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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [930,2,Mod(497,930)] 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("930.497"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(930, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([2, 1, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 930 = 2 \cdot 3 \cdot 5 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 930.j (of order \(4\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,0,-8,0,0,0,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(7.42608738798\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: 8.0.1698758656.6
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 18x^{6} + 97x^{4} + 176x^{2} + 64 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 683.1
Root \(-2.16053i\) of defining polynomial
Character \(\chi\) \(=\) 930.683
Dual form 930.2.j.c.497.1

$q$-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+(-0.707107 - 0.707107i) q^{2} +(-1.70711 + 0.292893i) q^{3} +1.00000i q^{4} +(-2.23483 - 0.0743018i) q^{5} +(1.41421 + 1.00000i) q^{6} +(0.218591 - 0.218591i) q^{7} +(0.707107 - 0.707107i) q^{8} +(2.82843 - 1.00000i) q^{9} +(1.52773 + 1.63280i) q^{10} -0.894921i q^{11} +(-0.292893 - 1.70711i) q^{12} +(3.57474 + 3.57474i) q^{13} -0.309135 q^{14} +(3.83686 - 0.527726i) q^{15} -1.00000 q^{16} +(-4.36459 - 4.36459i) q^{17} +(-2.70711 - 1.29289i) q^{18} -6.55178i q^{19} +(0.0743018 - 2.23483i) q^{20} +(-0.309135 + 0.437183i) q^{21} +(-0.632805 + 0.632805i) q^{22} +(-3.19562 + 3.19562i) q^{23} +(-1.00000 + 1.41421i) q^{24} +(4.98896 + 0.332104i) q^{25} -5.05545i q^{26} +(-4.53553 + 2.53553i) q^{27} +(0.218591 + 0.218591i) q^{28} +2.39124 q^{29} +(-3.08623 - 2.33991i) q^{30} +1.00000 q^{31} +(0.707107 + 0.707107i) q^{32} +(0.262116 + 1.52773i) q^{33} +6.17246i q^{34} +(-0.504757 + 0.472274i) q^{35} +(1.00000 + 2.82843i) q^{36} +(-2.00000 + 2.00000i) q^{37} +(-4.63280 + 4.63280i) q^{38} +(-7.14949 - 5.05545i) q^{39} +(-1.63280 + 1.52773i) q^{40} +4.71231i q^{41} +(0.527726 - 0.0905435i) q^{42} +(-3.37912 - 3.37912i) q^{43} +0.894921 q^{44} +(-6.39536 + 2.02468i) q^{45} +4.51929 q^{46} +(8.98896 + 8.98896i) q^{47} +(1.70711 - 0.292893i) q^{48} +6.90444i q^{49} +(-3.29289 - 3.76256i) q^{50} +(8.72918 + 6.17246i) q^{51} +(-3.57474 + 3.57474i) q^{52} +(-10.1579 + 10.1579i) q^{53} +(5.00000 + 1.41421i) q^{54} +(-0.0664943 + 2.00000i) q^{55} -0.309135i q^{56} +(1.91897 + 11.1846i) q^{57} +(-1.69087 - 1.69087i) q^{58} -6.79073 q^{59} +(0.527726 + 3.83686i) q^{60} -5.61916 q^{61} +(-0.707107 - 0.707107i) q^{62} +(0.399678 - 0.836861i) q^{63} -1.00000i q^{64} +(-7.72335 - 8.25457i) q^{65} +(0.894921 - 1.26561i) q^{66} +(-9.52512 + 9.52512i) q^{67} +(4.36459 - 4.36459i) q^{68} +(4.51929 - 6.39124i) q^{69} +(0.690865 + 0.0229693i) q^{70} -12.5518i q^{71} +(1.29289 - 2.70711i) q^{72} +(-7.83686 - 7.83686i) q^{73} +2.82843 q^{74} +(-8.61396 + 0.894295i) q^{75} +6.55178 q^{76} +(-0.195622 - 0.195622i) q^{77} +(1.48071 + 8.63020i) q^{78} +13.0830i q^{79} +(2.23483 + 0.0743018i) q^{80} +(7.00000 - 5.65685i) q^{81} +(3.33210 - 3.33210i) q^{82} +(-1.55913 + 1.55913i) q^{83} +(-0.437183 - 0.309135i) q^{84} +(9.42983 + 10.0784i) q^{85} +4.77880i q^{86} +(-4.08211 + 0.700379i) q^{87} +(-0.632805 - 0.632805i) q^{88} +1.83947 q^{89} +(5.95387 + 3.09054i) q^{90} +1.56282 q^{91} +(-3.19562 - 3.19562i) q^{92} +(-1.70711 + 0.292893i) q^{93} -12.7123i q^{94} +(-0.486809 + 14.6421i) q^{95} +(-1.41421 - 1.00000i) q^{96} +(-6.81739 + 6.81739i) q^{97} +(4.88217 - 4.88217i) q^{98} +(-0.894921 - 2.53122i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8 q^{3} + 4 q^{7} - 8 q^{12} - 4 q^{13} + 12 q^{14} + 4 q^{15} - 8 q^{16} + 4 q^{17} - 16 q^{18} + 4 q^{20} + 12 q^{21} + 4 q^{22} - 12 q^{23} - 8 q^{24} - 4 q^{25} - 8 q^{27} + 4 q^{28} - 8 q^{29}+ \cdots - 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/930\mathbb{Z}\right)^\times\).

\(n\) \(187\) \(311\) \(871\)
\(\chi(n)\) \(e\left(\frac{3}{4}\right)\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.707107 − 0.707107i −0.500000 − 0.500000i
\(3\) −1.70711 + 0.292893i −0.985599 + 0.169102i
\(4\) 1.00000i 0.500000i
\(5\) −2.23483 − 0.0743018i −0.999448 − 0.0332288i
\(6\) 1.41421 + 1.00000i 0.577350 + 0.408248i
\(7\) 0.218591 − 0.218591i 0.0826198 − 0.0826198i −0.664589 − 0.747209i \(-0.731393\pi\)
0.747209 + 0.664589i \(0.231393\pi\)
\(8\) 0.707107 − 0.707107i 0.250000 − 0.250000i
\(9\) 2.82843 − 1.00000i 0.942809 − 0.333333i
\(10\) 1.52773 + 1.63280i 0.483109 + 0.516338i
\(11\) − 0.894921i − 0.269829i −0.990857 − 0.134914i \(-0.956924\pi\)
0.990857 − 0.134914i \(-0.0430760\pi\)
\(12\) −0.292893 − 1.70711i −0.0845510 − 0.492799i
\(13\) 3.57474 + 3.57474i 0.991456 + 0.991456i 0.999964 − 0.00850795i \(-0.00270820\pi\)
−0.00850795 + 0.999964i \(0.502708\pi\)
\(14\) −0.309135 −0.0826198
\(15\) 3.83686 − 0.527726i 0.990673 − 0.136258i
\(16\) −1.00000 −0.250000
\(17\) −4.36459 − 4.36459i −1.05857 − 1.05857i −0.998175 − 0.0603934i \(-0.980764\pi\)
−0.0603934 − 0.998175i \(-0.519236\pi\)
\(18\) −2.70711 − 1.29289i −0.638071 − 0.304738i
\(19\) − 6.55178i − 1.50308i −0.659687 − 0.751540i \(-0.729311\pi\)
0.659687 − 0.751540i \(-0.270689\pi\)
\(20\) 0.0743018 − 2.23483i 0.0166144 − 0.499724i
\(21\) −0.309135 + 0.437183i −0.0674588 + 0.0954011i
\(22\) −0.632805 + 0.632805i −0.134914 + 0.134914i
\(23\) −3.19562 + 3.19562i −0.666333 + 0.666333i −0.956865 − 0.290532i \(-0.906168\pi\)
0.290532 + 0.956865i \(0.406168\pi\)
\(24\) −1.00000 + 1.41421i −0.204124 + 0.288675i
\(25\) 4.98896 + 0.332104i 0.997792 + 0.0664208i
\(26\) − 5.05545i − 0.991456i
\(27\) −4.53553 + 2.53553i −0.872864 + 0.487964i
\(28\) 0.218591 + 0.218591i 0.0413099 + 0.0413099i
\(29\) 2.39124 0.444043 0.222021 − 0.975042i \(-0.428734\pi\)
0.222021 + 0.975042i \(0.428734\pi\)
\(30\) −3.08623 − 2.33991i −0.563466 − 0.427207i
\(31\) 1.00000 0.179605
\(32\) 0.707107 + 0.707107i 0.125000 + 0.125000i
\(33\) 0.262116 + 1.52773i 0.0456286 + 0.265943i
\(34\) 6.17246i 1.05857i
\(35\) −0.504757 + 0.472274i −0.0853195 + 0.0798288i
\(36\) 1.00000 + 2.82843i 0.166667 + 0.471405i
\(37\) −2.00000 + 2.00000i −0.328798 + 0.328798i −0.852129 − 0.523331i \(-0.824689\pi\)
0.523331 + 0.852129i \(0.324689\pi\)
\(38\) −4.63280 + 4.63280i −0.751540 + 0.751540i
\(39\) −7.14949 − 5.05545i −1.14483 − 0.809520i
\(40\) −1.63280 + 1.52773i −0.258169 + 0.241555i
\(41\) 4.71231i 0.735939i 0.929838 + 0.367969i \(0.119947\pi\)
−0.929838 + 0.367969i \(0.880053\pi\)
\(42\) 0.527726 − 0.0905435i 0.0814299 − 0.0139712i
\(43\) −3.37912 − 3.37912i −0.515311 − 0.515311i 0.400838 − 0.916149i \(-0.368719\pi\)
−0.916149 + 0.400838i \(0.868719\pi\)
\(44\) 0.894921 0.134914
\(45\) −6.39536 + 2.02468i −0.953365 + 0.301821i
\(46\) 4.51929 0.666333
\(47\) 8.98896 + 8.98896i 1.31117 + 1.31117i 0.920550 + 0.390624i \(0.127741\pi\)
0.390624 + 0.920550i \(0.372259\pi\)
\(48\) 1.70711 − 0.292893i 0.246400 − 0.0422755i
\(49\) 6.90444i 0.986348i
\(50\) −3.29289 − 3.76256i −0.465685 − 0.532106i
\(51\) 8.72918 + 6.17246i 1.22233 + 0.864317i
\(52\) −3.57474 + 3.57474i −0.495728 + 0.495728i
\(53\) −10.1579 + 10.1579i −1.39530 + 1.39530i −0.582385 + 0.812913i \(0.697880\pi\)
−0.812913 + 0.582385i \(0.802120\pi\)
\(54\) 5.00000 + 1.41421i 0.680414 + 0.192450i
\(55\) −0.0664943 + 2.00000i −0.00896609 + 0.269680i
\(56\) − 0.309135i − 0.0413099i
\(57\) 1.91897 + 11.1846i 0.254174 + 1.48143i
\(58\) −1.69087 − 1.69087i −0.222021 − 0.222021i
\(59\) −6.79073 −0.884078 −0.442039 − 0.896996i \(-0.645745\pi\)
−0.442039 + 0.896996i \(0.645745\pi\)
\(60\) 0.527726 + 3.83686i 0.0681292 + 0.495337i
\(61\) −5.61916 −0.719459 −0.359730 − 0.933057i \(-0.617131\pi\)
−0.359730 + 0.933057i \(0.617131\pi\)
\(62\) −0.707107 − 0.707107i −0.0898027 − 0.0898027i
\(63\) 0.399678 − 0.836861i 0.0503548 − 0.105435i
\(64\) − 1.00000i − 0.125000i
\(65\) −7.72335 − 8.25457i −0.957963 − 1.02385i
\(66\) 0.894921 − 1.26561i 0.110157 − 0.155786i
\(67\) −9.52512 + 9.52512i −1.16368 + 1.16368i −0.180015 + 0.983664i \(0.557614\pi\)
−0.983664 + 0.180015i \(0.942386\pi\)
\(68\) 4.36459 − 4.36459i 0.529284 − 0.529284i
\(69\) 4.51929 − 6.39124i 0.544059 − 0.769415i
\(70\) 0.690865 + 0.0229693i 0.0825742 + 0.00274535i
\(71\) − 12.5518i − 1.48962i −0.667276 − 0.744811i \(-0.732540\pi\)
0.667276 − 0.744811i \(-0.267460\pi\)
\(72\) 1.29289 − 2.70711i 0.152369 − 0.319036i
\(73\) −7.83686 − 7.83686i −0.917235 − 0.917235i 0.0795923 − 0.996828i \(-0.474638\pi\)
−0.996828 + 0.0795923i \(0.974638\pi\)
\(74\) 2.82843 0.328798
\(75\) −8.61396 + 0.894295i −0.994654 + 0.103264i
\(76\) 6.55178 0.751540
\(77\) −0.195622 − 0.195622i −0.0222932 − 0.0222932i
\(78\) 1.48071 + 8.63020i 0.167657 + 0.977177i
\(79\) 13.0830i 1.47195i 0.677008 + 0.735976i \(0.263276\pi\)
−0.677008 + 0.735976i \(0.736724\pi\)
\(80\) 2.23483 + 0.0743018i 0.249862 + 0.00830719i
\(81\) 7.00000 − 5.65685i 0.777778 − 0.628539i
\(82\) 3.33210 − 3.33210i 0.367969 − 0.367969i
\(83\) −1.55913 + 1.55913i −0.171137 + 0.171137i −0.787479 − 0.616342i \(-0.788614\pi\)
0.616342 + 0.787479i \(0.288614\pi\)
\(84\) −0.437183 − 0.309135i −0.0477006 − 0.0337294i
\(85\) 9.42983 + 10.0784i 1.02281 + 1.09316i
\(86\) 4.77880i 0.515311i
\(87\) −4.08211 + 0.700379i −0.437648 + 0.0750885i
\(88\) −0.632805 − 0.632805i −0.0674572 − 0.0674572i
\(89\) 1.83947 0.194983 0.0974916 − 0.995236i \(-0.468918\pi\)
0.0974916 + 0.995236i \(0.468918\pi\)
\(90\) 5.95387 + 3.09054i 0.627593 + 0.325772i
\(91\) 1.56282 0.163828
\(92\) −3.19562 − 3.19562i −0.333167 − 0.333167i
\(93\) −1.70711 + 0.292893i −0.177019 + 0.0303716i
\(94\) − 12.7123i − 1.31117i
\(95\) −0.486809 + 14.6421i −0.0499455 + 1.50225i
\(96\) −1.41421 − 1.00000i −0.144338 − 0.102062i
\(97\) −6.81739 + 6.81739i −0.692201 + 0.692201i −0.962716 − 0.270515i \(-0.912806\pi\)
0.270515 + 0.962716i \(0.412806\pi\)
\(98\) 4.88217 − 4.88217i 0.493174 − 0.493174i
\(99\) −0.894921 − 2.53122i −0.0899430 − 0.254397i
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 930.2.j.c.683.1 yes 8
3.2 odd 2 930.2.j.f.683.4 yes 8
5.2 odd 4 930.2.j.f.497.4 yes 8
15.2 even 4 inner 930.2.j.c.497.1 ✓ 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
930.2.j.c.497.1 ✓ 8 15.2 even 4 inner
930.2.j.c.683.1 yes 8 1.1 even 1 trivial
930.2.j.f.497.4 yes 8 5.2 odd 4
930.2.j.f.683.4 yes 8 3.2 odd 2