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Properties

Label 765.6.a.g.1.4
Level $765$
Weight $6$
Character 765.1
Self dual yes
Analytic conductor $122.694$
Analytic rank $0$
Dimension $5$
CM no
Inner twists $1$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [765,6,Mod(1,765)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("765.1"); S:= CuspForms(chi, 6); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(765, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 6, names="a")
 
Level: \( N \) \(=\) \( 765 = 3^{2} \cdot 5 \cdot 17 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 765.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [5,7,0,43,-125,0,-204] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(122.693622157\)
Analytic rank: \(0\)
Dimension: \(5\)
Coefficient field: \(\mathbb{Q}[x]/(x^{5} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - 2x^{4} - 95x^{3} + 220x^{2} + 1668x - 4640 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 85)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(3.95319\) of defining polynomial
Character \(\chi\) \(=\) 765.1

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+4.95319 q^{2} -7.46587 q^{4} -25.0000 q^{5} +13.5050 q^{7} -195.482 q^{8} -123.830 q^{10} -742.882 q^{11} -179.248 q^{13} +66.8928 q^{14} -729.353 q^{16} -289.000 q^{17} -1855.97 q^{19} +186.647 q^{20} -3679.64 q^{22} +2557.44 q^{23} +625.000 q^{25} -887.850 q^{26} -100.826 q^{28} +1108.67 q^{29} -8774.81 q^{31} +2642.80 q^{32} -1431.47 q^{34} -337.625 q^{35} -5237.19 q^{37} -9192.99 q^{38} +4887.05 q^{40} +16598.4 q^{41} +14484.7 q^{43} +5546.26 q^{44} +12667.5 q^{46} -16088.0 q^{47} -16624.6 q^{49} +3095.75 q^{50} +1338.24 q^{52} -11804.4 q^{53} +18572.1 q^{55} -2639.98 q^{56} +5491.46 q^{58} +29390.1 q^{59} +29333.5 q^{61} -43463.3 q^{62} +36429.6 q^{64} +4481.20 q^{65} +4122.11 q^{67} +2157.64 q^{68} -1672.32 q^{70} +13649.5 q^{71} -33582.1 q^{73} -25940.8 q^{74} +13856.4 q^{76} -10032.6 q^{77} -86281.4 q^{79} +18233.8 q^{80} +82215.3 q^{82} +73620.8 q^{83} +7225.00 q^{85} +71745.7 q^{86} +145220. q^{88} +100199. q^{89} -2420.74 q^{91} -19093.5 q^{92} -79687.2 q^{94} +46399.3 q^{95} +4407.67 q^{97} -82344.9 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q + 7 q^{2} + 43 q^{4} - 125 q^{5} - 204 q^{7} + 63 q^{8} - 175 q^{10} + 792 q^{11} + 88 q^{13} - 860 q^{14} - 2365 q^{16} - 1445 q^{17} - 5160 q^{19} - 1075 q^{20} - 3058 q^{22} + 6140 q^{23} + 3125 q^{25}+ \cdots - 49329 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\). You can download additional coefficients here.



Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 4.95319 0.875609 0.437805 − 0.899070i \(-0.355756\pi\)
0.437805 + 0.899070i \(0.355756\pi\)
\(3\) 0 0
\(4\) −7.46587 −0.233308
\(5\) −25.0000 −0.447214
\(6\) 0 0
\(7\) 13.5050 0.104172 0.0520858 − 0.998643i \(-0.483413\pi\)
0.0520858 + 0.998643i \(0.483413\pi\)
\(8\) −195.482 −1.07990
\(9\) 0 0
\(10\) −123.830 −0.391584
\(11\) −742.882 −1.85114 −0.925568 − 0.378581i \(-0.876412\pi\)
−0.925568 + 0.378581i \(0.876412\pi\)
\(12\) 0 0
\(13\) −179.248 −0.294168 −0.147084 − 0.989124i \(-0.546989\pi\)
−0.147084 + 0.989124i \(0.546989\pi\)
\(14\) 66.8928 0.0912135
\(15\) 0 0
\(16\) −729.353 −0.712259
\(17\) −289.000 −0.242536
\(18\) 0 0
\(19\) −1855.97 −1.17947 −0.589735 − 0.807596i \(-0.700768\pi\)
−0.589735 + 0.807596i \(0.700768\pi\)
\(20\) 186.647 0.104339
\(21\) 0 0
\(22\) −3679.64 −1.62087
\(23\) 2557.44 1.00806 0.504029 − 0.863687i \(-0.331850\pi\)
0.504029 + 0.863687i \(0.331850\pi\)
\(24\) 0 0
\(25\) 625.000 0.200000
\(26\) −887.850 −0.257577
\(27\) 0 0
\(28\) −100.826 −0.0243041
\(29\) 1108.67 0.244798 0.122399 − 0.992481i \(-0.460941\pi\)
0.122399 + 0.992481i \(0.460941\pi\)
\(30\) 0 0
\(31\) −8774.81 −1.63996 −0.819980 − 0.572391i \(-0.806016\pi\)
−0.819980 + 0.572391i \(0.806016\pi\)
\(32\) 2642.80 0.456236
\(33\) 0 0
\(34\) −1431.47 −0.212366
\(35\) −337.625 −0.0465869
\(36\) 0 0
\(37\) −5237.19 −0.628919 −0.314459 − 0.949271i \(-0.601823\pi\)
−0.314459 + 0.949271i \(0.601823\pi\)
\(38\) −9192.99 −1.03276
\(39\) 0 0
\(40\) 4887.05 0.482944
\(41\) 16598.4 1.54208 0.771042 − 0.636785i \(-0.219736\pi\)
0.771042 + 0.636785i \(0.219736\pi\)
\(42\) 0 0
\(43\) 14484.7 1.19465 0.597324 − 0.802000i \(-0.296231\pi\)
0.597324 + 0.802000i \(0.296231\pi\)
\(44\) 5546.26 0.431886
\(45\) 0 0
\(46\) 12667.5 0.882665
\(47\) −16088.0 −1.06233 −0.531164 − 0.847269i \(-0.678245\pi\)
−0.531164 + 0.847269i \(0.678245\pi\)
\(48\) 0 0
\(49\) −16624.6 −0.989148
\(50\) 3095.75 0.175122
\(51\) 0 0
\(52\) 1338.24 0.0686319
\(53\) −11804.4 −0.577237 −0.288618 − 0.957444i \(-0.593196\pi\)
−0.288618 + 0.957444i \(0.593196\pi\)
\(54\) 0 0
\(55\) 18572.1 0.827853
\(56\) −2639.98 −0.112494
\(57\) 0 0
\(58\) 5491.46 0.214347
\(59\) 29390.1 1.09919 0.549593 − 0.835433i \(-0.314783\pi\)
0.549593 + 0.835433i \(0.314783\pi\)
\(60\) 0 0
\(61\) 29333.5 1.00934 0.504671 − 0.863311i \(-0.331614\pi\)
0.504671 + 0.863311i \(0.331614\pi\)
\(62\) −43463.3 −1.43597
\(63\) 0 0
\(64\) 36429.6 1.11174
\(65\) 4481.20 0.131556
\(66\) 0 0
\(67\) 4122.11 0.112184 0.0560922 − 0.998426i \(-0.482136\pi\)
0.0560922 + 0.998426i \(0.482136\pi\)
\(68\) 2157.64 0.0565856
\(69\) 0 0
\(70\) −1672.32 −0.0407919
\(71\) 13649.5 0.321343 0.160672 − 0.987008i \(-0.448634\pi\)
0.160672 + 0.987008i \(0.448634\pi\)
\(72\) 0 0
\(73\) −33582.1 −0.737566 −0.368783 − 0.929516i \(-0.620225\pi\)
−0.368783 + 0.929516i \(0.620225\pi\)
\(74\) −25940.8 −0.550687
\(75\) 0 0
\(76\) 13856.4 0.275180
\(77\) −10032.6 −0.192836
\(78\) 0 0
\(79\) −86281.4 −1.55543 −0.777713 − 0.628619i \(-0.783621\pi\)
−0.777713 + 0.628619i \(0.783621\pi\)
\(80\) 18233.8 0.318532
\(81\) 0 0
\(82\) 82215.3 1.35026
\(83\) 73620.8 1.17302 0.586510 − 0.809942i \(-0.300501\pi\)
0.586510 + 0.809942i \(0.300501\pi\)
\(84\) 0 0
\(85\) 7225.00 0.108465
\(86\) 71745.7 1.04604
\(87\) 0 0
\(88\) 145220. 1.99903
\(89\) 100199. 1.34087 0.670435 − 0.741968i \(-0.266107\pi\)
0.670435 + 0.741968i \(0.266107\pi\)
\(90\) 0 0
\(91\) −2420.74 −0.0306440
\(92\) −19093.5 −0.235188
\(93\) 0 0
\(94\) −79687.2 −0.930184
\(95\) 46399.3 0.527475
\(96\) 0 0
\(97\) 4407.67 0.0475642 0.0237821 − 0.999717i \(-0.492429\pi\)
0.0237821 + 0.999717i \(0.492429\pi\)
\(98\) −82344.9 −0.866107
\(99\) 0 0
Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 765.6.a.g.1.4 5
3.2 odd 2 85.6.a.a.1.2 ✓ 5
15.14 odd 2 425.6.a.d.1.4 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
85.6.a.a.1.2 ✓ 5 3.2 odd 2
425.6.a.d.1.4 5 15.14 odd 2
765.6.a.g.1.4 5 1.1 even 1 trivial