# Properties

 Label 234.6.b.e Level $234$ Weight $6$ Character orbit 234.b Analytic conductor $37.530$ Analytic rank $0$ Dimension $12$ Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [234,6,Mod(181,234)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(234, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("234.181");

S:= CuspForms(chi, 6);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$234 = 2 \cdot 3^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 234.b (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: $$37.5298138362$$ Analytic rank: $$0$$ Dimension: $$12$$ Coefficient field: $$\mathbb{Q}[x]/(x^{12} + \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{12} + 1616511x^{8} + 609939232149x^{4} + 125481622652164$$ x^12 + 1616511*x^8 + 609939232149*x^4 + 125481622652164 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{24}\cdot 3^{6}$$ 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 $$1,\beta_1,\ldots,\beta_{11}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta_{3} q^{2} - 16 q^{4} + (\beta_{4} - \beta_{3}) q^{5} - \beta_{7} q^{7} + 16 \beta_{3} q^{8}+O(q^{10})$$ q - b3 * q^2 - 16 * q^4 + (b4 - b3) * q^5 - b7 * q^7 + 16*b3 * q^8 $$q - \beta_{3} q^{2} - 16 q^{4} + (\beta_{4} - \beta_{3}) q^{5} - \beta_{7} q^{7} + 16 \beta_{3} q^{8} + ( - 2 \beta_1 - 18) q^{10} + (\beta_{5} + 3 \beta_{4} - 22 \beta_{3}) q^{11} + ( - \beta_{9} - \beta_{2} - \beta_1 - 37) q^{13} - \beta_{11} q^{14} + 256 q^{16} + (\beta_{8} - \beta_{6}) q^{17} + ( - \beta_{10} + 4 \beta_{9} + \cdots - 2) q^{19}+ \cdots + ( - 232 \beta_{5} + \cdots + 749 \beta_{3}) q^{98}+O(q^{100})$$ q - b3 * q^2 - 16 * q^4 + (b4 - b3) * q^5 - b7 * q^7 + 16*b3 * q^8 + (-2*b1 - 18) * q^10 + (b5 + 3*b4 - 22*b3) * q^11 + (-b9 - b2 - b1 - 37) * q^13 - b11 * q^14 + 256 * q^16 + (b8 - b6) * q^17 + (-b10 + 4*b9 - 10*b7 + 2*b1 - 2) * q^19 + (-16*b4 + 16*b3) * q^20 + (-2*b2 - 6*b1 - 358) * q^22 + (3*b11 - 5*b8 + 2*b6) * q^23 + (-5*b2 + 5*b1 - 566) * q^25 + (b11 - 2*b8 - 4*b6 - 8*b5 - 4*b4 + 37*b3) * q^26 + 16*b7 * q^28 + (5*b11 + 6*b8 + b6) * q^29 + (13*b10 + 6*b9 + 8*b7 + 3*b1 - 3) * q^31 - 256*b3 * q^32 + (-6*b10 + 4*b9 - 2*b7 + 2*b1 - 2) * q^34 + (-11*b11 + 15*b8 + 20*b6) * q^35 + (-17*b10 + 16*b9 + 45*b7 + 8*b1 - 8) * q^37 + (-13*b11 + 4*b8 + 16*b6) * q^38 + (32*b1 + 288) * q^40 + (-54*b5 - b4 + 1147*b3) * q^41 + (-5*b2 + 38*b1 + 3334) * q^43 + (-16*b5 - 48*b4 + 352*b3) * q^44 + (24*b10 - 8*b9 - 32*b7 - 4*b1 + 4) * q^46 + (-53*b5 + 111*b4 - 1024*b3) * q^47 + (-29*b2 - 34*b1 - 783) * q^49 + (-40*b5 + 40*b4 + 571*b3) * q^50 + (16*b9 + 16*b2 + 16*b1 + 592) * q^52 + (-37*b11 + 19*b8 + 78*b6) * q^53 + (-27*b2 - 96*b1 - 10728) * q^55 + 16*b11 * q^56 + (-22*b10 - 4*b9 - 106*b7 - 2*b1 + 2) * q^58 + (-81*b5 - 223*b4 - 398*b3) * q^59 + (5*b2 - 155*b1 + 5063) * q^61 + (-11*b11 + 64*b8 + 24*b6) * q^62 - 4096 * q^64 + (43*b11 + 18*b8 + 23*b6 - 58*b5 - 315*b4 + 577*b3) * q^65 + (-22*b10 + 46*b9 - 299*b7 + 23*b1 - 23) * q^67 + (-16*b8 + 16*b6) * q^68 + (-20*b10 - 80*b9 + 76*b7 - 40*b1 + 40) * q^70 + (-135*b5 - 51*b4 + 1800*b3) * q^71 + (63*b10 - 46*b9 + 125*b7 - 23*b1 + 23) * q^73 + (46*b11 - 36*b8 + 64*b6) * q^74 + (16*b10 - 64*b9 + 160*b7 - 32*b1 + 32) * q^76 + (-96*b11 + 57*b8 + 63*b6) * q^77 + (76*b2 + 350*b1 + 11010) * q^79 + (256*b4 - 256*b3) * q^80 + (108*b2 + 2*b1 + 18354) * q^82 + (-197*b5 - 875*b4 + 1594*b3) * q^83 + (-31*b10 + 14*b9 + 23*b7 + 7*b1 - 7) * q^85 + (-40*b5 + 304*b4 - 3296*b3) * q^86 + (32*b2 + 96*b1 + 5728) * q^88 + (-194*b5 + 1203*b4 + 2219*b3) * q^89 + (-26*b10 - 26*b9 + 377*b7 + 13*b2 + 481*b1 - 3705) * q^91 + (-48*b11 + 80*b8 - 32*b6) * q^92 + (106*b2 - 222*b1 - 16606) * q^94 + (-243*b11 + 69*b8 + 126*b6) * q^95 + (37*b10 + 2*b9 - 5*b7 + b1 - 1) * q^97 + (-232*b5 - 272*b4 + 749*b3) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12 q - 192 q^{4}+O(q^{10})$$ 12 * q - 192 * q^4 $$12 q - 192 q^{4} - 224 q^{10} - 452 q^{13} + 3072 q^{16} - 4320 q^{22} - 6772 q^{25} + 3584 q^{40} + 40160 q^{43} - 9532 q^{49} + 7232 q^{52} - 129120 q^{55} + 60136 q^{61} - 49152 q^{64} + 133520 q^{79} + 220256 q^{82} + 69120 q^{88} - 42640 q^{91} - 200160 q^{94}+O(q^{100})$$ 12 * q - 192 * q^4 - 224 * q^10 - 452 * q^13 + 3072 * q^16 - 4320 * q^22 - 6772 * q^25 + 3584 * q^40 + 40160 * q^43 - 9532 * q^49 + 7232 * q^52 - 129120 * q^55 + 60136 * q^61 - 49152 * q^64 + 133520 * q^79 + 220256 * q^82 + 69120 * q^88 - 42640 * q^91 - 200160 * q^94

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} + 1616511x^{8} + 609939232149x^{4} + 125481622652164$$ :

 $$\beta_{1}$$ $$=$$ $$( 4432\nu^{8} + 4986886340\nu^{4} + 630446369467633 ) / 4774958026695$$ (4432*v^8 + 4986886340*v^4 + 630446369467633) / 4774958026695 $$\beta_{2}$$ $$=$$ $$( -2956\nu^{8} - 1410744980\nu^{4} + 612633725585336 ) / 1591652675565$$ (-2956*v^8 - 1410744980*v^4 + 612633725585336) / 1591652675565 $$\beta_{3}$$ $$=$$ $$( 3594\nu^{10} + 5832144250\nu^{6} + 2246402667691266\nu^{2} ) / 8053056574977055$$ (3594*v^10 + 5832144250*v^6 + 2246402667691266*v^2) / 8053056574977055 $$\beta_{4}$$ $$=$$ $$( -215151856\nu^{10} - 340173743335370\nu^{6} - 123860603519620408774\nu^{2} ) / 26744200885498799655$$ (-215151856*v^10 - 340173743335370*v^6 - 123860603519620408774*v^2) / 26744200885498799655 $$\beta_{5}$$ $$=$$ $$( -36806026\nu^{10} - 61519337273576\nu^{6} - 23940387725169549598\nu^{2} ) / 1782946725699919977$$ (-36806026*v^10 - 61519337273576*v^6 - 23940387725169549598*v^2) / 1782946725699919977 $$\beta_{6}$$ $$=$$ $$( 10782\nu^{11} + 17496432750\nu^{7} + 6739208003073798\nu^{3} - 96636678899724660\nu ) / 8053056574977055$$ (10782*v^11 + 17496432750*v^7 + 6739208003073798*v^3 - 96636678899724660*v) / 8053056574977055 $$\beta_{7}$$ $$=$$ $$( 12347502607 \nu^{11} + 223353846662 \nu^{9} + \cdots + 10\!\cdots\!18 \nu ) / 31\!\cdots\!90$$ (12347502607*v^11 + 223353846662*v^9 + 19920327598602185*v^7 + 331775629974089890*v^5 + 7490526500868791939743*v^3 + 107477074621493401046618*v) / 3155815704488858359290 $$\beta_{8}$$ $$=$$ $$( 7767645982 \nu^{11} + 24890528476 \nu^{9} + \cdots - 69\!\cdots\!16 \nu ) / 15\!\cdots\!45$$ (7767645982*v^11 + 24890528476*v^9 + 12676608141445790*v^7 - 68542639376271520*v^5 + 4833079459526183568058*v^3 - 69317532856593021881516*v) / 1577907852244429179645 $$\beta_{9}$$ $$=$$ $$( - 24340377185 \nu^{11} - 198463318186 \nu^{9} - 1464575722352 \nu^{8} + \cdots - 20\!\cdots\!18 ) / 31\!\cdots\!90$$ (-24340377185*v^11 - 198463318186*v^9 - 1464575722352*v^8 - 39453402813252475*v^7 - 400318269350361410*v^5 - 1647940582963181740*v^4 - 14964553314223529320445*v^3 - 214664395931952723239614*v - 206756128015818404452718) / 3155815704488858359290 $$\beta_{10}$$ $$=$$ $$( - 9294264857 \nu^{11} - 57857596570 \nu^{9} + \cdots - 82\!\cdots\!50 \nu ) / 10\!\cdots\!30$$ (-9294264857*v^11 - 57857596570*v^9 - 15091181293831255*v^7 - 156286969575544310*v^5 - 5718895139973719691953*v^3 - 82037380111559814936550*v) / 1051938568162952786430 $$\beta_{11}$$ $$=$$ $$( - 24695005214 \nu^{11} + 446707693324 \nu^{9} + \cdots + 21\!\cdots\!36 \nu ) / 15\!\cdots\!45$$ (-24695005214*v^11 + 446707693324*v^9 - 39840655197204370*v^7 + 663551259948179780*v^5 - 14981053001737583879486*v^3 + 214954149242986802093236*v) / 1577907852244429179645
 $$\nu$$ $$=$$ $$( \beta_{10} - 2\beta_{9} - \beta_{7} - 2\beta_{6} - \beta _1 + 1 ) / 48$$ (b10 - 2*b9 - b7 - 2*b6 - b1 + 1) / 48 $$\nu^{2}$$ $$=$$ $$( 3\beta_{5} + 9\beta_{4} + 301\beta_{3} ) / 2$$ (3*b5 + 9*b4 + 301*b3) / 2 $$\nu^{3}$$ $$=$$ $$( 72\beta_{11} + 1085\beta_{10} - 1702\beta_{9} - 234\beta_{8} - 905\beta_{7} + 1702\beta_{6} - 851\beta _1 + 851 ) / 48$$ (72*b11 + 1085*b10 - 1702*b9 - 234*b8 - 905*b7 + 1702*b6 - 851*b1 + 851) / 48 $$\nu^{4}$$ $$=$$ $$( 3324\beta_{2} + 6651\beta _1 - 2157565 ) / 4$$ (3324*b2 + 6651*b1 - 2157565) / 4 $$\nu^{5}$$ $$=$$ $$( 9999 \beta_{11} - 564677 \beta_{10} + 770452 \beta_{9} - 179451 \beta_{8} + 245771 \beta_{7} + \cdots - 385226 ) / 24$$ (9999*b11 - 564677*b10 + 770452*b9 - 179451*b8 + 245771*b7 + 770452*b6 + 385226*b1 - 385226) / 24 $$\nu^{6}$$ $$=$$ $$( -3554181\beta_{5} - 4694706\beta_{4} - 249026899\beta_{3} ) / 2$$ (-3554181*b5 - 4694706*b4 - 249026899*b3) / 2 $$\nu^{7}$$ $$=$$ $$( 22120722 \beta_{11} - 1165762331 \beta_{10} + 1419002206 \beta_{9} + 456261228 \beta_{8} + \cdots - 709501103 ) / 48$$ (22120722*b11 - 1165762331*b10 + 1419002206*b9 + 456261228*b8 + 164756987*b7 - 1419002206*b6 + 709501103*b1 - 709501103) / 48 $$\nu^{8}$$ $$=$$ $$( -3740164755\beta_{2} - 3174176205\beta _1 + 1858697190499 ) / 4$$ (-3740164755*b2 - 3174176205*b1 + 1858697190499) / 4 $$\nu^{9}$$ $$=$$ $$( 55069770960 \beta_{11} + 1196375456191 \beta_{10} - 1326506127002 \beta_{9} + 533122392690 \beta_{8} + \cdots + 663253063501 ) / 48$$ (55069770960*b11 + 1196375456191*b10 - 1326506127002*b9 + 533122392690*b8 + 90148413029*b7 - 1326506127002*b6 - 663253063501*b1 + 663253063501) / 48 $$\nu^{10}$$ $$=$$ $$( 3892400745558\beta_{5} + 1992926709549\beta_{4} + 220450669803301\beta_{3} ) / 2$$ (3892400745558*b5 + 1992926709549*b4 + 220450669803301*b3) / 2 $$\nu^{11}$$ $$=$$ $$( - 40449670797429 \beta_{11} + 611263134037720 \beta_{10} - 628390744108166 \beta_{9} + \cdots + 314195372054083 ) / 24$$ (-40449670797429*b11 + 611263134037720*b10 - 628390744108166*b9 - 297067761983637*b8 + 144671073119270*b7 + 628390744108166*b6 - 314195372054083*b1 + 314195372054083) / 24

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/234\mathbb{Z}\right)^\times$$.

 $$n$$ $$145$$ $$209$$ $$\chi(n)$$ $$-1$$ $$1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
181.1
 2.67835 + 2.67835i −2.67835 − 2.67835i 22.4536 + 22.4536i −22.4536 − 22.4536i −19.6764 − 19.6764i 19.6764 + 19.6764i 19.6764 − 19.6764i −19.6764 + 19.6764i −22.4536 + 22.4536i 22.4536 − 22.4536i −2.67835 + 2.67835i 2.67835 − 2.67835i
4.00000i 0 −16.0000 70.4084i 0 182.316i 64.0000i 0 −281.634
181.2 4.00000i 0 −16.0000 70.4084i 0 182.316i 64.0000i 0 −281.634
181.3 4.00000i 0 −16.0000 19.3336i 0 14.8060i 64.0000i 0 −77.3346
181.4 4.00000i 0 −16.0000 19.3336i 0 14.8060i 64.0000i 0 −77.3346
181.5 4.00000i 0 −16.0000 75.7421i 0 139.088i 64.0000i 0 302.968
181.6 4.00000i 0 −16.0000 75.7421i 0 139.088i 64.0000i 0 302.968
181.7 4.00000i 0 −16.0000 75.7421i 0 139.088i 64.0000i 0 302.968
181.8 4.00000i 0 −16.0000 75.7421i 0 139.088i 64.0000i 0 302.968
181.9 4.00000i 0 −16.0000 19.3336i 0 14.8060i 64.0000i 0 −77.3346
181.10 4.00000i 0 −16.0000 19.3336i 0 14.8060i 64.0000i 0 −77.3346
181.11 4.00000i 0 −16.0000 70.4084i 0 182.316i 64.0000i 0 −281.634
181.12 4.00000i 0 −16.0000 70.4084i 0 182.316i 64.0000i 0 −281.634
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 181.12 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
13.b even 2 1 inner
39.d odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 234.6.b.e 12
3.b odd 2 1 inner 234.6.b.e 12
13.b even 2 1 inner 234.6.b.e 12
39.d odd 2 1 inner 234.6.b.e 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
234.6.b.e 12 1.a even 1 1 trivial
234.6.b.e 12 3.b odd 2 1 inner
234.6.b.e 12 13.b even 2 1 inner
234.6.b.e 12 39.d odd 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{6} + 11068T_{5}^{4} + 32437008T_{5}^{2} + 10630434816$$ acting on $$S_{6}^{\mathrm{new}}(234, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} + 16)^{6}$$
$3$ $$T^{12}$$
$5$ $$(T^{6} + 11068 T^{4} + \cdots + 10630434816)^{2}$$
$7$ $$(T^{6} + 52804 T^{4} + \cdots + 140963846400)^{2}$$
$11$ $$(T^{6} + \cdots + 5563560038400)^{2}$$
$13$ $$(T^{6} + \cdots + 51\!\cdots\!57)^{2}$$
$17$ $$(T^{6} + \cdots - 3416496537600)^{2}$$
$19$ $$(T^{6} + \cdots + 66\!\cdots\!00)^{2}$$
$23$ $$(T^{6} + \cdots - 98\!\cdots\!84)^{2}$$
$29$ $$(T^{6} + \cdots - 31\!\cdots\!00)^{2}$$
$31$ $$(T^{6} + \cdots + 22\!\cdots\!36)^{2}$$
$37$ $$(T^{6} + \cdots + 17\!\cdots\!84)^{2}$$
$41$ $$(T^{6} + \cdots + 44\!\cdots\!44)^{2}$$
$43$ $$(T^{3} - 10040 T^{2} + \cdots + 196255210112)^{4}$$
$47$ $$(T^{6} + \cdots + 10\!\cdots\!00)^{2}$$
$53$ $$(T^{6} + \cdots - 77\!\cdots\!16)^{2}$$
$59$ $$(T^{6} + \cdots + 40\!\cdots\!00)^{2}$$
$61$ $$(T^{3} + \cdots - 1133585028152)^{4}$$
$67$ $$(T^{6} + \cdots + 16\!\cdots\!00)^{2}$$
$71$ $$(T^{6} + \cdots + 64\!\cdots\!00)^{2}$$
$73$ $$(T^{6} + \cdots + 86\!\cdots\!96)^{2}$$
$79$ $$(T^{3} + \cdots - 60751859232960)^{4}$$
$83$ $$(T^{6} + \cdots + 28\!\cdots\!64)^{2}$$
$89$ $$(T^{6} + \cdots + 95\!\cdots\!84)^{2}$$
$97$ $$(T^{6} + \cdots + 35\!\cdots\!00)^{2}$$