LMFDB - Embedded newform 280.3.c.f.69.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [280,3,Mod(69,280)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(280, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("280.69");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 280 = 2^{3} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	280.c (of order $2$, degree $1$, 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:	$7.62944740209$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{14})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 49 $ x^4 + 49
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			69.3
Root			$1.87083 - 1.87083i$ of defining polynomial
Character	$\chi$	$=$	280.69
Dual form			280.3.c.f.69.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.00000i q^{2} -5.74166i q^{3} -4.00000 q^{4} +(1.12917 + 4.87083i) q^{5} +11.4833 q^{6} +7.00000i q^{7} -8.00000i q^{8} -23.9666 q^{9} +O(q^{10})$	\(q+2.00000i q^{2} -5.74166i q^{3} -4.00000 q^{4} +(1.12917 + 4.87083i) q^{5} +11.4833 q^{6} +7.00000i q^{7} -8.00000i q^{8} -23.9666 q^{9} +(-9.74166 + 2.25834i) q^{10} +22.9666i q^{12} +14.2583i q^{13} -14.0000 q^{14} +(27.9666 - 6.48331i) q^{15} +16.0000 q^{16} -47.9333i q^{18} +20.1916 q^{19} +(-4.51669 - 19.4833i) q^{20} +40.1916 q^{21} +44.8999i q^{23} -45.9333 q^{24} +(-22.4499 + 11.0000i) q^{25} -28.5167 q^{26} +85.9333i q^{27} -28.0000i q^{28} +(12.9666 + 55.9333i) q^{30} +32.0000i q^{32} +(-34.0958 + 7.90420i) q^{35} +95.8665 q^{36} +40.3832i q^{38} +81.8665 q^{39} +(38.9666 - 9.03337i) q^{40} +80.3832i q^{42} +(-27.0624 - 116.737i) q^{45} -89.7998 q^{46} -91.8665i q^{48} -49.0000 q^{49} +(-22.0000 - 44.8999i) q^{50} -57.0334i q^{52} -171.867 q^{54} +56.0000 q^{56} -115.933i q^{57} +117.608 q^{59} +(-111.867 + 25.9333i) q^{60} -92.0581 q^{61} -167.766i q^{63} -64.0000 q^{64} +(-69.4499 + 16.1001i) q^{65} +257.800 q^{69} +(-15.8084 - 68.1916i) q^{70} -89.7998 q^{71} +191.733i q^{72} +(63.1582 + 128.900i) q^{75} -80.7664 q^{76} +163.733i q^{78} -89.7998 q^{79} +(18.0667 + 77.9333i) q^{80} +277.700 q^{81} -97.9914i q^{83} -160.766 q^{84} +(233.475 - 54.1249i) q^{90} -99.8084 q^{91} -179.600i q^{92} +(22.7998 + 98.3498i) q^{95} +183.733 q^{96} -98.0000i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 16 q^{4} + 12 q^{5} + 16 q^{6} - 36 q^{9}+O(q^{10}) $ 4 * q - 16 * q^4 + 12 * q^5 + 16 * q^6 - 36 * q^9	$ 4 q - 16 q^{4} + 12 q^{5} + 16 q^{6} - 36 q^{9} - 24 q^{10} - 56 q^{14} + 52 q^{15} + 64 q^{16} - 24 q^{19} - 48 q^{20} + 56 q^{21} - 64 q^{24} - 144 q^{26} - 8 q^{30} - 84 q^{35} + 144 q^{36} + 88 q^{39} + 96 q^{40} + 4 q^{45} - 196 q^{49} - 88 q^{50} - 448 q^{54} + 224 q^{56} + 216 q^{59} - 208 q^{60} - 24 q^{61} - 256 q^{64} - 188 q^{65} + 672 q^{69} - 168 q^{70} + 88 q^{75} + 96 q^{76} + 192 q^{80} + 572 q^{81} - 224 q^{84} + 440 q^{90} - 504 q^{91} - 268 q^{95} + 256 q^{96}+O(q^{100}) $ 4 * q - 16 * q^4 + 12 * q^5 + 16 * q^6 - 36 * q^9 - 24 * q^10 - 56 * q^14 + 52 * q^15 + 64 * q^16 - 24 * q^19 - 48 * q^20 + 56 * q^21 - 64 * q^24 - 144 * q^26 - 8 * q^30 - 84 * q^35 + 144 * q^36 + 88 * q^39 + 96 * q^40 + 4 * q^45 - 196 * q^49 - 88 * q^50 - 448 * q^54 + 224 * q^56 + 216 * q^59 - 208 * q^60 - 24 * q^61 - 256 * q^64 - 188 * q^65 + 672 * q^69 - 168 * q^70 + 88 * q^75 + 96 * q^76 + 192 * q^80 + 572 * q^81 - 224 * q^84 + 440 * q^90 - 504 * q^91 - 268 * q^95 + 256 * q^96

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/280\mathbb{Z}\right)^\times$.

$n$	$57$	$71$	$141$	$241$
$\chi(n)$	$-1$	$1$	$-1$	$-1$

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 $

$2$			2.00000i			1.00000i
$2$			2.00000i			1.00000i
$3$		−	5.74166i		−	1.91389i	−0.290276	−	0.956943i	$-0.593747\pi$
$3$		−	5.74166i		−	1.91389i	0.290276	−	0.956943i	$-0.406253\pi$
$4$	−4.00000			−1.00000
$5$	1.12917	+	4.87083i	0.225834	+	0.974166i
$5$	1.12917	+	4.87083i	0.225834	+	0.974166i
$6$	11.4833			1.91389
$7$			7.00000i			1.00000i
$7$			7.00000i			1.00000i
$8$		−	8.00000i		−	1.00000i
$9$	−23.9666			−2.66296
$10$	−9.74166	+	2.25834i	−0.974166	+	0.225834i
$11$		0			0		1.00000			$0$
$11$		0			0		−1.00000			$\pi$
$12$			22.9666i			1.91389i
$13$			14.2583i			1.09680i	0.836218	+	0.548398i	$0.184762\pi$
$13$			14.2583i			1.09680i	−0.836218	+	0.548398i	$0.815238\pi$
$14$	−14.0000			−1.00000
$15$	27.9666	−	6.48331i	1.86444	−	0.432221i
$16$	16.0000			1.00000
$17$		0			0			−	1.00000i	$-0.5\pi$
$17$		0			0				1.00000i	$0.5\pi$
$18$		−	47.9333i		−	2.66296i
$19$	20.1916			1.06272			0.531358	−	0.847147i	$-0.321682\pi$
$19$	20.1916			1.06272			0.531358	+	0.847147i	$0.321682\pi$
$20$	−4.51669	−	19.4833i	−0.225834	−	0.974166i
$21$	40.1916			1.91389
$22$		0			0
$23$			44.8999i			1.95217i	0.217391	+	0.976085i	$0.430245\pi$
$23$			44.8999i			1.95217i	−0.217391	+	0.976085i	$0.569755\pi$
$24$	−45.9333			−1.91389
$25$	−22.4499	+	11.0000i	−0.897998	+	0.440000i
$26$	−28.5167			−1.09680
$27$			85.9333i			3.18271i
$28$		−	28.0000i		−	1.00000i
$29$		0			0		1.00000			$0$
$29$		0			0		−1.00000			$\pi$
$30$	12.9666	+	55.9333i	0.432221	+	1.86444i
$31$		0			0		1.00000			$0$
$31$		0			0		−1.00000			$\pi$
$32$			32.0000i			1.00000i
$33$		0			0
$34$		0			0
$35$	−34.0958	+	7.90420i	−0.974166	+	0.225834i
$36$	95.8665			2.66296
$37$		0			0			−	1.00000i	$-0.5\pi$
$37$		0			0				1.00000i	$0.5\pi$
$38$			40.3832i			1.06272i
$39$	81.8665			2.09914
$40$	38.9666	−	9.03337i	0.974166	−	0.225834i
$41$		0			0		1.00000			$0$
$41$		0			0		−1.00000			$\pi$
$42$			80.3832i			1.91389i
$43$		0			0			−	1.00000i	$-0.5\pi$
$43$		0			0				1.00000i	$0.5\pi$
$44$		0			0
$45$	−27.0624	−	116.737i	−0.601387	−	2.59416i
$46$	−89.7998			−1.95217
$47$		0			0			−	1.00000i	$-0.5\pi$
$47$		0			0				1.00000i	$0.5\pi$
$48$		−	91.8665i		−	1.91389i
$49$	−49.0000			−1.00000
$50$	−22.0000	−	44.8999i	−0.440000	−	0.897998i
$51$		0			0
$52$		−	57.0334i		−	1.09680i
$53$		0			0			−	1.00000i	$-0.5\pi$
$53$		0			0				1.00000i	$0.5\pi$
$54$	−171.867			−3.18271
$55$		0			0
$56$	56.0000			1.00000
$57$		−	115.933i		−	2.03392i
$58$		0			0
$59$	117.608			1.99336			0.996679	−	0.0814252i	$-0.0259472\pi$
$59$	117.608			1.99336			0.996679	+	0.0814252i	$0.0259472\pi$
$60$	−111.867	+	25.9333i	−1.86444	+	0.432221i
$61$	−92.0581			−1.50915			−0.754575	−	0.656214i	$-0.772157\pi$
$61$	−92.0581			−1.50915			−0.754575	+	0.656214i	$0.772157\pi$
$62$		0			0
$63$		−	167.766i		−	2.66296i
$64$	−64.0000			−1.00000
$65$	−69.4499	+	16.1001i	−1.06846	+	0.247694i
$66$		0			0
$67$		0			0			−	1.00000i	$-0.5\pi$
$67$		0			0				1.00000i	$0.5\pi$
$68$		0			0
$69$	257.800			3.73623
$70$	−15.8084	−	68.1916i	−0.225834	−	0.974166i
$71$	−89.7998			−1.26479			−0.632393	−	0.774648i	$-0.717927\pi$
$71$	−89.7998			−1.26479			−0.632393	+	0.774648i	$0.717927\pi$
$72$			191.733i			2.66296i
$73$		0			0			−	1.00000i	$-0.5\pi$
$73$		0			0				1.00000i	$0.5\pi$
$74$		0			0
$75$	63.1582	+	128.900i	0.842110	+	1.71867i
$76$	−80.7664			−1.06272
$77$		0			0
$78$			163.733i			2.09914i
$79$	−89.7998			−1.13671			−0.568353	−	0.822785i	$-0.692419\pi$
$79$	−89.7998			−1.13671			−0.568353	+	0.822785i	$0.692419\pi$
$80$	18.0667	+	77.9333i	0.225834	+	0.974166i
$81$	277.700			3.42839
$82$		0			0
$83$		−	97.9914i		−	1.18062i	−0.807177	−	0.590310i	$-0.799006\pi$
$83$		−	97.9914i		−	1.18062i	0.807177	−	0.590310i	$-0.200994\pi$
$84$	−160.766			−1.91389
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		1.00000			$0$
$89$		0			0		−1.00000			$\pi$
$90$	233.475	−	54.1249i	2.59416	−	0.601387i
$91$	−99.8084			−1.09680
$92$		−	179.600i		−	1.95217i
$93$		0			0
$94$		0			0
$95$	22.7998	+	98.3498i	0.239998	+	1.03526i
$96$	183.733			1.91389
$97$		0			0			−	1.00000i	$-0.5\pi$
$97$		0			0				1.00000i	$0.5\pi$
$98$		−	98.0000i		−	1.00000i
$99$		0			0
$100$	89.7998	−	44.0000i	0.897998	−	0.440000i
$101$	27.9419			0.276652			0.138326	−	0.990387i	$-0.455828\pi$
$101$	27.9419			0.276652			0.138326	+	0.990387i	$0.455828\pi$
$102$		0			0
$103$		0			0			−	1.00000i	$-0.5\pi$
$103$		0			0				1.00000i	$0.5\pi$
$104$	114.067			1.09680
$105$	45.3832	+	195.766i	0.432221	+	1.86444i
$106$		0			0
$107$		0			0			−	1.00000i	$-0.5\pi$
$107$		0			0				1.00000i	$0.5\pi$
$108$		−	343.733i		−	3.18271i
$109$		0			0		1.00000			$0$
$109$		0			0		−1.00000			$\pi$
$110$		0			0
$111$		0			0
$112$			112.000i			1.00000i
$113$		−	26.0000i		−	0.230088i	−0.993360	−	0.115044i	$-0.963299\pi$
$113$		−	26.0000i		−	0.230088i	0.993360	−	0.115044i	$-0.0367010\pi$
$114$	231.867			2.03392
$115$	−218.700	+	50.6997i	−1.90174	+	0.440867i
$116$		0			0
$117$		−	341.724i		−	2.92072i
$118$			235.216i			1.99336i
$119$		0			0
$120$	−51.8665	−	223.733i	−0.432221	−	1.86444i
$121$	121.000			1.00000
$122$		−	184.116i		−	1.50915i
$123$		0			0
$124$		0			0
$125$	−78.9289	−	96.9289i	−0.631432	−	0.775432i
$126$	335.533			2.66296
$127$			44.8999i			0.353542i	0.984252	+	0.176771i	$0.0565653\pi$
$127$			44.8999i			0.353542i	−0.984252	+	0.176771i	$0.943435\pi$
$128$		−	128.000i		−	1.00000i
$129$		0			0
$130$	−32.2002	−	138.900i	−0.247694	−	1.06846i
$131$	237.608			1.81380			0.906901	−	0.421343i	$-0.138441\pi$
$131$	237.608			1.81380			0.906901	+	0.421343i	$0.138441\pi$
$132$		0			0
$133$			141.341i			1.06272i
$134$		0			0
$135$	−418.566	+	97.0334i	−3.10049	+	0.718766i
$136$		0			0
$137$			269.399i			1.96642i	0.182482	+	0.983209i	$0.441587\pi$
$137$			269.399i			1.96642i	−0.182482	+	0.983209i	$0.558413\pi$
$138$			515.600i			3.73623i
$139$	−122.392			−0.880517			−0.440258	−	0.897871i	$-0.645113\pi$
$139$	−122.392			−0.880517			−0.440258	+	0.897871i	$0.645113\pi$
$140$	136.383	−	31.6168i	0.974166	−	0.225834i
$141$		0			0
$142$		−	179.600i		−	1.26479i
$143$		0			0
$144$	−383.466			−2.66296
$145$		0			0
$146$		0			0
$147$			281.341i			1.91389i
$148$		0			0
$149$		0			0		1.00000			$0$
$149$		0			0		−1.00000			$\pi$
$150$	−257.800	+	126.316i	−1.71867	+	0.842110i
$151$	202.000			1.33775			0.668874	−	0.743376i	$-0.266776\pi$
$151$	202.000			1.33775			0.668874	+	0.743376i	$0.266776\pi$
$152$		−	161.533i		−	1.06272i
$153$		0			0
$154$		0			0
$155$		0			0
$156$	−327.466			−2.09914
$157$		−	225.742i		−	1.43784i	−0.695090	−	0.718922i	$-0.744636\pi$
$157$		−	225.742i		−	1.43784i	0.695090	−	0.718922i	$-0.255364\pi$
$158$		−	179.600i		−	1.13671i
$159$		0			0
$160$	−155.867	+	36.1335i	−0.974166	+	0.225834i
$161$	−314.299			−1.95217
$162$			555.399i			3.42839i
$163$		0			0			−	1.00000i	$-0.5\pi$
$163$		0			0				1.00000i	$0.5\pi$
$164$		0			0
$165$		0			0
$166$	195.983			1.18062
$167$		0			0			−	1.00000i	$-0.5\pi$
$167$		0			0				1.00000i	$0.5\pi$
$168$		−	321.533i		−	1.91389i
$169$	−34.3003			−0.202961
$170$		0			0
$171$	−483.925			−2.82997
$172$		0			0
$173$			21.3412i			0.123360i	0.998096	+	0.0616798i	$0.0196458\pi$
$173$			21.3412i			0.123360i	−0.998096	+	0.0616798i	$0.980354\pi$
$174$		0			0
$175$	−77.0000	−	157.150i	−0.440000	−	0.897998i
$176$		0			0
$177$		−	675.266i		−	3.81506i
$178$		0			0
$179$		0			0		1.00000			$0$
$179$		0			0		−1.00000			$\pi$
$180$	108.250	+	466.949i	0.601387	+	2.59416i
$181$	−10.1421			−0.0560337			−0.0280169	−	0.999607i	$-0.508919\pi$
$181$	−10.1421			−0.0560337			−0.0280169	+	0.999607i	$0.508919\pi$
$182$		−	199.617i		−	1.09680i
$183$			528.566i			2.88834i
$184$	359.199			1.95217
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$	−601.533			−3.18271
$190$	−196.700	+	45.5996i	−1.03526	+	0.239998i
$191$	359.199			1.88062			0.940312	−	0.340314i	$-0.110533\pi$
$191$	359.199			1.88062			0.940312	+	0.340314i	$0.110533\pi$
$192$			367.466i			1.91389i
$193$			314.000i			1.62694i	0.581605	+	0.813472i	$0.302425\pi$
$193$			314.000i			1.62694i	−0.581605	+	0.813472i	$0.697575\pi$
$194$		0			0
$195$	92.4413	+	398.758i	0.474058	+	2.04491i
$196$	196.000			1.00000
$197$		0			0			−	1.00000i	$-0.5\pi$
$197$		0			0				1.00000i	$0.5\pi$
$198$		0			0
$199$		0			0		1.00000			$0$
$199$		0			0		−1.00000			$\pi$
$200$	88.0000	+	179.600i	0.440000	+	0.897998i
$201$		0			0
$202$			55.8838i			0.276652i
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		−	1076.10i		−	5.19855i
$208$			228.133i			1.09680i
$209$		0			0
$210$	−391.533	+	90.7664i	−1.86444	+	0.432221i
$211$		0			0		1.00000			$0$
$211$		0			0		−1.00000			$\pi$
$212$		0			0
$213$			515.600i			2.42066i
$214$		0			0
$215$		0			0
$216$	687.466			3.18271
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0			−	1.00000i	$-0.5\pi$
$223$		0			0				1.00000i	$0.5\pi$
$224$	−224.000			−1.00000
$225$	538.049	−	263.633i	2.39133	−	1.17170i
$226$	52.0000			0.230088
$227$		−	435.408i		−	1.91810i	−0.283242	−	0.959048i	$-0.591410\pi$
$227$		−	435.408i		−	1.91810i	0.283242	−	0.959048i	$-0.408590\pi$
$228$			463.733i			2.03392i
$229$	−99.1410			−0.432930			−0.216465	−	0.976290i	$-0.569453\pi$
$229$	−99.1410			−0.432930			−0.216465	+	0.976290i	$0.569453\pi$
$230$	−101.399	−	437.399i	−0.440867	−	1.90174i
$231$		0			0
$232$		0			0
$233$		−	179.600i		−	0.770814i	−0.922747	−	0.385407i	$-0.874061\pi$
$233$		−	179.600i		−	0.770814i	0.922747	−	0.385407i	$-0.125939\pi$
$234$	683.449			2.92072
$235$		0			0
$236$	−470.433			−1.99336
$237$			515.600i			2.17553i
$238$		0			0
$239$	422.000			1.76569			0.882845	−	0.469664i	$-0.155625\pi$
$239$	422.000			1.76569			0.882845	+	0.469664i	$0.155625\pi$
$240$	447.466	−	103.733i	1.86444	−	0.432221i
$241$		0			0		1.00000			$0$
$241$		0			0		−1.00000			$\pi$
$242$			242.000i			1.00000i
$243$		−	821.057i		−	3.37884i
$244$	368.232			1.50915
$245$	−55.3294	−	238.671i	−0.225834	−	0.974166i
$246$		0			0
$247$			287.899i			1.16558i
$248$		0			0
$249$	−562.633			−2.25957
$250$	193.858	−	157.858i	0.775432	−	0.631432i
$251$	380.192			1.51471			0.757354	−	0.653005i	$-0.226492\pi$
$251$	380.192			1.51471			0.757354	+	0.653005i	$0.226492\pi$
$252$			671.066i			2.66296i
$253$		0			0
$254$	−89.7998			−0.353542
$255$		0			0
$256$	256.000			1.00000
$257$		0			0			−	1.00000i	$-0.5\pi$
$257$		0			0				1.00000i	$0.5\pi$
$258$		0			0
$259$		0			0
$260$	277.800	−	64.4004i	1.06846	−	0.247694i
$261$		0			0
$262$			475.216i			1.81380i
$263$			274.000i			1.04183i	0.853610	+	0.520913i	$0.174408\pi$
$263$			274.000i			1.04183i	−0.853610	+	0.520913i	$0.825592\pi$
$264$		0			0
$265$		0			0
$266$	−282.682			−1.06272
$267$		0			0
$268$		0			0
$269$	476.941			1.77301			0.886507	−	0.462715i	$-0.153125\pi$
$269$	476.941			1.77301			0.886507	+	0.462715i	$0.153125\pi$
$270$	−194.067	−	837.132i	−0.718766	−	3.10049i
$271$		0			0		1.00000			$0$
$271$		0			0		−1.00000			$\pi$
$272$		0			0
$273$			573.066i			2.09914i
$274$	−538.799			−1.96642
$275$		0			0
$276$	−1031.20			−3.73623
$277$		0			0			−	1.00000i	$-0.5\pi$
$277$		0			0				1.00000i	$0.5\pi$
$278$		−	244.784i		−	0.880517i
$279$		0			0
$280$	63.2336	+	272.766i	0.225834	+	0.974166i
$281$	−338.000			−1.20285			−0.601423	−	0.798930i	$-0.705400\pi$
$281$	−338.000			−1.20285			−0.601423	+	0.798930i	$0.705400\pi$
$282$		0			0
$283$			111.008i			0.392253i	0.980579	+	0.196126i	$0.0628363\pi$
$283$			111.008i			0.392253i	−0.980579	+	0.196126i	$0.937164\pi$
$284$	359.199			1.26479
$285$	564.691	−	130.909i	1.98137	−	0.459328i
$286$		0			0
$287$		0			0
$288$		−	766.932i		−	2.66296i
$289$	−289.000			−1.00000
$290$		0			0
$291$		0			0
$292$		0			0
$293$			52.3423i			0.178643i	0.996003	+	0.0893214i	$0.0284698\pi$
$293$			52.3423i			0.178643i	−0.996003	+	0.0893214i	$0.971530\pi$
$294$	−562.682			−1.91389
$295$	132.800	+	572.849i	0.450169	+	1.94186i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−640.198			−2.14113
$300$	−252.633	−	515.600i	−0.842110	−	1.71867i
$301$		0			0
$302$			404.000i			1.33775i
$303$		−	160.433i		−	0.529481i
$304$	323.066			1.06272
$305$	−103.949	−	448.399i	−0.340818	−	1.47016i
$306$		0			0
$307$			613.591i			1.99867i	0.0364969	+	0.999334i	$0.488380\pi$
$307$			613.591i			1.99867i	−0.0364969	+	0.999334i	$0.511620\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		1.00000			$0$
$311$		0			0		−1.00000			$\pi$
$312$		−	654.932i		−	2.09914i
$313$		0			0			−	1.00000i	$-0.5\pi$
$313$		0			0				1.00000i	$0.5\pi$
$314$	451.483			1.43784
$315$	817.161	−	189.437i	2.59416	−	0.601387i
$316$	359.199			1.13671
$317$		0			0			−	1.00000i	$-0.5\pi$
$317$		0			0				1.00000i	$0.5\pi$
$318$		0			0
$319$		0			0
$320$	−72.2670	−	311.733i	−0.225834	−	0.974166i
$321$		0			0
$322$		−	628.598i		−	1.95217i
$323$		0			0
$324$	−1110.80			−3.42839
$325$	−156.842	−	320.099i	−0.482590	−	0.984920i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$		0			0		1.00000			$0$
$331$		0			0		−1.00000			$\pi$
$332$			391.966i			1.18062i
$333$		0			0
$334$		0			0
$335$		0			0
$336$	643.066			1.91389
$337$		−	26.0000i		−	0.0771513i	−0.999256	−	0.0385757i	$-0.987718\pi$
$337$		−	26.0000i		−	0.0771513i	0.999256	−	0.0385757i	$-0.0122821\pi$
$338$		−	68.6007i		−	0.202961i
$339$	−149.283			−0.440363
$340$		0			0
$341$		0			0
$342$		−	967.849i		−	2.82997i
$343$		−	343.000i		−	1.00000i
$344$		0			0
$345$	291.100	+	1255.70i	0.843768	+	3.63971i
$346$	−42.6824			−0.123360
$347$		0			0			−	1.00000i	$-0.5\pi$
$347$		0			0				1.00000i	$0.5\pi$
$348$		0			0
$349$	−572.058			−1.63914			−0.819568	−	0.572983i	$-0.805786\pi$
$349$	−572.058			−1.63914			−0.819568	+	0.572983i	$0.805786\pi$
$350$	314.299	−	154.000i	0.897998	−	0.440000i
$351$	−1225.27			−3.49079
$352$		0			0
$353$		0			0			−	1.00000i	$-0.5\pi$
$353$		0			0				1.00000i	$0.5\pi$
$354$	1350.53			3.81506
$355$	−101.399	−	437.399i	−0.285632	−	1.23211i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	682.000			1.89972			0.949861	−	0.312673i	$-0.101225\pi$
$359$	682.000			1.89972			0.949861	+	0.312673i	$0.101225\pi$
$360$	−933.899	+	216.499i	−2.59416	+	0.601387i
$361$	46.7008			0.129365
$362$		−	20.2842i		−	0.0560337i
$363$		−	694.741i		−	1.91389i
$364$	399.234			1.09680
$365$		0			0
$366$	−1057.13			−2.88834
$367$		0			0			−	1.00000i	$-0.5\pi$
$367$		0			0				1.00000i	$0.5\pi$
$368$			718.398i			1.95217i
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$		0			0			−	1.00000i	$-0.5\pi$
$373$		0			0				1.00000i	$0.5\pi$
$374$		0			0
$375$	−556.533	+	453.183i	−1.48409	+	1.20849i
$376$		0			0
$377$		0			0
$378$		−	1203.07i		−	3.18271i
$379$		0			0		1.00000			$0$
$379$		0			0		−1.00000			$\pi$
$380$	−91.1991	−	393.399i	−0.239998	−	1.03526i
$381$	257.800			0.676640
$382$			718.398i			1.88062i
$383$		0			0			−	1.00000i	$-0.5\pi$
$383$		0			0				1.00000i	$0.5\pi$
$384$	−734.932			−1.91389
$385$		0			0
$386$	−628.000			−1.62694
$387$		0			0
$388$		0			0
$389$		0			0		1.00000			$0$
$389$		0			0		−1.00000			$\pi$
$390$	−797.516	+	184.883i	−2.04491	+	0.474058i
$391$		0			0
$392$			392.000i			1.00000i
$393$		−	1364.26i		−	3.47141i
$394$		0			0
$395$	−101.399	−	437.399i	−0.256707	−	1.10734i
$396$		0			0
$397$			103.257i			0.260094i	0.991508	+	0.130047i	$0.0415128\pi$
$397$			103.257i			0.260094i	−0.991508	+	0.130047i	$0.958487\pi$
$398$		0			0
$399$	811.533			2.03392
$400$	−359.199	+	176.000i	−0.897998	+	0.440000i
$401$	583.699			1.45561			0.727804	−	0.685786i	$-0.240541\pi$
$401$	583.699			1.45561			0.727804	+	0.685786i	$0.240541\pi$
$402$		0			0
$403$		0			0
$404$	−111.768			−0.276652
$405$	313.570	+	1352.63i	0.774248	+	3.33982i
$406$		0			0
$407$		0			0
$408$		0			0
$409$		0			0		1.00000			$0$
$409$		0			0		−1.00000			$\pi$
$410$		0			0
$411$	1546.80			3.76350
$412$		0			0
$413$			823.257i			1.99336i
$414$	2152.20			5.19855
$415$	477.299	−	110.649i	1.15012	−	0.266624i
$416$	−456.267			−1.09680
$417$			702.732i			1.68521i
$418$		0			0
$419$	829.190			1.97897			0.989487	−	0.144619i	$-0.0461956\pi$
$419$	829.190			1.97897			0.989487	+	0.144619i	$0.0461956\pi$
$420$	−181.533	−	783.066i	−0.432221	−	1.86444i
$421$		0			0		1.00000			$0$
$421$		0			0		−1.00000			$\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$	−1031.20			−2.42066
$427$		−	644.407i		−	1.50915i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−538.000			−1.24826			−0.624130	−	0.781321i	$-0.714546\pi$
$431$	−538.000			−1.24826			−0.624130	+	0.781321i	$0.714546\pi$
$432$			1374.93i			3.18271i
$433$		0			0			−	1.00000i	$-0.5\pi$
$433$		0			0				1.00000i	$0.5\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$			906.601i			2.07460i
$438$		0			0
$439$		0			0		1.00000			$0$
$439$		0			0		−1.00000			$\pi$
$440$		0			0
$441$	1174.36			2.66296
$442$		0			0
$443$		0			0			−	1.00000i	$-0.5\pi$
$443$		0			0				1.00000i	$0.5\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		−	448.000i		−	1.00000i
$449$	2.00000			0.00445434			0.00222717	−	0.999998i	$-0.499291\pi$
$449$	2.00000			0.00445434			0.00222717	+	0.999998i	$0.499291\pi$
$450$	527.266	+	1076.10i	1.17170	+	2.39133i
$451$		0			0
$452$			104.000i			0.230088i
$453$		−	1159.81i		−	2.56030i
$454$	870.816			1.91810
$455$	−112.701	−	486.150i	−0.247694	−	1.06846i
$456$	−927.466			−2.03392
$457$		−	886.000i		−	1.93873i	−0.245623	−	0.969365i	$-0.578993\pi$
$457$		−	886.000i		−	1.93873i	0.245623	−	0.969365i	$-0.421007\pi$
$458$		−	198.282i		−	0.432930i
$459$		0			0
$460$	874.799	−	202.799i	1.90174	−	0.440867i
$461$	438.857			0.951967			0.475983	−	0.879454i	$-0.342092\pi$
$461$	438.857			0.951967			0.475983	+	0.879454i	$0.342092\pi$
$462$		0			0
$463$		−	226.000i		−	0.488121i	−0.969760	−	0.244060i	$-0.921520\pi$
$463$		−	226.000i		−	0.488121i	0.969760	−	0.244060i	$-0.0784795\pi$
$464$		0			0
$465$		0			0
$466$	359.199			0.770814
$467$		−	906.990i		−	1.94216i	−0.238748	−	0.971082i	$-0.576737\pi$
$467$		−	906.990i		−	1.94216i	0.238748	−	0.971082i	$-0.423263\pi$
$468$			1366.90i			2.92072i
$469$		0			0
$470$		0			0
$471$	−1296.13			−2.75187
$472$		−	940.865i		−	1.99336i
$473$		0			0
$474$	−1031.20			−2.17553
$475$	−453.300	+	222.108i	−0.954316	+	0.467595i
$476$		0			0
$477$		0			0
$478$			844.000i			1.76569i
$479$		0			0		1.00000			$0$
$479$		0			0		−1.00000			$\pi$
$480$	207.466	+	894.932i	0.432221	+	1.86444i
$481$		0			0
$482$		0			0
$483$			1804.60i			3.73623i
$484$	−484.000			−1.00000
$485$		0			0
$486$	1642.11			3.37884
$487$		−	853.098i		−	1.75174i	−0.482546	−	0.875871i	$-0.660288\pi$
$487$		−	853.098i		−	1.75174i	0.482546	−	0.875871i	$-0.339712\pi$
$488$			736.465i			1.50915i
$489$		0			0
$490$	477.341	−	110.659i	0.974166	−	0.225834i
$491$		0			0		1.00000			$0$
$491$		0			0		−1.00000			$\pi$
$492$		0			0
$493$		0			0
$494$	−575.798			−1.16558
$495$		0			0
$496$		0			0
$497$		−	628.598i		−	1.26479i
$498$		−	1125.27i		−	2.25957i
$499$		0			0		1.00000			$0$
$499$		0			0		−1.00000			$\pi$
$500$	315.716	+	387.716i	0.631432	+	0.775432i
$501$		0			0
$502$			760.383i			1.51471i
$503$		0			0			−	1.00000i	$-0.5\pi$
$503$		0			0				1.00000i	$0.5\pi$
$504$	−1342.13			−2.66296
$505$	31.5512	+	136.100i	0.0624776	+	0.269505i
$506$		0			0
$507$			196.941i			0.388443i
$508$		−	179.600i		−	0.353542i
$509$	−939.141			−1.84507			−0.922535	−	0.385913i	$-0.873887\pi$
$509$	−939.141			−1.84507			−0.922535	+	0.385913i	$0.873887\pi$
$510$		0			0
$511$		0			0
$512$			512.000i			1.00000i
$513$			1735.13i			3.38232i
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$	122.534			0.236096
$520$	128.801	+	555.600i	0.247694	+	1.06846i
$521$		0			0		1.00000			$0$
$521$		0			0		−1.00000			$\pi$
$522$		0			0
$523$			973.591i			1.86155i	0.365592	+	0.930775i	$0.380867\pi$
$523$			973.591i			1.86155i	−0.365592	+	0.930775i	$0.619133\pi$
$524$	−950.433			−1.81380
$525$	−902.299	+	442.108i	−1.71867	+	0.842110i
$526$	−548.000			−1.04183
$527$		0			0
$528$		0			0
$529$	−1487.00			−2.81096
$530$		0			0
$531$	−2818.67			−5.30823
$532$		−	565.365i		−	1.06272i
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$			953.882i			1.77301i
$539$		0			0
$540$	1674.26	−	388.133i	3.10049	−	0.718766i
$541$		0			0		1.00000			$0$
$541$		0			0		−1.00000			$\pi$
$542$		0			0
$543$			58.2325i			0.107242i
$544$		0			0
$545$		0			0
$546$	−1146.13			−2.09914
$547$		0			0			−	1.00000i	$-0.5\pi$
$547$		0			0				1.00000i	$0.5\pi$
$548$		−	1077.60i		−	1.96642i
$549$	2206.32			4.01880
$550$		0			0
$551$		0			0
$552$		−	2062.40i		−	3.73623i
$553$		−	628.598i		−	1.13671i
$554$		0			0
$555$		0			0
$556$	489.567			0.880517
$557$		0			0			−	1.00000i	$-0.5\pi$
$557$		0			0				1.00000i	$0.5\pi$
$558$		0			0
$559$		0			0
$560$	−545.533	+	126.467i	−0.974166	+	0.225834i
$561$		0			0
$562$		−	676.000i		−	1.20285i
$563$			462.590i			0.821652i	0.911714	+	0.410826i	$0.134760\pi$
$563$			462.590i			0.821652i	−0.911714	+	0.410826i	$0.865240\pi$
$564$		0			0
$565$	126.642	−	29.3585i	0.224144	−	0.0519619i
$566$	−222.015			−0.392253
$567$			1943.90i			3.42839i
$568$			718.398i			1.26479i
$569$	134.700			0.236731			0.118365	−	0.992970i	$-0.462235\pi$
$569$	134.700			0.236731			0.118365	+	0.992970i	$0.462235\pi$
$570$	261.817	+	1129.38i	0.459328	+	1.98137i
$571$		0			0		1.00000			$0$
$571$		0			0		−1.00000			$\pi$
$572$		0			0
$573$		−	2062.40i		−	3.59930i
$574$		0			0
$575$	−493.899	−	1008.00i	−0.858954	−	1.75304i
$576$	1533.86			2.66296
$577$		0			0			−	1.00000i	$-0.5\pi$
$577$		0			0				1.00000i	$0.5\pi$
$578$		−	578.000i		−	1.00000i
$579$	1802.88			3.11378
$580$		0			0
$581$	685.940			1.18062
$582$		0			0
$583$		0			0
$584$		0			0
$585$	1664.48	−	385.865i	2.84527	−	0.659599i
$586$	−104.685			−0.178643
$587$		−	937.991i		−	1.59794i	−0.601370	−	0.798971i	$-0.705378\pi$
$587$		−	937.991i		−	1.59794i	0.601370	−	0.798971i	$-0.294622\pi$
$588$		−	1125.36i		−	1.91389i
$589$		0			0
$590$	−1145.70	+	265.600i	−1.94186	+	0.450169i
$591$		0			0
$592$		0			0
$593$		0			0			−	1.00000i	$-0.5\pi$
$593$		0			0				1.00000i	$0.5\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		−	1280.40i		−	2.14113i
$599$	−538.799			−0.899497			−0.449748	−	0.893155i	$-0.648486\pi$
$599$	−538.799			−0.899497			−0.449748	+	0.893155i	$0.648486\pi$
$600$	1031.20	−	505.266i	1.71867	−	0.842110i
$601$		0			0		1.00000			$0$
$601$		0			0		−1.00000			$\pi$
$602$		0			0
$603$		0			0
$604$	−808.000			−1.33775
$605$	136.630	+	589.370i	0.225834	+	0.974166i
$606$	320.865			0.529481
$607$		0			0			−	1.00000i	$-0.5\pi$
$607$		0			0				1.00000i	$0.5\pi$
$608$			646.131i			1.06272i
$609$		0			0
$610$	896.799	−	207.899i	1.47016	−	0.340818i
$611$		0			0
$612$		0			0
$613$		0			0			−	1.00000i	$-0.5\pi$
$613$		0			0				1.00000i	$0.5\pi$
$614$	−1227.18			−1.99867
$615$		0			0
$616$		0			0
$617$			1034.00i			1.67585i	0.545785	+	0.837925i	$0.316232\pi$
$617$			1034.00i			1.67585i	−0.545785	+	0.837925i	$0.683768\pi$
$618$		0			0
$619$	−1171.39			−1.89239			−0.946196	−	0.323594i	$-0.895109\pi$
$619$	−1171.39			−1.89239			−0.946196	+	0.323594i	$0.895109\pi$
$620$		0			0
$621$	−3858.39			−6.21319
$622$		0			0
$623$		0			0
$624$	1309.86			2.09914
$625$	383.000	−	493.899i	0.612800	−	0.790238i
$626$		0			0
$627$		0			0
$628$			902.967i			1.43784i
$629$		0			0
$630$	378.874	+	1634.32i	0.601387	+	2.59416i
$631$	1257.20			1.99239			0.996194	−	0.0871632i	$-0.0277802\pi$
$631$	1257.20			1.99239			0.996194	+	0.0871632i	$0.0277802\pi$
$632$			718.398i			1.13671i
$633$		0			0
$634$		0			0
$635$	−218.700	+	50.6997i	−0.344409	+	0.0798420i
$636$		0			0
$637$		−	698.659i		−	1.09680i
$638$		0			0
$639$	2152.20			3.36807
$640$	623.466	−	144.534i	0.974166	−	0.225834i
$641$	−763.298			−1.19079			−0.595396	−	0.803432i	$-0.703005\pi$
$641$	−763.298			−1.19079			−0.595396	+	0.803432i	$0.703005\pi$
$642$		0			0
$643$			222.590i			0.346174i	0.984907	+	0.173087i	$0.0553742\pi$
$643$			222.590i			0.346174i	−0.984907	+	0.173087i	$0.944626\pi$
$644$	1257.20			1.95217
$645$		0			0
$646$		0			0
$647$		0			0			−	1.00000i	$-0.5\pi$
$647$		0			0				1.00000i	$0.5\pi$
$648$		−	2221.60i		−	3.42839i
$649$		0			0
$650$	640.198	−	313.684i	0.984920	−	0.482590i
$651$		0			0
$652$		0			0
$653$		0			0			−	1.00000i	$-0.5\pi$
$653$		0			0				1.00000i	$0.5\pi$
$654$		0			0
$655$	268.300	+	1157.35i	0.409619	+	1.76694i
$656$		0			0
$657$		0			0
$658$		0			0
$659$		0			0		1.00000			$0$
$659$		0			0		−1.00000			$\pi$
$660$		0			0
$661$	620.859			0.939272			0.469636	−	0.882860i	$-0.344385\pi$
$661$	620.859			0.939272			0.469636	+	0.882860i	$0.344385\pi$
$662$		0			0
$663$		0			0
$664$	−783.931			−1.18062
$665$	−688.449	+	159.598i	−1.03526	+	0.239998i
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$			1286.13i			1.91389i
$673$			1167.40i			1.73462i	0.497771	+	0.867308i	$0.334152\pi$
$673$			1167.40i			1.73462i	−0.497771	+	0.867308i	$0.665848\pi$
$674$	52.0000			0.0771513
$675$	−945.266	−	1929.20i	−1.40039	−	2.85807i
$676$	137.201			0.202961
$677$			552.256i			0.815740i	0.913040	+	0.407870i	$0.133728\pi$
$677$			552.256i			0.815740i	−0.913040	+	0.407870i	$0.866272\pi$
$678$		−	298.566i		−	0.440363i
$679$		0			0
$680$		0			0
$681$	−2499.96			−3.67102
$682$		0			0
$683$		0			0			−	1.00000i	$-0.5\pi$
$683$		0			0				1.00000i	$0.5\pi$
$684$	1935.70			2.82997
$685$	−1312.20	+	304.198i	−1.91562	+	0.444085i
$686$	686.000			1.00000
$687$			569.234i			0.828579i
$688$		0			0
$689$		0			0
$690$	−2511.40	+	582.200i	−3.63971	+	0.843768i
$691$	1135.61			1.64342			0.821712	−	0.569903i	$-0.193019\pi$
$691$	1135.61			1.64342			0.821712	+	0.569903i	$0.193019\pi$
$692$		−	85.3648i		−	0.123360i
$693$		0			0
$694$		0			0
$695$	−138.201	−	596.150i	−0.198851	−	0.857769i
$696$		0			0
$697$		0			0
$698$		−	1144.12i		−	1.63914i
$699$	−1031.20			−1.47525
$700$	308.000	+	628.598i	0.440000	+	0.897998i
$701$		0			0		1.00000			$0$
$701$		0			0		−1.00000			$\pi$
$702$		−	2450.53i		−	3.49079i
$703$		0			0
$704$		0			0
$705$		0			0
$706$		0			0
$707$			195.593i			0.276652i
$708$			2701.06i			3.81506i
$709$		0			0		1.00000			$0$
$709$		0			0		−1.00000			$\pi$
$710$	874.799	−	202.799i	1.23211	−	0.285632i
$711$	2152.20			3.02700
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		−	2422.98i		−	3.37933i
$718$			1364.00i			1.89972i
$719$		0			0		1.00000			$0$
$719$		0			0		−1.00000			$\pi$
$720$	−432.999	−	1867.80i	−0.601387	−	2.59416i
$721$		0			0
$722$			93.4016i			0.129365i
$723$		0			0
$724$	40.5684			0.0560337
$725$		0			0
$726$	1389.48			1.91389
$727$		0			0			−	1.00000i	$-0.5\pi$
$727$		0			0				1.00000i	$0.5\pi$
$728$			798.467i			1.09680i
$729$	−2214.93			−3.03831
$730$		0			0
$731$		0			0
$732$		−	2114.26i		−	2.88834i
$733$		−	1267.66i		−	1.72941i	−0.502280	−	0.864705i	$-0.667505\pi$
$733$		−	1267.66i		−	1.72941i	0.502280	−	0.864705i	$-0.332495\pi$
$734$		0			0
$735$	−1370.36	+	317.682i	−1.86444	+	0.432221i
$736$	−1436.80			−1.95217
$737$		0			0
$738$		0			0
$739$		0			0		1.00000			$0$
$739$		0			0		−1.00000			$\pi$
$740$		0			0
$741$	1653.02			2.23079
$742$		0			0
$743$		−	404.099i		−	0.543875i	−0.962315	−	0.271937i	$-0.912336\pi$
$743$		−	404.099i		−	0.543875i	0.962315	−	0.271937i	$-0.0876644\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$			2348.52i			3.14394i
$748$		0			0
$749$		0			0
$750$	−906.366	−	1113.07i	−1.20849	−	1.48409i
$751$	−998.000			−1.32889			−0.664447	−	0.747335i	$-0.731333\pi$
$751$	−998.000			−1.32889			−0.664447	+	0.747335i	$0.731333\pi$
$752$		0			0
$753$		−	2182.93i		−	2.89898i
$754$		0			0
$755$	228.093	+	983.907i	0.302109	+	1.30319i
$756$	2406.13			3.18271
$757$		0			0			−	1.00000i	$-0.5\pi$
$757$		0			0				1.00000i	$0.5\pi$
$758$		0			0
$759$		0			0
$760$	786.799	−	182.398i	1.03526	−	0.239998i
$761$		0			0		1.00000			$0$
$761$		0			0		−1.00000			$\pi$
$762$			515.600i			0.676640i
$763$		0			0
$764$	−1436.80			−1.88062
$765$		0			0
$766$		0			0
$767$			1676.90i			2.18631i
$768$		−	1469.86i		−	1.91389i
$769$		0			0		1.00000			$0$
$769$		0			0		−1.00000			$\pi$
$770$		0			0
$771$		0			0
$772$		−	1256.00i		−	1.62694i
$773$		−	283.739i		−	0.367063i	−0.983014	−	0.183531i	$-0.941247\pi$
$773$		−	283.739i		−	0.367063i	0.983014	−	0.183531i	$-0.0587529\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$	−369.765	−	1595.03i	−0.474058	−	2.04491i
$781$		0			0
$782$		0			0
$783$		0			0
$784$	−784.000			−1.00000
$785$	1099.55	−	254.901i	1.40070	−	0.324715i
$786$	2728.53			3.47141
$787$		−	728.992i		−	0.926293i	−0.886282	−	0.463146i	$-0.846720\pi$
$787$		−	728.992i		−	0.926293i	0.886282	−	0.463146i	$-0.153280\pi$
$788$		0			0
$789$	1573.21			1.99393
$790$	874.799	−	202.799i	1.10734	−	0.256707i
$791$	182.000			0.230088
$792$		0			0
$793$		−	1312.60i		−	1.65523i
$794$	−206.514			−0.260094
$795$		0			0
$796$		0			0
$797$			1399.34i			1.75576i	0.478883	+	0.877879i	$0.341042\pi$
$797$			1399.34i			1.75576i	−0.478883	+	0.877879i	$0.658958\pi$
$798$			1623.07i			2.03392i
$799$		0			0
$800$	−352.000	−	718.398i	−0.440000	−	0.897998i
$801$		0			0
$802$			1167.40i			1.45561i
$803$		0			0
$804$		0			0
$805$	−354.898	−	1530.90i	−0.440867	−	1.90174i
$806$		0			0
$807$		−	2738.43i		−	3.39335i
$808$		−	223.535i		−	0.276652i
$809$	1481.70			1.83152			0.915758	−	0.401731i	$-0.131591\pi$
$809$	1481.70			1.83152			0.915758	+	0.401731i	$0.131591\pi$
$810$	−2705.26	+	627.141i	−3.33982	+	0.774248i
$811$	−513.393			−0.633037			−0.316518	−	0.948586i	$-0.602514\pi$
$811$	−513.393			−0.633037			−0.316518	+	0.948586i	$0.602514\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$		0			0
$819$	2392.07			2.92072
$820$		0			0
$821$		0			0		1.00000			$0$
$821$		0			0		−1.00000			$\pi$
$822$			3093.60i			3.76350i
$823$		−	946.000i		−	1.14945i	−0.818345	−	0.574727i	$-0.805108\pi$
$823$		−	946.000i		−	1.14945i	0.818345	−	0.574727i	$-0.194892\pi$
$824$		0			0
$825$		0			0
$826$	−1646.51			−1.99336
$827$		0			0			−	1.00000i	$-0.5\pi$
$827$		0			0				1.00000i	$0.5\pi$
$828$			4304.40i			5.19855i
$829$	−1117.14			−1.34757			−0.673787	−	0.738926i	$-0.735333\pi$
$829$	−1117.14			−1.34757			−0.673787	+	0.738926i	$0.735333\pi$
$830$	221.298	+	954.598i	0.266624	+	1.15012i
$831$		0			0
$832$		−	912.534i		−	1.09680i
$833$		0			0
$834$	−1405.46			−1.68521
$835$		0			0
$836$		0			0
$837$		0			0
$838$			1658.38i			1.97897i
$839$		0			0		1.00000			$0$
$839$		0			0		−1.00000			$\pi$
$840$	1566.13	−	363.066i	1.86444	−	0.432221i
$841$	841.000			1.00000
$842$		0			0
$843$			1940.68i			2.30211i
$844$		0			0
$845$	−38.7310	−	167.071i	−0.0458354	−	0.197717i
$846$		0			0
$847$			847.000i			1.00000i
$848$		0			0
$849$	637.367			0.750727
$850$		0			0
$851$		0			0
$852$		−	2062.40i		−	2.42066i
$853$		−	845.655i		−	0.991390i	−0.868497	−	0.495695i	$-0.834913\pi$
$853$		−	845.655i		−	0.991390i	0.868497	−	0.495695i	$-0.165087\pi$
$854$	1288.81			1.50915
$855$	−546.434	−	2357.11i	−0.639104	−	2.75686i
$856$		0			0
$857$		0			0			−	1.00000i	$-0.5\pi$
$857$		0			0				1.00000i	$0.5\pi$
$858$		0			0
$859$	−1717.81			−1.99977			−0.999887	−	0.0150211i	$-0.995218\pi$
$859$	−1717.81			−1.99977			−0.999887	+	0.0150211i	$0.995218\pi$
$860$		0			0
$861$		0			0
$862$		−	1076.00i		−	1.24826i
$863$			1474.00i			1.70800i	0.520277	+	0.853998i	$0.325829\pi$
$863$			1474.00i			1.70800i	−0.520277	+	0.853998i	$0.674171\pi$
$864$	−2749.86			−3.18271
$865$	−103.949	+	24.0979i	−0.120173	+	0.0278588i
$866$		0			0
$867$			1659.34i			1.91389i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$		0			0
$874$	−1813.20			−2.07460
$875$	678.503	−	552.503i	0.775432	−	0.631432i
$876$		0			0
$877$		0			0			−	1.00000i	$-0.5\pi$
$877$		0			0				1.00000i	$0.5\pi$
$878$		0			0
$879$	300.532			0.341902
$880$		0			0
$881$		0			0		1.00000			$0$
$881$		0			0		−1.00000			$\pi$
$882$			2348.73i			2.66296i
$883$		0			0			−	1.00000i	$-0.5\pi$
$883$		0			0				1.00000i	$0.5\pi$
$884$		0			0
$885$	3289.10	−	762.491i	3.71650	−	0.861572i
$886$		0			0
$887$		0			0			−	1.00000i	$-0.5\pi$
$887$		0			0				1.00000i	$0.5\pi$
$888$		0			0
$889$	−314.299			−0.353542
$890$		0			0
$891$		0			0
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$	896.000			1.00000
$897$			3675.80i			4.09788i
$898$			4.00000i			0.00445434i
$899$		0			0
$900$	−2152.20	+	1054.53i	−2.39133	+	1.17170i
$901$		0			0
$902$		0			0
$903$		0			0
$904$	−208.000			−0.230088
$905$	−11.4522	−	49.4004i	−0.0126543	−	0.0545861i
$906$	2319.63			2.56030
$907$		0			0			−	1.00000i	$-0.5\pi$
$907$		0			0				1.00000i	$0.5\pi$
$908$			1741.63i			1.91810i
$909$	−669.673			−0.736714
$910$	972.299	−	225.402i	1.06846	−	0.247694i
$911$	922.000			1.01207			0.506037	−	0.862512i	$-0.331110\pi$
$911$	922.000			1.01207			0.506037	+	0.862512i	$0.331110\pi$
$912$		−	1854.93i		−	2.03392i
$913$		0			0
$914$	1772.00			1.93873
$915$	−2574.56	+	596.842i	−2.81372	+	0.652286i
$916$	396.564			0.432930
$917$			1663.26i			1.81380i
$918$		0			0
$919$	−987.798			−1.07486			−0.537431	−	0.843308i	$-0.680605\pi$
$919$	−987.798			−1.07486			−0.537431	+	0.843308i	$0.680605\pi$
$920$	405.597	+	1749.60i	0.440867	+	1.90174i
$921$	3523.03			3.82522
$922$			877.714i			0.951967i
$923$		−	1280.40i		−	1.38721i
$924$		0			0
$925$		0			0
$926$	452.000			0.488121
$927$		0			0
$928$		0			0
$929$		0			0		1.00000			$0$
$929$		0			0		−1.00000			$\pi$
$930$		0			0
$931$	−989.388			−1.06272
$932$			718.398i			0.770814i
$933$		0			0
$934$	1813.98			1.94216
$935$		0			0
$936$	−2733.80			−2.92072
$937$		0			0			−	1.00000i	$-0.5\pi$
$937$		0			0				1.00000i	$0.5\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	287.856			0.305904			0.152952	−	0.988234i	$-0.451122\pi$
$941$	287.856			0.305904			0.152952	+	0.988234i	$0.451122\pi$
$942$		−	2592.26i		−	2.75187i
$943$		0			0
$944$	1881.73			1.99336
$945$	−679.234	−	2929.96i	−0.718766	−	3.10049i
$946$		0			0
$947$		0			0			−	1.00000i	$-0.5\pi$
$947$		0			0				1.00000i	$0.5\pi$
$948$		−	2062.40i		−	2.17553i
$949$		0			0
$950$	−444.215	−	906.601i	−0.467595	−	0.954316i
$951$		0			0
$952$		0			0
$953$			1616.40i			1.69611i	0.529906	+	0.848057i	$0.322227\pi$
$953$			1616.40i			1.69611i	−0.529906	+	0.848057i	$0.677773\pi$
$954$		0			0
$955$	405.597	+	1749.60i	0.424709	+	1.83204i
$956$	−1688.00			−1.76569
$957$		0			0
$958$		0			0
$959$	−1885.80			−1.96642
$960$	−1789.86	+	414.932i	−1.86444	+	0.432221i
$961$	961.000			1.00000
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−1529.44	+	354.560i	−1.58491	+	0.367419i
$966$	−3609.20			−3.73623
$967$		−	1302.10i		−	1.34653i	−0.739400	−	0.673266i	$-0.764891\pi$
$967$		−	1302.10i		−	1.34653i	0.739400	−	0.673266i	$-0.235109\pi$
$968$		−	968.000i		−	1.00000i
$969$		0			0
$970$		0			0
$971$	1367.19			1.40802			0.704010	−	0.710190i	$-0.251391\pi$
$971$	1367.19			1.40802			0.704010	+	0.710190i	$0.251391\pi$
$972$			3284.23i			3.37884i
$973$		−	856.743i		−	0.880517i
$974$	1706.20			1.75174
$975$	−1837.90	+	900.532i	−1.88502	+	0.923622i
$976$	−1472.93			−1.50915
$977$		−	1077.60i		−	1.10297i	−0.834186	−	0.551483i	$-0.814062\pi$
$977$		−	1077.60i		−	1.10297i	0.834186	−	0.551483i	$-0.185938\pi$
$978$		0			0
$979$		0			0
$980$	221.318	+	954.682i	0.225834	+	0.974166i
$981$		0			0
$982$		0			0
$983$		0			0			−	1.00000i	$-0.5\pi$
$983$		0			0				1.00000i	$0.5\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		−	1151.60i		−	1.16558i
$989$		0			0
$990$		0			0
$991$	−1885.80			−1.90292			−0.951461	−	0.307770i	$-0.900417\pi$
$991$	−1885.80			−1.90292			−0.951461	+	0.307770i	$0.900417\pi$
$992$		0			0
$993$		0			0
$994$	1257.20			1.26479
$995$		0			0
$996$	2250.53			2.25957
$997$			1581.34i			1.58610i	0.609157	+	0.793050i	$0.291508\pi$
$997$			1581.34i			1.58610i	−0.609157	+	0.793050i	$0.708492\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	280.3.c.f.69.3	yes	4
4.3	odd	2		1120.3.c.f.209.4		4
5.4	even	2	inner	280.3.c.f.69.2	yes	4
7.6	odd	2		280.3.c.e.69.4	yes	4
8.3	odd	2		1120.3.c.e.209.1		4
8.5	even	2		280.3.c.e.69.4	yes	4
20.19	odd	2		1120.3.c.f.209.1		4
28.27	even	2		1120.3.c.e.209.1		4
35.34	odd	2		280.3.c.e.69.1	✓	4
40.19	odd	2		1120.3.c.e.209.4		4
40.29	even	2		280.3.c.e.69.1	✓	4
56.13	odd	2	CM	280.3.c.f.69.3	yes	4
56.27	even	2		1120.3.c.f.209.4		4
140.139	even	2		1120.3.c.e.209.4		4
280.69	odd	2	inner	280.3.c.f.69.2	yes	4
280.139	even	2		1120.3.c.f.209.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
280.3.c.e.69.1	✓	4	35.34	odd	2
280.3.c.e.69.1	✓	4	40.29	even	2
280.3.c.e.69.4	yes	4	7.6	odd	2
280.3.c.e.69.4	yes	4	8.5	even	2
280.3.c.f.69.2	yes	4	5.4	even	2	inner
280.3.c.f.69.2	yes	4	280.69	odd	2	inner
280.3.c.f.69.3	yes	4	1.1	even	1	trivial
280.3.c.f.69.3	yes	4	56.13	odd	2	CM
1120.3.c.e.209.1		4	8.3	odd	2
1120.3.c.e.209.1		4	28.27	even	2
1120.3.c.e.209.4		4	40.19	odd	2
1120.3.c.e.209.4		4	140.139	even	2
1120.3.c.f.209.1		4	20.19	odd	2
1120.3.c.f.209.1		4	280.139	even	2
1120.3.c.f.209.4		4	4.3	odd	2
1120.3.c.f.209.4		4	56.27	even	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:	\( N \)	\(=\)	\( 280 = 2^{3} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	280.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+2.00000i q^{2} -5.74166i q^{3} -4.00000 q^{4} +(1.12917 + 4.87083i) q^{5} +11.4833 q^{6} +7.00000i q^{7} -8.00000i q^{8} -23.9666 q^{9} +O(q^{10})\)	\(q+2.00000i q^{2} -5.74166i q^{3} -4.00000 q^{4} +(1.12917 + 4.87083i) q^{5} +11.4833 q^{6} +7.00000i q^{7} -8.00000i q^{8} -23.9666 q^{9} +(-9.74166 + 2.25834i) q^{10} +22.9666i q^{12} +14.2583i q^{13} -14.0000 q^{14} +(27.9666 - 6.48331i) q^{15} +16.0000 q^{16} -47.9333i q^{18} +20.1916 q^{19} +(-4.51669 - 19.4833i) q^{20} +40.1916 q^{21} +44.8999i q^{23} -45.9333 q^{24} +(-22.4499 + 11.0000i) q^{25} -28.5167 q^{26} +85.9333i q^{27} -28.0000i q^{28} +(12.9666 + 55.9333i) q^{30} +32.0000i q^{32} +(-34.0958 + 7.90420i) q^{35} +95.8665 q^{36} +40.3832i q^{38} +81.8665 q^{39} +(38.9666 - 9.03337i) q^{40} +80.3832i q^{42} +(-27.0624 - 116.737i) q^{45} -89.7998 q^{46} -91.8665i q^{48} -49.0000 q^{49} +(-22.0000 - 44.8999i) q^{50} -57.0334i q^{52} -171.867 q^{54} +56.0000 q^{56} -115.933i q^{57} +117.608 q^{59} +(-111.867 + 25.9333i) q^{60} -92.0581 q^{61} -167.766i q^{63} -64.0000 q^{64} +(-69.4499 + 16.1001i) q^{65} +257.800 q^{69} +(-15.8084 - 68.1916i) q^{70} -89.7998 q^{71} +191.733i q^{72} +(63.1582 + 128.900i) q^{75} -80.7664 q^{76} +163.733i q^{78} -89.7998 q^{79} +(18.0667 + 77.9333i) q^{80} +277.700 q^{81} -97.9914i q^{83} -160.766 q^{84} +(233.475 - 54.1249i) q^{90} -99.8084 q^{91} -179.600i q^{92} +(22.7998 + 98.3498i) q^{95} +183.733 q^{96} -98.0000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 16 q^{4} + 12 q^{5} + 16 q^{6} - 36 q^{9}+O(q^{10}) \) 4 * q - 16 * q^4 + 12 * q^5 + 16 * q^6 - 36 * q^9	\( 4 q - 16 q^{4} + 12 q^{5} + 16 q^{6} - 36 q^{9} - 24 q^{10} - 56 q^{14} + 52 q^{15} + 64 q^{16} - 24 q^{19} - 48 q^{20} + 56 q^{21} - 64 q^{24} - 144 q^{26} - 8 q^{30} - 84 q^{35} + 144 q^{36} + 88 q^{39} + 96 q^{40} + 4 q^{45} - 196 q^{49} - 88 q^{50} - 448 q^{54} + 224 q^{56} + 216 q^{59} - 208 q^{60} - 24 q^{61} - 256 q^{64} - 188 q^{65} + 672 q^{69} - 168 q^{70} + 88 q^{75} + 96 q^{76} + 192 q^{80} + 572 q^{81} - 224 q^{84} + 440 q^{90} - 504 q^{91} - 268 q^{95} + 256 q^{96}+O(q^{100}) \) 4 * q - 16 * q^4 + 12 * q^5 + 16 * q^6 - 36 * q^9 - 24 * q^10 - 56 * q^14 + 52 * q^15 + 64 * q^16 - 24 * q^19 - 48 * q^20 + 56 * q^21 - 64 * q^24 - 144 * q^26 - 8 * q^30 - 84 * q^35 + 144 * q^36 + 88 * q^39 + 96 * q^40 + 4 * q^45 - 196 * q^49 - 88 * q^50 - 448 * q^54 + 224 * q^56 + 216 * q^59 - 208 * q^60 - 24 * q^61 - 256 * q^64 - 188 * q^65 + 672 * q^69 - 168 * q^70 + 88 * q^75 + 96 * q^76 + 192 * q^80 + 572 * q^81 - 224 * q^84 + 440 * q^90 - 504 * q^91 - 268 * q^95 + 256 * q^96

Introduction

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