Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1840,4,Mod(1,1840)] 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("1840.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1840, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 1840 = 2^{4} \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1840.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [9,0,3,0,-45,0,-25] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(108.563514411\)
Analytic rank: \(1\)
Dimension: \(9\)
Coefficient field: \(\mathbb{Q}[x]/(x^{9} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{9} - 3x^{8} - 148x^{7} + 278x^{6} + 6502x^{5} - 4928x^{4} - 87343x^{3} + 42737x^{2} + 286800x + 53104 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{5} \)
Twist minimal: no (minimal twist has level 920)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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 \(1,\beta_1,\ldots,\beta_{8}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} - 5 q^{5} + (\beta_{3} - 3) q^{7} + (\beta_{4} - \beta_{3} + 2 \beta_1 + 6) q^{9} + (\beta_{7} + \beta_{6} - \beta_{5} + \cdots - 2) q^{11} + (\beta_{8} - \beta_{7} + \beta_{6} + \cdots + 4) q^{13}+ \cdots + ( - 20 \beta_{8} + 4 \beta_{7} + \cdots - 346) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9 q + 3 q^{3} - 45 q^{5} - 25 q^{7} + 62 q^{9} - 22 q^{11} + 23 q^{13} - 15 q^{15} - 135 q^{17} - 102 q^{19} + 54 q^{21} + 207 q^{23} + 225 q^{25} + 363 q^{27} + 280 q^{29} - 168 q^{31} + 28 q^{33} + 125 q^{35}+ \cdots - 3206 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{9} - 3x^{8} - 148x^{7} + 278x^{6} + 6502x^{5} - 4928x^{4} - 87343x^{3} + 42737x^{2} + 286800x + 53104 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 163 \nu^{8} + 4127 \nu^{7} - 40716 \nu^{6} - 544206 \nu^{5} + 2205470 \nu^{4} + 18403540 \nu^{3} + \cdots - 161016460 ) / 2417192 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 514256 \nu^{8} - 5815223 \nu^{7} + 100384680 \nu^{6} + 832440392 \nu^{5} + \cdots + 74795096684 ) / 2501793720 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 514256 \nu^{8} - 5815223 \nu^{7} + 100384680 \nu^{6} + 832440392 \nu^{5} - 5143221570 \nu^{4} + \cdots - 7764096076 ) / 2501793720 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 569393 \nu^{8} + 3148951 \nu^{7} + 79478880 \nu^{6} - 360717754 \nu^{5} + \cdots - 139395895828 ) / 2501793720 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 1230451 \nu^{8} - 4616552 \nu^{7} - 147932640 \nu^{6} + 312182138 \nu^{5} + 3965373360 \nu^{4} + \cdots - 170844106264 ) / 2501793720 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 1015897 \nu^{8} + 226071 \nu^{7} - 153500780 \nu^{6} - 184830594 \nu^{5} + 6448742270 \nu^{4} + \cdots + 14256990172 ) / 833931240 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 1382473 \nu^{8} - 2387616 \nu^{7} - 197866340 \nu^{6} + 90650274 \nu^{5} + 7829453240 \nu^{4} + \cdots + 35031968488 ) / 833931240 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{4} - \beta_{3} + 2\beta _1 + 33 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{8} + 3\beta_{7} - \beta_{6} + 3\beta_{5} + 2\beta_{4} + 2\beta_{3} + 66\beta _1 + 36 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -9\beta_{8} + 10\beta_{7} + 3\beta_{6} - 7\beta_{5} + 84\beta_{4} - 86\beta_{3} - 11\beta_{2} + 221\beta _1 + 2121 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 179 \beta_{8} + 390 \beta_{7} - 69 \beta_{6} + 221 \beta_{5} + 225 \beta_{4} + 247 \beta_{3} + \cdots + 4632 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 1566 \beta_{8} + 1799 \beta_{7} + 596 \beta_{6} - 1068 \beta_{5} + 6866 \beta_{4} - 6586 \beta_{3} + \cdots + 161221 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 22114 \beta_{8} + 41284 \beta_{7} - 3210 \beta_{6} + 13650 \beta_{5} + 22456 \beta_{4} + \cdots + 478470 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 194214 \beta_{8} + 232172 \beta_{7} + 72094 \beta_{6} - 118534 \beta_{5} + 570969 \beta_{4} + \cdots + 13022959 \) Copy content Toggle raw display

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−8.72630
−5.18969
−4.91244
−1.70964
−0.192858
2.99459
3.15779
7.95723
9.62132
0 −8.72630 0 −5.00000 0 −28.9030 0 49.1484 0
1.2 0 −5.18969 0 −5.00000 0 −12.0220 0 −0.0670810 0
1.3 0 −4.91244 0 −5.00000 0 33.9952 0 −2.86795 0
1.4 0 −1.70964 0 −5.00000 0 −20.8693 0 −24.0771 0
1.5 0 −0.192858 0 −5.00000 0 11.9366 0 −26.9628 0
1.6 0 2.99459 0 −5.00000 0 22.8972 0 −18.0324 0
1.7 0 3.15779 0 −5.00000 0 −11.8460 0 −17.0284 0
1.8 0 7.95723 0 −5.00000 0 −21.6920 0 36.3175 0
1.9 0 9.62132 0 −5.00000 0 1.50324 0 65.5699 0
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 1.9
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( +1 \)
\(5\) \( +1 \)
\(23\) \( -1 \)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1840.4.a.z 9
4.b odd 2 1 920.4.a.e ✓ 9
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
920.4.a.e ✓ 9 4.b odd 2 1
1840.4.a.z 9 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1840))\):

\( T_{3}^{9} - 3 T_{3}^{8} - 148 T_{3}^{7} + 278 T_{3}^{6} + 6502 T_{3}^{5} - 4928 T_{3}^{4} - 87343 T_{3}^{3} + \cdots + 53104 \) Copy content Toggle raw display
\( T_{7}^{9} + 25 T_{7}^{8} - 1613 T_{7}^{7} - 47585 T_{7}^{6} + 521969 T_{7}^{5} + 22839351 T_{7}^{4} + \cdots + 26025813504 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{9} \) Copy content Toggle raw display
$3$ \( T^{9} - 3 T^{8} + \cdots + 53104 \) Copy content Toggle raw display
$5$ \( (T + 5)^{9} \) Copy content Toggle raw display
$7$ \( T^{9} + \cdots + 26025813504 \) Copy content Toggle raw display
$11$ \( T^{9} + \cdots + 2775829222400 \) Copy content Toggle raw display
$13$ \( T^{9} + \cdots - 189030103898988 \) Copy content Toggle raw display
$17$ \( T^{9} + \cdots - 287072829439872 \) Copy content Toggle raw display
$19$ \( T^{9} + \cdots + 501210422257408 \) Copy content Toggle raw display
$23$ \( (T - 23)^{9} \) Copy content Toggle raw display
$29$ \( T^{9} + \cdots - 28\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{9} + \cdots + 26\!\cdots\!52 \) Copy content Toggle raw display
$37$ \( T^{9} + \cdots - 37\!\cdots\!68 \) Copy content Toggle raw display
$41$ \( T^{9} + \cdots - 24\!\cdots\!18 \) Copy content Toggle raw display
$43$ \( T^{9} + \cdots - 25\!\cdots\!08 \) Copy content Toggle raw display
$47$ \( T^{9} + \cdots - 52\!\cdots\!24 \) Copy content Toggle raw display
$53$ \( T^{9} + \cdots - 87\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( T^{9} + \cdots - 28\!\cdots\!52 \) Copy content Toggle raw display
$61$ \( T^{9} + \cdots + 80\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( T^{9} + \cdots - 71\!\cdots\!08 \) Copy content Toggle raw display
$71$ \( T^{9} + \cdots + 78\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( T^{9} + \cdots - 92\!\cdots\!44 \) Copy content Toggle raw display
$79$ \( T^{9} + \cdots + 93\!\cdots\!52 \) Copy content Toggle raw display
$83$ \( T^{9} + \cdots - 15\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( T^{9} + \cdots + 77\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{9} + \cdots - 12\!\cdots\!84 \) Copy content Toggle raw display
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