LMFDB - Embedded newform 6525.2.a.bm.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [6525,2,Mod(1,6525)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage: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")

magma://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:traces = [5,-2,0,4,0,0,6,-3,0,0,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	yes
Analytic conductor:	$52.1023873189$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	5.5.240881.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 2x^{4} - 5x^{3} + 9x^{2} + 5x - 7 $ x^5 - 2x^4 - 5x^3 + 9x^2 + 5x - 7
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 725)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$0.838718$ 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)

$f(q)$	$=$	$q-0.838718 q^{2} -1.29655 q^{4} +2.46832 q^{7} +2.76488 q^{8} +1.49072 q^{11} +0.929772 q^{13} -2.07023 q^{14} +0.274153 q^{16} -2.80583 q^{17} -0.0373385 q^{19} -1.25030 q^{22} -6.79703 q^{23} -0.779816 q^{26} -3.20031 q^{28} -1.00000 q^{29} +2.15767 q^{31} -5.75969 q^{32} +2.35330 q^{34} +7.43182 q^{37} +0.0313164 q^{38} +1.97399 q^{41} +9.61550 q^{43} -1.93280 q^{44} +5.70079 q^{46} +5.17430 q^{47} -0.907373 q^{49} -1.20550 q^{52} +1.33750 q^{53} +6.82462 q^{56} +0.838718 q^{58} -4.57758 q^{59} +12.1506 q^{61} -1.80968 q^{62} +4.28245 q^{64} -0.291367 q^{67} +3.63790 q^{68} +10.7486 q^{71} +0.0141028 q^{73} -6.23320 q^{74} +0.0484113 q^{76} +3.67959 q^{77} +5.37881 q^{79} -1.65562 q^{82} -10.1503 q^{83} -8.06469 q^{86} +4.12167 q^{88} +6.88136 q^{89} +2.29498 q^{91} +8.81271 q^{92} -4.33978 q^{94} -9.42386 q^{97} +0.761030 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 2 q^{2} + 4 q^{4} + 6 q^{7} - 3 q^{8} + 2 q^{11} + 4 q^{13} - 11 q^{14} - 10 q^{16} - 9 q^{17} + 2 q^{19} + 4 q^{22} - q^{23} + 16 q^{26} + 10 q^{28} - 5 q^{29} - q^{31} + 2 q^{32} + 3 q^{34} + 14 q^{37}+ \cdots - 3 q^{98}+O(q^{100}) $ 5 * q - 2 * q^2 + 4 * q^4 + 6 * q^7 - 3 * q^8 + 2 * q^11 + 4 * q^13 - 11 * q^14 - 10 * q^16 - 9 * q^17 + 2 * q^19 + 4 * q^22 - q^23 + 16 * q^26 + 10 * q^28 - 5 * q^29 - q^31 + 2 * q^32 + 3 * q^34 + 14 * q^37 + 3 * q^38 - 5 * q^41 + 28 * q^43 - 7 * q^44 - 20 * q^46 - 15 * q^47 - 3 * q^49 + 6 * q^52 + 8 * q^53 + 16 * q^56 + 2 * q^58 + 11 * q^59 - 5 * q^61 + 6 * q^62 - 5 * q^64 + 23 * q^67 - q^68 + 5 * q^71 + 16 * q^73 - 8 * q^74 + 16 * q^76 + 30 * q^77 - 10 * q^79 - 19 * q^82 - 9 * q^83 + 13 * q^86 + 35 * q^88 + 18 * q^89 + q^91 + 27 * q^92 - 13 * q^94 + 23 * q^97 - 3 * 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 $

$2$	−0.838718	−0.593063	−0.296532	−	0.955023i	$-0.595830\pi$
$2$	−0.838718	−0.593063	−0.296532	+	0.955023i	$0.595830\pi$
$3$	0	0
$3$	0	0
$4$	−1.29655	−0.648276
$5$	0	0
$5$	0	0
$6$	0	0
$7$	2.46832	0.932939	0.466470	−	0.884537i	$-0.345526\pi$
$7$	2.46832	0.932939	0.466470	+	0.884537i	$0.345526\pi$
$8$	2.76488	0.977532
$9$	0	0
$10$	0	0
$11$	1.49072	0.449470	0.224735	−	0.974420i	$-0.427848\pi$
$11$	1.49072	0.449470	0.224735	+	0.974420i	$0.427848\pi$
$12$	0	0
$13$	0.929772	0.257872	0.128936	−	0.991653i	$-0.458844\pi$
$13$	0.929772	0.257872	0.128936	+	0.991653i	$0.458844\pi$
$14$	−2.07023	−0.553292
$15$	0	0
$16$	0.274153	0.0685383
$17$	−2.80583	−0.680513	−0.340257	−	0.940333i	$-0.610514\pi$
$17$	−2.80583	−0.680513	−0.340257	+	0.940333i	$0.610514\pi$
$18$	0	0
$19$	−0.0373385	−0.00856603	−0.00428302	−	0.999991i	$-0.501363\pi$
$19$	−0.0373385	−0.00856603	−0.00428302	+	0.999991i	$0.501363\pi$
$20$	0	0
$21$	0	0
$22$	−1.25030	−0.266564
$23$	−6.79703	−1.41728	−0.708639	−	0.705571i	$-0.750691\pi$
$23$	−6.79703	−1.41728	−0.708639	+	0.705571i	$0.750691\pi$
$24$	0	0
$25$	0	0
$26$	−0.779816	−0.152935
$27$	0	0
$28$	−3.20031	−0.604802
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	0	0
$31$	2.15767	0.387529	0.193764	−	0.981048i	$-0.437930\pi$
$31$	2.15767	0.387529	0.193764	+	0.981048i	$0.437930\pi$
$32$	−5.75969	−1.01818
$33$	0	0
$34$	2.35330	0.403587
$35$	0	0
$36$	0	0
$37$	7.43182	1.22178	0.610892	−	0.791714i	$-0.290811\pi$
$37$	7.43182	1.22178	0.610892	+	0.791714i	$0.290811\pi$
$38$	0.0313164	0.00508020
$39$	0	0
$40$	0	0
$41$	1.97399	0.308285	0.154143	−	0.988049i	$-0.450738\pi$
$41$	1.97399	0.308285	0.154143	+	0.988049i	$0.450738\pi$
$42$	0	0
$43$	9.61550	1.46635	0.733175	−	0.680040i	$-0.238037\pi$
$43$	9.61550	1.46635	0.733175	+	0.680040i	$0.238037\pi$
$44$	−1.93280	−0.291381
$45$	0	0
$46$	5.70079	0.840536
$47$	5.17430	0.754749	0.377375	−	0.926061i	$-0.376827\pi$
$47$	5.17430	0.754749	0.377375	+	0.926061i	$0.376827\pi$
$48$	0	0
$49$	−0.907373	−0.129625
$50$	0	0
$51$	0	0
$52$	−1.20550	−0.167173
$53$	1.33750	0.183720	0.0918601	−	0.995772i	$-0.470719\pi$
$53$	1.33750	0.183720	0.0918601	+	0.995772i	$0.470719\pi$
$54$	0	0
$55$	0	0
$56$	6.82462	0.911978
$57$	0	0
$58$	0.838718	0.110129
$59$	−4.57758	−0.595951	−0.297975	−	0.954574i	$-0.596311\pi$
$59$	−4.57758	−0.595951	−0.297975	+	0.954574i	$0.596311\pi$
$60$	0	0
$61$	12.1506	1.55572	0.777860	−	0.628437i	$-0.216305\pi$
$61$	12.1506	1.55572	0.777860	+	0.628437i	$0.216305\pi$
$62$	−1.80968	−0.229829
$63$	0	0
$64$	4.28245	0.535306
$65$	0	0
$66$	0	0
$67$	−0.291367	−0.0355961	−0.0177981	−	0.999842i	$-0.505666\pi$
$67$	−0.291367	−0.0355961	−0.0177981	+	0.999842i	$0.505666\pi$
$68$	3.63790	0.441161
$69$	0	0
$70$	0	0
$71$	10.7486	1.27563	0.637813	−	0.770191i	$-0.279839\pi$
$71$	10.7486	1.27563	0.637813	+	0.770191i	$0.279839\pi$
$72$	0	0
$73$	0.0141028	0.00165061	0.000825303	−	1.00000i	$-0.499737\pi$
$73$	0.0141028	0.00165061	0.000825303		1.00000i	$0.499737\pi$
$74$	−6.23320	−0.724595
$75$	0	0
$76$	0.0484113	0.00555315
$77$	3.67959	0.419328
$78$	0	0
$79$	5.37881	0.605163	0.302582	−	0.953123i	$-0.402152\pi$
$79$	5.37881	0.605163	0.302582	+	0.953123i	$0.402152\pi$
$80$	0	0
$81$	0	0
$82$	−1.65562	−0.182833
$83$	−10.1503	−1.11414	−0.557072	−	0.830464i	$-0.688075\pi$
$83$	−10.1503	−1.11414	−0.557072	+	0.830464i	$0.688075\pi$
$84$	0	0
$85$	0	0
$86$	−8.06469	−0.869638
$87$	0	0
$88$	4.12167	0.439371
$89$	6.88136	0.729423	0.364711	−	0.931121i	$-0.381168\pi$
$89$	6.88136	0.729423	0.364711	+	0.931121i	$0.381168\pi$
$90$	0	0
$91$	2.29498	0.240579
$92$	8.81271	0.918788
$93$	0	0
$94$	−4.33978	−0.447614
$95$	0	0
$96$	0	0
$97$	−9.42386	−0.956848	−0.478424	−	0.878129i	$-0.658792\pi$
$97$	−9.42386	−0.956848	−0.478424	+	0.878129i	$0.658792\pi$
$98$	0.761030	0.0768756
$99$	0	0
$100$	0	0
$101$	−11.0204	−1.09657	−0.548284	−	0.836293i	$-0.684719\pi$
$101$	−11.0204	−1.09657	−0.548284	+	0.836293i	$0.684719\pi$
$102$	0	0
$103$	−4.51423	−0.444801	−0.222400	−	0.974955i	$-0.571389\pi$
$103$	−4.51423	−0.444801	−0.222400	+	0.974955i	$0.571389\pi$
$104$	2.57071	0.252078
$105$	0	0
$106$	−1.12179	−0.108958
$107$	0.277920	0.0268675	0.0134338	−	0.999910i	$-0.495724\pi$
$107$	0.277920	0.0268675	0.0134338	+	0.999910i	$0.495724\pi$
$108$	0	0
$109$	−16.6659	−1.59630	−0.798150	−	0.602459i	$-0.794188\pi$
$109$	−16.6659	−1.59630	−0.798150	+	0.602459i	$0.794188\pi$
$110$	0	0
$111$	0	0
$112$	0.676699	0.0639420
$113$	14.9017	1.40183	0.700916	−	0.713244i	$-0.252775\pi$
$113$	14.9017	1.40183	0.700916	+	0.713244i	$0.252775\pi$
$114$	0	0
$115$	0	0
$116$	1.29655	0.120382
$117$	0	0
$118$	3.83930	0.353436
$119$	−6.92570	−0.634877
$120$	0	0
$121$	−8.77774	−0.797977
$122$	−10.1909	−0.922640
$123$	0	0
$124$	−2.79753	−0.251226
$125$	0	0
$126$	0	0
$127$	14.2274	1.26248	0.631240	−	0.775587i	$-0.282546\pi$
$127$	14.2274	1.26248	0.631240	+	0.775587i	$0.282546\pi$
$128$	7.92762	0.700709
$129$	0	0
$130$	0	0
$131$	−19.7992	−1.72987	−0.864933	−	0.501888i	$-0.832639\pi$
$131$	−19.7992	−1.72987	−0.864933	+	0.501888i	$0.832639\pi$
$132$	0	0
$133$	−0.0921635	−0.00799159
$134$	0.244374	0.0211107
$135$	0	0
$136$	−7.75777	−0.665223
$137$	0.954829	0.0815765	0.0407883	−	0.999168i	$-0.487013\pi$
$137$	0.954829	0.0815765	0.0407883	+	0.999168i	$0.487013\pi$
$138$	0	0
$139$	−22.5382	−1.91166	−0.955831	−	0.293916i	$-0.905041\pi$
$139$	−22.5382	−1.91166	−0.955831	+	0.293916i	$0.905041\pi$
$140$	0	0
$141$	0	0
$142$	−9.01506	−0.756527
$143$	1.38603	0.115906
$144$	0	0
$145$	0	0
$146$	−0.0118282	−0.000978913 0
$147$	0	0
$148$	−9.63575	−0.792054
$149$	−8.56867	−0.701972	−0.350986	−	0.936381i	$-0.614154\pi$
$149$	−8.56867	−0.701972	−0.350986	+	0.936381i	$0.614154\pi$
$150$	0	0
$151$	13.2588	1.07898	0.539491	−	0.841991i	$-0.318617\pi$
$151$	13.2588	1.07898	0.539491	+	0.841991i	$0.318617\pi$
$152$	−0.103236	−0.00837357
$153$	0	0
$154$	−3.08614	−0.248688
$155$	0	0
$156$	0	0
$157$	14.1001	1.12531	0.562656	−	0.826691i	$-0.309780\pi$
$157$	14.1001	1.12531	0.562656	+	0.826691i	$0.309780\pi$
$158$	−4.51130	−0.358900
$159$	0	0
$160$	0	0
$161$	−16.7773	−1.32223
$162$	0	0
$163$	17.2267	1.34930	0.674650	−	0.738138i	$-0.264295\pi$
$163$	17.2267	1.34930	0.674650	+	0.738138i	$0.264295\pi$
$164$	−2.55938	−0.199854
$165$	0	0
$166$	8.51326	0.660757
$167$	14.4358	1.11707	0.558537	−	0.829480i	$-0.311363\pi$
$167$	14.4358	1.11707	0.558537	+	0.829480i	$0.311363\pi$
$168$	0	0
$169$	−12.1355	−0.933502
$170$	0	0
$171$	0	0
$172$	−12.4670	−0.950600
$173$	15.5680	1.18362	0.591808	−	0.806079i	$-0.298414\pi$
$173$	15.5680	1.18362	0.591808	+	0.806079i	$0.298414\pi$
$174$	0	0
$175$	0	0
$176$	0.408687	0.0308059
$177$	0	0
$178$	−5.77152	−0.432594
$179$	−20.5247	−1.53409	−0.767044	−	0.641594i	$-0.778273\pi$
$179$	−20.5247	−1.53409	−0.767044	+	0.641594i	$0.778273\pi$
$180$	0	0
$181$	17.5111	1.30159	0.650794	−	0.759255i	$-0.274436\pi$
$181$	17.5111	1.30159	0.650794	+	0.759255i	$0.274436\pi$
$182$	−1.92484	−0.142679
$183$	0	0
$184$	−18.7930	−1.38543
$185$	0	0
$186$	0	0
$187$	−4.18272	−0.305870
$188$	−6.70875	−0.489286
$189$	0	0
$190$	0	0
$191$	−15.5726	−1.12680	−0.563398	−	0.826186i	$-0.690506\pi$
$191$	−15.5726	−1.12680	−0.563398	+	0.826186i	$0.690506\pi$
$192$	0	0
$193$	−8.44449	−0.607847	−0.303924	−	0.952696i	$-0.598297\pi$
$193$	−8.44449	−0.607847	−0.303924	+	0.952696i	$0.598297\pi$
$194$	7.90396	0.567471
$195$	0	0
$196$	1.17646	0.0840326
$197$	10.1937	0.726272	0.363136	−	0.931736i	$-0.381706\pi$
$197$	10.1937	0.726272	0.363136	+	0.931736i	$0.381706\pi$
$198$	0	0
$199$	6.54034	0.463633	0.231816	−	0.972760i	$-0.425533\pi$
$199$	6.54034	0.463633	0.231816	+	0.972760i	$0.425533\pi$
$200$	0	0
$201$	0	0
$202$	9.24297	0.650333
$203$	−2.46832	−0.173242
$204$	0	0
$205$	0	0
$206$	3.78617	0.263795
$207$	0	0
$208$	0.254900	0.0176741
$209$	−0.0556614	−0.00385018
$210$	0	0
$211$	15.5988	1.07386	0.536932	−	0.843625i	$-0.319583\pi$
$211$	15.5988	1.07386	0.536932	+	0.843625i	$0.319583\pi$
$212$	−1.73414	−0.119101
$213$	0	0
$214$	−0.233096	−0.0159341
$215$	0	0
$216$	0	0
$217$	5.32583	0.361541
$218$	13.9780	0.946707
$219$	0	0
$220$	0	0
$221$	−2.60878	−0.175486
$222$	0	0
$223$	23.7662	1.59150	0.795751	−	0.605623i	$-0.207076\pi$
$223$	23.7662	1.59150	0.795751	+	0.605623i	$0.207076\pi$
$224$	−14.2168	−0.949899
$225$	0	0
$226$	−12.4983	−0.831375
$227$	9.10250	0.604154	0.302077	−	0.953284i	$-0.402320\pi$
$227$	9.10250	0.604154	0.302077	+	0.953284i	$0.402320\pi$
$228$	0	0
$229$	24.3822	1.61122	0.805611	−	0.592445i	$-0.201837\pi$
$229$	24.3822	1.61122	0.805611	+	0.592445i	$0.201837\pi$
$230$	0	0
$231$	0	0
$232$	−2.76488	−0.181523
$233$	−18.6324	−1.22065	−0.610325	−	0.792151i	$-0.708961\pi$
$233$	−18.6324	−1.22065	−0.610325	+	0.792151i	$0.708961\pi$
$234$	0	0
$235$	0	0
$236$	5.93508	0.386341
$237$	0	0
$238$	5.80870	0.376522
$239$	−1.69469	−0.109620	−0.0548101	−	0.998497i	$-0.517455\pi$
$239$	−1.69469	−0.109620	−0.0548101	+	0.998497i	$0.517455\pi$
$240$	0	0
$241$	8.57531	0.552384	0.276192	−	0.961102i	$-0.410927\pi$
$241$	8.57531	0.552384	0.276192	+	0.961102i	$0.410927\pi$
$242$	7.36205	0.473250
$243$	0	0
$244$	−15.7538	−1.00854
$245$	0	0
$246$	0	0
$247$	−0.0347163	−0.00220894
$248$	5.96569	0.378822
$249$	0	0
$250$	0	0
$251$	27.6230	1.74355	0.871774	−	0.489909i	$-0.162970\pi$
$251$	27.6230	1.74355	0.871774	+	0.489909i	$0.162970\pi$
$252$	0	0
$253$	−10.1325	−0.637025
$254$	−11.9328	−0.748731
$255$	0	0
$256$	−15.2139	−0.950871
$257$	−6.02021	−0.375530	−0.187765	−	0.982214i	$-0.560124\pi$
$257$	−6.02021	−0.375530	−0.187765	+	0.982214i	$0.560124\pi$
$258$	0	0
$259$	18.3442	1.13985
$260$	0	0
$261$	0	0
$262$	16.6060	1.02592
$263$	−8.13397	−0.501562	−0.250781	−	0.968044i	$-0.580687\pi$
$263$	−8.13397	−0.501562	−0.250781	+	0.968044i	$0.580687\pi$
$264$	0	0
$265$	0	0
$266$	0.0772991	0.00473951
$267$	0	0
$268$	0.377772	0.0230761
$269$	3.43905	0.209682	0.104841	−	0.994489i	$-0.466567\pi$
$269$	3.43905	0.209682	0.104841	+	0.994489i	$0.466567\pi$
$270$	0	0
$271$	−10.4822	−0.636748	−0.318374	−	0.947965i	$-0.603137\pi$
$271$	−10.4822	−0.636748	−0.318374	+	0.947965i	$0.603137\pi$
$272$	−0.769227	−0.0466412
$273$	0	0
$274$	−0.800832	−0.0483800
$275$	0	0
$276$	0	0
$277$	18.9262	1.13716	0.568582	−	0.822627i	$-0.307492\pi$
$277$	18.9262	1.13716	0.568582	+	0.822627i	$0.307492\pi$
$278$	18.9032	1.13374
$279$	0	0
$280$	0	0
$281$	24.6790	1.47223	0.736113	−	0.676859i	$-0.236659\pi$
$281$	24.6790	1.47223	0.736113	+	0.676859i	$0.236659\pi$
$282$	0	0
$283$	3.12685	0.185872	0.0929361	−	0.995672i	$-0.470375\pi$
$283$	3.12685	0.185872	0.0929361	+	0.995672i	$0.470375\pi$
$284$	−13.9361	−0.826958
$285$	0	0
$286$	−1.16249	−0.0687395
$287$	4.87244	0.287611
$288$	0	0
$289$	−9.12733	−0.536902
$290$	0	0
$291$	0	0
$292$	−0.0182850	−0.00107005
$293$	6.95025	0.406038	0.203019	−	0.979175i	$-0.434925\pi$
$293$	6.95025	0.406038	0.203019	+	0.979175i	$0.434925\pi$
$294$	0	0
$295$	0	0
$296$	20.5481	1.19433
$297$	0	0
$298$	7.18669	0.416314
$299$	−6.31969	−0.365477
$300$	0	0
$301$	23.7342	1.36802
$302$	−11.1203	−0.639904
$303$	0	0
$304$	−0.0102365	−0.000587101 0
$305$	0	0
$306$	0	0
$307$	15.4392	0.881160	0.440580	−	0.897713i	$-0.354773\pi$
$307$	15.4392	0.881160	0.440580	+	0.897713i	$0.354773\pi$
$308$	−4.77078	−0.271841
$309$	0	0
$310$	0	0
$311$	12.9889	0.736534	0.368267	−	0.929720i	$-0.379951\pi$
$311$	12.9889	0.736534	0.368267	+	0.929720i	$0.379951\pi$
$312$	0	0
$313$	24.8765	1.40611	0.703053	−	0.711138i	$-0.251820\pi$
$313$	24.8765	1.40611	0.703053	+	0.711138i	$0.251820\pi$
$314$	−11.8260	−0.667381
$315$	0	0
$316$	−6.97391	−0.392313
$317$	17.1132	0.961176	0.480588	−	0.876947i	$-0.340423\pi$
$317$	17.1132	0.961176	0.480588	+	0.876947i	$0.340423\pi$
$318$	0	0
$319$	−1.49072	−0.0834645
$320$	0	0
$321$	0	0
$322$	14.0714	0.784169
$323$	0.104765	0.00582930
$324$	0	0
$325$	0	0
$326$	−14.4483	−0.800219
$327$	0	0
$328$	5.45783	0.301359
$329$	12.7719	0.704135
$330$	0	0
$331$	−25.9203	−1.42471	−0.712353	−	0.701821i	$-0.752371\pi$
$331$	−25.9203	−1.42471	−0.712353	+	0.701821i	$0.752371\pi$
$332$	13.1604	0.722273
$333$	0	0
$334$	−12.1075	−0.662495
$335$	0	0
$336$	0	0
$337$	−18.9557	−1.03258	−0.516290	−	0.856414i	$-0.672687\pi$
$337$	−18.9557	−1.03258	−0.516290	+	0.856414i	$0.672687\pi$
$338$	10.1783	0.553625
$339$	0	0
$340$	0	0
$341$	3.21649	0.174183
$342$	0	0
$343$	−19.5180	−1.05387
$344$	26.5857	1.43340
$345$	0	0
$346$	−13.0572	−0.701959
$347$	−7.70815	−0.413795	−0.206898	−	0.978363i	$-0.566337\pi$
$347$	−7.70815	−0.413795	−0.206898	+	0.978363i	$0.566337\pi$
$348$	0	0
$349$	−15.0015	−0.803014	−0.401507	−	0.915856i	$-0.631513\pi$
$349$	−15.0015	−0.803014	−0.401507	+	0.915856i	$0.631513\pi$
$350$	0	0
$351$	0	0
$352$	−8.58611	−0.457641
$353$	16.9486	0.902085	0.451043	−	0.892502i	$-0.351052\pi$
$353$	16.9486	0.902085	0.451043	+	0.892502i	$0.351052\pi$
$354$	0	0
$355$	0	0
$356$	−8.92205	−0.472867
$357$	0	0
$358$	17.2144	0.909811
$359$	5.23876	0.276491	0.138246	−	0.990398i	$-0.455854\pi$
$359$	5.23876	0.276491	0.138246	+	0.990398i	$0.455854\pi$
$360$	0	0
$361$	−18.9986	−0.999927
$362$	−14.6868	−0.771923
$363$	0	0
$364$	−2.97556	−0.155962
$365$	0	0
$366$	0	0
$367$	15.2817	0.797701	0.398850	−	0.917016i	$-0.369409\pi$
$367$	15.2817	0.797701	0.398850	+	0.917016i	$0.369409\pi$
$368$	−1.86343	−0.0971378
$369$	0	0
$370$	0	0
$371$	3.30139	0.171400
$372$	0	0
$373$	24.3674	1.26169	0.630847	−	0.775907i	$-0.282708\pi$
$373$	24.3674	1.26169	0.630847	+	0.775907i	$0.282708\pi$
$374$	3.50812	0.181400
$375$	0	0
$376$	14.3063	0.737791
$377$	−0.929772	−0.0478857
$378$	0	0
$379$	−12.1058	−0.621832	−0.310916	−	0.950437i	$-0.600636\pi$
$379$	−12.1058	−0.621832	−0.310916	+	0.950437i	$0.600636\pi$
$380$	0	0
$381$	0	0
$382$	13.0610	0.668261
$383$	−21.0250	−1.07432	−0.537162	−	0.843479i	$-0.680504\pi$
$383$	−21.0250	−1.07432	−0.537162	+	0.843479i	$0.680504\pi$
$384$	0	0
$385$	0	0
$386$	7.08254	0.360492
$387$	0	0
$388$	12.2185	0.620302
$389$	10.3647	0.525509	0.262755	−	0.964863i	$-0.415369\pi$
$389$	10.3647	0.525509	0.262755	+	0.964863i	$0.415369\pi$
$390$	0	0
$391$	19.0713	0.964477
$392$	−2.50877	−0.126712
$393$	0	0
$394$	−8.54964	−0.430725
$395$	0	0
$396$	0	0
$397$	−17.7864	−0.892673	−0.446337	−	0.894865i	$-0.647272\pi$
$397$	−17.7864	−0.892673	−0.446337	+	0.894865i	$0.647272\pi$
$398$	−5.48550	−0.274963
$399$	0	0
$400$	0	0
$401$	29.9868	1.49747	0.748734	−	0.662870i	$-0.230662\pi$
$401$	29.9868	1.49747	0.748734	+	0.662870i	$0.230662\pi$
$402$	0	0
$403$	2.00614	0.0999330
$404$	14.2885	0.710878
$405$	0	0
$406$	2.07023	0.102744
$407$	11.0788	0.549156
$408$	0	0
$409$	30.6717	1.51662	0.758309	−	0.651896i	$-0.226026\pi$
$409$	30.6717	1.51662	0.758309	+	0.651896i	$0.226026\pi$
$410$	0	0
$411$	0	0
$412$	5.85294	0.288354
$413$	−11.2990	−0.555986
$414$	0	0
$415$	0	0
$416$	−5.35520	−0.262560
$417$	0	0
$418$	0.0466842	0.00228340
$419$	15.0395	0.734729	0.367364	−	0.930077i	$-0.380260\pi$
$419$	15.0395	0.734729	0.367364	+	0.930077i	$0.380260\pi$
$420$	0	0
$421$	−31.3625	−1.52852	−0.764258	−	0.644911i	$-0.776894\pi$
$421$	−31.3625	−1.52852	−0.764258	+	0.644911i	$0.776894\pi$
$422$	−13.0830	−0.636869
$423$	0	0
$424$	3.69803	0.179592
$425$	0	0
$426$	0	0
$427$	29.9915	1.45139
$428$	−0.360338	−0.0174176
$429$	0	0
$430$	0	0
$431$	28.7116	1.38299	0.691495	−	0.722381i	$-0.256952\pi$
$431$	28.7116	1.38299	0.691495	+	0.722381i	$0.256952\pi$
$432$	0	0
$433$	30.9086	1.48537	0.742686	−	0.669640i	$-0.233552\pi$
$433$	30.9086	1.48537	0.742686	+	0.669640i	$0.233552\pi$
$434$	−4.46687	−0.214417
$435$	0	0
$436$	21.6082	1.03484
$437$	0.253791	0.0121405
$438$	0	0
$439$	18.5828	0.886907	0.443454	−	0.896297i	$-0.353753\pi$
$439$	18.5828	0.886907	0.443454	+	0.896297i	$0.353753\pi$
$440$	0	0
$441$	0	0
$442$	2.18803	0.104074
$443$	32.4127	1.53997	0.769986	−	0.638061i	$-0.220263\pi$
$443$	32.4127	1.53997	0.769986	+	0.638061i	$0.220263\pi$
$444$	0	0
$445$	0	0
$446$	−19.9331	−0.943862
$447$	0	0
$448$	10.5705	0.499408
$449$	12.7486	0.601645	0.300822	−	0.953680i	$-0.402739\pi$
$449$	12.7486	0.601645	0.300822	+	0.953680i	$0.402739\pi$
$450$	0	0
$451$	2.94267	0.138565
$452$	−19.3208	−0.908775
$453$	0	0
$454$	−7.63443	−0.358301
$455$	0	0
$456$	0	0
$457$	−20.5383	−0.960742	−0.480371	−	0.877065i	$-0.659498\pi$
$457$	−20.5383	−0.960742	−0.480371	+	0.877065i	$0.659498\pi$
$458$	−20.4498	−0.955556
$459$	0	0
$460$	0	0
$461$	−15.0965	−0.703114	−0.351557	−	0.936167i	$-0.614348\pi$
$461$	−15.0965	−0.703114	−0.351557	+	0.936167i	$0.614348\pi$
$462$	0	0
$463$	−1.19501	−0.0555367	−0.0277684	−	0.999614i	$-0.508840\pi$
$463$	−1.19501	−0.0555367	−0.0277684	+	0.999614i	$0.508840\pi$
$464$	−0.274153	−0.0127272
$465$	0	0
$466$	15.6273	0.723922
$467$	7.08659	0.327929	0.163964	−	0.986466i	$-0.447572\pi$
$467$	7.08659	0.327929	0.163964	+	0.986466i	$0.447572\pi$
$468$	0	0
$469$	−0.719188	−0.0332090
$470$	0	0
$471$	0	0
$472$	−12.6565	−0.582561
$473$	14.3341	0.659081
$474$	0	0
$475$	0	0
$476$	8.97953	0.411576
$477$	0	0
$478$	1.42136	0.0650117
$479$	−38.8768	−1.77633	−0.888163	−	0.459528i	$-0.848019\pi$
$479$	−38.8768	−1.77633	−0.888163	+	0.459528i	$0.848019\pi$
$480$	0	0
$481$	6.90990	0.315064
$482$	−7.19226	−0.327599
$483$	0	0
$484$	11.3808	0.517309
$485$	0	0
$486$	0	0
$487$	30.3083	1.37340	0.686701	−	0.726940i	$-0.259058\pi$
$487$	30.3083	1.37340	0.686701	+	0.726940i	$0.259058\pi$
$488$	33.5948	1.52077
$489$	0	0
$490$	0	0
$491$	9.84537	0.444315	0.222158	−	0.975011i	$-0.428690\pi$
$491$	9.84537	0.444315	0.222158	+	0.975011i	$0.428690\pi$
$492$	0	0
$493$	2.80583	0.126368
$494$	0.0291171	0.00131004
$495$	0	0
$496$	0.591532	0.0265606
$497$	26.5311	1.19008
$498$	0	0
$499$	20.9719	0.938829	0.469415	−	0.882978i	$-0.344465\pi$
$499$	20.9719	0.938829	0.469415	+	0.882978i	$0.344465\pi$
$500$	0	0
$501$	0	0
$502$	−23.1679	−1.03403
$503$	−42.2017	−1.88168	−0.940841	−	0.338849i	$-0.889963\pi$
$503$	−42.2017	−1.88168	−0.940841	+	0.338849i	$0.889963\pi$
$504$	0	0
$505$	0	0
$506$	8.49831	0.377796
$507$	0	0
$508$	−18.4466	−0.818436
$509$	−7.00080	−0.310305	−0.155153	−	0.987891i	$-0.549587\pi$
$509$	−7.00080	−0.310305	−0.155153	+	0.987891i	$0.549587\pi$
$510$	0	0
$511$	0.0348102	0.00153991
$512$	−3.09504	−0.136783
$513$	0	0
$514$	5.04925	0.222713
$515$	0	0
$516$	0	0
$517$	7.71345	0.339237
$518$	−15.3856	−0.676003
$519$	0	0
$520$	0	0
$521$	−12.0767	−0.529091	−0.264545	−	0.964373i	$-0.585222\pi$
$521$	−12.0767	−0.529091	−0.264545	+	0.964373i	$0.585222\pi$
$522$	0	0
$523$	30.9879	1.35500	0.677502	−	0.735521i	$-0.263062\pi$
$523$	30.9879	1.35500	0.677502	+	0.735521i	$0.263062\pi$
$524$	25.6707	1.12143
$525$	0	0
$526$	6.82211	0.297458
$527$	−6.05405	−0.263719
$528$	0	0
$529$	23.1996	1.00868
$530$	0	0
$531$	0	0
$532$	0.119495	0.00518075
$533$	1.83536	0.0794982
$534$	0	0
$535$	0	0
$536$	−0.805593	−0.0347963
$537$	0	0
$538$	−2.88439	−0.124355
$539$	−1.35264	−0.0582624
$540$	0	0
$541$	−5.40732	−0.232479	−0.116240	−	0.993221i	$-0.537084\pi$
$541$	−5.40732	−0.232479	−0.116240	+	0.993221i	$0.537084\pi$
$542$	8.79161	0.377632
$543$	0	0
$544$	16.1607	0.692885
$545$	0	0
$546$	0	0
$547$	−0.766804	−0.0327862	−0.0163931	−	0.999866i	$-0.505218\pi$
$547$	−0.766804	−0.0327862	−0.0163931	+	0.999866i	$0.505218\pi$
$548$	−1.23799	−0.0528841
$549$	0	0
$550$	0	0
$551$	0.0373385	0.00159067
$552$	0	0
$553$	13.2766	0.564580
$554$	−15.8737	−0.674410
$555$	0	0
$556$	29.2219	1.23929
$557$	38.2414	1.62034	0.810170	−	0.586195i	$-0.199375\pi$
$557$	38.2414	1.62034	0.810170	+	0.586195i	$0.199375\pi$
$558$	0	0
$559$	8.94023	0.378131
$560$	0	0
$561$	0	0
$562$	−20.6987	−0.873123
$563$	−36.0317	−1.51855	−0.759277	−	0.650768i	$-0.774447\pi$
$563$	−36.0317	−1.51855	−0.759277	+	0.650768i	$0.774447\pi$
$564$	0	0
$565$	0	0
$566$	−2.62255	−0.110234
$567$	0	0
$568$	29.7186	1.24697
$569$	17.6536	0.740077	0.370038	−	0.929016i	$-0.379345\pi$
$569$	17.6536	0.740077	0.370038	+	0.929016i	$0.379345\pi$
$570$	0	0
$571$	−3.66464	−0.153360	−0.0766801	−	0.997056i	$-0.524432\pi$
$571$	−3.66464	−0.153360	−0.0766801	+	0.997056i	$0.524432\pi$
$572$	−1.79707	−0.0751391
$573$	0	0
$574$	−4.08661	−0.170572
$575$	0	0
$576$	0	0
$577$	−25.9865	−1.08183	−0.540917	−	0.841076i	$-0.681923\pi$
$577$	−25.9865	−1.08183	−0.540917	+	0.841076i	$0.681923\pi$
$578$	7.65525	0.318417
$579$	0	0
$580$	0	0
$581$	−25.0543	−1.03943
$582$	0	0
$583$	1.99385	0.0825768
$584$	0.0389924	0.00161352
$585$	0	0
$586$	−5.82930	−0.240806
$587$	26.3857	1.08905	0.544527	−	0.838744i	$-0.316709\pi$
$587$	26.3857	1.08905	0.544527	+	0.838744i	$0.316709\pi$
$588$	0	0
$589$	−0.0805641	−0.00331958
$590$	0	0
$591$	0	0
$592$	2.03746	0.0837390
$593$	37.1560	1.52581	0.762907	−	0.646508i	$-0.223771\pi$
$593$	37.1560	1.52581	0.762907	+	0.646508i	$0.223771\pi$
$594$	0	0
$595$	0	0
$596$	11.1097	0.455072
$597$	0	0
$598$	5.30044	0.216751
$599$	5.96103	0.243561	0.121781	−	0.992557i	$-0.461140\pi$
$599$	5.96103	0.243561	0.121781	+	0.992557i	$0.461140\pi$
$600$	0	0
$601$	9.40636	0.383693	0.191847	−	0.981425i	$-0.438552\pi$
$601$	9.40636	0.383693	0.191847	+	0.981425i	$0.438552\pi$
$602$	−19.9063	−0.811320
$603$	0	0
$604$	−17.1907	−0.699478
$605$	0	0
$606$	0	0
$607$	−11.5121	−0.467263	−0.233632	−	0.972325i	$-0.575061\pi$
$607$	−11.5121	−0.467263	−0.233632	+	0.972325i	$0.575061\pi$
$608$	0.215058	0.00872176
$609$	0	0
$610$	0	0
$611$	4.81092	0.194629
$612$	0	0
$613$	−33.9655	−1.37185	−0.685927	−	0.727670i	$-0.740603\pi$
$613$	−33.9655	−1.37185	−0.685927	+	0.727670i	$0.740603\pi$
$614$	−12.9491	−0.522583
$615$	0	0
$616$	10.1736	0.409907
$617$	−18.1241	−0.729650	−0.364825	−	0.931076i	$-0.618871\pi$
$617$	−18.1241	−0.729650	−0.364825	+	0.931076i	$0.618871\pi$
$618$	0	0
$619$	−43.7822	−1.75976	−0.879878	−	0.475200i	$-0.842376\pi$
$619$	−43.7822	−1.75976	−0.879878	+	0.475200i	$0.842376\pi$
$620$	0	0
$621$	0	0
$622$	−10.8940	−0.436811
$623$	16.9854	0.680507
$624$	0	0
$625$	0	0
$626$	−20.8644	−0.833909
$627$	0	0
$628$	−18.2815	−0.729513
$629$	−20.8524	−0.831440
$630$	0	0
$631$	−31.8836	−1.26927	−0.634633	−	0.772814i	$-0.718849\pi$
$631$	−31.8836	−1.26927	−0.634633	+	0.772814i	$0.718849\pi$
$632$	14.8717	0.591566
$633$	0	0
$634$	−14.3532	−0.570038
$635$	0	0
$636$	0	0
$637$	−0.843650	−0.0334266
$638$	1.25030	0.0494997
$639$	0	0
$640$	0	0
$641$	7.56055	0.298624	0.149312	−	0.988790i	$-0.452294\pi$
$641$	7.56055	0.298624	0.149312	+	0.988790i	$0.452294\pi$
$642$	0	0
$643$	−6.78946	−0.267750	−0.133875	−	0.990998i	$-0.542742\pi$
$643$	−6.78946	−0.267750	−0.133875	+	0.990998i	$0.542742\pi$
$644$	21.7526	0.857173
$645$	0	0
$646$	−0.0878685	−0.00345714
$647$	−32.4336	−1.27510	−0.637549	−	0.770410i	$-0.720051\pi$
$647$	−32.4336	−1.27510	−0.637549	+	0.770410i	$0.720051\pi$
$648$	0	0
$649$	−6.82391	−0.267862
$650$	0	0
$651$	0	0
$652$	−22.3353	−0.874718
$653$	−20.3067	−0.794661	−0.397331	−	0.917676i	$-0.630063\pi$
$653$	−20.3067	−0.794661	−0.397331	+	0.917676i	$0.630063\pi$
$654$	0	0
$655$	0	0
$656$	0.541175	0.0211293
$657$	0	0
$658$	−10.7120	−0.417597
$659$	47.3789	1.84562	0.922809	−	0.385257i	$-0.125887\pi$
$659$	47.3789	1.84562	0.922809	+	0.385257i	$0.125887\pi$
$660$	0	0
$661$	23.1097	0.898862	0.449431	−	0.893315i	$-0.351627\pi$
$661$	23.1097	0.898862	0.449431	+	0.893315i	$0.351627\pi$
$662$	21.7398	0.844941
$663$	0	0
$664$	−28.0644	−1.08911
$665$	0	0
$666$	0	0
$667$	6.79703	0.263182
$668$	−18.7167	−0.724172
$669$	0	0
$670$	0	0
$671$	18.1131	0.699250
$672$	0	0
$673$	0.691021	0.0266369	0.0133184	−	0.999911i	$-0.495760\pi$
$673$	0.691021	0.0266369	0.0133184	+	0.999911i	$0.495760\pi$
$674$	15.8984	0.612385
$675$	0	0
$676$	15.7343	0.605167
$677$	−10.2253	−0.392989	−0.196494	−	0.980505i	$-0.562956\pi$
$677$	−10.2253	−0.392989	−0.196494	+	0.980505i	$0.562956\pi$
$678$	0	0
$679$	−23.2611	−0.892681
$680$	0	0
$681$	0	0
$682$	−2.69773	−0.103301
$683$	9.74838	0.373012	0.186506	−	0.982454i	$-0.440284\pi$
$683$	9.74838	0.373012	0.186506	+	0.982454i	$0.440284\pi$
$684$	0	0
$685$	0	0
$686$	16.3701	0.625012
$687$	0	0
$688$	2.63612	0.100501
$689$	1.24357	0.0473764
$690$	0	0
$691$	−32.7638	−1.24639	−0.623197	−	0.782065i	$-0.714167\pi$
$691$	−32.7638	−1.24639	−0.623197	+	0.782065i	$0.714167\pi$
$692$	−20.1848	−0.767311
$693$	0	0
$694$	6.46496	0.245407
$695$	0	0
$696$	0	0
$697$	−5.53867	−0.209792
$698$	12.5821	0.476238
$699$	0	0
$700$	0	0
$701$	14.7894	0.558588	0.279294	−	0.960206i	$-0.409900\pi$
$701$	14.7894	0.558588	0.279294	+	0.960206i	$0.409900\pi$
$702$	0	0
$703$	−0.277493	−0.0104658
$704$	6.38395	0.240604
$705$	0	0
$706$	−14.2151	−0.534993
$707$	−27.2018	−1.02303
$708$	0	0
$709$	42.1665	1.58360	0.791799	−	0.610782i	$-0.209145\pi$
$709$	42.1665	1.58360	0.791799	+	0.610782i	$0.209145\pi$
$710$	0	0
$711$	0	0
$712$	19.0261	0.713034
$713$	−14.6657	−0.549236
$714$	0	0
$715$	0	0
$716$	26.6114	0.994513
$717$	0	0
$718$	−4.39384	−0.163977
$719$	16.2819	0.607211	0.303606	−	0.952798i	$-0.401809\pi$
$719$	16.2819	0.607211	0.303606	+	0.952798i	$0.401809\pi$
$720$	0	0
$721$	−11.1426	−0.414972
$722$	15.9345	0.593020
$723$	0	0
$724$	−22.7040	−0.843788
$725$	0	0
$726$	0	0
$727$	10.2431	0.379894	0.189947	−	0.981794i	$-0.439168\pi$
$727$	10.2431	0.379894	0.189947	+	0.981794i	$0.439168\pi$
$728$	6.34534	0.235174
$729$	0	0
$730$	0	0
$731$	−26.9795	−0.997871
$732$	0	0
$733$	1.34776	0.0497807	0.0248904	−	0.999690i	$-0.492076\pi$
$733$	1.34776	0.0497807	0.0248904	+	0.999690i	$0.492076\pi$
$734$	−12.8171	−0.473087
$735$	0	0
$736$	39.1488	1.44304
$737$	−0.434347	−0.0159994
$738$	0	0
$739$	−33.2111	−1.22169	−0.610844	−	0.791751i	$-0.709170\pi$
$739$	−33.2111	−1.22169	−0.610844	+	0.791751i	$0.709170\pi$
$740$	0	0
$741$	0	0
$742$	−2.76894	−0.101651
$743$	36.6826	1.34576	0.672878	−	0.739754i	$-0.265058\pi$
$743$	36.6826	1.34576	0.672878	+	0.739754i	$0.265058\pi$
$744$	0	0
$745$	0	0
$746$	−20.4373	−0.748264
$747$	0	0
$748$	5.42311	0.198289
$749$	0.685996	0.0250658
$750$	0	0
$751$	−16.0304	−0.584959	−0.292480	−	0.956272i	$-0.594480\pi$
$751$	−16.0304	−0.584959	−0.292480	+	0.956272i	$0.594480\pi$
$752$	1.41855	0.0517292
$753$	0	0
$754$	0.779816	0.0283992
$755$	0	0
$756$	0	0
$757$	−42.4991	−1.54466	−0.772328	−	0.635224i	$-0.780908\pi$
$757$	−42.4991	−1.54466	−0.772328	+	0.635224i	$0.780908\pi$
$758$	10.1533	0.368786
$759$	0	0
$760$	0	0
$761$	−24.7756	−0.898114	−0.449057	−	0.893503i	$-0.648240\pi$
$761$	−24.7756	−0.898114	−0.449057	+	0.893503i	$0.648240\pi$
$762$	0	0
$763$	−41.1368	−1.48925
$764$	20.1907	0.730475
$765$	0	0
$766$	17.6340	0.637142
$767$	−4.25611	−0.153679
$768$	0	0
$769$	22.9249	0.826694	0.413347	−	0.910573i	$-0.364360\pi$
$769$	22.9249	0.826694	0.413347	+	0.910573i	$0.364360\pi$
$770$	0	0
$771$	0	0
$772$	10.9487	0.394053
$773$	46.2921	1.66501	0.832506	−	0.554016i	$-0.186905\pi$
$773$	46.2921	1.66501	0.832506	+	0.554016i	$0.186905\pi$
$774$	0	0
$775$	0	0
$776$	−26.0558	−0.935349
$777$	0	0
$778$	−8.69302	−0.311660
$779$	−0.0737057	−0.00264078
$780$	0	0
$781$	16.0232	0.573356
$782$	−15.9954	−0.571996
$783$	0	0
$784$	−0.248759	−0.00888425
$785$	0	0
$786$	0	0
$787$	3.89849	0.138966	0.0694832	−	0.997583i	$-0.477865\pi$
$787$	3.89849	0.138966	0.0694832	+	0.997583i	$0.477865\pi$
$788$	−13.2167	−0.470825
$789$	0	0
$790$	0	0
$791$	36.7822	1.30782
$792$	0	0
$793$	11.2973	0.401177
$794$	14.9178	0.529411
$795$	0	0
$796$	−8.47990	−0.300562
$797$	−21.1877	−0.750506	−0.375253	−	0.926922i	$-0.622444\pi$
$797$	−21.1877	−0.750506	−0.375253	+	0.926922i	$0.622444\pi$
$798$	0	0
$799$	−14.5182	−0.513617
$800$	0	0
$801$	0	0
$802$	−25.1504	−0.888093
$803$	0.0210233	0.000741898 0
$804$	0	0
$805$	0	0
$806$	−1.68259	−0.0592666
$807$	0	0
$808$	−30.4699	−1.07193
$809$	−23.9959	−0.843651	−0.421826	−	0.906677i	$-0.638611\pi$
$809$	−23.9959	−0.843651	−0.421826	+	0.906677i	$0.638611\pi$
$810$	0	0
$811$	−8.15390	−0.286322	−0.143161	−	0.989699i	$-0.545727\pi$
$811$	−8.15390	−0.286322	−0.143161	+	0.989699i	$0.545727\pi$
$812$	3.20031	0.112309
$813$	0	0
$814$	−9.29198	−0.325684
$815$	0	0
$816$	0	0
$817$	−0.359028	−0.0125608
$818$	−25.7249	−0.899450
$819$	0	0
$820$	0	0
$821$	8.76657	0.305955	0.152978	−	0.988230i	$-0.451114\pi$
$821$	8.76657	0.305955	0.152978	+	0.988230i	$0.451114\pi$
$822$	0	0
$823$	49.1112	1.71191	0.855955	−	0.517050i	$-0.172970\pi$
$823$	49.1112	1.71191	0.855955	+	0.517050i	$0.172970\pi$
$824$	−12.4813	−0.434807
$825$	0	0
$826$	9.47664	0.329734
$827$	−34.1055	−1.18597	−0.592983	−	0.805215i	$-0.702050\pi$
$827$	−34.1055	−1.18597	−0.592983	+	0.805215i	$0.702050\pi$
$828$	0	0
$829$	1.20117	0.0417183	0.0208591	−	0.999782i	$-0.493360\pi$
$829$	1.20117	0.0417183	0.0208591	+	0.999782i	$0.493360\pi$
$830$	0	0
$831$	0	0
$832$	3.98170	0.138041
$833$	2.54593	0.0882113
$834$	0	0
$835$	0	0
$836$	0.0721679	0.00249598
$837$	0	0
$838$	−12.6139	−0.435740
$839$	1.87849	0.0648526	0.0324263	−	0.999474i	$-0.489677\pi$
$839$	1.87849	0.0648526	0.0324263	+	0.999474i	$0.489677\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	26.3043	0.906506
$843$	0	0
$844$	−20.2246	−0.696161
$845$	0	0
$846$	0	0
$847$	−21.6663	−0.744463
$848$	0.366681	0.0125919
$849$	0	0
$850$	0	0
$851$	−50.5143	−1.73161
$852$	0	0
$853$	−33.0917	−1.13304	−0.566519	−	0.824049i	$-0.691710\pi$
$853$	−33.0917	−1.13304	−0.566519	+	0.824049i	$0.691710\pi$
$854$	−25.1544	−0.860767
$855$	0	0
$856$	0.768414	0.0262639
$857$	47.6859	1.62892	0.814460	−	0.580220i	$-0.197034\pi$
$857$	47.6859	1.62892	0.814460	+	0.580220i	$0.197034\pi$
$858$	0	0
$859$	0.746091	0.0254563	0.0127281	−	0.999919i	$-0.495948\pi$
$859$	0.746091	0.0254563	0.0127281	+	0.999919i	$0.495948\pi$
$860$	0	0
$861$	0	0
$862$	−24.0810	−0.820201
$863$	46.8245	1.59392	0.796962	−	0.604030i	$-0.206439\pi$
$863$	46.8245	1.59392	0.796962	+	0.604030i	$0.206439\pi$
$864$	0	0
$865$	0	0
$866$	−25.9236	−0.880919
$867$	0	0
$868$	−6.90522	−0.234378
$869$	8.01832	0.272003
$870$	0	0
$871$	−0.270905	−0.00917925
$872$	−46.0791	−1.56043
$873$	0	0
$874$	−0.212859	−0.00720005
$875$	0	0
$876$	0	0
$877$	0.970257	0.0327632	0.0163816	−	0.999866i	$-0.494785\pi$
$877$	0.970257	0.0327632	0.0163816	+	0.999866i	$0.494785\pi$
$878$	−15.5857	−0.525992
$879$	0	0
$880$	0	0
$881$	−0.755979	−0.0254696	−0.0127348	−	0.999919i	$-0.504054\pi$
$881$	−0.755979	−0.0254696	−0.0127348	+	0.999919i	$0.504054\pi$
$882$	0	0
$883$	15.8839	0.534536	0.267268	−	0.963622i	$-0.413879\pi$
$883$	15.8839	0.534536	0.267268	+	0.963622i	$0.413879\pi$
$884$	3.38242	0.113763
$885$	0	0
$886$	−27.1851	−0.913300
$887$	12.5061	0.419915	0.209958	−	0.977710i	$-0.432667\pi$
$887$	12.5061	0.419915	0.209958	+	0.977710i	$0.432667\pi$
$888$	0	0
$889$	35.1179	1.17782
$890$	0	0
$891$	0	0
$892$	−30.8141	−1.03173
$893$	−0.193200	−0.00646521
$894$	0	0
$895$	0	0
$896$	19.5679	0.653719
$897$	0	0
$898$	−10.6925	−0.356813
$899$	−2.15767	−0.0719623
$900$	0	0
$901$	−3.75281	−0.125024
$902$	−2.46807	−0.0821778
$903$	0	0
$904$	41.2013	1.37034
$905$	0	0
$906$	0	0
$907$	0.966094	0.0320786	0.0160393	−	0.999871i	$-0.494894\pi$
$907$	0.966094	0.0320786	0.0160393	+	0.999871i	$0.494894\pi$
$908$	−11.8019	−0.391659
$909$	0	0
$910$	0	0
$911$	21.2288	0.703340	0.351670	−	0.936124i	$-0.385614\pi$
$911$	21.2288	0.703340	0.351670	+	0.936124i	$0.385614\pi$
$912$	0	0
$913$	−15.1313	−0.500774
$914$	17.2259	0.569781
$915$	0	0
$916$	−31.6128	−1.04452
$917$	−48.8709	−1.61386
$918$	0	0
$919$	16.9547	0.559284	0.279642	−	0.960104i	$-0.409784\pi$
$919$	16.9547	0.559284	0.279642	+	0.960104i	$0.409784\pi$
$920$	0	0
$921$	0	0
$922$	12.6617	0.416991
$923$	9.99377	0.328949
$924$	0	0
$925$	0	0
$926$	1.00227	0.0329368
$927$	0	0
$928$	5.75969	0.189071
$929$	26.0792	0.855632	0.427816	−	0.903866i	$-0.359283\pi$
$929$	26.0792	0.855632	0.427816	+	0.903866i	$0.359283\pi$
$930$	0	0
$931$	0.0338799	0.00111037
$932$	24.1579	0.791318
$933$	0	0
$934$	−5.94365	−0.194482
$935$	0	0
$936$	0	0
$937$	6.07903	0.198593	0.0992967	−	0.995058i	$-0.468341\pi$
$937$	6.07903	0.198593	0.0992967	+	0.995058i	$0.468341\pi$
$938$	0.603196	0.0196950
$939$	0	0
$940$	0	0
$941$	−49.5166	−1.61419	−0.807097	−	0.590419i	$-0.798963\pi$
$941$	−49.5166	−1.61419	−0.807097	+	0.590419i	$0.798963\pi$
$942$	0	0
$943$	−13.4173	−0.436926
$944$	−1.25496	−0.0408454
$945$	0	0
$946$	−12.0222	−0.390877
$947$	26.9125	0.874539	0.437269	−	0.899331i	$-0.355946\pi$
$947$	26.9125	0.874539	0.437269	+	0.899331i	$0.355946\pi$
$948$	0	0
$949$	0.0131124	0.000425645 0
$950$	0	0
$951$	0	0
$952$	−19.1487	−0.620613
$953$	−30.8022	−0.997781	−0.498891	−	0.866665i	$-0.666259\pi$
$953$	−30.8022	−0.997781	−0.498891	+	0.866665i	$0.666259\pi$
$954$	0	0
$955$	0	0
$956$	2.19725	0.0710642
$957$	0	0
$958$	32.6067	1.05347
$959$	2.35683	0.0761059
$960$	0	0
$961$	−26.3445	−0.849821
$962$	−5.79546	−0.186853
$963$	0	0
$964$	−11.1183	−0.358098
$965$	0	0
$966$	0	0
$967$	50.1110	1.61146	0.805730	−	0.592283i	$-0.201773\pi$
$967$	50.1110	1.61146	0.805730	+	0.592283i	$0.201773\pi$
$968$	−24.2694	−0.780047
$969$	0	0
$970$	0	0
$971$	13.0301	0.418156	0.209078	−	0.977899i	$-0.432954\pi$
$971$	13.0301	0.418156	0.209078	+	0.977899i	$0.432954\pi$
$972$	0	0
$973$	−55.6315	−1.78346
$974$	−25.4201	−0.814514
$975$	0	0
$976$	3.33111	0.106626
$977$	3.50116	0.112012	0.0560060	−	0.998430i	$-0.482163\pi$
$977$	3.50116	0.112012	0.0560060	+	0.998430i	$0.482163\pi$
$978$	0	0
$979$	10.2582	0.327854
$980$	0	0
$981$	0	0
$982$	−8.25748	−0.263507
$983$	41.2281	1.31497	0.657486	−	0.753467i	$-0.271620\pi$
$983$	41.2281	1.31497	0.657486	+	0.753467i	$0.271620\pi$
$984$	0	0
$985$	0	0
$986$	−2.35330	−0.0749443
$987$	0	0
$988$	0.0450115	0.00143201
$989$	−65.3569	−2.07823
$990$	0	0
$991$	3.17997	0.101015	0.0505076	−	0.998724i	$-0.483916\pi$
$991$	3.17997	0.101015	0.0505076	+	0.998724i	$0.483916\pi$
$992$	−12.4275	−0.394574
$993$	0	0
$994$	−22.2521	−0.705793
$995$	0	0
$996$	0	0
$997$	33.8545	1.07218	0.536092	−	0.844160i	$-0.319900\pi$
$997$	33.8545	1.07218	0.536092	+	0.844160i	$0.319900\pi$
$998$	−17.5895	−0.556785
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6525.2.a.bm.1.3		5
3.2	odd	2		725.2.a.k.1.3	yes	5
5.4	even	2		6525.2.a.bq.1.3		5
15.2	even	4		725.2.b.f.349.6		10
15.8	even	4		725.2.b.f.349.5		10
15.14	odd	2		725.2.a.h.1.3	✓	5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
725.2.a.h.1.3	✓	5	15.14	odd	2
725.2.a.k.1.3	yes	5	3.2	odd	2
725.2.b.f.349.5		10	15.8	even	4
725.2.b.f.349.6		10	15.2	even	4
6525.2.a.bm.1.3		5	1.1	even	1	trivial
6525.2.a.bq.1.3		5	5.4	even	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 6525 = 3^{2} \cdot 5^{2} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6525.a (trivial)

\(f(q)\)	\(=\)	\(q-0.838718 q^{2} -1.29655 q^{4} +2.46832 q^{7} +2.76488 q^{8} +1.49072 q^{11} +0.929772 q^{13} -2.07023 q^{14} +0.274153 q^{16} -2.80583 q^{17} -0.0373385 q^{19} -1.25030 q^{22} -6.79703 q^{23} -0.779816 q^{26} -3.20031 q^{28} -1.00000 q^{29} +2.15767 q^{31} -5.75969 q^{32} +2.35330 q^{34} +7.43182 q^{37} +0.0313164 q^{38} +1.97399 q^{41} +9.61550 q^{43} -1.93280 q^{44} +5.70079 q^{46} +5.17430 q^{47} -0.907373 q^{49} -1.20550 q^{52} +1.33750 q^{53} +6.82462 q^{56} +0.838718 q^{58} -4.57758 q^{59} +12.1506 q^{61} -1.80968 q^{62} +4.28245 q^{64} -0.291367 q^{67} +3.63790 q^{68} +10.7486 q^{71} +0.0141028 q^{73} -6.23320 q^{74} +0.0484113 q^{76} +3.67959 q^{77} +5.37881 q^{79} -1.65562 q^{82} -10.1503 q^{83} -8.06469 q^{86} +4.12167 q^{88} +6.88136 q^{89} +2.29498 q^{91} +8.81271 q^{92} -4.33978 q^{94} -9.42386 q^{97} +0.761030 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 2 q^{2} + 4 q^{4} + 6 q^{7} - 3 q^{8} + 2 q^{11} + 4 q^{13} - 11 q^{14} - 10 q^{16} - 9 q^{17} + 2 q^{19} + 4 q^{22} - q^{23} + 16 q^{26} + 10 q^{28} - 5 q^{29} - q^{31} + 2 q^{32} + 3 q^{34} + 14 q^{37}+ \cdots - 3 q^{98}+O(q^{100}) \) 5 * q - 2 * q^2 + 4 * q^4 + 6 * q^7 - 3 * q^8 + 2 * q^11 + 4 * q^13 - 11 * q^14 - 10 * q^16 - 9 * q^17 + 2 * q^19 + 4 * q^22 - q^23 + 16 * q^26 + 10 * q^28 - 5 * q^29 - q^31 + 2 * q^32 + 3 * q^34 + 14 * q^37 + 3 * q^38 - 5 * q^41 + 28 * q^43 - 7 * q^44 - 20 * q^46 - 15 * q^47 - 3 * q^49 + 6 * q^52 + 8 * q^53 + 16 * q^56 + 2 * q^58 + 11 * q^59 - 5 * q^61 + 6 * q^62 - 5 * q^64 + 23 * q^67 - q^68 + 5 * q^71 + 16 * q^73 - 8 * q^74 + 16 * q^76 + 30 * q^77 - 10 * q^79 - 19 * q^82 - 9 * q^83 + 13 * q^86 + 35 * q^88 + 18 * q^89 + q^91 + 27 * q^92 - 13 * q^94 + 23 * q^97 - 3 * q^98

Introduction

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