# Properties

 Label 95.11.d.c 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: 4.4.7600.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - 9x^{2} + 19$$ x^4 - 9*x^2 + 19 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{2}\cdot 5^{2}\cdot 11$$ 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.

 $$f(q)$$ $$=$$ $$q + (2 \beta_{2} + \beta_1) q^{2} + (17 \beta_{2} - 6 \beta_1) q^{3} + ( - 177 \beta_{3} + 1024) q^{4} + 3125 q^{5} + (395 \beta_{3} - 427) q^{6} + (2655 \beta_{2} + 2301 \beta_1) q^{8} + (10042 \beta_{3} + 59049) q^{9}+O(q^{10})$$ q + (2*b2 + b1) * q^2 + (17*b2 - 6*b1) * q^3 + (-177*b3 + 1024) * q^4 + 3125 * q^5 + (395*b3 - 427) * q^6 + (2655*b2 + 2301*b1) * q^8 + (10042*b3 + 59049) * q^9 $$q + (2 \beta_{2} + \beta_1) q^{2} + (17 \beta_{2} - 6 \beta_1) q^{3} + ( - 177 \beta_{3} + 1024) q^{4} + 3125 q^{5} + (395 \beta_{3} - 427) q^{6} + (2655 \beta_{2} + 2301 \beta_1) q^{8} + (10042 \beta_{3} + 59049) q^{9} + (6250 \beta_{2} + 3125 \beta_1) q^{10} + 3954 \beta_{3} q^{11} + ( - 24187 \beta_{2} + 582 \beta_1) q^{12} + (27053 \beta_{2} + 10090 \beta_1) q^{13} + (53125 \beta_{2} - 18750 \beta_1) q^{15} + ( - 181248 \beta_{3} + 2867549) q^{16} + ( - 32532 \beta_{2} - 71497 \beta_1) q^{18} - 2476099 q^{19} + ( - 553125 \beta_{3} + 3200000) q^{20} + ( - 59310 \beta_{2} - 51402 \beta_1) q^{22} + (75579 \beta_{3} - 8739375) q^{24} + 9765625 q^{25} + ( - 1944009 \beta_{3} + 23475377) q^{26} + (2359870 \beta_{2} - 381596 \beta_1) q^{27} + (1234375 \beta_{3} - 1334375) q^{30} + (5735098 \beta_{2} + 2867549 \beta_1) q^{32} + (929190 \beta_{2} - 150252 \beta_1) q^{33} + ( - 168665 \beta_{3} - 161713074) q^{36} + (485473 \beta_{2} - 2699446 \beta_1) q^{37} + ( - 4952198 \beta_{2} - 2476099 \beta_1) q^{38} + (6927194 \beta_{3} + 23073602) q^{39} + (8296875 \beta_{2} + 7190625 \beta_1) q^{40} + (4048896 \beta_{3} - 87482250) q^{44} + (31381250 \beta_{3} + 184528125) q^{45} + (6155053 \beta_{2} - 10317870 \beta_1) q^{48} + 282475249 q^{49} + (19531250 \beta_{2} + 9765625 \beta_1) q^{50} + (48408617 \beta_{2} + 38415334 \beta_1) q^{52} + ( - 39302663 \beta_{2} + 1469738 \beta_1) q^{53} + ( - 4287934 \beta_{3} + 495823750) q^{54} + 12356250 \beta_{3} q^{55} + ( - 42093683 \beta_{2} + 14856594 \beta_1) q^{57} + ( - 75584375 \beta_{2} + 1818750 \beta_1) q^{60} - 113882814 \beta_{3} q^{61} + ( - 321958221 \beta_{3} + 2936370176) q^{64} + (84540625 \beta_{2} + 31531250 \beta_1) q^{65} + ( - 1688358 \beta_{3} + 195228750) q^{66} + (60088517 \beta_{2} - 45183926 \beta_1) q^{67} + ( - 287583405 \beta_{2} - 86307501 \beta_1) q^{72} + (342461547 \beta_{3} - 3121760123) q^{74} + (166015625 \beta_{2} - 58593750 \beta_1) q^{75} + (438269523 \beta_{3} - 2535525376) q^{76} + ( - 57760706 \beta_{2} - 66979920 \beta_1) q^{78} + ( - 566400000 \beta_{3} + 8961090625) q^{80} + (592970058 \beta_{3} + 9118436099) q^{81} + ( - 174964500 \beta_{2} - 87482250 \beta_1) q^{88} + ( - 101662500 \beta_{2} - 223428125 \beta_1) q^{90} - 7737809375 q^{95} + (1132681855 \beta_{3} - 1224443423) q^{96} + (699355393 \beta_{2} - 155187022 \beta_1) q^{97} + (564950498 \beta_{2} + 282475249 \beta_1) q^{98} + (233479746 \beta_{3} + 4963258500) q^{99}+O(q^{100})$$ q + (2*b2 + b1) * q^2 + (17*b2 - 6*b1) * q^3 + (-177*b3 + 1024) * q^4 + 3125 * q^5 + (395*b3 - 427) * q^6 + (2655*b2 + 2301*b1) * q^8 + (10042*b3 + 59049) * q^9 + (6250*b2 + 3125*b1) * q^10 + 3954*b3 * q^11 + (-24187*b2 + 582*b1) * q^12 + (27053*b2 + 10090*b1) * q^13 + (53125*b2 - 18750*b1) * q^15 + (-181248*b3 + 2867549) * q^16 + (-32532*b2 - 71497*b1) * q^18 - 2476099 * q^19 + (-553125*b3 + 3200000) * q^20 + (-59310*b2 - 51402*b1) * q^22 + (75579*b3 - 8739375) * q^24 + 9765625 * q^25 + (-1944009*b3 + 23475377) * q^26 + (2359870*b2 - 381596*b1) * q^27 + (1234375*b3 - 1334375) * q^30 + (5735098*b2 + 2867549*b1) * q^32 + (929190*b2 - 150252*b1) * q^33 + (-168665*b3 - 161713074) * q^36 + (485473*b2 - 2699446*b1) * q^37 + (-4952198*b2 - 2476099*b1) * q^38 + (6927194*b3 + 23073602) * q^39 + (8296875*b2 + 7190625*b1) * q^40 + (4048896*b3 - 87482250) * q^44 + (31381250*b3 + 184528125) * q^45 + (6155053*b2 - 10317870*b1) * q^48 + 282475249 * q^49 + (19531250*b2 + 9765625*b1) * q^50 + (48408617*b2 + 38415334*b1) * q^52 + (-39302663*b2 + 1469738*b1) * q^53 + (-4287934*b3 + 495823750) * q^54 + 12356250*b3 * q^55 + (-42093683*b2 + 14856594*b1) * q^57 + (-75584375*b2 + 1818750*b1) * q^60 - 113882814*b3 * q^61 + (-321958221*b3 + 2936370176) * q^64 + (84540625*b2 + 31531250*b1) * q^65 + (-1688358*b3 + 195228750) * q^66 + (60088517*b2 - 45183926*b1) * q^67 + (-287583405*b2 - 86307501*b1) * q^72 + (342461547*b3 - 3121760123) * q^74 + (166015625*b2 - 58593750*b1) * q^75 + (438269523*b3 - 2535525376) * q^76 + (-57760706*b2 - 66979920*b1) * q^78 + (-566400000*b3 + 8961090625) * q^80 + (592970058*b3 + 9118436099) * q^81 + (-174964500*b2 - 87482250*b1) * q^88 + (-101662500*b2 - 223428125*b1) * q^90 - 7737809375 * q^95 + (1132681855*b3 - 1224443423) * q^96 + (699355393*b2 - 155187022*b1) * q^97 + (564950498*b2 + 282475249*b1) * q^98 + (233479746*b3 + 4963258500) * 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 $$4 q + 4096 q^{4} + 12500 q^{5} - 1708 q^{6} + 236196 q^{9} + 11470196 q^{16} - 9904396 q^{19} + 12800000 q^{20} - 34957500 q^{24} + 39062500 q^{25} + 93901508 q^{26} - 5337500 q^{30} - 646852296 q^{36} + 92294408 q^{39} - 349929000 q^{44} + 738112500 q^{45} + 1129900996 q^{49} + 1983295000 q^{54} + 11745480704 q^{64} + 780915000 q^{66} - 12487040492 q^{74} - 10142101504 q^{76} + 35844362500 q^{80} + 36473744396 q^{81} - 30951237500 q^{95} - 4897773692 q^{96} + 19853034000 q^{99}+O(q^{100})$$ 4 * q + 4096 * q^4 + 12500 * q^5 - 1708 * q^6 + 236196 * q^9 + 11470196 * q^16 - 9904396 * q^19 + 12800000 * q^20 - 34957500 * q^24 + 39062500 * q^25 + 93901508 * q^26 - 5337500 * q^30 - 646852296 * q^36 + 92294408 * q^39 - 349929000 * q^44 + 738112500 * q^45 + 1129900996 * q^49 + 1983295000 * q^54 + 11745480704 * q^64 + 780915000 * q^66 - 12487040492 * q^74 - 10142101504 * q^76 + 35844362500 * q^80 + 36473744396 * q^81 - 30951237500 * q^95 - 4897773692 * q^96 + 19853034000 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 9x^{2} + 19$$ :

 $$\beta_{1}$$ $$=$$ $$-5\nu^{3} + 40\nu$$ -5*v^3 + 40*v $$\beta_{2}$$ $$=$$ $$-6\nu^{3} + 26\nu$$ -6*v^3 + 26*v $$\beta_{3}$$ $$=$$ $$10\nu^{2} - 45$$ 10*v^2 - 45
 $$\nu$$ $$=$$ $$( -5\beta_{2} + 6\beta_1 ) / 110$$ (-5*b2 + 6*b1) / 110 $$\nu^{2}$$ $$=$$ $$( \beta_{3} + 45 ) / 10$$ (b3 + 45) / 10 $$\nu^{3}$$ $$=$$ $$( -20\beta_{2} + 13\beta_1 ) / 55$$ (-20*b2 + 13*b1) / 55

## 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
 −1.83901 2.37024 −2.37024 1.83901
−63.4580 76.3219 3002.92 3125.00 −4843.23 0 −125578. −53224.0 −198306.
94.2 −8.31143 −479.970 −954.920 3125.00 3989.23 0 16447.7 171322. −25973.2
94.3 8.31143 479.970 −954.920 3125.00 3989.23 0 −16447.7 171322. 25973.2
94.4 63.4580 −76.3219 3002.92 3125.00 −4843.23 0 125578. −53224.0 198306.
 $$n$$: 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.c 4
5.b even 2 1 inner 95.11.d.c 4
19.b odd 2 1 inner 95.11.d.c 4
95.d odd 2 1 CM 95.11.d.c 4

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{4} - 4096T_{2}^{2} + 278179$$ acting on $$S_{11}^{\mathrm{new}}(95, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} - 4096 T^{2} + 278179$$
$3$ $$T^{4} - 236196 T^{2} + \cdots + 1341917104$$
$5$ $$(T - 3125)^{4}$$
$7$ $$T^{4}$$
$11$ $$(T^{2} - 1954264500)^{2}$$
$13$ $$T^{4} - 551433967396 T^{2} + \cdots + 22\!\cdots\!04$$
$17$ $$T^{4}$$
$19$ $$(T + 2476099)^{4}$$
$23$ $$T^{4}$$
$29$ $$T^{4}$$
$31$ $$T^{4}$$
$37$ $$T^{4} + \cdots + 86\!\cdots\!04$$
$41$ $$T^{4}$$
$43$ $$T^{4}$$
$47$ $$T^{4}$$
$53$ $$T^{4} + \cdots + 70\!\cdots\!04$$
$59$ $$T^{4}$$
$61$ $$(T^{2} - 16\!\cdots\!00)^{2}$$
$67$ $$T^{4} + \cdots + 93\!\cdots\!04$$
$71$ $$T^{4}$$
$73$ $$T^{4}$$
$79$ $$T^{4}$$
$83$ $$T^{4}$$
$89$ $$T^{4}$$
$97$ $$T^{4} + \cdots + 16\!\cdots\!04$$
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