Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [75,13,Mod(74,75)] 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("75.74"); S:= CuspForms(chi, 13); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(75, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1])) N = Newforms(chi, 13, names="a")
 
Level: \( N \) \(=\) \( 75 = 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 13 \)
Character orbit: \([\chi]\) \(=\) 75.d (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,17312,0,101088] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(6)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(68.5495362957\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{26})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 169 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{3}\cdot 3^{4}\cdot 5^{4} \)
Twist minimal: no (minimal twist has level 3)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 74.4
Root \(-2.54951 + 2.54951i\) of defining polynomial
Character \(\chi\) \(=\) 75.74
Dual form 75.13.d.b.74.3

$q$-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)
 
\(f(q)\) \(=\) \(q+91.7824 q^{2} +(275.347 + 675.000i) q^{3} +4328.00 q^{4} +(25272.0 + 61953.1i) q^{6} +40250.0i q^{7} +21293.5 q^{8} +(-379809. + 371719. i) q^{9} +1.16105e6i q^{11} +(1.19170e6 + 2.92140e6i) q^{12} -1.28405e6i q^{13} +3.69424e6i q^{14} -1.57731e7 q^{16} +1.48445e7 q^{17} +(-3.48598e7 + 3.41172e7i) q^{18} -5.33436e7 q^{19} +(-2.71688e7 + 1.10827e7i) q^{21} +1.06564e8i q^{22} -1.07466e8 q^{23} +(5.86310e6 + 1.43731e7i) q^{24} -1.17853e8i q^{26} +(-3.55489e8 - 1.54019e8i) q^{27} +1.74202e8i q^{28} +1.20239e8i q^{29} +6.65262e7 q^{31} -1.53491e9 q^{32} +(-7.83707e8 + 3.19691e8i) q^{33} +1.36246e9 q^{34} +(-1.64381e9 + 1.60880e9i) q^{36} +2.22873e9i q^{37} -4.89600e9 q^{38} +(8.66734e8 - 3.53559e8i) q^{39} +8.21168e9i q^{41} +(-2.49361e9 + 1.01720e9i) q^{42} -8.97722e9i q^{43} +5.02501e9i q^{44} -9.86353e9 q^{46} -1.07692e9 q^{47} +(-4.34308e9 - 1.06469e10i) q^{48} +1.22212e10 q^{49} +(4.08739e9 + 1.00200e10i) q^{51} -5.55737e9i q^{52} -4.11442e10 q^{53} +(-3.26276e10 - 1.41363e10i) q^{54} +8.57064e8i q^{56} +(-1.46880e10 - 3.60069e10i) q^{57} +1.10359e10i q^{58} -4.61074e10i q^{59} -4.06799e10 q^{61} +6.10593e9 q^{62} +(-1.49617e10 - 1.52873e10i) q^{63} -7.62712e10 q^{64} +(-7.19304e10 + 2.93420e10i) q^{66} +1.21177e11i q^{67} +6.42470e10 q^{68} +(-2.95906e10 - 7.25399e10i) q^{69} -4.48565e10i q^{71} +(-8.08747e9 + 7.91519e9i) q^{72} +6.09562e10i q^{73} +2.04558e11i q^{74} -2.30871e11 q^{76} -4.67321e10 q^{77} +(7.95509e10 - 3.24505e10i) q^{78} +2.52325e11 q^{79} +(6.08022e9 - 2.82364e11i) q^{81} +7.53688e11i q^{82} +4.10810e11 q^{83} +(-1.17586e11 + 4.79660e10i) q^{84} -8.23950e11i q^{86} +(-8.11616e10 + 3.31076e10i) q^{87} +2.47228e10i q^{88} +1.12519e11i q^{89} +5.16830e10 q^{91} -4.65115e11 q^{92} +(1.83178e10 + 4.49052e10i) q^{93} -9.88427e10 q^{94} +(-4.22634e11 - 1.03607e12i) q^{96} +6.53818e11i q^{97} +1.12169e12 q^{98} +(-4.31583e11 - 4.40976e11i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 17312 q^{4} + 101088 q^{6} - 1519236 q^{9} - 63092480 q^{16} - 213374312 q^{19} - 108675000 q^{21} + 23452416 q^{24} + 266104808 q^{31} + 5449856256 q^{34} - 6575253408 q^{36} + 3466935000 q^{39}+ \cdots - 1726330320000 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/75\mathbb{Z}\right)^\times\).

\(n\) \(26\) \(52\)
\(\chi(n)\) \(-1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 91.7824 1.43410 0.717050 − 0.697022i \(-0.245492\pi\)
0.717050 + 0.697022i \(0.245492\pi\)
\(3\) 275.347 + 675.000i 0.377705 + 0.925926i
\(4\) 4328.00 1.05664
\(5\) 0 0
\(6\) 25272.0 + 61953.1i 0.541667 + 1.32787i
\(7\) 40250.0i 0.342119i 0.985261 + 0.171060i \(0.0547191\pi\)
−0.985261 + 0.171060i \(0.945281\pi\)
\(8\) 21293.5 0.0812283
\(9\) −379809. + 371719.i −0.714678 + 0.699454i
\(10\) 0 0
\(11\) 1.16105e6i 0.655381i 0.944785 + 0.327690i \(0.106270\pi\)
−0.944785 + 0.327690i \(0.893730\pi\)
\(12\) 1.19170e6 + 2.92140e6i 0.399099 + 0.978371i
\(13\) − 1.28405e6i − 0.266025i −0.991114 − 0.133012i \(-0.957535\pi\)
0.991114 − 0.133012i \(-0.0424650\pi\)
\(14\) 3.69424e6i 0.490633i
\(15\) 0 0
\(16\) −1.57731e7 −0.940151
\(17\) 1.48445e7 0.614996 0.307498 − 0.951549i \(-0.400508\pi\)
0.307498 + 0.951549i \(0.400508\pi\)
\(18\) −3.48598e7 + 3.41172e7i −1.02492 + 1.00309i
\(19\) −5.33436e7 −1.13386 −0.566931 − 0.823765i \(-0.691870\pi\)
−0.566931 + 0.823765i \(0.691870\pi\)
\(20\) 0 0
\(21\) −2.71688e7 + 1.10827e7i −0.316777 + 0.129220i
\(22\) 1.06564e8i 0.939881i
\(23\) −1.07466e8 −0.725949 −0.362974 − 0.931799i \(-0.618239\pi\)
−0.362974 + 0.931799i \(0.618239\pi\)
\(24\) 5.86310e6 + 1.43731e7i 0.0306803 + 0.0752114i
\(25\) 0 0
\(26\) − 1.17853e8i − 0.381506i
\(27\) −3.55489e8 − 1.54019e8i −0.917580 − 0.397551i
\(28\) 1.74202e8i 0.361497i
\(29\) 1.20239e8i 0.202143i 0.994879 + 0.101072i \(0.0322271\pi\)
−0.994879 + 0.101072i \(0.967773\pi\)
\(30\) 0 0
\(31\) 6.65262e7 0.0749588 0.0374794 − 0.999297i \(-0.488067\pi\)
0.0374794 + 0.999297i \(0.488067\pi\)
\(32\) −1.53491e9 −1.42950
\(33\) −7.83707e8 + 3.19691e8i −0.606834 + 0.247541i
\(34\) 1.36246e9 0.881965
\(35\) 0 0
\(36\) −1.64381e9 + 1.60880e9i −0.755157 + 0.739071i
\(37\) 2.22873e9i 0.868653i 0.900755 + 0.434327i \(0.143014\pi\)
−0.900755 + 0.434327i \(0.856986\pi\)
\(38\) −4.89600e9 −1.62607
\(39\) 8.66734e8 − 3.53559e8i 0.246319 − 0.100479i
\(40\) 0 0
\(41\) 8.21168e9i 1.72874i 0.502858 + 0.864369i \(0.332282\pi\)
−0.502858 + 0.864369i \(0.667718\pi\)
\(42\) −2.49361e9 + 1.01720e9i −0.454290 + 0.185315i
\(43\) − 8.97722e9i − 1.42014i −0.704131 − 0.710070i \(-0.748663\pi\)
0.704131 − 0.710070i \(-0.251337\pi\)
\(44\) 5.02501e9i 0.692502i
\(45\) 0 0
\(46\) −9.86353e9 −1.04108
\(47\) −1.07692e9 −0.0999075 −0.0499538 − 0.998752i \(-0.515907\pi\)
−0.0499538 + 0.998752i \(0.515907\pi\)
\(48\) −4.34308e9 − 1.06469e10i −0.355100 − 0.870510i
\(49\) 1.22212e10 0.882954
\(50\) 0 0
\(51\) 4.08739e9 + 1.00200e10i 0.232287 + 0.569441i
\(52\) − 5.55737e9i − 0.281092i
\(53\) −4.11442e10 −1.85632 −0.928160 − 0.372181i \(-0.878610\pi\)
−0.928160 + 0.372181i \(0.878610\pi\)
\(54\) −3.26276e10 − 1.41363e10i −1.31590 − 0.570128i
\(55\) 0 0
\(56\) 8.57064e8i 0.0277898i
\(57\) −1.46880e10 − 3.60069e10i −0.428266 − 1.04987i
\(58\) 1.10359e10i 0.289893i
\(59\) − 4.61074e10i − 1.09310i −0.837428 − 0.546548i \(-0.815942\pi\)
0.837428 − 0.546548i \(-0.184058\pi\)
\(60\) 0 0
\(61\) −4.06799e10 −0.789589 −0.394795 − 0.918769i \(-0.629184\pi\)
−0.394795 + 0.918769i \(0.629184\pi\)
\(62\) 6.10593e9 0.107498
\(63\) −1.49617e10 − 1.52873e10i −0.239297 − 0.244505i
\(64\) −7.62712e10 −1.10989
\(65\) 0 0
\(66\) −7.19304e10 + 2.93420e10i −0.870260 + 0.354998i
\(67\) 1.21177e11i 1.33959i 0.742548 + 0.669793i \(0.233617\pi\)
−0.742548 + 0.669793i \(0.766383\pi\)
\(68\) 6.42470e10 0.649830
\(69\) −2.95906e10 − 7.25399e10i −0.274195 − 0.672175i
\(70\) 0 0
\(71\) − 4.48565e10i − 0.350167i −0.984554 − 0.175083i \(-0.943980\pi\)
0.984554 − 0.175083i \(-0.0560195\pi\)
\(72\) −8.08747e9 + 7.91519e9i −0.0580520 + 0.0568154i
\(73\) 6.09562e10i 0.402792i 0.979510 + 0.201396i \(0.0645478\pi\)
−0.979510 + 0.201396i \(0.935452\pi\)
\(74\) 2.04558e11i 1.24573i
\(75\) 0 0
\(76\) −2.30871e11 −1.19809
\(77\) −4.67321e10 −0.224218
\(78\) 7.95509e10 − 3.24505e10i 0.353246 − 0.144097i
\(79\) 2.52325e11 1.03800 0.519000 − 0.854774i \(-0.326304\pi\)
0.519000 + 0.854774i \(0.326304\pi\)
\(80\) 0 0
\(81\) 6.08022e9 − 2.82364e11i 0.0215283 − 0.999768i
\(82\) 7.53688e11i 2.47918i
\(83\) 4.10810e11 1.25653 0.628264 − 0.778000i \(-0.283766\pi\)
0.628264 + 0.778000i \(0.283766\pi\)
\(84\) −1.17586e11 + 4.79660e10i −0.334720 + 0.136539i
\(85\) 0 0
\(86\) − 8.23950e11i − 2.03662i
\(87\) −8.11616e10 + 3.31076e10i −0.187170 + 0.0763505i
\(88\) 2.47228e10i 0.0532354i
\(89\) 1.12519e11i 0.226404i 0.993572 + 0.113202i \(0.0361108\pi\)
−0.993572 + 0.113202i \(0.963889\pi\)
\(90\) 0 0
\(91\) 5.16830e10 0.0910122
\(92\) −4.65115e11 −0.767067
\(93\) 1.83178e10 + 4.49052e10i 0.0283123 + 0.0694063i
\(94\) −9.88427e10 −0.143277
\(95\) 0 0
\(96\) −4.22634e11 − 1.03607e12i −0.539929 − 1.32361i
\(97\) 6.53818e11i 0.784922i 0.919769 + 0.392461i \(0.128376\pi\)
−0.919769 + 0.392461i \(0.871624\pi\)
\(98\) 1.12169e12 1.26624
\(99\) −4.31583e11 − 4.40976e11i −0.458409 − 0.468386i
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 75.13.d.b.74.4 4
3.2 odd 2 inner 75.13.d.b.74.2 4
5.2 odd 4 75.13.c.c.26.2 2
5.3 odd 4 3.13.b.b.2.1 ✓ 2
5.4 even 2 inner 75.13.d.b.74.1 4
15.2 even 4 75.13.c.c.26.1 2
15.8 even 4 3.13.b.b.2.2 yes 2
15.14 odd 2 inner 75.13.d.b.74.3 4
20.3 even 4 48.13.e.b.17.1 2
40.3 even 4 192.13.e.c.65.2 2
40.13 odd 4 192.13.e.d.65.1 2
45.13 odd 12 81.13.d.c.26.2 4
45.23 even 12 81.13.d.c.26.1 4
45.38 even 12 81.13.d.c.53.2 4
45.43 odd 12 81.13.d.c.53.1 4
60.23 odd 4 48.13.e.b.17.2 2
120.53 even 4 192.13.e.d.65.2 2
120.83 odd 4 192.13.e.c.65.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3.13.b.b.2.1 ✓ 2 5.3 odd 4
3.13.b.b.2.2 yes 2 15.8 even 4
48.13.e.b.17.1 2 20.3 even 4
48.13.e.b.17.2 2 60.23 odd 4
75.13.c.c.26.1 2 15.2 even 4
75.13.c.c.26.2 2 5.2 odd 4
75.13.d.b.74.1 4 5.4 even 2 inner
75.13.d.b.74.2 4 3.2 odd 2 inner
75.13.d.b.74.3 4 15.14 odd 2 inner
75.13.d.b.74.4 4 1.1 even 1 trivial
81.13.d.c.26.1 4 45.23 even 12
81.13.d.c.26.2 4 45.13 odd 12
81.13.d.c.53.1 4 45.43 odd 12
81.13.d.c.53.2 4 45.38 even 12
192.13.e.c.65.1 2 120.83 odd 4
192.13.e.c.65.2 2 40.3 even 4
192.13.e.d.65.1 2 40.13 odd 4
192.13.e.d.65.2 2 120.53 even 4