Properties

Label 256.12.a.o
Level 256256
Weight 1212
Character orbit 256.a
Self dual yes
Analytic conductor 196.696196.696
Analytic rank 11
Dimension 1010
Inner twists 22

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [256,12,Mod(1,256)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(256, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 12, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("256.1");
 
S:= CuspForms(chi, 12);
 
N := Newforms(S);
 
Level: N N == 256=28 256 = 2^{8}
Weight: k k == 12 12
Character orbit: [χ][\chi] == 256.a (trivial)

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: yes
Analytic conductor: 196.695854223196.695854223
Analytic rank: 11
Dimension: 1010
Coefficient field: Q[x]/(x10)\mathbb{Q}[x]/(x^{10} - \cdots)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: x105011x8+8617108x65521716928x4+760691642368x220440645894144 x^{10} - 5011x^{8} + 8617108x^{6} - 5521716928x^{4} + 760691642368x^{2} - 20440645894144 Copy content Toggle raw display
Coefficient ring: Z[a1,,a13]\Z[a_1, \ldots, a_{13}]
Coefficient ring index: 285327 2^{85}\cdot 3^{2}\cdot 7
Twist minimal: no (minimal twist has level 8)
Fricke sign: 1-1
Sato-Tate group: SU(2)\mathrm{SU}(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,,β91,\beta_1,\ldots,\beta_{9} for the coefficient ring described below. We also show the integral qq-expansion of the trace form.

f(q)f(q) == qβ1q3+(β4+2β1)q5+(β23362)q7+(β3β2+47239)q9+(β7+11β4+30β1)q11+(β8+β7++191β1)q13++(12519β9++36045633β1)q99+O(q100) q - \beta_1 q^{3} + (\beta_{4} + 2 \beta_1) q^{5} + ( - \beta_{2} - 3362) q^{7} + (\beta_{3} - \beta_{2} + 47239) q^{9} + ( - \beta_{7} + 11 \beta_{4} + 30 \beta_1) q^{11} + (\beta_{8} + \beta_{7} + \cdots + 191 \beta_1) q^{13}+ \cdots + (12519 \beta_{9} + \cdots + 36045633 \beta_1) q^{99}+O(q^{100}) Copy content Toggle raw display
Tr(f)(q)\operatorname{Tr}(f)(q) == 10q33616q7+472394q93391792q15+2639732q1745357808q23+41502654q25442035392q3154732216q33431459312q39+654907964q41+560226528q47+1143661722q49++60995282900q97+O(q100) 10 q - 33616 q^{7} + 472394 q^{9} - 3391792 q^{15} + 2639732 q^{17} - 45357808 q^{23} + 41502654 q^{25} - 442035392 q^{31} - 54732216 q^{33} - 431459312 q^{39} + 654907964 q^{41} + 560226528 q^{47} + 1143661722 q^{49}+ \cdots + 60995282900 q^{97}+O(q^{100}) Copy content Toggle raw display

Basis of coefficient ring in terms of a root ν\nu of x105011x8+8617108x65521716928x4+760691642368x220440645894144 x^{10} - 5011x^{8} + 8617108x^{6} - 5521716928x^{4} + 760691642368x^{2} - 20440645894144 : Copy content Toggle raw display

β1\beta_{1}== (149ν99960207ν7+31671582116ν527251267768256ν3+26 ⁣ ⁣88ν)/16750327136256 ( 149 \nu^{9} - 9960207 \nu^{7} + 31671582116 \nu^{5} - 27251267768256 \nu^{3} + 26\!\cdots\!88 \nu ) / 16750327136256 Copy content Toggle raw display
β2\beta_{2}== (463ν81593437ν6+1624201260ν4450735578944ν2+29780898726656)/412137600 ( 463\nu^{8} - 1593437\nu^{6} + 1624201260\nu^{4} - 450735578944\nu^{2} + 29780898726656 ) / 412137600 Copy content Toggle raw display
β3\beta_{3}== (1283029ν8+4129573071ν63210195094180ν4183379598326848ν2+55 ⁣ ⁣52)/233682019200 ( - 1283029 \nu^{8} + 4129573071 \nu^{6} - 3210195094180 \nu^{4} - 183379598326848 \nu^{2} + 55\!\cdots\!52 ) / 233682019200 Copy content Toggle raw display
β4\beta_{4}== (220573ν9+932839527ν71317203062660ν5+654642405138624ν346 ⁣ ⁣76ν)/26172386150400 ( - 220573 \nu^{9} + 932839527 \nu^{7} - 1317203062660 \nu^{5} + 654642405138624 \nu^{3} - 46\!\cdots\!76 \nu ) / 26172386150400 Copy content Toggle raw display
β5\beta_{5}== (1004951ν83552349749ν6+3286505129420ν4593318829165888ν2+29 ⁣ ⁣12)/77894006400 ( 1004951 \nu^{8} - 3552349749 \nu^{6} + 3286505129420 \nu^{4} - 593318829165888 \nu^{2} + 29\!\cdots\!12 ) / 77894006400 Copy content Toggle raw display
β6\beta_{6}== (642661ν82625370239ν6+3364324782820ν4++87 ⁣ ⁣32)/16691572800 ( 642661 \nu^{8} - 2625370239 \nu^{6} + 3364324782820 \nu^{4} + \cdots + 87\!\cdots\!32 ) / 16691572800 Copy content Toggle raw display
β7\beta_{7}== (209202013ν9892317094887ν7++86 ⁣ ⁣56ν)/418758178406400 ( 209202013 \nu^{9} - 892317094887 \nu^{7} + \cdots + 86\!\cdots\!56 \nu ) / 418758178406400 Copy content Toggle raw display
β8\beta_{8}== (15204743ν9+62583530757ν779676872043660ν5++29 ⁣ ⁣84ν)/23264343244800 ( - 15204743 \nu^{9} + 62583530757 \nu^{7} - 79676872043660 \nu^{5} + \cdots + 29\!\cdots\!84 \nu ) / 23264343244800 Copy content Toggle raw display
β9\beta_{9}== (474068719ν91650307867581ν7++40 ⁣ ⁣28ν)/418758178406400 ( 474068719 \nu^{9} - 1650307867581 \nu^{7} + \cdots + 40\!\cdots\!28 \nu ) / 418758178406400 Copy content Toggle raw display
ν\nu== (β9+β8β7β4+1420β1)/131072 ( \beta_{9} + \beta_{8} - \beta_{7} - \beta_{4} + 1420\beta_1 ) / 131072 Copy content Toggle raw display
ν2\nu^{2}== (3β616β545β3139β2+65680128)/65536 ( 3\beta_{6} - 16\beta_{5} - 45\beta_{3} - 139\beta_{2} + 65680128 ) / 65536 Copy content Toggle raw display
ν3\nu^{3}== (1553β9+1361β83409β796593β4+2290956β1)/131072 ( 1553\beta_{9} + 1361\beta_{8} - 3409\beta_{7} - 96593\beta_{4} + 2290956\beta_1 ) / 131072 Copy content Toggle raw display
ν4\nu^{4}== (3873β628368β550511β353817β2+103231047712)/65536 ( 3873\beta_{6} - 28368\beta_{5} - 50511\beta_{3} - 53817\beta_{2} + 103231047712 ) / 65536 Copy content Toggle raw display
ν5\nu^{5}== (2477263β9+2662543β86097039β7231320719β4+3610559092β1)/131072 ( 2477263\beta_{9} + 2662543\beta_{8} - 6097039\beta_{7} - 231320719\beta_{4} + 3610559092\beta_1 ) / 131072 Copy content Toggle raw display
ν6\nu^{6}== (2452407β647878192β556106201β3+191584289β2+168440485682784)/65536 ( 2452407\beta_{6} - 47878192\beta_{5} - 56106201\beta_{3} + 191584289\beta_{2} + 168440485682784 ) / 65536 Copy content Toggle raw display
ν7\nu^{7}== (3991931081β9+5154901769β810601277193β7486153544457β4+5493305137836β1)/131072 ( 3991931081 \beta_{9} + 5154901769 \beta_{8} - 10601277193 \beta_{7} - 486153544457 \beta_{4} + 5493305137836 \beta_1 ) / 131072 Copy content Toggle raw display
ν8\nu^{8}== (2225850303β680836523856β559707921839β3+771154942119β2+27 ⁣ ⁣08)/65536 ( - 2225850303 \beta_{6} - 80836523856 \beta_{5} - 59707921839 \beta_{3} + 771154942119 \beta_{2} + 27\!\cdots\!08 ) / 65536 Copy content Toggle raw display
ν9\nu^{9}== (6488027798911β9+9730139486527β818332159091007β7++81 ⁣ ⁣12β1)/131072 ( 6488027798911 \beta_{9} + 9730139486527 \beta_{8} - 18332159091007 \beta_{7} + \cdots + 81\!\cdots\!12 \beta_1 ) / 131072 Copy content Toggle raw display

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}
1.1
40.4634
5.96180
−11.7812
−42.0479
−37.8331
37.8331
42.0479
11.7812
−5.96180
−40.4634
0 −712.065 0 7122.63 0 46911.8 0 329890. 0
1.2 0 −614.710 0 −4397.69 0 −41556.6 0 200721. 0
1.3 0 −381.717 0 −8937.55 0 10175.0 0 −31439.0 0
1.4 0 −284.352 0 10467.0 0 −70503.2 0 −96290.8 0
1.5 0 −102.287 0 −2319.82 0 38165.0 0 −166684. 0
1.6 0 102.287 0 2319.82 0 38165.0 0 −166684. 0
1.7 0 284.352 0 −10467.0 0 −70503.2 0 −96290.8 0
1.8 0 381.717 0 8937.55 0 10175.0 0 −31439.0 0
1.9 0 614.710 0 4397.69 0 −41556.6 0 200721. 0
1.10 0 712.065 0 −7122.63 0 46911.8 0 329890. 0
nn: e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.10
Significant digits:
Format:

Atkin-Lehner signs

p p Sign
22 1 -1

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 256.12.a.o 10
4.b odd 2 1 256.12.a.p 10
8.b even 2 1 inner 256.12.a.o 10
8.d odd 2 1 256.12.a.p 10
16.e even 4 2 32.12.b.a 10
16.f odd 4 2 8.12.b.a 10
48.k even 4 2 72.12.d.b 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8.12.b.a 10 16.f odd 4 2
32.12.b.a 10 16.e even 4 2
72.12.d.b 10 48.k even 4 2
256.12.a.o 10 1.a even 1 1 trivial
256.12.a.o 10 8.b even 2 1 inner
256.12.a.p 10 4.b odd 2 1
256.12.a.p 10 8.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on S12new(Γ0(256))S_{12}^{\mathrm{new}}(\Gamma_0(256)):

T3101121932T38+415491272352T36+23 ⁣ ⁣64 T_{3}^{10} - 1121932 T_{3}^{8} + 415491272352 T_{3}^{6} + \cdots - 23\!\cdots\!64 Copy content Toggle raw display
T510264891952T58+46 ⁣ ⁣00 T_{5}^{10} - 264891952 T_{5}^{8} + \cdots - 46\!\cdots\!00 Copy content Toggle raw display
T75+16808T745087977856T73+342941398016T72+53 ⁣ ⁣92 T_{7}^{5} + 16808 T_{7}^{4} - 5087977856 T_{7}^{3} + 342941398016 T_{7}^{2} + \cdots - 53\!\cdots\!92 Copy content Toggle raw display

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 T10 T^{10} Copy content Toggle raw display
33 T10+23 ⁣ ⁣64 T^{10} + \cdots - 23\!\cdots\!64 Copy content Toggle raw display
55 T10+46 ⁣ ⁣00 T^{10} + \cdots - 46\!\cdots\!00 Copy content Toggle raw display
77 (T5+53 ⁣ ⁣92)2 (T^{5} + \cdots - 53\!\cdots\!92)^{2} Copy content Toggle raw display
1111 T10+23 ⁣ ⁣00 T^{10} + \cdots - 23\!\cdots\!00 Copy content Toggle raw display
1313 T10+24 ⁣ ⁣00 T^{10} + \cdots - 24\!\cdots\!00 Copy content Toggle raw display
1717 (T5+18 ⁣ ⁣08)2 (T^{5} + \cdots - 18\!\cdots\!08)^{2} Copy content Toggle raw display
1919 T10+57 ⁣ ⁣84 T^{10} + \cdots - 57\!\cdots\!84 Copy content Toggle raw display
2323 (T5+62 ⁣ ⁣32)2 (T^{5} + \cdots - 62\!\cdots\!32)^{2} Copy content Toggle raw display
2929 T10+15 ⁣ ⁣00 T^{10} + \cdots - 15\!\cdots\!00 Copy content Toggle raw display
3131 (T5++52 ⁣ ⁣84)2 (T^{5} + \cdots + 52\!\cdots\!84)^{2} Copy content Toggle raw display
3737 T10+12 ⁣ ⁣84 T^{10} + \cdots - 12\!\cdots\!84 Copy content Toggle raw display
4141 (T5++47 ⁣ ⁣00)2 (T^{5} + \cdots + 47\!\cdots\!00)^{2} Copy content Toggle raw display
4343 T10+19 ⁣ ⁣56 T^{10} + \cdots - 19\!\cdots\!56 Copy content Toggle raw display
4747 (T5++14 ⁣ ⁣72)2 (T^{5} + \cdots + 14\!\cdots\!72)^{2} Copy content Toggle raw display
5353 T10+45 ⁣ ⁣64 T^{10} + \cdots - 45\!\cdots\!64 Copy content Toggle raw display
5959 T10+15 ⁣ ⁣64 T^{10} + \cdots - 15\!\cdots\!64 Copy content Toggle raw display
6161 T10+22 ⁣ ⁣00 T^{10} + \cdots - 22\!\cdots\!00 Copy content Toggle raw display
6767 T10+13 ⁣ ⁣36 T^{10} + \cdots - 13\!\cdots\!36 Copy content Toggle raw display
7171 (T5++95 ⁣ ⁣24)2 (T^{5} + \cdots + 95\!\cdots\!24)^{2} Copy content Toggle raw display
7373 (T5++58 ⁣ ⁣68)2 (T^{5} + \cdots + 58\!\cdots\!68)^{2} Copy content Toggle raw display
7979 (T5+38 ⁣ ⁣20)2 (T^{5} + \cdots - 38\!\cdots\!20)^{2} Copy content Toggle raw display
8383 T10+61 ⁣ ⁣96 T^{10} + \cdots - 61\!\cdots\!96 Copy content Toggle raw display
8989 (T5+91 ⁣ ⁣60)2 (T^{5} + \cdots - 91\!\cdots\!60)^{2} Copy content Toggle raw display
9797 (T5++31 ⁣ ⁣32)2 (T^{5} + \cdots + 31\!\cdots\!32)^{2} Copy content Toggle raw display
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