Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [925,2,Mod(1,925)] 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("925.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(925, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 925 = 5^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 925.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [9,-5,-8] 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: \(7.38616218697\)
Analytic rank: \(1\)
Dimension: \(9\)
Coefficient field: \(\mathbb{Q}[x]/(x^{9} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{9} - 4x^{8} - 6x^{7} + 30x^{6} + 15x^{5} - 70x^{4} - 22x^{3} + 44x^{2} + 4x - 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 185)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(-0.356037\) of defining polynomial
Character \(\chi\) \(=\) 925.1

$q$-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)
 
\(f(q)\) \(=\) \(q-1.35604 q^{2} -1.13547 q^{3} -0.161165 q^{4} +1.53974 q^{6} -0.748140 q^{7} +2.93062 q^{8} -1.71070 q^{9} +5.40521 q^{11} +0.182998 q^{12} -5.16556 q^{13} +1.01451 q^{14} -3.65170 q^{16} -3.44151 q^{17} +2.31978 q^{18} +5.38802 q^{19} +0.849493 q^{21} -7.32966 q^{22} +3.48474 q^{23} -3.32764 q^{24} +7.00469 q^{26} +5.34887 q^{27} +0.120574 q^{28} +6.47076 q^{29} -2.23046 q^{31} -0.909404 q^{32} -6.13746 q^{33} +4.66682 q^{34} +0.275705 q^{36} +1.00000 q^{37} -7.30635 q^{38} +5.86535 q^{39} -6.44821 q^{41} -1.15194 q^{42} -2.71018 q^{43} -0.871130 q^{44} -4.72543 q^{46} -10.3708 q^{47} +4.14640 q^{48} -6.44029 q^{49} +3.90774 q^{51} +0.832507 q^{52} +4.64473 q^{53} -7.25327 q^{54} -2.19251 q^{56} -6.11794 q^{57} -8.77459 q^{58} -12.7207 q^{59} -10.3592 q^{61} +3.02459 q^{62} +1.27985 q^{63} +8.53658 q^{64} +8.32262 q^{66} +6.09691 q^{67} +0.554651 q^{68} -3.95682 q^{69} -2.80433 q^{71} -5.01342 q^{72} +1.51542 q^{73} -1.35604 q^{74} -0.868359 q^{76} -4.04385 q^{77} -7.95363 q^{78} -13.9090 q^{79} -0.941386 q^{81} +8.74401 q^{82} -1.14982 q^{83} -0.136908 q^{84} +3.67511 q^{86} -7.34737 q^{87} +15.8406 q^{88} -0.373310 q^{89} +3.86456 q^{91} -0.561617 q^{92} +2.53263 q^{93} +14.0632 q^{94} +1.03260 q^{96} -8.96381 q^{97} +8.73326 q^{98} -9.24670 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9 q - 5 q^{2} - 8 q^{3} + 11 q^{4} + 2 q^{6} - 8 q^{7} - 15 q^{8} + 13 q^{9} - 16 q^{12} - 6 q^{13} - 4 q^{14} + 11 q^{16} - 18 q^{17} + 3 q^{18} - 4 q^{19} + 4 q^{21} - 6 q^{22} - 16 q^{23} + 6 q^{24}+ \cdots + 8 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.35604 −0.958863 −0.479431 − 0.877579i \(-0.659157\pi\)
−0.479431 + 0.877579i \(0.659157\pi\)
\(3\) −1.13547 −0.655565 −0.327783 − 0.944753i \(-0.606301\pi\)
−0.327783 + 0.944753i \(0.606301\pi\)
\(4\) −0.161165 −0.0805825
\(5\) 0 0
\(6\) 1.53974 0.628597
\(7\) −0.748140 −0.282770 −0.141385 − 0.989955i \(-0.545156\pi\)
−0.141385 + 0.989955i \(0.545156\pi\)
\(8\) 2.93062 1.03613
\(9\) −1.71070 −0.570234
\(10\) 0 0
\(11\) 5.40521 1.62973 0.814866 − 0.579650i \(-0.196811\pi\)
0.814866 + 0.579650i \(0.196811\pi\)
\(12\) 0.182998 0.0528270
\(13\) −5.16556 −1.43267 −0.716334 − 0.697757i \(-0.754181\pi\)
−0.716334 + 0.697757i \(0.754181\pi\)
\(14\) 1.01451 0.271138
\(15\) 0 0
\(16\) −3.65170 −0.912924
\(17\) −3.44151 −0.834690 −0.417345 − 0.908748i \(-0.637039\pi\)
−0.417345 + 0.908748i \(0.637039\pi\)
\(18\) 2.31978 0.546776
\(19\) 5.38802 1.23610 0.618048 − 0.786140i \(-0.287924\pi\)
0.618048 + 0.786140i \(0.287924\pi\)
\(20\) 0 0
\(21\) 0.849493 0.185374
\(22\) −7.32966 −1.56269
\(23\) 3.48474 0.726618 0.363309 − 0.931669i \(-0.381647\pi\)
0.363309 + 0.931669i \(0.381647\pi\)
\(24\) −3.32764 −0.679251
\(25\) 0 0
\(26\) 7.00469 1.37373
\(27\) 5.34887 1.02939
\(28\) 0.120574 0.0227863
\(29\) 6.47076 1.20159 0.600795 − 0.799403i \(-0.294851\pi\)
0.600795 + 0.799403i \(0.294851\pi\)
\(30\) 0 0
\(31\) −2.23046 −0.400603 −0.200301 − 0.979734i \(-0.564192\pi\)
−0.200301 + 0.979734i \(0.564192\pi\)
\(32\) −0.909404 −0.160761
\(33\) −6.13746 −1.06840
\(34\) 4.66682 0.800353
\(35\) 0 0
\(36\) 0.275705 0.0459509
\(37\) 1.00000 0.164399
\(38\) −7.30635 −1.18525
\(39\) 5.86535 0.939207
\(40\) 0 0
\(41\) −6.44821 −1.00704 −0.503521 − 0.863983i \(-0.667962\pi\)
−0.503521 + 0.863983i \(0.667962\pi\)
\(42\) −1.15194 −0.177749
\(43\) −2.71018 −0.413299 −0.206649 − 0.978415i \(-0.566256\pi\)
−0.206649 + 0.978415i \(0.566256\pi\)
\(44\) −0.871130 −0.131328
\(45\) 0 0
\(46\) −4.72543 −0.696726
\(47\) −10.3708 −1.51274 −0.756370 − 0.654144i \(-0.773029\pi\)
−0.756370 + 0.654144i \(0.773029\pi\)
\(48\) 4.14640 0.598481
\(49\) −6.44029 −0.920041
\(50\) 0 0
\(51\) 3.90774 0.547193
\(52\) 0.832507 0.115448
\(53\) 4.64473 0.638002 0.319001 − 0.947754i \(-0.396653\pi\)
0.319001 + 0.947754i \(0.396653\pi\)
\(54\) −7.25327 −0.987044
\(55\) 0 0
\(56\) −2.19251 −0.292987
\(57\) −6.11794 −0.810341
\(58\) −8.77459 −1.15216
\(59\) −12.7207 −1.65609 −0.828046 − 0.560660i \(-0.810548\pi\)
−0.828046 + 0.560660i \(0.810548\pi\)
\(60\) 0 0
\(61\) −10.3592 −1.32635 −0.663177 − 0.748462i \(-0.730792\pi\)
−0.663177 + 0.748462i \(0.730792\pi\)
\(62\) 3.02459 0.384123
\(63\) 1.27985 0.161245
\(64\) 8.53658 1.06707
\(65\) 0 0
\(66\) 8.32262 1.02444
\(67\) 6.09691 0.744857 0.372428 − 0.928061i \(-0.378525\pi\)
0.372428 + 0.928061i \(0.378525\pi\)
\(68\) 0.554651 0.0672613
\(69\) −3.95682 −0.476345
\(70\) 0 0
\(71\) −2.80433 −0.332813 −0.166407 − 0.986057i \(-0.553216\pi\)
−0.166407 + 0.986057i \(0.553216\pi\)
\(72\) −5.01342 −0.590837
\(73\) 1.51542 0.177367 0.0886833 − 0.996060i \(-0.471734\pi\)
0.0886833 + 0.996060i \(0.471734\pi\)
\(74\) −1.35604 −0.157636
\(75\) 0 0
\(76\) −0.868359 −0.0996076
\(77\) −4.04385 −0.460840
\(78\) −7.95363 −0.900571
\(79\) −13.9090 −1.56489 −0.782444 − 0.622721i \(-0.786027\pi\)
−0.782444 + 0.622721i \(0.786027\pi\)
\(80\) 0 0
\(81\) −0.941386 −0.104598
\(82\) 8.74401 0.965614
\(83\) −1.14982 −0.126209 −0.0631047 − 0.998007i \(-0.520100\pi\)
−0.0631047 + 0.998007i \(0.520100\pi\)
\(84\) −0.136908 −0.0149379
\(85\) 0 0
\(86\) 3.67511 0.396297
\(87\) −7.34737 −0.787721
\(88\) 15.8406 1.68861
\(89\) −0.373310 −0.0395707 −0.0197854 − 0.999804i \(-0.506298\pi\)
−0.0197854 + 0.999804i \(0.506298\pi\)
\(90\) 0 0
\(91\) 3.86456 0.405116
\(92\) −0.561617 −0.0585526
\(93\) 2.53263 0.262621
\(94\) 14.0632 1.45051
\(95\) 0 0
\(96\) 1.03260 0.105390
\(97\) −8.96381 −0.910137 −0.455069 − 0.890456i \(-0.650385\pi\)
−0.455069 + 0.890456i \(0.650385\pi\)
\(98\) 8.73326 0.882193
\(99\) −9.24670 −0.929329
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 925.2.a.l.1.4 9
3.2 odd 2 8325.2.a.cr.1.6 9
5.2 odd 4 185.2.b.a.149.6 ✓ 18
5.3 odd 4 185.2.b.a.149.13 yes 18
5.4 even 2 925.2.a.m.1.6 9
15.2 even 4 1665.2.c.e.334.13 18
15.8 even 4 1665.2.c.e.334.6 18
15.14 odd 2 8325.2.a.cq.1.4 9
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
185.2.b.a.149.6 ✓ 18 5.2 odd 4
185.2.b.a.149.13 yes 18 5.3 odd 4
925.2.a.l.1.4 9 1.1 even 1 trivial
925.2.a.m.1.6 9 5.4 even 2
1665.2.c.e.334.6 18 15.8 even 4
1665.2.c.e.334.13 18 15.2 even 4
8325.2.a.cq.1.4 9 15.14 odd 2
8325.2.a.cr.1.6 9 3.2 odd 2