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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [234,2,Mod(11,234)] 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("234.11"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(234, base_ring=CyclotomicField(12)) chi = DirichletCharacter(H, H._module([2, 7])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 234 = 2 \cdot 3^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 234.y (of order \(12\), degree \(4\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(1.86849940730\)
Analytic rank: \(0\)
Dimension: \(56\)
Relative dimension: \(14\) over \(\Q(\zeta_{12})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label 149.7
Character \(\chi\) \(=\) 234.149
Dual form 234.2.y.a.11.7

$q$-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} +(1.67398 + 0.444746i) q^{3} +1.00000i q^{4} +(3.83221 - 1.02684i) q^{5} +(-0.869198 - 1.49816i) q^{6} +(-1.55176 + 0.415793i) q^{7} +(0.707107 - 0.707107i) q^{8} +(2.60440 + 1.48899i) q^{9} +(-3.43586 - 1.98370i) q^{10} +(-3.50976 + 3.50976i) q^{11} +(-0.444746 + 1.67398i) q^{12} +(-1.03908 - 3.45258i) q^{13} +(1.39127 + 0.803251i) q^{14} +(6.87171 - 0.0145410i) q^{15} -1.00000 q^{16} +(0.584080 + 1.01166i) q^{17} +(-0.788714 - 2.89447i) q^{18} +(-4.16646 - 1.11640i) q^{19} +(1.02684 + 3.83221i) q^{20} +(-2.78254 + 0.00588804i) q^{21} +4.96355 q^{22} +(1.63500 + 2.83190i) q^{23} +(1.49816 - 0.869198i) q^{24} +(9.30128 - 5.37010i) q^{25} +(-1.70661 + 3.17608i) q^{26} +(3.69749 + 3.65084i) q^{27} +(-0.415793 - 1.55176i) q^{28} -7.50570i q^{29} +(-4.86931 - 4.84875i) q^{30} +(-1.94388 - 7.25468i) q^{31} +(0.707107 + 0.707107i) q^{32} +(-7.43622 + 4.31431i) q^{33} +(0.302342 - 1.12836i) q^{34} +(-5.51972 + 3.18681i) q^{35} +(-1.48899 + 2.60440i) q^{36} +(-4.28775 + 1.14890i) q^{37} +(2.15672 + 3.73555i) q^{38} +(-0.203868 - 6.24167i) q^{39} +(1.98370 - 3.43586i) q^{40} +(-1.44991 + 5.41115i) q^{41} +(1.97171 + 1.96339i) q^{42} +(-0.770434 - 0.444811i) q^{43} +(-3.50976 - 3.50976i) q^{44} +(11.5096 + 3.03183i) q^{45} +(0.846338 - 3.15858i) q^{46} +(2.43773 + 0.653187i) q^{47} +(-1.67398 - 0.444746i) q^{48} +(-3.82710 + 2.20958i) q^{49} +(-10.3742 - 2.77977i) q^{50} +(0.527807 + 1.95326i) q^{51} +(3.45258 - 1.03908i) q^{52} +6.60181i q^{53} +(-0.0329860 - 5.19605i) q^{54} +(-9.84618 + 17.0541i) q^{55} +(-0.803251 + 1.39127i) q^{56} +(-6.47805 - 3.72185i) q^{57} +(-5.30733 + 5.30733i) q^{58} +(5.81273 - 5.81273i) q^{59} +(0.0145410 + 6.87171i) q^{60} +(-7.05927 + 12.2270i) q^{61} +(-3.75530 + 6.50436i) q^{62} +(-4.66052 - 1.22767i) q^{63} -1.00000i q^{64} +(-7.52719 - 12.1640i) q^{65} +(8.30888 + 2.20752i) q^{66} +(-3.19229 - 0.855370i) q^{67} +(-1.01166 + 0.584080i) q^{68} +(1.47747 + 5.46770i) q^{69} +(6.15644 + 1.64961i) q^{70} +(0.942701 - 3.51821i) q^{71} +(2.89447 - 0.788714i) q^{72} +(-3.33343 - 3.33343i) q^{73} +(3.84429 + 2.21950i) q^{74} +(17.9585 - 4.85271i) q^{75} +(1.11640 - 4.16646i) q^{76} +(3.98698 - 6.90565i) q^{77} +(-4.26937 + 4.55768i) q^{78} +(-1.16902 - 2.02480i) q^{79} +(-3.83221 + 1.02684i) q^{80} +(4.56581 + 7.75586i) q^{81} +(4.85151 - 2.80102i) q^{82} +(-2.82772 + 10.5532i) q^{83} +(-0.00588804 - 2.78254i) q^{84} +(3.27712 + 3.27712i) q^{85} +(0.230251 + 0.859308i) q^{86} +(3.33813 - 12.5644i) q^{87} +4.96355i q^{88} +(-1.78297 - 6.65415i) q^{89} +(-5.99466 - 10.2823i) q^{90} +(3.04796 + 4.92554i) q^{91} +(-2.83190 + 1.63500i) q^{92} +(-0.0275273 - 13.0087i) q^{93} +(-1.26186 - 2.18561i) q^{94} -17.1131 q^{95} +(0.869198 + 1.49816i) q^{96} +(-1.65585 - 6.17970i) q^{97} +(4.26857 + 1.14376i) q^{98} +(-14.3668 + 3.91482i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 56 q + 4 q^{7} + 24 q^{15} - 56 q^{16} - 8 q^{19} + 36 q^{27} - 4 q^{28} + 8 q^{31} - 36 q^{33} + 24 q^{35} - 24 q^{36} - 4 q^{37} - 36 q^{38} - 24 q^{39} - 48 q^{41} + 36 q^{42} + 12 q^{43} - 36 q^{45}+ \cdots - 24 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/234\mathbb{Z}\right)^\times\).

\(n\) \(145\) \(209\)
\(\chi(n)\) \(e\left(\frac{5}{12}\right)\) \(e\left(\frac{5}{6}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.707107 − 0.707107i −0.500000 − 0.500000i
\(3\) 1.67398 + 0.444746i 0.966471 + 0.256775i
\(4\) 1.00000i 0.500000i
\(5\) 3.83221 − 1.02684i 1.71381 − 0.459215i 0.737460 − 0.675391i \(-0.236025\pi\)
0.976354 + 0.216176i \(0.0693584\pi\)
\(6\) −0.869198 − 1.49816i −0.354848 − 0.611623i
\(7\) −1.55176 + 0.415793i −0.586511 + 0.157155i −0.539858 − 0.841756i \(-0.681522\pi\)
−0.0466526 + 0.998911i \(0.514855\pi\)
\(8\) 0.707107 − 0.707107i 0.250000 − 0.250000i
\(9\) 2.60440 + 1.48899i 0.868134 + 0.496330i
\(10\) −3.43586 − 1.98370i −1.08651 − 0.627300i
\(11\) −3.50976 + 3.50976i −1.05823 + 1.05823i −0.0600368 + 0.998196i \(0.519122\pi\)
−0.998196 + 0.0600368i \(0.980878\pi\)
\(12\) −0.444746 + 1.67398i −0.128387 + 0.483236i
\(13\) −1.03908 − 3.45258i −0.288188 − 0.957574i
\(14\) 1.39127 + 0.803251i 0.371833 + 0.214678i
\(15\) 6.87171 − 0.0145410i 1.77427 − 0.00375447i
\(16\) −1.00000 −0.250000
\(17\) 0.584080 + 1.01166i 0.141660 + 0.245363i 0.928122 − 0.372276i \(-0.121423\pi\)
−0.786462 + 0.617639i \(0.788089\pi\)
\(18\) −0.788714 − 2.89447i −0.185902 − 0.682232i
\(19\) −4.16646 − 1.11640i −0.955853 − 0.256120i −0.253009 − 0.967464i \(-0.581420\pi\)
−0.702844 + 0.711344i \(0.748087\pi\)
\(20\) 1.02684 + 3.83221i 0.229608 + 0.856907i
\(21\) −2.78254 + 0.00588804i −0.607199 + 0.00128488i
\(22\) 4.96355 1.05823
\(23\) 1.63500 + 2.83190i 0.340921 + 0.590492i 0.984604 − 0.174800i \(-0.0559278\pi\)
−0.643683 + 0.765292i \(0.722594\pi\)
\(24\) 1.49816 − 0.869198i 0.305811 − 0.177424i
\(25\) 9.30128 − 5.37010i 1.86026 − 1.07402i
\(26\) −1.70661 + 3.17608i −0.334693 + 0.622881i
\(27\) 3.69749 + 3.65084i 0.711581 + 0.702604i
\(28\) −0.415793 − 1.55176i −0.0785775 − 0.293255i
\(29\) − 7.50570i − 1.39377i −0.717182 − 0.696886i \(-0.754568\pi\)
0.717182 − 0.696886i \(-0.245432\pi\)
\(30\) −4.86931 − 4.84875i −0.889011 − 0.885256i
\(31\) −1.94388 − 7.25468i −0.349132 − 1.30298i −0.887710 − 0.460403i \(-0.847705\pi\)
0.538578 − 0.842576i \(-0.318962\pi\)
\(32\) 0.707107 + 0.707107i 0.125000 + 0.125000i
\(33\) −7.43622 + 4.31431i −1.29448 + 0.751025i
\(34\) 0.302342 − 1.12836i 0.0518513 − 0.193512i
\(35\) −5.51972 + 3.18681i −0.933002 + 0.538669i
\(36\) −1.48899 + 2.60440i −0.248165 + 0.434067i
\(37\) −4.28775 + 1.14890i −0.704902 + 0.188878i −0.593425 − 0.804889i \(-0.702225\pi\)
−0.111477 + 0.993767i \(0.535558\pi\)
\(38\) 2.15672 + 3.73555i 0.349866 + 0.605986i
\(39\) −0.203868 − 6.24167i −0.0326449 − 0.999467i
\(40\) 1.98370 − 3.43586i 0.313650 − 0.543257i
\(41\) −1.44991 + 5.41115i −0.226439 + 0.845080i 0.755384 + 0.655282i \(0.227450\pi\)
−0.981823 + 0.189799i \(0.939217\pi\)
\(42\) 1.97171 + 1.96339i 0.304242 + 0.302957i
\(43\) −0.770434 − 0.444811i −0.117490 − 0.0678330i 0.440103 − 0.897947i \(-0.354942\pi\)
−0.557593 + 0.830114i \(0.688275\pi\)
\(44\) −3.50976 − 3.50976i −0.529116 − 0.529116i
\(45\) 11.5096 + 3.03183i 1.71574 + 0.451958i
\(46\) 0.846338 − 3.15858i 0.124786 − 0.465707i
\(47\) 2.43773 + 0.653187i 0.355579 + 0.0952772i 0.432187 − 0.901784i \(-0.357742\pi\)
−0.0766074 + 0.997061i \(0.524409\pi\)
\(48\) −1.67398 − 0.444746i −0.241618 − 0.0641936i
\(49\) −3.82710 + 2.20958i −0.546728 + 0.315654i
\(50\) −10.3742 − 2.77977i −1.46714 − 0.393118i
\(51\) 0.527807 + 1.95326i 0.0739077 + 0.273511i
\(52\) 3.45258 − 1.03908i 0.478787 − 0.144094i
\(53\) 6.60181i 0.906828i 0.891300 + 0.453414i \(0.149794\pi\)
−0.891300 + 0.453414i \(0.850206\pi\)
\(54\) −0.0329860 − 5.19605i −0.00448882 − 0.707093i
\(55\) −9.84618 + 17.0541i −1.32766 + 2.29957i
\(56\) −0.803251 + 1.39127i −0.107339 + 0.185916i
\(57\) −6.47805 − 3.72185i −0.858039 − 0.492971i
\(58\) −5.30733 + 5.30733i −0.696886 + 0.696886i
\(59\) 5.81273 − 5.81273i 0.756753 − 0.756753i −0.218977 − 0.975730i \(-0.570272\pi\)
0.975730 + 0.218977i \(0.0702719\pi\)
\(60\) 0.0145410 + 6.87171i 0.00187724 + 0.887134i
\(61\) −7.05927 + 12.2270i −0.903847 + 1.56551i −0.0813898 + 0.996682i \(0.525936\pi\)
−0.822457 + 0.568827i \(0.807397\pi\)
\(62\) −3.75530 + 6.50436i −0.476923 + 0.826055i
\(63\) −4.66052 − 1.22767i −0.587170 − 0.154671i
\(64\) − 1.00000i − 0.125000i
\(65\) −7.52719 − 12.1640i −0.933633 − 1.50876i
\(66\) 8.30888 + 2.20752i 1.02275 + 0.271727i
\(67\) −3.19229 − 0.855370i −0.390000 − 0.104500i 0.0584902 − 0.998288i \(-0.481371\pi\)
−0.448490 + 0.893788i \(0.648038\pi\)
\(68\) −1.01166 + 0.584080i −0.122681 + 0.0708301i
\(69\) 1.47747 + 5.46770i 0.177867 + 0.658234i
\(70\) 6.15644 + 1.64961i 0.735836 + 0.197167i
\(71\) 0.942701 − 3.51821i 0.111878 − 0.417534i −0.887156 − 0.461469i \(-0.847323\pi\)
0.999034 + 0.0439344i \(0.0139893\pi\)
\(72\) 2.89447 − 0.788714i 0.341116 − 0.0929508i
\(73\) −3.33343 − 3.33343i −0.390149 − 0.390149i 0.484592 − 0.874741i \(-0.338968\pi\)
−0.874741 + 0.484592i \(0.838968\pi\)
\(74\) 3.84429 + 2.21950i 0.446890 + 0.258012i
\(75\) 17.9585 − 4.85271i 2.07366 − 0.560342i
\(76\) 1.11640 − 4.16646i 0.128060 − 0.477926i
\(77\) 3.98698 − 6.90565i 0.454358 − 0.786971i
\(78\) −4.26937 + 4.55768i −0.483411 + 0.516056i
\(79\) −1.16902 − 2.02480i −0.131525 − 0.227808i 0.792740 − 0.609560i \(-0.208654\pi\)
−0.924265 + 0.381753i \(0.875321\pi\)
\(80\) −3.83221 + 1.02684i −0.428454 + 0.114804i
\(81\) 4.56581 + 7.75586i 0.507312 + 0.861762i
\(82\) 4.85151 − 2.80102i 0.535760 − 0.309321i
\(83\) −2.82772 + 10.5532i −0.310383 + 1.15837i 0.617829 + 0.786313i \(0.288012\pi\)
−0.928212 + 0.372052i \(0.878654\pi\)
\(84\) −0.00588804 − 2.78254i −0.000642438 − 0.303600i
\(85\) 3.27712 + 3.27712i 0.355454 + 0.355454i
\(86\) 0.230251 + 0.859308i 0.0248286 + 0.0926616i
\(87\) 3.33813 − 12.5644i 0.357885 − 1.34704i
\(88\) 4.96355i 0.529116i
\(89\) −1.78297 − 6.65415i −0.188995 − 0.705338i −0.993740 − 0.111718i \(-0.964365\pi\)
0.804745 − 0.593620i \(-0.202302\pi\)
\(90\) −5.99466 − 10.2823i −0.631892 − 1.08385i
\(91\) 3.04796 + 4.92554i 0.319513 + 0.516337i
\(92\) −2.83190 + 1.63500i −0.295246 + 0.170460i
\(93\) −0.0275273 − 13.0087i −0.00285445 − 1.34894i
\(94\) −1.26186 − 2.18561i −0.130151 − 0.225428i
\(95\) −17.1131 −1.75577
\(96\) 0.869198 + 1.49816i 0.0887121 + 0.152906i
\(97\) −1.65585 − 6.17970i −0.168126 − 0.627453i −0.997621 − 0.0689395i \(-0.978038\pi\)
0.829495 − 0.558514i \(-0.188628\pi\)
\(98\) 4.26857 + 1.14376i 0.431191 + 0.115537i
\(99\) −14.3668 + 3.91482i −1.44392 + 0.393455i
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 234.2.y.a.149.7 yes 56
3.2 odd 2 702.2.bb.a.71.8 56
9.2 odd 6 234.2.z.a.227.10 yes 56
9.7 even 3 702.2.bc.a.305.1 56
13.11 odd 12 234.2.z.a.167.10 yes 56
39.11 even 12 702.2.bc.a.557.1 56
117.11 even 12 inner 234.2.y.a.11.7 ✓ 56
117.115 odd 12 702.2.bb.a.89.8 56
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
234.2.y.a.11.7 ✓ 56 117.11 even 12 inner
234.2.y.a.149.7 yes 56 1.1 even 1 trivial
234.2.z.a.167.10 yes 56 13.11 odd 12
234.2.z.a.227.10 yes 56 9.2 odd 6
702.2.bb.a.71.8 56 3.2 odd 2
702.2.bb.a.89.8 56 117.115 odd 12
702.2.bc.a.305.1 56 9.7 even 3
702.2.bc.a.557.1 56 39.11 even 12