LMFDB - Embedded newform 768.3.l.d.319.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [768,3,Mod(319,768)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(768, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([2, 1, 0]))

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

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

chi := DirichletCharacter("768.319");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 768 = 2^{8} \cdot 3 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	768.l (of order $4$, degree $2$, 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:	$20.9264843029$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(i)$
Coefficient field:	$\Q(\zeta_{24})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{4} + 1 $ x^8 - x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{10} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			319.2
Root			$0.258819 + 0.965926i$ of defining polynomial
Character	$\chi$	$=$	768.319
Dual form			768.3.l.d.703.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.22474 - 1.22474i) q^{3} +(2.00000 + 2.00000i) q^{5} +0.378937 q^{7} +3.00000i q^{9} +O(q^{10})$	\(q+(-1.22474 - 1.22474i) q^{3} +(2.00000 + 2.00000i) q^{5} +0.378937 q^{7} +3.00000i q^{9} +(10.5558 - 10.5558i) q^{11} +(-6.46410 + 6.46410i) q^{13} -4.89898i q^{15} -11.8564 q^{17} +(18.6622 + 18.6622i) q^{19} +(-0.464102 - 0.464102i) q^{21} +12.0716 q^{23} -17.0000i q^{25} +(3.67423 - 3.67423i) q^{27} +(-1.07180 + 1.07180i) q^{29} -6.38752i q^{31} -25.8564 q^{33} +(0.757875 + 0.757875i) q^{35} +(35.3923 + 35.3923i) q^{37} +15.8338 q^{39} +3.56922i q^{41} +(29.9759 - 29.9759i) q^{43} +(-6.00000 + 6.00000i) q^{45} +31.6675i q^{47} -48.8564 q^{49} +(14.5211 + 14.5211i) q^{51} +(68.7846 + 68.7846i) q^{53} +42.2233 q^{55} -45.7128i q^{57} +(27.8781 - 27.8781i) q^{59} +(62.8179 - 62.8179i) q^{61} +1.13681i q^{63} -25.8564 q^{65} +(30.5579 + 30.5579i) q^{67} +(-14.7846 - 14.7846i) q^{69} +131.975 q^{71} -50.1436i q^{73} +(-20.8207 + 20.8207i) q^{75} +(4.00000 - 4.00000i) q^{77} -114.219i q^{79} -9.00000 q^{81} +(1.92189 + 1.92189i) q^{83} +(-23.7128 - 23.7128i) q^{85} +2.62536 q^{87} -74.0000i q^{89} +(-2.44949 + 2.44949i) q^{91} +(-7.82309 + 7.82309i) q^{93} +74.6487i q^{95} -89.2820 q^{97} +(31.6675 + 31.6675i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 16 q^{5}+O(q^{10}) $ 8 * q + 16 * q^5	$ 8 q + 16 q^{5} - 24 q^{13} + 16 q^{17} + 24 q^{21} - 64 q^{29} - 96 q^{33} + 200 q^{37} - 48 q^{45} - 280 q^{49} + 384 q^{53} - 24 q^{61} - 96 q^{65} + 48 q^{69} + 32 q^{77} - 72 q^{81} + 32 q^{85} - 312 q^{93} - 160 q^{97}+O(q^{100}) $ 8 * q + 16 * q^5 - 24 * q^13 + 16 * q^17 + 24 * q^21 - 64 * q^29 - 96 * q^33 + 200 * q^37 - 48 * q^45 - 280 * q^49 + 384 * q^53 - 24 * q^61 - 96 * q^65 + 48 * q^69 + 32 * q^77 - 72 * q^81 + 32 * q^85 - 312 * q^93 - 160 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/768\mathbb{Z}\right)^\times$.

$n$	$257$	$511$	$517$
$\chi(n)$	$1$	$-1$	$e\left(\frac{1}{4}\right)$

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
$3$	−1.22474	−	1.22474i	−0.408248	−	0.408248i
$3$	−1.22474	−	1.22474i	−0.408248	−	0.408248i
$4$		0			0
$5$	2.00000	+	2.00000i	0.400000	+	0.400000i	0.878233	−	0.478233i	$-0.158723\pi$
$5$	2.00000	+	2.00000i	0.400000	+	0.400000i	−0.478233	+	0.878233i	$0.658723\pi$
$6$		0			0
$7$	0.378937			0.0541339			0.0270670	−	0.999634i	$-0.491383\pi$
$7$	0.378937			0.0541339			0.0270670	+	0.999634i	$0.491383\pi$
$8$		0			0
$9$			3.00000i			0.333333i
$10$		0			0
$11$	10.5558	−	10.5558i	0.959621	−	0.959621i	−0.0395946	−	0.999216i	$-0.512607\pi$
$11$	10.5558	−	10.5558i	0.959621	−	0.959621i	0.999216	+	0.0395946i	$0.0126066\pi$
$12$		0			0
$13$	−6.46410	+	6.46410i	−0.497239	+	0.497239i	−0.910577	−	0.413339i	$-0.864363\pi$
$13$	−6.46410	+	6.46410i	−0.497239	+	0.497239i	0.413339	+	0.910577i	$0.364363\pi$
$14$		0			0
$15$		−	4.89898i		−	0.326599i
$16$		0			0
$17$	−11.8564			−0.697436			−0.348718	−	0.937228i	$-0.613383\pi$
$17$	−11.8564			−0.697436			−0.348718	+	0.937228i	$0.613383\pi$
$18$		0			0
$19$	18.6622	+	18.6622i	0.982220	+	0.982220i	0.999845	−	0.0176248i	$-0.00561044\pi$
$19$	18.6622	+	18.6622i	0.982220	+	0.982220i	−0.0176248	+	0.999845i	$0.505610\pi$
$20$		0			0
$21$	−0.464102	−	0.464102i	−0.0221001	−	0.0221001i
$22$		0			0
$23$	12.0716			0.524851			0.262426	−	0.964952i	$-0.415478\pi$
$23$	12.0716			0.524851			0.262426	+	0.964952i	$0.415478\pi$
$24$		0			0
$25$		−	17.0000i		−	0.680000i
$26$		0			0
$27$	3.67423	−	3.67423i	0.136083	−	0.136083i
$28$		0			0
$29$	−1.07180	+	1.07180i	−0.0369585	+	0.0369585i	−0.725345	−	0.688386i	$-0.758320\pi$
$29$	−1.07180	+	1.07180i	−0.0369585	+	0.0369585i	0.688386	+	0.725345i	$0.258320\pi$
$30$		0			0
$31$		−	6.38752i		−	0.206049i	−0.994679	−	0.103025i	$-0.967148\pi$
$31$		−	6.38752i		−	0.206049i	0.994679	−	0.103025i	$-0.0328520\pi$
$32$		0			0
$33$	−25.8564			−0.783527
$34$		0			0
$35$	0.757875	+	0.757875i	0.0216536	+	0.0216536i
$36$		0			0
$37$	35.3923	+	35.3923i	0.956549	+	0.956549i	0.999095	−	0.0425457i	$-0.0135468\pi$
$37$	35.3923	+	35.3923i	0.956549	+	0.956549i	−0.0425457	+	0.999095i	$0.513547\pi$
$38$		0			0
$39$	15.8338			0.405994
$40$		0			0
$41$			3.56922i			0.0870541i	0.999052	+	0.0435271i	$0.0138595\pi$
$41$			3.56922i			0.0870541i	−0.999052	+	0.0435271i	$0.986141\pi$
$42$		0			0
$43$	29.9759	−	29.9759i	0.697114	−	0.697114i	−0.266673	−	0.963787i	$-0.585924\pi$
$43$	29.9759	−	29.9759i	0.697114	−	0.697114i	0.963787	+	0.266673i	$0.0859245\pi$
$44$		0			0
$45$	−6.00000	+	6.00000i	−0.133333	+	0.133333i
$46$		0			0
$47$			31.6675i			0.673777i	0.941545	+	0.336888i	$0.109374\pi$
$47$			31.6675i			0.673777i	−0.941545	+	0.336888i	$0.890626\pi$
$48$		0			0
$49$	−48.8564			−0.997070
$50$		0			0
$51$	14.5211	+	14.5211i	0.284727	+	0.284727i
$52$		0			0
$53$	68.7846	+	68.7846i	1.29782	+	1.29782i	0.929828	+	0.367995i	$0.119956\pi$
$53$	68.7846	+	68.7846i	1.29782	+	1.29782i	0.367995	+	0.929828i	$0.380044\pi$
$54$		0			0
$55$	42.2233			0.767697
$56$		0			0
$57$		−	45.7128i		−	0.801979i
$58$		0			0
$59$	27.8781	−	27.8781i	0.472511	−	0.472511i	−0.430216	−	0.902726i	$-0.641562\pi$
$59$	27.8781	−	27.8781i	0.472511	−	0.472511i	0.902726	+	0.430216i	$0.141562\pi$
$60$		0			0
$61$	62.8179	−	62.8179i	1.02980	−	1.02980i	0.0302601	−	0.999542i	$-0.490366\pi$
$61$	62.8179	−	62.8179i	1.02980	−	1.02980i	0.999542	−	0.0302601i	$-0.00963355\pi$
$62$		0			0
$63$			1.13681i			0.0180446i
$64$		0			0
$65$	−25.8564			−0.397791
$66$		0			0
$67$	30.5579	+	30.5579i	0.456088	+	0.456088i	0.897369	−	0.441281i	$-0.145476\pi$
$67$	30.5579	+	30.5579i	0.456088	+	0.456088i	−0.441281	+	0.897369i	$0.645476\pi$
$68$		0			0
$69$	−14.7846	−	14.7846i	−0.214270	−	0.214270i
$70$		0			0
$71$	131.975			1.85880			0.929402	−	0.369068i	$-0.120323\pi$
$71$	131.975			1.85880			0.929402	+	0.369068i	$0.120323\pi$
$72$		0			0
$73$		−	50.1436i		−	0.686899i	−0.939171	−	0.343449i	$-0.888405\pi$
$73$		−	50.1436i		−	0.686899i	0.939171	−	0.343449i	$-0.111595\pi$
$74$		0			0
$75$	−20.8207	+	20.8207i	−0.277609	+	0.277609i
$76$		0			0
$77$	4.00000	−	4.00000i	0.0519481	−	0.0519481i
$78$		0			0
$79$		−	114.219i		−	1.44582i	−0.690944	−	0.722908i	$-0.742805\pi$
$79$		−	114.219i		−	1.44582i	0.690944	−	0.722908i	$-0.257195\pi$
$80$		0			0
$81$	−9.00000			−0.111111
$82$		0			0
$83$	1.92189	+	1.92189i	0.0231553	+	0.0231553i	0.718590	−	0.695434i	$-0.244788\pi$
$83$	1.92189	+	1.92189i	0.0231553	+	0.0231553i	−0.695434	+	0.718590i	$0.744788\pi$
$84$		0			0
$85$	−23.7128	−	23.7128i	−0.278974	−	0.278974i
$86$		0			0
$87$	2.62536			0.0301765
$88$		0			0
$89$		−	74.0000i		−	0.831461i	−0.909488	−	0.415730i	$-0.863526\pi$
$89$		−	74.0000i		−	0.831461i	0.909488	−	0.415730i	$-0.136474\pi$
$90$		0			0
$91$	−2.44949	+	2.44949i	−0.0269175	+	0.0269175i
$92$		0			0
$93$	−7.82309	+	7.82309i	−0.0841192	+	0.0841192i
$94$		0			0
$95$			74.6487i			0.785776i
$96$		0			0
$97$	−89.2820			−0.920433			−0.460217	−	0.887807i	$-0.652228\pi$
$97$	−89.2820			−0.920433			−0.460217	+	0.887807i	$0.652228\pi$
$98$		0			0
$99$	31.6675	+	31.6675i	0.319874	+	0.319874i
$100$		0			0
$101$	123.923	+	123.923i	1.22696	+	1.22696i	0.965108	+	0.261853i	$0.0843336\pi$
$101$	123.923	+	123.923i	1.22696	+	1.22696i	0.261853	+	0.965108i	$0.415666\pi$
$102$		0			0
$103$	96.1393			0.933391			0.466696	−	0.884418i	$-0.345444\pi$
$103$	96.1393			0.933391			0.466696	+	0.884418i	$0.345444\pi$
$104$		0			0
$105$		−	1.85641i		−	0.0176801i
$106$		0			0
$107$	115.708	−	115.708i	1.08138	−	1.08138i	0.0850027	−	0.996381i	$-0.472910\pi$
$107$	115.708	−	115.708i	1.08138	−	1.08138i	0.996381	−	0.0850027i	$-0.0270899\pi$
$108$		0			0
$109$	−81.2487	+	81.2487i	−0.745401	+	0.745401i	−0.973612	−	0.228211i	$-0.926712\pi$
$109$	−81.2487	+	81.2487i	−0.745401	+	0.745401i	0.228211	+	0.973612i	$0.426712\pi$
$110$		0			0
$111$		−	86.6931i		−	0.781019i
$112$		0			0
$113$	164.851			1.45886			0.729430	−	0.684055i	$-0.239785\pi$
$113$	164.851			1.45886			0.729430	+	0.684055i	$0.239785\pi$
$114$		0			0
$115$	24.1432	+	24.1432i	0.209941	+	0.209941i
$116$		0			0
$117$	−19.3923	−	19.3923i	−0.165746	−	0.165746i
$118$		0			0
$119$	−4.49284			−0.0377549
$120$		0			0
$121$		−	101.851i		−	0.841746i
$122$		0			0
$123$	4.37138	−	4.37138i	0.0355397	−	0.0355397i
$124$		0			0
$125$	84.0000	−	84.0000i	0.672000	−	0.672000i
$126$		0			0
$127$			66.7454i			0.525555i	0.964857	+	0.262777i	$0.0846385\pi$
$127$			66.7454i			0.525555i	−0.964857	+	0.262777i	$0.915362\pi$
$128$		0			0
$129$	−73.4256			−0.569191
$130$		0			0
$131$	−136.116	−	136.116i	−1.03906	−	1.03906i	−0.999206	−	0.0398495i	$-0.987312\pi$
$131$	−136.116	−	136.116i	−1.03906	−	1.03906i	−0.0398495	−	0.999206i	$-0.512688\pi$
$132$		0			0
$133$	7.07180	+	7.07180i	0.0531714	+	0.0531714i
$134$		0			0
$135$	14.6969			0.108866
$136$		0			0
$137$			225.559i			1.64642i	0.567740	+	0.823208i	$0.307818\pi$
$137$			225.559i			1.64642i	−0.567740	+	0.823208i	$0.692182\pi$
$138$		0			0
$139$	−41.0593	+	41.0593i	−0.295391	+	0.295391i	−0.839205	−	0.543815i	$-0.816979\pi$
$139$	−41.0593	+	41.0593i	−0.295391	+	0.295391i	0.543815	+	0.839205i	$0.316979\pi$
$140$		0			0
$141$	38.7846	−	38.7846i	0.275068	−	0.275068i
$142$		0			0
$143$			136.468i			0.954321i
$144$		0			0
$145$	−4.28719			−0.0295668
$146$		0			0
$147$	59.8366	+	59.8366i	0.407052	+	0.407052i
$148$		0			0
$149$	−31.8564	−	31.8564i	−0.213801	−	0.213801i	0.592079	−	0.805880i	$-0.298307\pi$
$149$	−31.8564	−	31.8564i	−0.213801	−	0.213801i	−0.805880	+	0.592079i	$0.798307\pi$
$150$		0			0
$151$	−27.5536			−0.182474			−0.0912371	−	0.995829i	$-0.529082\pi$
$151$	−27.5536			−0.182474			−0.0912371	+	0.995829i	$0.529082\pi$
$152$		0			0
$153$		−	35.5692i		−	0.232479i
$154$		0			0
$155$	12.7750	−	12.7750i	0.0824196	−	0.0824196i
$156$		0			0
$157$	−65.5359	+	65.5359i	−0.417426	+	0.417426i	−0.884316	−	0.466890i	$-0.845375\pi$
$157$	−65.5359	+	65.5359i	−0.417426	+	0.417426i	0.466890	+	0.884316i	$0.345375\pi$
$158$		0			0
$159$		−	168.487i		−	1.05967i
$160$		0			0
$161$	4.57437			0.0284123
$162$		0			0
$163$	80.8332	+	80.8332i	0.495909	+	0.495909i	0.910162	−	0.414253i	$-0.135957\pi$
$163$	80.8332	+	80.8332i	0.495909	+	0.495909i	−0.414253	+	0.910162i	$0.635957\pi$
$164$		0			0
$165$	−51.7128	−	51.7128i	−0.313411	−	0.313411i
$166$		0			0
$167$	−252.528			−1.51214			−0.756071	−	0.654490i	$-0.772884\pi$
$167$	−252.528			−1.51214			−0.756071	+	0.654490i	$0.772884\pi$
$168$		0			0
$169$			85.4308i			0.505508i
$170$		0			0
$171$	−55.9865	+	55.9865i	−0.327407	+	0.327407i
$172$		0			0
$173$	−90.4205	+	90.4205i	−0.522662	+	0.522662i	−0.918374	−	0.395713i	$-0.870498\pi$
$173$	−90.4205	+	90.4205i	−0.522662	+	0.522662i	0.395713	+	0.918374i	$0.370498\pi$
$174$		0			0
$175$		−	6.44194i		−	0.0368111i
$176$		0			0
$177$	−68.2872			−0.385803
$178$		0			0
$179$	−9.09450	−	9.09450i	−0.0508072	−	0.0508072i	0.681247	−	0.732054i	$-0.261438\pi$
$179$	−9.09450	−	9.09450i	−0.0508072	−	0.0508072i	−0.732054	+	0.681247i	$0.761438\pi$
$180$		0			0
$181$	−214.310	−	214.310i	−1.18403	−	1.18403i	−0.978690	−	0.205344i	$-0.934169\pi$
$181$	−214.310	−	214.310i	−1.18403	−	1.18403i	−0.205344	−	0.978690i	$-0.565831\pi$
$182$		0			0
$183$	−153.872			−0.840830
$184$		0			0
$185$			141.569i			0.765239i
$186$		0			0
$187$	−125.154	+	125.154i	−0.669274	+	0.669274i
$188$		0			0
$189$	1.39230	−	1.39230i	0.00736669	−	0.00736669i
$190$		0			0
$191$			91.1043i			0.476986i	0.971144	+	0.238493i	$0.0766534\pi$
$191$			91.1043i			0.476986i	−0.971144	+	0.238493i	$0.923347\pi$
$192$		0			0
$193$	−191.990			−0.994765			−0.497383	−	0.867531i	$-0.665705\pi$
$193$	−191.990			−0.994765			−0.497383	+	0.867531i	$0.665705\pi$
$194$		0			0
$195$	31.6675	+	31.6675i	0.162397	+	0.162397i
$196$		0			0
$197$	252.631	+	252.631i	1.28239	+	1.28239i	0.939304	+	0.343085i	$0.111472\pi$
$197$	252.631	+	252.631i	1.28239	+	1.28239i	0.343085	+	0.939304i	$0.388528\pi$
$198$		0			0
$199$	−268.876			−1.35114			−0.675569	−	0.737297i	$-0.736102\pi$
$199$	−268.876			−1.35114			−0.675569	+	0.737297i	$0.736102\pi$
$200$		0			0
$201$		−	74.8513i		−	0.372394i
$202$		0			0
$203$	−0.406144	+	0.406144i	−0.00200071	+	0.00200071i
$204$		0			0
$205$	−7.13844	+	7.13844i	−0.0348217	+	0.0348217i
$206$		0			0
$207$			36.2147i			0.174950i
$208$		0			0
$209$	393.990			1.88512
$210$		0			0
$211$	−140.663	−	140.663i	−0.666652	−	0.666652i	0.290288	−	0.956939i	$-0.406249\pi$
$211$	−140.663	−	140.663i	−0.666652	−	0.666652i	−0.956939	+	0.290288i	$0.906249\pi$
$212$		0			0
$213$	−161.636	−	161.636i	−0.758854	−	0.758854i
$214$		0			0
$215$	119.904			0.557691
$216$		0			0
$217$		−	2.42047i		−	0.0111542i
$218$		0			0
$219$	−61.4131	+	61.4131i	−0.280425	+	0.280425i
$220$		0			0
$221$	76.6410	−	76.6410i	0.346792	−	0.346792i
$222$		0			0
$223$		−	135.331i		−	0.606866i	−0.952853	−	0.303433i	$-0.901867\pi$
$223$		−	135.331i		−	0.606866i	0.952853	−	0.303433i	$-0.0981329\pi$
$224$		0			0
$225$	51.0000			0.226667
$226$		0			0
$227$	−219.102	−	219.102i	−0.965205	−	0.965205i	0.0342095	−	0.999415i	$-0.489109\pi$
$227$	−219.102	−	219.102i	−0.965205	−	0.965205i	−0.999415	+	0.0342095i	$0.989109\pi$
$228$		0			0
$229$	−160.895	−	160.895i	−0.702598	−	0.702598i	0.262370	−	0.964967i	$-0.415496\pi$
$229$	−160.895	−	160.895i	−0.702598	−	0.702598i	−0.964967	+	0.262370i	$0.915496\pi$
$230$		0			0
$231$	−9.79796			−0.0424154
$232$		0			0
$233$			231.990i			0.995664i	0.867274	+	0.497832i	$0.165870\pi$
$233$			231.990i			0.995664i	−0.867274	+	0.497832i	$0.834130\pi$
$234$		0			0
$235$	−63.3350	+	63.3350i	−0.269511	+	0.269511i
$236$		0			0
$237$	−139.890	+	139.890i	−0.590252	+	0.590252i
$238$		0			0
$239$			454.550i			1.90188i	0.309372	+	0.950941i	$0.399881\pi$
$239$			454.550i			1.90188i	−0.309372	+	0.950941i	$0.600119\pi$
$240$		0			0
$241$	330.554			1.37159			0.685796	−	0.727794i	$-0.259454\pi$
$241$	330.554			1.37159			0.685796	+	0.727794i	$0.259454\pi$
$242$		0			0
$243$	11.0227	+	11.0227i	0.0453609	+	0.0453609i
$244$		0			0
$245$	−97.7128	−	97.7128i	−0.398828	−	0.398828i
$246$		0			0
$247$	−241.268			−0.976795
$248$		0			0
$249$		−	4.70766i		−	0.0189063i
$250$		0			0
$251$	−24.9010	+	24.9010i	−0.0992073	+	0.0992073i	−0.754968	−	0.655761i	$-0.772348\pi$
$251$	−24.9010	+	24.9010i	−0.0992073	+	0.0992073i	0.655761	+	0.754968i	$0.272348\pi$
$252$		0			0
$253$	127.426	−	127.426i	0.503659	−	0.503659i
$254$		0			0
$255$			58.0843i			0.227782i
$256$		0			0
$257$	277.138			1.07836			0.539180	−	0.842191i	$-0.318734\pi$
$257$	277.138			1.07836			0.539180	+	0.842191i	$0.318734\pi$
$258$		0			0
$259$	13.4115	+	13.4115i	0.0517817	+	0.0517817i
$260$		0			0
$261$	−3.21539	−	3.21539i	−0.0123195	−	0.0123195i
$262$		0			0
$263$	−119.200			−0.453232			−0.226616	−	0.973984i	$-0.572766\pi$
$263$	−119.200			−0.453232			−0.226616	+	0.973984i	$0.572766\pi$
$264$		0			0
$265$			275.138i			1.03826i
$266$		0			0
$267$	−90.6311	+	90.6311i	−0.339442	+	0.339442i
$268$		0			0
$269$	−151.061	+	151.061i	−0.561567	+	0.561567i	−0.929752	−	0.368185i	$-0.879979\pi$
$269$	−151.061	+	151.061i	−0.561567	+	0.561567i	0.368185	+	0.929752i	$0.379979\pi$
$270$		0			0
$271$		−	430.137i		−	1.58722i	−0.608427	−	0.793610i	$-0.708199\pi$
$271$		−	430.137i		−	1.58722i	0.608427	−	0.793610i	$-0.291801\pi$
$272$		0			0
$273$	6.00000			0.0219780
$274$		0			0
$275$	−179.449	−	179.449i	−0.652542	−	0.652542i
$276$		0			0
$277$	−213.249	−	213.249i	−0.769851	−	0.769851i	0.208229	−	0.978080i	$-0.433230\pi$
$277$	−213.249	−	213.249i	−0.769851	−	0.769851i	−0.978080	+	0.208229i	$0.933230\pi$
$278$		0			0
$279$	19.1626			0.0686830
$280$		0			0
$281$			78.0000i			0.277580i	0.990322	+	0.138790i	$0.0443213\pi$
$281$			78.0000i			0.277580i	−0.990322	+	0.138790i	$0.955679\pi$
$282$		0			0
$283$	−41.7628	+	41.7628i	−0.147572	+	0.147572i	−0.777032	−	0.629461i	$-0.783276\pi$
$283$	−41.7628	+	41.7628i	−0.147572	+	0.147572i	0.629461	+	0.777032i	$0.283276\pi$
$284$		0			0
$285$	91.4256	−	91.4256i	0.320792	−	0.320792i
$286$		0			0
$287$			1.35251i			0.00471258i
$288$		0			0
$289$	−148.426			−0.513583
$290$		0			0
$291$	109.348	+	109.348i	0.375765	+	0.375765i
$292$		0			0
$293$	−141.933	−	141.933i	−0.484414	−	0.484414i	0.422124	−	0.906538i	$-0.361285\pi$
$293$	−141.933	−	141.933i	−0.484414	−	0.484414i	−0.906538	+	0.422124i	$0.861285\pi$
$294$		0			0
$295$	111.513			0.378009
$296$		0			0
$297$		−	77.5692i		−	0.261176i
$298$		0			0
$299$	−78.0319	+	78.0319i	−0.260976	+	0.260976i
$300$		0			0
$301$	11.3590	−	11.3590i	0.0377375	−	0.0377375i
$302$		0			0
$303$		−	303.548i		−	1.00181i
$304$		0			0
$305$	251.272			0.823842
$306$		0			0
$307$	−16.3215	−	16.3215i	−0.0531645	−	0.0531645i	0.680025	−	0.733189i	$-0.261969\pi$
$307$	−16.3215	−	16.3215i	−0.0531645	−	0.0531645i	−0.733189	+	0.680025i	$0.761969\pi$
$308$		0			0
$309$	−117.746	−	117.746i	−0.381055	−	0.381055i
$310$		0			0
$311$	−111.404			−0.358211			−0.179106	−	0.983830i	$-0.557320\pi$
$311$	−111.404			−0.358211			−0.179106	+	0.983830i	$0.557320\pi$
$312$		0			0
$313$		−	502.267i		−	1.60469i	−0.596864	−	0.802343i	$-0.703587\pi$
$313$		−	502.267i		−	1.60469i	0.596864	−	0.802343i	$-0.296413\pi$
$314$		0			0
$315$	−2.27362	+	2.27362i	−0.00721785	+	0.00721785i
$316$		0			0
$317$	140.785	−	140.785i	0.444115	−	0.444115i	−0.449277	−	0.893393i	$-0.648318\pi$
$317$	140.785	−	140.785i	0.444115	−	0.444115i	0.893393	+	0.449277i	$0.148318\pi$
$318$		0			0
$319$			22.6274i			0.0709323i
$320$		0			0
$321$	−283.426			−0.882946
$322$		0			0
$323$	−221.266	−	221.266i	−0.685035	−	0.685035i
$324$		0			0
$325$	109.890	+	109.890i	0.338122	+	0.338122i
$326$		0			0
$327$	199.018			0.608617
$328$		0			0
$329$			12.0000i			0.0364742i
$330$		0			0
$331$	−302.682	+	302.682i	−0.914446	+	0.914446i	−0.996618	−	0.0821724i	$-0.973814\pi$
$331$	−302.682	+	302.682i	−0.914446	+	0.914446i	0.0821724	+	0.996618i	$0.473814\pi$
$332$		0			0
$333$	−106.177	+	106.177i	−0.318850	+	0.318850i
$334$		0			0
$335$			122.232i			0.364870i
$336$		0			0
$337$	435.559			1.29246			0.646230	−	0.763143i	$-0.276345\pi$
$337$	435.559			1.29246			0.646230	+	0.763143i	$0.276345\pi$
$338$		0			0
$339$	−201.901	−	201.901i	−0.595577	−	0.595577i
$340$		0			0
$341$	−67.4256	−	67.4256i	−0.197729	−	0.197729i
$342$		0			0
$343$	−37.0815			−0.108109
$344$		0			0
$345$		−	59.1384i		−	0.171416i
$346$		0			0
$347$	59.7885	−	59.7885i	0.172301	−	0.172301i	−0.615688	−	0.787990i	$-0.711122\pi$
$347$	59.7885	−	59.7885i	0.172301	−	0.172301i	0.787990	+	0.615688i	$0.211122\pi$
$348$		0			0
$349$	56.4538	−	56.4538i	0.161759	−	0.161759i	−0.621587	−	0.783345i	$-0.713512\pi$
$349$	56.4538	−	56.4538i	0.161759	−	0.161759i	0.783345	+	0.621587i	$0.213512\pi$
$350$		0			0
$351$			47.5013i			0.135331i
$352$		0			0
$353$	−87.9897			−0.249263			−0.124631	−	0.992203i	$-0.539775\pi$
$353$	−87.9897			−0.249263			−0.124631	+	0.992203i	$0.539775\pi$
$354$		0			0
$355$	263.950	+	263.950i	0.743522	+	0.743522i
$356$		0			0
$357$	5.50258	+	5.50258i	0.0154134	+	0.0154134i
$358$		0			0
$359$	−213.499			−0.594705			−0.297353	−	0.954768i	$-0.596104\pi$
$359$	−213.499			−0.594705			−0.297353	+	0.954768i	$0.596104\pi$
$360$		0			0
$361$			335.554i			0.929512i
$362$		0			0
$363$	−124.742	+	124.742i	−0.343641	+	0.343641i
$364$		0			0
$365$	100.287	−	100.287i	0.274759	−	0.274759i
$366$		0			0
$367$			537.102i			1.46349i	0.681577	+	0.731746i	$0.261294\pi$
$367$			537.102i			1.46349i	−0.681577	+	0.731746i	$0.738706\pi$
$368$		0			0
$369$	−10.7077			−0.0290180
$370$		0			0
$371$	26.0651	+	26.0651i	0.0702562	+	0.0702562i
$372$		0			0
$373$	120.895	+	120.895i	0.324115	+	0.324115i	0.850343	−	0.526228i	$-0.176394\pi$
$373$	120.895	+	120.895i	0.324115	+	0.324115i	−0.526228	+	0.850343i	$0.676394\pi$
$374$		0			0
$375$	−205.757			−0.548686
$376$		0			0
$377$		−	13.8564i		−	0.0367544i
$378$		0			0
$379$	178.922	−	178.922i	0.472089	−	0.472089i	−0.430501	−	0.902590i	$-0.641663\pi$
$379$	178.922	−	178.922i	0.472089	−	0.472089i	0.902590	+	0.430501i	$0.141663\pi$
$380$		0			0
$381$	81.7461	−	81.7461i	0.214557	−	0.214557i
$382$		0			0
$383$		−	61.1158i		−	0.159571i	−0.996812	−	0.0797856i	$-0.974576\pi$
$383$		−	61.1158i		−	0.159571i	0.996812	−	0.0797856i	$-0.0254236\pi$
$384$		0			0
$385$	16.0000			0.0415584
$386$		0			0
$387$	89.9277	+	89.9277i	0.232371	+	0.232371i
$388$		0			0
$389$	399.569	+	399.569i	1.02717	+	1.02717i	0.999620	+	0.0275498i	$0.00877048\pi$
$389$	399.569	+	399.569i	1.02717	+	1.02717i	0.0275498	+	0.999620i	$0.491230\pi$
$390$		0			0
$391$	−143.126			−0.366050
$392$		0			0
$393$			333.415i			0.848385i
$394$		0			0
$395$	228.439	−	228.439i	0.578327	−	0.578327i
$396$		0			0
$397$	−310.597	+	310.597i	−0.782361	+	0.782361i	−0.980229	−	0.197868i	$-0.936598\pi$
$397$	−310.597	+	310.597i	−0.782361	+	0.782361i	0.197868	+	0.980229i	$0.436598\pi$
$398$		0			0
$399$		−	17.3223i		−	0.0434143i
$400$		0			0
$401$	−447.856			−1.11685			−0.558424	−	0.829555i	$-0.688594\pi$
$401$	−447.856			−1.11685			−0.558424	+	0.829555i	$0.688594\pi$
$402$		0			0
$403$	41.2896	+	41.2896i	0.102456	+	0.102456i
$404$		0			0
$405$	−18.0000	−	18.0000i	−0.0444444	−	0.0444444i
$406$		0			0
$407$	747.191			1.83585
$408$		0			0
$409$		−	208.133i		−	0.508883i	−0.967088	−	0.254442i	$-0.918108\pi$
$409$		−	208.133i		−	0.508883i	0.967088	−	0.254442i	$-0.0818917\pi$
$410$		0			0
$411$	276.252	−	276.252i	0.672146	−	0.672146i
$412$		0			0
$413$	10.5641	−	10.5641i	0.0255788	−	0.0255788i
$414$		0			0
$415$			7.68757i			0.0185243i
$416$		0			0
$417$	100.574			0.241186
$418$		0			0
$419$	169.191	+	169.191i	0.403796	+	0.403796i	0.879569	−	0.475772i	$-0.157831\pi$
$419$	169.191	+	169.191i	0.403796	+	0.403796i	−0.475772	+	0.879569i	$0.657831\pi$
$420$		0			0
$421$	64.9615	+	64.9615i	0.154303	+	0.154303i	0.780037	−	0.625734i	$-0.215200\pi$
$421$	64.9615	+	64.9615i	0.154303	+	0.154303i	−0.625734	+	0.780037i	$0.715200\pi$
$422$		0			0
$423$	−95.0025			−0.224592
$424$		0			0
$425$			201.559i			0.474256i
$426$		0			0
$427$	23.8041	−	23.8041i	0.0557472	−	0.0557472i
$428$		0			0
$429$	167.138	−	167.138i	0.389600	−	0.389600i
$430$		0			0
$431$			104.206i			0.241777i	0.992666	+	0.120888i	$0.0385743\pi$
$431$			104.206i			0.241777i	−0.992666	+	0.120888i	$0.961426\pi$
$432$		0			0
$433$	−533.549			−1.23221			−0.616107	−	0.787663i	$-0.711291\pi$
$433$	−533.549			−1.23221			−0.616107	+	0.787663i	$0.711291\pi$
$434$		0			0
$435$	5.25071	+	5.25071i	0.0120706	+	0.0120706i
$436$		0			0
$437$	225.282	+	225.282i	0.515520	+	0.515520i
$438$		0			0
$439$	163.318			0.372023			0.186012	−	0.982548i	$-0.440444\pi$
$439$	163.318			0.372023			0.186012	+	0.982548i	$0.440444\pi$
$440$		0			0
$441$		−	146.569i		−	0.332357i
$442$		0			0
$443$	−238.589	+	238.589i	−0.538575	+	0.538575i	−0.923110	−	0.384535i	$-0.874362\pi$
$443$	−238.589	+	238.589i	−0.538575	+	0.538575i	0.384535	+	0.923110i	$0.374362\pi$
$444$		0			0
$445$	148.000	−	148.000i	0.332584	−	0.332584i
$446$		0			0
$447$			78.0319i			0.174568i
$448$		0			0
$449$	−333.559			−0.742893			−0.371446	−	0.928454i	$-0.621138\pi$
$449$	−333.559			−0.742893			−0.371446	+	0.928454i	$0.621138\pi$
$450$		0			0
$451$	37.6761	+	37.6761i	0.0835390	+	0.0835390i
$452$		0			0
$453$	33.7461	+	33.7461i	0.0744948	+	0.0744948i
$454$		0			0
$455$	−9.79796			−0.0215340
$456$		0			0
$457$		−	431.292i		−	0.943747i	−0.881666	−	0.471873i	$-0.843578\pi$
$457$		−	431.292i		−	0.943747i	0.881666	−	0.471873i	$-0.156422\pi$
$458$		0			0
$459$	−43.5632	+	43.5632i	−0.0949090	+	0.0949090i
$460$		0			0
$461$	−232.277	+	232.277i	−0.503854	+	0.503854i	−0.912633	−	0.408779i	$-0.865955\pi$
$461$	−232.277	+	232.277i	−0.503854	+	0.503854i	0.408779	+	0.912633i	$0.365955\pi$
$462$		0			0
$463$		−	470.681i		−	1.01659i	−0.861183	−	0.508295i	$-0.830276\pi$
$463$		−	470.681i		−	1.01659i	0.861183	−	0.508295i	$-0.169724\pi$
$464$		0			0
$465$	−31.2923			−0.0672954
$466$		0			0
$467$	412.950	+	412.950i	0.884262	+	0.884262i	0.993964	−	0.109702i	$-0.0349898\pi$
$467$	412.950	+	412.950i	0.884262	+	0.884262i	−0.109702	+	0.993964i	$0.534990\pi$
$468$		0			0
$469$	11.5795	+	11.5795i	0.0246898	+	0.0246898i
$470$		0			0
$471$	160.530			0.340827
$472$		0			0
$473$		−	632.841i		−	1.33793i
$474$		0			0
$475$	317.257	−	317.257i	0.667910	−	0.667910i
$476$		0			0
$477$	−206.354	+	206.354i	−0.432608	+	0.432608i
$478$		0			0
$479$		−	802.784i		−	1.67596i	−0.545703	−	0.837979i	$-0.683737\pi$
$479$		−	802.784i		−	1.67596i	0.545703	−	0.837979i	$-0.316263\pi$
$480$		0			0
$481$	−457.559			−0.951266
$482$		0			0
$483$	−5.60244	−	5.60244i	−0.0115993	−	0.0115993i
$484$		0			0
$485$	−178.564	−	178.564i	−0.368173	−	0.368173i
$486$		0			0
$487$	317.326			0.651593			0.325797	−	0.945440i	$-0.394367\pi$
$487$	317.326			0.651593			0.325797	+	0.945440i	$0.394367\pi$
$488$		0			0
$489$		−	198.000i		−	0.404908i
$490$		0			0
$491$	651.648	−	651.648i	1.32719	−	1.32719i	0.419369	−	0.907816i	$-0.362251\pi$
$491$	651.648	−	651.648i	1.32719	−	1.32719i	0.907816	−	0.419369i	$-0.137749\pi$
$492$		0			0
$493$	12.7077	−	12.7077i	0.0257762	−	0.0257762i
$494$		0			0
$495$			126.670i			0.255899i
$496$		0			0
$497$	50.0103			0.100624
$498$		0			0
$499$	38.9489	+	38.9489i	0.0780540	+	0.0780540i	0.745056	−	0.667002i	$-0.232423\pi$
$499$	38.9489	+	38.9489i	0.0780540	+	0.0780540i	−0.667002	+	0.745056i	$0.732423\pi$
$500$		0			0
$501$	309.282	+	309.282i	0.617329	+	0.617329i
$502$		0			0
$503$	−205.594			−0.408735			−0.204368	−	0.978894i	$-0.565514\pi$
$503$	−205.594			−0.408735			−0.204368	+	0.978894i	$0.565514\pi$
$504$		0			0
$505$			495.692i			0.981569i
$506$		0			0
$507$	104.631	−	104.631i	0.206373	−	0.206373i
$508$		0			0
$509$	−91.6565	+	91.6565i	−0.180072	+	0.180072i	−0.791387	−	0.611315i	$-0.790641\pi$
$509$	−91.6565	+	91.6565i	−0.180072	+	0.180072i	0.611315	+	0.791387i	$0.290641\pi$
$510$		0			0
$511$		−	19.0013i		−	0.0371845i
$512$		0			0
$513$	137.138			0.267326
$514$		0			0
$515$	192.279	+	192.279i	0.373357	+	0.373357i
$516$		0			0
$517$	334.277	+	334.277i	0.646570	+	0.646570i
$518$		0			0
$519$	221.484			0.426751
$520$		0			0
$521$			155.282i			0.298046i	0.988834	+	0.149023i	$0.0476129\pi$
$521$			155.282i			0.298046i	−0.988834	+	0.149023i	$0.952387\pi$
$522$		0			0
$523$	501.090	−	501.090i	0.958107	−	0.958107i	−0.0410496	−	0.999157i	$-0.513070\pi$
$523$	501.090	−	501.090i	0.958107	−	0.958107i	0.999157	+	0.0410496i	$0.0130702\pi$
$524$		0			0
$525$	−7.88973	+	7.88973i	−0.0150281	+	0.0150281i
$526$		0			0
$527$			75.7331i			0.143706i
$528$		0			0
$529$	−383.277			−0.724531
$530$		0			0
$531$	83.6344	+	83.6344i	0.157504	+	0.157504i
$532$		0			0
$533$	−23.0718	−	23.0718i	−0.0432867	−	0.0432867i
$534$		0			0
$535$	462.832			0.865107
$536$		0			0
$537$			22.2769i			0.0414839i
$538$		0			0
$539$	−515.720	+	515.720i	−0.956809	+	0.956809i
$540$		0			0
$541$	−1.53590	+	1.53590i	−0.00283900	+	0.00283900i	−0.708525	−	0.705686i	$-0.750639\pi$
$541$	−1.53590	+	1.53590i	−0.00283900	+	0.00283900i	0.705686	+	0.708525i	$0.250639\pi$
$542$		0			0
$543$			524.951i			0.966760i
$544$		0			0
$545$	−324.995			−0.596321
$546$		0			0
$547$	−456.891	−	456.891i	−0.835266	−	0.835266i	0.152965	−	0.988232i	$-0.451118\pi$
$547$	−456.891	−	456.891i	−0.835266	−	0.835266i	−0.988232	+	0.152965i	$0.951118\pi$
$548$		0			0
$549$	188.454	+	188.454i	0.343267	+	0.343267i
$550$		0			0
$551$	−40.0041			−0.0726028
$552$		0			0
$553$		−	43.2820i		−	0.0782677i
$554$		0			0
$555$	173.386	−	173.386i	0.312408	−	0.312408i
$556$		0			0
$557$	−9.86672	+	9.86672i	−0.0177140	+	0.0177140i	−0.715908	−	0.698194i	$-0.753987\pi$
$557$	−9.86672	+	9.86672i	−0.0177140	+	0.0177140i	0.698194	+	0.715908i	$0.253987\pi$
$558$		0			0
$559$			387.534i			0.693264i
$560$		0			0
$561$	306.564			0.546460
$562$		0			0
$563$	533.909	+	533.909i	0.948329	+	0.948329i	0.998729	−	0.0504003i	$-0.0160497\pi$
$563$	533.909	+	533.909i	0.948329	+	0.948329i	−0.0504003	+	0.998729i	$0.516050\pi$
$564$		0			0
$565$	329.703	+	329.703i	0.583544	+	0.583544i
$566$		0			0
$567$	−3.41044			−0.00601488
$568$		0			0
$569$		−	673.846i		−	1.18426i	−0.805841	−	0.592132i	$-0.798286\pi$
$569$		−	673.846i		−	1.18426i	0.805841	−	0.592132i	$-0.201714\pi$
$570$		0			0
$571$	490.467	−	490.467i	0.858962	−	0.858962i	−0.132254	−	0.991216i	$-0.542221\pi$
$571$	490.467	−	490.467i	0.858962	−	0.858962i	0.991216	+	0.132254i	$0.0422214\pi$
$572$		0			0
$573$	111.580	−	111.580i	0.194729	−	0.194729i
$574$		0			0
$575$		−	205.217i		−	0.356899i
$576$		0			0
$577$	403.005			0.698449			0.349225	−	0.937039i	$-0.386445\pi$
$577$	403.005			0.698449			0.349225	+	0.937039i	$0.386445\pi$
$578$		0			0
$579$	235.138	+	235.138i	0.406111	+	0.406111i
$580$		0			0
$581$	0.728277	+	0.728277i	0.00125349	+	0.00125349i
$582$		0			0
$583$	1452.16			2.49084
$584$		0			0
$585$		−	77.5692i		−	0.132597i
$586$		0			0
$587$	157.688	−	157.688i	0.268635	−	0.268635i	−0.559915	−	0.828550i	$-0.689166\pi$
$587$	157.688	−	157.688i	0.268635	−	0.268635i	0.828550	+	0.559915i	$0.189166\pi$
$588$		0			0
$589$	119.205	−	119.205i	0.202386	−	0.202386i
$590$		0			0
$591$		−	618.816i		−	1.04707i
$592$		0			0
$593$	30.0000			0.0505902			0.0252951	−	0.999680i	$-0.491947\pi$
$593$	30.0000			0.0505902			0.0252951	+	0.999680i	$0.491947\pi$
$594$		0			0
$595$	−8.98567	−	8.98567i	−0.0151020	−	0.0151020i
$596$		0			0
$597$	329.305	+	329.305i	0.551600	+	0.551600i
$598$		0			0
$599$	1139.76			1.90277			0.951384	−	0.308007i	$-0.0996622\pi$
$599$	1139.76			1.90277			0.951384	+	0.308007i	$0.0996622\pi$
$600$		0			0
$601$			327.559i			0.545023i	0.962153	+	0.272512i	$0.0878543\pi$
$601$			327.559i			0.545023i	−0.962153	+	0.272512i	$0.912146\pi$
$602$		0			0
$603$	−91.6737	+	91.6737i	−0.152029	+	0.152029i
$604$		0			0
$605$	203.703	−	203.703i	0.336698	−	0.336698i
$606$		0			0
$607$			484.540i			0.798254i	0.916896	+	0.399127i	$0.130687\pi$
$607$			484.540i			0.798254i	−0.916896	+	0.399127i	$0.869313\pi$
$608$		0			0
$609$	0.994845			0.00163357
$610$		0			0
$611$	−204.702	−	204.702i	−0.335028	−	0.335028i
$612$		0			0
$613$	463.105	+	463.105i	0.755473	+	0.755473i	0.975495	−	0.220022i	$-0.0706128\pi$
$613$	463.105	+	463.105i	0.755473	+	0.755473i	−0.220022	+	0.975495i	$0.570613\pi$
$614$		0			0
$615$	17.4855			0.0284318
$616$		0			0
$617$		−	1115.39i		−	1.80777i	−0.427774	−	0.903885i	$-0.640702\pi$
$617$		−	1115.39i		−	1.80777i	0.427774	−	0.903885i	$-0.359298\pi$
$618$		0			0
$619$	−590.884	+	590.884i	−0.954578	+	0.954578i	−0.999012	−	0.0444343i	$-0.985851\pi$
$619$	−590.884	+	590.884i	−0.954578	+	0.954578i	0.0444343	+	0.999012i	$0.485851\pi$
$620$		0			0
$621$	44.3538	−	44.3538i	0.0714232	−	0.0714232i
$622$		0			0
$623$		−	28.0414i		−	0.0450102i
$624$		0			0
$625$	−89.0000			−0.142400
$626$		0			0
$627$	−482.537	−	482.537i	−0.769596	−	0.769596i
$628$		0			0
$629$	−419.626	−	419.626i	−0.667131	−	0.667131i
$630$		0			0
$631$	−1221.55			−1.93590			−0.967949	−	0.251145i	$-0.919193\pi$
$631$	−1221.55			−1.93590			−0.967949	+	0.251145i	$0.919193\pi$
$632$		0			0
$633$			344.554i			0.544319i
$634$		0			0
$635$	−133.491	+	133.491i	−0.210222	+	0.210222i
$636$		0			0
$637$	315.813	−	315.813i	0.495781	−	0.495781i
$638$		0			0
$639$			395.925i			0.619602i
$640$		0			0
$641$	−1152.41			−1.79783			−0.898916	−	0.438121i	$-0.855644\pi$
$641$	−1152.41			−1.79783			−0.898916	+	0.438121i	$0.855644\pi$
$642$		0			0
$643$	177.649	+	177.649i	0.276281	+	0.276281i	0.831622	−	0.555341i	$-0.187412\pi$
$643$	177.649	+	177.649i	0.276281	+	0.276281i	−0.555341	+	0.831622i	$0.687412\pi$
$644$		0			0
$645$	−146.851	−	146.851i	−0.227676	−	0.227676i
$646$		0			0
$647$	−390.892			−0.604161			−0.302081	−	0.953282i	$-0.597681\pi$
$647$	−390.892			−0.604161			−0.302081	+	0.953282i	$0.597681\pi$
$648$		0			0
$649$		−	588.554i		−	0.906862i
$650$		0			0
$651$	−2.96446	+	2.96446i	−0.00455370	+	0.00455370i
$652$		0			0
$653$	441.538	−	441.538i	0.676169	−	0.676169i	−0.282962	−	0.959131i	$-0.591317\pi$
$653$	441.538	−	441.538i	0.676169	−	0.676169i	0.959131	+	0.282962i	$0.0913171\pi$
$654$		0			0
$655$		−	544.465i		−	0.831244i
$656$		0			0
$657$	150.431			0.228966
$658$		0			0
$659$	671.947	+	671.947i	1.01965	+	1.01965i	0.999803	+	0.0198437i	$0.00631685\pi$
$659$	671.947	+	671.947i	1.01965	+	1.01965i	0.0198437	+	0.999803i	$0.493683\pi$
$660$		0			0
$661$	−16.0333	−	16.0333i	−0.0242562	−	0.0242562i	0.694875	−	0.719131i	$-0.255460\pi$
$661$	−16.0333	−	16.0333i	−0.0242562	−	0.0242562i	−0.719131	+	0.694875i	$0.755460\pi$
$662$		0			0
$663$	−187.731			−0.283154
$664$		0			0
$665$			28.2872i			0.0425371i
$666$		0			0
$667$	−12.9383	+	12.9383i	−0.0193977	+	0.0193977i
$668$		0			0
$669$	−165.746	+	165.746i	−0.247752	+	0.247752i
$670$		0			0
$671$		−	1326.19i		−	1.97644i
$672$		0			0
$673$	512.954			0.762190			0.381095	−	0.924536i	$-0.375547\pi$
$673$	512.954			0.762190			0.381095	+	0.924536i	$0.375547\pi$
$674$		0			0
$675$	−62.4620	−	62.4620i	−0.0925363	−	0.0925363i
$676$		0			0
$677$	−282.000	−	282.000i	−0.416544	−	0.416544i	0.467467	−	0.884011i	$-0.345167\pi$
$677$	−282.000	−	282.000i	−0.416544	−	0.416544i	−0.884011	+	0.467467i	$0.845167\pi$
$678$		0			0
$679$	−33.8323			−0.0498267
$680$		0			0
$681$			536.687i			0.788087i
$682$		0			0
$683$	−307.660	+	307.660i	−0.450454	+	0.450454i	−0.895505	−	0.445051i	$-0.853186\pi$
$683$	−307.660	+	307.660i	−0.450454	+	0.450454i	0.445051	+	0.895505i	$0.353186\pi$
$684$		0			0
$685$	−451.118	+	451.118i	−0.658566	+	0.658566i
$686$		0			0
$687$			394.110i			0.573669i
$688$		0			0
$689$	−889.261			−1.29066
$690$		0			0
$691$	−599.422	−	599.422i	−0.867470	−	0.867470i	0.124722	−	0.992192i	$-0.460196\pi$
$691$	−599.422	−	599.422i	−0.867470	−	0.867470i	−0.992192	+	0.124722i	$0.960196\pi$
$692$		0			0
$693$	12.0000	+	12.0000i	0.0173160	+	0.0173160i
$694$		0			0
$695$	−164.237			−0.236313
$696$		0			0
$697$		−	42.3181i		−	0.0607147i
$698$		0			0
$699$	284.128	−	284.128i	0.406478	−	0.406478i
$700$		0			0
$701$	−63.3487	+	63.3487i	−0.0903690	+	0.0903690i	−0.750846	−	0.660477i	$-0.770354\pi$
$701$	−63.3487	+	63.3487i	−0.0903690	+	0.0903690i	0.660477	+	0.750846i	$0.270354\pi$
$702$		0			0
$703$			1320.99i			1.87908i
$704$		0			0
$705$	155.138			0.220055
$706$		0			0
$707$	46.9591	+	46.9591i	0.0664202	+	0.0664202i
$708$		0			0
$709$	−247.259	−	247.259i	−0.348743	−	0.348743i	0.510898	−	0.859641i	$-0.329313\pi$
$709$	−247.259	−	247.259i	−0.348743	−	0.348743i	−0.859641	+	0.510898i	$0.829313\pi$
$710$		0			0
$711$	342.658			0.481939
$712$		0			0
$713$		−	77.1075i		−	0.108145i
$714$		0			0
$715$	−272.936	+	272.936i	−0.381729	+	0.381729i
$716$		0			0
$717$	556.708	−	556.708i	0.776440	−	0.776440i
$718$		0			0
$719$		−	794.987i		−	1.10568i	−0.833286	−	0.552842i	$-0.813543\pi$
$719$		−	794.987i		−	1.10568i	0.833286	−	0.552842i	$-0.186457\pi$
$720$		0			0
$721$	36.4308			0.0505281
$722$		0			0
$723$	−404.844	−	404.844i	−0.559950	−	0.559950i
$724$		0			0
$725$	18.2205	+	18.2205i	0.0251318	+	0.0251318i
$726$		0			0
$727$	395.706			0.544300			0.272150	−	0.962255i	$-0.412265\pi$
$727$	395.706			0.544300			0.272150	+	0.962255i	$0.412265\pi$
$728$		0			0
$729$		−	27.0000i		−	0.0370370i
$730$		0			0
$731$	−355.406	+	355.406i	−0.486192	+	0.486192i
$732$		0			0
$733$	−486.443	+	486.443i	−0.663634	+	0.663634i	−0.956235	−	0.292601i	$-0.905479\pi$
$733$	−486.443	+	486.443i	−0.663634	+	0.663634i	0.292601	+	0.956235i	$0.405479\pi$
$734$		0			0
$735$			239.347i			0.325642i
$736$		0			0
$737$	645.128			0.875343
$738$		0			0
$739$	503.079	+	503.079i	0.680757	+	0.680757i	0.960171	−	0.279414i	$-0.0901402\pi$
$739$	503.079	+	503.079i	0.680757	+	0.680757i	−0.279414	+	0.960171i	$0.590140\pi$
$740$		0			0
$741$	295.492	+	295.492i	0.398775	+	0.398775i
$742$		0			0
$743$	−270.934			−0.364649			−0.182325	−	0.983238i	$-0.558362\pi$
$743$	−270.934			−0.364649			−0.182325	+	0.983238i	$0.558362\pi$
$744$		0			0
$745$		−	127.426i		−	0.171041i
$746$		0			0
$747$	−5.76568	+	5.76568i	−0.00771845	+	0.00771845i
$748$		0			0
$749$	43.8461	−	43.8461i	0.0585395	−	0.0585395i
$750$		0			0
$751$			348.232i			0.463691i	0.972753	+	0.231845i	$0.0744763\pi$
$751$			348.232i			0.463691i	−0.972753	+	0.231845i	$0.925524\pi$
$752$		0			0
$753$	60.9948			0.0810025
$754$		0			0
$755$	−55.1072	−	55.1072i	−0.0729897	−	0.0729897i
$756$		0			0
$757$	231.792	+	231.792i	0.306198	+	0.306198i	0.843433	−	0.537235i	$-0.180531\pi$
$757$	231.792	+	231.792i	0.306198	+	0.306198i	−0.537235	+	0.843433i	$0.680531\pi$
$758$		0			0
$759$	−312.128			−0.411236
$760$		0			0
$761$			1038.99i			1.36530i	0.730745	+	0.682651i	$0.239173\pi$
$761$			1038.99i			1.36530i	−0.730745	+	0.682651i	$0.760827\pi$
$762$		0			0
$763$	−30.7882	+	30.7882i	−0.0403515	+	0.0403515i
$764$		0			0
$765$	71.1384	−	71.1384i	0.0929914	−	0.0929914i
$766$		0			0
$767$			360.414i			0.469901i
$768$		0			0
$769$	1047.42			1.36205			0.681024	−	0.732261i	$-0.261535\pi$
$769$	1047.42			1.36205			0.681024	+	0.732261i	$0.261535\pi$
$770$		0			0
$771$	−339.424	−	339.424i	−0.440238	−	0.440238i
$772$		0			0
$773$	−863.041	−	863.041i	−1.11648	−	1.11648i	−0.992254	−	0.124229i	$-0.960354\pi$
$773$	−863.041	−	863.041i	−1.11648	−	1.11648i	−0.124229	−	0.992254i	$-0.539646\pi$
$774$		0			0
$775$	−108.588			−0.140113
$776$		0			0
$777$		−	32.8513i		−	0.0422796i
$778$		0			0
$779$	−66.6094	+	66.6094i	−0.0855063	+	0.0855063i
$780$		0			0
$781$	1393.11	−	1393.11i	1.78375	−	1.78375i
$782$		0			0
$783$			7.87607i			0.0100588i
$784$		0			0
$785$	−262.144			−0.333941
$786$		0			0
$787$	61.6142	+	61.6142i	0.0782900	+	0.0782900i	0.745167	−	0.666877i	$-0.232370\pi$
$787$	61.6142	+	61.6142i	0.0782900	+	0.0782900i	−0.666877	+	0.745167i	$0.732370\pi$
$788$		0			0
$789$	145.990	+	145.990i	0.185031	+	0.185031i
$790$		0			0
$791$	62.4683			0.0789738
$792$		0			0
$793$			812.123i			1.02411i
$794$		0			0
$795$	336.974	−	336.974i	0.423867	−	0.423867i
$796$		0			0
$797$	−998.379	+	998.379i	−1.25267	+	1.25267i	−0.298154	+	0.954518i	$0.596371\pi$
$797$	−998.379	+	998.379i	−1.25267	+	1.25267i	−0.954518	+	0.298154i	$0.903629\pi$
$798$		0			0
$799$		−	375.463i		−	0.469916i
$800$		0			0
$801$	222.000			0.277154
$802$		0			0
$803$	−529.307	−	529.307i	−0.659162	−	0.659162i
$804$		0			0
$805$	9.14875	+	9.14875i	0.0113649	+	0.0113649i
$806$		0			0
$807$	370.024			0.458517
$808$		0			0
$809$		−	400.390i		−	0.494919i	−0.968898	−	0.247460i	$-0.920404\pi$
$809$		−	400.390i		−	0.494919i	0.968898	−	0.247460i	$-0.0795957\pi$
$810$		0			0
$811$	504.096	−	504.096i	0.621574	−	0.621574i	−0.324360	−	0.945934i	$-0.605149\pi$
$811$	504.096	−	504.096i	0.621574	−	0.621574i	0.945934	+	0.324360i	$0.105149\pi$
$812$		0			0
$813$	−526.808	+	526.808i	−0.647980	+	0.647980i
$814$		0			0
$815$			323.333i			0.396727i
$816$		0			0
$817$	1118.83			1.36944
$818$		0			0
$819$	−7.34847	−	7.34847i	−0.00897249	−	0.00897249i
$820$		0			0
$821$	−596.477	−	596.477i	−0.726525	−	0.726525i	0.243401	−	0.969926i	$-0.421737\pi$
$821$	−596.477	−	596.477i	−0.726525	−	0.726525i	−0.969926	+	0.243401i	$0.921737\pi$
$822$		0			0
$823$	−1615.20			−1.96258			−0.981290	−	0.192534i	$-0.938329\pi$
$823$	−1615.20			−1.96258			−0.981290	+	0.192534i	$0.938329\pi$
$824$		0			0
$825$			439.559i			0.532799i
$826$		0			0
$827$	−1010.90	+	1010.90i	−1.22237	+	1.22237i	−0.255579	+	0.966788i	$0.582266\pi$
$827$	−1010.90	+	1010.90i	−1.22237	+	1.22237i	−0.966788	+	0.255579i	$0.917734\pi$
$828$		0			0
$829$	−933.561	+	933.561i	−1.12613	+	1.12613i	−0.135329	+	0.990801i	$0.543209\pi$
$829$	−933.561	+	933.561i	−1.12613	+	1.12613i	−0.990801	+	0.135329i	$0.956791\pi$
$830$		0			0
$831$			522.351i			0.628581i
$832$		0			0
$833$	579.261			0.695392
$834$		0			0
$835$	−505.055	−	505.055i	−0.604857	−	0.604857i
$836$		0			0
$837$	−23.4693	−	23.4693i	−0.0280397	−	0.0280397i
$838$		0			0
$839$	−1408.20			−1.67843			−0.839216	−	0.543798i	$-0.816986\pi$
$839$	−1408.20			−1.67843			−0.839216	+	0.543798i	$0.816986\pi$
$840$		0			0
$841$			838.703i			0.997268i
$842$		0			0
$843$	95.5301	−	95.5301i	0.113322	−	0.113322i
$844$		0			0
$845$	−170.862	+	170.862i	−0.202203	+	0.202203i
$846$		0			0
$847$		−	38.5952i		−	0.0455670i
$848$		0			0
$849$	102.297			0.120492
$850$		0			0
$851$	427.241	+	427.241i	0.502046	+	0.502046i
$852$		0			0
$853$	41.8691	+	41.8691i	0.0490845	+	0.0490845i	0.731223	−	0.682138i	$-0.238950\pi$
$853$	41.8691	+	41.8691i	0.0490845	+	0.0490845i	−0.682138	+	0.731223i	$0.738950\pi$
$854$		0			0
$855$	−223.946			−0.261925
$856$		0			0
$857$			418.441i			0.488263i	0.969742	+	0.244131i	$0.0785028\pi$
$857$			418.441i			0.488263i	−0.969742	+	0.244131i	$0.921497\pi$
$858$		0			0
$859$	272.814	−	272.814i	0.317595	−	0.317595i	−0.530247	−	0.847843i	$-0.677901\pi$
$859$	272.814	−	272.814i	0.317595	−	0.317595i	0.847843	+	0.530247i	$0.177901\pi$
$860$		0			0
$861$	1.65648	−	1.65648i	0.00192390	−	0.00192390i
$862$		0			0
$863$		−	988.405i		−	1.14531i	−0.819796	−	0.572656i	$-0.805913\pi$
$863$		−	988.405i		−	1.14531i	0.819796	−	0.572656i	$-0.194087\pi$
$864$		0			0
$865$	−361.682			−0.418129
$866$		0			0
$867$	181.784	+	181.784i	0.209670	+	0.209670i
$868$		0			0
$869$	−1205.68	−	1205.68i	−1.38744	−	1.38744i
$870$		0			0
$871$	−395.059			−0.453569
$872$		0			0
$873$		−	267.846i		−	0.306811i
$874$		0			0
$875$	31.8307	−	31.8307i	0.0363780	−	0.0363780i
$876$		0			0
$877$	−116.254	+	116.254i	−0.132559	+	0.132559i	−0.770273	−	0.637714i	$-0.779880\pi$
$877$	−116.254	+	116.254i	−0.132559	+	0.132559i	0.637714	+	0.770273i	$0.279880\pi$
$878$		0			0
$879$			347.664i			0.395523i
$880$		0			0
$881$	−727.395			−0.825647			−0.412823	−	0.910811i	$-0.635457\pi$
$881$	−727.395			−0.825647			−0.412823	+	0.910811i	$0.635457\pi$
$882$		0			0
$883$	44.6981	+	44.6981i	0.0506207	+	0.0506207i	0.731964	−	0.681343i	$-0.238604\pi$
$883$	44.6981	+	44.6981i	0.0506207	+	0.0506207i	−0.681343	+	0.731964i	$0.738604\pi$
$884$		0			0
$885$	−136.574	−	136.574i	−0.154321	−	0.154321i
$886$		0			0
$887$	−566.766			−0.638969			−0.319485	−	0.947591i	$-0.603510\pi$
$887$	−566.766			−0.638969			−0.319485	+	0.947591i	$0.603510\pi$
$888$		0			0
$889$			25.2923i			0.0284503i
$890$		0			0
$891$	−95.0025	+	95.0025i	−0.106625	+	0.106625i
$892$		0			0
$893$	−590.985	+	590.985i	−0.661797	+	0.661797i
$894$		0			0
$895$		−	36.3780i		−	0.0406458i
$896$		0			0
$897$	191.138			0.213086
$898$		0			0
$899$	6.84613	+	6.84613i	0.00761527	+	0.00761527i
$900$		0			0
$901$	−815.538	−	815.538i	−0.905148	−	0.905148i
$902$		0			0
$903$	−27.8237			−0.0308125
$904$		0			0
$905$		−	857.241i		−	0.947227i
$906$		0			0
$907$	−259.499	+	259.499i	−0.286107	+	0.286107i	−0.835539	−	0.549432i	$-0.814844\pi$
$907$	−259.499	+	259.499i	−0.286107	+	0.286107i	0.549432	+	0.835539i	$0.314844\pi$
$908$		0			0
$909$	−371.769	+	371.769i	−0.408987	+	0.408987i
$910$		0			0
$911$		−	1020.02i		−	1.11967i	−0.828605	−	0.559834i	$-0.810865\pi$
$911$		−	1020.02i		−	1.11967i	0.828605	−	0.559834i	$-0.189135\pi$
$912$		0			0
$913$	40.5744			0.0444407
$914$		0			0
$915$	−307.744	−	307.744i	−0.336332	−	0.336332i
$916$		0			0
$917$	−51.5795	−	51.5795i	−0.0562481	−	0.0562481i
$918$		0			0
$919$	−398.632			−0.433768			−0.216884	−	0.976197i	$-0.569589\pi$
$919$	−398.632			−0.433768			−0.216884	+	0.976197i	$0.569589\pi$
$920$		0			0
$921$			39.9794i			0.0434087i
$922$		0			0
$923$	−853.101	+	853.101i	−0.924269	+	0.924269i
$924$		0			0
$925$	601.669	−	601.669i	0.650453	−	0.650453i
$926$		0			0
$927$			288.418i			0.311130i
$928$		0			0
$929$	645.251			0.694565			0.347283	−	0.937761i	$-0.387104\pi$
$929$	645.251			0.694565			0.347283	+	0.937761i	$0.387104\pi$
$930$		0			0
$931$	−911.767	−	911.767i	−0.979342	−	0.979342i
$932$		0			0
$933$	136.441	+	136.441i	0.146239	+	0.146239i
$934$		0			0
$935$	−500.617			−0.535419
$936$		0			0
$937$			1279.72i			1.36577i	0.730528	+	0.682883i	$0.239274\pi$
$937$			1279.72i			1.36577i	−0.730528	+	0.682883i	$0.760726\pi$
$938$		0			0
$939$	−615.148	+	615.148i	−0.655110	+	0.655110i
$940$		0			0
$941$	796.144	−	796.144i	0.846061	−	0.846061i	−0.143578	−	0.989639i	$-0.545861\pi$
$941$	796.144	−	796.144i	0.846061	−	0.846061i	0.989639	+	0.143578i	$0.0458607\pi$
$942$		0			0
$943$			43.0861i			0.0456905i
$944$		0			0
$945$	5.56922			0.00589335
$946$		0			0
$947$	903.254	+	903.254i	0.953806	+	0.953806i	0.998979	−	0.0451730i	$-0.0143839\pi$
$947$	903.254	+	903.254i	0.953806	+	0.953806i	−0.0451730	+	0.998979i	$0.514384\pi$
$948$		0			0
$949$	324.133	+	324.133i	0.341552	+	0.341552i
$950$		0			0
$951$	−344.850			−0.362619
$952$		0			0
$953$		−	286.687i		−	0.300826i	−0.988623	−	0.150413i	$-0.951940\pi$
$953$		−	286.687i		−	0.300826i	0.988623	−	0.150413i	$-0.0480603\pi$
$954$		0			0
$955$	−182.209	+	182.209i	−0.190794	+	0.190794i
$956$		0			0
$957$	27.7128	−	27.7128i	0.0289580	−	0.0289580i
$958$		0			0
$959$			85.4727i			0.0891269i
$960$		0			0
$961$	920.200			0.957544
$962$		0			0
$963$	347.124	+	347.124i	0.360461	+	0.360461i
$964$		0			0
$965$	−383.979	−	383.979i	−0.397906	−	0.397906i
$966$		0			0
$967$	−153.089			−0.158313			−0.0791565	−	0.996862i	$-0.525223\pi$
$967$	−153.089			−0.158313			−0.0791565	+	0.996862i	$0.525223\pi$
$968$		0			0
$969$			541.990i			0.559329i
$970$		0			0
$971$	704.184	−	704.184i	0.725215	−	0.725215i	−0.244447	−	0.969663i	$-0.578606\pi$
$971$	704.184	−	704.184i	0.725215	−	0.725215i	0.969663	+	0.244447i	$0.0786065\pi$
$972$		0			0
$973$	−15.5589	+	15.5589i	−0.0159907	+	0.0159907i
$974$		0			0
$975$		−	269.174i		−	0.276076i
$976$		0			0
$977$	−521.230			−0.533501			−0.266751	−	0.963766i	$-0.585950\pi$
$977$	−521.230			−0.533501			−0.266751	+	0.963766i	$0.585950\pi$
$978$		0			0
$979$	−781.132	−	781.132i	−0.797887	−	0.797887i
$980$		0			0
$981$	−243.746	−	243.746i	−0.248467	−	0.248467i
$982$		0			0
$983$	952.730			0.969207			0.484603	−	0.874734i	$-0.338964\pi$
$983$	952.730			0.969207			0.484603	+	0.874734i	$0.338964\pi$
$984$		0			0
$985$			1010.52i			1.02591i
$986$		0			0
$987$	14.6969	−	14.6969i	0.0148905	−	0.0148905i
$988$		0			0
$989$	361.856	−	361.856i	0.365881	−	0.365881i
$990$		0			0
$991$		−	788.654i		−	0.795816i	−0.917425	−	0.397908i	$-0.869736\pi$
$991$		−	788.654i		−	0.795816i	0.917425	−	0.397908i	$-0.130264\pi$
$992$		0			0
$993$	741.415			0.746642
$994$		0			0
$995$	−537.753	−	537.753i	−0.540455	−	0.540455i
$996$		0			0
$997$	−370.177	−	370.177i	−0.371291	−	0.371291i	0.496656	−	0.867947i	$-0.334561\pi$
$997$	−370.177	−	370.177i	−0.371291	−	0.371291i	−0.867947	+	0.496656i	$0.834561\pi$
$998$		0			0
$999$	260.079			0.260340

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	768.3.l.d.319.2	yes	8
4.3	odd	2	inner	768.3.l.d.319.3	yes	8
8.3	odd	2		768.3.l.a.319.1	✓	8
8.5	even	2		768.3.l.a.319.4	yes	8
16.3	odd	4	inner	768.3.l.d.703.2	yes	8
16.5	even	4		768.3.l.a.703.1	yes	8
16.11	odd	4		768.3.l.a.703.4	yes	8
16.13	even	4	inner	768.3.l.d.703.3	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
768.3.l.a.319.1	✓	8	8.3	odd	2
768.3.l.a.319.4	yes	8	8.5	even	2
768.3.l.a.703.1	yes	8	16.5	even	4
768.3.l.a.703.4	yes	8	16.11	odd	4
768.3.l.d.319.2	yes	8	1.1	even	1	trivial
768.3.l.d.319.3	yes	8	4.3	odd	2	inner
768.3.l.d.703.2	yes	8	16.3	odd	4	inner
768.3.l.d.703.3	yes	8	16.13	even	4	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:	\( N \)	\(=\)	\( 768 = 2^{8} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	768.l (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.22474 - 1.22474i) q^{3} +(2.00000 + 2.00000i) q^{5} +0.378937 q^{7} +3.00000i q^{9} +O(q^{10})\)	\(q+(-1.22474 - 1.22474i) q^{3} +(2.00000 + 2.00000i) q^{5} +0.378937 q^{7} +3.00000i q^{9} +(10.5558 - 10.5558i) q^{11} +(-6.46410 + 6.46410i) q^{13} -4.89898i q^{15} -11.8564 q^{17} +(18.6622 + 18.6622i) q^{19} +(-0.464102 - 0.464102i) q^{21} +12.0716 q^{23} -17.0000i q^{25} +(3.67423 - 3.67423i) q^{27} +(-1.07180 + 1.07180i) q^{29} -6.38752i q^{31} -25.8564 q^{33} +(0.757875 + 0.757875i) q^{35} +(35.3923 + 35.3923i) q^{37} +15.8338 q^{39} +3.56922i q^{41} +(29.9759 - 29.9759i) q^{43} +(-6.00000 + 6.00000i) q^{45} +31.6675i q^{47} -48.8564 q^{49} +(14.5211 + 14.5211i) q^{51} +(68.7846 + 68.7846i) q^{53} +42.2233 q^{55} -45.7128i q^{57} +(27.8781 - 27.8781i) q^{59} +(62.8179 - 62.8179i) q^{61} +1.13681i q^{63} -25.8564 q^{65} +(30.5579 + 30.5579i) q^{67} +(-14.7846 - 14.7846i) q^{69} +131.975 q^{71} -50.1436i q^{73} +(-20.8207 + 20.8207i) q^{75} +(4.00000 - 4.00000i) q^{77} -114.219i q^{79} -9.00000 q^{81} +(1.92189 + 1.92189i) q^{83} +(-23.7128 - 23.7128i) q^{85} +2.62536 q^{87} -74.0000i q^{89} +(-2.44949 + 2.44949i) q^{91} +(-7.82309 + 7.82309i) q^{93} +74.6487i q^{95} -89.2820 q^{97} +(31.6675 + 31.6675i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 16 q^{5}+O(q^{10}) \) 8 * q + 16 * q^5	\( 8 q + 16 q^{5} - 24 q^{13} + 16 q^{17} + 24 q^{21} - 64 q^{29} - 96 q^{33} + 200 q^{37} - 48 q^{45} - 280 q^{49} + 384 q^{53} - 24 q^{61} - 96 q^{65} + 48 q^{69} + 32 q^{77} - 72 q^{81} + 32 q^{85} - 312 q^{93} - 160 q^{97}+O(q^{100}) \) 8 * q + 16 * q^5 - 24 * q^13 + 16 * q^17 + 24 * q^21 - 64 * q^29 - 96 * q^33 + 200 * q^37 - 48 * q^45 - 280 * q^49 + 384 * q^53 - 24 * q^61 - 96 * q^65 + 48 * q^69 + 32 * q^77 - 72 * q^81 + 32 * q^85 - 312 * q^93 - 160 * q^97

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