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,-9,0,-15,0,-34,0,60,0,-52] 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: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.568.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 6x - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 85)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.76156\) 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)
 
\(f(q)\) \(=\) \(q-9.28467 q^{3} -5.00000 q^{5} -32.4157 q^{7} +59.2051 q^{9} -41.8742 q^{11} -50.3621 q^{13} +46.4234 q^{15} -17.0000 q^{17} +113.150 q^{19} +300.969 q^{21} +57.7951 q^{23} +25.0000 q^{25} -299.014 q^{27} -110.333 q^{29} +25.4263 q^{31} +388.788 q^{33} +162.078 q^{35} -124.544 q^{37} +467.596 q^{39} -16.7597 q^{41} +344.435 q^{43} -296.026 q^{45} +188.485 q^{47} +707.775 q^{49} +157.839 q^{51} +394.600 q^{53} +209.371 q^{55} -1050.56 q^{57} -344.824 q^{59} +257.635 q^{61} -1919.17 q^{63} +251.811 q^{65} -282.255 q^{67} -536.608 q^{69} +696.547 q^{71} -493.004 q^{73} -232.117 q^{75} +1357.38 q^{77} +1173.97 q^{79} +1177.71 q^{81} -902.089 q^{83} +85.0000 q^{85} +1024.41 q^{87} +648.960 q^{89} +1632.52 q^{91} -236.075 q^{93} -565.751 q^{95} -530.420 q^{97} -2479.17 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 9 q^{3} - 15 q^{5} - 34 q^{7} + 60 q^{9} - 52 q^{11} + 19 q^{13} + 45 q^{15} - 51 q^{17} + 153 q^{19} + 286 q^{21} - 162 q^{23} + 75 q^{25} - 291 q^{27} + 45 q^{29} + 67 q^{31} + 244 q^{33} + 170 q^{35}+ \cdots - 2524 q^{99}+O(q^{100}) \) 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
\(3\) −9.28467 −1.78684 −0.893418 − 0.449226i \(-0.851700\pi\)
−0.893418 + 0.449226i \(0.851700\pi\)
\(4\) 0 0
\(5\) −5.00000 −0.447214
\(6\) 0 0
\(7\) −32.4157 −1.75028 −0.875141 − 0.483869i \(-0.839231\pi\)
−0.875141 + 0.483869i \(0.839231\pi\)
\(8\) 0 0
\(9\) 59.2051 2.19278
\(10\) 0 0
\(11\) −41.8742 −1.14778 −0.573889 − 0.818933i \(-0.694566\pi\)
−0.573889 + 0.818933i \(0.694566\pi\)
\(12\) 0 0
\(13\) −50.3621 −1.07446 −0.537229 − 0.843437i \(-0.680529\pi\)
−0.537229 + 0.843437i \(0.680529\pi\)
\(14\) 0 0
\(15\) 46.4234 0.799097
\(16\) 0 0
\(17\) −17.0000 −0.242536
\(18\) 0 0
\(19\) 113.150 1.36623 0.683116 − 0.730309i \(-0.260624\pi\)
0.683116 + 0.730309i \(0.260624\pi\)
\(20\) 0 0
\(21\) 300.969 3.12747
\(22\) 0 0
\(23\) 57.7951 0.523961 0.261981 − 0.965073i \(-0.415624\pi\)
0.261981 + 0.965073i \(0.415624\pi\)
\(24\) 0 0
\(25\) 25.0000 0.200000
\(26\) 0 0
\(27\) −299.014 −2.13131
\(28\) 0 0
\(29\) −110.333 −0.706494 −0.353247 − 0.935530i \(-0.614923\pi\)
−0.353247 + 0.935530i \(0.614923\pi\)
\(30\) 0 0
\(31\) 25.4263 0.147313 0.0736564 − 0.997284i \(-0.476533\pi\)
0.0736564 + 0.997284i \(0.476533\pi\)
\(32\) 0 0
\(33\) 388.788 2.05089
\(34\) 0 0
\(35\) 162.078 0.782750
\(36\) 0 0
\(37\) −124.544 −0.553377 −0.276689 − 0.960960i \(-0.589237\pi\)
−0.276689 + 0.960960i \(0.589237\pi\)
\(38\) 0 0
\(39\) 467.596 1.91988
\(40\) 0 0
\(41\) −16.7597 −0.0638398 −0.0319199 − 0.999490i \(-0.510162\pi\)
−0.0319199 + 0.999490i \(0.510162\pi\)
\(42\) 0 0
\(43\) 344.435 1.22153 0.610765 − 0.791812i \(-0.290862\pi\)
0.610765 + 0.791812i \(0.290862\pi\)
\(44\) 0 0
\(45\) −296.026 −0.980642
\(46\) 0 0
\(47\) 188.485 0.584966 0.292483 − 0.956271i \(-0.405519\pi\)
0.292483 + 0.956271i \(0.405519\pi\)
\(48\) 0 0
\(49\) 707.775 2.06348
\(50\) 0 0
\(51\) 157.839 0.433371
\(52\) 0 0
\(53\) 394.600 1.02269 0.511344 − 0.859376i \(-0.329148\pi\)
0.511344 + 0.859376i \(0.329148\pi\)
\(54\) 0 0
\(55\) 209.371 0.513302
\(56\) 0 0
\(57\) −1050.56 −2.44123
\(58\) 0 0
\(59\) −344.824 −0.760886 −0.380443 − 0.924804i \(-0.624228\pi\)
−0.380443 + 0.924804i \(0.624228\pi\)
\(60\) 0 0
\(61\) 257.635 0.540767 0.270383 − 0.962753i \(-0.412850\pi\)
0.270383 + 0.962753i \(0.412850\pi\)
\(62\) 0 0
\(63\) −1919.17 −3.83799
\(64\) 0 0
\(65\) 251.811 0.480512
\(66\) 0 0
\(67\) −282.255 −0.514671 −0.257335 − 0.966322i \(-0.582844\pi\)
−0.257335 + 0.966322i \(0.582844\pi\)
\(68\) 0 0
\(69\) −536.608 −0.936233
\(70\) 0 0
\(71\) 696.547 1.16430 0.582148 − 0.813083i \(-0.302213\pi\)
0.582148 + 0.813083i \(0.302213\pi\)
\(72\) 0 0
\(73\) −493.004 −0.790435 −0.395218 − 0.918588i \(-0.629331\pi\)
−0.395218 + 0.918588i \(0.629331\pi\)
\(74\) 0 0
\(75\) −232.117 −0.357367
\(76\) 0 0
\(77\) 1357.38 2.00893
\(78\) 0 0
\(79\) 1173.97 1.67192 0.835961 − 0.548789i \(-0.184911\pi\)
0.835961 + 0.548789i \(0.184911\pi\)
\(80\) 0 0
\(81\) 1177.71 1.61551
\(82\) 0 0
\(83\) −902.089 −1.19298 −0.596489 − 0.802622i \(-0.703438\pi\)
−0.596489 + 0.802622i \(0.703438\pi\)
\(84\) 0 0
\(85\) 85.0000 0.108465
\(86\) 0 0
\(87\) 1024.41 1.26239
\(88\) 0 0
\(89\) 648.960 0.772918 0.386459 − 0.922307i \(-0.373698\pi\)
0.386459 + 0.922307i \(0.373698\pi\)
\(90\) 0 0
\(91\) 1632.52 1.88060
\(92\) 0 0
\(93\) −236.075 −0.263224
\(94\) 0 0
\(95\) −565.751 −0.610998
\(96\) 0 0
\(97\) −530.420 −0.555216 −0.277608 − 0.960694i \(-0.589542\pi\)
−0.277608 + 0.960694i \(0.589542\pi\)
\(98\) 0 0
\(99\) −2479.17 −2.51683
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.p.1.1 3
4.3 odd 2 85.4.a.f.1.2 ✓ 3
12.11 even 2 765.4.a.k.1.2 3
20.3 even 4 425.4.b.h.324.3 6
20.7 even 4 425.4.b.h.324.4 6
20.19 odd 2 425.4.a.f.1.2 3
68.67 odd 2 1445.4.a.k.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
85.4.a.f.1.2 ✓ 3 4.3 odd 2
425.4.a.f.1.2 3 20.19 odd 2
425.4.b.h.324.3 6 20.3 even 4
425.4.b.h.324.4 6 20.7 even 4
765.4.a.k.1.2 3 12.11 even 2
1360.4.a.p.1.1 3 1.1 even 1 trivial
1445.4.a.k.1.2 3 68.67 odd 2