# Properties

 Label 36.9.d.b Level $36$ Weight $9$ Character orbit 36.d Analytic conductor $14.666$ 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] = [36,9,Mod(19,36)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(36, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("36.19");

S:= CuspForms(chi, 9);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$36 = 2^{2} \cdot 3^{2}$$ Weight: $$k$$ $$=$$ $$9$$ Character orbit: $$[\chi]$$ $$=$$ 36.d (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: $$14.6656299622$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-39})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x + 10$$ x^2 - x + 10 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 4) 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 $$\beta = 2\sqrt{-39}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - \beta + 10) q^{2} + ( - 20 \beta - 56) q^{4} - 610 q^{5} + 112 \beta q^{7} + ( - 144 \beta - 3680) q^{8}+O(q^{10})$$ q + (-b + 10) * q^2 + (-20*b - 56) * q^4 - 610 * q^5 + 112*b * q^7 + (-144*b - 3680) * q^8 $$q + ( - \beta + 10) q^{2} + ( - 20 \beta - 56) q^{4} - 610 q^{5} + 112 \beta q^{7} + ( - 144 \beta - 3680) q^{8} + (610 \beta - 6100) q^{10} + 1480 \beta q^{11} - 5470 q^{13} + (1120 \beta + 17472) q^{14} + (2240 \beta - 59264) q^{16} - 73090 q^{17} - 1560 \beta q^{19} + (12200 \beta + 34160) q^{20} + (14800 \beta + 230880) q^{22} + 18992 \beta q^{23} - 18525 q^{25} + (5470 \beta - 54700) q^{26} + ( - 6272 \beta + 349440) q^{28} + 128222 q^{29} - 5440 \beta q^{31} + (81664 \beta - 243200) q^{32} + (73090 \beta - 730900) q^{34} - 68320 \beta q^{35} - 3472030 q^{37} + ( - 15600 \beta - 243360) q^{38} + (87840 \beta + 2244800) q^{40} - 2146882 q^{41} - 474632 \beta q^{43} + ( - 82880 \beta + 4617600) q^{44} + (189920 \beta + 2962752) q^{46} - 610592 \beta q^{47} + 3807937 q^{49} + (18525 \beta - 185250) q^{50} + (109400 \beta + 306320) q^{52} - 824290 q^{53} - 902800 \beta q^{55} + ( - 412160 \beta + 2515968) q^{56} + ( - 128222 \beta + 1282220) q^{58} + 298280 \beta q^{59} - 14746078 q^{61} + ( - 54400 \beta - 848640) q^{62} + (1059840 \beta + 10307584) q^{64} + 3336700 q^{65} + 1221512 \beta q^{67} + (1461800 \beta + 4093040) q^{68} + ( - 683200 \beta - 10657920) q^{70} + 95760 \beta q^{71} - 5725630 q^{73} + (3472030 \beta - 34720300) q^{74} + (87360 \beta - 4867200) q^{76} - 25858560 q^{77} + 2875360 \beta q^{79} + ( - 1366400 \beta + 36151040) q^{80} + (2146882 \beta - 21468820) q^{82} + 4160152 \beta q^{83} + 44584900 q^{85} + ( - 4746320 \beta - 74042592) q^{86} + ( - 5446400 \beta + 33246720) q^{88} + 83324222 q^{89} - 612640 \beta q^{91} + ( - 1063552 \beta + 59255040) q^{92} + ( - 6105920 \beta - 95252352) q^{94} + 951600 \beta q^{95} + 120619010 q^{97} + ( - 3807937 \beta + 38079370) q^{98} +O(q^{100})$$ q + (-b + 10) * q^2 + (-20*b - 56) * q^4 - 610 * q^5 + 112*b * q^7 + (-144*b - 3680) * q^8 + (610*b - 6100) * q^10 + 1480*b * q^11 - 5470 * q^13 + (1120*b + 17472) * q^14 + (2240*b - 59264) * q^16 - 73090 * q^17 - 1560*b * q^19 + (12200*b + 34160) * q^20 + (14800*b + 230880) * q^22 + 18992*b * q^23 - 18525 * q^25 + (5470*b - 54700) * q^26 + (-6272*b + 349440) * q^28 + 128222 * q^29 - 5440*b * q^31 + (81664*b - 243200) * q^32 + (73090*b - 730900) * q^34 - 68320*b * q^35 - 3472030 * q^37 + (-15600*b - 243360) * q^38 + (87840*b + 2244800) * q^40 - 2146882 * q^41 - 474632*b * q^43 + (-82880*b + 4617600) * q^44 + (189920*b + 2962752) * q^46 - 610592*b * q^47 + 3807937 * q^49 + (18525*b - 185250) * q^50 + (109400*b + 306320) * q^52 - 824290 * q^53 - 902800*b * q^55 + (-412160*b + 2515968) * q^56 + (-128222*b + 1282220) * q^58 + 298280*b * q^59 - 14746078 * q^61 + (-54400*b - 848640) * q^62 + (1059840*b + 10307584) * q^64 + 3336700 * q^65 + 1221512*b * q^67 + (1461800*b + 4093040) * q^68 + (-683200*b - 10657920) * q^70 + 95760*b * q^71 - 5725630 * q^73 + (3472030*b - 34720300) * q^74 + (87360*b - 4867200) * q^76 - 25858560 * q^77 + 2875360*b * q^79 + (-1366400*b + 36151040) * q^80 + (2146882*b - 21468820) * q^82 + 4160152*b * q^83 + 44584900 * q^85 + (-4746320*b - 74042592) * q^86 + (-5446400*b + 33246720) * q^88 + 83324222 * q^89 - 612640*b * q^91 + (-1063552*b + 59255040) * q^92 + (-6105920*b - 95252352) * q^94 + 951600*b * q^95 + 120619010 * q^97 + (-3807937*b + 38079370) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 20 q^{2} - 112 q^{4} - 1220 q^{5} - 7360 q^{8}+O(q^{10})$$ 2 * q + 20 * q^2 - 112 * q^4 - 1220 * q^5 - 7360 * q^8 $$2 q + 20 q^{2} - 112 q^{4} - 1220 q^{5} - 7360 q^{8} - 12200 q^{10} - 10940 q^{13} + 34944 q^{14} - 118528 q^{16} - 146180 q^{17} + 68320 q^{20} + 461760 q^{22} - 37050 q^{25} - 109400 q^{26} + 698880 q^{28} + 256444 q^{29} - 486400 q^{32} - 1461800 q^{34} - 6944060 q^{37} - 486720 q^{38} + 4489600 q^{40} - 4293764 q^{41} + 9235200 q^{44} + 5925504 q^{46} + 7615874 q^{49} - 370500 q^{50} + 612640 q^{52} - 1648580 q^{53} + 5031936 q^{56} + 2564440 q^{58} - 29492156 q^{61} - 1697280 q^{62} + 20615168 q^{64} + 6673400 q^{65} + 8186080 q^{68} - 21315840 q^{70} - 11451260 q^{73} - 69440600 q^{74} - 9734400 q^{76} - 51717120 q^{77} + 72302080 q^{80} - 42937640 q^{82} + 89169800 q^{85} - 148085184 q^{86} + 66493440 q^{88} + 166648444 q^{89} + 118510080 q^{92} - 190504704 q^{94} + 241238020 q^{97} + 76158740 q^{98}+O(q^{100})$$ 2 * q + 20 * q^2 - 112 * q^4 - 1220 * q^5 - 7360 * q^8 - 12200 * q^10 - 10940 * q^13 + 34944 * q^14 - 118528 * q^16 - 146180 * q^17 + 68320 * q^20 + 461760 * q^22 - 37050 * q^25 - 109400 * q^26 + 698880 * q^28 + 256444 * q^29 - 486400 * q^32 - 1461800 * q^34 - 6944060 * q^37 - 486720 * q^38 + 4489600 * q^40 - 4293764 * q^41 + 9235200 * q^44 + 5925504 * q^46 + 7615874 * q^49 - 370500 * q^50 + 612640 * q^52 - 1648580 * q^53 + 5031936 * q^56 + 2564440 * q^58 - 29492156 * q^61 - 1697280 * q^62 + 20615168 * q^64 + 6673400 * q^65 + 8186080 * q^68 - 21315840 * q^70 - 11451260 * q^73 - 69440600 * q^74 - 9734400 * q^76 - 51717120 * q^77 + 72302080 * q^80 - 42937640 * q^82 + 89169800 * q^85 - 148085184 * q^86 + 66493440 * q^88 + 166648444 * q^89 + 118510080 * q^92 - 190504704 * q^94 + 241238020 * q^97 + 76158740 * q^98

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/36\mathbb{Z}\right)^\times$$.

 $$n$$ $$19$$ $$29$$ $$\chi(n)$$ $$-1$$ $$1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
19.1
 0.5 + 3.12250i 0.5 − 3.12250i
10.0000 12.4900i 0 −56.0000 249.800i −610.000 0 1398.88i −3680.00 1798.56i 0 −6100.00 + 7618.90i
19.2 10.0000 + 12.4900i 0 −56.0000 + 249.800i −610.000 0 1398.88i −3680.00 + 1798.56i 0 −6100.00 7618.90i
 $$n$$: 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
4.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 36.9.d.b 2
3.b odd 2 1 4.9.b.b 2
4.b odd 2 1 inner 36.9.d.b 2
12.b even 2 1 4.9.b.b 2
15.d odd 2 1 100.9.b.c 2
15.e even 4 2 100.9.d.b 4
24.f even 2 1 64.9.c.b 2
24.h odd 2 1 64.9.c.b 2
48.i odd 4 2 256.9.d.e 4
48.k even 4 2 256.9.d.e 4
60.h even 2 1 100.9.b.c 2
60.l odd 4 2 100.9.d.b 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4.9.b.b 2 3.b odd 2 1
4.9.b.b 2 12.b even 2 1
36.9.d.b 2 1.a even 1 1 trivial
36.9.d.b 2 4.b odd 2 1 inner
64.9.c.b 2 24.f even 2 1
64.9.c.b 2 24.h odd 2 1
100.9.b.c 2 15.d odd 2 1
100.9.b.c 2 60.h even 2 1
100.9.d.b 4 15.e even 4 2
100.9.d.b 4 60.l odd 4 2
256.9.d.e 4 48.i odd 4 2
256.9.d.e 4 48.k even 4 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5} + 610$$ acting on $$S_{9}^{\mathrm{new}}(36, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - 20T + 256$$
$3$ $$T^{2}$$
$5$ $$(T + 610)^{2}$$
$7$ $$T^{2} + 1956864$$
$11$ $$T^{2} + 341702400$$
$13$ $$(T + 5470)^{2}$$
$17$ $$(T + 73090)^{2}$$
$19$ $$T^{2} + 379641600$$
$23$ $$T^{2} + 56268585984$$
$29$ $$(T - 128222)^{2}$$
$31$ $$T^{2} + 4616601600$$
$37$ $$(T + 3472030)^{2}$$
$41$ $$(T + 2146882)^{2}$$
$43$ $$T^{2} + 35142983526144$$
$47$ $$T^{2} + 58160324112384$$
$53$ $$(T + 824290)^{2}$$
$59$ $$T^{2} + 13879469510400$$
$61$ $$(T + 14746078)^{2}$$
$67$ $$T^{2} + \cdots + 232766284318464$$
$71$ $$T^{2} + 1430516505600$$
$73$ $$(T + 5725630)^{2}$$
$79$ $$T^{2} + 12\!\cdots\!00$$
$83$ $$T^{2} + 26\!\cdots\!24$$
$89$ $$(T - 83324222)^{2}$$
$97$ $$(T - 120619010)^{2}$$