# 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 $2$

# Related objects

## 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: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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 $$S_{33}^{\mathrm{new}}(25, [\chi])$$.