↑

Properties

Label 828.3.d.a
Level $828$
Weight $3$
Character orbit 828.d
Analytic conductor $22.561$
Analytic rank $0$
Dimension $16$
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [828,3,Mod(737,828)] 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("828.737"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(828, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 828 = 2^{2} \cdot 3^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 828.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(22.5613658890\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 236 x^{14} + 21460 x^{12} + 962776 x^{10} + 22691080 x^{8} + 272328848 x^{6} + 1422566992 x^{4} + \cdots + 390615696 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{23}]\)
Coefficient ring index: \( 2^{10}\cdot 3^{4} \)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{5} - \beta_{11} q^{7} - \beta_{6} q^{11} + ( - \beta_{9} - 1) q^{13} - \beta_{4} q^{17} + ( - \beta_{12} + 2) q^{19} - \beta_{5} q^{23} + (\beta_{12} - \beta_{10} + \beta_{7} - 4) q^{25}+ \cdots + ( - \beta_{15} - 4 \beta_{14} + \cdots + 10) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 8 q^{13} + 40 q^{19} - 72 q^{25} - 32 q^{31} - 64 q^{37} + 48 q^{43} + 8 q^{49} + 144 q^{55} - 48 q^{61} - 40 q^{67} + 48 q^{73} - 24 q^{79} - 8 q^{85} - 24 q^{91} + 168 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 236 x^{14} + 21460 x^{12} + 962776 x^{10} + 22691080 x^{8} + 272328848 x^{6} + 1422566992 x^{4} + \cdots + 390615696 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 31543 \nu^{15} + 6744162 \nu^{13} + 515486422 \nu^{11} + 16367733116 \nu^{9} + \cdots - 77716511343216 \nu ) / 8503822760256 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 2146989821 \nu^{15} + 670150237869 \nu^{13} + 82023829719302 \nu^{11} + \cdots + 16\!\cdots\!48 \nu ) / 29\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 107498101 \nu^{15} + 28622929356 \nu^{13} + 3022807081795 \nu^{11} + \cdots + 26\!\cdots\!04 \nu ) / 74\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 98 \nu^{15} - 21891 \nu^{13} - 1835348 \nu^{11} - 72983278 \nu^{9} - 1439635968 \nu^{7} + \cdots - 71646725016 \nu ) / 6611295168 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 2459870041 \nu^{15} + 563941695744 \nu^{13} + 48962834098162 \nu^{11} + \cdots + 19\!\cdots\!28 \nu ) / 14\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 5779759733 \nu^{14} - 1210336768332 \nu^{12} - 88399954241006 \nu^{10} + \cdots + 36\!\cdots\!96 ) / 29\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 10187022263 \nu^{15} - 2434140623502 \nu^{13} - 224440468079726 \nu^{11} + \cdots - 12\!\cdots\!44 \nu ) / 29\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 1075881031 \nu^{14} + 271001675214 \nu^{12} + 26280193927222 \nu^{10} + \cdots + 16\!\cdots\!48 ) / 49\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 2234418482 \nu^{14} + 483830785953 \nu^{12} + 38437995644819 \nu^{10} + \cdots - 27\!\cdots\!24 ) / 74\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 2515654712 \nu^{14} - 527987070063 \nu^{12} - 39788106480539 \nu^{10} + \cdots + 12\!\cdots\!64 ) / 74\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 14717433661 \nu^{14} + 3145659912144 \nu^{12} + 242151936820282 \nu^{10} + \cdots - 59\!\cdots\!12 ) / 29\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 13753415089 \nu^{15} - 3161918540706 \nu^{13} - 276847951753138 \nu^{11} + \cdots - 55\!\cdots\!52 \nu ) / 98\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 9147839641 \nu^{14} - 1962935163234 \nu^{12} - 156133365111262 \nu^{10} + \cdots - 23\!\cdots\!28 ) / 98\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 28807145413 \nu^{14} - 6362021851272 \nu^{12} - 522334368152866 \nu^{10} + \cdots - 20\!\cdots\!84 ) / 14\!\cdots\!60 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{12} - \beta_{10} + \beta_{7} - 29 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{13} - 3\beta_{8} - 3\beta_{6} - 12\beta_{5} + 5\beta_{4} - 4\beta_{3} + \beta_{2} - 51\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2\beta_{15} - 12\beta_{14} - 66\beta_{12} + 46\beta_{11} + 72\beta_{10} - 2\beta_{9} - 102\beta_{7} + 1562 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 196 \beta_{13} + 186 \beta_{8} + 268 \beta_{6} + 1236 \beta_{5} - 544 \beta_{4} + 340 \beta_{3} + \cdots + 3258 \beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 586 \beta_{15} + 1664 \beta_{14} + 4384 \beta_{12} - 4850 \beta_{11} - 5294 \beta_{10} + \cdots - 102588 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 16928 \beta_{13} - 10538 \beta_{8} - 23050 \beta_{6} - 111504 \beta_{5} + 47438 \beta_{4} + \cdots - 230518 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 79932 \beta_{15} - 174592 \beta_{14} - 305980 \beta_{12} + 406020 \beta_{11} + 391656 \beta_{10} + \cdots + 7383896 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 1409776 \beta_{13} + 603376 \beta_{8} + 2037688 \beta_{6} + 9622080 \beta_{5} - 3899296 \beta_{4} + \cdots + 17129468 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 8626560 \beta_{15} + 16471920 \beta_{14} + 22219732 \beta_{12} - 31937184 \beta_{11} + \cdots - 555801884 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 115869896 \beta_{13} - 35092068 \beta_{8} - 179966460 \beta_{6} - 813495504 \beta_{5} + \cdots - 1307701260 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 835788464 \beta_{15} - 1473913968 \beta_{14} - 1659380784 \beta_{12} + 2475417232 \beta_{11} + \cdots + 42872345072 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 9474687856 \beta_{13} + 2051561496 \beta_{8} + 15673618432 \beta_{6} + 67999748016 \beta_{5} + \cdots + 101549807736 \beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 76333936216 \beta_{15} + 127953252704 \beta_{14} + 126445512304 \beta_{12} - 192176623736 \beta_{11} + \cdots - 3357186688944 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 773175637472 \beta_{13} - 119295016568 \beta_{8} - 1344412441192 \beta_{6} - 5643657919488 \beta_{5} + \cdots - 7980026716888 \beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/828\mathbb{Z}\right)^\times\).

\(n\) \(415\) \(461\) \(649\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
737.1
− 9.02853i
− 7.82949i
− 5.70213i
− 5.44642i
− 3.94297i
− 3.72779i
− 1.12594i
− 0.543986i
0.543986i
1.12594i
3.72779i
3.94297i
5.44642i
5.70213i
7.82949i
9.02853i
0 0 0 − 9.02853i 0 −2.91592 0 0 0
737.2 0 0 0 − 7.82949i 0 −3.84835 0 0 0
737.3 0 0 0 − 5.70213i 0 3.93523 0 0 0
737.4 0 0 0 − 5.44642i 0 12.3629 0 0 0
737.5 0 0 0 − 3.94297i 0 −5.13765 0 0 0
737.6 0 0 0 − 3.72779i 0 5.11959 0 0 0
737.7 0 0 0 − 1.12594i 0 2.53870 0 0 0
737.8 0 0 0 − 0.543986i 0 −12.0545 0 0 0
737.9 0 0 0 0.543986i 0 −12.0545 0 0 0
737.10 0 0 0 1.12594i 0 2.53870 0 0 0
737.11 0 0 0 3.72779i 0 5.11959 0 0 0
737.12 0 0 0 3.94297i 0 −5.13765 0 0 0
737.13 0 0 0 5.44642i 0 12.3629 0 0 0
737.14 0 0 0 5.70213i 0 3.93523 0 0 0
737.15 0 0 0 7.82949i 0 −3.84835 0 0 0
737.16 0 0 0 9.02853i 0 −2.91592 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 737.16
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 828.3.d.a ✓ 16
3.b odd 2 1 inner 828.3.d.a ✓ 16
4.b odd 2 1 3312.3.g.c 16
12.b even 2 1 3312.3.g.c 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
828.3.d.a ✓ 16 1.a even 1 1 trivial
828.3.d.a ✓ 16 3.b odd 2 1 inner
3312.3.g.c 16 4.b odd 2 1
3312.3.g.c 16 12.b even 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(828, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( T^{16} + \cdots + 390615696 \) Copy content Toggle raw display
$7$ \( (T^{8} - 198 T^{6} + \cdots + 439440)^{2} \) Copy content Toggle raw display
$11$ \( T^{16} + \cdots + 78364164096 \) Copy content Toggle raw display
$13$ \( (T^{8} + 4 T^{7} + \cdots + 75057280)^{2} \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 14\!\cdots\!64 \) Copy content Toggle raw display
$19$ \( (T^{8} - 20 T^{7} + \cdots - 44464884)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} + 23)^{8} \) Copy content Toggle raw display
$29$ \( T^{16} + \cdots + 13\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( (T^{8} + 16 T^{7} + \cdots - 69888960)^{2} \) Copy content Toggle raw display
$37$ \( (T^{8} + 32 T^{7} + \cdots + 515839047600)^{2} \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 13\!\cdots\!36 \) Copy content Toggle raw display
$43$ \( (T^{8} + \cdots - 2096572054452)^{2} \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 50\!\cdots\!04 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 32\!\cdots\!44 \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 27\!\cdots\!16 \) Copy content Toggle raw display
$61$ \( (T^{8} + 24 T^{7} + \cdots + 339452612528)^{2} \) Copy content Toggle raw display
$67$ \( (T^{8} + 20 T^{7} + \cdots - 914372126100)^{2} \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 11\!\cdots\!16 \) Copy content Toggle raw display
$73$ \( (T^{8} + \cdots + 59655980520000)^{2} \) Copy content Toggle raw display
$79$ \( (T^{8} + \cdots - 16\!\cdots\!08)^{2} \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 30\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 71\!\cdots\!16 \) Copy content Toggle raw display
$97$ \( (T^{8} + \cdots + 549053370504576)^{2} \) Copy content Toggle raw display
show more
show less