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Properties

Label 25.16.b.d
Level $25$
Weight $16$
Character orbit 25.b
Analytic conductor $35.673$
Analytic rank $0$
Dimension $10$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [25,16,Mod(24,25)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(25, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 16, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("25.24");
 
S:= CuspForms(chi, 16);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 25 = 5^{2} \)
Weight: \( k \) \(=\) \( 16 \)
Character orbit: \([\chi]\) \(=\) 25.b (of order \(2\), degree \(1\), 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: \(35.6733762750\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} + 296901 x^{8} + 30788485188 x^{6} + \cdots + 84\!\cdots\!36 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{16}\cdot 3^{2}\cdot 5^{24} \)
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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{5} q^{2} + (\beta_{7} - 3 \beta_{5}) q^{3} + (\beta_1 - 26673) q^{4} + (\beta_{4} - \beta_{3} + \beta_{2} + \cdots - 185623) q^{6}+ \cdots + (7 \beta_{4} - 8 \beta_{3} + \cdots - 7621982) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{5} q^{2} + (\beta_{7} - 3 \beta_{5}) q^{3} + (\beta_1 - 26673) q^{4} + (\beta_{4} - \beta_{3} + \beta_{2} + \cdots - 185623) q^{6}+ \cdots + ( - 36462247 \beta_{4} + \cdots + 213347931084506) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 266730 q^{4} - 1856230 q^{6} - 76219820 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q - 266730 q^{4} - 1856230 q^{6} - 76219820 q^{9} - 202222830 q^{11} - 1849648260 q^{14} + 5354682210 q^{16} + 9575578350 q^{19} + 32659074420 q^{21} + 44932386450 q^{24} - 156134493480 q^{26} - 184019857200 q^{29} + 117299769220 q^{31} + 2955627887590 q^{34} + 6682168662460 q^{36} + 3100307781560 q^{39} + 4115854217970 q^{41} + 18500144134590 q^{44} + 11858491358220 q^{46} + 17240281895870 q^{49} + 39117518326370 q^{51} + 118502009094850 q^{54} + 127745505326700 q^{56} - 20466863745600 q^{59} + 33569809228020 q^{61} - 70154893720930 q^{64} - 235803243673910 q^{66} - 273870639724140 q^{69} - 203260907547480 q^{71} - 4859057616660 q^{74} - 10\!\cdots\!50 q^{76}+ \cdots + 21\!\cdots\!60 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} + 296901 x^{8} + 30788485188 x^{6} + \cdots + 84\!\cdots\!36 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 281579 \nu^{8} + 67540167127 \nu^{6} + \cdots + 15\!\cdots\!68 ) / 24\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 7490510989 \nu^{8} + \cdots + 56\!\cdots\!12 ) / 95\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 62676150211 \nu^{8} + \cdots + 40\!\cdots\!12 ) / 43\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 494430991027 \nu^{8} + \cdots + 11\!\cdots\!28 ) / 15\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 669614459 \nu^{9} + 160264542486951 \nu^{7} + \cdots - 93\!\cdots\!68 \nu ) / 68\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 2092545184375 \nu^{9} + \cdots + 77\!\cdots\!00 \nu ) / 54\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 227609904349577 \nu^{9} + \cdots - 73\!\cdots\!08 \nu ) / 46\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 701982750694337 \nu^{9} + \cdots + 66\!\cdots\!16 \nu ) / 88\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 14\!\cdots\!85 \nu^{9} + \cdots + 13\!\cdots\!96 \nu ) / 52\!\cdots\!92 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -8\beta_{6} - 3125\beta_{5} ) / 3125 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -16\beta_{3} + 16\beta_{2} + 3125\beta _1 - 185563125 ) / 3125 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -3725\beta_{9} + 14425\beta_{8} - 367725\beta_{7} + 1550236\beta_{6} + 279568250\beta_{5} ) / 3125 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -114375\beta_{4} + 1162946\beta_{3} + 559174\beta_{2} - 75740710\beta _1 + 3321654178125 ) / 625 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 594085625 \beta_{9} - 2059744375 \beta_{8} + 83995048125 \beta_{7} - 186729942178 \beta_{6} - 27903739437500 \beta_{5} ) / 3125 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 121852659375 \beta_{4} - 850600848946 \beta_{3} - 437450280854 \beta_{2} + 42560390319650 \beta _1 - 16\!\cdots\!25 ) / 3125 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 73849796362025 \beta_{9} + 239707930766575 \beta_{8} + \cdots + 29\!\cdots\!00 \beta_{5} ) / 3125 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 37\!\cdots\!75 \beta_{4} + \cdots + 34\!\cdots\!25 ) / 625 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 85\!\cdots\!25 \beta_{9} + \cdots - 31\!\cdots\!00 \beta_{5} ) / 3125 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/25\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
24.1
− 322.501i
− 334.321i
− 195.595i
− 205.954i
− 21.1792i
21.1792i
205.954i
195.595i
334.321i
322.501i
− 330.501i 4779.94i −76463.2 0 1.57978e6 − 3.66541e6i 1.44413e7i −8.49891e6 0
24.2 − 326.321i − 6929.36i −73717.6 0 −2.26120e6 − 137864.i 1.33627e7i −3.36671e7 0
24.3 − 203.595i − 4593.94i −8682.90 0 −935303. 845401.i − 4.90360e6i −6.75536e6 0
24.4 − 197.954i 3046.17i −6417.88 0 603002. 706780.i − 5.21612e6i 5.06977e6 0
24.5 − 29.1792i 2933.81i 31916.6 0 85606.1 670229.i − 1.88744e6i 5.74167e6 0
24.6 29.1792i − 2933.81i 31916.6 0 85606.1 − 670229.i 1.88744e6i 5.74167e6 0
24.7 197.954i − 3046.17i −6417.88 0 603002. − 706780.i 5.21612e6i 5.06977e6 0
24.8 203.595i 4593.94i −8682.90 0 −935303. − 845401.i 4.90360e6i −6.75536e6 0
24.9 326.321i 6929.36i −73717.6 0 −2.26120e6 137864.i − 1.33627e7i −3.36671e7 0
24.10 330.501i − 4779.94i −76463.2 0 1.57978e6 3.66541e6i − 1.44413e7i −8.49891e6 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 24.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 25.16.b.d 10
5.b even 2 1 inner 25.16.b.d 10
5.c odd 4 1 25.16.a.d ✓ 5
5.c odd 4 1 25.16.a.e yes 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
25.16.a.d ✓ 5 5.c odd 4 1
25.16.a.e yes 5 5.c odd 4 1
25.16.b.d 10 1.a even 1 1 trivial
25.16.b.d 10 5.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{10} + 297205 T_{2}^{8} + 30902869860 T_{2}^{6} + \cdots + 16\!\cdots\!76 \) acting on \(S_{16}^{\mathrm{new}}(25, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} + \cdots + 16\!\cdots\!76 \) Copy content Toggle raw display
$3$ \( T^{10} + \cdots + 18\!\cdots\!49 \) Copy content Toggle raw display
$5$ \( T^{10} \) Copy content Toggle raw display
$7$ \( T^{10} + \cdots + 40\!\cdots\!76 \) Copy content Toggle raw display
$11$ \( (T^{5} + \cdots + 45\!\cdots\!43)^{2} \) Copy content Toggle raw display
$13$ \( T^{10} + \cdots + 11\!\cdots\!24 \) Copy content Toggle raw display
$17$ \( T^{10} + \cdots + 30\!\cdots\!01 \) Copy content Toggle raw display
$19$ \( (T^{5} + \cdots - 32\!\cdots\!75)^{2} \) Copy content Toggle raw display
$23$ \( T^{10} + \cdots + 13\!\cdots\!24 \) Copy content Toggle raw display
$29$ \( (T^{5} + \cdots + 36\!\cdots\!00)^{2} \) Copy content Toggle raw display
$31$ \( (T^{5} + \cdots - 13\!\cdots\!32)^{2} \) Copy content Toggle raw display
$37$ \( T^{10} + \cdots + 11\!\cdots\!76 \) Copy content Toggle raw display
$41$ \( (T^{5} + \cdots - 26\!\cdots\!57)^{2} \) Copy content Toggle raw display
$43$ \( T^{10} + \cdots + 13\!\cdots\!24 \) Copy content Toggle raw display
$47$ \( T^{10} + \cdots + 44\!\cdots\!76 \) Copy content Toggle raw display
$53$ \( T^{10} + \cdots + 17\!\cdots\!24 \) Copy content Toggle raw display
$59$ \( (T^{5} + \cdots + 91\!\cdots\!00)^{2} \) Copy content Toggle raw display
$61$ \( (T^{5} + \cdots - 98\!\cdots\!32)^{2} \) Copy content Toggle raw display
$67$ \( T^{10} + \cdots + 15\!\cdots\!01 \) Copy content Toggle raw display
$71$ \( (T^{5} + \cdots + 35\!\cdots\!68)^{2} \) Copy content Toggle raw display
$73$ \( T^{10} + \cdots + 13\!\cdots\!49 \) Copy content Toggle raw display
$79$ \( (T^{5} + \cdots - 65\!\cdots\!00)^{2} \) Copy content Toggle raw display
$83$ \( T^{10} + \cdots + 35\!\cdots\!49 \) Copy content Toggle raw display
$89$ \( (T^{5} + \cdots + 75\!\cdots\!75)^{2} \) Copy content Toggle raw display
$97$ \( T^{10} + \cdots + 25\!\cdots\!76 \) Copy content Toggle raw display
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