LMFDB - Newform orbit 400.4.n.g

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [400,4,Mod(143,400)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(400, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([2, 0, 3]))

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("400.143");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 400 = 2^{4} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	400.n (of order $4$, degree $2$, 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:	$23.6007640023$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$8$ over $\Q(i)$
Coefficient field:	16.0.11007531417600000000.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 7x^{12} + 48x^{8} - 7x^{4} + 1 $ x^16 - 7x^12 + 48x^8 - 7*x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{36}\cdot 3^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

$q$-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_{8} q^{3} + (2 \beta_{9} + \beta_{6}) q^{7} + (\beta_{13} - 22 \beta_{3}) q^{9}+O(q^{10}) $ q + b8 * q^3 + (2b9 + b6) q^7 + (b13 - 22b3) q^9	\( q + \beta_{8} q^{3} + (2 \beta_{9} + \beta_{6}) q^{7} + (\beta_{13} - 22 \beta_{3}) q^{9} + (\beta_{11} - 4 \beta_{7}) q^{11} + (\beta_{15} + 14 \beta_{4}) q^{13} + ( - 5 \beta_{14} + 11 \beta_{2}) q^{17} + ( - 9 \beta_{10} + 14 \beta_{5}) q^{19} + (3 \beta_{12} - 116) q^{21} + ( - 6 \beta_{8} + 14 \beta_1) q^{23} + (13 \beta_{9} + 31 \beta_{6}) q^{27} + (3 \beta_{13} - 54 \beta_{3}) q^{29} + (10 \beta_{11} - 54 \beta_{7}) q^{31} + ( - 2 \beta_{15} + 19 \beta_{4}) q^{33} + (3 \beta_{14} - 120 \beta_{2}) q^{37} + (20 \beta_{10} - 71 \beta_{5}) q^{39} + ( - 4 \beta_{12} - 123) q^{41} + (38 \beta_{8} + 41 \beta_1) q^{43} + ( - 24 \beta_{9} + 79 \beta_{6}) q^{47} + ( - 8 \beta_{13} - 39 \beta_{3}) q^{49} + ( - 41 \beta_{11} - 196 \beta_{7}) q^{51} + (2 \beta_{15} + 242 \beta_{4}) q^{53} + (23 \beta_{14} - 417 \beta_{2}) q^{57} + ( - 12 \beta_{10} + 9 \beta_{5}) q^{59} + ( - 13 \beta_{12} - 124) q^{61} + ( - 116 \beta_{8} + 66 \beta_1) q^{63} + ( - 83 \beta_{9} + 113 \beta_{6}) q^{67} + ( - 20 \beta_{13} + 546 \beta_{3}) q^{69} + (40 \beta_{11} - 133 \beta_{7}) q^{71} + ( - \beta_{15} - 103 \beta_{4}) q^{73} + (\beta_{14} - 4 \beta_{2}) q^{77} + (50 \beta_{10} - 165 \beta_{5}) q^{79} + (17 \beta_{12} - 601) q^{81} + (129 \beta_{8} + 83 \beta_1) q^{83} + (108 \beta_{9} + 93 \beta_{6}) q^{87} + (19 \beta_{13} - 813 \beta_{3}) q^{89} + ( - 52 \beta_{11} - 138 \beta_{7}) q^{91} + ( - 34 \beta_{15} + 106 \beta_{4}) q^{93} + ( - 20 \beta_{14} - 252 \beta_{2}) q^{97} + ( - 20 \beta_{10} - 33 \beta_{5}) q^{99}+O(q^{100}) \) q + b8 * q^3 + (2b9 + b6) q^7 + (b13 - 22b3) q^9 + (b11 - 4b7) q^11 + (b15 + 14b4) q^13 + (-5b14 + 11b2) * q^17 + (-9b10 + 14b5) * q^19 + (3b12 - 116) q^21 + (-6b8 + 14b1) * q^23 + (13b9 + 31b6) * q^27 + (3b13 - 54b3) * q^29 + (10b11 - 54b7) * q^31 + (-2b15 + 19b4) * q^33 + (3b14 - 120b2) * q^37 + (20b10 - 71b5) * q^39 + (-4b12 - 123) q^41 + (38b8 + 41b1) * q^43 + (-24b9 + 79b6) * q^47 + (-8b13 - 39b3) * q^49 + (-41b11 - 196b7) * q^51 + (2b15 + 242b4) * q^53 + (23b14 - 417b2) * q^57 + (-12b10 + 9b5) * q^59 + (-13b12 - 124) q^61 + (-116b8 + 66b1) * q^63 + (-83b9 + 113b6) * q^67 + (-20b13 + 546b3) * q^69 + (40b11 - 133b7) * q^71 + (-b15 - 103b4) q^73 + (b14 - 4b2) q^77 + (50b10 - 165b5) * q^79 + (17b12 - 601) q^81 + (129b8 + 83b1) * q^83 + (108b9 + 93b6) * q^87 + (19b13 - 813b3) * q^89 + (-52b11 - 138b7) * q^91 + (-34b15 + 106b4) * q^93 + (-20b14 - 252b2) * q^97 + (-20b10 - 33b5) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q+O(q^{10}) $ 16 * q	$ 16 q - 1856 q^{21} - 1968 q^{41} - 1984 q^{61} - 9616 q^{81}+O(q^{100}) $ 16 * q - 1856 * q^21 - 1968 * q^41 - 1984 * q^61 - 9616 * q^81

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{16} - 7x^{12} + 48x^{8} - 7x^{4} + 1 $ x^16 - 7*x^12 + 48*x^8 - 7*x^4 + 1 :

$\beta_{1}$	$=$	$ ( -\nu^{15} - 305\nu^{3} ) / 12 $ (-v^15 - 305*v^3) / 12
$\beta_{2}$	$=$	$ ( 3\nu^{13} - 20\nu^{9} + 136\nu^{5} + 19\nu ) / 12 $ (3v^13 - 20v^9 + 136v^5 + 19v) / 12
$\beta_{3}$	$=$	$ ( -7\nu^{14} + 48\nu^{10} - 330\nu^{6} + \nu^{2} ) / 18 $ (-7v^14 + 48v^10 - 330*v^6 + v^2) / 18
$\beta_{4}$	$=$	$ ( -15\nu^{15} + 104\nu^{11} - 712\nu^{7} + 53\nu^{3} ) / 12 $ (-15v^15 + 104v^11 - 712v^7 + 53v^3) / 12
$\beta_{5}$	$=$	$ ( 9\nu^{14} - 64\nu^{10} + 440\nu^{6} - 127\nu^{2} ) / 12 $ (9v^14 - 64v^10 + 440v^6 - 127v^2) / 12
$\beta_{6}$	$=$	$ ( 17\nu^{13} - 120\nu^{9} + 816\nu^{5} - 119\nu ) / 12 $ (17v^13 - 120v^9 + 816v^5 - 119v) / 12
$\beta_{7}$	$=$	$ ( -7\nu^{12} + 48\nu^{8} - 336\nu^{4} + 25 ) / 12 $ (-7v^12 + 48v^8 - 336*v^4 + 25) / 12
$\beta_{8}$	$=$	$ ( 3\nu^{15} - 126\nu^{14} - 2\nu^{12} + 864\nu^{10} - 5904\nu^{6} + 915\nu^{3} + 18\nu^{2} - 322 ) / 72 $ (3v^15 - 126v^14 - 2v^12 + 864v^10 - 5904v^6 + 915v^3 + 18*v^2 - 322) / 72
$\beta_{9}$	$=$	$ ( - 126 \nu^{14} + 51 \nu^{13} + 2 \nu^{12} + 864 \nu^{10} - 360 \nu^{9} - 5904 \nu^{6} + 2448 \nu^{5} + \cdots + 322 ) / 72 $ (-126v^14 + 51v^13 + 2v^12 + 864v^10 - 360v^9 - 5904v^6 + 2448v^5 + 18v^2 - 357*v + 322) / 72
$\beta_{10}$	$=$	$ ( 134 \nu^{15} + 9 \nu^{14} - 26 \nu^{13} - 928 \nu^{11} - 64 \nu^{10} + 176 \nu^{9} + 6368 \nu^{7} + \cdots - 170 \nu ) / 24 $ (134v^15 + 9v^14 - 26v^13 - 928v^11 - 64v^10 + 176v^9 + 6368v^7 + 440v^6 - 1216v^5 - 474v^3 - 127v^2 - 170v) / 24
$\beta_{11}$	$=$	$ ( 134 \nu^{15} + 26 \nu^{13} - 7 \nu^{12} - 928 \nu^{11} - 176 \nu^{9} + 48 \nu^{8} + 6368 \nu^{7} + \cdots + 25 ) / 24 $ (134v^15 + 26v^13 - 7v^12 - 928v^11 - 176v^9 + 48v^8 + 6368v^7 + 1216v^5 - 336v^4 - 474v^3 + 170*v + 25) / 24
$\beta_{12}$	$=$	$ ( \nu^{15} - 19\nu^{13} + 132\nu^{9} - 912\nu^{5} + 341\nu^{3} + 133\nu ) / 3 $ (v^15 - 19v^13 + 132v^9 - 912v^5 + 341v^3 + 133*v) / 3
$\beta_{13}$	$=$	$ ( -\nu^{15} - 19\nu^{13} + 132\nu^{9} - 912\nu^{5} - 341\nu^{3} + 133\nu ) / 3 $ (-v^15 - 19v^13 + 132v^9 - 912v^5 - 341v^3 + 133*v) / 3
$\beta_{14}$	$=$	$ ( -10\nu^{14} + 8\nu^{12} + 72\nu^{10} - 56\nu^{8} - 492\nu^{6} + 376\nu^{4} + 142\nu^{2} - 28 ) / 3 $ (-10v^14 + 8v^12 + 72v^10 - 56v^8 - 492v^6 + 376v^4 + 142*v^2 - 28) / 3
$\beta_{15}$	$=$	$ ( 10\nu^{14} + 8\nu^{12} - 72\nu^{10} - 56\nu^{8} + 492\nu^{6} + 376\nu^{4} - 142\nu^{2} - 28 ) / 3 $ (10v^14 + 8v^12 - 72v^10 - 56v^8 + 492v^6 + 376v^4 - 142*v^2 - 28) / 3

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

$\nu$	$=$	$ ( \beta_{13} + \beta_{12} + 6\beta_{11} - 6\beta_{10} - 3\beta_{7} + 4\beta_{6} + 3\beta_{5} - 24\beta_{2} ) / 96 $ (b13 + b12 + 6b11 - 6b10 - 3b7 + 4b6 + 3b5 - 24b2) / 96
$\nu^{2}$	$=$	$ ( \beta_{15} - \beta_{14} + 2\beta_{9} + 2\beta_{8} - \beta_{6} - 12\beta_{5} - 24\beta_{3} + \beta_1 ) / 32 $ (b15 - b14 + 2b9 + 2b8 - b6 - 12b5 - 24b3 + b1) / 32
$\nu^{3}$	$=$	$ ( -\beta_{13} + \beta_{12} + 8\beta_1 ) / 24 $ (-b13 + b12 + 8*b1) / 24
$\nu^{4}$	$=$	$ ( -3\beta_{15} - 3\beta_{14} - 6\beta_{9} + 6\beta_{8} - 28\beta_{7} + 3\beta_{6} + 3\beta _1 + 56 ) / 32 $ (-3b15 - 3b14 - 6b9 + 6b8 - 28b7 + 3b6 + 3*b1 + 56) / 32
$\nu^{5}$	$=$	$ ( - 5 \beta_{13} - 5 \beta_{12} + 30 \beta_{11} - 30 \beta_{10} - 15 \beta_{7} - 44 \beta_{6} + \cdots - 264 \beta_{2} ) / 96 $ (-5b13 - 5b12 + 30b11 - 30b10 - 15b7 - 44b6 + 15b5 - 264b2) / 96
$\nu^{6}$	$=$	$ ( 2\beta_{9} + 2\beta_{8} - \beta_{6} - 18\beta_{3} + \beta_1 ) / 2 $ (2b9 + 2b8 - b6 - 18*b3 + b1) / 2
$\nu^{7}$	$=$	$ ( - 13 \beta_{13} + 13 \beta_{12} + 78 \beta_{11} + 78 \beta_{10} - 39 \beta_{7} - 39 \beta_{5} + \cdots + 116 \beta_1 ) / 96 $ (-13b13 + 13b12 + 78b11 + 78b10 - 39b7 - 39b5 + 696b4 + 116b1) / 96
$\nu^{8}$	$=$	$ ( -21\beta_{15} - 21\beta_{14} + 42\beta_{9} - 42\beta_{8} - 188\beta_{7} - 21\beta_{6} - 21\beta _1 - 376 ) / 32 $ (-21b15 - 21b14 + 42b9 - 42b8 - 188b7 - 21b6 - 21*b1 - 376) / 32
$\nu^{9}$	$=$	$ ( -17\beta_{13} - 17\beta_{12} - 152\beta_{6} ) / 24 $ (-17b13 - 17b12 - 152*b6) / 24
$\nu^{10}$	$=$	$ ( - 55 \beta_{15} + 55 \beta_{14} + 110 \beta_{9} + 110 \beta_{8} - 55 \beta_{6} + 492 \beta_{5} + \cdots + 55 \beta_1 ) / 32 $ (-55b15 + 55b14 + 110b9 + 110b8 - 55b6 + 492b5 - 984b3 + 55b1) / 32
$\nu^{11}$	$=$	$ ( 89 \beta_{13} - 89 \beta_{12} + 534 \beta_{11} + 534 \beta_{10} - 267 \beta_{7} - 267 \beta_{5} + \cdots - 796 \beta_1 ) / 96 $ (89b13 - 89b12 + 534b11 + 534b10 - 267b7 - 267b5 + 4776b4 - 796b1) / 96
$\nu^{12}$	$=$	$ 18\beta_{9} - 18\beta_{8} - 9\beta_{6} - 9\beta _1 - 161 $ 18b9 - 18b8 - 9b6 - 9b1 - 161
$\nu^{13}$	$=$	$ ( - 233 \beta_{13} - 233 \beta_{12} - 1398 \beta_{11} + 1398 \beta_{10} + 699 \beta_{7} + \cdots + 12504 \beta_{2} ) / 96 $ (-233b13 - 233b12 - 1398b11 + 1398b10 + 699b7 - 2084b6 - 699b5 + 12504b2) / 96
$\nu^{14}$	$=$	$ ( - 377 \beta_{15} + 377 \beta_{14} - 754 \beta_{9} - 754 \beta_{8} + 377 \beta_{6} + 3372 \beta_{5} + \cdots - 377 \beta_1 ) / 32 $ (-377b15 + 377b14 - 754b9 - 754b8 + 377b6 + 3372b5 + 6744b3 - 377b1) / 32
$\nu^{15}$	$=$	$ ( 305\beta_{13} - 305\beta_{12} - 2728\beta_1 ) / 24 $ (305b13 - 305b12 - 2728*b1) / 24

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/400\mathbb{Z}\right)^\times$.

$n$	$101$	$177$	$351$
$\chi(n)$	$1$	$-\beta_{3}$	$-1$

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} $

143.1

−0.596975	+	0.159959i
0.159959	−	0.596975i
0.596975	−	0.159959i
−0.159959	+	0.596975i
1.56290	−	0.418778i
−0.418778	+	1.56290i
−1.56290	+	0.418778i
0.418778	−	1.56290i
0.159959	+	0.596975i
−0.596975	−	0.159959i
−0.159959	−	0.596975i
0.596975	+	0.159959i
−0.418778	−	1.56290i
1.56290	+	0.418778i
0.418778	+	1.56290i
−1.56290	−	0.418778i

−6.59346

6.59346i

17.4296

17.4296i

−

59.9473i

143.2

−6.59346

6.59346i

17.4296

17.4296i

−

59.9473i

143.3

−2.35082

2.35082i

0.458991

0.458991i

15.9473i

143.4

−2.35082

2.35082i

0.458991

0.458991i

15.9473i

143.5

2.35082

−

2.35082i

−0.458991

−

0.458991i

15.9473i

143.6

2.35082

−

2.35082i

−0.458991

−

0.458991i

15.9473i

143.7

6.59346

−

6.59346i

−17.4296

−

17.4296i

−

59.9473i

143.8

6.59346

−

6.59346i

−17.4296

−

17.4296i

−

59.9473i

207.1

−6.59346

−

6.59346i

17.4296

−

17.4296i

59.9473i

207.2

−6.59346

−

6.59346i

17.4296

−

17.4296i

59.9473i

207.3

−2.35082

−

2.35082i

0.458991

−

0.458991i

−

15.9473i

207.4

−2.35082

−

2.35082i

0.458991

−

0.458991i

−

15.9473i

207.5

2.35082

2.35082i

−0.458991

0.458991i

−

15.9473i

207.6

2.35082

2.35082i

−0.458991

0.458991i

−

15.9473i

207.7

6.59346

6.59346i

−17.4296

17.4296i

59.9473i

207.8

6.59346

6.59346i

−17.4296

17.4296i

59.9473i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	inner
5.b	even	2	1	inner
5.c	odd	4	2	inner
20.d	odd	2	1	inner
20.e	even	4	2	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	400.4.n.g	✓	16
4.b	odd	2	1	inner	400.4.n.g	✓	16
5.b	even	2	1	inner	400.4.n.g	✓	16
5.c	odd	4	2	inner	400.4.n.g	✓	16
20.d	odd	2	1	inner	400.4.n.g	✓	16
20.e	even	4	2	inner	400.4.n.g	✓	16

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
400.4.n.g	✓	16	1.a	even	1	1	trivial
400.4.n.g	✓	16	4.b	odd	2	1	inner
400.4.n.g	✓	16	5.b	even	2	1	inner
400.4.n.g	✓	16	5.c	odd	4	2	inner
400.4.n.g	✓	16	20.d	odd	2	1	inner
400.4.n.g	✓	16	20.e	even	4	2	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{3}^{8} + 7682T_{3}^{4} + 923521 $ T3^8 + 7682*T3^4 + 923521 acting on $S_{4}^{\mathrm{new}}(400, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{16} $ T^16
$3$	$ (T^{8} + 7682 T^{4} + 923521)^{2} $ (T^8 + 7682*T^4 + 923521)^2
$5$	$ T^{16} $ T^16
$7$	$ (T^{8} + 369152 T^{4} + 65536)^{2} $ (T^8 + 369152*T^4 + 65536)^2
$11$	$ (T^{4} + 534 T^{2} + 729)^{4} $ (T^4 + 534*T^2 + 729)^4
$13$	$ (T^{8} + 4539168 T^{4} + 136048896)^{2} $ (T^8 + 4539168*T^4 + 136048896)^2
$17$	$ (T^{8} + \cdots + 18\!\cdots\!61)^{2} $ (T^8 + 340535538*T^4 + 18338513836013361)^2
$19$	$ (T^{4} - 21606 T^{2} + 74597769)^{4} $ (T^4 - 21606*T^2 + 74597769)^4
$23$	$ (T^{8} + \cdots + 64\!\cdots\!16)^{2} $ (T^8 + 400414752*T^4 + 6456652178063616)^2
$29$	$ (T^{4} + 31752 T^{2} + 100881936)^{4} $ (T^4 + 31752*T^2 + 100881936)^4
$31$	$ (T^{4} + 81624 T^{2} + 282643344)^{4} $ (T^4 + 81624*T^2 + 282643344)^4
$37$	$ (T^{8} + \cdots + 22\!\cdots\!00)^{2} $ (T^8 + 6009292800*T^4 + 2285099025039360000)^2
$41$	$ (T^{2} + 246 T - 7911)^{8} $ (T^2 + 246*T - 7911)^8
$43$	$ (T^{8} + \cdots + 26\!\cdots\!16)^{2} $ (T^8 + 19356549632*T^4 + 2647105883634466816)^2
$47$	$ (T^{8} + \cdots + 36\!\cdots\!16)^{2} $ (T^8 + 97973662752*T^4 + 368639312486720553216)^2
$53$	$ (T^{8} + \cdots + 91\!\cdots\!56)^{2} $ (T^8 + 65790674208*T^4 + 911841173070969344256)^2
$59$	$ (T^{4} - 34776 T^{2} + 294877584)^{4} $ (T^4 - 34776*T^2 + 294877584)^4
$61$	$ (T^{2} + 248 T - 227984)^{8} $ (T^2 + 248*T - 227984)^8
$67$	$ (T^{8} + \cdots + 70\!\cdots\!21)^{2} $ (T^8 + 828180862082*T^4 + 70152808825804024321)^2
$71$	$ (T^{4} + 690456 T^{2} + 1503267984)^{4} $ (T^4 + 690456*T^2 + 1503267984)^4
$73$	$ (T^{8} + \cdots + 96\!\cdots\!81)^{2} $ (T^8 + 2209700178*T^4 + 965570381750779281)^2
$79$	$ (T^{4} - 1070400 T^{2} + 4199040000)^{4} $ (T^4 - 1070400*T^2 + 4199040000)^4
$83$	$ (T^{8} + \cdots + 18\!\cdots\!21)^{2} $ (T^8 + 984873038562*T^4 + 182180191634889963809121)^2
$89$	$ (T^{4} + 2361618 T^{2} + 19917394641)^{4} $ (T^4 + 2361618*T^2 + 19917394641)^4
$97$	$ (T^{8} + \cdots + 4902433652736)^{2} $ (T^8 + 585257292288*T^4 + 4902433652736)^2
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Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 400 = 2^{4} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	400.n (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\( q + \beta_{8} q^{3} + (2 \beta_{9} + \beta_{6}) q^{7} + (\beta_{13} - 22 \beta_{3}) q^{9}+O(q^{10}) \) q + b8 * q^3 + (2b9 + b6) q^7 + (b13 - 22b3) q^9	\( q + \beta_{8} q^{3} + (2 \beta_{9} + \beta_{6}) q^{7} + (\beta_{13} - 22 \beta_{3}) q^{9} + (\beta_{11} - 4 \beta_{7}) q^{11} + (\beta_{15} + 14 \beta_{4}) q^{13} + ( - 5 \beta_{14} + 11 \beta_{2}) q^{17} + ( - 9 \beta_{10} + 14 \beta_{5}) q^{19} + (3 \beta_{12} - 116) q^{21} + ( - 6 \beta_{8} + 14 \beta_1) q^{23} + (13 \beta_{9} + 31 \beta_{6}) q^{27} + (3 \beta_{13} - 54 \beta_{3}) q^{29} + (10 \beta_{11} - 54 \beta_{7}) q^{31} + ( - 2 \beta_{15} + 19 \beta_{4}) q^{33} + (3 \beta_{14} - 120 \beta_{2}) q^{37} + (20 \beta_{10} - 71 \beta_{5}) q^{39} + ( - 4 \beta_{12} - 123) q^{41} + (38 \beta_{8} + 41 \beta_1) q^{43} + ( - 24 \beta_{9} + 79 \beta_{6}) q^{47} + ( - 8 \beta_{13} - 39 \beta_{3}) q^{49} + ( - 41 \beta_{11} - 196 \beta_{7}) q^{51} + (2 \beta_{15} + 242 \beta_{4}) q^{53} + (23 \beta_{14} - 417 \beta_{2}) q^{57} + ( - 12 \beta_{10} + 9 \beta_{5}) q^{59} + ( - 13 \beta_{12} - 124) q^{61} + ( - 116 \beta_{8} + 66 \beta_1) q^{63} + ( - 83 \beta_{9} + 113 \beta_{6}) q^{67} + ( - 20 \beta_{13} + 546 \beta_{3}) q^{69} + (40 \beta_{11} - 133 \beta_{7}) q^{71} + ( - \beta_{15} - 103 \beta_{4}) q^{73} + (\beta_{14} - 4 \beta_{2}) q^{77} + (50 \beta_{10} - 165 \beta_{5}) q^{79} + (17 \beta_{12} - 601) q^{81} + (129 \beta_{8} + 83 \beta_1) q^{83} + (108 \beta_{9} + 93 \beta_{6}) q^{87} + (19 \beta_{13} - 813 \beta_{3}) q^{89} + ( - 52 \beta_{11} - 138 \beta_{7}) q^{91} + ( - 34 \beta_{15} + 106 \beta_{4}) q^{93} + ( - 20 \beta_{14} - 252 \beta_{2}) q^{97} + ( - 20 \beta_{10} - 33 \beta_{5}) q^{99}+O(q^{100}) \) q + b8 * q^3 + (2b9 + b6) q^7 + (b13 - 22b3) q^9 + (b11 - 4b7) q^11 + (b15 + 14b4) q^13 + (-5b14 + 11b2) * q^17 + (-9b10 + 14b5) * q^19 + (3b12 - 116) q^21 + (-6b8 + 14b1) * q^23 + (13b9 + 31b6) * q^27 + (3b13 - 54b3) * q^29 + (10b11 - 54b7) * q^31 + (-2b15 + 19b4) * q^33 + (3b14 - 120b2) * q^37 + (20b10 - 71b5) * q^39 + (-4b12 - 123) q^41 + (38b8 + 41b1) * q^43 + (-24b9 + 79b6) * q^47 + (-8b13 - 39b3) * q^49 + (-41b11 - 196b7) * q^51 + (2b15 + 242b4) * q^53 + (23b14 - 417b2) * q^57 + (-12b10 + 9b5) * q^59 + (-13b12 - 124) q^61 + (-116b8 + 66b1) * q^63 + (-83b9 + 113b6) * q^67 + (-20b13 + 546b3) * q^69 + (40b11 - 133b7) * q^71 + (-b15 - 103b4) q^73 + (b14 - 4b2) q^77 + (50b10 - 165b5) * q^79 + (17b12 - 601) q^81 + (129b8 + 83b1) * q^83 + (108b9 + 93b6) * q^87 + (19b13 - 813b3) * q^89 + (-52b11 - 138b7) * q^91 + (-34b15 + 106b4) * q^93 + (-20b14 - 252b2) * q^97 + (-20b10 - 33b5) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q+O(q^{10}) \) 16 * q	\( 16 q - 1856 q^{21} - 1968 q^{41} - 1984 q^{61} - 9616 q^{81}+O(q^{100}) \) 16 * q - 1856 * q^21 - 1968 * q^41 - 1984 * q^61 - 9616 * q^81

Introduction

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Newspace parameters

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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	400.4.n.g

Level	$400$
Weight	$4$
Character orbit	400.n
Analytic conductor	$23.601$
Analytic rank	$0$
Dimension	$16$
CM	no
Inner twists	$8$

Self dual:	no
Analytic conductor:	\(23.6007640023\)
Analytic rank:	\(0\)
Dimension:	\(16\)
Relative dimension:	\(8\) over \(\Q(i)\)
Coefficient field:	16.0.11007531417600000000.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{16} - 7x^{12} + 48x^{8} - 7x^{4} + 1 \) x^16 - 7x^12 + 48x^8 - 7*x^4 + 1
Coefficient ring:	\(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index:	\( 2^{36}\cdot 3^{4} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

\(\beta_{1}\)	\(=\)	\( ( -\nu^{15} - 305\nu^{3} ) / 12 \) (-v^15 - 305*v^3) / 12
\(\beta_{2}\)	\(=\)	\( ( 3\nu^{13} - 20\nu^{9} + 136\nu^{5} + 19\nu ) / 12 \) (3v^13 - 20v^9 + 136v^5 + 19v) / 12
\(\beta_{3}\)	\(=\)	\( ( -7\nu^{14} + 48\nu^{10} - 330\nu^{6} + \nu^{2} ) / 18 \) (-7v^14 + 48v^10 - 330*v^6 + v^2) / 18
\(\beta_{4}\)	\(=\)	\( ( -15\nu^{15} + 104\nu^{11} - 712\nu^{7} + 53\nu^{3} ) / 12 \) (-15v^15 + 104v^11 - 712v^7 + 53v^3) / 12
\(\beta_{5}\)	\(=\)	\( ( 9\nu^{14} - 64\nu^{10} + 440\nu^{6} - 127\nu^{2} ) / 12 \) (9v^14 - 64v^10 + 440v^6 - 127v^2) / 12
\(\beta_{6}\)	\(=\)	\( ( 17\nu^{13} - 120\nu^{9} + 816\nu^{5} - 119\nu ) / 12 \) (17v^13 - 120v^9 + 816v^5 - 119v) / 12
\(\beta_{7}\)	\(=\)	\( ( -7\nu^{12} + 48\nu^{8} - 336\nu^{4} + 25 ) / 12 \) (-7v^12 + 48v^8 - 336*v^4 + 25) / 12
\(\beta_{8}\)	\(=\)	\( ( 3\nu^{15} - 126\nu^{14} - 2\nu^{12} + 864\nu^{10} - 5904\nu^{6} + 915\nu^{3} + 18\nu^{2} - 322 ) / 72 \) (3v^15 - 126v^14 - 2v^12 + 864v^10 - 5904v^6 + 915v^3 + 18*v^2 - 322) / 72
\(\beta_{9}\)	\(=\)	\( ( - 126 \nu^{14} + 51 \nu^{13} + 2 \nu^{12} + 864 \nu^{10} - 360 \nu^{9} - 5904 \nu^{6} + 2448 \nu^{5} + \cdots + 322 ) / 72 \) (-126v^14 + 51v^13 + 2v^12 + 864v^10 - 360v^9 - 5904v^6 + 2448v^5 + 18v^2 - 357*v + 322) / 72
\(\beta_{10}\)	\(=\)	\( ( 134 \nu^{15} + 9 \nu^{14} - 26 \nu^{13} - 928 \nu^{11} - 64 \nu^{10} + 176 \nu^{9} + 6368 \nu^{7} + \cdots - 170 \nu ) / 24 \) (134v^15 + 9v^14 - 26v^13 - 928v^11 - 64v^10 + 176v^9 + 6368v^7 + 440v^6 - 1216v^5 - 474v^3 - 127v^2 - 170v) / 24
\(\beta_{11}\)	\(=\)	\( ( 134 \nu^{15} + 26 \nu^{13} - 7 \nu^{12} - 928 \nu^{11} - 176 \nu^{9} + 48 \nu^{8} + 6368 \nu^{7} + \cdots + 25 ) / 24 \) (134v^15 + 26v^13 - 7v^12 - 928v^11 - 176v^9 + 48v^8 + 6368v^7 + 1216v^5 - 336v^4 - 474v^3 + 170*v + 25) / 24
\(\beta_{12}\)	\(=\)	\( ( \nu^{15} - 19\nu^{13} + 132\nu^{9} - 912\nu^{5} + 341\nu^{3} + 133\nu ) / 3 \) (v^15 - 19v^13 + 132v^9 - 912v^5 + 341v^3 + 133*v) / 3
\(\beta_{13}\)	\(=\)	\( ( -\nu^{15} - 19\nu^{13} + 132\nu^{9} - 912\nu^{5} - 341\nu^{3} + 133\nu ) / 3 \) (-v^15 - 19v^13 + 132v^9 - 912v^5 - 341v^3 + 133*v) / 3
\(\beta_{14}\)	\(=\)	\( ( -10\nu^{14} + 8\nu^{12} + 72\nu^{10} - 56\nu^{8} - 492\nu^{6} + 376\nu^{4} + 142\nu^{2} - 28 ) / 3 \) (-10v^14 + 8v^12 + 72v^10 - 56v^8 - 492v^6 + 376v^4 + 142*v^2 - 28) / 3
\(\beta_{15}\)	\(=\)	\( ( 10\nu^{14} + 8\nu^{12} - 72\nu^{10} - 56\nu^{8} + 492\nu^{6} + 376\nu^{4} - 142\nu^{2} - 28 ) / 3 \) (10v^14 + 8v^12 - 72v^10 - 56v^8 + 492v^6 + 376v^4 - 142*v^2 - 28) / 3

\(\nu\)	\(=\)	\( ( \beta_{13} + \beta_{12} + 6\beta_{11} - 6\beta_{10} - 3\beta_{7} + 4\beta_{6} + 3\beta_{5} - 24\beta_{2} ) / 96 \) (b13 + b12 + 6b11 - 6b10 - 3b7 + 4b6 + 3b5 - 24b2) / 96
\(\nu^{2}\)	\(=\)	\( ( \beta_{15} - \beta_{14} + 2\beta_{9} + 2\beta_{8} - \beta_{6} - 12\beta_{5} - 24\beta_{3} + \beta_1 ) / 32 \) (b15 - b14 + 2b9 + 2b8 - b6 - 12b5 - 24b3 + b1) / 32
\(\nu^{3}\)	\(=\)	\( ( -\beta_{13} + \beta_{12} + 8\beta_1 ) / 24 \) (-b13 + b12 + 8*b1) / 24
\(\nu^{4}\)	\(=\)	\( ( -3\beta_{15} - 3\beta_{14} - 6\beta_{9} + 6\beta_{8} - 28\beta_{7} + 3\beta_{6} + 3\beta _1 + 56 ) / 32 \) (-3b15 - 3b14 - 6b9 + 6b8 - 28b7 + 3b6 + 3*b1 + 56) / 32
\(\nu^{5}\)	\(=\)	\( ( - 5 \beta_{13} - 5 \beta_{12} + 30 \beta_{11} - 30 \beta_{10} - 15 \beta_{7} - 44 \beta_{6} + \cdots - 264 \beta_{2} ) / 96 \) (-5b13 - 5b12 + 30b11 - 30b10 - 15b7 - 44b6 + 15b5 - 264b2) / 96
\(\nu^{6}\)	\(=\)	\( ( 2\beta_{9} + 2\beta_{8} - \beta_{6} - 18\beta_{3} + \beta_1 ) / 2 \) (2b9 + 2b8 - b6 - 18*b3 + b1) / 2
\(\nu^{7}\)	\(=\)	\( ( - 13 \beta_{13} + 13 \beta_{12} + 78 \beta_{11} + 78 \beta_{10} - 39 \beta_{7} - 39 \beta_{5} + \cdots + 116 \beta_1 ) / 96 \) (-13b13 + 13b12 + 78b11 + 78b10 - 39b7 - 39b5 + 696b4 + 116b1) / 96
\(\nu^{8}\)	\(=\)	\( ( -21\beta_{15} - 21\beta_{14} + 42\beta_{9} - 42\beta_{8} - 188\beta_{7} - 21\beta_{6} - 21\beta _1 - 376 ) / 32 \) (-21b15 - 21b14 + 42b9 - 42b8 - 188b7 - 21b6 - 21*b1 - 376) / 32
\(\nu^{9}\)	\(=\)	\( ( -17\beta_{13} - 17\beta_{12} - 152\beta_{6} ) / 24 \) (-17b13 - 17b12 - 152*b6) / 24
\(\nu^{10}\)	\(=\)	\( ( - 55 \beta_{15} + 55 \beta_{14} + 110 \beta_{9} + 110 \beta_{8} - 55 \beta_{6} + 492 \beta_{5} + \cdots + 55 \beta_1 ) / 32 \) (-55b15 + 55b14 + 110b9 + 110b8 - 55b6 + 492b5 - 984b3 + 55b1) / 32
\(\nu^{11}\)	\(=\)	\( ( 89 \beta_{13} - 89 \beta_{12} + 534 \beta_{11} + 534 \beta_{10} - 267 \beta_{7} - 267 \beta_{5} + \cdots - 796 \beta_1 ) / 96 \) (89b13 - 89b12 + 534b11 + 534b10 - 267b7 - 267b5 + 4776b4 - 796b1) / 96
\(\nu^{12}\)	\(=\)	\( 18\beta_{9} - 18\beta_{8} - 9\beta_{6} - 9\beta _1 - 161 \) 18b9 - 18b8 - 9b6 - 9b1 - 161
\(\nu^{13}\)	\(=\)	\( ( - 233 \beta_{13} - 233 \beta_{12} - 1398 \beta_{11} + 1398 \beta_{10} + 699 \beta_{7} + \cdots + 12504 \beta_{2} ) / 96 \) (-233b13 - 233b12 - 1398b11 + 1398b10 + 699b7 - 2084b6 - 699b5 + 12504b2) / 96
\(\nu^{14}\)	\(=\)	\( ( - 377 \beta_{15} + 377 \beta_{14} - 754 \beta_{9} - 754 \beta_{8} + 377 \beta_{6} + 3372 \beta_{5} + \cdots - 377 \beta_1 ) / 32 \) (-377b15 + 377b14 - 754b9 - 754b8 + 377b6 + 3372b5 + 6744b3 - 377b1) / 32
\(\nu^{15}\)	\(=\)	\( ( 305\beta_{13} - 305\beta_{12} - 2728\beta_1 ) / 24 \) (305b13 - 305b12 - 2728*b1) / 24

\(n\)	\(101\)	\(177\)	\(351\)
\(\chi(n)\)	\(1\)	\(-\beta_{3}\)	\(-1\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 143.8
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{16} \) T^16
$3$	\( (T^{8} + 7682 T^{4} + 923521)^{2} \) (T^8 + 7682*T^4 + 923521)^2
$5$	\( T^{16} \) T^16
$7$	\( (T^{8} + 369152 T^{4} + 65536)^{2} \) (T^8 + 369152*T^4 + 65536)^2
$11$	\( (T^{4} + 534 T^{2} + 729)^{4} \) (T^4 + 534*T^2 + 729)^4
$13$	\( (T^{8} + 4539168 T^{4} + 136048896)^{2} \) (T^8 + 4539168*T^4 + 136048896)^2
$17$	\( (T^{8} + \cdots + 18\!\cdots\!61)^{2} \) (T^8 + 340535538*T^4 + 18338513836013361)^2
$19$	\( (T^{4} - 21606 T^{2} + 74597769)^{4} \) (T^4 - 21606*T^2 + 74597769)^4
$23$	\( (T^{8} + \cdots + 64\!\cdots\!16)^{2} \) (T^8 + 400414752*T^4 + 6456652178063616)^2
$29$	\( (T^{4} + 31752 T^{2} + 100881936)^{4} \) (T^4 + 31752*T^2 + 100881936)^4
$31$	\( (T^{4} + 81624 T^{2} + 282643344)^{4} \) (T^4 + 81624*T^2 + 282643344)^4
$37$	\( (T^{8} + \cdots + 22\!\cdots\!00)^{2} \) (T^8 + 6009292800*T^4 + 2285099025039360000)^2
$41$	\( (T^{2} + 246 T - 7911)^{8} \) (T^2 + 246*T - 7911)^8
$43$	\( (T^{8} + \cdots + 26\!\cdots\!16)^{2} \) (T^8 + 19356549632*T^4 + 2647105883634466816)^2
$47$	\( (T^{8} + \cdots + 36\!\cdots\!16)^{2} \) (T^8 + 97973662752*T^4 + 368639312486720553216)^2
$53$	\( (T^{8} + \cdots + 91\!\cdots\!56)^{2} \) (T^8 + 65790674208*T^4 + 911841173070969344256)^2
$59$	\( (T^{4} - 34776 T^{2} + 294877584)^{4} \) (T^4 - 34776*T^2 + 294877584)^4
$61$	\( (T^{2} + 248 T - 227984)^{8} \) (T^2 + 248*T - 227984)^8
$67$	\( (T^{8} + \cdots + 70\!\cdots\!21)^{2} \) (T^8 + 828180862082*T^4 + 70152808825804024321)^2
$71$	\( (T^{4} + 690456 T^{2} + 1503267984)^{4} \) (T^4 + 690456*T^2 + 1503267984)^4
$73$	\( (T^{8} + \cdots + 96\!\cdots\!81)^{2} \) (T^8 + 2209700178*T^4 + 965570381750779281)^2
$79$	\( (T^{4} - 1070400 T^{2} + 4199040000)^{4} \) (T^4 - 1070400*T^2 + 4199040000)^4
$83$	\( (T^{8} + \cdots + 18\!\cdots\!21)^{2} \) (T^8 + 984873038562*T^4 + 182180191634889963809121)^2
$89$	\( (T^{4} + 2361618 T^{2} + 19917394641)^{4} \) (T^4 + 2361618*T^2 + 19917394641)^4
$97$	\( (T^{8} + \cdots + 4902433652736)^{2} \) (T^8 + 585257292288*T^4 + 4902433652736)^2
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