# Properties

 Label 5.35.c.a Level $5$ Weight $35$ Character orbit 5.c Analytic conductor $36.613$ Analytic rank $0$ Dimension $32$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$5$$ Weight: $$k$$ $$=$$ $$35$$ Character orbit: $$[\chi]$$ $$=$$ 5.c (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$36.6128270213$$ Analytic rank: $$0$$ Dimension: $$32$$ Relative dimension: $$16$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$32q + 131070q^{2} - 78988260q^{3} - 730721770860q^{5} - 44043049959816q^{6} + 5337954235100q^{7} - 4536119823236220q^{8} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$32q + 131070q^{2} - 78988260q^{3} - 730721770860q^{5} - 44043049959816q^{6} + 5337954235100q^{7} - 4536119823236220q^{8} + 141196165961877590q^{10} + 365981602487621544q^{11} - 5313878568511975560q^{12} - 11000397968713344520q^{13} +$$$$20\!\cdots\!80$$$$q^{15} -$$$$14\!\cdots\!48$$$$q^{16} +$$$$49\!\cdots\!60$$$$q^{17} +$$$$19\!\cdots\!90$$$$q^{18} +$$$$21\!\cdots\!20$$$$q^{20} -$$$$14\!\cdots\!16$$$$q^{21} +$$$$20\!\cdots\!40$$$$q^{22} -$$$$43\!\cdots\!80$$$$q^{23} +$$$$12\!\cdots\!00$$$$q^{25} -$$$$57\!\cdots\!96$$$$q^{26} +$$$$79\!\cdots\!60$$$$q^{27} +$$$$88\!\cdots\!80$$$$q^{28} +$$$$16\!\cdots\!80$$$$q^{30} +$$$$84\!\cdots\!04$$$$q^{31} -$$$$18\!\cdots\!80$$$$q^{32} +$$$$73\!\cdots\!80$$$$q^{33} -$$$$33\!\cdots\!60$$$$q^{35} +$$$$11\!\cdots\!32$$$$q^{36} -$$$$33\!\cdots\!20$$$$q^{37} -$$$$14\!\cdots\!80$$$$q^{38} +$$$$46\!\cdots\!00$$$$q^{40} +$$$$36\!\cdots\!84$$$$q^{41} -$$$$47\!\cdots\!40$$$$q^{42} +$$$$16\!\cdots\!00$$$$q^{43} -$$$$54\!\cdots\!20$$$$q^{45} +$$$$13\!\cdots\!24$$$$q^{46} -$$$$16\!\cdots\!60$$$$q^{47} +$$$$26\!\cdots\!20$$$$q^{48} -$$$$49\!\cdots\!50$$$$q^{50} +$$$$87\!\cdots\!84$$$$q^{51} -$$$$13\!\cdots\!00$$$$q^{52} +$$$$82\!\cdots\!40$$$$q^{53} -$$$$12\!\cdots\!20$$$$q^{55} +$$$$25\!\cdots\!20$$$$q^{56} -$$$$30\!\cdots\!80$$$$q^{57} +$$$$23\!\cdots\!80$$$$q^{58} -$$$$49\!\cdots\!60$$$$q^{60} +$$$$10\!\cdots\!44$$$$q^{61} -$$$$17\!\cdots\!60$$$$q^{62} +$$$$86\!\cdots\!60$$$$q^{63} -$$$$38\!\cdots\!20$$$$q^{65} -$$$$27\!\cdots\!72$$$$q^{66} +$$$$23\!\cdots\!60$$$$q^{67} -$$$$16\!\cdots\!60$$$$q^{68} +$$$$10\!\cdots\!40$$$$q^{70} -$$$$16\!\cdots\!76$$$$q^{71} +$$$$41\!\cdots\!80$$$$q^{72} -$$$$29\!\cdots\!80$$$$q^{73} +$$$$25\!\cdots\!00$$$$q^{75} -$$$$58\!\cdots\!60$$$$q^{76} +$$$$31\!\cdots\!00$$$$q^{77} +$$$$25\!\cdots\!00$$$$q^{78} -$$$$11\!\cdots\!60$$$$q^{80} +$$$$46\!\cdots\!32$$$$q^{81} -$$$$48\!\cdots\!60$$$$q^{82} -$$$$98\!\cdots\!40$$$$q^{83} +$$$$54\!\cdots\!40$$$$q^{85} +$$$$25\!\cdots\!64$$$$q^{86} -$$$$36\!\cdots\!20$$$$q^{87} +$$$$44\!\cdots\!60$$$$q^{88} -$$$$12\!\cdots\!70$$$$q^{90} +$$$$97\!\cdots\!04$$$$q^{91} -$$$$58\!\cdots\!40$$$$q^{92} +$$$$11\!\cdots\!80$$$$q^{93} +$$$$13\!\cdots\!00$$$$q^{95} +$$$$24\!\cdots\!04$$$$q^{96} -$$$$29\!\cdots\!60$$$$q^{97} +$$$$25\!\cdots\!70$$$$q^{98} + O(q^{100})$$

## 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 −169510. 169510.i −198644. + 198644.i 4.02874e10i −3.95922e11 + 6.52168e11i 6.73444e10 1.10969e14 + 1.10969e14i 3.91696e15 3.91696e15i 1.66771e16i 1.77662e17 4.34363e16i
2.2 −151262. 151262.i −1.00065e8 + 1.00065e8i 2.85804e10i 1.61018e11 7.45755e11i 3.02721e13 −1.52632e14 1.52632e14i 1.72447e15 1.72447e15i 3.34887e15i −1.37160e17 + 8.84484e16i
2.3 −134834. 134834.i 1.19387e8 1.19387e8i 1.91805e10i 6.75906e11 3.53875e11i −3.21948e13 2.58783e14 + 2.58783e14i 2.69752e14 2.69752e14i 1.18293e16i −1.38849e17 4.34207e16i
2.4 −115430. 115430.i 1.39345e8 1.39345e8i 9.46824e9i −7.62275e11 3.18319e10i −3.21691e13 −2.97793e14 2.97793e14i −8.90153e14 + 8.90153e14i 2.21567e16i 8.43150e16 + 9.16637e16i
2.5 −85294.4 85294.4i −1.68126e8 + 1.68126e8i 2.62960e9i −5.92285e11 + 4.80911e11i 2.86804e13 2.04951e14 + 2.04951e14i −1.68964e15 + 1.68964e15i 3.98554e16i 9.15376e16 + 9.49962e15i
2.6 −83255.7 83255.7i −2.65055e7 + 2.65055e7i 3.31684e9i 5.91968e11 + 4.81301e11i 4.41347e12 −1.23465e14 1.23465e14i −1.70647e15 + 1.70647e15i 1.52721e16i −9.21369e15 8.93558e16i
2.7 −39637.5 39637.5i 937048. 937048.i 1.40376e10i −4.84629e11 5.89246e11i −7.42844e10 1.00289e14 + 1.00289e14i −1.23738e15 + 1.23738e15i 1.66754e16i −4.14676e15 + 4.25657e16i
2.8 8626.63 + 8626.63i 9.89110e7 9.89110e7i 1.70310e10i −8.06672e10 + 7.58663e11i 1.70654e12 1.10848e14 + 1.10848e14i 2.95125e14 2.95125e14i 2.88960e15i −7.24059e15 + 5.84882e15i
2.9 26483.0 + 26483.0i −1.33847e8 + 1.33847e8i 1.57772e10i 6.94260e11 3.16354e11i −7.08935e12 1.17241e13 + 1.17241e13i 8.72802e14 8.72802e14i 1.91529e16i 2.67641e16 + 1.00081e16i
2.10 36635.8 + 36635.8i 1.35857e8 1.35857e8i 1.44955e10i 6.50279e11 3.99016e11i 9.95447e12 −1.47937e14 1.47937e14i 1.16045e15 1.16045e15i 2.02371e16i 3.84418e16 + 9.20523e15i
2.11 67956.3 + 67956.3i −7.73822e7 + 7.73822e7i 7.94376e9i −6.92309e11 + 3.20600e11i −1.05172e13 −2.59708e14 2.59708e14i 1.70731e15 1.70731e15i 4.70118e15i −6.88336e16 2.52599e16i
2.12 109228. + 109228.i 1.53588e7 1.53588e7i 6.68164e9i 1.05031e11 7.55675e11i 3.35523e12 5.89338e13 + 5.89338e13i 1.14670e15 1.14670e15i 1.62054e16i 9.40132e16 7.10686e16i
2.13 117370. + 117370.i −6.15595e7 + 6.15595e7i 1.03717e10i 3.42977e11 + 6.81501e11i −1.44505e13 2.43299e14 + 2.43299e14i 7.99081e14 7.99081e14i 9.09803e15i −3.97325e16 + 1.20243e17i
2.14 136964. + 136964.i 1.30224e8 1.30224e8i 2.03383e10i −7.62514e11 + 2.54690e10i 3.56719e13 1.04083e14 + 1.04083e14i −4.32593e14 + 4.32593e14i 1.72393e16i −1.07925e17 1.00949e17i
2.15 167195. + 167195.i 4.67349e7 4.67349e7i 3.87285e10i 6.83351e11 + 3.39275e11i 1.56277e13 −2.57154e14 2.57154e14i −3.60283e15 + 3.60283e15i 1.23089e16i 5.75278e16 + 1.70978e17i
2.16 174299. + 174299.i −1.58564e8 + 1.58564e8i 4.35807e10i −4.99549e11 5.76652e11i −5.52754e13 3.74774e13 + 3.74774e13i −4.60166e15 + 4.60166e15i 3.36082e16i 1.34389e16 1.87581e17i
3.1 −169510. + 169510.i −198644. 198644.i 4.02874e10i −3.95922e11 6.52168e11i 6.73444e10 1.10969e14 1.10969e14i 3.91696e15 + 3.91696e15i 1.66771e16i 1.77662e17 + 4.34363e16i
3.2 −151262. + 151262.i −1.00065e8 1.00065e8i 2.85804e10i 1.61018e11 + 7.45755e11i 3.02721e13 −1.52632e14 + 1.52632e14i 1.72447e15 + 1.72447e15i 3.34887e15i −1.37160e17 8.84484e16i
3.3 −134834. + 134834.i 1.19387e8 + 1.19387e8i 1.91805e10i 6.75906e11 + 3.53875e11i −3.21948e13 2.58783e14 2.58783e14i 2.69752e14 + 2.69752e14i 1.18293e16i −1.38849e17 + 4.34207e16i
3.4 −115430. + 115430.i 1.39345e8 + 1.39345e8i 9.46824e9i −7.62275e11 + 3.18319e10i −3.21691e13 −2.97793e14 + 2.97793e14i −8.90153e14 8.90153e14i 2.21567e16i 8.43150e16 9.16637e16i
See all 32 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 3.16 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 5.35.c.a 32
5.b even 2 1 25.35.c.b 32
5.c odd 4 1 inner 5.35.c.a 32
5.c odd 4 1 25.35.c.b 32

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5.35.c.a 32 1.a even 1 1 trivial
5.35.c.a 32 5.c odd 4 1 inner
25.35.c.b 32 5.b even 2 1
25.35.c.b 32 5.c odd 4 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{35}^{\mathrm{new}}(5, [\chi])$$.

## Hecke characteristic polynomials

There are no characteristic polynomials of Hecke operators in the database