Properties

Label 370.2.c.b
Level $370$
Weight $2$
Character orbit 370.c
Analytic conductor $2.954$
Analytic rank $0$
Dimension $10$
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.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(2.95446487479\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} + \cdots)\)
Defining polynomial: \(x^{10} + 19 x^{8} + 103 x^{6} + 210 x^{4} + 140 x^{2} + 16\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{10} + 19 x^{8} + 103 x^{6} + 210 x^{4} + 140 x^{2} + 16\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( -3 \nu^{9} - 45 \nu^{7} - 121 \nu^{5} - 26 \nu^{3} + 12 \nu \)\()/8\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{9} + 6 \nu^{8} + 15 \nu^{7} + 94 \nu^{6} + 39 \nu^{5} + 302 \nu^{4} - 10 \nu^{3} + 212 \nu^{2} - 44 \nu + 16 \)\()/8\)
\(\beta_{4}\)\(=\)\((\)\( 3 \nu^{9} + 2 \nu^{8} + 49 \nu^{7} + 34 \nu^{6} + 181 \nu^{5} + 142 \nu^{4} + 186 \nu^{3} + 196 \nu^{2} + 20 \nu + 64 \)\()/8\)
\(\beta_{5}\)\(=\)\((\)\( \nu^{9} - 6 \nu^{8} + 15 \nu^{7} - 94 \nu^{6} + 39 \nu^{5} - 302 \nu^{4} - 10 \nu^{3} - 212 \nu^{2} - 44 \nu - 16 \)\()/8\)
\(\beta_{6}\)\(=\)\((\)\( 3 \nu^{9} - 2 \nu^{8} + 49 \nu^{7} - 34 \nu^{6} + 181 \nu^{5} - 142 \nu^{4} + 186 \nu^{3} - 196 \nu^{2} + 20 \nu - 64 \)\()/8\)
\(\beta_{7}\)\(=\)\((\)\( 3 \nu^{8} + 49 \nu^{6} + 181 \nu^{4} + 186 \nu^{2} + 28 \)\()/4\)
\(\beta_{8}\)\(=\)\((\)\( 3 \nu^{8} + 49 \nu^{6} + 181 \nu^{4} + 190 \nu^{2} + 44 \)\()/4\)
\(\beta_{9}\)\(=\)\((\)\( -\nu^{9} - 17 \nu^{7} - 71 \nu^{5} - 100 \nu^{3} - 46 \nu \)\()/2\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{8} - \beta_{7} - 4\)
\(\nu^{3}\)\(=\)\(-\beta_{9} - \beta_{6} + \beta_{5} - \beta_{4} + \beta_{3} - 7 \beta_{1}\)
\(\nu^{4}\)\(=\)\(-14 \beta_{8} + 10 \beta_{7} - 3 \beta_{6} - \beta_{5} + 3 \beta_{4} + \beta_{3} + 32\)
\(\nu^{5}\)\(=\)\(14 \beta_{9} + 14 \beta_{6} - 17 \beta_{5} + 14 \beta_{4} - 17 \beta_{3} - 2 \beta_{2} + 68 \beta_{1}\)
\(\nu^{6}\)\(=\)\(170 \beta_{8} - 108 \beta_{7} + 45 \beta_{6} + 16 \beta_{5} - 45 \beta_{4} - 16 \beta_{3} - 330\)
\(\nu^{7}\)\(=\)\(-170 \beta_{9} - 169 \beta_{6} + 215 \beta_{5} - 169 \beta_{4} + 215 \beta_{3} + 32 \beta_{2} - 748 \beta_{1}\)
\(\nu^{8}\)\(=\)\(-1994 \beta_{8} + 1224 \beta_{7} - 554 \beta_{6} - 201 \beta_{5} + 554 \beta_{4} + 201 \beta_{3} + 3698\)
\(\nu^{9}\)\(=\)\(1994 \beta_{9} + 1979 \beta_{6} - 2548 \beta_{5} + 1979 \beta_{4} - 2548 \beta_{3} - 402 \beta_{2} + 8542 \beta_{1}\)

Character values

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

\(n\) \(261\) \(297\)
\(\chi(n)\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
369.1
3.40359i
1.78647i
1.76216i
0.987983i
0.377861i
0.377861i
0.987983i
1.76216i
1.78647i
3.40359i
1.00000 3.40359i 1.00000 1.28269 + 1.83159i 3.40359i 2.06225i 1.00000 −8.58443 1.28269 + 1.83159i
369.2 1.00000 1.78647i 1.00000 −2.21736 0.288618i 1.78647i 3.14934i 1.00000 −0.191472 −2.21736 0.288618i
369.3 1.00000 1.76216i 1.00000 1.62868 1.53213i 1.76216i 1.22131i 1.00000 −0.105209 1.62868 1.53213i
369.4 1.00000 0.987983i 1.00000 1.85396 + 1.25013i 0.987983i 4.78937i 1.00000 2.02389 1.85396 + 1.25013i
369.5 1.00000 0.377861i 1.00000 −1.04797 + 1.97529i 0.377861i 0.631751i 1.00000 2.85722 −1.04797 + 1.97529i
369.6 1.00000 0.377861i 1.00000 −1.04797 1.97529i 0.377861i 0.631751i 1.00000 2.85722 −1.04797 1.97529i
369.7 1.00000 0.987983i 1.00000 1.85396 1.25013i 0.987983i 4.78937i 1.00000 2.02389 1.85396 1.25013i
369.8 1.00000 1.76216i 1.00000 1.62868 + 1.53213i 1.76216i 1.22131i 1.00000 −0.105209 1.62868 + 1.53213i
369.9 1.00000 1.78647i 1.00000 −2.21736 + 0.288618i 1.78647i 3.14934i 1.00000 −0.191472 −2.21736 + 0.288618i
369.10 1.00000 3.40359i 1.00000 1.28269 1.83159i 3.40359i 2.06225i 1.00000 −8.58443 1.28269 1.83159i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 369.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
185.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 370.2.c.b yes 10
3.b odd 2 1 3330.2.e.c 10
5.b even 2 1 370.2.c.a 10
5.c odd 4 2 1850.2.d.i 20
15.d odd 2 1 3330.2.e.d 10
37.b even 2 1 370.2.c.a 10
111.d odd 2 1 3330.2.e.d 10
185.d even 2 1 inner 370.2.c.b yes 10
185.h odd 4 2 1850.2.d.i 20
555.b odd 2 1 3330.2.e.c 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
370.2.c.a 10 5.b even 2 1
370.2.c.a 10 37.b even 2 1
370.2.c.b yes 10 1.a even 1 1 trivial
370.2.c.b yes 10 185.d even 2 1 inner
1850.2.d.i 20 5.c odd 4 2
1850.2.d.i 20 185.h odd 4 2
3330.2.e.c 10 3.b odd 2 1
3330.2.e.c 10 555.b odd 2 1
3330.2.e.d 10 15.d odd 2 1
3330.2.e.d 10 111.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{13}^{5} - T_{13}^{4} - 39 T_{13}^{3} + 100 T_{13}^{2} + 160 T_{13} - 488 \) acting on \(S_{2}^{\mathrm{new}}(370, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -1 + T )^{10} \)
$3$ \( 16 + 140 T^{2} + 210 T^{4} + 103 T^{6} + 19 T^{8} + T^{10} \)
$5$ \( 3125 - 1875 T + 250 T^{2} + 400 T^{3} - 95 T^{4} + 22 T^{5} - 19 T^{6} + 16 T^{7} + 2 T^{8} - 3 T^{9} + T^{10} \)
$7$ \( 576 + 2048 T^{2} + 1684 T^{4} + 438 T^{6} + 39 T^{8} + T^{10} \)
$11$ \( ( -48 + 16 T + 51 T^{2} - 28 T^{3} + T^{5} )^{2} \)
$13$ \( ( -488 + 160 T + 100 T^{2} - 39 T^{3} - T^{4} + T^{5} )^{2} \)
$17$ \( ( 144 - 112 T - 108 T^{2} + 4 T^{3} + 9 T^{4} + T^{5} )^{2} \)
$19$ \( 9216 + 374336 T^{2} + 72240 T^{4} + 4668 T^{6} + 118 T^{8} + T^{10} \)
$23$ \( ( 768 + 256 T - 308 T^{2} - 63 T^{3} + 5 T^{4} + T^{5} )^{2} \)
$29$ \( 2262016 + 1251872 T^{2} + 194729 T^{4} + 10258 T^{6} + 182 T^{8} + T^{10} \)
$31$ \( 60516 + 346130 T^{2} + 68053 T^{4} + 4502 T^{6} + 116 T^{8} + T^{10} \)
$37$ \( 69343957 - 14993288 T + 4913341 T^{2} - 788544 T^{3} + 195434 T^{4} - 25584 T^{5} + 5282 T^{6} - 576 T^{7} + 97 T^{8} - 8 T^{9} + T^{10} \)
$41$ \( ( -36 - 8 T + 89 T^{2} - 44 T^{3} + 2 T^{4} + T^{5} )^{2} \)
$43$ \( ( 2624 + 1856 T + 76 T^{2} - 88 T^{3} - 5 T^{4} + T^{5} )^{2} \)
$47$ \( 4596736 + 10494016 T^{2} + 992816 T^{4} + 29148 T^{6} + 310 T^{8} + T^{10} \)
$53$ \( 39337984 + 10468352 T^{2} + 826000 T^{4} + 22964 T^{6} + 257 T^{8} + T^{10} \)
$59$ \( 21827584 + 7068992 T^{2} + 590448 T^{4} + 18844 T^{6} + 238 T^{8} + T^{10} \)
$61$ \( 82944 + 89792 T^{2} + 28169 T^{4} + 3402 T^{6} + 150 T^{8} + T^{10} \)
$67$ \( 559417104 + 70672748 T^{2} + 2839394 T^{4} + 48727 T^{6} + 367 T^{8} + T^{10} \)
$71$ \( ( 4608 - 2816 T - 1896 T^{2} - 164 T^{3} + 10 T^{4} + T^{5} )^{2} \)
$73$ \( 589824 + 785408 T^{2} + 189152 T^{4} + 10965 T^{6} + 189 T^{8} + T^{10} \)
$79$ \( 186486336 + 29601332 T^{2} + 1472598 T^{4} + 31019 T^{6} + 291 T^{8} + T^{10} \)
$83$ \( 1024 + 23736128 T^{2} + 2459520 T^{4} + 62212 T^{6} + 466 T^{8} + T^{10} \)
$89$ \( 1230045184 + 117901312 T^{2} + 3949632 T^{4} + 59888 T^{6} + 412 T^{8} + T^{10} \)
$97$ \( ( -1168 + 2824 T + 248 T^{2} - 178 T^{3} + T^{4} + T^{5} )^{2} \)
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