# Properties

 Label 400.10.a.u Level $400$ Weight $10$ Character orbit 400.a Self dual yes Analytic conductor $206.014$ Analytic rank $1$ Dimension $3$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$400 = 2^{4} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$10$$ Character orbit: $$[\chi]$$ $$=$$ 400.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$206.014334466$$ Analytic rank: $$1$$ Dimension: $$3$$ Coefficient field: $$\mathbb{Q}[x]/(x^{3} - \cdots)$$ Defining polynomial: $$x^{3} - x^{2} - 652 x + 4000$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{4}\cdot 5$$ Twist minimal: no (minimal twist has level 25) 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.

 $$f(q)$$ $$=$$ $$q + ( -30 + \beta_{1} ) q^{3} + ( -1744 - 26 \beta_{1} + 23 \beta_{2} ) q^{7} + ( 19422 - 32 \beta_{1} - 153 \beta_{2} ) q^{9} +O(q^{10})$$ $$q + ( -30 + \beta_{1} ) q^{3} + ( -1744 - 26 \beta_{1} + 23 \beta_{2} ) q^{7} + ( 19422 - 32 \beta_{1} - 153 \beta_{2} ) q^{9} + ( 18298 - 195 \beta_{1} + 95 \beta_{2} ) q^{11} + ( 71808 + 460 \beta_{1} + 209 \beta_{2} ) q^{13} + ( 111605 + 168 \beta_{1} + 835 \beta_{2} ) q^{17} + ( -273798 + 2549 \beta_{1} - 3254 \beta_{2} ) q^{19} + ( -790314 - 4452 \beta_{1} + 2667 \beta_{2} ) q^{21} + ( -1174476 - 3426 \beta_{1} - 1338 \beta_{2} ) q^{23} + ( -2217186 + 18163 \beta_{1} + 13617 \beta_{2} ) q^{27} + ( 727252 - 6276 \beta_{1} - 13429 \beta_{2} ) q^{29} + ( -1425052 + 1090 \beta_{1} - 7390 \beta_{2} ) q^{31} + ( -7376475 + 7288 \beta_{1} + 24420 \beta_{2} ) q^{33} + ( -3447538 + 37572 \beta_{1} + 3831 \beta_{2} ) q^{37} + ( 16789428 + 45808 \beta_{1} - 82293 \beta_{2} ) q^{39} + ( 1962517 + 38760 \beta_{1} + 110540 \beta_{2} ) q^{41} + ( 8142408 + 2732 \beta_{1} + 24439 \beta_{2} ) q^{43} + ( -22283108 - 9384 \beta_{1} + 83831 \beta_{2} ) q^{47} + ( -2224403 + 219280 \beta_{1} - 87880 \beta_{2} ) q^{49} + ( 8541210 + 11069 \beta_{1} - 73299 \beta_{2} ) q^{51} + ( 44311790 - 314856 \beta_{1} + 211780 \beta_{2} ) q^{53} + ( 84278277 + 111584 \beta_{1} - 204519 \beta_{2} ) q^{57} + ( -1775144 - 345528 \beta_{1} + 36413 \beta_{2} ) q^{59} + ( 41835342 - 199100 \beta_{1} - 533275 \beta_{2} ) q^{61} + ( -94577904 - 589692 \beta_{1} + 76428 \beta_{2} ) q^{63} + ( -29717410 + 322747 \beta_{1} - 1059625 \beta_{2} ) q^{67} + ( -104422626 - 1007064 \beta_{1} + 600444 \beta_{2} ) q^{69} + ( -98809132 - 1123200 \beta_{1} - 232175 \beta_{2} ) q^{71} + ( 60459531 - 56864 \beta_{1} - 1374247 \beta_{2} ) q^{73} + ( 186837678 + 701052 \beta_{1} - 70551 \beta_{2} ) q^{77} + ( 103098848 + 728626 \beta_{1} + 1813079 \beta_{2} ) q^{79} + ( 467368353 - 3257696 \beta_{1} - 543609 \beta_{2} ) q^{81} + ( 243751566 - 165897 \beta_{1} - 1002837 \beta_{2} ) q^{83} + ( -349578948 + 2351284 \beta_{1} + 1725681 \beta_{2} ) q^{87} + ( -367574409 - 1136808 \beta_{1} - 929457 \beta_{2} ) q^{89} + ( -393086120 - 3929296 \beta_{1} + 3554866 \beta_{2} ) q^{91} + ( 35975730 - 540432 \beta_{1} + 254460 \beta_{2} ) q^{93} + ( -110724742 - 62616 \beta_{1} + 4150494 \beta_{2} ) q^{97} + ( 299572596 - 6483266 \beta_{1} - 4376889 \beta_{2} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3q - 89q^{3} - 5258q^{7} + 58234q^{9} + O(q^{10})$$ $$3q - 89q^{3} - 5258q^{7} + 58234q^{9} + 54699q^{11} + 215884q^{13} + 334983q^{17} - 818845q^{19} - 2375394q^{21} - 3526854q^{23} - 6633395q^{27} + 2175480q^{29} - 4274066q^{31} - 22122137q^{33} - 10305042q^{37} + 50414092q^{39} + 5926311q^{41} + 24429956q^{43} - 66858708q^{47} - 6453929q^{49} + 25634699q^{51} + 132620514q^{53} + 252946415q^{57} - 5670960q^{59} + 125306926q^{61} - 284323404q^{63} - 88829483q^{67} - 314274942q^{69} - 297550596q^{71} + 181321729q^{73} + 561214086q^{77} + 310025170q^{79} + 1398847363q^{81} + 731088801q^{83} - 1046385560q^{87} - 1103860035q^{89} - 1183187656q^{91} + 107386758q^{93} - 332236842q^{97} + 892234522q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 652 x + 4000$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{2} + 63 \nu - 454$$$$)/6$$ $$\beta_{2}$$ $$=$$ $$($$$$-2 \nu^{2} - 6 \nu + 872$$$$)/3$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{2} + 4 \beta_{1} + 12$$$$)/40$$ $$\nu^{2}$$ $$=$$ $$($$$$-63 \beta_{2} - 12 \beta_{1} + 17404$$$$)/40$$

## 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
 −27.7229 6.48955 22.2334
0 −268.664 0 0 0 637.237 0 52497.3 0
1.2 0 −30.5073 0 0 0 4010.25 0 −18752.3 0
1.3 0 210.171 0 0 0 −9905.49 0 24489.0 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 400.10.a.u 3
4.b odd 2 1 25.10.a.d yes 3
5.b even 2 1 400.10.a.y 3
5.c odd 4 2 400.10.c.q 6
12.b even 2 1 225.10.a.m 3
20.d odd 2 1 25.10.a.c 3
20.e even 4 2 25.10.b.c 6
60.h even 2 1 225.10.a.p 3
60.l odd 4 2 225.10.b.m 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
25.10.a.c 3 20.d odd 2 1
25.10.a.d yes 3 4.b odd 2 1
25.10.b.c 6 20.e even 4 2
225.10.a.m 3 12.b even 2 1
225.10.a.p 3 60.h even 2 1
225.10.b.m 6 60.l odd 4 2
400.10.a.u 3 1.a even 1 1 trivial
400.10.a.y 3 5.b even 2 1
400.10.c.q 6 5.c odd 4 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{3} + 89 T_{3}^{2} - 54681 T_{3} - 1722609$$ acting on $$S_{10}^{\mathrm{new}}(\Gamma_0(400))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3}$$
$3$ $$-1722609 - 54681 T + 89 T^{2} + T^{3}$$
$5$ $$T^{3}$$
$7$ $$25313297688 - 43480164 T + 5258 T^{2} + T^{3}$$
$11$ $$75283351667163 - 1257655633 T - 54699 T^{2} + T^{3}$$
$13$ $$1483699912993984 - 588341776 T - 215884 T^{2} + T^{3}$$
$17$ $$1773847472341707 - 732967429 T - 334983 T^{2} + T^{3}$$
$19$ $$-226123024842854125 - 499889436625 T + 818845 T^{2} + T^{3}$$
$23$ $$381747315559395816 + 3297138740604 T + 3526854 T^{2} + T^{3}$$
$29$ $$3964526545895424000 - 11063731072000 T - 2175480 T^{2} + T^{3}$$
$31$ $$817098195664566648 + 3532627327452 T + 4274066 T^{2} + T^{3}$$
$37$ $$-$$$$17\!\cdots\!28$$$$- 49043023094484 T + 10305042 T^{2} + T^{3}$$
$41$ $$15\!\cdots\!47$$$$- 750102076057093 T - 5926311 T^{2} + T^{3}$$
$43$ $$-$$$$33\!\cdots\!84$$$$+ 168272432263664 T - 24429956 T^{2} + T^{3}$$
$47$ $$14\!\cdots\!08$$$$+ 1159874710756976 T + 66858708 T^{2} + T^{3}$$
$53$ $$40\!\cdots\!84$$$$- 690042287731156 T - 132620514 T^{2} + T^{3}$$
$59$ $$49\!\cdots\!00$$$$- 6651292273432000 T + 5670960 T^{2} + T^{3}$$
$61$ $$62\!\cdots\!12$$$$- 12890308075143508 T - 125306926 T^{2} + T^{3}$$
$67$ $$27\!\cdots\!93$$$$- 51057218268990129 T + 88829483 T^{2} + T^{3}$$
$71$ $$-$$$$11\!\cdots\!32$$$$- 50509406137014928 T + 297550596 T^{2} + T^{3}$$
$73$ $$13\!\cdots\!09$$$$- 82249507598185621 T - 181321729 T^{2} + T^{3}$$
$79$ $$26\!\cdots\!00$$$$- 183462234827962500 T - 310025170 T^{2} + T^{3}$$
$83$ $$83\!\cdots\!41$$$$+ 124612582244716359 T - 731088801 T^{2} + T^{3}$$
$89$ $$-$$$$19\!\cdots\!75$$$$+ 269595002285863875 T + 1103860035 T^{2} + T^{3}$$
$97$ $$-$$$$34\!\cdots\!08$$$$- 792948149717036724 T + 332236842 T^{2} + T^{3}$$