LMFDB - Embedded newform 400.4.n.e.143.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [400,4,Mod(143,400)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(400, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([2, 0, 3]))

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("400.143");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 400 = 2^{4} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	400.n (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:	$23.6007640023$
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}\cdot 5^{8} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			143.3
Root			$-0.258819 + 0.965926i$ of defining polynomial
Character	$\chi$	$=$	400.143
Dual form			400.4.n.e.207.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(3.53553 - 3.53553i) q^{3} +(14.1421 + 14.1421i) q^{7} +2.00000i q^{9} -43.3013i q^{11} +(-12.2474 - 12.2474i) q^{13} +(42.8661 - 42.8661i) q^{17} +129.904 q^{19} +100.000 q^{21} +(-63.6396 + 63.6396i) q^{23} +(102.530 + 102.530i) q^{27} +66.0000i q^{29} +86.6025i q^{31} +(-153.093 - 153.093i) q^{33} +(293.939 - 293.939i) q^{37} -86.6025 q^{39} +453.000 q^{41} +(155.563 - 155.563i) q^{43} +(-275.772 - 275.772i) q^{47} +57.0000i q^{49} -303.109i q^{51} +(-232.702 - 232.702i) q^{53} +(459.279 - 459.279i) q^{57} -779.423 q^{59} +668.000 q^{61} +(-28.2843 + 28.2843i) q^{63} +(-194.454 - 194.454i) q^{67} +450.000i q^{69} +1125.83i q^{71} +(-508.269 - 508.269i) q^{73} +(612.372 - 612.372i) q^{77} +692.820 q^{79} +671.000 q^{81} +(10.6066 - 10.6066i) q^{83} +(233.345 + 233.345i) q^{87} -549.000i q^{89} -346.410i q^{91} +(306.186 + 306.186i) q^{93} +(-73.4847 + 73.4847i) q^{97} +86.6025 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 800 q^{21} + 3624 q^{41} + 5344 q^{61} + 5368 q^{81}+O(q^{100}) $ 8 * q + 800 * q^21 + 3624 * q^41 + 5344 * q^61 + 5368 * q^81

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/400\mathbb{Z}\right)^\times$.

$n$	$101$	$177$	$351$
$\chi(n)$	$1$	$e\left(\frac{3}{4}\right)$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$	3.53553	−	3.53553i	0.680414	−	0.680414i	−0.279680	−	0.960093i	$-0.590228\pi$
$3$	3.53553	−	3.53553i	0.680414	−	0.680414i	0.960093	+	0.279680i	$0.0902282\pi$
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	14.1421	+	14.1421i	0.763604	+	0.763604i	0.976972	−	0.213368i	$-0.0684434\pi$
$7$	14.1421	+	14.1421i	0.763604	+	0.763604i	−0.213368	+	0.976972i	$0.568443\pi$
$8$		0			0
$9$			2.00000i			0.0740741i
$10$		0			0
$11$		−	43.3013i		−	1.18689i	−0.804873	−	0.593447i	$-0.797767\pi$
$11$		−	43.3013i		−	1.18689i	0.804873	−	0.593447i	$-0.202233\pi$
$12$		0			0
$13$	−12.2474	−	12.2474i	−0.261295	−	0.261295i	0.564285	−	0.825580i	$-0.309152\pi$
$13$	−12.2474	−	12.2474i	−0.261295	−	0.261295i	−0.825580	+	0.564285i	$0.809152\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	42.8661	−	42.8661i	0.611562	−	0.611562i	−0.331791	−	0.943353i	$-0.607653\pi$
$17$	42.8661	−	42.8661i	0.611562	−	0.611562i	0.943353	+	0.331791i	$0.107653\pi$
$18$		0			0
$19$	129.904			1.56853			0.784263	−	0.620429i	$-0.213042\pi$
$19$	129.904			1.56853			0.784263	+	0.620429i	$0.213042\pi$
$20$		0			0
$21$	100.000			1.03913
$22$		0			0
$23$	−63.6396	+	63.6396i	−0.576947	+	0.576947i	−0.934061	−	0.357114i	$-0.883761\pi$
$23$	−63.6396	+	63.6396i	−0.576947	+	0.576947i	0.357114	+	0.934061i	$0.383761\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$	102.530	+	102.530i	0.730815	+	0.730815i
$28$		0			0
$29$			66.0000i			0.422617i	0.977419	+	0.211308i	$0.0677725\pi$
$29$			66.0000i			0.422617i	−0.977419	+	0.211308i	$0.932228\pi$
$30$		0			0
$31$			86.6025i			0.501751i	0.968019	+	0.250875i	$0.0807184\pi$
$31$			86.6025i			0.501751i	−0.968019	+	0.250875i	$0.919282\pi$
$32$		0			0
$33$	−153.093	−	153.093i	−0.807578	−	0.807578i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	293.939	−	293.939i	1.30603	−	1.30603i	0.381780	−	0.924253i	$-0.375311\pi$
$37$	293.939	−	293.939i	1.30603	−	1.30603i	0.924253	−	0.381780i	$-0.124689\pi$
$38$		0			0
$39$	−86.6025			−0.355577
$40$		0			0
$41$	453.000			1.72553			0.862765	−	0.505605i	$-0.168731\pi$
$41$	453.000			1.72553			0.862765	+	0.505605i	$0.168731\pi$
$42$		0			0
$43$	155.563	−	155.563i	0.551703	−	0.551703i	−0.375229	−	0.926932i	$-0.622436\pi$
$43$	155.563	−	155.563i	0.551703	−	0.551703i	0.926932	+	0.375229i	$0.122436\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−275.772	−	275.772i	−0.855860	−	0.855860i	0.134987	−	0.990847i	$-0.456901\pi$
$47$	−275.772	−	275.772i	−0.855860	−	0.855860i	−0.990847	+	0.134987i	$0.956901\pi$
$48$		0			0
$49$			57.0000i			0.166181i
$50$		0			0
$51$		−	303.109i		−	0.832230i
$52$		0			0
$53$	−232.702	−	232.702i	−0.603095	−	0.603095i	0.338038	−	0.941132i	$-0.390237\pi$
$53$	−232.702	−	232.702i	−0.603095	−	0.603095i	−0.941132	+	0.338038i	$0.890237\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	459.279	−	459.279i	1.06725	−	1.06725i
$58$		0			0
$59$	−779.423			−1.71987			−0.859934	−	0.510405i	$-0.829495\pi$
$59$	−779.423			−1.71987			−0.859934	+	0.510405i	$0.829495\pi$
$60$		0			0
$61$	668.000			1.40211			0.701054	−	0.713108i	$-0.252713\pi$
$61$	668.000			1.40211			0.701054	+	0.713108i	$0.252713\pi$
$62$		0			0
$63$	−28.2843	+	28.2843i	−0.0565632	+	0.0565632i
$64$		0			0
$65$		0			0
$66$		0			0
$67$	−194.454	−	194.454i	−0.354573	−	0.354573i	0.507235	−	0.861808i	$-0.330668\pi$
$67$	−194.454	−	194.454i	−0.354573	−	0.354573i	−0.861808	+	0.507235i	$0.830668\pi$
$68$		0			0
$69$			450.000i			0.785125i
$70$		0			0
$71$			1125.83i			1.88186i	0.338606	+	0.940928i	$0.390045\pi$
$71$			1125.83i			1.88186i	−0.338606	+	0.940928i	$0.609955\pi$
$72$		0			0
$73$	−508.269	−	508.269i	−0.814910	−	0.814910i	0.170456	−	0.985365i	$-0.445476\pi$
$73$	−508.269	−	508.269i	−0.814910	−	0.814910i	−0.985365	+	0.170456i	$0.945476\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	612.372	−	612.372i	0.906316	−	0.906316i
$78$		0			0
$79$	692.820			0.986688			0.493344	−	0.869834i	$-0.335774\pi$
$79$	692.820			0.986688			0.493344	+	0.869834i	$0.335774\pi$
$80$		0			0
$81$	671.000			0.920439
$82$		0			0
$83$	10.6066	−	10.6066i	0.0140268	−	0.0140268i	−0.700059	−	0.714085i	$-0.746843\pi$
$83$	10.6066	−	10.6066i	0.0140268	−	0.0140268i	0.714085	+	0.700059i	$0.246843\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	233.345	+	233.345i	0.287554	+	0.287554i
$88$		0			0
$89$		−	549.000i		−	0.653864i	−0.945048	−	0.326932i	$-0.893985\pi$
$89$		−	549.000i		−	0.653864i	0.945048	−	0.326932i	$-0.106015\pi$
$90$		0			0
$91$		−	346.410i		−	0.399051i
$92$		0			0
$93$	306.186	+	306.186i	0.341398	+	0.341398i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−73.4847	+	73.4847i	−0.0769200	+	0.0769200i	−0.744520	−	0.667600i	$-0.767322\pi$
$97$	−73.4847	+	73.4847i	−0.0769200	+	0.0769200i	0.667600	+	0.744520i	$0.267322\pi$
$98$		0			0
$99$	86.6025			0.0879180
$100$		0			0
$101$	−48.0000			−0.0472889			−0.0236444	−	0.999720i	$-0.507527\pi$
$101$	−48.0000			−0.0472889			−0.0236444	+	0.999720i	$0.507527\pi$
$102$		0			0
$103$	−813.173	+	813.173i	−0.777906	+	0.777906i	−0.979474	−	0.201569i	$-0.935396\pi$
$103$	−813.173	+	813.173i	−0.777906	+	0.777906i	0.201569	+	0.979474i	$0.435396\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	201.525	+	201.525i	0.182077	+	0.182077i	0.792260	−	0.610183i	$-0.208904\pi$
$107$	201.525	+	201.525i	0.182077	+	0.182077i	−0.610183	+	0.792260i	$0.708904\pi$
$108$		0			0
$109$			904.000i			0.794381i	0.917736	+	0.397190i	$0.130015\pi$
$109$			904.000i			0.794381i	−0.917736	+	0.397190i	$0.869985\pi$
$110$		0			0
$111$		−	2078.46i		−	1.77729i
$112$		0			0
$113$	−728.723	−	728.723i	−0.606659	−	0.606659i	0.335412	−	0.942072i	$-0.391124\pi$
$113$	−728.723	−	728.723i	−0.606659	−	0.606659i	−0.942072	+	0.335412i	$0.891124\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	24.4949	−	24.4949i	0.0193552	−	0.0193552i
$118$		0			0
$119$	1212.44			0.933981
$120$		0			0
$121$	−544.000			−0.408715
$122$		0			0
$123$	1601.60	−	1601.60i	1.17407	−	1.17407i
$124$		0			0
$125$		0			0
$126$		0			0
$127$	1279.86	+	1279.86i	0.894248	+	0.894248i	0.994920	−	0.100672i	$-0.0320992\pi$
$127$	1279.86	+	1279.86i	0.894248	+	0.894248i	−0.100672	+	0.994920i	$0.532099\pi$
$128$		0			0
$129$		−	1100.00i		−	0.750772i
$130$		0			0
$131$		−	86.6025i		−	0.0577595i	−0.999583	−	0.0288798i	$-0.990806\pi$
$131$		−	86.6025i		−	0.0577595i	0.999583	−	0.0288798i	$-0.00919399\pi$
$132$		0			0
$133$	1837.12	+	1837.12i	1.19773	+	1.19773i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−1953.47	+	1953.47i	−1.21822	+	1.21822i	−0.249964	+	0.968255i	$0.580419\pi$
$137$	−1953.47	+	1953.47i	−1.21822	+	1.21822i	−0.968255	+	0.249964i	$0.919581\pi$
$138$		0			0
$139$	129.904			0.0792683			0.0396342	−	0.999214i	$-0.487381\pi$
$139$	129.904			0.0792683			0.0396342	+	0.999214i	$0.487381\pi$
$140$		0			0
$141$	−1950.00			−1.16468
$142$		0			0
$143$	−530.330	+	530.330i	−0.310129	+	0.310129i
$144$		0			0
$145$		0			0
$146$		0			0
$147$	201.525	+	201.525i	0.113072	+	0.113072i
$148$		0			0
$149$			114.000i			0.0626795i	0.999509	+	0.0313397i	$0.00997738\pi$
$149$			114.000i			0.0626795i	−0.999509	+	0.0313397i	$0.990023\pi$
$150$		0			0
$151$			2251.67i			1.21350i	0.794894	+	0.606748i	$0.207526\pi$
$151$			2251.67i			1.21350i	−0.794894	+	0.606748i	$0.792474\pi$
$152$		0			0
$153$	85.7321	+	85.7321i	0.0453009	+	0.0453009i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−477.650	+	477.650i	−0.242807	+	0.242807i	−0.818010	−	0.575204i	$-0.804923\pi$
$157$	−477.650	+	477.650i	−0.242807	+	0.242807i	0.575204	+	0.818010i	$0.304923\pi$
$158$		0			0
$159$	−1645.45			−0.820708
$160$		0			0
$161$	−1800.00			−0.881117
$162$		0			0
$163$	−2202.64	+	2202.64i	−1.05843	+	1.05843i	−0.0602452	+	0.998184i	$0.519188\pi$
$163$	−2202.64	+	2202.64i	−1.05843	+	1.05843i	−0.998184	+	0.0602452i	$0.980812\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−2269.81	−	2269.81i	−1.05176	−	1.05176i	−0.998585	−	0.0531714i	$-0.983067\pi$
$167$	−2269.81	−	2269.81i	−1.05176	−	1.05176i	−0.0531714	−	0.998585i	$-0.516933\pi$
$168$		0			0
$169$		−	1897.00i		−	0.863450i
$170$		0			0
$171$			259.808i			0.116187i
$172$		0			0
$173$	1371.71	+	1371.71i	0.602830	+	0.602830i	0.941062	−	0.338233i	$-0.109829\pi$
$173$	1371.71	+	1371.71i	0.602830	+	0.602830i	−0.338233	+	0.941062i	$0.609829\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	−2755.68	+	2755.68i	−1.17022	+	1.17022i
$178$		0			0
$179$	1602.15			0.668995			0.334497	−	0.942397i	$-0.391433\pi$
$179$	1602.15			0.668995			0.334497	+	0.942397i	$0.391433\pi$
$180$		0			0
$181$	−3358.00			−1.37900			−0.689498	−	0.724288i	$-0.742169\pi$
$181$	−3358.00			−1.37900			−0.689498	+	0.724288i	$0.742169\pi$
$182$		0			0
$183$	2361.74	−	2361.74i	0.954014	−	0.954014i
$184$		0			0
$185$		0			0
$186$		0			0
$187$	−1856.16	−	1856.16i	−0.725858	−	0.725858i
$188$		0			0
$189$			2900.00i			1.11611i
$190$		0			0
$191$			259.808i			0.0984242i	0.998788	+	0.0492121i	$0.0156710\pi$
$191$			259.808i			0.0984242i	−0.998788	+	0.0492121i	$0.984329\pi$
$192$		0			0
$193$	324.557	+	324.557i	0.121047	+	0.121047i	0.765036	−	0.643988i	$-0.222721\pi$
$193$	324.557	+	324.557i	0.121047	+	0.121047i	−0.643988	+	0.765036i	$0.722721\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−1922.85	+	1922.85i	−0.695418	+	0.695418i	−0.963419	−	0.268001i	$-0.913637\pi$
$197$	−1922.85	+	1922.85i	−0.695418	+	0.695418i	0.268001	+	0.963419i	$0.413637\pi$
$198$		0			0
$199$	−4503.33			−1.60418			−0.802092	−	0.597200i	$-0.796280\pi$
$199$	−4503.33			−1.60418			−0.802092	+	0.597200i	$0.796280\pi$
$200$		0			0
$201$	−1375.00			−0.482513
$202$		0			0
$203$	−933.381	+	933.381i	−0.322712	+	0.322712i
$204$		0			0
$205$		0			0
$206$		0			0
$207$	−127.279	−	127.279i	−0.0427368	−	0.0427368i
$208$		0			0
$209$		−	5625.00i		−	1.86167i
$210$		0			0
$211$			2035.16i			0.664010i	0.943278	+	0.332005i	$0.107725\pi$
$211$			2035.16i			0.664010i	−0.943278	+	0.332005i	$0.892275\pi$
$212$		0			0
$213$	3980.42	+	3980.42i	1.28044	+	1.28044i
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−1224.74	+	1224.74i	−0.383139	+	0.383139i
$218$		0			0
$219$	−3594.01			−1.10895
$220$		0			0
$221$	−1050.00			−0.319596
$222$		0			0
$223$	3238.55	−	3238.55i	0.972508	−	0.972508i	−0.0271241	−	0.999632i	$-0.508635\pi$
$223$	3238.55	−	3238.55i	0.972508	−	0.972508i	0.999632	+	0.0271241i	$0.00863492\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	3903.23	+	3903.23i	1.14126	+	1.14126i	0.988220	+	0.153042i	$0.0489070\pi$
$227$	3903.23	+	3903.23i	1.14126	+	1.14126i	0.153042	+	0.988220i	$0.451093\pi$
$228$		0			0
$229$			3716.00i			1.07232i	0.844118	+	0.536158i	$0.180125\pi$
$229$			3716.00i			1.07232i	−0.844118	+	0.536158i	$0.819875\pi$
$230$		0			0
$231$		−	4330.13i		−	1.23334i
$232$		0			0
$233$	−4482.57	−	4482.57i	−1.26035	−	1.26035i	−0.950921	−	0.309434i	$-0.899860\pi$
$233$	−4482.57	−	4482.57i	−1.26035	−	1.26035i	−0.309434	−	0.950921i	$-0.600140\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	2449.49	−	2449.49i	0.671356	−	0.671356i
$238$		0			0
$239$	3204.29			0.867232			0.433616	−	0.901098i	$-0.357237\pi$
$239$	3204.29			0.867232			0.433616	+	0.901098i	$0.357237\pi$
$240$		0			0
$241$	−947.000			−0.253119			−0.126559	−	0.991959i	$-0.540393\pi$
$241$	−947.000			−0.253119			−0.126559	+	0.991959i	$0.540393\pi$
$242$		0			0
$243$	−395.980	+	395.980i	−0.104535	+	0.104535i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−1590.99	−	1590.99i	−0.409847	−	0.409847i
$248$		0			0
$249$		−	75.0000i		−	0.0190881i
$250$		0			0
$251$			1342.34i			0.337561i	0.985654	+	0.168780i	$0.0539828\pi$
$251$			1342.34i			0.337561i	−0.985654	+	0.168780i	$0.946017\pi$
$252$		0			0
$253$	2755.68	+	2755.68i	0.684774	+	0.684774i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−3159.84	+	3159.84i	−0.766948	+	0.766948i	−0.977568	−	0.210620i	$-0.932452\pi$
$257$	−3159.84	+	3159.84i	−0.766948	+	0.766948i	0.210620	+	0.977568i	$0.432452\pi$
$258$		0			0
$259$	8313.84			1.99458
$260$		0			0
$261$	−132.000			−0.0313050
$262$		0			0
$263$	−3585.03	+	3585.03i	−0.840542	+	0.840542i	−0.988929	−	0.148387i	$-0.952592\pi$
$263$	−3585.03	+	3585.03i	−0.840542	+	0.840542i	0.148387	+	0.988929i	$0.452592\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	−1941.01	−	1941.01i	−0.444898	−	0.444898i
$268$		0			0
$269$		−	1656.00i		−	0.375346i	−0.982232	−	0.187673i	$-0.939905\pi$
$269$		−	1656.00i		−	0.375346i	0.982232	−	0.187673i	$-0.0600945\pi$
$270$		0			0
$271$		−	2165.06i		−	0.485307i	−0.970113	−	0.242654i	$-0.921982\pi$
$271$		−	2165.06i		−	0.485307i	0.970113	−	0.242654i	$-0.0780178\pi$
$272$		0			0
$273$	−1224.74	−	1224.74i	−0.271520	−	0.271520i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	3319.06	−	3319.06i	0.719938	−	0.719938i	−0.248654	−	0.968592i	$-0.579988\pi$
$277$	3319.06	−	3319.06i	0.719938	−	0.719938i	0.968592	+	0.248654i	$0.0799882\pi$
$278$		0			0
$279$	−173.205			−0.0371667
$280$		0			0
$281$	7158.00			1.51961			0.759805	−	0.650151i	$-0.225294\pi$
$281$	7158.00			1.51961			0.759805	+	0.650151i	$0.225294\pi$
$282$		0			0
$283$	−3164.30	+	3164.30i	−0.664658	+	0.664658i	−0.956474	−	0.291816i	$-0.905740\pi$
$283$	−3164.30	+	3164.30i	−0.664658	+	0.664658i	0.291816	+	0.956474i	$0.405740\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	6406.39	+	6406.39i	1.31762	+	1.31762i
$288$		0			0
$289$			1238.00i			0.251985i
$290$		0			0
$291$			519.615i			0.104675i
$292$		0			0
$293$	−2963.88	−	2963.88i	−0.590962	−	0.590962i	0.346929	−	0.937891i	$-0.387224\pi$
$293$	−2963.88	−	2963.88i	−0.590962	−	0.590962i	−0.937891	+	0.346929i	$0.887224\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	4439.70	−	4439.70i	0.867399	−	0.867399i
$298$		0			0
$299$	1558.85			0.301506
$300$		0			0
$301$	4400.00			0.842564
$302$		0			0
$303$	−169.706	+	169.706i	−0.0321760	+	0.0321760i
$304$		0			0
$305$		0			0
$306$		0			0
$307$	109.602	+	109.602i	0.0203755	+	0.0203755i	0.717221	−	0.696846i	$-0.245414\pi$
$307$	109.602	+	109.602i	0.0203755	+	0.0203755i	−0.696846	+	0.717221i	$0.745414\pi$
$308$		0			0
$309$			5750.00i			1.05860i
$310$		0			0
$311$		−	692.820i		−	0.126322i	−0.998003	−	0.0631612i	$-0.979882\pi$
$311$		−	692.820i		−	0.126322i	0.998003	−	0.0631612i	$-0.0201182\pi$
$312$		0			0
$313$	−2816.91	−	2816.91i	−0.508694	−	0.508694i	0.405431	−	0.914126i	$-0.367121\pi$
$313$	−2816.91	−	2816.91i	−0.508694	−	0.508694i	−0.914126	+	0.405431i	$0.867121\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	4041.66	−	4041.66i	0.716095	−	0.716095i	−0.251708	−	0.967803i	$-0.580992\pi$
$317$	4041.66	−	4041.66i	0.716095	−	0.716095i	0.967803	+	0.251708i	$0.0809923\pi$
$318$		0			0
$319$	2857.88			0.501601
$320$		0			0
$321$	1425.00			0.247775
$322$		0			0
$323$	5568.47	−	5568.47i	0.959250	−	0.959250i
$324$		0			0
$325$		0			0
$326$		0			0
$327$	3196.12	+	3196.12i	0.540508	+	0.540508i
$328$		0			0
$329$		−	7800.00i		−	1.30708i
$330$		0			0
$331$			2641.38i			0.438620i	0.975655	+	0.219310i	$0.0703806\pi$
$331$			2641.38i			0.438620i	−0.975655	+	0.219310i	$0.929619\pi$
$332$		0			0
$333$	587.878	+	587.878i	0.0967432	+	0.0967432i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	3459.90	−	3459.90i	0.559267	−	0.559267i	−0.369832	−	0.929099i	$-0.620585\pi$
$337$	3459.90	−	3459.90i	0.559267	−	0.559267i	0.929099	+	0.369832i	$0.120585\pi$
$338$		0			0
$339$	−5152.85			−0.825559
$340$		0			0
$341$	3750.00			0.595525
$342$		0			0
$343$	4044.65	−	4044.65i	0.636707	−	0.636707i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−7138.24	−	7138.24i	−1.10433	−	1.10433i	−0.993882	−	0.110443i	$-0.964773\pi$
$347$	−7138.24	−	7138.24i	−1.10433	−	1.10433i	−0.110443	−	0.993882i	$-0.535227\pi$
$348$		0			0
$349$			9436.00i			1.44727i	0.690182	+	0.723635i	$0.257530\pi$
$349$			9436.00i			1.44727i	−0.690182	+	0.723635i	$0.742470\pi$
$350$		0			0
$351$		−	2511.47i		−	0.381916i
$352$		0			0
$353$	−8205.79	−	8205.79i	−1.23725	−	1.23725i	−0.961119	−	0.276133i	$-0.910947\pi$
$353$	−8205.79	−	8205.79i	−1.23725	−	1.23725i	−0.276133	−	0.961119i	$-0.589053\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$	4286.61	−	4286.61i	0.635494	−	0.635494i
$358$		0			0
$359$	−8487.05			−1.24771			−0.623857	−	0.781539i	$-0.714435\pi$
$359$	−8487.05			−1.24771			−0.623857	+	0.781539i	$0.714435\pi$
$360$		0			0
$361$	10016.0			1.46027
$362$		0			0
$363$	−1923.33	+	1923.33i	−0.278096	+	0.278096i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	−2559.73	−	2559.73i	−0.364078	−	0.364078i	0.501234	−	0.865312i	$-0.332880\pi$
$367$	−2559.73	−	2559.73i	−0.364078	−	0.364078i	−0.865312	+	0.501234i	$0.832880\pi$
$368$		0			0
$369$			906.000i			0.127817i
$370$		0			0
$371$		−	6581.79i		−	0.921050i
$372$		0			0
$373$	3123.10	+	3123.10i	0.433533	+	0.433533i	0.889829	−	0.456295i	$-0.150824\pi$
$373$	3123.10	+	3123.10i	0.433533	+	0.433533i	−0.456295	+	0.889829i	$0.650824\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	808.332	−	808.332i	0.110428	−	0.110428i
$378$		0			0
$379$	−10175.8			−1.37914			−0.689572	−	0.724217i	$-0.742201\pi$
$379$	−10175.8			−1.37914			−0.689572	+	0.724217i	$0.742201\pi$
$380$		0			0
$381$	9050.00			1.21692
$382$		0			0
$383$	−5218.45	+	5218.45i	−0.696215	+	0.696215i	−0.963592	−	0.267377i	$-0.913843\pi$
$383$	−5218.45	+	5218.45i	−0.696215	+	0.696215i	0.267377	+	0.963592i	$0.413843\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	311.127	+	311.127i	0.0408669	+	0.0408669i
$388$		0			0
$389$		−	5226.00i		−	0.681154i	−0.940217	−	0.340577i	$-0.889378\pi$
$389$		−	5226.00i		−	0.681154i	0.940217	−	0.340577i	$-0.110622\pi$
$390$		0			0
$391$			5455.96i			0.705677i
$392$		0			0
$393$	−306.186	−	306.186i	−0.0393004	−	0.0393004i
$394$		0			0
$395$		0			0
$396$		0			0
$397$	7593.42	−	7593.42i	0.959957	−	0.959957i	−0.0392720	−	0.999229i	$-0.512504\pi$
$397$	7593.42	−	7593.42i	0.959957	−	0.959957i	0.999229	+	0.0392720i	$0.0125039\pi$
$398$		0			0
$399$	12990.4			1.62991
$400$		0			0
$401$	27.0000			0.00336238			0.00168119	−	0.999999i	$-0.499465\pi$
$401$	27.0000			0.00336238			0.00168119	+	0.999999i	$0.499465\pi$
$402$		0			0
$403$	1060.66	−	1060.66i	0.131105	−	0.131105i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−12727.9	−	12727.9i	−1.55012	−	1.55012i
$408$		0			0
$409$			4079.00i			0.493138i	0.969125	+	0.246569i	$0.0793032\pi$
$409$			4079.00i			0.493138i	−0.969125	+	0.246569i	$0.920697\pi$
$410$		0			0
$411$			13813.1i			1.65779i
$412$		0			0
$413$	−11022.7	−	11022.7i	−1.31330	−	1.31330i
$414$		0			0
$415$		0			0
$416$		0			0
$417$	459.279	−	459.279i	0.0539353	−	0.0539353i
$418$		0			0
$419$	−4633.24			−0.540211			−0.270105	−	0.962831i	$-0.587059\pi$
$419$	−4633.24			−0.540211			−0.270105	+	0.962831i	$0.587059\pi$
$420$		0			0
$421$	−9112.00			−1.05485			−0.527425	−	0.849602i	$-0.676842\pi$
$421$	−9112.00			−1.05485			−0.527425	+	0.849602i	$0.676842\pi$
$422$		0			0
$423$	551.543	−	551.543i	0.0633971	−	0.0633971i
$424$		0			0
$425$		0			0
$426$		0			0
$427$	9446.95	+	9446.95i	1.07066	+	1.07066i
$428$		0			0
$429$			3750.00i			0.422032i
$430$		0			0
$431$			15068.8i			1.68408i	0.539411	+	0.842042i	$0.318647\pi$
$431$			15068.8i			1.68408i	−0.539411	+	0.842042i	$0.681353\pi$
$432$		0			0
$433$	6778.96	+	6778.96i	0.752370	+	0.752370i	0.974921	−	0.222551i	$-0.0714384\pi$
$433$	6778.96	+	6778.96i	0.752370	+	0.752370i	−0.222551	+	0.974921i	$0.571438\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−8267.03	+	8267.03i	−0.904956	+	0.904956i
$438$		0			0
$439$	−3723.91			−0.404858			−0.202429	−	0.979297i	$-0.564883\pi$
$439$	−3723.91			−0.404858			−0.202429	+	0.979297i	$0.564883\pi$
$440$		0			0
$441$	−114.000			−0.0123097
$442$		0			0
$443$	5526.04	−	5526.04i	0.592664	−	0.592664i	−0.345686	−	0.938350i	$-0.612354\pi$
$443$	5526.04	−	5526.04i	0.592664	−	0.592664i	0.938350	+	0.345686i	$0.112354\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	403.051	+	403.051i	0.0426480	+	0.0426480i
$448$		0			0
$449$		−	6261.00i		−	0.658073i	−0.944317	−	0.329037i	$-0.893276\pi$
$449$		−	6261.00i		−	0.658073i	0.944317	−	0.329037i	$-0.106724\pi$
$450$		0			0
$451$		−	19615.5i		−	2.04802i
$452$		0			0
$453$	7960.84	+	7960.84i	0.825680	+	0.825680i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	5113.31	−	5113.31i	0.523393	−	0.523393i	−0.395202	−	0.918594i	$-0.629325\pi$
$457$	5113.31	−	5113.31i	0.523393	−	0.523393i	0.918594	+	0.395202i	$0.129325\pi$
$458$		0			0
$459$	8790.16			0.893877
$460$		0			0
$461$	9732.00			0.983220			0.491610	−	0.870815i	$-0.336409\pi$
$461$	9732.00			0.983220			0.491610	+	0.870815i	$0.336409\pi$
$462$		0			0
$463$	−7311.48	+	7311.48i	−0.733895	+	0.733895i	−0.971389	−	0.237494i	$-0.923674\pi$
$463$	−7311.48	+	7311.48i	−0.733895	+	0.733895i	0.237494	+	0.971389i	$0.423674\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	6066.98	+	6066.98i	0.601170	+	0.601170i	0.940623	−	0.339453i	$-0.110242\pi$
$467$	6066.98	+	6066.98i	0.601170	+	0.601170i	−0.339453	+	0.940623i	$0.610242\pi$
$468$		0			0
$469$		−	5500.00i		−	0.541506i
$470$		0			0
$471$			3377.50i			0.330418i
$472$		0			0
$473$	−6736.10	−	6736.10i	−0.654812	−	0.654812i
$474$		0			0
$475$		0			0
$476$		0			0
$477$	465.403	−	465.403i	0.0446737	−	0.0446737i
$478$		0			0
$479$	−2511.47			−0.239566			−0.119783	−	0.992800i	$-0.538220\pi$
$479$	−2511.47			−0.239566			−0.119783	+	0.992800i	$0.538220\pi$
$480$		0			0
$481$	−7200.00			−0.682519
$482$		0			0
$483$	−6363.96	+	6363.96i	−0.599524	+	0.599524i
$484$		0			0
$485$		0			0
$486$		0			0
$487$	5932.63	+	5932.63i	0.552018	+	0.552018i	0.927023	−	0.375005i	$-0.122359\pi$
$487$	5932.63	+	5932.63i	0.552018	+	0.552018i	−0.375005	+	0.927023i	$0.622359\pi$
$488$		0			0
$489$			15575.0i			1.44034i
$490$		0			0
$491$		−	18619.5i		−	1.71138i	−0.517487	−	0.855691i	$-0.673133\pi$
$491$		−	18619.5i		−	1.71138i	0.517487	−	0.855691i	$-0.326867\pi$
$492$		0			0
$493$	2829.16	+	2829.16i	0.258456	+	0.258456i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−15921.7	+	15921.7i	−1.43699	+	1.43699i
$498$		0			0
$499$	7880.83			0.707003			0.353501	−	0.935434i	$-0.384991\pi$
$499$	7880.83			0.707003			0.353501	+	0.935434i	$0.384991\pi$
$500$		0			0
$501$	−16050.0			−1.43126
$502$		0			0
$503$	10776.3	−	10776.3i	0.955252	−	0.955252i	−0.0437887	−	0.999041i	$-0.513943\pi$
$503$	10776.3	−	10776.3i	0.955252	−	0.955252i	0.999041	+	0.0437887i	$0.0139428\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−6706.91	−	6706.91i	−0.587503	−	0.587503i
$508$		0			0
$509$		−	17754.0i		−	1.54604i	−0.634384	−	0.773018i	$-0.718746\pi$
$509$		−	17754.0i		−	1.54604i	0.634384	−	0.773018i	$-0.281254\pi$
$510$		0			0
$511$		−	14376.0i		−	1.24454i
$512$		0			0
$513$	13319.1	+	13319.1i	1.14630	+	1.14630i
$514$		0			0
$515$		0			0
$516$		0			0
$517$	−11941.3	+	11941.3i	−1.01581	+	1.01581i
$518$		0			0
$519$	9699.48			0.820347
$520$		0			0
$521$	−11163.0			−0.938695			−0.469347	−	0.883014i	$-0.655511\pi$
$521$	−11163.0			−0.938695			−0.469347	+	0.883014i	$0.655511\pi$
$522$		0			0
$523$	9188.85	−	9188.85i	0.768261	−	0.768261i	−0.209539	−	0.977800i	$-0.567196\pi$
$523$	9188.85	−	9188.85i	0.768261	−	0.768261i	0.977800	+	0.209539i	$0.0671964\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	3712.31	+	3712.31i	0.306852	+	0.306852i
$528$		0			0
$529$			4067.00i			0.334265i
$530$		0			0
$531$		−	1558.85i		−	0.127398i
$532$		0			0
$533$	−5548.09	−	5548.09i	−0.450872	−	0.450872i
$534$		0			0
$535$		0			0
$536$		0			0
$537$	5664.45	−	5664.45i	0.455193	−	0.455193i
$538$		0			0
$539$	2468.17			0.197239
$540$		0			0
$541$	−9128.00			−0.725404			−0.362702	−	0.931905i	$-0.618146\pi$
$541$	−9128.00			−0.725404			−0.362702	+	0.931905i	$0.618146\pi$
$542$		0			0
$543$	−11872.3	+	11872.3i	−0.938288	+	0.938288i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	3857.27	+	3857.27i	0.301508	+	0.301508i	0.841604	−	0.540096i	$-0.181612\pi$
$547$	3857.27	+	3857.27i	0.301508	+	0.301508i	−0.540096	+	0.841604i	$0.681612\pi$
$548$		0			0
$549$			1336.00i			0.103860i
$550$		0			0
$551$			8573.65i			0.662885i
$552$		0			0
$553$	9797.96	+	9797.96i	0.753439	+	0.753439i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−9320.31	+	9320.31i	−0.709002	+	0.709002i	−0.966325	−	0.257323i	$-0.917159\pi$
$557$	−9320.31	+	9320.31i	−0.709002	+	0.709002i	0.257323	+	0.966325i	$0.417159\pi$
$558$		0			0
$559$	−3810.51			−0.288314
$560$		0			0
$561$	−13125.0			−0.987768
$562$		0			0
$563$	212.132	−	212.132i	0.0158798	−	0.0158798i	−0.699122	−	0.715002i	$-0.746426\pi$
$563$	212.132	−	212.132i	0.0158798	−	0.0158798i	0.715002	+	0.699122i	$0.246426\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	9489.37	+	9489.37i	0.702850	+	0.702850i
$568$		0			0
$569$		−	20031.0i		−	1.47582i	−0.674898	−	0.737911i	$-0.735812\pi$
$569$		−	20031.0i		−	1.47582i	0.674898	−	0.737911i	$-0.264188\pi$
$570$		0			0
$571$			14982.2i			1.09805i	0.835806	+	0.549026i	$0.185001\pi$
$571$			14982.2i			1.09805i	−0.835806	+	0.549026i	$0.814999\pi$
$572$		0			0
$573$	918.559	+	918.559i	0.0669692	+	0.0669692i
$574$		0			0
$575$		0			0
$576$		0			0
$577$	−7134.14	+	7134.14i	−0.514728	+	0.514728i	−0.915971	−	0.401243i	$-0.868578\pi$
$577$	−7134.14	+	7134.14i	−0.514728	+	0.514728i	0.401243	+	0.915971i	$0.368578\pi$
$578$		0			0
$579$	2294.97			0.164725
$580$		0			0
$581$	300.000			0.0214219
$582$		0			0
$583$	−10076.3	+	10076.3i	−0.715809	+	0.715809i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	6310.93	+	6310.93i	0.443748	+	0.443748i	0.893269	−	0.449522i	$-0.148406\pi$
$587$	6310.93	+	6310.93i	0.443748	+	0.443748i	−0.449522	+	0.893269i	$0.648406\pi$
$588$		0			0
$589$			11250.0i			0.787009i
$590$		0			0
$591$			13596.6i			0.946344i
$592$		0			0
$593$	9240.70	+	9240.70i	0.639916	+	0.639916i	0.950535	−	0.310619i	$-0.100536\pi$
$593$	9240.70	+	9240.70i	0.639916	+	0.639916i	−0.310619	+	0.950535i	$0.600536\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−15921.7	+	15921.7i	−1.09151	+	1.09151i
$598$		0			0
$599$	−3983.72			−0.271737			−0.135868	−	0.990727i	$-0.543382\pi$
$599$	−3983.72			−0.271737			−0.135868	+	0.990727i	$0.543382\pi$
$600$		0			0
$601$	−2027.00			−0.137576			−0.0687879	−	0.997631i	$-0.521913\pi$
$601$	−2027.00			−0.137576			−0.0687879	+	0.997631i	$0.521913\pi$
$602$		0			0
$603$	388.909	−	388.909i	0.0262647	−	0.0262647i
$604$		0			0
$605$		0			0
$606$		0			0
$607$	−20223.3	−	20223.3i	−1.35228	−	1.35228i	−0.883101	−	0.469184i	$-0.844548\pi$
$607$	−20223.3	−	20223.3i	−1.35228	−	1.35228i	−0.469184	−	0.883101i	$-0.655452\pi$
$608$		0			0
$609$			6600.00i			0.439155i
$610$		0			0
$611$			6755.00i			0.447263i
$612$		0			0
$613$	4984.71	+	4984.71i	0.328435	+	0.328435i	0.851991	−	0.523556i	$-0.175395\pi$
$613$	4984.71	+	4984.71i	0.328435	+	0.328435i	−0.523556	+	0.851991i	$0.675395\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−2596.46	+	2596.46i	−0.169416	+	0.169416i	−0.786723	−	0.617307i	$-0.788224\pi$
$617$	−2596.46	+	2596.46i	−0.169416	+	0.169416i	0.617307	+	0.786723i	$0.288224\pi$
$618$		0			0
$619$	−8400.45			−0.545464			−0.272732	−	0.962090i	$-0.587927\pi$
$619$	−8400.45			−0.545464			−0.272732	+	0.962090i	$0.587927\pi$
$620$		0			0
$621$	−13050.0			−0.843283
$622$		0			0
$623$	7764.03	−	7764.03i	0.499293	−	0.499293i
$624$		0			0
$625$		0			0
$626$		0			0
$627$	−19887.4	−	19887.4i	−1.26671	−	1.26671i
$628$		0			0
$629$		−	25200.0i		−	1.59744i
$630$		0			0
$631$		−	11344.9i		−	0.715744i	−0.933771	−	0.357872i	$-0.883502\pi$
$631$		−	11344.9i		−	0.715744i	0.933771	−	0.357872i	$-0.116498\pi$
$632$		0			0
$633$	7195.38	+	7195.38i	0.451802	+	0.451802i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	698.105	−	698.105i	0.0434222	−	0.0434222i
$638$		0			0
$639$	−2251.67			−0.139397
$640$		0			0
$641$	−23322.0			−1.43707			−0.718536	−	0.695489i	$-0.755188\pi$
$641$	−23322.0			−1.43707			−0.718536	+	0.695489i	$0.755188\pi$
$642$		0			0
$643$	9008.54	−	9008.54i	0.552507	−	0.552507i	−0.374656	−	0.927164i	$-0.622239\pi$
$643$	9008.54	−	9008.54i	0.552507	−	0.552507i	0.927164	+	0.374656i	$0.122239\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−2969.85	−	2969.85i	−0.180459	−	0.180459i	0.611097	−	0.791556i	$-0.290729\pi$
$647$	−2969.85	−	2969.85i	−0.180459	−	0.180459i	−0.791556	+	0.611097i	$0.790729\pi$
$648$		0			0
$649$			33750.0i			2.04130i
$650$		0			0
$651$			8660.25i			0.521386i
$652$		0			0
$653$	−23172.2	−	23172.2i	−1.38866	−	1.38866i	−0.828136	−	0.560528i	$-0.810598\pi$
$653$	−23172.2	−	23172.2i	−1.38866	−	1.38866i	−0.560528	−	0.828136i	$-0.689402\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	1016.54	−	1016.54i	0.0603637	−	0.0603637i
$658$		0			0
$659$	−15372.0			−0.908659			−0.454329	−	0.890834i	$-0.650121\pi$
$659$	−15372.0			−0.908659			−0.454329	+	0.890834i	$0.650121\pi$
$660$		0			0
$661$	−2882.00			−0.169587			−0.0847933	−	0.996399i	$-0.527023\pi$
$661$	−2882.00			−0.169587			−0.0847933	+	0.996399i	$0.527023\pi$
$662$		0			0
$663$	−3712.31	+	3712.31i	−0.217457	+	0.217457i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−4200.21	−	4200.21i	−0.243828	−	0.243828i
$668$		0			0
$669$		−	22900.0i		−	1.32342i
$670$		0			0
$671$		−	28925.2i		−	1.66415i
$672$		0			0
$673$	16264.6	+	16264.6i	0.931582	+	0.931582i	0.997805	−	0.0662228i	$-0.0210948\pi$
$673$	16264.6	+	16264.6i	0.931582	+	0.931582i	−0.0662228	+	0.997805i	$0.521095\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−10777.8	+	10777.8i	−0.611850	+	0.611850i	−0.943428	−	0.331578i	$-0.892419\pi$
$677$	−10777.8	+	10777.8i	−0.611850	+	0.611850i	0.331578	+	0.943428i	$0.392419\pi$
$678$		0			0
$679$	−2078.46			−0.117473
$680$		0			0
$681$	27600.0			1.55306
$682$		0			0
$683$	−19972.2	+	19972.2i	−1.11891	+	1.11891i	−0.127009	+	0.991902i	$0.540538\pi$
$683$	−19972.2	+	19972.2i	−1.11891	+	1.11891i	−0.991902	+	0.127009i	$0.959462\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	13138.0	+	13138.0i	0.729618	+	0.729618i
$688$		0			0
$689$			5700.00i			0.315171i
$690$		0			0
$691$			18229.8i			1.00361i	0.864980	+	0.501806i	$0.167331\pi$
$691$			18229.8i			1.00361i	−0.864980	+	0.501806i	$0.832669\pi$
$692$		0			0
$693$	1224.74	+	1224.74i	0.0671345	+	0.0671345i
$694$		0			0
$695$		0			0
$696$		0			0
$697$	19418.3	−	19418.3i	1.05527	−	1.05527i
$698$		0			0
$699$	−31696.5			−1.71513
$700$		0			0
$701$	25998.0			1.40076			0.700379	−	0.713771i	$-0.253014\pi$
$701$	25998.0			1.40076			0.700379	+	0.713771i	$0.253014\pi$
$702$		0			0
$703$	38183.8	−	38183.8i	2.04855	−	2.04855i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−678.823	−	678.823i	−0.0361100	−	0.0361100i
$708$		0			0
$709$		−	18004.0i		−	0.953673i	−0.878992	−	0.476837i	$-0.841783\pi$
$709$		−	18004.0i		−	0.953673i	0.878992	−	0.476837i	$-0.158217\pi$
$710$		0			0
$711$			1385.64i			0.0730880i
$712$		0			0
$713$	−5511.35	−	5511.35i	−0.289484	−	0.289484i
$714$		0			0
$715$		0			0
$716$		0			0
$717$	11328.9	−	11328.9i	0.590077	−	0.590077i
$718$		0			0
$719$	2424.87			0.125775			0.0628876	−	0.998021i	$-0.479969\pi$
$719$	2424.87			0.125775			0.0628876	+	0.998021i	$0.479969\pi$
$720$		0			0
$721$	−23000.0			−1.18802
$722$		0			0
$723$	−3348.15	+	3348.15i	−0.172226	+	0.172226i
$724$		0			0
$725$		0			0
$726$		0			0
$727$	5225.52	+	5225.52i	0.266580	+	0.266580i	0.827721	−	0.561140i	$-0.189637\pi$
$727$	5225.52	+	5225.52i	0.266580	+	0.266580i	−0.561140	+	0.827721i	$0.689637\pi$
$728$		0			0
$729$			20917.0i			1.06269i
$730$		0			0
$731$		−	13336.8i		−	0.674800i
$732$		0			0
$733$	−8511.98	−	8511.98i	−0.428918	−	0.428918i	0.459342	−	0.888260i	$-0.348085\pi$
$733$	−8511.98	−	8511.98i	−0.428918	−	0.428918i	−0.888260	+	0.459342i	$0.848085\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−8420.12	+	8420.12i	−0.420840	+	0.420840i
$738$		0			0
$739$	17753.5			0.883726			0.441863	−	0.897082i	$-0.354318\pi$
$739$	17753.5			0.883726			0.441863	+	0.897082i	$0.354318\pi$
$740$		0			0
$741$	−11250.0			−0.557732
$742$		0			0
$743$	1803.12	−	1803.12i	0.0890311	−	0.0890311i	−0.661189	−	0.750220i	$-0.729948\pi$
$743$	1803.12	−	1803.12i	0.0890311	−	0.0890311i	0.750220	+	0.661189i	$0.229948\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	21.2132	+	21.2132i	0.00103902	+	0.00103902i
$748$		0			0
$749$			5700.00i			0.278069i
$750$		0			0
$751$			17926.7i			0.871046i	0.900178	+	0.435523i	$0.143437\pi$
$751$			17926.7i			0.871046i	−0.900178	+	0.435523i	$0.856563\pi$
$752$		0			0
$753$	4745.89	+	4745.89i	0.229681	+	0.229681i
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−3698.73	+	3698.73i	−0.177586	+	0.177586i	−0.790303	−	0.612717i	$-0.790077\pi$
$757$	−3698.73	+	3698.73i	−0.177586	+	0.177586i	0.612717	+	0.790303i	$0.290077\pi$
$758$		0			0
$759$	19485.6			0.931860
$760$		0			0
$761$	30807.0			1.46748			0.733740	−	0.679430i	$-0.237773\pi$
$761$	30807.0			1.46748			0.733740	+	0.679430i	$0.237773\pi$
$762$		0			0
$763$	−12784.5	+	12784.5i	−0.606592	+	0.606592i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	9545.94	+	9545.94i	0.449392	+	0.449392i
$768$		0			0
$769$		−	3319.00i		−	0.155639i	−0.996967	−	0.0778194i	$-0.975204\pi$
$769$		−	3319.00i		−	0.155639i	0.996967	−	0.0778194i	$-0.0247957\pi$
$770$		0			0
$771$			22343.5i			1.04368i
$772$		0			0
$773$	−22915.0	−	22915.0i	−1.06623	−	1.06623i	−0.997645	−	0.0685828i	$-0.978152\pi$
$773$	−22915.0	−	22915.0i	−1.06623	−	1.06623i	−0.0685828	−	0.997645i	$-0.521848\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	29393.9	−	29393.9i	1.35714	−	1.35714i
$778$		0			0
$779$	58846.4			2.70654
$780$		0			0
$781$	48750.0			2.23356
$782$		0			0
$783$	−6767.01	+	6767.01i	−0.308855	+	0.308855i
$784$		0			0
$785$		0			0
$786$		0			0
$787$	4058.79	+	4058.79i	0.183838	+	0.183838i	0.793026	−	0.609188i	$-0.208505\pi$
$787$	4058.79	+	4058.79i	0.183838	+	0.183838i	−0.609188	+	0.793026i	$0.708505\pi$
$788$		0			0
$789$			25350.0i			1.14383i
$790$		0			0
$791$		−	20611.4i		−	0.926495i
$792$		0			0
$793$	−8181.30	−	8181.30i	−0.366364	−	0.366364i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	20906.4	−	20906.4i	0.929162	−	0.929162i	−0.0684894	−	0.997652i	$-0.521818\pi$
$797$	20906.4	−	20906.4i	0.929162	−	0.929162i	0.997652	+	0.0684894i	$0.0218179\pi$
$798$		0			0
$799$	−23642.5			−1.04682
$800$		0			0
$801$	1098.00			0.0484344
$802$		0			0
$803$	−22008.7	+	22008.7i	−0.967211	+	0.967211i
$804$		0			0
$805$		0			0
$806$		0			0
$807$	−5854.84	−	5854.84i	−0.255391	−	0.255391i
$808$		0			0
$809$		−	7554.00i		−	0.328287i	−0.986436	−	0.164144i	$-0.947514\pi$
$809$		−	7554.00i		−	0.328287i	0.986436	−	0.164144i	$-0.0524861\pi$
$810$		0			0
$811$		−	35420.4i		−	1.53364i	−0.641864	−	0.766819i	$-0.721839\pi$
$811$		−	35420.4i		−	1.53364i	0.641864	−	0.766819i	$-0.278161\pi$
$812$		0			0
$813$	−7654.66	−	7654.66i	−0.330210	−	0.330210i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	20208.3	−	20208.3i	0.865359	−	0.865359i
$818$		0			0
$819$	692.820			0.0295593
$820$		0			0
$821$	−8562.00			−0.363966			−0.181983	−	0.983302i	$-0.558252\pi$
$821$	−8562.00			−0.363966			−0.181983	+	0.983302i	$0.558252\pi$
$822$		0			0
$823$	−26233.7	+	26233.7i	−1.11112	+	1.11112i	−0.118116	+	0.993000i	$0.537685\pi$
$823$	−26233.7	+	26233.7i	−1.11112	+	1.11112i	−0.993000	+	0.118116i	$0.962315\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	18084.3	+	18084.3i	0.760400	+	0.760400i	0.976395	−	0.215994i	$-0.0692992\pi$
$827$	18084.3	+	18084.3i	0.760400	+	0.760400i	−0.215994	+	0.976395i	$0.569299\pi$
$828$		0			0
$829$		−	15766.0i		−	0.660526i	−0.943889	−	0.330263i	$-0.892863\pi$
$829$		−	15766.0i		−	0.660526i	0.943889	−	0.330263i	$-0.107137\pi$
$830$		0			0
$831$		−	23469.3i		−	0.979712i
$832$		0			0
$833$	2443.37	+	2443.37i	0.101630	+	0.101630i
$834$		0			0
$835$		0			0
$836$		0			0
$837$	−8879.40	+	8879.40i	−0.366687	+	0.366687i
$838$		0			0
$839$	4503.33			0.185307			0.0926533	−	0.995698i	$-0.470465\pi$
$839$	4503.33			0.185307			0.0926533	+	0.995698i	$0.470465\pi$
$840$		0			0
$841$	20033.0			0.821395
$842$		0			0
$843$	25307.4	−	25307.4i	1.03396	−	1.03396i
$844$		0			0
$845$		0			0
$846$		0			0
$847$	−7693.32	−	7693.32i	−0.312096	−	0.312096i
$848$		0			0
$849$			22375.0i			0.904485i
$850$		0			0
$851$			37412.3i			1.50702i
$852$		0			0
$853$	845.074	+	845.074i	0.0339212	+	0.0339212i	0.723864	−	0.689943i	$-0.242364\pi$
$853$	845.074	+	845.074i	0.0339212	+	0.0339212i	−0.689943	+	0.723864i	$0.742364\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	14176.4	−	14176.4i	0.565061	−	0.565061i	−0.365680	−	0.930741i	$-0.619163\pi$
$857$	14176.4	−	14176.4i	0.565061	−	0.565061i	0.930741	+	0.365680i	$0.119163\pi$
$858$		0			0
$859$	22993.0			0.913283			0.456642	−	0.889651i	$-0.349052\pi$
$859$	22993.0			0.913283			0.456642	+	0.889651i	$0.349052\pi$
$860$		0			0
$861$	45300.0			1.79305
$862$		0			0
$863$	20301.0	−	20301.0i	0.800759	−	0.800759i	−0.182455	−	0.983214i	$-0.558404\pi$
$863$	20301.0	−	20301.0i	0.800759	−	0.800759i	0.983214	+	0.182455i	$0.0584044\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	4376.99	+	4376.99i	0.171454	+	0.171454i
$868$		0			0
$869$		−	30000.0i		−	1.17109i
$870$		0			0
$871$			4763.14i			0.185296i
$872$		0			0
$873$	−146.969	−	146.969i	−0.00569778	−	0.00569778i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	13202.7	−	13202.7i	0.508353	−	0.508353i	−0.405668	−	0.914021i	$-0.632961\pi$
$877$	13202.7	−	13202.7i	0.508353	−	0.508353i	0.914021	+	0.405668i	$0.132961\pi$
$878$		0			0
$879$	−20957.8			−0.804197
$880$		0			0
$881$	12258.0			0.468766			0.234383	−	0.972144i	$-0.424693\pi$
$881$	12258.0			0.468766			0.234383	+	0.972144i	$0.424693\pi$
$882$		0			0
$883$	8460.53	−	8460.53i	0.322446	−	0.322446i	−0.527259	−	0.849705i	$-0.676780\pi$
$883$	8460.53	−	8460.53i	0.322446	−	0.322446i	0.849705	+	0.527259i	$0.176780\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	2036.47	+	2036.47i	0.0770889	+	0.0770889i	0.744600	−	0.667511i	$-0.232640\pi$
$887$	2036.47	+	2036.47i	0.0770889	+	0.0770889i	−0.667511	+	0.744600i	$0.732640\pi$
$888$		0			0
$889$			36200.0i			1.36570i
$890$		0			0
$891$		−	29055.2i		−	1.09246i
$892$		0			0
$893$	−35823.8	−	35823.8i	−1.34244	−	1.34244i
$894$		0			0
$895$		0			0
$896$		0			0
$897$	5511.35	−	5511.35i	0.205149	−	0.205149i
$898$		0			0
$899$	−5715.77			−0.212048
$900$		0			0
$901$	−19950.0			−0.737659
$902$		0			0
$903$	15556.3	−	15556.3i	0.573292	−	0.573292i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	14580.5	+	14580.5i	0.533781	+	0.533781i	0.921695	−	0.387915i	$-0.126804\pi$
$907$	14580.5	+	14580.5i	0.533781	+	0.533781i	−0.387915	+	0.921695i	$0.626804\pi$
$908$		0			0
$909$		−	96.0000i		−	0.00350288i
$910$		0			0
$911$			18879.4i			0.686609i	0.939224	+	0.343305i	$0.111546\pi$
$911$			18879.4i			0.686609i	−0.939224	+	0.343305i	$0.888454\pi$
$912$		0			0
$913$	−459.279	−	459.279i	−0.0166483	−	0.0166483i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	1224.74	−	1224.74i	0.0441054	−	0.0441054i
$918$		0			0
$919$	−12037.8			−0.432088			−0.216044	−	0.976384i	$-0.569315\pi$
$919$	−12037.8			−0.432088			−0.216044	+	0.976384i	$0.569315\pi$
$920$		0			0
$921$	775.000			0.0277276
$922$		0			0
$923$	13788.6	−	13788.6i	0.491719	−	0.491719i
$924$		0			0
$925$		0			0
$926$		0			0
$927$	−1626.35	−	1626.35i	−0.0576227	−	0.0576227i
$928$		0			0
$929$			7734.00i			0.273137i	0.990631	+	0.136569i	$0.0436074\pi$
$929$			7734.00i			0.273137i	−0.990631	+	0.136569i	$0.956393\pi$
$930$		0			0
$931$			7404.52i			0.260659i
$932$		0			0
$933$	−2449.49	−	2449.49i	−0.0859514	−	0.0859514i
$934$		0			0
$935$		0			0
$936$		0			0
$937$	18842.7	−	18842.7i	0.656952	−	0.656952i	−0.297706	−	0.954658i	$-0.596221\pi$
$937$	18842.7	−	18842.7i	0.656952	−	0.656952i	0.954658	+	0.297706i	$0.0962214\pi$
$938$		0			0
$939$	−19918.6			−0.692245
$940$		0			0
$941$	14622.0			0.506550			0.253275	−	0.967394i	$-0.418492\pi$
$941$	14622.0			0.506550			0.253275	+	0.967394i	$0.418492\pi$
$942$		0			0
$943$	−28828.7	+	28828.7i	−0.995539	+	0.995539i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−24649.7	−	24649.7i	−0.845838	−	0.845838i	0.143772	−	0.989611i	$-0.454077\pi$
$947$	−24649.7	−	24649.7i	−0.845838	−	0.845838i	−0.989611	+	0.143772i	$0.954077\pi$
$948$		0			0
$949$			12450.0i			0.425863i
$950$		0			0
$951$		−	28578.8i		−	0.974482i
$952$		0			0
$953$	8334.39	+	8334.39i	0.283292	+	0.283292i	0.834420	−	0.551128i	$-0.185803\pi$
$953$	8334.39	+	8334.39i	0.283292	+	0.283292i	−0.551128	+	0.834420i	$0.685803\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$	10104.1	−	10104.1i	0.341296	−	0.341296i
$958$		0			0
$959$	−55252.4			−1.86047
$960$		0			0
$961$	22291.0			0.748246
$962$		0			0
$963$	−403.051	+	403.051i	−0.0134872	+	0.0134872i
$964$		0			0
$965$		0			0
$966$		0			0
$967$	−10366.2	−	10366.2i	−0.344730	−	0.344730i	0.513412	−	0.858142i	$-0.328381\pi$
$967$	−10366.2	−	10366.2i	−0.344730	−	0.344730i	−0.858142	+	0.513412i	$0.828381\pi$
$968$		0			0
$969$		−	39375.0i		−	1.30537i
$970$		0			0
$971$			25331.2i			0.837197i	0.908171	+	0.418598i	$0.137479\pi$
$971$			25331.2i			0.837197i	−0.908171	+	0.418598i	$0.862521\pi$
$972$		0			0
$973$	1837.12	+	1837.12i	0.0605296	+	0.0605296i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−30502.3	+	30502.3i	−0.998827	+	0.998827i	−0.999999	−	0.00117218i	$-0.999627\pi$
$977$	−30502.3	+	30502.3i	−0.998827	+	0.998827i	0.00117218	+	0.999999i	$0.499627\pi$
$978$		0			0
$979$	−23772.4			−0.776067
$980$		0			0
$981$	−1808.00			−0.0588430
$982$		0			0
$983$	2969.85	−	2969.85i	0.0963616	−	0.0963616i	−0.657283	−	0.753644i	$-0.728294\pi$
$983$	2969.85	−	2969.85i	0.0963616	−	0.0963616i	0.753644	+	0.657283i	$0.228294\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	−27577.2	−	27577.2i	−0.889352	−	0.889352i
$988$		0			0
$989$			19800.0i			0.636606i
$990$		0			0
$991$			51615.1i			1.65450i	0.561835	+	0.827249i	$0.310096\pi$
$991$			51615.1i			1.65450i	−0.561835	+	0.827249i	$0.689904\pi$
$992$		0			0
$993$	9338.68	+	9338.68i	0.298443	+	0.298443i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−23515.1	+	23515.1i	−0.746972	+	0.746972i	−0.973909	−	0.226938i	$-0.927129\pi$
$997$	−23515.1	+	23515.1i	−0.746972	+	0.746972i	0.226938	+	0.973909i	$0.427129\pi$
$998$		0			0
$999$	60275.4			1.90894

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	400.4.n.e.143.3	yes	8
4.3	odd	2	inner	400.4.n.e.143.2	yes	8
5.2	odd	4	inner	400.4.n.e.207.1	yes	8
5.3	odd	4	inner	400.4.n.e.207.3	yes	8
5.4	even	2	inner	400.4.n.e.143.1	✓	8
20.3	even	4	inner	400.4.n.e.207.2	yes	8
20.7	even	4	inner	400.4.n.e.207.4	yes	8
20.19	odd	2	inner	400.4.n.e.143.4	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
400.4.n.e.143.1	✓	8	5.4	even	2	inner
400.4.n.e.143.2	yes	8	4.3	odd	2	inner
400.4.n.e.143.3	yes	8	1.1	even	1	trivial
400.4.n.e.143.4	yes	8	20.19	odd	2	inner
400.4.n.e.207.1	yes	8	5.2	odd	4	inner
400.4.n.e.207.2	yes	8	20.3	even	4	inner
400.4.n.e.207.3	yes	8	5.3	odd	4	inner
400.4.n.e.207.4	yes	8	20.7	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:	\( N \)	\(=\)	\( 400 = 2^{4} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	400.n (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(3.53553 - 3.53553i) q^{3} +(14.1421 + 14.1421i) q^{7} +2.00000i q^{9} -43.3013i q^{11} +(-12.2474 - 12.2474i) q^{13} +(42.8661 - 42.8661i) q^{17} +129.904 q^{19} +100.000 q^{21} +(-63.6396 + 63.6396i) q^{23} +(102.530 + 102.530i) q^{27} +66.0000i q^{29} +86.6025i q^{31} +(-153.093 - 153.093i) q^{33} +(293.939 - 293.939i) q^{37} -86.6025 q^{39} +453.000 q^{41} +(155.563 - 155.563i) q^{43} +(-275.772 - 275.772i) q^{47} +57.0000i q^{49} -303.109i q^{51} +(-232.702 - 232.702i) q^{53} +(459.279 - 459.279i) q^{57} -779.423 q^{59} +668.000 q^{61} +(-28.2843 + 28.2843i) q^{63} +(-194.454 - 194.454i) q^{67} +450.000i q^{69} +1125.83i q^{71} +(-508.269 - 508.269i) q^{73} +(612.372 - 612.372i) q^{77} +692.820 q^{79} +671.000 q^{81} +(10.6066 - 10.6066i) q^{83} +(233.345 + 233.345i) q^{87} -549.000i q^{89} -346.410i q^{91} +(306.186 + 306.186i) q^{93} +(-73.4847 + 73.4847i) q^{97} +86.6025 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 800 q^{21} + 3624 q^{41} + 5344 q^{61} + 5368 q^{81}+O(q^{100}) \) 8 * q + 800 * q^21 + 3624 * q^41 + 5344 * q^61 + 5368 * q^81

Introduction

L-functions

Modular forms

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