Properties

Label 5.33.c.a
Level $5$
Weight $33$
Character orbit 5.c
Analytic conductor $32.433$
Analytic rank $0$
Dimension $30$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 5 \)
Weight: \( k \) \(=\) \( 33 \)
Character orbit: \([\chi]\) \(=\) 5.c (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(32.4333275711\)
Analytic rank: \(0\)
Dimension: \(30\)
Relative dimension: \(15\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 30q - 2q^{2} - 2792232q^{3} + 229266409900q^{5} - 645476451240q^{6} + 21807690136848q^{7} + 340768936037220q^{8} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 30q - 2q^{2} - 2792232q^{3} + 229266409900q^{5} - 645476451240q^{6} + 21807690136848q^{7} + 340768936037220q^{8} - 17555754485145450q^{10} - 60908362837533640q^{11} + 444566273630869608q^{12} + 649759187023107138q^{13} + 6285624407445962400q^{15} - 46108906958970522120q^{16} - \)\(21\!\cdots\!02\)\(q^{17} + \)\(24\!\cdots\!58\)\(q^{18} - \)\(30\!\cdots\!00\)\(q^{20} + \)\(45\!\cdots\!60\)\(q^{21} - \)\(11\!\cdots\!44\)\(q^{22} + \)\(13\!\cdots\!08\)\(q^{23} + \)\(35\!\cdots\!50\)\(q^{25} - \)\(23\!\cdots\!40\)\(q^{26} + \)\(17\!\cdots\!60\)\(q^{27} + \)\(59\!\cdots\!12\)\(q^{28} + \)\(12\!\cdots\!00\)\(q^{30} - \)\(12\!\cdots\!40\)\(q^{31} + \)\(31\!\cdots\!68\)\(q^{32} - \)\(16\!\cdots\!04\)\(q^{33} - \)\(81\!\cdots\!00\)\(q^{35} + \)\(31\!\cdots\!80\)\(q^{36} - \)\(12\!\cdots\!02\)\(q^{37} + \)\(57\!\cdots\!80\)\(q^{38} + \)\(21\!\cdots\!00\)\(q^{40} - \)\(15\!\cdots\!40\)\(q^{41} + \)\(79\!\cdots\!36\)\(q^{42} - \)\(54\!\cdots\!52\)\(q^{43} + \)\(10\!\cdots\!50\)\(q^{45} - \)\(34\!\cdots\!40\)\(q^{46} + \)\(27\!\cdots\!48\)\(q^{47} - \)\(31\!\cdots\!92\)\(q^{48} + \)\(44\!\cdots\!50\)\(q^{50} - \)\(61\!\cdots\!40\)\(q^{51} + \)\(60\!\cdots\!28\)\(q^{52} - \)\(11\!\cdots\!82\)\(q^{53} - \)\(71\!\cdots\!00\)\(q^{55} - \)\(47\!\cdots\!00\)\(q^{56} + \)\(44\!\cdots\!20\)\(q^{57} - \)\(29\!\cdots\!80\)\(q^{58} + \)\(34\!\cdots\!00\)\(q^{60} + \)\(50\!\cdots\!60\)\(q^{61} - \)\(21\!\cdots\!24\)\(q^{62} + \)\(21\!\cdots\!08\)\(q^{63} - \)\(34\!\cdots\!50\)\(q^{65} + \)\(13\!\cdots\!20\)\(q^{66} - \)\(73\!\cdots\!52\)\(q^{67} + \)\(10\!\cdots\!12\)\(q^{68} - \)\(36\!\cdots\!00\)\(q^{70} + \)\(81\!\cdots\!60\)\(q^{71} - \)\(95\!\cdots\!20\)\(q^{72} - \)\(37\!\cdots\!42\)\(q^{73} + \)\(10\!\cdots\!00\)\(q^{75} - \)\(66\!\cdots\!00\)\(q^{76} + \)\(63\!\cdots\!56\)\(q^{77} - \)\(91\!\cdots\!84\)\(q^{78} + \)\(58\!\cdots\!00\)\(q^{80} - \)\(14\!\cdots\!70\)\(q^{81} + \)\(27\!\cdots\!36\)\(q^{82} + \)\(49\!\cdots\!28\)\(q^{83} - \)\(44\!\cdots\!50\)\(q^{85} + \)\(16\!\cdots\!60\)\(q^{86} + \)\(36\!\cdots\!80\)\(q^{87} - \)\(47\!\cdots\!60\)\(q^{88} + \)\(81\!\cdots\!50\)\(q^{90} - \)\(10\!\cdots\!40\)\(q^{91} - \)\(18\!\cdots\!52\)\(q^{92} + \)\(11\!\cdots\!16\)\(q^{93} - \)\(10\!\cdots\!00\)\(q^{95} + \)\(38\!\cdots\!60\)\(q^{96} - \)\(15\!\cdots\!02\)\(q^{97} + \)\(12\!\cdots\!02\)\(q^{98} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2.1 −86676.0 86676.0i 3.32299e7 3.32299e7i 1.07305e10i 1.18818e11 + 9.57363e10i −5.76047e12 −3.06435e13 3.06435e13i 5.57804e14 5.57804e14i 3.55432e14i −2.00059e15 1.85967e16i
2.2 −73528.8 73528.8i −4.29377e6 + 4.29377e6i 6.51799e9i −1.18376e11 9.62823e10i 6.31431e11 1.36489e13 + 1.36489e13i 1.63456e14 1.63456e14i 1.81615e15i 1.62449e15 + 1.57835e16i
2.3 −70960.9 70960.9i −4.83700e7 + 4.83700e7i 5.77593e9i 1.51118e11 + 2.11291e10i 6.86476e12 2.09740e13 + 2.09740e13i 1.05090e14 1.05090e14i 2.82629e15i −9.22412e15 1.22228e16i
2.4 −42029.2 42029.2i −2.23370e7 + 2.23370e7i 7.62067e8i −7.62027e10 + 1.32198e11i 1.87761e12 −2.39791e13 2.39791e13i −2.12543e14 + 2.12543e14i 8.55140e14i 8.75889e15 2.35342e15i
2.5 −39535.2 39535.2i 4.69567e7 4.69567e7i 1.16890e9i −7.61848e10 + 1.32208e11i −3.71289e12 2.85172e13 + 2.85172e13i −2.16015e14 + 2.16015e14i 2.55684e15i 8.23885e15 2.21489e15i
2.6 −37971.9 37971.9i 3.04306e7 3.04306e7i 1.41124e9i 7.42414e10 1.33309e11i −2.31101e12 −9.26812e12 9.26812e12i −2.16675e14 + 2.16675e14i 9.78102e11i −7.88108e15 + 2.24291e15i
2.7 −7096.95 7096.95i −4.70380e7 + 4.70380e7i 4.19423e9i −3.87281e10 1.47591e11i 6.67652e11 −3.04405e13 3.04405e13i −6.02475e13 + 6.02475e13i 2.57212e15i −7.72597e14 + 1.32230e15i
2.8 −5100.58 5100.58i −7.18185e6 + 7.18185e6i 4.24294e9i 1.52141e11 + 1.16729e10i 7.32632e10 2.17171e13 + 2.17171e13i −4.35482e13 + 4.35482e13i 1.74986e15i −7.16467e14 8.35545e14i
2.9 24352.5 + 24352.5i −2.67962e7 + 2.67962e7i 3.10888e9i −1.49880e11 + 2.86196e10i −1.30511e12 4.33877e13 + 4.33877e13i 1.80302e14 1.80302e14i 4.16950e14i −4.34691e15 2.95300e15i
2.10 25524.4 + 25524.4i 3.82280e7 3.82280e7i 2.99198e9i −1.26030e11 8.60210e10i 1.95149e12 −1.22489e13 1.22489e13i 1.85995e14 1.85995e14i 1.06974e15i −1.02119e15 5.41245e15i
2.11 32395.0 + 32395.0i 1.63684e7 1.63684e7i 2.19610e9i 5.30856e10 + 1.43056e11i 1.06051e12 −2.79912e13 2.79912e13i 2.10278e14 2.10278e14i 1.31717e15i −2.91458e15 + 6.35400e15i
2.12 61735.6 + 61735.6i −5.69775e7 + 5.69775e7i 3.32760e9i 7.47335e10 + 1.33034e11i −7.03508e12 −1.03601e13 1.03601e13i 5.97211e13 5.97211e13i 4.63985e15i −3.59920e15 + 1.28266e16i
2.13 65410.6 + 65410.6i −1.37699e7 + 1.37699e7i 4.26212e9i 5.68539e10 1.41600e11i −1.80139e12 −2.41000e12 2.41000e12i 2.14856e12 2.14856e12i 1.47380e15i 1.29810e16 5.54333e15i
2.14 68938.0 + 68938.0i 5.43018e7 5.43018e7i 5.20993e9i 1.52114e11 + 1.20131e10i 7.48691e12 3.40627e13 + 3.40627e13i −6.30755e13 + 6.30755e13i 4.04435e15i 9.65829e15 + 1.13146e16i
2.15 84542.4 + 84542.4i 5.85262e6 5.85262e6i 9.99987e9i −1.33071e11 + 7.46670e10i 9.89589e11 −4.06221e12 4.06221e12i −4.82306e14 + 4.82306e14i 1.78451e15i −1.75627e16 4.93762e15i
3.1 −86676.0 + 86676.0i 3.32299e7 + 3.32299e7i 1.07305e10i 1.18818e11 9.57363e10i −5.76047e12 −3.06435e13 + 3.06435e13i 5.57804e14 + 5.57804e14i 3.55432e14i −2.00059e15 + 1.85967e16i
3.2 −73528.8 + 73528.8i −4.29377e6 4.29377e6i 6.51799e9i −1.18376e11 + 9.62823e10i 6.31431e11 1.36489e13 1.36489e13i 1.63456e14 + 1.63456e14i 1.81615e15i 1.62449e15 1.57835e16i
3.3 −70960.9 + 70960.9i −4.83700e7 4.83700e7i 5.77593e9i 1.51118e11 2.11291e10i 6.86476e12 2.09740e13 2.09740e13i 1.05090e14 + 1.05090e14i 2.82629e15i −9.22412e15 + 1.22228e16i
3.4 −42029.2 + 42029.2i −2.23370e7 2.23370e7i 7.62067e8i −7.62027e10 1.32198e11i 1.87761e12 −2.39791e13 + 2.39791e13i −2.12543e14 2.12543e14i 8.55140e14i 8.75889e15 + 2.35342e15i
3.5 −39535.2 + 39535.2i 4.69567e7 + 4.69567e7i 1.16890e9i −7.61848e10 1.32208e11i −3.71289e12 2.85172e13 2.85172e13i −2.16015e14 2.16015e14i 2.55684e15i 8.23885e15 + 2.21489e15i
See all 30 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 3.15
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 5.33.c.a 30
5.b even 2 1 25.33.c.b 30
5.c odd 4 1 inner 5.33.c.a 30
5.c odd 4 1 25.33.c.b 30
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5.33.c.a 30 1.a even 1 1 trivial
5.33.c.a 30 5.c odd 4 1 inner
25.33.c.b 30 5.b even 2 1
25.33.c.b 30 5.c odd 4 1

Hecke kernels

This newform subspace is the entire newspace \(S_{33}^{\mathrm{new}}(5, [\chi])\).

Hecke characteristic polynomials

There are no characteristic polynomials of Hecke operators in the database