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Properties

Label 350.3.f.d
Level $350$
Weight $3$
Character orbit 350.f
Analytic conductor $9.537$
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] = [350,3,Mod(43,350)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(350, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([3, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("350.43");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 350 = 2 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 350.f (of order \(4\), degree \(2\), 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: \(9.53680925261\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: 8.0.664811929600.20
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 18x^{6} + 85x^{4} + 92x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 5^{2} \)
Twist minimal: no (minimal twist has level 70)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( - \beta_{3} - 1) q^{2} + (\beta_{3} - \beta_1) q^{3} + 2 \beta_{3} q^{4} + ( - \beta_{2} + \beta_1) q^{6} - \beta_{5} q^{7} + ( - 2 \beta_{3} + 2) q^{8} + ( - \beta_{7} + \beta_{5} + \cdots + \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{3} - 1) q^{2} + (\beta_{3} - \beta_1) q^{3} + 2 \beta_{3} q^{4} + ( - \beta_{2} + \beta_1) q^{6} - \beta_{5} q^{7} + ( - 2 \beta_{3} + 2) q^{8} + ( - \beta_{7} + \beta_{5} + \cdots + \beta_1) q^{9}+ \cdots + ( - 4 \beta_{7} + 4 \beta_{5} + \cdots + 6 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8 q^{2} - 4 q^{3} + 8 q^{6} + 16 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 8 q^{2} - 4 q^{3} + 8 q^{6} + 16 q^{8} + 20 q^{11} - 8 q^{12} + 16 q^{13} - 32 q^{16} - 8 q^{17} - 28 q^{18} - 28 q^{21} - 20 q^{22} - 24 q^{23} - 32 q^{26} + 116 q^{27} - 80 q^{31} + 32 q^{32} + 44 q^{33} + 56 q^{36} - 44 q^{37} - 72 q^{38} + 240 q^{41} + 28 q^{42} + 84 q^{43} + 48 q^{46} - 92 q^{47} + 16 q^{48} - 308 q^{51} + 32 q^{52} - 120 q^{53} - 252 q^{57} - 12 q^{58} + 96 q^{61} + 80 q^{62} + 56 q^{63} - 88 q^{66} + 172 q^{67} + 16 q^{68} + 176 q^{71} - 56 q^{72} - 60 q^{73} + 144 q^{76} + 276 q^{78} - 160 q^{81} - 240 q^{82} - 272 q^{83} - 168 q^{86} + 220 q^{87} + 40 q^{88} - 84 q^{91} - 48 q^{92} + 52 q^{93} - 32 q^{96} + 304 q^{97} + 56 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 18x^{6} + 85x^{4} + 92x^{2} + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{7} + 2\nu^{6} - 14\nu^{5} + 38\nu^{4} - 19\nu^{3} + 168\nu^{2} + 114\nu + 72 ) / 40 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{7} - 2\nu^{6} - 14\nu^{5} - 38\nu^{4} - 19\nu^{3} - 168\nu^{2} + 114\nu - 72 ) / 40 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -3\nu^{7} - 52\nu^{5} - 227\nu^{3} - 198\nu ) / 40 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 5\nu^{7} - \nu^{6} + 90\nu^{5} - 14\nu^{4} + 415\nu^{3} - 19\nu^{2} + 370\nu + 74 ) / 40 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -5\nu^{7} - \nu^{6} - 90\nu^{5} - 14\nu^{4} - 415\nu^{3} - 19\nu^{2} - 370\nu + 74 ) / 40 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -7\nu^{6} - 108\nu^{4} - 363\nu^{2} - 142 ) / 20 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 7\nu^{7} + 128\nu^{5} + 643\nu^{3} + 1022\nu ) / 40 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{7} + 3\beta_{3} - \beta_{2} - \beta_1 ) / 5 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{6} + 5\beta_{5} + 5\beta_{4} + \beta_{2} - \beta _1 - 22 ) / 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -7\beta_{7} + 5\beta_{5} - 5\beta_{4} - 41\beta_{3} + 12\beta_{2} + 12\beta_1 ) / 5 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 13\beta_{6} - 45\beta_{5} - 45\beta_{4} - 23\beta_{2} + 23\beta _1 + 176 ) / 5 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 13\beta_{7} - 17\beta_{5} + 17\beta_{4} + 103\beta_{3} - 24\beta_{2} - 24\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -163\beta_{6} + 435\beta_{5} + 435\beta_{4} + 303\beta_{2} - 303\beta _1 - 1676 ) / 5 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -663\beta_{7} + 1095\beta_{5} - 1095\beta_{4} - 6089\beta_{3} + 1238\beta_{2} + 1238\beta_1 ) / 5 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/350\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(127\)
\(\chi(n)\) \(1\) \(\beta_{3}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
43.1
− 3.32695i
0.212981i
2.32695i
− 1.21298i
3.32695i
− 0.212981i
− 2.32695i
1.21298i
−1.00000 + 1.00000i −3.89721 − 3.89721i − 2.00000i 0 7.79441 1.87083 − 1.87083i 2.00000 + 2.00000i 21.3764i 0
43.2 −1.00000 + 1.00000i −1.61143 − 1.61143i − 2.00000i 0 3.22287 −1.87083 + 1.87083i 2.00000 + 2.00000i − 3.80656i 0
43.3 −1.00000 + 1.00000i 1.02638 + 1.02638i − 2.00000i 0 −2.05275 1.87083 − 1.87083i 2.00000 + 2.00000i − 6.89310i 0
43.4 −1.00000 + 1.00000i 2.48226 + 2.48226i − 2.00000i 0 −4.96452 −1.87083 + 1.87083i 2.00000 + 2.00000i 3.32325i 0
57.1 −1.00000 − 1.00000i −3.89721 + 3.89721i 2.00000i 0 7.79441 1.87083 + 1.87083i 2.00000 − 2.00000i − 21.3764i 0
57.2 −1.00000 − 1.00000i −1.61143 + 1.61143i 2.00000i 0 3.22287 −1.87083 − 1.87083i 2.00000 − 2.00000i 3.80656i 0
57.3 −1.00000 − 1.00000i 1.02638 − 1.02638i 2.00000i 0 −2.05275 1.87083 + 1.87083i 2.00000 − 2.00000i 6.89310i 0
57.4 −1.00000 − 1.00000i 2.48226 − 2.48226i 2.00000i 0 −4.96452 −1.87083 − 1.87083i 2.00000 − 2.00000i − 3.32325i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 43.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 350.3.f.d 8
5.b even 2 1 70.3.f.b ✓ 8
5.c odd 4 1 70.3.f.b ✓ 8
5.c odd 4 1 inner 350.3.f.d 8
15.d odd 2 1 630.3.o.c 8
15.e even 4 1 630.3.o.c 8
20.d odd 2 1 560.3.bh.c 8
20.e even 4 1 560.3.bh.c 8
35.c odd 2 1 490.3.f.n 8
35.f even 4 1 490.3.f.n 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
70.3.f.b ✓ 8 5.b even 2 1
70.3.f.b ✓ 8 5.c odd 4 1
350.3.f.d 8 1.a even 1 1 trivial
350.3.f.d 8 5.c odd 4 1 inner
490.3.f.n 8 35.c odd 2 1
490.3.f.n 8 35.f even 4 1
560.3.bh.c 8 20.d odd 2 1
560.3.bh.c 8 20.e even 4 1
630.3.o.c 8 15.d odd 2 1
630.3.o.c 8 15.e even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{8} + 4T_{3}^{7} + 8T_{3}^{6} - 52T_{3}^{5} + 313T_{3}^{4} + 416T_{3}^{3} + 512T_{3}^{2} - 2048T_{3} + 4096 \) acting on \(S_{3}^{\mathrm{new}}(350, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 2 T + 2)^{4} \) Copy content Toggle raw display
$3$ \( T^{8} + 4 T^{7} + \cdots + 4096 \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( (T^{4} + 49)^{2} \) Copy content Toggle raw display
$11$ \( (T^{4} - 10 T^{3} + \cdots + 1600)^{2} \) Copy content Toggle raw display
$13$ \( T^{8} - 16 T^{7} + \cdots + 522488164 \) Copy content Toggle raw display
$17$ \( T^{8} + 8 T^{7} + \cdots + 442429156 \) Copy content Toggle raw display
$19$ \( T^{8} + 1188 T^{6} + \cdots + 167961600 \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 6400000000 \) Copy content Toggle raw display
$29$ \( T^{8} + \cdots + 1921945600 \) Copy content Toggle raw display
$31$ \( (T^{4} + 40 T^{3} + \cdots + 27392)^{2} \) Copy content Toggle raw display
$37$ \( T^{8} + \cdots + 18415575616 \) Copy content Toggle raw display
$41$ \( (T^{4} - 120 T^{3} + \cdots - 3003328)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} + \cdots + 16433188864 \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 138717512704 \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots + 1675191781264 \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 1057442022400 \) Copy content Toggle raw display
$61$ \( (T^{4} - 48 T^{3} + \cdots - 40960)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots + 117631898238976 \) Copy content Toggle raw display
$71$ \( (T^{4} - 88 T^{3} + \cdots + 2588800)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 421045254400 \) Copy content Toggle raw display
$79$ \( T^{8} + \cdots + 114187176505600 \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 516593985519616 \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots + 26\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots + 19822530062500 \) Copy content Toggle raw display
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