Properties

Label 370.2.g.c
Level $370$
Weight $2$
Character orbit 370.g
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.g (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(2.95446487479\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(\zeta_{8})\)
Defining polynomial: \(x^{4} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{8}\). We also show the integral \(q\)-expansion of the trace form.

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

Character values

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

\(n\) \(261\) \(297\)
\(\chi(n)\) \(\zeta_{8}^{2}\) \(\zeta_{8}^{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
43.1
−0.707107 + 0.707107i
0.707107 0.707107i
−0.707107 0.707107i
0.707107 + 0.707107i
1.00000i −1.41421 + 1.41421i −1.00000 −2.12132 + 0.707107i 1.41421 + 1.41421i 1.29289 1.29289i 1.00000i 1.00000i 0.707107 + 2.12132i
43.2 1.00000i 1.41421 1.41421i −1.00000 2.12132 0.707107i −1.41421 1.41421i 2.70711 2.70711i 1.00000i 1.00000i −0.707107 2.12132i
327.1 1.00000i −1.41421 1.41421i −1.00000 −2.12132 0.707107i 1.41421 1.41421i 1.29289 + 1.29289i 1.00000i 1.00000i 0.707107 2.12132i
327.2 1.00000i 1.41421 + 1.41421i −1.00000 2.12132 + 0.707107i −1.41421 + 1.41421i 2.70711 + 2.70711i 1.00000i 1.00000i −0.707107 + 2.12132i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
185.f even 4 1 inner

Twists

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} + 16 \) acting on \(S_{2}^{\mathrm{new}}(370, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T^{2} )^{2} \)
$3$ \( 16 + T^{4} \)
$5$ \( 25 - 8 T^{2} + T^{4} \)
$7$ \( 49 - 56 T + 32 T^{2} - 8 T^{3} + T^{4} \)
$11$ \( 49 + 18 T^{2} + T^{4} \)
$13$ \( 196 + 44 T^{2} + T^{4} \)
$17$ \( ( -7 - 2 T + T^{2} )^{2} \)
$19$ \( 4 + 24 T + 72 T^{2} + 12 T^{3} + T^{4} \)
$23$ \( 196 + 36 T^{2} + T^{4} \)
$29$ \( 5041 + 1704 T + 288 T^{2} + 24 T^{3} + T^{4} \)
$31$ \( 1 - 8 T + 32 T^{2} + 8 T^{3} + T^{4} \)
$37$ \( 1369 - 24 T^{2} + T^{4} \)
$41$ \( ( 49 + T^{2} )^{2} \)
$43$ \( ( 49 + T^{2} )^{2} \)
$47$ \( 16 - 64 T + 128 T^{2} + 16 T^{3} + T^{4} \)
$53$ \( 81 + T^{4} \)
$59$ \( 256 + 256 T + 128 T^{2} + 16 T^{3} + T^{4} \)
$61$ \( 39601 - 7960 T + 800 T^{2} - 40 T^{3} + T^{4} \)
$67$ \( 324 - 216 T + 72 T^{2} + 12 T^{3} + T^{4} \)
$71$ \( ( 4 + 12 T + T^{2} )^{2} \)
$73$ \( ( 72 - 12 T + T^{2} )^{2} \)
$79$ \( 1024 + 512 T + 128 T^{2} - 16 T^{3} + T^{4} \)
$83$ \( 64 + 64 T + 32 T^{2} - 8 T^{3} + T^{4} \)
$89$ \( 4624 - 1632 T + 288 T^{2} - 24 T^{3} + T^{4} \)
$97$ \( ( 7 + T )^{4} \)
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