# 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

## 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: $$x^{4} - 75x^{2} - 42x + 736$$ x^4 - 75*x^2 - 42*x + 736 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.

 $$f(q)$$ $$=$$ $$q + (\beta_1 - 1) q^{2} + (\beta_{3} - \beta_1 + 7) q^{4} + (2 \beta_{3} - \beta_{2} - 3 \beta_1 - 5) q^{5} + ( - \beta_{3} - 4 \beta_{2} + 12 \beta_1 - 18) q^{7} + ( - 2 \beta_{3} + 4 \beta_{2} - 9 \beta_1 - 17) q^{8}+O(q^{10})$$ q + (b1 - 1) * q^2 + (b3 - b1 + 7) * q^4 + (2*b3 - b2 - 3*b1 - 5) * q^5 + (-b3 - 4*b2 + 12*b1 - 18) * q^7 + (-2*b3 + 4*b2 - 9*b1 - 17) * q^8 $$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})$$ q + (b1 - 1) * q^2 + (b3 - b1 + 7) * q^4 + (2*b3 - b2 - 3*b1 - 5) * q^5 + (-b3 - 4*b2 + 12*b1 - 18) * q^7 + (-2*b3 + 4*b2 - 9*b1 - 17) * q^8 + (-10*b3 + 10*b2 + 24*b1 - 124) * q^10 + (-13*b3 - 3*b2 - 41*b1 + 261) * q^11 + (22*b2 - 22*b1 - 88) * q^13 + (-7*b3 + 4*b2 - 46*b1 + 450) * q^14 + (-19*b3 - 16*b2 - 5*b1 - 513) * q^16 + (-43*b3 + 34*b2 + 66*b1 - 22) * q^17 + (-8*b3 - 19*b2 + 43*b1 - 1605) * q^19 + (20*b3 - 28*b2 - 158*b1 + 1306) * q^20 + (-43*b3 - 46*b2 + 44*b1 - 1788) * q^22 + 529 * q^23 + (20*b3 - 10*b2 - 450*b1 + 321) * q^25 + (88*b3 - 44*b2 - 22*b1 - 594) * q^26 + (13*b3 + 92*b2 - 34*b1 - 1566) * q^28 + (120*b3 - 10*b2 - 2*b1 - 932) * q^29 + (76*b3 + 14*b2 - 770*b1 + 818) * q^31 + (-2*b3 - 172*b2 - 577*b1 + 831) * q^32 + (279*b3 - 240*b2 - 608*b1 + 2940) * q^34 + (-90*b3 + 282*b2 - 258*b1 - 230) * q^35 + (-33*b3 + 231*b2 - 11*b1 - 4267) * q^37 + (-44*b3 + 6*b2 - 1790*b1 + 3138) * q^38 + (2*b3 - 184*b2 + 774*b1 - 3618) * q^40 + (-4*b3 - 300*b2 + 248*b1 - 4822) * q^41 + (58*b3 + 451*b2 + 365*b1 - 6207) * q^43 + (273*b3 + 16*b2 - 1302*b1 - 5042) * q^44 + (529*b1 - 529) * q^46 + (-12*b3 + 424*b2 + 348*b1 - 4392) * q^47 + (-662*b3 + 40*b2 - 456*b1 + 1677) * q^49 + (-520*b3 + 100*b2 + 611*b1 - 17571) * q^50 + (-330*b3 - 264*b2 + 1386*b1 + 1914) * q^52 + (-16*b3 + 203*b2 - 1519*b1 + 6723) * q^53 + (906*b3 - 366*b2 - 646*b1 - 9942) * q^55 + (637*b3 - 260*b2 + 390*b1 - 13534) * q^56 + (-172*b3 + 500*b2 + 958*b1 + 306) * q^58 + (-132*b3 - 288*b2 + 1792*b1 + 3420) * q^59 + (233*b3 - 1087*b2 - 661*b1 - 3849) * q^61 + (-776*b3 + 276*b2 + 2076*b1 - 30284) * q^62 + (-827*b3 + 848*b2 + 443*b1 - 7537) * q^64 + (-220*b3 - 902*b2 + 3542*b1 - 10142) * q^65 + (-638*b3 + 497*b2 - 2001*b1 - 2845) * q^67 + (-711*b3 + 508*b2 + 4572*b1 - 28136) * q^68 + (1242*b3 - 924*b2 - 824*b1 - 7240) * q^70 + (-508*b3 + 1408*b2 + 3940*b1 + 14824) * q^71 + (754*b3 - 362*b2 + 1990*b1 - 27872) * q^73 + (1177*b3 - 594*b2 - 4102*b1 + 5598) * q^74 + (-1460*b3 + 420*b2 + 1076*b1 - 19580) * q^76 + (-518*b3 - 2538*b2 + 6514*b1 - 330) * q^77 + (-879*b3 - 116*b2 + 2484*b1 - 60686) * q^79 + (-788*b3 + 1272*b2 + 918*b1 - 10058) * q^80 + (-1248*b3 + 584*b2 - 5786*b1 + 12162) * q^82 + (2561*b3 - 629*b2 + 2161*b1 + 24283) * q^83 + (-826*b3 - 92*b2 + 12220*b1 - 81260) * q^85 + (2562*b3 - 670*b2 - 3926*b1 + 23002) * q^86 + (-119*b3 + 2532*b2 - 2034*b1 + 11802) * q^88 + (2301*b3 + 2260*b2 - 4520*b1 + 29176) * q^89 + (3410*b3 - 132*b2 - 396*b1 - 73744) * q^91 + (529*b3 - 529*b1 + 3703) * q^92 + (2480*b3 - 896*b2 - 3312*b1 + 20632) * q^94 + (-3348*b3 + 2772*b2 + 3416*b1 + 4372) * q^95 + (3368*b3 - 1646*b2 + 9542*b1 - 12408) * q^97 + (406*b3 - 2728*b2 - 8795*b1 - 16077) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 4 q^{2} + 26 q^{4} - 22 q^{5} - 62 q^{7} - 72 q^{8}+O(q^{10})$$ 4 * q - 4 * q^2 + 26 * q^4 - 22 * q^5 - 62 * q^7 - 72 * q^8 $$4 q - 4 q^{2} + 26 q^{4} - 22 q^{5} - 62 q^{7} - 72 q^{8} - 496 q^{10} + 1076 q^{11} - 396 q^{13} + 1806 q^{14} - 1982 q^{16} - 70 q^{17} - 6366 q^{19} + 5240 q^{20} - 6974 q^{22} + 2116 q^{23} + 1264 q^{25} - 2464 q^{26} - 6474 q^{28} - 3948 q^{29} + 3092 q^{31} + 3672 q^{32} + 11682 q^{34} - 1304 q^{35} - 17464 q^{37} + 12628 q^{38} - 14108 q^{40} - 18680 q^{41} - 25846 q^{43} - 20746 q^{44} - 2116 q^{46} - 18392 q^{47} + 7952 q^{49} - 69444 q^{50} + 8844 q^{52} + 26518 q^{53} - 40848 q^{55} - 54890 q^{56} + 568 q^{58} + 14520 q^{59} - 13688 q^{61} - 120136 q^{62} - 30190 q^{64} - 38324 q^{65} - 11098 q^{67} - 112138 q^{68} - 29596 q^{70} + 57496 q^{71} - 112272 q^{73} + 21226 q^{74} - 76240 q^{76} + 4792 q^{77} - 240754 q^{79} - 41200 q^{80} + 49976 q^{82} + 93268 q^{83} - 323204 q^{85} + 88224 q^{86} + 42382 q^{88} + 107582 q^{89} - 301532 q^{91} + 13754 q^{92} + 79360 q^{94} + 18640 q^{95} - 53076 q^{97} - 59664 q^{98}+O(q^{100})$$ 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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 75x^{2} - 42x + 736$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{3} - \nu^{2} - 54\nu + 4 ) / 4$$ (v^3 - v^2 - 54*v + 4) / 4 $$\beta_{3}$$ $$=$$ $$\nu^{2} - \nu - 38$$ v^2 - v - 38
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{3} + \beta _1 + 38$$ b3 + b1 + 38 $$\nu^{3}$$ $$=$$ $$\beta_{3} + 4\beta_{2} + 55\beta _1 + 34$$ b3 + 4*b2 + 55*b1 + 34

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

## Hecke characteristic polynomials

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