Properties

Label 2401.4.a.c
Level $2401$
Weight $4$
Character orbit 2401.a
Self dual yes
Analytic conductor $141.664$
Analytic rank $1$
Dimension $39$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2401 = 7^{4} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2401.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(141.663585924\)
Analytic rank: \(1\)
Dimension: \(39\)
Twist minimal: no (minimal twist has level 49)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 39 q + q^{2} - q^{3} + 145 q^{4} - 27 q^{5} - 41 q^{6} - 12 q^{8} + 312 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 39 q + q^{2} - q^{3} + 145 q^{4} - 27 q^{5} - 41 q^{6} - 12 q^{8} + 312 q^{9} - 78 q^{10} + q^{11} + 91 q^{12} - 77 q^{13} - 161 q^{15} + 461 q^{16} - 211 q^{17} + 8 q^{18} - 314 q^{19} - 476 q^{20} - 61 q^{22} - 69 q^{23} - 330 q^{24} + 606 q^{25} - 504 q^{26} + 50 q^{27} + 57 q^{29} + 42 q^{30} - 638 q^{31} - 1600 q^{32} - 1574 q^{33} - 1343 q^{34} + 782 q^{36} + 71 q^{37} - 1359 q^{38} - 84 q^{39} + 155 q^{40} - 1393 q^{41} - 125 q^{43} + 52 q^{44} - 1129 q^{45} - 1454 q^{46} - 1483 q^{47} + 974 q^{48} + 3074 q^{50} - 2044 q^{51} - 3899 q^{52} + 2213 q^{53} - 1142 q^{54} - 1604 q^{55} + 98 q^{57} + 2403 q^{58} - 2073 q^{59} - 1519 q^{60} - 2575 q^{61} - 1742 q^{62} + 1358 q^{64} + 1876 q^{65} + 48 q^{66} + 176 q^{67} - 3038 q^{68} + 638 q^{69} - 1259 q^{71} - 3799 q^{72} + 307 q^{73} + 3845 q^{74} - 131 q^{75} - 1974 q^{76} - 6041 q^{78} + 22 q^{79} + 804 q^{80} + 795 q^{81} - 8043 q^{82} - 6349 q^{83} + 3094 q^{85} + 1745 q^{86} - 9508 q^{87} + 1299 q^{88} - 2253 q^{89} - 11156 q^{90} + 1284 q^{92} - 3430 q^{93} - 2738 q^{94} - 3290 q^{95} - 3031 q^{96} + 770 q^{97} + 5384 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1 −5.52124 4.05315 22.4841 −14.0923 −22.3784 0 −79.9700 −10.5720 77.8068
1.2 −5.47147 8.31145 21.9370 5.94042 −45.4758 0 −76.2558 42.0802 −32.5028
1.3 −5.14785 −6.96013 18.5003 −3.24701 35.8297 0 −54.0542 21.4435 16.7151
1.4 −4.68909 −5.32877 13.9876 6.47646 24.9871 0 −28.0763 1.39583 −30.3687
1.5 −4.44388 −6.83693 11.7481 2.20264 30.3825 0 −16.6559 19.7437 −9.78829
1.6 −4.34611 2.32971 10.8887 2.24811 −10.1252 0 −12.5546 −21.5725 −9.77055
1.7 −3.93571 3.09696 7.48982 18.2977 −12.1887 0 2.00794 −17.4089 −72.0144
1.8 −3.74761 −2.12177 6.04460 −10.5286 7.95158 0 7.32810 −22.4981 39.4572
1.9 −3.57033 9.05210 4.74728 −15.7342 −32.3190 0 11.6133 54.9406 56.1764
1.10 −2.58597 −3.61260 −1.31274 −3.92216 9.34209 0 24.0825 −13.9491 10.1426
1.11 −2.46450 5.32585 −1.92624 −15.0765 −13.1256 0 24.4632 1.36464 37.1561
1.12 −2.34370 8.19575 −2.50708 −0.508590 −19.2083 0 24.6254 40.1702 1.19198
1.13 −2.11024 5.68890 −3.54689 12.6417 −12.0050 0 24.3667 5.36363 −26.6769
1.14 −2.08878 2.27918 −3.63702 20.5308 −4.76069 0 24.3071 −21.8054 −42.8842
1.15 −2.07454 −7.56379 −3.69627 6.53935 15.6914 0 24.2644 30.2110 −13.5662
1.16 −1.58052 −3.81461 −5.50196 −14.4403 6.02907 0 21.3401 −12.4487 22.8232
1.17 −1.28751 −1.89643 −6.34231 −17.6887 2.44167 0 18.4659 −23.4036 22.7744
1.18 −0.904464 −9.08614 −7.18194 3.14137 8.21809 0 13.7315 55.5579 −2.84125
1.19 −0.275911 −3.60983 −7.92387 3.79895 0.995991 0 4.39357 −13.9691 −1.04817
1.20 0.246994 −9.97745 −7.93899 15.8609 −2.46437 0 −3.93684 72.5495 3.91755
See all 39 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.39
Significant digits:
Format:

Atkin-Lehner signs

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

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 2401.4.a.c 39
7.b odd 2 1 2401.4.a.d 39
49.f odd 14 2 49.4.e.a 78
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
49.4.e.a 78 49.f odd 14 2
2401.4.a.c 39 1.a even 1 1 trivial
2401.4.a.d 39 7.b odd 2 1