# Properties

 Label 75.9.d.b Level $75$ Weight $9$ Character orbit 75.d Analytic conductor $30.553$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [75,9,Mod(74,75)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(75, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 9, names="a")

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

chi := DirichletCharacter("75.74");

S:= CuspForms(chi, 9);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$75 = 3 \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$9$$ Character orbit: $$[\chi]$$ $$=$$ 75.d (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: $$30.5533957546$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{14})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 49$$ x^4 + 49 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{3}\cdot 3^{2}\cdot 5^{2}$$ Twist minimal: no (minimal twist has level 3) 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 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.

 $$f(q)$$ $$=$$ $$q + \beta_{2} q^{2} + (3 \beta_{2} - 9 \beta_1) q^{3} + 248 q^{4} + ( - 9 \beta_{3} + 1512) q^{6} - 350 \beta_1 q^{7} - 8 \beta_{2} q^{8} + ( - 54 \beta_{3} + 2511) q^{9}+O(q^{10})$$ q + b2 * q^2 + (3*b2 - 9*b1) * q^3 + 248 * q^4 + (-9*b3 + 1512) * q^6 - 350*b1 * q^7 - 8*b2 * q^8 + (-54*b3 + 2511) * q^9 $$q + \beta_{2} q^{2} + (3 \beta_{2} - 9 \beta_1) q^{3} + 248 q^{4} + ( - 9 \beta_{3} + 1512) q^{6} - 350 \beta_1 q^{7} - 8 \beta_{2} q^{8} + ( - 54 \beta_{3} + 2511) q^{9} - 62 \beta_{3} q^{11} + (744 \beta_{2} - 2232 \beta_1) q^{12} - 5146 \beta_1 q^{13} - 350 \beta_{3} q^{14} - 67520 q^{16} + 3336 \beta_{2} q^{17} + (2511 \beta_{2} - 27216 \beta_1) q^{18} - 18938 q^{19} + ( - 1050 \beta_{3} - 78750) q^{21} - 31248 \beta_1 q^{22} + 20956 \beta_{2} q^{23} + (72 \beta_{3} - 12096) q^{24} - 5146 \beta_{3} q^{26} + ( - 4617 \beta_{2} - 104247 \beta_1) q^{27} - 86800 \beta_1 q^{28} + 4106 \beta_{3} q^{29} - 351478 q^{31} - 65472 \beta_{2} q^{32} + ( - 13950 \beta_{2} - 93744 \beta_1) q^{33} + 1681344 q^{34} + ( - 13392 \beta_{3} + 622728) q^{36} + 267034 \beta_1 q^{37} - 18938 \beta_{2} q^{38} + ( - 15438 \beta_{3} - 1157850) q^{39} - 16708 \beta_{3} q^{41} + ( - 78750 \beta_{2} - 529200 \beta_1) q^{42} + 705230 \beta_1 q^{43} - 15376 \beta_{3} q^{44} + 10561824 q^{46} - 181784 \beta_{2} q^{47} + ( - 202560 \beta_{2} + 607680 \beta_1) q^{48} + 2702301 q^{49} + ( - 30024 \beta_{3} + 5044032) q^{51} - 1276208 \beta_1 q^{52} + 294066 \beta_{2} q^{53} + ( - 104247 \beta_{3} - 2326968) q^{54} + 2800 \beta_{3} q^{56} + ( - 56814 \beta_{2} + 170442 \beta_1) q^{57} + 2069424 \beta_1 q^{58} + 122182 \beta_{3} q^{59} + 753602 q^{61} - 351478 \beta_{2} q^{62} + ( - 472500 \beta_{2} - 878850 \beta_1) q^{63} - 15712768 q^{64} + ( - 93744 \beta_{3} - 7030800) q^{66} + 453778 \beta_1 q^{67} + 827328 \beta_{2} q^{68} + ( - 188604 \beta_{3} + 31685472) q^{69} - 151644 \beta_{3} q^{71} + ( - 20088 \beta_{2} + 217728 \beta_1) q^{72} - 5534554 \beta_1 q^{73} + 267034 \beta_{3} q^{74} - 4696624 q^{76} - 542500 \beta_{2} q^{77} + ( - 1157850 \beta_{2} - 7780752 \beta_1) q^{78} + 22980982 q^{79} + ( - 271188 \beta_{3} - 30436479) q^{81} - 8420832 \beta_1 q^{82} + 2066606 \beta_{2} q^{83} + ( - 260400 \beta_{3} - 19530000) q^{84} + 705230 \beta_{3} q^{86} + (923850 \beta_{2} + 6208272 \beta_1) q^{87} + 249984 \beta_1 q^{88} + 646908 \beta_{3} q^{89} - 45027500 q^{91} + 5197088 \beta_{2} q^{92} + ( - 1054434 \beta_{2} + 3163302 \beta_1) q^{93} - 91619136 q^{94} + (589248 \beta_{3} - 98993664) q^{96} + 29454202 \beta_1 q^{97} + 2702301 \beta_{2} q^{98} + ( - 155682 \beta_{3} - 42184800) q^{99}+O(q^{100})$$ q + b2 * q^2 + (3*b2 - 9*b1) * q^3 + 248 * q^4 + (-9*b3 + 1512) * q^6 - 350*b1 * q^7 - 8*b2 * q^8 + (-54*b3 + 2511) * q^9 - 62*b3 * q^11 + (744*b2 - 2232*b1) * q^12 - 5146*b1 * q^13 - 350*b3 * q^14 - 67520 * q^16 + 3336*b2 * q^17 + (2511*b2 - 27216*b1) * q^18 - 18938 * q^19 + (-1050*b3 - 78750) * q^21 - 31248*b1 * q^22 + 20956*b2 * q^23 + (72*b3 - 12096) * q^24 - 5146*b3 * q^26 + (-4617*b2 - 104247*b1) * q^27 - 86800*b1 * q^28 + 4106*b3 * q^29 - 351478 * q^31 - 65472*b2 * q^32 + (-13950*b2 - 93744*b1) * q^33 + 1681344 * q^34 + (-13392*b3 + 622728) * q^36 + 267034*b1 * q^37 - 18938*b2 * q^38 + (-15438*b3 - 1157850) * q^39 - 16708*b3 * q^41 + (-78750*b2 - 529200*b1) * q^42 + 705230*b1 * q^43 - 15376*b3 * q^44 + 10561824 * q^46 - 181784*b2 * q^47 + (-202560*b2 + 607680*b1) * q^48 + 2702301 * q^49 + (-30024*b3 + 5044032) * q^51 - 1276208*b1 * q^52 + 294066*b2 * q^53 + (-104247*b3 - 2326968) * q^54 + 2800*b3 * q^56 + (-56814*b2 + 170442*b1) * q^57 + 2069424*b1 * q^58 + 122182*b3 * q^59 + 753602 * q^61 - 351478*b2 * q^62 + (-472500*b2 - 878850*b1) * q^63 - 15712768 * q^64 + (-93744*b3 - 7030800) * q^66 + 453778*b1 * q^67 + 827328*b2 * q^68 + (-188604*b3 + 31685472) * q^69 - 151644*b3 * q^71 + (-20088*b2 + 217728*b1) * q^72 - 5534554*b1 * q^73 + 267034*b3 * q^74 - 4696624 * q^76 - 542500*b2 * q^77 + (-1157850*b2 - 7780752*b1) * q^78 + 22980982 * q^79 + (-271188*b3 - 30436479) * q^81 - 8420832*b1 * q^82 + 2066606*b2 * q^83 + (-260400*b3 - 19530000) * q^84 + 705230*b3 * q^86 + (923850*b2 + 6208272*b1) * q^87 + 249984*b1 * q^88 + 646908*b3 * q^89 - 45027500 * q^91 + 5197088*b2 * q^92 + (-1054434*b2 + 3163302*b1) * q^93 - 91619136 * q^94 + (589248*b3 - 98993664) * q^96 + 29454202*b1 * q^97 + 2702301*b2 * q^98 + (-155682*b3 - 42184800) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 992 q^{4} + 6048 q^{6} + 10044 q^{9}+O(q^{10})$$ 4 * q + 992 * q^4 + 6048 * q^6 + 10044 * q^9 $$4 q + 992 q^{4} + 6048 q^{6} + 10044 q^{9} - 270080 q^{16} - 75752 q^{19} - 315000 q^{21} - 48384 q^{24} - 1405912 q^{31} + 6725376 q^{34} + 2490912 q^{36} - 4631400 q^{39} + 42247296 q^{46} + 10809204 q^{49} + 20176128 q^{51} - 9307872 q^{54} + 3014408 q^{61} - 62851072 q^{64} - 28123200 q^{66} + 126741888 q^{69} - 18786496 q^{76} + 91923928 q^{79} - 121745916 q^{81} - 78120000 q^{84} - 180110000 q^{91} - 366476544 q^{94} - 395974656 q^{96} - 168739200 q^{99}+O(q^{100})$$ 4 * q + 992 * q^4 + 6048 * q^6 + 10044 * q^9 - 270080 * q^16 - 75752 * q^19 - 315000 * q^21 - 48384 * q^24 - 1405912 * q^31 + 6725376 * q^34 + 2490912 * q^36 - 4631400 * q^39 + 42247296 * q^46 + 10809204 * q^49 + 20176128 * q^51 - 9307872 * q^54 + 3014408 * q^61 - 62851072 * q^64 - 28123200 * q^66 + 126741888 * q^69 - 18786496 * q^76 + 91923928 * q^79 - 121745916 * q^81 - 78120000 * q^84 - 180110000 * q^91 - 366476544 * q^94 - 395974656 * q^96 - 168739200 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 49$$ :

 $$\beta_{1}$$ $$=$$ $$( 5\nu^{2} ) / 7$$ (5*v^2) / 7 $$\beta_{2}$$ $$=$$ $$( -6\nu^{3} + 42\nu ) / 7$$ (-6*v^3 + 42*v) / 7 $$\beta_{3}$$ $$=$$ $$( 30\nu^{3} + 210\nu ) / 7$$ (30*v^3 + 210*v) / 7
 $$\nu$$ $$=$$ $$( \beta_{3} + 5\beta_{2} ) / 60$$ (b3 + 5*b2) / 60 $$\nu^{2}$$ $$=$$ $$( 7\beta_1 ) / 5$$ (7*b1) / 5 $$\nu^{3}$$ $$=$$ $$( 7\beta_{3} - 35\beta_{2} ) / 60$$ (7*b3 - 35*b2) / 60

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/75\mathbb{Z}\right)^\times$$.

 $$n$$ $$26$$ $$52$$ $$\chi(n)$$ $$-1$$ $$-1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
74.1
 −1.87083 − 1.87083i −1.87083 + 1.87083i 1.87083 + 1.87083i 1.87083 − 1.87083i
−22.4499 −67.3498 45.0000i 248.000 0 1512.00 + 1010.25i 1750.00i 179.600 2511.00 + 6061.48i 0
74.2 −22.4499 −67.3498 + 45.0000i 248.000 0 1512.00 1010.25i 1750.00i 179.600 2511.00 6061.48i 0
74.3 22.4499 67.3498 45.0000i 248.000 0 1512.00 1010.25i 1750.00i −179.600 2511.00 6061.48i 0
74.4 22.4499 67.3498 + 45.0000i 248.000 0 1512.00 + 1010.25i 1750.00i −179.600 2511.00 + 6061.48i 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
3.b odd 2 1 inner
5.b even 2 1 inner
15.d odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 75.9.d.b 4
3.b odd 2 1 inner 75.9.d.b 4
5.b even 2 1 inner 75.9.d.b 4
5.c odd 4 1 3.9.b.a 2
5.c odd 4 1 75.9.c.c 2
15.d odd 2 1 inner 75.9.d.b 4
15.e even 4 1 3.9.b.a 2
15.e even 4 1 75.9.c.c 2
20.e even 4 1 48.9.e.b 2
40.i odd 4 1 192.9.e.e 2
40.k even 4 1 192.9.e.f 2
45.k odd 12 2 81.9.d.d 4
45.l even 12 2 81.9.d.d 4
60.l odd 4 1 48.9.e.b 2
120.q odd 4 1 192.9.e.f 2
120.w even 4 1 192.9.e.e 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3.9.b.a 2 5.c odd 4 1
3.9.b.a 2 15.e even 4 1
48.9.e.b 2 20.e even 4 1
48.9.e.b 2 60.l odd 4 1
75.9.c.c 2 5.c odd 4 1
75.9.c.c 2 15.e even 4 1
75.9.d.b 4 1.a even 1 1 trivial
75.9.d.b 4 3.b odd 2 1 inner
75.9.d.b 4 5.b even 2 1 inner
75.9.d.b 4 15.d odd 2 1 inner
81.9.d.d 4 45.k odd 12 2
81.9.d.d 4 45.l even 12 2
192.9.e.e 2 40.i odd 4 1
192.9.e.e 2 120.w even 4 1
192.9.e.f 2 40.k even 4 1
192.9.e.f 2 120.q odd 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} - 504$$ acting on $$S_{9}^{\mathrm{new}}(75, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} - 504)^{2}$$
$3$ $$T^{4} - 5022 T^{2} + 43046721$$
$5$ $$T^{4}$$
$7$ $$(T^{2} + 3062500)^{2}$$
$11$ $$(T^{2} + 48434400)^{2}$$
$13$ $$(T^{2} + 662032900)^{2}$$
$17$ $$(T^{2} - 5608963584)^{2}$$
$19$ $$(T + 18938)^{4}$$
$23$ $$(T^{2} - 221333583744)^{2}$$
$29$ $$(T^{2} + 212426373600)^{2}$$
$31$ $$(T + 351478)^{4}$$
$37$ $$(T^{2} + 1782678928900)^{2}$$
$41$ $$(T^{2} + 3517381526400)^{2}$$
$43$ $$(T^{2} + 12433733822500)^{2}$$
$47$ $$(T^{2} - 16654893018624)^{2}$$
$53$ $$(T^{2} - 43583305427424)^{2}$$
$59$ $$(T^{2} + 188098358162400)^{2}$$
$61$ $$(T - 753602)^{4}$$
$67$ $$(T^{2} + 5147861832100)^{2}$$
$71$ $$(T^{2} + 289748374473600)^{2}$$
$73$ $$(T^{2} + 765782199472900)^{2}$$
$79$ $$(T - 22980982)^{4}$$
$83$ $$(T^{2} - 21\!\cdots\!44)^{2}$$
$89$ $$(T^{2} + 52\!\cdots\!00)^{2}$$
$97$ $$(T^{2} + 21\!\cdots\!00)^{2}$$