# Properties

 Label 300.9.g.d Level $300$ Weight $9$ Character orbit 300.g Analytic conductor $122.214$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [300,9,Mod(101,300)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(300, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("300.101");

S:= CuspForms(chi, 9);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$300 = 2^{2} \cdot 3 \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$9$$ Character orbit: $$[\chi]$$ $$=$$ 300.g (of order $$2$$, degree $$1$$, 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: $$122.213583018$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-110})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} + 110$$ x^2 + 110 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2\cdot 3$$ Twist minimal: no (minimal twist has level 12) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = 6\sqrt{-110}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta + 51) q^{3} + 3094 q^{7} + (102 \beta - 1359) q^{9}+O(q^{10})$$ q + (b + 51) * q^3 + 3094 * q^7 + (102*b - 1359) * q^9 $$q + (\beta + 51) q^{3} + 3094 q^{7} + (102 \beta - 1359) q^{9} + 18 \beta q^{11} + 7294 q^{13} + 936 \beta q^{17} - 80326 q^{19} + (3094 \beta + 157794) q^{21} - 1548 \beta q^{23} + (3843 \beta - 473229) q^{27} + 13734 \beta q^{29} + 435914 q^{31} + (918 \beta - 71280) q^{33} - 1159298 q^{37} + (7294 \beta + 371994) q^{39} + 43164 \beta q^{41} - 990266 q^{43} - 106488 \beta q^{47} + 3808035 q^{49} + (47736 \beta - 3706560) q^{51} + 160038 \beta q^{53} + ( - 80326 \beta - 4096626) q^{57} - 25398 \beta q^{59} + 19369154 q^{61} + (315588 \beta - 4204746) q^{63} + 28024294 q^{67} + ( - 78948 \beta + 6130080) q^{69} + 534852 \beta q^{71} + 25230142 q^{73} + 55692 \beta q^{77} - 63401398 q^{79} + ( - 277236 \beta - 39352959) q^{81} + 755514 \beta q^{83} + (700434 \beta - 54386640) q^{87} + 1244196 \beta q^{89} + 22567636 q^{91} + (435914 \beta + 22231614) q^{93} - 19550306 q^{97} + ( - 24462 \beta - 7270560) q^{99} +O(q^{100})$$ q + (b + 51) * q^3 + 3094 * q^7 + (102*b - 1359) * q^9 + 18*b * q^11 + 7294 * q^13 + 936*b * q^17 - 80326 * q^19 + (3094*b + 157794) * q^21 - 1548*b * q^23 + (3843*b - 473229) * q^27 + 13734*b * q^29 + 435914 * q^31 + (918*b - 71280) * q^33 - 1159298 * q^37 + (7294*b + 371994) * q^39 + 43164*b * q^41 - 990266 * q^43 - 106488*b * q^47 + 3808035 * q^49 + (47736*b - 3706560) * q^51 + 160038*b * q^53 + (-80326*b - 4096626) * q^57 - 25398*b * q^59 + 19369154 * q^61 + (315588*b - 4204746) * q^63 + 28024294 * q^67 + (-78948*b + 6130080) * q^69 + 534852*b * q^71 + 25230142 * q^73 + 55692*b * q^77 - 63401398 * q^79 + (-277236*b - 39352959) * q^81 + 755514*b * q^83 + (700434*b - 54386640) * q^87 + 1244196*b * q^89 + 22567636 * q^91 + (435914*b + 22231614) * q^93 - 19550306 * q^97 + (-24462*b - 7270560) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 102 q^{3} + 6188 q^{7} - 2718 q^{9}+O(q^{10})$$ 2 * q + 102 * q^3 + 6188 * q^7 - 2718 * q^9 $$2 q + 102 q^{3} + 6188 q^{7} - 2718 q^{9} + 14588 q^{13} - 160652 q^{19} + 315588 q^{21} - 946458 q^{27} + 871828 q^{31} - 142560 q^{33} - 2318596 q^{37} + 743988 q^{39} - 1980532 q^{43} + 7616070 q^{49} - 7413120 q^{51} - 8193252 q^{57} + 38738308 q^{61} - 8409492 q^{63} + 56048588 q^{67} + 12260160 q^{69} + 50460284 q^{73} - 126802796 q^{79} - 78705918 q^{81} - 108773280 q^{87} + 45135272 q^{91} + 44463228 q^{93} - 39100612 q^{97} - 14541120 q^{99}+O(q^{100})$$ 2 * q + 102 * q^3 + 6188 * q^7 - 2718 * q^9 + 14588 * q^13 - 160652 * q^19 + 315588 * q^21 - 946458 * q^27 + 871828 * q^31 - 142560 * q^33 - 2318596 * q^37 + 743988 * q^39 - 1980532 * q^43 + 7616070 * q^49 - 7413120 * q^51 - 8193252 * q^57 + 38738308 * q^61 - 8409492 * q^63 + 56048588 * q^67 + 12260160 * q^69 + 50460284 * q^73 - 126802796 * q^79 - 78705918 * q^81 - 108773280 * q^87 + 45135272 * q^91 + 44463228 * q^93 - 39100612 * q^97 - 14541120 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/300\mathbb{Z}\right)^\times$$.

 $$n$$ $$101$$ $$151$$ $$277$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
101.1
 − 10.4881i 10.4881i
0 51.0000 62.9285i 0 0 0 3094.00 0 −1359.00 6418.71i 0
101.2 0 51.0000 + 62.9285i 0 0 0 3094.00 0 −1359.00 + 6418.71i 0
 $$n$$: e.g. 2-40 or 990-1000 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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 300.9.g.d 2
3.b odd 2 1 inner 300.9.g.d 2
5.b even 2 1 12.9.c.b 2
5.c odd 4 2 300.9.b.c 4
15.d odd 2 1 12.9.c.b 2
15.e even 4 2 300.9.b.c 4
20.d odd 2 1 48.9.e.c 2
40.e odd 2 1 192.9.e.d 2
40.f even 2 1 192.9.e.g 2
45.h odd 6 2 324.9.g.f 4
45.j even 6 2 324.9.g.f 4
60.h even 2 1 48.9.e.c 2
120.i odd 2 1 192.9.e.g 2
120.m even 2 1 192.9.e.d 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
12.9.c.b 2 5.b even 2 1
12.9.c.b 2 15.d odd 2 1
48.9.e.c 2 20.d odd 2 1
48.9.e.c 2 60.h even 2 1
192.9.e.d 2 40.e odd 2 1
192.9.e.d 2 120.m even 2 1
192.9.e.g 2 40.f even 2 1
192.9.e.g 2 120.i odd 2 1
300.9.b.c 4 5.c odd 4 2
300.9.b.c 4 15.e even 4 2
300.9.g.d 2 1.a even 1 1 trivial
300.9.g.d 2 3.b odd 2 1 inner
324.9.g.f 4 45.h odd 6 2
324.9.g.f 4 45.j even 6 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7} - 3094$$ acting on $$S_{9}^{\mathrm{new}}(300, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} - 102T + 6561$$
$5$ $$T^{2}$$
$7$ $$(T - 3094)^{2}$$
$11$ $$T^{2} + 1283040$$
$13$ $$(T - 7294)^{2}$$
$17$ $$T^{2} + 3469340160$$
$19$ $$(T + 80326)^{2}$$
$23$ $$T^{2} + 9489363840$$
$29$ $$T^{2} + 746946113760$$
$31$ $$(T - 435914)^{2}$$
$37$ $$(T + 1159298)^{2}$$
$41$ $$T^{2} + 7377998348160$$
$43$ $$(T + 990266)^{2}$$
$47$ $$T^{2} + 44905188810240$$
$53$ $$T^{2} + \cdots + 101424159318240$$
$59$ $$T^{2} + 2554431279840$$
$61$ $$(T - 19369154)^{2}$$
$67$ $$(T - 28024294)^{2}$$
$71$ $$T^{2} + 11\!\cdots\!40$$
$73$ $$(T - 25230142)^{2}$$
$79$ $$(T + 63401398)^{2}$$
$83$ $$T^{2} + 22\!\cdots\!60$$
$89$ $$T^{2} + 61\!\cdots\!60$$
$97$ $$(T + 19550306)^{2}$$