LMFDB - Embedded newform 384.2.c.b.383.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [384,2,Mod(383,384)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(384, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("384.383");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$384 = 2^{7} \cdot 3$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	384.c (of order $2$ , degree $1$ , minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$3.06625543762$
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:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			383.1
Root			$-0.618034i$ of defining polynomial
Character	$\chi$	$=$	384.383
Dual form			384.2.c.b.383.2

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.61803 - 0.618034i) q^{3} +1.23607i q^{5} -3.23607i q^{7} +(2.23607 + 2.00000i) q^{9} +0.763932 q^{11} -4.47214 q^{13} +(0.763932 - 2.00000i) q^{15} -6.47214i q^{17} -5.23607i q^{19} +(-2.00000 + 5.23607i) q^{21} -6.47214 q^{23} +3.47214 q^{25} +(-2.38197 - 4.61803i) q^{27} -9.23607i q^{29} +0.763932i q^{31} +(-1.23607 - 0.472136i) q^{33} +4.00000 q^{35} +0.472136 q^{37} +(7.23607 + 2.76393i) q^{39} +2.47214i q^{41} +2.76393i q^{43} +(-2.47214 + 2.76393i) q^{45} -8.00000 q^{47} -3.47214 q^{49} +(-4.00000 + 10.4721i) q^{51} +1.23607i q^{53} +0.944272i q^{55} +(-3.23607 + 8.47214i) q^{57} +3.23607 q^{59} +8.47214 q^{61} +(6.47214 - 7.23607i) q^{63} -5.52786i q^{65} +3.70820i q^{67} +(10.4721 + 4.00000i) q^{69} +11.4164 q^{71} -2.00000 q^{73} +(-5.61803 - 2.14590i) q^{75} -2.47214i q^{77} +13.7082i q^{79} +(1.00000 + 8.94427i) q^{81} +7.23607 q^{83} +8.00000 q^{85} +(-5.70820 + 14.9443i) q^{87} -4.00000i q^{89} +14.4721i q^{91} +(0.472136 - 1.23607i) q^{93} +6.47214 q^{95} -8.47214 q^{97} +(1.70820 + 1.52786i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$4 q - 2 q^{3} + 12 q^{11} + 12 q^{15} - 8 q^{21} - 8 q^{23} - 4 q^{25} - 14 q^{27} + 4 q^{33} + 16 q^{35} - 16 q^{37} + 20 q^{39} + 8 q^{45} - 32 q^{47} + 4 q^{49} - 16 q^{51} - 4 q^{57} + 4 q^{59} + 16 q^{61}+ \cdots - 20 q^{99}+O(q^{100})$ 4 * q - 2 * q^3 + 12 * q^11 + 12 * q^15 - 8 * q^21 - 8 * q^23 - 4 * q^25 - 14 * q^27 + 4 * q^33 + 16 * q^35 - 16 * q^37 + 20 * q^39 + 8 * q^45 - 32 * q^47 + 4 * q^49 - 16 * q^51 - 4 * q^57 + 4 * q^59 + 16 * q^61 + 8 * q^63 + 24 * q^69 - 8 * q^71 - 8 * q^73 - 18 * q^75 + 4 * q^81 + 20 * q^83 + 32 * q^85 + 4 * q^87 - 16 * q^93 + 8 * q^95 - 16 * q^97 - 20 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/384\mathbb{Z}\right)^\times$ .

$n$	$127$	$133$	$257$
$\chi(n)$	$-1$	$1$	$-1$

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
$3$	−1.61803	−	0.618034i	−0.934172	−	0.356822i
$3$	−1.61803	−	0.618034i	−0.934172	−	0.356822i
$4$		0			0
$5$			1.23607i			0.552786i	0.961045	+	0.276393i	$0.0891392\pi$
$5$			1.23607i			0.552786i	−0.961045	+	0.276393i	$0.910861\pi$
$6$		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
$11$	0.763932			0.230334			0.115167	−	0.993346i	$-0.463260\pi$
$11$	0.763932			0.230334			0.115167	+	0.993346i	$0.463260\pi$
$12$		0			0
$13$	−4.47214			−1.24035			−0.620174	−	0.784465i	$-0.712938\pi$
$13$	−4.47214			−1.24035			−0.620174	+	0.784465i	$0.712938\pi$
$14$		0			0
$15$	0.763932	−	2.00000i	0.197246	−	0.516398i
$16$		0			0
$17$		−	6.47214i		−	1.56972i	−0.619671	−	0.784862i	$-0.712734\pi$
$17$		−	6.47214i		−	1.56972i	0.619671	−	0.784862i	$-0.287266\pi$
$18$		0			0
$19$		−	5.23607i		−	1.20124i	−0.799536	−	0.600618i	$-0.794921\pi$
$19$		−	5.23607i		−	1.20124i	0.799536	−	0.600618i	$-0.205079\pi$
$20$		0			0
$21$	−2.00000	+	5.23607i	−0.436436	+	1.14260i
$22$		0			0
$23$	−6.47214			−1.34953			−0.674767	−	0.738031i	$-0.735756\pi$
$23$	−6.47214			−1.34953			−0.674767	+	0.738031i	$0.735756\pi$
$24$		0			0
$25$	3.47214			0.694427
$26$		0			0
$27$	−2.38197	−	4.61803i	−0.458410	−	0.888741i
$28$		0			0
$29$		−	9.23607i		−	1.71509i	−0.514405	−	0.857547i	$-0.671987\pi$
$29$		−	9.23607i		−	1.71509i	0.514405	−	0.857547i	$-0.328013\pi$
$30$		0			0
$31$			0.763932i			0.137206i	0.997644	+	0.0686031i	$0.0218542\pi$
$31$			0.763932i			0.137206i	−0.997644	+	0.0686031i	$0.978146\pi$
$32$		0			0
$33$	−1.23607	−	0.472136i	−0.215172	−	0.0821883i
$34$		0			0
$35$	4.00000			0.676123
$36$		0			0
$37$	0.472136			0.0776187			0.0388093	−	0.999247i	$-0.487644\pi$
$37$	0.472136			0.0776187			0.0388093	+	0.999247i	$0.487644\pi$
$38$		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
$43$			2.76393i			0.421496i	0.977540	+	0.210748i	$0.0675899\pi$
$43$			2.76393i			0.421496i	−0.977540	+	0.210748i	$0.932410\pi$
$44$		0			0
$45$	−2.47214	+	2.76393i	−0.368524	+	0.412023i
$46$		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
$51$	−4.00000	+	10.4721i	−0.560112	+	1.46639i
$52$		0			0
$53$			1.23607i			0.169787i	0.996390	+	0.0848935i	$0.0270550\pi$
$53$			1.23607i			0.169787i	−0.996390	+	0.0848935i	$0.972945\pi$
$54$		0			0
$55$			0.944272i			0.127326i
$56$		0			0
$57$	−3.23607	+	8.47214i	−0.428628	+	1.12216i
$58$		0			0
$59$	3.23607			0.421300			0.210650	−	0.977562i	$-0.432442\pi$
$59$	3.23607			0.421300			0.210650	+	0.977562i	$0.432442\pi$
$60$		0			0
$61$	8.47214			1.08475			0.542373	−	0.840138i	$-0.317526\pi$
$61$	8.47214			1.08475			0.542373	+	0.840138i	$0.317526\pi$
$62$		0			0
$63$	6.47214	−	7.23607i	0.815412	−	0.911659i
$64$		0			0
$65$		−	5.52786i		−	0.685647i
$66$		0			0
$67$			3.70820i			0.453029i	0.974008	+	0.226515i	$0.0727331\pi$
$67$			3.70820i			0.453029i	−0.974008	+	0.226515i	$0.927267\pi$
$68$		0			0
$69$	10.4721	+	4.00000i	1.26070	+	0.481543i
$70$		0			0
$71$	11.4164			1.35488			0.677439	−	0.735579i	$-0.263090\pi$
$71$	11.4164			1.35488			0.677439	+	0.735579i	$0.263090\pi$
$72$		0			0
$73$	−2.00000			−0.234082			−0.117041	−	0.993127i	$-0.537341\pi$
$73$	−2.00000			−0.234082			−0.117041	+	0.993127i	$0.537341\pi$
$74$		0			0
$75$	−5.61803	−	2.14590i	−0.648715	−	0.247787i
$76$		0			0
$77$		−	2.47214i		−	0.281726i
$78$		0			0
$79$			13.7082i			1.54229i	0.636657	+	0.771147i	$0.280317\pi$
$79$			13.7082i			1.54229i	−0.636657	+	0.771147i	$0.719683\pi$
$80$		0			0
$81$	1.00000	+	8.94427i	0.111111	+	0.993808i
$82$		0			0
$83$	7.23607			0.794262			0.397131	−	0.917762i	$-0.370006\pi$
$83$	7.23607			0.794262			0.397131	+	0.917762i	$0.370006\pi$
$84$		0			0
$85$	8.00000			0.867722
$86$		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
$91$			14.4721i			1.51709i
$92$		0			0
$93$	0.472136	−	1.23607i	0.0489582	−	0.128174i
$94$		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
$99$	1.70820	+	1.52786i	0.171681	+	0.153556i
$100$		0			0
$101$		−	11.7082i		−	1.16501i	−0.812827	−	0.582505i	$-0.802073\pi$
$101$		−	11.7082i		−	1.16501i	0.812827	−	0.582505i	$-0.197927\pi$
$102$		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
$107$	8.18034			0.790823			0.395412	−	0.918504i	$-0.370602\pi$
$107$	8.18034			0.790823			0.395412	+	0.918504i	$0.370602\pi$
$108$		0			0
$109$	0.472136			0.0452224			0.0226112	−	0.999744i	$-0.492802\pi$
$109$	0.472136			0.0452224			0.0226112	+	0.999744i	$0.492802\pi$
$110$		0			0
$111$	−0.763932	−	0.291796i	−0.0725092	−	0.0276961i
$112$		0			0
$113$			8.00000i			0.752577i	0.926503	+	0.376288i	$0.122800\pi$
$113$			8.00000i			0.752577i	−0.926503	+	0.376288i	$0.877200\pi$
$114$		0			0
$115$		−	8.00000i		−	0.746004i
$116$		0			0
$117$	−10.0000	−	8.94427i	−0.924500	−	0.826898i
$118$		0			0
$119$	−20.9443			−1.91996
$120$		0			0
$121$	−10.4164			−0.946946
$122$		0			0
$123$	1.52786	−	4.00000i	0.137763	−	0.360668i
$124$		0			0
$125$			10.4721i			0.936656i
$126$		0			0
$127$			8.76393i			0.777673i	0.921307	+	0.388837i	$0.127123\pi$
$127$			8.76393i			0.777673i	−0.921307	+	0.388837i	$0.872877\pi$
$128$		0			0
$129$	1.70820	−	4.47214i	0.150399	−	0.393750i
$130$		0			0
$131$	−11.2361			−0.981700			−0.490850	−	0.871244i	$-0.663314\pi$
$131$	−11.2361			−0.981700			−0.490850	+	0.871244i	$0.663314\pi$
$132$		0			0
$133$	−16.9443			−1.46925
$134$		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
$139$			3.70820i			0.314526i	0.987557	+	0.157263i	$0.0502670\pi$
$139$			3.70820i			0.314526i	−0.987557	+	0.157263i	$0.949733\pi$
$140$		0			0
$141$	12.9443	+	4.94427i	1.09010	+	0.416383i
$142$		0			0
$143$	−3.41641			−0.285694
$144$		0			0
$145$	11.4164			0.948081
$146$		0			0
$147$	5.61803	+	2.14590i	0.463368	+	0.176991i
$148$		0			0
$149$		−	3.70820i		−	0.303788i	−0.988397	−	0.151894i	$-0.951463\pi$
$149$		−	3.70820i		−	0.303788i	0.988397	−	0.151894i	$-0.0485372\pi$
$150$		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
$155$	−0.944272			−0.0758457
$156$		0			0
$157$	−4.47214			−0.356915			−0.178458	−	0.983948i	$-0.557111\pi$
$157$	−4.47214			−0.356915			−0.178458	+	0.983948i	$0.557111\pi$
$158$		0			0
$159$	0.763932	−	2.00000i	0.0605838	−	0.158610i
$160$		0			0
$161$			20.9443i			1.65064i
$162$		0			0
$163$		−	18.1803i		−	1.42399i	−0.702182	−	0.711997i	$-0.747791\pi$
$163$		−	18.1803i		−	1.42399i	0.702182	−	0.711997i	$-0.252209\pi$
$164$		0			0
$165$	0.583592	−	1.52786i	0.0454326	−	0.118944i
$166$		0			0
$167$	22.4721			1.73895			0.869473	−	0.493980i	$-0.164459\pi$
$167$	22.4721			1.73895			0.869473	+	0.493980i	$0.164459\pi$
$168$		0			0
$169$	7.00000			0.538462
$170$		0			0
$171$	10.4721	−	11.7082i	0.800824	−	0.895349i
$172$		0			0
$173$			8.65248i			0.657836i	0.944359	+	0.328918i	$0.106684\pi$
$173$			8.65248i			0.657836i	−0.944359	+	0.328918i	$0.893316\pi$
$174$		0			0
$175$		−	11.2361i		−	0.849367i
$176$		0			0
$177$	−5.23607	−	2.00000i	−0.393567	−	0.150329i
$178$		0			0
$179$	−16.1803			−1.20938			−0.604688	−	0.796463i	$-0.706702\pi$
$179$	−16.1803			−1.20938			−0.604688	+	0.796463i	$0.706702\pi$
$180$		0			0
$181$	11.5279			0.856859			0.428430	−	0.903575i	$-0.359067\pi$
$181$	11.5279			0.856859			0.428430	+	0.903575i	$0.359067\pi$
$182$		0			0
$183$	−13.7082	−	5.23607i	−1.01334	−	0.387061i
$184$		0			0
$185$			0.583592i			0.0429065i
$186$		0			0
$187$		−	4.94427i		−	0.361561i
$188$		0			0
$189$	−14.9443	+	7.70820i	−1.08704	+	0.560689i
$190$		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
$195$	−3.41641	+	8.94427i	−0.244654	+	0.640513i
$196$		0			0
$197$			9.23607i			0.658043i	0.944323	+	0.329021i	$0.106719\pi$
$197$			9.23607i			0.658043i	−0.944323	+	0.329021i	$0.893281\pi$
$198$		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
$203$	−29.8885			−2.09776
$204$		0			0
$205$	−3.05573			−0.213421
$206$		0			0
$207$	−14.4721	−	12.9443i	−1.00588	−	0.899689i
$208$		0			0
$209$		−	4.00000i		−	0.276686i
$210$		0			0
$211$			6.76393i			0.465648i	0.972519	+	0.232824i	$0.0747967\pi$
$211$			6.76393i			0.465648i	−0.972519	+	0.232824i	$0.925203\pi$
$212$		0			0
$213$	−18.4721	−	7.05573i	−1.26569	−	0.483451i
$214$		0			0
$215$	−3.41641			−0.232997
$216$		0			0
$217$	2.47214			0.167820
$218$		0			0
$219$	3.23607	+	1.23607i	0.218673	+	0.0835257i
$220$		0			0
$221$			28.9443i			1.94700i
$222$		0			0
$223$		−	12.1803i		−	0.815656i	−0.913059	−	0.407828i	$-0.866286\pi$
$223$		−	12.1803i		−	0.815656i	0.913059	−	0.407828i	$-0.133714\pi$
$224$		0			0
$225$	7.76393	+	6.94427i	0.517595	+	0.462951i
$226$		0			0
$227$	20.1803			1.33942			0.669708	−	0.742624i	$-0.266419\pi$
$227$	20.1803			1.33942			0.669708	+	0.742624i	$0.266419\pi$
$228$		0			0
$229$	−20.4721			−1.35284			−0.676418	−	0.736518i	$-0.736469\pi$
$229$	−20.4721			−1.35284			−0.676418	+	0.736518i	$0.736469\pi$
$230$		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
$235$		−	9.88854i		−	0.645057i
$236$		0			0
$237$	8.47214	−	22.1803i	0.550324	−	1.44077i
$238$		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
$243$	3.90983	−	15.0902i	0.250816	−	0.968035i
$244$		0			0
$245$		−	4.29180i		−	0.274193i
$246$		0			0
$247$			23.4164i			1.48995i
$248$		0			0
$249$	−11.7082	−	4.47214i	−0.741977	−	0.283410i
$250$		0			0
$251$	−10.2918			−0.649612			−0.324806	−	0.945781i	$-0.605299\pi$
$251$	−10.2918			−0.649612			−0.324806	+	0.945781i	$0.605299\pi$
$252$		0			0
$253$	−4.94427			−0.310844
$254$		0			0
$255$	−12.9443	−	4.94427i	−0.810602	−	0.309622i
$256$		0			0
$257$			3.05573i			0.190611i	0.995448	+	0.0953055i	$0.0303828\pi$
$257$			3.05573i			0.190611i	−0.995448	+	0.0953055i	$0.969617\pi$
$258$		0			0
$259$		−	1.52786i		−	0.0949369i
$260$		0			0
$261$	18.4721	−	20.6525i	1.14340	−	1.27836i
$262$		0			0
$263$	1.52786			0.0942121			0.0471061	−	0.998890i	$-0.485000\pi$
$263$	1.52786			0.0942121			0.0471061	+	0.998890i	$0.485000\pi$
$264$		0			0
$265$	−1.52786			−0.0938559
$266$		0			0
$267$	−2.47214	+	6.47214i	−0.151292	+	0.396088i
$268$		0			0
$269$		−	6.18034i		−	0.376822i	−0.982090	−	0.188411i	$-0.939666\pi$
$269$		−	6.18034i		−	0.376822i	0.982090	−	0.188411i	$-0.0603337\pi$
$270$		0			0
$271$			13.7082i			0.832714i	0.909201	+	0.416357i	$0.136693\pi$
$271$			13.7082i			0.832714i	−0.909201	+	0.416357i	$0.863307\pi$
$272$		0			0
$273$	8.94427	−	23.4164i	0.541332	−	1.41723i
$274$		0			0
$275$	2.65248			0.159950
$276$		0			0
$277$	16.4721			0.989715			0.494857	−	0.868974i	$-0.335220\pi$
$277$	16.4721			0.989715			0.494857	+	0.868974i	$0.335220\pi$
$278$		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
$283$		−	25.2361i		−	1.50013i	−0.661365	−	0.750064i	$-0.730023\pi$
$283$		−	25.2361i		−	1.50013i	0.661365	−	0.750064i	$-0.269977\pi$
$284$		0			0
$285$	−10.4721	−	4.00000i	−0.620316	−	0.236940i
$286$		0			0
$287$	8.00000			0.472225
$288$		0			0
$289$	−24.8885			−1.46403
$290$		0			0
$291$	13.7082	+	5.23607i	0.803589	+	0.306944i
$292$		0			0
$293$			25.2361i			1.47431i	0.675725	+	0.737153i	$0.263831\pi$
$293$			25.2361i			1.47431i	−0.675725	+	0.737153i	$0.736169\pi$
$294$		0			0
$295$			4.00000i			0.232889i
$296$		0			0
$297$	−1.81966	−	3.52786i	−0.105587	−	0.204707i
$298$		0			0
$299$	28.9443			1.67389
$300$		0			0
$301$	8.94427			0.515539
$302$		0			0
$303$	−7.23607	+	18.9443i	−0.415701	+	1.08832i
$304$		0			0
$305$			10.4721i			0.599633i
$306$		0			0
$307$			21.5967i			1.23259i	0.787515	+	0.616296i	$0.211367\pi$
$307$			21.5967i			1.23259i	−0.787515	+	0.616296i	$0.788633\pi$
$308$		0			0
$309$	−5.05573	+	13.2361i	−0.287610	+	0.752974i
$310$		0			0
$311$	14.4721			0.820640			0.410320	−	0.911942i	$-0.365417\pi$
$311$	14.4721			0.820640			0.410320	+	0.911942i	$0.365417\pi$
$312$		0			0
$313$	8.47214			0.478873			0.239437	−	0.970912i	$-0.423037\pi$
$313$	8.47214			0.478873			0.239437	+	0.970912i	$0.423037\pi$
$314$		0			0
$315$	8.94427	+	8.00000i	0.503953	+	0.450749i
$316$		0			0
$317$		−	27.1246i		−	1.52347i	−0.647889	−	0.761735i	$-0.724348\pi$
$317$		−	27.1246i		−	1.52347i	0.647889	−	0.761735i	$-0.275652\pi$
$318$		0			0
$319$		−	7.05573i		−	0.395045i
$320$		0			0
$321$	−13.2361	−	5.05573i	−0.738765	−	0.282183i
$322$		0			0
$323$	−33.8885			−1.88561
$324$		0			0
$325$	−15.5279			−0.861331
$326$		0			0
$327$	−0.763932	−	0.291796i	−0.0422455	−	0.0161364i
$328$		0			0
$329$			25.8885i			1.42728i
$330$		0			0
$331$			34.5410i			1.89855i	0.314453	+	0.949273i	$0.398179\pi$
$331$			34.5410i			1.89855i	−0.314453	+	0.949273i	$0.601821\pi$
$332$		0			0
$333$	1.05573	+	0.944272i	0.0578535	+	0.0517458i
$334$		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
$339$	4.94427	−	12.9443i	0.268536	−	0.703036i
$340$		0			0
$341$			0.583592i			0.0316033i
$342$		0			0
$343$		−	11.4164i		−	0.616428i
$344$		0			0
$345$	−4.94427	+	12.9443i	−0.266191	+	0.696896i
$346$		0			0
$347$	−10.2918			−0.552493			−0.276246	−	0.961087i	$-0.589091\pi$
$347$	−10.2918			−0.552493			−0.276246	+	0.961087i	$0.589091\pi$
$348$		0			0
$349$	19.5279			1.04530			0.522651	−	0.852547i	$-0.324943\pi$
$349$	19.5279			1.04530			0.522651	+	0.852547i	$0.324943\pi$
$350$		0			0
$351$	10.6525	+	20.6525i	0.568587	+	1.10235i
$352$		0			0
$353$		−	25.8885i		−	1.37791i	−0.724805	−	0.688954i	$-0.758070\pi$
$353$		−	25.8885i		−	1.37791i	0.724805	−	0.688954i	$-0.241930\pi$
$354$		0			0
$355$			14.1115i			0.748958i
$356$		0			0
$357$	33.8885	+	12.9443i	1.79357	+	0.685084i
$358$		0			0
$359$	−4.58359			−0.241913			−0.120956	−	0.992658i	$-0.538596\pi$
$359$	−4.58359			−0.241913			−0.120956	+	0.992658i	$0.538596\pi$
$360$		0			0
$361$	−8.41641			−0.442969
$362$		0			0
$363$	16.8541	+	6.43769i	0.884611	+	0.337891i
$364$		0			0
$365$		−	2.47214i		−	0.129398i
$366$		0			0
$367$			0.763932i			0.0398769i	0.999801	+	0.0199385i	$0.00634703\pi$
$367$			0.763932i			0.0398769i	−0.999801	+	0.0199385i	$0.993653\pi$
$368$		0			0
$369$	−4.94427	+	5.52786i	−0.257389	+	0.287769i
$370$		0			0
$371$	4.00000			0.207670
$372$		0			0
$373$	29.4164			1.52312			0.761562	−	0.648092i	$-0.224433\pi$
$373$	29.4164			1.52312			0.761562	+	0.648092i	$0.224433\pi$
$374$		0			0
$375$	6.47214	−	16.9443i	0.334220	−	0.874998i
$376$		0			0
$377$			41.3050i			2.12731i
$378$		0			0
$379$			20.6525i			1.06085i	0.847733	+	0.530423i	$0.177967\pi$
$379$			20.6525i			1.06085i	−0.847733	+	0.530423i	$0.822033\pi$
$380$		0			0
$381$	5.41641	−	14.1803i	0.277491	−	0.726481i
$382$		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
$387$	−5.52786	+	6.18034i	−0.280997	+	0.314164i
$388$		0			0
$389$		−	9.81966i		−	0.497877i	−0.968519	−	0.248938i	$-0.919918\pi$
$389$		−	9.81966i		−	0.497877i	0.968519	−	0.248938i	$-0.0800816\pi$
$390$		0			0
$391$			41.8885i			2.11839i
$392$		0			0
$393$	18.1803	+	6.94427i	0.917077	+	0.350292i
$394$		0			0
$395$	−16.9443			−0.852559
$396$		0			0
$397$	−9.41641			−0.472596			−0.236298	−	0.971681i	$-0.575934\pi$
$397$	−9.41641			−0.472596			−0.236298	+	0.971681i	$0.575934\pi$
$398$		0			0
$399$	27.4164	+	10.4721i	1.37254	+	0.524263i
$400$		0			0
$401$			16.3607i			0.817013i	0.912755	+	0.408507i	$0.133950\pi$
$401$			16.3607i			0.817013i	−0.912755	+	0.408507i	$0.866050\pi$
$402$		0			0
$403$		−	3.41641i		−	0.170183i
$404$		0			0
$405$	−11.0557	+	1.23607i	−0.549364	+	0.0614207i
$406$		0			0
$407$	0.360680			0.0178782
$408$		0			0
$409$	3.88854			0.192276			0.0961381	−	0.995368i	$-0.469351\pi$
$409$	3.88854			0.192276			0.0961381	+	0.995368i	$0.469351\pi$
$410$		0			0
$411$	−6.47214	+	16.9443i	−0.319247	+	0.835799i
$412$		0			0
$413$		−	10.4721i		−	0.515300i
$414$		0			0
$415$			8.94427i			0.439057i
$416$		0			0
$417$	2.29180	−	6.00000i	0.112230	−	0.293821i
$418$		0			0
$419$	12.1803			0.595049			0.297524	−	0.954714i	$-0.403839\pi$
$419$	12.1803			0.595049			0.297524	+	0.954714i	$0.403839\pi$
$420$		0			0
$421$	5.41641			0.263980			0.131990	−	0.991251i	$-0.457863\pi$
$421$	5.41641			0.263980			0.131990	+	0.991251i	$0.457863\pi$
$422$		0			0
$423$	−17.8885	−	16.0000i	−0.869771	−	0.777947i
$424$		0			0
$425$		−	22.4721i		−	1.09006i
$426$		0			0
$427$		−	27.4164i		−	1.32677i
$428$		0			0
$429$	5.52786	+	2.11146i	0.266888	+	0.101942i
$430$		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
$435$	−18.4721	−	7.05573i	−0.885671	−	0.338296i
$436$		0			0
$437$			33.8885i			1.62111i
$438$		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
$443$	29.7082			1.41148			0.705740	−	0.708471i	$-0.250615\pi$
$443$	29.7082			1.41148			0.705740	+	0.708471i	$0.250615\pi$
$444$		0			0
$445$	4.94427			0.234381
$446$		0			0
$447$	−2.29180	+	6.00000i	−0.108398	+	0.283790i
$448$		0			0
$449$		−	27.4164i		−	1.29386i	−0.762549	−	0.646930i	$-0.776053\pi$
$449$		−	27.4164i		−	1.29386i	0.762549	−	0.646930i	$-0.223947\pi$
$450$		0			0
$451$			1.88854i			0.0889281i
$452$		0			0
$453$	−3.88854	+	10.1803i	−0.182700	+	0.478314i
$454$		0			0
$455$	−17.8885			−0.838628
$456$		0			0
$457$	−22.0000			−1.02912			−0.514558	−	0.857455i	$-0.672044\pi$
$457$	−22.0000			−1.02912			−0.514558	+	0.857455i	$0.672044\pi$
$458$		0			0
$459$	−29.8885	+	15.4164i	−1.39508	+	0.719576i
$460$		0			0
$461$		−	14.1803i		−	0.660444i	−0.943903	−	0.330222i	$-0.892876\pi$
$461$		−	14.1803i		−	0.660444i	0.943903	−	0.330222i	$-0.107124\pi$
$462$		0			0
$463$		−	39.2361i		−	1.82345i	−0.410796	−	0.911727i	$-0.634749\pi$
$463$		−	39.2361i		−	1.82345i	0.410796	−	0.911727i	$-0.365251\pi$
$464$		0			0
$465$	1.52786	+	0.583592i	0.0708530	+	0.0270634i
$466$		0			0
$467$	15.2361			0.705041			0.352521	−	0.935804i	$-0.385325\pi$
$467$	15.2361			0.705041			0.352521	+	0.935804i	$0.385325\pi$
$468$		0			0
$469$	12.0000			0.554109
$470$		0			0
$471$	7.23607	+	2.76393i	0.333420	+	0.127355i
$472$		0			0
$473$			2.11146i			0.0970849i
$474$		0			0
$475$		−	18.1803i		−	0.834171i
$476$		0			0
$477$	−2.47214	+	2.76393i	−0.113191	+	0.126552i
$478$		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
$483$	12.9443	−	33.8885i	0.588985	−	1.54198i
$484$		0			0
$485$		−	10.4721i		−	0.475515i
$486$		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
$491$	26.0689			1.17647			0.588236	−	0.808689i	$-0.299823\pi$
$491$	26.0689			1.17647			0.588236	+	0.808689i	$0.299823\pi$
$492$		0			0
$493$	−59.7771			−2.69222
$494$		0			0
$495$	−1.88854	+	2.11146i	−0.0848837	+	0.0949029i
$496$		0			0
$497$		−	36.9443i		−	1.65718i
$498$		0			0
$499$			22.7639i			1.01905i	0.860455	+	0.509527i	$0.170180\pi$
$499$			22.7639i			1.01905i	−0.860455	+	0.509527i	$0.829820\pi$
$500$		0			0
$501$	−36.3607	−	13.8885i	−1.62448	−	0.620494i
$502$		0			0
$503$	30.4721			1.35869			0.679343	−	0.733821i	$-0.262265\pi$
$503$	30.4721			1.35869			0.679343	+	0.733821i	$0.262265\pi$
$504$		0			0
$505$	14.4721			0.644002
$506$		0			0
$507$	−11.3262	−	4.32624i	−0.503016	−	0.192135i
$508$		0			0
$509$		−	12.2918i		−	0.544824i	−0.962181	−	0.272412i	$-0.912179\pi$
$509$		−	12.2918i		−	0.544824i	0.962181	−	0.272412i	$-0.0878214\pi$
$510$		0			0
$511$			6.47214i			0.286310i
$512$		0			0
$513$	−24.1803	+	12.4721i	−1.06759	+	0.550658i
$514$		0			0
$515$	10.1115			0.445564
$516$		0			0
$517$	−6.11146			−0.268782
$518$		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
$523$		−	34.1803i		−	1.49460i	−0.664486	−	0.747301i	$-0.731349\pi$
$523$		−	34.1803i		−	1.49460i	0.664486	−	0.747301i	$-0.268651\pi$
$524$		0			0
$525$	−6.94427	+	18.1803i	−0.303073	+	0.793455i
$526$		0			0
$527$	4.94427			0.215376
$528$		0			0
$529$	18.8885			0.821241
$530$		0			0
$531$	7.23607	+	6.47214i	0.314019	+	0.280867i
$532$		0			0
$533$		−	11.0557i		−	0.478877i
$534$		0			0
$535$			10.1115i			0.437156i
$536$		0			0
$537$	26.1803	+	10.0000i	1.12977	+	0.431532i
$538$		0			0
$539$	−2.65248			−0.114250
$540$		0			0
$541$	18.3607			0.789387			0.394694	−	0.918813i	$-0.370851\pi$
$541$	18.3607			0.789387			0.394694	+	0.918813i	$0.370851\pi$
$542$		0			0
$543$	−18.6525	−	7.12461i	−0.800454	−	0.305746i
$544$		0			0
$545$			0.583592i			0.0249983i
$546$		0			0
$547$			2.76393i			0.118177i	0.998253	+	0.0590886i	$0.0188194\pi$
$547$			2.76393i			0.118177i	−0.998253	+	0.0590886i	$0.981181\pi$
$548$		0			0
$549$	18.9443	+	16.9443i	0.808522	+	0.723164i
$550$		0			0
$551$	−48.3607			−2.06023
$552$		0			0
$553$	44.3607			1.88641
$554$		0			0
$555$	0.360680	−	0.944272i	0.0153100	−	0.0400821i
$556$		0			0
$557$			16.6525i			0.705588i	0.935701	+	0.352794i	$0.114768\pi$
$557$			16.6525i			0.705588i	−0.935701	+	0.352794i	$0.885232\pi$
$558$		0			0
$559$		−	12.3607i		−	0.522801i
$560$		0			0
$561$	−3.05573	+	8.00000i	−0.129013	+	0.337760i
$562$		0			0
$563$	−36.5410			−1.54002			−0.770010	−	0.638032i	$-0.779749\pi$
$563$	−36.5410			−1.54002			−0.770010	+	0.638032i	$0.779749\pi$
$564$		0			0
$565$	−9.88854			−0.416014
$566$		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
$571$		−	35.1246i		−	1.46992i	−0.678111	−	0.734960i	$-0.737201\pi$
$571$		−	35.1246i		−	1.46992i	0.678111	−	0.734960i	$-0.262799\pi$
$572$		0			0
$573$	−8.00000	−	3.05573i	−0.334205	−	0.127655i
$574$		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
$579$	−38.6525	−	14.7639i	−1.60634	−	0.613568i
$580$		0			0
$581$		−	23.4164i		−	0.971476i
$582$		0			0
$583$			0.944272i			0.0391077i
$584$		0			0
$585$	11.0557	−	12.3607i	0.457098	−	0.511051i
$586$		0			0
$587$	−27.5967			−1.13904			−0.569520	−	0.821978i	$-0.692871\pi$
$587$	−27.5967			−1.13904			−0.569520	+	0.821978i	$0.692871\pi$
$588$		0			0
$589$	4.00000			0.164817
$590$		0			0
$591$	5.70820	−	14.9443i	0.234804	−	0.614725i
$592$		0			0
$593$			11.0557i			0.454004i	0.973894	+	0.227002i	$0.0728924\pi$
$593$			11.0557i			0.454004i	−0.973894	+	0.227002i	$0.927108\pi$
$594$		0			0
$595$		−	25.8885i		−	1.06133i
$596$		0			0
$597$	−10.0000	+	26.1803i	−0.409273	+	1.07149i
$598$		0			0
$599$	19.4164			0.793333			0.396666	−	0.917963i	$-0.370167\pi$
$599$	19.4164			0.793333			0.396666	+	0.917963i	$0.370167\pi$
$600$		0			0
$601$	−37.7771			−1.54096			−0.770480	−	0.637464i	$-0.779983\pi$
$601$	−37.7771			−1.54096			−0.770480	+	0.637464i	$0.779983\pi$
$602$		0			0
$603$	−7.41641	+	8.29180i	−0.302019	+	0.337668i
$604$		0			0
$605$		−	12.8754i		−	0.523459i
$606$		0			0
$607$		−	39.2361i		−	1.59254i	−0.604940	−	0.796271i	$-0.706803\pi$
$607$		−	39.2361i		−	1.59254i	0.604940	−	0.796271i	$-0.293197\pi$
$608$		0			0
$609$	48.3607	+	18.4721i	1.95967	+	0.748529i
$610$		0			0
$611$	35.7771			1.44739
$612$		0			0
$613$	−43.3050			−1.74907			−0.874535	−	0.484962i	$-0.838833\pi$
$613$	−43.3050			−1.74907			−0.874535	+	0.484962i	$0.838833\pi$
$614$		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
$619$		−	11.1246i		−	0.447136i	−0.974688	−	0.223568i	$-0.928230\pi$
$619$		−	11.1246i		−	0.447136i	0.974688	−	0.223568i	$-0.0717705\pi$
$620$		0			0
$621$	15.4164	+	29.8885i	0.618639	+	1.19939i
$622$		0			0
$623$	−12.9443			−0.518601
$624$		0			0
$625$	4.41641			0.176656
$626$		0			0
$627$	−2.47214	+	6.47214i	−0.0987276	+	0.258472i
$628$		0			0
$629$		−	3.05573i		−	0.121840i
$630$		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
$635$	−10.8328			−0.429887
$636$		0			0
$637$	15.5279			0.615236
$638$		0			0
$639$	25.5279	+	22.8328i	1.00987	+	0.903252i
$640$		0			0
$641$			37.3050i			1.47346i	0.676189	+	0.736729i	$0.263630\pi$
$641$			37.3050i			1.47346i	−0.676189	+	0.736729i	$0.736370\pi$
$642$		0			0
$643$			44.6525i			1.76092i	0.474119	+	0.880461i	$0.342767\pi$
$643$			44.6525i			1.76092i	−0.474119	+	0.880461i	$0.657233\pi$
$644$		0			0
$645$	5.52786	+	2.11146i	0.217659	+	0.0831385i
$646$		0			0
$647$	−29.3050			−1.15210			−0.576048	−	0.817416i	$-0.695406\pi$
$647$	−29.3050			−1.15210			−0.576048	+	0.817416i	$0.695406\pi$
$648$		0			0
$649$	2.47214			0.0970398
$650$		0			0
$651$	−4.00000	−	1.52786i	−0.156772	−	0.0598817i
$652$		0			0
$653$			27.7082i			1.08431i	0.840280	+	0.542153i	$0.182391\pi$
$653$			27.7082i			1.08431i	−0.840280	+	0.542153i	$0.817609\pi$
$654$		0			0
$655$		−	13.8885i		−	0.542670i
$656$		0			0
$657$	−4.47214	−	4.00000i	−0.174475	−	0.156055i
$658$		0			0
$659$	41.7082			1.62472			0.812360	−	0.583156i	$-0.198182\pi$
$659$	41.7082			1.62472			0.812360	+	0.583156i	$0.198182\pi$
$660$		0			0
$661$	−11.3050			−0.439712			−0.219856	−	0.975532i	$-0.570559\pi$
$661$	−11.3050			−0.439712			−0.219856	+	0.975532i	$0.570559\pi$
$662$		0			0
$663$	17.8885	−	46.8328i	0.694733	−	1.81884i
$664$		0			0
$665$		−	20.9443i		−	0.812184i
$666$		0			0
$667$			59.7771i			2.31458i
$668$		0			0
$669$	−7.52786	+	19.7082i	−0.291044	+	0.761963i
$670$		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
$675$	−8.27051	−	16.0344i	−0.318332	−	0.617166i
$676$		0			0
$677$		−	27.7082i		−	1.06491i	−0.846457	−	0.532456i	$-0.821269\pi$
$677$		−	27.7082i		−	1.06491i	0.846457	−	0.532456i	$-0.178731\pi$
$678$		0			0
$679$			27.4164i			1.05215i
$680$		0			0
$681$	−32.6525	−	12.4721i	−1.25125	−	0.477933i
$682$		0			0
$683$	8.76393			0.335343			0.167671	−	0.985843i	$-0.446375\pi$
$683$	8.76393			0.335343			0.167671	+	0.985843i	$0.446375\pi$
$684$		0			0
$685$	12.9443			0.494575
$686$		0			0
$687$	33.1246	+	12.6525i	1.26378	+	0.482722i
$688$		0			0
$689$		−	5.52786i		−	0.210595i
$690$		0			0
$691$		−	16.2918i		−	0.619769i	−0.950774	−	0.309885i	$-0.899710\pi$
$691$		−	16.2918i		−	0.619769i	0.950774	−	0.309885i	$-0.100290\pi$
$692$		0			0
$693$	4.94427	−	5.52786i	0.187817	−	0.209986i
$694$		0			0
$695$	−4.58359			−0.173866
$696$		0			0
$697$	16.0000			0.606043
$698$		0			0
$699$	4.36068	−	11.4164i	0.164936	−	0.431808i
$700$		0			0
$701$			27.7082i			1.04652i	0.852172	+	0.523262i	$0.175285\pi$
$701$			27.7082i			1.04652i	−0.852172	+	0.523262i	$0.824715\pi$
$702$		0			0
$703$		−	2.47214i		−	0.0932384i
$704$		0			0
$705$	−6.11146	+	16.0000i	−0.230171	+	0.602595i
$706$		0			0
$707$	−37.8885			−1.42495
$708$		0			0
$709$	16.4721			0.618624			0.309312	−	0.950961i	$-0.399901\pi$
$709$	16.4721			0.618624			0.309312	+	0.950961i	$0.399901\pi$
$710$		0			0
$711$	−27.4164	+	30.6525i	−1.02820	+	1.14956i
$712$		0			0
$713$		−	4.94427i		−	0.185164i
$714$		0			0
$715$		−	4.22291i		−	0.157928i
$716$		0			0
$717$	−12.9443	−	4.94427i	−0.483413	−	0.184647i
$718$		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
$723$	3.23607	+	1.23607i	0.120351	+	0.0459699i
$724$		0			0
$725$		−	32.0689i		−	1.19101i
$726$		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
$731$	17.8885			0.661632
$732$		0			0
$733$	−2.58359			−0.0954272			−0.0477136	−	0.998861i	$-0.515193\pi$
$733$	−2.58359			−0.0954272			−0.0477136	+	0.998861i	$0.515193\pi$
$734$		0			0
$735$	−2.65248	+	6.94427i	−0.0978380	+	0.256143i
$736$		0			0
$737$			2.83282i			0.104348i
$738$		0			0
$739$		−	27.1246i		−	0.997795i	−0.866661	−	0.498897i	$-0.833738\pi$
$739$		−	27.1246i		−	0.997795i	0.866661	−	0.498897i	$-0.166262\pi$
$740$		0			0
$741$	14.4721	−	37.8885i	0.531647	−	1.39187i
$742$		0			0
$743$	−19.4164			−0.712319			−0.356159	−	0.934425i	$-0.615914\pi$
$743$	−19.4164			−0.712319			−0.356159	+	0.934425i	$0.615914\pi$
$744$		0			0
$745$	4.58359			0.167930
$746$		0			0
$747$	16.1803	+	14.4721i	0.592008	+	0.529508i
$748$		0			0
$749$		−	26.4721i		−	0.967271i
$750$		0			0
$751$			31.5967i			1.15298i	0.817104	+	0.576491i	$0.195578\pi$
$751$			31.5967i			1.15298i	−0.817104	+	0.576491i	$0.804422\pi$
$752$		0			0
$753$	16.6525	+	6.36068i	0.606850	+	0.231796i
$754$		0			0
$755$	7.77709			0.283037
$756$		0			0
$757$	21.4164			0.778393			0.389196	−	0.921155i	$-0.372753\pi$
$757$	21.4164			0.778393			0.389196	+	0.921155i	$0.372753\pi$
$758$		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
$763$		−	1.52786i		−	0.0553124i
$764$		0			0
$765$	17.8885	+	16.0000i	0.646762	+	0.578481i
$766$		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
$771$	1.88854	−	4.94427i	0.0680142	−	0.178064i
$772$		0			0
$773$			30.1803i			1.08551i	0.839891	+	0.542756i	$0.182619\pi$
$773$			30.1803i			1.08551i	−0.839891	+	0.542756i	$0.817381\pi$
$774$		0			0
$775$			2.65248i			0.0952797i
$776$		0			0
$777$	−0.944272	+	2.47214i	−0.0338756	+	0.0886874i
$778$		0			0
$779$	12.9443			0.463777
$780$		0			0
$781$	8.72136			0.312075
$782$		0			0
$783$	−42.6525	+	22.0000i	−1.52428	+	0.786216i
$784$		0			0
$785$		−	5.52786i		−	0.197298i
$786$		0			0
$787$			5.81966i			0.207448i	0.994606	+	0.103724i	$0.0330759\pi$
$787$			5.81966i			0.207448i	−0.994606	+	0.103724i	$0.966924\pi$
$788$		0			0
$789$	−2.47214	−	0.944272i	−0.0880104	−	0.0336170i
$790$		0			0
$791$	25.8885			0.920491
$792$		0			0
$793$	−37.8885			−1.34546
$794$		0			0
$795$	2.47214	+	0.944272i	0.0876776	+	0.0334899i
$796$		0			0
$797$		−	8.06888i		−	0.285815i	−0.989736	−	0.142907i	$-0.954355\pi$
$797$		−	8.06888i		−	0.285815i	0.989736	−	0.142907i	$-0.0456451\pi$
$798$		0			0
$799$			51.7771i			1.83174i
$800$		0			0
$801$	8.00000	−	8.94427i	0.282666	−	0.316030i
$802$		0			0
$803$	−1.52786			−0.0539172
$804$		0			0
$805$	−25.8885			−0.912451
$806$		0			0
$807$	−3.81966	+	10.0000i	−0.134458	+	0.352017i
$808$		0			0
$809$		−	23.4164i		−	0.823277i	−0.911347	−	0.411639i	$-0.864957\pi$
$809$		−	23.4164i		−	0.823277i	0.911347	−	0.411639i	$-0.135043\pi$
$810$		0			0
$811$			29.8197i			1.04711i	0.851992	+	0.523555i	$0.175395\pi$
$811$			29.8197i			1.04711i	−0.851992	+	0.523555i	$0.824605\pi$
$812$		0			0
$813$	8.47214	−	22.1803i	0.297131	−	0.777898i
$814$		0			0
$815$	22.4721			0.787165
$816$		0			0
$817$	14.4721			0.506316
$818$		0			0
$819$	−28.9443	+	32.3607i	−1.01139	+	1.13077i
$820$		0			0
$821$		−	52.4296i		−	1.82980i	−0.403676	−	0.914902i	$-0.632268\pi$
$821$		−	52.4296i		−	1.82980i	0.403676	−	0.914902i	$-0.367732\pi$
$822$		0			0
$823$			10.8754i			0.379092i	0.981872	+	0.189546i	$0.0607016\pi$
$823$			10.8754i			0.379092i	−0.981872	+	0.189546i	$0.939298\pi$
$824$		0			0
$825$	−4.29180	−	1.63932i	−0.149421	−	0.0570738i
$826$		0			0
$827$	21.1246			0.734575			0.367287	−	0.930108i	$-0.380287\pi$
$827$	21.1246			0.734575			0.367287	+	0.930108i	$0.380287\pi$
$828$		0			0
$829$	−4.47214			−0.155324			−0.0776619	−	0.996980i	$-0.524745\pi$
$829$	−4.47214			−0.155324			−0.0776619	+	0.996980i	$0.524745\pi$
$830$		0			0
$831$	−26.6525	−	10.1803i	−0.924564	−	0.353152i
$832$		0			0
$833$			22.4721i			0.778613i
$834$		0			0
$835$			27.7771i			0.961266i
$836$		0			0
$837$	3.52786	−	1.81966i	0.121941	−	0.0628967i
$838$		0			0
$839$	−4.58359			−0.158243			−0.0791216	−	0.996865i	$-0.525212\pi$
$839$	−4.58359			−0.158243			−0.0791216	+	0.996865i	$0.525212\pi$
$840$		0			0
$841$	−56.3050			−1.94155
$842$		0			0
$843$	−4.36068	+	11.4164i	−0.150190	+	0.393202i
$844$		0			0
$845$			8.65248i			0.297654i
$846$		0			0
$847$			33.7082i			1.15823i
$848$		0			0
$849$	−15.5967	+	40.8328i	−0.535279	+	1.40138i
$850$		0			0
$851$	−3.05573			−0.104749
$852$		0			0
$853$	23.3050			0.797946			0.398973	−	0.916963i	$-0.369367\pi$
$853$	23.3050			0.797946			0.398973	+	0.916963i	$0.369367\pi$
$854$		0			0
$855$	14.4721	+	12.9443i	0.494937	+	0.442685i
$856$		0			0
$857$			52.3607i			1.78861i	0.447461	+	0.894303i	$0.352328\pi$
$857$			52.3607i			1.78861i	−0.447461	+	0.894303i	$0.647672\pi$
$858$		0			0
$859$		−	21.2361i		−	0.724565i	−0.932068	−	0.362283i	$-0.881998\pi$
$859$		−	21.2361i		−	0.724565i	0.932068	−	0.362283i	$-0.118002\pi$
$860$		0			0
$861$	−12.9443	−	4.94427i	−0.441140	−	0.168500i
$862$		0			0
$863$	30.8328			1.04956			0.524781	−	0.851238i	$-0.324147\pi$
$863$	30.8328			1.04956			0.524781	+	0.851238i	$0.324147\pi$
$864$		0			0
$865$	−10.6950			−0.363643
$866$		0			0
$867$	40.2705	+	15.3820i	1.36766	+	0.522399i
$868$		0			0
$869$			10.4721i			0.355243i
$870$		0			0
$871$		−	16.5836i		−	0.561914i
$872$		0			0
$873$	−18.9443	−	16.9443i	−0.641166	−	0.573477i
$874$		0			0
$875$	33.8885			1.14564
$876$		0			0
$877$	45.4164			1.53360			0.766802	−	0.641884i	$-0.221847\pi$
$877$	45.4164			1.53360			0.766802	+	0.641884i	$0.221847\pi$
$878$		0			0
$879$	15.5967	−	40.8328i	0.526065	−	1.37726i
$880$		0			0
$881$			24.0000i			0.808581i	0.914631	+	0.404290i	$0.132481\pi$
$881$			24.0000i			0.808581i	−0.914631	+	0.404290i	$0.867519\pi$
$882$		0			0
$883$		−	5.23607i		−	0.176208i	−0.996111	−	0.0881039i	$-0.971919\pi$
$883$		−	5.23607i		−	0.176208i	0.996111	−	0.0881039i	$-0.0280807\pi$
$884$		0			0
$885$	2.47214	−	6.47214i	0.0830999	−	0.217558i
$886$		0			0
$887$	24.3607			0.817952			0.408976	−	0.912545i	$-0.365886\pi$
$887$	24.3607			0.817952			0.408976	+	0.912545i	$0.365886\pi$
$888$		0			0
$889$	28.3607			0.951187
$890$		0			0
$891$	0.763932	+	6.83282i	0.0255927	+	0.228908i
$892$		0			0
$893$			41.8885i			1.40175i
$894$		0			0
$895$		−	20.0000i		−	0.668526i
$896$		0			0
$897$	−46.8328	−	17.8885i	−1.56370	−	0.597281i
$898$		0			0
$899$	7.05573			0.235322
$900$		0			0
$901$	8.00000			0.266519
$902$		0			0
$903$	−14.4721	−	5.52786i	−0.481603	−	0.183956i
$904$		0			0
$905$			14.2492i			0.473660i
$906$		0			0
$907$			34.7639i			1.15432i	0.816632	+	0.577159i	$0.195839\pi$
$907$			34.7639i			1.15432i	−0.816632	+	0.577159i	$0.804161\pi$
$908$		0			0
$909$	23.4164	−	26.1803i	0.776673	−	0.868347i
$910$		0			0
$911$	−1.88854			−0.0625702			−0.0312851	−	0.999511i	$-0.509960\pi$
$911$	−1.88854			−0.0625702			−0.0312851	+	0.999511i	$0.509960\pi$
$912$		0			0
$913$	5.52786			0.182946
$914$		0			0
$915$	6.47214	−	16.9443i	0.213962	−	0.560160i
$916$		0			0
$917$			36.3607i			1.20074i
$918$		0			0
$919$			8.54102i			0.281742i	0.990028	+	0.140871i	$0.0449903\pi$
$919$			8.54102i			0.281742i	−0.990028	+	0.140871i	$0.955010\pi$
$920$		0			0
$921$	13.3475	−	34.9443i	0.439816	−	1.15145i
$922$		0			0
$923$	−51.0557			−1.68052
$924$		0			0
$925$	1.63932			0.0539005
$926$		0			0
$927$	16.3607	−	18.2918i	0.537355	−	0.600781i
$928$		0			0
$929$		−	20.5836i		−	0.675326i	−0.941267	−	0.337663i	$-0.890364\pi$
$929$		−	20.5836i		−	0.675326i	0.941267	−	0.337663i	$-0.109636\pi$
$930$		0			0
$931$			18.1803i			0.595837i
$932$		0			0
$933$	−23.4164	−	8.94427i	−0.766619	−	0.292822i
$934$		0			0
$935$	6.11146			0.199866
$936$		0			0
$937$	30.3607			0.991840			0.495920	−	0.868368i	$-0.334831\pi$
$937$	30.3607			0.991840			0.495920	+	0.868368i	$0.334831\pi$
$938$		0			0
$939$	−13.7082	−	5.23607i	−0.447350	−	0.170873i
$940$		0			0
$941$			11.7082i			0.381677i	0.981621	+	0.190838i	$0.0611206\pi$
$941$			11.7082i			0.381677i	−0.981621	+	0.190838i	$0.938879\pi$
$942$		0			0
$943$		−	16.0000i		−	0.521032i
$944$		0			0
$945$	−9.52786	−	18.4721i	−0.309941	−	0.600899i
$946$		0			0
$947$	−34.0689			−1.10709			−0.553545	−	0.832819i	$-0.686725\pi$
$947$	−34.0689			−1.10709			−0.553545	+	0.832819i	$0.686725\pi$
$948$		0			0
$949$	8.94427			0.290343
$950$		0			0
$951$	−16.7639	+	43.8885i	−0.543608	+	1.42318i
$952$		0			0
$953$			20.3607i			0.659547i	0.944060	+	0.329774i	$0.106972\pi$
$953$			20.3607i			0.659547i	−0.944060	+	0.329774i	$0.893028\pi$
$954$		0			0
$955$			6.11146i			0.197762i
$956$		0			0
$957$	−4.36068	+	11.4164i	−0.140961	+	0.369040i
$958$		0			0
$959$	−33.8885			−1.09432
$960$		0			0
$961$	30.4164			0.981174
$962$		0			0
$963$	18.2918	+	16.3607i	0.589445	+	0.527216i
$964$		0			0
$965$			29.5279i			0.950536i
$966$		0			0
$967$			4.76393i			0.153198i	0.997062	+	0.0765989i	$0.0244061\pi$
$967$			4.76393i			0.153198i	−0.997062	+	0.0765989i	$0.975594\pi$
$968$		0			0
$969$	54.8328	+	20.9443i	1.76148	+	0.672827i
$970$		0			0
$971$	−59.0132			−1.89382			−0.946911	−	0.321495i	$-0.895815\pi$
$971$	−59.0132			−1.89382			−0.946911	+	0.321495i	$0.895815\pi$
$972$		0			0
$973$	12.0000			0.384702
$974$		0			0
$975$	25.1246	+	9.59675i	0.804632	+	0.307342i
$976$		0			0
$977$			15.6393i			0.500346i	0.968201	+	0.250173i	$0.0804875\pi$
$977$			15.6393i			0.500346i	−0.968201	+	0.250173i	$0.919512\pi$
$978$		0			0
$979$		−	3.05573i		−	0.0976615i
$980$		0			0
$981$	1.05573	+	0.944272i	0.0337068	+	0.0301483i
$982$		0			0
$983$	30.4721			0.971910			0.485955	−	0.873984i	$-0.338472\pi$
$983$	30.4721			0.971910			0.485955	+	0.873984i	$0.338472\pi$
$984$		0			0
$985$	−11.4164			−0.363757
$986$		0			0
$987$	16.0000	−	41.8885i	0.509286	−	1.33333i
$988$		0			0
$989$		−	17.8885i		−	0.568823i
$990$		0			0
$991$			27.8197i			0.883721i	0.897084	+	0.441860i	$0.145681\pi$
$991$			27.8197i			0.883721i	−0.897084	+	0.441860i	$0.854319\pi$
$992$		0			0
$993$	21.3475	−	55.8885i	0.677443	−	1.77357i
$994$		0			0
$995$	20.0000			0.634043
$996$		0			0
$997$	26.3607			0.834851			0.417426	−	0.908711i	$-0.362932\pi$
$997$	26.3607			0.834851			0.417426	+	0.908711i	$0.362932\pi$
$998$		0			0
$999$	−1.12461	−	2.18034i	−0.0355811	−	0.0689829i

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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	384.2.c.b.383.1	yes	4
3.2	odd	2		384.2.c.c.383.3	yes	4
4.3	odd	2		384.2.c.c.383.4	yes	4
8.3	odd	2		384.2.c.a.383.1	✓	4
8.5	even	2		384.2.c.d.383.4	yes	4
12.11	even	2	inner	384.2.c.b.383.2	yes	4
16.3	odd	4		768.2.f.e.383.2		4
16.5	even	4		768.2.f.f.383.2		4
16.11	odd	4		768.2.f.c.383.3		4
16.13	even	4		768.2.f.b.383.3		4
24.5	odd	2		384.2.c.a.383.2	yes	4
24.11	even	2		384.2.c.d.383.3	yes	4
48.5	odd	4		768.2.f.e.383.1		4
48.11	even	4		768.2.f.b.383.4		4
48.29	odd	4		768.2.f.c.383.4		4
48.35	even	4		768.2.f.f.383.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
384.2.c.a.383.1	✓	4	8.3	odd	2
384.2.c.a.383.2	yes	4	24.5	odd	2
384.2.c.b.383.1	yes	4	1.1	even	1	trivial
384.2.c.b.383.2	yes	4	12.11	even	2	inner
384.2.c.c.383.3	yes	4	3.2	odd	2
384.2.c.c.383.4	yes	4	4.3	odd	2
384.2.c.d.383.3	yes	4	24.11	even	2
384.2.c.d.383.4	yes	4	8.5	even	2
768.2.f.b.383.3		4	16.13	even	4
768.2.f.b.383.4		4	48.11	even	4
768.2.f.c.383.3		4	16.11	odd	4
768.2.f.c.383.4		4	48.29	odd	4
768.2.f.e.383.1		4	48.5	odd	4
768.2.f.e.383.2		4	16.3	odd	4
768.2.f.f.383.1		4	48.35	even	4
768.2.f.f.383.2		4	16.5	even	4

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 $\F_{q}$

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

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Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Character values

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	384.2.c.b.383.1

Level	$384$
Weight	$2$
Character	384.383
Analytic conductor	$3.066$
Analytic rank	$0$
Dimension	$4$
Inner twists	$2$