Properties

Label 25.33.c.b
Level $25$
Weight $33$
Character orbit 25.c
Analytic conductor $162.167$
Analytic rank $0$
Dimension $30$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 25 = 5^{2} \)
Weight: \( k \) \(=\) \( 33 \)
Character orbit: \([\chi]\) \(=\) 25.c (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(162.166637856\)
Analytic rank: \(0\)
Dimension: \(30\)
Relative dimension: \(15\) over \(\Q(i)\)
Twist minimal: no (minimal twist has level 5)
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.

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

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
7.1 −84542.4 84542.4i −5.85262e6 + 5.85262e6i 9.99987e9i 0 9.89589e11 4.06221e12 + 4.06221e12i 4.82306e14 4.82306e14i 1.78451e15i 0
7.2 −68938.0 68938.0i −5.43018e7 + 5.43018e7i 5.20993e9i 0 7.48691e12 −3.40627e13 3.40627e13i 6.30755e13 6.30755e13i 4.04435e15i 0
7.3 −65410.6 65410.6i 1.37699e7 1.37699e7i 4.26212e9i 0 −1.80139e12 2.41000e12 + 2.41000e12i −2.14856e12 + 2.14856e12i 1.47380e15i 0
7.4 −61735.6 61735.6i 5.69775e7 5.69775e7i 3.32760e9i 0 −7.03508e12 1.03601e13 + 1.03601e13i −5.97211e13 + 5.97211e13i 4.63985e15i 0
7.5 −32395.0 32395.0i −1.63684e7 + 1.63684e7i 2.19610e9i 0 1.06051e12 2.79912e13 + 2.79912e13i −2.10278e14 + 2.10278e14i 1.31717e15i 0
7.6 −25524.4 25524.4i −3.82280e7 + 3.82280e7i 2.99198e9i 0 1.95149e12 1.22489e13 + 1.22489e13i −1.85995e14 + 1.85995e14i 1.06974e15i 0
7.7 −24352.5 24352.5i 2.67962e7 2.67962e7i 3.10888e9i 0 −1.30511e12 −4.33877e13 4.33877e13i −1.80302e14 + 1.80302e14i 4.16950e14i 0
7.8 5100.58 + 5100.58i 7.18185e6 7.18185e6i 4.24294e9i 0 7.32632e10 −2.17171e13 2.17171e13i 4.35482e13 4.35482e13i 1.74986e15i 0
7.9 7096.95 + 7096.95i 4.70380e7 4.70380e7i 4.19423e9i 0 6.67652e11 3.04405e13 + 3.04405e13i 6.02475e13 6.02475e13i 2.57212e15i 0
7.10 37971.9 + 37971.9i −3.04306e7 + 3.04306e7i 1.41124e9i 0 −2.31101e12 9.26812e12 + 9.26812e12i 2.16675e14 2.16675e14i 9.78102e11i 0
7.11 39535.2 + 39535.2i −4.69567e7 + 4.69567e7i 1.16890e9i 0 −3.71289e12 −2.85172e13 2.85172e13i 2.16015e14 2.16015e14i 2.55684e15i 0
7.12 42029.2 + 42029.2i 2.23370e7 2.23370e7i 7.62067e8i 0 1.87761e12 2.39791e13 + 2.39791e13i 2.12543e14 2.12543e14i 8.55140e14i 0
7.13 70960.9 + 70960.9i 4.83700e7 4.83700e7i 5.77593e9i 0 6.86476e12 −2.09740e13 2.09740e13i −1.05090e14 + 1.05090e14i 2.82629e15i 0
7.14 73528.8 + 73528.8i 4.29377e6 4.29377e6i 6.51799e9i 0 6.31431e11 −1.36489e13 1.36489e13i −1.63456e14 + 1.63456e14i 1.81615e15i 0
7.15 86676.0 + 86676.0i −3.32299e7 + 3.32299e7i 1.07305e10i 0 −5.76047e12 3.06435e13 + 3.06435e13i −5.57804e14 + 5.57804e14i 3.55432e14i 0
18.1 −84542.4 + 84542.4i −5.85262e6 5.85262e6i 9.99987e9i 0 9.89589e11 4.06221e12 4.06221e12i 4.82306e14 + 4.82306e14i 1.78451e15i 0
18.2 −68938.0 + 68938.0i −5.43018e7 5.43018e7i 5.20993e9i 0 7.48691e12 −3.40627e13 + 3.40627e13i 6.30755e13 + 6.30755e13i 4.04435e15i 0
18.3 −65410.6 + 65410.6i 1.37699e7 + 1.37699e7i 4.26212e9i 0 −1.80139e12 2.41000e12 2.41000e12i −2.14856e12 2.14856e12i 1.47380e15i 0
18.4 −61735.6 + 61735.6i 5.69775e7 + 5.69775e7i 3.32760e9i 0 −7.03508e12 1.03601e13 1.03601e13i −5.97211e13 5.97211e13i 4.63985e15i 0
18.5 −32395.0 + 32395.0i −1.63684e7 1.63684e7i 2.19610e9i 0 1.06051e12 2.79912e13 2.79912e13i −2.10278e14 2.10278e14i 1.31717e15i 0
See all 30 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 18.15
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 25.33.c.b 30
5.b even 2 1 5.33.c.a 30
5.c odd 4 1 5.33.c.a 30
5.c odd 4 1 inner 25.33.c.b 30
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5.33.c.a 30 5.b even 2 1
5.33.c.a 30 5.c odd 4 1
25.33.c.b 30 1.a even 1 1 trivial
25.33.c.b 30 5.c odd 4 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{2}^{30} - \cdots\) acting on \(S_{33}^{\mathrm{new}}(25, [\chi])\).