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Properties

Label 9.76.a.c.1.4
Level $9$
Weight $76$
Character 9.1
Self dual yes
Analytic conductor $320.606$
Analytic rank $1$
Dimension $6$
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] = [9,76,Mod(1,9)] 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("9.1"); S:= CuspForms(chi, 76); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(9, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 76, names="a")
 
Level: \( N \) \(=\) \( 9 = 3^{2} \)
Weight: \( k \) \(=\) \( 76 \)
Character orbit: \([\chi]\) \(=\) 9.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6,57080822040] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(320.605553540\)
Analytic rank: \(1\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 3 x^{5} + \cdots - 67\!\cdots\!50 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: multiple of \( 2^{58}\cdot 3^{36}\cdot 5^{7}\cdot 7^{3}\cdot 11\cdot 13\cdot 19 \)
Twist minimal: no (minimal twist has level 1)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(-1.60397e9\) of defining polynomial
Character \(\chi\) \(=\) 9.1

$q$-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)
 
\(f(q)\) \(=\) \(q+1.25000e11 q^{2} -2.21540e22 q^{4} -1.80927e26 q^{5} +2.76229e31 q^{7} -7.49160e33 q^{8} -2.26158e37 q^{10} -7.60456e38 q^{11} +2.57727e41 q^{13} +3.45284e42 q^{14} -9.94899e43 q^{16} +1.39330e46 q^{17} -1.16447e48 q^{19} +4.00827e48 q^{20} -9.50567e49 q^{22} -9.81970e50 q^{23} +6.26484e51 q^{25} +3.22158e52 q^{26} -6.11958e53 q^{28} -3.47815e54 q^{29} +1.19579e56 q^{31} +2.70588e56 q^{32} +1.74162e57 q^{34} -4.99772e57 q^{35} +1.03750e59 q^{37} -1.45558e59 q^{38} +1.35543e60 q^{40} -4.07748e60 q^{41} +2.20097e61 q^{43} +1.68472e61 q^{44} -1.22746e62 q^{46} +4.37785e62 q^{47} -1.64884e63 q^{49} +7.83102e62 q^{50} -5.70970e63 q^{52} -7.68796e64 q^{53} +1.37587e65 q^{55} -2.06939e65 q^{56} -4.34767e65 q^{58} +3.79588e66 q^{59} +5.02447e66 q^{61} +1.49473e67 q^{62} +3.75820e67 q^{64} -4.66298e67 q^{65} +3.39062e68 q^{67} -3.08672e68 q^{68} -6.24713e68 q^{70} +3.17915e69 q^{71} +7.11742e69 q^{73} +1.29687e70 q^{74} +2.57978e70 q^{76} -2.10060e70 q^{77} +9.47303e70 q^{79} +1.80004e70 q^{80} -5.09684e71 q^{82} +1.43323e71 q^{83} -2.52086e72 q^{85} +2.75120e72 q^{86} +5.69703e72 q^{88} -9.94475e72 q^{89} +7.11916e72 q^{91} +2.17546e73 q^{92} +5.47229e73 q^{94} +2.10685e74 q^{95} -7.42743e73 q^{97} -2.06105e74 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 57080822040 q^{2} + 17\!\cdots\!28 q^{4} + 38\!\cdots\!40 q^{5} + 19\!\cdots\!00 q^{7} - 44\!\cdots\!20 q^{8} + 13\!\cdots\!60 q^{10} + 94\!\cdots\!88 q^{11} + 53\!\cdots\!20 q^{13} - 82\!\cdots\!76 q^{14}+ \cdots + 16\!\cdots\!20 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.25000e11 0.643108 0.321554 − 0.946891i \(-0.395795\pi\)
0.321554 + 0.946891i \(0.395795\pi\)
\(3\) 0 0
\(4\) −2.21540e22 −0.586413
\(5\) −1.80927e26 −1.11206 −0.556030 − 0.831162i \(-0.687676\pi\)
−0.556030 + 0.831162i \(0.687676\pi\)
\(6\) 0 0
\(7\) 2.76229e31 0.562461 0.281230 − 0.959640i \(-0.409258\pi\)
0.281230 + 0.959640i \(0.409258\pi\)
\(8\) −7.49160e33 −1.02023
\(9\) 0 0
\(10\) −2.26158e37 −0.715175
\(11\) −7.60456e38 −0.674293 −0.337146 − 0.941452i \(-0.609462\pi\)
−0.337146 + 0.941452i \(0.609462\pi\)
\(12\) 0 0
\(13\) 2.57727e41 0.434795 0.217397 − 0.976083i \(-0.430243\pi\)
0.217397 + 0.976083i \(0.430243\pi\)
\(14\) 3.45284e42 0.361723
\(15\) 0 0
\(16\) −9.94899e43 −0.0697076
\(17\) 1.39330e46 1.00510 0.502551 − 0.864548i \(-0.332395\pi\)
0.502551 + 0.864548i \(0.332395\pi\)
\(18\) 0 0
\(19\) −1.16447e48 −1.29679 −0.648394 − 0.761305i \(-0.724559\pi\)
−0.648394 + 0.761305i \(0.724559\pi\)
\(20\) 4.00827e48 0.652126
\(21\) 0 0
\(22\) −9.50567e49 −0.433643
\(23\) −9.81970e50 −0.845872 −0.422936 − 0.906160i \(-0.639000\pi\)
−0.422936 + 0.906160i \(0.639000\pi\)
\(24\) 0 0
\(25\) 6.26484e51 0.236679
\(26\) 3.22158e52 0.279620
\(27\) 0 0
\(28\) −6.11958e53 −0.329834
\(29\) −3.47815e54 −0.502832 −0.251416 − 0.967879i \(-0.580896\pi\)
−0.251416 + 0.967879i \(0.580896\pi\)
\(30\) 0 0
\(31\) 1.19579e56 1.41773 0.708863 − 0.705346i \(-0.249209\pi\)
0.708863 + 0.705346i \(0.249209\pi\)
\(32\) 2.70588e56 0.975405
\(33\) 0 0
\(34\) 1.74162e57 0.646388
\(35\) −4.99772e57 −0.625490
\(36\) 0 0
\(37\) 1.03750e59 1.61593 0.807963 − 0.589233i \(-0.200570\pi\)
0.807963 + 0.589233i \(0.200570\pi\)
\(38\) −1.45558e59 −0.833974
\(39\) 0 0
\(40\) 1.35543e60 1.13456
\(41\) −4.07748e60 −1.35207 −0.676033 − 0.736872i \(-0.736302\pi\)
−0.676033 + 0.736872i \(0.736302\pi\)
\(42\) 0 0
\(43\) 2.20097e61 1.22334 0.611670 − 0.791113i \(-0.290498\pi\)
0.611670 + 0.791113i \(0.290498\pi\)
\(44\) 1.68472e61 0.395414
\(45\) 0 0
\(46\) −1.22746e62 −0.543987
\(47\) 4.37785e62 0.866146 0.433073 − 0.901359i \(-0.357429\pi\)
0.433073 + 0.901359i \(0.357429\pi\)
\(48\) 0 0
\(49\) −1.64884e63 −0.683638
\(50\) 7.83102e62 0.152210
\(51\) 0 0
\(52\) −5.70970e63 −0.254969
\(53\) −7.68796e64 −1.68061 −0.840303 − 0.542117i \(-0.817623\pi\)
−0.840303 + 0.542117i \(0.817623\pi\)
\(54\) 0 0
\(55\) 1.37587e65 0.749855
\(56\) −2.06939e65 −0.573841
\(57\) 0 0
\(58\) −4.34767e65 −0.323375
\(59\) 3.79588e66 1.48717 0.743587 − 0.668639i \(-0.233123\pi\)
0.743587 + 0.668639i \(0.233123\pi\)
\(60\) 0 0
\(61\) 5.02447e66 0.563925 0.281962 − 0.959425i \(-0.409015\pi\)
0.281962 + 0.959425i \(0.409015\pi\)
\(62\) 1.49473e67 0.911750
\(63\) 0 0
\(64\) 3.75820e67 0.696998
\(65\) −4.66298e67 −0.483518
\(66\) 0 0
\(67\) 3.39062e68 1.12843 0.564214 − 0.825629i \(-0.309179\pi\)
0.564214 + 0.825629i \(0.309179\pi\)
\(68\) −3.08672e68 −0.589404
\(69\) 0 0
\(70\) −6.24713e68 −0.402258
\(71\) 3.17915e69 1.20261 0.601303 − 0.799021i \(-0.294648\pi\)
0.601303 + 0.799021i \(0.294648\pi\)
\(72\) 0 0
\(73\) 7.11742e69 0.949982 0.474991 − 0.879991i \(-0.342451\pi\)
0.474991 + 0.879991i \(0.342451\pi\)
\(74\) 1.29687e70 1.03921
\(75\) 0 0
\(76\) 2.57978e70 0.760453
\(77\) −2.10060e70 −0.379263
\(78\) 0 0
\(79\) 9.47303e70 0.653842 0.326921 − 0.945052i \(-0.393989\pi\)
0.326921 + 0.945052i \(0.393989\pi\)
\(80\) 1.80004e70 0.0775190
\(81\) 0 0
\(82\) −5.09684e71 −0.869523
\(83\) 1.43323e71 0.155198 0.0775992 − 0.996985i \(-0.475275\pi\)
0.0775992 + 0.996985i \(0.475275\pi\)
\(84\) 0 0
\(85\) −2.52086e72 −1.11773
\(86\) 2.75120e72 0.786739
\(87\) 0 0
\(88\) 5.69703e72 0.687937
\(89\) −9.94475e72 −0.786083 −0.393042 − 0.919521i \(-0.628577\pi\)
−0.393042 + 0.919521i \(0.628577\pi\)
\(90\) 0 0
\(91\) 7.11916e72 0.244555
\(92\) 2.17546e73 0.496030
\(93\) 0 0
\(94\) 5.47229e73 0.557025
\(95\) 2.10685e74 1.44211
\(96\) 0 0
\(97\) −7.42743e73 −0.232755 −0.116377 − 0.993205i \(-0.537128\pi\)
−0.116377 + 0.993205i \(0.537128\pi\)
\(98\) −2.06105e74 −0.439653
\(99\) 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 9.76.a.c.1.4 6
3.2 odd 2 1.76.a.a.1.3 ✓ 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1.76.a.a.1.3 ✓ 6 3.2 odd 2
9.76.a.c.1.4 6 1.1 even 1 trivial