# Properties

 Label 25.7.c.a Level $25$ Weight $7$ Character orbit 25.c Analytic conductor $5.751$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$25 = 5^{2}$$ Weight: $$k$$ $$=$$ $$7$$ Character orbit: $$[\chi]$$ $$=$$ 25.c (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$5.75135209050$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{4}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -3 - 3 i ) q^{2} + ( 33 - 33 i ) q^{3} -46 i q^{4} -198 q^{6} + ( 207 + 207 i ) q^{7} + ( -330 + 330 i ) q^{8} -1449 i q^{9} +O(q^{10})$$ $$q + ( -3 - 3 i ) q^{2} + ( 33 - 33 i ) q^{3} -46 i q^{4} -198 q^{6} + ( 207 + 207 i ) q^{7} + ( -330 + 330 i ) q^{8} -1449 i q^{9} -1188 q^{11} + ( -1518 - 1518 i ) q^{12} + ( 1548 - 1548 i ) q^{13} -1242 i q^{14} -964 q^{16} + ( 3252 + 3252 i ) q^{17} + ( -4347 + 4347 i ) q^{18} + 5060 i q^{19} + 13662 q^{21} + ( 3564 + 3564 i ) q^{22} + ( 5313 - 5313 i ) q^{23} + 21780 i q^{24} -9288 q^{26} + ( -23760 - 23760 i ) q^{27} + ( 9522 - 9522 i ) q^{28} -8910 i q^{29} + 25432 q^{31} + ( 24012 + 24012 i ) q^{32} + ( -39204 + 39204 i ) q^{33} -19512 i q^{34} -66654 q^{36} + ( 20592 + 20592 i ) q^{37} + ( 15180 - 15180 i ) q^{38} -102168 i q^{39} -19008 q^{41} + ( -40986 - 40986 i ) q^{42} + ( 80343 - 80343 i ) q^{43} + 54648 i q^{44} -31878 q^{46} + ( 16137 + 16137 i ) q^{47} + ( -31812 + 31812 i ) q^{48} -31951 i q^{49} + 214632 q^{51} + ( -71208 - 71208 i ) q^{52} + ( -155892 + 155892 i ) q^{53} + 142560 i q^{54} -136620 q^{56} + ( 166980 + 166980 i ) q^{57} + ( -26730 + 26730 i ) q^{58} + 360180 i q^{59} + 178112 q^{61} + ( -76296 - 76296 i ) q^{62} + ( 299943 - 299943 i ) q^{63} -82376 i q^{64} + 235224 q^{66} + ( -240273 - 240273 i ) q^{67} + ( 149592 - 149592 i ) q^{68} -350658 i q^{69} -617328 q^{71} + ( 478170 + 478170 i ) q^{72} + ( -306612 + 306612 i ) q^{73} -123552 i q^{74} + 232760 q^{76} + ( -245916 - 245916 i ) q^{77} + ( -306504 + 306504 i ) q^{78} -232760 i q^{79} -511839 q^{81} + ( 57024 + 57024 i ) q^{82} + ( -134097 + 134097 i ) q^{83} -628452 i q^{84} -482058 q^{86} + ( -294030 - 294030 i ) q^{87} + ( 392040 - 392040 i ) q^{88} + 270270 i q^{89} + 640872 q^{91} + ( -244398 - 244398 i ) q^{92} + ( 839256 - 839256 i ) q^{93} -96822 i q^{94} + 1584792 q^{96} + ( 810612 + 810612 i ) q^{97} + ( -95853 + 95853 i ) q^{98} + 1721412 i q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 6q^{2} + 66q^{3} - 396q^{6} + 414q^{7} - 660q^{8} + O(q^{10})$$ $$2q - 6q^{2} + 66q^{3} - 396q^{6} + 414q^{7} - 660q^{8} - 2376q^{11} - 3036q^{12} + 3096q^{13} - 1928q^{16} + 6504q^{17} - 8694q^{18} + 27324q^{21} + 7128q^{22} + 10626q^{23} - 18576q^{26} - 47520q^{27} + 19044q^{28} + 50864q^{31} + 48024q^{32} - 78408q^{33} - 133308q^{36} + 41184q^{37} + 30360q^{38} - 38016q^{41} - 81972q^{42} + 160686q^{43} - 63756q^{46} + 32274q^{47} - 63624q^{48} + 429264q^{51} - 142416q^{52} - 311784q^{53} - 273240q^{56} + 333960q^{57} - 53460q^{58} + 356224q^{61} - 152592q^{62} + 599886q^{63} + 470448q^{66} - 480546q^{67} + 299184q^{68} - 1234656q^{71} + 956340q^{72} - 613224q^{73} + 465520q^{76} - 491832q^{77} - 613008q^{78} - 1023678q^{81} + 114048q^{82} - 268194q^{83} - 964116q^{86} - 588060q^{87} + 784080q^{88} + 1281744q^{91} - 488796q^{92} + 1678512q^{93} + 3169584q^{96} + 1621224q^{97} - 191706q^{98} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/25\mathbb{Z}\right)^\times$$.

 $$n$$ $$2$$ $$\chi(n)$$ $$i$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
7.1
 1.00000i − 1.00000i
−3.00000 3.00000i 33.0000 33.0000i 46.0000i 0 −198.000 207.000 + 207.000i −330.000 + 330.000i 1449.00i 0
18.1 −3.00000 + 3.00000i 33.0000 + 33.0000i 46.0000i 0 −198.000 207.000 207.000i −330.000 330.000i 1449.00i 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.c odd 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 25.7.c.a 2
3.b odd 2 1 225.7.g.b 2
5.b even 2 1 25.7.c.b yes 2
5.c odd 4 1 inner 25.7.c.a 2
5.c odd 4 1 25.7.c.b yes 2
15.d odd 2 1 225.7.g.a 2
15.e even 4 1 225.7.g.a 2
15.e even 4 1 225.7.g.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
25.7.c.a 2 1.a even 1 1 trivial
25.7.c.a 2 5.c odd 4 1 inner
25.7.c.b yes 2 5.b even 2 1
25.7.c.b yes 2 5.c odd 4 1
225.7.g.a 2 15.d odd 2 1
225.7.g.a 2 15.e even 4 1
225.7.g.b 2 3.b odd 2 1
225.7.g.b 2 15.e even 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} + 6 T_{2} + 18$$ acting on $$S_{7}^{\mathrm{new}}(25, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + 6 T + 18 T^{2} + 384 T^{3} + 4096 T^{4}$$
$3$ $$1 - 66 T + 2178 T^{2} - 48114 T^{3} + 531441 T^{4}$$
$5$ 1
$7$ $$1 - 414 T + 85698 T^{2} - 48706686 T^{3} + 13841287201 T^{4}$$
$11$ $$( 1 + 1188 T + 1771561 T^{2} )^{2}$$
$13$ $$1 - 3096 T + 4792608 T^{2} - 14943800664 T^{3} + 23298085122481 T^{4}$$
$17$ $$1 - 6504 T + 21151008 T^{2} - 156990748776 T^{3} + 582622237229761 T^{4}$$
$19$ $$1 - 68488162 T^{2} + 2213314919066161 T^{4}$$
$23$ $$1 - 10626 T + 56455938 T^{2} - 1573029356514 T^{3} + 21914624432020321 T^{4}$$
$29$ $$1 - 1110258542 T^{2} + 353814783205469041 T^{4}$$
$31$ $$( 1 - 25432 T + 887503681 T^{2} )^{2}$$
$37$ $$1 - 41184 T + 848060928 T^{2} - 105666876428256 T^{3} + 6582952005840035281 T^{4}$$
$41$ $$( 1 + 19008 T + 4750104241 T^{2} )^{2}$$
$43$ $$1 - 160686 T + 12909995298 T^{2} - 1015754542891614 T^{3} + 39959630797262576401 T^{4}$$
$47$ $$1 - 32274 T + 520805538 T^{2} - 347888395528146 T^{3} +$$$$11\!\cdots\!41$$$$T^{4}$$
$53$ $$1 + 311784 T + 48604631328 T^{2} + 6910493170244136 T^{3} +$$$$49\!\cdots\!41$$$$T^{4}$$
$59$ $$1 + 45368565118 T^{2} +$$$$17\!\cdots\!81$$$$T^{4}$$
$61$ $$( 1 - 178112 T + 51520374361 T^{2} )^{2}$$
$67$ $$1 + 480546 T + 115462229058 T^{2} + 43469413717784274 T^{3} +$$$$81\!\cdots\!61$$$$T^{4}$$
$71$ $$( 1 + 617328 T + 128100283921 T^{2} )^{2}$$
$73$ $$1 + 613224 T + 188021837088 T^{2} + 92801779581845736 T^{3} +$$$$22\!\cdots\!21$$$$T^{4}$$
$79$ $$1 - 431997693442 T^{2} +$$$$59\!\cdots\!41$$$$T^{4}$$
$83$ $$1 + 268194 T + 35964010818 T^{2} + 87683446495325586 T^{3} +$$$$10\!\cdots\!61$$$$T^{4}$$
$89$ $$1 - 920916709022 T^{2} +$$$$24\!\cdots\!21$$$$T^{4}$$
$97$ $$1 - 1621224 T + 1314183629088 T^{2} - 1350434205719013096 T^{3} +$$$$69\!\cdots\!41$$$$T^{4}$$