LMFDB - Embedded newform 9075.2.a.dt.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9075,2,Mod(1,9075)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9075, 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("9075.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9075 = 3 \cdot 5^{2} \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9075.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:	$72.4642398343$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{7} - 11x^{6} + 9x^{5} + 38x^{4} - 25x^{3} - 41x^{2} + 20x + 5 $ x^8 - x^7 - 11x^6 + 9x^5 + 38x^4 - 25x^3 - 41x^2 + 20x + 5
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 825)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-0.188059$ of defining polynomial
Character	$\chi$	$=$	9075.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.188059 q^{2} -1.00000 q^{3} -1.96463 q^{4} -0.188059 q^{6} -1.64886 q^{7} -0.745587 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+0.188059 q^{2} -1.00000 q^{3} -1.96463 q^{4} -0.188059 q^{6} -1.64886 q^{7} -0.745587 q^{8} +1.00000 q^{9} +1.96463 q^{12} +6.71928 q^{13} -0.310083 q^{14} +3.78905 q^{16} +1.16891 q^{17} +0.188059 q^{18} +2.24317 q^{19} +1.64886 q^{21} +8.49577 q^{23} +0.745587 q^{24} +1.26362 q^{26} -1.00000 q^{27} +3.23940 q^{28} +9.45126 q^{29} +3.73551 q^{31} +2.20374 q^{32} +0.219824 q^{34} -1.96463 q^{36} +9.66735 q^{37} +0.421849 q^{38} -6.71928 q^{39} -6.80663 q^{41} +0.310083 q^{42} -2.58363 q^{43} +1.59771 q^{46} -7.66886 q^{47} -3.78905 q^{48} -4.28127 q^{49} -1.16891 q^{51} -13.2009 q^{52} -1.03004 q^{53} -0.188059 q^{54} +1.22937 q^{56} -2.24317 q^{57} +1.77740 q^{58} +7.76816 q^{59} -1.99623 q^{61} +0.702499 q^{62} -1.64886 q^{63} -7.16367 q^{64} -8.37354 q^{67} -2.29648 q^{68} -8.49577 q^{69} -3.77056 q^{71} -0.745587 q^{72} -5.86523 q^{73} +1.81804 q^{74} -4.40701 q^{76} -1.26362 q^{78} +10.2974 q^{79} +1.00000 q^{81} -1.28005 q^{82} +2.47744 q^{83} -3.23940 q^{84} -0.485876 q^{86} -9.45126 q^{87} +11.3935 q^{89} -11.0791 q^{91} -16.6911 q^{92} -3.73551 q^{93} -1.44220 q^{94} -2.20374 q^{96} +4.27825 q^{97} -0.805133 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - q^{2} - 8 q^{3} + 7 q^{4} + q^{6} - 8 q^{7} - 3 q^{8} + 8 q^{9}+O(q^{10}) $ 8 * q - q^2 - 8 * q^3 + 7 * q^4 + q^6 - 8 * q^7 - 3 * q^8 + 8 * q^9	$ 8 q - q^{2} - 8 q^{3} + 7 q^{4} + q^{6} - 8 q^{7} - 3 q^{8} + 8 q^{9} - 7 q^{12} - 7 q^{13} - 8 q^{14} - 7 q^{16} + q^{17} - q^{18} + 6 q^{19} + 8 q^{21} - 7 q^{23} + 3 q^{24} - q^{26} - 8 q^{27} - 11 q^{28} + 28 q^{29} + 10 q^{31} - 12 q^{32} - 8 q^{34} + 7 q^{36} - 14 q^{37} - 19 q^{38} + 7 q^{39} + 14 q^{41} + 8 q^{42} - 11 q^{43} - 5 q^{47} + 7 q^{48} + 2 q^{49} - q^{51} - q^{52} + 5 q^{53} + q^{54} - 4 q^{56} - 6 q^{57} + 3 q^{58} - 2 q^{59} + 36 q^{61} + 17 q^{62} - 8 q^{63} - 23 q^{64} - 6 q^{67} + 3 q^{68} + 7 q^{69} - 17 q^{71} - 3 q^{72} - 32 q^{73} + 47 q^{74} + 5 q^{76} + q^{78} + 41 q^{79} + 8 q^{81} + 6 q^{83} + 11 q^{84} - 9 q^{86} - 28 q^{87} + 21 q^{89} - 9 q^{91} - 69 q^{92} - 10 q^{93} + 51 q^{94} + 12 q^{96} + 4 q^{97} + 27 q^{98}+O(q^{100}) $ 8 * q - q^2 - 8 * q^3 + 7 * q^4 + q^6 - 8 * q^7 - 3 * q^8 + 8 * q^9 - 7 * q^12 - 7 * q^13 - 8 * q^14 - 7 * q^16 + q^17 - q^18 + 6 * q^19 + 8 * q^21 - 7 * q^23 + 3 * q^24 - q^26 - 8 * q^27 - 11 * q^28 + 28 * q^29 + 10 * q^31 - 12 * q^32 - 8 * q^34 + 7 * q^36 - 14 * q^37 - 19 * q^38 + 7 * q^39 + 14 * q^41 + 8 * q^42 - 11 * q^43 - 5 * q^47 + 7 * q^48 + 2 * q^49 - q^51 - q^52 + 5 * q^53 + q^54 - 4 * q^56 - 6 * q^57 + 3 * q^58 - 2 * q^59 + 36 * q^61 + 17 * q^62 - 8 * q^63 - 23 * q^64 - 6 * q^67 + 3 * q^68 + 7 * q^69 - 17 * q^71 - 3 * q^72 - 32 * q^73 + 47 * q^74 + 5 * q^76 + q^78 + 41 * q^79 + 8 * q^81 + 6 * q^83 + 11 * q^84 - 9 * q^86 - 28 * q^87 + 21 * q^89 - 9 * q^91 - 69 * q^92 - 10 * q^93 + 51 * q^94 + 12 * q^96 + 4 * q^97 + 27 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0.188059	0.132978	0.0664891	−	0.997787i	$-0.478820\pi$
$2$	0.188059	0.132978	0.0664891	+	0.997787i	$0.478820\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	−1.96463	−0.982317
$5$	0	0
$5$	0	0
$6$	−0.188059	−0.0767750
$7$	−1.64886	−0.623210	−0.311605	−	0.950212i	$-0.600866\pi$
$7$	−1.64886	−0.623210	−0.311605	+	0.950212i	$0.600866\pi$
$8$	−0.745587	−0.263605
$9$	1.00000	0.333333
$10$	0	0
$11$	0	0
$11$	0	0
$12$	1.96463	0.567141
$13$	6.71928	1.86359	0.931797	−	0.362981i	$-0.118241\pi$
$13$	6.71928	1.86359	0.931797	+	0.362981i	$0.118241\pi$
$14$	−0.310083	−0.0828732
$15$	0	0
$16$	3.78905	0.947263
$17$	1.16891	0.283502	0.141751	−	0.989902i	$-0.454727\pi$
$17$	1.16891	0.283502	0.141751	+	0.989902i	$0.454727\pi$
$18$	0.188059	0.0443260
$19$	2.24317	0.514618	0.257309	−	0.966329i	$-0.417164\pi$
$19$	2.24317	0.514618	0.257309	+	0.966329i	$0.417164\pi$
$20$	0	0
$21$	1.64886	0.359810
$22$	0	0
$23$	8.49577	1.77149	0.885746	−	0.464171i	$-0.153648\pi$
$23$	8.49577	1.77149	0.885746	+	0.464171i	$0.153648\pi$
$24$	0.745587	0.152192
$25$	0	0
$26$	1.26362	0.247817
$27$	−1.00000	−0.192450
$28$	3.23940	0.612189
$29$	9.45126	1.75506	0.877528	−	0.479526i	$-0.159191\pi$
$29$	9.45126	1.75506	0.877528	+	0.479526i	$0.159191\pi$
$30$	0	0
$31$	3.73551	0.670918	0.335459	−	0.942055i	$-0.391109\pi$
$31$	3.73551	0.670918	0.335459	+	0.942055i	$0.391109\pi$
$32$	2.20374	0.389570
$33$	0	0
$34$	0.219824	0.0376996
$35$	0	0
$36$	−1.96463	−0.327439
$37$	9.66735	1.58930	0.794651	−	0.607066i	$-0.207654\pi$
$37$	9.66735	1.58930	0.794651	+	0.607066i	$0.207654\pi$
$38$	0.421849	0.0684330
$39$	−6.71928	−1.07595
$40$	0	0
$41$	−6.80663	−1.06302	−0.531509	−	0.847053i	$-0.678375\pi$
$41$	−6.80663	−1.06302	−0.531509	+	0.847053i	$0.678375\pi$
$42$	0.310083	0.0478469
$43$	−2.58363	−0.394000	−0.197000	−	0.980404i	$-0.563120\pi$
$43$	−2.58363	−0.394000	−0.197000	+	0.980404i	$0.563120\pi$
$44$	0	0
$45$	0	0
$46$	1.59771	0.235570
$47$	−7.66886	−1.11862	−0.559309	−	0.828959i	$-0.688934\pi$
$47$	−7.66886	−1.11862	−0.559309	+	0.828959i	$0.688934\pi$
$48$	−3.78905	−0.546903
$49$	−4.28127	−0.611610
$50$	0	0
$51$	−1.16891	−0.163680
$52$	−13.2009	−1.83064
$53$	−1.03004	−0.141488	−0.0707438	−	0.997495i	$-0.522537\pi$
$53$	−1.03004	−0.141488	−0.0707438	+	0.997495i	$0.522537\pi$
$54$	−0.188059	−0.0255917
$55$	0	0
$56$	1.22937	0.164281
$57$	−2.24317	−0.297115
$58$	1.77740	0.233384
$59$	7.76816	1.01133	0.505664	−	0.862730i	$-0.331247\pi$
$59$	7.76816	1.01133	0.505664	+	0.862730i	$0.331247\pi$
$60$	0	0
$61$	−1.99623	−0.255591	−0.127796	−	0.991801i	$-0.540790\pi$
$61$	−1.99623	−0.255591	−0.127796	+	0.991801i	$0.540790\pi$
$62$	0.702499	0.0892175
$63$	−1.64886	−0.207737
$64$	−7.16367	−0.895459
$65$	0	0
$66$	0	0
$67$	−8.37354	−1.02299	−0.511495	−	0.859286i	$-0.670908\pi$
$67$	−8.37354	−1.02299	−0.511495	+	0.859286i	$0.670908\pi$
$68$	−2.29648	−0.278489
$69$	−8.49577	−1.02277
$70$	0	0
$71$	−3.77056	−0.447484	−0.223742	−	0.974648i	$-0.571827\pi$
$71$	−3.77056	−0.447484	−0.223742	+	0.974648i	$0.571827\pi$
$72$	−0.745587	−0.0878683
$73$	−5.86523	−0.686473	−0.343237	−	0.939249i	$-0.611523\pi$
$73$	−5.86523	−0.686473	−0.343237	+	0.939249i	$0.611523\pi$
$74$	1.81804	0.211343
$75$	0	0
$76$	−4.40701	−0.505518
$77$	0	0
$78$	−1.26362	−0.143077
$79$	10.2974	1.15855	0.579273	−	0.815134i	$-0.303337\pi$
$79$	10.2974	1.15855	0.579273	+	0.815134i	$0.303337\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	−1.28005	−0.141358
$83$	2.47744	0.271934	0.135967	−	0.990713i	$-0.456586\pi$
$83$	2.47744	0.271934	0.135967	+	0.990713i	$0.456586\pi$
$84$	−3.23940	−0.353448
$85$	0	0
$86$	−0.485876	−0.0523933
$87$	−9.45126	−1.01328
$88$	0	0
$89$	11.3935	1.20771	0.603854	−	0.797095i	$-0.293631\pi$
$89$	11.3935	1.20771	0.603854	+	0.797095i	$0.293631\pi$
$90$	0	0
$91$	−11.0791	−1.16141
$92$	−16.6911	−1.74017
$93$	−3.73551	−0.387355
$94$	−1.44220	−0.148752
$95$	0	0
$96$	−2.20374	−0.224918
$97$	4.27825	0.434390	0.217195	−	0.976128i	$-0.430309\pi$
$97$	4.27825	0.434390	0.217195	+	0.976128i	$0.430309\pi$
$98$	−0.805133	−0.0813307
$99$	0	0
$100$	0	0
$101$	−13.4141	−1.33475	−0.667376	−	0.744721i	$-0.732583\pi$
$101$	−13.4141	−1.33475	−0.667376	+	0.744721i	$0.732583\pi$
$102$	−0.219824	−0.0217658
$103$	−9.96972	−0.982346	−0.491173	−	0.871062i	$-0.663432\pi$
$103$	−9.96972	−0.982346	−0.491173	+	0.871062i	$0.663432\pi$
$104$	−5.00981	−0.491252
$105$	0	0
$106$	−0.193710	−0.0188147
$107$	−2.73642	−0.264540	−0.132270	−	0.991214i	$-0.542227\pi$
$107$	−2.73642	−0.264540	−0.132270	+	0.991214i	$0.542227\pi$
$108$	1.96463	0.189047
$109$	15.3255	1.46791	0.733957	−	0.679196i	$-0.237671\pi$
$109$	15.3255	1.46791	0.733957	+	0.679196i	$0.237671\pi$
$110$	0	0
$111$	−9.66735	−0.917584
$112$	−6.24761	−0.590343
$113$	3.94504	0.371118	0.185559	−	0.982633i	$-0.440590\pi$
$113$	3.94504	0.371118	0.185559	+	0.982633i	$0.440590\pi$
$114$	−0.421849	−0.0395098
$115$	0	0
$116$	−18.5683	−1.72402
$117$	6.71928	0.621198
$118$	1.46088	0.134485
$119$	−1.92736	−0.176681
$120$	0	0
$121$	0	0
$122$	−0.375410	−0.0339880
$123$	6.80663	0.613733
$124$	−7.33892	−0.659054
$125$	0	0
$126$	−0.310083	−0.0276244
$127$	−6.44234	−0.571665	−0.285833	−	0.958280i	$-0.592270\pi$
$127$	−6.44234	−0.571665	−0.285833	+	0.958280i	$0.592270\pi$
$128$	−5.75468	−0.508647
$129$	2.58363	0.227476
$130$	0	0
$131$	14.2875	1.24831	0.624153	−	0.781302i	$-0.285444\pi$
$131$	14.2875	1.24831	0.624153	+	0.781302i	$0.285444\pi$
$132$	0	0
$133$	−3.69867	−0.320715
$134$	−1.57472	−0.136035
$135$	0	0
$136$	−0.871523	−0.0747325
$137$	11.7446	1.00341	0.501706	−	0.865038i	$-0.332706\pi$
$137$	11.7446	1.00341	0.501706	+	0.865038i	$0.332706\pi$
$138$	−1.59771	−0.136006
$139$	−16.1703	−1.37155	−0.685773	−	0.727815i	$-0.740536\pi$
$139$	−16.1703	−1.37155	−0.685773	+	0.727815i	$0.740536\pi$
$140$	0	0
$141$	7.66886	0.645835
$142$	−0.709090	−0.0595055
$143$	0	0
$144$	3.78905	0.315754
$145$	0	0
$146$	−1.10301	−0.0912859
$147$	4.28127	0.353113
$148$	−18.9928	−1.56120
$149$	13.4396	1.10102	0.550508	−	0.834830i	$-0.314434\pi$
$149$	13.4396	1.10102	0.550508	+	0.834830i	$0.314434\pi$
$150$	0	0
$151$	6.55130	0.533137	0.266569	−	0.963816i	$-0.414110\pi$
$151$	6.55130	0.533137	0.266569	+	0.963816i	$0.414110\pi$
$152$	−1.67248	−0.135656
$153$	1.16891	0.0945006
$154$	0	0
$155$	0	0
$156$	13.2009	1.05692
$157$	11.7201	0.935363	0.467682	−	0.883897i	$-0.345089\pi$
$157$	11.7201	0.935363	0.467682	+	0.883897i	$0.345089\pi$
$158$	1.93652	0.154061
$159$	1.03004	0.0816879
$160$	0	0
$161$	−14.0083	−1.10401
$162$	0.188059	0.0147753
$163$	3.08089	0.241314	0.120657	−	0.992694i	$-0.461500\pi$
$163$	3.08089	0.241314	0.120657	+	0.992694i	$0.461500\pi$
$164$	13.3725	1.04422
$165$	0	0
$166$	0.465906	0.0361613
$167$	−11.2494	−0.870505	−0.435252	−	0.900308i	$-0.643341\pi$
$167$	−11.2494	−0.870505	−0.435252	+	0.900308i	$0.643341\pi$
$168$	−1.22937	−0.0948477
$169$	32.1487	2.47298
$170$	0	0
$171$	2.24317	0.171539
$172$	5.07588	0.387032
$173$	14.4571	1.09915	0.549577	−	0.835443i	$-0.314789\pi$
$173$	14.4571	1.09915	0.549577	+	0.835443i	$0.314789\pi$
$174$	−1.77740	−0.134744
$175$	0	0
$176$	0	0
$177$	−7.76816	−0.583891
$178$	2.14265	0.160599
$179$	11.5086	0.860192	0.430096	−	0.902783i	$-0.358480\pi$
$179$	11.5086	0.860192	0.430096	+	0.902783i	$0.358480\pi$
$180$	0	0
$181$	−18.4098	−1.36839	−0.684195	−	0.729299i	$-0.739846\pi$
$181$	−18.4098	−1.36839	−0.684195	+	0.729299i	$0.739846\pi$
$182$	−2.08354	−0.154442
$183$	1.99623	0.147566
$184$	−6.33434	−0.466974
$185$	0	0
$186$	−0.702499	−0.0515097
$187$	0	0
$188$	15.0665	1.09884
$189$	1.64886	0.119937
$190$	0	0
$191$	14.2160	1.02864	0.514318	−	0.857600i	$-0.328045\pi$
$191$	14.2160	1.02864	0.514318	+	0.857600i	$0.328045\pi$
$192$	7.16367	0.516993
$193$	2.19986	0.158350	0.0791748	−	0.996861i	$-0.474771\pi$
$193$	2.19986	0.158350	0.0791748	+	0.996861i	$0.474771\pi$
$194$	0.804565	0.0577644
$195$	0	0
$196$	8.41112	0.600795
$197$	−19.0936	−1.36037	−0.680183	−	0.733043i	$-0.738100\pi$
$197$	−19.0936	−1.36037	−0.680183	+	0.733043i	$0.738100\pi$
$198$	0	0
$199$	1.78668	0.126654	0.0633270	−	0.997993i	$-0.479829\pi$
$199$	1.78668	0.126654	0.0633270	+	0.997993i	$0.479829\pi$
$200$	0	0
$201$	8.37354	0.590624
$202$	−2.52265	−0.177493
$203$	−15.5838	−1.09377
$204$	2.29648	0.160786
$205$	0	0
$206$	−1.87490	−0.130631
$207$	8.49577	0.590497
$208$	25.4597	1.76531
$209$	0	0
$210$	0	0
$211$	8.70636	0.599370	0.299685	−	0.954038i	$-0.403118\pi$
$211$	8.70636	0.599370	0.299685	+	0.954038i	$0.403118\pi$
$212$	2.02366	0.138986
$213$	3.77056	0.258355
$214$	−0.514610	−0.0351780
$215$	0	0
$216$	0.745587	0.0507308
$217$	−6.15933	−0.418123
$218$	2.88210	0.195201
$219$	5.86523	0.396336
$220$	0	0
$221$	7.85422	0.528332
$222$	−1.81804	−0.122019
$223$	−2.66436	−0.178419	−0.0892093	−	0.996013i	$-0.528434\pi$
$223$	−2.66436	−0.178419	−0.0892093	+	0.996013i	$0.528434\pi$
$224$	−3.63366	−0.242784
$225$	0	0
$226$	0.741902	0.0493506
$227$	−15.7660	−1.04642	−0.523212	−	0.852203i	$-0.675266\pi$
$227$	−15.7660	−1.04642	−0.523212	+	0.852203i	$0.675266\pi$
$228$	4.40701	0.291861
$229$	5.09266	0.336532	0.168266	−	0.985742i	$-0.446183\pi$
$229$	5.09266	0.336532	0.168266	+	0.985742i	$0.446183\pi$
$230$	0	0
$231$	0	0
$232$	−7.04674	−0.462641
$233$	12.0061	0.786545	0.393273	−	0.919422i	$-0.371343\pi$
$233$	12.0061	0.786545	0.393273	+	0.919422i	$0.371343\pi$
$234$	1.26362	0.0826057
$235$	0	0
$236$	−15.2616	−0.993445
$237$	−10.2974	−0.668887
$238$	−0.362459	−0.0234947
$239$	18.7618	1.21360	0.606800	−	0.794855i	$-0.292453\pi$
$239$	18.7618	1.21360	0.606800	+	0.794855i	$0.292453\pi$
$240$	0	0
$241$	3.88549	0.250286	0.125143	−	0.992139i	$-0.460061\pi$
$241$	3.88549	0.250286	0.125143	+	0.992139i	$0.460061\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	3.92186	0.251071
$245$	0	0
$246$	1.28005	0.0816131
$247$	15.0725	0.959039
$248$	−2.78515	−0.176857
$249$	−2.47744	−0.157001
$250$	0	0
$251$	−17.7497	−1.12035	−0.560174	−	0.828375i	$-0.689266\pi$
$251$	−17.7497	−1.12035	−0.560174	+	0.828375i	$0.689266\pi$
$252$	3.23940	0.204063
$253$	0	0
$254$	−1.21154	−0.0760190
$255$	0	0
$256$	13.2451	0.827820
$257$	−5.91917	−0.369227	−0.184614	−	0.982811i	$-0.559103\pi$
$257$	−5.91917	−0.369227	−0.184614	+	0.982811i	$0.559103\pi$
$258$	0.485876	0.0302493
$259$	−15.9401	−0.990469
$260$	0	0
$261$	9.45126	0.585019
$262$	2.68690	0.165997
$263$	1.73320	0.106873	0.0534367	−	0.998571i	$-0.482982\pi$
$263$	1.73320	0.106873	0.0534367	+	0.998571i	$0.482982\pi$
$264$	0	0
$265$	0	0
$266$	−0.695569	−0.0426481
$267$	−11.3935	−0.697271
$268$	16.4509	1.00490
$269$	−28.1477	−1.71620	−0.858099	−	0.513485i	$-0.828354\pi$
$269$	−28.1477	−1.71620	−0.858099	+	0.513485i	$0.828354\pi$
$270$	0	0
$271$	18.5941	1.12951	0.564755	−	0.825258i	$-0.308971\pi$
$271$	18.5941	1.12951	0.564755	+	0.825258i	$0.308971\pi$
$272$	4.42906	0.268551
$273$	11.0791	0.670540
$274$	2.20869	0.133432
$275$	0	0
$276$	16.6911	1.00469
$277$	−12.1890	−0.732365	−0.366182	−	0.930543i	$-0.619335\pi$
$277$	−12.1890	−0.732365	−0.366182	+	0.930543i	$0.619335\pi$
$278$	−3.04098	−0.182386
$279$	3.73551	0.223639
$280$	0	0
$281$	10.4234	0.621810	0.310905	−	0.950441i	$-0.399368\pi$
$281$	10.4234	0.621810	0.310905	+	0.950441i	$0.399368\pi$
$282$	1.44220	0.0858819
$283$	−16.4996	−0.980801	−0.490401	−	0.871497i	$-0.663150\pi$
$283$	−16.4996	−0.980801	−0.490401	+	0.871497i	$0.663150\pi$
$284$	7.40778	0.439571
$285$	0	0
$286$	0	0
$287$	11.2232	0.662483
$288$	2.20374	0.129857
$289$	−15.6337	−0.919627
$290$	0	0
$291$	−4.27825	−0.250795
$292$	11.5230	0.674334
$293$	−30.0307	−1.75441	−0.877205	−	0.480116i	$-0.840595\pi$
$293$	−30.0307	−1.75441	−0.877205	+	0.480116i	$0.840595\pi$
$294$	0.805133	0.0469563
$295$	0	0
$296$	−7.20785	−0.418948
$297$	0	0
$298$	2.52745	0.146411
$299$	57.0855	3.30134
$300$	0	0
$301$	4.26003	0.245544
$302$	1.23203	0.0708956
$303$	13.4141	0.770619
$304$	8.49948	0.487479
$305$	0	0
$306$	0.219824	0.0125665
$307$	−10.4942	−0.598934	−0.299467	−	0.954107i	$-0.596809\pi$
$307$	−10.4942	−0.598934	−0.299467	+	0.954107i	$0.596809\pi$
$308$	0	0
$309$	9.96972	0.567158
$310$	0	0
$311$	−30.7119	−1.74151	−0.870755	−	0.491716i	$-0.836370\pi$
$311$	−30.7119	−1.74151	−0.870755	+	0.491716i	$0.836370\pi$
$312$	5.00981	0.283624
$313$	8.45228	0.477751	0.238876	−	0.971050i	$-0.423221\pi$
$313$	8.45228	0.477751	0.238876	+	0.971050i	$0.423221\pi$
$314$	2.20407	0.124383
$315$	0	0
$316$	−20.2306	−1.13806
$317$	−17.4886	−0.982258	−0.491129	−	0.871087i	$-0.663416\pi$
$317$	−17.4886	−0.982258	−0.491129	+	0.871087i	$0.663416\pi$
$318$	0.193710	0.0108627
$319$	0	0
$320$	0	0
$321$	2.73642	0.152732
$322$	−2.63440	−0.146809
$323$	2.62206	0.145895
$324$	−1.96463	−0.109146
$325$	0	0
$326$	0.579391	0.0320895
$327$	−15.3255	−0.847501
$328$	5.07494	0.280216
$329$	12.6449	0.697134
$330$	0	0
$331$	6.42686	0.353252	0.176626	−	0.984278i	$-0.443482\pi$
$331$	6.42686	0.353252	0.176626	+	0.984278i	$0.443482\pi$
$332$	−4.86726	−0.267125
$333$	9.66735	0.529768
$334$	−2.11556	−0.115758
$335$	0	0
$336$	6.24761	0.340835
$337$	−1.68541	−0.0918103	−0.0459051	−	0.998946i	$-0.514617\pi$
$337$	−1.68541	−0.0918103	−0.0459051	+	0.998946i	$0.514617\pi$
$338$	6.04587	0.328852
$339$	−3.94504	−0.214265
$340$	0	0
$341$	0	0
$342$	0.421849	0.0228110
$343$	18.6012	1.00437
$344$	1.92632	0.103860
$345$	0	0
$346$	2.71880	0.146163
$347$	34.0259	1.82660	0.913302	−	0.407283i	$-0.133524\pi$
$347$	34.0259	1.82660	0.913302	+	0.407283i	$0.133524\pi$
$348$	18.5683	0.995364
$349$	29.5701	1.58285	0.791425	−	0.611267i	$-0.209340\pi$
$349$	29.5701	1.58285	0.791425	+	0.611267i	$0.209340\pi$
$350$	0	0
$351$	−6.71928	−0.358649
$352$	0	0
$353$	23.7493	1.26405	0.632025	−	0.774948i	$-0.282224\pi$
$353$	23.7493	1.26405	0.632025	+	0.774948i	$0.282224\pi$
$354$	−1.46088	−0.0776447
$355$	0	0
$356$	−22.3840	−1.18635
$357$	1.92736	0.102007
$358$	2.16430	0.114387
$359$	−3.30561	−0.174464	−0.0872318	−	0.996188i	$-0.527802\pi$
$359$	−3.30561	−0.174464	−0.0872318	+	0.996188i	$0.527802\pi$
$360$	0	0
$361$	−13.9682	−0.735168
$362$	−3.46214	−0.181966
$363$	0	0
$364$	21.7664	1.14087
$365$	0	0
$366$	0.375410	0.0196230
$367$	−12.9461	−0.675781	−0.337891	−	0.941185i	$-0.609713\pi$
$367$	−12.9461	−0.675781	−0.337891	+	0.941185i	$0.609713\pi$
$368$	32.1909	1.67807
$369$	−6.80663	−0.354339
$370$	0	0
$371$	1.69840	0.0881764
$372$	7.33892	0.380505
$373$	−11.1678	−0.578244	−0.289122	−	0.957292i	$-0.593363\pi$
$373$	−11.1678	−0.578244	−0.289122	+	0.957292i	$0.593363\pi$
$374$	0	0
$375$	0	0
$376$	5.71780	0.294873
$377$	63.5057	3.27071
$378$	0.310083	0.0159490
$379$	−20.9485	−1.07605	−0.538025	−	0.842929i	$-0.680830\pi$
$379$	−20.9485	−1.07605	−0.538025	+	0.842929i	$0.680830\pi$
$380$	0	0
$381$	6.44234	0.330051
$382$	2.67346	0.136786
$383$	−32.4887	−1.66010	−0.830048	−	0.557693i	$-0.811687\pi$
$383$	−32.4887	−1.66010	−0.830048	+	0.557693i	$0.811687\pi$
$384$	5.75468	0.293667
$385$	0	0
$386$	0.413705	0.0210570
$387$	−2.58363	−0.131333
$388$	−8.40519	−0.426709
$389$	−27.7626	−1.40762	−0.703811	−	0.710387i	$-0.748520\pi$
$389$	−27.7626	−1.40762	−0.703811	+	0.710387i	$0.748520\pi$
$390$	0	0
$391$	9.93078	0.502221
$392$	3.19206	0.161223
$393$	−14.2875	−0.720710
$394$	−3.59074	−0.180899
$395$	0	0
$396$	0	0
$397$	−5.04365	−0.253134	−0.126567	−	0.991958i	$-0.540396\pi$
$397$	−5.04365	−0.253134	−0.126567	+	0.991958i	$0.540396\pi$
$398$	0.336001	0.0168422
$399$	3.69867	0.185165
$400$	0	0
$401$	23.9811	1.19756	0.598779	−	0.800915i	$-0.295653\pi$
$401$	23.9811	1.19756	0.598779	+	0.800915i	$0.295653\pi$
$402$	1.57472	0.0785401
$403$	25.1000	1.25032
$404$	26.3538	1.31115
$405$	0	0
$406$	−2.93068	−0.145447
$407$	0	0
$408$	0.871523	0.0431468
$409$	−0.686396	−0.0339401	−0.0169700	−	0.999856i	$-0.505402\pi$
$409$	−0.686396	−0.0339401	−0.0169700	+	0.999856i	$0.505402\pi$
$410$	0	0
$411$	−11.7446	−0.579320
$412$	19.5869	0.964975
$413$	−12.8086	−0.630270
$414$	1.59771	0.0785232
$415$	0	0
$416$	14.8076	0.726000
$417$	16.1703	0.791862
$418$	0	0
$419$	15.7520	0.769536	0.384768	−	0.923013i	$-0.374281\pi$
$419$	15.7520	0.769536	0.384768	+	0.923013i	$0.374281\pi$
$420$	0	0
$421$	27.8048	1.35512	0.677562	−	0.735466i	$-0.263037\pi$
$421$	27.8048	1.35512	0.677562	+	0.735466i	$0.263037\pi$
$422$	1.63731	0.0797032
$423$	−7.66886	−0.372873
$424$	0.767988	0.0372968
$425$	0	0
$426$	0.709090	0.0343555
$427$	3.29150	0.159287
$428$	5.37606	0.259862
$429$	0	0
$430$	0	0
$431$	24.2052	1.16592	0.582961	−	0.812500i	$-0.301894\pi$
$431$	24.2052	1.16592	0.582961	+	0.812500i	$0.301894\pi$
$432$	−3.78905	−0.182301
$433$	17.1067	0.822096	0.411048	−	0.911614i	$-0.365163\pi$
$433$	17.1067	0.822096	0.411048	+	0.911614i	$0.365163\pi$
$434$	−1.15832	−0.0556012
$435$	0	0
$436$	−30.1090	−1.44196
$437$	19.0575	0.911642
$438$	1.10301	0.0527040
$439$	15.0666	0.719091	0.359545	−	0.933128i	$-0.382932\pi$
$439$	15.0666	0.719091	0.359545	+	0.933128i	$0.382932\pi$
$440$	0	0
$441$	−4.28127	−0.203870
$442$	1.47706	0.0702566
$443$	−30.4595	−1.44717	−0.723587	−	0.690234i	$-0.757508\pi$
$443$	−30.4595	−1.44717	−0.723587	+	0.690234i	$0.757508\pi$
$444$	18.9928	0.901359
$445$	0	0
$446$	−0.501058	−0.0237258
$447$	−13.4396	−0.635672
$448$	11.8119	0.558059
$449$	−4.86199	−0.229452	−0.114726	−	0.993397i	$-0.536599\pi$
$449$	−4.86199	−0.229452	−0.114726	+	0.993397i	$0.536599\pi$
$450$	0	0
$451$	0	0
$452$	−7.75056	−0.364556
$453$	−6.55130	−0.307807
$454$	−2.96494	−0.139151
$455$	0	0
$456$	1.67248	0.0783209
$457$	−12.7446	−0.596169	−0.298085	−	0.954539i	$-0.596348\pi$
$457$	−12.7446	−0.596169	−0.298085	+	0.954539i	$0.596348\pi$
$458$	0.957722	0.0447514
$459$	−1.16891	−0.0545600
$460$	0	0
$461$	22.7026	1.05736	0.528682	−	0.848820i	$-0.322686\pi$
$461$	22.7026	1.05736	0.528682	+	0.848820i	$0.322686\pi$
$462$	0	0
$463$	29.7884	1.38438	0.692192	−	0.721713i	$-0.256645\pi$
$463$	29.7884	1.38438	0.692192	+	0.721713i	$0.256645\pi$
$464$	35.8113	1.66250
$465$	0	0
$466$	2.25786	0.104593
$467$	−2.80924	−0.129996	−0.0649980	−	0.997885i	$-0.520704\pi$
$467$	−2.80924	−0.129996	−0.0649980	+	0.997885i	$0.520704\pi$
$468$	−13.2009	−0.610213
$469$	13.8068	0.637538
$470$	0	0
$471$	−11.7201	−0.540032
$472$	−5.79184	−0.266591
$473$	0	0
$474$	−1.93652	−0.0889473
$475$	0	0
$476$	3.78656	0.173557
$477$	−1.03004	−0.0471625
$478$	3.52833	0.161382
$479$	−5.92474	−0.270708	−0.135354	−	0.990797i	$-0.543217\pi$
$479$	−5.92474	−0.270708	−0.135354	+	0.990797i	$0.543217\pi$
$480$	0	0
$481$	64.9576	2.96181
$482$	0.730703	0.0332826
$483$	14.0083	0.637401
$484$	0	0
$485$	0	0
$486$	−0.188059	−0.00853055
$487$	−34.3024	−1.55439	−0.777194	−	0.629261i	$-0.783358\pi$
$487$	−34.3024	−1.55439	−0.777194	+	0.629261i	$0.783358\pi$
$488$	1.48836	0.0673750
$489$	−3.08089	−0.139323
$490$	0	0
$491$	6.57575	0.296760	0.148380	−	0.988930i	$-0.452594\pi$
$491$	6.57575	0.296760	0.148380	+	0.988930i	$0.452594\pi$
$492$	−13.3725	−0.602881
$493$	11.0477	0.497562
$494$	2.83452	0.127531
$495$	0	0
$496$	14.1541	0.635536
$497$	6.21712	0.278876
$498$	−0.465906	−0.0208777
$499$	−16.8275	−0.753300	−0.376650	−	0.926356i	$-0.622924\pi$
$499$	−16.8275	−0.753300	−0.376650	+	0.926356i	$0.622924\pi$
$500$	0	0
$501$	11.2494	0.502586
$502$	−3.33799	−0.148982
$503$	6.25374	0.278840	0.139420	−	0.990233i	$-0.455476\pi$
$503$	6.25374	0.278840	0.139420	+	0.990233i	$0.455476\pi$
$504$	1.22937	0.0547603
$505$	0	0
$506$	0	0
$507$	−32.1487	−1.42778
$508$	12.6568	0.561556
$509$	−15.7340	−0.697397	−0.348699	−	0.937235i	$-0.613376\pi$
$509$	−15.7340	−0.697397	−0.348699	+	0.937235i	$0.613376\pi$
$510$	0	0
$511$	9.67093	0.427817
$512$	14.0002	0.618728
$513$	−2.24317	−0.0990383
$514$	−1.11316	−0.0490992
$515$	0	0
$516$	−5.07588	−0.223453
$517$	0	0
$518$	−2.99768	−0.131711
$519$	−14.4571	−0.634597
$520$	0	0
$521$	13.6797	0.599318	0.299659	−	0.954046i	$-0.403127\pi$
$521$	13.6797	0.599318	0.299659	+	0.954046i	$0.403127\pi$
$522$	1.77740	0.0777947
$523$	−29.9003	−1.30745	−0.653724	−	0.756733i	$-0.726794\pi$
$523$	−29.9003	−1.30745	−0.653724	+	0.756733i	$0.726794\pi$
$524$	−28.0697	−1.22623
$525$	0	0
$526$	0.325944	0.0142118
$527$	4.36647	0.190207
$528$	0	0
$529$	49.1782	2.13818
$530$	0	0
$531$	7.76816	0.337110
$532$	7.26652	0.315044
$533$	−45.7357	−1.98103
$534$	−2.14265	−0.0927217
$535$	0	0
$536$	6.24320	0.269665
$537$	−11.5086	−0.496632
$538$	−5.29345	−0.228217
$539$	0	0
$540$	0	0
$541$	33.0430	1.42063	0.710314	−	0.703885i	$-0.248553\pi$
$541$	33.0430	1.42063	0.710314	+	0.703885i	$0.248553\pi$
$542$	3.49680	0.150200
$543$	18.4098	0.790041
$544$	2.57597	0.110444
$545$	0	0
$546$	2.08354	0.0891671
$547$	−0.195862	−0.00837447	−0.00418723	−	0.999991i	$-0.501333\pi$
$547$	−0.195862	−0.00837447	−0.00418723	+	0.999991i	$0.501333\pi$
$548$	−23.0739	−0.985668
$549$	−1.99623	−0.0851970
$550$	0	0
$551$	21.2008	0.903184
$552$	6.33434	0.269607
$553$	−16.9789	−0.722017
$554$	−2.29225	−0.0973885
$555$	0	0
$556$	31.7687	1.34729
$557$	7.64929	0.324111	0.162055	−	0.986782i	$-0.448188\pi$
$557$	7.64929	0.324111	0.162055	+	0.986782i	$0.448188\pi$
$558$	0.702499	0.0297392
$559$	−17.3601	−0.734255
$560$	0	0
$561$	0	0
$562$	1.96023	0.0826871
$563$	10.9546	0.461683	0.230841	−	0.972991i	$-0.425852\pi$
$563$	10.9546	0.461683	0.230841	+	0.972991i	$0.425852\pi$
$564$	−15.0665	−0.634414
$565$	0	0
$566$	−3.10291	−0.130425
$567$	−1.64886	−0.0692455
$568$	2.81128	0.117959
$569$	−7.31765	−0.306772	−0.153386	−	0.988166i	$-0.549018\pi$
$569$	−7.31765	−0.306772	−0.153386	+	0.988166i	$0.549018\pi$
$570$	0	0
$571$	19.7305	0.825694	0.412847	−	0.910800i	$-0.364534\pi$
$571$	19.7305	0.825694	0.412847	+	0.910800i	$0.364534\pi$
$572$	0	0
$573$	−14.2160	−0.593883
$574$	2.11062	0.0880957
$575$	0	0
$576$	−7.16367	−0.298486
$577$	−10.6683	−0.444125	−0.222063	−	0.975032i	$-0.571279\pi$
$577$	−10.6683	−0.444125	−0.222063	+	0.975032i	$0.571279\pi$
$578$	−2.94006	−0.122290
$579$	−2.19986	−0.0914232
$580$	0	0
$581$	−4.08494	−0.169472
$582$	−0.804565	−0.0333503
$583$	0	0
$584$	4.37304	0.180958
$585$	0	0
$586$	−5.64755	−0.233298
$587$	−26.4313	−1.09093	−0.545467	−	0.838132i	$-0.683648\pi$
$587$	−26.4313	−1.09093	−0.545467	+	0.838132i	$0.683648\pi$
$588$	−8.41112	−0.346869
$589$	8.37939	0.345267
$590$	0	0
$591$	19.0936	0.785407
$592$	36.6301	1.50549
$593$	48.2320	1.98065	0.990325	−	0.138765i	$-0.0443132\pi$
$593$	48.2320	1.98065	0.990325	+	0.138765i	$0.0443132\pi$
$594$	0	0
$595$	0	0
$596$	−26.4039	−1.08155
$597$	−1.78668	−0.0731238
$598$	10.7355	0.439006
$599$	1.10657	0.0452131	0.0226066	−	0.999744i	$-0.492803\pi$
$599$	1.10657	0.0452131	0.0226066	+	0.999744i	$0.492803\pi$
$600$	0	0
$601$	27.4264	1.11875	0.559374	−	0.828915i	$-0.311042\pi$
$601$	27.4264	1.11875	0.559374	+	0.828915i	$0.311042\pi$
$602$	0.801140	0.0326520
$603$	−8.37354	−0.340997
$604$	−12.8709	−0.523710
$605$	0	0
$606$	2.52265	0.102476
$607$	−29.2528	−1.18734	−0.593668	−	0.804710i	$-0.702321\pi$
$607$	−29.2528	−1.18734	−0.593668	+	0.804710i	$0.702321\pi$
$608$	4.94336	0.200480
$609$	15.5838	0.631487
$610$	0	0
$611$	−51.5292	−2.08465
$612$	−2.29648	−0.0928296
$613$	44.8755	1.81251	0.906253	−	0.422736i	$-0.138930\pi$
$613$	44.8755	1.81251	0.906253	+	0.422736i	$0.138930\pi$
$614$	−1.97353	−0.0796451
$615$	0	0
$616$	0	0
$617$	5.36793	0.216105	0.108052	−	0.994145i	$-0.465539\pi$
$617$	5.36793	0.216105	0.108052	+	0.994145i	$0.465539\pi$
$618$	1.87490	0.0754196
$619$	26.1888	1.05262	0.526308	−	0.850294i	$-0.323576\pi$
$619$	26.1888	1.05262	0.526308	+	0.850294i	$0.323576\pi$
$620$	0	0
$621$	−8.49577	−0.340924
$622$	−5.77566	−0.231583
$623$	−18.7862	−0.752655
$624$	−25.4597	−1.01920
$625$	0	0
$626$	1.58953	0.0635305
$627$	0	0
$628$	−23.0256	−0.918823
$629$	11.3002	0.450570
$630$	0	0
$631$	−7.31705	−0.291287	−0.145644	−	0.989337i	$-0.546525\pi$
$631$	−7.31705	−0.291287	−0.145644	+	0.989337i	$0.546525\pi$
$632$	−7.67759	−0.305398
$633$	−8.70636	−0.346047
$634$	−3.28890	−0.130619
$635$	0	0
$636$	−2.02366	−0.0802434
$637$	−28.7670	−1.13979
$638$	0	0
$639$	−3.77056	−0.149161
$640$	0	0
$641$	−33.7967	−1.33489	−0.667445	−	0.744659i	$-0.732612\pi$
$641$	−33.7967	−1.33489	−0.667445	+	0.744659i	$0.732612\pi$
$642$	0.514610	0.0203100
$643$	−35.1537	−1.38633	−0.693163	−	0.720781i	$-0.743783\pi$
$643$	−35.1537	−1.38633	−0.693163	+	0.720781i	$0.743783\pi$
$644$	27.5212	1.08449
$645$	0	0
$646$	0.493103	0.0194009
$647$	−41.1219	−1.61667	−0.808334	−	0.588724i	$-0.799630\pi$
$647$	−41.1219	−1.61667	−0.808334	+	0.588724i	$0.799630\pi$
$648$	−0.745587	−0.0292894
$649$	0	0
$650$	0	0
$651$	6.15933	0.241403
$652$	−6.05283	−0.237047
$653$	−9.82371	−0.384431	−0.192216	−	0.981353i	$-0.561567\pi$
$653$	−9.82371	−0.384431	−0.192216	+	0.981353i	$0.561567\pi$
$654$	−2.88210	−0.112699
$655$	0	0
$656$	−25.7907	−1.00696
$657$	−5.86523	−0.228824
$658$	2.37799	0.0927035
$659$	3.76123	0.146517	0.0732584	−	0.997313i	$-0.476660\pi$
$659$	3.76123	0.146517	0.0732584	+	0.997313i	$0.476660\pi$
$660$	0	0
$661$	−12.7212	−0.494798	−0.247399	−	0.968914i	$-0.579576\pi$
$661$	−12.7212	−0.494798	−0.247399	+	0.968914i	$0.579576\pi$
$662$	1.20863	0.0469748
$663$	−7.85422	−0.305033
$664$	−1.84715	−0.0716831
$665$	0	0
$666$	1.81804	0.0704475
$667$	80.2958	3.10907
$668$	22.1010	0.855112
$669$	2.66436	0.103010
$670$	0	0
$671$	0	0
$672$	3.63366	0.140171
$673$	−19.1791	−0.739300	−0.369650	−	0.929171i	$-0.620522\pi$
$673$	−19.1791	−0.739300	−0.369650	+	0.929171i	$0.620522\pi$
$674$	−0.316958	−0.0122088
$675$	0	0
$676$	−63.1605	−2.42925
$677$	33.8934	1.30263	0.651314	−	0.758808i	$-0.274218\pi$
$677$	33.8934	1.30263	0.651314	+	0.758808i	$0.274218\pi$
$678$	−0.741902	−0.0284926
$679$	−7.05422	−0.270716
$680$	0	0
$681$	15.7660	0.604153
$682$	0	0
$683$	−19.1578	−0.733054	−0.366527	−	0.930408i	$-0.619453\pi$
$683$	−19.1578	−0.733054	−0.366527	+	0.930408i	$0.619453\pi$
$684$	−4.40701	−0.168506
$685$	0	0
$686$	3.49813	0.133559
$687$	−5.09266	−0.194297
$688$	−9.78950	−0.373221
$689$	−6.92116	−0.263675
$690$	0	0
$691$	−17.7125	−0.673816	−0.336908	−	0.941538i	$-0.609381\pi$
$691$	−17.7125	−0.673816	−0.336908	+	0.941538i	$0.609381\pi$
$692$	−28.4029	−1.07972
$693$	0	0
$694$	6.39889	0.242898
$695$	0	0
$696$	7.04674	0.267106
$697$	−7.95633	−0.301367
$698$	5.56093	0.210484
$699$	−12.0061	−0.454112
$700$	0	0
$701$	10.0017	0.377759	0.188879	−	0.982000i	$-0.439515\pi$
$701$	10.0017	0.377759	0.188879	+	0.982000i	$0.439515\pi$
$702$	−1.26362	−0.0476924
$703$	21.6855	0.817884
$704$	0	0
$705$	0	0
$706$	4.46629	0.168091
$707$	22.1179	0.831830
$708$	15.2616	0.573566
$709$	−34.0416	−1.27846	−0.639230	−	0.769016i	$-0.720747\pi$
$709$	−34.0416	−1.27846	−0.639230	+	0.769016i	$0.720747\pi$
$710$	0	0
$711$	10.2974	0.386182
$712$	−8.49484	−0.318358
$713$	31.7361	1.18853
$714$	0.362459	0.0135647
$715$	0	0
$716$	−22.6101	−0.844981
$717$	−18.7618	−0.700672
$718$	−0.621652	−0.0231998
$719$	20.2917	0.756751	0.378376	−	0.925652i	$-0.376483\pi$
$719$	20.2917	0.756751	0.378376	+	0.925652i	$0.376483\pi$
$720$	0	0
$721$	16.4387	0.612207
$722$	−2.62685	−0.0977613
$723$	−3.88549	−0.144503
$724$	36.1685	1.34419
$725$	0	0
$726$	0	0
$727$	−18.5664	−0.688590	−0.344295	−	0.938862i	$-0.611882\pi$
$727$	−18.5664	−0.688590	−0.344295	+	0.938862i	$0.611882\pi$
$728$	8.26046	0.306153
$729$	1.00000	0.0370370
$730$	0	0
$731$	−3.02002	−0.111700
$732$	−3.92186	−0.144956
$733$	3.06859	0.113341	0.0566705	−	0.998393i	$-0.481952\pi$
$733$	3.06859	0.113341	0.0566705	+	0.998393i	$0.481952\pi$
$734$	−2.43464	−0.0898641
$735$	0	0
$736$	18.7225	0.690120
$737$	0	0
$738$	−1.28005	−0.0471194
$739$	21.9285	0.806652	0.403326	−	0.915056i	$-0.367854\pi$
$739$	21.9285	0.806652	0.403326	+	0.915056i	$0.367854\pi$
$740$	0	0
$741$	−15.0725	−0.553701
$742$	0.319400	0.0117255
$743$	1.01348	0.0371810	0.0185905	−	0.999827i	$-0.494082\pi$
$743$	1.01348	0.0371810	0.0185905	+	0.999827i	$0.494082\pi$
$744$	2.78515	0.102109
$745$	0	0
$746$	−2.10020	−0.0768939
$747$	2.47744	0.0906447
$748$	0	0
$749$	4.51197	0.164864
$750$	0	0
$751$	44.5293	1.62490	0.812449	−	0.583032i	$-0.198134\pi$
$751$	44.5293	1.62490	0.812449	+	0.583032i	$0.198134\pi$
$752$	−29.0577	−1.05963
$753$	17.7497	0.646833
$754$	11.9428	0.434933
$755$	0	0
$756$	−3.23940	−0.117816
$757$	−51.5432	−1.87337	−0.936686	−	0.350172i	$-0.886123\pi$
$757$	−51.5432	−1.87337	−0.936686	+	0.350172i	$0.886123\pi$
$758$	−3.93956	−0.143091
$759$	0	0
$760$	0	0
$761$	10.4611	0.379216	0.189608	−	0.981860i	$-0.439278\pi$
$761$	10.4611	0.379216	0.189608	+	0.981860i	$0.439278\pi$
$762$	1.21154	0.0438896
$763$	−25.2695	−0.914818
$764$	−27.9293	−1.01045
$765$	0	0
$766$	−6.10981	−0.220756
$767$	52.1965	1.88471
$768$	−13.2451	−0.477942
$769$	21.2997	0.768089	0.384044	−	0.923315i	$-0.374531\pi$
$769$	21.2997	0.768089	0.384044	+	0.923315i	$0.374531\pi$
$770$	0	0
$771$	5.91917	0.213174
$772$	−4.32192	−0.155549
$773$	33.1238	1.19138	0.595691	−	0.803214i	$-0.296878\pi$
$773$	33.1238	1.19138	0.595691	+	0.803214i	$0.296878\pi$
$774$	−0.485876	−0.0174644
$775$	0	0
$776$	−3.18980	−0.114507
$777$	15.9401	0.571847
$778$	−5.22103	−0.187183
$779$	−15.2684	−0.547048
$780$	0	0
$781$	0	0
$782$	1.86758	0.0667844
$783$	−9.45126	−0.337761
$784$	−16.2220	−0.579355
$785$	0	0
$786$	−2.68690	−0.0958387
$787$	−42.5368	−1.51627	−0.758137	−	0.652095i	$-0.773890\pi$
$787$	−42.5368	−1.51627	−0.758137	+	0.652095i	$0.773890\pi$
$788$	37.5120	1.33631
$789$	−1.73320	−0.0617034
$790$	0	0
$791$	−6.50481	−0.231284
$792$	0	0
$793$	−13.4132	−0.476318
$794$	−0.948506	−0.0336612
$795$	0	0
$796$	−3.51016	−0.124414
$797$	33.0475	1.17060	0.585301	−	0.810816i	$-0.300976\pi$
$797$	33.0475	1.17060	0.585301	+	0.810816i	$0.300976\pi$
$798$	0.695569	0.0246229
$799$	−8.96420	−0.317130
$800$	0	0
$801$	11.3935	0.402569
$802$	4.50987	0.159249
$803$	0	0
$804$	−16.4509	−0.580180
$805$	0	0
$806$	4.72029	0.166265
$807$	28.1477	0.990847
$808$	10.0014	0.351847
$809$	47.7436	1.67858	0.839288	−	0.543687i	$-0.182972\pi$
$809$	47.7436	1.67858	0.839288	+	0.543687i	$0.182972\pi$
$810$	0	0
$811$	−30.3565	−1.06596	−0.532981	−	0.846127i	$-0.678928\pi$
$811$	−30.3565	−1.06596	−0.532981	+	0.846127i	$0.678928\pi$
$812$	30.6164	1.07443
$813$	−18.5941	−0.652123
$814$	0	0
$815$	0	0
$816$	−4.42906	−0.155048
$817$	−5.79551	−0.202759
$818$	−0.129083	−0.00451329
$819$	−11.0791	−0.387136
$820$	0	0
$821$	18.0543	0.630100	0.315050	−	0.949075i	$-0.397979\pi$
$821$	18.0543	0.630100	0.315050	+	0.949075i	$0.397979\pi$
$822$	−2.20869	−0.0770369
$823$	−24.2551	−0.845479	−0.422739	−	0.906251i	$-0.638931\pi$
$823$	−24.2551	−0.845479	−0.422739	+	0.906251i	$0.638931\pi$
$824$	7.43330	0.258951
$825$	0	0
$826$	−2.40878	−0.0838121
$827$	35.5556	1.23639	0.618194	−	0.786025i	$-0.287865\pi$
$827$	35.5556	1.23639	0.618194	+	0.786025i	$0.287865\pi$
$828$	−16.6911	−0.580055
$829$	40.6182	1.41073	0.705364	−	0.708845i	$-0.250783\pi$
$829$	40.6182	1.41073	0.705364	+	0.708845i	$0.250783\pi$
$830$	0	0
$831$	12.1890	0.422831
$832$	−48.1347	−1.66877
$833$	−5.00441	−0.173393
$834$	3.04098	0.105300
$835$	0	0
$836$	0	0
$837$	−3.73551	−0.129118
$838$	2.96232	0.102332
$839$	28.6355	0.988608	0.494304	−	0.869289i	$-0.335423\pi$
$839$	28.6355	0.988608	0.494304	+	0.869289i	$0.335423\pi$
$840$	0	0
$841$	60.3264	2.08022
$842$	5.22896	0.180202
$843$	−10.4234	−0.359002
$844$	−17.1048	−0.588772
$845$	0	0
$846$	−1.44220	−0.0495839
$847$	0	0
$848$	−3.90289	−0.134026
$849$	16.4996	0.566266
$850$	0	0
$851$	82.1316	2.81544
$852$	−7.40778	−0.253786
$853$	25.6964	0.879827	0.439913	−	0.898040i	$-0.355009\pi$
$853$	25.6964	0.879827	0.439913	+	0.898040i	$0.355009\pi$
$854$	0.618998	0.0211817
$855$	0	0
$856$	2.04024	0.0697339
$857$	0.635176	0.0216972	0.0108486	−	0.999941i	$-0.496547\pi$
$857$	0.635176	0.0216972	0.0108486	+	0.999941i	$0.496547\pi$
$858$	0	0
$859$	−35.4262	−1.20873	−0.604363	−	0.796709i	$-0.706572\pi$
$859$	−35.4262	−1.20873	−0.604363	+	0.796709i	$0.706572\pi$
$860$	0	0
$861$	−11.2232	−0.382485
$862$	4.55202	0.155042
$863$	−8.61480	−0.293251	−0.146626	−	0.989192i	$-0.546841\pi$
$863$	−8.61480	−0.293251	−0.146626	+	0.989192i	$0.546841\pi$
$864$	−2.20374	−0.0749728
$865$	0	0
$866$	3.21708	0.109321
$867$	15.6337	0.530947
$868$	12.1008	0.410729
$869$	0	0
$870$	0	0
$871$	−56.2642	−1.90644
$872$	−11.4265	−0.386949
$873$	4.27825	0.144797
$874$	3.58393	0.121228
$875$	0	0
$876$	−11.5230	−0.389327
$877$	−33.2671	−1.12335	−0.561676	−	0.827358i	$-0.689843\pi$
$877$	−33.2671	−1.12335	−0.561676	+	0.827358i	$0.689843\pi$
$878$	2.83342	0.0956233
$879$	30.0307	1.01291
$880$	0	0
$881$	−19.6147	−0.660836	−0.330418	−	0.943835i	$-0.607190\pi$
$881$	−19.6147	−0.660836	−0.330418	+	0.943835i	$0.607190\pi$
$882$	−0.805133	−0.0271102
$883$	7.62230	0.256511	0.128255	−	0.991741i	$-0.459062\pi$
$883$	7.62230	0.256511	0.128255	+	0.991741i	$0.459062\pi$
$884$	−15.4307	−0.518990
$885$	0	0
$886$	−5.72819	−0.192442
$887$	0.131658	0.00442064	0.00221032	−	0.999998i	$-0.499296\pi$
$887$	0.131658	0.00442064	0.00221032	+	0.999998i	$0.499296\pi$
$888$	7.20785	0.241880
$889$	10.6225	0.356267
$890$	0	0
$891$	0	0
$892$	5.23448	0.175264
$893$	−17.2026	−0.575661
$894$	−2.52745	−0.0845304
$895$	0	0
$896$	9.48864	0.316993
$897$	−57.0855	−1.90603
$898$	−0.914344	−0.0305120
$899$	35.3053	1.17750
$900$	0	0
$901$	−1.20403	−0.0401120
$902$	0	0
$903$	−4.26003	−0.141765
$904$	−2.94137	−0.0978285
$905$	0	0
$906$	−1.23203	−0.0409316
$907$	21.3370	0.708485	0.354242	−	0.935154i	$-0.384739\pi$
$907$	21.3370	0.708485	0.354242	+	0.935154i	$0.384739\pi$
$908$	30.9743	1.02792
$909$	−13.4141	−0.444917
$910$	0	0
$911$	14.4414	0.478466	0.239233	−	0.970962i	$-0.423104\pi$
$911$	14.4414	0.478466	0.239233	+	0.970962i	$0.423104\pi$
$912$	−8.49948	−0.281446
$913$	0	0
$914$	−2.39675	−0.0792775
$915$	0	0
$916$	−10.0052	−0.330581
$917$	−23.5581	−0.777956
$918$	−0.219824	−0.00725528
$919$	−46.0459	−1.51891	−0.759457	−	0.650557i	$-0.774535\pi$
$919$	−46.0459	−1.51891	−0.759457	+	0.650557i	$0.774535\pi$
$920$	0	0
$921$	10.4942	0.345794
$922$	4.26943	0.140606
$923$	−25.3355	−0.833927
$924$	0	0
$925$	0	0
$926$	5.60199	0.184093
$927$	−9.96972	−0.327449
$928$	20.8281	0.683717
$929$	−1.34861	−0.0442466	−0.0221233	−	0.999755i	$-0.507043\pi$
$929$	−1.34861	−0.0442466	−0.0221233	+	0.999755i	$0.507043\pi$
$930$	0	0
$931$	−9.60361	−0.314746
$932$	−23.5876	−0.772636
$933$	30.7119	1.00546
$934$	−0.528304	−0.0172866
$935$	0	0
$936$	−5.00981	−0.163751
$937$	−37.7761	−1.23409	−0.617046	−	0.786927i	$-0.711671\pi$
$937$	−37.7761	−1.23409	−0.617046	+	0.786927i	$0.711671\pi$
$938$	2.59649	0.0847785
$939$	−8.45228	−0.275830
$940$	0	0
$941$	−18.4804	−0.602445	−0.301222	−	0.953554i	$-0.597395\pi$
$941$	−18.4804	−0.602445	−0.301222	+	0.953554i	$0.597395\pi$
$942$	−2.20407	−0.0718125
$943$	−57.8276	−1.88313
$944$	29.4340	0.957995
$945$	0	0
$946$	0	0
$947$	11.1495	0.362310	0.181155	−	0.983455i	$-0.442016\pi$
$947$	11.1495	0.362310	0.181155	+	0.983455i	$0.442016\pi$
$948$	20.2306	0.657059
$949$	−39.4101	−1.27931
$950$	0	0
$951$	17.4886	0.567107
$952$	1.43702	0.0465740
$953$	41.2343	1.33571	0.667854	−	0.744292i	$-0.267213\pi$
$953$	41.2343	1.33571	0.667854	+	0.744292i	$0.267213\pi$
$954$	−0.193710	−0.00627158
$955$	0	0
$956$	−36.8601	−1.19214
$957$	0	0
$958$	−1.11420	−0.0359983
$959$	−19.3652	−0.625336
$960$	0	0
$961$	−17.0459	−0.549869
$962$	12.2159	0.393856
$963$	−2.73642	−0.0881799
$964$	−7.63356	−0.245860
$965$	0	0
$966$	2.63440	0.0847603
$967$	2.72062	0.0874892	0.0437446	−	0.999043i	$-0.486071\pi$
$967$	2.72062	0.0874892	0.0437446	+	0.999043i	$0.486071\pi$
$968$	0	0
$969$	−2.62206	−0.0842327
$970$	0	0
$971$	12.1034	0.388417	0.194209	−	0.980960i	$-0.437786\pi$
$971$	12.1034	0.388417	0.194209	+	0.980960i	$0.437786\pi$
$972$	1.96463	0.0630157
$973$	26.6625	0.854761
$974$	−6.45088	−0.206700
$975$	0	0
$976$	−7.56382	−0.242112
$977$	24.5003	0.783834	0.391917	−	0.920001i	$-0.371812\pi$
$977$	24.5003	0.783834	0.391917	+	0.920001i	$0.371812\pi$
$978$	−0.579391	−0.0185269
$979$	0	0
$980$	0	0
$981$	15.3255	0.489305
$982$	1.23663	0.0394625
$983$	14.8847	0.474748	0.237374	−	0.971418i	$-0.423713\pi$
$983$	14.8847	0.474748	0.237374	+	0.971418i	$0.423713\pi$
$984$	−5.07494	−0.161783
$985$	0	0
$986$	2.07762	0.0661648
$987$	−12.6449	−0.402490
$988$	−29.6119	−0.942080
$989$	−21.9499	−0.697967
$990$	0	0
$991$	27.8902	0.885961	0.442980	−	0.896531i	$-0.353921\pi$
$991$	27.8902	0.885961	0.442980	+	0.896531i	$0.353921\pi$
$992$	8.23211	0.261370
$993$	−6.42686	−0.203950
$994$	1.16919	0.0370844
$995$	0	0
$996$	4.86726	0.154225
$997$	−19.4309	−0.615384	−0.307692	−	0.951486i	$-0.599557\pi$
$997$	−19.4309	−0.615384	−0.307692	+	0.951486i	$0.599557\pi$
$998$	−3.16456	−0.100172
$999$	−9.66735	−0.305861

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9075.2.a.dt.1.5		8
5.4	even	2		9075.2.a.dw.1.4		8
11.7	odd	10		825.2.n.n.676.2	yes	16
11.8	odd	10		825.2.n.n.526.2	yes	16
11.10	odd	2		9075.2.a.dv.1.4		8
55.7	even	20		825.2.bx.j.49.5		32
55.8	even	20		825.2.bx.j.724.5		32
55.18	even	20		825.2.bx.j.49.4		32
55.19	odd	10		825.2.n.m.526.3	✓	16
55.29	odd	10		825.2.n.m.676.3	yes	16
55.52	even	20		825.2.bx.j.724.4		32
55.54	odd	2		9075.2.a.du.1.5		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
825.2.n.m.526.3	✓	16	55.19	odd	10
825.2.n.m.676.3	yes	16	55.29	odd	10
825.2.n.n.526.2	yes	16	11.8	odd	10
825.2.n.n.676.2	yes	16	11.7	odd	10
825.2.bx.j.49.4		32	55.18	even	20
825.2.bx.j.49.5		32	55.7	even	20
825.2.bx.j.724.4		32	55.52	even	20
825.2.bx.j.724.5		32	55.8	even	20
9075.2.a.dt.1.5		8	1.1	even	1	trivial
9075.2.a.du.1.5		8	55.54	odd	2
9075.2.a.dv.1.4		8	11.10	odd	2
9075.2.a.dw.1.4		8	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 9075 = 3 \cdot 5^{2} \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9075.a (trivial)

\(f(q)\)	\(=\)	\(q+0.188059 q^{2} -1.00000 q^{3} -1.96463 q^{4} -0.188059 q^{6} -1.64886 q^{7} -0.745587 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+0.188059 q^{2} -1.00000 q^{3} -1.96463 q^{4} -0.188059 q^{6} -1.64886 q^{7} -0.745587 q^{8} +1.00000 q^{9} +1.96463 q^{12} +6.71928 q^{13} -0.310083 q^{14} +3.78905 q^{16} +1.16891 q^{17} +0.188059 q^{18} +2.24317 q^{19} +1.64886 q^{21} +8.49577 q^{23} +0.745587 q^{24} +1.26362 q^{26} -1.00000 q^{27} +3.23940 q^{28} +9.45126 q^{29} +3.73551 q^{31} +2.20374 q^{32} +0.219824 q^{34} -1.96463 q^{36} +9.66735 q^{37} +0.421849 q^{38} -6.71928 q^{39} -6.80663 q^{41} +0.310083 q^{42} -2.58363 q^{43} +1.59771 q^{46} -7.66886 q^{47} -3.78905 q^{48} -4.28127 q^{49} -1.16891 q^{51} -13.2009 q^{52} -1.03004 q^{53} -0.188059 q^{54} +1.22937 q^{56} -2.24317 q^{57} +1.77740 q^{58} +7.76816 q^{59} -1.99623 q^{61} +0.702499 q^{62} -1.64886 q^{63} -7.16367 q^{64} -8.37354 q^{67} -2.29648 q^{68} -8.49577 q^{69} -3.77056 q^{71} -0.745587 q^{72} -5.86523 q^{73} +1.81804 q^{74} -4.40701 q^{76} -1.26362 q^{78} +10.2974 q^{79} +1.00000 q^{81} -1.28005 q^{82} +2.47744 q^{83} -3.23940 q^{84} -0.485876 q^{86} -9.45126 q^{87} +11.3935 q^{89} -11.0791 q^{91} -16.6911 q^{92} -3.73551 q^{93} -1.44220 q^{94} -2.20374 q^{96} +4.27825 q^{97} -0.805133 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - q^{2} - 8 q^{3} + 7 q^{4} + q^{6} - 8 q^{7} - 3 q^{8} + 8 q^{9}+O(q^{10}) \) 8 * q - q^2 - 8 * q^3 + 7 * q^4 + q^6 - 8 * q^7 - 3 * q^8 + 8 * q^9	\( 8 q - q^{2} - 8 q^{3} + 7 q^{4} + q^{6} - 8 q^{7} - 3 q^{8} + 8 q^{9} - 7 q^{12} - 7 q^{13} - 8 q^{14} - 7 q^{16} + q^{17} - q^{18} + 6 q^{19} + 8 q^{21} - 7 q^{23} + 3 q^{24} - q^{26} - 8 q^{27} - 11 q^{28} + 28 q^{29} + 10 q^{31} - 12 q^{32} - 8 q^{34} + 7 q^{36} - 14 q^{37} - 19 q^{38} + 7 q^{39} + 14 q^{41} + 8 q^{42} - 11 q^{43} - 5 q^{47} + 7 q^{48} + 2 q^{49} - q^{51} - q^{52} + 5 q^{53} + q^{54} - 4 q^{56} - 6 q^{57} + 3 q^{58} - 2 q^{59} + 36 q^{61} + 17 q^{62} - 8 q^{63} - 23 q^{64} - 6 q^{67} + 3 q^{68} + 7 q^{69} - 17 q^{71} - 3 q^{72} - 32 q^{73} + 47 q^{74} + 5 q^{76} + q^{78} + 41 q^{79} + 8 q^{81} + 6 q^{83} + 11 q^{84} - 9 q^{86} - 28 q^{87} + 21 q^{89} - 9 q^{91} - 69 q^{92} - 10 q^{93} + 51 q^{94} + 12 q^{96} + 4 q^{97} + 27 q^{98}+O(q^{100}) \) 8 * q - q^2 - 8 * q^3 + 7 * q^4 + q^6 - 8 * q^7 - 3 * q^8 + 8 * q^9 - 7 * q^12 - 7 * q^13 - 8 * q^14 - 7 * q^16 + q^17 - q^18 + 6 * q^19 + 8 * q^21 - 7 * q^23 + 3 * q^24 - q^26 - 8 * q^27 - 11 * q^28 + 28 * q^29 + 10 * q^31 - 12 * q^32 - 8 * q^34 + 7 * q^36 - 14 * q^37 - 19 * q^38 + 7 * q^39 + 14 * q^41 + 8 * q^42 - 11 * q^43 - 5 * q^47 + 7 * q^48 + 2 * q^49 - q^51 - q^52 + 5 * q^53 + q^54 - 4 * q^56 - 6 * q^57 + 3 * q^58 - 2 * q^59 + 36 * q^61 + 17 * q^62 - 8 * q^63 - 23 * q^64 - 6 * q^67 + 3 * q^68 + 7 * q^69 - 17 * q^71 - 3 * q^72 - 32 * q^73 + 47 * q^74 + 5 * q^76 + q^78 + 41 * q^79 + 8 * q^81 + 6 * q^83 + 11 * q^84 - 9 * q^86 - 28 * q^87 + 21 * q^89 - 9 * q^91 - 69 * q^92 - 10 * q^93 + 51 * q^94 + 12 * q^96 + 4 * q^97 + 27 * q^98

Introduction

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