Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [75,11,Mod(26,75)] 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("75.26"); S:= CuspForms(chi, 11); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(75, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0])) N = Newforms(chi, 11, names="a")
 
Level: \( N \) \(=\) \( 75 = 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 11 \)
Character orbit: \([\chi]\) \(=\) 75.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [14,0,-44] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(47.6517939505\)
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: \( x^{14} + 11554 x^{12} + 52224391 x^{10} + 115670558124 x^{8} + 127683454012911 x^{6} + \cdots + 62\!\cdots\!00 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{11}\cdot 3^{22}\cdot 5^{21} \)
Twist minimal: no (minimal twist has level 15)
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.

\(f(q)\) \(=\) \( q + \beta_{2} q^{2} + (\beta_{3} - \beta_{2} - 3) q^{3} + ( - \beta_1 - 629) q^{4} + (\beta_{4} + 1563) q^{6} + (\beta_{5} - \beta_{4} - 2 \beta_{3} + \cdots + 3610) q^{7} + (\beta_{9} - \beta_{7} - \beta_{4} + \cdots - 4) q^{8}+ \cdots + ( - 53167 \beta_{13} + \cdots + 2592117009) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q - 44 q^{3} - 8802 q^{4} + 21886 q^{6} + 50548 q^{7} + 116362 q^{9} - 43756 q^{12} - 699408 q^{13} + 2871906 q^{16} + 3243880 q^{18} + 3814644 q^{19} - 2191008 q^{21} + 10493420 q^{22} + 9454542 q^{24}+ \cdots + 36258312560 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} + 11554 x^{12} + 52224391 x^{10} + 115670558124 x^{8} + 127683454012911 x^{6} + \cdots + 62\!\cdots\!00 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 570117271 \nu^{12} + 4883354422201 \nu^{10} + \cdots - 12\!\cdots\!60 ) / 75\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 9510660222317 \nu^{13} + \cdots + 92\!\cdots\!00 \nu ) / 49\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 31\!\cdots\!99 \nu^{13} + \cdots - 48\!\cdots\!00 ) / 23\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 47\!\cdots\!77 \nu^{13} + \cdots - 49\!\cdots\!00 ) / 14\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 22\!\cdots\!67 \nu^{13} + \cdots - 81\!\cdots\!00 ) / 64\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 23\!\cdots\!33 \nu^{13} + \cdots - 13\!\cdots\!00 ) / 38\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 36\!\cdots\!59 \nu^{13} + \cdots + 35\!\cdots\!00 ) / 58\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 74\!\cdots\!09 \nu^{13} + \cdots + 25\!\cdots\!00 ) / 11\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 40\!\cdots\!33 \nu^{13} + \cdots + 24\!\cdots\!00 ) / 58\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 97\!\cdots\!31 \nu^{13} + \cdots + 29\!\cdots\!00 ) / 11\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 29\!\cdots\!73 \nu^{13} + \cdots - 32\!\cdots\!00 ) / 23\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 93\!\cdots\!67 \nu^{13} + \cdots + 24\!\cdots\!00 ) / 38\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 11\!\cdots\!77 \nu^{13} + \cdots - 17\!\cdots\!00 ) / 29\!\cdots\!00 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{9} + 2\beta_{3} + 3742\beta_{2} ) / 3750 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{10} - \beta_{9} - 2\beta_{6} + \beta_{5} + 2\beta_{4} - 53\beta_{3} + 3\beta_{2} - 602\beta _1 - 1031786 ) / 625 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 90 \beta_{13} - 90 \beta_{12} + 630 \beta_{11} + 90 \beta_{10} - 503 \beta_{9} + 450 \beta_{8} + \cdots - 16620 ) / 3750 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 4175 \beta_{13} + 7975 \beta_{12} + 7975 \beta_{11} + 3622 \beta_{10} - 678 \beta_{9} + \cdots + 2594070992 ) / 625 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 120 \beta_{13} + 131880 \beta_{12} - 275160 \beta_{11} + 60120 \beta_{10} - 864739 \beta_{9} + \cdots + 10101840 ) / 750 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 20323700 \beta_{13} - 35983900 \beta_{12} - 45150900 \beta_{11} - 9823283 \beta_{10} + \cdots - 7074635118838 ) / 625 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 792903390 \beta_{13} - 2798594610 \beta_{12} + 112668270 \beta_{11} - 2530113390 \beta_{10} + \cdots - 119719231980 ) / 3750 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 78036660975 \beta_{13} + 130076391075 \beta_{12} + 179370600575 \beta_{11} + 25543589704 \beta_{10} + \cdots + 20\!\cdots\!44 ) / 625 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 4521340365480 \beta_{13} + 10769390183520 \beta_{12} + 12734985132360 \beta_{11} + \cdots + 244305007680360 ) / 3750 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 54591079891400 \beta_{13} - 87050628983800 \beta_{12} - 125949386273800 \beta_{11} + \cdots - 11\!\cdots\!46 ) / 125 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 19\!\cdots\!70 \beta_{13} + \cdots - 39\!\cdots\!40 ) / 3750 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 91\!\cdots\!75 \beta_{13} + \cdots + 16\!\cdots\!16 ) / 625 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 75\!\cdots\!60 \beta_{13} + \cdots + 27\!\cdots\!20 ) / 3750 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/75\mathbb{Z}\right)^\times\).

\(n\) \(26\) \(52\)
\(\chi(n)\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
26.1
− 55.5349i
− 54.9539i
− 49.8576i
− 42.9372i
− 29.7613i
− 15.0833i
− 2.70449i
2.70449i
15.0833i
29.7613i
42.9372i
49.8576i
54.9539i
55.5349i
− 57.7709i 60.6159 + 235.318i −2313.48 0 13594.6 − 3501.84i −22792.7 74494.6i −51700.4 + 28528.1i 0
26.2 − 52.7178i 210.660 − 121.125i −1755.17 0 −6385.47 − 11105.5i 8585.72 38545.6i 29706.2 − 51032.6i 0
26.3 − 52.0937i −196.615 − 142.800i −1689.75 0 −7438.96 + 10242.4i 32323.0 34681.6i 18265.6 + 56152.9i 0
26.4 − 40.7012i −230.652 + 76.4761i −632.586 0 3112.67 + 9387.81i −19744.7 − 15931.0i 47351.8 − 35278.8i 0
26.5 − 27.5253i 80.1400 + 229.405i 266.360 0 6314.43 − 2205.87i 24115.7 − 35517.5i −46204.2 + 36769.0i 0
26.6 − 17.3194i 236.663 + 55.1313i 724.039 0 954.839 − 4098.86i −2728.90 − 30275.0i 52970.1 + 26095.1i 0
26.7 − 4.94055i −182.813 + 160.089i 999.591 0 790.929 + 903.195i 5515.83 − 9997.66i 7791.87 − 58532.7i 0
26.8 4.94055i −182.813 − 160.089i 999.591 0 790.929 − 903.195i 5515.83 9997.66i 7791.87 + 58532.7i 0
26.9 17.3194i 236.663 − 55.1313i 724.039 0 954.839 + 4098.86i −2728.90 30275.0i 52970.1 − 26095.1i 0
26.10 27.5253i 80.1400 − 229.405i 266.360 0 6314.43 + 2205.87i 24115.7 35517.5i −46204.2 − 36769.0i 0
26.11 40.7012i −230.652 − 76.4761i −632.586 0 3112.67 − 9387.81i −19744.7 15931.0i 47351.8 + 35278.8i 0
26.12 52.0937i −196.615 + 142.800i −1689.75 0 −7438.96 − 10242.4i 32323.0 − 34681.6i 18265.6 − 56152.9i 0
26.13 52.7178i 210.660 + 121.125i −1755.17 0 −6385.47 + 11105.5i 8585.72 − 38545.6i 29706.2 + 51032.6i 0
26.14 57.7709i 60.6159 − 235.318i −2313.48 0 13594.6 + 3501.84i −22792.7 − 74494.6i −51700.4 − 28528.1i 0
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 26.14
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 75.11.c.g 14
3.b odd 2 1 inner 75.11.c.g 14
5.b even 2 1 15.11.c.a ✓ 14
5.c odd 4 2 75.11.d.d 28
15.d odd 2 1 15.11.c.a ✓ 14
15.e even 4 2 75.11.d.d 28
20.d odd 2 1 240.11.l.b 14
60.h even 2 1 240.11.l.b 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.11.c.a ✓ 14 5.b even 2 1
15.11.c.a ✓ 14 15.d odd 2 1
75.11.c.g 14 1.a even 1 1 trivial
75.11.c.g 14 3.b odd 2 1 inner
75.11.d.d 28 5.c odd 4 2
75.11.d.d 28 15.e even 4 2
240.11.l.b 14 20.d odd 2 1
240.11.l.b 14 60.h even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{11}^{\mathrm{new}}(75, [\chi])\):

\( T_{2}^{14} + 11569 T_{2}^{12} + 52102936 T_{2}^{10} + 114518599604 T_{2}^{8} + 125620895405696 T_{2}^{6} + \cdots + 23\!\cdots\!00 \) Copy content Toggle raw display
\( T_{7}^{7} - 25274 T_{7}^{6} - 1004258096 T_{7}^{5} + 21084027831504 T_{7}^{4} + \cdots + 45\!\cdots\!00 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{14} + \cdots + 23\!\cdots\!00 \) Copy content Toggle raw display
$3$ \( T^{14} + \cdots + 25\!\cdots\!49 \) Copy content Toggle raw display
$5$ \( T^{14} \) Copy content Toggle raw display
$7$ \( (T^{7} + \cdots + 45\!\cdots\!00)^{2} \) Copy content Toggle raw display
$11$ \( T^{14} + \cdots + 56\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( (T^{7} + \cdots - 17\!\cdots\!00)^{2} \) Copy content Toggle raw display
$17$ \( T^{14} + \cdots + 13\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( (T^{7} + \cdots + 11\!\cdots\!48)^{2} \) Copy content Toggle raw display
$23$ \( T^{14} + \cdots + 55\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( T^{14} + \cdots + 40\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( (T^{7} + \cdots + 13\!\cdots\!12)^{2} \) Copy content Toggle raw display
$37$ \( (T^{7} + \cdots - 32\!\cdots\!00)^{2} \) Copy content Toggle raw display
$41$ \( T^{14} + \cdots + 20\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( (T^{7} + \cdots - 49\!\cdots\!00)^{2} \) Copy content Toggle raw display
$47$ \( T^{14} + \cdots + 13\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{14} + \cdots + 23\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( T^{14} + \cdots + 26\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( (T^{7} + \cdots - 71\!\cdots\!08)^{2} \) Copy content Toggle raw display
$67$ \( (T^{7} + \cdots - 14\!\cdots\!00)^{2} \) Copy content Toggle raw display
$71$ \( T^{14} + \cdots + 21\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( (T^{7} + \cdots - 26\!\cdots\!00)^{2} \) Copy content Toggle raw display
$79$ \( (T^{7} + \cdots - 24\!\cdots\!12)^{2} \) Copy content Toggle raw display
$83$ \( T^{14} + \cdots + 57\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( T^{14} + \cdots + 59\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( (T^{7} + \cdots - 17\!\cdots\!00)^{2} \) Copy content Toggle raw display
show more
show less