Properties

Label 507.4.b.k
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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Newspace parameters

Level: \( N \) \(=\) \( 507 = 3 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 507.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(29.9139683729\)
Analytic rank: \(0\)
Dimension: \(18\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} + \cdots)\)
Defining polynomial: \( x^{18} + 112 x^{16} + 5026 x^{14} + 114847 x^{12} + 1397921 x^{10} + 8545747 x^{8} + 21033277 x^{6} + 6703200 x^{4} + 137781 x^{2} + 729 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 13^{8} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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.

\(f(q)\) \(=\) \( q - \beta_{9} q^{2} + 3 q^{3} + (\beta_{2} - 5) q^{4} - \beta_{15} q^{5} - 3 \beta_{9} q^{6} + (\beta_{17} - \beta_{16} - \beta_{15} + \beta_{9}) q^{7} + ( - \beta_{16} - 3 \beta_{15} + \beta_{14} - 2 \beta_{13} + \beta_{12} - \beta_{11} + 4 \beta_{9}) q^{8} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{9} q^{2} + 3 q^{3} + (\beta_{2} - 5) q^{4} - \beta_{15} q^{5} - 3 \beta_{9} q^{6} + (\beta_{17} - \beta_{16} - \beta_{15} + \beta_{9}) q^{7} + ( - \beta_{16} - 3 \beta_{15} + \beta_{14} - 2 \beta_{13} + \beta_{12} - \beta_{11} + 4 \beta_{9}) q^{8} + 9 q^{9} + ( - \beta_{8} - \beta_{4} - \beta_{3} - 2 \beta_{2} - \beta_1 + 6) q^{10} + (\beta_{15} + \beta_{14} + \beta_{11} + \beta_{10} - 5 \beta_{9}) q^{11} + (3 \beta_{2} - 15) q^{12} + (2 \beta_{8} - 3 \beta_{7} - \beta_{6} + 2 \beta_{5} - \beta_{4} + 2 \beta_{3} - 2 \beta_{2} + \cdots + 19) q^{14}+ \cdots + (9 \beta_{15} + 9 \beta_{14} + 9 \beta_{11} + 9 \beta_{10} - 45 \beta_{9}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q + 54 q^{3} - 88 q^{4} + 162 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 18 q + 54 q^{3} - 88 q^{4} + 162 q^{9} + 108 q^{10} - 264 q^{12} + 316 q^{14} + 432 q^{16} - 356 q^{17} - 1260 q^{22} - 300 q^{23} + 40 q^{25} + 486 q^{27} - 194 q^{29} + 324 q^{30} - 836 q^{35} - 792 q^{36} + 1320 q^{38} - 3012 q^{40} + 948 q^{42} - 484 q^{43} + 1296 q^{48} + 76 q^{49} - 1068 q^{51} - 302 q^{53} + 4128 q^{55} - 4552 q^{56} - 2680 q^{61} - 694 q^{62} - 1786 q^{64} - 3780 q^{66} + 5570 q^{68} - 900 q^{69} - 2382 q^{74} + 120 q^{75} + 4284 q^{77} - 3182 q^{79} + 1458 q^{81} - 3034 q^{82} - 582 q^{87} + 7432 q^{88} + 972 q^{90} + 1030 q^{92} - 1384 q^{94} - 8316 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{18} + 112 x^{16} + 5026 x^{14} + 114847 x^{12} + 1397921 x^{10} + 8545747 x^{8} + 21033277 x^{6} + 6703200 x^{4} + 137781 x^{2} + 729 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 16\!\cdots\!77 \nu^{16} + \cdots + 10\!\cdots\!68 ) / 30\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 17\!\cdots\!82 \nu^{16} + \cdots - 86\!\cdots\!26 ) / 30\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 20\!\cdots\!10 \nu^{16} + \cdots - 11\!\cdots\!75 ) / 30\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 10\!\cdots\!10 \nu^{16} + \cdots + 64\!\cdots\!52 ) / 15\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 98034109972 \nu^{16} + 10978375356709 \nu^{14} + 492560731054663 \nu^{12} + \cdots + 60\!\cdots\!96 ) / 50991629993928 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 79\!\cdots\!43 \nu^{16} + \cdots + 55\!\cdots\!28 ) / 30\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 12\!\cdots\!58 \nu^{16} + \cdots + 83\!\cdots\!89 ) / 33\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 13\!\cdots\!97 \nu^{16} + \cdots + 91\!\cdots\!10 ) / 30\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 35\!\cdots\!60 \nu^{17} + \cdots - 24\!\cdots\!24 \nu ) / 11\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 98\!\cdots\!55 \nu^{17} + \cdots + 47\!\cdots\!74 \nu ) / 11\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 15\!\cdots\!78 \nu^{17} + \cdots - 99\!\cdots\!97 \nu ) / 11\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 53\!\cdots\!73 \nu^{17} + \cdots - 44\!\cdots\!99 \nu ) / 39\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 19\!\cdots\!02 \nu^{17} + \cdots - 14\!\cdots\!49 \nu ) / 13\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 18\!\cdots\!32 \nu^{17} + \cdots - 11\!\cdots\!73 \nu ) / 11\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 21\!\cdots\!56 \nu^{17} + \cdots - 15\!\cdots\!25 \nu ) / 11\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 94\!\cdots\!88 \nu^{17} + \cdots + 63\!\cdots\!22 \nu ) / 39\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 32\!\cdots\!29 \nu^{17} + \cdots + 21\!\cdots\!19 \nu ) / 11\!\cdots\!48 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -\beta_{16} + \beta_{12} - \beta_{11} - \beta_{10} - 11\beta_{9} ) / 13 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 4\beta_{8} - 2\beta_{7} + 3\beta_{5} - 4\beta_{4} + 4\beta_{3} + 13\beta_{2} - 6\beta _1 - 164 ) / 13 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 12 \beta_{17} + 38 \beta_{16} + 5 \beta_{15} + \beta_{14} + \beta_{13} - 28 \beta_{12} + 37 \beta_{11} + 36 \beta_{10} + 218 \beta_{9} ) / 13 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 133 \beta_{8} + 86 \beta_{7} + 25 \beta_{6} - 119 \beta_{5} + 112 \beta_{4} - 134 \beta_{3} - 311 \beta_{2} + 172 \beta _1 + 3635 ) / 13 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 471 \beta_{17} - 1120 \beta_{16} - 108 \beta_{15} + 29 \beta_{14} - 127 \beta_{13} + 718 \beta_{12} - 1029 \beta_{11} - 987 \beta_{10} - 4973 \beta_{9} ) / 13 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 3951 \beta_{8} - 2869 \beta_{7} - 881 \beta_{6} + 3542 \beta_{5} - 2825 \beta_{4} + 4097 \beta_{3} + 7797 \beta_{2} - 4414 \beta _1 - 88251 ) / 13 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 15422 \beta_{17} + 31613 \beta_{16} + 830 \beta_{15} - 1824 \beta_{14} + 5066 \beta_{13} - 17616 \beta_{12} + 27096 \beta_{11} + 25574 \beta_{10} + 122423 \beta_{9} ) / 13 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 112659 \beta_{8} + 87424 \beta_{7} + 26613 \beta_{6} - 101131 \beta_{5} + 69455 \beta_{4} - 121120 \beta_{3} - 203673 \beta_{2} + 110654 \beta _1 + 2241983 ) / 13 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 472480 \beta_{17} - 880613 \beta_{16} + 49184 \beta_{15} + 66984 \beta_{14} - 164273 \beta_{13} + 422747 \beta_{12} - 715006 \beta_{11} - 657831 \beta_{10} - 3144277 \beta_{9} ) / 13 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 3151008 \beta_{8} - 2549244 \beta_{7} - 780149 \beta_{6} + 2870585 \beta_{5} - 1683812 \beta_{4} + 3534216 \beta_{3} + 5471940 \beta_{2} - 2772413 \beta _1 - 58513476 ) / 13 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 13985706 \beta_{17} + 24482374 \beta_{16} - 3433113 \beta_{15} - 2071596 \beta_{14} + 4997674 \beta_{13} - 10008957 \beta_{12} + 19183436 \beta_{11} + 17030993 \beta_{10} + 82894537 \beta_{9} ) / 13 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 6717553 \beta_{8} + 5587839 \beta_{7} + 1747024 \beta_{6} - 6267832 \beta_{5} + 3091124 \beta_{4} - 7887469 \beta_{3} - 11521438 \beta_{2} + 5376733 \beta _1 + 119635062 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 405866305 \beta_{17} - 682002933 \beta_{16} + 151906638 \beta_{15} + 58422208 \beta_{14} - 148945251 \beta_{13} + 234290681 \beta_{12} - 523892539 \beta_{11} + \cdots - 2224629301 \beta_{9} ) / 13 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 2411900560 \beta_{8} - 2044996748 \beta_{7} - 660895040 \beta_{6} + 2314331308 \beta_{5} - 938935300 \beta_{4} + 2966243492 \beta_{3} + 4152803408 \beta_{2} + \cdots - 41900390969 ) / 13 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 11641827508 \beta_{17} + 19058011049 \beta_{16} - 5749856744 \beta_{15} - 1543210564 \beta_{14} + 4412106180 \beta_{13} - 5414805541 \beta_{12} + \cdots + 60489950895 \beta_{9} ) / 13 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( ( - 66610447868 \beta_{8} + 57224896958 \beta_{7} + 19271064448 \beta_{6} - 65768623299 \beta_{5} + 21293864236 \beta_{4} - 85666003544 \beta_{3} + \cdots + 1140922643492 ) / 13 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( ( 331693568820 \beta_{17} - 534304309710 \beta_{16} + 201173682407 \beta_{15} + 38458587547 \beta_{14} - 130512482617 \beta_{13} + 123072569376 \beta_{12} + \cdots - 1661682958306 \beta_{9} ) / 13 \) Copy content Toggle raw display

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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
337.1
3.27560i
5.39246i
4.14324i
4.83218i
2.37150i
5.06791i
0.100291i
0.107680i
0.588238i
0.588238i
0.107680i
0.100291i
5.06791i
2.37150i
4.83218i
4.14324i
5.39246i
3.27560i
5.52257i 3.00000 −22.4988 6.08065i 16.5677i 20.2718i 80.0709i 9.00000 33.5808
337.2 4.83750i 3.00000 −15.4014 21.1983i 14.5125i 16.2806i 35.8043i 9.00000 102.547
337.3 4.69820i 3.00000 −14.0731 4.47249i 14.0946i 27.2096i 28.5326i 9.00000 −21.0127
337.4 4.03025i 3.00000 −8.24289 8.08864i 12.0907i 5.95078i 0.978887i 9.00000 −32.5992
337.5 3.17344i 3.00000 −2.07074 6.74147i 9.52033i 14.1726i 18.8162i 9.00000 21.3937
337.6 2.82093i 3.00000 0.0423641 3.41089i 8.46278i 13.3442i 22.6869i 9.00000 −9.62187
337.7 2.34727i 3.00000 2.49032 15.3991i 7.04181i 10.1317i 24.6236i 9.00000 −36.1458
337.8 0.447278i 3.00000 7.79994 1.93073i 1.34183i 8.14537i 7.06697i 9.00000 −0.863573
337.9 0.213700i 3.00000 7.95433 15.3391i 0.641100i 32.3928i 3.40944i 9.00000 −3.27797
337.10 0.213700i 3.00000 7.95433 15.3391i 0.641100i 32.3928i 3.40944i 9.00000 −3.27797
337.11 0.447278i 3.00000 7.79994 1.93073i 1.34183i 8.14537i 7.06697i 9.00000 −0.863573
337.12 2.34727i 3.00000 2.49032 15.3991i 7.04181i 10.1317i 24.6236i 9.00000 −36.1458
337.13 2.82093i 3.00000 0.0423641 3.41089i 8.46278i 13.3442i 22.6869i 9.00000 −9.62187
337.14 3.17344i 3.00000 −2.07074 6.74147i 9.52033i 14.1726i 18.8162i 9.00000 21.3937
337.15 4.03025i 3.00000 −8.24289 8.08864i 12.0907i 5.95078i 0.978887i 9.00000 −32.5992
337.16 4.69820i 3.00000 −14.0731 4.47249i 14.0946i 27.2096i 28.5326i 9.00000 −21.0127
337.17 4.83750i 3.00000 −15.4014 21.1983i 14.5125i 16.2806i 35.8043i 9.00000 102.547
337.18 5.52257i 3.00000 −22.4988 6.08065i 16.5677i 20.2718i 80.0709i 9.00000 33.5808
\(n\): 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.k 18
13.b even 2 1 inner 507.4.b.k 18
13.d odd 4 1 507.4.a.o 9
13.d odd 4 1 507.4.a.p yes 9
39.f even 4 1 1521.4.a.bf 9
39.f even 4 1 1521.4.a.bi 9
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
507.4.a.o 9 13.d odd 4 1
507.4.a.p yes 9 13.d odd 4 1
507.4.b.k 18 1.a even 1 1 trivial
507.4.b.k 18 13.b even 2 1 inner
1521.4.a.bf 9 39.f even 4 1
1521.4.a.bi 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 \(S_{4}^{\mathrm{new}}(507, [\chi])\):

\( T_{2}^{18} + 116 T_{2}^{16} + 5516 T_{2}^{14} + 138863 T_{2}^{12} + 1992090 T_{2}^{10} + 16267053 T_{2}^{8} + 70428381 T_{2}^{6} + 129478420 T_{2}^{4} + 28371296 T_{2}^{2} + 1032256 \) Copy content Toggle raw display
\( T_{5}^{18} + 1105 T_{5}^{16} + 449620 T_{5}^{14} + 86264149 T_{5}^{12} + 8372407387 T_{5}^{10} + 433913382945 T_{5}^{8} + 12132347525883 T_{5}^{6} + 174948957367794 T_{5}^{4} + \cdots + 23\!\cdots\!01 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{18} + 116 T^{16} + 5516 T^{14} + \cdots + 1032256 \) Copy content Toggle raw display
$3$ \( (T - 3)^{18} \) Copy content Toggle raw display
$5$ \( T^{18} + 1105 T^{16} + \cdots + 23\!\cdots\!01 \) Copy content Toggle raw display
$7$ \( T^{18} + 3049 T^{16} + \cdots + 72\!\cdots\!21 \) Copy content Toggle raw display
$11$ \( T^{18} + 9999 T^{16} + \cdots + 16\!\cdots\!21 \) Copy content Toggle raw display
$13$ \( T^{18} \) Copy content Toggle raw display
$17$ \( (T^{9} + 178 T^{8} + \cdots + 25\!\cdots\!76)^{2} \) Copy content Toggle raw display
$19$ \( T^{18} + 80158 T^{16} + \cdots + 30\!\cdots\!76 \) Copy content Toggle raw display
$23$ \( (T^{9} + 150 T^{8} + \cdots - 60\!\cdots\!44)^{2} \) Copy content Toggle raw display
$29$ \( (T^{9} + 97 T^{8} + \cdots + 34\!\cdots\!27)^{2} \) Copy content Toggle raw display
$31$ \( T^{18} + 293897 T^{16} + \cdots + 16\!\cdots\!09 \) Copy content Toggle raw display
$37$ \( T^{18} + 754028 T^{16} + \cdots + 64\!\cdots\!36 \) Copy content Toggle raw display
$41$ \( T^{18} + 724066 T^{16} + \cdots + 13\!\cdots\!24 \) Copy content Toggle raw display
$43$ \( (T^{9} + 242 T^{8} + \cdots - 31\!\cdots\!88)^{2} \) Copy content Toggle raw display
$47$ \( T^{18} + 1044094 T^{16} + \cdots + 14\!\cdots\!44 \) Copy content Toggle raw display
$53$ \( (T^{9} + 151 T^{8} + \cdots + 65\!\cdots\!51)^{2} \) Copy content Toggle raw display
$59$ \( T^{18} + 1229153 T^{16} + \cdots + 21\!\cdots\!76 \) Copy content Toggle raw display
$61$ \( (T^{9} + 1340 T^{8} + \cdots - 61\!\cdots\!68)^{2} \) Copy content Toggle raw display
$67$ \( T^{18} + 3423458 T^{16} + \cdots + 66\!\cdots\!64 \) Copy content Toggle raw display
$71$ \( T^{18} + 2022260 T^{16} + \cdots + 51\!\cdots\!44 \) Copy content Toggle raw display
$73$ \( T^{18} + 2980521 T^{16} + \cdots + 37\!\cdots\!29 \) Copy content Toggle raw display
$79$ \( (T^{9} + 1591 T^{8} + \cdots + 11\!\cdots\!83)^{2} \) Copy content Toggle raw display
$83$ \( T^{18} + 6278601 T^{16} + \cdots + 40\!\cdots\!09 \) Copy content Toggle raw display
$89$ \( T^{18} + 8842714 T^{16} + \cdots + 30\!\cdots\!84 \) Copy content Toggle raw display
$97$ \( T^{18} + 8699011 T^{16} + \cdots + 99\!\cdots\!21 \) Copy content Toggle raw display
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