LMFDB - Embedded newform 8400.2.a.dg.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8400,2,Mod(1,8400)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8400, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 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("8400.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8400 = 2^{4} \cdot 3 \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8400.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:	$67.0743376979$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.148.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 3x + 1 $ x^3 - x^2 - 3*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{23}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 105)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.17009$ of defining polynomial
Character	$\chi$	$=$	8400.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} -1.00000 q^{7} +1.00000 q^{9} -2.00000 q^{11} +0.921622 q^{13} +1.07838 q^{17} -3.07838 q^{19} +1.00000 q^{21} -2.34017 q^{23} -1.00000 q^{27} -6.68035 q^{29} +7.75872 q^{31} +2.00000 q^{33} +10.8371 q^{37} -0.921622 q^{39} +6.49693 q^{41} +6.52359 q^{43} -4.68035 q^{47} +1.00000 q^{49} -1.07838 q^{51} +3.75872 q^{53} +3.07838 q^{57} -10.5236 q^{59} -4.15676 q^{61} -1.00000 q^{63} -4.68035 q^{67} +2.34017 q^{69} -2.00000 q^{71} +7.07838 q^{73} +2.00000 q^{77} -6.15676 q^{79} +1.00000 q^{81} -6.83710 q^{83} +6.68035 q^{87} +8.34017 q^{89} -0.921622 q^{91} -7.75872 q^{93} -8.43907 q^{97} -2.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{3} - 3 q^{7} + 3 q^{9}+O(q^{10}) $ 3 * q - 3 * q^3 - 3 * q^7 + 3 * q^9	$ 3 q - 3 q^{3} - 3 q^{7} + 3 q^{9} - 6 q^{11} + 6 q^{13} - 6 q^{19} + 3 q^{21} + 4 q^{23} - 3 q^{27} + 2 q^{29} - 2 q^{31} + 6 q^{33} + 4 q^{37} - 6 q^{39} + 2 q^{41} + 4 q^{43} + 8 q^{47} + 3 q^{49} - 14 q^{53} + 6 q^{57} - 16 q^{59} - 6 q^{61} - 3 q^{63} + 8 q^{67} - 4 q^{69} - 6 q^{71} + 18 q^{73} + 6 q^{77} - 12 q^{79} + 3 q^{81} + 8 q^{83} - 2 q^{87} + 14 q^{89} - 6 q^{91} + 2 q^{93} + 22 q^{97} - 6 q^{99}+O(q^{100}) $ 3 * q - 3 * q^3 - 3 * q^7 + 3 * q^9 - 6 * q^11 + 6 * q^13 - 6 * q^19 + 3 * q^21 + 4 * q^23 - 3 * q^27 + 2 * q^29 - 2 * q^31 + 6 * q^33 + 4 * q^37 - 6 * q^39 + 2 * q^41 + 4 * q^43 + 8 * q^47 + 3 * q^49 - 14 * q^53 + 6 * q^57 - 16 * q^59 - 6 * q^61 - 3 * q^63 + 8 * q^67 - 4 * q^69 - 6 * q^71 + 18 * q^73 + 6 * q^77 - 12 * q^79 + 3 * q^81 + 8 * q^83 - 2 * q^87 + 14 * q^89 - 6 * q^91 + 2 * q^93 + 22 * q^97 - 6 * q^99

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	0
$2$	0	0
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−2.00000	−0.603023	−0.301511	−	0.953463i	$-0.597491\pi$
$11$	−2.00000	−0.603023	−0.301511	+	0.953463i	$0.597491\pi$
$12$	0	0
$13$	0.921622	0.255612	0.127806	−	0.991799i	$-0.459207\pi$
$13$	0.921622	0.255612	0.127806	+	0.991799i	$0.459207\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	1.07838	0.261545	0.130773	−	0.991412i	$-0.458254\pi$
$17$	1.07838	0.261545	0.130773	+	0.991412i	$0.458254\pi$
$18$	0	0
$19$	−3.07838	−0.706228	−0.353114	−	0.935580i	$-0.614877\pi$
$19$	−3.07838	−0.706228	−0.353114	+	0.935580i	$0.614877\pi$
$20$	0	0
$21$	1.00000	0.218218
$22$	0	0
$23$	−2.34017	−0.487960	−0.243980	−	0.969780i	$-0.578453\pi$
$23$	−2.34017	−0.487960	−0.243980	+	0.969780i	$0.578453\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−6.68035	−1.24051	−0.620255	−	0.784401i	$-0.712971\pi$
$29$	−6.68035	−1.24051	−0.620255	+	0.784401i	$0.712971\pi$
$30$	0	0
$31$	7.75872	1.39351	0.696754	−	0.717310i	$-0.254627\pi$
$31$	7.75872	1.39351	0.696754	+	0.717310i	$0.254627\pi$
$32$	0	0
$33$	2.00000	0.348155
$34$	0	0
$35$	0	0
$36$	0	0
$37$	10.8371	1.78161	0.890804	−	0.454387i	$-0.150142\pi$
$37$	10.8371	1.78161	0.890804	+	0.454387i	$0.150142\pi$
$38$	0	0
$39$	−0.921622	−0.147578
$40$	0	0
$41$	6.49693	1.01465	0.507325	−	0.861755i	$-0.330634\pi$
$41$	6.49693	1.01465	0.507325	+	0.861755i	$0.330634\pi$
$42$	0	0
$43$	6.52359	0.994838	0.497419	−	0.867510i	$-0.334281\pi$
$43$	6.52359	0.994838	0.497419	+	0.867510i	$0.334281\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−4.68035	−0.682699	−0.341349	−	0.939937i	$-0.610884\pi$
$47$	−4.68035	−0.682699	−0.341349	+	0.939937i	$0.610884\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−1.07838	−0.151003
$52$	0	0
$53$	3.75872	0.516300	0.258150	−	0.966105i	$-0.416887\pi$
$53$	3.75872	0.516300	0.258150	+	0.966105i	$0.416887\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	3.07838	0.407741
$58$	0	0
$59$	−10.5236	−1.37005	−0.685027	−	0.728517i	$-0.740210\pi$
$59$	−10.5236	−1.37005	−0.685027	+	0.728517i	$0.740210\pi$
$60$	0	0
$61$	−4.15676	−0.532218	−0.266109	−	0.963943i	$-0.585738\pi$
$61$	−4.15676	−0.532218	−0.266109	+	0.963943i	$0.585738\pi$
$62$	0	0
$63$	−1.00000	−0.125988
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−4.68035	−0.571795	−0.285898	−	0.958260i	$-0.592292\pi$
$67$	−4.68035	−0.571795	−0.285898	+	0.958260i	$0.592292\pi$
$68$	0	0
$69$	2.34017	0.281724
$70$	0	0
$71$	−2.00000	−0.237356	−0.118678	−	0.992933i	$-0.537866\pi$
$71$	−2.00000	−0.237356	−0.118678	+	0.992933i	$0.537866\pi$
$72$	0	0
$73$	7.07838	0.828461	0.414231	−	0.910172i	$-0.364051\pi$
$73$	7.07838	0.828461	0.414231	+	0.910172i	$0.364051\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	2.00000	0.227921
$78$	0	0
$79$	−6.15676	−0.692689	−0.346345	−	0.938107i	$-0.612577\pi$
$79$	−6.15676	−0.692689	−0.346345	+	0.938107i	$0.612577\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−6.83710	−0.750469	−0.375235	−	0.926930i	$-0.622438\pi$
$83$	−6.83710	−0.750469	−0.375235	+	0.926930i	$0.622438\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	6.68035	0.716208
$88$	0	0
$89$	8.34017	0.884057	0.442028	−	0.897001i	$-0.354259\pi$
$89$	8.34017	0.884057	0.442028	+	0.897001i	$0.354259\pi$
$90$	0	0
$91$	−0.921622	−0.0966123
$92$	0	0
$93$	−7.75872	−0.804542
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−8.43907	−0.856858	−0.428429	−	0.903576i	$-0.640933\pi$
$97$	−8.43907	−0.856858	−0.428429	+	0.903576i	$0.640933\pi$
$98$	0	0
$99$	−2.00000	−0.201008
$100$	0	0
$101$	−5.81658	−0.578772	−0.289386	−	0.957213i	$-0.593451\pi$
$101$	−5.81658	−0.578772	−0.289386	+	0.957213i	$0.593451\pi$
$102$	0	0
$103$	2.15676	0.212511	0.106256	−	0.994339i	$-0.466114\pi$
$103$	2.15676	0.212511	0.106256	+	0.994339i	$0.466114\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	16.4969	1.59482	0.797409	−	0.603439i	$-0.206203\pi$
$107$	16.4969	1.59482	0.797409	+	0.603439i	$0.206203\pi$
$108$	0	0
$109$	−12.8371	−1.22957	−0.614786	−	0.788694i	$-0.710757\pi$
$109$	−12.8371	−1.22957	−0.614786	+	0.788694i	$0.710757\pi$
$110$	0	0
$111$	−10.8371	−1.02861
$112$	0	0
$113$	−5.23513	−0.492480	−0.246240	−	0.969209i	$-0.579195\pi$
$113$	−5.23513	−0.492480	−0.246240	+	0.969209i	$0.579195\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0.921622	0.0852040
$118$	0	0
$119$	−1.07838	−0.0988547
$120$	0	0
$121$	−7.00000	−0.636364
$122$	0	0
$123$	−6.49693	−0.585808
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−1.84324	−0.163562	−0.0817808	−	0.996650i	$-0.526061\pi$
$127$	−1.84324	−0.163562	−0.0817808	+	0.996650i	$0.526061\pi$
$128$	0	0
$129$	−6.52359	−0.574370
$130$	0	0
$131$	−1.47641	−0.128995	−0.0644973	−	0.997918i	$-0.520544\pi$
$131$	−1.47641	−0.128995	−0.0644973	+	0.997918i	$0.520544\pi$
$132$	0	0
$133$	3.07838	0.266929
$134$	0	0
$135$	0	0
$136$	0	0
$137$	4.43907	0.379255	0.189628	−	0.981856i	$-0.439272\pi$
$137$	4.43907	0.379255	0.189628	+	0.981856i	$0.439272\pi$
$138$	0	0
$139$	−13.6020	−1.15370	−0.576852	−	0.816849i	$-0.695719\pi$
$139$	−13.6020	−1.15370	−0.576852	+	0.816849i	$0.695719\pi$
$140$	0	0
$141$	4.68035	0.394156
$142$	0	0
$143$	−1.84324	−0.154140
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−1.00000	−0.0824786
$148$	0	0
$149$	15.6742	1.28408	0.642040	−	0.766671i	$-0.278088\pi$
$149$	15.6742	1.28408	0.642040	+	0.766671i	$0.278088\pi$
$150$	0	0
$151$	−5.84324	−0.475516	−0.237758	−	0.971324i	$-0.576413\pi$
$151$	−5.84324	−0.475516	−0.237758	+	0.971324i	$0.576413\pi$
$152$	0	0
$153$	1.07838	0.0871817
$154$	0	0
$155$	0	0
$156$	0	0
$157$	4.92162	0.392788	0.196394	−	0.980525i	$-0.437077\pi$
$157$	4.92162	0.392788	0.196394	+	0.980525i	$0.437077\pi$
$158$	0	0
$159$	−3.75872	−0.298086
$160$	0	0
$161$	2.34017	0.184431
$162$	0	0
$163$	9.84324	0.770982	0.385491	−	0.922712i	$-0.374032\pi$
$163$	9.84324	0.770982	0.385491	+	0.922712i	$0.374032\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	19.2039	1.48605	0.743023	−	0.669266i	$-0.233391\pi$
$167$	19.2039	1.48605	0.743023	+	0.669266i	$0.233391\pi$
$168$	0	0
$169$	−12.1506	−0.934662
$170$	0	0
$171$	−3.07838	−0.235409
$172$	0	0
$173$	−22.4391	−1.70601	−0.853005	−	0.521902i	$-0.825223\pi$
$173$	−22.4391	−1.70601	−0.853005	+	0.521902i	$0.825223\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	10.5236	0.791001
$178$	0	0
$179$	−10.0000	−0.747435	−0.373718	−	0.927543i	$-0.621917\pi$
$179$	−10.0000	−0.747435	−0.373718	+	0.927543i	$0.621917\pi$
$180$	0	0
$181$	−8.52359	−0.633553	−0.316777	−	0.948500i	$-0.602601\pi$
$181$	−8.52359	−0.633553	−0.316777	+	0.948500i	$0.602601\pi$
$182$	0	0
$183$	4.15676	0.307276
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−2.15676	−0.157718
$188$	0	0
$189$	1.00000	0.0727393
$190$	0	0
$191$	−15.3607	−1.11146	−0.555730	−	0.831363i	$-0.687561\pi$
$191$	−15.3607	−1.11146	−0.555730	+	0.831363i	$0.687561\pi$
$192$	0	0
$193$	8.36683	0.602258	0.301129	−	0.953583i	$-0.402637\pi$
$193$	8.36683	0.602258	0.301129	+	0.953583i	$0.402637\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−11.7587	−0.837774	−0.418887	−	0.908038i	$-0.637580\pi$
$197$	−11.7587	−0.837774	−0.418887	+	0.908038i	$0.637580\pi$
$198$	0	0
$199$	22.5958	1.60178	0.800888	−	0.598814i	$-0.204361\pi$
$199$	22.5958	1.60178	0.800888	+	0.598814i	$0.204361\pi$
$200$	0	0
$201$	4.68035	0.330126
$202$	0	0
$203$	6.68035	0.468868
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−2.34017	−0.162653
$208$	0	0
$209$	6.15676	0.425872
$210$	0	0
$211$	13.6742	0.941371	0.470685	−	0.882301i	$-0.344007\pi$
$211$	13.6742	0.941371	0.470685	+	0.882301i	$0.344007\pi$
$212$	0	0
$213$	2.00000	0.137038
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−7.75872	−0.526696
$218$	0	0
$219$	−7.07838	−0.478312
$220$	0	0
$221$	0.993857	0.0668541
$222$	0	0
$223$	21.6742	1.45141	0.725706	−	0.688005i	$-0.241513\pi$
$223$	21.6742	1.45141	0.725706	+	0.688005i	$0.241513\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	11.5174	0.764440	0.382220	−	0.924071i	$-0.375160\pi$
$227$	11.5174	0.764440	0.382220	+	0.924071i	$0.375160\pi$
$228$	0	0
$229$	12.8371	0.848300	0.424150	−	0.905592i	$-0.360573\pi$
$229$	12.8371	0.848300	0.424150	+	0.905592i	$0.360573\pi$
$230$	0	0
$231$	−2.00000	−0.131590
$232$	0	0
$233$	−6.76487	−0.443181	−0.221591	−	0.975140i	$-0.571125\pi$
$233$	−6.76487	−0.443181	−0.221591	+	0.975140i	$0.571125\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	6.15676	0.399924
$238$	0	0
$239$	23.3607	1.51108	0.755539	−	0.655104i	$-0.227375\pi$
$239$	23.3607	1.51108	0.755539	+	0.655104i	$0.227375\pi$
$240$	0	0
$241$	−14.6803	−0.945644	−0.472822	−	0.881158i	$-0.656765\pi$
$241$	−14.6803	−0.945644	−0.472822	+	0.881158i	$0.656765\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−2.83710	−0.180520
$248$	0	0
$249$	6.83710	0.433284
$250$	0	0
$251$	−9.16290	−0.578357	−0.289179	−	0.957275i	$-0.593382\pi$
$251$	−9.16290	−0.578357	−0.289179	+	0.957275i	$0.593382\pi$
$252$	0	0
$253$	4.68035	0.294251
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−5.07838	−0.316781	−0.158390	−	0.987377i	$-0.550630\pi$
$257$	−5.07838	−0.316781	−0.158390	+	0.987377i	$0.550630\pi$
$258$	0	0
$259$	−10.8371	−0.673385
$260$	0	0
$261$	−6.68035	−0.413503
$262$	0	0
$263$	−5.65983	−0.349000	−0.174500	−	0.984657i	$-0.555831\pi$
$263$	−5.65983	−0.349000	−0.174500	+	0.984657i	$0.555831\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−8.34017	−0.510410
$268$	0	0
$269$	−27.8576	−1.69851	−0.849255	−	0.527984i	$-0.822948\pi$
$269$	−27.8576	−1.69851	−0.849255	+	0.527984i	$0.822948\pi$
$270$	0	0
$271$	−25.1194	−1.52590	−0.762948	−	0.646460i	$-0.776249\pi$
$271$	−25.1194	−1.52590	−0.762948	+	0.646460i	$0.776249\pi$
$272$	0	0
$273$	0.921622	0.0557791
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−28.1978	−1.69424	−0.847121	−	0.531401i	$-0.821666\pi$
$277$	−28.1978	−1.69424	−0.847121	+	0.531401i	$0.821666\pi$
$278$	0	0
$279$	7.75872	0.464503
$280$	0	0
$281$	−20.3545	−1.21425	−0.607125	−	0.794606i	$-0.707677\pi$
$281$	−20.3545	−1.21425	−0.607125	+	0.794606i	$0.707677\pi$
$282$	0	0
$283$	−23.5174	−1.39797	−0.698984	−	0.715138i	$-0.746364\pi$
$283$	−23.5174	−1.39797	−0.698984	+	0.715138i	$0.746364\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−6.49693	−0.383502
$288$	0	0
$289$	−15.8371	−0.931594
$290$	0	0
$291$	8.43907	0.494707
$292$	0	0
$293$	−2.92162	−0.170683	−0.0853415	−	0.996352i	$-0.527198\pi$
$293$	−2.92162	−0.170683	−0.0853415	+	0.996352i	$0.527198\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	2.00000	0.116052
$298$	0	0
$299$	−2.15676	−0.124728
$300$	0	0
$301$	−6.52359	−0.376014
$302$	0	0
$303$	5.81658	0.334154
$304$	0	0
$305$	0	0
$306$	0	0
$307$	10.4703	0.597570	0.298785	−	0.954321i	$-0.403419\pi$
$307$	10.4703	0.597570	0.298785	+	0.954321i	$0.403419\pi$
$308$	0	0
$309$	−2.15676	−0.122694
$310$	0	0
$311$	−23.8310	−1.35133	−0.675665	−	0.737209i	$-0.736143\pi$
$311$	−23.8310	−1.35133	−0.675665	+	0.737209i	$0.736143\pi$
$312$	0	0
$313$	32.7526	1.85129	0.925643	−	0.378399i	$-0.123525\pi$
$313$	32.7526	1.85129	0.925643	+	0.378399i	$0.123525\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−17.9155	−1.00623	−0.503117	−	0.864218i	$-0.667813\pi$
$317$	−17.9155	−1.00623	−0.503117	+	0.864218i	$0.667813\pi$
$318$	0	0
$319$	13.3607	0.748055
$320$	0	0
$321$	−16.4969	−0.920769
$322$	0	0
$323$	−3.31965	−0.184710
$324$	0	0
$325$	0	0
$326$	0	0
$327$	12.8371	0.709893
$328$	0	0
$329$	4.68035	0.258036
$330$	0	0
$331$	1.36069	0.0747904	0.0373952	−	0.999301i	$-0.488094\pi$
$331$	1.36069	0.0747904	0.0373952	+	0.999301i	$0.488094\pi$
$332$	0	0
$333$	10.8371	0.593870
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−25.3607	−1.38148	−0.690742	−	0.723101i	$-0.742716\pi$
$337$	−25.3607	−1.38148	−0.690742	+	0.723101i	$0.742716\pi$
$338$	0	0
$339$	5.23513	0.284333
$340$	0	0
$341$	−15.5174	−0.840317
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−16.8638	−0.905294	−0.452647	−	0.891690i	$-0.649520\pi$
$347$	−16.8638	−0.905294	−0.452647	+	0.891690i	$0.649520\pi$
$348$	0	0
$349$	9.51745	0.509457	0.254729	−	0.967013i	$-0.418014\pi$
$349$	9.51745	0.509457	0.254729	+	0.967013i	$0.418014\pi$
$350$	0	0
$351$	−0.921622	−0.0491926
$352$	0	0
$353$	−35.7998	−1.90543	−0.952715	−	0.303867i	$-0.901722\pi$
$353$	−35.7998	−1.90543	−0.952715	+	0.303867i	$0.901722\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	1.07838	0.0570738
$358$	0	0
$359$	22.3135	1.17766	0.588831	−	0.808256i	$-0.299588\pi$
$359$	22.3135	1.17766	0.588831	+	0.808256i	$0.299588\pi$
$360$	0	0
$361$	−9.52359	−0.501242
$362$	0	0
$363$	7.00000	0.367405
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−20.3135	−1.06036	−0.530178	−	0.847886i	$-0.677875\pi$
$367$	−20.3135	−1.06036	−0.530178	+	0.847886i	$0.677875\pi$
$368$	0	0
$369$	6.49693	0.338217
$370$	0	0
$371$	−3.75872	−0.195143
$372$	0	0
$373$	−16.0000	−0.828449	−0.414224	−	0.910175i	$-0.635947\pi$
$373$	−16.0000	−0.828449	−0.414224	+	0.910175i	$0.635947\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−6.15676	−0.317089
$378$	0	0
$379$	−6.15676	−0.316251	−0.158126	−	0.987419i	$-0.550545\pi$
$379$	−6.15676	−0.316251	−0.158126	+	0.987419i	$0.550545\pi$
$380$	0	0
$381$	1.84324	0.0944323
$382$	0	0
$383$	−26.8371	−1.37131	−0.685656	−	0.727926i	$-0.740485\pi$
$383$	−26.8371	−1.37131	−0.685656	+	0.727926i	$0.740485\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	6.52359	0.331613
$388$	0	0
$389$	−5.63317	−0.285613	−0.142806	−	0.989751i	$-0.545613\pi$
$389$	−5.63317	−0.285613	−0.142806	+	0.989751i	$0.545613\pi$
$390$	0	0
$391$	−2.52359	−0.127623
$392$	0	0
$393$	1.47641	0.0744750
$394$	0	0
$395$	0	0
$396$	0	0
$397$	37.7998	1.89712	0.948558	−	0.316604i	$-0.102543\pi$
$397$	37.7998	1.89712	0.948558	+	0.316604i	$0.102543\pi$
$398$	0	0
$399$	−3.07838	−0.154112
$400$	0	0
$401$	−13.6332	−0.680808	−0.340404	−	0.940279i	$-0.610564\pi$
$401$	−13.6332	−0.680808	−0.340404	+	0.940279i	$0.610564\pi$
$402$	0	0
$403$	7.15061	0.356197
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−21.6742	−1.07435
$408$	0	0
$409$	12.3545	0.610893	0.305447	−	0.952209i	$-0.401194\pi$
$409$	12.3545	0.610893	0.305447	+	0.952209i	$0.401194\pi$
$410$	0	0
$411$	−4.43907	−0.218963
$412$	0	0
$413$	10.5236	0.517832
$414$	0	0
$415$	0	0
$416$	0	0
$417$	13.6020	0.666091
$418$	0	0
$419$	−28.9939	−1.41644	−0.708221	−	0.705991i	$-0.750502\pi$
$419$	−28.9939	−1.41644	−0.708221	+	0.705991i	$0.750502\pi$
$420$	0	0
$421$	−15.1629	−0.738994	−0.369497	−	0.929232i	$-0.620470\pi$
$421$	−15.1629	−0.738994	−0.369497	+	0.929232i	$0.620470\pi$
$422$	0	0
$423$	−4.68035	−0.227566
$424$	0	0
$425$	0	0
$426$	0	0
$427$	4.15676	0.201159
$428$	0	0
$429$	1.84324	0.0889927
$430$	0	0
$431$	10.3135	0.496784	0.248392	−	0.968660i	$-0.420098\pi$
$431$	10.3135	0.496784	0.248392	+	0.968660i	$0.420098\pi$
$432$	0	0
$433$	20.4391	0.982239	0.491120	−	0.871092i	$-0.336588\pi$
$433$	20.4391	0.982239	0.491120	+	0.871092i	$0.336588\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	7.20394	0.344611
$438$	0	0
$439$	16.9216	0.807625	0.403812	−	0.914842i	$-0.367685\pi$
$439$	16.9216	0.807625	0.403812	+	0.914842i	$0.367685\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	12.8104	0.608642	0.304321	−	0.952569i	$-0.401570\pi$
$443$	12.8104	0.608642	0.304321	+	0.952569i	$0.401570\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−15.6742	−0.741364
$448$	0	0
$449$	−14.6270	−0.690292	−0.345146	−	0.938549i	$-0.612171\pi$
$449$	−14.6270	−0.690292	−0.345146	+	0.938549i	$0.612171\pi$
$450$	0	0
$451$	−12.9939	−0.611857
$452$	0	0
$453$	5.84324	0.274540
$454$	0	0
$455$	0	0
$456$	0	0
$457$	14.1568	0.662225	0.331113	−	0.943591i	$-0.392576\pi$
$457$	14.1568	0.662225	0.331113	+	0.943591i	$0.392576\pi$
$458$	0	0
$459$	−1.07838	−0.0503344
$460$	0	0
$461$	0.340173	0.0158434	0.00792172	−	0.999969i	$-0.497478\pi$
$461$	0.340173	0.0158434	0.00792172	+	0.999969i	$0.497478\pi$
$462$	0	0
$463$	9.84324	0.457454	0.228727	−	0.973491i	$-0.426544\pi$
$463$	9.84324	0.457454	0.228727	+	0.973491i	$0.426544\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	11.5174	0.532964	0.266482	−	0.963840i	$-0.414139\pi$
$467$	11.5174	0.532964	0.266482	+	0.963840i	$0.414139\pi$
$468$	0	0
$469$	4.68035	0.216118
$470$	0	0
$471$	−4.92162	−0.226776
$472$	0	0
$473$	−13.0472	−0.599910
$474$	0	0
$475$	0	0
$476$	0	0
$477$	3.75872	0.172100
$478$	0	0
$479$	19.5174	0.891775	0.445887	−	0.895089i	$-0.352888\pi$
$479$	19.5174	0.891775	0.445887	+	0.895089i	$0.352888\pi$
$480$	0	0
$481$	9.98771	0.455401
$482$	0	0
$483$	−2.34017	−0.106482
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−23.1506	−1.04905	−0.524527	−	0.851394i	$-0.675758\pi$
$487$	−23.1506	−1.04905	−0.524527	+	0.851394i	$0.675758\pi$
$488$	0	0
$489$	−9.84324	−0.445127
$490$	0	0
$491$	−2.00000	−0.0902587	−0.0451294	−	0.998981i	$-0.514370\pi$
$491$	−2.00000	−0.0902587	−0.0451294	+	0.998981i	$0.514370\pi$
$492$	0	0
$493$	−7.20394	−0.324449
$494$	0	0
$495$	0	0
$496$	0	0
$497$	2.00000	0.0897123
$498$	0	0
$499$	−27.2039	−1.21782	−0.608908	−	0.793241i	$-0.708392\pi$
$499$	−27.2039	−1.21782	−0.608908	+	0.793241i	$0.708392\pi$
$500$	0	0
$501$	−19.2039	−0.857969
$502$	0	0
$503$	18.8371	0.839905	0.419952	−	0.907546i	$-0.362047\pi$
$503$	18.8371	0.839905	0.419952	+	0.907546i	$0.362047\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	12.1506	0.539628
$508$	0	0
$509$	6.81044	0.301867	0.150934	−	0.988544i	$-0.451772\pi$
$509$	6.81044	0.301867	0.150934	+	0.988544i	$0.451772\pi$
$510$	0	0
$511$	−7.07838	−0.313129
$512$	0	0
$513$	3.07838	0.135914
$514$	0	0
$515$	0	0
$516$	0	0
$517$	9.36069	0.411683
$518$	0	0
$519$	22.4391	0.984966
$520$	0	0
$521$	−25.8166	−1.13105	−0.565523	−	0.824733i	$-0.691325\pi$
$521$	−25.8166	−1.13105	−0.565523	+	0.824733i	$0.691325\pi$
$522$	0	0
$523$	−4.00000	−0.174908	−0.0874539	−	0.996169i	$-0.527873\pi$
$523$	−4.00000	−0.174908	−0.0874539	+	0.996169i	$0.527873\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	8.36683	0.364465
$528$	0	0
$529$	−17.5236	−0.761895
$530$	0	0
$531$	−10.5236	−0.456685
$532$	0	0
$533$	5.98771	0.259357
$534$	0	0
$535$	0	0
$536$	0	0
$537$	10.0000	0.431532
$538$	0	0
$539$	−2.00000	−0.0861461
$540$	0	0
$541$	25.8843	1.11285	0.556426	−	0.830897i	$-0.312172\pi$
$541$	25.8843	1.11285	0.556426	+	0.830897i	$0.312172\pi$
$542$	0	0
$543$	8.52359	0.365782
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−11.3197	−0.483993	−0.241997	−	0.970277i	$-0.577802\pi$
$547$	−11.3197	−0.483993	−0.241997	+	0.970277i	$0.577802\pi$
$548$	0	0
$549$	−4.15676	−0.177406
$550$	0	0
$551$	20.5646	0.876083
$552$	0	0
$553$	6.15676	0.261812
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−26.6491	−1.12916	−0.564580	−	0.825378i	$-0.690962\pi$
$557$	−26.6491	−1.12916	−0.564580	+	0.825378i	$0.690962\pi$
$558$	0	0
$559$	6.01229	0.254293
$560$	0	0
$561$	2.15676	0.0910583
$562$	0	0
$563$	−46.3545	−1.95361	−0.976806	−	0.214128i	$-0.931309\pi$
$563$	−46.3545	−1.95361	−0.976806	+	0.214128i	$0.931309\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−1.00000	−0.0419961
$568$	0	0
$569$	−14.3668	−0.602289	−0.301145	−	0.953579i	$-0.597369\pi$
$569$	−14.3668	−0.602289	−0.301145	+	0.953579i	$0.597369\pi$
$570$	0	0
$571$	−38.7214	−1.62044	−0.810220	−	0.586126i	$-0.800652\pi$
$571$	−38.7214	−1.62044	−0.810220	+	0.586126i	$0.800652\pi$
$572$	0	0
$573$	15.3607	0.641702
$574$	0	0
$575$	0	0
$576$	0	0
$577$	43.4740	1.80984	0.904922	−	0.425577i	$-0.139929\pi$
$577$	43.4740	1.80984	0.904922	+	0.425577i	$0.139929\pi$
$578$	0	0
$579$	−8.36683	−0.347714
$580$	0	0
$581$	6.83710	0.283651
$582$	0	0
$583$	−7.51745	−0.311341
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−36.0288	−1.48707	−0.743533	−	0.668699i	$-0.766851\pi$
$587$	−36.0288	−1.48707	−0.743533	+	0.668699i	$0.766851\pi$
$588$	0	0
$589$	−23.8843	−0.984135
$590$	0	0
$591$	11.7587	0.483689
$592$	0	0
$593$	31.4863	1.29299	0.646493	−	0.762920i	$-0.276235\pi$
$593$	31.4863	1.29299	0.646493	+	0.762920i	$0.276235\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−22.5958	−0.924786
$598$	0	0
$599$	−29.0349	−1.18633	−0.593167	−	0.805080i	$-0.702123\pi$
$599$	−29.0349	−1.18633	−0.593167	+	0.805080i	$0.702123\pi$
$600$	0	0
$601$	15.3607	0.626576	0.313288	−	0.949658i	$-0.398570\pi$
$601$	15.3607	0.626576	0.313288	+	0.949658i	$0.398570\pi$
$602$	0	0
$603$	−4.68035	−0.190598
$604$	0	0
$605$	0	0
$606$	0	0
$607$	13.0472	0.529569	0.264784	−	0.964308i	$-0.414699\pi$
$607$	13.0472	0.529569	0.264784	+	0.964308i	$0.414699\pi$
$608$	0	0
$609$	−6.68035	−0.270701
$610$	0	0
$611$	−4.31351	−0.174506
$612$	0	0
$613$	−15.5174	−0.626744	−0.313372	−	0.949630i	$-0.601459\pi$
$613$	−15.5174	−0.626744	−0.313372	+	0.949630i	$0.601459\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−22.7649	−0.916479	−0.458240	−	0.888829i	$-0.651520\pi$
$617$	−22.7649	−0.916479	−0.458240	+	0.888829i	$0.651520\pi$
$618$	0	0
$619$	−7.92777	−0.318644	−0.159322	−	0.987227i	$-0.550931\pi$
$619$	−7.92777	−0.318644	−0.159322	+	0.987227i	$0.550931\pi$
$620$	0	0
$621$	2.34017	0.0939079
$622$	0	0
$623$	−8.34017	−0.334142
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−6.15676	−0.245877
$628$	0	0
$629$	11.6865	0.465971
$630$	0	0
$631$	−19.2039	−0.764497	−0.382248	−	0.924060i	$-0.624850\pi$
$631$	−19.2039	−0.764497	−0.382248	+	0.924060i	$0.624850\pi$
$632$	0	0
$633$	−13.6742	−0.543501
$634$	0	0
$635$	0	0
$636$	0	0
$637$	0.921622	0.0365160
$638$	0	0
$639$	−2.00000	−0.0791188
$640$	0	0
$641$	−5.94668	−0.234880	−0.117440	−	0.993080i	$-0.537469\pi$
$641$	−5.94668	−0.234880	−0.117440	+	0.993080i	$0.537469\pi$
$642$	0	0
$643$	30.8904	1.21820	0.609100	−	0.793094i	$-0.291531\pi$
$643$	30.8904	1.21820	0.609100	+	0.793094i	$0.291531\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	19.2039	0.754985	0.377492	−	0.926013i	$-0.376786\pi$
$647$	19.2039	0.754985	0.377492	+	0.926013i	$0.376786\pi$
$648$	0	0
$649$	21.0472	0.826174
$650$	0	0
$651$	7.75872	0.304088
$652$	0	0
$653$	−28.5548	−1.11744	−0.558718	−	0.829358i	$-0.688706\pi$
$653$	−28.5548	−1.11744	−0.558718	+	0.829358i	$0.688706\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	7.07838	0.276154
$658$	0	0
$659$	27.9877	1.09025	0.545123	−	0.838356i	$-0.316483\pi$
$659$	27.9877	1.09025	0.545123	+	0.838356i	$0.316483\pi$
$660$	0	0
$661$	−22.1445	−0.861320	−0.430660	−	0.902514i	$-0.641719\pi$
$661$	−22.1445	−0.861320	−0.430660	+	0.902514i	$0.641719\pi$
$662$	0	0
$663$	−0.993857	−0.0385982
$664$	0	0
$665$	0	0
$666$	0	0
$667$	15.6332	0.605319
$668$	0	0
$669$	−21.6742	−0.837973
$670$	0	0
$671$	8.31351	0.320940
$672$	0	0
$673$	2.21008	0.0851923	0.0425962	−	0.999092i	$-0.486437\pi$
$673$	2.21008	0.0851923	0.0425962	+	0.999092i	$0.486437\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	19.5486	0.751315	0.375658	−	0.926758i	$-0.377417\pi$
$677$	19.5486	0.751315	0.375658	+	0.926758i	$0.377417\pi$
$678$	0	0
$679$	8.43907	0.323862
$680$	0	0
$681$	−11.5174	−0.441350
$682$	0	0
$683$	−11.8166	−0.452149	−0.226074	−	0.974110i	$-0.572589\pi$
$683$	−11.8166	−0.452149	−0.226074	+	0.974110i	$0.572589\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−12.8371	−0.489766
$688$	0	0
$689$	3.46412	0.131973
$690$	0	0
$691$	−11.7587	−0.447323	−0.223661	−	0.974667i	$-0.571801\pi$
$691$	−11.7587	−0.447323	−0.223661	+	0.974667i	$0.571801\pi$
$692$	0	0
$693$	2.00000	0.0759737
$694$	0	0
$695$	0	0
$696$	0	0
$697$	7.00614	0.265377
$698$	0	0
$699$	6.76487	0.255871
$700$	0	0
$701$	9.94668	0.375681	0.187840	−	0.982200i	$-0.439851\pi$
$701$	9.94668	0.375681	0.187840	+	0.982200i	$0.439851\pi$
$702$	0	0
$703$	−33.3607	−1.25822
$704$	0	0
$705$	0	0
$706$	0	0
$707$	5.81658	0.218755
$708$	0	0
$709$	−11.0472	−0.414886	−0.207443	−	0.978247i	$-0.566514\pi$
$709$	−11.0472	−0.414886	−0.207443	+	0.978247i	$0.566514\pi$
$710$	0	0
$711$	−6.15676	−0.230896
$712$	0	0
$713$	−18.1568	−0.679976
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−23.3607	−0.872421
$718$	0	0
$719$	6.15676	0.229608	0.114804	−	0.993388i	$-0.463376\pi$
$719$	6.15676	0.229608	0.114804	+	0.993388i	$0.463376\pi$
$720$	0	0
$721$	−2.15676	−0.0803218
$722$	0	0
$723$	14.6803	0.545968
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−2.89043	−0.107200	−0.0536000	−	0.998562i	$-0.517070\pi$
$727$	−2.89043	−0.107200	−0.0536000	+	0.998562i	$0.517070\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	7.03489	0.260195
$732$	0	0
$733$	−25.7998	−0.952936	−0.476468	−	0.879192i	$-0.658083\pi$
$733$	−25.7998	−0.952936	−0.476468	+	0.879192i	$0.658083\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	9.36069	0.344806
$738$	0	0
$739$	1.04718	0.0385212	0.0192606	−	0.999814i	$-0.493869\pi$
$739$	1.04718	0.0385212	0.0192606	+	0.999814i	$0.493869\pi$
$740$	0	0
$741$	2.83710	0.104224
$742$	0	0
$743$	9.97334	0.365886	0.182943	−	0.983123i	$-0.441438\pi$
$743$	9.97334	0.365886	0.182943	+	0.983123i	$0.441438\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−6.83710	−0.250156
$748$	0	0
$749$	−16.4969	−0.602785
$750$	0	0
$751$	−3.26633	−0.119190	−0.0595950	−	0.998223i	$-0.518981\pi$
$751$	−3.26633	−0.119190	−0.0595950	+	0.998223i	$0.518981\pi$
$752$	0	0
$753$	9.16290	0.333915
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−49.9877	−1.81683	−0.908417	−	0.418065i	$-0.862708\pi$
$757$	−49.9877	−1.81683	−0.908417	+	0.418065i	$0.862708\pi$
$758$	0	0
$759$	−4.68035	−0.169886
$760$	0	0
$761$	2.61265	0.0947083	0.0473542	−	0.998878i	$-0.484921\pi$
$761$	2.61265	0.0947083	0.0473542	+	0.998878i	$0.484921\pi$
$762$	0	0
$763$	12.8371	0.464734
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−9.69878	−0.350202
$768$	0	0
$769$	−15.6742	−0.565226	−0.282613	−	0.959234i	$-0.591201\pi$
$769$	−15.6742	−0.565226	−0.282613	+	0.959234i	$0.591201\pi$
$770$	0	0
$771$	5.07838	0.182893
$772$	0	0
$773$	5.81205	0.209045	0.104522	−	0.994523i	$-0.466669\pi$
$773$	5.81205	0.209045	0.104522	+	0.994523i	$0.466669\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	10.8371	0.388779
$778$	0	0
$779$	−20.0000	−0.716574
$780$	0	0
$781$	4.00000	0.143131
$782$	0	0
$783$	6.68035	0.238736
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−39.3484	−1.40262	−0.701310	−	0.712857i	$-0.747401\pi$
$787$	−39.3484	−1.40262	−0.701310	+	0.712857i	$0.747401\pi$
$788$	0	0
$789$	5.65983	0.201495
$790$	0	0
$791$	5.23513	0.186140
$792$	0	0
$793$	−3.83096	−0.136041
$794$	0	0
$795$	0	0
$796$	0	0
$797$	28.2823	1.00181	0.500905	−	0.865502i	$-0.333000\pi$
$797$	28.2823	1.00181	0.500905	+	0.865502i	$0.333000\pi$
$798$	0	0
$799$	−5.04718	−0.178556
$800$	0	0
$801$	8.34017	0.294686
$802$	0	0
$803$	−14.1568	−0.499581
$804$	0	0
$805$	0	0
$806$	0	0
$807$	27.8576	0.980635
$808$	0	0
$809$	−15.6742	−0.551076	−0.275538	−	0.961290i	$-0.588856\pi$
$809$	−15.6742	−0.551076	−0.275538	+	0.961290i	$0.588856\pi$
$810$	0	0
$811$	42.1666	1.48067	0.740335	−	0.672238i	$-0.234667\pi$
$811$	42.1666	1.48067	0.740335	+	0.672238i	$0.234667\pi$
$812$	0	0
$813$	25.1194	0.880976
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−20.0821	−0.702583
$818$	0	0
$819$	−0.921622	−0.0322041
$820$	0	0
$821$	−39.0472	−1.36276	−0.681378	−	0.731932i	$-0.738619\pi$
$821$	−39.0472	−1.36276	−0.681378	+	0.731932i	$0.738619\pi$
$822$	0	0
$823$	36.5646	1.27456	0.637281	−	0.770631i	$-0.280059\pi$
$823$	36.5646	1.27456	0.637281	+	0.770631i	$0.280059\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−50.2245	−1.74648	−0.873238	−	0.487294i	$-0.837984\pi$
$827$	−50.2245	−1.74648	−0.873238	+	0.487294i	$0.837984\pi$
$828$	0	0
$829$	32.8371	1.14048	0.570240	−	0.821478i	$-0.306850\pi$
$829$	32.8371	1.14048	0.570240	+	0.821478i	$0.306850\pi$
$830$	0	0
$831$	28.1978	0.978171
$832$	0	0
$833$	1.07838	0.0373636
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−7.75872	−0.268181
$838$	0	0
$839$	13.3607	0.461262	0.230631	−	0.973041i	$-0.425921\pi$
$839$	13.3607	0.461262	0.230631	+	0.973041i	$0.425921\pi$
$840$	0	0
$841$	15.6270	0.538863
$842$	0	0
$843$	20.3545	0.701048
$844$	0	0
$845$	0	0
$846$	0	0
$847$	7.00000	0.240523
$848$	0	0
$849$	23.5174	0.807117
$850$	0	0
$851$	−25.3607	−0.869353
$852$	0	0
$853$	−39.6430	−1.35735	−0.678675	−	0.734438i	$-0.737446\pi$
$853$	−39.6430	−1.35735	−0.678675	+	0.734438i	$0.737446\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−29.7054	−1.01472	−0.507359	−	0.861735i	$-0.669378\pi$
$857$	−29.7054	−1.01472	−0.507359	+	0.861735i	$0.669378\pi$
$858$	0	0
$859$	3.07838	0.105033	0.0525164	−	0.998620i	$-0.483276\pi$
$859$	3.07838	0.105033	0.0525164	+	0.998620i	$0.483276\pi$
$860$	0	0
$861$	6.49693	0.221415
$862$	0	0
$863$	6.39350	0.217637	0.108819	−	0.994062i	$-0.465293\pi$
$863$	6.39350	0.217637	0.108819	+	0.994062i	$0.465293\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	15.8371	0.537856
$868$	0	0
$869$	12.3135	0.417707
$870$	0	0
$871$	−4.31351	−0.146158
$872$	0	0
$873$	−8.43907	−0.285619
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−1.21622	−0.0410689	−0.0205345	−	0.999789i	$-0.506537\pi$
$877$	−1.21622	−0.0410689	−0.0205345	+	0.999789i	$0.506537\pi$
$878$	0	0
$879$	2.92162	0.0985439
$880$	0	0
$881$	15.9733	0.538155	0.269078	−	0.963118i	$-0.413281\pi$
$881$	15.9733	0.538155	0.269078	+	0.963118i	$0.413281\pi$
$882$	0	0
$883$	−11.6865	−0.393282	−0.196641	−	0.980476i	$-0.563003\pi$
$883$	−11.6865	−0.393282	−0.196641	+	0.980476i	$0.563003\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	25.6209	0.860265	0.430132	−	0.902766i	$-0.358467\pi$
$887$	25.6209	0.860265	0.430132	+	0.902766i	$0.358467\pi$
$888$	0	0
$889$	1.84324	0.0618204
$890$	0	0
$891$	−2.00000	−0.0670025
$892$	0	0
$893$	14.4079	0.482141
$894$	0	0
$895$	0	0
$896$	0	0
$897$	2.15676	0.0720120
$898$	0	0
$899$	−51.8310	−1.72866
$900$	0	0
$901$	4.05332	0.135036
$902$	0	0
$903$	6.52359	0.217091
$904$	0	0
$905$	0	0
$906$	0	0
$907$	57.7563	1.91777	0.958883	−	0.283802i	$-0.0915959\pi$
$907$	57.7563	1.91777	0.958883	+	0.283802i	$0.0915959\pi$
$908$	0	0
$909$	−5.81658	−0.192924
$910$	0	0
$911$	35.9877	1.19233	0.596163	−	0.802863i	$-0.296691\pi$
$911$	35.9877	1.19233	0.596163	+	0.802863i	$0.296691\pi$
$912$	0	0
$913$	13.6742	0.452550
$914$	0	0
$915$	0	0
$916$	0	0
$917$	1.47641	0.0487553
$918$	0	0
$919$	−46.7214	−1.54120	−0.770598	−	0.637321i	$-0.780042\pi$
$919$	−46.7214	−1.54120	−0.770598	+	0.637321i	$0.780042\pi$
$920$	0	0
$921$	−10.4703	−0.345007
$922$	0	0
$923$	−1.84324	−0.0606711
$924$	0	0
$925$	0	0
$926$	0	0
$927$	2.15676	0.0708371
$928$	0	0
$929$	−53.0493	−1.74049	−0.870245	−	0.492619i	$-0.836040\pi$
$929$	−53.0493	−1.74049	−0.870245	+	0.492619i	$0.836040\pi$
$930$	0	0
$931$	−3.07838	−0.100890
$932$	0	0
$933$	23.8310	0.780191
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−16.1256	−0.526799	−0.263400	−	0.964687i	$-0.584844\pi$
$937$	−16.1256	−0.526799	−0.263400	+	0.964687i	$0.584844\pi$
$938$	0	0
$939$	−32.7526	−1.06884
$940$	0	0
$941$	24.7070	0.805425	0.402713	−	0.915326i	$-0.368067\pi$
$941$	24.7070	0.805425	0.402713	+	0.915326i	$0.368067\pi$
$942$	0	0
$943$	−15.2039	−0.495108
$944$	0	0
$945$	0	0
$946$	0	0
$947$	6.53797	0.212455	0.106228	−	0.994342i	$-0.466123\pi$
$947$	6.53797	0.212455	0.106228	+	0.994342i	$0.466123\pi$
$948$	0	0
$949$	6.52359	0.211765
$950$	0	0
$951$	17.9155	0.580949
$952$	0	0
$953$	6.11327	0.198028	0.0990142	−	0.995086i	$-0.468431\pi$
$953$	6.11327	0.198028	0.0990142	+	0.995086i	$0.468431\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−13.3607	−0.431890
$958$	0	0
$959$	−4.43907	−0.143345
$960$	0	0
$961$	29.1978	0.941864
$962$	0	0
$963$	16.4969	0.531606
$964$	0	0
$965$	0	0
$966$	0	0
$967$	25.6209	0.823912	0.411956	−	0.911204i	$-0.364846\pi$
$967$	25.6209	0.823912	0.411956	+	0.911204i	$0.364846\pi$
$968$	0	0
$969$	3.31965	0.106643
$970$	0	0
$971$	−4.05332	−0.130077	−0.0650387	−	0.997883i	$-0.520717\pi$
$971$	−4.05332	−0.130077	−0.0650387	+	0.997883i	$0.520717\pi$
$972$	0	0
$973$	13.6020	0.436059
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−3.81205	−0.121958	−0.0609791	−	0.998139i	$-0.519422\pi$
$977$	−3.81205	−0.121958	−0.0609791	+	0.998139i	$0.519422\pi$
$978$	0	0
$979$	−16.6803	−0.533106
$980$	0	0
$981$	−12.8371	−0.409857
$982$	0	0
$983$	−24.0000	−0.765481	−0.382741	−	0.923856i	$-0.625020\pi$
$983$	−24.0000	−0.765481	−0.382741	+	0.923856i	$0.625020\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−4.68035	−0.148977
$988$	0	0
$989$	−15.2663	−0.485441
$990$	0	0
$991$	42.4079	1.34713	0.673565	−	0.739128i	$-0.264762\pi$
$991$	42.4079	1.34713	0.673565	+	0.739128i	$0.264762\pi$
$992$	0	0
$993$	−1.36069	−0.0431803
$994$	0	0
$995$	0	0
$996$	0	0
$997$	43.4740	1.37683	0.688417	−	0.725315i	$-0.258306\pi$
$997$	43.4740	1.37683	0.688417	+	0.725315i	$0.258306\pi$
$998$	0	0
$999$	−10.8371	−0.342871

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8400.2.a.dg.1.2		3
4.3	odd	2		525.2.a.j.1.1		3
5.2	odd	4		1680.2.t.k.1009.6		6
5.3	odd	4		1680.2.t.k.1009.3		6
5.4	even	2		8400.2.a.dj.1.2		3
12.11	even	2		1575.2.a.x.1.3		3
15.2	even	4		5040.2.t.v.1009.2		6
15.8	even	4		5040.2.t.v.1009.1		6
20.3	even	4		105.2.d.b.64.6	yes	6
20.7	even	4		105.2.d.b.64.1	✓	6
20.19	odd	2		525.2.a.k.1.3		3
28.27	even	2		3675.2.a.bi.1.1		3
60.23	odd	4		315.2.d.e.64.1		6
60.47	odd	4		315.2.d.e.64.6		6
60.59	even	2		1575.2.a.w.1.1		3
140.3	odd	12		735.2.q.f.79.1		12
140.23	even	12		735.2.q.e.214.6		12
140.27	odd	4		735.2.d.b.589.1		6
140.47	odd	12		735.2.q.f.214.1		12
140.67	even	12		735.2.q.e.79.6		12
140.83	odd	4		735.2.d.b.589.6		6
140.87	odd	12		735.2.q.f.79.6		12
140.103	odd	12		735.2.q.f.214.6		12
140.107	even	12		735.2.q.e.214.1		12
140.123	even	12		735.2.q.e.79.1		12
140.139	even	2		3675.2.a.bj.1.3		3
420.83	even	4		2205.2.d.l.1324.1		6
420.167	even	4		2205.2.d.l.1324.6		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.2.d.b.64.1	✓	6	20.7	even	4
105.2.d.b.64.6	yes	6	20.3	even	4
315.2.d.e.64.1		6	60.23	odd	4
315.2.d.e.64.6		6	60.47	odd	4
525.2.a.j.1.1		3	4.3	odd	2
525.2.a.k.1.3		3	20.19	odd	2
735.2.d.b.589.1		6	140.27	odd	4
735.2.d.b.589.6		6	140.83	odd	4
735.2.q.e.79.1		12	140.123	even	12
735.2.q.e.79.6		12	140.67	even	12
735.2.q.e.214.1		12	140.107	even	12
735.2.q.e.214.6		12	140.23	even	12
735.2.q.f.79.1		12	140.3	odd	12
735.2.q.f.79.6		12	140.87	odd	12
735.2.q.f.214.1		12	140.47	odd	12
735.2.q.f.214.6		12	140.103	odd	12
1575.2.a.w.1.1		3	60.59	even	2
1575.2.a.x.1.3		3	12.11	even	2
1680.2.t.k.1009.3		6	5.3	odd	4
1680.2.t.k.1009.6		6	5.2	odd	4
2205.2.d.l.1324.1		6	420.83	even	4
2205.2.d.l.1324.6		6	420.167	even	4
3675.2.a.bi.1.1		3	28.27	even	2
3675.2.a.bj.1.3		3	140.139	even	2
5040.2.t.v.1009.1		6	15.8	even	4
5040.2.t.v.1009.2		6	15.2	even	4
8400.2.a.dg.1.2		3	1.1	even	1	trivial
8400.2.a.dj.1.2		3	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:	\( N \)	\(=\)	\( 8400 = 2^{4} \cdot 3 \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8400.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} -1.00000 q^{7} +1.00000 q^{9} -2.00000 q^{11} +0.921622 q^{13} +1.07838 q^{17} -3.07838 q^{19} +1.00000 q^{21} -2.34017 q^{23} -1.00000 q^{27} -6.68035 q^{29} +7.75872 q^{31} +2.00000 q^{33} +10.8371 q^{37} -0.921622 q^{39} +6.49693 q^{41} +6.52359 q^{43} -4.68035 q^{47} +1.00000 q^{49} -1.07838 q^{51} +3.75872 q^{53} +3.07838 q^{57} -10.5236 q^{59} -4.15676 q^{61} -1.00000 q^{63} -4.68035 q^{67} +2.34017 q^{69} -2.00000 q^{71} +7.07838 q^{73} +2.00000 q^{77} -6.15676 q^{79} +1.00000 q^{81} -6.83710 q^{83} +6.68035 q^{87} +8.34017 q^{89} -0.921622 q^{91} -7.75872 q^{93} -8.43907 q^{97} -2.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{3} - 3 q^{7} + 3 q^{9}+O(q^{10}) \) 3 * q - 3 * q^3 - 3 * q^7 + 3 * q^9	\( 3 q - 3 q^{3} - 3 q^{7} + 3 q^{9} - 6 q^{11} + 6 q^{13} - 6 q^{19} + 3 q^{21} + 4 q^{23} - 3 q^{27} + 2 q^{29} - 2 q^{31} + 6 q^{33} + 4 q^{37} - 6 q^{39} + 2 q^{41} + 4 q^{43} + 8 q^{47} + 3 q^{49} - 14 q^{53} + 6 q^{57} - 16 q^{59} - 6 q^{61} - 3 q^{63} + 8 q^{67} - 4 q^{69} - 6 q^{71} + 18 q^{73} + 6 q^{77} - 12 q^{79} + 3 q^{81} + 8 q^{83} - 2 q^{87} + 14 q^{89} - 6 q^{91} + 2 q^{93} + 22 q^{97} - 6 q^{99}+O(q^{100}) \) 3 * q - 3 * q^3 - 3 * q^7 + 3 * q^9 - 6 * q^11 + 6 * q^13 - 6 * q^19 + 3 * q^21 + 4 * q^23 - 3 * q^27 + 2 * q^29 - 2 * q^31 + 6 * q^33 + 4 * q^37 - 6 * q^39 + 2 * q^41 + 4 * q^43 + 8 * q^47 + 3 * q^49 - 14 * q^53 + 6 * q^57 - 16 * q^59 - 6 * q^61 - 3 * q^63 + 8 * q^67 - 4 * q^69 - 6 * q^71 + 18 * q^73 + 6 * q^77 - 12 * q^79 + 3 * q^81 + 8 * q^83 - 2 * q^87 + 14 * q^89 - 6 * q^91 + 2 * q^93 + 22 * q^97 - 6 * q^99

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