Properties

Label 507.4.b.f
Level $507$
Weight $4$
Character orbit 507.b
Analytic conductor $29.914$
Analytic rank $0$
Dimension $4$
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: \(4\)
Coefficient field: \(\Q(i, \sqrt{14})\)
Defining polynomial: \( x^{4} + 49 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 39)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{2} q^{2} - 3 q^{3} + ( - \beta_{3} - 7) q^{4} + ( - 2 \beta_{2} + 7 \beta_1) q^{5} - 3 \beta_{2} q^{6} + (2 \beta_{2} - \beta_1) q^{7} + ( - \beta_{2} - 13 \beta_1) q^{8} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{2} - 3 q^{3} + ( - \beta_{3} - 7) q^{4} + ( - 2 \beta_{2} + 7 \beta_1) q^{5} - 3 \beta_{2} q^{6} + (2 \beta_{2} - \beta_1) q^{7} + ( - \beta_{2} - 13 \beta_1) q^{8} + 9 q^{9} + ( - 5 \beta_{3} + 16) q^{10} + (12 \beta_{2} + 5 \beta_1) q^{11} + (3 \beta_{3} + 21) q^{12} + ( - \beta_{3} - 28) q^{14} + (6 \beta_{2} - 21 \beta_1) q^{15} + (6 \beta_{3} - 15) q^{16} + ( - 2 \beta_{3} - 82) q^{17} + 9 \beta_{2} q^{18} + (2 \beta_{2} + 11 \beta_1) q^{19} + ( - 10 \beta_{2} - 9 \beta_1) q^{20} + ( - 6 \beta_{2} + 3 \beta_1) q^{21} + ( - 17 \beta_{3} - 190) q^{22} + ( - 24 \beta_{3} - 4) q^{23} + (3 \beta_{2} + 39 \beta_1) q^{24} + (24 \beta_{3} - 75) q^{25} - 27 q^{27} + ( - 14 \beta_{2} - 21 \beta_1) q^{28} + ( - 12 \beta_{3} + 202) q^{29} + (15 \beta_{3} - 48) q^{30} + ( - 26 \beta_{2} + 23 \beta_1) q^{31} + ( - 11 \beta_{2} - 26 \beta_1) q^{32} + ( - 36 \beta_{2} - 15 \beta_1) q^{33} + ( - 86 \beta_{2} - 26 \beta_1) q^{34} + ( - 12 \beta_{3} + 56) q^{35} + ( - 9 \beta_{3} - 63) q^{36} + ( - 28 \beta_{2} + 39 \beta_1) q^{37} + ( - 13 \beta_{3} - 52) q^{38} + ( - 21 \beta_{3} + 296) q^{40} + (94 \beta_{2} + 3 \beta_1) q^{41} + (3 \beta_{3} + 84) q^{42} + ( - 26 \beta_{3} + 308) q^{43} + ( - 128 \beta_{2} - 181 \beta_1) q^{44} + ( - 18 \beta_{2} + 63 \beta_1) q^{45} + ( - 52 \beta_{2} - 312 \beta_1) q^{46} + ( - 32 \beta_{2} + 97 \beta_1) q^{47} + ( - 18 \beta_{3} + 45) q^{48} + 287 q^{49} + ( - 27 \beta_{2} + 312 \beta_1) q^{50} + (6 \beta_{3} + 246) q^{51} + ( - 60 \beta_{3} - 82) q^{53} - 27 \beta_{2} q^{54} + ( - 50 \beta_{3} + 72) q^{55} + (27 \beta_{3} + 28) q^{56} + ( - 6 \beta_{2} - 33 \beta_1) q^{57} + (178 \beta_{2} - 156 \beta_1) q^{58} + ( - 40 \beta_{2} - 15 \beta_1) q^{59} + (30 \beta_{2} + 27 \beta_1) q^{60} + (68 \beta_{3} + 314) q^{61} + (3 \beta_{3} + 344) q^{62} + (18 \beta_{2} - 9 \beta_1) q^{63} + (85 \beta_{3} + 97) q^{64} + (51 \beta_{3} + 570) q^{66} + ( - 170 \beta_{2} - 33 \beta_1) q^{67} + (96 \beta_{3} + 686) q^{68} + (72 \beta_{3} + 12) q^{69} + (32 \beta_{2} - 156 \beta_1) q^{70} + ( - 84 \beta_{2} + 149 \beta_1) q^{71} + ( - 9 \beta_{2} - 117 \beta_1) q^{72} + ( - 76 \beta_{2} + 263 \beta_1) q^{73} + ( - 11 \beta_{3} + 342) q^{74} + ( - 72 \beta_{3} + 225) q^{75} + ( - 62 \beta_{2} - 81 \beta_1) q^{76} + ( - 22 \beta_{3} - 336) q^{77} + ( - 44 \beta_{3} - 216) q^{79} + (174 \beta_{2} - 345 \beta_1) q^{80} + 81 q^{81} + ( - 97 \beta_{3} - 1416) q^{82} + (64 \beta_{2} - 379 \beta_1) q^{83} + (42 \beta_{2} + 63 \beta_1) q^{84} + (116 \beta_{2} - 494 \beta_1) q^{85} + (256 \beta_{2} - 338 \beta_1) q^{86} + (36 \beta_{3} - 606) q^{87} + (173 \beta_{3} + 762) q^{88} + (190 \beta_{2} - 335 \beta_1) q^{89} + ( - 45 \beta_{3} + 144) q^{90} + (172 \beta_{3} + 1372) q^{92} + (78 \beta_{2} - 69 \beta_1) q^{93} + ( - 65 \beta_{3} + 286) q^{94} + (12 \beta_{3} - 232) q^{95} + (33 \beta_{2} + 78 \beta_1) q^{96} + ( - 220 \beta_{2} - 23 \beta_1) q^{97} + 287 \beta_{2} q^{98} + (108 \beta_{2} + 45 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 12 q^{3} - 28 q^{4} + 36 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 12 q^{3} - 28 q^{4} + 36 q^{9} + 64 q^{10} + 84 q^{12} - 112 q^{14} - 60 q^{16} - 328 q^{17} - 760 q^{22} - 16 q^{23} - 300 q^{25} - 108 q^{27} + 808 q^{29} - 192 q^{30} + 224 q^{35} - 252 q^{36} - 208 q^{38} + 1184 q^{40} + 336 q^{42} + 1232 q^{43} + 180 q^{48} + 1148 q^{49} + 984 q^{51} - 328 q^{53} + 288 q^{55} + 112 q^{56} + 1256 q^{61} + 1376 q^{62} + 388 q^{64} + 2280 q^{66} + 2744 q^{68} + 48 q^{69} + 1368 q^{74} + 900 q^{75} - 1344 q^{77} - 864 q^{79} + 324 q^{81} - 5664 q^{82} - 2424 q^{87} + 3048 q^{88} + 576 q^{90} + 5488 q^{92} + 1144 q^{94} - 928 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 49 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 2\nu^{2} ) / 7 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} + \nu^{2} + 7\nu ) / 7 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -2\nu^{3} + 14\nu ) / 7 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{3} + 2\beta_{2} - \beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 7\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -7\beta_{3} + 14\beta_{2} - 7\beta_1 ) / 4 \) 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
1.87083 1.87083i
−1.87083 1.87083i
−1.87083 + 1.87083i
1.87083 + 1.87083i
4.74166i −3.00000 −14.4833 4.51669i 14.2250i 7.48331i 30.7417i 9.00000 −21.4166
337.2 2.74166i −3.00000 0.483315 19.4833i 8.22497i 7.48331i 23.2583i 9.00000 53.4166
337.3 2.74166i −3.00000 0.483315 19.4833i 8.22497i 7.48331i 23.2583i 9.00000 53.4166
337.4 4.74166i −3.00000 −14.4833 4.51669i 14.2250i 7.48331i 30.7417i 9.00000 −21.4166
\(n\): e.g. 2-40 or 990-1000
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.f 4
13.b even 2 1 inner 507.4.b.f 4
13.d odd 4 1 39.4.a.b 2
13.d odd 4 1 507.4.a.f 2
39.f even 4 1 117.4.a.c 2
39.f even 4 1 1521.4.a.s 2
52.f even 4 1 624.4.a.r 2
65.g odd 4 1 975.4.a.j 2
91.i even 4 1 1911.4.a.h 2
104.j odd 4 1 2496.4.a.bc 2
104.m even 4 1 2496.4.a.s 2
156.l odd 4 1 1872.4.a.t 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.4.a.b 2 13.d odd 4 1
117.4.a.c 2 39.f even 4 1
507.4.a.f 2 13.d odd 4 1
507.4.b.f 4 1.a even 1 1 trivial
507.4.b.f 4 13.b even 2 1 inner
624.4.a.r 2 52.f even 4 1
975.4.a.j 2 65.g odd 4 1
1521.4.a.s 2 39.f even 4 1
1872.4.a.t 2 156.l odd 4 1
1911.4.a.h 2 91.i even 4 1
2496.4.a.s 2 104.m even 4 1
2496.4.a.bc 2 104.j odd 4 1

Hecke kernels

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

\( T_{2}^{4} + 30T_{2}^{2} + 169 \) Copy content Toggle raw display
\( T_{5}^{4} + 400T_{5}^{2} + 7744 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 30T^{2} + 169 \) Copy content Toggle raw display
$3$ \( (T + 3)^{4} \) Copy content Toggle raw display
$5$ \( T^{4} + 400T^{2} + 7744 \) Copy content Toggle raw display
$7$ \( (T^{2} + 56)^{2} \) Copy content Toggle raw display
$11$ \( T^{4} + 5000 T^{2} + \cdots + 2347024 \) Copy content Toggle raw display
$13$ \( T^{4} \) Copy content Toggle raw display
$17$ \( (T^{2} + 164 T + 6500)^{2} \) Copy content Toggle raw display
$19$ \( T^{4} + 1264 T^{2} + 270400 \) Copy content Toggle raw display
$23$ \( (T^{2} + 8 T - 32240)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} - 404 T + 32740)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} + 19728 T^{2} + \cdots + 82156096 \) Copy content Toggle raw display
$37$ \( T^{4} + 26952 T^{2} + \cdots + 71842576 \) Copy content Toggle raw display
$41$ \( T^{4} + 267408 T^{2} + \cdots + 12928599616 \) Copy content Toggle raw display
$43$ \( (T^{2} - 616 T + 57008)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + 81160 T^{2} + \cdots + 141800464 \) Copy content Toggle raw display
$53$ \( (T^{2} + 164 T - 194876)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} + 54600 T^{2} + \cdots + 306250000 \) Copy content Toggle raw display
$61$ \( (T^{2} - 628 T - 160348)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} + 920592 T^{2} + \cdots + 121734001216 \) Copy content Toggle raw display
$71$ \( T^{4} + 289160 T^{2} + \cdots + 2807728144 \) Copy content Toggle raw display
$73$ \( T^{4} + 566728 T^{2} + \cdots + 14795316496 \) Copy content Toggle raw display
$79$ \( (T^{2} + 432 T - 61760)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 1077960 T^{2} + \cdots + 180023701264 \) Copy content Toggle raw display
$89$ \( T^{4} + 1471600 T^{2} + \cdots + 75625000000 \) Copy content Toggle raw display
$97$ \( T^{4} + 1496712 T^{2} + \cdots + 368259640336 \) Copy content Toggle raw display
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