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Properties

Label 90.11.h.a.11.7
Level $90$
Weight $11$
Character 90.11
Analytic conductor $57.182$
Analytic rank $0$
Dimension $80$
Inner twists $2$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [90,11,Mod(11,90)] 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("90.11"); S:= CuspForms(chi, 11); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(90, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([1, 0])) N = Newforms(chi, 11, names="a")
 
Level: \( N \) \(=\) \( 90 = 2 \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 11 \)
Character orbit: \([\chi]\) \(=\) 90.h (of order \(6\), degree \(2\), 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: \(57.1821527406\)
Analytic rank: \(0\)
Dimension: \(80\)
Relative dimension: \(40\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 11.7
Character \(\chi\) \(=\) 90.11
Dual form 90.11.h.a.41.7

$q$-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+(-19.5959 - 11.3137i) q^{2} +(-126.394 - 207.542i) q^{3} +(256.000 + 443.405i) q^{4} +(-1210.31 + 698.771i) q^{5} +(128.746 + 5496.95i) q^{6} +(6493.94 - 11247.8i) q^{7} -11585.2i q^{8} +(-27098.0 + 52464.1i) q^{9} +31622.8 q^{10} +(89770.6 + 51829.1i) q^{11} +(59668.1 - 109174. i) q^{12} +(-101631. - 176030. i) q^{13} +(-254509. + 146941. i) q^{14} +(298000. + 162868. i) q^{15} +(-131072. + 227023. i) q^{16} -636934. i q^{17} +(1.12457e6 - 721503. i) q^{18} +1.63516e6 q^{19} +(-619677. - 357771. i) q^{20} +(-3.15519e6 + 73898.5i) q^{21} +(-1.17276e6 - 2.03128e6i) q^{22} +(1.45501e6 - 840052. i) q^{23} +(-2.40442e6 + 1.46431e6i) q^{24} +(976562. - 1.69146e6i) q^{25} +4.59929e6i q^{26} +(1.43135e7 - 1.00720e6i) q^{27} +6.64979e6 q^{28} +(-3.31457e7 - 1.91367e7i) q^{29} +(-3.99694e6 - 6.56304e6i) q^{30} +(-2.47249e7 - 4.28248e7i) q^{31} +(5.13695e6 - 2.96582e6i) q^{32} +(-589795. - 2.51820e7i) q^{33} +(-7.20608e6 + 1.24813e7i) q^{34} +1.81511e7i q^{35} +(-3.01999e7 + 1.41542e6i) q^{36} -7.32260e7 q^{37} +(-3.20425e7 - 1.84997e7i) q^{38} +(-2.36880e7 + 4.33418e7i) q^{39} +(8.09543e6 + 1.40217e7i) q^{40} +(3.49881e7 - 2.02004e7i) q^{41} +(6.26649e7 + 3.42488e7i) q^{42} +(1.40217e8 - 2.42863e8i) q^{43} +5.30730e7i q^{44} +(-3.86349e6 - 8.24330e7i) q^{45} -3.80164e7 q^{46} +(2.72299e8 + 1.57212e8i) q^{47} +(6.36835e7 - 1.49155e6i) q^{48} +(5.68952e7 + 9.85453e7i) q^{49} +(-3.82733e7 + 2.20971e7i) q^{50} +(-1.32190e8 + 8.05047e7i) q^{51} +(5.20351e7 - 9.01274e7i) q^{52} +4.49386e7i q^{53} +(-2.91882e8 - 1.42202e8i) q^{54} -1.44867e8 q^{55} +(-1.30309e8 - 7.52338e7i) q^{56} +(-2.06675e8 - 3.39364e8i) q^{57} +(4.33014e8 + 7.50001e8i) q^{58} +(-4.02237e8 + 2.32232e8i) q^{59} +(4.07130e6 + 1.73829e8i) q^{60} +(-4.29710e8 + 7.44279e8i) q^{61} +1.11892e9i q^{62} +(4.14135e8 + 6.45492e8i) q^{63} -1.34218e8 q^{64} +(2.46009e8 + 1.42034e8i) q^{65} +(-2.73345e8 + 5.00138e8i) q^{66} +(-1.44938e8 - 2.51040e8i) q^{67} +(2.82420e8 - 1.63055e8i) q^{68} +(-3.58251e8 - 1.95798e8i) q^{69} +(2.05356e8 - 3.55688e8i) q^{70} +2.12427e9i q^{71} +(6.07809e8 + 3.13937e8i) q^{72} -2.76465e9 q^{73} +(1.43493e9 + 8.28457e8i) q^{74} +(-4.74479e8 + 1.11129e7i) q^{75} +(4.18601e8 + 7.25039e8i) q^{76} +(1.16593e9 - 6.73150e8i) q^{77} +(9.54544e8 - 5.81324e8i) q^{78} +(-1.95855e9 + 3.39231e9i) q^{79} -3.66357e8i q^{80} +(-2.01818e9 - 2.84335e9i) q^{81} -9.14167e8 q^{82} +(1.07679e9 + 6.21682e8i) q^{83} +(-8.40495e8 - 1.38011e9i) q^{84} +(4.45071e8 + 7.70885e8i) q^{85} +(-5.49537e9 + 3.17275e9i) q^{86} +(2.17768e8 + 9.29787e9i) q^{87} +(6.00452e8 - 1.04001e9i) q^{88} -1.36850e9i q^{89} +(-8.56914e8 + 1.65906e9i) q^{90} -2.63994e9 q^{91} +(7.44967e8 + 4.30107e8i) q^{92} +(-5.76284e9 + 1.05442e10i) q^{93} +(-3.55729e9 - 6.16141e9i) q^{94} +(-1.97905e9 + 1.14260e9i) q^{95} +(-1.26481e9 - 6.91269e8i) q^{96} +(1.85028e9 - 3.20478e9i) q^{97} -2.57478e9i q^{98} +(-5.15177e9 + 3.30527e9i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 80 q + 44 q^{3} + 20480 q^{4} - 13568 q^{6} - 24476 q^{7} - 103396 q^{9} + 1944 q^{11} + 45056 q^{12} + 561100 q^{13} - 175680 q^{14} - 687500 q^{15} - 10485760 q^{16} - 3888896 q^{18} + 5932480 q^{19}+ \cdots - 78067269744 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/90\mathbb{Z}\right)^\times\).

\(n\) \(11\) \(37\)
\(\chi(n)\) \(e\left(\frac{1}{6}\right)\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −19.5959 − 11.3137i −0.612372 − 0.353553i
\(3\) −126.394 − 207.542i −0.520141 − 0.854081i
\(4\) 256.000 + 443.405i 0.250000 + 0.433013i
\(5\) −1210.31 + 698.771i −0.387298 + 0.223607i
\(6\) 128.746 + 5496.95i 0.0165568 + 0.706913i
\(7\) 6493.94 − 11247.8i 0.386383 − 0.669235i −0.605577 − 0.795787i \(-0.707058\pi\)
0.991960 + 0.126552i \(0.0403910\pi\)
\(8\) − 11585.2i − 0.353553i
\(9\) −27098.0 + 52464.1i −0.458907 + 0.888484i
\(10\) 31622.8 0.316228
\(11\) 89770.6 + 51829.1i 0.557405 + 0.321818i 0.752103 − 0.659045i \(-0.229039\pi\)
−0.194698 + 0.980863i \(0.562373\pi\)
\(12\) 59668.1 − 109174.i 0.239793 − 0.438748i
\(13\) −101631. − 176030.i −0.273722 − 0.474100i 0.696090 − 0.717954i \(-0.254921\pi\)
−0.969812 + 0.243854i \(0.921588\pi\)
\(14\) −254509. + 146941.i −0.473220 + 0.273214i
\(15\) 298000. + 162868.i 0.392428 + 0.214477i
\(16\) −131072. + 227023.i −0.125000 + 0.216506i
\(17\) − 636934.i − 0.448590i −0.974521 − 0.224295i \(-0.927992\pi\)
0.974521 − 0.224295i \(-0.0720079\pi\)
\(18\) 1.12457e6 − 721503.i 0.595149 − 0.381835i
\(19\) 1.63516e6 0.660378 0.330189 − 0.943915i \(-0.392888\pi\)
0.330189 + 0.943915i \(0.392888\pi\)
\(20\) −619677. − 357771.i −0.193649 − 0.111803i
\(21\) −3.15519e6 + 73898.5i −0.772554 + 0.0180942i
\(22\) −1.17276e6 − 2.03128e6i −0.227560 − 0.394145i
\(23\) 1.45501e6 − 840052.i 0.226062 − 0.130517i −0.382692 − 0.923876i \(-0.625003\pi\)
0.608754 + 0.793359i \(0.291670\pi\)
\(24\) −2.40442e6 + 1.46431e6i −0.301963 + 0.183898i
\(25\) 976562. − 1.69146e6i 0.100000 − 0.173205i
\(26\) 4.59929e6i 0.387101i
\(27\) 1.43135e7 − 1.00720e6i 0.997533 − 0.0701932i
\(28\) 6.64979e6 0.386383
\(29\) −3.31457e7 − 1.91367e7i −1.61598 − 0.932989i −0.987945 − 0.154808i \(-0.950524\pi\)
−0.628040 − 0.778181i \(-0.716142\pi\)
\(30\) −3.99694e6 − 6.56304e6i −0.164483 − 0.270084i
\(31\) −2.47249e7 − 4.28248e7i −0.863626 − 1.49584i −0.868405 − 0.495856i \(-0.834854\pi\)
0.00477827 − 0.999989i \(-0.498479\pi\)
\(32\) 5.13695e6 − 2.96582e6i 0.153093 − 0.0883883i
\(33\) −589795. − 2.51820e7i −0.0150706 − 0.643459i
\(34\) −7.20608e6 + 1.24813e7i −0.158600 + 0.274704i
\(35\) 1.81511e7i 0.345591i
\(36\) −3.01999e7 + 1.41542e6i −0.499452 + 0.0234084i
\(37\) −7.32260e7 −1.05598 −0.527991 − 0.849250i \(-0.677055\pi\)
−0.527991 + 0.849250i \(0.677055\pi\)
\(38\) −3.20425e7 − 1.84997e7i −0.404397 − 0.233479i
\(39\) −2.36880e7 + 4.33418e7i −0.262546 + 0.480379i
\(40\) 8.09543e6 + 1.40217e7i 0.0790569 + 0.136931i
\(41\) 3.49881e7 − 2.02004e7i 0.301996 − 0.174358i −0.341343 − 0.939939i \(-0.610882\pi\)
0.643339 + 0.765581i \(0.277548\pi\)
\(42\) 6.26649e7 + 3.42488e7i 0.479488 + 0.262059i
\(43\) 1.40217e8 − 2.42863e8i 0.953803 − 1.65204i 0.216720 − 0.976234i \(-0.430464\pi\)
0.737084 − 0.675802i \(-0.236202\pi\)
\(44\) 5.30730e7i 0.321818i
\(45\) −3.86349e6 − 8.24330e7i −0.0209371 − 0.446723i
\(46\) −3.80164e7 −0.184579
\(47\) 2.72299e8 + 1.57212e8i 1.18729 + 0.685481i 0.957689 − 0.287804i \(-0.0929253\pi\)
0.229599 + 0.973285i \(0.426259\pi\)
\(48\) 6.36835e7 − 1.49155e6i 0.249931 − 0.00585371i
\(49\) 5.68952e7 + 9.85453e7i 0.201417 + 0.348864i
\(50\) −3.82733e7 + 2.20971e7i −0.122474 + 0.0707107i
\(51\) −1.32190e8 + 8.05047e7i −0.383132 + 0.233330i
\(52\) 5.20351e7 − 9.01274e7i 0.136861 − 0.237050i
\(53\) 4.49386e7i 0.107458i 0.998556 + 0.0537292i \(0.0171108\pi\)
−0.998556 + 0.0537292i \(0.982889\pi\)
\(54\) −2.91882e8 − 1.42202e8i −0.635679 − 0.309697i
\(55\) −1.44867e8 −0.287843
\(56\) −1.30309e8 − 7.52338e7i −0.236610 − 0.136607i
\(57\) −2.06675e8 − 3.39364e8i −0.343489 − 0.564016i
\(58\) 4.33014e8 + 7.50001e8i 0.659723 + 1.14267i
\(59\) −4.02237e8 + 2.32232e8i −0.562629 + 0.324834i −0.754200 − 0.656645i \(-0.771975\pi\)
0.191571 + 0.981479i \(0.438642\pi\)
\(60\) 4.07130e6 + 1.73829e8i 0.00523572 + 0.223545i
\(61\) −4.29710e8 + 7.44279e8i −0.508775 + 0.881225i 0.491173 + 0.871062i \(0.336568\pi\)
−0.999948 + 0.0101627i \(0.996765\pi\)
\(62\) 1.11892e9i 1.22135i
\(63\) 4.14135e8 + 6.45492e8i 0.417291 + 0.650412i
\(64\) −1.34218e8 −0.125000
\(65\) 2.46009e8 + 1.42034e8i 0.212024 + 0.122412i
\(66\) −2.73345e8 + 5.00138e8i −0.218268 + 0.399365i
\(67\) −1.44938e8 − 2.51040e8i −0.107352 − 0.185938i 0.807345 − 0.590080i \(-0.200904\pi\)
−0.914697 + 0.404141i \(0.867570\pi\)
\(68\) 2.82420e8 − 1.63055e8i 0.194245 − 0.112147i
\(69\) −3.58251e8 − 1.95798e8i −0.229056 − 0.125188i
\(70\) 2.05356e8 − 3.55688e8i 0.122185 − 0.211631i
\(71\) 2.12427e9i 1.17738i 0.808358 + 0.588691i \(0.200356\pi\)
−0.808358 + 0.588691i \(0.799644\pi\)
\(72\) 6.07809e8 + 3.13937e8i 0.314127 + 0.162248i
\(73\) −2.76465e9 −1.33360 −0.666801 − 0.745236i \(-0.732337\pi\)
−0.666801 + 0.745236i \(0.732337\pi\)
\(74\) 1.43493e9 + 8.28457e8i 0.646654 + 0.373346i
\(75\) −4.74479e8 + 1.11129e7i −0.199945 + 0.00468297i
\(76\) 4.18601e8 + 7.25039e8i 0.165094 + 0.285952i
\(77\) 1.16593e9 − 6.73150e8i 0.430743 − 0.248690i
\(78\) 9.54544e8 − 5.81324e8i 0.330615 − 0.201347i
\(79\) −1.95855e9 + 3.39231e9i −0.636502 + 1.10245i 0.349693 + 0.936864i \(0.386286\pi\)
−0.986195 + 0.165589i \(0.947048\pi\)
\(80\) − 3.66357e8i − 0.111803i
\(81\) −2.01818e9 − 2.84335e9i −0.578808 − 0.815464i
\(82\) −9.14167e8 −0.246579
\(83\) 1.07679e9 + 6.21682e8i 0.273362 + 0.157826i 0.630415 − 0.776259i \(-0.282885\pi\)
−0.357052 + 0.934084i \(0.616218\pi\)
\(84\) −8.40495e8 − 1.38011e9i −0.200973 − 0.330002i
\(85\) 4.45071e8 + 7.70885e8i 0.100308 + 0.173738i
\(86\) −5.49537e9 + 3.17275e9i −1.16817 + 0.674441i
\(87\) 2.17768e8 + 9.29787e9i 0.0436916 + 1.86547i
\(88\) 6.00452e8 − 1.04001e9i 0.113780 − 0.197072i
\(89\) − 1.36850e9i − 0.245072i −0.992464 − 0.122536i \(-0.960897\pi\)
0.992464 − 0.122536i \(-0.0391027\pi\)
\(90\) −8.56914e8 + 1.65906e9i −0.145119 + 0.280963i
\(91\) −2.63994e9 −0.423046
\(92\) 7.44967e8 + 4.30107e8i 0.113031 + 0.0652585i
\(93\) −5.76284e9 + 1.05442e10i −0.828365 + 1.51566i
\(94\) −3.55729e9 − 6.16141e9i −0.484708 − 0.839539i
\(95\) −1.97905e9 + 1.14260e9i −0.255763 + 0.147665i
\(96\) −1.26481e9 − 6.91269e8i −0.155121 − 0.0847795i
\(97\) 1.85028e9 − 3.20478e9i 0.215466 − 0.373198i −0.737951 − 0.674855i \(-0.764206\pi\)
0.953417 + 0.301657i \(0.0975396\pi\)
\(98\) − 2.57478e9i − 0.284846i
\(99\) −5.15177e9 + 3.30527e9i −0.541727 + 0.347561i
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 90.11.h.a.11.7 ✓ 80
3.2 odd 2 270.11.h.a.251.31 80
9.4 even 3 270.11.h.a.71.31 80
9.5 odd 6 inner 90.11.h.a.41.7 yes 80
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
90.11.h.a.11.7 ✓ 80 1.1 even 1 trivial
90.11.h.a.41.7 yes 80 9.5 odd 6 inner
270.11.h.a.71.31 80 9.4 even 3
270.11.h.a.251.31 80 3.2 odd 2