LMFDB - Embedded newform 1792.2.f.a.1791.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1792,2,Mod(1791,1792)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1792, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("1792.1791");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1792 = 2^{8} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1792.f (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:	$14.3091920422$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{12})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{2} + 1 $ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 896)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1791.3
Root			$-0.866025 + 0.500000i$ of defining polynomial
Character	$\chi$	$=$	1792.1791
Dual form			1792.2.f.a.1791.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.732051 q^{3} -2.73205i q^{5} +(-2.00000 - 1.73205i) q^{7} -2.46410 q^{9} +O(q^{10})$	$q+0.732051 q^{3} -2.73205i q^{5} +(-2.00000 - 1.73205i) q^{7} -2.46410 q^{9} +1.46410i q^{11} +1.26795i q^{13} -2.00000i q^{15} +4.00000i q^{17} -4.73205 q^{19} +(-1.46410 - 1.26795i) q^{21} -1.46410i q^{23} -2.46410 q^{25} -4.00000 q^{27} +6.92820 q^{29} -6.92820 q^{31} +1.07180i q^{33} +(-4.73205 + 5.46410i) q^{35} -4.00000 q^{37} +0.928203i q^{39} -10.9282i q^{41} +9.46410i q^{43} +6.73205i q^{45} +6.92820 q^{47} +(1.00000 + 6.92820i) q^{49} +2.92820i q^{51} -6.92820 q^{53} +4.00000 q^{55} -3.46410 q^{57} -14.1962 q^{59} +5.66025i q^{61} +(4.92820 + 4.26795i) q^{63} +3.46410 q^{65} +9.46410i q^{67} -1.07180i q^{69} +11.4641i q^{71} +6.92820i q^{73} -1.80385 q^{75} +(2.53590 - 2.92820i) q^{77} -3.46410i q^{79} +4.46410 q^{81} -7.26795 q^{83} +10.9282 q^{85} +5.07180 q^{87} -1.07180i q^{89} +(2.19615 - 2.53590i) q^{91} -5.07180 q^{93} +12.9282i q^{95} -12.0000i q^{97} -3.60770i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{3} - 8 q^{7} + 4 q^{9}+O(q^{10}) $ 4 * q - 4 * q^3 - 8 * q^7 + 4 * q^9	$ 4 q - 4 q^{3} - 8 q^{7} + 4 q^{9} - 12 q^{19} + 8 q^{21} + 4 q^{25} - 16 q^{27} - 12 q^{35} - 16 q^{37} + 4 q^{49} + 16 q^{55} - 36 q^{59} - 8 q^{63} - 28 q^{75} + 24 q^{77} + 4 q^{81} - 36 q^{83} + 16 q^{85} + 48 q^{87} - 12 q^{91} - 48 q^{93}+O(q^{100}) $ 4 * q - 4 * q^3 - 8 * q^7 + 4 * q^9 - 12 * q^19 + 8 * q^21 + 4 * q^25 - 16 * q^27 - 12 * q^35 - 16 * q^37 + 4 * q^49 + 16 * q^55 - 36 * q^59 - 8 * q^63 - 28 * q^75 + 24 * q^77 + 4 * q^81 - 36 * q^83 + 16 * q^85 + 48 * q^87 - 12 * q^91 - 48 * q^93

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1792\mathbb{Z}\right)^\times$.

$n$	$1023$	$1025$	$1541$
$\chi(n)$	$-1$	$-1$	$1$

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
$3$	0.732051			0.422650			0.211325	−	0.977416i	$-0.432222\pi$
$3$	0.732051			0.422650			0.211325	+	0.977416i	$0.432222\pi$
$4$		0			0
$5$		−	2.73205i		−	1.22181i	−0.791704	−	0.610905i	$-0.790806\pi$
$5$		−	2.73205i		−	1.22181i	0.791704	−	0.610905i	$-0.209194\pi$
$6$		0			0
$7$	−2.00000	−	1.73205i	−0.755929	−	0.654654i
$7$	−2.00000	−	1.73205i	−0.755929	−	0.654654i
$8$		0			0
$9$	−2.46410			−0.821367
$10$		0			0
$11$			1.46410i			0.441443i	0.975337	+	0.220722i	$0.0708412\pi$
$11$			1.46410i			0.441443i	−0.975337	+	0.220722i	$0.929159\pi$
$12$		0			0
$13$			1.26795i			0.351666i	0.984420	+	0.175833i	$0.0562618\pi$
$13$			1.26795i			0.351666i	−0.984420	+	0.175833i	$0.943738\pi$
$14$		0			0
$15$		−	2.00000i		−	0.516398i
$16$		0			0
$17$			4.00000i			0.970143i	0.874475	+	0.485071i	$0.161206\pi$
$17$			4.00000i			0.970143i	−0.874475	+	0.485071i	$0.838794\pi$
$18$		0			0
$19$	−4.73205			−1.08561			−0.542803	−	0.839860i	$-0.682637\pi$
$19$	−4.73205			−1.08561			−0.542803	+	0.839860i	$0.682637\pi$
$20$		0			0
$21$	−1.46410	−	1.26795i	−0.319493	−	0.276689i
$22$		0			0
$23$		−	1.46410i		−	0.305286i	−0.988281	−	0.152643i	$-0.951221\pi$
$23$		−	1.46410i		−	0.305286i	0.988281	−	0.152643i	$-0.0487785\pi$
$24$		0			0
$25$	−2.46410			−0.492820
$26$		0			0
$27$	−4.00000			−0.769800
$28$		0			0
$29$	6.92820			1.28654			0.643268	−	0.765641i	$-0.277578\pi$
$29$	6.92820			1.28654			0.643268	+	0.765641i	$0.277578\pi$
$30$		0			0
$31$	−6.92820			−1.24434			−0.622171	−	0.782881i	$-0.713749\pi$
$31$	−6.92820			−1.24434			−0.622171	+	0.782881i	$0.713749\pi$
$32$		0			0
$33$			1.07180i			0.186576i
$34$		0			0
$35$	−4.73205	+	5.46410i	−0.799863	+	0.923602i
$36$		0			0
$37$	−4.00000			−0.657596			−0.328798	−	0.944400i	$-0.606644\pi$
$37$	−4.00000			−0.657596			−0.328798	+	0.944400i	$0.606644\pi$
$38$		0			0
$39$			0.928203i			0.148631i
$40$		0			0
$41$		−	10.9282i		−	1.70670i	−0.521340	−	0.853349i	$-0.674568\pi$
$41$		−	10.9282i		−	1.70670i	0.521340	−	0.853349i	$-0.325432\pi$
$42$		0			0
$43$			9.46410i			1.44326i	0.692278	+	0.721631i	$0.256607\pi$
$43$			9.46410i			1.44326i	−0.692278	+	0.721631i	$0.743393\pi$
$44$		0			0
$45$			6.73205i			1.00355i
$46$		0			0
$47$	6.92820			1.01058			0.505291	−	0.862949i	$-0.331385\pi$
$47$	6.92820			1.01058			0.505291	+	0.862949i	$0.331385\pi$
$48$		0			0
$49$	1.00000	+	6.92820i	0.142857	+	0.989743i
$50$		0			0
$51$			2.92820i			0.410030i
$52$		0			0
$53$	−6.92820			−0.951662			−0.475831	−	0.879537i	$-0.657853\pi$
$53$	−6.92820			−0.951662			−0.475831	+	0.879537i	$0.657853\pi$
$54$		0			0
$55$	4.00000			0.539360
$56$		0			0
$57$	−3.46410			−0.458831
$58$		0			0
$59$	−14.1962			−1.84818			−0.924091	−	0.382173i	$-0.875176\pi$
$59$	−14.1962			−1.84818			−0.924091	+	0.382173i	$0.875176\pi$
$60$		0			0
$61$			5.66025i			0.724721i	0.932038	+	0.362361i	$0.118029\pi$
$61$			5.66025i			0.724721i	−0.932038	+	0.362361i	$0.881971\pi$
$62$		0			0
$63$	4.92820	+	4.26795i	0.620895	+	0.537711i
$64$		0			0
$65$	3.46410			0.429669
$66$		0			0
$67$			9.46410i			1.15622i	0.815957	+	0.578112i	$0.196210\pi$
$67$			9.46410i			1.15622i	−0.815957	+	0.578112i	$0.803790\pi$
$68$		0			0
$69$		−	1.07180i		−	0.129029i
$70$		0			0
$71$			11.4641i			1.36054i	0.732962	+	0.680269i	$0.238137\pi$
$71$			11.4641i			1.36054i	−0.732962	+	0.680269i	$0.761863\pi$
$72$		0			0
$73$			6.92820i			0.810885i	0.914121	+	0.405442i	$0.132883\pi$
$73$			6.92820i			0.810885i	−0.914121	+	0.405442i	$0.867117\pi$
$74$		0			0
$75$	−1.80385			−0.208290
$76$		0			0
$77$	2.53590	−	2.92820i	0.288992	−	0.333700i
$78$		0			0
$79$		−	3.46410i		−	0.389742i	−0.980829	−	0.194871i	$-0.937571\pi$
$79$		−	3.46410i		−	0.389742i	0.980829	−	0.194871i	$-0.0624288\pi$
$80$		0			0
$81$	4.46410			0.496011
$82$		0			0
$83$	−7.26795			−0.797761			−0.398881	−	0.917003i	$-0.630601\pi$
$83$	−7.26795			−0.797761			−0.398881	+	0.917003i	$0.630601\pi$
$84$		0			0
$85$	10.9282			1.18533
$86$		0			0
$87$	5.07180			0.543754
$88$		0			0
$89$		−	1.07180i		−	0.113610i	−0.998385	−	0.0568051i	$-0.981909\pi$
$89$		−	1.07180i		−	0.113610i	0.998385	−	0.0568051i	$-0.0180914\pi$
$90$		0			0
$91$	2.19615	−	2.53590i	0.230219	−	0.265834i
$92$		0			0
$93$	−5.07180			−0.525921
$94$		0			0
$95$			12.9282i			1.32641i
$96$		0			0
$97$		−	12.0000i		−	1.21842i	−0.793011	−	0.609208i	$-0.791488\pi$
$97$		−	12.0000i		−	1.21842i	0.793011	−	0.609208i	$-0.208512\pi$
$98$		0			0
$99$		−	3.60770i		−	0.362587i
$100$		0			0
$101$		−	4.19615i		−	0.417533i	−0.977966	−	0.208766i	$-0.933055\pi$
$101$		−	4.19615i		−	0.417533i	0.977966	−	0.208766i	$-0.0669448\pi$
$102$		0			0
$103$	−5.07180			−0.499739			−0.249869	−	0.968280i	$-0.580388\pi$
$103$	−5.07180			−0.499739			−0.249869	+	0.968280i	$0.580388\pi$
$104$		0			0
$105$	−3.46410	+	4.00000i	−0.338062	+	0.390360i
$106$		0			0
$107$			12.3923i			1.19801i	0.800746	+	0.599005i	$0.204437\pi$
$107$			12.3923i			1.19801i	−0.800746	+	0.599005i	$0.795563\pi$
$108$		0			0
$109$	−6.92820			−0.663602			−0.331801	−	0.943349i	$-0.607656\pi$
$109$	−6.92820			−0.663602			−0.331801	+	0.943349i	$0.607656\pi$
$110$		0			0
$111$	−2.92820			−0.277933
$112$		0			0
$113$	−2.53590			−0.238557			−0.119279	−	0.992861i	$-0.538058\pi$
$113$	−2.53590			−0.238557			−0.119279	+	0.992861i	$0.538058\pi$
$114$		0			0
$115$	−4.00000			−0.373002
$116$		0			0
$117$		−	3.12436i		−	0.288847i
$118$		0			0
$119$	6.92820	−	8.00000i	0.635107	−	0.733359i
$120$		0			0
$121$	8.85641			0.805128
$122$		0			0
$123$		−	8.00000i		−	0.721336i
$124$		0			0
$125$		−	6.92820i		−	0.619677i
$126$		0			0
$127$		−	14.5359i		−	1.28985i	−0.764245	−	0.644926i	$-0.776888\pi$
$127$		−	14.5359i		−	1.28985i	0.764245	−	0.644926i	$-0.223112\pi$
$128$		0			0
$129$			6.92820i			0.609994i
$130$		0			0
$131$	−7.26795			−0.635004			−0.317502	−	0.948258i	$-0.602844\pi$
$131$	−7.26795			−0.635004			−0.317502	+	0.948258i	$0.602844\pi$
$132$		0			0
$133$	9.46410	+	8.19615i	0.820642	+	0.710697i
$134$		0			0
$135$			10.9282i			0.940550i
$136$		0			0
$137$	7.85641			0.671218			0.335609	−	0.942001i	$-0.391058\pi$
$137$	7.85641			0.671218			0.335609	+	0.942001i	$0.391058\pi$
$138$		0			0
$139$	−6.19615			−0.525551			−0.262775	−	0.964857i	$-0.584638\pi$
$139$	−6.19615			−0.525551			−0.262775	+	0.964857i	$0.584638\pi$
$140$		0			0
$141$	5.07180			0.427122
$142$		0			0
$143$	−1.85641			−0.155241
$144$		0			0
$145$		−	18.9282i		−	1.57190i
$146$		0			0
$147$	0.732051	+	5.07180i	0.0603785	+	0.418315i
$148$		0			0
$149$	−12.0000			−0.983078			−0.491539	−	0.870855i	$-0.663566\pi$
$149$	−12.0000			−0.983078			−0.491539	+	0.870855i	$0.663566\pi$
$150$		0			0
$151$		−	23.3205i		−	1.89780i	−0.315583	−	0.948898i	$-0.602200\pi$
$151$		−	23.3205i		−	1.89780i	0.315583	−	0.948898i	$-0.397800\pi$
$152$		0			0
$153$		−	9.85641i		−	0.796843i
$154$		0			0
$155$			18.9282i			1.52035i
$156$		0			0
$157$		−	22.7321i		−	1.81422i	−0.420899	−	0.907108i	$-0.638285\pi$
$157$		−	22.7321i		−	1.81422i	0.420899	−	0.907108i	$-0.361715\pi$
$158$		0			0
$159$	−5.07180			−0.402220
$160$		0			0
$161$	−2.53590	+	2.92820i	−0.199857	+	0.230775i
$162$		0			0
$163$		−	4.39230i		−	0.344032i	−0.985094	−	0.172016i	$-0.944972\pi$
$163$		−	4.39230i		−	0.344032i	0.985094	−	0.172016i	$-0.0550281\pi$
$164$		0			0
$165$	2.92820			0.227960
$166$		0			0
$167$	−18.9282			−1.46471			−0.732354	−	0.680924i	$-0.761578\pi$
$167$	−18.9282			−1.46471			−0.732354	+	0.680924i	$0.761578\pi$
$168$		0			0
$169$	11.3923			0.876331
$170$		0			0
$171$	11.6603			0.891682
$172$		0			0
$173$		−	6.73205i		−	0.511828i	−0.966700	−	0.255914i	$-0.917624\pi$
$173$		−	6.73205i		−	0.511828i	0.966700	−	0.255914i	$-0.0823764\pi$
$174$		0			0
$175$	4.92820	+	4.26795i	0.372537	+	0.322627i
$176$		0			0
$177$	−10.3923			−0.781133
$178$		0			0
$179$			20.3923i			1.52419i	0.647464	+	0.762096i	$0.275830\pi$
$179$			20.3923i			1.52419i	−0.647464	+	0.762096i	$0.724170\pi$
$180$		0			0
$181$			13.2679i			0.986199i	0.869973	+	0.493099i	$0.164136\pi$
$181$			13.2679i			0.986199i	−0.869973	+	0.493099i	$0.835864\pi$
$182$		0			0
$183$			4.14359i			0.306303i
$184$		0			0
$185$			10.9282i			0.803457i
$186$		0			0
$187$	−5.85641			−0.428263
$188$		0			0
$189$	8.00000	+	6.92820i	0.581914	+	0.503953i
$190$		0			0
$191$		−	4.53590i		−	0.328206i	−0.986443	−	0.164103i	$-0.947527\pi$
$191$		−	4.53590i		−	0.328206i	0.986443	−	0.164103i	$-0.0524730\pi$
$192$		0			0
$193$	−16.3923			−1.17994			−0.589972	−	0.807424i	$-0.700861\pi$
$193$	−16.3923			−1.17994			−0.589972	+	0.807424i	$0.700861\pi$
$194$		0			0
$195$	2.53590			0.181599
$196$		0			0
$197$	12.0000			0.854965			0.427482	−	0.904024i	$-0.359401\pi$
$197$	12.0000			0.854965			0.427482	+	0.904024i	$0.359401\pi$
$198$		0			0
$199$	−24.7846			−1.75693			−0.878467	−	0.477803i	$-0.841433\pi$
$199$	−24.7846			−1.75693			−0.878467	+	0.477803i	$0.841433\pi$
$200$		0			0
$201$			6.92820i			0.488678i
$202$		0			0
$203$	−13.8564	−	12.0000i	−0.972529	−	0.842235i
$204$		0			0
$205$	−29.8564			−2.08526
$206$		0			0
$207$			3.60770i			0.250752i
$208$		0			0
$209$		−	6.92820i		−	0.479234i
$210$		0			0
$211$		−	28.3923i		−	1.95461i	−0.211844	−	0.977303i	$-0.567947\pi$
$211$		−	28.3923i		−	1.95461i	0.211844	−	0.977303i	$-0.432053\pi$
$212$		0			0
$213$			8.39230i			0.575031i
$214$		0			0
$215$	25.8564			1.76339
$216$		0			0
$217$	13.8564	+	12.0000i	0.940634	+	0.814613i
$218$		0			0
$219$			5.07180i			0.342720i
$220$		0			0
$221$	−5.07180			−0.341166
$222$		0			0
$223$	−8.00000			−0.535720			−0.267860	−	0.963458i	$-0.586316\pi$
$223$	−8.00000			−0.535720			−0.267860	+	0.963458i	$0.586316\pi$
$224$		0			0
$225$	6.07180			0.404786
$226$		0			0
$227$	−4.73205			−0.314077			−0.157039	−	0.987592i	$-0.550195\pi$
$227$	−4.73205			−0.314077			−0.157039	+	0.987592i	$0.550195\pi$
$228$		0			0
$229$			17.6603i			1.16702i	0.812105	+	0.583511i	$0.198322\pi$
$229$			17.6603i			1.16702i	−0.812105	+	0.583511i	$0.801678\pi$
$230$		0			0
$231$	1.85641	−	2.14359i	0.122143	−	0.141038i
$232$		0			0
$233$	−7.85641			−0.514690			−0.257345	−	0.966320i	$-0.582848\pi$
$233$	−7.85641			−0.514690			−0.257345	+	0.966320i	$0.582848\pi$
$234$		0			0
$235$		−	18.9282i		−	1.23474i
$236$		0			0
$237$		−	2.53590i		−	0.164724i
$238$		0			0
$239$		−	20.3923i		−	1.31907i	−0.751674	−	0.659534i	$-0.770754\pi$
$239$		−	20.3923i		−	1.31907i	0.751674	−	0.659534i	$-0.229246\pi$
$240$		0			0
$241$		−	12.0000i		−	0.772988i	−0.922292	−	0.386494i	$-0.873686\pi$
$241$		−	12.0000i		−	0.772988i	0.922292	−	0.386494i	$-0.126314\pi$
$242$		0			0
$243$	15.2679			0.979439
$244$		0			0
$245$	18.9282	−	2.73205i	1.20928	−	0.174544i
$246$		0			0
$247$		−	6.00000i		−	0.381771i
$248$		0			0
$249$	−5.32051			−0.337173
$250$		0			0
$251$	16.0526			1.01323			0.506614	−	0.862173i	$-0.330897\pi$
$251$	16.0526			1.01323			0.506614	+	0.862173i	$0.330897\pi$
$252$		0			0
$253$	2.14359			0.134767
$254$		0			0
$255$	8.00000			0.500979
$256$		0			0
$257$			5.85641i			0.365313i	0.983177	+	0.182656i	$0.0584696\pi$
$257$			5.85641i			0.365313i	−0.983177	+	0.182656i	$0.941530\pi$
$258$		0			0
$259$	8.00000	+	6.92820i	0.497096	+	0.430498i
$260$		0			0
$261$	−17.0718			−1.05672
$262$		0			0
$263$		−	4.53590i		−	0.279695i	−0.990173	−	0.139848i	$-0.955339\pi$
$263$		−	4.53590i		−	0.279695i	0.990173	−	0.139848i	$-0.0446613\pi$
$264$		0			0
$265$			18.9282i			1.16275i
$266$		0			0
$267$		−	0.784610i		−	0.0480173i
$268$		0			0
$269$			23.1244i			1.40992i	0.709249	+	0.704958i	$0.249034\pi$
$269$			23.1244i			1.40992i	−0.709249	+	0.704958i	$0.750966\pi$
$270$		0			0
$271$		0			0			−	1.00000i	$-0.5\pi$
$271$		0			0				1.00000i	$0.5\pi$
$272$		0			0
$273$	1.60770	−	1.85641i	0.0973021	−	0.112355i
$274$		0			0
$275$		−	3.60770i		−	0.217552i
$276$		0			0
$277$	12.7846			0.768153			0.384076	−	0.923301i	$-0.374520\pi$
$277$	12.7846			0.768153			0.384076	+	0.923301i	$0.374520\pi$
$278$		0			0
$279$	17.0718			1.02206
$280$		0			0
$281$	−6.00000			−0.357930			−0.178965	−	0.983855i	$-0.557275\pi$
$281$	−6.00000			−0.357930			−0.178965	+	0.983855i	$0.557275\pi$
$282$		0			0
$283$	26.1962			1.55720			0.778600	−	0.627521i	$-0.215930\pi$
$283$	26.1962			1.55720			0.778600	+	0.627521i	$0.215930\pi$
$284$		0			0
$285$			9.46410i			0.560605i
$286$		0			0
$287$	−18.9282	+	21.8564i	−1.11730	+	1.29014i
$288$		0			0
$289$	1.00000			0.0588235
$290$		0			0
$291$		−	8.78461i		−	0.514963i
$292$		0			0
$293$		−	12.1962i		−	0.712507i	−0.934389	−	0.356253i	$-0.884054\pi$
$293$		−	12.1962i		−	0.712507i	0.934389	−	0.356253i	$-0.115946\pi$
$294$		0			0
$295$			38.7846i			2.25813i
$296$		0			0
$297$		−	5.85641i		−	0.339823i
$298$		0			0
$299$	1.85641			0.107359
$300$		0			0
$301$	16.3923	−	18.9282i	0.944837	−	1.09100i
$302$		0			0
$303$		−	3.07180i		−	0.176470i
$304$		0			0
$305$	15.4641			0.885472
$306$		0			0
$307$	16.7321			0.954949			0.477474	−	0.878646i	$-0.341552\pi$
$307$	16.7321			0.954949			0.477474	+	0.878646i	$0.341552\pi$
$308$		0			0
$309$	−3.71281			−0.211215
$310$		0			0
$311$	−1.85641			−0.105267			−0.0526336	−	0.998614i	$-0.516762\pi$
$311$	−1.85641			−0.105267			−0.0526336	+	0.998614i	$0.516762\pi$
$312$		0			0
$313$		−	8.78461i		−	0.496535i	−0.968691	−	0.248268i	$-0.920139\pi$
$313$		−	8.78461i		−	0.496535i	0.968691	−	0.248268i	$-0.0798613\pi$
$314$		0			0
$315$	11.6603	−	13.4641i	0.656981	−	0.758616i
$316$		0			0
$317$	−25.8564			−1.45224			−0.726120	−	0.687568i	$-0.758678\pi$
$317$	−25.8564			−1.45224			−0.726120	+	0.687568i	$0.758678\pi$
$318$		0			0
$319$			10.1436i			0.567932i
$320$		0			0
$321$			9.07180i			0.506338i
$322$		0			0
$323$		−	18.9282i		−	1.05319i
$324$		0			0
$325$		−	3.12436i		−	0.173308i
$326$		0			0
$327$	−5.07180			−0.280471
$328$		0			0
$329$	−13.8564	−	12.0000i	−0.763928	−	0.661581i
$330$		0			0
$331$		−	23.3205i		−	1.28181i	−0.767620	−	0.640906i	$-0.778559\pi$
$331$		−	23.3205i		−	1.28181i	0.767620	−	0.640906i	$-0.221441\pi$
$332$		0			0
$333$	9.85641			0.540128
$334$		0			0
$335$	25.8564			1.41269
$336$		0			0
$337$	−24.3923			−1.32873			−0.664367	−	0.747407i	$-0.731299\pi$
$337$	−24.3923			−1.32873			−0.664367	+	0.747407i	$0.731299\pi$
$338$		0			0
$339$	−1.85641			−0.100826
$340$		0			0
$341$		−	10.1436i		−	0.549306i
$342$		0			0
$343$	10.0000	−	15.5885i	0.539949	−	0.841698i
$344$		0			0
$345$	−2.92820			−0.157649
$346$		0			0
$347$		−	6.53590i		−	0.350865i	−0.984491	−	0.175433i	$-0.943868\pi$
$347$		−	6.53590i		−	0.350865i	0.984491	−	0.175433i	$-0.0561324\pi$
$348$		0			0
$349$			5.66025i			0.302986i	0.988458	+	0.151493i	$0.0484082\pi$
$349$			5.66025i			0.302986i	−0.988458	+	0.151493i	$0.951592\pi$
$350$		0			0
$351$		−	5.07180i		−	0.270712i
$352$		0			0
$353$			2.14359i			0.114092i	0.998372	+	0.0570460i	$0.0181682\pi$
$353$			2.14359i			0.114092i	−0.998372	+	0.0570460i	$0.981832\pi$
$354$		0			0
$355$	31.3205			1.66232
$356$		0			0
$357$	5.07180	−	5.85641i	0.268428	−	0.309954i
$358$		0			0
$359$			20.3923i			1.07626i	0.842860	+	0.538132i	$0.180870\pi$
$359$			20.3923i			1.07626i	−0.842860	+	0.538132i	$0.819130\pi$
$360$		0			0
$361$	3.39230			0.178542
$362$		0			0
$363$	6.48334			0.340287
$364$		0			0
$365$	18.9282			0.990747
$366$		0			0
$367$	8.00000			0.417597			0.208798	−	0.977959i	$-0.433045\pi$
$367$	8.00000			0.417597			0.208798	+	0.977959i	$0.433045\pi$
$368$		0			0
$369$			26.9282i			1.40183i
$370$		0			0
$371$	13.8564	+	12.0000i	0.719389	+	0.623009i
$372$		0			0
$373$	−30.9282			−1.60140			−0.800701	−	0.599064i	$-0.795539\pi$
$373$	−30.9282			−1.60140			−0.800701	+	0.599064i	$0.795539\pi$
$374$		0			0
$375$		−	5.07180i		−	0.261906i
$376$		0			0
$377$			8.78461i			0.452430i
$378$		0			0
$379$			14.5359i			0.746659i	0.927699	+	0.373329i	$0.121784\pi$
$379$			14.5359i			0.746659i	−0.927699	+	0.373329i	$0.878216\pi$
$380$		0			0
$381$		−	10.6410i		−	0.545156i
$382$		0			0
$383$	6.92820			0.354015			0.177007	−	0.984210i	$-0.443358\pi$
$383$	6.92820			0.354015			0.177007	+	0.984210i	$0.443358\pi$
$384$		0			0
$385$	−8.00000	−	6.92820i	−0.407718	−	0.353094i
$386$		0			0
$387$		−	23.3205i		−	1.18545i
$388$		0			0
$389$	−25.8564			−1.31097			−0.655486	−	0.755207i	$-0.727536\pi$
$389$	−25.8564			−1.31097			−0.655486	+	0.755207i	$0.727536\pi$
$390$		0			0
$391$	5.85641			0.296171
$392$		0			0
$393$	−5.32051			−0.268384
$394$		0			0
$395$	−9.46410			−0.476191
$396$		0			0
$397$		−	35.9090i		−	1.80222i	−0.433591	−	0.901110i	$-0.642754\pi$
$397$		−	35.9090i		−	1.80222i	0.433591	−	0.901110i	$-0.357246\pi$
$398$		0			0
$399$	6.92820	+	6.00000i	0.346844	+	0.300376i
$400$		0			0
$401$	11.3205			0.565319			0.282660	−	0.959220i	$-0.408783\pi$
$401$	11.3205			0.565319			0.282660	+	0.959220i	$0.408783\pi$
$402$		0			0
$403$		−	8.78461i		−	0.437593i
$404$		0			0
$405$		−	12.1962i		−	0.606032i
$406$		0			0
$407$		−	5.85641i		−	0.290291i
$408$		0			0
$409$		−	18.9282i		−	0.935939i	−0.883745	−	0.467970i	$-0.844986\pi$
$409$		−	18.9282i		−	0.935939i	0.883745	−	0.467970i	$-0.155014\pi$
$410$		0			0
$411$	5.75129			0.283690
$412$		0			0
$413$	28.3923	+	24.5885i	1.39709	+	1.20992i
$414$		0			0
$415$			19.8564i			0.974713i
$416$		0			0
$417$	−4.53590			−0.222124
$418$		0			0
$419$	−21.1244			−1.03199			−0.515996	−	0.856591i	$-0.672578\pi$
$419$	−21.1244			−1.03199			−0.515996	+	0.856591i	$0.672578\pi$
$420$		0			0
$421$	20.0000			0.974740			0.487370	−	0.873195i	$-0.337956\pi$
$421$	20.0000			0.974740			0.487370	+	0.873195i	$0.337956\pi$
$422$		0			0
$423$	−17.0718			−0.830059
$424$		0			0
$425$		−	9.85641i		−	0.478106i
$426$		0			0
$427$	9.80385	−	11.3205i	0.474441	−	0.547838i
$428$		0			0
$429$	−1.35898			−0.0656124
$430$		0			0
$431$			1.46410i			0.0705233i	0.999378	+	0.0352616i	$0.0112265\pi$
$431$			1.46410i			0.0705233i	−0.999378	+	0.0352616i	$0.988774\pi$
$432$		0			0
$433$			25.8564i			1.24258i	0.783581	+	0.621290i	$0.213391\pi$
$433$			25.8564i			1.24258i	−0.783581	+	0.621290i	$0.786609\pi$
$434$		0			0
$435$		−	13.8564i		−	0.664364i
$436$		0			0
$437$			6.92820i			0.331421i
$438$		0			0
$439$	4.00000			0.190910			0.0954548	−	0.995434i	$-0.469569\pi$
$439$	4.00000			0.190910			0.0954548	+	0.995434i	$0.469569\pi$
$440$		0			0
$441$	−2.46410	−	17.0718i	−0.117338	−	0.812943i
$442$		0			0
$443$		−	1.46410i		−	0.0695616i	−0.999395	−	0.0347808i	$-0.988927\pi$
$443$		−	1.46410i		−	0.0695616i	0.999395	−	0.0347808i	$-0.0110733\pi$
$444$		0			0
$445$	−2.92820			−0.138810
$446$		0			0
$447$	−8.78461			−0.415498
$448$		0			0
$449$	31.8564			1.50340			0.751698	−	0.659507i	$-0.229235\pi$
$449$	31.8564			1.50340			0.751698	+	0.659507i	$0.229235\pi$
$450$		0			0
$451$	16.0000			0.753411
$452$		0			0
$453$		−	17.0718i		−	0.802103i
$454$		0			0
$455$	−6.92820	−	6.00000i	−0.324799	−	0.281284i
$456$		0			0
$457$	8.39230			0.392575			0.196288	−	0.980546i	$-0.437111\pi$
$457$	8.39230			0.392575			0.196288	+	0.980546i	$0.437111\pi$
$458$		0			0
$459$		−	16.0000i		−	0.746816i
$460$		0			0
$461$			27.5167i			1.28158i	0.767717	+	0.640789i	$0.221393\pi$
$461$			27.5167i			1.28158i	−0.767717	+	0.640789i	$0.778607\pi$
$462$		0			0
$463$		−	20.5359i		−	0.954384i	−0.878799	−	0.477192i	$-0.841655\pi$
$463$		−	20.5359i		−	0.954384i	0.878799	−	0.477192i	$-0.158345\pi$
$464$		0			0
$465$			13.8564i			0.642575i
$466$		0			0
$467$	−18.5885			−0.860171			−0.430086	−	0.902788i	$-0.641517\pi$
$467$	−18.5885			−0.860171			−0.430086	+	0.902788i	$0.641517\pi$
$468$		0			0
$469$	16.3923	−	18.9282i	0.756926	−	0.874023i
$470$		0			0
$471$		−	16.6410i		−	0.766778i
$472$		0			0
$473$	−13.8564			−0.637118
$474$		0			0
$475$	11.6603			0.535009
$476$		0			0
$477$	17.0718			0.781664
$478$		0			0
$479$	−30.9282			−1.41315			−0.706573	−	0.707640i	$-0.749760\pi$
$479$	−30.9282			−1.41315			−0.706573	+	0.707640i	$0.749760\pi$
$480$		0			0
$481$		−	5.07180i		−	0.231254i
$482$		0			0
$483$	−1.85641	+	2.14359i	−0.0844694	+	0.0975369i
$484$		0			0
$485$	−32.7846			−1.48867
$486$		0			0
$487$			4.39230i			0.199034i	0.995036	+	0.0995172i	$0.0317298\pi$
$487$			4.39230i			0.199034i	−0.995036	+	0.0995172i	$0.968270\pi$
$488$		0			0
$489$		−	3.21539i		−	0.145405i
$490$		0			0
$491$			15.3205i			0.691405i	0.938344	+	0.345702i	$0.112359\pi$
$491$			15.3205i			0.691405i	−0.938344	+	0.345702i	$0.887641\pi$
$492$		0			0
$493$			27.7128i			1.24812i
$494$		0			0
$495$	−9.85641			−0.443013
$496$		0			0
$497$	19.8564	−	22.9282i	0.890682	−	1.02847i
$498$		0			0
$499$		−	4.39230i		−	0.196627i	−0.995156	−	0.0983133i	$-0.968655\pi$
$499$		−	4.39230i		−	0.196627i	0.995156	−	0.0983133i	$-0.0313447\pi$
$500$		0			0
$501$	−13.8564			−0.619059
$502$		0			0
$503$	−12.0000			−0.535054			−0.267527	−	0.963550i	$-0.586206\pi$
$503$	−12.0000			−0.535054			−0.267527	+	0.963550i	$0.586206\pi$
$504$		0			0
$505$	−11.4641			−0.510146
$506$		0			0
$507$	8.33975			0.370381
$508$		0			0
$509$			31.8038i			1.40968i	0.709366	+	0.704840i	$0.248981\pi$
$509$			31.8038i			1.40968i	−0.709366	+	0.704840i	$0.751019\pi$
$510$		0			0
$511$	12.0000	−	13.8564i	0.530849	−	0.612971i
$512$		0			0
$513$	18.9282			0.835701
$514$		0			0
$515$			13.8564i			0.610586i
$516$		0			0
$517$			10.1436i			0.446115i
$518$		0			0
$519$		−	4.92820i		−	0.216324i
$520$		0			0
$521$		−	26.9282i		−	1.17975i	−0.807496	−	0.589873i	$-0.799178\pi$
$521$		−	26.9282i		−	1.17975i	0.807496	−	0.589873i	$-0.200822\pi$
$522$		0			0
$523$	43.3731			1.89657			0.948286	−	0.317417i	$-0.102816\pi$
$523$	43.3731			1.89657			0.948286	+	0.317417i	$0.102816\pi$
$524$		0			0
$525$	3.60770	+	3.12436i	0.157453	+	0.136358i
$526$		0			0
$527$		−	27.7128i		−	1.20719i
$528$		0			0
$529$	20.8564			0.906800
$530$		0			0
$531$	34.9808			1.51804
$532$		0			0
$533$	13.8564			0.600188
$534$		0			0
$535$	33.8564			1.46374
$536$		0			0
$537$			14.9282i			0.644200i
$538$		0			0
$539$	−10.1436	+	1.46410i	−0.436916	+	0.0630633i
$540$		0			0
$541$	−28.0000			−1.20381			−0.601907	−	0.798566i	$-0.705592\pi$
$541$	−28.0000			−1.20381			−0.601907	+	0.798566i	$0.705592\pi$
$542$		0			0
$543$			9.71281i			0.416817i
$544$		0			0
$545$			18.9282i			0.810795i
$546$		0			0
$547$		−	14.5359i		−	0.621510i	−0.950490	−	0.310755i	$-0.899418\pi$
$547$		−	14.5359i		−	0.621510i	0.950490	−	0.310755i	$-0.100582\pi$
$548$		0			0
$549$		−	13.9474i		−	0.595262i
$550$		0			0
$551$	−32.7846			−1.39667
$552$		0			0
$553$	−6.00000	+	6.92820i	−0.255146	+	0.294617i
$554$		0			0
$555$			8.00000i			0.339581i
$556$		0			0
$557$	−17.0718			−0.723355			−0.361678	−	0.932303i	$-0.617796\pi$
$557$	−17.0718			−0.723355			−0.361678	+	0.932303i	$0.617796\pi$
$558$		0			0
$559$	−12.0000			−0.507546
$560$		0			0
$561$	−4.28719			−0.181005
$562$		0			0
$563$	22.9808			0.968524			0.484262	−	0.874923i	$-0.339088\pi$
$563$	22.9808			0.968524			0.484262	+	0.874923i	$0.339088\pi$
$564$		0			0
$565$			6.92820i			0.291472i
$566$		0			0
$567$	−8.92820	−	7.73205i	−0.374949	−	0.324716i
$568$		0			0
$569$	−7.60770			−0.318931			−0.159466	−	0.987203i	$-0.550977\pi$
$569$	−7.60770			−0.318931			−0.159466	+	0.987203i	$0.550977\pi$
$570$		0			0
$571$		−	18.2487i		−	0.763685i	−0.924227	−	0.381842i	$-0.875290\pi$
$571$		−	18.2487i		−	0.763685i	0.924227	−	0.381842i	$-0.124710\pi$
$572$		0			0
$573$		−	3.32051i		−	0.138716i
$574$		0			0
$575$			3.60770i			0.150451i
$576$		0			0
$577$		−	37.8564i		−	1.57598i	−0.615686	−	0.787991i	$-0.711121\pi$
$577$		−	37.8564i		−	1.57598i	0.615686	−	0.787991i	$-0.288879\pi$
$578$		0			0
$579$	−12.0000			−0.498703
$580$		0			0
$581$	14.5359	+	12.5885i	0.603051	+	0.522257i
$582$		0			0
$583$		−	10.1436i		−	0.420105i
$584$		0			0
$585$	−8.53590			−0.352916
$586$		0			0
$587$	−14.1962			−0.585938			−0.292969	−	0.956122i	$-0.594643\pi$
$587$	−14.1962			−0.585938			−0.292969	+	0.956122i	$0.594643\pi$
$588$		0			0
$589$	32.7846			1.35087
$590$		0			0
$591$	8.78461			0.361351
$592$		0			0
$593$			35.7128i			1.46655i	0.679932	+	0.733275i	$0.262009\pi$
$593$			35.7128i			1.46655i	−0.679932	+	0.733275i	$0.737991\pi$
$594$		0			0
$595$	−21.8564	−	18.9282i	−0.896025	−	0.775981i
$596$		0			0
$597$	−18.1436			−0.742568
$598$		0			0
$599$		−	19.4641i		−	0.795282i	−0.917541	−	0.397641i	$-0.869829\pi$
$599$		−	19.4641i		−	0.795282i	0.917541	−	0.397641i	$-0.130171\pi$
$600$		0			0
$601$			34.6410i			1.41304i	0.707695	+	0.706518i	$0.249735\pi$
$601$			34.6410i			1.41304i	−0.707695	+	0.706518i	$0.750265\pi$
$602$		0			0
$603$		−	23.3205i		−	0.949685i
$604$		0			0
$605$		−	24.1962i		−	0.983713i
$606$		0			0
$607$	−8.00000			−0.324710			−0.162355	−	0.986732i	$-0.551909\pi$
$607$	−8.00000			−0.324710			−0.162355	+	0.986732i	$0.551909\pi$
$608$		0			0
$609$	−10.1436	−	8.78461i	−0.411039	−	0.355970i
$610$		0			0
$611$			8.78461i			0.355387i
$612$		0			0
$613$	−1.85641			−0.0749796			−0.0374898	−	0.999297i	$-0.511936\pi$
$613$	−1.85641			−0.0749796			−0.0374898	+	0.999297i	$0.511936\pi$
$614$		0			0
$615$	−21.8564			−0.881335
$616$		0			0
$617$	16.3923			0.659929			0.329965	−	0.943993i	$-0.392963\pi$
$617$	16.3923			0.659929			0.329965	+	0.943993i	$0.392963\pi$
$618$		0			0
$619$	−13.8038			−0.554823			−0.277412	−	0.960751i	$-0.589477\pi$
$619$	−13.8038			−0.554823			−0.277412	+	0.960751i	$0.589477\pi$
$620$		0			0
$621$			5.85641i			0.235009i
$622$		0			0
$623$	−1.85641	+	2.14359i	−0.0743754	+	0.0858813i
$624$		0			0
$625$	−31.2487			−1.24995
$626$		0			0
$627$		−	5.07180i		−	0.202548i
$628$		0			0
$629$		−	16.0000i		−	0.637962i
$630$		0			0
$631$			24.2487i			0.965326i	0.875806	+	0.482663i	$0.160330\pi$
$631$			24.2487i			0.965326i	−0.875806	+	0.482663i	$0.839670\pi$
$632$		0			0
$633$		−	20.7846i		−	0.826114i
$634$		0			0
$635$	−39.7128			−1.57595
$636$		0			0
$637$	−8.78461	+	1.26795i	−0.348059	+	0.0502380i
$638$		0			0
$639$		−	28.2487i		−	1.11750i
$640$		0			0
$641$	−26.5359			−1.04810			−0.524052	−	0.851686i	$-0.675580\pi$
$641$	−26.5359			−1.04810			−0.524052	+	0.851686i	$0.675580\pi$
$642$		0			0
$643$	19.2679			0.759854			0.379927	−	0.925017i	$-0.375949\pi$
$643$	19.2679			0.759854			0.379927	+	0.925017i	$0.375949\pi$
$644$		0			0
$645$	18.9282			0.745297
$646$		0			0
$647$	46.6410			1.83365			0.916824	−	0.399292i	$-0.130744\pi$
$647$	46.6410			1.83365			0.916824	+	0.399292i	$0.130744\pi$
$648$		0			0
$649$		−	20.7846i		−	0.815867i
$650$		0			0
$651$	10.1436	+	8.78461i	0.397559	+	0.344296i
$652$		0			0
$653$	−12.0000			−0.469596			−0.234798	−	0.972044i	$-0.575443\pi$
$653$	−12.0000			−0.469596			−0.234798	+	0.972044i	$0.575443\pi$
$654$		0			0
$655$			19.8564i			0.775854i
$656$		0			0
$657$		−	17.0718i		−	0.666034i
$658$		0			0
$659$			40.1051i			1.56227i	0.624360	+	0.781137i	$0.285360\pi$
$659$			40.1051i			1.56227i	−0.624360	+	0.781137i	$0.714640\pi$
$660$		0			0
$661$			3.80385i			0.147953i	0.997260	+	0.0739763i	$0.0235689\pi$
$661$			3.80385i			0.147953i	−0.997260	+	0.0739763i	$0.976431\pi$
$662$		0			0
$663$	−3.71281			−0.144194
$664$		0			0
$665$	22.3923	−	25.8564i	0.868336	−	1.00267i
$666$		0			0
$667$		−	10.1436i		−	0.392762i
$668$		0			0
$669$	−5.85641			−0.226422
$670$		0			0
$671$	−8.28719			−0.319923
$672$		0			0
$673$	30.0000			1.15642			0.578208	−	0.815890i	$-0.303752\pi$
$673$	30.0000			1.15642			0.578208	+	0.815890i	$0.303752\pi$
$674$		0			0
$675$	9.85641			0.379373
$676$		0			0
$677$			43.1244i			1.65740i	0.559690	+	0.828702i	$0.310920\pi$
$677$			43.1244i			1.65740i	−0.559690	+	0.828702i	$0.689080\pi$
$678$		0			0
$679$	−20.7846	+	24.0000i	−0.797640	+	0.921035i
$680$		0			0
$681$	−3.46410			−0.132745
$682$		0			0
$683$			22.5359i			0.862312i	0.902277	+	0.431156i	$0.141894\pi$
$683$			22.5359i			0.862312i	−0.902277	+	0.431156i	$0.858106\pi$
$684$		0			0
$685$		−	21.4641i		−	0.820101i
$686$		0			0
$687$			12.9282i			0.493242i
$688$		0			0
$689$		−	8.78461i		−	0.334667i
$690$		0			0
$691$	−18.9808			−0.722062			−0.361031	−	0.932554i	$-0.617575\pi$
$691$	−18.9808			−0.722062			−0.361031	+	0.932554i	$0.617575\pi$
$692$		0			0
$693$	−6.24871	+	7.21539i	−0.237369	+	0.274090i
$694$		0			0
$695$			16.9282i			0.642123i
$696$		0			0
$697$	43.7128			1.65574
$698$		0			0
$699$	−5.75129			−0.217534
$700$		0			0
$701$	25.8564			0.976583			0.488291	−	0.872681i	$-0.337620\pi$
$701$	25.8564			0.976583			0.488291	+	0.872681i	$0.337620\pi$
$702$		0			0
$703$	18.9282			0.713891
$704$		0			0
$705$		−	13.8564i		−	0.521862i
$706$		0			0
$707$	−7.26795	+	8.39230i	−0.273339	+	0.315625i
$708$		0			0
$709$	−17.0718			−0.641145			−0.320572	−	0.947224i	$-0.603875\pi$
$709$	−17.0718			−0.641145			−0.320572	+	0.947224i	$0.603875\pi$
$710$		0			0
$711$			8.53590i			0.320121i
$712$		0			0
$713$			10.1436i			0.379881i
$714$		0			0
$715$			5.07180i			0.189674i
$716$		0			0
$717$		−	14.9282i		−	0.557504i
$718$		0			0
$719$	30.9282			1.15343			0.576714	−	0.816946i	$-0.304335\pi$
$719$	30.9282			1.15343			0.576714	+	0.816946i	$0.304335\pi$
$720$		0			0
$721$	10.1436	+	8.78461i	0.377767	+	0.327156i
$722$		0			0
$723$		−	8.78461i		−	0.326703i
$724$		0			0
$725$	−17.0718			−0.634031
$726$		0			0
$727$	32.7846			1.21591			0.607957	−	0.793970i	$-0.291989\pi$
$727$	32.7846			1.21591			0.607957	+	0.793970i	$0.291989\pi$
$728$		0			0
$729$	−2.21539			−0.0820515
$730$		0			0
$731$	−37.8564			−1.40017
$732$		0			0
$733$		−	8.19615i		−	0.302732i	−0.988478	−	0.151366i	$-0.951633\pi$
$733$		−	8.19615i		−	0.302732i	0.988478	−	0.151366i	$-0.0483672\pi$
$734$		0			0
$735$	13.8564	−	2.00000i	0.511101	−	0.0737711i
$736$		0			0
$737$	−13.8564			−0.510407
$738$		0			0
$739$			23.3205i			0.857859i	0.903338	+	0.428929i	$0.141109\pi$
$739$			23.3205i			0.857859i	−0.903338	+	0.428929i	$0.858891\pi$
$740$		0			0
$741$		−	4.39230i		−	0.161355i
$742$		0			0
$743$		−	17.4641i		−	0.640696i	−0.947300	−	0.320348i	$-0.896200\pi$
$743$		−	17.4641i		−	0.640696i	0.947300	−	0.320348i	$-0.103800\pi$
$744$		0			0
$745$			32.7846i			1.20114i
$746$		0			0
$747$	17.9090			0.655255
$748$		0			0
$749$	21.4641	−	24.7846i	0.784281	−	0.905610i
$750$		0			0
$751$			33.4641i			1.22112i	0.791969	+	0.610561i	$0.209056\pi$
$751$			33.4641i			1.22112i	−0.791969	+	0.610561i	$0.790944\pi$
$752$		0			0
$753$	11.7513			0.428241
$754$		0			0
$755$	−63.7128			−2.31875
$756$		0			0
$757$	53.5692			1.94701			0.973503	−	0.228673i	$-0.0734388\pi$
$757$	53.5692			1.94701			0.973503	+	0.228673i	$0.0734388\pi$
$758$		0			0
$759$	1.56922			0.0569591
$760$		0			0
$761$			24.7846i			0.898441i	0.893421	+	0.449221i	$0.148298\pi$
$761$			24.7846i			0.898441i	−0.893421	+	0.449221i	$0.851702\pi$
$762$		0			0
$763$	13.8564	+	12.0000i	0.501636	+	0.434429i
$764$		0			0
$765$	−26.9282			−0.973591
$766$		0			0
$767$		−	18.0000i		−	0.649942i
$768$		0			0
$769$			25.8564i			0.932406i	0.884678	+	0.466203i	$0.154378\pi$
$769$			25.8564i			0.932406i	−0.884678	+	0.466203i	$0.845622\pi$
$770$		0			0
$771$			4.28719i			0.154399i
$772$		0			0
$773$		−	22.4449i		−	0.807286i	−0.914917	−	0.403643i	$-0.867744\pi$
$773$		−	22.4449i		−	0.807286i	0.914917	−	0.403643i	$-0.132256\pi$
$774$		0			0
$775$	17.0718			0.613237
$776$		0			0
$777$	5.85641	+	5.07180i	0.210097	+	0.181950i
$778$		0			0
$779$			51.7128i			1.85280i
$780$		0			0
$781$	−16.7846			−0.600601
$782$		0			0
$783$	−27.7128			−0.990375
$784$		0			0
$785$	−62.1051			−2.21663
$786$		0			0
$787$	−10.5885			−0.377438			−0.188719	−	0.982031i	$-0.560433\pi$
$787$	−10.5885			−0.377438			−0.188719	+	0.982031i	$0.560433\pi$
$788$		0			0
$789$		−	3.32051i		−	0.118213i
$790$		0			0
$791$	5.07180	+	4.39230i	0.180332	+	0.156172i
$792$		0			0
$793$	−7.17691			−0.254860
$794$		0			0
$795$			13.8564i			0.491436i
$796$		0			0
$797$		−	16.8756i		−	0.597766i	−0.954290	−	0.298883i	$-0.903386\pi$
$797$		−	16.8756i		−	0.597766i	0.954290	−	0.298883i	$-0.0966140\pi$
$798$		0			0
$799$			27.7128i			0.980409i
$800$		0			0
$801$			2.64102i			0.0933157i
$802$		0			0
$803$	−10.1436			−0.357960
$804$		0			0
$805$	8.00000	+	6.92820i	0.281963	+	0.244187i
$806$		0			0
$807$			16.9282i			0.595901i
$808$		0			0
$809$	−25.1769			−0.885173			−0.442587	−	0.896726i	$-0.645939\pi$
$809$	−25.1769			−0.885173			−0.442587	+	0.896726i	$0.645939\pi$
$810$		0			0
$811$	−23.3731			−0.820739			−0.410370	−	0.911919i	$-0.634600\pi$
$811$	−23.3731			−0.820739			−0.410370	+	0.911919i	$0.634600\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−12.0000			−0.420342
$816$		0			0
$817$		−	44.7846i		−	1.56682i
$818$		0			0
$819$	−5.41154	+	6.24871i	−0.189095	+	0.218348i
$820$		0			0
$821$	6.92820			0.241796			0.120898	−	0.992665i	$-0.461423\pi$
$821$	6.92820			0.241796			0.120898	+	0.992665i	$0.461423\pi$
$822$		0			0
$823$			27.4641i			0.957338i	0.877995	+	0.478669i	$0.158881\pi$
$823$			27.4641i			0.957338i	−0.877995	+	0.478669i	$0.841119\pi$
$824$		0			0
$825$		−	2.64102i		−	0.0919484i
$826$		0			0
$827$		−	8.67949i		−	0.301816i	−0.988548	−	0.150908i	$-0.951780\pi$
$827$		−	8.67949i		−	0.301816i	0.988548	−	0.150908i	$-0.0482196\pi$
$828$		0			0
$829$		−	8.19615i		−	0.284664i	−0.989819	−	0.142332i	$-0.954540\pi$
$829$		−	8.19615i		−	0.284664i	0.989819	−	0.142332i	$-0.0454601\pi$
$830$		0			0
$831$	9.35898			0.324660
$832$		0			0
$833$	−27.7128	+	4.00000i	−0.960192	+	0.138592i
$834$		0			0
$835$			51.7128i			1.78960i
$836$		0			0
$837$	27.7128			0.957895
$838$		0			0
$839$	18.9282			0.653474			0.326737	−	0.945115i	$-0.394051\pi$
$839$	18.9282			0.653474			0.326737	+	0.945115i	$0.394051\pi$
$840$		0			0
$841$	19.0000			0.655172
$842$		0			0
$843$	−4.39230			−0.151279
$844$		0			0
$845$		−	31.1244i		−	1.07071i
$846$		0			0
$847$	−17.7128	−	15.3397i	−0.608619	−	0.527080i
$848$		0			0
$849$	19.1769			0.658150
$850$		0			0
$851$			5.85641i			0.200755i
$852$		0			0
$853$			21.3731i			0.731800i	0.930654	+	0.365900i	$0.119239\pi$
$853$			21.3731i			0.731800i	−0.930654	+	0.365900i	$0.880761\pi$
$854$		0			0
$855$		−	31.8564i		−	1.08947i
$856$		0			0
$857$			34.9282i			1.19312i	0.802567	+	0.596562i	$0.203467\pi$
$857$			34.9282i			1.19312i	−0.802567	+	0.596562i	$0.796533\pi$
$858$		0			0
$859$	−25.5167			−0.870617			−0.435309	−	0.900281i	$-0.643361\pi$
$859$	−25.5167			−0.870617			−0.435309	+	0.900281i	$0.643361\pi$
$860$		0			0
$861$	−13.8564	+	16.0000i	−0.472225	+	0.545279i
$862$		0			0
$863$			23.1769i			0.788951i	0.918906	+	0.394476i	$0.129074\pi$
$863$			23.1769i			0.788951i	−0.918906	+	0.394476i	$0.870926\pi$
$864$		0			0
$865$	−18.3923			−0.625357
$866$		0			0
$867$	0.732051			0.0248617
$868$		0			0
$869$	5.07180			0.172049
$870$		0			0
$871$	−12.0000			−0.406604
$872$		0			0
$873$			29.5692i			1.00077i
$874$		0			0
$875$	−12.0000	+	13.8564i	−0.405674	+	0.468432i
$876$		0			0
$877$	−53.5692			−1.80890			−0.904452	−	0.426575i	$-0.859720\pi$
$877$	−53.5692			−1.80890			−0.904452	+	0.426575i	$0.859720\pi$
$878$		0			0
$879$		−	8.92820i		−	0.301141i
$880$		0			0
$881$		−	19.7128i		−	0.664142i	−0.943254	−	0.332071i	$-0.892253\pi$
$881$		−	19.7128i		−	0.664142i	0.943254	−	0.332071i	$-0.107747\pi$
$882$		0			0
$883$		−	9.46410i		−	0.318492i	−0.987239	−	0.159246i	$-0.949094\pi$
$883$		−	9.46410i		−	0.318492i	0.987239	−	0.159246i	$-0.0509064\pi$
$884$		0			0
$885$			28.3923i			0.954397i
$886$		0			0
$887$	−42.9282			−1.44139			−0.720694	−	0.693253i	$-0.756177\pi$
$887$	−42.9282			−1.44139			−0.720694	+	0.693253i	$0.756177\pi$
$888$		0			0
$889$	−25.1769	+	29.0718i	−0.844407	+	0.975037i
$890$		0			0
$891$			6.53590i			0.218961i
$892$		0			0
$893$	−32.7846			−1.09710
$894$		0			0
$895$	55.7128			1.86227
$896$		0			0
$897$	1.35898			0.0453751
$898$		0			0
$899$	−48.0000			−1.60089
$900$		0			0
$901$		−	27.7128i		−	0.923248i
$902$		0			0
$903$	12.0000	−	13.8564i	0.399335	−	0.461112i
$904$		0			0
$905$	36.2487			1.20495
$906$		0			0
$907$			32.1051i			1.06603i	0.846105	+	0.533016i	$0.178942\pi$
$907$			32.1051i			1.06603i	−0.846105	+	0.533016i	$0.821058\pi$
$908$		0			0
$909$			10.3397i			0.342948i
$910$		0			0
$911$			11.6077i			0.384580i	0.981338	+	0.192290i	$0.0615914\pi$
$911$			11.6077i			0.384580i	−0.981338	+	0.192290i	$0.938409\pi$
$912$		0			0
$913$		−	10.6410i		−	0.352166i
$914$		0			0
$915$	11.3205			0.374244
$916$		0			0
$917$	14.5359	+	12.5885i	0.480018	+	0.415707i
$918$		0			0
$919$		−	20.5359i		−	0.677417i	−0.940891	−	0.338708i	$-0.890010\pi$
$919$		−	20.5359i		−	0.677417i	0.940891	−	0.338708i	$-0.109990\pi$
$920$		0			0
$921$	12.2487			0.403609
$922$		0			0
$923$	−14.5359			−0.478455
$924$		0			0
$925$	9.85641			0.324077
$926$		0			0
$927$	12.4974			0.410469
$928$		0			0
$929$		−	4.00000i		−	0.131236i	−0.997845	−	0.0656179i	$-0.979098\pi$
$929$		−	4.00000i		−	0.131236i	0.997845	−	0.0656179i	$-0.0209018\pi$
$930$		0			0
$931$	−4.73205	−	32.7846i	−0.155087	−	1.07447i
$932$		0			0
$933$	−1.35898			−0.0444911
$934$		0			0
$935$			16.0000i			0.523256i
$936$		0			0
$937$			17.0718i			0.557711i	0.960333	+	0.278856i	$0.0899551\pi$
$937$			17.0718i			0.557711i	−0.960333	+	0.278856i	$0.910045\pi$
$938$		0			0
$939$		−	6.43078i		−	0.209861i
$940$		0			0
$941$			7.12436i			0.232247i	0.993235	+	0.116124i	$0.0370469\pi$
$941$			7.12436i			0.232247i	−0.993235	+	0.116124i	$0.962953\pi$
$942$		0			0
$943$	−16.0000			−0.521032
$944$		0			0
$945$	18.9282	−	21.8564i	0.615734	−	0.710989i
$946$		0			0
$947$			6.53590i			0.212388i	0.994345	+	0.106194i	$0.0338665\pi$
$947$			6.53590i			0.212388i	−0.994345	+	0.106194i	$0.966134\pi$
$948$		0			0
$949$	−8.78461			−0.285160
$950$		0			0
$951$	−18.9282			−0.613789
$952$		0			0
$953$	−21.7128			−0.703347			−0.351674	−	0.936123i	$-0.614387\pi$
$953$	−21.7128			−0.703347			−0.351674	+	0.936123i	$0.614387\pi$
$954$		0			0
$955$	−12.3923			−0.401006
$956$		0			0
$957$			7.42563i			0.240036i
$958$		0			0
$959$	−15.7128	−	13.6077i	−0.507393	−	0.439415i
$960$		0			0
$961$	17.0000			0.548387
$962$		0			0
$963$		−	30.5359i		−	0.984006i
$964$		0			0
$965$			44.7846i			1.44167i
$966$		0			0
$967$		−	23.3205i		−	0.749937i	−0.927038	−	0.374968i	$-0.877654\pi$
$967$		−	23.3205i		−	0.749937i	0.927038	−	0.374968i	$-0.122346\pi$
$968$		0			0
$969$		−	13.8564i		−	0.445132i
$970$		0			0
$971$	−10.4833			−0.336426			−0.168213	−	0.985751i	$-0.553800\pi$
$971$	−10.4833			−0.336426			−0.168213	+	0.985751i	$0.553800\pi$
$972$		0			0
$973$	12.3923	+	10.7321i	0.397279	+	0.344054i
$974$		0			0
$975$		−	2.28719i		−	0.0732486i
$976$		0			0
$977$	4.14359			0.132565			0.0662827	−	0.997801i	$-0.478886\pi$
$977$	4.14359			0.132565			0.0662827	+	0.997801i	$0.478886\pi$
$978$		0			0
$979$	1.56922			0.0501525
$980$		0			0
$981$	17.0718			0.545061
$982$		0			0
$983$	−18.9282			−0.603716			−0.301858	−	0.953353i	$-0.597607\pi$
$983$	−18.9282			−0.603716			−0.301858	+	0.953353i	$0.597607\pi$
$984$		0			0
$985$		−	32.7846i		−	1.04460i
$986$		0			0
$987$	−10.1436	−	8.78461i	−0.322874	−	0.279617i
$988$		0			0
$989$	13.8564			0.440608
$990$		0			0
$991$		−	31.1769i		−	0.990367i	−0.868788	−	0.495184i	$-0.835101\pi$
$991$		−	31.1769i		−	0.990367i	0.868788	−	0.495184i	$-0.164899\pi$
$992$		0			0
$993$		−	17.0718i		−	0.541757i
$994$		0			0
$995$			67.7128i			2.14664i
$996$		0			0
$997$			27.1244i			0.859037i	0.903058	+	0.429518i	$0.141317\pi$
$997$			27.1244i			0.859037i	−0.903058	+	0.429518i	$0.858683\pi$
$998$		0			0
$999$	16.0000			0.506218

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1792.2.f.a.1791.3		4
4.3	odd	2		1792.2.f.i.1791.1		4
7.6	odd	2		1792.2.f.i.1791.2		4
8.3	odd	2		1792.2.f.b.1791.4		4
8.5	even	2		1792.2.f.h.1791.2		4
16.3	odd	4		896.2.e.a.447.2	✓	4
16.5	even	4		896.2.e.f.447.2	yes	4
16.11	odd	4		896.2.e.e.447.3	yes	4
16.13	even	4		896.2.e.b.447.3	yes	4
28.27	even	2	inner	1792.2.f.a.1791.4		4
56.13	odd	2		1792.2.f.b.1791.3		4
56.27	even	2		1792.2.f.h.1791.1		4
112.13	odd	4		896.2.e.e.447.2	yes	4
112.27	even	4		896.2.e.b.447.2	yes	4
112.69	odd	4		896.2.e.a.447.3	yes	4
112.83	even	4		896.2.e.f.447.3	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
896.2.e.a.447.2	✓	4	16.3	odd	4
896.2.e.a.447.3	yes	4	112.69	odd	4
896.2.e.b.447.2	yes	4	112.27	even	4
896.2.e.b.447.3	yes	4	16.13	even	4
896.2.e.e.447.2	yes	4	112.13	odd	4
896.2.e.e.447.3	yes	4	16.11	odd	4
896.2.e.f.447.2	yes	4	16.5	even	4
896.2.e.f.447.3	yes	4	112.83	even	4
1792.2.f.a.1791.3		4	1.1	even	1	trivial
1792.2.f.a.1791.4		4	28.27	even	2	inner
1792.2.f.b.1791.3		4	56.13	odd	2
1792.2.f.b.1791.4		4	8.3	odd	2
1792.2.f.h.1791.1		4	56.27	even	2
1792.2.f.h.1791.2		4	8.5	even	2
1792.2.f.i.1791.1		4	4.3	odd	2
1792.2.f.i.1791.2		4	7.6	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1792 = 2^{8} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1792.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+0.732051 q^{3} -2.73205i q^{5} +(-2.00000 - 1.73205i) q^{7} -2.46410 q^{9} +O(q^{10})\)	\(q+0.732051 q^{3} -2.73205i q^{5} +(-2.00000 - 1.73205i) q^{7} -2.46410 q^{9} +1.46410i q^{11} +1.26795i q^{13} -2.00000i q^{15} +4.00000i q^{17} -4.73205 q^{19} +(-1.46410 - 1.26795i) q^{21} -1.46410i q^{23} -2.46410 q^{25} -4.00000 q^{27} +6.92820 q^{29} -6.92820 q^{31} +1.07180i q^{33} +(-4.73205 + 5.46410i) q^{35} -4.00000 q^{37} +0.928203i q^{39} -10.9282i q^{41} +9.46410i q^{43} +6.73205i q^{45} +6.92820 q^{47} +(1.00000 + 6.92820i) q^{49} +2.92820i q^{51} -6.92820 q^{53} +4.00000 q^{55} -3.46410 q^{57} -14.1962 q^{59} +5.66025i q^{61} +(4.92820 + 4.26795i) q^{63} +3.46410 q^{65} +9.46410i q^{67} -1.07180i q^{69} +11.4641i q^{71} +6.92820i q^{73} -1.80385 q^{75} +(2.53590 - 2.92820i) q^{77} -3.46410i q^{79} +4.46410 q^{81} -7.26795 q^{83} +10.9282 q^{85} +5.07180 q^{87} -1.07180i q^{89} +(2.19615 - 2.53590i) q^{91} -5.07180 q^{93} +12.9282i q^{95} -12.0000i q^{97} -3.60770i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{3} - 8 q^{7} + 4 q^{9}+O(q^{10}) \) 4 * q - 4 * q^3 - 8 * q^7 + 4 * q^9	\( 4 q - 4 q^{3} - 8 q^{7} + 4 q^{9} - 12 q^{19} + 8 q^{21} + 4 q^{25} - 16 q^{27} - 12 q^{35} - 16 q^{37} + 4 q^{49} + 16 q^{55} - 36 q^{59} - 8 q^{63} - 28 q^{75} + 24 q^{77} + 4 q^{81} - 36 q^{83} + 16 q^{85} + 48 q^{87} - 12 q^{91} - 48 q^{93}+O(q^{100}) \) 4 * q - 4 * q^3 - 8 * q^7 + 4 * q^9 - 12 * q^19 + 8 * q^21 + 4 * q^25 - 16 * q^27 - 12 * q^35 - 16 * q^37 + 4 * q^49 + 16 * q^55 - 36 * q^59 - 8 * q^63 - 28 * q^75 + 24 * q^77 + 4 * q^81 - 36 * q^83 + 16 * q^85 + 48 * q^87 - 12 * q^91 - 48 * q^93

Introduction

L-functions

Modular forms

Varieties

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Database

Properties

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