Properties

Label 370.2.n.b
Level $370$
Weight $2$
Character orbit 370.n
Analytic conductor $2.954$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 370 = 2 \cdot 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 370.n (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(2.95446487479\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \(x^{4} - x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \zeta_{12} q^{2} + \zeta_{12}^{2} q^{4} + ( -\zeta_{12} + 2 \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{5} + ( 4 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{7} + \zeta_{12}^{3} q^{8} + ( -3 + 3 \zeta_{12}^{2} ) q^{9} +O(q^{10})\) \( q + \zeta_{12} q^{2} + \zeta_{12}^{2} q^{4} + ( -\zeta_{12} + 2 \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{5} + ( 4 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{7} + \zeta_{12}^{3} q^{8} + ( -3 + 3 \zeta_{12}^{2} ) q^{9} + ( -1 + 2 \zeta_{12}^{3} ) q^{10} -3 q^{11} + ( \zeta_{12} - \zeta_{12}^{3} ) q^{13} + 4 q^{14} + ( -1 + \zeta_{12}^{2} ) q^{16} + 6 \zeta_{12} q^{17} + ( -3 \zeta_{12} + 3 \zeta_{12}^{3} ) q^{18} -3 \zeta_{12}^{2} q^{19} + ( -2 - \zeta_{12} + 2 \zeta_{12}^{2} ) q^{20} -3 \zeta_{12} q^{22} -\zeta_{12}^{3} q^{23} + ( -3 - 4 \zeta_{12} + 3 \zeta_{12}^{2} ) q^{25} + q^{26} + 4 \zeta_{12} q^{28} + 6 q^{29} -4 q^{31} + ( -\zeta_{12} + \zeta_{12}^{3} ) q^{32} + 6 \zeta_{12}^{2} q^{34} + ( -4 + 8 \zeta_{12} + 4 \zeta_{12}^{2} ) q^{35} -3 q^{36} + ( -3 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{37} -3 \zeta_{12}^{3} q^{38} + ( -2 \zeta_{12} - \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{40} + 10 \zeta_{12}^{2} q^{41} -2 \zeta_{12}^{3} q^{43} -3 \zeta_{12}^{2} q^{44} + ( -6 - 3 \zeta_{12}^{3} ) q^{45} + ( 1 - \zeta_{12}^{2} ) q^{46} -11 \zeta_{12}^{3} q^{47} + ( 9 - 9 \zeta_{12}^{2} ) q^{49} + ( -3 \zeta_{12} - 4 \zeta_{12}^{2} + 3 \zeta_{12}^{3} ) q^{50} + \zeta_{12} q^{52} -10 \zeta_{12} q^{53} + ( 3 \zeta_{12} - 6 \zeta_{12}^{2} - 3 \zeta_{12}^{3} ) q^{55} + 4 \zeta_{12}^{2} q^{56} + 6 \zeta_{12} q^{58} + ( 15 - 15 \zeta_{12}^{2} ) q^{59} -12 \zeta_{12}^{2} q^{61} -4 \zeta_{12} q^{62} + 12 \zeta_{12}^{3} q^{63} - q^{64} + ( -1 + 2 \zeta_{12} + \zeta_{12}^{2} ) q^{65} + ( 2 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{67} + 6 \zeta_{12}^{3} q^{68} + ( -4 \zeta_{12} + 8 \zeta_{12}^{2} + 4 \zeta_{12}^{3} ) q^{70} + 6 \zeta_{12}^{2} q^{71} -3 \zeta_{12} q^{72} -2 \zeta_{12}^{3} q^{73} + ( 4 - 7 \zeta_{12}^{2} ) q^{74} + ( 3 - 3 \zeta_{12}^{2} ) q^{76} + ( -12 \zeta_{12} + 12 \zeta_{12}^{3} ) q^{77} + 4 \zeta_{12}^{2} q^{79} + ( -2 - \zeta_{12}^{3} ) q^{80} -9 \zeta_{12}^{2} q^{81} + 10 \zeta_{12}^{3} q^{82} -6 \zeta_{12} q^{83} + ( -6 + 12 \zeta_{12}^{3} ) q^{85} + ( 2 - 2 \zeta_{12}^{2} ) q^{86} -3 \zeta_{12}^{3} q^{88} + ( -15 + 15 \zeta_{12}^{2} ) q^{89} + ( 3 - 6 \zeta_{12} - 3 \zeta_{12}^{2} ) q^{90} + ( 4 - 4 \zeta_{12}^{2} ) q^{91} + ( \zeta_{12} - \zeta_{12}^{3} ) q^{92} + ( 11 - 11 \zeta_{12}^{2} ) q^{94} + ( 6 + 3 \zeta_{12} - 6 \zeta_{12}^{2} ) q^{95} + 2 \zeta_{12}^{3} q^{97} + ( 9 \zeta_{12} - 9 \zeta_{12}^{3} ) q^{98} + ( 9 - 9 \zeta_{12}^{2} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 2q^{4} + 4q^{5} - 6q^{9} + O(q^{10}) \) \( 4q + 2q^{4} + 4q^{5} - 6q^{9} - 4q^{10} - 12q^{11} + 16q^{14} - 2q^{16} - 6q^{19} - 4q^{20} - 6q^{25} + 4q^{26} + 24q^{29} - 16q^{31} + 12q^{34} - 8q^{35} - 12q^{36} - 2q^{40} + 20q^{41} - 6q^{44} - 24q^{45} + 2q^{46} + 18q^{49} - 8q^{50} - 12q^{55} + 8q^{56} + 30q^{59} - 24q^{61} - 4q^{64} - 2q^{65} + 16q^{70} + 12q^{71} + 2q^{74} + 6q^{76} + 8q^{79} - 8q^{80} - 18q^{81} - 24q^{85} + 4q^{86} - 30q^{89} + 6q^{90} + 8q^{91} + 22q^{94} + 12q^{95} + 18q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/370\mathbb{Z}\right)^\times\).

\(n\) \(261\) \(297\)
\(\chi(n)\) \(-\zeta_{12}^{2}\) \(-1\)

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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
269.1
−0.866025 0.500000i
0.866025 + 0.500000i
−0.866025 + 0.500000i
0.866025 0.500000i
−0.866025 0.500000i 0 0.500000 + 0.866025i 1.86603 + 1.23205i 0 −3.46410 + 2.00000i 1.00000i −1.50000 + 2.59808i −1.00000 2.00000i
269.2 0.866025 + 0.500000i 0 0.500000 + 0.866025i 0.133975 + 2.23205i 0 3.46410 2.00000i 1.00000i −1.50000 + 2.59808i −1.00000 + 2.00000i
359.1 −0.866025 + 0.500000i 0 0.500000 0.866025i 1.86603 1.23205i 0 −3.46410 2.00000i 1.00000i −1.50000 2.59808i −1.00000 + 2.00000i
359.2 0.866025 0.500000i 0 0.500000 0.866025i 0.133975 2.23205i 0 3.46410 + 2.00000i 1.00000i −1.50000 2.59808i −1.00000 2.00000i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
37.c even 3 1 inner
185.n even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 370.2.n.b 4
5.b even 2 1 inner 370.2.n.b 4
37.c even 3 1 inner 370.2.n.b 4
185.n even 6 1 inner 370.2.n.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
370.2.n.b 4 1.a even 1 1 trivial
370.2.n.b 4 5.b even 2 1 inner
370.2.n.b 4 37.c even 3 1 inner
370.2.n.b 4 185.n even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(370, [\chi])\):

\( T_{3} \)
\( T_{7}^{4} - 16 T_{7}^{2} + 256 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - T^{2} + T^{4} \)
$3$ \( T^{4} \)
$5$ \( 25 - 20 T + 11 T^{2} - 4 T^{3} + T^{4} \)
$7$ \( 256 - 16 T^{2} + T^{4} \)
$11$ \( ( 3 + T )^{4} \)
$13$ \( 1 - T^{2} + T^{4} \)
$17$ \( 1296 - 36 T^{2} + T^{4} \)
$19$ \( ( 9 + 3 T + T^{2} )^{2} \)
$23$ \( ( 1 + T^{2} )^{2} \)
$29$ \( ( -6 + T )^{4} \)
$31$ \( ( 4 + T )^{4} \)
$37$ \( 1369 + 47 T^{2} + T^{4} \)
$41$ \( ( 100 - 10 T + T^{2} )^{2} \)
$43$ \( ( 4 + T^{2} )^{2} \)
$47$ \( ( 121 + T^{2} )^{2} \)
$53$ \( 10000 - 100 T^{2} + T^{4} \)
$59$ \( ( 225 - 15 T + T^{2} )^{2} \)
$61$ \( ( 144 + 12 T + T^{2} )^{2} \)
$67$ \( 16 - 4 T^{2} + T^{4} \)
$71$ \( ( 36 - 6 T + T^{2} )^{2} \)
$73$ \( ( 4 + T^{2} )^{2} \)
$79$ \( ( 16 - 4 T + T^{2} )^{2} \)
$83$ \( 1296 - 36 T^{2} + T^{4} \)
$89$ \( ( 225 + 15 T + T^{2} )^{2} \)
$97$ \( ( 4 + T^{2} )^{2} \)
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