# Properties

 Label 32.9.c.b Level $32$ Weight $9$ Character orbit 32.c Analytic conductor $13.036$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [32,9,Mod(31,32)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(32, 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("32.31");

S:= CuspForms(chi, 9);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$32 = 2^{5}$$ Weight: $$k$$ $$=$$ $$9$$ Character orbit: $$[\chi]$$ $$=$$ 32.c (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: $$13.0361155220$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{19})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - 9x^{2} + 25$$ x^4 - 9*x^2 + 25 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{18}$$ Twist minimal: yes 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 a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{3} q^{3} + (3 \beta_{2} + 266) q^{5} + ( - 42 \beta_{3} - 7 \beta_1) q^{7} + ( - 10 \beta_{2} + 1297) q^{9}+O(q^{10})$$ q + b3 * q^3 + (3*b2 + 266) * q^5 + (-42*b3 - 7*b1) * q^7 + (-10*b2 + 1297) * q^9 $$q + \beta_{3} q^{3} + (3 \beta_{2} + 266) q^{5} + ( - 42 \beta_{3} - 7 \beta_1) q^{7} + ( - 10 \beta_{2} + 1297) q^{9} + (87 \beta_{3} - 98 \beta_1) q^{11} + (39 \beta_{2} + 23146) q^{13} + (506 \beta_{3} - 837 \beta_1) q^{15} + ( - 258 \beta_{2} + 62130) q^{17} + ( - 1299 \beta_{3} - 1350 \beta_1) q^{19} + (308 \beta_{2} + 212128) q^{21} + ( - 1998 \beta_{3} - 6169 \beta_1) q^{23} + (1596 \beta_{2} + 380547) q^{25} + (7058 \beta_{3} + 2790 \beta_1) q^{27} + ( - 3261 \beta_{2} - 122678) q^{29} + ( - 168 \beta_{3} - 11656 \beta_1) q^{31} + ( - 2438 \beta_{2} - 583408) q^{33} + ( - 15876 \beta_{3} + 34972 \beta_1) q^{35} + (6519 \beta_{2} - 795222) q^{37} + (26266 \beta_{3} - 10881 \beta_1) q^{39} + (2076 \beta_{2} - 1741022) q^{41} + (1311 \beta_{3} + 88772 \beta_1) q^{43} + (1231 \beta_{2} - 1989718) q^{45} + ( - 88140 \beta_{3} - 29638 \beta_1) q^{47} + ( - 8232 \beta_{2} - 2968959) q^{49} + (41490 \beta_{3} + 71982 \beta_1) q^{51} + ( - 9057 \beta_{2} - 581526) q^{53} + (119286 \beta_{3} - 75367 \beta_1) q^{55} + ( - 8610 \beta_{2} + 5109936) q^{57} + (60699 \beta_{3} - 85256 \beta_1) q^{59} + (13095 \beta_{2} + 9745642) q^{61} + ( - 38794 \beta_{3} - 131859 \beta_1) q^{63} + (79812 \beta_{2} + 15262244) q^{65} + ( - 456159 \beta_{3} - 265174 \beta_1) q^{67} + ( - 78724 \beta_{2} + 2621152) q^{69} + ( - 47898 \beta_{3} + 26557 \beta_1) q^{71} + ( - 42450 \beta_{2} + 7444338) q^{73} + (508227 \beta_{3} - 445284 \beta_1) q^{75} + (92652 \beta_{2} + 20913760) q^{77} + (100620 \beta_{3} + 621826 \beta_1) q^{79} + ( - 91550 \beta_{2} - 25072495) q^{81} + (503085 \beta_{3} - 386560 \beta_1) q^{83} + (117762 \beta_{2} - 43709196) q^{85} + ( - 383558 \beta_{3} + 909819 \beta_1) q^{87} + ( - 106818 \beta_{2} - 11811726) q^{89} + ( - 1033284 \beta_{3} + 316820 \beta_1) q^{91} + ( - 184816 \beta_{2} - 14035328) q^{93} + (379506 \beta_{3} + 1052163 \beta_1) q^{95} + (126054 \beta_{2} - 98994830) q^{97} + ( - 207641 \beta_{3} + 37224 \beta_1) q^{99}+O(q^{100})$$ q + b3 * q^3 + (3*b2 + 266) * q^5 + (-42*b3 - 7*b1) * q^7 + (-10*b2 + 1297) * q^9 + (87*b3 - 98*b1) * q^11 + (39*b2 + 23146) * q^13 + (506*b3 - 837*b1) * q^15 + (-258*b2 + 62130) * q^17 + (-1299*b3 - 1350*b1) * q^19 + (308*b2 + 212128) * q^21 + (-1998*b3 - 6169*b1) * q^23 + (1596*b2 + 380547) * q^25 + (7058*b3 + 2790*b1) * q^27 + (-3261*b2 - 122678) * q^29 + (-168*b3 - 11656*b1) * q^31 + (-2438*b2 - 583408) * q^33 + (-15876*b3 + 34972*b1) * q^35 + (6519*b2 - 795222) * q^37 + (26266*b3 - 10881*b1) * q^39 + (2076*b2 - 1741022) * q^41 + (1311*b3 + 88772*b1) * q^43 + (1231*b2 - 1989718) * q^45 + (-88140*b3 - 29638*b1) * q^47 + (-8232*b2 - 2968959) * q^49 + (41490*b3 + 71982*b1) * q^51 + (-9057*b2 - 581526) * q^53 + (119286*b3 - 75367*b1) * q^55 + (-8610*b2 + 5109936) * q^57 + (60699*b3 - 85256*b1) * q^59 + (13095*b2 + 9745642) * q^61 + (-38794*b3 - 131859*b1) * q^63 + (79812*b2 + 15262244) * q^65 + (-456159*b3 - 265174*b1) * q^67 + (-78724*b2 + 2621152) * q^69 + (-47898*b3 + 26557*b1) * q^71 + (-42450*b2 + 7444338) * q^73 + (508227*b3 - 445284*b1) * q^75 + (92652*b2 + 20913760) * q^77 + (100620*b3 + 621826*b1) * q^79 + (-91550*b2 - 25072495) * q^81 + (503085*b3 - 386560*b1) * q^83 + (117762*b2 - 43709196) * q^85 + (-383558*b3 + 909819*b1) * q^87 + (-106818*b2 - 11811726) * q^89 + (-1033284*b3 + 316820*b1) * q^91 + (-184816*b2 - 14035328) * q^93 + (379506*b3 + 1052163*b1) * q^95 + (126054*b2 - 98994830) * q^97 + (-207641*b3 + 37224*b1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 1064 q^{5} + 5188 q^{9}+O(q^{10})$$ 4 * q + 1064 * q^5 + 5188 * q^9 $$4 q + 1064 q^{5} + 5188 q^{9} + 92584 q^{13} + 248520 q^{17} + 848512 q^{21} + 1522188 q^{25} - 490712 q^{29} - 2333632 q^{33} - 3180888 q^{37} - 6964088 q^{41} - 7958872 q^{45} - 11875836 q^{49} - 2326104 q^{53} + 20439744 q^{57} + 38982568 q^{61} + 61048976 q^{65} + 10484608 q^{69} + 29777352 q^{73} + 83655040 q^{77} - 100289980 q^{81} - 174836784 q^{85} - 47246904 q^{89} - 56141312 q^{93} - 395979320 q^{97}+O(q^{100})$$ 4 * q + 1064 * q^5 + 5188 * q^9 + 92584 * q^13 + 248520 * q^17 + 848512 * q^21 + 1522188 * q^25 - 490712 * q^29 - 2333632 * q^33 - 3180888 * q^37 - 6964088 * q^41 - 7958872 * q^45 - 11875836 * q^49 - 2326104 * q^53 + 20439744 * q^57 + 38982568 * q^61 + 61048976 * q^65 + 10484608 * q^69 + 29777352 * q^73 + 83655040 * q^77 - 100289980 * q^81 - 174836784 * q^85 - 47246904 * q^89 - 56141312 * q^93 - 395979320 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 9x^{2} + 25$$ :

 $$\beta_{1}$$ $$=$$ $$( -64\nu^{3} + 256\nu ) / 5$$ (-64*v^3 + 256*v) / 5 $$\beta_{2}$$ $$=$$ $$( -64\nu^{3} + 896\nu ) / 5$$ (-64*v^3 + 896*v) / 5 $$\beta_{3}$$ $$=$$ $$4\nu^{3} + 32\nu^{2} - 16\nu - 144$$ 4*v^3 + 32*v^2 - 16*v - 144
 $$\nu$$ $$=$$ $$( \beta_{2} - \beta_1 ) / 128$$ (b2 - b1) / 128 $$\nu^{2}$$ $$=$$ $$( 16\beta_{3} + 5\beta _1 + 2304 ) / 512$$ (16*b3 + 5*b1 + 2304) / 512 $$\nu^{3}$$ $$=$$ $$( 2\beta_{2} - 7\beta_1 ) / 64$$ (2*b2 - 7*b1) / 64

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/32\mathbb{Z}\right)^\times$$.

 $$n$$ $$5$$ $$31$$ $$\chi(n)$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
31.1
 2.17945 − 0.500000i −2.17945 + 0.500000i −2.17945 − 0.500000i 2.17945 + 0.500000i
0 89.7424i 0 1102.91 0 3321.18i 0 −1492.70 0
31.2 0 49.7424i 0 −570.909 0 2537.18i 0 4086.70 0
31.3 0 49.7424i 0 −570.909 0 2537.18i 0 4086.70 0
31.4 0 89.7424i 0 1102.91 0 3321.18i 0 −1492.70 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
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 32.9.c.b 4
3.b odd 2 1 288.9.g.a 4
4.b odd 2 1 inner 32.9.c.b 4
8.b even 2 1 64.9.c.e 4
8.d odd 2 1 64.9.c.e 4
12.b even 2 1 288.9.g.a 4
16.e even 4 1 256.9.d.c 4
16.e even 4 1 256.9.d.g 4
16.f odd 4 1 256.9.d.c 4
16.f odd 4 1 256.9.d.g 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
32.9.c.b 4 1.a even 1 1 trivial
32.9.c.b 4 4.b odd 2 1 inner
64.9.c.e 4 8.b even 2 1
64.9.c.e 4 8.d odd 2 1
256.9.d.c 4 16.e even 4 1
256.9.d.c 4 16.f odd 4 1
256.9.d.g 4 16.e even 4 1
256.9.d.g 4 16.f odd 4 1
288.9.g.a 4 3.b odd 2 1
288.9.g.a 4 12.b even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{4} + 10528T_{3}^{2} + 19927296$$ acting on $$S_{9}^{\mathrm{new}}(32, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4} + 10528 T^{2} + \cdots + 19927296$$
$5$ $$(T^{2} - 532 T - 629660)^{2}$$
$7$ $$T^{4} + 17467520 T^{2} + \cdots + 71004756250624$$
$11$ $$T^{4} + \cdots + 749474285334784$$
$13$ $$(T^{2} - 46292 T + 417367012)^{2}$$
$17$ $$(T^{2} - 124260 T - 1320139836)^{2}$$
$19$ $$T^{4} + 23716189728 T^{2} + \cdots + 20\!\cdots\!96$$
$23$ $$T^{4} + 290679776384 T^{2} + \cdots + 11\!\cdots\!00$$
$29$ $$(T^{2} + 245356 T - 812539941020)^{2}$$
$31$ $$T^{4} + 1103255373824 T^{2} + \cdots + 30\!\cdots\!00$$
$37$ $$(T^{2} + 1590444 T - 2674936593180)^{2}$$
$41$ $$(T^{2} + 3482044 T + 2695753597060)^{2}$$
$43$ $$T^{4} + 63979022348576 T^{2} + \cdots + 10\!\cdots\!00$$
$47$ $$T^{4} + 75609449742848 T^{2} + \cdots + 14\!\cdots\!76$$
$53$ $$(T^{2} + 1163052 T - 6045671785500)^{2}$$
$59$ $$T^{4} + 124829049535520 T^{2} + \cdots + 70\!\cdots\!24$$
$61$ $$(T^{2} - 19491284 T + 81632354350564)^{2}$$
$67$ $$T^{4} + \cdots + 90\!\cdots\!04$$
$71$ $$T^{4} + 36443917197440 T^{2} + \cdots + 16\!\cdots\!64$$
$73$ $$(T^{2} - 14888676 T - 84820874301756)^{2}$$
$79$ $$T^{4} + \cdots + 19\!\cdots\!16$$
$83$ $$T^{4} + \cdots + 39\!\cdots\!00$$
$89$ $$(T^{2} + 23623452 T - 748461593591100)^{2}$$
$97$ $$(T^{2} + 197989660 T + 85\!\cdots\!16)^{2}$$