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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 299.7
Character \(\chi\) \(=\) 925.299
Dual form 925.2.ba.c.99.7

$q$-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+(0.113027 + 0.0948413i) q^{2} +(-1.90530 - 2.27065i) q^{3} +(-0.343516 - 1.94818i) q^{4} -0.437347i q^{6} +(0.370651 - 1.01836i) q^{7} +(0.293488 - 0.508336i) q^{8} +(-1.00473 + 5.69811i) q^{9} +(-2.73043 + 4.72924i) q^{11} +(-3.76912 + 4.49186i) q^{12} +(0.190904 + 1.08267i) q^{13} +(0.138476 - 0.0799492i) q^{14} +(-3.63647 + 1.32357i) q^{16} +(-0.343537 + 1.94830i) q^{17} +(-0.653979 + 0.548753i) q^{18} +(4.07265 + 4.85359i) q^{19} +(-3.01853 + 1.09866i) q^{21} +(-0.757141 + 0.275577i) q^{22} +(-2.00254 - 3.46850i) q^{23} +(-1.71343 + 0.302125i) q^{24} +(-0.0811047 + 0.140477i) q^{26} +(7.15171 - 4.12904i) q^{27} +(-2.11126 - 0.372272i) q^{28} +(-3.78731 - 2.18661i) q^{29} +7.01816i q^{31} +(-1.63970 - 0.596803i) q^{32} +(15.9407 - 2.81078i) q^{33} +(-0.223608 + 0.187629i) q^{34} +11.4461 q^{36} +(1.22092 - 5.95897i) q^{37} +0.934844i q^{38} +(2.09464 - 2.49629i) q^{39} +(-1.32170 - 7.49572i) q^{41} +(-0.445375 - 0.162103i) q^{42} -1.79786 q^{43} +(10.1513 + 3.69479i) q^{44} +(0.102615 - 0.581959i) q^{46} +(-10.6029 + 6.12161i) q^{47} +(9.93393 + 5.73536i) q^{48} +(4.46264 + 3.74460i) q^{49} +(5.07844 - 2.93204i) q^{51} +(2.04366 - 0.743831i) q^{52} +(0.721413 + 1.98207i) q^{53} +(1.19994 + 0.211582i) q^{54} +(-0.408885 - 0.487290i) q^{56} +(3.26118 - 18.4951i) q^{57} +(-0.220690 - 0.606340i) q^{58} +(-2.01822 - 5.54501i) q^{59} +(5.79589 - 1.02197i) q^{61} +(-0.665612 + 0.793245i) q^{62} +(5.43030 + 3.13519i) q^{63} +(3.74112 + 6.47982i) q^{64} +(2.06832 + 1.19414i) q^{66} +(-1.83502 + 5.04168i) q^{67} +3.91363 q^{68} +(-4.06031 + 11.1556i) q^{69} +(-3.02256 + 2.53623i) q^{71} +(2.60168 + 2.18307i) q^{72} +10.8633i q^{73} +(0.703154 - 0.557734i) q^{74} +(8.05663 - 9.60152i) q^{76} +(3.80402 + 4.53345i) q^{77} +(0.473504 - 0.0834915i) q^{78} +(-5.45707 + 14.9932i) q^{79} +(-6.69053 - 2.43515i) q^{81} +(0.561516 - 0.972574i) q^{82} +(-5.18161 - 0.913657i) q^{83} +(3.17729 + 5.50323i) q^{84} +(-0.203207 - 0.170511i) q^{86} +(2.25095 + 12.7658i) q^{87} +(1.60270 + 2.77595i) q^{88} +(-3.35854 - 9.22753i) q^{89} +(1.17331 + 0.206885i) q^{91} +(-6.06934 + 5.09278i) q^{92} +(15.9358 - 13.3717i) q^{93} +(-1.77900 - 0.313687i) q^{94} +(1.76900 + 4.86028i) q^{96} +(6.10262 + 10.5700i) q^{97} +(0.149258 + 0.846486i) q^{98} +(-24.2044 - 20.3099i) 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 \(\left(\mathbb{Z}/925\mathbb{Z}\right)^\times\).

\(n\) \(76\) \(852\)
\(\chi(n)\) \(e\left(\frac{13}{18}\right)\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.113027 + 0.0948413i 0.0799225 + 0.0670629i 0.681874 − 0.731470i \(-0.261165\pi\)
−0.601951 + 0.798533i \(0.705610\pi\)
\(3\) −1.90530 − 2.27065i −1.10003 − 1.31096i −0.946460 − 0.322821i \(-0.895369\pi\)
−0.153566 − 0.988138i \(-0.549076\pi\)
\(4\) −0.343516 − 1.94818i −0.171758 − 0.974088i
\(5\) 0 0
\(6\) − 0.437347i − 0.178546i
\(7\) 0.370651 − 1.01836i 0.140093 − 0.384903i −0.849728 − 0.527222i \(-0.823234\pi\)
0.989821 + 0.142319i \(0.0454560\pi\)
\(8\) 0.293488 − 0.508336i 0.103764 − 0.179724i
\(9\) −1.00473 + 5.69811i −0.334910 + 1.89937i
\(10\) 0 0
\(11\) −2.73043 + 4.72924i −0.823256 + 1.42592i 0.0799897 + 0.996796i \(0.474511\pi\)
−0.903245 + 0.429125i \(0.858822\pi\)
\(12\) −3.76912 + 4.49186i −1.08805 + 1.29669i
\(13\) 0.190904 + 1.08267i 0.0529474 + 0.300279i 0.999769 − 0.0214786i \(-0.00683737\pi\)
−0.946822 + 0.321758i \(0.895726\pi\)
\(14\) 0.138476 − 0.0799492i 0.0370093 − 0.0213673i
\(15\) 0 0
\(16\) −3.63647 + 1.32357i −0.909118 + 0.330892i
\(17\) −0.343537 + 1.94830i −0.0833200 + 0.472531i 0.914386 + 0.404842i \(0.132674\pi\)
−0.997706 + 0.0676888i \(0.978437\pi\)
\(18\) −0.653979 + 0.548753i −0.154144 + 0.129342i
\(19\) 4.07265 + 4.85359i 0.934329 + 1.11349i 0.993339 + 0.115232i \(0.0367613\pi\)
−0.0590096 + 0.998257i \(0.518794\pi\)
\(20\) 0 0
\(21\) −3.01853 + 1.09866i −0.658698 + 0.239746i
\(22\) −0.757141 + 0.275577i −0.161423 + 0.0587532i
\(23\) −2.00254 − 3.46850i −0.417558 − 0.723232i 0.578135 − 0.815941i \(-0.303781\pi\)
−0.995693 + 0.0927090i \(0.970447\pi\)
\(24\) −1.71343 + 0.302125i −0.349753 + 0.0616709i
\(25\) 0 0
\(26\) −0.0811047 + 0.140477i −0.0159059 + 0.0275499i
\(27\) 7.15171 − 4.12904i 1.37635 − 0.794634i
\(28\) −2.11126 − 0.372272i −0.398991 − 0.0703529i
\(29\) −3.78731 − 2.18661i −0.703286 − 0.406042i 0.105284 − 0.994442i \(-0.466425\pi\)
−0.808570 + 0.588400i \(0.799758\pi\)
\(30\) 0 0
\(31\) 7.01816i 1.26050i 0.776393 + 0.630250i \(0.217047\pi\)
−0.776393 + 0.630250i \(0.782953\pi\)
\(32\) −1.63970 − 0.596803i −0.289861 − 0.105501i
\(33\) 15.9407 − 2.81078i 2.77493 − 0.489294i
\(34\) −0.223608 + 0.187629i −0.0383485 + 0.0321782i
\(35\) 0 0
\(36\) 11.4461 1.90768
\(37\) 1.22092 − 5.95897i 0.200718 − 0.979649i
\(38\) 0.934844i 0.151652i
\(39\) 2.09464 − 2.49629i 0.335411 − 0.399727i
\(40\) 0 0
\(41\) −1.32170 − 7.49572i −0.206415 − 1.17064i −0.895198 − 0.445668i \(-0.852966\pi\)
0.688784 − 0.724967i \(-0.258145\pi\)
\(42\) −0.445375 − 0.162103i −0.0687228 − 0.0250131i
\(43\) −1.79786 −0.274171 −0.137085 − 0.990559i \(-0.543773\pi\)
−0.137085 + 0.990559i \(0.543773\pi\)
\(44\) 10.1513 + 3.69479i 1.53037 + 0.557010i
\(45\) 0 0
\(46\) 0.102615 − 0.581959i 0.0151298 − 0.0858052i
\(47\) −10.6029 + 6.12161i −1.54660 + 0.892928i −0.548199 + 0.836348i \(0.684686\pi\)
−0.998398 + 0.0565807i \(0.981980\pi\)
\(48\) 9.93393 + 5.73536i 1.43384 + 0.827827i
\(49\) 4.46264 + 3.74460i 0.637521 + 0.534943i
\(50\) 0 0
\(51\) 5.07844 − 2.93204i 0.711124 − 0.410567i
\(52\) 2.04366 − 0.743831i 0.283404 − 0.103151i
\(53\) 0.721413 + 1.98207i 0.0990937 + 0.272258i 0.979327 − 0.202283i \(-0.0648362\pi\)
−0.880233 + 0.474541i \(0.842614\pi\)
\(54\) 1.19994 + 0.211582i 0.163292 + 0.0287927i
\(55\) 0 0
\(56\) −0.408885 − 0.487290i −0.0546396 − 0.0651169i
\(57\) 3.26118 − 18.4951i 0.431954 − 2.44973i
\(58\) −0.220690 − 0.606340i −0.0289780 − 0.0796163i
\(59\) −2.01822 − 5.54501i −0.262750 − 0.721899i −0.998979 − 0.0451670i \(-0.985618\pi\)
0.736230 − 0.676732i \(-0.236604\pi\)
\(60\) 0 0
\(61\) 5.79589 − 1.02197i 0.742088 − 0.130850i 0.210192 − 0.977660i \(-0.432591\pi\)
0.531896 + 0.846810i \(0.321480\pi\)
\(62\) −0.665612 + 0.793245i −0.0845328 + 0.100742i
\(63\) 5.43030 + 3.13519i 0.684154 + 0.394996i
\(64\) 3.74112 + 6.47982i 0.467640 + 0.809977i
\(65\) 0 0
\(66\) 2.06832 + 1.19414i 0.254593 + 0.146989i
\(67\) −1.83502 + 5.04168i −0.224183 + 0.615939i −0.999885 − 0.0151610i \(-0.995174\pi\)
0.775702 + 0.631100i \(0.217396\pi\)
\(68\) 3.91363 0.474598
\(69\) −4.06031 + 11.1556i −0.488803 + 1.34298i
\(70\) 0 0
\(71\) −3.02256 + 2.53623i −0.358712 + 0.300995i −0.804277 − 0.594254i \(-0.797447\pi\)
0.445565 + 0.895250i \(0.353003\pi\)
\(72\) 2.60168 + 2.18307i 0.306611 + 0.257277i
\(73\) 10.8633i 1.27146i 0.771913 + 0.635729i \(0.219300\pi\)
−0.771913 + 0.635729i \(0.780700\pi\)
\(74\) 0.703154 − 0.557734i 0.0817400 − 0.0648353i
\(75\) 0 0
\(76\) 8.05663 − 9.60152i 0.924159 − 1.10137i
\(77\) 3.80402 + 4.53345i 0.433508 + 0.516635i
\(78\) 0.473504 − 0.0834915i 0.0536137 − 0.00945355i
\(79\) −5.45707 + 14.9932i −0.613968 + 1.68686i 0.107317 + 0.994225i \(0.465774\pi\)
−0.721285 + 0.692638i \(0.756448\pi\)
\(80\) 0 0
\(81\) −6.69053 − 2.43515i −0.743392 − 0.270573i
\(82\) 0.561516 − 0.972574i 0.0620091 − 0.107403i
\(83\) −5.18161 − 0.913657i −0.568755 − 0.100287i −0.118127 − 0.992999i \(-0.537689\pi\)
−0.450628 + 0.892712i \(0.648800\pi\)
\(84\) 3.17729 + 5.50323i 0.346671 + 0.600451i
\(85\) 0 0
\(86\) −0.203207 − 0.170511i −0.0219124 − 0.0183867i
\(87\) 2.25095 + 12.7658i 0.241328 + 1.36864i
\(88\) 1.60270 + 2.77595i 0.170848 + 0.295917i
\(89\) −3.35854 − 9.22753i −0.356005 − 0.978116i −0.980402 − 0.197008i \(-0.936877\pi\)
0.624397 − 0.781107i \(-0.285345\pi\)
\(90\) 0 0
\(91\) 1.17331 + 0.206885i 0.122996 + 0.0216875i
\(92\) −6.06934 + 5.09278i −0.632773 + 0.530960i
\(93\) 15.9358 − 13.3717i 1.65246 − 1.38658i
\(94\) −1.77900 − 0.313687i −0.183490 − 0.0323543i
\(95\) 0 0
\(96\) 1.76900 + 4.86028i 0.180547 + 0.496050i
\(97\) 6.10262 + 10.5700i 0.619627 + 1.07323i 0.989554 + 0.144165i \(0.0460495\pi\)
−0.369926 + 0.929061i \(0.620617\pi\)
\(98\) 0.149258 + 0.846486i 0.0150774 + 0.0855080i
\(99\) −24.2044 − 20.3099i −2.43263 − 2.04122i
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.299.7 72
5.2 odd 4 925.2.bb.b.151.6 72
5.3 odd 4 185.2.w.a.151.7 yes 72
5.4 even 2 925.2.ba.b.299.6 72
37.25 even 18 925.2.ba.b.99.6 72
185.62 odd 36 925.2.bb.b.876.6 72
185.99 even 18 inner 925.2.ba.c.99.7 72
185.143 even 36 6845.2.a.t.1.20 36
185.153 even 36 6845.2.a.u.1.17 36
185.173 odd 36 185.2.w.a.136.7 ✓ 72
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
185.2.w.a.136.7 ✓ 72 185.173 odd 36
185.2.w.a.151.7 yes 72 5.3 odd 4
925.2.ba.b.99.6 72 37.25 even 18
925.2.ba.b.299.6 72 5.4 even 2
925.2.ba.c.99.7 72 185.99 even 18 inner
925.2.ba.c.299.7 72 1.1 even 1 trivial
925.2.bb.b.151.6 72 5.2 odd 4
925.2.bb.b.876.6 72 185.62 odd 36
6845.2.a.t.1.20 36 185.143 even 36
6845.2.a.u.1.17 36 185.153 even 36