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Properties

Label 425.2.r.a.69.1
Level $425$
Weight $2$
Character 425.69
Analytic conductor $3.394$
Analytic rank $0$
Dimension $160$
Inner twists $2$

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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] = [425,2,Mod(69,425)] 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("425.69"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(425, base_ring=CyclotomicField(10)) chi = DirichletCharacter(H, H._module([9, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 425 = 5^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 425.r (of order \(10\), degree \(4\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(3.39364208590\)
Analytic rank: \(0\)
Dimension: \(160\)
Relative dimension: \(40\) over \(\Q(\zeta_{10})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

Embedding invariants

Embedding label 69.1
Character \(\chi\) \(=\) 425.69
Dual form 425.2.r.a.154.1

$q$-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+(-2.60604 + 0.846752i) q^{2} +(-1.86275 - 2.56385i) q^{3} +(4.45640 - 3.23776i) q^{4} +(0.0540158 + 2.23542i) q^{5} +(7.02533 + 5.10420i) q^{6} -2.47810i q^{7} +(-5.65071 + 7.77753i) q^{8} +(-2.17645 + 6.69843i) q^{9} +(-2.03361 - 5.77983i) q^{10} +(-1.18679 - 3.65255i) q^{11} +(-16.6023 - 5.39440i) q^{12} +(-1.34575 - 0.437262i) q^{13} +(2.09834 + 6.45802i) q^{14} +(5.63065 - 4.30250i) q^{15} +(4.73592 - 14.5757i) q^{16} +(0.587785 - 0.809017i) q^{17} -19.2993i q^{18} +(-0.679976 - 0.494032i) q^{19} +(7.47846 + 9.78701i) q^{20} +(-6.35348 + 4.61608i) q^{21} +(6.18561 + 8.51377i) q^{22} +(-1.51412 + 0.491967i) q^{23} +30.4663 q^{24} +(-4.99416 + 0.241496i) q^{25} +3.87733 q^{26} +(12.1860 - 3.95947i) q^{27} +(-8.02351 - 11.0434i) q^{28} +(-4.61529 + 3.35321i) q^{29} +(-11.0305 + 15.9802i) q^{30} +(-3.88614 - 2.82345i) q^{31} +22.7677i q^{32} +(-7.15392 + 9.84652i) q^{33} +(-0.846752 + 2.60604i) q^{34} +(5.53959 - 0.133857i) q^{35} +(11.9888 + 36.8977i) q^{36} +(4.34079 + 1.41041i) q^{37} +(2.19036 + 0.711693i) q^{38} +(1.38572 + 4.26482i) q^{39} +(-17.6912 - 12.2116i) q^{40} +(-0.386186 + 1.18856i) q^{41} +(12.6487 - 17.4095i) q^{42} +8.80279i q^{43} +(-17.1149 - 12.4347i) q^{44} +(-15.0913 - 4.50345i) q^{45} +(3.52927 - 2.56417i) q^{46} +(2.80817 + 3.86511i) q^{47} +(-46.1916 + 15.0086i) q^{48} +0.859006 q^{49} +(12.8105 - 4.85817i) q^{50} -3.16909 q^{51} +(-7.41296 + 2.40862i) q^{52} +(-1.10303 - 1.51819i) q^{53} +(-28.4044 + 20.6370i) q^{54} +(8.10087 - 2.85026i) q^{55} +(19.2735 + 14.0030i) q^{56} +2.66361i q^{57} +(9.18828 - 12.6466i) q^{58} +(-1.27514 + 3.92447i) q^{59} +(11.1619 - 37.4044i) q^{60} +(3.05333 + 9.39720i) q^{61} +(12.5182 + 4.06740i) q^{62} +(16.5994 + 5.39347i) q^{63} +(-9.80678 - 30.1822i) q^{64} +(0.904770 - 3.03194i) q^{65} +(10.3058 - 31.7180i) q^{66} +(-9.16910 + 12.6202i) q^{67} -5.50841i q^{68} +(4.08175 + 2.96556i) q^{69} +(-14.3230 + 5.03950i) q^{70} +(0.0639051 - 0.0464298i) q^{71} +(-39.7988 - 54.7783i) q^{72} +(3.40877 - 1.10758i) q^{73} -12.5065 q^{74} +(9.92202 + 12.3544i) q^{75} -4.62980 q^{76} +(-9.05140 + 2.94098i) q^{77} +(-7.22249 - 9.94090i) q^{78} +(0.953265 - 0.692588i) q^{79} +(32.8385 + 9.79943i) q^{80} +(-15.7568 - 11.4480i) q^{81} -3.42443i q^{82} +(-5.68451 + 7.82406i) q^{83} +(-13.3679 + 41.1421i) q^{84} +(1.84024 + 1.27024i) q^{85} +(-7.45378 - 22.9404i) q^{86} +(17.1942 + 5.58674i) q^{87} +(35.1140 + 11.4092i) q^{88} +(1.37884 + 4.24364i) q^{89} +(43.1419 - 1.04247i) q^{90} +(-1.08358 + 3.33492i) q^{91} +(-5.15465 + 7.09476i) q^{92} +15.2228i q^{93} +(-10.5910 - 7.69480i) q^{94} +(1.06764 - 1.54671i) q^{95} +(58.3730 - 42.4105i) q^{96} +(0.481530 + 0.662769i) q^{97} +(-2.23860 + 0.727365i) q^{98} +27.0494 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 160 q + 40 q^{4} - 8 q^{5} + 4 q^{6} - 30 q^{8} + 36 q^{9} - 6 q^{10} + 8 q^{11} - 40 q^{12} - 20 q^{14} - 40 q^{15} - 64 q^{16} + 6 q^{19} + 2 q^{20} - 50 q^{22} + 20 q^{23} + 20 q^{24} + 32 q^{25} + 20 q^{26}+ \cdots + 44 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/425\mathbb{Z}\right)^\times\).

\(n\) \(52\) \(326\)
\(\chi(n)\) \(e\left(\frac{9}{10}\right)\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.60604 + 0.846752i −1.84275 + 0.598744i −0.844771 + 0.535128i \(0.820263\pi\)
−0.997974 + 0.0636161i \(0.979737\pi\)
\(3\) −1.86275 − 2.56385i −1.07546 − 1.48024i −0.864426 − 0.502760i \(-0.832318\pi\)
−0.211031 − 0.977479i \(-0.567682\pi\)
\(4\) 4.45640 − 3.23776i 2.22820 − 1.61888i
\(5\) 0.0540158 + 2.23542i 0.0241566 + 0.999708i
\(6\) 7.02533 + 5.10420i 2.86808 + 2.08378i
\(7\) − 2.47810i − 0.936635i −0.883560 − 0.468317i \(-0.844860\pi\)
0.883560 − 0.468317i \(-0.155140\pi\)
\(8\) −5.65071 + 7.77753i −1.99783 + 2.74977i
\(9\) −2.17645 + 6.69843i −0.725484 + 2.23281i
\(10\) −2.03361 − 5.77983i −0.643084 − 1.82774i
\(11\) −1.18679 − 3.65255i −0.357830 − 1.10129i −0.954351 − 0.298689i \(-0.903451\pi\)
0.596521 − 0.802597i \(-0.296549\pi\)
\(12\) −16.6023 − 5.39440i −4.79266 − 1.55723i
\(13\) −1.34575 − 0.437262i −0.373245 − 0.121275i 0.116387 − 0.993204i \(-0.462869\pi\)
−0.489632 + 0.871929i \(0.662869\pi\)
\(14\) 2.09834 + 6.45802i 0.560805 + 1.72598i
\(15\) 5.63065 − 4.30250i 1.45383 − 1.11090i
\(16\) 4.73592 − 14.5757i 1.18398 − 3.64392i
\(17\) 0.587785 − 0.809017i 0.142559 − 0.196215i
\(18\) − 19.2993i − 4.54888i
\(19\) −0.679976 − 0.494032i −0.155997 − 0.113339i 0.507048 − 0.861918i \(-0.330737\pi\)
−0.663046 + 0.748579i \(0.730737\pi\)
\(20\) 7.47846 + 9.78701i 1.67223 + 2.18844i
\(21\) −6.35348 + 4.61608i −1.38644 + 1.00731i
\(22\) 6.18561 + 8.51377i 1.31878 + 1.81514i
\(23\) −1.51412 + 0.491967i −0.315716 + 0.102582i −0.462588 − 0.886573i \(-0.653079\pi\)
0.146873 + 0.989155i \(0.453079\pi\)
\(24\) 30.4663 6.21890
\(25\) −4.99416 + 0.241496i −0.998833 + 0.0482991i
\(26\) 3.87733 0.760408
\(27\) 12.1860 − 3.95947i 2.34519 − 0.762000i
\(28\) −8.02351 − 11.0434i −1.51630 − 2.08701i
\(29\) −4.61529 + 3.35321i −0.857038 + 0.622675i −0.927078 − 0.374870i \(-0.877687\pi\)
0.0700394 + 0.997544i \(0.477687\pi\)
\(30\) −11.0305 + 15.9802i −2.01389 + 2.91758i
\(31\) −3.88614 − 2.82345i −0.697972 − 0.507106i 0.181299 − 0.983428i \(-0.441970\pi\)
−0.879271 + 0.476322i \(0.841970\pi\)
\(32\) 22.7677i 4.02480i
\(33\) −7.15392 + 9.84652i −1.24534 + 1.71406i
\(34\) −0.846752 + 2.60604i −0.145217 + 0.446931i
\(35\) 5.53959 − 0.133857i 0.936362 − 0.0226259i
\(36\) 11.9888 + 36.8977i 1.99813 + 6.14962i
\(37\) 4.34079 + 1.41041i 0.713621 + 0.231870i 0.643255 − 0.765652i \(-0.277583\pi\)
0.0703659 + 0.997521i \(0.477583\pi\)
\(38\) 2.19036 + 0.711693i 0.355324 + 0.115452i
\(39\) 1.38572 + 4.26482i 0.221893 + 0.682917i
\(40\) −17.6912 − 12.2116i −2.79723 − 1.93082i
\(41\) −0.386186 + 1.18856i −0.0603122 + 0.185622i −0.976673 − 0.214731i \(-0.931112\pi\)
0.916361 + 0.400353i \(0.131112\pi\)
\(42\) 12.6487 − 17.4095i 1.95174 − 2.68634i
\(43\) 8.80279i 1.34241i 0.741271 + 0.671206i \(0.234224\pi\)
−0.741271 + 0.671206i \(0.765776\pi\)
\(44\) −17.1149 − 12.4347i −2.58017 − 1.87460i
\(45\) −15.0913 − 4.50345i −2.24968 − 0.671335i
\(46\) 3.52927 − 2.56417i 0.520363 − 0.378066i
\(47\) 2.80817 + 3.86511i 0.409614 + 0.563785i 0.963124 − 0.269057i \(-0.0867121\pi\)
−0.553511 + 0.832842i \(0.686712\pi\)
\(48\) −46.1916 + 15.0086i −6.66719 + 2.16630i
\(49\) 0.859006 0.122715
\(50\) 12.8105 − 4.85817i 1.81168 − 0.687048i
\(51\) −3.16909 −0.443762
\(52\) −7.41296 + 2.40862i −1.02799 + 0.334015i
\(53\) −1.10303 − 1.51819i −0.151513 − 0.208540i 0.726513 − 0.687153i \(-0.241140\pi\)
−0.878026 + 0.478613i \(0.841140\pi\)
\(54\) −28.4044 + 20.6370i −3.86535 + 2.80834i
\(55\) 8.10087 − 2.85026i 1.09232 − 0.384328i
\(56\) 19.2735 + 14.0030i 2.57553 + 1.87124i
\(57\) 2.66361i 0.352804i
\(58\) 9.18828 − 12.6466i 1.20648 − 1.66058i
\(59\) −1.27514 + 3.92447i −0.166009 + 0.510923i −0.999109 − 0.0421992i \(-0.986564\pi\)
0.833100 + 0.553122i \(0.186564\pi\)
\(60\) 11.1619 − 37.4044i 1.44100 − 4.82888i
\(61\) 3.05333 + 9.39720i 0.390939 + 1.20319i 0.932079 + 0.362255i \(0.117993\pi\)
−0.541140 + 0.840933i \(0.682007\pi\)
\(62\) 12.5182 + 4.06740i 1.58981 + 0.516561i
\(63\) 16.5994 + 5.39347i 2.09133 + 0.679514i
\(64\) −9.80678 − 30.1822i −1.22585 − 3.77277i
\(65\) 0.904770 − 3.03194i 0.112223 − 0.376065i
\(66\) 10.3058 − 31.7180i 1.26856 − 3.90421i
\(67\) −9.16910 + 12.6202i −1.12018 + 1.54180i −0.314691 + 0.949194i \(0.601901\pi\)
−0.805492 + 0.592606i \(0.798099\pi\)
\(68\) − 5.50841i − 0.667993i
\(69\) 4.08175 + 2.96556i 0.491385 + 0.357012i
\(70\) −14.3230 + 5.03950i −1.71193 + 0.602335i
\(71\) 0.0639051 − 0.0464298i 0.00758414 − 0.00551020i −0.583987 − 0.811763i \(-0.698508\pi\)
0.591571 + 0.806253i \(0.298508\pi\)
\(72\) −39.7988 − 54.7783i −4.69033 − 6.45569i
\(73\) 3.40877 − 1.10758i 0.398966 − 0.129632i −0.102660 − 0.994717i \(-0.532735\pi\)
0.501626 + 0.865085i \(0.332735\pi\)
\(74\) −12.5065 −1.45385
\(75\) 9.92202 + 12.3544i 1.14570 + 1.42657i
\(76\) −4.62980 −0.531075
\(77\) −9.05140 + 2.94098i −1.03150 + 0.335156i
\(78\) −7.22249 − 9.94090i −0.817785 − 1.12559i
\(79\) 0.953265 − 0.692588i 0.107251 − 0.0779222i −0.532867 − 0.846199i \(-0.678885\pi\)
0.640118 + 0.768277i \(0.278885\pi\)
\(80\) 32.8385 + 9.79943i 3.67145 + 1.09561i
\(81\) −15.7568 − 11.4480i −1.75075 − 1.27200i
\(82\) − 3.42443i − 0.378165i
\(83\) −5.68451 + 7.82406i −0.623956 + 0.858802i −0.997633 − 0.0687565i \(-0.978097\pi\)
0.373677 + 0.927559i \(0.378097\pi\)
\(84\) −13.3679 + 41.1421i −1.45856 + 4.48897i
\(85\) 1.84024 + 1.27024i 0.199602 + 0.137777i
\(86\) −7.45378 − 22.9404i −0.803762 − 2.47372i
\(87\) 17.1942 + 5.58674i 1.84341 + 0.598962i
\(88\) 35.1140 + 11.4092i 3.74317 + 1.21623i
\(89\) 1.37884 + 4.24364i 0.146157 + 0.449825i 0.997158 − 0.0753389i \(-0.0240039\pi\)
−0.851001 + 0.525164i \(0.824004\pi\)
\(90\) 43.1419 − 1.04247i 4.54755 − 0.109886i
\(91\) −1.08358 + 3.33492i −0.113590 + 0.349594i
\(92\) −5.15465 + 7.09476i −0.537409 + 0.739680i
\(93\) 15.2228i 1.57854i
\(94\) −10.5910 − 7.69480i −1.09238 − 0.793658i
\(95\) 1.06764 − 1.54671i 0.109537 − 0.158690i
\(96\) 58.3730 − 42.4105i 5.95767 − 4.32850i
\(97\) 0.481530 + 0.662769i 0.0488920 + 0.0672940i 0.832763 − 0.553630i \(-0.186758\pi\)
−0.783871 + 0.620924i \(0.786758\pi\)
\(98\) −2.23860 + 0.727365i −0.226133 + 0.0734750i
\(99\) 27.0494 2.71856
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 425.2.r.a.69.1 ✓ 160
25.4 even 10 inner 425.2.r.a.154.1 yes 160
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
425.2.r.a.69.1 ✓ 160 1.1 even 1 trivial
425.2.r.a.154.1 yes 160 25.4 even 10 inner