Properties

Label 76.7.b.a
Level $76$
Weight $7$
Character orbit 76.b
Analytic conductor $17.484$
Analytic rank $0$
Dimension $54$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 76 = 2^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 76.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(17.4841103551\)
Analytic rank: \(0\)
Dimension: \(54\)
Twist minimal: yes
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.

\(\operatorname{Tr}(f)(q) = \) \( 54q - 10q^{2} + 90q^{4} - 44q^{5} - 510q^{6} + 440q^{8} - 13122q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 54q - 10q^{2} + 90q^{4} - 44q^{5} - 510q^{6} + 440q^{8} - 13122q^{9} - 2088q^{10} - 2384q^{12} - 828q^{13} + 4116q^{14} - 7782q^{16} + 12220q^{17} + 15790q^{18} - 7000q^{20} - 16048q^{21} - 51864q^{22} + 19294q^{24} + 218346q^{25} - 18110q^{26} + 97926q^{28} - 101996q^{29} + 50076q^{30} - 235160q^{32} + 9072q^{33} - 51156q^{34} - 48028q^{36} - 63420q^{37} + 157260q^{40} - 135700q^{41} + 302290q^{42} + 137784q^{44} - 50348q^{45} - 177084q^{46} + 111320q^{48} - 1099338q^{49} + 485770q^{50} + 232980q^{52} + 892932q^{53} - 730010q^{54} - 298644q^{56} - 702534q^{58} + 1231628q^{60} + 323700q^{61} + 1573116q^{62} - 1065942q^{64} + 587512q^{65} - 74604q^{66} - 1235974q^{68} + 217376q^{69} + 1163796q^{70} - 1228320q^{72} - 739428q^{73} - 1852264q^{74} - 563144q^{77} - 2516852q^{78} - 1597436q^{80} + 4660886q^{81} - 919236q^{82} + 4928476q^{84} - 737904q^{85} - 1444124q^{86} + 2679960q^{88} - 3415940q^{89} + 4702636q^{90} - 3230670q^{92} - 5021584q^{93} + 1881960q^{94} + 1838870q^{96} + 3483660q^{97} - 3419858q^{98} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
39.1 −7.98783 0.441092i 34.6865i 63.6109 + 7.04673i 112.076 −15.2999 + 277.070i 328.667i −505.005 84.3464i −474.152 −895.242 49.4357i
39.2 −7.98783 + 0.441092i 34.6865i 63.6109 7.04673i 112.076 −15.2999 277.070i 328.667i −505.005 + 84.3464i −474.152 −895.242 + 49.4357i
39.3 −7.93032 1.05361i 19.7809i 61.7798 + 16.7110i 65.3551 −20.8415 + 156.869i 574.962i −472.326 197.615i 337.714 −518.287 68.8591i
39.4 −7.93032 + 1.05361i 19.7809i 61.7798 16.7110i 65.3551 −20.8415 156.869i 574.962i −472.326 + 197.615i 337.714 −518.287 + 68.8591i
39.5 −7.73539 2.04054i 48.2920i 55.6724 + 31.5687i −82.2790 98.5416 373.557i 359.057i −366.231 357.798i −1603.11 636.459 + 167.893i
39.6 −7.73539 + 2.04054i 48.2920i 55.6724 31.5687i −82.2790 98.5416 + 373.557i 359.057i −366.231 + 357.798i −1603.11 636.459 167.893i
39.7 −7.53857 2.67768i 14.2612i 49.6600 + 40.3718i −201.264 38.1869 107.509i 304.241i −266.263 437.319i 525.619 1517.25 + 538.922i
39.8 −7.53857 + 2.67768i 14.2612i 49.6600 40.3718i −201.264 38.1869 + 107.509i 304.241i −266.263 + 437.319i 525.619 1517.25 538.922i
39.9 −7.15940 3.56973i 19.3279i 38.5141 + 51.1142i −43.0893 −68.9953 + 138.376i 282.338i −93.2744 503.432i 355.433 308.494 + 153.817i
39.10 −7.15940 + 3.56973i 19.3279i 38.5141 51.1142i −43.0893 −68.9953 138.376i 282.338i −93.2744 + 503.432i 355.433 308.494 153.817i
39.11 −6.52919 4.62273i 11.8519i 21.2607 + 60.3654i 179.762 54.7881 77.3834i 39.4677i 140.237 492.420i 588.532 −1173.70 830.991i
39.12 −6.52919 + 4.62273i 11.8519i 21.2607 60.3654i 179.762 54.7881 + 77.3834i 39.4677i 140.237 + 492.420i 588.532 −1173.70 + 830.991i
39.13 −5.59318 5.71982i 42.2777i −1.43269 + 63.9840i −135.842 −241.821 + 236.467i 128.069i 373.990 349.679i −1058.40 759.791 + 776.994i
39.14 −5.59318 + 5.71982i 42.2777i −1.43269 63.9840i −135.842 −241.821 236.467i 128.069i 373.990 + 349.679i −1058.40 759.791 776.994i
39.15 −5.09424 6.16837i 49.6987i −12.0975 + 62.8462i 46.2569 306.560 253.177i 483.548i 449.286 245.532i −1740.96 −235.643 285.329i
39.16 −5.09424 + 6.16837i 49.6987i −12.0975 62.8462i 46.2569 306.560 + 253.177i 483.548i 449.286 + 245.532i −1740.96 −235.643 + 285.329i
39.17 −4.87964 6.33949i 47.1905i −16.3782 + 61.8689i 222.735 −299.163 + 230.273i 51.2381i 472.136 198.069i −1497.94 −1086.87 1412.03i
39.18 −4.87964 + 6.33949i 47.1905i −16.3782 61.8689i 222.735 −299.163 230.273i 51.2381i 472.136 + 198.069i −1497.94 −1086.87 + 1412.03i
39.19 −4.60097 6.54454i 21.0346i −21.6621 + 60.2225i −46.6232 137.662 96.7797i 184.040i 493.796 135.314i 286.545 214.512 + 305.128i
39.20 −4.60097 + 6.54454i 21.0346i −21.6621 60.2225i −46.6232 137.662 + 96.7797i 184.040i 493.796 + 135.314i 286.545 214.512 305.128i
See all 54 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 39.54
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 76.7.b.a 54
4.b odd 2 1 inner 76.7.b.a 54
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
76.7.b.a 54 1.a even 1 1 trivial
76.7.b.a 54 4.b odd 2 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{7}^{\mathrm{new}}(76, [\chi])\).