# Properties

 Label 95.11.d.b Level $95$ Weight $11$ Character orbit 95.d Self dual yes Analytic conductor $60.359$ Analytic rank $0$ Dimension $4$ CM discriminant -95 Inner twists $4$

# Learn more

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [95,11,Mod(94,95)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(95, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 11, names="a")

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

chi := DirichletCharacter("95.94");

S:= CuspForms(chi, 11);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$95 = 5 \cdot 19$$ Weight: $$k$$ $$=$$ $$11$$ Character orbit: $$[\chi]$$ $$=$$ 95.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: yes Analytic conductor: $$60.3589390040$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: 4.4.462080.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - 16x^{2} + 45$$ x^4 - 16*x^2 + 45 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$5$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + (12 \beta_{2} + \beta_1) q^{2} + (43 \beta_{2} - 23 \beta_1) q^{3} + (121 \beta_{3} + 1024) q^{4} - 3125 q^{5} + ( - 3421 \beta_{3} + 427) q^{6} + ( - 1089 \beta_{2} + 1694 \beta_1) q^{8} + ( - 8404 \beta_{3} + 59049) q^{9}+O(q^{10})$$ q + (12*b2 + b1) * q^2 + (43*b2 - 23*b1) * q^3 + (121*b3 + 1024) * q^4 - 3125 * q^5 + (-3421*b3 + 427) * q^6 + (-1089*b2 + 1694*b1) * q^8 + (-8404*b3 + 59049) * q^9 $$q + (12 \beta_{2} + \beta_1) q^{2} + (43 \beta_{2} - 23 \beta_1) q^{3} + (121 \beta_{3} + 1024) q^{4} - 3125 q^{5} + ( - 3421 \beta_{3} + 427) q^{6} + ( - 1089 \beta_{2} + 1694 \beta_1) q^{8} + ( - 8404 \beta_{3} + 59049) q^{9} + ( - 37500 \beta_{2} - 3125 \beta_1) q^{10} - 73196 \beta_{3} q^{11} + ( - 8119 \beta_{2} - 23915 \beta_1) q^{12} + (127953 \beta_{2} + 16963 \beta_1) q^{13} + ( - 134375 \beta_{2} + 71875 \beta_1) q^{15} + (123904 \beta_{3} - 770397) q^{16} + (784224 \beta_{2} - 58607 \beta_1) q^{18} + 2476099 q^{19} + ( - 378125 \beta_{3} - 3200000) q^{20} + (658764 \beta_{2} - 1024744 \beta_1) q^{22} + (51667 \beta_{3} - 7864879) q^{24} + 9765625 q^{25} + (2203729 \beta_{3} + 23475377) q^{26} + (3622124 \beta_{2} + 25212 \beta_1) q^{27} + (10690625 \beta_{3} - 1334375) q^{30} + ( - 9244764 \beta_{2} - 770397 \beta_1) q^{32} + (31547476 \beta_{2} + 219588 \beta_1) q^{33} + ( - 1460767 \beta_{3} + 41145380) q^{36} + (26508987 \beta_{2} - 3184919 \beta_1) q^{37} + (29713188 \beta_{2} + 2476099 \beta_1) q^{38} + ( - 37390804 \beta_{3} - 23073602) q^{39} + (3403125 \beta_{2} - 5293750 \beta_1) q^{40} + ( - 74952704 \beta_{3} - 168277604) q^{44} + (26262500 \beta_{3} - 184528125) q^{45} + ( - 86529695 \beta_{2} + 17347419 \beta_1) q^{48} + 282475249 q^{49} + (117187500 \beta_{2} + 9765625 \beta_1) q^{50} + (130847091 \beta_{2} + 36957471 \beta_1) q^{52} + (24605283 \beta_{2} + 40772401 \beta_1) q^{53} + ( - 3588508 \beta_{3} + 546251596) q^{54} + 228737500 \beta_{3} q^{55} + (106472257 \beta_{2} - 56950277 \beta_1) q^{57} + (25371875 \beta_{2} + 74734375 \beta_1) q^{60} + 254662804 \beta_{3} q^{61} + ( - 220095733 \beta_{3} - 788886528) q^{64} + ( - 399853125 \beta_{2} - 53009375 \beta_1) q^{65} + ( - 31254692 \beta_{3} + 4757666804) q^{66} + ( - 391750743 \beta_{2} + 105272443 \beta_1) q^{67} + ( - 296153913 \beta_{2} + 80708210 \beta_1) q^{72} + ( - 514831229 \beta_{3} + 3121760123) q^{74} + (419921875 \beta_{2} - 224609375 \beta_1) q^{75} + (299607979 \beta_{3} + 2535525376) q^{76} + (59634012 \beta_{2} - 546544858 \beta_1) q^{78} + ( - 387200000 \beta_{3} + 2407490625) q^{80} + ( - 496247796 \beta_{3} - 2144867297) q^{81} + ( - 2019331248 \beta_{2} - 168277604 \beta_1) q^{88} + ( - 2450700000 \beta_{2} + 183146875 \beta_1) q^{90} - 7737809375 q^{95} + (2635528137 \beta_{3} - 328959519) q^{96} + (852514827 \beta_{2} - 854542415 \beta_1) q^{97} + (3389702988 \beta_{2} + 282475249 \beta_1) q^{98} + ( - 4322150604 \beta_{3} + 11687644496) q^{99}+O(q^{100})$$ q + (12*b2 + b1) * q^2 + (43*b2 - 23*b1) * q^3 + (121*b3 + 1024) * q^4 - 3125 * q^5 + (-3421*b3 + 427) * q^6 + (-1089*b2 + 1694*b1) * q^8 + (-8404*b3 + 59049) * q^9 + (-37500*b2 - 3125*b1) * q^10 - 73196*b3 * q^11 + (-8119*b2 - 23915*b1) * q^12 + (127953*b2 + 16963*b1) * q^13 + (-134375*b2 + 71875*b1) * q^15 + (123904*b3 - 770397) * q^16 + (784224*b2 - 58607*b1) * q^18 + 2476099 * q^19 + (-378125*b3 - 3200000) * q^20 + (658764*b2 - 1024744*b1) * q^22 + (51667*b3 - 7864879) * q^24 + 9765625 * q^25 + (2203729*b3 + 23475377) * q^26 + (3622124*b2 + 25212*b1) * q^27 + (10690625*b3 - 1334375) * q^30 + (-9244764*b2 - 770397*b1) * q^32 + (31547476*b2 + 219588*b1) * q^33 + (-1460767*b3 + 41145380) * q^36 + (26508987*b2 - 3184919*b1) * q^37 + (29713188*b2 + 2476099*b1) * q^38 + (-37390804*b3 - 23073602) * q^39 + (3403125*b2 - 5293750*b1) * q^40 + (-74952704*b3 - 168277604) * q^44 + (26262500*b3 - 184528125) * q^45 + (-86529695*b2 + 17347419*b1) * q^48 + 282475249 * q^49 + (117187500*b2 + 9765625*b1) * q^50 + (130847091*b2 + 36957471*b1) * q^52 + (24605283*b2 + 40772401*b1) * q^53 + (-3588508*b3 + 546251596) * q^54 + 228737500*b3 * q^55 + (106472257*b2 - 56950277*b1) * q^57 + (25371875*b2 + 74734375*b1) * q^60 + 254662804*b3 * q^61 + (-220095733*b3 - 788886528) * q^64 + (-399853125*b2 - 53009375*b1) * q^65 + (-31254692*b3 + 4757666804) * q^66 + (-391750743*b2 + 105272443*b1) * q^67 + (-296153913*b2 + 80708210*b1) * q^72 + (-514831229*b3 + 3121760123) * q^74 + (419921875*b2 - 224609375*b1) * q^75 + (299607979*b3 + 2535525376) * q^76 + (59634012*b2 - 546544858*b1) * q^78 + (-387200000*b3 + 2407490625) * q^80 + (-496247796*b3 - 2144867297) * q^81 + (-2019331248*b2 - 168277604*b1) * q^88 + (-2450700000*b2 + 183146875*b1) * q^90 - 7737809375 * q^95 + (2635528137*b3 - 328959519) * q^96 + (852514827*b2 - 854542415*b1) * q^97 + (3389702988*b2 + 282475249*b1) * q^98 + (-4322150604*b3 + 11687644496) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 4096 q^{4} - 12500 q^{5} + 1708 q^{6} + 236196 q^{9}+O(q^{10})$$ 4 * q + 4096 * q^4 - 12500 * q^5 + 1708 * q^6 + 236196 * q^9 $$4 q + 4096 q^{4} - 12500 q^{5} + 1708 q^{6} + 236196 q^{9} - 3081588 q^{16} + 9904396 q^{19} - 12800000 q^{20} - 31459516 q^{24} + 39062500 q^{25} + 93901508 q^{26} - 5337500 q^{30} + 164581520 q^{36} - 92294408 q^{39} - 673110416 q^{44} - 738112500 q^{45} + 1129900996 q^{49} + 2185006384 q^{54} - 3155546112 q^{64} + 19030667216 q^{66} + 12487040492 q^{74} + 10142101504 q^{76} + 9629962500 q^{80} - 8579469188 q^{81} - 30951237500 q^{95} - 1315838076 q^{96} + 46750577984 q^{99}+O(q^{100})$$ 4 * q + 4096 * q^4 - 12500 * q^5 + 1708 * q^6 + 236196 * q^9 - 3081588 * q^16 + 9904396 * q^19 - 12800000 * q^20 - 31459516 * q^24 + 39062500 * q^25 + 93901508 * q^26 - 5337500 * q^30 + 164581520 * q^36 - 92294408 * q^39 - 673110416 * q^44 - 738112500 * q^45 + 1129900996 * q^49 + 2185006384 * q^54 - 3155546112 * q^64 + 19030667216 * q^66 + 12487040492 * q^74 + 10142101504 * q^76 + 9629962500 * q^80 - 8579469188 * q^81 - 30951237500 * q^95 - 1315838076 * q^96 + 46750577984 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 16x^{2} + 45$$ :

 $$\beta_{1}$$ $$=$$ $$5\nu$$ 5*v $$\beta_{2}$$ $$=$$ $$( \nu^{3} - 10\nu ) / 3$$ (v^3 - 10*v) / 3 $$\beta_{3}$$ $$=$$ $$\nu^{2} - 8$$ v^2 - 8
 $$\nu$$ $$=$$ $$( \beta_1 ) / 5$$ (b1) / 5 $$\nu^{2}$$ $$=$$ $$\beta_{3} + 8$$ b3 + 8 $$\nu^{3}$$ $$=$$ $$3\beta_{2} + 2\beta_1$$ 3*b2 + 2*b1

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/95\mathbb{Z}\right)^\times$$.

 $$n$$ $$21$$ $$77$$ $$\chi(n)$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
94.1
 −3.51552 1.90817 −1.90817 3.51552
−50.7487 285.422 1551.43 −3125.00 −14484.8 0 −26766.2 22416.8 158590.
94.2 −38.9945 −393.358 496.573 −3125.00 15338.8 0 20566.8 95681.2 121858.
94.3 38.9945 393.358 496.573 −3125.00 15338.8 0 −20566.8 95681.2 −121858.
94.4 50.7487 −285.422 1551.43 −3125.00 −14484.8 0 26766.2 22416.8 −158590.
 $$n$$: 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
95.d odd 2 1 CM by $$\Q(\sqrt{-95})$$
5.b even 2 1 inner
19.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 95.11.d.b 4
5.b even 2 1 inner 95.11.d.b 4
19.b odd 2 1 inner 95.11.d.b 4
95.d odd 2 1 CM 95.11.d.b 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
95.11.d.b 4 1.a even 1 1 trivial
95.11.d.b 4 5.b even 2 1 inner
95.11.d.b 4 19.b odd 2 1 inner
95.11.d.b 4 95.d odd 2 1 CM

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{4} - 4096T_{2}^{2} + 3916125$$ acting on $$S_{11}^{\mathrm{new}}(95, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} - 4096 T^{2} + \cdots + 3916125$$
$3$ $$T^{4} - 236196 T^{2} + \cdots + 12605220500$$
$5$ $$(T + 3125)^{4}$$
$7$ $$T^{4}$$
$11$ $$(T^{2} - 101795433904)^{2}$$
$13$ $$T^{4} - 551433967396 T^{2} + \cdots + 53\!\cdots\!00$$
$17$ $$T^{4}$$
$19$ $$(T - 2476099)^{4}$$
$23$ $$T^{4}$$
$29$ $$T^{4}$$
$31$ $$T^{4}$$
$37$ $$T^{4} + \cdots + 56\!\cdots\!00$$
$41$ $$T^{4}$$
$43$ $$T^{4}$$
$47$ $$T^{4}$$
$53$ $$T^{4} + \cdots + 51\!\cdots\!00$$
$59$ $$T^{4}$$
$61$ $$(T^{2} - 12\!\cdots\!04)^{2}$$
$67$ $$T^{4} + \cdots + 39\!\cdots\!00$$
$71$ $$T^{4}$$
$73$ $$T^{4}$$
$79$ $$T^{4}$$
$83$ $$T^{4}$$
$89$ $$T^{4}$$
$97$ $$T^{4} + \cdots + 21\!\cdots\!00$$
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