# Properties

 Label 25.6.a.d Level 25 Weight 6 Character orbit 25.a Self dual yes Analytic conductor 4.010 Analytic rank 0 Dimension 2 CM no Inner twists 1

# Learn more about

## Newspace parameters

 Level: $$N$$ $$=$$ $$25 = 5^{2}$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 25.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$4.00959549532$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{241})$$ Defining polynomial: $$x^{2} - x - 60$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \frac{1}{2}(1 + \sqrt{241})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 3 - \beta ) q^{2} + ( 9 + 2 \beta ) q^{3} + ( 37 - 5 \beta ) q^{4} + ( -93 - 5 \beta ) q^{6} + ( 98 + 4 \beta ) q^{7} + ( 315 - 15 \beta ) q^{8} + ( 78 + 40 \beta ) q^{9} +O(q^{10})$$ $$q + ( 3 - \beta ) q^{2} + ( 9 + 2 \beta ) q^{3} + ( 37 - 5 \beta ) q^{4} + ( -93 - 5 \beta ) q^{6} + ( 98 + 4 \beta ) q^{7} + ( 315 - 15 \beta ) q^{8} + ( 78 + 40 \beta ) q^{9} + ( -123 + 50 \beta ) q^{11} + ( -267 + 19 \beta ) q^{12} + ( -196 + 32 \beta ) q^{13} + ( 54 - 90 \beta ) q^{14} + ( 661 - 185 \beta ) q^{16} + ( 813 - 136 \beta ) q^{17} + ( -2166 + 2 \beta ) q^{18} + ( -1555 - 70 \beta ) q^{19} + ( 1362 + 240 \beta ) q^{21} + ( -3369 + 223 \beta ) q^{22} + ( 774 + 12 \beta ) q^{23} + ( 1035 + 465 \beta ) q^{24} + ( -2508 + 260 \beta ) q^{26} + ( 3315 + 110 \beta ) q^{27} + ( 2426 - 362 \beta ) q^{28} + ( -1920 - 80 \beta ) q^{29} + ( 2 - 1100 \beta ) q^{31} + ( 3003 - 551 \beta ) q^{32} + ( 4893 + 304 \beta ) q^{33} + ( 10599 - 1085 \beta ) q^{34} + ( -9114 + 890 \beta ) q^{36} + ( 818 + 384 \beta ) q^{37} + ( -465 + 1415 \beta ) q^{38} + ( 2076 - 40 \beta ) q^{39} + ( 13677 + 400 \beta ) q^{41} + ( -10314 - 882 \beta ) q^{42} + ( -436 - 2128 \beta ) q^{43} + ( -19551 + 2215 \beta ) q^{44} + ( 1602 - 750 \beta ) q^{46} + ( 12108 + 1544 \beta ) q^{47} + ( -16251 - 713 \beta ) q^{48} + ( -6243 + 800 \beta ) q^{49} + ( -9003 + 130 \beta ) q^{51} + ( -16852 + 2004 \beta ) q^{52} + ( -13866 + 752 \beta ) q^{53} + ( 3345 - 3095 \beta ) q^{54} + ( 27270 - 270 \beta ) q^{56} + ( -22395 - 3880 \beta ) q^{57} + ( -960 + 1760 \beta ) q^{58} + ( 6960 - 1960 \beta ) q^{59} + ( -13198 + 2000 \beta ) q^{61} + ( 66006 - 2202 \beta ) q^{62} + ( 17244 + 4392 \beta ) q^{63} + ( 20917 + 1815 \beta ) q^{64} + ( -3561 - 4285 \beta ) q^{66} + ( -19237 - 1586 \beta ) q^{67} + ( 70881 - 8417 \beta ) q^{68} + ( 8406 + 1680 \beta ) q^{69} + ( -44148 + 1000 \beta ) q^{71} + ( -11430 + 10830 \beta ) q^{72} + ( -35701 + 1112 \beta ) q^{73} + ( -20586 - 50 \beta ) q^{74} + ( -36535 + 5535 \beta ) q^{76} + ( -54 + 4608 \beta ) q^{77} + ( 8628 - 2156 \beta ) q^{78} + ( 30230 + 5020 \beta ) q^{79} + ( 24081 - 1880 \beta ) q^{81} + ( 17031 - 12877 \beta ) q^{82} + ( 46719 - 858 \beta ) q^{83} + ( -21606 + 870 \beta ) q^{84} + ( 126372 - 3820 \beta ) q^{86} + ( -26880 - 4720 \beta ) q^{87} + ( -83745 + 16845 \beta ) q^{88} + ( -31185 - 10440 \beta ) q^{89} + ( -11528 + 2480 \beta ) q^{91} + ( 25038 - 3486 \beta ) q^{92} + ( -131982 - 12096 \beta ) q^{93} + ( -56316 - 9020 \beta ) q^{94} + ( -39093 - 55 \beta ) q^{96} + ( -68542 + 10944 \beta ) q^{97} + ( -66729 + 7843 \beta ) q^{98} + ( 110406 + 980 \beta ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 5q^{2} + 20q^{3} + 69q^{4} - 191q^{6} + 200q^{7} + 615q^{8} + 196q^{9} + O(q^{10})$$ $$2q + 5q^{2} + 20q^{3} + 69q^{4} - 191q^{6} + 200q^{7} + 615q^{8} + 196q^{9} - 196q^{11} - 515q^{12} - 360q^{13} + 18q^{14} + 1137q^{16} + 1490q^{17} - 4330q^{18} - 3180q^{19} + 2964q^{21} - 6515q^{22} + 1560q^{23} + 2535q^{24} - 4756q^{26} + 6740q^{27} + 4490q^{28} - 3920q^{29} - 1096q^{31} + 5455q^{32} + 10090q^{33} + 20113q^{34} - 17338q^{36} + 2020q^{37} + 485q^{38} + 4112q^{39} + 27754q^{41} - 21510q^{42} - 3000q^{43} - 36887q^{44} + 2454q^{46} + 25760q^{47} - 33215q^{48} - 11686q^{49} - 17876q^{51} - 31700q^{52} - 26980q^{53} + 3595q^{54} + 54270q^{56} - 48670q^{57} - 160q^{58} + 11960q^{59} - 24396q^{61} + 129810q^{62} + 38880q^{63} + 43649q^{64} - 11407q^{66} - 40060q^{67} + 133345q^{68} + 18492q^{69} - 87296q^{71} - 12030q^{72} - 70290q^{73} - 41222q^{74} - 67535q^{76} + 4500q^{77} + 15100q^{78} + 65480q^{79} + 46282q^{81} + 21185q^{82} + 92580q^{83} - 42342q^{84} + 248924q^{86} - 58480q^{87} - 150645q^{88} - 72810q^{89} - 20576q^{91} + 46590q^{92} - 276060q^{93} - 121652q^{94} - 78241q^{96} - 126140q^{97} - 125615q^{98} + 221792q^{99} + O(q^{100})$$

## 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
 8.26209 −7.26209
−5.26209 25.5242 −4.31044 0 −134.310 131.048 191.069 408.483 0
1.2 10.2621 −5.52417 73.3104 0 −56.6896 68.9517 423.931 −212.483 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

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

## 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 25.6.a.d yes 2
3.b odd 2 1 225.6.a.l 2
4.b odd 2 1 400.6.a.o 2
5.b even 2 1 25.6.a.b 2
5.c odd 4 2 25.6.b.b 4
15.d odd 2 1 225.6.a.s 2
15.e even 4 2 225.6.b.i 4
20.d odd 2 1 400.6.a.w 2
20.e even 4 2 400.6.c.n 4

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} - 5 T_{2} - 54$$ acting on $$S_{6}^{\mathrm{new}}(\Gamma_0(25))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - 5 T + 10 T^{2} - 160 T^{3} + 1024 T^{4}$$
$3$ $$1 - 20 T + 345 T^{2} - 4860 T^{3} + 59049 T^{4}$$
$5$ 1
$7$ $$1 - 200 T + 42650 T^{2} - 3361400 T^{3} + 282475249 T^{4}$$
$11$ $$1 + 196 T + 181081 T^{2} + 31565996 T^{3} + 25937424601 T^{4}$$
$13$ $$1 + 360 T + 713290 T^{2} + 133665480 T^{3} + 137858491849 T^{4}$$
$17$ $$1 - 1490 T + 2280355 T^{2} - 2115586930 T^{3} + 2015993900449 T^{4}$$
$19$ $$1 + 3180 T + 7185073 T^{2} + 7873994820 T^{3} + 6131066257801 T^{4}$$
$23$ $$1 - 1560 T + 13472410 T^{2} - 10040695080 T^{3} + 41426511213649 T^{4}$$
$29$ $$1 + 3920 T + 44478298 T^{2} + 80403704080 T^{3} + 420707233300201 T^{4}$$
$31$ $$1 + 1096 T - 15343894 T^{2} + 31377549496 T^{3} + 819628286980801 T^{4}$$
$37$ $$1 - 2020 T + 130823790 T^{2} - 140074793140 T^{3} + 4808584372417849 T^{4}$$
$41$ $$1 - 27754 T + 414643531 T^{2} - 3215473002554 T^{3} + 13422659310152401 T^{4}$$
$43$ $$1 + 3000 T + 23431750 T^{2} + 441025329000 T^{3} + 21611482313284249 T^{4}$$
$47$ $$1 - 25760 T + 480952270 T^{2} - 5907927380320 T^{3} + 52599132235830049 T^{4}$$
$53$ $$1 + 26980 T + 984299470 T^{2} + 11282914401140 T^{3} + 174887470365513049 T^{4}$$
$59$ $$1 - 11960 T + 1234152598 T^{2} - 8550494616040 T^{3} + 511116753300641401 T^{4}$$
$61$ $$1 + 24396 T + 1596983806 T^{2} + 20604771359196 T^{3} + 713342911662882601 T^{4}$$
$67$ $$1 + 40060 T + 2949898505 T^{2} + 54086011786420 T^{3} + 1822837804551761449 T^{4}$$
$71$ $$1 + 87296 T + 5453356606 T^{2} + 157502005424896 T^{3} + 3255243551009881201 T^{4}$$
$73$ $$1 + 70290 T + 5306812435 T^{2} + 145716202271970 T^{3} + 4297625829703557649 T^{4}$$
$79$ $$1 - 65480 T + 5707696298 T^{2} - 201485653006520 T^{3} + 9468276082626847201 T^{4}$$
$83$ $$1 - 92580 T + 9976491505 T^{2} - 364676382728940 T^{3} + 15516041187205853449 T^{4}$$
$89$ $$1 + 72810 T + 5926578523 T^{2} + 406575368481690 T^{3} + 31181719929966183601 T^{4}$$
$97$ $$1 + 126140 T + 13936294470 T^{2} + 1083207100017980 T^{3} + 73742412689492826049 T^{4}$$
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