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Properties

Label 225.7.c.e
Level $225$
Weight $7$
Character orbit 225.c
Analytic conductor $51.762$
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] = [225,7,Mod(26,225)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(225, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0]))
 
N = Newforms(chi, 7, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("225.26");
 
S:= CuspForms(chi, 7);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 225 = 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 225.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: \(51.7621688145\)
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: \( x^{12} + 654x^{10} + 151557x^{8} + 15450132x^{6} + 718595460x^{4} + 14140615200x^{2} + 82024960000 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{29}]\)
Coefficient ring index: \( 2^{9}\cdot 3^{20}\cdot 5^{4} \)
Twist minimal: no (minimal twist has level 45)
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.

\(f(q)\) \(=\) \( q + \beta_{3} q^{2} + ( - \beta_1 - 43) q^{4} - \beta_{2} q^{7} + (\beta_{8} - 54 \beta_{3}) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{3} q^{2} + ( - \beta_1 - 43) q^{4} - \beta_{2} q^{7} + (\beta_{8} - 54 \beta_{3}) q^{8} + (3 \beta_{10} + 5 \beta_{9} + \beta_{7}) q^{11} + (\beta_{5} + \beta_{4} - \beta_{2}) q^{13} + (5 \beta_{10} + 8 \beta_{9} - 2 \beta_{7}) q^{14} + (\beta_{6} + 71 \beta_1 + 3031) q^{16} + ( - 6 \beta_{11} + 3 \beta_{8} + 85 \beta_{3}) q^{17} + ( - 2 \beta_{6} + 8 \beta_1 - 360) q^{19} + ( - 3 \beta_{5} - 12 \beta_{4} + 22 \beta_{2}) q^{22} + (11 \beta_{11} - 11 \beta_{8} + 836 \beta_{3}) q^{23} + (63 \beta_{10} + 96 \beta_{9} + 4 \beta_{7}) q^{26} + ( - 5 \beta_{5} - 16 \beta_{4} + 82 \beta_{2}) q^{28} + (41 \beta_{10} + 6 \beta_{9} - \beta_{7}) q^{29} + ( - \beta_{6} + 184 \beta_1 + 5016) q^{31} + (32 \beta_{11} - 33 \beta_{8} + 4888 \beta_{3}) q^{32} + (9 \beta_{6} + 68 \beta_1 - 8858) q^{34} + (14 \beta_{5} - 54 \beta_{4} + 55 \beta_{2}) q^{37} + ( - 64 \beta_{11} + \cdots + 264 \beta_{3}) q^{38}+ \cdots + ( - 160 \beta_{11} + \cdots - 39433 \beta_{3}) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 516 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 516 q^{4} + 36372 q^{16} - 4320 q^{19} + 60192 q^{31} - 106296 q^{34} - 1078968 q^{46} - 711516 q^{49} - 449784 q^{61} - 3964572 q^{64} - 584400 q^{76} - 4324608 q^{79} + 631152 q^{91} - 5793408 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 654x^{10} + 151557x^{8} + 15450132x^{6} + 718595460x^{4} + 14140615200x^{2} + 82024960000 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 26311 \nu^{10} + 15856929 \nu^{8} + 3199968417 \nu^{6} + 253984572697 \nu^{4} + \cdots + 5968060692750 ) / 460284062050 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 34748973 \nu^{10} - 20163785372 \nu^{8} - 3725052586281 \nu^{6} - 236599946400946 \nu^{4} + \cdots + 60\!\cdots\!00 ) / 176749079827200 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 37917519 \nu^{11} + 23290963346 \nu^{9} + 4838380533963 \nu^{7} + 402536482736748 \nu^{5} + \cdots + 16\!\cdots\!00 \nu ) / 52\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 239343309 \nu^{10} - 143467265276 \nu^{8} - 28608006996873 \nu^{6} + \cdots - 43\!\cdots\!00 ) / 88374539913600 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 129946419 \nu^{10} + 80605158436 \nu^{8} + 16980482564343 \nu^{6} + \cdots + 39\!\cdots\!80 ) / 17674907982720 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 4113619 \nu^{10} - 2531648991 \nu^{8} - 523744119293 \nu^{6} - 42281850692613 \nu^{4} + \cdots - 60\!\cdots\!00 ) / 460284062050 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 761173811 \nu^{11} + 1406071056874 \nu^{9} + 634438971664447 \nu^{7} + \cdots + 91\!\cdots\!00 \nu ) / 15\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 115193559 \nu^{11} + 73375936146 \nu^{9} + 16051296166803 \nu^{7} + \cdots + 10\!\cdots\!00 \nu ) / 26\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 3071319039 \nu^{11} - 1886568031026 \nu^{9} - 391908823251003 \nu^{7} + \cdots - 91\!\cdots\!00 \nu ) / 52\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 799149903 \nu^{11} + 496116322482 \nu^{9} + 104678300343051 \nu^{7} + \cdots + 27\!\cdots\!00 \nu ) / 52\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 12599309561 \nu^{11} - 7652041563524 \nu^{9} + \cdots + 15\!\cdots\!00 \nu ) / 52\!\cdots\!00 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{9} + 81\beta_{3} ) / 81 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -2\beta_{4} + 4\beta_{2} - 81\beta _1 - 8829 ) / 81 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -81\beta_{10} - 338\beta_{9} + 81\beta_{8} - 15228\beta_{3} ) / 81 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 27\beta_{6} + 36\beta_{5} + 224\beta_{4} - 808\beta_{2} + 7425\beta _1 + 560709 ) / 27 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 2592\beta_{11} + 35235\beta_{10} + 104924\beta_{9} - 25029\beta_{8} - 2430\beta_{7} + 3471012\beta_{3} ) / 81 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -28431\beta_{6} - 42714\beta_{5} - 210232\beta_{4} + 940916\beta_{2} - 5675751\beta _1 - 387744651 ) / 81 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 940896 \beta_{11} - 11864853 \beta_{10} - 32414656 \beta_{9} + 6702345 \beta_{8} + \cdots - 852362676 \beta_{3} ) / 81 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 2672865 \beta_{6} + 4595112 \beta_{5} + 21328352 \beta_{4} - 102500944 \beta_{2} + 484860141 \beta _1 + 32083980201 ) / 27 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 270488160 \beta_{11} + 3659281839 \beta_{10} + 9744849112 \beta_{9} - 1759669029 \beta_{8} + \cdots + 217438372836 \beta_{3} ) / 81 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 2156692635 \beta_{6} - 4155675462 \beta_{5} - 18953854664 \beta_{4} + 93234795724 \beta_{2} + \cdots - 24782449483179 ) / 81 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 73604274912 \beta_{11} - 1080668531745 \beta_{10} - 2854661880344 \beta_{9} + \cdots - 56788468364916 \beta_{3} ) / 81 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/225\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(127\)
\(\chi(n)\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
26.1
− 13.7655i
− 16.5939i
− 9.79216i
− 6.96373i
− 3.10099i
− 5.92942i
3.10099i
5.92942i
9.79216i
6.96373i
13.7655i
16.5939i
− 15.1797i 0 −166.423 0 0 −291.826 1554.75i 0 0
26.2 − 15.1797i 0 −166.423 0 0 291.826 1554.75i 0 0
26.3 − 8.37795i 0 −6.18997 0 0 −270.176 − 484.329i 0 0
26.4 − 8.37795i 0 −6.18997 0 0 270.176 − 484.329i 0 0
26.5 − 4.51521i 0 43.6129 0 0 −130.042 − 485.894i 0 0
26.6 − 4.51521i 0 43.6129 0 0 130.042 − 485.894i 0 0
26.7 4.51521i 0 43.6129 0 0 −130.042 485.894i 0 0
26.8 4.51521i 0 43.6129 0 0 130.042 485.894i 0 0
26.9 8.37795i 0 −6.18997 0 0 −270.176 484.329i 0 0
26.10 8.37795i 0 −6.18997 0 0 270.176 484.329i 0 0
26.11 15.1797i 0 −166.423 0 0 −291.826 − 1554.75i 0 0
26.12 15.1797i 0 −166.423 0 0 291.826 − 1554.75i 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 26.12
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
15.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 225.7.c.e 12
3.b odd 2 1 inner 225.7.c.e 12
5.b even 2 1 inner 225.7.c.e 12
5.c odd 4 2 45.7.d.a ✓ 12
15.d odd 2 1 inner 225.7.c.e 12
15.e even 4 2 45.7.d.a ✓ 12
20.e even 4 2 720.7.c.a 12
60.l odd 4 2 720.7.c.a 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
45.7.d.a ✓ 12 5.c odd 4 2
45.7.d.a ✓ 12 15.e even 4 2
225.7.c.e 12 1.a even 1 1 trivial
225.7.c.e 12 3.b odd 2 1 inner
225.7.c.e 12 5.b even 2 1 inner
225.7.c.e 12 15.d odd 2 1 inner
720.7.c.a 12 20.e even 4 2
720.7.c.a 12 60.l odd 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{7}^{\mathrm{new}}(225, [\chi])\):

\( T_{2}^{6} + 321T_{2}^{4} + 22302T_{2}^{2} + 329728 \) Copy content Toggle raw display
\( T_{7}^{6} - 175068T_{7}^{4} + 8890993188T_{7}^{2} - 105125674360896 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{6} + 321 T^{4} + \cdots + 329728)^{2} \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( T^{12} \) Copy content Toggle raw display
$7$ \( (T^{6} + \cdots - 105125674360896)^{2} \) Copy content Toggle raw display
$11$ \( (T^{6} + \cdots + 68\!\cdots\!52)^{2} \) Copy content Toggle raw display
$13$ \( (T^{6} + \cdots - 27\!\cdots\!16)^{2} \) Copy content Toggle raw display
$17$ \( (T^{6} + \cdots + 31\!\cdots\!28)^{2} \) Copy content Toggle raw display
$19$ \( (T^{3} + 1080 T^{2} + \cdots - 129822352000)^{4} \) Copy content Toggle raw display
$23$ \( (T^{6} + \cdots + 21\!\cdots\!32)^{2} \) Copy content Toggle raw display
$29$ \( (T^{6} + \cdots + 59\!\cdots\!28)^{2} \) Copy content Toggle raw display
$31$ \( (T^{3} + \cdots + 1211143755184)^{4} \) Copy content Toggle raw display
$37$ \( (T^{6} + \cdots - 13\!\cdots\!44)^{2} \) Copy content Toggle raw display
$41$ \( (T^{6} + \cdots + 15\!\cdots\!68)^{2} \) Copy content Toggle raw display
$43$ \( (T^{6} + \cdots - 25\!\cdots\!04)^{2} \) Copy content Toggle raw display
$47$ \( (T^{6} + \cdots + 12\!\cdots\!52)^{2} \) Copy content Toggle raw display
$53$ \( (T^{6} + \cdots + 49\!\cdots\!92)^{2} \) Copy content Toggle raw display
$59$ \( (T^{6} + \cdots + 16\!\cdots\!72)^{2} \) Copy content Toggle raw display
$61$ \( (T^{3} + \cdots + 53\!\cdots\!28)^{4} \) Copy content Toggle raw display
$67$ \( (T^{6} + \cdots - 57\!\cdots\!24)^{2} \) Copy content Toggle raw display
$71$ \( (T^{6} + \cdots + 70\!\cdots\!08)^{2} \) Copy content Toggle raw display
$73$ \( (T^{6} + \cdots - 33\!\cdots\!96)^{2} \) Copy content Toggle raw display
$79$ \( (T^{3} + \cdots - 12\!\cdots\!16)^{4} \) Copy content Toggle raw display
$83$ \( (T^{6} + \cdots + 60\!\cdots\!92)^{2} \) Copy content Toggle raw display
$89$ \( (T^{6} + \cdots + 13\!\cdots\!32)^{2} \) Copy content Toggle raw display
$97$ \( (T^{6} + \cdots - 18\!\cdots\!04)^{2} \) Copy content Toggle raw display
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