# Properties

 Label 300.2.r.a Level $300$ Weight $2$ Character orbit 300.r Analytic conductor $2.396$ Analytic rank $0$ Dimension $224$ CM no Inner twists $8$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [300,2,Mod(59,300)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(300, base_ring=CyclotomicField(10))

chi = DirichletCharacter(H, H._module([5, 5, 7]))

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

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

chi := DirichletCharacter("300.59");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$300 = 2^{2} \cdot 3 \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 300.r (of order $$10$$, degree $$4$$, 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: $$2.39551206064$$ Analytic rank: $$0$$ Dimension: $$224$$ Relative dimension: $$56$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$224 q - 6 q^{4} - 7 q^{6} - 6 q^{9}+O(q^{10})$$ 224 * q - 6 * q^4 - 7 * q^6 - 6 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$224 q - 6 q^{4} - 7 q^{6} - 6 q^{9} - 5 q^{12} - 20 q^{13} + 6 q^{16} - 24 q^{21} - 10 q^{22} - 28 q^{24} - 20 q^{25} - 50 q^{28} - 25 q^{30} - 10 q^{33} - 14 q^{34} - 25 q^{36} - 60 q^{37} - 20 q^{40} - 85 q^{42} - 30 q^{45} + 6 q^{46} - 70 q^{48} + 88 q^{49} + 40 q^{52} - 46 q^{54} + 50 q^{58} + 10 q^{60} - 28 q^{61} + 18 q^{64} + 64 q^{66} - 42 q^{69} - 40 q^{70} - 5 q^{72} - 20 q^{73} - 80 q^{76} - 85 q^{78} + 18 q^{81} - 48 q^{84} - 20 q^{85} + 50 q^{88} - 60 q^{90} + 60 q^{94} + 44 q^{96} - 120 q^{97}+O(q^{100})$$ 224 * q - 6 * q^4 - 7 * q^6 - 6 * q^9 - 5 * q^12 - 20 * q^13 + 6 * q^16 - 24 * q^21 - 10 * q^22 - 28 * q^24 - 20 * q^25 - 50 * q^28 - 25 * q^30 - 10 * q^33 - 14 * q^34 - 25 * q^36 - 60 * q^37 - 20 * q^40 - 85 * q^42 - 30 * q^45 + 6 * q^46 - 70 * q^48 + 88 * q^49 + 40 * q^52 - 46 * q^54 + 50 * q^58 + 10 * q^60 - 28 * q^61 + 18 * q^64 + 64 * q^66 - 42 * q^69 - 40 * q^70 - 5 * q^72 - 20 * q^73 - 80 * q^76 - 85 * q^78 + 18 * q^81 - 48 * q^84 - 20 * q^85 + 50 * q^88 - 60 * q^90 + 60 * q^94 + 44 * q^96 - 120 * q^97

## 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   $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
59.1 −1.41059 0.101112i −1.18137 + 1.26664i 1.97955 + 0.285257i −0.243022 + 2.22282i 1.79450 1.66726i −4.47645 −2.76350 0.602540i −0.208752 2.99273i 0.567561 3.11093i
59.2 −1.41029 0.105278i 1.45041 0.946744i 1.97783 + 0.296946i −2.23180 0.138020i −2.14516 + 1.18249i −1.71470 −2.75806 0.627002i 1.20735 2.74632i 3.13296 + 0.429608i
59.3 −1.40665 + 0.146030i 1.00033 + 1.41398i 1.95735 0.410828i −1.66487 1.49272i −1.61361 1.84290i −0.571505 −2.69332 + 0.863724i −0.998661 + 2.82890i 2.55988 + 1.85662i
59.4 −1.37885 0.314289i 1.03587 1.38816i 1.80244 + 0.866714i 1.94623 1.10099i −1.86458 + 1.58850i 2.63015 −2.21290 1.76156i −0.853964 2.87589i −3.02959 + 0.906413i
59.5 −1.37668 + 0.323633i −1.44316 0.957755i 1.79052 0.891082i 2.16605 + 0.555182i 2.29674 + 0.851472i 0.146941 −2.17660 + 1.80621i 1.16541 + 2.76438i −3.16164 0.0633046i
59.6 −1.32465 + 0.495274i 1.60045 + 0.662243i 1.50941 1.31213i 0.462356 + 2.18774i −2.44803 0.0845826i 2.28606 −1.34958 + 2.48569i 2.12287 + 2.11977i −1.69599 2.66901i
59.7 −1.30486 0.545281i 0.0653839 + 1.73082i 1.40534 + 1.42303i 2.14307 + 0.638169i 0.858463 2.29413i 4.06778 −1.05782 2.62317i −2.99145 + 0.226335i −2.44843 2.00130i
59.8 −1.29016 0.579209i −1.71582 0.236546i 1.32903 + 1.49455i −0.101169 2.23378i 2.07668 + 1.29900i −0.903444 −0.849014 2.69799i 2.88809 + 0.811741i −1.16330 + 2.94053i
59.9 −1.28571 0.589029i −0.760326 1.55625i 1.30609 + 1.51464i −1.70605 + 1.44547i 0.0608822 + 2.44873i 1.06519 −0.787085 2.71671i −1.84381 + 2.36651i 3.04491 0.853547i
59.10 −1.26137 + 0.639492i −0.689827 + 1.58875i 1.18210 1.61327i 1.00094 1.99953i −0.145871 2.44514i −0.923130 −0.459388 + 2.79087i −2.04828 2.19193i 0.0161381 + 3.16224i
59.11 −1.19645 + 0.753994i −0.269355 1.71098i 0.862987 1.80423i −1.27460 1.83722i 1.61234 + 1.84401i 4.55385 0.327858 + 2.80936i −2.85490 + 0.921722i 2.91025 + 1.23711i
59.12 −1.12183 + 0.861108i 1.64941 0.528616i 0.516985 1.93203i 1.99646 1.00704i −1.39516 + 2.01334i −3.00727 1.08372 + 2.61258i 2.44113 1.74381i −1.37251 + 2.84890i
59.13 −1.06241 + 0.933425i 0.154412 1.72515i 0.257435 1.98336i −0.778418 + 2.09620i 1.44625 + 1.97695i −3.14870 1.57782 + 2.34744i −2.95231 0.532768i −1.12965 2.95362i
59.14 −0.957505 1.04076i 0.760326 + 1.55625i −0.166367 + 1.99307i −1.70605 + 1.44547i 0.891666 2.28143i −1.06519 2.23361 1.73523i −1.84381 + 2.36651i 3.13794 + 0.391539i
59.15 −0.949542 1.04803i 1.71582 + 0.236546i −0.196739 + 1.99030i −0.101169 2.23378i −1.38134 2.02285i 0.903444 2.27271 1.68369i 2.88809 + 0.811741i −2.24500 + 2.22710i
59.16 −0.932419 + 1.06329i −1.73059 + 0.0711045i −0.261191 1.98287i −1.84864 1.25798i 1.53803 1.90643i −2.80461 2.35192 + 1.57114i 2.98989 0.246105i 3.06132 0.792685i
59.17 −0.921818 1.07250i −0.0653839 1.73082i −0.300505 + 1.97730i 2.14307 + 0.638169i −1.79603 + 1.66562i −4.06778 2.39766 1.50042i −2.99145 + 0.226335i −1.29108 2.88671i
59.18 −0.724994 1.21424i −1.03587 + 1.38816i −0.948767 + 1.76064i 1.94623 1.10099i 2.43656 + 0.251385i −2.63015 2.82569 0.124420i −0.853964 2.87589i −2.74787 1.56499i
59.19 −0.723120 + 1.21536i −1.44187 + 0.959691i −0.954195 1.75770i 1.84864 + 1.25798i −0.123723 2.44636i 2.80461 2.82623 + 0.111339i 1.15799 2.76750i −2.86569 + 1.33709i
59.20 −0.559437 + 1.29886i 1.13894 + 1.30492i −1.37406 1.45326i 0.778418 2.09620i −2.33207 + 0.749304i 3.14870 2.65627 0.971702i −0.405623 + 2.97245i 2.28719 + 2.18375i
See next 80 embeddings (of 224 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 59.56 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
4.b odd 2 1 inner
12.b even 2 1 inner
25.e even 10 1 inner
75.h odd 10 1 inner
100.h odd 10 1 inner
300.r even 10 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 300.2.r.a 224
3.b odd 2 1 inner 300.2.r.a 224
4.b odd 2 1 inner 300.2.r.a 224
12.b even 2 1 inner 300.2.r.a 224
25.e even 10 1 inner 300.2.r.a 224
75.h odd 10 1 inner 300.2.r.a 224
100.h odd 10 1 inner 300.2.r.a 224
300.r even 10 1 inner 300.2.r.a 224

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
300.2.r.a 224 1.a even 1 1 trivial
300.2.r.a 224 3.b odd 2 1 inner
300.2.r.a 224 4.b odd 2 1 inner
300.2.r.a 224 12.b even 2 1 inner
300.2.r.a 224 25.e even 10 1 inner
300.2.r.a 224 75.h odd 10 1 inner
300.2.r.a 224 100.h odd 10 1 inner
300.2.r.a 224 300.r even 10 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(300, [\chi])$$.