Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [575,4,Mod(1,575)] 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("575.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(575, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 575 = 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 575.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [5,-6,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(33.9260982533\)
Analytic rank: \(1\)
Dimension: \(5\)
Coefficient field: \(\mathbb{Q}[x]/(x^{5} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - x^{4} - 27x^{3} + 7x^{2} + 168x + 92 \) 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 1.3
Root \(-0.595043\) of defining polynomial
Character \(\chi\) \(=\) 575.1

$q$-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)
 
\(f(q)\) \(=\) \(q-0.404957 q^{2} +7.11323 q^{3} -7.83601 q^{4} -2.88055 q^{6} -13.7888 q^{7} +6.41290 q^{8} +23.5981 q^{9} +24.2317 q^{11} -55.7394 q^{12} -3.05016 q^{13} +5.58389 q^{14} +60.0911 q^{16} -63.1126 q^{17} -9.55621 q^{18} -2.07770 q^{19} -98.0832 q^{21} -9.81282 q^{22} +23.0000 q^{23} +45.6165 q^{24} +1.23518 q^{26} -24.1987 q^{27} +108.049 q^{28} -8.16397 q^{29} -156.989 q^{31} -75.6376 q^{32} +172.366 q^{33} +25.5579 q^{34} -184.915 q^{36} -302.801 q^{37} +0.841380 q^{38} -21.6965 q^{39} -42.7514 q^{41} +39.7195 q^{42} -215.265 q^{43} -189.880 q^{44} -9.31401 q^{46} -247.096 q^{47} +427.442 q^{48} -152.868 q^{49} -448.935 q^{51} +23.9010 q^{52} -600.400 q^{53} +9.79943 q^{54} -88.4265 q^{56} -14.7792 q^{57} +3.30606 q^{58} +92.2014 q^{59} +532.635 q^{61} +63.5740 q^{62} -325.390 q^{63} -450.099 q^{64} -69.8009 q^{66} -30.3010 q^{67} +494.551 q^{68} +163.604 q^{69} -736.349 q^{71} +151.332 q^{72} -349.936 q^{73} +122.622 q^{74} +16.2809 q^{76} -334.127 q^{77} +8.78614 q^{78} +301.545 q^{79} -809.279 q^{81} +17.3125 q^{82} -139.488 q^{83} +768.581 q^{84} +87.1732 q^{86} -58.0722 q^{87} +155.396 q^{88} +859.551 q^{89} +42.0581 q^{91} -180.228 q^{92} -1116.70 q^{93} +100.063 q^{94} -538.028 q^{96} +927.475 q^{97} +61.9051 q^{98} +571.823 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q - 6 q^{2} - 4 q^{3} + 22 q^{4} + 19 q^{6} + 3 q^{7} - 138 q^{8} + 77 q^{9} + 23 q^{11} - 47 q^{12} - 132 q^{13} + 93 q^{14} + 282 q^{16} - 23 q^{17} + 15 q^{18} - 161 q^{19} - 60 q^{21} - 193 q^{22}+ \cdots + 2021 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.404957 −0.143174 −0.0715870 − 0.997434i \(-0.522806\pi\)
−0.0715870 + 0.997434i \(0.522806\pi\)
\(3\) 7.11323 1.36894 0.684471 − 0.729040i \(-0.260033\pi\)
0.684471 + 0.729040i \(0.260033\pi\)
\(4\) −7.83601 −0.979501
\(5\) 0 0
\(6\) −2.88055 −0.195997
\(7\) −13.7888 −0.744527 −0.372263 − 0.928127i \(-0.621418\pi\)
−0.372263 + 0.928127i \(0.621418\pi\)
\(8\) 6.41290 0.283413
\(9\) 23.5981 0.874003
\(10\) 0 0
\(11\) 24.2317 0.664195 0.332098 − 0.943245i \(-0.392244\pi\)
0.332098 + 0.943245i \(0.392244\pi\)
\(12\) −55.7394 −1.34088
\(13\) −3.05016 −0.0650739 −0.0325370 − 0.999471i \(-0.510359\pi\)
−0.0325370 + 0.999471i \(0.510359\pi\)
\(14\) 5.58389 0.106597
\(15\) 0 0
\(16\) 60.0911 0.938924
\(17\) −63.1126 −0.900415 −0.450208 − 0.892924i \(-0.648650\pi\)
−0.450208 + 0.892924i \(0.648650\pi\)
\(18\) −9.55621 −0.125134
\(19\) −2.07770 −0.0250872 −0.0125436 − 0.999921i \(-0.503993\pi\)
−0.0125436 + 0.999921i \(0.503993\pi\)
\(20\) 0 0
\(21\) −98.0832 −1.01921
\(22\) −9.81282 −0.0950955
\(23\) 23.0000 0.208514
\(24\) 45.6165 0.387976
\(25\) 0 0
\(26\) 1.23518 0.00931689
\(27\) −24.1987 −0.172483
\(28\) 108.049 0.729265
\(29\) −8.16397 −0.0522762 −0.0261381 − 0.999658i \(-0.508321\pi\)
−0.0261381 + 0.999658i \(0.508321\pi\)
\(30\) 0 0
\(31\) −156.989 −0.909553 −0.454776 − 0.890606i \(-0.650281\pi\)
−0.454776 + 0.890606i \(0.650281\pi\)
\(32\) −75.6376 −0.417842
\(33\) 172.366 0.909245
\(34\) 25.5579 0.128916
\(35\) 0 0
\(36\) −184.915 −0.856087
\(37\) −302.801 −1.34541 −0.672706 − 0.739910i \(-0.734868\pi\)
−0.672706 + 0.739910i \(0.734868\pi\)
\(38\) 0.841380 0.00359184
\(39\) −21.6965 −0.0890824
\(40\) 0 0
\(41\) −42.7514 −0.162845 −0.0814225 − 0.996680i \(-0.525946\pi\)
−0.0814225 + 0.996680i \(0.525946\pi\)
\(42\) 39.7195 0.145925
\(43\) −215.265 −0.763434 −0.381717 − 0.924279i \(-0.624667\pi\)
−0.381717 + 0.924279i \(0.624667\pi\)
\(44\) −189.880 −0.650580
\(45\) 0 0
\(46\) −9.31401 −0.0298538
\(47\) −247.096 −0.766866 −0.383433 − 0.923569i \(-0.625258\pi\)
−0.383433 + 0.923569i \(0.625258\pi\)
\(48\) 427.442 1.28533
\(49\) −152.868 −0.445680
\(50\) 0 0
\(51\) −448.935 −1.23262
\(52\) 23.9010 0.0637400
\(53\) −600.400 −1.55606 −0.778031 − 0.628225i \(-0.783782\pi\)
−0.778031 + 0.628225i \(0.783782\pi\)
\(54\) 9.79943 0.0246951
\(55\) 0 0
\(56\) −88.4265 −0.211009
\(57\) −14.7792 −0.0343430
\(58\) 3.30606 0.00748460
\(59\) 92.2014 0.203451 0.101725 − 0.994813i \(-0.467564\pi\)
0.101725 + 0.994813i \(0.467564\pi\)
\(60\) 0 0
\(61\) 532.635 1.11798 0.558991 − 0.829174i \(-0.311189\pi\)
0.558991 + 0.829174i \(0.311189\pi\)
\(62\) 63.5740 0.130224
\(63\) −325.390 −0.650719
\(64\) −450.099 −0.879100
\(65\) 0 0
\(66\) −69.8009 −0.130180
\(67\) −30.3010 −0.0552515 −0.0276258 − 0.999618i \(-0.508795\pi\)
−0.0276258 + 0.999618i \(0.508795\pi\)
\(68\) 494.551 0.881958
\(69\) 163.604 0.285444
\(70\) 0 0
\(71\) −736.349 −1.23083 −0.615413 − 0.788205i \(-0.711011\pi\)
−0.615413 + 0.788205i \(0.711011\pi\)
\(72\) 151.332 0.247704
\(73\) −349.936 −0.561053 −0.280527 − 0.959846i \(-0.590509\pi\)
−0.280527 + 0.959846i \(0.590509\pi\)
\(74\) 122.622 0.192628
\(75\) 0 0
\(76\) 16.2809 0.0245730
\(77\) −334.127 −0.494511
\(78\) 8.78614 0.0127543
\(79\) 301.545 0.429449 0.214725 − 0.976675i \(-0.431115\pi\)
0.214725 + 0.976675i \(0.431115\pi\)
\(80\) 0 0
\(81\) −809.279 −1.11012
\(82\) 17.3125 0.0233152
\(83\) −139.488 −0.184468 −0.0922340 − 0.995737i \(-0.529401\pi\)
−0.0922340 + 0.995737i \(0.529401\pi\)
\(84\) 768.581 0.998322
\(85\) 0 0
\(86\) 87.1732 0.109304
\(87\) −58.0722 −0.0715631
\(88\) 155.396 0.188242
\(89\) 859.551 1.02373 0.511866 − 0.859065i \(-0.328954\pi\)
0.511866 + 0.859065i \(0.328954\pi\)
\(90\) 0 0
\(91\) 42.0581 0.0484493
\(92\) −180.228 −0.204240
\(93\) −1116.70 −1.24513
\(94\) 100.063 0.109795
\(95\) 0 0
\(96\) −538.028 −0.572002
\(97\) 927.475 0.970833 0.485417 − 0.874283i \(-0.338668\pi\)
0.485417 + 0.874283i \(0.338668\pi\)
\(98\) 61.9051 0.0638097
\(99\) 571.823 0.580508
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 575.4.a.j.1.3 5
5.2 odd 4 575.4.b.i.24.5 10
5.3 odd 4 575.4.b.i.24.6 10
5.4 even 2 115.4.a.e.1.3 ✓ 5
15.14 odd 2 1035.4.a.k.1.3 5
20.19 odd 2 1840.4.a.n.1.5 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
115.4.a.e.1.3 ✓ 5 5.4 even 2
575.4.a.j.1.3 5 1.1 even 1 trivial
575.4.b.i.24.5 10 5.2 odd 4
575.4.b.i.24.6 10 5.3 odd 4
1035.4.a.k.1.3 5 15.14 odd 2
1840.4.a.n.1.5 5 20.19 odd 2