LMFDB - Embedded newform 1280.3.h.i.1279.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1280,3,Mod(1279,1280)]

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([1, 0, 1]))

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

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

chi := DirichletCharacter("1280.1279");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1280 = 2^{8} \cdot 5 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1280.h (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:	$34.8774738381$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.691798081536.3
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 4x^{7} + 26x^{6} - 64x^{5} + 229x^{4} - 356x^{3} + 164x^{2} + 4x + 985 $ x^8 - 4x^7 + 26x^6 - 64x^5 + 229x^4 - 356x^3 + 164x^2 + 4*x + 985
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{10} $
Twist minimal:	no (minimal twist has level 320)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1279.6
Root			$1.72474 - 0.954705i$ of defining polynomial
Character	$\chi$	$=$	1280.1279
Dual form			1280.3.h.i.1279.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.44949 q^{3} +(-4.35890 + 2.44949i) q^{5} -10.6771 q^{7} -3.00000 q^{9} +O(q^{10})$	$q+2.44949 q^{3} +(-4.35890 + 2.44949i) q^{5} -10.6771 q^{7} -3.00000 q^{9} +8.71780i q^{11} +19.5959i q^{13} +(-10.6771 + 6.00000i) q^{15} -21.3542i q^{17} -26.1534i q^{19} -26.1534 q^{21} +10.6771 q^{23} +(13.0000 - 21.3542i) q^{25} -29.3939 q^{27} +34.8712 q^{29} +4.00000i q^{31} +21.3542i q^{33} +(46.5403 - 26.1534i) q^{35} -14.6969i q^{37} +48.0000i q^{39} +24.0000 q^{41} +56.3383 q^{43} +(13.0767 - 7.34847i) q^{45} -10.6771 q^{47} +65.0000 q^{49} -52.3068i q^{51} -48.9898i q^{53} +(-21.3542 - 38.0000i) q^{55} -64.0625i q^{57} -43.5890i q^{59} -26.1534 q^{61} +32.0312 q^{63} +(-48.0000 - 85.4166i) q^{65} +7.34847 q^{67} +26.1534 q^{69} -84.0000i q^{71} +106.771i q^{73} +(31.8434 - 52.3068i) q^{75} -93.0806i q^{77} -100.000i q^{79} -45.0000 q^{81} +17.1464 q^{83} +(52.3068 + 93.0806i) q^{85} +85.4166 q^{87} -150.000 q^{89} -209.227i q^{91} +9.79796i q^{93} +(64.0625 + 114.000i) q^{95} +21.3542i q^{97} -26.1534i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 24 q^{9}+O(q^{10}) $ 8 * q - 24 * q^9	$ 8 q - 24 q^{9} + 104 q^{25} + 192 q^{41} + 520 q^{49} - 384 q^{65} - 360 q^{81} - 1200 q^{89}+O(q^{100}) $ 8 * q - 24 * q^9 + 104 * q^25 + 192 * q^41 + 520 * q^49 - 384 * q^65 - 360 * q^81 - 1200 * q^89

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
$3$	2.44949			0.816497			0.408248	−	0.912871i	$-0.366140\pi$
$3$	2.44949			0.816497			0.408248	+	0.912871i	$0.366140\pi$
$4$		0			0
$5$	−4.35890	+	2.44949i	−0.871780	+	0.489898i
$5$	−4.35890	+	2.44949i	−0.871780	+	0.489898i
$6$		0			0
$7$	−10.6771			−1.52530			−0.762648	−	0.646813i	$-0.776101\pi$
$7$	−10.6771			−1.52530			−0.762648	+	0.646813i	$0.776101\pi$
$8$		0			0
$9$	−3.00000			−0.333333
$10$		0			0
$11$			8.71780i			0.792527i	0.918137	+	0.396264i	$0.129693\pi$
$11$			8.71780i			0.792527i	−0.918137	+	0.396264i	$0.870307\pi$
$12$		0			0
$13$			19.5959i			1.50738i	0.657231	+	0.753689i	$0.271728\pi$
$13$			19.5959i			1.50738i	−0.657231	+	0.753689i	$0.728272\pi$
$14$		0			0
$15$	−10.6771	+	6.00000i	−0.711805	+	0.400000i
$16$		0			0
$17$		−	21.3542i		−	1.25613i	−0.778162	−	0.628063i	$-0.783848\pi$
$17$		−	21.3542i		−	1.25613i	0.778162	−	0.628063i	$-0.216152\pi$
$18$		0			0
$19$		−	26.1534i		−	1.37649i	−0.725476	−	0.688247i	$-0.758380\pi$
$19$		−	26.1534i		−	1.37649i	0.725476	−	0.688247i	$-0.241620\pi$
$20$		0			0
$21$	−26.1534			−1.24540
$22$		0			0
$23$	10.6771			0.464221			0.232110	−	0.972689i	$-0.425437\pi$
$23$	10.6771			0.464221			0.232110	+	0.972689i	$0.425437\pi$
$24$		0			0
$25$	13.0000	−	21.3542i	0.520000	−	0.854166i
$26$		0			0
$27$	−29.3939			−1.08866
$28$		0			0
$29$	34.8712			1.20245			0.601227	−	0.799078i	$-0.294679\pi$
$29$	34.8712			1.20245			0.601227	+	0.799078i	$0.294679\pi$
$30$		0			0
$31$			4.00000i			0.129032i	0.997917	+	0.0645161i	$0.0205504\pi$
$31$			4.00000i			0.129032i	−0.997917	+	0.0645161i	$0.979450\pi$
$32$		0			0
$33$			21.3542i			0.647096i
$34$		0			0
$35$	46.5403	−	26.1534i	1.32972	−	0.747240i
$36$		0			0
$37$		−	14.6969i		−	0.397215i	−0.980079	−	0.198607i	$-0.936358\pi$
$37$		−	14.6969i		−	0.397215i	0.980079	−	0.198607i	$-0.0636418\pi$
$38$		0			0
$39$			48.0000i			1.23077i
$40$		0			0
$41$	24.0000			0.585366			0.292683	−	0.956210i	$-0.405452\pi$
$41$	24.0000			0.585366			0.292683	+	0.956210i	$0.405452\pi$
$42$		0			0
$43$	56.3383			1.31019			0.655096	−	0.755546i	$-0.272628\pi$
$43$	56.3383			1.31019			0.655096	+	0.755546i	$0.272628\pi$
$44$		0			0
$45$	13.0767	−	7.34847i	0.290593	−	0.163299i
$46$		0			0
$47$	−10.6771			−0.227172			−0.113586	−	0.993528i	$-0.536234\pi$
$47$	−10.6771			−0.227172			−0.113586	+	0.993528i	$0.536234\pi$
$48$		0			0
$49$	65.0000			1.32653
$50$		0			0
$51$		−	52.3068i		−	1.02562i
$52$		0			0
$53$		−	48.9898i		−	0.924336i	−0.886792	−	0.462168i	$-0.847072\pi$
$53$		−	48.9898i		−	0.924336i	0.886792	−	0.462168i	$-0.152928\pi$
$54$		0			0
$55$	−21.3542	−	38.0000i	−0.388257	−	0.690909i
$56$		0			0
$57$		−	64.0625i		−	1.12390i
$58$		0			0
$59$		−	43.5890i		−	0.738796i	−0.929271	−	0.369398i	$-0.879564\pi$
$59$		−	43.5890i		−	0.738796i	0.929271	−	0.369398i	$-0.120436\pi$
$60$		0			0
$61$	−26.1534			−0.428744			−0.214372	−	0.976752i	$-0.568771\pi$
$61$	−26.1534			−0.428744			−0.214372	+	0.976752i	$0.568771\pi$
$62$		0			0
$63$	32.0312			0.508432
$64$		0			0
$65$	−48.0000	−	85.4166i	−0.738462	−	1.31410i
$66$		0			0
$67$	7.34847			0.109679			0.0548393	−	0.998495i	$-0.482535\pi$
$67$	7.34847			0.109679			0.0548393	+	0.998495i	$0.482535\pi$
$68$		0			0
$69$	26.1534			0.379035
$70$		0			0
$71$		−	84.0000i		−	1.18310i	−0.806269	−	0.591549i	$-0.798517\pi$
$71$		−	84.0000i		−	1.18310i	0.806269	−	0.591549i	$-0.201483\pi$
$72$		0			0
$73$			106.771i			1.46261i	0.682049	+	0.731307i	$0.261089\pi$
$73$			106.771i			1.46261i	−0.682049	+	0.731307i	$0.738911\pi$
$74$		0			0
$75$	31.8434	−	52.3068i	0.424578	−	0.697424i
$76$		0			0
$77$		−	93.0806i		−	1.20884i
$78$		0			0
$79$		−	100.000i		−	1.26582i	−0.774224	−	0.632911i	$-0.781860\pi$
$79$		−	100.000i		−	1.26582i	0.774224	−	0.632911i	$-0.218140\pi$
$80$		0			0
$81$	−45.0000			−0.555556
$82$		0			0
$83$	17.1464			0.206583			0.103292	−	0.994651i	$-0.467062\pi$
$83$	17.1464			0.206583			0.103292	+	0.994651i	$0.467062\pi$
$84$		0			0
$85$	52.3068	+	93.0806i	0.615374	+	1.09507i
$86$		0			0
$87$	85.4166			0.981800
$88$		0			0
$89$	−150.000			−1.68539			−0.842697	−	0.538389i	$-0.819033\pi$
$89$	−150.000			−1.68539			−0.842697	+	0.538389i	$0.819033\pi$
$90$		0			0
$91$		−	209.227i		−	2.29920i
$92$		0			0
$93$			9.79796i			0.105354i
$94$		0			0
$95$	64.0625	+	114.000i	0.674342	+	1.20000i
$96$		0			0
$97$			21.3542i			0.220146i	0.993924	+	0.110073i	$0.0351085\pi$
$97$			21.3542i			0.220146i	−0.993924	+	0.110073i	$0.964892\pi$
$98$		0			0
$99$		−	26.1534i		−	0.264176i
$100$		0			0
$101$	69.7424			0.690519			0.345259	−	0.938507i	$-0.387791\pi$
$101$	69.7424			0.690519			0.345259	+	0.938507i	$0.387791\pi$
$102$		0			0
$103$	−32.0312			−0.310983			−0.155491	−	0.987837i	$-0.549696\pi$
$103$	−32.0312			−0.310983			−0.155491	+	0.987837i	$0.549696\pi$
$104$		0			0
$105$	114.000	−	64.0625i	1.08571	−	0.610119i
$106$		0			0
$107$	149.419			1.39644			0.698219	−	0.715884i	$-0.253976\pi$
$107$	149.419			1.39644			0.698219	+	0.715884i	$0.253976\pi$
$108$		0			0
$109$	−183.074			−1.67958			−0.839788	−	0.542915i	$-0.817321\pi$
$109$	−183.074			−1.67958			−0.839788	+	0.542915i	$0.817321\pi$
$110$		0			0
$111$		−	36.0000i		−	0.324324i
$112$		0			0
$113$			170.833i			1.51180i	0.654688	+	0.755899i	$0.272800\pi$
$113$			170.833i			1.51180i	−0.654688	+	0.755899i	$0.727200\pi$
$114$		0			0
$115$	−46.5403	+	26.1534i	−0.404698	+	0.227421i
$116$		0			0
$117$		−	58.7878i		−	0.502459i
$118$		0			0
$119$			228.000i			1.91597i
$120$		0			0
$121$	45.0000			0.371901
$122$		0			0
$123$	58.7878			0.477949
$124$		0			0
$125$	−4.35890	+	124.924i	−0.0348712	+	0.999392i
$126$		0			0
$127$	53.3854			0.420357			0.210179	−	0.977663i	$-0.432595\pi$
$127$	53.3854			0.420357			0.210179	+	0.977663i	$0.432595\pi$
$128$		0			0
$129$	138.000			1.06977
$130$		0			0
$131$		−	95.8958i		−	0.732029i	−0.930609	−	0.366014i	$-0.880722\pi$
$131$		−	95.8958i		−	0.732029i	0.930609	−	0.366014i	$-0.119278\pi$
$132$		0			0
$133$			279.242i			2.09956i
$134$		0			0
$135$	128.125	−	72.0000i	0.949074	−	0.533333i
$136$		0			0
$137$			128.125i			0.935219i	0.883935	+	0.467609i	$0.154885\pi$
$137$			128.125i			0.935219i	−0.883935	+	0.467609i	$0.845115\pi$
$138$		0			0
$139$		−	235.381i		−	1.69339i	−0.532082	−	0.846693i	$-0.678590\pi$
$139$		−	235.381i		−	1.69339i	0.532082	−	0.846693i	$-0.321410\pi$
$140$		0			0
$141$	−26.1534			−0.185485
$142$		0			0
$143$	−170.833			−1.19464
$144$		0			0
$145$	−152.000	+	85.4166i	−1.04828	+	0.589080i
$146$		0			0
$147$	159.217			1.08311
$148$		0			0
$149$	−95.8958			−0.643596			−0.321798	−	0.946808i	$-0.604287\pi$
$149$	−95.8958			−0.643596			−0.321798	+	0.946808i	$0.604287\pi$
$150$		0			0
$151$		−	160.000i		−	1.05960i	−0.848122	−	0.529801i	$-0.822266\pi$
$151$		−	160.000i		−	1.05960i	0.848122	−	0.529801i	$-0.177734\pi$
$152$		0			0
$153$			64.0625i			0.418709i
$154$		0			0
$155$	−9.79796	−	17.4356i	−0.0632126	−	0.112488i
$156$		0			0
$157$		−	34.2929i		−	0.218426i	−0.994018	−	0.109213i	$-0.965167\pi$
$157$		−	34.2929i		−	0.218426i	0.994018	−	0.109213i	$-0.0348330\pi$
$158$		0			0
$159$		−	120.000i		−	0.754717i
$160$		0			0
$161$	−114.000			−0.708075
$162$		0			0
$163$	120.025			0.736350			0.368175	−	0.929757i	$-0.379983\pi$
$163$	120.025			0.736350			0.368175	+	0.929757i	$0.379983\pi$
$164$		0			0
$165$	−52.3068	−	93.0806i	−0.317011	−	0.564125i
$166$		0			0
$167$	−288.281			−1.72623			−0.863117	−	0.505004i	$-0.831491\pi$
$167$	−288.281			−1.72623			−0.863117	+	0.505004i	$0.831491\pi$
$168$		0			0
$169$	−215.000			−1.27219
$170$		0			0
$171$			78.4602i			0.458831i
$172$		0			0
$173$		−	24.4949i		−	0.141589i	−0.997491	−	0.0707945i	$-0.977447\pi$
$173$		−	24.4949i		−	0.141589i	0.997491	−	0.0707945i	$-0.0225535\pi$
$174$		0			0
$175$	−138.802	+	228.000i	−0.793154	+	1.30286i
$176$		0			0
$177$		−	106.771i		−	0.603225i
$178$		0			0
$179$		−	252.816i		−	1.41238i	−0.708022	−	0.706190i	$-0.750412\pi$
$179$		−	252.816i		−	1.41238i	0.708022	−	0.706190i	$-0.249588\pi$
$180$		0			0
$181$		0			0			−	1.00000i	$-0.5\pi$
$181$		0			0				1.00000i	$0.5\pi$
$182$		0			0
$183$	−64.0625			−0.350068
$184$		0			0
$185$	36.0000	+	64.0625i	0.194595	+	0.346284i
$186$		0			0
$187$	186.161			0.995515
$188$		0			0
$189$	313.841			1.66053
$190$		0			0
$191$			24.0000i			0.125654i	0.998024	+	0.0628272i	$0.0200117\pi$
$191$			24.0000i			0.125654i	−0.998024	+	0.0628272i	$0.979988\pi$
$192$		0			0
$193$		−	106.771i		−	0.553216i	−0.960983	−	0.276608i	$-0.910790\pi$
$193$		−	106.771i		−	0.553216i	0.960983	−	0.276608i	$-0.0892104\pi$
$194$		0			0
$195$	−117.576	−	209.227i	−0.602951	−	1.07296i
$196$		0			0
$197$		−	342.929i		−	1.74075i	−0.492386	−	0.870377i	$-0.663875\pi$
$197$		−	342.929i		−	1.74075i	0.492386	−	0.870377i	$-0.336125\pi$
$198$		0			0
$199$			188.000i			0.944724i	0.881405	+	0.472362i	$0.156598\pi$
$199$			188.000i			0.944724i	−0.881405	+	0.472362i	$0.843402\pi$
$200$		0			0
$201$	18.0000			0.0895522
$202$		0			0
$203$	−372.322			−1.83410
$204$		0			0
$205$	−104.614	+	58.7878i	−0.510310	+	0.286770i
$206$		0			0
$207$	−32.0312			−0.154740
$208$		0			0
$209$	228.000			1.09091
$210$		0			0
$211$		−	26.1534i		−	0.123950i	−0.998078	−	0.0619749i	$-0.980260\pi$
$211$		−	26.1534i		−	0.123950i	0.998078	−	0.0619749i	$-0.0197399\pi$
$212$		0			0
$213$		−	205.757i		−	0.965996i
$214$		0			0
$215$	−245.573	+	138.000i	−1.14220	+	0.641860i
$216$		0			0
$217$		−	42.7083i		−	0.196813i
$218$		0			0
$219$			261.534i			1.19422i
$220$		0			0
$221$	418.454			1.89346
$222$		0			0
$223$	224.219			1.00546			0.502732	−	0.864442i	$-0.332328\pi$
$223$	224.219			1.00546			0.502732	+	0.864442i	$0.332328\pi$
$224$		0			0
$225$	−39.0000	+	64.0625i	−0.173333	+	0.284722i
$226$		0			0
$227$	80.8332			0.356093			0.178047	−	0.984022i	$-0.443022\pi$
$227$	80.8332			0.356093			0.178047	+	0.984022i	$0.443022\pi$
$228$		0			0
$229$	104.614			0.456828			0.228414	−	0.973564i	$-0.426646\pi$
$229$	104.614			0.456828			0.228414	+	0.973564i	$0.426646\pi$
$230$		0			0
$231$		−	228.000i		−	0.987013i
$232$		0			0
$233$		−	320.312i		−	1.37473i	−0.726312	−	0.687366i	$-0.758767\pi$
$233$		−	320.312i		−	1.37473i	0.726312	−	0.687366i	$-0.241233\pi$
$234$		0			0
$235$	46.5403	−	26.1534i	0.198044	−	0.111291i
$236$		0			0
$237$		−	244.949i		−	1.03354i
$238$		0			0
$239$		−	108.000i		−	0.451883i	−0.974141	−	0.225941i	$-0.927454\pi$
$239$		−	108.000i		−	0.451883i	0.974141	−	0.225941i	$-0.0725458\pi$
$240$		0			0
$241$	−28.0000			−0.116183			−0.0580913	−	0.998311i	$-0.518501\pi$
$241$	−28.0000			−0.116183			−0.0580913	+	0.998311i	$0.518501\pi$
$242$		0			0
$243$	154.318			0.635053
$244$		0			0
$245$	−283.328	+	159.217i	−1.15644	+	0.649865i
$246$		0			0
$247$	512.500			2.07490
$248$		0			0
$249$	42.0000			0.168675
$250$		0			0
$251$			252.816i			1.00724i	0.863927	+	0.503618i	$0.167998\pi$
$251$			252.816i			1.00724i	−0.863927	+	0.503618i	$0.832002\pi$
$252$		0			0
$253$			93.0806i			0.367908i
$254$		0			0
$255$	128.125	+	228.000i	0.502451	+	0.894118i
$256$		0			0
$257$			298.958i			1.16326i	0.813453	+	0.581631i	$0.197585\pi$
$257$			298.958i			1.16326i	−0.813453	+	0.581631i	$0.802415\pi$
$258$		0			0
$259$			156.920i			0.605870i
$260$		0			0
$261$	−104.614			−0.400818
$262$		0			0
$263$	288.281			1.09613			0.548063	−	0.836437i	$-0.315365\pi$
$263$	288.281			1.09613			0.548063	+	0.836437i	$0.315365\pi$
$264$		0			0
$265$	120.000	+	213.542i	0.452830	+	0.805817i
$266$		0			0
$267$	−367.423			−1.37612
$268$		0			0
$269$	−61.0246			−0.226857			−0.113429	−	0.993546i	$-0.536183\pi$
$269$	−61.0246			−0.226857			−0.113429	+	0.993546i	$0.536183\pi$
$270$		0			0
$271$		−	428.000i		−	1.57934i	−0.613535	−	0.789668i	$-0.710253\pi$
$271$		−	428.000i		−	1.57934i	0.613535	−	0.789668i	$-0.289747\pi$
$272$		0			0
$273$		−	512.500i		−	1.87729i
$274$		0			0
$275$	186.161	+	113.331i	0.676950	+	0.412114i
$276$		0			0
$277$		−	73.4847i		−	0.265288i	−0.991164	−	0.132644i	$-0.957653\pi$
$277$		−	73.4847i		−	0.265288i	0.991164	−	0.132644i	$-0.0423467\pi$
$278$		0			0
$279$		−	12.0000i		−	0.0430108i
$280$		0			0
$281$	204.000			0.725979			0.362989	−	0.931793i	$-0.381756\pi$
$281$	204.000			0.725979			0.362989	+	0.931793i	$0.381756\pi$
$282$		0			0
$283$	404.166			1.42815			0.714074	−	0.700070i	$-0.246848\pi$
$283$	404.166			1.42815			0.714074	+	0.700070i	$0.246848\pi$
$284$		0			0
$285$	156.920	+	279.242i	0.550598	+	0.979796i
$286$		0			0
$287$	−256.250			−0.892857
$288$		0			0
$289$	−167.000			−0.577855
$290$		0			0
$291$			52.3068i			0.179748i
$292$		0			0
$293$			161.666i			0.551762i	0.961192	+	0.275881i	$0.0889696\pi$
$293$			161.666i			0.551762i	−0.961192	+	0.275881i	$0.911030\pi$
$294$		0			0
$295$	106.771	+	190.000i	0.361935	+	0.644068i
$296$		0			0
$297$		−	256.250i		−	0.862794i
$298$		0			0
$299$			209.227i			0.699756i
$300$		0			0
$301$	−601.528			−1.99843
$302$		0			0
$303$	170.833			0.563806
$304$		0			0
$305$	114.000	−	64.0625i	0.373770	−	0.210041i
$306$		0			0
$307$	−502.145			−1.63565			−0.817826	−	0.575465i	$-0.804821\pi$
$307$	−502.145			−1.63565			−0.817826	+	0.575465i	$0.804821\pi$
$308$		0			0
$309$	−78.4602			−0.253916
$310$		0			0
$311$		−	408.000i		−	1.31190i	−0.754806	−	0.655949i	$-0.772269\pi$
$311$		−	408.000i		−	1.31190i	0.754806	−	0.655949i	$-0.227731\pi$
$312$		0			0
$313$		−	298.958i		−	0.955138i	−0.878594	−	0.477569i	$-0.841518\pi$
$313$		−	298.958i		−	0.955138i	0.878594	−	0.477569i	$-0.158482\pi$
$314$		0			0
$315$	−139.621	+	78.4602i	−0.443241	+	0.249080i
$316$		0			0
$317$			186.161i			0.587259i	0.955919	+	0.293630i	$0.0948633\pi$
$317$			186.161i			0.587259i	−0.955919	+	0.293630i	$0.905137\pi$
$318$		0			0
$319$			304.000i			0.952978i
$320$		0			0
$321$	366.000			1.14019
$322$		0			0
$323$	−558.484			−1.72905
$324$		0			0
$325$	418.454	+	254.747i	1.28755	+	0.783837i
$326$		0			0
$327$	−448.437			−1.37137
$328$		0			0
$329$	114.000			0.346505
$330$		0			0
$331$		−	287.687i		−	0.869146i	−0.900637	−	0.434573i	$-0.856899\pi$
$331$		−	287.687i		−	0.869146i	0.900637	−	0.434573i	$-0.143101\pi$
$332$		0			0
$333$			44.0908i			0.132405i
$334$		0			0
$335$	−32.0312	+	18.0000i	−0.0956156	+	0.0537313i
$336$		0			0
$337$		0			0		1.00000			$0$
$337$		0			0		−1.00000			$\pi$
$338$		0			0
$339$			418.454i			1.23438i
$340$		0			0
$341$	−34.8712			−0.102262
$342$		0			0
$343$	−170.833			−0.498056
$344$		0			0
$345$	−114.000	+	64.0625i	−0.330435	+	0.185688i
$346$		0			0
$347$	−413.964			−1.19298			−0.596490	−	0.802621i	$-0.703438\pi$
$347$	−413.964			−1.19298			−0.596490	+	0.802621i	$0.703438\pi$
$348$		0			0
$349$	−209.227			−0.599505			−0.299752	−	0.954017i	$-0.596904\pi$
$349$	−209.227			−0.599505			−0.299752	+	0.954017i	$0.596904\pi$
$350$		0			0
$351$		−	576.000i		−	1.64103i
$352$		0			0
$353$		−	213.542i		−	0.604934i	−0.953160	−	0.302467i	$-0.902190\pi$
$353$		−	213.542i		−	0.604934i	0.953160	−	0.302467i	$-0.0978101\pi$
$354$		0			0
$355$	205.757	+	366.148i	0.579598	+	1.03140i
$356$		0			0
$357$			558.484i			1.56438i
$358$		0			0
$359$		−	72.0000i		−	0.200557i	−0.994959	−	0.100279i	$-0.968027\pi$
$359$		−	72.0000i		−	0.200557i	0.994959	−	0.100279i	$-0.0319734\pi$
$360$		0			0
$361$	−323.000			−0.894737
$362$		0			0
$363$	110.227			0.303656
$364$		0			0
$365$	−261.534	−	465.403i	−0.716531	−	1.27508i
$366$		0			0
$367$	523.177			1.42555			0.712775	−	0.701393i	$-0.247438\pi$
$367$	523.177			1.42555			0.712775	+	0.701393i	$0.247438\pi$
$368$		0			0
$369$	−72.0000			−0.195122
$370$		0			0
$371$			523.068i			1.40989i
$372$		0			0
$373$		−	93.0806i		−	0.249546i	−0.992185	−	0.124773i	$-0.960180\pi$
$373$		−	93.0806i		−	0.249546i	0.992185	−	0.124773i	$-0.0398202\pi$
$374$		0			0
$375$	−10.6771	+	306.000i	−0.0284722	+	0.816000i
$376$		0			0
$377$			683.333i			1.81255i
$378$		0			0
$379$		−	339.994i		−	0.897082i	−0.893762	−	0.448541i	$-0.851944\pi$
$379$		−	339.994i		−	0.897082i	0.893762	−	0.448541i	$-0.148056\pi$
$380$		0			0
$381$	130.767			0.343220
$382$		0			0
$383$	−245.573			−0.641182			−0.320591	−	0.947218i	$-0.603882\pi$
$383$	−245.573			−0.641182			−0.320591	+	0.947218i	$0.603882\pi$
$384$		0			0
$385$	228.000	+	405.729i	0.592208	+	1.05384i
$386$		0			0
$387$	−169.015			−0.436731
$388$		0			0
$389$	148.203			0.380983			0.190492	−	0.981689i	$-0.438992\pi$
$389$	148.203			0.380983			0.190492	+	0.981689i	$0.438992\pi$
$390$		0			0
$391$		−	228.000i		−	0.583120i
$392$		0			0
$393$		−	234.896i		−	0.597699i
$394$		0			0
$395$	244.949	+	435.890i	0.620124	+	1.10352i
$396$		0			0
$397$		−	460.504i		−	1.15996i	−0.814631	−	0.579980i	$-0.803060\pi$
$397$		−	460.504i		−	1.15996i	0.814631	−	0.579980i	$-0.196940\pi$
$398$		0			0
$399$			684.000i			1.71429i
$400$		0			0
$401$	−330.000			−0.822943			−0.411471	−	0.911423i	$-0.634985\pi$
$401$	−330.000			−0.822943			−0.411471	+	0.911423i	$0.634985\pi$
$402$		0			0
$403$	−78.3837			−0.194500
$404$		0			0
$405$	196.150	−	110.227i	0.484322	−	0.272166i
$406$		0			0
$407$	128.125			0.314803
$408$		0			0
$409$	536.000			1.31051			0.655257	−	0.755406i	$-0.272560\pi$
$409$	536.000			1.31051			0.655257	+	0.755406i	$0.272560\pi$
$410$		0			0
$411$			313.841i			0.763603i
$412$		0			0
$413$			465.403i			1.12688i
$414$		0			0
$415$	−74.7395	+	42.0000i	−0.180095	+	0.101205i
$416$		0			0
$417$		−	576.562i		−	1.38264i
$418$		0			0
$419$			479.479i			1.14434i	0.820135	+	0.572171i	$0.193898\pi$
$419$			479.479i			1.14434i	−0.820135	+	0.572171i	$0.806102\pi$
$420$		0			0
$421$	496.914			1.18032			0.590160	−	0.807287i	$-0.299065\pi$
$421$	496.914			1.18032			0.590160	+	0.807287i	$0.299065\pi$
$422$		0			0
$423$	32.0312			0.0757240
$424$		0			0
$425$	−456.000	−	277.604i	−1.07294	−	0.653186i
$426$		0			0
$427$	279.242			0.653962
$428$		0			0
$429$	−418.454			−0.975418
$430$		0			0
$431$		−	180.000i		−	0.417633i	−0.977955	−	0.208817i	$-0.933039\pi$
$431$		−	180.000i		−	0.417633i	0.977955	−	0.208817i	$-0.0669612\pi$
$432$		0			0
$433$			363.021i			0.838385i	0.907897	+	0.419192i	$0.137687\pi$
$433$			363.021i			0.838385i	−0.907897	+	0.419192i	$0.862313\pi$
$434$		0			0
$435$	−372.322	+	209.227i	−0.855914	+	0.480982i
$436$		0			0
$437$		−	279.242i		−	0.638997i
$438$		0			0
$439$		−	136.000i		−	0.309795i	−0.987931	−	0.154897i	$-0.950495\pi$
$439$		−	136.000i		−	0.309795i	0.987931	−	0.154897i	$-0.0495047\pi$
$440$		0			0
$441$	−195.000			−0.442177
$442$		0			0
$443$	−183.712			−0.414699			−0.207350	−	0.978267i	$-0.566484\pi$
$443$	−183.712			−0.414699			−0.207350	+	0.978267i	$0.566484\pi$
$444$		0			0
$445$	653.835	−	367.423i	1.46929	−	0.825671i
$446$		0			0
$447$	−234.896			−0.525494
$448$		0			0
$449$	−684.000			−1.52339			−0.761693	−	0.647939i	$-0.775631\pi$
$449$	−684.000			−1.52339			−0.761693	+	0.647939i	$0.775631\pi$
$450$		0			0
$451$			209.227i			0.463918i
$452$		0			0
$453$		−	391.918i		−	0.865162i
$454$		0			0
$455$	512.500	+	912.000i	1.12637	+	2.00440i
$456$		0			0
$457$		−	85.4166i		−	0.186907i	−0.995624	−	0.0934536i	$-0.970209\pi$
$457$		−	85.4166i		−	0.186907i	0.995624	−	0.0934536i	$-0.0297907\pi$
$458$		0			0
$459$			627.681i			1.36750i
$460$		0			0
$461$	−348.712			−0.756425			−0.378212	−	0.925719i	$-0.623461\pi$
$461$	−348.712			−0.756425			−0.378212	+	0.925719i	$0.623461\pi$
$462$		0			0
$463$	−565.885			−1.22221			−0.611107	−	0.791548i	$-0.709276\pi$
$463$	−565.885			−1.22221			−0.611107	+	0.791548i	$0.709276\pi$
$464$		0			0
$465$	−24.0000	−	42.7083i	−0.0516129	−	0.0918458i
$466$		0			0
$467$	−105.328			−0.225542			−0.112771	−	0.993621i	$-0.535973\pi$
$467$	−105.328			−0.225542			−0.112771	+	0.993621i	$0.535973\pi$
$468$		0			0
$469$	−78.4602			−0.167292
$470$		0			0
$471$		−	84.0000i		−	0.178344i
$472$		0			0
$473$			491.146i			1.03836i
$474$		0			0
$475$	−558.484	−	339.994i	−1.17576	−	0.715777i
$476$		0			0
$477$			146.969i			0.308112i
$478$		0			0
$479$			552.000i			1.15240i	0.817308	+	0.576200i	$0.195465\pi$
$479$			552.000i			1.15240i	−0.817308	+	0.576200i	$0.804535\pi$
$480$		0			0
$481$	288.000			0.598753
$482$		0			0
$483$	−279.242			−0.578140
$484$		0			0
$485$	−52.3068	−	93.0806i	−0.107849	−	0.191919i
$486$		0			0
$487$	−224.219			−0.460408			−0.230204	−	0.973142i	$-0.573939\pi$
$487$	−224.219			−0.460408			−0.230204	+	0.973142i	$0.573939\pi$
$488$		0			0
$489$	294.000			0.601227
$490$		0			0
$491$			479.479i			0.976535i	0.872694	+	0.488268i	$0.162371\pi$
$491$			479.479i			0.976535i	−0.872694	+	0.488268i	$0.837629\pi$
$492$		0			0
$493$		−	744.645i		−	1.51044i
$494$		0			0
$495$	64.0625	+	114.000i	0.129419	+	0.230303i
$496$		0			0
$497$			896.875i			1.80458i
$498$		0			0
$499$		−	287.687i		−	0.576528i	−0.957551	−	0.288264i	$-0.906922\pi$
$499$		−	287.687i		−	0.576528i	0.957551	−	0.288264i	$-0.0930780\pi$
$500$		0			0
$501$	−706.142			−1.40946
$502$		0			0
$503$	−715.364			−1.42220			−0.711098	−	0.703093i	$-0.751802\pi$
$503$	−715.364			−1.42220			−0.711098	+	0.703093i	$0.751802\pi$
$504$		0			0
$505$	−304.000	+	170.833i	−0.601980	+	0.338284i
$506$		0			0
$507$	−526.640			−1.03874
$508$		0			0
$509$	802.037			1.57571			0.787856	−	0.615860i	$-0.211191\pi$
$509$	802.037			1.57571			0.787856	+	0.615860i	$0.211191\pi$
$510$		0			0
$511$		−	1140.00i		−	2.23092i
$512$		0			0
$513$			768.750i			1.49854i
$514$		0			0
$515$	139.621	−	78.4602i	0.271109	−	0.152350i
$516$		0			0
$517$		−	93.0806i		−	0.180040i
$518$		0			0
$519$		−	60.0000i		−	0.115607i
$520$		0			0
$521$	102.000			0.195777			0.0978887	−	0.995197i	$-0.468791\pi$
$521$	102.000			0.195777			0.0978887	+	0.995197i	$0.468791\pi$
$522$		0			0
$523$	−575.630			−1.10063			−0.550316	−	0.834957i	$-0.685493\pi$
$523$	−575.630			−1.10063			−0.550316	+	0.834957i	$0.685493\pi$
$524$		0			0
$525$	−339.994	+	558.484i	−0.647608	+	1.06378i
$526$		0			0
$527$	85.4166			0.162081
$528$		0			0
$529$	−415.000			−0.784499
$530$		0			0
$531$			130.767i			0.246265i
$532$		0			0
$533$			470.302i			0.882368i
$534$		0			0
$535$	−651.302	+	366.000i	−1.21739	+	0.684112i
$536$		0			0
$537$		−	619.271i		−	1.15320i
$538$		0			0
$539$			566.657i			1.05131i
$540$		0			0
$541$	836.909			1.54697			0.773483	−	0.633817i	$-0.218513\pi$
$541$	836.909			1.54697			0.773483	+	0.633817i	$0.218513\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	798.000	−	448.437i	1.46422	−	0.822821i
$546$		0			0
$547$	560.933			1.02547			0.512736	−	0.858546i	$-0.328632\pi$
$547$	560.933			1.02547			0.512736	+	0.858546i	$0.328632\pi$
$548$		0			0
$549$	78.4602			0.142915
$550$		0			0
$551$		−	912.000i		−	1.65517i
$552$		0			0
$553$			1067.71i			1.93076i
$554$		0			0
$555$	88.1816	+	156.920i	0.158886	+	0.282739i
$556$		0			0
$557$			788.736i			1.41604i	0.706191	+	0.708021i	$0.250412\pi$
$557$			788.736i			1.41604i	−0.706191	+	0.708021i	$0.749588\pi$
$558$		0			0
$559$			1104.00i			1.97496i
$560$		0			0
$561$	456.000			0.812834
$562$		0			0
$563$	149.419			0.265398			0.132699	−	0.991156i	$-0.457636\pi$
$563$	149.419			0.265398			0.132699	+	0.991156i	$0.457636\pi$
$564$		0			0
$565$	−418.454	−	744.645i	−0.740627	−	1.31796i
$566$		0			0
$567$	480.469			0.847387
$568$		0			0
$569$	144.000			0.253076			0.126538	−	0.991962i	$-0.459614\pi$
$569$	144.000			0.253076			0.126538	+	0.991962i	$0.459614\pi$
$570$		0			0
$571$			496.914i			0.870253i	0.900369	+	0.435127i	$0.143296\pi$
$571$			496.914i			0.870253i	−0.900369	+	0.435127i	$0.856704\pi$
$572$		0			0
$573$			58.7878i			0.102596i
$574$		0			0
$575$	138.802	−	228.000i	0.241395	−	0.396522i
$576$		0			0
$577$		−	384.375i		−	0.666161i	−0.942898	−	0.333080i	$-0.891912\pi$
$577$		−	384.375i		−	0.666161i	0.942898	−	0.333080i	$-0.108088\pi$
$578$		0			0
$579$		−	261.534i		−	0.451699i
$580$		0			0
$581$	−183.074			−0.315101
$582$		0			0
$583$	427.083			0.732561
$584$		0			0
$585$	144.000	+	256.250i	0.246154	+	0.438034i
$586$		0			0
$587$	869.569			1.48138			0.740689	−	0.671848i	$-0.234499\pi$
$587$	869.569			1.48138			0.740689	+	0.671848i	$0.234499\pi$
$588$		0			0
$589$	104.614			0.177612
$590$		0			0
$591$		−	840.000i		−	1.42132i
$592$		0			0
$593$		−	213.542i		−	0.360104i	−0.983657	−	0.180052i	$-0.942373\pi$
$593$		−	213.542i		−	0.360104i	0.983657	−	0.180052i	$-0.0576266\pi$
$594$		0			0
$595$	−558.484	−	993.829i	−0.938628	−	1.67030i
$596$		0			0
$597$			460.504i			0.771364i
$598$		0			0
$599$		−	300.000i		−	0.500835i	−0.968138	−	0.250417i	$-0.919432\pi$
$599$		−	300.000i		−	0.500835i	0.968138	−	0.250417i	$-0.0805678\pi$
$600$		0			0
$601$	−740.000			−1.23128			−0.615641	−	0.788027i	$-0.711103\pi$
$601$	−740.000			−1.23128			−0.615641	+	0.788027i	$0.711103\pi$
$602$		0			0
$603$	−22.0454			−0.0365595
$604$		0			0
$605$	−196.150	+	110.227i	−0.324216	+	0.182193i
$606$		0			0
$607$	−608.593			−1.00263			−0.501313	−	0.865266i	$-0.667149\pi$
$607$	−608.593			−1.00263			−0.501313	+	0.865266i	$0.667149\pi$
$608$		0			0
$609$	−912.000			−1.49754
$610$		0			0
$611$		−	209.227i		−	0.342434i
$612$		0			0
$613$		−	431.110i		−	0.703279i	−0.936135	−	0.351640i	$-0.885624\pi$
$613$		−	431.110i		−	0.703279i	0.936135	−	0.351640i	$-0.114376\pi$
$614$		0			0
$615$	−256.250	+	144.000i	−0.416666	+	0.234146i
$616$		0			0
$617$			106.771i			0.173048i	0.996250	+	0.0865241i	$0.0275760\pi$
$617$			106.771i			0.173048i	−0.996250	+	0.0865241i	$0.972424\pi$
$618$		0			0
$619$		−	183.074i		−	0.295757i	−0.989006	−	0.147879i	$-0.952755\pi$
$619$		−	183.074i		−	0.295757i	0.989006	−	0.147879i	$-0.0472445\pi$
$620$		0			0
$621$	−313.841			−0.505380
$622$		0			0
$623$	1601.56			2.57073
$624$		0			0
$625$	−287.000	−	555.208i	−0.459200	−	0.888333i
$626$		0			0
$627$	558.484			0.890724
$628$		0			0
$629$	−313.841			−0.498952
$630$		0			0
$631$		−	320.000i		−	0.507132i	−0.967318	−	0.253566i	$-0.918397\pi$
$631$		−	320.000i		−	0.507132i	0.967318	−	0.253566i	$-0.0816034\pi$
$632$		0			0
$633$		−	64.0625i		−	0.101205i
$634$		0			0
$635$	−232.702	+	130.767i	−0.366459	+	0.205932i
$636$		0			0
$637$			1273.73i			1.99958i
$638$		0			0
$639$			252.000i			0.394366i
$640$		0			0
$641$	−420.000			−0.655226			−0.327613	−	0.944812i	$-0.606244\pi$
$641$	−420.000			−0.655226			−0.327613	+	0.944812i	$0.606244\pi$
$642$		0			0
$643$	1045.93			1.62664			0.813322	−	0.581814i	$-0.197657\pi$
$643$	1045.93			1.62664			0.813322	+	0.581814i	$0.197657\pi$
$644$		0			0
$645$	−601.528	+	338.030i	−0.932602	+	0.524077i
$646$		0			0
$647$	544.531			0.841624			0.420812	−	0.907148i	$-0.361745\pi$
$647$	544.531			0.841624			0.420812	+	0.907148i	$0.361745\pi$
$648$		0			0
$649$	380.000			0.585516
$650$		0			0
$651$		−	104.614i		−	0.160697i
$652$		0			0
$653$		−	509.494i		−	0.780236i	−0.920765	−	0.390118i	$-0.872434\pi$
$653$		−	509.494i		−	0.780236i	0.920765	−	0.390118i	$-0.127566\pi$
$654$		0			0
$655$	234.896	+	418.000i	0.358619	+	0.638168i
$656$		0			0
$657$		−	320.312i		−	0.487538i
$658$		0			0
$659$			514.350i			0.780501i	0.920709	+	0.390250i	$0.127612\pi$
$659$			514.350i			0.780501i	−0.920709	+	0.390250i	$0.872388\pi$
$660$		0			0
$661$	26.1534			0.0395664			0.0197832	−	0.999804i	$-0.493702\pi$
$661$	26.1534			0.0395664			0.0197832	+	0.999804i	$0.493702\pi$
$662$		0			0
$663$	1025.00			1.54600
$664$		0			0
$665$	−684.000	−	1217.19i	−1.02857	−	1.83036i
$666$		0			0
$667$	372.322			0.558205
$668$		0			0
$669$	549.221			0.820959
$670$		0			0
$671$		−	228.000i		−	0.339791i
$672$		0			0
$673$			533.854i			0.793245i	0.917982	+	0.396623i	$0.129818\pi$
$673$			533.854i			0.793245i	−0.917982	+	0.396623i	$0.870182\pi$
$674$		0			0
$675$	−382.120	+	627.681i	−0.566104	+	0.929898i
$676$		0			0
$677$			1116.97i			1.64988i	0.565222	+	0.824939i	$0.308791\pi$
$677$			1116.97i			1.64988i	−0.565222	+	0.824939i	$0.691209\pi$
$678$		0			0
$679$		−	228.000i		−	0.335788i
$680$		0			0
$681$	198.000			0.290749
$682$		0			0
$683$	−1256.59			−1.83981			−0.919904	−	0.392145i	$-0.871733\pi$
$683$	−1256.59			−1.83981			−0.919904	+	0.392145i	$0.871733\pi$
$684$		0			0
$685$	−313.841	−	558.484i	−0.458162	−	0.815305i
$686$		0			0
$687$	256.250			0.372998
$688$		0			0
$689$	960.000			1.39332
$690$		0			0
$691$		−	1072.29i		−	1.55179i	−0.630860	−	0.775897i	$-0.717298\pi$
$691$		−	1072.29i		−	1.55179i	0.630860	−	0.775897i	$-0.282702\pi$
$692$		0			0
$693$			279.242i			0.402946i
$694$		0			0
$695$	576.562	+	1026.00i	0.829586	+	1.47626i
$696$		0			0
$697$		−	512.500i		−	0.735294i
$698$		0			0
$699$		−	784.602i		−	1.12246i
$700$		0			0
$701$	−43.5890			−0.0621812			−0.0310906	−	0.999517i	$-0.509898\pi$
$701$	−43.5890			−0.0621812			−0.0310906	+	0.999517i	$0.509898\pi$
$702$		0			0
$703$	−384.375			−0.546764
$704$		0			0
$705$	114.000	−	64.0625i	0.161702	−	0.0908688i
$706$		0			0
$707$	−744.645			−1.05325
$708$		0			0
$709$	−209.227			−0.295102			−0.147551	−	0.989054i	$-0.547139\pi$
$709$	−209.227			−0.295102			−0.147551	+	0.989054i	$0.547139\pi$
$710$		0			0
$711$			300.000i			0.421941i
$712$		0			0
$713$			42.7083i			0.0598995i
$714$		0			0
$715$	744.645	−	418.454i	1.04146	−	0.585251i
$716$		0			0
$717$		−	264.545i		−	0.368961i
$718$		0			0
$719$		−	612.000i		−	0.851182i	−0.904916	−	0.425591i	$-0.860066\pi$
$719$		−	612.000i		−	0.851182i	0.904916	−	0.425591i	$-0.139934\pi$
$720$		0			0
$721$	342.000			0.474341
$722$		0			0
$723$	−68.5857			−0.0948627
$724$		0			0
$725$	453.325	−	744.645i	0.625277	−	1.02710i
$726$		0			0
$727$	−1334.63			−1.83581			−0.917906	−	0.396799i	$-0.870121\pi$
$727$	−1334.63			−1.83581			−0.917906	+	0.396799i	$0.870121\pi$
$728$		0			0
$729$	783.000			1.07407
$730$		0			0
$731$		−	1203.06i		−	1.64577i
$732$		0			0
$733$			220.454i			0.300756i	0.988629	+	0.150378i	$0.0480491\pi$
$733$			220.454i			0.300756i	−0.988629	+	0.150378i	$0.951951\pi$
$734$		0			0
$735$	−694.010	+	390.000i	−0.944231	+	0.530612i
$736$		0			0
$737$			64.0625i			0.0869233i
$738$		0			0
$739$		−	1438.44i		−	1.94646i	−0.229826	−	0.973232i	$-0.573816\pi$
$739$		−	1438.44i		−	1.94646i	0.229826	−	0.973232i	$-0.426184\pi$
$740$		0			0
$741$	1255.36			1.69415
$742$		0			0
$743$	−138.802			−0.186813			−0.0934065	−	0.995628i	$-0.529776\pi$
$743$	−138.802			−0.186813			−0.0934065	+	0.995628i	$0.529776\pi$
$744$		0			0
$745$	418.000	−	234.896i	0.561074	−	0.315296i
$746$		0			0
$747$	−51.4393			−0.0688612
$748$		0			0
$749$	−1595.36			−2.12998
$750$		0			0
$751$		−	608.000i		−	0.809587i	−0.914408	−	0.404794i	$-0.867343\pi$
$751$		−	608.000i		−	0.809587i	0.914408	−	0.404794i	$-0.132657\pi$
$752$		0			0
$753$			619.271i			0.822404i
$754$		0			0
$755$	391.918	+	697.424i	0.519097	+	0.923740i
$756$		0			0
$757$		−	592.777i		−	0.783060i	−0.920165	−	0.391530i	$-0.871946\pi$
$757$		−	592.777i		−	0.783060i	0.920165	−	0.391530i	$-0.128054\pi$
$758$		0			0
$759$			228.000i			0.300395i
$760$		0			0
$761$	642.000			0.843627			0.421813	−	0.906683i	$-0.361394\pi$
$761$	642.000			0.843627			0.421813	+	0.906683i	$0.361394\pi$
$762$		0			0
$763$	1954.69			2.56185
$764$		0			0
$765$	−156.920	−	279.242i	−0.205125	−	0.365022i
$766$		0			0
$767$	854.166			1.11365
$768$		0			0
$769$	−230.000			−0.299090			−0.149545	−	0.988755i	$-0.547781\pi$
$769$	−230.000			−0.299090			−0.149545	+	0.988755i	$0.547781\pi$
$770$		0			0
$771$			732.295i			0.949799i
$772$		0			0
$773$		−	88.1816i		−	0.114077i	−0.998372	−	0.0570386i	$-0.981834\pi$
$773$		−	88.1816i		−	0.114077i	0.998372	−	0.0570386i	$-0.0181658\pi$
$774$		0			0
$775$	85.4166	+	52.0000i	0.110215	+	0.0670968i
$776$		0			0
$777$			384.375i			0.494691i
$778$		0			0
$779$		−	627.681i		−	0.805753i
$780$		0			0
$781$	732.295			0.937638
$782$		0			0
$783$	−1025.00			−1.30907
$784$		0			0
$785$	84.0000	+	149.479i	0.107006	+	0.190419i
$786$		0			0
$787$	232.702			0.295682			0.147841	−	0.989011i	$-0.452768\pi$
$787$	232.702			0.295682			0.147841	+	0.989011i	$0.452768\pi$
$788$		0			0
$789$	706.142			0.894983
$790$		0			0
$791$		−	1824.00i		−	2.30594i
$792$		0			0
$793$		−	512.500i		−	0.646280i
$794$		0			0
$795$	293.939	+	523.068i	0.369734	+	0.657947i
$796$		0			0
$797$		−	989.594i		−	1.24165i	−0.783950	−	0.620824i	$-0.786798\pi$
$797$		−	989.594i		−	1.24165i	0.783950	−	0.620824i	$-0.213202\pi$
$798$		0			0
$799$			228.000i			0.285357i
$800$		0			0
$801$	450.000			0.561798
$802$		0			0
$803$	−930.806			−1.15916
$804$		0			0
$805$	496.914	−	279.242i	0.617285	−	0.346884i
$806$		0			0
$807$	−149.479			−0.185228
$808$		0			0
$809$	−906.000			−1.11990			−0.559951	−	0.828526i	$-0.689180\pi$
$809$	−906.000			−1.11990			−0.559951	+	0.828526i	$0.689180\pi$
$810$		0			0
$811$		−	1281.52i		−	1.58017i	−0.612999	−	0.790084i	$-0.710037\pi$
$811$		−	1281.52i		−	1.58017i	0.612999	−	0.790084i	$-0.289963\pi$
$812$		0			0
$813$		−	1048.38i		−	1.28952i
$814$		0			0
$815$	−523.177	+	294.000i	−0.641935	+	0.360736i
$816$		0			0
$817$		−	1473.44i		−	1.80347i
$818$		0			0
$819$			627.681i			0.766400i
$820$		0			0
$821$	409.737			0.499070			0.249535	−	0.968366i	$-0.419722\pi$
$821$	409.737			0.499070			0.249535	+	0.968366i	$0.419722\pi$
$822$		0			0
$823$	−373.698			−0.454068			−0.227034	−	0.973887i	$-0.572903\pi$
$823$	−373.698			−0.454068			−0.227034	+	0.973887i	$0.572903\pi$
$824$		0			0
$825$	456.000	+	277.604i	0.552727	+	0.336490i
$826$		0			0
$827$	−169.015			−0.204371			−0.102185	−	0.994765i	$-0.532584\pi$
$827$	−169.015			−0.204371			−0.102185	+	0.994765i	$0.532584\pi$
$828$		0			0
$829$	−130.767			−0.157741			−0.0788703	−	0.996885i	$-0.525131\pi$
$829$	−130.767			−0.157741			−0.0788703	+	0.996885i	$0.525131\pi$
$830$		0			0
$831$		−	180.000i		−	0.216606i
$832$		0			0
$833$		−	1388.02i		−	1.66629i
$834$		0			0
$835$	1256.59	−	706.142i	1.50490	−	0.845679i
$836$		0			0
$837$		−	117.576i		−	0.140473i
$838$		0			0
$839$			816.000i			0.972586i	0.873796	+	0.486293i	$0.161651\pi$
$839$			816.000i			0.972586i	−0.873796	+	0.486293i	$0.838349\pi$
$840$		0			0
$841$	375.000			0.445898
$842$		0			0
$843$	499.696			0.592759
$844$		0			0
$845$	937.163	−	526.640i	1.10907	−	0.623243i
$846$		0			0
$847$	−480.469			−0.567259
$848$		0			0
$849$	990.000			1.16608
$850$		0			0
$851$		−	156.920i		−	0.184395i
$852$		0			0
$853$			450.706i			0.528378i	0.964471	+	0.264189i	$0.0851042\pi$
$853$			450.706i			0.528378i	−0.964471	+	0.264189i	$0.914896\pi$
$854$		0			0
$855$	−192.187	−	342.000i	−0.224781	−	0.400000i
$856$		0			0
$857$			512.500i			0.598016i	0.954251	+	0.299008i	$0.0966557\pi$
$857$			512.500i			0.598016i	−0.954251	+	0.299008i	$0.903344\pi$
$858$		0			0
$859$		−	496.914i		−	0.578480i	−0.957257	−	0.289240i	$-0.906597\pi$
$859$		−	496.914i		−	0.578480i	0.957257	−	0.289240i	$-0.0934026\pi$
$860$		0			0
$861$	−627.681			−0.729014
$862$		0			0
$863$	−53.3854			−0.0618602			−0.0309301	−	0.999522i	$-0.509847\pi$
$863$	−53.3854			−0.0618602			−0.0309301	+	0.999522i	$0.509847\pi$
$864$		0			0
$865$	60.0000	+	106.771i	0.0693642	+	0.123434i
$866$		0			0
$867$	−409.065			−0.471816
$868$		0			0
$869$	871.780			1.00320
$870$		0			0
$871$			144.000i			0.165327i
$872$		0			0
$873$		−	64.0625i		−	0.0733820i
$874$		0			0
$875$	46.5403	−	1333.82i	0.0531889	−	1.52437i
$876$		0			0
$877$			230.252i			0.262545i	0.991346	+	0.131273i	$0.0419063\pi$
$877$			230.252i			0.262545i	−0.991346	+	0.131273i	$0.958094\pi$
$878$		0			0
$879$			396.000i			0.450512i
$880$		0			0
$881$	−1320.00			−1.49830			−0.749149	−	0.662402i	$-0.769537\pi$
$881$	−1320.00			−1.49830			−0.749149	+	0.662402i	$0.769537\pi$
$882$		0			0
$883$	1050.83			1.19007			0.595035	−	0.803700i	$-0.297138\pi$
$883$	1050.83			1.19007			0.595035	+	0.803700i	$0.297138\pi$
$884$		0			0
$885$	261.534	+	465.403i	0.295519	+	0.525879i
$886$		0			0
$887$	117.448			0.132410			0.0662051	−	0.997806i	$-0.478911\pi$
$887$	117.448			0.132410			0.0662051	+	0.997806i	$0.478911\pi$
$888$		0			0
$889$	−570.000			−0.641170
$890$		0			0
$891$		−	392.301i		−	0.440293i
$892$		0			0
$893$			279.242i			0.312701i
$894$		0			0
$895$	619.271	+	1102.00i	0.691922	+	1.23128i
$896$		0			0
$897$			512.500i			0.571349i
$898$		0			0
$899$			139.485i			0.155155i
$900$		0			0
$901$	−1046.14			−1.16108
$902$		0			0
$903$	−1473.44			−1.63171
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−585.428			−0.645455			−0.322728	−	0.946492i	$-0.604600\pi$
$907$	−585.428			−0.645455			−0.322728	+	0.946492i	$0.604600\pi$
$908$		0			0
$909$	−209.227			−0.230173
$910$		0			0
$911$			564.000i			0.619100i	0.950883	+	0.309550i	$0.100178\pi$
$911$			564.000i			0.619100i	−0.950883	+	0.309550i	$0.899822\pi$
$912$		0			0
$913$			149.479i			0.163723i
$914$		0			0
$915$	279.242	−	156.920i	0.305182	−	0.171498i
$916$		0			0
$917$			1023.89i			1.11656i
$918$		0			0
$919$			1124.00i			1.22307i	0.791218	+	0.611534i	$0.209447\pi$
$919$			1124.00i			1.22307i	−0.791218	+	0.611534i	$0.790553\pi$
$920$		0			0
$921$	−1230.00			−1.33550
$922$		0			0
$923$	1646.06			1.78338
$924$		0			0
$925$	−313.841	−	191.060i	−0.339287	−	0.206552i
$926$		0			0
$927$	96.0937			0.103661
$928$		0			0
$929$	−276.000			−0.297094			−0.148547	−	0.988905i	$-0.547460\pi$
$929$	−276.000			−0.297094			−0.148547	+	0.988905i	$0.547460\pi$
$930$		0			0
$931$		−	1699.97i		−	1.82596i
$932$		0			0
$933$		−	999.392i		−	1.07116i
$934$		0			0
$935$	−811.458	+	456.000i	−0.867869	+	0.487701i
$936$		0			0
$937$		−	619.271i		−	0.660908i	−0.943822	−	0.330454i	$-0.892798\pi$
$937$		−	619.271i		−	0.660908i	0.943822	−	0.330454i	$-0.107202\pi$
$938$		0			0
$939$		−	732.295i		−	0.779867i
$940$		0			0
$941$	−1115.88			−1.18584			−0.592921	−	0.805260i	$-0.702026\pi$
$941$	−1115.88			−1.18584			−0.592921	+	0.805260i	$0.702026\pi$
$942$		0			0
$943$	256.250			0.271739
$944$		0			0
$945$	−1368.00	+	768.750i	−1.44762	+	0.813492i
$946$		0			0
$947$	−364.974			−0.385400			−0.192700	−	0.981258i	$-0.561724\pi$
$947$	−364.974			−0.385400			−0.192700	+	0.981258i	$0.561724\pi$
$948$		0			0
$949$	−2092.27			−2.20471
$950$		0			0
$951$			456.000i			0.479495i
$952$		0			0
$953$		−	341.667i		−	0.358517i	−0.983802	−	0.179258i	$-0.942630\pi$
$953$		−	341.667i		−	0.358517i	0.983802	−	0.179258i	$-0.0573698\pi$
$954$		0			0
$955$	−58.7878	−	104.614i	−0.0615579	−	0.109543i
$956$		0			0
$957$			744.645i			0.778103i
$958$		0			0
$959$		−	1368.00i		−	1.42649i
$960$		0			0
$961$	945.000			0.983351
$962$		0			0
$963$	−448.257			−0.465479
$964$		0			0
$965$	261.534	+	465.403i	0.271020	+	0.482283i
$966$		0			0
$967$	1548.18			1.60101			0.800505	−	0.599326i	$-0.204565\pi$
$967$	1548.18			1.60101			0.800505	+	0.599326i	$0.204565\pi$
$968$		0			0
$969$	−1368.00			−1.41176
$970$		0			0
$971$			828.191i			0.852926i	0.904505	+	0.426463i	$0.140241\pi$
$971$			828.191i			0.852926i	−0.904505	+	0.426463i	$0.859759\pi$
$972$		0			0
$973$			2513.18i			2.58292i
$974$		0			0
$975$	1025.00	+	624.000i	1.05128	+	0.640000i
$976$		0			0
$977$			1558.85i			1.59555i	0.602955	+	0.797776i	$0.293990\pi$
$977$			1558.85i			1.59555i	−0.602955	+	0.797776i	$0.706010\pi$
$978$		0			0
$979$		−	1307.67i		−	1.33572i
$980$		0			0
$981$	549.221			0.559859
$982$		0			0
$983$	843.489			0.858076			0.429038	−	0.903286i	$-0.358853\pi$
$983$	843.489			0.858076			0.429038	+	0.903286i	$0.358853\pi$
$984$		0			0
$985$	840.000	+	1494.79i	0.852792	+	1.51755i
$986$		0			0
$987$	279.242			0.282920
$988$		0			0
$989$	601.528			0.608218
$990$		0			0
$991$		−	208.000i		−	0.209889i	−0.994478	−	0.104945i	$-0.966534\pi$
$991$		−	208.000i		−	0.209889i	0.994478	−	0.104945i	$-0.0334665\pi$
$992$		0			0
$993$		−	704.687i		−	0.709655i
$994$		0			0
$995$	−460.504	−	819.473i	−0.462818	−	0.823591i
$996$		0			0
$997$			950.402i			0.953262i	0.879104	+	0.476631i	$0.158142\pi$
$997$			950.402i			0.953262i	−0.879104	+	0.476631i	$0.841858\pi$
$998$		0			0
$999$			432.000i			0.432432i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1280.3.h.i.1279.6		8
4.3	odd	2	inner	1280.3.h.i.1279.2		8
5.4	even	2	inner	1280.3.h.i.1279.1		8
8.3	odd	2	inner	1280.3.h.i.1279.7		8
8.5	even	2	inner	1280.3.h.i.1279.3		8
16.3	odd	4		320.3.e.a.159.4	yes	8
16.5	even	4		320.3.e.a.159.1	✓	8
16.11	odd	4		320.3.e.a.159.5	yes	8
16.13	even	4		320.3.e.a.159.8	yes	8
20.19	odd	2	inner	1280.3.h.i.1279.5		8
40.19	odd	2	inner	1280.3.h.i.1279.4		8
40.29	even	2	inner	1280.3.h.i.1279.8		8
80.3	even	4		1600.3.g.h.351.7		8
80.13	odd	4		1600.3.g.h.351.2		8
80.19	odd	4		320.3.e.a.159.6	yes	8
80.27	even	4		1600.3.g.h.351.6		8
80.29	even	4		320.3.e.a.159.2	yes	8
80.37	odd	4		1600.3.g.h.351.3		8
80.43	even	4		1600.3.g.h.351.4		8
80.53	odd	4		1600.3.g.h.351.5		8
80.59	odd	4		320.3.e.a.159.3	yes	8
80.67	even	4		1600.3.g.h.351.1		8
80.69	even	4		320.3.e.a.159.7	yes	8
80.77	odd	4		1600.3.g.h.351.8		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
320.3.e.a.159.1	✓	8	16.5	even	4
320.3.e.a.159.2	yes	8	80.29	even	4
320.3.e.a.159.3	yes	8	80.59	odd	4
320.3.e.a.159.4	yes	8	16.3	odd	4
320.3.e.a.159.5	yes	8	16.11	odd	4
320.3.e.a.159.6	yes	8	80.19	odd	4
320.3.e.a.159.7	yes	8	80.69	even	4
320.3.e.a.159.8	yes	8	16.13	even	4
1280.3.h.i.1279.1		8	5.4	even	2	inner
1280.3.h.i.1279.2		8	4.3	odd	2	inner
1280.3.h.i.1279.3		8	8.5	even	2	inner
1280.3.h.i.1279.4		8	40.19	odd	2	inner
1280.3.h.i.1279.5		8	20.19	odd	2	inner
1280.3.h.i.1279.6		8	1.1	even	1	trivial
1280.3.h.i.1279.7		8	8.3	odd	2	inner
1280.3.h.i.1279.8		8	40.29	even	2	inner
1600.3.g.h.351.1		8	80.67	even	4
1600.3.g.h.351.2		8	80.13	odd	4
1600.3.g.h.351.3		8	80.37	odd	4
1600.3.g.h.351.4		8	80.43	even	4
1600.3.g.h.351.5		8	80.53	odd	4
1600.3.g.h.351.6		8	80.27	even	4
1600.3.g.h.351.7		8	80.3	even	4
1600.3.g.h.351.8		8	80.77	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:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1280.h (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+2.44949 q^{3} +(-4.35890 + 2.44949i) q^{5} -10.6771 q^{7} -3.00000 q^{9} +O(q^{10})\)	\(q+2.44949 q^{3} +(-4.35890 + 2.44949i) q^{5} -10.6771 q^{7} -3.00000 q^{9} +8.71780i q^{11} +19.5959i q^{13} +(-10.6771 + 6.00000i) q^{15} -21.3542i q^{17} -26.1534i q^{19} -26.1534 q^{21} +10.6771 q^{23} +(13.0000 - 21.3542i) q^{25} -29.3939 q^{27} +34.8712 q^{29} +4.00000i q^{31} +21.3542i q^{33} +(46.5403 - 26.1534i) q^{35} -14.6969i q^{37} +48.0000i q^{39} +24.0000 q^{41} +56.3383 q^{43} +(13.0767 - 7.34847i) q^{45} -10.6771 q^{47} +65.0000 q^{49} -52.3068i q^{51} -48.9898i q^{53} +(-21.3542 - 38.0000i) q^{55} -64.0625i q^{57} -43.5890i q^{59} -26.1534 q^{61} +32.0312 q^{63} +(-48.0000 - 85.4166i) q^{65} +7.34847 q^{67} +26.1534 q^{69} -84.0000i q^{71} +106.771i q^{73} +(31.8434 - 52.3068i) q^{75} -93.0806i q^{77} -100.000i q^{79} -45.0000 q^{81} +17.1464 q^{83} +(52.3068 + 93.0806i) q^{85} +85.4166 q^{87} -150.000 q^{89} -209.227i q^{91} +9.79796i q^{93} +(64.0625 + 114.000i) q^{95} +21.3542i q^{97} -26.1534i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 24 q^{9}+O(q^{10}) \) 8 * q - 24 * q^9	\( 8 q - 24 q^{9} + 104 q^{25} + 192 q^{41} + 520 q^{49} - 384 q^{65} - 360 q^{81} - 1200 q^{89}+O(q^{100}) \) 8 * q - 24 * q^9 + 104 * q^25 + 192 * q^41 + 520 * q^49 - 384 * q^65 - 360 * q^81 - 1200 * q^89

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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