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

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Newspace parameters

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

Newform invariants

Self dual: No
Analytic conductor: \(32.4333275711\)
Analytic rank: \(0\)
Dimension: \(30\)
Relative dimension: \(15\) over \(\Q(i)\)
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.

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

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

This newform does not have CM; other inner twists have not been computed.

Hecke kernels

There are no other newforms in \(S_{33}^{\mathrm{new}}(5, [\chi])\).