# Properties

 Label 7.6.c.a Level $7$ Weight $6$ Character orbit 7.c Analytic conductor $1.123$ 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] = [7,6,Mod(2,7)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([2]))

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

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

chi := DirichletCharacter("7.2");

S:= CuspForms(chi, 6);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$7$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 7.c (of order $$3$$, degree $$2$$, 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: $$1.12268673869$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{37})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{3} + 10x^{2} + 9x + 81$$ x^4 - x^3 + 10*x^2 + 9*x + 81 Coefficient ring: $$\Z[a_1, \ldots, a_{4}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( - \beta_{2} - \beta_1) q^{2} + ( - \beta_{3} - \beta_{2} - 4 \beta_1 + 4) q^{3} + (2 \beta_{3} + 2 \beta_{2} + 6 \beta_1 - 6) q^{4} + (10 \beta_{2} + 19 \beta_1) q^{5} + (5 \beta_{3} - 41) q^{6} + ( - 21 \beta_{3} - 14 \beta_{2} + \cdots - 14) q^{7}+ \cdots + ( - 8 \beta_{2} + 190 \beta_1) q^{9}+O(q^{10})$$ q + (-b2 - b1) * q^2 + (-b3 - b2 - 4*b1 + 4) * q^3 + (2*b3 + 2*b2 + 6*b1 - 6) * q^4 + (10*b2 + 19*b1) * q^5 + (5*b3 - 41) * q^6 + (-21*b3 - 14*b2 - 56*b1 - 14) * q^7 + (24*b3 + 48) * q^8 + (-8*b2 + 190*b1) * q^9 $$q + ( - \beta_{2} - \beta_1) q^{2} + ( - \beta_{3} - \beta_{2} - 4 \beta_1 + 4) q^{3} + (2 \beta_{3} + 2 \beta_{2} + 6 \beta_1 - 6) q^{4} + (10 \beta_{2} + 19 \beta_1) q^{5} + (5 \beta_{3} - 41) q^{6} + ( - 21 \beta_{3} - 14 \beta_{2} + \cdots - 14) q^{7}+ \cdots + (2674 \beta_{3} - 33472) q^{99}+O(q^{100})$$ q + (-b2 - b1) * q^2 + (-b3 - b2 - 4*b1 + 4) * q^3 + (2*b3 + 2*b2 + 6*b1 - 6) * q^4 + (10*b2 + 19*b1) * q^5 + (5*b3 - 41) * q^6 + (-21*b3 - 14*b2 - 56*b1 - 14) * q^7 + (24*b3 + 48) * q^8 + (-8*b2 + 190*b1) * q^9 + (-29*b3 - 29*b2 - 389*b1 + 389) * q^10 + (23*b3 + 23*b2 + 212*b1 - 212) * q^11 + (14*b2 + 98*b1) * q^12 + (-28*b3 - 462) * q^13 + (70*b3 + 63*b2 - 189*b1 - 574) * q^14 + (-59*b3 + 446) * q^15 + (40*b2 + 1032*b1) * q^16 + (132*b3 + 132*b2 - 1173*b1 + 1173) * q^17 + (-182*b3 - 182*b2 + 106*b1 - 106) * q^18 + (-277*b2 + 180*b1) * q^19 + (98*b3 - 854) * q^20 + (42*b3 - 70*b2 - 721*b1 - 21) * q^21 + (-235*b3 + 1063) * q^22 + (69*b2 + 6*b1) * q^23 + (48*b3 + 48*b2 + 696*b1 - 696) * q^24 + (380*b3 + 380*b2 + 936*b1 - 936) * q^25 + (434*b2 - 574*b1) * q^26 + (-401*b3 + 1436) * q^27 + (-98*b3 + 98*b2 + 1470*b1 - 98) * q^28 + (700*b3 - 3526) * q^29 + (-505*b2 - 2629*b1) * q^30 + (-715*b3 - 715*b2 + 1774*b1 - 1774) * q^31 + (-304*b3 - 304*b2 - 4048*b1 + 4048) * q^32 + (304*b2 + 1699*b1) * q^33 + (1041*b3 + 3711) * q^34 + (-826*b3 - 567*b2 + 1260*b1 + 6244) * q^35 + (332*b3 - 548) * q^36 + (790*b2 - 5545*b1) * q^37 + (97*b3 + 97*b2 + 10069*b1 - 10069) * q^38 + (350*b3 + 350*b2 + 812*b1 - 812) * q^39 + (24*b2 - 7968*b1) * q^40 + (-868*b3 + 1750) * q^41 + (791*b3 + 854*b2 + 4886*b1 - 3311) * q^42 + (-1344*b3 - 6340) * q^43 + (-562*b2 - 2974*b1) * q^44 + (1748*b3 + 1748*b2 + 650*b1 - 650) * q^45 + (-75*b3 - 75*b2 - 2559*b1 + 2559) * q^46 + (-1635*b2 + 11478*b1) * q^47 + (-1192*b3 + 5608) * q^48 + (2156*b3 - 392*b2 - 9800*b1 + 6125) * q^49 + (-1316*b3 + 14996) * q^50 + (-645*b2 + 192*b1) * q^51 + (-756*b3 - 756*b2 - 700*b1 + 700) * q^52 + (-1818*b3 - 1818*b2 + 1521*b1 - 1521) * q^53 + (-1837*b2 - 16273*b1) * q^54 + (2557*b3 - 12538) * q^55 + (-1344*b3 + 672*b2 + 9744*b1 - 19320) * q^56 + (928*b3 - 9529) * q^57 + (4226*b2 + 29426*b1) * q^58 + (531*b3 + 531*b2 - 32904*b1 + 32904) * q^59 + (1246*b3 + 1246*b2 + 7042*b1 - 7042) * q^60 + (4154*b2 + 21243*b1) * q^61 + (-1059*b3 - 24681) * q^62 + (-2212*b3 + 1890*b2 - 15372*b1 + 6496) * q^63 + (3072*b3 + 17728) * q^64 + (-4088*b2 + 1582*b1) * q^65 + (-2003*b3 - 2003*b2 - 12947*b1 + 12947) * q^66 + (919*b3 + 919*b2 + 21156*b1 - 21156) * q^67 + (1554*b2 - 2730*b1) * q^68 + (-282*b3 + 2577) * q^69 + (-693*b3 - 7763*b2 - 17087*b1 - 19719) * q^70 + (2184*b3 - 1104) * q^71 + (-4944*b2 + 16224*b1) * q^72 + (-7372*b3 - 7372*b2 - 25253*b1 + 25253) * q^73 + (4755*b3 + 4755*b2 - 23685*b1 + 23685) * q^74 + (2456*b2 + 17804*b1) * q^75 + (-1302*b3 + 19418) * q^76 + (-126*b3 + 4130*b2 + 14903*b1 + 8883) * q^77 + (-1162*b3 + 13762) * q^78 + (5193*b2 - 4502*b1) * q^79 + (11080*b3 + 11080*b2 + 34408*b1 - 34408) * q^80 + (-4984*b3 - 4984*b2 + 25589*b1 - 25589) * q^81 + (-2618*b2 - 33866*b1) * q^82 + (4536*b3 - 52164) * q^83 + (-2156*b3 - 294*b2 - 3234*b1 + 12740) * q^84 + (-9222*b3 - 26553) * q^85 + (4996*b2 - 43388*b1) * q^86 + (6326*b3 + 6326*b2 + 40004*b1 - 40004) * q^87 + (-3984*b3 - 3984*b2 - 10248*b1 + 10248) * q^88 + (-9356*b2 + 13333*b1) * q^89 + (-2398*b3 + 65326) * q^90 + (10094*b3 + 4900*b2 + 11368*b1 + 28224) * q^91 + (426*b3 - 5142) * q^92 + (-1086*b2 - 19359*b1) * q^93 + (-9843*b3 - 9843*b2 + 49017*b1 - 49017) * q^94 + (-3463*b3 - 3463*b2 - 99070*b1 + 99070) * q^95 + (-5264*b2 - 27440*b1) * q^96 + (196*b3 + 104566) * q^97 + (10192*b3 + 6223*b2 + 97951*b1 - 24304) * q^98 + (2674*b3 - 33472) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 2 q^{2} + 8 q^{3} - 12 q^{4} + 38 q^{5} - 164 q^{6} - 168 q^{7} + 192 q^{8} + 380 q^{9}+O(q^{10})$$ 4 * q - 2 * q^2 + 8 * q^3 - 12 * q^4 + 38 * q^5 - 164 * q^6 - 168 * q^7 + 192 * q^8 + 380 * q^9 $$4 q - 2 q^{2} + 8 q^{3} - 12 q^{4} + 38 q^{5} - 164 q^{6} - 168 q^{7} + 192 q^{8} + 380 q^{9} + 778 q^{10} - 424 q^{11} + 196 q^{12} - 1848 q^{13} - 2674 q^{14} + 1784 q^{15} + 2064 q^{16} + 2346 q^{17} - 212 q^{18} + 360 q^{19} - 3416 q^{20} - 1526 q^{21} + 4252 q^{22} + 12 q^{23} - 1392 q^{24} - 1872 q^{25} - 1148 q^{26} + 5744 q^{27} + 2548 q^{28} - 14104 q^{29} - 5258 q^{30} - 3548 q^{31} + 8096 q^{32} + 3398 q^{33} + 14844 q^{34} + 27496 q^{35} - 2192 q^{36} - 11090 q^{37} - 20138 q^{38} - 1624 q^{39} - 15936 q^{40} + 7000 q^{41} - 3472 q^{42} - 25360 q^{43} - 5948 q^{44} - 1300 q^{45} + 5118 q^{46} + 22956 q^{47} + 22432 q^{48} + 4900 q^{49} + 59984 q^{50} + 384 q^{51} + 1400 q^{52} - 3042 q^{53} - 32546 q^{54} - 50152 q^{55} - 57792 q^{56} - 38116 q^{57} + 58852 q^{58} + 65808 q^{59} - 14084 q^{60} + 42486 q^{61} - 98724 q^{62} - 4760 q^{63} + 70912 q^{64} + 3164 q^{65} + 25894 q^{66} - 42312 q^{67} - 5460 q^{68} + 10308 q^{69} - 113050 q^{70} - 4416 q^{71} + 32448 q^{72} + 50506 q^{73} + 47370 q^{74} + 35608 q^{75} + 77672 q^{76} + 65338 q^{77} + 55048 q^{78} - 9004 q^{79} - 68816 q^{80} - 51178 q^{81} - 67732 q^{82} - 208656 q^{83} + 44492 q^{84} - 106212 q^{85} - 86776 q^{86} - 80008 q^{87} + 20496 q^{88} + 26666 q^{89} + 261304 q^{90} + 135632 q^{91} - 20568 q^{92} - 38718 q^{93} - 98034 q^{94} + 198140 q^{95} - 54880 q^{96} + 418264 q^{97} + 98686 q^{98} - 133888 q^{99}+O(q^{100})$$ 4 * q - 2 * q^2 + 8 * q^3 - 12 * q^4 + 38 * q^5 - 164 * q^6 - 168 * q^7 + 192 * q^8 + 380 * q^9 + 778 * q^10 - 424 * q^11 + 196 * q^12 - 1848 * q^13 - 2674 * q^14 + 1784 * q^15 + 2064 * q^16 + 2346 * q^17 - 212 * q^18 + 360 * q^19 - 3416 * q^20 - 1526 * q^21 + 4252 * q^22 + 12 * q^23 - 1392 * q^24 - 1872 * q^25 - 1148 * q^26 + 5744 * q^27 + 2548 * q^28 - 14104 * q^29 - 5258 * q^30 - 3548 * q^31 + 8096 * q^32 + 3398 * q^33 + 14844 * q^34 + 27496 * q^35 - 2192 * q^36 - 11090 * q^37 - 20138 * q^38 - 1624 * q^39 - 15936 * q^40 + 7000 * q^41 - 3472 * q^42 - 25360 * q^43 - 5948 * q^44 - 1300 * q^45 + 5118 * q^46 + 22956 * q^47 + 22432 * q^48 + 4900 * q^49 + 59984 * q^50 + 384 * q^51 + 1400 * q^52 - 3042 * q^53 - 32546 * q^54 - 50152 * q^55 - 57792 * q^56 - 38116 * q^57 + 58852 * q^58 + 65808 * q^59 - 14084 * q^60 + 42486 * q^61 - 98724 * q^62 - 4760 * q^63 + 70912 * q^64 + 3164 * q^65 + 25894 * q^66 - 42312 * q^67 - 5460 * q^68 + 10308 * q^69 - 113050 * q^70 - 4416 * q^71 + 32448 * q^72 + 50506 * q^73 + 47370 * q^74 + 35608 * q^75 + 77672 * q^76 + 65338 * q^77 + 55048 * q^78 - 9004 * q^79 - 68816 * q^80 - 51178 * q^81 - 67732 * q^82 - 208656 * q^83 + 44492 * q^84 - 106212 * q^85 - 86776 * q^86 - 80008 * q^87 + 20496 * q^88 + 26666 * q^89 + 261304 * q^90 + 135632 * q^91 - 20568 * q^92 - 38718 * q^93 - 98034 * q^94 + 198140 * q^95 - 54880 * q^96 + 418264 * q^97 + 98686 * q^98 - 133888 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} + 10x^{2} + 9x + 81$$ :

 $$\beta_{1}$$ $$=$$ $$( -\nu^{3} + 10\nu^{2} - 10\nu + 81 ) / 90$$ (-v^3 + 10*v^2 - 10*v + 81) / 90 $$\beta_{2}$$ $$=$$ $$( \nu^{3} - 10\nu^{2} + 190\nu - 81 ) / 90$$ (v^3 - 10*v^2 + 190*v - 81) / 90 $$\beta_{3}$$ $$=$$ $$( \nu^{3} + 14 ) / 5$$ (v^3 + 14) / 5
 $$\nu$$ $$=$$ $$( \beta_{2} + \beta_1 ) / 2$$ (b2 + b1) / 2 $$\nu^{2}$$ $$=$$ $$( \beta_{3} + \beta_{2} + 19\beta _1 - 19 ) / 2$$ (b3 + b2 + 19*b1 - 19) / 2 $$\nu^{3}$$ $$=$$ $$5\beta_{3} - 14$$ 5*b3 - 14

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/7\mathbb{Z}\right)^\times$$.

 $$n$$ $$3$$ $$\chi(n)$$ $$-\beta_{1}$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
2.1
 1.77069 − 3.06693i −1.27069 + 2.20090i 1.77069 + 3.06693i −1.27069 − 2.20090i
−3.54138 + 6.13385i 5.04138 + 8.73193i −9.08276 15.7318i 39.9138 69.1328i −71.4138 43.1587 + 122.247i −97.9863 70.6689 122.402i 282.700 + 489.651i
2.2 2.54138 4.40180i −1.04138 1.80373i 3.08276 + 5.33950i −20.9138 + 36.2238i −10.5862 −127.159 25.2522i 193.986 119.331 206.687i 106.300 + 184.117i
4.1 −3.54138 6.13385i 5.04138 8.73193i −9.08276 + 15.7318i 39.9138 + 69.1328i −71.4138 43.1587 122.247i −97.9863 70.6689 + 122.402i 282.700 489.651i
4.2 2.54138 + 4.40180i −1.04138 + 1.80373i 3.08276 5.33950i −20.9138 36.2238i −10.5862 −127.159 + 25.2522i 193.986 119.331 + 206.687i 106.300 184.117i
 $$n$$: 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
7.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 7.6.c.a 4
3.b odd 2 1 63.6.e.d 4
4.b odd 2 1 112.6.i.c 4
7.b odd 2 1 49.6.c.f 4
7.c even 3 1 inner 7.6.c.a 4
7.c even 3 1 49.6.a.d 2
7.d odd 6 1 49.6.a.e 2
7.d odd 6 1 49.6.c.f 4
21.g even 6 1 441.6.a.m 2
21.h odd 6 1 63.6.e.d 4
21.h odd 6 1 441.6.a.n 2
28.f even 6 1 784.6.a.t 2
28.g odd 6 1 112.6.i.c 4
28.g odd 6 1 784.6.a.ba 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7.6.c.a 4 1.a even 1 1 trivial
7.6.c.a 4 7.c even 3 1 inner
49.6.a.d 2 7.c even 3 1
49.6.a.e 2 7.d odd 6 1
49.6.c.f 4 7.b odd 2 1
49.6.c.f 4 7.d odd 6 1
63.6.e.d 4 3.b odd 2 1
63.6.e.d 4 21.h odd 6 1
112.6.i.c 4 4.b odd 2 1
112.6.i.c 4 28.g odd 6 1
441.6.a.m 2 21.g even 6 1
441.6.a.n 2 21.h odd 6 1
784.6.a.t 2 28.f even 6 1
784.6.a.ba 2 28.g odd 6 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{6}^{\mathrm{new}}(7, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} + 2 T^{3} + \cdots + 1296$$
$3$ $$T^{4} - 8 T^{3} + \cdots + 441$$
$5$ $$T^{4} - 38 T^{3} + \cdots + 11148921$$
$7$ $$T^{4} + 168 T^{3} + \cdots + 282475249$$
$11$ $$T^{4} + 424 T^{3} + \cdots + 643687641$$
$13$ $$(T^{2} + 924 T + 184436)^{2}$$
$17$ $$T^{4} + \cdots + 534713400081$$
$19$ $$T^{4} + \cdots + 7876852004329$$
$23$ $$T^{4} + \cdots + 31018606641$$
$29$ $$(T^{2} + 7052 T - 5697324)^{2}$$
$31$ $$T^{4} + \cdots + 248637676526001$$
$37$ $$T^{4} + \cdots + 58604000855625$$
$41$ $$(T^{2} - 3500 T - 24814188)^{2}$$
$43$ $$(T^{2} + 12680 T - 26638832)^{2}$$
$47$ $$T^{4} + \cdots + 10\!\cdots\!81$$
$53$ $$T^{4} + \cdots + 14\!\cdots\!09$$
$59$ $$T^{4} + \cdots + 11\!\cdots\!81$$
$61$ $$T^{4} + \cdots + 35\!\cdots\!49$$
$67$ $$T^{4} + \cdots + 17\!\cdots\!41$$
$71$ $$(T^{2} + 2208 T - 175265856)^{2}$$
$73$ $$T^{4} + \cdots + 18\!\cdots\!01$$
$79$ $$T^{4} + \cdots + 95\!\cdots\!81$$
$83$ $$(T^{2} + 104328 T + 1959796944)^{2}$$
$89$ $$T^{4} + \cdots + 93\!\cdots\!49$$
$97$ $$(T^{2} - 209132 T + 10932626964)^{2}$$