↑

Properties

Label 28.8.a.b.1.1
Level $28$
Weight $8$
Character 28.1
Self dual yes
Analytic conductor $8.747$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [28,8,Mod(1,28)] 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("28.1"); S:= CuspForms(chi, 8); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(28, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 8, names="a")
 
Level: \( N \) \(=\) \( 28 = 2^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 28.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,14] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(8.74678071356\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{1009}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 252 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(16.3824\) of defining polynomial
Character \(\chi\) \(=\) 28.1

$q$-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)
 
\(f(q)\) \(=\) \(q-24.7648 q^{3} +202.412 q^{5} -343.000 q^{7} -1573.71 q^{9} -7527.19 q^{11} -1700.28 q^{13} -5012.69 q^{15} -2481.86 q^{17} -39816.0 q^{19} +8494.31 q^{21} +25560.1 q^{23} -37154.2 q^{25} +93133.0 q^{27} +148830. q^{29} -33538.4 q^{31} +186409. q^{33} -69427.4 q^{35} +400557. q^{37} +42107.1 q^{39} -362750. q^{41} -324652. q^{43} -318538. q^{45} -708899. q^{47} +117649. q^{49} +61462.7 q^{51} -185924. q^{53} -1.52360e6 q^{55} +986034. q^{57} -1.19372e6 q^{59} +2.51204e6 q^{61} +539781. q^{63} -344158. q^{65} -2.85983e6 q^{67} -632989. q^{69} +3.22083e6 q^{71} +5.01287e6 q^{73} +920116. q^{75} +2.58182e6 q^{77} +5.94208e6 q^{79} +1.13528e6 q^{81} -1.02006e7 q^{83} -502359. q^{85} -3.68573e6 q^{87} +1.85373e6 q^{89} +583197. q^{91} +830571. q^{93} -8.05925e6 q^{95} -1.52142e7 q^{97} +1.18456e7 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 14 q^{3} - 294 q^{5} - 686 q^{7} - 2258 q^{9} - 3492 q^{11} - 16170 q^{13} - 24256 q^{15} - 29232 q^{17} - 3206 q^{19} - 4802 q^{21} - 9360 q^{23} + 131146 q^{25} - 18172 q^{27} + 184704 q^{29} + 165060 q^{31}+ \cdots + 9084332 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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
\(3\) −24.7648 −0.529553 −0.264777 − 0.964310i \(-0.585298\pi\)
−0.264777 + 0.964310i \(0.585298\pi\)
\(4\) 0 0
\(5\) 202.412 0.724172 0.362086 − 0.932145i \(-0.382065\pi\)
0.362086 + 0.932145i \(0.382065\pi\)
\(6\) 0 0
\(7\) −343.000 −0.377964
\(8\) 0 0
\(9\) −1573.71 −0.719573
\(10\) 0 0
\(11\) −7527.19 −1.70513 −0.852567 − 0.522619i \(-0.824955\pi\)
−0.852567 + 0.522619i \(0.824955\pi\)
\(12\) 0 0
\(13\) −1700.28 −0.214644 −0.107322 − 0.994224i \(-0.534228\pi\)
−0.107322 + 0.994224i \(0.534228\pi\)
\(14\) 0 0
\(15\) −5012.69 −0.383488
\(16\) 0 0
\(17\) −2481.86 −0.122520 −0.0612599 − 0.998122i \(-0.519512\pi\)
−0.0612599 + 0.998122i \(0.519512\pi\)
\(18\) 0 0
\(19\) −39816.0 −1.33174 −0.665871 − 0.746067i \(-0.731940\pi\)
−0.665871 + 0.746067i \(0.731940\pi\)
\(20\) 0 0
\(21\) 8494.31 0.200152
\(22\) 0 0
\(23\) 25560.1 0.438041 0.219020 − 0.975720i \(-0.429714\pi\)
0.219020 + 0.975720i \(0.429714\pi\)
\(24\) 0 0
\(25\) −37154.2 −0.475574
\(26\) 0 0
\(27\) 93133.0 0.910606
\(28\) 0 0
\(29\) 148830. 1.13317 0.566587 − 0.824002i \(-0.308263\pi\)
0.566587 + 0.824002i \(0.308263\pi\)
\(30\) 0 0
\(31\) −33538.4 −0.202198 −0.101099 − 0.994876i \(-0.532236\pi\)
−0.101099 + 0.994876i \(0.532236\pi\)
\(32\) 0 0
\(33\) 186409. 0.902959
\(34\) 0 0
\(35\) −69427.4 −0.273711
\(36\) 0 0
\(37\) 400557. 1.30005 0.650023 − 0.759915i \(-0.274759\pi\)
0.650023 + 0.759915i \(0.274759\pi\)
\(38\) 0 0
\(39\) 42107.1 0.113666
\(40\) 0 0
\(41\) −362750. −0.821985 −0.410992 − 0.911639i \(-0.634818\pi\)
−0.410992 + 0.911639i \(0.634818\pi\)
\(42\) 0 0
\(43\) −324652. −0.622699 −0.311350 − 0.950295i \(-0.600781\pi\)
−0.311350 + 0.950295i \(0.600781\pi\)
\(44\) 0 0
\(45\) −318538. −0.521095
\(46\) 0 0
\(47\) −708899. −0.995960 −0.497980 − 0.867188i \(-0.665925\pi\)
−0.497980 + 0.867188i \(0.665925\pi\)
\(48\) 0 0
\(49\) 117649. 0.142857
\(50\) 0 0
\(51\) 61462.7 0.0648808
\(52\) 0 0
\(53\) −185924. −0.171542 −0.0857709 − 0.996315i \(-0.527335\pi\)
−0.0857709 + 0.996315i \(0.527335\pi\)
\(54\) 0 0
\(55\) −1.52360e6 −1.23481
\(56\) 0 0
\(57\) 986034. 0.705228
\(58\) 0 0
\(59\) −1.19372e6 −0.756692 −0.378346 − 0.925664i \(-0.623507\pi\)
−0.378346 + 0.925664i \(0.623507\pi\)
\(60\) 0 0
\(61\) 2.51204e6 1.41701 0.708504 − 0.705707i \(-0.249370\pi\)
0.708504 + 0.705707i \(0.249370\pi\)
\(62\) 0 0
\(63\) 539781. 0.271973
\(64\) 0 0
\(65\) −344158. −0.155440
\(66\) 0 0
\(67\) −2.85983e6 −1.16166 −0.580829 − 0.814025i \(-0.697272\pi\)
−0.580829 + 0.814025i \(0.697272\pi\)
\(68\) 0 0
\(69\) −632989. −0.231966
\(70\) 0 0
\(71\) 3.22083e6 1.06798 0.533990 − 0.845491i \(-0.320692\pi\)
0.533990 + 0.845491i \(0.320692\pi\)
\(72\) 0 0
\(73\) 5.01287e6 1.50819 0.754095 − 0.656765i \(-0.228076\pi\)
0.754095 + 0.656765i \(0.228076\pi\)
\(74\) 0 0
\(75\) 920116. 0.251842
\(76\) 0 0
\(77\) 2.58182e6 0.644480
\(78\) 0 0
\(79\) 5.94208e6 1.35595 0.677975 − 0.735085i \(-0.262858\pi\)
0.677975 + 0.735085i \(0.262858\pi\)
\(80\) 0 0
\(81\) 1.13528e6 0.237359
\(82\) 0 0
\(83\) −1.02006e7 −1.95819 −0.979094 − 0.203407i \(-0.934799\pi\)
−0.979094 + 0.203407i \(0.934799\pi\)
\(84\) 0 0
\(85\) −502359. −0.0887255
\(86\) 0 0
\(87\) −3.68573e6 −0.600076
\(88\) 0 0
\(89\) 1.85373e6 0.278729 0.139365 − 0.990241i \(-0.455494\pi\)
0.139365 + 0.990241i \(0.455494\pi\)
\(90\) 0 0
\(91\) 583197. 0.0811280
\(92\) 0 0
\(93\) 830571. 0.107075
\(94\) 0 0
\(95\) −8.05925e6 −0.964411
\(96\) 0 0
\(97\) −1.52142e7 −1.69257 −0.846287 − 0.532727i \(-0.821167\pi\)
−0.846287 + 0.532727i \(0.821167\pi\)
\(98\) 0 0
\(99\) 1.18456e7 1.22697
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 28.8.a.b.1.1 ✓ 2
3.2 odd 2 252.8.a.f.1.1 2
4.3 odd 2 112.8.a.h.1.2 2
7.2 even 3 196.8.e.b.165.2 4
7.3 odd 6 196.8.e.c.177.1 4
7.4 even 3 196.8.e.b.177.2 4
7.5 odd 6 196.8.e.c.165.1 4
7.6 odd 2 196.8.a.a.1.2 2
8.3 odd 2 448.8.a.q.1.1 2
8.5 even 2 448.8.a.o.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
28.8.a.b.1.1 ✓ 2 1.1 even 1 trivial
112.8.a.h.1.2 2 4.3 odd 2
196.8.a.a.1.2 2 7.6 odd 2
196.8.e.b.165.2 4 7.2 even 3
196.8.e.b.177.2 4 7.4 even 3
196.8.e.c.165.1 4 7.5 odd 6
196.8.e.c.177.1 4 7.3 odd 6
252.8.a.f.1.1 2 3.2 odd 2
448.8.a.o.1.2 2 8.5 even 2
448.8.a.q.1.1 2 8.3 odd 2