LMFDB - Embedded newform 276.8.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [276,8,Mod(1,276)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(276, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 8, names="a")

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

chi := DirichletCharacter("276.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 276 = 2^{2} \cdot 3 \cdot 23 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	276.a (trivial)

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:	yes
Analytic conductor:	$86.2182670336$
Analytic rank:	$1$
Dimension:	$5$
Coefficient field:	$\mathbb{Q}[x]/(x^{5} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 82397x^{3} + 7346801x^{2} + 425424304x - 21076942492 $ x^5 - x^4 - 82397x^3 + 7346801x^2 + 425424304*x - 21076942492
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{4}\cdot 3 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-62.3694$ of defining polynomial
Character	$\chi$	$=$	276.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-27.0000 q^{3} -386.322 q^{5} -1140.12 q^{7} +729.000 q^{9} +O(q^{10})$	$q-27.0000 q^{3} -386.322 q^{5} -1140.12 q^{7} +729.000 q^{9} +932.227 q^{11} -3778.85 q^{13} +10430.7 q^{15} +35079.1 q^{17} +38980.1 q^{19} +30783.3 q^{21} -12167.0 q^{23} +71119.6 q^{25} -19683.0 q^{27} -97548.6 q^{29} +115987. q^{31} -25170.1 q^{33} +440455. q^{35} +398847. q^{37} +102029. q^{39} -347416. q^{41} -16634.5 q^{43} -281629. q^{45} -666822. q^{47} +476339. q^{49} -947136. q^{51} +656028. q^{53} -360140. q^{55} -1.05246e6 q^{57} +566943. q^{59} -2.74032e6 q^{61} -831150. q^{63} +1.45985e6 q^{65} -1.67050e6 q^{67} +328509. q^{69} +1.60434e6 q^{71} +4.30426e6 q^{73} -1.92023e6 q^{75} -1.06285e6 q^{77} +5.13482e6 q^{79} +531441. q^{81} +2.63957e6 q^{83} -1.35518e7 q^{85} +2.63381e6 q^{87} -6.63888e6 q^{89} +4.30835e6 q^{91} -3.13165e6 q^{93} -1.50589e7 q^{95} -7.96456e6 q^{97} +679593. q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 135 q^{3} + 252 q^{5} + 414 q^{7} + 3645 q^{9}+O(q^{10}) $ 5 * q - 135 * q^3 + 252 * q^5 + 414 * q^7 + 3645 * q^9	$ 5 q - 135 q^{3} + 252 q^{5} + 414 q^{7} + 3645 q^{9} - 1040 q^{11} - 15626 q^{13} - 6804 q^{15} + 7776 q^{17} + 9650 q^{19} - 11178 q^{21} - 60835 q^{23} + 12979 q^{25} - 98415 q^{27} - 85958 q^{29} + 226044 q^{31} + 28080 q^{33} + 591236 q^{35} + 821430 q^{37} + 421902 q^{39} - 57334 q^{41} + 109706 q^{43} + 183708 q^{45} + 24284 q^{47} + 101813 q^{49} - 209952 q^{51} + 904096 q^{53} + 556632 q^{55} - 260550 q^{57} - 377448 q^{59} - 2834138 q^{61} + 301806 q^{63} - 488120 q^{65} - 1411266 q^{67} + 1642545 q^{69} + 1860048 q^{71} - 940078 q^{73} - 350433 q^{75} - 7845032 q^{77} - 1217086 q^{79} + 2657205 q^{81} - 4542112 q^{83} - 23930564 q^{85} + 2320866 q^{87} - 21715884 q^{89} - 932204 q^{91} - 6103188 q^{93} - 36809060 q^{95} - 30249906 q^{97} - 758160 q^{99}+O(q^{100}) $ 5 * q - 135 * q^3 + 252 * q^5 + 414 * q^7 + 3645 * q^9 - 1040 * q^11 - 15626 * q^13 - 6804 * q^15 + 7776 * q^17 + 9650 * q^19 - 11178 * q^21 - 60835 * q^23 + 12979 * q^25 - 98415 * q^27 - 85958 * q^29 + 226044 * q^31 + 28080 * q^33 + 591236 * q^35 + 821430 * q^37 + 421902 * q^39 - 57334 * q^41 + 109706 * q^43 + 183708 * q^45 + 24284 * q^47 + 101813 * q^49 - 209952 * q^51 + 904096 * q^53 + 556632 * q^55 - 260550 * q^57 - 377448 * q^59 - 2834138 * q^61 + 301806 * q^63 - 488120 * q^65 - 1411266 * q^67 + 1642545 * q^69 + 1860048 * q^71 - 940078 * q^73 - 350433 * q^75 - 7845032 * q^77 - 1217086 * q^79 + 2657205 * q^81 - 4542112 * q^83 - 23930564 * q^85 + 2320866 * q^87 - 21715884 * q^89 - 932204 * q^91 - 6103188 * q^93 - 36809060 * q^95 - 30249906 * q^97 - 758160 * q^99

Display 100 coefficients 10 coefficients

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
$3$	−27.0000	−0.577350
$3$	−27.0000	−0.577350
$4$	0	0
$5$	−386.322	−1.38215	−0.691074	−	0.722784i	$-0.742862\pi$
$5$	−386.322	−1.38215	−0.691074	+	0.722784i	$0.742862\pi$
$6$	0	0
$7$	−1140.12	−1.25634	−0.628172	−	0.778074i	$-0.716197\pi$
$7$	−1140.12	−1.25634	−0.628172	+	0.778074i	$0.716197\pi$
$8$	0	0
$9$	729.000	0.333333
$10$	0	0
$11$	932.227	0.211177	0.105589	−	0.994410i	$-0.466327\pi$
$11$	932.227	0.211177	0.105589	+	0.994410i	$0.466327\pi$
$12$	0	0
$13$	−3778.85	−0.477043	−0.238522	−	0.971137i	$-0.576663\pi$
$13$	−3778.85	−0.477043	−0.238522	+	0.971137i	$0.576663\pi$
$14$	0	0
$15$	10430.7	0.797983
$16$	0	0
$17$	35079.1	1.73172	0.865860	−	0.500287i	$-0.166772\pi$
$17$	35079.1	1.73172	0.865860	+	0.500287i	$0.166772\pi$
$18$	0	0
$19$	38980.1	1.30378	0.651892	−	0.758312i	$-0.273976\pi$
$19$	38980.1	1.30378	0.651892	+	0.758312i	$0.273976\pi$
$20$	0	0
$21$	30783.3	0.725351
$22$	0	0
$23$	−12167.0	−0.208514
$23$	−12167.0	−0.208514
$24$	0	0
$25$	71119.6	0.910331
$26$	0	0
$27$	−19683.0	−0.192450
$28$	0	0
$29$	−97548.6	−0.742725	−0.371363	−	0.928488i	$-0.621109\pi$
$29$	−97548.6	−0.742725	−0.371363	+	0.928488i	$0.621109\pi$
$30$	0	0
$31$	115987.	0.699267	0.349633	−	0.936887i	$-0.386306\pi$
$31$	115987.	0.699267	0.349633	+	0.936887i	$0.386306\pi$
$32$	0	0
$33$	−25170.1	−0.121923
$34$	0	0
$35$	440455.	1.73645
$36$	0	0
$37$	398847.	1.29450	0.647248	−	0.762280i	$-0.275920\pi$
$37$	398847.	1.29450	0.647248	+	0.762280i	$0.275920\pi$
$38$	0	0
$39$	102029.	0.275421
$40$	0	0
$41$	−347416.	−0.787238	−0.393619	−	0.919274i	$-0.628777\pi$
$41$	−347416.	−0.787238	−0.393619	+	0.919274i	$0.628777\pi$
$42$	0	0
$43$	−16634.5	−0.0319059	−0.0159530	−	0.999873i	$-0.505078\pi$
$43$	−16634.5	−0.0319059	−0.0159530	+	0.999873i	$0.505078\pi$
$44$	0	0
$45$	−281629.	−0.460716
$46$	0	0
$47$	−666822.	−0.936844	−0.468422	−	0.883505i	$-0.655177\pi$
$47$	−666822.	−0.936844	−0.468422	+	0.883505i	$0.655177\pi$
$48$	0	0
$49$	476339.	0.578402
$50$	0	0
$51$	−947136.	−0.999809
$52$	0	0
$53$	656028.	0.605281	0.302640	−	0.953105i	$-0.402132\pi$
$53$	656028.	0.605281	0.302640	+	0.953105i	$0.402132\pi$
$54$	0	0
$55$	−360140.	−0.291878
$56$	0	0
$57$	−1.05246e6	−0.752740
$58$	0	0
$59$	566943.	0.359383	0.179692	−	0.983723i	$-0.442490\pi$
$59$	566943.	0.359383	0.179692	+	0.983723i	$0.442490\pi$
$60$	0	0
$61$	−2.74032e6	−1.54578	−0.772888	−	0.634543i	$-0.781188\pi$
$61$	−2.74032e6	−1.54578	−0.772888	+	0.634543i	$0.781188\pi$
$62$	0	0
$63$	−831150.	−0.418782
$64$	0	0
$65$	1.45985e6	0.659344
$66$	0	0
$67$	−1.67050e6	−0.678555	−0.339278	−	0.940686i	$-0.610183\pi$
$67$	−1.67050e6	−0.678555	−0.339278	+	0.940686i	$0.610183\pi$
$68$	0	0
$69$	328509.	0.120386
$70$	0	0
$71$	1.60434e6	0.531977	0.265988	−	0.963976i	$-0.414302\pi$
$71$	1.60434e6	0.531977	0.265988	+	0.963976i	$0.414302\pi$
$72$	0	0
$73$	4.30426e6	1.29500	0.647499	−	0.762067i	$-0.275815\pi$
$73$	4.30426e6	1.29500	0.647499	+	0.762067i	$0.275815\pi$
$74$	0	0
$75$	−1.92023e6	−0.525580
$76$	0	0
$77$	−1.06285e6	−0.265311
$78$	0	0
$79$	5.13482e6	1.17174	0.585868	−	0.810406i	$-0.300754\pi$
$79$	5.13482e6	1.17174	0.585868	+	0.810406i	$0.300754\pi$
$80$	0	0
$81$	531441.	0.111111
$82$	0	0
$83$	2.63957e6	0.506711	0.253355	−	0.967373i	$-0.418466\pi$
$83$	2.63957e6	0.506711	0.253355	+	0.967373i	$0.418466\pi$
$84$	0	0
$85$	−1.35518e7	−2.39349
$86$	0	0
$87$	2.63381e6	0.428813
$88$	0	0
$89$	−6.63888e6	−0.998228	−0.499114	−	0.866536i	$-0.666341\pi$
$89$	−6.63888e6	−0.998228	−0.499114	+	0.866536i	$0.666341\pi$
$90$	0	0
$91$	4.30835e6	0.599331
$92$	0	0
$93$	−3.13165e6	−0.403722
$94$	0	0
$95$	−1.50589e7	−1.80202
$96$	0	0
$97$	−7.96456e6	−0.886055	−0.443027	−	0.896508i	$-0.646096\pi$
$97$	−7.96456e6	−0.886055	−0.443027	+	0.896508i	$0.646096\pi$
$98$	0	0
$99$	679593.	0.0703924
$100$	0	0
$101$	1.88096e7	1.81659	0.908293	−	0.418335i	$-0.137386\pi$
$101$	1.88096e7	1.81659	0.908293	+	0.418335i	$0.137386\pi$
$102$	0	0
$103$	−1.59860e7	−1.44149	−0.720744	−	0.693202i	$-0.756199\pi$
$103$	−1.59860e7	−1.44149	−0.720744	+	0.693202i	$0.756199\pi$
$104$	0	0
$105$	−1.18923e7	−1.00254
$106$	0	0
$107$	5.54484e6	0.437568	0.218784	−	0.975773i	$-0.429791\pi$
$107$	5.54484e6	0.437568	0.218784	+	0.975773i	$0.429791\pi$
$108$	0	0
$109$	−810325.	−0.0599331	−0.0299666	−	0.999551i	$-0.509540\pi$
$109$	−810325.	−0.0599331	−0.0299666	+	0.999551i	$0.509540\pi$
$110$	0	0
$111$	−1.07689e7	−0.747377
$112$	0	0
$113$	1.38865e7	0.905355	0.452677	−	0.891674i	$-0.350469\pi$
$113$	1.38865e7	0.905355	0.452677	+	0.891674i	$0.350469\pi$
$114$	0	0
$115$	4.70038e6	0.288198
$116$	0	0
$117$	−2.75478e6	−0.159014
$118$	0	0
$119$	−3.99945e7	−2.17564
$120$	0	0
$121$	−1.86181e7	−0.955404
$122$	0	0
$123$	9.38022e6	0.454512
$124$	0	0
$125$	2.70634e6	0.123936
$126$	0	0
$127$	−9.91017e6	−0.429307	−0.214654	−	0.976690i	$-0.568862\pi$
$127$	−9.91017e6	−0.429307	−0.214654	+	0.976690i	$0.568862\pi$
$128$	0	0
$129$	449133.	0.0184209
$130$	0	0
$131$	1.82273e7	0.708390	0.354195	−	0.935172i	$-0.384755\pi$
$131$	1.82273e7	0.708390	0.354195	+	0.935172i	$0.384755\pi$
$132$	0	0
$133$	−4.44421e7	−1.63800
$134$	0	0
$135$	7.60397e6	0.265994
$136$	0	0
$137$	−2.24792e6	−0.0746895	−0.0373448	−	0.999302i	$-0.511890\pi$
$137$	−2.24792e6	−0.0746895	−0.0373448	+	0.999302i	$0.511890\pi$
$138$	0	0
$139$	−4.77969e7	−1.50955	−0.754775	−	0.655983i	$-0.772254\pi$
$139$	−4.77969e7	−1.50955	−0.754775	+	0.655983i	$0.772254\pi$
$140$	0	0
$141$	1.80042e7	0.540887
$142$	0	0
$143$	−3.52274e6	−0.100741
$144$	0	0
$145$	3.76852e7	1.02656
$146$	0	0
$147$	−1.28611e7	−0.333940
$148$	0	0
$149$	−6.35196e7	−1.57310	−0.786549	−	0.617528i	$-0.788134\pi$
$149$	−6.35196e7	−1.57310	−0.786549	+	0.617528i	$0.788134\pi$
$150$	0	0
$151$	−4.67865e7	−1.10586	−0.552932	−	0.833227i	$-0.686491\pi$
$151$	−4.67865e7	−1.10586	−0.552932	+	0.833227i	$0.686491\pi$
$152$	0	0
$153$	2.55727e7	0.577240
$154$	0	0
$155$	−4.48083e7	−0.966490
$156$	0	0
$157$	2.66238e7	0.549062	0.274531	−	0.961578i	$-0.411477\pi$
$157$	2.66238e7	0.549062	0.274531	+	0.961578i	$0.411477\pi$
$158$	0	0
$159$	−1.77128e7	−0.349459
$160$	0	0
$161$	1.38719e7	0.261966
$162$	0	0
$163$	−2.77746e7	−0.502332	−0.251166	−	0.967944i	$-0.580814\pi$
$163$	−2.77746e7	−0.502332	−0.251166	+	0.967944i	$0.580814\pi$
$164$	0	0
$165$	9.72377e6	0.168516
$166$	0	0
$167$	1.36477e7	0.226752	0.113376	−	0.993552i	$-0.463834\pi$
$167$	1.36477e7	0.226752	0.113376	+	0.993552i	$0.463834\pi$
$168$	0	0
$169$	−4.84688e7	−0.772430
$170$	0	0
$171$	2.84165e7	0.434595
$172$	0	0
$173$	−8.79997e7	−1.29217	−0.646086	−	0.763265i	$-0.723595\pi$
$173$	−8.79997e7	−1.29217	−0.646086	+	0.763265i	$0.723595\pi$
$174$	0	0
$175$	−8.10851e7	−1.14369
$176$	0	0
$177$	−1.53075e7	−0.207490
$178$	0	0
$179$	9.19112e7	1.19780	0.598898	−	0.800826i	$-0.295606\pi$
$179$	9.19112e7	1.19780	0.598898	+	0.800826i	$0.295606\pi$
$180$	0	0
$181$	6.55401e7	0.821547	0.410773	−	0.911737i	$-0.365259\pi$
$181$	6.55401e7	0.821547	0.410773	+	0.911737i	$0.365259\pi$
$182$	0	0
$183$	7.39886e7	0.892454
$184$	0	0
$185$	−1.54083e8	−1.78918
$186$	0	0
$187$	3.27017e7	0.365700
$188$	0	0
$189$	2.24411e7	0.241784
$190$	0	0
$191$	−1.05724e8	−1.09789	−0.548944	−	0.835859i	$-0.684970\pi$
$191$	−1.05724e8	−1.09789	−0.548944	+	0.835859i	$0.684970\pi$
$192$	0	0
$193$	1.25689e8	1.25848	0.629242	−	0.777209i	$-0.283365\pi$
$193$	1.25689e8	1.25848	0.629242	+	0.777209i	$0.283365\pi$
$194$	0	0
$195$	−3.94160e7	−0.380672
$196$	0	0
$197$	−1.01808e8	−0.948748	−0.474374	−	0.880323i	$-0.657326\pi$
$197$	−1.01808e8	−0.948748	−0.474374	+	0.880323i	$0.657326\pi$
$198$	0	0
$199$	3.72028e7	0.334649	0.167325	−	0.985902i	$-0.446487\pi$
$199$	3.72028e7	0.334649	0.167325	+	0.985902i	$0.446487\pi$
$200$	0	0
$201$	4.51035e7	0.391764
$202$	0	0
$203$	1.11217e8	0.933119
$204$	0	0
$205$	1.34214e8	1.08808
$206$	0	0
$207$	−8.86974e6	−0.0695048
$208$	0	0
$209$	3.63383e7	0.275329
$210$	0	0
$211$	−1.02297e8	−0.749679	−0.374840	−	0.927090i	$-0.622302\pi$
$211$	−1.02297e8	−0.749679	−0.374840	+	0.927090i	$0.622302\pi$
$212$	0	0
$213$	−4.33172e7	−0.307137
$214$	0	0
$215$	6.42629e6	0.0440987
$216$	0	0
$217$	−1.32239e8	−0.878520
$218$	0	0
$219$	−1.16215e8	−0.747667
$220$	0	0
$221$	−1.32559e8	−0.826105
$222$	0	0
$223$	1.19172e8	0.719625	0.359813	−	0.933025i	$-0.382841\pi$
$223$	1.19172e8	0.719625	0.359813	+	0.933025i	$0.382841\pi$
$224$	0	0
$225$	5.18462e7	0.303444
$226$	0	0
$227$	2.56106e8	1.45321	0.726606	−	0.687055i	$-0.241097\pi$
$227$	2.56106e8	1.45321	0.726606	+	0.687055i	$0.241097\pi$
$228$	0	0
$229$	−8.41116e7	−0.462841	−0.231421	−	0.972854i	$-0.574337\pi$
$229$	−8.41116e7	−0.462841	−0.231421	+	0.972854i	$0.574337\pi$
$230$	0	0
$231$	2.86970e7	0.153178
$232$	0	0
$233$	−1.90836e7	−0.0988357	−0.0494179	−	0.998778i	$-0.515737\pi$
$233$	−1.90836e7	−0.0988357	−0.0494179	+	0.998778i	$0.515737\pi$
$234$	0	0
$235$	2.57608e8	1.29486
$236$	0	0
$237$	−1.38640e8	−0.676503
$238$	0	0
$239$	1.27027e8	0.601872	0.300936	−	0.953644i	$-0.402701\pi$
$239$	1.27027e8	0.601872	0.300936	+	0.953644i	$0.402701\pi$
$240$	0	0
$241$	−9.56957e7	−0.440385	−0.220193	−	0.975456i	$-0.570669\pi$
$241$	−9.56957e7	−0.440385	−0.220193	+	0.975456i	$0.570669\pi$
$242$	0	0
$243$	−1.43489e7	−0.0641500
$244$	0	0
$245$	−1.84020e8	−0.799436
$246$	0	0
$247$	−1.47300e8	−0.621961
$248$	0	0
$249$	−7.12685e7	−0.292550
$250$	0	0
$251$	3.88164e8	1.54938	0.774688	−	0.632344i	$-0.217907\pi$
$251$	3.88164e8	1.54938	0.774688	+	0.632344i	$0.217907\pi$
$252$	0	0
$253$	−1.13424e7	−0.0440335
$254$	0	0
$255$	3.65900e8	1.38188
$256$	0	0
$257$	−2.51104e8	−0.922758	−0.461379	−	0.887203i	$-0.652645\pi$
$257$	−2.51104e8	−0.922758	−0.461379	+	0.887203i	$0.652645\pi$
$258$	0	0
$259$	−4.54735e8	−1.62633
$260$	0	0
$261$	−7.11129e7	−0.247575
$262$	0	0
$263$	3.45366e8	1.17067	0.585335	−	0.810792i	$-0.300963\pi$
$263$	3.45366e8	1.17067	0.585335	+	0.810792i	$0.300963\pi$
$264$	0	0
$265$	−2.53438e8	−0.836587
$266$	0	0
$267$	1.79250e8	0.576327
$268$	0	0
$269$	−3.92699e8	−1.23006	−0.615030	−	0.788504i	$-0.710856\pi$
$269$	−3.92699e8	−1.23006	−0.615030	+	0.788504i	$0.710856\pi$
$270$	0	0
$271$	−5.33998e8	−1.62985	−0.814924	−	0.579568i	$-0.803221\pi$
$271$	−5.33998e8	−1.62985	−0.814924	+	0.579568i	$0.803221\pi$
$272$	0	0
$273$	−1.16326e8	−0.346024
$274$	0	0
$275$	6.62996e7	0.192241
$276$	0	0
$277$	−3.67834e8	−1.03986	−0.519928	−	0.854210i	$-0.674041\pi$
$277$	−3.67834e8	−1.03986	−0.519928	+	0.854210i	$0.674041\pi$
$278$	0	0
$279$	8.45545e7	0.233089
$280$	0	0
$281$	3.15344e7	0.0847836	0.0423918	−	0.999101i	$-0.486502\pi$
$281$	3.15344e7	0.0847836	0.0423918	+	0.999101i	$0.486502\pi$
$282$	0	0
$283$	4.88658e8	1.28160	0.640800	−	0.767708i	$-0.278603\pi$
$283$	4.88658e8	1.28160	0.640800	+	0.767708i	$0.278603\pi$
$284$	0	0
$285$	4.06590e8	1.04040
$286$	0	0
$287$	3.96097e8	0.989042
$288$	0	0
$289$	8.20206e8	1.99885
$290$	0	0
$291$	2.15043e8	0.511564
$292$	0	0
$293$	3.64216e7	0.0845906	0.0422953	−	0.999105i	$-0.486533\pi$
$293$	3.64216e7	0.0845906	0.0422953	+	0.999105i	$0.486533\pi$
$294$	0	0
$295$	−2.19023e8	−0.496720
$296$	0	0
$297$	−1.83490e7	−0.0406411
$298$	0	0
$299$	4.59772e7	0.0994704
$300$	0	0
$301$	1.89654e7	0.0400848
$302$	0	0
$303$	−5.07860e8	−1.04881
$304$	0	0
$305$	1.05864e9	2.13649
$306$	0	0
$307$	−1.10846e8	−0.218643	−0.109321	−	0.994006i	$-0.534868\pi$
$307$	−1.10846e8	−0.218643	−0.109321	+	0.994006i	$0.534868\pi$
$308$	0	0
$309$	4.31623e8	0.832243
$310$	0	0
$311$	4.68967e8	0.884057	0.442029	−	0.897001i	$-0.354259\pi$
$311$	4.68967e8	0.884057	0.442029	+	0.897001i	$0.354259\pi$
$312$	0	0
$313$	−4.55497e8	−0.839615	−0.419807	−	0.907613i	$-0.637902\pi$
$313$	−4.55497e8	−0.839615	−0.419807	+	0.907613i	$0.637902\pi$
$314$	0	0
$315$	3.21091e8	0.578818
$316$	0	0
$317$	7.61877e8	1.34331	0.671657	−	0.740863i	$-0.265583\pi$
$317$	7.61877e8	1.34331	0.671657	+	0.740863i	$0.265583\pi$
$318$	0	0
$319$	−9.09374e7	−0.156847
$320$	0	0
$321$	−1.49711e8	−0.252630
$322$	0	0
$323$	1.36739e9	2.25779
$324$	0	0
$325$	−2.68750e8	−0.434267
$326$	0	0
$327$	2.18788e7	0.0346024
$328$	0	0
$329$	7.60259e8	1.17700
$330$	0	0
$331$	6.46629e8	0.980070	0.490035	−	0.871703i	$-0.336984\pi$
$331$	6.46629e8	0.980070	0.490035	+	0.871703i	$0.336984\pi$
$332$	0	0
$333$	2.90760e8	0.431499
$334$	0	0
$335$	6.45351e8	0.937863
$336$	0	0
$337$	−1.75588e8	−0.249914	−0.124957	−	0.992162i	$-0.539879\pi$
$337$	−1.75588e8	−0.249914	−0.124957	+	0.992162i	$0.539879\pi$
$338$	0	0
$339$	−3.74936e8	−0.522707
$340$	0	0
$341$	1.08126e8	0.147669
$342$	0	0
$343$	3.95856e8	0.529673
$344$	0	0
$345$	−1.26910e8	−0.166391
$346$	0	0
$347$	8.63741e8	1.10976	0.554882	−	0.831929i	$-0.312764\pi$
$347$	8.63741e8	1.10976	0.554882	+	0.831929i	$0.312764\pi$
$348$	0	0
$349$	1.87858e8	0.236560	0.118280	−	0.992980i	$-0.462262\pi$
$349$	1.87858e8	0.236560	0.118280	+	0.992980i	$0.462262\pi$
$350$	0	0
$351$	7.43791e7	0.0918070
$352$	0	0
$353$	6.16831e8	0.746371	0.373186	−	0.927757i	$-0.378265\pi$
$353$	6.16831e8	0.746371	0.373186	+	0.927757i	$0.378265\pi$
$354$	0	0
$355$	−6.19792e8	−0.735270
$356$	0	0
$357$	1.07985e9	1.25610
$358$	0	0
$359$	3.20164e8	0.365209	0.182605	−	0.983186i	$-0.441547\pi$
$359$	3.20164e8	0.365209	0.182605	+	0.983186i	$0.441547\pi$
$360$	0	0
$361$	6.25577e8	0.699851
$362$	0	0
$363$	5.02689e8	0.551603
$364$	0	0
$365$	−1.66283e9	−1.78988
$366$	0	0
$367$	−1.50839e9	−1.59288	−0.796438	−	0.604720i	$-0.793285\pi$
$367$	−1.50839e9	−1.59288	−0.796438	+	0.604720i	$0.793285\pi$
$368$	0	0
$369$	−2.53266e8	−0.262413
$370$	0	0
$371$	−7.47953e8	−0.760441
$372$	0	0
$373$	−1.75709e9	−1.75313	−0.876565	−	0.481283i	$-0.840171\pi$
$373$	−1.75709e9	−1.75313	−0.876565	+	0.481283i	$0.840171\pi$
$374$	0	0
$375$	−7.30712e7	−0.0715544
$376$	0	0
$377$	3.68621e8	0.354312
$378$	0	0
$379$	−4.17325e8	−0.393765	−0.196883	−	0.980427i	$-0.563082\pi$
$379$	−4.17325e8	−0.393765	−0.196883	+	0.980427i	$0.563082\pi$
$380$	0	0
$381$	2.67575e8	0.247861
$382$	0	0
$383$	−6.24218e8	−0.567728	−0.283864	−	0.958864i	$-0.591616\pi$
$383$	−6.24218e8	−0.567728	−0.283864	+	0.958864i	$0.591616\pi$
$384$	0	0
$385$	4.10604e8	0.366699
$386$	0	0
$387$	−1.21266e7	−0.0106353
$388$	0	0
$389$	−8.26279e8	−0.711710	−0.355855	−	0.934541i	$-0.615810\pi$
$389$	−8.26279e8	−0.711710	−0.355855	+	0.934541i	$0.615810\pi$
$390$	0	0
$391$	−4.26808e8	−0.361088
$392$	0	0
$393$	−4.92137e8	−0.408989
$394$	0	0
$395$	−1.98369e9	−1.61951
$396$	0	0
$397$	−1.53014e7	−0.0122733	−0.00613667	−	0.999981i	$-0.501953\pi$
$397$	−1.53014e7	−0.0122733	−0.00613667	+	0.999981i	$0.501953\pi$
$398$	0	0
$399$	1.19994e9	0.945701
$400$	0	0
$401$	−1.07538e9	−0.832832	−0.416416	−	0.909174i	$-0.636714\pi$
$401$	−1.07538e9	−0.832832	−0.416416	+	0.909174i	$0.636714\pi$
$402$	0	0
$403$	−4.38297e8	−0.333581
$404$	0	0
$405$	−2.05307e8	−0.153572
$406$	0	0
$407$	3.71816e8	0.273368
$408$	0	0
$409$	7.95803e8	0.575141	0.287570	−	0.957760i	$-0.407152\pi$
$409$	7.95803e8	0.575141	0.287570	+	0.957760i	$0.407152\pi$
$410$	0	0
$411$	6.06940e7	0.0431220
$412$	0	0
$413$	−6.46385e8	−0.451509
$414$	0	0
$415$	−1.01972e9	−0.700349
$416$	0	0
$417$	1.29052e9	0.871540
$418$	0	0
$419$	−1.72767e9	−1.14739	−0.573695	−	0.819069i	$-0.694490\pi$
$419$	−1.72767e9	−1.14739	−0.573695	+	0.819069i	$0.694490\pi$
$420$	0	0
$421$	−2.70780e9	−1.76860	−0.884299	−	0.466921i	$-0.845363\pi$
$421$	−2.70780e9	−1.76860	−0.884299	+	0.466921i	$0.845363\pi$
$422$	0	0
$423$	−4.86113e8	−0.312281
$424$	0	0
$425$	2.49481e9	1.57644
$426$	0	0
$427$	3.12430e9	1.94203
$428$	0	0
$429$	9.51140e7	0.0581626
$430$	0	0
$431$	−1.01796e9	−0.612436	−0.306218	−	0.951961i	$-0.599064\pi$
$431$	−1.01796e9	−0.612436	−0.306218	+	0.951961i	$0.599064\pi$
$432$	0	0
$433$	−1.04837e9	−0.620594	−0.310297	−	0.950640i	$-0.600428\pi$
$433$	−1.04837e9	−0.620594	−0.310297	+	0.950640i	$0.600428\pi$
$434$	0	0
$435$	−1.01750e9	−0.592682
$436$	0	0
$437$	−4.74271e8	−0.271858
$438$	0	0
$439$	−2.02197e9	−1.14064	−0.570320	−	0.821423i	$-0.693181\pi$
$439$	−2.02197e9	−1.14064	−0.570320	+	0.821423i	$0.693181\pi$
$440$	0	0
$441$	3.47251e8	0.192801
$442$	0	0
$443$	2.75110e9	1.50347	0.751733	−	0.659468i	$-0.229218\pi$
$443$	2.75110e9	1.50347	0.751733	+	0.659468i	$0.229218\pi$
$444$	0	0
$445$	2.56474e9	1.37970
$446$	0	0
$447$	1.71503e9	0.908229
$448$	0	0
$449$	−1.41557e9	−0.738024	−0.369012	−	0.929425i	$-0.620304\pi$
$449$	−1.41557e9	−0.738024	−0.369012	+	0.929425i	$0.620304\pi$
$450$	0	0
$451$	−3.23870e8	−0.166247
$452$	0	0
$453$	1.26324e9	0.638471
$454$	0	0
$455$	−1.66441e9	−0.828363
$456$	0	0
$457$	3.84914e8	0.188650	0.0943249	−	0.995541i	$-0.469931\pi$
$457$	3.84914e8	0.188650	0.0943249	+	0.995541i	$0.469931\pi$
$458$	0	0
$459$	−6.90462e8	−0.333270
$460$	0	0
$461$	−4.09299e9	−1.94575	−0.972876	−	0.231329i	$-0.925693\pi$
$461$	−4.09299e9	−1.94575	−0.972876	+	0.231329i	$0.925693\pi$
$462$	0	0
$463$	−3.11228e9	−1.45729	−0.728643	−	0.684894i	$-0.759849\pi$
$463$	−3.11228e9	−1.45729	−0.728643	+	0.684894i	$0.759849\pi$
$464$	0	0
$465$	1.20982e9	0.558003
$466$	0	0
$467$	−8.61214e8	−0.391293	−0.195646	−	0.980674i	$-0.562681\pi$
$467$	−8.61214e8	−0.391293	−0.195646	+	0.980674i	$0.562681\pi$
$468$	0	0
$469$	1.90458e9	0.852499
$470$	0	0
$471$	−7.18843e8	−0.317001
$472$	0	0
$473$	−1.55072e7	−0.00673780
$474$	0	0
$475$	2.77225e9	1.18687
$476$	0	0
$477$	4.78244e8	0.201760
$478$	0	0
$479$	3.49752e9	1.45407	0.727035	−	0.686600i	$-0.240898\pi$
$479$	3.49752e9	1.45407	0.727035	+	0.686600i	$0.240898\pi$
$480$	0	0
$481$	−1.50718e9	−0.617530
$482$	0	0
$483$	−3.74541e8	−0.151246
$484$	0	0
$485$	3.07688e9	1.22466
$486$	0	0
$487$	−1.45097e9	−0.569255	−0.284628	−	0.958638i	$-0.591870\pi$
$487$	−1.45097e9	−0.569255	−0.284628	+	0.958638i	$0.591870\pi$
$488$	0	0
$489$	7.49913e8	0.290021
$490$	0	0
$491$	1.03432e9	0.394339	0.197170	−	0.980369i	$-0.436825\pi$
$491$	1.03432e9	0.394339	0.197170	+	0.980369i	$0.436825\pi$
$492$	0	0
$493$	−3.42192e9	−1.28619
$494$	0	0
$495$	−2.62542e8	−0.0972927
$496$	0	0
$497$	−1.82915e9	−0.668346
$498$	0	0
$499$	−1.12992e9	−0.407094	−0.203547	−	0.979065i	$-0.565247\pi$
$499$	−1.12992e9	−0.407094	−0.203547	+	0.979065i	$0.565247\pi$
$500$	0	0
$501$	−3.68487e8	−0.130915
$502$	0	0
$503$	5.72161e8	0.200461	0.100231	−	0.994964i	$-0.468042\pi$
$503$	5.72161e8	0.200461	0.100231	+	0.994964i	$0.468042\pi$
$504$	0	0
$505$	−7.26658e9	−2.51079
$506$	0	0
$507$	1.30866e9	0.445963
$508$	0	0
$509$	1.96761e9	0.661344	0.330672	−	0.943746i	$-0.392725\pi$
$509$	1.96761e9	0.661344	0.330672	+	0.943746i	$0.392725\pi$
$510$	0	0
$511$	−4.90739e9	−1.62696
$512$	0	0
$513$	−7.67246e8	−0.250913
$514$	0	0
$515$	6.17576e9	1.99235
$516$	0	0
$517$	−6.21629e8	−0.197840
$518$	0	0
$519$	2.37599e9	0.746035
$520$	0	0
$521$	−5.90281e9	−1.82863	−0.914316	−	0.405001i	$-0.867271\pi$
$521$	−5.90281e9	−1.82863	−0.914316	+	0.405001i	$0.867271\pi$
$522$	0	0
$523$	−2.27685e9	−0.695951	−0.347975	−	0.937504i	$-0.613131\pi$
$523$	−2.27685e9	−0.695951	−0.347975	+	0.937504i	$0.613131\pi$
$524$	0	0
$525$	2.18930e9	0.660309
$526$	0	0
$527$	4.06872e9	1.21093
$528$	0	0
$529$	1.48036e8	0.0434783
$530$	0	0
$531$	4.13302e8	0.119794
$532$	0	0
$533$	1.31283e9	0.375546
$534$	0	0
$535$	−2.14209e9	−0.604783
$536$	0	0
$537$	−2.48160e9	−0.691547
$538$	0	0
$539$	4.44056e8	0.122145
$540$	0	0
$541$	−5.81625e9	−1.57926	−0.789629	−	0.613585i	$-0.789727\pi$
$541$	−5.81625e9	−1.57926	−0.789629	+	0.613585i	$0.789727\pi$
$542$	0	0
$543$	−1.76958e9	−0.474320
$544$	0	0
$545$	3.13046e8	0.0828364
$546$	0	0
$547$	4.01140e9	1.04795	0.523975	−	0.851734i	$-0.324449\pi$
$547$	4.01140e9	1.04795	0.523975	+	0.851734i	$0.324449\pi$
$548$	0	0
$549$	−1.99769e9	−0.515258
$550$	0	0
$551$	−3.80246e9	−0.968353
$552$	0	0
$553$	−5.85432e9	−1.47211
$554$	0	0
$555$	4.16025e9	1.03299
$556$	0	0
$557$	−5.04583e9	−1.23720	−0.618600	−	0.785706i	$-0.712300\pi$
$557$	−5.04583e9	−1.23720	−0.618600	+	0.785706i	$0.712300\pi$
$558$	0	0
$559$	6.28594e7	0.0152205
$560$	0	0
$561$	−8.82946e8	−0.211137
$562$	0	0
$563$	5.42106e9	1.28028	0.640140	−	0.768258i	$-0.278876\pi$
$563$	5.42106e9	1.28028	0.640140	+	0.768258i	$0.278876\pi$
$564$	0	0
$565$	−5.36466e9	−1.25133
$566$	0	0
$567$	−6.05908e8	−0.139594
$568$	0	0
$569$	−8.77026e9	−1.99581	−0.997905	−	0.0646923i	$-0.979393\pi$
$569$	−8.77026e9	−1.99581	−0.997905	+	0.0646923i	$0.979393\pi$
$570$	0	0
$571$	3.61343e8	0.0812257	0.0406128	−	0.999175i	$-0.487069\pi$
$571$	3.61343e8	0.0812257	0.0406128	+	0.999175i	$0.487069\pi$
$572$	0	0
$573$	2.85456e9	0.633866
$574$	0	0
$575$	−8.65312e8	−0.189817
$576$	0	0
$577$	−3.75627e9	−0.814032	−0.407016	−	0.913421i	$-0.633431\pi$
$577$	−3.75627e9	−0.814032	−0.407016	+	0.913421i	$0.633431\pi$
$578$	0	0
$579$	−3.39361e9	−0.726586
$580$	0	0
$581$	−3.00944e9	−0.636604
$582$	0	0
$583$	6.11567e8	0.127821
$584$	0	0
$585$	1.06423e9	0.219781
$586$	0	0
$587$	7.98967e9	1.63041	0.815203	−	0.579175i	$-0.196625\pi$
$587$	7.98967e9	1.63041	0.815203	+	0.579175i	$0.196625\pi$
$588$	0	0
$589$	4.52118e9	0.911693
$590$	0	0
$591$	2.74882e9	0.547760
$592$	0	0
$593$	7.75480e9	1.52714	0.763570	−	0.645725i	$-0.223445\pi$
$593$	7.75480e9	1.52714	0.763570	+	0.645725i	$0.223445\pi$
$594$	0	0
$595$	1.54508e10	3.00705
$596$	0	0
$597$	−1.00448e9	−0.193210
$598$	0	0
$599$	−7.99111e9	−1.51919	−0.759597	−	0.650394i	$-0.774604\pi$
$599$	−7.99111e9	−1.51919	−0.759597	+	0.650394i	$0.774604\pi$
$600$	0	0
$601$	−1.00290e10	−1.88451	−0.942253	−	0.334903i	$-0.891296\pi$
$601$	−1.00290e10	−1.88451	−0.942253	+	0.334903i	$0.891296\pi$
$602$	0	0
$603$	−1.21780e9	−0.226185
$604$	0	0
$605$	7.19259e9	1.32051
$606$	0	0
$607$	9.54835e9	1.73288	0.866439	−	0.499283i	$-0.166403\pi$
$607$	9.54835e9	1.73288	0.866439	+	0.499283i	$0.166403\pi$
$608$	0	0
$609$	−3.00287e9	−0.538736
$610$	0	0
$611$	2.51982e9	0.446915
$612$	0	0
$613$	−2.94379e9	−0.516173	−0.258087	−	0.966122i	$-0.583092\pi$
$613$	−2.94379e9	−0.516173	−0.258087	+	0.966122i	$0.583092\pi$
$614$	0	0
$615$	−3.62379e9	−0.628202
$616$	0	0
$617$	−9.00119e9	−1.54277	−0.771386	−	0.636368i	$-0.780436\pi$
$617$	−9.00119e9	−1.54277	−0.771386	+	0.636368i	$0.780436\pi$
$618$	0	0
$619$	−6.10081e9	−1.03388	−0.516940	−	0.856022i	$-0.672929\pi$
$619$	−6.10081e9	−1.03388	−0.516940	+	0.856022i	$0.672929\pi$
$620$	0	0
$621$	2.39483e8	0.0401286
$622$	0	0
$623$	7.56914e9	1.25412
$624$	0	0
$625$	−6.60174e9	−1.08163
$626$	0	0
$627$	−9.81134e8	−0.158961
$628$	0	0
$629$	1.39912e10	2.24170
$630$	0	0
$631$	7.72064e9	1.22335	0.611675	−	0.791109i	$-0.290496\pi$
$631$	7.72064e9	1.22335	0.611675	+	0.791109i	$0.290496\pi$
$632$	0	0
$633$	2.76203e9	0.432827
$634$	0	0
$635$	3.82852e9	0.593366
$636$	0	0
$637$	−1.80001e9	−0.275923
$638$	0	0
$639$	1.16957e9	0.177326
$640$	0	0
$641$	5.52841e9	0.829081	0.414541	−	0.910031i	$-0.363942\pi$
$641$	5.52841e9	0.829081	0.414541	+	0.910031i	$0.363942\pi$
$642$	0	0
$643$	2.42023e9	0.359019	0.179510	−	0.983756i	$-0.442549\pi$
$643$	2.42023e9	0.359019	0.179510	+	0.983756i	$0.442549\pi$
$644$	0	0
$645$	−1.73510e8	−0.0254604
$646$	0	0
$647$	−1.28197e9	−0.186086	−0.0930430	−	0.995662i	$-0.529659\pi$
$647$	−1.28197e9	−0.186086	−0.0930430	+	0.995662i	$0.529659\pi$
$648$	0	0
$649$	5.28520e8	0.0758935
$650$	0	0
$651$	3.57046e9	0.507214
$652$	0	0
$653$	−2.11169e9	−0.296780	−0.148390	−	0.988929i	$-0.547409\pi$
$653$	−2.11169e9	−0.296780	−0.148390	+	0.988929i	$0.547409\pi$
$654$	0	0
$655$	−7.04160e9	−0.979100
$656$	0	0
$657$	3.13781e9	0.431666
$658$	0	0
$659$	−8.13028e9	−1.10664	−0.553320	−	0.832969i	$-0.686639\pi$
$659$	−8.13028e9	−1.10664	−0.553320	+	0.832969i	$0.686639\pi$
$660$	0	0
$661$	1.19282e10	1.60645	0.803227	−	0.595674i	$-0.203115\pi$
$661$	1.19282e10	1.60645	0.803227	+	0.595674i	$0.203115\pi$
$662$	0	0
$663$	3.57908e9	0.476952
$664$	0	0
$665$	1.71690e10	2.26396
$666$	0	0
$667$	1.18687e9	0.154869
$668$	0	0
$669$	−3.21764e9	−0.415476
$670$	0	0
$671$	−2.55460e9	−0.326433
$672$	0	0
$673$	−8.55668e9	−1.08206	−0.541032	−	0.841002i	$-0.681966\pi$
$673$	−8.55668e9	−1.08206	−0.541032	+	0.841002i	$0.681966\pi$
$674$	0	0
$675$	−1.39985e9	−0.175193
$676$	0	0
$677$	−2.71171e9	−0.335879	−0.167939	−	0.985797i	$-0.553711\pi$
$677$	−2.71171e9	−0.335879	−0.167939	+	0.985797i	$0.553711\pi$
$678$	0	0
$679$	9.08058e9	1.11319
$680$	0	0
$681$	−6.91485e9	−0.839012
$682$	0	0
$683$	−5.01237e9	−0.601964	−0.300982	−	0.953630i	$-0.597314\pi$
$683$	−5.01237e9	−0.601964	−0.300982	+	0.953630i	$0.597314\pi$
$684$	0	0
$685$	8.68423e8	0.103232
$686$	0	0
$687$	2.27101e9	0.267221
$688$	0	0
$689$	−2.47903e9	−0.288745
$690$	0	0
$691$	−6.92473e9	−0.798417	−0.399208	−	0.916860i	$-0.630715\pi$
$691$	−6.92473e9	−0.798417	−0.399208	+	0.916860i	$0.630715\pi$
$692$	0	0
$693$	−7.74820e8	−0.0884371
$694$	0	0
$695$	1.84650e10	2.08642
$696$	0	0
$697$	−1.21870e10	−1.36327
$698$	0	0
$699$	5.15256e8	0.0570628
$700$	0	0
$701$	−9.23924e9	−1.01303	−0.506516	−	0.862231i	$-0.669067\pi$
$701$	−9.23924e9	−1.01303	−0.506516	+	0.862231i	$0.669067\pi$
$702$	0	0
$703$	1.55471e10	1.68774
$704$	0	0
$705$	−6.95541e9	−0.747585
$706$	0	0
$707$	−2.14453e10	−2.28226
$708$	0	0
$709$	1.83216e10	1.93064	0.965321	−	0.261067i	$-0.0840744\pi$
$709$	1.83216e10	1.93064	0.965321	+	0.261067i	$0.0840744\pi$
$710$	0	0
$711$	3.74328e9	0.390579
$712$	0	0
$713$	−1.41121e9	−0.145807
$714$	0	0
$715$	1.36091e9	0.139238
$716$	0	0
$717$	−3.42974e9	−0.347491
$718$	0	0
$719$	−2.29207e9	−0.229973	−0.114987	−	0.993367i	$-0.536682\pi$
$719$	−2.29207e9	−0.229973	−0.114987	+	0.993367i	$0.536682\pi$
$720$	0	0
$721$	1.82261e10	1.81100
$722$	0	0
$723$	2.58378e9	0.254257
$724$	0	0
$725$	−6.93762e9	−0.676126
$726$	0	0
$727$	−7.08806e9	−0.684159	−0.342080	−	0.939671i	$-0.611131\pi$
$727$	−7.08806e9	−0.684159	−0.342080	+	0.939671i	$0.611131\pi$
$728$	0	0
$729$	3.87420e8	0.0370370
$730$	0	0
$731$	−5.83525e8	−0.0552521
$732$	0	0
$733$	−2.69156e9	−0.252429	−0.126215	−	0.992003i	$-0.540283\pi$
$733$	−2.69156e9	−0.252429	−0.126215	+	0.992003i	$0.540283\pi$
$734$	0	0
$735$	4.96854e9	0.461555
$736$	0	0
$737$	−1.55729e9	−0.143295
$738$	0	0
$739$	6.42829e9	0.585922	0.292961	−	0.956124i	$-0.405359\pi$
$739$	6.42829e9	0.585922	0.292961	+	0.956124i	$0.405359\pi$
$740$	0	0
$741$	3.97710e9	0.359089
$742$	0	0
$743$	−3.65165e9	−0.326609	−0.163304	−	0.986576i	$-0.552215\pi$
$743$	−3.65165e9	−0.326609	−0.163304	+	0.986576i	$0.552215\pi$
$744$	0	0
$745$	2.45390e10	2.17425
$746$	0	0
$747$	1.92425e9	0.168904
$748$	0	0
$749$	−6.32180e9	−0.549736
$750$	0	0
$751$	6.45024e9	0.555694	0.277847	−	0.960625i	$-0.410379\pi$
$751$	6.45024e9	0.555694	0.277847	+	0.960625i	$0.410379\pi$
$752$	0	0
$753$	−1.04804e10	−0.894533
$754$	0	0
$755$	1.80747e10	1.52847
$756$	0	0
$757$	2.27767e10	1.90834	0.954169	−	0.299268i	$-0.0967424\pi$
$757$	2.27767e10	1.90834	0.954169	+	0.299268i	$0.0967424\pi$
$758$	0	0
$759$	3.06245e8	0.0254227
$760$	0	0
$761$	−2.99181e9	−0.246086	−0.123043	−	0.992401i	$-0.539265\pi$
$761$	−2.99181e9	−0.246086	−0.123043	+	0.992401i	$0.539265\pi$
$762$	0	0
$763$	9.23871e8	0.0752966
$764$	0	0
$765$	−9.87929e9	−0.797830
$766$	0	0
$767$	−2.14239e9	−0.171441
$768$	0	0
$769$	−1.30214e9	−0.103256	−0.0516279	−	0.998666i	$-0.516441\pi$
$769$	−1.30214e9	−0.103256	−0.0516279	+	0.998666i	$0.516441\pi$
$770$	0	0
$771$	6.77981e9	0.532755
$772$	0	0
$773$	3.60764e9	0.280928	0.140464	−	0.990086i	$-0.455141\pi$
$773$	3.60764e9	0.280928	0.140464	+	0.990086i	$0.455141\pi$
$774$	0	0
$775$	8.24894e9	0.636564
$776$	0	0
$777$	1.22778e10	0.938964
$778$	0	0
$779$	−1.35423e10	−1.02639
$780$	0	0
$781$	1.49561e9	0.112341
$782$	0	0
$783$	1.92005e9	0.142938
$784$	0	0
$785$	−1.02854e10	−0.758885
$786$	0	0
$787$	−2.15458e9	−0.157562	−0.0787809	−	0.996892i	$-0.525103\pi$
$787$	−2.15458e9	−0.157562	−0.0787809	+	0.996892i	$0.525103\pi$
$788$	0	0
$789$	−9.32488e9	−0.675886
$790$	0	0
$791$	−1.58323e10	−1.13744
$792$	0	0
$793$	1.03552e10	0.737402
$794$	0	0
$795$	6.84282e9	0.483004
$796$	0	0
$797$	6.12829e9	0.428780	0.214390	−	0.976748i	$-0.431224\pi$
$797$	6.12829e9	0.428780	0.214390	+	0.976748i	$0.431224\pi$
$798$	0	0
$799$	−2.33915e10	−1.62235
$800$	0	0
$801$	−4.83974e9	−0.332743
$802$	0	0
$803$	4.01255e9	0.273474
$804$	0	0
$805$	−5.35901e9	−0.362076
$806$	0	0
$807$	1.06029e10	0.710176
$808$	0	0
$809$	−1.11433e10	−0.739938	−0.369969	−	0.929044i	$-0.620632\pi$
$809$	−1.11433e10	−0.739938	−0.369969	+	0.929044i	$0.620632\pi$
$810$	0	0
$811$	7.98553e9	0.525691	0.262846	−	0.964838i	$-0.415339\pi$
$811$	7.98553e9	0.525691	0.262846	+	0.964838i	$0.415339\pi$
$812$	0	0
$813$	1.44179e10	0.940993
$814$	0	0
$815$	1.07299e10	0.694296
$816$	0	0
$817$	−6.48416e8	−0.0415984
$818$	0	0
$819$	3.14079e9	0.199777
$820$	0	0
$821$	1.84292e10	1.16227	0.581134	−	0.813808i	$-0.302609\pi$
$821$	1.84292e10	1.16227	0.581134	+	0.813808i	$0.302609\pi$
$822$	0	0
$823$	−5.28514e9	−0.330489	−0.165244	−	0.986253i	$-0.552841\pi$
$823$	−5.28514e9	−0.330489	−0.165244	+	0.986253i	$0.552841\pi$
$824$	0	0
$825$	−1.79009e9	−0.110990
$826$	0	0
$827$	2.20880e10	1.35796	0.678979	−	0.734157i	$-0.262423\pi$
$827$	2.20880e10	1.35796	0.678979	+	0.734157i	$0.262423\pi$
$828$	0	0
$829$	9.35771e9	0.570465	0.285232	−	0.958458i	$-0.407929\pi$
$829$	9.35771e9	0.570465	0.285232	+	0.958458i	$0.407929\pi$
$830$	0	0
$831$	9.93152e9	0.600361
$832$	0	0
$833$	1.67095e10	1.00163
$834$	0	0
$835$	−5.27239e9	−0.313405
$836$	0	0
$837$	−2.28297e9	−0.134574
$838$	0	0
$839$	−1.92901e9	−0.112763	−0.0563815	−	0.998409i	$-0.517956\pi$
$839$	−1.92901e9	−0.112763	−0.0563815	+	0.998409i	$0.517956\pi$
$840$	0	0
$841$	−7.73415e9	−0.448359
$842$	0	0
$843$	−8.51428e8	−0.0489498
$844$	0	0
$845$	1.87246e10	1.06761
$846$	0	0
$847$	2.12270e10	1.20032
$848$	0	0
$849$	−1.31938e10	−0.739932
$850$	0	0
$851$	−4.85277e9	−0.269921
$852$	0	0
$853$	−2.25735e10	−1.24531	−0.622655	−	0.782497i	$-0.713946\pi$
$853$	−2.25735e10	−1.24531	−0.622655	+	0.782497i	$0.713946\pi$
$854$	0	0
$855$	−1.09779e10	−0.600674
$856$	0	0
$857$	1.00830e10	0.547213	0.273607	−	0.961842i	$-0.411783\pi$
$857$	1.00830e10	0.547213	0.273607	+	0.961842i	$0.411783\pi$
$858$	0	0
$859$	2.42564e10	1.30572	0.652862	−	0.757477i	$-0.273568\pi$
$859$	2.42564e10	1.30572	0.652862	+	0.757477i	$0.273568\pi$
$860$	0	0
$861$	−1.06946e10	−0.571024
$862$	0	0
$863$	−1.27819e10	−0.676951	−0.338475	−	0.940975i	$-0.609911\pi$
$863$	−1.27819e10	−0.676951	−0.338475	+	0.940975i	$0.609911\pi$
$864$	0	0
$865$	3.39962e10	1.78597
$866$	0	0
$867$	−2.21456e10	−1.15404
$868$	0	0
$869$	4.78681e9	0.247444
$870$	0	0
$871$	6.31257e9	0.323700
$872$	0	0
$873$	−5.80616e9	−0.295352
$874$	0	0
$875$	−3.08556e9	−0.155706
$876$	0	0
$877$	−2.06695e10	−1.03474	−0.517371	−	0.855761i	$-0.673089\pi$
$877$	−2.06695e10	−1.03474	−0.517371	+	0.855761i	$0.673089\pi$
$878$	0	0
$879$	−9.83383e8	−0.0488384
$880$	0	0
$881$	−2.61893e10	−1.29035	−0.645175	−	0.764035i	$-0.723216\pi$
$881$	−2.61893e10	−1.29035	−0.645175	+	0.764035i	$0.723216\pi$
$882$	0	0
$883$	−3.17941e10	−1.55412	−0.777060	−	0.629427i	$-0.783290\pi$
$883$	−3.17941e10	−1.55412	−0.777060	+	0.629427i	$0.783290\pi$
$884$	0	0
$885$	5.91361e9	0.286782
$886$	0	0
$887$	−5.22762e9	−0.251519	−0.125760	−	0.992061i	$-0.540137\pi$
$887$	−5.22762e9	−0.251519	−0.125760	+	0.992061i	$0.540137\pi$
$888$	0	0
$889$	1.12988e10	0.539358
$890$	0	0
$891$	4.95423e8	0.0234641
$892$	0	0
$893$	−2.59928e10	−1.22144
$894$	0	0
$895$	−3.55073e10	−1.65553
$896$	0	0
$897$	−1.24139e9	−0.0574292
$898$	0	0
$899$	−1.13144e10	−0.519363
$900$	0	0
$901$	2.30129e10	1.04818
$902$	0	0
$903$	−5.12067e8	−0.0231430
$904$	0	0
$905$	−2.53196e10	−1.13550
$906$	0	0
$907$	−2.74180e10	−1.22014	−0.610070	−	0.792347i	$-0.708859\pi$
$907$	−2.74180e10	−1.22014	−0.610070	+	0.792347i	$0.708859\pi$
$908$	0	0
$909$	1.37122e10	0.605528
$910$	0	0
$911$	2.41618e10	1.05880	0.529402	−	0.848371i	$-0.322417\pi$
$911$	2.41618e10	1.05880	0.529402	+	0.848371i	$0.322417\pi$
$912$	0	0
$913$	2.46068e9	0.107006
$914$	0	0
$915$	−2.85834e10	−1.23350
$916$	0	0
$917$	−2.07814e10	−0.889982
$918$	0	0
$919$	−4.27301e10	−1.81606	−0.908029	−	0.418908i	$-0.862413\pi$
$919$	−4.27301e10	−1.81606	−0.908029	+	0.418908i	$0.862413\pi$
$920$	0	0
$921$	2.99283e9	0.126233
$922$	0	0
$923$	−6.06256e9	−0.253776
$924$	0	0
$925$	2.83659e10	1.17842
$926$	0	0
$927$	−1.16538e10	−0.480496
$928$	0	0
$929$	4.37039e9	0.178840	0.0894201	−	0.995994i	$-0.471499\pi$
$929$	4.37039e9	0.178840	0.0894201	+	0.995994i	$0.471499\pi$
$930$	0	0
$931$	1.85677e10	0.754111
$932$	0	0
$933$	−1.26621e10	−0.510411
$934$	0	0
$935$	−1.26334e10	−0.505451
$936$	0	0
$937$	−3.32817e10	−1.32165	−0.660826	−	0.750539i	$-0.729794\pi$
$937$	−3.32817e10	−1.32165	−0.660826	+	0.750539i	$0.729794\pi$
$938$	0	0
$939$	1.22984e10	0.484752
$940$	0	0
$941$	2.85629e10	1.11748	0.558739	−	0.829344i	$-0.311285\pi$
$941$	2.85629e10	1.11748	0.558739	+	0.829344i	$0.311285\pi$
$942$	0	0
$943$	4.22701e9	0.164150
$944$	0	0
$945$	−8.66947e9	−0.334181
$946$	0	0
$947$	6.07030e9	0.232266	0.116133	−	0.993234i	$-0.462950\pi$
$947$	6.07030e9	0.232266	0.116133	+	0.993234i	$0.462950\pi$
$948$	0	0
$949$	−1.62652e10	−0.617770
$950$	0	0
$951$	−2.05707e10	−0.775562
$952$	0	0
$953$	−2.67399e10	−1.00077	−0.500386	−	0.865802i	$-0.666809\pi$
$953$	−2.67399e10	−1.00077	−0.500386	+	0.865802i	$0.666809\pi$
$954$	0	0
$955$	4.08436e10	1.51744
$956$	0	0
$957$	2.45531e9	0.0905554
$958$	0	0
$959$	2.56291e9	0.0938358
$960$	0	0
$961$	−1.40597e10	−0.511026
$962$	0	0
$963$	4.04219e9	0.145856
$964$	0	0
$965$	−4.85565e10	−1.73941
$966$	0	0
$967$	4.14235e10	1.47317	0.736587	−	0.676343i	$-0.236436\pi$
$967$	4.14235e10	1.47317	0.736587	+	0.676343i	$0.236436\pi$
$968$	0	0
$969$	−3.69195e10	−1.30353
$970$	0	0
$971$	−2.88858e10	−1.01255	−0.506276	−	0.862372i	$-0.668978\pi$
$971$	−2.88858e10	−1.01255	−0.506276	+	0.862372i	$0.668978\pi$
$972$	0	0
$973$	5.44943e10	1.89652
$974$	0	0
$975$	7.25625e9	0.250724
$976$	0	0
$977$	−7.53761e9	−0.258585	−0.129292	−	0.991607i	$-0.541271\pi$
$977$	−7.53761e9	−0.258585	−0.129292	+	0.991607i	$0.541271\pi$
$978$	0	0
$979$	−6.18894e9	−0.210803
$980$	0	0
$981$	−5.90727e8	−0.0199777
$982$	0	0
$983$	−1.47825e10	−0.496375	−0.248188	−	0.968712i	$-0.579835\pi$
$983$	−1.47825e10	−0.496375	−0.248188	+	0.968712i	$0.579835\pi$
$984$	0	0
$985$	3.93307e10	1.31131
$986$	0	0
$987$	−2.05270e10	−0.679540
$988$	0	0
$989$	2.02392e8	0.00665284
$990$	0	0
$991$	2.90508e10	0.948201	0.474101	−	0.880471i	$-0.342773\pi$
$991$	2.90508e10	0.948201	0.474101	+	0.880471i	$0.342773\pi$
$992$	0	0
$993$	−1.74590e10	−0.565844
$994$	0	0
$995$	−1.43723e10	−0.462534
$996$	0	0
$997$	1.38617e8	0.00442979	0.00221490	−	0.999998i	$-0.499295\pi$
$997$	1.38617e8	0.00442979	0.00221490	+	0.999998i	$0.499295\pi$
$998$	0	0
$999$	−7.85051e9	−0.249126

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	276.8.a.a.1.1	✓	5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
276.8.a.a.1.1	✓	5	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 276 = 2^{2} \cdot 3 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	276.a (trivial)

\(f(q)\)	\(=\)	\(q-27.0000 q^{3} -386.322 q^{5} -1140.12 q^{7} +729.000 q^{9} +O(q^{10})\)	\(q-27.0000 q^{3} -386.322 q^{5} -1140.12 q^{7} +729.000 q^{9} +932.227 q^{11} -3778.85 q^{13} +10430.7 q^{15} +35079.1 q^{17} +38980.1 q^{19} +30783.3 q^{21} -12167.0 q^{23} +71119.6 q^{25} -19683.0 q^{27} -97548.6 q^{29} +115987. q^{31} -25170.1 q^{33} +440455. q^{35} +398847. q^{37} +102029. q^{39} -347416. q^{41} -16634.5 q^{43} -281629. q^{45} -666822. q^{47} +476339. q^{49} -947136. q^{51} +656028. q^{53} -360140. q^{55} -1.05246e6 q^{57} +566943. q^{59} -2.74032e6 q^{61} -831150. q^{63} +1.45985e6 q^{65} -1.67050e6 q^{67} +328509. q^{69} +1.60434e6 q^{71} +4.30426e6 q^{73} -1.92023e6 q^{75} -1.06285e6 q^{77} +5.13482e6 q^{79} +531441. q^{81} +2.63957e6 q^{83} -1.35518e7 q^{85} +2.63381e6 q^{87} -6.63888e6 q^{89} +4.30835e6 q^{91} -3.13165e6 q^{93} -1.50589e7 q^{95} -7.96456e6 q^{97} +679593. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 135 q^{3} + 252 q^{5} + 414 q^{7} + 3645 q^{9}+O(q^{10}) \) 5 * q - 135 * q^3 + 252 * q^5 + 414 * q^7 + 3645 * q^9	\( 5 q - 135 q^{3} + 252 q^{5} + 414 q^{7} + 3645 q^{9} - 1040 q^{11} - 15626 q^{13} - 6804 q^{15} + 7776 q^{17} + 9650 q^{19} - 11178 q^{21} - 60835 q^{23} + 12979 q^{25} - 98415 q^{27} - 85958 q^{29} + 226044 q^{31} + 28080 q^{33} + 591236 q^{35} + 821430 q^{37} + 421902 q^{39} - 57334 q^{41} + 109706 q^{43} + 183708 q^{45} + 24284 q^{47} + 101813 q^{49} - 209952 q^{51} + 904096 q^{53} + 556632 q^{55} - 260550 q^{57} - 377448 q^{59} - 2834138 q^{61} + 301806 q^{63} - 488120 q^{65} - 1411266 q^{67} + 1642545 q^{69} + 1860048 q^{71} - 940078 q^{73} - 350433 q^{75} - 7845032 q^{77} - 1217086 q^{79} + 2657205 q^{81} - 4542112 q^{83} - 23930564 q^{85} + 2320866 q^{87} - 21715884 q^{89} - 932204 q^{91} - 6103188 q^{93} - 36809060 q^{95} - 30249906 q^{97} - 758160 q^{99}+O(q^{100}) \) 5 * q - 135 * q^3 + 252 * q^5 + 414 * q^7 + 3645 * q^9 - 1040 * q^11 - 15626 * q^13 - 6804 * q^15 + 7776 * q^17 + 9650 * q^19 - 11178 * q^21 - 60835 * q^23 + 12979 * q^25 - 98415 * q^27 - 85958 * q^29 + 226044 * q^31 + 28080 * q^33 + 591236 * q^35 + 821430 * q^37 + 421902 * q^39 - 57334 * q^41 + 109706 * q^43 + 183708 * q^45 + 24284 * q^47 + 101813 * q^49 - 209952 * q^51 + 904096 * q^53 + 556632 * q^55 - 260550 * q^57 - 377448 * q^59 - 2834138 * q^61 + 301806 * q^63 - 488120 * q^65 - 1411266 * q^67 + 1642545 * q^69 + 1860048 * q^71 - 940078 * q^73 - 350433 * q^75 - 7845032 * q^77 - 1217086 * q^79 + 2657205 * q^81 - 4542112 * q^83 - 23930564 * q^85 + 2320866 * q^87 - 21715884 * q^89 - 932204 * q^91 - 6103188 * q^93 - 36809060 * q^95 - 30249906 * q^97 - 758160 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more