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Properties

Label 400.2.bi.c
Level $400$
Weight $2$
Character orbit 400.bi
Analytic conductor $3.194$
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] = [400,2,Mod(47,400)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(400, base_ring=CyclotomicField(20))
 
chi = DirichletCharacter(H, H._module([10, 0, 17]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("400.47");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 400 = 2^{4} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 400.bi (of order \(20\), degree \(8\), 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: \(3.19401608085\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(2\) over \(\Q(\zeta_{20})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 8 x^{15} + 32 x^{14} - 64 x^{13} + 66 x^{12} - 28 x^{11} + 160 x^{10} - 392 x^{9} + 419 x^{8} + \cdots + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{20}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( - \beta_{15} + \beta_{13}) q^{3} + ( - 2 \beta_{5} + 2 \beta_{3} - 1) q^{5} + ( - \beta_{15} + \beta_{14} + \cdots - \beta_1) q^{7}+ \cdots + (2 \beta_{8} + 2 \beta_{6} + \beta_{3} + \cdots + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{15} + \beta_{13}) q^{3} + ( - 2 \beta_{5} + 2 \beta_{3} - 1) q^{5} + ( - \beta_{15} + \beta_{14} + \cdots - \beta_1) q^{7}+ \cdots + (4 \beta_{15} - \beta_{14} + \cdots + 3 \beta_{7}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 20 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 20 q^{9} + 20 q^{21} + 80 q^{25} + 20 q^{29} - 20 q^{33} - 40 q^{37} - 12 q^{41} + 20 q^{45} - 40 q^{53} + 20 q^{57} - 12 q^{61} - 60 q^{69} - 40 q^{73} - 100 q^{77} - 24 q^{81} - 60 q^{89} - 100 q^{93} - 80 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 8 x^{15} + 32 x^{14} - 64 x^{13} + 66 x^{12} - 28 x^{11} + 160 x^{10} - 392 x^{9} + 419 x^{8} + \cdots + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 35312983332889 \nu^{15} + 365935926611786 \nu^{14} + \cdots + 13\!\cdots\!57 ) / 33\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 105935192248106 \nu^{15} + 840235881864435 \nu^{14} + \cdots - 44\!\cdots\!49 ) / 33\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 117209906825 \nu^{15} - 924139326112 \nu^{14} + 3631112773384 \nu^{13} + \cdots + 3493669457165 ) / 2236954634084 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 218207872659542 \nu^{15} + \cdots - 52\!\cdots\!11 ) / 33\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 180815089545 \nu^{15} - 1457184870880 \nu^{14} + 5865587619720 \nu^{13} - 11865271567808 \nu^{12} + \cdots - 58649296127 ) / 2236954634084 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 166819737348942 \nu^{15} + \cdots - 20\!\cdots\!34 ) / 16\!\cdots\!02 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 4335591935556 \nu^{15} - 35033507449881 \nu^{14} + 141652275765242 \nu^{13} + \cdots + 53828129372136 ) / 40401583086322 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 355901287700060 \nu^{15} + \cdots + 15\!\cdots\!39 ) / 33\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 249695867990 \nu^{15} + 2022173337513 \nu^{14} - 8193254035631 \nu^{13} + \cdots - 1984753724642 ) / 2236954634084 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 6199579052327 \nu^{15} + 49079694422172 \nu^{14} - 194515097033337 \nu^{13} + \cdots - 272518996468363 ) / 54310324804564 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 405023439724493 \nu^{15} + \cdots - 11\!\cdots\!48 ) / 33\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 493087238017905 \nu^{15} + \cdots - 11\!\cdots\!30 ) / 33\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 634567973983286 \nu^{15} + \cdots - 97\!\cdots\!84 ) / 33\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 716948764061689 \nu^{15} + \cdots - 16\!\cdots\!90 ) / 33\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 791578643039920 \nu^{15} + \cdots - 18\!\cdots\!17 ) / 33\!\cdots\!04 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{15} - \beta_{14} - \beta_{12} + \beta_{11} + \beta_{7} + \beta_{6} + \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -2\beta_{9} + \beta_{7} + 3\beta_{6} - \beta_{5} + \beta_{3} - 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 5 \beta_{15} + 6 \beta_{14} + 5 \beta_{12} - 6 \beta_{11} + 2 \beta_{10} - 6 \beta_{9} - 3 \beta_{8} + \cdots - 11 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 8 \beta_{15} + 10 \beta_{14} - 2 \beta_{13} + 8 \beta_{12} - 10 \beta_{11} + 2 \beta_{10} - 8 \beta_{8} + \cdots - 16 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 32 \beta_{15} + 43 \beta_{14} - 22 \beta_{13} + 34 \beta_{12} - 43 \beta_{11} + 60 \beta_{9} + \cdots - 24 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 3 \beta_{15} - 23 \beta_{13} + 3 \beta_{12} - 23 \beta_{10} + 126 \beta_{9} + 15 \beta_{8} - 61 \beta_{7} + \cdots + 63 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 261 \beta_{15} - 325 \beta_{14} - 223 \beta_{12} + 323 \beta_{11} - 204 \beta_{10} + 490 \beta_{9} + \cdots + 617 ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 468 \beta_{15} - 628 \beta_{14} + 212 \beta_{13} - 468 \beta_{12} + 628 \beta_{11} - 212 \beta_{10} + \cdots + 701 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 1609 \beta_{15} - 2442 \beta_{14} + 1778 \beta_{13} - 2083 \beta_{12} + 2498 \beta_{11} - 3816 \beta_{9} + \cdots + 721 ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 549 \beta_{15} + 80 \beta_{14} + 1829 \beta_{13} - 549 \beta_{12} + 80 \beta_{11} + 1829 \beta_{10} + \cdots - 3750 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 16784 \beta_{15} + 19325 \beta_{14} + 11806 \beta_{12} - 18475 \beta_{11} + 15038 \beta_{10} + \cdots - 33582 ) / 2 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 28190 \beta_{15} + 36988 \beta_{14} - 15348 \beta_{13} + 28190 \beta_{12} - 36988 \beta_{11} + \cdots - 36029 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 87475 \beta_{15} + 139949 \beta_{14} - 125036 \beta_{13} + 135303 \beta_{12} - 149993 \beta_{11} + \cdots - 24959 ) / 2 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 51308 \beta_{15} - 11572 \beta_{14} - 126900 \beta_{13} + 51308 \beta_{12} - 11572 \beta_{11} + \cdots + 220563 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 1088461 \beta_{15} - 1166632 \beta_{14} - 652493 \beta_{12} + 1062472 \beta_{11} - 1028278 \beta_{10} + \cdots + 1872787 ) / 2 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(101\) \(177\) \(351\)
\(\chi(n)\) \(1\) \(\beta_{4}\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
47.1
1.98360 − 1.98360i
−0.983596 + 0.983596i
1.31377 + 1.31377i
−0.313766 − 0.313766i
1.31377 − 1.31377i
−0.313766 + 0.313766i
1.85565 + 1.85565i
−0.855649 − 0.855649i
1.85565 − 1.85565i
−0.855649 + 0.855649i
1.04676 + 1.04676i
−0.0467572 − 0.0467572i
1.04676 − 1.04676i
−0.0467572 + 0.0467572i
1.98360 + 1.98360i
−0.983596 − 0.983596i
0 −0.145225 − 0.916913i 0 2.23607 0 −2.96719 + 2.96719i 0 2.03353 − 0.660734i 0
47.2 0 0.145225 + 0.916913i 0 2.23607 0 2.96719 − 2.96719i 0 2.03353 − 0.660734i 0
63.1 0 −2.58417 − 1.31670i 0 −2.23607 0 −1.62753 − 1.62753i 0 3.18088 + 4.37811i 0
63.2 0 2.58417 + 1.31670i 0 −2.23607 0 1.62753 + 1.62753i 0 3.18088 + 4.37811i 0
127.1 0 −2.58417 + 1.31670i 0 −2.23607 0 −1.62753 + 1.62753i 0 3.18088 − 4.37811i 0
127.2 0 2.58417 − 1.31670i 0 −2.23607 0 1.62753 − 1.62753i 0 3.18088 − 4.37811i 0
223.1 0 −1.11764 − 2.19349i 0 −2.23607 0 −2.71130 − 2.71130i 0 −1.79892 + 2.47599i 0
223.2 0 1.11764 + 2.19349i 0 −2.23607 0 2.71130 + 2.71130i 0 −1.79892 + 2.47599i 0
287.1 0 −1.11764 + 2.19349i 0 −2.23607 0 −2.71130 + 2.71130i 0 −1.79892 − 2.47599i 0
287.2 0 1.11764 − 2.19349i 0 −2.23607 0 2.71130 − 2.71130i 0 −1.79892 − 2.47599i 0
303.1 0 −2.13351 + 0.337915i 0 2.23607 0 −1.09351 − 1.09351i 0 1.58450 − 0.514836i 0
303.2 0 2.13351 − 0.337915i 0 2.23607 0 1.09351 + 1.09351i 0 1.58450 − 0.514836i 0
367.1 0 −2.13351 − 0.337915i 0 2.23607 0 −1.09351 + 1.09351i 0 1.58450 + 0.514836i 0
367.2 0 2.13351 + 0.337915i 0 2.23607 0 1.09351 − 1.09351i 0 1.58450 + 0.514836i 0
383.1 0 −0.145225 + 0.916913i 0 2.23607 0 −2.96719 − 2.96719i 0 2.03353 + 0.660734i 0
383.2 0 0.145225 − 0.916913i 0 2.23607 0 2.96719 + 2.96719i 0 2.03353 + 0.660734i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 47.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
25.f odd 20 1 inner
100.l even 20 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 400.2.bi.c ✓ 16
4.b odd 2 1 inner 400.2.bi.c ✓ 16
25.f odd 20 1 inner 400.2.bi.c ✓ 16
100.l even 20 1 inner 400.2.bi.c ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
400.2.bi.c ✓ 16 1.a even 1 1 trivial
400.2.bi.c ✓ 16 4.b odd 2 1 inner
400.2.bi.c ✓ 16 25.f odd 20 1 inner
400.2.bi.c ✓ 16 100.l even 20 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{16} - 10T_{3}^{14} + 65T_{3}^{12} - 120T_{3}^{10} + 1810T_{3}^{8} - 16650T_{3}^{6} + 25400T_{3}^{4} + 77900T_{3}^{2} + 42025 \) acting on \(S_{2}^{\mathrm{new}}(400, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( T^{16} - 10 T^{14} + \cdots + 42025 \) Copy content Toggle raw display
$5$ \( (T^{2} - 5)^{8} \) Copy content Toggle raw display
$7$ \( T^{16} + 560 T^{12} + \cdots + 10758400 \) Copy content Toggle raw display
$11$ \( T^{16} - 50 T^{14} + \cdots + 26265625 \) Copy content Toggle raw display
$13$ \( (T^{8} + 35 T^{6} + \cdots + 25)^{2} \) Copy content Toggle raw display
$17$ \( (T^{8} + 15 T^{6} + \cdots + 25)^{2} \) Copy content Toggle raw display
$19$ \( T^{16} + 30 T^{14} + \cdots + 26265625 \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 5476740025 \) Copy content Toggle raw display
$29$ \( (T^{8} - 10 T^{7} + \cdots + 57121)^{2} \) Copy content Toggle raw display
$31$ \( T^{16} + \cdots + 363669667515625 \) Copy content Toggle raw display
$37$ \( (T^{8} + 20 T^{7} + \cdots + 255025)^{2} \) Copy content Toggle raw display
$41$ \( (T^{8} + 6 T^{7} + \cdots + 606841)^{2} \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 1402045446400 \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 509232096025 \) Copy content Toggle raw display
$53$ \( (T^{8} + 20 T^{7} + \cdots + 33582025)^{2} \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 3422962515625 \) Copy content Toggle raw display
$61$ \( (T^{8} + 6 T^{7} + \cdots + 39601)^{2} \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 16416015625 \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 11\!\cdots\!25 \) Copy content Toggle raw display
$73$ \( (T^{8} + 20 T^{7} + \cdots + 1428025)^{2} \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 74220378765625 \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 118752606025 \) Copy content Toggle raw display
$89$ \( (T^{8} + 30 T^{7} + \cdots + 32041)^{2} \) Copy content Toggle raw display
$97$ \( (T^{8} + 40 T^{7} + \cdots + 6225025)^{2} \) Copy content Toggle raw display
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