↑

Properties

Label 400.9.b.m
Level $400$
Weight $9$
Character orbit 400.b
Analytic conductor $162.951$
Analytic rank $0$
Dimension $16$
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [400,9,Mod(351,400)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(400, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 0]))
 
N = Newforms(chi, 9, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("400.351");
 
S:= CuspForms(chi, 9);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 400 = 2^{4} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 400.b (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: \(162.951444024\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 1394 x^{14} + 1332306 x^{12} + 657883370 x^{10} + 233333110300 x^{8} + 45644912663250 x^{6} + \cdots + 23\!\cdots\!00 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{29}]\)
Coefficient ring index: \( 2^{100}\cdot 3^{9}\cdot 5^{12} \)
Twist minimal: no (minimal twist has level 80)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{3} + ( - \beta_{7} + 4 \beta_1) q^{7} + ( - \beta_{3} - 2636) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{3} + ( - \beta_{7} + 4 \beta_1) q^{7} + ( - \beta_{3} - 2636) q^{9} + \beta_{4} q^{11} + \beta_{14} q^{13} + ( - 3 \beta_{14} - \beta_{11} - 2 \beta_{10}) q^{17} + (\beta_{9} + \beta_{6} + \cdots + 7 \beta_{2}) q^{19}+ \cdots + ( - 365 \beta_{9} + \cdots + 2673 \beta_{2}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 42176 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 42176 q^{9} + 619344 q^{21} + 3497568 q^{29} - 2169168 q^{41} - 26174912 q^{49} + 22772656 q^{61} - 8461392 q^{69} - 224999456 q^{81} + 94250976 q^{89}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 1394 x^{14} + 1332306 x^{12} + 657883370 x^{10} + 233333110300 x^{8} + 45644912663250 x^{6} + \cdots + 23\!\cdots\!00 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 79\!\cdots\!89 \nu^{15} + \cdots + 15\!\cdots\!00 \nu ) / 73\!\cdots\!75 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 22\!\cdots\!96 \nu^{14} + \cdots + 59\!\cdots\!00 ) / 65\!\cdots\!25 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 30\!\cdots\!22 \nu^{14} + \cdots + 22\!\cdots\!75 ) / 49\!\cdots\!25 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 39\!\cdots\!16 \nu^{14} + \cdots + 87\!\cdots\!00 ) / 50\!\cdots\!75 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 12\!\cdots\!86 \nu^{14} + \cdots - 89\!\cdots\!75 ) / 15\!\cdots\!75 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 74\!\cdots\!28 \nu^{14} + \cdots - 20\!\cdots\!00 ) / 35\!\cdots\!25 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 47\!\cdots\!23 \nu^{15} + \cdots + 21\!\cdots\!00 \nu ) / 73\!\cdots\!75 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 42\!\cdots\!14 \nu^{14} + \cdots - 82\!\cdots\!25 ) / 39\!\cdots\!25 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 65\!\cdots\!36 \nu^{14} + \cdots + 19\!\cdots\!00 ) / 35\!\cdots\!25 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 32\!\cdots\!64 \nu^{15} + \cdots - 49\!\cdots\!00 \nu ) / 19\!\cdots\!25 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 13\!\cdots\!84 \nu^{15} + \cdots + 18\!\cdots\!00 \nu ) / 10\!\cdots\!75 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 10\!\cdots\!44 \nu^{15} + \cdots + 15\!\cdots\!00 \nu ) / 31\!\cdots\!25 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 78\!\cdots\!39 \nu^{15} + \cdots + 40\!\cdots\!00 \nu ) / 14\!\cdots\!75 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 20\!\cdots\!24 \nu^{15} + \cdots + 31\!\cdots\!00 \nu ) / 34\!\cdots\!75 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 11\!\cdots\!32 \nu^{15} + \cdots + 49\!\cdots\!00 \nu ) / 73\!\cdots\!75 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( 6\beta_{15} - 12\beta_{13} + 15\beta_{12} - 80\beta_{11} + 99\beta_{10} + 1596\beta_{7} - 6504\beta_1 ) / 307200 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( - 104 \beta_{9} - 80 \beta_{8} - 308 \beta_{6} + 256 \beta_{5} - 1716 \beta_{4} + 2320 \beta_{3} + \cdots - 35686400 ) / 204800 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -1200\beta_{14} - 8805\beta_{12} + 18760\beta_{11} - 83913\beta_{10} ) / 76800 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 69896 \beta_{9} - 59120 \beta_{8} + 299492 \beta_{6} + 847744 \beta_{5} + 1189284 \beta_{4} + \cdots - 18467225600 ) / 204800 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 3758106 \beta_{15} + 2385600 \beta_{14} + 8709012 \beta_{13} + 11758515 \beta_{12} + \cdots + 4983511704 \beta_1 ) / 307200 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 608045\beta_{8} - 13775854\beta_{5} - 28892755\beta_{3} + 177836519600 ) / 1600 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 2746601994 \beta_{15} + 2179874400 \beta_{14} - 6583141188 \beta_{13} + 7603900485 \beta_{12} + \cdots - 2795702556696 \beta_1 ) / 307200 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 34713140696 \beta_{9} - 25879283120 \beta_{8} - 182877978092 \beta_{6} + 735914738944 \beta_{5} + \cdots - 75\!\cdots\!00 ) / 204800 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 453453491400 \beta_{14} - 1262339032785 \beta_{12} + 2435503413620 \beta_{11} - 23802921471081 \beta_{10} ) / 38400 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 25144311256904 \beta_{9} - 17640274888880 \beta_{8} + 135191766887108 \beta_{6} + \cdots - 51\!\cdots\!00 ) / 204800 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 14\!\cdots\!94 \beta_{15} + \cdots + 10\!\cdots\!96 \beta_1 ) / 307200 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 383216752502785 \beta_{8} + \cdots + 11\!\cdots\!00 ) / 3200 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 10\!\cdots\!06 \beta_{15} + \cdots - 66\!\cdots\!04 \beta_1 ) / 307200 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 13\!\cdots\!04 \beta_{9} + \cdots - 25\!\cdots\!00 ) / 204800 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 39\!\cdots\!00 \beta_{14} + \cdots - 18\!\cdots\!63 \beta_{10} ) / 76800 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/400\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(177\) \(351\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
351.1
−10.1328 − 17.5505i
10.1328 − 17.5505i
−4.96896 + 8.60649i
4.96896 + 8.60649i
−6.45800 + 11.1856i
6.45800 + 11.1856i
−13.3951 − 23.2011i
13.3951 − 23.2011i
13.3951 + 23.2011i
−13.3951 + 23.2011i
6.45800 − 11.1856i
−6.45800 − 11.1856i
4.96896 − 8.60649i
−4.96896 − 8.60649i
10.1328 + 17.5505i
−10.1328 + 17.5505i
0 − 125.551i 0 0 0 4128.49i 0 −9202.10 0
351.2 0 − 125.551i 0 0 0 4128.49i 0 −9202.10 0
351.3 0 − 120.512i 0 0 0 − 1534.67i 0 −7962.26 0
351.4 0 − 120.512i 0 0 0 − 1534.67i 0 −7962.26 0
351.5 0 − 77.4652i 0 0 0 − 2766.32i 0 560.136 0
351.6 0 − 77.4652i 0 0 0 − 2766.32i 0 560.136 0
351.7 0 − 22.3781i 0 0 0 1597.12i 0 6060.22 0
351.8 0 − 22.3781i 0 0 0 1597.12i 0 6060.22 0
351.9 0 22.3781i 0 0 0 − 1597.12i 0 6060.22 0
351.10 0 22.3781i 0 0 0 − 1597.12i 0 6060.22 0
351.11 0 77.4652i 0 0 0 2766.32i 0 560.136 0
351.12 0 77.4652i 0 0 0 2766.32i 0 560.136 0
351.13 0 120.512i 0 0 0 1534.67i 0 −7962.26 0
351.14 0 120.512i 0 0 0 1534.67i 0 −7962.26 0
351.15 0 125.551i 0 0 0 − 4128.49i 0 −9202.10 0
351.16 0 125.551i 0 0 0 − 4128.49i 0 −9202.10 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 351.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
5.b even 2 1 inner
20.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 400.9.b.m 16
4.b odd 2 1 inner 400.9.b.m 16
5.b even 2 1 inner 400.9.b.m 16
5.c odd 4 2 80.9.h.d ✓ 16
20.d odd 2 1 inner 400.9.b.m 16
20.e even 4 2 80.9.h.d ✓ 16
40.i odd 4 2 320.9.h.f 16
40.k even 4 2 320.9.h.f 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
80.9.h.d ✓ 16 5.c odd 4 2
80.9.h.d ✓ 16 20.e even 4 2
320.9.h.f 16 40.i odd 4 2
320.9.h.f 16 40.k even 4 2
400.9.b.m 16 1.a even 1 1 trivial
400.9.b.m 16 4.b odd 2 1 inner
400.9.b.m 16 5.b even 2 1 inner
400.9.b.m 16 20.d odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{9}^{\mathrm{new}}(400, [\chi])\):

\( T_{3}^{8} + 36788T_{3}^{6} + 428847744T_{3}^{4} + 1579444964352T_{3}^{2} + 687963618902016 \) Copy content Toggle raw display
\( T_{13}^{8} - 3005461056 T_{13}^{6} + \cdots + 45\!\cdots\!76 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( (T^{8} + \cdots + 687963618902016)^{2} \) Copy content Toggle raw display
$5$ \( T^{16} \) Copy content Toggle raw display
$7$ \( (T^{8} + \cdots + 78\!\cdots\!36)^{2} \) Copy content Toggle raw display
$11$ \( (T^{8} + \cdots + 56\!\cdots\!56)^{2} \) Copy content Toggle raw display
$13$ \( (T^{8} + \cdots + 45\!\cdots\!76)^{2} \) Copy content Toggle raw display
$17$ \( (T^{8} + \cdots + 64\!\cdots\!96)^{2} \) Copy content Toggle raw display
$19$ \( (T^{8} + \cdots + 15\!\cdots\!16)^{2} \) Copy content Toggle raw display
$23$ \( (T^{8} + \cdots + 65\!\cdots\!56)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} + \cdots + 16\!\cdots\!56)^{4} \) Copy content Toggle raw display
$31$ \( (T^{8} + \cdots + 21\!\cdots\!56)^{2} \) Copy content Toggle raw display
$37$ \( (T^{8} + \cdots + 15\!\cdots\!00)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} + \cdots - 14\!\cdots\!44)^{4} \) Copy content Toggle raw display
$43$ \( (T^{8} + \cdots + 37\!\cdots\!76)^{2} \) Copy content Toggle raw display
$47$ \( (T^{8} + \cdots + 18\!\cdots\!96)^{2} \) Copy content Toggle raw display
$53$ \( (T^{8} + \cdots + 58\!\cdots\!56)^{2} \) Copy content Toggle raw display
$59$ \( (T^{8} + \cdots + 26\!\cdots\!36)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} + \cdots + 44\!\cdots\!76)^{4} \) Copy content Toggle raw display
$67$ \( (T^{8} + \cdots + 15\!\cdots\!56)^{2} \) Copy content Toggle raw display
$71$ \( (T^{8} + \cdots + 35\!\cdots\!56)^{2} \) Copy content Toggle raw display
$73$ \( (T^{8} + \cdots + 32\!\cdots\!56)^{2} \) Copy content Toggle raw display
$79$ \( (T^{8} + \cdots + 16\!\cdots\!00)^{2} \) Copy content Toggle raw display
$83$ \( (T^{8} + \cdots + 22\!\cdots\!36)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} + \cdots + 22\!\cdots\!56)^{4} \) Copy content Toggle raw display
$97$ \( (T^{8} + \cdots + 68\!\cdots\!56)^{2} \) Copy content Toggle raw display
show more
show less