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Properties

Label 800.2.o.h.143.1
Level $800$
Weight $2$
Character 800.143
Analytic conductor $6.388$
Analytic rank $0$
Dimension $8$
Inner twists $8$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [800,2,Mod(143,800)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("800.143"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(800, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([2, 2, 3])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 800 = 2^{5} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 800.o (of order \(4\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,0,0,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(6.38803216170\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: \(\Q(i, \sqrt{6}, \sqrt{10})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 7x^{4} + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{8} \)
Twist minimal: no (minimal twist has level 200)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 143.1
Root \(-0.178197 + 1.40294i\) of defining polynomial
Character \(\chi\) \(=\) 800.143
Dual form 800.2.o.h.207.1

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-1.22474 - 1.22474i) q^{3} +(-3.16228 - 3.16228i) q^{7} +1.00000 q^{11} +(3.16228 - 3.16228i) q^{13} +(-3.67423 + 3.67423i) q^{17} +3.00000i q^{19} +7.74597i q^{21} +(-3.67423 + 3.67423i) q^{27} -7.74597 q^{29} +(-1.22474 - 1.22474i) q^{33} +(-3.16228 - 3.16228i) q^{37} -7.74597 q^{39} -1.00000 q^{41} +(2.44949 + 2.44949i) q^{43} +(-3.16228 - 3.16228i) q^{47} +13.0000i q^{49} +9.00000 q^{51} +(-6.32456 + 6.32456i) q^{53} +(3.67423 - 3.67423i) q^{57} +4.00000i q^{59} -7.74597i q^{61} +(3.67423 - 3.67423i) q^{67} +7.74597i q^{71} +(1.22474 + 1.22474i) q^{73} +(-3.16228 - 3.16228i) q^{77} -7.74597 q^{79} +9.00000 q^{81} +(-1.22474 - 1.22474i) q^{83} +(9.48683 + 9.48683i) q^{87} -13.0000i q^{89} -20.0000 q^{91} +(-4.89898 + 4.89898i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{11} - 8 q^{41} + 72 q^{51} + 72 q^{81} - 160 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/800\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(351\) \(577\)
\(\chi(n)\) \(-1\) \(-1\) \(e\left(\frac{3}{4}\right)\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\). You can download additional coefficients here.



Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −1.22474 − 1.22474i −0.707107 − 0.707107i 0.258819 − 0.965926i \(-0.416667\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) −3.16228 − 3.16228i −1.19523 − 1.19523i −0.975579 − 0.219650i \(-0.929509\pi\)
−0.219650 − 0.975579i \(-0.570491\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 1.00000 0.301511 0.150756 − 0.988571i \(-0.451829\pi\)
0.150756 + 0.988571i \(0.451829\pi\)
\(12\) 0 0
\(13\) 3.16228 − 3.16228i 0.877058 − 0.877058i −0.116171 − 0.993229i \(-0.537062\pi\)
0.993229 + 0.116171i \(0.0370621\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −3.67423 + 3.67423i −0.891133 + 0.891133i −0.994630 − 0.103497i \(-0.966997\pi\)
0.103497 + 0.994630i \(0.466997\pi\)
\(18\) 0 0
\(19\) 3.00000i 0.688247i 0.938924 + 0.344124i \(0.111824\pi\)
−0.938924 + 0.344124i \(0.888176\pi\)
\(20\) 0 0
\(21\) 7.74597i 1.69031i
\(22\) 0 0
\(23\) 0 0 −0.707107 − 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) −3.67423 + 3.67423i −0.707107 + 0.707107i
\(28\) 0 0
\(29\) −7.74597 −1.43839 −0.719195 − 0.694808i \(-0.755489\pi\)
−0.719195 + 0.694808i \(0.755489\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(32\) 0 0
\(33\) −1.22474 − 1.22474i −0.213201 − 0.213201i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −3.16228 − 3.16228i −0.519875 − 0.519875i 0.397658 − 0.917534i \(-0.369823\pi\)
−0.917534 + 0.397658i \(0.869823\pi\)
\(38\) 0 0
\(39\) −7.74597 −1.24035
\(40\) 0 0
\(41\) −1.00000 −0.156174 −0.0780869 − 0.996947i \(-0.524881\pi\)
−0.0780869 + 0.996947i \(0.524881\pi\)
\(42\) 0 0
\(43\) 2.44949 + 2.44949i 0.373544 + 0.373544i 0.868766 − 0.495222i \(-0.164913\pi\)
−0.495222 + 0.868766i \(0.664913\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −3.16228 − 3.16228i −0.461266 − 0.461266i 0.437805 − 0.899070i \(-0.355756\pi\)
−0.899070 + 0.437805i \(0.855756\pi\)
\(48\) 0 0
\(49\) 13.0000i 1.85714i
\(50\) 0 0
\(51\) 9.00000 1.26025
\(52\) 0 0
\(53\) −6.32456 + 6.32456i −0.868744 + 0.868744i −0.992333 − 0.123589i \(-0.960560\pi\)
0.123589 + 0.992333i \(0.460560\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 3.67423 − 3.67423i 0.486664 − 0.486664i
\(58\) 0 0
\(59\) 4.00000i 0.520756i 0.965507 + 0.260378i \(0.0838471\pi\)
−0.965507 + 0.260378i \(0.916153\pi\)
\(60\) 0 0
\(61\) − 7.74597i − 0.991769i −0.868388 − 0.495885i \(-0.834844\pi\)
0.868388 − 0.495885i \(-0.165156\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 3.67423 − 3.67423i 0.448879 − 0.448879i −0.446103 − 0.894982i \(-0.647188\pi\)
0.894982 + 0.446103i \(0.147188\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 7.74597i 0.919277i 0.888106 + 0.459639i \(0.152021\pi\)
−0.888106 + 0.459639i \(0.847979\pi\)
\(72\) 0 0
\(73\) 1.22474 + 1.22474i 0.143346 + 0.143346i 0.775138 − 0.631792i \(-0.217680\pi\)
−0.631792 + 0.775138i \(0.717680\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −3.16228 − 3.16228i −0.360375 − 0.360375i
\(78\) 0 0
\(79\) −7.74597 −0.871489 −0.435745 − 0.900070i \(-0.643515\pi\)
−0.435745 + 0.900070i \(0.643515\pi\)
\(80\) 0 0
\(81\) 9.00000 1.00000
\(82\) 0 0
\(83\) −1.22474 − 1.22474i −0.134433 − 0.134433i 0.636688 − 0.771121i \(-0.280304\pi\)
−0.771121 + 0.636688i \(0.780304\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 9.48683 + 9.48683i 1.01710 + 1.01710i
\(88\) 0 0
\(89\) − 13.0000i − 1.37800i −0.724763 − 0.688999i \(-0.758051\pi\)
0.724763 − 0.688999i \(-0.241949\pi\)
\(90\) 0 0
\(91\) −20.0000 −2.09657
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −4.89898 + 4.89898i −0.497416 + 0.497416i −0.910633 − 0.413217i \(-0.864405\pi\)
0.413217 + 0.910633i \(0.364405\pi\)
\(98\) 0 0
\(99\) 0 0
Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 800.2.o.h.143.1 8
4.3 odd 2 200.2.k.g.43.2 yes 8
5.2 odd 4 inner 800.2.o.h.207.2 8
5.3 odd 4 inner 800.2.o.h.207.3 8
5.4 even 2 inner 800.2.o.h.143.4 8
8.3 odd 2 inner 800.2.o.h.143.2 8
8.5 even 2 200.2.k.g.43.4 yes 8
20.3 even 4 200.2.k.g.107.1 yes 8
20.7 even 4 200.2.k.g.107.4 yes 8
20.19 odd 2 200.2.k.g.43.3 yes 8
40.3 even 4 inner 800.2.o.h.207.4 8
40.13 odd 4 200.2.k.g.107.3 yes 8
40.19 odd 2 inner 800.2.o.h.143.3 8
40.27 even 4 inner 800.2.o.h.207.1 8
40.29 even 2 200.2.k.g.43.1 ✓ 8
40.37 odd 4 200.2.k.g.107.2 yes 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
200.2.k.g.43.1 ✓ 8 40.29 even 2
200.2.k.g.43.2 yes 8 4.3 odd 2
200.2.k.g.43.3 yes 8 20.19 odd 2
200.2.k.g.43.4 yes 8 8.5 even 2
200.2.k.g.107.1 yes 8 20.3 even 4
200.2.k.g.107.2 yes 8 40.37 odd 4
200.2.k.g.107.3 yes 8 40.13 odd 4
200.2.k.g.107.4 yes 8 20.7 even 4
800.2.o.h.143.1 8 1.1 even 1 trivial
800.2.o.h.143.2 8 8.3 odd 2 inner
800.2.o.h.143.3 8 40.19 odd 2 inner
800.2.o.h.143.4 8 5.4 even 2 inner
800.2.o.h.207.1 8 40.27 even 4 inner
800.2.o.h.207.2 8 5.2 odd 4 inner
800.2.o.h.207.3 8 5.3 odd 4 inner
800.2.o.h.207.4 8 40.3 even 4 inner