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Properties

Label 464.3.d.b
Level $464$
Weight $3$
Character orbit 464.d
Analytic conductor $12.643$
Analytic rank $0$
Dimension $20$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [464,3,Mod(175,464)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(464, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("464.175");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 464 = 2^{4} \cdot 29 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 464.d (of order \(2\), degree \(1\), 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: \(12.6430842663\)
Analytic rank: \(0\)
Dimension: \(20\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} + 69 x^{18} + 1795 x^{16} + 24222 x^{14} + 189561 x^{12} + 892623 x^{10} + 2508433 x^{8} + \cdots + 21609 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{29}]\)
Coefficient ring index: \( 2^{36}\cdot 7^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{11} q^{3} - \beta_{3} q^{5} - \beta_{12} q^{7} + (\beta_{2} - 3) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{11} q^{3} - \beta_{3} q^{5} - \beta_{12} q^{7} + (\beta_{2} - 3) q^{9} + \beta_{16} q^{11} + ( - \beta_{8} + 1) q^{13} + ( - \beta_{18} + \beta_{16} + \cdots - \beta_{4}) q^{15}+ \cdots + (3 \beta_{19} - 3 \beta_{18} + \cdots - 4 \beta_{4}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - 8 q^{5} - 68 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q - 8 q^{5} - 68 q^{9} + 16 q^{13} + 40 q^{17} - 48 q^{21} + 188 q^{25} - 120 q^{33} - 80 q^{37} - 72 q^{41} + 72 q^{45} - 28 q^{49} + 96 q^{53} + 104 q^{57} - 96 q^{61} - 80 q^{65} + 352 q^{69} - 312 q^{73} - 192 q^{77} + 60 q^{81} + 336 q^{85} - 152 q^{89} - 472 q^{93} + 56 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} + 69 x^{18} + 1795 x^{16} + 24222 x^{14} + 189561 x^{12} + 892623 x^{10} + 2508433 x^{8} + \cdots + 21609 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 269119 \nu^{18} + 14955825 \nu^{16} + 264806302 \nu^{14} + 1929025386 \nu^{12} + \cdots - 24957173283 ) / 3750443928 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 538883 \nu^{18} - 36402591 \nu^{16} - 918073114 \nu^{14} - 11927037014 \nu^{12} + \cdots - 29241914583 ) / 4643406768 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 13214681 \nu^{18} + 897749793 \nu^{16} + 22671980714 \nu^{14} + 290379259542 \nu^{12} + \cdots - 132028188327 ) / 32503847376 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 10936 \nu^{19} + 747381 \nu^{17} + 19101116 \nu^{15} + 250205022 \nu^{13} + \cdots + 10598814513 \nu ) / 236023788 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 757289 \nu^{18} - 50166367 \nu^{16} - 1225295662 \nu^{14} - 15217333392 \nu^{12} + \cdots - 41589924285 ) / 1250147976 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 13082613 \nu^{18} - 855428506 \nu^{16} - 20436920204 \nu^{14} - 245686535916 \nu^{12} + \cdots - 12352318272 ) / 16251923688 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 3168803 \nu^{18} - 217019291 \nu^{16} - 5536671654 \nu^{14} - 71565124038 \nu^{12} + \cdots - 75153189783 ) / 2954895216 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 37977677 \nu^{18} - 2535952577 \nu^{16} - 62516866638 \nu^{14} - 780334603670 \nu^{12} + \cdots - 1132793231829 ) / 32503847376 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 11254547 \nu^{18} + 730259037 \nu^{16} + 17257669802 \nu^{14} + 205149355410 \nu^{12} + \cdots + 183966243909 ) / 8864685648 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 155501111 \nu^{18} + 10347255573 \nu^{16} + 253965882410 \nu^{14} + 3159391510218 \nu^{12} + \cdots + 1264306787709 ) / 97511542128 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 108346853 \nu^{19} + 7258096644 \nu^{17} + 179293861268 \nu^{15} + 2227767857166 \nu^{13} + \cdots - 16844008902672 \nu ) / 682580794896 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 24959967 \nu^{19} + 1740535793 \nu^{17} + 45661735278 \nu^{15} + 612950649536 \nu^{13} + \cdots + 4647445493355 \nu ) / 113763465816 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 60588499 \nu^{19} - 4013927296 \nu^{17} - 97597388088 \nu^{15} - 1191456127314 \nu^{13} + \cdots - 4009400719128 \nu ) / 227526931632 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 258162071 \nu^{19} - 17021686536 \nu^{17} - 412592166284 \nu^{15} + \cdots + 35240748098808 \nu ) / 682580794896 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 60980495 \nu^{19} + 4117881696 \nu^{17} + 103406444000 \nu^{15} + 1326071703002 \nu^{13} + \cdots + 19264652092644 \nu ) / 113763465816 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 389833673 \nu^{19} - 25576963008 \nu^{17} - 613696544504 \nu^{15} + \cdots + 40430927254620 \nu ) / 682580794896 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 165753053 \nu^{19} - 11335372240 \nu^{17} - 288959154860 \nu^{15} + \cdots - 13092847367712 \nu ) / 227526931632 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( - 687333589 \nu^{19} - 45371705136 \nu^{17} - 1102288771984 \nu^{15} + \cdots - 63775765633188 \nu ) / 682580794896 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( - 67516814 \nu^{19} - 4481300403 \nu^{17} - 109749087416 \nu^{15} - 1367135984622 \nu^{13} + \cdots - 8694964918311 \nu ) / 48755771064 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( - 3 \beta_{18} - \beta_{17} + 5 \beta_{16} - 3 \beta_{15} + \beta_{14} - 2 \beta_{13} + \cdots + \beta_{4} ) / 28 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( - \beta_{10} - \beta_{9} - 2 \beta_{8} - 3 \beta_{7} + 4 \beta_{6} - 3 \beta_{5} - 3 \beta_{3} + \cdots - 91 ) / 14 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 28 \beta_{19} + 27 \beta_{18} - 5 \beta_{17} - 101 \beta_{16} + 55 \beta_{15} - 23 \beta_{14} + \cdots - 44 \beta_{4} ) / 28 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 8 \beta_{10} + 50 \beta_{9} + 16 \beta_{8} + 94 \beta_{7} - 88 \beta_{6} + 108 \beta_{5} + 80 \beta_{3} + \cdots + 1477 ) / 14 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 952 \beta_{19} - 514 \beta_{18} + 370 \beta_{17} + 2700 \beta_{16} - 1368 \beta_{15} + \cdots + 1359 \beta_{4} ) / 28 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 19 \beta_{10} - 240 \beta_{9} - 23 \beta_{8} - 407 \beta_{7} + 331 \beta_{6} - 493 \beta_{5} + \cdots - 5235 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 29246 \beta_{19} + 13773 \beta_{18} - 12755 \beta_{17} - 79249 \beta_{16} + 38987 \beta_{15} + \cdots - 41019 \beta_{4} ) / 28 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 3478 \beta_{10} + 52240 \beta_{9} + 2336 \beta_{8} + 86398 \beta_{7} - 67028 \beta_{6} + 106040 \beta_{5} + \cdots + 1051337 ) / 14 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 886956 \beta_{19} - 405119 \beta_{18} + 398267 \beta_{17} + 2381551 \beta_{16} + \cdots + 1238732 \beta_{4} ) / 28 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 102570 \beta_{10} - 1592555 \beta_{9} - 49551 \beta_{8} - 2618396 \beta_{7} + 2002311 \beta_{6} + \cdots - 31386026 ) / 14 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 26853946 \beta_{19} + 12180984 \beta_{18} - 12155844 \beta_{17} - 71971874 \beta_{16} + \cdots - 37456957 \beta_{4} ) / 28 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 220990 \beta_{10} + 3449872 \beta_{9} + 94163 \beta_{8} + 5664594 \beta_{7} - 4313480 \beta_{6} + \cdots + 67621484 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 812917196 \beta_{19} - 368149435 \beta_{18} + 368819275 \beta_{17} + 2177913421 \beta_{16} + \cdots + 1133409677 \beta_{4} ) / 28 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 93646799 \beta_{10} - 1462797155 \beta_{9} - 38326360 \beta_{8} - 2401197039 \beta_{7} + \cdots - 28632047717 ) / 14 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 24608798564 \beta_{19} + 11140467351 \beta_{18} - 11172193765 \beta_{17} - 65925563581 \beta_{16} + \cdots - 34305851260 \beta_{4} ) / 28 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( ( 2835454462 \beta_{10} + 44288251036 \beta_{9} + 1146402476 \beta_{8} + 72695641994 \beta_{7} + \cdots + 866556678029 ) / 14 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( ( - 744973789700 \beta_{19} - 337220701814 \beta_{18} + 338273969270 \beta_{17} + \cdots + 1038478608483 \beta_{4} ) / 28 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( ( - 12263760671 \beta_{10} - 191539148568 \beta_{9} - 4940489665 \beta_{8} - 314393408887 \beta_{7} + \cdots - 3747349638765 ) / 2 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( ( 22552482932770 \beta_{19} + 10208403611445 \beta_{18} - 10241049289423 \beta_{17} + \cdots - 31437203065743 \beta_{4} ) / 28 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/464\mathbb{Z}\right)^\times\).

\(n\) \(117\) \(175\) \(321\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
175.1
5.50209i
1.44310i
− 1.53049i
− 2.95207i
− 0.166656i
2.79820i
− 2.72620i
0.682100i
2.50448i
1.88675i
− 1.88675i
− 2.50448i
− 0.682100i
2.72620i
− 2.79820i
0.166656i
2.95207i
1.53049i
− 1.44310i
− 5.50209i
0 − 5.45563i 0 −3.94765 0 2.61861i 0 −20.7639 0
175.2 0 − 5.12416i 0 −8.13026 0 − 11.0093i 0 −17.2570 0
175.3 0 − 4.55005i 0 8.26335 0 − 0.295091i 0 −11.7029 0
175.4 0 − 3.94782i 0 −0.459342 0 4.40863i 0 −6.58528 0
175.5 0 − 3.92832i 0 7.62576 0 5.54000i 0 −6.43170 0
175.6 0 − 2.34966i 0 −1.82181 0 − 0.690645i 0 3.47908 0
175.7 0 − 2.28467i 0 1.85233 0 − 13.1301i 0 3.78028 0
175.8 0 − 2.23726i 0 −3.52607 0 6.82457i 0 3.99465 0
175.9 0 − 0.707016i 0 −9.31226 0 − 6.08820i 0 8.50013 0
175.10 0 − 0.115341i 0 5.45596 0 8.31894i 0 8.98670 0
175.11 0 0.115341i 0 5.45596 0 − 8.31894i 0 8.98670 0
175.12 0 0.707016i 0 −9.31226 0 6.08820i 0 8.50013 0
175.13 0 2.23726i 0 −3.52607 0 − 6.82457i 0 3.99465 0
175.14 0 2.28467i 0 1.85233 0 13.1301i 0 3.78028 0
175.15 0 2.34966i 0 −1.82181 0 0.690645i 0 3.47908 0
175.16 0 3.92832i 0 7.62576 0 − 5.54000i 0 −6.43170 0
175.17 0 3.94782i 0 −0.459342 0 − 4.40863i 0 −6.58528 0
175.18 0 4.55005i 0 8.26335 0 0.295091i 0 −11.7029 0
175.19 0 5.12416i 0 −8.13026 0 11.0093i 0 −17.2570 0
175.20 0 5.45563i 0 −3.94765 0 − 2.61861i 0 −20.7639 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 175.20
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 464.3.d.b ✓ 20
4.b odd 2 1 inner 464.3.d.b ✓ 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
464.3.d.b ✓ 20 1.a even 1 1 trivial
464.3.d.b ✓ 20 4.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{20} + 124 T_{3}^{18} + 6404 T_{3}^{16} + 178968 T_{3}^{14} + 2944622 T_{3}^{12} + 29101248 T_{3}^{10} + \cdots + 3732624 \) acting on \(S_{3}^{\mathrm{new}}(464, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} \) Copy content Toggle raw display
$3$ \( T^{20} + 124 T^{18} + \cdots + 3732624 \) Copy content Toggle raw display
$5$ \( (T^{10} + 4 T^{9} + \cdots + 561636)^{2} \) Copy content Toggle raw display
$7$ \( T^{20} + \cdots + 424144797696 \) Copy content Toggle raw display
$11$ \( T^{20} + \cdots + 10\!\cdots\!56 \) Copy content Toggle raw display
$13$ \( (T^{10} - 8 T^{9} + \cdots + 17020989468)^{2} \) Copy content Toggle raw display
$17$ \( (T^{10} - 20 T^{9} + \cdots - 2764274688)^{2} \) Copy content Toggle raw display
$19$ \( T^{20} + \cdots + 13\!\cdots\!76 \) Copy content Toggle raw display
$23$ \( T^{20} + \cdots + 36\!\cdots\!76 \) Copy content Toggle raw display
$29$ \( (T^{2} - 29)^{10} \) Copy content Toggle raw display
$31$ \( T^{20} + \cdots + 76\!\cdots\!24 \) Copy content Toggle raw display
$37$ \( (T^{10} + 40 T^{9} + \cdots + 428081152)^{2} \) Copy content Toggle raw display
$41$ \( (T^{10} + \cdots - 3072101280768)^{2} \) Copy content Toggle raw display
$43$ \( T^{20} + \cdots + 98\!\cdots\!76 \) Copy content Toggle raw display
$47$ \( T^{20} + \cdots + 82\!\cdots\!36 \) Copy content Toggle raw display
$53$ \( (T^{10} + \cdots - 25\!\cdots\!00)^{2} \) Copy content Toggle raw display
$59$ \( T^{20} + \cdots + 25\!\cdots\!64 \) Copy content Toggle raw display
$61$ \( (T^{10} + \cdots - 52\!\cdots\!32)^{2} \) Copy content Toggle raw display
$67$ \( T^{20} + \cdots + 11\!\cdots\!36 \) Copy content Toggle raw display
$71$ \( T^{20} + \cdots + 27\!\cdots\!16 \) Copy content Toggle raw display
$73$ \( (T^{10} + \cdots + 22\!\cdots\!72)^{2} \) Copy content Toggle raw display
$79$ \( T^{20} + \cdots + 33\!\cdots\!96 \) Copy content Toggle raw display
$83$ \( T^{20} + \cdots + 43\!\cdots\!96 \) Copy content Toggle raw display
$89$ \( (T^{10} + \cdots + 75\!\cdots\!08)^{2} \) Copy content Toggle raw display
$97$ \( (T^{10} + \cdots + 81\!\cdots\!24)^{2} \) Copy content Toggle raw display
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