Properties

Label 225.6.f.b
Level $225$
Weight $6$
Character orbit 225.f
Analytic conductor $36.086$
Analytic rank $0$
Dimension $20$
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] = [225,6,Mod(107,225)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(225, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 1]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("225.107");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 225 = 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 225.f (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: \(36.0863594579\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(10\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} + 19978x^{16} + 11248353x^{12} + 1386043201x^{8} + 1477627450x^{4} + 332150625 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{29}]\)
Coefficient ring index: \( 2^{29}\cdot 3^{24}\cdot 5^{4} \)
Twist minimal: no (minimal twist has level 45)
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 \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{14} q^{2} + ( - \beta_{5} + 14 \beta_{2}) q^{4} + (\beta_{7} - \beta_{5} + 8 \beta_{2} + \beta_1 - 8) q^{7} + (\beta_{18} + \beta_{17} + \beta_{13} + \beta_{12} - \beta_{4} - 16 \beta_{3}) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{14} q^{2} + ( - \beta_{5} + 14 \beta_{2}) q^{4} + (\beta_{7} - \beta_{5} + 8 \beta_{2} + \beta_1 - 8) q^{7} + (\beta_{18} + \beta_{17} + \beta_{13} + \beta_{12} - \beta_{4} - 16 \beta_{3}) q^{8} + (\beta_{19} + 2 \beta_{17} - 3 \beta_{16} + 7 \beta_{14} - 2 \beta_{12} - 2 \beta_{4} + 4 \beta_{3}) q^{11} + ( - \beta_{8} + 8 \beta_{5} - 92 \beta_{2} + 8 \beta_1 - 92) q^{13} + ( - \beta_{19} + 3 \beta_{18} + 2 \beta_{15} + 35 \beta_{14} + 6 \beta_{13} + \beta_{4} - 35 \beta_{3}) q^{14} + ( - 3 \beta_{11} + 3 \beta_{10} - \beta_{9} + \beta_{8} - \beta_{7} - \beta_{6} + 25 \beta_1 - 316) q^{16} + ( - \beta_{19} - 4 \beta_{18} + 4 \beta_{17} + 3 \beta_{16} - 3 \beta_{15} - 17 \beta_{14} + \cdots + 3 \beta_{3}) q^{17}+ \cdots + (16 \beta_{18} + 16 \beta_{17} - 300 \beta_{16} - 300 \beta_{15} + \cdots + 1977 \beta_{3}) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - 152 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 20 q - 152 q^{7} - 1844 q^{13} - 6280 q^{16} + 6512 q^{22} - 29728 q^{28} + 43120 q^{31} + 37516 q^{37} - 5408 q^{43} + 118720 q^{46} + 285256 q^{52} - 274752 q^{58} + 163360 q^{61} + 302704 q^{67} - 146324 q^{73} + 760800 q^{76} + 305744 q^{82} - 355296 q^{88} + 499120 q^{91} + 362524 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} + 19978x^{16} + 11248353x^{12} + 1386043201x^{8} + 1477627450x^{4} + 332150625 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 4854966023630 \nu^{16} + \cdots - 62\!\cdots\!75 ) / 59\!\cdots\!25 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 159785243 \nu^{18} - 3192258575999 \nu^{14} + \cdots - 34\!\cdots\!75 \nu^{2} ) / 15\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 208235594721671 \nu^{17} + \cdots + 16\!\cdots\!25 \nu ) / 53\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 88723261303277 \nu^{17} + \cdots + 43\!\cdots\!35 \nu ) / 21\!\cdots\!10 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 79\!\cdots\!07 \nu^{18} + \cdots + 93\!\cdots\!25 \nu^{2} ) / 48\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 21\!\cdots\!63 \nu^{18} + \cdots - 91\!\cdots\!25 ) / 72\!\cdots\!75 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 21\!\cdots\!63 \nu^{18} + \cdots - 91\!\cdots\!25 ) / 72\!\cdots\!75 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 24\!\cdots\!99 \nu^{18} + \cdots - 12\!\cdots\!00 ) / 28\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 24\!\cdots\!99 \nu^{18} + \cdots + 12\!\cdots\!00 ) / 28\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 67\!\cdots\!53 \nu^{18} + \cdots - 31\!\cdots\!50 ) / 72\!\cdots\!75 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 67\!\cdots\!53 \nu^{18} + \cdots + 31\!\cdots\!50 ) / 72\!\cdots\!75 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 72\!\cdots\!76 \nu^{19} + \cdots - 59\!\cdots\!75 \nu ) / 14\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 72\!\cdots\!76 \nu^{19} + \cdots - 59\!\cdots\!75 \nu ) / 14\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 58\!\cdots\!56 \nu^{19} + \cdots + 50\!\cdots\!50 \nu^{3} ) / 97\!\cdots\!25 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 10\!\cdots\!39 \nu^{19} + \cdots + 15\!\cdots\!25 \nu ) / 86\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 54\!\cdots\!95 \nu^{19} + \cdots - 94\!\cdots\!75 \nu ) / 43\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 11\!\cdots\!12 \nu^{19} + \cdots - 91\!\cdots\!25 \nu ) / 39\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( 11\!\cdots\!12 \nu^{19} + \cdots - 91\!\cdots\!25 \nu ) / 39\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( 97\!\cdots\!14 \nu^{19} + \cdots + 50\!\cdots\!50 \nu^{3} ) / 19\!\cdots\!25 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( 15 \beta_{18} + 15 \beta_{17} - 18 \beta_{16} - 18 \beta_{15} + 2 \beta_{13} + 2 \beta_{12} - 36 \beta_{4} + 384 \beta_{3} ) / 1080 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( - 71 \beta_{11} - 71 \beta_{10} + 18 \beta_{9} + 18 \beta_{8} - 252 \beta_{7} + 252 \beta_{6} - 518 \beta_{5} - 37736 \beta_{2} ) / 1080 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 909 \beta_{19} - 555 \beta_{18} + 555 \beta_{17} - 1134 \beta_{16} + 1134 \beta_{15} + 22731 \beta_{14} - 2756 \beta_{13} + 2756 \beta_{12} - 1134 \beta_{4} - 1134 \beta_{3} ) / 540 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 11851 \beta_{11} - 11851 \beta_{10} - 3123 \beta_{9} + 3123 \beta_{8} + 36837 \beta_{7} + 36837 \beta_{6} - 56548 \beta _1 - 4336486 ) / 1080 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 136365 \beta_{18} - 136365 \beta_{17} + 319698 \beta_{16} + 319698 \beta_{15} - 838022 \beta_{13} - 838022 \beta_{12} + 554076 \beta_{4} - 5803224 \beta_{3} ) / 1080 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 1685186 \beta_{11} + 1685186 \beta_{10} - 450513 \beta_{9} - 450513 \beta_{8} + 5149557 \beta_{7} - 5149557 \beta_{6} + 7612538 \beta_{5} + 588914126 \beta_{2} ) / 1080 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 32198508 \beta_{19} + 18699735 \beta_{18} - 18699735 \beta_{17} + 44695908 \beta_{16} - 44695908 \beta_{15} - 848755272 \beta_{14} + 117885322 \beta_{13} + \cdots + 44695908 \beta_{3} ) / 1080 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 235339081 \beta_{11} + 235339081 \beta_{10} + 63105813 \beta_{9} - 63105813 \beta_{8} - 717465447 \beta_{7} - 717465447 \beta_{6} + 1055963188 \beta _1 + 81761679466 ) / 1080 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 1299893745 \beta_{18} + 1299893745 \beta_{17} - 3115094769 \beta_{16} - 3115094769 \beta_{15} + 8220146116 \beta_{13} + 8220146116 \beta_{12} + \cdots + 55956490797 \beta_{3} ) / 540 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 32791620941 \beta_{11} - 32791620941 \beta_{10} + 8797697028 \beta_{9} + 8797697028 \beta_{8} - 99937410642 \beta_{7} + 99937410642 \beta_{6} + \cdots - 11383726373756 \beta_{2} ) / 1080 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 623122741548 \beta_{19} - 362041420485 \beta_{18} + 362041420485 \beta_{17} - 867912621948 \beta_{16} + 867912621948 \beta_{15} + \cdots - 867912621948 \beta_{3} ) / 1080 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 1141944263059 \beta_{11} - 1141944263059 \beta_{10} - 306399587382 \beta_{9} + 306399587382 \beta_{8} + 3480087500883 \beta_{7} + \cdots - 396389006418874 ) / 270 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 50427516797115 \beta_{18} - 50427516797115 \beta_{17} + 120894535915578 \beta_{16} + 120894535915578 \beta_{15} - 319029519819242 \beta_{13} + \cdots - 21\!\cdots\!64 \beta_{3} ) / 1080 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 636253236999971 \beta_{11} + 636253236999971 \beta_{10} - 170717870654118 \beta_{9} - 170717870654118 \beta_{8} + \cdots + 22\!\cdots\!36 \beta_{2} ) / 1080 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 60\!\cdots\!19 \beta_{19} + \cdots + 84\!\cdots\!94 \beta_{3} ) / 540 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( ( - 88\!\cdots\!91 \beta_{11} + \cdots + 30\!\cdots\!26 ) / 1080 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( ( 97\!\cdots\!65 \beta_{18} + \cdots + 42\!\cdots\!84 \beta_{3} ) / 1080 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( ( - 12\!\cdots\!26 \beta_{11} + \cdots - 42\!\cdots\!66 \beta_{2} ) / 1080 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( ( - 23\!\cdots\!28 \beta_{19} + \cdots - 32\!\cdots\!28 \beta_{3} ) / 1080 \) Copy content Toggle raw display

Character values

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

\(n\) \(101\) \(127\)
\(\chi(n)\) \(-1\) \(-\beta_{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
107.1
−0.532441 0.532441i
0.658471 + 0.658471i
−3.15495 3.15495i
−8.34538 8.34538i
−2.58530 2.58530i
2.58530 + 2.58530i
8.34538 + 8.34538i
3.15495 + 3.15495i
−0.658471 0.658471i
0.532441 + 0.532441i
−0.532441 + 0.532441i
0.658471 0.658471i
−3.15495 + 3.15495i
−8.34538 + 8.34538i
−2.58530 + 2.58530i
2.58530 2.58530i
8.34538 8.34538i
3.15495 3.15495i
−0.658471 + 0.658471i
0.532441 0.532441i
−7.55546 + 7.55546i 0 82.1700i 0 0 −97.1320 97.1320i 379.058 + 379.058i 0 0
107.2 −5.73598 + 5.73598i 0 33.8030i 0 0 92.1304 + 92.1304i 10.3420 + 10.3420i 0 0
107.3 −3.49760 + 3.49760i 0 7.53358i 0 0 −24.1207 24.1207i −138.273 138.273i 0 0
107.4 −3.44473 + 3.44473i 0 8.26762i 0 0 −137.623 137.623i −138.711 138.711i 0 0
107.5 −0.956082 + 0.956082i 0 30.1718i 0 0 128.745 + 128.745i −59.4413 59.4413i 0 0
107.6 0.956082 0.956082i 0 30.1718i 0 0 128.745 + 128.745i 59.4413 + 59.4413i 0 0
107.7 3.44473 3.44473i 0 8.26762i 0 0 −137.623 137.623i 138.711 + 138.711i 0 0
107.8 3.49760 3.49760i 0 7.53358i 0 0 −24.1207 24.1207i 138.273 + 138.273i 0 0
107.9 5.73598 5.73598i 0 33.8030i 0 0 92.1304 + 92.1304i −10.3420 10.3420i 0 0
107.10 7.55546 7.55546i 0 82.1700i 0 0 −97.1320 97.1320i −379.058 379.058i 0 0
143.1 −7.55546 7.55546i 0 82.1700i 0 0 −97.1320 + 97.1320i 379.058 379.058i 0 0
143.2 −5.73598 5.73598i 0 33.8030i 0 0 92.1304 92.1304i 10.3420 10.3420i 0 0
143.3 −3.49760 3.49760i 0 7.53358i 0 0 −24.1207 + 24.1207i −138.273 + 138.273i 0 0
143.4 −3.44473 3.44473i 0 8.26762i 0 0 −137.623 + 137.623i −138.711 + 138.711i 0 0
143.5 −0.956082 0.956082i 0 30.1718i 0 0 128.745 128.745i −59.4413 + 59.4413i 0 0
143.6 0.956082 + 0.956082i 0 30.1718i 0 0 128.745 128.745i 59.4413 59.4413i 0 0
143.7 3.44473 + 3.44473i 0 8.26762i 0 0 −137.623 + 137.623i 138.711 138.711i 0 0
143.8 3.49760 + 3.49760i 0 7.53358i 0 0 −24.1207 + 24.1207i 138.273 138.273i 0 0
143.9 5.73598 + 5.73598i 0 33.8030i 0 0 92.1304 92.1304i −10.3420 + 10.3420i 0 0
143.10 7.55546 + 7.55546i 0 82.1700i 0 0 −97.1320 + 97.1320i −379.058 + 379.058i 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 107.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.c odd 4 1 inner
15.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 225.6.f.b 20
3.b odd 2 1 inner 225.6.f.b 20
5.b even 2 1 45.6.f.a 20
5.c odd 4 1 45.6.f.a 20
5.c odd 4 1 inner 225.6.f.b 20
15.d odd 2 1 45.6.f.a 20
15.e even 4 1 45.6.f.a 20
15.e even 4 1 inner 225.6.f.b 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
45.6.f.a 20 5.b even 2 1
45.6.f.a 20 5.c odd 4 1
45.6.f.a 20 15.d odd 2 1
45.6.f.a 20 15.e even 4 1
225.6.f.b 20 1.a even 1 1 trivial
225.6.f.b 20 3.b odd 2 1 inner
225.6.f.b 20 5.c odd 4 1 inner
225.6.f.b 20 15.e even 4 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{20} + 18530T_{2}^{16} + 77015185T_{2}^{12} + 71686822960T_{2}^{8} + 19267872346880T_{2}^{4} + 63600242200576 \) acting on \(S_{6}^{\mathrm{new}}(225, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} + 18530 T^{16} + \cdots + 63600242200576 \) Copy content Toggle raw display
$3$ \( T^{20} \) Copy content Toggle raw display
$5$ \( T^{20} \) Copy content Toggle raw display
$7$ \( (T^{10} + 76 T^{9} + \cdots + 46\!\cdots\!68)^{2} \) Copy content Toggle raw display
$11$ \( (T^{10} + 708040 T^{8} + \cdots + 10\!\cdots\!68)^{2} \) Copy content Toggle raw display
$13$ \( (T^{10} + 922 T^{9} + \cdots + 23\!\cdots\!32)^{2} \) Copy content Toggle raw display
$17$ \( T^{20} + 17415572890880 T^{16} + \cdots + 47\!\cdots\!76 \) Copy content Toggle raw display
$19$ \( (T^{10} + 11822400 T^{8} + \cdots + 11\!\cdots\!00)^{2} \) Copy content Toggle raw display
$23$ \( T^{20} + 328215392257280 T^{16} + \cdots + 61\!\cdots\!76 \) Copy content Toggle raw display
$29$ \( (T^{10} - 94027290 T^{8} + \cdots - 27\!\cdots\!68)^{2} \) Copy content Toggle raw display
$31$ \( (T^{5} - 10780 T^{4} + \cdots + 91\!\cdots\!24)^{4} \) Copy content Toggle raw display
$37$ \( (T^{10} - 18758 T^{9} + \cdots + 12\!\cdots\!32)^{2} \) Copy content Toggle raw display
$41$ \( (T^{10} + 662486410 T^{8} + \cdots + 10\!\cdots\!32)^{2} \) Copy content Toggle raw display
$43$ \( (T^{10} + 2704 T^{9} + \cdots + 43\!\cdots\!68)^{2} \) Copy content Toggle raw display
$47$ \( T^{20} + \cdots + 15\!\cdots\!76 \) Copy content Toggle raw display
$53$ \( T^{20} + \cdots + 44\!\cdots\!76 \) Copy content Toggle raw display
$59$ \( (T^{10} - 3051809160 T^{8} + \cdots - 66\!\cdots\!32)^{2} \) Copy content Toggle raw display
$61$ \( (T^{5} - 40840 T^{4} + \cdots - 44\!\cdots\!68)^{4} \) Copy content Toggle raw display
$67$ \( (T^{10} - 151352 T^{9} + \cdots + 17\!\cdots\!32)^{2} \) Copy content Toggle raw display
$71$ \( (T^{10} + 13132819360 T^{8} + \cdots + 17\!\cdots\!32)^{2} \) Copy content Toggle raw display
$73$ \( (T^{10} + 73162 T^{9} + \cdots + 17\!\cdots\!32)^{2} \) Copy content Toggle raw display
$79$ \( (T^{10} + 14898562320 T^{8} + \cdots + 21\!\cdots\!24)^{2} \) Copy content Toggle raw display
$83$ \( T^{20} + \cdots + 30\!\cdots\!76 \) Copy content Toggle raw display
$89$ \( (T^{10} - 10367885610 T^{8} + \cdots - 12\!\cdots\!32)^{2} \) Copy content Toggle raw display
$97$ \( (T^{10} - 181262 T^{9} + \cdots + 53\!\cdots\!32)^{2} \) Copy content Toggle raw display
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