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Properties

Label 1456.2.r.p.417.2
Level $1456$
Weight $2$
Character 1456.417
Analytic conductor $11.626$
Analytic rank $0$
Dimension $10$
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 = [10,0,0,0,-2,0,-1] 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: \(10\)
Relative dimension: \(5\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - x^{9} + 8x^{8} + 7x^{7} + 41x^{6} + 18x^{5} + 58x^{4} + 28x^{3} + 64x^{2} + 16x + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2}\cdot 3 \)
Twist minimal: no (minimal twist has level 91)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 417.2
Root \(-0.862625 - 1.49411i\) of defining polynomial
Character \(\chi\) \(=\) 1456.417
Dual form 1456.2.r.p.625.2

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.673208 + 1.16603i) q^{3} +(-1.09358 - 1.89414i) q^{5} +(2.19729 + 1.47375i) q^{7} +(0.593582 + 1.02811i) q^{9} +(-0.524077 + 0.907729i) q^{11} +1.00000 q^{13} +2.94483 q^{15} +(2.64562 - 4.58236i) q^{17} +(0.378453 + 0.655500i) q^{19} +(-3.19767 + 1.56996i) q^{21} +(0.326792 + 0.566020i) q^{23} +(0.108157 - 0.187333i) q^{25} -5.63766 q^{27} -3.10408 q^{29} +(0.513956 - 0.890198i) q^{31} +(-0.705626 - 1.22218i) q^{33} +(0.388575 - 5.77363i) q^{35} +(5.44661 + 9.43381i) q^{37} +(-0.673208 + 1.16603i) q^{39} +7.32040 q^{41} -0.887771 q^{43} +(1.29826 - 2.24865i) q^{45} +(1.16875 + 2.02434i) q^{47} +(2.65613 + 6.47650i) q^{49} +(3.56211 + 6.16976i) q^{51} +(-2.44407 + 4.23325i) q^{53} +2.29249 q^{55} -1.01911 q^{57} +(-0.524077 + 0.907729i) q^{59} +(6.24989 + 10.8251i) q^{61} +(-0.210913 + 3.13385i) q^{63} +(-1.09358 - 1.89414i) q^{65} +(2.23944 - 3.87883i) q^{67} -0.879996 q^{69} +6.60274 q^{71} +(4.14174 - 7.17370i) q^{73} +(0.145624 + 0.252229i) q^{75} +(-2.48931 + 1.22218i) q^{77} +(1.07007 + 1.85342i) q^{79} +(2.01457 - 3.48935i) q^{81} +6.66558 q^{83} -11.5728 q^{85} +(2.08969 - 3.61946i) q^{87} +(2.88388 + 4.99503i) q^{89} +(2.19729 + 1.47375i) q^{91} +(0.691998 + 1.19858i) q^{93} +(0.827739 - 1.43369i) q^{95} -2.88777 q^{97} -1.24433 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 2 q^{5} - q^{7} - 3 q^{9} + 11 q^{11} + 10 q^{13} + 5 q^{17} + 9 q^{19} + 2 q^{21} + 10 q^{23} - 9 q^{25} - 6 q^{29} - 6 q^{31} - 8 q^{33} + 4 q^{35} - 4 q^{37} + 28 q^{41} - 4 q^{43} + 32 q^{45}+ \cdots - 22 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(561\) \(911\) \(1093\) \(1249\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(e\left(\frac{2}{3}\right)\)

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
\(3\) −0.673208 + 1.16603i −0.388677 + 0.673208i −0.992272 − 0.124083i \(-0.960401\pi\)
0.603595 + 0.797291i \(0.293734\pi\)
\(4\) 0 0
\(5\) −1.09358 − 1.89414i −0.489065 − 0.847085i 0.510856 − 0.859666i \(-0.329328\pi\)
−0.999921 + 0.0125813i \(0.995995\pi\)
\(6\) 0 0
\(7\) 2.19729 + 1.47375i 0.830496 + 0.557025i
\(8\) 0 0
\(9\) 0.593582 + 1.02811i 0.197861 + 0.342705i
\(10\) 0 0
\(11\) −0.524077 + 0.907729i −0.158015 + 0.273691i −0.934153 − 0.356873i \(-0.883843\pi\)
0.776138 + 0.630564i \(0.217176\pi\)
\(12\) 0 0
\(13\) 1.00000 0.277350
\(14\) 0 0
\(15\) 2.94483 0.760352
\(16\) 0 0
\(17\) 2.64562 − 4.58236i 0.641658 − 1.11138i −0.343404 − 0.939188i \(-0.611580\pi\)
0.985063 − 0.172197i \(-0.0550865\pi\)
\(18\) 0 0
\(19\) 0.378453 + 0.655500i 0.0868231 + 0.150382i 0.906167 − 0.422921i \(-0.138995\pi\)
−0.819344 + 0.573303i \(0.805662\pi\)
\(20\) 0 0
\(21\) −3.19767 + 1.56996i −0.697788 + 0.342594i
\(22\) 0 0
\(23\) 0.326792 + 0.566020i 0.0681408 + 0.118023i 0.898083 − 0.439826i \(-0.144960\pi\)
−0.829942 + 0.557850i \(0.811627\pi\)
\(24\) 0 0
\(25\) 0.108157 − 0.187333i 0.0216314 − 0.0374667i
\(26\) 0 0
\(27\) −5.63766 −1.08497
\(28\) 0 0
\(29\) −3.10408 −0.576414 −0.288207 − 0.957568i \(-0.593059\pi\)
−0.288207 + 0.957568i \(0.593059\pi\)
\(30\) 0 0
\(31\) 0.513956 − 0.890198i 0.0923092 − 0.159884i −0.816173 − 0.577807i \(-0.803909\pi\)
0.908482 + 0.417923i \(0.137242\pi\)
\(32\) 0 0
\(33\) −0.705626 − 1.22218i −0.122834 − 0.212754i
\(34\) 0 0
\(35\) 0.388575 − 5.77363i 0.0656811 − 0.975922i
\(36\) 0 0
\(37\) 5.44661 + 9.43381i 0.895418 + 1.55091i 0.833287 + 0.552841i \(0.186456\pi\)
0.0621309 + 0.998068i \(0.480210\pi\)
\(38\) 0 0
\(39\) −0.673208 + 1.16603i −0.107800 + 0.186714i
\(40\) 0 0
\(41\) 7.32040 1.14325 0.571627 − 0.820514i \(-0.306312\pi\)
0.571627 + 0.820514i \(0.306312\pi\)
\(42\) 0 0
\(43\) −0.887771 −0.135384 −0.0676919 − 0.997706i \(-0.521563\pi\)
−0.0676919 + 0.997706i \(0.521563\pi\)
\(44\) 0 0
\(45\) 1.29826 − 2.24865i 0.193533 − 0.335210i
\(46\) 0 0
\(47\) 1.16875 + 2.02434i 0.170480 + 0.295281i 0.938588 − 0.345040i \(-0.112135\pi\)
−0.768108 + 0.640321i \(0.778801\pi\)
\(48\) 0 0
\(49\) 2.65613 + 6.47650i 0.379447 + 0.925214i
\(50\) 0 0
\(51\) 3.56211 + 6.16976i 0.498795 + 0.863939i
\(52\) 0 0
\(53\) −2.44407 + 4.23325i −0.335719 + 0.581482i −0.983623 − 0.180240i \(-0.942313\pi\)
0.647904 + 0.761722i \(0.275646\pi\)
\(54\) 0 0
\(55\) 2.29249 0.309119
\(56\) 0 0
\(57\) −1.01911 −0.134985
\(58\) 0 0
\(59\) −0.524077 + 0.907729i −0.0682291 + 0.118176i −0.898122 − 0.439747i \(-0.855068\pi\)
0.829893 + 0.557923i \(0.188402\pi\)
\(60\) 0 0
\(61\) 6.24989 + 10.8251i 0.800217 + 1.38602i 0.919473 + 0.393153i \(0.128616\pi\)
−0.119256 + 0.992864i \(0.538051\pi\)
\(62\) 0 0
\(63\) −0.210913 + 3.13385i −0.0265726 + 0.394828i
\(64\) 0 0
\(65\) −1.09358 − 1.89414i −0.135642 − 0.234939i
\(66\) 0 0
\(67\) 2.23944 − 3.87883i 0.273592 − 0.473875i −0.696187 − 0.717860i \(-0.745122\pi\)
0.969779 + 0.243986i \(0.0784550\pi\)
\(68\) 0 0
\(69\) −0.879996 −0.105939
\(70\) 0 0
\(71\) 6.60274 0.783601 0.391801 − 0.920050i \(-0.371852\pi\)
0.391801 + 0.920050i \(0.371852\pi\)
\(72\) 0 0
\(73\) 4.14174 − 7.17370i 0.484754 − 0.839618i −0.515093 − 0.857134i \(-0.672243\pi\)
0.999847 + 0.0175164i \(0.00557593\pi\)
\(74\) 0 0
\(75\) 0.145624 + 0.252229i 0.0168152 + 0.0291249i
\(76\) 0 0
\(77\) −2.48931 + 1.22218i −0.283683 + 0.139280i
\(78\) 0 0
\(79\) 1.07007 + 1.85342i 0.120392 + 0.208526i 0.919922 − 0.392100i \(-0.128251\pi\)
−0.799530 + 0.600626i \(0.794918\pi\)
\(80\) 0 0
\(81\) 2.01457 − 3.48935i 0.223842 − 0.387705i
\(82\) 0 0
\(83\) 6.66558 0.731642 0.365821 − 0.930685i \(-0.380788\pi\)
0.365821 + 0.930685i \(0.380788\pi\)
\(84\) 0 0
\(85\) −11.5728 −1.25525
\(86\) 0 0
\(87\) 2.08969 − 3.61946i 0.224039 − 0.388047i
\(88\) 0 0
\(89\) 2.88388 + 4.99503i 0.305691 + 0.529472i 0.977415 − 0.211329i \(-0.0677792\pi\)
−0.671724 + 0.740802i \(0.734446\pi\)
\(90\) 0 0
\(91\) 2.19729 + 1.47375i 0.230338 + 0.154491i
\(92\) 0 0
\(93\) 0.691998 + 1.19858i 0.0717569 + 0.124287i
\(94\) 0 0
\(95\) 0.827739 − 1.43369i 0.0849242 − 0.147093i
\(96\) 0 0
\(97\) −2.88777 −0.293209 −0.146604 − 0.989195i \(-0.546834\pi\)
−0.146604 + 0.989195i \(0.546834\pi\)
\(98\) 0 0
\(99\) −1.24433 −0.125060
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.p.417.2 10
4.3 odd 2 91.2.e.c.53.1 ✓ 10
7.2 even 3 inner 1456.2.r.p.625.2 10
12.11 even 2 819.2.j.h.235.5 10
28.3 even 6 637.2.a.k.1.5 5
28.11 odd 6 637.2.a.l.1.5 5
28.19 even 6 637.2.e.m.79.1 10
28.23 odd 6 91.2.e.c.79.1 yes 10
28.27 even 2 637.2.e.m.508.1 10
52.51 odd 2 1183.2.e.f.508.5 10
84.11 even 6 5733.2.a.bl.1.1 5
84.23 even 6 819.2.j.h.352.5 10
84.59 odd 6 5733.2.a.bm.1.1 5
364.51 odd 6 1183.2.e.f.170.5 10
364.207 odd 6 8281.2.a.bw.1.1 5
364.311 even 6 8281.2.a.bx.1.1 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
91.2.e.c.53.1 ✓ 10 4.3 odd 2
91.2.e.c.79.1 yes 10 28.23 odd 6
637.2.a.k.1.5 5 28.3 even 6
637.2.a.l.1.5 5 28.11 odd 6
637.2.e.m.79.1 10 28.19 even 6
637.2.e.m.508.1 10 28.27 even 2
819.2.j.h.235.5 10 12.11 even 2
819.2.j.h.352.5 10 84.23 even 6
1183.2.e.f.170.5 10 364.51 odd 6
1183.2.e.f.508.5 10 52.51 odd 2
1456.2.r.p.417.2 10 1.1 even 1 trivial
1456.2.r.p.625.2 10 7.2 even 3 inner
5733.2.a.bl.1.1 5 84.11 even 6
5733.2.a.bm.1.1 5 84.59 odd 6
8281.2.a.bw.1.1 5 364.207 odd 6
8281.2.a.bx.1.1 5 364.311 even 6