LMFDB - Newform orbit 1575.2.b.h

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1575,2,Mod(251,1575)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1575, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0, 1]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("1575.251");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1575 = 3^{2} \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1575.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:	$12.5764383184$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 24x^{14} + 184x^{12} - 240x^{10} + 228x^{8} + 912x^{6} + 976x^{4} + 480x^{2} + 100 $ x^16 - 24x^14 + 184x^12 - 240x^10 + 228x^8 + 912x^6 + 976x^4 + 480*x^2 + 100
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{16} $
Twist minimal:	no (minimal twist has level 315)
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 a basis $1,\beta_1,\ldots,\beta_{15}$ for the coefficient ring described below. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q + \beta_{3} q^{2} + ( - \beta_1 - 1) q^{4} - \beta_{8} q^{7} + ( - \beta_{3} - \beta_{2}) q^{8}+O(q^{10}) $ q + b3 * q^2 + (-b1 - 1) * q^4 - b8 * q^7 + (-b3 - b2) * q^8	$ q + \beta_{3} q^{2} + ( - \beta_1 - 1) q^{4} - \beta_{8} q^{7} + ( - \beta_{3} - \beta_{2}) q^{8} - \beta_{6} q^{11} - \beta_{5} q^{13} + ( - \beta_{14} - \beta_{10}) q^{14} + q^{16} + (\beta_{13} - \beta_{11}) q^{17} - \beta_{7} q^{19} + \beta_{12} q^{22} + (2 \beta_{3} - \beta_{2}) q^{23} + \beta_{4} q^{26} + (\beta_{12} - 2 \beta_{9} + \cdots + \beta_{5}) q^{28}+ \cdots + ( - 2 \beta_{13} + \beta_{11} + \cdots - 2 \beta_{2}) q^{98}+O(q^{100}) $ q + b3 * q^2 + (-b1 - 1) * q^4 - b8 * q^7 + (-b3 - b2) * q^8 - b6 * q^11 - b5 * q^13 + (-b14 - b10) * q^14 + q^16 + (b13 - b11) * q^17 - b7 * q^19 + b12 * q^22 + (2b3 - b2) q^23 + b4 * q^26 + (b12 - 2b9 + b8 + b5) q^28 + (2b14 + b6) q^29 + (b15 - b7) * q^31 + (-b3 - 2b2) q^32 + (2b15 - b7) q^34 + (-2b12 - b9 - b8) q^37 - 3b13 q^38 + (-b14 - 2b10 - b4) q^41 + (2b12 - b9 - b8) q^43 + (b14 + b6) * q^44 + (-b1 - 6) * q^46 + b11 * q^47 + (b15 - 2b1 - 1) q^49 + (-b9 + b8 - b5) * q^52 - 3b2 q^53 + (b10 + 3b6 - b4) q^56 + (-5b12 - 2b9 - 2b8) q^58 + (-b14 - 2b10) q^59 + 2b15 q^61 + (-5b13 + b11) q^62 + (3b1 + 5) q^64 + (2b12 + 2b9 + 2b8) q^67 - 5b13 q^68 + (2b14 - b6) q^71 + (-2b9 + 2b8 + b5) * q^73 + (-3b14 - 6b6) * q^74 + (-3b15 + b7) q^76 + (-b13 + b2) * q^77 + 4 * q^79 + (-4b9 + 4b8 + b5) * q^82 - b11 * q^83 + (b14 + 6b6) q^86 + (-b12 - b9 - b8) * q^88 - 3b4 q^89 + (-b15 - b7 - 2b1 + 2) q^91 + (-4b3 - 3b2) * q^92 - b15 * q^94 + b5 * q^97 + (-2b13 + b11 - 5b3 - 2b2) q^98
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 16 q^{4}+O(q^{10}) $ 16 * q - 16 * q^4	$ 16 q - 16 q^{4} + 16 q^{16} - 96 q^{46} - 16 q^{49} + 80 q^{64} + 64 q^{79} + 32 q^{91}+O(q^{100}) $ 16 * q - 16 * q^4 + 16 * q^16 - 96 * q^46 - 16 * q^49 + 80 * q^64 + 64 * q^79 + 32 * q^91

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{16} - 24x^{14} + 184x^{12} - 240x^{10} + 228x^{8} + 912x^{6} + 976x^{4} + 480x^{2} + 100 $ x^16 - 24*x^14 + 184*x^12 - 240*x^10 + 228*x^8 + 912*x^6 + 976*x^4 + 480*x^2 + 100 :

$\beta_{1}$	$=$	$ ( 3594 \nu^{14} - 85793 \nu^{12} + 648758 \nu^{10} - 741984 \nu^{8} + 416572 \nu^{6} + 3970770 \nu^{4} + \cdots + 1105576 ) / 136036 $ (3594v^14 - 85793v^12 + 648758v^10 - 741984v^8 + 416572v^6 + 3970770v^4 + 3153500*v^2 + 1105576) / 136036
$\beta_{2}$	$=$	$ ( 85775 \nu^{14} - 2122031 \nu^{12} + 17330261 \nu^{10} - 32853787 \nu^{8} + 39175106 \nu^{6} + \cdots + 4773434 ) / 2584684 $ (85775v^14 - 2122031v^12 + 17330261v^10 - 32853787v^8 + 39175106v^6 + 59596242v^4 + 30152810*v^2 + 4773434) / 2584684
$\beta_{3}$	$=$	$ ( 173747 \nu^{14} - 4257795 \nu^{12} + 34120732 \nu^{10} - 58911722 \nu^{8} + 69119662 \nu^{6} + \cdots + 29892812 ) / 5169368 $ (173747v^14 - 4257795v^12 + 34120732v^10 - 58911722v^8 + 69119662v^6 + 123588066v^4 + 105387680*v^2 + 29892812) / 5169368
$\beta_{4}$	$=$	$ ( - 940391 \nu^{14} + 22482464 \nu^{12} - 170494524 \nu^{10} + 198564690 \nu^{8} + \cdots - 271464140 ) / 25846840 $ (-940391v^14 + 22482464v^12 - 170494524v^10 + 198564690v^8 - 103391638v^6 - 1038560992v^4 - 826644096*v^2 - 271464140) / 25846840
$\beta_{5}$	$=$	$ ( - 63329 \nu^{14} + 1567596 \nu^{12} - 12812766 \nu^{10} + 24351000 \nu^{8} - 28824842 \nu^{6} + \cdots - 2150160 ) / 1360360 $ (-63329v^14 + 1567596v^12 - 12812766v^10 + 24351000v^8 - 28824842v^6 - 42777408v^4 - 18926324*v^2 - 2150160) / 1360360
$\beta_{6}$	$=$	$ ( - 511052 \nu^{15} + 12187053 \nu^{13} - 92206563 \nu^{11} + 109383925 \nu^{9} + \cdots - 441391230 \nu ) / 129234200 $ (-511052v^15 + 12187053v^13 - 92206563v^11 + 109383925v^9 - 105356856v^7 - 481433634v^5 - 585568942v^3 - 441391230v) / 129234200
$\beta_{7}$	$=$	$ ( 12153 \nu^{15} - 302512 \nu^{13} + 2488197 \nu^{11} - 4701810 \nu^{9} + 3543414 \nu^{7} + \cdots - 11106940 \nu ) / 1360360 $ (12153v^15 - 302512v^13 + 2488197v^11 - 4701810v^9 + 3543414v^7 + 12883176v^5 - 594822v^3 - 11106940v) / 1360360
$\beta_{8}$	$=$	$ ( 287208 \nu^{15} - 465460 \nu^{14} - 7186487 \nu^{13} + 11397040 \nu^{12} + 60057352 \nu^{11} + \cdots - 86889400 ) / 13603600 $ (287208v^15 - 465460v^14 - 7186487v^13 + 11397040v^12 + 60057352v^11 - 91188090v^10 - 126964950v^9 + 156205900v^8 + 167095424v^7 - 183510080v^6 + 152461586v^5 - 334366720v^4 + 48715768v^3 - 297976660v^2 - 1352580*v - 86889400) / 13603600
$\beta_{9}$	$=$	$ ( 287208 \nu^{15} + 465460 \nu^{14} - 7186487 \nu^{13} - 11397040 \nu^{12} + 60057352 \nu^{11} + \cdots + 86889400 ) / 13603600 $ (287208v^15 + 465460v^14 - 7186487v^13 - 11397040v^12 + 60057352v^11 + 91188090v^10 - 126964950v^9 - 156205900v^8 + 167095424v^7 + 183510080v^6 + 152461586v^5 + 334366720v^4 + 48715768v^3 + 297976660v^2 - 1352580*v + 86889400) / 13603600
$\beta_{10}$	$=$	$ ( 9284457 \nu^{15} + 6984990 \nu^{14} - 224357373 \nu^{13} - 166660710 \nu^{12} + \cdots + 3032257600 ) / 258468400 $ (9284457v^15 + 6984990v^14 - 224357373v^13 - 166660710v^12 + 1743369358v^11 + 1259177310v^10 - 2467352950v^9 - 1432095600v^8 + 2128759346v^7 + 822195220v^6 + 8881994494v^5 + 7718027380v^4 + 6547004572v^3 + 6125354140v^2 + 2143247180*v + 3032257600) / 258468400
$\beta_{11}$	$=$	$ ( 1378141 \nu^{15} - 34703964 \nu^{13} + 293481144 \nu^{11} - 650229910 \nu^{9} + \cdots - 69778380 \nu ) / 25846840 $ (1378141v^15 - 34703964v^13 + 293481144v^11 - 650229910v^9 + 860274378v^7 + 654286872v^5 + 61963256v^3 - 69778380v) / 25846840
$\beta_{12}$	$=$	$ ( 434447 \nu^{15} - 10659458 \nu^{13} + 85636768 \nu^{11} - 149861325 \nu^{9} + 177156866 \nu^{7} + \cdots + 66348030 \nu ) / 6801800 $ (434447v^15 - 10659458v^13 + 85636768v^11 - 149861325v^9 + 177156866v^7 + 304460624v^5 + 256836912v^3 + 66348030v) / 6801800
$\beta_{13}$	$=$	$ ( - 1753109 \nu^{15} + 42943351 \nu^{13} - 343861031 \nu^{11} + 591349820 \nu^{9} + \cdots - 288707080 \nu ) / 25846840 $ (-1753109v^15 + 42943351v^13 - 343861031v^11 + 591349820v^9 - 694267462v^7 - 1253237098v^5 - 1093094054v^3 - 288707080v) / 25846840
$\beta_{14}$	$=$	$ ( - 273 \nu^{15} + 6597 \nu^{13} - 51262 \nu^{11} + 72550 \nu^{9} - 62594 \nu^{7} - 261166 \nu^{5} + \cdots - 63020 \nu ) / 3800 $ (-273v^15 + 6597v^13 - 51262v^11 + 72550v^9 - 62594v^7 - 261166v^5 - 192508v^3 - 63020v) / 3800
$\beta_{15}$	$=$	$ ( 130579 \nu^{15} - 3158076 \nu^{13} + 24582166 \nu^{11} - 35162080 \nu^{9} + 30225862 \nu^{7} + \cdots + 27855680 \nu ) / 1360360 $ (130579v^15 - 3158076v^13 + 24582166v^11 - 35162080v^9 + 30225862v^7 + 125735248v^5 + 90912484v^3 + 27855680v) / 1360360

Display $\nu^j$ in terms of $\beta_i$

$\nu$	$=$	$ ( \beta_{13} + \beta_{12} - \beta_{6} ) / 2 $ (b13 + b12 - b6) / 2
$\nu^{2}$	$=$	$ ( -\beta_{14} - 2\beta_{10} + \beta_{9} - \beta_{8} - 2\beta_{3} + 2\beta _1 + 6 ) / 2 $ (-b14 - 2b10 + b9 - b8 - 2b3 + 2*b1 + 6) / 2
$\nu^{3}$	$=$	$ ( - 3 \beta_{15} - 3 \beta_{14} + 9 \beta_{13} + 10 \beta_{12} - 2 \beta_{11} + 2 \beta_{9} + \cdots - 10 \beta_{6} ) / 2 $ (-3b15 - 3b14 + 9b13 + 10b12 - 2b11 + 2b9 + 2b8 + 3b7 - 10*b6) / 2
$\nu^{4}$	$=$	$ - 4 \beta_{14} - 8 \beta_{10} + 10 \beta_{9} - 10 \beta_{8} - 4 \beta_{5} + 2 \beta_{4} - 22 \beta_{3} + \cdots + 26 $ -4b14 - 8b10 + 10b9 - 10b8 - 4b5 + 2b4 - 22b3 - 4b2 + 11*b1 + 26
$\nu^{5}$	$=$	$ ( - 60 \beta_{15} - 65 \beta_{14} + 72 \beta_{13} + 78 \beta_{12} - 17 \beta_{11} + 19 \beta_{9} + \cdots - 164 \beta_{6} ) / 2 $ (-60b15 - 65b14 + 72b13 + 78b12 - 17b11 + 19b9 + 19b8 + 50b7 - 164*b6) / 2
$\nu^{6}$	$=$	$ - 25 \beta_{14} - 50 \beta_{10} + 159 \beta_{9} - 159 \beta_{8} - 71 \beta_{5} + 13 \beta_{4} + \cdots + 162 $ -25b14 - 50b10 + 159b9 - 159b8 - 71b5 + 13b4 - 346b3 - 77b2 + 69*b1 + 162
$\nu^{7}$	$=$	$ ( - 875 \beta_{15} - 952 \beta_{14} + 258 \beta_{13} + 280 \beta_{12} - 61 \beta_{11} + \cdots - 2330 \beta_{6} ) / 2 $ (-875b15 - 952b14 + 258b13 + 280b12 - 61b11 + 68b9 + 68b8 + 714b7 - 2330*b6) / 2
$\nu^{8}$	$=$	$ 44 \beta_{14} + 88 \beta_{10} + 2096 \beta_{9} - 2096 \beta_{8} - 944 \beta_{5} - 16 \beta_{4} + \cdots - 290 $ 44b14 + 88b10 + 2096b9 - 2096b8 - 944b5 - 16b4 - 4560b3 - 1028b2 - 112*b1 - 290
$\nu^{9}$	$=$	$ - 5310 \beta_{15} - 5778 \beta_{14} - 2589 \beta_{13} - 2817 \beta_{12} + 580 \beta_{11} + \cdots - 14135 \beta_{6} $ -5310b15 - 5778b14 - 2589b13 - 2817b12 + 580b11 - 630b9 - 630b8 + 4332b7 - 14135*b6
$\nu^{10}$	$=$	$ 4869 \beta_{14} + 9738 \beta_{10} + 23359 \beta_{9} - 23359 \beta_{8} - 10508 \beta_{5} - 2188 \beta_{4} + \cdots - 31782 $ 4869b14 + 9738b10 + 23359b9 - 23359b8 - 10508b5 - 2188b4 - 50822b3 - 11436b2 - 12974*b1 - 31782
$\nu^{11}$	$=$	$ - 53779 \beta_{15} - 58509 \beta_{14} - 78757 \beta_{13} - 85678 \beta_{12} + 17704 \beta_{11} + \cdots - 143270 \beta_{6} $ -53779b15 - 58509b14 - 78757b13 - 85678b12 + 17704b11 - 19262b9 - 19262b8 + 43901b7 - 143270*b6
$\nu^{12}$	$=$	$ 104168 \beta_{14} + 208336 \beta_{10} + 210396 \beta_{9} - 210396 \beta_{8} - 94584 \beta_{5} + \cdots - 679924 $ 104168b14 + 208336b10 + 210396b9 - 210396b8 - 94584b5 - 46844b4 - 457772b3 - 102904b2 - 277614*b1 - 679924
$\nu^{13}$	$=$	$ - 410436 \beta_{15} - 446511 \beta_{14} - 1395656 \beta_{13} - 1518314 \beta_{12} + \cdots - 1093696 \beta_{6} $ -410436b15 - 446511b14 - 1395656b13 - 1518314b12 + 313689b11 - 341271b9 - 341271b8 + 335114b7 - 1093696*b6
$\nu^{14}$	$=$	$ 1624542 \beta_{14} + 3249084 \beta_{10} + 1208638 \beta_{9} - 1208638 \beta_{8} - 543258 \beta_{5} + \cdots - 10603884 $ 1624542b14 + 3249084b10 + 1208638b9 - 1208638b8 - 543258b5 - 730282b4 - 2629732b3 - 590998b2 - 4329130*b1 - 10603884
$\nu^{15}$	$=$	$ - 1100949 \beta_{15} - 1197696 \beta_{14} - 19692702 \beta_{13} - 21423472 \beta_{12} + \cdots - 2933950 \beta_{6} $ -1100949b15 - 1197696b14 - 19692702b13 - 21423472b12 + 4425877b11 - 4814888b9 - 4814888b8 + 898962b7 - 2933950*b6

Display $\beta_i$ in terms of $\nu^j$

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1575\mathbb{Z}\right)^\times$.

$n$	$127$	$451$	$1226$
$\chi(n)$	$1$	$-1$	$-1$

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} $

251.1

−0.145077	+	0.707107i
3.44644	+	0.707107i
0.145077	−	0.707107i
−3.44644	−	0.707107i
0.355838	−	0.707107i
−1.40513	−	0.707107i
−0.355838	+	0.707107i
1.40513	+	0.707107i
−1.40513	+	0.707107i
0.355838	+	0.707107i
1.40513	−	0.707107i
−0.355838	−	0.707107i
3.44644	−	0.707107i
−0.145077	−	0.707107i
−3.44644	+	0.707107i
0.145077	+	0.707107i

−

2.33441i

−3.44949

−0.741964

−

2.53958i

3.38371i

251.2

−

2.33441i

−3.44949

−0.741964

2.53958i

3.38371i

251.3

−

2.33441i

−3.44949

0.741964

−

2.53958i

3.38371i

251.4

−

2.33441i

−3.44949

0.741964

2.53958i

3.38371i

251.5

−

0.741964i

1.44949

−2.33441

−

1.24519i

−

2.55940i

251.6

−

0.741964i

1.44949

−2.33441

1.24519i

−

2.55940i

251.7

−

0.741964i

1.44949

2.33441

−

1.24519i

−

2.55940i

251.8

−

0.741964i

1.44949

2.33441

1.24519i

−

2.55940i

251.9

0.741964i

1.44949

−2.33441

−

1.24519i

2.55940i

251.10

0.741964i

1.44949

−2.33441

1.24519i

2.55940i

251.11

0.741964i

1.44949

2.33441

−

1.24519i

2.55940i

251.12

0.741964i

1.44949

2.33441

1.24519i

2.55940i

251.13

2.33441i

−3.44949

−0.741964

−

2.53958i

−

3.38371i

251.14

2.33441i

−3.44949

−0.741964

2.53958i

−

3.38371i

251.15

2.33441i

−3.44949

0.741964

−

2.53958i

−

3.38371i

251.16

2.33441i

−3.44949

0.741964

2.53958i

−

3.38371i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
5.b	even	2	1	inner
7.b	odd	2	1	inner
15.d	odd	2	1	inner
21.c	even	2	1	inner
35.c	odd	2	1	inner
105.g	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1575.2.b.h		16
3.b	odd	2	1	inner	1575.2.b.h		16
5.b	even	2	1	inner	1575.2.b.h		16
5.c	odd	4	2		315.2.g.a	✓	16
7.b	odd	2	1	inner	1575.2.b.h		16
15.d	odd	2	1	inner	1575.2.b.h		16
15.e	even	4	2		315.2.g.a	✓	16
20.e	even	4	2		5040.2.k.g		16
21.c	even	2	1	inner	1575.2.b.h		16
35.c	odd	2	1	inner	1575.2.b.h		16
35.f	even	4	2		315.2.g.a	✓	16
60.l	odd	4	2		5040.2.k.g		16
105.g	even	2	1	inner	1575.2.b.h		16
105.k	odd	4	2		315.2.g.a	✓	16
140.j	odd	4	2		5040.2.k.g		16
420.w	even	4	2		5040.2.k.g		16

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
315.2.g.a	✓	16	5.c	odd	4	2
315.2.g.a	✓	16	15.e	even	4	2
315.2.g.a	✓	16	35.f	even	4	2
315.2.g.a	✓	16	105.k	odd	4	2
1575.2.b.h		16	1.a	even	1	1	trivial
1575.2.b.h		16	3.b	odd	2	1	inner
1575.2.b.h		16	5.b	even	2	1	inner
1575.2.b.h		16	7.b	odd	2	1	inner
1575.2.b.h		16	15.d	odd	2	1	inner
1575.2.b.h		16	21.c	even	2	1	inner
1575.2.b.h		16	35.c	odd	2	1	inner
1575.2.b.h		16	105.g	even	2	1	inner
5040.2.k.g		16	20.e	even	4	2
5040.2.k.g		16	60.l	odd	4	2
5040.2.k.g		16	140.j	odd	4	2
5040.2.k.g		16	420.w	even	4	2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(1575, [\chi])$:

$ T_{2}^{4} + 6T_{2}^{2} + 3 $

$ T_{37}^{4} - 72T_{37}^{2} + 432 $

$ T_{47}^{4} - 64T_{47}^{2} + 160 $

$ T_{67}^{4} - 144T_{67}^{2} + 4800 $

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ (T^{4} + 6 T^{2} + 3)^{4} $ (T^4 + 6*T^2 + 3)^4
$3$	$ T^{16} $ T^16
$5$	$ T^{16} $ T^16
$7$	$ (T^{8} + 4 T^{6} + \cdots + 2401)^{2} $ (T^8 + 4T^6 + 6T^4 + 196*T^2 + 2401)^2
$11$	$ (T^{2} + 2)^{8} $ (T^2 + 2)^8
$13$	$ (T^{4} + 32 T^{2} + 40)^{4} $ (T^4 + 32*T^2 + 40)^4
$17$	$ (T^{4} - 64 T^{2} + 1000)^{4} $ (T^4 - 64*T^2 + 1000)^4
$19$	$ (T^{4} + 72 T^{2} + 1080)^{4} $ (T^4 + 72*T^2 + 1080)^4
$23$	$ (T^{4} + 36 T^{2} + 300)^{4} $ (T^4 + 36*T^2 + 300)^4
$29$	$ (T^{4} + 100 T^{2} + 2116)^{4} $ (T^4 + 100*T^2 + 2116)^4
$31$	$ (T^{4} + 72 T^{2} + 120)^{4} $ (T^4 + 72*T^2 + 120)^4
$37$	$ (T^{4} - 72 T^{2} + 432)^{4} $ (T^4 - 72*T^2 + 432)^4
$41$	$ (T^{4} - 120 T^{2} + 3000)^{4} $ (T^4 - 120*T^2 + 3000)^4
$43$	$ (T^{4} - 72 T^{2} + 1200)^{4} $ (T^4 - 72*T^2 + 1200)^4
$47$	$ (T^{4} - 64 T^{2} + 160)^{4} $ (T^4 - 64*T^2 + 160)^4
$53$	$ (T^{4} + 108 T^{2} + 972)^{4} $ (T^4 + 108*T^2 + 972)^4
$59$	$ (T^{4} - 144 T^{2} + 480)^{4} $ (T^4 - 144*T^2 + 480)^4
$61$	$ (T^{4} + 192 T^{2} + 7680)^{4} $ (T^4 + 192*T^2 + 7680)^4
$67$	$ (T^{4} - 144 T^{2} + 4800)^{4} $ (T^4 - 144*T^2 + 4800)^4
$71$	$ (T^{4} + 100 T^{2} + 2116)^{4} $ (T^4 + 100*T^2 + 2116)^4
$73$	$ (T^{4} + 128 T^{2} + 40)^{4} $ (T^4 + 128*T^2 + 40)^4
$79$	$ (T - 4)^{16} $ (T - 4)^16
$83$	$ (T^{4} - 64 T^{2} + 160)^{4} $ (T^4 - 64*T^2 + 160)^4
$89$	$ (T^{4} - 216 T^{2} + 9720)^{4} $ (T^4 - 216*T^2 + 9720)^4
$97$	$ (T^{4} + 32 T^{2} + 40)^{4} $ (T^4 + 32*T^2 + 40)^4
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Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1575 = 3^{2} \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1575.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\( q + \beta_{3} q^{2} + ( - \beta_1 - 1) q^{4} - \beta_{8} q^{7} + ( - \beta_{3} - \beta_{2}) q^{8}+O(q^{10}) \) q + b3 * q^2 + (-b1 - 1) * q^4 - b8 * q^7 + (-b3 - b2) * q^8	\( q + \beta_{3} q^{2} + ( - \beta_1 - 1) q^{4} - \beta_{8} q^{7} + ( - \beta_{3} - \beta_{2}) q^{8} - \beta_{6} q^{11} - \beta_{5} q^{13} + ( - \beta_{14} - \beta_{10}) q^{14} + q^{16} + (\beta_{13} - \beta_{11}) q^{17} - \beta_{7} q^{19} + \beta_{12} q^{22} + (2 \beta_{3} - \beta_{2}) q^{23} + \beta_{4} q^{26} + (\beta_{12} - 2 \beta_{9} + \cdots + \beta_{5}) q^{28}+ \cdots + ( - 2 \beta_{13} + \beta_{11} + \cdots - 2 \beta_{2}) q^{98}+O(q^{100}) \) q + b3 * q^2 + (-b1 - 1) * q^4 - b8 * q^7 + (-b3 - b2) * q^8 - b6 * q^11 - b5 * q^13 + (-b14 - b10) * q^14 + q^16 + (b13 - b11) * q^17 - b7 * q^19 + b12 * q^22 + (2b3 - b2) q^23 + b4 * q^26 + (b12 - 2b9 + b8 + b5) q^28 + (2b14 + b6) q^29 + (b15 - b7) * q^31 + (-b3 - 2b2) q^32 + (2b15 - b7) q^34 + (-2b12 - b9 - b8) q^37 - 3b13 q^38 + (-b14 - 2b10 - b4) q^41 + (2b12 - b9 - b8) q^43 + (b14 + b6) * q^44 + (-b1 - 6) * q^46 + b11 * q^47 + (b15 - 2b1 - 1) q^49 + (-b9 + b8 - b5) * q^52 - 3b2 q^53 + (b10 + 3b6 - b4) q^56 + (-5b12 - 2b9 - 2b8) q^58 + (-b14 - 2b10) q^59 + 2b15 q^61 + (-5b13 + b11) q^62 + (3b1 + 5) q^64 + (2b12 + 2b9 + 2b8) q^67 - 5b13 q^68 + (2b14 - b6) q^71 + (-2b9 + 2b8 + b5) * q^73 + (-3b14 - 6b6) * q^74 + (-3b15 + b7) q^76 + (-b13 + b2) * q^77 + 4 * q^79 + (-4b9 + 4b8 + b5) * q^82 - b11 * q^83 + (b14 + 6b6) q^86 + (-b12 - b9 - b8) * q^88 - 3b4 q^89 + (-b15 - b7 - 2b1 + 2) q^91 + (-4b3 - 3b2) * q^92 - b15 * q^94 + b5 * q^97 + (-2b13 + b11 - 5b3 - 2b2) q^98
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 16 q^{4}+O(q^{10}) \) 16 * q - 16 * q^4	\( 16 q - 16 q^{4} + 16 q^{16} - 96 q^{46} - 16 q^{49} + 80 q^{64} + 64 q^{79} + 32 q^{91}+O(q^{100}) \) 16 * q - 16 * q^4 + 16 * q^16 - 96 * q^46 - 16 * q^49 + 80 * q^64 + 64 * q^79 + 32 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more

Newspace parameters

Newform invariants

$q$-expansion

Character values

Embeddings

Inner twists

Twists

Hecke kernels

Hecke characteristic polynomials

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	1575.2.b.h

Level	$1575$
Weight	$2$
Character orbit	1575.b
Analytic conductor	$12.576$
Analytic rank	$0$
Dimension	$16$
CM	no
Inner twists	$8$

Self dual:	no
Analytic conductor:	\(12.5764383184\)
Analytic rank:	\(0\)
Dimension:	\(16\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{16} - 24x^{14} + 184x^{12} - 240x^{10} + 228x^{8} + 912x^{6} + 976x^{4} + 480x^{2} + 100 \) x^16 - 24x^14 + 184x^12 - 240x^10 + 228x^8 + 912x^6 + 976x^4 + 480*x^2 + 100
Coefficient ring:	\(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index:	\( 2^{16} \)
Twist minimal:	no (minimal twist has level 315)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

\(\beta_{1}\)	\(=\)	\( ( 3594 \nu^{14} - 85793 \nu^{12} + 648758 \nu^{10} - 741984 \nu^{8} + 416572 \nu^{6} + 3970770 \nu^{4} + \cdots + 1105576 ) / 136036 \) (3594v^14 - 85793v^12 + 648758v^10 - 741984v^8 + 416572v^6 + 3970770v^4 + 3153500*v^2 + 1105576) / 136036
\(\beta_{2}\)	\(=\)	\( ( 85775 \nu^{14} - 2122031 \nu^{12} + 17330261 \nu^{10} - 32853787 \nu^{8} + 39175106 \nu^{6} + \cdots + 4773434 ) / 2584684 \) (85775v^14 - 2122031v^12 + 17330261v^10 - 32853787v^8 + 39175106v^6 + 59596242v^4 + 30152810*v^2 + 4773434) / 2584684
\(\beta_{3}\)	\(=\)	\( ( 173747 \nu^{14} - 4257795 \nu^{12} + 34120732 \nu^{10} - 58911722 \nu^{8} + 69119662 \nu^{6} + \cdots + 29892812 ) / 5169368 \) (173747v^14 - 4257795v^12 + 34120732v^10 - 58911722v^8 + 69119662v^6 + 123588066v^4 + 105387680*v^2 + 29892812) / 5169368
\(\beta_{4}\)	\(=\)	\( ( - 940391 \nu^{14} + 22482464 \nu^{12} - 170494524 \nu^{10} + 198564690 \nu^{8} + \cdots - 271464140 ) / 25846840 \) (-940391v^14 + 22482464v^12 - 170494524v^10 + 198564690v^8 - 103391638v^6 - 1038560992v^4 - 826644096*v^2 - 271464140) / 25846840
\(\beta_{5}\)	\(=\)	\( ( - 63329 \nu^{14} + 1567596 \nu^{12} - 12812766 \nu^{10} + 24351000 \nu^{8} - 28824842 \nu^{6} + \cdots - 2150160 ) / 1360360 \) (-63329v^14 + 1567596v^12 - 12812766v^10 + 24351000v^8 - 28824842v^6 - 42777408v^4 - 18926324*v^2 - 2150160) / 1360360
\(\beta_{6}\)	\(=\)	\( ( - 511052 \nu^{15} + 12187053 \nu^{13} - 92206563 \nu^{11} + 109383925 \nu^{9} + \cdots - 441391230 \nu ) / 129234200 \) (-511052v^15 + 12187053v^13 - 92206563v^11 + 109383925v^9 - 105356856v^7 - 481433634v^5 - 585568942v^3 - 441391230v) / 129234200
\(\beta_{7}\)	\(=\)	\( ( 12153 \nu^{15} - 302512 \nu^{13} + 2488197 \nu^{11} - 4701810 \nu^{9} + 3543414 \nu^{7} + \cdots - 11106940 \nu ) / 1360360 \) (12153v^15 - 302512v^13 + 2488197v^11 - 4701810v^9 + 3543414v^7 + 12883176v^5 - 594822v^3 - 11106940v) / 1360360
\(\beta_{8}\)	\(=\)	\( ( 287208 \nu^{15} - 465460 \nu^{14} - 7186487 \nu^{13} + 11397040 \nu^{12} + 60057352 \nu^{11} + \cdots - 86889400 ) / 13603600 \) (287208v^15 - 465460v^14 - 7186487v^13 + 11397040v^12 + 60057352v^11 - 91188090v^10 - 126964950v^9 + 156205900v^8 + 167095424v^7 - 183510080v^6 + 152461586v^5 - 334366720v^4 + 48715768v^3 - 297976660v^2 - 1352580*v - 86889400) / 13603600
\(\beta_{9}\)	\(=\)	\( ( 287208 \nu^{15} + 465460 \nu^{14} - 7186487 \nu^{13} - 11397040 \nu^{12} + 60057352 \nu^{11} + \cdots + 86889400 ) / 13603600 \) (287208v^15 + 465460v^14 - 7186487v^13 - 11397040v^12 + 60057352v^11 + 91188090v^10 - 126964950v^9 - 156205900v^8 + 167095424v^7 + 183510080v^6 + 152461586v^5 + 334366720v^4 + 48715768v^3 + 297976660v^2 - 1352580*v + 86889400) / 13603600
\(\beta_{10}\)	\(=\)	\( ( 9284457 \nu^{15} + 6984990 \nu^{14} - 224357373 \nu^{13} - 166660710 \nu^{12} + \cdots + 3032257600 ) / 258468400 \) (9284457v^15 + 6984990v^14 - 224357373v^13 - 166660710v^12 + 1743369358v^11 + 1259177310v^10 - 2467352950v^9 - 1432095600v^8 + 2128759346v^7 + 822195220v^6 + 8881994494v^5 + 7718027380v^4 + 6547004572v^3 + 6125354140v^2 + 2143247180*v + 3032257600) / 258468400
\(\beta_{11}\)	\(=\)	\( ( 1378141 \nu^{15} - 34703964 \nu^{13} + 293481144 \nu^{11} - 650229910 \nu^{9} + \cdots - 69778380 \nu ) / 25846840 \) (1378141v^15 - 34703964v^13 + 293481144v^11 - 650229910v^9 + 860274378v^7 + 654286872v^5 + 61963256v^3 - 69778380v) / 25846840
\(\beta_{12}\)	\(=\)	\( ( 434447 \nu^{15} - 10659458 \nu^{13} + 85636768 \nu^{11} - 149861325 \nu^{9} + 177156866 \nu^{7} + \cdots + 66348030 \nu ) / 6801800 \) (434447v^15 - 10659458v^13 + 85636768v^11 - 149861325v^9 + 177156866v^7 + 304460624v^5 + 256836912v^3 + 66348030v) / 6801800
\(\beta_{13}\)	\(=\)	\( ( - 1753109 \nu^{15} + 42943351 \nu^{13} - 343861031 \nu^{11} + 591349820 \nu^{9} + \cdots - 288707080 \nu ) / 25846840 \) (-1753109v^15 + 42943351v^13 - 343861031v^11 + 591349820v^9 - 694267462v^7 - 1253237098v^5 - 1093094054v^3 - 288707080v) / 25846840
\(\beta_{14}\)	\(=\)	\( ( - 273 \nu^{15} + 6597 \nu^{13} - 51262 \nu^{11} + 72550 \nu^{9} - 62594 \nu^{7} - 261166 \nu^{5} + \cdots - 63020 \nu ) / 3800 \) (-273v^15 + 6597v^13 - 51262v^11 + 72550v^9 - 62594v^7 - 261166v^5 - 192508v^3 - 63020v) / 3800
\(\beta_{15}\)	\(=\)	\( ( 130579 \nu^{15} - 3158076 \nu^{13} + 24582166 \nu^{11} - 35162080 \nu^{9} + 30225862 \nu^{7} + \cdots + 27855680 \nu ) / 1360360 \) (130579v^15 - 3158076v^13 + 24582166v^11 - 35162080v^9 + 30225862v^7 + 125735248v^5 + 90912484v^3 + 27855680v) / 1360360

\(\nu\)	\(=\)	\( ( \beta_{13} + \beta_{12} - \beta_{6} ) / 2 \) (b13 + b12 - b6) / 2
\(\nu^{2}\)	\(=\)	\( ( -\beta_{14} - 2\beta_{10} + \beta_{9} - \beta_{8} - 2\beta_{3} + 2\beta _1 + 6 ) / 2 \) (-b14 - 2b10 + b9 - b8 - 2b3 + 2*b1 + 6) / 2
\(\nu^{3}\)	\(=\)	\( ( - 3 \beta_{15} - 3 \beta_{14} + 9 \beta_{13} + 10 \beta_{12} - 2 \beta_{11} + 2 \beta_{9} + \cdots - 10 \beta_{6} ) / 2 \) (-3b15 - 3b14 + 9b13 + 10b12 - 2b11 + 2b9 + 2b8 + 3b7 - 10*b6) / 2
\(\nu^{4}\)	\(=\)	\( - 4 \beta_{14} - 8 \beta_{10} + 10 \beta_{9} - 10 \beta_{8} - 4 \beta_{5} + 2 \beta_{4} - 22 \beta_{3} + \cdots + 26 \) -4b14 - 8b10 + 10b9 - 10b8 - 4b5 + 2b4 - 22b3 - 4b2 + 11*b1 + 26
\(\nu^{5}\)	\(=\)	\( ( - 60 \beta_{15} - 65 \beta_{14} + 72 \beta_{13} + 78 \beta_{12} - 17 \beta_{11} + 19 \beta_{9} + \cdots - 164 \beta_{6} ) / 2 \) (-60b15 - 65b14 + 72b13 + 78b12 - 17b11 + 19b9 + 19b8 + 50b7 - 164*b6) / 2
\(\nu^{6}\)	\(=\)	\( - 25 \beta_{14} - 50 \beta_{10} + 159 \beta_{9} - 159 \beta_{8} - 71 \beta_{5} + 13 \beta_{4} + \cdots + 162 \) -25b14 - 50b10 + 159b9 - 159b8 - 71b5 + 13b4 - 346b3 - 77b2 + 69*b1 + 162
\(\nu^{7}\)	\(=\)	\( ( - 875 \beta_{15} - 952 \beta_{14} + 258 \beta_{13} + 280 \beta_{12} - 61 \beta_{11} + \cdots - 2330 \beta_{6} ) / 2 \) (-875b15 - 952b14 + 258b13 + 280b12 - 61b11 + 68b9 + 68b8 + 714b7 - 2330*b6) / 2
\(\nu^{8}\)	\(=\)	\( 44 \beta_{14} + 88 \beta_{10} + 2096 \beta_{9} - 2096 \beta_{8} - 944 \beta_{5} - 16 \beta_{4} + \cdots - 290 \) 44b14 + 88b10 + 2096b9 - 2096b8 - 944b5 - 16b4 - 4560b3 - 1028b2 - 112*b1 - 290
\(\nu^{9}\)	\(=\)	\( - 5310 \beta_{15} - 5778 \beta_{14} - 2589 \beta_{13} - 2817 \beta_{12} + 580 \beta_{11} + \cdots - 14135 \beta_{6} \) -5310b15 - 5778b14 - 2589b13 - 2817b12 + 580b11 - 630b9 - 630b8 + 4332b7 - 14135*b6
\(\nu^{10}\)	\(=\)	\( 4869 \beta_{14} + 9738 \beta_{10} + 23359 \beta_{9} - 23359 \beta_{8} - 10508 \beta_{5} - 2188 \beta_{4} + \cdots - 31782 \) 4869b14 + 9738b10 + 23359b9 - 23359b8 - 10508b5 - 2188b4 - 50822b3 - 11436b2 - 12974*b1 - 31782
\(\nu^{11}\)	\(=\)	\( - 53779 \beta_{15} - 58509 \beta_{14} - 78757 \beta_{13} - 85678 \beta_{12} + 17704 \beta_{11} + \cdots - 143270 \beta_{6} \) -53779b15 - 58509b14 - 78757b13 - 85678b12 + 17704b11 - 19262b9 - 19262b8 + 43901b7 - 143270*b6
\(\nu^{12}\)	\(=\)	\( 104168 \beta_{14} + 208336 \beta_{10} + 210396 \beta_{9} - 210396 \beta_{8} - 94584 \beta_{5} + \cdots - 679924 \) 104168b14 + 208336b10 + 210396b9 - 210396b8 - 94584b5 - 46844b4 - 457772b3 - 102904b2 - 277614*b1 - 679924
\(\nu^{13}\)	\(=\)	\( - 410436 \beta_{15} - 446511 \beta_{14} - 1395656 \beta_{13} - 1518314 \beta_{12} + \cdots - 1093696 \beta_{6} \) -410436b15 - 446511b14 - 1395656b13 - 1518314b12 + 313689b11 - 341271b9 - 341271b8 + 335114b7 - 1093696*b6
\(\nu^{14}\)	\(=\)	\( 1624542 \beta_{14} + 3249084 \beta_{10} + 1208638 \beta_{9} - 1208638 \beta_{8} - 543258 \beta_{5} + \cdots - 10603884 \) 1624542b14 + 3249084b10 + 1208638b9 - 1208638b8 - 543258b5 - 730282b4 - 2629732b3 - 590998b2 - 4329130*b1 - 10603884
\(\nu^{15}\)	\(=\)	\( - 1100949 \beta_{15} - 1197696 \beta_{14} - 19692702 \beta_{13} - 21423472 \beta_{12} + \cdots - 2933950 \beta_{6} \) -1100949b15 - 1197696b14 - 19692702b13 - 21423472b12 + 4425877b11 - 4814888b9 - 4814888b8 + 898962b7 - 2933950*b6

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 251.16
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( (T^{4} + 6 T^{2} + 3)^{4} \) (T^4 + 6*T^2 + 3)^4
$3$	\( T^{16} \) T^16
$5$	\( T^{16} \) T^16
$7$	\( (T^{8} + 4 T^{6} + \cdots + 2401)^{2} \) (T^8 + 4T^6 + 6T^4 + 196*T^2 + 2401)^2
$11$	\( (T^{2} + 2)^{8} \) (T^2 + 2)^8
$13$	\( (T^{4} + 32 T^{2} + 40)^{4} \) (T^4 + 32*T^2 + 40)^4
$17$	\( (T^{4} - 64 T^{2} + 1000)^{4} \) (T^4 - 64*T^2 + 1000)^4
$19$	\( (T^{4} + 72 T^{2} + 1080)^{4} \) (T^4 + 72*T^2 + 1080)^4
$23$	\( (T^{4} + 36 T^{2} + 300)^{4} \) (T^4 + 36*T^2 + 300)^4
$29$	\( (T^{4} + 100 T^{2} + 2116)^{4} \) (T^4 + 100*T^2 + 2116)^4
$31$	\( (T^{4} + 72 T^{2} + 120)^{4} \) (T^4 + 72*T^2 + 120)^4
$37$	\( (T^{4} - 72 T^{2} + 432)^{4} \) (T^4 - 72*T^2 + 432)^4
$41$	\( (T^{4} - 120 T^{2} + 3000)^{4} \) (T^4 - 120*T^2 + 3000)^4
$43$	\( (T^{4} - 72 T^{2} + 1200)^{4} \) (T^4 - 72*T^2 + 1200)^4
$47$	\( (T^{4} - 64 T^{2} + 160)^{4} \) (T^4 - 64*T^2 + 160)^4
$53$	\( (T^{4} + 108 T^{2} + 972)^{4} \) (T^4 + 108*T^2 + 972)^4
$59$	\( (T^{4} - 144 T^{2} + 480)^{4} \) (T^4 - 144*T^2 + 480)^4
$61$	\( (T^{4} + 192 T^{2} + 7680)^{4} \) (T^4 + 192*T^2 + 7680)^4
$67$	\( (T^{4} - 144 T^{2} + 4800)^{4} \) (T^4 - 144*T^2 + 4800)^4
$71$	\( (T^{4} + 100 T^{2} + 2116)^{4} \) (T^4 + 100*T^2 + 2116)^4
$73$	\( (T^{4} + 128 T^{2} + 40)^{4} \) (T^4 + 128*T^2 + 40)^4
$79$	\( (T - 4)^{16} \) (T - 4)^16
$83$	\( (T^{4} - 64 T^{2} + 160)^{4} \) (T^4 - 64*T^2 + 160)^4
$89$	\( (T^{4} - 216 T^{2} + 9720)^{4} \) (T^4 - 216*T^2 + 9720)^4
$97$	\( (T^{4} + 32 T^{2} + 40)^{4} \) (T^4 + 32*T^2 + 40)^4
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\(n\)	\(127\)	\(451\)	\(1226\)
\(\chi(n)\)	\(1\)	\(-1\)	\(-1\)