# Properties

 Label 400.2.u.e Level $400$ Weight $2$ Character orbit 400.u Analytic conductor $3.194$ Analytic rank $0$ Dimension $8$ CM no Inner twists $2$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$3.19401608085$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$2$$ over $$\Q(\zeta_{5})$$ Coefficient field: 8.0.58140625.2 Defining polynomial: $$x^{8} - 3x^{7} + 4x^{6} - 7x^{5} + 11x^{4} + 5x^{3} - 10x^{2} - 25x + 25$$ x^8 - 3*x^7 + 4*x^6 - 7*x^5 + 11*x^4 + 5*x^3 - 10*x^2 - 25*x + 25 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$5$$ Twist minimal: no (minimal twist has level 200) Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - \beta_{3} + 1) q^{3} + (\beta_{6} - \beta_{4}) q^{5} + ( - \beta_{5} - \beta_{4} + \beta_1 - 1) q^{7} + (\beta_{6} + 2 \beta_{3} + \beta_{2}) q^{9}+O(q^{10})$$ q + (-b3 + 1) * q^3 + (b6 - b4) * q^5 + (-b5 - b4 + b1 - 1) * q^7 + (b6 + 2*b3 + b2) * q^9 $$q + ( - \beta_{3} + 1) q^{3} + (\beta_{6} - \beta_{4}) q^{5} + ( - \beta_{5} - \beta_{4} + \beta_1 - 1) q^{7} + (\beta_{6} + 2 \beta_{3} + \beta_{2}) q^{9} + ( - \beta_{7} + \beta_{5} + \beta_{4} + 3 \beta_{2}) q^{11} + ( - \beta_{7} + 2 \beta_{6} + 2 \beta_{5} + \beta_{3} + 2 \beta_{2} - \beta_1) q^{13} + (\beta_{6} - \beta_{4} + \beta_{2} + \beta_1) q^{15} + (2 \beta_{7} + 2 \beta_{6} + 2 \beta_{3} - \beta_{2} + 1) q^{17} + (2 \beta_{7} + 3 \beta_{6} - \beta_{4} + 3 \beta_{3} + 2 \beta_{2} + \beta_1 - 2) q^{19} + (\beta_{7} - 2 \beta_{5} - \beta_{4} + \beta_{3} + 2 \beta_1 - 1) q^{21} + (\beta_{7} + 2 \beta_{6} - \beta_{5} - \beta_{4} + \beta_{2} - 2) q^{23} + ( - \beta_{7} - 4 \beta_{6} - \beta_{4} - \beta_{3}) q^{25} + ( - 4 \beta_{6} - 3 \beta_{2} + 4) q^{27} + ( - \beta_{7} - 3 \beta_{6} - \beta_{5} + \beta_{4} - 2 \beta_{3} + \beta_1 + 2) q^{29} + ( - \beta_{7} + \beta_{6} + 2 \beta_{4} + \beta_{3} - \beta_{2} - 2 \beta_1 + 1) q^{31} + ( - \beta_{7} + 3 \beta_{2} - 3) q^{33} + ( - 5 \beta_{6} - \beta_{5} - 4 \beta_{2} + \beta_1 + 4) q^{35} + (2 \beta_{7} + 3 \beta_{6} - 3 \beta_{3} + 3 \beta_{2} + 2 \beta_1) q^{37} + ( - 2 \beta_{7} + \beta_{6} + 2 \beta_{5} - \beta_{4} + 3 \beta_{2} - 1) q^{39} + (\beta_{7} - 2 \beta_{5} - 3 \beta_{3} + \beta_1) q^{41} + (\beta_{5} + \beta_{4} - 5 \beta_1 + 1) q^{43} + (\beta_{6} - \beta_{5} - \beta_{4} - \beta_{2} - \beta_1 - 1) q^{45} + (\beta_{7} + 5 \beta_{6} - \beta_{5} - \beta_{4} + \beta_1) q^{47} + (\beta_{5} + \beta_{4} - 4 \beta_{3} - 4 \beta_{2} - \beta_1 + 2) q^{49} + (2 \beta_{5} + 2 \beta_{4} - \beta_{3} - \beta_{2} - 2 \beta_1 + 4) q^{51} + ( - 3 \beta_{7} - 4 \beta_{6} + 2 \beta_{5} + 3 \beta_{4} - 2 \beta_1) q^{53} + (2 \beta_{7} + 7 \beta_{6} - 2 \beta_{5} - 2 \beta_{4} + 7 \beta_{3} + 3 \beta_{2} + \cdots - 7) q^{55}+ \cdots + (2 \beta_{5} + 2 \beta_{4} + 3 \beta_{3} + 3 \beta_{2} + \beta_1 + 3) q^{99}+O(q^{100})$$ q + (-b3 + 1) * q^3 + (b6 - b4) * q^5 + (-b5 - b4 + b1 - 1) * q^7 + (b6 + 2*b3 + b2) * q^9 + (-b7 + b5 + b4 + 3*b2) * q^11 + (-b7 + 2*b6 + 2*b5 + b3 + 2*b2 - b1) * q^13 + (b6 - b4 + b2 + b1) * q^15 + (2*b7 + 2*b6 + 2*b3 - b2 + 1) * q^17 + (2*b7 + 3*b6 - b4 + 3*b3 + 2*b2 + b1 - 2) * q^19 + (b7 - 2*b5 - b4 + b3 + 2*b1 - 1) * q^21 + (b7 + 2*b6 - b5 - b4 + b2 - 2) * q^23 + (-b7 - 4*b6 - b4 - b3) * q^25 + (-4*b6 - 3*b2 + 4) * q^27 + (-b7 - 3*b6 - b5 + b4 - 2*b3 + b1 + 2) * q^29 + (-b7 + b6 + 2*b4 + b3 - b2 - 2*b1 + 1) * q^31 + (-b7 + 3*b2 - 3) * q^33 + (-5*b6 - b5 - 4*b2 + b1 + 4) * q^35 + (2*b7 + 3*b6 - 3*b3 + 3*b2 + 2*b1) * q^37 + (-2*b7 + b6 + 2*b5 - b4 + 3*b2 - 1) * q^39 + (b7 - 2*b5 - 3*b3 + b1) * q^41 + (b5 + b4 - 5*b1 + 1) * q^43 + (b6 - b5 - b4 - b2 - b1 - 1) * q^45 + (b7 + 5*b6 - b5 - b4 + b1) * q^47 + (b5 + b4 - 4*b3 - 4*b2 - b1 + 2) * q^49 + (2*b5 + 2*b4 - b3 - b2 - 2*b1 + 4) * q^51 + (-3*b7 - 4*b6 + 2*b5 + 3*b4 - 2*b1) * q^53 + (2*b7 + 7*b6 - 2*b5 - 2*b4 + 7*b3 + 3*b2 + 3*b1 - 7) * q^55 + (b5 + b4 + 2*b3 + 2*b2 - 1) * q^57 + (-4*b6 + 3*b5 - 2*b3 - 4*b2) * q^59 + (-b7 - 5*b6 + b5 - 7*b2 + 5) * q^61 + (-b7 - b6 - 2*b3 - b2 - b1) * q^63 + (-b7 + 2*b6 - 2*b4 + 4*b3 + 5*b2 - 10) * q^65 + (b7 + b6 + b3 + 5*b2 - 5) * q^67 + (b7 + 2*b6 + 2*b3 + 3*b2 - 3) * q^69 + (-3*b7 - 3*b6 - b5 + 3*b4 - 4*b3 + b1 + 4) * q^71 + (-2*b6 + b4 - 4*b2 + 2) * q^73 + (-3*b6 - b5 - 2*b4 - 3*b2 + 2*b1 - 1) * q^75 + (3*b7 - 3*b5 - 7*b2) * q^77 + (-b7 + 8*b6 + 2*b5 + b4 + 6*b3 - 2*b1 - 6) * q^79 + (2*b6 + 2*b3 - 4*b2 + 4) * q^81 + (-b7 - 11*b6 + 2*b4 - 11*b3 - 5*b2 - 2*b1 + 5) * q^83 + (-b7 + b5 - 11*b3 - 3*b2 - 3*b1 + 1) * q^85 + (b7 - b6 - 3*b5 - 2*b3 - b2 + b1) * q^87 + (-2*b7 + 2*b5 + 3*b4 + 2*b2) * q^89 + (b7 - 10*b6 - 3*b5 + 3*b3 - 10*b2 + b1) * q^91 + (b5 + b4 - b3 - b2 - 3*b1 + 3) * q^93 + (b7 - 4*b6 - 2*b5 - b4 - 9*b3 - 5*b2 - 2) * q^95 + (2*b7 + b6 + b5 - 2*b4 + b3 - b1 - 1) * q^97 + (2*b5 + 2*b4 + 3*b3 + 3*b2 + b1 + 3) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + 6 q^{3} + 5 q^{5} - 4 q^{7} + 8 q^{9}+O(q^{10})$$ 8 * q + 6 * q^3 + 5 * q^5 - 4 * q^7 + 8 * q^9 $$8 q + 6 q^{3} + 5 q^{5} - 4 q^{7} + 8 q^{9} + 2 q^{11} + 8 q^{13} + 5 q^{15} + 10 q^{17} - 3 q^{19} - 3 q^{21} - 6 q^{23} - 5 q^{25} + 18 q^{27} + 6 q^{29} + 10 q^{31} - 16 q^{33} + 15 q^{35} - 2 q^{37} + q^{39} - 4 q^{41} + 12 q^{43} + 12 q^{47} - 4 q^{49} + 20 q^{51} - 13 q^{53} - 20 q^{55} - 6 q^{57} - 29 q^{59} + 15 q^{61} - 4 q^{63} - 50 q^{65} - 28 q^{67} - 12 q^{69} + 16 q^{71} + q^{73} - 15 q^{75} - 11 q^{77} - 23 q^{79} + 32 q^{81} - 14 q^{83} - 15 q^{85} - 3 q^{87} - 7 q^{89} - 29 q^{91} + 20 q^{93} - 45 q^{95} - 3 q^{97} + 22 q^{99}+O(q^{100})$$ 8 * q + 6 * q^3 + 5 * q^5 - 4 * q^7 + 8 * q^9 + 2 * q^11 + 8 * q^13 + 5 * q^15 + 10 * q^17 - 3 * q^19 - 3 * q^21 - 6 * q^23 - 5 * q^25 + 18 * q^27 + 6 * q^29 + 10 * q^31 - 16 * q^33 + 15 * q^35 - 2 * q^37 + q^39 - 4 * q^41 + 12 * q^43 + 12 * q^47 - 4 * q^49 + 20 * q^51 - 13 * q^53 - 20 * q^55 - 6 * q^57 - 29 * q^59 + 15 * q^61 - 4 * q^63 - 50 * q^65 - 28 * q^67 - 12 * q^69 + 16 * q^71 + q^73 - 15 * q^75 - 11 * q^77 - 23 * q^79 + 32 * q^81 - 14 * q^83 - 15 * q^85 - 3 * q^87 - 7 * q^89 - 29 * q^91 + 20 * q^93 - 45 * q^95 - 3 * q^97 + 22 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 3x^{7} + 4x^{6} - 7x^{5} + 11x^{4} + 5x^{3} - 10x^{2} - 25x + 25$$ :

 $$\beta_{1}$$ $$=$$ $$( -64\nu^{7} + 206\nu^{6} - 318\nu^{5} + 399\nu^{4} - 732\nu^{3} - 126\nu^{2} + 2175\nu + 235 ) / 1355$$ (-64*v^7 + 206*v^6 - 318*v^5 + 399*v^4 - 732*v^3 - 126*v^2 + 2175*v + 235) / 1355 $$\beta_{2}$$ $$=$$ $$( 406\nu^{7} - 714\nu^{6} + 747\nu^{5} - 1896\nu^{4} + 2103\nu^{3} + 4949\nu^{2} + 1065\nu - 7800 ) / 1355$$ (406*v^7 - 714*v^6 + 747*v^5 - 1896*v^4 + 2103*v^3 + 4949*v^2 + 1065*v - 7800) / 1355 $$\beta_{3}$$ $$=$$ $$( -420\nu^{7} + 776\nu^{6} - 698\nu^{5} + 1924\nu^{4} - 2297\nu^{3} - 5129\nu^{2} - 1055\nu + 10265 ) / 1355$$ (-420*v^7 + 776*v^6 - 698*v^5 + 1924*v^4 - 2297*v^3 - 5129*v^2 - 1055*v + 10265) / 1355 $$\beta_{4}$$ $$=$$ $$( 504\nu^{7} - 877\nu^{6} + 946\nu^{5} - 2363\nu^{4} + 2919\nu^{3} + 5125\nu^{2} + 3705\nu - 11505 ) / 1355$$ (504*v^7 - 877*v^6 + 946*v^5 - 2363*v^4 + 2919*v^3 + 5125*v^2 + 3705*v - 11505) / 1355 $$\beta_{5}$$ $$=$$ $$( -613\nu^{7} + 1321\nu^{6} - 1513\nu^{5} + 3394\nu^{4} - 4352\nu^{3} - 6178\nu^{2} - 530\nu + 12985 ) / 1355$$ (-613*v^7 + 1321*v^6 - 1513*v^5 + 3394*v^4 - 4352*v^3 - 6178*v^2 - 530*v + 12985) / 1355 $$\beta_{6}$$ $$=$$ $$( -714\nu^{7} + 1265\nu^{6} - 1295\nu^{5} + 3325\nu^{4} - 3390\nu^{3} - 8367\nu^{2} - 2200\nu + 14605 ) / 1355$$ (-714*v^7 + 1265*v^6 - 1295*v^5 + 3325*v^4 - 3390*v^3 - 8367*v^2 - 2200*v + 14605) / 1355 $$\beta_{7}$$ $$=$$ $$( -1020\nu^{7} + 1962\nu^{6} - 2121\nu^{5} + 5563\nu^{4} - 6314\nu^{3} - 10443\nu^{2} - 3530\nu + 21445 ) / 1355$$ (-1020*v^7 + 1962*v^6 - 2121*v^5 + 5563*v^4 - 6314*v^3 - 10443*v^2 - 3530*v + 21445) / 1355
 $$\nu$$ $$=$$ $$( \beta_{7} - \beta_{6} - 2\beta_{5} + \beta_{4} + \beta_{3} - 2\beta_{2} + 3\beta _1 + 3 ) / 5$$ (b7 - b6 - 2*b5 + b4 + b3 - 2*b2 + 3*b1 + 3) / 5 $$\nu^{2}$$ $$=$$ $$( 2\beta_{7} + 3\beta_{6} - 4\beta_{5} - 3\beta_{4} + 2\beta_{3} + 11\beta_{2} + 6\beta _1 - 4 ) / 5$$ (2*b7 + 3*b6 - 4*b5 - 3*b4 + 2*b3 + 11*b2 + 6*b1 - 4) / 5 $$\nu^{3}$$ $$=$$ $$( -\beta_{7} + 16\beta_{6} + 2\beta_{5} + 4\beta_{4} - 6\beta_{3} + 17\beta_{2} - 3\beta _1 + 2 ) / 5$$ (-b7 + 16*b6 + 2*b5 + 4*b4 - 6*b3 + 17*b2 - 3*b1 + 2) / 5 $$\nu^{4}$$ $$=$$ $$( 13\beta_{7} - 3\beta_{6} - 6\beta_{5} + 13\beta_{4} - 7\beta_{3} - 6\beta_{2} - 6\beta _1 + 14 ) / 5$$ (13*b7 - 3*b6 - 6*b5 + 13*b4 - 7*b3 - 6*b2 - 6*b1 + 14) / 5 $$\nu^{5}$$ $$=$$ $$( 16\beta_{7} - 6\beta_{6} - 22\beta_{5} + 6\beta_{4} + 51\beta_{3} + 43\beta_{2} + 8\beta _1 - 67 ) / 5$$ (16*b7 - 6*b6 - 22*b5 + 6*b4 + 51*b3 + 43*b2 + 8*b1 - 67) / 5 $$\nu^{6}$$ $$=$$ $$( -23\beta_{7} + 63\beta_{6} + 46\beta_{5} + 12\beta_{4} + 52\beta_{3} + 156\beta_{2} - 34\beta _1 - 144 ) / 5$$ (-23*b7 + 63*b6 + 46*b5 + 12*b4 + 52*b3 + 156*b2 - 34*b1 - 144) / 5 $$\nu^{7}$$ $$=$$ $$( -31\beta_{7} - 9\beta_{6} + 137\beta_{5} + 84\beta_{4} - 31\beta_{3} - 33\beta_{2} - 168\beta _1 + 62 ) / 5$$ (-31*b7 - 9*b6 + 137*b5 + 84*b4 - 31*b3 - 33*b2 - 168*b1 + 62) / 5

## Character values

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

 $$n$$ $$101$$ $$177$$ $$351$$ $$\chi(n)$$ $$1$$ $$-\beta_{6}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
81.1
 −0.357358 + 1.86824i 1.66637 − 0.917186i 1.17421 − 0.0566033i −0.983224 + 0.644389i 1.17421 + 0.0566033i −0.983224 − 0.644389i −0.357358 − 1.86824i 1.66637 + 0.917186i
0 1.30902 0.951057i 0 0.279141 2.21858i 0 −3.77447 0 −0.118034 + 0.363271i 0
81.2 0 1.30902 0.951057i 0 1.52988 + 1.63079i 0 2.77447 0 −0.118034 + 0.363271i 0
161.1 0 0.190983 0.587785i 0 −1.39991 + 1.74363i 0 −1.83337 0 2.11803 + 1.53884i 0
161.2 0 0.190983 0.587785i 0 2.09089 0.792578i 0 0.833366 0 2.11803 + 1.53884i 0
241.1 0 0.190983 + 0.587785i 0 −1.39991 1.74363i 0 −1.83337 0 2.11803 1.53884i 0
241.2 0 0.190983 + 0.587785i 0 2.09089 + 0.792578i 0 0.833366 0 2.11803 1.53884i 0
321.1 0 1.30902 + 0.951057i 0 0.279141 + 2.21858i 0 −3.77447 0 −0.118034 0.363271i 0
321.2 0 1.30902 + 0.951057i 0 1.52988 1.63079i 0 2.77447 0 −0.118034 0.363271i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 321.2 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
25.d even 5 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 400.2.u.e 8
4.b odd 2 1 200.2.m.b 8
20.d odd 2 1 1000.2.m.b 8
20.e even 4 2 1000.2.q.b 16
25.d even 5 1 inner 400.2.u.e 8
25.d even 5 1 10000.2.a.q 4
25.e even 10 1 10000.2.a.z 4
100.h odd 10 1 1000.2.m.b 8
100.h odd 10 1 5000.2.a.f 4
100.j odd 10 1 200.2.m.b 8
100.j odd 10 1 5000.2.a.i 4
100.l even 20 2 1000.2.q.b 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
200.2.m.b 8 4.b odd 2 1
200.2.m.b 8 100.j odd 10 1
400.2.u.e 8 1.a even 1 1 trivial
400.2.u.e 8 25.d even 5 1 inner
1000.2.m.b 8 20.d odd 2 1
1000.2.m.b 8 100.h odd 10 1
1000.2.q.b 16 20.e even 4 2
1000.2.q.b 16 100.l even 20 2
5000.2.a.f 4 100.h odd 10 1
5000.2.a.i 4 100.j odd 10 1
10000.2.a.q 4 25.d even 5 1
10000.2.a.z 4 25.e even 10 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{4} - 3T_{3}^{3} + 4T_{3}^{2} - 2T_{3} + 1$$ acting on $$S_{2}^{\mathrm{new}}(400, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$(T^{4} - 3 T^{3} + 4 T^{2} - 2 T + 1)^{2}$$
$5$ $$T^{8} - 5 T^{7} + 15 T^{6} - 35 T^{5} + \cdots + 625$$
$7$ $$(T^{4} + 2 T^{3} - 11 T^{2} - 12 T + 16)^{2}$$
$11$ $$T^{8} - 2 T^{7} + 8 T^{6} - 24 T^{5} + \cdots + 3481$$
$13$ $$T^{8} - 8 T^{7} + 78 T^{6} + \cdots + 6241$$
$17$ $$T^{8} - 10 T^{7} + 75 T^{6} + \cdots + 24025$$
$19$ $$T^{8} + 3 T^{7} + 58 T^{6} + \cdots + 2401$$
$23$ $$T^{8} + 6 T^{7} + 22 T^{6} + \cdots + 2401$$
$29$ $$T^{8} - 6 T^{7} + 52 T^{6} + \cdots + 116281$$
$31$ $$T^{8} - 10 T^{7} + 100 T^{6} + \cdots + 15625$$
$37$ $$T^{8} + 2 T^{7} + 83 T^{6} + \cdots + 4532641$$
$41$ $$T^{8} + 4 T^{7} + 72 T^{6} + \cdots + 7921$$
$43$ $$(T^{4} - 6 T^{3} - 179 T^{2} + 904 T + 5056)^{2}$$
$47$ $$T^{8} - 12 T^{7} + 88 T^{6} + \cdots + 96721$$
$53$ $$T^{8} + 13 T^{7} + 168 T^{6} + \cdots + 495616$$
$59$ $$T^{8} + 29 T^{7} + 472 T^{6} + \cdots + 1936$$
$61$ $$T^{8} - 15 T^{7} + 310 T^{6} + \cdots + 21025$$
$67$ $$T^{8} + 28 T^{7} + 368 T^{6} + \cdots + 961$$
$71$ $$T^{8} - 16 T^{7} + 312 T^{6} + \cdots + 15768841$$
$73$ $$T^{8} - T^{7} + 42 T^{6} + 142 T^{5} + \cdots + 1936$$
$79$ $$T^{8} + 23 T^{7} + 478 T^{6} + \cdots + 9759376$$
$83$ $$T^{8} + 14 T^{7} + 392 T^{6} + \cdots + 8288641$$
$89$ $$T^{8} + 7 T^{7} - 22 T^{6} + \cdots + 80656$$
$97$ $$T^{8} + 3 T^{7} + 98 T^{6} + \cdots + 4330561$$