Properties

Label 207.6.a.e
Level $207$
Weight $6$
Character orbit 207.a
Self dual yes
Analytic conductor $33.199$
Analytic rank $1$
Dimension $4$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 207 = 3^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 207.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(33.1994507013\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
Defining polynomial: \( x^{4} - 75x^{2} - 42x + 736 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 69)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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.

\(f(q)\) \(=\) \( q + (\beta_1 - 1) q^{2} + (\beta_{3} - \beta_1 + 7) q^{4} + (2 \beta_{3} - \beta_{2} - 3 \beta_1 - 5) q^{5} + ( - \beta_{3} - 4 \beta_{2} + 12 \beta_1 - 18) q^{7} + ( - 2 \beta_{3} + 4 \beta_{2} - 9 \beta_1 - 17) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_1 - 1) q^{2} + (\beta_{3} - \beta_1 + 7) q^{4} + (2 \beta_{3} - \beta_{2} - 3 \beta_1 - 5) q^{5} + ( - \beta_{3} - 4 \beta_{2} + 12 \beta_1 - 18) q^{7} + ( - 2 \beta_{3} + 4 \beta_{2} - 9 \beta_1 - 17) q^{8} + ( - 10 \beta_{3} + 10 \beta_{2} + 24 \beta_1 - 124) q^{10} + ( - 13 \beta_{3} - 3 \beta_{2} - 41 \beta_1 + 261) q^{11} + (22 \beta_{2} - 22 \beta_1 - 88) q^{13} + ( - 7 \beta_{3} + 4 \beta_{2} - 46 \beta_1 + 450) q^{14} + ( - 19 \beta_{3} - 16 \beta_{2} - 5 \beta_1 - 513) q^{16} + ( - 43 \beta_{3} + 34 \beta_{2} + 66 \beta_1 - 22) q^{17} + ( - 8 \beta_{3} - 19 \beta_{2} + 43 \beta_1 - 1605) q^{19} + (20 \beta_{3} - 28 \beta_{2} - 158 \beta_1 + 1306) q^{20} + ( - 43 \beta_{3} - 46 \beta_{2} + 44 \beta_1 - 1788) q^{22} + 529 q^{23} + (20 \beta_{3} - 10 \beta_{2} - 450 \beta_1 + 321) q^{25} + (88 \beta_{3} - 44 \beta_{2} - 22 \beta_1 - 594) q^{26} + (13 \beta_{3} + 92 \beta_{2} - 34 \beta_1 - 1566) q^{28} + (120 \beta_{3} - 10 \beta_{2} - 2 \beta_1 - 932) q^{29} + (76 \beta_{3} + 14 \beta_{2} - 770 \beta_1 + 818) q^{31} + ( - 2 \beta_{3} - 172 \beta_{2} - 577 \beta_1 + 831) q^{32} + (279 \beta_{3} - 240 \beta_{2} - 608 \beta_1 + 2940) q^{34} + ( - 90 \beta_{3} + 282 \beta_{2} - 258 \beta_1 - 230) q^{35} + ( - 33 \beta_{3} + 231 \beta_{2} - 11 \beta_1 - 4267) q^{37} + ( - 44 \beta_{3} + 6 \beta_{2} - 1790 \beta_1 + 3138) q^{38} + (2 \beta_{3} - 184 \beta_{2} + 774 \beta_1 - 3618) q^{40} + ( - 4 \beta_{3} - 300 \beta_{2} + 248 \beta_1 - 4822) q^{41} + (58 \beta_{3} + 451 \beta_{2} + 365 \beta_1 - 6207) q^{43} + (273 \beta_{3} + 16 \beta_{2} - 1302 \beta_1 - 5042) q^{44} + (529 \beta_1 - 529) q^{46} + ( - 12 \beta_{3} + 424 \beta_{2} + 348 \beta_1 - 4392) q^{47} + ( - 662 \beta_{3} + 40 \beta_{2} - 456 \beta_1 + 1677) q^{49} + ( - 520 \beta_{3} + 100 \beta_{2} + 611 \beta_1 - 17571) q^{50} + ( - 330 \beta_{3} - 264 \beta_{2} + 1386 \beta_1 + 1914) q^{52} + ( - 16 \beta_{3} + 203 \beta_{2} - 1519 \beta_1 + 6723) q^{53} + (906 \beta_{3} - 366 \beta_{2} - 646 \beta_1 - 9942) q^{55} + (637 \beta_{3} - 260 \beta_{2} + 390 \beta_1 - 13534) q^{56} + ( - 172 \beta_{3} + 500 \beta_{2} + 958 \beta_1 + 306) q^{58} + ( - 132 \beta_{3} - 288 \beta_{2} + 1792 \beta_1 + 3420) q^{59} + (233 \beta_{3} - 1087 \beta_{2} - 661 \beta_1 - 3849) q^{61} + ( - 776 \beta_{3} + 276 \beta_{2} + 2076 \beta_1 - 30284) q^{62} + ( - 827 \beta_{3} + 848 \beta_{2} + 443 \beta_1 - 7537) q^{64} + ( - 220 \beta_{3} - 902 \beta_{2} + 3542 \beta_1 - 10142) q^{65} + ( - 638 \beta_{3} + 497 \beta_{2} - 2001 \beta_1 - 2845) q^{67} + ( - 711 \beta_{3} + 508 \beta_{2} + 4572 \beta_1 - 28136) q^{68} + (1242 \beta_{3} - 924 \beta_{2} - 824 \beta_1 - 7240) q^{70} + ( - 508 \beta_{3} + 1408 \beta_{2} + 3940 \beta_1 + 14824) q^{71} + (754 \beta_{3} - 362 \beta_{2} + 1990 \beta_1 - 27872) q^{73} + (1177 \beta_{3} - 594 \beta_{2} - 4102 \beta_1 + 5598) q^{74} + ( - 1460 \beta_{3} + 420 \beta_{2} + 1076 \beta_1 - 19580) q^{76} + ( - 518 \beta_{3} - 2538 \beta_{2} + 6514 \beta_1 - 330) q^{77} + ( - 879 \beta_{3} - 116 \beta_{2} + 2484 \beta_1 - 60686) q^{79} + ( - 788 \beta_{3} + 1272 \beta_{2} + 918 \beta_1 - 10058) q^{80} + ( - 1248 \beta_{3} + 584 \beta_{2} - 5786 \beta_1 + 12162) q^{82} + (2561 \beta_{3} - 629 \beta_{2} + 2161 \beta_1 + 24283) q^{83} + ( - 826 \beta_{3} - 92 \beta_{2} + 12220 \beta_1 - 81260) q^{85} + (2562 \beta_{3} - 670 \beta_{2} - 3926 \beta_1 + 23002) q^{86} + ( - 119 \beta_{3} + 2532 \beta_{2} - 2034 \beta_1 + 11802) q^{88} + (2301 \beta_{3} + 2260 \beta_{2} - 4520 \beta_1 + 29176) q^{89} + (3410 \beta_{3} - 132 \beta_{2} - 396 \beta_1 - 73744) q^{91} + (529 \beta_{3} - 529 \beta_1 + 3703) q^{92} + (2480 \beta_{3} - 896 \beta_{2} - 3312 \beta_1 + 20632) q^{94} + ( - 3348 \beta_{3} + 2772 \beta_{2} + 3416 \beta_1 + 4372) q^{95} + (3368 \beta_{3} - 1646 \beta_{2} + 9542 \beta_1 - 12408) q^{97} + (406 \beta_{3} - 2728 \beta_{2} - 8795 \beta_1 - 16077) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{2} + 26 q^{4} - 22 q^{5} - 62 q^{7} - 72 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{2} + 26 q^{4} - 22 q^{5} - 62 q^{7} - 72 q^{8} - 496 q^{10} + 1076 q^{11} - 396 q^{13} + 1806 q^{14} - 1982 q^{16} - 70 q^{17} - 6366 q^{19} + 5240 q^{20} - 6974 q^{22} + 2116 q^{23} + 1264 q^{25} - 2464 q^{26} - 6474 q^{28} - 3948 q^{29} + 3092 q^{31} + 3672 q^{32} + 11682 q^{34} - 1304 q^{35} - 17464 q^{37} + 12628 q^{38} - 14108 q^{40} - 18680 q^{41} - 25846 q^{43} - 20746 q^{44} - 2116 q^{46} - 18392 q^{47} + 7952 q^{49} - 69444 q^{50} + 8844 q^{52} + 26518 q^{53} - 40848 q^{55} - 54890 q^{56} + 568 q^{58} + 14520 q^{59} - 13688 q^{61} - 120136 q^{62} - 30190 q^{64} - 38324 q^{65} - 11098 q^{67} - 112138 q^{68} - 29596 q^{70} + 57496 q^{71} - 112272 q^{73} + 21226 q^{74} - 76240 q^{76} + 4792 q^{77} - 240754 q^{79} - 41200 q^{80} + 49976 q^{82} + 93268 q^{83} - 323204 q^{85} + 88224 q^{86} + 42382 q^{88} + 107582 q^{89} - 301532 q^{91} + 13754 q^{92} + 79360 q^{94} + 18640 q^{95} - 53076 q^{97} - 59664 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 75x^{2} - 42x + 736 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} - \nu^{2} - 54\nu + 4 ) / 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{2} - \nu - 38 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + \beta _1 + 38 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + 4\beta_{2} + 55\beta _1 + 34 \) Copy content Toggle raw display

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−7.50608
−3.86863
3.04157
8.33314
−8.50608 0 40.3535 86.6918 0 −64.0051 −71.0553 0 −737.408
1.2 −4.86863 0 −8.29644 −66.7344 0 −185.299 196.188 0 324.905
1.3 2.04157 0 −27.8320 −42.3660 0 191.647 −122.151 0 −86.4934
1.4 7.33314 0 21.7749 0.408582 0 −4.34307 −74.9818 0 2.99619
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(23\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 207.6.a.e 4
3.b odd 2 1 69.6.a.d 4
12.b even 2 1 1104.6.a.o 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
69.6.a.d 4 3.b odd 2 1
207.6.a.e 4 1.a even 1 1 trivial
1104.6.a.o 4 12.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} + 4T_{2}^{3} - 69T_{2}^{2} - 188T_{2} + 620 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(207))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 4 T^{3} - 69 T^{2} - 188 T + 620 \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} + 22 T^{3} - 6640 T^{2} + \cdots + 100144 \) Copy content Toggle raw display
$7$ \( T^{4} + 62 T^{3} - 35668 T^{2} + \cdots - 9871616 \) Copy content Toggle raw display
$11$ \( T^{4} - 1076 T^{3} + \cdots - 45159083072 \) Copy content Toggle raw display
$13$ \( T^{4} + 396 T^{3} + \cdots + 16813724400 \) Copy content Toggle raw display
$17$ \( T^{4} + 70 T^{3} + \cdots - 95458629376 \) Copy content Toggle raw display
$19$ \( T^{4} + 6366 T^{3} + \cdots + 3908943190016 \) Copy content Toggle raw display
$23$ \( (T - 529)^{4} \) Copy content Toggle raw display
$29$ \( T^{4} + 3948 T^{3} + \cdots + 61820529282864 \) Copy content Toggle raw display
$31$ \( T^{4} + \cdots + 378047008189440 \) Copy content Toggle raw display
$37$ \( T^{4} + \cdots - 684323468629888 \) Copy content Toggle raw display
$41$ \( T^{4} + 18680 T^{3} + \cdots - 12\!\cdots\!64 \) Copy content Toggle raw display
$43$ \( T^{4} + 25846 T^{3} + \cdots + 13\!\cdots\!84 \) Copy content Toggle raw display
$47$ \( T^{4} + 18392 T^{3} + \cdots + 12\!\cdots\!92 \) Copy content Toggle raw display
$53$ \( T^{4} - 26518 T^{3} + \cdots + 40\!\cdots\!32 \) Copy content Toggle raw display
$59$ \( T^{4} - 14520 T^{3} + \cdots + 23\!\cdots\!56 \) Copy content Toggle raw display
$61$ \( T^{4} + 13688 T^{3} + \cdots + 60\!\cdots\!56 \) Copy content Toggle raw display
$67$ \( T^{4} + 11098 T^{3} + \cdots - 73\!\cdots\!28 \) Copy content Toggle raw display
$71$ \( T^{4} - 57496 T^{3} + \cdots + 12\!\cdots\!76 \) Copy content Toggle raw display
$73$ \( T^{4} + 112272 T^{3} + \cdots - 14\!\cdots\!88 \) Copy content Toggle raw display
$79$ \( T^{4} + 240754 T^{3} + \cdots + 75\!\cdots\!80 \) Copy content Toggle raw display
$83$ \( T^{4} - 93268 T^{3} + \cdots + 12\!\cdots\!52 \) Copy content Toggle raw display
$89$ \( T^{4} - 107582 T^{3} + \cdots - 75\!\cdots\!28 \) Copy content Toggle raw display
$97$ \( T^{4} + 53076 T^{3} + \cdots + 20\!\cdots\!20 \) Copy content Toggle raw display
show more
show less