# Properties

 Label 150.7.b.a Level $150$ Weight $7$ Character orbit 150.b Analytic conductor $34.508$ 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] = [150,7,Mod(149,150)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(150, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 7, names="a")

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

chi := DirichletCharacter("150.149");

S:= CuspForms(chi, 7);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$150 = 2 \cdot 3 \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$7$$ Character orbit: $$[\chi]$$ $$=$$ 150.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: $$34.5081125430$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{8})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 1$$ x^4 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{5}$$ Twist minimal: no (minimal twist has level 6) 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.

 $$f(q)$$ $$=$$ $$q + \beta_{3} q^{2} + ( - 3 \beta_{3} - 21 \beta_1) q^{3} + 32 q^{4} + ( - 21 \beta_{2} - 96) q^{6} + 2 \beta_1 q^{7} + 32 \beta_{3} q^{8} + (126 \beta_{2} - 153) q^{9}+O(q^{10})$$ q + b3 * q^2 + (-3*b3 - 21*b1) * q^3 + 32 * q^4 + (-21*b2 - 96) * q^6 + 2*b1 * q^7 + 32*b3 * q^8 + (126*b2 - 153) * q^9 $$q + \beta_{3} q^{2} + ( - 3 \beta_{3} - 21 \beta_1) q^{3} + 32 q^{4} + ( - 21 \beta_{2} - 96) q^{6} + 2 \beta_1 q^{7} + 32 \beta_{3} q^{8} + (126 \beta_{2} - 153) q^{9} + 6 \beta_{2} q^{11} + ( - 96 \beta_{3} - 672 \beta_1) q^{12} + 2950 \beta_1 q^{13} + 2 \beta_{2} q^{14} + 1024 q^{16} + 792 \beta_{3} q^{17} + ( - 153 \beta_{3} + 4032 \beta_1) q^{18} - 5258 q^{19} + ( - 6 \beta_{2} + 42) q^{21} + 192 \beta_1 q^{22} + 1812 \beta_{3} q^{23} + ( - 672 \beta_{2} - 3072) q^{24} + 2950 \beta_{2} q^{26} + (3105 \beta_{3} - 8883 \beta_1) q^{27} + 64 \beta_1 q^{28} + 390 \beta_{2} q^{29} + 22898 q^{31} + 1024 \beta_{3} q^{32} + (126 \beta_{3} - 576 \beta_1) q^{33} + 25344 q^{34} + (4032 \beta_{2} - 4896) q^{36} + 34058 \beta_1 q^{37} - 5258 \beta_{3} q^{38} + ( - 8850 \beta_{2} + 61950) q^{39} + 2964 \beta_{2} q^{41} + (42 \beta_{3} - 192 \beta_1) q^{42} + 6406 \beta_1 q^{43} + 192 \beta_{2} q^{44} + 57984 q^{46} + 31800 \beta_{3} q^{47} + ( - 3072 \beta_{3} - 21504 \beta_1) q^{48} + 117645 q^{49} + ( - 16632 \beta_{2} - 76032) q^{51} + 94400 \beta_1 q^{52} + 34038 \beta_{3} q^{53} + ( - 8883 \beta_{2} + 99360) q^{54} + 64 \beta_{2} q^{56} + (15774 \beta_{3} + 110418 \beta_1) q^{57} + 12480 \beta_1 q^{58} - 57774 \beta_{2} q^{59} - 62566 q^{61} + 22898 \beta_{3} q^{62} + ( - 252 \beta_{3} - 306 \beta_1) q^{63} + 32768 q^{64} + ( - 576 \beta_{2} + 4032) q^{66} + 438698 \beta_1 q^{67} + 25344 \beta_{3} q^{68} + ( - 38052 \beta_{2} - 173952) q^{69} + 12060 \beta_{2} q^{71} + ( - 4896 \beta_{3} + 129024 \beta_1) q^{72} + 730510 \beta_1 q^{73} + 34058 \beta_{2} q^{74} - 168256 q^{76} - 12 \beta_{3} q^{77} + (61950 \beta_{3} - 283200 \beta_1) q^{78} - 340562 q^{79} + ( - 38556 \beta_{2} - 484623) q^{81} + 94848 \beta_1 q^{82} - 87726 \beta_{3} q^{83} + ( - 192 \beta_{2} + 1344) q^{84} + 6406 \beta_{2} q^{86} + (8190 \beta_{3} - 37440 \beta_1) q^{87} + 6144 \beta_1 q^{88} - 68364 \beta_{2} q^{89} - 5900 q^{91} + 57984 \beta_{3} q^{92} + ( - 68694 \beta_{3} - 480858 \beta_1) q^{93} + 1017600 q^{94} + ( - 21504 \beta_{2} - 98304) q^{96} - 281086 \beta_1 q^{97} + 117645 \beta_{3} q^{98} + ( - 918 \beta_{2} - 24192) q^{99}+O(q^{100})$$ q + b3 * q^2 + (-3*b3 - 21*b1) * q^3 + 32 * q^4 + (-21*b2 - 96) * q^6 + 2*b1 * q^7 + 32*b3 * q^8 + (126*b2 - 153) * q^9 + 6*b2 * q^11 + (-96*b3 - 672*b1) * q^12 + 2950*b1 * q^13 + 2*b2 * q^14 + 1024 * q^16 + 792*b3 * q^17 + (-153*b3 + 4032*b1) * q^18 - 5258 * q^19 + (-6*b2 + 42) * q^21 + 192*b1 * q^22 + 1812*b3 * q^23 + (-672*b2 - 3072) * q^24 + 2950*b2 * q^26 + (3105*b3 - 8883*b1) * q^27 + 64*b1 * q^28 + 390*b2 * q^29 + 22898 * q^31 + 1024*b3 * q^32 + (126*b3 - 576*b1) * q^33 + 25344 * q^34 + (4032*b2 - 4896) * q^36 + 34058*b1 * q^37 - 5258*b3 * q^38 + (-8850*b2 + 61950) * q^39 + 2964*b2 * q^41 + (42*b3 - 192*b1) * q^42 + 6406*b1 * q^43 + 192*b2 * q^44 + 57984 * q^46 + 31800*b3 * q^47 + (-3072*b3 - 21504*b1) * q^48 + 117645 * q^49 + (-16632*b2 - 76032) * q^51 + 94400*b1 * q^52 + 34038*b3 * q^53 + (-8883*b2 + 99360) * q^54 + 64*b2 * q^56 + (15774*b3 + 110418*b1) * q^57 + 12480*b1 * q^58 - 57774*b2 * q^59 - 62566 * q^61 + 22898*b3 * q^62 + (-252*b3 - 306*b1) * q^63 + 32768 * q^64 + (-576*b2 + 4032) * q^66 + 438698*b1 * q^67 + 25344*b3 * q^68 + (-38052*b2 - 173952) * q^69 + 12060*b2 * q^71 + (-4896*b3 + 129024*b1) * q^72 + 730510*b1 * q^73 + 34058*b2 * q^74 - 168256 * q^76 - 12*b3 * q^77 + (61950*b3 - 283200*b1) * q^78 - 340562 * q^79 + (-38556*b2 - 484623) * q^81 + 94848*b1 * q^82 - 87726*b3 * q^83 + (-192*b2 + 1344) * q^84 + 6406*b2 * q^86 + (8190*b3 - 37440*b1) * q^87 + 6144*b1 * q^88 - 68364*b2 * q^89 - 5900 * q^91 + 57984*b3 * q^92 + (-68694*b3 - 480858*b1) * q^93 + 1017600 * q^94 + (-21504*b2 - 98304) * q^96 - 281086*b1 * q^97 + 117645*b3 * q^98 + (-918*b2 - 24192) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 128 q^{4} - 384 q^{6} - 612 q^{9}+O(q^{10})$$ 4 * q + 128 * q^4 - 384 * q^6 - 612 * q^9 $$4 q + 128 q^{4} - 384 q^{6} - 612 q^{9} + 4096 q^{16} - 21032 q^{19} + 168 q^{21} - 12288 q^{24} + 91592 q^{31} + 101376 q^{34} - 19584 q^{36} + 247800 q^{39} + 231936 q^{46} + 470580 q^{49} - 304128 q^{51} + 397440 q^{54} - 250264 q^{61} + 131072 q^{64} + 16128 q^{66} - 695808 q^{69} - 673024 q^{76} - 1362248 q^{79} - 1938492 q^{81} + 5376 q^{84} - 23600 q^{91} + 4070400 q^{94} - 393216 q^{96} - 96768 q^{99}+O(q^{100})$$ 4 * q + 128 * q^4 - 384 * q^6 - 612 * q^9 + 4096 * q^16 - 21032 * q^19 + 168 * q^21 - 12288 * q^24 + 91592 * q^31 + 101376 * q^34 - 19584 * q^36 + 247800 * q^39 + 231936 * q^46 + 470580 * q^49 - 304128 * q^51 + 397440 * q^54 - 250264 * q^61 + 131072 * q^64 + 16128 * q^66 - 695808 * q^69 - 673024 * q^76 - 1362248 * q^79 - 1938492 * q^81 + 5376 * q^84 - 23600 * q^91 + 4070400 * q^94 - 393216 * q^96 - 96768 * q^99

Basis of coefficient ring

 $$\beta_{1}$$ $$=$$ $$\zeta_{8}^{2}$$ v^2 $$\beta_{2}$$ $$=$$ $$4\zeta_{8}^{3} + 4\zeta_{8}$$ 4*v^3 + 4*v $$\beta_{3}$$ $$=$$ $$-4\zeta_{8}^{3} + 4\zeta_{8}$$ -4*v^3 + 4*v
 $$\zeta_{8}$$ $$=$$ $$( \beta_{3} + \beta_{2} ) / 8$$ (b3 + b2) / 8 $$\zeta_{8}^{2}$$ $$=$$ $$\beta_1$$ b1 $$\zeta_{8}^{3}$$ $$=$$ $$( -\beta_{3} + \beta_{2} ) / 8$$ (-b3 + b2) / 8

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/150\mathbb{Z}\right)^\times$$.

 $$n$$ $$101$$ $$127$$ $$\chi(n)$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
149.1
 −0.707107 − 0.707107i −0.707107 + 0.707107i 0.707107 + 0.707107i 0.707107 − 0.707107i
−5.65685 16.9706 21.0000i 32.0000 0 −96.0000 + 118.794i 2.00000i −181.019 −153.000 712.764i 0
149.2 −5.65685 16.9706 + 21.0000i 32.0000 0 −96.0000 118.794i 2.00000i −181.019 −153.000 + 712.764i 0
149.3 5.65685 −16.9706 21.0000i 32.0000 0 −96.0000 118.794i 2.00000i 181.019 −153.000 + 712.764i 0
149.4 5.65685 −16.9706 + 21.0000i 32.0000 0 −96.0000 + 118.794i 2.00000i 181.019 −153.000 712.764i 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 150.7.b.a 4
3.b odd 2 1 inner 150.7.b.a 4
5.b even 2 1 inner 150.7.b.a 4
5.c odd 4 1 6.7.b.a 2
5.c odd 4 1 150.7.d.a 2
15.d odd 2 1 inner 150.7.b.a 4
15.e even 4 1 6.7.b.a 2
15.e even 4 1 150.7.d.a 2
20.e even 4 1 48.7.e.b 2
35.f even 4 1 294.7.b.a 2
40.i odd 4 1 192.7.e.c 2
40.k even 4 1 192.7.e.f 2
45.k odd 12 2 162.7.d.b 4
45.l even 12 2 162.7.d.b 4
60.l odd 4 1 48.7.e.b 2
105.k odd 4 1 294.7.b.a 2
120.q odd 4 1 192.7.e.f 2
120.w even 4 1 192.7.e.c 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6.7.b.a 2 5.c odd 4 1
6.7.b.a 2 15.e even 4 1
48.7.e.b 2 20.e even 4 1
48.7.e.b 2 60.l odd 4 1
150.7.b.a 4 1.a even 1 1 trivial
150.7.b.a 4 3.b odd 2 1 inner
150.7.b.a 4 5.b even 2 1 inner
150.7.b.a 4 15.d odd 2 1 inner
150.7.d.a 2 5.c odd 4 1
150.7.d.a 2 15.e even 4 1
162.7.d.b 4 45.k odd 12 2
162.7.d.b 4 45.l even 12 2
192.7.e.c 2 40.i odd 4 1
192.7.e.c 2 120.w even 4 1
192.7.e.f 2 40.k even 4 1
192.7.e.f 2 120.q odd 4 1
294.7.b.a 2 35.f even 4 1
294.7.b.a 2 105.k odd 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}^{2} + 4$$ acting on $$S_{7}^{\mathrm{new}}(150, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} - 32)^{2}$$
$3$ $$T^{4} + 306 T^{2} + 531441$$
$5$ $$T^{4}$$
$7$ $$(T^{2} + 4)^{2}$$
$11$ $$(T^{2} + 1152)^{2}$$
$13$ $$(T^{2} + 8702500)^{2}$$
$17$ $$(T^{2} - 20072448)^{2}$$
$19$ $$(T + 5258)^{4}$$
$23$ $$(T^{2} - 105067008)^{2}$$
$29$ $$(T^{2} + 4867200)^{2}$$
$31$ $$(T - 22898)^{4}$$
$37$ $$(T^{2} + 1159947364)^{2}$$
$41$ $$(T^{2} + 281129472)^{2}$$
$43$ $$(T^{2} + 41036836)^{2}$$
$47$ $$(T^{2} - 32359680000)^{2}$$
$53$ $$(T^{2} - 37074734208)^{2}$$
$59$ $$(T^{2} + 106810722432)^{2}$$
$61$ $$(T + 62566)^{4}$$
$67$ $$(T^{2} + 192455935204)^{2}$$
$71$ $$(T^{2} + 4654195200)^{2}$$
$73$ $$(T^{2} + 533644860100)^{2}$$
$79$ $$(T + 340562)^{4}$$
$83$ $$(T^{2} - 246267234432)^{2}$$
$89$ $$(T^{2} + 149556367872)^{2}$$
$97$ $$(T^{2} + 79009339396)^{2}$$