LMFDB - Embedded newform 6525.2.a.p.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6525,2,Mod(1,6525)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6525, 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("6525.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6525 = 3^{2} \cdot 5^{2} \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6525.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:	$52.1023873189$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 145)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.41421$ of defining polynomial
Character	$\chi$	$=$	6525.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.414214 q^{2} -1.82843 q^{4} +4.82843 q^{7} -1.58579 q^{8} -0.828427 q^{11} +2.00000 q^{13} +2.00000 q^{14} +3.00000 q^{16} +2.82843 q^{17} -4.82843 q^{19} -0.343146 q^{22} -3.17157 q^{23} +0.828427 q^{26} -8.82843 q^{28} -1.00000 q^{29} +6.48528 q^{31} +4.41421 q^{32} +1.17157 q^{34} +8.48528 q^{37} -2.00000 q^{38} +6.00000 q^{41} +6.00000 q^{43} +1.51472 q^{44} -1.31371 q^{46} -11.6569 q^{47} +16.3137 q^{49} -3.65685 q^{52} -3.65685 q^{53} -7.65685 q^{56} -0.414214 q^{58} -3.65685 q^{61} +2.68629 q^{62} -4.17157 q^{64} -6.48528 q^{67} -5.17157 q^{68} +15.3137 q^{71} -8.48528 q^{73} +3.51472 q^{74} +8.82843 q^{76} -4.00000 q^{77} -2.48528 q^{79} +2.48528 q^{82} +7.17157 q^{83} +2.48528 q^{86} +1.31371 q^{88} +7.65685 q^{89} +9.65685 q^{91} +5.79899 q^{92} -4.82843 q^{94} +12.4853 q^{97} +6.75736 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} + 2 q^{4} + 4 q^{7} - 6 q^{8} + 4 q^{11} + 4 q^{13} + 4 q^{14} + 6 q^{16} - 4 q^{19} - 12 q^{22} - 12 q^{23} - 4 q^{26} - 12 q^{28} - 2 q^{29} - 4 q^{31} + 6 q^{32} + 8 q^{34} - 4 q^{38} + 12 q^{41}+ \cdots + 22 q^{98}+O(q^{100}) $ 2 * q - 2 * q^2 + 2 * q^4 + 4 * q^7 - 6 * q^8 + 4 * q^11 + 4 * q^13 + 4 * q^14 + 6 * q^16 - 4 * q^19 - 12 * q^22 - 12 * q^23 - 4 * q^26 - 12 * q^28 - 2 * q^29 - 4 * q^31 + 6 * q^32 + 8 * q^34 - 4 * q^38 + 12 * q^41 + 12 * q^43 + 20 * q^44 + 20 * q^46 - 12 * q^47 + 10 * q^49 + 4 * q^52 + 4 * q^53 - 4 * q^56 + 2 * q^58 + 4 * q^61 + 28 * q^62 - 14 * q^64 + 4 * q^67 - 16 * q^68 + 8 * q^71 + 24 * q^74 + 12 * q^76 - 8 * q^77 + 12 * q^79 - 12 * q^82 + 20 * q^83 - 12 * q^86 - 20 * q^88 + 4 * q^89 + 8 * q^91 - 28 * q^92 - 4 * q^94 + 8 * q^97 + 22 * q^98

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 $

$2$	0.414214	0.292893	0.146447	−	0.989219i	$-0.453216\pi$
$2$	0.414214	0.292893	0.146447	+	0.989219i	$0.453216\pi$
$3$	0	0
$3$	0	0
$4$	−1.82843	−0.914214
$5$	0	0
$5$	0	0
$6$	0	0
$7$	4.82843	1.82497	0.912487	−	0.409106i	$-0.134159\pi$
$7$	4.82843	1.82497	0.912487	+	0.409106i	$0.134159\pi$
$8$	−1.58579	−0.560660
$9$	0	0
$10$	0	0
$11$	−0.828427	−0.249780	−0.124890	−	0.992171i	$-0.539858\pi$
$11$	−0.828427	−0.249780	−0.124890	+	0.992171i	$0.539858\pi$
$12$	0	0
$13$	2.00000	0.554700	0.277350	−	0.960769i	$-0.410544\pi$
$13$	2.00000	0.554700	0.277350	+	0.960769i	$0.410544\pi$
$14$	2.00000	0.534522
$15$	0	0
$16$	3.00000	0.750000
$17$	2.82843	0.685994	0.342997	−	0.939336i	$-0.388558\pi$
$17$	2.82843	0.685994	0.342997	+	0.939336i	$0.388558\pi$
$18$	0	0
$19$	−4.82843	−1.10772	−0.553859	−	0.832611i	$-0.686845\pi$
$19$	−4.82843	−1.10772	−0.553859	+	0.832611i	$0.686845\pi$
$20$	0	0
$21$	0	0
$22$	−0.343146	−0.0731589
$23$	−3.17157	−0.661319	−0.330659	−	0.943750i	$-0.607271\pi$
$23$	−3.17157	−0.661319	−0.330659	+	0.943750i	$0.607271\pi$
$24$	0	0
$25$	0	0
$26$	0.828427	0.162468
$27$	0	0
$28$	−8.82843	−1.66842
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	0	0
$31$	6.48528	1.16479	0.582395	−	0.812906i	$-0.302116\pi$
$31$	6.48528	1.16479	0.582395	+	0.812906i	$0.302116\pi$
$32$	4.41421	0.780330
$33$	0	0
$34$	1.17157	0.200923
$35$	0	0
$36$	0	0
$37$	8.48528	1.39497	0.697486	−	0.716599i	$-0.254302\pi$
$37$	8.48528	1.39497	0.697486	+	0.716599i	$0.254302\pi$
$38$	−2.00000	−0.324443
$39$	0	0
$40$	0	0
$41$	6.00000	0.937043	0.468521	−	0.883452i	$-0.344787\pi$
$41$	6.00000	0.937043	0.468521	+	0.883452i	$0.344787\pi$
$42$	0	0
$43$	6.00000	0.914991	0.457496	−	0.889212i	$-0.348747\pi$
$43$	6.00000	0.914991	0.457496	+	0.889212i	$0.348747\pi$
$44$	1.51472	0.228352
$45$	0	0
$46$	−1.31371	−0.193696
$47$	−11.6569	−1.70033	−0.850163	−	0.526519i	$-0.823497\pi$
$47$	−11.6569	−1.70033	−0.850163	+	0.526519i	$0.823497\pi$
$48$	0	0
$49$	16.3137	2.33053
$50$	0	0
$51$	0	0
$52$	−3.65685	−0.507114
$53$	−3.65685	−0.502308	−0.251154	−	0.967947i	$-0.580810\pi$
$53$	−3.65685	−0.502308	−0.251154	+	0.967947i	$0.580810\pi$
$54$	0	0
$55$	0	0
$56$	−7.65685	−1.02319
$57$	0	0
$58$	−0.414214	−0.0543889
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	0	0
$61$	−3.65685	−0.468212	−0.234106	−	0.972211i	$-0.575216\pi$
$61$	−3.65685	−0.468212	−0.234106	+	0.972211i	$0.575216\pi$
$62$	2.68629	0.341159
$63$	0	0
$64$	−4.17157	−0.521447
$65$	0	0
$66$	0	0
$67$	−6.48528	−0.792303	−0.396152	−	0.918185i	$-0.629655\pi$
$67$	−6.48528	−0.792303	−0.396152	+	0.918185i	$0.629655\pi$
$68$	−5.17157	−0.627145
$69$	0	0
$70$	0	0
$71$	15.3137	1.81740	0.908701	−	0.417447i	$-0.137075\pi$
$71$	15.3137	1.81740	0.908701	+	0.417447i	$0.137075\pi$
$72$	0	0
$73$	−8.48528	−0.993127	−0.496564	−	0.868000i	$-0.665405\pi$
$73$	−8.48528	−0.993127	−0.496564	+	0.868000i	$0.665405\pi$
$74$	3.51472	0.408578
$75$	0	0
$76$	8.82843	1.01269
$77$	−4.00000	−0.455842
$78$	0	0
$79$	−2.48528	−0.279616	−0.139808	−	0.990179i	$-0.544649\pi$
$79$	−2.48528	−0.279616	−0.139808	+	0.990179i	$0.544649\pi$
$80$	0	0
$81$	0	0
$82$	2.48528	0.274453
$83$	7.17157	0.787182	0.393591	−	0.919286i	$-0.371233\pi$
$83$	7.17157	0.787182	0.393591	+	0.919286i	$0.371233\pi$
$84$	0	0
$85$	0	0
$86$	2.48528	0.267995
$87$	0	0
$88$	1.31371	0.140042
$89$	7.65685	0.811625	0.405812	−	0.913956i	$-0.366989\pi$
$89$	7.65685	0.811625	0.405812	+	0.913956i	$0.366989\pi$
$90$	0	0
$91$	9.65685	1.01231
$92$	5.79899	0.604586
$93$	0	0
$94$	−4.82843	−0.498014
$95$	0	0
$96$	0	0
$97$	12.4853	1.26769	0.633844	−	0.773461i	$-0.281476\pi$
$97$	12.4853	1.26769	0.633844	+	0.773461i	$0.281476\pi$
$98$	6.75736	0.682596
$99$	0	0
$100$	0	0
$101$	−15.6569	−1.55792	−0.778958	−	0.627077i	$-0.784251\pi$
$101$	−15.6569	−1.55792	−0.778958	+	0.627077i	$0.784251\pi$
$102$	0	0
$103$	−16.1421	−1.59053	−0.795266	−	0.606261i	$-0.792669\pi$
$103$	−16.1421	−1.59053	−0.795266	+	0.606261i	$0.792669\pi$
$104$	−3.17157	−0.310998
$105$	0	0
$106$	−1.51472	−0.147122
$107$	20.1421	1.94721	0.973607	−	0.228232i	$-0.0732943\pi$
$107$	20.1421	1.94721	0.973607	+	0.228232i	$0.0732943\pi$
$108$	0	0
$109$	2.00000	0.191565	0.0957826	−	0.995402i	$-0.469465\pi$
$109$	2.00000	0.191565	0.0957826	+	0.995402i	$0.469465\pi$
$110$	0	0
$111$	0	0
$112$	14.4853	1.36873
$113$	−2.82843	−0.266076	−0.133038	−	0.991111i	$-0.542473\pi$
$113$	−2.82843	−0.266076	−0.133038	+	0.991111i	$0.542473\pi$
$114$	0	0
$115$	0	0
$116$	1.82843	0.169765
$117$	0	0
$118$	0	0
$119$	13.6569	1.25192
$120$	0	0
$121$	−10.3137	−0.937610
$122$	−1.51472	−0.137136
$123$	0	0
$124$	−11.8579	−1.06487
$125$	0	0
$126$	0	0
$127$	−6.00000	−0.532414	−0.266207	−	0.963916i	$-0.585770\pi$
$127$	−6.00000	−0.532414	−0.266207	+	0.963916i	$0.585770\pi$
$128$	−10.5563	−0.933058
$129$	0	0
$130$	0	0
$131$	12.1421	1.06086	0.530432	−	0.847728i	$-0.322030\pi$
$131$	12.1421	1.06086	0.530432	+	0.847728i	$0.322030\pi$
$132$	0	0
$133$	−23.3137	−2.02155
$134$	−2.68629	−0.232060
$135$	0	0
$136$	−4.48528	−0.384610
$137$	−5.17157	−0.441837	−0.220919	−	0.975292i	$-0.570906\pi$
$137$	−5.17157	−0.441837	−0.220919	+	0.975292i	$0.570906\pi$
$138$	0	0
$139$	21.6569	1.83691	0.918455	−	0.395525i	$-0.129437\pi$
$139$	21.6569	1.83691	0.918455	+	0.395525i	$0.129437\pi$
$140$	0	0
$141$	0	0
$142$	6.34315	0.532305
$143$	−1.65685	−0.138553
$144$	0	0
$145$	0	0
$146$	−3.51472	−0.290880
$147$	0	0
$148$	−15.5147	−1.27530
$149$	−9.31371	−0.763009	−0.381504	−	0.924367i	$-0.624594\pi$
$149$	−9.31371	−0.763009	−0.381504	+	0.924367i	$0.624594\pi$
$150$	0	0
$151$	−12.0000	−0.976546	−0.488273	−	0.872691i	$-0.662373\pi$
$151$	−12.0000	−0.976546	−0.488273	+	0.872691i	$0.662373\pi$
$152$	7.65685	0.621053
$153$	0	0
$154$	−1.65685	−0.133513
$155$	0	0
$156$	0	0
$157$	−0.485281	−0.0387297	−0.0193648	−	0.999812i	$-0.506164\pi$
$157$	−0.485281	−0.0387297	−0.0193648	+	0.999812i	$0.506164\pi$
$158$	−1.02944	−0.0818976
$159$	0	0
$160$	0	0
$161$	−15.3137	−1.20689
$162$	0	0
$163$	8.34315	0.653486	0.326743	−	0.945113i	$-0.394049\pi$
$163$	8.34315	0.653486	0.326743	+	0.945113i	$0.394049\pi$
$164$	−10.9706	−0.856657
$165$	0	0
$166$	2.97056	0.230560
$167$	−2.48528	−0.192317	−0.0961584	−	0.995366i	$-0.530656\pi$
$167$	−2.48528	−0.192317	−0.0961584	+	0.995366i	$0.530656\pi$
$168$	0	0
$169$	−9.00000	−0.692308
$170$	0	0
$171$	0	0
$172$	−10.9706	−0.836498
$173$	17.3137	1.31634	0.658168	−	0.752871i	$-0.271331\pi$
$173$	17.3137	1.31634	0.658168	+	0.752871i	$0.271331\pi$
$174$	0	0
$175$	0	0
$176$	−2.48528	−0.187335
$177$	0	0
$178$	3.17157	0.237719
$179$	23.3137	1.74255	0.871274	−	0.490797i	$-0.163294\pi$
$179$	23.3137	1.74255	0.871274	+	0.490797i	$0.163294\pi$
$180$	0	0
$181$	−6.00000	−0.445976	−0.222988	−	0.974821i	$-0.571581\pi$
$181$	−6.00000	−0.445976	−0.222988	+	0.974821i	$0.571581\pi$
$182$	4.00000	0.296500
$183$	0	0
$184$	5.02944	0.370775
$185$	0	0
$186$	0	0
$187$	−2.34315	−0.171348
$188$	21.3137	1.55446
$189$	0	0
$190$	0	0
$191$	20.8284	1.50709	0.753546	−	0.657395i	$-0.228342\pi$
$191$	20.8284	1.50709	0.753546	+	0.657395i	$0.228342\pi$
$192$	0	0
$193$	−4.48528	−0.322858	−0.161429	−	0.986884i	$-0.551610\pi$
$193$	−4.48528	−0.322858	−0.161429	+	0.986884i	$0.551610\pi$
$194$	5.17157	0.371297
$195$	0	0
$196$	−29.8284	−2.13060
$197$	−19.6569	−1.40049	−0.700246	−	0.713901i	$-0.746927\pi$
$197$	−19.6569	−1.40049	−0.700246	+	0.713901i	$0.746927\pi$
$198$	0	0
$199$	12.0000	0.850657	0.425329	−	0.905039i	$-0.360158\pi$
$199$	12.0000	0.850657	0.425329	+	0.905039i	$0.360158\pi$
$200$	0	0
$201$	0	0
$202$	−6.48528	−0.456303
$203$	−4.82843	−0.338889
$204$	0	0
$205$	0	0
$206$	−6.68629	−0.465856
$207$	0	0
$208$	6.00000	0.416025
$209$	4.00000	0.276686
$210$	0	0
$211$	−0.828427	−0.0570313	−0.0285156	−	0.999593i	$-0.509078\pi$
$211$	−0.828427	−0.0570313	−0.0285156	+	0.999593i	$0.509078\pi$
$212$	6.68629	0.459216
$213$	0	0
$214$	8.34315	0.570326
$215$	0	0
$216$	0	0
$217$	31.3137	2.12571
$218$	0.828427	0.0561082
$219$	0	0
$220$	0	0
$221$	5.65685	0.380521
$222$	0	0
$223$	17.7990	1.19191	0.595954	−	0.803018i	$-0.296774\pi$
$223$	17.7990	1.19191	0.595954	+	0.803018i	$0.296774\pi$
$224$	21.3137	1.42408
$225$	0	0
$226$	−1.17157	−0.0779319
$227$	20.1421	1.33688	0.668440	−	0.743766i	$-0.266962\pi$
$227$	20.1421	1.33688	0.668440	+	0.743766i	$0.266962\pi$
$228$	0	0
$229$	−2.00000	−0.132164	−0.0660819	−	0.997814i	$-0.521050\pi$
$229$	−2.00000	−0.132164	−0.0660819	+	0.997814i	$0.521050\pi$
$230$	0	0
$231$	0	0
$232$	1.58579	0.104112
$233$	18.0000	1.17922	0.589610	−	0.807688i	$-0.299282\pi$
$233$	18.0000	1.17922	0.589610	+	0.807688i	$0.299282\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	5.65685	0.366679
$239$	0.686292	0.0443925	0.0221963	−	0.999754i	$-0.492934\pi$
$239$	0.686292	0.0443925	0.0221963	+	0.999754i	$0.492934\pi$
$240$	0	0
$241$	10.0000	0.644157	0.322078	−	0.946713i	$-0.395619\pi$
$241$	10.0000	0.644157	0.322078	+	0.946713i	$0.395619\pi$
$242$	−4.27208	−0.274620
$243$	0	0
$244$	6.68629	0.428046
$245$	0	0
$246$	0	0
$247$	−9.65685	−0.614451
$248$	−10.2843	−0.653052
$249$	0	0
$250$	0	0
$251$	−8.82843	−0.557245	−0.278623	−	0.960401i	$-0.589878\pi$
$251$	−8.82843	−0.557245	−0.278623	+	0.960401i	$0.589878\pi$
$252$	0	0
$253$	2.62742	0.165184
$254$	−2.48528	−0.155940
$255$	0	0
$256$	3.97056	0.248160
$257$	6.68629	0.417079	0.208540	−	0.978014i	$-0.433129\pi$
$257$	6.68629	0.417079	0.208540	+	0.978014i	$0.433129\pi$
$258$	0	0
$259$	40.9706	2.54579
$260$	0	0
$261$	0	0
$262$	5.02944	0.310720
$263$	−19.6569	−1.21209	−0.606047	−	0.795429i	$-0.707246\pi$
$263$	−19.6569	−1.21209	−0.606047	+	0.795429i	$0.707246\pi$
$264$	0	0
$265$	0	0
$266$	−9.65685	−0.592100
$267$	0	0
$268$	11.8579	0.724334
$269$	21.3137	1.29952	0.649760	−	0.760140i	$-0.274869\pi$
$269$	21.3137	1.29952	0.649760	+	0.760140i	$0.274869\pi$
$270$	0	0
$271$	−9.79899	−0.595246	−0.297623	−	0.954683i	$-0.596194\pi$
$271$	−9.79899	−0.595246	−0.297623	+	0.954683i	$0.596194\pi$
$272$	8.48528	0.514496
$273$	0	0
$274$	−2.14214	−0.129411
$275$	0	0
$276$	0	0
$277$	3.65685	0.219719	0.109860	−	0.993947i	$-0.464960\pi$
$277$	3.65685	0.219719	0.109860	+	0.993947i	$0.464960\pi$
$278$	8.97056	0.538019
$279$	0	0
$280$	0	0
$281$	29.3137	1.74871	0.874355	−	0.485288i	$-0.161285\pi$
$281$	29.3137	1.74871	0.874355	+	0.485288i	$0.161285\pi$
$282$	0	0
$283$	−4.82843	−0.287020	−0.143510	−	0.989649i	$-0.545839\pi$
$283$	−4.82843	−0.287020	−0.143510	+	0.989649i	$0.545839\pi$
$284$	−28.0000	−1.66149
$285$	0	0
$286$	−0.686292	−0.0405813
$287$	28.9706	1.71008
$288$	0	0
$289$	−9.00000	−0.529412
$290$	0	0
$291$	0	0
$292$	15.5147	0.907930
$293$	8.48528	0.495715	0.247858	−	0.968796i	$-0.420273\pi$
$293$	8.48528	0.495715	0.247858	+	0.968796i	$0.420273\pi$
$294$	0	0
$295$	0	0
$296$	−13.4558	−0.782105
$297$	0	0
$298$	−3.85786	−0.223480
$299$	−6.34315	−0.366834
$300$	0	0
$301$	28.9706	1.66984
$302$	−4.97056	−0.286024
$303$	0	0
$304$	−14.4853	−0.830788
$305$	0	0
$306$	0	0
$307$	22.9706	1.31100	0.655500	−	0.755195i	$-0.272458\pi$
$307$	22.9706	1.31100	0.655500	+	0.755195i	$0.272458\pi$
$308$	7.31371	0.416737
$309$	0	0
$310$	0	0
$311$	−14.4853	−0.821385	−0.410692	−	0.911774i	$-0.634713\pi$
$311$	−14.4853	−0.821385	−0.410692	+	0.911774i	$0.634713\pi$
$312$	0	0
$313$	6.00000	0.339140	0.169570	−	0.985518i	$-0.445762\pi$
$313$	6.00000	0.339140	0.169570	+	0.985518i	$0.445762\pi$
$314$	−0.201010	−0.0113437
$315$	0	0
$316$	4.54416	0.255629
$317$	2.82843	0.158860	0.0794301	−	0.996840i	$-0.474690\pi$
$317$	2.82843	0.158860	0.0794301	+	0.996840i	$0.474690\pi$
$318$	0	0
$319$	0.828427	0.0463830
$320$	0	0
$321$	0	0
$322$	−6.34315	−0.353490
$323$	−13.6569	−0.759888
$324$	0	0
$325$	0	0
$326$	3.45584	0.191402
$327$	0	0
$328$	−9.51472	−0.525362
$329$	−56.2843	−3.10305
$330$	0	0
$331$	21.7990	1.19818	0.599090	−	0.800681i	$-0.295529\pi$
$331$	21.7990	1.19818	0.599090	+	0.800681i	$0.295529\pi$
$332$	−13.1127	−0.719653
$333$	0	0
$334$	−1.02944	−0.0563283
$335$	0	0
$336$	0	0
$337$	1.17157	0.0638196	0.0319098	−	0.999491i	$-0.489841\pi$
$337$	1.17157	0.0638196	0.0319098	+	0.999491i	$0.489841\pi$
$338$	−3.72792	−0.202772
$339$	0	0
$340$	0	0
$341$	−5.37258	−0.290942
$342$	0	0
$343$	44.9706	2.42818
$344$	−9.51472	−0.512999
$345$	0	0
$346$	7.17157	0.385546
$347$	−8.14214	−0.437093	−0.218546	−	0.975827i	$-0.570131\pi$
$347$	−8.14214	−0.437093	−0.218546	+	0.975827i	$0.570131\pi$
$348$	0	0
$349$	20.6274	1.10416	0.552080	−	0.833791i	$-0.313834\pi$
$349$	20.6274	1.10416	0.552080	+	0.833791i	$0.313834\pi$
$350$	0	0
$351$	0	0
$352$	−3.65685	−0.194911
$353$	−4.34315	−0.231162	−0.115581	−	0.993298i	$-0.536873\pi$
$353$	−4.34315	−0.231162	−0.115581	+	0.993298i	$0.536873\pi$
$354$	0	0
$355$	0	0
$356$	−14.0000	−0.741999
$357$	0	0
$358$	9.65685	0.510381
$359$	3.85786	0.203610	0.101805	−	0.994804i	$-0.467538\pi$
$359$	3.85786	0.203610	0.101805	+	0.994804i	$0.467538\pi$
$360$	0	0
$361$	4.31371	0.227037
$362$	−2.48528	−0.130623
$363$	0	0
$364$	−17.6569	−0.925471
$365$	0	0
$366$	0	0
$367$	−18.0000	−0.939592	−0.469796	−	0.882775i	$-0.655673\pi$
$367$	−18.0000	−0.939592	−0.469796	+	0.882775i	$0.655673\pi$
$368$	−9.51472	−0.495989
$369$	0	0
$370$	0	0
$371$	−17.6569	−0.916698
$372$	0	0
$373$	6.97056	0.360922	0.180461	−	0.983582i	$-0.442241\pi$
$373$	6.97056	0.360922	0.180461	+	0.983582i	$0.442241\pi$
$374$	−0.970563	−0.0501866
$375$	0	0
$376$	18.4853	0.953306
$377$	−2.00000	−0.103005
$378$	0	0
$379$	22.4853	1.15499	0.577496	−	0.816394i	$-0.304030\pi$
$379$	22.4853	1.15499	0.577496	+	0.816394i	$0.304030\pi$
$380$	0	0
$381$	0	0
$382$	8.62742	0.441417
$383$	2.48528	0.126992	0.0634960	−	0.997982i	$-0.479775\pi$
$383$	2.48528	0.126992	0.0634960	+	0.997982i	$0.479775\pi$
$384$	0	0
$385$	0	0
$386$	−1.85786	−0.0945628
$387$	0	0
$388$	−22.8284	−1.15894
$389$	29.3137	1.48626	0.743132	−	0.669145i	$-0.233339\pi$
$389$	29.3137	1.48626	0.743132	+	0.669145i	$0.233339\pi$
$390$	0	0
$391$	−8.97056	−0.453661
$392$	−25.8701	−1.30664
$393$	0	0
$394$	−8.14214	−0.410195
$395$	0	0
$396$	0	0
$397$	19.6569	0.986549	0.493275	−	0.869874i	$-0.335800\pi$
$397$	19.6569	0.986549	0.493275	+	0.869874i	$0.335800\pi$
$398$	4.97056	0.249152
$399$	0	0
$400$	0	0
$401$	6.68629	0.333897	0.166949	−	0.985966i	$-0.446609\pi$
$401$	6.68629	0.333897	0.166949	+	0.985966i	$0.446609\pi$
$402$	0	0
$403$	12.9706	0.646110
$404$	28.6274	1.42427
$405$	0	0
$406$	−2.00000	−0.0992583
$407$	−7.02944	−0.348436
$408$	0	0
$409$	−2.97056	−0.146885	−0.0734424	−	0.997299i	$-0.523399\pi$
$409$	−2.97056	−0.146885	−0.0734424	+	0.997299i	$0.523399\pi$
$410$	0	0
$411$	0	0
$412$	29.5147	1.45409
$413$	0	0
$414$	0	0
$415$	0	0
$416$	8.82843	0.432849
$417$	0	0
$418$	1.65685	0.0810394
$419$	−28.9706	−1.41530	−0.707652	−	0.706561i	$-0.750246\pi$
$419$	−28.9706	−1.41530	−0.707652	+	0.706561i	$0.750246\pi$
$420$	0	0
$421$	18.9706	0.924569	0.462284	−	0.886732i	$-0.347030\pi$
$421$	18.9706	0.924569	0.462284	+	0.886732i	$0.347030\pi$
$422$	−0.343146	−0.0167041
$423$	0	0
$424$	5.79899	0.281624
$425$	0	0
$426$	0	0
$427$	−17.6569	−0.854475
$428$	−36.8284	−1.78017
$429$	0	0
$430$	0	0
$431$	−3.31371	−0.159616	−0.0798079	−	0.996810i	$-0.525431\pi$
$431$	−3.31371	−0.159616	−0.0798079	+	0.996810i	$0.525431\pi$
$432$	0	0
$433$	29.1716	1.40190	0.700948	−	0.713212i	$-0.252760\pi$
$433$	29.1716	1.40190	0.700948	+	0.713212i	$0.252760\pi$
$434$	12.9706	0.622607
$435$	0	0
$436$	−3.65685	−0.175132
$437$	15.3137	0.732554
$438$	0	0
$439$	−10.3431	−0.493651	−0.246826	−	0.969060i	$-0.579388\pi$
$439$	−10.3431	−0.493651	−0.246826	+	0.969060i	$0.579388\pi$
$440$	0	0
$441$	0	0
$442$	2.34315	0.111452
$443$	−7.65685	−0.363788	−0.181894	−	0.983318i	$-0.558223\pi$
$443$	−7.65685	−0.363788	−0.181894	+	0.983318i	$0.558223\pi$
$444$	0	0
$445$	0	0
$446$	7.37258	0.349102
$447$	0	0
$448$	−20.1421	−0.951626
$449$	−11.6569	−0.550121	−0.275060	−	0.961427i	$-0.588698\pi$
$449$	−11.6569	−0.550121	−0.275060	+	0.961427i	$0.588698\pi$
$450$	0	0
$451$	−4.97056	−0.234055
$452$	5.17157	0.243250
$453$	0	0
$454$	8.34315	0.391563
$455$	0	0
$456$	0	0
$457$	−19.6569	−0.919509	−0.459754	−	0.888046i	$-0.652063\pi$
$457$	−19.6569	−0.919509	−0.459754	+	0.888046i	$0.652063\pi$
$458$	−0.828427	−0.0387099
$459$	0	0
$460$	0	0
$461$	35.6569	1.66071	0.830353	−	0.557238i	$-0.188139\pi$
$461$	35.6569	1.66071	0.830353	+	0.557238i	$0.188139\pi$
$462$	0	0
$463$	−21.7990	−1.01308	−0.506542	−	0.862215i	$-0.669077\pi$
$463$	−21.7990	−1.01308	−0.506542	+	0.862215i	$0.669077\pi$
$464$	−3.00000	−0.139272
$465$	0	0
$466$	7.45584	0.345385
$467$	10.9706	0.507657	0.253829	−	0.967249i	$-0.418310\pi$
$467$	10.9706	0.507657	0.253829	+	0.967249i	$0.418310\pi$
$468$	0	0
$469$	−31.3137	−1.44593
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−4.97056	−0.228547
$474$	0	0
$475$	0	0
$476$	−24.9706	−1.14452
$477$	0	0
$478$	0.284271	0.0130023
$479$	−7.17157	−0.327678	−0.163839	−	0.986487i	$-0.552388\pi$
$479$	−7.17157	−0.327678	−0.163839	+	0.986487i	$0.552388\pi$
$480$	0	0
$481$	16.9706	0.773791
$482$	4.14214	0.188669
$483$	0	0
$484$	18.8579	0.857176
$485$	0	0
$486$	0	0
$487$	−9.79899	−0.444035	−0.222017	−	0.975043i	$-0.571264\pi$
$487$	−9.79899	−0.444035	−0.222017	+	0.975043i	$0.571264\pi$
$488$	5.79899	0.262508
$489$	0	0
$490$	0	0
$491$	7.45584	0.336478	0.168239	−	0.985746i	$-0.446192\pi$
$491$	7.45584	0.336478	0.168239	+	0.985746i	$0.446192\pi$
$492$	0	0
$493$	−2.82843	−0.127386
$494$	−4.00000	−0.179969
$495$	0	0
$496$	19.4558	0.873593
$497$	73.9411	3.31671
$498$	0	0
$499$	36.0000	1.61158	0.805791	−	0.592200i	$-0.201741\pi$
$499$	36.0000	1.61158	0.805791	+	0.592200i	$0.201741\pi$
$500$	0	0
$501$	0	0
$502$	−3.65685	−0.163213
$503$	−30.0000	−1.33763	−0.668817	−	0.743427i	$-0.733199\pi$
$503$	−30.0000	−1.33763	−0.668817	+	0.743427i	$0.733199\pi$
$504$	0	0
$505$	0	0
$506$	1.08831	0.0483814
$507$	0	0
$508$	10.9706	0.486740
$509$	−0.627417	−0.0278098	−0.0139049	−	0.999903i	$-0.504426\pi$
$509$	−0.627417	−0.0278098	−0.0139049	+	0.999903i	$0.504426\pi$
$510$	0	0
$511$	−40.9706	−1.81243
$512$	22.7574	1.00574
$513$	0	0
$514$	2.76955	0.122160
$515$	0	0
$516$	0	0
$517$	9.65685	0.424708
$518$	16.9706	0.745644
$519$	0	0
$520$	0	0
$521$	21.3137	0.933771	0.466885	−	0.884318i	$-0.345376\pi$
$521$	21.3137	0.933771	0.466885	+	0.884318i	$0.345376\pi$
$522$	0	0
$523$	−2.48528	−0.108674	−0.0543369	−	0.998523i	$-0.517304\pi$
$523$	−2.48528	−0.108674	−0.0543369	+	0.998523i	$0.517304\pi$
$524$	−22.2010	−0.969856
$525$	0	0
$526$	−8.14214	−0.355014
$527$	18.3431	0.799040
$528$	0	0
$529$	−12.9411	−0.562658
$530$	0	0
$531$	0	0
$532$	42.6274	1.84813
$533$	12.0000	0.519778
$534$	0	0
$535$	0	0
$536$	10.2843	0.444213
$537$	0	0
$538$	8.82843	0.380621
$539$	−13.5147	−0.582120
$540$	0	0
$541$	−5.02944	−0.216232	−0.108116	−	0.994138i	$-0.534482\pi$
$541$	−5.02944	−0.216232	−0.108116	+	0.994138i	$0.534482\pi$
$542$	−4.05887	−0.174344
$543$	0	0
$544$	12.4853	0.535302
$545$	0	0
$546$	0	0
$547$	2.48528	0.106263	0.0531315	−	0.998588i	$-0.483080\pi$
$547$	2.48528	0.106263	0.0531315	+	0.998588i	$0.483080\pi$
$548$	9.45584	0.403934
$549$	0	0
$550$	0	0
$551$	4.82843	0.205698
$552$	0	0
$553$	−12.0000	−0.510292
$554$	1.51472	0.0643542
$555$	0	0
$556$	−39.5980	−1.67933
$557$	−27.9411	−1.18390	−0.591952	−	0.805973i	$-0.701642\pi$
$557$	−27.9411	−1.18390	−0.591952	+	0.805973i	$0.701642\pi$
$558$	0	0
$559$	12.0000	0.507546
$560$	0	0
$561$	0	0
$562$	12.1421	0.512185
$563$	7.65685	0.322698	0.161349	−	0.986897i	$-0.448416\pi$
$563$	7.65685	0.322698	0.161349	+	0.986897i	$0.448416\pi$
$564$	0	0
$565$	0	0
$566$	−2.00000	−0.0840663
$567$	0	0
$568$	−24.2843	−1.01895
$569$	−27.6569	−1.15944	−0.579718	−	0.814817i	$-0.696837\pi$
$569$	−27.6569	−1.15944	−0.579718	+	0.814817i	$0.696837\pi$
$570$	0	0
$571$	28.0000	1.17176	0.585882	−	0.810397i	$-0.300748\pi$
$571$	28.0000	1.17176	0.585882	+	0.810397i	$0.300748\pi$
$572$	3.02944	0.126667
$573$	0	0
$574$	12.0000	0.500870
$575$	0	0
$576$	0	0
$577$	−23.7990	−0.990765	−0.495382	−	0.868675i	$-0.664972\pi$
$577$	−23.7990	−0.990765	−0.495382	+	0.868675i	$0.664972\pi$
$578$	−3.72792	−0.155061
$579$	0	0
$580$	0	0
$581$	34.6274	1.43659
$582$	0	0
$583$	3.02944	0.125466
$584$	13.4558	0.556807
$585$	0	0
$586$	3.51472	0.145192
$587$	−29.7990	−1.22994	−0.614968	−	0.788552i	$-0.710831\pi$
$587$	−29.7990	−1.22994	−0.614968	+	0.788552i	$0.710831\pi$
$588$	0	0
$589$	−31.3137	−1.29026
$590$	0	0
$591$	0	0
$592$	25.4558	1.04623
$593$	−7.65685	−0.314429	−0.157215	−	0.987564i	$-0.550251\pi$
$593$	−7.65685	−0.314429	−0.157215	+	0.987564i	$0.550251\pi$
$594$	0	0
$595$	0	0
$596$	17.0294	0.697553
$597$	0	0
$598$	−2.62742	−0.107443
$599$	37.7990	1.54442	0.772212	−	0.635364i	$-0.219150\pi$
$599$	37.7990	1.54442	0.772212	+	0.635364i	$0.219150\pi$
$600$	0	0
$601$	2.00000	0.0815817	0.0407909	−	0.999168i	$-0.487012\pi$
$601$	2.00000	0.0815817	0.0407909	+	0.999168i	$0.487012\pi$
$602$	12.0000	0.489083
$603$	0	0
$604$	21.9411	0.892772
$605$	0	0
$606$	0	0
$607$	−9.02944	−0.366494	−0.183247	−	0.983067i	$-0.558661\pi$
$607$	−9.02944	−0.366494	−0.183247	+	0.983067i	$0.558661\pi$
$608$	−21.3137	−0.864385
$609$	0	0
$610$	0	0
$611$	−23.3137	−0.943172
$612$	0	0
$613$	2.00000	0.0807792	0.0403896	−	0.999184i	$-0.487140\pi$
$613$	2.00000	0.0807792	0.0403896	+	0.999184i	$0.487140\pi$
$614$	9.51472	0.383983
$615$	0	0
$616$	6.34315	0.255573
$617$	−9.17157	−0.369234	−0.184617	−	0.982811i	$-0.559104\pi$
$617$	−9.17157	−0.369234	−0.184617	+	0.982811i	$0.559104\pi$
$618$	0	0
$619$	−9.79899	−0.393855	−0.196927	−	0.980418i	$-0.563096\pi$
$619$	−9.79899	−0.393855	−0.196927	+	0.980418i	$0.563096\pi$
$620$	0	0
$621$	0	0
$622$	−6.00000	−0.240578
$623$	36.9706	1.48119
$624$	0	0
$625$	0	0
$626$	2.48528	0.0993318
$627$	0	0
$628$	0.887302	0.0354072
$629$	24.0000	0.956943
$630$	0	0
$631$	36.9706	1.47177	0.735887	−	0.677104i	$-0.236765\pi$
$631$	36.9706	1.47177	0.735887	+	0.677104i	$0.236765\pi$
$632$	3.94113	0.156770
$633$	0	0
$634$	1.17157	0.0465291
$635$	0	0
$636$	0	0
$637$	32.6274	1.29275
$638$	0.343146	0.0135853
$639$	0	0
$640$	0	0
$641$	−0.627417	−0.0247815	−0.0123907	−	0.999923i	$-0.503944\pi$
$641$	−0.627417	−0.0247815	−0.0123907	+	0.999923i	$0.503944\pi$
$642$	0	0
$643$	−19.4558	−0.767264	−0.383632	−	0.923486i	$-0.625327\pi$
$643$	−19.4558	−0.767264	−0.383632	+	0.923486i	$0.625327\pi$
$644$	28.0000	1.10335
$645$	0	0
$646$	−5.65685	−0.222566
$647$	−41.1127	−1.61631	−0.808153	−	0.588972i	$-0.799533\pi$
$647$	−41.1127	−1.61631	−0.808153	+	0.588972i	$0.799533\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	0	0
$652$	−15.2548	−0.597425
$653$	−17.1716	−0.671976	−0.335988	−	0.941866i	$-0.609070\pi$
$653$	−17.1716	−0.671976	−0.335988	+	0.941866i	$0.609070\pi$
$654$	0	0
$655$	0	0
$656$	18.0000	0.702782
$657$	0	0
$658$	−23.3137	−0.908863
$659$	−1.79899	−0.0700787	−0.0350393	−	0.999386i	$-0.511156\pi$
$659$	−1.79899	−0.0700787	−0.0350393	+	0.999386i	$0.511156\pi$
$660$	0	0
$661$	26.0000	1.01128	0.505641	−	0.862744i	$-0.331256\pi$
$661$	26.0000	1.01128	0.505641	+	0.862744i	$0.331256\pi$
$662$	9.02944	0.350939
$663$	0	0
$664$	−11.3726	−0.441342
$665$	0	0
$666$	0	0
$667$	3.17157	0.122804
$668$	4.54416	0.175819
$669$	0	0
$670$	0	0
$671$	3.02944	0.116950
$672$	0	0
$673$	−22.9706	−0.885450	−0.442725	−	0.896657i	$-0.645988\pi$
$673$	−22.9706	−0.885450	−0.442725	+	0.896657i	$0.645988\pi$
$674$	0.485281	0.0186923
$675$	0	0
$676$	16.4558	0.632917
$677$	−36.7696	−1.41317	−0.706584	−	0.707629i	$-0.749765\pi$
$677$	−36.7696	−1.41317	−0.706584	+	0.707629i	$0.749765\pi$
$678$	0	0
$679$	60.2843	2.31350
$680$	0	0
$681$	0	0
$682$	−2.22540	−0.0852148
$683$	11.8579	0.453729	0.226864	−	0.973926i	$-0.427153\pi$
$683$	11.8579	0.453729	0.226864	+	0.973926i	$0.427153\pi$
$684$	0	0
$685$	0	0
$686$	18.6274	0.711198
$687$	0	0
$688$	18.0000	0.686244
$689$	−7.31371	−0.278630
$690$	0	0
$691$	−44.9706	−1.71076	−0.855380	−	0.518000i	$-0.826677\pi$
$691$	−44.9706	−1.71076	−0.855380	+	0.518000i	$0.826677\pi$
$692$	−31.6569	−1.20341
$693$	0	0
$694$	−3.37258	−0.128022
$695$	0	0
$696$	0	0
$697$	16.9706	0.642806
$698$	8.54416	0.323401
$699$	0	0
$700$	0	0
$701$	−6.68629	−0.252538	−0.126269	−	0.991996i	$-0.540300\pi$
$701$	−6.68629	−0.252538	−0.126269	+	0.991996i	$0.540300\pi$
$702$	0	0
$703$	−40.9706	−1.54523
$704$	3.45584	0.130247
$705$	0	0
$706$	−1.79899	−0.0677059
$707$	−75.5980	−2.84315
$708$	0	0
$709$	−22.0000	−0.826227	−0.413114	−	0.910679i	$-0.635559\pi$
$709$	−22.0000	−0.826227	−0.413114	+	0.910679i	$0.635559\pi$
$710$	0	0
$711$	0	0
$712$	−12.1421	−0.455046
$713$	−20.5685	−0.770298
$714$	0	0
$715$	0	0
$716$	−42.6274	−1.59306
$717$	0	0
$718$	1.59798	0.0596361
$719$	34.6274	1.29138	0.645692	−	0.763598i	$-0.276569\pi$
$719$	34.6274	1.29138	0.645692	+	0.763598i	$0.276569\pi$
$720$	0	0
$721$	−77.9411	−2.90268
$722$	1.78680	0.0664977
$723$	0	0
$724$	10.9706	0.407718
$725$	0	0
$726$	0	0
$727$	23.9411	0.887927	0.443964	−	0.896045i	$-0.353572\pi$
$727$	23.9411	0.887927	0.443964	+	0.896045i	$0.353572\pi$
$728$	−15.3137	−0.567564
$729$	0	0
$730$	0	0
$731$	16.9706	0.627679
$732$	0	0
$733$	22.8284	0.843187	0.421594	−	0.906785i	$-0.361471\pi$
$733$	22.8284	0.843187	0.421594	+	0.906785i	$0.361471\pi$
$734$	−7.45584	−0.275200
$735$	0	0
$736$	−14.0000	−0.516047
$737$	5.37258	0.197902
$738$	0	0
$739$	14.4853	0.532850	0.266425	−	0.963856i	$-0.414158\pi$
$739$	14.4853	0.532850	0.266425	+	0.963856i	$0.414158\pi$
$740$	0	0
$741$	0	0
$742$	−7.31371	−0.268495
$743$	−52.6274	−1.93071	−0.965356	−	0.260935i	$-0.915969\pi$
$743$	−52.6274	−1.93071	−0.965356	+	0.260935i	$0.915969\pi$
$744$	0	0
$745$	0	0
$746$	2.88730	0.105712
$747$	0	0
$748$	4.28427	0.156648
$749$	97.2548	3.55361
$750$	0	0
$751$	16.1421	0.589035	0.294517	−	0.955646i	$-0.404841\pi$
$751$	16.1421	0.589035	0.294517	+	0.955646i	$0.404841\pi$
$752$	−34.9706	−1.27525
$753$	0	0
$754$	−0.828427	−0.0301695
$755$	0	0
$756$	0	0
$757$	−19.5147	−0.709275	−0.354637	−	0.935004i	$-0.615396\pi$
$757$	−19.5147	−0.709275	−0.354637	+	0.935004i	$0.615396\pi$
$758$	9.31371	0.338289
$759$	0	0
$760$	0	0
$761$	−8.62742	−0.312744	−0.156372	−	0.987698i	$-0.549980\pi$
$761$	−8.62742	−0.312744	−0.156372	+	0.987698i	$0.549980\pi$
$762$	0	0
$763$	9.65685	0.349602
$764$	−38.0833	−1.37780
$765$	0	0
$766$	1.02944	0.0371951
$767$	0	0
$768$	0	0
$769$	−15.6569	−0.564601	−0.282300	−	0.959326i	$-0.591097\pi$
$769$	−15.6569	−0.564601	−0.282300	+	0.959326i	$0.591097\pi$
$770$	0	0
$771$	0	0
$772$	8.20101	0.295161
$773$	−8.48528	−0.305194	−0.152597	−	0.988288i	$-0.548764\pi$
$773$	−8.48528	−0.305194	−0.152597	+	0.988288i	$0.548764\pi$
$774$	0	0
$775$	0	0
$776$	−19.7990	−0.710742
$777$	0	0
$778$	12.1421	0.435317
$779$	−28.9706	−1.03798
$780$	0	0
$781$	−12.6863	−0.453951
$782$	−3.71573	−0.132874
$783$	0	0
$784$	48.9411	1.74790
$785$	0	0
$786$	0	0
$787$	−17.7990	−0.634465	−0.317233	−	0.948348i	$-0.602754\pi$
$787$	−17.7990	−0.634465	−0.317233	+	0.948348i	$0.602754\pi$
$788$	35.9411	1.28035
$789$	0	0
$790$	0	0
$791$	−13.6569	−0.485582
$792$	0	0
$793$	−7.31371	−0.259717
$794$	8.14214	0.288954
$795$	0	0
$796$	−21.9411	−0.777683
$797$	−5.85786	−0.207496	−0.103748	−	0.994604i	$-0.533084\pi$
$797$	−5.85786	−0.207496	−0.103748	+	0.994604i	$0.533084\pi$
$798$	0	0
$799$	−32.9706	−1.16641
$800$	0	0
$801$	0	0
$802$	2.76955	0.0977963
$803$	7.02944	0.248063
$804$	0	0
$805$	0	0
$806$	5.37258	0.189241
$807$	0	0
$808$	24.8284	0.873461
$809$	−42.2843	−1.48664	−0.743318	−	0.668938i	$-0.766749\pi$
$809$	−42.2843	−1.48664	−0.743318	+	0.668938i	$0.766749\pi$
$810$	0	0
$811$	37.6569	1.32231	0.661155	−	0.750249i	$-0.270066\pi$
$811$	37.6569	1.32231	0.661155	+	0.750249i	$0.270066\pi$
$812$	8.82843	0.309817
$813$	0	0
$814$	−2.91169	−0.102055
$815$	0	0
$816$	0	0
$817$	−28.9706	−1.01355
$818$	−1.23045	−0.0430216
$819$	0	0
$820$	0	0
$821$	22.6863	0.791757	0.395879	−	0.918303i	$-0.370440\pi$
$821$	22.6863	0.791757	0.395879	+	0.918303i	$0.370440\pi$
$822$	0	0
$823$	30.9706	1.07957	0.539783	−	0.841804i	$-0.318506\pi$
$823$	30.9706	1.07957	0.539783	+	0.841804i	$0.318506\pi$
$824$	25.5980	0.891748
$825$	0	0
$826$	0	0
$827$	17.3137	0.602057	0.301028	−	0.953615i	$-0.402670\pi$
$827$	17.3137	0.602057	0.301028	+	0.953615i	$0.402670\pi$
$828$	0	0
$829$	20.6274	0.716420	0.358210	−	0.933641i	$-0.383387\pi$
$829$	20.6274	0.716420	0.358210	+	0.933641i	$0.383387\pi$
$830$	0	0
$831$	0	0
$832$	−8.34315	−0.289247
$833$	46.1421	1.59873
$834$	0	0
$835$	0	0
$836$	−7.31371	−0.252950
$837$	0	0
$838$	−12.0000	−0.414533
$839$	−2.48528	−0.0858014	−0.0429007	−	0.999079i	$-0.513660\pi$
$839$	−2.48528	−0.0858014	−0.0429007	+	0.999079i	$0.513660\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	7.85786	0.270800
$843$	0	0
$844$	1.51472	0.0521388
$845$	0	0
$846$	0	0
$847$	−49.7990	−1.71111
$848$	−10.9706	−0.376731
$849$	0	0
$850$	0	0
$851$	−26.9117	−0.922521
$852$	0	0
$853$	−51.1127	−1.75007	−0.875033	−	0.484064i	$-0.839160\pi$
$853$	−51.1127	−1.75007	−0.875033	+	0.484064i	$0.839160\pi$
$854$	−7.31371	−0.250270
$855$	0	0
$856$	−31.9411	−1.09173
$857$	3.37258	0.115205	0.0576026	−	0.998340i	$-0.481654\pi$
$857$	3.37258	0.115205	0.0576026	+	0.998340i	$0.481654\pi$
$858$	0	0
$859$	−56.4264	−1.92524	−0.962622	−	0.270848i	$-0.912696\pi$
$859$	−56.4264	−1.92524	−0.962622	+	0.270848i	$0.912696\pi$
$860$	0	0
$861$	0	0
$862$	−1.37258	−0.0467504
$863$	−36.1421	−1.23029	−0.615146	−	0.788413i	$-0.710903\pi$
$863$	−36.1421	−1.23029	−0.615146	+	0.788413i	$0.710903\pi$
$864$	0	0
$865$	0	0
$866$	12.0833	0.410606
$867$	0	0
$868$	−57.2548	−1.94336
$869$	2.05887	0.0698425
$870$	0	0
$871$	−12.9706	−0.439491
$872$	−3.17157	−0.107403
$873$	0	0
$874$	6.34315	0.214560
$875$	0	0
$876$	0	0
$877$	−38.2843	−1.29277	−0.646384	−	0.763012i	$-0.723720\pi$
$877$	−38.2843	−1.29277	−0.646384	+	0.763012i	$0.723720\pi$
$878$	−4.28427	−0.144587
$879$	0	0
$880$	0	0
$881$	−29.3137	−0.987604	−0.493802	−	0.869574i	$-0.664393\pi$
$881$	−29.3137	−0.987604	−0.493802	+	0.869574i	$0.664393\pi$
$882$	0	0
$883$	14.4853	0.487469	0.243734	−	0.969842i	$-0.421628\pi$
$883$	14.4853	0.487469	0.243734	+	0.969842i	$0.421628\pi$
$884$	−10.3431	−0.347878
$885$	0	0
$886$	−3.17157	−0.106551
$887$	−6.68629	−0.224504	−0.112252	−	0.993680i	$-0.535806\pi$
$887$	−6.68629	−0.224504	−0.112252	+	0.993680i	$0.535806\pi$
$888$	0	0
$889$	−28.9706	−0.971641
$890$	0	0
$891$	0	0
$892$	−32.5442	−1.08966
$893$	56.2843	1.88348
$894$	0	0
$895$	0	0
$896$	−50.9706	−1.70281
$897$	0	0
$898$	−4.82843	−0.161127
$899$	−6.48528	−0.216296
$900$	0	0
$901$	−10.3431	−0.344580
$902$	−2.05887	−0.0685530
$903$	0	0
$904$	4.48528	0.149178
$905$	0	0
$906$	0	0
$907$	10.0000	0.332045	0.166022	−	0.986122i	$-0.446908\pi$
$907$	10.0000	0.332045	0.166022	+	0.986122i	$0.446908\pi$
$908$	−36.8284	−1.22219
$909$	0	0
$910$	0	0
$911$	−32.1421	−1.06492	−0.532458	−	0.846456i	$-0.678732\pi$
$911$	−32.1421	−1.06492	−0.532458	+	0.846456i	$0.678732\pi$
$912$	0	0
$913$	−5.94113	−0.196623
$914$	−8.14214	−0.269318
$915$	0	0
$916$	3.65685	0.120826
$917$	58.6274	1.93605
$918$	0	0
$919$	−36.0000	−1.18753	−0.593765	−	0.804638i	$-0.702359\pi$
$919$	−36.0000	−1.18753	−0.593765	+	0.804638i	$0.702359\pi$
$920$	0	0
$921$	0	0
$922$	14.7696	0.486409
$923$	30.6274	1.00811
$924$	0	0
$925$	0	0
$926$	−9.02944	−0.296726
$927$	0	0
$928$	−4.41421	−0.144904
$929$	4.62742	0.151821	0.0759103	−	0.997115i	$-0.475814\pi$
$929$	4.62742	0.151821	0.0759103	+	0.997115i	$0.475814\pi$
$930$	0	0
$931$	−78.7696	−2.58157
$932$	−32.9117	−1.07806
$933$	0	0
$934$	4.54416	0.148689
$935$	0	0
$936$	0	0
$937$	−19.6569	−0.642161	−0.321081	−	0.947052i	$-0.604046\pi$
$937$	−19.6569	−0.642161	−0.321081	+	0.947052i	$0.604046\pi$
$938$	−12.9706	−0.423504
$939$	0	0
$940$	0	0
$941$	27.9411	0.910855	0.455427	−	0.890273i	$-0.349486\pi$
$941$	27.9411	0.910855	0.455427	+	0.890273i	$0.349486\pi$
$942$	0	0
$943$	−19.0294	−0.619684
$944$	0	0
$945$	0	0
$946$	−2.05887	−0.0669398
$947$	44.9117	1.45943	0.729717	−	0.683749i	$-0.239652\pi$
$947$	44.9117	1.45943	0.729717	+	0.683749i	$0.239652\pi$
$948$	0	0
$949$	−16.9706	−0.550888
$950$	0	0
$951$	0	0
$952$	−21.6569	−0.701903
$953$	29.3137	0.949564	0.474782	−	0.880103i	$-0.342527\pi$
$953$	29.3137	0.949564	0.474782	+	0.880103i	$0.342527\pi$
$954$	0	0
$955$	0	0
$956$	−1.25483	−0.0405842
$957$	0	0
$958$	−2.97056	−0.0959745
$959$	−24.9706	−0.806342
$960$	0	0
$961$	11.0589	0.356738
$962$	7.02944	0.226638
$963$	0	0
$964$	−18.2843	−0.588897
$965$	0	0
$966$	0	0
$967$	−14.9706	−0.481421	−0.240710	−	0.970597i	$-0.577380\pi$
$967$	−14.9706	−0.481421	−0.240710	+	0.970597i	$0.577380\pi$
$968$	16.3553	0.525681
$969$	0	0
$970$	0	0
$971$	−28.1421	−0.903124	−0.451562	−	0.892240i	$-0.649133\pi$
$971$	−28.1421	−0.903124	−0.451562	+	0.892240i	$0.649133\pi$
$972$	0	0
$973$	104.569	3.35231
$974$	−4.05887	−0.130055
$975$	0	0
$976$	−10.9706	−0.351159
$977$	−2.68629	−0.0859421	−0.0429710	−	0.999076i	$-0.513682\pi$
$977$	−2.68629	−0.0859421	−0.0429710	+	0.999076i	$0.513682\pi$
$978$	0	0
$979$	−6.34315	−0.202728
$980$	0	0
$981$	0	0
$982$	3.08831	0.0985520
$983$	−9.31371	−0.297061	−0.148531	−	0.988908i	$-0.547454\pi$
$983$	−9.31371	−0.297061	−0.148531	+	0.988908i	$0.547454\pi$
$984$	0	0
$985$	0	0
$986$	−1.17157	−0.0373105
$987$	0	0
$988$	17.6569	0.561739
$989$	−19.0294	−0.605101
$990$	0	0
$991$	52.0000	1.65183	0.825917	−	0.563791i	$-0.190658\pi$
$991$	52.0000	1.65183	0.825917	+	0.563791i	$0.190658\pi$
$992$	28.6274	0.908921
$993$	0	0
$994$	30.6274	0.971443
$995$	0	0
$996$	0	0
$997$	−6.82843	−0.216258	−0.108129	−	0.994137i	$-0.534486\pi$
$997$	−6.82843	−0.216258	−0.108129	+	0.994137i	$0.534486\pi$
$998$	14.9117	0.472021
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6525.2.a.p.1.2		2
3.2	odd	2		725.2.a.c.1.1		2
5.4	even	2		1305.2.a.n.1.1		2
15.2	even	4		725.2.b.c.349.2		4
15.8	even	4		725.2.b.c.349.3		4
15.14	odd	2		145.2.a.b.1.2	✓	2
60.59	even	2		2320.2.a.k.1.2		2
105.104	even	2		7105.2.a.e.1.2		2
120.29	odd	2		9280.2.a.be.1.1		2
120.59	even	2		9280.2.a.w.1.2		2
435.434	odd	2		4205.2.a.d.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
145.2.a.b.1.2	✓	2	15.14	odd	2
725.2.a.c.1.1		2	3.2	odd	2
725.2.b.c.349.2		4	15.2	even	4
725.2.b.c.349.3		4	15.8	even	4
1305.2.a.n.1.1		2	5.4	even	2
2320.2.a.k.1.2		2	60.59	even	2
4205.2.a.d.1.1		2	435.434	odd	2
6525.2.a.p.1.2		2	1.1	even	1	trivial
7105.2.a.e.1.2		2	105.104	even	2
9280.2.a.w.1.2		2	120.59	even	2
9280.2.a.be.1.1		2	120.29	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:	\( N \)	\(=\)	\( 6525 = 3^{2} \cdot 5^{2} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6525.a (trivial)

\(f(q)\)	\(=\)	\(q+0.414214 q^{2} -1.82843 q^{4} +4.82843 q^{7} -1.58579 q^{8} -0.828427 q^{11} +2.00000 q^{13} +2.00000 q^{14} +3.00000 q^{16} +2.82843 q^{17} -4.82843 q^{19} -0.343146 q^{22} -3.17157 q^{23} +0.828427 q^{26} -8.82843 q^{28} -1.00000 q^{29} +6.48528 q^{31} +4.41421 q^{32} +1.17157 q^{34} +8.48528 q^{37} -2.00000 q^{38} +6.00000 q^{41} +6.00000 q^{43} +1.51472 q^{44} -1.31371 q^{46} -11.6569 q^{47} +16.3137 q^{49} -3.65685 q^{52} -3.65685 q^{53} -7.65685 q^{56} -0.414214 q^{58} -3.65685 q^{61} +2.68629 q^{62} -4.17157 q^{64} -6.48528 q^{67} -5.17157 q^{68} +15.3137 q^{71} -8.48528 q^{73} +3.51472 q^{74} +8.82843 q^{76} -4.00000 q^{77} -2.48528 q^{79} +2.48528 q^{82} +7.17157 q^{83} +2.48528 q^{86} +1.31371 q^{88} +7.65685 q^{89} +9.65685 q^{91} +5.79899 q^{92} -4.82843 q^{94} +12.4853 q^{97} +6.75736 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 2 q^{4} + 4 q^{7} - 6 q^{8} + 4 q^{11} + 4 q^{13} + 4 q^{14} + 6 q^{16} - 4 q^{19} - 12 q^{22} - 12 q^{23} - 4 q^{26} - 12 q^{28} - 2 q^{29} - 4 q^{31} + 6 q^{32} + 8 q^{34} - 4 q^{38} + 12 q^{41}+ \cdots + 22 q^{98}+O(q^{100}) \) 2 * q - 2 * q^2 + 2 * q^4 + 4 * q^7 - 6 * q^8 + 4 * q^11 + 4 * q^13 + 4 * q^14 + 6 * q^16 - 4 * q^19 - 12 * q^22 - 12 * q^23 - 4 * q^26 - 12 * q^28 - 2 * q^29 - 4 * q^31 + 6 * q^32 + 8 * q^34 - 4 * q^38 + 12 * q^41 + 12 * q^43 + 20 * q^44 + 20 * q^46 - 12 * q^47 + 10 * q^49 + 4 * q^52 + 4 * q^53 - 4 * q^56 + 2 * q^58 + 4 * q^61 + 28 * q^62 - 14 * q^64 + 4 * q^67 - 16 * q^68 + 8 * q^71 + 24 * q^74 + 12 * q^76 - 8 * q^77 + 12 * q^79 - 12 * q^82 + 20 * q^83 - 12 * q^86 - 20 * q^88 + 4 * q^89 + 8 * q^91 - 28 * q^92 - 4 * q^94 + 8 * q^97 + 22 * q^98

Introduction

L-functions

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