# Properties

 Label 100.9.d.c Level $100$ Weight $9$ Character orbit 100.d Analytic conductor $40.738$ Analytic rank $0$ Dimension $32$ CM no Inner twists $4$

# Learn more

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [100,9,Mod(99,100)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(100, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 1]))

N = Newforms(chi, 9, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("100.99");

S:= CuspForms(chi, 9);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$100 = 2^{2} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$9$$ Character orbit: $$[\chi]$$ $$=$$ 100.d (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: no Analytic conductor: $$40.7378610061$$ Analytic rank: $$0$$ Dimension: $$32$$ Twist minimal: no (minimal twist has level 20) 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.

 $$\operatorname{Tr}(f)(q) =$$ $$32 q + 104 q^{4} + 8736 q^{6} + 77600 q^{9}+O(q^{10})$$ 32 * q + 104 * q^4 + 8736 * q^6 + 77600 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$32 q + 104 q^{4} + 8736 q^{6} + 77600 q^{9} - 136944 q^{14} - 162848 q^{16} + 828992 q^{21} - 327584 q^{24} + 2074248 q^{26} - 5529792 q^{29} - 7587928 q^{34} - 10937832 q^{36} - 17152896 q^{41} - 33842400 q^{44} - 15948304 q^{46} + 65607200 q^{49} - 17797408 q^{54} + 68269536 q^{56} + 16743424 q^{61} + 95087744 q^{64} + 38716000 q^{66} - 15054528 q^{69} + 276421752 q^{74} + 5140800 q^{76} + 281173344 q^{81} - 139523648 q^{84} - 203449344 q^{86} - 213294912 q^{89} - 110529264 q^{94} - 906779904 q^{96}+O(q^{100})$$ 32 * q + 104 * q^4 + 8736 * q^6 + 77600 * q^9 - 136944 * q^14 - 162848 * q^16 + 828992 * q^21 - 327584 * q^24 + 2074248 * q^26 - 5529792 * q^29 - 7587928 * q^34 - 10937832 * q^36 - 17152896 * q^41 - 33842400 * q^44 - 15948304 * q^46 + 65607200 * q^49 - 17797408 * q^54 + 68269536 * q^56 + 16743424 * q^61 + 95087744 * q^64 + 38716000 * q^66 - 15054528 * q^69 + 276421752 * q^74 + 5140800 * q^76 + 281173344 * q^81 - 139523648 * q^84 - 203449344 * q^86 - 213294912 * q^89 - 110529264 * q^94 - 906779904 * q^96

## 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.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label   $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
99.1 −15.9375 1.41320i −39.9624 252.006 + 45.0455i 0 636.899 + 56.4746i 2633.20 −3952.68 1074.04i −4964.01 0
99.2 −15.9375 + 1.41320i −39.9624 252.006 45.0455i 0 636.899 56.4746i 2633.20 −3952.68 + 1074.04i −4964.01 0
99.3 −15.3124 4.64016i 75.7492 212.938 + 142.104i 0 −1159.90 351.488i −210.345 −2601.20 3164.01i −823.060 0
99.4 −15.3124 + 4.64016i 75.7492 212.938 142.104i 0 −1159.90 + 351.488i −210.345 −2601.20 + 3164.01i −823.060 0
99.5 −14.2118 7.35022i −110.171 147.949 + 208.919i 0 1565.72 + 809.778i −3540.70 −567.011 4056.56i 5576.56 0
99.6 −14.2118 + 7.35022i −110.171 147.949 208.919i 0 1565.72 809.778i −3540.70 −567.011 + 4056.56i 5576.56 0
99.7 −11.2775 11.3498i −137.297 −1.63618 + 255.995i 0 1548.37 + 1558.29i 3940.57 2923.94 2868.41i 12289.4 0
99.8 −11.2775 + 11.3498i −137.297 −1.63618 255.995i 0 1548.37 1558.29i 3940.57 2923.94 + 2868.41i 12289.4 0
99.9 −11.0540 11.5676i −27.2434 −11.6196 + 255.736i 0 301.148 + 315.141i −3325.58 3086.70 2692.49i −5818.80 0
99.10 −11.0540 + 11.5676i −27.2434 −11.6196 255.736i 0 301.148 315.141i −3325.58 3086.70 + 2692.49i −5818.80 0
99.11 −6.70489 14.5274i 150.211 −166.089 + 194.809i 0 −1007.15 2182.17i 2626.96 3943.67 + 1106.66i 16002.3 0
99.12 −6.70489 + 14.5274i 150.211 −166.089 194.809i 0 −1007.15 + 2182.17i 2626.96 3943.67 1106.66i 16002.3 0
99.13 −5.65855 14.9660i 25.1248 −191.962 + 169.372i 0 −142.170 376.017i 2973.76 3621.04 + 1914.50i −5929.74 0
99.14 −5.65855 + 14.9660i 25.1248 −191.962 169.372i 0 −142.170 + 376.017i 2973.76 3621.04 1914.50i −5929.74 0
99.15 −4.49522 15.3556i −98.1237 −215.586 + 138.053i 0 441.088 + 1506.74i 820.952 3088.99 + 2689.86i 3067.27 0
99.16 −4.49522 + 15.3556i −98.1237 −215.586 138.053i 0 441.088 1506.74i 820.952 3088.99 2689.86i 3067.27 0
99.17 4.49522 15.3556i 98.1237 −215.586 138.053i 0 441.088 1506.74i −820.952 −3088.99 + 2689.86i 3067.27 0
99.18 4.49522 + 15.3556i 98.1237 −215.586 + 138.053i 0 441.088 + 1506.74i −820.952 −3088.99 2689.86i 3067.27 0
99.19 5.65855 14.9660i −25.1248 −191.962 169.372i 0 −142.170 + 376.017i −2973.76 −3621.04 + 1914.50i −5929.74 0
99.20 5.65855 + 14.9660i −25.1248 −191.962 + 169.372i 0 −142.170 376.017i −2973.76 −3621.04 1914.50i −5929.74 0
See all 32 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 99.32 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
5.b even 2 1 inner
20.d odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 100.9.d.c 32
4.b odd 2 1 inner 100.9.d.c 32
5.b even 2 1 inner 100.9.d.c 32
5.c odd 4 1 20.9.b.a 16
5.c odd 4 1 100.9.b.d 16
15.e even 4 1 180.9.c.a 16
20.d odd 2 1 inner 100.9.d.c 32
20.e even 4 1 20.9.b.a 16
20.e even 4 1 100.9.b.d 16
40.i odd 4 1 320.9.b.d 16
40.k even 4 1 320.9.b.d 16
60.l odd 4 1 180.9.c.a 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
20.9.b.a 16 5.c odd 4 1
20.9.b.a 16 20.e even 4 1
100.9.b.d 16 5.c odd 4 1
100.9.b.d 16 20.e even 4 1
100.9.d.c 32 1.a even 1 1 trivial
100.9.d.c 32 4.b odd 2 1 inner
100.9.d.c 32 5.b even 2 1 inner
100.9.d.c 32 20.d odd 2 1 inner
180.9.c.a 16 15.e even 4 1
180.9.c.a 16 60.l odd 4 1
320.9.b.d 16 40.i odd 4 1
320.9.b.d 16 40.k even 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{16} - 71888 T_{3}^{14} + 2013496736 T_{3}^{12} - 27929868057600 T_{3}^{10} + \cdots + 21\!\cdots\!00$$ acting on $$S_{9}^{\mathrm{new}}(100, [\chi])$$.