# Properties

 Label 400.10.a.l Level 400 Weight 10 Character orbit 400.a Self dual yes Analytic conductor 206.014 Analytic rank 0 Dimension 2 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: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{79})$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{4}$$ Twist minimal: no (minimal twist has level 20) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = 16\sqrt{79}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -130 + \beta ) q^{3} + ( -190 + 69 \beta ) q^{7} + ( 17441 - 260 \beta ) q^{9} +O(q^{10})$$ $$q + ( -130 + \beta ) q^{3} + ( -190 + 69 \beta ) q^{7} + ( 17441 - 260 \beta ) q^{9} + ( -51360 - 30 \beta ) q^{11} + ( -89570 - 564 \beta ) q^{13} + ( -158010 - 348 \beta ) q^{17} + ( -68636 - 5160 \beta ) q^{19} + ( 1420156 - 9160 \beta ) q^{21} + ( -332730 - 6393 \beta ) q^{23} + ( -4966780 + 31558 \beta ) q^{27} + ( -3446874 - 5880 \beta ) q^{29} + ( -145916 + 26250 \beta ) q^{31} + ( 6070080 - 47460 \beta ) q^{33} + ( -5630690 + 59976 \beta ) q^{37} + ( 237764 - 16250 \beta ) q^{39} + ( 14886726 + 72180 \beta ) q^{41} + ( -5854090 - 99615 \beta ) q^{43} + ( 31246650 + 42573 \beta ) q^{47} + ( 55968957 - 26220 \beta ) q^{49} + ( 13503348 - 112770 \beta ) q^{51} + ( -4708890 + 212532 \beta ) q^{53} + ( -95433160 + 602164 \beta ) q^{57} + ( 46465428 + 373980 \beta ) q^{59} + ( 97836962 + 721440 \beta ) q^{61} + ( -366132350 + 1252829 \beta ) q^{63} + ( -109883710 + 203139 \beta ) q^{67} + ( -86037132 + 498360 \beta ) q^{69} + ( -155603508 + 1397610 \beta ) q^{71} + ( 49612030 + 2047716 \beta ) q^{73} + ( -32105280 - 3538140 \beta ) q^{77} + ( -271130888 - 70020 \beta ) q^{79} + ( 940619189 - 3951740 \beta ) q^{81} + ( -628457850 + 192957 \beta ) q^{83} + ( 329176500 - 2682474 \beta ) q^{87} + ( -231145926 + 5421960 \beta ) q^{89} + ( -770018884 - 6073170 \beta ) q^{91} + ( 549849080 - 3558416 \beta ) q^{93} + ( -835858370 + 2902956 \beta ) q^{97} + ( -738022560 + 12830370 \beta ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 260q^{3} - 380q^{7} + 34882q^{9} + O(q^{10})$$ $$2q - 260q^{3} - 380q^{7} + 34882q^{9} - 102720q^{11} - 179140q^{13} - 316020q^{17} - 137272q^{19} + 2840312q^{21} - 665460q^{23} - 9933560q^{27} - 6893748q^{29} - 291832q^{31} + 12140160q^{33} - 11261380q^{37} + 475528q^{39} + 29773452q^{41} - 11708180q^{43} + 62493300q^{47} + 111937914q^{49} + 27006696q^{51} - 9417780q^{53} - 190866320q^{57} + 92930856q^{59} + 195673924q^{61} - 732264700q^{63} - 219767420q^{67} - 172074264q^{69} - 311207016q^{71} + 99224060q^{73} - 64210560q^{77} - 542261776q^{79} + 1881238378q^{81} - 1256915700q^{83} + 658353000q^{87} - 462291852q^{89} - 1540037768q^{91} + 1099698160q^{93} - 1671716740q^{97} - 1476045120q^{99} + O(q^{100})$$

## 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
 −8.88819 8.88819
0 −272.211 0 0 0 −10002.6 0 54415.9 0
1.2 0 12.2111 0 0 0 9622.57 0 −19533.9 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 400.10.a.l 2
4.b odd 2 1 100.10.a.c 2
5.b even 2 1 80.10.a.j 2
5.c odd 4 2 400.10.c.l 4
20.d odd 2 1 20.10.a.b 2
20.e even 4 2 100.10.c.c 4
40.e odd 2 1 320.10.a.t 2
40.f even 2 1 320.10.a.l 2
60.h even 2 1 180.10.a.e 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
20.10.a.b 2 20.d odd 2 1
80.10.a.j 2 5.b even 2 1
100.10.a.c 2 4.b odd 2 1
100.10.c.c 4 20.e even 4 2
180.10.a.e 2 60.h even 2 1
320.10.a.l 2 40.f even 2 1
320.10.a.t 2 40.e odd 2 1
400.10.a.l 2 1.a even 1 1 trivial
400.10.c.l 4 5.c odd 4 2

## Atkin-Lehner signs

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{2} + 260 T_{3} - 3324$$ acting on $$S_{10}^{\mathrm{new}}(\Gamma_0(400))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$1 + 260 T + 36042 T^{2} + 5117580 T^{3} + 387420489 T^{4}$$
$5$ 1
$7$ $$1 + 380 T - 15543150 T^{2} + 15334370660 T^{3} + 1628413597910449 T^{4}$$
$11$ $$1 + 102720 T + 7335543382 T^{2} + 242208386819520 T^{3} + 5559917313492231481 T^{4}$$
$13$ $$1 + 179140 T + 22798610142 T^{2} + 1899690017679220 T^{3} +$$$$11\!\cdots\!29$$$$T^{4}$$
$17$ $$1 + 316020 T + 259693705798 T^{2} + 37476140730581940 T^{3} +$$$$14\!\cdots\!09$$$$T^{4}$$
$19$ $$1 + 137272 T + 111610161654 T^{2} + 44295985649518888 T^{3} +$$$$10\!\cdots\!41$$$$T^{4}$$
$23$ $$1 + 665460 T + 2886450615250 T^{2} + 1198595050097167980 T^{3} +$$$$32\!\cdots\!69$$$$T^{4}$$
$29$ $$1 + 6893748 T + 40195999658014 T^{2} +$$$$10\!\cdots\!12$$$$T^{3} +$$$$21\!\cdots\!61$$$$T^{4}$$
$31$ $$1 + 291832 T + 38964935800398 T^{2} + 7715927814392939272 T^{3} +$$$$69\!\cdots\!41$$$$T^{4}$$
$37$ $$1 + 11261380 T + 218879982937230 T^{2} +$$$$14\!\cdots\!60$$$$T^{3} +$$$$16\!\cdots\!29$$$$T^{4}$$
$41$ $$1 - 29773452 T + 771012402449398 T^{2} -$$$$97\!\cdots\!72$$$$T^{3} +$$$$10\!\cdots\!21$$$$T^{4}$$
$43$ $$1 + 11708180 T + 838769843899386 T^{2} +$$$$58\!\cdots\!40$$$$T^{3} +$$$$25\!\cdots\!49$$$$T^{4}$$
$47$ $$1 - 62493300 T + 3177958884734338 T^{2} -$$$$69\!\cdots\!00$$$$T^{3} +$$$$12\!\cdots\!89$$$$T^{4}$$
$53$ $$1 + 9417780 T + 5708185761526990 T^{2} +$$$$31\!\cdots\!40$$$$T^{3} +$$$$10\!\cdots\!89$$$$T^{4}$$
$59$ $$1 - 92930856 T + 16656477955483462 T^{2} -$$$$80\!\cdots\!84$$$$T^{3} +$$$$75\!\cdots\!21$$$$T^{4}$$
$61$ $$1 - 195673924 T + 22434263296171326 T^{2} -$$$$22\!\cdots\!84$$$$T^{3} +$$$$13\!\cdots\!81$$$$T^{4}$$
$67$ $$1 + 219767420 T + 65652945987990090 T^{2} +$$$$59\!\cdots\!40$$$$T^{3} +$$$$74\!\cdots\!09$$$$T^{4}$$
$71$ $$1 + 311207016 T + 76405636625293726 T^{2} +$$$$14\!\cdots\!96$$$$T^{3} +$$$$21\!\cdots\!61$$$$T^{4}$$
$73$ $$1 - 99224060 T + 35402447061205782 T^{2} -$$$$58\!\cdots\!80$$$$T^{3} +$$$$34\!\cdots\!69$$$$T^{4}$$
$79$ $$1 + 542261776 T + 313115996157615582 T^{2} +$$$$64\!\cdots\!44$$$$T^{3} +$$$$14\!\cdots\!61$$$$T^{4}$$
$83$ $$1 + 1256915700 T + 768086791626261130 T^{2} +$$$$23\!\cdots\!00$$$$T^{3} +$$$$34\!\cdots\!09$$$$T^{4}$$
$89$ $$1 + 462291852 T + 159603168035249494 T^{2} +$$$$16\!\cdots\!68$$$$T^{3} +$$$$12\!\cdots\!81$$$$T^{4}$$
$97$ $$1 + 1671716740 T + 2048690578856969670 T^{2} +$$$$12\!\cdots\!80$$$$T^{3} +$$$$57\!\cdots\!89$$$$T^{4}$$