# Properties

 Label 207.6.a.c Level $207$ Weight $6$ Character orbit 207.a Self dual yes Analytic conductor $33.199$ Analytic rank $0$ 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: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.5333.1 Defining polynomial: $$x^{3} - x^{2} - 11x + 8$$ x^3 - x^2 - 11*x + 8 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 a basis $$1,\beta_1,\beta_2$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - \beta_{2} + 3) q^{2} + ( - 9 \beta_{2} - 4 \beta_1 + 9) q^{4} + ( - 6 \beta_{2} - 11 \beta_1 + 17) q^{5} + ( - 8 \beta_{2} - 26 \beta_1 - 44) q^{7} + ( - 35 \beta_{2} - 32 \beta_1 + 171) q^{8}+O(q^{10})$$ q + (-b2 + 3) * q^2 + (-9*b2 - 4*b1 + 9) * q^4 + (-6*b2 - 11*b1 + 17) * q^5 + (-8*b2 - 26*b1 - 44) * q^7 + (-35*b2 - 32*b1 + 171) * q^8 $$q + ( - \beta_{2} + 3) q^{2} + ( - 9 \beta_{2} - 4 \beta_1 + 9) q^{4} + ( - 6 \beta_{2} - 11 \beta_1 + 17) q^{5} + ( - 8 \beta_{2} - 26 \beta_1 - 44) q^{7} + ( - 35 \beta_{2} - 32 \beta_1 + 171) q^{8} + ( - 64 \beta_{2} - 13 \beta_1 + 111) q^{10} + (42 \beta_{2} - 49 \beta_1 + 95) q^{11} + ( - 132 \beta_{2} - 66 \beta_1 - 264) q^{13} + ( - 30 \beta_{2} - 6 \beta_1 - 188) q^{14} + ( - 125 \beta_{2} + 20 \beta_1 + 961) q^{16} + ( - 72 \beta_{2} + 68 \beta_1 + 896) q^{17} + ( - 10 \beta_{2} - 143 \beta_1 - 993) q^{19} + ( - 316 \beta_{2} + 109 \beta_1 + 1681) q^{20} + (108 \beta_{2} + 217 \beta_1 - 1647) q^{22} - 529 q^{23} + (40 \beta_{2} + 10 \beta_1 + 241) q^{25} + ( - 594 \beta_{2} - 462 \beta_1 + 2640) q^{26} + (258 \beta_{2} + 718 \beta_1 + 1732) q^{28} + ( - 200 \beta_{2} + 254 \beta_1 + 5608) q^{29} + (560 \beta_{2} - 158 \beta_1 - 5158) q^{31} + ( - 571 \beta_{2} + 504 \beta_1 + 1651) q^{32} + ( - 1260 \beta_{2} - 356 \beta_1 + 5808) q^{34} + (884 \beta_{2} + 826 \beta_1 + 6154) q^{35} + (370 \beta_{2} - 105 \beta_1 + 5133) q^{37} + (790 \beta_{2} + 103 \beta_1 - 4375) q^{38} + ( - 1420 \beta_{2} - 957 \beta_1 + 12911) q^{40} + ( - 1204 \beta_{2} - 1048 \beta_1 - 4150) q^{41} + (1538 \beta_{2} + 163 \beta_1 + 593) q^{43} + (1168 \beta_{2} + 1783 \beta_1 - 8833) q^{44} + (529 \beta_{2} - 1587) q^{46} + ( - 284 \beta_{2} - 2000 \beta_1 - 10548) q^{47} + (2800 \beta_{2} + 3696 \beta_1 + 1789) q^{49} + (9 \beta_{2} + 150 \beta_1 - 437) q^{50} + ( - 2442 \beta_{2} + 198 \beta_1 + 29832) q^{52} + ( - 3166 \beta_{2} - 5619 \beta_1 + 13877) q^{53} + (3224 \beta_{2} - 1542 \beta_1 + 11198) q^{55} + (1494 \beta_{2} + 506 \beta_1 + 11572) q^{56} + ( - 6554 \beta_{2} - 1054 \beta_1 + 26272) q^{58} + (752 \beta_{2} + 4164 \beta_1 + 10904) q^{59} + ( - 986 \beta_{2} + 1545 \beta_1 + 18847) q^{61} + (8360 \beta_{2} + 2398 \beta_1 - 35290) q^{62} + ( - 573 \beta_{2} - 3428 \beta_1 - 1479) q^{64} + ( - 1980 \beta_{2} + 6006 \beta_1 + 19734) q^{65} + ( - 6138 \beta_{2} - 9559 \beta_1 + 13323) q^{67} + ( - 11420 \beta_{2} - 6860 \beta_1 + 24800) q^{68} + ( - 24 \beta_{2} + 2710 \beta_1 + 86) q^{70} + (1364 \beta_{2} - 4708 \beta_1 - 9832) q^{71} + (7808 \beta_{2} + 8490 \beta_1 - 30240) q^{73} + ( - 3018 \beta_{2} + 1585 \beta_1 + 2299) q^{74} + (9538 \beta_{2} + 7633 \beta_1 - 5393) q^{76} + (4196 \beta_{2} - 1770 \beta_1 + 30414) q^{77} + (32 \beta_{2} - 9526 \beta_1 - 19652) q^{79} + ( - 12276 \beta_{2} - 8211 \beta_1 + 18897) q^{80} + ( - 4122 \beta_{2} - 3768 \beta_1 + 13502) q^{82} + (16006 \beta_{2} + 6321 \beta_1 + 31417) q^{83} + ( - 11272 \beta_{2} - 8892 \beta_1 + 2756) q^{85} + (8798 \beta_{2} + 5989 \beta_1 - 45481) q^{86} + (14168 \beta_{2} - 4055 \beta_1 + 10225) q^{88} + (8868 \beta_{2} - 1622 \beta_1 - 35534) q^{89} + (7656 \beta_{2} + 21384 \beta_1 + 47652) q^{91} + (4761 \beta_{2} + 2116 \beta_1 - 4761) q^{92} + (6844 \beta_{2} + 864 \beta_1 - 46556) q^{94} + (10932 \beta_{2} + 12124 \beta_1 + 19040) q^{95} + ( - 10068 \beta_{2} + 7014 \beta_1 - 39412) q^{97} + (18707 \beta_{2} + 7504 \beta_1 - 39881) q^{98}+O(q^{100})$$ q + (-b2 + 3) * q^2 + (-9*b2 - 4*b1 + 9) * q^4 + (-6*b2 - 11*b1 + 17) * q^5 + (-8*b2 - 26*b1 - 44) * q^7 + (-35*b2 - 32*b1 + 171) * q^8 + (-64*b2 - 13*b1 + 111) * q^10 + (42*b2 - 49*b1 + 95) * q^11 + (-132*b2 - 66*b1 - 264) * q^13 + (-30*b2 - 6*b1 - 188) * q^14 + (-125*b2 + 20*b1 + 961) * q^16 + (-72*b2 + 68*b1 + 896) * q^17 + (-10*b2 - 143*b1 - 993) * q^19 + (-316*b2 + 109*b1 + 1681) * q^20 + (108*b2 + 217*b1 - 1647) * q^22 - 529 * q^23 + (40*b2 + 10*b1 + 241) * q^25 + (-594*b2 - 462*b1 + 2640) * q^26 + (258*b2 + 718*b1 + 1732) * q^28 + (-200*b2 + 254*b1 + 5608) * q^29 + (560*b2 - 158*b1 - 5158) * q^31 + (-571*b2 + 504*b1 + 1651) * q^32 + (-1260*b2 - 356*b1 + 5808) * q^34 + (884*b2 + 826*b1 + 6154) * q^35 + (370*b2 - 105*b1 + 5133) * q^37 + (790*b2 + 103*b1 - 4375) * q^38 + (-1420*b2 - 957*b1 + 12911) * q^40 + (-1204*b2 - 1048*b1 - 4150) * q^41 + (1538*b2 + 163*b1 + 593) * q^43 + (1168*b2 + 1783*b1 - 8833) * q^44 + (529*b2 - 1587) * q^46 + (-284*b2 - 2000*b1 - 10548) * q^47 + (2800*b2 + 3696*b1 + 1789) * q^49 + (9*b2 + 150*b1 - 437) * q^50 + (-2442*b2 + 198*b1 + 29832) * q^52 + (-3166*b2 - 5619*b1 + 13877) * q^53 + (3224*b2 - 1542*b1 + 11198) * q^55 + (1494*b2 + 506*b1 + 11572) * q^56 + (-6554*b2 - 1054*b1 + 26272) * q^58 + (752*b2 + 4164*b1 + 10904) * q^59 + (-986*b2 + 1545*b1 + 18847) * q^61 + (8360*b2 + 2398*b1 - 35290) * q^62 + (-573*b2 - 3428*b1 - 1479) * q^64 + (-1980*b2 + 6006*b1 + 19734) * q^65 + (-6138*b2 - 9559*b1 + 13323) * q^67 + (-11420*b2 - 6860*b1 + 24800) * q^68 + (-24*b2 + 2710*b1 + 86) * q^70 + (1364*b2 - 4708*b1 - 9832) * q^71 + (7808*b2 + 8490*b1 - 30240) * q^73 + (-3018*b2 + 1585*b1 + 2299) * q^74 + (9538*b2 + 7633*b1 - 5393) * q^76 + (4196*b2 - 1770*b1 + 30414) * q^77 + (32*b2 - 9526*b1 - 19652) * q^79 + (-12276*b2 - 8211*b1 + 18897) * q^80 + (-4122*b2 - 3768*b1 + 13502) * q^82 + (16006*b2 + 6321*b1 + 31417) * q^83 + (-11272*b2 - 8892*b1 + 2756) * q^85 + (8798*b2 + 5989*b1 - 45481) * q^86 + (14168*b2 - 4055*b1 + 10225) * q^88 + (8868*b2 - 1622*b1 - 35534) * q^89 + (7656*b2 + 21384*b1 + 47652) * q^91 + (4761*b2 + 2116*b1 - 4761) * q^92 + (6844*b2 + 864*b1 - 46556) * q^94 + (10932*b2 + 12124*b1 + 19040) * q^95 + (-10068*b2 + 7014*b1 - 39412) * q^97 + (18707*b2 + 7504*b1 - 39881) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + 8 q^{2} + 22 q^{4} + 56 q^{5} - 114 q^{7} + 510 q^{8}+O(q^{10})$$ 3 * q + 8 * q^2 + 22 * q^4 + 56 * q^5 - 114 * q^7 + 510 * q^8 $$3 q + 8 q^{2} + 22 q^{4} + 56 q^{5} - 114 q^{7} + 510 q^{8} + 282 q^{10} + 376 q^{11} - 858 q^{13} - 588 q^{14} + 2738 q^{16} + 2548 q^{17} - 2846 q^{19} + 4618 q^{20} - 5050 q^{22} - 1587 q^{23} + 753 q^{25} + 7788 q^{26} + 4736 q^{28} + 16370 q^{29} - 14756 q^{31} + 3878 q^{32} + 16520 q^{34} + 18520 q^{35} + 15874 q^{37} - 12438 q^{38} + 38270 q^{40} - 12606 q^{41} + 3154 q^{43} - 27114 q^{44} - 4232 q^{46} - 29928 q^{47} + 4471 q^{49} - 1452 q^{50} + 86856 q^{52} + 44084 q^{53} + 38360 q^{55} + 35704 q^{56} + 73316 q^{58} + 29300 q^{59} + 54010 q^{61} - 99908 q^{62} - 1582 q^{64} + 51216 q^{65} + 43390 q^{67} + 69840 q^{68} - 2476 q^{70} - 23424 q^{71} - 91402 q^{73} + 2294 q^{74} - 14274 q^{76} + 97208 q^{77} - 49398 q^{79} + 52626 q^{80} + 40152 q^{82} + 103936 q^{83} + 5888 q^{85} - 133634 q^{86} + 48898 q^{88} - 96112 q^{89} + 129228 q^{91} - 11638 q^{92} - 133688 q^{94} + 55928 q^{95} - 135318 q^{97} - 108440 q^{98}+O(q^{100})$$ 3 * q + 8 * q^2 + 22 * q^4 + 56 * q^5 - 114 * q^7 + 510 * q^8 + 282 * q^10 + 376 * q^11 - 858 * q^13 - 588 * q^14 + 2738 * q^16 + 2548 * q^17 - 2846 * q^19 + 4618 * q^20 - 5050 * q^22 - 1587 * q^23 + 753 * q^25 + 7788 * q^26 + 4736 * q^28 + 16370 * q^29 - 14756 * q^31 + 3878 * q^32 + 16520 * q^34 + 18520 * q^35 + 15874 * q^37 - 12438 * q^38 + 38270 * q^40 - 12606 * q^41 + 3154 * q^43 - 27114 * q^44 - 4232 * q^46 - 29928 * q^47 + 4471 * q^49 - 1452 * q^50 + 86856 * q^52 + 44084 * q^53 + 38360 * q^55 + 35704 * q^56 + 73316 * q^58 + 29300 * q^59 + 54010 * q^61 - 99908 * q^62 - 1582 * q^64 + 51216 * q^65 + 43390 * q^67 + 69840 * q^68 - 2476 * q^70 - 23424 * q^71 - 91402 * q^73 + 2294 * q^74 - 14274 * q^76 + 97208 * q^77 - 49398 * q^79 + 52626 * q^80 + 40152 * q^82 + 103936 * q^83 + 5888 * q^85 - 133634 * q^86 + 48898 * q^88 - 96112 * q^89 + 129228 * q^91 - 11638 * q^92 - 133688 * q^94 + 55928 * q^95 - 135318 * q^97 - 108440 * q^98

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 11x + 8$$ :

 $$\beta_{1}$$ $$=$$ $$2\nu - 1$$ 2*v - 1 $$\beta_{2}$$ $$=$$ $$\nu^{2} - \nu - 7$$ v^2 - v - 7
 $$\nu$$ $$=$$ $$( \beta _1 + 1 ) / 2$$ (b1 + 1) / 2 $$\nu^{2}$$ $$=$$ $$( 2\beta_{2} + \beta _1 + 15 ) / 2$$ (2*b2 + b1 + 15) / 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
 −3.20733 3.49331 0.714018
−3.49429 0 −19.7900 59.5955 0 96.8268 180.969 0 −208.244
1.2 1.29009 0 −30.3357 −59.1123 0 −213.331 −80.4187 0 −76.2602
1.3 10.2042 0 72.1256 55.5168 0 2.50462 409.450 0 566.504
 $$n$$: 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.c 3
3.b odd 2 1 69.6.a.b 3
12.b even 2 1 1104.6.a.i 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
69.6.a.b 3 3.b odd 2 1
207.6.a.c 3 1.a even 1 1 trivial
1104.6.a.i 3 12.b even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{3} - 8T_{2}^{2} - 27T_{2} + 46$$ acting on $$S_{6}^{\mathrm{new}}(\Gamma_0(207))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3} - 8 T^{2} - 27 T + 46$$
$3$ $$T^{3}$$
$5$ $$T^{3} - 56 T^{2} - 3496 T + 195576$$
$7$ $$T^{3} + 114 T^{2} - 20948 T + 51736$$
$11$ $$T^{3} - 376 T^{2} + \cdots - 21141352$$
$13$ $$T^{3} + 858 T^{2} + \cdots - 368282376$$
$17$ $$T^{3} - 2548 T^{2} + \cdots + 129112640$$
$19$ $$T^{3} + 2846 T^{2} + \cdots - 4313168$$
$23$ $$(T + 529)^{3}$$
$29$ $$T^{3} - 16370 T^{2} + \cdots - 117835741080$$
$31$ $$T^{3} + 14756 T^{2} + \cdots + 16664141952$$
$37$ $$T^{3} - 15874 T^{2} + \cdots - 103469473312$$
$41$ $$T^{3} + 12606 T^{2} + \cdots - 213582513784$$
$43$ $$T^{3} - 3154 T^{2} + \cdots + 409945701888$$
$47$ $$T^{3} + 29928 T^{2} + \cdots - 524672802816$$
$53$ $$T^{3} - 44084 T^{2} + \cdots + 30187666172280$$
$59$ $$T^{3} - 29300 T^{2} + \cdots + 4070512924224$$
$61$ $$T^{3} - 54010 T^{2} + \cdots - 694379910768$$
$67$ $$T^{3} + \cdots + 128918418373088$$
$71$ $$T^{3} + 23424 T^{2} + \cdots - 26245560332032$$
$73$ $$T^{3} + \cdots - 119459239092680$$
$79$ $$T^{3} + 49398 T^{2} + \cdots - 93978622829240$$
$83$ $$T^{3} + \cdots + 694250483317800$$
$89$ $$T^{3} + \cdots - 102645237296960$$
$97$ $$T^{3} + 135318 T^{2} + \cdots - 82905816948920$$