LMFDB - Newform orbit 75.11.c.d

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [75,11,Mod(26,75)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("75.26"); S:= CuspForms(chi, 11); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(75, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0])) N = Newforms(chi, 11, names="a")

Level:	$ N $	$=$	$ 75 = 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 11 $
Character orbit:	$[\chi]$	$=$	75.c (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,54] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$47.6517939505$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{-5}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 5 $ x^2 + 5
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2^{2}\cdot 3 $
Twist minimal:	no (minimal twist has level 3)
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 $\beta = 12\sqrt{-5}$. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q + \beta q^{2} + (9 \beta + 27) q^{3} + 304 q^{4} + (27 \beta - 6480) q^{6} - 17234 q^{7} + 1328 \beta q^{8} + (486 \beta - 57591) q^{9} + 6962 \beta q^{11} + (2736 \beta + 8208) q^{12} + 169654 q^{13}+ \cdots + ( - 400948542 \beta - 2436143040) q^{99}+O(q^{100}) $ q + b * q^2 + (9b + 27) q^3 + 304 * q^4 + (27b - 6480) q^6 - 17234 * q^7 + 1328b q^8 + (486b - 57591) q^9 + 6962b q^11 + (2736b + 8208) q^12 + 169654 * q^13 - 17234b q^14 - 644864 * q^16 - 12792b q^17 + (-57591b - 349920) q^18 - 949462 * q^19 + (-155106b - 465318) q^21 - 5012640 * q^22 + 99044b q^23 + (35856b - 8605440) q^24 + 169654b q^26 + (-505197b - 4704237) q^27 - 5239136 * q^28 - 118594b q^29 - 29793118 * q^31 + 715008b q^32 + (187974b - 45113760) q^33 + 9210240 * q^34 + (147744b - 17507664) q^36 + 60811846 * q^37 - 949462b q^38 + (1526886b + 4580658) q^39 - 6770372b q^41 + (-465318b + 111676320) q^42 - 107419706 * q^43 + 2116448b q^44 - 71311680 * q^46 - 9987608b q^47 + (-5803776b - 17411328) q^48 + 14535507 * q^49 + (-345384b + 82892160) q^51 + 51574816 * q^52 + 7158798b q^53 + (-4704237b + 363741840) q^54 - 22886752b q^56 + (-8545158b - 25635474) q^57 + 85387680 * q^58 + 24192682b q^59 + 1030793642 * q^61 - 29793118b q^62 + (-8375724b + 992523294) q^63 - 1175146496 * q^64 + (-45113760b - 135341280) q^66 - 1876742474 * q^67 - 3888768b q^68 + (2674188b - 641805120) q^69 - 100003596b q^71 + (-76480848b - 464693760) q^72 + 2846528494 * q^73 + 60811846b q^74 - 288636448 * q^76 - 119983108b q^77 + (4580658b - 1099357920) q^78 + 1488647618 * q^79 + (-55978452b + 3146662161) q^81 + 4874667840 * q^82 + 47175562b q^83 + (-47152224b - 141456672) q^84 - 107419706b q^86 + (-3202038b + 768489120) q^87 - 6656785920 * q^88 + 224371428b q^89 - 2923817036 * q^91 + 30109376b q^92 + (-268138062b - 804414186) q^93 + 7191077760 * q^94 + (19305216b - 4633251840) q^96 + 1592948926 * q^97 + 14535507b q^98 + (-400948542b - 2436143040) q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 54 q^{3} + 608 q^{4} - 12960 q^{6} - 34468 q^{7} - 115182 q^{9} + 16416 q^{12} + 339308 q^{13} - 1289728 q^{16} - 699840 q^{18} - 1898924 q^{19} - 930636 q^{21} - 10025280 q^{22} - 17210880 q^{24}+ \cdots - 4872286080 q^{99}+O(q^{100}) $ 2 * q + 54 * q^3 + 608 * q^4 - 12960 * q^6 - 34468 * q^7 - 115182 * q^9 + 16416 * q^12 + 339308 * q^13 - 1289728 * q^16 - 699840 * q^18 - 1898924 * q^19 - 930636 * q^21 - 10025280 * q^22 - 17210880 * q^24 - 9408474 * q^27 - 10478272 * q^28 - 59586236 * q^31 - 90227520 * q^33 + 18420480 * q^34 - 35015328 * q^36 + 121623692 * q^37 + 9161316 * q^39 + 223352640 * q^42 - 214839412 * q^43 - 142623360 * q^46 - 34822656 * q^48 + 29071014 * q^49 + 165784320 * q^51 + 103149632 * q^52 + 727483680 * q^54 - 51270948 * q^57 + 170775360 * q^58 + 2061587284 * q^61 + 1985046588 * q^63 - 2350292992 * q^64 - 270682560 * q^66 - 3753484948 * q^67 - 1283610240 * q^69 - 929387520 * q^72 + 5693056988 * q^73 - 577272896 * q^76 - 2198715840 * q^78 + 2977295236 * q^79 + 6293324322 * q^81 + 9749335680 * q^82 - 282913344 * q^84 + 1536978240 * q^87 - 13313571840 * q^88 - 5847634072 * q^91 - 1608828372 * q^93 + 14382155520 * q^94 - 9266503680 * q^96 + 3185897852 * q^97 - 4872286080 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/75\mathbb{Z}\right)^\times$.

$n$	$26$	$52$
$\chi(n)$	$-1$	$1$

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} $

26.1

	−	2.23607i
		2.23607i

−

26.8328i

27.0000

−

241.495i

304.000

−6480.00

−

724.486i

−17234.0

−

35634.0i

−57591.0

−

13040.7i

26.2

26.8328i

27.0000

241.495i

304.000

−6480.00

724.486i

−17234.0

35634.0i

−57591.0

13040.7i

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	75.11.c.d		2
3.b	odd	2	1	inner	75.11.c.d		2
5.b	even	2	1		3.11.b.a	✓	2
5.c	odd	4	2		75.11.d.b		4
15.d	odd	2	1		3.11.b.a	✓	2
15.e	even	4	2		75.11.d.b		4
20.d	odd	2	1		48.11.e.c		2
40.e	odd	2	1		192.11.e.d		2
40.f	even	2	1		192.11.e.e		2
45.h	odd	6	2		81.11.d.d		4
45.j	even	6	2		81.11.d.d		4
60.h	even	2	1		48.11.e.c		2
120.i	odd	2	1		192.11.e.e		2
120.m	even	2	1		192.11.e.d		2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
3.11.b.a	✓	2	5.b	even	2	1
3.11.b.a	✓	2	15.d	odd	2	1
48.11.e.c		2	20.d	odd	2	1
48.11.e.c		2	60.h	even	2	1
75.11.c.d		2	1.a	even	1	1	trivial
75.11.c.d		2	3.b	odd	2	1	inner
75.11.d.b		4	5.c	odd	4	2
75.11.d.b		4	15.e	even	4	2
81.11.d.d		4	45.h	odd	6	2
81.11.d.d		4	45.j	even	6	2
192.11.e.d		2	40.e	odd	2	1
192.11.e.d		2	120.m	even	2	1
192.11.e.e		2	40.f	even	2	1
192.11.e.e		2	120.i	odd	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{11}^{\mathrm{new}}(75, [\chi])$:

$ T_{2}^{2} + 720 $

T2^2 + 720

$ T_{7} + 17234 $

T7 + 17234

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{2} + 720 $ T^2 + 720
$3$	$ T^{2} - 54T + 59049 $ T^2 - 54*T + 59049
$5$	$ T^{2} $ T^2
$7$	$ (T + 17234)^{2} $ (T + 17234)^2
$11$	$ T^{2} + 34897999680 $ T^2 + 34897999680
$13$	$ (T - 169654)^{2} $ (T - 169654)^2
$17$	$ T^{2} + 117817390080 $ T^2 + 117817390080
$19$	$ (T + 949462)^{2} $ (T + 949462)^2
$23$	$ T^{2} + 7062994033920 $ T^2 + 7062994033920
$29$	$ T^{2} + 10126466521920 $ T^2 + 10126466521920
$31$	$ (T + 29793118)^{2} $ (T + 29793118)^2
$37$	$ (T - 60811846)^{2} $ (T - 60811846)^2
$41$	$ T^{2} + 33\!\cdots\!80 $ T^2 + 33003314653236480
$43$	$ (T + 107419706)^{2} $ (T + 107419706)^2
$47$	$ T^{2} + 71\!\cdots\!80 $ T^2 + 71821665764398080
$53$	$ T^{2} + 36\!\cdots\!80 $ T^2 + 36898839939458880
$59$	$ T^{2} + 42\!\cdots\!80 $ T^2 + 421405820894249280
$61$	$ (T - 1030793642)^{2} $ (T - 1030793642)^2
$67$	$ (T + 1876742474)^{2} $ (T + 1876742474)^2
$71$	$ T^{2} + 72\!\cdots\!20 $ T^2 + 7200517833310475520
$73$	$ (T - 2846528494)^{2} $ (T - 2846528494)^2
$79$	$ (T - 1488647618)^{2} $ (T - 1488647618)^2
$83$	$ T^{2} + 16\!\cdots\!80 $ T^2 + 1602384228011407680
$89$	$ T^{2} + 36\!\cdots\!80 $ T^2 + 36246627145986612480
$97$	$ (T - 1592948926)^{2} $ (T - 1592948926)^2
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Elliptic curves over $\Q$
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Genus 2 curves over $\Q$
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Belyi maps

Level:	\( N \)	\(=\)	\( 75 = 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 11 \)
Character orbit:	\([\chi]\)	\(=\)	75.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\( q + \beta q^{2} + (9 \beta + 27) q^{3} + 304 q^{4} + (27 \beta - 6480) q^{6} - 17234 q^{7} + 1328 \beta q^{8} + (486 \beta - 57591) q^{9} + 6962 \beta q^{11} + (2736 \beta + 8208) q^{12} + 169654 q^{13}+ \cdots + ( - 400948542 \beta - 2436143040) q^{99}+O(q^{100}) \) q + b * q^2 + (9b + 27) q^3 + 304 * q^4 + (27b - 6480) q^6 - 17234 * q^7 + 1328b q^8 + (486b - 57591) q^9 + 6962b q^11 + (2736b + 8208) q^12 + 169654 * q^13 - 17234b q^14 - 644864 * q^16 - 12792b q^17 + (-57591b - 349920) q^18 - 949462 * q^19 + (-155106b - 465318) q^21 - 5012640 * q^22 + 99044b q^23 + (35856b - 8605440) q^24 + 169654b q^26 + (-505197b - 4704237) q^27 - 5239136 * q^28 - 118594b q^29 - 29793118 * q^31 + 715008b q^32 + (187974b - 45113760) q^33 + 9210240 * q^34 + (147744b - 17507664) q^36 + 60811846 * q^37 - 949462b q^38 + (1526886b + 4580658) q^39 - 6770372b q^41 + (-465318b + 111676320) q^42 - 107419706 * q^43 + 2116448b q^44 - 71311680 * q^46 - 9987608b q^47 + (-5803776b - 17411328) q^48 + 14535507 * q^49 + (-345384b + 82892160) q^51 + 51574816 * q^52 + 7158798b q^53 + (-4704237b + 363741840) q^54 - 22886752b q^56 + (-8545158b - 25635474) q^57 + 85387680 * q^58 + 24192682b q^59 + 1030793642 * q^61 - 29793118b q^62 + (-8375724b + 992523294) q^63 - 1175146496 * q^64 + (-45113760b - 135341280) q^66 - 1876742474 * q^67 - 3888768b q^68 + (2674188b - 641805120) q^69 - 100003596b q^71 + (-76480848b - 464693760) q^72 + 2846528494 * q^73 + 60811846b q^74 - 288636448 * q^76 - 119983108b q^77 + (4580658b - 1099357920) q^78 + 1488647618 * q^79 + (-55978452b + 3146662161) q^81 + 4874667840 * q^82 + 47175562b q^83 + (-47152224b - 141456672) q^84 - 107419706b q^86 + (-3202038b + 768489120) q^87 - 6656785920 * q^88 + 224371428b q^89 - 2923817036 * q^91 + 30109376b q^92 + (-268138062b - 804414186) q^93 + 7191077760 * q^94 + (19305216b - 4633251840) q^96 + 1592948926 * q^97 + 14535507b q^98 + (-400948542b - 2436143040) q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 54 q^{3} + 608 q^{4} - 12960 q^{6} - 34468 q^{7} - 115182 q^{9} + 16416 q^{12} + 339308 q^{13} - 1289728 q^{16} - 699840 q^{18} - 1898924 q^{19} - 930636 q^{21} - 10025280 q^{22} - 17210880 q^{24}+ \cdots - 4872286080 q^{99}+O(q^{100}) \) 2 * q + 54 * q^3 + 608 * q^4 - 12960 * q^6 - 34468 * q^7 - 115182 * q^9 + 16416 * q^12 + 339308 * q^13 - 1289728 * q^16 - 699840 * q^18 - 1898924 * q^19 - 930636 * q^21 - 10025280 * q^22 - 17210880 * q^24 - 9408474 * q^27 - 10478272 * q^28 - 59586236 * q^31 - 90227520 * q^33 + 18420480 * q^34 - 35015328 * q^36 + 121623692 * q^37 + 9161316 * q^39 + 223352640 * q^42 - 214839412 * q^43 - 142623360 * q^46 - 34822656 * q^48 + 29071014 * q^49 + 165784320 * q^51 + 103149632 * q^52 + 727483680 * q^54 - 51270948 * q^57 + 170775360 * q^58 + 2061587284 * q^61 + 1985046588 * q^63 - 2350292992 * q^64 - 270682560 * q^66 - 3753484948 * q^67 - 1283610240 * q^69 - 929387520 * q^72 + 5693056988 * q^73 - 577272896 * q^76 - 2198715840 * q^78 + 2977295236 * q^79 + 6293324322 * q^81 + 9749335680 * q^82 - 282913344 * q^84 + 1536978240 * q^87 - 13313571840 * q^88 - 5847634072 * q^91 - 1608828372 * q^93 + 14382155520 * q^94 - 9266503680 * q^96 + 3185897852 * q^97 - 4872286080 * q^99

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