Properties

Label 1152.4.c.b
Level 11521152
Weight 44
Character orbit 1152.c
Analytic conductor 67.97067.970
Analytic rank 00
Dimension 1212
Inner twists 22

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1152,4,Mod(1151,1152)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1152, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1152.1151");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: N N == 1152=2732 1152 = 2^{7} \cdot 3^{2}
Weight: k k == 4 4
Character orbit: [χ][\chi] == 1152.c (of order 22, degree 11, 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: 67.970200326667.9702003266
Analytic rank: 00
Dimension: 1212
Coefficient field: Q[x]/(x12)\mathbb{Q}[x]/(x^{12} - \cdots)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: x124x114x10+12x9+88x8+356x7+1278x6+3320x5+8177x4++98 x^{12} - 4 x^{11} - 4 x^{10} + 12 x^{9} + 88 x^{8} + 356 x^{7} + 1278 x^{6} + 3320 x^{5} + 8177 x^{4} + \cdots + 98 Copy content Toggle raw display
Coefficient ring: Z[a1,,a17]\Z[a_1, \ldots, a_{17}]
Coefficient ring index: 23334 2^{33}\cdot 3^{4}
Twist minimal: yes
Sato-Tate group: SU(2)[C2]\mathrm{SU}(2)[C_{2}]

qq-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the qq-expansion are expressed in terms of a basis 1,β1,,β111,\beta_1,\ldots,\beta_{11} for the coefficient ring described below. We also show the integral qq-expansion of the trace form.

f(q)f(q) == qβ9q5β7q7+(β4+β2)q11+(β5β4++β1)q13+(β11+β8++2β6)q17+(β104β9+5β6)q19++(6β5+10β4+21β3++8)q97+O(q100) q - \beta_{9} q^{5} - \beta_{7} q^{7} + (\beta_{4} + \beta_{2}) q^{11} + ( - \beta_{5} - \beta_{4} + \cdots + \beta_1) q^{13} + ( - \beta_{11} + \beta_{8} + \cdots + 2 \beta_{6}) q^{17} + ( - \beta_{10} - 4 \beta_{9} + \cdots - 5 \beta_{6}) q^{19}+ \cdots + (6 \beta_{5} + 10 \beta_{4} + 21 \beta_{3} + \cdots + 8) q^{97}+O(q^{100}) Copy content Toggle raw display
Tr(f)(q)\operatorname{Tr}(f)(q) == 12q240q23300q25+864q35264q37624q47+132q49+1632q59312q614080q71+432q73+3744q83+1704q855856q95+96q97+O(q100) 12 q - 240 q^{23} - 300 q^{25} + 864 q^{35} - 264 q^{37} - 624 q^{47} + 132 q^{49} + 1632 q^{59} - 312 q^{61} - 4080 q^{71} + 432 q^{73} + 3744 q^{83} + 1704 q^{85} - 5856 q^{95} + 96 q^{97}+O(q^{100}) Copy content Toggle raw display

Basis of coefficient ring in terms of a root ν\nu of x124x114x10+12x9+88x8+356x7+1278x6+3320x5+8177x4++98 x^{12} - 4 x^{11} - 4 x^{10} + 12 x^{9} + 88 x^{8} + 356 x^{7} + 1278 x^{6} + 3320 x^{5} + 8177 x^{4} + \cdots + 98 : Copy content Toggle raw display

β1\beta_{1}== (31 ⁣ ⁣24ν11+15 ⁣ ⁣52)/77 ⁣ ⁣53 ( - 31\!\cdots\!24 \nu^{11} + \cdots - 15\!\cdots\!52 ) / 77\!\cdots\!53 Copy content Toggle raw display
β2\beta_{2}== (58 ⁣ ⁣86ν11+39 ⁣ ⁣38)/77 ⁣ ⁣53 ( - 58\!\cdots\!86 \nu^{11} + \cdots - 39\!\cdots\!38 ) / 77\!\cdots\!53 Copy content Toggle raw display
β3\beta_{3}== (205422544184ν11+976420265916ν10+327105386208ν9++10 ⁣ ⁣36)/32247935360533 ( - 205422544184 \nu^{11} + 976420265916 \nu^{10} + 327105386208 \nu^{9} + \cdots + 10\!\cdots\!36 ) / 32247935360533 Copy content Toggle raw display
β4\beta_{4}== (52 ⁣ ⁣20ν11+11 ⁣ ⁣44)/77 ⁣ ⁣53 ( 52\!\cdots\!20 \nu^{11} + \cdots - 11\!\cdots\!44 ) / 77\!\cdots\!53 Copy content Toggle raw display
β5\beta_{5}== (65 ⁣ ⁣76ν11++28 ⁣ ⁣12)/77 ⁣ ⁣53 ( - 65\!\cdots\!76 \nu^{11} + \cdots + 28\!\cdots\!12 ) / 77\!\cdots\!53 Copy content Toggle raw display
β6\beta_{6}== (54512090104ν11211105124627ν10248574560260ν9+637387911231ν8+47487255309242)/5192575793062 ( 54512090104 \nu^{11} - 211105124627 \nu^{10} - 248574560260 \nu^{9} + 637387911231 \nu^{8} + \cdots - 47487255309242 ) / 5192575793062 Copy content Toggle raw display
β7\beta_{7}== (57 ⁣ ⁣54ν11++52 ⁣ ⁣34)/54 ⁣ ⁣71 ( - 57\!\cdots\!54 \nu^{11} + \cdots + 52\!\cdots\!34 ) / 54\!\cdots\!71 Copy content Toggle raw display
β8\beta_{8}== (682713622216ν112695113490060ν102824084391672ν9+612625091208704)/35302542901367 ( 682713622216 \nu^{11} - 2695113490060 \nu^{10} - 2824084391672 \nu^{9} + \cdots - 612625091208704 ) / 35302542901367 Copy content Toggle raw display
β9\beta_{9}== (61 ⁣ ⁣12ν11+53 ⁣ ⁣98)/15 ⁣ ⁣06 ( 61\!\cdots\!12 \nu^{11} + \cdots - 53\!\cdots\!98 ) / 15\!\cdots\!06 Copy content Toggle raw display
β10\beta_{10}== (14 ⁣ ⁣52ν11+13 ⁣ ⁣86)/10 ⁣ ⁣42 ( 14\!\cdots\!52 \nu^{11} + \cdots - 13\!\cdots\!86 ) / 10\!\cdots\!42 Copy content Toggle raw display
β11\beta_{11}== (24 ⁣ ⁣84ν11++22 ⁣ ⁣78)/10 ⁣ ⁣42 ( - 24\!\cdots\!84 \nu^{11} + \cdots + 22\!\cdots\!78 ) / 10\!\cdots\!42 Copy content Toggle raw display
ν\nu== (2β114β104β9+8β88β7+2β612β5++64)/192 ( - 2 \beta_{11} - 4 \beta_{10} - 4 \beta_{9} + 8 \beta_{8} - 8 \beta_{7} + 2 \beta_{6} - 12 \beta_{5} + \cdots + 64 ) / 192 Copy content Toggle raw display
ν2\nu^{2}== (2β108β9+11β8+2β7+38β66β52β4++64)/32 ( - 2 \beta_{10} - 8 \beta_{9} + 11 \beta_{8} + 2 \beta_{7} + 38 \beta_{6} - 6 \beta_{5} - 2 \beta_{4} + \cdots + 64 ) / 32 Copy content Toggle raw display
ν3\nu^{3}== (22β1192β10188β9+526β8+224β7+678β6++1216)/192 ( - 22 \beta_{11} - 92 \beta_{10} - 188 \beta_{9} + 526 \beta_{8} + 224 \beta_{7} + 678 \beta_{6} + \cdots + 1216 ) / 192 Copy content Toggle raw display
ν4\nu^{4}== (β1119β1044β9+144β8+41β7+166β6)/8 ( -\beta_{11} - 19\beta_{10} - 44\beta_{9} + 144\beta_{8} + 41\beta_{7} + 166\beta_{6} ) / 8 Copy content Toggle raw display
ν5\nu^{5}== (370β112216β104964β9+13708β8+4580β7+18718β6+33856)/192 ( - 370 \beta_{11} - 2216 \beta_{10} - 4964 \beta_{9} + 13708 \beta_{8} + 4580 \beta_{7} + 18718 \beta_{6} + \cdots - 33856 ) / 192 Copy content Toggle raw display
ν6\nu^{6}== (192β111386β103320β9+8681β8+2578β7+12894β6+54880)/32 ( - 192 \beta_{11} - 1386 \beta_{10} - 3320 \beta_{9} + 8681 \beta_{8} + 2578 \beta_{7} + 12894 \beta_{6} + \cdots - 54880 ) / 32 Copy content Toggle raw display
ν7\nu^{7}== (3530β1122900β1051364β9+138242β8+41488β7+2149888)/192 ( - 3530 \beta_{11} - 22900 \beta_{10} - 51364 \beta_{9} + 138242 \beta_{8} + 41488 \beta_{7} + \cdots - 2149888 ) / 192 Copy content Toggle raw display
ν8\nu^{8}== (63370β5+38960β440293β3+6628β248064β1974432)/16 ( 63370\beta_{5} + 38960\beta_{4} - 40293\beta_{3} + 6628\beta_{2} - 48064\beta _1 - 974432 ) / 16 Copy content Toggle raw display
ν9\nu^{9}== (80698β11+551936β10+1264580β93442072β81055132β7+52910144)/192 ( 80698 \beta_{11} + 551936 \beta_{10} + 1264580 \beta_{9} - 3442072 \beta_{8} - 1055132 \beta_{7} + \cdots - 52910144 ) / 192 Copy content Toggle raw display
ν10\nu^{10}== (119680β11+836914β10+1932720β95276455β81623818β7+33424736)/32 ( 119680 \beta_{11} + 836914 \beta_{10} + 1932720 \beta_{9} - 5276455 \beta_{8} - 1623818 \beta_{7} + \cdots - 33424736 ) / 32 Copy content Toggle raw display
ν11\nu^{11}== (4664114β11+32492956β10+74847748β9204979694β863330232β7+536969600)/192 ( 4664114 \beta_{11} + 32492956 \beta_{10} + 74847748 \beta_{9} - 204979694 \beta_{8} - 63330232 \beta_{7} + \cdots - 536969600 ) / 192 Copy content Toggle raw display

Character values

We give the values of χ\chi on generators for (Z/1152Z)×\left(\mathbb{Z}/1152\mathbb{Z}\right)^\times.

nn 127127 641641 901901
χ(n)\chi(n) 1-1 1-1 11

Embeddings

For each embedding ιm\iota_m of the coefficient field, the values ιm(an)\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   ιm(ν)\iota_m(\nu) a2 a_{2} a3 a_{3} a4 a_{4} a5 a_{5} a6 a_{6} a7 a_{7} a8 a_{8} a9 a_{9} a10 a_{10}
1151.1
−0.843160 + 2.03557i
0.0928179 + 0.0384464i
−0.552426 1.33367i
4.57521 1.89511i
−2.25381 + 0.933560i
0.981372 2.36924i
0.981372 + 2.36924i
−2.25381 0.933560i
4.57521 + 1.89511i
−0.552426 + 1.33367i
0.0928179 0.0384464i
−0.843160 2.03557i
0 0 0 19.4649i 0 22.2014i 0 0 0
1151.2 0 0 0 15.9893i 0 4.39346i 0 0 0
1151.3 0 0 0 12.9056i 0 14.2604i 0 0 0
1151.4 0 0 0 9.77328i 0 16.6123i 0 0 0
1151.5 0 0 0 1.63027i 0 15.3904i 0 0 0
1151.6 0 0 0 0.854915i 0 27.6333i 0 0 0
1151.7 0 0 0 0.854915i 0 27.6333i 0 0 0
1151.8 0 0 0 1.63027i 0 15.3904i 0 0 0
1151.9 0 0 0 9.77328i 0 16.6123i 0 0 0
1151.10 0 0 0 12.9056i 0 14.2604i 0 0 0
1151.11 0 0 0 15.9893i 0 4.39346i 0 0 0
1151.12 0 0 0 19.4649i 0 22.2014i 0 0 0
nn: e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1151.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
12.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1152.4.c.b yes 12
3.b odd 2 1 1152.4.c.c yes 12
4.b odd 2 1 1152.4.c.c yes 12
8.b even 2 1 1152.4.c.a 12
8.d odd 2 1 1152.4.c.d yes 12
12.b even 2 1 inner 1152.4.c.b yes 12
16.e even 4 1 2304.4.f.j 12
16.e even 4 1 2304.4.f.l 12
16.f odd 4 1 2304.4.f.i 12
16.f odd 4 1 2304.4.f.k 12
24.f even 2 1 1152.4.c.a 12
24.h odd 2 1 1152.4.c.d yes 12
48.i odd 4 1 2304.4.f.i 12
48.i odd 4 1 2304.4.f.k 12
48.k even 4 1 2304.4.f.j 12
48.k even 4 1 2304.4.f.l 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1152.4.c.a 12 8.b even 2 1
1152.4.c.a 12 24.f even 2 1
1152.4.c.b yes 12 1.a even 1 1 trivial
1152.4.c.b yes 12 12.b even 2 1 inner
1152.4.c.c yes 12 3.b odd 2 1
1152.4.c.c yes 12 4.b odd 2 1
1152.4.c.d yes 12 8.d odd 2 1
1152.4.c.d yes 12 24.h odd 2 1
2304.4.f.i 12 16.f odd 4 1
2304.4.f.i 12 48.i odd 4 1
2304.4.f.j 12 16.e even 4 1
2304.4.f.j 12 48.k even 4 1
2304.4.f.k 12 16.f odd 4 1
2304.4.f.k 12 48.i odd 4 1
2304.4.f.l 12 16.e even 4 1
2304.4.f.l 12 48.k even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on S4new(1152,[χ])S_{4}^{\mathrm{new}}(1152, [\chi]):

T1164440T114+18432T113+2362560T112+17547264T119179648 T_{11}^{6} - 4440T_{11}^{4} + 18432T_{11}^{3} + 2362560T_{11}^{2} + 17547264T_{11} - 9179648 Copy content Toggle raw display
T1365976T13424576T133+4408512T132+14352384T13433607168 T_{13}^{6} - 5976T_{13}^{4} - 24576T_{13}^{3} + 4408512T_{13}^{2} + 14352384T_{13} - 433607168 Copy content Toggle raw display

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 T12 T^{12} Copy content Toggle raw display
33 T12 T^{12} Copy content Toggle raw display
55 T12++2993402944 T^{12} + \cdots + 2993402944 Copy content Toggle raw display
77 T12++96575117725696 T^{12} + \cdots + 96575117725696 Copy content Toggle raw display
1111 (T64440T4+9179648)2 (T^{6} - 4440 T^{4} + \cdots - 9179648)^{2} Copy content Toggle raw display
1313 (T65976T4+433607168)2 (T^{6} - 5976 T^{4} + \cdots - 433607168)^{2} Copy content Toggle raw display
1717 T12++16 ⁣ ⁣96 T^{12} + \cdots + 16\!\cdots\!96 Copy content Toggle raw display
1919 T12++18 ⁣ ⁣96 T^{12} + \cdots + 18\!\cdots\!96 Copy content Toggle raw display
2323 (T6+120T5+705453121024)2 (T^{6} + 120 T^{5} + \cdots - 705453121024)^{2} Copy content Toggle raw display
2929 T12++39 ⁣ ⁣04 T^{12} + \cdots + 39\!\cdots\!04 Copy content Toggle raw display
3131 T12++12 ⁣ ⁣76 T^{12} + \cdots + 12\!\cdots\!76 Copy content Toggle raw display
3737 (T6+17955598011328)2 (T^{6} + \cdots - 17955598011328)^{2} Copy content Toggle raw display
4141 T12++21 ⁣ ⁣36 T^{12} + \cdots + 21\!\cdots\!36 Copy content Toggle raw display
4343 T12++34 ⁣ ⁣84 T^{12} + \cdots + 34\!\cdots\!84 Copy content Toggle raw display
4747 (T6++3631402471936)2 (T^{6} + \cdots + 3631402471936)^{2} Copy content Toggle raw display
5353 T12++36 ⁣ ⁣04 T^{12} + \cdots + 36\!\cdots\!04 Copy content Toggle raw display
5959 (T6+55328206716928)2 (T^{6} + \cdots - 55328206716928)^{2} Copy content Toggle raw display
6161 (T6++14 ⁣ ⁣04)2 (T^{6} + \cdots + 14\!\cdots\!04)^{2} Copy content Toggle raw display
6767 T12++13 ⁣ ⁣04 T^{12} + \cdots + 13\!\cdots\!04 Copy content Toggle raw display
7171 (T6++14 ⁣ ⁣36)2 (T^{6} + \cdots + 14\!\cdots\!36)^{2} Copy content Toggle raw display
7373 (T6+433942746165248)2 (T^{6} + \cdots - 433942746165248)^{2} Copy content Toggle raw display
7979 T12++83 ⁣ ⁣64 T^{12} + \cdots + 83\!\cdots\!64 Copy content Toggle raw display
8383 (T6+53 ⁣ ⁣16)2 (T^{6} + \cdots - 53\!\cdots\!16)^{2} Copy content Toggle raw display
8989 T12++68 ⁣ ⁣16 T^{12} + \cdots + 68\!\cdots\!16 Copy content Toggle raw display
9797 (T6+17 ⁣ ⁣44)2 (T^{6} + \cdots - 17\!\cdots\!44)^{2} Copy content Toggle raw display
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