Properties

Label 605.4.a.q
Level 605605
Weight 44
Character orbit 605.a
Self dual yes
Analytic conductor 35.69635.696
Analytic rank 11
Dimension 1212
CM no
Inner twists 11

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [605,4,Mod(1,605)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(605, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("605.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: N N == 605=5112 605 = 5 \cdot 11^{2}
Weight: k k == 4 4
Character orbit: [χ][\chi] == 605.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: 35.696155553535.6961555535
Analytic rank: 11
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: x12x1167x10+65x9+1558x81475x714915x6+12951x5++27856 x^{12} - x^{11} - 67 x^{10} + 65 x^{9} + 1558 x^{8} - 1475 x^{7} - 14915 x^{6} + 12951 x^{5} + \cdots + 27856 Copy content Toggle raw display
Coefficient ring: Z[a1,,a7]\Z[a_1, \ldots, a_{7}]
Coefficient ring index: 5112 5\cdot 11^{2}
Twist minimal: no (minimal twist has level 55)
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,,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) == qβ1q2+β4q3+(β2+3)q45q5+(β81)q6+(β11β9β6+4)q7+(β11+β10+β7++1)q8++(54β11+13β10++382)q98+O(q100) q - \beta_1 q^{2} + \beta_{4} q^{3} + (\beta_{2} + 3) q^{4} - 5 q^{5} + ( - \beta_{8} - 1) q^{6} + (\beta_{11} - \beta_{9} - \beta_{6} + \cdots - 4) q^{7} + ( - \beta_{11} + \beta_{10} + \beta_{7} + \cdots + 1) q^{8}+ \cdots + ( - 54 \beta_{11} + 13 \beta_{10} + \cdots + 382) q^{98}+O(q^{100}) Copy content Toggle raw display
Tr(f)(q)\operatorname{Tr}(f)(q) == 12qq2+q3+39q460q510q641q7+9q8+131q9+5q1029q12109q13247q145q15+283q16+167q17135q18332q19195q20++4209q98+O(q100) 12 q - q^{2} + q^{3} + 39 q^{4} - 60 q^{5} - 10 q^{6} - 41 q^{7} + 9 q^{8} + 131 q^{9} + 5 q^{10} - 29 q^{12} - 109 q^{13} - 247 q^{14} - 5 q^{15} + 283 q^{16} + 167 q^{17} - 135 q^{18} - 332 q^{19} - 195 q^{20}+ \cdots + 4209 q^{98}+O(q^{100}) Copy content Toggle raw display

Basis of coefficient ring in terms of a root ν\nu of x12x1167x10+65x9+1558x81475x714915x6+12951x5++27856 x^{12} - x^{11} - 67 x^{10} + 65 x^{9} + 1558 x^{8} - 1475 x^{7} - 14915 x^{6} + 12951 x^{5} + \cdots + 27856 : Copy content Toggle raw display

β1\beta_{1}== ν \nu Copy content Toggle raw display
β2\beta_{2}== ν211 \nu^{2} - 11 Copy content Toggle raw display
β3\beta_{3}== (1107391ν11+7673220ν10+70067177ν9487545538ν8+25808536144)/3033701760 ( - 1107391 \nu^{11} + 7673220 \nu^{10} + 70067177 \nu^{9} - 487545538 \nu^{8} + \cdots - 25808536144 ) / 3033701760 Copy content Toggle raw display
β4\beta_{4}== (1547143ν115364860ν1098589101ν9+346707534ν8+2080085148ν7++54810001872)/1516850880 ( 1547143 \nu^{11} - 5364860 \nu^{10} - 98589101 \nu^{9} + 346707534 \nu^{8} + 2080085148 \nu^{7} + \cdots + 54810001872 ) / 1516850880 Copy content Toggle raw display
β5\beta_{5}== (3213147ν115066460ν10213226829ν9+325869666ν8++58802064848)/3033701760 ( 3213147 \nu^{11} - 5066460 \nu^{10} - 213226829 \nu^{9} + 325869666 \nu^{8} + \cdots + 58802064848 ) / 3033701760 Copy content Toggle raw display
β6\beta_{6}== (4650909ν114351580ν10298006803ν9+269668622ν8+115681745104)/3033701760 ( 4650909 \nu^{11} - 4351580 \nu^{10} - 298006803 \nu^{9} + 269668622 \nu^{8} + \cdots - 115681745104 ) / 3033701760 Copy content Toggle raw display
β7\beta_{7}== (732784ν111688045ν1046784388ν9+107260687ν8+990422074ν7++11388669576)/379212720 ( 732784 \nu^{11} - 1688045 \nu^{10} - 46784388 \nu^{9} + 107260687 \nu^{8} + 990422074 \nu^{7} + \cdots + 11388669576 ) / 379212720 Copy content Toggle raw display
β8\beta_{8}== (3817717ν11+5069480ν10+246143239ν9330363646ν8+44614066288)/1516850880 ( - 3817717 \nu^{11} + 5069480 \nu^{10} + 246143239 \nu^{9} - 330363646 \nu^{8} + \cdots - 44614066288 ) / 1516850880 Copy content Toggle raw display
β9\beta_{9}== (4071101ν115076600ν10264479807ν9+326978918ν8++57175693584)/1516850880 ( 4071101 \nu^{11} - 5076600 \nu^{10} - 264479807 \nu^{9} + 326978918 \nu^{8} + \cdots + 57175693584 ) / 1516850880 Copy content Toggle raw display
β10\beta_{10}== (551436ν111451975ν1035084837ν9+93506228ν8+742205606ν7++12332638144)/189606360 ( 551436 \nu^{11} - 1451975 \nu^{10} - 35084837 \nu^{9} + 93506228 \nu^{8} + 742205606 \nu^{7} + \cdots + 12332638144 ) / 189606360 Copy content Toggle raw display
β11\beta_{11}== (10483571ν1118333020ν10668387117ν9+1173224538ν8++156036728784)/3033701760 ( 10483571 \nu^{11} - 18333020 \nu^{10} - 668387117 \nu^{9} + 1173224538 \nu^{8} + \cdots + 156036728784 ) / 3033701760 Copy content Toggle raw display
ν\nu== β1 \beta_1 Copy content Toggle raw display
ν2\nu^{2}== β2+11 \beta_{2} + 11 Copy content Toggle raw display
ν3\nu^{3}== β11β10β7+β4β3+21β11 \beta_{11} - \beta_{10} - \beta_{7} + \beta_{4} - \beta_{3} + 21\beta _1 - 1 Copy content Toggle raw display
ν4\nu^{4}== 2β10+3β9+β8+β7β63β5+3β4+27β2β1+225 -2\beta_{10} + 3\beta_{9} + \beta_{8} + \beta_{7} - \beta_{6} - 3\beta_{5} + 3\beta_{4} + 27\beta_{2} - \beta _1 + 225 Copy content Toggle raw display
ν5\nu^{5}== 40β1135β105β9+6β840β7+4β6+9β5+20 40 \beta_{11} - 35 \beta_{10} - 5 \beta_{9} + 6 \beta_{8} - 40 \beta_{7} + 4 \beta_{6} + 9 \beta_{5} + \cdots - 20 Copy content Toggle raw display
ν6\nu^{6}== 22β1182β10+154β9+50β8+42β726β6++5342 - 22 \beta_{11} - 82 \beta_{10} + 154 \beta_{9} + 50 \beta_{8} + 42 \beta_{7} - 26 \beta_{6} + \cdots + 5342 Copy content Toggle raw display
ν7\nu^{7}== 1268β111042β10168β9+308β81320β7+204β6++203 1268 \beta_{11} - 1042 \beta_{10} - 168 \beta_{9} + 308 \beta_{8} - 1320 \beta_{7} + 204 \beta_{6} + \cdots + 203 Copy content Toggle raw display
ν8\nu^{8}== 1177β112701β10+5728β9+1752β8+1249β7568β6++135553 - 1177 \beta_{11} - 2701 \beta_{10} + 5728 \beta_{9} + 1752 \beta_{8} + 1249 \beta_{7} - 568 \beta_{6} + \cdots + 135553 Copy content Toggle raw display
ν9\nu^{9}== 37662β1129810β104155β9+11547β840611β7+7485β6++28119 37662 \beta_{11} - 29810 \beta_{10} - 4155 \beta_{9} + 11547 \beta_{8} - 40611 \beta_{7} + 7485 \beta_{6} + \cdots + 28119 Copy content Toggle raw display
ν10\nu^{10}== 43872β1183631β10+188539β9+54844β8+32146β7++3583236 - 43872 \beta_{11} - 83631 \beta_{10} + 188539 \beta_{9} + 54844 \beta_{8} + 32146 \beta_{7} + \cdots + 3583236 Copy content Toggle raw display
ν11\nu^{11}== 1089982β11845122β1085396β9+384822β81207126β7++1372748 1089982 \beta_{11} - 845122 \beta_{10} - 85396 \beta_{9} + 384822 \beta_{8} - 1207126 \beta_{7} + \cdots + 1372748 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
5.39954
4.36383
3.48072
2.17093
1.85253
0.696762
−0.747441
−1.25858
−1.43938
−3.94598
−4.31191
−5.26103
−5.39954 6.27988 21.1550 −5.00000 −33.9084 1.92176 −71.0308 12.4368 26.9977
1.2 −4.36383 −6.47441 11.0430 −5.00000 28.2532 24.2107 −13.2791 14.9180 21.8191
1.3 −3.48072 −4.49803 4.11544 −5.00000 15.6564 0.859797 13.5211 −6.76775 17.4036
1.4 −2.17093 8.94509 −3.28705 −5.00000 −19.4192 −13.6976 24.5034 53.0147 10.8547
1.5 −1.85253 −5.02253 −4.56812 −5.00000 9.30441 −33.4856 23.2829 −1.77419 9.26267
1.6 −0.696762 2.36331 −7.51452 −5.00000 −1.64666 26.2619 10.8099 −21.4148 3.48381
1.7 0.747441 1.20483 −7.44133 −5.00000 0.900538 18.4546 −11.5415 −25.5484 −3.73720
1.8 1.25858 −9.09998 −6.41597 −5.00000 −11.4531 −16.3005 −18.1437 55.8097 −6.29291
1.9 1.43938 9.32704 −5.92820 −5.00000 13.4251 −14.6375 −20.0479 59.9937 −7.19688
1.10 3.94598 5.15950 7.57072 −5.00000 20.3593 −18.9445 −1.69392 −0.379584 −19.7299
1.11 4.31191 −6.66658 10.5926 −5.00000 −28.7457 17.4967 11.1791 17.4433 −21.5596
1.12 5.26103 −0.518116 19.6784 −5.00000 −2.72582 −33.1399 61.4406 −26.7316 −26.3051
nn: e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.12
Significant digits:
Format:

Atkin-Lehner signs

p p Sign
55 +1 +1
1111 1 -1

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 605.4.a.q 12
11.b odd 2 1 605.4.a.s 12
11.d odd 10 2 55.4.g.a 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
55.4.g.a 24 11.d odd 10 2
605.4.a.q 12 1.a even 1 1 trivial
605.4.a.s 12 11.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on S4new(Γ0(605))S_{4}^{\mathrm{new}}(\Gamma_0(605)):

T212+T21167T21065T29+1558T28+1475T2714915T26++27856 T_{2}^{12} + T_{2}^{11} - 67 T_{2}^{10} - 65 T_{2}^{9} + 1558 T_{2}^{8} + 1475 T_{2}^{7} - 14915 T_{2}^{6} + \cdots + 27856 Copy content Toggle raw display
T312T311227T310+98T39+18918T38+620T37++35387420 T_{3}^{12} - T_{3}^{11} - 227 T_{3}^{10} + 98 T_{3}^{9} + 18918 T_{3}^{8} + 620 T_{3}^{7} + \cdots + 35387420 Copy content Toggle raw display

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 T12+T11++27856 T^{12} + T^{11} + \cdots + 27856 Copy content Toggle raw display
33 T12T11++35387420 T^{12} - T^{11} + \cdots + 35387420 Copy content Toggle raw display
55 (T+5)12 (T + 5)^{12} Copy content Toggle raw display
77 T12++23307327589120 T^{12} + \cdots + 23307327589120 Copy content Toggle raw display
1111 T12 T^{12} Copy content Toggle raw display
1313 T12++60 ⁣ ⁣00 T^{12} + \cdots + 60\!\cdots\!00 Copy content Toggle raw display
1717 T12++20 ⁣ ⁣16 T^{12} + \cdots + 20\!\cdots\!16 Copy content Toggle raw display
1919 T12+41 ⁣ ⁣25 T^{12} + \cdots - 41\!\cdots\!25 Copy content Toggle raw display
2323 T12+43 ⁣ ⁣96 T^{12} + \cdots - 43\!\cdots\!96 Copy content Toggle raw display
2929 T12++38 ⁣ ⁣00 T^{12} + \cdots + 38\!\cdots\!00 Copy content Toggle raw display
3131 T12++53 ⁣ ⁣44 T^{12} + \cdots + 53\!\cdots\!44 Copy content Toggle raw display
3737 T12++54 ⁣ ⁣84 T^{12} + \cdots + 54\!\cdots\!84 Copy content Toggle raw display
4141 T12++44 ⁣ ⁣01 T^{12} + \cdots + 44\!\cdots\!01 Copy content Toggle raw display
4343 T12+37 ⁣ ⁣76 T^{12} + \cdots - 37\!\cdots\!76 Copy content Toggle raw display
4747 T12+14 ⁣ ⁣04 T^{12} + \cdots - 14\!\cdots\!04 Copy content Toggle raw display
5353 T12++65 ⁣ ⁣76 T^{12} + \cdots + 65\!\cdots\!76 Copy content Toggle raw display
5959 T12+97 ⁣ ⁣25 T^{12} + \cdots - 97\!\cdots\!25 Copy content Toggle raw display
6161 T12+21 ⁣ ⁣80 T^{12} + \cdots - 21\!\cdots\!80 Copy content Toggle raw display
6767 T12+51 ⁣ ⁣36 T^{12} + \cdots - 51\!\cdots\!36 Copy content Toggle raw display
7171 T12++12 ⁣ ⁣84 T^{12} + \cdots + 12\!\cdots\!84 Copy content Toggle raw display
7373 T12++83 ⁣ ⁣64 T^{12} + \cdots + 83\!\cdots\!64 Copy content Toggle raw display
7979 T12+42 ⁣ ⁣00 T^{12} + \cdots - 42\!\cdots\!00 Copy content Toggle raw display
8383 T12++10 ⁣ ⁣96 T^{12} + \cdots + 10\!\cdots\!96 Copy content Toggle raw display
8989 T12+60 ⁣ ⁣75 T^{12} + \cdots - 60\!\cdots\!75 Copy content Toggle raw display
9797 T12++13 ⁣ ⁣00 T^{12} + \cdots + 13\!\cdots\!00 Copy content Toggle raw display
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