# Properties

 Label 740.2.u.a Level $740$ Weight $2$ Character orbit 740.u Analytic conductor $5.909$ Analytic rank $0$ Dimension $152$ Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [740,2,Mod(31,740)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(740, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([2, 0, 1]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("740.31");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$740 = 2^{2} \cdot 5 \cdot 37$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 740.u (of order $$4$$, degree $$2$$, 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: $$5.90892974957$$ Analytic rank: $$0$$ Dimension: $$152$$ Relative dimension: $$76$$ over $$\Q(i)$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-expansion

The algebraic $$q$$-expansion of this newform has not been computed, but we have computed the trace expansion.

 $$\operatorname{Tr}(f)(q) =$$ $$152 q - 4 q^{2} - 4 q^{8} + 152 q^{9}+O(q^{10})$$ 152 * q - 4 * q^2 - 4 * q^8 + 152 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$152 q - 4 q^{2} - 4 q^{8} + 152 q^{9} - 16 q^{13} + 24 q^{14} + 40 q^{16} + 16 q^{22} - 36 q^{24} + 32 q^{29} - 4 q^{32} - 16 q^{37} - 40 q^{38} - 20 q^{42} - 72 q^{44} + 8 q^{46} - 248 q^{49} - 4 q^{50} - 20 q^{52} + 8 q^{54} - 4 q^{56} + 48 q^{57} + 28 q^{60} + 16 q^{61} + 16 q^{66} - 28 q^{68} - 128 q^{72} - 80 q^{74} + 76 q^{76} + 152 q^{81} + 16 q^{82} - 56 q^{84} - 96 q^{86} - 72 q^{88} + 8 q^{89} - 152 q^{92} + 16 q^{93} + 28 q^{94} + 48 q^{96} - 48 q^{97} + 24 q^{98}+O(q^{100})$$ 152 * q - 4 * q^2 - 4 * q^8 + 152 * q^9 - 16 * q^13 + 24 * q^14 + 40 * q^16 + 16 * q^22 - 36 * q^24 + 32 * q^29 - 4 * q^32 - 16 * q^37 - 40 * q^38 - 20 * q^42 - 72 * q^44 + 8 * q^46 - 248 * q^49 - 4 * q^50 - 20 * q^52 + 8 * q^54 - 4 * q^56 + 48 * q^57 + 28 * q^60 + 16 * q^61 + 16 * q^66 - 28 * q^68 - 128 * q^72 - 80 * q^74 + 76 * q^76 + 152 * q^81 + 16 * q^82 - 56 * q^84 - 96 * q^86 - 72 * q^88 + 8 * q^89 - 152 * q^92 + 16 * q^93 + 28 * q^94 + 48 * q^96 - 48 * q^97 + 24 * 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   $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
31.1 −1.40873 + 0.124456i 3.04334 1.96902 0.350649i 0.707107 0.707107i −4.28724 + 0.378763i 3.54875i −2.73017 + 0.739026i 6.26194 −0.908116 + 1.08412i
31.2 −1.40814 0.130950i 2.09062 1.96570 + 0.368792i −0.707107 + 0.707107i −2.94388 0.273767i 2.92000i −2.71969 0.776720i 1.37069 1.08830 0.903108i
31.3 −1.40804 0.132013i −1.51447 1.96515 + 0.371758i −0.707107 + 0.707107i 2.13243 + 0.199929i 1.47727i −2.71792 0.782873i −0.706387 1.08898 0.902287i
31.4 −1.40327 0.175606i −0.390097 1.93833 + 0.492844i −0.707107 + 0.707107i 0.547410 + 0.0685032i 0.0448530i −2.63344 1.03197i −2.84782 1.11643 0.868089i
31.5 −1.39860 0.209573i −2.88896 1.91216 + 0.586216i 0.707107 0.707107i 4.04050 + 0.605448i 2.24958i −2.55149 1.22062i 5.34611 −1.13715 + 0.840769i
31.6 −1.39661 0.222436i −1.94167 1.90104 + 0.621313i 0.707107 0.707107i 2.71176 + 0.431897i 4.49715i −2.51682 1.29059i 0.770091 −1.14484 + 0.830267i
31.7 −1.38578 0.282160i 3.03129 1.84077 + 0.782022i −0.707107 + 0.707107i −4.20069 0.855307i 4.17536i −2.33025 1.60310i 6.18869 1.17941 0.780377i
31.8 −1.37626 + 0.325442i −0.313272 1.78817 0.895785i 0.707107 0.707107i 0.431143 0.101952i 0.0788742i −2.16946 + 1.81478i −2.90186 −0.743039 + 1.20328i
31.9 −1.35901 + 0.391281i 0.907469 1.69380 1.06351i 0.707107 0.707107i −1.23326 + 0.355075i 4.09672i −1.88576 + 2.10806i −2.17650 −0.684286 + 1.23764i
31.10 −1.34065 0.450162i 2.21520 1.59471 + 1.20702i 0.707107 0.707107i −2.96982 0.997201i 1.86106i −1.59459 2.33608i 1.90712 −1.26630 + 0.629673i
31.11 −1.32904 + 0.483377i 1.02884 1.53269 1.28485i −0.707107 + 0.707107i −1.36736 + 0.497316i 1.37227i −1.41594 + 2.44849i −1.94150 0.597974 1.28157i
31.12 −1.31189 + 0.528162i −3.30999 1.44209 1.38578i −0.707107 + 0.707107i 4.34233 1.74821i 5.02019i −1.15994 + 2.57964i 7.95603 0.554177 1.30111i
31.13 −1.29954 + 0.557859i −2.62771 1.37759 1.44991i −0.707107 + 0.707107i 3.41481 1.46589i 5.20761i −0.981377 + 2.65272i 3.90487 0.524445 1.31338i
31.14 −1.24845 0.664363i 0.997645 1.11724 + 1.65884i 0.707107 0.707107i −1.24551 0.662798i 0.344735i −0.292747 2.81324i −2.00470 −1.35256 + 0.413011i
31.15 −1.10727 0.879744i −0.686367 0.452103 + 1.94823i −0.707107 + 0.707107i 0.759995 + 0.603827i 4.62323i 1.21334 2.55496i −2.52890 1.40503 0.160887i
31.16 −1.08074 0.912141i −1.37212 0.335997 + 1.97157i 0.707107 0.707107i 1.48290 + 1.25156i 1.94978i 1.43523 2.43724i −1.11730 −1.40918 + 0.119217i
31.17 −1.07586 + 0.917891i 3.19932 0.314953 1.97505i 0.707107 0.707107i −3.44203 + 2.93663i 1.85618i 1.47403 + 2.41397i 7.23567 −0.111702 + 1.40980i
31.18 −1.07503 + 0.918861i 0.615294 0.311387 1.97561i 0.707107 0.707107i −0.661461 + 0.565370i 3.79949i 1.48056 + 2.40997i −2.62141 −0.110429 + 1.40990i
31.19 −1.07092 0.923652i −2.12677 0.293735 + 1.97831i −0.707107 + 0.707107i 2.27760 + 1.96440i 1.56565i 1.51271 2.38992i 1.52316 1.41037 0.104134i
31.20 −0.987927 + 1.01193i 0.651424 −0.0480006 1.99942i −0.707107 + 0.707107i −0.643559 + 0.659195i 2.59929i 2.07070 + 1.92671i −2.57565 −0.0169720 1.41411i
See next 80 embeddings (of 152 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 31.76 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
37.d odd 4 1 inner
148.g even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 740.2.u.a 152
4.b odd 2 1 inner 740.2.u.a 152
37.d odd 4 1 inner 740.2.u.a 152
148.g even 4 1 inner 740.2.u.a 152

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
740.2.u.a 152 1.a even 1 1 trivial
740.2.u.a 152 4.b odd 2 1 inner
740.2.u.a 152 37.d odd 4 1 inner
740.2.u.a 152 148.g even 4 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(740, [\chi])$$.