# Properties

 Label 7.7.b.b Level $7$ Weight $7$ Character orbit 7.b Analytic conductor $1.610$ 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] = [7,7,Mod(6,7)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 7, names="a")

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

chi := DirichletCharacter("7.6");

S:= CuspForms(chi, 7);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$7$$ Weight: $$k$$ $$=$$ $$7$$ Character orbit: $$[\chi]$$ $$=$$ 7.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: $$1.61037858534$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-510})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} + 510$$ x^2 + 510 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2$$ 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 $$\beta = 2\sqrt{-510}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - 8 q^{2} + \beta q^{3} - \beta q^{5} - 8 \beta q^{6} + (7 \beta + 133) q^{7} + 512 q^{8} - 1311 q^{9} +O(q^{10})$$ q - 8 * q^2 + b * q^3 - b * q^5 - 8*b * q^6 + (7*b + 133) * q^7 + 512 * q^8 - 1311 * q^9 $$q - 8 q^{2} + \beta q^{3} - \beta q^{5} - 8 \beta q^{6} + (7 \beta + 133) q^{7} + 512 q^{8} - 1311 q^{9} + 8 \beta q^{10} + 874 q^{11} + 49 \beta q^{13} + ( - 56 \beta - 1064) q^{14} + 2040 q^{15} - 4096 q^{16} - 132 \beta q^{17} + 10488 q^{18} + 69 \beta q^{19} + (133 \beta - 14280) q^{21} - 6992 q^{22} + 4738 q^{23} + 512 \beta q^{24} + 13585 q^{25} - 392 \beta q^{26} - 582 \beta q^{27} + 11146 q^{29} - 16320 q^{30} - 608 \beta q^{31} + 874 \beta q^{33} + 1056 \beta q^{34} + ( - 133 \beta + 14280) q^{35} + 3002 q^{37} - 552 \beta q^{38} - 99960 q^{39} - 512 \beta q^{40} + 1274 \beta q^{41} + ( - 1064 \beta + 114240) q^{42} + 31418 q^{43} + 1311 \beta q^{45} - 37904 q^{46} - 1604 \beta q^{47} - 4096 \beta q^{48} + (1862 \beta - 82271) q^{49} - 108680 q^{50} + 269280 q^{51} - 76406 q^{53} + 4656 \beta q^{54} - 874 \beta q^{55} + (3584 \beta + 68096) q^{56} - 140760 q^{57} - 89168 q^{58} + 2507 \beta q^{59} - 6091 \beta q^{61} + 4864 \beta q^{62} + ( - 9177 \beta - 174363) q^{63} + 262144 q^{64} + 99960 q^{65} - 6992 \beta q^{66} + 495242 q^{67} + 4738 \beta q^{69} + (1064 \beta - 114240) q^{70} - 184406 q^{71} - 671232 q^{72} - 1350 \beta q^{73} - 24016 q^{74} + 13585 \beta q^{75} + (6118 \beta + 116242) q^{77} + 799680 q^{78} - 534934 q^{79} + 4096 \beta q^{80} + 231561 q^{81} - 10192 \beta q^{82} - 15827 \beta q^{83} - 269280 q^{85} - 251344 q^{86} + 11146 \beta q^{87} + 447488 q^{88} - 13938 \beta q^{89} - 10488 \beta q^{90} + (6517 \beta - 699720) q^{91} + 1240320 q^{93} + 12832 \beta q^{94} + 140760 q^{95} + 18032 \beta q^{97} + ( - 14896 \beta + 658168) q^{98} - 1145814 q^{99} +O(q^{100})$$ q - 8 * q^2 + b * q^3 - b * q^5 - 8*b * q^6 + (7*b + 133) * q^7 + 512 * q^8 - 1311 * q^9 + 8*b * q^10 + 874 * q^11 + 49*b * q^13 + (-56*b - 1064) * q^14 + 2040 * q^15 - 4096 * q^16 - 132*b * q^17 + 10488 * q^18 + 69*b * q^19 + (133*b - 14280) * q^21 - 6992 * q^22 + 4738 * q^23 + 512*b * q^24 + 13585 * q^25 - 392*b * q^26 - 582*b * q^27 + 11146 * q^29 - 16320 * q^30 - 608*b * q^31 + 874*b * q^33 + 1056*b * q^34 + (-133*b + 14280) * q^35 + 3002 * q^37 - 552*b * q^38 - 99960 * q^39 - 512*b * q^40 + 1274*b * q^41 + (-1064*b + 114240) * q^42 + 31418 * q^43 + 1311*b * q^45 - 37904 * q^46 - 1604*b * q^47 - 4096*b * q^48 + (1862*b - 82271) * q^49 - 108680 * q^50 + 269280 * q^51 - 76406 * q^53 + 4656*b * q^54 - 874*b * q^55 + (3584*b + 68096) * q^56 - 140760 * q^57 - 89168 * q^58 + 2507*b * q^59 - 6091*b * q^61 + 4864*b * q^62 + (-9177*b - 174363) * q^63 + 262144 * q^64 + 99960 * q^65 - 6992*b * q^66 + 495242 * q^67 + 4738*b * q^69 + (1064*b - 114240) * q^70 - 184406 * q^71 - 671232 * q^72 - 1350*b * q^73 - 24016 * q^74 + 13585*b * q^75 + (6118*b + 116242) * q^77 + 799680 * q^78 - 534934 * q^79 + 4096*b * q^80 + 231561 * q^81 - 10192*b * q^82 - 15827*b * q^83 - 269280 * q^85 - 251344 * q^86 + 11146*b * q^87 + 447488 * q^88 - 13938*b * q^89 - 10488*b * q^90 + (6517*b - 699720) * q^91 + 1240320 * q^93 + 12832*b * q^94 + 140760 * q^95 + 18032*b * q^97 + (-14896*b + 658168) * q^98 - 1145814 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 16 q^{2} + 266 q^{7} + 1024 q^{8} - 2622 q^{9}+O(q^{10})$$ 2 * q - 16 * q^2 + 266 * q^7 + 1024 * q^8 - 2622 * q^9 $$2 q - 16 q^{2} + 266 q^{7} + 1024 q^{8} - 2622 q^{9} + 1748 q^{11} - 2128 q^{14} + 4080 q^{15} - 8192 q^{16} + 20976 q^{18} - 28560 q^{21} - 13984 q^{22} + 9476 q^{23} + 27170 q^{25} + 22292 q^{29} - 32640 q^{30} + 28560 q^{35} + 6004 q^{37} - 199920 q^{39} + 228480 q^{42} + 62836 q^{43} - 75808 q^{46} - 164542 q^{49} - 217360 q^{50} + 538560 q^{51} - 152812 q^{53} + 136192 q^{56} - 281520 q^{57} - 178336 q^{58} - 348726 q^{63} + 524288 q^{64} + 199920 q^{65} + 990484 q^{67} - 228480 q^{70} - 368812 q^{71} - 1342464 q^{72} - 48032 q^{74} + 232484 q^{77} + 1599360 q^{78} - 1069868 q^{79} + 463122 q^{81} - 538560 q^{85} - 502688 q^{86} + 894976 q^{88} - 1399440 q^{91} + 2480640 q^{93} + 281520 q^{95} + 1316336 q^{98} - 2291628 q^{99}+O(q^{100})$$ 2 * q - 16 * q^2 + 266 * q^7 + 1024 * q^8 - 2622 * q^9 + 1748 * q^11 - 2128 * q^14 + 4080 * q^15 - 8192 * q^16 + 20976 * q^18 - 28560 * q^21 - 13984 * q^22 + 9476 * q^23 + 27170 * q^25 + 22292 * q^29 - 32640 * q^30 + 28560 * q^35 + 6004 * q^37 - 199920 * q^39 + 228480 * q^42 + 62836 * q^43 - 75808 * q^46 - 164542 * q^49 - 217360 * q^50 + 538560 * q^51 - 152812 * q^53 + 136192 * q^56 - 281520 * q^57 - 178336 * q^58 - 348726 * q^63 + 524288 * q^64 + 199920 * q^65 + 990484 * q^67 - 228480 * q^70 - 368812 * q^71 - 1342464 * q^72 - 48032 * q^74 + 232484 * q^77 + 1599360 * q^78 - 1069868 * q^79 + 463122 * q^81 - 538560 * q^85 - 502688 * q^86 + 894976 * q^88 - 1399440 * q^91 + 2480640 * q^93 + 281520 * q^95 + 1316336 * q^98 - 2291628 * q^99

## Character values

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

 $$n$$ $$3$$ $$\chi(n)$$ $$-1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
6.1
 − 22.5832i 22.5832i
−8.00000 45.1664i 0 45.1664i 361.331i 133.000 316.165i 512.000 −1311.00 361.331i
6.2 −8.00000 45.1664i 0 45.1664i 361.331i 133.000 + 316.165i 512.000 −1311.00 361.331i
 $$n$$: 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.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 7.7.b.b 2
3.b odd 2 1 63.7.d.d 2
4.b odd 2 1 112.7.c.b 2
5.b even 2 1 175.7.d.e 2
5.c odd 4 2 175.7.c.c 4
7.b odd 2 1 inner 7.7.b.b 2
7.c even 3 2 49.7.d.d 4
7.d odd 6 2 49.7.d.d 4
8.b even 2 1 448.7.c.d 2
8.d odd 2 1 448.7.c.c 2
21.c even 2 1 63.7.d.d 2
28.d even 2 1 112.7.c.b 2
35.c odd 2 1 175.7.d.e 2
35.f even 4 2 175.7.c.c 4
56.e even 2 1 448.7.c.c 2
56.h odd 2 1 448.7.c.d 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7.7.b.b 2 1.a even 1 1 trivial
7.7.b.b 2 7.b odd 2 1 inner
49.7.d.d 4 7.c even 3 2
49.7.d.d 4 7.d odd 6 2
63.7.d.d 2 3.b odd 2 1
63.7.d.d 2 21.c even 2 1
112.7.c.b 2 4.b odd 2 1
112.7.c.b 2 28.d even 2 1
175.7.c.c 4 5.c odd 4 2
175.7.c.c 4 35.f even 4 2
175.7.d.e 2 5.b even 2 1
175.7.d.e 2 35.c odd 2 1
448.7.c.c 2 8.d odd 2 1
448.7.c.c 2 56.e even 2 1
448.7.c.d 2 8.b even 2 1
448.7.c.d 2 56.h odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2} + 8$$ acting on $$S_{7}^{\mathrm{new}}(7, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T + 8)^{2}$$
$3$ $$T^{2} + 2040$$
$5$ $$T^{2} + 2040$$
$7$ $$T^{2} - 266T + 117649$$
$11$ $$(T - 874)^{2}$$
$13$ $$T^{2} + 4898040$$
$17$ $$T^{2} + 35544960$$
$19$ $$T^{2} + 9712440$$
$23$ $$(T - 4738)^{2}$$
$29$ $$(T - 11146)^{2}$$
$31$ $$T^{2} + 754114560$$
$37$ $$(T - 3002)^{2}$$
$41$ $$T^{2} + 3311075040$$
$43$ $$(T - 31418)^{2}$$
$47$ $$T^{2} + 5248544640$$
$53$ $$(T + 76406)^{2}$$
$59$ $$T^{2} + 12821499960$$
$61$ $$T^{2} + 75684573240$$
$67$ $$(T - 495242)^{2}$$
$71$ $$(T + 184406)^{2}$$
$73$ $$T^{2} + 3717900000$$
$79$ $$(T + 534934)^{2}$$
$83$ $$T^{2} + 511007615160$$
$89$ $$T^{2} + 396306401760$$
$97$ $$T^{2} + 663312168960$$