\pi$
$151$	−7.24266	−	12.5447i	−0.589400	−	1.02087i	0.404912	−	0.914356i	$-0.367302\pi$
$152$		0			0
$153$	−8.08160	−	13.9977i	−0.653359	−	1.13165i
$154$		0			0
$155$	−26.4204			−2.12214
$156$		0			0
$157$	−6.45437	−	11.1793i	−0.515115	−	0.892205i	−0.999846	−	0.0175421i	$-0.994416\pi$
$157$	−6.45437	−	11.1793i	−0.515115	−	0.892205i	0.484731	−	0.874663i	$-0.338917\pi$
$158$		0			0
$159$	−3.63933			−0.288618
$160$		0			0
$161$		0			0
$162$		0			0
$163$	8.94692	+	15.4965i	0.700777	+	1.21378i	0.968194	+	0.250201i	$0.0804966\pi$
$163$	8.94692	+	15.4965i	0.700777	+	1.21378i	−0.267417	+	0.963581i	$0.586170\pi$
$164$		0			0
$165$	9.14408			0.711866
$166$		0			0
$167$	0.668433	+	1.15776i	0.0517249	+	0.0895901i	0.890729	−	0.454536i	$-0.150195\pi$
$167$	0.668433	+	1.15776i	0.0517249	+	0.0895901i	−0.839004	+	0.544126i	$0.816861\pi$
$168$		0			0
$169$	12.9204	−	1.43653i	0.993876	−	0.110502i
$170$		0			0
$171$	0.766355	−	1.32737i	0.0586046	−	0.101506i
$172$		0			0
$173$	11.8514	+	20.5271i	0.901042	+	1.56065i	0.826144	+	0.563459i	$0.190530\pi$
$173$	11.8514	+	20.5271i	0.901042	+	1.56065i	0.0748978	+	0.997191i	$0.476137\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	1.19833	−	2.07557i	0.0900719	−	0.156009i
$178$		0			0
$179$	3.23080	−	5.59591i	0.241481	−	0.418258i	−0.719655	−	0.694332i	$-0.755700\pi$
$179$	3.23080	−	5.59591i	0.241481	−	0.418258i	0.961136	+	0.276074i	$0.0890334\pi$
$180$		0			0
$181$	−14.5851			−1.08410			−0.542050	−	0.840346i	$-0.682352\pi$
$181$	−14.5851			−1.08410			−0.542050	+	0.840346i	$0.682352\pi$
$182$		0			0
$183$	4.94692			0.365687
$184$		0			0
$185$	4.07942	−	7.06577i	0.299925	−	0.519486i
$186$		0			0
$187$	16.5633	−	28.6885i	1.21123	−	2.09791i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	0.0661096	+	0.114505i	0.00478352	+	0.00828531i	0.868407	−	0.495852i	$-0.165144\pi$
$191$	0.0661096	+	0.114505i	0.00478352	+	0.00828531i	−0.863624	+	0.504137i	$0.831811\pi$
$192$		0			0
$193$	1.20426	−	2.08584i	0.0866846	−	0.150142i	−0.819423	−	0.573189i	$-0.805706\pi$
$193$	1.20426	−	2.08584i	0.0866846	−	0.150142i	0.906108	+	0.423047i	$0.139039\pi$
$194$		0			0
$195$	−3.29580	−	5.04186i	−0.236017	−	0.361055i
$196$		0			0
$197$	−3.53957	−	6.13072i	−0.252184	−	0.436796i	0.711943	−	0.702237i	$-0.247816\pi$
$197$	−3.53957	−	6.13072i	−0.252184	−	0.436796i	−0.964127	+	0.265442i	$0.914482\pi$
$198$		0			0
$199$	−19.3458			−1.37139			−0.685695	−	0.727889i	$-0.740502\pi$
$199$	−19.3458			−1.37139			−0.685695	+	0.727889i	$0.740502\pi$
$200$		0			0
$201$	4.15667	+	7.19957i	0.293189	+	0.507819i
$202$		0			0
$203$		0			0
$204$		0			0
$205$	30.4323			2.12548
$206$		0			0
$207$	7.13222	+	12.3534i	0.495724	+	0.858618i
$208$		0			0
$209$	3.14130			0.217288
$210$		0			0
$211$	5.43982	+	9.42205i	0.374493	+	0.648641i	0.990251	−	0.139295i	$-0.0444835\pi$
$211$	5.43982	+	9.42205i	0.374493	+	0.648641i	−0.615758	+	0.787935i	$0.711150\pi$
$212$		0			0
$213$	3.52618	+	6.10753i	0.241610	+	0.418481i
$214$		0			0
$215$	−10.5904	−	18.3431i	−0.722257	−	1.25099i
$216$		0			0
$217$		0			0
$218$		0			0
$219$	−1.73556			−0.117278
$220$		0			0
$221$	−21.7882	+	1.20752i	−1.46563	+	0.0812263i
$222$		0			0
$223$	0.192440	+	0.333316i	0.0128867	+	0.0223205i	0.872397	−	0.488798i	$-0.162565\pi$
$223$	0.192440	+	0.333316i	0.0128867	+	0.0223205i	−0.859510	+	0.511119i	$0.829231\pi$
$224$		0			0
$225$	4.63815	−	8.03351i	0.309210	−	0.535568i
$226$		0			0
$227$	1.64586			0.109240			0.0546199	−	0.998507i	$-0.482605\pi$
$227$	1.64586			0.109240			0.0546199	+	0.998507i	$0.482605\pi$
$228$		0			0
$229$	4.10010	−	7.10158i	0.270942	−	0.469285i	−0.698161	−	0.715941i	$-0.745998\pi$
$229$	4.10010	−	7.10158i	0.270942	−	0.469285i	0.969103	+	0.246655i	$0.0793315\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−1.93389	+	3.34960i	−0.126693	+	0.219439i	−0.922394	−	0.386251i	$-0.873770\pi$
$233$	−1.93389	+	3.34960i	−0.126693	+	0.219439i	0.795700	+	0.605691i	$0.207103\pi$
$234$		0			0
$235$	3.73673	+	6.47221i	0.243757	+	0.422200i
$236$		0			0
$237$	0.958795	+	1.66068i	0.0622804	+	0.107873i
$238$		0			0
$239$	9.53840			0.616988			0.308494	−	0.951226i	$-0.400175\pi$
$239$	9.53840			0.616988			0.308494	+	0.951226i	$0.400175\pi$
$240$		0			0
$241$	5.93907			0.382569			0.191285	−	0.981535i	$-0.438735\pi$
$241$	5.93907			0.382569			0.191285	+	0.981535i	$0.438735\pi$
$242$		0			0
$243$	−6.64477	−	11.5091i	−0.426262	−	0.738308i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−1.13222	−	1.73205i	−0.0720414	−	0.110208i
$248$		0			0
$249$	−3.88081	+	6.72177i	−0.245937	+	0.425975i
$250$		0			0
$251$	9.60887	−	16.6430i	0.606506	−	1.05050i	−0.385305	−	0.922789i	$-0.625904\pi$
$251$	9.60887	−	16.6430i	0.606506	−	1.05050i	0.991811	−	0.127711i	$-0.0407628\pi$
$252$		0			0
$253$	−14.6175	+	25.3183i	−0.918997	+	1.59175i
$254$		0			0
$255$	5.05549	+	8.75636i	0.316587	+	0.548345i
$256$		0			0
$257$	2.94819			0.183903			0.0919516	−	0.995763i	$-0.470690\pi$
$257$	2.94819			0.183903			0.0919516	+	0.995763i	$0.470690\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	−11.0514	+	19.1416i	−0.684067	+	1.18484i
$262$		0			0
$263$	−6.24266	+	10.8126i	−0.384939	+	0.666734i	−0.991761	−	0.128104i	$-0.959111\pi$
$263$	−6.24266	+	10.8126i	−0.384939	+	0.666734i	0.606822	+	0.794838i	$0.292444\pi$
$264$		0			0
$265$	−18.4589			−1.13392
$266$		0			0
$267$	4.29691	−	7.44246i	0.262967	−	0.455471i
$268$		0			0
$269$	5.43696			0.331497			0.165749	−	0.986168i	$-0.446996\pi$
$269$	5.43696			0.331497			0.165749	+	0.986168i	$0.446996\pi$
$270$		0			0
$271$	4.71943			0.286685			0.143343	−	0.989673i	$-0.454215\pi$
$271$	4.71943			0.286685			0.143343	+	0.989673i	$0.454215\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	19.0119			1.14646
$276$		0			0
$277$	−18.2114			−1.09421			−0.547107	−	0.837062i	$-0.684271\pi$
$277$	−18.2114			−1.09421			−0.547107	+	0.837062i	$0.684271\pi$
$278$		0			0
$279$	−12.1197	+	20.9919i	−0.725586	+	1.25675i
$280$		0			0
$281$	−11.4853			−0.685157			−0.342579	−	0.939489i	$-0.611300\pi$
$281$	−11.4853			−0.685157			−0.342579	+	0.939489i	$0.611300\pi$
$282$		0			0
$283$	2.68054	−	4.64283i	0.159342	−	0.275988i	−0.775290	−	0.631606i	$-0.782396\pi$
$283$	2.68054	−	4.64283i	0.159342	−	0.275988i	0.934631	+	0.355618i	$0.115730\pi$
$284$		0			0
$285$	−0.479397	+	0.830341i	−0.0283971	+	0.0491851i
$286$		0			0
$287$		0			0
$288$		0			0
$289$	19.6294			1.15467
$290$		0			0
$291$	2.37488	+	4.11342i	0.139218	+	0.241133i
$292$		0			0
$293$	6.75996	−	11.7086i	0.394921	−	0.684024i	−0.598170	−	0.801369i	$-0.704105\pi$
$293$	6.75996	−	11.7086i	0.394921	−	0.684024i	0.993091	+	0.117346i	$0.0374385\pi$
$294$		0			0
$295$	6.07797	−	10.5274i	0.353873	−	0.612927i
$296$		0			0
$297$	8.90657	−	15.4266i	0.516812	−	0.895144i
$298$		0			0
$299$	19.2286	−	1.06566i	1.11202	−	0.0616289i
$300$		0			0
$301$		0			0
$302$		0			0
$303$	−4.83531	−	8.37500i	−0.277781	−	0.481131i
$304$		0			0
$305$	25.0910			1.43671
$306$		0			0
$307$	24.0503			1.37262			0.686312	−	0.727307i	$-0.259228\pi$
$307$	24.0503			1.37262			0.686312	+	0.727307i	$0.259228\pi$
$308$		0			0
$309$	1.72487	+	2.98756i	0.0981242	+	0.169956i
$310$		0			0
$311$	12.2969	+	21.2988i	0.697292	+	1.20775i	0.969402	+	0.245480i	$0.0789455\pi$
$311$	12.2969	+	21.2988i	0.697292	+	1.20775i	−0.272109	+	0.962266i	$0.587721\pi$
$312$		0			0
$313$	−14.2111	+	24.6143i	−0.803257	+	1.39128i	0.114204	+	0.993457i	$0.463568\pi$
$313$	−14.2111	+	24.6143i	−0.803257	+	1.39128i	−0.917461	+	0.397825i	$0.869765\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	5.03840	−	8.72677i	0.282985	−	0.490144i	−0.689134	−	0.724634i	$-0.742009\pi$
$317$	5.03840	−	8.72677i	0.282985	−	0.490144i	0.972119	+	0.234490i	$0.0753420\pi$
$318$		0			0
$319$	−45.3000			−2.53631
$320$		0			0
$321$	4.13804	−	7.16729i	0.230963	−	0.400039i
$322$		0			0
$323$	1.73673	+	3.00811i	0.0966343	+	0.167376i
$324$		0			0
$325$	−6.85244	−	10.4828i	−0.380105	−	0.581479i
$326$		0			0
$327$	2.56058			0.141600
$328$		0			0
$329$		0			0
$330$		0			0
$331$	1.13105	+	1.95904i	0.0621681	+	0.107678i	0.895434	−	0.445194i	$-0.146865\pi$
$331$	1.13105	+	1.95904i	0.0621681	+	0.107678i	−0.833266	+	0.552872i	$0.813532\pi$
$332$		0			0
$333$	−3.74266	−	6.48248i	−0.205097	−	0.355238i
$334$		0			0
$335$	21.0828	+	36.5165i	1.15188	+	1.99511i
$336$		0			0
$337$	−0.382454			−0.0208336			−0.0104168	−	0.999946i	$-0.503316\pi$
$337$	−0.382454			−0.0208336			−0.0104168	+	0.999946i	$0.503316\pi$
$338$		0			0
$339$	−1.81967	−	3.15176i	−0.0988308	−	0.171180i
$340$		0			0
$341$	−49.6788			−2.69026
$342$		0			0
$343$		0			0
$344$		0			0
$345$	−4.46160	−	7.72771i	−0.240204	−	0.416046i
$346$		0			0
$347$	2.78864			0.149702			0.0748509	−	0.997195i	$-0.476152\pi$
$347$	2.78864			0.149702			0.0748509	+	0.997195i	$0.476152\pi$
$348$		0			0
$349$	2.50333	+	4.33590i	0.134000	+	0.232095i	0.925215	−	0.379443i	$-0.123884\pi$
$349$	2.50333	+	4.33590i	0.134000	+	0.232095i	−0.791215	+	0.611538i	$0.790551\pi$
$350$		0			0
$351$	−11.7161	+	0.649316i	−0.625361	+	0.0346579i
$352$		0			0
$353$	12.4839	−	21.6227i	0.664450	−	1.15086i	−0.314984	−	0.949097i	$-0.601999\pi$
$353$	12.4839	−	21.6227i	0.664450	−	1.15086i	0.979434	−	0.201764i	$-0.0646675\pi$
$354$		0			0
$355$	17.8849	+	30.9776i	0.949234	+	1.64412i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−5.83531	+	10.1071i	−0.307976	+	0.533430i	−0.977919	−	0.208982i	$-0.932985\pi$
$359$	−5.83531	+	10.1071i	−0.307976	+	0.533430i	0.669944	+	0.742412i	$0.266318\pi$
$360$		0			0
$361$	9.33531	−	16.1692i	0.491332	−	0.851012i
$362$		0			0
$363$	10.8807			0.571090
$364$		0			0
$365$	−8.80284			−0.460762
$366$		0			0
$367$	2.06528	−	3.57717i	0.107807	−	0.186727i	−0.807075	−	0.590449i	$-0.798951\pi$
$367$	2.06528	−	3.57717i	0.107807	−	0.186727i	0.914881	+	0.403723i	$0.132284\pi$
$368$		0			0
$369$	13.9600	−	24.1795i	0.726730	−	1.25873i
$370$		0			0
$371$		0			0
$372$		0			0
$373$	−8.05144	−	13.9455i	−0.416888	−	0.722070i	0.578737	−	0.815514i	$-0.303546\pi$
$373$	−8.05144	−	13.9455i	−0.416888	−	0.722070i	−0.995625	+	0.0934438i	$0.970212\pi$
$374$		0			0
$375$	1.27513	−	2.20860i	0.0658476	−	0.114051i
$376$		0			0
$377$	16.3275	+	24.9775i	0.840908	+	1.28641i
$378$		0			0
$379$	15.3424	+	26.5738i	0.788087	+	1.36501i	0.927138	+	0.374721i	$0.122262\pi$
$379$	15.3424	+	26.5738i	0.788087	+	1.36501i	−0.139051	+	0.990285i	$0.544405\pi$
$380$		0			0
$381$	−3.10269			−0.158956
$382$		0			0
$383$	−4.55338	−	7.88668i	−0.232667	−	0.402991i	0.725925	−	0.687774i	$-0.241412\pi$
$383$	−4.55338	−	7.88668i	−0.232667	−	0.402991i	−0.958592	+	0.284783i	$0.908078\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−19.4323			−0.987797
$388$		0			0
$389$	12.9398	+	22.4124i	0.656075	+	1.13636i	0.981623	+	0.190830i	$0.0611179\pi$
$389$	12.9398	+	22.4124i	0.656075	+	1.13636i	−0.325548	+	0.945525i	$0.605549\pi$
$390$		0			0
$391$	−32.3264			−1.63482
$392$		0			0
$393$	−4.11044	−	7.11950i	−0.207344	−	0.359131i
$394$		0			0
$395$	4.86305	+	8.42305i	0.244686	+	0.423809i
$396$		0			0
$397$	12.6784	+	21.9596i	0.636309	+	1.10212i	0.986236	+	0.165342i	$0.0528729\pi$
$397$	12.6784	+	21.9596i	0.636309	+	1.10212i	−0.349927	+	0.936777i	$0.613794\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	3.35311			0.167446			0.0837231	−	0.996489i	$-0.473319\pi$
$401$	3.35311			0.167446			0.0837231	+	0.996489i	$0.473319\pi$
$402$		0			0
$403$	17.9057	+	27.3919i	0.891947	+	1.36449i
$404$		0			0
$405$	−8.94247	−	15.4888i	−0.444355	−	0.769645i
$406$		0			0
$407$	7.67062	−	13.2859i	0.380219	−	0.658558i
$408$		0			0
$409$	−15.4680			−0.764842			−0.382421	−	0.923988i	$-0.624910\pi$
$409$	−15.4680			−0.764842			−0.382421	+	0.923988i	$0.624910\pi$
$410$		0			0
$411$	−0.0792872	+	0.137330i	−0.00391095	+	0.00677396i
$412$		0			0
$413$		0			0
$414$		0			0
$415$	−19.6837	+	34.0931i	−0.966233	+	1.67356i
$416$		0			0
$417$	1.88956	+	3.27281i	0.0925319	+	0.160270i
$418$		0			0
$419$	9.74881	+	16.8854i	0.476260	+	0.824907i	0.999630	−	0.0271986i	$-0.00865865\pi$
$419$	9.74881	+	16.8854i	0.476260	+	0.824907i	−0.523370	+	0.852106i	$0.675325\pi$
$420$		0			0
$421$	11.1972			0.545716			0.272858	−	0.962054i	$-0.412031\pi$
$421$	11.1972			0.545716			0.272858	+	0.962054i	$0.412031\pi$
$422$		0			0
$423$	6.85652			0.333375
$424$		0			0
$425$	10.5111	+	18.2057i	0.509863	+	0.883108i
$426$		0			0
$427$		0			0
$428$		0			0
$429$	−6.19716	−	9.48031i	−0.299202	−	0.457714i
$430$		0			0
$431$	13.1116	−	22.7100i	0.631564	−	1.09390i	−0.355668	−	0.934612i	$-0.615747\pi$
$431$	13.1116	−	22.7100i	0.631564	−	1.09390i	0.987232	−	0.159289i	$-0.0509201\pi$
$432$		0			0
$433$	−15.2851	+	26.4745i	−0.734553	+	1.27228i	0.220366	+	0.975417i	$0.429275\pi$
$433$	−15.2851	+	26.4745i	−0.734553	+	1.27228i	−0.954919	+	0.296866i	$0.904059\pi$
$434$		0			0
$435$	6.91328	−	11.9742i	0.331467	−	0.574117i
$436$		0			0
$437$	−1.53271	−	2.65473i	−0.0733195	−	0.126993i
$438$		0			0
$439$	24.2848			1.15905			0.579525	−	0.814955i	$-0.303238\pi$
$439$	24.2848			1.15905			0.579525	+	0.814955i	$0.303238\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	2.81471	−	4.87521i	0.133731	−	0.231628i	−0.791381	−	0.611323i	$-0.790638\pi$
$443$	2.81471	−	4.87521i	0.133731	−	0.231628i	0.925112	+	0.379695i	$0.123971\pi$
$444$		0			0
$445$	21.7941	−	37.7485i	1.03314	−	1.78945i
$446$		0			0
$447$	0.233115			0.0110259
$448$		0			0
$449$	5.39432	−	9.34323i	0.254574	−	0.440934i	−0.710206	−	0.703994i	$-0.751398\pi$
$449$	5.39432	−	9.34323i	0.254574	−	0.440934i	0.964780	+	0.263059i	$0.0847315\pi$
$450$		0			0
$451$	57.2224			2.69450
$452$		0			0
$453$	8.31335			0.390595
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−18.0142			−0.842669			−0.421334	−	0.906905i	$-0.638438\pi$
$457$	−18.0142			−0.842669			−0.421334	+	0.906905i	$0.638438\pi$
$458$		0			0
$459$	19.6967			0.919363
$460$		0			0
$461$	−7.35319	+	12.7361i	−0.342472	+	0.593179i	−0.984891	−	0.173175i	$-0.944598\pi$
$461$	−7.35319	+	12.7361i	−0.342472	+	0.593179i	0.642419	+	0.766353i	$0.277931\pi$
$462$		0			0
$463$	34.3792			1.59774			0.798868	−	0.601507i	$-0.205433\pi$
$463$	34.3792			1.59774			0.798868	+	0.601507i	$0.205433\pi$
$464$		0			0
$465$	7.58153	−	13.1316i	0.351585	−	0.608963i
$466$		0			0
$467$	−2.93363	+	5.08120i	−0.135752	+	0.235130i	−0.925885	−	0.377806i	$-0.876678\pi$
$467$	−2.93363	+	5.08120i	−0.135752	+	0.235130i	0.790132	+	0.612936i	$0.210012\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	7.40852			0.341367
$472$		0			0
$473$	−19.9133	−	34.4908i	−0.915614	−	1.58589i
$474$		0			0
$475$	−0.996736	+	1.72640i	−0.0457334	+	0.0792126i
$476$		0			0
$477$	−8.46753	+	14.6662i	−0.387702	+	0.671519i
$478$		0			0
$479$	5.91976	−	10.2533i	0.270481	−	0.468487i	−0.698504	−	0.715606i	$-0.746151\pi$
$479$	5.91976	−	10.2533i	0.270481	−	0.468487i	0.968985	+	0.247119i	$0.0794840\pi$
$480$		0			0
$481$	−10.0903	+	0.559211i	−0.460078	+	0.0254978i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	12.0455	+	20.8634i	0.546958	+	0.947359i
$486$		0			0
$487$	−3.14642			−0.142578			−0.0712890	−	0.997456i	$-0.522711\pi$
$487$	−3.14642			−0.142578			−0.0712890	+	0.997456i	$0.522711\pi$
$488$		0			0
$489$	−10.2695			−0.464405
$490$		0			0
$491$	17.4410	+	30.2087i	0.787101	+	1.36330i	0.927736	+	0.373237i	$0.121752\pi$
$491$	17.4410	+	30.2087i	0.787101	+	1.36330i	−0.140635	+	0.990062i	$0.544914\pi$
$492$		0			0
$493$	−25.0450	−	43.3792i	−1.12797	−	1.95370i
$494$		0			0
$495$	21.2753	−	36.8498i	0.956252	−	1.65628i
$496$		0			0
$497$		0			0
$498$		0			0
$499$	−0.505932	+	0.876300i	−0.0226486	+	0.0392286i	−0.877128	−	0.480257i	$-0.840543\pi$
$499$	−0.505932	+	0.876300i	−0.0226486	+	0.0392286i	0.854479	+	0.519486i	$0.173877\pi$
$500$		0			0
$501$	−0.767247			−0.0342781
$502$		0			0
$503$	3.99810	−	6.92491i	0.178266	−	0.308767i	−0.763020	−	0.646374i	$-0.776284\pi$
$503$	3.99810	−	6.92491i	0.178266	−	0.308767i	0.941287	+	0.337608i	$0.109618\pi$
$504$		0			0
$505$	−24.5249	−	42.4784i	−1.09134	−	1.89026i
$506$		0			0
$507$	−2.99361	+	6.83398i	−0.132951	+	0.303508i
$508$		0			0
$509$	1.83082			0.0811498			0.0405749	−	0.999177i	$-0.487081\pi$
$509$	1.83082			0.0811498			0.0405749	+	0.999177i	$0.487081\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$	0.933890	+	1.61755i	0.0412323	+	0.0714164i
$514$		0			0
$515$	8.74860	+	15.1530i	0.385509	+	0.667721i
$516$		0			0
$517$	7.02625	+	12.1698i	0.309014	+	0.535228i
$518$		0			0
$519$	−13.6033			−0.597120
$520$		0			0
$521$	−11.5058	−	19.9286i	−0.504078	−	0.873088i	−0.999989	−	0.00471473i	$-0.998499\pi$
$521$	−11.5058	−	19.9286i	−0.504078	−	0.873088i	0.495911	−	0.868373i	$-0.334834\pi$
$522$		0			0
$523$	43.8874			1.91906			0.959531	−	0.281604i	$-0.0908664\pi$
$523$	43.8874			1.91906			0.959531	+	0.281604i	$0.0908664\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−27.4659	−	47.5723i	−1.19643	−	2.07228i
$528$		0			0
$529$	5.52888			0.240386
$530$		0			0
$531$	−5.57623	−	9.65832i	−0.241988	−	0.419135i
$532$		0			0
$533$	−20.6246	−	31.5513i	−0.893353	−	1.36664i
$534$		0			0
$535$	20.9883	−	36.3528i	0.907404	−	1.57167i
$536$		0			0
$537$	1.85420	+	3.21158i	0.0800148	+	0.138590i
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−4.20543	+	7.28402i	−0.180806	+	0.313164i	−0.942155	−	0.335177i	$-0.891204\pi$
$541$	−4.20543	+	7.28402i	−0.180806	+	0.313164i	0.761350	+	0.648342i	$0.224537\pi$
$542$		0			0
$543$	4.18529	−	7.24914i	0.179608	−	0.311091i
$544$		0			0
$545$	12.9874			0.556317
$546$		0			0
$547$	−22.4853			−0.961403			−0.480702	−	0.876884i	$-0.659618\pi$
$547$	−22.4853			−0.961403			−0.480702	+	0.876884i	$0.659618\pi$
$548$		0			0
$549$	11.5099	−	19.9357i	0.491229	−	0.850833i
$550$		0			0
$551$	2.37495	−	4.11353i	0.101176	−	0.175242i
$552$		0			0
$553$		0			0
$554$		0			0
$555$	2.34124	+	4.05515i	0.0993802	+	0.172132i
$556$		0			0
$557$	−22.3159	+	38.6522i	−0.945554	+	1.63775i	−0.190916	+	0.981606i	$0.561146\pi$
$557$	−22.3159	+	38.6522i	−0.945554	+	1.63775i	−0.754638	+	0.656141i	$0.772188\pi$
$558$		0			0
$559$	−11.8402	+	23.4113i	−0.500787	+	0.990193i
$560$		0			0
$561$	9.50593	+	16.4648i	0.401341	+	0.695143i
$562$		0			0
$563$	22.5176			0.949004			0.474502	−	0.880254i	$-0.342628\pi$
$563$	22.5176			0.949004			0.474502	+	0.880254i	$0.342628\pi$
$564$		0			0
$565$	−9.22943	−	15.9858i	−0.388285	−	0.672529i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−29.5233			−1.23768			−0.618840	−	0.785517i	$-0.712397\pi$
$569$	−29.5233			−1.23768			−0.618840	+	0.785517i	$0.712397\pi$
$570$		0			0
$571$	16.6045	+	28.7599i	0.694877	+	1.20356i	0.970222	+	0.242217i	$0.0778746\pi$
$571$	16.6045	+	28.7599i	0.694877	+	1.20356i	−0.275345	+	0.961346i	$0.588792\pi$
$572$		0			0
$573$	−0.0758826			−0.00317004
$574$		0			0
$575$	−9.27630	−	16.0670i	−0.386849	−	0.670041i
$576$		0			0
$577$	1.91963	+	3.32489i	0.0799151	+	0.138417i	0.903213	−	0.429192i	$-0.141202\pi$
$577$	1.91963	+	3.32489i	0.0799151	+	0.138417i	−0.823298	+	0.567609i	$0.807868\pi$
$578$		0			0
$579$	0.691144	+	1.19710i	0.0287229	+	0.0497496i
$580$		0			0
$581$		0			0
$582$		0			0
$583$	−34.7086			−1.43748
$584$		0			0
$585$	−27.9865	+	1.55103i	−1.15710	+	0.0641272i
$586$		0			0
$587$	12.0292	+	20.8352i	0.496500	+	0.859963i	0.999992	−	0.00403710i	$-0.00128505\pi$
$587$	12.0292	+	20.8352i	0.496500	+	0.859963i	−0.503492	+	0.864000i	$0.667952\pi$
$588$		0			0
$589$	2.60451	−	4.51115i	0.107317	−	0.185879i
$590$		0			0
$591$	4.06283			0.167122
$592$		0			0
$593$	−13.9566	+	24.1736i	−0.573129	+	0.992689i	0.423113	+	0.906077i	$0.360937\pi$
$593$	−13.9566	+	24.1736i	−0.573129	+	0.992689i	−0.996242	+	0.0866122i	$0.972396\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	5.55144	−	9.61537i	0.227205	−	0.393531i
$598$		0			0
$599$	−13.5851	−	23.5300i	−0.555071	−	0.961411i	−0.997898	−	0.0648043i	$-0.979358\pi$
$599$	−13.5851	−	23.5300i	−0.555071	−	0.961411i	0.442827	−	0.896607i	$-0.353976\pi$
$600$		0			0
$601$	3.83381	+	6.64036i	0.156384	+	0.270866i	0.933562	−	0.358415i	$-0.116683\pi$
$601$	3.83381	+	6.64036i	0.156384	+	0.270866i	−0.777178	+	0.629281i	$0.783349\pi$
$602$		0			0
$603$	38.6848			1.57537
$604$		0			0
$605$	55.1875			2.24369
$606$		0			0
$607$	5.45967	+	9.45643i	0.221601	+	0.383825i	0.955294	−	0.295656i	$-0.0955383\pi$
$607$	5.45967	+	9.45643i	0.221601	+	0.383825i	−0.733693	+	0.679481i	$0.762205\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	4.17772	−	8.26050i	0.169013	−	0.334184i
$612$		0			0
$613$	2.61919	−	4.53656i	0.105788	−	0.183230i	−0.808272	−	0.588809i	$-0.799597\pi$
$613$	2.61919	−	4.53656i	0.105788	−	0.183230i	0.914060	+	0.405579i	$0.132930\pi$
$614$		0			0
$615$	−8.73276	+	15.1256i	−0.352139	+	0.609923i
$616$		0			0
$617$	−22.7209	+	39.3537i	−0.914709	+	1.58432i	−0.107381	+	0.994218i	$0.534246\pi$
$617$	−22.7209	+	39.3537i	−0.914709	+	1.58432i	−0.807327	+	0.590104i	$0.799087\pi$
$618$		0			0
$619$	14.0979	+	24.4183i	0.566643	+	0.981455i	0.996895	+	0.0787458i	$0.0250915\pi$
$619$	14.0979	+	24.4183i	0.566643	+	0.981455i	−0.430252	+	0.902709i	$0.641575\pi$
$620$		0			0
$621$	−17.3828			−0.697549
$622$		0			0
$623$		0			0
$624$		0			0
$625$	15.1512	−	26.2426i	0.606047	−	1.04970i
$626$		0			0
$627$	−0.901420	+	1.56131i	−0.0359993	+	0.0623526i
$628$		0			0
$629$	16.9634			0.676376
$630$		0			0
$631$	−3.48650	+	6.03879i	−0.138795	+	0.240400i	−0.927041	−	0.374960i	$-0.877656\pi$
$631$	−3.48650	+	6.03879i	−0.138795	+	0.240400i	0.788246	+	0.615361i	$0.210990\pi$
$632$		0			0
$633$	−6.24399			−0.248176
$634$		0			0
$635$	−15.7370			−0.624503
$636$		0			0
$637$		0			0
$638$		0			0
$639$	32.8170			1.29822
$640$		0			0
$641$	−0.997661			−0.0394052			−0.0197026	−	0.999806i	$-0.506272\pi$
$641$	−0.997661			−0.0394052			−0.0197026	+	0.999806i	$0.506272\pi$
$642$		0			0
$643$	−1.81626	+	3.14586i	−0.0716264	+	0.124061i	−0.899614	−	0.436685i	$-0.856152\pi$
$643$	−1.81626	+	3.14586i	−0.0716264	+	0.124061i	0.827988	+	0.560746i	$0.189486\pi$
$644$		0			0
$645$	12.1559			0.478640
$646$		0			0
$647$	−16.8503	+	29.1855i	−0.662452	+	1.14740i	0.317517	+	0.948253i	$0.397151\pi$
$647$	−16.8503	+	29.1855i	−0.662452	+	1.14740i	−0.979969	+	0.199149i	$0.936182\pi$
$648$		0			0
$649$	11.4285	−	19.7948i	0.448609	−	0.777014i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	17.2763			0.676074			0.338037	−	0.941133i	$-0.390237\pi$
$653$	17.2763			0.676074			0.338037	+	0.941133i	$0.390237\pi$
$654$		0			0
$655$	−20.8483	−	36.1104i	−0.814612	−	1.41095i
$656$		0			0
$657$	−4.03808	+	6.99416i	−0.157540	+	0.272868i
$658$		0			0
$659$	−5.34124	+	9.25130i	−0.208065	+	0.360380i	−0.951105	−	0.308868i	$-0.900050\pi$
$659$	−5.34124	+	9.25130i	−0.208065	+	0.360380i	0.743040	+	0.669247i	$0.233383\pi$
$660$		0			0
$661$	5.07276	−	8.78627i	0.197307	−	0.341746i	−0.750347	−	0.661044i	$-0.770114\pi$
$661$	5.07276	−	8.78627i	0.197307	−	0.341746i	0.947654	+	0.319298i	$0.103447\pi$
$662$		0			0
$663$	5.65211	−	11.1758i	0.219510	−	0.434031i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	22.1029	+	38.2833i	0.855827	+	1.48234i
$668$		0			0
$669$	−0.220888			−0.00854004
$670$		0			0
$671$	47.1791			1.82133
$672$		0			0
$673$	7.69716	+	13.3319i	0.296704	+	0.513906i	0.975380	−	0.220532i	$-0.0707793\pi$
$673$	7.69716	+	13.3319i	0.296704	+	0.513906i	−0.678676	+	0.734438i	$0.737446\pi$
$674$		0			0
$675$	5.65211	+	9.78975i	0.217550	+	0.376808i
$676$		0			0
$677$	−7.48701	+	12.9679i	−0.287749	+	0.498396i	−0.973272	−	0.229655i	$-0.926240\pi$
$677$	−7.48701	+	12.9679i	−0.287749	+	0.498396i	0.685523	+	0.728051i	$0.259574\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$	−0.472293	+	0.818035i	−0.0180983	+	0.0313472i
$682$		0			0
$683$	−32.8559			−1.25720			−0.628598	−	0.777730i	$-0.716371\pi$
$683$	−32.8559			−1.25720			−0.628598	+	0.777730i	$0.716371\pi$
$684$		0			0
$685$	−0.402148	+	0.696541i	−0.0153653	+	0.0266135i
$686$		0			0
$687$	2.35311	+	4.07570i	0.0897767	+	0.155498i
$688$		0			0
$689$	12.5100	+	19.1376i	0.476593	+	0.729084i
$690$		0			0
$691$	6.16945			0.234697			0.117349	−	0.993091i	$-0.462561\pi$
$691$	6.16945			0.234697			0.117349	+	0.993091i	$0.462561\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	9.58391	+	16.5998i	0.363538	+	0.629667i
$696$		0			0
$697$	31.6365	+	54.7960i	1.19832	+	2.07555i
$698$		0			0
$699$	−1.10989	−	1.92238i	−0.0419799	−	0.0727112i
$700$		0			0
$701$	9.52326			0.359689			0.179844	−	0.983695i	$-0.442441\pi$
$701$	9.52326			0.359689			0.179844	+	0.983695i	$0.442441\pi$
$702$		0			0
$703$	0.804296	+	1.39308i	0.0303346	+	0.0525411i
$704$		0			0
$705$	−4.28913			−0.161538
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−16.4121	−	28.4266i	−0.616370	−	1.06758i	−0.990143	−	0.140064i	$-0.955269\pi$
$709$	−16.4121	−	28.4266i	−0.616370	−	1.06758i	0.373773	−	0.927520i	$-0.378064\pi$
$710$		0			0
$711$	8.92320			0.334646
$712$		0			0
$713$	24.2394	+	41.9838i	0.907771	+	1.57231i
$714$		0			0
$715$	−31.4323	−	48.0846i	−1.17550	−	1.79826i
$716$		0			0
$717$	−2.73712	+	4.74082i	−0.102219	+	0.177049i
$718$		0			0
$719$	1.62723	+	2.81844i	0.0606853	+	0.105110i	0.894772	−	0.446523i	$-0.147338\pi$
$719$	1.62723	+	2.81844i	0.0606853	+	0.105110i	−0.834087	+	0.551634i	$0.814005\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$	−1.70426	+	2.95187i	−0.0633822	+	0.109781i
$724$		0			0
$725$	14.3737	−	24.8960i	0.533826	−	0.924614i
$726$		0			0
$727$	−21.3325			−0.791179			−0.395589	−	0.918427i	$-0.629460\pi$
$727$	−21.3325			−0.791179			−0.395589	+	0.918427i	$0.629460\pi$
$728$		0			0
$729$	−10.8052			−0.400192
$730$		0			0
$731$	22.0189	−	38.1379i	0.814399	−	1.41058i
$732$		0			0
$733$	−7.68489	+	13.3106i	−0.283848	+	0.491639i	−0.972329	−	0.233615i	$-0.924944\pi$
$733$	−7.68489	+	13.3106i	−0.283848	+	0.491639i	0.688481	+	0.725254i	$0.258278\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	39.6424	+	68.6627i	1.46025	+	2.52922i
$738$		0			0
$739$	1.87012	−	3.23914i	0.0687935	−	0.119154i	−0.829577	−	0.558392i	$-0.811418\pi$
$739$	1.87012	−	3.23914i	0.0687935	−	0.119154i	0.898370	+	0.439239i	$0.144752\pi$
$740$		0			0
$741$	1.18577	−	0.0657163i	0.0435604	−	0.00241415i
$742$		0			0
$743$	10.1647	+	17.6058i	0.372906	+	0.645893i	0.990011	−	0.140988i	$-0.0450280\pi$
$743$	10.1647	+	17.6058i	0.372906	+	0.645893i	−0.617105	+	0.786881i	$0.711695\pi$
$744$		0			0
$745$	1.18237			0.0433186
$746$		0			0
$747$	18.0587	+	31.2787i	0.660735	+	1.14443i
$748$		0			0
$749$		0			0
$750$		0			0
$751$	35.9611			1.31224			0.656120	−	0.754656i	$-0.272196\pi$
$751$	35.9611			1.31224			0.656120	+	0.754656i	$0.272196\pi$
$752$		0			0
$753$	5.51467	+	9.55169i	0.200966	+	0.348083i
$754$		0			0
$755$	42.1656			1.53456
$756$		0			0
$757$	−6.73790	−	11.6704i	−0.244893	−	0.424167i	0.717208	−	0.696859i	$-0.245420\pi$
$757$	−6.73790	−	11.6704i	−0.244893	−	0.424167i	−0.962102	+	0.272691i	$0.912086\pi$
$758$		0			0
$759$	−8.38923	−	14.5306i	−0.304510	−	0.527426i
$760$		0			0
$761$	12.0211	+	20.8211i	0.435764	+	0.754765i	0.997358	−	0.0726478i	$-0.0231449\pi$
$761$	12.0211	+	20.8211i	0.435764	+	0.754765i	−0.561594	+	0.827413i	$0.689812\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$	47.0498			1.70109
$766$		0			0
$767$	−15.0336	+	0.833175i	−0.542833	+	0.0300842i
$768$		0			0
$769$	−6.95104	−	12.0396i	−0.250661	−	0.434157i	0.713047	−	0.701116i	$-0.247315\pi$
$769$	−6.95104	−	12.0396i	−0.250661	−	0.434157i	−0.963708	+	0.266959i	$0.913981\pi$
$770$		0			0
$771$	−0.846006	+	1.46532i	−0.0304682	+	0.0527724i
$772$		0			0
$773$	37.0377			1.33215			0.666076	−	0.745884i	$-0.267972\pi$
$773$	37.0377			1.33215			0.666076	+	0.745884i	$0.267972\pi$
$774$		0			0
$775$	15.7631	−	27.3025i	0.566227	−	0.980734i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−3.00000	+	5.19615i	−0.107486	+	0.186171i
$780$		0			0
$781$	33.6294	+	58.2478i	1.20335	+	2.08427i
$782$		0			0
$783$	−13.4674	−	23.3263i	−0.481287	−	0.833613i
$784$		0			0
$785$	37.5763			1.34116
$786$		0			0
$787$	−45.1179			−1.60828			−0.804140	−	0.594440i	$-0.797374\pi$
$787$	−45.1179			−1.60828			−0.804140	+	0.594440i	$0.797374\pi$
$788$		0			0
$789$	−3.58276	−	6.20552i	−0.127550	−	0.220922i
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−17.0048	−	26.0136i	−0.603857	−	0.923770i
$794$		0			0
$795$	5.29691	−	9.17451i	0.187862	−	0.325387i
$796$		0			0
$797$	2.52197	−	4.36817i	0.0893326	−	0.154729i	−0.817897	−	0.575365i	$-0.804860\pi$
$797$	2.52197	−	4.36817i	0.0893326	−	0.154729i	0.907229	+	0.420637i	$0.138193\pi$
$798$		0			0
$799$	−7.76920	+	13.4567i	−0.274855	+	0.476062i
$800$		0			0
$801$	−19.9950	−	34.6323i	−0.706488	−	1.22367i
$802$		0			0
$803$	−16.5522			−0.584113
$804$		0			0
$805$		0			0
$806$		0			0
$807$	−1.56018	+	2.70231i	−0.0549208	+	0.0951257i
$808$		0			0
$809$	15.7763	−	27.3254i	0.554665	−	0.960708i	−0.443264	−	0.896391i	$-0.646180\pi$
$809$	15.7763	−	27.3254i	0.554665	−	0.960708i	0.997929	−	0.0643174i	$-0.0204870\pi$
$810$		0			0
$811$	−10.7526			−0.377576			−0.188788	−	0.982018i	$-0.560456\pi$
$811$	−10.7526			−0.377576			−0.188788	+	0.982018i	$0.560456\pi$
$812$		0			0
$813$	−1.35428	+	2.34568i	−0.0474966	+	0.0822665i
$814$		0			0
$815$	−52.0876			−1.82455
$816$		0			0
$817$	4.17598			0.146099
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−3.14408			−0.109729			−0.0548646	−	0.998494i	$-0.517473\pi$
$821$	−3.14408			−0.109729			−0.0548646	+	0.998494i	$0.517473\pi$
$822$		0			0
$823$	−45.8503			−1.59824			−0.799120	−	0.601171i	$-0.794701\pi$
$823$	−45.8503			−1.59824			−0.799120	+	0.601171i	$0.794701\pi$
$824$		0			0
$825$	−5.45560	+	9.44937i	−0.189940	+	0.328985i
$826$		0			0
$827$	31.5882			1.09843			0.549215	−	0.835681i	$-0.314927\pi$
$827$	31.5882			1.09843			0.549215	+	0.835681i	$0.314927\pi$
$828$		0			0
$829$	22.3340	−	38.6837i	0.775692	−	1.34354i	−0.158712	−	0.987325i	$-0.550734\pi$
$829$	22.3340	−	38.6837i	0.775692	−	1.34354i	0.934404	−	0.356214i	$-0.115933\pi$
$830$		0			0
$831$	5.22589	−	9.05150i	0.181284	−	0.313993i
$832$		0			0
$833$		0			0
$834$		0			0
$835$	−3.89151			−0.134671
$836$		0			0
$837$	−14.7692	−	25.5810i	−0.510498	−	0.884209i
$838$		0			0
$839$	−17.5262	+	30.3562i	−0.605071	+	1.04801i	0.386969	+	0.922093i	$0.373522\pi$
$839$	−17.5262	+	30.3562i	−0.605071	+	1.04801i	−0.992040	+	0.125921i	$0.959811\pi$
$840$		0			0
$841$	−19.7486	+	34.2056i	−0.680986	+	1.17950i
$842$		0			0
$843$	3.29580	−	5.70849i	0.113513	−	0.196611i
$844$		0			0
$845$	−15.1837	+	34.6622i	−0.522336	+	1.19242i
$846$		0			0
$847$		0			0
$848$		0			0
$849$	1.53840	+	2.66459i	0.0527978	+	0.0914485i
$850$		0			0
$851$	−14.9707			−0.513187
$852$		0			0
$853$	−40.4398			−1.38463			−0.692317	−	0.721594i	$-0.743410\pi$
$853$	−40.4398			−1.38463			−0.692317	+	0.721594i	$0.743410\pi$
$854$		0			0
$855$	2.23080	+	3.86386i	0.0762917	+	0.132141i
$856$		0			0
$857$	8.28927	+	14.3574i	0.283156	+	0.490441i	0.972160	−	0.234317i	$-0.0752852\pi$
$857$	8.28927	+	14.3574i	0.283156	+	0.490441i	−0.689004	+	0.724757i	$0.741952\pi$
$858$		0			0
$859$	−5.82592	+	10.0908i	−0.198778	+	0.344293i	−0.948132	−	0.317876i	$-0.897031\pi$
$859$	−5.82592	+	10.0908i	−0.198778	+	0.344293i	0.749355	+	0.662169i	$0.230364\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	5.18529	−	8.98119i	0.176509	−	0.305723i	−0.764173	−	0.645011i	$-0.776853\pi$
$863$	5.18529	−	8.98119i	0.176509	−	0.305723i	0.940683	+	0.339288i	$0.110186\pi$
$864$		0			0
$865$	−68.9967			−2.34596
$866$		0			0
$867$	−5.63281	+	9.75631i	−0.191300	+	0.331342i
$868$		0			0
$869$	9.14408	+	15.8380i	0.310192	+	0.537268i
$870$		0			0
$871$	23.5709	−	46.6061i	0.798670	−	1.57919i
$872$		0			0
$873$	22.1023			0.748048
$874$		0			0
$875$		0			0
$876$		0			0
$877$	8.63222	+	14.9514i	0.291489	+	0.504874i	0.974162	−	0.225850i	$-0.0725159\pi$
$877$	8.63222	+	14.9514i	0.291489	+	0.504874i	−0.682673	+	0.730724i	$0.739183\pi$
$878$		0			0
$879$	3.87965	+	6.71974i	0.130857	+	0.226651i
$880$		0			0
$881$	−20.0309	−	34.6945i	−0.674857	−	1.16889i	−0.976511	−	0.215470i	$-0.930872\pi$
$881$	−20.0309	−	34.6945i	−0.674857	−	1.16889i	0.301653	−	0.953418i	$-0.402462\pi$
$882$		0			0
$883$	−46.3554			−1.55998			−0.779992	−	0.625789i	$-0.784777\pi$
$883$	−46.3554			−1.55998			−0.779992	+	0.625789i	$0.784777\pi$
$884$		0			0
$885$	3.48824	+	6.04181i	0.117256	+	0.203093i
$886$		0			0
$887$	27.8787			0.936075			0.468037	−	0.883709i	$-0.344961\pi$
$887$	27.8787			0.936075			0.468037	+	0.883709i	$0.344961\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	−16.8147	−	29.1239i	−0.563314	−	0.975688i
$892$		0			0
$893$	−1.47346			−0.0493075
$894$		0			0
$895$	9.40460	+	16.2892i	0.314361	+	0.544489i
$896$		0			0
$897$	−4.98814	+	9.86290i	−0.166549	+	0.329313i
$898$		0			0
$899$	−37.5591	+	65.0543i	−1.25267	+	2.16968i
$900$		0			0
$901$	−19.1893	−	33.2369i	−0.639289	−	1.10728i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	21.2280	−	36.7680i	0.705642	−	1.22221i
$906$		0			0
$907$	20.8796	−	36.1646i	0.693297	−	1.20083i	−0.277454	−	0.960739i	$-0.589491\pi$
$907$	20.8796	−	36.1646i	0.693297	−	1.20083i	0.970751	−	0.240087i	$-0.0771760\pi$
$908$		0			0
$909$	−45.0007			−1.49258
$910$		0			0
$911$	−42.8384			−1.41930			−0.709650	−	0.704554i	$-0.751147\pi$
$911$	−42.8384			−1.41930			−0.709650	+	0.704554i	$0.751147\pi$
$912$		0			0
$913$	−37.0116	+	64.1059i	−1.22490	+	2.12160i
$914$		0			0
$915$	−7.20005	+	12.4709i	−0.238026	+	0.412274i
$916$		0			0
$917$		0			0
$918$		0			0
$919$	−23.2232	−	40.2238i	−0.766064	−	1.32686i	−0.939682	−	0.342048i	$-0.888879\pi$
$919$	−23.2232	−	40.2238i	−0.766064	−	1.32686i	0.173619	−	0.984813i	$-0.444454\pi$
$920$		0			0
$921$	−6.90142	+	11.9536i	−0.227410	+	0.393885i
$922$		0			0
$923$	19.9956	−	39.5368i	0.658165	−	1.30137i
$924$		0			0
$925$	4.86778	+	8.43124i	0.160052	+	0.277218i
$926$		0			0
$927$	16.0528			0.527242
$928$		0			0
$929$	12.4873	+	21.6286i	0.409695	+	0.709612i	0.994855	−	0.101305i	$-0.0323019\pi$
$929$	12.4873	+	21.6286i	0.409695	+	0.709612i	−0.585161	+	0.810917i	$0.698969\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$	−14.1147			−0.462096
$934$		0			0
$935$	48.2145	+	83.5099i	1.57678	+	2.73107i
$936$		0			0
$937$	−14.1697			−0.462905			−0.231452	−	0.972846i	$-0.574348\pi$
$937$	−14.1697			−0.462905			−0.231452	+	0.972846i	$0.574348\pi$
$938$		0			0
$939$	−8.15595	−	14.1265i	−0.266159	−	0.461001i
$940$		0			0
$941$	−13.3841	−	23.1819i	−0.436308	−	0.755708i	0.561093	−	0.827753i	$-0.310381\pi$
$941$	−13.3841	−	23.1819i	−0.436308	−	0.755708i	−0.997401	+	0.0720448i	$0.977048\pi$
$942$		0			0
$943$	−27.9200	−	48.3589i	−0.909201	−	1.57478i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−2.36825			−0.0769578			−0.0384789	−	0.999259i	$-0.512251\pi$
$947$	−2.36825			−0.0769578			−0.0384789	+	0.999259i	$0.512251\pi$
$948$		0			0
$949$	5.96589	+	9.12652i	0.193661	+	0.296260i
$950$		0			0
$951$	2.89161	+	5.00842i	0.0937670	+	0.162409i
$952$		0			0
$953$	22.8483	−	39.5745i	0.740130	−	1.28194i	−0.212305	−	0.977203i	$-0.568097\pi$
$953$	22.8483	−	39.5745i	0.740130	−	1.28194i	0.952436	−	0.304740i	$-0.0985696\pi$
$954$		0			0
$955$	−0.384880			−0.0124544
$956$		0			0
$957$	12.9992	−	22.5152i	0.420204	−	0.727815i
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−25.6896	+	44.4957i	−0.828696	+	1.43534i
$962$		0			0
$963$	−19.2557	−	33.3518i	−0.620506	−	1.07475i
$964$		0			0
$965$	3.50551	+	6.07172i	0.112846	+	0.195456i
$966$		0			0
$967$	−44.2968			−1.42449			−0.712244	−	0.701932i	$-0.752321\pi$
$967$	−44.2968			−1.42449			−0.712244	+	0.701932i	$0.752321\pi$
$968$		0			0
$969$	−1.99347			−0.0640396
$970$		0			0
$971$	−9.06174	−	15.6954i	−0.290805	−	0.503689i	0.683195	−	0.730236i	$-0.260590\pi$
$971$	−9.06174	−	15.6954i	−0.290805	−	0.503689i	−0.974000	+	0.226547i	$0.927256\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$	7.17655	−	0.397730i	0.229834	−	0.0127375i
$976$		0			0
$977$	20.7090	−	35.8691i	0.662540	−	1.14755i	−0.317406	−	0.948290i	$-0.602812\pi$
$977$	20.7090	−	35.8691i	0.662540	−	1.14755i	0.979946	−	0.199264i	$-0.0638550\pi$
$978$		0			0
$979$	40.9799	−	70.9792i	1.30972	−	2.26851i
$980$		0			0
$981$	5.95762	−	10.3189i	0.190212	−	0.329457i
$982$		0			0
$983$	−3.87039	−	6.70371i	−0.123446	−	0.213815i	0.797678	−	0.603083i	$-0.206061\pi$
$983$	−3.87039	−	6.70371i	−0.123446	−	0.213815i	−0.921124	+	0.389268i	$0.872728\pi$
$984$		0			0
$985$	20.6068			0.656588
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−19.4323	+	33.6576i	−0.617910	+	1.07025i
$990$		0			0
$991$	−8.68366	+	15.0405i	−0.275845	+	0.477778i	−0.970348	−	0.241712i	$-0.922291\pi$
$991$	−8.68366	+	15.0405i	−0.275845	+	0.477778i	0.694503	+	0.719490i	$0.255624\pi$
$992$		0			0
$993$	−1.29825			−0.0411988
$994$		0			0
$995$	28.1571	−	48.7696i	0.892641	−	1.54610i
$996$		0			0
$997$	4.20917			0.133306			0.0666529	−	0.997776i	$-0.478768\pi$
$997$	4.20917			0.133306			0.0666529	+	0.997776i	$0.478768\pi$
$998$		0			0
$999$	9.12172			0.288598

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2548.2.i.m.1745.3		12
7.2	even	3		2548.2.k.g.393.3	✓	12
7.3	odd	6		2548.2.l.m.1537.3		12
7.4	even	3		2548.2.l.m.1537.4		12
7.5	odd	6		2548.2.k.g.393.4	yes	12
7.6	odd	2	inner	2548.2.i.m.1745.4		12
13.9	even	3		2548.2.l.m.373.4		12
91.9	even	3		2548.2.k.g.1569.3	yes	12
91.48	odd	6		2548.2.l.m.373.3		12
91.61	odd	6		2548.2.k.g.1569.4	yes	12
91.74	even	3	inner	2548.2.i.m.165.3		12
91.87	odd	6	inner	2548.2.i.m.165.4		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2548.2.i.m.165.3		12	91.74	even	3	inner
2548.2.i.m.165.4		12	91.87	odd	6	inner
2548.2.i.m.1745.3		12	1.1	even	1	trivial
2548.2.i.m.1745.4		12	7.6	odd	2	inner
2548.2.k.g.393.3	✓	12	7.2	even	3
2548.2.k.g.393.4	yes	12	7.5	odd	6
2548.2.k.g.1569.3	yes	12	91.9	even	3
2548.2.k.g.1569.4	yes	12	91.61	odd	6
2548.2.l.m.373.3		12	91.48	odd	6
2548.2.l.m.373.4		12	13.9	even	3
2548.2.l.m.1537.3		12	7.3	odd	6
2548.2.l.m.1537.4		12	7.4	even	3

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:	\( N \)	\(=\)	\( 2548 = 2^{2} \cdot 7^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2548.i (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-0.286958 + 0.497025i) q^{3} +(-1.45546 + 2.52093i) q^{5} +(1.33531 + 2.31283i) q^{9} +(-2.73673 + 4.74016i) q^{11} +(3.60003 - 0.199516i) q^{13} +(-0.835311 - 1.44680i) q^{15} -6.05222 q^{17} +(-0.286958 - 0.497025i) q^{19} +5.34124 q^{23} +(-1.73673 - 3.00811i) q^{25} -3.25446 q^{27} +(4.13815 + 7.16749i) q^{29} +(4.53815 + 7.86030i) q^{31} +(-1.57065 - 2.72045i) q^{33} -2.80284 q^{37} +(-0.933890 + 1.84656i) q^{39} +(-5.22725 - 9.05387i) q^{41} +(-3.63815 + 6.30146i) q^{43} -7.77397 q^{45} +(1.28369 - 2.22342i) q^{47} +(1.73673 - 3.00811i) q^{51} +(3.17062 + 5.49168i) q^{53} +(-7.96641 - 13.7982i) q^{55} +0.329378 q^{57} -4.17598 q^{59} +(-4.30981 - 7.46480i) q^{61} +(-4.73673 + 9.36581i) q^{65} +(7.24266 - 12.5447i) q^{67} +(-1.53271 + 2.65473i) q^{69} +(6.14408 - 10.6419i) q^{71} +(1.51204 + 2.61892i) q^{73} +1.99347 q^{75} +(1.67062 - 2.89360i) q^{79} +(-3.07204 + 5.32093i) q^{81} +13.5240 q^{83} +(8.80877 - 15.2572i) q^{85} -4.74989 q^{87} -14.9740 q^{89} -5.20902 q^{93} +1.67062 q^{95} +(4.13804 - 7.16729i) q^{97} -14.6175 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 6 q^{11} + 6 q^{15} + 6 q^{25} + 12 q^{29} - 12 q^{37} - 6 q^{39} - 6 q^{43} - 6 q^{51} + 6 q^{53} + 36 q^{57} - 30 q^{65} + 12 q^{67} - 12 q^{71} - 12 q^{79} + 6 q^{81} + 36 q^{85} + 12 q^{93} - 12 q^{95}+ \cdots - 36 q^{99}+O(q^{100}) \) 12 * q - 6 * q^11 + 6 * q^15 + 6 * q^25 + 12 * q^29 - 12 * q^37 - 6 * q^39 - 6 * q^43 - 6 * q^51 + 6 * q^53 + 36 * q^57 - 30 * q^65 + 12 * q^67 - 12 * q^71 - 12 * q^79 + 6 * q^81 + 36 * q^85 + 12 * q^93 - 12 * q^95 - 36 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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