LMFDB - Embedded newform 6050.2.a.de.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6050,2,Mod(1,6050)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6050, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("6050.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6050 = 2 \cdot 5^{2} \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6050.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$48.3094932229$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.35136.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 8x^{2} + 6x + 9 $ x^4 - 2x^3 - 8x^2 + 6*x + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-2.13633$ of defining polynomial
Character	$\chi$	$=$	6050.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} -0.404278 q^{3} +1.00000 q^{4} +0.404278 q^{6} +1.40428 q^{7} -1.00000 q^{8} -2.83656 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -0.404278 q^{3} +1.00000 q^{4} +0.404278 q^{6} +1.40428 q^{7} -1.00000 q^{8} -2.83656 q^{9} -0.404278 q^{12} -0.327773 q^{13} -1.40428 q^{14} +1.00000 q^{16} -3.83656 q^{17} +2.83656 q^{18} +2.56861 q^{19} -0.567719 q^{21} +5.24084 q^{23} +0.404278 q^{24} +0.327773 q^{26} +2.35959 q^{27} +1.40428 q^{28} -2.62372 q^{29} -4.57243 q^{31} -1.00000 q^{32} +3.83656 q^{34} -2.83656 q^{36} -0.808556 q^{37} -2.56861 q^{38} +0.132511 q^{39} -4.16815 q^{41} +0.567719 q^{42} +11.3325 q^{43} -5.24084 q^{46} +5.10069 q^{47} -0.404278 q^{48} -5.02800 q^{49} +1.55104 q^{51} -0.327773 q^{52} +1.16815 q^{53} -2.35959 q^{54} -1.40428 q^{56} -1.03843 q^{57} +2.62372 q^{58} +8.84127 q^{59} -10.7767 q^{61} +4.57243 q^{62} -3.98332 q^{63} +1.00000 q^{64} -5.16815 q^{67} -3.83656 q^{68} -2.11876 q^{69} +13.0607 q^{71} +2.83656 q^{72} -4.26795 q^{73} +0.808556 q^{74} +2.56861 q^{76} -0.132511 q^{78} +0.272658 q^{79} +7.55575 q^{81} +4.16815 q^{82} +3.19144 q^{83} -0.567719 q^{84} -11.3325 q^{86} +1.06071 q^{87} -5.80856 q^{89} -0.460284 q^{91} +5.24084 q^{92} +1.84853 q^{93} -5.10069 q^{94} +0.404278 q^{96} -17.5377 q^{97} +5.02800 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{2} + 2 q^{3} + 4 q^{4} - 2 q^{6} + 2 q^{7} - 4 q^{8} + 8 q^{9}+O(q^{10}) $ 4 * q - 4 * q^2 + 2 * q^3 + 4 * q^4 - 2 * q^6 + 2 * q^7 - 4 * q^8 + 8 * q^9	$ 4 q - 4 q^{2} + 2 q^{3} + 4 q^{4} - 2 q^{6} + 2 q^{7} - 4 q^{8} + 8 q^{9} + 2 q^{12} + 2 q^{13} - 2 q^{14} + 4 q^{16} + 4 q^{17} - 8 q^{18} - 16 q^{19} - 18 q^{21} - 2 q^{23} - 2 q^{24} - 2 q^{26} + 8 q^{27} + 2 q^{28} - 2 q^{29} - 6 q^{31} - 4 q^{32} - 4 q^{34} + 8 q^{36} + 4 q^{37} + 16 q^{38} - 24 q^{39} - 8 q^{41} + 18 q^{42} + 14 q^{43} + 2 q^{46} - 6 q^{47} + 2 q^{48} - 8 q^{49} + 12 q^{51} + 2 q^{52} - 4 q^{53} - 8 q^{54} - 2 q^{56} - 12 q^{57} + 2 q^{58} - 12 q^{59} - 34 q^{61} + 6 q^{62} - 6 q^{63} + 4 q^{64} - 12 q^{67} + 4 q^{68} - 30 q^{69} - 8 q^{72} - 24 q^{73} - 4 q^{74} - 16 q^{76} + 24 q^{78} - 20 q^{79} + 8 q^{81} + 8 q^{82} + 20 q^{83} - 18 q^{84} - 14 q^{86} - 48 q^{87} - 16 q^{89} + 26 q^{91} - 2 q^{92} + 26 q^{93} + 6 q^{94} - 2 q^{96} + 8 q^{98}+O(q^{100}) $ 4 * q - 4 * q^2 + 2 * q^3 + 4 * q^4 - 2 * q^6 + 2 * q^7 - 4 * q^8 + 8 * q^9 + 2 * q^12 + 2 * q^13 - 2 * q^14 + 4 * q^16 + 4 * q^17 - 8 * q^18 - 16 * q^19 - 18 * q^21 - 2 * q^23 - 2 * q^24 - 2 * q^26 + 8 * q^27 + 2 * q^28 - 2 * q^29 - 6 * q^31 - 4 * q^32 - 4 * q^34 + 8 * q^36 + 4 * q^37 + 16 * q^38 - 24 * q^39 - 8 * q^41 + 18 * q^42 + 14 * q^43 + 2 * q^46 - 6 * q^47 + 2 * q^48 - 8 * q^49 + 12 * q^51 + 2 * q^52 - 4 * q^53 - 8 * q^54 - 2 * q^56 - 12 * q^57 + 2 * q^58 - 12 * q^59 - 34 * q^61 + 6 * q^62 - 6 * q^63 + 4 * q^64 - 12 * q^67 + 4 * q^68 - 30 * q^69 - 8 * q^72 - 24 * q^73 - 4 * q^74 - 16 * q^76 + 24 * q^78 - 20 * q^79 + 8 * q^81 + 8 * q^82 + 20 * q^83 - 18 * q^84 - 14 * q^86 - 48 * q^87 - 16 * q^89 + 26 * q^91 - 2 * q^92 + 26 * q^93 + 6 * q^94 - 2 * q^96 + 8 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	−0.404278	−0.233410	−0.116705	−	0.993167i	$-0.537233\pi$
$3$	−0.404278	−0.233410	−0.116705	+	0.993167i	$0.537233\pi$
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	0.404278	0.165046
$7$	1.40428	0.530767	0.265384	−	0.964143i	$-0.414501\pi$
$7$	1.40428	0.530767	0.265384	+	0.964143i	$0.414501\pi$
$8$	−1.00000	−0.353553
$9$	−2.83656	−0.945520
$10$	0	0
$11$	0	0
$11$	0	0
$12$	−0.404278	−0.116705
$13$	−0.327773	−0.0909078	−0.0454539	−	0.998966i	$-0.514473\pi$
$13$	−0.327773	−0.0909078	−0.0454539	+	0.998966i	$0.514473\pi$
$14$	−1.40428	−0.375309
$15$	0	0
$16$	1.00000	0.250000
$17$	−3.83656	−0.930502	−0.465251	−	0.885179i	$-0.654036\pi$
$17$	−3.83656	−0.930502	−0.465251	+	0.885179i	$0.654036\pi$
$18$	2.83656	0.668583
$19$	2.56861	0.589280	0.294640	−	0.955608i	$-0.404800\pi$
$19$	2.56861	0.589280	0.294640	+	0.955608i	$0.404800\pi$
$20$	0	0
$21$	−0.567719	−0.123886
$22$	0	0
$23$	5.24084	1.09279	0.546395	−	0.837528i	$-0.316000\pi$
$23$	5.24084	1.09279	0.546395	+	0.837528i	$0.316000\pi$
$24$	0.404278	0.0825229
$25$	0	0
$26$	0.327773	0.0642815
$27$	2.35959	0.454104
$28$	1.40428	0.265384
$29$	−2.62372	−0.487213	−0.243607	−	0.969874i	$-0.578331\pi$
$29$	−2.62372	−0.487213	−0.243607	+	0.969874i	$0.578331\pi$
$30$	0	0
$31$	−4.57243	−0.821232	−0.410616	−	0.911808i	$-0.634686\pi$
$31$	−4.57243	−0.821232	−0.410616	+	0.911808i	$0.634686\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	3.83656	0.657964
$35$	0	0
$36$	−2.83656	−0.472760
$37$	−0.808556	−0.132926	−0.0664629	−	0.997789i	$-0.521171\pi$
$37$	−0.808556	−0.132926	−0.0664629	+	0.997789i	$0.521171\pi$
$38$	−2.56861	−0.416684
$39$	0.132511	0.0212188
$40$	0	0
$41$	−4.16815	−0.650956	−0.325478	−	0.945550i	$-0.605525\pi$
$41$	−4.16815	−0.650956	−0.325478	+	0.945550i	$0.605525\pi$
$42$	0.567719	0.0876009
$43$	11.3325	1.72819	0.864094	−	0.503331i	$-0.167892\pi$
$43$	11.3325	1.72819	0.864094	+	0.503331i	$0.167892\pi$
$44$	0	0
$45$	0	0
$46$	−5.24084	−0.772719
$47$	5.10069	0.744012	0.372006	−	0.928230i	$-0.378670\pi$
$47$	5.10069	0.744012	0.372006	+	0.928230i	$0.378670\pi$
$48$	−0.404278	−0.0583525
$49$	−5.02800	−0.718286
$50$	0	0
$51$	1.55104	0.217189
$52$	−0.327773	−0.0454539
$53$	1.16815	0.160458	0.0802288	−	0.996776i	$-0.474435\pi$
$53$	1.16815	0.160458	0.0802288	+	0.996776i	$0.474435\pi$
$54$	−2.35959	−0.321100
$55$	0	0
$56$	−1.40428	−0.187655
$57$	−1.03843	−0.137544
$58$	2.62372	0.344512
$59$	8.84127	1.15104	0.575518	−	0.817789i	$-0.304801\pi$
$59$	8.84127	1.15104	0.575518	+	0.817789i	$0.304801\pi$
$60$	0	0
$61$	−10.7767	−1.37982	−0.689910	−	0.723895i	$-0.742350\pi$
$61$	−10.7767	−1.37982	−0.689910	+	0.723895i	$0.742350\pi$
$62$	4.57243	0.580699
$63$	−3.98332	−0.501851
$64$	1.00000	0.125000
$65$	0	0
$66$	0	0
$67$	−5.16815	−0.631390	−0.315695	−	0.948861i	$-0.602238\pi$
$67$	−5.16815	−0.631390	−0.315695	+	0.948861i	$0.602238\pi$
$68$	−3.83656	−0.465251
$69$	−2.11876	−0.255068
$70$	0	0
$71$	13.0607	1.55002	0.775011	−	0.631948i	$-0.217744\pi$
$71$	13.0607	1.55002	0.775011	+	0.631948i	$0.217744\pi$
$72$	2.83656	0.334292
$73$	−4.26795	−0.499526	−0.249763	−	0.968307i	$-0.580353\pi$
$73$	−4.26795	−0.499526	−0.249763	+	0.968307i	$0.580353\pi$
$74$	0.808556	0.0939928
$75$	0	0
$76$	2.56861	0.294640
$77$	0	0
$78$	−0.132511	−0.0150040
$79$	0.272658	0.0306764	0.0153382	−	0.999882i	$-0.495118\pi$
$79$	0.272658	0.0306764	0.0153382	+	0.999882i	$0.495118\pi$
$80$	0	0
$81$	7.55575	0.839527
$82$	4.16815	0.460295
$83$	3.19144	0.350306	0.175153	−	0.984541i	$-0.443958\pi$
$83$	3.19144	0.350306	0.175153	+	0.984541i	$0.443958\pi$
$84$	−0.567719	−0.0619432
$85$	0	0
$86$	−11.3325	−1.22201
$87$	1.06071	0.113721
$88$	0	0
$89$	−5.80856	−0.615706	−0.307853	−	0.951434i	$-0.599610\pi$
$89$	−5.80856	−0.615706	−0.307853	+	0.951434i	$0.599610\pi$
$90$	0	0
$91$	−0.460284	−0.0482509
$92$	5.24084	0.546395
$93$	1.84853	0.191684
$94$	−5.10069	−0.526096
$95$	0	0
$96$	0.404278	0.0412615
$97$	−17.5377	−1.78068	−0.890341	−	0.455295i	$-0.849534\pi$
$97$	−17.5377	−1.78068	−0.890341	+	0.455295i	$0.849534\pi$
$98$	5.02800	0.507905
$99$	0	0
$100$	0	0
$101$	−9.97671	−0.992719	−0.496360	−	0.868117i	$-0.665330\pi$
$101$	−9.97671	−0.992719	−0.496360	+	0.868117i	$0.665330\pi$
$102$	−1.55104	−0.153576
$103$	−4.95532	−0.488262	−0.244131	−	0.969742i	$-0.578503\pi$
$103$	−4.95532	−0.488262	−0.244131	+	0.969742i	$0.578503\pi$
$104$	0.327773	0.0321408
$105$	0	0
$106$	−1.16815	−0.113461
$107$	5.61711	0.543027	0.271513	−	0.962435i	$-0.412476\pi$
$107$	5.61711	0.543027	0.271513	+	0.962435i	$0.412476\pi$
$108$	2.35959	0.227052
$109$	−13.6731	−1.30965	−0.654824	−	0.755782i	$-0.727257\pi$
$109$	−13.6731	−1.30965	−0.654824	+	0.755782i	$0.727257\pi$
$110$	0	0
$111$	0.326882	0.0310262
$112$	1.40428	0.132692
$113$	18.4537	1.73598	0.867988	−	0.496586i	$-0.165413\pi$
$113$	18.4537	1.73598	0.867988	+	0.496586i	$0.165413\pi$
$114$	1.03843	0.0972581
$115$	0	0
$116$	−2.62372	−0.243607
$117$	0.929747	0.0859551
$118$	−8.84127	−0.813905
$119$	−5.38760	−0.493880
$120$	0	0
$121$	0	0
$122$	10.7767	0.975680
$123$	1.68509	0.151940
$124$	−4.57243	−0.410616
$125$	0	0
$126$	3.98332	0.354862
$127$	−11.4808	−1.01875	−0.509377	−	0.860543i	$-0.670124\pi$
$127$	−11.4808	−1.01875	−0.509377	+	0.860543i	$0.670124\pi$
$128$	−1.00000	−0.0883883
$129$	−4.58147	−0.403376
$130$	0	0
$131$	−16.9092	−1.47737	−0.738684	−	0.674052i	$-0.764552\pi$
$131$	−16.9092	−1.47737	−0.738684	+	0.674052i	$0.764552\pi$
$132$	0	0
$133$	3.60704	0.312770
$134$	5.16815	0.446460
$135$	0	0
$136$	3.83656	0.328982
$137$	18.1729	1.55261	0.776306	−	0.630356i	$-0.217091\pi$
$137$	18.1729	1.55261	0.776306	+	0.630356i	$0.217091\pi$
$138$	2.11876	0.180360
$139$	−6.41560	−0.544164	−0.272082	−	0.962274i	$-0.587712\pi$
$139$	−6.41560	−0.544164	−0.272082	+	0.962274i	$0.587712\pi$
$140$	0	0
$141$	−2.06210	−0.173660
$142$	−13.0607	−1.09603
$143$	0	0
$144$	−2.83656	−0.236380
$145$	0	0
$146$	4.26795	0.353218
$147$	2.03271	0.167655
$148$	−0.808556	−0.0664629
$149$	−18.6331	−1.52649	−0.763243	−	0.646111i	$-0.776394\pi$
$149$	−18.6331	−1.52649	−0.763243	+	0.646111i	$0.776394\pi$
$150$	0	0
$151$	−15.3501	−1.24917	−0.624585	−	0.780957i	$-0.714732\pi$
$151$	−15.3501	−1.24917	−0.624585	+	0.780957i	$0.714732\pi$
$152$	−2.56861	−0.208342
$153$	10.8826	0.879808
$154$	0	0
$155$	0	0
$156$	0.132511	0.0106094
$157$	−7.02610	−0.560744	−0.280372	−	0.959891i	$-0.590458\pi$
$157$	−7.02610	−0.560744	−0.280372	+	0.959891i	$0.590458\pi$
$158$	−0.272658	−0.0216915
$159$	−0.472257	−0.0374524
$160$	0	0
$161$	7.35959	0.580017
$162$	−7.55575	−0.593635
$163$	6.79658	0.532349	0.266175	−	0.963925i	$-0.414240\pi$
$163$	6.79658	0.532349	0.266175	+	0.963925i	$0.414240\pi$
$164$	−4.16815	−0.325478
$165$	0	0
$166$	−3.19144	−0.247704
$167$	−12.8812	−0.996781	−0.498390	−	0.866953i	$-0.666075\pi$
$167$	−12.8812	−0.996781	−0.498390	+	0.866953i	$0.666075\pi$
$168$	0.567719	0.0438005
$169$	−12.8926	−0.991736
$170$	0	0
$171$	−7.28601	−0.557175
$172$	11.3325	0.864094
$173$	−9.40899	−0.715352	−0.357676	−	0.933846i	$-0.616431\pi$
$173$	−9.40899	−0.715352	−0.357676	+	0.933846i	$0.616431\pi$
$174$	−1.06071	−0.0804126
$175$	0	0
$176$	0	0
$177$	−3.57433	−0.268663
$178$	5.80856	0.435370
$179$	−3.14676	−0.235200	−0.117600	−	0.993061i	$-0.537520\pi$
$179$	−3.14676	−0.235200	−0.117600	+	0.993061i	$0.537520\pi$
$180$	0	0
$181$	25.1776	1.87143	0.935717	−	0.352752i	$-0.114754\pi$
$181$	25.1776	1.87143	0.935717	+	0.352752i	$0.114754\pi$
$182$	0.460284	0.0341185
$183$	4.35680	0.322064
$184$	−5.24084	−0.386360
$185$	0	0
$186$	−1.84853	−0.135541
$187$	0	0
$188$	5.10069	0.372006
$189$	3.31353	0.241023
$190$	0	0
$191$	18.8180	1.36162	0.680810	−	0.732460i	$-0.261628\pi$
$191$	18.8180	1.36162	0.680810	+	0.732460i	$0.261628\pi$
$192$	−0.404278	−0.0291763
$193$	22.8056	1.64159	0.820793	−	0.571226i	$-0.193532\pi$
$193$	22.8056	1.64159	0.820793	+	0.571226i	$0.193532\pi$
$194$	17.5377	1.25913
$195$	0	0
$196$	−5.02800	−0.359143
$197$	−1.03998	−0.0740952	−0.0370476	−	0.999314i	$-0.511795\pi$
$197$	−1.03998	−0.0740952	−0.0370476	+	0.999314i	$0.511795\pi$
$198$	0	0
$199$	−13.2408	−0.938618	−0.469309	−	0.883034i	$-0.655497\pi$
$199$	−13.2408	−0.938618	−0.469309	+	0.883034i	$0.655497\pi$
$200$	0	0
$201$	2.08937	0.147373
$202$	9.97671	0.701959
$203$	−3.68444	−0.258597
$204$	1.55104	0.108594
$205$	0	0
$206$	4.95532	0.345253
$207$	−14.8659	−1.03325
$208$	−0.327773	−0.0227269
$209$	0	0
$210$	0	0
$211$	−12.4603	−0.857801	−0.428901	−	0.903352i	$-0.641099\pi$
$211$	−12.4603	−0.857801	−0.428901	+	0.903352i	$0.641099\pi$
$212$	1.16815	0.0802288
$213$	−5.28016	−0.361791
$214$	−5.61711	−0.383978
$215$	0	0
$216$	−2.35959	−0.160550
$217$	−6.42096	−0.435883
$218$	13.6731	0.926060
$219$	1.72544	0.116594
$220$	0	0
$221$	1.25752	0.0845899
$222$	−0.326882	−0.0219389
$223$	11.0541	0.740237	0.370119	−	0.928984i	$-0.379317\pi$
$223$	11.0541	0.740237	0.370119	+	0.928984i	$0.379317\pi$
$224$	−1.40428	−0.0938273
$225$	0	0
$226$	−18.4537	−1.22752
$227$	−9.50445	−0.630832	−0.315416	−	0.948953i	$-0.602144\pi$
$227$	−9.50445	−0.630832	−0.315416	+	0.948953i	$0.602144\pi$
$228$	−1.03843	−0.0687719
$229$	−14.4156	−0.952610	−0.476305	−	0.879280i	$-0.658024\pi$
$229$	−14.4156	−0.952610	−0.476305	+	0.879280i	$0.658024\pi$
$230$	0	0
$231$	0	0
$232$	2.62372	0.172256
$233$	4.55104	0.298148	0.149074	−	0.988826i	$-0.452371\pi$
$233$	4.55104	0.298148	0.149074	+	0.988826i	$0.452371\pi$
$234$	−0.929747	−0.0607794
$235$	0	0
$236$	8.84127	0.575518
$237$	−0.110230	−0.00716018
$238$	5.38760	0.349226
$239$	20.0821	1.29900	0.649502	−	0.760360i	$-0.274978\pi$
$239$	20.0821	1.29900	0.649502	+	0.760360i	$0.274978\pi$
$240$	0	0
$241$	−27.7881	−1.78999	−0.894993	−	0.446080i	$-0.852820\pi$
$241$	−27.7881	−1.78999	−0.894993	+	0.446080i	$0.852820\pi$
$242$	0	0
$243$	−10.1334	−0.650058
$244$	−10.7767	−0.689910
$245$	0	0
$246$	−1.68509	−0.107438
$247$	−0.841920	−0.0535701
$248$	4.57243	0.290349
$249$	−1.29023	−0.0817651
$250$	0	0
$251$	−25.6939	−1.62178	−0.810891	−	0.585197i	$-0.801017\pi$
$251$	−25.6939	−1.62178	−0.810891	+	0.585197i	$0.801017\pi$
$252$	−3.98332	−0.250925
$253$	0	0
$254$	11.4808	0.720368
$255$	0	0
$256$	1.00000	0.0625000
$257$	3.74719	0.233743	0.116872	−	0.993147i	$-0.462713\pi$
$257$	3.74719	0.233743	0.116872	+	0.993147i	$0.462713\pi$
$258$	4.58147	0.285230
$259$	−1.13544	−0.0705527
$260$	0	0
$261$	7.44235	0.460670
$262$	16.9092	1.04466
$263$	26.0821	1.60829	0.804146	−	0.594432i	$-0.202623\pi$
$263$	26.0821	1.60829	0.804146	+	0.594432i	$0.202623\pi$
$264$	0	0
$265$	0	0
$266$	−3.60704	−0.221162
$267$	2.34827	0.143712
$268$	−5.16815	−0.315695
$269$	20.3063	1.23809	0.619047	−	0.785354i	$-0.287519\pi$
$269$	20.3063	1.23809	0.619047	+	0.785354i	$0.287519\pi$
$270$	0	0
$271$	22.1381	1.34479	0.672397	−	0.740190i	$-0.265265\pi$
$271$	22.1381	1.34479	0.672397	+	0.740190i	$0.265265\pi$
$272$	−3.83656	−0.232626
$273$	0.186083	0.0112622
$274$	−18.1729	−1.09786
$275$	0	0
$276$	−2.11876	−0.127534
$277$	2.20049	0.132215	0.0661073	−	0.997813i	$-0.478942\pi$
$277$	2.20049	0.132215	0.0661073	+	0.997813i	$0.478942\pi$
$278$	6.41560	0.384782
$279$	12.9700	0.776491
$280$	0	0
$281$	−1.27610	−0.0761260	−0.0380630	−	0.999275i	$-0.512119\pi$
$281$	−1.27610	−0.0761260	−0.0380630	+	0.999275i	$0.512119\pi$
$282$	2.06210	0.122796
$283$	15.4565	0.918792	0.459396	−	0.888232i	$-0.348066\pi$
$283$	15.4565	0.918792	0.459396	+	0.888232i	$0.348066\pi$
$284$	13.0607	0.775011
$285$	0	0
$286$	0	0
$287$	−5.85324	−0.345506
$288$	2.83656	0.167146
$289$	−2.28081	−0.134166
$290$	0	0
$291$	7.09010	0.415629
$292$	−4.26795	−0.249763
$293$	−18.8507	−1.10127	−0.550634	−	0.834747i	$-0.685614\pi$
$293$	−18.8507	−1.10127	−0.550634	+	0.834747i	$0.685614\pi$
$294$	−2.03271	−0.118550
$295$	0	0
$296$	0.808556	0.0469964
$297$	0	0
$298$	18.6331	1.07939
$299$	−1.71780	−0.0993431
$300$	0	0
$301$	15.9140	0.917265
$302$	15.3501	0.883296
$303$	4.03336	0.231711
$304$	2.56861	0.147320
$305$	0	0
$306$	−10.8826	−0.622118
$307$	−0.276996	−0.0158090	−0.00790450	−	0.999969i	$-0.502516\pi$
$307$	−0.276996	−0.0158090	−0.00790450	+	0.999969i	$0.502516\pi$
$308$	0	0
$309$	2.00333	0.113965
$310$	0	0
$311$	−24.4417	−1.38596	−0.692981	−	0.720956i	$-0.743703\pi$
$311$	−24.4417	−1.38596	−0.692981	+	0.720956i	$0.743703\pi$
$312$	−0.132511	−0.00750198
$313$	−15.1268	−0.855017	−0.427508	−	0.904011i	$-0.640609\pi$
$313$	−15.1268	−0.855017	−0.427508	+	0.904011i	$0.640609\pi$
$314$	7.02610	0.396506
$315$	0	0
$316$	0.272658	0.0153382
$317$	−24.1609	−1.35701	−0.678505	−	0.734596i	$-0.737372\pi$
$317$	−24.1609	−1.35701	−0.678505	+	0.734596i	$0.737372\pi$
$318$	0.472257	0.0264829
$319$	0	0
$320$	0	0
$321$	−2.27088	−0.126748
$322$	−7.35959	−0.410134
$323$	−9.85462	−0.548326
$324$	7.55575	0.419764
$325$	0	0
$326$	−6.79658	−0.376428
$327$	5.52774	0.305685
$328$	4.16815	0.230148
$329$	7.16279	0.394897
$330$	0	0
$331$	4.45087	0.244642	0.122321	−	0.992491i	$-0.460966\pi$
$331$	4.45087	0.244642	0.122321	+	0.992491i	$0.460966\pi$
$332$	3.19144	0.175153
$333$	2.29352	0.125684
$334$	12.8812	0.704830
$335$	0	0
$336$	−0.567719	−0.0309716
$337$	−17.2996	−0.942372	−0.471186	−	0.882034i	$-0.656174\pi$
$337$	−17.2996	−0.942372	−0.471186	+	0.882034i	$0.656174\pi$
$338$	12.8926	0.701263
$339$	−7.46042	−0.405194
$340$	0	0
$341$	0	0
$342$	7.28601	0.393983
$343$	−16.8907	−0.912010
$344$	−11.3325	−0.611006
$345$	0	0
$346$	9.40899	0.505830
$347$	−31.6492	−1.69902	−0.849508	−	0.527575i	$-0.823101\pi$
$347$	−31.6492	−1.69902	−0.849508	+	0.527575i	$0.823101\pi$
$348$	1.06071	0.0568603
$349$	22.5068	1.20476	0.602379	−	0.798210i	$-0.294219\pi$
$349$	22.5068	1.20476	0.602379	+	0.798210i	$0.294219\pi$
$350$	0	0
$351$	−0.773410	−0.0412816
$352$	0	0
$353$	−14.2289	−0.757326	−0.378663	−	0.925535i	$-0.623616\pi$
$353$	−14.2289	−0.757326	−0.378663	+	0.925535i	$0.623616\pi$
$354$	3.57433	0.189974
$355$	0	0
$356$	−5.80856	−0.307853
$357$	2.17809	0.115277
$358$	3.14676	0.166311
$359$	−2.60514	−0.137494	−0.0687470	−	0.997634i	$-0.521900\pi$
$359$	−2.60514	−0.137494	−0.0687470	+	0.997634i	$0.521900\pi$
$360$	0	0
$361$	−12.4022	−0.652750
$362$	−25.1776	−1.32330
$363$	0	0
$364$	−0.460284	−0.0241254
$365$	0	0
$366$	−4.35680	−0.227734
$367$	13.0207	0.679677	0.339839	−	0.940484i	$-0.389628\pi$
$367$	13.0207	0.679677	0.339839	+	0.940484i	$0.389628\pi$
$368$	5.24084	0.273198
$369$	11.8232	0.615491
$370$	0	0
$371$	1.64041	0.0851657
$372$	1.84853	0.0958419
$373$	−1.73587	−0.0898799	−0.0449399	−	0.998990i	$-0.514310\pi$
$373$	−1.73587	−0.0898799	−0.0449399	+	0.998990i	$0.514310\pi$
$374$	0	0
$375$	0	0
$376$	−5.10069	−0.263048
$377$	0.859985	0.0442915
$378$	−3.31353	−0.170429
$379$	19.8298	1.01859	0.509295	−	0.860592i	$-0.329906\pi$
$379$	19.8298	1.01859	0.509295	+	0.860592i	$0.329906\pi$
$380$	0	0
$381$	4.64143	0.237788
$382$	−18.8180	−0.962811
$383$	10.0094	0.511457	0.255729	−	0.966749i	$-0.417685\pi$
$383$	10.0094	0.511457	0.255729	+	0.966749i	$0.417685\pi$
$384$	0.404278	0.0206307
$385$	0	0
$386$	−22.8056	−1.16078
$387$	−32.1453	−1.63404
$388$	−17.5377	−0.890341
$389$	−5.15199	−0.261216	−0.130608	−	0.991434i	$-0.541693\pi$
$389$	−5.15199	−0.261216	−0.130608	+	0.991434i	$0.541693\pi$
$390$	0	0
$391$	−20.1068	−1.01684
$392$	5.02800	0.253952
$393$	6.83604	0.344833
$394$	1.03998	0.0523932
$395$	0	0
$396$	0	0
$397$	−37.4977	−1.88196	−0.940978	−	0.338468i	$-0.890091\pi$
$397$	−37.4977	−1.88196	−0.940978	+	0.338468i	$0.890091\pi$
$398$	13.2408	0.663703
$399$	−1.45825	−0.0730037
$400$	0	0
$401$	4.86404	0.242899	0.121449	−	0.992598i	$-0.461246\pi$
$401$	4.86404	0.242899	0.121449	+	0.992598i	$0.461246\pi$
$402$	−2.08937	−0.104208
$403$	1.49872	0.0746564
$404$	−9.97671	−0.496360
$405$	0	0
$406$	3.68444	0.182856
$407$	0	0
$408$	−1.55104	−0.0767878
$409$	−24.1216	−1.19273	−0.596367	−	0.802712i	$-0.703390\pi$
$409$	−24.1216	−1.19273	−0.596367	+	0.802712i	$0.703390\pi$
$410$	0	0
$411$	−7.34689	−0.362395
$412$	−4.95532	−0.244131
$413$	12.4156	0.610932
$414$	14.8659	0.730621
$415$	0	0
$416$	0.327773	0.0160704
$417$	2.59369	0.127013
$418$	0	0
$419$	−6.49300	−0.317204	−0.158602	−	0.987343i	$-0.550699\pi$
$419$	−6.49300	−0.317204	−0.158602	+	0.987343i	$0.550699\pi$
$420$	0	0
$421$	6.65644	0.324415	0.162207	−	0.986757i	$-0.448139\pi$
$421$	6.65644	0.324415	0.162207	+	0.986757i	$0.448139\pi$
$422$	12.4603	0.606557
$423$	−14.4684	−0.703478
$424$	−1.16815	−0.0567304
$425$	0	0
$426$	5.28016	0.255825
$427$	−15.1335	−0.732363
$428$	5.61711	0.271513
$429$	0	0
$430$	0	0
$431$	3.88072	0.186928	0.0934639	−	0.995623i	$-0.470206\pi$
$431$	3.88072	0.186928	0.0934639	+	0.995623i	$0.470206\pi$
$432$	2.35959	0.113526
$433$	−17.4297	−0.837619	−0.418810	−	0.908074i	$-0.637553\pi$
$433$	−17.4297	−0.837619	−0.418810	+	0.908074i	$0.637553\pi$
$434$	6.42096	0.308216
$435$	0	0
$436$	−13.6731	−0.654824
$437$	13.4617	0.643959
$438$	−1.72544	−0.0824446
$439$	−8.78246	−0.419164	−0.209582	−	0.977791i	$-0.567210\pi$
$439$	−8.78246	−0.419164	−0.209582	+	0.977791i	$0.567210\pi$
$440$	0	0
$441$	14.2622	0.679154
$442$	−1.25752	−0.0598141
$443$	−30.5824	−1.45301	−0.726506	−	0.687160i	$-0.758857\pi$
$443$	−30.5824	−1.45301	−0.726506	+	0.687160i	$0.758857\pi$
$444$	0.326882	0.0155131
$445$	0	0
$446$	−11.0541	−0.523427
$447$	7.53297	0.356297
$448$	1.40428	0.0663459
$449$	−11.0374	−0.520888	−0.260444	−	0.965489i	$-0.583869\pi$
$449$	−11.0374	−0.520888	−0.260444	+	0.965489i	$0.583869\pi$
$450$	0	0
$451$	0	0
$452$	18.4537	0.867988
$453$	6.20569	0.291569
$454$	9.50445	0.446066
$455$	0	0
$456$	1.03843	0.0486291
$457$	−13.7244	−0.642001	−0.321001	−	0.947079i	$-0.604019\pi$
$457$	−13.7244	−0.642001	−0.321001	+	0.947079i	$0.604019\pi$
$458$	14.4156	0.673597
$459$	−9.05272	−0.422545
$460$	0	0
$461$	−30.3784	−1.41486	−0.707432	−	0.706781i	$-0.750147\pi$
$461$	−30.3784	−1.41486	−0.707432	+	0.706781i	$0.750147\pi$
$462$	0	0
$463$	6.86737	0.319154	0.159577	−	0.987186i	$-0.448987\pi$
$463$	6.86737	0.319154	0.159577	+	0.987186i	$0.448987\pi$
$464$	−2.62372	−0.121803
$465$	0	0
$466$	−4.55104	−0.210823
$467$	3.20342	0.148236	0.0741182	−	0.997249i	$-0.476386\pi$
$467$	3.20342	0.148236	0.0741182	+	0.997249i	$0.476386\pi$
$468$	0.929747	0.0429776
$469$	−7.25752	−0.335121
$470$	0	0
$471$	2.84050	0.130883
$472$	−8.84127	−0.406952
$473$	0	0
$474$	0.110230	0.00506301
$475$	0	0
$476$	−5.38760	−0.246940
$477$	−3.31353	−0.151716
$478$	−20.0821	−0.918534
$479$	−9.07740	−0.414757	−0.207378	−	0.978261i	$-0.566493\pi$
$479$	−9.07740	−0.414757	−0.207378	+	0.978261i	$0.566493\pi$
$480$	0	0
$481$	0.265023	0.0120840
$482$	27.7881	1.26571
$483$	−2.97532	−0.135382
$484$	0	0
$485$	0	0
$486$	10.1334	0.459660
$487$	−20.1948	−0.915112	−0.457556	−	0.889181i	$-0.651275\pi$
$487$	−20.1948	−0.915112	−0.457556	+	0.889181i	$0.651275\pi$
$488$	10.7767	0.487840
$489$	−2.74771	−0.124256
$490$	0	0
$491$	28.5024	1.28630	0.643148	−	0.765742i	$-0.277628\pi$
$491$	28.5024	1.28630	0.643148	+	0.765742i	$0.277628\pi$
$492$	1.68509	0.0759698
$493$	10.0661	0.453353
$494$	0.841920	0.0378798
$495$	0	0
$496$	−4.57243	−0.205308
$497$	18.3409	0.822701
$498$	1.29023	0.0578166
$499$	−9.61646	−0.430492	−0.215246	−	0.976560i	$-0.569055\pi$
$499$	−9.61646	−0.430492	−0.215246	+	0.976560i	$0.569055\pi$
$500$	0	0
$501$	5.20761	0.232659
$502$	25.6939	1.14677
$503$	−8.51504	−0.379667	−0.189833	−	0.981816i	$-0.560795\pi$
$503$	−8.51504	−0.379667	−0.189833	+	0.981816i	$0.560795\pi$
$504$	3.98332	0.177431
$505$	0	0
$506$	0	0
$507$	5.21218	0.231481
$508$	−11.4808	−0.509377
$509$	0.152120	0.00674259	0.00337130	−	0.999994i	$-0.498927\pi$
$509$	0.152120	0.00674259	0.00337130	+	0.999994i	$0.498927\pi$
$510$	0	0
$511$	−5.99339	−0.265132
$512$	−1.00000	−0.0441942
$513$	6.06087	0.267594
$514$	−3.74719	−0.165281
$515$	0	0
$516$	−4.58147	−0.201688
$517$	0	0
$518$	1.13544	0.0498883
$519$	3.80385	0.166970
$520$	0	0
$521$	29.9766	1.31330	0.656649	−	0.754197i	$-0.271973\pi$
$521$	29.9766	1.31330	0.656649	+	0.754197i	$0.271973\pi$
$522$	−7.44235	−0.325743
$523$	−30.9204	−1.35206	−0.676028	−	0.736876i	$-0.736300\pi$
$523$	−30.9204	−1.35206	−0.676028	+	0.736876i	$0.736300\pi$
$524$	−16.9092	−0.738684
$525$	0	0
$526$	−26.0821	−1.13723
$527$	17.5424	0.764158
$528$	0	0
$529$	4.46638	0.194190
$530$	0	0
$531$	−25.0788	−1.08833
$532$	3.60704	0.156385
$533$	1.36621	0.0591769
$534$	−2.34827	−0.101620
$535$	0	0
$536$	5.16815	0.223230
$537$	1.27217	0.0548980
$538$	−20.3063	−0.875465
$539$	0	0
$540$	0	0
$541$	7.61800	0.327524	0.163762	−	0.986500i	$-0.447637\pi$
$541$	7.61800	0.327524	0.163762	+	0.986500i	$0.447637\pi$
$542$	−22.1381	−0.950913
$543$	−10.1787	−0.436812
$544$	3.83656	0.164491
$545$	0	0
$546$	−0.186083	−0.00796361
$547$	23.4350	1.00201	0.501003	−	0.865445i	$-0.332965\pi$
$547$	23.4350	1.00201	0.501003	+	0.865445i	$0.332965\pi$
$548$	18.1729	0.776306
$549$	30.5688	1.30465
$550$	0	0
$551$	−6.73933	−0.287105
$552$	2.11876	0.0901802
$553$	0.382887	0.0162820
$554$	−2.20049	−0.0934899
$555$	0	0
$556$	−6.41560	−0.272082
$557$	−42.2529	−1.79031	−0.895157	−	0.445751i	$-0.852937\pi$
$557$	−42.2529	−1.79031	−0.895157	+	0.445751i	$0.852937\pi$
$558$	−12.9700	−0.549062
$559$	−3.71448	−0.157106
$560$	0	0
$561$	0	0
$562$	1.27610	0.0538292
$563$	42.2202	1.77937	0.889685	−	0.456575i	$-0.150924\pi$
$563$	42.2202	1.77937	0.889685	+	0.456575i	$0.150924\pi$
$564$	−2.06210	−0.0868300
$565$	0	0
$566$	−15.4565	−0.649684
$567$	10.6104	0.445594
$568$	−13.0607	−0.548015
$569$	−19.8973	−0.834137	−0.417069	−	0.908875i	$-0.636943\pi$
$569$	−19.8973	−0.834137	−0.417069	+	0.908875i	$0.636943\pi$
$570$	0	0
$571$	−23.6498	−0.989714	−0.494857	−	0.868974i	$-0.664780\pi$
$571$	−23.6498	−0.989714	−0.494857	+	0.868974i	$0.664780\pi$
$572$	0	0
$573$	−7.60770	−0.317816
$574$	5.85324	0.244310
$575$	0	0
$576$	−2.83656	−0.118190
$577$	−1.96258	−0.0817033	−0.0408516	−	0.999165i	$-0.513007\pi$
$577$	−1.96258	−0.0817033	−0.0408516	+	0.999165i	$0.513007\pi$
$578$	2.28081	0.0948693
$579$	−9.21982	−0.383163
$580$	0	0
$581$	4.48167	0.185931
$582$	−7.09010	−0.293894
$583$	0	0
$584$	4.26795	0.176609
$585$	0	0
$586$	18.8507	0.778715
$587$	−40.8406	−1.68567	−0.842836	−	0.538170i	$-0.819116\pi$
$587$	−40.8406	−1.68567	−0.842836	+	0.538170i	$0.819116\pi$
$588$	2.03271	0.0838276
$589$	−11.7448	−0.483935
$590$	0	0
$591$	0.420440	0.0172946
$592$	−0.808556	−0.0332315
$593$	−17.9248	−0.736082	−0.368041	−	0.929810i	$-0.619971\pi$
$593$	−17.9248	−0.736082	−0.368041	+	0.929810i	$0.619971\pi$
$594$	0	0
$595$	0	0
$596$	−18.6331	−0.763243
$597$	5.35298	0.219083
$598$	1.71780	0.0702462
$599$	7.38479	0.301734	0.150867	−	0.988554i	$-0.451793\pi$
$599$	7.38479	0.301734	0.150867	+	0.988554i	$0.451793\pi$
$600$	0	0
$601$	−23.1677	−0.945028	−0.472514	−	0.881323i	$-0.656653\pi$
$601$	−23.1677	−0.945028	−0.472514	+	0.881323i	$0.656653\pi$
$602$	−15.9140	−0.648604
$603$	14.6598	0.596992
$604$	−15.3501	−0.624585
$605$	0	0
$606$	−4.03336	−0.163844
$607$	−4.80129	−0.194878	−0.0974392	−	0.995241i	$-0.531065\pi$
$607$	−4.80129	−0.194878	−0.0974392	+	0.995241i	$0.531065\pi$
$608$	−2.56861	−0.104171
$609$	1.48954	0.0603591
$610$	0	0
$611$	−1.67187	−0.0676365
$612$	10.8826	0.439904
$613$	32.2720	1.30345	0.651727	−	0.758454i	$-0.274045\pi$
$613$	32.2720	1.30345	0.651727	+	0.758454i	$0.274045\pi$
$614$	0.276996	0.0111786
$615$	0	0
$616$	0	0
$617$	14.0374	0.565125	0.282563	−	0.959249i	$-0.408815\pi$
$617$	14.0374	0.565125	0.282563	+	0.959249i	$0.408815\pi$
$618$	−2.00333	−0.0805856
$619$	46.6144	1.87359	0.936796	−	0.349876i	$-0.113776\pi$
$619$	46.6144	1.87359	0.936796	+	0.349876i	$0.113776\pi$
$620$	0	0
$621$	12.3662	0.496240
$622$	24.4417	0.980023
$623$	−8.15683	−0.326796
$624$	0.132511	0.00530470
$625$	0	0
$626$	15.1268	0.604588
$627$	0	0
$628$	−7.02610	−0.280372
$629$	3.10207	0.123688
$630$	0	0
$631$	−24.7686	−0.986022	−0.493011	−	0.870023i	$-0.664104\pi$
$631$	−24.7686	−0.986022	−0.493011	+	0.870023i	$0.664104\pi$
$632$	−0.272658	−0.0108457
$633$	5.03742	0.200219
$634$	24.1609	0.959551
$635$	0	0
$636$	−0.472257	−0.0187262
$637$	1.64804	0.0652978
$638$	0	0
$639$	−37.0475	−1.46558
$640$	0	0
$641$	−7.00942	−0.276855	−0.138428	−	0.990373i	$-0.544205\pi$
$641$	−7.00942	−0.276855	−0.138428	+	0.990373i	$0.544205\pi$
$642$	2.27088	0.0896243
$643$	−29.3148	−1.15606	−0.578031	−	0.816015i	$-0.696179\pi$
$643$	−29.3148	−1.15606	−0.578031	+	0.816015i	$0.696179\pi$
$644$	7.35959	0.290009
$645$	0	0
$646$	9.85462	0.387725
$647$	−21.9520	−0.863023	−0.431512	−	0.902107i	$-0.642020\pi$
$647$	−21.9520	−0.863023	−0.431512	+	0.902107i	$0.642020\pi$
$648$	−7.55575	−0.296818
$649$	0	0
$650$	0	0
$651$	2.59585	0.101740
$652$	6.79658	0.266175
$653$	24.0981	0.943033	0.471516	−	0.881857i	$-0.343707\pi$
$653$	24.0981	0.943033	0.471516	+	0.881857i	$0.343707\pi$
$654$	−5.52774	−0.216152
$655$	0	0
$656$	−4.16815	−0.162739
$657$	12.1063	0.472311
$658$	−7.16279	−0.279235
$659$	18.3382	0.714355	0.357177	−	0.934037i	$-0.383739\pi$
$659$	18.3382	0.714355	0.357177	+	0.934037i	$0.383739\pi$
$660$	0	0
$661$	3.80194	0.147878	0.0739392	−	0.997263i	$-0.476443\pi$
$661$	3.80194	0.147878	0.0739392	+	0.997263i	$0.476443\pi$
$662$	−4.45087	−0.172988
$663$	−0.508388	−0.0197441
$664$	−3.19144	−0.123852
$665$	0	0
$666$	−2.29352	−0.0888720
$667$	−13.7505	−0.532422
$668$	−12.8812	−0.498390
$669$	−4.46893	−0.172779
$670$	0	0
$671$	0	0
$672$	0.567719	0.0219002
$673$	−1.07281	−0.0413538	−0.0206769	−	0.999786i	$-0.506582\pi$
$673$	−1.07281	−0.0413538	−0.0206769	+	0.999786i	$0.506582\pi$
$674$	17.2996	0.666358
$675$	0	0
$676$	−12.8926	−0.495868
$677$	12.5050	0.480605	0.240302	−	0.970698i	$-0.422753\pi$
$677$	12.5050	0.480605	0.240302	+	0.970698i	$0.422753\pi$
$678$	7.46042	0.286516
$679$	−24.6278	−0.945127
$680$	0	0
$681$	3.84244	0.147243
$682$	0	0
$683$	−13.2455	−0.506827	−0.253413	−	0.967358i	$-0.581553\pi$
$683$	−13.2455	−0.506827	−0.253413	+	0.967358i	$0.581553\pi$
$684$	−7.28601	−0.278588
$685$	0	0
$686$	16.8907	0.644888
$687$	5.82791	0.222349
$688$	11.3325	0.432047
$689$	−0.382887	−0.0145869
$690$	0	0
$691$	46.2882	1.76089	0.880443	−	0.474152i	$-0.157245\pi$
$691$	46.2882	1.76089	0.880443	+	0.474152i	$0.157245\pi$
$692$	−9.40899	−0.357676
$693$	0	0
$694$	31.6492	1.20139
$695$	0	0
$696$	−1.06071	−0.0402063
$697$	15.9914	0.605716
$698$	−22.5068	−0.851893
$699$	−1.83988	−0.0695908
$700$	0	0
$701$	2.34343	0.0885102	0.0442551	−	0.999020i	$-0.485909\pi$
$701$	2.34343	0.0885102	0.0442551	+	0.999020i	$0.485909\pi$
$702$	0.773410	0.0291905
$703$	−2.07687	−0.0783305
$704$	0	0
$705$	0	0
$706$	14.2289	0.535510
$707$	−14.0101	−0.526903
$708$	−3.57433	−0.134332
$709$	−34.0188	−1.27760	−0.638802	−	0.769371i	$-0.720570\pi$
$709$	−34.0188	−1.27760	−0.638802	+	0.769371i	$0.720570\pi$
$710$	0	0
$711$	−0.773410	−0.0290051
$712$	5.80856	0.217685
$713$	−23.9633	−0.897434
$714$	−2.17809	−0.0815129
$715$	0	0
$716$	−3.14676	−0.117600
$717$	−8.11876	−0.303200
$718$	2.60514	0.0972229
$719$	18.7352	0.698706	0.349353	−	0.936991i	$-0.386401\pi$
$719$	18.7352	0.698706	0.349353	+	0.936991i	$0.386401\pi$
$720$	0	0
$721$	−6.95864	−0.259153
$722$	12.4022	0.461564
$723$	11.2341	0.417801
$724$	25.1776	0.935717
$725$	0	0
$726$	0	0
$727$	−50.3617	−1.86781	−0.933907	−	0.357516i	$-0.883624\pi$
$727$	−50.3617	−1.86781	−0.933907	+	0.357516i	$0.883624\pi$
$728$	0.460284	0.0170593
$729$	−18.5705	−0.687797
$730$	0	0
$731$	−43.4777	−1.60808
$732$	4.35680	0.161032
$733$	−47.9655	−1.77165	−0.885823	−	0.464023i	$-0.846405\pi$
$733$	−47.9655	−1.77165	−0.885823	+	0.464023i	$0.846405\pi$
$734$	−13.0207	−0.480604
$735$	0	0
$736$	−5.24084	−0.193180
$737$	0	0
$738$	−11.8232	−0.435218
$739$	−26.0484	−0.958205	−0.479102	−	0.877759i	$-0.659038\pi$
$739$	−26.0484	−0.958205	−0.479102	+	0.877759i	$0.659038\pi$
$740$	0	0
$741$	0.340370	0.0125038
$742$	−1.64041	−0.0602212
$743$	−37.7385	−1.38449	−0.692246	−	0.721661i	$-0.743379\pi$
$743$	−37.7385	−1.38449	−0.692246	+	0.721661i	$0.743379\pi$
$744$	−1.84853	−0.0677705
$745$	0	0
$746$	1.73587	0.0635547
$747$	−9.05272	−0.331222
$748$	0	0
$749$	7.88799	0.288221
$750$	0	0
$751$	16.8900	0.616325	0.308163	−	0.951334i	$-0.400286\pi$
$751$	16.8900	0.616325	0.308163	+	0.951334i	$0.400286\pi$
$752$	5.10069	0.186003
$753$	10.3875	0.378540
$754$	−0.859985	−0.0313188
$755$	0	0
$756$	3.31353	0.120512
$757$	12.2270	0.444396	0.222198	−	0.975002i	$-0.428677\pi$
$757$	12.2270	0.444396	0.222198	+	0.975002i	$0.428677\pi$
$758$	−19.8298	−0.720251
$759$	0	0
$760$	0	0
$761$	21.0279	0.762260	0.381130	−	0.924522i	$-0.375535\pi$
$761$	21.0279	0.762260	0.381130	+	0.924522i	$0.375535\pi$
$762$	−4.64143	−0.168141
$763$	−19.2009	−0.695118
$764$	18.8180	0.680810
$765$	0	0
$766$	−10.0094	−0.361655
$767$	−2.89793	−0.104638
$768$	−0.404278	−0.0145881
$769$	1.89448	0.0683167	0.0341583	−	0.999416i	$-0.489125\pi$
$769$	1.89448	0.0683167	0.0341583	+	0.999416i	$0.489125\pi$
$770$	0	0
$771$	−1.51491	−0.0545580
$772$	22.8056	0.820793
$773$	9.68319	0.348280	0.174140	−	0.984721i	$-0.444285\pi$
$773$	9.68319	0.348280	0.174140	+	0.984721i	$0.444285\pi$
$774$	32.1453	1.15544
$775$	0	0
$776$	17.5377	0.629566
$777$	0.459033	0.0164677
$778$	5.15199	0.184708
$779$	−10.7064	−0.383595
$780$	0	0
$781$	0	0
$782$	20.1068	0.719017
$783$	−6.19092	−0.221246
$784$	−5.02800	−0.179572
$785$	0	0
$786$	−6.83604	−0.243833
$787$	6.89219	0.245680	0.122840	−	0.992426i	$-0.460800\pi$
$787$	6.89219	0.245680	0.122840	+	0.992426i	$0.460800\pi$
$788$	−1.03998	−0.0370476
$789$	−10.5444	−0.375391
$790$	0	0
$791$	25.9141	0.921399
$792$	0	0
$793$	3.53232	0.125436
$794$	37.4977	1.33074
$795$	0	0
$796$	−13.2408	−0.469309
$797$	13.2309	0.468663	0.234331	−	0.972157i	$-0.424710\pi$
$797$	13.2309	0.468663	0.234331	+	0.972157i	$0.424710\pi$
$798$	1.45825	0.0516214
$799$	−19.5691	−0.692305
$800$	0	0
$801$	16.4763	0.582162
$802$	−4.86404	−0.171755
$803$	0	0
$804$	2.08937	0.0736864
$805$	0	0
$806$	−1.49872	−0.0527900
$807$	−8.20938	−0.288984
$808$	9.97671	0.350979
$809$	−26.6259	−0.936116	−0.468058	−	0.883698i	$-0.655046\pi$
$809$	−26.6259	−0.936116	−0.468058	+	0.883698i	$0.655046\pi$
$810$	0	0
$811$	−4.40428	−0.154655	−0.0773276	−	0.997006i	$-0.524639\pi$
$811$	−4.40428	−0.154655	−0.0773276	+	0.997006i	$0.524639\pi$
$812$	−3.68444	−0.129298
$813$	−8.94995	−0.313889
$814$	0	0
$815$	0	0
$816$	1.55104	0.0542972
$817$	29.1087	1.01839
$818$	24.1216	0.843391
$819$	1.30562	0.0456222
$820$	0	0
$821$	−15.7159	−0.548489	−0.274244	−	0.961660i	$-0.588428\pi$
$821$	−15.7159	−0.548489	−0.274244	+	0.961660i	$0.588428\pi$
$822$	7.34689	0.256252
$823$	−46.1802	−1.60974	−0.804871	−	0.593450i	$-0.797766\pi$
$823$	−46.1802	−1.60974	−0.804871	+	0.593450i	$0.797766\pi$
$824$	4.95532	0.172627
$825$	0	0
$826$	−12.4156	−0.431994
$827$	54.8582	1.90761	0.953803	−	0.300432i	$-0.0971308\pi$
$827$	54.8582	1.90761	0.953803	+	0.300432i	$0.0971308\pi$
$828$	−14.8659	−0.516627
$829$	−47.1442	−1.63739	−0.818693	−	0.574232i	$-0.805301\pi$
$829$	−47.1442	−1.63739	−0.818693	+	0.574232i	$0.805301\pi$
$830$	0	0
$831$	−0.889610	−0.0308602
$832$	−0.327773	−0.0113635
$833$	19.2902	0.668367
$834$	−2.59369	−0.0898120
$835$	0	0
$836$	0	0
$837$	−10.7891	−0.372925
$838$	6.49300	0.224297
$839$	−8.24835	−0.284765	−0.142382	−	0.989812i	$-0.545476\pi$
$839$	−8.24835	−0.284765	−0.142382	+	0.989812i	$0.545476\pi$
$840$	0	0
$841$	−22.1161	−0.762623
$842$	−6.65644	−0.229396
$843$	0.515901	0.0177686
$844$	−12.4603	−0.428901
$845$	0	0
$846$	14.4684	0.497434
$847$	0	0
$848$	1.16815	0.0401144
$849$	−6.24871	−0.214455
$850$	0	0
$851$	−4.23751	−0.145260
$852$	−5.28016	−0.180895
$853$	15.1124	0.517438	0.258719	−	0.965953i	$-0.416700\pi$
$853$	15.1124	0.517438	0.258719	+	0.965953i	$0.416700\pi$
$854$	15.1335	0.517859
$855$	0	0
$856$	−5.61711	−0.191989
$857$	11.5839	0.395698	0.197849	−	0.980233i	$-0.436604\pi$
$857$	11.5839	0.395698	0.197849	+	0.980233i	$0.436604\pi$
$858$	0	0
$859$	25.1656	0.858639	0.429319	−	0.903153i	$-0.358753\pi$
$859$	25.1656	0.858639	0.429319	+	0.903153i	$0.358753\pi$
$860$	0	0
$861$	2.36634	0.0806446
$862$	−3.88072	−0.132178
$863$	19.1703	0.652565	0.326282	−	0.945272i	$-0.394204\pi$
$863$	19.1703	0.652565	0.326282	+	0.945272i	$0.394204\pi$
$864$	−2.35959	−0.0802750
$865$	0	0
$866$	17.4297	0.592286
$867$	0.922083	0.0313156
$868$	−6.42096	−0.217942
$869$	0	0
$870$	0	0
$871$	1.69398	0.0573983
$872$	13.6731	0.463030
$873$	49.7467	1.68367
$874$	−13.4617	−0.455348
$875$	0	0
$876$	1.72544	0.0582972
$877$	10.6741	0.360440	0.180220	−	0.983626i	$-0.442319\pi$
$877$	10.6741	0.360440	0.180220	+	0.983626i	$0.442319\pi$
$878$	8.78246	0.296394
$879$	7.62092	0.257047
$880$	0	0
$881$	1.74390	0.0587536	0.0293768	−	0.999568i	$-0.490648\pi$
$881$	1.74390	0.0587536	0.0293768	+	0.999568i	$0.490648\pi$
$882$	−14.2622	−0.480234
$883$	−35.8726	−1.20721	−0.603604	−	0.797284i	$-0.706269\pi$
$883$	−35.8726	−1.20721	−0.603604	+	0.797284i	$0.706269\pi$
$884$	1.25752	0.0422950
$885$	0	0
$886$	30.5824	1.02743
$887$	−8.75851	−0.294082	−0.147041	−	0.989130i	$-0.546975\pi$
$887$	−8.75851	−0.294082	−0.147041	+	0.989130i	$0.546975\pi$
$888$	−0.326882	−0.0109694
$889$	−16.1222	−0.540722
$890$	0	0
$891$	0	0
$892$	11.0541	0.370119
$893$	13.1017	0.438431
$894$	−7.53297	−0.251940
$895$	0	0
$896$	−1.40428	−0.0469136
$897$	0.694470	0.0231877
$898$	11.0374	0.368323
$899$	11.9968	0.400115
$900$	0	0
$901$	−4.48167	−0.149306
$902$	0	0
$903$	−6.43366	−0.214099
$904$	−18.4537	−0.613760
$905$	0	0
$906$	−6.20569	−0.206170
$907$	−31.0521	−1.03107	−0.515533	−	0.856869i	$-0.672406\pi$
$907$	−31.0521	−1.03107	−0.515533	+	0.856869i	$0.672406\pi$
$908$	−9.50445	−0.315416
$909$	28.2995	0.938636
$910$	0	0
$911$	11.9694	0.396565	0.198283	−	0.980145i	$-0.436464\pi$
$911$	11.9694	0.396565	0.198283	+	0.980145i	$0.436464\pi$
$912$	−1.03843	−0.0343859
$913$	0	0
$914$	13.7244	0.453963
$915$	0	0
$916$	−14.4156	−0.476305
$917$	−23.7453	−0.784138
$918$	9.05272	0.298784
$919$	42.3865	1.39820	0.699100	−	0.715024i	$-0.253584\pi$
$919$	42.3865	1.39820	0.699100	+	0.715024i	$0.253584\pi$
$920$	0	0
$921$	0.111983	0.00368998
$922$	30.3784	1.00046
$923$	−4.28095	−0.140909
$924$	0	0
$925$	0	0
$926$	−6.86737	−0.225676
$927$	14.0560	0.461661
$928$	2.62372	0.0861280
$929$	12.4542	0.408609	0.204304	−	0.978907i	$-0.434507\pi$
$929$	12.4542	0.408609	0.204304	+	0.978907i	$0.434507\pi$
$930$	0	0
$931$	−12.9150	−0.423271
$932$	4.55104	0.149074
$933$	9.88124	0.323497
$934$	−3.20342	−0.104819
$935$	0	0
$936$	−0.929747	−0.0303897
$937$	−5.08872	−0.166241	−0.0831206	−	0.996539i	$-0.526489\pi$
$937$	−5.08872	−0.166241	−0.0831206	+	0.996539i	$0.526489\pi$
$938$	7.25752	0.236966
$939$	6.11543	0.199570
$940$	0	0
$941$	55.1637	1.79828	0.899142	−	0.437656i	$-0.144191\pi$
$941$	55.1637	1.79828	0.899142	+	0.437656i	$0.144191\pi$
$942$	−2.84050	−0.0925484
$943$	−21.8446	−0.711358
$944$	8.84127	0.287759
$945$	0	0
$946$	0	0
$947$	25.2707	0.821189	0.410594	−	0.911818i	$-0.365321\pi$
$947$	25.2707	0.821189	0.410594	+	0.911818i	$0.365321\pi$
$948$	−0.110230	−0.00358009
$949$	1.39892	0.0454108
$950$	0	0
$951$	9.76772	0.316740
$952$	5.38760	0.174613
$953$	−19.5598	−0.633604	−0.316802	−	0.948492i	$-0.602609\pi$
$953$	−19.5598	−0.633604	−0.316802	+	0.948492i	$0.602609\pi$
$954$	3.31353	0.107279
$955$	0	0
$956$	20.0821	0.649502
$957$	0	0
$958$	9.07740	0.293277
$959$	25.5197	0.824076
$960$	0	0
$961$	−10.0929	−0.325578
$962$	−0.265023	−0.00854467
$963$	−15.9333	−0.513443
$964$	−27.7881	−0.894993
$965$	0	0
$966$	2.97532	0.0957294
$967$	38.8778	1.25023	0.625113	−	0.780534i	$-0.285053\pi$
$967$	38.8778	1.25023	0.625113	+	0.780534i	$0.285053\pi$
$968$	0	0
$969$	3.98401	0.127985
$970$	0	0
$971$	−8.90990	−0.285932	−0.142966	−	0.989728i	$-0.545664\pi$
$971$	−8.90990	−0.285932	−0.142966	+	0.989728i	$0.545664\pi$
$972$	−10.1334	−0.325029
$973$	−9.00929	−0.288824
$974$	20.1948	0.647082
$975$	0	0
$976$	−10.7767	−0.344955
$977$	30.1357	0.964126	0.482063	−	0.876137i	$-0.339888\pi$
$977$	30.1357	0.964126	0.482063	+	0.876137i	$0.339888\pi$
$978$	2.74771	0.0878621
$979$	0	0
$980$	0	0
$981$	38.7846	1.23830
$982$	−28.5024	−0.909549
$983$	−35.2850	−1.12542	−0.562708	−	0.826655i	$-0.690241\pi$
$983$	−35.2850	−1.12542	−0.562708	+	0.826655i	$0.690241\pi$
$984$	−1.68509	−0.0537188
$985$	0	0
$986$	−10.0661	−0.320569
$987$	−2.89576	−0.0921730
$988$	−0.841920	−0.0267850
$989$	59.3917	1.88855
$990$	0	0
$991$	39.9187	1.26806	0.634029	−	0.773309i	$-0.281400\pi$
$991$	39.9187	1.26806	0.634029	+	0.773309i	$0.281400\pi$
$992$	4.57243	0.145175
$993$	−1.79939	−0.0571019
$994$	−18.3409	−0.581737
$995$	0	0
$996$	−1.29023	−0.0408825
$997$	32.2052	1.01995	0.509975	−	0.860189i	$-0.329655\pi$
$997$	32.2052	1.01995	0.509975	+	0.860189i	$0.329655\pi$
$998$	9.61646	0.304404
$999$	−1.90786	−0.0603621

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6050.2.a.de.1.2	yes	4
5.4	even	2		6050.2.a.dj.1.3	yes	4
11.10	odd	2		6050.2.a.dm.1.2	yes	4
55.54	odd	2		6050.2.a.db.1.3	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6050.2.a.db.1.3	✓	4	55.54	odd	2
6050.2.a.de.1.2	yes	4	1.1	even	1	trivial
6050.2.a.dj.1.3	yes	4	5.4	even	2
6050.2.a.dm.1.2	yes	4	11.10	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 6050 = 2 \cdot 5^{2} \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6050.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -0.404278 q^{3} +1.00000 q^{4} +0.404278 q^{6} +1.40428 q^{7} -1.00000 q^{8} -2.83656 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -0.404278 q^{3} +1.00000 q^{4} +0.404278 q^{6} +1.40428 q^{7} -1.00000 q^{8} -2.83656 q^{9} -0.404278 q^{12} -0.327773 q^{13} -1.40428 q^{14} +1.00000 q^{16} -3.83656 q^{17} +2.83656 q^{18} +2.56861 q^{19} -0.567719 q^{21} +5.24084 q^{23} +0.404278 q^{24} +0.327773 q^{26} +2.35959 q^{27} +1.40428 q^{28} -2.62372 q^{29} -4.57243 q^{31} -1.00000 q^{32} +3.83656 q^{34} -2.83656 q^{36} -0.808556 q^{37} -2.56861 q^{38} +0.132511 q^{39} -4.16815 q^{41} +0.567719 q^{42} +11.3325 q^{43} -5.24084 q^{46} +5.10069 q^{47} -0.404278 q^{48} -5.02800 q^{49} +1.55104 q^{51} -0.327773 q^{52} +1.16815 q^{53} -2.35959 q^{54} -1.40428 q^{56} -1.03843 q^{57} +2.62372 q^{58} +8.84127 q^{59} -10.7767 q^{61} +4.57243 q^{62} -3.98332 q^{63} +1.00000 q^{64} -5.16815 q^{67} -3.83656 q^{68} -2.11876 q^{69} +13.0607 q^{71} +2.83656 q^{72} -4.26795 q^{73} +0.808556 q^{74} +2.56861 q^{76} -0.132511 q^{78} +0.272658 q^{79} +7.55575 q^{81} +4.16815 q^{82} +3.19144 q^{83} -0.567719 q^{84} -11.3325 q^{86} +1.06071 q^{87} -5.80856 q^{89} -0.460284 q^{91} +5.24084 q^{92} +1.84853 q^{93} -5.10069 q^{94} +0.404278 q^{96} -17.5377 q^{97} +5.02800 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{2} + 2 q^{3} + 4 q^{4} - 2 q^{6} + 2 q^{7} - 4 q^{8} + 8 q^{9}+O(q^{10}) \) 4 * q - 4 * q^2 + 2 * q^3 + 4 * q^4 - 2 * q^6 + 2 * q^7 - 4 * q^8 + 8 * q^9	\( 4 q - 4 q^{2} + 2 q^{3} + 4 q^{4} - 2 q^{6} + 2 q^{7} - 4 q^{8} + 8 q^{9} + 2 q^{12} + 2 q^{13} - 2 q^{14} + 4 q^{16} + 4 q^{17} - 8 q^{18} - 16 q^{19} - 18 q^{21} - 2 q^{23} - 2 q^{24} - 2 q^{26} + 8 q^{27} + 2 q^{28} - 2 q^{29} - 6 q^{31} - 4 q^{32} - 4 q^{34} + 8 q^{36} + 4 q^{37} + 16 q^{38} - 24 q^{39} - 8 q^{41} + 18 q^{42} + 14 q^{43} + 2 q^{46} - 6 q^{47} + 2 q^{48} - 8 q^{49} + 12 q^{51} + 2 q^{52} - 4 q^{53} - 8 q^{54} - 2 q^{56} - 12 q^{57} + 2 q^{58} - 12 q^{59} - 34 q^{61} + 6 q^{62} - 6 q^{63} + 4 q^{64} - 12 q^{67} + 4 q^{68} - 30 q^{69} - 8 q^{72} - 24 q^{73} - 4 q^{74} - 16 q^{76} + 24 q^{78} - 20 q^{79} + 8 q^{81} + 8 q^{82} + 20 q^{83} - 18 q^{84} - 14 q^{86} - 48 q^{87} - 16 q^{89} + 26 q^{91} - 2 q^{92} + 26 q^{93} + 6 q^{94} - 2 q^{96} + 8 q^{98}+O(q^{100}) \) 4 * q - 4 * q^2 + 2 * q^3 + 4 * q^4 - 2 * q^6 + 2 * q^7 - 4 * q^8 + 8 * q^9 + 2 * q^12 + 2 * q^13 - 2 * q^14 + 4 * q^16 + 4 * q^17 - 8 * q^18 - 16 * q^19 - 18 * q^21 - 2 * q^23 - 2 * q^24 - 2 * q^26 + 8 * q^27 + 2 * q^28 - 2 * q^29 - 6 * q^31 - 4 * q^32 - 4 * q^34 + 8 * q^36 + 4 * q^37 + 16 * q^38 - 24 * q^39 - 8 * q^41 + 18 * q^42 + 14 * q^43 + 2 * q^46 - 6 * q^47 + 2 * q^48 - 8 * q^49 + 12 * q^51 + 2 * q^52 - 4 * q^53 - 8 * q^54 - 2 * q^56 - 12 * q^57 + 2 * q^58 - 12 * q^59 - 34 * q^61 + 6 * q^62 - 6 * q^63 + 4 * q^64 - 12 * q^67 + 4 * q^68 - 30 * q^69 - 8 * q^72 - 24 * q^73 - 4 * q^74 - 16 * q^76 + 24 * q^78 - 20 * q^79 + 8 * q^81 + 8 * q^82 + 20 * q^83 - 18 * q^84 - 14 * q^86 - 48 * q^87 - 16 * q^89 + 26 * q^91 - 2 * q^92 + 26 * q^93 + 6 * q^94 - 2 * q^96 + 8 * q^98

Introduction

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