Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [961,2,Mod(374,961)] 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("961.374"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(961, base_ring=CyclotomicField(10)) chi = DirichletCharacter(H, H._module([8])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 961 = 31^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 961.d (of order \(5\), degree \(4\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,-3,6,3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(4)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(7.67362363425\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{10})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} + x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 31)
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

Embedding invariants

Embedding label 374.1
Root \(0.809017 + 0.587785i\) of defining polynomial
Character \(\chi\) \(=\) 961.374
Dual form 961.2.d.c.388.1

$q$-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+(-1.30902 - 0.951057i) q^{2} +(2.61803 - 1.90211i) q^{3} +(0.190983 + 0.587785i) q^{4} +1.00000 q^{5} -5.23607 q^{6} +(0.0729490 + 0.224514i) q^{7} +(-0.690983 + 2.12663i) q^{8} +(2.30902 - 7.10642i) q^{9} +(-1.30902 - 0.951057i) q^{10} +(0.618034 + 1.90211i) q^{11} +(1.61803 + 1.17557i) q^{12} +(2.61803 - 1.90211i) q^{13} +(0.118034 - 0.363271i) q^{14} +(2.61803 - 1.90211i) q^{15} +(3.92705 - 2.85317i) q^{16} +(0.236068 - 0.726543i) q^{17} +(-9.78115 + 7.10642i) q^{18} +(1.80902 + 1.31433i) q^{19} +(0.190983 + 0.587785i) q^{20} +(0.618034 + 0.449028i) q^{21} +(1.00000 - 3.07768i) q^{22} +(1.76393 - 5.42882i) q^{23} +(2.23607 + 6.88191i) q^{24} -4.00000 q^{25} -5.23607 q^{26} +(-4.47214 - 13.7638i) q^{27} +(-0.118034 + 0.0857567i) q^{28} +(-2.23607 - 1.62460i) q^{29} -5.23607 q^{30} -3.38197 q^{32} +(5.23607 + 3.80423i) q^{33} +(-1.00000 + 0.726543i) q^{34} +(0.0729490 + 0.224514i) q^{35} +4.61803 q^{36} -2.00000 q^{37} +(-1.11803 - 3.44095i) q^{38} +(3.23607 - 9.95959i) q^{39} +(-0.690983 + 2.12663i) q^{40} +(-5.66312 - 4.11450i) q^{41} +(-0.381966 - 1.17557i) q^{42} +(-1.00000 - 0.726543i) q^{43} +(-1.00000 + 0.726543i) q^{44} +(2.30902 - 7.10642i) q^{45} +(-7.47214 + 5.42882i) q^{46} +(-2.00000 + 1.45309i) q^{47} +(4.85410 - 14.9394i) q^{48} +(5.61803 - 4.08174i) q^{49} +(5.23607 + 3.80423i) q^{50} +(-0.763932 - 2.35114i) q^{51} +(1.61803 + 1.17557i) q^{52} +(-3.23607 + 9.95959i) q^{53} +(-7.23607 + 22.2703i) q^{54} +(0.618034 + 1.90211i) q^{55} -0.527864 q^{56} +7.23607 q^{57} +(1.38197 + 4.25325i) q^{58} +(-1.80902 + 1.31433i) q^{59} +(1.61803 + 1.17557i) q^{60} +8.18034 q^{61} +1.76393 q^{63} +(-3.42705 - 2.48990i) q^{64} +(2.61803 - 1.90211i) q^{65} +(-3.23607 - 9.95959i) q^{66} +8.00000 q^{67} +0.472136 q^{68} +(-5.70820 - 17.5680i) q^{69} +(0.118034 - 0.363271i) q^{70} +(-2.83688 + 8.73102i) q^{71} +(13.5172 + 9.82084i) q^{72} +(2.61803 + 8.05748i) q^{73} +(2.61803 + 1.90211i) q^{74} +(-10.4721 + 7.60845i) q^{75} +(-0.427051 + 1.31433i) q^{76} +(-0.381966 + 0.277515i) q^{77} +(-13.7082 + 9.95959i) q^{78} +(-3.61803 + 11.1352i) q^{79} +(3.92705 - 2.85317i) q^{80} +(-19.7533 - 14.3516i) q^{81} +(3.50000 + 10.7719i) q^{82} +(12.0902 + 8.78402i) q^{83} +(-0.145898 + 0.449028i) q^{84} +(0.236068 - 0.726543i) q^{85} +(0.618034 + 1.90211i) q^{86} -8.94427 q^{87} -4.47214 q^{88} +(3.61803 + 11.1352i) q^{89} +(-9.78115 + 7.10642i) q^{90} +(0.618034 + 0.449028i) q^{91} +3.52786 q^{92} +4.00000 q^{94} +(1.80902 + 1.31433i) q^{95} +(-8.85410 + 6.43288i) q^{96} +(-4.92705 - 15.1639i) q^{97} -11.2361 q^{98} +14.9443 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 3 q^{2} + 6 q^{3} + 3 q^{4} + 4 q^{5} - 12 q^{6} + 7 q^{7} - 5 q^{8} + 7 q^{9} - 3 q^{10} - 2 q^{11} + 2 q^{12} + 6 q^{13} - 4 q^{14} + 6 q^{15} + 9 q^{16} - 8 q^{17} - 19 q^{18} + 5 q^{19} + 3 q^{20}+ \cdots + 24 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/961\mathbb{Z}\right)^\times\).

\(n\) \(3\)
\(\chi(n)\) \(e\left(\frac{4}{5}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.30902 − 0.951057i −0.925615 − 0.672499i 0.0193004 − 0.999814i \(-0.493856\pi\)
−0.944915 + 0.327315i \(0.893856\pi\)
\(3\) 2.61803 − 1.90211i 1.51152 − 1.09819i 0.546027 − 0.837768i \(-0.316140\pi\)
0.965496 − 0.260418i \(-0.0838602\pi\)
\(4\) 0.190983 + 0.587785i 0.0954915 + 0.293893i
\(5\) 1.00000 0.447214 0.223607 − 0.974679i \(-0.428217\pi\)
0.223607 + 0.974679i \(0.428217\pi\)
\(6\) −5.23607 −2.13762
\(7\) 0.0729490 + 0.224514i 0.0275721 + 0.0848583i 0.963896 − 0.266280i \(-0.0857946\pi\)
−0.936324 + 0.351138i \(0.885795\pi\)
\(8\) −0.690983 + 2.12663i −0.244299 + 0.751876i
\(9\) 2.30902 − 7.10642i 0.769672 − 2.36881i
\(10\) −1.30902 − 0.951057i −0.413948 − 0.300750i
\(11\) 0.618034 + 1.90211i 0.186344 + 0.573509i 0.999969 − 0.00788181i \(-0.00250889\pi\)
−0.813625 + 0.581390i \(0.802509\pi\)
\(12\) 1.61803 + 1.17557i 0.467086 + 0.339358i
\(13\) 2.61803 − 1.90211i 0.726112 − 0.527551i −0.162219 − 0.986755i \(-0.551865\pi\)
0.888331 + 0.459204i \(0.151865\pi\)
\(14\) 0.118034 − 0.363271i 0.0315459 − 0.0970883i
\(15\) 2.61803 − 1.90211i 0.675973 − 0.491123i
\(16\) 3.92705 − 2.85317i 0.981763 − 0.713292i
\(17\) 0.236068 − 0.726543i 0.0572549 − 0.176212i −0.918339 − 0.395794i \(-0.870469\pi\)
0.975594 + 0.219582i \(0.0704693\pi\)
\(18\) −9.78115 + 7.10642i −2.30544 + 1.67500i
\(19\) 1.80902 + 1.31433i 0.415017 + 0.301527i 0.775630 − 0.631188i \(-0.217432\pi\)
−0.360613 + 0.932716i \(0.617432\pi\)
\(20\) 0.190983 + 0.587785i 0.0427051 + 0.131433i
\(21\) 0.618034 + 0.449028i 0.134866 + 0.0979859i
\(22\) 1.00000 − 3.07768i 0.213201 − 0.656164i
\(23\) 1.76393 − 5.42882i 0.367805 − 1.13199i −0.580400 − 0.814331i \(-0.697104\pi\)
0.948206 − 0.317657i \(-0.102896\pi\)
\(24\) 2.23607 + 6.88191i 0.456435 + 1.40476i
\(25\) −4.00000 −0.800000
\(26\) −5.23607 −1.02688
\(27\) −4.47214 − 13.7638i −0.860663 − 2.64885i
\(28\) −0.118034 + 0.0857567i −0.0223063 + 0.0162065i
\(29\) −2.23607 − 1.62460i −0.415227 − 0.301680i 0.360487 − 0.932764i \(-0.382610\pi\)
−0.775715 + 0.631084i \(0.782610\pi\)
\(30\) −5.23607 −0.955971
\(31\) 0 0
\(32\) −3.38197 −0.597853
\(33\) 5.23607 + 3.80423i 0.911482 + 0.662231i
\(34\) −1.00000 + 0.726543i −0.171499 + 0.124601i
\(35\) 0.0729490 + 0.224514i 0.0123306 + 0.0379498i
\(36\) 4.61803 0.769672
\(37\) −2.00000 −0.328798 −0.164399 − 0.986394i \(-0.552568\pi\)
−0.164399 + 0.986394i \(0.552568\pi\)
\(38\) −1.11803 − 3.44095i −0.181369 − 0.558197i
\(39\) 3.23607 − 9.95959i 0.518186 − 1.59481i
\(40\) −0.690983 + 2.12663i −0.109254 + 0.336249i
\(41\) −5.66312 − 4.11450i −0.884431 − 0.642576i 0.0499893 − 0.998750i \(-0.484081\pi\)
−0.934420 + 0.356173i \(0.884081\pi\)
\(42\) −0.381966 − 1.17557i −0.0589386 − 0.181394i
\(43\) −1.00000 − 0.726543i −0.152499 − 0.110797i 0.508920 − 0.860814i \(-0.330045\pi\)
−0.661418 + 0.750017i \(0.730045\pi\)
\(44\) −1.00000 + 0.726543i −0.150756 + 0.109530i
\(45\) 2.30902 − 7.10642i 0.344208 − 1.05936i
\(46\) −7.47214 + 5.42882i −1.10171 + 0.800437i
\(47\) −2.00000 + 1.45309i −0.291730 + 0.211954i −0.724018 − 0.689782i \(-0.757707\pi\)
0.432288 + 0.901736i \(0.357707\pi\)
\(48\) 4.85410 − 14.9394i 0.700629 − 2.15632i
\(49\) 5.61803 − 4.08174i 0.802576 − 0.583106i
\(50\) 5.23607 + 3.80423i 0.740492 + 0.537999i
\(51\) −0.763932 − 2.35114i −0.106972 − 0.329226i
\(52\) 1.61803 + 1.17557i 0.224381 + 0.163022i
\(53\) −3.23607 + 9.95959i −0.444508 + 1.36806i 0.438514 + 0.898724i \(0.355505\pi\)
−0.883022 + 0.469331i \(0.844495\pi\)
\(54\) −7.23607 + 22.2703i −0.984704 + 3.03061i
\(55\) 0.618034 + 1.90211i 0.0833357 + 0.256481i
\(56\) −0.527864 −0.0705388
\(57\) 7.23607 0.958441
\(58\) 1.38197 + 4.25325i 0.181461 + 0.558480i
\(59\) −1.80902 + 1.31433i −0.235514 + 0.171111i −0.699282 − 0.714846i \(-0.746497\pi\)
0.463768 + 0.885956i \(0.346497\pi\)
\(60\) 1.61803 + 1.17557i 0.208887 + 0.151765i
\(61\) 8.18034 1.04739 0.523693 − 0.851907i \(-0.324554\pi\)
0.523693 + 0.851907i \(0.324554\pi\)
\(62\) 0 0
\(63\) 1.76393 0.222235
\(64\) −3.42705 − 2.48990i −0.428381 − 0.311237i
\(65\) 2.61803 − 1.90211i 0.324727 − 0.235928i
\(66\) −3.23607 − 9.95959i −0.398332 − 1.22594i
\(67\) 8.00000 0.977356 0.488678 − 0.872464i \(-0.337479\pi\)
0.488678 + 0.872464i \(0.337479\pi\)
\(68\) 0.472136 0.0572549
\(69\) −5.70820 − 17.5680i −0.687187 − 2.11494i
\(70\) 0.118034 − 0.363271i 0.0141078 − 0.0434192i
\(71\) −2.83688 + 8.73102i −0.336676 + 1.03618i 0.629215 + 0.777231i \(0.283377\pi\)
−0.965891 + 0.258950i \(0.916623\pi\)
\(72\) 13.5172 + 9.82084i 1.59302 + 1.15740i
\(73\) 2.61803 + 8.05748i 0.306418 + 0.943057i 0.979144 + 0.203166i \(0.0651230\pi\)
−0.672727 + 0.739891i \(0.734877\pi\)
\(74\) 2.61803 + 1.90211i 0.304340 + 0.221116i
\(75\) −10.4721 + 7.60845i −1.20922 + 0.878548i
\(76\) −0.427051 + 1.31433i −0.0489861 + 0.150764i
\(77\) −0.381966 + 0.277515i −0.0435291 + 0.0316257i
\(78\) −13.7082 + 9.95959i −1.55215 + 1.12770i
\(79\) −3.61803 + 11.1352i −0.407061 + 1.25280i 0.512102 + 0.858925i \(0.328867\pi\)
−0.919162 + 0.393879i \(0.871133\pi\)
\(80\) 3.92705 − 2.85317i 0.439058 − 0.318994i
\(81\) −19.7533 − 14.3516i −2.19481 − 1.59462i
\(82\) 3.50000 + 10.7719i 0.386510 + 1.18956i
\(83\) 12.0902 + 8.78402i 1.32707 + 0.964172i 0.999815 + 0.0192352i \(0.00612313\pi\)
0.327254 + 0.944937i \(0.393877\pi\)
\(84\) −0.145898 + 0.449028i −0.0159188 + 0.0489930i
\(85\) 0.236068 − 0.726543i 0.0256052 − 0.0788046i
\(86\) 0.618034 + 1.90211i 0.0666443 + 0.205110i
\(87\) −8.94427 −0.958927
\(88\) −4.47214 −0.476731
\(89\) 3.61803 + 11.1352i 0.383511 + 1.18032i 0.937555 + 0.347838i \(0.113084\pi\)
−0.554044 + 0.832487i \(0.686916\pi\)
\(90\) −9.78115 + 7.10642i −1.03102 + 0.749083i
\(91\) 0.618034 + 0.449028i 0.0647876 + 0.0470709i
\(92\) 3.52786 0.367805
\(93\) 0 0
\(94\) 4.00000 0.412568
\(95\) 1.80902 + 1.31433i 0.185601 + 0.134847i
\(96\) −8.85410 + 6.43288i −0.903668 + 0.656553i
\(97\) −4.92705 − 15.1639i −0.500266 − 1.53966i −0.808586 − 0.588378i \(-0.799767\pi\)
0.308320 − 0.951283i \(-0.400233\pi\)
\(98\) −11.2361 −1.13501
\(99\) 14.9443 1.50196
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 961.2.d.c.374.1 4
31.2 even 5 961.2.d.d.628.1 4
31.3 odd 30 961.2.g.d.235.1 8
31.4 even 5 31.2.a.a.1.2 ✓ 2
31.5 even 3 961.2.g.a.732.1 8
31.6 odd 6 961.2.g.d.816.1 8
31.7 even 15 961.2.c.e.439.2 4
31.8 even 5 961.2.d.d.531.1 4
31.9 even 15 961.2.g.h.844.1 8
31.10 even 15 961.2.g.h.448.1 8
31.11 odd 30 961.2.c.c.521.2 4
31.12 odd 30 961.2.g.e.846.1 8
31.13 odd 30 961.2.g.d.338.1 8
31.14 even 15 961.2.g.h.547.1 8
31.15 odd 10 961.2.d.a.388.1 4
31.16 even 5 inner 961.2.d.c.388.1 4
31.17 odd 30 961.2.g.e.547.1 8
31.18 even 15 961.2.g.a.338.1 8
31.19 even 15 961.2.g.h.846.1 8
31.20 even 15 961.2.c.e.521.2 4
31.21 odd 30 961.2.g.e.448.1 8
31.22 odd 30 961.2.g.e.844.1 8
31.23 odd 10 961.2.d.g.531.1 4
31.24 odd 30 961.2.c.c.439.2 4
31.25 even 3 961.2.g.a.816.1 8
31.26 odd 6 961.2.g.d.732.1 8
31.27 odd 10 961.2.a.f.1.2 2
31.28 even 15 961.2.g.a.235.1 8
31.29 odd 10 961.2.d.g.628.1 4
31.30 odd 2 961.2.d.a.374.1 4
93.35 odd 10 279.2.a.a.1.1 2
93.89 even 10 8649.2.a.c.1.1 2
124.35 odd 10 496.2.a.i.1.2 2
155.4 even 10 775.2.a.d.1.1 2
155.97 odd 20 775.2.b.d.249.4 4
155.128 odd 20 775.2.b.d.249.1 4
217.97 odd 10 1519.2.a.a.1.2 2
248.35 odd 10 1984.2.a.n.1.1 2
248.221 even 10 1984.2.a.r.1.2 2
341.252 odd 10 3751.2.a.b.1.1 2
372.35 even 10 4464.2.a.bf.1.1 2
403.376 even 10 5239.2.a.f.1.1 2
465.314 odd 10 6975.2.a.y.1.2 2
527.407 even 10 8959.2.a.b.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
31.2.a.a.1.2 ✓ 2 31.4 even 5
279.2.a.a.1.1 2 93.35 odd 10
496.2.a.i.1.2 2 124.35 odd 10
775.2.a.d.1.1 2 155.4 even 10
775.2.b.d.249.1 4 155.128 odd 20
775.2.b.d.249.4 4 155.97 odd 20
961.2.a.f.1.2 2 31.27 odd 10
961.2.c.c.439.2 4 31.24 odd 30
961.2.c.c.521.2 4 31.11 odd 30
961.2.c.e.439.2 4 31.7 even 15
961.2.c.e.521.2 4 31.20 even 15
961.2.d.a.374.1 4 31.30 odd 2
961.2.d.a.388.1 4 31.15 odd 10
961.2.d.c.374.1 4 1.1 even 1 trivial
961.2.d.c.388.1 4 31.16 even 5 inner
961.2.d.d.531.1 4 31.8 even 5
961.2.d.d.628.1 4 31.2 even 5
961.2.d.g.531.1 4 31.23 odd 10
961.2.d.g.628.1 4 31.29 odd 10
961.2.g.a.235.1 8 31.28 even 15
961.2.g.a.338.1 8 31.18 even 15
961.2.g.a.732.1 8 31.5 even 3
961.2.g.a.816.1 8 31.25 even 3
961.2.g.d.235.1 8 31.3 odd 30
961.2.g.d.338.1 8 31.13 odd 30
961.2.g.d.732.1 8 31.26 odd 6
961.2.g.d.816.1 8 31.6 odd 6
961.2.g.e.448.1 8 31.21 odd 30
961.2.g.e.547.1 8 31.17 odd 30
961.2.g.e.844.1 8 31.22 odd 30
961.2.g.e.846.1 8 31.12 odd 30
961.2.g.h.448.1 8 31.10 even 15
961.2.g.h.547.1 8 31.14 even 15
961.2.g.h.844.1 8 31.9 even 15
961.2.g.h.846.1 8 31.19 even 15
1519.2.a.a.1.2 2 217.97 odd 10
1984.2.a.n.1.1 2 248.35 odd 10
1984.2.a.r.1.2 2 248.221 even 10
3751.2.a.b.1.1 2 341.252 odd 10
4464.2.a.bf.1.1 2 372.35 even 10
5239.2.a.f.1.1 2 403.376 even 10
6975.2.a.y.1.2 2 465.314 odd 10
8649.2.a.c.1.1 2 93.89 even 10
8959.2.a.b.1.2 2 527.407 even 10