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Properties

Label 1456.2.r.j.625.2
Level $1456$
Weight $2$
Character 1456.625
Analytic conductor $11.626$
Analytic rank $0$
Dimension $4$
Inner twists $2$

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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] = [1456,2,Mod(417,1456)] 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("1456.417"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1456, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 4, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1456 = 2^{4} \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1456.r (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,0,0,0,8] 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: \(11.6262185343\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{5})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} + 2x^{2} + x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 91)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 625.2
Root \(-0.309017 - 0.535233i\) of defining polynomial
Character \(\chi\) \(=\) 1456.625
Dual form 1456.2.r.j.417.2

$q$-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)
 
\(f(q)\) \(=\) \(q+(1.11803 + 1.93649i) q^{3} +(1.11803 - 1.93649i) q^{5} +(2.00000 + 1.73205i) q^{7} +(-1.00000 + 1.73205i) q^{9} +(-1.50000 - 2.59808i) q^{11} -1.00000 q^{13} +5.00000 q^{15} +(3.73607 + 6.47106i) q^{17} +(1.50000 - 2.59808i) q^{19} +(-1.11803 + 5.80948i) q^{21} +(-1.88197 + 3.25966i) q^{23} +2.23607 q^{27} -4.47214 q^{29} +(2.50000 + 4.33013i) q^{31} +(3.35410 - 5.80948i) q^{33} +(5.59017 - 1.93649i) q^{35} +(4.35410 - 7.54153i) q^{37} +(-1.11803 - 1.93649i) q^{39} +4.47214 q^{41} +8.00000 q^{43} +(2.23607 + 3.87298i) q^{45} +(0.736068 - 1.27491i) q^{47} +(1.00000 + 6.92820i) q^{49} +(-8.35410 + 14.4697i) q^{51} +(-0.736068 - 1.27491i) q^{53} -6.70820 q^{55} +6.70820 q^{57} +(3.73607 + 6.47106i) q^{59} +(-1.50000 + 2.59808i) q^{61} +(-5.00000 + 1.73205i) q^{63} +(-1.11803 + 1.93649i) q^{65} +(-1.50000 - 2.59808i) q^{67} -8.41641 q^{69} -8.94427 q^{71} +(-5.35410 - 9.27358i) q^{73} +(1.50000 - 7.79423i) q^{77} +(5.35410 - 9.27358i) q^{79} +(5.50000 + 9.52628i) q^{81} +16.7082 q^{85} +(-5.00000 - 8.66025i) q^{87} +(1.11803 - 1.93649i) q^{89} +(-2.00000 - 1.73205i) q^{91} +(-5.59017 + 9.68246i) q^{93} +(-3.35410 - 5.80948i) q^{95} -17.4164 q^{97} +6.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 8 q^{7} - 4 q^{9} - 6 q^{11} - 4 q^{13} + 20 q^{15} + 6 q^{17} + 6 q^{19} - 12 q^{23} + 10 q^{31} + 4 q^{37} + 32 q^{43} - 6 q^{47} + 4 q^{49} - 20 q^{51} + 6 q^{53} + 6 q^{59} - 6 q^{61} - 20 q^{63}+ \cdots + 24 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1456\mathbb{Z}\right)^\times\).

\(n\) \(561\) \(911\) \(1093\) \(1249\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(e\left(\frac{1}{3}\right)\)

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
\(3\) 1.11803 + 1.93649i 0.645497 + 1.11803i 0.984186 + 0.177136i \(0.0566831\pi\)
−0.338689 + 0.940898i \(0.609984\pi\)
\(4\) 0 0
\(5\) 1.11803 − 1.93649i 0.500000 − 0.866025i −0.500000 − 0.866025i \(-0.666667\pi\)
1.00000 \(0\)
\(6\) 0 0
\(7\) 2.00000 + 1.73205i 0.755929 + 0.654654i
\(8\) 0 0
\(9\) −1.00000 + 1.73205i −0.333333 + 0.577350i
\(10\) 0 0
\(11\) −1.50000 − 2.59808i −0.452267 − 0.783349i 0.546259 − 0.837616i \(-0.316051\pi\)
−0.998526 + 0.0542666i \(0.982718\pi\)
\(12\) 0 0
\(13\) −1.00000 −0.277350
\(14\) 0 0
\(15\) 5.00000 1.29099
\(16\) 0 0
\(17\) 3.73607 + 6.47106i 0.906130 + 1.56946i 0.819394 + 0.573231i \(0.194310\pi\)
0.0867359 + 0.996231i \(0.472356\pi\)
\(18\) 0 0
\(19\) 1.50000 − 2.59808i 0.344124 − 0.596040i −0.641071 − 0.767482i \(-0.721509\pi\)
0.985194 + 0.171442i \(0.0548427\pi\)
\(20\) 0 0
\(21\) −1.11803 + 5.80948i −0.243975 + 1.26773i
\(22\) 0 0
\(23\) −1.88197 + 3.25966i −0.392417 + 0.679686i −0.992768 − 0.120051i \(-0.961694\pi\)
0.600351 + 0.799737i \(0.295028\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 2.23607 0.430331
\(28\) 0 0
\(29\) −4.47214 −0.830455 −0.415227 − 0.909718i \(-0.636298\pi\)
−0.415227 + 0.909718i \(0.636298\pi\)
\(30\) 0 0
\(31\) 2.50000 + 4.33013i 0.449013 + 0.777714i 0.998322 − 0.0579057i \(-0.0184423\pi\)
−0.549309 + 0.835619i \(0.685109\pi\)
\(32\) 0 0
\(33\) 3.35410 − 5.80948i 0.583874 − 1.01130i
\(34\) 0 0
\(35\) 5.59017 − 1.93649i 0.944911 − 0.327327i
\(36\) 0 0
\(37\) 4.35410 − 7.54153i 0.715810 − 1.23982i −0.246836 − 0.969057i \(-0.579391\pi\)
0.962646 − 0.270762i \(-0.0872757\pi\)
\(38\) 0 0
\(39\) −1.11803 − 1.93649i −0.179029 − 0.310087i
\(40\) 0 0
\(41\) 4.47214 0.698430 0.349215 − 0.937043i \(-0.386448\pi\)
0.349215 + 0.937043i \(0.386448\pi\)
\(42\) 0 0
\(43\) 8.00000 1.21999 0.609994 − 0.792406i \(-0.291172\pi\)
0.609994 + 0.792406i \(0.291172\pi\)
\(44\) 0 0
\(45\) 2.23607 + 3.87298i 0.333333 + 0.577350i
\(46\) 0 0
\(47\) 0.736068 − 1.27491i 0.107367 − 0.185964i −0.807336 − 0.590092i \(-0.799091\pi\)
0.914703 + 0.404128i \(0.132425\pi\)
\(48\) 0 0
\(49\) 1.00000 + 6.92820i 0.142857 + 0.989743i
\(50\) 0 0
\(51\) −8.35410 + 14.4697i −1.16981 + 2.02617i
\(52\) 0 0
\(53\) −0.736068 − 1.27491i −0.101107 − 0.175122i 0.811034 − 0.584999i \(-0.198905\pi\)
−0.912141 + 0.409877i \(0.865572\pi\)
\(54\) 0 0
\(55\) −6.70820 −0.904534
\(56\) 0 0
\(57\) 6.70820 0.888523
\(58\) 0 0
\(59\) 3.73607 + 6.47106i 0.486395 + 0.842460i 0.999878 − 0.0156395i \(-0.00497842\pi\)
−0.513483 + 0.858100i \(0.671645\pi\)
\(60\) 0 0
\(61\) −1.50000 + 2.59808i −0.192055 + 0.332650i −0.945931 − 0.324367i \(-0.894849\pi\)
0.753876 + 0.657017i \(0.228182\pi\)
\(62\) 0 0
\(63\) −5.00000 + 1.73205i −0.629941 + 0.218218i
\(64\) 0 0
\(65\) −1.11803 + 1.93649i −0.138675 + 0.240192i
\(66\) 0 0
\(67\) −1.50000 − 2.59808i −0.183254 − 0.317406i 0.759733 − 0.650236i \(-0.225330\pi\)
−0.942987 + 0.332830i \(0.891996\pi\)
\(68\) 0 0
\(69\) −8.41641 −1.01322
\(70\) 0 0
\(71\) −8.94427 −1.06149 −0.530745 − 0.847532i \(-0.678088\pi\)
−0.530745 + 0.847532i \(0.678088\pi\)
\(72\) 0 0
\(73\) −5.35410 − 9.27358i −0.626650 − 1.08539i −0.988219 − 0.153045i \(-0.951092\pi\)
0.361569 − 0.932345i \(-0.382241\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 1.50000 − 7.79423i 0.170941 − 0.888235i
\(78\) 0 0
\(79\) 5.35410 − 9.27358i 0.602384 − 1.04336i −0.390076 − 0.920783i \(-0.627551\pi\)
0.992459 − 0.122576i \(-0.0391155\pi\)
\(80\) 0 0
\(81\) 5.50000 + 9.52628i 0.611111 + 1.05848i
\(82\) 0 0
\(83\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 0 0
\(85\) 16.7082 1.81226
\(86\) 0 0
\(87\) −5.00000 − 8.66025i −0.536056 − 0.928477i
\(88\) 0 0
\(89\) 1.11803 − 1.93649i 0.118511 − 0.205268i −0.800667 − 0.599110i \(-0.795521\pi\)
0.919178 + 0.393842i \(0.128854\pi\)
\(90\) 0 0
\(91\) −2.00000 − 1.73205i −0.209657 − 0.181568i
\(92\) 0 0
\(93\) −5.59017 + 9.68246i −0.579674 + 1.00402i
\(94\) 0 0
\(95\) −3.35410 − 5.80948i −0.344124 − 0.596040i
\(96\) 0 0
\(97\) −17.4164 −1.76837 −0.884184 − 0.467139i \(-0.845285\pi\)
−0.884184 + 0.467139i \(0.845285\pi\)
\(98\) 0 0
\(99\) 6.00000 0.603023
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 1456.2.r.j.625.2 4
4.3 odd 2 91.2.e.b.79.1 yes 4
7.4 even 3 inner 1456.2.r.j.417.2 4
12.11 even 2 819.2.j.c.352.2 4
28.3 even 6 637.2.e.h.508.1 4
28.11 odd 6 91.2.e.b.53.1 ✓ 4
28.19 even 6 637.2.a.e.1.2 2
28.23 odd 6 637.2.a.f.1.2 2
28.27 even 2 637.2.e.h.79.1 4
52.51 odd 2 1183.2.e.d.170.2 4
84.11 even 6 819.2.j.c.235.2 4
84.23 even 6 5733.2.a.v.1.1 2
84.47 odd 6 5733.2.a.w.1.1 2
364.51 odd 6 8281.2.a.z.1.1 2
364.103 even 6 8281.2.a.ba.1.1 2
364.207 odd 6 1183.2.e.d.508.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
91.2.e.b.53.1 ✓ 4 28.11 odd 6
91.2.e.b.79.1 yes 4 4.3 odd 2
637.2.a.e.1.2 2 28.19 even 6
637.2.a.f.1.2 2 28.23 odd 6
637.2.e.h.79.1 4 28.27 even 2
637.2.e.h.508.1 4 28.3 even 6
819.2.j.c.235.2 4 84.11 even 6
819.2.j.c.352.2 4 12.11 even 2
1183.2.e.d.170.2 4 52.51 odd 2
1183.2.e.d.508.2 4 364.207 odd 6
1456.2.r.j.417.2 4 7.4 even 3 inner
1456.2.r.j.625.2 4 1.1 even 1 trivial
5733.2.a.v.1.1 2 84.23 even 6
5733.2.a.w.1.1 2 84.47 odd 6
8281.2.a.z.1.1 2 364.51 odd 6
8281.2.a.ba.1.1 2 364.103 even 6