# Properties

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

# Related objects

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: 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.

 $$\operatorname{Tr}(f)(q) =$$ $$32 q + 662 q^{4} - 966 q^{6} + 66176 q^{9}+O(q^{10})$$ 32 * q + 662 * q^4 - 966 * q^6 + 66176 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$32 q + 662 q^{4} - 966 q^{6} + 66176 q^{9} - 40716 q^{14} + 254722 q^{16} + 98048 q^{21} + 1468354 q^{24} - 2237400 q^{26} + 867264 q^{29} - 4453930 q^{34} + 3743556 q^{36} + 3480864 q^{41} + 947910 q^{44} + 11027204 q^{46} + 11002976 q^{49} + 32852018 q^{54} - 10823436 q^{56} + 65056384 q^{61} - 11392498 q^{64} + 24651010 q^{66} + 93440448 q^{69} - 30344220 q^{74} + 105668010 q^{76} - 135528480 q^{81} + 268030468 q^{84} - 193355376 q^{86} - 161219616 q^{89} - 411820536 q^{94} + 479746254 q^{96}+O(q^{100})$$ 32 * q + 662 * q^4 - 966 * q^6 + 66176 * q^9 - 40716 * q^14 + 254722 * q^16 + 98048 * q^21 + 1468354 * q^24 - 2237400 * q^26 + 867264 * q^29 - 4453930 * q^34 + 3743556 * q^36 + 3480864 * q^41 + 947910 * q^44 + 11027204 * q^46 + 11002976 * q^49 + 32852018 * q^54 - 10823436 * q^56 + 65056384 * q^61 - 11392498 * q^64 + 24651010 * q^66 + 93440448 * q^69 - 30344220 * q^74 + 105668010 * q^76 - 135528480 * q^81 + 268030468 * q^84 - 193355376 * q^86 - 161219616 * q^89 - 411820536 * q^94 + 479746254 * 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 −15.9719 0.948150i 19.5136 254.202 + 30.2875i 0 −311.670 18.5019i −3691.10 −4031.37 724.770i −6180.22 0
99.2 −15.9719 + 0.948150i 19.5136 254.202 30.2875i 0 −311.670 + 18.5019i −3691.10 −4031.37 + 724.770i −6180.22 0
99.3 −15.7341 2.90464i −140.214 239.126 + 91.4039i 0 2206.15 + 407.271i 132.843 −3496.95 2132.74i 13099.0 0
99.4 −15.7341 + 2.90464i −140.214 239.126 91.4039i 0 2206.15 407.271i 132.843 −3496.95 + 2132.74i 13099.0 0
99.5 −15.1177 5.23963i 107.117 201.093 + 158.423i 0 −1619.37 561.253i 4212.76 −2209.99 3448.65i 4913.04 0
99.6 −15.1177 + 5.23963i 107.117 201.093 158.423i 0 −1619.37 + 561.253i 4212.76 −2209.99 + 3448.65i 4913.04 0
99.7 −12.4780 10.0150i −48.3156 55.3985 + 249.934i 0 602.880 + 483.882i 2732.84 1811.84 3673.48i −4226.60 0
99.8 −12.4780 + 10.0150i −48.3156 55.3985 249.934i 0 602.880 483.882i 2732.84 1811.84 + 3673.48i −4226.60 0
99.9 −10.9610 11.6558i 122.786 −15.7140 + 255.517i 0 −1345.85 1431.16i −1671.99 3150.49 2617.56i 8515.34 0
99.10 −10.9610 + 11.6558i 122.786 −15.7140 255.517i 0 −1345.85 + 1431.16i −1671.99 3150.49 + 2617.56i 8515.34 0
99.11 −9.58313 12.8126i −17.7267 −72.3273 + 245.570i 0 169.877 + 227.125i −536.344 3839.52 1426.63i −6246.77 0
99.12 −9.58313 + 12.8126i −17.7267 −72.3273 245.570i 0 169.877 227.125i −536.344 3839.52 + 1426.63i −6246.77 0
99.13 −2.55014 15.7955i −76.1896 −242.994 + 80.5612i 0 194.294 + 1203.45i −2180.40 1892.17 + 3632.76i −756.150 0
99.14 −2.55014 + 15.7955i −76.1896 −242.994 80.5612i 0 194.294 1203.45i −2180.40 1892.17 3632.76i −756.150 0
99.15 −1.16525 15.9575i 118.268 −253.284 + 37.1889i 0 −137.812 1887.26i −1474.52 888.582 + 3998.45i 7426.34 0
99.16 −1.16525 + 15.9575i 118.268 −253.284 37.1889i 0 −137.812 + 1887.26i −1474.52 888.582 3998.45i 7426.34 0
99.17 1.16525 15.9575i −118.268 −253.284 37.1889i 0 −137.812 + 1887.26i 1474.52 −888.582 + 3998.45i 7426.34 0
99.18 1.16525 + 15.9575i −118.268 −253.284 + 37.1889i 0 −137.812 1887.26i 1474.52 −888.582 3998.45i 7426.34 0
99.19 2.55014 15.7955i 76.1896 −242.994 80.5612i 0 194.294 1203.45i 2180.40 −1892.17 + 3632.76i −756.150 0
99.20 2.55014 + 15.7955i 76.1896 −242.994 + 80.5612i 0 194.294 + 1203.45i 2180.40 −1892.17 3632.76i −756.150 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.d 32
4.b odd 2 1 inner 100.9.d.d 32
5.b even 2 1 inner 100.9.d.d 32
5.c odd 4 1 100.9.b.e 16
5.c odd 4 1 100.9.b.f yes 16
20.d odd 2 1 inner 100.9.d.d 32
20.e even 4 1 100.9.b.e 16
20.e even 4 1 100.9.b.f yes 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
100.9.b.e 16 5.c odd 4 1
100.9.b.e 16 20.e even 4 1
100.9.b.f yes 16 5.c odd 4 1
100.9.b.f yes 16 20.e even 4 1
100.9.d.d 32 1.a even 1 1 trivial
100.9.d.d 32 4.b odd 2 1 inner
100.9.d.d 32 5.b even 2 1 inner
100.9.d.d 32 20.d odd 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{16} - 69032 T_{3}^{14} + 1892458028 T_{3}^{12} - 26145166260984 T_{3}^{10} + \cdots + 77\!\cdots\!25$$ acting on $$S_{9}^{\mathrm{new}}(100, [\chi])$$.