Properties

Label 630.2.bo.a
Level 630630
Weight 22
Character orbit 630.bo
Analytic conductor 5.0315.031
Analytic rank 00
Dimension 1616
Inner twists 44

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [630,2,Mod(89,630)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(630, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 3, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("630.89");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: N N == 630=23257 630 = 2 \cdot 3^{2} \cdot 5 \cdot 7
Weight: k k == 2 2
Character orbit: [χ][\chi] == 630.bo (of order 66, degree 22, 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: 5.030575327345.03057532734
Analytic rank: 00
Dimension: 1616
Relative dimension: 88 over Q(ζ6)\Q(\zeta_{6})
Coefficient field: Q[x]/(x16)\mathbb{Q}[x]/(x^{16} - \cdots)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: x166x15+21x1454x13+113x12168x11+186x1084x9++390625 x^{16} - 6 x^{15} + 21 x^{14} - 54 x^{13} + 113 x^{12} - 168 x^{11} + 186 x^{10} - 84 x^{9} + \cdots + 390625 Copy content Toggle raw display
Coefficient ring: Z[a1,,a7]\Z[a_1, \ldots, a_{7}]
Coefficient ring index: 1 1
Twist minimal: yes
Sato-Tate group: SU(2)[C6]\mathrm{SU}(2)[C_{6}]

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,,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) == qβ8q2+(β81)q4β7q5+(β13β10)q7+q8+(β8+β7β41)q10+(β13β5+β3)q11++(β12+β112β9++2)q98+O(q100) q - \beta_{8} q^{2} + (\beta_{8} - 1) q^{4} - \beta_{7} q^{5} + ( - \beta_{13} - \beta_{10}) q^{7} + q^{8} + (\beta_{8} + \beta_{7} - \beta_{4} - 1) q^{10} + (\beta_{13} - \beta_{5} + \beta_{3}) q^{11}+ \cdots + (\beta_{12} + \beta_{11} - 2 \beta_{9} + \cdots + 2) q^{98}+O(q^{100}) Copy content Toggle raw display
Tr(f)(q)\operatorname{Tr}(f)(q) == 16q8q28q46q5+16q8+6q108q16+24q198q236q25+12q318q32+4q3524q386q408q46+60q4728q49+12q50++20q98+O(q100) 16 q - 8 q^{2} - 8 q^{4} - 6 q^{5} + 16 q^{8} + 6 q^{10} - 8 q^{16} + 24 q^{19} - 8 q^{23} - 6 q^{25} + 12 q^{31} - 8 q^{32} + 4 q^{35} - 24 q^{38} - 6 q^{40} - 8 q^{46} + 60 q^{47} - 28 q^{49} + 12 q^{50}+ \cdots + 20 q^{98}+O(q^{100}) Copy content Toggle raw display

Basis of coefficient ring in terms of a root ν\nu of x166x15+21x1454x13+113x12168x11+186x1084x9++390625 x^{16} - 6 x^{15} + 21 x^{14} - 54 x^{13} + 113 x^{12} - 168 x^{11} + 186 x^{10} - 84 x^{9} + \cdots + 390625 : Copy content Toggle raw display

β1\beta_{1}== ν \nu Copy content Toggle raw display
β2\beta_{2}== (384ν15+11789ν1467399ν13+175846ν12241632ν11++608281250)/568125000 ( - 384 \nu^{15} + 11789 \nu^{14} - 67399 \nu^{13} + 175846 \nu^{12} - 241632 \nu^{11} + \cdots + 608281250 ) / 568125000 Copy content Toggle raw display
β3\beta_{3}== (8097ν15+5902ν1424132ν13+294133ν12550416ν11++3815703125)/3408750000 ( - 8097 \nu^{15} + 5902 \nu^{14} - 24132 \nu^{13} + 294133 \nu^{12} - 550416 \nu^{11} + \cdots + 3815703125 ) / 3408750000 Copy content Toggle raw display
β4\beta_{4}== (6137ν15+20678ν1442523ν13+210377ν12519294ν11++2128046875)/1704375000 ( 6137 \nu^{15} + 20678 \nu^{14} - 42523 \nu^{13} + 210377 \nu^{12} - 519294 \nu^{11} + \cdots + 2128046875 ) / 1704375000 Copy content Toggle raw display
β5\beta_{5}== (6613ν15+7932ν1414237ν13+93033ν12+60504ν11++91640625)/1704375000 ( 6613 \nu^{15} + 7932 \nu^{14} - 14237 \nu^{13} + 93033 \nu^{12} + 60504 \nu^{11} + \cdots + 91640625 ) / 1704375000 Copy content Toggle raw display
β6\beta_{6}== (17122ν154902ν14+66757ν1347583ν12+118266ν11++1565390625)/1704375000 ( 17122 \nu^{15} - 4902 \nu^{14} + 66757 \nu^{13} - 47583 \nu^{12} + 118266 \nu^{11} + \cdots + 1565390625 ) / 1704375000 Copy content Toggle raw display
β7\beta_{7}== (ν15+6ν1421ν13+54ν12113ν11+168ν10186ν9++468750)/78125 ( - \nu^{15} + 6 \nu^{14} - 21 \nu^{13} + 54 \nu^{12} - 113 \nu^{11} + 168 \nu^{10} - 186 \nu^{9} + \cdots + 468750 ) / 78125 Copy content Toggle raw display
β8\beta_{8}== (22146ν15+79646ν14238236ν13+546479ν12982878ν11++3913984375)/1704375000 ( - 22146 \nu^{15} + 79646 \nu^{14} - 238236 \nu^{13} + 546479 \nu^{12} - 982878 \nu^{11} + \cdots + 3913984375 ) / 1704375000 Copy content Toggle raw display
β9\beta_{9}== (31084ν15+147184ν14411169ν13+973141ν121435662ν11++6441171875)/1704375000 ( - 31084 \nu^{15} + 147184 \nu^{14} - 411169 \nu^{13} + 973141 \nu^{12} - 1435662 \nu^{11} + \cdots + 6441171875 ) / 1704375000 Copy content Toggle raw display
β10\beta_{10}== (29284ν15117899ν14+303959ν13699881ν12+1218072ν11+2969453125)/1136250000 ( 29284 \nu^{15} - 117899 \nu^{14} + 303959 \nu^{13} - 699881 \nu^{12} + 1218072 \nu^{11} + \cdots - 2969453125 ) / 1136250000 Copy content Toggle raw display
β11\beta_{11}== (52692ν15+298312ν14764517ν13+2102878ν123542736ν11++14823593750)/1704375000 ( - 52692 \nu^{15} + 298312 \nu^{14} - 764517 \nu^{13} + 2102878 \nu^{12} - 3542736 \nu^{11} + \cdots + 14823593750 ) / 1704375000 Copy content Toggle raw display
β12\beta_{12}== (53838ν15173323ν14+537918ν131115872ν12+1770924ν11+6022343750)/1704375000 ( 53838 \nu^{15} - 173323 \nu^{14} + 537918 \nu^{13} - 1115872 \nu^{12} + 1770924 \nu^{11} + \cdots - 6022343750 ) / 1704375000 Copy content Toggle raw display
β13\beta_{13}== (115983ν15499393ν14+1698663ν134301152ν12+7691484ν11+27148437500)/3408750000 ( 115983 \nu^{15} - 499393 \nu^{14} + 1698663 \nu^{13} - 4301152 \nu^{12} + 7691484 \nu^{11} + \cdots - 27148437500 ) / 3408750000 Copy content Toggle raw display
β14\beta_{14}== (74084ν15+358884ν141026644ν13+2840991ν124526112ν11++16796953125)/1704375000 ( - 74084 \nu^{15} + 358884 \nu^{14} - 1026644 \nu^{13} + 2840991 \nu^{12} - 4526112 \nu^{11} + \cdots + 16796953125 ) / 1704375000 Copy content Toggle raw display
β15\beta_{15}== (182946ν15+901171ν142551911ν13+6076429ν1210614828ν11++36593515625)/3408750000 ( - 182946 \nu^{15} + 901171 \nu^{14} - 2551911 \nu^{13} + 6076429 \nu^{12} - 10614828 \nu^{11} + \cdots + 36593515625 ) / 3408750000 Copy content Toggle raw display
ν\nu== β1 \beta_1 Copy content Toggle raw display
ν2\nu^{2}== β15β12β10+β7+2β6β4+β12 -\beta_{15} - \beta_{12} - \beta_{10} + \beta_{7} + 2\beta_{6} - \beta_{4} + \beta _1 - 2 Copy content Toggle raw display
ν3\nu^{3}== β15+β14+4β132β12+β103β9+3β8+1 \beta_{15} + \beta_{14} + 4 \beta_{13} - 2 \beta_{12} + \beta_{10} - 3 \beta_{9} + 3 \beta_{8} + \cdots - 1 Copy content Toggle raw display
ν4\nu^{4}== β153β14+2β13+3β12+6β11+β10+4β9+9 - \beta_{15} - 3 \beta_{14} + 2 \beta_{13} + 3 \beta_{12} + 6 \beta_{11} + \beta_{10} + 4 \beta_{9} + \cdots - 9 Copy content Toggle raw display
ν5\nu^{5}== 6β15+8β142β134β126β11+12β108β9++4 6 \beta_{15} + 8 \beta_{14} - 2 \beta_{13} - 4 \beta_{12} - 6 \beta_{11} + 12 \beta_{10} - 8 \beta_{9} + \cdots + 4 Copy content Toggle raw display
ν6\nu^{6}== 24β15+16β1422β1348β11+2β106β928β8++39 24 \beta_{15} + 16 \beta_{14} - 22 \beta_{13} - 48 \beta_{11} + 2 \beta_{10} - 6 \beta_{9} - 28 \beta_{8} + \cdots + 39 Copy content Toggle raw display
ν7\nu^{7}== 8β1528β1440β1328β12+6β11+8β1042β8+2 8 \beta_{15} - 28 \beta_{14} - 40 \beta_{13} - 28 \beta_{12} + 6 \beta_{11} + 8 \beta_{10} - 42 \beta_{8} + \cdots - 2 Copy content Toggle raw display
ν8\nu^{8}== 55β15+58β133β12+42β11+113β10+20β9+190 55 \beta_{15} + 58 \beta_{13} - 3 \beta_{12} + 42 \beta_{11} + 113 \beta_{10} + 20 \beta_{9} + \cdots - 190 Copy content Toggle raw display
ν9\nu^{9}== 33β1577β14142β13+154β12+45β1055β9+403 - 33 \beta_{15} - 77 \beta_{14} - 142 \beta_{13} + 154 \beta_{12} + 45 \beta_{10} - 55 \beta_{9} + \cdots - 403 Copy content Toggle raw display
ν10\nu^{10}== 311β15121β14+334β13+121β12108β11+311β10+2293 - 311 \beta_{15} - 121 \beta_{14} + 334 \beta_{13} + 121 \beta_{12} - 108 \beta_{11} + 311 \beta_{10} + \cdots - 2293 Copy content Toggle raw display
ν11\nu^{11}== 564β15+384β14+756β13192β12252β11+768β10++1752 - 564 \beta_{15} + 384 \beta_{14} + 756 \beta_{13} - 192 \beta_{12} - 252 \beta_{11} + 768 \beta_{10} + \cdots + 1752 Copy content Toggle raw display
ν12\nu^{12}== 1152β15+3456β14+2196β13+1728β11+1044β102004β9++4945 - 1152 \beta_{15} + 3456 \beta_{14} + 2196 \beta_{13} + 1728 \beta_{11} + 1044 \beta_{10} - 2004 \beta_{9} + \cdots + 4945 Copy content Toggle raw display
ν13\nu^{13}== 15024β1596β142304β1396β12+14220β1115024β10+7908 - 15024 \beta_{15} - 96 \beta_{14} - 2304 \beta_{13} - 96 \beta_{12} + 14220 \beta_{11} - 15024 \beta_{10} + \cdots - 7908 Copy content Toggle raw display
ν14\nu^{14}== 6947β15+10116β133169β12+9612β1110873β108472β9++21406 6947 \beta_{15} + 10116 \beta_{13} - 3169 \beta_{12} + 9612 \beta_{11} - 10873 \beta_{10} - 8472 \beta_{9} + \cdots + 21406 Copy content Toggle raw display
ν15\nu^{15}== 39073β1525031β1411048β13+50062β1259β10+23433β9++47975 39073 \beta_{15} - 25031 \beta_{14} - 11048 \beta_{13} + 50062 \beta_{12} - 59 \beta_{10} + 23433 \beta_{9} + \cdots + 47975 Copy content Toggle raw display

Character values

We give the values of χ\chi on generators for (Z/630Z)×\left(\mathbb{Z}/630\mathbb{Z}\right)^\times.

nn 127127 281281 451451
χ(n)\chi(n) 1-1 1-1 β8\beta_{8}

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}
89.1
2.11423 0.728019i
1.98669 + 1.02619i
1.68760 1.46697i
0.948234 + 2.02506i
0.104634 2.23362i
−0.442358 + 2.19188i
−1.27963 1.83372i
−2.11940 0.712845i
2.11423 + 0.728019i
1.98669 1.02619i
1.68760 + 1.46697i
0.948234 2.02506i
0.104634 + 2.23362i
−0.442358 2.19188i
−1.27963 + 1.83372i
−2.11940 + 0.712845i
−0.500000 + 0.866025i 0 −0.500000 0.866025i −2.11423 0.728019i 0 2.30608 1.29693i 1.00000 0 1.68760 1.46697i
89.2 −0.500000 + 0.866025i 0 −0.500000 0.866025i −1.98669 + 1.02619i 0 1.39924 2.24547i 1.00000 0 0.104634 2.23362i
89.3 −0.500000 + 0.866025i 0 −0.500000 0.866025i −1.68760 1.46697i 0 −2.30608 + 1.29693i 1.00000 0 2.11423 0.728019i
89.4 −0.500000 + 0.866025i 0 −0.500000 0.866025i −0.948234 + 2.02506i 0 0.732536 + 2.54232i 1.00000 0 −1.27963 1.83372i
89.5 −0.500000 + 0.866025i 0 −0.500000 0.866025i −0.104634 2.23362i 0 −1.39924 + 2.24547i 1.00000 0 1.98669 + 1.02619i
89.6 −0.500000 + 0.866025i 0 −0.500000 0.866025i 0.442358 + 2.19188i 0 −1.63937 2.07665i 1.00000 0 −2.11940 0.712845i
89.7 −0.500000 + 0.866025i 0 −0.500000 0.866025i 1.27963 1.83372i 0 −0.732536 2.54232i 1.00000 0 0.948234 + 2.02506i
89.8 −0.500000 + 0.866025i 0 −0.500000 0.866025i 2.11940 0.712845i 0 1.63937 + 2.07665i 1.00000 0 −0.442358 + 2.19188i
269.1 −0.500000 0.866025i 0 −0.500000 + 0.866025i −2.11423 + 0.728019i 0 2.30608 + 1.29693i 1.00000 0 1.68760 + 1.46697i
269.2 −0.500000 0.866025i 0 −0.500000 + 0.866025i −1.98669 1.02619i 0 1.39924 + 2.24547i 1.00000 0 0.104634 + 2.23362i
269.3 −0.500000 0.866025i 0 −0.500000 + 0.866025i −1.68760 + 1.46697i 0 −2.30608 1.29693i 1.00000 0 2.11423 + 0.728019i
269.4 −0.500000 0.866025i 0 −0.500000 + 0.866025i −0.948234 2.02506i 0 0.732536 2.54232i 1.00000 0 −1.27963 + 1.83372i
269.5 −0.500000 0.866025i 0 −0.500000 + 0.866025i −0.104634 + 2.23362i 0 −1.39924 2.24547i 1.00000 0 1.98669 1.02619i
269.6 −0.500000 0.866025i 0 −0.500000 + 0.866025i 0.442358 2.19188i 0 −1.63937 + 2.07665i 1.00000 0 −2.11940 + 0.712845i
269.7 −0.500000 0.866025i 0 −0.500000 + 0.866025i 1.27963 + 1.83372i 0 −0.732536 + 2.54232i 1.00000 0 0.948234 2.02506i
269.8 −0.500000 0.866025i 0 −0.500000 + 0.866025i 2.11940 + 0.712845i 0 1.63937 2.07665i 1.00000 0 −0.442358 2.19188i
nn: e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 89.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.d odd 6 1 inner
15.d odd 2 1 inner
105.p even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 630.2.bo.a 16
3.b odd 2 1 630.2.bo.b yes 16
5.b even 2 1 630.2.bo.b yes 16
5.c odd 4 2 3150.2.bf.f 32
7.c even 3 1 4410.2.d.b 16
7.d odd 6 1 inner 630.2.bo.a 16
7.d odd 6 1 4410.2.d.b 16
15.d odd 2 1 inner 630.2.bo.a 16
15.e even 4 2 3150.2.bf.f 32
21.g even 6 1 630.2.bo.b yes 16
21.g even 6 1 4410.2.d.a 16
21.h odd 6 1 4410.2.d.a 16
35.i odd 6 1 630.2.bo.b yes 16
35.i odd 6 1 4410.2.d.a 16
35.j even 6 1 4410.2.d.a 16
35.k even 12 2 3150.2.bf.f 32
105.o odd 6 1 4410.2.d.b 16
105.p even 6 1 inner 630.2.bo.a 16
105.p even 6 1 4410.2.d.b 16
105.w odd 12 2 3150.2.bf.f 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
630.2.bo.a 16 1.a even 1 1 trivial
630.2.bo.a 16 7.d odd 6 1 inner
630.2.bo.a 16 15.d odd 2 1 inner
630.2.bo.a 16 105.p even 6 1 inner
630.2.bo.b yes 16 3.b odd 2 1
630.2.bo.b yes 16 5.b even 2 1
630.2.bo.b yes 16 21.g even 6 1
630.2.bo.b yes 16 35.i odd 6 1
3150.2.bf.f 32 5.c odd 4 2
3150.2.bf.f 32 15.e even 4 2
3150.2.bf.f 32 35.k even 12 2
3150.2.bf.f 32 105.w odd 12 2
4410.2.d.a 16 21.g even 6 1
4410.2.d.a 16 21.h odd 6 1
4410.2.d.a 16 35.i odd 6 1
4410.2.d.a 16 35.j even 6 1
4410.2.d.b 16 7.c even 3 1
4410.2.d.b 16 7.d odd 6 1
4410.2.d.b 16 105.o odd 6 1
4410.2.d.b 16 105.p even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T17842T176+1516T1743024T1738688T172+17856T17+61504 T_{17}^{8} - 42T_{17}^{6} + 1516T_{17}^{4} - 3024T_{17}^{3} - 8688T_{17}^{2} + 17856T_{17} + 61504 acting on S2new(630,[χ])S_{2}^{\mathrm{new}}(630, [\chi]). Copy content Toggle raw display

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 (T2+T+1)8 (T^{2} + T + 1)^{8} Copy content Toggle raw display
33 T16 T^{16} Copy content Toggle raw display
55 T16+6T15++390625 T^{16} + 6 T^{15} + \cdots + 390625 Copy content Toggle raw display
77 T16+14T14++5764801 T^{16} + 14 T^{14} + \cdots + 5764801 Copy content Toggle raw display
1111 T1650T14++923521 T^{16} - 50 T^{14} + \cdots + 923521 Copy content Toggle raw display
1313 (T872T6++1156)2 (T^{8} - 72 T^{6} + \cdots + 1156)^{2} Copy content Toggle raw display
1717 (T842T6++61504)2 (T^{8} - 42 T^{6} + \cdots + 61504)^{2} Copy content Toggle raw display
1919 (T812T7++19044)2 (T^{8} - 12 T^{7} + \cdots + 19044)^{2} Copy content Toggle raw display
2323 (T8+4T7++86436)2 (T^{8} + 4 T^{7} + \cdots + 86436)^{2} Copy content Toggle raw display
2929 (T8+122T6++33856)2 (T^{8} + 122 T^{6} + \cdots + 33856)^{2} Copy content Toggle raw display
3131 (T86T7++6533136)2 (T^{8} - 6 T^{7} + \cdots + 6533136)^{2} Copy content Toggle raw display
3737 T16112T14++3111696 T^{16} - 112 T^{14} + \cdots + 3111696 Copy content Toggle raw display
4141 (T8244T6++12787776)2 (T^{8} - 244 T^{6} + \cdots + 12787776)^{2} Copy content Toggle raw display
4343 (T8+148T6++553536)2 (T^{8} + 148 T^{6} + \cdots + 553536)^{2} Copy content Toggle raw display
4747 (T830T7++10404)2 (T^{8} - 30 T^{7} + \cdots + 10404)^{2} Copy content Toggle raw display
5353 (T88T7++690561)2 (T^{8} - 8 T^{7} + \cdots + 690561)^{2} Copy content Toggle raw display
5959 T16+46T14++104976 T^{16} + 46 T^{14} + \cdots + 104976 Copy content Toggle raw display
6161 (T812T7++78428736)2 (T^{8} - 12 T^{7} + \cdots + 78428736)^{2} Copy content Toggle raw display
6767 T16++429981696 T^{16} + \cdots + 429981696 Copy content Toggle raw display
7171 (T8+620T6++352538176)2 (T^{8} + 620 T^{6} + \cdots + 352538176)^{2} Copy content Toggle raw display
7373 T16++347892350976 T^{16} + \cdots + 347892350976 Copy content Toggle raw display
7979 (T82T7++86044176)2 (T^{8} - 2 T^{7} + \cdots + 86044176)^{2} Copy content Toggle raw display
8383 (T8+414T6++364816)2 (T^{8} + 414 T^{6} + \cdots + 364816)^{2} Copy content Toggle raw display
8989 T16++306402103296 T^{16} + \cdots + 306402103296 Copy content Toggle raw display
9797 (T8338T6++27123264)2 (T^{8} - 338 T^{6} + \cdots + 27123264)^{2} Copy content Toggle raw display
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