# Properties

 Label 7.22.c.a Level $7$ Weight $22$ Character orbit 7.c Analytic conductor $19.563$ Analytic rank $0$ Dimension $26$ CM no Inner twists $2$

# Learn more

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7,22,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, 22, names="a")

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

chi := DirichletCharacter("7.2");

S:= CuspForms(chi, 22);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$7$$ Weight: $$k$$ $$=$$ $$22$$ 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: $$19.5634141001$$ Analytic rank: $$0$$ Dimension: $$26$$ Relative dimension: $$13$$ over $$\Q(\zeta_{3})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

The dimension is sufficiently large that we do not compute an algebraic $$q$$-expansion, but we have computed the trace expansion.

 $$\operatorname{Tr}(f)(q) =$$ $$26 q + 286 q^{2} + 118097 q^{3} - 11780748 q^{4} + 19296893 q^{5} - 457302020 q^{6} + 1244099388 q^{7} - 3356984640 q^{8} - 36260337262 q^{9}+O(q^{10})$$ 26 * q + 286 * q^2 + 118097 * q^3 - 11780748 * q^4 + 19296893 * q^5 - 457302020 * q^6 + 1244099388 * q^7 - 3356984640 * q^8 - 36260337262 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$26 q + 286 q^{2} + 118097 q^{3} - 11780748 q^{4} + 19296893 q^{5} - 457302020 q^{6} + 1244099388 q^{7} - 3356984640 q^{8} - 36260337262 q^{9} + 45908292458 q^{10} - 96908527507 q^{11} + 703726516612 q^{12} + 573150555348 q^{13} - 519063760642 q^{14} - 1250393933326 q^{15} - 13121838202992 q^{16} + 3631296873225 q^{17} - 26768119563764 q^{18} + 56849486179647 q^{19} - 211235752093016 q^{20} + 287520772660135 q^{21} - 564206457341956 q^{22} - 56010101087361 q^{23} - 151975129265904 q^{24} - 672811740581052 q^{25} - 13\!\cdots\!76 q^{26}+ \cdots - 23\!\cdots\!52 q^{99}+O(q^{100})$$ 26 * q + 286 * q^2 + 118097 * q^3 - 11780748 * q^4 + 19296893 * q^5 - 457302020 * q^6 + 1244099388 * q^7 - 3356984640 * q^8 - 36260337262 * q^9 + 45908292458 * q^10 - 96908527507 * q^11 + 703726516612 * q^12 + 573150555348 * q^13 - 519063760642 * q^14 - 1250393933326 * q^15 - 13121838202992 * q^16 + 3631296873225 * q^17 - 26768119563764 * q^18 + 56849486179647 * q^19 - 211235752093016 * q^20 + 287520772660135 * q^21 - 564206457341956 * q^22 - 56010101087361 * q^23 - 151975129265904 * q^24 - 672811740581052 * q^25 - 1358522589413276 * q^26 - 2144448281577238 * q^27 - 4826527281886156 * q^28 + 11056108922426564 * q^29 - 6230983254308858 * q^30 + 2915918111714909 * q^31 + 6031769195105696 * q^32 + 14438281026776999 * q^33 - 79038877574366724 * q^34 + 40323248296500001 * q^35 + 92042127456862576 * q^36 - 56690306381981455 * q^37 + 69816975237291142 * q^38 + 192986544424759946 * q^39 + 327938659319189184 * q^40 - 485968527188727716 * q^41 - 97625587586165296 * q^42 + 198454074593197048 * q^43 - 545860052256697724 * q^44 - 37395922860484930 * q^45 + 1062080303131000686 * q^46 + 545845114015708227 * q^47 - 4323454314028504928 * q^48 - 834179171614979974 * q^49 + 3759707778375489104 * q^50 + 220241766654585435 * q^51 - 2470837410767658632 * q^52 + 1066530642992301045 * q^53 + 355603765001582734 * q^54 - 4587818432221886582 * q^55 + 6023120323352639808 * q^56 - 2436016836137617690 * q^57 - 2647477665953695612 * q^58 + 7474478541444602961 * q^59 - 21033489989317069124 * q^60 - 2848450223489054583 * q^61 + 63246313374361735548 * q^62 - 2628048802502234510 * q^63 + 13199726305469796608 * q^64 + 14969380626438713594 * q^65 - 34922560371104300090 * q^66 + 6260905331410186617 * q^67 - 60830600544042602196 * q^68 + 55070614676968282518 * q^69 - 66171982028236816970 * q^70 - 19829692806806168640 * q^71 + 39591517263878049216 * q^72 - 22389609053337927163 * q^73 - 109307996679538553910 * q^74 + 84570010377850408348 * q^75 + 71802870184863662824 * q^76 - 67599586966274830733 * q^77 - 201547269737063893496 * q^78 - 152765407297011897977 * q^79 + 226029126485786363824 * q^80 + 284650712144159176523 * q^81 + 126519604886693145484 * q^82 - 1235016850633512172968 * q^83 + 262424412632001748940 * q^84 + 672057383707197482898 * q^85 + 385414116959388791048 * q^86 + 367683551076552943442 * q^87 + 575858305808050147728 * q^88 + 31327966399121384405 * q^89 - 1322424440541914944136 * q^90 + 318458511806028001240 * q^91 + 1274558836326262037544 * q^92 - 39704754034417637049 * q^93 + 279079228503504697758 * q^94 + 529353291415579422425 * q^95 + 81057395943474767264 * q^96 + 234002595919330799036 * q^97 - 5157755886664145781506 * q^98 - 237991978233977177452 * q^99

## 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   $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
2.1 −1330.60 + 2304.67i 72122.8 + 124920.i −2.49243e6 4.31702e6i 1.64217e7 2.84432e7i −3.83867e8 6.58019e8 + 3.54341e8i 7.68480e9 −5.17323e9 + 8.96030e9i 4.37015e10 + 7.56932e10i
2.2 −1089.84 + 1887.65i −65571.0 113572.i −1.32691e6 2.29828e6i 3.92338e6 6.79550e6i 2.85847e8 4.31432e8 6.10256e8i 1.21336e9 −3.36895e9 + 5.83519e9i 8.55170e9 + 1.48120e10i
2.3 −1071.21 + 1855.40i −769.406 1332.65i −1.24642e6 2.15887e6i −9.82127e6 + 1.70109e7i 3.29679e6 −6.30322e8 + 4.01546e8i 8.47744e8 5.22899e9 9.05688e9i −2.10414e10 3.64447e10i
2.4 −618.250 + 1070.84i 88565.1 + 153399.i 284110. + 492093.i −1.58690e7 + 2.74859e7i −2.19021e8 4.82606e7 7.45799e8i −3.29573e9 −1.04574e10 + 1.81127e10i −1.96220e10 3.39863e10i
2.5 −351.499 + 608.815i −5803.13 10051.3i 801472. + 1.38819e6i −459458. + 795804.i 8.15919e6 6.00451e8 + 4.44977e8i −2.60116e9 5.16282e9 8.94227e9i −3.22998e8 5.59450e8i
2.6 −346.683 + 600.472i 35430.0 + 61366.5i 808198. + 1.39984e6i 1.75720e7 3.04357e7i −4.91319e7 −5.82353e8 4.68414e8i −2.57485e9 2.71961e9 4.71050e9i 1.21838e10 + 2.11030e10i
2.7 −90.6842 + 157.070i −89485.1 154993.i 1.03213e6 + 1.78770e6i 3.79465e6 6.57253e6i 3.24596e7 −6.07843e8 + 4.34825e8i −7.54748e8 −1.07850e10 + 1.86802e10i 6.88230e8 + 1.19205e9i
2.8 356.792 617.981i −36627.4 63440.5i 793975. + 1.37521e6i −1.72220e7 + 2.98293e7i −5.22734e7 2.12395e8 7.16543e8i 2.62963e9 2.54705e9 4.41162e9i 1.22893e10 + 2.12857e10i
2.9 464.062 803.780i 69924.4 + 121113.i 617868. + 1.07018e6i −547104. + 947613.i 1.29797e8 3.31595e8 + 6.69769e8i 3.09334e9 −4.54866e9 + 7.87851e9i 5.07781e8 + 8.79503e8i
2.10 793.007 1373.53i 27102.5 + 46943.0i −209144. 362248.i 259504. 449475.i 8.59700e7 −7.41186e8 9.58592e7i 2.66270e9 3.76108e9 6.51438e9i −4.11578e8 7.12873e8i
2.11 857.595 1485.40i −46198.3 80017.8i −422364. 731556.i 2.01337e7 3.48725e7i −1.58478e8 6.24130e8 4.11105e8i 2.14815e9 9.61607e8 1.66555e9i −3.45331e10 5.98130e10i
2.12 1211.86 2098.99i −55093.7 95425.0i −1.88861e6 3.27117e6i −7.61674e6 + 1.31926e7i −2.67062e8 −1.78785e8 + 7.25660e8i −4.07200e9 −8.40444e8 + 1.45569e9i 1.84608e10 + 3.19750e10i
2.13 1358.46 2352.92i 65451.7 + 113366.i −2.64224e6 4.57650e6i −920929. + 1.59510e6i 3.55654e8 4.56257e8 5.91925e8i −8.65972e9 −3.33767e9 + 5.78102e9i 2.50209e9 + 4.33374e9i
4.1 −1330.60 2304.67i 72122.8 124920.i −2.49243e6 + 4.31702e6i 1.64217e7 + 2.84432e7i −3.83867e8 6.58019e8 3.54341e8i 7.68480e9 −5.17323e9 8.96030e9i 4.37015e10 7.56932e10i
4.2 −1089.84 1887.65i −65571.0 + 113572.i −1.32691e6 + 2.29828e6i 3.92338e6 + 6.79550e6i 2.85847e8 4.31432e8 + 6.10256e8i 1.21336e9 −3.36895e9 5.83519e9i 8.55170e9 1.48120e10i
4.3 −1071.21 1855.40i −769.406 + 1332.65i −1.24642e6 + 2.15887e6i −9.82127e6 1.70109e7i 3.29679e6 −6.30322e8 4.01546e8i 8.47744e8 5.22899e9 + 9.05688e9i −2.10414e10 + 3.64447e10i
4.4 −618.250 1070.84i 88565.1 153399.i 284110. 492093.i −1.58690e7 2.74859e7i −2.19021e8 4.82606e7 + 7.45799e8i −3.29573e9 −1.04574e10 1.81127e10i −1.96220e10 + 3.39863e10i
4.5 −351.499 608.815i −5803.13 + 10051.3i 801472. 1.38819e6i −459458. 795804.i 8.15919e6 6.00451e8 4.44977e8i −2.60116e9 5.16282e9 + 8.94227e9i −3.22998e8 + 5.59450e8i
4.6 −346.683 600.472i 35430.0 61366.5i 808198. 1.39984e6i 1.75720e7 + 3.04357e7i −4.91319e7 −5.82353e8 + 4.68414e8i −2.57485e9 2.71961e9 + 4.71050e9i 1.21838e10 2.11030e10i
4.7 −90.6842 157.070i −89485.1 + 154993.i 1.03213e6 1.78770e6i 3.79465e6 + 6.57253e6i 3.24596e7 −6.07843e8 4.34825e8i −7.54748e8 −1.07850e10 1.86802e10i 6.88230e8 1.19205e9i
See all 26 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 2.13 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 7.22.c.a 26
7.c even 3 1 inner 7.22.c.a 26
7.c even 3 1 49.22.a.f 13
7.d odd 6 1 49.22.a.g 13

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7.22.c.a 26 1.a even 1 1 trivial
7.22.c.a 26 7.c even 3 1 inner
49.22.a.f 13 7.c even 3 1
49.22.a.g 13 7.d odd 6 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{22}^{\mathrm{new}}(7, [\chi])$$.