# Properties

 Label 400.6.c.o Level $400$ Weight $6$ Character orbit 400.c Analytic conductor $64.154$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$400 = 2^{4} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 400.c (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$64.1535279252$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{241})$$ Defining polynomial: $$x^{4} + 121x^{2} + 3600$$ x^4 + 121*x^2 + 3600 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 200) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( - \beta_{2} - 4 \beta_1) q^{3} + ( - 2 \beta_{2} - 4 \beta_1) q^{7} + ( - 8 \beta_{3} - 14) q^{9}+O(q^{10})$$ q + (-b2 - 4*b1) * q^3 + (-2*b2 - 4*b1) * q^7 + (-8*b3 - 14) * q^9 $$q + ( - \beta_{2} - 4 \beta_1) q^{3} + ( - 2 \beta_{2} - 4 \beta_1) q^{7} + ( - 8 \beta_{3} - 14) q^{9} + (21 \beta_{3} + 100) q^{11} + ( - 52 \beta_{2} - 296 \beta_1) q^{13} + (16 \beta_{2} - 139 \beta_1) q^{17} + (59 \beta_{3} - 420) q^{19} + ( - 12 \beta_{3} - 498) q^{21} + (98 \beta_{2} + 976 \beta_1) q^{23} + ( - 197 \beta_{2} + 1012 \beta_1) q^{27} + ( - 148 \beta_{3} + 2340) q^{29} + (354 \beta_{3} + 2504) q^{31} + ( - 184 \beta_{2} - 5461 \beta_1) q^{33} + ( - 484 \beta_{2} - 6250 \beta_1) q^{37} + ( - 504 \beta_{3} - 13716) q^{39} + (936 \beta_{3} - 2667) q^{41} + ( - 1076 \beta_{2} - 112 \beta_1) q^{43} + 13036 \beta_1 q^{47} + ( - 16 \beta_{3} + 15827) q^{49} + ( - 75 \beta_{3} + 3300) q^{51} + (360 \beta_{2} - 23406 \beta_1) q^{53} + (184 \beta_{2} - 12539 \beta_1) q^{57} + ( - 416 \beta_{3} - 40888) q^{59} + (44 \beta_{3} - 23466) q^{61} + (60 \beta_{2} + 3912 \beta_1) q^{63} + (323 \beta_{2} + 34404 \beta_1) q^{67} + (1368 \beta_{3} + 27522) q^{69} + (344 \beta_{3} - 3724) q^{71} + (72 \beta_{2} - 54411 \beta_1) q^{73} + ( - 284 \beta_{2} - 10522 \beta_1) q^{77} + (3062 \beta_{3} - 54052) q^{79} + ( - 1720 \beta_{3} - 46831) q^{81} + (3841 \beta_{2} - 13612 \beta_1) q^{83} + ( - 1748 \beta_{2} + 26308 \beta_1) q^{87} + ( - 5024 \beta_{3} - 35495) q^{89} + ( - 800 \beta_{3} - 26248) q^{91} + ( - 3920 \beta_{2} - 95330 \beta_1) q^{93} + (3800 \beta_{2} + 48426 \beta_1) q^{97} + ( - 1094 \beta_{3} - 41888) q^{99}+O(q^{100})$$ q + (-b2 - 4*b1) * q^3 + (-2*b2 - 4*b1) * q^7 + (-8*b3 - 14) * q^9 + (21*b3 + 100) * q^11 + (-52*b2 - 296*b1) * q^13 + (16*b2 - 139*b1) * q^17 + (59*b3 - 420) * q^19 + (-12*b3 - 498) * q^21 + (98*b2 + 976*b1) * q^23 + (-197*b2 + 1012*b1) * q^27 + (-148*b3 + 2340) * q^29 + (354*b3 + 2504) * q^31 + (-184*b2 - 5461*b1) * q^33 + (-484*b2 - 6250*b1) * q^37 + (-504*b3 - 13716) * q^39 + (936*b3 - 2667) * q^41 + (-1076*b2 - 112*b1) * q^43 + 13036*b1 * q^47 + (-16*b3 + 15827) * q^49 + (-75*b3 + 3300) * q^51 + (360*b2 - 23406*b1) * q^53 + (184*b2 - 12539*b1) * q^57 + (-416*b3 - 40888) * q^59 + (44*b3 - 23466) * q^61 + (60*b2 + 3912*b1) * q^63 + (323*b2 + 34404*b1) * q^67 + (1368*b3 + 27522) * q^69 + (344*b3 - 3724) * q^71 + (72*b2 - 54411*b1) * q^73 + (-284*b2 - 10522*b1) * q^77 + (3062*b3 - 54052) * q^79 + (-1720*b3 - 46831) * q^81 + (3841*b2 - 13612*b1) * q^83 + (-1748*b2 + 26308*b1) * q^87 + (-5024*b3 - 35495) * q^89 + (-800*b3 - 26248) * q^91 + (-3920*b2 - 95330*b1) * q^93 + (3800*b2 + 48426*b1) * q^97 + (-1094*b3 - 41888) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 56 q^{9}+O(q^{10})$$ 4 * q - 56 * q^9 $$4 q - 56 q^{9} + 400 q^{11} - 1680 q^{19} - 1992 q^{21} + 9360 q^{29} + 10016 q^{31} - 54864 q^{39} - 10668 q^{41} + 63308 q^{49} + 13200 q^{51} - 163552 q^{59} - 93864 q^{61} + 110088 q^{69} - 14896 q^{71} - 216208 q^{79} - 187324 q^{81} - 141980 q^{89} - 104992 q^{91} - 167552 q^{99}+O(q^{100})$$ 4 * q - 56 * q^9 + 400 * q^11 - 1680 * q^19 - 1992 * q^21 + 9360 * q^29 + 10016 * q^31 - 54864 * q^39 - 10668 * q^41 + 63308 * q^49 + 13200 * q^51 - 163552 * q^59 - 93864 * q^61 + 110088 * q^69 - 14896 * q^71 - 216208 * q^79 - 187324 * q^81 - 141980 * q^89 - 104992 * q^91 - 167552 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 121x^{2} + 3600$$ :

 $$\beta_{1}$$ $$=$$ $$( \nu^{3} + 61\nu ) / 60$$ (v^3 + 61*v) / 60 $$\beta_{2}$$ $$=$$ $$( \nu^{3} + 181\nu ) / 60$$ (v^3 + 181*v) / 60 $$\beta_{3}$$ $$=$$ $$2\nu^{2} + 121$$ 2*v^2 + 121
 $$\nu$$ $$=$$ $$( \beta_{2} - \beta_1 ) / 2$$ (b2 - b1) / 2 $$\nu^{2}$$ $$=$$ $$( \beta_{3} - 121 ) / 2$$ (b3 - 121) / 2 $$\nu^{3}$$ $$=$$ $$( -61\beta_{2} + 181\beta_1 ) / 2$$ (-61*b2 + 181*b1) / 2

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/400\mathbb{Z}\right)^\times$$.

 $$n$$ $$101$$ $$177$$ $$351$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
49.1
 7.26209i 8.26209i − 8.26209i − 7.26209i
0 19.5242i 0 0 0 35.0483i 0 −138.193 0
49.2 0 11.5242i 0 0 0 27.0483i 0 110.193 0
49.3 0 11.5242i 0 0 0 27.0483i 0 110.193 0
49.4 0 19.5242i 0 0 0 35.0483i 0 −138.193 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 400.6.c.o 4
4.b odd 2 1 200.6.c.f 4
5.b even 2 1 inner 400.6.c.o 4
5.c odd 4 1 400.6.a.r 2
5.c odd 4 1 400.6.a.u 2
20.d odd 2 1 200.6.c.f 4
20.e even 4 1 200.6.a.e 2
20.e even 4 1 200.6.a.f yes 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
200.6.a.e 2 20.e even 4 1
200.6.a.f yes 2 20.e even 4 1
200.6.c.f 4 4.b odd 2 1
200.6.c.f 4 20.d odd 2 1
400.6.a.r 2 5.c odd 4 1
400.6.a.u 2 5.c odd 4 1
400.6.c.o 4 1.a even 1 1 trivial
400.6.c.o 4 5.b even 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{4} + 514T_{3}^{2} + 50625$$ acting on $$S_{6}^{\mathrm{new}}(400, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4} + 514 T^{2} + 50625$$
$5$ $$T^{4}$$
$7$ $$T^{4} + 1960 T^{2} + 898704$$
$11$ $$(T^{2} - 200 T - 96281)^{2}$$
$13$ $$T^{4} + 1478560 T^{2} + \cdots + 318150146304$$
$17$ $$T^{4} + 162034 T^{2} + \cdots + 1795640625$$
$19$ $$(T^{2} + 840 T - 662521)^{2}$$
$23$ $$T^{4} + 6534280 T^{2} + \cdots + 1855011312144$$
$29$ $$(T^{2} - 4680 T + 196736)^{2}$$
$31$ $$(T^{2} - 5008 T - 23931140)^{2}$$
$37$ $$T^{4} + \cdots + 302523267094416$$
$41$ $$(T^{2} + 5334 T - 204026247)^{2}$$
$43$ $$T^{4} + 558073120 T^{2} + \cdots + 77\!\cdots\!84$$
$47$ $$(T^{2} + 169937296)^{2}$$
$53$ $$T^{4} + 1158148872 T^{2} + \cdots + 26\!\cdots\!96$$
$59$ $$(T^{2} + 81776 T + 1630122048)^{2}$$
$61$ $$(T^{2} + 46932 T + 550186580)^{2}$$
$67$ $$T^{4} + 2417557010 T^{2} + \cdots + 13\!\cdots\!29$$
$71$ $$(T^{2} + 7448 T - 14650800)^{2}$$
$73$ $$T^{4} + 5923612530 T^{2} + \cdots + 87\!\cdots\!29$$
$79$ $$(T^{2} + 108104 T + 662040300)^{2}$$
$83$ $$T^{4} + 7481654530 T^{2} + \cdots + 11\!\cdots\!29$$
$89$ $$(T^{2} + 70990 T - 4823083791)^{2}$$
$97$ $$T^{4} + 11650234952 T^{2} + \cdots + 12\!\cdots\!76$$