↑

Properties

Label 225.4.k.b
Level $225$
Weight $4$
Character orbit 225.k
Analytic conductor $13.275$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [225,4,Mod(49,225)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(225, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([2, 3]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("225.49");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 225 = 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 225.k (of order \(6\), degree \(2\), not 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: \(13.2754297513\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: 8.0.303595776.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 5x^{6} + 16x^{4} + 45x^{2} + 81 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3^{2} \)
Twist minimal: no (minimal twist has level 9)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{7} + \beta_1) q^{2} + ( - \beta_{7} - \beta_{5} + \beta_{4}) q^{3} + ( - 3 \beta_{6} - 4 \beta_{3} + 3 \beta_{2} - 3) q^{4} + ( - 3 \beta_{6} - 18 \beta_{3} + 3 \beta_{2} - 9) q^{6} + ( - 3 \beta_{7} + 2 \beta_1) q^{7} + (16 \beta_{5} - \beta_{4} + 16 \beta_1) q^{8} + ( - 6 \beta_{6} - 24 \beta_{3} + 3 \beta_{2} - 27) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{7} + \beta_1) q^{2} + ( - \beta_{7} - \beta_{5} + \beta_{4}) q^{3} + ( - 3 \beta_{6} - 4 \beta_{3} + 3 \beta_{2} - 3) q^{4} + ( - 3 \beta_{6} - 18 \beta_{3} + 3 \beta_{2} - 9) q^{6} + ( - 3 \beta_{7} + 2 \beta_1) q^{7} + (16 \beta_{5} - \beta_{4} + 16 \beta_1) q^{8} + ( - 6 \beta_{6} - 24 \beta_{3} + 3 \beta_{2} - 27) q^{9} + ( - 29 \beta_{3} - 8 \beta_{2} - 29) q^{11} + ( - 2 \beta_{7} + 52 \beta_{5} + 5 \beta_{4} + 30 \beta_1) q^{12} + (15 \beta_{7} + 13 \beta_{5} + 15 \beta_{4} + 15 \beta_1) q^{13} + ( - 8 \beta_{6} - 34 \beta_{3} + 8 \beta_{2} - 8) q^{14} + (8 \beta_{3} - 9 \beta_{2} + 8) q^{16} + ( - 54 \beta_{5} - 9 \beta_{4} - 54 \beta_1) q^{17} + (18 \beta_{7} + 72 \beta_{5} + 27 \beta_{4} + 72 \beta_1) q^{18} + ( - 27 \beta_{6} + 25) q^{19} + ( - 8 \beta_{6} - 53 \beta_{3} + 7 \beta_{2} - 27) q^{21} + (21 \beta_{7} - 6 \beta_{5} + 21 \beta_{4} + 21 \beta_1) q^{22} + (19 \beta_{7} - 7 \beta_{5} + 19 \beta_{4} + 19 \beta_1) q^{23} + (18 \beta_{6} - 24 \beta_{3} + 15 \beta_{2} + 18) q^{24} + (28 \beta_{6} - 118) q^{26} + (36 \beta_{7} + 117 \beta_{5} + 18 \beta_{4} + 135 \beta_1) q^{27} + (92 \beta_{5} + 18 \beta_{4} + 92 \beta_1) q^{28} + ( - 25 \beta_{3} - \beta_{2} - 25) q^{29} + (3 \beta_{6} + 23 \beta_{3} - 3 \beta_{2} + 3) q^{31} + ( - 9 \beta_{7} - 216 \beta_{5} - 9 \beta_{4} - 9 \beta_1) q^{32} + (66 \beta_{7} - 6 \beta_{5} + 21 \beta_{4} + 159 \beta_1) q^{33} + (180 \beta_{3} - 63 \beta_{2} + 180) q^{34} + (60 \beta_{6} - 156 \beta_{3} + 15 \beta_{2} - 108) q^{36} + ( - 2 \beta_{5} - 54 \beta_{4} - 2 \beta_1) q^{37} + ( - 79 \beta_{7} + 241 \beta_1) q^{38} + (11 \beta_{6} + 107 \beta_{3} + 17 \beta_{2} - 135) q^{39} + (98 \beta_{6} + 115 \beta_{3} - 98 \beta_{2} + 98) q^{41} + (18 \beta_{7} + 162 \beta_{5} + 60 \beta_{4} + 150 \beta_1) q^{42} + ( - 6 \beta_{7} - 47 \beta_1) q^{43} + (79 \beta_{6} + 155) q^{44} + (12 \beta_{6} - 126) q^{46} + ( - 91 \beta_{7} + 154 \beta_1) q^{47} + ( - 7 \beta_{7} - 88 \beta_{5} - 17 \beta_{4} + 57 \beta_1) q^{48} + ( - 21 \beta_{6} + 246 \beta_{3} + 21 \beta_{2} - 21) q^{49} + ( - 36 \beta_{6} + 180 \beta_{3} - 63 \beta_{2} + 162) q^{51} + (54 \beta_{7} - 358 \beta_1) q^{52} + ( - 54 \beta_{5} - 162 \beta_{4} - 54 \beta_1) q^{53} + (54 \beta_{6} - 54 \beta_{3} + 81 \beta_{2} - 324) q^{54} + ( - 56 \beta_{3} + 46 \beta_{2} - 56) q^{56} + ( - 79 \beta_{7} + 164 \beta_{5} - 2 \beta_{4} + 405 \beta_1) q^{57} + (24 \beta_{7} + 42 \beta_{5} + 24 \beta_{4} + 24 \beta_1) q^{58} + (136 \beta_{6} - 331 \beta_{3} - 136 \beta_{2} + 136) q^{59} + ( - 167 \beta_{3} - 105 \beta_{2} - 167) q^{61} + ( - 70 \beta_{5} - 26 \beta_{4} - 70 \beta_1) q^{62} + (57 \beta_{7} + 192 \beta_{5} + 78 \beta_{4} + 216 \beta_1) q^{63} + ( - 153 \beta_{6} + 287) q^{64} + ( - 33 \beta_{6} + 174 \beta_{3} + 48 \beta_{2} - 189) q^{66} + (66 \beta_{7} + 527 \beta_{5} + 66 \beta_{4} + 66 \beta_1) q^{67} + ( - 171 \beta_{7} - 432 \beta_{5} - 171 \beta_{4} - 171 \beta_1) q^{68} + ( - 33 \beta_{6} + 159 \beta_{3} + 45 \beta_{2} - 171) q^{69} + ( - 144 \beta_{6} + 612) q^{71} + (27 \beta_{7} + 99 \beta_{4} - 333 \beta_1) q^{72} + ( - 106 \beta_{5} + 243 \beta_{4} - 106 \beta_1) q^{73} + (436 \beta_{3} - 56 \beta_{2} + 436) q^{74} + ( - 183 \beta_{6} - 856 \beta_{3} + 183 \beta_{2} - 183) q^{76} + (71 \beta_{7} - 47 \beta_{5} + 71 \beta_{4} + 71 \beta_1) q^{77} + (174 \beta_{7} - 78 \beta_{5} - 90 \beta_{4} - 420 \beta_1) q^{78} + ( - 247 \beta_{3} - 309 \beta_{2} - 247) q^{79} + (135 \beta_{6} + 216 \beta_{3} + 135 \beta_{2}) q^{81} + ( - 1014 \beta_{5} - 213 \beta_{4} - 1014 \beta_1) q^{82} + ( - 107 \beta_{7} - 460 \beta_1) q^{83} + (56 \beta_{6} - 328 \beta_{3} + 110 \beta_{2} - 324) q^{84} + (35 \beta_{6} + 34 \beta_{3} - 35 \beta_{2} + 35) q^{86} + (51 \beta_{7} + 42 \beta_{5} + 24 \beta_{4} + 84 \beta_1) q^{87} + ( - 165 \beta_{7} - 693 \beta_1) q^{88} + (72 \beta_{6} + 234) q^{89} + (69 \beta_{6} - 356) q^{91} + ( - 2 \beta_{7} - 430 \beta_1) q^{92} + ( - 17 \beta_{7} - 71 \beta_{5} - 43 \beta_{4} - 87 \beta_1) q^{93} + ( - 336 \beta_{6} - 1218 \beta_{3} + 336 \beta_{2} - 336) q^{94} + ( - 423 \beta_{6} + 144 \beta_{3} + 198 \beta_{2} + 81) q^{96} + ( - 102 \beta_{7} + 317 \beta_1) q^{97} + ( - 324 \beta_{5} - 225 \beta_{4} - 324 \beta_1) q^{98} + ( - 81 \beta_{6} + 801 \beta_{3} + 279 \beta_{2} + 216) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 10 q^{4} + 18 q^{6} - 90 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 10 q^{4} + 18 q^{6} - 90 q^{9} - 132 q^{11} + 120 q^{14} + 14 q^{16} + 308 q^{19} + 42 q^{21} + 198 q^{24} - 1056 q^{26} - 102 q^{29} - 86 q^{31} + 594 q^{34} - 450 q^{36} - 1518 q^{39} - 264 q^{41} + 924 q^{44} - 1056 q^{46} - 1026 q^{49} + 594 q^{51} - 2430 q^{54} - 132 q^{56} + 1596 q^{59} - 878 q^{61} + 2908 q^{64} - 1980 q^{66} - 1782 q^{69} + 5472 q^{71} + 1632 q^{74} + 3058 q^{76} - 1606 q^{79} - 1134 q^{81} - 1284 q^{84} - 66 q^{86} + 1584 q^{89} - 3124 q^{91} + 4200 q^{94} + 2160 q^{96} - 594 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 5x^{6} + 16x^{4} + 45x^{2} + 81 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{7} + 32\nu^{5} + 16\nu^{3} + 45\nu ) / 432 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{6} - 16\nu^{4} - 32\nu^{2} - 51 ) / 48 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -5\nu^{6} - 16\nu^{4} - 80\nu^{2} - 225 ) / 144 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{7} + 8\nu^{5} + 40\nu^{3} + 165\nu ) / 72 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{7} + 13\nu ) / 48 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -\nu^{6} - 5\nu^{4} - 7\nu^{2} - 27 ) / 9 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{7} + 2\nu^{5} + 10\nu^{3} - 3\nu ) / 18 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{7} + 2\beta_{5} + \beta_{4} ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2\beta_{6} - 7\beta_{3} - \beta_{2} - 6 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 4\beta_{7} - 11\beta_{5} + 2\beta_{4} - 9\beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -5\beta_{6} + 13\beta_{3} - 5\beta_{2} ) / 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -\beta_{7} - \beta_{5} - 2\beta_{4} + 45\beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -16\beta_{6} - 16\beta_{3} + 32\beta_{2} - 39 ) / 3 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 13\beta_{7} + 118\beta_{5} - 13\beta_{4} ) / 3 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(101\) \(127\)
\(\chi(n)\) \(\beta_{3}\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
49.1
−0.396143 + 1.68614i
−1.26217 + 1.18614i
1.26217 − 1.18614i
0.396143 − 1.68614i
−0.396143 − 1.68614i
−1.26217 − 1.18614i
1.26217 + 1.18614i
0.396143 + 1.68614i
−3.78651 + 2.18614i −3.78651 + 3.55842i 5.55842 − 9.62747i 0 6.55842 − 21.7518i −10.4935 + 6.05842i 13.6277i 1.67527 − 26.9480i 0
49.2 −1.18843 + 0.686141i −1.18843 + 5.05842i −3.05842 + 5.29734i 0 −2.05842 − 6.82701i −4.43132 + 2.55842i − 19.3723i −24.1753 − 12.0232i 0
49.3 1.18843 − 0.686141i 1.18843 − 5.05842i −3.05842 + 5.29734i 0 −2.05842 − 6.82701i 4.43132 − 2.55842i 19.3723i −24.1753 − 12.0232i 0
49.4 3.78651 − 2.18614i 3.78651 − 3.55842i 5.55842 − 9.62747i 0 6.55842 − 21.7518i 10.4935 − 6.05842i − 13.6277i 1.67527 − 26.9480i 0
124.1 −3.78651 − 2.18614i −3.78651 − 3.55842i 5.55842 + 9.62747i 0 6.55842 + 21.7518i −10.4935 − 6.05842i − 13.6277i 1.67527 + 26.9480i 0
124.2 −1.18843 − 0.686141i −1.18843 − 5.05842i −3.05842 − 5.29734i 0 −2.05842 + 6.82701i −4.43132 − 2.55842i 19.3723i −24.1753 + 12.0232i 0
124.3 1.18843 + 0.686141i 1.18843 + 5.05842i −3.05842 − 5.29734i 0 −2.05842 + 6.82701i 4.43132 + 2.55842i − 19.3723i −24.1753 + 12.0232i 0
124.4 3.78651 + 2.18614i 3.78651 + 3.55842i 5.55842 + 9.62747i 0 6.55842 + 21.7518i 10.4935 + 6.05842i 13.6277i 1.67527 + 26.9480i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 124.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
9.c even 3 1 inner
45.j even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 225.4.k.b 8
5.b even 2 1 inner 225.4.k.b 8
5.c odd 4 1 9.4.c.a ✓ 4
5.c odd 4 1 225.4.e.b 4
9.c even 3 1 inner 225.4.k.b 8
15.e even 4 1 27.4.c.a 4
20.e even 4 1 144.4.i.c 4
45.j even 6 1 inner 225.4.k.b 8
45.k odd 12 1 9.4.c.a ✓ 4
45.k odd 12 1 81.4.a.d 2
45.k odd 12 1 225.4.e.b 4
45.k odd 12 1 2025.4.a.g 2
45.l even 12 1 27.4.c.a 4
45.l even 12 1 81.4.a.a 2
45.l even 12 1 2025.4.a.n 2
60.l odd 4 1 432.4.i.c 4
180.v odd 12 1 432.4.i.c 4
180.v odd 12 1 1296.4.a.i 2
180.x even 12 1 144.4.i.c 4
180.x even 12 1 1296.4.a.u 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
9.4.c.a ✓ 4 5.c odd 4 1
9.4.c.a ✓ 4 45.k odd 12 1
27.4.c.a 4 15.e even 4 1
27.4.c.a 4 45.l even 12 1
81.4.a.a 2 45.l even 12 1
81.4.a.d 2 45.k odd 12 1
144.4.i.c 4 20.e even 4 1
144.4.i.c 4 180.x even 12 1
225.4.e.b 4 5.c odd 4 1
225.4.e.b 4 45.k odd 12 1
225.4.k.b 8 1.a even 1 1 trivial
225.4.k.b 8 5.b even 2 1 inner
225.4.k.b 8 9.c even 3 1 inner
225.4.k.b 8 45.j even 6 1 inner
432.4.i.c 4 60.l odd 4 1
432.4.i.c 4 180.v odd 12 1
1296.4.a.i 2 180.v odd 12 1
1296.4.a.u 2 180.x even 12 1
2025.4.a.g 2 45.k odd 12 1
2025.4.a.n 2 45.l even 12 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{8} - 21T_{2}^{6} + 405T_{2}^{4} - 756T_{2}^{2} + 1296 \) acting on \(S_{4}^{\mathrm{new}}(225, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} - 21 T^{6} + 405 T^{4} + \cdots + 1296 \) Copy content Toggle raw display
$3$ \( T^{8} + 45 T^{6} + 1296 T^{4} + \cdots + 531441 \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( T^{8} - 173 T^{6} + \cdots + 14776336 \) Copy content Toggle raw display
$11$ \( (T^{4} + 66 T^{3} + 3795 T^{2} + \cdots + 314721)^{2} \) Copy content Toggle raw display
$13$ \( T^{8} - 3773 T^{6} + \cdots + 11117396444176 \) Copy content Toggle raw display
$17$ \( (T^{4} + 6237 T^{2} + 3175524)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} - 77 T - 4532)^{4} \) Copy content Toggle raw display
$23$ \( T^{8} - 6501 T^{6} + \cdots + 53618068974096 \) Copy content Toggle raw display
$29$ \( (T^{4} + 51 T^{3} + 1959 T^{2} + \cdots + 412164)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} + 43 T^{3} + 1461 T^{2} + \cdots + 150544)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} + 49364 T^{2} + \cdots + 549058624)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} + 132 T^{3} + 92301 T^{2} + \cdots + 5606565129)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} - 4466 T^{6} + \cdots + 7216320515041 \) Copy content Toggle raw display
$47$ \( T^{8} - 216237 T^{6} + \cdots + 66\!\cdots\!76 \) Copy content Toggle raw display
$53$ \( (T^{4} + 434484 T^{2} + \cdots + 46562734656)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} - 798 T^{3} + 630195 T^{2} + \cdots + 43678881)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} + 439 T^{3} + 235497 T^{2} + \cdots + 1829786176)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} - 559946 T^{6} + \cdots + 18\!\cdots\!01 \) Copy content Toggle raw display
$71$ \( (T^{2} - 1368 T + 296784)^{4} \) Copy content Toggle raw display
$73$ \( (T^{4} + 1077821 T^{2} + \cdots + 189571418404)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} + 803 T^{3} + \cdots + 392522298256)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} - 519393 T^{6} + \cdots + 25\!\cdots\!36 \) Copy content Toggle raw display
$89$ \( (T^{2} - 396 T - 3564)^{4} \) Copy content Toggle raw display
$97$ \( T^{8} - 442514 T^{6} + \cdots + 60\!\cdots\!61 \) Copy content Toggle raw display
show more
show less