Properties

Label 207.6.a.a
Level $207$
Weight $6$
Character orbit 207.a
Self dual yes
Analytic conductor $33.199$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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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: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{29}) \)
Defining polynomial: \( x^{2} - x - 7 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
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 \(\beta = \sqrt{29}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 2 \beta q^{2} + 84 q^{4} + ( - \beta - 47) q^{5} + ( - 11 \beta - 59) q^{7} - 104 \beta q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q - 2 \beta q^{2} + 84 q^{4} + ( - \beta - 47) q^{5} + ( - 11 \beta - 59) q^{7} - 104 \beta q^{8} + (94 \beta + 58) q^{10} + ( - 108 \beta - 160) q^{11} + ( - 74 \beta - 144) q^{13} + (118 \beta + 638) q^{14} + 3344 q^{16} + ( - 13 \beta + 905) q^{17} + ( - 311 \beta + 365) q^{19} + ( - 84 \beta - 3948) q^{20} + (320 \beta + 6264) q^{22} - 529 q^{23} + (94 \beta - 887) q^{25} + (288 \beta + 4292) q^{26} + ( - 924 \beta - 4956) q^{28} + ( - 266 \beta - 4104) q^{29} + ( - 1514 \beta + 886) q^{31} - 3360 \beta q^{32} + ( - 1810 \beta + 754) q^{34} + (576 \beta + 3092) q^{35} + (206 \beta - 11556) q^{37} + ( - 730 \beta + 18038) q^{38} + (4888 \beta + 3016) q^{40} + (1372 \beta - 2758) q^{41} + ( - 2667 \beta + 5161) q^{43} + ( - 9072 \beta - 13440) q^{44} + 1058 \beta q^{46} + (1152 \beta - 21476) q^{47} + (1298 \beta - 9817) q^{49} + (1774 \beta - 5452) q^{50} + ( - 6216 \beta - 12096) q^{52} + (4369 \beta + 12675) q^{53} + (5236 \beta + 10652) q^{55} + (6136 \beta + 33176) q^{56} + (8208 \beta + 15428) q^{58} + (508 \beta - 9172) q^{59} + ( - 3946 \beta + 18612) q^{61} + ( - 1772 \beta + 87812) q^{62} + 87872 q^{64} + (3622 \beta + 8914) q^{65} + (4607 \beta - 3741) q^{67} + ( - 1092 \beta + 76020) q^{68} + ( - 6184 \beta - 33408) q^{70} + ( - 2088 \beta - 63424) q^{71} + (2212 \beta + 68830) q^{73} + (23112 \beta - 11948) q^{74} + ( - 26124 \beta + 30660) q^{76} + (8132 \beta + 43892) q^{77} + (1479 \beta + 31143) q^{79} + ( - 3344 \beta - 157168) q^{80} + (5516 \beta - 79576) q^{82} + ( - 11124 \beta - 41560) q^{83} + ( - 294 \beta - 42158) q^{85} + ( - 10322 \beta + 154686) q^{86} + (16640 \beta + 325728) q^{88} + (2317 \beta - 34885) q^{89} + (5950 \beta + 32102) q^{91} - 44436 q^{92} + (42952 \beta - 66816) q^{94} + (14252 \beta - 8136) q^{95} + (10274 \beta - 85052) q^{97} + (19634 \beta - 75284) q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 168 q^{4} - 94 q^{5} - 118 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 168 q^{4} - 94 q^{5} - 118 q^{7} + 116 q^{10} - 320 q^{11} - 288 q^{13} + 1276 q^{14} + 6688 q^{16} + 1810 q^{17} + 730 q^{19} - 7896 q^{20} + 12528 q^{22} - 1058 q^{23} - 1774 q^{25} + 8584 q^{26} - 9912 q^{28} - 8208 q^{29} + 1772 q^{31} + 1508 q^{34} + 6184 q^{35} - 23112 q^{37} + 36076 q^{38} + 6032 q^{40} - 5516 q^{41} + 10322 q^{43} - 26880 q^{44} - 42952 q^{47} - 19634 q^{49} - 10904 q^{50} - 24192 q^{52} + 25350 q^{53} + 21304 q^{55} + 66352 q^{56} + 30856 q^{58} - 18344 q^{59} + 37224 q^{61} + 175624 q^{62} + 175744 q^{64} + 17828 q^{65} - 7482 q^{67} + 152040 q^{68} - 66816 q^{70} - 126848 q^{71} + 137660 q^{73} - 23896 q^{74} + 61320 q^{76} + 87784 q^{77} + 62286 q^{79} - 314336 q^{80} - 159152 q^{82} - 83120 q^{83} - 84316 q^{85} + 309372 q^{86} + 651456 q^{88} - 69770 q^{89} + 64204 q^{91} - 88872 q^{92} - 133632 q^{94} - 16272 q^{95} - 170104 q^{97} - 150568 q^{98}+O(q^{100}) \) 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
3.19258
−2.19258
−10.7703 0 84.0000 −52.3852 0 −118.237 −560.057 0 564.205
1.2 10.7703 0 84.0000 −41.6148 0 0.236813 560.057 0 −448.205
\(n\): 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.a 2
3.b odd 2 1 69.6.a.a 2
12.b even 2 1 1104.6.a.h 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
69.6.a.a 2 3.b odd 2 1
207.6.a.a 2 1.a even 1 1 trivial
1104.6.a.h 2 12.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} - 116 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(207))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - 116 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + 94T + 2180 \) Copy content Toggle raw display
$7$ \( T^{2} + 118T - 28 \) Copy content Toggle raw display
$11$ \( T^{2} + 320T - 312656 \) Copy content Toggle raw display
$13$ \( T^{2} + 288T - 138068 \) Copy content Toggle raw display
$17$ \( T^{2} - 1810 T + 814124 \) Copy content Toggle raw display
$19$ \( T^{2} - 730 T - 2671684 \) Copy content Toggle raw display
$23$ \( (T + 529)^{2} \) Copy content Toggle raw display
$29$ \( T^{2} + 8208 T + 14790892 \) Copy content Toggle raw display
$31$ \( T^{2} - 1772 T - 65688688 \) Copy content Toggle raw display
$37$ \( T^{2} + 23112 T + 132310492 \) Copy content Toggle raw display
$41$ \( T^{2} + 5516 T - 46982572 \) Copy content Toggle raw display
$43$ \( T^{2} - 10322 T - 179637860 \) Copy content Toggle raw display
$47$ \( T^{2} + 42952 T + 422732560 \) Copy content Toggle raw display
$53$ \( T^{2} - 25350 T - 392901044 \) Copy content Toggle raw display
$59$ \( T^{2} + 18344 T + 76641728 \) Copy content Toggle raw display
$61$ \( T^{2} - 37224 T - 105150020 \) Copy content Toggle raw display
$67$ \( T^{2} + 7482 T - 601513940 \) Copy content Toggle raw display
$71$ \( T^{2} + 126848 T + 3896171200 \) Copy content Toggle raw display
$73$ \( T^{2} - 137660 T + 4595673524 \) Copy content Toggle raw display
$79$ \( T^{2} - 62286 T + 906450660 \) Copy content Toggle raw display
$83$ \( T^{2} + 83120 T - 1861324304 \) Copy content Toggle raw display
$89$ \( T^{2} + 69770 T + 1061277044 \) Copy content Toggle raw display
$97$ \( T^{2} + 170104 T + 4172745500 \) Copy content Toggle raw display
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