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Properties

Label 825.2.f.f
Level $825$
Weight $2$
Character orbit 825.f
Analytic conductor $6.588$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [825,2,Mod(626,825)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(825, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("825.626");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 825 = 3 \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 825.f (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: \(6.58765816676\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 32x^{14} + 384x^{12} + 2192x^{10} + 6394x^{8} + 9216x^{6} + 5376x^{4} + 432x^{2} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{11} \)
Twist minimal: no (minimal twist has level 165)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{6} q^{2} - \beta_{5} q^{3} + ( - \beta_{2} + 2) q^{4} + \beta_{14} q^{6} + \beta_{9} q^{7} + (\beta_{6} - \beta_1) q^{8} + ( - \beta_{3} - \beta_{2}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{6} q^{2} - \beta_{5} q^{3} + ( - \beta_{2} + 2) q^{4} + \beta_{14} q^{6} + \beta_{9} q^{7} + (\beta_{6} - \beta_1) q^{8} + ( - \beta_{3} - \beta_{2}) q^{9} + ( - \beta_{10} - \beta_{8}) q^{11} + (\beta_{12} - 2 \beta_{5} + \beta_{4}) q^{12} + \beta_{9} q^{13} + (4 \beta_{10} + \beta_{3}) q^{14} + ( - 2 \beta_{2} - 1) q^{16} - \beta_1 q^{17} + (\beta_{13} - \beta_{12} + \cdots - \beta_1) q^{18}+ \cdots + ( - 2 \beta_{15} + 2 \beta_{11} + \cdots - 3) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 32 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 32 q^{4} - 16 q^{16} - 16 q^{34} + 48 q^{36} - 16 q^{49} - 32 q^{64} - 48 q^{66} + 48 q^{69} - 48 q^{81} - 128 q^{91} - 48 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 32x^{14} + 384x^{12} + 2192x^{10} + 6394x^{8} + 9216x^{6} + 5376x^{4} + 432x^{2} + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 65 \nu^{14} + 2284 \nu^{12} + 30759 \nu^{10} + 199882 \nu^{8} + 652049 \nu^{6} + 1006380 \nu^{4} + \cdots + 12522 ) / 3828 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -8\nu^{14} - 253\nu^{12} - 2988\nu^{10} - 16720\nu^{8} - 47840\nu^{6} - 67707\nu^{4} - 38052\nu^{2} - 1584 ) / 174 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{15} + 31\nu^{13} + 355\nu^{11} + 1895\nu^{9} + 5093\nu^{7} + 6731\nu^{5} + 3711\nu^{3} + 435\nu ) / 24 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 281 \nu^{15} - 40 \nu^{14} + 8912 \nu^{13} - 1381 \nu^{12} + 105461 \nu^{11} - 18217 \nu^{10} + \cdots - 23754 ) / 7656 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 281 \nu^{15} + 40 \nu^{14} + 8912 \nu^{13} + 1381 \nu^{12} + 105461 \nu^{11} + 18217 \nu^{10} + \cdots + 23754 ) / 7656 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 256 \nu^{14} + 8009 \nu^{12} + 92919 \nu^{10} + 504677 \nu^{8} + 1379152 \nu^{6} + 1831761 \nu^{4} + \cdots + 41205 ) / 3828 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 1213 \nu^{15} + 37676 \nu^{13} + 432654 \nu^{11} + 2317235 \nu^{9} + 6230185 \nu^{7} + \cdots + 20151 \nu ) / 22968 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 139 \nu^{15} + 39 \nu^{14} - 4472 \nu^{13} + 1179 \nu^{12} - 54048 \nu^{11} + 12957 \nu^{10} + \cdots - 2979 ) / 2088 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 2329 \nu^{15} + 74579 \nu^{13} + 896025 \nu^{11} + 5125739 \nu^{9} + 15002653 \nu^{7} + \cdots + 836199 \nu ) / 22968 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 365 \nu^{15} + 11641 \nu^{13} + 138981 \nu^{11} + 787123 \nu^{9} + 2269265 \nu^{7} + 3207789 \nu^{5} + \cdots + 75663 \nu ) / 2088 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 380 \nu^{15} - 12148 \nu^{13} - 145497 \nu^{11} - 827347 \nu^{9} - 2395244 \nu^{7} + \cdots - 128223 \nu ) / 2088 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 1504 \nu^{15} - 369 \nu^{14} + 48289 \nu^{13} - 11695 \nu^{12} + 581986 \nu^{11} - 138329 \nu^{10} + \cdots - 62709 ) / 7656 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 1504 \nu^{15} - 369 \nu^{14} - 48289 \nu^{13} - 11695 \nu^{12} - 581986 \nu^{11} - 138329 \nu^{10} + \cdots - 62709 ) / 7656 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 176 \nu^{15} - 21 \nu^{14} + 5624 \nu^{13} - 733 \nu^{12} + 67331 \nu^{11} - 9830 \nu^{10} + \cdots - 9726 ) / 696 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 176 \nu^{15} - 21 \nu^{14} - 5624 \nu^{13} - 733 \nu^{12} - 67331 \nu^{11} - 9830 \nu^{10} + \cdots - 9726 ) / 696 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{10} + \beta_{9} + \beta_{5} + \beta_{4} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{13} + \beta_{12} - \beta_{10} - 2\beta_{8} + 2\beta_{6} + \beta_{3} - 8 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 3 \beta_{15} - 3 \beta_{14} - 2 \beta_{13} + 2 \beta_{12} + 11 \beta_{10} - 7 \beta_{9} + \cdots - 8 \beta_{4} ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 7 \beta_{13} - 7 \beta_{12} + 5 \beta_{10} + 10 \beta_{8} - 12 \beta_{6} + 2 \beta_{5} - 2 \beta_{4} + \cdots + 32 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 40 \beta_{15} + 40 \beta_{14} + 24 \beta_{13} - 24 \beta_{12} - 124 \beta_{10} + 63 \beta_{9} + \cdots + 5 \beta_{3} ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 2 \beta_{15} + 2 \beta_{14} + 163 \beta_{13} + 163 \beta_{12} - 103 \beta_{10} - 206 \beta_{8} + \cdots - 620 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 455 \beta_{15} - 455 \beta_{14} - 255 \beta_{13} + 255 \beta_{12} - 14 \beta_{11} + 1359 \beta_{10} + \cdots - 98 \beta_{3} ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 28 \beta_{15} - 28 \beta_{14} - 898 \beta_{13} - 898 \beta_{12} + 546 \beta_{10} + 1092 \beta_{8} + \cdots + 3203 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 4992 \beta_{15} + 4992 \beta_{14} + 2672 \beta_{13} - 2672 \beta_{12} + 336 \beta_{11} + \cdots + 1464 \beta_{3} ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 1016 \beta_{15} + 1016 \beta_{14} + 19463 \beta_{13} + 19463 \beta_{12} - 11727 \beta_{10} + \cdots - 67568 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 54241 \beta_{15} - 54241 \beta_{14} - 28006 \beta_{13} + 28006 \beta_{12} - 5632 \beta_{11} + \cdots - 19800 \beta_{3} ) / 2 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 7720 \beta_{15} - 7720 \beta_{14} - 104975 \beta_{13} - 104975 \beta_{12} + 63293 \beta_{10} + \cdots + 358880 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 587912 \beta_{15} + 587912 \beta_{14} + 294032 \beta_{13} - 294032 \beta_{12} + \cdots + 254423 \beta_{3} ) / 2 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 213894 \beta_{15} + 213894 \beta_{14} + 2262869 \beta_{13} + 2262869 \beta_{12} - 1369465 \beta_{10} + \cdots - 7647452 ) / 2 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 6370349 \beta_{15} - 6370349 \beta_{14} - 3091709 \beta_{13} + 3091709 \beta_{12} + \cdots - 3166886 \beta_{3} ) / 2 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/825\mathbb{Z}\right)^\times\).

\(n\) \(376\) \(551\) \(727\)
\(\chi(n)\) \(-1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
626.1
3.19627i
− 3.19627i
− 1.60382i
1.60382i
− 0.233800i
0.233800i
3.31018i
− 3.31018i
1.89596i
− 1.89596i
1.18041i
− 1.18041i
− 0.189602i
0.189602i
1.78205i
− 1.78205i
−2.39417 −1.53819 − 0.796225i 3.73205 0 3.68269 + 1.90630i 3.38587i −4.14682 1.73205 + 2.44949i 0
626.2 −2.39417 −1.53819 + 0.796225i 3.73205 0 3.68269 − 1.90630i − 3.38587i −4.14682 1.73205 − 2.44949i 0
626.3 −2.39417 1.53819 − 0.796225i 3.73205 0 −3.68269 + 1.90630i − 3.38587i −4.14682 1.73205 − 2.44949i 0
626.4 −2.39417 1.53819 + 0.796225i 3.73205 0 −3.68269 − 1.90630i 3.38587i −4.14682 1.73205 + 2.44949i 0
626.5 −1.50597 −0.796225 − 1.53819i 0.267949 0 1.19909 + 2.31647i − 2.12976i 2.60842 −1.73205 + 2.44949i 0
626.6 −1.50597 −0.796225 + 1.53819i 0.267949 0 1.19909 − 2.31647i 2.12976i 2.60842 −1.73205 − 2.44949i 0
626.7 −1.50597 0.796225 − 1.53819i 0.267949 0 −1.19909 + 2.31647i 2.12976i 2.60842 −1.73205 − 2.44949i 0
626.8 −1.50597 0.796225 + 1.53819i 0.267949 0 −1.19909 − 2.31647i − 2.12976i 2.60842 −1.73205 + 2.44949i 0
626.9 1.50597 −0.796225 − 1.53819i 0.267949 0 −1.19909 − 2.31647i 2.12976i −2.60842 −1.73205 + 2.44949i 0
626.10 1.50597 −0.796225 + 1.53819i 0.267949 0 −1.19909 + 2.31647i − 2.12976i −2.60842 −1.73205 − 2.44949i 0
626.11 1.50597 0.796225 − 1.53819i 0.267949 0 1.19909 − 2.31647i − 2.12976i −2.60842 −1.73205 − 2.44949i 0
626.12 1.50597 0.796225 + 1.53819i 0.267949 0 1.19909 + 2.31647i 2.12976i −2.60842 −1.73205 + 2.44949i 0
626.13 2.39417 −1.53819 − 0.796225i 3.73205 0 −3.68269 − 1.90630i − 3.38587i 4.14682 1.73205 + 2.44949i 0
626.14 2.39417 −1.53819 + 0.796225i 3.73205 0 −3.68269 + 1.90630i 3.38587i 4.14682 1.73205 − 2.44949i 0
626.15 2.39417 1.53819 − 0.796225i 3.73205 0 3.68269 − 1.90630i 3.38587i 4.14682 1.73205 − 2.44949i 0
626.16 2.39417 1.53819 + 0.796225i 3.73205 0 3.68269 + 1.90630i − 3.38587i 4.14682 1.73205 + 2.44949i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 626.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.b even 2 1 inner
11.b odd 2 1 inner
15.d odd 2 1 inner
33.d even 2 1 inner
55.d odd 2 1 inner
165.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 825.2.f.f 16
3.b odd 2 1 inner 825.2.f.f 16
5.b even 2 1 inner 825.2.f.f 16
5.c odd 4 2 165.2.d.c ✓ 16
11.b odd 2 1 inner 825.2.f.f 16
15.d odd 2 1 inner 825.2.f.f 16
15.e even 4 2 165.2.d.c ✓ 16
33.d even 2 1 inner 825.2.f.f 16
55.d odd 2 1 inner 825.2.f.f 16
55.e even 4 2 165.2.d.c ✓ 16
165.d even 2 1 inner 825.2.f.f 16
165.l odd 4 2 165.2.d.c ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
165.2.d.c ✓ 16 5.c odd 4 2
165.2.d.c ✓ 16 15.e even 4 2
165.2.d.c ✓ 16 55.e even 4 2
165.2.d.c ✓ 16 165.l odd 4 2
825.2.f.f 16 1.a even 1 1 trivial
825.2.f.f 16 3.b odd 2 1 inner
825.2.f.f 16 5.b even 2 1 inner
825.2.f.f 16 11.b odd 2 1 inner
825.2.f.f 16 15.d odd 2 1 inner
825.2.f.f 16 33.d even 2 1 inner
825.2.f.f 16 55.d odd 2 1 inner
825.2.f.f 16 165.d 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 \(S_{2}^{\mathrm{new}}(825, [\chi])\):

\( T_{2}^{4} - 8T_{2}^{2} + 13 \) Copy content Toggle raw display
\( T_{23}^{4} + 36T_{23}^{2} + 216 \) Copy content Toggle raw display
\( T_{37}^{4} - 72T_{37}^{2} + 96 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - 8 T^{2} + 13)^{4} \) Copy content Toggle raw display
$3$ \( (T^{8} + 6 T^{4} + 81)^{2} \) Copy content Toggle raw display
$5$ \( T^{16} \) Copy content Toggle raw display
$7$ \( (T^{4} + 16 T^{2} + 52)^{4} \) Copy content Toggle raw display
$11$ \( (T^{8} - 28 T^{6} + \cdots + 14641)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} + 16 T^{2} + 52)^{4} \) Copy content Toggle raw display
$17$ \( (T^{4} - 20 T^{2} + 52)^{4} \) Copy content Toggle raw display
$19$ \( (T^{4} + 36 T^{2} + 312)^{4} \) Copy content Toggle raw display
$23$ \( (T^{4} + 36 T^{2} + 216)^{4} \) Copy content Toggle raw display
$29$ \( (T^{4} - 84 T^{2} + 312)^{4} \) Copy content Toggle raw display
$31$ \( (T^{2} - 12)^{8} \) Copy content Toggle raw display
$37$ \( (T^{4} - 72 T^{2} + 96)^{4} \) Copy content Toggle raw display
$41$ \( (T^{4} - 60 T^{2} + 312)^{4} \) Copy content Toggle raw display
$43$ \( (T^{4} + 160 T^{2} + 6292)^{4} \) Copy content Toggle raw display
$47$ \( (T^{4} + 12 T^{2} + 24)^{4} \) Copy content Toggle raw display
$53$ \( (T^{4} + 96 T^{2} + 1536)^{4} \) Copy content Toggle raw display
$59$ \( (T^{2} + 24)^{8} \) Copy content Toggle raw display
$61$ \( (T^{4} + 120 T^{2} + 1248)^{4} \) Copy content Toggle raw display
$67$ \( (T^{4} - 324 T^{2} + 26136)^{4} \) Copy content Toggle raw display
$71$ \( (T^{4} + 192 T^{2} + 2304)^{4} \) Copy content Toggle raw display
$73$ \( (T^{4} + 16 T^{2} + 52)^{4} \) Copy content Toggle raw display
$79$ \( (T^{4} + 324 T^{2} + 25272)^{4} \) Copy content Toggle raw display
$83$ \( (T^{4} - 188 T^{2} + 8788)^{4} \) Copy content Toggle raw display
$89$ \( (T^{4} + 112 T^{2} + 1936)^{4} \) Copy content Toggle raw display
$97$ \( (T^{4} - 72 T^{2} + 96)^{4} \) Copy content Toggle raw display
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