Properties

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

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Newspace parameters

Level: \( N \) \(=\) \( 370 = 2 \cdot 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 370.q (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(2.95446487479\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-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.

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

Character values

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

\(n\) \(261\) \(297\)
\(\chi(n)\) \(\zeta_{12}\) \(\zeta_{12}^{3}\)

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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
97.1
−0.866025 + 0.500000i
−0.866025 0.500000i
0.866025 + 0.500000i
0.866025 0.500000i
−0.500000 + 0.866025i 0.133975 0.500000i −0.500000 0.866025i −1.86603 1.23205i 0.366025 + 0.366025i −0.732051 + 2.73205i 1.00000 2.36603 + 1.36603i 2.00000 1.00000i
103.1 −0.500000 0.866025i 0.133975 + 0.500000i −0.500000 + 0.866025i −1.86603 + 1.23205i 0.366025 0.366025i −0.732051 2.73205i 1.00000 2.36603 1.36603i 2.00000 + 1.00000i
267.1 −0.500000 0.866025i 1.86603 0.500000i −0.500000 + 0.866025i −0.133975 + 2.23205i −1.36603 1.36603i 2.73205 0.732051i 1.00000 0.633975 0.366025i 2.00000 1.00000i
273.1 −0.500000 + 0.866025i 1.86603 + 0.500000i −0.500000 0.866025i −0.133975 2.23205i −1.36603 + 1.36603i 2.73205 + 0.732051i 1.00000 0.633975 + 0.366025i 2.00000 + 1.00000i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
185.p even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 370.2.q.a 4
5.c odd 4 1 370.2.r.a yes 4
37.g odd 12 1 370.2.r.a yes 4
185.p even 12 1 inner 370.2.q.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
370.2.q.a 4 1.a even 1 1 trivial
370.2.q.a 4 185.p even 12 1 inner
370.2.r.a yes 4 5.c odd 4 1
370.2.r.a yes 4 37.g odd 12 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} - 4T_{3}^{3} + 5T_{3}^{2} - 2T_{3} + 1 \) acting on \(S_{2}^{\mathrm{new}}(370, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + T + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{4} - 4 T^{3} + 5 T^{2} - 2 T + 1 \) Copy content Toggle raw display
$5$ \( T^{4} + 4 T^{3} + 11 T^{2} + 20 T + 25 \) Copy content Toggle raw display
$7$ \( T^{4} - 4 T^{3} + 8 T^{2} - 32 T + 64 \) Copy content Toggle raw display
$11$ \( (T^{2} + 4)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} - 2 T^{3} + 15 T^{2} + 22 T + 121 \) Copy content Toggle raw display
$17$ \( T^{4} - 18 T^{3} + 134 T^{2} + \cdots + 676 \) Copy content Toggle raw display
$19$ \( T^{4} - 10 T^{3} + 74 T^{2} + \cdots + 484 \) Copy content Toggle raw display
$23$ \( (T - 6)^{4} \) Copy content Toggle raw display
$29$ \( T^{4} + 12 T^{3} + 72 T^{2} + \cdots + 144 \) Copy content Toggle raw display
$31$ \( T^{4} - 10 T^{3} + 50 T^{2} + \cdots + 625 \) Copy content Toggle raw display
$37$ \( (T^{2} + 11 T + 37)^{2} \) Copy content Toggle raw display
$41$ \( T^{4} + 12 T^{3} + 51 T^{2} + 36 T + 9 \) Copy content Toggle raw display
$43$ \( (T + 11)^{4} \) Copy content Toggle raw display
$47$ \( T^{4} + 4 T^{3} + 8 T^{2} - 208 T + 2704 \) Copy content Toggle raw display
$53$ \( T^{4} - 16 T^{3} + 233 T^{2} + \cdots + 3721 \) Copy content Toggle raw display
$59$ \( T^{4} - 2 T^{3} + 50 T^{2} - 364 T + 676 \) Copy content Toggle raw display
$61$ \( T^{4} + 12 T^{3} + 36 T^{2} + \cdots + 144 \) Copy content Toggle raw display
$67$ \( T^{4} - 30 T^{3} + 306 T^{2} + \cdots + 6084 \) Copy content Toggle raw display
$71$ \( T^{4} + 8 T^{3} + 96 T^{2} + \cdots + 1024 \) Copy content Toggle raw display
$73$ \( T^{4} \) Copy content Toggle raw display
$79$ \( T^{4} + 10 T^{3} + 50 T^{2} + \cdots + 2500 \) Copy content Toggle raw display
$83$ \( T^{4} + 38 T^{3} + 650 T^{2} + \cdots + 45796 \) Copy content Toggle raw display
$89$ \( T^{4} - 26 T^{3} + 194 T^{2} + \cdots + 484 \) Copy content Toggle raw display
$97$ \( T^{4} + 32T^{2} + 64 \) Copy content Toggle raw display
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