Properties

Label 350.10.a.c
Level $350$
Weight $10$
Character orbit 350.a
Self dual yes
Analytic conductor $180.263$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 350 = 2 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 350.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(180.262542657\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 14)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + 16 q^{2} + 6 q^{3} + 256 q^{4} + 96 q^{6} + 2401 q^{7} + 4096 q^{8} - 19647 q^{9} + O(q^{10}) \) \( q + 16 q^{2} + 6 q^{3} + 256 q^{4} + 96 q^{6} + 2401 q^{7} + 4096 q^{8} - 19647 q^{9} - 54152 q^{11} + 1536 q^{12} + 113172 q^{13} + 38416 q^{14} + 65536 q^{16} - 6262 q^{17} - 314352 q^{18} + 257078 q^{19} + 14406 q^{21} - 866432 q^{22} + 266000 q^{23} + 24576 q^{24} + 1810752 q^{26} - 235980 q^{27} + 614656 q^{28} + 1574714 q^{29} - 4637484 q^{31} + 1048576 q^{32} - 324912 q^{33} - 100192 q^{34} - 5029632 q^{36} + 11946238 q^{37} + 4113248 q^{38} + 679032 q^{39} + 21909126 q^{41} + 230496 q^{42} - 27520592 q^{43} - 13862912 q^{44} + 4256000 q^{46} - 52927836 q^{47} + 393216 q^{48} + 5764801 q^{49} - 37572 q^{51} + 28972032 q^{52} - 16221222 q^{53} - 3775680 q^{54} + 9834496 q^{56} + 1542468 q^{57} + 25195424 q^{58} - 140509618 q^{59} - 202963560 q^{61} - 74199744 q^{62} - 47172447 q^{63} + 16777216 q^{64} - 5198592 q^{66} - 153734572 q^{67} - 1603072 q^{68} + 1596000 q^{69} + 279655936 q^{71} - 80474112 q^{72} + 404022830 q^{73} + 191139808 q^{74} + 65811968 q^{76} - 130018952 q^{77} + 10864512 q^{78} - 130689816 q^{79} + 385296021 q^{81} + 350546016 q^{82} - 420134014 q^{83} + 3687936 q^{84} - 440329472 q^{86} + 9448284 q^{87} - 221806592 q^{88} - 469542390 q^{89} + 271725972 q^{91} + 68096000 q^{92} - 27824904 q^{93} - 846845376 q^{94} + 6291456 q^{96} + 872501690 q^{97} + 92236816 q^{98} + 1063924344 q^{99} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
0
16.0000 6.00000 256.000 0 96.0000 2401.00 4096.00 −19647.0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(5\) \(1\)
\(7\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 350.10.a.c 1
5.b even 2 1 14.10.a.a 1
5.c odd 4 2 350.10.c.b 2
15.d odd 2 1 126.10.a.e 1
20.d odd 2 1 112.10.a.b 1
35.c odd 2 1 98.10.a.a 1
35.i odd 6 2 98.10.c.e 2
35.j even 6 2 98.10.c.f 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
14.10.a.a 1 5.b even 2 1
98.10.a.a 1 35.c odd 2 1
98.10.c.e 2 35.i odd 6 2
98.10.c.f 2 35.j even 6 2
112.10.a.b 1 20.d odd 2 1
126.10.a.e 1 15.d odd 2 1
350.10.a.c 1 1.a even 1 1 trivial
350.10.c.b 2 5.c odd 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} - 6 \) acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(350))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -16 + T \)
$3$ \( -6 + T \)
$5$ \( T \)
$7$ \( -2401 + T \)
$11$ \( 54152 + T \)
$13$ \( -113172 + T \)
$17$ \( 6262 + T \)
$19$ \( -257078 + T \)
$23$ \( -266000 + T \)
$29$ \( -1574714 + T \)
$31$ \( 4637484 + T \)
$37$ \( -11946238 + T \)
$41$ \( -21909126 + T \)
$43$ \( 27520592 + T \)
$47$ \( 52927836 + T \)
$53$ \( 16221222 + T \)
$59$ \( 140509618 + T \)
$61$ \( 202963560 + T \)
$67$ \( 153734572 + T \)
$71$ \( -279655936 + T \)
$73$ \( -404022830 + T \)
$79$ \( 130689816 + T \)
$83$ \( 420134014 + T \)
$89$ \( 469542390 + T \)
$97$ \( -872501690 + T \)
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