Properties

Label 310.2.q.d
Level $310$
Weight $2$
Character orbit 310.q
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(41,310)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(310, base_ring=CyclotomicField(30))
 
chi = DirichletCharacter(H, H._module([0, 14]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("310.41");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 310 = 2 \cdot 5 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 310.q (of order \(15\), degree \(8\), 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: \(8\)
Coefficient field: \(\Q(\zeta_{15})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{7} + x^{5} - x^{4} + x^{3} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{15}]$

$q$-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 primitive root of unity \(\zeta_{15}\). We also show the integral \(q\)-expansion of the trace form.

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

Character values

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

\(n\) \(187\) \(251\)
\(\chi(n)\) \(1\) \(1 - \zeta_{15}^{2} + \zeta_{15}^{3} - \zeta_{15}^{4} + \zeta_{15}^{6} - \zeta_{15}^{7}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
41.1
−0.978148 0.207912i
−0.104528 0.994522i
0.913545 0.406737i
0.669131 0.743145i
0.669131 + 0.743145i
−0.978148 + 0.207912i
0.913545 + 0.406737i
−0.104528 + 0.994522i
0.809017 + 0.587785i −0.226341 2.15349i 0.309017 + 0.951057i 0.500000 + 0.866025i 1.08268 1.87525i 2.43444 + 0.517457i −0.309017 + 0.951057i −1.65185 + 0.351111i −0.104528 + 0.994522i
51.1 −0.309017 0.951057i −0.755585 0.839162i −0.809017 + 0.587785i 0.500000 + 0.866025i −0.564602 + 0.977920i −0.186415 1.77362i 0.809017 + 0.587785i 0.180301 1.71545i 0.669131 0.743145i
71.1 −0.309017 + 0.951057i −0.126381 0.0268631i −0.809017 0.587785i 0.500000 + 0.866025i 0.0646021 0.111894i −3.24064 + 1.44282i 0.809017 0.587785i −2.72539 1.21342i −0.978148 + 0.207912i
81.1 0.809017 + 0.587785i −2.89169 1.28746i 0.309017 + 0.951057i 0.500000 0.866025i −1.58268 2.74128i −2.50739 + 2.78474i −0.309017 + 0.951057i 4.69693 + 5.21647i 0.913545 0.406737i
111.1 0.809017 0.587785i −2.89169 + 1.28746i 0.309017 0.951057i 0.500000 + 0.866025i −1.58268 + 2.74128i −2.50739 2.78474i −0.309017 0.951057i 4.69693 5.21647i 0.913545 + 0.406737i
121.1 0.809017 0.587785i −0.226341 + 2.15349i 0.309017 0.951057i 0.500000 0.866025i 1.08268 + 1.87525i 2.43444 0.517457i −0.309017 0.951057i −1.65185 0.351111i −0.104528 0.994522i
131.1 −0.309017 0.951057i −0.126381 + 0.0268631i −0.809017 + 0.587785i 0.500000 0.866025i 0.0646021 + 0.111894i −3.24064 1.44282i 0.809017 + 0.587785i −2.72539 + 1.21342i −0.978148 0.207912i
231.1 −0.309017 + 0.951057i −0.755585 + 0.839162i −0.809017 0.587785i 0.500000 0.866025i −0.564602 0.977920i −0.186415 + 1.77362i 0.809017 0.587785i 0.180301 + 1.71545i 0.669131 + 0.743145i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 41.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
31.g even 15 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 310.2.q.d 8
31.g even 15 1 inner 310.2.q.d 8
31.g even 15 1 9610.2.a.bc 4
31.h odd 30 1 9610.2.a.bg 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
310.2.q.d 8 1.a even 1 1 trivial
310.2.q.d 8 31.g even 15 1 inner
9610.2.a.bc 4 31.g even 15 1
9610.2.a.bg 4 31.h odd 30 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{8} + 8T_{3}^{7} + 30T_{3}^{6} + 73T_{3}^{5} + 134T_{3}^{4} + 142T_{3}^{3} + 90T_{3}^{2} + 17T_{3} + 1 \) 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
$3$ \( T^{8} + 8 T^{7} + 30 T^{6} + 73 T^{5} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( (T^{2} - T + 1)^{4} \) Copy content Toggle raw display
$7$ \( T^{8} + 7 T^{7} + 15 T^{6} + \cdots + 3481 \) Copy content Toggle raw display
$11$ \( T^{8} - 6 T^{7} - 30 T^{6} + \cdots + 68121 \) Copy content Toggle raw display
$13$ \( T^{8} + 10 T^{7} + 45 T^{6} + 115 T^{5} + \cdots + 25 \) Copy content Toggle raw display
$17$ \( T^{8} - 18 T^{7} + 135 T^{6} + \cdots + 81 \) Copy content Toggle raw display
$19$ \( T^{8} + 13 T^{7} + 30 T^{6} + \cdots + 841 \) Copy content Toggle raw display
$23$ \( T^{8} + 12 T^{7} + 108 T^{6} + \cdots + 20736 \) Copy content Toggle raw display
$29$ \( T^{8} + 9 T^{7} + 117 T^{6} + \cdots + 77841 \) Copy content Toggle raw display
$31$ \( T^{8} + 11 T^{7} + 30 T^{6} + \cdots + 923521 \) Copy content Toggle raw display
$37$ \( T^{8} - 8 T^{7} + 50 T^{6} - 98 T^{5} + \cdots + 1 \) Copy content Toggle raw display
$41$ \( T^{8} + 3 T^{7} - 15 T^{6} + \cdots + 77841 \) Copy content Toggle raw display
$43$ \( T^{8} - 21 T^{7} + 140 T^{6} + \cdots + 7778521 \) Copy content Toggle raw display
$47$ \( T^{8} + 9 T^{7} + 102 T^{6} + \cdots + 641601 \) Copy content Toggle raw display
$53$ \( T^{8} + 12 T^{7} + 612 T^{5} + \cdots + 68121 \) Copy content Toggle raw display
$59$ \( T^{8} + 30 T^{7} + 450 T^{6} + \cdots + 1703025 \) Copy content Toggle raw display
$61$ \( (T^{4} - 34 T^{3} + 386 T^{2} - 1499 T + 421)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} - 23 T^{7} + 395 T^{6} + \cdots + 1100401 \) Copy content Toggle raw display
$71$ \( T^{8} + 15 T^{6} - 45 T^{5} + \cdots + 2025 \) Copy content Toggle raw display
$73$ \( T^{8} - 17 T^{7} + 75 T^{6} + \cdots + 4044121 \) Copy content Toggle raw display
$79$ \( T^{8} - 51 T^{7} + 1295 T^{6} + \cdots + 12117361 \) Copy content Toggle raw display
$83$ \( T^{8} - 6 T^{7} + 240 T^{6} + \cdots + 281961 \) Copy content Toggle raw display
$89$ \( T^{8} + 21 T^{7} + 387 T^{6} + \cdots + 84437721 \) Copy content Toggle raw display
$97$ \( T^{8} + 6 T^{7} - 58 T^{6} + \cdots + 1666681 \) Copy content Toggle raw display
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