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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 324.1
Character \(\chi\) \(=\) 925.324
Dual form 925.2.ba.c.374.1

$q$-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+(-0.424099 - 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 \(-0.723836\pi\)
0.346780 − 0.937947i \(-0.387275\pi\)
\(3\) −3.06850 − 0.541060i −1.77160 − 0.312381i −0.809918 − 0.586543i \(-0.800488\pi\)
−0.961683 + 0.274162i \(0.911599\pi\)
\(4\) −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 \(0.492436\pi\)
−0.988656 + 0.150201i \(0.952008\pi\)
\(8\) 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 −0.190973 − 0.981595i \(-0.561164\pi\)
0.484663 + 0.874701i \(0.338942\pi\)
\(14\) 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 0.147084 − 0.989124i \(-0.453011\pi\)
−0.523124 + 0.852256i \(0.675234\pi\)
\(18\) 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 −0.612319 − 0.790611i \(-0.709763\pi\)
0.990849 + 0.134978i \(0.0430965\pi\)
\(24\) −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 \(-0.387514\pi\)
0.346075 + 0.938207i \(0.387514\pi\)
\(44\) −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 0.650290 − 0.759686i \(-0.274647\pi\)
−0.332762 + 0.943011i \(0.607981\pi\)
\(48\) −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 0.810700 − 0.585462i \(-0.199087\pi\)
−0.717344 + 0.696719i \(0.754643\pi\)
\(54\) −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 −0.598147 − 0.801386i \(-0.704096\pi\)
0.893079 + 0.449900i \(0.148541\pi\)
\(68\) 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 \(0.0149931\pi\)
−0.998891 + 0.0470849i \(0.985007\pi\)
\(74\) 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 \(0.272876\pi\)
−0.987365 + 0.158462i \(0.949346\pi\)
\(84\) −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 −0.862121 − 0.506702i \(-0.830864\pi\)
0.869877 + 0.493268i \(0.164198\pi\)
\(98\) −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.c.324.1 72
5.2 odd 4 185.2.w.a.176.12 yes 72
5.3 odd 4 925.2.bb.b.176.1 72
5.4 even 2 925.2.ba.b.324.12 72
37.4 even 18 925.2.ba.b.374.12 72
185.2 even 36 6845.2.a.u.1.33 36
185.4 even 18 inner 925.2.ba.c.374.1 72
185.72 even 36 6845.2.a.t.1.4 36
185.78 odd 36 925.2.bb.b.226.1 72
185.152 odd 36 185.2.w.a.41.12 ✓ 72
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
185.2.w.a.41.12 ✓ 72 185.152 odd 36
185.2.w.a.176.12 yes 72 5.2 odd 4
925.2.ba.b.324.12 72 5.4 even 2
925.2.ba.b.374.12 72 37.4 even 18
925.2.ba.c.324.1 72 1.1 even 1 trivial
925.2.ba.c.374.1 72 185.4 even 18 inner
925.2.bb.b.176.1 72 5.3 odd 4
925.2.bb.b.226.1 72 185.78 odd 36
6845.2.a.t.1.4 36 185.72 even 36
6845.2.a.u.1.33 36 185.2 even 36