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Properties

Label 725.2.e.b
Level $725$
Weight $2$
Character orbit 725.e
Analytic conductor $5.789$
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] = [725,2,Mod(157,725)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(725, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("725.157");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 725 = 5^{2} \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 725.e (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: \(5.78915414654\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 28x^{14} + 288x^{12} + 1372x^{10} + 3184x^{8} + 3696x^{6} + 2076x^{4} + 504x^{2} + 36 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: yes
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{7} q^{2} + \beta_{9} q^{3} - \beta_{2} q^{4} + (\beta_{14} - \beta_{12} + \cdots + \beta_{4}) q^{6}+ \cdots + (\beta_{5} + \beta_{2} + 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{7} q^{2} + \beta_{9} q^{3} - \beta_{2} q^{4} + (\beta_{14} - \beta_{12} + \cdots + \beta_{4}) q^{6}+ \cdots + ( - 3 \beta_{14} + \beta_{13} + \cdots + 5) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 8 q^{4} + 40 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 8 q^{4} + 40 q^{9} - 4 q^{11} - 20 q^{14} - 16 q^{16} + 12 q^{19} - 32 q^{21} - 40 q^{26} + 4 q^{29} + 20 q^{31} - 80 q^{34} - 104 q^{36} - 16 q^{39} - 28 q^{44} + 44 q^{46} + 36 q^{56} + 24 q^{61} + 32 q^{64} + 80 q^{66} + 40 q^{69} - 36 q^{76} - 52 q^{79} + 40 q^{81} + 136 q^{84} - 40 q^{89} + 92 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 28x^{14} + 288x^{12} + 1372x^{10} + 3184x^{8} + 3696x^{6} + 2076x^{4} + 504x^{2} + 36 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 30 \nu^{14} - 863 \nu^{12} - 9206 \nu^{10} - 45753 \nu^{8} - 108620 \nu^{6} - 113912 \nu^{4} + \cdots - 1362 ) / 1596 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -30\nu^{14} - 863\nu^{12} - 9206\nu^{10} - 45753\nu^{8} - 108620\nu^{6} - 113912\nu^{4} - 42564\nu^{2} - 3756 ) / 798 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 90 \nu^{14} + 2380 \nu^{12} + 22165 \nu^{10} + 87612 \nu^{8} + 137266 \nu^{6} + 65590 \nu^{4} + \cdots - 6972 ) / 1596 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 313 \nu^{15} - 8446 \nu^{13} - 81570 \nu^{11} - 346642 \nu^{9} - 643474 \nu^{7} - 486936 \nu^{5} + \cdots + 720 \nu ) / 9576 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 66 \nu^{14} - 1834 \nu^{12} - 18604 \nu^{10} - 86247 \nu^{8} - 188956 \nu^{6} - 194488 \nu^{4} + \cdots - 10794 ) / 798 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 493 \nu^{15} + 13624 \nu^{13} + 136806 \nu^{11} + 621160 \nu^{9} + 1295194 \nu^{7} + 1170408 \nu^{5} + \cdots - 2124 \nu ) / 9576 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 493 \nu^{15} - 13624 \nu^{13} - 136806 \nu^{11} - 621160 \nu^{9} - 1295194 \nu^{7} + \cdots + 11700 \nu ) / 9576 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 349 \nu^{15} + 9721 \nu^{13} + 99081 \nu^{11} + 464029 \nu^{9} + 1039552 \nu^{7} + 1115358 \nu^{5} + \cdots + 56934 \nu ) / 4788 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 109 \nu^{14} - 2969 \nu^{12} - 29157 \nu^{10} - 128006 \nu^{8} - 255446 \nu^{6} - 229940 \nu^{4} + \cdots - 7240 ) / 532 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 1441 \nu^{15} + 21 \nu^{14} + 39295 \nu^{13} + 264 \nu^{12} + 386592 \nu^{11} - 2520 \nu^{10} + \cdots - 16272 ) / 9576 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 1441 \nu^{15} + 201 \nu^{14} - 39295 \nu^{13} + 5442 \nu^{12} - 386592 \nu^{11} + 52716 \nu^{10} + \cdots - 8100 ) / 9576 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 1420 \nu^{15} - 1212 \nu^{14} + 39031 \nu^{13} - 33330 \nu^{12} + 389112 \nu^{11} - 332448 \nu^{10} + \cdots - 95472 ) / 9576 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 316 \nu^{15} + 8766 \nu^{13} + 88715 \nu^{11} + 410047 \nu^{9} + 895332 \nu^{7} + 919808 \nu^{5} + \cdots + 54786 \nu ) / 1596 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 1420 \nu^{15} - 1212 \nu^{14} - 39031 \nu^{13} - 33330 \nu^{12} - 389112 \nu^{11} - 332448 \nu^{10} + \cdots - 95472 ) / 9576 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 406 \nu^{15} - 11146 \nu^{13} - 110880 \nu^{11} - 497659 \nu^{9} - 1032598 \nu^{7} + \cdots - 47814 \nu ) / 1596 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_{7} + \beta_{6} \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{2} + 2\beta _1 - 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{15} + \beta_{11} - \beta_{10} - 6\beta_{7} - 7\beta_{6} - 3\beta_{4} + \beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{14} - \beta_{12} + 4\beta_{11} + 4\beta_{10} + \beta_{5} - 4\beta_{3} + 11\beta_{2} - 20\beta _1 + 21 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 17 \beta_{15} + 5 \beta_{14} + 5 \beta_{13} - 5 \beta_{12} - 17 \beta_{11} + 17 \beta_{10} + \cdots - 17 \beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 23 \beta_{14} + 23 \beta_{12} - 62 \beta_{11} - 62 \beta_{10} - 6 \beta_{9} - 22 \beta_{5} + 62 \beta_{3} + \cdots - 188 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 222 \beta_{15} - 91 \beta_{14} - 84 \beta_{13} + 91 \beta_{12} + 223 \beta_{11} - 223 \beta_{10} + \cdots + 223 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 344 \beta_{14} - 344 \beta_{12} + 776 \beta_{11} + 776 \beta_{10} + 128 \beta_{9} + 306 \beta_{5} + \cdots + 1838 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 2624 \beta_{15} + 1248 \beta_{14} + 1074 \beta_{13} - 1248 \beta_{12} - 2670 \beta_{11} + \cdots - 2670 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 4428 \beta_{14} + 4428 \beta_{12} - 9084 \beta_{11} - 9084 \beta_{10} - 1932 \beta_{9} - 3698 \beta_{5} + \cdots - 18828 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 29764 \beta_{15} - 15444 \beta_{14} - 12562 \beta_{13} + 15444 \beta_{12} + 30714 \beta_{11} + \cdots + 30714 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 53248 \beta_{14} - 53248 \beta_{12} + 103336 \beta_{11} + 103336 \beta_{10} + 25416 \beta_{9} + \cdots + 198446 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 331778 \beta_{15} + 182000 \beta_{14} + 141830 \beta_{13} - 182000 \beta_{12} - 346532 \beta_{11} + \cdots - 346532 \beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 618118 \beta_{14} + 618118 \beta_{12} - 1159240 \beta_{11} - 1159240 \beta_{10} - 311756 \beta_{9} + \cdots - 2129000 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 3671800 \beta_{15} - 2089114 \beta_{14} - 1577924 \beta_{13} + 2089114 \beta_{12} + \cdots + 3871234 \beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/725\mathbb{Z}\right)^\times\).

\(n\) \(176\) \(552\)
\(\chi(n)\) \(-\beta_{6}\) \(-\beta_{6}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
157.1
− 3.32267i
− 2.66264i
− 2.34301i
− 1.19281i
− 0.807187i
0.343006i
0.662643i
1.32267i
− 1.32267i
− 0.662643i
− 0.343006i
0.807187i
1.19281i
2.34301i
2.66264i
3.32267i
− 2.32267i 3.11277 −3.39477 0 − 7.22993i −2.50245 − 2.50245i 3.23960i 6.68935 0
157.2 − 1.66264i −2.84124 −0.764383 0 4.72396i 0.890023 + 0.890023i − 2.05439i 5.07263 0
157.3 − 1.34301i 0.842805 0.196335 0 − 1.13189i −0.720606 − 0.720606i − 2.94969i −2.28968 0
157.4 − 0.192813i 1.87822 1.96282 0 − 0.362145i 1.55767 + 1.55767i − 0.764084i 0.527707 0
157.5 0.192813i −1.87822 1.96282 0 − 0.362145i −1.55767 − 1.55767i 0.764084i 0.527707 0
157.6 1.34301i −0.842805 0.196335 0 − 1.13189i 0.720606 + 0.720606i 2.94969i −2.28968 0
157.7 1.66264i 2.84124 −0.764383 0 4.72396i −0.890023 − 0.890023i 2.05439i 5.07263 0
157.8 2.32267i −3.11277 −3.39477 0 − 7.22993i 2.50245 + 2.50245i − 3.23960i 6.68935 0
568.1 − 2.32267i −3.11277 −3.39477 0 7.22993i 2.50245 − 2.50245i 3.23960i 6.68935 0
568.2 − 1.66264i 2.84124 −0.764383 0 − 4.72396i −0.890023 + 0.890023i − 2.05439i 5.07263 0
568.3 − 1.34301i −0.842805 0.196335 0 1.13189i 0.720606 − 0.720606i − 2.94969i −2.28968 0
568.4 − 0.192813i −1.87822 1.96282 0 0.362145i −1.55767 + 1.55767i − 0.764084i 0.527707 0
568.5 0.192813i 1.87822 1.96282 0 0.362145i 1.55767 − 1.55767i 0.764084i 0.527707 0
568.6 1.34301i 0.842805 0.196335 0 1.13189i −0.720606 + 0.720606i 2.94969i −2.28968 0
568.7 1.66264i −2.84124 −0.764383 0 − 4.72396i 0.890023 − 0.890023i 2.05439i 5.07263 0
568.8 2.32267i 3.11277 −3.39477 0 7.22993i −2.50245 + 2.50245i − 3.23960i 6.68935 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 157.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
145.e even 4 1 inner
145.j even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 725.2.e.b ✓ 16
5.b even 2 1 inner 725.2.e.b ✓ 16
5.c odd 4 2 725.2.j.b yes 16
29.c odd 4 1 725.2.j.b yes 16
145.e even 4 1 inner 725.2.e.b ✓ 16
145.f odd 4 1 725.2.j.b yes 16
145.j even 4 1 inner 725.2.e.b ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
725.2.e.b ✓ 16 1.a even 1 1 trivial
725.2.e.b ✓ 16 5.b even 2 1 inner
725.2.e.b ✓ 16 145.e even 4 1 inner
725.2.e.b ✓ 16 145.j even 4 1 inner
725.2.j.b yes 16 5.c odd 4 2
725.2.j.b yes 16 29.c odd 4 1
725.2.j.b yes 16 145.f odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{8} + 10T_{2}^{6} + 30T_{2}^{4} + 28T_{2}^{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(725, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{8} + 10 T^{6} + 30 T^{4} + \cdots + 1)^{2} \) Copy content Toggle raw display
$3$ \( (T^{8} - 22 T^{6} + \cdots + 196)^{2} \) Copy content Toggle raw display
$5$ \( T^{16} \) Copy content Toggle raw display
$7$ \( T^{16} + 184 T^{12} + \cdots + 10000 \) Copy content Toggle raw display
$11$ \( (T^{8} + 2 T^{7} + \cdots + 2116)^{2} \) Copy content Toggle raw display
$13$ \( T^{16} + \cdots + 723394816 \) Copy content Toggle raw display
$17$ \( (T^{8} + 58 T^{6} + \cdots + 9604)^{2} \) Copy content Toggle raw display
$19$ \( (T^{8} - 6 T^{7} + 18 T^{6} + \cdots + 36)^{2} \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 178506250000 \) Copy content Toggle raw display
$29$ \( (T^{8} - 2 T^{7} + \cdots + 707281)^{2} \) Copy content Toggle raw display
$31$ \( (T^{8} - 10 T^{7} + \cdots + 196)^{2} \) Copy content Toggle raw display
$37$ \( (T^{8} - 226 T^{6} + \cdots + 2149156)^{2} \) Copy content Toggle raw display
$41$ \( (T^{8} + 72 T^{5} + \cdots + 576)^{2} \) Copy content Toggle raw display
$43$ \( (T^{8} - 198 T^{6} + \cdots + 2965284)^{2} \) Copy content Toggle raw display
$47$ \( (T^{8} - 414 T^{6} + \cdots + 104407524)^{2} \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 62500000000 \) Copy content Toggle raw display
$59$ \( (T^{8} + 204 T^{6} + \cdots + 7056)^{2} \) Copy content Toggle raw display
$61$ \( (T^{8} - 12 T^{7} + \cdots + 451584)^{2} \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 1003875856 \) Copy content Toggle raw display
$71$ \( (T^{8} + 300 T^{6} + \cdots + 90000)^{2} \) Copy content Toggle raw display
$73$ \( (T^{8} + 238 T^{6} + \cdots + 2365444)^{2} \) Copy content Toggle raw display
$79$ \( (T^{8} + 26 T^{7} + \cdots + 3062500)^{2} \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 138458410000 \) Copy content Toggle raw display
$89$ \( (T^{8} + 20 T^{7} + \cdots + 8976016)^{2} \) Copy content Toggle raw display
$97$ \( (T^{8} - 438 T^{6} + \cdots + 5560164)^{2} \) Copy content Toggle raw display
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