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 628.1
Root \(-0.309017 + 0.951057i\) of defining polynomial
Character \(\chi\) \(=\) 961.628
Dual form 961.2.d.c.531.1

$q$-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+(-0.190983 + 0.587785i) q^{2} +(0.381966 + 1.17557i) q^{3} +(1.30902 + 0.951057i) q^{4} +1.00000 q^{5} -0.763932 q^{6} +(3.42705 + 2.48990i) q^{7} +(-1.80902 + 1.31433i) q^{8} +(1.19098 - 0.865300i) q^{9} +(-0.190983 + 0.587785i) q^{10} +(-1.61803 - 1.17557i) q^{11} +(-0.618034 + 1.90211i) q^{12} +(0.381966 + 1.17557i) q^{13} +(-2.11803 + 1.53884i) q^{14} +(0.381966 + 1.17557i) q^{15} +(0.572949 + 1.76336i) q^{16} +(-4.23607 + 3.07768i) q^{17} +(0.281153 + 0.865300i) q^{18} +(0.690983 - 2.12663i) q^{19} +(1.30902 + 0.951057i) q^{20} +(-1.61803 + 4.97980i) q^{21} +(1.00000 - 0.726543i) q^{22} +(6.23607 - 4.53077i) q^{23} +(-2.23607 - 1.62460i) q^{24} -4.00000 q^{25} -0.763932 q^{26} +(4.47214 + 3.24920i) q^{27} +(2.11803 + 6.51864i) q^{28} +(2.23607 - 6.88191i) q^{29} -0.763932 q^{30} -5.61803 q^{32} +(0.763932 - 2.35114i) q^{33} +(-1.00000 - 3.07768i) q^{34} +(3.42705 + 2.48990i) q^{35} +2.38197 q^{36} -2.00000 q^{37} +(1.11803 + 0.812299i) q^{38} +(-1.23607 + 0.898056i) q^{39} +(-1.80902 + 1.31433i) q^{40} +(2.16312 - 6.65740i) q^{41} +(-2.61803 - 1.90211i) q^{42} +(-1.00000 + 3.07768i) q^{43} +(-1.00000 - 3.07768i) q^{44} +(1.19098 - 0.865300i) q^{45} +(1.47214 + 4.53077i) q^{46} +(-2.00000 - 6.15537i) q^{47} +(-1.85410 + 1.34708i) q^{48} +(3.38197 + 10.4086i) q^{49} +(0.763932 - 2.35114i) q^{50} +(-5.23607 - 3.80423i) q^{51} +(-0.618034 + 1.90211i) q^{52} +(1.23607 - 0.898056i) q^{53} +(-2.76393 + 2.00811i) q^{54} +(-1.61803 - 1.17557i) q^{55} -9.47214 q^{56} +2.76393 q^{57} +(3.61803 + 2.62866i) q^{58} +(-0.690983 - 2.12663i) q^{59} +(-0.618034 + 1.90211i) q^{60} -14.1803 q^{61} +6.23607 q^{63} +(-0.0729490 + 0.224514i) q^{64} +(0.381966 + 1.17557i) q^{65} +(1.23607 + 0.898056i) q^{66} +8.00000 q^{67} -8.47214 q^{68} +(7.70820 + 5.60034i) q^{69} +(-2.11803 + 1.53884i) q^{70} +(-10.6631 + 7.74721i) q^{71} +(-1.01722 + 3.13068i) q^{72} +(0.381966 + 0.277515i) q^{73} +(0.381966 - 1.17557i) q^{74} +(-1.52786 - 4.70228i) q^{75} +(2.92705 - 2.12663i) q^{76} +(-2.61803 - 8.05748i) q^{77} +(-0.291796 - 0.898056i) q^{78} +(-1.38197 + 1.00406i) q^{79} +(0.572949 + 1.76336i) q^{80} +(-0.746711 + 2.29814i) q^{81} +(3.50000 + 2.54290i) q^{82} +(0.909830 - 2.80017i) q^{83} +(-6.85410 + 4.97980i) q^{84} +(-4.23607 + 3.07768i) q^{85} +(-1.61803 - 1.17557i) q^{86} +8.94427 q^{87} +4.47214 q^{88} +(1.38197 + 1.00406i) q^{89} +(0.281153 + 0.865300i) q^{90} +(-1.61803 + 4.97980i) q^{91} +12.4721 q^{92} +4.00000 q^{94} +(0.690983 - 2.12663i) q^{95} +(-2.14590 - 6.60440i) q^{96} +(-1.57295 - 1.14281i) q^{97} -6.76393 q^{98} -2.94427 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 \(\left(\mathbb{Z}/961\mathbb{Z}\right)^\times\).

\(n\) \(3\)
\(\chi(n)\) \(e\left(\frac{2}{5}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.190983 + 0.587785i −0.135045 + 0.415627i −0.995597 − 0.0937362i \(-0.970119\pi\)
0.860552 + 0.509363i \(0.170119\pi\)
\(3\) 0.381966 + 1.17557i 0.220528 + 0.678716i 0.998715 + 0.0506828i \(0.0161398\pi\)
−0.778187 + 0.628033i \(0.783860\pi\)
\(4\) 1.30902 + 0.951057i 0.654508 + 0.475528i
\(5\) 1.00000 0.447214 0.223607 − 0.974679i \(-0.428217\pi\)
0.223607 + 0.974679i \(0.428217\pi\)
\(6\) −0.763932 −0.311874
\(7\) 3.42705 + 2.48990i 1.29530 + 0.941093i 0.999898 − 0.0142789i \(-0.00454526\pi\)
0.295405 + 0.955372i \(0.404545\pi\)
\(8\) −1.80902 + 1.31433i −0.639584 + 0.464685i
\(9\) 1.19098 − 0.865300i 0.396994 − 0.288433i
\(10\) −0.190983 + 0.587785i −0.0603941 + 0.185874i
\(11\) −1.61803 − 1.17557i −0.487856 − 0.354448i 0.316503 − 0.948591i \(-0.397491\pi\)
−0.804359 + 0.594144i \(0.797491\pi\)
\(12\) −0.618034 + 1.90211i −0.178411 + 0.549093i
\(13\) 0.381966 + 1.17557i 0.105938 + 0.326045i 0.989950 − 0.141421i \(-0.0451671\pi\)
−0.884011 + 0.467466i \(0.845167\pi\)
\(14\) −2.11803 + 1.53884i −0.566068 + 0.411273i
\(15\) 0.381966 + 1.17557i 0.0986232 + 0.303531i
\(16\) 0.572949 + 1.76336i 0.143237 + 0.440839i
\(17\) −4.23607 + 3.07768i −1.02740 + 0.746448i −0.967786 − 0.251774i \(-0.918986\pi\)
−0.0596113 + 0.998222i \(0.518986\pi\)
\(18\) 0.281153 + 0.865300i 0.0662684 + 0.203953i
\(19\) 0.690983 − 2.12663i 0.158522 − 0.487882i −0.839978 − 0.542620i \(-0.817432\pi\)
0.998501 + 0.0547382i \(0.0174324\pi\)
\(20\) 1.30902 + 0.951057i 0.292705 + 0.212663i
\(21\) −1.61803 + 4.97980i −0.353084 + 1.08668i
\(22\) 1.00000 − 0.726543i 0.213201 − 0.154899i
\(23\) 6.23607 − 4.53077i 1.30031 − 0.944731i 0.300351 − 0.953829i \(-0.402896\pi\)
0.999959 + 0.00909805i \(0.00289604\pi\)
\(24\) −2.23607 − 1.62460i −0.456435 − 0.331620i
\(25\) −4.00000 −0.800000
\(26\) −0.763932 −0.149819
\(27\) 4.47214 + 3.24920i 0.860663 + 0.625308i
\(28\) 2.11803 + 6.51864i 0.400271 + 1.23191i
\(29\) 2.23607 − 6.88191i 0.415227 − 1.27794i −0.496820 − 0.867854i \(-0.665499\pi\)
0.912047 − 0.410085i \(-0.134501\pi\)
\(30\) −0.763932 −0.139474
\(31\) 0 0
\(32\) −5.61803 −0.993137
\(33\) 0.763932 − 2.35114i 0.132983 − 0.409281i
\(34\) −1.00000 − 3.07768i −0.171499 − 0.527818i
\(35\) 3.42705 + 2.48990i 0.579277 + 0.420870i
\(36\) 2.38197 0.396994
\(37\) −2.00000 −0.328798 −0.164399 − 0.986394i \(-0.552568\pi\)
−0.164399 + 0.986394i \(0.552568\pi\)
\(38\) 1.11803 + 0.812299i 0.181369 + 0.131772i
\(39\) −1.23607 + 0.898056i −0.197929 + 0.143804i
\(40\) −1.80902 + 1.31433i −0.286031 + 0.207813i
\(41\) 2.16312 − 6.65740i 0.337822 − 1.03971i −0.627493 − 0.778623i \(-0.715919\pi\)
0.965315 − 0.261088i \(-0.0840813\pi\)
\(42\) −2.61803 − 1.90211i −0.403971 − 0.293502i
\(43\) −1.00000 + 3.07768i −0.152499 + 0.469342i −0.997899 − 0.0647909i \(-0.979362\pi\)
0.845400 + 0.534133i \(0.179362\pi\)
\(44\) −1.00000 − 3.07768i −0.150756 − 0.463978i
\(45\) 1.19098 − 0.865300i 0.177541 − 0.128991i
\(46\) 1.47214 + 4.53077i 0.217055 + 0.668025i
\(47\) −2.00000 − 6.15537i −0.291730 − 0.897853i −0.984300 − 0.176502i \(-0.943522\pi\)
0.692570 − 0.721350i \(-0.256478\pi\)
\(48\) −1.85410 + 1.34708i −0.267617 + 0.194435i
\(49\) 3.38197 + 10.4086i 0.483138 + 1.48695i
\(50\) 0.763932 − 2.35114i 0.108036 − 0.332502i
\(51\) −5.23607 − 3.80423i −0.733196 − 0.532698i
\(52\) −0.618034 + 1.90211i −0.0857059 + 0.263776i
\(53\) 1.23607 − 0.898056i 0.169787 − 0.123357i −0.499647 − 0.866229i \(-0.666537\pi\)
0.669434 + 0.742872i \(0.266537\pi\)
\(54\) −2.76393 + 2.00811i −0.376124 + 0.273270i
\(55\) −1.61803 − 1.17557i −0.218176 − 0.158514i
\(56\) −9.47214 −1.26577
\(57\) 2.76393 0.366092
\(58\) 3.61803 + 2.62866i 0.475071 + 0.345159i
\(59\) −0.690983 − 2.12663i −0.0899583 − 0.276863i 0.895949 − 0.444158i \(-0.146497\pi\)
−0.985907 + 0.167294i \(0.946497\pi\)
\(60\) −0.618034 + 1.90211i −0.0797878 + 0.245562i
\(61\) −14.1803 −1.81561 −0.907803 − 0.419396i \(-0.862242\pi\)
−0.907803 + 0.419396i \(0.862242\pi\)
\(62\) 0 0
\(63\) 6.23607 0.785671
\(64\) −0.0729490 + 0.224514i −0.00911863 + 0.0280642i
\(65\) 0.381966 + 1.17557i 0.0473771 + 0.145812i
\(66\) 1.23607 + 0.898056i 0.152149 + 0.110543i
\(67\) 8.00000 0.977356 0.488678 − 0.872464i \(-0.337479\pi\)
0.488678 + 0.872464i \(0.337479\pi\)
\(68\) −8.47214 −1.02740
\(69\) 7.70820 + 5.60034i 0.927959 + 0.674201i
\(70\) −2.11803 + 1.53884i −0.253153 + 0.183927i
\(71\) −10.6631 + 7.74721i −1.26548 + 0.919425i −0.999013 − 0.0444196i \(-0.985856\pi\)
−0.266466 + 0.963844i \(0.585856\pi\)
\(72\) −1.01722 + 3.13068i −0.119881 + 0.368955i
\(73\) 0.381966 + 0.277515i 0.0447057 + 0.0324806i 0.609914 − 0.792468i \(-0.291204\pi\)
−0.565208 + 0.824948i \(0.691204\pi\)
\(74\) 0.381966 − 1.17557i 0.0444026 − 0.136657i
\(75\) −1.52786 − 4.70228i −0.176423 − 0.542973i
\(76\) 2.92705 − 2.12663i 0.335756 − 0.243941i
\(77\) −2.61803 − 8.05748i −0.298353 − 0.918235i
\(78\) −0.291796 − 0.898056i −0.0330394 − 0.101685i
\(79\) −1.38197 + 1.00406i −0.155483 + 0.112965i −0.662807 − 0.748790i \(-0.730635\pi\)
0.507323 + 0.861756i \(0.330635\pi\)
\(80\) 0.572949 + 1.76336i 0.0640576 + 0.197149i
\(81\) −0.746711 + 2.29814i −0.0829679 + 0.255349i
\(82\) 3.50000 + 2.54290i 0.386510 + 0.280816i
\(83\) 0.909830 − 2.80017i 0.0998668 − 0.307358i −0.888625 − 0.458635i \(-0.848338\pi\)
0.988491 + 0.151277i \(0.0483385\pi\)
\(84\) −6.85410 + 4.97980i −0.747844 + 0.543340i
\(85\) −4.23607 + 3.07768i −0.459466 + 0.333822i
\(86\) −1.61803 − 1.17557i −0.174477 − 0.126765i
\(87\) 8.94427 0.958927
\(88\) 4.47214 0.476731
\(89\) 1.38197 + 1.00406i 0.146488 + 0.106430i 0.658615 − 0.752480i \(-0.271142\pi\)
−0.512127 + 0.858910i \(0.671142\pi\)
\(90\) 0.281153 + 0.865300i 0.0296361 + 0.0912106i
\(91\) −1.61803 + 4.97980i −0.169616 + 0.522025i
\(92\) 12.4721 1.30031
\(93\) 0 0
\(94\) 4.00000 0.412568
\(95\) 0.690983 − 2.12663i 0.0708934 − 0.218187i
\(96\) −2.14590 − 6.60440i −0.219015 − 0.674058i
\(97\) −1.57295 − 1.14281i −0.159709 − 0.116035i 0.505061 − 0.863084i \(-0.331470\pi\)
−0.664769 + 0.747049i \(0.731470\pi\)
\(98\) −6.76393 −0.683260
\(99\) −2.94427 −0.295910
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.628.1 4
31.2 even 5 31.2.a.a.1.1 ✓ 2
31.3 odd 30 961.2.g.e.816.1 8
31.4 even 5 inner 961.2.d.c.531.1 4
31.5 even 3 961.2.g.a.448.1 8
31.6 odd 6 961.2.g.d.846.1 8
31.7 even 15 961.2.g.a.547.1 8
31.8 even 5 961.2.d.d.388.1 4
31.9 even 15 961.2.g.h.338.1 8
31.10 even 15 961.2.c.e.521.1 4
31.11 odd 30 961.2.g.d.844.1 8
31.12 odd 30 961.2.c.c.439.1 4
31.13 odd 30 961.2.g.e.732.1 8
31.14 even 15 961.2.g.h.235.1 8
31.15 odd 10 961.2.d.g.374.1 4
31.16 even 5 961.2.d.d.374.1 4
31.17 odd 30 961.2.g.e.235.1 8
31.18 even 15 961.2.g.h.732.1 8
31.19 even 15 961.2.c.e.439.1 4
31.20 even 15 961.2.g.a.844.1 8
31.21 odd 30 961.2.c.c.521.1 4
31.22 odd 30 961.2.g.e.338.1 8
31.23 odd 10 961.2.d.g.388.1 4
31.24 odd 30 961.2.g.d.547.1 8
31.25 even 3 961.2.g.a.846.1 8
31.26 odd 6 961.2.g.d.448.1 8
31.27 odd 10 961.2.d.a.531.1 4
31.28 even 15 961.2.g.h.816.1 8
31.29 odd 10 961.2.a.f.1.1 2
31.30 odd 2 961.2.d.a.628.1 4
93.2 odd 10 279.2.a.a.1.2 2
93.29 even 10 8649.2.a.c.1.2 2
124.95 odd 10 496.2.a.i.1.1 2
155.2 odd 20 775.2.b.d.249.2 4
155.33 odd 20 775.2.b.d.249.3 4
155.64 even 10 775.2.a.d.1.2 2
217.188 odd 10 1519.2.a.a.1.1 2
248.157 even 10 1984.2.a.r.1.1 2
248.219 odd 10 1984.2.a.n.1.2 2
341.219 odd 10 3751.2.a.b.1.2 2
372.95 even 10 4464.2.a.bf.1.2 2
403.64 even 10 5239.2.a.f.1.2 2
465.374 odd 10 6975.2.a.y.1.1 2
527.33 even 10 8959.2.a.b.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
31.2.a.a.1.1 ✓ 2 31.2 even 5
279.2.a.a.1.2 2 93.2 odd 10
496.2.a.i.1.1 2 124.95 odd 10
775.2.a.d.1.2 2 155.64 even 10
775.2.b.d.249.2 4 155.2 odd 20
775.2.b.d.249.3 4 155.33 odd 20
961.2.a.f.1.1 2 31.29 odd 10
961.2.c.c.439.1 4 31.12 odd 30
961.2.c.c.521.1 4 31.21 odd 30
961.2.c.e.439.1 4 31.19 even 15
961.2.c.e.521.1 4 31.10 even 15
961.2.d.a.531.1 4 31.27 odd 10
961.2.d.a.628.1 4 31.30 odd 2
961.2.d.c.531.1 4 31.4 even 5 inner
961.2.d.c.628.1 4 1.1 even 1 trivial
961.2.d.d.374.1 4 31.16 even 5
961.2.d.d.388.1 4 31.8 even 5
961.2.d.g.374.1 4 31.15 odd 10
961.2.d.g.388.1 4 31.23 odd 10
961.2.g.a.448.1 8 31.5 even 3
961.2.g.a.547.1 8 31.7 even 15
961.2.g.a.844.1 8 31.20 even 15
961.2.g.a.846.1 8 31.25 even 3
961.2.g.d.448.1 8 31.26 odd 6
961.2.g.d.547.1 8 31.24 odd 30
961.2.g.d.844.1 8 31.11 odd 30
961.2.g.d.846.1 8 31.6 odd 6
961.2.g.e.235.1 8 31.17 odd 30
961.2.g.e.338.1 8 31.22 odd 30
961.2.g.e.732.1 8 31.13 odd 30
961.2.g.e.816.1 8 31.3 odd 30
961.2.g.h.235.1 8 31.14 even 15
961.2.g.h.338.1 8 31.9 even 15
961.2.g.h.732.1 8 31.18 even 15
961.2.g.h.816.1 8 31.28 even 15
1519.2.a.a.1.1 2 217.188 odd 10
1984.2.a.n.1.2 2 248.219 odd 10
1984.2.a.r.1.1 2 248.157 even 10
3751.2.a.b.1.2 2 341.219 odd 10
4464.2.a.bf.1.2 2 372.95 even 10
5239.2.a.f.1.2 2 403.64 even 10
6975.2.a.y.1.1 2 465.374 odd 10
8649.2.a.c.1.2 2 93.29 even 10
8959.2.a.b.1.1 2 527.33 even 10