LMFDB - Embedded newform 320.6.f.c.289.14

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [320,6,Mod(289,320)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(320, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("320.289");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 320 = 2^{6} \cdot 5 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	320.f (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:	$51.3228223402$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 39122x^{12} + 391243971x^{8} + 75462898750x^{4} + 18538406640625 $ x^16 - 39122x^12 + 391243971x^8 + 75462898750*x^4 + 18538406640625
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{42}\cdot 5^{6} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			289.14
Root			$-11.8760 + 0.526271i$ of defining polynomial
Character	$\chi$	$=$	320.289
Dual form			320.6.f.c.289.13

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+24.8046 q^{3} +(-53.4447 + 16.3911i) q^{5} +100.281i q^{7} +372.267 q^{9} +O(q^{10})$	$q+24.8046 q^{3} +(-53.4447 + 16.3911i) q^{5} +100.281i q^{7} +372.267 q^{9} +5.36667i q^{11} -506.854 q^{13} +(-1325.67 + 406.573i) q^{15} +1454.37i q^{17} -722.433i q^{19} +2487.43i q^{21} -3086.22i q^{23} +(2587.67 - 1752.03i) q^{25} +3206.40 q^{27} +7298.39i q^{29} -8609.68 q^{31} +133.118i q^{33} +(-1643.72 - 5359.50i) q^{35} -5276.52 q^{37} -12572.3 q^{39} -18175.5 q^{41} -8340.95 q^{43} +(-19895.7 + 6101.84i) q^{45} +22120.2i q^{47} +6750.67 q^{49} +36075.1i q^{51} -13363.7 q^{53} +(-87.9654 - 286.820i) q^{55} -17919.6i q^{57} -10958.8i q^{59} -10982.0i q^{61} +37331.4i q^{63} +(27088.7 - 8307.88i) q^{65} -13007.5 q^{67} -76552.4i q^{69} +36009.0 q^{71} +64286.0i q^{73} +(64186.0 - 43458.3i) q^{75} -538.177 q^{77} +48638.8 q^{79} -10927.3 q^{81} -49784.4 q^{83} +(-23838.7 - 77728.6i) q^{85} +181033. i q^{87} +7764.40 q^{89} -50828.0i q^{91} -213559. q^{93} +(11841.4 + 38610.2i) q^{95} -36848.8i q^{97} +1997.83i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 1904 q^{9}+O(q^{10}) $ 16 * q + 1904 * q^9	$ 16 q + 1904 q^{9} + 880 q^{25} - 100352 q^{41} + 128272 q^{49} + 149760 q^{65} - 154576 q^{81} - 459296 q^{89}+O(q^{100}) $ 16 * q + 1904 * q^9 + 880 * q^25 - 100352 * q^41 + 128272 * q^49 + 149760 * q^65 - 154576 * q^81 - 459296 * q^89

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/320\mathbb{Z}\right)^\times$.

$n$	$191$	$257$	$261$
$\chi(n)$	$1$	$-1$	$-1$

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
$3$	24.8046			1.59121			0.795607	−	0.605813i	$-0.207152\pi$
$3$	24.8046			1.59121			0.795607	+	0.605813i	$0.207152\pi$
$4$		0			0
$5$	−53.4447	+	16.3911i	−0.956047	+	0.293212i
$5$	−53.4447	+	16.3911i	−0.956047	+	0.293212i
$6$		0			0
$7$			100.281i			0.773526i	0.922179	+	0.386763i	$0.126407\pi$
$7$			100.281i			0.773526i	−0.922179	+	0.386763i	$0.873593\pi$
$8$		0			0
$9$	372.267			1.53196
$10$		0			0
$11$			5.36667i			0.0133728i	0.999978	+	0.00668641i	$0.00212837\pi$
$11$			5.36667i			0.0133728i	−0.999978	+	0.00668641i	$0.997872\pi$
$12$		0			0
$13$	−506.854			−0.831811			−0.415906	−	0.909408i	$-0.636535\pi$
$13$	−506.854			−0.831811			−0.415906	+	0.909408i	$0.636535\pi$
$14$		0			0
$15$	−1325.67	+	406.573i	−1.52128	+	0.466563i
$16$		0			0
$17$			1454.37i			1.22055i	0.792191	+	0.610273i	$0.208940\pi$
$17$			1454.37i			1.22055i	−0.792191	+	0.610273i	$0.791060\pi$
$18$		0			0
$19$		−	722.433i		−	0.459107i	−0.973296	−	0.229553i	$-0.926273\pi$
$19$		−	722.433i		−	0.459107i	0.973296	−	0.229553i	$-0.0737266\pi$
$20$		0			0
$21$			2487.43i			1.23084i
$22$		0			0
$23$		−	3086.22i		−	1.21649i	−0.793750	−	0.608244i	$-0.791874\pi$
$23$		−	3086.22i		−	1.21649i	0.793750	−	0.608244i	$-0.208126\pi$
$24$		0			0
$25$	2587.67	−	1752.03i	0.828053	−	0.560649i
$26$		0			0
$27$	3206.40			0.846465
$28$		0			0
$29$			7298.39i			1.61151i	0.592251	+	0.805753i	$0.298239\pi$
$29$			7298.39i			1.61151i	−0.592251	+	0.805753i	$0.701761\pi$
$30$		0			0
$31$	−8609.68			−1.60910			−0.804549	−	0.593886i	$-0.797593\pi$
$31$	−8609.68			−1.60910			−0.804549	+	0.593886i	$0.797593\pi$
$32$		0			0
$33$			133.118i			0.0212790i
$34$		0			0
$35$	−1643.72	−	5359.50i	−0.226807	−	0.739527i
$36$		0			0
$37$	−5276.52			−0.633641			−0.316821	−	0.948485i	$-0.602615\pi$
$37$	−5276.52			−0.633641			−0.316821	+	0.948485i	$0.602615\pi$
$38$		0			0
$39$	−12572.3			−1.32359
$40$		0			0
$41$	−18175.5			−1.68860			−0.844301	−	0.535869i	$-0.819984\pi$
$41$	−18175.5			−1.68860			−0.844301	+	0.535869i	$0.819984\pi$
$42$		0			0
$43$	−8340.95			−0.687930			−0.343965	−	0.938982i	$-0.611770\pi$
$43$	−8340.95			−0.687930			−0.343965	+	0.938982i	$0.611770\pi$
$44$		0			0
$45$	−19895.7	+	6101.84i	−1.46463	+	0.449190i
$46$		0			0
$47$			22120.2i			1.46064i	0.683103	+	0.730322i	$0.260630\pi$
$47$			22120.2i			1.46064i	−0.683103	+	0.730322i	$0.739370\pi$
$48$		0			0
$49$	6750.67			0.401658
$50$		0			0
$51$			36075.1i			1.94215i
$52$		0			0
$53$	−13363.7			−0.653489			−0.326745	−	0.945113i	$-0.605952\pi$
$53$	−13363.7			−0.653489			−0.326745	+	0.945113i	$0.605952\pi$
$54$		0			0
$55$	−87.9654	−	286.820i	−0.00392108	−	0.0127851i
$56$		0			0
$57$		−	17919.6i		−	0.730537i
$58$		0			0
$59$		−	10958.8i		−	0.409859i	−0.978777	−	0.204929i	$-0.934304\pi$
$59$		−	10958.8i		−	0.409859i	0.978777	−	0.204929i	$-0.0656965\pi$
$60$		0			0
$61$		−	10982.0i		−	0.377883i	−0.981988	−	0.188941i	$-0.939494\pi$
$61$		−	10982.0i		−	0.377883i	0.981988	−	0.188941i	$-0.0605056\pi$
$62$		0			0
$63$			37331.4i			1.18501i
$64$		0			0
$65$	27088.7	−	8307.88i	0.795251	−	0.243897i
$66$		0			0
$67$	−13007.5			−0.354003			−0.177002	−	0.984211i	$-0.556640\pi$
$67$	−13007.5			−0.354003			−0.177002	+	0.984211i	$0.556640\pi$
$68$		0			0
$69$		−	76552.4i		−	1.93569i
$70$		0			0
$71$	36009.0			0.847744			0.423872	−	0.905722i	$-0.360671\pi$
$71$	36009.0			0.847744			0.423872	+	0.905722i	$0.360671\pi$
$72$		0			0
$73$			64286.0i			1.41192i	0.708253	+	0.705959i	$0.249484\pi$
$73$			64286.0i			1.41192i	−0.708253	+	0.705959i	$0.750516\pi$
$74$		0			0
$75$	64186.0	−	43458.3i	1.31761	−	0.892113i
$76$		0			0
$77$	−538.177			−0.0103442
$78$		0			0
$79$	48638.8			0.876830			0.438415	−	0.898773i	$-0.355540\pi$
$79$	48638.8			0.876830			0.438415	+	0.898773i	$0.355540\pi$
$80$		0			0
$81$	−10927.3			−0.185055
$82$		0			0
$83$	−49784.4			−0.793228			−0.396614	−	0.917985i	$-0.629815\pi$
$83$	−49784.4			−0.793228			−0.396614	+	0.917985i	$0.629815\pi$
$84$		0			0
$85$	−23838.7	−	77728.6i	−0.357879	−	1.16690i
$86$		0			0
$87$			181033.i			2.56425i
$88$		0			0
$89$	7764.40			0.103904			0.0519521	−	0.998650i	$-0.483456\pi$
$89$	7764.40			0.103904			0.0519521	+	0.998650i	$0.483456\pi$
$90$		0			0
$91$		−	50828.0i		−	0.643427i
$92$		0			0
$93$	−213559.			−2.56042
$94$		0			0
$95$	11841.4	+	38610.2i	0.134616	+	0.438928i
$96$		0			0
$97$		−	36848.8i		−	0.397644i	−0.980036	−	0.198822i	$-0.936288\pi$
$97$		−	36848.8i		−	0.397644i	0.980036	−	0.198822i	$-0.0637116\pi$
$98$		0			0
$99$			1997.83i			0.0204867i
$100$		0			0
$101$		−	174557.i		−	1.70268i	−0.524611	−	0.851342i	$-0.675789\pi$
$101$		−	174557.i		−	1.70268i	0.524611	−	0.851342i	$-0.324211\pi$
$102$		0			0
$103$			172505.i			1.60217i	0.598551	+	0.801085i	$0.295743\pi$
$103$			172505.i			1.60217i	−0.598551	+	0.801085i	$0.704257\pi$
$104$		0			0
$105$	−40771.7	−	132940.i	−0.360899	−	1.17675i
$106$		0			0
$107$	155176.			1.31028			0.655140	−	0.755507i	$-0.272609\pi$
$107$	155176.			1.31028			0.655140	+	0.755507i	$0.272609\pi$
$108$		0			0
$109$			111055.i			0.895306i	0.894207	+	0.447653i	$0.147740\pi$
$109$			111055.i			0.895306i	−0.894207	+	0.447653i	$0.852260\pi$
$110$		0			0
$111$	−130882.			−1.00826
$112$		0			0
$113$			199647.i			1.47084i	0.677609	+	0.735422i	$0.263016\pi$
$113$			199647.i			1.47084i	−0.677609	+	0.735422i	$0.736984\pi$
$114$		0			0
$115$	50586.4	+	164942.i	0.356689	+	1.16302i
$116$		0			0
$117$	−188685.			−1.27430
$118$		0			0
$119$	−145847.			−0.944123
$120$		0			0
$121$	161022.			0.999821
$122$		0			0
$123$	−450836.			−2.68693
$124$		0			0
$125$	−109579.	+	136051.i	−0.627269	+	0.778803i
$126$		0			0
$127$		−	93300.5i		−	0.513304i	−0.966504	−	0.256652i	$-0.917381\pi$
$127$		−	93300.5i		−	0.513304i	0.966504	−	0.256652i	$-0.0826194\pi$
$128$		0			0
$129$	−206894.			−1.09464
$130$		0			0
$131$		−	174939.i		−	0.890655i	−0.895368	−	0.445328i	$-0.853087\pi$
$131$		−	174939.i		−	0.890655i	0.895368	−	0.445328i	$-0.146913\pi$
$132$		0			0
$133$	72446.5			0.355131
$134$		0			0
$135$	−171365.	+	52556.3i	−0.809260	+	0.248194i
$136$		0			0
$137$		−	127441.i		−	0.580106i	−0.957011	−	0.290053i	$-0.906327\pi$
$137$		−	127441.i		−	0.580106i	0.957011	−	0.290053i	$-0.0936730\pi$
$138$		0			0
$139$			393660.i			1.72816i	0.503352	+	0.864082i	$0.332100\pi$
$139$			393660.i			1.72816i	−0.503352	+	0.864082i	$0.667900\pi$
$140$		0			0
$141$			548682.i			2.32420i
$142$		0			0
$143$		−	2720.12i		−	0.0111237i
$144$		0			0
$145$	−119628.	−	390060.i	−0.472513	−	1.54068i
$146$		0			0
$147$	167447.			0.639124
$148$		0			0
$149$		−	167112.i		−	0.616656i	−0.951280	−	0.308328i	$-0.900231\pi$
$149$		−	167112.i		−	0.616656i	0.951280	−	0.308328i	$-0.0997694\pi$
$150$		0			0
$151$	−184305.			−0.657803			−0.328901	−	0.944364i	$-0.606678\pi$
$151$	−184305.			−0.657803			−0.328901	+	0.944364i	$0.606678\pi$
$152$		0			0
$153$			541415.i			1.86983i
$154$		0			0
$155$	460141.	−	141122.i	1.53837	−	0.471807i
$156$		0			0
$157$	−403676.			−1.30702			−0.653512	−	0.756916i	$-0.726705\pi$
$157$	−403676.			−1.30702			−0.653512	+	0.756916i	$0.726705\pi$
$158$		0			0
$159$	−331482.			−1.03984
$160$		0			0
$161$	309490.			0.940984
$162$		0			0
$163$	−357436.			−1.05373			−0.526864	−	0.849949i	$-0.676632\pi$
$163$	−357436.			−1.05373			−0.526864	+	0.849949i	$0.676632\pi$
$164$		0			0
$165$	−2181.94	−	7114.45i	−0.00623927	−	0.0203438i
$166$		0			0
$167$			38221.8i			0.106052i	0.998593	+	0.0530262i	$0.0168867\pi$
$167$			38221.8i			0.106052i	−0.998593	+	0.0530262i	$0.983113\pi$
$168$		0			0
$169$	−114392.			−0.308090
$170$		0			0
$171$		−	268938.i		−	0.703334i
$172$		0			0
$173$	134258.			0.341056			0.170528	−	0.985353i	$-0.445453\pi$
$173$	134258.			0.341056			0.170528	+	0.985353i	$0.445453\pi$
$174$		0			0
$175$	175696.	+	259494.i	0.433677	+	0.640520i
$176$		0			0
$177$		−	271829.i		−	0.652173i
$178$		0			0
$179$		−	566014.i		−	1.32037i	−0.751105	−	0.660183i	$-0.770479\pi$
$179$		−	566014.i		−	1.32037i	0.751105	−	0.660183i	$-0.229521\pi$
$180$		0			0
$181$		−	572156.i		−	1.29813i	−0.760733	−	0.649065i	$-0.775160\pi$
$181$		−	572156.i		−	1.29813i	0.760733	−	0.649065i	$-0.224840\pi$
$182$		0			0
$183$		−	272404.i		−	0.601292i
$184$		0			0
$185$	282002.	−	86487.8i	0.605791	−	0.185791i
$186$		0			0
$187$	−7805.15			−0.0163221
$188$		0			0
$189$			321542.i			0.654762i
$190$		0			0
$191$	685564.			1.35977			0.679883	−	0.733320i	$-0.262030\pi$
$191$	685564.			1.35977			0.679883	+	0.733320i	$0.262030\pi$
$192$		0			0
$193$			931503.i			1.80008i	0.435810	+	0.900039i	$0.356462\pi$
$193$			931503.i			1.80008i	−0.435810	+	0.900039i	$0.643538\pi$
$194$		0			0
$195$	671923.	−	206073.i	1.26541	−	0.388093i
$196$		0			0
$197$	692118.			1.27062			0.635308	−	0.772259i	$-0.280873\pi$
$197$	692118.			1.27062			0.635308	+	0.772259i	$0.280873\pi$
$198$		0			0
$199$	969167.			1.73487			0.867433	−	0.497554i	$-0.165768\pi$
$199$	969167.			1.73487			0.867433	+	0.497554i	$0.165768\pi$
$200$		0			0
$201$	−322646.			−0.563295
$202$		0			0
$203$	−731892.			−1.24654
$204$		0			0
$205$	971385.	−	297916.i	1.61438	−	0.495119i
$206$		0			0
$207$		−	1.14890e6i		−	1.86361i
$208$		0			0
$209$	3877.06			0.00613956
$210$		0			0
$211$			800177.i			1.23731i	0.785661	+	0.618657i	$0.212323\pi$
$211$			800177.i			1.23731i	−0.785661	+	0.618657i	$0.787677\pi$
$212$		0			0
$213$	893187.			1.34894
$214$		0			0
$215$	445779.	−	136717.i	0.657694	−	0.201709i
$216$		0			0
$217$		−	863389.i		−	1.24468i
$218$		0			0
$219$			1.59459e6i			2.24666i
$220$		0			0
$221$		−	737156.i		−	1.01526i
$222$		0			0
$223$		−	619923.i		−	0.834787i	−0.908726	−	0.417394i	$-0.862944\pi$
$223$		−	619923.i		−	0.834787i	0.908726	−	0.417394i	$-0.137056\pi$
$224$		0			0
$225$	963302.	−	652222.i	1.26855	−	0.858893i
$226$		0			0
$227$	547541.			0.705264			0.352632	−	0.935762i	$-0.385287\pi$
$227$	547541.			0.705264			0.352632	+	0.935762i	$0.385287\pi$
$228$		0			0
$229$		−	368527.i		−	0.464387i	−0.972670	−	0.232194i	$-0.925410\pi$
$229$		−	368527.i		−	0.464387i	0.972670	−	0.232194i	$-0.0745903\pi$
$230$		0			0
$231$	−13349.2			−0.0164599
$232$		0			0
$233$			347048.i			0.418794i	0.977831	+	0.209397i	$0.0671500\pi$
$233$			347048.i			0.418794i	−0.977831	+	0.209397i	$0.932850\pi$
$234$		0			0
$235$	−362573.	−	1.18221e6i	−0.428278	−	1.39644i
$236$		0			0
$237$	1.20646e6			1.39522
$238$		0			0
$239$	1.17224e6			1.32746			0.663731	−	0.747971i	$-0.268972\pi$
$239$	1.17224e6			1.32746			0.663731	+	0.747971i	$0.268972\pi$
$240$		0			0
$241$	−1.10163e6			−1.22178			−0.610889	−	0.791716i	$-0.709188\pi$
$241$	−1.10163e6			−1.22178			−0.610889	+	0.791716i	$0.709188\pi$
$242$		0			0
$243$	−1.05020e6			−1.14093
$244$		0			0
$245$	−360787.	+	110651.i	−0.384004	+	0.117771i
$246$		0			0
$247$			366168.i			0.381890i
$248$		0			0
$249$	−1.23488e6			−1.26220
$250$		0			0
$251$			97443.4i			0.0976265i	0.998808	+	0.0488133i	$0.0155439\pi$
$251$			97443.4i			0.0976265i	−0.998808	+	0.0488133i	$0.984456\pi$
$252$		0			0
$253$	16562.7			0.0162679
$254$		0			0
$255$	−591310.	−	1.92802e6i	−0.569462	−	1.85679i
$256$		0			0
$257$			1.35298e6i			1.27779i	0.769296	+	0.638893i	$0.220607\pi$
$257$			1.35298e6i			1.27779i	−0.769296	+	0.638893i	$0.779393\pi$
$258$		0			0
$259$		−	529136.i		−	0.490138i
$260$		0			0
$261$			2.71695e6i			2.46877i
$262$		0			0
$263$			892943.i			0.796039i	0.917377	+	0.398019i	$0.130302\pi$
$263$			892943.i			0.796039i	−0.917377	+	0.398019i	$0.869698\pi$
$264$		0			0
$265$	714221.	−	219046.i	0.624766	−	0.191611i
$266$		0			0
$267$	192593.			0.165334
$268$		0			0
$269$		−	395573.i		−	0.333308i	−0.986015	−	0.166654i	$-0.946704\pi$
$269$		−	395573.i		−	0.333308i	0.986015	−	0.166654i	$-0.0532963\pi$
$270$		0			0
$271$	548411.			0.453610			0.226805	−	0.973940i	$-0.427172\pi$
$271$	548411.			0.453610			0.226805	+	0.973940i	$0.427172\pi$
$272$		0			0
$273$		−	1.26077e6i		−	1.02383i
$274$		0			0
$275$	9402.56	+	13887.2i	0.00749747	+	0.0110734i
$276$		0			0
$277$	84078.7			0.0658395			0.0329198	−	0.999458i	$-0.489519\pi$
$277$	84078.7			0.0658395			0.0329198	+	0.999458i	$0.489519\pi$
$278$		0			0
$279$	−3.20509e6			−2.46508
$280$		0			0
$281$	869957.			0.657252			0.328626	−	0.944460i	$-0.393414\pi$
$281$	869957.			0.657252			0.328626	+	0.944460i	$0.393414\pi$
$282$		0			0
$283$	1.80499e6			1.33970			0.669852	−	0.742495i	$-0.266358\pi$
$283$	1.80499e6			1.33970			0.669852	+	0.742495i	$0.266358\pi$
$284$		0			0
$285$	293722.	+	957710.i	0.214202	+	0.698428i
$286$		0			0
$287$		−	1.82267e6i		−	1.30618i
$288$		0			0
$289$	−695348.			−0.489731
$290$		0			0
$291$		−	914020.i		−	0.632737i
$292$		0			0
$293$	734707.			0.499972			0.249986	−	0.968250i	$-0.419574\pi$
$293$	734707.			0.499972			0.249986	+	0.968250i	$0.419574\pi$
$294$		0			0
$295$	179627.	+	585691.i	0.120176	+	0.391844i
$296$		0			0
$297$			17207.7i			0.0113196i
$298$		0			0
$299$			1.56427e6i			1.01189i
$300$		0			0
$301$		−	836441.i		−	0.532132i
$302$		0			0
$303$		−	4.32981e6i		−	2.70933i
$304$		0			0
$305$	180007.	+	586930.i	0.110800	+	0.361274i
$306$		0			0
$307$	−568071.			−0.343999			−0.171999	−	0.985097i	$-0.555023\pi$
$307$	−568071.			−0.343999			−0.171999	+	0.985097i	$0.555023\pi$
$308$		0			0
$309$			4.27891e6i			2.54939i
$310$		0			0
$311$	−1.69667e6			−0.994712			−0.497356	−	0.867547i	$-0.665696\pi$
$311$	−1.69667e6			−0.994712			−0.497356	+	0.867547i	$0.665696\pi$
$312$		0			0
$313$			349961.i			0.201910i	0.994891	+	0.100955i	$0.0321899\pi$
$313$			349961.i			0.201910i	−0.994891	+	0.100955i	$0.967810\pi$
$314$		0			0
$315$	−611901.	−	1.99516e6i	−0.347460	−	1.13293i
$316$		0			0
$317$	2.16674e6			1.21104			0.605520	−	0.795830i	$-0.292965\pi$
$317$	2.16674e6			1.21104			0.605520	+	0.795830i	$0.292965\pi$
$318$		0			0
$319$	−39168.1			−0.0215504
$320$		0			0
$321$	3.84907e6			2.08494
$322$		0			0
$323$	1.05069e6			0.560361
$324$		0			0
$325$	−1.31157e6	+	888024.i	−0.688784	+	0.466354i
$326$		0			0
$327$			2.75467e6i			1.42462i
$328$		0			0
$329$	−2.21824e6			−1.12985
$330$		0			0
$331$		−	3.09909e6i		−	1.55476i	−0.629031	−	0.777380i	$-0.716548\pi$
$331$		−	3.09909e6i		−	1.55476i	0.629031	−	0.777380i	$-0.283452\pi$
$332$		0			0
$333$	−1.96427e6			−0.970714
$334$		0			0
$335$	695182.	−	213207.i	0.338444	−	0.103798i
$336$		0			0
$337$			913351.i			0.438089i	0.975715	+	0.219045i	$0.0702941\pi$
$337$			913351.i			0.438089i	−0.975715	+	0.219045i	$0.929706\pi$
$338$		0			0
$339$			4.95216e6i			2.34043i
$340$		0			0
$341$		−	46205.3i		−	0.0215182i
$342$		0			0
$343$			2.36239e6i			1.08422i
$344$		0			0
$345$	1.25478e6	+	4.09132e6i	0.567568	+	1.85061i
$346$		0			0
$347$	−1.81927e6			−0.811100			−0.405550	−	0.914073i	$-0.632920\pi$
$347$	−1.81927e6			−0.811100			−0.405550	+	0.914073i	$0.632920\pi$
$348$		0			0
$349$		−	3.66132e6i		−	1.60907i	−0.593908	−	0.804533i	$-0.702416\pi$
$349$		−	3.66132e6i		−	1.60907i	0.593908	−	0.804533i	$-0.297584\pi$
$350$		0			0
$351$	−1.62518e6			−0.704099
$352$		0			0
$353$		−	4.00606e6i		−	1.71112i	−0.517703	−	0.855560i	$-0.673213\pi$
$353$		−	4.00606e6i		−	1.71112i	0.517703	−	0.855560i	$-0.326787\pi$
$354$		0			0
$355$	−1.92449e6	+	590225.i	−0.810483	+	0.248569i
$356$		0			0
$357$	−3.61766e6			−1.50230
$358$		0			0
$359$	3.37100e6			1.38046			0.690228	−	0.723592i	$-0.257510\pi$
$359$	3.37100e6			1.38046			0.690228	+	0.723592i	$0.257510\pi$
$360$		0			0
$361$	1.95419e6			0.789221
$362$		0			0
$363$	3.99409e6			1.59093
$364$		0			0
$365$	−1.05372e6	−	3.43574e6i	−0.413991	−	1.34986i
$366$		0			0
$367$			3.30503e6i			1.28089i	0.768005	+	0.640444i	$0.221249\pi$
$367$			3.30503e6i			1.28089i	−0.768005	+	0.640444i	$0.778751\pi$
$368$		0			0
$369$	−6.76614e6			−2.58687
$370$		0			0
$371$		−	1.34013e6i		−	0.505491i
$372$		0			0
$373$	−582132.			−0.216646			−0.108323	−	0.994116i	$-0.534548\pi$
$373$	−582132.			−0.216646			−0.108323	+	0.994116i	$0.534548\pi$
$374$		0			0
$375$	−2.71807e6	+	3.37469e6i	−0.998119	+	1.23924i
$376$		0			0
$377$		−	3.69922e6i		−	1.34047i
$378$		0			0
$379$			2.24404e6i			0.802478i	0.915973	+	0.401239i	$0.131420\pi$
$379$			2.24404e6i			0.802478i	−0.915973	+	0.401239i	$0.868580\pi$
$380$		0			0
$381$		−	2.31428e6i		−	0.816776i
$382$		0			0
$383$			2.03163e6i			0.707696i	0.935303	+	0.353848i	$0.115127\pi$
$383$			2.03163e6i			0.707696i	−0.935303	+	0.353848i	$0.884873\pi$
$384$		0			0
$385$	28762.7	−	8821.28i	0.00988957	−	0.00303305i
$386$		0			0
$387$	−3.10506e6			−1.05388
$388$		0			0
$389$		−	3.29458e6i		−	1.10389i	−0.833881	−	0.551945i	$-0.813886\pi$
$389$		−	3.29458e6i		−	1.10389i	0.833881	−	0.551945i	$-0.186114\pi$
$390$		0			0
$391$	4.48852e6			1.48478
$392$		0			0
$393$		−	4.33930e6i		−	1.41722i
$394$		0			0
$395$	−2.59949e6	+	797242.i	−0.838291	+	0.257097i
$396$		0			0
$397$	−2.66575e6			−0.848875			−0.424438	−	0.905457i	$-0.639528\pi$
$397$	−2.66575e6			−0.848875			−0.424438	+	0.905457i	$0.639528\pi$
$398$		0			0
$399$	1.79700e6			0.565089
$400$		0			0
$401$	−1.41349e6			−0.438965			−0.219483	−	0.975616i	$-0.570437\pi$
$401$	−1.41349e6			−0.438965			−0.219483	+	0.975616i	$0.570437\pi$
$402$		0			0
$403$	4.36385e6			1.33847
$404$		0			0
$405$	584008.	−	179111.i	0.176922	−	0.0542605i
$406$		0			0
$407$		−	28317.4i		−	0.00847358i
$408$		0			0
$409$	−1.65117e6			−0.488072			−0.244036	−	0.969766i	$-0.578472\pi$
$409$	−1.65117e6			−0.488072			−0.244036	+	0.969766i	$0.578472\pi$
$410$		0			0
$411$		−	3.16112e6i		−	0.923073i
$412$		0			0
$413$	1.09897e6			0.317036
$414$		0			0
$415$	2.66071e6	−	816019.i	0.758364	−	0.232584i
$416$		0			0
$417$			9.76458e6i			2.74988i
$418$		0			0
$419$			4.58123e6i			1.27482i	0.770527	+	0.637408i	$0.219993\pi$
$419$			4.58123e6i			1.27482i	−0.770527	+	0.637408i	$0.780007\pi$
$420$		0			0
$421$		−	3.54620e6i		−	0.975120i	−0.873089	−	0.487560i	$-0.837887\pi$
$421$		−	3.54620e6i		−	0.975120i	0.873089	−	0.487560i	$-0.162113\pi$
$422$		0			0
$423$			8.23461e6i			2.23765i
$424$		0			0
$425$	2.54811e6	+	3.76344e6i	0.684298	+	1.01068i
$426$		0			0
$427$	1.10129e6			0.292302
$428$		0			0
$429$		−	67471.4i		−	0.0177001i
$430$		0			0
$431$	3.24667e6			0.841871			0.420935	−	0.907091i	$-0.361702\pi$
$431$	3.24667e6			0.841871			0.420935	+	0.907091i	$0.361702\pi$
$432$		0			0
$433$		−	4.93639e6i		−	1.26529i	−0.774443	−	0.632644i	$-0.781970\pi$
$433$		−	4.93639e6i		−	1.26529i	0.774443	−	0.632644i	$-0.218030\pi$
$434$		0			0
$435$	−2.96733e6	−	9.67527e6i	−0.751870	−	2.45155i
$436$		0			0
$437$	−2.22959e6			−0.558498
$438$		0			0
$439$	−5.91592e6			−1.46508			−0.732539	−	0.680725i	$-0.761665\pi$
$439$	−5.91592e6			−1.46508			−0.732539	+	0.680725i	$0.761665\pi$
$440$		0			0
$441$	2.51305e6			0.615325
$442$		0			0
$443$	723323.			0.175115			0.0875574	−	0.996159i	$-0.472094\pi$
$443$	723323.			0.175115			0.0875574	+	0.996159i	$0.472094\pi$
$444$		0			0
$445$	−414966.	+	127267.i	−0.0993373	+	0.0304660i
$446$		0			0
$447$		−	4.14515e6i		−	0.981231i
$448$		0			0
$449$	−1.61902e6			−0.378999			−0.189499	−	0.981881i	$-0.560686\pi$
$449$	−1.61902e6			−0.378999			−0.189499	+	0.981881i	$0.560686\pi$
$450$		0			0
$451$		−	97542.1i		−	0.0225814i
$452$		0			0
$453$	−4.57162e6			−1.04670
$454$		0			0
$455$	833125.	+	2.71649e6i	0.188661	+	0.615147i
$456$		0			0
$457$		−	419909.i		−	0.0940512i	−0.998894	−	0.0470256i	$-0.985026\pi$
$457$		−	419909.i		−	0.0940512i	0.998894	−	0.0470256i	$-0.0149742\pi$
$458$		0			0
$459$			4.66331e6i			1.03315i
$460$		0			0
$461$			3.17942e6i			0.696780i	0.937350	+	0.348390i	$0.113272\pi$
$461$			3.17942e6i			0.696780i	−0.937350	+	0.348390i	$0.886728\pi$
$462$		0			0
$463$			620530.i			0.134527i	0.997735	+	0.0672636i	$0.0214269\pi$
$463$			620530.i			0.134527i	−0.997735	+	0.0672636i	$0.978573\pi$
$464$		0			0
$465$	1.14136e7	−	3.50046e6i	2.44788	−	0.750746i
$466$		0			0
$467$	1.96565e6			0.417075			0.208538	−	0.978014i	$-0.433130\pi$
$467$	1.96565e6			0.417075			0.208538	+	0.978014i	$0.433130\pi$
$468$		0			0
$469$		−	1.30441e6i		−	0.273831i
$470$		0			0
$471$	−1.00130e7			−2.07975
$472$		0			0
$473$		−	44763.1i		−	0.00919957i
$474$		0			0
$475$	−1.26572e6	−	1.86942e6i	−0.257398	−	0.380165i
$476$		0			0
$477$	−4.97487e6			−1.00112
$478$		0			0
$479$	−4.57977e6			−0.912022			−0.456011	−	0.889974i	$-0.650722\pi$
$479$	−4.57977e6			−0.912022			−0.456011	+	0.889974i	$0.650722\pi$
$480$		0			0
$481$	2.67443e6			0.527070
$482$		0			0
$483$	7.67677e6			1.49731
$484$		0			0
$485$	603991.	+	1.96937e6i	0.116594	+	0.380167i
$486$		0			0
$487$			9.22794e6i			1.76312i	0.472070	+	0.881561i	$0.343507\pi$
$487$			9.22794e6i			1.76312i	−0.472070	+	0.881561i	$0.656493\pi$
$488$		0			0
$489$	−8.86603e6			−1.67671
$490$		0			0
$491$			6.77438e6i			1.26814i	0.773277	+	0.634068i	$0.218616\pi$
$491$			6.77438e6i			1.26814i	−0.773277	+	0.634068i	$0.781384\pi$
$492$		0			0
$493$	−1.06146e7			−1.96692
$494$		0			0
$495$	−32746.6	−	106774.i	−0.00600694	−	0.0195862i
$496$		0			0
$497$			3.61102e6i			0.655752i
$498$		0			0
$499$			3.54380e6i			0.637115i	0.947903	+	0.318558i	$0.103198\pi$
$499$			3.54380e6i			0.637115i	−0.947903	+	0.318558i	$0.896802\pi$
$500$		0			0
$501$			948077.i			0.168752i
$502$		0			0
$503$			582992.i			0.102741i	0.998680	+	0.0513704i	$0.0163589\pi$
$503$			582992.i			0.102741i	−0.998680	+	0.0513704i	$0.983641\pi$
$504$		0			0
$505$	2.86118e6	+	9.32915e6i	0.499248	+	1.62785i
$506$		0			0
$507$	−2.83744e6			−0.490237
$508$		0			0
$509$			1.00694e7i			1.72270i	0.508013	+	0.861349i	$0.330380\pi$
$509$			1.00694e7i			1.72270i	−0.508013	+	0.861349i	$0.669620\pi$
$510$		0			0
$511$	−6.44668e6			−1.09215
$512$		0			0
$513$		−	2.31641e6i		−	0.388618i
$514$		0			0
$515$	−2.82754e6	−	9.21947e6i	−0.469775	−	1.53175i
$516$		0			0
$517$	−118712.			−0.0195329
$518$		0			0
$519$	3.33022e6			0.542693
$520$		0			0
$521$	−6.76200e6			−1.09139			−0.545696	−	0.837983i	$-0.683735\pi$
$521$	−6.76200e6			−1.09139			−0.545696	+	0.837983i	$0.683735\pi$
$522$		0			0
$523$	−6.39905e6			−1.02297			−0.511483	−	0.859293i	$-0.670904\pi$
$523$	−6.39905e6			−1.02297			−0.511483	+	0.859293i	$0.670904\pi$
$524$		0			0
$525$	4.35806e6	+	6.43665e6i	0.690072	+	1.01921i
$526$		0			0
$527$		−	1.25217e7i		−	1.96398i
$528$		0			0
$529$	−3.08843e6			−0.479842
$530$		0			0
$531$		−	4.07961e6i		−	0.627888i
$532$		0			0
$533$	9.21235e6			1.40460
$534$		0			0
$535$	−8.29332e6	+	2.54349e6i	−1.25269	+	0.384190i
$536$		0			0
$537$		−	1.40397e7i		−	2.10098i
$538$		0			0
$539$			36228.6i			0.00537130i
$540$		0			0
$541$			9.42388e6i			1.38432i	0.721744	+	0.692160i	$0.243341\pi$
$541$			9.42388e6i			1.38432i	−0.721744	+	0.692160i	$0.756659\pi$
$542$		0			0
$543$		−	1.41921e7i		−	2.06560i
$544$		0			0
$545$	−1.82031e6	−	5.93529e6i	−0.262514	−	0.855955i
$546$		0			0
$547$	−1.39418e6			−0.199228			−0.0996138	−	0.995026i	$-0.531761\pi$
$547$	−1.39418e6			−0.199228			−0.0996138	+	0.995026i	$0.531761\pi$
$548$		0			0
$549$		−	4.08824e6i		−	0.578902i
$550$		0			0
$551$	5.27260e6			0.739854
$552$		0			0
$553$			4.87756e6i			0.678250i
$554$		0			0
$555$	6.99494e6	−	2.14529e6i	0.963943	−	0.295634i
$556$		0			0
$557$	1.27617e6			0.174289			0.0871445	−	0.996196i	$-0.472226\pi$
$557$	1.27617e6			0.174289			0.0871445	+	0.996196i	$0.472226\pi$
$558$		0			0
$559$	4.22765e6			0.572228
$560$		0			0
$561$	−193603.			−0.0259720
$562$		0			0
$563$	−1.87381e6			−0.249146			−0.124573	−	0.992210i	$-0.539756\pi$
$563$	−1.87381e6			−0.249146			−0.124573	+	0.992210i	$0.539756\pi$
$564$		0			0
$565$	−3.27243e6	−	1.06701e7i	−0.431269	−	1.40620i
$566$		0			0
$567$		−	1.09581e6i		−	0.143145i
$568$		0			0
$569$	−6.96458e6			−0.901808			−0.450904	−	0.892572i	$-0.648898\pi$
$569$	−6.96458e6			−0.901808			−0.450904	+	0.892572i	$0.648898\pi$
$570$		0			0
$571$			344717.i			0.0442458i	0.999755	+	0.0221229i	$0.00704251\pi$
$571$			344717.i			0.0442458i	−0.999755	+	0.0221229i	$0.992957\pi$
$572$		0			0
$573$	1.70051e7			2.16368
$574$		0			0
$575$	−5.40715e6	−	7.98612e6i	−0.682023	−	1.00732i
$576$		0			0
$577$			5.35769e6i			0.669944i	0.942228	+	0.334972i	$0.108727\pi$
$577$			5.35769e6i			0.669944i	−0.942228	+	0.334972i	$0.891273\pi$
$578$		0			0
$579$			2.31055e7i			2.86431i
$580$		0			0
$581$		−	4.99245e6i		−	0.613583i
$582$		0			0
$583$		−	71718.8i		−	0.00873900i
$584$		0			0
$585$	1.00842e7	−	3.09275e6i	1.21829	−	0.373641i
$586$		0			0
$587$	−1.33004e6			−0.159320			−0.0796599	−	0.996822i	$-0.525383\pi$
$587$	−1.33004e6			−0.159320			−0.0796599	+	0.996822i	$0.525383\pi$
$588$		0			0
$589$			6.21992e6i			0.738748i
$590$		0			0
$591$	1.71677e7			2.02182
$592$		0			0
$593$			4.87875e6i			0.569734i	0.958567	+	0.284867i	$0.0919494\pi$
$593$			4.87875e6i			0.569734i	−0.958567	+	0.284867i	$0.908051\pi$
$594$		0			0
$595$	7.79472e6	−	2.39058e6i	0.902627	−	0.276828i
$596$		0			0
$597$	2.40398e7			2.76054
$598$		0			0
$599$	−2.71017e6			−0.308624			−0.154312	−	0.988022i	$-0.549316\pi$
$599$	−2.71017e6			−0.308624			−0.154312	+	0.988022i	$0.549316\pi$
$600$		0			0
$601$	1.02295e7			1.15523			0.577615	−	0.816309i	$-0.303983\pi$
$601$	1.02295e7			1.15523			0.577615	+	0.816309i	$0.303983\pi$
$602$		0			0
$603$	−4.84226e6			−0.542319
$604$		0			0
$605$	−8.60578e6	+	2.63932e6i	−0.955876	+	0.293160i
$606$		0			0
$607$			6.63212e6i			0.730602i	0.930890	+	0.365301i	$0.119034\pi$
$607$			6.63212e6i			0.730602i	−0.930890	+	0.365301i	$0.880966\pi$
$608$		0			0
$609$	−1.81543e7			−1.98351
$610$		0			0
$611$		−	1.12117e7i		−	1.21498i
$612$		0			0
$613$	−7.17868e6			−0.771602			−0.385801	−	0.922582i	$-0.626075\pi$
$613$	−7.17868e6			−0.771602			−0.385801	+	0.922582i	$0.626075\pi$
$614$		0			0
$615$	2.40948e7	−	7.38968e6i	2.56883	−	0.787840i
$616$		0			0
$617$		−	1.22481e7i		−	1.29526i	−0.761956	−	0.647629i	$-0.775761\pi$
$617$		−	1.22481e7i		−	1.29526i	0.761956	−	0.647629i	$-0.224239\pi$
$618$		0			0
$619$			2.10012e6i			0.220302i	0.993915	+	0.110151i	$0.0351334\pi$
$619$			2.10012e6i			0.220302i	−0.993915	+	0.110151i	$0.964867\pi$
$620$		0			0
$621$		−	9.89568e6i		−	1.02971i
$622$		0			0
$623$			778624.i			0.0803725i
$624$		0			0
$625$	3.62641e6	−	9.06734e6i	0.371345	−	0.928495i
$626$		0			0
$627$	96168.9			0.00976935
$628$		0			0
$629$		−	7.67404e6i		−	0.773388i
$630$		0			0
$631$	−5.94323e6			−0.594223			−0.297111	−	0.954843i	$-0.596023\pi$
$631$	−5.94323e6			−0.594223			−0.297111	+	0.954843i	$0.596023\pi$
$632$		0			0
$633$			1.98481e7i			1.96883i
$634$		0			0
$635$	1.52929e6	+	4.98641e6i	0.150507	+	0.490743i
$636$		0			0
$637$	−3.42160e6			−0.334104
$638$		0			0
$639$	1.34049e7			1.29871
$640$		0			0
$641$	1.73665e7			1.66943			0.834713	−	0.550686i	$-0.185634\pi$
$641$	1.73665e7			1.66943			0.834713	+	0.550686i	$0.185634\pi$
$642$		0			0
$643$	−1.88563e7			−1.79858			−0.899290	−	0.437353i	$-0.855916\pi$
$643$	−1.88563e7			−1.79858			−0.899290	+	0.437353i	$0.855916\pi$
$644$		0			0
$645$	1.10574e7	−	3.39121e6i	1.04653	−	0.320963i
$646$		0			0
$647$		−	2.16987e6i		−	0.203785i	−0.994795	−	0.101893i	$-0.967510\pi$
$647$		−	2.16987e6i		−	0.203785i	0.994795	−	0.101893i	$-0.0324898\pi$
$648$		0			0
$649$	58812.5			0.00548097
$650$		0			0
$651$		−	2.14160e7i		−	1.98055i
$652$		0			0
$653$	1.97704e7			1.81440			0.907198	−	0.420703i	$-0.138217\pi$
$653$	1.97704e7			1.81440			0.907198	+	0.420703i	$0.138217\pi$
$654$		0			0
$655$	2.86744e6	+	9.34958e6i	0.261151	+	0.851509i
$656$		0			0
$657$			2.39315e7i			2.16300i
$658$		0			0
$659$			3.61872e6i			0.324595i	0.986742	+	0.162297i	$0.0518904\pi$
$659$			3.61872e6i			0.324595i	−0.986742	+	0.162297i	$0.948110\pi$
$660$		0			0
$661$		−	3.75269e6i		−	0.334071i	−0.985951	−	0.167036i	$-0.946580\pi$
$661$		−	3.75269e6i		−	0.334071i	0.985951	−	0.167036i	$-0.0534195\pi$
$662$		0			0
$663$		−	1.82848e7i		−	1.61550i
$664$		0			0
$665$	−3.87188e6	+	1.18748e6i	−0.339522	+	0.104129i
$666$		0			0
$667$	2.25245e7			1.96038
$668$		0			0
$669$		−	1.53769e7i		−	1.32833i
$670$		0			0
$671$	58936.8			0.00505336
$672$		0			0
$673$		−	1.93077e6i		−	0.164321i	−0.996619	−	0.0821606i	$-0.973818\pi$
$673$		−	1.93077e6i		−	0.164321i	0.996619	−	0.0821606i	$-0.0261820\pi$
$674$		0			0
$675$	8.29710e6	−	5.61771e6i	0.700918	−	0.474570i
$676$		0			0
$677$	7.53721e6			0.632032			0.316016	−	0.948754i	$-0.397655\pi$
$677$	7.53721e6			0.632032			0.316016	+	0.948754i	$0.397655\pi$
$678$		0			0
$679$	3.69525e6			0.307588
$680$		0			0
$681$	1.35815e7			1.12223
$682$		0			0
$683$	−2.28279e6			−0.187247			−0.0936234	−	0.995608i	$-0.529845\pi$
$683$	−2.28279e6			−0.187247			−0.0936234	+	0.995608i	$0.529845\pi$
$684$		0			0
$685$	2.08889e6	+	6.81104e6i	0.170094	+	0.554609i
$686$		0			0
$687$		−	9.14114e6i		−	0.738939i
$688$		0			0
$689$	6.77347e6			0.543580
$690$		0			0
$691$			4.89825e6i			0.390252i	0.980778	+	0.195126i	$0.0625116\pi$
$691$			4.89825e6i			0.390252i	−0.980778	+	0.195126i	$0.937488\pi$
$692$		0			0
$693$	−200345.			−0.0158470
$694$		0			0
$695$	−6.45251e6	−	2.10391e7i	−0.506718	−	1.65221i
$696$		0			0
$697$		−	2.64340e7i		−	2.06102i
$698$		0			0
$699$			8.60838e6i			0.666390i
$700$		0			0
$701$			1.36266e7i			1.04735i	0.851917	+	0.523676i	$0.175440\pi$
$701$			1.36266e7i			1.04735i	−0.851917	+	0.523676i	$0.824560\pi$
$702$		0			0
$703$			3.81194e6i			0.290909i
$704$		0			0
$705$	−8.99347e6	−	2.93241e7i	−0.681483	−	2.22204i
$706$		0			0
$707$	1.75048e7			1.31707
$708$		0			0
$709$			1.03111e7i			0.770350i	0.922843	+	0.385175i	$0.125859\pi$
$709$			1.03111e7i			0.770350i	−0.922843	+	0.385175i	$0.874141\pi$
$710$		0			0
$711$	1.81066e7			1.34327
$712$		0			0
$713$			2.65714e7i			1.95745i
$714$		0			0
$715$	44585.6	+	145376.i	0.00326159	+	0.0106348i
$716$		0			0
$717$	2.90769e7			2.11228
$718$		0			0
$719$	−9.38661e6			−0.677153			−0.338576	−	0.940939i	$-0.609945\pi$
$719$	−9.38661e6			−0.677153			−0.338576	+	0.940939i	$0.609945\pi$
$720$		0			0
$721$	−1.72990e7			−1.23932
$722$		0			0
$723$	−2.73254e7			−1.94411
$724$		0			0
$725$	1.27870e7	+	1.88858e7i	0.903490	+	1.33441i
$726$		0			0
$727$		−	4.62442e6i		−	0.324505i	−0.986749	−	0.162252i	$-0.948124\pi$
$727$		−	4.62442e6i		−	0.324505i	0.986749	−	0.162252i	$-0.0518759\pi$
$728$		0			0
$729$	−2.33945e7			−1.63040
$730$		0			0
$731$		−	1.21309e7i		−	0.839650i
$732$		0			0
$733$	−2.68859e7			−1.84827			−0.924134	−	0.382068i	$-0.875212\pi$
$733$	−2.68859e7			−1.84827			−0.924134	+	0.382068i	$0.875212\pi$
$734$		0			0
$735$	−8.94917e6	+	2.74464e6i	−0.611033	+	0.187399i
$736$		0			0
$737$		−	69807.0i		−	0.00473403i
$738$		0			0
$739$		−	686127.i		−	0.0462161i	−0.999733	−	0.0231081i	$-0.992644\pi$
$739$		−	686127.i		−	0.0462161i	0.999733	−	0.0231081i	$-0.00735618\pi$
$740$		0			0
$741$			9.08265e6i			0.607669i
$742$		0			0
$743$		−	1.09400e7i		−	0.727017i	−0.931591	−	0.363508i	$-0.881579\pi$
$743$		−	1.09400e7i		−	0.727017i	0.931591	−	0.363508i	$-0.118421\pi$
$744$		0			0
$745$	2.73915e6	+	8.93126e6i	0.180811	+	0.589552i
$746$		0			0
$747$	−1.85331e7			−1.21520
$748$		0			0
$749$			1.55612e7i			1.01354i
$750$		0			0
$751$	−2.37534e7			−1.53683			−0.768416	−	0.639951i	$-0.778955\pi$
$751$	−2.37534e7			−1.53683			−0.768416	+	0.639951i	$0.778955\pi$
$752$		0			0
$753$			2.41704e6i			0.155345i
$754$		0			0
$755$	9.85014e6	−	3.02096e6i	0.628891	−	0.192876i
$756$		0			0
$757$	−2.95506e7			−1.87424			−0.937122	−	0.349003i	$-0.886520\pi$
$757$	−2.95506e7			−1.87424			−0.937122	+	0.349003i	$0.886520\pi$
$758$		0			0
$759$	410832.			0.0258857
$760$		0			0
$761$	2.38225e6			0.149116			0.0745582	−	0.997217i	$-0.476245\pi$
$761$	2.38225e6			0.149116			0.0745582	+	0.997217i	$0.476245\pi$
$762$		0			0
$763$	−1.11367e7			−0.692542
$764$		0			0
$765$	−8.87437e6	−	2.89358e7i	−0.548256	−	1.78764i
$766$		0			0
$767$			5.55453e6i			0.340925i
$768$		0			0
$769$	−1.58003e7			−0.963495			−0.481747	−	0.876310i	$-0.659998\pi$
$769$	−1.58003e7			−0.963495			−0.481747	+	0.876310i	$0.659998\pi$
$770$		0			0
$771$			3.35600e7i			2.03323i
$772$		0			0
$773$	−8.84834e6			−0.532615			−0.266307	−	0.963888i	$-0.585804\pi$
$773$	−8.84834e6			−0.532615			−0.266307	+	0.963888i	$0.585804\pi$
$774$		0			0
$775$	−2.22790e7	+	1.50844e7i	−1.33242	+	0.902140i
$776$		0			0
$777$		−	1.31250e7i		−	0.779914i
$778$		0			0
$779$			1.31306e7i			0.775249i
$780$		0			0
$781$			193248.i			0.0113367i
$782$		0			0
$783$			2.34016e7i			1.36408i
$784$		0			0
$785$	2.15743e7	−	6.61667e6i	1.24958	−	0.383235i
$786$		0			0
$787$	−5.30753e6			−0.305461			−0.152730	−	0.988268i	$-0.548807\pi$
$787$	−5.30753e6			−0.305461			−0.152730	+	0.988268i	$0.548807\pi$
$788$		0			0
$789$			2.21491e7i			1.26667i
$790$		0			0
$791$	−2.00209e7			−1.13774
$792$		0			0
$793$			5.56628e6i			0.314327i
$794$		0			0
$795$	1.77159e7	−	5.43334e6i	0.994137	−	0.304894i
$796$		0			0
$797$	−1.99099e7			−1.11026			−0.555128	−	0.831765i	$-0.687331\pi$
$797$	−1.99099e7			−1.11026			−0.555128	+	0.831765i	$0.687331\pi$
$798$		0			0
$799$	−3.21710e7			−1.78278
$800$		0			0
$801$	2.89043e6			0.159177
$802$		0			0
$803$	−345002.			−0.0188813
$804$		0			0
$805$	−1.65406e7	+	5.07287e6i	−0.899626	+	0.275908i
$806$		0			0
$807$		−	9.81202e6i		−	0.530365i
$808$		0			0
$809$	2.14513e7			1.15234			0.576172	−	0.817329i	$-0.304546\pi$
$809$	2.14513e7			1.15234			0.576172	+	0.817329i	$0.304546\pi$
$810$		0			0
$811$		−	2.07036e6i		−	0.110534i	−0.998472	−	0.0552668i	$-0.982399\pi$
$811$		−	2.07036e6i		−	0.110534i	0.998472	−	0.0552668i	$-0.0176009\pi$
$812$		0			0
$813$	1.36031e7			0.721791
$814$		0			0
$815$	1.91030e7	−	5.85875e6i	1.00741	−	0.308966i
$816$		0			0
$817$			6.02578e6i			0.315833i
$818$		0			0
$819$		−	1.89216e7i		−	0.985706i
$820$		0			0
$821$			1.97968e7i			1.02503i	0.858677	+	0.512517i	$0.171287\pi$
$821$			1.97968e7i			1.02503i	−0.858677	+	0.512517i	$0.828713\pi$
$822$		0			0
$823$			2.89602e7i			1.49040i	0.666842	+	0.745199i	$0.267645\pi$
$823$			2.89602e7i			1.49040i	−0.666842	+	0.745199i	$0.732355\pi$
$824$		0			0
$825$	233227.	+	344465.i	0.0119301	+	0.0176202i
$826$		0			0
$827$	1.81070e6			0.0920624			0.0460312	−	0.998940i	$-0.485343\pi$
$827$	1.81070e6			0.0920624			0.0460312	+	0.998940i	$0.485343\pi$
$828$		0			0
$829$		−	1.35089e7i		−	0.682706i	−0.939935	−	0.341353i	$-0.889115\pi$
$829$		−	1.35089e7i		−	0.682706i	0.939935	−	0.341353i	$-0.110885\pi$
$830$		0			0
$831$	2.08554e6			0.104765
$832$		0			0
$833$			9.81800e6i			0.490242i
$834$		0			0
$835$	−626496.	−	2.04275e6i	−0.0310959	−	0.101391i
$836$		0			0
$837$	−2.76061e7			−1.36204
$838$		0			0
$839$	1.09747e7			0.538252			0.269126	−	0.963105i	$-0.413265\pi$
$839$	1.09747e7			0.538252			0.269126	+	0.963105i	$0.413265\pi$
$840$		0			0
$841$	−3.27554e7			−1.59695
$842$		0			0
$843$	2.15789e7			1.04583
$844$		0			0
$845$	6.11363e6	−	1.87500e6i	0.294549	−	0.0903357i
$846$		0			0
$847$			1.61475e7i			0.773387i
$848$		0			0
$849$	4.47720e7			2.13176
$850$		0			0
$851$			1.62845e7i			0.770817i
$852$		0			0
$853$	−1.28394e6			−0.0604186			−0.0302093	−	0.999544i	$-0.509617\pi$
$853$	−1.28394e6			−0.0604186			−0.0302093	+	0.999544i	$0.509617\pi$
$854$		0			0
$855$	4.40817e6	+	1.43733e7i	0.206226	+	0.672421i
$856$		0			0
$857$		−	1.48910e7i		−	0.692582i	−0.938127	−	0.346291i	$-0.887441\pi$
$857$		−	1.48910e7i		−	0.692582i	0.938127	−	0.346291i	$-0.112559\pi$
$858$		0			0
$859$			1.77820e6i			0.0822239i	0.999155	+	0.0411120i	$0.0130900\pi$
$859$			1.77820e6i			0.0822239i	−0.999155	+	0.0411120i	$0.986910\pi$
$860$		0			0
$861$		−	4.52104e7i		−	2.07841i
$862$		0			0
$863$			601133.i			0.0274754i	0.999906	+	0.0137377i	$0.00437298\pi$
$863$			601133.i			0.0274754i	−0.999906	+	0.0137377i	$0.995627\pi$
$864$		0			0
$865$	−7.17539e6	+	2.20064e6i	−0.326066	+	0.100002i
$866$		0			0
$867$	−1.72478e7			−0.779267
$868$		0			0
$869$			261029.i			0.0117257i
$870$		0			0
$871$	6.59292e6			0.294464
$872$		0			0
$873$		−	1.37176e7i		−	0.609175i
$874$		0			0
$875$	−1.36434e7	−	1.09888e7i	−0.602424	−	0.485209i
$876$		0			0
$877$	−2.57080e7			−1.12868			−0.564338	−	0.825544i	$-0.690868\pi$
$877$	−2.57080e7			−1.12868			−0.564338	+	0.825544i	$0.690868\pi$
$878$		0			0
$879$	1.82241e7			0.795562
$880$		0			0
$881$	−1.82816e7			−0.793551			−0.396775	−	0.917916i	$-0.629871\pi$
$881$	−1.82816e7			−0.793551			−0.396775	+	0.917916i	$0.629871\pi$
$882$		0			0
$883$	8.27706e6			0.357252			0.178626	−	0.983917i	$-0.442835\pi$
$883$	8.27706e6			0.357252			0.178626	+	0.983917i	$0.442835\pi$
$884$		0			0
$885$	4.45557e6	+	1.45278e7i	0.191225	+	0.623508i
$886$		0			0
$887$		−	1.11992e7i		−	0.477947i	−0.971026	−	0.238973i	$-0.923189\pi$
$887$		−	1.11992e7i		−	0.477947i	0.971026	−	0.238973i	$-0.0768109\pi$
$888$		0			0
$889$	9.35629e6			0.397054
$890$		0			0
$891$		−	58643.4i		−	0.00247471i
$892$		0			0
$893$	1.59804e7			0.670591
$894$		0			0
$895$	9.27756e6	+	3.02504e7i	0.387147	+	1.26233i
$896$		0			0
$897$			3.88009e7i			1.61013i
$898$		0			0
$899$		−	6.28368e7i		−	2.59307i
$900$		0			0
$901$		−	1.94359e7i		−	0.797613i
$902$		0			0
$903$		−	2.07476e7i		−	0.846735i
$904$		0			0
$905$	9.37824e6	+	3.05787e7i	0.380627	+	1.24107i
$906$		0			0
$907$	2.21915e7			0.895712			0.447856	−	0.894106i	$-0.352188\pi$
$907$	2.21915e7			0.895712			0.447856	+	0.894106i	$0.352188\pi$
$908$		0			0
$909$		−	6.49818e7i		−	2.60845i
$910$		0			0
$911$	−2.01968e6			−0.0806281			−0.0403140	−	0.999187i	$-0.512836\pi$
$911$	−2.01968e6			−0.0806281			−0.0403140	+	0.999187i	$0.512836\pi$
$912$		0			0
$913$		−	267177.i		−	0.0106077i
$914$		0			0
$915$	4.46499e6	+	1.45585e7i	0.176306	+	0.574864i
$916$		0			0
$917$	1.75432e7			0.688945
$918$		0			0
$919$	−1.22871e6			−0.0479909			−0.0239955	−	0.999712i	$-0.507639\pi$
$919$	−1.22871e6			−0.0479909			−0.0239955	+	0.999712i	$0.507639\pi$
$920$		0			0
$921$	−1.40908e7			−0.547375
$922$		0			0
$923$	−1.82513e7			−0.705163
$924$		0			0
$925$	−1.36539e7	+	9.24462e6i	−0.524689	+	0.355251i
$926$		0			0
$927$			6.42178e7i			2.45446i
$928$		0			0
$929$	−2.87347e7			−1.09236			−0.546182	−	0.837666i	$-0.683919\pi$
$929$	−2.87347e7			−1.09236			−0.546182	+	0.837666i	$0.683919\pi$
$930$		0			0
$931$		−	4.87691e6i		−	0.184404i
$932$		0			0
$933$	−4.20852e7			−1.58280
$934$		0			0
$935$	417144.	−	127935.i	0.0156047	−	0.00478585i
$936$		0			0
$937$		−	1.96219e7i		−	0.730117i	−0.930985	−	0.365059i	$-0.881049\pi$
$937$		−	1.96219e7i		−	0.730117i	0.930985	−	0.365059i	$-0.118951\pi$
$938$		0			0
$939$			8.68063e6i			0.321283i
$940$		0			0
$941$		−	1.28766e7i		−	0.474054i	−0.971503	−	0.237027i	$-0.923827\pi$
$941$		−	1.28766e7i		−	0.474054i	0.971503	−	0.237027i	$-0.0761731\pi$
$942$		0			0
$943$			5.60937e7i			2.05416i
$944$		0			0
$945$	−5.27042e6	−	1.71847e7i	−0.191984	−	0.625984i
$946$		0			0
$947$	3.29327e7			1.19331			0.596654	−	0.802498i	$-0.296496\pi$
$947$	3.29327e7			1.19331			0.596654	+	0.802498i	$0.296496\pi$
$948$		0			0
$949$		−	3.25836e7i		−	1.17445i
$950$		0			0
$951$	5.37451e7			1.92702
$952$		0			0
$953$		−	3.23485e7i		−	1.15378i	−0.816823	−	0.576889i	$-0.804267\pi$
$953$		−	3.23485e7i		−	1.15378i	0.816823	−	0.576889i	$-0.195733\pi$
$954$		0			0
$955$	−3.66397e7	+	1.12371e7i	−1.30000	+	0.398700i
$956$		0			0
$957$	−971547.			−0.0342913
$958$		0			0
$959$	1.27799e7			0.448727
$960$		0			0
$961$	4.54974e7			1.58920
$962$		0			0
$963$	5.77667e7			2.00730
$964$		0			0
$965$	−1.52683e7	−	4.97839e7i	−0.527804	−	1.72096i
$966$		0			0
$967$			5.94417e6i			0.204421i	0.994763	+	0.102210i	$0.0325915\pi$
$967$			5.94417e6i			0.204421i	−0.994763	+	0.102210i	$0.967409\pi$
$968$		0			0
$969$	2.60619e7			0.891654
$970$		0			0
$971$			1.42360e7i			0.484551i	0.970208	+	0.242275i	$0.0778938\pi$
$971$			1.42360e7i			0.484551i	−0.970208	+	0.242275i	$0.922106\pi$
$972$		0			0
$973$	−3.94768e7			−1.33678
$974$		0			0
$975$	−3.25329e7	+	2.20270e7i	−1.09600	+	0.742070i
$976$		0			0
$977$			1.01878e7i			0.341464i	0.985318	+	0.170732i	$0.0546132\pi$
$977$			1.01878e7i			0.341464i	−0.985318	+	0.170732i	$0.945387\pi$
$978$		0			0
$979$			41669.0i			0.00138949i
$980$		0			0
$981$			4.13420e7i			1.37157i
$982$		0			0
$983$		−	1.72803e7i		−	0.570384i	−0.958470	−	0.285192i	$-0.907943\pi$
$983$		−	1.72803e7i		−	0.570384i	0.958470	−	0.285192i	$-0.0920574\pi$
$984$		0			0
$985$	−3.69900e7	+	1.13445e7i	−1.21477	+	0.372560i
$986$		0			0
$987$	−5.50225e7			−1.79783
$988$		0			0
$989$			2.57420e7i			0.836858i
$990$		0			0
$991$	−7.41374e6			−0.239802			−0.119901	−	0.992786i	$-0.538258\pi$
$991$	−7.41374e6			−0.239802			−0.119901	+	0.992786i	$0.538258\pi$
$992$		0			0
$993$		−	7.68715e7i		−	2.47396i
$994$		0			0
$995$	−5.17968e7	+	1.58857e7i	−1.65861	+	0.508684i
$996$		0			0
$997$	−2.79839e7			−0.891601			−0.445800	−	0.895132i	$-0.647081\pi$
$997$	−2.79839e7			−0.891601			−0.445800	+	0.895132i	$0.647081\pi$
$998$		0			0
$999$	−1.69187e7			−0.536355

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	320.6.f.c.289.14	yes	16
4.3	odd	2	inner	320.6.f.c.289.2	yes	16
5.4	even	2	inner	320.6.f.c.289.4	yes	16
8.3	odd	2	inner	320.6.f.c.289.15	yes	16
8.5	even	2	inner	320.6.f.c.289.3	yes	16
20.19	odd	2	inner	320.6.f.c.289.16	yes	16
40.19	odd	2	inner	320.6.f.c.289.1	✓	16
40.29	even	2	inner	320.6.f.c.289.13	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
320.6.f.c.289.1	✓	16	40.19	odd	2	inner
320.6.f.c.289.2	yes	16	4.3	odd	2	inner
320.6.f.c.289.3	yes	16	8.5	even	2	inner
320.6.f.c.289.4	yes	16	5.4	even	2	inner
320.6.f.c.289.13	yes	16	40.29	even	2	inner
320.6.f.c.289.14	yes	16	1.1	even	1	trivial
320.6.f.c.289.15	yes	16	8.3	odd	2	inner
320.6.f.c.289.16	yes	16	20.19	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 320 = 2^{6} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	320.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+24.8046 q^{3} +(-53.4447 + 16.3911i) q^{5} +100.281i q^{7} +372.267 q^{9} +O(q^{10})\)	\(q+24.8046 q^{3} +(-53.4447 + 16.3911i) q^{5} +100.281i q^{7} +372.267 q^{9} +5.36667i q^{11} -506.854 q^{13} +(-1325.67 + 406.573i) q^{15} +1454.37i q^{17} -722.433i q^{19} +2487.43i q^{21} -3086.22i q^{23} +(2587.67 - 1752.03i) q^{25} +3206.40 q^{27} +7298.39i q^{29} -8609.68 q^{31} +133.118i q^{33} +(-1643.72 - 5359.50i) q^{35} -5276.52 q^{37} -12572.3 q^{39} -18175.5 q^{41} -8340.95 q^{43} +(-19895.7 + 6101.84i) q^{45} +22120.2i q^{47} +6750.67 q^{49} +36075.1i q^{51} -13363.7 q^{53} +(-87.9654 - 286.820i) q^{55} -17919.6i q^{57} -10958.8i q^{59} -10982.0i q^{61} +37331.4i q^{63} +(27088.7 - 8307.88i) q^{65} -13007.5 q^{67} -76552.4i q^{69} +36009.0 q^{71} +64286.0i q^{73} +(64186.0 - 43458.3i) q^{75} -538.177 q^{77} +48638.8 q^{79} -10927.3 q^{81} -49784.4 q^{83} +(-23838.7 - 77728.6i) q^{85} +181033. i q^{87} +7764.40 q^{89} -50828.0i q^{91} -213559. q^{93} +(11841.4 + 38610.2i) q^{95} -36848.8i q^{97} +1997.83i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 1904 q^{9}+O(q^{10}) \) 16 * q + 1904 * q^9	\( 16 q + 1904 q^{9} + 880 q^{25} - 100352 q^{41} + 128272 q^{49} + 149760 q^{65} - 154576 q^{81} - 459296 q^{89}+O(q^{100}) \) 16 * q + 1904 * q^9 + 880 * q^25 - 100352 * q^41 + 128272 * q^49 + 149760 * q^65 - 154576 * q^81 - 459296 * q^89

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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