Properties

Label 1512.2.k.a
Level 15121512
Weight 22
Character orbit 1512.k
Analytic conductor 12.07312.073
Analytic rank 00
Dimension 1616
Inner twists 44

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1512,2,Mod(377,1512)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1512, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1, 1])) N = Newforms(chi, 2, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1512.377"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: N N == 1512=23337 1512 = 2^{3} \cdot 3^{3} \cdot 7
Weight: k k == 2 2
Character orbit: [χ][\chi] == 1512.k (of order 22, degree 11, minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [16,0,0,0,0,0,-2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(25)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: 12.073380785612.0733807856
Analytic rank: 00
Dimension: 1616
Coefficient field: Q[x]/(x16)\mathbb{Q}[x]/(x^{16} - \cdots)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: x1624x14+230x121052x10+2139x81244x6+1134x4104x2+169 x^{16} - 24x^{14} + 230x^{12} - 1052x^{10} + 2139x^{8} - 1244x^{6} + 1134x^{4} - 104x^{2} + 169 Copy content Toggle raw display
Coefficient ring: Z[a1,,a11]\Z[a_1, \ldots, a_{11}]
Coefficient ring index: 220 2^{20}
Twist minimal: yes
Sato-Tate group: SU(2)[C2]\mathrm{SU}(2)[C_{2}]

qq-expansion

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

Coefficients of the qq-expansion are expressed in terms of a basis 1,β1,,β151,\beta_1,\ldots,\beta_{15} for the coefficient ring described below. We also show the integral qq-expansion of the trace form.

f(q)f(q) == qβ9q5β1q7β10q11β5q13+β11q17β3q19+(β15β13)q23+(β81)q25+(β15β10)q29++(3β6+2β5++3β1)q97+O(q100) q - \beta_{9} q^{5} - \beta_1 q^{7} - \beta_{10} q^{11} - \beta_{5} q^{13} + \beta_{11} q^{17} - \beta_{3} q^{19} + ( - \beta_{15} - \beta_{13}) q^{23} + (\beta_{8} - 1) q^{25} + ( - \beta_{15} - \beta_{10}) q^{29}+ \cdots + (3 \beta_{6} + 2 \beta_{5} + \cdots + 3 \beta_1) q^{97}+O(q^{100}) Copy content Toggle raw display
Tr(f)(q)\operatorname{Tr}(f)(q) == 16q2q712q258q37+8q43+2q4928q6744q79+16q85+18q91+O(q100) 16 q - 2 q^{7} - 12 q^{25} - 8 q^{37} + 8 q^{43} + 2 q^{49} - 28 q^{67} - 44 q^{79} + 16 q^{85} + 18 q^{91}+O(q^{100}) Copy content Toggle raw display

Basis of coefficient ring in terms of a root ν\nu of x1624x14+230x121052x10+2139x81244x6+1134x4104x2+169 x^{16} - 24x^{14} + 230x^{12} - 1052x^{10} + 2139x^{8} - 1244x^{6} + 1134x^{4} - 104x^{2} + 169 : Copy content Toggle raw display

β1\beta_{1}== (305389ν14+7316240ν1269689225ν10+319235369ν8695756145ν6++237296969)/192947092 ( - 305389 \nu^{14} + 7316240 \nu^{12} - 69689225 \nu^{10} + 319235369 \nu^{8} - 695756145 \nu^{6} + \cdots + 237296969 ) / 192947092 Copy content Toggle raw display
β2\beta_{2}== (2694ν14+61469ν12552649ν10+2285409ν83789335ν6+10061233)/1663337 ( - 2694 \nu^{14} + 61469 \nu^{12} - 552649 \nu^{10} + 2285409 \nu^{8} - 3789335 \nu^{6} + \cdots - 10061233 ) / 1663337 Copy content Toggle raw display
β3\beta_{3}== (2694ν1461469ν12+552649ν102285409ν8+3789335ν6++81211)/1663337 ( 2694 \nu^{14} - 61469 \nu^{12} + 552649 \nu^{10} - 2285409 \nu^{8} + 3789335 \nu^{6} + \cdots + 81211 ) / 1663337 Copy content Toggle raw display
β4\beta_{4}== (11520ν14+293628ν123040296ν10+15624254ν839217560ν6++6370193)/3710521 ( - 11520 \nu^{14} + 293628 \nu^{12} - 3040296 \nu^{10} + 15624254 \nu^{8} - 39217560 \nu^{6} + \cdots + 6370193 ) / 3710521 Copy content Toggle raw display
β5\beta_{5}== (1139534ν14+28712659ν12295900503ν10+1529685268ν8++691999581)/192947092 ( - 1139534 \nu^{14} + 28712659 \nu^{12} - 295900503 \nu^{10} + 1529685268 \nu^{8} + \cdots + 691999581 ) / 192947092 Copy content Toggle raw display
β6\beta_{6}== (1429545ν14+33944011ν12320576374ν10+1427099575ν8+44292248)/192947092 ( - 1429545 \nu^{14} + 33944011 \nu^{12} - 320576374 \nu^{10} + 1427099575 \nu^{8} + \cdots - 44292248 ) / 192947092 Copy content Toggle raw display
β7\beta_{7}== (458582ν15+16643678ν13243643736ν11+1832626360ν9++2564873142ν)/627078049 ( - 458582 \nu^{15} + 16643678 \nu^{13} - 243643736 \nu^{11} + 1832626360 \nu^{9} + \cdots + 2564873142 \nu ) / 627078049 Copy content Toggle raw display
β8\beta_{8}== (88700ν142103563ν12+19736021ν1086408618ν8+154953525ν6++15626169)/6653348 ( 88700 \nu^{14} - 2103563 \nu^{12} + 19736021 \nu^{10} - 86408618 \nu^{8} + 154953525 \nu^{6} + \cdots + 15626169 ) / 6653348 Copy content Toggle raw display
β9\beta_{9}== (4302859ν15113264303ν13+1228806350ν116828867199ν9+10467201680ν)/2508312196 ( 4302859 \nu^{15} - 113264303 \nu^{13} + 1228806350 \nu^{11} - 6828867199 \nu^{9} + \cdots - 10467201680 \nu ) / 2508312196 Copy content Toggle raw display
β10\beta_{10}== (403473ν15+10071623ν13101747496ν11+504132793ν9++225114682ν)/86493524 ( - 403473 \nu^{15} + 10071623 \nu^{13} - 101747496 \nu^{11} + 504132793 \nu^{9} + \cdots + 225114682 \nu ) / 86493524 Copy content Toggle raw display
β11\beta_{11}== (3300067ν1575367115ν13+666756659ν112588222763ν9++1131065416ν)/627078049 ( 3300067 \nu^{15} - 75367115 \nu^{13} + 666756659 \nu^{11} - 2588222763 \nu^{9} + \cdots + 1131065416 \nu ) / 627078049 Copy content Toggle raw display
β12\beta_{12}== (6600134ν15150734230ν13+1333513318ν115176445526ν9++4770443028ν)/627078049 ( 6600134 \nu^{15} - 150734230 \nu^{13} + 1333513318 \nu^{11} - 5176445526 \nu^{9} + \cdots + 4770443028 \nu ) / 627078049 Copy content Toggle raw display
β13\beta_{13}== (248063ν15+6021151ν1358426783ν11+271352435ν9+8003320ν)/21623381 ( - 248063 \nu^{15} + 6021151 \nu^{13} - 58426783 \nu^{11} + 271352435 \nu^{9} + \cdots - 8003320 \nu ) / 21623381 Copy content Toggle raw display
β14\beta_{14}== (10755599ν15+252302706ν132332684002ν11+9941641297ν9++1098391515ν)/627078049 ( - 10755599 \nu^{15} + 252302706 \nu^{13} - 2332684002 \nu^{11} + 9941641297 \nu^{9} + \cdots + 1098391515 \nu ) / 627078049 Copy content Toggle raw display
β15\beta_{15}== (1723809ν15+41505147ν13399318104ν11+1835444329ν9+140493626ν)/86493524 ( - 1723809 \nu^{15} + 41505147 \nu^{13} - 399318104 \nu^{11} + 1835444329 \nu^{9} + \cdots - 140493626 \nu ) / 86493524 Copy content Toggle raw display
ν\nu== (β122β11)/4 ( \beta_{12} - 2\beta_{11} ) / 4 Copy content Toggle raw display
ν2\nu^{2}== (β3+β2+6)/2 ( \beta_{3} + \beta_{2} + 6 ) / 2 Copy content Toggle raw display
ν3\nu^{3}== (3β15β14+7β1210β113β10+4β9β7)/4 ( 3\beta_{15} - \beta_{14} + 7\beta_{12} - 10\beta_{11} - 3\beta_{10} + 4\beta_{9} - \beta_{7} ) / 4 Copy content Toggle raw display
ν4\nu^{4}== (β8+4β6+β54β4+13β3+5β2+2β1+29)/2 ( \beta_{8} + 4\beta_{6} + \beta_{5} - 4\beta_{4} + 13\beta_{3} + 5\beta_{2} + 2\beta _1 + 29 ) / 2 Copy content Toggle raw display
ν5\nu^{5}== (30β152β145β13+51β1243β1130β10+24β920β7)/4 ( 30\beta_{15} - 2\beta_{14} - 5\beta_{13} + 51\beta_{12} - 43\beta_{11} - 30\beta_{10} + 24\beta_{9} - 20\beta_{7} ) / 4 Copy content Toggle raw display
ν6\nu^{6}== (13β8+78β6+35β5113β4+231β3+35β2+36β1+212)/4 ( 13\beta_{8} + 78\beta_{6} + 35\beta_{5} - 113\beta_{4} + 231\beta_{3} + 35\beta_{2} + 36\beta _1 + 212 ) / 4 Copy content Toggle raw display
ν7\nu^{7}== (245β15+7β1477β13+314β1299β11217β10+217β7)/4 ( 245 \beta_{15} + 7 \beta_{14} - 77 \beta_{13} + 314 \beta_{12} - 99 \beta_{11} - 217 \beta_{10} + \cdots - 217 \beta_{7} ) / 4 Copy content Toggle raw display
ν8\nu^{8}== (20β8+266β6+194β5559β4+864β3+2β2+178β1+29)/2 ( 20\beta_{8} + 266\beta_{6} + 194\beta_{5} - 559\beta_{4} + 864\beta_{3} + 2\beta_{2} + 178\beta _1 + 29 ) / 2 Copy content Toggle raw display
ν9\nu^{9}== (1728β15+88β14771β13+1556β12+629β11+1868β7)/4 ( 1728 \beta_{15} + 88 \beta_{14} - 771 \beta_{13} + 1556 \beta_{12} + 629 \beta_{11} + \cdots - 1868 \beta_{7} ) / 4 Copy content Toggle raw display
ν10\nu^{10}== (319β8+2600β6+3285β58921β4+11245β31473β2+8758)/4 ( - 319 \beta_{8} + 2600 \beta_{6} + 3285 \beta_{5} - 8921 \beta_{4} + 11245 \beta_{3} - 1473 \beta_{2} + \cdots - 8758 ) / 4 Copy content Toggle raw display
ν11\nu^{11}== (10560β15+286β145995β13+4951β12+12093β11+14166β7)/4 ( 10560 \beta_{15} + 286 \beta_{14} - 5995 \beta_{13} + 4951 \beta_{12} + 12093 \beta_{11} + \cdots - 14166 \beta_{7} ) / 4 Copy content Toggle raw display
ν12\nu^{12}== (3605β8+2573β6+11182β529241β4+31239β39280β2+56567)/2 ( - 3605 \beta_{8} + 2573 \beta_{6} + 11182 \beta_{5} - 29241 \beta_{4} + 31239 \beta_{3} - 9280 \beta_{2} + \cdots - 56567 ) / 2 Copy content Toggle raw display
ν13\nu^{13}== (53248β152884β1436868β1310003β12+123074β11+97052β7)/4 ( 53248 \beta_{15} - 2884 \beta_{14} - 36868 \beta_{13} - 10003 \beta_{12} + 123074 \beta_{11} + \cdots - 97052 \beta_{7} ) / 4 Copy content Toggle raw display
ν14\nu^{14}== (41422β836017β6+58447β5148066β4+133427β3+517223)/2 ( - 41422 \beta_{8} - 36017 \beta_{6} + 58447 \beta_{5} - 148066 \beta_{4} + 133427 \beta_{3} + \cdots - 517223 ) / 2 Copy content Toggle raw display
ν15\nu^{15}== (176793β1558799β14158604β13357861β12+1003586β11+595917β7)/4 ( 176793 \beta_{15} - 58799 \beta_{14} - 158604 \beta_{13} - 357861 \beta_{12} + 1003586 \beta_{11} + \cdots - 595917 \beta_{7} ) / 4 Copy content Toggle raw display

Character values

We give the values of χ\chi on generators for (Z/1512Z)×\left(\mathbb{Z}/1512\mathbb{Z}\right)^\times.

nn 757757 785785 10811081 11351135
χ(n)\chi(n) 11 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.

Copy content comment:embeddings in the coefficient field
 
Copy content 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}
377.1
2.62616 0.500000i
2.62616 + 0.500000i
−0.415570 + 0.500000i
−0.415570 0.500000i
−2.32849 + 0.500000i
−2.32849 0.500000i
−0.713245 + 0.500000i
−0.713245 0.500000i
0.713245 0.500000i
0.713245 + 0.500000i
2.32849 0.500000i
2.32849 + 0.500000i
0.415570 0.500000i
0.415570 + 0.500000i
−2.62616 + 0.500000i
−2.62616 0.500000i
0 0 0 −2.86833 0 2.43500 1.03478i 0 0 0
377.2 0 0 0 −2.86833 0 2.43500 + 1.03478i 0 0 0
377.3 0 0 0 −2.70790 0 −0.946562 2.47063i 0 0 0
377.4 0 0 0 −2.70790 0 −0.946562 + 2.47063i 0 0 0
377.5 0 0 0 −1.10598 0 −2.64465 0.0763047i 0 0 0
377.6 0 0 0 −1.10598 0 −2.64465 + 0.0763047i 0 0 0
377.7 0 0 0 −0.465643 0 0.656211 2.56308i 0 0 0
377.8 0 0 0 −0.465643 0 0.656211 + 2.56308i 0 0 0
377.9 0 0 0 0.465643 0 0.656211 2.56308i 0 0 0
377.10 0 0 0 0.465643 0 0.656211 + 2.56308i 0 0 0
377.11 0 0 0 1.10598 0 −2.64465 0.0763047i 0 0 0
377.12 0 0 0 1.10598 0 −2.64465 + 0.0763047i 0 0 0
377.13 0 0 0 2.70790 0 −0.946562 2.47063i 0 0 0
377.14 0 0 0 2.70790 0 −0.946562 + 2.47063i 0 0 0
377.15 0 0 0 2.86833 0 2.43500 1.03478i 0 0 0
377.16 0 0 0 2.86833 0 2.43500 + 1.03478i 0 0 0
nn: e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 377.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.b odd 2 1 inner
21.c even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1512.2.k.a 16
3.b odd 2 1 inner 1512.2.k.a 16
4.b odd 2 1 3024.2.k.k 16
7.b odd 2 1 inner 1512.2.k.a 16
12.b even 2 1 3024.2.k.k 16
21.c even 2 1 inner 1512.2.k.a 16
28.d even 2 1 3024.2.k.k 16
84.h odd 2 1 3024.2.k.k 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1512.2.k.a 16 1.a even 1 1 trivial
1512.2.k.a 16 3.b odd 2 1 inner
1512.2.k.a 16 7.b odd 2 1 inner
1512.2.k.a 16 21.c even 2 1 inner
3024.2.k.k 16 4.b odd 2 1
3024.2.k.k 16 12.b even 2 1
3024.2.k.k 16 28.d even 2 1
3024.2.k.k 16 84.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T5817T56+83T5491T52+16 T_{5}^{8} - 17T_{5}^{6} + 83T_{5}^{4} - 91T_{5}^{2} + 16 acting on S2new(1512,[χ])S_{2}^{\mathrm{new}}(1512, [\chi]). Copy content Toggle raw display

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 T16 T^{16} Copy content Toggle raw display
33 T16 T^{16} Copy content Toggle raw display
55 (T817T6++16)2 (T^{8} - 17 T^{6} + \cdots + 16)^{2} Copy content Toggle raw display
77 (T8+T7+5T5++2401)2 (T^{8} + T^{7} + 5 T^{5} + \cdots + 2401)^{2} Copy content Toggle raw display
1111 (T8+51T6++784)2 (T^{8} + 51 T^{6} + \cdots + 784)^{2} Copy content Toggle raw display
1313 (T8+55T6++3136)2 (T^{8} + 55 T^{6} + \cdots + 3136)^{2} Copy content Toggle raw display
1717 (T864T6++4096)2 (T^{8} - 64 T^{6} + \cdots + 4096)^{2} Copy content Toggle raw display
1919 (T8+52T6++841)2 (T^{8} + 52 T^{6} + \cdots + 841)^{2} Copy content Toggle raw display
2323 (T8+163T6++559504)2 (T^{8} + 163 T^{6} + \cdots + 559504)^{2} Copy content Toggle raw display
2929 (T8+60T6++4096)2 (T^{8} + 60 T^{6} + \cdots + 4096)^{2} Copy content Toggle raw display
3131 (T8+61T6++3844)2 (T^{8} + 61 T^{6} + \cdots + 3844)^{2} Copy content Toggle raw display
3737 (T4+2T3+161)4 (T^{4} + 2 T^{3} + \cdots - 161)^{4} Copy content Toggle raw display
4141 (T8193T6++1827904)2 (T^{8} - 193 T^{6} + \cdots + 1827904)^{2} Copy content Toggle raw display
4343 (T42T3132T2++64)4 (T^{4} - 2 T^{3} - 132 T^{2} + \cdots + 64)^{4} Copy content Toggle raw display
4747 (T8356T6++30647296)2 (T^{8} - 356 T^{6} + \cdots + 30647296)^{2} Copy content Toggle raw display
5353 (T8+240T6++1048576)2 (T^{8} + 240 T^{6} + \cdots + 1048576)^{2} Copy content Toggle raw display
5959 (T8324T6++4064256)2 (T^{8} - 324 T^{6} + \cdots + 4064256)^{2} Copy content Toggle raw display
6161 (T8+171T6++331776)2 (T^{8} + 171 T^{6} + \cdots + 331776)^{2} Copy content Toggle raw display
6767 (T4+7T3+2744)4 (T^{4} + 7 T^{3} + \cdots - 2744)^{4} Copy content Toggle raw display
7171 (T8+227T6++8479744)2 (T^{8} + 227 T^{6} + \cdots + 8479744)^{2} Copy content Toggle raw display
7373 (T8+263T6++10291264)2 (T^{8} + 263 T^{6} + \cdots + 10291264)^{2} Copy content Toggle raw display
7979 (T4+11T3++184)4 (T^{4} + 11 T^{3} + \cdots + 184)^{4} Copy content Toggle raw display
8383 (T8532T6++51380224)2 (T^{8} - 532 T^{6} + \cdots + 51380224)^{2} Copy content Toggle raw display
8989 (T8361T6++8737936)2 (T^{8} - 361 T^{6} + \cdots + 8737936)^{2} Copy content Toggle raw display
9797 (T8+491T6++177209344)2 (T^{8} + 491 T^{6} + \cdots + 177209344)^{2} Copy content Toggle raw display
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