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Properties

Label 425.2.r.a.154.1
Level $425$
Weight $2$
Character 425.154
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 154.1
Character \(\chi\) \(=\) 425.154
Dual form 425.2.r.a.69.1

$q$-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 \(\left(\mathbb{Z}/425\mathbb{Z}\right)^\times\).

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

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 −0.997974 − 0.0636161i \(-0.979737\pi\)
−0.844771 − 0.535128i \(-0.820263\pi\)
\(3\) −1.86275 + 2.56385i −1.07546 + 1.48024i −0.211031 + 0.977479i \(0.567682\pi\)
−0.864426 + 0.502760i \(0.832318\pi\)
\(4\) 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 \(0.155140\pi\)
−0.883560 + 0.468317i \(0.844860\pi\)
\(8\) −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 0.596521 + 0.802597i \(0.296549\pi\)
−0.954351 + 0.298689i \(0.903451\pi\)
\(12\) −16.6023 + 5.39440i −4.79266 + 1.55723i
\(13\) −1.34575 + 0.437262i −0.373245 + 0.121275i −0.489632 − 0.871929i \(-0.662869\pi\)
0.116387 + 0.993204i \(0.462869\pi\)
\(14\) 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 −0.663046 − 0.748579i \(-0.730737\pi\)
0.507048 + 0.861918i \(0.330737\pi\)
\(20\) 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 0.146873 − 0.989155i \(-0.453079\pi\)
−0.462588 + 0.886573i \(0.653079\pi\)
\(24\) 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 0.0700394 − 0.997544i \(-0.477687\pi\)
−0.927078 + 0.374870i \(0.877687\pi\)
\(30\) −11.0305 − 15.9802i −2.01389 − 2.91758i
\(31\) −3.88614 + 2.82345i −0.697972 + 0.507106i −0.879271 − 0.476322i \(-0.841970\pi\)
0.181299 + 0.983428i \(0.441970\pi\)
\(32\) − 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 0.0703659 − 0.997521i \(-0.477583\pi\)
0.643255 + 0.765652i \(0.277583\pi\)
\(38\) 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 0.916361 − 0.400353i \(-0.131112\pi\)
−0.976673 + 0.214731i \(0.931112\pi\)
\(42\) 12.6487 + 17.4095i 1.95174 + 2.68634i
\(43\) − 8.80279i − 1.34241i −0.741271 − 0.671206i \(-0.765776\pi\)
0.741271 − 0.671206i \(-0.234224\pi\)
\(44\) −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 −0.553511 − 0.832842i \(-0.686712\pi\)
0.963124 + 0.269057i \(0.0867121\pi\)
\(48\) −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 −0.878026 − 0.478613i \(-0.841140\pi\)
0.726513 + 0.687153i \(0.241140\pi\)
\(54\) −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 0.833100 − 0.553122i \(-0.186564\pi\)
−0.999109 + 0.0421992i \(0.986564\pi\)
\(60\) 11.1619 + 37.4044i 1.44100 + 4.82888i
\(61\) 3.05333 − 9.39720i 0.390939 − 1.20319i −0.541140 − 0.840933i \(-0.682007\pi\)
0.932079 − 0.362255i \(-0.117993\pi\)
\(62\) 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 −0.805492 − 0.592606i \(-0.798099\pi\)
−0.314691 − 0.949194i \(-0.601901\pi\)
\(68\) 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 0.591571 − 0.806253i \(-0.298508\pi\)
−0.583987 + 0.811763i \(0.698508\pi\)
\(72\) −39.7988 + 54.7783i −4.69033 + 6.45569i
\(73\) 3.40877 + 1.10758i 0.398966 + 0.129632i 0.501626 − 0.865085i \(-0.332735\pi\)
−0.102660 + 0.994717i \(0.532735\pi\)
\(74\) −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 0.640118 − 0.768277i \(-0.278885\pi\)
−0.532867 + 0.846199i \(0.678885\pi\)
\(80\) 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 0.373677 − 0.927559i \(-0.378097\pi\)
−0.997633 + 0.0687565i \(0.978097\pi\)
\(84\) −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 −0.851001 − 0.525164i \(-0.824004\pi\)
0.997158 + 0.0753389i \(0.0240039\pi\)
\(90\) 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 −0.783871 − 0.620924i \(-0.786758\pi\)
0.832763 + 0.553630i \(0.186758\pi\)
\(98\) −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.154.1 yes 160
25.19 even 10 inner 425.2.r.a.69.1 ✓ 160
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
425.2.r.a.69.1 ✓ 160 25.19 even 10 inner
425.2.r.a.154.1 yes 160 1.1 even 1 trivial