Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1840,4,Mod(1,1840)] 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("1840.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1840, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 1840 = 2^{4} \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1840.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,3,0,10,0,-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: \(108.563514411\)
Analytic rank: \(1\)
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: \(\Z[a_1, a_2, a_3]\)
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 \(\chi\) \(=\) 1840.1

$q$-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)
 
\(f(q)\) \(=\) \(q+6.72015 q^{3} +5.00000 q^{5} -26.6008 q^{7} +18.1605 q^{9} +39.6008 q^{11} -23.1605 q^{13} +33.6008 q^{15} -2.95893 q^{17} -32.3620 q^{19} -178.761 q^{21} +23.0000 q^{23} +25.0000 q^{25} -59.4031 q^{27} -162.798 q^{29} +241.243 q^{31} +266.123 q^{33} -133.004 q^{35} -180.164 q^{37} -155.642 q^{39} -353.922 q^{41} -365.761 q^{43} +90.8023 q^{45} +195.291 q^{47} +364.601 q^{49} -19.8844 q^{51} -461.687 q^{53} +198.004 q^{55} -217.478 q^{57} +290.888 q^{59} -301.049 q^{61} -483.082 q^{63} -115.802 q^{65} +366.732 q^{67} +154.564 q^{69} +8.14513 q^{71} +360.650 q^{73} +168.004 q^{75} -1053.41 q^{77} -1243.87 q^{79} -889.530 q^{81} +1481.70 q^{83} -14.7946 q^{85} -1094.03 q^{87} +829.628 q^{89} +616.086 q^{91} +1621.19 q^{93} -161.810 q^{95} -390.191 q^{97} +719.168 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 3 q^{3} + 10 q^{5} - q^{7} + 5 q^{9} + 27 q^{11} - 15 q^{13} + 15 q^{15} - 79 q^{17} + 71 q^{19} - 274 q^{21} + 46 q^{23} + 50 q^{25} + 90 q^{27} - 430 q^{29} + 305 q^{31} + 313 q^{33} - 5 q^{35}+ \cdots + 885 q^{99}+O(q^{100}) \) 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
\(3\) 6.72015 1.29329 0.646647 − 0.762789i \(-0.276171\pi\)
0.646647 + 0.762789i \(0.276171\pi\)
\(4\) 0 0
\(5\) 5.00000 0.447214
\(6\) 0 0
\(7\) −26.6008 −1.43631 −0.718153 − 0.695885i \(-0.755012\pi\)
−0.718153 + 0.695885i \(0.755012\pi\)
\(8\) 0 0
\(9\) 18.1605 0.672610
\(10\) 0 0
\(11\) 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 \(0.579464\pi\)
\(14\) 0 0
\(15\) 33.6008 0.578379
\(16\) 0 0
\(17\) −2.95893 −0.0422144 −0.0211072 − 0.999777i \(-0.506719\pi\)
−0.0211072 + 0.999777i \(0.506719\pi\)
\(18\) 0 0
\(19\) −32.3620 −0.390755 −0.195378 − 0.980728i \(-0.562593\pi\)
−0.195378 + 0.980728i \(0.562593\pi\)
\(20\) 0 0
\(21\) −178.761 −1.85757
\(22\) 0 0
\(23\) 23.0000 0.208514
\(24\) 0 0
\(25\) 25.0000 0.200000
\(26\) 0 0
\(27\) −59.4031 −0.423412
\(28\) 0 0
\(29\) −162.798 −1.04245 −0.521223 − 0.853421i \(-0.674524\pi\)
−0.521223 + 0.853421i \(0.674524\pi\)
\(30\) 0 0
\(31\) 241.243 1.39769 0.698846 − 0.715272i \(-0.253697\pi\)
0.698846 + 0.715272i \(0.253697\pi\)
\(32\) 0 0
\(33\) 266.123 1.40382
\(34\) 0 0
\(35\) −133.004 −0.642336
\(36\) 0 0
\(37\) −180.164 −0.800509 −0.400254 − 0.916404i \(-0.631078\pi\)
−0.400254 + 0.916404i \(0.631078\pi\)
\(38\) 0 0
\(39\) −155.642 −0.639042
\(40\) 0 0
\(41\) −353.922 −1.34813 −0.674064 − 0.738673i \(-0.735453\pi\)
−0.674064 + 0.738673i \(0.735453\pi\)
\(42\) 0 0
\(43\) −365.761 −1.29716 −0.648582 − 0.761145i \(-0.724638\pi\)
−0.648582 + 0.761145i \(0.724638\pi\)
\(44\) 0 0
\(45\) 90.8023 0.300800
\(46\) 0 0
\(47\) 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
\(50\) 0 0
\(51\) −19.8844 −0.0545957
\(52\) 0 0
\(53\) −461.687 −1.19656 −0.598279 − 0.801288i \(-0.704148\pi\)
−0.598279 + 0.801288i \(0.704148\pi\)
\(54\) 0 0
\(55\) 198.004 0.485433
\(56\) 0 0
\(57\) −217.478 −0.505361
\(58\) 0 0
\(59\) 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 \(0.602322\pi\)
\(62\) 0 0
\(63\) −483.082 −0.966073
\(64\) 0 0
\(65\) −115.802 −0.220977
\(66\) 0 0
\(67\) 366.732 0.668707 0.334354 − 0.942448i \(-0.391482\pi\)
0.334354 + 0.942448i \(0.391482\pi\)
\(68\) 0 0
\(69\) 154.564 0.269670
\(70\) 0 0
\(71\) 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 \(0.406639\pi\)
\(74\) 0 0
\(75\) 168.004 0.258659
\(76\) 0 0
\(77\) −1053.41 −1.55906
\(78\) 0 0
\(79\) −1243.87 −1.77147 −0.885734 − 0.464194i \(-0.846344\pi\)
−0.885734 + 0.464194i \(0.846344\pi\)
\(80\) 0 0
\(81\) −889.530 −1.22021
\(82\) 0 0
\(83\) 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
\(86\) 0 0
\(87\) −1094.03 −1.34819
\(88\) 0 0
\(89\) 829.628 0.988094 0.494047 − 0.869435i \(-0.335517\pi\)
0.494047 + 0.869435i \(0.335517\pi\)
\(90\) 0 0
\(91\) 616.086 0.709707
\(92\) 0 0
\(93\) 1621.19 1.80763
\(94\) 0 0
\(95\) −161.810 −0.174751
\(96\) 0 0
\(97\) −390.191 −0.408432 −0.204216 − 0.978926i \(-0.565464\pi\)
−0.204216 + 0.978926i \(0.565464\pi\)
\(98\) 0 0
\(99\) 719.168 0.730092
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 1840.4.a.h.1.2 2
4.3 odd 2 115.4.a.c.1.1 ✓ 2
12.11 even 2 1035.4.a.g.1.2 2
20.3 even 4 575.4.b.f.24.3 4
20.7 even 4 575.4.b.f.24.2 4
20.19 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 4.3 odd 2
575.4.a.h.1.2 2 20.19 odd 2
575.4.b.f.24.2 4 20.7 even 4
575.4.b.f.24.3 4 20.3 even 4
1035.4.a.g.1.2 2 12.11 even 2
1840.4.a.h.1.2 2 1.1 even 1 trivial