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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [75,7,Mod(26,75)] 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("75.26"); S:= CuspForms(chi, 7); 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, 0])) N = Newforms(chi, 7, names="a")
 
Level: \( N \) \(=\) \( 75 = 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 75.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0] 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: no
Analytic conductor: \(17.2540562715\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) 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 15)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 26.1
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 75.26
Dual form 75.7.c.b.26.2

$q$-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)
 
\(f(q)\) \(=\) \(q-11.0000i q^{2} +27.0000i q^{3} -57.0000 q^{4} +297.000 q^{6} -77.0000i q^{8} -729.000 q^{9} -1539.00i q^{12} -4495.00 q^{16} +9394.00i q^{17} +8019.00i q^{18} -13178.0 q^{19} +14654.0i q^{23} +2079.00 q^{24} -19683.0i q^{27} -5758.00 q^{31} +44517.0i q^{32} +103334. q^{34} +41553.0 q^{36} +144958. i q^{38} +161194. q^{46} +90034.0i q^{47} -121365. i q^{48} -117649. q^{49} -253638. q^{51} -88666.0i q^{53} -216513. q^{54} -355806. i q^{57} -325798. q^{61} +63338.0i q^{62} +202007. q^{64} -535458. i q^{68} -395658. q^{69} +56133.0i q^{72} +751146. q^{76} +893662. q^{79} +531441. q^{81} -469546. i q^{83} -835278. i q^{92} -155466. i q^{93} +990374. q^{94} -1.20196e6 q^{96} +1.29414e6i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 114 q^{4} + 594 q^{6} - 1458 q^{9} - 8990 q^{16} - 26356 q^{19} + 4158 q^{24} - 11516 q^{31} + 206668 q^{34} + 83106 q^{36} + 322388 q^{46} - 235298 q^{49} - 507276 q^{51} - 433026 q^{54} - 651596 q^{61}+ \cdots - 2403918 q^{96}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 11.0000i − 1.37500i −0.726184 − 0.687500i \(-0.758708\pi\)
0.726184 − 0.687500i \(-0.241292\pi\)
\(3\) 27.0000i 1.00000i
\(4\) −57.0000 −0.890625
\(5\) 0 0
\(6\) 297.000 1.37500
\(7\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(8\) − 77.0000i − 0.150391i
\(9\) −729.000 −1.00000
\(10\) 0 0
\(11\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(12\) − 1539.00i − 0.890625i
\(13\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −4495.00 −1.09741
\(17\) 9394.00i 1.91207i 0.293253 + 0.956035i \(0.405262\pi\)
−0.293253 + 0.956035i \(0.594738\pi\)
\(18\) 8019.00i 1.37500i
\(19\) −13178.0 −1.92127 −0.960636 − 0.277811i \(-0.910391\pi\)
−0.960636 + 0.277811i \(0.910391\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 14654.0i 1.20441i 0.798343 + 0.602203i \(0.205710\pi\)
−0.798343 + 0.602203i \(0.794290\pi\)
\(24\) 2079.00 0.150391
\(25\) 0 0
\(26\) 0 0
\(27\) − 19683.0i − 1.00000i
\(28\) 0 0
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 0 0
\(31\) −5758.00 −0.193280 −0.0966399 − 0.995319i \(-0.530810\pi\)
−0.0966399 + 0.995319i \(0.530810\pi\)
\(32\) 44517.0i 1.35855i
\(33\) 0 0
\(34\) 103334. 2.62910
\(35\) 0 0
\(36\) 41553.0 0.890625
\(37\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(38\) 144958.i 2.64175i
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(42\) 0 0
\(43\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 161194. 1.65606
\(47\) 90034.0i 0.867187i 0.901108 + 0.433594i \(0.142755\pi\)
−0.901108 + 0.433594i \(0.857245\pi\)
\(48\) − 121365.i − 1.09741i
\(49\) −117649. −1.00000
\(50\) 0 0
\(51\) −253638. −1.91207
\(52\) 0 0
\(53\) − 88666.0i − 0.595565i −0.954634 − 0.297783i \(-0.903753\pi\)
0.954634 − 0.297783i \(-0.0962471\pi\)
\(54\) −216513. −1.37500
\(55\) 0 0
\(56\) 0 0
\(57\) − 355806.i − 1.92127i
\(58\) 0 0
\(59\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(60\) 0 0
\(61\) −325798. −1.43535 −0.717677 − 0.696376i \(-0.754794\pi\)
−0.717677 + 0.696376i \(0.754794\pi\)
\(62\) 63338.0i 0.265760i
\(63\) 0 0
\(64\) 202007. 0.770596
\(65\) 0 0
\(66\) 0 0
\(67\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(68\) − 535458.i − 1.70294i
\(69\) −395658. −1.20441
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 56133.0i 0.150391i
\(73\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 751146. 1.71113
\(77\) 0 0
\(78\) 0 0
\(79\) 893662. 1.81256 0.906279 − 0.422680i \(-0.138910\pi\)
0.906279 + 0.422680i \(0.138910\pi\)
\(80\) 0 0
\(81\) 531441. 1.00000
\(82\) 0 0
\(83\) − 469546.i − 0.821190i −0.911818 − 0.410595i \(-0.865321\pi\)
0.911818 − 0.410595i \(-0.134679\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) − 835278.i − 1.07267i
\(93\) − 155466.i − 0.193280i
\(94\) 990374. 1.19238
\(95\) 0 0
\(96\) −1.20196e6 −1.35855
\(97\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(98\) 1.29414e6i 1.37500i
\(99\) 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 75.7.c.b.26.1 2
3.2 odd 2 inner 75.7.c.b.26.2 2
5.2 odd 4 15.7.d.b.14.1 yes 1
5.3 odd 4 15.7.d.a.14.1 ✓ 1
5.4 even 2 inner 75.7.c.b.26.2 2
15.2 even 4 15.7.d.a.14.1 ✓ 1
15.8 even 4 15.7.d.b.14.1 yes 1
15.14 odd 2 CM 75.7.c.b.26.1 2
20.3 even 4 240.7.c.b.209.1 1
20.7 even 4 240.7.c.a.209.1 1
60.23 odd 4 240.7.c.a.209.1 1
60.47 odd 4 240.7.c.b.209.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
15.7.d.a.14.1 ✓ 1 5.3 odd 4
15.7.d.a.14.1 ✓ 1 15.2 even 4
15.7.d.b.14.1 yes 1 5.2 odd 4
15.7.d.b.14.1 yes 1 15.8 even 4
75.7.c.b.26.1 2 1.1 even 1 trivial
75.7.c.b.26.1 2 15.14 odd 2 CM
75.7.c.b.26.2 2 3.2 odd 2 inner
75.7.c.b.26.2 2 5.4 even 2 inner
240.7.c.a.209.1 1 20.7 even 4
240.7.c.a.209.1 1 60.23 odd 4
240.7.c.b.209.1 1 20.3 even 4
240.7.c.b.209.1 1 60.47 odd 4