LMFDB - Embedded newform 1280.4.d.l.641.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1280,4,Mod(641,1280)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1280, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 0]))

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

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

chi := DirichletCharacter("1280.641");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1280 = 2^{8} \cdot 5 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1280.d (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$75.5224448073$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 5)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			641.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	1280.641
Dual form			1280.4.d.l.641.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.00000i q^{3} +5.00000i q^{5} +6.00000 q^{7} +23.0000 q^{9} +O(q^{10})$	$q-2.00000i q^{3} +5.00000i q^{5} +6.00000 q^{7} +23.0000 q^{9} +32.0000i q^{11} -38.0000i q^{13} +10.0000 q^{15} +26.0000 q^{17} -100.000i q^{19} -12.0000i q^{21} -78.0000 q^{23} -25.0000 q^{25} -100.000i q^{27} -50.0000i q^{29} +108.000 q^{31} +64.0000 q^{33} +30.0000i q^{35} -266.000i q^{37} -76.0000 q^{39} -22.0000 q^{41} +442.000i q^{43} +115.000i q^{45} +514.000 q^{47} -307.000 q^{49} -52.0000i q^{51} -2.00000i q^{53} -160.000 q^{55} -200.000 q^{57} +500.000i q^{59} -518.000i q^{61} +138.000 q^{63} +190.000 q^{65} -126.000i q^{67} +156.000i q^{69} +412.000 q^{71} +878.000 q^{73} +50.0000i q^{75} +192.000i q^{77} -600.000 q^{79} +421.000 q^{81} -282.000i q^{83} +130.000i q^{85} -100.000 q^{87} +150.000 q^{89} -228.000i q^{91} -216.000i q^{93} +500.000 q^{95} +386.000 q^{97} +736.000i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 12 q^{7} + 46 q^{9}+O(q^{10}) $ 2 * q + 12 * q^7 + 46 * q^9	$ 2 q + 12 q^{7} + 46 q^{9} + 20 q^{15} + 52 q^{17} - 156 q^{23} - 50 q^{25} + 216 q^{31} + 128 q^{33} - 152 q^{39} - 44 q^{41} + 1028 q^{47} - 614 q^{49} - 320 q^{55} - 400 q^{57} + 276 q^{63} + 380 q^{65} + 824 q^{71} + 1756 q^{73} - 1200 q^{79} + 842 q^{81} - 200 q^{87} + 300 q^{89} + 1000 q^{95} + 772 q^{97}+O(q^{100}) $ 2 * q + 12 * q^7 + 46 * q^9 + 20 * q^15 + 52 * q^17 - 156 * q^23 - 50 * q^25 + 216 * q^31 + 128 * q^33 - 152 * q^39 - 44 * q^41 + 1028 * q^47 - 614 * q^49 - 320 * q^55 - 400 * q^57 + 276 * q^63 + 380 * q^65 + 824 * q^71 + 1756 * q^73 - 1200 * q^79 + 842 * q^81 - 200 * q^87 + 300 * q^89 + 1000 * q^95 + 772 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1280\mathbb{Z}\right)^\times$.

$n$	$257$	$261$	$511$
$\chi(n)$	$1$	$-1$	$1$

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
$3$	− 2.00000i	− 0.384900i	−0.981307	−	0.192450i	$-0.938357\pi$
$3$	− 2.00000i	− 0.384900i	0.981307	−	0.192450i	$-0.0616434\pi$
$4$	0	0
$5$	5.00000i	0.447214i
$5$	5.00000i	0.447214i
$6$	0	0
$7$	6.00000	0.323970	0.161985	−	0.986793i	$-0.448210\pi$
$7$	6.00000	0.323970	0.161985	+	0.986793i	$0.448210\pi$
$8$	0	0
$9$	23.0000	0.851852
$10$	0	0
$11$	32.0000i	0.877124i	0.898701	+	0.438562i	$0.144512\pi$
$11$	32.0000i	0.877124i	−0.898701	+	0.438562i	$0.855488\pi$
$12$	0	0
$13$	− 38.0000i	− 0.810716i	−0.914158	−	0.405358i	$-0.867147\pi$
$13$	− 38.0000i	− 0.810716i	0.914158	−	0.405358i	$-0.132853\pi$
$14$	0	0
$15$	10.0000	0.172133
$16$	0	0
$17$	26.0000	0.370937	0.185468	−	0.982650i	$-0.440620\pi$
$17$	26.0000	0.370937	0.185468	+	0.982650i	$0.440620\pi$
$18$	0	0
$19$	− 100.000i	− 1.20745i	−0.797192	−	0.603726i	$-0.793682\pi$
$19$	− 100.000i	− 1.20745i	0.797192	−	0.603726i	$-0.206318\pi$
$20$	0	0
$21$	− 12.0000i	− 0.124696i
$22$	0	0
$23$	−78.0000	−0.707136	−0.353568	−	0.935409i	$-0.615032\pi$
$23$	−78.0000	−0.707136	−0.353568	+	0.935409i	$0.615032\pi$
$24$	0	0
$25$	−25.0000	−0.200000
$26$	0	0
$27$	− 100.000i	− 0.712778i
$28$	0	0
$29$	− 50.0000i	− 0.320164i	−0.987104	−	0.160082i	$-0.948824\pi$
$29$	− 50.0000i	− 0.320164i	0.987104	−	0.160082i	$-0.0511759\pi$
$30$	0	0
$31$	108.000	0.625722	0.312861	−	0.949799i	$-0.398713\pi$
$31$	108.000	0.625722	0.312861	+	0.949799i	$0.398713\pi$
$32$	0	0
$33$	64.0000	0.337605
$34$	0	0
$35$	30.0000i	0.144884i
$36$	0	0
$37$	− 266.000i	− 1.18190i	−0.806710	−	0.590948i	$-0.798754\pi$
$37$	− 266.000i	− 1.18190i	0.806710	−	0.590948i	$-0.201246\pi$
$38$	0	0
$39$	−76.0000	−0.312045
$40$	0	0
$41$	−22.0000	−0.0838006	−0.0419003	−	0.999122i	$-0.513341\pi$
$41$	−22.0000	−0.0838006	−0.0419003	+	0.999122i	$0.513341\pi$
$42$	0	0
$43$	442.000i	1.56754i	0.621049	+	0.783772i	$0.286707\pi$
$43$	442.000i	1.56754i	−0.621049	+	0.783772i	$0.713293\pi$
$44$	0	0
$45$	115.000i	0.380960i
$46$	0	0
$47$	514.000	1.59520	0.797602	−	0.603184i	$-0.206101\pi$
$47$	514.000	1.59520	0.797602	+	0.603184i	$0.206101\pi$
$48$	0	0
$49$	−307.000	−0.895044
$50$	0	0
$51$	− 52.0000i	− 0.142774i
$52$	0	0
$53$	− 2.00000i	− 0.00518342i	−0.999997	−	0.00259171i	$-0.999175\pi$
$53$	− 2.00000i	− 0.00518342i	0.999997	−	0.00259171i	$-0.000824967\pi$
$54$	0	0
$55$	−160.000	−0.392262
$56$	0	0
$57$	−200.000	−0.464748
$58$	0	0
$59$	500.000i	1.10330i	0.834077	+	0.551648i	$0.186001\pi$
$59$	500.000i	1.10330i	−0.834077	+	0.551648i	$0.813999\pi$
$60$	0	0
$61$	− 518.000i	− 1.08726i	−0.839324	−	0.543632i	$-0.817049\pi$
$61$	− 518.000i	− 1.08726i	0.839324	−	0.543632i	$-0.182951\pi$
$62$	0	0
$63$	138.000	0.275974
$64$	0	0
$65$	190.000	0.362563
$66$	0	0
$67$	− 126.000i	− 0.229751i	−0.993380	−	0.114876i	$-0.963353\pi$
$67$	− 126.000i	− 0.229751i	0.993380	−	0.114876i	$-0.0366470\pi$
$68$	0	0
$69$	156.000i	0.272177i
$70$	0	0
$71$	412.000	0.688668	0.344334	−	0.938847i	$-0.388105\pi$
$71$	412.000	0.688668	0.344334	+	0.938847i	$0.388105\pi$
$72$	0	0
$73$	878.000	1.40770	0.703850	−	0.710348i	$-0.251463\pi$
$73$	878.000	1.40770	0.703850	+	0.710348i	$0.251463\pi$
$74$	0	0
$75$	50.0000i	0.0769800i
$76$	0	0
$77$	192.000i	0.284161i
$78$	0	0
$79$	−600.000	−0.854497	−0.427249	−	0.904134i	$-0.640517\pi$
$79$	−600.000	−0.854497	−0.427249	+	0.904134i	$0.640517\pi$
$80$	0	0
$81$	421.000	0.577503
$82$	0	0
$83$	− 282.000i	− 0.372934i	−0.982461	−	0.186467i	$-0.940296\pi$
$83$	− 282.000i	− 0.372934i	0.982461	−	0.186467i	$-0.0597037\pi$
$84$	0	0
$85$	130.000i	0.165888i
$86$	0	0
$87$	−100.000	−0.123231
$88$	0	0
$89$	150.000	0.178651	0.0893257	−	0.996002i	$-0.471529\pi$
$89$	150.000	0.178651	0.0893257	+	0.996002i	$0.471529\pi$
$90$	0	0
$91$	− 228.000i	− 0.262647i
$92$	0	0
$93$	− 216.000i	− 0.240840i
$94$	0	0
$95$	500.000	0.539989
$96$	0	0
$97$	386.000	0.404045	0.202022	−	0.979381i	$-0.435249\pi$
$97$	386.000	0.404045	0.202022	+	0.979381i	$0.435249\pi$
$98$	0	0
$99$	736.000i	0.747180i
$100$	0	0
$101$	− 702.000i	− 0.691600i	−0.938308	−	0.345800i	$-0.887608\pi$
$101$	− 702.000i	− 0.691600i	0.938308	−	0.345800i	$-0.112392\pi$
$102$	0	0
$103$	−598.000	−0.572065	−0.286032	−	0.958220i	$-0.592337\pi$
$103$	−598.000	−0.572065	−0.286032	+	0.958220i	$0.592337\pi$
$104$	0	0
$105$	60.0000	0.0557657
$106$	0	0
$107$	− 1194.00i	− 1.07877i	−0.842059	−	0.539385i	$-0.818657\pi$
$107$	− 1194.00i	− 1.07877i	0.842059	−	0.539385i	$-0.181343\pi$
$108$	0	0
$109$	− 550.000i	− 0.483307i	−0.970363	−	0.241653i	$-0.922310\pi$
$109$	− 550.000i	− 0.483307i	0.970363	−	0.241653i	$-0.0776897\pi$
$110$	0	0
$111$	−532.000	−0.454912
$112$	0	0
$113$	1562.00	1.30036	0.650180	−	0.759781i	$-0.274694\pi$
$113$	1562.00	1.30036	0.650180	+	0.759781i	$0.274694\pi$
$114$	0	0
$115$	− 390.000i	− 0.316241i
$116$	0	0
$117$	− 874.000i	− 0.690610i
$118$	0	0
$119$	156.000	0.120172
$120$	0	0
$121$	307.000	0.230654
$122$	0	0
$123$	44.0000i	0.0322548i
$124$	0	0
$125$	− 125.000i	− 0.0894427i
$126$	0	0
$127$	−1846.00	−1.28981	−0.644906	−	0.764262i	$-0.723103\pi$
$127$	−1846.00	−1.28981	−0.644906	+	0.764262i	$0.723103\pi$
$128$	0	0
$129$	884.000	0.603348
$130$	0	0
$131$	2208.00i	1.47262i	0.676642	+	0.736312i	$0.263435\pi$
$131$	2208.00i	1.47262i	−0.676642	+	0.736312i	$0.736565\pi$
$132$	0	0
$133$	− 600.000i	− 0.391177i
$134$	0	0
$135$	500.000	0.318764
$136$	0	0
$137$	2334.00	1.45553	0.727763	−	0.685829i	$-0.240560\pi$
$137$	2334.00	1.45553	0.727763	+	0.685829i	$0.240560\pi$
$138$	0	0
$139$	− 700.000i	− 0.427146i	−0.976927	−	0.213573i	$-0.931490\pi$
$139$	− 700.000i	− 0.427146i	0.976927	−	0.213573i	$-0.0685100\pi$
$140$	0	0
$141$	− 1028.00i	− 0.613994i
$142$	0	0
$143$	1216.00	0.711098
$144$	0	0
$145$	250.000	0.143182
$146$	0	0
$147$	614.000i	0.344502i
$148$	0	0
$149$	− 2050.00i	− 1.12713i	−0.826071	−	0.563566i	$-0.809429\pi$
$149$	− 2050.00i	− 1.12713i	0.826071	−	0.563566i	$-0.190571\pi$
$150$	0	0
$151$	1852.00	0.998103	0.499052	−	0.866572i	$-0.333682\pi$
$151$	1852.00	0.998103	0.499052	+	0.866572i	$0.333682\pi$
$152$	0	0
$153$	598.000	0.315983
$154$	0	0
$155$	540.000i	0.279831i
$156$	0	0
$157$	− 2494.00i	− 1.26779i	−0.773420	−	0.633894i	$-0.781455\pi$
$157$	− 2494.00i	− 1.26779i	0.773420	−	0.633894i	$-0.218545\pi$
$158$	0	0
$159$	−4.00000	−0.00199510
$160$	0	0
$161$	−468.000	−0.229090
$162$	0	0
$163$	− 2762.00i	− 1.32722i	−0.748080	−	0.663609i	$-0.769024\pi$
$163$	− 2762.00i	− 1.32722i	0.748080	−	0.663609i	$-0.230976\pi$
$164$	0	0
$165$	320.000i	0.150982i
$166$	0	0
$167$	3126.00	1.44849	0.724243	−	0.689545i	$-0.242189\pi$
$167$	3126.00	1.44849	0.724243	+	0.689545i	$0.242189\pi$
$168$	0	0
$169$	753.000	0.342740
$170$	0	0
$171$	− 2300.00i	− 1.02857i
$172$	0	0
$173$	− 78.0000i	− 0.0342788i	−0.999853	−	0.0171394i	$-0.994544\pi$
$173$	− 78.0000i	− 0.0342788i	0.999853	−	0.0171394i	$-0.00545591\pi$
$174$	0	0
$175$	−150.000	−0.0647939
$176$	0	0
$177$	1000.00	0.424659
$178$	0	0
$179$	1300.00i	0.542830i	0.962462	+	0.271415i	$0.0874916\pi$
$179$	1300.00i	0.542830i	−0.962462	+	0.271415i	$0.912508\pi$
$180$	0	0
$181$	− 1742.00i	− 0.715369i	−0.933842	−	0.357685i	$-0.883566\pi$
$181$	− 1742.00i	− 0.715369i	0.933842	−	0.357685i	$-0.116434\pi$
$182$	0	0
$183$	−1036.00	−0.418488
$184$	0	0
$185$	1330.00	0.528560
$186$	0	0
$187$	832.000i	0.325358i
$188$	0	0
$189$	− 600.000i	− 0.230918i
$190$	0	0
$191$	−3772.00	−1.42897	−0.714483	−	0.699653i	$-0.753338\pi$
$191$	−3772.00	−1.42897	−0.714483	+	0.699653i	$0.753338\pi$
$192$	0	0
$193$	−358.000	−0.133520	−0.0667601	−	0.997769i	$-0.521266\pi$
$193$	−358.000	−0.133520	−0.0667601	+	0.997769i	$0.521266\pi$
$194$	0	0
$195$	− 380.000i	− 0.139551i
$196$	0	0
$197$	2214.00i	0.800716i	0.916359	+	0.400358i	$0.131114\pi$
$197$	2214.00i	0.800716i	−0.916359	+	0.400358i	$0.868886\pi$
$198$	0	0
$199$	−2600.00	−0.926176	−0.463088	−	0.886312i	$-0.653259\pi$
$199$	−2600.00	−0.926176	−0.463088	+	0.886312i	$0.653259\pi$
$200$	0	0
$201$	−252.000	−0.0884314
$202$	0	0
$203$	− 300.000i	− 0.103724i
$204$	0	0
$205$	− 110.000i	− 0.0374767i
$206$	0	0
$207$	−1794.00	−0.602375
$208$	0	0
$209$	3200.00	1.05908
$210$	0	0
$211$	1168.00i	0.381083i	0.981679	+	0.190541i	$0.0610243\pi$
$211$	1168.00i	0.381083i	−0.981679	+	0.190541i	$0.938976\pi$
$212$	0	0
$213$	− 824.000i	− 0.265068i
$214$	0	0
$215$	−2210.00	−0.701027
$216$	0	0
$217$	648.000	0.202715
$218$	0	0
$219$	− 1756.00i	− 0.541824i
$220$	0	0
$221$	− 988.000i	− 0.300724i
$222$	0	0
$223$	6478.00	1.94529	0.972643	−	0.232303i	$-0.0746262\pi$
$223$	6478.00	1.94529	0.972643	+	0.232303i	$0.0746262\pi$
$224$	0	0
$225$	−575.000	−0.170370
$226$	0	0
$227$	− 646.000i	− 0.188883i	−0.995530	−	0.0944417i	$-0.969893\pi$
$227$	− 646.000i	− 0.188883i	0.995530	−	0.0944417i	$-0.0301066\pi$
$228$	0	0
$229$	− 3750.00i	− 1.08213i	−0.840982	−	0.541063i	$-0.818022\pi$
$229$	− 3750.00i	− 1.08213i	0.840982	−	0.541063i	$-0.181978\pi$
$230$	0	0
$231$	384.000	0.109374
$232$	0	0
$233$	−1482.00	−0.416691	−0.208346	−	0.978055i	$-0.566808\pi$
$233$	−1482.00	−0.416691	−0.208346	+	0.978055i	$0.566808\pi$
$234$	0	0
$235$	2570.00i	0.713397i
$236$	0	0
$237$	1200.00i	0.328896i
$238$	0	0
$239$	−1400.00	−0.378906	−0.189453	−	0.981890i	$-0.560671\pi$
$239$	−1400.00	−0.378906	−0.189453	+	0.981890i	$0.560671\pi$
$240$	0	0
$241$	3022.00	0.807735	0.403867	−	0.914817i	$-0.367666\pi$
$241$	3022.00	0.807735	0.403867	+	0.914817i	$0.367666\pi$
$242$	0	0
$243$	− 3542.00i	− 0.935059i
$244$	0	0
$245$	− 1535.00i	− 0.400276i
$246$	0	0
$247$	−3800.00	−0.978900
$248$	0	0
$249$	−564.000	−0.143542
$250$	0	0
$251$	− 1248.00i	− 0.313837i	−0.987612	−	0.156918i	$-0.949844\pi$
$251$	− 1248.00i	− 0.313837i	0.987612	−	0.156918i	$-0.0501560\pi$
$252$	0	0
$253$	− 2496.00i	− 0.620246i
$254$	0	0
$255$	260.000	0.0638503
$256$	0	0
$257$	2106.00	0.511162	0.255581	−	0.966788i	$-0.417733\pi$
$257$	2106.00	0.511162	0.255581	+	0.966788i	$0.417733\pi$
$258$	0	0
$259$	− 1596.00i	− 0.382898i
$260$	0	0
$261$	− 1150.00i	− 0.272733i
$262$	0	0
$263$	−3638.00	−0.852961	−0.426480	−	0.904497i	$-0.640247\pi$
$263$	−3638.00	−0.852961	−0.426480	+	0.904497i	$0.640247\pi$
$264$	0	0
$265$	10.0000	0.00231809
$266$	0	0
$267$	− 300.000i	− 0.0687629i
$268$	0	0
$269$	− 6550.00i	− 1.48461i	−0.670061	−	0.742306i	$-0.733732\pi$
$269$	− 6550.00i	− 1.48461i	0.670061	−	0.742306i	$-0.266268\pi$
$270$	0	0
$271$	4388.00	0.983587	0.491793	−	0.870712i	$-0.336342\pi$
$271$	4388.00	0.983587	0.491793	+	0.870712i	$0.336342\pi$
$272$	0	0
$273$	−456.000	−0.101093
$274$	0	0
$275$	− 800.000i	− 0.175425i
$276$	0	0
$277$	− 546.000i	− 0.118433i	−0.998245	−	0.0592165i	$-0.981140\pi$
$277$	− 546.000i	− 0.118433i	0.998245	−	0.0592165i	$-0.0188602\pi$
$278$	0	0
$279$	2484.00	0.533022
$280$	0	0
$281$	6858.00	1.45592	0.727961	−	0.685619i	$-0.240468\pi$
$281$	6858.00	1.45592	0.727961	+	0.685619i	$0.240468\pi$
$282$	0	0
$283$	9282.00i	1.94967i	0.222920	+	0.974837i	$0.428441\pi$
$283$	9282.00i	1.94967i	−0.222920	+	0.974837i	$0.571559\pi$
$284$	0	0
$285$	− 1000.00i	− 0.207842i
$286$	0	0
$287$	−132.000	−0.0271488
$288$	0	0
$289$	−4237.00	−0.862406
$290$	0	0
$291$	− 772.000i	− 0.155517i
$292$	0	0
$293$	− 4842.00i	− 0.965436i	−0.875776	−	0.482718i	$-0.839650\pi$
$293$	− 4842.00i	− 0.965436i	0.875776	−	0.482718i	$-0.160350\pi$
$294$	0	0
$295$	−2500.00	−0.493409
$296$	0	0
$297$	3200.00	0.625195
$298$	0	0
$299$	2964.00i	0.573286i
$300$	0	0
$301$	2652.00i	0.507836i
$302$	0	0
$303$	−1404.00	−0.266197
$304$	0	0
$305$	2590.00	0.486239
$306$	0	0
$307$	2594.00i	0.482239i	0.970495	+	0.241120i	$0.0775146\pi$
$307$	2594.00i	0.482239i	−0.970495	+	0.241120i	$0.922485\pi$
$308$	0	0
$309$	1196.00i	0.220188i
$310$	0	0
$311$	7332.00	1.33685	0.668424	−	0.743781i	$-0.266969\pi$
$311$	7332.00	1.33685	0.668424	+	0.743781i	$0.266969\pi$
$312$	0	0
$313$	−1562.00	−0.282075	−0.141037	−	0.990004i	$-0.545044\pi$
$313$	−1562.00	−0.282075	−0.141037	+	0.990004i	$0.545044\pi$
$314$	0	0
$315$	690.000i	0.123419i
$316$	0	0
$317$	1426.00i	0.252657i	0.991988	+	0.126328i	$0.0403193\pi$
$317$	1426.00i	0.252657i	−0.991988	+	0.126328i	$0.959681\pi$
$318$	0	0
$319$	1600.00	0.280824
$320$	0	0
$321$	−2388.00	−0.415219
$322$	0	0
$323$	− 2600.00i	− 0.447888i
$324$	0	0
$325$	950.000i	0.162143i
$326$	0	0
$327$	−1100.00	−0.186025
$328$	0	0
$329$	3084.00	0.516798
$330$	0	0
$331$	− 4008.00i	− 0.665558i	−0.943005	−	0.332779i	$-0.892014\pi$
$331$	− 4008.00i	− 0.665558i	0.943005	−	0.332779i	$-0.107986\pi$
$332$	0	0
$333$	− 6118.00i	− 1.00680i
$334$	0	0
$335$	630.000	0.102748
$336$	0	0
$337$	8866.00	1.43312	0.716561	−	0.697525i	$-0.245715\pi$
$337$	8866.00	1.43312	0.716561	+	0.697525i	$0.245715\pi$
$338$	0	0
$339$	− 3124.00i	− 0.500509i
$340$	0	0
$341$	3456.00i	0.548835i
$342$	0	0
$343$	−3900.00	−0.613936
$344$	0	0
$345$	−780.000	−0.121721
$346$	0	0
$347$	− 1714.00i	− 0.265165i	−0.991172	−	0.132583i	$-0.957673\pi$
$347$	− 1714.00i	− 0.265165i	0.991172	−	0.132583i	$-0.0423270\pi$
$348$	0	0
$349$	1150.00i	0.176384i	0.996103	+	0.0881921i	$0.0281089\pi$
$349$	1150.00i	0.176384i	−0.996103	+	0.0881921i	$0.971891\pi$
$350$	0	0
$351$	−3800.00	−0.577860
$352$	0	0
$353$	−4398.00	−0.663122	−0.331561	−	0.943434i	$-0.607575\pi$
$353$	−4398.00	−0.663122	−0.331561	+	0.943434i	$0.607575\pi$
$354$	0	0
$355$	2060.00i	0.307982i
$356$	0	0
$357$	− 312.000i	− 0.0462543i
$358$	0	0
$359$	1800.00	0.264625	0.132312	−	0.991208i	$-0.457760\pi$
$359$	1800.00	0.264625	0.132312	+	0.991208i	$0.457760\pi$
$360$	0	0
$361$	−3141.00	−0.457938
$362$	0	0
$363$	− 614.000i	− 0.0887786i
$364$	0	0
$365$	4390.00i	0.629543i
$366$	0	0
$367$	5874.00	0.835478	0.417739	−	0.908567i	$-0.362823\pi$
$367$	5874.00	0.835478	0.417739	+	0.908567i	$0.362823\pi$
$368$	0	0
$369$	−506.000	−0.0713857
$370$	0	0
$371$	− 12.0000i	− 0.00167927i
$372$	0	0
$373$	2078.00i	0.288458i	0.989544	+	0.144229i	$0.0460702\pi$
$373$	2078.00i	0.288458i	−0.989544	+	0.144229i	$0.953930\pi$
$374$	0	0
$375$	−250.000	−0.0344265
$376$	0	0
$377$	−1900.00	−0.259562
$378$	0	0
$379$	7900.00i	1.07070i	0.844630	+	0.535351i	$0.179821\pi$
$379$	7900.00i	1.07070i	−0.844630	+	0.535351i	$0.820179\pi$
$380$	0	0
$381$	3692.00i	0.496449i
$382$	0	0
$383$	7518.00	1.00301	0.501504	−	0.865155i	$-0.332780\pi$
$383$	7518.00	1.00301	0.501504	+	0.865155i	$0.332780\pi$
$384$	0	0
$385$	−960.000	−0.127081
$386$	0	0
$387$	10166.0i	1.33531i
$388$	0	0
$389$	1950.00i	0.254162i	0.991892	+	0.127081i	$0.0405608\pi$
$389$	1950.00i	0.254162i	−0.991892	+	0.127081i	$0.959439\pi$
$390$	0	0
$391$	−2028.00	−0.262303
$392$	0	0
$393$	4416.00	0.566814
$394$	0	0
$395$	− 3000.00i	− 0.382143i
$396$	0	0
$397$	13786.0i	1.74282i	0.490555	+	0.871410i	$0.336794\pi$
$397$	13786.0i	1.74282i	−0.490555	+	0.871410i	$0.663206\pi$
$398$	0	0
$399$	−1200.00	−0.150564
$400$	0	0
$401$	6402.00	0.797258	0.398629	−	0.917112i	$-0.369486\pi$
$401$	6402.00	0.797258	0.398629	+	0.917112i	$0.369486\pi$
$402$	0	0
$403$	− 4104.00i	− 0.507282i
$404$	0	0
$405$	2105.00i	0.258267i
$406$	0	0
$407$	8512.00	1.03667
$408$	0	0
$409$	−11150.0	−1.34800	−0.674000	−	0.738731i	$-0.735425\pi$
$409$	−11150.0	−1.34800	−0.674000	+	0.738731i	$0.735425\pi$
$410$	0	0
$411$	− 4668.00i	− 0.560232i
$412$	0	0
$413$	3000.00i	0.357434i
$414$	0	0
$415$	1410.00	0.166781
$416$	0	0
$417$	−1400.00	−0.164408
$418$	0	0
$419$	13700.0i	1.59735i	0.601764	+	0.798674i	$0.294465\pi$
$419$	13700.0i	1.59735i	−0.601764	+	0.798674i	$0.705535\pi$
$420$	0	0
$421$	5438.00i	0.629529i	0.949170	+	0.314765i	$0.101926\pi$
$421$	5438.00i	0.629529i	−0.949170	+	0.314765i	$0.898074\pi$
$422$	0	0
$423$	11822.0	1.35888
$424$	0	0
$425$	−650.000	−0.0741874
$426$	0	0
$427$	− 3108.00i	− 0.352240i
$428$	0	0
$429$	− 2432.00i	− 0.273702i
$430$	0	0
$431$	−7692.00	−0.859653	−0.429827	−	0.902911i	$-0.641425\pi$
$431$	−7692.00	−0.859653	−0.429827	+	0.902911i	$0.641425\pi$
$432$	0	0
$433$	−1118.00	−0.124082	−0.0620412	−	0.998074i	$-0.519761\pi$
$433$	−1118.00	−0.124082	−0.0620412	+	0.998074i	$0.519761\pi$
$434$	0	0
$435$	− 500.000i	− 0.0551107i
$436$	0	0
$437$	7800.00i	0.853832i
$438$	0	0
$439$	−2600.00	−0.282668	−0.141334	−	0.989962i	$-0.545139\pi$
$439$	−2600.00	−0.282668	−0.141334	+	0.989962i	$0.545139\pi$
$440$	0	0
$441$	−7061.00	−0.762445
$442$	0	0
$443$	− 11958.0i	− 1.28249i	−0.767337	−	0.641243i	$-0.778419\pi$
$443$	− 11958.0i	− 1.28249i	0.767337	−	0.641243i	$-0.221581\pi$
$444$	0	0
$445$	750.000i	0.0798953i
$446$	0	0
$447$	−4100.00	−0.433833
$448$	0	0
$449$	−17050.0	−1.79207	−0.896035	−	0.443984i	$-0.853565\pi$
$449$	−17050.0	−1.79207	−0.896035	+	0.443984i	$0.853565\pi$
$450$	0	0
$451$	− 704.000i	− 0.0735035i
$452$	0	0
$453$	− 3704.00i	− 0.384170i
$454$	0	0
$455$	1140.00	0.117459
$456$	0	0
$457$	9494.00	0.971796	0.485898	−	0.874016i	$-0.338493\pi$
$457$	9494.00	0.971796	0.485898	+	0.874016i	$0.338493\pi$
$458$	0	0
$459$	− 2600.00i	− 0.264396i
$460$	0	0
$461$	− 11418.0i	− 1.15356i	−0.816901	−	0.576778i	$-0.804310\pi$
$461$	− 11418.0i	− 1.15356i	0.816901	−	0.576778i	$-0.195690\pi$
$462$	0	0
$463$	−7962.00	−0.799191	−0.399596	−	0.916692i	$-0.630849\pi$
$463$	−7962.00	−0.799191	−0.399596	+	0.916692i	$0.630849\pi$
$464$	0	0
$465$	1080.00	0.107707
$466$	0	0
$467$	− 6526.00i	− 0.646654i	−0.946287	−	0.323327i	$-0.895199\pi$
$467$	− 6526.00i	− 0.646654i	0.946287	−	0.323327i	$-0.104801\pi$
$468$	0	0
$469$	− 756.000i	− 0.0744325i
$470$	0	0
$471$	−4988.00	−0.487972
$472$	0	0
$473$	−14144.0	−1.37493
$474$	0	0
$475$	2500.00i	0.241490i
$476$	0	0
$477$	− 46.0000i	− 0.00441550i
$478$	0	0
$479$	−17400.0	−1.65976	−0.829881	−	0.557940i	$-0.811592\pi$
$479$	−17400.0	−1.65976	−0.829881	+	0.557940i	$0.811592\pi$
$480$	0	0
$481$	−10108.0	−0.958181
$482$	0	0
$483$	936.000i	0.0881770i
$484$	0	0
$485$	1930.00i	0.180694i
$486$	0	0
$487$	1166.00	0.108494	0.0542469	−	0.998528i	$-0.482724\pi$
$487$	1166.00	0.108494	0.0542469	+	0.998528i	$0.482724\pi$
$488$	0	0
$489$	−5524.00	−0.510846
$490$	0	0
$491$	7072.00i	0.650010i	0.945712	+	0.325005i	$0.105366\pi$
$491$	7072.00i	0.650010i	−0.945712	+	0.325005i	$0.894634\pi$
$492$	0	0
$493$	− 1300.00i	− 0.118761i
$494$	0	0
$495$	−3680.00	−0.334149
$496$	0	0
$497$	2472.00	0.223107
$498$	0	0
$499$	− 100.000i	− 0.00897117i	−0.999990	−	0.00448559i	$-0.998572\pi$
$499$	− 100.000i	− 0.00897117i	0.999990	−	0.00448559i	$-0.00142781\pi$
$500$	0	0
$501$	− 6252.00i	− 0.557522i
$502$	0	0
$503$	2602.00	0.230651	0.115325	−	0.993328i	$-0.463209\pi$
$503$	2602.00	0.230651	0.115325	+	0.993328i	$0.463209\pi$
$504$	0	0
$505$	3510.00	0.309293
$506$	0	0
$507$	− 1506.00i	− 0.131921i
$508$	0	0
$509$	11150.0i	0.970953i	0.874250	+	0.485476i	$0.161354\pi$
$509$	11150.0i	0.970953i	−0.874250	+	0.485476i	$0.838646\pi$
$510$	0	0
$511$	5268.00	0.456052
$512$	0	0
$513$	−10000.0	−0.860645
$514$	0	0
$515$	− 2990.00i	− 0.255835i
$516$	0	0
$517$	16448.0i	1.39919i
$518$	0	0
$519$	−156.000	−0.0131939
$520$	0	0
$521$	3638.00	0.305919	0.152959	−	0.988232i	$-0.451120\pi$
$521$	3638.00	0.305919	0.152959	+	0.988232i	$0.451120\pi$
$522$	0	0
$523$	− 2078.00i	− 0.173737i	−0.996220	−	0.0868686i	$-0.972314\pi$
$523$	− 2078.00i	− 0.173737i	0.996220	−	0.0868686i	$-0.0276860\pi$
$524$	0	0
$525$	300.000i	0.0249392i
$526$	0	0
$527$	2808.00	0.232103
$528$	0	0
$529$	−6083.00	−0.499959
$530$	0	0
$531$	11500.0i	0.939845i
$532$	0	0
$533$	836.000i	0.0679384i
$534$	0	0
$535$	5970.00	0.482440
$536$	0	0
$537$	2600.00	0.208935
$538$	0	0
$539$	− 9824.00i	− 0.785064i
$540$	0	0
$541$	5622.00i	0.446781i	0.974729	+	0.223391i	$0.0717126\pi$
$541$	5622.00i	0.446781i	−0.974729	+	0.223391i	$0.928287\pi$
$542$	0	0
$543$	−3484.00	−0.275346
$544$	0	0
$545$	2750.00	0.216141
$546$	0	0
$547$	− 16486.0i	− 1.28865i	−0.764753	−	0.644324i	$-0.777139\pi$
$547$	− 16486.0i	− 1.28865i	0.764753	−	0.644324i	$-0.222861\pi$
$548$	0	0
$549$	− 11914.0i	− 0.926188i
$550$	0	0
$551$	−5000.00	−0.386583
$552$	0	0
$553$	−3600.00	−0.276831
$554$	0	0
$555$	− 2660.00i	− 0.203443i
$556$	0	0
$557$	11706.0i	0.890483i	0.895410	+	0.445242i	$0.146882\pi$
$557$	11706.0i	0.890483i	−0.895410	+	0.445242i	$0.853118\pi$
$558$	0	0
$559$	16796.0	1.27083
$560$	0	0
$561$	1664.00	0.125230
$562$	0	0
$563$	25038.0i	1.87429i	0.348939	+	0.937146i	$0.386542\pi$
$563$	25038.0i	1.87429i	−0.348939	+	0.937146i	$0.613458\pi$
$564$	0	0
$565$	7810.00i	0.581538i
$566$	0	0
$567$	2526.00	0.187094
$568$	0	0
$569$	−17550.0	−1.29303	−0.646515	−	0.762901i	$-0.723774\pi$
$569$	−17550.0	−1.29303	−0.646515	+	0.762901i	$0.723774\pi$
$570$	0	0
$571$	10712.0i	0.785084i	0.919734	+	0.392542i	$0.128404\pi$
$571$	10712.0i	0.785084i	−0.919734	+	0.392542i	$0.871596\pi$
$572$	0	0
$573$	7544.00i	0.550009i
$574$	0	0
$575$	1950.00	0.141427
$576$	0	0
$577$	−13654.0	−0.985136	−0.492568	−	0.870274i	$-0.663942\pi$
$577$	−13654.0	−0.985136	−0.492568	+	0.870274i	$0.663942\pi$
$578$	0	0
$579$	716.000i	0.0513920i
$580$	0	0
$581$	− 1692.00i	− 0.120819i
$582$	0	0
$583$	64.0000	0.00454650
$584$	0	0
$585$	4370.00	0.308850
$586$	0	0
$587$	14166.0i	0.996071i	0.867157	+	0.498035i	$0.165945\pi$
$587$	14166.0i	0.996071i	−0.867157	+	0.498035i	$0.834055\pi$
$588$	0	0
$589$	− 10800.0i	− 0.755528i
$590$	0	0
$591$	4428.00	0.308196
$592$	0	0
$593$	17842.0	1.23555	0.617777	−	0.786354i	$-0.288034\pi$
$593$	17842.0	1.23555	0.617777	+	0.786354i	$0.288034\pi$
$594$	0	0
$595$	780.000i	0.0537427i
$596$	0	0
$597$	5200.00i	0.356485i
$598$	0	0
$599$	−17600.0	−1.20053	−0.600264	−	0.799802i	$-0.704938\pi$
$599$	−17600.0	−1.20053	−0.600264	+	0.799802i	$0.704938\pi$
$600$	0	0
$601$	−27302.0	−1.85303	−0.926516	−	0.376256i	$-0.877211\pi$
$601$	−27302.0	−1.85303	−0.926516	+	0.376256i	$0.877211\pi$
$602$	0	0
$603$	− 2898.00i	− 0.195714i
$604$	0	0
$605$	1535.00i	0.103151i
$606$	0	0
$607$	3794.00	0.253696	0.126848	−	0.991922i	$-0.459514\pi$
$607$	3794.00	0.253696	0.126848	+	0.991922i	$0.459514\pi$
$608$	0	0
$609$	−600.000	−0.0399232
$610$	0	0
$611$	− 19532.0i	− 1.29326i
$612$	0	0
$613$	13238.0i	0.872231i	0.899891	+	0.436116i	$0.143646\pi$
$613$	13238.0i	0.872231i	−0.899891	+	0.436116i	$0.856354\pi$
$614$	0	0
$615$	−220.000	−0.0144248
$616$	0	0
$617$	11574.0	0.755189	0.377595	−	0.925971i	$-0.376751\pi$
$617$	11574.0	0.755189	0.377595	+	0.925971i	$0.376751\pi$
$618$	0	0
$619$	8300.00i	0.538942i	0.963008	+	0.269471i	$0.0868489\pi$
$619$	8300.00i	0.538942i	−0.963008	+	0.269471i	$0.913151\pi$
$620$	0	0
$621$	7800.00i	0.504031i
$622$	0	0
$623$	900.000	0.0578776
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	− 6400.00i	− 0.407642i
$628$	0	0
$629$	− 6916.00i	− 0.438409i
$630$	0	0
$631$	−7508.00	−0.473675	−0.236837	−	0.971549i	$-0.576111\pi$
$631$	−7508.00	−0.473675	−0.236837	+	0.971549i	$0.576111\pi$
$632$	0	0
$633$	2336.00	0.146679
$634$	0	0
$635$	− 9230.00i	− 0.576821i
$636$	0	0
$637$	11666.0i	0.725626i
$638$	0	0
$639$	9476.00	0.586643
$640$	0	0
$641$	−27378.0	−1.68700	−0.843499	−	0.537130i	$-0.819508\pi$
$641$	−27378.0	−1.68700	−0.843499	+	0.537130i	$0.819508\pi$
$642$	0	0
$643$	− 1842.00i	− 0.112973i	−0.998403	−	0.0564863i	$-0.982010\pi$
$643$	− 1842.00i	− 0.112973i	0.998403	−	0.0564863i	$-0.0179897\pi$
$644$	0	0
$645$	4420.00i	0.269825i
$646$	0	0
$647$	−10114.0	−0.614563	−0.307282	−	0.951619i	$-0.599419\pi$
$647$	−10114.0	−0.614563	−0.307282	+	0.951619i	$0.599419\pi$
$648$	0	0
$649$	−16000.0	−0.967727
$650$	0	0
$651$	− 1296.00i	− 0.0780250i
$652$	0	0
$653$	10402.0i	0.623372i	0.950185	+	0.311686i	$0.100894\pi$
$653$	10402.0i	0.623372i	−0.950185	+	0.311686i	$0.899106\pi$
$654$	0	0
$655$	−11040.0	−0.658578
$656$	0	0
$657$	20194.0	1.19915
$658$	0	0
$659$	− 7100.00i	− 0.419692i	−0.977734	−	0.209846i	$-0.932704\pi$
$659$	− 7100.00i	− 0.419692i	0.977734	−	0.209846i	$-0.0672962\pi$
$660$	0	0
$661$	7118.00i	0.418847i	0.977825	+	0.209424i	$0.0671588\pi$
$661$	7118.00i	0.418847i	−0.977825	+	0.209424i	$0.932841\pi$
$662$	0	0
$663$	−1976.00	−0.115749
$664$	0	0
$665$	3000.00	0.174940
$666$	0	0
$667$	3900.00i	0.226400i
$668$	0	0
$669$	− 12956.0i	− 0.748741i
$670$	0	0
$671$	16576.0	0.953665
$672$	0	0
$673$	−31278.0	−1.79150	−0.895749	−	0.444560i	$-0.853360\pi$
$673$	−31278.0	−1.79150	−0.895749	+	0.444560i	$0.853360\pi$
$674$	0	0
$675$	2500.00i	0.142556i
$676$	0	0
$677$	30054.0i	1.70616i	0.521782	+	0.853079i	$0.325268\pi$
$677$	30054.0i	1.70616i	−0.521782	+	0.853079i	$0.674732\pi$
$678$	0	0
$679$	2316.00	0.130898
$680$	0	0
$681$	−1292.00	−0.0727012
$682$	0	0
$683$	− 4518.00i	− 0.253113i	−0.991959	−	0.126557i	$-0.959607\pi$
$683$	− 4518.00i	− 0.253113i	0.991959	−	0.126557i	$-0.0403926\pi$
$684$	0	0
$685$	11670.0i	0.650931i
$686$	0	0
$687$	−7500.00	−0.416511
$688$	0	0
$689$	−76.0000	−0.00420228
$690$	0	0
$691$	− 29272.0i	− 1.61152i	−0.592243	−	0.805759i	$-0.701758\pi$
$691$	− 29272.0i	− 1.61152i	0.592243	−	0.805759i	$-0.298242\pi$
$692$	0	0
$693$	4416.00i	0.242063i
$694$	0	0
$695$	3500.00	0.191025
$696$	0	0
$697$	−572.000	−0.0310847
$698$	0	0
$699$	2964.00i	0.160385i
$700$	0	0
$701$	− 5798.00i	− 0.312393i	−0.987726	−	0.156196i	$-0.950077\pi$
$701$	− 5798.00i	− 0.312393i	0.987726	−	0.156196i	$-0.0499233\pi$
$702$	0	0
$703$	−26600.0	−1.42708
$704$	0	0
$705$	5140.00	0.274587
$706$	0	0
$707$	− 4212.00i	− 0.224057i
$708$	0	0
$709$	− 8950.00i	− 0.474082i	−0.971500	−	0.237041i	$-0.923822\pi$
$709$	− 8950.00i	− 0.474082i	0.971500	−	0.237041i	$-0.0761776\pi$
$710$	0	0
$711$	−13800.0	−0.727905
$712$	0	0
$713$	−8424.00	−0.442470
$714$	0	0
$715$	6080.00i	0.318013i
$716$	0	0
$717$	2800.00i	0.145841i
$718$	0	0
$719$	−7800.00	−0.404577	−0.202289	−	0.979326i	$-0.564838\pi$
$719$	−7800.00	−0.404577	−0.202289	+	0.979326i	$0.564838\pi$
$720$	0	0
$721$	−3588.00	−0.185332
$722$	0	0
$723$	− 6044.00i	− 0.310897i
$724$	0	0
$725$	1250.00i	0.0640329i
$726$	0	0
$727$	−8554.00	−0.436383	−0.218191	−	0.975906i	$-0.570016\pi$
$727$	−8554.00	−0.436383	−0.218191	+	0.975906i	$0.570016\pi$
$728$	0	0
$729$	4283.00	0.217599
$730$	0	0
$731$	11492.0i	0.581460i
$732$	0	0
$733$	2882.00i	0.145224i	0.997360	+	0.0726119i	$0.0231335\pi$
$733$	2882.00i	0.145224i	−0.997360	+	0.0726119i	$0.976867\pi$
$734$	0	0
$735$	−3070.00	−0.154066
$736$	0	0
$737$	4032.00	0.201521
$738$	0	0
$739$	− 18700.0i	− 0.930840i	−0.885090	−	0.465420i	$-0.845903\pi$
$739$	− 18700.0i	− 0.930840i	0.885090	−	0.465420i	$-0.154097\pi$
$740$	0	0
$741$	7600.00i	0.376779i
$742$	0	0
$743$	12242.0	0.604462	0.302231	−	0.953235i	$-0.402269\pi$
$743$	12242.0	0.604462	0.302231	+	0.953235i	$0.402269\pi$
$744$	0	0
$745$	10250.0	0.504068
$746$	0	0
$747$	− 6486.00i	− 0.317685i
$748$	0	0
$749$	− 7164.00i	− 0.349488i
$750$	0	0
$751$	31148.0	1.51346	0.756729	−	0.653729i	$-0.226796\pi$
$751$	31148.0	1.51346	0.756729	+	0.653729i	$0.226796\pi$
$752$	0	0
$753$	−2496.00	−0.120796
$754$	0	0
$755$	9260.00i	0.446365i
$756$	0	0
$757$	7694.00i	0.369410i	0.982794	+	0.184705i	$0.0591329\pi$
$757$	7694.00i	0.369410i	−0.982794	+	0.184705i	$0.940867\pi$
$758$	0	0
$759$	−4992.00	−0.238733
$760$	0	0
$761$	4518.00	0.215213	0.107607	−	0.994194i	$-0.465681\pi$
$761$	4518.00	0.215213	0.107607	+	0.994194i	$0.465681\pi$
$762$	0	0
$763$	− 3300.00i	− 0.156577i
$764$	0	0
$765$	2990.00i	0.141312i
$766$	0	0
$767$	19000.0	0.894459
$768$	0	0
$769$	−39550.0	−1.85463	−0.927314	−	0.374283i	$-0.877889\pi$
$769$	−39550.0	−1.85463	−0.927314	+	0.374283i	$0.877889\pi$
$770$	0	0
$771$	− 4212.00i	− 0.196746i
$772$	0	0
$773$	− 22122.0i	− 1.02933i	−0.857391	−	0.514666i	$-0.827916\pi$
$773$	− 22122.0i	− 1.02933i	0.857391	−	0.514666i	$-0.172084\pi$
$774$	0	0
$775$	−2700.00	−0.125144
$776$	0	0
$777$	−3192.00	−0.147378
$778$	0	0
$779$	2200.00i	0.101185i
$780$	0	0
$781$	13184.0i	0.604047i
$782$	0	0
$783$	−5000.00	−0.228206
$784$	0	0
$785$	12470.0	0.566972
$786$	0	0
$787$	16634.0i	0.753416i	0.926332	+	0.376708i	$0.122944\pi$
$787$	16634.0i	0.753416i	−0.926332	+	0.376708i	$0.877056\pi$
$788$	0	0
$789$	7276.00i	0.328305i
$790$	0	0
$791$	9372.00	0.421277
$792$	0	0
$793$	−19684.0	−0.881462
$794$	0	0
$795$	− 20.0000i	0 0.000892235i
$796$	0	0
$797$	27586.0i	1.22603i	0.790071	+	0.613015i	$0.210044\pi$
$797$	27586.0i	1.22603i	−0.790071	+	0.613015i	$0.789956\pi$
$798$	0	0
$799$	13364.0	0.591720
$800$	0	0
$801$	3450.00	0.152184
$802$	0	0
$803$	28096.0i	1.23473i
$804$	0	0
$805$	− 2340.00i	− 0.102452i
$806$	0	0
$807$	−13100.0	−0.571427
$808$	0	0
$809$	−3850.00	−0.167316	−0.0836581	−	0.996495i	$-0.526660\pi$
$809$	−3850.00	−0.167316	−0.0836581	+	0.996495i	$0.526660\pi$
$810$	0	0
$811$	10032.0i	0.434366i	0.976131	+	0.217183i	$0.0696869\pi$
$811$	10032.0i	0.434366i	−0.976131	+	0.217183i	$0.930313\pi$
$812$	0	0
$813$	− 8776.00i	− 0.378583i
$814$	0	0
$815$	13810.0	0.593550
$816$	0	0
$817$	44200.0	1.89273
$818$	0	0
$819$	− 5244.00i	− 0.223736i
$820$	0	0
$821$	− 20562.0i	− 0.874079i	−0.899442	−	0.437039i	$-0.856027\pi$
$821$	− 20562.0i	− 0.874079i	0.899442	−	0.437039i	$-0.143973\pi$
$822$	0	0
$823$	10322.0	0.437184	0.218592	−	0.975816i	$-0.429854\pi$
$823$	10322.0	0.437184	0.218592	+	0.975816i	$0.429854\pi$
$824$	0	0
$825$	−1600.00	−0.0675210
$826$	0	0
$827$	8846.00i	0.371954i	0.982554	+	0.185977i	$0.0595449\pi$
$827$	8846.00i	0.371954i	−0.982554	+	0.185977i	$0.940455\pi$
$828$	0	0
$829$	− 25350.0i	− 1.06205i	−0.847355	−	0.531026i	$-0.821806\pi$
$829$	− 25350.0i	− 1.06205i	0.847355	−	0.531026i	$-0.178194\pi$
$830$	0	0
$831$	−1092.00	−0.0455849
$832$	0	0
$833$	−7982.00	−0.332005
$834$	0	0
$835$	15630.0i	0.647783i
$836$	0	0
$837$	− 10800.0i	− 0.446001i
$838$	0	0
$839$	46000.0	1.89284	0.946422	−	0.322932i	$-0.104669\pi$
$839$	46000.0	1.89284	0.946422	+	0.322932i	$0.104669\pi$
$840$	0	0
$841$	21889.0	0.897495
$842$	0	0
$843$	− 13716.0i	− 0.560385i
$844$	0	0
$845$	3765.00i	0.153278i
$846$	0	0
$847$	1842.00	0.0747248
$848$	0	0
$849$	18564.0	0.750430
$850$	0	0
$851$	20748.0i	0.835761i
$852$	0	0
$853$	16998.0i	0.682298i	0.940009	+	0.341149i	$0.110816\pi$
$853$	16998.0i	0.682298i	−0.940009	+	0.341149i	$0.889184\pi$
$854$	0	0
$855$	11500.0	0.459990
$856$	0	0
$857$	26494.0	1.05603	0.528015	−	0.849235i	$-0.322936\pi$
$857$	26494.0	1.05603	0.528015	+	0.849235i	$0.322936\pi$
$858$	0	0
$859$	− 21500.0i	− 0.853982i	−0.904256	−	0.426991i	$-0.859574\pi$
$859$	− 21500.0i	− 0.853982i	0.904256	−	0.426991i	$-0.140426\pi$
$860$	0	0
$861$	264.000i	0.0104496i
$862$	0	0
$863$	−25762.0	−1.01616	−0.508082	−	0.861309i	$-0.669645\pi$
$863$	−25762.0	−1.01616	−0.508082	+	0.861309i	$0.669645\pi$
$864$	0	0
$865$	390.000	0.0153299
$866$	0	0
$867$	8474.00i	0.331940i
$868$	0	0
$869$	− 19200.0i	− 0.749500i
$870$	0	0
$871$	−4788.00	−0.186263
$872$	0	0
$873$	8878.00	0.344186
$874$	0	0
$875$	− 750.000i	− 0.0289767i
$876$	0	0
$877$	30546.0i	1.17613i	0.808814	+	0.588064i	$0.200110\pi$
$877$	30546.0i	1.17613i	−0.808814	+	0.588064i	$0.799890\pi$
$878$	0	0
$879$	−9684.00	−0.371596
$880$	0	0
$881$	32942.0	1.25976	0.629878	−	0.776694i	$-0.283105\pi$
$881$	32942.0	1.25976	0.629878	+	0.776694i	$0.283105\pi$
$882$	0	0
$883$	27118.0i	1.03351i	0.856132	+	0.516757i	$0.172861\pi$
$883$	27118.0i	1.03351i	−0.856132	+	0.516757i	$0.827139\pi$
$884$	0	0
$885$	5000.00i	0.189913i
$886$	0	0
$887$	−38634.0	−1.46246	−0.731230	−	0.682131i	$-0.761054\pi$
$887$	−38634.0	−1.46246	−0.731230	+	0.682131i	$0.761054\pi$
$888$	0	0
$889$	−11076.0	−0.417860
$890$	0	0
$891$	13472.0i	0.506542i
$892$	0	0
$893$	− 51400.0i	− 1.92613i
$894$	0	0
$895$	−6500.00	−0.242761
$896$	0	0
$897$	5928.00	0.220658
$898$	0	0
$899$	− 5400.00i	− 0.200334i
$900$	0	0
$901$	− 52.0000i	− 0.00192272i
$902$	0	0
$903$	5304.00	0.195466
$904$	0	0
$905$	8710.00	0.319923
$906$	0	0
$907$	− 1794.00i	− 0.0656767i	−0.999461	−	0.0328384i	$-0.989545\pi$
$907$	− 1794.00i	− 0.0656767i	0.999461	−	0.0328384i	$-0.0104547\pi$
$908$	0	0
$909$	− 16146.0i	− 0.589141i
$910$	0	0
$911$	−41732.0	−1.51772	−0.758860	−	0.651254i	$-0.774243\pi$
$911$	−41732.0	−1.51772	−0.758860	+	0.651254i	$0.774243\pi$
$912$	0	0
$913$	9024.00	0.327109
$914$	0	0
$915$	− 5180.00i	− 0.187154i
$916$	0	0
$917$	13248.0i	0.477086i
$918$	0	0
$919$	29200.0	1.04812	0.524058	−	0.851682i	$-0.324417\pi$
$919$	29200.0	1.04812	0.524058	+	0.851682i	$0.324417\pi$
$920$	0	0
$921$	5188.00	0.185614
$922$	0	0
$923$	− 15656.0i	− 0.558314i
$924$	0	0
$925$	6650.00i	0.236379i
$926$	0	0
$927$	−13754.0	−0.487315
$928$	0	0
$929$	−48650.0	−1.71814	−0.859071	−	0.511856i	$-0.828958\pi$
$929$	−48650.0	−1.71814	−0.859071	+	0.511856i	$0.828958\pi$
$930$	0	0
$931$	30700.0i	1.08072i
$932$	0	0
$933$	− 14664.0i	− 0.514553i
$934$	0	0
$935$	−4160.00	−0.145504
$936$	0	0
$937$	11334.0	0.395161	0.197580	−	0.980287i	$-0.436692\pi$
$937$	11334.0	0.395161	0.197580	+	0.980287i	$0.436692\pi$
$938$	0	0
$939$	3124.00i	0.108571i
$940$	0	0
$941$	− 31178.0i	− 1.08010i	−0.841633	−	0.540050i	$-0.818405\pi$
$941$	− 31178.0i	− 1.08010i	0.841633	−	0.540050i	$-0.181595\pi$
$942$	0	0
$943$	1716.00	0.0592584
$944$	0	0
$945$	3000.00	0.103270
$946$	0	0
$947$	− 4686.00i	− 0.160797i	−0.996763	−	0.0803984i	$-0.974381\pi$
$947$	− 4686.00i	− 0.160797i	0.996763	−	0.0803984i	$-0.0256193\pi$
$948$	0	0
$949$	− 33364.0i	− 1.14124i
$950$	0	0
$951$	2852.00	0.0972476
$952$	0	0
$953$	598.000	0.0203265	0.0101632	−	0.999948i	$-0.496765\pi$
$953$	598.000	0.0203265	0.0101632	+	0.999948i	$0.496765\pi$
$954$	0	0
$955$	− 18860.0i	− 0.639053i
$956$	0	0
$957$	− 3200.00i	− 0.108089i
$958$	0	0
$959$	14004.0	0.471546
$960$	0	0
$961$	−18127.0	−0.608472
$962$	0	0
$963$	− 27462.0i	− 0.918952i
$964$	0	0
$965$	− 1790.00i	− 0.0597121i
$966$	0	0
$967$	41726.0	1.38761	0.693804	−	0.720163i	$-0.255933\pi$
$967$	41726.0	1.38761	0.693804	+	0.720163i	$0.255933\pi$
$968$	0	0
$969$	−5200.00	−0.172392
$970$	0	0
$971$	24312.0i	0.803511i	0.915747	+	0.401756i	$0.131600\pi$
$971$	24312.0i	0.803511i	−0.915747	+	0.401756i	$0.868400\pi$
$972$	0	0
$973$	− 4200.00i	− 0.138382i
$974$	0	0
$975$	1900.00	0.0624089
$976$	0	0
$977$	40946.0	1.34082	0.670409	−	0.741992i	$-0.266119\pi$
$977$	40946.0	1.34082	0.670409	+	0.741992i	$0.266119\pi$
$978$	0	0
$979$	4800.00i	0.156699i
$980$	0	0
$981$	− 12650.0i	− 0.411706i
$982$	0	0
$983$	42282.0	1.37191	0.685954	−	0.727645i	$-0.259385\pi$
$983$	42282.0	1.37191	0.685954	+	0.727645i	$0.259385\pi$
$984$	0	0
$985$	−11070.0	−0.358091
$986$	0	0
$987$	− 6168.00i	− 0.198916i
$988$	0	0
$989$	− 34476.0i	− 1.10847i
$990$	0	0
$991$	−1172.00	−0.0375679	−0.0187840	−	0.999824i	$-0.505979\pi$
$991$	−1172.00	−0.0375679	−0.0187840	+	0.999824i	$0.505979\pi$
$992$	0	0
$993$	−8016.00	−0.256173
$994$	0	0
$995$	− 13000.0i	− 0.414199i
$996$	0	0
$997$	31614.0i	1.00424i	0.864798	+	0.502119i	$0.167446\pi$
$997$	31614.0i	1.00424i	−0.864798	+	0.502119i	$0.832554\pi$
$998$	0	0
$999$	−26600.0	−0.842429

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1280.4.d.l.641.1		2
4.3	odd	2		1280.4.d.e.641.2		2
8.3	odd	2		1280.4.d.e.641.1		2
8.5	even	2	inner	1280.4.d.l.641.2		2
16.3	odd	4		320.4.a.g.1.1		1
16.5	even	4		80.4.a.d.1.1		1
16.11	odd	4		5.4.a.a.1.1	✓	1
16.13	even	4		320.4.a.h.1.1		1
48.5	odd	4		720.4.a.u.1.1		1
48.11	even	4		45.4.a.d.1.1		1
80.19	odd	4		1600.4.a.bi.1.1		1
80.27	even	4		25.4.b.a.24.1		2
80.29	even	4		1600.4.a.s.1.1		1
80.37	odd	4		400.4.c.k.49.2		2
80.43	even	4		25.4.b.a.24.2		2
80.53	odd	4		400.4.c.k.49.1		2
80.59	odd	4		25.4.a.c.1.1		1
80.69	even	4		400.4.a.m.1.1		1
112.11	odd	12		245.4.e.f.226.1		2
112.27	even	4		245.4.a.a.1.1		1
112.59	even	12		245.4.e.g.226.1		2
112.75	even	12		245.4.e.g.116.1		2
112.107	odd	12		245.4.e.f.116.1		2
144.11	even	12		405.4.e.c.271.1		2
144.43	odd	12		405.4.e.l.271.1		2
144.59	even	12		405.4.e.c.136.1		2
144.139	odd	12		405.4.e.l.136.1		2
176.43	even	4		605.4.a.d.1.1		1
208.155	odd	4		845.4.a.b.1.1		1
240.59	even	4		225.4.a.b.1.1		1
240.107	odd	4		225.4.b.c.199.2		2
240.203	odd	4		225.4.b.c.199.1		2
272.203	odd	4		1445.4.a.a.1.1		1
304.75	even	4		1805.4.a.h.1.1		1
336.251	odd	4		2205.4.a.q.1.1		1
560.139	even	4		1225.4.a.k.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5.4.a.a.1.1	✓	1	16.11	odd	4
25.4.a.c.1.1		1	80.59	odd	4
25.4.b.a.24.1		2	80.27	even	4
25.4.b.a.24.2		2	80.43	even	4
45.4.a.d.1.1		1	48.11	even	4
80.4.a.d.1.1		1	16.5	even	4
225.4.a.b.1.1		1	240.59	even	4
225.4.b.c.199.1		2	240.203	odd	4
225.4.b.c.199.2		2	240.107	odd	4
245.4.a.a.1.1		1	112.27	even	4
245.4.e.f.116.1		2	112.107	odd	12
245.4.e.f.226.1		2	112.11	odd	12
245.4.e.g.116.1		2	112.75	even	12
245.4.e.g.226.1		2	112.59	even	12
320.4.a.g.1.1		1	16.3	odd	4
320.4.a.h.1.1		1	16.13	even	4
400.4.a.m.1.1		1	80.69	even	4
400.4.c.k.49.1		2	80.53	odd	4
400.4.c.k.49.2		2	80.37	odd	4
405.4.e.c.136.1		2	144.59	even	12
405.4.e.c.271.1		2	144.11	even	12
405.4.e.l.136.1		2	144.139	odd	12
405.4.e.l.271.1		2	144.43	odd	12
605.4.a.d.1.1		1	176.43	even	4
720.4.a.u.1.1		1	48.5	odd	4
845.4.a.b.1.1		1	208.155	odd	4
1225.4.a.k.1.1		1	560.139	even	4
1280.4.d.e.641.1		2	8.3	odd	2
1280.4.d.e.641.2		2	4.3	odd	2
1280.4.d.l.641.1		2	1.1	even	1	trivial
1280.4.d.l.641.2		2	8.5	even	2	inner
1445.4.a.a.1.1		1	272.203	odd	4
1600.4.a.s.1.1		1	80.29	even	4
1600.4.a.bi.1.1		1	80.19	odd	4
1805.4.a.h.1.1		1	304.75	even	4
2205.4.a.q.1.1		1	336.251	odd	4

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:	\( N \)	\(=\)	\( 1280 = 2^{8} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1280.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-2.00000i q^{3} +5.00000i q^{5} +6.00000 q^{7} +23.0000 q^{9} +O(q^{10})\)	\(q-2.00000i q^{3} +5.00000i q^{5} +6.00000 q^{7} +23.0000 q^{9} +32.0000i q^{11} -38.0000i q^{13} +10.0000 q^{15} +26.0000 q^{17} -100.000i q^{19} -12.0000i q^{21} -78.0000 q^{23} -25.0000 q^{25} -100.000i q^{27} -50.0000i q^{29} +108.000 q^{31} +64.0000 q^{33} +30.0000i q^{35} -266.000i q^{37} -76.0000 q^{39} -22.0000 q^{41} +442.000i q^{43} +115.000i q^{45} +514.000 q^{47} -307.000 q^{49} -52.0000i q^{51} -2.00000i q^{53} -160.000 q^{55} -200.000 q^{57} +500.000i q^{59} -518.000i q^{61} +138.000 q^{63} +190.000 q^{65} -126.000i q^{67} +156.000i q^{69} +412.000 q^{71} +878.000 q^{73} +50.0000i q^{75} +192.000i q^{77} -600.000 q^{79} +421.000 q^{81} -282.000i q^{83} +130.000i q^{85} -100.000 q^{87} +150.000 q^{89} -228.000i q^{91} -216.000i q^{93} +500.000 q^{95} +386.000 q^{97} +736.000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 12 q^{7} + 46 q^{9}+O(q^{10}) \) 2 * q + 12 * q^7 + 46 * q^9	\( 2 q + 12 q^{7} + 46 q^{9} + 20 q^{15} + 52 q^{17} - 156 q^{23} - 50 q^{25} + 216 q^{31} + 128 q^{33} - 152 q^{39} - 44 q^{41} + 1028 q^{47} - 614 q^{49} - 320 q^{55} - 400 q^{57} + 276 q^{63} + 380 q^{65} + 824 q^{71} + 1756 q^{73} - 1200 q^{79} + 842 q^{81} - 200 q^{87} + 300 q^{89} + 1000 q^{95} + 772 q^{97}+O(q^{100}) \) 2 * q + 12 * q^7 + 46 * q^9 + 20 * q^15 + 52 * q^17 - 156 * q^23 - 50 * q^25 + 216 * q^31 + 128 * q^33 - 152 * q^39 - 44 * q^41 + 1028 * q^47 - 614 * q^49 - 320 * q^55 - 400 * q^57 + 276 * q^63 + 380 * q^65 + 824 * q^71 + 1756 * q^73 - 1200 * q^79 + 842 * q^81 - 200 * q^87 + 300 * q^89 + 1000 * q^95 + 772 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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