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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [925,2,Mod(99,925)] 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("925.99"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(925, base_ring=CyclotomicField(18)) chi = DirichletCharacter(H, H._module([9, 5])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 925 = 5^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 925.ba (of order \(18\), degree \(6\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [72,0,0,6,0,0,-6] 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: \(7.38616218697\)
Analytic rank: \(0\)
Dimension: \(72\)
Relative dimension: \(12\) over \(\Q(\zeta_{18})\)
Twist minimal: no (minimal twist has level 185)
Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label 324.12
Character \(\chi\) \(=\) 925.324
Dual form 925.2.ba.b.374.12

$q$-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+(0.424099 + 2.40518i) q^{2} +(3.06850 + 0.541060i) q^{3} +(-3.72567 + 1.35603i) q^{4} +7.60978i q^{6} +(2.55288 - 3.04240i) q^{7} +(-2.39927 - 4.15566i) q^{8} +(6.30389 + 2.29443i) q^{9} +(0.733441 + 1.27036i) q^{11} +(-12.1659 + 2.14518i) q^{12} +(-1.05891 + 0.385414i) q^{13} +(8.40020 + 4.84986i) q^{14} +(2.90321 - 2.43609i) q^{16} +(1.55045 + 0.564319i) q^{17} +(-2.84505 + 16.1351i) q^{18} +(-7.43945 - 1.31178i) q^{19} +(9.47963 - 7.95435i) q^{21} +(-2.74439 + 2.30282i) q^{22} +(-1.81536 + 3.14430i) q^{23} +(-5.11371 - 14.0498i) q^{24} +(-1.37608 - 2.38343i) q^{26} +(10.0069 + 5.77748i) q^{27} +(-5.38557 + 14.7967i) q^{28} +(5.38072 - 3.10656i) q^{29} +8.99026i q^{31} +(-0.261307 - 0.219263i) q^{32} +(1.56323 + 4.29493i) q^{33} +(-0.699746 + 3.96846i) q^{34} -26.5975 q^{36} +(3.95345 - 4.62280i) q^{37} -18.4496i q^{38} +(-3.45782 + 0.609706i) q^{39} +(2.47011 - 0.899047i) q^{41} +(23.1520 + 19.4268i) q^{42} -4.53873 q^{43} +(-4.45520 - 3.73836i) q^{44} +(-8.33252 - 3.03279i) q^{46} +(-2.17686 - 1.25681i) q^{47} +(10.2266 - 5.90432i) q^{48} +(-1.52348 - 8.64007i) q^{49} +(4.45225 + 2.57051i) q^{51} +(3.42253 - 2.87185i) q^{52} +(-0.679638 - 0.809961i) q^{53} +(-9.65200 + 26.5186i) q^{54} +(-18.7682 - 3.30934i) q^{56} +(-22.1182 - 8.05038i) q^{57} +(9.75381 + 11.6241i) q^{58} +(-7.80224 - 9.29834i) q^{59} +(-4.20357 - 11.5492i) q^{61} +(-21.6232 + 3.81276i) q^{62} +(23.0736 - 13.3216i) q^{63} +(4.20642 - 7.28574i) q^{64} +(-9.66714 + 5.58133i) q^{66} +(-2.41412 + 2.87704i) q^{67} -6.54171 q^{68} +(-7.27171 + 8.66609i) q^{69} +(0.0803616 - 0.455753i) q^{71} +(-5.58988 - 31.7018i) q^{72} -0.804587i q^{73} +(12.7953 + 7.54825i) q^{74} +(29.4957 - 5.20090i) q^{76} +(5.73732 + 1.01164i) q^{77} +(-2.93291 - 8.05811i) q^{78} +(0.506288 - 0.603371i) q^{79} +(12.1633 + 10.2062i) q^{81} +(3.20994 + 5.55979i) q^{82} +(3.03247 - 8.33165i) q^{83} +(-24.5316 + 42.4899i) q^{84} +(-1.92487 - 10.9165i) q^{86} +(18.1916 - 6.62120i) q^{87} +(3.51945 - 6.09586i) q^{88} +(0.979173 + 1.16693i) q^{89} +(-1.53070 + 4.20555i) q^{91} +(2.49967 - 14.1763i) q^{92} +(-4.86427 + 27.5866i) q^{93} +(2.09966 - 5.76877i) q^{94} +(-0.683188 - 0.814192i) q^{96} +(-0.0763897 + 0.132311i) q^{97} +(20.1349 - 7.32849i) q^{98} +(1.70879 + 9.69102i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 72 q + 6 q^{4} - 6 q^{7} - 18 q^{8} - 42 q^{12} - 18 q^{13} + 18 q^{14} - 18 q^{16} + 12 q^{17} + 24 q^{18} + 18 q^{19} + 24 q^{22} - 24 q^{23} + 30 q^{24} - 48 q^{28} - 18 q^{29} + 24 q^{33} - 66 q^{34}+ \cdots + 12 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/925\mathbb{Z}\right)^\times\).

\(n\) \(76\) \(852\)
\(\chi(n)\) \(e\left(\frac{17}{18}\right)\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.424099 + 2.40518i 0.299883 + 1.70072i 0.646663 + 0.762776i \(0.276164\pi\)
−0.346780 + 0.937947i \(0.612725\pi\)
\(3\) 3.06850 + 0.541060i 1.77160 + 0.312381i 0.961683 − 0.274162i \(-0.0884006\pi\)
0.809918 + 0.586543i \(0.199512\pi\)
\(4\) −3.72567 + 1.35603i −1.86283 + 0.678016i
\(5\) 0 0
\(6\) 7.60978i 3.10668i
\(7\) 2.55288 − 3.04240i 0.964896 − 1.14992i −0.0237594 − 0.999718i \(-0.507564\pi\)
0.988656 − 0.150201i \(-0.0479920\pi\)
\(8\) −2.39927 − 4.15566i −0.848270 − 1.46925i
\(9\) 6.30389 + 2.29443i 2.10130 + 0.764809i
\(10\) 0 0
\(11\) 0.733441 + 1.27036i 0.221141 + 0.383027i 0.955155 − 0.296107i \(-0.0956886\pi\)
−0.734014 + 0.679135i \(0.762355\pi\)
\(12\) −12.1659 + 2.14518i −3.51200 + 0.619260i
\(13\) −1.05891 + 0.385414i −0.293690 + 0.106894i −0.484663 − 0.874701i \(-0.661058\pi\)
0.190973 + 0.981595i \(0.438836\pi\)
\(14\) 8.40020 + 4.84986i 2.24505 + 1.29618i
\(15\) 0 0
\(16\) 2.90321 − 2.43609i 0.725803 − 0.609021i
\(17\) 1.55045 + 0.564319i 0.376041 + 0.136868i 0.523124 − 0.852256i \(-0.324766\pi\)
−0.147084 + 0.989124i \(0.546989\pi\)
\(18\) −2.84505 + 16.1351i −0.670585 + 3.80307i
\(19\) −7.43945 − 1.31178i −1.70673 − 0.300942i −0.766691 − 0.642016i \(-0.778098\pi\)
−0.940037 + 0.341074i \(0.889209\pi\)
\(20\) 0 0
\(21\) 9.47963 − 7.95435i 2.06862 − 1.73578i
\(22\) −2.74439 + 2.30282i −0.585106 + 0.490963i
\(23\) −1.81536 + 3.14430i −0.378530 + 0.655633i −0.990849 − 0.134978i \(-0.956904\pi\)
0.612319 + 0.790611i \(0.290237\pi\)
\(24\) −5.11371 − 14.0498i −1.04383 − 2.86790i
\(25\) 0 0
\(26\) −1.37608 − 2.38343i −0.269871 − 0.467430i
\(27\) 10.0069 + 5.77748i 1.92583 + 1.11188i
\(28\) −5.38557 + 14.7967i −1.01778 + 2.79632i
\(29\) 5.38072 − 3.10656i 0.999175 − 0.576874i 0.0911708 − 0.995835i \(-0.470939\pi\)
0.908004 + 0.418961i \(0.137606\pi\)
\(30\) 0 0
\(31\) 8.99026i 1.61470i 0.590074 + 0.807349i \(0.299099\pi\)
−0.590074 + 0.807349i \(0.700901\pi\)
\(32\) −0.261307 − 0.219263i −0.0461931 − 0.0387606i
\(33\) 1.56323 + 4.29493i 0.272123 + 0.747652i
\(34\) −0.699746 + 3.96846i −0.120005 + 0.680585i
\(35\) 0 0
\(36\) −26.5975 −4.43292
\(37\) 3.95345 − 4.62280i 0.649943 − 0.759983i
\(38\) − 18.4496i − 2.99292i
\(39\) −3.45782 + 0.609706i −0.553694 + 0.0976311i
\(40\) 0 0
\(41\) 2.47011 − 0.899047i 0.385766 − 0.140407i −0.141854 − 0.989888i \(-0.545306\pi\)
0.527621 + 0.849480i \(0.323084\pi\)
\(42\) 23.1520 + 19.4268i 3.57243 + 2.99762i
\(43\) −4.53873 −0.692150 −0.346075 − 0.938207i \(-0.612486\pi\)
−0.346075 + 0.938207i \(0.612486\pi\)
\(44\) −4.45520 − 3.73836i −0.671647 − 0.563579i
\(45\) 0 0
\(46\) −8.33252 − 3.03279i −1.22856 − 0.447161i
\(47\) −2.17686 − 1.25681i −0.317528 − 0.183325i 0.332762 − 0.943011i \(-0.392019\pi\)
−0.650290 + 0.759686i \(0.725353\pi\)
\(48\) 10.2266 − 5.90432i 1.47608 − 0.852216i
\(49\) −1.52348 − 8.64007i −0.217640 − 1.23430i
\(50\) 0 0
\(51\) 4.45225 + 2.57051i 0.623439 + 0.359943i
\(52\) 3.42253 − 2.87185i 0.474620 − 0.398253i
\(53\) −0.679638 − 0.809961i −0.0933554 − 0.111257i 0.717344 − 0.696719i \(-0.245357\pi\)
−0.810700 + 0.585462i \(0.800913\pi\)
\(54\) −9.65200 + 26.5186i −1.31347 + 3.60873i
\(55\) 0 0
\(56\) −18.7682 − 3.30934i −2.50801 − 0.442229i
\(57\) −22.1182 − 8.05038i −2.92963 − 1.06630i
\(58\) 9.75381 + 11.6241i 1.28074 + 1.52632i
\(59\) −7.80224 − 9.29834i −1.01576 − 1.21054i −0.977426 − 0.211276i \(-0.932238\pi\)
−0.0383383 − 0.999265i \(-0.512206\pi\)
\(60\) 0 0
\(61\) −4.20357 − 11.5492i −0.538213 − 1.47873i −0.849075 − 0.528272i \(-0.822840\pi\)
0.310863 − 0.950455i \(-0.399382\pi\)
\(62\) −21.6232 + 3.81276i −2.74615 + 0.484221i
\(63\) 23.0736 − 13.3216i 2.90700 − 1.67836i
\(64\) 4.20642 − 7.28574i 0.525803 − 0.910717i
\(65\) 0 0
\(66\) −9.66714 + 5.58133i −1.18994 + 0.687014i
\(67\) −2.41412 + 2.87704i −0.294932 + 0.351486i −0.893079 − 0.449900i \(-0.851459\pi\)
0.598147 + 0.801386i \(0.295904\pi\)
\(68\) −6.54171 −0.793299
\(69\) −7.27171 + 8.66609i −0.875411 + 1.04327i
\(70\) 0 0
\(71\) 0.0803616 − 0.455753i 0.00953716 − 0.0540879i −0.979668 − 0.200627i \(-0.935702\pi\)
0.989205 + 0.146539i \(0.0468133\pi\)
\(72\) −5.58988 − 31.7018i −0.658773 − 3.73609i
\(73\) − 0.804587i − 0.0941698i −0.998891 − 0.0470849i \(-0.985007\pi\)
0.998891 − 0.0470849i \(-0.0149931\pi\)
\(74\) 12.7953 + 7.54825i 1.48743 + 0.877466i
\(75\) 0 0
\(76\) 29.4957 − 5.20090i 3.38339 − 0.596584i
\(77\) 5.73732 + 1.01164i 0.653828 + 0.115288i
\(78\) −2.93291 − 8.05811i −0.332087 − 0.912401i
\(79\) 0.506288 − 0.603371i 0.0569619 − 0.0678846i −0.736811 − 0.676099i \(-0.763669\pi\)
0.793773 + 0.608214i \(0.208114\pi\)
\(80\) 0 0
\(81\) 12.1633 + 10.2062i 1.35148 + 1.13402i
\(82\) 3.20994 + 5.55979i 0.354479 + 0.613976i
\(83\) 3.03247 − 8.33165i 0.332857 − 0.914517i −0.654508 − 0.756055i \(-0.727124\pi\)
0.987365 − 0.158462i \(-0.0506536\pi\)
\(84\) −24.5316 + 42.4899i −2.67661 + 4.63603i
\(85\) 0 0
\(86\) −1.92487 − 10.9165i −0.207564 − 1.17715i
\(87\) 18.1916 − 6.62120i 1.95034 − 0.709867i
\(88\) 3.51945 − 6.09586i 0.375174 − 0.649821i
\(89\) 0.979173 + 1.16693i 0.103792 + 0.123695i 0.815440 − 0.578842i \(-0.196495\pi\)
−0.711648 + 0.702537i \(0.752051\pi\)
\(90\) 0 0
\(91\) −1.53070 + 4.20555i −0.160461 + 0.440862i
\(92\) 2.49967 − 14.1763i 0.260608 − 1.47798i
\(93\) −4.86427 + 27.5866i −0.504401 + 2.86060i
\(94\) 2.09966 − 5.76877i 0.216564 − 0.595003i
\(95\) 0 0
\(96\) −0.683188 − 0.814192i −0.0697276 − 0.0830982i
\(97\) −0.0763897 + 0.132311i −0.00775620 + 0.0134341i −0.869877 − 0.493268i \(-0.835802\pi\)
0.862121 + 0.506702i \(0.169136\pi\)
\(98\) 20.1349 − 7.32849i 2.03393 − 0.740289i
\(99\) 1.70879 + 9.69102i 0.171740 + 0.973984i
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 925.2.ba.b.324.12 72
5.2 odd 4 925.2.bb.b.176.1 72
5.3 odd 4 185.2.w.a.176.12 yes 72
5.4 even 2 925.2.ba.c.324.1 72
37.4 even 18 925.2.ba.c.374.1 72
185.4 even 18 inner 925.2.ba.b.374.12 72
185.78 odd 36 185.2.w.a.41.12 ✓ 72
185.113 even 36 6845.2.a.u.1.33 36
185.152 odd 36 925.2.bb.b.226.1 72
185.183 even 36 6845.2.a.t.1.4 36
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
185.2.w.a.41.12 ✓ 72 185.78 odd 36
185.2.w.a.176.12 yes 72 5.3 odd 4
925.2.ba.b.324.12 72 1.1 even 1 trivial
925.2.ba.b.374.12 72 185.4 even 18 inner
925.2.ba.c.324.1 72 5.4 even 2
925.2.ba.c.374.1 72 37.4 even 18
925.2.bb.b.176.1 72 5.2 odd 4
925.2.bb.b.226.1 72 185.152 odd 36
6845.2.a.t.1.4 36 185.183 even 36
6845.2.a.u.1.33 36 185.113 even 36