Properties

Label 350.3.d.a
Level $350$
Weight $3$
Character orbit 350.d
Analytic conductor $9.537$
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] = [350,3,Mod(349,350)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(350, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("350.349");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 350 = 2 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 350.d (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: \(9.53680925261\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.205520896.4
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 4x^{7} + 12x^{6} - 12x^{5} - 8x^{4} + 12x^{3} + 12x^{2} + 4x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{5} \)
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.

\(f(q)\) \(=\) \( q + \beta_{2} q^{2} - \beta_{7} q^{3} - 2 q^{4} + ( - \beta_{6} - \beta_{5}) q^{6} + ( - \beta_{7} - 2 \beta_{4} + \beta_{2} - 3 \beta_1) q^{7} - 2 \beta_{2} q^{8} + ( - 2 \beta_{3} - 3) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{2} - \beta_{7} q^{3} - 2 q^{4} + ( - \beta_{6} - \beta_{5}) q^{6} + ( - \beta_{7} - 2 \beta_{4} + \beta_{2} - 3 \beta_1) q^{7} - 2 \beta_{2} q^{8} + ( - 2 \beta_{3} - 3) q^{9} + ( - 3 \beta_{3} + 5) q^{11} + 2 \beta_{7} q^{12} + ( - 4 \beta_{7} - 3 \beta_{4}) q^{13} + ( - 3 \beta_{6} + \beta_{5} + 3 \beta_{3} - 2) q^{14} + 4 q^{16} + (6 \beta_{7} - 4 \beta_{4}) q^{17} + ( - 3 \beta_{2} - 4 \beta_1) q^{18} - 8 \beta_{6} q^{19} + ( - 4 \beta_{6} - \beta_{5} - 10 \beta_{3} + 2) q^{21} + (5 \beta_{2} - 6 \beta_1) q^{22} + (\beta_{2} + 9 \beta_1) q^{23} + (2 \beta_{6} + 2 \beta_{5}) q^{24} + ( - 7 \beta_{6} - \beta_{5}) q^{26} + (10 \beta_{7} - 2 \beta_{4}) q^{27} + (2 \beta_{7} + 4 \beta_{4} - 2 \beta_{2} + 6 \beta_1) q^{28} + ( - 16 \beta_{3} + 21) q^{29} + ( - 11 \beta_{6} - 2 \beta_{5}) q^{31} + 4 \beta_{2} q^{32} + ( - 8 \beta_{7} - 3 \beta_{4}) q^{33} + (2 \beta_{6} + 10 \beta_{5}) q^{34} + (4 \beta_{3} + 6) q^{36} + (22 \beta_{2} + 27 \beta_1) q^{37} + (8 \beta_{7} + 8 \beta_{4}) q^{38} + ( - 20 \beta_{3} + 18) q^{39} + (2 \beta_{6} + 17 \beta_{5}) q^{41} + (5 \beta_{7} + 3 \beta_{4} + 2 \beta_{2} - 20 \beta_1) q^{42} + (5 \beta_{2} + 19 \beta_1) q^{43} + (6 \beta_{3} - 10) q^{44} + ( - 9 \beta_{3} - 2) q^{46} + (8 \beta_{7} - 6 \beta_{4}) q^{47} - 4 \beta_{7} q^{48} + ( - 12 \beta_{6} - 10 \beta_{5} + 12 \beta_{3} + 27) q^{49} + ( - 4 \beta_{3} - 44) q^{51} + (8 \beta_{7} + 6 \beta_{4}) q^{52} + ( - 38 \beta_{2} - 4 \beta_1) q^{53} + (8 \beta_{6} + 12 \beta_{5}) q^{54} + (6 \beta_{6} - 2 \beta_{5} - 6 \beta_{3} + 4) q^{56} + (16 \beta_{2} - 48 \beta_1) q^{57} + (21 \beta_{2} - 32 \beta_1) q^{58} + ( - 15 \beta_{6} - 20 \beta_{5}) q^{59} + (22 \beta_{6} + 15 \beta_{5}) q^{61} + (13 \beta_{7} + 9 \beta_{4}) q^{62} + ( - 3 \beta_{7} + 8 \beta_{4} + 3 \beta_{2} + 5 \beta_1) q^{63} - 8 q^{64} + ( - 11 \beta_{6} - 5 \beta_{5}) q^{66} + ( - 57 \beta_{2} + 21 \beta_1) q^{67} + ( - 12 \beta_{7} + 8 \beta_{4}) q^{68} + (8 \beta_{6} - \beta_{5}) q^{69} + (41 \beta_{3} + 23) q^{71} + (6 \beta_{2} + 8 \beta_1) q^{72} + (32 \beta_{7} + 13 \beta_{4}) q^{73} + ( - 27 \beta_{3} - 44) q^{74} + 16 \beta_{6} q^{76} + ( - 14 \beta_{7} - 7 \beta_{4} + 14 \beta_{2} - 21 \beta_1) q^{77} + (18 \beta_{2} - 40 \beta_1) q^{78} + (55 \beta_{3} - 51) q^{79} + (30 \beta_{3} - 37) q^{81} + ( - 19 \beta_{7} + 15 \beta_{4}) q^{82} + ( - 37 \beta_{7} - 40 \beta_{4}) q^{83} + (8 \beta_{6} + 2 \beta_{5} + 20 \beta_{3} - 4) q^{84} + ( - 19 \beta_{3} - 10) q^{86} + ( - 37 \beta_{7} - 16 \beta_{4}) q^{87} + ( - 10 \beta_{2} + 12 \beta_1) q^{88} + (4 \beta_{6} - 21 \beta_{5}) q^{89} + ( - 19 \beta_{6} - 10 \beta_{5} - 16 \beta_{3} + 62) q^{91} + ( - 2 \beta_{2} - 18 \beta_1) q^{92} + (30 \beta_{2} - 62 \beta_1) q^{93} + (2 \beta_{6} + 14 \beta_{5}) q^{94} + ( - 4 \beta_{6} - 4 \beta_{5}) q^{96} + ( - 44 \beta_{7} + 17 \beta_{4}) q^{97} + (22 \beta_{7} + 2 \beta_{4} + 27 \beta_{2} + 24 \beta_1) q^{98} + ( - \beta_{3} - 3) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 16 q^{4} - 24 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 16 q^{4} - 24 q^{9} + 40 q^{11} - 16 q^{14} + 32 q^{16} + 16 q^{21} + 168 q^{29} + 48 q^{36} + 144 q^{39} - 80 q^{44} - 16 q^{46} + 216 q^{49} - 352 q^{51} + 32 q^{56} - 64 q^{64} + 184 q^{71} - 352 q^{74} - 408 q^{79} - 296 q^{81} - 32 q^{84} - 80 q^{86} + 496 q^{91} - 24 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 4x^{7} + 12x^{6} - 12x^{5} - 8x^{4} + 12x^{3} + 12x^{2} + 4x + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 6\nu^{7} - 14\nu^{6} + 26\nu^{5} + 79\nu^{4} - 262\nu^{3} + 134\nu^{2} + 182\nu + 44 ) / 51 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 32\nu^{7} - 137\nu^{6} + 422\nu^{5} - 508\nu^{4} - 94\nu^{3} + 335\nu^{2} + 336\nu + 76 ) / 51 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -2\nu^{7} + 9\nu^{6} - 28\nu^{5} + 36\nu^{4} + 4\nu^{3} - 33\nu^{2} - 10\nu ) / 3 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -35\nu^{7} + 161\nu^{6} - 503\nu^{5} + 664\nu^{4} + 55\nu^{3} - 623\nu^{2} - 189\nu + 140 ) / 51 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -3\nu^{7} + 11\nu^{6} - 31\nu^{5} + 20\nu^{4} + 49\nu^{3} - 43\nu^{2} - 51\nu - 12 ) / 3 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 3\nu^{7} - 13\nu^{6} + 41\nu^{5} - 52\nu^{4} + \nu^{3} + 29\nu^{2} + 27\nu + 6 ) / 3 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 61\nu^{7} - 267\nu^{6} + 831\nu^{5} - 1030\nu^{4} - 159\nu^{3} + 909\nu^{2} + 275\nu + 28 ) / 51 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -\beta_{7} + \beta_{6} - \beta_{5} - \beta_{4} - 2\beta_{2} - 2\beta _1 + 2 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{7} + \beta_{6} - \beta_{5} + \beta_{4} - 3\beta_{3} - 3\beta_{2} - 2\beta _1 - 2 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -5\beta_{7} + \beta_{6} + 5\beta_{5} + 9\beta_{4} - 18\beta_{3} + 4\beta_{2} + 14\beta _1 - 22 ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -2\beta_{6} + 8\beta_{5} + 12\beta_{2} + 21\beta_1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 37\beta_{7} - 7\beta_{6} + 37\beta_{5} - 81\beta_{4} + 150\beta_{3} + 52\beta_{2} + 98\beta _1 + 202 ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 43\beta_{7} + 43\beta_{6} - 91\beta_{5} - 91\beta_{4} + 171\beta_{3} - 171\beta_{2} - 226\beta _1 + 226 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -103\beta_{7} + 305\beta_{6} - 713\beta_{5} + 305\beta_{4} - 504\beta_{3} - 1286\beta_{2} - 1790\beta _1 - 782 ) / 4 \) Copy content Toggle raw display

Character values

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

\(n\) \(101\) \(127\)
\(\chi(n)\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
349.1
−0.550501 0.228025i
1.12964 + 2.72719i
−0.129640 0.312979i
1.55050 + 0.642239i
−0.550501 + 0.228025i
1.12964 2.72719i
−0.129640 + 0.312979i
1.55050 0.642239i
1.41421i −2.97127 −2.00000 0 4.20201i −5.43275 4.41421i 2.82843i −0.171573 0
349.2 1.41421i −1.78089 −2.00000 0 2.51856i 6.81801 + 1.58579i 2.82843i −5.82843 0
349.3 1.41421i 1.78089 −2.00000 0 2.51856i −6.81801 + 1.58579i 2.82843i −5.82843 0
349.4 1.41421i 2.97127 −2.00000 0 4.20201i 5.43275 4.41421i 2.82843i −0.171573 0
349.5 1.41421i −2.97127 −2.00000 0 4.20201i −5.43275 + 4.41421i 2.82843i −0.171573 0
349.6 1.41421i −1.78089 −2.00000 0 2.51856i 6.81801 1.58579i 2.82843i −5.82843 0
349.7 1.41421i 1.78089 −2.00000 0 2.51856i −6.81801 1.58579i 2.82843i −5.82843 0
349.8 1.41421i 2.97127 −2.00000 0 4.20201i 5.43275 + 4.41421i 2.82843i −0.171573 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 349.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
7.b odd 2 1 inner
35.c odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 350.3.d.a 8
5.b even 2 1 inner 350.3.d.a 8
5.c odd 4 1 350.3.b.a 4
5.c odd 4 1 350.3.b.b yes 4
7.b odd 2 1 inner 350.3.d.a 8
35.c odd 2 1 inner 350.3.d.a 8
35.f even 4 1 350.3.b.a 4
35.f even 4 1 350.3.b.b yes 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
350.3.b.a 4 5.c odd 4 1
350.3.b.a 4 35.f even 4 1
350.3.b.b yes 4 5.c odd 4 1
350.3.b.b yes 4 35.f even 4 1
350.3.d.a 8 1.a even 1 1 trivial
350.3.d.a 8 5.b even 2 1 inner
350.3.d.a 8 7.b odd 2 1 inner
350.3.d.a 8 35.c odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} - 12T_{3}^{2} + 28 \) acting on \(S_{3}^{\mathrm{new}}(350, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 2)^{4} \) Copy content Toggle raw display
$3$ \( (T^{4} - 12 T^{2} + 28)^{2} \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( T^{8} - 108 T^{6} + 6566 T^{4} + \cdots + 5764801 \) Copy content Toggle raw display
$11$ \( (T^{2} - 10 T + 7)^{4} \) Copy content Toggle raw display
$13$ \( (T^{4} - 276 T^{2} + 8092)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} - 944 T^{2} + 129472)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} + 768 T^{2} + 114688)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} + 166 T^{2} + 6241)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} - 42 T - 71)^{4} \) Copy content Toggle raw display
$31$ \( (T^{4} + 1356 T^{2} + 149212)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} + 3394 T^{2} + 57121)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} + 5556 T^{2} + 3489052)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} + 822 T^{2} + 96721)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} - 1872 T^{2} + 430528)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} + 5808 T^{2} + 8248384)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} + 8300 T^{2} + 16817500)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} + 7668 T^{2} + 4502428)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} + 13878 T^{2} + 36687249)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} - 46 T - 2833)^{4} \) Copy content Toggle raw display
$73$ \( (T^{4} - 12340 T^{2} + 14812)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} + 102 T - 3449)^{4} \) Copy content Toggle raw display
$83$ \( (T^{4} - 36588 T^{2} + \cdots + 285109468)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} + 9684 T^{2} + 1849372)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} - 34996 T^{2} + \cdots + 276596572)^{2} \) Copy content Toggle raw display
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