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Properties

Label 925.2.bb.e.176.3
Level $925$
Weight $2$
Character 925.176
Analytic conductor $7.386$
Analytic rank $0$
Dimension $96$
Inner twists $4$

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Show commands: Magma / Pari/GP / SageMath

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 = [96,0] 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: \(96\)
Relative dimension: \(16\) over \(\Q(\zeta_{18})\)
Twist minimal: no (minimal twist has level 185)
Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label 176.3
Character \(\chi\) \(=\) 925.176
Dual form 925.2.bb.e.226.3

$q$-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+(-2.22054 + 0.391541i) q^{2} +(-0.457886 + 2.59680i) q^{3} +(2.89811 - 1.05483i) q^{4} -5.94558i q^{6} +(0.761600 + 0.639059i) q^{7} +(-2.11694 + 1.22222i) q^{8} +(-3.71464 - 1.35202i) q^{9} +(-2.23710 - 3.87477i) q^{11} +(1.41217 + 8.00880i) q^{12} +(0.156821 + 0.430863i) q^{13} +(-1.94138 - 1.12086i) q^{14} +(-0.502918 + 0.421999i) q^{16} +(1.31622 - 3.61628i) q^{17} +(8.77787 + 1.54777i) q^{18} +(6.33129 + 1.11638i) q^{19} +(-2.00823 + 1.68511i) q^{21} +(6.48469 + 7.72816i) q^{22} +(-1.59935 - 0.923384i) q^{23} +(-2.20453 - 6.05691i) q^{24} +(-0.516929 - 0.895348i) q^{26} +(1.25651 - 2.17633i) q^{27} +(2.88130 + 1.04871i) q^{28} +(6.71853 - 3.87895i) q^{29} -1.28733i q^{31} +(4.09402 - 4.87906i) q^{32} +(11.0863 - 4.03509i) q^{33} +(-1.50679 + 8.54544i) q^{34} -12.1916 q^{36} +(-5.40445 + 2.79140i) q^{37} -14.4960 q^{38} +(-1.19067 + 0.209948i) q^{39} +(10.2048 - 3.71426i) q^{41} +(3.79957 - 4.52816i) q^{42} -4.21419i q^{43} +(-10.5706 - 8.86975i) q^{44} +(3.91296 + 1.42420i) q^{46} +(6.11134 - 10.5851i) q^{47} +(-0.865567 - 1.49921i) q^{48} +(-1.04390 - 5.92024i) q^{49} +(8.78807 + 5.07379i) q^{51} +(0.908971 + 1.08327i) q^{52} +(-2.19000 + 1.83762i) q^{53} +(-1.93800 + 5.32461i) q^{54} +(-2.39333 - 0.422009i) q^{56} +(-5.79802 + 15.9299i) q^{57} +(-13.4000 + 11.2439i) q^{58} +(4.99958 + 5.95827i) q^{59} +(-0.980592 - 2.69415i) q^{61} +(0.504044 + 2.85858i) q^{62} +(-1.96505 - 3.40357i) q^{63} +(-6.52407 + 11.3000i) q^{64} +(-23.0377 + 13.3008i) q^{66} +(-1.35393 - 1.13608i) q^{67} -11.8687i q^{68} +(3.13016 - 3.73038i) q^{69} +(-1.98450 + 11.2547i) q^{71} +(9.51611 - 1.67795i) q^{72} +12.7316 q^{73} +(10.9079 - 8.31448i) q^{74} +(19.5263 - 3.44302i) q^{76} +(0.772428 - 4.38066i) q^{77} +(2.56173 - 0.932395i) q^{78} +(-9.83583 + 11.7219i) q^{79} +(-4.00843 - 3.36347i) q^{81} +(-21.2060 + 12.2433i) q^{82} +(2.25275 + 0.819933i) q^{83} +(-4.04259 + 7.00196i) q^{84} +(1.65003 + 9.35777i) q^{86} +(6.99653 + 19.2228i) q^{87} +(9.47160 + 5.46843i) q^{88} +(0.0452057 + 0.0538740i) q^{89} +(-0.155912 + 0.428364i) q^{91} +(-5.60910 - 0.989035i) q^{92} +(3.34295 + 0.589452i) q^{93} +(-9.42595 + 25.8976i) q^{94} +(10.7954 + 12.8654i) q^{96} +(11.3598 + 6.55858i) q^{97} +(4.63604 + 12.7374i) q^{98} +(3.07125 + 17.4179i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 96 q + 12 q^{4} + 6 q^{9} - 30 q^{11} + 36 q^{14} + 18 q^{19} - 24 q^{21} - 96 q^{24} + 48 q^{26} + 18 q^{29} + 54 q^{34} + 24 q^{36} + 36 q^{39} + 72 q^{41} + 84 q^{44} - 18 q^{46} + 6 q^{49} - 18 q^{51}+ \cdots + 24 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.22054 + 0.391541i −1.57016 + 0.276861i −0.889916 − 0.456125i \(-0.849237\pi\)
−0.680243 + 0.732986i \(0.738126\pi\)
\(3\) −0.457886 + 2.59680i −0.264361 + 1.49926i 0.506489 + 0.862246i \(0.330943\pi\)
−0.770850 + 0.637017i \(0.780168\pi\)
\(4\) 2.89811 − 1.05483i 1.44905 − 0.527413i
\(5\) 0 0
\(6\) − 5.94558i − 2.42727i
\(7\) 0.761600 + 0.639059i 0.287858 + 0.241541i 0.775269 − 0.631631i \(-0.217614\pi\)
−0.487411 + 0.873173i \(0.662059\pi\)
\(8\) −2.11694 + 1.22222i −0.748451 + 0.432119i
\(9\) −3.71464 − 1.35202i −1.23821 − 0.450672i
\(10\) 0 0
\(11\) −2.23710 − 3.87477i −0.674510 − 1.16829i −0.976612 − 0.215010i \(-0.931022\pi\)
0.302102 − 0.953276i \(-0.402312\pi\)
\(12\) 1.41217 + 8.00880i 0.407658 + 2.31194i
\(13\) 0.156821 + 0.430863i 0.0434945 + 0.119500i 0.959538 − 0.281578i \(-0.0908578\pi\)
−0.916044 + 0.401078i \(0.868636\pi\)
\(14\) −1.94138 − 1.12086i −0.518856 − 0.299562i
\(15\) 0 0
\(16\) −0.502918 + 0.421999i −0.125730 + 0.105500i
\(17\) 1.31622 − 3.61628i 0.319229 − 0.877076i −0.671473 − 0.741029i \(-0.734338\pi\)
0.990702 − 0.136047i \(-0.0434397\pi\)
\(18\) 8.77787 + 1.54777i 2.06896 + 0.364814i
\(19\) 6.33129 + 1.11638i 1.45250 + 0.256114i 0.843530 − 0.537081i \(-0.180473\pi\)
0.608967 + 0.793196i \(0.291584\pi\)
\(20\) 0 0
\(21\) −2.00823 + 1.68511i −0.438232 + 0.367721i
\(22\) 6.48469 + 7.72816i 1.38254 + 1.64765i
\(23\) −1.59935 − 0.923384i −0.333487 − 0.192539i 0.323901 − 0.946091i \(-0.395006\pi\)
−0.657388 + 0.753552i \(0.728339\pi\)
\(24\) −2.20453 − 6.05691i −0.449998 − 1.23636i
\(25\) 0 0
\(26\) −0.516929 − 0.895348i −0.101378 − 0.175592i
\(27\) 1.25651 − 2.17633i 0.241815 − 0.418835i
\(28\) 2.88130 + 1.04871i 0.544514 + 0.198187i
\(29\) 6.71853 − 3.87895i 1.24760 − 0.720302i 0.276970 − 0.960879i \(-0.410670\pi\)
0.970630 + 0.240576i \(0.0773363\pi\)
\(30\) 0 0
\(31\) − 1.28733i − 0.231212i −0.993295 − 0.115606i \(-0.963119\pi\)
0.993295 − 0.115606i \(-0.0368810\pi\)
\(32\) 4.09402 − 4.87906i 0.723728 − 0.862505i
\(33\) 11.0863 − 4.03509i 1.92988 − 0.702420i
\(34\) −1.50679 + 8.54544i −0.258413 + 1.46553i
\(35\) 0 0
\(36\) −12.1916 −2.03193
\(37\) −5.40445 + 2.79140i −0.888486 + 0.458903i
\(38\) −14.4960 −2.35156
\(39\) −1.19067 + 0.209948i −0.190660 + 0.0336185i
\(40\) 0 0
\(41\) 10.2048 − 3.71426i 1.59373 − 0.580070i 0.615598 − 0.788060i \(-0.288915\pi\)
0.978131 + 0.207990i \(0.0666923\pi\)
\(42\) 3.79957 − 4.52816i 0.586287 − 0.698710i
\(43\) − 4.21419i − 0.642658i −0.946968 − 0.321329i \(-0.895871\pi\)
0.946968 − 0.321329i \(-0.104129\pi\)
\(44\) −10.5706 − 8.86975i −1.59357 − 1.33716i
\(45\) 0 0
\(46\) 3.91296 + 1.42420i 0.576935 + 0.209987i
\(47\) 6.11134 − 10.5851i 0.891430 − 1.54400i 0.0532682 − 0.998580i \(-0.483036\pi\)
0.838162 − 0.545422i \(-0.183630\pi\)
\(48\) −0.865567 − 1.49921i −0.124934 − 0.216392i
\(49\) −1.04390 − 5.92024i −0.149128 − 0.845749i
\(50\) 0 0
\(51\) 8.78807 + 5.07379i 1.23058 + 0.710473i
\(52\) 0.908971 + 1.08327i 0.126052 + 0.150223i
\(53\) −2.19000 + 1.83762i −0.300819 + 0.252417i −0.780685 − 0.624925i \(-0.785130\pi\)
0.479866 + 0.877342i \(0.340685\pi\)
\(54\) −1.93800 + 5.32461i −0.263728 + 0.724587i
\(55\) 0 0
\(56\) −2.39333 − 0.422009i −0.319822 − 0.0563933i
\(57\) −5.79802 + 15.9299i −0.767966 + 2.10997i
\(58\) −13.4000 + 11.2439i −1.75951 + 1.47640i
\(59\) 4.99958 + 5.95827i 0.650890 + 0.775700i 0.986048 − 0.166462i \(-0.0532342\pi\)
−0.335158 + 0.942162i \(0.608790\pi\)
\(60\) 0 0
\(61\) −0.980592 − 2.69415i −0.125552 − 0.344951i 0.860953 − 0.508685i \(-0.169868\pi\)
−0.986504 + 0.163734i \(0.947646\pi\)
\(62\) 0.504044 + 2.85858i 0.0640137 + 0.363040i
\(63\) −1.96505 − 3.40357i −0.247573 − 0.428809i
\(64\) −6.52407 + 11.3000i −0.815509 + 1.41250i
\(65\) 0 0
\(66\) −23.0377 + 13.3008i −2.83575 + 1.63722i
\(67\) −1.35393 − 1.13608i −0.165409 − 0.138794i 0.556326 − 0.830964i \(-0.312211\pi\)
−0.721735 + 0.692170i \(0.756655\pi\)
\(68\) − 11.8687i − 1.43930i
\(69\) 3.13016 − 3.73038i 0.376828 − 0.449086i
\(70\) 0 0
\(71\) −1.98450 + 11.2547i −0.235517 + 1.33569i 0.606004 + 0.795461i \(0.292771\pi\)
−0.841521 + 0.540224i \(0.818340\pi\)
\(72\) 9.51611 − 1.67795i 1.12148 − 0.197748i
\(73\) 12.7316 1.49012 0.745059 − 0.666999i \(-0.232422\pi\)
0.745059 + 0.666999i \(0.232422\pi\)
\(74\) 10.9079 − 8.31448i 1.26801 − 0.966539i
\(75\) 0 0
\(76\) 19.5263 − 3.44302i 2.23983 − 0.394942i
\(77\) 0.772428 − 4.38066i 0.0880264 − 0.499222i
\(78\) 2.56173 − 0.932395i 0.290059 − 0.105573i
\(79\) −9.83583 + 11.7219i −1.10662 + 1.31881i −0.163429 + 0.986555i \(0.552255\pi\)
−0.943188 + 0.332260i \(0.892189\pi\)
\(80\) 0 0
\(81\) −4.00843 − 3.36347i −0.445381 − 0.373719i
\(82\) −21.2060 + 12.2433i −2.34181 + 1.35204i
\(83\) 2.25275 + 0.819933i 0.247271 + 0.0899993i 0.462682 − 0.886524i \(-0.346887\pi\)
−0.215411 + 0.976523i \(0.569109\pi\)
\(84\) −4.04259 + 7.00196i −0.441082 + 0.763977i
\(85\) 0 0
\(86\) 1.65003 + 9.35777i 0.177927 + 1.00907i
\(87\) 6.99653 + 19.2228i 0.750107 + 2.06090i
\(88\) 9.47160 + 5.46843i 1.00968 + 0.582937i
\(89\) 0.0452057 + 0.0538740i 0.00479179 + 0.00571064i 0.768435 − 0.639928i \(-0.221036\pi\)
−0.763643 + 0.645638i \(0.776591\pi\)
\(90\) 0 0
\(91\) −0.155912 + 0.428364i −0.0163440 + 0.0449047i
\(92\) −5.60910 − 0.989035i −0.584789 − 0.103114i
\(93\) 3.34295 + 0.589452i 0.346648 + 0.0611234i
\(94\) −9.42595 + 25.8976i −0.972212 + 2.67113i
\(95\) 0 0
\(96\) 10.7954 + 12.8654i 1.10180 + 1.31307i
\(97\) 11.3598 + 6.55858i 1.15341 + 0.665923i 0.949717 − 0.313111i \(-0.101371\pi\)
0.203696 + 0.979034i \(0.434704\pi\)
\(98\) 4.63604 + 12.7374i 0.468310 + 1.28667i
\(99\) 3.07125 + 17.4179i 0.308672 + 1.75057i
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.e.176.3 96
5.2 odd 4 185.2.v.a.139.3 yes 96
5.3 odd 4 185.2.v.a.139.14 yes 96
5.4 even 2 inner 925.2.bb.e.176.14 96
37.4 even 18 inner 925.2.bb.e.226.3 96
185.4 even 18 inner 925.2.bb.e.226.14 96
185.78 odd 36 185.2.v.a.4.3 ✓ 96
185.152 odd 36 185.2.v.a.4.14 yes 96
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
185.2.v.a.4.3 ✓ 96 185.78 odd 36
185.2.v.a.4.14 yes 96 185.152 odd 36
185.2.v.a.139.3 yes 96 5.2 odd 4
185.2.v.a.139.14 yes 96 5.3 odd 4
925.2.bb.e.176.3 96 1.1 even 1 trivial
925.2.bb.e.176.14 96 5.4 even 2 inner
925.2.bb.e.226.3 96 37.4 even 18 inner
925.2.bb.e.226.14 96 185.4 even 18 inner