# Properties

 Label 150.7.d.a Level $150$ Weight $7$ Character orbit 150.d Analytic conductor $34.508$ 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] = [150,7,Mod(101,150)]

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, 0]))

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

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

chi := DirichletCharacter("150.101");

S:= CuspForms(chi, 7);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$150 = 2 \cdot 3 \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$7$$ Character orbit: $$[\chi]$$ $$=$$ 150.d (of order $$2$$, degree $$1$$, 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: $$2$$ Coefficient field: $$\Q(\sqrt{-2})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} + 2$$ x^2 + 2 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$2^{2}$$ 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 $$\beta = 4\sqrt{-2}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta q^{2} + ( - 3 \beta - 21) q^{3} - 32 q^{4} + (21 \beta - 96) q^{6} - 2 q^{7} + 32 \beta q^{8} + (126 \beta + 153) q^{9} +O(q^{10})$$ q - b * q^2 + (-3*b - 21) * q^3 - 32 * q^4 + (21*b - 96) * q^6 - 2 * q^7 + 32*b * q^8 + (126*b + 153) * q^9 $$q - \beta q^{2} + ( - 3 \beta - 21) q^{3} - 32 q^{4} + (21 \beta - 96) q^{6} - 2 q^{7} + 32 \beta q^{8} + (126 \beta + 153) q^{9} - 6 \beta q^{11} + (96 \beta + 672) q^{12} + 2950 q^{13} + 2 \beta q^{14} + 1024 q^{16} - 792 \beta q^{17} + ( - 153 \beta + 4032) q^{18} + 5258 q^{19} + (6 \beta + 42) q^{21} - 192 q^{22} + 1812 \beta q^{23} + ( - 672 \beta + 3072) q^{24} - 2950 \beta q^{26} + ( - 3105 \beta + 8883) q^{27} + 64 q^{28} + 390 \beta q^{29} + 22898 q^{31} - 1024 \beta q^{32} + (126 \beta - 576) q^{33} - 25344 q^{34} + ( - 4032 \beta - 4896) q^{36} - 34058 q^{37} - 5258 \beta q^{38} + ( - 8850 \beta - 61950) q^{39} - 2964 \beta q^{41} + ( - 42 \beta + 192) q^{42} + 6406 q^{43} + 192 \beta q^{44} + 57984 q^{46} - 31800 \beta q^{47} + ( - 3072 \beta - 21504) q^{48} - 117645 q^{49} + (16632 \beta - 76032) q^{51} - 94400 q^{52} + 34038 \beta q^{53} + ( - 8883 \beta - 99360) q^{54} - 64 \beta q^{56} + ( - 15774 \beta - 110418) q^{57} + 12480 q^{58} - 57774 \beta q^{59} - 62566 q^{61} - 22898 \beta q^{62} + ( - 252 \beta - 306) q^{63} - 32768 q^{64} + (576 \beta + 4032) q^{66} - 438698 q^{67} + 25344 \beta q^{68} + ( - 38052 \beta + 173952) q^{69} - 12060 \beta q^{71} + (4896 \beta - 129024) q^{72} + 730510 q^{73} + 34058 \beta q^{74} - 168256 q^{76} + 12 \beta q^{77} + (61950 \beta - 283200) q^{78} + 340562 q^{79} + (38556 \beta - 484623) q^{81} - 94848 q^{82} - 87726 \beta q^{83} + ( - 192 \beta - 1344) q^{84} - 6406 \beta q^{86} + ( - 8190 \beta + 37440) q^{87} + 6144 q^{88} - 68364 \beta q^{89} - 5900 q^{91} - 57984 \beta q^{92} + ( - 68694 \beta - 480858) q^{93} - 1017600 q^{94} + (21504 \beta - 98304) q^{96} + 281086 q^{97} + 117645 \beta q^{98} + ( - 918 \beta + 24192) q^{99} +O(q^{100})$$ q - b * q^2 + (-3*b - 21) * q^3 - 32 * q^4 + (21*b - 96) * q^6 - 2 * q^7 + 32*b * q^8 + (126*b + 153) * q^9 - 6*b * q^11 + (96*b + 672) * q^12 + 2950 * q^13 + 2*b * q^14 + 1024 * q^16 - 792*b * q^17 + (-153*b + 4032) * q^18 + 5258 * q^19 + (6*b + 42) * q^21 - 192 * q^22 + 1812*b * q^23 + (-672*b + 3072) * q^24 - 2950*b * q^26 + (-3105*b + 8883) * q^27 + 64 * q^28 + 390*b * q^29 + 22898 * q^31 - 1024*b * q^32 + (126*b - 576) * q^33 - 25344 * q^34 + (-4032*b - 4896) * q^36 - 34058 * q^37 - 5258*b * q^38 + (-8850*b - 61950) * q^39 - 2964*b * q^41 + (-42*b + 192) * q^42 + 6406 * q^43 + 192*b * q^44 + 57984 * q^46 - 31800*b * q^47 + (-3072*b - 21504) * q^48 - 117645 * q^49 + (16632*b - 76032) * q^51 - 94400 * q^52 + 34038*b * q^53 + (-8883*b - 99360) * q^54 - 64*b * q^56 + (-15774*b - 110418) * q^57 + 12480 * q^58 - 57774*b * q^59 - 62566 * q^61 - 22898*b * q^62 + (-252*b - 306) * q^63 - 32768 * q^64 + (576*b + 4032) * q^66 - 438698 * q^67 + 25344*b * q^68 + (-38052*b + 173952) * q^69 - 12060*b * q^71 + (4896*b - 129024) * q^72 + 730510 * q^73 + 34058*b * q^74 - 168256 * q^76 + 12*b * q^77 + (61950*b - 283200) * q^78 + 340562 * q^79 + (38556*b - 484623) * q^81 - 94848 * q^82 - 87726*b * q^83 + (-192*b - 1344) * q^84 - 6406*b * q^86 + (-8190*b + 37440) * q^87 + 6144 * q^88 - 68364*b * q^89 - 5900 * q^91 - 57984*b * q^92 + (-68694*b - 480858) * q^93 - 1017600 * q^94 + (21504*b - 98304) * q^96 + 281086 * q^97 + 117645*b * q^98 + (-918*b + 24192) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 42 q^{3} - 64 q^{4} - 192 q^{6} - 4 q^{7} + 306 q^{9}+O(q^{10})$$ 2 * q - 42 * q^3 - 64 * q^4 - 192 * q^6 - 4 * q^7 + 306 * q^9 $$2 q - 42 q^{3} - 64 q^{4} - 192 q^{6} - 4 q^{7} + 306 q^{9} + 1344 q^{12} + 5900 q^{13} + 2048 q^{16} + 8064 q^{18} + 10516 q^{19} + 84 q^{21} - 384 q^{22} + 6144 q^{24} + 17766 q^{27} + 128 q^{28} + 45796 q^{31} - 1152 q^{33} - 50688 q^{34} - 9792 q^{36} - 68116 q^{37} - 123900 q^{39} + 384 q^{42} + 12812 q^{43} + 115968 q^{46} - 43008 q^{48} - 235290 q^{49} - 152064 q^{51} - 188800 q^{52} - 198720 q^{54} - 220836 q^{57} + 24960 q^{58} - 125132 q^{61} - 612 q^{63} - 65536 q^{64} + 8064 q^{66} - 877396 q^{67} + 347904 q^{69} - 258048 q^{72} + 1461020 q^{73} - 336512 q^{76} - 566400 q^{78} + 681124 q^{79} - 969246 q^{81} - 189696 q^{82} - 2688 q^{84} + 74880 q^{87} + 12288 q^{88} - 11800 q^{91} - 961716 q^{93} - 2035200 q^{94} - 196608 q^{96} + 562172 q^{97} + 48384 q^{99}+O(q^{100})$$ 2 * q - 42 * q^3 - 64 * q^4 - 192 * q^6 - 4 * q^7 + 306 * q^9 + 1344 * q^12 + 5900 * q^13 + 2048 * q^16 + 8064 * q^18 + 10516 * q^19 + 84 * q^21 - 384 * q^22 + 6144 * q^24 + 17766 * q^27 + 128 * q^28 + 45796 * q^31 - 1152 * q^33 - 50688 * q^34 - 9792 * q^36 - 68116 * q^37 - 123900 * q^39 + 384 * q^42 + 12812 * q^43 + 115968 * q^46 - 43008 * q^48 - 235290 * q^49 - 152064 * q^51 - 188800 * q^52 - 198720 * q^54 - 220836 * q^57 + 24960 * q^58 - 125132 * q^61 - 612 * q^63 - 65536 * q^64 + 8064 * q^66 - 877396 * q^67 + 347904 * q^69 - 258048 * q^72 + 1461020 * q^73 - 336512 * q^76 - 566400 * q^78 + 681124 * q^79 - 969246 * q^81 - 189696 * q^82 - 2688 * q^84 + 74880 * q^87 + 12288 * q^88 - 11800 * q^91 - 961716 * q^93 - 2035200 * q^94 - 196608 * q^96 + 562172 * q^97 + 48384 * q^99

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
101.1
 1.41421i − 1.41421i
5.65685i −21.0000 16.9706i −32.0000 0 −96.0000 + 118.794i −2.00000 181.019i 153.000 + 712.764i 0
101.2 5.65685i −21.0000 + 16.9706i −32.0000 0 −96.0000 118.794i −2.00000 181.019i 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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 150.7.d.a 2
3.b odd 2 1 inner 150.7.d.a 2
5.b even 2 1 6.7.b.a 2
5.c odd 4 2 150.7.b.a 4
15.d odd 2 1 6.7.b.a 2
15.e even 4 2 150.7.b.a 4
20.d odd 2 1 48.7.e.b 2
35.c odd 2 1 294.7.b.a 2
40.e odd 2 1 192.7.e.f 2
40.f even 2 1 192.7.e.c 2
45.h odd 6 2 162.7.d.b 4
45.j even 6 2 162.7.d.b 4
60.h even 2 1 48.7.e.b 2
105.g even 2 1 294.7.b.a 2
120.i odd 2 1 192.7.e.c 2
120.m even 2 1 192.7.e.f 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6.7.b.a 2 5.b even 2 1
6.7.b.a 2 15.d odd 2 1
48.7.e.b 2 20.d odd 2 1
48.7.e.b 2 60.h even 2 1
150.7.b.a 4 5.c odd 4 2
150.7.b.a 4 15.e even 4 2
150.7.d.a 2 1.a even 1 1 trivial
150.7.d.a 2 3.b odd 2 1 inner
162.7.d.b 4 45.h odd 6 2
162.7.d.b 4 45.j even 6 2
192.7.e.c 2 40.f even 2 1
192.7.e.c 2 120.i odd 2 1
192.7.e.f 2 40.e odd 2 1
192.7.e.f 2 120.m even 2 1
294.7.b.a 2 35.c odd 2 1
294.7.b.a 2 105.g even 2 1

## Hecke kernels

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 32$$
$3$ $$T^{2} + 42T + 729$$
$5$ $$T^{2}$$
$7$ $$(T + 2)^{2}$$
$11$ $$T^{2} + 1152$$
$13$ $$(T - 2950)^{2}$$
$17$ $$T^{2} + 20072448$$
$19$ $$(T - 5258)^{2}$$
$23$ $$T^{2} + 105067008$$
$29$ $$T^{2} + 4867200$$
$31$ $$(T - 22898)^{2}$$
$37$ $$(T + 34058)^{2}$$
$41$ $$T^{2} + 281129472$$
$43$ $$(T - 6406)^{2}$$
$47$ $$T^{2} + 32359680000$$
$53$ $$T^{2} + 37074734208$$
$59$ $$T^{2} + 106810722432$$
$61$ $$(T + 62566)^{2}$$
$67$ $$(T + 438698)^{2}$$
$71$ $$T^{2} + 4654195200$$
$73$ $$(T - 730510)^{2}$$
$79$ $$(T - 340562)^{2}$$
$83$ $$T^{2} + 246267234432$$
$89$ $$T^{2} + 149556367872$$
$97$ $$(T - 281086)^{2}$$