Properties

Label 304.7.e.f
Level $304$
Weight $7$
Character orbit 304.e
Analytic conductor $69.936$
Analytic rank $0$
Dimension $30$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 304 = 2^{4} \cdot 19 \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 304.e (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(69.9364414204\)
Analytic rank: \(0\)
Dimension: \(30\)
Twist minimal: no (minimal twist has level 152)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 30q + 720q^{7} - 8670q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 30q + 720q^{7} - 8670q^{9} + 2524q^{11} + 9700q^{17} - 4014q^{19} + 39376q^{23} + 110742q^{25} + 19976q^{35} + 266500q^{39} + 106788q^{43} - 91360q^{45} - 222756q^{47} + 593586q^{49} - 540936q^{55} - 545972q^{57} - 242640q^{61} + 377716q^{63} + 545964q^{73} - 272356q^{77} + 2189926q^{81} - 1542652q^{83} - 826908q^{85} + 2729572q^{87} - 2139912q^{93} - 2142716q^{95} + 293012q^{99} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
113.1 0 50.6470i 0 207.318 0 −216.532 0 −1836.12 0
113.2 0 48.7582i 0 −200.006 0 536.711 0 −1648.37 0
113.3 0 47.0707i 0 −22.4613 0 196.082 0 −1486.65 0
113.4 0 43.2780i 0 −100.391 0 −640.715 0 −1143.99 0
113.5 0 36.8625i 0 9.31317 0 −333.649 0 −629.840 0
113.6 0 33.4715i 0 71.5877 0 −51.8109 0 −391.339 0
113.7 0 33.1604i 0 225.972 0 256.290 0 −370.612 0
113.8 0 28.4003i 0 32.4520 0 263.602 0 −77.5754 0
113.9 0 27.9511i 0 −141.368 0 −302.893 0 −52.2651 0
113.10 0 20.4841i 0 −148.208 0 486.917 0 309.402 0
113.11 0 17.1584i 0 166.964 0 600.271 0 434.588 0
113.12 0 11.4888i 0 −77.3334 0 256.225 0 597.007 0
113.13 0 9.57785i 0 39.7533 0 24.5800 0 637.265 0
113.14 0 9.20554i 0 −204.194 0 −264.373 0 644.258 0
113.15 0 7.05423i 0 140.601 0 −450.706 0 679.238 0
113.16 0 7.05423i 0 140.601 0 −450.706 0 679.238 0
113.17 0 9.20554i 0 −204.194 0 −264.373 0 644.258 0
113.18 0 9.57785i 0 39.7533 0 24.5800 0 637.265 0
113.19 0 11.4888i 0 −77.3334 0 256.225 0 597.007 0
113.20 0 17.1584i 0 166.964 0 600.271 0 434.588 0
See all 30 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 113.30
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 304.7.e.f 30
4.b odd 2 1 152.7.e.a 30
19.b odd 2 1 inner 304.7.e.f 30
76.d even 2 1 152.7.e.a 30
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
152.7.e.a 30 4.b odd 2 1
152.7.e.a 30 76.d even 2 1
304.7.e.f 30 1.a even 1 1 trivial
304.7.e.f 30 19.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{7}^{\mathrm{new}}(304, [\chi])\):

\(10\!\cdots\!99\)\( T_{3}^{22} + \)\(18\!\cdots\!10\)\( T_{3}^{20} + \)\(22\!\cdots\!17\)\( T_{3}^{18} + \)\(20\!\cdots\!20\)\( T_{3}^{16} + \)\(12\!\cdots\!52\)\( T_{3}^{14} + \)\(53\!\cdots\!12\)\( T_{3}^{12} + \)\(15\!\cdots\!28\)\( T_{3}^{10} + \)\(30\!\cdots\!52\)\( T_{3}^{8} + \)\(37\!\cdots\!96\)\( T_{3}^{6} + \)\(27\!\cdots\!08\)\( T_{3}^{4} + \)\(10\!\cdots\!80\)\( T_{3}^{2} + \)\(16\!\cdots\!68\)\( \)">\(T_{3}^{30} + \cdots\)
\(20\!\cdots\!35\)\( T_{5}^{9} - \)\(44\!\cdots\!50\)\( T_{5}^{8} + \)\(26\!\cdots\!00\)\( T_{5}^{7} + \)\(51\!\cdots\!00\)\( T_{5}^{6} - \)\(15\!\cdots\!00\)\( T_{5}^{5} - \)\(12\!\cdots\!00\)\( T_{5}^{4} + \)\(34\!\cdots\!00\)\( T_{5}^{3} - \)\(28\!\cdots\!00\)\( T_{5}^{2} - \)\(15\!\cdots\!00\)\( T_{5} + \)\(14\!\cdots\!00\)\( \)">\(T_{5}^{15} - \cdots\)