Properties

Label 300.3.c.f
Level $300$
Weight $3$
Character orbit 300.c
Analytic conductor $8.174$
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] = [300,3,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: 8.0.6080256576.2
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 3x^{7} + 7x^{6} - 12x^{5} + 12x^{4} - 48x^{3} + 112x^{2} - 192x + 256 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{8} \)
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.

\(f(q)\) \(=\) \( q + \beta_{5} q^{2} - \beta_{2} q^{3} + ( - \beta_{7} + 1) q^{4} + ( - \beta_{6} - 1) q^{6} + (\beta_{5} - \beta_{3} + 3 \beta_{2}) q^{7} + (\beta_{5} - \beta_{3} + \beta_{2} - \beta_1) q^{8} - 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{5} q^{2} - \beta_{2} q^{3} + ( - \beta_{7} + 1) q^{4} + ( - \beta_{6} - 1) q^{6} + (\beta_{5} - \beta_{3} + 3 \beta_{2}) q^{7} + (\beta_{5} - \beta_{3} + \beta_{2} - \beta_1) q^{8} - 3 q^{9} + ( - 3 \beta_{7} + \beta_{6} + \beta_{4} - 1) q^{11} + ( - \beta_{5} + \beta_{3} - \beta_{2} - \beta_1) q^{12} + (3 \beta_{5} + \beta_{3} + \beta_{2}) q^{13} + ( - 2 \beta_{7} + 4 \beta_{6} - 2) q^{14} + ( - 3 \beta_{7} + 2 \beta_{6} - 2 \beta_{4} - 5) q^{16} + ( - \beta_{5} + \beta_{3} + \beta_{2} - 4 \beta_1) q^{17} - 3 \beta_{5} q^{18} + (2 \beta_{7} - 6 \beta_{6} + 2 \beta_{4} - 2) q^{19} + ( - \beta_{7} - \beta_{6} - \beta_{4} + 9) q^{21} + ( - 2 \beta_{5} - 6 \beta_{3} - 2 \beta_{2} - 2 \beta_1) q^{22} + ( - 2 \beta_{5} + 2 \beta_{3} + 6 \beta_{2}) q^{23} + (\beta_{7} - 2 \beta_{6} - 2 \beta_{4} + 3) q^{24} + ( - 2 \beta_{7} + 10) q^{26} + 3 \beta_{2} q^{27} + ( - 2 \beta_{5} - 6 \beta_{3} + 6 \beta_{2} + 2 \beta_1) q^{28} + (3 \beta_{7} + 3 \beta_{6} + 3 \beta_{4} + 23) q^{29} + (8 \beta_{7} - 8 \beta_{6}) q^{31} + ( - 3 \beta_{5} - \beta_{3} + 17 \beta_{2} - \beta_1) q^{32} + ( - \beta_{5} + \beta_{3} + \beta_{2} - 4 \beta_1) q^{33} + ( - 2 \beta_{7} - 8 \beta_{4} + 2) q^{34} + (3 \beta_{7} - 3) q^{36} + (9 \beta_{5} + 3 \beta_{3} + 3 \beta_{2}) q^{37} + ( - 4 \beta_{5} + 4 \beta_{3} - 20 \beta_{2} - 4 \beta_1) q^{38} + (\beta_{7} - 3 \beta_{6} + \beta_{4} - 1) q^{39} + (6 \beta_{7} + 6 \beta_{6} + 6 \beta_{4} - 32) q^{41} + (10 \beta_{5} + 2 \beta_{3} + 6 \beta_{2} - 2 \beta_1) q^{42} + (8 \beta_{5} - 8 \beta_{3} + 4 \beta_{2}) q^{43} + ( - 6 \beta_{7} + 4 \beta_{6} - 4 \beta_{4} - 42) q^{44} + (4 \beta_{7} + 4 \beta_{6} + 16) q^{46} + (8 \beta_{5} - 8 \beta_{3} - 12 \beta_{2}) q^{47} + (5 \beta_{5} + 7 \beta_{3} + 9 \beta_{2} - \beta_1) q^{48} + (6 \beta_{7} + 6 \beta_{6} + 6 \beta_{4} + 3) q^{49} + (9 \beta_{7} - 3 \beta_{6} - 3 \beta_{4} + 3) q^{51} + (10 \beta_{5} - 2 \beta_{3} + 2 \beta_{2} - 2 \beta_1) q^{52} + ( - 9 \beta_{5} - 3 \beta_{3} - 3 \beta_{2}) q^{53} + (3 \beta_{6} + 3) q^{54} + ( - 2 \beta_{7} + 12 \beta_{6} + 4 \beta_{4} - 30) q^{56} + ( - 18 \beta_{5} - 6 \beta_{3} - 6 \beta_{2}) q^{57} + (20 \beta_{5} - 6 \beta_{3} - 18 \beta_{2} + 6 \beta_1) q^{58} + (5 \beta_{7} - 23 \beta_{6} + 9 \beta_{4} - 9) q^{59} + 38 q^{61} + (16 \beta_{3} - 16 \beta_{2}) q^{62} + ( - 3 \beta_{5} + 3 \beta_{3} - 9 \beta_{2}) q^{63} + (\beta_{7} + 18 \beta_{6} - 2 \beta_{4} + 7) q^{64} + ( - 2 \beta_{7} - 8 \beta_{4} + 2) q^{66} + ( - 14 \beta_{5} + 14 \beta_{3} + 18 \beta_{2}) q^{67} + (10 \beta_{5} + 14 \beta_{3} + 50 \beta_{2} - 2 \beta_1) q^{68} + (2 \beta_{7} + 2 \beta_{6} + 2 \beta_{4} + 18) q^{69} + ( - 14 \beta_{7} - 6 \beta_{6} + 10 \beta_{4} - 10) q^{71} + ( - 3 \beta_{5} + 3 \beta_{3} - 3 \beta_{2} + 3 \beta_1) q^{72} + (2 \beta_{5} - 2 \beta_{3} - 2 \beta_{2} + 8 \beta_1) q^{73} + ( - 6 \beta_{7} + 30) q^{74} + (4 \beta_{7} - 24 \beta_{6} - 8 \beta_{4} - 4) q^{76} + ( - 22 \beta_{5} - 10 \beta_{3} - 10 \beta_{2} + 8 \beta_1) q^{77} + ( - 2 \beta_{5} + 2 \beta_{3} - 10 \beta_{2} - 2 \beta_1) q^{78} + (8 \beta_{7} - 8 \beta_{6}) q^{79} + 9 q^{81} + ( - 38 \beta_{5} - 12 \beta_{3} - 36 \beta_{2} + 12 \beta_1) q^{82} + (4 \beta_{5} - 4 \beta_{3} - 24 \beta_{2}) q^{83} + ( - 10 \beta_{7} + 4 \beta_{6} - 4 \beta_{4} + 26) q^{84} + ( - 16 \beta_{7} + 12 \beta_{6} - 36) q^{86} + (9 \beta_{5} - 9 \beta_{3} - 23 \beta_{2}) q^{87} + ( - 38 \beta_{5} - 2 \beta_{3} + 34 \beta_{2} - 2 \beta_1) q^{88} + (4 \beta_{7} + 4 \beta_{6} + 4 \beta_{4} - 70) q^{89} + ( - 10 \beta_{7} + 14 \beta_{6} - 2 \beta_{4} + 2) q^{91} + (16 \beta_{5} + 8 \beta_1) q^{92} + ( - 16 \beta_{5} - 8 \beta_{3} - 8 \beta_{2} + 8 \beta_1) q^{93} + ( - 16 \beta_{7} - 4 \beta_{6} - 52) q^{94} + (\beta_{7} + 2 \beta_{6} - 2 \beta_{4} + 55) q^{96} + (68 \beta_{5} + 20 \beta_{3} + 20 \beta_{2} + 8 \beta_1) q^{97} + ( - 3 \beta_{5} - 12 \beta_{3} - 36 \beta_{2} + 12 \beta_1) q^{98} + (9 \beta_{7} - 3 \beta_{6} - 3 \beta_{4} + 3) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 10 q^{4} - 6 q^{6} - 24 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 10 q^{4} - 6 q^{6} - 24 q^{9} - 20 q^{14} - 46 q^{16} + 72 q^{21} + 18 q^{24} + 84 q^{26} + 184 q^{29} - 12 q^{34} - 30 q^{36} - 256 q^{41} - 348 q^{44} + 112 q^{46} + 24 q^{49} + 18 q^{54} - 244 q^{56} + 304 q^{61} + 10 q^{64} - 12 q^{66} + 144 q^{69} + 252 q^{74} - 24 q^{76} + 72 q^{81} + 204 q^{84} - 280 q^{86} - 560 q^{89} - 376 q^{94} + 426 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 3x^{7} + 7x^{6} - 12x^{5} + 12x^{4} - 48x^{3} + 112x^{2} - 192x + 256 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -3\nu^{7} - 13\nu^{6} + 5\nu^{5} - 30\nu^{4} + 108\nu^{3} + 264\nu^{2} + 368\nu - 160 ) / 224 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 3\nu^{7} - \nu^{6} - 19\nu^{5} - 12\nu^{4} + 4\nu^{3} - 96\nu^{2} + 80\nu + 608 ) / 224 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{7} - 12\nu^{6} + 10\nu^{5} - 11\nu^{4} - 8\nu^{3} - 4\nu^{2} + 400\nu - 320 ) / 112 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{7} + 9\nu^{6} - 11\nu^{5} + 10\nu^{4} - 22\nu^{3} + 24\nu^{2} - 160\nu + 408 ) / 56 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 3\nu^{7} - 15\nu^{6} + 23\nu^{5} + 2\nu^{4} + 60\nu^{3} - 152\nu^{2} + 528\nu - 736 ) / 224 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -5\nu^{7} + 18\nu^{6} - 36\nu^{5} + 41\nu^{4} - 72\nu^{3} + 244\nu^{2} - 656\nu + 928 ) / 112 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -11\nu^{7} + 20\nu^{6} - 26\nu^{5} + 37\nu^{4} - 52\nu^{3} + 324\nu^{2} - 592\nu + 384 ) / 112 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{7} - \beta_{6} + \beta_{5} + \beta_{4} + \beta_{3} + \beta_{2} + 1 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{7} + \beta_{6} - \beta_{5} + \beta_{4} + \beta_{3} - 3\beta_{2} + 2\beta _1 - 3 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 3\beta_{5} - 3\beta_{3} + \beta_{2} + 2\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -\beta_{7} + 9\beta_{6} + 25\beta_{5} + \beta_{4} - \beta_{3} - 5\beta_{2} - 2\beta _1 + 13 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -3\beta_{7} - 5\beta_{6} - 7\beta_{5} + 3\beta_{4} - \beta_{3} - 37\beta_{2} - 2\beta _1 + 103 ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 13\beta_{7} - 21\beta_{6} + 19\beta_{4} - 9 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -73\beta_{7} + 49\beta_{6} - 11\beta_{5} + 41\beta_{4} + 3\beta_{3} - 81\beta_{2} + 38\beta _1 - 235 ) / 4 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
151.1
−1.51328 1.30766i
−1.51328 + 1.30766i
0.670410 + 1.88429i
0.670410 1.88429i
1.96705 0.361553i
1.96705 + 0.361553i
0.375825 + 1.96437i
0.375825 1.96437i
−1.88911 0.656712i 1.73205i 3.13746 + 2.48120i 0 1.13746 3.27203i 9.55505i −4.29756 6.74766i −3.00000 0
151.2 −1.88911 + 0.656712i 1.73205i 3.13746 2.48120i 0 1.13746 + 3.27203i 9.55505i −4.29756 + 6.74766i −3.00000 0
151.3 −1.29664 1.52274i 1.73205i −0.637459 + 3.94888i 0 −2.63746 + 2.24584i 0.837253i 6.83966 4.14959i −3.00000 0
151.4 −1.29664 + 1.52274i 1.73205i −0.637459 3.94888i 0 −2.63746 2.24584i 0.837253i 6.83966 + 4.14959i −3.00000 0
151.5 1.29664 1.52274i 1.73205i −0.637459 3.94888i 0 −2.63746 2.24584i 0.837253i −6.83966 4.14959i −3.00000 0
151.6 1.29664 + 1.52274i 1.73205i −0.637459 + 3.94888i 0 −2.63746 + 2.24584i 0.837253i −6.83966 + 4.14959i −3.00000 0
151.7 1.88911 0.656712i 1.73205i 3.13746 2.48120i 0 1.13746 + 3.27203i 9.55505i 4.29756 6.74766i −3.00000 0
151.8 1.88911 + 0.656712i 1.73205i 3.13746 + 2.48120i 0 1.13746 3.27203i 9.55505i 4.29756 + 6.74766i −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
5.b even 2 1 inner
20.d 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.f 8
3.b odd 2 1 900.3.c.r 8
4.b odd 2 1 inner 300.3.c.f 8
5.b even 2 1 inner 300.3.c.f 8
5.c odd 4 2 60.3.f.b 8
12.b even 2 1 900.3.c.r 8
15.d odd 2 1 900.3.c.r 8
15.e even 4 2 180.3.f.h 8
20.d odd 2 1 inner 300.3.c.f 8
20.e even 4 2 60.3.f.b 8
40.i odd 4 2 960.3.j.e 8
40.k even 4 2 960.3.j.e 8
60.h even 2 1 900.3.c.r 8
60.l odd 4 2 180.3.f.h 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
60.3.f.b 8 5.c odd 4 2
60.3.f.b 8 20.e even 4 2
180.3.f.h 8 15.e even 4 2
180.3.f.h 8 60.l odd 4 2
300.3.c.f 8 1.a even 1 1 trivial
300.3.c.f 8 4.b odd 2 1 inner
300.3.c.f 8 5.b even 2 1 inner
300.3.c.f 8 20.d odd 2 1 inner
900.3.c.r 8 3.b odd 2 1
900.3.c.r 8 12.b even 2 1
900.3.c.r 8 15.d odd 2 1
900.3.c.r 8 60.h even 2 1
960.3.j.e 8 40.i odd 4 2
960.3.j.e 8 40.k even 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(300, [\chi])\):

\( T_{7}^{4} + 92T_{7}^{2} + 64 \) Copy content Toggle raw display
\( T_{13}^{4} - 84T_{13}^{2} + 1536 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} - 5 T^{6} + 24 T^{4} - 80 T^{2} + \cdots + 256 \) Copy content Toggle raw display
$3$ \( (T^{2} + 3)^{4} \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( (T^{4} + 92 T^{2} + 64)^{2} \) Copy content Toggle raw display
$11$ \( (T^{4} + 348 T^{2} + 24576)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} - 84 T^{2} + 1536)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} - 1044 T^{2} + 221184)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} + 1008 T^{2} + 221184)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} + 368 T^{2} + 1024)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} - 46 T + 16)^{4} \) Copy content Toggle raw display
$31$ \( (T^{4} + 2304 T^{2} + 393216)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} - 756 T^{2} + 124416)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} + 64 T - 1028)^{4} \) Copy content Toggle raw display
$43$ \( (T^{4} + 2528 T^{2} + 1364224)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} + 3296 T^{2} + 614656)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} - 756 T^{2} + 124416)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} + 16668 T^{2} + 69033984)^{2} \) Copy content Toggle raw display
$61$ \( (T - 38)^{8} \) Copy content Toggle raw display
$67$ \( (T^{4} + 9392 T^{2} + 7573504)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + 17136 T^{2} + 884736)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} - 4176 T^{2} + 3538944)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} + 2304 T^{2} + 393216)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} + 4064 T^{2} + 2027776)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + 140 T + 3988)^{4} \) Copy content Toggle raw display
$97$ \( (T^{4} - 45888 T^{2} + \cdots + 495550464)^{2} \) Copy content Toggle raw display
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