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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [575,2,Mod(24,575)] 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("575.24"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(575, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 575 = 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 575.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [14,0,0,-22] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(4)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(4.59139811622\)
Analytic rank: \(0\)
Dimension: \(14\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} + \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{14} + 25x^{12} + 248x^{10} + 1239x^{8} + 3259x^{6} + 4248x^{4} + 2149x^{2} + 81 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
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 \(1,\beta_1,\ldots,\beta_{13}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} - \beta_{10} q^{3} + (\beta_{2} - 2) q^{4} + ( - \beta_{13} - \beta_{9} - \beta_{3} + \cdots + 1) q^{6} + ( - \beta_{8} - \beta_{4} + \beta_1) q^{7} + (\beta_{6} + \beta_{5} - \beta_1) q^{8}+ \cdots + ( - 3 \beta_{13} - \beta_{11} + \cdots + 8) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q - 22 q^{4} + 10 q^{6} - 30 q^{9} - 2 q^{11} - 14 q^{14} + 14 q^{16} - 30 q^{19} + 4 q^{21} - 36 q^{24} - 40 q^{26} - 6 q^{29} + 28 q^{31} - 40 q^{34} + 16 q^{39} + 38 q^{41} + 6 q^{44} + 2 q^{46} - 80 q^{49}+ \cdots + 106 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} + 25x^{12} + 248x^{10} + 1239x^{8} + 3259x^{6} + 4248x^{4} + 2149x^{2} + 81 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{12} - 21\nu^{10} - 164\nu^{8} - 583\nu^{6} - 927\nu^{4} - 565\nu^{2} - 89 ) / 25 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 2\nu^{13} + 47\nu^{11} + 463\nu^{9} + 2421\nu^{7} + 6749\nu^{5} + 8535\nu^{3} + 3158\nu ) / 225 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 2\nu^{13} + 44\nu^{11} + 367\nu^{9} + 1428\nu^{7} + 2552\nu^{5} + 1752\nu^{3} + 290\nu ) / 45 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -2\nu^{13} - 44\nu^{11} - 367\nu^{9} - 1428\nu^{7} - 2552\nu^{5} - 1707\nu^{3} - 65\nu ) / 45 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 2\nu^{13} + 47\nu^{11} + 418\nu^{9} + 1746\nu^{7} + 3464\nu^{5} + 3000\nu^{3} + 998\nu ) / 45 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -19\nu^{13} - 439\nu^{11} - 3821\nu^{9} - 15342\nu^{7} - 27523\nu^{5} - 16875\nu^{3} - 106\nu ) / 225 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -2\nu^{12} - 43\nu^{10} - 350\nu^{8} - 1322\nu^{6} - 2248\nu^{4} - 1336\nu^{2} - 54 ) / 15 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 28\nu^{13} + 613\nu^{11} + 5042\nu^{9} + 18999\nu^{7} + 31306\nu^{5} + 15720\nu^{3} - 2408\nu ) / 225 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 13\nu^{12} + 293\nu^{10} + 2497\nu^{8} + 9874\nu^{6} + 17681\nu^{4} + 11390\nu^{2} + 627 ) / 75 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( -\nu^{12} - 23\nu^{10} - 198\nu^{8} - 780\nu^{6} - 1360\nu^{4} - 812\nu^{2} - 31 ) / 5 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( -16\nu^{12} - 356\nu^{10} - 2989\nu^{8} - 11623\nu^{6} - 20387\nu^{4} - 12560\nu^{2} - 444 ) / 75 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{6} + \beta_{5} - 5\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{13} + \beta_{11} - \beta_{3} - 7\beta_{2} + 22 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -\beta_{10} - 9\beta_{6} - 6\beta_{5} - \beta_{4} + 29\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -11\beta_{13} + \beta_{12} - 9\beta_{11} + 2\beta_{9} + 8\beta_{3} + 46\beta_{2} - 132 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 13\beta_{10} + 2\beta_{8} + 3\beta_{7} + 66\beta_{6} + 28\beta_{5} + 12\beta_{4} - 180\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 91\beta_{13} - 11\beta_{12} + 68\beta_{11} - 26\beta_{9} - 49\beta_{3} - 297\beta_{2} + 822 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( -122\beta_{10} - 30\beta_{8} - 46\beta_{7} - 456\beta_{6} - 105\beta_{5} - 102\beta_{4} + 1150\beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( -680\beta_{13} + 86\beta_{12} - 486\beta_{11} + 245\beta_{9} + 274\beta_{3} + 1910\beta_{2} - 5212 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 1000\beta_{10} + 298\beta_{8} + 479\beta_{7} + 3076\beta_{6} + 210\beta_{5} + 766\beta_{4} - 7442\beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 4842\beta_{13} - 585\beta_{12} + 3374\beta_{11} - 2047\beta_{9} - 1480\beta_{3} - 12296\beta_{2} + 33377 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( -7619\beta_{10} - 2479\beta_{8} - 4239\beta_{7} - 20512\beta_{6} + 1458\beta_{5} - 5427\beta_{4} + 48450\beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/575\mathbb{Z}\right)^\times\).

\(n\) \(51\) \(277\)
\(\chi(n)\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
24.1
− 2.58128i
− 2.53289i
− 2.27220i
− 1.69496i
− 1.63662i
− 1.07994i
− 0.202227i
0.202227i
1.07994i
1.63662i
1.69496i
2.27220i
2.53289i
2.58128i
− 2.58128i 0.785406i −4.66299 0 2.02735 1.95865i 6.87392i 2.38314 0
24.2 − 2.53289i − 0.345624i −4.41555 0 −0.875428 − 5.12894i 6.11832i 2.88054 0
24.3 − 2.27220i 3.13672i −3.16289 0 7.12726 4.34930i 2.64233i −6.83902 0
24.4 − 1.69496i − 3.30905i −0.872898 0 −5.60872 0.852729i − 1.91040i −7.94982 0
24.5 − 1.63662i 2.46212i −0.678510 0 4.02954 − 4.74965i − 2.16277i −3.06202 0
24.6 − 1.07994i − 1.06928i 0.833738 0 −1.15476 − 2.95289i − 3.06026i 1.85663 0
24.7 − 0.202227i − 2.69619i 1.95910 0 −0.545243 2.81698i − 0.800639i −4.26945 0
24.8 0.202227i 2.69619i 1.95910 0 −0.545243 − 2.81698i 0.800639i −4.26945 0
24.9 1.07994i 1.06928i 0.833738 0 −1.15476 2.95289i 3.06026i 1.85663 0
24.10 1.63662i − 2.46212i −0.678510 0 4.02954 4.74965i 2.16277i −3.06202 0
24.11 1.69496i 3.30905i −0.872898 0 −5.60872 − 0.852729i 1.91040i −7.94982 0
24.12 2.27220i − 3.13672i −3.16289 0 7.12726 − 4.34930i − 2.64233i −6.83902 0
24.13 2.53289i 0.345624i −4.41555 0 −0.875428 5.12894i − 6.11832i 2.88054 0
24.14 2.58128i − 0.785406i −4.66299 0 2.02735 − 1.95865i − 6.87392i 2.38314 0
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 24.14
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 575.2.b.f 14
5.b even 2 1 inner 575.2.b.f 14
5.c odd 4 1 575.2.a.k ✓ 7
5.c odd 4 1 575.2.a.l yes 7
15.e even 4 1 5175.2.a.cb 7
15.e even 4 1 5175.2.a.cg 7
20.e even 4 1 9200.2.a.da 7
20.e even 4 1 9200.2.a.db 7
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
575.2.a.k ✓ 7 5.c odd 4 1
575.2.a.l yes 7 5.c odd 4 1
575.2.b.f 14 1.a even 1 1 trivial
575.2.b.f 14 5.b even 2 1 inner
5175.2.a.cb 7 15.e even 4 1
5175.2.a.cg 7 15.e even 4 1
9200.2.a.da 7 20.e even 4 1
9200.2.a.db 7 20.e even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{14} + 25T_{2}^{12} + 248T_{2}^{10} + 1239T_{2}^{8} + 3259T_{2}^{6} + 4248T_{2}^{4} + 2149T_{2}^{2} + 81 \) acting on \(S_{2}^{\mathrm{new}}(575, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{14} + 25 T^{12} + \cdots + 81 \) Copy content Toggle raw display
$3$ \( T^{14} + 36 T^{12} + \cdots + 400 \) Copy content Toggle raw display
$5$ \( T^{14} \) Copy content Toggle raw display
$7$ \( T^{14} + 89 T^{12} + \cdots + 2166784 \) Copy content Toggle raw display
$11$ \( (T^{7} + T^{6} - 58 T^{5} + \cdots - 4800)^{2} \) Copy content Toggle raw display
$13$ \( T^{14} + 101 T^{12} + \cdots + 2679769 \) Copy content Toggle raw display
$17$ \( T^{14} + \cdots + 2123366400 \) Copy content Toggle raw display
$19$ \( (T^{7} + 15 T^{6} + \cdots + 1600)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} + 1)^{7} \) Copy content Toggle raw display
$29$ \( (T^{7} + 3 T^{6} + \cdots + 3375)^{2} \) Copy content Toggle raw display
$31$ \( (T^{7} - 14 T^{6} + \cdots - 24350)^{2} \) Copy content Toggle raw display
$37$ \( T^{14} + 332 T^{12} + \cdots + 84934656 \) Copy content Toggle raw display
$41$ \( (T^{7} - 19 T^{6} + \cdots + 14217)^{2} \) Copy content Toggle raw display
$43$ \( T^{14} + \cdots + 207360000 \) Copy content Toggle raw display
$47$ \( T^{14} + \cdots + 971568900 \) Copy content Toggle raw display
$53$ \( T^{14} + 264 T^{12} + \cdots + 11943936 \) Copy content Toggle raw display
$59$ \( (T^{7} - 16 T^{6} + \cdots - 149340)^{2} \) Copy content Toggle raw display
$61$ \( (T^{7} - 40 T^{6} + \cdots + 79616)^{2} \) Copy content Toggle raw display
$67$ \( T^{14} + \cdots + 2123366400 \) Copy content Toggle raw display
$71$ \( (T^{7} + 14 T^{6} + \cdots + 31170)^{2} \) Copy content Toggle raw display
$73$ \( T^{14} + \cdots + 364793048361 \) Copy content Toggle raw display
$79$ \( (T^{7} - T^{6} + \cdots - 22720)^{2} \) Copy content Toggle raw display
$83$ \( T^{14} + 369 T^{12} + \cdots + 36864 \) Copy content Toggle raw display
$89$ \( (T^{7} + 16 T^{6} + \cdots - 2478720)^{2} \) Copy content Toggle raw display
$97$ \( T^{14} + 568 T^{12} + \cdots + 802816 \) Copy content Toggle raw display
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