LMFDB - Embedded newform 2400.2.w.l.607.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2400,2,Mod(607,2400)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2400, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([2, 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("2400.607");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2400 = 2^{5} \cdot 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2400.w (of order $4$, degree $2$, 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:	$19.1640964851$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(i)$
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:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			607.2
Root			$-0.258819 - 0.965926i$ of defining polynomial
Character	$\chi$	$=$	2400.607
Dual form			2400.2.w.l.2143.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.707107 - 0.707107i) q^{3} +(2.02494 - 2.02494i) q^{7} +1.00000i q^{9} -0.964724i q^{11} +(2.63896 - 2.63896i) q^{13} +(-2.14626 - 2.14626i) q^{17} +4.76733 q^{19} -2.86370 q^{21} +(0.282561 + 0.282561i) q^{23} +(0.707107 - 0.707107i) q^{27} +7.32780i q^{29} -8.42418i q^{31} +(-0.682163 + 0.682163i) q^{33} +(1.36433 + 1.36433i) q^{37} -3.73205 q^{39} -8.05646 q^{41} +(3.30286 + 3.30286i) q^{43} +(6.87832 - 6.87832i) q^{47} -1.20080i q^{49} +3.03528i q^{51} +(2.33245 - 2.33245i) q^{53} +(-3.37101 - 3.37101i) q^{57} -3.46410 q^{59} -10.0504 q^{61} +(2.02494 + 2.02494i) q^{63} +(1.03956 - 1.03956i) q^{67} -0.399602i q^{69} +4.84252i q^{71} +(9.46410 - 9.46410i) q^{73} +(-1.95351 - 1.95351i) q^{77} +1.53465 q^{79} -1.00000 q^{81} +(8.94887 + 8.94887i) q^{83} +(5.18154 - 5.18154i) q^{87} -14.6556i q^{89} -10.6875i q^{91} +(-5.95680 + 5.95680i) q^{93} +(-12.9108 - 12.9108i) q^{97} +0.964724 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 8 q^{7} + 8 q^{17} + 16 q^{19} + 8 q^{21} + 8 q^{23} - 8 q^{33} + 16 q^{37} - 16 q^{39} - 16 q^{41} - 24 q^{43} + 16 q^{47} + 8 q^{53} + 8 q^{57} - 24 q^{61} + 8 q^{63} - 8 q^{67} + 48 q^{73} - 8 q^{77}+ \cdots + 16 q^{99}+O(q^{100}) $ 8 * q + 8 * q^7 + 8 * q^17 + 16 * q^19 + 8 * q^21 + 8 * q^23 - 8 * q^33 + 16 * q^37 - 16 * q^39 - 16 * q^41 - 24 * q^43 + 16 * q^47 + 8 * q^53 + 8 * q^57 - 24 * q^61 + 8 * q^63 - 8 * q^67 + 48 * q^73 - 8 * q^77 - 32 * q^79 - 8 * q^81 + 16 * q^83 + 8 * q^87 - 24 * q^93 - 32 * q^97 + 16 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2400\mathbb{Z}\right)^\times$.

$n$	$577$	$901$	$1601$	$1951$
$\chi(n)$	$e\left(\frac{1}{4}\right)$	$1$	$1$	$-1$

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
$3$	−0.707107	−	0.707107i	−0.408248	−	0.408248i
$3$	−0.707107	−	0.707107i	−0.408248	−	0.408248i
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	2.02494	−	2.02494i	0.765357	−	0.765357i	−0.211928	−	0.977285i	$-0.567974\pi$
$7$	2.02494	−	2.02494i	0.765357	−	0.765357i	0.977285	+	0.211928i	$0.0679743\pi$
$8$		0			0
$9$			1.00000i			0.333333i
$10$		0			0
$11$		−	0.964724i		−	0.290875i	−0.989367	−	0.145438i	$-0.953541\pi$
$11$		−	0.964724i		−	0.290875i	0.989367	−	0.145438i	$-0.0464590\pi$
$12$		0			0
$13$	2.63896	−	2.63896i	0.731915	−	0.731915i	−0.239084	−	0.970999i	$-0.576847\pi$
$13$	2.63896	−	2.63896i	0.731915	−	0.731915i	0.970999	+	0.239084i	$0.0768470\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−2.14626	−	2.14626i	−0.520546	−	0.520546i	0.397191	−	0.917736i	$-0.369985\pi$
$17$	−2.14626	−	2.14626i	−0.520546	−	0.520546i	−0.917736	+	0.397191i	$0.869985\pi$
$18$		0			0
$19$	4.76733			1.09370			0.546850	−	0.837231i	$-0.315827\pi$
$19$	4.76733			1.09370			0.546850	+	0.837231i	$0.315827\pi$
$20$		0			0
$21$	−2.86370			−0.624911
$22$		0			0
$23$	0.282561	+	0.282561i	0.0589181	+	0.0589181i	0.735952	−	0.677034i	$-0.236735\pi$
$23$	0.282561	+	0.282561i	0.0589181	+	0.0589181i	−0.677034	+	0.735952i	$0.736735\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$	0.707107	−	0.707107i	0.136083	−	0.136083i
$28$		0			0
$29$			7.32780i			1.36074i	0.732869	+	0.680370i	$0.238181\pi$
$29$			7.32780i			1.36074i	−0.732869	+	0.680370i	$0.761819\pi$
$30$		0			0
$31$		−	8.42418i		−	1.51303i	−0.653978	−	0.756514i	$-0.726901\pi$
$31$		−	8.42418i		−	1.51303i	0.653978	−	0.756514i	$-0.273099\pi$
$32$		0			0
$33$	−0.682163	+	0.682163i	−0.118749	+	0.118749i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	1.36433	+	1.36433i	0.224294	+	0.224294i	0.810304	−	0.586010i	$-0.199302\pi$
$37$	1.36433	+	1.36433i	0.224294	+	0.224294i	−0.586010	+	0.810304i	$0.699302\pi$
$38$		0			0
$39$	−3.73205			−0.597606
$40$		0			0
$41$	−8.05646			−1.25821			−0.629104	−	0.777322i	$-0.716578\pi$
$41$	−8.05646			−1.25821			−0.629104	+	0.777322i	$0.716578\pi$
$42$		0			0
$43$	3.30286	+	3.30286i	0.503682	+	0.503682i	0.912580	−	0.408898i	$-0.134087\pi$
$43$	3.30286	+	3.30286i	0.503682	+	0.503682i	−0.408898	+	0.912580i	$0.634087\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	6.87832	−	6.87832i	1.00331	−	1.00331i	0.00331087	−	0.999995i	$-0.498946\pi$
$47$	6.87832	−	6.87832i	1.00331	−	1.00331i	0.999995	−	0.00331087i	$-0.00105388\pi$
$48$		0			0
$49$		−	1.20080i		−	0.171542i
$50$		0			0
$51$			3.03528i			0.425024i
$52$		0			0
$53$	2.33245	−	2.33245i	0.320387	−	0.320387i	−0.528529	−	0.848915i	$-0.677256\pi$
$53$	2.33245	−	2.33245i	0.320387	−	0.320387i	0.848915	+	0.528529i	$0.177256\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	−3.37101	−	3.37101i	−0.446501	−	0.446501i
$58$		0			0
$59$	−3.46410			−0.450988			−0.225494	−	0.974245i	$-0.572400\pi$
$59$	−3.46410			−0.450988			−0.225494	+	0.974245i	$0.572400\pi$
$60$		0			0
$61$	−10.0504			−1.28682			−0.643411	−	0.765521i	$-0.722481\pi$
$61$	−10.0504			−1.28682			−0.643411	+	0.765521i	$0.722481\pi$
$62$		0			0
$63$	2.02494	+	2.02494i	0.255119	+	0.255119i
$64$		0			0
$65$		0			0
$66$		0			0
$67$	1.03956	−	1.03956i	0.127002	−	0.127002i	−0.640749	−	0.767751i	$-0.721376\pi$
$67$	1.03956	−	1.03956i	0.127002	−	0.127002i	0.767751	+	0.640749i	$0.221376\pi$
$68$		0			0
$69$		−	0.399602i		−	0.0481064i
$70$		0			0
$71$			4.84252i			0.574702i	0.957825	+	0.287351i	$0.0927746\pi$
$71$			4.84252i			0.574702i	−0.957825	+	0.287351i	$0.907225\pi$
$72$		0			0
$73$	9.46410	−	9.46410i	1.10769	−	1.10769i	0.114236	−	0.993454i	$-0.463558\pi$
$73$	9.46410	−	9.46410i	1.10769	−	1.10769i	0.993454	−	0.114236i	$-0.0364419\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−1.95351	−	1.95351i	−0.222623	−	0.222623i
$78$		0			0
$79$	1.53465			0.172662			0.0863310	−	0.996267i	$-0.472486\pi$
$79$	1.53465			0.172662			0.0863310	+	0.996267i	$0.472486\pi$
$80$		0			0
$81$	−1.00000			−0.111111
$82$		0			0
$83$	8.94887	+	8.94887i	0.982266	+	0.982266i	0.999845	−	0.0175796i	$-0.00559606\pi$
$83$	8.94887	+	8.94887i	0.982266	+	0.982266i	−0.0175796	+	0.999845i	$0.505596\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	5.18154	−	5.18154i	0.555519	−	0.555519i
$88$		0			0
$89$		−	14.6556i		−	1.55349i	−0.629814	−	0.776746i	$-0.716869\pi$
$89$		−	14.6556i		−	1.55349i	0.629814	−	0.776746i	$-0.283131\pi$
$90$		0			0
$91$		−	10.6875i		−	1.12035i
$92$		0			0
$93$	−5.95680	+	5.95680i	−0.617691	+	0.617691i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−12.9108	−	12.9108i	−1.31090	−	1.31090i	−0.920756	−	0.390140i	$-0.872427\pi$
$97$	−12.9108	−	12.9108i	−1.31090	−	1.31090i	−0.390140	−	0.920756i	$-0.627573\pi$
$98$		0			0
$99$	0.964724			0.0969584
$100$		0			0
$101$	−12.2560			−1.21952			−0.609759	−	0.792587i	$-0.708734\pi$
$101$	−12.2560			−1.21952			−0.609759	+	0.792587i	$0.708734\pi$
$102$		0			0
$103$	0.292529	+	0.292529i	0.0288237	+	0.0288237i	0.721372	−	0.692548i	$-0.243512\pi$
$103$	0.292529	+	0.292529i	0.0288237	+	0.0288237i	−0.692548	+	0.721372i	$0.743512\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	10.6955	−	10.6955i	1.03398	−	1.03398i	0.0345738	−	0.999402i	$-0.488993\pi$
$107$	10.6955	−	10.6955i	1.03398	−	1.03398i	0.999402	−	0.0345738i	$-0.0110074\pi$
$108$		0			0
$109$			17.1197i			1.63977i	0.572528	+	0.819885i	$0.305963\pi$
$109$			17.1197i			1.63977i	−0.572528	+	0.819885i	$0.694037\pi$
$110$		0			0
$111$		−	1.92945i		−	0.183135i
$112$		0			0
$113$	−8.00000	+	8.00000i	−0.752577	+	0.752577i	−0.974959	−	0.222383i	$-0.928617\pi$
$113$	−8.00000	+	8.00000i	−0.752577	+	0.752577i	0.222383	+	0.974959i	$0.428617\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	2.63896	+	2.63896i	0.243972	+	0.243972i
$118$		0			0
$119$	−8.69213			−0.796806
$120$		0			0
$121$	10.0693			0.915392
$122$		0			0
$123$	5.69677	+	5.69677i	0.513661	+	0.513661i
$124$		0			0
$125$		0			0
$126$		0			0
$127$	2.36308	−	2.36308i	0.209690	−	0.209690i	−0.594446	−	0.804136i	$-0.702629\pi$
$127$	2.36308	−	2.36308i	0.209690	−	0.209690i	0.804136	+	0.594446i	$0.202629\pi$
$128$		0			0
$129$		−	4.67095i		−	0.411254i
$130$		0			0
$131$		−	21.8131i		−	1.90582i	−0.303256	−	0.952909i	$-0.598074\pi$
$131$		−	21.8131i		−	1.90582i	0.303256	−	0.952909i	$-0.401926\pi$
$132$		0			0
$133$	9.65357	−	9.65357i	0.837071	−	0.837071i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−12.3989	−	12.3989i	−1.05931	−	1.05931i	−0.998127	−	0.0611805i	$-0.980513\pi$
$137$	−12.3989	−	12.3989i	−1.05931	−	1.05931i	−0.0611805	−	0.998127i	$-0.519487\pi$
$138$		0			0
$139$	0.535898			0.0454543			0.0227272	−	0.999742i	$-0.492765\pi$
$139$	0.535898			0.0454543			0.0227272	+	0.999742i	$0.492765\pi$
$140$		0			0
$141$	−9.72741			−0.819195
$142$		0			0
$143$	−2.54587	−	2.54587i	−0.212896	−	0.212896i
$144$		0			0
$145$		0			0
$146$		0			0
$147$	−0.849091	+	0.849091i	−0.0700319	+	0.0700319i
$148$		0			0
$149$			4.78710i			0.392174i	0.980586	+	0.196087i	$0.0628236\pi$
$149$			4.78710i			0.392174i	−0.980586	+	0.196087i	$0.937176\pi$
$150$		0			0
$151$		−	11.5665i		−	0.941271i	−0.882328	−	0.470635i	$-0.844025\pi$
$151$		−	11.5665i		−	0.941271i	0.882328	−	0.470635i	$-0.155975\pi$
$152$		0			0
$153$	2.14626	−	2.14626i	0.173515	−	0.173515i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−6.06727	−	6.06727i	−0.484221	−	0.484221i	0.422256	−	0.906477i	$-0.361238\pi$
$157$	−6.06727	−	6.06727i	−0.484221	−	0.484221i	−0.906477	+	0.422256i	$0.861238\pi$
$158$		0			0
$159$	−3.29858			−0.261595
$160$		0			0
$161$	1.14434			0.0901867
$162$		0			0
$163$	−14.9804	−	14.9804i	−1.17335	−	1.17335i	−0.981405	−	0.191949i	$-0.938519\pi$
$163$	−14.9804	−	14.9804i	−1.17335	−	1.17335i	−0.191949	−	0.981405i	$-0.561481\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−14.0306	+	14.0306i	−1.08572	+	1.08572i	−0.0897594	+	0.995963i	$0.528610\pi$
$167$	−14.0306	+	14.0306i	−1.08572	+	1.08572i	−0.995963	+	0.0897594i	$0.971390\pi$
$168$		0			0
$169$		−	0.928203i		−	0.0714002i
$170$		0			0
$171$			4.76733i			0.364567i
$172$		0			0
$173$	10.7727	−	10.7727i	0.819029	−	0.819029i	−0.166938	−	0.985967i	$-0.553388\pi$
$173$	10.7727	−	10.7727i	0.819029	−	0.819029i	0.985967	+	0.166938i	$0.0533880\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	2.44949	+	2.44949i	0.184115	+	0.184115i
$178$		0			0
$179$	−5.06251			−0.378390			−0.189195	−	0.981940i	$-0.560588\pi$
$179$	−5.06251			−0.378390			−0.189195	+	0.981940i	$0.560588\pi$
$180$		0			0
$181$	7.67700			0.570627			0.285314	−	0.958434i	$-0.407902\pi$
$181$	7.67700			0.570627			0.285314	+	0.958434i	$0.407902\pi$
$182$		0			0
$183$	7.10671	+	7.10671i	0.525343	+	0.525343i
$184$		0			0
$185$		0			0
$186$		0			0
$187$	−2.07055	+	2.07055i	−0.151414	+	0.151414i
$188$		0			0
$189$		−	2.86370i		−	0.208304i
$190$		0			0
$191$			0.906276i			0.0655759i	0.999462	+	0.0327879i	$0.0104386\pi$
$191$			0.906276i			0.0655759i	−0.999462	+	0.0327879i	$0.989561\pi$
$192$		0			0
$193$	2.58250	−	2.58250i	0.185893	−	0.185893i	−0.608025	−	0.793918i	$-0.708038\pi$
$193$	2.58250	−	2.58250i	0.185893	−	0.185893i	0.793918	+	0.608025i	$0.208038\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	12.2991	+	12.2991i	0.876274	+	0.876274i	0.993147	−	0.116873i	$-0.0372869\pi$
$197$	12.2991	+	12.2991i	0.876274	+	0.876274i	−0.116873	+	0.993147i	$0.537287\pi$
$198$		0			0
$199$	−1.43223			−0.101528			−0.0507639	−	0.998711i	$-0.516166\pi$
$199$	−1.43223			−0.101528			−0.0507639	+	0.998711i	$0.516166\pi$
$200$		0			0
$201$	−1.47015			−0.103697
$202$		0			0
$203$	14.8384	+	14.8384i	1.04145	+	1.04145i
$204$		0			0
$205$		0			0
$206$		0			0
$207$	−0.282561	+	0.282561i	−0.0196394	+	0.0196394i
$208$		0			0
$209$		−	4.59915i		−	0.318130i
$210$		0			0
$211$		−	19.1503i		−	1.31836i	−0.751983	−	0.659182i	$-0.770903\pi$
$211$		−	19.1503i		−	1.31836i	0.751983	−	0.659182i	$-0.229097\pi$
$212$		0			0
$213$	3.42418	−	3.42418i	0.234621	−	0.234621i
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−17.0585	−	17.0585i	−1.15801	−	1.15801i
$218$		0			0
$219$	−13.3843			−0.904425
$220$		0			0
$221$	−11.3278			−0.761991
$222$		0			0
$223$	−14.4603	−	14.4603i	−0.968336	−	0.968336i	0.0311780	−	0.999514i	$-0.490074\pi$
$223$	−14.4603	−	14.4603i	−0.968336	−	0.968336i	−0.999514	+	0.0311780i	$0.990074\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	12.2939	−	12.2939i	0.815977	−	0.815977i	−0.169545	−	0.985522i	$-0.554230\pi$
$227$	12.2939	−	12.2939i	0.815977	−	0.815977i	0.985522	+	0.169545i	$0.0542299\pi$
$228$		0			0
$229$			14.7342i			0.973664i	0.873496	+	0.486832i	$0.161848\pi$
$229$			14.7342i			0.973664i	−0.873496	+	0.486832i	$0.838152\pi$
$230$		0			0
$231$			2.76268i			0.181771i
$232$		0			0
$233$	−1.06110	+	1.06110i	−0.0695150	+	0.0695150i	−0.741010	−	0.671495i	$-0.765653\pi$
$233$	−1.06110	+	1.06110i	−0.0695150	+	0.0695150i	0.671495	+	0.741010i	$0.265653\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	−1.08516	−	1.08516i	−0.0704890	−	0.0704890i
$238$		0			0
$239$	12.0151			0.777194			0.388597	−	0.921408i	$-0.372960\pi$
$239$	12.0151			0.777194			0.388597	+	0.921408i	$0.372960\pi$
$240$		0			0
$241$	25.5241			1.64415			0.822077	−	0.569377i	$-0.192815\pi$
$241$	25.5241			1.64415			0.822077	+	0.569377i	$0.192815\pi$
$242$		0			0
$243$	0.707107	+	0.707107i	0.0453609	+	0.0453609i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	12.5808	−	12.5808i	0.800496	−	0.800496i
$248$		0			0
$249$		−	12.6556i		−	0.802017i
$250$		0			0
$251$			21.3112i			1.34515i	0.740028	+	0.672576i	$0.234812\pi$
$251$			21.3112i			1.34515i	−0.740028	+	0.672576i	$0.765188\pi$
$252$		0			0
$253$	0.272593	−	0.272593i	0.0171378	−	0.0171378i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−8.75787	−	8.75787i	−0.546301	−	0.546301i	0.379068	−	0.925369i	$-0.376245\pi$
$257$	−8.75787	−	8.75787i	−0.546301	−	0.546301i	−0.925369	+	0.379068i	$0.876245\pi$
$258$		0			0
$259$	5.52537			0.343330
$260$		0			0
$261$	−7.32780			−0.453580
$262$		0			0
$263$	12.7180	+	12.7180i	0.784223	+	0.784223i	0.980540	−	0.196318i	$-0.0628984\pi$
$263$	12.7180	+	12.7180i	0.784223	+	0.784223i	−0.196318	+	0.980540i	$0.562898\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	−10.3631	+	10.3631i	−0.634210	+	0.634210i
$268$		0			0
$269$		−	1.47140i		−	0.0897127i	−0.998993	−	0.0448564i	$-0.985717\pi$
$269$		−	1.47140i		−	0.0897127i	0.998993	−	0.0448564i	$-0.0142830\pi$
$270$		0			0
$271$		−	23.0600i		−	1.40080i	−0.713752	−	0.700398i	$-0.753006\pi$
$271$		−	23.0600i		−	1.40080i	0.713752	−	0.700398i	$-0.246994\pi$
$272$		0			0
$273$	−7.55719	+	7.55719i	−0.457382	+	0.457382i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	12.9156	+	12.9156i	0.776025	+	0.776025i	0.979152	−	0.203127i	$-0.0651105\pi$
$277$	12.9156	+	12.9156i	0.776025	+	0.776025i	−0.203127	+	0.979152i	$0.565111\pi$
$278$		0			0
$279$	8.42418			0.504343
$280$		0			0
$281$	−10.0846			−0.601600			−0.300800	−	0.953687i	$-0.597254\pi$
$281$	−10.0846			−0.601600			−0.300800	+	0.953687i	$0.597254\pi$
$282$		0			0
$283$	−8.18191	−	8.18191i	−0.486364	−	0.486364i	0.420793	−	0.907157i	$-0.361752\pi$
$283$	−8.18191	−	8.18191i	−0.486364	−	0.486364i	−0.907157	+	0.420793i	$0.861752\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−16.3139	+	16.3139i	−0.962977	+	0.962977i
$288$		0			0
$289$		−	7.78710i		−	0.458065i
$290$		0			0
$291$			18.2587i			1.07034i
$292$		0			0
$293$	−20.3472	+	20.3472i	−1.18870	+	1.18870i	−0.211270	+	0.977428i	$0.567760\pi$
$293$	−20.3472	+	20.3472i	−1.18870	+	1.18870i	−0.977428	+	0.211270i	$0.932240\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	−0.682163	−	0.682163i	−0.0395831	−	0.0395831i
$298$		0			0
$299$	1.49133			0.0862461
$300$		0			0
$301$	13.3762			0.770992
$302$		0			0
$303$	8.66631	+	8.66631i	0.497866	+	0.497866i
$304$		0			0
$305$		0			0
$306$		0			0
$307$	−7.81809	+	7.81809i	−0.446202	+	0.446202i	−0.894090	−	0.447888i	$-0.852176\pi$
$307$	−7.81809	+	7.81809i	−0.446202	+	0.446202i	0.447888	+	0.894090i	$0.352176\pi$
$308$		0			0
$309$		−	0.413698i		−	0.0235345i
$310$		0			0
$311$			29.3205i			1.66261i	0.555814	+	0.831307i	$0.312407\pi$
$311$			29.3205i			1.66261i	−0.555814	+	0.831307i	$0.687593\pi$
$312$		0			0
$313$	−3.06778	+	3.06778i	−0.173401	+	0.173401i	−0.788472	−	0.615071i	$-0.789127\pi$
$313$	−3.06778	+	3.06778i	−0.173401	+	0.173401i	0.615071	+	0.788472i	$0.289127\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	6.02922	+	6.02922i	0.338635	+	0.338635i	0.855853	−	0.517218i	$-0.173033\pi$
$317$	6.02922	+	6.02922i	0.338635	+	0.338635i	−0.517218	+	0.855853i	$0.673033\pi$
$318$		0			0
$319$	7.06931			0.395805
$320$		0			0
$321$	−15.1258			−0.844238
$322$		0			0
$323$	−10.2319	−	10.2319i	−0.569321	−	0.569321i
$324$		0			0
$325$		0			0
$326$		0			0
$327$	12.1055	−	12.1055i	0.669434	−	0.669434i
$328$		0			0
$329$		−	27.8564i		−	1.53577i
$330$		0			0
$331$			21.1794i			1.16413i	0.813144	+	0.582063i	$0.197754\pi$
$331$			21.1794i			1.16413i	−0.813144	+	0.582063i	$0.802246\pi$
$332$		0			0
$333$	−1.36433	+	1.36433i	−0.0747646	+	0.0747646i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	15.9602	+	15.9602i	0.869407	+	0.869407i	0.992407	−	0.122999i	$-0.0392514\pi$
$337$	15.9602	+	15.9602i	0.869407	+	0.869407i	−0.122999	+	0.992407i	$0.539251\pi$
$338$		0			0
$339$	11.3137			0.614476
$340$		0			0
$341$	−8.12701			−0.440102
$342$		0			0
$343$	11.7431	+	11.7431i	0.634066	+	0.634066i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	7.81862	−	7.81862i	0.419726	−	0.419726i	−0.465383	−	0.885109i	$-0.654084\pi$
$347$	7.81862	−	7.81862i	0.419726	−	0.419726i	0.885109	+	0.465383i	$0.154084\pi$
$348$		0			0
$349$			25.4548i			1.36256i	0.732021	+	0.681282i	$0.238577\pi$
$349$			25.4548i			1.36256i	−0.732021	+	0.681282i	$0.761423\pi$
$350$		0			0
$351$		−	3.73205i		−	0.199202i
$352$		0			0
$353$	1.71263	−	1.71263i	0.0911541	−	0.0911541i	−0.660059	−	0.751213i	$-0.729469\pi$
$353$	1.71263	−	1.71263i	0.0911541	−	0.0911541i	0.751213	+	0.660059i	$0.229469\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$	6.14626	+	6.14626i	0.325295	+	0.325295i
$358$		0			0
$359$	−18.3981			−0.971017			−0.485508	−	0.874232i	$-0.661365\pi$
$359$	−18.3981			−0.971017			−0.485508	+	0.874232i	$0.661365\pi$
$360$		0			0
$361$	3.72741			0.196179
$362$		0			0
$363$	−7.12008	−	7.12008i	−0.373707	−	0.373707i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	−6.36736	+	6.36736i	−0.332374	+	0.332374i	−0.853487	−	0.521114i	$-0.825517\pi$
$367$	−6.36736	+	6.36736i	−0.332374	+	0.332374i	0.521114	+	0.853487i	$0.325517\pi$
$368$		0			0
$369$		−	8.05646i		−	0.419402i
$370$		0			0
$371$		−	9.44616i		−	0.490420i
$372$		0			0
$373$	10.1602	−	10.1602i	0.526078	−	0.526078i	−0.393323	−	0.919400i	$-0.628675\pi$
$373$	10.1602	−	10.1602i	0.526078	−	0.526078i	0.919400	+	0.393323i	$0.128675\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	19.3378	+	19.3378i	0.995946	+	0.995946i
$378$		0			0
$379$	−6.16088			−0.316463			−0.158231	−	0.987402i	$-0.550579\pi$
$379$	−6.16088			−0.316463			−0.158231	+	0.987402i	$0.550579\pi$
$380$		0			0
$381$	−3.34190			−0.171211
$382$		0			0
$383$	−10.8484	−	10.8484i	−0.554325	−	0.554325i	0.373361	−	0.927686i	$-0.378205\pi$
$383$	−10.8484	−	10.8484i	−0.554325	−	0.554325i	−0.927686	+	0.373361i	$0.878205\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−3.30286	+	3.30286i	−0.167894	+	0.167894i
$388$		0			0
$389$			13.0834i			0.663355i	0.943393	+	0.331677i	$0.107615\pi$
$389$			13.0834i			0.663355i	−0.943393	+	0.331677i	$0.892385\pi$
$390$		0			0
$391$		−	1.21290i		−	0.0613391i
$392$		0			0
$393$	−15.4242	+	15.4242i	−0.778047	+	0.778047i
$394$		0			0
$395$		0			0
$396$		0			0
$397$	16.7745	+	16.7745i	0.841889	+	0.841889i	0.989104	−	0.147215i	$-0.0470310\pi$
$397$	16.7745	+	16.7745i	0.841889	+	0.841889i	−0.147215	+	0.989104i	$0.547031\pi$
$398$		0			0
$399$	−13.6522			−0.683465
$400$		0			0
$401$	38.0258			1.89892			0.949458	−	0.313893i	$-0.101633\pi$
$401$	38.0258			1.89892			0.949458	+	0.313893i	$0.101633\pi$
$402$		0			0
$403$	−22.2311	−	22.2311i	−1.10741	−	1.10741i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	1.31620	−	1.31620i	0.0652415	−	0.0652415i
$408$		0			0
$409$			19.2540i			0.952050i	0.879432	+	0.476025i	$0.157923\pi$
$409$			19.2540i			0.952050i	−0.879432	+	0.476025i	$0.842077\pi$
$410$		0			0
$411$			17.5347i			0.864921i
$412$		0			0
$413$	−7.01461	+	7.01461i	−0.345166	+	0.345166i
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−0.378937	−	0.378937i	−0.0185566	−	0.0185566i
$418$		0			0
$419$	−19.0260			−0.929480			−0.464740	−	0.885447i	$-0.653852\pi$
$419$	−19.0260			−0.929480			−0.464740	+	0.885447i	$0.653852\pi$
$420$		0			0
$421$	21.0411			1.02548			0.512741	−	0.858544i	$-0.328630\pi$
$421$	21.0411			1.02548			0.512741	+	0.858544i	$0.328630\pi$
$422$		0			0
$423$	6.87832	+	6.87832i	0.334435	+	0.334435i
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−20.3515	+	20.3515i	−0.984878	+	0.984878i
$428$		0			0
$429$			3.60040i			0.173829i
$430$		0			0
$431$			33.2187i			1.60009i	0.599940	+	0.800045i	$0.295191\pi$
$431$			33.2187i			1.60009i	−0.599940	+	0.800045i	$0.704809\pi$
$432$		0			0
$433$	4.93878	−	4.93878i	0.237343	−	0.237343i	−0.578406	−	0.815749i	$-0.696325\pi$
$433$	4.93878	−	4.93878i	0.237343	−	0.237343i	0.815749	+	0.578406i	$0.196325\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	1.34706	+	1.34706i	0.0644387	+	0.0644387i
$438$		0			0
$439$	34.8072			1.66126			0.830628	−	0.556827i	$-0.187981\pi$
$439$	34.8072			1.66126			0.830628	+	0.556827i	$0.187981\pi$
$440$		0			0
$441$	1.20080			0.0571808
$442$		0			0
$443$	7.82038	+	7.82038i	0.371558	+	0.371558i	0.868044	−	0.496487i	$-0.165377\pi$
$443$	7.82038	+	7.82038i	0.371558	+	0.371558i	−0.496487	+	0.868044i	$0.665377\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	3.38499	−	3.38499i	0.160105	−	0.160105i
$448$		0			0
$449$			12.0000i			0.566315i	0.959073	+	0.283158i	$0.0913819\pi$
$449$			12.0000i			0.566315i	−0.959073	+	0.283158i	$0.908618\pi$
$450$		0			0
$451$			7.77225i			0.365981i
$452$		0			0
$453$	−8.17877	+	8.17877i	−0.384272	+	0.384272i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−5.17157	−	5.17157i	−0.241916	−	0.241916i	0.575726	−	0.817642i	$-0.304719\pi$
$457$	−5.17157	−	5.17157i	−0.241916	−	0.241916i	−0.817642	+	0.575726i	$0.804719\pi$
$458$		0			0
$459$	−3.03528			−0.141675
$460$		0			0
$461$	−25.1963			−1.17351			−0.586755	−	0.809765i	$-0.699595\pi$
$461$	−25.1963			−1.17351			−0.586755	+	0.809765i	$0.699595\pi$
$462$		0			0
$463$	−14.0492	−	14.0492i	−0.652920	−	0.652920i	0.300775	−	0.953695i	$-0.402755\pi$
$463$	−14.0492	−	14.0492i	−0.652920	−	0.652920i	−0.953695	+	0.300775i	$0.902755\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−7.96864	+	7.96864i	−0.368745	+	0.368745i	−0.867019	−	0.498275i	$-0.833967\pi$
$467$	−7.96864	+	7.96864i	−0.368745	+	0.368745i	0.498275	+	0.867019i	$0.333967\pi$
$468$		0			0
$469$		−	4.21009i		−	0.194404i
$470$		0			0
$471$			8.58041i			0.395365i
$472$		0			0
$473$	3.18635	−	3.18635i	0.146508	−	0.146508i
$474$		0			0
$475$		0			0
$476$		0			0
$477$	2.33245	+	2.33245i	0.106796	+	0.106796i
$478$		0			0
$479$	40.3032			1.84150			0.920750	−	0.390154i	$-0.127578\pi$
$479$	40.3032			1.84150			0.920750	+	0.390154i	$0.127578\pi$
$480$		0			0
$481$	7.20080			0.328328
$482$		0			0
$483$	−0.809171	−	0.809171i	−0.0368186	−	0.0368186i
$484$		0			0
$485$		0			0
$486$		0			0
$487$	−13.4384	+	13.4384i	−0.608953	+	0.608953i	−0.942673	−	0.333719i	$-0.891696\pi$
$487$	−13.4384	+	13.4384i	−0.608953	+	0.608953i	0.333719	+	0.942673i	$0.391696\pi$
$488$		0			0
$489$			21.1855i			0.958039i
$490$		0			0
$491$			2.24191i			0.101176i	0.998720	+	0.0505880i	$0.0161095\pi$
$491$			2.24191i			0.101176i	−0.998720	+	0.0505880i	$0.983890\pi$
$492$		0			0
$493$	15.7274	−	15.7274i	0.708327	−	0.708327i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	9.80584	+	9.80584i	0.439852	+	0.439852i
$498$		0			0
$499$	42.6144			1.90768			0.953842	−	0.300308i	$-0.0970895\pi$
$499$	42.6144			1.90768			0.953842	+	0.300308i	$0.0970895\pi$
$500$		0			0
$501$	19.8423			0.886489
$502$		0			0
$503$	−13.5653	−	13.5653i	−0.604846	−	0.604846i	0.336748	−	0.941595i	$-0.390673\pi$
$503$	−13.5653	−	13.5653i	−0.604846	−	0.604846i	−0.941595	+	0.336748i	$0.890673\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−0.656339	+	0.656339i	−0.0291490	+	0.0291490i
$508$		0			0
$509$		−	21.3233i		−	0.945140i	−0.881293	−	0.472570i	$-0.843326\pi$
$509$		−	21.3233i		−	0.945140i	0.881293	−	0.472570i	$-0.156674\pi$
$510$		0			0
$511$		−	38.3286i		−	1.69556i
$512$		0			0
$513$	3.37101	−	3.37101i	0.148834	−	0.148834i
$514$		0			0
$515$		0			0
$516$		0			0
$517$	−6.63567	−	6.63567i	−0.291837	−	0.291837i
$518$		0			0
$519$	−15.2348			−0.668735
$520$		0			0
$521$	20.5710			0.901230			0.450615	−	0.892718i	$-0.351205\pi$
$521$	20.5710			0.901230			0.450615	+	0.892718i	$0.351205\pi$
$522$		0			0
$523$	−11.0086	−	11.0086i	−0.481371	−	0.481371i	0.424198	−	0.905569i	$-0.360556\pi$
$523$	−11.0086	−	11.0086i	−0.481371	−	0.481371i	−0.905569	+	0.424198i	$0.860556\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−18.0805	+	18.0805i	−0.787600	+	0.787600i
$528$		0			0
$529$		−	22.8403i		−	0.993057i
$530$		0			0
$531$		−	3.46410i		−	0.150329i
$532$		0			0
$533$	−21.2607	+	21.2607i	−0.920901	+	0.920901i
$534$		0			0
$535$		0			0
$536$		0			0
$537$	3.57973	+	3.57973i	0.154477	+	0.154477i
$538$		0			0
$539$	−1.15844			−0.0498974
$540$		0			0
$541$	19.9189			0.856381			0.428191	−	0.903688i	$-0.359151\pi$
$541$	19.9189			0.856381			0.428191	+	0.903688i	$0.359151\pi$
$542$		0			0
$543$	−5.42846	−	5.42846i	−0.232958	−	0.232958i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−10.0905	+	10.0905i	−0.431438	+	0.431438i	−0.889117	−	0.457679i	$-0.848681\pi$
$547$	−10.0905	+	10.0905i	−0.431438	+	0.431438i	0.457679	+	0.889117i	$0.348681\pi$
$548$		0			0
$549$		−	10.0504i		−	0.428941i
$550$		0			0
$551$			34.9340i			1.48824i
$552$		0			0
$553$	3.10759	−	3.10759i	0.132148	−	0.132148i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	24.6398	+	24.6398i	1.04402	+	1.04402i	0.998985	+	0.0450344i	$0.0143398\pi$
$557$	24.6398	+	24.6398i	1.04402	+	1.04402i	0.0450344	+	0.998985i	$0.485660\pi$
$558$		0			0
$559$	17.4322			0.737305
$560$		0			0
$561$	2.92820			0.123629
$562$		0			0
$563$	2.57406	+	2.57406i	0.108484	+	0.108484i	0.759265	−	0.650781i	$-0.225559\pi$
$563$	2.57406	+	2.57406i	0.108484	+	0.108484i	−0.650781	+	0.759265i	$0.725559\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	−2.02494	+	2.02494i	−0.0850397	+	0.0850397i
$568$		0			0
$569$			34.5685i			1.44919i	0.689177	+	0.724593i	$0.257972\pi$
$569$			34.5685i			1.44919i	−0.689177	+	0.724593i	$0.742028\pi$
$570$		0			0
$571$			31.3310i			1.31116i	0.755125	+	0.655581i	$0.227576\pi$
$571$			31.3310i			1.31116i	−0.755125	+	0.655581i	$0.772424\pi$
$572$		0			0
$573$	0.640834	−	0.640834i	0.0267712	−	0.0267712i
$574$		0			0
$575$		0			0
$576$		0			0
$577$	14.8151	+	14.8151i	0.616762	+	0.616762i	0.944699	−	0.327938i	$-0.106354\pi$
$577$	14.8151	+	14.8151i	0.616762	+	0.616762i	−0.327938	+	0.944699i	$0.606354\pi$
$578$		0			0
$579$	−3.65221			−0.151781
$580$		0			0
$581$	36.2419			1.50357
$582$		0			0
$583$	−2.25017	−	2.25017i	−0.0931925	−	0.0931925i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	31.1224	−	31.1224i	1.28456	−	1.28456i	0.346512	−	0.938046i	$-0.387366\pi$
$587$	31.1224	−	31.1224i	1.28456	−	1.28456i	0.938046	−	0.346512i	$-0.112634\pi$
$588$		0			0
$589$		−	40.1608i		−	1.65480i
$590$		0			0
$591$		−	17.3935i		−	0.715475i
$592$		0			0
$593$	−21.4149	+	21.4149i	−0.879404	+	0.879404i	−0.993473	−	0.114069i	$-0.963612\pi$
$593$	−21.4149	+	21.4149i	−0.879404	+	0.879404i	0.114069	+	0.993473i	$0.463612\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	1.01274	+	1.01274i	0.0414485	+	0.0414485i
$598$		0			0
$599$	−18.7383			−0.765625			−0.382812	−	0.923826i	$-0.625044\pi$
$599$	−18.7383			−0.765625			−0.382812	+	0.923826i	$0.625044\pi$
$600$		0			0
$601$	20.4266			0.833219			0.416610	−	0.909085i	$-0.363218\pi$
$601$	20.4266			0.833219			0.416610	+	0.909085i	$0.363218\pi$
$602$		0			0
$603$	1.03956	+	1.03956i	0.0423340	+	0.0423340i
$604$		0			0
$605$		0			0
$606$		0			0
$607$	−12.7165	+	12.7165i	−0.516149	+	0.516149i	−0.916404	−	0.400255i	$-0.868922\pi$
$607$	−12.7165	+	12.7165i	−0.516149	+	0.516149i	0.400255	+	0.916404i	$0.368922\pi$
$608$		0			0
$609$		−	20.9847i		−	0.850341i
$610$		0			0
$611$		−	36.3032i		−	1.46867i
$612$		0			0
$613$	2.55583	−	2.55583i	0.103229	−	0.103229i	−0.653606	−	0.756835i	$-0.726745\pi$
$613$	2.55583	−	2.55583i	0.103229	−	0.103229i	0.756835	+	0.653606i	$0.226745\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−7.11172	−	7.11172i	−0.286307	−	0.286307i	0.549311	−	0.835618i	$-0.314890\pi$
$617$	−7.11172	−	7.11172i	−0.286307	−	0.286307i	−0.835618	+	0.549311i	$0.814890\pi$
$618$		0			0
$619$	−0.0810355			−0.00325709			−0.00162855	−	0.999999i	$-0.500518\pi$
$619$	−0.0810355			−0.00325709			−0.00162855	+	0.999999i	$0.500518\pi$
$620$		0			0
$621$	0.399602			0.0160355
$622$		0			0
$623$	−29.6768	−	29.6768i	−1.18898	−	1.18898i
$624$		0			0
$625$		0			0
$626$		0			0
$627$	−3.25209	+	3.25209i	−0.129876	+	0.129876i
$628$		0			0
$629$		−	5.85641i		−	0.233510i
$630$		0			0
$631$			45.5213i			1.81217i	0.423093	+	0.906086i	$0.360944\pi$
$631$			45.5213i			1.81217i	−0.423093	+	0.906086i	$0.639056\pi$
$632$		0			0
$633$	−13.5413	+	13.5413i	−0.538220	+	0.538220i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−3.16885	−	3.16885i	−0.125555	−	0.125555i
$638$		0			0
$639$	−4.84252			−0.191567
$640$		0			0
$641$	−32.6531			−1.28972			−0.644860	−	0.764300i	$-0.723084\pi$
$641$	−32.6531			−1.28972			−0.644860	+	0.764300i	$0.723084\pi$
$642$		0			0
$643$	1.02015	+	1.02015i	0.0402307	+	0.0402307i	0.726936	−	0.686705i	$-0.240944\pi$
$643$	1.02015	+	1.02015i	0.0402307	+	0.0402307i	−0.686705	+	0.726936i	$0.740944\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−29.7766	+	29.7766i	−1.17064	+	1.17064i	−0.188580	+	0.982058i	$0.560388\pi$
$647$	−29.7766	+	29.7766i	−1.17064	+	1.17064i	−0.982058	+	0.188580i	$0.939612\pi$
$648$		0			0
$649$			3.34190i			0.131181i
$650$		0			0
$651$			24.1244i			0.945508i
$652$		0			0
$653$	3.93498	−	3.93498i	0.153988	−	0.153988i	−0.625909	−	0.779896i	$-0.715272\pi$
$653$	3.93498	−	3.93498i	0.153988	−	0.153988i	0.779896	+	0.625909i	$0.215272\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	9.46410	+	9.46410i	0.369230	+	0.369230i
$658$		0			0
$659$	−3.81558			−0.148634			−0.0743169	−	0.997235i	$-0.523678\pi$
$659$	−3.81558			−0.148634			−0.0743169	+	0.997235i	$0.523678\pi$
$660$		0			0
$661$	2.66771			0.103762			0.0518810	−	0.998653i	$-0.483478\pi$
$661$	2.66771			0.103762			0.0518810	+	0.998653i	$0.483478\pi$
$662$		0			0
$663$	8.00997	+	8.00997i	0.311081	+	0.311081i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−2.07055	+	2.07055i	−0.0801721	+	0.0801721i
$668$		0			0
$669$			20.4500i			0.790643i
$670$		0			0
$671$			9.69586i			0.374305i
$672$		0			0
$673$	12.1928	−	12.1928i	0.469996	−	0.469996i	−0.431917	−	0.901913i	$-0.642163\pi$
$673$	12.1928	−	12.1928i	0.469996	−	0.469996i	0.901913	+	0.431917i	$0.142163\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−9.36020	−	9.36020i	−0.359742	−	0.359742i	0.503976	−	0.863718i	$-0.331870\pi$
$677$	−9.36020	−	9.36020i	−0.359742	−	0.359742i	−0.863718	+	0.503976i	$0.831870\pi$
$678$		0			0
$679$	−52.2874			−2.00661
$680$		0			0
$681$	−17.3863			−0.666243
$682$		0			0
$683$	28.6463	+	28.6463i	1.09612	+	1.09612i	0.994860	+	0.101261i	$0.0322878\pi$
$683$	28.6463	+	28.6463i	1.09612	+	1.09612i	0.101261	+	0.994860i	$0.467712\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	10.4187	−	10.4187i	0.397497	−	0.397497i
$688$		0			0
$689$		−	12.3105i		−	0.468992i
$690$		0			0
$691$		−	21.1608i		−	0.804996i	−0.915421	−	0.402498i	$-0.868142\pi$
$691$		−	21.1608i		−	0.804996i	0.915421	−	0.402498i	$-0.131858\pi$
$692$		0			0
$693$	1.95351	−	1.95351i	0.0742078	−	0.0742078i
$694$		0			0
$695$		0			0
$696$		0			0
$697$	17.2913	+	17.2913i	0.654954	+	0.654954i
$698$		0			0
$699$	1.50062			0.0567587
$700$		0			0
$701$	17.8423			0.673895			0.336947	−	0.941523i	$-0.390606\pi$
$701$	17.8423			0.673895			0.336947	+	0.941523i	$0.390606\pi$
$702$		0			0
$703$	6.50419	+	6.50419i	0.245310	+	0.245310i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−24.8177	+	24.8177i	−0.933367	+	0.933367i
$708$		0			0
$709$			16.9625i			0.637041i	0.947916	+	0.318520i	$0.103186\pi$
$709$			16.9625i			0.637041i	−0.947916	+	0.318520i	$0.896814\pi$
$710$		0			0
$711$			1.53465i			0.0575540i
$712$		0			0
$713$	2.38035	−	2.38035i	0.0891446	−	0.0891446i
$714$		0			0
$715$		0			0
$716$		0			0
$717$	−8.49598	−	8.49598i	−0.317288	−	0.317288i
$718$		0			0
$719$	12.6191			0.470613			0.235306	−	0.971921i	$-0.424391\pi$
$719$	12.6191			0.470613			0.235306	+	0.971921i	$0.424391\pi$
$720$		0			0
$721$	1.18471			0.0441209
$722$		0			0
$723$	−18.0483	−	18.0483i	−0.671223	−	0.671223i
$724$		0			0
$725$		0			0
$726$		0			0
$727$	14.0149	−	14.0149i	0.519785	−	0.519785i	−0.397721	−	0.917506i	$-0.630199\pi$
$727$	14.0149	−	14.0149i	0.519785	−	0.519785i	0.917506	+	0.397721i	$0.130199\pi$
$728$		0			0
$729$		−	1.00000i		−	0.0370370i
$730$		0			0
$731$		−	14.1776i		−	0.524378i
$732$		0			0
$733$	−17.5254	+	17.5254i	−0.647314	+	0.647314i	−0.952343	−	0.305029i	$-0.901334\pi$
$733$	−17.5254	+	17.5254i	−0.647314	+	0.647314i	0.305029	+	0.952343i	$0.401334\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−1.00288	−	1.00288i	−0.0369417	−	0.0369417i
$738$		0			0
$739$	7.10885			0.261503			0.130752	−	0.991415i	$-0.458261\pi$
$739$	7.10885			0.261503			0.130752	+	0.991415i	$0.458261\pi$
$740$		0			0
$741$	−17.7919			−0.653602
$742$		0			0
$743$	14.6075	+	14.6075i	0.535897	+	0.535897i	0.922321	−	0.386425i	$-0.126290\pi$
$743$	14.6075	+	14.6075i	0.535897	+	0.535897i	−0.386425	+	0.922321i	$0.626290\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	−8.94887	+	8.94887i	−0.327422	+	0.327422i
$748$		0			0
$749$		−	43.3157i		−	1.58272i
$750$		0			0
$751$		−	7.21970i		−	0.263451i	−0.991286	−	0.131725i	$-0.957948\pi$
$751$		−	7.21970i		−	0.263451i	0.991286	−	0.131725i	$-0.0420517\pi$
$752$		0			0
$753$	15.0693	−	15.0693i	0.549156	−	0.549156i
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−1.93273	−	1.93273i	−0.0702463	−	0.0702463i	0.671111	−	0.741357i	$-0.265817\pi$
$757$	−1.93273	−	1.93273i	−0.0702463	−	0.0702463i	−0.741357	+	0.671111i	$0.765817\pi$
$758$		0			0
$759$	−0.385505			−0.0139930
$760$		0			0
$761$	−6.23230			−0.225921			−0.112960	−	0.993600i	$-0.536033\pi$
$761$	−6.23230			−0.225921			−0.112960	+	0.993600i	$0.536033\pi$
$762$		0			0
$763$	34.6665	+	34.6665i	1.25501	+	1.25501i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−9.14162	+	9.14162i	−0.330085	+	0.330085i
$768$		0			0
$769$			26.1419i			0.942702i	0.881946	+	0.471351i	$0.156233\pi$
$769$			26.1419i			0.942702i	−0.881946	+	0.471351i	$0.843767\pi$
$770$		0			0
$771$			12.3855i			0.446053i
$772$		0			0
$773$	−11.7580	+	11.7580i	−0.422907	+	0.422907i	−0.886203	−	0.463296i	$-0.846667\pi$
$773$	−11.7580	+	11.7580i	−0.422907	+	0.422907i	0.463296	+	0.886203i	$0.346667\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	−3.90702	−	3.90702i	−0.140164	−	0.140164i
$778$		0			0
$779$	−38.4078			−1.37610
$780$		0			0
$781$	4.67170			0.167166
$782$		0			0
$783$	5.18154	+	5.18154i	0.185173	+	0.185173i
$784$		0			0
$785$		0			0
$786$		0			0
$787$	−13.1008	+	13.1008i	−0.466994	+	0.466994i	−0.900939	−	0.433945i	$-0.857121\pi$
$787$	−13.1008	+	13.1008i	−0.466994	+	0.466994i	0.433945	+	0.900939i	$0.357121\pi$
$788$		0			0
$789$		−	17.9859i		−	0.640315i
$790$		0			0
$791$			32.3991i			1.15198i
$792$		0			0
$793$	−26.5226	+	26.5226i	−0.941845	+	0.941845i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−18.9282	−	18.9282i	−0.670471	−	0.670471i	0.287353	−	0.957825i	$-0.407225\pi$
$797$	−18.9282	−	18.9282i	−0.670471	−	0.670471i	−0.957825	+	0.287353i	$0.907225\pi$
$798$		0			0
$799$	−29.5254			−1.04453
$800$		0			0
$801$	14.6556			0.517831
$802$		0			0
$803$	−9.13024	−	9.13024i	−0.322199	−	0.322199i
$804$		0			0
$805$		0			0
$806$		0			0
$807$	−1.04044	+	1.04044i	−0.0366251	+	0.0366251i
$808$		0			0
$809$		−	2.08714i		−	0.0733799i	−0.999327	−	0.0366899i	$-0.988319\pi$
$809$		−	2.08714i		−	0.0733799i	0.999327	−	0.0366899i	$-0.0116814\pi$
$810$		0			0
$811$		−	9.00244i		−	0.316118i	−0.987430	−	0.158059i	$-0.949476\pi$
$811$		−	9.00244i		−	0.316118i	0.987430	−	0.158059i	$-0.0505237\pi$
$812$		0			0
$813$	−16.3059	+	16.3059i	−0.571873	+	0.571873i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	15.7458	+	15.7458i	0.550876	+	0.550876i
$818$		0			0
$819$	10.6875			0.373451
$820$		0			0
$821$	4.24191			0.148044			0.0740219	−	0.997257i	$-0.476417\pi$
$821$	4.24191			0.148044			0.0740219	+	0.997257i	$0.476417\pi$
$822$		0			0
$823$	−2.96525	−	2.96525i	−0.103362	−	0.103362i	0.653535	−	0.756897i	$-0.273285\pi$
$823$	−2.96525	−	2.96525i	−0.103362	−	0.103362i	−0.756897	+	0.653535i	$0.773285\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	15.7873	−	15.7873i	0.548977	−	0.548977i	−0.377168	−	0.926145i	$-0.623102\pi$
$827$	15.7873	−	15.7873i	0.548977	−	0.548977i	0.926145	+	0.377168i	$0.123102\pi$
$828$		0			0
$829$		−	3.69672i		−	0.128393i	−0.997937	−	0.0641963i	$-0.979552\pi$
$829$		−	3.69672i		−	0.128393i	0.997937	−	0.0641963i	$-0.0204484\pi$
$830$		0			0
$831$		−	18.2655i		−	0.633622i
$832$		0			0
$833$	−2.57723	+	2.57723i	−0.0892956	+	0.0892956i
$834$		0			0
$835$		0			0
$836$		0			0
$837$	−5.95680	−	5.95680i	−0.205897	−	0.205897i
$838$		0			0
$839$	2.50468			0.0864713			0.0432356	−	0.999065i	$-0.486233\pi$
$839$	2.50468			0.0864713			0.0432356	+	0.999065i	$0.486233\pi$
$840$		0			0
$841$	−24.6967			−0.851611
$842$		0			0
$843$	7.13092	+	7.13092i	0.245602	+	0.245602i
$844$		0			0
$845$		0			0
$846$		0			0
$847$	20.3898	−	20.3898i	0.700601	−	0.700601i
$848$		0			0
$849$			11.5710i			0.397114i
$850$		0			0
$851$			0.771011i			0.0264299i
$852$		0			0
$853$	−29.7252	+	29.7252i	−1.01777	+	1.01777i	−0.0179306	+	0.999839i	$0.505708\pi$
$853$	−29.7252	+	29.7252i	−1.01777	+	1.01777i	−0.999839	+	0.0179306i	$0.994292\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	24.8191	+	24.8191i	0.847806	+	0.847806i	0.989859	−	0.142053i	$-0.0453705\pi$
$857$	24.8191	+	24.8191i	0.847806	+	0.847806i	−0.142053	+	0.989859i	$0.545370\pi$
$858$		0			0
$859$	−17.3205			−0.590968			−0.295484	−	0.955348i	$-0.595481\pi$
$859$	−17.3205			−0.590968			−0.295484	+	0.955348i	$0.595481\pi$
$860$		0			0
$861$	23.0713			0.786268
$862$		0			0
$863$	−20.6463	−	20.6463i	−0.702809	−	0.702809i	0.262204	−	0.965013i	$-0.415551\pi$
$863$	−20.6463	−	20.6463i	−0.702809	−	0.702809i	−0.965013	+	0.262204i	$0.915551\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	−5.50631	+	5.50631i	−0.187004	+	0.187004i
$868$		0			0
$869$		−	1.48052i		−	0.0502231i
$870$		0			0
$871$		−	5.48669i		−	0.185909i
$872$		0			0
$873$	12.9108	−	12.9108i	0.436965	−	0.436965i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−40.8595	−	40.8595i	−1.37973	−	1.37973i	−0.845069	−	0.534658i	$-0.820440\pi$
$877$	−40.8595	−	40.8595i	−1.37973	−	1.37973i	−0.534658	−	0.845069i	$-0.679560\pi$
$878$		0			0
$879$	28.7753			0.970567
$880$		0			0
$881$	−22.5145			−0.758533			−0.379266	−	0.925287i	$-0.623824\pi$
$881$	−22.5145			−0.758533			−0.379266	+	0.925287i	$0.623824\pi$
$882$		0			0
$883$	−34.0215	−	34.0215i	−1.14491	−	1.14491i	−0.987539	−	0.157376i	$-0.949697\pi$
$883$	−34.0215	−	34.0215i	−1.14491	−	1.14491i	−0.157376	−	0.987539i	$-0.550303\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	0.959187	−	0.959187i	0.0322063	−	0.0322063i	−0.690820	−	0.723027i	$-0.742750\pi$
$887$	0.959187	−	0.959187i	0.0322063	−	0.0322063i	0.723027	+	0.690820i	$0.242750\pi$
$888$		0			0
$889$		−	9.57021i		−	0.320975i
$890$		0			0
$891$			0.964724i			0.0323195i
$892$		0			0
$893$	32.7912	−	32.7912i	1.09731	−	1.09731i
$894$		0			0
$895$		0			0
$896$		0			0
$897$	−1.05453	−	1.05453i	−0.0352098	−	0.0352098i
$898$		0			0
$899$	61.7308			2.05884
$900$		0			0
$901$	−10.0121			−0.333552
$902$		0			0
$903$	−9.45841	−	9.45841i	−0.314756	−	0.314756i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−2.24337	+	2.24337i	−0.0744899	+	0.0744899i	−0.743370	−	0.668880i	$-0.766774\pi$
$907$	−2.24337	+	2.24337i	−0.0744899	+	0.0744899i	0.668880	+	0.743370i	$0.266774\pi$
$908$		0			0
$909$		−	12.2560i		−	0.406506i
$910$		0			0
$911$		−	17.4238i		−	0.577276i	−0.957438	−	0.288638i	$-0.906798\pi$
$911$		−	17.4238i		−	0.577276i	0.957438	−	0.288638i	$-0.0932025\pi$
$912$		0			0
$913$	8.63319	−	8.63319i	0.285717	−	0.285717i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−44.1703	−	44.1703i	−1.45863	−	1.45863i
$918$		0			0
$919$	−1.74038			−0.0574098			−0.0287049	−	0.999588i	$-0.509138\pi$
$919$	−1.74038			−0.0574098			−0.0287049	+	0.999588i	$0.509138\pi$
$920$		0			0
$921$	11.0565			0.364323
$922$		0			0
$923$	12.7792	+	12.7792i	0.420633	+	0.420633i
$924$		0			0
$925$		0			0
$926$		0			0
$927$	−0.292529	+	0.292529i	−0.00960790	+	0.00960790i
$928$		0			0
$929$			32.7403i			1.07417i	0.843527	+	0.537087i	$0.180475\pi$
$929$			32.7403i			1.07417i	−0.843527	+	0.537087i	$0.819525\pi$
$930$		0			0
$931$		−	5.72459i		−	0.187616i
$932$		0			0
$933$	20.7327	−	20.7327i	0.678759	−	0.678759i
$934$		0			0
$935$		0			0
$936$		0			0
$937$	0.418742	+	0.418742i	0.0136797	+	0.0136797i	0.713914	−	0.700234i	$-0.246921\pi$
$937$	0.418742	+	0.418742i	0.0136797	+	0.0136797i	−0.700234	+	0.713914i	$0.746921\pi$
$938$		0			0
$939$	4.33850			0.141582
$940$		0			0
$941$	18.1507			0.591697			0.295848	−	0.955235i	$-0.404398\pi$
$941$	18.1507			0.591697			0.295848	+	0.955235i	$0.404398\pi$
$942$		0			0
$943$	−2.27644	−	2.27644i	−0.0741311	−	0.0741311i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	32.8359	−	32.8359i	1.06702	−	1.06702i	0.0694373	−	0.997586i	$-0.477880\pi$
$947$	32.8359	−	32.8359i	1.06702	−	1.06702i	0.997586	−	0.0694373i	$-0.0221204\pi$
$948$		0			0
$949$		−	49.9507i		−	1.62147i
$950$		0			0
$951$		−	8.52661i		−	0.276494i
$952$		0			0
$953$	30.8615	−	30.8615i	0.999702	−	0.999702i	−0.000297967	−	1.00000i	$-0.500095\pi$
$953$	30.8615	−	30.8615i	0.999702	−	0.999702i	1.00000		0.000297967i	$9.48460e-5\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$	−4.99876	−	4.99876i	−0.161587	−	0.161587i
$958$		0			0
$959$	−50.2140			−1.62150
$960$		0			0
$961$	−39.9668			−1.28925
$962$		0			0
$963$	10.6955	+	10.6955i	0.344659	+	0.344659i
$964$		0			0
$965$		0			0
$966$		0			0
$967$	25.1399	−	25.1399i	0.808443	−	0.808443i	−0.175955	−	0.984398i	$-0.556301\pi$
$967$	25.1399	−	25.1399i	0.808443	−	0.808443i	0.984398	+	0.175955i	$0.0563013\pi$
$968$		0			0
$969$			14.4702i			0.464848i
$970$		0			0
$971$			6.84587i			0.219695i	0.993948	+	0.109847i	$0.0350362\pi$
$971$			6.84587i			0.219695i	−0.993948	+	0.109847i	$0.964964\pi$
$972$		0			0
$973$	1.08516	−	1.08516i	0.0347888	−	0.0347888i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	1.62209	+	1.62209i	0.0518954	+	0.0518954i	0.732578	−	0.680683i	$-0.238317\pi$
$977$	1.62209	+	1.62209i	0.0518954	+	0.0518954i	−0.680683	+	0.732578i	$0.738317\pi$
$978$		0			0
$979$	−14.1386			−0.451872
$980$		0			0
$981$	−17.1197			−0.546590
$982$		0			0
$983$	1.61553	+	1.61553i	0.0515273	+	0.0515273i	0.732401	−	0.680874i	$-0.238400\pi$
$983$	1.61553	+	1.61553i	0.0515273	+	0.0515273i	−0.680874	+	0.732401i	$0.738400\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	−19.6975	+	19.6975i	−0.626977	+	0.626977i
$988$		0			0
$989$			1.86652i			0.0593519i
$990$		0			0
$991$		−	45.1617i		−	1.43461i	−0.696760	−	0.717304i	$-0.745376\pi$
$991$		−	45.1617i		−	1.43461i	0.696760	−	0.717304i	$-0.254624\pi$
$992$		0			0
$993$	14.9761	−	14.9761i	0.475252	−	0.475252i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	5.54394	+	5.54394i	0.175578	+	0.175578i	0.789425	−	0.613847i	$-0.210379\pi$
$997$	5.54394	+	5.54394i	0.175578	+	0.175578i	−0.613847	+	0.789425i	$0.710379\pi$
$998$		0			0
$999$	1.92945			0.0610450

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2400.2.w.l.607.2	yes	8
4.3	odd	2		2400.2.w.h.607.3	yes	8
5.2	odd	4		2400.2.w.k.2143.2	yes	8
5.3	odd	4		2400.2.w.h.2143.3	yes	8
5.4	even	2		2400.2.w.g.607.3	✓	8
20.3	even	4	inner	2400.2.w.l.2143.2	yes	8
20.7	even	4		2400.2.w.g.2143.3	yes	8
20.19	odd	2		2400.2.w.k.607.2	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2400.2.w.g.607.3	✓	8	5.4	even	2
2400.2.w.g.2143.3	yes	8	20.7	even	4
2400.2.w.h.607.3	yes	8	4.3	odd	2
2400.2.w.h.2143.3	yes	8	5.3	odd	4
2400.2.w.k.607.2	yes	8	20.19	odd	2
2400.2.w.k.2143.2	yes	8	5.2	odd	4
2400.2.w.l.607.2	yes	8	1.1	even	1	trivial
2400.2.w.l.2143.2	yes	8	20.3	even	4	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:	\( N \)	\(=\)	\( 2400 = 2^{5} \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2400.w (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.707107 - 0.707107i) q^{3} +(2.02494 - 2.02494i) q^{7} +1.00000i q^{9} -0.964724i q^{11} +(2.63896 - 2.63896i) q^{13} +(-2.14626 - 2.14626i) q^{17} +4.76733 q^{19} -2.86370 q^{21} +(0.282561 + 0.282561i) q^{23} +(0.707107 - 0.707107i) q^{27} +7.32780i q^{29} -8.42418i q^{31} +(-0.682163 + 0.682163i) q^{33} +(1.36433 + 1.36433i) q^{37} -3.73205 q^{39} -8.05646 q^{41} +(3.30286 + 3.30286i) q^{43} +(6.87832 - 6.87832i) q^{47} -1.20080i q^{49} +3.03528i q^{51} +(2.33245 - 2.33245i) q^{53} +(-3.37101 - 3.37101i) q^{57} -3.46410 q^{59} -10.0504 q^{61} +(2.02494 + 2.02494i) q^{63} +(1.03956 - 1.03956i) q^{67} -0.399602i q^{69} +4.84252i q^{71} +(9.46410 - 9.46410i) q^{73} +(-1.95351 - 1.95351i) q^{77} +1.53465 q^{79} -1.00000 q^{81} +(8.94887 + 8.94887i) q^{83} +(5.18154 - 5.18154i) q^{87} -14.6556i q^{89} -10.6875i q^{91} +(-5.95680 + 5.95680i) q^{93} +(-12.9108 - 12.9108i) q^{97} +0.964724 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{7} + 8 q^{17} + 16 q^{19} + 8 q^{21} + 8 q^{23} - 8 q^{33} + 16 q^{37} - 16 q^{39} - 16 q^{41} - 24 q^{43} + 16 q^{47} + 8 q^{53} + 8 q^{57} - 24 q^{61} + 8 q^{63} - 8 q^{67} + 48 q^{73} - 8 q^{77}+ \cdots + 16 q^{99}+O(q^{100}) \) 8 * q + 8 * q^7 + 8 * q^17 + 16 * q^19 + 8 * q^21 + 8 * q^23 - 8 * q^33 + 16 * q^37 - 16 * q^39 - 16 * q^41 - 24 * q^43 + 16 * q^47 + 8 * q^53 + 8 * q^57 - 24 * q^61 + 8 * q^63 - 8 * q^67 + 48 * q^73 - 8 * q^77 - 32 * q^79 - 8 * q^81 + 16 * q^83 + 8 * q^87 - 24 * q^93 - 32 * q^97 + 16 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more