# Properties

 Label 25.12.b.c Level 25 Weight 12 Character orbit 25.b Analytic conductor 19.209 Analytic rank 0 Dimension 4 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ = $$25 = 5^{2}$$ Weight: $$k$$ = $$12$$ Character orbit: $$[\chi]$$ = 25.b (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$19.2085795140$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{151})$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{6}\cdot 5^{2}$$ Twist minimal: no (minimal twist has level 5) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( \beta_{1} - 3 \beta_{3} ) q^{2} + ( -11 \beta_{1} + 16 \beta_{3} ) q^{3} + ( -3488 + 6 \beta_{2} ) q^{4} + ( 30092 - 49 \beta_{2} ) q^{6} + ( -2895 \beta_{1} - 528 \beta_{3} ) q^{7} + ( -12312 \beta_{1} + 4920 \beta_{3} ) q^{8} + ( 10423 + 352 \beta_{2} ) q^{9} +O(q^{10})$$ $$q + ( \beta_{1} - 3 \beta_{3} ) q^{2} + ( -11 \beta_{1} + 16 \beta_{3} ) q^{3} + ( -3488 + 6 \beta_{2} ) q^{4} + ( 30092 - 49 \beta_{2} ) q^{6} + ( -2895 \beta_{1} - 528 \beta_{3} ) q^{7} + ( -12312 \beta_{1} + 4920 \beta_{3} ) q^{8} + ( 10423 + 352 \beta_{2} ) q^{9} + ( -309088 - 2640 \beta_{2} ) q^{11} + ( 96352 \beta_{1} - 62408 \beta_{3} ) q^{12} + ( 170713 \beta_{1} - 12864 \beta_{3} ) q^{13} + ( -667236 - 8157 \beta_{2} ) q^{14} + ( 3002816 - 29568 \beta_{2} ) q^{16} + ( -65897 \beta_{1} - 126528 \beta_{3} ) q^{17} + ( -627401 \beta_{1} + 3931 \beta_{3} ) q^{18} + ( -2662660 - 27456 \beta_{2} ) q^{19} + ( 1918092 + 40512 \beta_{2} ) q^{21} + ( 4474592 \beta_{1} + 663264 \beta_{3} ) q^{22} + ( 2947197 \beta_{1} + 33456 \beta_{3} ) q^{23} + ( -61090080 + 251112 \beta_{2} ) q^{24} + ( -40380868 + 525003 \beta_{2} ) q^{26} + ( 1338458 \beta_{1} + 2613920 \beta_{3} ) q^{27} + ( 8184288 \beta_{1} + 104664 \beta_{3} ) q^{28} + ( -47070190 - 229824 \beta_{2} ) q^{29} + ( 122271732 - 720720 \beta_{2} ) q^{31} + ( 31365056 \beta_{1} - 1889088 \beta_{3} ) q^{32} + ( -22112992 \beta_{1} - 2041408 \beta_{3} ) q^{33} + ( -222679036 - 71163 \beta_{2} ) q^{34} + ( 91209376 - 1165238 \beta_{2} ) q^{36} + ( -1050161 \beta_{1} - 19033728 \beta_{3} ) q^{37} + ( 47087612 \beta_{1} + 5242380 \beta_{3} ) q^{38} + ( 312101996 - 2872912 \beta_{2} ) q^{39} + ( -372871658 - 2265120 \beta_{2} ) q^{41} + ( -71489652 \beta_{1} - 1703076 \beta_{3} ) q^{42} + ( 31497505 \beta_{1} - 13909104 \beta_{3} ) q^{43} + ( 121362944 + 7353792 \beta_{2} ) q^{44} + ( -234097428 + 8808135 \beta_{2} ) q^{46} + ( 70103077 \beta_{1} + 20505072 \beta_{3} ) q^{47} + ( -318776128 \beta_{1} + 80569856 \beta_{3} ) q^{48} + ( 970838707 - 3057120 \beta_{2} ) q^{49} + ( 1150279892 - 337456 \beta_{2} ) q^{51} + ( -642066080 \beta_{1} + 147297432 \beta_{3} ) q^{52} + ( 56916029 \beta_{1} - 186753984 \beta_{3} ) q^{53} + ( 4602577240 + 1401454 \beta_{2} ) q^{54} + ( -1995276960 + 7742664 \beta_{2} ) q^{56} + ( -236045524 \beta_{1} - 12400960 \beta_{3} ) q^{57} + ( 369370898 \beta_{1} + 118228170 \beta_{3} ) q^{58} + ( -3658757780 - 17581728 \beta_{2} ) q^{59} + ( -758212838 - 5356800 \beta_{2} ) q^{61} + ( 1428216372 \beta_{1} - 438887196 \beta_{3} ) q^{62} + ( -142431609 \beta_{1} - 107407344 \beta_{3} ) q^{63} + ( -409765888 + 35428992 \beta_{2} ) q^{64} + ( -1487732096 - 64297568 \beta_{2} ) q^{66} + ( -786714507 \beta_{1} + 91691472 \beta_{3} ) q^{67} + ( -228688736 \beta_{1} + 401791464 \beta_{3} ) q^{68} + ( 2918597916 - 46787136 \beta_{2} ) q^{69} + ( 16469235772 - 5480400 \beta_{2} ) q^{71} + ( 917703384 \beta_{1} - 382101240 \beta_{3} ) q^{72} + ( -1499142443 \beta_{1} + 339617856 \beta_{3} ) q^{73} + ( -34384099036 + 15883245 \beta_{2} ) q^{74} + ( -662696320 + 79790568 \beta_{2} ) q^{76} + ( 1736737440 \beta_{1} + 927478464 \beta_{3} ) q^{77} + ( 5517818540 \beta_{1} - 1223597188 \beta_{3} ) q^{78} + ( 1651411560 + 57563616 \beta_{2} ) q^{79} + ( -21942215899 + 69693536 \beta_{2} ) q^{81} + ( 3731525782 \beta_{1} + 892102974 \beta_{3} ) q^{82} + ( 664955121 \beta_{1} - 1100818224 \beta_{3} ) q^{83} + ( 7991243904 - 129797304 \beta_{2} ) q^{84} + ( -28353046948 + 108401619 \beta_{2} ) q^{86} + ( -1703247046 \beta_{1} - 500316640 \beta_{3} ) q^{87} + ( -4039743744 \beta_{1} + 1729655040 \beta_{3} ) q^{88} + ( 6337385430 + 145528128 \beta_{2} ) q^{89} + ( 45318929532 + 52895184 \beta_{2} ) q^{91} + ( -10158578592 \beta_{1} + 1651623672 \beta_{3} ) q^{92} + ( -8310027132 \beta_{1} + 2749139712 \beta_{3} ) q^{93} + ( 30144882764 + 189804159 \beta_{2} ) q^{94} + ( 52757708032 - 522620864 \beta_{2} ) q^{96} + ( 154035187 \beta_{1} - 4545870528 \beta_{3} ) q^{97} + ( 6510340147 \beta_{1} - 3218228121 \beta_{3} ) q^{98} + ( -59350136224 - 136315696 \beta_{2} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 13952q^{4} + 120368q^{6} + 41692q^{9} + O(q^{10})$$ $$4q - 13952q^{4} + 120368q^{6} + 41692q^{9} - 1236352q^{11} - 2668944q^{14} + 12011264q^{16} - 10650640q^{19} + 7672368q^{21} - 244360320q^{24} - 161523472q^{26} - 188280760q^{29} + 489086928q^{31} - 890716144q^{34} + 364837504q^{36} + 1248407984q^{39} - 1491486632q^{41} + 485451776q^{44} - 936389712q^{46} + 3883354828q^{49} + 4601119568q^{51} + 18410308960q^{54} - 7981107840q^{56} - 14635031120q^{59} - 3032851352q^{61} - 1639063552q^{64} - 5950928384q^{66} + 11674391664q^{69} + 65876943088q^{71} - 137536396144q^{74} - 2650785280q^{76} + 6605646240q^{79} - 87768863596q^{81} + 31964975616q^{84} - 113412187792q^{86} + 25349541720q^{89} + 181275718128q^{91} + 120579531056q^{94} + 211030832128q^{96} - 237400544896q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 75 x^{2} + 1444$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$5 \nu^{3} - 185 \nu$$$$)/19$$ $$\beta_{2}$$ $$=$$ $$($$$$-10 \nu^{3} + 1130 \nu$$$$)/19$$ $$\beta_{3}$$ $$=$$ $$4 \nu^{2} - 150$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{2} + 2 \beta_{1}$$$$)/40$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{3} + 150$$$$)/4$$ $$\nu^{3}$$ $$=$$ $$($$$$37 \beta_{2} + 226 \beta_{1}$$$$)/40$$

## Character values

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

 $$n$$ $$2$$ $$\chi(n)$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
24.1
 −6.14410 − 0.500000i 6.14410 + 0.500000i 6.14410 − 0.500000i −6.14410 + 0.500000i
83.7292i 503.223i −4962.58 0 42134.4 15973.7i 244036.i −76086.0 0
24.2 63.7292i 283.223i −2013.42 0 18049.6 41926.3i 2204.06i 96932.0 0
24.3 63.7292i 283.223i −2013.42 0 18049.6 41926.3i 2204.06i 96932.0 0
24.4 83.7292i 503.223i −4962.58 0 42134.4 15973.7i 244036.i −76086.0 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 25.12.b.c 4
3.b odd 2 1 225.12.b.f 4
5.b even 2 1 inner 25.12.b.c 4
5.c odd 4 1 5.12.a.b 2
5.c odd 4 1 25.12.a.c 2
15.d odd 2 1 225.12.b.f 4
15.e even 4 1 45.12.a.d 2
15.e even 4 1 225.12.a.h 2
20.e even 4 1 80.12.a.j 2
35.f even 4 1 245.12.a.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5.12.a.b 2 5.c odd 4 1
25.12.a.c 2 5.c odd 4 1
25.12.b.c 4 1.a even 1 1 trivial
25.12.b.c 4 5.b even 2 1 inner
45.12.a.d 2 15.e even 4 1
80.12.a.j 2 20.e even 4 1
225.12.a.h 2 15.e even 4 1
225.12.b.f 4 3.b odd 2 1
225.12.b.f 4 15.d odd 2 1
245.12.a.b 2 35.f even 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{4} + 11072 T_{2}^{2} + 28472896$$ acting on $$S_{12}^{\mathrm{new}}(25, [\chi])$$.

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$1 + 2880 T^{2} + 8287808 T^{4} + 12079595520 T^{6} + 17592186044416 T^{8}$$
$3$ $$1 - 375140 T^{2} + 90460822518 T^{4} - 11772290701720260 T^{6} +$$$$98\!\cdots\!81$$$$T^{8}$$
$5$ 1
$7$ $$1 - 5896330900 T^{2} + 15946824262997918598 T^{4} -$$$$23\!\cdots\!00$$$$T^{6} +$$$$15\!\cdots\!01$$$$T^{8}$$
$11$ $$( 1 + 618176 T + 245194892966 T^{2} + 176372827291625536 T^{3} +$$$$81\!\cdots\!21$$$$T^{4} )^{2}$$
$13$ $$1 - 1140153047180 T^{2} +$$$$55\!\cdots\!38$$$$T^{4} -$$$$36\!\cdots\!20$$$$T^{6} +$$$$10\!\cdots\!61$$$$T^{8}$$
$17$ $$1 - 116879825889660 T^{2} +$$$$57\!\cdots\!78$$$$T^{4} -$$$$13\!\cdots\!40$$$$T^{6} +$$$$13\!\cdots\!21$$$$T^{8}$$
$19$ $$( 1 + 5325320 T + 194538827137638 T^{2} +$$$$62\!\cdots\!80$$$$T^{3} +$$$$13\!\cdots\!61$$$$T^{4} )^{2}$$
$23$ $$1 - 2072692881139220 T^{2} +$$$$28\!\cdots\!58$$$$T^{4} -$$$$18\!\cdots\!80$$$$T^{6} +$$$$82\!\cdots\!41$$$$T^{8}$$
$29$ $$( 1 + 94140380 T + 23426350431097358 T^{2} +$$$$11\!\cdots\!20$$$$T^{3} +$$$$14\!\cdots\!41$$$$T^{4} )^{2}$$
$31$ $$( 1 - 244543464 T + 34393316207729486 T^{2} -$$$$62\!\cdots\!84$$$$T^{3} +$$$$64\!\cdots\!61$$$$T^{4} )^{2}$$
$37$ $$1 - 273812295186452780 T^{2} +$$$$81\!\cdots\!38$$$$T^{4} -$$$$86\!\cdots\!20$$$$T^{6} +$$$$10\!\cdots\!61$$$$T^{8}$$
$41$ $$( 1 + 745743316 T + 929792912462405846 T^{2} +$$$$41\!\cdots\!56$$$$T^{3} +$$$$30\!\cdots\!81$$$$T^{4} )^{2}$$
$43$ $$1 - 3285052879347844100 T^{2} +$$$$43\!\cdots\!98$$$$T^{4} -$$$$28\!\cdots\!00$$$$T^{6} +$$$$74\!\cdots\!01$$$$T^{8}$$
$47$ $$1 - 8397835342270441140 T^{2} +$$$$29\!\cdots\!18$$$$T^{4} -$$$$51\!\cdots\!60$$$$T^{6} +$$$$37\!\cdots\!81$$$$T^{8}$$
$53$ $$1 + 5703220206102687060 T^{2} +$$$$15\!\cdots\!18$$$$T^{4} +$$$$48\!\cdots\!40$$$$T^{6} +$$$$73\!\cdots\!81$$$$T^{8}$$
$59$ $$( 1 + 7317515560 T + 55027608950440780118 T^{2} +$$$$22\!\cdots\!40$$$$T^{3} +$$$$90\!\cdots\!81$$$$T^{4} )^{2}$$
$61$ $$( 1 + 1516425676 T + 85869525433683691566 T^{2} +$$$$65\!\cdots\!36$$$$T^{3} +$$$$18\!\cdots\!21$$$$T^{4} )^{2}$$
$67$ $$1 -$$$$35\!\cdots\!60$$$$T^{2} +$$$$60\!\cdots\!78$$$$T^{4} -$$$$52\!\cdots\!40$$$$T^{6} +$$$$22\!\cdots\!21$$$$T^{8}$$
$71$ $$( 1 - 32938471544 T +$$$$73\!\cdots\!26$$$$T^{2} -$$$$76\!\cdots\!24$$$$T^{3} +$$$$53\!\cdots\!41$$$$T^{4} )^{2}$$
$73$ $$1 -$$$$66\!\cdots\!20$$$$T^{2} +$$$$24\!\cdots\!58$$$$T^{4} -$$$$65\!\cdots\!80$$$$T^{6} +$$$$96\!\cdots\!41$$$$T^{8}$$
$79$ $$( 1 - 3302823120 T +$$$$12\!\cdots\!58$$$$T^{2} -$$$$24\!\cdots\!80$$$$T^{3} +$$$$55\!\cdots\!41$$$$T^{4} )^{2}$$
$83$ $$1 -$$$$35\!\cdots\!60$$$$T^{2} +$$$$64\!\cdots\!78$$$$T^{4} -$$$$59\!\cdots\!40$$$$T^{6} +$$$$27\!\cdots\!21$$$$T^{8}$$
$89$ $$( 1 - 12674770860 T +$$$$43\!\cdots\!78$$$$T^{2} -$$$$35\!\cdots\!40$$$$T^{3} +$$$$77\!\cdots\!21$$$$T^{4} )^{2}$$
$97$ $$1 -$$$$36\!\cdots\!40$$$$T^{2} +$$$$10\!\cdots\!18$$$$T^{4} -$$$$18\!\cdots\!60$$$$T^{6} +$$$$26\!\cdots\!81$$$$T^{8}$$