# Properties

 Label 25.6.b.b Level 25 Weight 6 Character orbit 25.b Analytic conductor 4.010 Analytic rank 0 Dimension 4 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$25 = 5^{2}$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 25.b (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.00959549532$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{241})$$ Defining polynomial: $$x^{4} + 121 x^{2} + 3600$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$5^{2}$$ Twist minimal: yes 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.

 $$f(q)$$ $$=$$ $$q + ( -\beta_{1} + \beta_{2} ) q^{2} + ( -2 \beta_{1} - \beta_{2} ) q^{3} + ( -35 + \beta_{3} ) q^{4} + ( -95 - \beta_{3} ) q^{6} + ( 4 \beta_{1} + 18 \beta_{2} ) q^{7} + ( 15 \beta_{1} - 69 \beta_{2} ) q^{8} + ( -94 - 8 \beta_{3} ) q^{9} +O(q^{10})$$ $$q + ( -\beta_{1} + \beta_{2} ) q^{2} + ( -2 \beta_{1} - \beta_{2} ) q^{3} + ( -35 + \beta_{3} ) q^{4} + ( -95 - \beta_{3} ) q^{6} + ( 4 \beta_{1} + 18 \beta_{2} ) q^{7} + ( 15 \beta_{1} - 69 \beta_{2} ) q^{8} + ( -94 - 8 \beta_{3} ) q^{9} + ( -103 + 10 \beta_{3} ) q^{11} + ( 19 \beta_{1} - 61 \beta_{2} ) q^{12} + ( -32 \beta_{1} + 52 \beta_{2} ) q^{13} + ( -18 + 18 \beta_{3} ) q^{14} + ( 587 - 37 \beta_{3} ) q^{16} + ( -136 \beta_{1} + 217 \beta_{2} ) q^{17} + ( -2 \beta_{1} + 434 \beta_{2} ) q^{18} + ( 1583 + 14 \beta_{3} ) q^{19} + ( 1458 + 48 \beta_{3} ) q^{21} + ( 223 \beta_{1} - 763 \beta_{2} ) q^{22} + ( -12 \beta_{1} - 150 \beta_{2} ) q^{23} + ( -1221 - 93 \beta_{3} ) q^{24} + ( -2404 + 52 \beta_{3} ) q^{26} + ( 110 \beta_{1} + 619 \beta_{2} ) q^{27} + ( 362 \beta_{1} - 630 \beta_{2} ) q^{28} + ( 1952 + 16 \beta_{3} ) q^{29} + ( -438 - 220 \beta_{3} ) q^{31} + ( -551 \beta_{1} + 821 \beta_{2} ) q^{32} + ( -304 \beta_{1} - 857 \beta_{2} ) q^{33} + ( -10165 + 217 \beta_{3} ) q^{34} + ( -8758 + 178 \beta_{3} ) q^{36} + ( 384 \beta_{1} + 10 \beta_{2} ) q^{37} + ( -1415 \beta_{1} + 659 \beta_{2} ) q^{38} + ( -2060 + 8 \beta_{3} ) q^{39} + ( 13837 + 80 \beta_{3} ) q^{41} + ( -882 \beta_{1} - 1710 \beta_{2} ) q^{42} + ( 2128 \beta_{1} - 764 \beta_{2} ) q^{43} + ( 18665 - 443 \beta_{3} ) q^{44} + ( 1302 - 150 \beta_{3} ) q^{46} + ( 1544 \beta_{1} + 1804 \beta_{2} ) q^{47} + ( 713 \beta_{1} + 2965 \beta_{2} ) q^{48} + ( 5923 - 160 \beta_{3} ) q^{49} + ( -8951 + 26 \beta_{3} ) q^{51} + ( 2004 \beta_{1} - 4172 \beta_{2} ) q^{52} + ( -752 \beta_{1} + 3074 \beta_{2} ) q^{53} + ( -2107 + 619 \beta_{3} ) q^{54} + ( 27162 - 54 \beta_{3} ) q^{56} + ( -3880 \beta_{1} - 2927 \beta_{2} ) q^{57} + ( -1760 \beta_{1} + 896 \beta_{2} ) q^{58} + ( -6176 + 392 \beta_{3} ) q^{59} + ( -12398 + 400 \beta_{3} ) q^{61} + ( -2202 \beta_{1} + 14082 \beta_{2} ) q^{62} + ( -4392 \beta_{1} - 1692 \beta_{2} ) q^{63} + ( -21643 - 363 \beta_{3} ) q^{64} + ( -5275 - 857 \beta_{3} ) q^{66} + ( -1586 \beta_{1} - 3213 \beta_{2} ) q^{67} + ( 8417 \beta_{1} - 17543 \beta_{2} ) q^{68} + ( -9078 - 336 \beta_{3} ) q^{69} + ( -43748 + 200 \beta_{3} ) q^{71} + ( 10830 \beta_{1} - 6618 \beta_{2} ) q^{72} + ( -1112 \beta_{1} + 7585 \beta_{2} ) q^{73} + ( 20606 + 10 \beta_{3} ) q^{74} + ( -34321 + 1107 \beta_{3} ) q^{76} + ( 4608 \beta_{1} - 1854 \beta_{2} ) q^{77} + ( 2156 \beta_{1} - 2588 \beta_{2} ) q^{78} + ( -32238 - 1004 \beta_{3} ) q^{79} + ( 23329 - 376 \beta_{3} ) q^{81} + ( -12877 \beta_{1} + 8557 \beta_{2} ) q^{82} + ( 858 \beta_{1} - 9687 \beta_{2} ) q^{83} + ( 21258 - 174 \beta_{3} ) q^{84} + ( 124844 - 764 \beta_{3} ) q^{86} + ( -4720 \beta_{1} - 3488 \beta_{2} ) q^{87} + ( -16845 \beta_{1} + 23487 \beta_{2} ) q^{88} + ( 35361 + 2088 \beta_{3} ) q^{89} + ( -10536 + 496 \beta_{3} ) q^{91} + ( -3486 \beta_{1} + 6402 \beta_{2} ) q^{92} + ( 12096 \beta_{1} + 21558 \beta_{2} ) q^{93} + ( 59924 + 1804 \beta_{3} ) q^{94} + ( -39115 - 11 \beta_{3} ) q^{96} + ( 10944 \beta_{1} - 18086 \beta_{2} ) q^{97} + ( -7843 \beta_{1} + 16483 \beta_{2} ) q^{98} + ( -110798 - 196 \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 138q^{4} - 382q^{6} - 392q^{9} + O(q^{10})$$ $$4q - 138q^{4} - 382q^{6} - 392q^{9} - 392q^{11} - 36q^{14} + 2274q^{16} + 6360q^{19} + 5928q^{21} - 5070q^{24} - 9512q^{26} + 7840q^{29} - 2192q^{31} - 40226q^{34} - 34676q^{36} - 8224q^{39} + 55508q^{41} + 73774q^{44} + 4908q^{46} + 23372q^{49} - 35752q^{51} - 7190q^{54} + 108540q^{56} - 23920q^{59} - 48792q^{61} - 87298q^{64} - 22814q^{66} - 36984q^{69} - 174592q^{71} + 82444q^{74} - 135070q^{76} - 130960q^{79} + 92564q^{81} + 84684q^{84} + 497848q^{86} + 145620q^{89} - 41152q^{91} + 243304q^{94} - 156482q^{96} - 443584q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 121 x^{2} + 3600$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-\nu^{3} - 81 \nu$$$$)/20$$ $$\beta_{2}$$ $$=$$ $$($$$$-\nu^{3} - 61 \nu$$$$)/12$$ $$\beta_{3}$$ $$=$$ $$5 \nu^{2} + 303$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$3 \beta_{2} - 5 \beta_{1}$$$$)/5$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{3} - 303$$$$)/5$$ $$\nu^{3}$$ $$=$$ $$($$$$-243 \beta_{2} + 305 \beta_{1}$$$$)/5$$

## 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
 − 8.26209i − 7.26209i 7.26209i 8.26209i
10.2621i 5.52417i −73.3104 0 −56.6896 68.9517i 423.931i 212.483 0
24.2 5.26209i 25.5242i 4.31044 0 −134.310 131.048i 191.069i −408.483 0
24.3 5.26209i 25.5242i 4.31044 0 −134.310 131.048i 191.069i −408.483 0
24.4 10.2621i 5.52417i −73.3104 0 −56.6896 68.9517i 423.931i 212.483 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 25.6.b.b 4
3.b odd 2 1 225.6.b.i 4
4.b odd 2 1 400.6.c.n 4
5.b even 2 1 inner 25.6.b.b 4
5.c odd 4 1 25.6.a.b 2
5.c odd 4 1 25.6.a.d yes 2
15.d odd 2 1 225.6.b.i 4
15.e even 4 1 225.6.a.l 2
15.e even 4 1 225.6.a.s 2
20.d odd 2 1 400.6.c.n 4
20.e even 4 1 400.6.a.o 2
20.e even 4 1 400.6.a.w 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
25.6.a.b 2 5.c odd 4 1
25.6.a.d yes 2 5.c odd 4 1
25.6.b.b 4 1.a even 1 1 trivial
25.6.b.b 4 5.b even 2 1 inner
225.6.a.l 2 15.e even 4 1
225.6.a.s 2 15.e even 4 1
225.6.b.i 4 3.b odd 2 1
225.6.b.i 4 15.d odd 2 1
400.6.a.o 2 20.e even 4 1
400.6.a.w 2 20.e even 4 1
400.6.c.n 4 4.b odd 2 1
400.6.c.n 4 20.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{4} + 133 T_{2}^{2} + 2916$$ acting on $$S_{6}^{\mathrm{new}}(25, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + 5 T^{2} + 548 T^{4} + 5120 T^{6} + 1048576 T^{8}$$
$3$ $$1 - 290 T^{2} + 42723 T^{4} - 17124210 T^{6} + 3486784401 T^{8}$$
$5$ 1
$7$ $$1 - 45300 T^{2} + 1039412998 T^{4} - 12796128779700 T^{6} + 79792266297612001 T^{8}$$
$11$ $$( 1 + 196 T + 181081 T^{2} + 31565996 T^{3} + 25937424601 T^{4} )^{2}$$
$13$ $$1 - 1296980 T^{2} + 688260462198 T^{4} - 178799706758316020 T^{6} +$$$$19\!\cdots\!01$$$$T^{8}$$
$17$ $$1 - 2340610 T^{2} + 2927557675523 T^{4} - 4718655483329933890 T^{6} +$$$$40\!\cdots\!01$$$$T^{8}$$
$19$ $$( 1 - 3180 T + 7185073 T^{2} - 7873994820 T^{3} + 6131066257801 T^{4} )^{2}$$
$23$ $$1 - 24511220 T^{2} + 233031884985798 T^{4} -$$$$10\!\cdots\!80$$$$T^{6} +$$$$17\!\cdots\!01$$$$T^{8}$$
$29$ $$( 1 - 3920 T + 44478298 T^{2} - 80403704080 T^{3} + 420707233300201 T^{4} )^{2}$$
$31$ $$( 1 + 1096 T - 15343894 T^{2} + 31377549496 T^{3} + 819628286980801 T^{4} )^{2}$$
$37$ $$1 - 257567180 T^{2} + 26166130610514198 T^{4} -$$$$12\!\cdots\!20$$$$T^{6} +$$$$23\!\cdots\!01$$$$T^{8}$$
$41$ $$( 1 - 27754 T + 414643531 T^{2} - 3215473002554 T^{3} + 13422659310152401 T^{4} )^{2}$$
$43$ $$1 - 37863500 T^{2} + 41125859560630998 T^{4} -$$$$81\!\cdots\!00$$$$T^{6} +$$$$46\!\cdots\!01$$$$T^{8}$$
$47$ $$1 - 298326940 T^{2} + 32136931855726598 T^{4} -$$$$15\!\cdots\!60$$$$T^{6} +$$$$27\!\cdots\!01$$$$T^{8}$$
$53$ $$1 - 1240678540 T^{2} + 709794326287792598 T^{4} -$$$$21\!\cdots\!60$$$$T^{6} +$$$$30\!\cdots\!01$$$$T^{8}$$
$59$ $$( 1 + 11960 T + 1234152598 T^{2} + 8550494616040 T^{3} + 511116753300641401 T^{4} )^{2}$$
$61$ $$( 1 + 24396 T + 1596983806 T^{2} + 20604771359196 T^{3} + 713342911662882601 T^{4} )^{2}$$
$67$ $$1 - 4294993410 T^{2} + 8014205534576787523 T^{4} -$$$$78\!\cdots\!90$$$$T^{6} +$$$$33\!\cdots\!01$$$$T^{8}$$
$71$ $$( 1 + 87296 T + 5453356606 T^{2} + 157502005424896 T^{3} + 3255243551009881201 T^{4} )^{2}$$
$73$ $$1 - 5672940770 T^{2} + 16272726164284201923 T^{4} -$$$$24\!\cdots\!30$$$$T^{6} +$$$$18\!\cdots\!01$$$$T^{8}$$
$79$ $$( 1 + 65480 T + 5707696298 T^{2} + 201485653006520 T^{3} + 9468276082626847201 T^{4} )^{2}$$
$83$ $$1 - 11381926610 T^{2} + 63038986097658341523 T^{4} -$$$$17\!\cdots\!90$$$$T^{6} +$$$$24\!\cdots\!01$$$$T^{8}$$
$89$ $$( 1 - 72810 T + 5926578523 T^{2} - 406575368481690 T^{3} + 31181719929966183601 T^{4} )^{2}$$
$97$ $$1 - 11961289340 T^{2} + 68433641741002238598 T^{4} -$$$$88\!\cdots\!60$$$$T^{6} +$$$$54\!\cdots\!01$$$$T^{8}$$