# Properties

 Label 400.6.a.ba Level $400$ Weight $6$ Character orbit 400.a Self dual yes Analytic conductor $64.154$ Analytic rank $0$ Dimension $4$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$400 = 2^{4} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 400.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$64.1535279252$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: 4.4.1595208.1 Defining polynomial: $$x^{4} - x^{3} - 20x^{2} + 33x - 3$$ x^4 - x^3 - 20*x^2 + 33*x - 3 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{8}\cdot 5^{2}$$ Twist minimal: no (minimal twist has level 40) 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.

 $$f(q)$$ $$=$$ $$q + (\beta_1 + 1) q^{3} + ( - \beta_{3} + \beta_1 + 37) q^{7} + ( - 2 \beta_{3} - \beta_{2} + 7 \beta_1 + 125) q^{9}+O(q^{10})$$ q + (b1 + 1) * q^3 + (-b3 + b1 + 37) * q^7 + (-2*b3 - b2 + 7*b1 + 125) * q^9 $$q + (\beta_1 + 1) q^{3} + ( - \beta_{3} + \beta_1 + 37) q^{7} + ( - 2 \beta_{3} - \beta_{2} + 7 \beta_1 + 125) q^{9} + ( - 2 \beta_{2} - 18 \beta_1 + 92) q^{11} + ( - 3 \beta_{3} + 2 \beta_{2} - 110) q^{13} + ( - 2 \beta_{3} - 4 \beta_{2} + 44 \beta_1 + 168) q^{17} + ( - 8 \beta_{3} + 2 \beta_{2} - 46 \beta_1 + 172) q^{19} + (6 \beta_{3} - 3 \beta_{2} + 149 \beta_1 + 248) q^{21} + ( - 5 \beta_{3} + 8 \beta_{2} + 23 \beta_1 + 1123) q^{23} + ( - 2 \beta_{3} - 16 \beta_{2} + 222 \beta_1 + 2038) q^{27} + (20 \beta_{2} - 12 \beta_1 - 734) q^{29} + (24 \beta_{3} + 12 \beta_{2} - 84 \beta_1 - 528) q^{31} + (28 \beta_{3} + 8 \beta_{2} + 156 \beta_1 - 6716) q^{33} + ( - 17 \beta_{3} - 2 \beta_{2} + 428 \beta_1 - 2198) q^{37} + (32 \beta_{3} + 4 \beta_{2} + 36 \beta_1 - 376) q^{39} + (2 \beta_{3} - 3 \beta_{2} - 491 \beta_1 + 2950) q^{41} + (54 \beta_{3} + 189 \beta_1 + 12069) q^{43} + (19 \beta_{3} + 56 \beta_{2} + 485 \beta_1 + 3681) q^{47} + (6 \beta_{3} - 69 \beta_{2} - 285 \beta_1 + 5625) q^{49} + ( - 88 \beta_{3} - 68 \beta_{2} + 988 \beta_1 + 15600) q^{51} + ( - 53 \beta_{3} - 34 \beta_{2} - 64 \beta_1 - 21074) q^{53} + (164 \beta_{3} + 40 \beta_{2} + 572 \beta_1 - 17756) q^{57} + ( - 88 \beta_{3} - 26 \beta_{2} + 598 \beta_1 + 11460) q^{59} + ( - 82 \beta_{3} + 61 \beta_{2} - 683 \beta_1 + 15482) q^{61} + ( - 115 \beta_{3} - 152 \beta_{2} + 521 \beta_1 + 46573) q^{63} + ( - 26 \beta_{3} - 32 \beta_{2} - 73 \beta_1 + 18175) q^{67} + (26 \beta_{3} + 7 \beta_{2} + 1103 \beta_1 + 9592) q^{69} + ( - 24 \beta_{3} + 232 \beta_{2} - 1560 \beta_1 + 15704) q^{71} + ( - 48 \beta_{3} - 80 \beta_{2} + 1212 \beta_1 - 33268) q^{73} + ( - 440 \beta_{3} + 56 \beta_{2} - 1972 \beta_1 - 2860) q^{77} + (328 \beta_{3} - 100 \beta_{2} - 196 \beta_1 + 5408) q^{79} + ( - 6 \beta_{3} - 63 \beta_{2} + 3257 \beta_1 + 51209) q^{81} + ( - 166 \beta_{3} + 352 \beta_{2} - 1679 \beta_1 + 18665) q^{83} + (104 \beta_{3} + 112 \beta_{2} - 2526 \beta_1 - 3118) q^{87} + (96 \beta_{3} + 300 \beta_{2} - 564 \beta_1 + 5238) q^{89} + (440 \beta_{3} - 200 \beta_{2} - 2120 \beta_1 + 60952) q^{91} + (24 \beta_{3} + 192 \beta_{2} - 4608 \beta_1 - 26400) q^{93} + (402 \beta_{3} - 172 \beta_{2} - 1860 \beta_1 - 14864) q^{97} + ( - 504 \beta_{3} + 426 \beta_{2} - 5062 \beta_1 + 33356) q^{99}+O(q^{100})$$ q + (b1 + 1) * q^3 + (-b3 + b1 + 37) * q^7 + (-2*b3 - b2 + 7*b1 + 125) * q^9 + (-2*b2 - 18*b1 + 92) * q^11 + (-3*b3 + 2*b2 - 110) * q^13 + (-2*b3 - 4*b2 + 44*b1 + 168) * q^17 + (-8*b3 + 2*b2 - 46*b1 + 172) * q^19 + (6*b3 - 3*b2 + 149*b1 + 248) * q^21 + (-5*b3 + 8*b2 + 23*b1 + 1123) * q^23 + (-2*b3 - 16*b2 + 222*b1 + 2038) * q^27 + (20*b2 - 12*b1 - 734) * q^29 + (24*b3 + 12*b2 - 84*b1 - 528) * q^31 + (28*b3 + 8*b2 + 156*b1 - 6716) * q^33 + (-17*b3 - 2*b2 + 428*b1 - 2198) * q^37 + (32*b3 + 4*b2 + 36*b1 - 376) * q^39 + (2*b3 - 3*b2 - 491*b1 + 2950) * q^41 + (54*b3 + 189*b1 + 12069) * q^43 + (19*b3 + 56*b2 + 485*b1 + 3681) * q^47 + (6*b3 - 69*b2 - 285*b1 + 5625) * q^49 + (-88*b3 - 68*b2 + 988*b1 + 15600) * q^51 + (-53*b3 - 34*b2 - 64*b1 - 21074) * q^53 + (164*b3 + 40*b2 + 572*b1 - 17756) * q^57 + (-88*b3 - 26*b2 + 598*b1 + 11460) * q^59 + (-82*b3 + 61*b2 - 683*b1 + 15482) * q^61 + (-115*b3 - 152*b2 + 521*b1 + 46573) * q^63 + (-26*b3 - 32*b2 - 73*b1 + 18175) * q^67 + (26*b3 + 7*b2 + 1103*b1 + 9592) * q^69 + (-24*b3 + 232*b2 - 1560*b1 + 15704) * q^71 + (-48*b3 - 80*b2 + 1212*b1 - 33268) * q^73 + (-440*b3 + 56*b2 - 1972*b1 - 2860) * q^77 + (328*b3 - 100*b2 - 196*b1 + 5408) * q^79 + (-6*b3 - 63*b2 + 3257*b1 + 51209) * q^81 + (-166*b3 + 352*b2 - 1679*b1 + 18665) * q^83 + (104*b3 + 112*b2 - 2526*b1 - 3118) * q^87 + (96*b3 + 300*b2 - 564*b1 + 5238) * q^89 + (440*b3 - 200*b2 - 2120*b1 + 60952) * q^91 + (24*b3 + 192*b2 - 4608*b1 - 26400) * q^93 + (402*b3 - 172*b2 - 1860*b1 - 14864) * q^97 + (-504*b3 + 426*b2 - 5062*b1 + 33356) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 4 q^{3} + 148 q^{7} + 500 q^{9}+O(q^{10})$$ 4 * q + 4 * q^3 + 148 * q^7 + 500 * q^9 $$4 q + 4 q^{3} + 148 q^{7} + 500 q^{9} + 368 q^{11} - 440 q^{13} + 672 q^{17} + 688 q^{19} + 992 q^{21} + 4492 q^{23} + 8152 q^{27} - 2936 q^{29} - 2112 q^{31} - 26864 q^{33} - 8792 q^{37} - 1504 q^{39} + 11800 q^{41} + 48276 q^{43} + 14724 q^{47} + 22500 q^{49} + 62400 q^{51} - 84296 q^{53} - 71024 q^{57} + 45840 q^{59} + 61928 q^{61} + 186292 q^{63} + 72700 q^{67} + 38368 q^{69} + 62816 q^{71} - 133072 q^{73} - 11440 q^{77} + 21632 q^{79} + 204836 q^{81} + 74660 q^{83} - 12472 q^{87} + 20952 q^{89} + 243808 q^{91} - 105600 q^{93} - 59456 q^{97} + 133424 q^{99}+O(q^{100})$$ 4 * q + 4 * q^3 + 148 * q^7 + 500 * q^9 + 368 * q^11 - 440 * q^13 + 672 * q^17 + 688 * q^19 + 992 * q^21 + 4492 * q^23 + 8152 * q^27 - 2936 * q^29 - 2112 * q^31 - 26864 * q^33 - 8792 * q^37 - 1504 * q^39 + 11800 * q^41 + 48276 * q^43 + 14724 * q^47 + 22500 * q^49 + 62400 * q^51 - 84296 * q^53 - 71024 * q^57 + 45840 * q^59 + 61928 * q^61 + 186292 * q^63 + 72700 * q^67 + 38368 * q^69 + 62816 * q^71 - 133072 * q^73 - 11440 * q^77 + 21632 * q^79 + 204836 * q^81 + 74660 * q^83 - 12472 * q^87 + 20952 * q^89 + 243808 * q^91 - 105600 * q^93 - 59456 * q^97 + 133424 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} - 20x^{2} + 33x - 3$$ :

 $$\beta_{1}$$ $$=$$ $$-2\nu^{3} + 40\nu - 29$$ -2*v^3 + 40*v - 29 $$\beta_{2}$$ $$=$$ $$( 22\nu^{3} + 40\nu^{2} - 240\nu - 141 ) / 3$$ (22*v^3 + 40*v^2 - 240*v - 141) / 3 $$\beta_{3}$$ $$=$$ $$( -8\nu^{3} + 40\nu^{2} + 120\nu - 516 ) / 3$$ (-8*v^3 + 40*v^2 + 120*v - 516) / 3
 $$\nu$$ $$=$$ $$( -\beta_{3} + \beta_{2} + 5\beta _1 + 20 ) / 80$$ (-b3 + b2 + 5*b1 + 20) / 80 $$\nu^{2}$$ $$=$$ $$( 5\beta_{3} + \beta_{2} - 3\beta _1 + 820 ) / 80$$ (5*b3 + b2 - 3*b1 + 820) / 80 $$\nu^{3}$$ $$=$$ $$( -\beta_{3} + \beta_{2} + 3\beta _1 - 38 ) / 4$$ (-b3 + b2 + 3*b1 - 38) / 4

## 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
 0.0965878 −4.73066 3.98753 1.64654
0 −24.1383 0 0 0 179.876 0 339.657 0
1.2 0 −5.49000 0 0 0 −188.968 0 −212.860 0
1.3 0 4.69449 0 0 0 10.2635 0 −220.962 0
1.4 0 28.9338 0 0 0 146.828 0 594.165 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$1$$
$$5$$ $$-1$$

## 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 400.6.a.ba 4
4.b odd 2 1 200.6.a.j 4
5.b even 2 1 400.6.a.z 4
5.c odd 4 2 80.6.c.d 8
15.e even 4 2 720.6.f.n 8
20.d odd 2 1 200.6.a.k 4
20.e even 4 2 40.6.c.a 8
40.i odd 4 2 320.6.c.i 8
40.k even 4 2 320.6.c.j 8
60.l odd 4 2 360.6.f.b 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
40.6.c.a 8 20.e even 4 2
80.6.c.d 8 5.c odd 4 2
200.6.a.j 4 4.b odd 2 1
200.6.a.k 4 20.d odd 2 1
320.6.c.i 8 40.i odd 4 2
320.6.c.j 8 40.k even 4 2
360.6.f.b 8 60.l odd 4 2
400.6.a.z 4 5.b even 2 1
400.6.a.ba 4 1.a even 1 1 trivial
720.6.f.n 8 15.e even 4 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{4} - 4T_{3}^{3} - 728T_{3}^{2} - 432T_{3} + 18000$$ acting on $$S_{6}^{\mathrm{new}}(\Gamma_0(400))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4} - 4 T^{3} - 728 T^{2} + \cdots + 18000$$
$5$ $$T^{4}$$
$7$ $$T^{4} - 148 T^{3} + \cdots - 51222832$$
$11$ $$T^{4} - 368 T^{3} + \cdots + 37397137664$$
$13$ $$T^{4} + 440 T^{3} + \cdots + 10683053312$$
$17$ $$T^{4} - 672 T^{3} + \cdots + 22118400000$$
$19$ $$T^{4} - 688 T^{3} + \cdots + 1042985883904$$
$23$ $$T^{4} - 4492 T^{3} + \cdots - 5643370924592$$
$29$ $$T^{4} + 2936 T^{3} + \cdots - 97147517576176$$
$31$ $$T^{4} + \cdots + 244229603328000$$
$37$ $$T^{4} + 8792 T^{3} + \cdots - 16\!\cdots\!36$$
$41$ $$T^{4} + \cdots - 296811236945008$$
$43$ $$T^{4} - 48276 T^{3} + \cdots - 74\!\cdots\!72$$
$47$ $$T^{4} - 14724 T^{3} + \cdots + 11\!\cdots\!96$$
$53$ $$T^{4} + 84296 T^{3} + \cdots + 11\!\cdots\!76$$
$59$ $$T^{4} - 45840 T^{3} + \cdots - 11\!\cdots\!48$$
$61$ $$T^{4} - 61928 T^{3} + \cdots + 31\!\cdots\!00$$
$67$ $$T^{4} - 72700 T^{3} + \cdots + 71\!\cdots\!32$$
$71$ $$T^{4} - 62816 T^{3} + \cdots + 25\!\cdots\!00$$
$73$ $$T^{4} + 133072 T^{3} + \cdots - 15\!\cdots\!32$$
$79$ $$T^{4} - 21632 T^{3} + \cdots - 96\!\cdots\!00$$
$83$ $$T^{4} - 74660 T^{3} + \cdots + 25\!\cdots\!52$$
$89$ $$T^{4} - 20952 T^{3} + \cdots - 93\!\cdots\!76$$
$97$ $$T^{4} + 59456 T^{3} + \cdots - 14\!\cdots\!84$$