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Properties

Label 300.3.c.g
Level $300$
Weight $3$
Character orbit 300.c
Analytic conductor $8.174$
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] = [300,3,Mod(151,300)]
 
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, 0, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("300.151");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 300 = 2^{2} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 300.c (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: \(8.17440793081\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.4069419264.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 7x^{6} + 50x^{4} - 84x^{3} + 55x^{2} - 12x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{8} \)
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.

\(f(q)\) \(=\) \( q - \beta_{4} q^{2} + \beta_{5} q^{3} + (\beta_{7} - 1) q^{4} + ( - \beta_{2} - 1) q^{6} + ( - \beta_{7} - \beta_{6} - \beta_{5}) q^{7} + ( - \beta_{6} + 2 \beta_{5} + \beta_{4} + \cdots + 3) q^{8}+ \cdots - 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{4} q^{2} + \beta_{5} q^{3} + (\beta_{7} - 1) q^{4} + ( - \beta_{2} - 1) q^{6} + ( - \beta_{7} - \beta_{6} - \beta_{5}) q^{7} + ( - \beta_{6} + 2 \beta_{5} + \beta_{4} + \cdots + 3) q^{8}+ \cdots + (3 \beta_{6} + 3 \beta_{5} - 3 \beta_{4} + \cdots + 3) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 2 q^{2} - 8 q^{4} - 6 q^{6} + 20 q^{8} - 24 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 2 q^{2} - 8 q^{4} - 6 q^{6} + 20 q^{8} - 24 q^{9} + 8 q^{13} + 22 q^{14} + 40 q^{16} - 6 q^{18} + 24 q^{21} + 4 q^{22} - 36 q^{24} - 66 q^{26} + 104 q^{28} - 32 q^{29} + 112 q^{32} + 124 q^{34} + 24 q^{36} - 176 q^{37} - 170 q^{38} - 16 q^{41} + 54 q^{42} + 40 q^{44} - 76 q^{46} + 24 q^{48} + 16 q^{49} + 56 q^{52} - 304 q^{53} + 18 q^{54} - 172 q^{56} + 72 q^{57} - 12 q^{58} + 136 q^{61} - 238 q^{62} + 16 q^{64} - 108 q^{66} + 88 q^{68} - 96 q^{69} - 60 q^{72} + 240 q^{73} - 108 q^{74} + 120 q^{76} - 384 q^{77} + 150 q^{78} + 72 q^{81} + 320 q^{82} - 144 q^{84} + 214 q^{86} - 200 q^{88} + 128 q^{89} + 312 q^{92} + 72 q^{93} + 12 q^{94} + 96 q^{96} + 216 q^{97} + 60 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 7x^{6} + 50x^{4} - 84x^{3} + 55x^{2} - 12x + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -73\nu^{7} - 599\nu^{6} - 16\nu^{5} + 3897\nu^{4} + 291\nu^{3} - 20680\nu^{2} + 20620\nu - 5669 ) / 515 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -291\nu^{7} + 32\nu^{6} + 1968\nu^{5} - 381\nu^{4} - 14163\nu^{3} + 27350\nu^{2} - 21000\nu + 3582 ) / 515 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 347\nu^{7} - 24\nu^{6} - 2506\nu^{5} + 157\nu^{4} + 17961\nu^{3} - 30040\nu^{2} + 17810\nu - 1399 ) / 515 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 347\nu^{7} - 24\nu^{6} - 2506\nu^{5} + 157\nu^{4} + 17961\nu^{3} - 30040\nu^{2} + 16780\nu - 1399 ) / 515 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -486\nu^{7} - 143\nu^{6} + 3308\nu^{5} + 914\nu^{4} - 23728\nu^{3} + 34050\nu^{2} - 19070\nu + 2372 ) / 515 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 924\nu^{7} + 132\nu^{6} - 6302\nu^{5} - 606\nu^{4} + 45672\nu^{3} - 72710\nu^{2} + 44700\nu - 5953 ) / 515 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 1134\nu^{7} + 162\nu^{6} - 8062\nu^{5} - 1446\nu^{4} + 57082\nu^{3} - 85630\nu^{2} + 45870\nu - 5363 ) / 515 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{4} + \beta_{3} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{7} - 2\beta_{6} - 7\beta_{5} + \beta_{4} - 2\beta_{3} + \beta _1 + 7 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{7} + 4\beta_{6} + 6\beta_{5} - 6\beta_{4} + 3\beta_{3} + 3\beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -7\beta_{7} - 11\beta_{6} - 22\beta_{5} + 11\beta_{4} + 7\beta_{3} - 4\beta_{2} - 22 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 19\beta_{7} + 58\beta_{6} + 55\beta_{5} - 19\beta_{4} - 88\beta_{3} + 36\beta_{2} + 11\beta _1 + 195 ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -25\beta_{7} - 26\beta_{6} + 29\beta_{5} - 23\beta_{4} + 190\beta_{3} - 22\beta_{2} - 29\beta _1 - 407 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -25\beta_{7} - 81\beta_{6} - 403\beta_{5} + 330\beta_{4} - 598\beta_{3} - 25\beta_{2} + 81\beta _1 + 997 ) / 2 \) Copy content Toggle raw display

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

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
151.1
0.151747 + 0.0876113i
0.151747 − 0.0876113i
0.845613 − 0.488215i
0.845613 + 0.488215i
−2.65095 − 1.53053i
−2.65095 + 1.53053i
1.65359 + 0.954702i
1.65359 − 0.954702i
−1.33290 − 1.49110i − 1.73205i −0.446749 + 3.97497i 0 −2.58266 + 2.30865i − 6.56834i 6.52255 − 4.63210i −3.00000 0
151.2 −1.33290 + 1.49110i 1.73205i −0.446749 − 3.97497i 0 −2.58266 − 2.30865i 6.56834i 6.52255 + 4.63210i −3.00000 0
151.3 −0.177680 − 1.99209i 1.73205i −3.93686 + 0.707911i 0 3.45040 − 0.307751i − 1.19501i 2.10973 + 7.71680i −3.00000 0
151.4 −0.177680 + 1.99209i − 1.73205i −3.93686 − 0.707911i 0 3.45040 + 0.307751i 1.19501i 2.10973 − 7.71680i −3.00000 0
151.5 0.534079 − 1.92737i − 1.73205i −3.42952 − 2.05874i 0 −3.33830 − 0.925051i 11.9716i −5.79958 + 5.51043i −3.00000 0
151.6 0.534079 + 1.92737i 1.73205i −3.42952 + 2.05874i 0 −3.33830 + 0.925051i − 11.9716i −5.79958 − 5.51043i −3.00000 0
151.7 1.97650 − 0.305673i − 1.73205i 3.81313 − 1.20833i 0 −0.529441 − 3.42340i 0.329898i 7.16731 − 3.55383i −3.00000 0
151.8 1.97650 + 0.305673i 1.73205i 3.81313 + 1.20833i 0 −0.529441 + 3.42340i − 0.329898i 7.16731 + 3.55383i −3.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 151.8
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 300.3.c.g yes 8
3.b odd 2 1 900.3.c.n 8
4.b odd 2 1 inner 300.3.c.g yes 8
5.b even 2 1 300.3.c.e ✓ 8
5.c odd 4 2 300.3.f.c 16
12.b even 2 1 900.3.c.n 8
15.d odd 2 1 900.3.c.t 8
15.e even 4 2 900.3.f.h 16
20.d odd 2 1 300.3.c.e ✓ 8
20.e even 4 2 300.3.f.c 16
60.h even 2 1 900.3.c.t 8
60.l odd 4 2 900.3.f.h 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
300.3.c.e ✓ 8 5.b even 2 1
300.3.c.e ✓ 8 20.d odd 2 1
300.3.c.g yes 8 1.a even 1 1 trivial
300.3.c.g yes 8 4.b odd 2 1 inner
300.3.f.c 16 5.c odd 4 2
300.3.f.c 16 20.e even 4 2
900.3.c.n 8 3.b odd 2 1
900.3.c.n 8 12.b even 2 1
900.3.c.t 8 15.d odd 2 1
900.3.c.t 8 60.h even 2 1
900.3.f.h 16 15.e even 4 2
900.3.f.h 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 \(S_{3}^{\mathrm{new}}(300, [\chi])\):

\( T_{7}^{8} + 188T_{7}^{6} + 6470T_{7}^{4} + 9532T_{7}^{2} + 961 \) Copy content Toggle raw display
\( T_{13}^{4} - 4T_{13}^{3} - 418T_{13}^{2} + 3916T_{13} - 1559 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} - 2 T^{7} + \cdots + 256 \) Copy content Toggle raw display
$3$ \( (T^{2} + 3)^{4} \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( T^{8} + 188 T^{6} + \cdots + 961 \) Copy content Toggle raw display
$11$ \( T^{8} + 704 T^{6} + \cdots + 30824704 \) Copy content Toggle raw display
$13$ \( (T^{4} - 4 T^{3} + \cdots - 1559)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} - 832 T^{2} + \cdots + 132400)^{2} \) Copy content Toggle raw display
$19$ \( T^{8} + \cdots + 1099651921 \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 1611540736 \) Copy content Toggle raw display
$29$ \( (T^{4} + 16 T^{3} + \cdots + 93616)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} + \cdots + 819736484449 \) Copy content Toggle raw display
$37$ \( (T^{4} + 88 T^{3} + \cdots - 4548464)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} + 8 T^{3} + \cdots + 3504448)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} + \cdots + 974581609681 \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 13610196640000 \) Copy content Toggle raw display
$53$ \( (T^{4} + 152 T^{3} + \cdots - 4946624)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 195562066176 \) Copy content Toggle raw display
$61$ \( (T^{4} - 68 T^{3} + \cdots + 9745129)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots + 23066205847441 \) Copy content Toggle raw display
$71$ \( T^{8} + \cdots + 11235904000000 \) Copy content Toggle raw display
$73$ \( (T^{4} - 120 T^{3} + \cdots - 43602032)^{2} \) Copy content Toggle raw display
$79$ \( T^{8} + \cdots + 2278988775424 \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 120362665464064 \) Copy content Toggle raw display
$89$ \( (T^{4} - 64 T^{3} + \cdots - 1507328)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} - 108 T^{3} + \cdots + 15618033)^{2} \) Copy content Toggle raw display
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