# Properties

 Label 150.3.k.a Level $150$ Weight $3$ Character orbit 150.k Analytic conductor $4.087$ Analytic rank $0$ Dimension $32$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [150,3,Mod(13,150)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(150, base_ring=CyclotomicField(20))

chi = DirichletCharacter(H, H._module([0, 19]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("150.13");

S:= CuspForms(chi, 3);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$150 = 2 \cdot 3 \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 150.k (of order $$20$$, degree $$8$$, minimal)

## Newform invariants

comment: select newform

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

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

 Self dual: no Analytic conductor: $$4.08720396540$$ Analytic rank: $$0$$ Dimension: $$32$$ Relative dimension: $$4$$ over $$\Q(\zeta_{20})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{20}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$32 q + 8 q^{2} + 8 q^{7} - 16 q^{8}+O(q^{10})$$ 32 * q + 8 * q^2 + 8 * q^7 - 16 * q^8 $$\operatorname{Tr}(f)(q) =$$ $$32 q + 8 q^{2} + 8 q^{7} - 16 q^{8} + 20 q^{10} + 32 q^{11} - 16 q^{13} - 60 q^{14} + 32 q^{16} + 148 q^{17} - 96 q^{18} + 180 q^{19} + 40 q^{20} - 36 q^{21} + 48 q^{22} + 48 q^{23} - 160 q^{25} - 8 q^{26} - 56 q^{28} - 200 q^{29} - 120 q^{30} + 120 q^{31} + 128 q^{32} - 156 q^{33} - 100 q^{34} - 180 q^{35} - 48 q^{36} + 444 q^{37} + 32 q^{38} - 120 q^{39} - 304 q^{41} - 24 q^{42} + 216 q^{43} + 40 q^{44} + 60 q^{45} - 16 q^{46} + 32 q^{47} + 40 q^{50} + 24 q^{51} - 32 q^{52} - 340 q^{53} + 80 q^{55} + 72 q^{56} - 24 q^{57} - 192 q^{58} - 560 q^{59} + 312 q^{61} + 40 q^{62} + 24 q^{63} - 520 q^{65} - 108 q^{66} + 688 q^{67} - 16 q^{68} + 180 q^{69} + 80 q^{70} + 212 q^{71} + 48 q^{72} - 376 q^{73} + 120 q^{75} - 64 q^{76} - 176 q^{77} - 48 q^{78} + 440 q^{79} + 80 q^{80} + 72 q^{81} - 256 q^{82} - 96 q^{83} - 240 q^{85} + 408 q^{86} + 264 q^{87} + 184 q^{88} - 560 q^{89} - 516 q^{91} + 216 q^{92} + 48 q^{93} + 80 q^{94} + 520 q^{95} - 716 q^{97} - 108 q^{98}+O(q^{100})$$ 32 * q + 8 * q^2 + 8 * q^7 - 16 * q^8 + 20 * q^10 + 32 * q^11 - 16 * q^13 - 60 * q^14 + 32 * q^16 + 148 * q^17 - 96 * q^18 + 180 * q^19 + 40 * q^20 - 36 * q^21 + 48 * q^22 + 48 * q^23 - 160 * q^25 - 8 * q^26 - 56 * q^28 - 200 * q^29 - 120 * q^30 + 120 * q^31 + 128 * q^32 - 156 * q^33 - 100 * q^34 - 180 * q^35 - 48 * q^36 + 444 * q^37 + 32 * q^38 - 120 * q^39 - 304 * q^41 - 24 * q^42 + 216 * q^43 + 40 * q^44 + 60 * q^45 - 16 * q^46 + 32 * q^47 + 40 * q^50 + 24 * q^51 - 32 * q^52 - 340 * q^53 + 80 * q^55 + 72 * q^56 - 24 * q^57 - 192 * q^58 - 560 * q^59 + 312 * q^61 + 40 * q^62 + 24 * q^63 - 520 * q^65 - 108 * q^66 + 688 * q^67 - 16 * q^68 + 180 * q^69 + 80 * q^70 + 212 * q^71 + 48 * q^72 - 376 * q^73 + 120 * q^75 - 64 * q^76 - 176 * q^77 - 48 * q^78 + 440 * q^79 + 80 * q^80 + 72 * q^81 - 256 * q^82 - 96 * q^83 - 240 * q^85 + 408 * q^86 + 264 * q^87 + 184 * q^88 - 560 * q^89 - 516 * q^91 + 216 * q^92 + 48 * q^93 + 80 * q^94 + 520 * q^95 - 716 * q^97 - 108 * q^98

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
13.1 1.39680 0.221232i −0.786335 + 1.54327i 1.90211 0.618034i 1.86127 + 4.64065i −0.756934 + 2.32960i −6.90199 + 6.90199i 2.52015 1.28408i −1.76336 2.42705i 3.62649 + 6.07030i
13.2 1.39680 0.221232i −0.786335 + 1.54327i 1.90211 0.618034i 1.98098 4.59083i −0.756934 + 2.32960i 3.83711 3.83711i 2.52015 1.28408i −1.76336 2.42705i 1.75141 6.85074i
13.3 1.39680 0.221232i 0.786335 1.54327i 1.90211 0.618034i −3.06706 3.94881i 0.756934 2.32960i 2.82371 2.82371i 2.52015 1.28408i −1.76336 2.42705i −5.15768 4.83718i
13.4 1.39680 0.221232i 0.786335 1.54327i 1.90211 0.618034i 4.42349 + 2.33083i 0.756934 2.32960i 0.284481 0.284481i 2.52015 1.28408i −1.76336 2.42705i 6.69439 + 2.27709i
37.1 0.221232 + 1.39680i −1.54327 0.786335i −1.90211 + 0.618034i −1.48914 4.77310i 0.756934 2.32960i 9.35986 + 9.35986i −1.28408 2.52015i 1.76336 + 2.42705i 6.33763 3.13599i
37.2 0.221232 + 1.39680i −1.54327 0.786335i −1.90211 + 0.618034i 3.55879 3.51212i 0.756934 2.32960i −5.48204 5.48204i −1.28408 2.52015i 1.76336 + 2.42705i 5.69306 + 4.19393i
37.3 0.221232 + 1.39680i 1.54327 + 0.786335i −1.90211 + 0.618034i −3.00822 + 3.99382i −0.756934 + 2.32960i 4.19292 + 4.19292i −1.28408 2.52015i 1.76336 + 2.42705i −6.24409 3.31834i
37.4 0.221232 + 1.39680i 1.54327 + 0.786335i −1.90211 + 0.618034i 4.68416 1.74889i −0.756934 + 2.32960i 2.83023 + 2.83023i −1.28408 2.52015i 1.76336 + 2.42705i 3.47914 + 6.15594i
67.1 0.642040 1.26007i −0.270952 + 1.71073i −1.17557 1.61803i −3.75559 3.30084i 1.98168 + 1.43977i −5.04938 5.04938i −2.79360 + 0.442463i −2.85317 0.927051i −6.57054 + 2.61306i
67.2 0.642040 1.26007i −0.270952 + 1.71073i −1.17557 1.61803i −2.43941 + 4.36455i 1.98168 + 1.43977i 4.46938 + 4.46938i −2.79360 + 0.442463i −2.85317 0.927051i 3.93346 + 5.87604i
67.3 0.642040 1.26007i 0.270952 1.71073i −1.17557 1.61803i −3.90603 + 3.12137i −1.98168 1.43977i −6.00700 6.00700i −2.79360 + 0.442463i −2.85317 0.927051i 1.42533 + 6.92592i
67.4 0.642040 1.26007i 0.270952 1.71073i −1.17557 1.61803i 2.55121 4.30016i −1.98168 1.43977i −1.41591 1.41591i −2.79360 + 0.442463i −2.85317 0.927051i −3.78054 5.97558i
73.1 0.221232 1.39680i −1.54327 + 0.786335i −1.90211 0.618034i −1.48914 + 4.77310i 0.756934 + 2.32960i 9.35986 9.35986i −1.28408 + 2.52015i 1.76336 2.42705i 6.33763 + 3.13599i
73.2 0.221232 1.39680i −1.54327 + 0.786335i −1.90211 0.618034i 3.55879 + 3.51212i 0.756934 + 2.32960i −5.48204 + 5.48204i −1.28408 + 2.52015i 1.76336 2.42705i 5.69306 4.19393i
73.3 0.221232 1.39680i 1.54327 0.786335i −1.90211 0.618034i −3.00822 3.99382i −0.756934 2.32960i 4.19292 4.19292i −1.28408 + 2.52015i 1.76336 2.42705i −6.24409 + 3.31834i
73.4 0.221232 1.39680i 1.54327 0.786335i −1.90211 0.618034i 4.68416 + 1.74889i −0.756934 2.32960i 2.83023 2.83023i −1.28408 + 2.52015i 1.76336 2.42705i 3.47914 6.15594i
97.1 −1.26007 + 0.642040i −1.71073 + 0.270952i 1.17557 1.61803i −4.55798 2.05543i 1.98168 1.43977i 3.78399 + 3.78399i −0.442463 + 2.79360i 2.85317 0.927051i 7.06306 0.336405i
97.2 −1.26007 + 0.642040i −1.71073 + 0.270952i 1.17557 1.61803i −0.888938 + 4.92034i 1.98168 1.43977i −2.71277 2.71277i −0.442463 + 2.79360i 2.85317 0.927051i −2.03893 6.77073i
97.3 −1.26007 + 0.642040i 1.71073 0.270952i 1.17557 1.61803i 0.270329 + 4.99269i −1.98168 + 1.43977i 0.734007 + 0.734007i −0.442463 + 2.79360i 2.85317 0.927051i −3.54614 6.11759i
97.4 −1.26007 + 0.642040i 1.71073 0.270952i 1.17557 1.61803i 3.78214 3.27039i −1.98168 + 1.43977i −0.746595 0.746595i −0.442463 + 2.79360i 2.85317 0.927051i −2.66605 + 6.54921i
See all 32 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 133.4 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.f odd 20 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 150.3.k.a 32
25.f odd 20 1 inner 150.3.k.a 32

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
150.3.k.a 32 1.a even 1 1 trivial
150.3.k.a 32 25.f odd 20 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}^{32} - 8 T_{7}^{31} + 32 T_{7}^{30} + 284 T_{7}^{29} + 25448 T_{7}^{28} - 142288 T_{7}^{27} + 364296 T_{7}^{26} - 1854300 T_{7}^{25} + 166638564 T_{7}^{24} - 1025374732 T_{7}^{23} + \cdots + 12\!\cdots\!61$$ acting on $$S_{3}^{\mathrm{new}}(150, [\chi])$$.