Properties

Label 74.4.f.a
Level $74$
Weight $4$
Character orbit 74.f
Analytic conductor $4.366$
Analytic rank $0$
Dimension $24$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 74 = 2 \cdot 37 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 74.f (of order \(9\), degree \(6\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(4.36614134042\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(4\) over \(\Q(\zeta_{9})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 24 q + 6 q^{3} - 9 q^{5} + 36 q^{6} - 75 q^{7} + 96 q^{8} - 48 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q + 6 q^{3} - 9 q^{5} + 36 q^{6} - 75 q^{7} + 96 q^{8} - 48 q^{9} + 6 q^{10} + 87 q^{11} + 24 q^{12} - 45 q^{13} + 6 q^{14} - 141 q^{15} + 120 q^{17} - 192 q^{18} - 177 q^{19} - 36 q^{20} + 159 q^{21} - 42 q^{22} + 45 q^{23} - 48 q^{24} - 321 q^{25} + 150 q^{26} + 435 q^{27} + 96 q^{28} - 153 q^{29} + 282 q^{30} + 1002 q^{31} - 210 q^{33} - 456 q^{34} + 3 q^{35} + 216 q^{36} - 1239 q^{37} + 780 q^{38} - 474 q^{39} - 144 q^{40} - 1437 q^{41} - 318 q^{42} + 504 q^{43} + 84 q^{44} - 171 q^{45} + 714 q^{46} + 396 q^{47} + 144 q^{48} + 399 q^{49} - 1212 q^{50} - 972 q^{51} + 504 q^{52} + 1173 q^{53} + 360 q^{54} + 1755 q^{55} - 408 q^{56} - 3525 q^{57} + 24 q^{58} + 1260 q^{59} - 108 q^{60} + 2946 q^{61} - 1380 q^{62} + 2514 q^{63} - 768 q^{64} + 1599 q^{65} - 252 q^{66} - 645 q^{67} + 1200 q^{68} - 5037 q^{69} - 366 q^{70} - 753 q^{71} - 768 q^{72} - 876 q^{73} - 1338 q^{74} + 6960 q^{75} - 708 q^{76} + 1197 q^{77} - 1590 q^{78} - 3276 q^{79} + 96 q^{80} + 36 q^{81} - 216 q^{82} - 1110 q^{83} + 504 q^{84} + 1992 q^{85} - 192 q^{86} + 1125 q^{87} - 696 q^{88} + 3045 q^{89} + 4590 q^{90} + 5046 q^{91} + 2604 q^{92} - 4917 q^{93} + 78 q^{94} + 2274 q^{95} + 384 q^{96} - 624 q^{97} + 1902 q^{98} - 5904 q^{99}+O(q^{100}) \) 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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
7.1 1.87939 0.684040i −4.40012 1.60151i 3.06418 2.57115i −0.0856491 0.485740i −9.36501 −4.85295 27.5224i 4.00000 6.92820i −3.88702 3.26159i −0.493234 0.854306i
7.2 1.87939 0.684040i −1.71848 0.625476i 3.06418 2.57115i −3.13691 17.7903i −3.65754 1.67925 + 9.52347i 4.00000 6.92820i −18.1212 15.2055i −18.0648 31.2891i
7.3 1.87939 0.684040i 2.73860 + 0.996769i 3.06418 2.57115i 3.22893 + 18.3122i 5.82871 2.30707 + 13.0840i 4.00000 6.92820i −14.1768 11.8958i 18.5947 + 32.2069i
7.4 1.87939 0.684040i 6.85178 + 2.49384i 3.06418 2.57115i −0.811775 4.60380i 14.5830 −0.602158 3.41501i 4.00000 6.92820i 20.0444 + 16.8192i −4.67483 8.09704i
9.1 −0.347296 1.96962i −1.43352 + 8.12991i −3.75877 + 1.36808i −2.21967 1.86252i 16.5106 −21.7866 18.2811i 4.00000 + 6.92820i −38.6687 14.0742i −2.89757 + 5.01874i
9.2 −0.347296 1.96962i −0.666426 + 3.77949i −3.75877 + 1.36808i −4.75553 3.99036i 7.67559 19.2298 + 16.1357i 4.00000 + 6.92820i 11.5313 + 4.19704i −6.20790 + 10.7524i
9.3 −0.347296 1.96962i −0.0459350 + 0.260510i −3.75877 + 1.36808i 12.6348 + 10.6018i 0.529058 0.748056 + 0.627693i 4.00000 + 6.92820i 25.3059 + 9.21061i 16.4935 28.5676i
9.4 −0.347296 1.96962i 1.09285 6.19786i −3.75877 + 1.36808i −4.09539 3.43644i −12.5869 −9.53718 8.00265i 4.00000 + 6.92820i −11.8474 4.31210i −5.34616 + 9.25982i
33.1 −0.347296 + 1.96962i −1.43352 8.12991i −3.75877 1.36808i −2.21967 + 1.86252i 16.5106 −21.7866 + 18.2811i 4.00000 6.92820i −38.6687 + 14.0742i −2.89757 5.01874i
33.2 −0.347296 + 1.96962i −0.666426 3.77949i −3.75877 1.36808i −4.75553 + 3.99036i 7.67559 19.2298 16.1357i 4.00000 6.92820i 11.5313 4.19704i −6.20790 10.7524i
33.3 −0.347296 + 1.96962i −0.0459350 0.260510i −3.75877 1.36808i 12.6348 10.6018i 0.529058 0.748056 0.627693i 4.00000 6.92820i 25.3059 9.21061i 16.4935 + 28.5676i
33.4 −0.347296 + 1.96962i 1.09285 + 6.19786i −3.75877 1.36808i −4.09539 + 3.43644i −12.5869 −9.53718 + 8.00265i 4.00000 6.92820i −11.8474 + 4.31210i −5.34616 9.25982i
49.1 −1.53209 + 1.28558i −6.58177 5.52276i 0.694593 3.93923i −2.85390 + 1.03873i 17.1838 −0.561982 + 0.204545i 4.00000 + 6.92820i 8.13033 + 46.1094i 3.03706 5.26034i
49.2 −1.53209 + 1.28558i −1.34104 1.12526i 0.694593 3.93923i 6.03617 2.19699i 3.50120 −28.8243 + 10.4912i 4.00000 + 6.92820i −4.15634 23.5718i −6.42356 + 11.1259i
49.3 −1.53209 + 1.28558i 3.31240 + 2.77944i 0.694593 3.93923i 10.2218 3.72042i −8.64807 12.8394 4.67315i 4.00000 + 6.92820i −1.44175 8.17656i −10.8778 + 18.8409i
49.4 −1.53209 + 1.28558i 5.19166 + 4.35632i 0.694593 3.93923i −18.6628 + 6.79271i −13.5545 −8.13840 + 2.96214i 4.00000 + 6.92820i 3.28731 + 18.6432i 19.8606 34.3995i
53.1 1.87939 + 0.684040i −4.40012 + 1.60151i 3.06418 + 2.57115i −0.0856491 + 0.485740i −9.36501 −4.85295 + 27.5224i 4.00000 + 6.92820i −3.88702 + 3.26159i −0.493234 + 0.854306i
53.2 1.87939 + 0.684040i −1.71848 + 0.625476i 3.06418 + 2.57115i −3.13691 + 17.7903i −3.65754 1.67925 9.52347i 4.00000 + 6.92820i −18.1212 + 15.2055i −18.0648 + 31.2891i
53.3 1.87939 + 0.684040i 2.73860 0.996769i 3.06418 + 2.57115i 3.22893 18.3122i 5.82871 2.30707 13.0840i 4.00000 + 6.92820i −14.1768 + 11.8958i 18.5947 32.2069i
53.4 1.87939 + 0.684040i 6.85178 2.49384i 3.06418 + 2.57115i −0.811775 + 4.60380i 14.5830 −0.602158 + 3.41501i 4.00000 + 6.92820i 20.0444 16.8192i −4.67483 + 8.09704i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 71.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
37.f even 9 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 74.4.f.a 24
37.f even 9 1 inner 74.4.f.a 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
74.4.f.a 24 1.a even 1 1 trivial
74.4.f.a 24 37.f even 9 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{24} - 6 T_{3}^{23} + 42 T_{3}^{22} - 487 T_{3}^{21} + 3579 T_{3}^{20} - 22725 T_{3}^{19} + 334902 T_{3}^{18} - 2355801 T_{3}^{17} + 9595461 T_{3}^{16} - 1079640 T_{3}^{15} - 201808932 T_{3}^{14} + \cdots + 17897307931441 \) acting on \(S_{4}^{\mathrm{new}}(74, [\chi])\). Copy content Toggle raw display