# Properties

 Label 100.4.i.a Level $100$ Weight $4$ Character orbit 100.i Analytic conductor $5.900$ Analytic rank $0$ Dimension $32$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$100 = 2^{2} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 100.i (of order $$10$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$5.90019100057$$ Analytic rank: $$0$$ Dimension: $$32$$ Relative dimension: $$8$$ over $$\Q(\zeta_{10})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

## $q$-expansion

The dimension is sufficiently large that we do not compute an algebraic $$q$$-expansion, but we have computed the trace expansion.

 $$\operatorname{Tr}(f)(q) =$$ $$32 q + 6 q^{5} + 122 q^{9}+O(q^{10})$$ 32 * q + 6 * q^5 + 122 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$32 q + 6 q^{5} + 122 q^{9} + 20 q^{11} + 68 q^{15} - 160 q^{17} + 2 q^{19} - 108 q^{21} + 290 q^{23} + 654 q^{25} + 600 q^{27} + 62 q^{29} - 378 q^{31} - 1280 q^{33} - 278 q^{35} + 680 q^{37} + 592 q^{39} - 528 q^{41} - 1044 q^{45} - 1810 q^{47} - 2796 q^{49} + 1664 q^{51} - 510 q^{53} - 1350 q^{55} + 144 q^{59} - 1346 q^{61} + 1660 q^{63} + 1142 q^{65} + 1890 q^{67} + 956 q^{69} + 786 q^{71} + 3720 q^{73} - 78 q^{75} + 2160 q^{77} + 896 q^{79} + 348 q^{81} + 570 q^{83} + 224 q^{85} + 3240 q^{87} - 2512 q^{89} - 2212 q^{91} + 1536 q^{95} - 2250 q^{97} - 2540 q^{99}+O(q^{100})$$ 32 * q + 6 * q^5 + 122 * q^9 + 20 * q^11 + 68 * q^15 - 160 * q^17 + 2 * q^19 - 108 * q^21 + 290 * q^23 + 654 * q^25 + 600 * q^27 + 62 * q^29 - 378 * q^31 - 1280 * q^33 - 278 * q^35 + 680 * q^37 + 592 * q^39 - 528 * q^41 - 1044 * q^45 - 1810 * q^47 - 2796 * q^49 + 1664 * q^51 - 510 * q^53 - 1350 * q^55 + 144 * q^59 - 1346 * q^61 + 1660 * q^63 + 1142 * q^65 + 1890 * q^67 + 956 * q^69 + 786 * q^71 + 3720 * q^73 - 78 * q^75 + 2160 * q^77 + 896 * q^79 + 348 * q^81 + 570 * q^83 + 224 * q^85 + 3240 * q^87 - 2512 * q^89 - 2212 * q^91 + 1536 * q^95 - 2250 * q^97 - 2540 * q^99

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
9.1 0 −9.31487 + 3.02658i 0 −11.0317 1.81673i 0 0.760275i 0 55.7631 40.5142i 0
9.2 0 −5.01325 + 1.62890i 0 9.21137 6.33645i 0 9.22290i 0 0.635846 0.461969i 0
9.3 0 −4.71349 + 1.53150i 0 3.20187 + 10.7121i 0 16.8592i 0 −1.97201 + 1.43275i 0
9.4 0 −0.776927 + 0.252439i 0 −4.16996 10.3736i 0 1.56595i 0 −21.3036 + 15.4779i 0
9.5 0 2.14386 0.696582i 0 −7.24969 + 8.51129i 0 16.0147i 0 −17.7325 + 12.8835i 0
9.6 0 3.55368 1.15466i 0 10.9812 + 2.10091i 0 30.1378i 0 −10.5481 + 7.66362i 0
9.7 0 8.14260 2.64569i 0 11.1773 0.261414i 0 36.0715i 0 37.4588 27.2154i 0
9.8 0 8.21445 2.66904i 0 −9.50227 5.89126i 0 24.1794i 0 38.5100 27.9792i 0
29.1 0 −5.78214 + 7.95843i 0 10.1359 + 4.71846i 0 24.7439i 0 −21.5601 66.3550i 0
29.2 0 −4.13050 + 5.68515i 0 −10.7351 + 3.12376i 0 21.6979i 0 −6.91642 21.2866i 0
29.3 0 −1.72442 + 2.37346i 0 6.46300 9.12303i 0 7.35306i 0 5.68377 + 17.4928i 0
29.4 0 −0.991787 + 1.36508i 0 −6.73237 8.92610i 0 19.2807i 0 7.46366 + 22.9708i 0
29.5 0 −0.618939 + 0.851897i 0 3.28989 + 10.6853i 0 15.9472i 0 8.00082 + 24.6240i 0
29.6 0 2.66799 3.67217i 0 −0.873009 + 11.1462i 0 36.0979i 0 1.97677 + 6.08388i 0
29.7 0 4.08679 5.62498i 0 −11.1568 + 0.725356i 0 19.2903i 0 −6.59510 20.2976i 0
29.8 0 4.25695 5.85918i 0 9.99045 5.01906i 0 4.97648i 0 −7.86498 24.2059i 0
69.1 0 −5.78214 7.95843i 0 10.1359 4.71846i 0 24.7439i 0 −21.5601 + 66.3550i 0
69.2 0 −4.13050 5.68515i 0 −10.7351 3.12376i 0 21.6979i 0 −6.91642 + 21.2866i 0
69.3 0 −1.72442 2.37346i 0 6.46300 + 9.12303i 0 7.35306i 0 5.68377 17.4928i 0
69.4 0 −0.991787 1.36508i 0 −6.73237 + 8.92610i 0 19.2807i 0 7.46366 22.9708i 0
See all 32 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 89.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
25.e even 10 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 100.4.i.a 32
5.b even 2 1 500.4.i.a 32
5.c odd 4 2 500.4.g.b 64
25.d even 5 1 500.4.i.a 32
25.e even 10 1 inner 100.4.i.a 32
25.f odd 20 2 500.4.g.b 64
25.f odd 20 2 2500.4.a.g 32

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
100.4.i.a 32 1.a even 1 1 trivial
100.4.i.a 32 25.e even 10 1 inner
500.4.g.b 64 5.c odd 4 2
500.4.g.b 64 25.f odd 20 2
500.4.i.a 32 5.b even 2 1
500.4.i.a 32 25.d even 5 1
2500.4.a.g 32 25.f odd 20 2

## Hecke kernels

This newform subspace is the entire newspace $$S_{4}^{\mathrm{new}}(100, [\chi])$$.