LMFDB - Embedded newform 2112.3.j.j.769.24

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2112,3,Mod(769,2112)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2112, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2112.769");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$N$	$=$	$2112 = 2^{6} \cdot 3 \cdot 11$
Weight:	$k$	$=$	$3$
Character orbit:	$[\chi]$	$=$	2112.j (of order $2$ , degree $1$ , not minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$57.5478318329$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	no (minimal twist has level 1056)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			769.24
Character	$\chi$	$=$	2112.769
Dual form			2112.3.j.j.769.13

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.73205 q^{3} -0.740283 q^{5} -5.81384i q^{7} +3.00000 q^{9} +(-10.7767 - 2.20496i) q^{11} +25.1602i q^{13} -1.28221 q^{15} +5.95546i q^{17} -23.4509i q^{19} -10.0699i q^{21} +30.8910 q^{23} -24.4520 q^{25} +5.19615 q^{27} +11.5273i q^{29} -25.7324 q^{31} +(-18.6659 - 3.81910i) q^{33} +4.30389i q^{35} +15.8320 q^{37} +43.5787i q^{39} -60.6729i q^{41} -17.2796i q^{43} -2.22085 q^{45} -59.5226 q^{47} +15.1993 q^{49} +10.3152i q^{51} -83.7233 q^{53} +(7.97784 + 1.63230i) q^{55} -40.6182i q^{57} -4.10146 q^{59} +47.0527i q^{61} -17.4415i q^{63} -18.6256i q^{65} -99.9412 q^{67} +53.5048 q^{69} -85.7022 q^{71} -61.1241i q^{73} -42.3521 q^{75} +(-12.8193 + 62.6543i) q^{77} -54.6537i q^{79} +9.00000 q^{81} +122.674i q^{83} -4.40873i q^{85} +19.9658i q^{87} -156.036 q^{89} +146.277 q^{91} -44.5699 q^{93} +17.3603i q^{95} -111.167 q^{97} +(-32.3302 - 6.61488i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$24 q + 72 q^{9} + 200 q^{25} - 24 q^{33} - 160 q^{37} - 24 q^{49} + 32 q^{53} - 96 q^{69} + 160 q^{77} + 216 q^{81} + 368 q^{89} + 96 q^{93} - 352 q^{97}+O(q^{100})$ 24 * q + 72 * q^9 + 200 * q^25 - 24 * q^33 - 160 * q^37 - 24 * q^49 + 32 * q^53 - 96 * q^69 + 160 * q^77 + 216 * q^81 + 368 * q^89 + 96 * q^93 - 352 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2112\mathbb{Z}\right)^\times$ .

$n$	$133$	$1409$	$1729$	$2047$
$\chi(n)$	$1$	$1$	$-1$	$1$

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
$3$	1.73205			0.577350
$3$	1.73205			0.577350
$4$		0			0
$5$	−0.740283			−0.148057			−0.0740283	−	0.997256i	$-0.523586\pi$
$5$	−0.740283			−0.148057			−0.0740283	+	0.997256i	$0.523586\pi$
$6$		0			0
$7$		−	5.81384i		−	0.830549i	−0.909696	−	0.415274i	$-0.863686\pi$
$7$		−	5.81384i		−	0.830549i	0.909696	−	0.415274i	$-0.136314\pi$
$8$		0			0
$9$	3.00000			0.333333
$10$		0			0
$11$	−10.7767	−	2.20496i	−0.979704	−	0.200451i
$11$	−10.7767	−	2.20496i	−0.979704	−	0.200451i
$12$		0			0
$13$			25.1602i			1.93540i	0.252110	+	0.967699i	$0.418876\pi$
$13$			25.1602i			1.93540i	−0.252110	+	0.967699i	$0.581124\pi$
$14$		0			0
$15$	−1.28221			−0.0854805
$16$		0			0
$17$			5.95546i			0.350321i	0.984540	+	0.175161i	$0.0560444\pi$
$17$			5.95546i			0.350321i	−0.984540	+	0.175161i	$0.943956\pi$
$18$		0			0
$19$		−	23.4509i		−	1.23426i	−0.786861	−	0.617130i	$-0.788295\pi$
$19$		−	23.4509i		−	1.23426i	0.786861	−	0.617130i	$-0.211705\pi$
$20$		0			0
$21$		−	10.0699i		−	0.479517i
$22$		0			0
$23$	30.8910			1.34309			0.671544	−	0.740965i	$-0.265632\pi$
$23$	30.8910			1.34309			0.671544	+	0.740965i	$0.265632\pi$
$24$		0			0
$25$	−24.4520			−0.978079
$26$		0			0
$27$	5.19615			0.192450
$28$		0			0
$29$			11.5273i			0.397492i	0.980051	+	0.198746i	$0.0636869\pi$
$29$			11.5273i			0.397492i	−0.980051	+	0.198746i	$0.936313\pi$
$30$		0			0
$31$	−25.7324			−0.830078			−0.415039	−	0.909804i	$-0.636232\pi$
$31$	−25.7324			−0.830078			−0.415039	+	0.909804i	$0.636232\pi$
$32$		0			0
$33$	−18.6659	−	3.81910i	−0.565632	−	0.115730i
$34$		0			0
$35$			4.30389i			0.122968i
$36$		0			0
$37$	15.8320			0.427891			0.213945	−	0.976846i	$-0.431369\pi$
$37$	15.8320			0.427891			0.213945	+	0.976846i	$0.431369\pi$
$38$		0			0
$39$			43.5787i			1.11740i
$40$		0			0
$41$		−	60.6729i		−	1.47983i	−0.672702	−	0.739913i	$-0.734866\pi$
$41$		−	60.6729i		−	1.47983i	0.672702	−	0.739913i	$-0.265134\pi$
$42$		0			0
$43$		−	17.2796i		−	0.401851i	−0.979607	−	0.200926i	$-0.935605\pi$
$43$		−	17.2796i		−	0.401851i	0.979607	−	0.200926i	$-0.0643949\pi$
$44$		0			0
$45$	−2.22085			−0.0493522
$46$		0			0
$47$	−59.5226			−1.26644			−0.633220	−	0.773972i	$-0.718267\pi$
$47$	−59.5226			−1.26644			−0.633220	+	0.773972i	$0.718267\pi$
$48$		0			0
$49$	15.1993			0.310189
$50$		0			0
$51$			10.3152i			0.202258i
$52$		0			0
$53$	−83.7233			−1.57968			−0.789842	−	0.613310i	$-0.789837\pi$
$53$	−83.7233			−1.57968			−0.789842	+	0.613310i	$0.789837\pi$
$54$		0			0
$55$	7.97784	+	1.63230i	0.145052	+	0.0296781i
$56$		0			0
$57$		−	40.6182i		−	0.712600i
$58$		0			0
$59$	−4.10146			−0.0695163			−0.0347581	−	0.999396i	$-0.511066\pi$
$59$	−4.10146			−0.0695163			−0.0347581	+	0.999396i	$0.511066\pi$
$60$		0			0
$61$			47.0527i			0.771356i	0.922634	+	0.385678i	$0.126032\pi$
$61$			47.0527i			0.771356i	−0.922634	+	0.385678i	$0.873968\pi$
$62$		0			0
$63$		−	17.4415i		−	0.276850i
$64$		0			0
$65$		−	18.6256i		−	0.286548i
$66$		0			0
$67$	−99.9412			−1.49166			−0.745830	−	0.666137i	$-0.767947\pi$
$67$	−99.9412			−1.49166			−0.745830	+	0.666137i	$0.767947\pi$
$68$		0			0
$69$	53.5048			0.775432
$70$		0			0
$71$	−85.7022			−1.20707			−0.603537	−	0.797335i	$-0.706242\pi$
$71$	−85.7022			−1.20707			−0.603537	+	0.797335i	$0.706242\pi$
$72$		0			0
$73$		−	61.1241i		−	0.837317i	−0.908144	−	0.418658i	$-0.862501\pi$
$73$		−	61.1241i		−	0.837317i	0.908144	−	0.418658i	$-0.137499\pi$
$74$		0			0
$75$	−42.3521			−0.564694
$76$		0			0
$77$	−12.8193	+	62.6543i	−0.166484	+	0.813692i
$78$		0			0
$79$		−	54.6537i		−	0.691818i	−0.938268	−	0.345909i	$-0.887571\pi$
$79$		−	54.6537i		−	0.691818i	0.938268	−	0.345909i	$-0.112429\pi$
$80$		0			0
$81$	9.00000			0.111111
$82$		0			0
$83$			122.674i			1.47800i	0.673704	+	0.739001i	$0.264702\pi$
$83$			122.674i			1.47800i	−0.673704	+	0.739001i	$0.735298\pi$
$84$		0			0
$85$		−	4.40873i		−	0.0518674i
$86$		0			0
$87$			19.9658i			0.229492i
$88$		0			0
$89$	−156.036			−1.75321			−0.876605	−	0.481210i	$-0.840197\pi$
$89$	−156.036			−1.75321			−0.876605	+	0.481210i	$0.840197\pi$
$90$		0			0
$91$	146.277			1.60744
$92$		0			0
$93$	−44.5699			−0.479246
$94$		0			0
$95$			17.3603i			0.182740i
$96$		0			0
$97$	−111.167			−1.14605			−0.573026	−	0.819537i	$-0.694230\pi$
$97$	−111.167			−1.14605			−0.573026	+	0.819537i	$0.694230\pi$
$98$		0			0
$99$	−32.3302	−	6.61488i	−0.326568	−	0.0668170i
$100$		0			0
$101$			109.931i			1.08842i	0.838949	+	0.544211i	$0.183171\pi$
$101$			109.931i			1.08842i	−0.838949	+	0.544211i	$0.816829\pi$
$102$		0			0
$103$	133.186			1.29307			0.646533	−	0.762886i	$-0.276218\pi$
$103$	133.186			1.29307			0.646533	+	0.762886i	$0.276218\pi$
$104$		0			0
$105$			7.45455i			0.0709958i
$106$		0			0
$107$		−	16.9391i		−	0.158309i	−0.996862	−	0.0791546i	$-0.974778\pi$
$107$		−	16.9391i		−	0.158309i	0.996862	−	0.0791546i	$-0.0252221\pi$
$108$		0			0
$109$			155.107i			1.42300i	0.702686	+	0.711500i	$0.251984\pi$
$109$			155.107i			1.42300i	−0.702686	+	0.711500i	$0.748016\pi$
$110$		0			0
$111$	27.4218			0.247043
$112$		0			0
$113$	−127.523			−1.12852			−0.564261	−	0.825596i	$-0.690839\pi$
$113$	−127.523			−1.12852			−0.564261	+	0.825596i	$0.690839\pi$
$114$		0			0
$115$	−22.8681			−0.198853
$116$		0			0
$117$			75.4805i			0.645132i
$118$		0			0
$119$	34.6241			0.290959
$120$		0			0
$121$	111.276	+	47.5246i	0.919639	+	0.392765i
$122$		0			0
$123$		−	105.089i		−	0.854378i
$124$		0			0
$125$	36.6085			0.292868
$126$		0			0
$127$			213.652i			1.68230i	0.540803	+	0.841150i	$0.318121\pi$
$127$			213.652i			1.68230i	−0.540803	+	0.841150i	$0.681879\pi$
$128$		0			0
$129$		−	29.9291i		−	0.232009i
$130$		0			0
$131$			147.173i			1.12346i	0.827321	+	0.561729i	$0.189864\pi$
$131$			147.173i			1.12346i	−0.827321	+	0.561729i	$0.810136\pi$
$132$		0			0
$133$	−136.340			−1.02511
$134$		0			0
$135$	−3.84662			−0.0284935
$136$		0			0
$137$	67.5490			0.493059			0.246529	−	0.969135i	$-0.420710\pi$
$137$	67.5490			0.493059			0.246529	+	0.969135i	$0.420710\pi$
$138$		0			0
$139$			82.2145i			0.591471i	0.955270	+	0.295736i	$0.0955648\pi$
$139$			82.2145i			0.591471i	−0.955270	+	0.295736i	$0.904435\pi$
$140$		0			0
$141$	−103.096			−0.731179
$142$		0			0
$143$	55.4772	−	271.145i	0.387952	−	1.89612i
$144$		0			0
$145$		−	8.53345i		−	0.0588514i
$146$		0			0
$147$	26.3259			0.179088
$148$		0			0
$149$			114.431i			0.767991i	0.923335	+	0.383995i	$0.125452\pi$
$149$			114.431i			0.767991i	−0.923335	+	0.383995i	$0.874548\pi$
$150$		0			0
$151$		−	37.0134i		−	0.245122i	−0.992461	−	0.122561i	$-0.960889\pi$
$151$		−	37.0134i		−	0.245122i	0.992461	−	0.122561i	$-0.0391106\pi$
$152$		0			0
$153$			17.8664i			0.116774i
$154$		0			0
$155$	19.0493			0.122899
$156$		0			0
$157$	−227.242			−1.44740			−0.723700	−	0.690114i	$-0.757560\pi$
$157$	−227.242			−1.44740			−0.723700	+	0.690114i	$0.757560\pi$
$158$		0			0
$159$	−145.013			−0.912031
$160$		0			0
$161$		−	179.595i		−	1.11550i
$162$		0			0
$163$	194.789			1.19503			0.597514	−	0.801859i	$-0.296155\pi$
$163$	194.789			1.19503			0.597514	+	0.801859i	$0.296155\pi$
$164$		0			0
$165$	13.8180	+	2.82722i	0.0837456	+	0.0171347i
$166$		0			0
$167$		−	203.014i		−	1.21565i	−0.794069	−	0.607827i	$-0.792041\pi$
$167$		−	203.014i		−	1.21565i	0.794069	−	0.607827i	$-0.207959\pi$
$168$		0			0
$169$	−464.034			−2.74576
$170$		0			0
$171$		−	70.3528i		−	0.411420i
$172$		0			0
$173$		−	243.217i		−	1.40588i	−0.711251	−	0.702938i	$-0.751871\pi$
$173$		−	243.217i		−	1.40588i	0.711251	−	0.702938i	$-0.248129\pi$
$174$		0			0
$175$			142.160i			0.812342i
$176$		0			0
$177$	−7.10394			−0.0401352
$178$		0			0
$179$	−148.593			−0.830131			−0.415065	−	0.909792i	$-0.636241\pi$
$179$	−148.593			−0.830131			−0.415065	+	0.909792i	$0.636241\pi$
$180$		0			0
$181$	−47.5814			−0.262881			−0.131440	−	0.991324i	$-0.541960\pi$
$181$	−47.5814			−0.262881			−0.131440	+	0.991324i	$0.541960\pi$
$182$		0			0
$183$			81.4976i			0.445342i
$184$		0			0
$185$	−11.7201			−0.0633521
$186$		0			0
$187$	13.1316	−	64.1804i	0.0702222	−	0.343211i
$188$		0			0
$189$		−	30.2096i		−	0.159839i
$190$		0			0
$191$	−139.935			−0.732647			−0.366323	−	0.930488i	$-0.619384\pi$
$191$	−139.935			−0.732647			−0.366323	+	0.930488i	$0.619384\pi$
$192$		0			0
$193$		−	306.144i		−	1.58624i	−0.609065	−	0.793120i	$-0.708455\pi$
$193$		−	306.144i		−	1.58624i	0.609065	−	0.793120i	$-0.291545\pi$
$194$		0			0
$195$		−	32.2606i		−	0.165439i
$196$		0			0
$197$		−	22.3476i		−	0.113439i	−0.998390	−	0.0567197i	$-0.981936\pi$
$197$		−	22.3476i		−	0.113439i	0.998390	−	0.0567197i	$-0.0180641\pi$
$198$		0			0
$199$	−189.395			−0.951734			−0.475867	−	0.879517i	$-0.657866\pi$
$199$	−189.395			−0.951734			−0.475867	+	0.879517i	$0.657866\pi$
$200$		0			0
$201$	−173.103			−0.861210
$202$		0			0
$203$	67.0178			0.330137
$204$		0			0
$205$			44.9151i			0.219098i
$206$		0			0
$207$	92.6730			0.447696
$208$		0			0
$209$	−51.7084	+	252.725i	−0.247408	+	1.20921i
$210$		0			0
$211$		−	280.132i		−	1.32764i	−0.747892	−	0.663821i	$-0.768934\pi$
$211$		−	280.132i		−	1.32764i	0.747892	−	0.663821i	$-0.231066\pi$
$212$		0			0
$213$	−148.441			−0.696904
$214$		0			0
$215$			12.7918i			0.0594967i
$216$		0			0
$217$			149.604i			0.689420i
$218$		0			0
$219$		−	105.870i		−	0.483425i
$220$		0			0
$221$	−149.840			−0.678010
$222$		0			0
$223$	258.461			1.15902			0.579510	−	0.814965i	$-0.303244\pi$
$223$	258.461			1.15902			0.579510	+	0.814965i	$0.303244\pi$
$224$		0			0
$225$	−73.3559			−0.326026
$226$		0			0
$227$			38.7456i			0.170685i	0.996352	+	0.0853427i	$0.0271985\pi$
$227$			38.7456i			0.170685i	−0.996352	+	0.0853427i	$0.972801\pi$
$228$		0			0
$229$	308.466			1.34701			0.673507	−	0.739181i	$-0.264787\pi$
$229$	308.466			1.34701			0.673507	+	0.739181i	$0.264787\pi$
$230$		0			0
$231$	−22.2037	+	108.520i	−0.0961197	+	0.469785i
$232$		0			0
$233$		−	90.7893i		−	0.389654i	−0.980838	−	0.194827i	$-0.937586\pi$
$233$		−	90.7893i		−	0.389654i	0.980838	−	0.194827i	$-0.0624145\pi$
$234$		0			0
$235$	44.0636			0.187505
$236$		0			0
$237$		−	94.6629i		−	0.399422i
$238$		0			0
$239$		−	290.303i		−	1.21466i	−0.794451	−	0.607329i	$-0.792241\pi$
$239$		−	290.303i		−	1.21466i	0.794451	−	0.607329i	$-0.207759\pi$
$240$		0			0
$241$			318.014i			1.31956i	0.751458	+	0.659781i	$0.229351\pi$
$241$			318.014i			1.31956i	−0.751458	+	0.659781i	$0.770649\pi$
$242$		0			0
$243$	15.5885			0.0641500
$244$		0			0
$245$	−11.2518			−0.0459255
$246$		0			0
$247$	590.029			2.38878
$248$		0			0
$249$			212.478i			0.853325i
$250$		0			0
$251$	−109.561			−0.436498			−0.218249	−	0.975893i	$-0.570035\pi$
$251$	−109.561			−0.436498			−0.218249	+	0.975893i	$0.570035\pi$
$252$		0			0
$253$	−332.904	−	68.1135i	−1.31583	−	0.269223i
$254$		0			0
$255$		−	7.63614i		−	0.0299456i
$256$		0			0
$257$	−7.59616			−0.0295571			−0.0147785	−	0.999891i	$-0.504704\pi$
$257$	−7.59616			−0.0295571			−0.0147785	+	0.999891i	$0.504704\pi$
$258$		0			0
$259$		−	92.0445i		−	0.355384i
$260$		0			0
$261$			34.5818i			0.132497i
$262$		0			0
$263$			247.418i			0.940751i	0.882466	+	0.470376i	$0.155882\pi$
$263$			247.418i			0.940751i	−0.882466	+	0.470376i	$0.844118\pi$
$264$		0			0
$265$	61.9789			0.233883
$266$		0			0
$267$	−270.262			−1.01222
$268$		0			0
$269$	−248.684			−0.924474			−0.462237	−	0.886756i	$-0.652953\pi$
$269$	−248.684			−0.924474			−0.462237	+	0.886756i	$0.652953\pi$
$270$		0			0
$271$			164.791i			0.608084i	0.952659	+	0.304042i	$0.0983363\pi$
$271$			164.791i			0.608084i	−0.952659	+	0.304042i	$0.901664\pi$
$272$		0			0
$273$	253.359			0.928057
$274$		0			0
$275$	263.513	+	53.9157i	0.958228	+	0.196057i
$276$		0			0
$277$			55.1722i			0.199177i	0.995029	+	0.0995887i	$0.0317527\pi$
$277$			55.1722i			0.199177i	−0.995029	+	0.0995887i	$0.968247\pi$
$278$		0			0
$279$	−77.1973			−0.276693
$280$		0			0
$281$			409.691i			1.45797i	0.684527	+	0.728987i	$0.260009\pi$
$281$			409.691i			1.45797i	−0.684527	+	0.728987i	$0.739991\pi$
$282$		0			0
$283$		−	145.590i		−	0.514453i	−0.966351	−	0.257226i	$-0.917191\pi$
$283$		−	145.590i		−	0.514453i	0.966351	−	0.257226i	$-0.0828086\pi$
$284$		0			0
$285$			30.0690i			0.105505i
$286$		0			0
$287$	−352.743			−1.22907
$288$		0			0
$289$	253.533			0.877275
$290$		0			0
$291$	−192.547			−0.661673
$292$		0			0
$293$		−	87.9832i		−	0.300284i	−0.988664	−	0.150142i	$-0.952027\pi$
$293$		−	87.9832i		−	0.300284i	0.988664	−	0.150142i	$-0.0479731\pi$
$294$		0			0
$295$	3.03624			0.0102923
$296$		0			0
$297$	−55.9976	−	11.4573i	−0.188544	−	0.0385768i
$298$		0			0
$299$			777.223i			2.59941i
$300$		0			0
$301$	−100.461			−0.333757
$302$		0			0
$303$			190.405i			0.628400i
$304$		0			0
$305$		−	34.8323i		−	0.114204i
$306$		0			0
$307$		−	239.625i		−	0.780536i	−0.920701	−	0.390268i	$-0.872382\pi$
$307$		−	239.625i		−	0.780536i	0.920701	−	0.390268i	$-0.127618\pi$
$308$		0			0
$309$	230.685			0.746552
$310$		0			0
$311$	−29.8243			−0.0958982			−0.0479491	−	0.998850i	$-0.515269\pi$
$311$	−29.8243			−0.0958982			−0.0479491	+	0.998850i	$0.515269\pi$
$312$		0			0
$313$	−220.768			−0.705329			−0.352665	−	0.935750i	$-0.614724\pi$
$313$	−220.768			−0.705329			−0.352665	+	0.935750i	$0.614724\pi$
$314$		0			0
$315$			12.9117i			0.0409894i
$316$		0			0
$317$	−219.625			−0.692825			−0.346412	−	0.938082i	$-0.612600\pi$
$317$	−219.625			−0.692825			−0.346412	+	0.938082i	$0.612600\pi$
$318$		0			0
$319$	25.4172	−	124.227i	0.0796777	−	0.389425i
$320$		0			0
$321$		−	29.3394i		−	0.0913998i
$322$		0			0
$323$	139.661			0.432387
$324$		0			0
$325$		−	615.216i		−	1.89297i
$326$		0			0
$327$			268.653i			0.821570i
$328$		0			0
$329$			346.055i			1.05184i
$330$		0			0
$331$	419.468			1.26727			0.633637	−	0.773631i	$-0.281561\pi$
$331$	419.468			1.26727			0.633637	+	0.773631i	$0.281561\pi$
$332$		0			0
$333$	47.4959			0.142630
$334$		0			0
$335$	73.9848			0.220850
$336$		0			0
$337$			472.656i			1.40254i	0.712896	+	0.701269i	$0.247383\pi$
$337$			472.656i			1.40254i	−0.712896	+	0.701269i	$0.752617\pi$
$338$		0			0
$339$	−220.876			−0.651552
$340$		0			0
$341$	277.312	+	56.7390i	0.813230	+	0.166390i
$342$		0			0
$343$		−	373.244i		−	1.08818i
$344$		0			0
$345$	−39.6087			−0.114808
$346$		0			0
$347$		−	175.998i		−	0.507200i	−0.967309	−	0.253600i	$-0.918385\pi$
$347$		−	175.998i		−	0.507200i	0.967309	−	0.253600i	$-0.0816147\pi$
$348$		0			0
$349$		−	296.588i		−	0.849822i	−0.905235	−	0.424911i	$-0.860305\pi$
$349$		−	296.588i		−	0.849822i	0.905235	−	0.424911i	$-0.139695\pi$
$350$		0			0
$351$			130.736i			0.372467i
$352$		0			0
$353$	−54.5261			−0.154465			−0.0772324	−	0.997013i	$-0.524608\pi$
$353$	−54.5261			−0.154465			−0.0772324	+	0.997013i	$0.524608\pi$
$354$		0			0
$355$	63.4439			0.178715
$356$		0			0
$357$	59.9707			0.167985
$358$		0			0
$359$		−	628.688i		−	1.75122i	−0.483018	−	0.875611i	$-0.660459\pi$
$359$		−	628.688i		−	1.75122i	0.483018	−	0.875611i	$-0.339541\pi$
$360$		0			0
$361$	−188.946			−0.523396
$362$		0			0
$363$	192.736	+	82.3150i	0.530954	+	0.226763i
$364$		0			0
$365$			45.2492i			0.123970i
$366$		0			0
$367$	296.903			0.808999			0.404500	−	0.914538i	$-0.367446\pi$
$367$	296.903			0.808999			0.404500	+	0.914538i	$0.367446\pi$
$368$		0			0
$369$		−	182.019i		−	0.493276i
$370$		0			0
$371$			486.754i			1.31200i
$372$		0			0
$373$		−	504.228i		−	1.35182i	−0.736985	−	0.675909i	$-0.763751\pi$
$373$		−	504.228i		−	1.35182i	0.736985	−	0.675909i	$-0.236249\pi$
$374$		0			0
$375$	63.4077			0.169087
$376$		0			0
$377$	−290.028			−0.769306
$378$		0			0
$379$	176.680			0.466174			0.233087	−	0.972456i	$-0.425117\pi$
$379$	176.680			0.466174			0.233087	+	0.972456i	$0.425117\pi$
$380$		0			0
$381$			370.056i			0.971276i
$382$		0			0
$383$	−41.7516			−0.109012			−0.0545060	−	0.998513i	$-0.517358\pi$
$383$	−41.7516			−0.109012			−0.0545060	+	0.998513i	$0.517358\pi$
$384$		0			0
$385$	9.48991	−	46.3819i	0.0246491	−	0.120472i
$386$		0			0
$387$		−	51.8388i		−	0.133950i
$388$		0			0
$389$	−535.501			−1.37661			−0.688304	−	0.725422i	$-0.741645\pi$
$389$	−535.501			−1.37661			−0.688304	+	0.725422i	$0.741645\pi$
$390$		0			0
$391$			183.970i			0.470512i
$392$		0			0
$393$			254.911i			0.648629i
$394$		0			0
$395$			40.4592i			0.102428i
$396$		0			0
$397$	80.5944			0.203009			0.101504	−	0.994835i	$-0.467634\pi$
$397$	80.5944			0.203009			0.101504	+	0.994835i	$0.467634\pi$
$398$		0			0
$399$	−236.148			−0.591849
$400$		0			0
$401$	195.742			0.488135			0.244067	−	0.969758i	$-0.421518\pi$
$401$	195.742			0.488135			0.244067	+	0.969758i	$0.421518\pi$
$402$		0			0
$403$		−	647.432i		−	1.60653i
$404$		0			0
$405$	−6.66255			−0.0164507
$406$		0			0
$407$	−170.617	−	34.9089i	−0.419206	−	0.0857711i
$408$		0			0
$409$		−	494.222i		−	1.20837i	−0.796845	−	0.604184i	$-0.793499\pi$
$409$		−	494.222i		−	1.20837i	0.796845	−	0.604184i	$-0.206501\pi$
$410$		0			0
$411$	116.998			0.284668
$412$		0			0
$413$			23.8452i			0.0577367i
$414$		0			0
$415$		−	90.8137i		−	0.218828i
$416$		0			0
$417$			142.400i			0.341486i
$418$		0			0
$419$	−589.299			−1.40644			−0.703221	−	0.710971i	$-0.748256\pi$
$419$	−589.299			−1.40644			−0.703221	+	0.710971i	$0.748256\pi$
$420$		0			0
$421$	−124.809			−0.296459			−0.148230	−	0.988953i	$-0.547357\pi$
$421$	−124.809			−0.296459			−0.148230	+	0.988953i	$0.547357\pi$
$422$		0			0
$423$	−178.568			−0.422146
$424$		0			0
$425$		−	145.623i		−	0.342642i
$426$		0			0
$427$	273.557			0.640648
$428$		0			0
$429$	96.0893	−	469.636i	0.223984	−	1.09472i
$430$		0			0
$431$			283.340i			0.657401i	0.944434	+	0.328701i	$0.106611\pi$
$431$			283.340i			0.657401i	−0.944434	+	0.328701i	$0.893389\pi$
$432$		0			0
$433$	430.103			0.993309			0.496655	−	0.867948i	$-0.334562\pi$
$433$	430.103			0.993309			0.496655	+	0.867948i	$0.334562\pi$
$434$		0			0
$435$		−	14.7804i		−	0.0339779i
$436$		0			0
$437$		−	724.423i		−	1.65772i
$438$		0			0
$439$			208.578i			0.475121i	0.971373	+	0.237561i	$0.0763478\pi$
$439$			208.578i			0.475121i	−0.971373	+	0.237561i	$0.923652\pi$
$440$		0			0
$441$	45.5978			0.103396
$442$		0			0
$443$	−249.349			−0.562864			−0.281432	−	0.959581i	$-0.590809\pi$
$443$	−249.349			−0.562864			−0.281432	+	0.959581i	$0.590809\pi$
$444$		0			0
$445$	115.511			0.259574
$446$		0			0
$447$			198.200i			0.443400i
$448$		0			0
$449$	445.210			0.991560			0.495780	−	0.868448i	$-0.334882\pi$
$449$	445.210			0.991560			0.495780	+	0.868448i	$0.334882\pi$
$450$		0			0
$451$	−133.781	+	653.856i	−0.296633	+	1.44979i
$452$		0			0
$453$		−	64.1090i		−	0.141521i
$454$		0			0
$455$	−108.287			−0.237992
$456$		0			0
$457$		−	166.724i		−	0.364822i	−0.983222	−	0.182411i	$-0.941610\pi$
$457$		−	166.724i		−	0.364822i	0.983222	−	0.182411i	$-0.0583901\pi$
$458$		0			0
$459$			30.9455i			0.0674193i
$460$		0			0
$461$			308.357i			0.668888i	0.942416	+	0.334444i	$0.108549\pi$
$461$			308.357i			0.668888i	−0.942416	+	0.334444i	$0.891451\pi$
$462$		0			0
$463$	−149.012			−0.321840			−0.160920	−	0.986967i	$-0.551446\pi$
$463$	−149.012			−0.321840			−0.160920	+	0.986967i	$0.551446\pi$
$464$		0			0
$465$	32.9943			0.0709555
$466$		0			0
$467$	−189.102			−0.404930			−0.202465	−	0.979290i	$-0.564895\pi$
$467$	−189.102			−0.404930			−0.202465	+	0.979290i	$0.564895\pi$
$468$		0			0
$469$			581.042i			1.23890i
$470$		0			0
$471$	−393.594			−0.835657
$472$		0			0
$473$	−38.1008	+	186.218i	−0.0805515	+	0.393695i
$474$		0			0
$475$			573.422i			1.20720i
$476$		0			0
$477$	−251.170			−0.526561
$478$		0			0
$479$			40.9470i			0.0854843i	0.999086	+	0.0427422i	$0.0136094\pi$
$479$			40.9470i			0.0854843i	−0.999086	+	0.0427422i	$0.986391\pi$
$480$		0			0
$481$			398.335i			0.828139i
$482$		0			0
$483$		−	311.068i		−	0.644034i
$484$		0			0
$485$	82.2951			0.169681
$486$		0			0
$487$	213.967			0.439358			0.219679	−	0.975572i	$-0.429499\pi$
$487$	213.967			0.439358			0.219679	+	0.975572i	$0.429499\pi$
$488$		0			0
$489$	337.385			0.689949
$490$		0			0
$491$			290.091i			0.590817i	0.955371	+	0.295408i	$0.0954557\pi$
$491$			290.091i			0.590817i	−0.955371	+	0.295408i	$0.904544\pi$
$492$		0			0
$493$	−68.6502			−0.139250
$494$		0			0
$495$	23.9335	+	4.89689i	0.0483506	+	0.00989270i
$496$		0			0
$497$			498.259i			1.00253i
$498$		0			0
$499$	−41.3330			−0.0828317			−0.0414159	−	0.999142i	$-0.513187\pi$
$499$	−41.3330			−0.0828317			−0.0414159	+	0.999142i	$0.513187\pi$
$500$		0			0
$501$		−	351.631i		−	0.701858i
$502$		0			0
$503$		−	473.440i		−	0.941233i	−0.882338	−	0.470616i	$-0.844032\pi$
$503$		−	473.440i		−	0.941233i	0.882338	−	0.470616i	$-0.155968\pi$
$504$		0			0
$505$		−	81.3798i		−	0.161148i
$506$		0			0
$507$	−803.730			−1.58527
$508$		0			0
$509$	−153.948			−0.302451			−0.151226	−	0.988499i	$-0.548322\pi$
$509$	−153.948			−0.302451			−0.151226	+	0.988499i	$0.548322\pi$
$510$		0			0
$511$	−355.366			−0.695432
$512$		0			0
$513$		−	121.855i		−	0.237533i
$514$		0			0
$515$	−98.5952			−0.191447
$516$		0			0
$517$	641.460	+	131.245i	1.24074	+	0.253859i
$518$		0			0
$519$		−	421.263i		−	0.811683i
$520$		0			0
$521$	−238.909			−0.458559			−0.229280	−	0.973361i	$-0.573637\pi$
$521$	−238.909			−0.458559			−0.229280	+	0.973361i	$0.573637\pi$
$522$		0			0
$523$		−	1006.06i		−	1.92363i	−0.273705	−	0.961814i	$-0.588249\pi$
$523$		−	1006.06i		−	1.92363i	0.273705	−	0.961814i	$-0.411751\pi$
$524$		0			0
$525$			246.228i			0.469006i
$526$		0			0
$527$		−	153.248i		−	0.290794i
$528$		0			0
$529$	425.254			0.803883
$530$		0			0
$531$	−12.3044			−0.0231721
$532$		0			0
$533$	1526.54			2.86405
$534$		0			0
$535$			12.5397i			0.0234387i
$536$		0			0
$537$	−257.371			−0.479276
$538$		0			0
$539$	−163.798	−	33.5138i	−0.303893	−	0.0621777i
$540$		0			0
$541$			204.776i			0.378514i	0.981928	+	0.189257i	$0.0606079\pi$
$541$			204.776i			0.378514i	−0.981928	+	0.189257i	$0.939392\pi$
$542$		0			0
$543$	−82.4134			−0.151774
$544$		0			0
$545$		−	114.823i		−	0.210685i
$546$		0			0
$547$			1087.33i			1.98781i	0.110236	+	0.993905i	$0.464839\pi$
$547$			1087.33i			1.98781i	−0.110236	+	0.993905i	$0.535161\pi$
$548$		0			0
$549$			141.158i			0.257119i
$550$		0			0
$551$	270.325			0.490609
$552$		0			0
$553$	−317.748			−0.574589
$554$		0			0
$555$	−20.2999			−0.0365763
$556$		0			0
$557$		−	332.563i		−	0.597061i	−0.954400	−	0.298530i	$-0.903504\pi$
$557$		−	332.563i		−	0.597061i	0.954400	−	0.298530i	$-0.0964964\pi$
$558$		0			0
$559$	434.758			0.777742
$560$		0			0
$561$	22.7445	−	111.164i	0.0405428	−	0.198153i
$562$		0			0
$563$			477.131i			0.847479i	0.905784	+	0.423740i	$0.139283\pi$
$563$			477.131i			0.847479i	−0.905784	+	0.423740i	$0.860717\pi$
$564$		0			0
$565$	94.4031			0.167085
$566$		0			0
$567$		−	52.3246i		−	0.0922832i
$568$		0			0
$569$		−	618.296i		−	1.08664i	−0.839527	−	0.543318i	$-0.817168\pi$
$569$		−	618.296i		−	1.08664i	0.839527	−	0.543318i	$-0.182832\pi$
$570$		0			0
$571$		−	1012.13i		−	1.77256i	−0.463147	−	0.886281i	$-0.653280\pi$
$571$		−	1012.13i		−	1.77256i	0.463147	−	0.886281i	$-0.346720\pi$
$572$		0			0
$573$	−242.375			−0.422994
$574$		0			0
$575$	−755.346			−1.31365
$576$		0			0
$577$	392.108			0.679564			0.339782	−	0.940504i	$-0.389647\pi$
$577$	392.108			0.679564			0.339782	+	0.940504i	$0.389647\pi$
$578$		0			0
$579$		−	530.258i		−	0.915816i
$580$		0			0
$581$	713.208			1.22755
$582$		0			0
$583$	902.264	+	184.606i	1.54762	+	0.316649i
$584$		0			0
$585$		−	55.8769i		−	0.0955161i
$586$		0			0
$587$	380.064			0.647468			0.323734	−	0.946148i	$-0.395062\pi$
$587$	380.064			0.647468			0.323734	+	0.946148i	$0.395062\pi$
$588$		0			0
$589$			603.449i			1.02453i
$590$		0			0
$591$		−	38.7071i		−	0.0654943i
$592$		0			0
$593$		−	437.825i		−	0.738323i	−0.929365	−	0.369161i	$-0.879645\pi$
$593$		−	437.825i		−	0.738323i	0.929365	−	0.369161i	$-0.120355\pi$
$594$		0			0
$595$	−25.6316			−0.0430784
$596$		0			0
$597$	−328.042			−0.549484
$598$		0			0
$599$	−644.797			−1.07646			−0.538228	−	0.842800i	$-0.680906\pi$
$599$	−644.797			−1.07646			−0.538228	+	0.842800i	$0.680906\pi$
$600$		0			0
$601$			503.757i			0.838199i	0.907940	+	0.419099i	$0.137654\pi$
$601$			503.757i			0.838199i	−0.907940	+	0.419099i	$0.862346\pi$
$602$		0			0
$603$	−299.824			−0.497220
$604$		0			0
$605$	−82.3760	−	35.1817i	−0.136159	−	0.0581515i
$606$		0			0
$607$			4.22221i			0.00695586i	0.999994	+	0.00347793i	$0.00110706\pi$
$607$			4.22221i			0.00695586i	−0.999994	+	0.00347793i	$0.998893\pi$
$608$		0			0
$609$	116.078			0.190605
$610$		0			0
$611$		−	1497.60i		−	2.45106i
$612$		0			0
$613$			340.694i			0.555781i	0.960613	+	0.277891i	$0.0896353\pi$
$613$			340.694i			0.555781i	−0.960613	+	0.277891i	$0.910365\pi$
$614$		0			0
$615$			77.7953i			0.126496i
$616$		0			0
$617$	121.489			0.196903			0.0984514	−	0.995142i	$-0.468611\pi$
$617$	121.489			0.196903			0.0984514	+	0.995142i	$0.468611\pi$
$618$		0			0
$619$	785.117			1.26836			0.634182	−	0.773184i	$-0.281337\pi$
$619$	785.117			1.26836			0.634182	+	0.773184i	$0.281337\pi$
$620$		0			0
$621$	160.514			0.258477
$622$		0			0
$623$			907.167i			1.45613i
$624$		0			0
$625$	584.199			0.934718
$626$		0			0
$627$	−89.5615	+	437.732i	−0.142841	+	0.698137i
$628$		0			0
$629$			94.2866i			0.149899i
$630$		0			0
$631$	1158.83			1.83649			0.918246	−	0.396010i	$-0.129605\pi$
$631$	1158.83			1.83649			0.918246	+	0.396010i	$0.129605\pi$
$632$		0			0
$633$		−	485.204i		−	0.766514i
$634$		0			0
$635$		−	158.163i		−	0.249076i
$636$		0			0
$637$			382.416i			0.600339i
$638$		0			0
$639$	−257.107			−0.402358
$640$		0			0
$641$	1070.51			1.67007			0.835033	−	0.550200i	$-0.185449\pi$
$641$	1070.51			1.67007			0.835033	+	0.550200i	$0.185449\pi$
$642$		0			0
$643$	1130.61			1.75833			0.879165	−	0.476517i	$-0.158101\pi$
$643$	1130.61			1.75833			0.879165	+	0.476517i	$0.158101\pi$
$644$		0			0
$645$			22.1560i			0.0343505i
$646$		0			0
$647$	−816.041			−1.26127			−0.630635	−	0.776080i	$-0.717205\pi$
$647$	−816.041			−1.26127			−0.630635	+	0.776080i	$0.717205\pi$
$648$		0			0
$649$	44.2004	+	9.04356i	0.0681054	+	0.0139346i
$650$		0			0
$651$			259.122i			0.398037i
$652$		0			0
$653$	20.8470			0.0319249			0.0159625	−	0.999873i	$-0.494919\pi$
$653$	20.8470			0.0319249			0.0159625	+	0.999873i	$0.494919\pi$
$654$		0			0
$655$		−	108.950i		−	0.166335i
$656$		0			0
$657$		−	183.372i		−	0.279106i
$658$		0			0
$659$		−	766.327i		−	1.16286i	−0.813595	−	0.581431i	$-0.802493\pi$
$659$		−	766.327i		−	1.16286i	0.813595	−	0.581431i	$-0.197507\pi$
$660$		0			0
$661$	427.215			0.646316			0.323158	−	0.946345i	$-0.395255\pi$
$661$	427.215			0.646316			0.323158	+	0.946345i	$0.395255\pi$
$662$		0			0
$663$	−259.531			−0.391449
$664$		0			0
$665$	100.930			0.151775
$666$		0			0
$667$			356.089i			0.533867i
$668$		0			0
$669$	447.668			0.669160
$670$		0			0
$671$	103.749	−	507.075i	0.154619	−	0.755700i
$672$		0			0
$673$		−	464.204i		−	0.689753i	−0.938648	−	0.344877i	$-0.887921\pi$
$673$		−	464.204i		−	0.689753i	0.938648	−	0.344877i	$-0.112079\pi$
$674$		0			0
$675$	−127.056			−0.188231
$676$		0			0
$677$			723.452i			1.06861i	0.845290	+	0.534307i	$0.179427\pi$
$677$			723.452i			1.06861i	−0.845290	+	0.534307i	$0.820573\pi$
$678$		0			0
$679$			646.307i			0.951851i
$680$		0			0
$681$			67.1093i			0.0985452i
$682$		0			0
$683$	−745.282			−1.09119			−0.545595	−	0.838049i	$-0.683696\pi$
$683$	−745.282			−1.09119			−0.545595	+	0.838049i	$0.683696\pi$
$684$		0			0
$685$	−50.0054			−0.0730006
$686$		0			0
$687$	534.279			0.777699
$688$		0			0
$689$		−	2106.49i		−	3.05732i
$690$		0			0
$691$	−134.098			−0.194064			−0.0970320	−	0.995281i	$-0.530935\pi$
$691$	−134.098			−0.194064			−0.0970320	+	0.995281i	$0.530935\pi$
$692$		0			0
$693$	−38.4579	+	187.963i	−0.0554948	+	0.271231i
$694$		0			0
$695$		−	60.8620i		−	0.0875713i
$696$		0			0
$697$	361.335			0.518414
$698$		0			0
$699$		−	157.252i		−	0.224967i
$700$		0			0
$701$			431.684i			0.615812i	0.951417	+	0.307906i	$0.0996282\pi$
$701$			431.684i			0.615812i	−0.951417	+	0.307906i	$0.900372\pi$
$702$		0			0
$703$		−	371.274i		−	0.528128i
$704$		0			0
$705$	76.3204			0.108256
$706$		0			0
$707$	639.119			0.903987
$708$		0			0
$709$	538.938			0.760138			0.380069	−	0.924958i	$-0.375900\pi$
$709$	538.938			0.760138			0.380069	+	0.924958i	$0.375900\pi$
$710$		0			0
$711$		−	163.961i		−	0.230606i
$712$		0			0
$713$	−794.900			−1.11487
$714$		0			0
$715$	−41.0688	+	200.724i	−0.0574389	+	0.280733i
$716$		0			0
$717$		−	502.820i		−	0.701283i
$718$		0			0
$719$	1165.01			1.62031			0.810157	−	0.586212i	$-0.199382\pi$
$719$	1165.01			1.62031			0.810157	+	0.586212i	$0.199382\pi$
$720$		0			0
$721$		−	774.321i		−	1.07395i
$722$		0			0
$723$			550.817i			0.761849i
$724$		0			0
$725$		−	281.865i		−	0.388779i
$726$		0			0
$727$	−255.805			−0.351864			−0.175932	−	0.984402i	$-0.556294\pi$
$727$	−255.805			−0.351864			−0.175932	+	0.984402i	$0.556294\pi$
$728$		0			0
$729$	27.0000			0.0370370
$730$		0			0
$731$	102.908			0.140777
$732$		0			0
$733$		−	654.876i		−	0.893419i	−0.894679	−	0.446709i	$-0.852596\pi$
$733$		−	654.876i		−	0.893419i	0.894679	−	0.446709i	$-0.147404\pi$
$734$		0			0
$735$	−19.4886			−0.0265151
$736$		0			0
$737$	1077.04	+	220.366i	1.46138	+	0.299005i
$738$		0			0
$739$			971.649i			1.31482i	0.753535	+	0.657408i	$0.228347\pi$
$739$			971.649i			1.31482i	−0.753535	+	0.657408i	$0.771653\pi$
$740$		0			0
$741$	1021.96			1.37916
$742$		0			0
$743$			1194.28i			1.60738i	0.595051	+	0.803688i	$0.297132\pi$
$743$			1194.28i			1.60738i	−0.595051	+	0.803688i	$0.702868\pi$
$744$		0			0
$745$		−	84.7111i		−	0.113706i
$746$		0			0
$747$			368.023i			0.492667i
$748$		0			0
$749$	−98.4811			−0.131483
$750$		0			0
$751$	−222.283			−0.295983			−0.147992	−	0.988989i	$-0.547281\pi$
$751$	−222.283			−0.295983			−0.147992	+	0.988989i	$0.547281\pi$
$752$		0			0
$753$	−189.765			−0.252012
$754$		0			0
$755$			27.4004i			0.0362919i
$756$		0			0
$757$	331.870			0.438402			0.219201	−	0.975680i	$-0.429655\pi$
$757$	331.870			0.438402			0.219201	+	0.975680i	$0.429655\pi$
$758$		0			0
$759$	−576.607	−	117.976i	−0.759693	−	0.155436i
$760$		0			0
$761$		−	638.455i		−	0.838968i	−0.907763	−	0.419484i	$-0.862211\pi$
$761$		−	638.455i		−	0.838968i	0.907763	−	0.419484i	$-0.137789\pi$
$762$		0			0
$763$	901.767			1.18187
$764$		0			0
$765$		−	13.2262i		−	0.0172891i
$766$		0			0
$767$		−	103.193i		−	0.134542i
$768$		0			0
$769$		−	494.108i		−	0.642533i	−0.946989	−	0.321266i	$-0.895892\pi$
$769$		−	494.108i		−	0.642533i	0.946989	−	0.321266i	$-0.104108\pi$
$770$		0			0
$771$	−13.1569			−0.0170648
$772$		0			0
$773$	461.903			0.597546			0.298773	−	0.954324i	$-0.403423\pi$
$773$	461.903			0.597546			0.298773	+	0.954324i	$0.403423\pi$
$774$		0			0
$775$	629.209			0.811882
$776$		0			0
$777$		−	159.426i		−	0.205181i
$778$		0			0
$779$	−1422.84			−1.82649
$780$		0			0
$781$	923.591	+	188.970i	1.18257	+	0.241959i
$782$		0			0
$783$			59.8975i			0.0764974i
$784$		0			0
$785$	168.223			0.214297
$786$		0			0
$787$		−	885.555i		−	1.12523i	−0.826719	−	0.562614i	$-0.809796\pi$
$787$		−	885.555i		−	1.12523i	0.826719	−	0.562614i	$-0.190204\pi$
$788$		0			0
$789$			428.540i			0.543143i
$790$		0			0
$791$			741.398i			0.937292i
$792$		0			0
$793$	−1183.85			−1.49288
$794$		0			0
$795$	107.351			0.135032
$796$		0			0
$797$	−274.767			−0.344751			−0.172376	−	0.985031i	$-0.555144\pi$
$797$	−274.767			−0.344751			−0.172376	+	0.985031i	$0.555144\pi$
$798$		0			0
$799$		−	354.485i		−	0.443660i
$800$		0			0
$801$	−468.107			−0.584403
$802$		0			0
$803$	−134.776	+	658.719i	−0.167841	+	0.820322i
$804$		0			0
$805$			132.951i			0.165157i
$806$		0			0
$807$	−430.732			−0.533745
$808$		0			0
$809$			1216.12i			1.50324i	0.659597	+	0.751619i	$0.270727\pi$
$809$			1216.12i			1.50324i	−0.659597	+	0.751619i	$0.729273\pi$
$810$		0			0
$811$		−	510.558i		−	0.629541i	−0.949168	−	0.314770i	$-0.898072\pi$
$811$		−	510.558i		−	0.629541i	0.949168	−	0.314770i	$-0.101928\pi$
$812$		0			0
$813$			285.426i			0.351078i
$814$		0			0
$815$	−144.199			−0.176932
$816$		0			0
$817$	−405.223			−0.495989
$818$		0			0
$819$	438.832			0.535814
$820$		0			0
$821$		−	124.568i		−	0.151728i	−0.997118	−	0.0758638i	$-0.975829\pi$
$821$		−	124.568i		−	0.151728i	0.997118	−	0.0758638i	$-0.0241714\pi$
$822$		0			0
$823$	−394.528			−0.479378			−0.239689	−	0.970850i	$-0.577046\pi$
$823$	−394.528			−0.479378			−0.239689	+	0.970850i	$0.577046\pi$
$824$		0			0
$825$	456.417	+	93.3847i	0.553233	+	0.113194i
$826$		0			0
$827$			1446.95i			1.74963i	0.484454	+	0.874816i	$0.339018\pi$
$827$			1446.95i			1.74963i	−0.484454	+	0.874816i	$0.660982\pi$
$828$		0			0
$829$	−842.715			−1.01654			−0.508272	−	0.861197i	$-0.669715\pi$
$829$	−842.715			−1.01654			−0.508272	+	0.861197i	$0.669715\pi$
$830$		0			0
$831$			95.5610i			0.114995i
$832$		0			0
$833$			90.5186i			0.108666i
$834$		0			0
$835$			150.288i			0.179986i
$836$		0			0
$837$	−133.710			−0.159749
$838$		0			0
$839$	634.515			0.756275			0.378137	−	0.925749i	$-0.376565\pi$
$839$	634.515			0.756275			0.378137	+	0.925749i	$0.376565\pi$
$840$		0			0
$841$	708.122			0.842000
$842$		0			0
$843$			709.605i			0.841762i
$844$		0			0
$845$	343.516			0.406528
$846$		0			0
$847$	276.300	−	646.943i	0.326211	−	0.763805i
$848$		0			0
$849$		−	252.170i		−	0.297019i
$850$		0			0
$851$	489.065			0.574695
$852$		0			0
$853$			865.739i			1.01493i	0.861671	+	0.507467i	$0.169418\pi$
$853$			865.739i			1.01493i	−0.861671	+	0.507467i	$0.830582\pi$
$854$		0			0
$855$			52.0810i			0.0609134i
$856$		0			0
$857$		−	670.769i		−	0.782694i	−0.920243	−	0.391347i	$-0.872009\pi$
$857$		−	670.769i		−	0.782694i	0.920243	−	0.391347i	$-0.127991\pi$
$858$		0			0
$859$	735.734			0.856500			0.428250	−	0.903660i	$-0.359130\pi$
$859$	735.734			0.856500			0.428250	+	0.903660i	$0.359130\pi$
$860$		0			0
$861$	−610.968			−0.709603
$862$		0			0
$863$	1154.53			1.33781			0.668907	−	0.743346i	$-0.266762\pi$
$863$	1154.53			1.33781			0.668907	+	0.743346i	$0.266762\pi$
$864$		0			0
$865$			180.049i			0.208149i
$866$		0			0
$867$	439.131			0.506495
$868$		0			0
$869$	−120.509	+	588.988i	−0.138676	+	0.677777i
$870$		0			0
$871$		−	2514.54i		−	2.88695i
$872$		0			0
$873$	−333.501			−0.382017
$874$		0			0
$875$		−	212.836i		−	0.243241i
$876$		0			0
$877$		−	102.172i		−	0.116502i	−0.998302	−	0.0582511i	$-0.981448\pi$
$877$		−	102.172i		−	0.116502i	0.998302	−	0.0582511i	$-0.0185524\pi$
$878$		0			0
$879$		−	152.391i		−	0.173369i
$880$		0			0
$881$	706.707			0.802164			0.401082	−	0.916042i	$-0.368634\pi$
$881$	706.707			0.802164			0.401082	+	0.916042i	$0.368634\pi$
$882$		0			0
$883$	−1269.05			−1.43720			−0.718600	−	0.695424i	$-0.755217\pi$
$883$	−1269.05			−1.43720			−0.718600	+	0.695424i	$0.755217\pi$
$884$		0			0
$885$	5.25893			0.00594229
$886$		0			0
$887$			1058.50i			1.19334i	0.802485	+	0.596672i	$0.203511\pi$
$887$			1058.50i			1.19334i	−0.802485	+	0.596672i	$0.796489\pi$
$888$		0			0
$889$	1242.14			1.39723
$890$		0			0
$891$	−96.9907	−	19.8446i	−0.108856	−	0.0222723i
$892$		0			0
$893$			1395.86i			1.56311i
$894$		0			0
$895$	110.001			0.122906
$896$		0			0
$897$			1346.19i			1.50077i
$898$		0			0
$899$		−	296.625i		−	0.329950i
$900$		0			0
$901$		−	498.610i		−	0.553397i
$902$		0			0
$903$	−174.003			−0.192695
$904$		0			0
$905$	35.2237			0.0389212
$906$		0			0
$907$	−423.578			−0.467009			−0.233505	−	0.972356i	$-0.575019\pi$
$907$	−423.578			−0.467009			−0.233505	+	0.972356i	$0.575019\pi$
$908$		0			0
$909$			329.792i			0.362807i
$910$		0			0
$911$	−771.322			−0.846676			−0.423338	−	0.905972i	$-0.639142\pi$
$911$	−771.322			−0.846676			−0.423338	+	0.905972i	$0.639142\pi$
$912$		0			0
$913$	270.492	−	1322.03i	0.296267	−	1.44800i
$914$		0			0
$915$		−	60.3313i		−	0.0659359i
$916$		0			0
$917$	855.640			0.933086
$918$		0			0
$919$			899.729i			0.979031i	0.871995	+	0.489515i	$0.162826\pi$
$919$			899.729i			0.979031i	−0.871995	+	0.489515i	$0.837174\pi$
$920$		0			0
$921$		−	415.042i		−	0.450643i
$922$		0			0
$923$		−	2156.28i		−	2.33617i
$924$		0			0
$925$	−387.123			−0.418511
$926$		0			0
$927$	399.557			0.431022
$928$		0			0
$929$	522.732			0.562683			0.281341	−	0.959608i	$-0.409221\pi$
$929$	522.732			0.562683			0.281341	+	0.959608i	$0.409221\pi$
$930$		0			0
$931$		−	356.437i		−	0.382854i
$932$		0			0
$933$	−51.6573			−0.0553668
$934$		0			0
$935$	−9.72107	+	47.5117i	−0.0103969	+	0.0508146i
$936$		0			0
$937$			709.280i			0.756970i	0.925608	+	0.378485i	$0.123555\pi$
$937$			709.280i			0.756970i	−0.925608	+	0.378485i	$0.876445\pi$
$938$		0			0
$939$	−382.381			−0.407222
$940$		0			0
$941$		−	837.255i		−	0.889750i	−0.895593	−	0.444875i	$-0.853248\pi$
$941$		−	837.255i		−	0.889750i	0.895593	−	0.444875i	$-0.146752\pi$
$942$		0			0
$943$		−	1874.25i		−	1.98754i
$944$		0			0
$945$			22.3637i			0.0236653i
$946$		0			0
$947$	−22.9863			−0.0242727			−0.0121364	−	0.999926i	$-0.503863\pi$
$947$	−22.9863			−0.0242727			−0.0121364	+	0.999926i	$0.503863\pi$
$948$		0			0
$949$	1537.89			1.62054
$950$		0			0
$951$	−380.402			−0.400002
$952$		0			0
$953$			819.715i			0.860141i	0.902795	+	0.430071i	$0.141511\pi$
$953$			819.715i			0.860141i	−0.902795	+	0.430071i	$0.858489\pi$
$954$		0			0
$955$	103.592			0.108473
$956$		0			0
$957$	44.0239	−	215.167i	0.0460020	−	0.224834i
$958$		0			0
$959$		−	392.719i		−	0.409509i
$960$		0			0
$961$	−298.843			−0.310971
$962$		0			0
$963$		−	50.8172i		−	0.0527697i
$964$		0			0
$965$			226.634i			0.234853i
$966$		0			0
$967$			1351.29i			1.39741i	0.715412	+	0.698703i	$0.246239\pi$
$967$			1351.29i			1.39741i	−0.715412	+	0.698703i	$0.753761\pi$
$968$		0			0
$969$	241.900			0.249639
$970$		0			0
$971$	−1205.17			−1.24117			−0.620583	−	0.784141i	$-0.713104\pi$
$971$	−1205.17			−1.24117			−0.620583	+	0.784141i	$0.713104\pi$
$972$		0			0
$973$	477.982			0.491246
$974$		0			0
$975$		−	1065.59i		−	1.09291i
$976$		0			0
$977$	554.965			0.568029			0.284015	−	0.958820i	$-0.408334\pi$
$977$	554.965			0.568029			0.284015	+	0.958820i	$0.408334\pi$
$978$		0			0
$979$	1681.56	+	344.053i	1.71763	+	0.351433i
$980$		0			0
$981$			465.321i			0.474333i
$982$		0			0
$983$	−458.586			−0.466517			−0.233258	−	0.972415i	$-0.574939\pi$
$983$	−458.586			−0.466517			−0.233258	+	0.972415i	$0.574939\pi$
$984$		0			0
$985$			16.5435i			0.0167955i
$986$		0			0
$987$			599.385i			0.607280i
$988$		0			0
$989$		−	533.784i		−	0.539721i
$990$		0			0
$991$	−218.105			−0.220086			−0.110043	−	0.993927i	$-0.535099\pi$
$991$	−218.105			−0.220086			−0.110043	+	0.993927i	$0.535099\pi$
$992$		0			0
$993$	726.539			0.731661
$994$		0			0
$995$	140.206			0.140911
$996$		0			0
$997$		−	1061.71i		−	1.06490i	−0.846460	−	0.532452i	$-0.821271\pi$
$997$		−	1061.71i		−	1.06490i	0.846460	−	0.532452i	$-0.178729\pi$
$998$		0			0
$999$	82.2653			0.0823476

Display

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a_n

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a_n

instead) (See

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a_n

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)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2112.3.j.j.769.24		24
4.3	odd	2	inner	2112.3.j.j.769.11		24
8.3	odd	2		1056.3.j.b.769.13	yes	24
8.5	even	2		1056.3.j.b.769.2	✓	24
11.10	odd	2	inner	2112.3.j.j.769.13		24
44.43	even	2	inner	2112.3.j.j.769.2		24
88.21	odd	2		1056.3.j.b.769.11	yes	24
88.43	even	2		1056.3.j.b.769.24	yes	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1056.3.j.b.769.2	✓	24	8.5	even	2
1056.3.j.b.769.11	yes	24	88.21	odd	2
1056.3.j.b.769.13	yes	24	8.3	odd	2
1056.3.j.b.769.24	yes	24	88.43	even	2
2112.3.j.j.769.2		24	44.43	even	2	inner
2112.3.j.j.769.11		24	4.3	odd	2	inner
2112.3.j.j.769.13		24	11.10	odd	2	inner
2112.3.j.j.769.24		24	1.1	even	1	trivial

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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

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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	2112.3.j.j.769.24

Level	$2112$
Weight	$3$
Character	2112.769
Analytic conductor	$57.548$
Analytic rank	$0$
Dimension	$24$
Inner twists	$4$