LMFDB - Embedded newform 8550.2.a.bu.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8550,2,Mod(1,8550)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8550, 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("8550.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8550 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8550.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:	$68.2720937282$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{6}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 6 $ x^2 - 6
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 2850)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.44949$ of defining polynomial
Character	$\chi$	$=$	8550.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} +4.44949 q^{7} -1.00000 q^{8} +O(q^{10})$	$q-1.00000 q^{2} +1.00000 q^{4} +4.44949 q^{7} -1.00000 q^{8} -3.44949 q^{11} -2.44949 q^{13} -4.44949 q^{14} +1.00000 q^{16} -4.44949 q^{17} +1.00000 q^{19} +3.44949 q^{22} +1.00000 q^{23} +2.44949 q^{26} +4.44949 q^{28} +4.34847 q^{29} -3.00000 q^{31} -1.00000 q^{32} +4.44949 q^{34} -7.79796 q^{37} -1.00000 q^{38} +0.898979 q^{41} -2.44949 q^{43} -3.44949 q^{44} -1.00000 q^{46} +7.79796 q^{47} +12.7980 q^{49} -2.44949 q^{52} +7.44949 q^{53} -4.44949 q^{56} -4.34847 q^{58} -6.44949 q^{59} -9.44949 q^{61} +3.00000 q^{62} +1.00000 q^{64} +15.2474 q^{67} -4.44949 q^{68} +1.55051 q^{71} -1.00000 q^{73} +7.79796 q^{74} +1.00000 q^{76} -15.3485 q^{77} -5.00000 q^{79} -0.898979 q^{82} -8.34847 q^{83} +2.44949 q^{86} +3.44949 q^{88} -2.10102 q^{89} -10.8990 q^{91} +1.00000 q^{92} -7.79796 q^{94} +1.55051 q^{97} -12.7980 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} + 2 q^{4} + 4 q^{7} - 2 q^{8}+O(q^{10}) $ 2 * q - 2 * q^2 + 2 * q^4 + 4 * q^7 - 2 * q^8	$ 2 q - 2 q^{2} + 2 q^{4} + 4 q^{7} - 2 q^{8} - 2 q^{11} - 4 q^{14} + 2 q^{16} - 4 q^{17} + 2 q^{19} + 2 q^{22} + 2 q^{23} + 4 q^{28} - 6 q^{29} - 6 q^{31} - 2 q^{32} + 4 q^{34} + 4 q^{37} - 2 q^{38} - 8 q^{41} - 2 q^{44} - 2 q^{46} - 4 q^{47} + 6 q^{49} + 10 q^{53} - 4 q^{56} + 6 q^{58} - 8 q^{59} - 14 q^{61} + 6 q^{62} + 2 q^{64} + 6 q^{67} - 4 q^{68} + 8 q^{71} - 2 q^{73} - 4 q^{74} + 2 q^{76} - 16 q^{77} - 10 q^{79} + 8 q^{82} - 2 q^{83} + 2 q^{88} - 14 q^{89} - 12 q^{91} + 2 q^{92} + 4 q^{94} + 8 q^{97} - 6 q^{98}+O(q^{100}) $ 2 * q - 2 * q^2 + 2 * q^4 + 4 * q^7 - 2 * q^8 - 2 * q^11 - 4 * q^14 + 2 * q^16 - 4 * q^17 + 2 * q^19 + 2 * q^22 + 2 * q^23 + 4 * q^28 - 6 * q^29 - 6 * q^31 - 2 * q^32 + 4 * q^34 + 4 * q^37 - 2 * q^38 - 8 * q^41 - 2 * q^44 - 2 * q^46 - 4 * q^47 + 6 * q^49 + 10 * q^53 - 4 * q^56 + 6 * q^58 - 8 * q^59 - 14 * q^61 + 6 * q^62 + 2 * q^64 + 6 * q^67 - 4 * q^68 + 8 * q^71 - 2 * q^73 - 4 * q^74 + 2 * q^76 - 16 * q^77 - 10 * q^79 + 8 * q^82 - 2 * q^83 + 2 * q^88 - 14 * q^89 - 12 * q^91 + 2 * q^92 + 4 * q^94 + 8 * q^97 - 6 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	0	0
$7$	4.44949	1.68175	0.840875	−	0.541230i	$-0.182041\pi$
$7$	4.44949	1.68175	0.840875	+	0.541230i	$0.182041\pi$
$8$	−1.00000	−0.353553
$9$	0	0
$10$	0	0
$11$	−3.44949	−1.04006	−0.520030	−	0.854148i	$-0.674079\pi$
$11$	−3.44949	−1.04006	−0.520030	+	0.854148i	$0.674079\pi$
$12$	0	0
$13$	−2.44949	−0.679366	−0.339683	−	0.940540i	$-0.610320\pi$
$13$	−2.44949	−0.679366	−0.339683	+	0.940540i	$0.610320\pi$
$14$	−4.44949	−1.18918
$15$	0	0
$16$	1.00000	0.250000
$17$	−4.44949	−1.07916	−0.539580	−	0.841934i	$-0.681417\pi$
$17$	−4.44949	−1.07916	−0.539580	+	0.841934i	$0.681417\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	0	0
$22$	3.44949	0.735434
$23$	1.00000	0.208514	0.104257	−	0.994550i	$-0.466753\pi$
$23$	1.00000	0.208514	0.104257	+	0.994550i	$0.466753\pi$
$24$	0	0
$25$	0	0
$26$	2.44949	0.480384
$27$	0	0
$28$	4.44949	0.840875
$29$	4.34847	0.807490	0.403745	−	0.914871i	$-0.367708\pi$
$29$	4.34847	0.807490	0.403745	+	0.914871i	$0.367708\pi$
$30$	0	0
$31$	−3.00000	−0.538816	−0.269408	−	0.963026i	$-0.586828\pi$
$31$	−3.00000	−0.538816	−0.269408	+	0.963026i	$0.586828\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	4.44949	0.763081
$35$	0	0
$36$	0	0
$37$	−7.79796	−1.28198	−0.640988	−	0.767551i	$-0.721475\pi$
$37$	−7.79796	−1.28198	−0.640988	+	0.767551i	$0.721475\pi$
$38$	−1.00000	−0.162221
$39$	0	0
$40$	0	0
$41$	0.898979	0.140397	0.0701985	−	0.997533i	$-0.477637\pi$
$41$	0.898979	0.140397	0.0701985	+	0.997533i	$0.477637\pi$
$42$	0	0
$43$	−2.44949	−0.373544	−0.186772	−	0.982403i	$-0.559803\pi$
$43$	−2.44949	−0.373544	−0.186772	+	0.982403i	$0.559803\pi$
$44$	−3.44949	−0.520030
$45$	0	0
$46$	−1.00000	−0.147442
$47$	7.79796	1.13745	0.568725	−	0.822528i	$-0.307437\pi$
$47$	7.79796	1.13745	0.568725	+	0.822528i	$0.307437\pi$
$48$	0	0
$49$	12.7980	1.82828
$50$	0	0
$51$	0	0
$52$	−2.44949	−0.339683
$53$	7.44949	1.02327	0.511633	−	0.859204i	$-0.329041\pi$
$53$	7.44949	1.02327	0.511633	+	0.859204i	$0.329041\pi$
$54$	0	0
$55$	0	0
$56$	−4.44949	−0.594588
$57$	0	0
$58$	−4.34847	−0.570982
$59$	−6.44949	−0.839652	−0.419826	−	0.907605i	$-0.637909\pi$
$59$	−6.44949	−0.839652	−0.419826	+	0.907605i	$0.637909\pi$
$60$	0	0
$61$	−9.44949	−1.20988	−0.604942	−	0.796270i	$-0.706804\pi$
$61$	−9.44949	−1.20988	−0.604942	+	0.796270i	$0.706804\pi$
$62$	3.00000	0.381000
$63$	0	0
$64$	1.00000	0.125000
$65$	0	0
$66$	0	0
$67$	15.2474	1.86277	0.931386	−	0.364033i	$-0.118600\pi$
$67$	15.2474	1.86277	0.931386	+	0.364033i	$0.118600\pi$
$68$	−4.44949	−0.539580
$69$	0	0
$70$	0	0
$71$	1.55051	0.184012	0.0920059	−	0.995758i	$-0.470672\pi$
$71$	1.55051	0.184012	0.0920059	+	0.995758i	$0.470672\pi$
$72$	0	0
$73$	−1.00000	−0.117041	−0.0585206	−	0.998286i	$-0.518638\pi$
$73$	−1.00000	−0.117041	−0.0585206	+	0.998286i	$0.518638\pi$
$74$	7.79796	0.906494
$75$	0	0
$76$	1.00000	0.114708
$77$	−15.3485	−1.74912
$78$	0	0
$79$	−5.00000	−0.562544	−0.281272	−	0.959628i	$-0.590756\pi$
$79$	−5.00000	−0.562544	−0.281272	+	0.959628i	$0.590756\pi$
$80$	0	0
$81$	0	0
$82$	−0.898979	−0.0992757
$83$	−8.34847	−0.916364	−0.458182	−	0.888859i	$-0.651499\pi$
$83$	−8.34847	−0.916364	−0.458182	+	0.888859i	$0.651499\pi$
$84$	0	0
$85$	0	0
$86$	2.44949	0.264135
$87$	0	0
$88$	3.44949	0.367717
$89$	−2.10102	−0.222708	−0.111354	−	0.993781i	$-0.535519\pi$
$89$	−2.10102	−0.222708	−0.111354	+	0.993781i	$0.535519\pi$
$90$	0	0
$91$	−10.8990	−1.14252
$92$	1.00000	0.104257
$93$	0	0
$94$	−7.79796	−0.804298
$95$	0	0
$96$	0	0
$97$	1.55051	0.157430	0.0787152	−	0.996897i	$-0.474918\pi$
$97$	1.55051	0.157430	0.0787152	+	0.996897i	$0.474918\pi$
$98$	−12.7980	−1.29279
$99$	0	0
$100$	0	0
$101$	1.55051	0.154282	0.0771408	−	0.997020i	$-0.475421\pi$
$101$	1.55051	0.154282	0.0771408	+	0.997020i	$0.475421\pi$
$102$	0	0
$103$	1.89898	0.187112	0.0935560	−	0.995614i	$-0.470177\pi$
$103$	1.89898	0.187112	0.0935560	+	0.995614i	$0.470177\pi$
$104$	2.44949	0.240192
$105$	0	0
$106$	−7.44949	−0.723558
$107$	−17.3485	−1.67714	−0.838570	−	0.544794i	$-0.816608\pi$
$107$	−17.3485	−1.67714	−0.838570	+	0.544794i	$0.816608\pi$
$108$	0	0
$109$	−2.89898	−0.277672	−0.138836	−	0.990315i	$-0.544336\pi$
$109$	−2.89898	−0.277672	−0.138836	+	0.990315i	$0.544336\pi$
$110$	0	0
$111$	0	0
$112$	4.44949	0.420437
$113$	16.7980	1.58022	0.790110	−	0.612966i	$-0.210024\pi$
$113$	16.7980	1.58022	0.790110	+	0.612966i	$0.210024\pi$
$114$	0	0
$115$	0	0
$116$	4.34847	0.403745
$117$	0	0
$118$	6.44949	0.593724
$119$	−19.7980	−1.81488
$120$	0	0
$121$	0.898979	0.0817254
$122$	9.44949	0.855517
$123$	0	0
$124$	−3.00000	−0.269408
$125$	0	0
$126$	0	0
$127$	0.101021	0.00896412	0.00448206	−	0.999990i	$-0.498573\pi$
$127$	0.101021	0.00896412	0.00448206	+	0.999990i	$0.498573\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	0	0
$131$	−9.24745	−0.807953	−0.403977	−	0.914769i	$-0.632372\pi$
$131$	−9.24745	−0.807953	−0.403977	+	0.914769i	$0.632372\pi$
$132$	0	0
$133$	4.44949	0.385820
$134$	−15.2474	−1.31718
$135$	0	0
$136$	4.44949	0.381541
$137$	4.89898	0.418548	0.209274	−	0.977857i	$-0.432890\pi$
$137$	4.89898	0.418548	0.209274	+	0.977857i	$0.432890\pi$
$138$	0	0
$139$	−22.2474	−1.88700	−0.943502	−	0.331367i	$-0.892490\pi$
$139$	−22.2474	−1.88700	−0.943502	+	0.331367i	$0.892490\pi$
$140$	0	0
$141$	0	0
$142$	−1.55051	−0.130116
$143$	8.44949	0.706582
$144$	0	0
$145$	0	0
$146$	1.00000	0.0827606
$147$	0	0
$148$	−7.79796	−0.640988
$149$	−5.79796	−0.474987	−0.237494	−	0.971389i	$-0.576326\pi$
$149$	−5.79796	−0.474987	−0.237494	+	0.971389i	$0.576326\pi$
$150$	0	0
$151$	−23.7980	−1.93665	−0.968325	−	0.249692i	$-0.919671\pi$
$151$	−23.7980	−1.93665	−0.968325	+	0.249692i	$0.919671\pi$
$152$	−1.00000	−0.0811107
$153$	0	0
$154$	15.3485	1.23681
$155$	0	0
$156$	0	0
$157$	−4.89898	−0.390981	−0.195491	−	0.980706i	$-0.562630\pi$
$157$	−4.89898	−0.390981	−0.195491	+	0.980706i	$0.562630\pi$
$158$	5.00000	0.397779
$159$	0	0
$160$	0	0
$161$	4.44949	0.350669
$162$	0	0
$163$	19.7980	1.55070	0.775348	−	0.631534i	$-0.217575\pi$
$163$	19.7980	1.55070	0.775348	+	0.631534i	$0.217575\pi$
$164$	0.898979	0.0701985
$165$	0	0
$166$	8.34847	0.647967
$167$	−18.0000	−1.39288	−0.696441	−	0.717614i	$-0.745234\pi$
$167$	−18.0000	−1.39288	−0.696441	+	0.717614i	$0.745234\pi$
$168$	0	0
$169$	−7.00000	−0.538462
$170$	0	0
$171$	0	0
$172$	−2.44949	−0.186772
$173$	1.65153	0.125564	0.0627818	−	0.998027i	$-0.480003\pi$
$173$	1.65153	0.125564	0.0627818	+	0.998027i	$0.480003\pi$
$174$	0	0
$175$	0	0
$176$	−3.44949	−0.260015
$177$	0	0
$178$	2.10102	0.157478
$179$	−25.7980	−1.92823	−0.964115	−	0.265485i	$-0.914468\pi$
$179$	−25.7980	−1.92823	−0.964115	+	0.265485i	$0.914468\pi$
$180$	0	0
$181$	−14.4495	−1.07402	−0.537011	−	0.843575i	$-0.680447\pi$
$181$	−14.4495	−1.07402	−0.537011	+	0.843575i	$0.680447\pi$
$182$	10.8990	0.807886
$183$	0	0
$184$	−1.00000	−0.0737210
$185$	0	0
$186$	0	0
$187$	15.3485	1.12239
$188$	7.79796	0.568725
$189$	0	0
$190$	0	0
$191$	0.101021	0.00730959	0.00365479	−	0.999993i	$-0.498837\pi$
$191$	0.101021	0.00730959	0.00365479	+	0.999993i	$0.498837\pi$
$192$	0	0
$193$	−15.3485	−1.10481	−0.552403	−	0.833577i	$-0.686289\pi$
$193$	−15.3485	−1.10481	−0.552403	+	0.833577i	$0.686289\pi$
$194$	−1.55051	−0.111320
$195$	0	0
$196$	12.7980	0.914140
$197$	−27.3485	−1.94850	−0.974249	−	0.225475i	$-0.927606\pi$
$197$	−27.3485	−1.94850	−0.974249	+	0.225475i	$0.927606\pi$
$198$	0	0
$199$	−10.0000	−0.708881	−0.354441	−	0.935079i	$-0.615329\pi$
$199$	−10.0000	−0.708881	−0.354441	+	0.935079i	$0.615329\pi$
$200$	0	0
$201$	0	0
$202$	−1.55051	−0.109094
$203$	19.3485	1.35800
$204$	0	0
$205$	0	0
$206$	−1.89898	−0.132308
$207$	0	0
$208$	−2.44949	−0.169842
$209$	−3.44949	−0.238606
$210$	0	0
$211$	−3.65153	−0.251382	−0.125691	−	0.992069i	$-0.540115\pi$
$211$	−3.65153	−0.251382	−0.125691	+	0.992069i	$0.540115\pi$
$212$	7.44949	0.511633
$213$	0	0
$214$	17.3485	1.18592
$215$	0	0
$216$	0	0
$217$	−13.3485	−0.906153
$218$	2.89898	0.196344
$219$	0	0
$220$	0	0
$221$	10.8990	0.733145
$222$	0	0
$223$	16.1010	1.07820	0.539102	−	0.842240i	$-0.318764\pi$
$223$	16.1010	1.07820	0.539102	+	0.842240i	$0.318764\pi$
$224$	−4.44949	−0.297294
$225$	0	0
$226$	−16.7980	−1.11738
$227$	−29.5959	−1.96435	−0.982175	−	0.187969i	$-0.939810\pi$
$227$	−29.5959	−1.96435	−0.982175	+	0.187969i	$0.939810\pi$
$228$	0	0
$229$	7.24745	0.478925	0.239462	−	0.970906i	$-0.423029\pi$
$229$	7.24745	0.478925	0.239462	+	0.970906i	$0.423029\pi$
$230$	0	0
$231$	0	0
$232$	−4.34847	−0.285491
$233$	−6.24745	−0.409284	−0.204642	−	0.978837i	$-0.565603\pi$
$233$	−6.24745	−0.409284	−0.204642	+	0.978837i	$0.565603\pi$
$234$	0	0
$235$	0	0
$236$	−6.44949	−0.419826
$237$	0	0
$238$	19.7980	1.28331
$239$	8.69694	0.562558	0.281279	−	0.959626i	$-0.409241\pi$
$239$	8.69694	0.562558	0.281279	+	0.959626i	$0.409241\pi$
$240$	0	0
$241$	−4.44949	−0.286617	−0.143308	−	0.989678i	$-0.545774\pi$
$241$	−4.44949	−0.286617	−0.143308	+	0.989678i	$0.545774\pi$
$242$	−0.898979	−0.0577886
$243$	0	0
$244$	−9.44949	−0.604942
$245$	0	0
$246$	0	0
$247$	−2.44949	−0.155857
$248$	3.00000	0.190500
$249$	0	0
$250$	0	0
$251$	18.0000	1.13615	0.568075	−	0.822977i	$-0.307688\pi$
$251$	18.0000	1.13615	0.568075	+	0.822977i	$0.307688\pi$
$252$	0	0
$253$	−3.44949	−0.216868
$254$	−0.101021	−0.00633859
$255$	0	0
$256$	1.00000	0.0625000
$257$	21.4949	1.34081	0.670407	−	0.741993i	$-0.266119\pi$
$257$	21.4949	1.34081	0.670407	+	0.741993i	$0.266119\pi$
$258$	0	0
$259$	−34.6969	−2.15596
$260$	0	0
$261$	0	0
$262$	9.24745	0.571309
$263$	6.79796	0.419180	0.209590	−	0.977789i	$-0.432787\pi$
$263$	6.79796	0.419180	0.209590	+	0.977789i	$0.432787\pi$
$264$	0	0
$265$	0	0
$266$	−4.44949	−0.272816
$267$	0	0
$268$	15.2474	0.931386
$269$	2.89898	0.176754	0.0883769	−	0.996087i	$-0.471832\pi$
$269$	2.89898	0.176754	0.0883769	+	0.996087i	$0.471832\pi$
$270$	0	0
$271$	22.9444	1.39377	0.696886	−	0.717182i	$-0.254568\pi$
$271$	22.9444	1.39377	0.696886	+	0.717182i	$0.254568\pi$
$272$	−4.44949	−0.269790
$273$	0	0
$274$	−4.89898	−0.295958
$275$	0	0
$276$	0	0
$277$	21.0454	1.26450	0.632248	−	0.774766i	$-0.282132\pi$
$277$	21.0454	1.26450	0.632248	+	0.774766i	$0.282132\pi$
$278$	22.2474	1.33431
$279$	0	0
$280$	0	0
$281$	−7.00000	−0.417585	−0.208792	−	0.977960i	$-0.566953\pi$
$281$	−7.00000	−0.417585	−0.208792	+	0.977960i	$0.566953\pi$
$282$	0	0
$283$	29.7980	1.77130	0.885652	−	0.464349i	$-0.153712\pi$
$283$	29.7980	1.77130	0.885652	+	0.464349i	$0.153712\pi$
$284$	1.55051	0.0920059
$285$	0	0
$286$	−8.44949	−0.499629
$287$	4.00000	0.236113
$288$	0	0
$289$	2.79796	0.164586
$290$	0	0
$291$	0	0
$292$	−1.00000	−0.0585206
$293$	23.2474	1.35813	0.679065	−	0.734078i	$-0.262385\pi$
$293$	23.2474	1.35813	0.679065	+	0.734078i	$0.262385\pi$
$294$	0	0
$295$	0	0
$296$	7.79796	0.453247
$297$	0	0
$298$	5.79796	0.335867
$299$	−2.44949	−0.141658
$300$	0	0
$301$	−10.8990	−0.628207
$302$	23.7980	1.36942
$303$	0	0
$304$	1.00000	0.0573539
$305$	0	0
$306$	0	0
$307$	−16.3485	−0.933056	−0.466528	−	0.884506i	$-0.654495\pi$
$307$	−16.3485	−0.933056	−0.466528	+	0.884506i	$0.654495\pi$
$308$	−15.3485	−0.874560
$309$	0	0
$310$	0	0
$311$	23.7980	1.34946	0.674729	−	0.738065i	$-0.264260\pi$
$311$	23.7980	1.34946	0.674729	+	0.738065i	$0.264260\pi$
$312$	0	0
$313$	31.8990	1.80304	0.901518	−	0.432741i	$-0.142453\pi$
$313$	31.8990	1.80304	0.901518	+	0.432741i	$0.142453\pi$
$314$	4.89898	0.276465
$315$	0	0
$316$	−5.00000	−0.281272
$317$	−15.2474	−0.856382	−0.428191	−	0.903688i	$-0.640849\pi$
$317$	−15.2474	−0.856382	−0.428191	+	0.903688i	$0.640849\pi$
$318$	0	0
$319$	−15.0000	−0.839839
$320$	0	0
$321$	0	0
$322$	−4.44949	−0.247960
$323$	−4.44949	−0.247576
$324$	0	0
$325$	0	0
$326$	−19.7980	−1.09651
$327$	0	0
$328$	−0.898979	−0.0496378
$329$	34.6969	1.91290
$330$	0	0
$331$	−22.3485	−1.22838	−0.614191	−	0.789157i	$-0.710518\pi$
$331$	−22.3485	−1.22838	−0.614191	+	0.789157i	$0.710518\pi$
$332$	−8.34847	−0.458182
$333$	0	0
$334$	18.0000	0.984916
$335$	0	0
$336$	0	0
$337$	−24.8990	−1.35633	−0.678167	−	0.734908i	$-0.737225\pi$
$337$	−24.8990	−1.35633	−0.678167	+	0.734908i	$0.737225\pi$
$338$	7.00000	0.380750
$339$	0	0
$340$	0	0
$341$	10.3485	0.560401
$342$	0	0
$343$	25.7980	1.39296
$344$	2.44949	0.132068
$345$	0	0
$346$	−1.65153	−0.0887868
$347$	23.5959	1.26670	0.633348	−	0.773867i	$-0.281680\pi$
$347$	23.5959	1.26670	0.633348	+	0.773867i	$0.281680\pi$
$348$	0	0
$349$	−25.9444	−1.38877	−0.694386	−	0.719603i	$-0.744324\pi$
$349$	−25.9444	−1.38877	−0.694386	+	0.719603i	$0.744324\pi$
$350$	0	0
$351$	0	0
$352$	3.44949	0.183858
$353$	18.2474	0.971214	0.485607	−	0.874177i	$-0.338599\pi$
$353$	18.2474	0.971214	0.485607	+	0.874177i	$0.338599\pi$
$354$	0	0
$355$	0	0
$356$	−2.10102	−0.111354
$357$	0	0
$358$	25.7980	1.36346
$359$	22.8990	1.20856	0.604281	−	0.796771i	$-0.293460\pi$
$359$	22.8990	1.20856	0.604281	+	0.796771i	$0.293460\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	14.4495	0.759448
$363$	0	0
$364$	−10.8990	−0.571262
$365$	0	0
$366$	0	0
$367$	−5.55051	−0.289734	−0.144867	−	0.989451i	$-0.546275\pi$
$367$	−5.55051	−0.289734	−0.144867	+	0.989451i	$0.546275\pi$
$368$	1.00000	0.0521286
$369$	0	0
$370$	0	0
$371$	33.1464	1.72088
$372$	0	0
$373$	−22.4495	−1.16239	−0.581195	−	0.813764i	$-0.697415\pi$
$373$	−22.4495	−1.16239	−0.581195	+	0.813764i	$0.697415\pi$
$374$	−15.3485	−0.793650
$375$	0	0
$376$	−7.79796	−0.402149
$377$	−10.6515	−0.548582
$378$	0	0
$379$	−1.30306	−0.0669338	−0.0334669	−	0.999440i	$-0.510655\pi$
$379$	−1.30306	−0.0669338	−0.0334669	+	0.999440i	$0.510655\pi$
$380$	0	0
$381$	0	0
$382$	−0.101021	−0.00516866
$383$	−16.2474	−0.830206	−0.415103	−	0.909774i	$-0.636254\pi$
$383$	−16.2474	−0.830206	−0.415103	+	0.909774i	$0.636254\pi$
$384$	0	0
$385$	0	0
$386$	15.3485	0.781217
$387$	0	0
$388$	1.55051	0.0787152
$389$	−21.5959	−1.09496	−0.547478	−	0.836820i	$-0.684412\pi$
$389$	−21.5959	−1.09496	−0.547478	+	0.836820i	$0.684412\pi$
$390$	0	0
$391$	−4.44949	−0.225020
$392$	−12.7980	−0.646395
$393$	0	0
$394$	27.3485	1.37780
$395$	0	0
$396$	0	0
$397$	23.9444	1.20173	0.600867	−	0.799349i	$-0.294822\pi$
$397$	23.9444	1.20173	0.600867	+	0.799349i	$0.294822\pi$
$398$	10.0000	0.501255
$399$	0	0
$400$	0	0
$401$	−11.2020	−0.559403	−0.279702	−	0.960087i	$-0.590236\pi$
$401$	−11.2020	−0.559403	−0.279702	+	0.960087i	$0.590236\pi$
$402$	0	0
$403$	7.34847	0.366053
$404$	1.55051	0.0771408
$405$	0	0
$406$	−19.3485	−0.960248
$407$	26.8990	1.33333
$408$	0	0
$409$	−22.8990	−1.13228	−0.566141	−	0.824309i	$-0.691564\pi$
$409$	−22.8990	−1.13228	−0.566141	+	0.824309i	$0.691564\pi$
$410$	0	0
$411$	0	0
$412$	1.89898	0.0935560
$413$	−28.6969	−1.41208
$414$	0	0
$415$	0	0
$416$	2.44949	0.120096
$417$	0	0
$418$	3.44949	0.168720
$419$	0	0		−	1.00000i	$-0.5\pi$
$419$	0	0			1.00000i	$0.5\pi$
$420$	0	0
$421$	22.0000	1.07221	0.536107	−	0.844150i	$-0.319894\pi$
$421$	22.0000	1.07221	0.536107	+	0.844150i	$0.319894\pi$
$422$	3.65153	0.177754
$423$	0	0
$424$	−7.44949	−0.361779
$425$	0	0
$426$	0	0
$427$	−42.0454	−2.03472
$428$	−17.3485	−0.838570
$429$	0	0
$430$	0	0
$431$	5.10102	0.245708	0.122854	−	0.992425i	$-0.460795\pi$
$431$	5.10102	0.245708	0.122854	+	0.992425i	$0.460795\pi$
$432$	0	0
$433$	4.65153	0.223538	0.111769	−	0.993734i	$-0.464348\pi$
$433$	4.65153	0.223538	0.111769	+	0.993734i	$0.464348\pi$
$434$	13.3485	0.640747
$435$	0	0
$436$	−2.89898	−0.138836
$437$	1.00000	0.0478365
$438$	0	0
$439$	−12.1010	−0.577550	−0.288775	−	0.957397i	$-0.593248\pi$
$439$	−12.1010	−0.577550	−0.288775	+	0.957397i	$0.593248\pi$
$440$	0	0
$441$	0	0
$442$	−10.8990	−0.518412
$443$	−21.2474	−1.00950	−0.504748	−	0.863267i	$-0.668415\pi$
$443$	−21.2474	−1.00950	−0.504748	+	0.863267i	$0.668415\pi$
$444$	0	0
$445$	0	0
$446$	−16.1010	−0.762405
$447$	0	0
$448$	4.44949	0.210219
$449$	15.0000	0.707894	0.353947	−	0.935266i	$-0.384839\pi$
$449$	15.0000	0.707894	0.353947	+	0.935266i	$0.384839\pi$
$450$	0	0
$451$	−3.10102	−0.146021
$452$	16.7980	0.790110
$453$	0	0
$454$	29.5959	1.38901
$455$	0	0
$456$	0	0
$457$	−24.8990	−1.16473	−0.582363	−	0.812929i	$-0.697872\pi$
$457$	−24.8990	−1.16473	−0.582363	+	0.812929i	$0.697872\pi$
$458$	−7.24745	−0.338651
$459$	0	0
$460$	0	0
$461$	−12.0000	−0.558896	−0.279448	−	0.960161i	$-0.590151\pi$
$461$	−12.0000	−0.558896	−0.279448	+	0.960161i	$0.590151\pi$
$462$	0	0
$463$	−14.6969	−0.683025	−0.341512	−	0.939877i	$-0.610939\pi$
$463$	−14.6969	−0.683025	−0.341512	+	0.939877i	$0.610939\pi$
$464$	4.34847	0.201873
$465$	0	0
$466$	6.24745	0.289407
$467$	6.34847	0.293772	0.146886	−	0.989153i	$-0.453075\pi$
$467$	6.34847	0.293772	0.146886	+	0.989153i	$0.453075\pi$
$468$	0	0
$469$	67.8434	3.13272
$470$	0	0
$471$	0	0
$472$	6.44949	0.296862
$473$	8.44949	0.388508
$474$	0	0
$475$	0	0
$476$	−19.7980	−0.907438
$477$	0	0
$478$	−8.69694	−0.397789
$479$	−16.5959	−0.758287	−0.379143	−	0.925338i	$-0.623781\pi$
$479$	−16.5959	−0.758287	−0.379143	+	0.925338i	$0.623781\pi$
$480$	0	0
$481$	19.1010	0.870932
$482$	4.44949	0.202669
$483$	0	0
$484$	0.898979	0.0408627
$485$	0	0
$486$	0	0
$487$	−32.0000	−1.45006	−0.725029	−	0.688718i	$-0.758174\pi$
$487$	−32.0000	−1.45006	−0.725029	+	0.688718i	$0.758174\pi$
$488$	9.44949	0.427758
$489$	0	0
$490$	0	0
$491$	−43.5959	−1.96746	−0.983728	−	0.179664i	$-0.942499\pi$
$491$	−43.5959	−1.96746	−0.983728	+	0.179664i	$0.942499\pi$
$492$	0	0
$493$	−19.3485	−0.871411
$494$	2.44949	0.110208
$495$	0	0
$496$	−3.00000	−0.134704
$497$	6.89898	0.309462
$498$	0	0
$499$	−23.8434	−1.06738	−0.533688	−	0.845682i	$-0.679194\pi$
$499$	−23.8434	−1.06738	−0.533688	+	0.845682i	$0.679194\pi$
$500$	0	0
$501$	0	0
$502$	−18.0000	−0.803379
$503$	−29.7980	−1.32863	−0.664313	−	0.747455i	$-0.731276\pi$
$503$	−29.7980	−1.32863	−0.664313	+	0.747455i	$0.731276\pi$
$504$	0	0
$505$	0	0
$506$	3.44949	0.153349
$507$	0	0
$508$	0.101021	0.00448206
$509$	30.1464	1.33622	0.668108	−	0.744064i	$-0.267104\pi$
$509$	30.1464	1.33622	0.668108	+	0.744064i	$0.267104\pi$
$510$	0	0
$511$	−4.44949	−0.196834
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	−21.4949	−0.948099
$515$	0	0
$516$	0	0
$517$	−26.8990	−1.18302
$518$	34.6969	1.52450
$519$	0	0
$520$	0	0
$521$	−15.6969	−0.687695	−0.343848	−	0.939025i	$-0.611730\pi$
$521$	−15.6969	−0.687695	−0.343848	+	0.939025i	$0.611730\pi$
$522$	0	0
$523$	−33.3939	−1.46021	−0.730106	−	0.683334i	$-0.760529\pi$
$523$	−33.3939	−1.46021	−0.730106	+	0.683334i	$0.760529\pi$
$524$	−9.24745	−0.403977
$525$	0	0
$526$	−6.79796	−0.296405
$527$	13.3485	0.581468
$528$	0	0
$529$	−22.0000	−0.956522
$530$	0	0
$531$	0	0
$532$	4.44949	0.192910
$533$	−2.20204	−0.0953810
$534$	0	0
$535$	0	0
$536$	−15.2474	−0.658589
$537$	0	0
$538$	−2.89898	−0.124984
$539$	−44.1464	−1.90152
$540$	0	0
$541$	22.1464	0.952149	0.476075	−	0.879405i	$-0.342059\pi$
$541$	22.1464	0.952149	0.476075	+	0.879405i	$0.342059\pi$
$542$	−22.9444	−0.985546
$543$	0	0
$544$	4.44949	0.190770
$545$	0	0
$546$	0	0
$547$	−46.6413	−1.99424	−0.997120	−	0.0758461i	$-0.975834\pi$
$547$	−46.6413	−1.99424	−0.997120	+	0.0758461i	$0.975834\pi$
$548$	4.89898	0.209274
$549$	0	0
$550$	0	0
$551$	4.34847	0.185251
$552$	0	0
$553$	−22.2474	−0.946058
$554$	−21.0454	−0.894134
$555$	0	0
$556$	−22.2474	−0.943502
$557$	−27.3485	−1.15879	−0.579396	−	0.815046i	$-0.696711\pi$
$557$	−27.3485	−1.15879	−0.579396	+	0.815046i	$0.696711\pi$
$558$	0	0
$559$	6.00000	0.253773
$560$	0	0
$561$	0	0
$562$	7.00000	0.295277
$563$	−6.24745	−0.263299	−0.131649	−	0.991296i	$-0.542027\pi$
$563$	−6.24745	−0.263299	−0.131649	+	0.991296i	$0.542027\pi$
$564$	0	0
$565$	0	0
$566$	−29.7980	−1.25250
$567$	0	0
$568$	−1.55051	−0.0650580
$569$	33.1918	1.39147	0.695737	−	0.718297i	$-0.255078\pi$
$569$	33.1918	1.39147	0.695737	+	0.718297i	$0.255078\pi$
$570$	0	0
$571$	−10.2474	−0.428842	−0.214421	−	0.976741i	$-0.568787\pi$
$571$	−10.2474	−0.428842	−0.214421	+	0.976741i	$0.568787\pi$
$572$	8.44949	0.353291
$573$	0	0
$574$	−4.00000	−0.166957
$575$	0	0
$576$	0	0
$577$	0.101021	0.00420554	0.00210277	−	0.999998i	$-0.499331\pi$
$577$	0.101021	0.00420554	0.00210277	+	0.999998i	$0.499331\pi$
$578$	−2.79796	−0.116380
$579$	0	0
$580$	0	0
$581$	−37.1464	−1.54109
$582$	0	0
$583$	−25.6969	−1.06426
$584$	1.00000	0.0413803
$585$	0	0
$586$	−23.2474	−0.960343
$587$	33.4495	1.38061	0.690304	−	0.723519i	$-0.257477\pi$
$587$	33.4495	1.38061	0.690304	+	0.723519i	$0.257477\pi$
$588$	0	0
$589$	−3.00000	−0.123613
$590$	0	0
$591$	0	0
$592$	−7.79796	−0.320494
$593$	−8.49490	−0.348844	−0.174422	−	0.984671i	$-0.555806\pi$
$593$	−8.49490	−0.348844	−0.174422	+	0.984671i	$0.555806\pi$
$594$	0	0
$595$	0	0
$596$	−5.79796	−0.237494
$597$	0	0
$598$	2.44949	0.100167
$599$	−16.4495	−0.672108	−0.336054	−	0.941843i	$-0.609092\pi$
$599$	−16.4495	−0.672108	−0.336054	+	0.941843i	$0.609092\pi$
$600$	0	0
$601$	17.1464	0.699417	0.349709	−	0.936858i	$-0.386281\pi$
$601$	17.1464	0.699417	0.349709	+	0.936858i	$0.386281\pi$
$602$	10.8990	0.444209
$603$	0	0
$604$	−23.7980	−0.968325
$605$	0	0
$606$	0	0
$607$	46.1918	1.87487	0.937434	−	0.348162i	$-0.113194\pi$
$607$	46.1918	1.87487	0.937434	+	0.348162i	$0.113194\pi$
$608$	−1.00000	−0.0405554
$609$	0	0
$610$	0	0
$611$	−19.1010	−0.772745
$612$	0	0
$613$	37.1918	1.50216	0.751082	−	0.660209i	$-0.229532\pi$
$613$	37.1918	1.50216	0.751082	+	0.660209i	$0.229532\pi$
$614$	16.3485	0.659771
$615$	0	0
$616$	15.3485	0.618407
$617$	−28.2929	−1.13903	−0.569514	−	0.821982i	$-0.692868\pi$
$617$	−28.2929	−1.13903	−0.569514	+	0.821982i	$0.692868\pi$
$618$	0	0
$619$	45.1464	1.81459	0.907294	−	0.420497i	$-0.138144\pi$
$619$	45.1464	1.81459	0.907294	+	0.420497i	$0.138144\pi$
$620$	0	0
$621$	0	0
$622$	−23.7980	−0.954211
$623$	−9.34847	−0.374539
$624$	0	0
$625$	0	0
$626$	−31.8990	−1.27494
$627$	0	0
$628$	−4.89898	−0.195491
$629$	34.6969	1.38346
$630$	0	0
$631$	−28.0000	−1.11466	−0.557331	−	0.830290i	$-0.688175\pi$
$631$	−28.0000	−1.11466	−0.557331	+	0.830290i	$0.688175\pi$
$632$	5.00000	0.198889
$633$	0	0
$634$	15.2474	0.605554
$635$	0	0
$636$	0	0
$637$	−31.3485	−1.24207
$638$	15.0000	0.593856
$639$	0	0
$640$	0	0
$641$	−37.7980	−1.49293	−0.746465	−	0.665425i	$-0.768250\pi$
$641$	−37.7980	−1.49293	−0.746465	+	0.665425i	$0.768250\pi$
$642$	0	0
$643$	−0.202041	−0.00796772	−0.00398386	−	0.999992i	$-0.501268\pi$
$643$	−0.202041	−0.00796772	−0.00398386	+	0.999992i	$0.501268\pi$
$644$	4.44949	0.175334
$645$	0	0
$646$	4.44949	0.175063
$647$	−25.8990	−1.01819	−0.509097	−	0.860709i	$-0.670021\pi$
$647$	−25.8990	−1.01819	−0.509097	+	0.860709i	$0.670021\pi$
$648$	0	0
$649$	22.2474	0.873289
$650$	0	0
$651$	0	0
$652$	19.7980	0.775348
$653$	14.6969	0.575136	0.287568	−	0.957760i	$-0.407153\pi$
$653$	14.6969	0.575136	0.287568	+	0.957760i	$0.407153\pi$
$654$	0	0
$655$	0	0
$656$	0.898979	0.0350993
$657$	0	0
$658$	−34.6969	−1.35263
$659$	18.0454	0.702949	0.351475	−	0.936197i	$-0.385680\pi$
$659$	18.0454	0.702949	0.351475	+	0.936197i	$0.385680\pi$
$660$	0	0
$661$	23.5959	0.917775	0.458887	−	0.888494i	$-0.348248\pi$
$661$	23.5959	0.917775	0.458887	+	0.888494i	$0.348248\pi$
$662$	22.3485	0.868598
$663$	0	0
$664$	8.34847	0.323983
$665$	0	0
$666$	0	0
$667$	4.34847	0.168373
$668$	−18.0000	−0.696441
$669$	0	0
$670$	0	0
$671$	32.5959	1.25835
$672$	0	0
$673$	−45.3485	−1.74806	−0.874028	−	0.485876i	$-0.838501\pi$
$673$	−45.3485	−1.74806	−0.874028	+	0.485876i	$0.838501\pi$
$674$	24.8990	0.959073
$675$	0	0
$676$	−7.00000	−0.269231
$677$	−22.3485	−0.858921	−0.429461	−	0.903086i	$-0.641296\pi$
$677$	−22.3485	−0.858921	−0.429461	+	0.903086i	$0.641296\pi$
$678$	0	0
$679$	6.89898	0.264759
$680$	0	0
$681$	0	0
$682$	−10.3485	−0.396263
$683$	35.6413	1.36378	0.681889	−	0.731456i	$-0.261159\pi$
$683$	35.6413	1.36378	0.681889	+	0.731456i	$0.261159\pi$
$684$	0	0
$685$	0	0
$686$	−25.7980	−0.984971
$687$	0	0
$688$	−2.44949	−0.0933859
$689$	−18.2474	−0.695172
$690$	0	0
$691$	9.75255	0.371005	0.185502	−	0.982644i	$-0.440609\pi$
$691$	9.75255	0.371005	0.185502	+	0.982644i	$0.440609\pi$
$692$	1.65153	0.0627818
$693$	0	0
$694$	−23.5959	−0.895689
$695$	0	0
$696$	0	0
$697$	−4.00000	−0.151511
$698$	25.9444	0.982010
$699$	0	0
$700$	0	0
$701$	32.4949	1.22732	0.613658	−	0.789572i	$-0.289698\pi$
$701$	32.4949	1.22732	0.613658	+	0.789572i	$0.289698\pi$
$702$	0	0
$703$	−7.79796	−0.294106
$704$	−3.44949	−0.130008
$705$	0	0
$706$	−18.2474	−0.686752
$707$	6.89898	0.259463
$708$	0	0
$709$	−12.7526	−0.478932	−0.239466	−	0.970905i	$-0.576972\pi$
$709$	−12.7526	−0.478932	−0.239466	+	0.970905i	$0.576972\pi$
$710$	0	0
$711$	0	0
$712$	2.10102	0.0787391
$713$	−3.00000	−0.112351
$714$	0	0
$715$	0	0
$716$	−25.7980	−0.964115
$717$	0	0
$718$	−22.8990	−0.854582
$719$	19.2020	0.716115	0.358058	−	0.933699i	$-0.383439\pi$
$719$	19.2020	0.716115	0.358058	+	0.933699i	$0.383439\pi$
$720$	0	0
$721$	8.44949	0.314675
$722$	−1.00000	−0.0372161
$723$	0	0
$724$	−14.4495	−0.537011
$725$	0	0
$726$	0	0
$727$	22.4949	0.834290	0.417145	−	0.908840i	$-0.363031\pi$
$727$	22.4949	0.834290	0.417145	+	0.908840i	$0.363031\pi$
$728$	10.8990	0.403943
$729$	0	0
$730$	0	0
$731$	10.8990	0.403113
$732$	0	0
$733$	4.14643	0.153152	0.0765759	−	0.997064i	$-0.475601\pi$
$733$	4.14643	0.153152	0.0765759	+	0.997064i	$0.475601\pi$
$734$	5.55051	0.204873
$735$	0	0
$736$	−1.00000	−0.0368605
$737$	−52.5959	−1.93740
$738$	0	0
$739$	32.8990	1.21021	0.605104	−	0.796146i	$-0.293131\pi$
$739$	32.8990	1.21021	0.605104	+	0.796146i	$0.293131\pi$
$740$	0	0
$741$	0	0
$742$	−33.1464	−1.21684
$743$	53.3939	1.95883	0.979416	−	0.201854i	$-0.0646966\pi$
$743$	53.3939	1.95883	0.979416	+	0.201854i	$0.0646966\pi$
$744$	0	0
$745$	0	0
$746$	22.4495	0.821934
$747$	0	0
$748$	15.3485	0.561196
$749$	−77.1918	−2.82053
$750$	0	0
$751$	−23.7980	−0.868400	−0.434200	−	0.900817i	$-0.642969\pi$
$751$	−23.7980	−0.868400	−0.434200	+	0.900817i	$0.642969\pi$
$752$	7.79796	0.284362
$753$	0	0
$754$	10.6515	0.387906
$755$	0	0
$756$	0	0
$757$	48.1464	1.74991	0.874956	−	0.484203i	$-0.160890\pi$
$757$	48.1464	1.74991	0.874956	+	0.484203i	$0.160890\pi$
$758$	1.30306	0.0473293
$759$	0	0
$760$	0	0
$761$	−51.3485	−1.86138	−0.930690	−	0.365808i	$-0.880793\pi$
$761$	−51.3485	−1.86138	−0.930690	+	0.365808i	$0.880793\pi$
$762$	0	0
$763$	−12.8990	−0.466974
$764$	0.101021	0.00365479
$765$	0	0
$766$	16.2474	0.587044
$767$	15.7980	0.570431
$768$	0	0
$769$	−7.89898	−0.284844	−0.142422	−	0.989806i	$-0.545489\pi$
$769$	−7.89898	−0.284844	−0.142422	+	0.989806i	$0.545489\pi$
$770$	0	0
$771$	0	0
$772$	−15.3485	−0.552403
$773$	−18.4949	−0.665215	−0.332608	−	0.943065i	$-0.607928\pi$
$773$	−18.4949	−0.665215	−0.332608	+	0.943065i	$0.607928\pi$
$774$	0	0
$775$	0	0
$776$	−1.55051	−0.0556601
$777$	0	0
$778$	21.5959	0.774251
$779$	0.898979	0.0322093
$780$	0	0
$781$	−5.34847	−0.191383
$782$	4.44949	0.159113
$783$	0	0
$784$	12.7980	0.457070
$785$	0	0
$786$	0	0
$787$	−33.4495	−1.19235	−0.596173	−	0.802856i	$-0.703313\pi$
$787$	−33.4495	−1.19235	−0.596173	+	0.802856i	$0.703313\pi$
$788$	−27.3485	−0.974249
$789$	0	0
$790$	0	0
$791$	74.7423	2.65753
$792$	0	0
$793$	23.1464	0.821954
$794$	−23.9444	−0.849755
$795$	0	0
$796$	−10.0000	−0.354441
$797$	−3.50510	−0.124157	−0.0620786	−	0.998071i	$-0.519773\pi$
$797$	−3.50510	−0.124157	−0.0620786	+	0.998071i	$0.519773\pi$
$798$	0	0
$799$	−34.6969	−1.22749
$800$	0	0
$801$	0	0
$802$	11.2020	0.395558
$803$	3.44949	0.121730
$804$	0	0
$805$	0	0
$806$	−7.34847	−0.258839
$807$	0	0
$808$	−1.55051	−0.0545468
$809$	3.55051	0.124829	0.0624146	−	0.998050i	$-0.480120\pi$
$809$	3.55051	0.124829	0.0624146	+	0.998050i	$0.480120\pi$
$810$	0	0
$811$	−19.4495	−0.682964	−0.341482	−	0.939888i	$-0.610929\pi$
$811$	−19.4495	−0.682964	−0.341482	+	0.939888i	$0.610929\pi$
$812$	19.3485	0.678998
$813$	0	0
$814$	−26.8990	−0.942809
$815$	0	0
$816$	0	0
$817$	−2.44949	−0.0856968
$818$	22.8990	0.800644
$819$	0	0
$820$	0	0
$821$	23.1464	0.807816	0.403908	−	0.914800i	$-0.367652\pi$
$821$	23.1464	0.807816	0.403908	+	0.914800i	$0.367652\pi$
$822$	0	0
$823$	−31.7980	−1.10841	−0.554204	−	0.832381i	$-0.686977\pi$
$823$	−31.7980	−1.10841	−0.554204	+	0.832381i	$0.686977\pi$
$824$	−1.89898	−0.0661541
$825$	0	0
$826$	28.6969	0.998494
$827$	−0.247449	−0.00860463	−0.00430232	−	0.999991i	$-0.501369\pi$
$827$	−0.247449	−0.00860463	−0.00430232	+	0.999991i	$0.501369\pi$
$828$	0	0
$829$	−39.6413	−1.37680	−0.688400	−	0.725331i	$-0.741687\pi$
$829$	−39.6413	−1.37680	−0.688400	+	0.725331i	$0.741687\pi$
$830$	0	0
$831$	0	0
$832$	−2.44949	−0.0849208
$833$	−56.9444	−1.97301
$834$	0	0
$835$	0	0
$836$	−3.44949	−0.119303
$837$	0	0
$838$	0	0
$839$	−10.6515	−0.367732	−0.183866	−	0.982951i	$-0.558861\pi$
$839$	−10.6515	−0.367732	−0.183866	+	0.982951i	$0.558861\pi$
$840$	0	0
$841$	−10.0908	−0.347959
$842$	−22.0000	−0.758170
$843$	0	0
$844$	−3.65153	−0.125691
$845$	0	0
$846$	0	0
$847$	4.00000	0.137442
$848$	7.44949	0.255817
$849$	0	0
$850$	0	0
$851$	−7.79796	−0.267311
$852$	0	0
$853$	1.10102	0.0376982	0.0188491	−	0.999822i	$-0.494000\pi$
$853$	1.10102	0.0376982	0.0188491	+	0.999822i	$0.494000\pi$
$854$	42.0454	1.43876
$855$	0	0
$856$	17.3485	0.592958
$857$	36.4949	1.24664	0.623321	−	0.781966i	$-0.285783\pi$
$857$	36.4949	1.24664	0.623321	+	0.781966i	$0.285783\pi$
$858$	0	0
$859$	−4.20204	−0.143372	−0.0716859	−	0.997427i	$-0.522838\pi$
$859$	−4.20204	−0.143372	−0.0716859	+	0.997427i	$0.522838\pi$
$860$	0	0
$861$	0	0
$862$	−5.10102	−0.173741
$863$	17.3031	0.589003	0.294502	−	0.955651i	$-0.404846\pi$
$863$	17.3031	0.589003	0.294502	+	0.955651i	$0.404846\pi$
$864$	0	0
$865$	0	0
$866$	−4.65153	−0.158065
$867$	0	0
$868$	−13.3485	−0.453077
$869$	17.2474	0.585080
$870$	0	0
$871$	−37.3485	−1.26550
$872$	2.89898	0.0981718
$873$	0	0
$874$	−1.00000	−0.0338255
$875$	0	0
$876$	0	0
$877$	−9.10102	−0.307320	−0.153660	−	0.988124i	$-0.549106\pi$
$877$	−9.10102	−0.307320	−0.153660	+	0.988124i	$0.549106\pi$
$878$	12.1010	0.408390
$879$	0	0
$880$	0	0
$881$	−50.6969	−1.70802	−0.854012	−	0.520254i	$-0.825837\pi$
$881$	−50.6969	−1.70802	−0.854012	+	0.520254i	$0.825837\pi$
$882$	0	0
$883$	34.6515	1.16612	0.583058	−	0.812430i	$-0.301856\pi$
$883$	34.6515	1.16612	0.583058	+	0.812430i	$0.301856\pi$
$884$	10.8990	0.366572
$885$	0	0
$886$	21.2474	0.713822
$887$	4.89898	0.164492	0.0822458	−	0.996612i	$-0.473791\pi$
$887$	4.89898	0.164492	0.0822458	+	0.996612i	$0.473791\pi$
$888$	0	0
$889$	0.449490	0.0150754
$890$	0	0
$891$	0	0
$892$	16.1010	0.539102
$893$	7.79796	0.260949
$894$	0	0
$895$	0	0
$896$	−4.44949	−0.148647
$897$	0	0
$898$	−15.0000	−0.500556
$899$	−13.0454	−0.435089
$900$	0	0
$901$	−33.1464	−1.10427
$902$	3.10102	0.103253
$903$	0	0
$904$	−16.7980	−0.558692
$905$	0	0
$906$	0	0
$907$	−22.0000	−0.730498	−0.365249	−	0.930910i	$-0.619016\pi$
$907$	−22.0000	−0.730498	−0.365249	+	0.930910i	$0.619016\pi$
$908$	−29.5959	−0.982175
$909$	0	0
$910$	0	0
$911$	−6.49490	−0.215186	−0.107593	−	0.994195i	$-0.534314\pi$
$911$	−6.49490	−0.215186	−0.107593	+	0.994195i	$0.534314\pi$
$912$	0	0
$913$	28.7980	0.953073
$914$	24.8990	0.823585
$915$	0	0
$916$	7.24745	0.239462
$917$	−41.1464	−1.35877
$918$	0	0
$919$	23.8434	0.786520	0.393260	−	0.919427i	$-0.371347\pi$
$919$	23.8434	0.786520	0.393260	+	0.919427i	$0.371347\pi$
$920$	0	0
$921$	0	0
$922$	12.0000	0.395199
$923$	−3.79796	−0.125011
$924$	0	0
$925$	0	0
$926$	14.6969	0.482971
$927$	0	0
$928$	−4.34847	−0.142745
$929$	31.5959	1.03663	0.518314	−	0.855190i	$-0.326560\pi$
$929$	31.5959	1.03663	0.518314	+	0.855190i	$0.326560\pi$
$930$	0	0
$931$	12.7980	0.419436
$932$	−6.24745	−0.204642
$933$	0	0
$934$	−6.34847	−0.207728
$935$	0	0
$936$	0	0
$937$	−19.3939	−0.633570	−0.316785	−	0.948497i	$-0.602603\pi$
$937$	−19.3939	−0.633570	−0.316785	+	0.948497i	$0.602603\pi$
$938$	−67.8434	−2.21516
$939$	0	0
$940$	0	0
$941$	−36.6413	−1.19447	−0.597237	−	0.802065i	$-0.703735\pi$
$941$	−36.6413	−1.19447	−0.597237	+	0.802065i	$0.703735\pi$
$942$	0	0
$943$	0.898979	0.0292748
$944$	−6.44949	−0.209913
$945$	0	0
$946$	−8.44949	−0.274717
$947$	−8.00000	−0.259965	−0.129983	−	0.991516i	$-0.541492\pi$
$947$	−8.00000	−0.259965	−0.129983	+	0.991516i	$0.541492\pi$
$948$	0	0
$949$	2.44949	0.0795138
$950$	0	0
$951$	0	0
$952$	19.7980	0.641656
$953$	15.4949	0.501929	0.250964	−	0.967996i	$-0.419252\pi$
$953$	15.4949	0.501929	0.250964	+	0.967996i	$0.419252\pi$
$954$	0	0
$955$	0	0
$956$	8.69694	0.281279
$957$	0	0
$958$	16.5959	0.536190
$959$	21.7980	0.703893
$960$	0	0
$961$	−22.0000	−0.709677
$962$	−19.1010	−0.615842
$963$	0	0
$964$	−4.44949	−0.143308
$965$	0	0
$966$	0	0
$967$	−34.8990	−1.12228	−0.561138	−	0.827722i	$-0.689636\pi$
$967$	−34.8990	−1.12228	−0.561138	+	0.827722i	$0.689636\pi$
$968$	−0.898979	−0.0288943
$969$	0	0
$970$	0	0
$971$	−41.6413	−1.33633	−0.668167	−	0.744011i	$-0.732921\pi$
$971$	−41.6413	−1.33633	−0.668167	+	0.744011i	$0.732921\pi$
$972$	0	0
$973$	−98.9898	−3.17347
$974$	32.0000	1.02535
$975$	0	0
$976$	−9.44949	−0.302471
$977$	−19.5959	−0.626929	−0.313464	−	0.949600i	$-0.601490\pi$
$977$	−19.5959	−0.626929	−0.313464	+	0.949600i	$0.601490\pi$
$978$	0	0
$979$	7.24745	0.231629
$980$	0	0
$981$	0	0
$982$	43.5959	1.39120
$983$	−50.4495	−1.60909	−0.804544	−	0.593892i	$-0.797590\pi$
$983$	−50.4495	−1.60909	−0.804544	+	0.593892i	$0.797590\pi$
$984$	0	0
$985$	0	0
$986$	19.3485	0.616181
$987$	0	0
$988$	−2.44949	−0.0779287
$989$	−2.44949	−0.0778892
$990$	0	0
$991$	44.1010	1.40092	0.700458	−	0.713694i	$-0.252979\pi$
$991$	44.1010	1.40092	0.700458	+	0.713694i	$0.252979\pi$
$992$	3.00000	0.0952501
$993$	0	0
$994$	−6.89898	−0.218822
$995$	0	0
$996$	0	0
$997$	39.4495	1.24938	0.624689	−	0.780874i	$-0.285226\pi$
$997$	39.4495	1.24938	0.624689	+	0.780874i	$0.285226\pi$
$998$	23.8434	0.754749
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8550.2.a.bu.1.2		2
3.2	odd	2		2850.2.a.bj.1.2	yes	2
5.4	even	2		8550.2.a.bv.1.1		2
15.2	even	4		2850.2.d.w.799.4		4
15.8	even	4		2850.2.d.w.799.1		4
15.14	odd	2		2850.2.a.bc.1.1	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2850.2.a.bc.1.1	✓	2	15.14	odd	2
2850.2.a.bj.1.2	yes	2	3.2	odd	2
2850.2.d.w.799.1		4	15.8	even	4
2850.2.d.w.799.4		4	15.2	even	4
8550.2.a.bu.1.2		2	1.1	even	1	trivial
8550.2.a.bv.1.1		2	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:	\( N \)	\(=\)	\( 8550 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8550.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} +4.44949 q^{7} -1.00000 q^{8} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{4} +4.44949 q^{7} -1.00000 q^{8} -3.44949 q^{11} -2.44949 q^{13} -4.44949 q^{14} +1.00000 q^{16} -4.44949 q^{17} +1.00000 q^{19} +3.44949 q^{22} +1.00000 q^{23} +2.44949 q^{26} +4.44949 q^{28} +4.34847 q^{29} -3.00000 q^{31} -1.00000 q^{32} +4.44949 q^{34} -7.79796 q^{37} -1.00000 q^{38} +0.898979 q^{41} -2.44949 q^{43} -3.44949 q^{44} -1.00000 q^{46} +7.79796 q^{47} +12.7980 q^{49} -2.44949 q^{52} +7.44949 q^{53} -4.44949 q^{56} -4.34847 q^{58} -6.44949 q^{59} -9.44949 q^{61} +3.00000 q^{62} +1.00000 q^{64} +15.2474 q^{67} -4.44949 q^{68} +1.55051 q^{71} -1.00000 q^{73} +7.79796 q^{74} +1.00000 q^{76} -15.3485 q^{77} -5.00000 q^{79} -0.898979 q^{82} -8.34847 q^{83} +2.44949 q^{86} +3.44949 q^{88} -2.10102 q^{89} -10.8990 q^{91} +1.00000 q^{92} -7.79796 q^{94} +1.55051 q^{97} -12.7980 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 2 q^{4} + 4 q^{7} - 2 q^{8}+O(q^{10}) \) 2 * q - 2 * q^2 + 2 * q^4 + 4 * q^7 - 2 * q^8	\( 2 q - 2 q^{2} + 2 q^{4} + 4 q^{7} - 2 q^{8} - 2 q^{11} - 4 q^{14} + 2 q^{16} - 4 q^{17} + 2 q^{19} + 2 q^{22} + 2 q^{23} + 4 q^{28} - 6 q^{29} - 6 q^{31} - 2 q^{32} + 4 q^{34} + 4 q^{37} - 2 q^{38} - 8 q^{41} - 2 q^{44} - 2 q^{46} - 4 q^{47} + 6 q^{49} + 10 q^{53} - 4 q^{56} + 6 q^{58} - 8 q^{59} - 14 q^{61} + 6 q^{62} + 2 q^{64} + 6 q^{67} - 4 q^{68} + 8 q^{71} - 2 q^{73} - 4 q^{74} + 2 q^{76} - 16 q^{77} - 10 q^{79} + 8 q^{82} - 2 q^{83} + 2 q^{88} - 14 q^{89} - 12 q^{91} + 2 q^{92} + 4 q^{94} + 8 q^{97} - 6 q^{98}+O(q^{100}) \) 2 * q - 2 * q^2 + 2 * q^4 + 4 * q^7 - 2 * q^8 - 2 * q^11 - 4 * q^14 + 2 * q^16 - 4 * q^17 + 2 * q^19 + 2 * q^22 + 2 * q^23 + 4 * q^28 - 6 * q^29 - 6 * q^31 - 2 * q^32 + 4 * q^34 + 4 * q^37 - 2 * q^38 - 8 * q^41 - 2 * q^44 - 2 * q^46 - 4 * q^47 + 6 * q^49 + 10 * q^53 - 4 * q^56 + 6 * q^58 - 8 * q^59 - 14 * q^61 + 6 * q^62 + 2 * q^64 + 6 * q^67 - 4 * q^68 + 8 * q^71 - 2 * q^73 - 4 * q^74 + 2 * q^76 - 16 * q^77 - 10 * q^79 + 8 * q^82 - 2 * q^83 + 2 * q^88 - 14 * q^89 - 12 * q^91 + 2 * q^92 + 4 * q^94 + 8 * q^97 - 6 * q^98

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