↑

Properties

Label 828.2.c.f
Level $828$
Weight $2$
Character orbit 828.c
Analytic conductor $6.612$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [828,2,Mod(323,828)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(828, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("828.323");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 828 = 2^{2} \cdot 3^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 828.c (of order \(2\), degree \(1\), 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: \(6.61161328736\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 6 x^{11} + 7 x^{10} + 20 x^{9} + 49 x^{8} - 382 x^{7} - 235 x^{6} + 2192 x^{5} + 668 x^{4} + \cdots + 1516 \) 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

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.

\(f(q)\) \(=\) \( q - \beta_{6} q^{2} + (\beta_{7} + \beta_{2}) q^{4} + ( - \beta_{6} - \beta_{3} - \beta_{2}) q^{5} + (\beta_{9} + \beta_{8} + \beta_{7} + \cdots - 1) q^{7}+ \cdots + ( - \beta_{7} + \beta_{5} + \beta_{2}) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{6} q^{2} + (\beta_{7} + \beta_{2}) q^{4} + ( - \beta_{6} - \beta_{3} - \beta_{2}) q^{5} + (\beta_{9} + \beta_{8} + \beta_{7} + \cdots - 1) q^{7}+ \cdots + ( - 2 \beta_{11} - 2 \beta_{10} + \cdots + 2) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 4 q^{2} + 4 q^{4} + 4 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 4 q^{2} + 4 q^{4} + 4 q^{8} - 8 q^{10} + 12 q^{11} - 4 q^{13} - 12 q^{14} - 12 q^{16} + 20 q^{20} + 4 q^{22} + 12 q^{23} - 40 q^{25} + 8 q^{26} - 24 q^{28} + 44 q^{32} - 28 q^{34} + 40 q^{35} - 32 q^{37} - 32 q^{38} - 12 q^{40} + 4 q^{44} + 4 q^{46} - 32 q^{49} - 20 q^{50} + 32 q^{52} - 8 q^{56} - 16 q^{58} - 24 q^{59} - 56 q^{61} + 28 q^{64} + 16 q^{68} + 64 q^{70} - 56 q^{71} + 8 q^{73} - 36 q^{74} - 28 q^{76} + 28 q^{80} + 24 q^{82} - 28 q^{83} - 12 q^{85} - 8 q^{86} + 4 q^{88} + 4 q^{92} + 16 q^{94} + 8 q^{95} - 4 q^{97} - 20 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 6 x^{11} + 7 x^{10} + 20 x^{9} + 49 x^{8} - 382 x^{7} - 235 x^{6} + 2192 x^{5} + 668 x^{4} + \cdots + 1516 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 88442 \nu^{11} + 1105913 \nu^{10} - 5247742 \nu^{9} - 4946728 \nu^{8} + 38222699 \nu^{7} + \cdots - 4154576737 ) / 1067332921 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 589208 \nu^{11} - 3840274 \nu^{10} + 6401729 \nu^{9} + 5949918 \nu^{8} + 32090090 \nu^{7} + \cdots - 1742855360 ) / 2134665842 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 1551363 \nu^{11} - 15642992 \nu^{10} + 43816423 \nu^{9} + 14197330 \nu^{8} - 67206299 \nu^{7} + \cdots + 18089930960 ) / 2134665842 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 1834094 \nu^{11} + 11372482 \nu^{10} - 18465355 \nu^{9} - 24462718 \nu^{8} + \cdots - 7917342992 ) / 2134665842 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 1085283 \nu^{11} - 6619103 \nu^{10} + 8530376 \nu^{9} + 17414378 \nu^{8} + 57335395 \nu^{7} + \cdots - 984295302 ) / 1067332921 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 2759774 \nu^{11} + 14478294 \nu^{10} - 10461551 \nu^{9} - 55150274 \nu^{8} + \cdots - 3027893328 ) / 2134665842 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 58542 \nu^{11} - 321981 \nu^{10} + 311847 \nu^{9} + 1011546 \nu^{8} + 3610340 \nu^{7} + \cdots + 37559816 ) / 40276714 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 1674491 \nu^{11} - 9260117 \nu^{10} + 8935805 \nu^{9} + 29441909 \nu^{8} + 101092833 \nu^{7} + \cdots + 2587702295 ) / 1067332921 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 4311137 \nu^{11} + 19871151 \nu^{10} - 3027299 \nu^{9} - 92248346 \nu^{8} - 315974843 \nu^{7} + \cdots - 79861492 ) / 2134665842 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 4525947 \nu^{11} + 30819072 \nu^{10} - 57015518 \nu^{9} - 46763110 \nu^{8} + \cdots - 12677532482 ) / 2134665842 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 4525947 \nu^{11} + 17767085 \nu^{10} + 8244417 \nu^{9} - 86251826 \nu^{8} + \cdots + 7646123486 ) / 2134665842 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{11} + \beta_{10} + \beta_{8} + \beta_{7} - \beta_{4} + \beta_{2} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( - \beta_{11} + 3 \beta_{10} + 2 \beta_{9} + 3 \beta_{8} + \beta_{7} + 2 \beta_{6} + 2 \beta_{5} + \cdots + 2 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 5 \beta_{11} + \beta_{10} - 4 \beta_{9} + \beta_{8} - \beta_{7} + 8 \beta_{6} - 6 \beta_{5} + \cdots + 6 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 9 \beta_{11} - \beta_{10} - 22 \beta_{9} + 19 \beta_{8} - 11 \beta_{7} + 48 \beta_{6} - 12 \beta_{5} + \cdots - 30 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 45 \beta_{11} - 35 \beta_{10} - 88 \beta_{9} - 55 \beta_{8} + 31 \beta_{7} + 160 \beta_{6} + \cdots + 16 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 25 \beta_{11} - 239 \beta_{10} - 510 \beta_{9} - 431 \beta_{8} + 19 \beta_{7} + 400 \beta_{6} + \cdots + 418 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 773 \beta_{11} - 179 \beta_{10} - 836 \beta_{9} - 807 \beta_{8} + 881 \beta_{7} + 634 \beta_{6} + \cdots + 906 ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 2251 \beta_{11} + 1099 \beta_{10} + 42 \beta_{9} - 2783 \beta_{8} + 3601 \beta_{7} - 1352 \beta_{6} + \cdots + 5568 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 6535 \beta_{11} + 7093 \beta_{10} + 6572 \beta_{9} - 379 \beta_{8} + 6295 \beta_{7} - 11280 \beta_{6} + \cdots + 12140 ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 4049 \beta_{11} + 37521 \beta_{10} + 49986 \beta_{9} + 39761 \beta_{8} + 9903 \beta_{7} - 37844 \beta_{6} + \cdots - 20542 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 87835 \beta_{11} + 80597 \beta_{10} + 140724 \beta_{9} + 133865 \beta_{8} - 54751 \beta_{7} + \cdots - 134414 ) / 2 \) 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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
323.1
2.66767 + 0.569132i
−1.66767 + 0.569132i
−1.66767 − 0.569132i
2.66767 − 0.569132i
1.48099 − 0.275067i
−0.480986 − 0.275067i
−0.480986 + 0.275067i
1.48099 + 0.275067i
−1.64216 − 2.25841i
2.64216 − 2.25841i
2.64216 + 2.25841i
−1.64216 + 2.25841i
−1.06244 − 0.933389i 0 0.257569 + 1.98335i − 3.19064i 0 − 2.72270i 1.57758 − 2.34760i 0 −2.97811 + 3.38987i
323.2 −1.06244 − 0.933389i 0 0.257569 + 1.98335i − 1.22862i 0 3.40840i 1.57758 − 2.34760i 0 −1.14678 + 1.30533i
323.3 −1.06244 + 0.933389i 0 0.257569 − 1.98335i 1.22862i 0 − 3.40840i 1.57758 + 2.34760i 0 −1.14678 − 1.30533i
323.4 −1.06244 + 0.933389i 0 0.257569 − 1.98335i 3.19064i 0 2.72270i 1.57758 + 2.34760i 0 −2.97811 − 3.38987i
323.5 0.681664 − 1.23909i 0 −1.07067 − 1.68928i − 4.07529i 0 − 0.833936i −2.82300 + 0.175128i 0 −5.04963 − 2.77798i
323.6 0.681664 − 1.23909i 0 −1.07067 − 1.68928i 3.56146i 0 − 3.60858i −2.82300 + 0.175128i 0 4.41296 + 2.42772i
323.7 0.681664 + 1.23909i 0 −1.07067 + 1.68928i − 3.56146i 0 3.60858i −2.82300 − 0.175128i 0 4.41296 − 2.42772i
323.8 0.681664 + 1.23909i 0 −1.07067 + 1.68928i 4.07529i 0 0.833936i −2.82300 − 0.175128i 0 −5.04963 + 2.77798i
323.9 1.38078 − 0.305697i 0 1.81310 − 0.844199i − 0.474164i 0 1.17248i 2.24542 − 1.71991i 0 −0.144950 − 0.654715i
323.10 1.38078 − 0.305697i 0 1.81310 − 0.844199i 2.96538i 0 − 4.88648i 2.24542 − 1.71991i 0 0.906508 + 4.09454i
323.11 1.38078 + 0.305697i 0 1.81310 + 0.844199i − 2.96538i 0 4.88648i 2.24542 + 1.71991i 0 0.906508 − 4.09454i
323.12 1.38078 + 0.305697i 0 1.81310 + 0.844199i 0.474164i 0 − 1.17248i 2.24542 + 1.71991i 0 −0.144950 + 0.654715i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 323.12
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 828.2.c.f yes 12
3.b odd 2 1 828.2.c.e ✓ 12
4.b odd 2 1 828.2.c.e ✓ 12
12.b even 2 1 inner 828.2.c.f yes 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
828.2.c.e ✓ 12 3.b odd 2 1
828.2.c.e ✓ 12 4.b odd 2 1
828.2.c.f yes 12 1.a even 1 1 trivial
828.2.c.f yes 12 12.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(828, [\chi])\):

\( T_{5}^{12} + 50T_{5}^{10} + 940T_{5}^{8} + 8120T_{5}^{6} + 30628T_{5}^{4} + 34952T_{5}^{2} + 6400 \) Copy content Toggle raw display
\( T_{11}^{6} - 6T_{11}^{5} - 24T_{11}^{4} + 136T_{11}^{3} + 278T_{11}^{2} - 844T_{11} - 1616 \) Copy content Toggle raw display
\( T_{47}^{3} - 10T_{47} - 8 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{6} - 2 T^{5} + T^{4} + \cdots + 8)^{2} \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( T^{12} + 50 T^{10} + \cdots + 6400 \) Copy content Toggle raw display
$7$ \( T^{12} + 58 T^{10} + \cdots + 25600 \) Copy content Toggle raw display
$11$ \( (T^{6} - 6 T^{5} + \cdots - 1616)^{2} \) Copy content Toggle raw display
$13$ \( (T^{6} + 2 T^{5} + \cdots + 1280)^{2} \) Copy content Toggle raw display
$17$ \( T^{12} + 90 T^{10} + \cdots + 123904 \) Copy content Toggle raw display
$19$ \( T^{12} + \cdots + 166513216 \) Copy content Toggle raw display
$23$ \( (T - 1)^{12} \) Copy content Toggle raw display
$29$ \( T^{12} + \cdots + 283316224 \) Copy content Toggle raw display
$31$ \( T^{12} + 160 T^{10} + \cdots + 1048576 \) Copy content Toggle raw display
$37$ \( (T^{6} + 16 T^{5} + \cdots + 2488)^{2} \) Copy content Toggle raw display
$41$ \( (T^{6} + 66 T^{4} + \cdots + 512)^{2} \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots + 18725185600 \) Copy content Toggle raw display
$47$ \( (T^{3} - 10 T - 8)^{4} \) Copy content Toggle raw display
$53$ \( T^{12} + 298 T^{10} + \cdots + 19009600 \) Copy content Toggle raw display
$59$ \( (T^{6} + 12 T^{5} + \cdots - 8192)^{2} \) Copy content Toggle raw display
$61$ \( (T^{6} + 28 T^{5} + \cdots - 108184)^{2} \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 413878336 \) Copy content Toggle raw display
$71$ \( (T^{6} + 28 T^{5} + \cdots + 403840)^{2} \) Copy content Toggle raw display
$73$ \( (T^{6} - 4 T^{5} + \cdots - 48512)^{2} \) Copy content Toggle raw display
$79$ \( T^{12} + 298 T^{10} + \cdots + 3564544 \) Copy content Toggle raw display
$83$ \( (T^{6} + 14 T^{5} + \cdots - 8656)^{2} \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 2372079616 \) Copy content Toggle raw display
$97$ \( (T^{6} + 2 T^{5} + \cdots + 107392)^{2} \) Copy content Toggle raw display
show more
show less