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Properties

Label 507.4.b.j
Level $507$
Weight $4$
Character orbit 507.b
Analytic conductor $29.914$
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] = [507,4,Mod(337,507)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(507, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("507.337");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 507 = 3 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 507.b (of order \(2\), degree \(1\), not 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: \(29.9139683729\)
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: \( x^{18} + 97 x^{16} + 3906 x^{14} + 84743 x^{12} + 1077128 x^{10} + 8187552 x^{8} + 36483705 x^{6} + \cdots + 26460736 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 13^{10} \)
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.

\(f(q)\) \(=\) \( q + \beta_{10} q^{2} - 3 q^{3} + (\beta_{2} - 4) q^{4} + ( - \beta_{12} + 2 \beta_{10}) q^{5} - 3 \beta_{10} q^{6} + ( - \beta_{14} + \beta_{11} + \cdots - \beta_{9}) q^{7}+ \cdots + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{10} q^{2} - 3 q^{3} + (\beta_{2} - 4) q^{4} + ( - \beta_{12} + 2 \beta_{10}) q^{5} - 3 \beta_{10} q^{6} + ( - \beta_{14} + \beta_{11} + \cdots - \beta_{9}) q^{7}+ \cdots + (9 \beta_{17} + 9 \beta_{16} + \cdots - 27 \beta_{10}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q - 54 q^{3} - 64 q^{4} + 162 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 18 q - 54 q^{3} - 64 q^{4} + 162 q^{9} - 396 q^{10} + 192 q^{12} + 196 q^{14} + 64 q^{16} + 268 q^{17} + 548 q^{22} - 452 q^{23} - 1224 q^{25} - 486 q^{27} - 1094 q^{29} + 1188 q^{30} + 276 q^{35} - 576 q^{36} + 832 q^{38} + 2684 q^{40} - 588 q^{42} - 316 q^{43} - 192 q^{48} - 1284 q^{49} - 804 q^{51} + 2798 q^{53} - 2816 q^{55} + 1232 q^{56} + 4184 q^{61} + 586 q^{62} - 4962 q^{64} - 1644 q^{66} - 3158 q^{68} + 1356 q^{69} + 2074 q^{74} + 3672 q^{75} + 3372 q^{77} - 230 q^{79} + 1458 q^{81} + 10294 q^{82} + 3282 q^{87} + 968 q^{88} - 3564 q^{90} + 4174 q^{92} - 936 q^{94} + 444 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{18} + 97 x^{16} + 3906 x^{14} + 84743 x^{12} + 1077128 x^{10} + 8187552 x^{8} + 36483705 x^{6} + \cdots + 26460736 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 337699465 \nu^{16} - 32409147909 \nu^{14} - 1282598456190 \nu^{12} + \cdots - 80\!\cdots\!56 ) / 84345031640064 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 121443 \nu^{16} + 11006175 \nu^{14} + 406207162 \nu^{12} + 7852438557 \nu^{10} + \cdots + 737331060496 ) / 11028377568 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 7448055 \nu^{16} + 701380891 \nu^{14} + 27024992866 \nu^{12} + 548115320953 \nu^{10} + \cdots + 68875575658864 ) / 499083027456 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 188631741 \nu^{16} + 17499465577 \nu^{14} + 664217725590 \nu^{12} + 13287339141715 \nu^{10} + \cdots + 13\!\cdots\!76 ) / 6488079356928 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 1323696165 \nu^{16} + 123712051601 \nu^{14} + 4721212524742 \nu^{12} + \cdots + 97\!\cdots\!32 ) / 42172515820032 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 3050639663 \nu^{16} - 279441123251 \nu^{14} - 10408004776370 \nu^{12} + \cdots - 14\!\cdots\!68 ) / 84345031640064 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 3573406425 \nu^{16} + 324617832117 \nu^{14} + 11960996341246 \nu^{12} + \cdots + 13\!\cdots\!92 ) / 84345031640064 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 992878663 \nu^{16} + 91675637363 \nu^{14} + 3455510144330 \nu^{12} + 68407025915321 \nu^{10} + \cdots + 64\!\cdots\!08 ) / 21086257910016 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 3209941781 \nu^{17} - 208707089855 \nu^{15} - 31724147958522 \nu^{13} + \cdots + 53\!\cdots\!60 \nu ) / 10\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 59877421 \nu^{17} - 5495758441 \nu^{15} - 205573324326 \nu^{13} + \cdots - 172360206360464 \nu ) / 56729974209792 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 558686229419 \nu^{17} - 52805876911935 \nu^{15} + \cdots - 80\!\cdots\!24 \nu ) / 10\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 315204832455 \nu^{17} + 29976384016219 \nu^{15} + \cdots + 32\!\cdots\!52 \nu ) / 54\!\cdots\!52 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 11333387329 \nu^{17} - 1052085600443 \nu^{15} - 39989575616904 \nu^{13} + \cdots - 10\!\cdots\!80 \nu ) / 16\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 1101606689877 \nu^{17} + 100119548308961 \nu^{15} + \cdots + 56\!\cdots\!40 \nu ) / 10\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 59877421 \nu^{17} + 5495758441 \nu^{15} + 205573324326 \nu^{13} + 4029427467139 \nu^{11} + \cdots + 233817678421072 \nu ) / 4727497850816 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 694871382623 \nu^{17} - 64734895114835 \nu^{15} + \cdots - 24\!\cdots\!84 \nu ) / 54\!\cdots\!52 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 942769046151 \nu^{17} - 86802481993355 \nu^{15} + \cdots - 80\!\cdots\!00 \nu ) / 54\!\cdots\!52 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{15} + 12\beta_{10} ) / 13 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -2\beta_{8} + 2\beta_{6} + 4\beta_{5} - 2\beta_{4} + 9\beta_{2} - 143 ) / 13 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 7 \beta_{17} + \beta_{16} - 28 \beta_{15} + 9 \beta_{14} + 10 \beta_{13} + 16 \beta_{12} + \cdots - 2 \beta_{9} ) / 13 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 56 \beta_{8} - 7 \beta_{7} - 57 \beta_{6} - 104 \beta_{5} + 27 \beta_{4} + 16 \beta_{3} - 201 \beta_{2} + \cdots + 2370 ) / 13 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 202 \beta_{17} - 15 \beta_{16} + 657 \beta_{15} - 264 \beta_{14} - 239 \beta_{13} + \cdots + 185 \beta_{9} ) / 13 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 1395 \beta_{8} + 411 \beta_{7} + 1554 \beta_{6} + 2363 \beta_{5} - 255 \beta_{4} - 464 \beta_{3} + \cdots - 44810 ) / 13 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 4822 \beta_{17} + 121 \beta_{16} - 15304 \beta_{15} + 6881 \beta_{14} + 4578 \beta_{13} + \cdots - 6528 \beta_{9} ) / 13 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 32087 \beta_{8} - 14276 \beta_{7} - 40359 \beta_{6} - 52468 \beta_{5} + 459 \beta_{4} + 12180 \beta_{3} + \cdots + 896522 ) / 13 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 105570 \beta_{17} + 1662 \beta_{16} + 356170 \beta_{15} - 176667 \beta_{14} - 79473 \beta_{13} + \cdots + 186585 \beta_{9} ) / 13 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 711946 \beta_{8} + 413713 \beta_{7} + 1010363 \beta_{6} + 1163967 \beta_{5} + 69984 \beta_{4} + \cdots - 18500373 ) / 13 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 2199377 \beta_{17} - 115740 \beta_{16} - 8283196 \beta_{15} + 4502785 \beta_{14} + \cdots - 4919299 \beta_{9} ) / 13 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 1199519 \beta_{8} - 848722 \beta_{7} - 1898215 \beta_{6} - 1996255 \beta_{5} - 219890 \beta_{4} + \cdots + 29983267 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 44331911 \beta_{17} + 3986983 \beta_{16} + 192525009 \beta_{15} - 113654892 \beta_{14} + \cdots + 124653997 \beta_{9} ) / 13 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 340897078 \beta_{8} + 281393164 \beta_{7} + 592974020 \beta_{6} + 582481917 \beta_{5} + \cdots - 8342751040 ) / 13 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 871319300 \beta_{17} - 115322050 \beta_{16} - 4473577903 \beta_{15} + 2837436494 \beta_{14} + \cdots - 3087034420 \beta_{9} ) / 13 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( ( 7476471422 \beta_{8} - 6987092704 \beta_{7} - 14096453402 \beta_{6} - 13163855848 \beta_{5} + \cdots + 180847969527 ) / 13 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( ( 16749505559 \beta_{17} + 3103545651 \beta_{16} + 103949299448 \beta_{15} - 70086310441 \beta_{14} + \cdots + 75325281294 \beta_{9} ) / 13 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/507\mathbb{Z}\right)^\times\).

\(n\) \(170\) \(340\)
\(\chi(n)\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
337.1
− 4.48584i
− 3.76649i
− 4.82618i
− 2.37739i
− 4.23649i
− 1.73419i
− 2.86460i
− 0.614643i
− 2.05129i
2.05129i
0.614643i
2.86460i
1.73419i
4.23649i
2.37739i
4.82618i
3.76649i
4.48584i
− 5.48584i −3.00000 −22.0945 − 13.3185i 16.4575i 21.4234i 77.3200i 9.00000 −73.0635
337.2 − 4.76649i −3.00000 −14.7194 − 18.8390i 14.2995i − 23.8593i 32.0282i 9.00000 −89.7961
337.3 − 3.82618i −3.00000 −6.63963 0.275426i 11.4785i − 0.0981245i − 5.20502i 9.00000 1.05383
337.4 − 3.37739i −3.00000 −3.40677 15.7127i 10.1322i − 17.1681i − 15.5131i 9.00000 53.0679
337.5 − 3.23649i −3.00000 −2.47490 − 13.5815i 9.70948i − 1.42933i − 17.8820i 9.00000 −43.9566
337.6 − 2.73419i −3.00000 0.524213 − 21.1246i 8.20257i 25.8618i − 23.3068i 9.00000 −57.7586
337.7 − 1.86460i −3.00000 4.52327 − 2.36060i 5.59380i 4.86461i − 23.3509i 9.00000 −4.40158
337.8 − 1.61464i −3.00000 5.39293 − 1.20859i 4.84393i 28.2769i − 21.6248i 9.00000 −1.95145
337.9 − 1.05129i −3.00000 6.89480 17.8886i 3.15386i 30.1975i − 15.6587i 9.00000 18.8061
337.10 1.05129i −3.00000 6.89480 − 17.8886i − 3.15386i − 30.1975i 15.6587i 9.00000 18.8061
337.11 1.61464i −3.00000 5.39293 1.20859i − 4.84393i − 28.2769i 21.6248i 9.00000 −1.95145
337.12 1.86460i −3.00000 4.52327 2.36060i − 5.59380i − 4.86461i 23.3509i 9.00000 −4.40158
337.13 2.73419i −3.00000 0.524213 21.1246i − 8.20257i − 25.8618i 23.3068i 9.00000 −57.7586
337.14 3.23649i −3.00000 −2.47490 13.5815i − 9.70948i 1.42933i 17.8820i 9.00000 −43.9566
337.15 3.37739i −3.00000 −3.40677 − 15.7127i − 10.1322i 17.1681i 15.5131i 9.00000 53.0679
337.16 3.82618i −3.00000 −6.63963 − 0.275426i − 11.4785i 0.0981245i 5.20502i 9.00000 1.05383
337.17 4.76649i −3.00000 −14.7194 18.8390i − 14.2995i 23.8593i − 32.0282i 9.00000 −89.7961
337.18 5.48584i −3.00000 −22.0945 13.3185i − 16.4575i − 21.4234i − 77.3200i 9.00000 −73.0635
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 337.18
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 507.4.b.j 18
13.b even 2 1 inner 507.4.b.j 18
13.d odd 4 1 507.4.a.n ✓ 9
13.d odd 4 1 507.4.a.q yes 9
39.f even 4 1 1521.4.a.be 9
39.f even 4 1 1521.4.a.bj 9
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
507.4.a.n ✓ 9 13.d odd 4 1
507.4.a.q yes 9 13.d odd 4 1
507.4.b.j 18 1.a even 1 1 trivial
507.4.b.j 18 13.b even 2 1 inner
1521.4.a.be 9 39.f even 4 1
1521.4.a.bj 9 39.f even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(507, [\chi])\):

\( T_{2}^{18} + 104 T_{2}^{16} + 4432 T_{2}^{14} + 101051 T_{2}^{12} + 1349342 T_{2}^{10} + 10831309 T_{2}^{8} + \cdots + 89567296 \) Copy content Toggle raw display
\( T_{5}^{18} + 1737 T_{5}^{16} + 1231596 T_{5}^{14} + 456756349 T_{5}^{12} + 93726274635 T_{5}^{10} + \cdots + 252800110296721 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{18} + 104 T^{16} + \cdots + 89567296 \) Copy content Toggle raw display
$3$ \( (T + 3)^{18} \) Copy content Toggle raw display
$5$ \( T^{18} + \cdots + 252800110296721 \) Copy content Toggle raw display
$7$ \( T^{18} + \cdots + 17\!\cdots\!41 \) Copy content Toggle raw display
$11$ \( T^{18} + \cdots + 96\!\cdots\!89 \) Copy content Toggle raw display
$13$ \( T^{18} \) Copy content Toggle raw display
$17$ \( (T^{9} + \cdots - 7231752990904)^{2} \) Copy content Toggle raw display
$19$ \( T^{18} + \cdots + 68\!\cdots\!24 \) Copy content Toggle raw display
$23$ \( (T^{9} + \cdots - 91\!\cdots\!88)^{2} \) Copy content Toggle raw display
$29$ \( (T^{9} + \cdots - 12\!\cdots\!83)^{2} \) Copy content Toggle raw display
$31$ \( T^{18} + \cdots + 46\!\cdots\!21 \) Copy content Toggle raw display
$37$ \( T^{18} + \cdots + 27\!\cdots\!16 \) Copy content Toggle raw display
$41$ \( T^{18} + \cdots + 27\!\cdots\!04 \) Copy content Toggle raw display
$43$ \( (T^{9} + \cdots + 14\!\cdots\!52)^{2} \) Copy content Toggle raw display
$47$ \( T^{18} + \cdots + 72\!\cdots\!84 \) Copy content Toggle raw display
$53$ \( (T^{9} + \cdots + 26\!\cdots\!69)^{2} \) Copy content Toggle raw display
$59$ \( T^{18} + \cdots + 42\!\cdots\!84 \) Copy content Toggle raw display
$61$ \( (T^{9} + \cdots - 18\!\cdots\!04)^{2} \) Copy content Toggle raw display
$67$ \( T^{18} + \cdots + 15\!\cdots\!96 \) Copy content Toggle raw display
$71$ \( T^{18} + \cdots + 23\!\cdots\!56 \) Copy content Toggle raw display
$73$ \( T^{18} + \cdots + 25\!\cdots\!29 \) Copy content Toggle raw display
$79$ \( (T^{9} + \cdots - 63\!\cdots\!89)^{2} \) Copy content Toggle raw display
$83$ \( T^{18} + \cdots + 30\!\cdots\!01 \) Copy content Toggle raw display
$89$ \( T^{18} + \cdots + 10\!\cdots\!24 \) Copy content Toggle raw display
$97$ \( T^{18} + \cdots + 65\!\cdots\!61 \) Copy content Toggle raw display
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