LMFDB - Embedded newform 525.2.a.k.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [525,2,Mod(1,525)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(525, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("525.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 525 = 3 \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	525.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:	$4.19214610612$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.148.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 3x + 1 $ x^3 - x^2 - 3*x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 105)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.48119$ of defining polynomial
Character	$\chi$	$=$	525.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+0.193937 q^{2} -1.00000 q^{3} -1.96239 q^{4} -0.193937 q^{6} -1.00000 q^{7} -0.768452 q^{8} +1.00000 q^{9} +2.00000 q^{11} +1.96239 q^{12} +1.35026 q^{13} -0.193937 q^{14} +3.77575 q^{16} -3.35026 q^{17} +0.193937 q^{18} +5.35026 q^{19} +1.00000 q^{21} +0.387873 q^{22} +4.96239 q^{23} +0.768452 q^{24} +0.261865 q^{26} -1.00000 q^{27} +1.96239 q^{28} +7.92478 q^{29} +4.57452 q^{31} +2.26916 q^{32} -2.00000 q^{33} -0.649738 q^{34} -1.96239 q^{36} -0.775746 q^{37} +1.03761 q^{38} -1.35026 q^{39} +3.73813 q^{41} +0.193937 q^{42} -12.6253 q^{43} -3.92478 q^{44} +0.962389 q^{46} +9.92478 q^{47} -3.77575 q^{48} +1.00000 q^{49} +3.35026 q^{51} -2.64974 q^{52} +8.57452 q^{53} -0.193937 q^{54} +0.768452 q^{56} -5.35026 q^{57} +1.53690 q^{58} -8.62530 q^{59} -8.70052 q^{61} +0.887166 q^{62} -1.00000 q^{63} -7.11142 q^{64} -0.387873 q^{66} +9.92478 q^{67} +6.57452 q^{68} -4.96239 q^{69} +2.00000 q^{71} -0.768452 q^{72} -9.35026 q^{73} -0.150446 q^{74} -10.4993 q^{76} -2.00000 q^{77} -0.261865 q^{78} +10.7005 q^{79} +1.00000 q^{81} +0.724961 q^{82} +3.22425 q^{83} -1.96239 q^{84} -2.44851 q^{86} -7.92478 q^{87} -1.53690 q^{88} +1.03761 q^{89} -1.35026 q^{91} -9.73813 q^{92} -4.57452 q^{93} +1.92478 q^{94} -2.26916 q^{96} -18.4993 q^{97} +0.193937 q^{98} +2.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + q^{2} - 3 q^{3} + 5 q^{4} - q^{6} - 3 q^{7} + 9 q^{8} + 3 q^{9} + 6 q^{11} - 5 q^{12} - 6 q^{13} - q^{14} + 13 q^{16} + q^{18} + 6 q^{19} + 3 q^{21} + 2 q^{22} + 4 q^{23} - 9 q^{24} + 10 q^{26}+ \cdots + 6 q^{99}+O(q^{100}) $ 3 * q + q^2 - 3 * q^3 + 5 * q^4 - q^6 - 3 * q^7 + 9 * q^8 + 3 * q^9 + 6 * q^11 - 5 * q^12 - 6 * q^13 - q^14 + 13 * q^16 + q^18 + 6 * q^19 + 3 * q^21 + 2 * q^22 + 4 * q^23 - 9 * q^24 + 10 * q^26 - 3 * q^27 - 5 * q^28 + 2 * q^29 + 2 * q^31 + 29 * q^32 - 6 * q^33 - 12 * q^34 + 5 * q^36 - 4 * q^37 + 14 * q^38 + 6 * q^39 + 2 * q^41 + q^42 + 4 * q^43 + 10 * q^44 - 8 * q^46 + 8 * q^47 - 13 * q^48 + 3 * q^49 - 18 * q^52 + 14 * q^53 - q^54 - 9 * q^56 - 6 * q^57 - 18 * q^58 + 16 * q^59 - 6 * q^61 - 30 * q^62 - 3 * q^63 + 13 * q^64 - 2 * q^66 + 8 * q^67 + 8 * q^68 - 4 * q^69 + 6 * q^71 + 9 * q^72 - 18 * q^73 - 44 * q^74 + 2 * q^76 - 6 * q^77 - 10 * q^78 + 12 * q^79 + 3 * q^81 + 34 * q^82 + 8 * q^83 + 5 * q^84 - 4 * q^86 - 2 * q^87 + 18 * q^88 + 14 * q^89 + 6 * q^91 - 20 * q^92 - 2 * q^93 - 16 * q^94 - 29 * q^96 - 22 * q^97 + q^98 + 6 * q^99

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 $

$2$	0.193937	0.137134	0.0685669	−	0.997647i	$-0.478157\pi$
$2$	0.193937	0.137134	0.0685669	+	0.997647i	$0.478157\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	−1.96239	−0.981194
$5$	0	0
$5$	0	0
$6$	−0.193937	−0.0791743
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	−0.768452	−0.271689
$9$	1.00000	0.333333
$10$	0	0
$11$	2.00000	0.603023	0.301511	−	0.953463i	$-0.402509\pi$
$11$	2.00000	0.603023	0.301511	+	0.953463i	$0.402509\pi$
$12$	1.96239	0.566493
$13$	1.35026	0.374495	0.187248	−	0.982313i	$-0.440043\pi$
$13$	1.35026	0.374495	0.187248	+	0.982313i	$0.440043\pi$
$14$	−0.193937	−0.0518317
$15$	0	0
$16$	3.77575	0.943937
$17$	−3.35026	−0.812558	−0.406279	−	0.913749i	$-0.633174\pi$
$17$	−3.35026	−0.812558	−0.406279	+	0.913749i	$0.633174\pi$
$18$	0.193937	0.0457113
$19$	5.35026	1.22743	0.613717	−	0.789526i	$-0.289674\pi$
$19$	5.35026	1.22743	0.613717	+	0.789526i	$0.289674\pi$
$20$	0	0
$21$	1.00000	0.218218
$22$	0.387873	0.0826948
$23$	4.96239	1.03473	0.517365	−	0.855765i	$-0.326913\pi$
$23$	4.96239	1.03473	0.517365	+	0.855765i	$0.326913\pi$
$24$	0.768452	0.156860
$25$	0	0
$26$	0.261865	0.0513560
$27$	−1.00000	−0.192450
$28$	1.96239	0.370857
$29$	7.92478	1.47159	0.735797	−	0.677202i	$-0.236808\pi$
$29$	7.92478	1.47159	0.735797	+	0.677202i	$0.236808\pi$
$30$	0	0
$31$	4.57452	0.821607	0.410804	−	0.911724i	$-0.365248\pi$
$31$	4.57452	0.821607	0.410804	+	0.911724i	$0.365248\pi$
$32$	2.26916	0.401134
$33$	−2.00000	−0.348155
$34$	−0.649738	−0.111429
$35$	0	0
$36$	−1.96239	−0.327065
$37$	−0.775746	−0.127532	−0.0637660	−	0.997965i	$-0.520311\pi$
$37$	−0.775746	−0.127532	−0.0637660	+	0.997965i	$0.520311\pi$
$38$	1.03761	0.168323
$39$	−1.35026	−0.216215
$40$	0	0
$41$	3.73813	0.583799	0.291899	−	0.956449i	$-0.405713\pi$
$41$	3.73813	0.583799	0.291899	+	0.956449i	$0.405713\pi$
$42$	0.193937	0.0299251
$43$	−12.6253	−1.92534	−0.962670	−	0.270677i	$-0.912752\pi$
$43$	−12.6253	−1.92534	−0.962670	+	0.270677i	$0.912752\pi$
$44$	−3.92478	−0.591682
$45$	0	0
$46$	0.962389	0.141896
$47$	9.92478	1.44768	0.723839	−	0.689969i	$-0.242376\pi$
$47$	9.92478	1.44768	0.723839	+	0.689969i	$0.242376\pi$
$48$	−3.77575	−0.544982
$49$	1.00000	0.142857
$50$	0	0
$51$	3.35026	0.469130
$52$	−2.64974	−0.367453
$53$	8.57452	1.17780	0.588900	−	0.808206i	$-0.299561\pi$
$53$	8.57452	1.17780	0.588900	+	0.808206i	$0.299561\pi$
$54$	−0.193937	−0.0263914
$55$	0	0
$56$	0.768452	0.102689
$57$	−5.35026	−0.708659
$58$	1.53690	0.201805
$59$	−8.62530	−1.12292	−0.561459	−	0.827504i	$-0.689760\pi$
$59$	−8.62530	−1.12292	−0.561459	+	0.827504i	$0.689760\pi$
$60$	0	0
$61$	−8.70052	−1.11399	−0.556994	−	0.830517i	$-0.688045\pi$
$61$	−8.70052	−1.11399	−0.556994	+	0.830517i	$0.688045\pi$
$62$	0.887166	0.112670
$63$	−1.00000	−0.125988
$64$	−7.11142	−0.888927
$65$	0	0
$66$	−0.387873	−0.0477439
$67$	9.92478	1.21250	0.606252	−	0.795272i	$-0.292672\pi$
$67$	9.92478	1.21250	0.606252	+	0.795272i	$0.292672\pi$
$68$	6.57452	0.797277
$69$	−4.96239	−0.597401
$70$	0	0
$71$	2.00000	0.237356	0.118678	−	0.992933i	$-0.462134\pi$
$71$	2.00000	0.237356	0.118678	+	0.992933i	$0.462134\pi$
$72$	−0.768452	−0.0905629
$73$	−9.35026	−1.09437	−0.547183	−	0.837013i	$-0.684300\pi$
$73$	−9.35026	−1.09437	−0.547183	+	0.837013i	$0.684300\pi$
$74$	−0.150446	−0.0174889
$75$	0	0
$76$	−10.4993	−1.20435
$77$	−2.00000	−0.227921
$78$	−0.261865	−0.0296504
$79$	10.7005	1.20390	0.601951	−	0.798533i	$-0.294390\pi$
$79$	10.7005	1.20390	0.601951	+	0.798533i	$0.294390\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0.724961	0.0800586
$83$	3.22425	0.353908	0.176954	−	0.984219i	$-0.443376\pi$
$83$	3.22425	0.353908	0.176954	+	0.984219i	$0.443376\pi$
$84$	−1.96239	−0.214114
$85$	0	0
$86$	−2.44851	−0.264029
$87$	−7.92478	−0.849625
$88$	−1.53690	−0.163835
$89$	1.03761	0.109987	0.0549933	−	0.998487i	$-0.482486\pi$
$89$	1.03761	0.109987	0.0549933	+	0.998487i	$0.482486\pi$
$90$	0	0
$91$	−1.35026	−0.141546
$92$	−9.73813	−1.01527
$93$	−4.57452	−0.474355
$94$	1.92478	0.198526
$95$	0	0
$96$	−2.26916	−0.231595
$97$	−18.4993	−1.87832	−0.939159	−	0.343482i	$-0.888394\pi$
$97$	−18.4993	−1.87832	−0.939159	+	0.343482i	$0.888394\pi$
$98$	0.193937	0.0195906
$99$	2.00000	0.201008
$100$	0	0
$101$	−17.6629	−1.75753	−0.878763	−	0.477259i	$-0.841630\pi$
$101$	−17.6629	−1.75753	−0.878763	+	0.477259i	$0.841630\pi$
$102$	0.649738	0.0643337
$103$	6.70052	0.660222	0.330111	−	0.943942i	$-0.392914\pi$
$103$	6.70052	0.660222	0.330111	+	0.943942i	$0.392914\pi$
$104$	−1.03761	−0.101746
$105$	0	0
$106$	1.66291	0.161516
$107$	13.7381	1.32812	0.664058	−	0.747681i	$-0.268833\pi$
$107$	13.7381	1.32812	0.664058	+	0.747681i	$0.268833\pi$
$108$	1.96239	0.188831
$109$	−2.77575	−0.265868	−0.132934	−	0.991125i	$-0.542440\pi$
$109$	−2.77575	−0.265868	−0.132934	+	0.991125i	$0.542440\pi$
$110$	0	0
$111$	0.775746	0.0736306
$112$	−3.77575	−0.356774
$113$	12.0508	1.13364	0.566821	−	0.823841i	$-0.308173\pi$
$113$	12.0508	1.13364	0.566821	+	0.823841i	$0.308173\pi$
$114$	−1.03761	−0.0971812
$115$	0	0
$116$	−15.5515	−1.44392
$117$	1.35026	0.124832
$118$	−1.67276	−0.153990
$119$	3.35026	0.307118
$120$	0	0
$121$	−7.00000	−0.636364
$122$	−1.68735	−0.152765
$123$	−3.73813	−0.337056
$124$	−8.97698	−0.806156
$125$	0	0
$126$	−0.193937	−0.0172772
$127$	2.70052	0.239633	0.119816	−	0.992796i	$-0.461769\pi$
$127$	2.70052	0.239633	0.119816	+	0.992796i	$0.461769\pi$
$128$	−5.91748	−0.523037
$129$	12.6253	1.11160
$130$	0	0
$131$	20.6253	1.80204	0.901020	−	0.433777i	$-0.142819\pi$
$131$	20.6253	1.80204	0.901020	+	0.433777i	$0.142819\pi$
$132$	3.92478	0.341608
$133$	−5.35026	−0.463927
$134$	1.92478	0.166275
$135$	0	0
$136$	2.57452	0.220763
$137$	22.4993	1.92224	0.961122	−	0.276124i	$-0.0890499\pi$
$137$	22.4993	1.92224	0.961122	+	0.276124i	$0.0890499\pi$
$138$	−0.962389	−0.0819240
$139$	−3.27504	−0.277785	−0.138893	−	0.990307i	$-0.544354\pi$
$139$	−3.27504	−0.277785	−0.138893	+	0.990307i	$0.544354\pi$
$140$	0	0
$141$	−9.92478	−0.835817
$142$	0.387873	0.0325496
$143$	2.70052	0.225829
$144$	3.77575	0.314646
$145$	0	0
$146$	−1.81336	−0.150075
$147$	−1.00000	−0.0824786
$148$	1.52232	0.125134
$149$	−4.44851	−0.364436	−0.182218	−	0.983258i	$-0.558328\pi$
$149$	−4.44851	−0.364436	−0.182218	+	0.983258i	$0.558328\pi$
$150$	0	0
$151$	1.29948	0.105750	0.0528749	−	0.998601i	$-0.483162\pi$
$151$	1.29948	0.105750	0.0528749	+	0.998601i	$0.483162\pi$
$152$	−4.11142	−0.333480
$153$	−3.35026	−0.270853
$154$	−0.387873	−0.0312557
$155$	0	0
$156$	2.64974	0.212149
$157$	−2.64974	−0.211472	−0.105736	−	0.994394i	$-0.533720\pi$
$157$	−2.64974	−0.211472	−0.105736	+	0.994394i	$0.533720\pi$
$158$	2.07522	0.165096
$159$	−8.57452	−0.680003
$160$	0	0
$161$	−4.96239	−0.391091
$162$	0.193937	0.0152371
$163$	5.29948	0.415087	0.207544	−	0.978226i	$-0.433453\pi$
$163$	5.29948	0.415087	0.207544	+	0.978226i	$0.433453\pi$
$164$	−7.33567	−0.572820
$165$	0	0
$166$	0.625301	0.0485327
$167$	−14.5501	−1.12592	−0.562959	−	0.826485i	$-0.690337\pi$
$167$	−14.5501	−1.12592	−0.562959	+	0.826485i	$0.690337\pi$
$168$	−0.768452	−0.0592874
$169$	−11.1768	−0.859753
$170$	0	0
$171$	5.35026	0.409145
$172$	24.7757	1.88913
$173$	−4.49929	−0.342075	−0.171037	−	0.985265i	$-0.554712\pi$
$173$	−4.49929	−0.342075	−0.171037	+	0.985265i	$0.554712\pi$
$174$	−1.53690	−0.116512
$175$	0	0
$176$	7.55149	0.569215
$177$	8.62530	0.648317
$178$	0.201231	0.0150829
$179$	10.0000	0.747435	0.373718	−	0.927543i	$-0.378083\pi$
$179$	10.0000	0.747435	0.373718	+	0.927543i	$0.378083\pi$
$180$	0	0
$181$	10.6253	0.789772	0.394886	−	0.918730i	$-0.370784\pi$
$181$	10.6253	0.789772	0.394886	+	0.918730i	$0.370784\pi$
$182$	−0.261865	−0.0194107
$183$	8.70052	0.643161
$184$	−3.81336	−0.281124
$185$	0	0
$186$	−0.887166	−0.0650502
$187$	−6.70052	−0.489991
$188$	−19.4763	−1.42045
$189$	1.00000	0.0727393
$190$	0	0
$191$	−13.8496	−1.00212	−0.501059	−	0.865413i	$-0.667056\pi$
$191$	−13.8496	−1.00212	−0.501059	+	0.865413i	$0.667056\pi$
$192$	7.11142	0.513222
$193$	15.3258	1.10318	0.551588	−	0.834116i	$-0.314022\pi$
$193$	15.3258	1.10318	0.551588	+	0.834116i	$0.314022\pi$
$194$	−3.58769	−0.257581
$195$	0	0
$196$	−1.96239	−0.140171
$197$	−0.574515	−0.0409325	−0.0204663	−	0.999791i	$-0.506515\pi$
$197$	−0.574515	−0.0409325	−0.0204663	+	0.999791i	$0.506515\pi$
$198$	0.387873	0.0275649
$199$	−0.201231	−0.0142649	−0.00713244	−	0.999975i	$-0.502270\pi$
$199$	−0.201231	−0.0142649	−0.00713244	+	0.999975i	$0.502270\pi$
$200$	0	0
$201$	−9.92478	−0.700040
$202$	−3.42548	−0.241016
$203$	−7.92478	−0.556210
$204$	−6.57452	−0.460308
$205$	0	0
$206$	1.29948	0.0905388
$207$	4.96239	0.344910
$208$	5.09825	0.353500
$209$	10.7005	0.740171
$210$	0	0
$211$	6.44851	0.443934	0.221967	−	0.975054i	$-0.428752\pi$
$211$	6.44851	0.443934	0.221967	+	0.975054i	$0.428752\pi$
$212$	−16.8265	−1.15565
$213$	−2.00000	−0.137038
$214$	2.66433	0.182130
$215$	0	0
$216$	0.768452	0.0522865
$217$	−4.57452	−0.310538
$218$	−0.538319	−0.0364595
$219$	9.35026	0.631832
$220$	0	0
$221$	−4.52373	−0.304299
$222$	0.150446	0.0100972
$223$	1.55149	0.103896	0.0519478	−	0.998650i	$-0.483457\pi$
$223$	1.55149	0.103896	0.0519478	+	0.998650i	$0.483457\pi$
$224$	−2.26916	−0.151615
$225$	0	0
$226$	2.33709	0.155461
$227$	−13.1490	−0.872732	−0.436366	−	0.899769i	$-0.643735\pi$
$227$	−13.1490	−0.872732	−0.436366	+	0.899769i	$0.643735\pi$
$228$	10.4993	0.695333
$229$	2.77575	0.183426	0.0917132	−	0.995785i	$-0.470766\pi$
$229$	2.77575	0.183426	0.0917132	+	0.995785i	$0.470766\pi$
$230$	0	0
$231$	2.00000	0.131590
$232$	−6.08981	−0.399816
$233$	−0.0507852	−0.00332705	−0.00166353	−	0.999999i	$-0.500530\pi$
$233$	−0.0507852	−0.00332705	−0.00166353	+	0.999999i	$0.500530\pi$
$234$	0.261865	0.0171187
$235$	0	0
$236$	16.9262	1.10180
$237$	−10.7005	−0.695074
$238$	0.649738	0.0421163
$239$	5.84955	0.378376	0.189188	−	0.981941i	$-0.439414\pi$
$239$	5.84955	0.378376	0.189188	+	0.981941i	$0.439414\pi$
$240$	0	0
$241$	−0.0752228	−0.00484553	−0.00242276	−	0.999997i	$-0.500771\pi$
$241$	−0.0752228	−0.00484553	−0.00242276	+	0.999997i	$0.500771\pi$
$242$	−1.35756	−0.0872670
$243$	−1.00000	−0.0641500
$244$	17.0738	1.09304
$245$	0	0
$246$	−0.724961	−0.0462218
$247$	7.22425	0.459668
$248$	−3.51530	−0.223222
$249$	−3.22425	−0.204329
$250$	0	0
$251$	19.2243	1.21342	0.606712	−	0.794922i	$-0.292488\pi$
$251$	19.2243	1.21342	0.606712	+	0.794922i	$0.292488\pi$
$252$	1.96239	0.123619
$253$	9.92478	0.623965
$254$	0.523730	0.0328618
$255$	0	0
$256$	13.0752	0.817201
$257$	7.35026	0.458497	0.229248	−	0.973368i	$-0.426373\pi$
$257$	7.35026	0.458497	0.229248	+	0.973368i	$0.426373\pi$
$258$	2.44851	0.152437
$259$	0.775746	0.0482025
$260$	0	0
$261$	7.92478	0.490531
$262$	4.00000	0.247121
$263$	−12.9624	−0.799295	−0.399648	−	0.916669i	$-0.630867\pi$
$263$	−12.9624	−0.799295	−0.399648	+	0.916669i	$0.630867\pi$
$264$	1.53690	0.0945899
$265$	0	0
$266$	−1.03761	−0.0636200
$267$	−1.03761	−0.0635008
$268$	−19.4763	−1.18970
$269$	4.11142	0.250678	0.125339	−	0.992114i	$-0.459998\pi$
$269$	4.11142	0.250678	0.125339	+	0.992114i	$0.459998\pi$
$270$	0	0
$271$	−16.4241	−0.997691	−0.498846	−	0.866691i	$-0.666243\pi$
$271$	−16.4241	−0.997691	−0.498846	+	0.866691i	$0.666243\pi$
$272$	−12.6497	−0.767003
$273$	1.35026	0.0817216
$274$	4.36344	0.263605
$275$	0	0
$276$	9.73813	0.586167
$277$	−11.0738	−0.665361	−0.332680	−	0.943040i	$-0.607953\pi$
$277$	−11.0738	−0.665361	−0.332680	+	0.943040i	$0.607953\pi$
$278$	−0.635150	−0.0380938
$279$	4.57452	0.273869
$280$	0	0
$281$	14.3733	0.857438	0.428719	−	0.903438i	$-0.358965\pi$
$281$	14.3733	0.857438	0.428719	+	0.903438i	$0.358965\pi$
$282$	−1.92478	−0.114619
$283$	1.14903	0.0683028	0.0341514	−	0.999417i	$-0.489127\pi$
$283$	1.14903	0.0683028	0.0341514	+	0.999417i	$0.489127\pi$
$284$	−3.92478	−0.232893
$285$	0	0
$286$	0.523730	0.0309688
$287$	−3.73813	−0.220655
$288$	2.26916	0.133711
$289$	−5.77575	−0.339750
$290$	0	0
$291$	18.4993	1.08445
$292$	18.3488	1.07379
$293$	0.649738	0.0379581	0.0189791	−	0.999820i	$-0.493958\pi$
$293$	0.649738	0.0379581	0.0189791	+	0.999820i	$0.493958\pi$
$294$	−0.193937	−0.0113106
$295$	0	0
$296$	0.596124	0.0346490
$297$	−2.00000	−0.116052
$298$	−0.862728	−0.0499765
$299$	6.70052	0.387501
$300$	0	0
$301$	12.6253	0.727710
$302$	0.252016	0.0145019
$303$	17.6629	1.01471
$304$	20.2012	1.15862
$305$	0	0
$306$	−0.649738	−0.0371431
$307$	24.1016	1.37555	0.687775	−	0.725924i	$-0.258588\pi$
$307$	24.1016	1.37555	0.687775	+	0.725924i	$0.258588\pi$
$308$	3.92478	0.223635
$309$	−6.70052	−0.381179
$310$	0	0
$311$	8.25202	0.467929	0.233964	−	0.972245i	$-0.424830\pi$
$311$	8.25202	0.467929	0.233964	+	0.972245i	$0.424830\pi$
$312$	1.03761	0.0587432
$313$	−14.9018	−0.842297	−0.421148	−	0.906992i	$-0.638373\pi$
$313$	−14.9018	−0.842297	−0.421148	+	0.906992i	$0.638373\pi$
$314$	−0.513881	−0.0290000
$315$	0	0
$316$	−20.9986	−1.18126
$317$	10.1260	0.568733	0.284367	−	0.958716i	$-0.408217\pi$
$317$	10.1260	0.568733	0.284367	+	0.958716i	$0.408217\pi$
$318$	−1.66291	−0.0932515
$319$	15.8496	0.887405
$320$	0	0
$321$	−13.7381	−0.766788
$322$	−0.962389	−0.0536318
$323$	−17.9248	−0.997361
$324$	−1.96239	−0.109022
$325$	0	0
$326$	1.02776	0.0569225
$327$	2.77575	0.153499
$328$	−2.87258	−0.158612
$329$	−9.92478	−0.547171
$330$	0	0
$331$	27.8496	1.53075	0.765375	−	0.643585i	$-0.222554\pi$
$331$	27.8496	1.53075	0.765375	+	0.643585i	$0.222554\pi$
$332$	−6.32724	−0.347252
$333$	−0.775746	−0.0425106
$334$	−2.82179	−0.154402
$335$	0	0
$336$	3.77575	0.205984
$337$	−3.84955	−0.209699	−0.104849	−	0.994488i	$-0.533436\pi$
$337$	−3.84955	−0.209699	−0.104849	+	0.994488i	$0.533436\pi$
$338$	−2.16759	−0.117901
$339$	−12.0508	−0.654509
$340$	0	0
$341$	9.14903	0.495448
$342$	1.03761	0.0561076
$343$	−1.00000	−0.0539949
$344$	9.70194	0.523093
$345$	0	0
$346$	−0.872577	−0.0469101
$347$	9.58769	0.514694	0.257347	−	0.966319i	$-0.417152\pi$
$347$	9.58769	0.514694	0.257347	+	0.966319i	$0.417152\pi$
$348$	15.5515	0.833648
$349$	−15.1490	−0.810909	−0.405455	−	0.914115i	$-0.632887\pi$
$349$	−15.1490	−0.810909	−0.405455	+	0.914115i	$0.632887\pi$
$350$	0	0
$351$	−1.35026	−0.0720716
$352$	4.53832	0.241893
$353$	−20.3488	−1.08306	−0.541530	−	0.840681i	$-0.682155\pi$
$353$	−20.3488	−1.08306	−0.541530	+	0.840681i	$0.682155\pi$
$354$	1.67276	0.0889063
$355$	0	0
$356$	−2.03620	−0.107918
$357$	−3.35026	−0.177315
$358$	1.93937	0.102499
$359$	−31.4010	−1.65728	−0.828642	−	0.559779i	$-0.810886\pi$
$359$	−31.4010	−1.65728	−0.828642	+	0.559779i	$0.810886\pi$
$360$	0	0
$361$	9.62530	0.506595
$362$	2.06063	0.108305
$363$	7.00000	0.367405
$364$	2.64974	0.138884
$365$	0	0
$366$	1.68735	0.0881992
$367$	−29.4010	−1.53472	−0.767361	−	0.641215i	$-0.778431\pi$
$367$	−29.4010	−1.53472	−0.767361	+	0.641215i	$0.778431\pi$
$368$	18.7367	0.976719
$369$	3.73813	0.194600
$370$	0	0
$371$	−8.57452	−0.445167
$372$	8.97698	0.465435
$373$	16.0000	0.828449	0.414224	−	0.910175i	$-0.364053\pi$
$373$	16.0000	0.828449	0.414224	+	0.910175i	$0.364053\pi$
$374$	−1.29948	−0.0671943
$375$	0	0
$376$	−7.62672	−0.393318
$377$	10.7005	0.551105
$378$	0.193937	0.00997502
$379$	10.7005	0.549649	0.274824	−	0.961494i	$-0.411380\pi$
$379$	10.7005	0.549649	0.274824	+	0.961494i	$0.411380\pi$
$380$	0	0
$381$	−2.70052	−0.138352
$382$	−2.68594	−0.137424
$383$	−16.7757	−0.857201	−0.428600	−	0.903494i	$-0.640993\pi$
$383$	−16.7757	−0.857201	−0.428600	+	0.903494i	$0.640993\pi$
$384$	5.91748	0.301975
$385$	0	0
$386$	2.97224	0.151283
$387$	−12.6253	−0.641780
$388$	36.3028	1.84300
$389$	−29.3258	−1.48688	−0.743439	−	0.668804i	$-0.766807\pi$
$389$	−29.3258	−1.48688	−0.743439	+	0.668804i	$0.766807\pi$
$390$	0	0
$391$	−16.6253	−0.840778
$392$	−0.768452	−0.0388127
$393$	−20.6253	−1.04041
$394$	−0.111420	−0.00561324
$395$	0	0
$396$	−3.92478	−0.197227
$397$	18.3488	0.920902	0.460451	−	0.887685i	$-0.347688\pi$
$397$	18.3488	0.920902	0.460451	+	0.887685i	$0.347688\pi$
$398$	−0.0390260	−0.00195620
$399$	5.35026	0.267848
$400$	0	0
$401$	−37.3258	−1.86396	−0.931981	−	0.362506i	$-0.881921\pi$
$401$	−37.3258	−1.86396	−0.931981	+	0.362506i	$0.881921\pi$
$402$	−1.92478	−0.0959992
$403$	6.17679	0.307688
$404$	34.6615	1.72447
$405$	0	0
$406$	−1.53690	−0.0762753
$407$	−1.55149	−0.0769046
$408$	−2.57452	−0.127458
$409$	−22.3733	−1.10629	−0.553144	−	0.833086i	$-0.686572\pi$
$409$	−22.3733	−1.10629	−0.553144	+	0.833086i	$0.686572\pi$
$410$	0	0
$411$	−22.4993	−1.10981
$412$	−13.1490	−0.647806
$413$	8.62530	0.424423
$414$	0.962389	0.0472988
$415$	0	0
$416$	3.06396	0.150223
$417$	3.27504	0.160379
$418$	2.07522	0.101502
$419$	23.4763	1.14689	0.573445	−	0.819244i	$-0.305606\pi$
$419$	23.4763	1.14689	0.573445	+	0.819244i	$0.305606\pi$
$420$	0	0
$421$	−25.2243	−1.22935	−0.614677	−	0.788779i	$-0.710714\pi$
$421$	−25.2243	−1.22935	−0.614677	+	0.788779i	$0.710714\pi$
$422$	1.25060	0.0608783
$423$	9.92478	0.482559
$424$	−6.58910	−0.319995
$425$	0	0
$426$	−0.387873	−0.0187925
$427$	8.70052	0.421048
$428$	−26.9596	−1.30314
$429$	−2.70052	−0.130383
$430$	0	0
$431$	−19.4010	−0.934516	−0.467258	−	0.884121i	$-0.654758\pi$
$431$	−19.4010	−0.934516	−0.467258	+	0.884121i	$0.654758\pi$
$432$	−3.77575	−0.181661
$433$	6.49929	0.312336	0.156168	−	0.987731i	$-0.450086\pi$
$433$	6.49929	0.312336	0.156168	+	0.987731i	$0.450086\pi$
$434$	−0.887166	−0.0425853
$435$	0	0
$436$	5.44709	0.260868
$437$	26.5501	1.27006
$438$	1.81336	0.0866456
$439$	−14.6497	−0.699194	−0.349597	−	0.936900i	$-0.613681\pi$
$439$	−14.6497	−0.699194	−0.349597	+	0.936900i	$0.613681\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	−0.877317	−0.0417297
$443$	19.1392	0.909330	0.454665	−	0.890663i	$-0.349759\pi$
$443$	19.1392	0.909330	0.454665	+	0.890663i	$0.349759\pi$
$444$	−1.52232	−0.0722459
$445$	0	0
$446$	0.300891	0.0142476
$447$	4.44851	0.210407
$448$	7.11142	0.335983
$449$	−32.8021	−1.54803	−0.774013	−	0.633169i	$-0.781754\pi$
$449$	−32.8021	−1.54803	−0.774013	+	0.633169i	$0.781754\pi$
$450$	0	0
$451$	7.47627	0.352044
$452$	−23.6483	−1.11232
$453$	−1.29948	−0.0610547
$454$	−2.55008	−0.119681
$455$	0	0
$456$	4.11142	0.192535
$457$	−18.7005	−0.874774	−0.437387	−	0.899273i	$-0.644096\pi$
$457$	−18.7005	−0.874774	−0.437387	+	0.899273i	$0.644096\pi$
$458$	0.538319	0.0251540
$459$	3.35026	0.156377
$460$	0	0
$461$	−6.96239	−0.324271	−0.162135	−	0.986769i	$-0.551838\pi$
$461$	−6.96239	−0.324271	−0.162135	+	0.986769i	$0.551838\pi$
$462$	0.387873	0.0180455
$463$	5.29948	0.246288	0.123144	−	0.992389i	$-0.460702\pi$
$463$	5.29948	0.246288	0.123144	+	0.992389i	$0.460702\pi$
$464$	29.9219	1.38909
$465$	0	0
$466$	−0.00984911	−0.000456251 0
$467$	−13.1490	−0.608465	−0.304232	−	0.952598i	$-0.598400\pi$
$467$	−13.1490	−0.608465	−0.304232	+	0.952598i	$0.598400\pi$
$468$	−2.64974	−0.122484
$469$	−9.92478	−0.458284
$470$	0	0
$471$	2.64974	0.122093
$472$	6.62813	0.305084
$473$	−25.2506	−1.16102
$474$	−2.07522	−0.0953181
$475$	0	0
$476$	−6.57452	−0.301342
$477$	8.57452	0.392600
$478$	1.13444	0.0518882
$479$	5.14903	0.235265	0.117633	−	0.993057i	$-0.462469\pi$
$479$	5.14903	0.235265	0.117633	+	0.993057i	$0.462469\pi$
$480$	0	0
$481$	−1.04746	−0.0477601
$482$	−0.0145884	−0.000664486 0
$483$	4.96239	0.225797
$484$	13.7367	0.624396
$485$	0	0
$486$	−0.193937	−0.00879714
$487$	−22.1768	−1.00493	−0.502463	−	0.864599i	$-0.667573\pi$
$487$	−22.1768	−1.00493	−0.502463	+	0.864599i	$0.667573\pi$
$488$	6.68594	0.302658
$489$	−5.29948	−0.239651
$490$	0	0
$491$	2.00000	0.0902587	0.0451294	−	0.998981i	$-0.485630\pi$
$491$	2.00000	0.0902587	0.0451294	+	0.998981i	$0.485630\pi$
$492$	7.33567	0.330718
$493$	−26.5501	−1.19576
$494$	1.40105	0.0630361
$495$	0	0
$496$	17.2722	0.775545
$497$	−2.00000	−0.0897123
$498$	−0.625301	−0.0280204
$499$	−6.55008	−0.293222	−0.146611	−	0.989194i	$-0.546837\pi$
$499$	−6.55008	−0.293222	−0.146611	+	0.989194i	$0.546837\pi$
$500$	0	0
$501$	14.5501	0.650050
$502$	3.72829	0.166402
$503$	8.77575	0.391291	0.195646	−	0.980675i	$-0.437320\pi$
$503$	8.77575	0.391291	0.195646	+	0.980675i	$0.437320\pi$
$504$	0.768452	0.0342296
$505$	0	0
$506$	1.92478	0.0855668
$507$	11.1768	0.496379
$508$	−5.29948	−0.235126
$509$	13.1392	0.582384	0.291192	−	0.956665i	$-0.405948\pi$
$509$	13.1392	0.582384	0.291192	+	0.956665i	$0.405948\pi$
$510$	0	0
$511$	9.35026	0.413631
$512$	14.3707	0.635103
$513$	−5.35026	−0.236220
$514$	1.42548	0.0628754
$515$	0	0
$516$	−24.7757	−1.09069
$517$	19.8496	0.872982
$518$	0.150446	0.00661020
$519$	4.49929	0.197497
$520$	0	0
$521$	−37.6629	−1.65004	−0.825021	−	0.565102i	$-0.808837\pi$
$521$	−37.6629	−1.65004	−0.825021	+	0.565102i	$0.808837\pi$
$522$	1.53690	0.0672685
$523$	−4.00000	−0.174908	−0.0874539	−	0.996169i	$-0.527873\pi$
$523$	−4.00000	−0.174908	−0.0874539	+	0.996169i	$0.527873\pi$
$524$	−40.4749	−1.76815
$525$	0	0
$526$	−2.51388	−0.109610
$527$	−15.3258	−0.667603
$528$	−7.55149	−0.328637
$529$	1.62530	0.0706652
$530$	0	0
$531$	−8.62530	−0.374306
$532$	10.4993	0.455202
$533$	5.04746	0.218630
$534$	−0.201231	−0.00870811
$535$	0	0
$536$	−7.62672	−0.329424
$537$	−10.0000	−0.431532
$538$	0.797355	0.0343764
$539$	2.00000	0.0861461
$540$	0	0
$541$	−22.4749	−0.966269	−0.483135	−	0.875546i	$-0.660502\pi$
$541$	−22.4749	−0.966269	−0.483135	+	0.875546i	$0.660502\pi$
$542$	−3.18523	−0.136817
$543$	−10.6253	−0.455975
$544$	−7.60228	−0.325945
$545$	0	0
$546$	0.261865	0.0112068
$547$	−25.9248	−1.10846	−0.554232	−	0.832362i	$-0.686988\pi$
$547$	−25.9248	−1.10846	−0.554232	+	0.832362i	$0.686988\pi$
$548$	−44.1524	−1.88610
$549$	−8.70052	−0.371329
$550$	0	0
$551$	42.3996	1.80629
$552$	3.81336	0.162307
$553$	−10.7005	−0.455033
$554$	−2.14762	−0.0912435
$555$	0	0
$556$	6.42690	0.272561
$557$	−28.5256	−1.20867	−0.604335	−	0.796730i	$-0.706561\pi$
$557$	−28.5256	−1.20867	−0.604335	+	0.796730i	$0.706561\pi$
$558$	0.887166	0.0375567
$559$	−17.0475	−0.721031
$560$	0	0
$561$	6.70052	0.282896
$562$	2.78751	0.117584
$563$	−11.6267	−0.490008	−0.245004	−	0.969522i	$-0.578789\pi$
$563$	−11.6267	−0.490008	−0.245004	+	0.969522i	$0.578789\pi$
$564$	19.4763	0.820099
$565$	0	0
$566$	0.222839	0.00936663
$567$	−1.00000	−0.0419961
$568$	−1.53690	−0.0644871
$569$	9.32582	0.390959	0.195479	−	0.980708i	$-0.437374\pi$
$569$	9.32582	0.390959	0.195479	+	0.980708i	$0.437374\pi$
$570$	0	0
$571$	−19.6991	−0.824382	−0.412191	−	0.911097i	$-0.635236\pi$
$571$	−19.6991	−0.824382	−0.412191	+	0.911097i	$0.635236\pi$
$572$	−5.29948	−0.221582
$573$	13.8496	0.578573
$574$	−0.724961	−0.0302593
$575$	0	0
$576$	−7.11142	−0.296309
$577$	32.7974	1.36537	0.682686	−	0.730712i	$-0.260812\pi$
$577$	32.7974	1.36537	0.682686	+	0.730712i	$0.260812\pi$
$578$	−1.12013	−0.0465912
$579$	−15.3258	−0.636920
$580$	0	0
$581$	−3.22425	−0.133765
$582$	3.58769	0.148715
$583$	17.1490	0.710240
$584$	7.18523	0.297327
$585$	0	0
$586$	0.126008	0.00520534
$587$	18.8218	0.776859	0.388429	−	0.921479i	$-0.373018\pi$
$587$	18.8218	0.776859	0.388429	+	0.921479i	$0.373018\pi$
$588$	1.96239	0.0809275
$589$	24.4749	1.00847
$590$	0	0
$591$	0.574515	0.0236324
$592$	−2.92902	−0.120382
$593$	33.7499	1.38594	0.692971	−	0.720965i	$-0.256301\pi$
$593$	33.7499	1.38594	0.692971	+	0.720965i	$0.256301\pi$
$594$	−0.387873	−0.0159146
$595$	0	0
$596$	8.72970	0.357582
$597$	0.201231	0.00823583
$598$	1.29948	0.0531395
$599$	−20.2981	−0.829356	−0.414678	−	0.909968i	$-0.636106\pi$
$599$	−20.2981	−0.829356	−0.414678	+	0.909968i	$0.636106\pi$
$600$	0	0
$601$	−13.8496	−0.564935	−0.282468	−	0.959277i	$-0.591153\pi$
$601$	−13.8496	−0.564935	−0.282468	+	0.959277i	$0.591153\pi$
$602$	2.44851	0.0997937
$603$	9.92478	0.404168
$604$	−2.55008	−0.103761
$605$	0	0
$606$	3.42548	0.139151
$607$	−25.2506	−1.02489	−0.512445	−	0.858720i	$-0.671260\pi$
$607$	−25.2506	−1.02489	−0.512445	+	0.858720i	$0.671260\pi$
$608$	12.1406	0.492366
$609$	7.92478	0.321128
$610$	0	0
$611$	13.4010	0.542148
$612$	6.57452	0.265759
$613$	−9.14903	−0.369526	−0.184763	−	0.982783i	$-0.559152\pi$
$613$	−9.14903	−0.369526	−0.184763	+	0.982783i	$0.559152\pi$
$614$	4.67418	0.188634
$615$	0	0
$616$	1.53690	0.0619236
$617$	15.9492	0.642091	0.321046	−	0.947064i	$-0.395966\pi$
$617$	15.9492	0.642091	0.321046	+	0.947064i	$0.395966\pi$
$618$	−1.29948	−0.0522726
$619$	11.1735	0.449100	0.224550	−	0.974463i	$-0.427909\pi$
$619$	11.1735	0.449100	0.224550	+	0.974463i	$0.427909\pi$
$620$	0	0
$621$	−4.96239	−0.199134
$622$	1.60037	0.0641689
$623$	−1.03761	−0.0415710
$624$	−5.09825	−0.204093
$625$	0	0
$626$	−2.89000	−0.115507
$627$	−10.7005	−0.427338
$628$	5.19982	0.207495
$629$	2.59895	0.103627
$630$	0	0
$631$	−14.5501	−0.579229	−0.289615	−	0.957143i	$-0.593527\pi$
$631$	−14.5501	−0.579229	−0.289615	+	0.957143i	$0.593527\pi$
$632$	−8.22284	−0.327087
$633$	−6.44851	−0.256305
$634$	1.96380	0.0779926
$635$	0	0
$636$	16.8265	0.667215
$637$	1.35026	0.0534993
$638$	3.07381	0.121693
$639$	2.00000	0.0791188
$640$	0	0
$641$	−38.7269	−1.52962	−0.764810	−	0.644256i	$-0.777167\pi$
$641$	−38.7269	−1.52962	−0.764810	+	0.644256i	$0.777167\pi$
$642$	−2.66433	−0.105153
$643$	−11.9511	−0.471306	−0.235653	−	0.971837i	$-0.575723\pi$
$643$	−11.9511	−0.471306	−0.235653	+	0.971837i	$0.575723\pi$
$644$	9.73813	0.383736
$645$	0	0
$646$	−3.47627	−0.136772
$647$	−14.5501	−0.572023	−0.286011	−	0.958226i	$-0.592329\pi$
$647$	−14.5501	−0.572023	−0.286011	+	0.958226i	$0.592329\pi$
$648$	−0.768452	−0.0301876
$649$	−17.2506	−0.677145
$650$	0	0
$651$	4.57452	0.179289
$652$	−10.3996	−0.407281
$653$	49.9756	1.95569	0.977847	−	0.209319i	$-0.0671247\pi$
$653$	49.9756	1.95569	0.977847	+	0.209319i	$0.0671247\pi$
$654$	0.538319	0.0210499
$655$	0	0
$656$	14.1142	0.551069
$657$	−9.35026	−0.364788
$658$	−1.92478	−0.0750356
$659$	−16.9525	−0.660377	−0.330189	−	0.943915i	$-0.607112\pi$
$659$	−16.9525	−0.660377	−0.330189	+	0.943915i	$0.607112\pi$
$660$	0	0
$661$	−15.6531	−0.608834	−0.304417	−	0.952539i	$-0.598462\pi$
$661$	−15.6531	−0.608834	−0.304417	+	0.952539i	$0.598462\pi$
$662$	5.40105	0.209918
$663$	4.52373	0.175687
$664$	−2.47768	−0.0961528
$665$	0	0
$666$	−0.150446	−0.00582965
$667$	39.3258	1.52270
$668$	28.5529	1.10475
$669$	−1.55149	−0.0599842
$670$	0	0
$671$	−17.4010	−0.671760
$672$	2.26916	0.0875347
$673$	26.0263	1.00324	0.501621	−	0.865088i	$-0.332737\pi$
$673$	26.0263	1.00324	0.501621	+	0.865088i	$0.332737\pi$
$674$	−0.746569	−0.0287568
$675$	0	0
$676$	21.9332	0.843585
$677$	−35.4518	−1.36252	−0.681262	−	0.732039i	$-0.738569\pi$
$677$	−35.4518	−1.36252	−0.681262	+	0.732039i	$0.738569\pi$
$678$	−2.33709	−0.0897553
$679$	18.4993	0.709938
$680$	0	0
$681$	13.1490	0.503872
$682$	1.77433	0.0679427
$683$	−23.6629	−0.905436	−0.452718	−	0.891654i	$-0.649546\pi$
$683$	−23.6629	−0.905436	−0.452718	+	0.891654i	$0.649546\pi$
$684$	−10.4993	−0.401450
$685$	0	0
$686$	−0.193937	−0.00740453
$687$	−2.77575	−0.105901
$688$	−47.6699	−1.81740
$689$	11.5778	0.441081
$690$	0	0
$691$	−0.574515	−0.0218556	−0.0109278	−	0.999940i	$-0.503478\pi$
$691$	−0.574515	−0.0218556	−0.0109278	+	0.999940i	$0.503478\pi$
$692$	8.82936	0.335642
$693$	−2.00000	−0.0759737
$694$	1.85940	0.0705820
$695$	0	0
$696$	6.08981	0.230834
$697$	−12.5237	−0.474370
$698$	−2.93795	−0.111203
$699$	0.0507852	0.00192087
$700$	0	0
$701$	42.7269	1.61377	0.806886	−	0.590707i	$-0.201151\pi$
$701$	42.7269	1.61377	0.806886	+	0.590707i	$0.201151\pi$
$702$	−0.261865	−0.00988346
$703$	−4.15045	−0.156537
$704$	−14.2228	−0.536043
$705$	0	0
$706$	−3.94639	−0.148524
$707$	17.6629	0.664282
$708$	−16.9262	−0.636125
$709$	27.2506	1.02342	0.511709	−	0.859159i	$-0.329013\pi$
$709$	27.2506	1.02342	0.511709	+	0.859159i	$0.329013\pi$
$710$	0	0
$711$	10.7005	0.401301
$712$	−0.797355	−0.0298821
$713$	22.7005	0.850141
$714$	−0.649738	−0.0243158
$715$	0	0
$716$	−19.6239	−0.733379
$717$	−5.84955	−0.218456
$718$	−6.08981	−0.227270
$719$	−10.7005	−0.399062	−0.199531	−	0.979891i	$-0.563942\pi$
$719$	−10.7005	−0.399062	−0.199531	+	0.979891i	$0.563942\pi$
$720$	0	0
$721$	−6.70052	−0.249541
$722$	1.86670	0.0694713
$723$	0.0752228	0.00279757
$724$	−20.8510	−0.774920
$725$	0	0
$726$	1.35756	0.0503836
$727$	39.9511	1.48171	0.740853	−	0.671668i	$-0.234422\pi$
$727$	39.9511	1.48171	0.740853	+	0.671668i	$0.234422\pi$
$728$	1.03761	0.0384564
$729$	1.00000	0.0370370
$730$	0	0
$731$	42.2981	1.56445
$732$	−17.0738	−0.631066
$733$	−30.3488	−1.12096	−0.560480	−	0.828168i	$-0.689383\pi$
$733$	−30.3488	−1.12096	−0.560480	+	0.828168i	$0.689383\pi$
$734$	−5.70194	−0.210462
$735$	0	0
$736$	11.2605	0.415066
$737$	19.8496	0.731168
$738$	0.724961	0.0266862
$739$	37.2506	1.37029	0.685143	−	0.728409i	$-0.259740\pi$
$739$	37.2506	1.37029	0.685143	+	0.728409i	$0.259740\pi$
$740$	0	0
$741$	−7.22425	−0.265390
$742$	−1.66291	−0.0610474
$743$	26.3634	0.967181	0.483590	−	0.875294i	$-0.339332\pi$
$743$	26.3634	0.967181	0.483590	+	0.875294i	$0.339332\pi$
$744$	3.51530	0.128877
$745$	0	0
$746$	3.10299	0.113608
$747$	3.22425	0.117969
$748$	13.1490	0.480776
$749$	−13.7381	−0.501981
$750$	0	0
$751$	50.6516	1.84830	0.924152	−	0.382024i	$-0.124773\pi$
$751$	50.6516	1.84830	0.924152	+	0.382024i	$0.124773\pi$
$752$	37.4734	1.36652
$753$	−19.2243	−0.700571
$754$	2.07522	0.0755752
$755$	0	0
$756$	−1.96239	−0.0713714
$757$	38.9525	1.41575	0.707877	−	0.706336i	$-0.249653\pi$
$757$	38.9525	1.41575	0.707877	+	0.706336i	$0.249653\pi$
$758$	2.07522	0.0753755
$759$	−9.92478	−0.360247
$760$	0	0
$761$	48.2130	1.74772	0.873860	−	0.486178i	$-0.161609\pi$
$761$	48.2130	1.74772	0.873860	+	0.486178i	$0.161609\pi$
$762$	−0.523730	−0.0189727
$763$	2.77575	0.100489
$764$	27.1782	0.983273
$765$	0	0
$766$	−3.25343	−0.117551
$767$	−11.6464	−0.420528
$768$	−13.0752	−0.471811
$769$	4.44851	0.160417	0.0802086	−	0.996778i	$-0.474441\pi$
$769$	4.44851	0.160417	0.0802086	+	0.996778i	$0.474441\pi$
$770$	0	0
$771$	−7.35026	−0.264713
$772$	−30.0752	−1.08243
$773$	39.3014	1.41357	0.706786	−	0.707427i	$-0.250144\pi$
$773$	39.3014	1.41357	0.706786	+	0.707427i	$0.250144\pi$
$774$	−2.44851	−0.0880098
$775$	0	0
$776$	14.2158	0.510318
$777$	−0.775746	−0.0278297
$778$	−5.68735	−0.203901
$779$	20.0000	0.716574
$780$	0	0
$781$	4.00000	0.143131
$782$	−3.22425	−0.115299
$783$	−7.92478	−0.283208
$784$	3.77575	0.134848
$785$	0	0
$786$	−4.00000	−0.142675
$787$	0.897015	0.0319751	0.0159876	−	0.999872i	$-0.494911\pi$
$787$	0.897015	0.0319751	0.0159876	+	0.999872i	$0.494911\pi$
$788$	1.12742	0.0401628
$789$	12.9624	0.461473
$790$	0	0
$791$	−12.0508	−0.428477
$792$	−1.53690	−0.0546115
$793$	−11.7480	−0.417183
$794$	3.55851	0.126287
$795$	0	0
$796$	0.394893	0.0139966
$797$	3.19982	0.113343	0.0566717	−	0.998393i	$-0.481951\pi$
$797$	3.19982	0.113343	0.0566717	+	0.998393i	$0.481951\pi$
$798$	1.03761	0.0367310
$799$	−33.2506	−1.17632
$800$	0	0
$801$	1.03761	0.0366622
$802$	−7.23884	−0.255612
$803$	−18.7005	−0.659927
$804$	19.4763	0.686875
$805$	0	0
$806$	1.19791	0.0421944
$807$	−4.11142	−0.144729
$808$	13.5731	0.477500
$809$	4.44851	0.156401	0.0782006	−	0.996938i	$-0.475083\pi$
$809$	4.44851	0.156401	0.0782006	+	0.996938i	$0.475083\pi$
$810$	0	0
$811$	37.6747	1.32294	0.661468	−	0.749973i	$-0.269934\pi$
$811$	37.6747	1.32294	0.661468	+	0.749973i	$0.269934\pi$
$812$	15.5515	0.545750
$813$	16.4241	0.576017
$814$	−0.300891	−0.0105462
$815$	0	0
$816$	12.6497	0.442829
$817$	−67.5487	−2.36323
$818$	−4.33900	−0.151710
$819$	−1.35026	−0.0471820
$820$	0	0
$821$	−0.749399	−0.0261542	−0.0130771	−	0.999914i	$-0.504163\pi$
$821$	−0.749399	−0.0261542	−0.0130771	+	0.999914i	$0.504163\pi$
$822$	−4.36344	−0.152192
$823$	−26.3996	−0.920233	−0.460117	−	0.887858i	$-0.652192\pi$
$823$	−26.3996	−0.920233	−0.460117	+	0.887858i	$0.652192\pi$
$824$	−5.14903	−0.179375
$825$	0	0
$826$	1.67276	0.0582028
$827$	5.43724	0.189071	0.0945357	−	0.995521i	$-0.469863\pi$
$827$	5.43724	0.189071	0.0945357	+	0.995521i	$0.469863\pi$
$828$	−9.73813	−0.338424
$829$	22.7757	0.791034	0.395517	−	0.918459i	$-0.370565\pi$
$829$	22.7757	0.791034	0.395517	+	0.918459i	$0.370565\pi$
$830$	0	0
$831$	11.0738	0.384146
$832$	−9.60228	−0.332899
$833$	−3.35026	−0.116080
$834$	0.635150	0.0219934
$835$	0	0
$836$	−20.9986	−0.726251
$837$	−4.57452	−0.158118
$838$	4.55291	0.157278
$839$	15.8496	0.547187	0.273594	−	0.961845i	$-0.411788\pi$
$839$	15.8496	0.547187	0.273594	+	0.961845i	$0.411788\pi$
$840$	0	0
$841$	33.8021	1.16559
$842$	−4.89191	−0.168586
$843$	−14.3733	−0.495042
$844$	−12.6545	−0.435585
$845$	0	0
$846$	1.92478	0.0661752
$847$	7.00000	0.240523
$848$	32.3752	1.11177
$849$	−1.14903	−0.0394346
$850$	0	0
$851$	−3.84955	−0.131961
$852$	3.92478	0.134461
$853$	−21.0494	−0.720717	−0.360358	−	0.932814i	$-0.617346\pi$
$853$	−21.0494	−0.720717	−0.360358	+	0.932814i	$0.617346\pi$
$854$	1.68735	0.0577399
$855$	0	0
$856$	−10.5571	−0.360834
$857$	50.1524	1.71317	0.856586	−	0.516004i	$-0.172581\pi$
$857$	50.1524	1.71317	0.856586	+	0.516004i	$0.172581\pi$
$858$	−0.523730	−0.0178799
$859$	−5.35026	−0.182549	−0.0912743	−	0.995826i	$-0.529094\pi$
$859$	−5.35026	−0.182549	−0.0912743	+	0.995826i	$0.529094\pi$
$860$	0	0
$861$	3.73813	0.127395
$862$	−3.76257	−0.128154
$863$	−33.6893	−1.14680	−0.573398	−	0.819277i	$-0.694375\pi$
$863$	−33.6893	−1.14680	−0.573398	+	0.819277i	$0.694375\pi$
$864$	−2.26916	−0.0771984
$865$	0	0
$866$	1.26045	0.0428319
$867$	5.77575	0.196155
$868$	8.97698	0.304698
$869$	21.4010	0.725981
$870$	0	0
$871$	13.4010	0.454077
$872$	2.13303	0.0722334
$873$	−18.4993	−0.626106
$874$	5.14903	0.174169
$875$	0	0
$876$	−18.3488	−0.619950
$877$	−21.5026	−0.726092	−0.363046	−	0.931771i	$-0.618263\pi$
$877$	−21.5026	−0.726092	−0.363046	+	0.931771i	$0.618263\pi$
$878$	−2.84112	−0.0958832
$879$	−0.649738	−0.0219151
$880$	0	0
$881$	32.3634	1.09035	0.545176	−	0.838322i	$-0.316463\pi$
$881$	32.3634	1.09035	0.545176	+	0.838322i	$0.316463\pi$
$882$	0.193937	0.00653018
$883$	−2.59895	−0.0874617	−0.0437309	−	0.999043i	$-0.513924\pi$
$883$	−2.59895	−0.0874617	−0.0437309	+	0.999043i	$0.513924\pi$
$884$	8.87732	0.298576
$885$	0	0
$886$	3.71179	0.124700
$887$	38.2784	1.28526	0.642631	−	0.766176i	$-0.277843\pi$
$887$	38.2784	1.28526	0.642631	+	0.766176i	$0.277843\pi$
$888$	−0.596124	−0.0200046
$889$	−2.70052	−0.0905727
$890$	0	0
$891$	2.00000	0.0670025
$892$	−3.04463	−0.101942
$893$	53.1002	1.77693
$894$	0.862728	0.0288539
$895$	0	0
$896$	5.91748	0.197689
$897$	−6.70052	−0.223724
$898$	−6.36153	−0.212287
$899$	36.2520	1.20907
$900$	0	0
$901$	−28.7269	−0.957031
$902$	1.44992	0.0482771
$903$	−12.6253	−0.420144
$904$	−9.26045	−0.307998
$905$	0	0
$906$	−0.252016	−0.00837267
$907$	−49.9972	−1.66013	−0.830064	−	0.557668i	$-0.811696\pi$
$907$	−49.9972	−1.66013	−0.830064	+	0.557668i	$0.811696\pi$
$908$	25.8035	0.856320
$909$	−17.6629	−0.585842
$910$	0	0
$911$	−24.9525	−0.826715	−0.413357	−	0.910569i	$-0.635644\pi$
$911$	−24.9525	−0.826715	−0.413357	+	0.910569i	$0.635644\pi$
$912$	−20.2012	−0.668930
$913$	6.44851	0.213414
$914$	−3.62672	−0.119961
$915$	0	0
$916$	−5.44709	−0.179977
$917$	−20.6253	−0.681107
$918$	0.649738	0.0214446
$919$	−11.6991	−0.385918	−0.192959	−	0.981207i	$-0.561808\pi$
$919$	−11.6991	−0.385918	−0.192959	+	0.981207i	$0.561808\pi$
$920$	0	0
$921$	−24.1016	−0.794174
$922$	−1.35026	−0.0444685
$923$	2.70052	0.0888888
$924$	−3.92478	−0.129116
$925$	0	0
$926$	1.02776	0.0337744
$927$	6.70052	0.220074
$928$	17.9826	0.590307
$929$	23.7090	0.777866	0.388933	−	0.921266i	$-0.372844\pi$
$929$	23.7090	0.777866	0.388933	+	0.921266i	$0.372844\pi$
$930$	0	0
$931$	5.35026	0.175348
$932$	0.0996603	0.00326448
$933$	−8.25202	−0.270159
$934$	−2.55008	−0.0834411
$935$	0	0
$936$	−1.03761	−0.0339154
$937$	−19.9003	−0.650116	−0.325058	−	0.945694i	$-0.605384\pi$
$937$	−19.9003	−0.650116	−0.325058	+	0.945694i	$0.605384\pi$
$938$	−1.92478	−0.0628462
$939$	14.9018	0.486300
$940$	0	0
$941$	−6.28821	−0.204990	−0.102495	−	0.994734i	$-0.532683\pi$
$941$	−6.28821	−0.204990	−0.102495	+	0.994734i	$0.532683\pi$
$942$	0.513881	0.0167432
$943$	18.5501	0.604074
$944$	−32.5669	−1.05996
$945$	0	0
$946$	−4.89701	−0.159216
$947$	−40.0362	−1.30100	−0.650501	−	0.759506i	$-0.725441\pi$
$947$	−40.0362	−1.30100	−0.650501	+	0.759506i	$0.725441\pi$
$948$	20.9986	0.682002
$949$	−12.6253	−0.409835
$950$	0	0
$951$	−10.1260	−0.328358
$952$	−2.57452	−0.0834405
$953$	40.9478	1.32643	0.663215	−	0.748429i	$-0.269192\pi$
$953$	40.9478	1.32643	0.663215	+	0.748429i	$0.269192\pi$
$954$	1.66291	0.0538388
$955$	0	0
$956$	−11.4791	−0.371261
$957$	−15.8496	−0.512343
$958$	0.998585	0.0322628
$959$	−22.4993	−0.726540
$960$	0	0
$961$	−10.0738	−0.324962
$962$	−0.203141	−0.00654953
$963$	13.7381	0.442705
$964$	0.147616	0.00475440
$965$	0	0
$966$	0.962389	0.0309643
$967$	38.2784	1.23095	0.615475	−	0.788157i	$-0.288964\pi$
$967$	38.2784	1.23095	0.615475	+	0.788157i	$0.288964\pi$
$968$	5.37916	0.172893
$969$	17.9248	0.575827
$970$	0	0
$971$	−28.7269	−0.921889	−0.460945	−	0.887429i	$-0.652489\pi$
$971$	−28.7269	−0.921889	−0.460945	+	0.887429i	$0.652489\pi$
$972$	1.96239	0.0629436
$973$	3.27504	0.104993
$974$	−4.30089	−0.137809
$975$	0	0
$976$	−32.8510	−1.05153
$977$	−41.3014	−1.32135	−0.660674	−	0.750673i	$-0.729730\pi$
$977$	−41.3014	−1.32135	−0.660674	+	0.750673i	$0.729730\pi$
$978$	−1.02776	−0.0328642
$979$	2.07522	0.0663244
$980$	0	0
$981$	−2.77575	−0.0886228
$982$	0.387873	0.0123775
$983$	−24.0000	−0.765481	−0.382741	−	0.923856i	$-0.625020\pi$
$983$	−24.0000	−0.765481	−0.382741	+	0.923856i	$0.625020\pi$
$984$	2.87258	0.0915744
$985$	0	0
$986$	−5.14903	−0.163979
$987$	9.92478	0.315909
$988$	−14.1768	−0.451024
$989$	−62.6516	−1.99221
$990$	0	0
$991$	25.1002	0.797333	0.398666	−	0.917096i	$-0.369473\pi$
$991$	25.1002	0.797333	0.398666	+	0.917096i	$0.369473\pi$
$992$	10.3803	0.329575
$993$	−27.8496	−0.883779
$994$	−0.387873	−0.0123026
$995$	0	0
$996$	6.32724	0.200486
$997$	32.7974	1.03870	0.519351	−	0.854561i	$-0.326174\pi$
$997$	32.7974	1.03870	0.519351	+	0.854561i	$0.326174\pi$
$998$	−1.27030	−0.0402106
$999$	0.775746	0.0245435

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	525.2.a.k.1.2		3
3.2	odd	2		1575.2.a.w.1.2		3
4.3	odd	2		8400.2.a.dj.1.3		3
5.2	odd	4		105.2.d.b.64.4	yes	6
5.3	odd	4		105.2.d.b.64.3	✓	6
5.4	even	2		525.2.a.j.1.2		3
7.6	odd	2		3675.2.a.bj.1.2		3
15.2	even	4		315.2.d.e.64.3		6
15.8	even	4		315.2.d.e.64.4		6
15.14	odd	2		1575.2.a.x.1.2		3
20.3	even	4		1680.2.t.k.1009.4		6
20.7	even	4		1680.2.t.k.1009.1		6
20.19	odd	2		8400.2.a.dg.1.1		3
35.2	odd	12		735.2.q.e.214.4		12
35.3	even	12		735.2.q.f.79.4		12
35.12	even	12		735.2.q.f.214.4		12
35.13	even	4		735.2.d.b.589.3		6
35.17	even	12		735.2.q.f.79.3		12
35.18	odd	12		735.2.q.e.79.4		12
35.23	odd	12		735.2.q.e.214.3		12
35.27	even	4		735.2.d.b.589.4		6
35.32	odd	12		735.2.q.e.79.3		12
35.33	even	12		735.2.q.f.214.3		12
35.34	odd	2		3675.2.a.bi.1.2		3
60.23	odd	4		5040.2.t.v.1009.6		6
60.47	odd	4		5040.2.t.v.1009.5		6
105.62	odd	4		2205.2.d.l.1324.3		6
105.83	odd	4		2205.2.d.l.1324.4		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.2.d.b.64.3	✓	6	5.3	odd	4
105.2.d.b.64.4	yes	6	5.2	odd	4
315.2.d.e.64.3		6	15.2	even	4
315.2.d.e.64.4		6	15.8	even	4
525.2.a.j.1.2		3	5.4	even	2
525.2.a.k.1.2		3	1.1	even	1	trivial
735.2.d.b.589.3		6	35.13	even	4
735.2.d.b.589.4		6	35.27	even	4
735.2.q.e.79.3		12	35.32	odd	12
735.2.q.e.79.4		12	35.18	odd	12
735.2.q.e.214.3		12	35.23	odd	12
735.2.q.e.214.4		12	35.2	odd	12
735.2.q.f.79.3		12	35.17	even	12
735.2.q.f.79.4		12	35.3	even	12
735.2.q.f.214.3		12	35.33	even	12
735.2.q.f.214.4		12	35.12	even	12
1575.2.a.w.1.2		3	3.2	odd	2
1575.2.a.x.1.2		3	15.14	odd	2
1680.2.t.k.1009.1		6	20.7	even	4
1680.2.t.k.1009.4		6	20.3	even	4
2205.2.d.l.1324.3		6	105.62	odd	4
2205.2.d.l.1324.4		6	105.83	odd	4
3675.2.a.bi.1.2		3	35.34	odd	2
3675.2.a.bj.1.2		3	7.6	odd	2
5040.2.t.v.1009.5		6	60.47	odd	4
5040.2.t.v.1009.6		6	60.23	odd	4
8400.2.a.dg.1.1		3	20.19	odd	2
8400.2.a.dj.1.3		3	4.3	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:	\( N \)	\(=\)	\( 525 = 3 \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	525.a (trivial)

\(f(q)\)	\(=\)	\(q+0.193937 q^{2} -1.00000 q^{3} -1.96239 q^{4} -0.193937 q^{6} -1.00000 q^{7} -0.768452 q^{8} +1.00000 q^{9} +2.00000 q^{11} +1.96239 q^{12} +1.35026 q^{13} -0.193937 q^{14} +3.77575 q^{16} -3.35026 q^{17} +0.193937 q^{18} +5.35026 q^{19} +1.00000 q^{21} +0.387873 q^{22} +4.96239 q^{23} +0.768452 q^{24} +0.261865 q^{26} -1.00000 q^{27} +1.96239 q^{28} +7.92478 q^{29} +4.57452 q^{31} +2.26916 q^{32} -2.00000 q^{33} -0.649738 q^{34} -1.96239 q^{36} -0.775746 q^{37} +1.03761 q^{38} -1.35026 q^{39} +3.73813 q^{41} +0.193937 q^{42} -12.6253 q^{43} -3.92478 q^{44} +0.962389 q^{46} +9.92478 q^{47} -3.77575 q^{48} +1.00000 q^{49} +3.35026 q^{51} -2.64974 q^{52} +8.57452 q^{53} -0.193937 q^{54} +0.768452 q^{56} -5.35026 q^{57} +1.53690 q^{58} -8.62530 q^{59} -8.70052 q^{61} +0.887166 q^{62} -1.00000 q^{63} -7.11142 q^{64} -0.387873 q^{66} +9.92478 q^{67} +6.57452 q^{68} -4.96239 q^{69} +2.00000 q^{71} -0.768452 q^{72} -9.35026 q^{73} -0.150446 q^{74} -10.4993 q^{76} -2.00000 q^{77} -0.261865 q^{78} +10.7005 q^{79} +1.00000 q^{81} +0.724961 q^{82} +3.22425 q^{83} -1.96239 q^{84} -2.44851 q^{86} -7.92478 q^{87} -1.53690 q^{88} +1.03761 q^{89} -1.35026 q^{91} -9.73813 q^{92} -4.57452 q^{93} +1.92478 q^{94} -2.26916 q^{96} -18.4993 q^{97} +0.193937 q^{98} +2.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + q^{2} - 3 q^{3} + 5 q^{4} - q^{6} - 3 q^{7} + 9 q^{8} + 3 q^{9} + 6 q^{11} - 5 q^{12} - 6 q^{13} - q^{14} + 13 q^{16} + q^{18} + 6 q^{19} + 3 q^{21} + 2 q^{22} + 4 q^{23} - 9 q^{24} + 10 q^{26}+ \cdots + 6 q^{99}+O(q^{100}) \) 3 * q + q^2 - 3 * q^3 + 5 * q^4 - q^6 - 3 * q^7 + 9 * q^8 + 3 * q^9 + 6 * q^11 - 5 * q^12 - 6 * q^13 - q^14 + 13 * q^16 + q^18 + 6 * q^19 + 3 * q^21 + 2 * q^22 + 4 * q^23 - 9 * q^24 + 10 * q^26 - 3 * q^27 - 5 * q^28 + 2 * q^29 + 2 * q^31 + 29 * q^32 - 6 * q^33 - 12 * q^34 + 5 * q^36 - 4 * q^37 + 14 * q^38 + 6 * q^39 + 2 * q^41 + q^42 + 4 * q^43 + 10 * q^44 - 8 * q^46 + 8 * q^47 - 13 * q^48 + 3 * q^49 - 18 * q^52 + 14 * q^53 - q^54 - 9 * q^56 - 6 * q^57 - 18 * q^58 + 16 * q^59 - 6 * q^61 - 30 * q^62 - 3 * q^63 + 13 * q^64 - 2 * q^66 + 8 * q^67 + 8 * q^68 - 4 * q^69 + 6 * q^71 + 9 * q^72 - 18 * q^73 - 44 * q^74 + 2 * q^76 - 6 * q^77 - 10 * q^78 + 12 * q^79 + 3 * q^81 + 34 * q^82 + 8 * q^83 + 5 * q^84 - 4 * q^86 - 2 * q^87 + 18 * q^88 + 14 * q^89 + 6 * q^91 - 20 * q^92 - 2 * q^93 - 16 * q^94 - 29 * q^96 - 22 * q^97 + q^98 + 6 * q^99

Introduction

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