Properties

Label 1400.2.q.m
Level $1400$
Weight $2$
Character orbit 1400.q
Analytic conductor $11.179$
Analytic rank $0$
Dimension $8$
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] = [1400,2,Mod(401,1400)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1400, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1400.401");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1400 = 2^{3} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1400.q (of order \(3\), 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: \(11.1790562830\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{7} + 8x^{6} + 3x^{5} + 50x^{4} - 12x^{3} + 11x^{2} + 2x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{5} - \beta_1) q^{3} + ( - \beta_{4} + \beta_{3} + \beta_1) q^{7} + (\beta_{7} - 2 \beta_{5} - \beta_{3} - \beta_1 - 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{5} - \beta_1) q^{3} + ( - \beta_{4} + \beta_{3} + \beta_1) q^{7} + (\beta_{7} - 2 \beta_{5} - \beta_{3} - \beta_1 - 2) q^{9} + (\beta_{7} + \beta_{4} + \beta_{2} - 1) q^{11} + (\beta_{6} + \beta_{2} + 1) q^{13} + (2 \beta_{5} + \beta_{4} - \beta_1) q^{17} + ( - \beta_{7} + 2 \beta_{6} + 4 \beta_{5} + 2 \beta_{4} + 4) q^{19} + ( - \beta_{6} - \beta_{5} - \beta_{4} + \beta_{3} + \beta_{2} + \beta_1 + 2) q^{21} + (3 \beta_{6} + \beta_{5} + 3 \beta_{4} - 2 \beta_{3} - 2 \beta_1 + 1) q^{23} + (\beta_{6} - 2 \beta_{3} - 2 \beta_{2} - 2) q^{27} + (2 \beta_{6} + \beta_{2} + 1) q^{29} + ( - \beta_{7} - 3 \beta_{5} - \beta_{2} + 1) q^{31} + (\beta_{7} + 2 \beta_{6} + 2 \beta_{4} - 4 \beta_{3} - 4 \beta_1) q^{33} + ( - \beta_{7} + \beta_{6} + \beta_{4} - 2 \beta_{3} - 2 \beta_1) q^{37} + (\beta_{7} - 4 \beta_{5} + \beta_{2} - 4 \beta_1 - 1) q^{39} + ( - 3 \beta_{6} - 2 \beta_{3} - 3) q^{41} + (2 \beta_{6} + \beta_{3} - \beta_{2} - 2) q^{43} + ( - \beta_{7} - 4 \beta_{5} - 3 \beta_{3} - 3 \beta_1 - 4) q^{47} + ( - 2 \beta_{7} - 2 \beta_{3} - \beta_{2} - \beta_1 - 1) q^{49} + (\beta_{7} + \beta_{6} - \beta_{5} + \beta_{4} + \beta_{3} + \beta_1 - 1) q^{51} + (2 \beta_{5} - \beta_{4}) q^{53} + (\beta_{6} + 5 \beta_{3} + \beta_{2} + 6) q^{57} + ( - \beta_{5} - 3 \beta_1) q^{59} + ( - 3 \beta_{6} + 5 \beta_{5} - 3 \beta_{4} + \beta_{3} + \beta_1 + 5) q^{61} + (\beta_{7} + 2 \beta_{6} - 5 \beta_{5} + \beta_{4} + 2 \beta_{2} - 4 \beta_1) q^{63} + (2 \beta_{7} - 2 \beta_{4} + 2 \beta_{2} + 4 \beta_1 - 2) q^{67} + (3 \beta_{6} - 2 \beta_{3} - 2 \beta_{2} - 2) q^{69} + ( - 2 \beta_{6} + 4 \beta_{3} - 5) q^{71} + ( - 2 \beta_{7} - 2 \beta_{4} - 2 \beta_{2} - 2 \beta_1 + 2) q^{73} + (3 \beta_{6} + 7 \beta_{5} + 4 \beta_{4} + 3 \beta_{3} - \beta_1 + 7) q^{77} + (3 \beta_{6} + \beta_{5} + 3 \beta_{4} + 1) q^{79} + ( - \beta_{7} + 7 \beta_{5} - 3 \beta_{4} - \beta_{2} + 8 \beta_1 + 1) q^{81} + (2 \beta_{6} - \beta_{2} - 5) q^{83} + (\beta_{7} - 5 \beta_{5} - \beta_{4} + \beta_{2} - 3 \beta_1 - 1) q^{87} + (\beta_{7} - 2 \beta_{6} + 4 \beta_{5} - 2 \beta_{4} + 3 \beta_{3} + 3 \beta_1 + 4) q^{89} + ( - 2 \beta_{7} - \beta_{6} + \beta_{5} + \beta_{4} + 4 \beta_{3} - 2 \beta_{2} + 3 \beta_1 + 4) q^{91} + ( - \beta_{7} - \beta_{6} - 2 \beta_{5} - \beta_{4} - 2) q^{93} + ( - \beta_{6} - 2 \beta_{3} + \beta_{2}) q^{97} + ( - 5 \beta_{3} - 2 \beta_{2} - 13) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 3 q^{3} - 3 q^{7} - 5 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 3 q^{3} - 3 q^{7} - 5 q^{9} + 8 q^{13} - 7 q^{17} + 10 q^{19} + 25 q^{21} - 24 q^{27} + 4 q^{29} + 14 q^{31} + 2 q^{33} - 2 q^{37} + 10 q^{39} - 8 q^{41} - 30 q^{43} - 15 q^{47} - 13 q^{49} - 5 q^{51} - 10 q^{53} + 38 q^{57} + q^{59} + 25 q^{61} + 20 q^{63} - 4 q^{67} - 32 q^{69} - 40 q^{71} - 2 q^{73} + 17 q^{77} - 2 q^{79} - 24 q^{81} - 52 q^{83} + 13 q^{87} + 19 q^{89} + 17 q^{91} - 8 q^{93} + 12 q^{97} - 102 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - x^{7} + 8x^{6} + 3x^{5} + 50x^{4} - 12x^{3} + 11x^{2} + 2x + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 84\nu^{7} - 31\nu^{6} + 636\nu^{5} + 456\nu^{4} + 4685\nu^{3} + 156\nu^{2} + 36\nu - 3755 ) / 1381 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 371\nu^{7} - 252\nu^{6} + 2809\nu^{5} + 2014\nu^{4} + 19196\nu^{3} + 689\nu^{2} + 159\nu + 793 ) / 4143 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 634\nu^{7} - 1056\nu^{6} + 5984\nu^{5} - 1885\nu^{4} + 34144\nu^{3} - 26048\nu^{2} + 36375\nu - 2464 ) / 4143 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 793\nu^{7} - 1164\nu^{6} + 6596\nu^{5} - 430\nu^{4} + 37636\nu^{3} - 28712\nu^{2} + 8034\nu - 2716 ) / 4143 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 2716\nu^{7} - 1923\nu^{6} + 20564\nu^{5} + 14744\nu^{4} + 135370\nu^{3} + 5044\nu^{2} + 1164\nu + 9323 ) / 4143 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 2801\nu^{7} - 4404\nu^{6} + 23575\nu^{5} - 3734\nu^{4} + 131348\nu^{3} - 111394\nu^{2} + 27834\nu + 4915 ) / 4143 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{7} - 4\beta_{5} + \beta_{3} + \beta _1 - 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{6} + 8\beta_{3} - \beta_{2} - 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -8\beta_{7} + 30\beta_{5} + \beta_{4} - 8\beta_{2} - 13\beta _1 + 8 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -13\beta_{7} + 8\beta_{6} + 43\beta_{5} + 8\beta_{4} - 66\beta_{3} - 66\beta _1 + 43 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 13\beta_{6} - 140\beta_{3} + 66\beta_{2} + 177 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 140\beta_{7} - 481\beta_{5} - 66\beta_{4} + 140\beta_{2} + 568\beta _1 - 140 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1400\mathbb{Z}\right)^\times\).

\(n\) \(351\) \(701\) \(801\) \(1177\)
\(\chi(n)\) \(1\) \(1\) \(-1 - \beta_{5}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
401.1
1.52336 + 2.63854i
0.267083 + 0.462601i
−0.132681 0.229810i
−1.15777 2.00531i
1.52336 2.63854i
0.267083 0.462601i
−0.132681 + 0.229810i
−1.15777 + 2.00531i
0 −1.02336 1.77252i 0 0 0 −2.18747 + 1.48827i 0 −0.594550 + 1.02979i 0
401.2 0 0.232917 + 0.403424i 0 0 0 −1.70312 2.02469i 0 1.39150 2.41015i 0
401.3 0 0.632681 + 1.09584i 0 0 0 1.51690 + 2.16772i 0 0.699429 1.21145i 0
401.4 0 1.65777 + 2.87134i 0 0 0 0.873699 2.49733i 0 −3.99638 + 6.92193i 0
1201.1 0 −1.02336 + 1.77252i 0 0 0 −2.18747 1.48827i 0 −0.594550 1.02979i 0
1201.2 0 0.232917 0.403424i 0 0 0 −1.70312 + 2.02469i 0 1.39150 + 2.41015i 0
1201.3 0 0.632681 1.09584i 0 0 0 1.51690 2.16772i 0 0.699429 + 1.21145i 0
1201.4 0 1.65777 2.87134i 0 0 0 0.873699 + 2.49733i 0 −3.99638 6.92193i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 401.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1400.2.q.m yes 8
5.b even 2 1 1400.2.q.l 8
5.c odd 4 2 1400.2.bh.j 16
7.c even 3 1 inner 1400.2.q.m yes 8
7.c even 3 1 9800.2.a.cj 4
7.d odd 6 1 9800.2.a.cu 4
35.i odd 6 1 9800.2.a.ck 4
35.j even 6 1 1400.2.q.l 8
35.j even 6 1 9800.2.a.ct 4
35.l odd 12 2 1400.2.bh.j 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1400.2.q.l 8 5.b even 2 1
1400.2.q.l 8 35.j even 6 1
1400.2.q.m yes 8 1.a even 1 1 trivial
1400.2.q.m yes 8 7.c even 3 1 inner
1400.2.bh.j 16 5.c odd 4 2
1400.2.bh.j 16 35.l odd 12 2
9800.2.a.cj 4 7.c even 3 1
9800.2.a.ck 4 35.i odd 6 1
9800.2.a.ct 4 35.j even 6 1
9800.2.a.cu 4 7.d odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1400, [\chi])\):

\( T_{3}^{8} - 3T_{3}^{7} + 13T_{3}^{6} - 10T_{3}^{5} + 53T_{3}^{4} - 68T_{3}^{3} + 105T_{3}^{2} - 44T_{3} + 16 \) Copy content Toggle raw display
\( T_{11}^{8} + 37T_{11}^{6} - 98T_{11}^{5} + 1235T_{11}^{4} - 1813T_{11}^{3} + 7359T_{11}^{2} + 6566T_{11} + 17956 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} - 3 T^{7} + 13 T^{6} - 10 T^{5} + \cdots + 16 \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( T^{8} + 3 T^{7} + 11 T^{6} + \cdots + 2401 \) Copy content Toggle raw display
$11$ \( T^{8} + 37 T^{6} - 98 T^{5} + \cdots + 17956 \) Copy content Toggle raw display
$13$ \( (T^{4} - 4 T^{3} - 23 T^{2} + 105 T - 84)^{2} \) Copy content Toggle raw display
$17$ \( T^{8} + 7 T^{7} + 42 T^{6} + 85 T^{5} + \cdots + 25 \) Copy content Toggle raw display
$19$ \( T^{8} - 10 T^{7} + 113 T^{6} + \cdots + 678976 \) Copy content Toggle raw display
$23$ \( T^{8} + 79 T^{6} + 182 T^{5} + \cdots + 908209 \) Copy content Toggle raw display
$29$ \( (T^{4} - 2 T^{3} - 49 T^{2} + 219 T - 222)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} - 14 T^{7} + 147 T^{6} + \cdots + 625 \) Copy content Toggle raw display
$37$ \( T^{8} + 2 T^{7} + 47 T^{6} + \cdots + 8464 \) Copy content Toggle raw display
$41$ \( (T^{4} + 4 T^{3} - 127 T^{2} - 125 T + 3319)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} + 15 T^{3} + 4 T^{2} - 621 T - 1956)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + 15 T^{7} + 224 T^{6} + \cdots + 9090225 \) Copy content Toggle raw display
$53$ \( T^{8} + 10 T^{7} + 71 T^{6} + \cdots + 1024 \) Copy content Toggle raw display
$59$ \( T^{8} - T^{7} + 67 T^{6} - 88 T^{5} + \cdots + 4900 \) Copy content Toggle raw display
$61$ \( T^{8} - 25 T^{7} + 461 T^{6} + \cdots + 4536900 \) Copy content Toggle raw display
$67$ \( T^{8} + 4 T^{7} + 188 T^{6} + \cdots + 2166784 \) Copy content Toggle raw display
$71$ \( (T^{4} + 20 T^{3} + 34 T^{2} - 732 T - 1035)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} + 2 T^{7} + 188 T^{6} + \cdots + 1327104 \) Copy content Toggle raw display
$79$ \( T^{8} + 2 T^{7} + 79 T^{6} + \cdots + 22201 \) Copy content Toggle raw display
$83$ \( (T^{4} + 26 T^{3} + 187 T^{2} - 15 T - 2826)^{2} \) Copy content Toggle raw display
$89$ \( T^{8} - 19 T^{7} + 308 T^{6} + \cdots + 2064969 \) Copy content Toggle raw display
$97$ \( (T^{4} - 6 T^{3} - 57 T^{2} + 81 T - 27)^{2} \) Copy content Toggle raw display
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