# Properties

 Label 207.6.a.b Level $207$ Weight $6$ Character orbit 207.a Self dual yes Analytic conductor $33.199$ Analytic rank $1$ Dimension $3$ 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: $$1$$ Dimension: $$3$$ Coefficient field: 3.3.7925.1 Defining polynomial: $$x^{3} - x^{2} - 13x + 12$$ x^3 - x^2 - 13*x + 12 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 23) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q + ( - \beta_1 + 1) q^{2} + (4 \beta_{2} - 6 \beta_1 + 2) q^{4} + ( - 2 \beta_{2} + 3 \beta_1 + 21) q^{5} + ( - 4 \beta_{2} + 17 \beta_1 - 87) q^{7} + (16 \beta_{2} - 8 \beta_1 + 112) q^{8}+O(q^{10})$$ q + (-b1 + 1) * q^2 + (4*b2 - 6*b1 + 2) * q^4 + (-2*b2 + 3*b1 + 21) * q^5 + (-4*b2 + 17*b1 - 87) * q^7 + (16*b2 - 8*b1 + 112) * q^8 $$q + ( - \beta_1 + 1) q^{2} + (4 \beta_{2} - 6 \beta_1 + 2) q^{4} + ( - 2 \beta_{2} + 3 \beta_1 + 21) q^{5} + ( - 4 \beta_{2} + 17 \beta_1 - 87) q^{7} + (16 \beta_{2} - 8 \beta_1 + 112) q^{8} + ( - 8 \beta_{2} - 2 \beta_1 - 50) q^{10} + ( - 28 \beta_{2} - 3 \beta_1 - 37) q^{11} + ( - 99 \beta_{2} + 45 \beta_1 - 324) q^{13} + ( - 60 \beta_{2} + 180 \beta_1 - 592) q^{14} + ( - 128 \beta_{2} + 8 \beta_1 + 88) q^{16} + (146 \beta_{2} + 57 \beta_1 + 269) q^{17} + (200 \beta_{2} + 40 \beta_1 + 498) q^{19} + (88 \beta_{2} - 40 \beta_1 - 544) q^{20} + (68 \beta_{2} + 78 \beta_1 + 454) q^{22} + 529 q^{23} + ( - 88 \beta_{2} + 86 \beta_1 - 2391) q^{25} + (18 \beta_{2} + 747 \beta_1 - 423) q^{26} + ( - 472 \beta_{2} + 1068 \beta_1 - 2908) q^{28} + ( - 523 \beta_{2} + 745 \beta_1 + 704) q^{29} + (479 \beta_{2} - 914 \beta_1 - 1471) q^{31} + ( - 288 \beta_{2} + 464 \beta_1 - 1968) q^{32} + ( - 520 \beta_{2} - 276 \beta_1 - 3656) q^{34} + (148 \beta_{2} - 160 \beta_1 - 460) q^{35} + (1662 \beta_{2} - 1117 \beta_1 + 2053) q^{37} + ( - 560 \beta_{2} - 698 \beta_1 - 3622) q^{38} + (240 \beta_{2} + 232 \beta_1 + 1144) q^{40} + (1847 \beta_{2} - 1171 \beta_1 + 3258) q^{41} + ( - 1150 \beta_{2} + 2050 \beta_1 - 4510) q^{43} + (448 \beta_{2} - 104 \beta_1 - 1888) q^{44} + ( - 529 \beta_1 + 529) q^{46} + (1021 \beta_{2} + 682 \beta_1 - 7613) q^{47} + (1428 \beta_{2} - 4306 \beta_1 - 949) q^{49} + ( - 168 \beta_{2} + 2997 \beta_1 - 3997) q^{50} + (144 \beta_{2} + 2682 \beta_1 - 14958) q^{52} + ( - 534 \beta_{2} - 4444 \beta_1 - 7006) q^{53} + ( - 840 \beta_{2} - 14 \beta_1 + 130) q^{55} + ( - 1408 \beta_{2} + 3432 \beta_1 - 12600) q^{56} + ( - 1934 \beta_{2} + 4067 \beta_1 - 16559) q^{58} + ( - 1980 \beta_{2} - 934 \beta_1 - 17710) q^{59} + ( - 42 \beta_{2} + 1322 \beta_1 - 23320) q^{61} + (2698 \beta_{2} - 4057 \beta_1 + 21985) q^{62} + (2816 \beta_{2} + 4608 \beta_1 - 16064) q^{64} + ( - 2250 \beta_{2} - 351 \beta_1 + 351) q^{65} + ( - 1436 \beta_{2} - 4323 \beta_1 - 21949) q^{67} + ( - 2528 \beta_{2} + 1492 \beta_1 + 4124) q^{68} + (344 \beta_{2} - 636 \beta_1 + 2748) q^{70} + ( - 2227 \beta_{2} + 1628 \beta_1 - 31515) q^{71} + ( - 21 \beta_{2} + 3241 \beta_1 - 26170) q^{73} + (1144 \beta_{2} - 10962 \beta_1 + 15646) q^{74} + ( - 2488 \beta_{2} - 28 \beta_1 + 11316) q^{76} + (876 \beta_{2} - 532 \beta_1 - 368) q^{77} + ( - 6662 \beta_{2} - 4778 \beta_1 + 20036) q^{79} + ( - 4224 \beta_{2} + 816 \beta_1 + 7536) q^{80} + (990 \beta_{2} - 12807 \beta_1 + 16043) q^{82} + (13758 \beta_{2} - 5867 \beta_1 - 9839) q^{83} + (4476 \beta_{2} + 508 \beta_1 + 3856) q^{85} + ( - 5900 \beta_{2} + 17060 \beta_1 - 56060) q^{86} + ( - 2656 \beta_{2} - 2024 \beta_1 - 19256) q^{88} + ( - 8666 \beta_{2} + 5964 \beta_1 - 4304) q^{89} + (6984 \beta_{2} - 14229 \beta_1 + 43587) q^{91} + (2116 \beta_{2} - 3174 \beta_1 + 1058) q^{92} + ( - 4770 \beta_{2} + 8981 \beta_1 - 44413) q^{94} + (5644 \beta_{2} + 894 \beta_1 + 5298) q^{95} + (618 \beta_{2} + 3351 \beta_1 - 90313) q^{97} + (14368 \beta_{2} - 23437 \beta_1 + 121157) q^{98}+O(q^{100})$$ q + (-b1 + 1) * q^2 + (4*b2 - 6*b1 + 2) * q^4 + (-2*b2 + 3*b1 + 21) * q^5 + (-4*b2 + 17*b1 - 87) * q^7 + (16*b2 - 8*b1 + 112) * q^8 + (-8*b2 - 2*b1 - 50) * q^10 + (-28*b2 - 3*b1 - 37) * q^11 + (-99*b2 + 45*b1 - 324) * q^13 + (-60*b2 + 180*b1 - 592) * q^14 + (-128*b2 + 8*b1 + 88) * q^16 + (146*b2 + 57*b1 + 269) * q^17 + (200*b2 + 40*b1 + 498) * q^19 + (88*b2 - 40*b1 - 544) * q^20 + (68*b2 + 78*b1 + 454) * q^22 + 529 * q^23 + (-88*b2 + 86*b1 - 2391) * q^25 + (18*b2 + 747*b1 - 423) * q^26 + (-472*b2 + 1068*b1 - 2908) * q^28 + (-523*b2 + 745*b1 + 704) * q^29 + (479*b2 - 914*b1 - 1471) * q^31 + (-288*b2 + 464*b1 - 1968) * q^32 + (-520*b2 - 276*b1 - 3656) * q^34 + (148*b2 - 160*b1 - 460) * q^35 + (1662*b2 - 1117*b1 + 2053) * q^37 + (-560*b2 - 698*b1 - 3622) * q^38 + (240*b2 + 232*b1 + 1144) * q^40 + (1847*b2 - 1171*b1 + 3258) * q^41 + (-1150*b2 + 2050*b1 - 4510) * q^43 + (448*b2 - 104*b1 - 1888) * q^44 + (-529*b1 + 529) * q^46 + (1021*b2 + 682*b1 - 7613) * q^47 + (1428*b2 - 4306*b1 - 949) * q^49 + (-168*b2 + 2997*b1 - 3997) * q^50 + (144*b2 + 2682*b1 - 14958) * q^52 + (-534*b2 - 4444*b1 - 7006) * q^53 + (-840*b2 - 14*b1 + 130) * q^55 + (-1408*b2 + 3432*b1 - 12600) * q^56 + (-1934*b2 + 4067*b1 - 16559) * q^58 + (-1980*b2 - 934*b1 - 17710) * q^59 + (-42*b2 + 1322*b1 - 23320) * q^61 + (2698*b2 - 4057*b1 + 21985) * q^62 + (2816*b2 + 4608*b1 - 16064) * q^64 + (-2250*b2 - 351*b1 + 351) * q^65 + (-1436*b2 - 4323*b1 - 21949) * q^67 + (-2528*b2 + 1492*b1 + 4124) * q^68 + (344*b2 - 636*b1 + 2748) * q^70 + (-2227*b2 + 1628*b1 - 31515) * q^71 + (-21*b2 + 3241*b1 - 26170) * q^73 + (1144*b2 - 10962*b1 + 15646) * q^74 + (-2488*b2 - 28*b1 + 11316) * q^76 + (876*b2 - 532*b1 - 368) * q^77 + (-6662*b2 - 4778*b1 + 20036) * q^79 + (-4224*b2 + 816*b1 + 7536) * q^80 + (990*b2 - 12807*b1 + 16043) * q^82 + (13758*b2 - 5867*b1 - 9839) * q^83 + (4476*b2 + 508*b1 + 3856) * q^85 + (-5900*b2 + 17060*b1 - 56060) * q^86 + (-2656*b2 - 2024*b1 - 19256) * q^88 + (-8666*b2 + 5964*b1 - 4304) * q^89 + (6984*b2 - 14229*b1 + 43587) * q^91 + (2116*b2 - 3174*b1 + 1058) * q^92 + (-4770*b2 + 8981*b1 - 44413) * q^94 + (5644*b2 + 894*b1 + 5298) * q^95 + (618*b2 + 3351*b1 - 90313) * q^97 + (14368*b2 - 23437*b1 + 121157) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + 4 q^{2} + 16 q^{4} + 58 q^{5} - 282 q^{7} + 360 q^{8}+O(q^{10})$$ 3 * q + 4 * q^2 + 16 * q^4 + 58 * q^5 - 282 * q^7 + 360 * q^8 $$3 q + 4 q^{2} + 16 q^{4} + 58 q^{5} - 282 q^{7} + 360 q^{8} - 156 q^{10} - 136 q^{11} - 1116 q^{13} - 2016 q^{14} + 128 q^{16} + 896 q^{17} + 1654 q^{19} - 1504 q^{20} + 1352 q^{22} + 1587 q^{23} - 7347 q^{25} - 1998 q^{26} - 10264 q^{28} + 844 q^{29} - 3020 q^{31} - 6656 q^{32} - 11212 q^{34} - 1072 q^{35} + 8938 q^{37} - 10728 q^{38} + 3440 q^{40} + 12792 q^{41} - 16730 q^{43} - 5112 q^{44} + 2116 q^{46} - 22500 q^{47} + 2887 q^{49} - 15156 q^{50} - 47412 q^{52} - 17108 q^{53} - 436 q^{55} - 42640 q^{56} - 55678 q^{58} - 54176 q^{59} - 71324 q^{61} + 72710 q^{62} - 49984 q^{64} - 846 q^{65} - 62960 q^{67} + 8352 q^{68} + 9224 q^{70} - 98400 q^{71} - 81772 q^{73} + 59044 q^{74} + 31488 q^{76} + 304 q^{77} + 58224 q^{79} + 17568 q^{80} + 61926 q^{82} - 9892 q^{83} + 15536 q^{85} - 191140 q^{86} - 58400 q^{88} - 27542 q^{89} + 151974 q^{91} + 8464 q^{92} - 146990 q^{94} + 20644 q^{95} - 273672 q^{97} + 401276 q^{98}+O(q^{100})$$ 3 * q + 4 * q^2 + 16 * q^4 + 58 * q^5 - 282 * q^7 + 360 * q^8 - 156 * q^10 - 136 * q^11 - 1116 * q^13 - 2016 * q^14 + 128 * q^16 + 896 * q^17 + 1654 * q^19 - 1504 * q^20 + 1352 * q^22 + 1587 * q^23 - 7347 * q^25 - 1998 * q^26 - 10264 * q^28 + 844 * q^29 - 3020 * q^31 - 6656 * q^32 - 11212 * q^34 - 1072 * q^35 + 8938 * q^37 - 10728 * q^38 + 3440 * q^40 + 12792 * q^41 - 16730 * q^43 - 5112 * q^44 + 2116 * q^46 - 22500 * q^47 + 2887 * q^49 - 15156 * q^50 - 47412 * q^52 - 17108 * q^53 - 436 * q^55 - 42640 * q^56 - 55678 * q^58 - 54176 * q^59 - 71324 * q^61 + 72710 * q^62 - 49984 * q^64 - 846 * q^65 - 62960 * q^67 + 8352 * q^68 + 9224 * q^70 - 98400 * q^71 - 81772 * q^73 + 59044 * q^74 + 31488 * q^76 + 304 * q^77 + 58224 * q^79 + 17568 * q^80 + 61926 * q^82 - 9892 * q^83 + 15536 * q^85 - 191140 * q^86 - 58400 * q^88 - 27542 * q^89 + 151974 * q^91 + 8464 * q^92 - 146990 * q^94 + 20644 * q^95 - 273672 * q^97 + 401276 * q^98

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 13x + 12$$ :

 $$\beta_{1}$$ $$=$$ $$2\nu - 1$$ 2*v - 1 $$\beta_{2}$$ $$=$$ $$\nu^{2} + \nu - 9$$ v^2 + v - 9
 $$\nu$$ $$=$$ $$( \beta _1 + 1 ) / 2$$ (b1 + 1) / 2 $$\nu^{2}$$ $$=$$ $$( 2\beta_{2} - \beta _1 + 17 ) / 2$$ (2*b2 - b1 + 17) / 2

## 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
 3.65736 0.917748 −3.57511
−5.31473 0 −3.75366 23.8768 0 −11.7843 190.021 0 −126.899
1.2 0.164504 0 −31.9729 37.9865 0 −43.8366 −10.5238 0 6.24894
1.3 9.15022 0 51.7266 −3.86330 0 −226.379 180.503 0 −35.3501
 $$n$$: 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.b 3
3.b odd 2 1 23.6.a.a 3
12.b even 2 1 368.6.a.e 3
15.d odd 2 1 575.6.a.b 3
69.c even 2 1 529.6.a.a 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
23.6.a.a 3 3.b odd 2 1
207.6.a.b 3 1.a even 1 1 trivial
368.6.a.e 3 12.b even 2 1
529.6.a.a 3 69.c even 2 1
575.6.a.b 3 15.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{3} - 4T_{2}^{2} - 48T_{2} + 8$$ acting on $$S_{6}^{\mathrm{new}}(\Gamma_0(207))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3} - 4 T^{2} - 48 T + 8$$
$3$ $$T^{3}$$
$5$ $$T^{3} - 58 T^{2} + 668 T + 3504$$
$7$ $$T^{3} + 282 T^{2} + 13108 T + 116944$$
$11$ $$T^{3} + 136 T^{2} - 43688 T - 840152$$
$13$ $$T^{3} + 1116 T^{2} + \cdots - 255615102$$
$17$ $$T^{3} - 896 T^{2} + \cdots - 220718408$$
$19$ $$T^{3} - 1654 T^{2} + \cdots + 460771768$$
$23$ $$(T - 529)^{3}$$
$29$ $$T^{3} - 844 T^{2} + \cdots + 33789223458$$
$31$ $$T^{3} + 3020 T^{2} + \cdots - 117638912880$$
$37$ $$T^{3} - 8938 T^{2} + \cdots + 1048082031344$$
$41$ $$T^{3} - 12792 T^{2} + \cdots + 1564944049486$$
$43$ $$T^{3} + 16730 T^{2} + \cdots - 95315904000$$
$47$ $$T^{3} + 22500 T^{2} + \cdots - 916008439440$$
$53$ $$T^{3} + 17108 T^{2} + \cdots - 7849670295504$$
$59$ $$T^{3} + 54176 T^{2} + \cdots + 1725012447168$$
$61$ $$T^{3} + 71324 T^{2} + \cdots + 11439907465152$$
$67$ $$T^{3} + 62960 T^{2} + \cdots - 11971711378840$$
$71$ $$T^{3} + 98400 T^{2} + \cdots + 24837760695040$$
$73$ $$T^{3} + 81772 T^{2} + \cdots + 7199078503954$$
$79$ $$T^{3} + \cdots + 235690469012368$$
$83$ $$T^{3} + \cdots + 297029282761704$$
$89$ $$T^{3} + \cdots - 125799322340896$$
$97$ $$T^{3} + \cdots + 693755159518744$$