LMFDB - Embedded newform 768.3.e.g.257.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [768,3,Mod(257,768)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(768, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("768.257");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 768 = 2^{8} \cdot 3 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	768.e (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:	$20.9264843029$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{3}, \sqrt{-5})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + x^{2} + 4 $ x^4 + x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 192)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			257.3
Root			$0.866025 - 1.11803i$ of defining polynomial
Character	$\chi$	$=$	768.257
Dual form			768.3.e.g.257.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-2.00000 + 2.23607i) q^{3} -7.74597i q^{5} -3.46410 q^{7} +(-1.00000 - 8.94427i) q^{9} +O(q^{10})$	$q+(-2.00000 + 2.23607i) q^{3} -7.74597i q^{5} -3.46410 q^{7} +(-1.00000 - 8.94427i) q^{9} +13.4164i q^{11} +20.7846 q^{13} +(17.3205 + 15.4919i) q^{15} -4.00000 q^{19} +(6.92820 - 7.74597i) q^{21} -30.9839i q^{23} -35.0000 q^{25} +(22.0000 + 15.6525i) q^{27} +7.74597i q^{29} -24.2487 q^{31} +(-30.0000 - 26.8328i) q^{33} +26.8328i q^{35} -34.6410 q^{37} +(-41.5692 + 46.4758i) q^{39} -53.6656i q^{41} -52.0000 q^{43} +(-69.2820 + 7.74597i) q^{45} -61.9677i q^{47} -37.0000 q^{49} +54.2218i q^{53} +103.923 q^{55} +(8.00000 - 8.94427i) q^{57} -40.2492i q^{59} -6.92820 q^{61} +(3.46410 + 30.9839i) q^{63} -160.997i q^{65} +28.0000 q^{67} +(69.2820 + 61.9677i) q^{69} +30.9839i q^{71} -74.0000 q^{73} +(70.0000 - 78.2624i) q^{75} -46.4758i q^{77} -51.9615 q^{79} +(-79.0000 + 17.8885i) q^{81} -120.748i q^{83} +(-17.3205 - 15.4919i) q^{87} -53.6656i q^{89} -72.0000 q^{91} +(48.4974 - 54.2218i) q^{93} +30.9839i q^{95} -62.0000 q^{97} +(120.000 - 13.4164i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 8 q^{3} - 4 q^{9}+O(q^{10}) $ 4 * q - 8 * q^3 - 4 * q^9	$ 4 q - 8 q^{3} - 4 q^{9} - 16 q^{19} - 140 q^{25} + 88 q^{27} - 120 q^{33} - 208 q^{43} - 148 q^{49} + 32 q^{57} + 112 q^{67} - 296 q^{73} + 280 q^{75} - 316 q^{81} - 288 q^{91} - 248 q^{97} + 480 q^{99}+O(q^{100}) $ 4 * q - 8 * q^3 - 4 * q^9 - 16 * q^19 - 140 * q^25 + 88 * q^27 - 120 * q^33 - 208 * q^43 - 148 * q^49 + 32 * q^57 + 112 * q^67 - 296 * q^73 + 280 * q^75 - 316 * q^81 - 288 * q^91 - 248 * q^97 + 480 * q^99

Display 100 coefficients 10 coefficients

Character values

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

$n$	$257$	$511$	$517$
$\chi(n)$	$-1$	$1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$	−2.00000	+	2.23607i	−0.666667	+	0.745356i
$3$	−2.00000	+	2.23607i	−0.666667	+	0.745356i
$4$		0			0
$5$		−	7.74597i		−	1.54919i	−0.632456	−	0.774597i	$-0.717953\pi$
$5$		−	7.74597i		−	1.54919i	0.632456	−	0.774597i	$-0.282047\pi$
$6$		0			0
$7$	−3.46410			−0.494872			−0.247436	−	0.968904i	$-0.579588\pi$
$7$	−3.46410			−0.494872			−0.247436	+	0.968904i	$0.579588\pi$
$8$		0			0
$9$	−1.00000	−	8.94427i	−0.111111	−	0.993808i
$10$		0			0
$11$			13.4164i			1.21967i	0.792527	+	0.609837i	$0.208765\pi$
$11$			13.4164i			1.21967i	−0.792527	+	0.609837i	$0.791235\pi$
$12$		0			0
$13$	20.7846			1.59882			0.799408	−	0.600788i	$-0.205147\pi$
$13$	20.7846			1.59882			0.799408	+	0.600788i	$0.205147\pi$
$14$		0			0
$15$	17.3205	+	15.4919i	1.15470	+	1.03280i
$16$		0			0
$17$		0			0		1.00000			$0$
$17$		0			0		−1.00000			$\pi$
$18$		0			0
$19$	−4.00000			−0.210526			−0.105263	−	0.994444i	$-0.533568\pi$
$19$	−4.00000			−0.210526			−0.105263	+	0.994444i	$0.533568\pi$
$20$		0			0
$21$	6.92820	−	7.74597i	0.329914	−	0.368856i
$22$		0			0
$23$		−	30.9839i		−	1.34712i	−0.739130	−	0.673562i	$-0.764763\pi$
$23$		−	30.9839i		−	1.34712i	0.739130	−	0.673562i	$-0.235237\pi$
$24$		0			0
$25$	−35.0000			−1.40000
$26$		0			0
$27$	22.0000	+	15.6525i	0.814815	+	0.579721i
$28$		0			0
$29$			7.74597i			0.267102i	0.991042	+	0.133551i	$0.0426380\pi$
$29$			7.74597i			0.267102i	−0.991042	+	0.133551i	$0.957362\pi$
$30$		0			0
$31$	−24.2487			−0.782216			−0.391108	−	0.920345i	$-0.627908\pi$
$31$	−24.2487			−0.782216			−0.391108	+	0.920345i	$0.627908\pi$
$32$		0			0
$33$	−30.0000	−	26.8328i	−0.909091	−	0.813116i
$34$		0			0
$35$			26.8328i			0.766652i
$36$		0			0
$37$	−34.6410			−0.936244			−0.468122	−	0.883664i	$-0.655069\pi$
$37$	−34.6410			−0.936244			−0.468122	+	0.883664i	$0.655069\pi$
$38$		0			0
$39$	−41.5692	+	46.4758i	−1.06588	+	1.19169i
$40$		0			0
$41$		−	53.6656i		−	1.30892i	−0.756098	−	0.654459i	$-0.772896\pi$
$41$		−	53.6656i		−	1.30892i	0.756098	−	0.654459i	$-0.227104\pi$
$42$		0			0
$43$	−52.0000			−1.20930			−0.604651	−	0.796490i	$-0.706687\pi$
$43$	−52.0000			−1.20930			−0.604651	+	0.796490i	$0.706687\pi$
$44$		0			0
$45$	−69.2820	+	7.74597i	−1.53960	+	0.172133i
$46$		0			0
$47$		−	61.9677i		−	1.31846i	−0.751940	−	0.659231i	$-0.770882\pi$
$47$		−	61.9677i		−	1.31846i	0.751940	−	0.659231i	$-0.229118\pi$
$48$		0			0
$49$	−37.0000			−0.755102
$50$		0			0
$51$		0			0
$52$		0			0
$53$			54.2218i			1.02305i	0.859268	+	0.511526i	$0.170920\pi$
$53$			54.2218i			1.02305i	−0.859268	+	0.511526i	$0.829080\pi$
$54$		0			0
$55$	103.923			1.88951
$56$		0			0
$57$	8.00000	−	8.94427i	0.140351	−	0.156917i
$58$		0			0
$59$		−	40.2492i		−	0.682190i	−0.940029	−	0.341095i	$-0.889202\pi$
$59$		−	40.2492i		−	0.682190i	0.940029	−	0.341095i	$-0.110798\pi$
$60$		0			0
$61$	−6.92820			−0.113577			−0.0567886	−	0.998386i	$-0.518086\pi$
$61$	−6.92820			−0.113577			−0.0567886	+	0.998386i	$0.518086\pi$
$62$		0			0
$63$	3.46410	+	30.9839i	0.0549857	+	0.491807i
$64$		0			0
$65$		−	160.997i		−	2.47688i
$66$		0			0
$67$	28.0000			0.417910			0.208955	−	0.977925i	$-0.432994\pi$
$67$	28.0000			0.417910			0.208955	+	0.977925i	$0.432994\pi$
$68$		0			0
$69$	69.2820	+	61.9677i	1.00409	+	0.898083i
$70$		0			0
$71$			30.9839i			0.436392i	0.975905	+	0.218196i	$0.0700173\pi$
$71$			30.9839i			0.436392i	−0.975905	+	0.218196i	$0.929983\pi$
$72$		0			0
$73$	−74.0000			−1.01370			−0.506849	−	0.862035i	$-0.669190\pi$
$73$	−74.0000			−1.01370			−0.506849	+	0.862035i	$0.669190\pi$
$74$		0			0
$75$	70.0000	−	78.2624i	0.933333	−	1.04350i
$76$		0			0
$77$		−	46.4758i		−	0.603582i
$78$		0			0
$79$	−51.9615			−0.657741			−0.328870	−	0.944375i	$-0.606668\pi$
$79$	−51.9615			−0.657741			−0.328870	+	0.944375i	$0.606668\pi$
$80$		0			0
$81$	−79.0000	+	17.8885i	−0.975309	+	0.220846i
$82$		0			0
$83$		−	120.748i		−	1.45479i	−0.686218	−	0.727396i	$-0.740731\pi$
$83$		−	120.748i		−	1.45479i	0.686218	−	0.727396i	$-0.259269\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	−17.3205	−	15.4919i	−0.199086	−	0.178068i
$88$		0			0
$89$		−	53.6656i		−	0.602985i	−0.953469	−	0.301492i	$-0.902515\pi$
$89$		−	53.6656i		−	0.602985i	0.953469	−	0.301492i	$-0.0974848\pi$
$90$		0			0
$91$	−72.0000			−0.791209
$92$		0			0
$93$	48.4974	−	54.2218i	0.521478	−	0.583030i
$94$		0			0
$95$			30.9839i			0.326146i
$96$		0			0
$97$	−62.0000			−0.639175			−0.319588	−	0.947557i	$-0.603544\pi$
$97$	−62.0000			−0.639175			−0.319588	+	0.947557i	$0.603544\pi$
$98$		0			0
$99$	120.000	−	13.4164i	1.21212	−	0.135519i
$100$		0			0
$101$		−	131.681i		−	1.30378i	−0.758315	−	0.651888i	$-0.773977\pi$
$101$		−	131.681i		−	1.30378i	0.758315	−	0.651888i	$-0.226023\pi$
$102$		0			0
$103$	−86.6025			−0.840801			−0.420401	−	0.907339i	$-0.638110\pi$
$103$	−86.6025			−0.840801			−0.420401	+	0.907339i	$0.638110\pi$
$104$		0			0
$105$	−60.0000	−	53.6656i	−0.571429	−	0.511101i
$106$		0			0
$107$		−	40.2492i		−	0.376161i	−0.982154	−	0.188080i	$-0.939773\pi$
$107$		−	40.2492i		−	0.376161i	0.982154	−	0.188080i	$-0.0602266\pi$
$108$		0			0
$109$	76.2102			0.699176			0.349588	−	0.936903i	$-0.386321\pi$
$109$	76.2102			0.699176			0.349588	+	0.936903i	$0.386321\pi$
$110$		0			0
$111$	69.2820	−	77.4597i	0.624162	−	0.697835i
$112$		0			0
$113$			107.331i			0.949834i	0.880031	+	0.474917i	$0.157522\pi$
$113$			107.331i			0.949834i	−0.880031	+	0.474917i	$0.842478\pi$
$114$		0			0
$115$	−240.000			−2.08696
$116$		0			0
$117$	−20.7846	−	185.903i	−0.177646	−	1.58892i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−59.0000			−0.487603
$122$		0			0
$123$	120.000	+	107.331i	0.975610	+	0.872612i
$124$		0			0
$125$			77.4597i			0.619677i
$126$		0			0
$127$	86.6025			0.681910			0.340955	−	0.940080i	$-0.389250\pi$
$127$	86.6025			0.681910			0.340955	+	0.940080i	$0.389250\pi$
$128$		0			0
$129$	104.000	−	116.276i	0.806202	−	0.901361i
$130$		0			0
$131$			147.580i			1.12657i	0.826263	+	0.563284i	$0.190462\pi$
$131$			147.580i			1.12657i	−0.826263	+	0.563284i	$0.809538\pi$
$132$		0			0
$133$	13.8564			0.104184
$134$		0			0
$135$	121.244	−	170.411i	0.898100	−	1.26231i
$136$		0			0
$137$		−	53.6656i		−	0.391720i	−0.980632	−	0.195860i	$-0.937250\pi$
$137$		−	53.6656i		−	0.391720i	0.980632	−	0.195860i	$-0.0627498\pi$
$138$		0			0
$139$	−68.0000			−0.489209			−0.244604	−	0.969623i	$-0.578658\pi$
$139$	−68.0000			−0.489209			−0.244604	+	0.969623i	$0.578658\pi$
$140$		0			0
$141$	138.564	+	123.935i	0.982724	+	0.878975i
$142$		0			0
$143$			278.855i			1.95003i
$144$		0			0
$145$	60.0000			0.413793
$146$		0			0
$147$	74.0000	−	82.7345i	0.503401	−	0.562820i
$148$		0			0
$149$			116.190i			0.779795i	0.920858	+	0.389898i	$0.127490\pi$
$149$			116.190i			0.779795i	−0.920858	+	0.389898i	$0.872510\pi$
$150$		0			0
$151$	−252.879			−1.67470			−0.837349	−	0.546669i	$-0.815896\pi$
$151$	−252.879			−1.67470			−0.837349	+	0.546669i	$0.815896\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$			187.830i			1.21180i
$156$		0			0
$157$	−117.779			−0.750188			−0.375094	−	0.926987i	$-0.622390\pi$
$157$	−117.779			−0.750188			−0.375094	+	0.926987i	$0.622390\pi$
$158$		0			0
$159$	−121.244	−	108.444i	−0.762538	−	0.682035i
$160$		0			0
$161$			107.331i			0.666654i
$162$		0			0
$163$	−52.0000			−0.319018			−0.159509	−	0.987196i	$-0.550991\pi$
$163$	−52.0000			−0.319018			−0.159509	+	0.987196i	$0.550991\pi$
$164$		0			0
$165$	−207.846	+	232.379i	−1.25967	+	1.40836i
$166$		0			0
$167$			30.9839i			0.185532i	0.995688	+	0.0927661i	$0.0295709\pi$
$167$			30.9839i			0.185532i	−0.995688	+	0.0927661i	$0.970429\pi$
$168$		0			0
$169$	263.000			1.55621
$170$		0			0
$171$	4.00000	+	35.7771i	0.0233918	+	0.209223i
$172$		0			0
$173$			7.74597i			0.0447744i	0.999749	+	0.0223872i	$0.00712666\pi$
$173$			7.74597i			0.0447744i	−0.999749	+	0.0223872i	$0.992873\pi$
$174$		0			0
$175$	121.244			0.692820
$176$		0			0
$177$	90.0000	+	80.4984i	0.508475	+	0.454793i
$178$		0			0
$179$		−	67.0820i		−	0.374760i	−0.982288	−	0.187380i	$-0.940000\pi$
$179$		−	67.0820i		−	0.374760i	0.982288	−	0.187380i	$-0.0599996\pi$
$180$		0			0
$181$	−6.92820			−0.0382774			−0.0191387	−	0.999817i	$-0.506092\pi$
$181$	−6.92820			−0.0382774			−0.0191387	+	0.999817i	$0.506092\pi$
$182$		0			0
$183$	13.8564	−	15.4919i	0.0757181	−	0.0846554i
$184$		0			0
$185$			268.328i			1.45042i
$186$		0			0
$187$		0			0
$188$		0			0
$189$	−76.2102	−	54.2218i	−0.403229	−	0.286888i
$190$		0			0
$191$		−	123.935i		−	0.648877i	−0.945907	−	0.324438i	$-0.894825\pi$
$191$		−	123.935i		−	0.648877i	0.945907	−	0.324438i	$-0.105175\pi$
$192$		0			0
$193$	−22.0000			−0.113990			−0.0569948	−	0.998374i	$-0.518152\pi$
$193$	−22.0000			−0.113990			−0.0569948	+	0.998374i	$0.518152\pi$
$194$		0			0
$195$	360.000	+	321.994i	1.84615	+	1.65125i
$196$		0			0
$197$		−	69.7137i		−	0.353877i	−0.984222	−	0.176938i	$-0.943381\pi$
$197$		−	69.7137i		−	0.353877i	0.984222	−	0.176938i	$-0.0566193\pi$
$198$		0			0
$199$	301.377			1.51446			0.757228	−	0.653150i	$-0.226553\pi$
$199$	301.377			1.51446			0.757228	+	0.653150i	$0.226553\pi$
$200$		0			0
$201$	−56.0000	+	62.6099i	−0.278607	+	0.311492i
$202$		0			0
$203$		−	26.8328i		−	0.132181i
$204$		0			0
$205$	−415.692			−2.02777
$206$		0			0
$207$	−277.128	+	30.9839i	−1.33878	+	0.149681i
$208$		0			0
$209$		−	53.6656i		−	0.256773i
$210$		0			0
$211$	−404.000			−1.91469			−0.957346	−	0.288944i	$-0.906696\pi$
$211$	−404.000			−1.91469			−0.957346	+	0.288944i	$0.906696\pi$
$212$		0			0
$213$	−69.2820	−	61.9677i	−0.325268	−	0.290928i
$214$		0			0
$215$			402.790i			1.87344i
$216$		0			0
$217$	84.0000			0.387097
$218$		0			0
$219$	148.000	−	165.469i	0.675799	−	0.755566i
$220$		0			0
$221$		0			0
$222$		0			0
$223$	169.741			0.761170			0.380585	−	0.924746i	$-0.375723\pi$
$223$	169.741			0.761170			0.380585	+	0.924746i	$0.375723\pi$
$224$		0			0
$225$	35.0000	+	313.050i	0.155556	+	1.39133i
$226$		0			0
$227$			308.577i			1.35937i	0.733503	+	0.679686i	$0.237884\pi$
$227$			308.577i			1.35937i	−0.733503	+	0.679686i	$0.762116\pi$
$228$		0			0
$229$	408.764			1.78500			0.892498	−	0.451052i	$-0.148951\pi$
$229$	408.764			1.78500			0.892498	+	0.451052i	$0.148951\pi$
$230$		0			0
$231$	103.923	+	92.9516i	0.449883	+	0.402388i
$232$		0			0
$233$			268.328i			1.15162i	0.817582	+	0.575811i	$0.195314\pi$
$233$			268.328i			1.15162i	−0.817582	+	0.575811i	$0.804686\pi$
$234$		0			0
$235$	−480.000			−2.04255
$236$		0			0
$237$	103.923	−	116.190i	0.438494	−	0.490251i
$238$		0			0
$239$		−	309.839i		−	1.29640i	−0.761472	−	0.648198i	$-0.775523\pi$
$239$		−	309.839i		−	1.29640i	0.761472	−	0.648198i	$-0.224477\pi$
$240$		0			0
$241$	74.0000			0.307054			0.153527	−	0.988144i	$-0.450937\pi$
$241$	74.0000			0.307054			0.153527	+	0.988144i	$0.450937\pi$
$242$		0			0
$243$	118.000	−	212.426i	0.485597	−	0.874183i
$244$		0			0
$245$			286.601i			1.16980i
$246$		0			0
$247$	−83.1384			−0.336593
$248$		0			0
$249$	270.000	+	241.495i	1.08434	+	0.969861i
$250$		0			0
$251$			13.4164i			0.0534518i	0.999643	+	0.0267259i	$0.00850814\pi$
$251$			13.4164i			0.0534518i	−0.999643	+	0.0267259i	$0.991492\pi$
$252$		0			0
$253$	415.692			1.64305
$254$		0			0
$255$		0			0
$256$		0			0
$257$			214.663i			0.835263i	0.908617	+	0.417631i	$0.137140\pi$
$257$			214.663i			0.835263i	−0.908617	+	0.417631i	$0.862860\pi$
$258$		0			0
$259$	120.000			0.463320
$260$		0			0
$261$	69.2820	−	7.74597i	0.265448	−	0.0296780i
$262$		0			0
$263$		−	464.758i		−	1.76714i	−0.468298	−	0.883570i	$-0.655133\pi$
$263$		−	464.758i		−	1.76714i	0.468298	−	0.883570i	$-0.344867\pi$
$264$		0			0
$265$	420.000			1.58491
$266$		0			0
$267$	120.000	+	107.331i	0.449438	+	0.401990i
$268$		0			0
$269$		−	302.093i		−	1.12302i	−0.827470	−	0.561511i	$-0.810220\pi$
$269$		−	302.093i		−	1.12302i	0.827470	−	0.561511i	$-0.189780\pi$
$270$		0			0
$271$	−329.090			−1.21435			−0.607176	−	0.794567i	$-0.707698\pi$
$271$	−329.090			−1.21435			−0.607176	+	0.794567i	$0.707698\pi$
$272$		0			0
$273$	144.000	−	160.997i	0.527473	−	0.589732i
$274$		0			0
$275$		−	469.574i		−	1.70754i
$276$		0			0
$277$	159.349			0.575266			0.287633	−	0.957741i	$-0.407132\pi$
$277$	159.349			0.575266			0.287633	+	0.957741i	$0.407132\pi$
$278$		0			0
$279$	24.2487	+	216.887i	0.0869129	+	0.777373i
$280$		0			0
$281$		−	268.328i		−	0.954904i	−0.878658	−	0.477452i	$-0.841560\pi$
$281$		−	268.328i		−	0.954904i	0.878658	−	0.477452i	$-0.158440\pi$
$282$		0			0
$283$	172.000			0.607774			0.303887	−	0.952708i	$-0.401715\pi$
$283$	172.000			0.607774			0.303887	+	0.952708i	$0.401715\pi$
$284$		0			0
$285$	−69.2820	−	61.9677i	−0.243095	−	0.217431i
$286$		0			0
$287$			185.903i			0.647746i
$288$		0			0
$289$	289.000			1.00000
$290$		0			0
$291$	124.000	−	138.636i	0.426117	−	0.476413i
$292$		0			0
$293$		−	317.585i		−	1.08391i	−0.840409	−	0.541953i	$-0.817685\pi$
$293$		−	317.585i		−	1.08391i	0.840409	−	0.541953i	$-0.182315\pi$
$294$		0			0
$295$	−311.769			−1.05684
$296$		0			0
$297$	−210.000	+	295.161i	−0.707071	+	0.993808i
$298$		0			0
$299$		−	643.988i		−	2.15380i
$300$		0			0
$301$	180.133			0.598449
$302$		0			0
$303$	294.449	+	263.363i	0.971778	+	0.869184i
$304$		0			0
$305$			53.6656i			0.175953i
$306$		0			0
$307$	−212.000			−0.690554			−0.345277	−	0.938501i	$-0.612215\pi$
$307$	−212.000			−0.690554			−0.345277	+	0.938501i	$0.612215\pi$
$308$		0			0
$309$	173.205	−	193.649i	0.560534	−	0.626696i
$310$		0			0
$311$		−	526.726i		−	1.69365i	−0.531870	−	0.846826i	$-0.678511\pi$
$311$		−	526.726i		−	1.69365i	0.531870	−	0.846826i	$-0.321489\pi$
$312$		0			0
$313$	−202.000			−0.645367			−0.322684	−	0.946507i	$-0.604585\pi$
$313$	−202.000			−0.645367			−0.322684	+	0.946507i	$0.604585\pi$
$314$		0			0
$315$	240.000	−	26.8328i	0.761905	−	0.0851835i
$316$		0			0
$317$			69.7137i			0.219917i	0.993936	+	0.109959i	$0.0350718\pi$
$317$			69.7137i			0.219917i	−0.993936	+	0.109959i	$0.964928\pi$
$318$		0			0
$319$	−103.923			−0.325778
$320$		0			0
$321$	90.0000	+	80.4984i	0.280374	+	0.250774i
$322$		0			0
$323$		0			0
$324$		0			0
$325$	−727.461			−2.23834
$326$		0			0
$327$	−152.420	+	170.411i	−0.466118	+	0.521135i
$328$		0			0
$329$			214.663i			0.652470i
$330$		0			0
$331$	28.0000			0.0845921			0.0422961	−	0.999105i	$-0.486533\pi$
$331$	28.0000			0.0845921			0.0422961	+	0.999105i	$0.486533\pi$
$332$		0			0
$333$	34.6410	+	309.839i	0.104027	+	0.930446i
$334$		0			0
$335$		−	216.887i		−	0.647424i
$336$		0			0
$337$	298.000			0.884273			0.442136	−	0.896948i	$-0.354221\pi$
$337$	298.000			0.884273			0.442136	+	0.896948i	$0.354221\pi$
$338$		0			0
$339$	−240.000	−	214.663i	−0.707965	−	0.633223i
$340$		0			0
$341$		−	325.331i		−	0.954049i
$342$		0			0
$343$	297.913			0.868550
$344$		0			0
$345$	480.000	−	536.656i	1.39130	−	1.55553i
$346$		0			0
$347$			442.741i			1.27591i	0.770073	+	0.637956i	$0.220220\pi$
$347$			442.741i			1.27591i	−0.770073	+	0.637956i	$0.779780\pi$
$348$		0			0
$349$	−173.205			−0.496290			−0.248145	−	0.968723i	$-0.579821\pi$
$349$	−173.205			−0.496290			−0.248145	+	0.968723i	$0.579821\pi$
$350$		0			0
$351$	457.261	+	325.331i	1.30274	+	0.926868i
$352$		0			0
$353$			643.988i			1.82433i	0.409826	+	0.912164i	$0.365589\pi$
$353$			643.988i			1.82433i	−0.409826	+	0.912164i	$0.634411\pi$
$354$		0			0
$355$	240.000			0.676056
$356$		0			0
$357$		0			0
$358$		0			0
$359$			402.790i			1.12198i	0.827823	+	0.560989i	$0.189579\pi$
$359$			402.790i			1.12198i	−0.827823	+	0.560989i	$0.810421\pi$
$360$		0			0
$361$	−345.000			−0.955679
$362$		0			0
$363$	118.000	−	131.928i	0.325069	−	0.363438i
$364$		0			0
$365$			573.202i			1.57042i
$366$		0			0
$367$	419.156			1.14212			0.571058	−	0.820910i	$-0.306533\pi$
$367$	419.156			1.14212			0.571058	+	0.820910i	$0.306533\pi$
$368$		0			0
$369$	−480.000	+	53.6656i	−1.30081	+	0.145435i
$370$		0			0
$371$		−	187.830i		−	0.506280i
$372$		0			0
$373$	−394.908			−1.05873			−0.529367	−	0.848393i	$-0.677570\pi$
$373$	−394.908			−1.05873			−0.529367	+	0.848393i	$0.677570\pi$
$374$		0			0
$375$	−173.205	−	154.919i	−0.461880	−	0.413118i
$376$		0			0
$377$			160.997i			0.427047i
$378$		0			0
$379$	−484.000			−1.27704			−0.638522	−	0.769603i	$-0.720454\pi$
$379$	−484.000			−1.27704			−0.638522	+	0.769603i	$0.720454\pi$
$380$		0			0
$381$	−173.205	+	193.649i	−0.454607	+	0.508266i
$382$		0			0
$383$		−	123.935i		−	0.323591i	−0.986824	−	0.161796i	$-0.948271\pi$
$383$		−	123.935i		−	0.323591i	0.986824	−	0.161796i	$-0.0517285\pi$
$384$		0			0
$385$	−360.000			−0.935065
$386$		0			0
$387$	52.0000	+	465.102i	0.134367	+	1.20181i
$388$		0			0
$389$			426.028i			1.09519i	0.836744	+	0.547594i	$0.184456\pi$
$389$			426.028i			1.09519i	−0.836744	+	0.547594i	$0.815544\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$	−330.000	−	295.161i	−0.839695	−	0.751046i
$394$		0			0
$395$			402.492i			1.01897i
$396$		0			0
$397$	464.190			1.16924			0.584622	−	0.811306i	$-0.301243\pi$
$397$	464.190			1.16924			0.584622	+	0.811306i	$0.301243\pi$
$398$		0			0
$399$	−27.7128	+	30.9839i	−0.0694557	+	0.0776538i
$400$		0			0
$401$		−	643.988i		−	1.60595i	−0.596010	−	0.802977i	$-0.703248\pi$
$401$		−	643.988i		−	1.60595i	0.596010	−	0.802977i	$-0.296752\pi$
$402$		0			0
$403$	−504.000			−1.25062
$404$		0			0
$405$	138.564	+	611.931i	0.342133	+	1.51094i
$406$		0			0
$407$		−	464.758i		−	1.14191i
$408$		0			0
$409$	−386.000			−0.943765			−0.471883	−	0.881661i	$-0.656425\pi$
$409$	−386.000			−0.943765			−0.471883	+	0.881661i	$0.656425\pi$
$410$		0			0
$411$	120.000	+	107.331i	0.291971	+	0.261147i
$412$		0			0
$413$			139.427i			0.337597i
$414$		0			0
$415$	−935.307			−2.25375
$416$		0			0
$417$	136.000	−	152.053i	0.326139	−	0.364635i
$418$		0			0
$419$			308.577i			0.736462i	0.929734	+	0.368231i	$0.120036\pi$
$419$			308.577i			0.736462i	−0.929734	+	0.368231i	$0.879964\pi$
$420$		0			0
$421$	−34.6410			−0.0822827			−0.0411413	−	0.999153i	$-0.513099\pi$
$421$	−34.6410			−0.0822827			−0.0411413	+	0.999153i	$0.513099\pi$
$422$		0			0
$423$	−554.256	+	61.9677i	−1.31030	+	0.146496i
$424$		0			0
$425$		0			0
$426$		0			0
$427$	24.0000			0.0562061
$428$		0			0
$429$	−623.538	−	557.710i	−1.45347	−	1.30002i
$430$		0			0
$431$			433.774i			1.00644i	0.864159	+	0.503218i	$0.167851\pi$
$431$			433.774i			1.00644i	−0.864159	+	0.503218i	$0.832149\pi$
$432$		0			0
$433$	322.000			0.743649			0.371824	−	0.928303i	$-0.378732\pi$
$433$	322.000			0.743649			0.371824	+	0.928303i	$0.378732\pi$
$434$		0			0
$435$	−120.000	+	134.164i	−0.275862	+	0.308423i
$436$		0			0
$437$			123.935i			0.283605i
$438$		0			0
$439$	633.931			1.44403			0.722017	−	0.691876i	$-0.243215\pi$
$439$	633.931			1.44403			0.722017	+	0.691876i	$0.243215\pi$
$440$		0			0
$441$	37.0000	+	330.938i	0.0839002	+	0.750426i
$442$		0			0
$443$			13.4164i			0.0302853i	0.999885	+	0.0151427i	$0.00482025\pi$
$443$			13.4164i			0.0302853i	−0.999885	+	0.0151427i	$0.995180\pi$
$444$		0			0
$445$	−415.692			−0.934140
$446$		0			0
$447$	−259.808	−	232.379i	−0.581225	−	0.519864i
$448$		0			0
$449$		−	536.656i		−	1.19523i	−0.801785	−	0.597613i	$-0.796116\pi$
$449$		−	536.656i		−	1.19523i	0.801785	−	0.597613i	$-0.203884\pi$
$450$		0			0
$451$	720.000			1.59645
$452$		0			0
$453$	505.759	−	565.456i	1.11647	−	1.24825i
$454$		0			0
$455$			557.710i			1.22574i
$456$		0			0
$457$	574.000			1.25602			0.628009	−	0.778206i	$-0.283870\pi$
$457$	574.000			1.25602			0.628009	+	0.778206i	$0.283870\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		−	487.996i		−	1.05856i	−0.848447	−	0.529280i	$-0.822462\pi$
$461$		−	487.996i		−	1.05856i	0.848447	−	0.529280i	$-0.177538\pi$
$462$		0			0
$463$	363.731			0.785595			0.392798	−	0.919625i	$-0.371507\pi$
$463$	363.731			0.785595			0.392798	+	0.919625i	$0.371507\pi$
$464$		0			0
$465$	−420.000	−	375.659i	−0.903226	−	0.807870i
$466$		0			0
$467$			308.577i			0.660765i	0.943847	+	0.330383i	$0.107178\pi$
$467$			308.577i			0.660765i	−0.943847	+	0.330383i	$0.892822\pi$
$468$		0			0
$469$	−96.9948			−0.206812
$470$		0			0
$471$	235.559	−	263.363i	0.500125	−	0.559157i
$472$		0			0
$473$		−	697.653i		−	1.47495i
$474$		0			0
$475$	140.000			0.294737
$476$		0			0
$477$	484.974	−	54.2218i	1.01672	−	0.113672i
$478$		0			0
$479$		−	371.806i		−	0.776214i	−0.921614	−	0.388107i	$-0.873129\pi$
$479$		−	371.806i		−	0.776214i	0.921614	−	0.388107i	$-0.126871\pi$
$480$		0			0
$481$	−720.000			−1.49688
$482$		0			0
$483$	−240.000	−	214.663i	−0.496894	−	0.444436i
$484$		0			0
$485$			480.250i			0.990206i
$486$		0			0
$487$	384.515			0.789559			0.394780	−	0.918776i	$-0.370821\pi$
$487$	384.515			0.789559			0.394780	+	0.918776i	$0.370821\pi$
$488$		0			0
$489$	104.000	−	116.276i	0.212679	−	0.237782i
$490$		0			0
$491$		−	254.912i		−	0.519169i	−0.965720	−	0.259584i	$-0.916414\pi$
$491$		−	254.912i		−	0.519169i	0.965720	−	0.259584i	$-0.0835855\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$	−103.923	−	929.516i	−0.209946	−	1.87781i
$496$		0			0
$497$		−	107.331i		−	0.215958i
$498$		0			0
$499$	844.000			1.69138			0.845691	−	0.533672i	$-0.179188\pi$
$499$	844.000			1.69138			0.845691	+	0.533672i	$0.179188\pi$
$500$		0			0
$501$	−69.2820	−	61.9677i	−0.138287	−	0.123688i
$502$		0			0
$503$			712.629i			1.41676i	0.705833	+	0.708379i	$0.250573\pi$
$503$			712.629i			1.41676i	−0.705833	+	0.708379i	$0.749427\pi$
$504$		0			0
$505$	−1020.00			−2.01980
$506$		0			0
$507$	−526.000	+	588.086i	−1.03748	+	1.15993i
$508$		0			0
$509$		−	983.738i		−	1.93269i	−0.257255	−	0.966344i	$-0.582818\pi$
$509$		−	983.738i		−	1.93269i	0.257255	−	0.966344i	$-0.417182\pi$
$510$		0			0
$511$	256.344			0.501651
$512$		0			0
$513$	−88.0000	−	62.6099i	−0.171540	−	0.122047i
$514$		0			0
$515$			670.820i			1.30256i
$516$		0			0
$517$	831.384			1.60809
$518$		0			0
$519$	−17.3205	−	15.4919i	−0.0333728	−	0.0298496i
$520$		0			0
$521$		−	912.316i		−	1.75109i	−0.483140	−	0.875543i	$-0.660504\pi$
$521$		−	912.316i		−	1.75109i	0.483140	−	0.875543i	$-0.339496\pi$
$522$		0			0
$523$	−52.0000			−0.0994264			−0.0497132	−	0.998764i	$-0.515831\pi$
$523$	−52.0000			−0.0994264			−0.0497132	+	0.998764i	$0.515831\pi$
$524$		0			0
$525$	−242.487	+	271.109i	−0.461880	+	0.516398i
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−431.000			−0.814745
$530$		0			0
$531$	−360.000	+	40.2492i	−0.677966	+	0.0757989i
$532$		0			0
$533$		−	1115.42i		−	2.09272i
$534$		0			0
$535$	−311.769			−0.582746
$536$		0			0
$537$	150.000	+	134.164i	0.279330	+	0.249840i
$538$		0			0
$539$		−	496.407i		−	0.920978i
$540$		0			0
$541$	491.902			0.909247			0.454623	−	0.890684i	$-0.349774\pi$
$541$	491.902			0.909247			0.454623	+	0.890684i	$0.349774\pi$
$542$		0			0
$543$	13.8564	−	15.4919i	0.0255182	−	0.0285303i
$544$		0			0
$545$		−	590.322i		−	1.08316i
$546$		0			0
$547$	−724.000			−1.32358			−0.661792	−	0.749688i	$-0.730204\pi$
$547$	−724.000			−1.32358			−0.661792	+	0.749688i	$0.730204\pi$
$548$		0			0
$549$	6.92820	+	61.9677i	0.0126197	+	0.112874i
$550$		0			0
$551$		−	30.9839i		−	0.0562321i
$552$		0			0
$553$	180.000			0.325497
$554$		0			0
$555$	−600.000	−	536.656i	−1.08108	−	0.966948i
$556$		0			0
$557$			441.520i			0.792675i	0.918105	+	0.396338i	$0.129719\pi$
$557$			441.520i			0.792675i	−0.918105	+	0.396338i	$0.870281\pi$
$558$		0			0
$559$	−1080.80			−1.93345
$560$		0			0
$561$		0			0
$562$		0			0
$563$		−	120.748i		−	0.214472i	−0.994234	−	0.107236i	$-0.965800\pi$
$563$		−	120.748i		−	0.214472i	0.994234	−	0.107236i	$-0.0342000\pi$
$564$		0			0
$565$	831.384			1.47148
$566$		0			0
$567$	273.664	−	61.9677i	0.482653	−	0.109291i
$568$		0			0
$569$			482.991i			0.848841i	0.905465	+	0.424421i	$0.139522\pi$
$569$			482.991i			0.848841i	−0.905465	+	0.424421i	$0.860478\pi$
$570$		0			0
$571$	1036.00			1.81436			0.907180	−	0.420742i	$-0.138230\pi$
$571$	1036.00			1.81436			0.907180	+	0.420742i	$0.138230\pi$
$572$		0			0
$573$	277.128	+	247.871i	0.483644	+	0.432585i
$574$		0			0
$575$			1084.44i			1.88597i
$576$		0			0
$577$	−182.000			−0.315425			−0.157712	−	0.987485i	$-0.550412\pi$
$577$	−182.000			−0.315425			−0.157712	+	0.987485i	$0.550412\pi$
$578$		0			0
$579$	44.0000	−	49.1935i	0.0759931	−	0.0849629i
$580$		0			0
$581$			418.282i			0.719935i
$582$		0			0
$583$	−727.461			−1.24779
$584$		0			0
$585$	−1440.00	+	160.997i	−2.46154	+	0.275208i
$586$		0			0
$587$			1033.06i			1.75990i	0.475063	+	0.879952i	$0.342425\pi$
$587$			1033.06i			1.75990i	−0.475063	+	0.879952i	$0.657575\pi$
$588$		0			0
$589$	96.9948			0.164677
$590$		0			0
$591$	155.885	+	139.427i	0.263764	+	0.235918i
$592$		0			0
$593$			107.331i			0.180997i	0.995897	+	0.0904985i	$0.0288460\pi$
$593$			107.331i			0.180997i	−0.995897	+	0.0904985i	$0.971154\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−602.754	+	673.899i	−1.00964	+	1.12881i
$598$		0			0
$599$		−	30.9839i		−	0.0517260i	−0.999665	−	0.0258630i	$-0.991767\pi$
$599$		−	30.9839i		−	0.0517260i	0.999665	−	0.0258630i	$-0.00823336\pi$
$600$		0			0
$601$	−554.000			−0.921797			−0.460899	−	0.887453i	$-0.652473\pi$
$601$	−554.000			−0.921797			−0.460899	+	0.887453i	$0.652473\pi$
$602$		0			0
$603$	−28.0000	−	250.440i	−0.0464345	−	0.415323i
$604$		0			0
$605$			457.012i			0.755392i
$606$		0			0
$607$	917.987			1.51233			0.756167	−	0.654379i	$-0.227070\pi$
$607$	917.987			1.51233			0.756167	+	0.654379i	$0.227070\pi$
$608$		0			0
$609$	60.0000	+	53.6656i	0.0985222	+	0.0881209i
$610$		0			0
$611$		−	1287.98i		−	2.10798i
$612$		0			0
$613$	−866.025			−1.41277			−0.706383	−	0.707830i	$-0.749674\pi$
$613$	−866.025			−1.41277			−0.706383	+	0.707830i	$0.749674\pi$
$614$		0			0
$615$	831.384	−	929.516i	1.35184	−	1.51141i
$616$		0			0
$617$		−	375.659i		−	0.608848i	−0.952537	−	0.304424i	$-0.901536\pi$
$617$		−	375.659i		−	0.608848i	0.952537	−	0.304424i	$-0.0984640\pi$
$618$		0			0
$619$	124.000			0.200323			0.100162	−	0.994971i	$-0.468064\pi$
$619$	124.000			0.200323			0.100162	+	0.994971i	$0.468064\pi$
$620$		0			0
$621$	484.974	−	681.645i	0.780957	−	1.09766i
$622$		0			0
$623$			185.903i			0.298400i
$624$		0			0
$625$	−275.000			−0.440000
$626$		0			0
$627$	120.000	+	107.331i	0.191388	+	0.171182i
$628$		0			0
$629$		0			0
$630$		0			0
$631$	−890.274			−1.41089			−0.705447	−	0.708763i	$-0.749254\pi$
$631$	−890.274			−1.41089			−0.705447	+	0.708763i	$0.749254\pi$
$632$		0			0
$633$	808.000	−	903.371i	1.27646	−	1.42713i
$634$		0			0
$635$		−	670.820i		−	1.05641i
$636$		0			0
$637$	−769.031			−1.20727
$638$		0			0
$639$	277.128	−	30.9839i	0.433690	−	0.0484881i
$640$		0			0
$641$		−	751.319i		−	1.17210i	−0.810273	−	0.586052i	$-0.800681\pi$
$641$		−	751.319i		−	1.17210i	0.810273	−	0.586052i	$-0.199319\pi$
$642$		0			0
$643$	524.000			0.814930			0.407465	−	0.913221i	$-0.366413\pi$
$643$	524.000			0.814930			0.407465	+	0.913221i	$0.366413\pi$
$644$		0			0
$645$	−900.666	−	805.581i	−1.39638	−	1.24896i
$646$		0			0
$647$		−	340.823i		−	0.526774i	−0.964690	−	0.263387i	$-0.915160\pi$
$647$		−	340.823i		−	0.526774i	0.964690	−	0.263387i	$-0.0848395\pi$
$648$		0			0
$649$	540.000			0.832049
$650$		0			0
$651$	−168.000	+	187.830i	−0.258065	+	0.288525i
$652$		0			0
$653$			627.423i			0.960832i	0.877041	+	0.480416i	$0.159514\pi$
$653$			627.423i			0.960832i	−0.877041	+	0.480416i	$0.840486\pi$
$654$		0			0
$655$	1143.15			1.74527
$656$		0			0
$657$	74.0000	+	661.876i	0.112633	+	1.00742i
$658$		0			0
$659$		−	1140.39i		−	1.73049i	−0.501347	−	0.865246i	$-0.667162\pi$
$659$		−	1140.39i		−	1.73049i	0.501347	−	0.865246i	$-0.332838\pi$
$660$		0			0
$661$	−450.333			−0.681291			−0.340645	−	0.940192i	$-0.610646\pi$
$661$	−450.333			−0.681291			−0.340645	+	0.940192i	$0.610646\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		−	107.331i		−	0.161400i
$666$		0			0
$667$	240.000			0.359820
$668$		0			0
$669$	−339.482	+	379.552i	−0.507447	+	0.567343i
$670$		0			0
$671$		−	92.9516i		−	0.138527i
$672$		0			0
$673$	226.000			0.335810			0.167905	−	0.985803i	$-0.446300\pi$
$673$	226.000			0.335810			0.167905	+	0.985803i	$0.446300\pi$
$674$		0			0
$675$	−770.000	−	547.837i	−1.14074	−	0.811610i
$676$		0			0
$677$		−	7.74597i		−	0.0114416i	−0.999984	−	0.00572080i	$-0.998179\pi$
$677$		−	7.74597i		−	0.0114416i	0.999984	−	0.00572080i	$-0.00182100\pi$
$678$		0			0
$679$	214.774			0.316310
$680$		0			0
$681$	−690.000	−	617.155i	−1.01322	−	0.906248i
$682$		0			0
$683$		−	415.909i		−	0.608944i	−0.952521	−	0.304472i	$-0.901520\pi$
$683$		−	415.909i		−	0.608944i	0.952521	−	0.304472i	$-0.0984800\pi$
$684$		0			0
$685$	−415.692			−0.606850
$686$		0			0
$687$	−817.528	+	914.024i	−1.19000	+	1.33046i
$688$		0			0
$689$			1126.98i			1.63567i
$690$		0			0
$691$	572.000			0.827786			0.413893	−	0.910326i	$-0.364169\pi$
$691$	572.000			0.827786			0.413893	+	0.910326i	$0.364169\pi$
$692$		0			0
$693$	−415.692	+	46.4758i	−0.599844	+	0.0670646i
$694$		0			0
$695$			526.726i			0.757879i
$696$		0			0
$697$		0			0
$698$		0			0
$699$	−600.000	−	536.656i	−0.858369	−	0.767749i
$700$		0			0
$701$			441.520i			0.629843i	0.949118	+	0.314922i	$0.101978\pi$
$701$			441.520i			0.629843i	−0.949118	+	0.314922i	$0.898022\pi$
$702$		0			0
$703$	138.564			0.197104
$704$		0			0
$705$	960.000	−	1073.31i	1.36170	−	1.52243i
$706$		0			0
$707$			456.158i			0.645202i
$708$		0			0
$709$	353.338			0.498362			0.249181	−	0.968457i	$-0.419839\pi$
$709$	353.338			0.498362			0.249181	+	0.968457i	$0.419839\pi$
$710$		0			0
$711$	51.9615	+	464.758i	0.0730823	+	0.653668i
$712$		0			0
$713$			751.319i			1.05374i
$714$		0			0
$715$	2160.00			3.02098
$716$		0			0
$717$	692.820	+	619.677i	0.966277	+	0.864264i
$718$		0			0
$719$		−	557.710i		−	0.775674i	−0.921728	−	0.387837i	$-0.873222\pi$
$719$		−	557.710i		−	0.775674i	0.921728	−	0.387837i	$-0.126778\pi$
$720$		0			0
$721$	300.000			0.416089
$722$		0			0
$723$	−148.000	+	165.469i	−0.204703	+	0.228864i
$724$		0			0
$725$		−	271.109i		−	0.373943i
$726$		0			0
$727$	−696.284			−0.957750			−0.478875	−	0.877883i	$-0.658955\pi$
$727$	−696.284			−0.957750			−0.478875	+	0.877883i	$0.658955\pi$
$728$		0			0
$729$	239.000	+	688.709i	0.327846	+	0.944731i
$730$		0			0
$731$		0			0
$732$		0			0
$733$	−173.205			−0.236296			−0.118148	−	0.992996i	$-0.537696\pi$
$733$	−173.205			−0.236296			−0.118148	+	0.992996i	$0.537696\pi$
$734$		0			0
$735$	−640.859	−	573.202i	−0.871917	−	0.779866i
$736$		0			0
$737$			375.659i			0.509714i
$738$		0			0
$739$	−548.000			−0.741543			−0.370771	−	0.928724i	$-0.620907\pi$
$739$	−548.000			−0.741543			−0.370771	+	0.928724i	$0.620907\pi$
$740$		0			0
$741$	166.277	−	185.903i	0.224395	−	0.250882i
$742$		0			0
$743$		−	1456.24i		−	1.95995i	−0.199124	−	0.979974i	$-0.563810\pi$
$743$		−	1456.24i		−	1.95995i	0.199124	−	0.979974i	$-0.436190\pi$
$744$		0			0
$745$	900.000			1.20805
$746$		0			0
$747$	−1080.00	+	120.748i	−1.44578	+	0.161643i
$748$		0			0
$749$			139.427i			0.186151i
$750$		0			0
$751$	−329.090			−0.438202			−0.219101	−	0.975702i	$-0.570312\pi$
$751$	−329.090			−0.438202			−0.219101	+	0.975702i	$0.570312\pi$
$752$		0			0
$753$	−30.0000	−	26.8328i	−0.0398406	−	0.0356345i
$754$		0			0
$755$			1958.80i			2.59443i
$756$		0			0
$757$	325.626			0.430153			0.215076	−	0.976597i	$-0.431000\pi$
$757$	325.626			0.430153			0.215076	+	0.976597i	$0.431000\pi$
$758$		0			0
$759$	−831.384	+	929.516i	−1.09537	+	1.22466i
$760$		0			0
$761$		−	160.997i		−	0.211560i	−0.994390	−	0.105780i	$-0.966266\pi$
$761$		−	160.997i		−	0.211560i	0.994390	−	0.105780i	$-0.0337339\pi$
$762$		0			0
$763$	−264.000			−0.346003
$764$		0			0
$765$		0			0
$766$		0			0
$767$		−	836.564i		−	1.09070i
$768$		0			0
$769$	−214.000			−0.278283			−0.139142	−	0.990272i	$-0.544434\pi$
$769$	−214.000			−0.278283			−0.139142	+	0.990272i	$0.544434\pi$
$770$		0			0
$771$	−480.000	−	429.325i	−0.622568	−	0.556842i
$772$		0			0
$773$			611.931i			0.791632i	0.918330	+	0.395816i	$0.129538\pi$
$773$			611.931i			0.791632i	−0.918330	+	0.395816i	$0.870462\pi$
$774$		0			0
$775$	848.705			1.09510
$776$		0			0
$777$	−240.000	+	268.328i	−0.308880	+	0.345339i
$778$		0			0
$779$			214.663i			0.275562i
$780$		0			0
$781$	−415.692			−0.532256
$782$		0			0
$783$	−121.244	+	170.411i	−0.154845	+	0.217639i
$784$		0			0
$785$			912.316i			1.16219i
$786$		0			0
$787$	1052.00			1.33672			0.668361	−	0.743837i	$-0.266996\pi$
$787$	1052.00			1.33672			0.668361	+	0.743837i	$0.266996\pi$
$788$		0			0
$789$	1039.23	+	929.516i	1.31715	+	1.17809i
$790$		0			0
$791$		−	371.806i		−	0.470046i
$792$		0			0
$793$	−144.000			−0.181589
$794$		0			0
$795$	−840.000	+	939.149i	−1.05660	+	1.18132i
$796$		0			0
$797$		−	1293.58i		−	1.62306i	−0.584313	−	0.811529i	$-0.698636\pi$
$797$		−	1293.58i		−	1.62306i	0.584313	−	0.811529i	$-0.301364\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$	−480.000	+	53.6656i	−0.599251	+	0.0669983i
$802$		0			0
$803$		−	992.814i		−	1.23638i
$804$		0			0
$805$	831.384			1.03278
$806$		0			0
$807$	675.500	+	604.185i	0.837051	+	0.748681i
$808$		0			0
$809$			375.659i			0.464350i	0.972674	+	0.232175i	$0.0745843\pi$
$809$			375.659i			0.464350i	−0.972674	+	0.232175i	$0.925416\pi$
$810$		0			0
$811$	−916.000			−1.12947			−0.564735	−	0.825272i	$-0.691022\pi$
$811$	−916.000			−1.12947			−0.564735	+	0.825272i	$0.691022\pi$
$812$		0			0
$813$	658.179	−	735.867i	0.809569	−	0.905125i
$814$		0			0
$815$			402.790i			0.494221i
$816$		0			0
$817$	208.000			0.254590
$818$		0			0
$819$	72.0000	+	643.988i	0.0879121	+	0.786310i
$820$		0			0
$821$			364.060i			0.443435i	0.975111	+	0.221718i	$0.0711663\pi$
$821$			364.060i			0.443435i	−0.975111	+	0.221718i	$0.928834\pi$
$822$		0			0
$823$	384.515			0.467212			0.233606	−	0.972331i	$-0.424947\pi$
$823$	384.515			0.467212			0.233606	+	0.972331i	$0.424947\pi$
$824$		0			0
$825$	1050.00	+	939.149i	1.27273	+	1.13836i
$826$		0			0
$827$			389.076i			0.470467i	0.971939	+	0.235233i	$0.0755854\pi$
$827$			389.076i			0.470467i	−0.971939	+	0.235233i	$0.924415\pi$
$828$		0			0
$829$	935.307			1.12824			0.564118	−	0.825694i	$-0.309216\pi$
$829$	935.307			1.12824			0.564118	+	0.825694i	$0.309216\pi$
$830$		0			0
$831$	−318.697	+	356.314i	−0.383511	+	0.428778i
$832$		0			0
$833$		0			0
$834$		0			0
$835$	240.000			0.287425
$836$		0			0
$837$	−533.472	−	379.552i	−0.637362	−	0.453468i
$838$		0			0
$839$			526.726i			0.627802i	0.949456	+	0.313901i	$0.101636\pi$
$839$			526.726i			0.627802i	−0.949456	+	0.313901i	$0.898364\pi$
$840$		0			0
$841$	781.000			0.928656
$842$		0			0
$843$	600.000	+	536.656i	0.711744	+	0.636603i
$844$		0			0
$845$		−	2037.19i		−	2.41087i
$846$		0			0
$847$	204.382			0.241301
$848$		0			0
$849$	−344.000	+	384.604i	−0.405183	+	0.453008i
$850$		0			0
$851$			1073.31i			1.26124i
$852$		0			0
$853$	325.626			0.381742			0.190871	−	0.981615i	$-0.438869\pi$
$853$	325.626			0.381742			0.190871	+	0.981615i	$0.438869\pi$
$854$		0			0
$855$	277.128	−	30.9839i	0.324126	−	0.0362384i
$856$		0			0
$857$			1556.30i			1.81599i	0.418981	+	0.907995i	$0.362387\pi$
$857$			1556.30i			1.81599i	−0.418981	+	0.907995i	$0.637613\pi$
$858$		0			0
$859$	188.000			0.218859			0.109430	−	0.993995i	$-0.465098\pi$
$859$	188.000			0.218859			0.109430	+	0.993995i	$0.465098\pi$
$860$		0			0
$861$	−415.692	−	371.806i	−0.482802	−	0.431831i
$862$		0			0
$863$		−	1239.35i		−	1.43610i	−0.695991	−	0.718050i	$-0.745035\pi$
$863$		−	1239.35i		−	1.43610i	0.695991	−	0.718050i	$-0.254965\pi$
$864$		0			0
$865$	60.0000			0.0693642
$866$		0			0
$867$	−578.000	+	646.224i	−0.666667	+	0.745356i
$868$		0			0
$869$		−	697.137i		−	0.802229i
$870$		0			0
$871$	581.969			0.668162
$872$		0			0
$873$	62.0000	+	554.545i	0.0710195	+	0.635217i
$874$		0			0
$875$		−	268.328i		−	0.306661i
$876$		0			0
$877$	−1087.73			−1.24028			−0.620141	−	0.784490i	$-0.712925\pi$
$877$	−1087.73			−1.24028			−0.620141	+	0.784490i	$0.712925\pi$
$878$		0			0
$879$	710.141	+	635.169i	0.807896	+	0.722604i
$880$		0			0
$881$		−	751.319i		−	0.852802i	−0.904534	−	0.426401i	$-0.859781\pi$
$881$		−	751.319i		−	0.852802i	0.904534	−	0.426401i	$-0.140219\pi$
$882$		0			0
$883$	668.000			0.756512			0.378256	−	0.925701i	$-0.376524\pi$
$883$	668.000			0.756512			0.378256	+	0.925701i	$0.376524\pi$
$884$		0			0
$885$	623.538	−	697.137i	0.704563	−	0.787725i
$886$		0			0
$887$			464.758i			0.523966i	0.965072	+	0.261983i	$0.0843765\pi$
$887$			464.758i			0.523966i	−0.965072	+	0.261983i	$0.915624\pi$
$888$		0			0
$889$	−300.000			−0.337458
$890$		0			0
$891$	−240.000	−	1059.90i	−0.269360	−	1.18956i
$892$		0			0
$893$			247.871i			0.277571i
$894$		0			0
$895$	−519.615			−0.580576
$896$		0			0
$897$	1440.00	+	1287.98i	1.60535	+	1.43587i
$898$		0			0
$899$		−	187.830i		−	0.208932i
$900$		0			0
$901$		0			0
$902$		0			0
$903$	−360.267	+	402.790i	−0.398966	+	0.446058i
$904$		0			0
$905$			53.6656i			0.0592990i
$906$		0			0
$907$	1196.00			1.31863			0.659316	−	0.751866i	$-0.270846\pi$
$907$	1196.00			1.31863			0.659316	+	0.751866i	$0.270846\pi$
$908$		0			0
$909$	−1177.79	+	131.681i	−1.29570	+	0.144864i
$910$		0			0
$911$		−	433.774i		−	0.476152i	−0.971247	−	0.238076i	$-0.923483\pi$
$911$		−	433.774i		−	0.476152i	0.971247	−	0.238076i	$-0.0765167\pi$
$912$		0			0
$913$	1620.00			1.77437
$914$		0			0
$915$	−120.000	−	107.331i	−0.131148	−	0.117302i
$916$		0			0
$917$		−	511.234i		−	0.557507i
$918$		0			0
$919$	1049.62			1.14214			0.571068	−	0.820903i	$-0.306529\pi$
$919$	1049.62			1.14214			0.571068	+	0.820903i	$0.306529\pi$
$920$		0			0
$921$	424.000	−	474.046i	0.460369	−	0.514708i
$922$		0			0
$923$			643.988i			0.697711i
$924$		0			0
$925$	1212.44			1.31074
$926$		0			0
$927$	86.6025	+	774.597i	0.0934224	+	0.835595i
$928$		0			0
$929$		−	107.331i		−	0.115534i	−0.998330	−	0.0577671i	$-0.981602\pi$
$929$		−	107.331i		−	0.115534i	0.998330	−	0.0577671i	$-0.0183981\pi$
$930$		0			0
$931$	148.000			0.158969
$932$		0			0
$933$	1177.79	+	1053.45i	1.26237	+	1.12910i
$934$		0			0
$935$		0			0
$936$		0			0
$937$	1502.00			1.60299			0.801494	−	0.598003i	$-0.204039\pi$
$937$	1502.00			1.60299			0.801494	+	0.598003i	$0.204039\pi$
$938$		0			0
$939$	404.000	−	451.686i	0.430245	−	0.481028i
$940$		0			0
$941$			1866.78i			1.98382i	0.126929	+	0.991912i	$0.459488\pi$
$941$			1866.78i			1.98382i	−0.126929	+	0.991912i	$0.540512\pi$
$942$		0			0
$943$	−1662.77			−1.76328
$944$		0			0
$945$	−420.000	+	590.322i	−0.444444	+	0.624679i
$946$		0			0
$947$			1220.89i			1.28922i	0.764511	+	0.644611i	$0.222981\pi$
$947$			1220.89i			1.28922i	−0.764511	+	0.644611i	$0.777019\pi$
$948$		0			0
$949$	−1538.06			−1.62072
$950$		0			0
$951$	−155.885	−	139.427i	−0.163916	−	0.146611i
$952$		0			0
$953$			53.6656i			0.0563123i	0.999604	+	0.0281562i	$0.00896357\pi$
$953$			53.6656i			0.0563123i	−0.999604	+	0.0281562i	$0.991036\pi$
$954$		0			0
$955$	−960.000			−1.00524
$956$		0			0
$957$	207.846	−	232.379i	0.217185	−	0.242820i
$958$		0			0
$959$			185.903i			0.193851i
$960$		0			0
$961$	−373.000			−0.388137
$962$		0			0
$963$	−360.000	+	40.2492i	−0.373832	+	0.0417957i
$964$		0			0
$965$			170.411i			0.176592i
$966$		0			0
$967$	−1887.94			−1.95236			−0.976182	−	0.216955i	$-0.930388\pi$
$967$	−1887.94			−1.95236			−0.976182	+	0.216955i	$0.930388\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$			872.067i			0.898112i	0.893504	+	0.449056i	$0.148240\pi$
$971$			872.067i			0.898112i	−0.893504	+	0.449056i	$0.851760\pi$
$972$		0			0
$973$	235.559			0.242095
$974$		0			0
$975$	1454.92	−	1626.65i	1.49223	−	1.66836i
$976$		0			0
$977$			429.325i			0.439432i	0.975564	+	0.219716i	$0.0705131\pi$
$977$			429.325i			0.439432i	−0.975564	+	0.219716i	$0.929487\pi$
$978$		0			0
$979$	720.000			0.735444
$980$		0			0
$981$	−76.2102	−	681.645i	−0.0776863	−	0.694847i
$982$		0			0
$983$		−	650.661i		−	0.661914i	−0.943646	−	0.330957i	$-0.892629\pi$
$983$		−	650.661i		−	0.661914i	0.943646	−	0.330957i	$-0.107371\pi$
$984$		0			0
$985$	−540.000			−0.548223
$986$		0			0
$987$	−480.000	−	429.325i	−0.486322	−	0.434980i
$988$		0			0
$989$			1611.16i			1.62908i
$990$		0			0
$991$	668.572			0.674643			0.337322	−	0.941389i	$-0.390479\pi$
$991$	668.572			0.674643			0.337322	+	0.941389i	$0.390479\pi$
$992$		0			0
$993$	−56.0000	+	62.6099i	−0.0563948	+	0.0630513i
$994$		0			0
$995$		−	2334.45i		−	2.34619i
$996$		0			0
$997$	−1143.15			−1.14659			−0.573297	−	0.819348i	$-0.694336\pi$
$997$	−1143.15			−1.14659			−0.573297	+	0.819348i	$0.694336\pi$
$998$		0			0
$999$	−762.102	−	542.218i	−0.762865	−	0.542760i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	768.3.e.g.257.3		4
3.2	odd	2	inner	768.3.e.g.257.2		4
4.3	odd	2		768.3.e.n.257.1		4
8.3	odd	2	inner	768.3.e.g.257.4		4
8.5	even	2		768.3.e.n.257.2		4
12.11	even	2		768.3.e.n.257.4		4
16.3	odd	4		192.3.h.c.161.7	yes	8
16.5	even	4		192.3.h.c.161.8	yes	8
16.11	odd	4		192.3.h.c.161.2	yes	8
16.13	even	4		192.3.h.c.161.1	✓	8
24.5	odd	2		768.3.e.n.257.3		4
24.11	even	2	inner	768.3.e.g.257.1		4
48.5	odd	4		192.3.h.c.161.3	yes	8
48.11	even	4		192.3.h.c.161.5	yes	8
48.29	odd	4		192.3.h.c.161.6	yes	8
48.35	even	4		192.3.h.c.161.4	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
192.3.h.c.161.1	✓	8	16.13	even	4
192.3.h.c.161.2	yes	8	16.11	odd	4
192.3.h.c.161.3	yes	8	48.5	odd	4
192.3.h.c.161.4	yes	8	48.35	even	4
192.3.h.c.161.5	yes	8	48.11	even	4
192.3.h.c.161.6	yes	8	48.29	odd	4
192.3.h.c.161.7	yes	8	16.3	odd	4
192.3.h.c.161.8	yes	8	16.5	even	4
768.3.e.g.257.1		4	24.11	even	2	inner
768.3.e.g.257.2		4	3.2	odd	2	inner
768.3.e.g.257.3		4	1.1	even	1	trivial
768.3.e.g.257.4		4	8.3	odd	2	inner
768.3.e.n.257.1		4	4.3	odd	2
768.3.e.n.257.2		4	8.5	even	2
768.3.e.n.257.3		4	24.5	odd	2
768.3.e.n.257.4		4	12.11	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 768 = 2^{8} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	768.e (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+(-2.00000 + 2.23607i) q^{3} -7.74597i q^{5} -3.46410 q^{7} +(-1.00000 - 8.94427i) q^{9} +O(q^{10})\)	\(q+(-2.00000 + 2.23607i) q^{3} -7.74597i q^{5} -3.46410 q^{7} +(-1.00000 - 8.94427i) q^{9} +13.4164i q^{11} +20.7846 q^{13} +(17.3205 + 15.4919i) q^{15} -4.00000 q^{19} +(6.92820 - 7.74597i) q^{21} -30.9839i q^{23} -35.0000 q^{25} +(22.0000 + 15.6525i) q^{27} +7.74597i q^{29} -24.2487 q^{31} +(-30.0000 - 26.8328i) q^{33} +26.8328i q^{35} -34.6410 q^{37} +(-41.5692 + 46.4758i) q^{39} -53.6656i q^{41} -52.0000 q^{43} +(-69.2820 + 7.74597i) q^{45} -61.9677i q^{47} -37.0000 q^{49} +54.2218i q^{53} +103.923 q^{55} +(8.00000 - 8.94427i) q^{57} -40.2492i q^{59} -6.92820 q^{61} +(3.46410 + 30.9839i) q^{63} -160.997i q^{65} +28.0000 q^{67} +(69.2820 + 61.9677i) q^{69} +30.9839i q^{71} -74.0000 q^{73} +(70.0000 - 78.2624i) q^{75} -46.4758i q^{77} -51.9615 q^{79} +(-79.0000 + 17.8885i) q^{81} -120.748i q^{83} +(-17.3205 - 15.4919i) q^{87} -53.6656i q^{89} -72.0000 q^{91} +(48.4974 - 54.2218i) q^{93} +30.9839i q^{95} -62.0000 q^{97} +(120.000 - 13.4164i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 8 q^{3} - 4 q^{9}+O(q^{10}) \) 4 * q - 8 * q^3 - 4 * q^9	\( 4 q - 8 q^{3} - 4 q^{9} - 16 q^{19} - 140 q^{25} + 88 q^{27} - 120 q^{33} - 208 q^{43} - 148 q^{49} + 32 q^{57} + 112 q^{67} - 296 q^{73} + 280 q^{75} - 316 q^{81} - 288 q^{91} - 248 q^{97} + 480 q^{99}+O(q^{100}) \) 4 * q - 8 * q^3 - 4 * q^9 - 16 * q^19 - 140 * q^25 + 88 * q^27 - 120 * q^33 - 208 * q^43 - 148 * q^49 + 32 * q^57 + 112 * q^67 - 296 * q^73 + 280 * q^75 - 316 * q^81 - 288 * q^91 - 248 * q^97 + 480 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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