LMFDB - Embedded newform 2400.2.w.h.2143.3

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			2143.3
Root			$0.258819 - 0.965926i$ of defining polynomial
Character	$\chi$	$=$	2400.2143
Dual form			2400.2.w.h.607.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.707107 - 0.707107i) q^{3} +(-2.02494 - 2.02494i) q^{7} -1.00000i q^{9} -0.964724i q^{11} +(2.63896 + 2.63896i) q^{13} +(-2.14626 + 2.14626i) q^{17} -4.76733 q^{19} -2.86370 q^{21} +(-0.282561 + 0.282561i) q^{23} +(-0.707107 - 0.707107i) q^{27} -7.32780i q^{29} -8.42418i q^{31} +(-0.682163 - 0.682163i) q^{33} +(1.36433 - 1.36433i) q^{37} +3.73205 q^{39} -8.05646 q^{41} +(-3.30286 + 3.30286i) q^{43} +(-6.87832 - 6.87832i) q^{47} +1.20080i q^{49} +3.03528i q^{51} +(2.33245 + 2.33245i) q^{53} +(-3.37101 + 3.37101i) q^{57} +3.46410 q^{59} -10.0504 q^{61} +(-2.02494 + 2.02494i) q^{63} +(-1.03956 - 1.03956i) q^{67} +0.399602i q^{69} +4.84252i q^{71} +(9.46410 + 9.46410i) q^{73} +(-1.95351 + 1.95351i) q^{77} -1.53465 q^{79} -1.00000 q^{81} +(-8.94887 + 8.94887i) q^{83} +(-5.18154 - 5.18154i) q^{87} +14.6556i q^{89} -10.6875i q^{91} +(-5.95680 - 5.95680i) q^{93} +(-12.9108 + 12.9108i) q^{97} -0.964724 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $\left(\mathbb{Z}/2400\mathbb{Z}\right)^\times$.

$n$	$577$	$901$	$1601$	$1951$
$\chi(n)$	$e\left(\frac{3}{4}\right)$	$1$	$1$	$-1$

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	$-0.432026\pi$
$7$	−2.02494	−	2.02494i	−0.765357	−	0.765357i	−0.977285	+	0.211928i	$0.932026\pi$
$8$		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	0.970999	−	0.239084i	$-0.0768470\pi$
$13$	2.63896	+	2.63896i	0.731915	+	0.731915i	−0.239084	+	0.970999i	$0.576847\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−2.14626	+	2.14626i	−0.520546	+	0.520546i	−0.917736	−	0.397191i	$-0.869985\pi$
$17$	−2.14626	+	2.14626i	−0.520546	+	0.520546i	0.397191	+	0.917736i	$0.369985\pi$
$18$		0			0
$19$	−4.76733			−1.09370			−0.546850	−	0.837231i	$-0.684173\pi$
$19$	−4.76733			−1.09370			−0.546850	+	0.837231i	$0.684173\pi$
$20$		0			0
$21$	−2.86370			−0.624911
$22$		0			0
$23$	−0.282561	+	0.282561i	−0.0589181	+	0.0589181i	−0.735952	−	0.677034i	$-0.763265\pi$
$23$	−0.282561	+	0.282561i	−0.0589181	+	0.0589181i	0.677034	+	0.735952i	$0.263265\pi$
$24$		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	$-0.761819\pi$
$29$		−	7.32780i		−	1.36074i	0.732869	−	0.680370i	$-0.238181\pi$
$30$		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	−0.586010	−	0.810304i	$-0.699302\pi$
$37$	1.36433	−	1.36433i	0.224294	−	0.224294i	0.810304	+	0.586010i	$0.199302\pi$
$38$		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	$-0.865913\pi$
$43$	−3.30286	+	3.30286i	−0.503682	+	0.503682i	0.408898	+	0.912580i	$0.365913\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−6.87832	−	6.87832i	−1.00331	−	1.00331i	−0.999995	−	0.00331087i	$-0.998946\pi$
$47$	−6.87832	−	6.87832i	−1.00331	−	1.00331i	−0.00331087	−	0.999995i	$-0.501054\pi$
$48$		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	0.848915	−	0.528529i	$-0.177256\pi$
$53$	2.33245	+	2.33245i	0.320387	+	0.320387i	−0.528529	+	0.848915i	$0.677256\pi$
$54$		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	$-0.427600\pi$
$59$	3.46410			0.450988			0.225494	+	0.974245i	$0.427600\pi$
$60$		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	$-0.278624\pi$
$67$	−1.03956	−	1.03956i	−0.127002	−	0.127002i	−0.767751	+	0.640749i	$0.778624\pi$
$68$		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	0.993454	+	0.114236i	$0.0364419\pi$
$73$	9.46410	+	9.46410i	1.10769	+	1.10769i	0.114236	+	0.993454i	$0.463558\pi$
$74$		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	$-0.527514\pi$
$79$	−1.53465			−0.172662			−0.0863310	+	0.996267i	$0.527514\pi$
$80$		0			0
$81$	−1.00000			−0.111111
$82$		0			0
$83$	−8.94887	+	8.94887i	−0.982266	+	0.982266i	−0.999845	−	0.0175796i	$-0.994404\pi$
$83$	−8.94887	+	8.94887i	−0.982266	+	0.982266i	0.0175796	+	0.999845i	$0.494404\pi$
$84$		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	$0.283131\pi$
$89$			14.6556i			1.55349i	−0.629814	+	0.776746i	$0.716869\pi$
$90$		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	−0.390140	+	0.920756i	$0.627573\pi$
$97$	−12.9108	+	12.9108i	−1.31090	+	1.31090i	−0.920756	+	0.390140i	$0.872427\pi$
$98$		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	$-0.756488\pi$
$103$	−0.292529	+	0.292529i	−0.0288237	+	0.0288237i	0.692548	+	0.721372i	$0.256488\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−10.6955	−	10.6955i	−1.03398	−	1.03398i	−0.999402	−	0.0345738i	$-0.988993\pi$
$107$	−10.6955	−	10.6955i	−1.03398	−	1.03398i	−0.0345738	−	0.999402i	$-0.511007\pi$
$108$		0			0
$109$		−	17.1197i		−	1.63977i	−0.572528	−	0.819885i	$-0.694037\pi$
$109$		−	17.1197i		−	1.63977i	0.572528	−	0.819885i	$-0.305963\pi$
$110$		0			0
$111$		−	1.92945i		−	0.183135i
$112$		0			0
$113$	−8.00000	−	8.00000i	−0.752577	−	0.752577i	0.222383	−	0.974959i	$-0.428617\pi$
$113$	−8.00000	−	8.00000i	−0.752577	−	0.752577i	−0.974959	+	0.222383i	$0.928617\pi$
$114$		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	$-0.297371\pi$
$127$	−2.36308	−	2.36308i	−0.209690	−	0.209690i	−0.804136	+	0.594446i	$0.797371\pi$
$128$		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	−0.0611805	+	0.998127i	$0.519487\pi$
$137$	−12.3989	+	12.3989i	−1.05931	+	1.05931i	−0.998127	+	0.0611805i	$0.980513\pi$
$138$		0			0
$139$	−0.535898			−0.0454543			−0.0227272	−	0.999742i	$-0.507235\pi$
$139$	−0.535898			−0.0454543			−0.0227272	+	0.999742i	$0.507235\pi$
$140$		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	$-0.937176\pi$
$149$		−	4.78710i		−	0.392174i	0.980586	−	0.196087i	$-0.0628236\pi$
$150$		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	−0.906477	−	0.422256i	$-0.861238\pi$
$157$	−6.06727	+	6.06727i	−0.484221	+	0.484221i	0.422256	+	0.906477i	$0.361238\pi$
$158$		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	0.191949	−	0.981405i	$-0.438519\pi$
$163$	14.9804	−	14.9804i	1.17335	−	1.17335i	0.981405	−	0.191949i	$-0.0614808\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	14.0306	+	14.0306i	1.08572	+	1.08572i	0.995963	+	0.0897594i	$0.0286098\pi$
$167$	14.0306	+	14.0306i	1.08572	+	1.08572i	0.0897594	+	0.995963i	$0.471390\pi$
$168$		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	0.985967	−	0.166938i	$-0.0533880\pi$
$173$	10.7727	+	10.7727i	0.819029	+	0.819029i	−0.166938	+	0.985967i	$0.553388\pi$
$174$		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	$-0.439412\pi$
$179$	5.06251			0.378390			0.189195	+	0.981940i	$0.439412\pi$
$180$		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	0.793918	−	0.608025i	$-0.208038\pi$
$193$	2.58250	+	2.58250i	0.185893	+	0.185893i	−0.608025	+	0.793918i	$0.708038\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	12.2991	−	12.2991i	0.876274	−	0.876274i	−0.116873	−	0.993147i	$-0.537287\pi$
$197$	12.2991	−	12.2991i	0.876274	−	0.876274i	0.993147	+	0.116873i	$0.0372869\pi$
$198$		0			0
$199$	1.43223			0.101528			0.0507639	−	0.998711i	$-0.483834\pi$
$199$	1.43223			0.101528			0.0507639	+	0.998711i	$0.483834\pi$
$200$		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	$-0.509926\pi$
$223$	14.4603	−	14.4603i	0.968336	−	0.968336i	0.999514	+	0.0311780i	$0.00992589\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−12.2939	−	12.2939i	−0.815977	−	0.815977i	0.169545	−	0.985522i	$-0.445770\pi$
$227$	−12.2939	−	12.2939i	−0.815977	−	0.815977i	−0.985522	+	0.169545i	$0.945770\pi$
$228$		0			0
$229$		−	14.7342i		−	0.973664i	−0.873496	−	0.486832i	$-0.838152\pi$
$229$		−	14.7342i		−	0.973664i	0.873496	−	0.486832i	$-0.161848\pi$
$230$		0			0
$231$			2.76268i			0.181771i
$232$		0			0
$233$	−1.06110	−	1.06110i	−0.0695150	−	0.0695150i	0.671495	−	0.741010i	$-0.265653\pi$
$233$	−1.06110	−	1.06110i	−0.0695150	−	0.0695150i	−0.741010	+	0.671495i	$0.765653\pi$
$234$		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	$-0.627040\pi$
$239$	−12.0151			−0.777194			−0.388597	+	0.921408i	$0.627040\pi$
$240$		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	−0.925369	−	0.379068i	$-0.876245\pi$
$257$	−8.75787	+	8.75787i	−0.546301	+	0.546301i	0.379068	+	0.925369i	$0.376245\pi$
$258$		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	$-0.937102\pi$
$263$	−12.7180	+	12.7180i	−0.784223	+	0.784223i	0.196318	+	0.980540i	$0.437102\pi$
$264$		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	$0.0142830\pi$
$269$			1.47140i			0.0897127i	−0.998993	+	0.0448564i	$0.985717\pi$
$270$		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	−0.203127	−	0.979152i	$-0.565111\pi$
$277$	12.9156	−	12.9156i	0.776025	−	0.776025i	0.979152	+	0.203127i	$0.0651105\pi$
$278$		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	$-0.638248\pi$
$283$	8.18191	−	8.18191i	0.486364	−	0.486364i	0.907157	+	0.420793i	$0.138248\pi$
$284$		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	−0.977428	−	0.211270i	$-0.932240\pi$
$293$	−20.3472	−	20.3472i	−1.18870	−	1.18870i	−0.211270	−	0.977428i	$-0.567760\pi$
$294$		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	$-0.147824\pi$
$307$	7.81809	+	7.81809i	0.446202	+	0.446202i	−0.447888	+	0.894090i	$0.647824\pi$
$308$		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	0.615071	−	0.788472i	$-0.289127\pi$
$313$	−3.06778	−	3.06778i	−0.173401	−	0.173401i	−0.788472	+	0.615071i	$0.789127\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	6.02922	−	6.02922i	0.338635	−	0.338635i	−0.517218	−	0.855853i	$-0.673033\pi$
$317$	6.02922	−	6.02922i	0.338635	−	0.338635i	0.855853	+	0.517218i	$0.173033\pi$
$318$		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	−0.122999	−	0.992407i	$-0.539251\pi$
$337$	15.9602	−	15.9602i	0.869407	−	0.869407i	0.992407	+	0.122999i	$0.0392514\pi$
$338$		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	$-0.345916\pi$
$347$	−7.81862	−	7.81862i	−0.419726	−	0.419726i	−0.885109	+	0.465383i	$0.845916\pi$
$348$		0			0
$349$		−	25.4548i		−	1.36256i	−0.732021	−	0.681282i	$-0.761423\pi$
$349$		−	25.4548i		−	1.36256i	0.732021	−	0.681282i	$-0.238577\pi$
$350$		0			0
$351$		−	3.73205i		−	0.199202i
$352$		0			0
$353$	1.71263	+	1.71263i	0.0911541	+	0.0911541i	0.751213	−	0.660059i	$-0.229469\pi$
$353$	1.71263	+	1.71263i	0.0911541	+	0.0911541i	−0.660059	+	0.751213i	$0.729469\pi$
$354$		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	$-0.338635\pi$
$359$	18.3981			0.971017			0.485508	+	0.874232i	$0.338635\pi$
$360$		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	$-0.174483\pi$
$367$	6.36736	+	6.36736i	0.332374	+	0.332374i	−0.521114	+	0.853487i	$0.674483\pi$
$368$		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	0.919400	−	0.393323i	$-0.128675\pi$
$373$	10.1602	+	10.1602i	0.526078	+	0.526078i	−0.393323	+	0.919400i	$0.628675\pi$
$374$		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	$-0.449421\pi$
$379$	6.16088			0.316463			0.158231	+	0.987402i	$0.449421\pi$
$380$		0			0
$381$	−3.34190			−0.171211
$382$		0			0
$383$	10.8484	−	10.8484i	0.554325	−	0.554325i	−0.373361	−	0.927686i	$-0.621795\pi$
$383$	10.8484	−	10.8484i	0.554325	−	0.554325i	0.927686	+	0.373361i	$0.121795\pi$
$384$		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	$-0.892385\pi$
$389$		−	13.0834i		−	0.663355i	0.943393	−	0.331677i	$-0.107615\pi$
$390$		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	−0.147215	−	0.989104i	$-0.547031\pi$
$397$	16.7745	−	16.7745i	0.841889	−	0.841889i	0.989104	+	0.147215i	$0.0470310\pi$
$398$		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	$-0.842077\pi$
$409$		−	19.2540i		−	0.952050i	0.879432	−	0.476025i	$-0.157923\pi$
$410$		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	$-0.346148\pi$
$419$	19.0260			0.929480			0.464740	+	0.885447i	$0.346148\pi$
$420$		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	0.815749	−	0.578406i	$-0.196325\pi$
$433$	4.93878	+	4.93878i	0.237343	+	0.237343i	−0.578406	+	0.815749i	$0.696325\pi$
$434$		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	$-0.812019\pi$
$439$	−34.8072			−1.66126			−0.830628	+	0.556827i	$0.812019\pi$
$440$		0			0
$441$	1.20080			0.0571808
$442$		0			0
$443$	−7.82038	+	7.82038i	−0.371558	+	0.371558i	−0.868044	−	0.496487i	$-0.834623\pi$
$443$	−7.82038	+	7.82038i	−0.371558	+	0.371558i	0.496487	+	0.868044i	$0.334623\pi$
$444$		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	$-0.908618\pi$
$449$		−	12.0000i		−	0.566315i	0.959073	−	0.283158i	$-0.0913819\pi$
$450$		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	−0.817642	−	0.575726i	$-0.804719\pi$
$457$	−5.17157	+	5.17157i	−0.241916	+	0.241916i	0.575726	+	0.817642i	$0.304719\pi$
$458$		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	$-0.597245\pi$
$463$	14.0492	−	14.0492i	0.652920	−	0.652920i	0.953695	+	0.300775i	$0.0972454\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	7.96864	+	7.96864i	0.368745	+	0.368745i	0.867019	−	0.498275i	$-0.166033\pi$
$467$	7.96864	+	7.96864i	0.368745	+	0.368745i	−0.498275	+	0.867019i	$0.666033\pi$
$468$		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	$-0.872422\pi$
$479$	−40.3032			−1.84150			−0.920750	+	0.390154i	$0.872422\pi$
$480$		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	$-0.108304\pi$
$487$	13.4384	+	13.4384i	0.608953	+	0.608953i	−0.333719	+	0.942673i	$0.608304\pi$
$488$		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	$-0.902911\pi$
$499$	−42.6144			−1.90768			−0.953842	+	0.300308i	$0.902911\pi$
$500$		0			0
$501$	19.8423			0.886489
$502$		0			0
$503$	13.5653	−	13.5653i	0.604846	−	0.604846i	−0.336748	−	0.941595i	$-0.609327\pi$
$503$	13.5653	−	13.5653i	0.604846	−	0.604846i	0.941595	+	0.336748i	$0.109327\pi$
$504$		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	$0.156674\pi$
$509$			21.3233i			0.945140i	−0.881293	+	0.472570i	$0.843326\pi$
$510$		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	$-0.639444\pi$
$523$	11.0086	−	11.0086i	0.481371	−	0.481371i	0.905569	+	0.424198i	$0.139444\pi$
$524$		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	$-0.151319\pi$
$547$	10.0905	+	10.0905i	0.431438	+	0.431438i	−0.457679	+	0.889117i	$0.651319\pi$
$548$		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	0.0450344	−	0.998985i	$-0.485660\pi$
$557$	24.6398	−	24.6398i	1.04402	−	1.04402i	0.998985	−	0.0450344i	$-0.0143398\pi$
$558$		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	$-0.774441\pi$
$563$	−2.57406	+	2.57406i	−0.108484	+	0.108484i	0.650781	+	0.759265i	$0.274441\pi$
$564$		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	$-0.742028\pi$
$569$		−	34.5685i		−	1.44919i	0.689177	−	0.724593i	$-0.257972\pi$
$570$		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	−0.327938	−	0.944699i	$-0.606354\pi$
$577$	14.8151	−	14.8151i	0.616762	−	0.616762i	0.944699	+	0.327938i	$0.106354\pi$
$578$		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	−0.938046	−	0.346512i	$-0.887366\pi$
$587$	−31.1224	−	31.1224i	−1.28456	−	1.28456i	−0.346512	−	0.938046i	$-0.612634\pi$
$588$		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	0.114069	−	0.993473i	$-0.463612\pi$
$593$	−21.4149	−	21.4149i	−0.879404	−	0.879404i	−0.993473	+	0.114069i	$0.963612\pi$
$594$		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	$-0.374956\pi$
$599$	18.7383			0.765625			0.382812	+	0.923826i	$0.374956\pi$
$600$		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	$-0.131078\pi$
$607$	12.7165	+	12.7165i	0.516149	+	0.516149i	−0.400255	+	0.916404i	$0.631078\pi$
$608$		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	0.756835	−	0.653606i	$-0.226745\pi$
$613$	2.55583	+	2.55583i	0.103229	+	0.103229i	−0.653606	+	0.756835i	$0.726745\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−7.11172	+	7.11172i	−0.286307	+	0.286307i	−0.835618	−	0.549311i	$-0.814890\pi$
$617$	−7.11172	+	7.11172i	−0.286307	+	0.286307i	0.549311	+	0.835618i	$0.314890\pi$
$618$		0			0
$619$	0.0810355			0.00325709			0.00162855	−	0.999999i	$-0.499482\pi$
$619$	0.0810355			0.00325709			0.00162855	+	0.999999i	$0.499482\pi$
$620$		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	$-0.759056\pi$
$643$	−1.02015	+	1.02015i	−0.0402307	+	0.0402307i	0.686705	+	0.726936i	$0.259056\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	29.7766	+	29.7766i	1.17064	+	1.17064i	0.982058	+	0.188580i	$0.0603884\pi$
$647$	29.7766	+	29.7766i	1.17064	+	1.17064i	0.188580	+	0.982058i	$0.439612\pi$
$648$		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	0.779896	−	0.625909i	$-0.215272\pi$
$653$	3.93498	+	3.93498i	0.153988	+	0.153988i	−0.625909	+	0.779896i	$0.715272\pi$
$654$		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	$-0.476322\pi$
$659$	3.81558			0.148634			0.0743169	+	0.997235i	$0.476322\pi$
$660$		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	0.901913	−	0.431917i	$-0.142163\pi$
$673$	12.1928	+	12.1928i	0.469996	+	0.469996i	−0.431917	+	0.901913i	$0.642163\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−9.36020	+	9.36020i	−0.359742	+	0.359742i	−0.863718	−	0.503976i	$-0.831870\pi$
$677$	−9.36020	+	9.36020i	−0.359742	+	0.359742i	0.503976	+	0.863718i	$0.331870\pi$
$678$		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	−0.101261	+	0.994860i	$0.532288\pi$
$683$	−28.6463	+	28.6463i	−1.09612	+	1.09612i	−0.994860	+	0.101261i	$0.967712\pi$
$684$		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	$-0.896814\pi$
$709$		−	16.9625i		−	0.637041i	0.947916	−	0.318520i	$-0.103186\pi$
$710$		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	$-0.575609\pi$
$719$	−12.6191			−0.470613			−0.235306	+	0.971921i	$0.575609\pi$
$720$		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	$-0.369801\pi$
$727$	−14.0149	−	14.0149i	−0.519785	−	0.519785i	−0.917506	+	0.397721i	$0.869801\pi$
$728$		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	0.305029	−	0.952343i	$-0.401334\pi$
$733$	−17.5254	−	17.5254i	−0.647314	−	0.647314i	−0.952343	+	0.305029i	$0.901334\pi$
$734$		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	$-0.541739\pi$
$739$	−7.10885			−0.261503			−0.130752	+	0.991415i	$0.541739\pi$
$740$		0			0
$741$	−17.7919			−0.653602
$742$		0			0
$743$	−14.6075	+	14.6075i	−0.535897	+	0.535897i	−0.922321	−	0.386425i	$-0.873710\pi$
$743$	−14.6075	+	14.6075i	−0.535897	+	0.535897i	0.386425	+	0.922321i	$0.373710\pi$
$744$		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	−0.741357	−	0.671111i	$-0.765817\pi$
$757$	−1.93273	+	1.93273i	−0.0702463	+	0.0702463i	0.671111	+	0.741357i	$0.265817\pi$
$758$		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	$-0.843767\pi$
$769$		−	26.1419i		−	0.942702i	0.881946	−	0.471351i	$-0.156233\pi$
$770$		0			0
$771$			12.3855i			0.446053i
$772$		0			0
$773$	−11.7580	−	11.7580i	−0.422907	−	0.422907i	0.463296	−	0.886203i	$-0.346667\pi$
$773$	−11.7580	−	11.7580i	−0.422907	−	0.422907i	−0.886203	+	0.463296i	$0.846667\pi$
$774$		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	$-0.142879\pi$
$787$	13.1008	+	13.1008i	0.466994	+	0.466994i	−0.433945	+	0.900939i	$0.642879\pi$
$788$		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	−0.957825	−	0.287353i	$-0.907225\pi$
$797$	−18.9282	+	18.9282i	−0.670471	+	0.670471i	0.287353	+	0.957825i	$0.407225\pi$
$798$		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	$0.0116814\pi$
$809$			2.08714i			0.0733799i	−0.999327	+	0.0366899i	$0.988319\pi$
$810$		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	$-0.726715\pi$
$823$	2.96525	−	2.96525i	0.103362	−	0.103362i	0.756897	+	0.653535i	$0.226715\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−15.7873	−	15.7873i	−0.548977	−	0.548977i	0.377168	−	0.926145i	$-0.376898\pi$
$827$	−15.7873	−	15.7873i	−0.548977	−	0.548977i	−0.926145	+	0.377168i	$0.876898\pi$
$828$		0			0
$829$			3.69672i			0.128393i	0.997937	+	0.0641963i	$0.0204484\pi$
$829$			3.69672i			0.128393i	−0.997937	+	0.0641963i	$0.979552\pi$
$830$		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	$-0.513767\pi$
$839$	−2.50468			−0.0864713			−0.0432356	+	0.999065i	$0.513767\pi$
$840$		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	−0.999839	−	0.0179306i	$-0.994292\pi$
$853$	−29.7252	−	29.7252i	−1.01777	−	1.01777i	−0.0179306	−	0.999839i	$-0.505708\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	24.8191	−	24.8191i	0.847806	−	0.847806i	−0.142053	−	0.989859i	$-0.545370\pi$
$857$	24.8191	−	24.8191i	0.847806	−	0.847806i	0.989859	+	0.142053i	$0.0453705\pi$
$858$		0			0
$859$	17.3205			0.590968			0.295484	−	0.955348i	$-0.404519\pi$
$859$	17.3205			0.590968			0.295484	+	0.955348i	$0.404519\pi$
$860$		0			0
$861$	23.0713			0.786268
$862$		0			0
$863$	20.6463	−	20.6463i	0.702809	−	0.702809i	−0.262204	−	0.965013i	$-0.584449\pi$
$863$	20.6463	−	20.6463i	0.702809	−	0.702809i	0.965013	+	0.262204i	$0.0844492\pi$
$864$		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	−0.534658	+	0.845069i	$0.679560\pi$
$877$	−40.8595	+	40.8595i	−1.37973	+	1.37973i	−0.845069	+	0.534658i	$0.820440\pi$
$878$		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	0.157376	−	0.987539i	$-0.449697\pi$
$883$	34.0215	−	34.0215i	1.14491	−	1.14491i	0.987539	−	0.157376i	$-0.0503033\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−0.959187	−	0.959187i	−0.0322063	−	0.0322063i	0.690820	−	0.723027i	$-0.257250\pi$
$887$	−0.959187	−	0.959187i	−0.0322063	−	0.0322063i	−0.723027	+	0.690820i	$0.757250\pi$
$888$		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	$-0.233226\pi$
$907$	2.24337	+	2.24337i	0.0744899	+	0.0744899i	−0.668880	+	0.743370i	$0.733226\pi$
$908$		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	$-0.490862\pi$
$919$	1.74038			0.0574098			0.0287049	+	0.999588i	$0.490862\pi$
$920$		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	$-0.819525\pi$
$929$		−	32.7403i		−	1.07417i	0.843527	−	0.537087i	$-0.180475\pi$
$930$		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	−0.700234	−	0.713914i	$-0.746921\pi$
$937$	0.418742	−	0.418742i	0.0136797	−	0.0136797i	0.713914	+	0.700234i	$0.246921\pi$
$938$		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	−0.997586	−	0.0694373i	$-0.977880\pi$
$947$	−32.8359	−	32.8359i	−1.06702	−	1.06702i	−0.0694373	−	0.997586i	$-0.522120\pi$
$948$		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	1.00000		0.000297967i	$-9.48460e-5\pi$
$953$	30.8615	+	30.8615i	0.999702	+	0.999702i	−0.000297967		1.00000i	$0.500095\pi$
$954$		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	$-0.443699\pi$
$967$	−25.1399	−	25.1399i	−0.808443	−	0.808443i	−0.984398	+	0.175955i	$0.943699\pi$
$968$		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	−0.680683	−	0.732578i	$-0.738317\pi$
$977$	1.62209	−	1.62209i	0.0518954	−	0.0518954i	0.732578	+	0.680683i	$0.238317\pi$
$978$		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	$-0.761600\pi$
$983$	−1.61553	+	1.61553i	−0.0515273	+	0.0515273i	0.680874	+	0.732401i	$0.261600\pi$
$984$		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	−0.613847	−	0.789425i	$-0.710379\pi$
$997$	5.54394	−	5.54394i	0.175578	−	0.175578i	0.789425	+	0.613847i	$0.210379\pi$
$998$		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.h.2143.3	yes	8
4.3	odd	2		2400.2.w.l.2143.2	yes	8
5.2	odd	4		2400.2.w.l.607.2	yes	8
5.3	odd	4		2400.2.w.g.607.3	✓	8
5.4	even	2		2400.2.w.k.2143.2	yes	8
20.3	even	4		2400.2.w.k.607.2	yes	8
20.7	even	4	inner	2400.2.w.h.607.3	yes	8
20.19	odd	2		2400.2.w.g.2143.3	yes	8

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

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