LMFDB - Newform orbit 1375.2.b.d

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1375,2,Mod(749,1375)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1375, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1375.749");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1375 = 5^{3} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1375.b (of order $2$, degree $1$, not 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:	$10.9794302779$
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} + 28x^{14} + 316x^{12} + 1843x^{10} + 5944x^{8} + 10643x^{6} + 10306x^{4} + 4898x^{2} + 841 $ x^16 + 28x^14 + 316x^12 + 1843x^10 + 5944x^8 + 10643x^6 + 10306x^4 + 4898*x^2 + 841
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
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_1 q^{2} + \beta_{14} q^{3} + (\beta_{2} - 2) q^{4} + ( - \beta_{15} - \beta_{8} + \cdots + \beta_{2}) q^{6}+ \cdots + ( - \beta_{12} - 2) q^{9}+O(q^{10}) $ q + b1 * q^2 + b14 * q^3 + (b2 - 2) * q^4 + (-b15 - b8 - b7 + b2) * q^6 + (b13 + b1) * q^7 + (-b14 + b10 + b5 - b1) * q^8 + (-b12 - 2) * q^9	$ q + \beta_1 q^{2} + \beta_{14} q^{3} + (\beta_{2} - 2) q^{4} + ( - \beta_{15} - \beta_{8} + \cdots + \beta_{2}) q^{6}+ \cdots + ( - \beta_{12} - 2) q^{99}+O(q^{100}) $ q + b1 * q^2 + b14 * q^3 + (b2 - 2) * q^4 + (-b15 - b8 - b7 + b2) * q^6 + (b13 + b1) * q^7 + (-b14 + b10 + b5 - b1) * q^8 + (-b12 - 2) * q^9 + q^11 + (-b14 - b13 - 2b9 - b4 + b3 - 2b1) * q^12 + (b10 - b3) * q^13 + (b11 - b8 + b2 - 2) * q^14 + (b8 + b7 - 2b2 + 3) q^16 + (-b10 - b9 + b5 - b1) * q^17 + (b14 - 2b10 - 2b9 + b5 + 2b3 - 3b1) * q^18 + (-b15 + b12 - b8 + b7 - b6 + 2) * q^19 + (-b15 + b12 + b11 - 2b8 - b6 + 2b2 + 1) * q^21 + b1 * q^22 + (b14 - b13 - b10 - 2b9 - b5 - b4 - 2b1) * q^23 + (b12 - b11 + 5b8 + b7 - 2b2) * q^24 + (-b15 - b12 + b8 - b7 - b6) * q^26 + (-4b14 + b13 + 2b10 + b5 + b4 - b3 + 2b1) q^27 + (-2b14 - 2b9 + b5 - b4 + b3 - 4b1) q^28 + (-b15 - b11 - 2b7 - b6 + 2b2 - 4) * q^29 + (-b15 - b11 + b8 + b2) * q^31 + (b14 + b13 + b9 + b5 + 2b4 - b3 + 4b1) * q^32 + b14 * q^33 + (2b15 + 4b8 + 2b6 - b2 - 1) q^34 + (3b15 + 5b8 + b7 + 3b6 - 3b2 + 1) * q^36 + (-b13 + b4 - b3) * q^37 + (b13 - b10 + 2b5 + 3b4 - 2b3 + 2b1) * q^38 + (-2b15 - b11 - b8 - b7 + b6 + b2 - 2) q^39 + (-b12 - b11 - b8 - b7 + 2) * q^41 + (-4b14 - 2b13 - 2b9 + 3b5 - 4b1) q^42 + (b13 - b5 + 2b4 + b1) q^43 + (b2 - 2) * q^44 + (b15 - b11 + 6b8 + b2 - 5) q^46 + (b14 - 2b5 - 2b4 - 2b1) q^47 + (b14 + b13 + b10 + 4b9 - 2b5 + b4 - 2b3 + 6b1) * q^48 + (-2b15 + b11 - b7 - 3) q^49 + (-b11 + 5b8 + b7 - b6 - b2 - 2) q^51 + (-b13 - 3b10 - 2b9 + 2b5 - b4 + 2b3) * q^52 + (-2b3 + 2b1) * q^53 + (3b15 - b12 + b11 + 6b8 + 2b7 - b6 - 4b2 - 3) * q^54 + (3b15 + b12 + 2b11 + 5b8 + 4b7 + b6 - 6b2 + 7) q^56 + (4b14 - 3b13 - 2b9 - 4b5 - 3b4 + b3 - 2b1) * q^57 + (-3b14 + 2b5 - 2b4 + 2b3 - 4b1) q^58 + (b12 + b11 + b7 - b6 + b2 - 2) * q^59 + (b15 + b11 + 2b7 - b2 + 1) q^61 + (-b14 + 2b13 + b10 + b9 + b5 + 2b4 - b3 + 2b1) q^62 + (3b14 - 2b13 - b10 - 6b9 - 3b4 + 2b3 - 7b1) * q^63 + (-b12 + b11 + b8 - 2b7 + b6 + b2 - 5) q^64 + (-b15 - b8 - b7 + b2) * q^66 + (b10 + 2b4 - b3 + 2b1) * q^67 + (b14 + 3b10 + 4b9 - 3b5 - 2b4 - 2b3 + 2b1) * q^68 + (2b15 - 2b12 + 3b6 - 3b2 + 1) * q^69 + (2b15 - b12 + b8 + b6 + b2) q^71 + (6b14 + b13 + 2b10 + 4b9 - 6b5 - b4 + 3b1) q^72 + (-b14 + b10 + 2b9 - b5 + b4 + 3b1) * q^73 + (b15 - b12 - b11 + 4b8 - 2b7 + b2 - 4) * q^74 + (2b15 + b11 + 5b8 - 3b7 + b6 + 3b2 - 6) * q^76 + (b13 + b1) * q^77 + (-4b14 + b13 + 2b10 - 2b9 - 2b5 + b4 - b3 - 2b1) q^78 + (b15 + b12 + 3b8 - b7 - b6 - 2b2) * q^79 + (-4b15 + 2b12 - b11 - b8 - b7 + 2b2 + 8) q^81 + (b14 + b13 - b10 - 2b9 - b4 + 2b3 + 2b1) q^82 + (b14 - b13 + 3b9 + b4 + 3b1) * q^83 + (4b15 + 2b12 + 10b8 + 4b7 + b6 - 7b2 + 9) q^84 + (2b15 + b11 + 3b8 - 2b7 - b6 + 2b2 - 3) * q^86 + (b14 - b10 - 6b9 - 3b5 - 2b4 + 3b3 - 8b1) q^87 + (-b14 + b10 + b5 - b1) * q^88 + (-2b15 - b8 - b6 + b2 - 7) q^89 + (b15 + b12 + 2b11 - 5b8 + b7 - b6 + 4) * q^91 + (3b14 + b10 + 4b9 - b5 - 2b4 - b3 - 2b1) * q^92 + (4b14 - b10 - 2b9 - 4b5 - 2b4 + 3b3 - 2b1) * q^93 + (-3b15 - 5b8 + b7 - 2b6 + b2 + 6) q^94 + (-5b15 - b11 - 4b8 - 2b7 - 3b6 + 6b2 - 10) q^96 + (-2b14 + 4b9 + 2b4 - 2b3) * q^97 + (-4b14 - 3b13 - 3b10 - 3b9 - b5 - b4 + 2b3 - 5b1) * q^98 + (-b12 - 2) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 24 q^{4} + 4 q^{6} - 34 q^{9}+O(q^{10}) $ 16 * q - 24 * q^4 + 4 * q^6 - 34 * q^9	$ 16 q - 24 q^{4} + 4 q^{6} - 34 q^{9} + 16 q^{11} - 36 q^{14} + 32 q^{16} + 10 q^{19} + 6 q^{21} + 22 q^{24} + 6 q^{26} - 36 q^{29} + 16 q^{31} + 24 q^{34} + 48 q^{36} - 24 q^{39} + 34 q^{41} - 24 q^{44} - 16 q^{46} - 52 q^{49} - 8 q^{51} - 46 q^{54} + 82 q^{56} - 38 q^{59} - 8 q^{61} - 50 q^{64} + 4 q^{66} + 8 q^{69} + 26 q^{71} - 2 q^{74} + 18 q^{79} + 136 q^{81} + 160 q^{84} + 8 q^{86} - 124 q^{89} + 10 q^{91} + 36 q^{94} - 156 q^{96} - 34 q^{99}+O(q^{100}) $ 16 * q - 24 * q^4 + 4 * q^6 - 34 * q^9 + 16 * q^11 - 36 * q^14 + 32 * q^16 + 10 * q^19 + 6 * q^21 + 22 * q^24 + 6 * q^26 - 36 * q^29 + 16 * q^31 + 24 * q^34 + 48 * q^36 - 24 * q^39 + 34 * q^41 - 24 * q^44 - 16 * q^46 - 52 * q^49 - 8 * q^51 - 46 * q^54 + 82 * q^56 - 38 * q^59 - 8 * q^61 - 50 * q^64 + 4 * q^66 + 8 * q^69 + 26 * q^71 - 2 * q^74 + 18 * q^79 + 136 * q^81 + 160 * q^84 + 8 * q^86 - 124 * q^89 + 10 * q^91 + 36 * q^94 - 156 * q^96 - 34 * q^99

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{16} + 28x^{14} + 316x^{12} + 1843x^{10} + 5944x^{8} + 10643x^{6} + 10306x^{4} + 4898x^{2} + 841 $ x^16 + 28*x^14 + 316*x^12 + 1843*x^10 + 5944*x^8 + 10643*x^6 + 10306*x^4 + 4898*x^2 + 841 :

$\beta_{1}$	$=$	$ \nu $ v
$\beta_{2}$	$=$	$ \nu^{2} + 4 $ v^2 + 4
$\beta_{3}$	$=$	$ ( - 175 \nu^{15} - 4581 \nu^{13} - 46919 \nu^{11} - 236076 \nu^{9} - 598936 \nu^{7} - 708441 \nu^{5} + \cdots - 43671 \nu ) / 1334 $ (-175v^15 - 4581v^13 - 46919v^11 - 236076v^9 - 598936v^7 - 708441v^5 - 342791v^3 - 43671v) / 1334
$\beta_{4}$	$=$	$ ( - 77 \nu^{15} - 2069 \nu^{13} - 21925 \nu^{11} - 115666 \nu^{9} - 315211 \nu^{7} - 419021 \nu^{5} + \cdots - 38745 \nu ) / 667 $ (-77v^15 - 2069v^13 - 21925v^11 - 115666v^9 - 315211v^7 - 419021v^5 - 239459v^3 - 38745v) / 667
$\beta_{5}$	$=$	$ ( - 421 \nu^{15} - 11295 \nu^{13} - 119841 \nu^{11} - 636297 \nu^{9} - 1763533 \nu^{7} + \cdots - 308312 \nu ) / 667 $ (-421v^15 - 11295v^13 - 119841v^11 - 636297v^9 - 1763533v^7 - 2439596v^5 - 1523738v^3 - 308312v) / 667
$\beta_{6}$	$=$	$ ( 17\nu^{14} + 455\nu^{12} + 4814\nu^{10} + 25479\nu^{8} + 70383\nu^{6} + 97072\nu^{4} + 60497\nu^{2} + 12278 ) / 23 $ (17v^14 + 455v^12 + 4814v^10 + 25479v^8 + 70383v^6 + 97072v^4 + 60497*v^2 + 12278) / 23
$\beta_{7}$	$=$	$ ( - 37 \nu^{14} - 993 \nu^{12} - 10533 \nu^{10} - 55848 \nu^{8} - 154254 \nu^{6} - 211871 \nu^{4} + \cdots - 26007 ) / 46 $ (-37v^14 - 993v^12 - 10533v^10 - 55848v^8 - 154254v^6 - 211871v^4 - 130777*v^2 - 26007) / 46
$\beta_{8}$	$=$	$ ( 37 \nu^{14} + 993 \nu^{12} + 10533 \nu^{10} + 55848 \nu^{8} + 154254 \nu^{6} + 211917 \nu^{4} + \cdots + 26421 ) / 46 $ (37v^14 + 993v^12 + 10533v^10 + 55848v^8 + 154254v^6 + 211917v^4 + 131145*v^2 + 26421) / 46
$\beta_{9}$	$=$	$ ( 37 \nu^{15} + 993 \nu^{13} + 10533 \nu^{11} + 55848 \nu^{9} + 154254 \nu^{7} + 211917 \nu^{5} + \cdots + 26375 \nu ) / 46 $ (37v^15 + 993v^13 + 10533v^11 + 55848v^9 + 154254v^7 + 211917v^5 + 131145v^3 + 26375v) / 46
$\beta_{10}$	$=$	$ ( - 1715 \nu^{15} - 45961 \nu^{13} - 486753 \nu^{11} - 2576076 \nu^{9} - 7097920 \nu^{7} + \cdots - 1188089 \nu ) / 1334 $ (-1715v^15 - 45961v^13 - 486753v^11 - 2576076v^9 - 7097920v^7 - 9715841v^5 - 5977727v^3 - 1188089v) / 1334
$\beta_{11}$	$=$	$ ( - 30 \nu^{14} - 807 \nu^{12} - 8590 \nu^{10} - 45795 \nu^{8} - 127589 \nu^{6} - 177592 \nu^{4} + \cdots - 22491 ) / 23 $ (-30v^14 - 807v^12 - 8590v^10 - 45795v^8 - 127589v^6 - 177592v^4 - 111308*v^2 - 22491) / 23
$\beta_{12}$	$=$	$ ( 85 \nu^{14} + 2275 \nu^{12} + 24047 \nu^{10} + 126912 \nu^{8} + 348304 \nu^{6} + 474159 \nu^{4} + \cdots + 57595 ) / 46 $ (85v^14 + 2275v^12 + 24047v^10 + 126912v^8 + 348304v^6 + 474159v^4 + 290019*v^2 + 57595) / 46
$\beta_{13}$	$=$	$ ( 2459 \nu^{15} + 66039 \nu^{13} + 701441 \nu^{11} + 3728260 \nu^{9} + 10340594 \nu^{7} + \cdots + 1785113 \nu ) / 1334 $ (2459v^15 + 66039v^13 + 701441v^11 + 3728260v^9 + 10340594v^7 + 14297609v^5 + 8896525v^3 + 1785113v) / 1334
$\beta_{14}$	$=$	$ ( - 2557 \nu^{15} - 68551 \nu^{13} - 726435 \nu^{11} - 3848670 \nu^{9} - 10624986 \nu^{7} + \cdots - 1811383 \nu ) / 1334 $ (-2557v^15 - 68551v^13 - 726435v^11 - 3848670v^9 - 10624986v^7 - 14595033v^5 - 9026537v^3 - 1811383v) / 1334
$\beta_{15}$	$=$	$ ( - 105 \nu^{14} - 2813 \nu^{12} - 29789 \nu^{10} - 157718 \nu^{8} - 435142 \nu^{6} - 597491 \nu^{4} + \cdots - 74383 ) / 46 $ (-105v^14 - 2813v^12 - 29789v^10 - 157718v^8 - 435142v^6 - 597491v^4 - 369729*v^2 - 74383) / 46

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

$\nu$	$=$	$ \beta_1 $ b1
$\nu^{2}$	$=$	$ \beta_{2} - 4 $ b2 - 4
$\nu^{3}$	$=$	$ -\beta_{14} + \beta_{10} + \beta_{5} - 5\beta_1 $ -b14 + b10 + b5 - 5*b1
$\nu^{4}$	$=$	$ \beta_{8} + \beta_{7} - 8\beta_{2} + 23 $ b8 + b7 - 8*b2 + 23
$\nu^{5}$	$=$	$ 9\beta_{14} + \beta_{13} - 8\beta_{10} + \beta_{9} - 7\beta_{5} + 2\beta_{4} - \beta_{3} + 32\beta_1 $ 9b14 + b13 - 8b10 + b9 - 7b5 + 2b4 - b3 + 32*b1
$\nu^{6}$	$=$	$ -\beta_{12} + \beta_{11} - 9\beta_{8} - 12\beta_{7} + \beta_{6} + 57\beta_{2} - 147 $ -b12 + b11 - 9b8 - 12b7 + b6 + 57*b2 - 147
$\nu^{7}$	$=$	$ -70\beta_{14} - 14\beta_{13} + 56\beta_{10} - 12\beta_{9} + 44\beta_{5} - 25\beta_{4} + 14\beta_{3} - 216\beta_1 $ -70b14 - 14b13 + 56b10 - 12b9 + 44b5 - 25b4 + 14b3 - 216b1
$\nu^{8}$	$=$	$ \beta_{15} + 14\beta_{12} - 14\beta_{11} + 68\beta_{8} + 109\beta_{7} - 12\beta_{6} - 400\beta_{2} + 972 $ b15 + 14b12 - 14b11 + 68b8 + 109b7 - 12b6 - 400b2 + 972
$\nu^{9}$	$=$	$ 522 \beta_{14} + 137 \beta_{13} - 381 \beta_{10} + 111 \beta_{9} - 281 \beta_{5} + 231 \beta_{4} + \cdots + 1482 \beta_1 $ 522b14 + 137b13 - 381b10 + 111b9 - 281b5 + 231b4 - 139b3 + 1482b1
$\nu^{10}$	$=$	$ -21\beta_{15} - 139\beta_{12} + 137\beta_{11} - 502\beta_{8} - 892\beta_{7} + 100\beta_{6} + 2805\beta_{2} - 6531 $ -21b15 - 139b12 + 137b11 - 502b8 - 892b7 + 100b6 + 2805*b2 - 6531
$\nu^{11}$	$=$	$ - 3816 \beta_{14} - 1166 \beta_{13} + 2569 \beta_{10} - 938 \beta_{9} + 1852 \beta_{5} + \cdots - 10251 \beta_1 $ -3816b14 - 1166b13 + 2569b10 - 938b9 + 1852b5 - 1900b4 + 1207b3 - 10251b1
$\nu^{12}$	$=$	$ 285 \beta_{15} + 1207 \beta_{12} - 1166 \beta_{11} + 3727 \beta_{8} + 6923 \beta_{7} - 717 \beta_{6} + \cdots + 44324 $ 285b15 + 1207b12 - 1166b11 + 3727b8 + 6923b7 - 717b6 - 19695*b2 + 44324
$\nu^{13}$	$=$	$ 27579 \beta_{14} + 9255 \beta_{13} - 17264 \beta_{10} + 7592 \beta_{9} - 12545 \beta_{5} + \cdots + 71285 \beta_1 $ 27579b14 + 9255b13 - 17264b10 + 7592b9 - 12545b5 + 14727b4 - 9786b3 + 71285b1
$\nu^{14}$	$=$	$ - 3180 \beta_{15} - 9786 \beta_{12} + 9255 \beta_{11} - 27962 \beta_{8} - 52092 \beta_{7} + \cdots - 302907 $ -3180b15 - 9786b12 + 9255b11 - 27962b8 - 52092b7 + 4719b6 + 138459*b2 - 302907
$\nu^{15}$	$=$	$ - 197919 \beta_{14} - 70602 \beta_{13} + 115890 \beta_{10} - 59969 \beta_{9} + 86715 \beta_{5} + \cdots - 497624 \beta_1 $ -197919b14 - 70602b13 + 115890b10 - 59969b9 + 86715b5 - 110259b4 + 76200b3 - 497624b1

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1375\mathbb{Z}\right)^\times$.

$n$	$376$	$1002$
$\chi(n)$	$1$	$-1$

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} $

749.1

	−	2.66606i
	−	2.61932i
	−	2.30823i
	−	2.14646i
	−	1.17331i
	−	1.09283i
	−	1.07481i
	−	0.608189i
		0.608189i
		1.07481i
		1.09283i
		1.17331i
		2.14646i
		2.30823i
		2.61932i
		2.66606i

−

2.66606i

−

2.88353i

−5.10789

−7.68768

−

4.48479i

8.28584i

−5.31476

749.2

−

2.61932i

0.222853i

−4.86083

0.583723

−

1.17943i

7.49344i

2.95034

749.3

−

2.30823i

1.33780i

−3.32793

3.08796

−

0.0711572i

3.06517i

1.21028

749.4

−

2.14646i

3.26422i

−2.60729

7.00653

0.0149681i

1.30352i

−7.65516

749.5

−

1.17331i

−

1.28122i

0.623347

−1.50326

−

5.12261i

−

3.07800i

1.35848

749.6

−

1.09283i

3.35614i

0.805718

3.66769

−

2.84255i

−

3.06618i

−8.26365

749.7

−

1.07481i

−

2.65247i

0.844776

−2.85092

4.22372i

−

3.05760i

−4.03562

749.8

−

0.608189i

−

0.499909i

1.63011

−0.304039

2.88607i

−

2.20779i

2.75009

749.9