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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [825,4,Mod(199,825)] 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("825.199"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(825, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 825 = 3 \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 825.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [10,0,0,-6,0,-6,0,0,-90,0,-110,0,0,-16] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(14)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(48.6765757547\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} + \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} + 43x^{8} + 631x^{6} + 3625x^{4} + 7104x^{2} + 900 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + 3 \beta_{2} q^{3} + (\beta_{3} - 1) q^{4} - 3 \beta_{4} q^{6} + ( - \beta_{9} - \beta_{7} + 4 \beta_{2}) q^{7} + (\beta_{5} + 2 \beta_1) q^{8} - 9 q^{9} - 11 q^{11} + (3 \beta_{7} - 3 \beta_{2}) q^{12}+ \cdots + 99 q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 6 q^{4} - 6 q^{6} - 90 q^{9} - 110 q^{11} - 16 q^{14} - 250 q^{16} + 114 q^{19} - 108 q^{21} - 18 q^{24} - 250 q^{26} + 214 q^{29} - 590 q^{31} - 68 q^{34} + 54 q^{36} + 186 q^{39} - 256 q^{41} + 66 q^{44}+ \cdots + 990 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} + 43x^{8} + 631x^{6} + 3625x^{4} + 7104x^{2} + 900 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{9} + 37\nu^{7} + 445\nu^{5} + 1891\nu^{3} + 2202\nu ) / 720 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{2} + 9 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{8} + 31\nu^{6} + 289\nu^{4} + 817\nu^{2} + 150 ) / 120 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( \nu^{3} + 14\nu \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{4} + 19\nu^{2} + 46 ) / 2 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{9} + 49\nu^{7} + 757\nu^{5} + 4039\nu^{3} + 6306\nu ) / 240 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -\nu^{8} - 46\nu^{6} - 619\nu^{4} - 2242\nu^{2} - 600 ) / 60 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( \nu^{9} + 55\nu^{7} + 1093\nu^{5} + 8893\nu^{3} + 22398\nu ) / 360 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} - 9 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{5} - 14\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2\beta_{6} - 19\beta_{3} + 125 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 2\beta_{9} - 2\beta_{7} - 21\beta_{5} + 2\beta_{2} + 216\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -4\beta_{8} - 44\beta_{6} - 8\beta_{4} + 323\beta_{3} - 1925 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -52\beta_{9} + 72\beta_{7} + 367\beta_{5} - 112\beta_{2} - 3452\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 124\beta_{8} + 786\beta_{6} + 368\beta_{4} - 5339\beta_{3} + 30753 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 1034\beta_{9} - 1774\beta_{7} - 6125\beta_{5} + 3974\beta_{2} + 55876\beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/825\mathbb{Z}\right)^\times\).

\(n\) \(376\) \(551\) \(727\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
199.1
− 4.08549i
− 3.98707i
− 2.51748i
− 1.98454i
− 0.368634i
0.368634i
1.98454i
2.51748i
3.98707i
4.08549i
− 4.08549i − 3.00000i −8.69126 0 −12.2565 − 7.95915i 2.82414i −9.00000 0
199.2 − 3.98707i 3.00000i −7.89672 0 11.9612 12.5627i − 0.411800i −9.00000 0
199.3 − 2.51748i − 3.00000i 1.66228 0 −7.55245 − 18.4627i − 24.3246i −9.00000 0
199.4 − 1.98454i 3.00000i 4.06159 0 5.95363 5.59777i − 23.9367i −9.00000 0
199.5 − 0.368634i − 3.00000i 7.86411 0 −1.10590 26.5824i − 5.84806i −9.00000 0
199.6 0.368634i 3.00000i 7.86411 0 −1.10590 − 26.5824i 5.84806i −9.00000 0
199.7 1.98454i − 3.00000i 4.06159 0 5.95363 − 5.59777i 23.9367i −9.00000 0
199.8 2.51748i 3.00000i 1.66228 0 −7.55245 18.4627i 24.3246i −9.00000 0
199.9 3.98707i − 3.00000i −7.89672 0 11.9612 − 12.5627i 0.411800i −9.00000 0
199.10 4.08549i 3.00000i −8.69126 0 −12.2565 7.95915i − 2.82414i −9.00000 0
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 199.10
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 825.4.c.s 10
5.b even 2 1 inner 825.4.c.s 10
5.c odd 4 1 825.4.a.x ✓ 5
5.c odd 4 1 825.4.a.y yes 5
15.e even 4 1 2475.4.a.bi 5
15.e even 4 1 2475.4.a.bj 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
825.4.a.x ✓ 5 5.c odd 4 1
825.4.a.y yes 5 5.c odd 4 1
825.4.c.s 10 1.a even 1 1 trivial
825.4.c.s 10 5.b even 2 1 inner
2475.4.a.bi 5 15.e even 4 1
2475.4.a.bj 5 15.e even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(825, [\chi])\):

\( T_{2}^{10} + 43T_{2}^{8} + 631T_{2}^{6} + 3625T_{2}^{4} + 7104T_{2}^{2} + 900 \) Copy content Toggle raw display
\( T_{7}^{10} + 1300T_{7}^{8} + 522294T_{7}^{6} + 78865816T_{7}^{4} + 4405598425T_{7}^{2} + 75458991204 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} + 43 T^{8} + \cdots + 900 \) Copy content Toggle raw display
$3$ \( (T^{2} + 9)^{5} \) Copy content Toggle raw display
$5$ \( T^{10} \) Copy content Toggle raw display
$7$ \( T^{10} + \cdots + 75458991204 \) Copy content Toggle raw display
$11$ \( (T + 11)^{10} \) Copy content Toggle raw display
$13$ \( T^{10} + \cdots + 25\!\cdots\!24 \) Copy content Toggle raw display
$17$ \( T^{10} + \cdots + 89\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( (T^{5} - 57 T^{4} + \cdots + 92396817)^{2} \) Copy content Toggle raw display
$23$ \( T^{10} + \cdots + 13\!\cdots\!16 \) Copy content Toggle raw display
$29$ \( (T^{5} - 107 T^{4} + \cdots - 16041054816)^{2} \) Copy content Toggle raw display
$31$ \( (T^{5} + 295 T^{4} + \cdots + 9419016168)^{2} \) Copy content Toggle raw display
$37$ \( T^{10} + \cdots + 13\!\cdots\!16 \) Copy content Toggle raw display
$41$ \( (T^{5} + 128 T^{4} + \cdots + 144243119304)^{2} \) Copy content Toggle raw display
$43$ \( T^{10} + \cdots + 11\!\cdots\!04 \) Copy content Toggle raw display
$47$ \( T^{10} + \cdots + 98\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{10} + \cdots + 42\!\cdots\!24 \) Copy content Toggle raw display
$59$ \( (T^{5} - 1152 T^{4} + \cdots + 849531990720)^{2} \) Copy content Toggle raw display
$61$ \( (T^{5} + 344 T^{4} + \cdots + 105472451720)^{2} \) Copy content Toggle raw display
$67$ \( T^{10} + \cdots + 11\!\cdots\!00 \) Copy content Toggle raw display
$71$ \( (T^{5} + \cdots + 13467667404912)^{2} \) Copy content Toggle raw display
$73$ \( T^{10} + \cdots + 79\!\cdots\!56 \) Copy content Toggle raw display
$79$ \( (T^{5} + \cdots - 57405986771584)^{2} \) Copy content Toggle raw display
$83$ \( T^{10} + \cdots + 49\!\cdots\!56 \) Copy content Toggle raw display
$89$ \( (T^{5} + \cdots + 205516733649744)^{2} \) Copy content Toggle raw display
$97$ \( T^{10} + \cdots + 21\!\cdots\!25 \) Copy content Toggle raw display
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