Properties

Label 300.2.h.a
Level $300$
Weight $2$
Character orbit 300.h
Analytic conductor $2.396$
Analytic rank $0$
Dimension $8$
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] = [300,2,Mod(299,300)]
 
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, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("300.299");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 300 = 2^{2} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 300.h (of order \(2\), degree \(1\), not 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: 8.0.342102016.5
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + x^{6} + 4x^{4} + 4x^{2} + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 60)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} - \beta_{4} q^{3} + ( - \beta_{4} + \beta_{2} - \beta_1) q^{4} + ( - \beta_{6} + \beta_{2} - 1) q^{6} + (\beta_{7} - \beta_{5}) q^{7} + ( - \beta_{6} + \beta_{5} + \beta_1 - 2) q^{8} + ( - \beta_{6} - \beta_{3} - \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} - \beta_{4} q^{3} + ( - \beta_{4} + \beta_{2} - \beta_1) q^{4} + ( - \beta_{6} + \beta_{2} - 1) q^{6} + (\beta_{7} - \beta_{5}) q^{7} + ( - \beta_{6} + \beta_{5} + \beta_1 - 2) q^{8} + ( - \beta_{6} - \beta_{3} - \beta_1) q^{9} + ( - \beta_{7} + \beta_{5} - \beta_{4} - \beta_{2}) q^{11} + ( - \beta_{3} - \beta_{2} - \beta_1 - 2) q^{12} + (\beta_{6} + \beta_{5} - \beta_{4} + 2 \beta_{3} + \beta_{2}) q^{13} + ( - \beta_{6} - \beta_{5} - \beta_{3} - \beta_{2}) q^{14} + (\beta_{6} - \beta_{5} - 3 \beta_1 - 2) q^{16} + 2 q^{17} + ( - 2 \beta_{7} - \beta_{6} + \beta_{5} + \beta_{4} - \beta_{2} + \beta_1 - 2) q^{18} + (\beta_{6} - \beta_{5} - \beta_{4} + \beta_{2} - 2 \beta_1) q^{19} + ( - \beta_{7} - \beta_{6} - \beta_{5} + \beta_{4} - \beta_{3} - \beta_{2} + \beta_1 + 1) q^{21} + (\beta_{4} + \beta_{3} + 2 \beta_{2}) q^{22} + (2 \beta_{6} - 2 \beta_{5} - \beta_{4} + \beta_{2} - 4 \beta_1) q^{23} + ( - 2 \beta_{7} - \beta_{6} + \beta_{5} + \beta_{4} + \beta_{3} - \beta_1) q^{24} + (4 \beta_{7} + 2 \beta_{6} - 2 \beta_{5}) q^{26} + ( - 2 \beta_{7} + \beta_{6} + \beta_{5} - \beta_{4} - 2 \beta_1) q^{27} + ( - 2 \beta_{7} - 2 \beta_{6} + \beta_{4} - \beta_{3}) q^{28} + ( - \beta_{7} + \beta_{5} - \beta_{4} + 2 \beta_{3} + \beta_{2}) q^{29} + ( - \beta_{6} + \beta_{5} - \beta_{4} + \beta_{2} + 2 \beta_1) q^{31} + ( - \beta_{6} + \beta_{5} + 2 \beta_{4} - 2 \beta_{2} + \beta_1 + 2) q^{32} + (\beta_{7} + \beta_{5} - \beta_{4} + \beta_{2} - 2 \beta_1 + 2) q^{33} + 2 \beta_1 q^{34} + (2 \beta_{6} + \beta_{4} + \beta_{2} - 3 \beta_1 + 2) q^{36} + (\beta_{6} + \beta_{5} - \beta_{4} + 2 \beta_{3} + \beta_{2}) q^{37} + ( - 2 \beta_{6} + 2 \beta_{5} + 2 \beta_1) q^{38} + (2 \beta_{7} - \beta_{6} - \beta_{5} + \beta_{4} - 3 \beta_{2} + 2 \beta_1) q^{39} + ( - 2 \beta_{7} - \beta_{6} + \beta_{5} - \beta_{4} + 2 \beta_{3} + \beta_{2}) q^{41} + ( - 2 \beta_{7} - \beta_{6} + \beta_{5} - \beta_{3} + \beta_{2} + 2) q^{42} + (2 \beta_{7} - 2 \beta_{5} - \beta_{4} - \beta_{2}) q^{43} + (2 \beta_{7} + 2 \beta_{6} - \beta_{4} - \beta_{3} - 2 \beta_{2}) q^{44} + ( - 3 \beta_{6} + 3 \beta_{5} + \beta_{4} - \beta_{2} + 4 \beta_1 + 2) q^{46} + ( - \beta_{4} + \beta_{2}) q^{47} + (2 \beta_{7} + 3 \beta_{6} - \beta_{5} + 3 \beta_{4} - \beta_{3} - 2 \beta_{2} + \beta_1 + 2) q^{48} + (\beta_{6} - \beta_{5} + \beta_{4} - \beta_{2} + 2 \beta_1 - 1) q^{49} - 2 \beta_{4} q^{51} + ( - 2 \beta_{6} - 2 \beta_{5} - 2 \beta_{4} - 2 \beta_{2}) q^{52} + (2 \beta_{6} - 2 \beta_{5} + 2 \beta_{4} - 2 \beta_{2} + 4 \beta_1 + 6) q^{53} + (3 \beta_{5} + \beta_{4} + 2 \beta_{3} + 2 \beta_{2} + 2 \beta_1 + 1) q^{54} + ( - 2 \beta_{7} + 2 \beta_{5} + \beta_{4} - \beta_{3}) q^{56} + (2 \beta_{7} + \beta_{6} - \beta_{5} + \beta_{4} - 2 \beta_{3} - \beta_{2} - 2) q^{57} + (4 \beta_{7} + 2 \beta_{6} - 2 \beta_{5} + \beta_{4} - \beta_{3}) q^{58} + ( - 3 \beta_{7} + 3 \beta_{5} + 3 \beta_{4} + 3 \beta_{2}) q^{59} + (\beta_{6} - \beta_{5} + \beta_{4} - \beta_{2} + 2 \beta_1) q^{61} + ( - 2 \beta_{4} + 2 \beta_{2} - 2 \beta_1 - 4) q^{62} + ( - \beta_{7} - \beta_{6} + 2 \beta_{5} - \beta_{4} + 3 \beta_{2} + 2 \beta_1) q^{63} + (3 \beta_{6} - 3 \beta_{5} + 2 \beta_{4} - 2 \beta_{2} + \beta_1 + 2) q^{64} + (\beta_{4} + \beta_{3} - 2 \beta_{2} + 4 \beta_1 - 4) q^{66} + (2 \beta_{7} - 2 \beta_{5} + \beta_{4} + \beta_{2}) q^{67} + ( - 2 \beta_{4} + 2 \beta_{2} - 2 \beta_1) q^{68} + (4 \beta_{7} + 3 \beta_{6} - 2 \beta_{5} + 2 \beta_{4} - 3 \beta_{3} - 2 \beta_{2} + \cdots - 1) q^{69}+ \cdots + ( - \beta_{7} + 2 \beta_{6} - \beta_{5} - 3 \beta_{4} - 3 \beta_{2} - 4 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 2 q^{2} + 2 q^{4} - 6 q^{6} - 14 q^{8} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 2 q^{2} + 2 q^{4} - 6 q^{6} - 14 q^{8} + 4 q^{9} - 14 q^{12} - 14 q^{16} + 16 q^{17} - 18 q^{18} + 4 q^{21} + 2 q^{24} + 18 q^{32} + 24 q^{33} - 4 q^{34} + 18 q^{36} + 4 q^{38} + 16 q^{42} + 20 q^{46} + 10 q^{48} - 16 q^{49} + 32 q^{53} + 10 q^{54} - 16 q^{57} - 8 q^{61} - 28 q^{62} + 2 q^{64} - 40 q^{66} + 4 q^{68} - 12 q^{69} + 10 q^{72} - 36 q^{76} - 32 q^{77} + 8 q^{78} - 16 q^{84} - 44 q^{92} - 24 q^{93} - 12 q^{94} + 42 q^{96} + 38 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( ( -\nu^{6} - 3\nu^{4} - 2\nu^{2} - 8 ) / 8 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{7} - \nu^{6} - \nu^{5} - 3\nu^{4} + 6\nu^{3} - 10\nu^{2} - 8\nu - 8 ) / 16 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{7} + \nu^{6} - \nu^{5} + 3\nu^{4} + 6\nu^{3} + 10\nu^{2} + 24\nu + 8 ) / 16 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{7} + \nu^{6} - \nu^{5} + 3\nu^{4} + 6\nu^{3} + 10\nu^{2} - 8\nu + 8 ) / 16 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 3\nu^{6} + 4\nu^{5} + \nu^{4} + 4\nu^{3} + 6\nu^{2} + 8 ) / 16 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -3\nu^{6} + 4\nu^{5} - \nu^{4} + 4\nu^{3} - 6\nu^{2} - 8 ) / 16 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -2\nu^{7} + 3\nu^{6} - 2\nu^{5} + \nu^{4} + 6\nu^{2} + 8 ) / 16 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -\beta_{4} + \beta_{3} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{4} - \beta_{2} + \beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 2\beta_{7} + 2\beta_{6} + \beta_{4} + \beta_{3} + 2\beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{6} - \beta_{5} - 3\beta _1 - 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -2\beta_{7} + 2\beta_{6} + 4\beta_{5} - \beta_{4} - \beta_{3} - 2\beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -3\beta_{6} + 3\beta_{5} - 2\beta_{4} + 2\beta_{2} - \beta _1 - 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -14\beta_{7} - 10\beta_{6} + 4\beta_{5} + \beta_{4} + \beta_{3} + 2\beta_{2} ) / 2 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
299.1
−0.599676 + 1.28078i
0.599676 1.28078i
−0.599676 1.28078i
0.599676 + 1.28078i
−1.17915 0.780776i
1.17915 + 0.780776i
−1.17915 + 0.780776i
1.17915 0.780776i
−1.28078 0.599676i −1.66757 + 0.468213i 1.28078 + 1.53610i 0 2.41656 + 0.400324i 0.936426 −0.719224 2.73546i 2.56155 1.56155i 0
299.2 −1.28078 0.599676i 1.66757 + 0.468213i 1.28078 + 1.53610i 0 −1.85500 1.59968i −0.936426 −0.719224 2.73546i 2.56155 + 1.56155i 0
299.3 −1.28078 + 0.599676i −1.66757 0.468213i 1.28078 1.53610i 0 2.41656 0.400324i 0.936426 −0.719224 + 2.73546i 2.56155 + 1.56155i 0
299.4 −1.28078 + 0.599676i 1.66757 0.468213i 1.28078 1.53610i 0 −1.85500 + 1.59968i −0.936426 −0.719224 + 2.73546i 2.56155 1.56155i 0
299.5 0.780776 1.17915i −0.848071 1.51022i −0.780776 1.84130i 0 −2.44293 0.179147i −3.02045 −2.78078 0.516994i −1.56155 + 2.56155i 0
299.6 0.780776 1.17915i 0.848071 1.51022i −0.780776 1.84130i 0 −1.11862 2.17915i 3.02045 −2.78078 0.516994i −1.56155 2.56155i 0
299.7 0.780776 + 1.17915i −0.848071 + 1.51022i −0.780776 + 1.84130i 0 −2.44293 + 0.179147i −3.02045 −2.78078 + 0.516994i −1.56155 2.56155i 0
299.8 0.780776 + 1.17915i 0.848071 + 1.51022i −0.780776 + 1.84130i 0 −1.11862 + 2.17915i 3.02045 −2.78078 + 0.516994i −1.56155 + 2.56155i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 299.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
15.d odd 2 1 inner
60.h 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.h.a 8
3.b odd 2 1 300.2.h.b 8
4.b odd 2 1 inner 300.2.h.a 8
5.b even 2 1 300.2.h.b 8
5.c odd 4 1 60.2.e.a 8
5.c odd 4 1 300.2.e.c 8
12.b even 2 1 300.2.h.b 8
15.d odd 2 1 inner 300.2.h.a 8
15.e even 4 1 60.2.e.a 8
15.e even 4 1 300.2.e.c 8
20.d odd 2 1 300.2.h.b 8
20.e even 4 1 60.2.e.a 8
20.e even 4 1 300.2.e.c 8
40.i odd 4 1 960.2.h.g 8
40.k even 4 1 960.2.h.g 8
60.h even 2 1 inner 300.2.h.a 8
60.l odd 4 1 60.2.e.a 8
60.l odd 4 1 300.2.e.c 8
120.q odd 4 1 960.2.h.g 8
120.w even 4 1 960.2.h.g 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
60.2.e.a 8 5.c odd 4 1
60.2.e.a 8 15.e even 4 1
60.2.e.a 8 20.e even 4 1
60.2.e.a 8 60.l odd 4 1
300.2.e.c 8 5.c odd 4 1
300.2.e.c 8 15.e even 4 1
300.2.e.c 8 20.e even 4 1
300.2.e.c 8 60.l odd 4 1
300.2.h.a 8 1.a even 1 1 trivial
300.2.h.a 8 4.b odd 2 1 inner
300.2.h.a 8 15.d odd 2 1 inner
300.2.h.a 8 60.h even 2 1 inner
300.2.h.b 8 3.b odd 2 1
300.2.h.b 8 5.b even 2 1
300.2.h.b 8 12.b even 2 1
300.2.h.b 8 20.d odd 2 1
960.2.h.g 8 40.i odd 4 1
960.2.h.g 8 40.k even 4 1
960.2.h.g 8 120.q odd 4 1
960.2.h.g 8 120.w even 4 1

Hecke kernels

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

\( T_{7}^{4} - 10T_{7}^{2} + 8 \) Copy content Toggle raw display
\( T_{17} - 2 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} + T^{3} + 2 T + 4)^{2} \) Copy content Toggle raw display
$3$ \( T^{8} - 2 T^{6} + 2 T^{4} - 18 T^{2} + \cdots + 81 \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( (T^{4} - 10 T^{2} + 8)^{2} \) Copy content Toggle raw display
$11$ \( (T^{4} - 20 T^{2} + 32)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} + 36 T^{2} + 256)^{2} \) Copy content Toggle raw display
$17$ \( (T - 2)^{8} \) Copy content Toggle raw display
$19$ \( (T^{4} + 20 T^{2} + 32)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} + 58 T^{2} + 8)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} + 36 T^{2} + 256)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} + 28 T^{2} + 128)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} + 36 T^{2} + 256)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} + 52 T^{2} + 64)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} - 62 T^{2} + 128)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} + 10 T^{2} + 8)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} - 8 T - 52)^{4} \) Copy content Toggle raw display
$59$ \( (T^{4} - 252 T^{2} + 10368)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + 2 T - 16)^{4} \) Copy content Toggle raw display
$67$ \( (T^{4} - 46 T^{2} + 512)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} - 56 T^{2} + 512)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} + 68)^{4} \) Copy content Toggle raw display
$79$ \( (T^{4} + 148 T^{2} + 5408)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} + 250 T^{2} + 5000)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} + 144 T^{2} + 4096)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + 36)^{4} \) Copy content Toggle raw display
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