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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [925,2,Mod(151,925)] 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("925.151"); 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([0, 13])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 925 = 5^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 925.bb (of order \(18\), degree \(6\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [72,-6] 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: \(7.38616218697\)
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 176.1
Character \(\chi\) \(=\) 925.176
Dual form 925.2.bb.b.226.1

$q$-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+(-2.40518 + 0.424099i) q^{2} +(0.541060 - 3.06850i) q^{3} +(3.72567 - 1.35603i) q^{4} +7.60978i q^{6} +(3.04240 + 2.55288i) q^{7} +(-4.15566 + 2.39927i) q^{8} +(-6.30389 - 2.29443i) q^{9} +(0.733441 + 1.27036i) q^{11} +(-2.14518 - 12.1659i) q^{12} +(0.385414 + 1.05891i) q^{13} +(-8.40020 - 4.84986i) q^{14} +(2.90321 - 2.43609i) q^{16} +(-0.564319 + 1.55045i) q^{17} +(16.1351 + 2.84505i) q^{18} +(7.43945 + 1.31178i) q^{19} +(9.47963 - 7.95435i) q^{21} +(-2.30282 - 2.74439i) q^{22} +(3.14430 + 1.81536i) q^{23} +(5.11371 + 14.0498i) q^{24} +(-1.37608 - 2.38343i) q^{26} +(-5.77748 + 10.0069i) q^{27} +(14.7967 + 5.38557i) q^{28} +(-5.38072 + 3.10656i) q^{29} +8.99026i q^{31} +(0.219263 - 0.261307i) q^{32} +(4.29493 - 1.56323i) q^{33} +(0.699746 - 3.96846i) q^{34} -26.5975 q^{36} +(4.62280 + 3.95345i) q^{37} -18.4496 q^{38} +(3.45782 - 0.609706i) q^{39} +(2.47011 - 0.899047i) q^{41} +(-19.4268 + 23.1520i) q^{42} +4.53873i q^{43} +(4.45520 + 3.73836i) q^{44} +(-8.33252 - 3.03279i) q^{46} +(1.25681 - 2.17686i) q^{47} +(-5.90432 - 10.2266i) q^{48} +(1.52348 + 8.64007i) q^{49} +(4.45225 + 2.57051i) q^{51} +(2.87185 + 3.42253i) q^{52} +(-0.809961 + 0.679638i) q^{53} +(9.65200 - 26.5186i) q^{54} +(-18.7682 - 3.30934i) q^{56} +(8.05038 - 22.1182i) q^{57} +(11.6241 - 9.75381i) q^{58} +(7.80224 + 9.29834i) q^{59} +(-4.20357 - 11.5492i) q^{61} +(-3.81276 - 21.6232i) q^{62} +(-13.3216 - 23.0736i) q^{63} +(-4.20642 + 7.28574i) q^{64} +(-9.66714 + 5.58133i) q^{66} +(-2.87704 - 2.41412i) q^{67} +6.54171i q^{68} +(7.27171 - 8.66609i) q^{69} +(0.0803616 - 0.455753i) q^{71} +(31.7018 - 5.58988i) q^{72} -0.804587 q^{73} +(-12.7953 - 7.54825i) q^{74} +(29.4957 - 5.20090i) q^{76} +(-1.01164 + 5.73732i) q^{77} +(-8.05811 + 2.93291i) q^{78} +(-0.506288 + 0.603371i) q^{79} +(12.1633 + 10.2062i) q^{81} +(-5.55979 + 3.20994i) q^{82} +(-8.33165 - 3.03247i) q^{83} +(24.5316 - 42.4899i) q^{84} +(-1.92487 - 10.9165i) q^{86} +(6.62120 + 18.1916i) q^{87} +(-6.09586 - 3.51945i) q^{88} +(-0.979173 - 1.16693i) q^{89} +(-1.53070 + 4.20555i) q^{91} +(14.1763 + 2.49967i) q^{92} +(27.5866 + 4.86427i) q^{93} +(-2.09966 + 5.76877i) q^{94} +(-0.683188 - 0.814192i) q^{96} +(-0.132311 - 0.0763897i) q^{97} +(-7.32849 - 20.1349i) q^{98} +(-1.70879 - 9.69102i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 72 q - 6 q^{2} - 6 q^{4} + 6 q^{7} + 18 q^{8} - 30 q^{12} + 30 q^{13} - 18 q^{14} - 18 q^{16} + 30 q^{18} - 18 q^{19} - 30 q^{22} - 30 q^{24} + 24 q^{27} - 30 q^{28} + 18 q^{29} + 36 q^{32} + 12 q^{33}+ \cdots - 12 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.40518 + 0.424099i −1.70072 + 0.299883i −0.937947 − 0.346780i \(-0.887275\pi\)
−0.762776 + 0.646663i \(0.776164\pi\)
\(3\) 0.541060 − 3.06850i 0.312381 − 1.77160i −0.274162 − 0.961683i \(-0.588401\pi\)
0.586543 − 0.809918i \(-0.300488\pi\)
\(4\) 3.72567 − 1.35603i 1.86283 − 0.678016i
\(5\) 0 0
\(6\) 7.60978i 3.10668i
\(7\) 3.04240 + 2.55288i 1.14992 + 0.964896i 0.999718 − 0.0237594i \(-0.00756356\pi\)
0.150201 + 0.988656i \(0.452008\pi\)
\(8\) −4.15566 + 2.39927i −1.46925 + 0.848270i
\(9\) −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 \(0.762355\pi\)
\(12\) −2.14518 − 12.1659i −0.619260 − 3.51200i
\(13\) 0.385414 + 1.05891i 0.106894 + 0.293690i 0.981595 − 0.190973i \(-0.0611643\pi\)
−0.874701 + 0.484663i \(0.838942\pi\)
\(14\) −8.40020 − 4.84986i −2.24505 − 1.29618i
\(15\) 0 0
\(16\) 2.90321 − 2.43609i 0.725803 − 0.609021i
\(17\) −0.564319 + 1.55045i −0.136868 + 0.376041i −0.989124 − 0.147084i \(-0.953011\pi\)
0.852256 + 0.523124i \(0.175234\pi\)
\(18\) 16.1351 + 2.84505i 3.80307 + 0.670585i
\(19\) 7.43945 + 1.31178i 1.70673 + 0.300942i 0.940037 − 0.341074i \(-0.110791\pi\)
0.766691 + 0.642016i \(0.221902\pi\)
\(20\) 0 0
\(21\) 9.47963 − 7.95435i 2.06862 − 1.73578i
\(22\) −2.30282 − 2.74439i −0.490963 − 0.585106i
\(23\) 3.14430 + 1.81536i 0.655633 + 0.378530i 0.790611 − 0.612319i \(-0.209763\pi\)
−0.134978 + 0.990849i \(0.543096\pi\)
\(24\) 5.11371 + 14.0498i 1.04383 + 2.86790i
\(25\) 0 0
\(26\) −1.37608 − 2.38343i −0.269871 − 0.467430i
\(27\) −5.77748 + 10.0069i −1.11188 + 1.92583i
\(28\) 14.7967 + 5.38557i 2.79632 + 1.01778i
\(29\) −5.38072 + 3.10656i −0.999175 + 0.576874i −0.908004 − 0.418961i \(-0.862394\pi\)
−0.0911708 + 0.995835i \(0.529061\pi\)
\(30\) 0 0
\(31\) 8.99026i 1.61470i 0.590074 + 0.807349i \(0.299099\pi\)
−0.590074 + 0.807349i \(0.700901\pi\)
\(32\) 0.219263 − 0.261307i 0.0387606 − 0.0461931i
\(33\) 4.29493 − 1.56323i 0.747652 − 0.272123i
\(34\) 0.699746 − 3.96846i 0.120005 − 0.680585i
\(35\) 0 0
\(36\) −26.5975 −4.43292
\(37\) 4.62280 + 3.95345i 0.759983 + 0.649943i
\(38\) −18.4496 −2.99292
\(39\) 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 \(0.323084\pi\)
\(42\) −19.4268 + 23.1520i −2.99762 + 3.57243i
\(43\) 4.53873i 0.692150i 0.938207 + 0.346075i \(0.112486\pi\)
−0.938207 + 0.346075i \(0.887514\pi\)
\(44\) 4.45520 + 3.73836i 0.671647 + 0.563579i
\(45\) 0 0
\(46\) −8.33252 − 3.03279i −1.22856 − 0.447161i
\(47\) 1.25681 − 2.17686i 0.183325 − 0.317528i −0.759686 − 0.650290i \(-0.774647\pi\)
0.943011 + 0.332762i \(0.107981\pi\)
\(48\) −5.90432 − 10.2266i −0.852216 − 1.47608i
\(49\) 1.52348 + 8.64007i 0.217640 + 1.23430i
\(50\) 0 0
\(51\) 4.45225 + 2.57051i 0.623439 + 0.359943i
\(52\) 2.87185 + 3.42253i 0.398253 + 0.474620i
\(53\) −0.809961 + 0.679638i −0.111257 + 0.0933554i −0.696719 − 0.717344i \(-0.745357\pi\)
0.585462 + 0.810700i \(0.300913\pi\)
\(54\) 9.65200 − 26.5186i 1.31347 − 3.60873i
\(55\) 0 0
\(56\) −18.7682 − 3.30934i −2.50801 − 0.442229i
\(57\) 8.05038 − 22.1182i 1.06630 − 2.92963i
\(58\) 11.6241 − 9.75381i 1.52632 − 1.28074i
\(59\) 7.80224 + 9.29834i 1.01576 + 1.21054i 0.977426 + 0.211276i \(0.0677620\pi\)
0.0383383 + 0.999265i \(0.487794\pi\)
\(60\) 0 0
\(61\) −4.20357 − 11.5492i −0.538213 − 1.47873i −0.849075 − 0.528272i \(-0.822840\pi\)
0.310863 − 0.950455i \(-0.399382\pi\)
\(62\) −3.81276 − 21.6232i −0.484221 − 2.74615i
\(63\) −13.3216 − 23.0736i −1.67836 − 2.90700i
\(64\) −4.20642 + 7.28574i −0.525803 + 0.910717i
\(65\) 0 0
\(66\) −9.66714 + 5.58133i −1.18994 + 0.687014i
\(67\) −2.87704 − 2.41412i −0.351486 − 0.294932i 0.449900 − 0.893079i \(-0.351459\pi\)
−0.801386 + 0.598147i \(0.795904\pi\)
\(68\) 6.54171i 0.793299i
\(69\) 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 \(0.0468133\pi\)
\(72\) 31.7018 − 5.58988i 3.73609 − 0.658773i
\(73\) −0.804587 −0.0941698 −0.0470849 − 0.998891i \(-0.514993\pi\)
−0.0470849 + 0.998891i \(0.514993\pi\)
\(74\) −12.7953 − 7.54825i −1.48743 − 0.877466i
\(75\) 0 0
\(76\) 29.4957 − 5.20090i 3.38339 − 0.596584i
\(77\) −1.01164 + 5.73732i −0.115288 + 0.653828i
\(78\) −8.05811 + 2.93291i −0.912401 + 0.332087i
\(79\) −0.506288 + 0.603371i −0.0569619 + 0.0678846i −0.793773 − 0.608214i \(-0.791886\pi\)
0.736811 + 0.676099i \(0.236331\pi\)
\(80\) 0 0
\(81\) 12.1633 + 10.2062i 1.35148 + 1.13402i
\(82\) −5.55979 + 3.20994i −0.613976 + 0.354479i
\(83\) −8.33165 − 3.03247i −0.914517 − 0.332857i −0.158462 − 0.987365i \(-0.550654\pi\)
−0.756055 + 0.654508i \(0.772876\pi\)
\(84\) 24.5316 − 42.4899i 2.67661 − 4.63603i
\(85\) 0 0
\(86\) −1.92487 − 10.9165i −0.207564 − 1.17715i
\(87\) 6.62120 + 18.1916i 0.709867 + 1.95034i
\(88\) −6.09586 − 3.51945i −0.649821 − 0.375174i
\(89\) −0.979173 − 1.16693i −0.103792 − 0.123695i 0.711648 − 0.702537i \(-0.247949\pi\)
−0.815440 + 0.578842i \(0.803505\pi\)
\(90\) 0 0
\(91\) −1.53070 + 4.20555i −0.160461 + 0.440862i
\(92\) 14.1763 + 2.49967i 1.47798 + 0.260608i
\(93\) 27.5866 + 4.86427i 2.86060 + 0.504401i
\(94\) −2.09966 + 5.76877i −0.216564 + 0.595003i
\(95\) 0 0
\(96\) −0.683188 − 0.814192i −0.0697276 − 0.0830982i
\(97\) −0.132311 − 0.0763897i −0.0134341 − 0.00775620i 0.493268 − 0.869877i \(-0.335802\pi\)
−0.506702 + 0.862121i \(0.669136\pi\)
\(98\) −7.32849 − 20.1349i −0.740289 − 2.03393i
\(99\) −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.bb.b.176.1 72
5.2 odd 4 925.2.ba.c.324.1 72
5.3 odd 4 925.2.ba.b.324.12 72
5.4 even 2 185.2.w.a.176.12 yes 72
37.4 even 18 inner 925.2.bb.b.226.1 72
185.4 even 18 185.2.w.a.41.12 ✓ 72
185.39 odd 36 6845.2.a.u.1.33 36
185.78 odd 36 925.2.ba.c.374.1 72
185.109 odd 36 6845.2.a.t.1.4 36
185.152 odd 36 925.2.ba.b.374.12 72
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
185.2.w.a.41.12 ✓ 72 185.4 even 18
185.2.w.a.176.12 yes 72 5.4 even 2
925.2.ba.b.324.12 72 5.3 odd 4
925.2.ba.b.374.12 72 185.152 odd 36
925.2.ba.c.324.1 72 5.2 odd 4
925.2.ba.c.374.1 72 185.78 odd 36
925.2.bb.b.176.1 72 1.1 even 1 trivial
925.2.bb.b.226.1 72 37.4 even 18 inner
6845.2.a.t.1.4 36 185.109 odd 36
6845.2.a.u.1.33 36 185.39 odd 36