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Properties

Label 145.4.b.a
Level $145$
Weight $4$
Character orbit 145.b
Analytic conductor $8.555$
Analytic rank $0$
Dimension $18$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [145,4,Mod(59,145)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(145, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("145.59");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 145 = 5 \cdot 29 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 145.b (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: \(8.55527695083\)
Analytic rank: \(0\)
Dimension: \(18\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{18} + 93 x^{16} + 3543 x^{14} + 71263 x^{12} + 812415 x^{10} + 5221259 x^{8} + 17554073 x^{6} + \cdots + 3013696 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{6} \)
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 \(1,\beta_1,\ldots,\beta_{17}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} - \beta_{13} q^{3} + (\beta_{2} - 2) q^{4} - \beta_{8} q^{5} + (\beta_{6} + \beta_{2} - 1) q^{6} + ( - \beta_{16} - \beta_1) q^{7} + (\beta_{14} + \beta_{13} - \beta_1) q^{8} + (\beta_{6} + \beta_{4} - 4) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} - \beta_{13} q^{3} + (\beta_{2} - 2) q^{4} - \beta_{8} q^{5} + (\beta_{6} + \beta_{2} - 1) q^{6} + ( - \beta_{16} - \beta_1) q^{7} + (\beta_{14} + \beta_{13} - \beta_1) q^{8} + (\beta_{6} + \beta_{4} - 4) q^{9} + (\beta_{14} - \beta_{13} - \beta_{12} + \cdots + 2) q^{10}+ \cdots + ( - 11 \beta_{11} + 11 \beta_{10} + \cdots + 430) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q - 42 q^{4} - 8 q^{5} - 30 q^{6} - 76 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 18 q - 42 q^{4} - 8 q^{5} - 30 q^{6} - 76 q^{9} + 62 q^{10} - 144 q^{11} + 170 q^{14} + 112 q^{15} - 186 q^{16} + 468 q^{19} + 72 q^{20} - 196 q^{21} + 454 q^{24} - 36 q^{25} - 170 q^{26} + 522 q^{29} - 494 q^{30} - 434 q^{31} + 978 q^{34} - 554 q^{35} - 396 q^{36} + 1414 q^{39} - 610 q^{40} - 336 q^{41} + 1028 q^{44} - 1354 q^{45} + 174 q^{46} + 840 q^{49} - 1130 q^{50} - 1856 q^{51} + 3124 q^{54} - 982 q^{55} - 1018 q^{56} + 2610 q^{59} - 1900 q^{60} - 1898 q^{61} + 3534 q^{64} - 2504 q^{65} - 608 q^{66} + 3590 q^{69} - 3836 q^{70} - 1652 q^{71} + 2064 q^{74} - 2098 q^{75} - 472 q^{76} + 4254 q^{79} - 4432 q^{80} - 942 q^{81} + 6412 q^{84} - 2042 q^{85} + 174 q^{86} + 2404 q^{89} - 5998 q^{90} - 4532 q^{91} + 9128 q^{94} - 1788 q^{95} + 850 q^{96} + 8016 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{18} + 93 x^{16} + 3543 x^{14} + 71263 x^{12} + 812415 x^{10} + 5221259 x^{8} + 17554073 x^{6} + \cdots + 3013696 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 10 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 909731 \nu^{16} - 59927876 \nu^{14} - 1175562721 \nu^{12} + 1846613204 \nu^{10} + \cdots + 4397273237312 ) / 39174062080 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 675571 \nu^{16} + 57674756 \nu^{14} + 1953301841 \nu^{12} + 33209310316 \nu^{10} + \cdots - 100575765312 ) / 19587031040 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 3077117 \nu^{16} + 258383292 \nu^{14} + 8563758207 \nu^{12} + 141569433172 \nu^{10} + \cdots - 46195321024 ) / 19587031040 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 7812401 \nu^{16} - 651473196 \nu^{14} - 21399069371 \nu^{12} - 349430176676 \nu^{10} + \cdots - 1851274538048 ) / 39174062080 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 338063 \nu^{16} + 28657068 \nu^{14} + 964397533 \nu^{12} + 16363466428 \nu^{10} + \cdots + 217145449664 ) / 1224189440 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 4333943 \nu^{17} - 21260792 \nu^{16} + 415715828 \nu^{15} - 1681663200 \nu^{14} + \cdots + 28428343691776 ) / 850077147136 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 4333943 \nu^{17} + 21260792 \nu^{16} + 415715828 \nu^{15} + 1681663200 \nu^{14} + \cdots - 28428343691776 ) / 850077147136 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 98335794 \nu^{17} - 1103468653 \nu^{16} + 9467214744 \nu^{15} - 93039176188 \nu^{14} + \cdots - 815325292133184 ) / 8500771471360 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 98335794 \nu^{17} + 1103468653 \nu^{16} + 9467214744 \nu^{15} + 93039176188 \nu^{14} + \cdots + 815325292133184 ) / 8500771471360 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 3626987 \nu^{17} + 333285092 \nu^{15} + 12441930297 \nu^{13} + 241840340812 \nu^{11} + \cdots + 8339109855936 \nu ) / 303598981120 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 107913881 \nu^{17} - 8340699916 \nu^{15} - 240969196851 \nu^{13} + \cdots + 337871801063872 \nu ) / 8500771471360 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 107913881 \nu^{17} + 8340699916 \nu^{15} + 240969196851 \nu^{13} + \cdots - 193358686050752 \nu ) / 8500771471360 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 190642643 \nu^{17} - 16600262788 \nu^{15} - 580626792433 \nu^{13} + \cdots - 625226569075904 \nu ) / 8500771471360 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 375829677 \nu^{17} - 32669242492 \nu^{15} - 1140334612367 \nu^{13} + \cdots - 13\!\cdots\!16 \nu ) / 8500771471360 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 5814011 \nu^{17} - 499335476 \nu^{15} - 17114085081 \nu^{13} - 298173923756 \nu^{11} + \cdots - 3661080919488 \nu ) / 75899745280 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 10 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{14} + \beta_{13} - 17\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -2\beta_{9} + 2\beta_{8} - \beta_{6} - \beta_{5} + \beta_{4} - \beta_{3} - 23\beta_{2} + 164 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{17} - 3 \beta_{16} + 2 \beta_{15} - 30 \beta_{14} - 29 \beta_{13} + 4 \beta_{12} + \cdots + 324 \beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 2 \beta_{11} - 2 \beta_{10} + 73 \beta_{9} - 73 \beta_{8} + 5 \beta_{7} + 41 \beta_{6} + 27 \beta_{5} + \cdots - 3050 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 47 \beta_{17} + 109 \beta_{16} - 62 \beta_{15} + 739 \beta_{14} + 724 \beta_{13} - 166 \beta_{12} + \cdots - 6516 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 70 \beta_{11} + 70 \beta_{10} - 2023 \beta_{9} + 2023 \beta_{8} - 265 \beta_{7} - 1248 \beta_{6} + \cdots + 60520 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 1586 \beta_{17} - 3074 \beta_{16} + 1508 \beta_{15} - 17130 \beta_{14} - 17260 \beta_{13} + \cdots + 135117 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 1808 \beta_{11} - 1808 \beta_{10} + 50928 \beta_{9} - 50928 \beta_{8} + 9544 \beta_{7} + 34060 \beta_{6} + \cdots - 1246234 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 46612 \beta_{17} + 79588 \beta_{16} - 34888 \beta_{15} + 387981 \beta_{14} + 404273 \beta_{13} + \cdots - 2853469 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 42056 \beta_{11} + 42056 \beta_{10} - 1226158 \beta_{9} + 1226158 \beta_{8} - 291588 \beta_{7} + \cdots + 26224668 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 1272753 \beta_{17} - 1982619 \beta_{16} + 808130 \beta_{15} - 8705730 \beta_{14} - 9398825 \beta_{13} + \cdots + 60969644 \beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 938338 \beta_{11} - 938338 \beta_{10} + 28813561 \beta_{9} - 28813561 \beta_{8} + 8150013 \beta_{7} + \cdots - 559318770 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 33243731 \beta_{17} + 48375185 \beta_{16} - 18987878 \beta_{15} + 194613847 \beta_{14} + \cdots - 1313394992 \beta_1 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( - 20626526 \beta_{11} + 20626526 \beta_{10} - 667225827 \beta_{9} + 667225827 \beta_{8} + \cdots + 12038142640 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( 843379386 \beta_{17} - 1165337778 \beta_{16} + 452386644 \beta_{15} - 4344633030 \beta_{14} + \cdots + 28467598013 \beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/145\mathbb{Z}\right)^\times\).

\(n\) \(31\) \(117\)
\(\chi(n)\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
59.1
− 4.76861i
− 4.60799i
− 4.22453i
− 3.20734i
− 3.05211i
− 3.00262i
− 1.13787i
− 0.965195i
− 0.579316i
0.579316i
0.965195i
1.13787i
3.00262i
3.05211i
3.20734i
4.22453i
4.60799i
4.76861i
− 4.76861i − 6.53467i −14.7397 −6.25269 + 9.26844i −31.1613 − 4.88520i 32.1389i −15.7019 44.1976 + 29.8166i
59.2 − 4.60799i 1.99045i −13.2336 −11.1417 − 0.928220i 9.17198 28.9591i 24.1162i 23.0381 −4.27723 + 51.3410i
59.3 − 4.22453i − 0.249438i −9.84666 10.9931 − 2.03769i −1.05376 − 13.3523i 7.80126i 26.9378 −8.60827 − 46.4406i
59.4 − 3.20734i 3.95870i −2.28701 1.52163 − 11.0763i 12.6969 12.9765i − 18.3235i 11.3287 −35.5255 − 4.88038i
59.5 − 3.05211i − 9.49341i −1.31541 10.5024 + 3.83407i −28.9750 − 7.00540i − 20.4022i −63.1249 11.7020 − 32.0545i
59.6 − 3.00262i 7.55695i −1.01575 1.28307 + 11.1065i 22.6907 13.2985i − 20.9711i −30.1075 33.3486 − 3.85257i
59.7 − 1.13787i 2.50246i 6.70526 −10.0649 + 4.86814i 2.84747 − 4.94229i − 16.7326i 20.7377 5.53929 + 11.4524i
59.8 − 0.965195i − 5.04196i 7.06840 4.75167 − 10.1204i −4.86647 − 3.36745i − 14.5439i 1.57864 −9.76812 − 4.58629i
59.9 − 0.579316i 6.29973i 7.66439 −5.59255 − 9.68109i 3.64953 − 34.5852i − 9.07463i −12.6866 −5.60841 + 3.23985i
59.10 0.579316i − 6.29973i 7.66439 −5.59255 + 9.68109i 3.64953 34.5852i 9.07463i −12.6866 −5.60841 − 3.23985i
59.11 0.965195i 5.04196i 7.06840 4.75167 + 10.1204i −4.86647 3.36745i 14.5439i 1.57864 −9.76812 + 4.58629i
59.12 1.13787i − 2.50246i 6.70526 −10.0649 − 4.86814i 2.84747 4.94229i 16.7326i 20.7377 5.53929 − 11.4524i
59.13 3.00262i − 7.55695i −1.01575 1.28307 − 11.1065i 22.6907 − 13.2985i 20.9711i −30.1075 33.3486 + 3.85257i
59.14 3.05211i 9.49341i −1.31541 10.5024 − 3.83407i −28.9750 7.00540i 20.4022i −63.1249 11.7020 + 32.0545i
59.15 3.20734i − 3.95870i −2.28701 1.52163 + 11.0763i 12.6969 − 12.9765i 18.3235i 11.3287 −35.5255 + 4.88038i
59.16 4.22453i 0.249438i −9.84666 10.9931 + 2.03769i −1.05376 13.3523i − 7.80126i 26.9378 −8.60827 + 46.4406i
59.17 4.60799i − 1.99045i −13.2336 −11.1417 + 0.928220i 9.17198 − 28.9591i − 24.1162i 23.0381 −4.27723 − 51.3410i
59.18 4.76861i 6.53467i −14.7397 −6.25269 − 9.26844i −31.1613 4.88520i − 32.1389i −15.7019 44.1976 − 29.8166i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 59.18
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 145.4.b.a ✓ 18
5.b even 2 1 inner 145.4.b.a ✓ 18
5.c odd 4 2 725.4.a.l 18
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
145.4.b.a ✓ 18 1.a even 1 1 trivial
145.4.b.a ✓ 18 5.b even 2 1 inner
725.4.a.l 18 5.c odd 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{18} + 93 T_{2}^{16} + 3543 T_{2}^{14} + 71263 T_{2}^{12} + 812415 T_{2}^{10} + 5221259 T_{2}^{8} + \cdots + 3013696 \) acting on \(S_{4}^{\mathrm{new}}(145, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{18} + 93 T^{16} + \cdots + 3013696 \) Copy content Toggle raw display
$3$ \( T^{18} + \cdots + 5364097600 \) Copy content Toggle raw display
$5$ \( T^{18} + \cdots + 74\!\cdots\!25 \) Copy content Toggle raw display
$7$ \( T^{18} + \cdots + 17\!\cdots\!56 \) Copy content Toggle raw display
$11$ \( (T^{9} + \cdots + 1257993949248)^{2} \) Copy content Toggle raw display
$13$ \( T^{18} + \cdots + 76\!\cdots\!64 \) Copy content Toggle raw display
$17$ \( T^{18} + \cdots + 12\!\cdots\!64 \) Copy content Toggle raw display
$19$ \( (T^{9} + \cdots + 84\!\cdots\!80)^{2} \) Copy content Toggle raw display
$23$ \( T^{18} + \cdots + 72\!\cdots\!56 \) Copy content Toggle raw display
$29$ \( (T - 29)^{18} \) Copy content Toggle raw display
$31$ \( (T^{9} + \cdots + 50\!\cdots\!32)^{2} \) Copy content Toggle raw display
$37$ \( T^{18} + \cdots + 94\!\cdots\!00 \) Copy content Toggle raw display
$41$ \( (T^{9} + \cdots - 15\!\cdots\!00)^{2} \) Copy content Toggle raw display
$43$ \( T^{18} + \cdots + 23\!\cdots\!64 \) Copy content Toggle raw display
$47$ \( T^{18} + \cdots + 32\!\cdots\!64 \) Copy content Toggle raw display
$53$ \( T^{18} + \cdots + 38\!\cdots\!36 \) Copy content Toggle raw display
$59$ \( (T^{9} + \cdots - 24\!\cdots\!00)^{2} \) Copy content Toggle raw display
$61$ \( (T^{9} + \cdots - 10\!\cdots\!36)^{2} \) Copy content Toggle raw display
$67$ \( T^{18} + \cdots + 12\!\cdots\!56 \) Copy content Toggle raw display
$71$ \( (T^{9} + \cdots + 84\!\cdots\!84)^{2} \) Copy content Toggle raw display
$73$ \( T^{18} + \cdots + 20\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( (T^{9} + \cdots + 11\!\cdots\!00)^{2} \) Copy content Toggle raw display
$83$ \( T^{18} + \cdots + 21\!\cdots\!84 \) Copy content Toggle raw display
$89$ \( (T^{9} + \cdots + 16\!\cdots\!60)^{2} \) Copy content Toggle raw display
$97$ \( T^{18} + \cdots + 49\!\cdots\!56 \) Copy content Toggle raw display
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