Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1035,4,Mod(1,1035)] 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("1035.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1035, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 1035 = 3^{2} \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1035.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,6,0,2,-10,0,1] 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: yes
Analytic conductor: \(61.0669768559\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{109}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 27 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 115)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(5.72015\) of defining polynomial
Character \(\chi\) \(=\) 1035.1

$q$-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)
 
\(f(q)\) \(=\) \(q+3.00000 q^{2} +1.00000 q^{4} -5.00000 q^{5} +26.6008 q^{7} -21.0000 q^{8} -15.0000 q^{10} +39.6008 q^{11} -23.1605 q^{13} +79.8023 q^{14} -71.0000 q^{16} +2.95893 q^{17} +32.3620 q^{19} -5.00000 q^{20} +118.802 q^{22} +23.0000 q^{23} +25.0000 q^{25} -69.4814 q^{26} +26.6008 q^{28} +162.798 q^{29} -241.243 q^{31} -45.0000 q^{32} +8.87678 q^{34} -133.004 q^{35} -180.164 q^{37} +97.0860 q^{38} +105.000 q^{40} +353.922 q^{41} +365.761 q^{43} +39.6008 q^{44} +69.0000 q^{46} +195.291 q^{47} +364.601 q^{49} +75.0000 q^{50} -23.1605 q^{52} +461.687 q^{53} -198.004 q^{55} -558.616 q^{56} +488.395 q^{58} +290.888 q^{59} -301.049 q^{61} -723.728 q^{62} +433.000 q^{64} +115.802 q^{65} -366.732 q^{67} +2.95893 q^{68} -399.011 q^{70} +8.14513 q^{71} +360.650 q^{73} -540.493 q^{74} +32.3620 q^{76} +1053.41 q^{77} +1243.87 q^{79} +355.000 q^{80} +1061.77 q^{82} +1481.70 q^{83} -14.7946 q^{85} +1097.28 q^{86} -831.616 q^{88} -829.628 q^{89} -616.086 q^{91} +23.0000 q^{92} +585.874 q^{94} -161.810 q^{95} -390.191 q^{97} +1093.80 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 6 q^{2} + 2 q^{4} - 10 q^{5} + q^{7} - 42 q^{8} - 30 q^{10} + 27 q^{11} - 15 q^{13} + 3 q^{14} - 142 q^{16} + 79 q^{17} - 71 q^{19} - 10 q^{20} + 81 q^{22} + 46 q^{23} + 50 q^{25} - 45 q^{26} + q^{28}+ \cdots + 2031 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 3.00000 1.06066 0.530330 − 0.847791i \(-0.322068\pi\)
0.530330 + 0.847791i \(0.322068\pi\)
\(3\) 0 0
\(4\) 1.00000 0.125000
\(5\) −5.00000 −0.447214
\(6\) 0 0
\(7\) 26.6008 1.43631 0.718153 − 0.695885i \(-0.244988\pi\)
0.718153 + 0.695885i \(0.244988\pi\)
\(8\) −21.0000 −0.928078
\(9\) 0 0
\(10\) −15.0000 −0.474342
\(11\) 39.6008 1.08546 0.542731 − 0.839907i \(-0.317390\pi\)
0.542731 + 0.839907i \(0.317390\pi\)
\(12\) 0 0
\(13\) −23.1605 −0.494120 −0.247060 − 0.969000i \(-0.579464\pi\)
−0.247060 + 0.969000i \(0.579464\pi\)
\(14\) 79.8023 1.52343
\(15\) 0 0
\(16\) −71.0000 −1.10938
\(17\) 2.95893 0.0422144 0.0211072 − 0.999777i \(-0.493281\pi\)
0.0211072 + 0.999777i \(0.493281\pi\)
\(18\) 0 0
\(19\) 32.3620 0.390755 0.195378 − 0.980728i \(-0.437407\pi\)
0.195378 + 0.980728i \(0.437407\pi\)
\(20\) −5.00000 −0.0559017
\(21\) 0 0
\(22\) 118.802 1.15131
\(23\) 23.0000 0.208514
\(24\) 0 0
\(25\) 25.0000 0.200000
\(26\) −69.4814 −0.524093
\(27\) 0 0
\(28\) 26.6008 0.179538
\(29\) 162.798 1.04245 0.521223 − 0.853421i \(-0.325476\pi\)
0.521223 + 0.853421i \(0.325476\pi\)
\(30\) 0 0
\(31\) −241.243 −1.39769 −0.698846 − 0.715272i \(-0.746303\pi\)
−0.698846 + 0.715272i \(0.746303\pi\)
\(32\) −45.0000 −0.248592
\(33\) 0 0
\(34\) 8.87678 0.0447752
\(35\) −133.004 −0.642336
\(36\) 0 0
\(37\) −180.164 −0.800509 −0.400254 − 0.916404i \(-0.631078\pi\)
−0.400254 + 0.916404i \(0.631078\pi\)
\(38\) 97.0860 0.414459
\(39\) 0 0
\(40\) 105.000 0.415049
\(41\) 353.922 1.34813 0.674064 − 0.738673i \(-0.264547\pi\)
0.674064 + 0.738673i \(0.264547\pi\)
\(42\) 0 0
\(43\) 365.761 1.29716 0.648582 − 0.761145i \(-0.275362\pi\)
0.648582 + 0.761145i \(0.275362\pi\)
\(44\) 39.6008 0.135683
\(45\) 0 0
\(46\) 69.0000 0.221163
\(47\) 195.291 0.606089 0.303044 − 0.952976i \(-0.401997\pi\)
0.303044 + 0.952976i \(0.401997\pi\)
\(48\) 0 0
\(49\) 364.601 1.06298
\(50\) 75.0000 0.212132
\(51\) 0 0
\(52\) −23.1605 −0.0617650
\(53\) 461.687 1.19656 0.598279 − 0.801288i \(-0.295852\pi\)
0.598279 + 0.801288i \(0.295852\pi\)
\(54\) 0 0
\(55\) −198.004 −0.485433
\(56\) −558.616 −1.33300
\(57\) 0 0
\(58\) 488.395 1.10568
\(59\) 290.888 0.641872 0.320936 − 0.947101i \(-0.396003\pi\)
0.320936 + 0.947101i \(0.396003\pi\)
\(60\) 0 0
\(61\) −301.049 −0.631891 −0.315945 − 0.948777i \(-0.602322\pi\)
−0.315945 + 0.948777i \(0.602322\pi\)
\(62\) −723.728 −1.48248
\(63\) 0 0
\(64\) 433.000 0.845703
\(65\) 115.802 0.220977
\(66\) 0 0
\(67\) −366.732 −0.668707 −0.334354 − 0.942448i \(-0.608518\pi\)
−0.334354 + 0.942448i \(0.608518\pi\)
\(68\) 2.95893 0.00527680
\(69\) 0 0
\(70\) −399.011 −0.681300
\(71\) 8.14513 0.0136148 0.00680739 − 0.999977i \(-0.497833\pi\)
0.00680739 + 0.999977i \(0.497833\pi\)
\(72\) 0 0
\(73\) 360.650 0.578231 0.289115 − 0.957294i \(-0.406639\pi\)
0.289115 + 0.957294i \(0.406639\pi\)
\(74\) −540.493 −0.849068
\(75\) 0 0
\(76\) 32.3620 0.0488444
\(77\) 1053.41 1.55906
\(78\) 0 0
\(79\) 1243.87 1.77147 0.885734 − 0.464194i \(-0.153656\pi\)
0.885734 + 0.464194i \(0.153656\pi\)
\(80\) 355.000 0.496128
\(81\) 0 0
\(82\) 1061.77 1.42991
\(83\) 1481.70 1.95949 0.979747 − 0.200241i \(-0.0641727\pi\)
0.979747 + 0.200241i \(0.0641727\pi\)
\(84\) 0 0
\(85\) −14.7946 −0.0188789
\(86\) 1097.28 1.37585
\(87\) 0 0
\(88\) −831.616 −1.00739
\(89\) −829.628 −0.988094 −0.494047 − 0.869435i \(-0.664483\pi\)
−0.494047 + 0.869435i \(0.664483\pi\)
\(90\) 0 0
\(91\) −616.086 −0.709707
\(92\) 23.0000 0.0260643
\(93\) 0 0
\(94\) 585.874 0.642854
\(95\) −161.810 −0.174751
\(96\) 0 0
\(97\) −390.191 −0.408432 −0.204216 − 0.978926i \(-0.565464\pi\)
−0.204216 + 0.978926i \(0.565464\pi\)
\(98\) 1093.80 1.12746
\(99\) 0 0
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 1035.4.a.g.1.2 2
3.2 odd 2 115.4.a.c.1.1 ✓ 2
12.11 even 2 1840.4.a.h.1.2 2
15.2 even 4 575.4.b.f.24.2 4
15.8 even 4 575.4.b.f.24.3 4
15.14 odd 2 575.4.a.h.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
115.4.a.c.1.1 ✓ 2 3.2 odd 2
575.4.a.h.1.2 2 15.14 odd 2
575.4.b.f.24.2 4 15.2 even 4
575.4.b.f.24.3 4 15.8 even 4
1035.4.a.g.1.2 2 1.1 even 1 trivial
1840.4.a.h.1.2 2 12.11 even 2