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Properties

Label 819.2.y.h
Level $819$
Weight $2$
Character orbit 819.y
Analytic conductor $6.540$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [819,2,Mod(307,819)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(819, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 2, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("819.307");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 819 = 3^{2} \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 819.y (of order \(4\), degree \(2\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(6.53974792554\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 35x^{8} + 295x^{4} + 169 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 91)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{7} q^{2} + ( - \beta_{8} - \beta_{7} + \cdots + \beta_{4}) q^{4}+ \cdots + ( - \beta_{6} - \beta_{3}) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{7} q^{2} + ( - \beta_{8} - \beta_{7} + \cdots + \beta_{4}) q^{4}+ \cdots + ( - 2 \beta_{10} - 2 \beta_{9} + \cdots - 2 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 4 q^{2} - 8 q^{7} - 4 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 4 q^{2} - 8 q^{7} - 4 q^{8} + 8 q^{11} - 8 q^{14} + 16 q^{16} - 16 q^{22} - 20 q^{28} + 4 q^{29} + 16 q^{32} + 4 q^{35} + 12 q^{37} - 12 q^{44} + 24 q^{46} + 28 q^{50} + 12 q^{53} - 44 q^{58} + 40 q^{65} + 60 q^{67} + 4 q^{70} + 48 q^{74} - 4 q^{79} + 12 q^{85} - 36 q^{86} - 32 q^{91} - 24 q^{92} + 28 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 35x^{8} + 295x^{4} + 169 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{9} + 58\nu^{5} + 659\nu ) / 194 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -5\nu^{8} - 96\nu^{4} + 3 ) / 194 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -9\nu^{10} - 328\nu^{6} - 3409\nu^{2} ) / 2522 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -9\nu^{11} - 328\nu^{7} - 3409\nu^{3} ) / 2522 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 7\nu^{10} + 115\nu^{6} - 431\nu^{2} ) / 1261 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -25\nu^{10} - 39\nu^{8} - 771\nu^{6} - 1001\nu^{4} - 5126\nu^{2} - 4264 ) / 2522 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -25\nu^{10} + 39\nu^{8} - 771\nu^{6} + 1001\nu^{4} - 5126\nu^{2} + 4264 ) / 2522 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 25\nu^{11} - 39\nu^{9} + 771\nu^{7} - 1001\nu^{5} + 5126\nu^{3} - 4264\nu ) / 2522 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( -25\nu^{11} - 39\nu^{9} - 771\nu^{7} - 1001\nu^{5} - 5126\nu^{3} - 4264\nu ) / 2522 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( -18\nu^{11} - 656\nu^{7} - 5557\nu^{3} ) / 1261 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{8} + \beta_{7} + \beta_{6} - 4\beta_{4} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{11} - 4\beta_{5} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 5\beta_{8} - 5\beta_{7} + 6\beta_{3} - 17 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{10} + \beta_{9} + 6\beta_{2} - 17\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -22\beta_{8} - 22\beta_{7} - 31\beta_{6} + 74\beta_{4} \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -31\beta_{11} + 9\beta_{10} - 9\beta_{9} + 74\beta_{5} \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( -96\beta_{8} + 96\beta_{7} - 154\beta_{3} + 327 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( -58\beta_{10} - 58\beta_{9} - 154\beta_{2} + 327\beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 423\beta_{8} + 423\beta_{7} + 751\beta_{6} - 1462\beta_{4} \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 751\beta_{11} - 328\beta_{10} + 328\beta_{9} - 1462\beta_{5} \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/819\mathbb{Z}\right)^\times\).

\(n\) \(92\) \(379\) \(703\)
\(\chi(n)\) \(1\) \(-\beta_{4}\) \(-1\)

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
307.1
1.33026 + 1.33026i
−1.33026 − 1.33026i
−1.52891 − 1.52891i
1.52891 + 1.52891i
0.626770 + 0.626770i
−0.626770 − 0.626770i
1.33026 − 1.33026i
−1.33026 + 1.33026i
−1.52891 + 1.52891i
1.52891 − 1.52891i
0.626770 − 0.626770i
−0.626770 + 0.626770i
−0.854638 − 0.854638i 0 − 0.539189i −0.612999 + 0.612999i 0 −2.64571 − 0.0148122i −2.17009 + 2.17009i 0 1.04778
307.2 −0.854638 − 0.854638i 0 − 0.539189i 0.612999 − 0.612999i 0 0.0148122 + 2.64571i −2.17009 + 2.17009i 0 −1.04778
307.3 0.403032 + 0.403032i 0 − 1.67513i −1.03221 + 1.03221i 0 2.60707 + 0.450747i 1.48119 − 1.48119i 0 −0.832030
307.4 0.403032 + 0.403032i 0 − 1.67513i 1.03221 − 1.03221i 0 −0.450747 − 2.60707i 1.48119 − 1.48119i 0 0.832030
307.5 1.45161 + 1.45161i 0 2.21432i −2.01464 + 2.01464i 0 −2.38948 + 1.13594i −0.311108 + 0.311108i 0 −5.84892
307.6 1.45161 + 1.45161i 0 2.21432i 2.01464 − 2.01464i 0 −1.13594 + 2.38948i −0.311108 + 0.311108i 0 5.84892
811.1 −0.854638 + 0.854638i 0 0.539189i −0.612999 − 0.612999i 0 −2.64571 + 0.0148122i −2.17009 − 2.17009i 0 1.04778
811.2 −0.854638 + 0.854638i 0 0.539189i 0.612999 + 0.612999i 0 0.0148122 − 2.64571i −2.17009 − 2.17009i 0 −1.04778
811.3 0.403032 − 0.403032i 0 1.67513i −1.03221 − 1.03221i 0 2.60707 − 0.450747i 1.48119 + 1.48119i 0 −0.832030
811.4 0.403032 − 0.403032i 0 1.67513i 1.03221 + 1.03221i 0 −0.450747 + 2.60707i 1.48119 + 1.48119i 0 0.832030
811.5 1.45161 − 1.45161i 0 − 2.21432i −2.01464 − 2.01464i 0 −2.38948 − 1.13594i −0.311108 − 0.311108i 0 −5.84892
811.6 1.45161 − 1.45161i 0 − 2.21432i 2.01464 + 2.01464i 0 −1.13594 − 2.38948i −0.311108 − 0.311108i 0 5.84892
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 307.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
13.d odd 4 1 inner
91.i even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 819.2.y.h 12
3.b odd 2 1 91.2.i.a ✓ 12
7.b odd 2 1 inner 819.2.y.h 12
13.d odd 4 1 inner 819.2.y.h 12
21.c even 2 1 91.2.i.a ✓ 12
21.g even 6 2 637.2.bc.a 24
21.h odd 6 2 637.2.bc.a 24
39.f even 4 1 91.2.i.a ✓ 12
91.i even 4 1 inner 819.2.y.h 12
273.o odd 4 1 91.2.i.a ✓ 12
273.cb odd 12 2 637.2.bc.a 24
273.cd even 12 2 637.2.bc.a 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.2.i.a ✓ 12 3.b odd 2 1
91.2.i.a ✓ 12 21.c even 2 1
91.2.i.a ✓ 12 39.f even 4 1
91.2.i.a ✓ 12 273.o odd 4 1
637.2.bc.a 24 21.g even 6 2
637.2.bc.a 24 21.h odd 6 2
637.2.bc.a 24 273.cb odd 12 2
637.2.bc.a 24 273.cd even 12 2
819.2.y.h 12 1.a even 1 1 trivial
819.2.y.h 12 7.b odd 2 1 inner
819.2.y.h 12 13.d odd 4 1 inner
819.2.y.h 12 91.i even 4 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(819, [\chi])\):

\( T_{2}^{6} - 2T_{2}^{5} + 2T_{2}^{4} + 2T_{2}^{3} + 4T_{2}^{2} - 4T_{2} + 2 \) Copy content Toggle raw display
\( T_{5}^{12} + 71T_{5}^{8} + 339T_{5}^{4} + 169 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{6} - 2 T^{5} + 2 T^{4} + \cdots + 2)^{2} \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( T^{12} + 71 T^{8} + \cdots + 169 \) Copy content Toggle raw display
$7$ \( T^{12} + 8 T^{11} + \cdots + 117649 \) Copy content Toggle raw display
$11$ \( (T^{6} - 4 T^{5} + 8 T^{4} + \cdots + 2)^{2} \) Copy content Toggle raw display
$13$ \( T^{12} - 20 T^{10} + \cdots + 4826809 \) Copy content Toggle raw display
$17$ \( (T^{6} - 82 T^{4} + \cdots - 9386)^{2} \) Copy content Toggle raw display
$19$ \( T^{12} + 1203 T^{8} + \cdots + 169 \) Copy content Toggle raw display
$23$ \( (T^{6} + 63 T^{4} + \cdots + 729)^{2} \) Copy content Toggle raw display
$29$ \( (T^{3} - T^{2} - 33 T + 85)^{4} \) Copy content Toggle raw display
$31$ \( T^{12} + \cdots + 316733209 \) Copy content Toggle raw display
$37$ \( (T^{6} - 6 T^{5} + \cdots + 1250)^{2} \) Copy content Toggle raw display
$41$ \( T^{12} + 160 T^{8} + \cdots + 43264 \) Copy content Toggle raw display
$43$ \( (T^{6} + 195 T^{4} + \cdots + 83521)^{2} \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 2339947129 \) Copy content Toggle raw display
$53$ \( (T^{3} - 3 T^{2} + \cdots - 113)^{4} \) Copy content Toggle raw display
$59$ \( T^{12} + 28060 T^{8} + \cdots + 2704 \) Copy content Toggle raw display
$61$ \( (T^{6} + 196 T^{4} + \cdots + 260000)^{2} \) Copy content Toggle raw display
$67$ \( (T^{6} - 30 T^{5} + \cdots + 35912)^{2} \) Copy content Toggle raw display
$71$ \( (T^{6} - 134 T^{3} + \cdots + 8978)^{2} \) Copy content Toggle raw display
$73$ \( T^{12} + 33867 T^{8} + \cdots + 105625 \) Copy content Toggle raw display
$79$ \( (T^{3} + T^{2} - 261 T - 1165)^{4} \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 6276893869129 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 19021098889 \) Copy content Toggle raw display
$97$ \( T^{12} + 6947 T^{8} + \cdots + 105625 \) Copy content Toggle raw display
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