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Properties

Label 407.2.q.a
Level $407$
Weight $2$
Character orbit 407.q
Analytic conductor $3.250$
Analytic rank $0$
Dimension $8$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [407,2,Mod(208,407)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(407, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([6, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("407.208");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 407 = 11 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 407.q (of order \(12\), degree \(4\), 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: \(3.24991136227\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{12})\)
Coefficient field: 8.0.22581504.2
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 4x^{7} + 5x^{6} + 2x^{5} - 11x^{4} + 4x^{3} + 20x^{2} - 32x + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-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.

\(f(q)\) \(=\) \( q + (\beta_{7} + \beta_{6} + \beta_{4} + \cdots - 1) q^{2}+ \cdots + ( - 2 \beta_{3} - 2 \beta_1 + 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{7} + \beta_{6} + \beta_{4} + \cdots - 1) q^{2}+ \cdots + (4 \beta_{7} + 2 \beta_{6} - 4 \beta_{4} + \cdots + 6) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 6 q^{3} - 6 q^{4} - 2 q^{5} - 6 q^{6} + 12 q^{8} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 6 q^{3} - 6 q^{4} - 2 q^{5} - 6 q^{6} + 12 q^{8} + 4 q^{9} - 4 q^{11} - 10 q^{12} - 18 q^{13} - 6 q^{14} + 16 q^{15} + 10 q^{16} - 4 q^{17} + 12 q^{18} - 8 q^{19} - 40 q^{20} - 2 q^{21} - 18 q^{22} + 4 q^{23} + 18 q^{24} - 36 q^{26} - 4 q^{28} - 4 q^{29} - 18 q^{30} - 12 q^{31} - 54 q^{32} - 16 q^{33} + 12 q^{34} - 2 q^{35} + 48 q^{37} - 12 q^{38} + 12 q^{39} + 42 q^{40} + 4 q^{41} + 6 q^{42} + 12 q^{43} + 34 q^{44} - 28 q^{45} - 6 q^{46} + 16 q^{47} - 24 q^{49} + 30 q^{50} - 8 q^{51} + 6 q^{52} + 12 q^{53} + 6 q^{54} + 12 q^{55} + 12 q^{56} - 22 q^{57} + 42 q^{58} - 8 q^{59} - 2 q^{60} + 16 q^{61} + 48 q^{65} + 42 q^{66} - 12 q^{67} - 10 q^{68} - 26 q^{69} - 6 q^{70} - 30 q^{71} - 72 q^{72} - 48 q^{73} - 6 q^{74} - 48 q^{75} + 28 q^{76} + 14 q^{77} + 36 q^{78} - 32 q^{79} + 38 q^{80} + 8 q^{81} - 42 q^{82} - 18 q^{83} + 44 q^{85} + 42 q^{86} + 20 q^{87} + 12 q^{88} + 6 q^{89} + 72 q^{90} + 18 q^{91} - 4 q^{92} + 18 q^{93} + 24 q^{94} + 46 q^{95} + 24 q^{96} + 36 q^{97} + 36 q^{98} + 28 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 4x^{7} + 5x^{6} + 2x^{5} - 11x^{4} + 4x^{3} + 20x^{2} - 32x + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{7} - 2\nu^{6} + \nu^{5} + 4\nu^{4} - 3\nu^{3} - 2\nu^{2} + 8\nu - 8 ) / 8 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{7} + 2\nu^{6} - \nu^{5} - 4\nu^{4} + 3\nu^{3} + 10\nu^{2} - 16\nu + 8 ) / 8 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{7} - 3\nu^{6} + 3\nu^{5} + 3\nu^{4} - 7\nu^{3} - 3\nu^{2} + 18\nu - 16 ) / 4 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 2\nu^{7} - 5\nu^{6} + 2\nu^{5} + 7\nu^{4} - 8\nu^{3} - 9\nu^{2} + 28\nu - 20 ) / 4 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -3\nu^{7} + 7\nu^{6} - 3\nu^{5} - 11\nu^{4} + 15\nu^{3} + 11\nu^{2} - 40\nu + 32 ) / 4 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 7\nu^{7} - 20\nu^{6} + 11\nu^{5} + 30\nu^{4} - 45\nu^{3} - 28\nu^{2} + 116\nu - 88 ) / 8 \) Copy content Toggle raw display

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

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

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/407\mathbb{Z}\right)^\times\).

\(n\) \(112\) \(298\)
\(\chi(n)\) \(-1\) \(\beta_{7}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
208.1
1.40994 + 0.109843i
−1.27597 − 0.609843i
1.20036 + 0.747754i
0.665665 − 1.24775i
1.40994 − 0.109843i
−1.27597 + 0.609843i
1.20036 − 0.747754i
0.665665 + 1.24775i
−0.909941 + 0.243818i 1.38581 − 0.800098i −0.963505 + 0.556280i −0.946436 + 3.53215i −1.06593 + 1.06593i −0.866025 + 0.500000i 2.07335 − 2.07335i −0.219687 + 0.380509i − 3.44481i
208.2 1.77597 − 0.475869i −2.01978 + 1.16612i 1.19556 − 0.690254i −0.419589 + 1.56593i −3.03215 + 3.03215i −0.866025 + 0.500000i −0.805399 + 0.805399i 1.21969 − 2.11256i 2.98070i
230.1 −0.700360 + 2.61378i 0.0820885 + 0.0473938i −4.60929 − 2.66117i 2.90893 − 0.779445i −0.181368 + 0.181368i 0.866025 + 0.500000i 6.35704 − 6.35704i −1.49551 − 2.59030i 8.14918i
230.2 −0.165665 + 0.618272i −2.44811 − 1.41342i 1.37724 + 0.795148i −2.54290 + 0.681368i 1.27944 − 1.27944i 0.866025 + 0.500000i −1.62499 + 1.62499i 2.49551 + 4.32235i − 1.68508i
362.1 −0.909941 − 0.243818i 1.38581 + 0.800098i −0.963505 − 0.556280i −0.946436 − 3.53215i −1.06593 − 1.06593i −0.866025 − 0.500000i 2.07335 + 2.07335i −0.219687 − 0.380509i 3.44481i
362.2 1.77597 + 0.475869i −2.01978 − 1.16612i 1.19556 + 0.690254i −0.419589 − 1.56593i −3.03215 − 3.03215i −0.866025 − 0.500000i −0.805399 − 0.805399i 1.21969 + 2.11256i − 2.98070i
384.1 −0.700360 − 2.61378i 0.0820885 − 0.0473938i −4.60929 + 2.66117i 2.90893 + 0.779445i −0.181368 − 0.181368i 0.866025 − 0.500000i 6.35704 + 6.35704i −1.49551 + 2.59030i − 8.14918i
384.2 −0.165665 − 0.618272i −2.44811 + 1.41342i 1.37724 − 0.795148i −2.54290 − 0.681368i 1.27944 + 1.27944i 0.866025 − 0.500000i −1.62499 − 1.62499i 2.49551 − 4.32235i 1.68508i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 208.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
407.q even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 407.2.q.a ✓ 8
11.b odd 2 1 407.2.q.b yes 8
37.g odd 12 1 407.2.q.b yes 8
407.q even 12 1 inner 407.2.q.a ✓ 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
407.2.q.a ✓ 8 1.a even 1 1 trivial
407.2.q.a ✓ 8 407.q even 12 1 inner
407.2.q.b yes 8 11.b odd 2 1
407.2.q.b yes 8 37.g odd 12 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{8} + 3T_{2}^{6} - 12T_{2}^{5} - 12T_{2}^{4} + 18T_{2}^{3} + 27T_{2}^{2} + 18T_{2} + 9 \) acting on \(S_{2}^{\mathrm{new}}(407, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} + 3 T^{6} + \cdots + 9 \) Copy content Toggle raw display
$3$ \( T^{8} + 6 T^{7} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( T^{8} + 2 T^{7} + \cdots + 2209 \) Copy content Toggle raw display
$7$ \( (T^{4} - T^{2} + 1)^{2} \) Copy content Toggle raw display
$11$ \( T^{8} + 4 T^{7} + \cdots + 14641 \) Copy content Toggle raw display
$13$ \( T^{8} + 18 T^{7} + \cdots + 178929 \) Copy content Toggle raw display
$17$ \( T^{8} + 4 T^{7} + \cdots + 52441 \) Copy content Toggle raw display
$19$ \( T^{8} + 8 T^{7} + \cdots + 169 \) Copy content Toggle raw display
$23$ \( T^{8} - 4 T^{7} + \cdots + 929296 \) Copy content Toggle raw display
$29$ \( T^{8} + 4 T^{7} + \cdots + 839056 \) Copy content Toggle raw display
$31$ \( T^{8} + 12 T^{7} + \cdots + 24336 \) Copy content Toggle raw display
$37$ \( (T^{2} - 12 T + 37)^{4} \) Copy content Toggle raw display
$41$ \( T^{8} - 4 T^{7} + \cdots + 4068289 \) Copy content Toggle raw display
$43$ \( T^{8} - 12 T^{7} + \cdots + 2862864 \) Copy content Toggle raw display
$47$ \( (T^{4} - 8 T^{3} + \cdots - 368)^{2} \) Copy content Toggle raw display
$53$ \( T^{8} - 12 T^{7} + \cdots + 19881 \) Copy content Toggle raw display
$59$ \( T^{8} + 8 T^{7} + \cdots + 687241 \) Copy content Toggle raw display
$61$ \( T^{8} - 16 T^{7} + \cdots + 744769 \) Copy content Toggle raw display
$67$ \( T^{8} + 12 T^{7} + \cdots + 167281 \) Copy content Toggle raw display
$71$ \( T^{8} + 30 T^{7} + \cdots + 471969 \) Copy content Toggle raw display
$73$ \( (T^{4} + 24 T^{3} + \cdots + 564)^{2} \) Copy content Toggle raw display
$79$ \( T^{8} + 32 T^{7} + \cdots + 1 \) Copy content Toggle raw display
$83$ \( T^{8} + 18 T^{7} + \cdots + 23648769 \) Copy content Toggle raw display
$89$ \( T^{8} - 6 T^{7} + \cdots + 12981609 \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots + 2217279744 \) Copy content Toggle raw display
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