# Properties

 Label 25.16.b.a Level $25$ Weight $16$ Character orbit 25.b Analytic conductor $35.673$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [25,16,Mod(24,25)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(25, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1]))

N = Newforms(chi, 16, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("25.24");

S:= CuspForms(chi, 16);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$25 = 5^{2}$$ Weight: $$k$$ $$=$$ $$16$$ Character orbit: $$[\chi]$$ $$=$$ 25.b (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: no Analytic conductor: $$35.6733762750$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 1) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = 2i$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 108 \beta q^{2} + 1674 \beta q^{3} - 13888 q^{4} - 723168 q^{6} + 1411228 \beta q^{7} + 2039040 \beta q^{8} + 3139803 q^{9}+O(q^{10})$$ q + 108*b * q^2 + 1674*b * q^3 - 13888 * q^4 - 723168 * q^6 + 1411228*b * q^7 + 2039040*b * q^8 + 3139803 * q^9 $$q + 108 \beta q^{2} + 1674 \beta q^{3} - 13888 q^{4} - 723168 q^{6} + 1411228 \beta q^{7} + 2039040 \beta q^{8} + 3139803 q^{9} + 20586852 q^{11} - 23248512 \beta q^{12} + 95036669 \beta q^{13} - 609650496 q^{14} - 1335947264 q^{16} + 823263993 \beta q^{17} + 339098724 \beta q^{18} - 1563257180 q^{19} - 9449582688 q^{21} + 2223380016 \beta q^{22} - 4725558036 \beta q^{23} - 13653411840 q^{24} - 41055841008 q^{26} + 29276100540 \beta q^{27} - 19599134464 \beta q^{28} + 36902568330 q^{29} + 71588483552 q^{31} - 77467041792 \beta q^{32} + 34462390248 \beta q^{33} - 355650044976 q^{34} - 43605584064 q^{36} - 516826040777 \beta q^{37} - 168831775440 \beta q^{38} - 636365535624 q^{39} + 1641974018202 q^{41} - 1020554930304 \beta q^{42} + 246201554654 \beta q^{43} - 285910200576 q^{44} + 2041441071552 q^{46} - 1705342476312 \beta q^{47} - 2236375719936 \beta q^{48} - 3218696361993 q^{49} - 5512575697128 q^{51} - 1319869259072 \beta q^{52} - 3398575827951 \beta q^{53} - 12647275433280 q^{54} - 11510201364480 q^{56} - 2616892519320 \beta q^{57} + 3985477379640 \beta q^{58} - 9858856815540 q^{59} + 4931842626902 q^{61} + 7731556223616 \beta q^{62} + 4430977908084 \beta q^{63} - 10310557892608 q^{64} - 14887752587136 q^{66} - 14418913312682 \beta q^{67} - 11433490334784 \beta q^{68} + 31642336609056 q^{69} + 125050114914552 q^{71} + 6402183909120 \beta q^{72} + 41085727756739 \beta q^{73} + 223268849615664 q^{74} + 21710515715840 q^{76} + 29052741974256 \beta q^{77} - 68727477847392 \beta q^{78} + 25413078694480 q^{79} - 150980027970519 q^{81} + 177333193965816 \beta q^{82} + 140868365445234 \beta q^{83} + 131235804370944 q^{84} - 106359071610528 q^{86} + 61774899384420 \beta q^{87} + 41977414702080 \beta q^{88} - 715618564776810 q^{89} - 536473633278128 q^{91} + 65628550003968 \beta q^{92} + 119839121466048 \beta q^{93} + 736707949766784 q^{94} + 518719311839232 q^{96} + 306393068040913 \beta q^{97} - 347619207095244 \beta q^{98} + 64638659670156 q^{99} +O(q^{100})$$ q + 108*b * q^2 + 1674*b * q^3 - 13888 * q^4 - 723168 * q^6 + 1411228*b * q^7 + 2039040*b * q^8 + 3139803 * q^9 + 20586852 * q^11 - 23248512*b * q^12 + 95036669*b * q^13 - 609650496 * q^14 - 1335947264 * q^16 + 823263993*b * q^17 + 339098724*b * q^18 - 1563257180 * q^19 - 9449582688 * q^21 + 2223380016*b * q^22 - 4725558036*b * q^23 - 13653411840 * q^24 - 41055841008 * q^26 + 29276100540*b * q^27 - 19599134464*b * q^28 + 36902568330 * q^29 + 71588483552 * q^31 - 77467041792*b * q^32 + 34462390248*b * q^33 - 355650044976 * q^34 - 43605584064 * q^36 - 516826040777*b * q^37 - 168831775440*b * q^38 - 636365535624 * q^39 + 1641974018202 * q^41 - 1020554930304*b * q^42 + 246201554654*b * q^43 - 285910200576 * q^44 + 2041441071552 * q^46 - 1705342476312*b * q^47 - 2236375719936*b * q^48 - 3218696361993 * q^49 - 5512575697128 * q^51 - 1319869259072*b * q^52 - 3398575827951*b * q^53 - 12647275433280 * q^54 - 11510201364480 * q^56 - 2616892519320*b * q^57 + 3985477379640*b * q^58 - 9858856815540 * q^59 + 4931842626902 * q^61 + 7731556223616*b * q^62 + 4430977908084*b * q^63 - 10310557892608 * q^64 - 14887752587136 * q^66 - 14418913312682*b * q^67 - 11433490334784*b * q^68 + 31642336609056 * q^69 + 125050114914552 * q^71 + 6402183909120*b * q^72 + 41085727756739*b * q^73 + 223268849615664 * q^74 + 21710515715840 * q^76 + 29052741974256*b * q^77 - 68727477847392*b * q^78 + 25413078694480 * q^79 - 150980027970519 * q^81 + 177333193965816*b * q^82 + 140868365445234*b * q^83 + 131235804370944 * q^84 - 106359071610528 * q^86 + 61774899384420*b * q^87 + 41977414702080*b * q^88 - 715618564776810 * q^89 - 536473633278128 * q^91 + 65628550003968*b * q^92 + 119839121466048*b * q^93 + 736707949766784 * q^94 + 518719311839232 * q^96 + 306393068040913*b * q^97 - 347619207095244*b * q^98 + 64638659670156 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 27776 q^{4} - 1446336 q^{6} + 6279606 q^{9}+O(q^{10})$$ 2 * q - 27776 * q^4 - 1446336 * q^6 + 6279606 * q^9 $$2 q - 27776 q^{4} - 1446336 q^{6} + 6279606 q^{9} + 41173704 q^{11} - 1219300992 q^{14} - 2671894528 q^{16} - 3126514360 q^{19} - 18899165376 q^{21} - 27306823680 q^{24} - 82111682016 q^{26} + 73805136660 q^{29} + 143176967104 q^{31} - 711300089952 q^{34} - 87211168128 q^{36} - 1272731071248 q^{39} + 3283948036404 q^{41} - 571820401152 q^{44} + 4082882143104 q^{46} - 6437392723986 q^{49} - 11025151394256 q^{51} - 25294550866560 q^{54} - 23020402728960 q^{56} - 19717713631080 q^{59} + 9863685253804 q^{61} - 20621115785216 q^{64} - 29775505174272 q^{66} + 63284673218112 q^{69} + 250100229829104 q^{71} + 446537699231328 q^{74} + 43421031431680 q^{76} + 50826157388960 q^{79} - 301960055941038 q^{81} + 262471608741888 q^{84} - 212718143221056 q^{86} - 14\!\cdots\!20 q^{89}+ \cdots + 129277319340312 q^{99}+O(q^{100})$$ 2 * q - 27776 * q^4 - 1446336 * q^6 + 6279606 * q^9 + 41173704 * q^11 - 1219300992 * q^14 - 2671894528 * q^16 - 3126514360 * q^19 - 18899165376 * q^21 - 27306823680 * q^24 - 82111682016 * q^26 + 73805136660 * q^29 + 143176967104 * q^31 - 711300089952 * q^34 - 87211168128 * q^36 - 1272731071248 * q^39 + 3283948036404 * q^41 - 571820401152 * q^44 + 4082882143104 * q^46 - 6437392723986 * q^49 - 11025151394256 * q^51 - 25294550866560 * q^54 - 23020402728960 * q^56 - 19717713631080 * q^59 + 9863685253804 * q^61 - 20621115785216 * q^64 - 29775505174272 * q^66 + 63284673218112 * q^69 + 250100229829104 * q^71 + 446537699231328 * q^74 + 43421031431680 * q^76 + 50826157388960 * q^79 - 301960055941038 * q^81 + 262471608741888 * q^84 - 212718143221056 * q^86 - 1431237129553620 * q^89 - 1072947266556256 * q^91 + 1473415899533568 * q^94 + 1037438623678464 * q^96 + 129277319340312 * q^99

## 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.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
24.1
 − 1.00000i 1.00000i
216.000i 3348.00i −13888.0 0 −723168. 2.82246e6i 4.07808e6i 3.13980e6 0
24.2 216.000i 3348.00i −13888.0 0 −723168. 2.82246e6i 4.07808e6i 3.13980e6 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 25.16.b.a 2
5.b even 2 1 inner 25.16.b.a 2
5.c odd 4 1 1.16.a.a 1
5.c odd 4 1 25.16.a.a 1
15.e even 4 1 9.16.a.a 1
20.e even 4 1 16.16.a.d 1
35.f even 4 1 49.16.a.a 1
35.k even 12 2 49.16.c.b 2
35.l odd 12 2 49.16.c.c 2
40.i odd 4 1 64.16.a.i 1
40.k even 4 1 64.16.a.c 1
55.e even 4 1 121.16.a.a 1
60.l odd 4 1 144.16.a.f 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1.16.a.a 1 5.c odd 4 1
9.16.a.a 1 15.e even 4 1
16.16.a.d 1 20.e even 4 1
25.16.a.a 1 5.c odd 4 1
25.16.b.a 2 1.a even 1 1 trivial
25.16.b.a 2 5.b even 2 1 inner
49.16.a.a 1 35.f even 4 1
49.16.c.b 2 35.k even 12 2
49.16.c.c 2 35.l odd 12 2
64.16.a.c 1 40.k even 4 1
64.16.a.i 1 40.i odd 4 1
121.16.a.a 1 55.e even 4 1
144.16.a.f 1 60.l odd 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} + 46656$$ acting on $$S_{16}^{\mathrm{new}}(25, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 46656$$
$3$ $$T^{2} + 11209104$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 7966257871936$$
$11$ $$(T - 20586852)^{2}$$
$13$ $$T^{2} + 36\!\cdots\!44$$
$17$ $$T^{2} + 27\!\cdots\!96$$
$19$ $$(T + 1563257180)^{2}$$
$23$ $$T^{2} + 89\!\cdots\!84$$
$29$ $$(T - 36902568330)^{2}$$
$31$ $$(T - 71588483552)^{2}$$
$37$ $$T^{2} + 10\!\cdots\!16$$
$41$ $$(T - 1641974018202)^{2}$$
$43$ $$T^{2} + 24\!\cdots\!64$$
$47$ $$T^{2} + 11\!\cdots\!76$$
$53$ $$T^{2} + 46\!\cdots\!04$$
$59$ $$(T + 9858856815540)^{2}$$
$61$ $$(T - 4931842626902)^{2}$$
$67$ $$T^{2} + 83\!\cdots\!96$$
$71$ $$(T - 125050114914552)^{2}$$
$73$ $$T^{2} + 67\!\cdots\!84$$
$79$ $$(T - 25413078694480)^{2}$$
$83$ $$T^{2} + 79\!\cdots\!24$$
$89$ $$(T + 715618564776810)^{2}$$
$97$ $$T^{2} + 37\!\cdots\!76$$