# Properties

 Label 294.7.b.a Level $294$ Weight $7$ Character orbit 294.b Analytic conductor $67.636$ 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] = [294,7,Mod(197,294)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(294, 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("294.197");

S:= CuspForms(chi, 7);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$294 = 2 \cdot 3 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$7$$ Character orbit: $$[\chi]$$ $$=$$ 294.b (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: $$67.6359005842$$ 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.

 $$f(q)$$ $$=$$ $$q + \beta q^{2} + ( - 3 \beta - 21) q^{3} - 32 q^{4} + 30 \beta q^{5} + ( - 21 \beta + 96) q^{6} - 32 \beta q^{8} + (126 \beta + 153) q^{9} +O(q^{10})$$ q + b * q^2 + (-3*b - 21) * q^3 - 32 * q^4 + 30*b * q^5 + (-21*b + 96) * q^6 - 32*b * q^8 + (126*b + 153) * q^9 $$q + \beta q^{2} + ( - 3 \beta - 21) q^{3} - 32 q^{4} + 30 \beta q^{5} + ( - 21 \beta + 96) q^{6} - 32 \beta q^{8} + (126 \beta + 153) q^{9} - 960 q^{10} - 6 \beta q^{11} + (96 \beta + 672) q^{12} + 2950 q^{13} + ( - 630 \beta + 2880) q^{15} + 1024 q^{16} - 792 \beta q^{17} + (153 \beta - 4032) q^{18} - 5258 q^{19} - 960 \beta q^{20} + 192 q^{22} - 1812 \beta q^{23} + (672 \beta - 3072) q^{24} - 13175 q^{25} + 2950 \beta q^{26} + ( - 3105 \beta + 8883) q^{27} + 390 \beta q^{29} + (2880 \beta + 20160) q^{30} - 22898 q^{31} + 1024 \beta q^{32} + (126 \beta - 576) q^{33} + 25344 q^{34} + ( - 4032 \beta - 4896) q^{36} + 34058 q^{37} - 5258 \beta q^{38} + ( - 8850 \beta - 61950) q^{39} + 30720 q^{40} + 2964 \beta q^{41} - 6406 q^{43} + 192 \beta q^{44} + (4590 \beta - 120960) q^{45} + 57984 q^{46} - 31800 \beta q^{47} + ( - 3072 \beta - 21504) q^{48} - 13175 \beta q^{50} + (16632 \beta - 76032) q^{51} - 94400 q^{52} - 34038 \beta q^{53} + (8883 \beta + 99360) q^{54} + 5760 q^{55} + (15774 \beta + 110418) q^{57} - 12480 q^{58} + 57774 \beta q^{59} + (20160 \beta - 92160) q^{60} + 62566 q^{61} - 22898 \beta q^{62} - 32768 q^{64} + 88500 \beta q^{65} + ( - 576 \beta - 4032) q^{66} + 438698 q^{67} + 25344 \beta q^{68} + (38052 \beta - 173952) q^{69} - 12060 \beta q^{71} + ( - 4896 \beta + 129024) q^{72} + 730510 q^{73} + 34058 \beta q^{74} + (39525 \beta + 276675) q^{75} + 168256 q^{76} + ( - 61950 \beta + 283200) q^{78} + 340562 q^{79} + 30720 \beta q^{80} + (38556 \beta - 484623) q^{81} - 94848 q^{82} - 87726 \beta q^{83} + 760320 q^{85} - 6406 \beta q^{86} + ( - 8190 \beta + 37440) q^{87} - 6144 q^{88} + 68364 \beta q^{89} + ( - 120960 \beta - 146880) q^{90} + 57984 \beta q^{92} + (68694 \beta + 480858) q^{93} + 1017600 q^{94} - 157740 \beta q^{95} + ( - 21504 \beta + 98304) q^{96} + 281086 q^{97} + ( - 918 \beta + 24192) q^{99} +O(q^{100})$$ q + b * q^2 + (-3*b - 21) * q^3 - 32 * q^4 + 30*b * q^5 + (-21*b + 96) * q^6 - 32*b * q^8 + (126*b + 153) * q^9 - 960 * q^10 - 6*b * q^11 + (96*b + 672) * q^12 + 2950 * q^13 + (-630*b + 2880) * q^15 + 1024 * q^16 - 792*b * q^17 + (153*b - 4032) * q^18 - 5258 * q^19 - 960*b * q^20 + 192 * q^22 - 1812*b * q^23 + (672*b - 3072) * q^24 - 13175 * q^25 + 2950*b * q^26 + (-3105*b + 8883) * q^27 + 390*b * q^29 + (2880*b + 20160) * q^30 - 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 + 30720 * q^40 + 2964*b * q^41 - 6406 * q^43 + 192*b * q^44 + (4590*b - 120960) * q^45 + 57984 * q^46 - 31800*b * q^47 + (-3072*b - 21504) * q^48 - 13175*b * q^50 + (16632*b - 76032) * q^51 - 94400 * q^52 - 34038*b * q^53 + (8883*b + 99360) * q^54 + 5760 * q^55 + (15774*b + 110418) * q^57 - 12480 * q^58 + 57774*b * q^59 + (20160*b - 92160) * q^60 + 62566 * q^61 - 22898*b * q^62 - 32768 * q^64 + 88500*b * q^65 + (-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 + (39525*b + 276675) * q^75 + 168256 * q^76 + (-61950*b + 283200) * q^78 + 340562 * q^79 + 30720*b * q^80 + (38556*b - 484623) * q^81 - 94848 * q^82 - 87726*b * q^83 + 760320 * q^85 - 6406*b * q^86 + (-8190*b + 37440) * q^87 - 6144 * q^88 + 68364*b * q^89 + (-120960*b - 146880) * q^90 + 57984*b * q^92 + (68694*b + 480858) * q^93 + 1017600 * q^94 - 157740*b * q^95 + (-21504*b + 98304) * q^96 + 281086 * q^97 + (-918*b + 24192) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 42 q^{3} - 64 q^{4} + 192 q^{6} + 306 q^{9}+O(q^{10})$$ 2 * q - 42 * q^3 - 64 * q^4 + 192 * q^6 + 306 * q^9 $$2 q - 42 q^{3} - 64 q^{4} + 192 q^{6} + 306 q^{9} - 1920 q^{10} + 1344 q^{12} + 5900 q^{13} + 5760 q^{15} + 2048 q^{16} - 8064 q^{18} - 10516 q^{19} + 384 q^{22} - 6144 q^{24} - 26350 q^{25} + 17766 q^{27} + 40320 q^{30} - 45796 q^{31} - 1152 q^{33} + 50688 q^{34} - 9792 q^{36} + 68116 q^{37} - 123900 q^{39} + 61440 q^{40} - 12812 q^{43} - 241920 q^{45} + 115968 q^{46} - 43008 q^{48} - 152064 q^{51} - 188800 q^{52} + 198720 q^{54} + 11520 q^{55} + 220836 q^{57} - 24960 q^{58} - 184320 q^{60} + 125132 q^{61} - 65536 q^{64} - 8064 q^{66} + 877396 q^{67} - 347904 q^{69} + 258048 q^{72} + 1461020 q^{73} + 553350 q^{75} + 336512 q^{76} + 566400 q^{78} + 681124 q^{79} - 969246 q^{81} - 189696 q^{82} + 1520640 q^{85} + 74880 q^{87} - 12288 q^{88} - 293760 q^{90} + 961716 q^{93} + 2035200 q^{94} + 196608 q^{96} + 562172 q^{97} + 48384 q^{99}+O(q^{100})$$ 2 * q - 42 * q^3 - 64 * q^4 + 192 * q^6 + 306 * q^9 - 1920 * q^10 + 1344 * q^12 + 5900 * q^13 + 5760 * q^15 + 2048 * q^16 - 8064 * q^18 - 10516 * q^19 + 384 * q^22 - 6144 * q^24 - 26350 * q^25 + 17766 * q^27 + 40320 * q^30 - 45796 * q^31 - 1152 * q^33 + 50688 * q^34 - 9792 * q^36 + 68116 * q^37 - 123900 * q^39 + 61440 * q^40 - 12812 * q^43 - 241920 * q^45 + 115968 * q^46 - 43008 * q^48 - 152064 * q^51 - 188800 * q^52 + 198720 * q^54 + 11520 * q^55 + 220836 * q^57 - 24960 * q^58 - 184320 * q^60 + 125132 * q^61 - 65536 * q^64 - 8064 * q^66 + 877396 * q^67 - 347904 * q^69 + 258048 * q^72 + 1461020 * q^73 + 553350 * q^75 + 336512 * q^76 + 566400 * q^78 + 681124 * q^79 - 969246 * q^81 - 189696 * q^82 + 1520640 * q^85 + 74880 * q^87 - 12288 * q^88 - 293760 * q^90 + 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 $$\left(\mathbb{Z}/294\mathbb{Z}\right)^\times$$.

 $$n$$ $$197$$ $$199$$ $$\chi(n)$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
197.1
 − 1.41421i 1.41421i
5.65685i −21.0000 + 16.9706i −32.0000 169.706i 96.0000 + 118.794i 0 181.019i 153.000 712.764i −960.000
197.2 5.65685i −21.0000 16.9706i −32.0000 169.706i 96.0000 118.794i 0 181.019i 153.000 + 712.764i −960.000
 $$n$$: 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 294.7.b.a 2
3.b odd 2 1 inner 294.7.b.a 2
7.b odd 2 1 6.7.b.a 2
21.c even 2 1 6.7.b.a 2
28.d even 2 1 48.7.e.b 2
35.c odd 2 1 150.7.d.a 2
35.f even 4 2 150.7.b.a 4
56.e even 2 1 192.7.e.f 2
56.h odd 2 1 192.7.e.c 2
63.l odd 6 2 162.7.d.b 4
63.o even 6 2 162.7.d.b 4
84.h odd 2 1 48.7.e.b 2
105.g even 2 1 150.7.d.a 2
105.k odd 4 2 150.7.b.a 4
168.e odd 2 1 192.7.e.f 2
168.i even 2 1 192.7.e.c 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6.7.b.a 2 7.b odd 2 1
6.7.b.a 2 21.c even 2 1
48.7.e.b 2 28.d even 2 1
48.7.e.b 2 84.h odd 2 1
150.7.b.a 4 35.f even 4 2
150.7.b.a 4 105.k odd 4 2
150.7.d.a 2 35.c odd 2 1
150.7.d.a 2 105.g even 2 1
162.7.d.b 4 63.l odd 6 2
162.7.d.b 4 63.o even 6 2
192.7.e.c 2 56.h odd 2 1
192.7.e.c 2 168.i even 2 1
192.7.e.f 2 56.e even 2 1
192.7.e.f 2 168.e odd 2 1
294.7.b.a 2 1.a even 1 1 trivial
294.7.b.a 2 3.b odd 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{7}^{\mathrm{new}}(294, [\chi])$$:

 $$T_{5}^{2} + 28800$$ T5^2 + 28800 $$T_{13} - 2950$$ T13 - 2950

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 32$$
$3$ $$T^{2} + 42T + 729$$
$5$ $$T^{2} + 28800$$
$7$ $$T^{2}$$
$11$ $$T^{2} + 1152$$
$13$ $$(T - 2950)^{2}$$
$17$ $$T^{2} + 20072448$$
$19$ $$(T + 5258)^{2}$$
$23$ $$T^{2} + 105067008$$
$29$ $$T^{2} + 4867200$$
$31$ $$(T + 22898)^{2}$$
$37$ $$(T - 34058)^{2}$$
$41$ $$T^{2} + 281129472$$
$43$ $$(T + 6406)^{2}$$
$47$ $$T^{2} + 32359680000$$
$53$ $$T^{2} + 37074734208$$
$59$ $$T^{2} + 106810722432$$
$61$ $$(T - 62566)^{2}$$
$67$ $$(T - 438698)^{2}$$
$71$ $$T^{2} + 4654195200$$
$73$ $$(T - 730510)^{2}$$
$79$ $$(T - 340562)^{2}$$
$83$ $$T^{2} + 246267234432$$
$89$ $$T^{2} + 149556367872$$
$97$ $$(T - 281086)^{2}$$