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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [90,11,Mod(7,90)] 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("90.7"); S:= CuspForms(chi, 11); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(90, base_ring=CyclotomicField(12)) chi = DirichletCharacter(H, H._module([8, 3])) N = Newforms(chi, 11, names="a")
 
Level: \( N \) \(=\) \( 90 = 2 \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 11 \)
Character orbit: \([\chi]\) \(=\) 90.k (of order \(12\), degree \(4\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [120,960] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(57.1821527406\)
Analytic rank: \(0\)
Dimension: \(120\)
Relative dimension: \(30\) over \(\Q(\zeta_{12})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label 7.14
Character \(\chi\) \(=\) 90.7
Dual form 90.11.k.b.13.14

$q$-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+(21.8564 - 5.85641i) q^{2} +(-54.6201 - 236.782i) q^{3} +(443.405 - 256.000i) q^{4} +(1139.02 - 2910.03i) q^{5} +(-2580.49 - 4855.32i) q^{6} +(-2572.53 + 689.308i) q^{7} +(8192.00 - 8192.00i) q^{8} +(-53082.3 + 25866.1i) q^{9} +(7852.62 - 70273.3i) q^{10} +(134766. - 233422. i) q^{11} +(-84835.0 - 91007.5i) q^{12} +(690735. + 185082. i) q^{13} +(-52189.5 + 30131.6i) q^{14} +(-751255. - 110754. i) q^{15} +(131072. - 227023. i) q^{16} +(-454842. - 454842. i) q^{17} +(-1.00871e6 + 876211. i) q^{18} -4.74420e6i q^{19} +(-239919. - 1.58191e6i) q^{20} +(303728. + 571479. i) q^{21} +(1.57849e6 - 5.89102e6i) q^{22} +(-6.38107e6 - 1.70980e6i) q^{23} +(-2.38716e6 - 1.49227e6i) q^{24} +(-7.17089e6 - 6.62917e6i) q^{25} +1.61809e7 q^{26} +(9.02398e6 + 1.11561e7i) q^{27} +(-964211. + 964211. i) q^{28} +(1.57898e7 + 9.11626e6i) q^{29} +(-1.70684e7 + 1.97898e6i) q^{30} +(2.20989e6 + 3.82764e6i) q^{31} +(1.53522e6 - 5.72953e6i) q^{32} +(-6.26311e7 - 1.91607e7i) q^{33} +(-1.26049e7 - 7.27746e6i) q^{34} +(-924265. + 8.27128e6i) q^{35} +(-1.69152e7 + 2.50582e7i) q^{36} +(4.18995e7 + 4.18995e7i) q^{37} +(-2.77840e7 - 1.03691e8i) q^{38} +(6.09602e6 - 1.73663e8i) q^{39} +(-1.45081e7 - 3.31698e7i) q^{40} +(9.89085e7 + 1.71315e8i) q^{41} +(9.98521e6 + 1.07117e7i) q^{42} +(-3.10770e7 - 1.15981e8i) q^{43} -1.38001e8i q^{44} +(1.48091e7 + 1.83933e8i) q^{45} -1.49481e8 q^{46} +(-1.41214e8 + 3.78382e7i) q^{47} +(-6.09142e7 - 1.86354e7i) q^{48} +(-2.38488e8 + 1.37691e8i) q^{49} +(-1.95553e8 - 1.02894e8i) q^{50} +(-8.28547e7 + 1.32542e8i) q^{51} +(3.53656e8 - 9.47619e7i) q^{52} +(-2.83458e8 + 2.83458e8i) q^{53} +(2.62567e8 + 1.90984e8i) q^{54} +(-5.25763e8 - 6.58047e8i) q^{55} +(-1.54274e7 + 2.67210e7i) q^{56} +(-1.12334e9 + 2.59129e8i) q^{57} +(3.98497e8 + 1.06777e8i) q^{58} +(-8.88843e8 + 5.13174e8i) q^{59} +(-3.61463e8 + 1.43213e8i) q^{60} +(3.81130e8 - 6.60136e8i) q^{61} +(7.07165e7 + 7.07165e7i) q^{62} +(1.18726e8 - 1.03131e8i) q^{63} -1.34218e8i q^{64} +(1.32536e9 - 1.79925e9i) q^{65} +(-1.48110e9 - 5.19907e7i) q^{66} +(-2.03351e8 + 7.58916e8i) q^{67} +(-3.18118e8 - 8.52396e7i) q^{68} +(-5.63156e7 + 1.60431e9i) q^{69} +(2.82388e7 + 1.86193e8i) q^{70} +1.13464e9 q^{71} +(-2.22955e8 + 6.46745e8i) q^{72} +(-4.13089e8 + 4.13089e8i) q^{73} +(1.16115e9 + 6.70393e8i) q^{74} +(-1.17799e9 + 2.06002e9i) q^{75} +(-1.21452e9 - 2.10360e9i) q^{76} +(-1.85791e8 + 6.93382e8i) q^{77} +(-8.83802e8 - 3.83134e9i) q^{78} +(-1.32772e9 - 7.66562e8i) q^{79} +(-5.11350e8 - 6.40007e8i) q^{80} +(2.14867e9 - 2.74606e9i) q^{81} +(3.16507e9 + 3.16507e9i) q^{82} +(-7.08790e8 - 2.64524e9i) q^{83} +(2.80973e8 + 1.75642e8i) q^{84} +(-1.84168e9 + 8.05527e8i) q^{85} +(-1.35846e9 - 2.35293e9i) q^{86} +(1.29612e9 - 4.23667e9i) q^{87} +(-8.08189e8 - 3.01620e9i) q^{88} -2.32697e9i q^{89} +(1.40086e9 + 3.93338e9i) q^{90} -1.90452e9 q^{91} +(-3.26711e9 + 8.75419e8i) q^{92} +(7.85611e8 - 7.32328e8i) q^{93} +(-2.86484e9 + 1.65402e9i) q^{94} +(-1.38058e10 - 5.40375e9i) q^{95} +(-1.44050e9 - 5.05655e7i) q^{96} +(-1.05946e9 + 2.83880e8i) q^{97} +(-4.40611e9 + 4.40611e9i) q^{98} +(-1.11599e9 + 1.58765e10i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 120 q + 960 q^{2} - 128 q^{3} - 8192 q^{6} + 1830 q^{7} + 983040 q^{8} + 105792 q^{10} + 217740 q^{11} - 65536 q^{12} - 3385658 q^{15} + 15728640 q^{16} + 2438244 q^{17} + 2534464 q^{18} + 1692672 q^{20}+ \cdots + 76966514304 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(11\) \(37\)
\(\chi(n)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{1}{4}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 21.8564 − 5.85641i 0.683013 − 0.183013i
\(3\) −54.6201 − 236.782i −0.224774 − 0.974411i
\(4\) 443.405 − 256.000i 0.433013 − 0.250000i
\(5\) 1139.02 − 2910.03i 0.364487 − 0.931209i
\(6\) −2580.49 − 4855.32i −0.331853 − 0.624399i
\(7\) −2572.53 + 689.308i −0.153063 + 0.0410132i −0.334537 − 0.942383i \(-0.608580\pi\)
0.181474 + 0.983396i \(0.441913\pi\)
\(8\) 8192.00 − 8192.00i 0.250000 − 0.250000i
\(9\) −53082.3 + 25866.1i −0.898953 + 0.438045i
\(10\) 7852.62 − 70273.3i 0.0785262 − 0.702733i
\(11\) 134766. − 233422.i 0.836793 − 1.44937i −0.0557687 − 0.998444i \(-0.517761\pi\)
0.892562 − 0.450925i \(-0.148906\pi\)
\(12\) −84835.0 − 91007.5i −0.340933 − 0.365739i
\(13\) 690735. + 185082.i 1.86035 + 0.498479i 0.999938 − 0.0111196i \(-0.00353956\pi\)
0.860412 + 0.509599i \(0.170206\pi\)
\(14\) −52189.5 + 30131.6i −0.0970382 + 0.0560250i
\(15\) −751255. − 110754.i −0.989307 − 0.145848i
\(16\) 131072. − 227023.i 0.125000 − 0.216506i
\(17\) −454842. − 454842.i −0.320343 − 0.320343i 0.528556 − 0.848899i \(-0.322734\pi\)
−0.848899 + 0.528556i \(0.822734\pi\)
\(18\) −1.00871e6 + 876211.i −0.533829 + 0.463710i
\(19\) − 4.74420e6i − 1.91600i −0.286770 − 0.958000i \(-0.592581\pi\)
0.286770 − 0.958000i \(-0.407419\pi\)
\(20\) −239919. − 1.58191e6i −0.0749747 − 0.494347i
\(21\) 303728. + 571479.i 0.0743683 + 0.139928i
\(22\) 1.57849e6 − 5.89102e6i 0.306288 − 1.14308i
\(23\) −6.38107e6 − 1.70980e6i −0.991412 − 0.265648i −0.273569 − 0.961852i \(-0.588204\pi\)
−0.717844 + 0.696204i \(0.754871\pi\)
\(24\) −2.38716e6 − 1.49227e6i −0.299796 − 0.187409i
\(25\) −7.17089e6 − 6.62917e6i −0.734299 − 0.678827i
\(26\) 1.61809e7 1.36187
\(27\) 9.02398e6 + 1.11561e7i 0.628897 + 0.777489i
\(28\) −964211. + 964211.i −0.0560250 + 0.0560250i
\(29\) 1.57898e7 + 9.11626e6i 0.769816 + 0.444454i 0.832809 − 0.553560i \(-0.186731\pi\)
−0.0629926 + 0.998014i \(0.520064\pi\)
\(30\) −1.70684e7 + 1.97898e6i −0.702401 + 0.0814394i
\(31\) 2.20989e6 + 3.82764e6i 0.0771902 + 0.133697i 0.902037 − 0.431660i \(-0.142072\pi\)
−0.824846 + 0.565357i \(0.808738\pi\)
\(32\) 1.53522e6 − 5.72953e6i 0.0457532 − 0.170753i
\(33\) −6.26311e7 − 1.91607e7i −1.60037 − 0.489600i
\(34\) −1.26049e7 − 7.27746e6i −0.277425 − 0.160172i
\(35\) −924265. + 8.27128e6i −0.0175977 + 0.157482i
\(36\) −1.69152e7 + 2.50582e7i −0.279747 + 0.414417i
\(37\) 4.18995e7 + 4.18995e7i 0.604228 + 0.604228i 0.941432 − 0.337204i \(-0.109481\pi\)
−0.337204 + 0.941432i \(0.609481\pi\)
\(38\) −2.77840e7 − 1.03691e8i −0.350652 − 1.30865i
\(39\) 6.09602e6 − 1.73663e8i 0.0675653 − 1.92479i
\(40\) −1.45081e7 − 3.31698e7i −0.141680 − 0.323924i
\(41\) 9.89085e7 + 1.71315e8i 0.853718 + 1.47868i 0.877829 + 0.478974i \(0.158991\pi\)
−0.0241111 + 0.999709i \(0.507676\pi\)
\(42\) 9.98521e6 + 1.07117e7i 0.0764030 + 0.0819621i
\(43\) −3.10770e7 − 1.15981e8i −0.211396 − 0.788941i −0.987404 − 0.158218i \(-0.949425\pi\)
0.776008 − 0.630723i \(-0.217241\pi\)
\(44\) − 1.38001e8i − 0.836793i
\(45\) 1.48091e7 + 1.83933e8i 0.0802541 + 0.996774i
\(46\) −1.49481e8 −0.725764
\(47\) −1.41214e8 + 3.78382e7i −0.615728 + 0.164984i −0.553185 − 0.833058i \(-0.686588\pi\)
−0.0625431 + 0.998042i \(0.519921\pi\)
\(48\) −6.09142e7 − 1.86354e7i −0.239063 − 0.0731364i
\(49\) −2.38488e8 + 1.37691e8i −0.844279 + 0.487445i
\(50\) −1.95553e8 − 1.02894e8i −0.625769 − 0.329261i
\(51\) −8.28547e7 + 1.32542e8i −0.240141 + 0.384151i
\(52\) 3.53656e8 − 9.47619e7i 0.930175 − 0.249240i
\(53\) −2.83458e8 + 2.83458e8i −0.677812 + 0.677812i −0.959505 − 0.281692i \(-0.909104\pi\)
0.281692 + 0.959505i \(0.409104\pi\)
\(54\) 2.62567e8 + 1.90984e8i 0.571835 + 0.415939i
\(55\) −5.25763e8 − 6.58047e8i −1.04466 − 1.30750i
\(56\) −1.54274e7 + 2.67210e7i −0.0280125 + 0.0485191i
\(57\) −1.12334e9 + 2.59129e8i −1.86697 + 0.430667i
\(58\) 3.98497e8 + 1.06777e8i 0.607135 + 0.162681i
\(59\) −8.88843e8 + 5.13174e8i −1.24327 + 0.717802i −0.969758 − 0.244067i \(-0.921518\pi\)
−0.273511 + 0.961869i \(0.588185\pi\)
\(60\) −3.61463e8 + 1.43213e8i −0.464845 + 0.184172i
\(61\) 3.81130e8 − 6.60136e8i 0.451257 − 0.781600i −0.547207 − 0.836997i \(-0.684309\pi\)
0.998464 + 0.0553970i \(0.0176425\pi\)
\(62\) 7.07165e7 + 7.07165e7i 0.0771902 + 0.0771902i
\(63\) 1.18726e8 − 1.03131e8i 0.119631 − 0.103917i
\(64\) − 1.34218e8i − 0.125000i
\(65\) 1.32536e9 − 1.79925e9i 1.14226 − 1.55068i
\(66\) −1.48110e9 − 5.19907e7i −1.18268 − 0.0415151i
\(67\) −2.03351e8 + 7.58916e8i −0.150616 + 0.562108i 0.848825 + 0.528675i \(0.177311\pi\)
−0.999441 + 0.0334335i \(0.989356\pi\)
\(68\) −3.18118e8 − 8.52396e7i −0.218798 − 0.0586269i
\(69\) −5.63156e7 + 1.60431e9i −0.0360067 + 1.02575i
\(70\) 2.82388e7 + 1.86193e8i 0.0168018 + 0.110783i
\(71\) 1.13464e9 0.628876 0.314438 − 0.949278i \(-0.398184\pi\)
0.314438 + 0.949278i \(0.398184\pi\)
\(72\) −2.22955e8 + 6.46745e8i −0.115227 + 0.334249i
\(73\) −4.13089e8 + 4.13089e8i −0.199264 + 0.199264i −0.799685 − 0.600420i \(-0.795000\pi\)
0.600420 + 0.799685i \(0.295000\pi\)
\(74\) 1.16115e9 + 6.70393e8i 0.523277 + 0.302114i
\(75\) −1.17799e9 + 2.06002e9i −0.496405 + 0.868091i
\(76\) −1.21452e9 − 2.10360e9i −0.479000 − 0.829652i
\(77\) −1.85791e8 + 6.93382e8i −0.0686391 + 0.256164i
\(78\) −8.83802e8 − 3.83134e9i −0.306113 − 1.32702i
\(79\) −1.32772e9 − 7.66562e8i −0.431492 − 0.249122i 0.268490 − 0.963282i \(-0.413475\pi\)
−0.699982 + 0.714161i \(0.746809\pi\)
\(80\) −5.11350e8 − 6.40007e8i −0.156052 − 0.195315i
\(81\) 2.14867e9 − 2.74606e9i 0.616234 − 0.787563i
\(82\) 3.16507e9 + 3.16507e9i 0.853718 + 0.853718i
\(83\) −7.08790e8 − 2.64524e9i −0.179940 − 0.671544i −0.995657 − 0.0930940i \(-0.970324\pi\)
0.815718 − 0.578450i \(-0.196342\pi\)
\(84\) 2.80973e8 + 1.75642e8i 0.0671843 + 0.0419984i
\(85\) −1.84168e9 + 8.05527e8i −0.415067 + 0.181545i
\(86\) −1.35846e9 − 2.35293e9i −0.288772 − 0.500168i
\(87\) 1.29612e9 − 4.23667e9i 0.260046 − 0.850019i
\(88\) −8.08189e8 − 3.01620e9i −0.153144 − 0.571540i
\(89\) − 2.32697e9i − 0.416716i −0.978053 − 0.208358i \(-0.933188\pi\)
0.978053 − 0.208358i \(-0.0668120\pi\)
\(90\) 1.40086e9 + 3.93338e9i 0.237237 + 0.666122i
\(91\) −1.90452e9 −0.305195
\(92\) −3.26711e9 + 8.75419e8i −0.495706 + 0.132824i
\(93\) 7.85611e8 − 7.32328e8i 0.112926 − 0.105267i
\(94\) −2.86484e9 + 1.65402e9i −0.390356 + 0.225372i
\(95\) −1.38058e10 − 5.40375e9i −1.78419 − 0.698357i
\(96\) −1.44050e9 − 5.05655e7i −0.176668 − 0.00620151i
\(97\) −1.05946e9 + 2.83880e8i −0.123374 + 0.0330580i −0.319978 − 0.947425i \(-0.603675\pi\)
0.196604 + 0.980483i \(0.437009\pi\)
\(98\) −4.40611e9 + 4.40611e9i −0.487445 + 0.487445i
\(99\) −1.11599e9 + 1.58765e10i −0.117350 + 1.66947i
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.k.b.7.14 ✓ 120
5.3 odd 4 inner 90.11.k.b.43.29 yes 120
9.4 even 3 inner 90.11.k.b.67.29 yes 120
45.13 odd 12 inner 90.11.k.b.13.14 yes 120
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
90.11.k.b.7.14 ✓ 120 1.1 even 1 trivial
90.11.k.b.13.14 yes 120 45.13 odd 12 inner
90.11.k.b.43.29 yes 120 5.3 odd 4 inner
90.11.k.b.67.29 yes 120 9.4 even 3 inner