Properties

Label 300.2.e.e
Level $300$
Weight $2$
Character orbit 300.e
Analytic conductor $2.396$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [300,2,Mod(251,300)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(300, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 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("300.251");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 300 = 2^{2} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 300.e (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: \(2.39551206064\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.4521217600.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + x^{6} - 2x^{4} + 4x^{2} + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{6} q^{2} + \beta_{3} q^{3} + ( - \beta_{3} - \beta_{2} + \beta_1) q^{4} + ( - \beta_{4} + \beta_1 + 1) q^{6} + (\beta_{7} - \beta_{4} + \beta_1 + 1) q^{7} + ( - \beta_{6} + \beta_{5} + \beta_{3} - \beta_{2}) q^{8} + (\beta_{6} - \beta_{5} + \beta_{2} - \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{6} q^{2} + \beta_{3} q^{3} + ( - \beta_{3} - \beta_{2} + \beta_1) q^{4} + ( - \beta_{4} + \beta_1 + 1) q^{6} + (\beta_{7} - \beta_{4} + \beta_1 + 1) q^{7} + ( - \beta_{6} + \beta_{5} + \beta_{3} - \beta_{2}) q^{8} + (\beta_{6} - \beta_{5} + \beta_{2} - \beta_1) q^{9} + ( - \beta_{7} + \beta_{6} + \beta_{5} - \beta_{4}) q^{11} + ( - \beta_{7} + \beta_{5} + 1) q^{12} + (\beta_{3} + \beta_{2} - 2 \beta_1 - 1) q^{13} + (\beta_{5} - \beta_{3} + \beta_{2}) q^{14} + (\beta_{7} - \beta_{4} + 2) q^{16} + ( - \beta_{7} - \beta_{6} - \beta_{5} - \beta_{4}) q^{17} + (\beta_{6} - \beta_{5} + \beta_{4} - \beta_{3} + \beta_{2} + \beta_1 - 1) q^{18} + (\beta_{7} - \beta_{4} + \beta_{3} + \beta_{2} + \beta_1 + 1) q^{19} + (\beta_{7} + 3 \beta_{6} - \beta_{5} + \beta_{4} - 2 \beta_{3} + 2 \beta_1 + 1) q^{21} + ( - \beta_{7} + \beta_{4} - 2 \beta_{3} - 2 \beta_{2} + 2) q^{22} + (\beta_{7} - \beta_{6} - \beta_{5} + \beta_{4} - \beta_{3} + \beta_{2}) q^{23} + ( - \beta_{7} + \beta_{6} - 2 \beta_{2} - \beta_1 + 1) q^{24} + (\beta_{7} + \beta_{6} - 2 \beta_{5} + \beta_{4} - 2 \beta_{3} + 2 \beta_{2}) q^{26} + (2 \beta_{7} - \beta_{6} - \beta_{5} + 2 \beta_{2} + \beta_1 + 1) q^{27} + ( - \beta_{7} + \beta_{4} - \beta_{3} - \beta_{2} - 3 \beta_1 - 2) q^{28} + ( - \beta_{7} - 3 \beta_{6} + \beta_{5} - \beta_{4} + \beta_{3} - \beta_{2}) q^{29} + ( - \beta_{7} + \beta_{4} - 2 \beta_{3} - 2 \beta_{2} - \beta_1 - 1) q^{31} + (\beta_{7} + 2 \beta_{6} + \beta_{4} - 2 \beta_{3} + 2 \beta_{2}) q^{32} + ( - \beta_{7} - 2 \beta_{6} - \beta_{4} - \beta_{3} - 2 \beta_{2} + 3 \beta_1 - 1) q^{33} + ( - \beta_{7} + \beta_{4} + 2 \beta_{3} + 2 \beta_{2} + 2) q^{34} + ( - 2 \beta_{7} - 2 \beta_{6} + \beta_{5} - \beta_{4} + 2 \beta_{3} - 2 \beta_{2} - 3) q^{36} + (2 \beta_{3} + 2 \beta_{2} - 4 \beta_1 - 2) q^{37} + ( - \beta_{7} + \beta_{5} - \beta_{4} - \beta_{3} + \beta_{2}) q^{38} + (2 \beta_{7} - \beta_{6} - \beta_{5} - \beta_{3} - \beta_{2} + \beta_1 + 1) q^{39} + ( - \beta_{7} - 5 \beta_{6} + 3 \beta_{5} - \beta_{4} + 2 \beta_{3} - 2 \beta_{2}) q^{41} + ( - \beta_{7} - \beta_{6} + 2 \beta_{5} - \beta_{4} - 4 \beta_{2} + 2 \beta_1 - 4) q^{42} + (\beta_{7} - \beta_{4} - 2 \beta_{3} - 2 \beta_{2} + \beta_1 + 1) q^{43} + (\beta_{7} + 2 \beta_{6} + \beta_{4} + 2 \beta_{3} - 2 \beta_{2}) q^{44} + (2 \beta_{3} + 2 \beta_{2} - 2 \beta_1 - 4) q^{46} + (2 \beta_{3} - 2 \beta_{2}) q^{47} + (2 \beta_{7} + 2 \beta_{6} - \beta_{5} + \beta_{4} - \beta_{3} - \beta_{2} + 3 \beta_1 + 3) q^{48} + ( - \beta_{3} - \beta_{2} + 2 \beta_1 - 2) q^{49} + (\beta_{7} - 2 \beta_{6} - 2 \beta_{5} + 3 \beta_{4} - \beta_{3} + 4 \beta_{2} - \beta_1 - 1) q^{51} + ( - \beta_{7} + \beta_{4} + \beta_{3} + \beta_{2} - \beta_1 - 6) q^{52} + (\beta_{7} + 3 \beta_{6} - \beta_{5} + \beta_{4} - \beta_{3} + \beta_{2}) q^{53} + ( - \beta_{7} + \beta_{5} - \beta_{4} + \beta_{3} + 3 \beta_{2} - 2 \beta_1 - 4) q^{54} + (3 \beta_{7} + \beta_{6} - 3 \beta_{5} + 3 \beta_{4} - \beta_{3} + \beta_{2}) q^{56} + (\beta_{7} + 4 \beta_{6} - 2 \beta_{5} + \beta_{4} - 2 \beta_{3} + \beta_{2} + \beta_1 - 2) q^{57} + (2 \beta_{3} + 2 \beta_{2} - 2 \beta_1 + 4) q^{58} + (\beta_{3} + \beta_{2} - 2 \beta_1 - 3) q^{61} + (2 \beta_{7} - \beta_{5} + 2 \beta_{4} + \beta_{3} - \beta_{2}) q^{62} + ( - \beta_{7} + 2 \beta_{6} + 2 \beta_{5} - 3 \beta_{4} + 2 \beta_{3} + 2 \beta_{2} + \cdots + 1) q^{63}+ \cdots + ( - 3 \beta_{7} + 3 \beta_{4} - 2 \beta_{3} + 3 \beta_{2} - 3 \beta_1 - 3) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 2 q^{4} + 3 q^{6} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 2 q^{4} + 3 q^{6} + 2 q^{9} + 11 q^{12} - 4 q^{13} + 10 q^{16} - 7 q^{18} + 4 q^{21} + 22 q^{22} + 13 q^{24} - 4 q^{28} - 14 q^{33} + 22 q^{34} - 21 q^{36} - 8 q^{37} - 36 q^{42} - 28 q^{46} + 15 q^{48} - 20 q^{49} - 40 q^{52} - 28 q^{54} - 18 q^{57} + 36 q^{58} - 20 q^{61} - 38 q^{64} + 29 q^{66} - 12 q^{69} + 51 q^{72} + 36 q^{73} + 18 q^{76} - 22 q^{78} - 30 q^{81} + 50 q^{82} + 40 q^{84} - 14 q^{88} + 40 q^{93} + 12 q^{94} - 39 q^{96} + 12 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( \nu^{2} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{7} + 2\nu^{6} - 3\nu^{5} + 2\nu^{4} + 2\nu^{3} + 4\nu^{2} + 12\nu + 8 ) / 16 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{7} + 2\nu^{6} + 3\nu^{5} + 2\nu^{4} - 2\nu^{3} + 4\nu^{2} - 12\nu + 8 ) / 16 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{7} - 2\nu^{6} + 3\nu^{5} + 6\nu^{4} + 6\nu^{3} + 12\nu^{2} + 4\nu - 8 ) / 16 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{7} + \nu^{5} + 6\nu^{3} + 4\nu ) / 8 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{7} + \nu^{5} - 2\nu^{3} + 4\nu ) / 8 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -\nu^{7} + 2\nu^{6} + 3\nu^{5} - 6\nu^{4} + 6\nu^{3} - 12\nu^{2} + 4\nu + 8 ) / 16 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{7} + \beta_{6} - \beta_{5} + \beta_{4} - \beta_{3} + \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{6} + \beta_{5} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{7} + \beta_{4} + \beta_{3} + \beta_{2} - 2\beta_1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{7} + 2\beta_{6} + \beta_{4} + \beta_{3} - \beta_{2} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( \beta_{7} - \beta_{4} + 3\beta_{3} + 3\beta_{2} - 4 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -3\beta_{7} + 2\beta_{6} + 4\beta_{5} - 3\beta_{4} + \beta_{3} - \beta_{2} \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/300\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(151\) \(277\)
\(\chi(n)\) \(-1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
251.1
1.29437 0.569745i
1.29437 + 0.569745i
0.273147 1.38758i
0.273147 + 1.38758i
−0.273147 1.38758i
−0.273147 + 1.38758i
−1.29437 0.569745i
−1.29437 + 0.569745i
−1.29437 0.569745i −0.908080 1.47492i 1.35078 + 1.47492i 0 0.335062 + 2.42647i 2.50967i −0.908080 2.67869i −1.35078 + 2.67869i 0
251.2 −1.29437 + 0.569745i −0.908080 + 1.47492i 1.35078 1.47492i 0 0.335062 2.42647i 2.50967i −0.908080 + 2.67869i −1.35078 2.67869i 0
251.3 −0.273147 1.38758i 1.55737 0.758030i −1.85078 + 0.758030i 0 −1.47722 1.95392i 3.56393i 1.55737 + 2.36106i 1.85078 2.36106i 0
251.4 −0.273147 + 1.38758i 1.55737 + 0.758030i −1.85078 0.758030i 0 −1.47722 + 1.95392i 3.56393i 1.55737 2.36106i 1.85078 + 2.36106i 0
251.5 0.273147 1.38758i −1.55737 + 0.758030i −1.85078 0.758030i 0 0.626440 + 2.36803i 3.56393i −1.55737 + 2.36106i 1.85078 2.36106i 0
251.6 0.273147 + 1.38758i −1.55737 0.758030i −1.85078 + 0.758030i 0 0.626440 2.36803i 3.56393i −1.55737 2.36106i 1.85078 + 2.36106i 0
251.7 1.29437 0.569745i 0.908080 + 1.47492i 1.35078 1.47492i 0 2.01572 + 1.39172i 2.50967i 0.908080 2.67869i −1.35078 + 2.67869i 0
251.8 1.29437 + 0.569745i 0.908080 1.47492i 1.35078 + 1.47492i 0 2.01572 1.39172i 2.50967i 0.908080 + 2.67869i −1.35078 2.67869i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 251.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
4.b odd 2 1 inner
12.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 300.2.e.e yes 8
3.b odd 2 1 inner 300.2.e.e yes 8
4.b odd 2 1 inner 300.2.e.e yes 8
5.b even 2 1 300.2.e.d 8
5.c odd 4 2 300.2.h.c 16
12.b even 2 1 inner 300.2.e.e yes 8
15.d odd 2 1 300.2.e.d 8
15.e even 4 2 300.2.h.c 16
20.d odd 2 1 300.2.e.d 8
20.e even 4 2 300.2.h.c 16
60.h even 2 1 300.2.e.d 8
60.l odd 4 2 300.2.h.c 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
300.2.e.d 8 5.b even 2 1
300.2.e.d 8 15.d odd 2 1
300.2.e.d 8 20.d odd 2 1
300.2.e.d 8 60.h even 2 1
300.2.e.e yes 8 1.a even 1 1 trivial
300.2.e.e yes 8 3.b odd 2 1 inner
300.2.e.e yes 8 4.b odd 2 1 inner
300.2.e.e yes 8 12.b even 2 1 inner
300.2.h.c 16 5.c odd 4 2
300.2.h.c 16 15.e even 4 2
300.2.h.c 16 20.e even 4 2
300.2.h.c 16 60.l odd 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(300, [\chi])\):

\( T_{7}^{4} + 19T_{7}^{2} + 80 \) Copy content Toggle raw display
\( T_{13}^{2} + T_{13} - 10 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} + T^{6} - 2 T^{4} + 4 T^{2} + \cdots + 16 \) Copy content Toggle raw display
$3$ \( T^{8} - T^{6} + 8 T^{4} - 9 T^{2} + \cdots + 81 \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( (T^{4} + 19 T^{2} + 80)^{2} \) Copy content Toggle raw display
$11$ \( (T^{4} - 29 T^{2} + 200)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + T - 10)^{4} \) Copy content Toggle raw display
$17$ \( (T^{4} + 59 T^{2} + 40)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} + 26 T^{2} + 5)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} - 28 T^{2} + 32)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} + 36 T^{2} + 160)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} + 55 T^{2} + 500)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 2 T - 40)^{4} \) Copy content Toggle raw display
$41$ \( (T^{4} + 115 T^{2} + 1000)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} + 71 T^{2} + 20)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} - 52 T^{2} + 512)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} + 36 T^{2} + 160)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} \) Copy content Toggle raw display
$61$ \( (T^{2} + 5 T - 4)^{4} \) Copy content Toggle raw display
$67$ \( (T^{4} + 34 T^{2} + 125)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} - 284 T^{2} + 20000)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} - 9 T + 10)^{4} \) Copy content Toggle raw display
$79$ \( (T^{4} + 76 T^{2} + 1280)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} - 13 T^{2} + 32)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} + 51 T^{2} + 640)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} - 3 T - 90)^{4} \) Copy content Toggle raw display
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