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Properties

Label 1568.3.c.e
Level $1568$
Weight $3$
Character orbit 1568.c
Analytic conductor $42.725$
Analytic rank $0$
Dimension $12$
Inner twists $2$

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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] = [1568,3,Mod(97,1568)] 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("1568.97"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1568, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 1568 = 2^{5} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1568.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [12,0,0,0,0,0,0,0,-36,0,-48,0,0,0,0,0,0,0,0,0,0,0,128,0,-124, 0,0,0,0,0,0,0,0,0,0,0,-128,0,-160,0,0,0,464,0,0,0,0,0,0,0,400,0,16] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(53)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(42.7249054517\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 64x^{10} + 1436x^{8} + 13392x^{6} + 45220x^{4} + 30880x^{2} + 1568 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{5}\cdot 7^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{6} q^{3} + \beta_{11} q^{5} + (\beta_{3} + \beta_1 - 3) q^{9} + ( - \beta_{4} + \beta_1 - 4) q^{11} + ( - \beta_{9} - \beta_{8} + \cdots - \beta_{5}) q^{13} + ( - \beta_{10} + 6 \beta_{3} + \cdots + 2 \beta_1) q^{15}+ \cdots + (5 \beta_{10} - 7 \beta_{4} + \cdots + 112) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 36 q^{9} - 48 q^{11} + 128 q^{23} - 124 q^{25} - 128 q^{37} - 160 q^{39} + 464 q^{43} + 400 q^{51} + 16 q^{53} + 224 q^{57} - 64 q^{65} - 64 q^{67} + 544 q^{71} + 288 q^{79} - 180 q^{81} - 272 q^{85}+ \cdots + 1328 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 64x^{10} + 1436x^{8} + 13392x^{6} + 45220x^{4} + 30880x^{2} + 1568 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -201\nu^{10} - 29690\nu^{8} - 1065310\nu^{6} - 12252764\nu^{4} - 30595592\nu^{2} + 14344064 ) / 3016048 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 124\nu^{10} + 24881\nu^{8} + 1013580\nu^{6} + 13769202\nu^{4} + 55307552\nu^{2} + 12810952 ) / 1508024 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -121\nu^{10} - 7557\nu^{8} - 162077\nu^{6} - 1387086\nu^{4} - 3876314\nu^{2} - 1275960 ) / 754012 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 2406\nu^{10} + 133129\nu^{8} + 2409576\nu^{6} + 16330250\nu^{4} + 26770848\nu^{2} - 47412400 ) / 6032096 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -5545\nu^{11} - 334070\nu^{9} - 6800034\nu^{7} - 52855492\nu^{5} - 99592112\nu^{3} + 163374176\nu ) / 12064192 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -2985\nu^{11} - 191323\nu^{9} - 4286998\nu^{7} - 39558118\nu^{5} - 128277280\nu^{3} - 80102512\nu ) / 6032096 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -3905\nu^{11} - 243885\nu^{9} - 5316350\nu^{7} - 48260922\nu^{5} - 162354272\nu^{3} - 131523056\nu ) / 6032096 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 8495\nu^{11} + 547688\nu^{9} + 12346318\nu^{7} + 114322952\nu^{5} + 367793344\nu^{3} + 193484928\nu ) / 12064192 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -4413\nu^{11} - 278060\nu^{9} - 6098290\nu^{7} - 55006208\nu^{5} - 176326288\nu^{3} - 113004416\nu ) / 6032096 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( -8467\nu^{10} - 511666\nu^{8} - 10682390\nu^{6} - 91500012\nu^{4} - 272314576\nu^{2} - 89846400 ) / 6032096 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( -8423\nu^{11} - 535847\nu^{9} - 11945422\nu^{7} - 110732126\nu^{5} - 369377016\nu^{3} - 210063504\nu ) / 6032096 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( -4\beta_{8} + \beta_{7} - 7\beta_{6} ) / 7 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -2\beta_{10} - 8\beta_{4} - 3\beta_{3} - 2\beta_{2} - \beta _1 - 76 ) / 7 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 14\beta_{11} + 21\beta_{9} + 94\beta_{8} - 48\beta_{7} + 126\beta_{6} ) / 7 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 68\beta_{10} + 188\beta_{4} - 122\beta_{3} + 61\beta_{2} + 62\beta _1 + 1471 ) / 7 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -434\beta_{11} - 966\beta_{9} - 2342\beta_{8} + 1464\beta_{7} - 2660\beta_{6} + 70\beta_{5} ) / 7 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -2116\beta_{10} - 4544\beta_{4} + 7214\beta_{3} - 1584\beta_{2} - 1954\beta _1 - 32276 ) / 7 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 11620\beta_{11} + 32270\beta_{9} + 59468\beta_{8} - 41544\beta_{7} + 61124\beta_{6} - 2856\beta_{5} ) / 7 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 61640\beta_{10} + 113224\beta_{4} - 259556\beta_{3} + 40990\beta_{2} + 54956\beta _1 + 759242 ) / 7 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( -305116\beta_{11} - 962444\beta_{9} - 1531380\beta_{8} + 1147616\beta_{7} - 1482040\beta_{6} + 92036\beta_{5} ) / 7 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( -1730808\beta_{10} - 2883616\beta_{4} + 7998484\beta_{3} - 1073480\beta_{2} - 1493612\beta _1 - 18687024 ) / 7 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 8017800 \beta_{11} + 27241396 \beta_{9} + 39851416 \beta_{8} - 31257008 \beta_{7} + \cdots - 2724960 \beta_{5} ) / 7 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1568\mathbb{Z}\right)^\times\).

\(n\) \(197\) \(1471\) \(1473\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
97.1
− 3.86019i
− 2.31026i
− 4.03404i
− 0.234863i
− 5.15825i
0.908546i
− 0.908546i
5.15825i
0.234863i
4.03404i
2.31026i
3.86019i
0 − 4.94258i 0 0.228417i 0 0 0 −15.4291 0
97.2 0 − 4.92339i 0 − 9.52370i 0 0 0 −15.2397 0
97.3 0 − 2.95165i 0 7.05411i 0 0 0 0.287786 0
97.4 0 − 2.84799i 0 7.39866i 0 0 0 0.888961 0
97.5 0 − 2.54512i 0 1.25344i 0 0 0 2.52235 0
97.6 0 − 0.173846i 0 3.89554i 0 0 0 8.96978 0
97.7 0 0.173846i 0 − 3.89554i 0 0 0 8.96978 0
97.8 0 2.54512i 0 − 1.25344i 0 0 0 2.52235 0
97.9 0 2.84799i 0 − 7.39866i 0 0 0 0.888961 0
97.10 0 2.95165i 0 − 7.05411i 0 0 0 0.287786 0
97.11 0 4.92339i 0 9.52370i 0 0 0 −15.2397 0
97.12 0 4.94258i 0 − 0.228417i 0 0 0 −15.4291 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 97.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1568.3.c.e ✓ 12
4.b odd 2 1 1568.3.c.f yes 12
7.b odd 2 1 inner 1568.3.c.e ✓ 12
28.d even 2 1 1568.3.c.f yes 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1568.3.c.e ✓ 12 1.a even 1 1 trivial
1568.3.c.e ✓ 12 7.b odd 2 1 inner
1568.3.c.f yes 12 4.b odd 2 1
1568.3.c.f yes 12 28.d even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(1568, [\chi])\):

\( T_{3}^{12} + 72T_{3}^{10} + 1908T_{3}^{8} + 23056T_{3}^{6} + 129348T_{3}^{4} + 274944T_{3}^{2} + 8192 \) Copy content Toggle raw display
\( T_{5}^{12} + 212T_{5}^{10} + 15506T_{5}^{8} + 456864T_{5}^{6} + 4452064T_{5}^{4} + 6121472T_{5}^{2} + 307328 \) Copy content Toggle raw display
\( T_{11}^{6} + 24T_{11}^{5} - 252T_{11}^{4} - 8624T_{11}^{3} - 14460T_{11}^{2} + 578496T_{11} + 2518784 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( T^{12} + 72 T^{10} + \cdots + 8192 \) Copy content Toggle raw display
$5$ \( T^{12} + 212 T^{10} + \cdots + 307328 \) Copy content Toggle raw display
$7$ \( T^{12} \) Copy content Toggle raw display
$11$ \( (T^{6} + 24 T^{5} + \cdots + 2518784)^{2} \) Copy content Toggle raw display
$13$ \( T^{12} + \cdots + 383141570688 \) Copy content Toggle raw display
$17$ \( T^{12} + \cdots + 241092902408 \) Copy content Toggle raw display
$19$ \( T^{12} + \cdots + 46519820288 \) Copy content Toggle raw display
$23$ \( (T^{6} - 64 T^{5} + \cdots + 2571264)^{2} \) Copy content Toggle raw display
$29$ \( (T^{6} - 3074 T^{4} + \cdots + 3709952)^{2} \) Copy content Toggle raw display
$31$ \( T^{12} + \cdots + 32\!\cdots\!88 \) Copy content Toggle raw display
$37$ \( (T^{6} + 64 T^{5} + \cdots + 344382336)^{2} \) Copy content Toggle raw display
$41$ \( T^{12} + \cdots + 82\!\cdots\!28 \) Copy content Toggle raw display
$43$ \( (T^{6} - 232 T^{5} + \cdots - 6122215168)^{2} \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 21\!\cdots\!88 \) Copy content Toggle raw display
$53$ \( (T^{6} - 8 T^{5} + \cdots - 6160006144)^{2} \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 15\!\cdots\!52 \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 18\!\cdots\!52 \) Copy content Toggle raw display
$67$ \( (T^{6} + 32 T^{5} + \cdots - 1195638784)^{2} \) Copy content Toggle raw display
$71$ \( (T^{6} - 272 T^{5} + \cdots - 291071655936)^{2} \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 25\!\cdots\!48 \) Copy content Toggle raw display
$79$ \( (T^{6} - 144 T^{5} + \cdots - 7068393472)^{2} \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 25\!\cdots\!88 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 14\!\cdots\!32 \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 75\!\cdots\!68 \) Copy content Toggle raw display
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