Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1360,4,Mod(1,1360)] 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("1360.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1360, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 1360 = 2^{4} \cdot 5 \cdot 17 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1360.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [3,0,4,0,-15,0,-8,0,39,0,118] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(80.2425976078\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.1304.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - 11x - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 85)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-3.22168\) of defining polynomial
Character \(\chi\) \(=\) 1360.1

$q$-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)
 
\(f(q)\) \(=\) \(q+1.15753 q^{3} -5.00000 q^{5} -14.6743 q^{7} -25.6601 q^{9} +71.3743 q^{11} -28.4035 q^{13} -5.78764 q^{15} +17.0000 q^{17} -85.9592 q^{19} -16.9860 q^{21} +7.17872 q^{23} +25.0000 q^{25} -60.9556 q^{27} -27.1610 q^{29} +15.6170 q^{31} +82.6178 q^{33} +73.3717 q^{35} -67.0036 q^{37} -32.8779 q^{39} -38.5701 q^{41} +251.495 q^{43} +128.301 q^{45} -57.9121 q^{47} -127.663 q^{49} +19.6780 q^{51} -677.807 q^{53} -356.872 q^{55} -99.5002 q^{57} -598.645 q^{59} +346.063 q^{61} +376.546 q^{63} +142.018 q^{65} +849.923 q^{67} +8.30957 q^{69} +911.836 q^{71} +704.352 q^{73} +28.9382 q^{75} -1047.37 q^{77} -55.8414 q^{79} +622.266 q^{81} +1235.19 q^{83} -85.0000 q^{85} -31.4396 q^{87} -72.8069 q^{89} +416.803 q^{91} +18.0771 q^{93} +429.796 q^{95} +972.939 q^{97} -1831.47 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 4 q^{3} - 15 q^{5} - 8 q^{7} + 39 q^{9} + 118 q^{11} - 10 q^{13} - 20 q^{15} + 51 q^{17} + 160 q^{19} - 472 q^{21} + 92 q^{23} + 75 q^{25} + 280 q^{27} + 374 q^{29} - 70 q^{31} - 8 q^{33} + 40 q^{35}+ \cdots - 770 q^{99}+O(q^{100}) \) 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
\(3\) 1.15753 0.222766 0.111383 − 0.993778i \(-0.464472\pi\)
0.111383 + 0.993778i \(0.464472\pi\)
\(4\) 0 0
\(5\) −5.00000 −0.447214
\(6\) 0 0
\(7\) −14.6743 −0.792340 −0.396170 − 0.918177i \(-0.629661\pi\)
−0.396170 + 0.918177i \(0.629661\pi\)
\(8\) 0 0
\(9\) −25.6601 −0.950375
\(10\) 0 0
\(11\) 71.3743 1.95638 0.978189 − 0.207716i \(-0.0666030\pi\)
0.978189 + 0.207716i \(0.0666030\pi\)
\(12\) 0 0
\(13\) −28.4035 −0.605979 −0.302989 − 0.952994i \(-0.597985\pi\)
−0.302989 + 0.952994i \(0.597985\pi\)
\(14\) 0 0
\(15\) −5.78764 −0.0996241
\(16\) 0 0
\(17\) 17.0000 0.242536
\(18\) 0 0
\(19\) −85.9592 −1.03792 −0.518958 − 0.854800i \(-0.673680\pi\)
−0.518958 + 0.854800i \(0.673680\pi\)
\(20\) 0 0
\(21\) −16.9860 −0.176507
\(22\) 0 0
\(23\) 7.17872 0.0650811 0.0325406 − 0.999470i \(-0.489640\pi\)
0.0325406 + 0.999470i \(0.489640\pi\)
\(24\) 0 0
\(25\) 25.0000 0.200000
\(26\) 0 0
\(27\) −60.9556 −0.434478
\(28\) 0 0
\(29\) −27.1610 −0.173919 −0.0869597 − 0.996212i \(-0.527715\pi\)
−0.0869597 + 0.996212i \(0.527715\pi\)
\(30\) 0 0
\(31\) 15.6170 0.0904803 0.0452401 − 0.998976i \(-0.485595\pi\)
0.0452401 + 0.998976i \(0.485595\pi\)
\(32\) 0 0
\(33\) 82.6178 0.435815
\(34\) 0 0
\(35\) 73.3717 0.354345
\(36\) 0 0
\(37\) −67.0036 −0.297712 −0.148856 − 0.988859i \(-0.547559\pi\)
−0.148856 + 0.988859i \(0.547559\pi\)
\(38\) 0 0
\(39\) −32.8779 −0.134992
\(40\) 0 0
\(41\) −38.5701 −0.146918 −0.0734590 − 0.997298i \(-0.523404\pi\)
−0.0734590 + 0.997298i \(0.523404\pi\)
\(42\) 0 0
\(43\) 251.495 0.891920 0.445960 − 0.895053i \(-0.352862\pi\)
0.445960 + 0.895053i \(0.352862\pi\)
\(44\) 0 0
\(45\) 128.301 0.425021
\(46\) 0 0
\(47\) −57.9121 −0.179731 −0.0898654 − 0.995954i \(-0.528644\pi\)
−0.0898654 + 0.995954i \(0.528644\pi\)
\(48\) 0 0
\(49\) −127.663 −0.372197
\(50\) 0 0
\(51\) 19.6780 0.0540288
\(52\) 0 0
\(53\) −677.807 −1.75668 −0.878339 − 0.478039i \(-0.841348\pi\)
−0.878339 + 0.478039i \(0.841348\pi\)
\(54\) 0 0
\(55\) −356.872 −0.874919
\(56\) 0 0
\(57\) −99.5002 −0.231213
\(58\) 0 0
\(59\) −598.645 −1.32097 −0.660483 − 0.750841i \(-0.729648\pi\)
−0.660483 + 0.750841i \(0.729648\pi\)
\(60\) 0 0
\(61\) 346.063 0.726375 0.363187 − 0.931716i \(-0.381689\pi\)
0.363187 + 0.931716i \(0.381689\pi\)
\(62\) 0 0
\(63\) 376.546 0.753021
\(64\) 0 0
\(65\) 142.018 0.271002
\(66\) 0 0
\(67\) 849.923 1.54977 0.774885 − 0.632102i \(-0.217808\pi\)
0.774885 + 0.632102i \(0.217808\pi\)
\(68\) 0 0
\(69\) 8.30957 0.0144979
\(70\) 0 0
\(71\) 911.836 1.52415 0.762077 − 0.647486i \(-0.224180\pi\)
0.762077 + 0.647486i \(0.224180\pi\)
\(72\) 0 0
\(73\) 704.352 1.12929 0.564645 − 0.825334i \(-0.309013\pi\)
0.564645 + 0.825334i \(0.309013\pi\)
\(74\) 0 0
\(75\) 28.9382 0.0445533
\(76\) 0 0
\(77\) −1047.37 −1.55012
\(78\) 0 0
\(79\) −55.8414 −0.0795272 −0.0397636 − 0.999209i \(-0.512660\pi\)
−0.0397636 + 0.999209i \(0.512660\pi\)
\(80\) 0 0
\(81\) 622.266 0.853588
\(82\) 0 0
\(83\) 1235.19 1.63350 0.816748 − 0.576995i \(-0.195775\pi\)
0.816748 + 0.576995i \(0.195775\pi\)
\(84\) 0 0
\(85\) −85.0000 −0.108465
\(86\) 0 0
\(87\) −31.4396 −0.0387434
\(88\) 0 0
\(89\) −72.8069 −0.0867136 −0.0433568 − 0.999060i \(-0.513805\pi\)
−0.0433568 + 0.999060i \(0.513805\pi\)
\(90\) 0 0
\(91\) 416.803 0.480141
\(92\) 0 0
\(93\) 18.0771 0.0201560
\(94\) 0 0
\(95\) 429.796 0.464170
\(96\) 0 0
\(97\) 972.939 1.01842 0.509211 − 0.860642i \(-0.329937\pi\)
0.509211 + 0.860642i \(0.329937\pi\)
\(98\) 0 0
\(99\) −1831.47 −1.85929
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 1360.4.a.s.1.2 3
4.3 odd 2 85.4.a.e.1.3 ✓ 3
12.11 even 2 765.4.a.l.1.1 3
20.3 even 4 425.4.b.g.324.3 6
20.7 even 4 425.4.b.g.324.4 6
20.19 odd 2 425.4.a.h.1.1 3
68.67 odd 2 1445.4.a.j.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
85.4.a.e.1.3 ✓ 3 4.3 odd 2
425.4.a.h.1.1 3 20.19 odd 2
425.4.b.g.324.3 6 20.3 even 4
425.4.b.g.324.4 6 20.7 even 4
765.4.a.l.1.1 3 12.11 even 2
1360.4.a.s.1.2 3 1.1 even 1 trivial
1445.4.a.j.1.3 3 68.67 odd 2