# Properties

 Label 400.6.a.q Level 400 Weight 6 Character orbit 400.a Self dual yes Analytic conductor 64.154 Analytic rank 1 Dimension 2 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: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{129})$$ Defining polynomial: $$x^{2} - x - 32$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{2}$$ 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 $$\beta = 2\sqrt{129}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -6 - \beta ) q^{3} + ( 26 - 3 \beta ) q^{7} + ( 309 + 12 \beta ) q^{9} +O(q^{10})$$ $$q + ( -6 - \beta ) q^{3} + ( 26 - 3 \beta ) q^{7} + ( 309 + 12 \beta ) q^{9} + ( -280 - 6 \beta ) q^{11} + ( -694 - 12 \beta ) q^{13} + ( -74 + 84 \beta ) q^{17} + ( 500 - 36 \beta ) q^{19} + ( 1392 - 8 \beta ) q^{21} + ( -1226 + 123 \beta ) q^{23} + ( -6588 - 138 \beta ) q^{27} + ( 670 + 312 \beta ) q^{29} + ( 1124 - 54 \beta ) q^{31} + ( 4776 + 316 \beta ) q^{33} + ( 2970 - 216 \beta ) q^{37} + ( 10356 + 766 \beta ) q^{39} + ( 11538 - 156 \beta ) q^{41} + ( 8842 + 339 \beta ) q^{43} + ( -1454 + 885 \beta ) q^{47} + ( -11487 - 156 \beta ) q^{49} + ( -42900 - 430 \beta ) q^{51} + ( 2706 + 540 \beta ) q^{53} + ( 15576 - 284 \beta ) q^{57} + ( -31292 + 504 \beta ) q^{59} + ( 7054 - 1104 \beta ) q^{61} + ( -10542 - 615 \beta ) q^{63} + ( -42706 - 543 \beta ) q^{67} + ( -56112 + 488 \beta ) q^{69} + ( -23604 + 546 \beta ) q^{71} + ( 33726 - 1308 \beta ) q^{73} + ( 2008 + 684 \beta ) q^{77} + ( 32952 - 2508 \beta ) q^{79} + ( 35649 + 4500 \beta ) q^{81} + ( 54362 + 711 \beta ) q^{83} + ( -165012 - 2542 \beta ) q^{87} + ( -27510 - 1464 \beta ) q^{89} + ( 532 + 1770 \beta ) q^{91} + ( 21120 - 800 \beta ) q^{93} + ( -73834 + 4620 \beta ) q^{97} + ( -123672 - 5214 \beta ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 12q^{3} + 52q^{7} + 618q^{9} + O(q^{10})$$ $$2q - 12q^{3} + 52q^{7} + 618q^{9} - 560q^{11} - 1388q^{13} - 148q^{17} + 1000q^{19} + 2784q^{21} - 2452q^{23} - 13176q^{27} + 1340q^{29} + 2248q^{31} + 9552q^{33} + 5940q^{37} + 20712q^{39} + 23076q^{41} + 17684q^{43} - 2908q^{47} - 22974q^{49} - 85800q^{51} + 5412q^{53} + 31152q^{57} - 62584q^{59} + 14108q^{61} - 21084q^{63} - 85412q^{67} - 112224q^{69} - 47208q^{71} + 67452q^{73} + 4016q^{77} + 65904q^{79} + 71298q^{81} + 108724q^{83} - 330024q^{87} - 55020q^{89} + 1064q^{91} + 42240q^{93} - 147668q^{97} - 247344q^{99} + O(q^{100})$$

## 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
 6.17891 −5.17891
0 −28.7156 0 0 0 −42.1469 0 581.588 0
1.2 0 16.7156 0 0 0 94.1469 0 36.4124 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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.q 2
4.b odd 2 1 200.6.a.g 2
5.b even 2 1 80.6.a.i 2
5.c odd 4 2 400.6.c.l 4
15.d odd 2 1 720.6.a.z 2
20.d odd 2 1 40.6.a.d 2
20.e even 4 2 200.6.c.e 4
40.e odd 2 1 320.6.a.w 2
40.f even 2 1 320.6.a.q 2
60.h even 2 1 360.6.a.l 2

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

## Atkin-Lehner signs

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{2} + 12 T_{3} - 480$$ acting on $$S_{6}^{\mathrm{new}}(\Gamma_0(400))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$1 + 12 T + 6 T^{2} + 2916 T^{3} + 59049 T^{4}$$
$5$ 1
$7$ $$1 - 52 T + 29646 T^{2} - 873964 T^{3} + 282475249 T^{4}$$
$11$ $$1 + 560 T + 381926 T^{2} + 90188560 T^{3} + 25937424601 T^{4}$$
$13$ $$1 + 1388 T + 1149918 T^{2} + 515354684 T^{3} + 137858491849 T^{4}$$
$17$ $$1 + 148 T - 795706 T^{2} + 210138836 T^{3} + 2015993900449 T^{4}$$
$19$ $$1 - 1000 T + 4533462 T^{2} - 2476099000 T^{3} + 6131066257801 T^{4}$$
$23$ $$1 + 2452 T + 6569198 T^{2} + 15781913036 T^{3} + 41426511213649 T^{4}$$
$29$ $$1 - 1340 T - 8758306 T^{2} - 27484939660 T^{3} + 420707233300201 T^{4}$$
$31$ $$1 - 2248 T + 57017022 T^{2} - 64358331448 T^{3} + 819628286980801 T^{4}$$
$37$ $$1 - 5940 T + 123434318 T^{2} - 411903104580 T^{3} + 4808584372417849 T^{4}$$
$41$ $$1 - 23076 T + 352280470 T^{2} - 2673497694276 T^{3} + 13422659310152401 T^{4}$$
$43$ $$1 - 17684 T + 312898614 T^{2} - 2599697306012 T^{3} + 21611482313284249 T^{4}$$
$47$ $$1 + 2908 T + 56660030 T^{2} + 666935280356 T^{3} + 52599132235830049 T^{4}$$
$53$ $$1 - 5412 T + 693247822 T^{2} - 2263274008116 T^{3} + 174887470365513049 T^{4}$$
$59$ $$1 + 62584 T + 2277965606 T^{2} + 44742822328616 T^{3} + 511116753300641401 T^{4}$$
$61$ $$1 - 14108 T + 1110042462 T^{2} - 11915564614508 T^{3} + 713342911662882601 T^{4}$$
$67$ $$1 + 85412 T + 4371910566 T^{2} + 115316885639084 T^{3} + 1822837804551761449 T^{4}$$
$71$ $$1 + 47208 T + 4011779662 T^{2} + 85174059202008 T^{3} + 3255243551009881201 T^{4}$$
$73$ $$1 - 67452 T + 4400780438 T^{2} - 139832825091036 T^{3} + 4297625829703557649 T^{4}$$
$79$ $$1 - 65904 T + 3994274078 T^{2} - 202790324919696 T^{3} + 9468276082626847201 T^{4}$$
$83$ $$1 - 108724 T + 10572459494 T^{2} - 428268254869532 T^{3} + 15516041187205853449 T^{4}$$
$89$ $$1 + 55020 T + 10818978262 T^{2} + 307234950883980 T^{3} + 31181719929966183601 T^{4}$$
$97$ $$1 + 147668 T + 11612429670 T^{2} + 1268075361070676 T^{3} + 73742412689492826049 T^{4}$$