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Properties

Label 704.6.a.q.1.2
Level $704$
Weight $6$
Character 704.1
Self dual yes
Analytic conductor $112.910$
Analytic rank $1$
Dimension $3$
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] = [704,6,Mod(1,704)] 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("704.1"); S:= CuspForms(chi, 6); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(704, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 6, names="a")
 
Level: \( N \) \(=\) \( 704 = 2^{6} \cdot 11 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 704.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [3,0,-34,0,-24] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(112.910209148\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.54492.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 52x - 38 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 11)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(8.04796\) of defining polynomial
Character \(\chi\) \(=\) 704.1

$q$-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)
 
\(f(q)\) \(=\) \(q-16.8394 q^{3} -75.2230 q^{5} -225.525 q^{7} +40.5643 q^{9} -121.000 q^{11} -455.465 q^{13} +1266.71 q^{15} +190.657 q^{17} +135.393 q^{19} +3797.69 q^{21} +2796.65 q^{23} +2533.51 q^{25} +3408.89 q^{27} +2608.58 q^{29} -1056.76 q^{31} +2037.56 q^{33} +16964.7 q^{35} -12536.8 q^{37} +7669.74 q^{39} +1130.09 q^{41} +14671.0 q^{43} -3051.37 q^{45} -16882.2 q^{47} +34054.4 q^{49} -3210.54 q^{51} -3313.02 q^{53} +9101.99 q^{55} -2279.93 q^{57} -11454.0 q^{59} +28227.5 q^{61} -9148.26 q^{63} +34261.4 q^{65} +51431.0 q^{67} -47093.8 q^{69} -16218.0 q^{71} -10168.8 q^{73} -42662.6 q^{75} +27288.5 q^{77} +60841.2 q^{79} -67260.7 q^{81} -45770.6 q^{83} -14341.8 q^{85} -43926.9 q^{87} -82267.9 q^{89} +102719. q^{91} +17795.1 q^{93} -10184.7 q^{95} +53097.0 q^{97} -4908.28 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 34 q^{3} - 24 q^{5} + 84 q^{7} - 7 q^{9} - 363 q^{11} - 486 q^{13} + 1654 q^{15} + 1086 q^{17} - 1380 q^{19} + 908 q^{21} - 3066 q^{23} - 57 q^{25} + 2990 q^{27} + 3426 q^{29} - 4098 q^{31} + 4114 q^{33}+ \cdots + 847 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −16.8394 −1.08025 −0.540123 − 0.841586i \(-0.681622\pi\)
−0.540123 + 0.841586i \(0.681622\pi\)
\(4\) 0 0
\(5\) −75.2230 −1.34563 −0.672815 − 0.739810i \(-0.734915\pi\)
−0.672815 + 0.739810i \(0.734915\pi\)
\(6\) 0 0
\(7\) −225.525 −1.73960 −0.869799 − 0.493406i \(-0.835752\pi\)
−0.869799 + 0.493406i \(0.835752\pi\)
\(8\) 0 0
\(9\) 40.5643 0.166931
\(10\) 0 0
\(11\) −121.000 −0.301511
\(12\) 0 0
\(13\) −455.465 −0.747474 −0.373737 − 0.927535i \(-0.621924\pi\)
−0.373737 + 0.927535i \(0.621924\pi\)
\(14\) 0 0
\(15\) 1266.71 1.45361
\(16\) 0 0
\(17\) 190.657 0.160003 0.0800017 − 0.996795i \(-0.474507\pi\)
0.0800017 + 0.996795i \(0.474507\pi\)
\(18\) 0 0
\(19\) 135.393 0.0860424 0.0430212 − 0.999074i \(-0.486302\pi\)
0.0430212 + 0.999074i \(0.486302\pi\)
\(20\) 0 0
\(21\) 3797.69 1.87919
\(22\) 0 0
\(23\) 2796.65 1.10235 0.551173 − 0.834391i \(-0.314180\pi\)
0.551173 + 0.834391i \(0.314180\pi\)
\(24\) 0 0
\(25\) 2533.51 0.810722
\(26\) 0 0
\(27\) 3408.89 0.899919
\(28\) 0 0
\(29\) 2608.58 0.575983 0.287991 − 0.957633i \(-0.407013\pi\)
0.287991 + 0.957633i \(0.407013\pi\)
\(30\) 0 0
\(31\) −1056.76 −0.197502 −0.0987510 − 0.995112i \(-0.531485\pi\)
−0.0987510 + 0.995112i \(0.531485\pi\)
\(32\) 0 0
\(33\) 2037.56 0.325706
\(34\) 0 0
\(35\) 16964.7 2.34086
\(36\) 0 0
\(37\) −12536.8 −1.50550 −0.752752 − 0.658304i \(-0.771274\pi\)
−0.752752 + 0.658304i \(0.771274\pi\)
\(38\) 0 0
\(39\) 7669.74 0.807456
\(40\) 0 0
\(41\) 1130.09 0.104991 0.0524954 − 0.998621i \(-0.483282\pi\)
0.0524954 + 0.998621i \(0.483282\pi\)
\(42\) 0 0
\(43\) 14671.0 1.21001 0.605005 − 0.796222i \(-0.293171\pi\)
0.605005 + 0.796222i \(0.293171\pi\)
\(44\) 0 0
\(45\) −3051.37 −0.224628
\(46\) 0 0
\(47\) −16882.2 −1.11477 −0.557383 − 0.830256i \(-0.688194\pi\)
−0.557383 + 0.830256i \(0.688194\pi\)
\(48\) 0 0
\(49\) 34054.4 2.02620
\(50\) 0 0
\(51\) −3210.54 −0.172843
\(52\) 0 0
\(53\) −3313.02 −0.162007 −0.0810035 − 0.996714i \(-0.525813\pi\)
−0.0810035 + 0.996714i \(0.525813\pi\)
\(54\) 0 0
\(55\) 9101.99 0.405723
\(56\) 0 0
\(57\) −2279.93 −0.0929469
\(58\) 0 0
\(59\) −11454.0 −0.428378 −0.214189 − 0.976792i \(-0.568711\pi\)
−0.214189 + 0.976792i \(0.568711\pi\)
\(60\) 0 0
\(61\) 28227.5 0.971286 0.485643 − 0.874157i \(-0.338585\pi\)
0.485643 + 0.874157i \(0.338585\pi\)
\(62\) 0 0
\(63\) −9148.26 −0.290394
\(64\) 0 0
\(65\) 34261.4 1.00582
\(66\) 0 0
\(67\) 51431.0 1.39971 0.699855 − 0.714285i \(-0.253248\pi\)
0.699855 + 0.714285i \(0.253248\pi\)
\(68\) 0 0
\(69\) −47093.8 −1.19081
\(70\) 0 0
\(71\) −16218.0 −0.381814 −0.190907 − 0.981608i \(-0.561143\pi\)
−0.190907 + 0.981608i \(0.561143\pi\)
\(72\) 0 0
\(73\) −10168.8 −0.223337 −0.111669 − 0.993745i \(-0.535620\pi\)
−0.111669 + 0.993745i \(0.535620\pi\)
\(74\) 0 0
\(75\) −42662.6 −0.875779
\(76\) 0 0
\(77\) 27288.5 0.524509
\(78\) 0 0
\(79\) 60841.2 1.09681 0.548404 − 0.836214i \(-0.315236\pi\)
0.548404 + 0.836214i \(0.315236\pi\)
\(80\) 0 0
\(81\) −67260.7 −1.13907
\(82\) 0 0
\(83\) −45770.6 −0.729275 −0.364638 − 0.931150i \(-0.618807\pi\)
−0.364638 + 0.931150i \(0.618807\pi\)
\(84\) 0 0
\(85\) −14341.8 −0.215306
\(86\) 0 0
\(87\) −43926.9 −0.622203
\(88\) 0 0
\(89\) −82267.9 −1.10092 −0.550460 − 0.834862i \(-0.685548\pi\)
−0.550460 + 0.834862i \(0.685548\pi\)
\(90\) 0 0
\(91\) 102719. 1.30031
\(92\) 0 0
\(93\) 17795.1 0.213351
\(94\) 0 0
\(95\) −10184.7 −0.115781
\(96\) 0 0
\(97\) 53097.0 0.572981 0.286491 − 0.958083i \(-0.407511\pi\)
0.286491 + 0.958083i \(0.407511\pi\)
\(98\) 0 0
\(99\) −4908.28 −0.0503317
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 704.6.a.q.1.2 3
4.3 odd 2 704.6.a.t.1.2 3
8.3 odd 2 176.6.a.i.1.2 3
8.5 even 2 11.6.a.b.1.2 ✓ 3
24.5 odd 2 99.6.a.g.1.2 3
40.13 odd 4 275.6.b.b.199.3 6
40.29 even 2 275.6.a.b.1.2 3
40.37 odd 4 275.6.b.b.199.4 6
56.13 odd 2 539.6.a.e.1.2 3
88.21 odd 2 121.6.a.d.1.2 3
264.197 even 2 1089.6.a.r.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
11.6.a.b.1.2 ✓ 3 8.5 even 2
99.6.a.g.1.2 3 24.5 odd 2
121.6.a.d.1.2 3 88.21 odd 2
176.6.a.i.1.2 3 8.3 odd 2
275.6.a.b.1.2 3 40.29 even 2
275.6.b.b.199.3 6 40.13 odd 4
275.6.b.b.199.4 6 40.37 odd 4
539.6.a.e.1.2 3 56.13 odd 2
704.6.a.q.1.2 3 1.1 even 1 trivial
704.6.a.t.1.2 3 4.3 odd 2
1089.6.a.r.1.2 3 264.197 even 2