# Properties

 Label 315.8.a.c Level $315$ Weight $8$ Character orbit 315.a Self dual yes Analytic conductor $98.401$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [315,8,Mod(1,315)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(315, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 8, names="a")

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

chi := DirichletCharacter("315.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$315 = 3^{2} \cdot 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$8$$ Character orbit: $$[\chi]$$ $$=$$ 315.a (trivial)

## 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: yes Analytic conductor: $$98.4012830275$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{11})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 11$$ x^2 - 11 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 35) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-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 $$\beta = 2\sqrt{11}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta - 8) q^{2} + ( - 16 \beta - 20) q^{4} - 125 q^{5} - 343 q^{7} + ( - 20 \beta + 480) q^{8}+O(q^{10})$$ q + (b - 8) * q^2 + (-16*b - 20) * q^4 - 125 * q^5 - 343 * q^7 + (-20*b + 480) * q^8 $$q + (\beta - 8) q^{2} + ( - 16 \beta - 20) q^{4} - 125 q^{5} - 343 q^{7} + ( - 20 \beta + 480) q^{8} + ( - 125 \beta + 1000) q^{10} + ( - 380 \beta + 3953) q^{11} + (418 \beta - 8909) q^{13} + ( - 343 \beta + 2744) q^{14} + (2688 \beta - 2160) q^{16} + (2210 \beta + 1199) q^{17} + ( - 5542 \beta - 1806) q^{19} + (2000 \beta + 2500) q^{20} + (6993 \beta - 48344) q^{22} + (10690 \beta - 6922) q^{23} + 15625 q^{25} + ( - 12253 \beta + 89664) q^{26} + (5488 \beta + 6860) q^{28} + ( - 23772 \beta + 63449) q^{29} + (22554 \beta + 126384) q^{31} + ( - 21104 \beta + 74112) q^{32} + ( - 16481 \beta + 87648) q^{34} + 42875 q^{35} + (43638 \beta - 132930) q^{37} + (42530 \beta - 229400) q^{38} + (2500 \beta - 60000) q^{40} + (37354 \beta + 55960) q^{41} + (24578 \beta + 473786) q^{43} + ( - 55648 \beta + 188460) q^{44} + ( - 92442 \beta + 525736) q^{46} + ( - 56742 \beta - 135637) q^{47} + 117649 q^{49} + (15625 \beta - 125000) q^{50} + (134184 \beta - 116092) q^{52} + ( - 65224 \beta + 633896) q^{53} + (47500 \beta - 494125) q^{55} + (6860 \beta - 164640) q^{56} + (253625 \beta - 1553560) q^{58} + (170640 \beta + 680060) q^{59} + ( - 11334 \beta - 906840) q^{61} + ( - 54048 \beta - 18696) q^{62} + ( - 101120 \beta - 1244992) q^{64} + ( - 52250 \beta + 1113625) q^{65} + ( - 506344 \beta - 1094656) q^{67} + ( - 63384 \beta - 1579820) q^{68} + (42875 \beta - 343000) q^{70} + ( - 222048 \beta + 747464) q^{71} + ( - 212396 \beta + 3584894) q^{73} + ( - 482034 \beta + 2983512) q^{74} + (139736 \beta + 3937688) q^{76} + (130340 \beta - 1355879) q^{77} + ( - 187504 \beta - 3971487) q^{79} + ( - 336000 \beta + 270000) q^{80} + ( - 242872 \beta + 1195896) q^{82} + ( - 939444 \beta + 152356) q^{83} + ( - 276250 \beta - 149875) q^{85} + (277162 \beta - 2708856) q^{86} + ( - 261460 \beta + 2231840) q^{88} + (247538 \beta + 8971764) q^{89} + ( - 143374 \beta + 3055787) q^{91} + ( - 103048 \beta - 7387320) q^{92} + (318299 \beta - 1411552) q^{94} + (692750 \beta + 225750) q^{95} + (680782 \beta + 2129037) q^{97} + (117649 \beta - 941192) q^{98}+O(q^{100})$$ q + (b - 8) * q^2 + (-16*b - 20) * q^4 - 125 * q^5 - 343 * q^7 + (-20*b + 480) * q^8 + (-125*b + 1000) * q^10 + (-380*b + 3953) * q^11 + (418*b - 8909) * q^13 + (-343*b + 2744) * q^14 + (2688*b - 2160) * q^16 + (2210*b + 1199) * q^17 + (-5542*b - 1806) * q^19 + (2000*b + 2500) * q^20 + (6993*b - 48344) * q^22 + (10690*b - 6922) * q^23 + 15625 * q^25 + (-12253*b + 89664) * q^26 + (5488*b + 6860) * q^28 + (-23772*b + 63449) * q^29 + (22554*b + 126384) * q^31 + (-21104*b + 74112) * q^32 + (-16481*b + 87648) * q^34 + 42875 * q^35 + (43638*b - 132930) * q^37 + (42530*b - 229400) * q^38 + (2500*b - 60000) * q^40 + (37354*b + 55960) * q^41 + (24578*b + 473786) * q^43 + (-55648*b + 188460) * q^44 + (-92442*b + 525736) * q^46 + (-56742*b - 135637) * q^47 + 117649 * q^49 + (15625*b - 125000) * q^50 + (134184*b - 116092) * q^52 + (-65224*b + 633896) * q^53 + (47500*b - 494125) * q^55 + (6860*b - 164640) * q^56 + (253625*b - 1553560) * q^58 + (170640*b + 680060) * q^59 + (-11334*b - 906840) * q^61 + (-54048*b - 18696) * q^62 + (-101120*b - 1244992) * q^64 + (-52250*b + 1113625) * q^65 + (-506344*b - 1094656) * q^67 + (-63384*b - 1579820) * q^68 + (42875*b - 343000) * q^70 + (-222048*b + 747464) * q^71 + (-212396*b + 3584894) * q^73 + (-482034*b + 2983512) * q^74 + (139736*b + 3937688) * q^76 + (130340*b - 1355879) * q^77 + (-187504*b - 3971487) * q^79 + (-336000*b + 270000) * q^80 + (-242872*b + 1195896) * q^82 + (-939444*b + 152356) * q^83 + (-276250*b - 149875) * q^85 + (277162*b - 2708856) * q^86 + (-261460*b + 2231840) * q^88 + (247538*b + 8971764) * q^89 + (-143374*b + 3055787) * q^91 + (-103048*b - 7387320) * q^92 + (318299*b - 1411552) * q^94 + (692750*b + 225750) * q^95 + (680782*b + 2129037) * q^97 + (117649*b - 941192) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 16 q^{2} - 40 q^{4} - 250 q^{5} - 686 q^{7} + 960 q^{8}+O(q^{10})$$ 2 * q - 16 * q^2 - 40 * q^4 - 250 * q^5 - 686 * q^7 + 960 * q^8 $$2 q - 16 q^{2} - 40 q^{4} - 250 q^{5} - 686 q^{7} + 960 q^{8} + 2000 q^{10} + 7906 q^{11} - 17818 q^{13} + 5488 q^{14} - 4320 q^{16} + 2398 q^{17} - 3612 q^{19} + 5000 q^{20} - 96688 q^{22} - 13844 q^{23} + 31250 q^{25} + 179328 q^{26} + 13720 q^{28} + 126898 q^{29} + 252768 q^{31} + 148224 q^{32} + 175296 q^{34} + 85750 q^{35} - 265860 q^{37} - 458800 q^{38} - 120000 q^{40} + 111920 q^{41} + 947572 q^{43} + 376920 q^{44} + 1051472 q^{46} - 271274 q^{47} + 235298 q^{49} - 250000 q^{50} - 232184 q^{52} + 1267792 q^{53} - 988250 q^{55} - 329280 q^{56} - 3107120 q^{58} + 1360120 q^{59} - 1813680 q^{61} - 37392 q^{62} - 2489984 q^{64} + 2227250 q^{65} - 2189312 q^{67} - 3159640 q^{68} - 686000 q^{70} + 1494928 q^{71} + 7169788 q^{73} + 5967024 q^{74} + 7875376 q^{76} - 2711758 q^{77} - 7942974 q^{79} + 540000 q^{80} + 2391792 q^{82} + 304712 q^{83} - 299750 q^{85} - 5417712 q^{86} + 4463680 q^{88} + 17943528 q^{89} + 6111574 q^{91} - 14774640 q^{92} - 2823104 q^{94} + 451500 q^{95} + 4258074 q^{97} - 1882384 q^{98}+O(q^{100})$$ 2 * q - 16 * q^2 - 40 * q^4 - 250 * q^5 - 686 * q^7 + 960 * q^8 + 2000 * q^10 + 7906 * q^11 - 17818 * q^13 + 5488 * q^14 - 4320 * q^16 + 2398 * q^17 - 3612 * q^19 + 5000 * q^20 - 96688 * q^22 - 13844 * q^23 + 31250 * q^25 + 179328 * q^26 + 13720 * q^28 + 126898 * q^29 + 252768 * q^31 + 148224 * q^32 + 175296 * q^34 + 85750 * q^35 - 265860 * q^37 - 458800 * q^38 - 120000 * q^40 + 111920 * q^41 + 947572 * q^43 + 376920 * q^44 + 1051472 * q^46 - 271274 * q^47 + 235298 * q^49 - 250000 * q^50 - 232184 * q^52 + 1267792 * q^53 - 988250 * q^55 - 329280 * q^56 - 3107120 * q^58 + 1360120 * q^59 - 1813680 * q^61 - 37392 * q^62 - 2489984 * q^64 + 2227250 * q^65 - 2189312 * q^67 - 3159640 * q^68 - 686000 * q^70 + 1494928 * q^71 + 7169788 * q^73 + 5967024 * q^74 + 7875376 * q^76 - 2711758 * q^77 - 7942974 * q^79 + 540000 * q^80 + 2391792 * q^82 + 304712 * q^83 - 299750 * q^85 - 5417712 * q^86 + 4463680 * q^88 + 17943528 * q^89 + 6111574 * q^91 - 14774640 * q^92 - 2823104 * q^94 + 451500 * q^95 + 4258074 * q^97 - 1882384 * q^98

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 −3.31662 3.31662
−14.6332 0 86.1320 −125.000 0 −343.000 612.665 0 1829.16
1.2 −1.36675 0 −126.132 −125.000 0 −343.000 347.335 0 170.844
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$-1$$
$$5$$ $$+1$$
$$7$$ $$+1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 315.8.a.c 2
3.b odd 2 1 35.8.a.a 2
12.b even 2 1 560.8.a.i 2
15.d odd 2 1 175.8.a.b 2
15.e even 4 2 175.8.b.c 4
21.c even 2 1 245.8.a.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.8.a.a 2 3.b odd 2 1
175.8.a.b 2 15.d odd 2 1
175.8.b.c 4 15.e even 4 2
245.8.a.b 2 21.c even 2 1
315.8.a.c 2 1.a even 1 1 trivial
560.8.a.i 2 12.b even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} + 16T_{2} + 20$$ acting on $$S_{8}^{\mathrm{new}}(\Gamma_0(315))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 16T + 20$$
$3$ $$T^{2}$$
$5$ $$(T + 125)^{2}$$
$7$ $$(T + 343)^{2}$$
$11$ $$T^{2} - 7906 T + 9272609$$
$13$ $$T^{2} + 17818 T + 71682425$$
$17$ $$T^{2} - 2398 T - 213462799$$
$19$ $$T^{2} + \cdots - 1348143980$$
$23$ $$T^{2} + \cdots - 4980234316$$
$29$ $$T^{2} + \cdots - 20838975695$$
$31$ $$T^{2} + \cdots - 6409132848$$
$37$ $$T^{2} + \cdots - 66117717036$$
$41$ $$T^{2} + \cdots - 58262616304$$
$43$ $$T^{2} + \cdots + 197893738100$$
$47$ $$T^{2} + \cdots - 123267405047$$
$53$ $$T^{2} + \cdots + 214640651072$$
$59$ $$T^{2} + \cdots - 818710818800$$
$61$ $$T^{2} + \cdots + 816706565136$$
$67$ $$T^{2} + \cdots - 10082635080448$$
$71$ $$T^{2} + \cdots - 1610731398080$$
$73$ $$T^{2} + \cdots + 10866534315332$$
$79$ $$T^{2} + \cdots + 14225767990465$$
$83$ $$T^{2} + \cdots - 38809208931248$$
$89$ $$T^{2} + \cdots + 77796446568160$$
$97$ $$T^{2} + \cdots - 15859623239687$$