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Properties

Label 400.2.l.f
Level $400$
Weight $2$
Character orbit 400.l
Analytic conductor $3.194$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [400,2,Mod(101,400)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(400, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("400.101");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 400 = 2^{4} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 400.l (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: \(3.19401608085\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(i)\)
Coefficient field: 12.0.4767670494822400.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 4 x^{11} + 7 x^{10} - 4 x^{9} - 8 x^{8} + 24 x^{7} - 38 x^{6} + 48 x^{5} - 32 x^{4} - 32 x^{3} + \cdots + 64 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} - \beta_{6} q^{3} + \beta_{2} q^{4} + ( - \beta_{10} - \beta_{7} + \cdots + \beta_1) q^{6}+ \cdots + ( - \beta_{11} + \beta_{10} + \beta_{5} + \cdots + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} - \beta_{6} q^{3} + \beta_{2} q^{4} + ( - \beta_{10} - \beta_{7} + \cdots + \beta_1) q^{6}+ \cdots + ( - \beta_{11} - \beta_{10} + 2 \beta_{6} + \cdots + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 4 q^{2} - 2 q^{3} + 2 q^{4} + 6 q^{6} + 8 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 4 q^{2} - 2 q^{3} + 2 q^{4} + 6 q^{6} + 8 q^{8} - 2 q^{11} - 8 q^{12} + 4 q^{13} + 14 q^{14} + 2 q^{16} + 8 q^{17} - 18 q^{18} - 14 q^{19} - 20 q^{21} - 2 q^{22} - 14 q^{24} - 16 q^{26} + 10 q^{27} - 26 q^{28} - 4 q^{31} + 16 q^{32} - 28 q^{33} - 6 q^{34} + 2 q^{36} - 8 q^{37} - 10 q^{38} - 10 q^{42} - 44 q^{44} - 10 q^{46} - 8 q^{47} + 28 q^{48} + 4 q^{49} + 10 q^{51} + 12 q^{52} + 16 q^{53} + 10 q^{54} + 6 q^{56} + 60 q^{58} + 20 q^{59} + 4 q^{61} + 18 q^{62} + 8 q^{63} + 38 q^{64} + 32 q^{66} - 50 q^{67} + 60 q^{68} + 14 q^{72} + 10 q^{74} + 60 q^{76} + 8 q^{77} - 4 q^{78} + 12 q^{79} - 8 q^{81} - 42 q^{82} + 2 q^{83} + 34 q^{84} + 6 q^{86} - 30 q^{88} + 2 q^{92} + 44 q^{93} + 32 q^{94} - 34 q^{96} - 64 q^{98} + 12 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 4 x^{11} + 7 x^{10} - 4 x^{9} - 8 x^{8} + 24 x^{7} - 38 x^{6} + 48 x^{5} - 32 x^{4} - 32 x^{3} + \cdots + 64 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{11} + 2\nu^{10} + \nu^{9} - 2\nu^{8} + 6\nu^{5} - 4\nu^{4} - 16\nu^{3} + 16\nu^{2} + 32\nu - 32 ) / 16 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - \nu^{11} + 2 \nu^{10} + \nu^{9} - 6 \nu^{8} + 8 \nu^{7} - 12 \nu^{6} + 14 \nu^{5} - 4 \nu^{4} + \cdots + 16 ) / 16 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{11} - 2 \nu^{10} - 5 \nu^{9} + 18 \nu^{8} - 12 \nu^{7} + 10 \nu^{5} - 28 \nu^{4} + 88 \nu^{3} + \cdots + 128 ) / 32 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -\nu^{11} + 2\nu^{10} - 4\nu^{8} + 5\nu^{7} - 6\nu^{6} + 10\nu^{5} - 4\nu^{4} - 18\nu^{3} + 36\nu^{2} - 16\nu - 8 ) / 8 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{11} - 6 \nu^{10} + 11 \nu^{9} - 2 \nu^{8} - 12 \nu^{7} + 24 \nu^{6} - 38 \nu^{5} + 60 \nu^{4} + \cdots - 64 ) / 32 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - \nu^{11} + 4 \nu^{10} - 3 \nu^{9} - 4 \nu^{8} + 12 \nu^{7} - 16 \nu^{6} + 22 \nu^{5} - 24 \nu^{4} + \cdots + 32 ) / 8 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 9 \nu^{11} - 22 \nu^{10} + 15 \nu^{9} + 22 \nu^{8} - 56 \nu^{7} + 80 \nu^{6} - 118 \nu^{5} + 124 \nu^{4} + \cdots - 64 ) / 32 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 3 \nu^{11} - 14 \nu^{10} + 21 \nu^{9} + 6 \nu^{8} - 56 \nu^{7} + 88 \nu^{6} - 114 \nu^{5} + 124 \nu^{4} + \cdots - 320 ) / 32 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 7 \nu^{11} + 30 \nu^{10} - 41 \nu^{9} + 2 \nu^{8} + 80 \nu^{7} - 144 \nu^{6} + 202 \nu^{5} + \cdots + 480 ) / 32 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{10} + \beta_{8} + \beta_{6} + \beta_{5} - 2\beta_{3} + \beta_{2} + \beta _1 - 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{11} + \beta_{9} - \beta_{8} + \beta_{7} + 2\beta_{6} - \beta_{5} - \beta_{4} \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -\beta_{11} - 2\beta_{10} + \beta_{9} - \beta_{7} + 2\beta_{6} + \beta_{5} - \beta_{4} + 2\beta_{3} - \beta_{2} + 2\beta _1 - 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -2\beta_{11} - 3\beta_{10} + \beta_{8} + \beta_{6} + \beta_{5} - 4\beta_{4} + 3\beta_{3} - \beta_{2} + 3 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -\beta_{11} - 2\beta_{10} - \beta_{9} + 2\beta_{8} + 5\beta_{7} + \beta_{5} - 5\beta_{4} - 3\beta_{2} + 6\beta _1 - 4 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 2 \beta_{11} - 3 \beta_{10} - 3 \beta_{8} + 8 \beta_{7} - 3 \beta_{6} - 3 \beta_{5} - 4 \beta_{4} + \cdots - 5 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - \beta_{11} + 4 \beta_{10} - \beta_{9} + 8 \beta_{8} + 5 \beta_{7} - 2 \beta_{6} + 7 \beta_{5} + \cdots - 10 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 8 \beta_{11} + 5 \beta_{10} + 6 \beta_{9} - \beta_{8} + 6 \beta_{7} + \beta_{6} - 9 \beta_{5} + \cdots + 11 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( \beta_{11} - 8 \beta_{10} + 13 \beta_{9} - 9 \beta_{7} - 6 \beta_{6} - 11 \beta_{5} + 5 \beta_{4} + \cdots - 6 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(101\) \(177\) \(351\)
\(\chi(n)\) \(-\beta_{7}\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
101.1
1.35979 − 0.388551i
1.22306 + 0.710021i
0.719139 − 1.21772i
0.618969 + 1.27156i
−0.507829 − 1.31989i
−1.41313 − 0.0554252i
1.35979 + 0.388551i
1.22306 − 0.710021i
0.719139 + 1.21772i
0.618969 − 1.27156i
−0.507829 + 1.31989i
−1.41313 + 0.0554252i
−1.35979 + 0.388551i −1.03997 + 1.03997i 1.69806 − 1.05670i 0 1.01006 − 1.81822i − 1.49668i −1.89842 + 2.09667i 0.836925i 0
101.2 −1.22306 − 0.710021i 1.09156 − 1.09156i 0.991741 + 1.73679i 0 −2.11008 + 0.560012i 0.973926i 0.0202025 − 2.82835i 0.616985i 0
101.3 −0.719139 + 1.21772i 1.66783 − 1.66783i −0.965679 − 1.75142i 0 0.831547 + 3.23035i − 1.87372i 2.82719 + 0.0835873i − 2.56332i 0
101.4 −0.618969 − 1.27156i −2.16859 + 2.16859i −1.23375 + 1.57412i 0 4.09979 + 1.41521i 3.30519i 2.76525 + 0.594467i − 6.40553i 0
101.5 0.507829 + 1.31989i −0.0623209 + 0.0623209i −1.48422 + 1.34056i 0 −0.113905 − 0.0506084i 0.375877i −2.52312 − 1.27824i 2.99223i 0
101.6 1.41313 + 0.0554252i −0.488516 + 0.488516i 1.99386 + 0.156646i 0 −0.717411 + 0.663259i 4.71540i 2.80889 + 0.331870i 2.52270i 0
301.1 −1.35979 − 0.388551i −1.03997 − 1.03997i 1.69806 + 1.05670i 0 1.01006 + 1.81822i 1.49668i −1.89842 − 2.09667i − 0.836925i 0
301.2 −1.22306 + 0.710021i 1.09156 + 1.09156i 0.991741 − 1.73679i 0 −2.11008 − 0.560012i − 0.973926i 0.0202025 + 2.82835i − 0.616985i 0
301.3 −0.719139 − 1.21772i 1.66783 + 1.66783i −0.965679 + 1.75142i 0 0.831547 − 3.23035i 1.87372i 2.82719 − 0.0835873i 2.56332i 0
301.4 −0.618969 + 1.27156i −2.16859 − 2.16859i −1.23375 − 1.57412i 0 4.09979 − 1.41521i − 3.30519i 2.76525 − 0.594467i 6.40553i 0
301.5 0.507829 − 1.31989i −0.0623209 − 0.0623209i −1.48422 − 1.34056i 0 −0.113905 + 0.0506084i − 0.375877i −2.52312 + 1.27824i − 2.99223i 0
301.6 1.41313 − 0.0554252i −0.488516 − 0.488516i 1.99386 − 0.156646i 0 −0.717411 − 0.663259i − 4.71540i 2.80889 − 0.331870i − 2.52270i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 101.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
16.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 400.2.l.f ✓ 12
4.b odd 2 1 1600.2.l.g 12
5.b even 2 1 400.2.l.g yes 12
5.c odd 4 1 400.2.q.e 12
5.c odd 4 1 400.2.q.f 12
16.e even 4 1 inner 400.2.l.f ✓ 12
16.f odd 4 1 1600.2.l.g 12
20.d odd 2 1 1600.2.l.f 12
20.e even 4 1 1600.2.q.e 12
20.e even 4 1 1600.2.q.f 12
80.i odd 4 1 400.2.q.e 12
80.j even 4 1 1600.2.q.f 12
80.k odd 4 1 1600.2.l.f 12
80.q even 4 1 400.2.l.g yes 12
80.s even 4 1 1600.2.q.e 12
80.t odd 4 1 400.2.q.f 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
400.2.l.f ✓ 12 1.a even 1 1 trivial
400.2.l.f ✓ 12 16.e even 4 1 inner
400.2.l.g yes 12 5.b even 2 1
400.2.l.g yes 12 80.q even 4 1
400.2.q.e 12 5.c odd 4 1
400.2.q.e 12 80.i odd 4 1
400.2.q.f 12 5.c odd 4 1
400.2.q.f 12 80.t odd 4 1
1600.2.l.f 12 20.d odd 2 1
1600.2.l.f 12 80.k odd 4 1
1600.2.l.g 12 4.b odd 2 1
1600.2.l.g 12 16.f odd 4 1
1600.2.q.e 12 20.e even 4 1
1600.2.q.e 12 80.s even 4 1
1600.2.q.f 12 20.e even 4 1
1600.2.q.f 12 80.j even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(400, [\chi])\):

\( T_{3}^{12} + 2 T_{3}^{11} + 2 T_{3}^{10} - 6 T_{3}^{9} + 51 T_{3}^{8} + 60 T_{3}^{7} + 36 T_{3}^{6} + \cdots + 1 \) Copy content Toggle raw display
\( T_{7}^{12} + 40T_{7}^{10} + 484T_{7}^{8} + 2144T_{7}^{6} + 3776T_{7}^{4} + 2304T_{7}^{2} + 256 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} + 4 T^{11} + \cdots + 64 \) Copy content Toggle raw display
$3$ \( T^{12} + 2 T^{11} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( T^{12} \) Copy content Toggle raw display
$7$ \( T^{12} + 40 T^{10} + \cdots + 256 \) Copy content Toggle raw display
$11$ \( T^{12} + 2 T^{11} + \cdots + 85849 \) Copy content Toggle raw display
$13$ \( T^{12} - 4 T^{11} + \cdots + 256 \) Copy content Toggle raw display
$17$ \( (T^{6} - 4 T^{5} + \cdots - 823)^{2} \) Copy content Toggle raw display
$19$ \( T^{12} + 14 T^{11} + \cdots + 29997529 \) Copy content Toggle raw display
$23$ \( T^{12} + 184 T^{10} + \cdots + 8248384 \) Copy content Toggle raw display
$29$ \( T^{12} + \cdots + 428655616 \) Copy content Toggle raw display
$31$ \( (T^{6} + 2 T^{5} + \cdots + 2152)^{2} \) Copy content Toggle raw display
$37$ \( T^{12} + 8 T^{11} + \cdots + 6801664 \) Copy content Toggle raw display
$41$ \( T^{12} + 262 T^{10} + \cdots + 86397025 \) Copy content Toggle raw display
$43$ \( T^{12} - 128 T^{9} + \cdots + 26214400 \) Copy content Toggle raw display
$47$ \( (T^{6} + 4 T^{5} + \cdots - 22016)^{2} \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 7225000000 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 56712564736 \) Copy content Toggle raw display
$61$ \( T^{12} - 4 T^{11} + \cdots + 473344 \) Copy content Toggle raw display
$67$ \( T^{12} + 50 T^{11} + \cdots + 38626225 \) Copy content Toggle raw display
$71$ \( T^{12} + 256 T^{10} + \cdots + 95257600 \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 192626641 \) Copy content Toggle raw display
$79$ \( (T^{6} - 6 T^{5} + \cdots + 1250320)^{2} \) Copy content Toggle raw display
$83$ \( T^{12} - 2 T^{11} + \cdots + 4583881 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 2165692369 \) Copy content Toggle raw display
$97$ \( (T^{6} - 324 T^{4} + \cdots + 37504)^{2} \) Copy content Toggle raw display
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