LMFDB - Embedded newform 768.2.f.b.383.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [768,2,Mod(383,768)]

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([1, 1, 1]))

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

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

chi := DirichletCharacter("768.383");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$6.13251087523$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{5})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 3x^{2} + 1 $ x^4 + 3*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 384)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			383.4
Root			$1.61803i$ of defining polynomial
Character	$\chi$	$=$	768.383
Dual form			768.2.f.b.383.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.618034 + 1.61803i) q^{3} +1.23607 q^{5} -3.23607i q^{7} +(-2.23607 + 2.00000i) q^{9} +O(q^{10})$	\(q+(0.618034 + 1.61803i) q^{3} +1.23607 q^{5} -3.23607i q^{7} +(-2.23607 + 2.00000i) q^{9} +0.763932i q^{11} +4.47214i q^{13} +(0.763932 + 2.00000i) q^{15} +6.47214i q^{17} +5.23607 q^{19} +(5.23607 - 2.00000i) q^{21} +6.47214 q^{23} -3.47214 q^{25} +(-4.61803 - 2.38197i) q^{27} +9.23607 q^{29} -0.763932i q^{31} +(-1.23607 + 0.472136i) q^{33} -4.00000i q^{35} +0.472136i q^{37} +(-7.23607 + 2.76393i) q^{39} +2.47214i q^{41} +2.76393 q^{43} +(-2.76393 + 2.47214i) q^{45} -8.00000 q^{47} -3.47214 q^{49} +(-10.4721 + 4.00000i) q^{51} +1.23607 q^{53} +0.944272i q^{55} +(3.23607 + 8.47214i) q^{57} +3.23607i q^{59} -8.47214i q^{61} +(6.47214 + 7.23607i) q^{63} +5.52786i q^{65} -3.70820 q^{67} +(4.00000 + 10.4721i) q^{69} -11.4164 q^{71} +2.00000 q^{73} +(-2.14590 - 5.61803i) q^{75} +2.47214 q^{77} -13.7082i q^{79} +(1.00000 - 8.94427i) q^{81} -7.23607i q^{83} +8.00000i q^{85} +(5.70820 + 14.9443i) q^{87} -4.00000i q^{89} +14.4721 q^{91} +(1.23607 - 0.472136i) q^{93} +6.47214 q^{95} -8.47214 q^{97} +(-1.52786 - 1.70820i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 2 q^{3} - 4 q^{5}+O(q^{10}) $ 4 * q - 2 * q^3 - 4 * q^5	$ 4 q - 2 q^{3} - 4 q^{5} + 12 q^{15} + 12 q^{19} + 12 q^{21} + 8 q^{23} + 4 q^{25} - 14 q^{27} + 28 q^{29} + 4 q^{33} - 20 q^{39} + 20 q^{43} - 20 q^{45} - 32 q^{47} + 4 q^{49} - 24 q^{51} - 4 q^{53} + 4 q^{57} + 8 q^{63} + 12 q^{67} + 16 q^{69} + 8 q^{71} + 8 q^{73} - 22 q^{75} - 8 q^{77} + 4 q^{81} - 4 q^{87} + 40 q^{91} - 4 q^{93} + 8 q^{95} - 16 q^{97} - 24 q^{99}+O(q^{100}) $ 4 * q - 2 * q^3 - 4 * q^5 + 12 * q^15 + 12 * q^19 + 12 * q^21 + 8 * q^23 + 4 * q^25 - 14 * q^27 + 28 * q^29 + 4 * q^33 - 20 * q^39 + 20 * q^43 - 20 * q^45 - 32 * q^47 + 4 * q^49 - 24 * q^51 - 4 * q^53 + 4 * q^57 + 8 * q^63 + 12 * q^67 + 16 * q^69 + 8 * q^71 + 8 * q^73 - 22 * q^75 - 8 * q^77 + 4 * q^81 - 4 * q^87 + 40 * q^91 - 4 * q^93 + 8 * q^95 - 16 * q^97 - 24 * 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
$3$	0.618034	+	1.61803i	0.356822	+	0.934172i
$3$	0.618034	+	1.61803i	0.356822	+	0.934172i
$4$		0			0
$5$	1.23607			0.552786			0.276393	−	0.961045i	$-0.410861\pi$
$5$	1.23607			0.552786			0.276393	+	0.961045i	$0.410861\pi$
$6$		0			0
$7$		−	3.23607i		−	1.22312i	−0.791199	−	0.611559i	$-0.790543\pi$
$7$		−	3.23607i		−	1.22312i	0.791199	−	0.611559i	$-0.209457\pi$
$8$		0			0
$9$	−2.23607	+	2.00000i	−0.745356	+	0.666667i
$10$		0			0
$11$			0.763932i			0.230334i	0.993346	+	0.115167i	$0.0367403\pi$
$11$			0.763932i			0.230334i	−0.993346	+	0.115167i	$0.963260\pi$
$12$		0			0
$13$			4.47214i			1.24035i	0.784465	+	0.620174i	$0.212938\pi$
$13$			4.47214i			1.24035i	−0.784465	+	0.620174i	$0.787062\pi$
$14$		0			0
$15$	0.763932	+	2.00000i	0.197246	+	0.516398i
$16$		0			0
$17$			6.47214i			1.56972i	0.619671	+	0.784862i	$0.287266\pi$
$17$			6.47214i			1.56972i	−0.619671	+	0.784862i	$0.712734\pi$
$18$		0			0
$19$	5.23607			1.20124			0.600618	−	0.799536i	$-0.294921\pi$
$19$	5.23607			1.20124			0.600618	+	0.799536i	$0.294921\pi$
$20$		0			0
$21$	5.23607	−	2.00000i	1.14260	−	0.436436i
$22$		0			0
$23$	6.47214			1.34953			0.674767	−	0.738031i	$-0.264244\pi$
$23$	6.47214			1.34953			0.674767	+	0.738031i	$0.264244\pi$
$24$		0			0
$25$	−3.47214			−0.694427
$26$		0			0
$27$	−4.61803	−	2.38197i	−0.888741	−	0.458410i
$28$		0			0
$29$	9.23607			1.71509			0.857547	−	0.514405i	$-0.171987\pi$
$29$	9.23607			1.71509			0.857547	+	0.514405i	$0.171987\pi$
$30$		0			0
$31$		−	0.763932i		−	0.137206i	−0.997644	−	0.0686031i	$-0.978146\pi$
$31$		−	0.763932i		−	0.137206i	0.997644	−	0.0686031i	$-0.0218542\pi$
$32$		0			0
$33$	−1.23607	+	0.472136i	−0.215172	+	0.0821883i
$34$		0			0
$35$		−	4.00000i		−	0.676123i
$36$		0			0
$37$			0.472136i			0.0776187i	0.999247	+	0.0388093i	$0.0123565\pi$
$37$			0.472136i			0.0776187i	−0.999247	+	0.0388093i	$0.987644\pi$
$38$		0			0
$39$	−7.23607	+	2.76393i	−1.15870	+	0.442583i
$40$		0			0
$41$			2.47214i			0.386083i	0.981191	+	0.193041i	$0.0618352\pi$
$41$			2.47214i			0.386083i	−0.981191	+	0.193041i	$0.938165\pi$
$42$		0			0
$43$	2.76393			0.421496			0.210748	−	0.977540i	$-0.432410\pi$
$43$	2.76393			0.421496			0.210748	+	0.977540i	$0.432410\pi$
$44$		0			0
$45$	−2.76393	+	2.47214i	−0.412023	+	0.368524i
$46$		0			0
$47$	−8.00000			−1.16692			−0.583460	−	0.812142i	$-0.698301\pi$
$47$	−8.00000			−1.16692			−0.583460	+	0.812142i	$0.698301\pi$
$48$		0			0
$49$	−3.47214			−0.496019
$50$		0			0
$51$	−10.4721	+	4.00000i	−1.46639	+	0.560112i
$52$		0			0
$53$	1.23607			0.169787			0.0848935	−	0.996390i	$-0.472945\pi$
$53$	1.23607			0.169787			0.0848935	+	0.996390i	$0.472945\pi$
$54$		0			0
$55$			0.944272i			0.127326i
$56$		0			0
$57$	3.23607	+	8.47214i	0.428628	+	1.12216i
$58$		0			0
$59$			3.23607i			0.421300i	0.977562	+	0.210650i	$0.0675581\pi$
$59$			3.23607i			0.421300i	−0.977562	+	0.210650i	$0.932442\pi$
$60$		0			0
$61$		−	8.47214i		−	1.08475i	−0.840138	−	0.542373i	$-0.817526\pi$
$61$		−	8.47214i		−	1.08475i	0.840138	−	0.542373i	$-0.182474\pi$
$62$		0			0
$63$	6.47214	+	7.23607i	0.815412	+	0.911659i
$64$		0			0
$65$			5.52786i			0.685647i
$66$		0			0
$67$	−3.70820			−0.453029			−0.226515	−	0.974008i	$-0.572733\pi$
$67$	−3.70820			−0.453029			−0.226515	+	0.974008i	$0.572733\pi$
$68$		0			0
$69$	4.00000	+	10.4721i	0.481543	+	1.26070i
$70$		0			0
$71$	−11.4164			−1.35488			−0.677439	−	0.735579i	$-0.736910\pi$
$71$	−11.4164			−1.35488			−0.677439	+	0.735579i	$0.736910\pi$
$72$		0			0
$73$	2.00000			0.234082			0.117041	−	0.993127i	$-0.462659\pi$
$73$	2.00000			0.234082			0.117041	+	0.993127i	$0.462659\pi$
$74$		0			0
$75$	−2.14590	−	5.61803i	−0.247787	−	0.648715i
$76$		0			0
$77$	2.47214			0.281726
$78$		0			0
$79$		−	13.7082i		−	1.54229i	−0.636657	−	0.771147i	$-0.719683\pi$
$79$		−	13.7082i		−	1.54229i	0.636657	−	0.771147i	$-0.280317\pi$
$80$		0			0
$81$	1.00000	−	8.94427i	0.111111	−	0.993808i
$82$		0			0
$83$		−	7.23607i		−	0.794262i	−0.917762	−	0.397131i	$-0.870006\pi$
$83$		−	7.23607i		−	0.794262i	0.917762	−	0.397131i	$-0.129994\pi$
$84$		0			0
$85$			8.00000i			0.867722i
$86$		0			0
$87$	5.70820	+	14.9443i	0.611984	+	1.60219i
$88$		0			0
$89$		−	4.00000i		−	0.423999i	−0.977270	−	0.212000i	$-0.932002\pi$
$89$		−	4.00000i		−	0.423999i	0.977270	−	0.212000i	$-0.0679975\pi$
$90$		0			0
$91$	14.4721			1.51709
$92$		0			0
$93$	1.23607	−	0.472136i	0.128174	−	0.0489582i
$94$		0			0
$95$	6.47214			0.664027
$96$		0			0
$97$	−8.47214			−0.860215			−0.430108	−	0.902778i	$-0.641524\pi$
$97$	−8.47214			−0.860215			−0.430108	+	0.902778i	$0.641524\pi$
$98$		0			0
$99$	−1.52786	−	1.70820i	−0.153556	−	0.171681i
$100$		0			0
$101$	−11.7082			−1.16501			−0.582505	−	0.812827i	$-0.697927\pi$
$101$	−11.7082			−1.16501			−0.582505	+	0.812827i	$0.697927\pi$
$102$		0			0
$103$		−	8.18034i		−	0.806033i	−0.915193	−	0.403016i	$-0.867962\pi$
$103$		−	8.18034i		−	0.806033i	0.915193	−	0.403016i	$-0.132038\pi$
$104$		0			0
$105$	6.47214	−	2.47214i	0.631616	−	0.241256i
$106$		0			0
$107$			8.18034i			0.790823i	0.918504	+	0.395412i	$0.129398\pi$
$107$			8.18034i			0.790823i	−0.918504	+	0.395412i	$0.870602\pi$
$108$		0			0
$109$		−	0.472136i		−	0.0452224i	−0.999744	−	0.0226112i	$-0.992802\pi$
$109$		−	0.472136i		−	0.0452224i	0.999744	−	0.0226112i	$-0.00719799\pi$
$110$		0			0
$111$	−0.763932	+	0.291796i	−0.0725092	+	0.0276961i
$112$		0			0
$113$		−	8.00000i		−	0.752577i	−0.926503	−	0.376288i	$-0.877200\pi$
$113$		−	8.00000i		−	0.752577i	0.926503	−	0.376288i	$-0.122800\pi$
$114$		0			0
$115$	8.00000			0.746004
$116$		0			0
$117$	−8.94427	−	10.0000i	−0.826898	−	0.924500i
$118$		0			0
$119$	20.9443			1.91996
$120$		0			0
$121$	10.4164			0.946946
$122$		0			0
$123$	−4.00000	+	1.52786i	−0.360668	+	0.137763i
$124$		0			0
$125$	−10.4721			−0.936656
$126$		0			0
$127$		−	8.76393i		−	0.777673i	−0.921307	−	0.388837i	$-0.872877\pi$
$127$		−	8.76393i		−	0.777673i	0.921307	−	0.388837i	$-0.127123\pi$
$128$		0			0
$129$	1.70820	+	4.47214i	0.150399	+	0.393750i
$130$		0			0
$131$			11.2361i			0.981700i	0.871244	+	0.490850i	$0.163314\pi$
$131$			11.2361i			0.981700i	−0.871244	+	0.490850i	$0.836686\pi$
$132$		0			0
$133$		−	16.9443i		−	1.46925i
$134$		0			0
$135$	−5.70820	−	2.94427i	−0.491284	−	0.253403i
$136$		0			0
$137$		−	10.4721i		−	0.894695i	−0.894360	−	0.447347i	$-0.852369\pi$
$137$		−	10.4721i		−	0.894695i	0.894360	−	0.447347i	$-0.147631\pi$
$138$		0			0
$139$	3.70820			0.314526			0.157263	−	0.987557i	$-0.449733\pi$
$139$	3.70820			0.314526			0.157263	+	0.987557i	$0.449733\pi$
$140$		0			0
$141$	−4.94427	−	12.9443i	−0.416383	−	1.09010i
$142$		0			0
$143$	−3.41641			−0.285694
$144$		0			0
$145$	11.4164			0.948081
$146$		0			0
$147$	−2.14590	−	5.61803i	−0.176991	−	0.463368i
$148$		0			0
$149$	−3.70820			−0.303788			−0.151894	−	0.988397i	$-0.548537\pi$
$149$	−3.70820			−0.303788			−0.151894	+	0.988397i	$0.548537\pi$
$150$		0			0
$151$		−	6.29180i		−	0.512019i	−0.966674	−	0.256010i	$-0.917592\pi$
$151$		−	6.29180i		−	0.512019i	0.966674	−	0.256010i	$-0.0824079\pi$
$152$		0			0
$153$	−12.9443	−	14.4721i	−1.04648	−	1.17000i
$154$		0			0
$155$		−	0.944272i		−	0.0758457i
$156$		0			0
$157$			4.47214i			0.356915i	0.983948	+	0.178458i	$0.0571108\pi$
$157$			4.47214i			0.356915i	−0.983948	+	0.178458i	$0.942889\pi$
$158$		0			0
$159$	0.763932	+	2.00000i	0.0605838	+	0.158610i
$160$		0			0
$161$		−	20.9443i		−	1.65064i
$162$		0			0
$163$	18.1803			1.42399			0.711997	−	0.702182i	$-0.247791\pi$
$163$	18.1803			1.42399			0.711997	+	0.702182i	$0.247791\pi$
$164$		0			0
$165$	−1.52786	+	0.583592i	−0.118944	+	0.0454326i
$166$		0			0
$167$	−22.4721			−1.73895			−0.869473	−	0.493980i	$-0.835541\pi$
$167$	−22.4721			−1.73895			−0.869473	+	0.493980i	$0.835541\pi$
$168$		0			0
$169$	−7.00000			−0.538462
$170$		0			0
$171$	−11.7082	+	10.4721i	−0.895349	+	0.800824i
$172$		0			0
$173$	−8.65248			−0.657836			−0.328918	−	0.944359i	$-0.606684\pi$
$173$	−8.65248			−0.657836			−0.328918	+	0.944359i	$0.606684\pi$
$174$		0			0
$175$			11.2361i			0.849367i
$176$		0			0
$177$	−5.23607	+	2.00000i	−0.393567	+	0.150329i
$178$		0			0
$179$			16.1803i			1.20938i	0.796463	+	0.604688i	$0.206702\pi$
$179$			16.1803i			1.20938i	−0.796463	+	0.604688i	$0.793298\pi$
$180$		0			0
$181$			11.5279i			0.856859i	0.903575	+	0.428430i	$0.140933\pi$
$181$			11.5279i			0.856859i	−0.903575	+	0.428430i	$0.859067\pi$
$182$		0			0
$183$	13.7082	−	5.23607i	1.01334	−	0.387061i
$184$		0			0
$185$			0.583592i			0.0429065i
$186$		0			0
$187$	−4.94427			−0.361561
$188$		0			0
$189$	−7.70820	+	14.9443i	−0.560689	+	1.08704i
$190$		0			0
$191$	4.94427			0.357755			0.178877	−	0.983871i	$-0.442753\pi$
$191$	4.94427			0.357755			0.178877	+	0.983871i	$0.442753\pi$
$192$		0			0
$193$	23.8885			1.71954			0.859768	−	0.510686i	$-0.170608\pi$
$193$	23.8885			1.71954			0.859768	+	0.510686i	$0.170608\pi$
$194$		0			0
$195$	−8.94427	+	3.41641i	−0.640513	+	0.244654i
$196$		0			0
$197$	9.23607			0.658043			0.329021	−	0.944323i	$-0.393281\pi$
$197$	9.23607			0.658043			0.329021	+	0.944323i	$0.393281\pi$
$198$		0			0
$199$		−	16.1803i		−	1.14699i	−0.819208	−	0.573497i	$-0.805586\pi$
$199$		−	16.1803i		−	1.14699i	0.819208	−	0.573497i	$-0.194414\pi$
$200$		0			0
$201$	−2.29180	−	6.00000i	−0.161651	−	0.423207i
$202$		0			0
$203$		−	29.8885i		−	2.09776i
$204$		0			0
$205$			3.05573i			0.213421i
$206$		0			0
$207$	−14.4721	+	12.9443i	−1.00588	+	0.899689i
$208$		0			0
$209$			4.00000i			0.276686i
$210$		0			0
$211$	−6.76393			−0.465648			−0.232824	−	0.972519i	$-0.574797\pi$
$211$	−6.76393			−0.465648			−0.232824	+	0.972519i	$0.574797\pi$
$212$		0			0
$213$	−7.05573	−	18.4721i	−0.483451	−	1.26569i
$214$		0			0
$215$	3.41641			0.232997
$216$		0			0
$217$	−2.47214			−0.167820
$218$		0			0
$219$	1.23607	+	3.23607i	0.0835257	+	0.218673i
$220$		0			0
$221$	−28.9443			−1.94700
$222$		0			0
$223$			12.1803i			0.815656i	0.913059	+	0.407828i	$0.133714\pi$
$223$			12.1803i			0.815656i	−0.913059	+	0.407828i	$0.866286\pi$
$224$		0			0
$225$	7.76393	−	6.94427i	0.517595	−	0.462951i
$226$		0			0
$227$		−	20.1803i		−	1.33942i	−0.742624	−	0.669708i	$-0.766419\pi$
$227$		−	20.1803i		−	1.33942i	0.742624	−	0.669708i	$-0.233581\pi$
$228$		0			0
$229$		−	20.4721i		−	1.35284i	−0.736518	−	0.676418i	$-0.763531\pi$
$229$		−	20.4721i		−	1.35284i	0.736518	−	0.676418i	$-0.236469\pi$
$230$		0			0
$231$	1.52786	+	4.00000i	0.100526	+	0.263181i
$232$		0			0
$233$			7.05573i			0.462236i	0.972926	+	0.231118i	$0.0742384\pi$
$233$			7.05573i			0.462236i	−0.972926	+	0.231118i	$0.925762\pi$
$234$		0			0
$235$	−9.88854			−0.645057
$236$		0			0
$237$	22.1803	−	8.47214i	1.44077	−	0.550324i
$238$		0			0
$239$	8.00000			0.517477			0.258738	−	0.965947i	$-0.416693\pi$
$239$	8.00000			0.517477			0.258738	+	0.965947i	$0.416693\pi$
$240$		0			0
$241$	−2.00000			−0.128831			−0.0644157	−	0.997923i	$-0.520518\pi$
$241$	−2.00000			−0.128831			−0.0644157	+	0.997923i	$0.520518\pi$
$242$		0			0
$243$	15.0902	−	3.90983i	0.968035	−	0.250816i
$244$		0			0
$245$	−4.29180			−0.274193
$246$		0			0
$247$			23.4164i			1.48995i
$248$		0			0
$249$	11.7082	−	4.47214i	0.741977	−	0.283410i
$250$		0			0
$251$		−	10.2918i		−	0.649612i	−0.945781	−	0.324806i	$-0.894701\pi$
$251$		−	10.2918i		−	0.649612i	0.945781	−	0.324806i	$-0.105299\pi$
$252$		0			0
$253$			4.94427i			0.310844i
$254$		0			0
$255$	−12.9443	+	4.94427i	−0.810602	+	0.309622i
$256$		0			0
$257$		−	3.05573i		−	0.190611i	−0.995448	−	0.0953055i	$-0.969617\pi$
$257$		−	3.05573i		−	0.190611i	0.995448	−	0.0953055i	$-0.0303828\pi$
$258$		0			0
$259$	1.52786			0.0949369
$260$		0			0
$261$	−20.6525	+	18.4721i	−1.27836	+	1.14340i
$262$		0			0
$263$	−1.52786			−0.0942121			−0.0471061	−	0.998890i	$-0.515000\pi$
$263$	−1.52786			−0.0942121			−0.0471061	+	0.998890i	$0.515000\pi$
$264$		0			0
$265$	1.52786			0.0938559
$266$		0			0
$267$	6.47214	−	2.47214i	0.396088	−	0.151292i
$268$		0			0
$269$	6.18034			0.376822			0.188411	−	0.982090i	$-0.439666\pi$
$269$	6.18034			0.376822			0.188411	+	0.982090i	$0.439666\pi$
$270$		0			0
$271$		−	13.7082i		−	0.832714i	−0.909201	−	0.416357i	$-0.863307\pi$
$271$		−	13.7082i		−	0.832714i	0.909201	−	0.416357i	$-0.136693\pi$
$272$		0			0
$273$	8.94427	+	23.4164i	0.541332	+	1.41723i
$274$		0			0
$275$		−	2.65248i		−	0.159950i
$276$		0			0
$277$			16.4721i			0.989715i	0.868974	+	0.494857i	$0.164780\pi$
$277$			16.4721i			0.989715i	−0.868974	+	0.494857i	$0.835220\pi$
$278$		0			0
$279$	1.52786	+	1.70820i	0.0914708	+	0.102267i
$280$		0			0
$281$		−	7.05573i		−	0.420909i	−0.977604	−	0.210455i	$-0.932506\pi$
$281$		−	7.05573i		−	0.420909i	0.977604	−	0.210455i	$-0.0674945\pi$
$282$		0			0
$283$	−25.2361			−1.50013			−0.750064	−	0.661365i	$-0.769977\pi$
$283$	−25.2361			−1.50013			−0.750064	+	0.661365i	$0.769977\pi$
$284$		0			0
$285$	4.00000	+	10.4721i	0.236940	+	0.620316i
$286$		0			0
$287$	8.00000			0.472225
$288$		0			0
$289$	−24.8885			−1.46403
$290$		0			0
$291$	−5.23607	−	13.7082i	−0.306944	−	0.803589i
$292$		0			0
$293$	25.2361			1.47431			0.737153	−	0.675725i	$-0.236169\pi$
$293$	25.2361			1.47431			0.737153	+	0.675725i	$0.236169\pi$
$294$		0			0
$295$			4.00000i			0.232889i
$296$		0			0
$297$	1.81966	−	3.52786i	0.105587	−	0.204707i
$298$		0			0
$299$			28.9443i			1.67389i
$300$		0			0
$301$		−	8.94427i		−	0.515539i
$302$		0			0
$303$	−7.23607	−	18.9443i	−0.415701	−	1.08832i
$304$		0			0
$305$		−	10.4721i		−	0.599633i
$306$		0			0
$307$	−21.5967			−1.23259			−0.616296	−	0.787515i	$-0.711367\pi$
$307$	−21.5967			−1.23259			−0.616296	+	0.787515i	$0.711367\pi$
$308$		0			0
$309$	13.2361	−	5.05573i	0.752974	−	0.287610i
$310$		0			0
$311$	−14.4721			−0.820640			−0.410320	−	0.911942i	$-0.634583\pi$
$311$	−14.4721			−0.820640			−0.410320	+	0.911942i	$0.634583\pi$
$312$		0			0
$313$	−8.47214			−0.478873			−0.239437	−	0.970912i	$-0.576963\pi$
$313$	−8.47214			−0.478873			−0.239437	+	0.970912i	$0.576963\pi$
$314$		0			0
$315$	8.00000	+	8.94427i	0.450749	+	0.503953i
$316$		0			0
$317$	27.1246			1.52347			0.761735	−	0.647889i	$-0.224348\pi$
$317$	27.1246			1.52347			0.761735	+	0.647889i	$0.224348\pi$
$318$		0			0
$319$			7.05573i			0.395045i
$320$		0			0
$321$	−13.2361	+	5.05573i	−0.738765	+	0.282183i
$322$		0			0
$323$			33.8885i			1.88561i
$324$		0			0
$325$		−	15.5279i		−	0.861331i
$326$		0			0
$327$	0.763932	−	0.291796i	0.0422455	−	0.0161364i
$328$		0			0
$329$			25.8885i			1.42728i
$330$		0			0
$331$	34.5410			1.89855			0.949273	−	0.314453i	$-0.101821\pi$
$331$	34.5410			1.89855			0.949273	+	0.314453i	$0.101821\pi$
$332$		0			0
$333$	−0.944272	−	1.05573i	−0.0517458	−	0.0578535i
$334$		0			0
$335$	−4.58359			−0.250428
$336$		0			0
$337$	−22.3607			−1.21806			−0.609032	−	0.793146i	$-0.708442\pi$
$337$	−22.3607			−1.21806			−0.609032	+	0.793146i	$0.708442\pi$
$338$		0			0
$339$	12.9443	−	4.94427i	0.703036	−	0.268536i
$340$		0			0
$341$	0.583592			0.0316033
$342$		0			0
$343$		−	11.4164i		−	0.616428i
$344$		0			0
$345$	4.94427	+	12.9443i	0.266191	+	0.696896i
$346$		0			0
$347$		−	10.2918i		−	0.552493i	−0.961087	−	0.276246i	$-0.910909\pi$
$347$		−	10.2918i		−	0.552493i	0.961087	−	0.276246i	$-0.0890905\pi$
$348$		0			0
$349$		−	19.5279i		−	1.04530i	−0.852547	−	0.522651i	$-0.824943\pi$
$349$		−	19.5279i		−	1.04530i	0.852547	−	0.522651i	$-0.175057\pi$
$350$		0			0
$351$	10.6525	−	20.6525i	0.568587	−	1.10235i
$352$		0			0
$353$			25.8885i			1.37791i	0.724805	+	0.688954i	$0.241930\pi$
$353$			25.8885i			1.37791i	−0.724805	+	0.688954i	$0.758070\pi$
$354$		0			0
$355$	−14.1115			−0.748958
$356$		0			0
$357$	12.9443	+	33.8885i	0.685084	+	1.79357i
$358$		0			0
$359$	4.58359			0.241913			0.120956	−	0.992658i	$-0.461404\pi$
$359$	4.58359			0.241913			0.120956	+	0.992658i	$0.461404\pi$
$360$		0			0
$361$	8.41641			0.442969
$362$		0			0
$363$	6.43769	+	16.8541i	0.337891	+	0.884611i
$364$		0			0
$365$	2.47214			0.129398
$366$		0			0
$367$		−	0.763932i		−	0.0398769i	−0.999801	−	0.0199385i	$-0.993653\pi$
$367$		−	0.763932i		−	0.0398769i	0.999801	−	0.0199385i	$-0.00634703\pi$
$368$		0			0
$369$	−4.94427	−	5.52786i	−0.257389	−	0.287769i
$370$		0			0
$371$		−	4.00000i		−	0.207670i
$372$		0			0
$373$			29.4164i			1.52312i	0.648092	+	0.761562i	$0.275567\pi$
$373$			29.4164i			1.52312i	−0.648092	+	0.761562i	$0.724433\pi$
$374$		0			0
$375$	−6.47214	−	16.9443i	−0.334220	−	0.874998i
$376$		0			0
$377$			41.3050i			2.12731i
$378$		0			0
$379$	20.6525			1.06085			0.530423	−	0.847733i	$-0.322033\pi$
$379$	20.6525			1.06085			0.530423	+	0.847733i	$0.322033\pi$
$380$		0			0
$381$	14.1803	−	5.41641i	0.726481	−	0.277491i
$382$		0			0
$383$	11.0557			0.564921			0.282461	−	0.959279i	$-0.408849\pi$
$383$	11.0557			0.564921			0.282461	+	0.959279i	$0.408849\pi$
$384$		0			0
$385$	3.05573			0.155734
$386$		0			0
$387$	−6.18034	+	5.52786i	−0.314164	+	0.280997i
$388$		0			0
$389$	−9.81966			−0.497877			−0.248938	−	0.968519i	$-0.580082\pi$
$389$	−9.81966			−0.497877			−0.248938	+	0.968519i	$0.580082\pi$
$390$		0			0
$391$			41.8885i			2.11839i
$392$		0			0
$393$	−18.1803	+	6.94427i	−0.917077	+	0.350292i
$394$		0			0
$395$		−	16.9443i		−	0.852559i
$396$		0			0
$397$			9.41641i			0.472596i	0.971681	+	0.236298i	$0.0759341\pi$
$397$			9.41641i			0.472596i	−0.971681	+	0.236298i	$0.924066\pi$
$398$		0			0
$399$	27.4164	−	10.4721i	1.37254	−	0.524263i
$400$		0			0
$401$		−	16.3607i		−	0.817013i	−0.912755	−	0.408507i	$-0.866050\pi$
$401$		−	16.3607i		−	0.817013i	0.912755	−	0.408507i	$-0.133950\pi$
$402$		0			0
$403$	3.41641			0.170183
$404$		0			0
$405$	1.23607	−	11.0557i	0.0614207	−	0.549364i
$406$		0			0
$407$	−0.360680			−0.0178782
$408$		0			0
$409$	−3.88854			−0.192276			−0.0961381	−	0.995368i	$-0.530649\pi$
$409$	−3.88854			−0.192276			−0.0961381	+	0.995368i	$0.530649\pi$
$410$		0			0
$411$	16.9443	−	6.47214i	0.835799	−	0.319247i
$412$		0			0
$413$	10.4721			0.515300
$414$		0			0
$415$		−	8.94427i		−	0.439057i
$416$		0			0
$417$	2.29180	+	6.00000i	0.112230	+	0.293821i
$418$		0			0
$419$		−	12.1803i		−	0.595049i	−0.954714	−	0.297524i	$-0.903839\pi$
$419$		−	12.1803i		−	0.595049i	0.954714	−	0.297524i	$-0.0961609\pi$
$420$		0			0
$421$			5.41641i			0.263980i	0.991251	+	0.131990i	$0.0421366\pi$
$421$			5.41641i			0.263980i	−0.991251	+	0.131990i	$0.957863\pi$
$422$		0			0
$423$	17.8885	−	16.0000i	0.869771	−	0.777947i
$424$		0			0
$425$		−	22.4721i		−	1.09006i
$426$		0			0
$427$	−27.4164			−1.32677
$428$		0			0
$429$	−2.11146	−	5.52786i	−0.101942	−	0.266888i
$430$		0			0
$431$	8.00000			0.385346			0.192673	−	0.981263i	$-0.438284\pi$
$431$	8.00000			0.385346			0.192673	+	0.981263i	$0.438284\pi$
$432$		0			0
$433$	7.52786			0.361766			0.180883	−	0.983505i	$-0.442104\pi$
$433$	7.52786			0.361766			0.180883	+	0.983505i	$0.442104\pi$
$434$		0			0
$435$	7.05573	+	18.4721i	0.338296	+	0.885671i
$436$		0			0
$437$	33.8885			1.62111
$438$		0			0
$439$		−	0.180340i		−	0.00860715i	−0.999991	−	0.00430358i	$-0.998630\pi$
$439$		−	0.180340i		−	0.00860715i	0.999991	−	0.00430358i	$-0.00136988\pi$
$440$		0			0
$441$	7.76393	−	6.94427i	0.369711	−	0.330680i
$442$		0			0
$443$			29.7082i			1.41148i	0.708471	+	0.705740i	$0.249385\pi$
$443$			29.7082i			1.41148i	−0.708471	+	0.705740i	$0.750615\pi$
$444$		0			0
$445$		−	4.94427i		−	0.234381i
$446$		0			0
$447$	−2.29180	−	6.00000i	−0.108398	−	0.283790i
$448$		0			0
$449$			27.4164i			1.29386i	0.762549	+	0.646930i	$0.223947\pi$
$449$			27.4164i			1.29386i	−0.762549	+	0.646930i	$0.776053\pi$
$450$		0			0
$451$	−1.88854			−0.0889281
$452$		0			0
$453$	10.1803	−	3.88854i	0.478314	−	0.182700i
$454$		0			0
$455$	17.8885			0.838628
$456$		0			0
$457$	22.0000			1.02912			0.514558	−	0.857455i	$-0.327956\pi$
$457$	22.0000			1.02912			0.514558	+	0.857455i	$0.327956\pi$
$458$		0			0
$459$	15.4164	−	29.8885i	0.719576	−	1.39508i
$460$		0			0
$461$	14.1803			0.660444			0.330222	−	0.943903i	$-0.392876\pi$
$461$	14.1803			0.660444			0.330222	+	0.943903i	$0.392876\pi$
$462$		0			0
$463$			39.2361i			1.82345i	0.410796	+	0.911727i	$0.365251\pi$
$463$			39.2361i			1.82345i	−0.410796	+	0.911727i	$0.634749\pi$
$464$		0			0
$465$	1.52786	−	0.583592i	0.0708530	−	0.0270634i
$466$		0			0
$467$		−	15.2361i		−	0.705041i	−0.935804	−	0.352521i	$-0.885325\pi$
$467$		−	15.2361i		−	0.705041i	0.935804	−	0.352521i	$-0.114675\pi$
$468$		0			0
$469$			12.0000i			0.554109i
$470$		0			0
$471$	−7.23607	+	2.76393i	−0.333420	+	0.127355i
$472$		0			0
$473$			2.11146i			0.0970849i
$474$		0			0
$475$	−18.1803			−0.834171
$476$		0			0
$477$	−2.76393	+	2.47214i	−0.126552	+	0.113191i
$478$		0			0
$479$	−41.8885			−1.91394			−0.956968	−	0.290193i	$-0.906281\pi$
$479$	−41.8885			−1.91394			−0.956968	+	0.290193i	$0.906281\pi$
$480$		0			0
$481$	−2.11146			−0.0962741
$482$		0			0
$483$	33.8885	−	12.9443i	1.54198	−	0.588985i
$484$		0			0
$485$	−10.4721			−0.475515
$486$		0			0
$487$		−	1.34752i		−	0.0610621i	−0.999534	−	0.0305311i	$-0.990280\pi$
$487$		−	1.34752i		−	0.0610621i	0.999534	−	0.0305311i	$-0.00971985\pi$
$488$		0			0
$489$	11.2361	+	29.4164i	0.508113	+	1.33026i
$490$		0			0
$491$			26.0689i			1.17647i	0.808689	+	0.588236i	$0.200177\pi$
$491$			26.0689i			1.17647i	−0.808689	+	0.588236i	$0.799823\pi$
$492$		0			0
$493$			59.7771i			2.69222i
$494$		0			0
$495$	−1.88854	−	2.11146i	−0.0848837	−	0.0949029i
$496$		0			0
$497$			36.9443i			1.65718i
$498$		0			0
$499$	−22.7639			−1.01905			−0.509527	−	0.860455i	$-0.670180\pi$
$499$	−22.7639			−1.01905			−0.509527	+	0.860455i	$0.670180\pi$
$500$		0			0
$501$	−13.8885	−	36.3607i	−0.620494	−	1.62448i
$502$		0			0
$503$	−30.4721			−1.35869			−0.679343	−	0.733821i	$-0.737735\pi$
$503$	−30.4721			−1.35869			−0.679343	+	0.733821i	$0.737735\pi$
$504$		0			0
$505$	−14.4721			−0.644002
$506$		0			0
$507$	−4.32624	−	11.3262i	−0.192135	−	0.503016i
$508$		0			0
$509$	12.2918			0.544824			0.272412	−	0.962181i	$-0.412179\pi$
$509$	12.2918			0.544824			0.272412	+	0.962181i	$0.412179\pi$
$510$		0			0
$511$		−	6.47214i		−	0.286310i
$512$		0			0
$513$	−24.1803	−	12.4721i	−1.06759	−	0.550658i
$514$		0			0
$515$		−	10.1115i		−	0.445564i
$516$		0			0
$517$		−	6.11146i		−	0.268782i
$518$		0			0
$519$	−5.34752	−	14.0000i	−0.234730	−	0.614532i
$520$		0			0
$521$		−	20.3607i		−	0.892018i	−0.895029	−	0.446009i	$-0.852845\pi$
$521$		−	20.3607i		−	0.892018i	0.895029	−	0.446009i	$-0.147155\pi$
$522$		0			0
$523$	−34.1803			−1.49460			−0.747301	−	0.664486i	$-0.768651\pi$
$523$	−34.1803			−1.49460			−0.747301	+	0.664486i	$0.768651\pi$
$524$		0			0
$525$	−18.1803	+	6.94427i	−0.793455	+	0.303073i
$526$		0			0
$527$	4.94427			0.215376
$528$		0			0
$529$	18.8885			0.821241
$530$		0			0
$531$	−6.47214	−	7.23607i	−0.280867	−	0.314019i
$532$		0			0
$533$	−11.0557			−0.478877
$534$		0			0
$535$			10.1115i			0.437156i
$536$		0			0
$537$	−26.1803	+	10.0000i	−1.12977	+	0.431532i
$538$		0			0
$539$		−	2.65248i		−	0.114250i
$540$		0			0
$541$		−	18.3607i		−	0.789387i	−0.918813	−	0.394694i	$-0.870851\pi$
$541$		−	18.3607i		−	0.789387i	0.918813	−	0.394694i	$-0.129149\pi$
$542$		0			0
$543$	−18.6525	+	7.12461i	−0.800454	+	0.305746i
$544$		0			0
$545$		−	0.583592i		−	0.0249983i
$546$		0			0
$547$	−2.76393			−0.118177			−0.0590886	−	0.998253i	$-0.518819\pi$
$547$	−2.76393			−0.118177			−0.0590886	+	0.998253i	$0.518819\pi$
$548$		0			0
$549$	16.9443	+	18.9443i	0.723164	+	0.808522i
$550$		0			0
$551$	48.3607			2.06023
$552$		0			0
$553$	−44.3607			−1.88641
$554$		0			0
$555$	−0.944272	+	0.360680i	−0.0400821	+	0.0153100i
$556$		0			0
$557$	−16.6525			−0.705588			−0.352794	−	0.935701i	$-0.614768\pi$
$557$	−16.6525			−0.705588			−0.352794	+	0.935701i	$0.614768\pi$
$558$		0			0
$559$			12.3607i			0.522801i
$560$		0			0
$561$	−3.05573	−	8.00000i	−0.129013	−	0.337760i
$562$		0			0
$563$			36.5410i			1.54002i	0.638032	+	0.770010i	$0.279749\pi$
$563$			36.5410i			1.54002i	−0.638032	+	0.770010i	$0.720251\pi$
$564$		0			0
$565$		−	9.88854i		−	0.416014i
$566$		0			0
$567$	−28.9443	−	3.23607i	−1.21555	−	0.135902i
$568$		0			0
$569$		−	37.5279i		−	1.57325i	−0.617431	−	0.786625i	$-0.711827\pi$
$569$		−	37.5279i		−	1.57325i	0.617431	−	0.786625i	$-0.288173\pi$
$570$		0			0
$571$	−35.1246			−1.46992			−0.734960	−	0.678111i	$-0.762799\pi$
$571$	−35.1246			−1.46992			−0.734960	+	0.678111i	$0.762799\pi$
$572$		0			0
$573$	3.05573	+	8.00000i	0.127655	+	0.334205i
$574$		0			0
$575$	−22.4721			−0.937153
$576$		0			0
$577$	19.5279			0.812956			0.406478	−	0.913661i	$-0.366757\pi$
$577$	19.5279			0.812956			0.406478	+	0.913661i	$0.366757\pi$
$578$		0			0
$579$	14.7639	+	38.6525i	0.613568	+	1.60634i
$580$		0			0
$581$	−23.4164			−0.971476
$582$		0			0
$583$			0.944272i			0.0391077i
$584$		0			0
$585$	−11.0557	−	12.3607i	−0.457098	−	0.511051i
$586$		0			0
$587$		−	27.5967i		−	1.13904i	−0.821978	−	0.569520i	$-0.807129\pi$
$587$		−	27.5967i		−	1.13904i	0.821978	−	0.569520i	$-0.192871\pi$
$588$		0			0
$589$		−	4.00000i		−	0.164817i
$590$		0			0
$591$	5.70820	+	14.9443i	0.234804	+	0.614725i
$592$		0			0
$593$		−	11.0557i		−	0.454004i	−0.973894	−	0.227002i	$-0.927108\pi$
$593$		−	11.0557i		−	0.454004i	0.973894	−	0.227002i	$-0.0728924\pi$
$594$		0			0
$595$	25.8885			1.06133
$596$		0			0
$597$	26.1803	−	10.0000i	1.07149	−	0.409273i
$598$		0			0
$599$	−19.4164			−0.793333			−0.396666	−	0.917963i	$-0.629833\pi$
$599$	−19.4164			−0.793333			−0.396666	+	0.917963i	$0.629833\pi$
$600$		0			0
$601$	37.7771			1.54096			0.770480	−	0.637464i	$-0.220017\pi$
$601$	37.7771			1.54096			0.770480	+	0.637464i	$0.220017\pi$
$602$		0			0
$603$	8.29180	−	7.41641i	0.337668	−	0.302019i
$604$		0			0
$605$	12.8754			0.523459
$606$		0			0
$607$			39.2361i			1.59254i	0.604940	+	0.796271i	$0.293197\pi$
$607$			39.2361i			1.59254i	−0.604940	+	0.796271i	$0.706803\pi$
$608$		0			0
$609$	48.3607	−	18.4721i	1.95967	−	0.748529i
$610$		0			0
$611$		−	35.7771i		−	1.44739i
$612$		0			0
$613$		−	43.3050i		−	1.74907i	−0.484962	−	0.874535i	$-0.661167\pi$
$613$		−	43.3050i		−	1.74907i	0.484962	−	0.874535i	$-0.338833\pi$
$614$		0			0
$615$	−4.94427	+	1.88854i	−0.199372	+	0.0761534i
$616$		0			0
$617$			13.8885i			0.559132i	0.960127	+	0.279566i	$0.0901905\pi$
$617$			13.8885i			0.559132i	−0.960127	+	0.279566i	$0.909809\pi$
$618$		0			0
$619$	−11.1246			−0.447136			−0.223568	−	0.974688i	$-0.571770\pi$
$619$	−11.1246			−0.447136			−0.223568	+	0.974688i	$0.571770\pi$
$620$		0			0
$621$	−29.8885	−	15.4164i	−1.19939	−	0.618639i
$622$		0			0
$623$	−12.9443			−0.518601
$624$		0			0
$625$	4.41641			0.176656
$626$		0			0
$627$	−6.47214	+	2.47214i	−0.258472	+	0.0987276i
$628$		0			0
$629$	−3.05573			−0.121840
$630$		0			0
$631$		−	21.1246i		−	0.840958i	−0.907302	−	0.420479i	$-0.861862\pi$
$631$		−	21.1246i		−	0.840958i	0.907302	−	0.420479i	$-0.138138\pi$
$632$		0			0
$633$	−4.18034	−	10.9443i	−0.166154	−	0.434996i
$634$		0			0
$635$		−	10.8328i		−	0.429887i
$636$		0			0
$637$		−	15.5279i		−	0.615236i
$638$		0			0
$639$	25.5279	−	22.8328i	1.00987	−	0.903252i
$640$		0			0
$641$		−	37.3050i		−	1.47346i	−0.676189	−	0.736729i	$-0.736370\pi$
$641$		−	37.3050i		−	1.47346i	0.676189	−	0.736729i	$-0.263630\pi$
$642$		0			0
$643$	−44.6525			−1.76092			−0.880461	−	0.474119i	$-0.842767\pi$
$643$	−44.6525			−1.76092			−0.880461	+	0.474119i	$0.842767\pi$
$644$		0			0
$645$	2.11146	+	5.52786i	0.0831385	+	0.217659i
$646$		0			0
$647$	29.3050			1.15210			0.576048	−	0.817416i	$-0.304594\pi$
$647$	29.3050			1.15210			0.576048	+	0.817416i	$0.304594\pi$
$648$		0			0
$649$	−2.47214			−0.0970398
$650$		0			0
$651$	−1.52786	−	4.00000i	−0.0598817	−	0.156772i
$652$		0			0
$653$	−27.7082			−1.08431			−0.542153	−	0.840280i	$-0.682391\pi$
$653$	−27.7082			−1.08431			−0.542153	+	0.840280i	$0.682391\pi$
$654$		0			0
$655$			13.8885i			0.542670i
$656$		0			0
$657$	−4.47214	+	4.00000i	−0.174475	+	0.156055i
$658$		0			0
$659$		−	41.7082i		−	1.62472i	−0.583156	−	0.812360i	$-0.698182\pi$
$659$		−	41.7082i		−	1.62472i	0.583156	−	0.812360i	$-0.301818\pi$
$660$		0			0
$661$		−	11.3050i		−	0.439712i	−0.975532	−	0.219856i	$-0.929441\pi$
$661$		−	11.3050i		−	0.439712i	0.975532	−	0.219856i	$-0.0705587\pi$
$662$		0			0
$663$	−17.8885	−	46.8328i	−0.694733	−	1.81884i
$664$		0			0
$665$		−	20.9443i		−	0.812184i
$666$		0			0
$667$	59.7771			2.31458
$668$		0			0
$669$	−19.7082	+	7.52786i	−0.761963	+	0.291044i
$670$		0			0
$671$	6.47214			0.249854
$672$		0			0
$673$	1.41641			0.0545985			0.0272993	−	0.999627i	$-0.491309\pi$
$673$	1.41641			0.0545985			0.0272993	+	0.999627i	$0.491309\pi$
$674$		0			0
$675$	16.0344	+	8.27051i	0.617166	+	0.318332i
$676$		0			0
$677$	−27.7082			−1.06491			−0.532456	−	0.846457i	$-0.678731\pi$
$677$	−27.7082			−1.06491			−0.532456	+	0.846457i	$0.678731\pi$
$678$		0			0
$679$			27.4164i			1.05215i
$680$		0			0
$681$	32.6525	−	12.4721i	1.25125	−	0.477933i
$682$		0			0
$683$			8.76393i			0.335343i	0.985843	+	0.167671i	$0.0536247\pi$
$683$			8.76393i			0.335343i	−0.985843	+	0.167671i	$0.946375\pi$
$684$		0			0
$685$		−	12.9443i		−	0.494575i
$686$		0			0
$687$	33.1246	−	12.6525i	1.26378	−	0.482722i
$688$		0			0
$689$			5.52786i			0.210595i
$690$		0			0
$691$	16.2918			0.619769			0.309885	−	0.950774i	$-0.399710\pi$
$691$	16.2918			0.619769			0.309885	+	0.950774i	$0.399710\pi$
$692$		0			0
$693$	−5.52786	+	4.94427i	−0.209986	+	0.187817i
$694$		0			0
$695$	4.58359			0.173866
$696$		0			0
$697$	−16.0000			−0.606043
$698$		0			0
$699$	−11.4164	+	4.36068i	−0.431808	+	0.164936i
$700$		0			0
$701$	−27.7082			−1.04652			−0.523262	−	0.852172i	$-0.675285\pi$
$701$	−27.7082			−1.04652			−0.523262	+	0.852172i	$0.675285\pi$
$702$		0			0
$703$			2.47214i			0.0932384i
$704$		0			0
$705$	−6.11146	−	16.0000i	−0.230171	−	0.602595i
$706$		0			0
$707$			37.8885i			1.42495i
$708$		0			0
$709$			16.4721i			0.618624i	0.950961	+	0.309312i	$0.100099\pi$
$709$			16.4721i			0.618624i	−0.950961	+	0.309312i	$0.899901\pi$
$710$		0			0
$711$	27.4164	+	30.6525i	1.02820	+	1.14956i
$712$		0			0
$713$		−	4.94427i		−	0.185164i
$714$		0			0
$715$	−4.22291			−0.157928
$716$		0			0
$717$	4.94427	+	12.9443i	0.184647	+	0.483413i
$718$		0			0
$719$	22.8328			0.851520			0.425760	−	0.904836i	$-0.360007\pi$
$719$	22.8328			0.851520			0.425760	+	0.904836i	$0.360007\pi$
$720$		0			0
$721$	−26.4721			−0.985874
$722$		0			0
$723$	−1.23607	−	3.23607i	−0.0459699	−	0.120351i
$724$		0			0
$725$	−32.0689			−1.19101
$726$		0			0
$727$		−	8.18034i		−	0.303392i	−0.988427	−	0.151696i	$-0.951527\pi$
$727$		−	8.18034i		−	0.303392i	0.988427	−	0.151696i	$-0.0484735\pi$
$728$		0			0
$729$	15.6525	+	22.0000i	0.579721	+	0.814815i
$730$		0			0
$731$			17.8885i			0.661632i
$732$		0			0
$733$			2.58359i			0.0954272i	0.998861	+	0.0477136i	$0.0151935\pi$
$733$			2.58359i			0.0954272i	−0.998861	+	0.0477136i	$0.984807\pi$
$734$		0			0
$735$	−2.65248	−	6.94427i	−0.0978380	−	0.256143i
$736$		0			0
$737$		−	2.83282i		−	0.104348i
$738$		0			0
$739$	27.1246			0.997795			0.498897	−	0.866661i	$-0.333738\pi$
$739$	27.1246			0.997795			0.498897	+	0.866661i	$0.333738\pi$
$740$		0			0
$741$	−37.8885	+	14.4721i	−1.39187	+	0.531647i
$742$		0			0
$743$	19.4164			0.712319			0.356159	−	0.934425i	$-0.384086\pi$
$743$	19.4164			0.712319			0.356159	+	0.934425i	$0.384086\pi$
$744$		0			0
$745$	−4.58359			−0.167930
$746$		0			0
$747$	14.4721	+	16.1803i	0.529508	+	0.592008i
$748$		0			0
$749$	26.4721			0.967271
$750$		0			0
$751$		−	31.5967i		−	1.15298i	−0.817104	−	0.576491i	$-0.804422\pi$
$751$		−	31.5967i		−	1.15298i	0.817104	−	0.576491i	$-0.195578\pi$
$752$		0			0
$753$	16.6525	−	6.36068i	0.606850	−	0.231796i
$754$		0			0
$755$		−	7.77709i		−	0.283037i
$756$		0			0
$757$			21.4164i			0.778393i	0.921155	+	0.389196i	$0.127247\pi$
$757$			21.4164i			0.778393i	−0.921155	+	0.389196i	$0.872753\pi$
$758$		0			0
$759$	−8.00000	+	3.05573i	−0.290382	+	0.110916i
$760$		0			0
$761$		−	38.2492i		−	1.38653i	−0.720681	−	0.693267i	$-0.756171\pi$
$761$		−	38.2492i		−	1.38653i	0.720681	−	0.693267i	$-0.243829\pi$
$762$		0			0
$763$	−1.52786			−0.0553124
$764$		0			0
$765$	−16.0000	−	17.8885i	−0.578481	−	0.646762i
$766$		0			0
$767$	−14.4721			−0.522559
$768$		0			0
$769$	−21.7771			−0.785302			−0.392651	−	0.919688i	$-0.628442\pi$
$769$	−21.7771			−0.785302			−0.392651	+	0.919688i	$0.628442\pi$
$770$		0			0
$771$	4.94427	−	1.88854i	0.178064	−	0.0680142i
$772$		0			0
$773$	30.1803			1.08551			0.542756	−	0.839891i	$-0.317381\pi$
$773$	30.1803			1.08551			0.542756	+	0.839891i	$0.317381\pi$
$774$		0			0
$775$			2.65248i			0.0952797i
$776$		0			0
$777$	0.944272	+	2.47214i	0.0338756	+	0.0886874i
$778$		0			0
$779$			12.9443i			0.463777i
$780$		0			0
$781$		−	8.72136i		−	0.312075i
$782$		0			0
$783$	−42.6525	−	22.0000i	−1.52428	−	0.786216i
$784$		0			0
$785$			5.52786i			0.197298i
$786$		0			0
$787$	−5.81966			−0.207448			−0.103724	−	0.994606i	$-0.533076\pi$
$787$	−5.81966			−0.207448			−0.103724	+	0.994606i	$0.533076\pi$
$788$		0			0
$789$	−0.944272	−	2.47214i	−0.0336170	−	0.0880104i
$790$		0			0
$791$	−25.8885			−0.920491
$792$		0			0
$793$	37.8885

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:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	768.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+(0.618034 + 1.61803i) q^{3} +1.23607 q^{5} -3.23607i q^{7} +(-2.23607 + 2.00000i) q^{9} +O(q^{10})\)	\(q+(0.618034 + 1.61803i) q^{3} +1.23607 q^{5} -3.23607i q^{7} +(-2.23607 + 2.00000i) q^{9} +0.763932i q^{11} +4.47214i q^{13} +(0.763932 + 2.00000i) q^{15} +6.47214i q^{17} +5.23607 q^{19} +(5.23607 - 2.00000i) q^{21} +6.47214 q^{23} -3.47214 q^{25} +(-4.61803 - 2.38197i) q^{27} +9.23607 q^{29} -0.763932i q^{31} +(-1.23607 + 0.472136i) q^{33} -4.00000i q^{35} +0.472136i q^{37} +(-7.23607 + 2.76393i) q^{39} +2.47214i q^{41} +2.76393 q^{43} +(-2.76393 + 2.47214i) q^{45} -8.00000 q^{47} -3.47214 q^{49} +(-10.4721 + 4.00000i) q^{51} +1.23607 q^{53} +0.944272i q^{55} +(3.23607 + 8.47214i) q^{57} +3.23607i q^{59} -8.47214i q^{61} +(6.47214 + 7.23607i) q^{63} +5.52786i q^{65} -3.70820 q^{67} +(4.00000 + 10.4721i) q^{69} -11.4164 q^{71} +2.00000 q^{73} +(-2.14590 - 5.61803i) q^{75} +2.47214 q^{77} -13.7082i q^{79} +(1.00000 - 8.94427i) q^{81} -7.23607i q^{83} +8.00000i q^{85} +(5.70820 + 14.9443i) q^{87} -4.00000i q^{89} +14.4721 q^{91} +(1.23607 - 0.472136i) q^{93} +6.47214 q^{95} -8.47214 q^{97} +(-1.52786 - 1.70820i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 2 q^{3} - 4 q^{5}+O(q^{10}) \) 4 * q - 2 * q^3 - 4 * q^5	\( 4 q - 2 q^{3} - 4 q^{5} + 12 q^{15} + 12 q^{19} + 12 q^{21} + 8 q^{23} + 4 q^{25} - 14 q^{27} + 28 q^{29} + 4 q^{33} - 20 q^{39} + 20 q^{43} - 20 q^{45} - 32 q^{47} + 4 q^{49} - 24 q^{51} - 4 q^{53} + 4 q^{57} + 8 q^{63} + 12 q^{67} + 16 q^{69} + 8 q^{71} + 8 q^{73} - 22 q^{75} - 8 q^{77} + 4 q^{81} - 4 q^{87} + 40 q^{91} - 4 q^{93} + 8 q^{95} - 16 q^{97} - 24 q^{99}+O(q^{100}) \) 4 * q - 2 * q^3 - 4 * q^5 + 12 * q^15 + 12 * q^19 + 12 * q^21 + 8 * q^23 + 4 * q^25 - 14 * q^27 + 28 * q^29 + 4 * q^33 - 20 * q^39 + 20 * q^43 - 20 * q^45 - 32 * q^47 + 4 * q^49 - 24 * q^51 - 4 * q^53 + 4 * q^57 + 8 * q^63 + 12 * q^67 + 16 * q^69 + 8 * q^71 + 8 * q^73 - 22 * q^75 - 8 * q^77 + 4 * q^81 - 4 * q^87 + 40 * q^91 - 4 * q^93 + 8 * q^95 - 16 * q^97 - 24 * q^99

Introduction

L-functions

Modular forms

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