# Properties

 Label 400.10.a.ba Level 400 Weight 10 Character orbit 400.a Self dual yes Analytic conductor 206.014 Analytic rank 1 Dimension 4 CM no Inner twists 2

# 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: $$4$$ Coefficient field: 4.4.49740556.1 Coefficient ring: $$\Z[a_1, \ldots, a_{17}]$$ Coefficient ring index: $$2^{8}\cdot 3^{2}\cdot 5^{2}$$ Twist minimal: no (minimal twist has level 5) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{3} + ( 49 \beta_{1} + 21 \beta_{2} ) q^{7} + ( -2907 + 9 \beta_{3} ) q^{9} +O(q^{10})$$ $$q + \beta_{1} q^{3} + ( 49 \beta_{1} + 21 \beta_{2} ) q^{7} + ( -2907 + 9 \beta_{3} ) q^{9} + ( -27492 - 10 \beta_{3} ) q^{11} + ( 110 \beta_{1} - 507 \beta_{2} ) q^{13} + ( -1632 \beta_{1} + 2270 \beta_{2} ) q^{17} + ( -159220 - 238 \beta_{3} ) q^{19} + ( 880992 + 399 \beta_{3} ) q^{21} + ( 5199 \beta_{1} - 5551 \beta_{2} ) q^{23} + ( -7362 \beta_{1} - 6966 \beta_{2} ) q^{27} + ( 882930 - 988 \beta_{3} ) q^{29} + ( 2646928 - 380 \beta_{3} ) q^{31} + ( -44412 \beta_{1} + 7740 \beta_{2} ) q^{33} + ( -51378 \beta_{1} + 96687 \beta_{2} ) q^{37} + ( 421704 + 2004 \beta_{3} ) q^{39} + ( -4197138 - 12445 \beta_{3} ) q^{41} + ( -132643 \beta_{1} - 146622 \beta_{2} ) q^{43} + ( -214259 \beta_{1} - 199463 \beta_{2} ) q^{47} + ( 11730257 + 13965 \beta_{3} ) q^{49} + ( -21004272 - 19228 \beta_{3} ) q^{51} + ( 259794 \beta_{1} - 402365 \beta_{2} ) q^{53} + ( -561916 \beta_{1} + 184212 \beta_{2} ) q^{57} + ( -115207260 + 26486 \beta_{3} ) q^{59} + ( 90122642 + 21575 \beta_{3} ) q^{61} + ( 591633 \beta_{1} - 722169 \beta_{2} ) q^{63} + ( -1448953 \beta_{1} + 2029650 \beta_{2} ) q^{67} + ( 71631216 + 57893 \beta_{3} ) q^{69} + ( 11902968 - 45600 \beta_{3} ) q^{71} + ( -90564 \beta_{1} - 439344 \beta_{2} ) q^{73} + ( -2162748 \beta_{1} + 157248 \beta_{2} ) q^{77} + ( -182010880 + 133988 \beta_{3} ) q^{79} + ( -85846959 - 229473 \beta_{3} ) q^{81} + ( 1180353 \beta_{1} + 2889326 \beta_{2} ) q^{83} + ( -788766 \beta_{1} + 764712 \beta_{2} ) q^{87} + ( 395675190 + 92796 \beta_{3} ) q^{89} + ( -118330632 + 178752 \beta_{3} ) q^{91} + ( 2003968 \beta_{1} + 294120 \beta_{2} ) q^{93} + ( -1954216 \beta_{1} - 3208062 \beta_{2} ) q^{97} + ( -182196756 - 218358 \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 11628q^{9} + O(q^{10})$$ $$4q - 11628q^{9} - 109968q^{11} - 636880q^{19} + 3523968q^{21} + 3531720q^{29} + 10587712q^{31} + 1686816q^{39} - 16788552q^{41} + 46921028q^{49} - 84017088q^{51} - 460829040q^{59} + 360490568q^{61} + 286524864q^{69} + 47611872q^{71} - 728043520q^{79} - 343387836q^{81} + 1582700760q^{89} - 473322528q^{91} - 728787024q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 45 x^{2} + 304$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$3 \nu^{3} - 51 \nu$$$$)/2$$ $$\beta_{2}$$ $$=$$ $$2 \nu^{3} - 74 \nu$$ $$\beta_{3}$$ $$=$$ $$120 \nu^{2} - 2700$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-3 \beta_{2} + 4 \beta_{1}$$$$)/120$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{3} + 2700$$$$)/120$$ $$\nu^{3}$$ $$=$$ $$($$$$-51 \beta_{2} + 148 \beta_{1}$$$$)/120$$

## 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
 −6.05982 2.87724 −2.87724 6.05982
0 −179.263 0 0 0 −8712.99 0 12452.2 0
1.2 0 −37.6407 0 0 0 −5315.22 0 −18266.2 0
1.3 0 37.6407 0 0 0 5315.22 0 −18266.2 0
1.4 0 179.263 0 0 0 8712.99 0 12452.2 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 400.10.a.ba 4
4.b odd 2 1 25.10.a.e 4
5.b even 2 1 inner 400.10.a.ba 4
5.c odd 4 2 80.10.c.c 4
12.b even 2 1 225.10.a.s 4
20.d odd 2 1 25.10.a.e 4
20.e even 4 2 5.10.b.a 4
60.h even 2 1 225.10.a.s 4
60.l odd 4 2 45.10.b.b 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5.10.b.a 4 20.e even 4 2
25.10.a.e 4 4.b odd 2 1
25.10.a.e 4 20.d odd 2 1
45.10.b.b 4 60.l odd 4 2
80.10.c.c 4 5.c odd 4 2
225.10.a.s 4 12.b even 2 1
225.10.a.s 4 60.h even 2 1
400.10.a.ba 4 1.a even 1 1 trivial
400.10.a.ba 4 5.b even 2 1 inner

## Atkin-Lehner signs

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{4} - 33552 T_{3}^{2} + 45529776$$ acting on $$S_{10}^{\mathrm{new}}(\Gamma_0(400))$$.

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$1 + 45180 T^{2} + 1049244678 T^{4} + 17503657693020 T^{6} + 150094635296999121 T^{8}$$
$5$ 1
$7$ $$1 + 57246700 T^{2} + 3508143545353398 T^{4} +$$$$93\!\cdots\!00$$$$T^{6} +$$$$26\!\cdots\!01$$$$T^{8}$$
$11$ $$( 1 + 54984 T + 5180465446 T^{2} + 129649395841944 T^{3} + 5559917313492231481 T^{4} )^{2}$$
$13$ $$1 + 35613791860 T^{2} +$$$$53\!\cdots\!58$$$$T^{4} +$$$$40\!\cdots\!40$$$$T^{6} +$$$$12\!\cdots\!41$$$$T^{8}$$
$17$ $$1 + 285780369220 T^{2} +$$$$48\!\cdots\!18$$$$T^{4} +$$$$40\!\cdots\!80$$$$T^{6} +$$$$19\!\cdots\!81$$$$T^{8}$$
$19$ $$( 1 + 318440 T + 505756418358 T^{2} + 102756670480744760 T^{3} +$$$$10\!\cdots\!41$$$$T^{4} )^{2}$$
$23$ $$1 + 5779790962540 T^{2} +$$$$14\!\cdots\!38$$$$T^{4} +$$$$18\!\cdots\!60$$$$T^{6} +$$$$10\!\cdots\!61$$$$T^{8}$$
$29$ $$( 1 - 1765860 T + 26950935551038 T^{2} - 25617588792948032340 T^{3} +$$$$21\!\cdots\!61$$$$T^{4} )^{2}$$
$31$ $$( 1 - 5293856 T + 59464921598526 T^{2} -$$$$13\!\cdots\!76$$$$T^{3} +$$$$69\!\cdots\!41$$$$T^{4} )^{2}$$
$37$ $$1 + 231603274936660 T^{2} +$$$$44\!\cdots\!58$$$$T^{4} +$$$$39\!\cdots\!40$$$$T^{6} +$$$$28\!\cdots\!41$$$$T^{8}$$
$41$ $$( 1 + 8394276 T + 221313076168966 T^{2} +$$$$27\!\cdots\!36$$$$T^{3} +$$$$10\!\cdots\!21$$$$T^{4} )^{2}$$
$43$ $$1 + 614109141147100 T^{2} +$$$$57\!\cdots\!98$$$$T^{4} +$$$$15\!\cdots\!00$$$$T^{6} +$$$$63\!\cdots\!01$$$$T^{8}$$
$47$ $$1 + 1368976020813580 T^{2} +$$$$29\!\cdots\!78$$$$T^{4} +$$$$17\!\cdots\!20$$$$T^{6} +$$$$15\!\cdots\!21$$$$T^{8}$$
$53$ $$1 + 7684297973864980 T^{2} +$$$$36\!\cdots\!78$$$$T^{4} +$$$$83\!\cdots\!20$$$$T^{6} +$$$$11\!\cdots\!21$$$$T^{8}$$
$59$ $$( 1 + 230414520 T + 28555631923987078 T^{2} +$$$$19\!\cdots\!80$$$$T^{3} +$$$$75\!\cdots\!21$$$$T^{4} )^{2}$$
$61$ $$( 1 - 180245284 T + 30154717014478446 T^{2} -$$$$21\!\cdots\!44$$$$T^{3} +$$$$13\!\cdots\!81$$$$T^{4} )^{2}$$
$67$ $$1 - 41160407446058180 T^{2} +$$$$18\!\cdots\!18$$$$T^{4} -$$$$30\!\cdots\!20$$$$T^{6} +$$$$54\!\cdots\!81$$$$T^{8}$$
$71$ $$( 1 - 23805936 T + 85782754020107086 T^{2} -$$$$10\!\cdots\!16$$$$T^{3} +$$$$21\!\cdots\!61$$$$T^{4} )^{2}$$
$73$ $$1 + 229489314868712740 T^{2} +$$$$20\!\cdots\!38$$$$T^{4} +$$$$79\!\cdots\!60$$$$T^{6} +$$$$12\!\cdots\!61$$$$T^{8}$$
$79$ $$( 1 + 364021760 T + 220545463862625438 T^{2} +$$$$43\!\cdots\!40$$$$T^{3} +$$$$14\!\cdots\!61$$$$T^{4} )^{2}$$
$83$ $$1 + 434569632367965820 T^{2} +$$$$10\!\cdots\!18$$$$T^{4} +$$$$15\!\cdots\!80$$$$T^{6} +$$$$12\!\cdots\!81$$$$T^{8}$$
$89$ $$( 1 - 791350380 T + 832192702699668118 T^{2} -$$$$27\!\cdots\!20$$$$T^{3} +$$$$12\!\cdots\!81$$$$T^{4} )^{2}$$
$97$ $$1 + 2561123777205326980 T^{2} +$$$$27\!\cdots\!78$$$$T^{4} +$$$$14\!\cdots\!20$$$$T^{6} +$$$$33\!\cdots\!21$$$$T^{8}$$