Properties

Label 310.2.h.d
Level $310$
Weight $2$
Character orbit 310.h
Analytic conductor $2.475$
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] = [310,2,Mod(101,310)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(310, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([0, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("310.101");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 310 = 2 \cdot 5 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 310.h (of order \(5\), degree \(4\), 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: \(2.47536246266\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{5})\)
Coefficient field: 8.0.64000000.2
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 2x^{6} + 4x^{4} + 8x^{2} + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( - \beta_{6} - \beta_{4} - \beta_{2} - 1) q^{2} + ( - \beta_{7} - \beta_{2}) q^{3} + \beta_{6} q^{4} + q^{5} + ( - \beta_{5} - 1) q^{6} + ( - \beta_{7} + \beta_{6} - \beta_{5}) q^{7} + \beta_{4} q^{8} + ( - 2 \beta_{7} - 2 \beta_{5} - 2 \beta_{3} - 2 \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{6} - \beta_{4} - \beta_{2} - 1) q^{2} + ( - \beta_{7} - \beta_{2}) q^{3} + \beta_{6} q^{4} + q^{5} + ( - \beta_{5} - 1) q^{6} + ( - \beta_{7} + \beta_{6} - \beta_{5}) q^{7} + \beta_{4} q^{8} + ( - 2 \beta_{7} - 2 \beta_{5} - 2 \beta_{3} - 2 \beta_1) q^{9} + ( - \beta_{6} - \beta_{4} - \beta_{2} - 1) q^{10} - \beta_{6} q^{11} + (\beta_{6} + \beta_{4} - \beta_{3} + \beta_{2} + 1) q^{12} + ( - 2 \beta_{7} - \beta_{3} - \beta_{2} - \beta_1) q^{13} + ( - \beta_{5} + \beta_{4} - \beta_{3}) q^{14} + ( - \beta_{7} - \beta_{2}) q^{15} + \beta_{2} q^{16} + (\beta_{6} - \beta_{5} - \beta_{4} - \beta_{3} + \beta_{2}) q^{17} + 2 \beta_{7} q^{18} + (2 \beta_{6} - 2 \beta_{5} - 2 \beta_1 + 2) q^{19} + \beta_{6} q^{20} + (\beta_{6} - \beta_{5} + 3 \beta_{4} - 2 \beta_{3} + 3 \beta_{2} - \beta_1 + 1) q^{21} - \beta_{4} q^{22} + (\beta_{7} - \beta_{6} + 2 \beta_{5} - \beta_{4} + 2 \beta_{3} - \beta_{2} + \beta_1) q^{23} + ( - \beta_{6} - \beta_1) q^{24} + q^{25} + (\beta_{7} - \beta_{5} + \beta_{3} - 1) q^{26} + ( - \beta_{6} + \beta_1) q^{27} + ( - \beta_{3} + \beta_{2} - \beta_1) q^{28} + (\beta_{6} - 3 \beta_{5} + \beta_{4} + \beta_{2} - 3 \beta_1 + 1) q^{29} + ( - \beta_{5} - 1) q^{30} + (3 \beta_{7} + 2 \beta_{6} + 2 \beta_{5} - \beta_{4} + 4 \beta_{3} + \beta_1) q^{31} + q^{32} + ( - \beta_{6} - \beta_{4} + \beta_{3} - \beta_{2} - 1) q^{33} + (\beta_{4} - \beta_{3} - \beta_{2} - \beta_1 + 1) q^{34} + ( - \beta_{7} + \beta_{6} - \beta_{5}) q^{35} + 2 \beta_{5} q^{36} + ( - 3 \beta_{7} + \beta_{5} - 3 \beta_{3} - 3) q^{37} + (2 \beta_{7} - 2 \beta_{6} + 2 \beta_{5} - 2 \beta_{2} + 2 \beta_1 - 2) q^{38} + ( - 3 \beta_{7} - 2 \beta_{6} - 2 \beta_{5} + 3 \beta_{4} - 2 \beta_{3} - 2 \beta_{2} - 3 \beta_1) q^{39} + \beta_{4} q^{40} + ( - 2 \beta_{6} + \beta_{4} + \beta_{2} - 2) q^{41} + (\beta_{7} - \beta_{6} + \beta_{5} + 2 \beta_{2} - \beta_1 + 2) q^{42} + (2 \beta_{6} + 2 \beta_{5} + 2 \beta_{4} + 4 \beta_{3} + 2 \beta_{2} + 2 \beta_1 + 2) q^{43} - \beta_{2} q^{44} + ( - 2 \beta_{7} - 2 \beta_{5} - 2 \beta_{3} - 2 \beta_1) q^{45} + ( - \beta_{7} - \beta_{4} + \beta_{3} - \beta_{2} + \beta_1 - 1) q^{46} + (4 \beta_{7} + 5 \beta_{3} + \beta_{2} + 5 \beta_1) q^{47} + (\beta_{7} + \beta_{5} - \beta_{4} + \beta_{3} + \beta_1) q^{48} + (2 \beta_{4} - 2 \beta_{3} - 2 \beta_{2} - 2 \beta_1 + 2) q^{49} + ( - \beta_{6} - \beta_{4} - \beta_{2} - 1) q^{50} + (2 \beta_{7} + 2 \beta_{6} + 2 \beta_{5} + 3 \beta_{2} + 2 \beta_1 + 3) q^{51} + (\beta_{6} + \beta_{5} + \beta_{4} - \beta_{3} + \beta_{2} + \beta_1 + 1) q^{52} + (4 \beta_{7} + 6 \beta_{5} - 2 \beta_{4} + 6 \beta_{3} + 4 \beta_1) q^{53} + ( - \beta_{7} - \beta_{5} - \beta_{4} - \beta_{3} - \beta_1) q^{54} - \beta_{6} q^{55} + (\beta_{7} + \beta_{5} + \beta_{3} + 1) q^{56} + ( - 2 \beta_{6} - 2 \beta_{4} - 2) q^{57} + (3 \beta_{7} - \beta_{6} + 3 \beta_{5} + 3 \beta_1) q^{58} + ( - 2 \beta_{7} - 2 \beta_{4} + 2 \beta_{3} - 6 \beta_{2} + 2 \beta_1 - 2) q^{59} + (\beta_{6} + \beta_{4} - \beta_{3} + \beta_{2} + 1) q^{60} + ( - 3 \beta_{6} + 5 \beta_{5} - 3 \beta_{4} - 4) q^{61} + ( - \beta_{7} + 2 \beta_{5} + 2 \beta_{4} + \beta_{3} - \beta_{2} + 3 \beta_1) q^{62} + ( - 4 \beta_{6} + 2 \beta_{5} - 4 \beta_{4}) q^{63} + ( - \beta_{6} - \beta_{4} - \beta_{2} - 1) q^{64} + ( - 2 \beta_{7} - \beta_{3} - \beta_{2} - \beta_1) q^{65} + (\beta_{6} + \beta_1) q^{66} + (\beta_{7} - \beta_{6} + 4 \beta_{5} - \beta_{4} + \beta_{3} - 3) q^{67} + (\beta_{7} - \beta_{6} + \beta_{5} - \beta_{4} + \beta_{3} - 2) q^{68} + ( - 2 \beta_{7} + 2 \beta_{6} - 2 \beta_{5} - 3 \beta_{2} + \beta_1 - 3) q^{69} + ( - \beta_{5} + \beta_{4} - \beta_{3}) q^{70} + (6 \beta_{7} - 3 \beta_{6} + 5 \beta_{5} - 6 \beta_{4} + 5 \beta_{3} - 3 \beta_{2} + 6 \beta_1) q^{71} + 2 \beta_{3} q^{72} + (4 \beta_{7} + 4 \beta_{5} + 4 \beta_{2} + 4 \beta_1 + 4) q^{73} + (3 \beta_{6} - 3 \beta_{5} + 3 \beta_{4} + \beta_{3} + 3 \beta_{2} - 3 \beta_1 + 3) q^{74} + ( - \beta_{7} - \beta_{2}) q^{75} + ( - 2 \beta_{7} + 2 \beta_{6} + 2 \beta_{2} - 2 \beta_1) q^{76} + (\beta_{3} - \beta_{2} + \beta_1) q^{77} + (3 \beta_{7} - 2 \beta_{4} + \beta_{3} + 3 \beta_{2} + \beta_1 - 2) q^{78} + (\beta_{7} + 7 \beta_{6} - 3 \beta_{5} + \beta_{4} - 3 \beta_{3} + 7 \beta_{2} + \beta_1) q^{79} + \beta_{2} q^{80} + (\beta_{6} + \beta_{4} - 6 \beta_{3} + \beta_{2} + 1) q^{81} + (2 \beta_{6} + 3 \beta_{2} + 3) q^{82} + ( - \beta_{6} + 2 \beta_{5} + 5 \beta_{4} + \beta_{3} + 5 \beta_{2} + 2 \beta_1 - 1) q^{83} + (\beta_{7} - 2 \beta_{6} + 2 \beta_{5} - 3 \beta_{4} + 2 \beta_{3} - 2 \beta_{2} + \beta_1) q^{84} + (\beta_{6} - \beta_{5} - \beta_{4} - \beta_{3} + \beta_{2}) q^{85} + ( - 2 \beta_{7} - 2 \beta_{6} - 2 \beta_{5} + 2 \beta_1) q^{86} + (3 \beta_{7} - 6 \beta_{6} + \beta_{5} - 6 \beta_{4} + 3 \beta_{3} - 5) q^{87} - q^{88} + ( - 2 \beta_{7} - 6 \beta_{6} - 2 \beta_{5} - 2 \beta_1) q^{89} + 2 \beta_{7} q^{90} + ( - \beta_{6} + \beta_{4} - 2 \beta_{3} + \beta_{2} - 1) q^{91} + ( - \beta_{7} + \beta_{6} - 2 \beta_{5} + \beta_{4} - \beta_{3}) q^{92} + (\beta_{7} + 5 \beta_{6} - \beta_{5} - 2 \beta_{4} + 4 \beta_1 - 4) q^{93} + ( - 5 \beta_{7} - \beta_{5} - 5 \beta_{3} + 1) q^{94} + (2 \beta_{6} - 2 \beta_{5} - 2 \beta_1 + 2) q^{95} + ( - \beta_{7} - \beta_{2}) q^{96} + (12 \beta_{6} + 2 \beta_{2} + 2) q^{97} + (2 \beta_{7} - 2 \beta_{6} + 2 \beta_{5} - 2 \beta_{4} + 2 \beta_{3} - 4) q^{98} - 2 \beta_{5} q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 2 q^{2} + 2 q^{3} - 2 q^{4} + 8 q^{5} - 8 q^{6} - 2 q^{7} - 2 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 2 q^{2} + 2 q^{3} - 2 q^{4} + 8 q^{5} - 8 q^{6} - 2 q^{7} - 2 q^{8} - 2 q^{10} + 2 q^{11} + 2 q^{12} + 2 q^{13} - 2 q^{14} + 2 q^{15} - 2 q^{16} - 2 q^{17} + 12 q^{19} - 2 q^{20} - 6 q^{21} + 2 q^{22} + 6 q^{23} + 2 q^{24} + 8 q^{25} - 8 q^{26} + 2 q^{27} - 2 q^{28} + 2 q^{29} - 8 q^{30} - 2 q^{31} + 8 q^{32} - 2 q^{33} + 8 q^{34} - 2 q^{35} - 24 q^{37} - 8 q^{38} + 2 q^{39} - 2 q^{40} - 16 q^{41} + 14 q^{42} + 4 q^{43} + 2 q^{44} - 4 q^{46} - 2 q^{47} + 2 q^{48} + 16 q^{49} - 2 q^{50} + 14 q^{51} + 2 q^{52} + 4 q^{53} + 2 q^{54} + 2 q^{55} + 8 q^{56} - 8 q^{57} + 2 q^{58} + 2 q^{60} - 20 q^{61} - 2 q^{62} + 16 q^{63} - 2 q^{64} + 2 q^{65} - 2 q^{66} - 20 q^{67} - 12 q^{68} - 22 q^{69} - 2 q^{70} + 24 q^{71} + 24 q^{73} + 6 q^{74} + 2 q^{75} - 8 q^{76} + 2 q^{77} - 18 q^{78} - 30 q^{79} - 2 q^{80} + 2 q^{81} + 14 q^{82} - 26 q^{83} + 14 q^{84} - 2 q^{85} + 4 q^{86} - 16 q^{87} - 8 q^{88} + 12 q^{89} - 10 q^{91} - 4 q^{92} - 38 q^{93} + 8 q^{94} + 12 q^{95} + 2 q^{96} - 12 q^{97} - 24 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} ) / 2 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{4} ) / 4 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{5} ) / 4 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{6} ) / 8 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{7} ) / 8 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 2\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{3} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 4\beta_{4} \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 4\beta_{5} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 8\beta_{6} \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 8\beta_{7} \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/310\mathbb{Z}\right)^\times\).

\(n\) \(187\) \(251\)
\(\chi(n)\) \(1\) \(\beta_{4}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
101.1
−1.14412 0.831254i
1.14412 + 0.831254i
−0.437016 + 1.34500i
0.437016 1.34500i
−1.14412 + 0.831254i
1.14412 0.831254i
−0.437016 1.34500i
0.437016 + 1.34500i
0.309017 0.951057i −0.746033 2.29605i −0.809017 0.587785i 1.00000 −2.41421 −2.66025 1.93278i −0.809017 + 0.587785i −2.28825 + 1.66251i 0.309017 0.951057i
101.2 0.309017 0.951057i 0.127999 + 0.393941i −0.809017 0.587785i 1.00000 0.414214 1.04221 + 0.757212i −0.809017 + 0.587785i 2.28825 1.66251i 0.309017 0.951057i
171.1 −0.809017 + 0.587785i −0.335106 0.243469i 0.309017 0.951057i 1.00000 0.414214 0.579108 1.78231i 0.309017 + 0.951057i −0.874032 2.68999i −0.809017 + 0.587785i
171.2 −0.809017 + 0.587785i 1.95314 + 1.41904i 0.309017 0.951057i 1.00000 −2.41421 0.0389262 0.119803i 0.309017 + 0.951057i 0.874032 + 2.68999i −0.809017 + 0.587785i
221.1 0.309017 + 0.951057i −0.746033 + 2.29605i −0.809017 + 0.587785i 1.00000 −2.41421 −2.66025 + 1.93278i −0.809017 0.587785i −2.28825 1.66251i 0.309017 + 0.951057i
221.2 0.309017 + 0.951057i 0.127999 0.393941i −0.809017 + 0.587785i 1.00000 0.414214 1.04221 0.757212i −0.809017 0.587785i 2.28825 + 1.66251i 0.309017 + 0.951057i
281.1 −0.809017 0.587785i −0.335106 + 0.243469i 0.309017 + 0.951057i 1.00000 0.414214 0.579108 + 1.78231i 0.309017 0.951057i −0.874032 + 2.68999i −0.809017 0.587785i
281.2 −0.809017 0.587785i 1.95314 1.41904i 0.309017 + 0.951057i 1.00000 −2.41421 0.0389262 + 0.119803i 0.309017 0.951057i 0.874032 2.68999i −0.809017 0.587785i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 101.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
31.d even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 310.2.h.d 8
31.d even 5 1 inner 310.2.h.d 8
31.d even 5 1 9610.2.a.bl 4
31.f odd 10 1 9610.2.a.by 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
310.2.h.d 8 1.a even 1 1 trivial
310.2.h.d 8 31.d even 5 1 inner
9610.2.a.bl 4 31.d even 5 1
9610.2.a.by 4 31.f odd 10 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{8} - 2T_{3}^{7} + 5T_{3}^{6} - 12T_{3}^{5} + 29T_{3}^{4} + 12T_{3}^{3} + 5T_{3}^{2} + 2T_{3} + 1 \) acting on \(S_{2}^{\mathrm{new}}(310, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} + T^{3} + T^{2} + T + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{8} - 2 T^{7} + 5 T^{6} - 12 T^{5} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( (T - 1)^{8} \) Copy content Toggle raw display
$7$ \( T^{8} + 2 T^{7} + T^{6} - 4 T^{5} + 39 T^{4} + \cdots + 1 \) Copy content Toggle raw display
$11$ \( (T^{4} - T^{3} + T^{2} - T + 1)^{2} \) Copy content Toggle raw display
$13$ \( T^{8} - 2 T^{7} - 5 T^{6} + 28 T^{5} + \cdots + 81 \) Copy content Toggle raw display
$17$ \( T^{8} + 2 T^{7} + 11 T^{6} - 4 T^{5} + \cdots + 961 \) Copy content Toggle raw display
$19$ \( T^{8} - 12 T^{7} + 60 T^{6} + \cdots + 256 \) Copy content Toggle raw display
$23$ \( T^{8} - 6 T^{7} + 9 T^{6} + 48 T^{5} + \cdots + 1 \) Copy content Toggle raw display
$29$ \( T^{8} - 2 T^{7} - 15 T^{6} + \cdots + 73441 \) Copy content Toggle raw display
$31$ \( T^{8} + 2 T^{7} - 57 T^{6} + \cdots + 923521 \) Copy content Toggle raw display
$37$ \( (T^{4} + 12 T^{3} - 16 T^{2} - 312 T - 449)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} + 8 T^{3} + 34 T^{2} + 77 T + 121)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} - 4 T^{7} + 100 T^{6} + \cdots + 20736 \) Copy content Toggle raw display
$47$ \( T^{8} + 2 T^{7} - 5 T^{6} + \cdots + 10439361 \) Copy content Toggle raw display
$53$ \( T^{8} - 4 T^{7} - 28 T^{6} + \cdots + 1290496 \) Copy content Toggle raw display
$59$ \( T^{8} + 168 T^{6} - 1680 T^{5} + \cdots + 256 \) Copy content Toggle raw display
$61$ \( (T^{4} + 10 T^{3} - 85 T^{2} - 550 T + 775)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} + 10 T^{3} - 19 T^{2} - 290 T - 71)^{2} \) Copy content Toggle raw display
$71$ \( T^{8} - 24 T^{7} + 370 T^{6} + \cdots + 4036081 \) Copy content Toggle raw display
$73$ \( T^{8} - 24 T^{7} + 400 T^{6} + \cdots + 5308416 \) Copy content Toggle raw display
$79$ \( T^{8} + 30 T^{7} + 607 T^{6} + \cdots + 6765201 \) Copy content Toggle raw display
$83$ \( T^{8} + 26 T^{7} + 457 T^{6} + \cdots + 641601 \) Copy content Toggle raw display
$89$ \( T^{8} - 12 T^{7} + 140 T^{6} + \cdots + 246016 \) Copy content Toggle raw display
$97$ \( (T^{4} + 6 T^{3} + 136 T^{2} + 1856 T + 13456)^{2} \) Copy content Toggle raw display
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