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Properties

Label 175.2.x.a.122.9
Level $175$
Weight $2$
Character 175.122
Analytic conductor $1.397$
Analytic rank $0$
Dimension $288$
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] = [175,2,Mod(3,175)] 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("175.3"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(175, base_ring=CyclotomicField(60)) chi = DirichletCharacter(H, H._module([21, 10])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 175 = 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 175.x (of order \(60\), degree \(16\), 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: \(1.39738203537\)
Analytic rank: \(0\)
Dimension: \(288\)
Relative dimension: \(18\) over \(\Q(\zeta_{60})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{60}]$

Embedding invariants

Embedding label 122.9
Character \(\chi\) \(=\) 175.122
Dual form 175.2.x.a.33.9

$q$-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+(0.0144040 + 0.00935405i) q^{2} +(1.05595 + 0.405343i) q^{3} +(-0.813353 - 1.82682i) q^{4} +(-0.770525 - 2.09912i) q^{5} +(0.0114183 + 0.0157160i) q^{6} +(-0.822022 - 2.51481i) q^{7} +(0.0107461 - 0.0678483i) q^{8} +(-1.27870 - 1.15134i) q^{9} +(0.00853662 - 0.0374431i) q^{10} +(3.39365 + 3.76903i) q^{11} +(-0.118375 - 2.25873i) q^{12} +(-0.145799 + 0.286147i) q^{13} +(0.0116833 - 0.0439125i) q^{14} +(0.0372227 - 2.52890i) q^{15} +(-2.67534 + 2.97126i) q^{16} +(2.90424 + 2.35181i) q^{17} +(-0.00764859 - 0.0285449i) q^{18} +(6.10342 + 2.71742i) q^{19} +(-3.20800 + 3.11494i) q^{20} +(0.151343 - 2.98873i) q^{21} +(0.0136263 + 0.0860334i) q^{22} +(1.86170 - 2.86676i) q^{23} +(0.0388492 - 0.0672888i) q^{24} +(-3.81258 + 3.23484i) q^{25} +(-0.00477673 + 0.00275784i) q^{26} +(-2.42406 - 4.75748i) q^{27} +(-3.92552 + 3.54712i) q^{28} +(1.99592 - 2.74715i) q^{29} +(0.0241916 - 0.0360780i) q^{30} +(0.746861 - 0.0784982i) q^{31} +(-0.199035 + 0.0533314i) q^{32} +(2.05579 + 5.35552i) q^{33} +(0.0198337 + 0.0610417i) q^{34} +(-4.64549 + 3.66325i) q^{35} +(-1.06327 + 3.27240i) q^{36} +(-3.29137 + 0.172493i) q^{37} +(0.0624947 + 0.0962334i) q^{38} +(-0.269945 + 0.243060i) q^{39} +(-0.150702 + 0.0297215i) q^{40} +(7.88534 - 2.56210i) q^{41} +(0.0301367 - 0.0416339i) q^{42} +(-3.99562 + 3.99562i) q^{43} +(4.12511 - 9.26514i) q^{44} +(-1.43154 + 3.57127i) q^{45} +(0.0536317 - 0.0238784i) q^{46} +(-3.96152 - 4.89207i) q^{47} +(-4.02942 + 2.05309i) q^{48} +(-5.64856 + 4.13446i) q^{49} +(-0.0851752 + 0.0109315i) q^{50} +(2.11346 + 3.66061i) q^{51} +(0.641326 + 0.0336105i) q^{52} +(-4.04766 + 10.5445i) q^{53} +(0.00958565 - 0.0912013i) q^{54} +(5.29674 - 10.0278i) q^{55} +(-0.179459 + 0.0287484i) q^{56} +(5.34345 + 5.34345i) q^{57} +(0.0544462 - 0.0208999i) q^{58} +(9.35373 + 1.98820i) q^{59} +(-4.65012 + 1.98889i) q^{60} +(0.570023 + 2.68175i) q^{61} +(0.0114920 + 0.00585548i) q^{62} +(-1.84430 + 4.16211i) q^{63} +(7.60172 + 2.46995i) q^{64} +(0.712999 + 0.0855661i) q^{65} +(-0.0204842 + 0.0963707i) q^{66} +(-4.10889 + 5.07406i) q^{67} +(1.93416 - 7.21837i) q^{68} +(3.12789 - 2.27255i) q^{69} +(-0.101180 + 0.00931112i) q^{70} +(-3.66851 - 2.66533i) q^{71} +(-0.0918577 + 0.0743849i) q^{72} +(0.528792 - 10.0900i) q^{73} +(-0.0490223 - 0.0283030i) q^{74} +(-5.33714 + 1.87045i) q^{75} -13.3601i q^{76} +(6.68874 - 11.6326i) q^{77} +(-0.00616188 + 0.000975946i) q^{78} +(-8.16023 - 0.857674i) q^{79} +(8.29845 + 3.32641i) q^{80} +(-0.0917101 - 0.872564i) q^{81} +(0.137546 + 0.0368554i) q^{82} +(-1.28282 - 0.203179i) q^{83} +(-5.58297 + 2.15442i) q^{84} +(2.69893 - 7.90846i) q^{85} +(-0.0949280 + 0.0201776i) q^{86} +(3.22114 - 2.09183i) q^{87} +(0.292191 - 0.189751i) q^{88} +(-0.969707 + 0.206118i) q^{89} +(-0.0540257 + 0.0380499i) q^{90} +(0.839457 + 0.131438i) q^{91} +(-6.75128 - 1.06930i) q^{92} +(0.820470 + 0.219844i) q^{93} +(-0.0113010 - 0.107521i) q^{94} +(1.00134 - 14.9056i) q^{95} +(-0.231790 - 0.0243621i) q^{96} +(-12.2690 + 1.94322i) q^{97} +(-0.120036 + 0.00671581i) q^{98} -8.72671i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 288 q - 8 q^{2} - 24 q^{3} - 10 q^{4} - 30 q^{5} - 10 q^{7} - 36 q^{8} - 10 q^{9} - 36 q^{10} - 6 q^{11} - 36 q^{12} - 20 q^{14} - 28 q^{15} - 30 q^{16} - 42 q^{17} - 14 q^{18} - 30 q^{19} - 12 q^{21} + 32 q^{22}+ \cdots + 222 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/175\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(127\)
\(\chi(n)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{17}{20}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.0144040 + 0.00935405i 0.0101851 + 0.00661431i 0.549722 − 0.835348i \(-0.314734\pi\)
−0.539536 + 0.841962i \(0.681400\pi\)
\(3\) 1.05595 + 0.405343i 0.609656 + 0.234025i 0.643554 − 0.765400i \(-0.277459\pi\)
−0.0338987 + 0.999425i \(0.510792\pi\)
\(4\) −0.813353 − 1.82682i −0.406677 − 0.913411i
\(5\) −0.770525 − 2.09912i −0.344589 − 0.938754i
\(6\) 0.0114183 + 0.0157160i 0.00466152 + 0.00641603i
\(7\) −0.822022 − 2.51481i −0.310695 − 0.950510i
\(8\) 0.0107461 − 0.0678483i 0.00379932 − 0.0239880i
\(9\) −1.27870 − 1.15134i −0.426232 − 0.383781i
\(10\) 0.00853662 − 0.0374431i 0.00269952 − 0.0118406i
\(11\) 3.39365 + 3.76903i 1.02322 + 1.13641i 0.990580 + 0.136938i \(0.0437262\pi\)
0.0326443 + 0.999467i \(0.489607\pi\)
\(12\) −0.118375 − 2.25873i −0.0341719 − 0.652039i
\(13\) −0.145799 + 0.286147i −0.0404375 + 0.0793630i −0.910341 − 0.413859i \(-0.864181\pi\)
0.869904 + 0.493222i \(0.164181\pi\)
\(14\) 0.0116833 − 0.0439125i 0.00312249 − 0.0117361i
\(15\) 0.0372227 − 2.52890i 0.00961085 − 0.652959i
\(16\) −2.67534 + 2.97126i −0.668835 + 0.742816i
\(17\) 2.90424 + 2.35181i 0.704381 + 0.570397i 0.913177 − 0.407562i \(-0.133621\pi\)
−0.208796 + 0.977959i \(0.566954\pi\)
\(18\) −0.00764859 − 0.0285449i −0.00180279 − 0.00672810i
\(19\) 6.10342 + 2.71742i 1.40022 + 0.623419i 0.961400 − 0.275155i \(-0.0887293\pi\)
0.438822 + 0.898574i \(0.355396\pi\)
\(20\) −3.20800 + 3.11494i −0.717331 + 0.696521i
\(21\) 0.151343 − 2.98873i 0.0330257 − 0.652194i
\(22\) 0.0136263 + 0.0860334i 0.00290515 + 0.0183424i
\(23\) 1.86170 − 2.86676i 0.388191 − 0.597762i −0.589511 − 0.807761i \(-0.700679\pi\)
0.977702 + 0.209999i \(0.0673461\pi\)
\(24\) 0.0388492 − 0.0672888i 0.00793006 − 0.0137353i
\(25\) −3.81258 + 3.23484i −0.762516 + 0.646969i
\(26\) −0.00477673 + 0.00275784i −0.000936793 + 0.000540858i
\(27\) −2.42406 − 4.75748i −0.466510 − 0.915577i
\(28\) −3.92552 + 3.54712i −0.741853 + 0.670342i
\(29\) 1.99592 − 2.74715i 0.370633 − 0.510133i −0.582440 − 0.812874i \(-0.697902\pi\)
0.953073 + 0.302741i \(0.0979018\pi\)
\(30\) 0.0241916 − 0.0360780i 0.00441676 − 0.00658691i
\(31\) 0.746861 − 0.0784982i 0.134140 − 0.0140987i −0.0372204 − 0.999307i \(-0.511850\pi\)
0.171361 + 0.985208i \(0.445184\pi\)
\(32\) −0.199035 + 0.0533314i −0.0351848 + 0.00942775i
\(33\) 2.05579 + 5.35552i 0.357867 + 0.932276i
\(34\) 0.0198337 + 0.0610417i 0.00340145 + 0.0104686i
\(35\) −4.64549 + 3.66325i −0.785232 + 0.619202i
\(36\) −1.06327 + 3.27240i −0.177211 + 0.545400i
\(37\) −3.29137 + 0.172493i −0.541098 + 0.0283577i −0.320928 − 0.947104i \(-0.603995\pi\)
−0.220170 + 0.975461i \(0.570661\pi\)
\(38\) 0.0624947 + 0.0962334i 0.0101380 + 0.0156111i
\(39\) −0.269945 + 0.243060i −0.0432258 + 0.0389207i
\(40\) −0.150702 + 0.0297215i −0.0238280 + 0.00469937i
\(41\) 7.88534 − 2.56210i 1.23148 − 0.400133i 0.380232 − 0.924891i \(-0.375844\pi\)
0.851252 + 0.524758i \(0.175844\pi\)
\(42\) 0.0301367 − 0.0416339i 0.00465019 − 0.00642425i
\(43\) −3.99562 + 3.99562i −0.609326 + 0.609326i −0.942770 − 0.333444i \(-0.891789\pi\)
0.333444 + 0.942770i \(0.391789\pi\)
\(44\) 4.12511 − 9.26514i 0.621883 − 1.39677i
\(45\) −1.43154 + 3.57127i −0.213401 + 0.532374i
\(46\) 0.0536317 − 0.0238784i 0.00790756 − 0.00352067i
\(47\) −3.96152 − 4.89207i −0.577847 − 0.713581i 0.401480 − 0.915868i \(-0.368496\pi\)
−0.979326 + 0.202287i \(0.935163\pi\)
\(48\) −4.02942 + 2.05309i −0.581596 + 0.296338i
\(49\) −5.64856 + 4.13446i −0.806937 + 0.590638i
\(50\) −0.0851752 + 0.0109315i −0.0120456 + 0.00154595i
\(51\) 2.11346 + 3.66061i 0.295943 + 0.512588i
\(52\) 0.641326 + 0.0336105i 0.0889360 + 0.00466094i
\(53\) −4.04766 + 10.5445i −0.555988 + 1.44840i 0.311164 + 0.950356i \(0.399281\pi\)
−0.867152 + 0.498043i \(0.834052\pi\)
\(54\) 0.00958565 − 0.0912013i 0.00130444 − 0.0124109i
\(55\) 5.29674 − 10.0278i 0.714212 − 1.35215i
\(56\) −0.179459 + 0.0287484i −0.0239812 + 0.00384166i
\(57\) 5.34345 + 5.34345i 0.707757 + 0.707757i
\(58\) 0.0544462 − 0.0208999i 0.00714913 − 0.00274429i
\(59\) 9.35373 + 1.98820i 1.21775 + 0.258841i 0.771595 − 0.636114i \(-0.219459\pi\)
0.446157 + 0.894955i \(0.352792\pi\)
\(60\) −4.65012 + 1.98889i −0.600328 + 0.256765i
\(61\) 0.570023 + 2.68175i 0.0729840 + 0.343363i 0.999455 − 0.0330028i \(-0.0105070\pi\)
−0.926471 + 0.376365i \(0.877174\pi\)
\(62\) 0.0114920 + 0.00585548i 0.00145949 + 0.000743647i
\(63\) −1.84430 + 4.16211i −0.232359 + 0.524377i
\(64\) 7.60172 + 2.46995i 0.950215 + 0.308744i
\(65\) 0.712999 + 0.0855661i 0.0884366 + 0.0106132i
\(66\) −0.0204842 + 0.0963707i −0.00252143 + 0.0118624i
\(67\) −4.10889 + 5.07406i −0.501981 + 0.619895i −0.963653 − 0.267155i \(-0.913916\pi\)
0.461672 + 0.887051i \(0.347250\pi\)
\(68\) 1.93416 − 7.21837i 0.234551 − 0.875357i
\(69\) 3.12789 − 2.27255i 0.376554 − 0.273582i
\(70\) −0.101180 + 0.00931112i −0.0120933 + 0.00111289i
\(71\) −3.66851 − 2.66533i −0.435372 − 0.316316i 0.348422 − 0.937338i \(-0.386718\pi\)
−0.783793 + 0.621022i \(0.786718\pi\)
\(72\) −0.0918577 + 0.0743849i −0.0108255 + 0.00876635i
\(73\) 0.528792 − 10.0900i 0.0618904 − 1.18094i −0.774416 − 0.632677i \(-0.781956\pi\)
0.836306 − 0.548263i \(-0.184711\pi\)
\(74\) −0.0490223 − 0.0283030i −0.00569873 − 0.00329016i
\(75\) −5.33714 + 1.87045i −0.616279 + 0.215980i
\(76\) − 13.3601i − 1.53251i
\(77\) 6.68874 − 11.6326i 0.762253 − 1.32566i
\(78\) −0.00616188 0.000975946i −0.000697695 0.000110504i
\(79\) −8.16023 − 0.857674i −0.918097 − 0.0964959i −0.366321 − 0.930488i \(-0.619383\pi\)
−0.551776 + 0.833992i \(0.686050\pi\)
\(80\) 8.29845 + 3.32641i 0.927794 + 0.371904i
\(81\) −0.0917101 − 0.872564i −0.0101900 − 0.0969515i
\(82\) 0.137546 + 0.0368554i 0.0151894 + 0.00407000i
\(83\) −1.28282 − 0.203179i −0.140808 − 0.0223018i 0.0856324 − 0.996327i \(-0.472709\pi\)
−0.226441 + 0.974025i \(0.572709\pi\)
\(84\) −5.58297 + 2.15442i −0.609152 + 0.235066i
\(85\) 2.69893 − 7.90846i 0.292740 − 0.857793i
\(86\) −0.0949280 + 0.0201776i −0.0102363 + 0.00217580i
\(87\) 3.22114 − 2.09183i 0.345342 − 0.224268i
\(88\) 0.292191 − 0.189751i 0.0311476 − 0.0202275i
\(89\) −0.969707 + 0.206118i −0.102789 + 0.0218484i −0.259019 − 0.965872i \(-0.583399\pi\)
0.156230 + 0.987721i \(0.450066\pi\)
\(90\) −0.0540257 + 0.0380499i −0.00569481 + 0.00401081i
\(91\) 0.839457 + 0.131438i 0.0879990 + 0.0137785i
\(92\) −6.75128 − 1.06930i −0.703870 − 0.111482i
\(93\) 0.820470 + 0.219844i 0.0850787 + 0.0227968i
\(94\) −0.0113010 − 0.107521i −0.00116560 − 0.0110900i
\(95\) 1.00134 − 14.9056i 0.102735 − 1.52929i
\(96\) −0.231790 − 0.0243621i −0.0236570 − 0.00248645i
\(97\) −12.2690 + 1.94322i −1.24573 + 0.197304i −0.744270 − 0.667879i \(-0.767202\pi\)
−0.501457 + 0.865183i \(0.667202\pi\)
\(98\) −0.120036 + 0.00671581i −0.0121254 + 0.000678400i
\(99\) − 8.72671i − 0.877067i
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 175.2.x.a.122.9 yes 288
5.2 odd 4 875.2.bb.a.493.10 288
5.3 odd 4 875.2.bb.b.493.9 288
5.4 even 2 875.2.bb.c.507.10 288
7.5 odd 6 inner 175.2.x.a.47.9 yes 288
25.6 even 5 875.2.bb.b.857.10 288
25.8 odd 20 inner 175.2.x.a.108.9 yes 288
25.17 odd 20 875.2.bb.c.143.10 288
25.19 even 10 875.2.bb.a.857.9 288
35.12 even 12 875.2.bb.a.243.9 288
35.19 odd 6 875.2.bb.c.257.10 288
35.33 even 12 875.2.bb.b.243.10 288
175.19 odd 30 875.2.bb.a.607.10 288
175.33 even 60 inner 175.2.x.a.33.9 ✓ 288
175.117 even 60 875.2.bb.c.768.10 288
175.131 odd 30 875.2.bb.b.607.9 288
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
175.2.x.a.33.9 ✓ 288 175.33 even 60 inner
175.2.x.a.47.9 yes 288 7.5 odd 6 inner
175.2.x.a.108.9 yes 288 25.8 odd 20 inner
175.2.x.a.122.9 yes 288 1.1 even 1 trivial
875.2.bb.a.243.9 288 35.12 even 12
875.2.bb.a.493.10 288 5.2 odd 4
875.2.bb.a.607.10 288 175.19 odd 30
875.2.bb.a.857.9 288 25.19 even 10
875.2.bb.b.243.10 288 35.33 even 12
875.2.bb.b.493.9 288 5.3 odd 4
875.2.bb.b.607.9 288 175.131 odd 30
875.2.bb.b.857.10 288 25.6 even 5
875.2.bb.c.143.10 288 25.17 odd 20
875.2.bb.c.257.10 288 35.19 odd 6
875.2.bb.c.507.10 288 5.4 even 2
875.2.bb.c.768.10 288 175.117 even 60