# Properties

 Label 25.34.b.d Level $25$ Weight $34$ Character orbit 25.b Analytic conductor $172.457$ Analytic rank $0$ Dimension $22$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$25 = 5^{2}$$ Weight: $$k$$ $$=$$ $$34$$ Character orbit: $$[\chi]$$ $$=$$ 25.b (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$172.457072203$$ Analytic rank: $$0$$ Dimension: $$22$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$22q - 98676413354q^{4} - 35567955353446q^{6} - 48599275949251316q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$22q - 98676413354q^{4} - 35567955353446q^{6} - 48599275949251316q^{9} - 388975069134227046q^{11} - 19390008595334902068q^{14} +$$$$57\!\cdots\!02$$$$q^{16} +$$$$46\!\cdots\!30$$$$q^{19} +$$$$79\!\cdots\!24$$$$q^{21} +$$$$40\!\cdots\!90$$$$q^{24} +$$$$46\!\cdots\!84$$$$q^{26} -$$$$33\!\cdots\!80$$$$q^{29} -$$$$37\!\cdots\!76$$$$q^{31} +$$$$89\!\cdots\!02$$$$q^{34} +$$$$38\!\cdots\!12$$$$q^{36} +$$$$71\!\cdots\!88$$$$q^{39} +$$$$10\!\cdots\!34$$$$q^{41} +$$$$59\!\cdots\!22$$$$q^{44} +$$$$33\!\cdots\!64$$$$q^{46} +$$$$50\!\cdots\!46$$$$q^{49} +$$$$13\!\cdots\!14$$$$q^{51} +$$$$42\!\cdots\!30$$$$q^{54} +$$$$67\!\cdots\!20$$$$q^{56} +$$$$13\!\cdots\!40$$$$q^{59} -$$$$68\!\cdots\!96$$$$q^{61} -$$$$28\!\cdots\!74$$$$q^{64} -$$$$32\!\cdots\!22$$$$q^{66} -$$$$14\!\cdots\!52$$$$q^{69} -$$$$11\!\cdots\!36$$$$q^{71} -$$$$16\!\cdots\!08$$$$q^{74} -$$$$75\!\cdots\!10$$$$q^{76} -$$$$56\!\cdots\!80$$$$q^{79} -$$$$90\!\cdots\!58$$$$q^{81} +$$$$59\!\cdots\!32$$$$q^{84} +$$$$91\!\cdots\!24$$$$q^{86} +$$$$48\!\cdots\!10$$$$q^{89} -$$$$69\!\cdots\!96$$$$q^{91} -$$$$46\!\cdots\!88$$$$q^{94} -$$$$36\!\cdots\!06$$$$q^{96} -$$$$26\!\cdots\!12$$$$q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
24.1 177167.i 1.20911e8i −2.27982e10 0 −2.14214e13 1.11259e14i 2.51723e15i −9.06036e15 0
24.2 171393.i 3.66048e7i −2.07856e10 0 6.27381e12 4.77732e13i 2.09024e15i 4.21915e15 0
24.3 153994.i 7.94763e7i −1.51243e10 0 1.22389e13 4.85062e13i 1.00626e15i −7.57418e14 0
24.4 140551.i 1.29136e8i −1.11647e10 0 −1.81502e13 4.00085e13i 3.61884e14i −1.11170e16 0
24.5 109239.i 1.96178e7i −3.34318e9 0 −2.14302e12 7.24227e12i 5.73150e14i 5.17420e15 0
24.6 104621.i 2.08087e7i −2.35554e9 0 −2.17702e12 5.34780e13i 6.52247e14i 5.12606e15 0
24.7 100295.i 1.18201e8i −1.46921e9 0 1.18550e13 1.07417e13i 7.14175e14i −8.41244e15 0
24.8 61242.8i 1.16397e8i 4.83925e9 0 −7.12847e12 1.52684e14i 8.22441e14i −7.98915e15 0
24.9 41215.3i 1.10688e8i 6.89123e9 0 4.56203e12 7.02715e13i 6.38061e14i −6.69270e15 0
24.10 34748.6i 4.98225e7i 7.38247e9 0 −1.73126e12 1.47813e14i 5.55018e14i 3.07678e15 0
24.11 642.905i 5.85306e7i 8.58952e9 0 3.76296e10 1.03914e14i 1.10448e13i 2.13323e15 0
24.12 642.905i 5.85306e7i 8.58952e9 0 3.76296e10 1.03914e14i 1.10448e13i 2.13323e15 0
24.13 34748.6i 4.98225e7i 7.38247e9 0 −1.73126e12 1.47813e14i 5.55018e14i 3.07678e15 0
24.14 41215.3i 1.10688e8i 6.89123e9 0 4.56203e12 7.02715e13i 6.38061e14i −6.69270e15 0
24.15 61242.8i 1.16397e8i 4.83925e9 0 −7.12847e12 1.52684e14i 8.22441e14i −7.98915e15 0
24.16 100295.i 1.18201e8i −1.46921e9 0 1.18550e13 1.07417e13i 7.14175e14i −8.41244e15 0
24.17 104621.i 2.08087e7i −2.35554e9 0 −2.17702e12 5.34780e13i 6.52247e14i 5.12606e15 0
24.18 109239.i 1.96178e7i −3.34318e9 0 −2.14302e12 7.24227e12i 5.73150e14i 5.17420e15 0
24.19 140551.i 1.29136e8i −1.11647e10 0 −1.81502e13 4.00085e13i 3.61884e14i −1.11170e16 0
24.20 153994.i 7.94763e7i −1.51243e10 0 1.22389e13 4.85062e13i 1.00626e15i −7.57418e14 0
See all 22 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 24.22 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.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 25.34.b.d 22
5.b even 2 1 inner 25.34.b.d 22
5.c odd 4 1 25.34.a.d 11
5.c odd 4 1 25.34.a.e yes 11

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
25.34.a.d 11 5.c odd 4 1
25.34.a.e yes 11 5.c odd 4 1
25.34.b.d 22 1.a even 1 1 trivial
25.34.b.d 22 5.b even 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$87\!\cdots\!80$$$$T_{2}^{18} +$$$$29\!\cdots\!60$$$$T_{2}^{16} +$$$$59\!\cdots\!80$$$$T_{2}^{14} +$$$$75\!\cdots\!88$$$$T_{2}^{12} +$$$$59\!\cdots\!12$$$$T_{2}^{10} +$$$$28\!\cdots\!20$$$$T_{2}^{8} +$$$$73\!\cdots\!40$$$$T_{2}^{6} +$$$$92\!\cdots\!20$$$$T_{2}^{4} +$$$$43\!\cdots\!36$$$$T_{2}^{2} +$$$$18\!\cdots\!24$$">$$T_{2}^{22} + \cdots$$ acting on $$S_{34}^{\mathrm{new}}(25, [\chi])$$.