LMFDB - Embedded newform 7744.2.a.dr.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7744,2,Mod(1,7744)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7744, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("7744.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$7744 = 2^{6} \cdot 11^{2}$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	7744.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:	$61.8361513253$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.5225.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{4} - x^{3} - 8x^{2} + x + 11$ x^4 - x^3 - 8*x^2 + x + 11
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 88)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$1.26498$ of defining polynomial
Character	$\chi$	$=$	7744.1

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.04678 q^{3} +0.163765 q^{5} -3.50105 q^{7} +1.18929 q^{9} -2.69372 q^{13} -0.335190 q^{15} -7.57673 q^{17} +4.61803 q^{19} +7.16586 q^{21} +0.706114 q^{23} -4.97318 q^{25} +3.70611 q^{27} +1.02891 q^{29} -5.39983 q^{31} -0.573348 q^{35} -11.5946 q^{37} +5.51344 q^{39} -0.280754 q^{41} +4.31175 q^{43} +0.194764 q^{45} -5.82857 q^{47} +5.25732 q^{49} +15.5079 q^{51} -9.87197 q^{53} -9.45208 q^{57} -3.09017 q^{59} -6.93188 q^{61} -4.16376 q^{63} -0.441137 q^{65} +1.91665 q^{67} -1.44526 q^{69} -12.1679 q^{71} -11.3764 q^{73} +10.1790 q^{75} +1.72736 q^{79} -11.1535 q^{81} +4.70950 q^{83} -1.24080 q^{85} -2.10595 q^{87} +6.31175 q^{89} +9.43083 q^{91} +11.0522 q^{93} +0.756272 q^{95} +7.00547 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$4 q + 2 q^{3} - q^{5} - q^{7} + 6 q^{9} - q^{13} - 16 q^{15} - 12 q^{17} + 14 q^{19} + q^{21} + 2 q^{23} + 11 q^{25} + 14 q^{27} + 9 q^{29} - 11 q^{31} + 18 q^{35} - 13 q^{37} + 18 q^{39} - 8 q^{41}+ \cdots + 4 q^{97}+O(q^{100})$ 4 * q + 2 * q^3 - q^5 - q^7 + 6 * q^9 - q^13 - 16 * q^15 - 12 * q^17 + 14 * q^19 + q^21 + 2 * q^23 + 11 * q^25 + 14 * q^27 + 9 * q^29 - 11 * q^31 + 18 * q^35 - 13 * q^37 + 18 * q^39 - 8 * q^41 + 3 * q^43 - 22 * q^45 - 7 * q^47 - q^49 + 14 * q^51 - 11 * q^53 + 7 * q^57 + 10 * q^59 + 17 * q^61 - 15 * q^63 - 5 * q^65 - 5 * q^67 + 4 * q^69 + 5 * q^71 - 6 * q^73 + 7 * q^75 - 7 * q^79 - 8 * q^81 + 20 * q^83 + 13 * q^85 + 3 * q^87 + 11 * q^89 + 6 * q^91 + 10 * q^93 - 6 * q^95 + 4 * q^97

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
$3$	−2.04678	−1.18171	−0.590853	−	0.806779i	$-0.701209\pi$
$3$	−2.04678	−1.18171	−0.590853	+	0.806779i	$0.701209\pi$
$4$	0	0
$5$	0.163765	0.0732379	0.0366189	−	0.999329i	$-0.488341\pi$
$5$	0.163765	0.0732379	0.0366189	+	0.999329i	$0.488341\pi$
$6$	0	0
$7$	−3.50105	−1.32327	−0.661635	−	0.749826i	$-0.730137\pi$
$7$	−3.50105	−1.32327	−0.661635	+	0.749826i	$0.730137\pi$
$8$	0	0
$9$	1.18929	0.396431
$10$	0	0
$11$	0	0
$11$	0	0
$12$	0	0
$13$	−2.69372	−0.747103	−0.373552	−	0.927609i	$-0.621860\pi$
$13$	−2.69372	−0.747103	−0.373552	+	0.927609i	$0.621860\pi$
$14$	0	0
$15$	−0.335190	−0.0865457
$16$	0	0
$17$	−7.57673	−1.83763	−0.918814	−	0.394692i	$-0.870851\pi$
$17$	−7.57673	−1.83763	−0.918814	+	0.394692i	$0.870851\pi$
$18$	0	0
$19$	4.61803	1.05945	0.529725	−	0.848170i	$-0.322295\pi$
$19$	4.61803	1.05945	0.529725	+	0.848170i	$0.322295\pi$
$20$	0	0
$21$	7.16586	1.56372
$22$	0	0
$23$	0.706114	0.147235	0.0736174	−	0.997287i	$-0.476546\pi$
$23$	0.706114	0.147235	0.0736174	+	0.997287i	$0.476546\pi$
$24$	0	0
$25$	−4.97318	−0.994636
$26$	0	0
$27$	3.70611	0.713242
$28$	0	0
$29$	1.02891	0.191064	0.0955318	−	0.995426i	$-0.469545\pi$
$29$	1.02891	0.191064	0.0955318	+	0.995426i	$0.469545\pi$
$30$	0	0
$31$	−5.39983	−0.969839	−0.484919	−	0.874559i	$-0.661151\pi$
$31$	−5.39983	−0.969839	−0.484919	+	0.874559i	$0.661151\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−0.573348	−0.0969135
$36$	0	0
$37$	−11.5946	−1.90614	−0.953070	−	0.302750i	$-0.902095\pi$
$37$	−11.5946	−1.90614	−0.953070	+	0.302750i	$0.902095\pi$
$38$	0	0
$39$	5.51344	0.882857
$40$	0	0
$41$	−0.280754	−0.0438464	−0.0219232	−	0.999760i	$-0.506979\pi$
$41$	−0.280754	−0.0438464	−0.0219232	+	0.999760i	$0.506979\pi$
$42$	0	0
$43$	4.31175	0.657536	0.328768	−	0.944411i	$-0.393367\pi$
$43$	4.31175	0.657536	0.328768	+	0.944411i	$0.393367\pi$
$44$	0	0
$45$	0.194764	0.0290337
$46$	0	0
$47$	−5.82857	−0.850185	−0.425093	−	0.905150i	$-0.639758\pi$
$47$	−5.82857	−0.850185	−0.425093	+	0.905150i	$0.639758\pi$
$48$	0	0
$49$	5.25732	0.751045
$50$	0	0
$51$	15.5079	2.17154
$52$	0	0
$53$	−9.87197	−1.35602	−0.678010	−	0.735053i	$-0.737157\pi$
$53$	−9.87197	−1.35602	−0.678010	+	0.735053i	$0.737157\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−9.45208	−1.25196
$58$	0	0
$59$	−3.09017	−0.402306	−0.201153	−	0.979560i	$-0.564469\pi$
$59$	−3.09017	−0.402306	−0.201153	+	0.979560i	$0.564469\pi$
$60$	0	0
$61$	−6.93188	−0.887536	−0.443768	−	0.896142i	$-0.646359\pi$
$61$	−6.93188	−0.887536	−0.443768	+	0.896142i	$0.646359\pi$
$62$	0	0
$63$	−4.16376	−0.524585
$64$	0	0
$65$	−0.441137	−0.0547163
$66$	0	0
$67$	1.91665	0.234157	0.117078	−	0.993123i	$-0.462647\pi$
$67$	1.91665	0.234157	0.117078	+	0.993123i	$0.462647\pi$
$68$	0	0
$69$	−1.44526	−0.173988
$70$	0	0
$71$	−12.1679	−1.44407	−0.722035	−	0.691857i	$-0.756793\pi$
$71$	−12.1679	−1.44407	−0.722035	+	0.691857i	$0.756793\pi$
$72$	0	0
$73$	−11.3764	−1.33151	−0.665753	−	0.746172i	$-0.731890\pi$
$73$	−11.3764	−1.33151	−0.665753	+	0.746172i	$0.731890\pi$
$74$	0	0
$75$	10.1790	1.17537
$76$	0	0
$77$	0	0
$78$	0	0
$79$	1.72736	0.194343	0.0971717	−	0.995268i	$-0.469020\pi$
$79$	1.72736	0.194343	0.0971717	+	0.995268i	$0.469020\pi$
$80$	0	0
$81$	−11.1535	−1.23927
$82$	0	0
$83$	4.70950	0.516934	0.258467	−	0.966020i	$-0.416783\pi$
$83$	4.70950	0.516934	0.258467	+	0.966020i	$0.416783\pi$
$84$	0	0
$85$	−1.24080	−0.134584
$86$	0	0
$87$	−2.10595	−0.225781
$88$	0	0
$89$	6.31175	0.669044	0.334522	−	0.942388i	$-0.391425\pi$
$89$	6.31175	0.669044	0.334522	+	0.942388i	$0.391425\pi$
$90$	0	0
$91$	9.43083	0.988620
$92$	0	0
$93$	11.0522	1.14606
$94$	0	0
$95$	0.756272	0.0775918
$96$	0	0
$97$	7.00547	0.711298	0.355649	−	0.934620i	$-0.384260\pi$
$97$	7.00547	0.711298	0.355649	+	0.934620i	$0.384260\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−4.02334	−0.400337	−0.200169	−	0.979761i	$-0.564149\pi$
$101$	−4.02334	−0.400337	−0.200169	+	0.979761i	$0.564149\pi$
$102$	0	0
$103$	−15.0646	−1.48436	−0.742182	−	0.670199i	$-0.766209\pi$
$103$	−15.0646	−1.48436	−0.742182	+	0.670199i	$0.766209\pi$
$104$	0	0
$105$	1.17352	0.114523
$106$	0	0
$107$	−0.518912	−0.0501651	−0.0250826	−	0.999685i	$-0.507985\pi$
$107$	−0.518912	−0.0501651	−0.0250826	+	0.999685i	$0.507985\pi$
$108$	0	0
$109$	0.285032	0.0273011	0.0136506	−	0.999907i	$-0.495655\pi$
$109$	0.285032	0.0273011	0.0136506	+	0.999907i	$0.495655\pi$
$110$	0	0
$111$	23.7315	2.25250
$112$	0	0
$113$	−11.5691	−1.08833	−0.544163	−	0.838979i	$-0.683153\pi$
$113$	−11.5691	−1.08833	−0.544163	+	0.838979i	$0.683153\pi$
$114$	0	0
$115$	0.115637	0.0107832
$116$	0	0
$117$	−3.20362	−0.296175
$118$	0	0
$119$	26.5265	2.43168
$120$	0	0
$121$	0	0
$122$	0	0
$123$	0.574640	0.0518135
$124$	0	0
$125$	−1.63326	−0.146083
$126$	0	0
$127$	17.1190	1.51906	0.759532	−	0.650470i	$-0.225428\pi$
$127$	17.1190	1.51906	0.759532	+	0.650470i	$0.225428\pi$
$128$	0	0
$129$	−8.82519	−0.777015
$130$	0	0
$131$	5.02344	0.438900	0.219450	−	0.975624i	$-0.429574\pi$
$131$	5.02344	0.438900	0.219450	+	0.975624i	$0.429574\pi$
$132$	0	0
$133$	−16.1679	−1.40194
$134$	0	0
$135$	0.606931	0.0522363
$136$	0	0
$137$	3.00547	0.256775	0.128387	−	0.991724i	$-0.459020\pi$
$137$	3.00547	0.256775	0.128387	+	0.991724i	$0.459020\pi$
$138$	0	0
$139$	−5.36062	−0.454682	−0.227341	−	0.973815i	$-0.573003\pi$
$139$	−5.36062	−0.454682	−0.227341	+	0.973815i	$0.573003\pi$
$140$	0	0
$141$	11.9298	1.00467
$142$	0	0
$143$	0	0
$144$	0	0
$145$	0.168499	0.0139931
$146$	0	0
$147$	−10.7605	−0.887515
$148$	0	0
$149$	−5.82091	−0.476868	−0.238434	−	0.971159i	$-0.576634\pi$
$149$	−5.82091	−0.476868	−0.238434	+	0.971159i	$0.576634\pi$
$150$	0	0
$151$	−3.64565	−0.296679	−0.148339	−	0.988937i	$-0.547393\pi$
$151$	−3.64565	−0.296679	−0.148339	+	0.988937i	$0.547393\pi$
$152$	0	0
$153$	−9.01094	−0.728492
$154$	0	0
$155$	−0.884303	−0.0710289
$156$	0	0
$157$	−0.145160	−0.0115850	−0.00579252	−	0.999983i	$-0.501844\pi$
$157$	−0.145160	−0.0115850	−0.00579252	+	0.999983i	$0.501844\pi$
$158$	0	0
$159$	20.2057	1.60242
$160$	0	0
$161$	−2.47214	−0.194832
$162$	0	0
$163$	−14.4384	−1.13090	−0.565451	−	0.824782i	$-0.691298\pi$
$163$	−14.4384	−1.13090	−0.565451	+	0.824782i	$0.691298\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−17.4397	−1.34953	−0.674764	−	0.738034i	$-0.735755\pi$
$167$	−17.4397	−1.34953	−0.674764	+	0.738034i	$0.735755\pi$
$168$	0	0
$169$	−5.74388	−0.441837
$170$	0	0
$171$	5.49219	0.419998
$172$	0	0
$173$	18.2429	1.38698	0.693491	−	0.720466i	$-0.256072\pi$
$173$	18.2429	1.38698	0.693491	+	0.720466i	$0.256072\pi$
$174$	0	0
$175$	17.4113	1.31617
$176$	0	0
$177$	6.32489	0.475408
$178$	0	0
$179$	−9.14799	−0.683753	−0.341876	−	0.939745i	$-0.611062\pi$
$179$	−9.14799	−0.683753	−0.341876	+	0.939745i	$0.611062\pi$
$180$	0	0
$181$	16.6401	1.23685	0.618424	−	0.785845i	$-0.287772\pi$
$181$	16.6401	1.23685	0.618424	+	0.785845i	$0.287772\pi$
$182$	0	0
$183$	14.1880	1.04881
$184$	0	0
$185$	−1.89879	−0.139602
$186$	0	0
$187$	0	0
$188$	0	0
$189$	−12.9753	−0.943812
$190$	0	0
$191$	7.67720	0.555503	0.277752	−	0.960653i	$-0.410411\pi$
$191$	7.67720	0.555503	0.277752	+	0.960653i	$0.410411\pi$
$192$	0	0
$193$	19.6695	1.41584	0.707922	−	0.706290i	$-0.249633\pi$
$193$	19.6695	1.41584	0.707922	+	0.706290i	$0.249633\pi$
$194$	0	0
$195$	0.902908	0.0646586
$196$	0	0
$197$	−22.7460	−1.62059	−0.810294	−	0.586024i	$-0.800692\pi$
$197$	−22.7460	−1.62059	−0.810294	+	0.586024i	$0.800692\pi$
$198$	0	0
$199$	−6.85748	−0.486114	−0.243057	−	0.970012i	$-0.578150\pi$
$199$	−6.85748	−0.486114	−0.243057	+	0.970012i	$0.578150\pi$
$200$	0	0
$201$	−3.92296	−0.276704
$202$	0	0
$203$	−3.60226	−0.252829
$204$	0	0
$205$	−0.0459776	−0.00321122
$206$	0	0
$207$	0.839775	0.0583684
$208$	0	0
$209$	0	0
$210$	0	0
$211$	−11.5024	−0.791858	−0.395929	−	0.918281i	$-0.629577\pi$
$211$	−11.5024	−0.791858	−0.395929	+	0.918281i	$0.629577\pi$
$212$	0	0
$213$	24.9051	1.70647
$214$	0	0
$215$	0.706114	0.0481566
$216$	0	0
$217$	18.9051	1.28336
$218$	0	0
$219$	23.2849	1.57345
$220$	0	0
$221$	20.4096	1.37290
$222$	0	0
$223$	−16.2692	−1.08946	−0.544732	−	0.838610i	$-0.683368\pi$
$223$	−16.2692	−1.08946	−0.544732	+	0.838610i	$0.683368\pi$
$224$	0	0
$225$	−5.91456	−0.394304
$226$	0	0
$227$	6.20790	0.412033	0.206016	−	0.978549i	$-0.433950\pi$
$227$	6.20790	0.412033	0.206016	+	0.978549i	$0.433950\pi$
$228$	0	0
$229$	20.6401	1.36393	0.681967	−	0.731382i	$-0.261125\pi$
$229$	20.6401	1.36393	0.681967	+	0.731382i	$0.261125\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−10.2731	−0.673013	−0.336506	−	0.941681i	$-0.609245\pi$
$233$	−10.2731	−0.673013	−0.336506	+	0.941681i	$0.609245\pi$
$234$	0	0
$235$	−0.954516	−0.0622657
$236$	0	0
$237$	−3.53552	−0.229657
$238$	0	0
$239$	20.0419	1.29641	0.648203	−	0.761468i	$-0.275521\pi$
$239$	20.0419	1.29641	0.648203	+	0.761468i	$0.275521\pi$
$240$	0	0
$241$	−6.87625	−0.442938	−0.221469	−	0.975167i	$-0.571085\pi$
$241$	−6.87625	−0.442938	−0.221469	+	0.975167i	$0.571085\pi$
$242$	0	0
$243$	11.7103	0.751216
$244$	0	0
$245$	0.860964	0.0550050
$246$	0	0
$247$	−12.4397	−0.791518
$248$	0	0
$249$	−9.63928	−0.610865
$250$	0	0
$251$	5.20159	0.328321	0.164161	−	0.986434i	$-0.447508\pi$
$251$	5.20159	0.328321	0.164161	+	0.986434i	$0.447508\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	2.53964	0.159039
$256$	0	0
$257$	18.5971	1.16006	0.580029	−	0.814596i	$-0.303041\pi$
$257$	18.5971	1.16006	0.580029	+	0.814596i	$0.303041\pi$
$258$	0	0
$259$	40.5932	2.52234
$260$	0	0
$261$	1.22367	0.0757435
$262$	0	0
$263$	−1.27857	−0.0788397	−0.0394199	−	0.999223i	$-0.512551\pi$
$263$	−1.27857	−0.0788397	−0.0394199	+	0.999223i	$0.512551\pi$
$264$	0	0
$265$	−1.61668	−0.0993120
$266$	0	0
$267$	−12.9187	−0.790614
$268$	0	0
$269$	−2.69372	−0.164239	−0.0821195	−	0.996622i	$-0.526169\pi$
$269$	−2.69372	−0.164239	−0.0821195	+	0.996622i	$0.526169\pi$
$270$	0	0
$271$	10.3875	0.630996	0.315498	−	0.948926i	$-0.397829\pi$
$271$	10.3875	0.630996	0.315498	+	0.948926i	$0.397829\pi$
$272$	0	0
$273$	−19.3028	−1.16826
$274$	0	0
$275$	0	0
$276$	0	0
$277$	7.31046	0.439243	0.219622	−	0.975585i	$-0.429518\pi$
$277$	7.31046	0.439243	0.219622	+	0.975585i	$0.429518\pi$
$278$	0	0
$279$	−6.42198	−0.384474
$280$	0	0
$281$	−16.7452	−0.998937	−0.499468	−	0.866332i	$-0.666471\pi$
$281$	−16.7452	−0.998937	−0.499468	+	0.866332i	$0.666471\pi$
$282$	0	0
$283$	−20.4783	−1.21731	−0.608656	−	0.793434i	$-0.708291\pi$
$283$	−20.4783	−1.21731	−0.608656	+	0.793434i	$0.708291\pi$
$284$	0	0
$285$	−1.54792	−0.0916908
$286$	0	0
$287$	0.982932	0.0580206
$288$	0	0
$289$	40.4068	2.37687
$290$	0	0
$291$	−14.3386	−0.840546
$292$	0	0
$293$	18.6613	1.09020	0.545102	−	0.838370i	$-0.316491\pi$
$293$	18.6613	1.09020	0.545102	+	0.838370i	$0.316491\pi$
$294$	0	0
$295$	−0.506061	−0.0294640
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−1.90207	−0.110000
$300$	0	0
$301$	−15.0956	−0.870098
$302$	0	0
$303$	8.23487	0.473081
$304$	0	0
$305$	−1.13520	−0.0650012
$306$	0	0
$307$	−16.2802	−0.929160	−0.464580	−	0.885531i	$-0.653795\pi$
$307$	−16.2802	−0.929160	−0.464580	+	0.885531i	$0.653795\pi$
$308$	0	0
$309$	30.8339	1.75408
$310$	0	0
$311$	−10.4521	−0.592683	−0.296342	−	0.955082i	$-0.595767\pi$
$311$	−10.4521	−0.592683	−0.296342	+	0.955082i	$0.595767\pi$
$312$	0	0
$313$	8.42327	0.476111	0.238056	−	0.971252i	$-0.423490\pi$
$313$	8.42327	0.476111	0.238056	+	0.971252i	$0.423490\pi$
$314$	0	0
$315$	−0.681878	−0.0384195
$316$	0	0
$317$	−21.4686	−1.20580	−0.602898	−	0.797818i	$-0.705988\pi$
$317$	−21.4686	−1.20580	−0.602898	+	0.797818i	$0.705988\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	1.06210	0.0592804
$322$	0	0
$323$	−34.9896	−1.94687
$324$	0	0
$325$	13.3964	0.743096
$326$	0	0
$327$	−0.583397	−0.0322619
$328$	0	0
$329$	20.4061	1.12502
$330$	0	0
$331$	21.5452	1.18423	0.592117	−	0.805852i	$-0.298292\pi$
$331$	21.5452	1.18423	0.592117	+	0.805852i	$0.298292\pi$
$332$	0	0
$333$	−13.7894	−0.755652
$334$	0	0
$335$	0.313881	0.0171491
$336$	0	0
$337$	11.2058	0.610419	0.305210	−	0.952285i	$-0.401273\pi$
$337$	11.2058	0.610419	0.305210	+	0.952285i	$0.401273\pi$
$338$	0	0
$339$	23.6793	1.28608
$340$	0	0
$341$	0	0
$342$	0	0
$343$	6.10121	0.329434
$344$	0	0
$345$	−0.236682	−0.0127425
$346$	0	0
$347$	12.5363	0.672985	0.336493	−	0.941686i	$-0.390759\pi$
$347$	12.5363	0.672985	0.336493	+	0.941686i	$0.390759\pi$
$348$	0	0
$349$	−16.0821	−0.860853	−0.430426	−	0.902626i	$-0.641637\pi$
$349$	−16.0821	−0.860853	−0.430426	+	0.902626i	$0.641637\pi$
$350$	0	0
$351$	−9.98323	−0.532865
$352$	0	0
$353$	−16.9450	−0.901892	−0.450946	−	0.892551i	$-0.648913\pi$
$353$	−16.9450	−0.901892	−0.450946	+	0.892551i	$0.648913\pi$
$354$	0	0
$355$	−1.99268	−0.105761
$356$	0	0
$357$	−54.2938	−2.87353
$358$	0	0
$359$	24.6740	1.30224	0.651122	−	0.758973i	$-0.274299\pi$
$359$	24.6740	1.30224	0.651122	+	0.758973i	$0.274299\pi$
$360$	0	0
$361$	2.32624	0.122434
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−1.86305	−0.0975167
$366$	0	0
$367$	−30.9490	−1.61552	−0.807762	−	0.589508i	$-0.799322\pi$
$367$	−30.9490	−1.61552	−0.807762	+	0.589508i	$0.799322\pi$
$368$	0	0
$369$	−0.333898	−0.0173820
$370$	0	0
$371$	34.5622	1.79438
$372$	0	0
$373$	16.6908	0.864217	0.432108	−	0.901822i	$-0.357770\pi$
$373$	16.6908	0.864217	0.432108	+	0.901822i	$0.357770\pi$
$374$	0	0
$375$	3.34291	0.172627
$376$	0	0
$377$	−2.77159	−0.142744
$378$	0	0
$379$	−16.6090	−0.853146	−0.426573	−	0.904453i	$-0.640279\pi$
$379$	−16.6090	−0.853146	−0.426573	+	0.904453i	$0.640279\pi$
$380$	0	0
$381$	−35.0387	−1.79509
$382$	0	0
$383$	−2.72945	−0.139469	−0.0697343	−	0.997566i	$-0.522215\pi$
$383$	−2.72945	−0.139469	−0.0697343	+	0.997566i	$0.522215\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	5.12793	0.260668
$388$	0	0
$389$	15.2712	0.774283	0.387141	−	0.922020i	$-0.373462\pi$
$389$	15.2712	0.774283	0.387141	+	0.922020i	$0.373462\pi$
$390$	0	0
$391$	−5.35003	−0.270563
$392$	0	0
$393$	−10.2819	−0.518651
$394$	0	0
$395$	0.282881	0.0142333
$396$	0	0
$397$	−21.4810	−1.07810	−0.539050	−	0.842274i	$-0.681217\pi$
$397$	−21.4810	−1.07810	−0.539050	+	0.842274i	$0.681217\pi$
$398$	0	0
$399$	33.0922	1.65668
$400$	0	0
$401$	−35.7778	−1.78666	−0.893328	−	0.449405i	$-0.851636\pi$
$401$	−35.7778	−1.78666	−0.893328	+	0.449405i	$0.851636\pi$
$402$	0	0
$403$	14.5456	0.724570
$404$	0	0
$405$	−1.82655	−0.0907618
$406$	0	0
$407$	0	0
$408$	0	0
$409$	10.5601	0.522161	0.261081	−	0.965317i	$-0.415921\pi$
$409$	10.5601	0.522161	0.261081	+	0.965317i	$0.415921\pi$
$410$	0	0
$411$	−6.15153	−0.303433
$412$	0	0
$413$	10.8188	0.532360
$414$	0	0
$415$	0.771250	0.0378592
$416$	0	0
$417$	10.9720	0.537301
$418$	0	0
$419$	29.9952	1.46536	0.732680	−	0.680573i	$-0.238269\pi$
$419$	29.9952	1.46536	0.732680	+	0.680573i	$0.238269\pi$
$420$	0	0
$421$	21.2049	1.03346	0.516731	−	0.856148i	$-0.327149\pi$
$421$	21.2049	1.03346	0.516731	+	0.856148i	$0.327149\pi$
$422$	0	0
$423$	−6.93188	−0.337039
$424$	0	0
$425$	37.6805	1.82777
$426$	0	0
$427$	24.2688	1.17445
$428$	0	0
$429$	0	0
$430$	0	0
$431$	13.4577	0.648232	0.324116	−	0.946017i	$-0.394933\pi$
$431$	13.4577	0.648232	0.324116	+	0.946017i	$0.394933\pi$
$432$	0	0
$433$	29.9161	1.43768	0.718838	−	0.695178i	$-0.244674\pi$
$433$	29.9161	1.43768	0.718838	+	0.695178i	$0.244674\pi$
$434$	0	0
$435$	−0.344880	−0.0165357
$436$	0	0
$437$	3.26086	0.155988
$438$	0	0
$439$	33.1644	1.58285	0.791425	−	0.611266i	$-0.209339\pi$
$439$	33.1644	1.58285	0.791425	+	0.611266i	$0.209339\pi$
$440$	0	0
$441$	6.25248	0.297737
$442$	0	0
$443$	−27.8197	−1.32175	−0.660877	−	0.750495i	$-0.729815\pi$
$443$	−27.8197	−1.32175	−0.660877	+	0.750495i	$0.729815\pi$
$444$	0	0
$445$	1.03364	0.0489994
$446$	0	0
$447$	11.9141	0.563518
$448$	0	0
$449$	−7.29588	−0.344314	−0.172157	−	0.985070i	$-0.555074\pi$
$449$	−7.29588	−0.344314	−0.172157	+	0.985070i	$0.555074\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	7.46183	0.350587
$454$	0	0
$455$	1.54444	0.0724044
$456$	0	0
$457$	−16.1301	−0.754534	−0.377267	−	0.926105i	$-0.623136\pi$
$457$	−16.1301	−0.754534	−0.377267	+	0.926105i	$0.623136\pi$
$458$	0	0
$459$	−28.0802	−1.31067
$460$	0	0
$461$	14.4253	0.671851	0.335926	−	0.941888i	$-0.390951\pi$
$461$	14.4253	0.671851	0.335926	+	0.941888i	$0.390951\pi$
$462$	0	0
$463$	19.7906	0.919748	0.459874	−	0.887984i	$-0.347895\pi$
$463$	19.7906	0.919748	0.459874	+	0.887984i	$0.347895\pi$
$464$	0	0
$465$	1.80997	0.0839354
$466$	0	0
$467$	37.8183	1.75002	0.875012	−	0.484102i	$-0.160853\pi$
$467$	37.8183	1.75002	0.875012	+	0.484102i	$0.160853\pi$
$468$	0	0
$469$	−6.71029	−0.309853
$470$	0	0
$471$	0.297110	0.0136901
$472$	0	0
$473$	0	0
$474$	0	0
$475$	−22.9663	−1.05377
$476$	0	0
$477$	−11.7407	−0.537568
$478$	0	0
$479$	38.3330	1.75148	0.875739	−	0.482785i	$-0.160375\pi$
$479$	38.3330	1.75148	0.875739	+	0.482785i	$0.160375\pi$
$480$	0	0
$481$	31.2326	1.42408
$482$	0	0
$483$	5.05991	0.230234
$484$	0	0
$485$	1.14725	0.0520940
$486$	0	0
$487$	11.4584	0.519227	0.259614	−	0.965713i	$-0.416405\pi$
$487$	11.4584	0.519227	0.259614	+	0.965713i	$0.416405\pi$
$488$	0	0
$489$	29.5522	1.33639
$490$	0	0
$491$	24.4874	1.10510	0.552550	−	0.833480i	$-0.313655\pi$
$491$	24.4874	1.10510	0.552550	+	0.833480i	$0.313655\pi$
$492$	0	0
$493$	−7.79577	−0.351104
$494$	0	0
$495$	0	0
$496$	0	0
$497$	42.6005	1.91089
$498$	0	0
$499$	15.2088	0.680839	0.340420	−	0.940274i	$-0.389431\pi$
$499$	15.2088	0.680839	0.340420	+	0.940274i	$0.389431\pi$
$500$	0	0
$501$	35.6953	1.59475
$502$	0	0
$503$	−26.6392	−1.18778	−0.593891	−	0.804545i	$-0.702409\pi$
$503$	−26.6392	−1.18778	−0.593891	+	0.804545i	$0.702409\pi$
$504$	0	0
$505$	−0.658882	−0.0293198
$506$	0	0
$507$	11.7564	0.522121
$508$	0	0
$509$	−15.2919	−0.677800	−0.338900	−	0.940822i	$-0.610055\pi$
$509$	−15.2919	−0.677800	−0.338900	+	0.940822i	$0.610055\pi$
$510$	0	0
$511$	39.8293	1.76194
$512$	0	0
$513$	17.1150	0.755644
$514$	0	0
$515$	−2.46706	−0.108712
$516$	0	0
$517$	0	0
$518$	0	0
$519$	−37.3391	−1.63901
$520$	0	0
$521$	−34.6400	−1.51761	−0.758804	−	0.651319i	$-0.774216\pi$
$521$	−34.6400	−1.51761	−0.758804	+	0.651319i	$0.774216\pi$
$522$	0	0
$523$	−16.7107	−0.730707	−0.365354	−	0.930869i	$-0.619052\pi$
$523$	−16.7107	−0.730707	−0.365354	+	0.930869i	$0.619052\pi$
$524$	0	0
$525$	−35.6371	−1.55533
$526$	0	0
$527$	40.9131	1.78220
$528$	0	0
$529$	−22.5014	−0.978322
$530$	0	0
$531$	−3.67511	−0.159486
$532$	0	0
$533$	0.756272	0.0327578
$534$	0	0
$535$	−0.0849796	−0.00367399
$536$	0	0
$537$	18.7239	0.807995
$538$	0	0
$539$	0	0
$540$	0	0
$541$	12.9233	0.555617	0.277808	−	0.960636i	$-0.410392\pi$
$541$	12.9233	0.555617	0.277808	+	0.960636i	$0.410392\pi$
$542$	0	0
$543$	−34.0585	−1.46159
$544$	0	0
$545$	0.0466782	0.00199948
$546$	0	0
$547$	5.22204	0.223278	0.111639	−	0.993749i	$-0.464390\pi$
$547$	5.22204	0.223278	0.111639	+	0.993749i	$0.464390\pi$
$548$	0	0
$549$	−8.24403	−0.351846
$550$	0	0
$551$	4.75154	0.202422
$552$	0	0
$553$	−6.04757	−0.257169
$554$	0	0
$555$	3.88639	0.164968
$556$	0	0
$557$	11.8023	0.500080	0.250040	−	0.968236i	$-0.419556\pi$
$557$	11.8023	0.500080	0.250040	+	0.968236i	$0.419556\pi$
$558$	0	0
$559$	−11.6147	−0.491247
$560$	0	0
$561$	0	0
$562$	0	0
$563$	36.8583	1.55339	0.776696	−	0.629876i	$-0.216894\pi$
$563$	36.8583	1.55339	0.776696	+	0.629876i	$0.216894\pi$
$564$	0	0
$565$	−1.89461	−0.0797067
$566$	0	0
$567$	39.0488	1.63989
$568$	0	0
$569$	−26.9651	−1.13043	−0.565217	−	0.824942i	$-0.691208\pi$
$569$	−26.9651	−1.13043	−0.565217	+	0.824942i	$0.691208\pi$
$570$	0	0
$571$	−14.3565	−0.600801	−0.300400	−	0.953813i	$-0.597120\pi$
$571$	−14.3565	−0.600801	−0.300400	+	0.953813i	$0.597120\pi$
$572$	0	0
$573$	−15.7135	−0.656442
$574$	0	0
$575$	−3.51163	−0.146445
$576$	0	0
$577$	10.5941	0.441040	0.220520	−	0.975382i	$-0.429225\pi$
$577$	10.5941	0.441040	0.220520	+	0.975382i	$0.429225\pi$
$578$	0	0
$579$	−40.2592	−1.67311
$580$	0	0
$581$	−16.4882	−0.684044
$582$	0	0
$583$	0	0
$584$	0	0
$585$	−0.524640	−0.0216912
$586$	0	0
$587$	19.9943	0.825254	0.412627	−	0.910900i	$-0.364611\pi$
$587$	19.9943	0.825254	0.412627	+	0.910900i	$0.364611\pi$
$588$	0	0
$589$	−24.9366	−1.02750
$590$	0	0
$591$	46.5560	1.91506
$592$	0	0
$593$	12.8414	0.527334	0.263667	−	0.964614i	$-0.415068\pi$
$593$	12.8414	0.527334	0.263667	+	0.964614i	$0.415068\pi$
$594$	0	0
$595$	4.34410	0.178091
$596$	0	0
$597$	14.0357	0.574444
$598$	0	0
$599$	−18.3058	−0.747954	−0.373977	−	0.927438i	$-0.622006\pi$
$599$	−18.3058	−0.747954	−0.373977	+	0.927438i	$0.622006\pi$
$600$	0	0
$601$	16.1250	0.657753	0.328876	−	0.944373i	$-0.393330\pi$
$601$	16.1250	0.657753	0.328876	+	0.944373i	$0.393330\pi$
$602$	0	0
$603$	2.27946	0.0928269
$604$	0	0
$605$	0	0
$606$	0	0
$607$	9.99855	0.405829	0.202914	−	0.979196i	$-0.434959\pi$
$607$	9.99855	0.405829	0.202914	+	0.979196i	$0.434959\pi$
$608$	0	0
$609$	7.37301	0.298770
$610$	0	0
$611$	15.7005	0.635176
$612$	0	0
$613$	−39.9553	−1.61378	−0.806889	−	0.590703i	$-0.798851\pi$
$613$	−39.9553	−1.61378	−0.806889	+	0.590703i	$0.798851\pi$
$614$	0	0
$615$	0.0941059	0.00379471
$616$	0	0
$617$	−28.5485	−1.14932	−0.574660	−	0.818392i	$-0.694866\pi$
$617$	−28.5485	−1.14932	−0.574660	+	0.818392i	$0.694866\pi$
$618$	0	0
$619$	13.0229	0.523434	0.261717	−	0.965145i	$-0.415711\pi$
$619$	13.0229	0.523434	0.261717	+	0.965145i	$0.415711\pi$
$620$	0	0
$621$	2.61694	0.105014
$622$	0	0
$623$	−22.0977	−0.885327
$624$	0	0
$625$	24.5984	0.983937
$626$	0	0
$627$	0	0
$628$	0	0
$629$	87.8491	3.50277
$630$	0	0
$631$	19.4294	0.773471	0.386736	−	0.922191i	$-0.373603\pi$
$631$	19.4294	0.773471	0.386736	+	0.922191i	$0.373603\pi$
$632$	0	0
$633$	23.5428	0.935744
$634$	0	0
$635$	2.80349	0.111253
$636$	0	0
$637$	−14.1617	−0.561108
$638$	0	0
$639$	−14.4712	−0.572473
$640$	0	0
$641$	−14.5658	−0.575314	−0.287657	−	0.957734i	$-0.592876\pi$
$641$	−14.5658	−0.575314	−0.287657	+	0.957734i	$0.592876\pi$
$642$	0	0
$643$	−3.61127	−0.142415	−0.0712073	−	0.997462i	$-0.522685\pi$
$643$	−3.61127	−0.142415	−0.0712073	+	0.997462i	$0.522685\pi$
$644$	0	0
$645$	−1.44526	−0.0569069
$646$	0	0
$647$	6.65811	0.261757	0.130879	−	0.991398i	$-0.458220\pi$
$647$	6.65811	0.261757	0.130879	+	0.991398i	$0.458220\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	−38.6944	−1.51655
$652$	0	0
$653$	−2.63680	−0.103186	−0.0515929	−	0.998668i	$-0.516430\pi$
$653$	−2.63680	−0.103186	−0.0515929	+	0.998668i	$0.516430\pi$
$654$	0	0
$655$	0.822663	0.0321441
$656$	0	0
$657$	−13.5299	−0.527850
$658$	0	0
$659$	18.2059	0.709200	0.354600	−	0.935018i	$-0.384617\pi$
$659$	18.2059	0.709200	0.354600	+	0.935018i	$0.384617\pi$
$660$	0	0
$661$	−37.6750	−1.46539	−0.732693	−	0.680559i	$-0.761737\pi$
$661$	−37.6750	−1.46539	−0.732693	+	0.680559i	$0.761737\pi$
$662$	0	0
$663$	−41.7738	−1.62236
$664$	0	0
$665$	−2.64774	−0.102675
$666$	0	0
$667$	0.726527	0.0281312
$668$	0	0
$669$	33.2993	1.28743
$670$	0	0
$671$	0	0
$672$	0	0
$673$	17.3044	0.667037	0.333518	−	0.942744i	$-0.391764\pi$
$673$	17.3044	0.667037	0.333518	+	0.942744i	$0.391764\pi$
$674$	0	0
$675$	−18.4312	−0.709416
$676$	0	0
$677$	−9.61579	−0.369565	−0.184782	−	0.982779i	$-0.559158\pi$
$677$	−9.61579	−0.369565	−0.184782	+	0.982779i	$0.559158\pi$
$678$	0	0
$679$	−24.5265	−0.941240
$680$	0	0
$681$	−12.7062	−0.486902
$682$	0	0
$683$	−33.2554	−1.27248	−0.636241	−	0.771491i	$-0.719511\pi$
$683$	−33.2554	−1.27248	−0.636241	+	0.771491i	$0.719511\pi$
$684$	0	0
$685$	0.492191	0.0188056
$686$	0	0
$687$	−42.2456	−1.61177
$688$	0	0
$689$	26.5923	1.01309
$690$	0	0
$691$	17.0510	0.648649	0.324324	−	0.945946i	$-0.394863\pi$
$691$	17.0510	0.648649	0.324324	+	0.945946i	$0.394863\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−0.877881	−0.0332999
$696$	0	0
$697$	2.12720	0.0805733
$698$	0	0
$699$	21.0267	0.795304
$700$	0	0
$701$	36.6213	1.38317	0.691583	−	0.722297i	$-0.256913\pi$
$701$	36.6213	1.38317	0.691583	+	0.722297i	$0.256913\pi$
$702$	0	0
$703$	−53.5442	−2.01946
$704$	0	0
$705$	1.95368	0.0735798
$706$	0	0
$707$	14.0859	0.529754
$708$	0	0
$709$	−19.5539	−0.734362	−0.367181	−	0.930149i	$-0.619677\pi$
$709$	−19.5539	−0.734362	−0.367181	+	0.930149i	$0.619677\pi$
$710$	0	0
$711$	2.05434	0.0770437
$712$	0	0
$713$	−3.81290	−0.142794
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−41.0214	−1.53197
$718$	0	0
$719$	−3.53407	−0.131799	−0.0658994	−	0.997826i	$-0.520992\pi$
$719$	−3.53407	−0.131799	−0.0658994	+	0.997826i	$0.520992\pi$
$720$	0	0
$721$	52.7420	1.96421
$722$	0	0
$723$	14.0741	0.523423
$724$	0	0
$725$	−5.11695	−0.190039
$726$	0	0
$727$	−5.02836	−0.186491	−0.0932457	−	0.995643i	$-0.529724\pi$
$727$	−5.02836	−0.186491	−0.0932457	+	0.995643i	$0.529724\pi$
$728$	0	0
$729$	9.49203	0.351557
$730$	0	0
$731$	−32.6690	−1.20831
$732$	0	0
$733$	11.4073	0.421338	0.210669	−	0.977557i	$-0.432436\pi$
$733$	11.4073	0.421338	0.210669	+	0.977557i	$0.432436\pi$
$734$	0	0
$735$	−1.76220	−0.0649997
$736$	0	0
$737$	0	0
$738$	0	0
$739$	4.52359	0.166403	0.0832014	−	0.996533i	$-0.473486\pi$
$739$	4.52359	0.166403	0.0832014	+	0.996533i	$0.473486\pi$
$740$	0	0
$741$	25.4613	0.935342
$742$	0	0
$743$	−24.1484	−0.885921	−0.442960	−	0.896541i	$-0.646072\pi$
$743$	−24.1484	−0.885921	−0.442960	+	0.896541i	$0.646072\pi$
$744$	0	0
$745$	−0.953261	−0.0349248
$746$	0	0
$747$	5.60097	0.204929
$748$	0	0
$749$	1.81673	0.0663820
$750$	0	0
$751$	−14.7617	−0.538662	−0.269331	−	0.963048i	$-0.586803\pi$
$751$	−14.7617	−0.538662	−0.269331	+	0.963048i	$0.586803\pi$
$752$	0	0
$753$	−10.6465	−0.387980
$754$	0	0
$755$	−0.597030	−0.0217281
$756$	0	0
$757$	−12.6153	−0.458511	−0.229255	−	0.973366i	$-0.573629\pi$
$757$	−12.6153	−0.458511	−0.229255	+	0.973366i	$0.573629\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−24.7387	−0.896779	−0.448389	−	0.893838i	$-0.648002\pi$
$761$	−24.7387	−0.896779	−0.448389	+	0.893838i	$0.648002\pi$
$762$	0	0
$763$	−0.997910	−0.0361268
$764$	0	0
$765$	−1.47568	−0.0533532
$766$	0	0
$767$	8.32405	0.300564
$768$	0	0
$769$	−5.37741	−0.193914	−0.0969571	−	0.995289i	$-0.530911\pi$
$769$	−5.37741	−0.193914	−0.0969571	+	0.995289i	$0.530911\pi$
$770$	0	0
$771$	−38.0642	−1.37085
$772$	0	0
$773$	−16.2677	−0.585108	−0.292554	−	0.956249i	$-0.594505\pi$
$773$	−16.2677	−0.585108	−0.292554	+	0.956249i	$0.594505\pi$
$774$	0	0
$775$	26.8543	0.964637
$776$	0	0
$777$	−83.0852	−2.98067
$778$	0	0
$779$	−1.29653	−0.0464530
$780$	0	0
$781$	0	0
$782$	0	0
$783$	3.81325	0.136275
$784$	0	0
$785$	−0.0237721	−0.000848463 0
$786$	0	0
$787$	−18.5653	−0.661781	−0.330891	−	0.943669i	$-0.607349\pi$
$787$	−18.5653	−0.661781	−0.330891	+	0.943669i	$0.607349\pi$
$788$	0	0
$789$	2.61694	0.0931654
$790$	0	0
$791$	40.5038	1.44015
$792$	0	0
$793$	18.6725	0.663081
$794$	0	0
$795$	3.30899	0.117358
$796$	0	0
$797$	32.1721	1.13960	0.569798	−	0.821785i	$-0.307022\pi$
$797$	32.1721	1.13960	0.569798	+	0.821785i	$0.307022\pi$
$798$	0	0
$799$	44.1615	1.56232
$800$	0	0
$801$	7.50652	0.265230
$802$	0	0
$803$	0	0
$804$	0	0
$805$	−0.404849	−0.0142691
$806$	0	0
$807$	5.51344	0.194082
$808$	0	0
$809$	−28.9394	−1.01745	−0.508727	−	0.860928i	$-0.669884\pi$
$809$	−28.9394	−1.01745	−0.508727	+	0.860928i	$0.669884\pi$
$810$	0	0
$811$	10.0382	0.352490	0.176245	−	0.984346i	$-0.443605\pi$
$811$	10.0382	0.352490	0.176245	+	0.984346i	$0.443605\pi$
$812$	0	0
$813$	−21.2609	−0.745652
$814$	0	0
$815$	−2.36450	−0.0828249
$816$	0	0
$817$	19.9118	0.696626
$818$	0	0
$819$	11.2160	0.391919
$820$	0	0
$821$	30.0266	1.04794	0.523968	−	0.851738i	$-0.324451\pi$
$821$	30.0266	1.04794	0.523968	+	0.851738i	$0.324451\pi$
$822$	0	0
$823$	30.5575	1.06517	0.532583	−	0.846378i	$-0.321221\pi$
$823$	30.5575	1.06517	0.532583	+	0.846378i	$0.321221\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−5.87708	−0.204366	−0.102183	−	0.994766i	$-0.532583\pi$
$827$	−5.87708	−0.204366	−0.102183	+	0.994766i	$0.532583\pi$
$828$	0	0
$829$	10.4742	0.363783	0.181891	−	0.983319i	$-0.441778\pi$
$829$	10.4742	0.363783	0.181891	+	0.983319i	$0.441778\pi$
$830$	0	0
$831$	−14.9629	−0.519056
$832$	0	0
$833$	−39.8333	−1.38014
$834$	0	0
$835$	−2.85602	−0.0988366
$836$	0	0
$837$	−20.0124	−0.691730
$838$	0	0
$839$	33.3400	1.15102	0.575512	−	0.817793i	$-0.304803\pi$
$839$	33.3400	1.15102	0.575512	+	0.817793i	$0.304803\pi$
$840$	0	0
$841$	−27.9413	−0.963495
$842$	0	0
$843$	34.2737	1.18045
$844$	0	0
$845$	−0.940645	−0.0323592
$846$	0	0
$847$	0	0
$848$	0	0
$849$	41.9146	1.43850
$850$	0	0
$851$	−8.18710	−0.280650
$852$	0	0
$853$	27.6967	0.948318	0.474159	−	0.880439i	$-0.342752\pi$
$853$	27.6967	0.948318	0.474159	+	0.880439i	$0.342752\pi$
$854$	0	0
$855$	0.899428	0.0307598
$856$	0	0
$857$	−2.13818	−0.0730387	−0.0365194	−	0.999333i	$-0.511627\pi$
$857$	−2.13818	−0.0730387	−0.0365194	+	0.999333i	$0.511627\pi$
$858$	0	0
$859$	−6.06982	−0.207099	−0.103550	−	0.994624i	$-0.533020\pi$
$859$	−6.06982	−0.207099	−0.103550	+	0.994624i	$0.533020\pi$
$860$	0	0
$861$	−2.01184	−0.0685633
$862$	0	0
$863$	−29.7979	−1.01433	−0.507166	−	0.861848i	$-0.669307\pi$
$863$	−29.7979	−1.01433	−0.507166	+	0.861848i	$0.669307\pi$
$864$	0	0
$865$	2.98755	0.101580
$866$	0	0
$867$	−82.7038	−2.80877
$868$	0	0
$869$	0	0
$870$	0	0
$871$	−5.16293	−0.174939
$872$	0	0
$873$	8.33155	0.281980
$874$	0	0
$875$	5.71811	0.193307
$876$	0	0
$877$	37.0577	1.25135	0.625675	−	0.780084i	$-0.284824\pi$
$877$	37.0577	1.25135	0.625675	+	0.780084i	$0.284824\pi$
$878$	0	0
$879$	−38.1954	−1.28830
$880$	0	0
$881$	−10.6956	−0.360345	−0.180172	−	0.983635i	$-0.557666\pi$
$881$	−10.6956	−0.360345	−0.180172	+	0.983635i	$0.557666\pi$
$882$	0	0
$883$	−16.0960	−0.541675	−0.270837	−	0.962625i	$-0.587301\pi$
$883$	−16.0960	−0.541675	−0.270837	+	0.962625i	$0.587301\pi$
$884$	0	0
$885$	1.03579	0.0348178
$886$	0	0
$887$	−26.2843	−0.882540	−0.441270	−	0.897374i	$-0.645472\pi$
$887$	−26.2843	−0.882540	−0.441270	+	0.897374i	$0.645472\pi$
$888$	0	0
$889$	−59.9343	−2.01013
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−26.9166	−0.900728
$894$	0	0
$895$	−1.49812	−0.0500766
$896$	0	0
$897$	3.89312	0.129987
$898$	0	0
$899$	−5.55594	−0.185301
$900$	0	0
$901$	74.7972	2.49186
$902$	0	0
$903$	30.8974	1.02820
$904$	0	0
$905$	2.72506	0.0905841
$906$	0	0
$907$	−6.59235	−0.218895	−0.109448	−	0.993993i	$-0.534908\pi$
$907$	−6.59235	−0.218895	−0.109448	+	0.993993i	$0.534908\pi$
$908$	0	0
$909$	−4.78492	−0.158706
$910$	0	0
$911$	44.1679	1.46335	0.731675	−	0.681654i	$-0.238739\pi$
$911$	44.1679	1.46335	0.731675	+	0.681654i	$0.238739\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	2.32350	0.0768124
$916$	0	0
$917$	−17.5873	−0.580783
$918$	0	0
$919$	−31.4754	−1.03828	−0.519138	−	0.854690i	$-0.673747\pi$
$919$	−31.4754	−1.03828	−0.519138	+	0.854690i	$0.673747\pi$
$920$	0	0
$921$	33.3219	1.09799
$922$	0	0
$923$	32.7770	1.07887
$924$	0	0
$925$	57.6620	1.89592
$926$	0	0
$927$	−17.9163	−0.588447
$928$	0	0
$929$	−32.4685	−1.06526	−0.532628	−	0.846349i	$-0.678796\pi$
$929$	−32.4685	−1.06526	−0.532628	+	0.846349i	$0.678796\pi$
$930$	0	0
$931$	24.2785	0.795695
$932$	0	0
$933$	21.3931	0.700378
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−29.0094	−0.947695	−0.473847	−	0.880607i	$-0.657135\pi$
$937$	−29.0094	−0.947695	−0.473847	+	0.880607i	$0.657135\pi$
$938$	0	0
$939$	−17.2405	−0.562624
$940$	0	0
$941$	−5.88999	−0.192008	−0.0960042	−	0.995381i	$-0.530606\pi$
$941$	−5.88999	−0.192008	−0.0960042	+	0.995381i	$0.530606\pi$
$942$	0	0
$943$	−0.198244	−0.00645572
$944$	0	0
$945$	−2.12489	−0.0691228
$946$	0	0
$947$	52.2319	1.69731	0.848655	−	0.528947i	$-0.177413\pi$
$947$	52.2319	1.69731	0.848655	+	0.528947i	$0.177413\pi$
$948$	0	0
$949$	30.6448	0.994773
$950$	0	0
$951$	43.9414	1.42490
$952$	0	0
$953$	−38.4195	−1.24453	−0.622265	−	0.782807i	$-0.713787\pi$
$953$	−38.4195	−1.24453	−0.622265	+	0.782807i	$0.713787\pi$
$954$	0	0
$955$	1.25726	0.0406839
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−10.5223	−0.339783
$960$	0	0
$961$	−1.84181	−0.0594131
$962$	0	0
$963$	−0.617138	−0.0198870
$964$	0	0
$965$	3.22118	0.103693
$966$	0	0
$967$	−50.8436	−1.63502	−0.817509	−	0.575915i	$-0.804646\pi$
$967$	−50.8436	−1.63502	−0.817509	+	0.575915i	$0.804646\pi$
$968$	0	0
$969$	71.6159	2.30063
$970$	0	0
$971$	−7.89992	−0.253521	−0.126760	−	0.991933i	$-0.540458\pi$
$971$	−7.89992	−0.253521	−0.126760	+	0.991933i	$0.540458\pi$
$972$	0	0
$973$	18.7678	0.601667
$974$	0	0
$975$	−27.4193	−0.878121
$976$	0	0
$977$	−16.0190	−0.512492	−0.256246	−	0.966612i	$-0.582486\pi$
$977$	−16.0190	−0.512492	−0.256246	+	0.966612i	$0.582486\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	0.338986	0.0108230
$982$	0	0
$983$	58.4809	1.86525	0.932625	−	0.360848i	$-0.117513\pi$
$983$	58.4809	1.86525	0.932625	+	0.360848i	$0.117513\pi$
$984$	0	0
$985$	−3.72500	−0.118688
$986$	0	0
$987$	−41.7667	−1.32945
$988$	0	0
$989$	3.04459	0.0968123
$990$	0	0
$991$	17.1741	0.545552	0.272776	−	0.962078i	$-0.412058\pi$
$991$	17.1741	0.545552	0.272776	+	0.962078i	$0.412058\pi$
$992$	0	0
$993$	−44.0983	−1.39942
$994$	0	0
$995$	−1.12302	−0.0356020
$996$	0	0
$997$	41.5451	1.31575	0.657874	−	0.753128i	$-0.271456\pi$
$997$	41.5451	1.31575	0.657874	+	0.753128i	$0.271456\pi$
$998$	0	0
$999$	−42.9709	−1.35954

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	7744.2.a.dr.1.1		4
4.3	odd	2		7744.2.a.di.1.4		4
8.3	odd	2		968.2.a.n.1.1		4
8.5	even	2		1936.2.a.bb.1.4		4
11.5	even	5		704.2.m.i.641.1		8
11.9	even	5		704.2.m.i.257.1		8
11.10	odd	2		7744.2.a.ds.1.1		4
24.11	even	2		8712.2.a.ce.1.3		4
44.27	odd	10		704.2.m.l.641.2		8
44.31	odd	10		704.2.m.l.257.2		8
44.43	even	2		7744.2.a.dh.1.4		4
88.3	odd	10		968.2.i.s.9.2		8
88.5	even	10		176.2.m.d.113.2		8
88.19	even	10		968.2.i.t.9.2		8
88.21	odd	2		1936.2.a.bc.1.4		4
88.27	odd	10		88.2.i.b.25.1	✓	8
88.35	even	10		968.2.i.p.81.1		8
88.43	even	2		968.2.a.m.1.1		4
88.51	even	10		968.2.i.t.753.2		8
88.53	even	10		176.2.m.d.81.2		8
88.59	odd	10		968.2.i.s.753.2		8
88.75	odd	10		88.2.i.b.81.1	yes	8
88.83	even	10		968.2.i.p.729.1		8
264.131	odd	2		8712.2.a.cd.1.3		4
264.203	even	10		792.2.r.g.289.1		8
264.251	even	10		792.2.r.g.433.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
88.2.i.b.25.1	✓	8	88.27	odd	10
88.2.i.b.81.1	yes	8	88.75	odd	10
176.2.m.d.81.2		8	88.53	even	10
176.2.m.d.113.2		8	88.5	even	10
704.2.m.i.257.1		8	11.9	even	5
704.2.m.i.641.1		8	11.5	even	5
704.2.m.l.257.2		8	44.31	odd	10
704.2.m.l.641.2		8	44.27	odd	10
792.2.r.g.289.1		8	264.203	even	10
792.2.r.g.433.1		8	264.251	even	10
968.2.a.m.1.1		4	88.43	even	2
968.2.a.n.1.1		4	8.3	odd	2
968.2.i.p.81.1		8	88.35	even	10
968.2.i.p.729.1		8	88.83	even	10
968.2.i.s.9.2		8	88.3	odd	10
968.2.i.s.753.2		8	88.59	odd	10
968.2.i.t.9.2		8	88.19	even	10
968.2.i.t.753.2		8	88.51	even	10
1936.2.a.bb.1.4		4	8.5	even	2
1936.2.a.bc.1.4		4	88.21	odd	2
7744.2.a.dh.1.4		4	44.43	even	2
7744.2.a.di.1.4		4	4.3	odd	2
7744.2.a.dr.1.1		4	1.1	even	1	trivial
7744.2.a.ds.1.1		4	11.10	odd	2
8712.2.a.cd.1.3		4	264.131	odd	2
8712.2.a.ce.1.3		4	24.11	even	2

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

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	7744.2.a.dr.1.1

Level	$7744$
Weight	$2$
Character	7744.1
Self dual	yes
Analytic conductor	$61.836$
Analytic rank	$0$
Dimension	$4$
CM	no
Inner twists	$1$