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Properties

Label 825.2.f.e
Level $825$
Weight $2$
Character orbit 825.f
Analytic conductor $6.588$
Analytic rank $0$
Dimension $16$
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] = [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: \( x^{16} - 2 x^{15} - 17 x^{14} + 26 x^{13} + 191 x^{12} - 390 x^{11} - 539 x^{10} + 1484 x^{9} + \cdots + 102940 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{3} q^{2} + \beta_{4} q^{3} + (\beta_1 + 1) q^{4} + \beta_{5} q^{6} + \beta_{11} q^{7} + ( - \beta_{9} - \beta_{8} + \cdots + 3 \beta_{3}) q^{8}+ \cdots + ( - \beta_{12} - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{3} q^{2} + \beta_{4} q^{3} + (\beta_1 + 1) q^{4} + \beta_{5} q^{6} + \beta_{11} q^{7} + ( - \beta_{9} - \beta_{8} + \cdots + 3 \beta_{3}) q^{8}+ \cdots + ( - \beta_{15} - \beta_{14} + 2 \beta_{13} + \cdots + 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 2 q^{3} + 16 q^{4} - 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 2 q^{3} + 16 q^{4} - 10 q^{9} - 6 q^{12} + 32 q^{16} - 28 q^{22} - 2 q^{27} - 12 q^{31} + 20 q^{33} + 28 q^{34} - 36 q^{36} - 4 q^{37} - 58 q^{42} + 40 q^{48} - 28 q^{49} + 16 q^{58} + 60 q^{64} - 30 q^{66} + 44 q^{67} + 44 q^{69} + 44 q^{78} - 26 q^{81} + 8 q^{82} - 76 q^{88} + 64 q^{91} + 14 q^{93} + 108 q^{97} + 14 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 2 x^{15} - 17 x^{14} + 26 x^{13} + 191 x^{12} - 390 x^{11} - 539 x^{10} + 1484 x^{9} + \cdots + 102940 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 27\!\cdots\!66 \nu^{15} + \cdots + 78\!\cdots\!40 ) / 13\!\cdots\!05 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 11\!\cdots\!13 \nu^{15} + \cdots + 35\!\cdots\!20 ) / 54\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 11\!\cdots\!13 \nu^{15} + \cdots - 35\!\cdots\!20 ) / 54\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 12\!\cdots\!19 \nu^{15} + \cdots - 31\!\cdots\!20 ) / 54\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 22\!\cdots\!59 \nu^{15} + \cdots + 29\!\cdots\!20 ) / 54\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 23\!\cdots\!27 \nu^{15} + \cdots + 59\!\cdots\!40 ) / 54\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 23\!\cdots\!19 \nu^{15} + \cdots - 54\!\cdots\!00 ) / 54\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 28\!\cdots\!99 \nu^{15} + \cdots - 18\!\cdots\!80 ) / 54\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 81\!\cdots\!38 \nu^{15} + \cdots - 54\!\cdots\!60 ) / 13\!\cdots\!30 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 34\!\cdots\!77 \nu^{15} + \cdots + 36\!\cdots\!00 ) / 54\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 40\!\cdots\!44 \nu^{15} + \cdots + 10\!\cdots\!20 ) / 54\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 55\!\cdots\!17 \nu^{15} + \cdots - 18\!\cdots\!24 ) / 54\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 58\!\cdots\!97 \nu^{15} + \cdots - 29\!\cdots\!00 ) / 54\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 86\!\cdots\!72 \nu^{15} + \cdots - 33\!\cdots\!60 ) / 54\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 45\!\cdots\!53 \nu^{15} + \cdots - 32\!\cdots\!20 ) / 27\!\cdots\!60 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_{3} + \beta_{2} \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{13} - 2\beta_{10} + \beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( - 2 \beta_{15} - \beta_{13} - 3 \beta_{12} - 3 \beta_{11} - 3 \beta_{10} - 2 \beta_{9} - \beta_{7} + \cdots + \beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 5 \beta_{15} - 16 \beta_{13} + 2 \beta_{12} - 8 \beta_{11} - 20 \beta_{10} - 11 \beta_{9} + 3 \beta_{8} + \cdots - 14 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - \beta_{15} + 5 \beta_{14} - 19 \beta_{13} + 6 \beta_{12} - 55 \beta_{11} - 42 \beta_{10} - 18 \beta_{9} + \cdots + 52 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 50 \beta_{15} + 26 \beta_{14} - 151 \beta_{13} + 48 \beta_{12} - 180 \beta_{11} - 42 \beta_{10} + \cdots - 314 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 326 \beta_{15} + 42 \beta_{14} - 306 \beta_{13} + 567 \beta_{12} - 609 \beta_{11} - 400 \beta_{10} + \cdots + 746 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 186 \beta_{15} + 576 \beta_{14} - 406 \beta_{13} + 815 \beta_{12} - 2424 \beta_{11} + 1008 \beta_{10} + \cdots - 2166 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 4969 \beta_{15} - 354 \beta_{14} - 3350 \beta_{13} + 8145 \beta_{12} - 5487 \beta_{11} - 1497 \beta_{10} + \cdots - 1048 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 6900 \beta_{15} + 5546 \beta_{14} + 10836 \beta_{13} + 14832 \beta_{12} - 22428 \beta_{11} + \cdots + 1976 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 29007 \beta_{15} - 4422 \beta_{14} - 8754 \beta_{13} + 65627 \beta_{12} - 38808 \beta_{11} + \cdots - 162404 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 10297 \beta_{15} + 7308 \beta_{14} + 199391 \beta_{13} + 249015 \beta_{12} - 94112 \beta_{11} + \cdots + 134930 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 229420 \beta_{15} + 64155 \beta_{14} + 526846 \beta_{13} + 258192 \beta_{12} - 81575 \beta_{11} + \cdots - 2457372 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 1102449 \beta_{15} - 623936 \beta_{14} + 2068227 \beta_{13} + 2836998 \beta_{12} + 1353008 \beta_{11} + \cdots - 995036 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 7028932 \beta_{15} + 1488311 \beta_{14} + 13275783 \beta_{13} - 802414 \beta_{12} + 4961154 \beta_{11} + \cdots - 15540582 \) 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
−3.00055 − 1.69650i
−3.00055 + 1.69650i
−0.411184 − 0.840091i
−0.411184 + 0.840091i
−2.22599 − 1.07196i
−2.22599 + 1.07196i
−0.0205765 − 1.58652i
−0.0205765 + 1.58652i
1.41047 − 1.58652i
1.41047 + 1.58652i
−0.494962 − 1.07196i
−0.494962 + 1.07196i
3.44054 − 0.840091i
3.44054 + 0.840091i
2.30226 − 1.69650i
2.30226 + 1.69650i
−2.65141 0.349146 − 1.69650i 5.02997 0 −0.925729 + 4.49810i 3.33404i −8.03369 −2.75619 − 1.18465i 0
626.2 −2.65141 0.349146 + 1.69650i 5.02997 0 −0.925729 − 4.49810i − 3.33404i −8.03369 −2.75619 + 1.18465i 0
626.3 −1.92586 −1.51468 − 0.840091i 1.70894 0 2.91706 + 1.61790i − 3.14352i 0.560539 1.58849 + 2.54493i 0
626.4 −1.92586 −1.51468 + 0.840091i 1.70894 0 2.91706 − 1.61790i 3.14352i 0.560539 1.58849 − 2.54493i 0
626.5 −0.865515 1.36048 − 1.07196i −1.25088 0 −1.17751 + 0.927799i 0.393056i 2.81369 0.701798 − 2.91676i 0
626.6 −0.865515 1.36048 + 1.07196i −1.25088 0 −1.17751 − 0.927799i − 0.393056i 2.81369 0.701798 + 2.91676i 0
626.7 −0.715523 −0.694946 − 1.58652i −1.48803 0 0.497250 + 1.13519i 3.72129i 2.49576 −2.03410 + 2.20509i 0
626.8 −0.715523 −0.694946 + 1.58652i −1.48803 0 0.497250 − 1.13519i − 3.72129i 2.49576 −2.03410 − 2.20509i 0
626.9 0.715523 −0.694946 − 1.58652i −1.48803 0 −0.497250 − 1.13519i − 3.72129i −2.49576 −2.03410 + 2.20509i 0
626.10 0.715523 −0.694946 + 1.58652i −1.48803 0 −0.497250 + 1.13519i 3.72129i −2.49576 −2.03410 − 2.20509i 0
626.11 0.865515 1.36048 − 1.07196i −1.25088 0 1.17751 − 0.927799i − 0.393056i −2.81369 0.701798 − 2.91676i 0
626.12 0.865515 1.36048 + 1.07196i −1.25088 0 1.17751 + 0.927799i 0.393056i −2.81369 0.701798 + 2.91676i 0
626.13 1.92586 −1.51468 − 0.840091i 1.70894 0 −2.91706 − 1.61790i 3.14352i −0.560539 1.58849 + 2.54493i 0
626.14 1.92586 −1.51468 + 0.840091i 1.70894 0 −2.91706 + 1.61790i − 3.14352i −0.560539 1.58849 − 2.54493i 0
626.15 2.65141 0.349146 − 1.69650i 5.02997 0 0.925729 − 4.49810i − 3.33404i 8.03369 −2.75619 − 1.18465i 0
626.16 2.65141 0.349146 + 1.69650i 5.02997 0 0.925729 + 4.49810i 3.33404i 8.03369 −2.75619 + 1.18465i 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
11.b odd 2 1 inner
33.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.e ✓ 16
3.b odd 2 1 inner 825.2.f.e ✓ 16
5.b even 2 1 825.2.f.g yes 16
5.c odd 4 2 825.2.d.f 32
11.b odd 2 1 inner 825.2.f.e ✓ 16
15.d odd 2 1 825.2.f.g yes 16
15.e even 4 2 825.2.d.f 32
33.d even 2 1 inner 825.2.f.e ✓ 16
55.d odd 2 1 825.2.f.g yes 16
55.e even 4 2 825.2.d.f 32
165.d even 2 1 825.2.f.g yes 16
165.l odd 4 2 825.2.d.f 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
825.2.d.f 32 5.c odd 4 2
825.2.d.f 32 15.e even 4 2
825.2.d.f 32 55.e even 4 2
825.2.d.f 32 165.l odd 4 2
825.2.f.e ✓ 16 1.a even 1 1 trivial
825.2.f.e ✓ 16 3.b odd 2 1 inner
825.2.f.e ✓ 16 11.b odd 2 1 inner
825.2.f.e ✓ 16 33.d even 2 1 inner
825.2.f.g yes 16 5.b even 2 1
825.2.f.g yes 16 15.d odd 2 1
825.2.f.g yes 16 55.d odd 2 1
825.2.f.g yes 16 165.d even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(825, [\chi])\):

\( T_{2}^{8} - 12T_{2}^{6} + 40T_{2}^{4} - 37T_{2}^{2} + 10 \) Copy content Toggle raw display
\( T_{23}^{8} + 88T_{23}^{6} + 2124T_{23}^{4} + 15563T_{23}^{2} + 376 \) Copy content Toggle raw display
\( T_{37}^{4} + T_{37}^{3} - 97T_{37}^{2} - 133T_{37} + 164 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{8} - 12 T^{6} + \cdots + 10)^{2} \) Copy content Toggle raw display
$3$ \( (T^{8} + T^{7} + 3 T^{6} + \cdots + 81)^{2} \) Copy content Toggle raw display
$5$ \( T^{16} \) Copy content Toggle raw display
$7$ \( (T^{8} + 35 T^{6} + \cdots + 235)^{2} \) Copy content Toggle raw display
$11$ \( T^{16} + \cdots + 214358881 \) Copy content Toggle raw display
$13$ \( (T^{8} + 77 T^{6} + \cdots + 3760)^{2} \) Copy content Toggle raw display
$17$ \( (T^{8} - 44 T^{6} + \cdots + 640)^{2} \) Copy content Toggle raw display
$19$ \( (T^{8} + 122 T^{6} + \cdots + 395035)^{2} \) Copy content Toggle raw display
$23$ \( (T^{8} + 88 T^{6} + \cdots + 376)^{2} \) Copy content Toggle raw display
$29$ \( (T^{8} - 109 T^{6} + \cdots + 64000)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} + 3 T^{3} + \cdots + 1370)^{4} \) Copy content Toggle raw display
$37$ \( (T^{4} + T^{3} - 97 T^{2} + \cdots + 164)^{4} \) Copy content Toggle raw display
$41$ \( (T^{8} - 89 T^{6} + \cdots + 250)^{2} \) Copy content Toggle raw display
$43$ \( (T^{8} + 181 T^{6} + \cdots + 3850240)^{2} \) Copy content Toggle raw display
$47$ \( (T^{8} + 115 T^{6} + \cdots + 94)^{2} \) Copy content Toggle raw display
$53$ \( (T^{8} + 203 T^{6} + \cdots + 632056)^{2} \) Copy content Toggle raw display
$59$ \( (T^{8} + 275 T^{6} + \cdots + 647566)^{2} \) Copy content Toggle raw display
$61$ \( (T^{8} + 190 T^{6} + \cdots + 587500)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} - 11 T^{3} + \cdots - 664)^{4} \) Copy content Toggle raw display
$71$ \( (T^{8} + 227 T^{6} + \cdots + 3760000)^{2} \) Copy content Toggle raw display
$73$ \( (T^{8} + 217 T^{6} + \cdots + 376000)^{2} \) Copy content Toggle raw display
$79$ \( (T^{8} + 340 T^{6} + \cdots + 4219660)^{2} \) Copy content Toggle raw display
$83$ \( (T^{8} - 209 T^{6} + \cdots + 3091360)^{2} \) Copy content Toggle raw display
$89$ \( (T^{8} + 343 T^{6} + \cdots + 7905400)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} - 27 T^{3} + \cdots + 281)^{4} \) Copy content Toggle raw display
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