# 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 $1$

# Related objects

## 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: $$x^{2} - x - 7$$ x^2 - x - 7 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.

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

## 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: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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 $$S_{6}^{\mathrm{new}}(\Gamma_0(207))$$.

## Hecke characteristic polynomials

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