LMFDB - Embedded newform 384.2.c.a.383.2

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			383.2
Root			$0.618034i$ of defining polynomial
Character	$\chi$	$=$	384.383
Dual form			384.2.c.a.383.1

$q$ -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	$-0.287062\pi$
$13$	4.47214			1.24035			0.620174	+	0.784465i	$0.287062\pi$
$14$		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
$19$			5.23607i			1.20124i	0.799536	+	0.600618i	$0.205079\pi$
$19$			5.23607i			1.20124i	−0.799536	+	0.600618i	$0.794921\pi$
$20$		0			0
$21$	2.00000	+	5.23607i	0.436436	+	1.14260i
$22$		0			0
$23$	6.47214			1.34953			0.674767	−	0.738031i	$-0.264244\pi$
$23$	6.47214			1.34953			0.674767	+	0.738031i	$0.264244\pi$
$24$		0			0
$25$	3.47214			0.694427
$26$		0			0
$27$	−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	$-0.512356\pi$
$37$	−0.472136			−0.0776187			−0.0388093	+	0.999247i	$0.512356\pi$
$38$		0			0
$39$	−7.23607	+	2.76393i	−1.15870	+	0.442583i
$40$		0			0
$41$		−	2.47214i		−	0.386083i	−0.981191	−	0.193041i	$-0.938165\pi$
$41$		−	2.47214i		−	0.386083i	0.981191	−	0.193041i	$-0.0618352\pi$
$42$		0			0
$43$		−	2.76393i		−	0.421496i	−0.977540	−	0.210748i	$-0.932410\pi$
$43$		−	2.76393i		−	0.421496i	0.977540	−	0.210748i	$-0.0675899\pi$
$44$		0			0
$45$	2.47214	+	2.76393i	0.368524	+	0.412023i
$46$		0			0
$47$	8.00000			1.16692			0.583460	−	0.812142i	$-0.301699\pi$
$47$	8.00000			1.16692			0.583460	+	0.812142i	$0.301699\pi$
$48$		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	$-0.682474\pi$
$61$	−8.47214			−1.08475			−0.542373	+	0.840138i	$0.682474\pi$
$62$		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	$-0.927267\pi$
$67$		−	3.70820i		−	0.453029i	0.974008	−	0.226515i	$-0.0727331\pi$
$68$		0			0
$69$	−10.4721	+	4.00000i	−1.26070	+	0.481543i
$70$		0			0
$71$	−11.4164			−1.35488			−0.677439	−	0.735579i	$-0.736910\pi$
$71$	−11.4164			−1.35488			−0.677439	+	0.735579i	$0.736910\pi$
$72$		0			0
$73$	−2.00000			−0.234082			−0.117041	−	0.993127i	$-0.537341\pi$
$73$	−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	$0.0679975\pi$
$89$			4.00000i			0.423999i	−0.977270	+	0.212000i	$0.932002\pi$
$90$		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	$-0.507198\pi$
$109$	−0.472136			−0.0452224			−0.0226112	+	0.999744i	$0.507198\pi$
$110$		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
$115$			8.00000i			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	$0.147631\pi$
$137$			10.4721i			0.894695i	−0.894360	+	0.447347i	$0.852369\pi$
$138$		0			0
$139$		−	3.70820i		−	0.314526i	−0.987557	−	0.157263i	$-0.949733\pi$
$139$		−	3.70820i		−	0.314526i	0.987557	−	0.157263i	$-0.0502670\pi$
$140$		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	$-0.442889\pi$
$157$	4.47214			0.356915			0.178458	+	0.983948i	$0.442889\pi$
$158$		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	$0.252209\pi$
$163$			18.1803i			1.42399i	−0.702182	+	0.711997i	$0.747791\pi$
$164$		0			0
$165$	−0.583592	−	1.52786i	−0.0454326	−	0.118944i
$166$		0			0
$167$	−22.4721			−1.73895			−0.869473	−	0.493980i	$-0.835541\pi$
$167$	−22.4721			−1.73895			−0.869473	+	0.493980i	$0.835541\pi$
$168$		0			0
$169$	7.00000			0.538462
$170$		0			0
$171$	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	$-0.640933\pi$
$181$	−11.5279			−0.856859			−0.428430	+	0.903575i	$0.640933\pi$
$182$		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	$-0.557247\pi$
$191$	−4.94427			−0.357755			−0.178877	+	0.983871i	$0.557247\pi$
$192$		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	$-0.925203\pi$
$211$		−	6.76393i		−	0.465648i	0.972519	−	0.232824i	$-0.0747967\pi$
$212$		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	$-0.263531\pi$
$229$	20.4721			1.35284			0.676418	+	0.736518i	$0.263531\pi$
$230$		0			0
$231$	1.52786	+	4.00000i	0.100526	+	0.263181i
$232$		0			0
$233$		−	7.05573i		−	0.462236i	−0.972926	−	0.231118i	$-0.925762\pi$
$233$		−	7.05573i		−	0.462236i	0.972926	−	0.231118i	$-0.0742384\pi$
$234$		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	$-0.583307\pi$
$239$	−8.00000			−0.517477			−0.258738	+	0.965947i	$0.583307\pi$
$240$		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	$-0.969617\pi$
$257$		−	3.05573i		−	0.190611i	0.995448	−	0.0953055i	$-0.0303828\pi$
$258$		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	$-0.515000\pi$
$263$	−1.52786			−0.0942121			−0.0471061	+	0.998890i	$0.515000\pi$
$264$		0			0
$265$	−1.52786			−0.0938559
$266$		0			0
$267$	−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	$-0.664780\pi$
$277$	−16.4721			−0.989715			−0.494857	+	0.868974i	$0.664780\pi$
$278$		0			0
$279$	1.52786	+	1.70820i	0.0914708	+	0.102267i
$280$		0			0
$281$			7.05573i			0.420909i	0.977604	+	0.210455i	$0.0674945\pi$
$281$			7.05573i			0.420909i	−0.977604	+	0.210455i	$0.932506\pi$
$282$		0			0
$283$			25.2361i			1.50013i	0.661365	+	0.750064i	$0.269977\pi$
$283$			25.2361i			1.50013i	−0.661365	+	0.750064i	$0.730023\pi$
$284$		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	$-0.788633\pi$
$307$		−	21.5967i		−	1.23259i	0.787515	−	0.616296i	$-0.211367\pi$
$308$		0			0
$309$	5.05573	+	13.2361i	0.287610	+	0.752974i
$310$		0			0
$311$	−14.4721			−0.820640			−0.410320	−	0.911942i	$-0.634583\pi$
$311$	−14.4721			−0.820640			−0.410320	+	0.911942i	$0.634583\pi$
$312$		0			0
$313$	8.47214			0.478873			0.239437	−	0.970912i	$-0.423037\pi$
$313$	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	$-0.601821\pi$
$331$		−	34.5410i		−	1.89855i	0.314453	−	0.949273i	$-0.398179\pi$
$332$		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	$-0.675057\pi$
$349$	−19.5279			−1.04530			−0.522651	+	0.852547i	$0.675057\pi$
$350$		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
$355$		−	14.1115i		−	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	$-0.461404\pi$
$359$	4.58359			0.241913			0.120956	+	0.992658i	$0.461404\pi$
$360$		0			0
$361$	−8.41641			−0.442969
$362$		0			0
$363$	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	$-0.775567\pi$
$373$	−29.4164			−1.52312			−0.761562	+	0.648092i	$0.775567\pi$
$374$		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	$-0.822033\pi$
$379$		−	20.6525i		−	1.06085i	0.847733	−	0.530423i	$-0.177967\pi$
$380$		0			0
$381$	−5.41641	−	14.1803i	−0.277491	−	0.726481i
$382$		0			0
$383$	−11.0557			−0.564921			−0.282461	−	0.959279i	$-0.591151\pi$
$383$	−11.0557			−0.564921			−0.282461	+	0.959279i	$0.591151\pi$
$384$		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	$-0.424066\pi$
$397$	9.41641			0.472596			0.236298	+	0.971681i	$0.424066\pi$
$398$		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
$403$			3.41641i			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	$-0.542137\pi$
$421$	−5.41641			−0.263980			−0.131990	+	0.991251i	$0.542137\pi$
$422$		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	$-0.561716\pi$
$431$	−8.00000			−0.385346			−0.192673	+	0.981263i	$0.561716\pi$
$432$		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	$0.223947\pi$
$449$			27.4164i			1.29386i	−0.762549	+	0.646930i	$0.776053\pi$
$450$		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	$-0.0937194\pi$
$479$	41.8885			1.91394			0.956968	+	0.290193i	$0.0937194\pi$
$480$		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	$-0.829820\pi$
$499$		−	22.7639i		−	1.01905i	0.860455	−	0.509527i	$-0.170180\pi$
$500$		0			0
$501$	36.3607	−	13.8885i	1.62448	−	0.620494i
$502$		0			0
$503$	−30.4721			−1.35869			−0.679343	−	0.733821i	$-0.737735\pi$
$503$	−30.4721			−1.35869			−0.679343	+	0.733821i	$0.737735\pi$
$504$		0			0
$505$	14.4721			0.644002
$506$		0			0
$507$	−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	$0.147155\pi$
$521$			20.3607i			0.892018i	−0.895029	+	0.446009i	$0.852845\pi$
$522$		0			0
$523$			34.1803i			1.49460i	0.664486	+	0.747301i	$0.268651\pi$
$523$			34.1803i			1.49460i	−0.664486	+	0.747301i	$0.731349\pi$
$524$		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	$-0.629149\pi$
$541$	−18.3607			−0.789387			−0.394694	+	0.918813i	$0.629149\pi$
$542$		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	$-0.981181\pi$
$547$		−	2.76393i		−	0.118177i	0.998253	−	0.0590886i	$-0.0188194\pi$
$548$		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	$0.288173\pi$
$569$			37.5279i			1.57325i	−0.617431	+	0.786625i	$0.711827\pi$
$570$		0			0
$571$			35.1246i			1.46992i	0.678111	+	0.734960i	$0.262799\pi$
$571$			35.1246i			1.46992i	−0.678111	+	0.734960i	$0.737201\pi$
$572$		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	$-0.927108\pi$
$593$		−	11.0557i		−	0.454004i	0.973894	−	0.227002i	$-0.0728924\pi$
$594$		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	$-0.629833\pi$
$599$	−19.4164			−0.793333			−0.396666	+	0.917963i	$0.629833\pi$
$600$		0			0
$601$	−37.7771			−1.54096			−0.770480	−	0.637464i	$-0.779983\pi$
$601$	−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	$-0.161167\pi$
$613$	43.3050			1.74907			0.874535	+	0.484962i	$0.161167\pi$
$614$		0			0
$615$	−4.94427	+	1.88854i	−0.199372	+	0.0761534i
$616$		0			0
$617$		−	13.8885i		−	0.559132i	−0.960127	−	0.279566i	$-0.909809\pi$
$617$		−	13.8885i		−	0.559132i	0.960127	−	0.279566i	$-0.0901905\pi$
$618$		0			0
$619$			11.1246i			0.447136i	0.974688	+	0.223568i	$0.0717705\pi$
$619$			11.1246i			0.447136i	−0.974688	+	0.223568i	$0.928230\pi$
$620$		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	$-0.736370\pi$
$641$		−	37.3050i		−	1.47346i	0.676189	−	0.736729i	$-0.263630\pi$
$642$		0			0
$643$		−	44.6525i		−	1.76092i	−0.474119	−	0.880461i	$-0.657233\pi$
$643$		−	44.6525i		−	1.76092i	0.474119	−	0.880461i	$-0.342767\pi$
$644$		0			0
$645$	−5.52786	+	2.11146i	−0.217659	+	0.0831385i
$646$		0			0
$647$	29.3050			1.15210			0.576048	−	0.817416i	$-0.304594\pi$
$647$	29.3050			1.15210			0.576048	+	0.817416i	$0.304594\pi$
$648$		0			0
$649$	2.47214			0.0970398
$650$		0			0
$651$	−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	$-0.429441\pi$
$661$	11.3050			0.439712			0.219856	+	0.975532i	$0.429441\pi$
$662$		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	$0.100290\pi$
$691$			16.2918i			0.619769i	−0.950774	+	0.309885i	$0.899710\pi$
$692$		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	$-0.600099\pi$
$709$	−16.4721			−0.618624			−0.309312	+	0.950961i	$0.600099\pi$
$710$		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	$-0.639993\pi$
$719$	−22.8328			−0.851520			−0.425760	+	0.904836i	$0.639993\pi$
$720$		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	$-0.484807\pi$
$733$	2.58359			0.0954272			0.0477136	+	0.998861i	$0.484807\pi$
$734$		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	$0.166262\pi$
$739$			27.1246i			0.997795i	−0.866661	+	0.498897i	$0.833738\pi$
$740$		0			0
$741$	−14.4721	−	37.8885i	−0.531647	−	1.39187i
$742$		0			0
$743$	19.4164			0.712319			0.356159	−	0.934425i	$-0.384086\pi$
$743$	19.4164			0.712319			0.356159	+	0.934425i	$0.384086\pi$
$744$		0			0
$745$	4.58359			0.167930
$746$		0			0
$747$	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	$-0.627247\pi$
$757$	−21.4164			−0.778393			−0.389196	+	0.921155i	$0.627247\pi$
$758$		0			0
$759$	−8.00000	+	3.05573i	−0.290382	+	0.110916i
$760$		0			0
$761$			38.2492i			1.38653i	0.720681	+	0.693267i	$0.243829\pi$
$761$			38.2492i			1.38653i	−0.720681	+	0.693267i	$0.756171\pi$
$762$		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	$-0.966924\pi$
$787$		−	5.81966i		−	0.207448i	0.994606	−	0.103724i	$-0.0330759\pi$
$788$		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	$0.135043\pi$
$809$			23.4164i			0.823277i	−0.911347	+	0.411639i	$0.864957\pi$
$810$		0			0
$811$		−	29.8197i		−	1.04711i	−0.851992	−	0.523555i	$-0.824605\pi$
$811$		−	29.8197i		−	1.04711i	0.851992	−	0.523555i	$-0.175395\pi$
$812$		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	$-0.475255\pi$
$829$	4.47214			0.155324			0.0776619	+	0.996980i	$0.475255\pi$
$830$		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	$-0.474788\pi$
$839$	4.58359			0.158243			0.0791216	+	0.996865i	$0.474788\pi$
$840$		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	$-0.630633\pi$
$853$	−23.3050			−0.797946			−0.398973	+	0.916963i	$0.630633\pi$
$854$		0			0
$855$	−14.4721	+	12.9443i	−0.494937	+	0.442685i
$856$		0			0
$857$		−	52.3607i		−	1.78861i	−0.447461	−	0.894303i	$-0.647672\pi$
$857$		−	52.3607i		−	1.78861i	0.447461	−	0.894303i	$-0.352328\pi$
$858$		0			0
$859$			21.2361i			0.724565i	0.932068	+	0.362283i	$0.118002\pi$
$859$			21.2361i			0.724565i	−0.932068	+	0.362283i	$0.881998\pi$
$860$		0			0
$861$	12.9443	−	4.94427i	0.441140	−	0.168500i
$862$		0			0
$863$	−30.8328			−1.04956			−0.524781	−	0.851238i	$-0.675853\pi$
$863$	−30.8328			−1.04956			−0.524781	+	0.851238i	$0.675853\pi$
$864$		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	$-0.778153\pi$
$877$	−45.4164			−1.53360			−0.766802	+	0.641884i	$0.778153\pi$
$878$		0			0
$879$	−15.5967	−	40.8328i	−0.526065	−	1.37726i
$880$		0			0
$881$		−	24.0000i		−	0.808581i	−0.914631	−	0.404290i	$-0.867519\pi$
$881$		−	24.0000i		−	0.808581i	0.914631	−	0.404290i	$-0.132481\pi$
$882$		0			0
$883$			5.23607i			0.176208i	0.996111	+	0.0881039i	$0.0280807\pi$
$883$			5.23607i			0.176208i	−0.996111	+	0.0881039i	$0.971919\pi$
$884$		0			0
$885$	−2.47214	−	6.47214i	−0.0830999	−	0.217558i
$886$		0			0
$887$	−24.3607			−0.817952			−0.408976	−	0.912545i	$-0.634114\pi$
$887$	−24.3607			−0.817952			−0.408976	+	0.912545i	$0.634114\pi$
$888$		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	$-0.804161\pi$
$907$		−	34.7639i		−	1.15432i	0.816632	−	0.577159i	$-0.195839\pi$
$908$		0			0
$909$	−23.4164	−	26.1803i	−0.776673	−	0.868347i
$910$		0			0
$911$	1.88854			0.0625702			0.0312851	−	0.999511i	$-0.490040\pi$
$911$	1.88854			0.0625702			0.0312851	+	0.999511i	$0.490040\pi$
$912$		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	$0.109636\pi$
$929$			20.5836i			0.675326i	−0.941267	+	0.337663i	$0.890364\pi$
$930$		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	$-0.893028\pi$
$953$		−	20.3607i		−	0.659547i	0.944060	−	0.329774i	$-0.106972\pi$
$954$		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	$-0.919512\pi$
$977$		−	15.6393i		−	0.500346i	0.968201	−	0.250173i	$-0.0804875\pi$
$978$		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	$-0.661528\pi$
$983$	−30.4721			−0.971910			−0.485955	+	0.873984i	$0.661528\pi$
$984$		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	$-0.637068\pi$
$997$	−26.3607			−0.834851			−0.417426	+	0.908711i	$0.637068\pi$
$998$		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.a.383.2	yes	4
3.2	odd	2		384.2.c.d.383.4	yes	4
4.3	odd	2		384.2.c.d.383.3	yes	4
8.3	odd	2		384.2.c.b.383.2	yes	4
8.5	even	2		384.2.c.c.383.3	yes	4
12.11	even	2	inner	384.2.c.a.383.1	✓	4
16.3	odd	4		768.2.f.b.383.4		4
16.5	even	4		768.2.f.c.383.4		4
16.11	odd	4		768.2.f.f.383.1		4
16.13	even	4		768.2.f.e.383.1		4
24.5	odd	2		384.2.c.b.383.1	yes	4
24.11	even	2		384.2.c.c.383.4	yes	4
48.5	odd	4		768.2.f.b.383.3		4
48.11	even	4		768.2.f.e.383.2		4
48.29	odd	4		768.2.f.f.383.2		4
48.35	even	4		768.2.f.c.383.3		4

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

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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.a.383.2

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