LMFDB - Embedded newform 2352.4.a.br.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2352,4,Mod(1,2352)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2352, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("2352.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2352 = 2^{4} \cdot 3 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2352.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:	$138.772492334$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{337}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 84 $ x^2 - x - 84
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 168)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$9.67878$ of defining polynomial
Character	$\chi$	$=$	2352.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.00000 q^{3} +21.3576 q^{5} +9.00000 q^{9} +O(q^{10})$	$q-3.00000 q^{3} +21.3576 q^{5} +9.00000 q^{9} -31.3576 q^{11} -84.7151 q^{13} -64.0727 q^{15} +16.0727 q^{17} +20.0000 q^{19} +80.7878 q^{23} +331.145 q^{25} -27.0000 q^{27} +102.000 q^{29} -245.721 q^{31} +94.0727 q^{33} +215.285 q^{37} +254.145 q^{39} -150.933 q^{41} -441.721 q^{43} +192.218 q^{45} -206.424 q^{47} -48.2180 q^{51} +426.436 q^{53} -669.721 q^{55} -60.0000 q^{57} +363.006 q^{59} +343.721 q^{61} -1809.31 q^{65} -69.2849 q^{67} -242.363 q^{69} +468.933 q^{71} -747.576 q^{73} -993.436 q^{75} -1298.74 q^{79} +81.0000 q^{81} -1293.16 q^{83} +343.273 q^{85} -306.000 q^{87} +563.346 q^{89} +737.163 q^{93} +427.151 q^{95} -1485.03 q^{97} -282.218 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 6 q^{3} + 6 q^{5} + 18 q^{9}+O(q^{10}) $ 2 * q - 6 * q^3 + 6 * q^5 + 18 * q^9	$ 2 q - 6 q^{3} + 6 q^{5} + 18 q^{9} - 26 q^{11} - 96 q^{13} - 18 q^{15} - 78 q^{17} + 40 q^{19} - 22 q^{23} + 442 q^{25} - 54 q^{27} + 204 q^{29} + 96 q^{31} + 78 q^{33} + 504 q^{37} + 288 q^{39} + 102 q^{41} - 296 q^{43} + 54 q^{45} - 780 q^{47} + 234 q^{51} + 192 q^{53} - 752 q^{55} - 120 q^{57} + 212 q^{59} + 100 q^{61} - 1636 q^{65} - 212 q^{67} + 66 q^{69} + 534 q^{71} - 1128 q^{73} - 1326 q^{75} - 468 q^{79} + 162 q^{81} - 824 q^{83} + 1788 q^{85} - 612 q^{87} + 2118 q^{89} - 288 q^{93} + 120 q^{95} - 400 q^{97} - 234 q^{99}+O(q^{100}) $ 2 * q - 6 * q^3 + 6 * q^5 + 18 * q^9 - 26 * q^11 - 96 * q^13 - 18 * q^15 - 78 * q^17 + 40 * q^19 - 22 * q^23 + 442 * q^25 - 54 * q^27 + 204 * q^29 + 96 * q^31 + 78 * q^33 + 504 * q^37 + 288 * q^39 + 102 * q^41 - 296 * q^43 + 54 * q^45 - 780 * q^47 + 234 * q^51 + 192 * q^53 - 752 * q^55 - 120 * q^57 + 212 * q^59 + 100 * q^61 - 1636 * q^65 - 212 * q^67 + 66 * q^69 + 534 * q^71 - 1128 * q^73 - 1326 * q^75 - 468 * q^79 + 162 * q^81 - 824 * q^83 + 1788 * q^85 - 612 * q^87 + 2118 * q^89 - 288 * q^93 + 120 * q^95 - 400 * q^97 - 234 * 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
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	0	0
$5$	21.3576	1.91028	0.955139	−	0.296158i	$-0.0957053\pi$
$5$	21.3576	1.91028	0.955139	+	0.296158i	$0.0957053\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	−31.3576	−0.859515	−0.429757	−	0.902944i	$-0.641401\pi$
$11$	−31.3576	−0.859515	−0.429757	+	0.902944i	$0.641401\pi$
$12$	0	0
$13$	−84.7151	−1.80737	−0.903683	−	0.428203i	$-0.859147\pi$
$13$	−84.7151	−1.80737	−0.903683	+	0.428203i	$0.859147\pi$
$14$	0	0
$15$	−64.0727	−1.10290
$16$	0	0
$17$	16.0727	0.229306	0.114653	−	0.993406i	$-0.463424\pi$
$17$	16.0727	0.229306	0.114653	+	0.993406i	$0.463424\pi$
$18$	0	0
$19$	20.0000	0.241490	0.120745	−	0.992684i	$-0.461472\pi$
$19$	20.0000	0.241490	0.120745	+	0.992684i	$0.461472\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	80.7878	0.732410	0.366205	−	0.930534i	$-0.380657\pi$
$23$	80.7878	0.732410	0.366205	+	0.930534i	$0.380657\pi$
$24$	0	0
$25$	331.145	2.64916
$26$	0	0
$27$	−27.0000	−0.192450
$28$	0	0
$29$	102.000	0.653135	0.326568	−	0.945174i	$-0.394108\pi$
$29$	102.000	0.653135	0.326568	+	0.945174i	$0.394108\pi$
$30$	0	0
$31$	−245.721	−1.42364	−0.711819	−	0.702363i	$-0.752128\pi$
$31$	−245.721	−1.42364	−0.711819	+	0.702363i	$0.752128\pi$
$32$	0	0
$33$	94.0727	0.496241
$34$	0	0
$35$	0	0
$36$	0	0
$37$	215.285	0.956557	0.478279	−	0.878208i	$-0.341261\pi$
$37$	215.285	0.956557	0.478279	+	0.878208i	$0.341261\pi$
$38$	0	0
$39$	254.145	1.04348
$40$	0	0
$41$	−150.933	−0.574922	−0.287461	−	0.957792i	$-0.592811\pi$
$41$	−150.933	−0.574922	−0.287461	+	0.957792i	$0.592811\pi$
$42$	0	0
$43$	−441.721	−1.56655	−0.783277	−	0.621673i	$-0.786453\pi$
$43$	−441.721	−1.56655	−0.783277	+	0.621673i	$0.786453\pi$
$44$	0	0
$45$	192.218	0.636759
$46$	0	0
$47$	−206.424	−0.640640	−0.320320	−	0.947309i	$-0.603790\pi$
$47$	−206.424	−0.640640	−0.320320	+	0.947309i	$0.603790\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	−48.2180	−0.132390
$52$	0	0
$53$	426.436	1.10520	0.552599	−	0.833447i	$-0.313636\pi$
$53$	426.436	1.10520	0.552599	+	0.833447i	$0.313636\pi$
$54$	0	0
$55$	−669.721	−1.64191
$56$	0	0
$57$	−60.0000	−0.139424
$58$	0	0
$59$	363.006	0.801006	0.400503	−	0.916296i	$-0.368835\pi$
$59$	363.006	0.801006	0.400503	+	0.916296i	$0.368835\pi$
$60$	0	0
$61$	343.721	0.721458	0.360729	−	0.932671i	$-0.382528\pi$
$61$	343.721	0.721458	0.360729	+	0.932671i	$0.382528\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−1809.31	−3.45257
$66$	0	0
$67$	−69.2849	−0.126336	−0.0631679	−	0.998003i	$-0.520120\pi$
$67$	−69.2849	−0.126336	−0.0631679	+	0.998003i	$0.520120\pi$
$68$	0	0
$69$	−242.363	−0.422857
$70$	0	0
$71$	468.933	0.783833	0.391916	−	0.920001i	$-0.371812\pi$
$71$	468.933	0.783833	0.391916	+	0.920001i	$0.371812\pi$
$72$	0	0
$73$	−747.576	−1.19859	−0.599295	−	0.800528i	$-0.704552\pi$
$73$	−747.576	−1.19859	−0.599295	+	0.800528i	$0.704552\pi$
$74$	0	0
$75$	−993.436	−1.52949
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−1298.74	−1.84961	−0.924807	−	0.380437i	$-0.875774\pi$
$79$	−1298.74	−1.84961	−0.924807	+	0.380437i	$0.875774\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	−1293.16	−1.71016	−0.855079	−	0.518498i	$-0.826491\pi$
$83$	−1293.16	−1.71016	−0.855079	+	0.518498i	$0.826491\pi$
$84$	0	0
$85$	343.273	0.438038
$86$	0	0
$87$	−306.000	−0.377088
$88$	0	0
$89$	563.346	0.670950	0.335475	−	0.942049i	$-0.391103\pi$
$89$	563.346	0.670950	0.335475	+	0.942049i	$0.391103\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	737.163	0.821938
$94$	0	0
$95$	427.151	0.461314
$96$	0	0
$97$	−1485.03	−1.55445	−0.777226	−	0.629221i	$-0.783374\pi$
$97$	−1485.03	−1.55445	−0.777226	+	0.629221i	$0.783374\pi$
$98$	0	0
$99$	−282.218	−0.286505
$100$	0	0
$101$	−1526.25	−1.50364	−0.751821	−	0.659367i	$-0.770824\pi$
$101$	−1526.25	−1.50364	−0.751821	+	0.659367i	$0.770824\pi$
$102$	0	0
$103$	−297.988	−0.285065	−0.142532	−	0.989790i	$-0.545524\pi$
$103$	−297.988	−0.285065	−0.142532	+	0.989790i	$0.545524\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−259.916	−0.234832	−0.117416	−	0.993083i	$-0.537461\pi$
$107$	−259.916	−0.234832	−0.117416	+	0.993083i	$0.537461\pi$
$108$	0	0
$109$	−1436.33	−1.26216	−0.631078	−	0.775719i	$-0.717387\pi$
$109$	−1436.33	−1.26216	−0.631078	+	0.775719i	$0.717387\pi$
$110$	0	0
$111$	−645.855	−0.552269
$112$	0	0
$113$	121.128	0.100838	0.0504192	−	0.998728i	$-0.483944\pi$
$113$	121.128	0.100838	0.0504192	+	0.998728i	$0.483944\pi$
$114$	0	0
$115$	1725.43	1.39911
$116$	0	0
$117$	−762.436	−0.602455
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−347.703	−0.261235
$122$	0	0
$123$	452.799	0.331931
$124$	0	0
$125$	4402.76	3.15036
$126$	0	0
$127$	−496.413	−0.346847	−0.173423	−	0.984847i	$-0.555483\pi$
$127$	−496.413	−0.346847	−0.173423	+	0.984847i	$0.555483\pi$
$128$	0	0
$129$	1325.16	0.904450
$130$	0	0
$131$	2674.59	1.78382	0.891909	−	0.452214i	$-0.149366\pi$
$131$	2674.59	1.78382	0.891909	+	0.452214i	$0.149366\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−576.654	−0.367633
$136$	0	0
$137$	2370.46	1.47826	0.739131	−	0.673561i	$-0.235236\pi$
$137$	2370.46	1.47826	0.739131	+	0.673561i	$0.235236\pi$
$138$	0	0
$139$	−95.4419	−0.0582394	−0.0291197	−	0.999576i	$-0.509270\pi$
$139$	−95.4419	−0.0582394	−0.0291197	+	0.999576i	$0.509270\pi$
$140$	0	0
$141$	619.273	0.369874
$142$	0	0
$143$	2656.46	1.55346
$144$	0	0
$145$	2178.47	1.24767
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−2261.56	−1.24345	−0.621727	−	0.783234i	$-0.713569\pi$
$149$	−2261.56	−1.24345	−0.621727	+	0.783234i	$0.713569\pi$
$150$	0	0
$151$	−126.884	−0.0683818	−0.0341909	−	0.999415i	$-0.510885\pi$
$151$	−126.884	−0.0683818	−0.0341909	+	0.999415i	$0.510885\pi$
$152$	0	0
$153$	144.654	0.0764352
$154$	0	0
$155$	−5248.00	−2.71955
$156$	0	0
$157$	−2977.73	−1.51369	−0.756844	−	0.653596i	$-0.773260\pi$
$157$	−2977.73	−1.51369	−0.756844	+	0.653596i	$0.773260\pi$
$158$	0	0
$159$	−1279.31	−0.638086
$160$	0	0
$161$	0	0
$162$	0	0
$163$	303.029	0.145614	0.0728070	−	0.997346i	$-0.476804\pi$
$163$	303.029	0.145614	0.0728070	+	0.997346i	$0.476804\pi$
$164$	0	0
$165$	2009.16	0.947958
$166$	0	0
$167$	3547.90	1.64398	0.821991	−	0.569501i	$-0.192864\pi$
$167$	3547.90	1.64398	0.821991	+	0.569501i	$0.192864\pi$
$168$	0	0
$169$	4979.65	2.26657
$170$	0	0
$171$	180.000	0.0804967
$172$	0	0
$173$	−283.102	−0.124415	−0.0622076	−	0.998063i	$-0.519814\pi$
$173$	−283.102	−0.124415	−0.0622076	+	0.998063i	$0.519814\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−1089.02	−0.462461
$178$	0	0
$179$	−1611.50	−0.672902	−0.336451	−	0.941701i	$-0.609227\pi$
$179$	−1611.50	−0.672902	−0.336451	+	0.941701i	$0.609227\pi$
$180$	0	0
$181$	2794.99	1.14779	0.573896	−	0.818928i	$-0.305431\pi$
$181$	2794.99	1.14779	0.573896	+	0.818928i	$0.305431\pi$
$182$	0	0
$183$	−1031.16	−0.416534
$184$	0	0
$185$	4597.96	1.82729
$186$	0	0
$187$	−504.000	−0.197092
$188$	0	0
$189$	0	0
$190$	0	0
$191$	2786.99	1.05581	0.527905	−	0.849304i	$-0.322978\pi$
$191$	2786.99	1.05581	0.527905	+	0.849304i	$0.322978\pi$
$192$	0	0
$193$	−1751.87	−0.653379	−0.326689	−	0.945132i	$-0.605933\pi$
$193$	−1751.87	−0.653379	−0.326689	+	0.945132i	$0.605933\pi$
$194$	0	0
$195$	5427.92	1.99334
$196$	0	0
$197$	964.715	0.348899	0.174450	−	0.984666i	$-0.444185\pi$
$197$	964.715	0.348899	0.174450	+	0.984666i	$0.444185\pi$
$198$	0	0
$199$	−5284.36	−1.88240	−0.941202	−	0.337844i	$-0.890302\pi$
$199$	−5284.36	−1.88240	−0.941202	+	0.337844i	$0.890302\pi$
$200$	0	0
$201$	207.855	0.0729400
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−3223.56	−1.09826
$206$	0	0
$207$	727.090	0.244137
$208$	0	0
$209$	−627.151	−0.207564
$210$	0	0
$211$	1998.04	0.651898	0.325949	−	0.945387i	$-0.394316\pi$
$211$	1998.04	0.651898	0.325949	+	0.945387i	$0.394316\pi$
$212$	0	0
$213$	−1406.80	−0.452546
$214$	0	0
$215$	−9434.08	−2.99255
$216$	0	0
$217$	0	0
$218$	0	0
$219$	2242.73	0.692007
$220$	0	0
$221$	−1361.60	−0.414439
$222$	0	0
$223$	−3969.41	−1.19198	−0.595989	−	0.802993i	$-0.703240\pi$
$223$	−3969.41	−1.19198	−0.595989	+	0.802993i	$0.703240\pi$
$224$	0	0
$225$	2980.31	0.883054
$226$	0	0
$227$	−2175.32	−0.636040	−0.318020	−	0.948084i	$-0.603018\pi$
$227$	−2175.32	−0.636040	−0.318020	+	0.948084i	$0.603018\pi$
$228$	0	0
$229$	−2567.33	−0.740847	−0.370424	−	0.928863i	$-0.620787\pi$
$229$	−2567.33	−0.740847	−0.370424	+	0.928863i	$0.620787\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−2114.68	−0.594581	−0.297290	−	0.954787i	$-0.596083\pi$
$233$	−2114.68	−0.594581	−0.297290	+	0.954787i	$0.596083\pi$
$234$	0	0
$235$	−4408.72	−1.22380
$236$	0	0
$237$	3896.22	1.06788
$238$	0	0
$239$	−5418.12	−1.46640	−0.733199	−	0.680014i	$-0.761974\pi$
$239$	−5418.12	−1.46640	−0.733199	+	0.680014i	$0.761974\pi$
$240$	0	0
$241$	−376.378	−0.100600	−0.0503000	−	0.998734i	$-0.516018\pi$
$241$	−376.378	−0.100600	−0.0503000	+	0.998734i	$0.516018\pi$
$242$	0	0
$243$	−243.000	−0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−1694.30	−0.436461
$248$	0	0
$249$	3879.49	0.987360
$250$	0	0
$251$	1833.94	0.461183	0.230592	−	0.973051i	$-0.425934\pi$
$251$	1833.94	0.461183	0.230592	+	0.973051i	$0.425934\pi$
$252$	0	0
$253$	−2533.31	−0.629517
$254$	0	0
$255$	−1029.82	−0.252901
$256$	0	0
$257$	−4606.35	−1.11804	−0.559020	−	0.829154i	$-0.688823\pi$
$257$	−4606.35	−1.11804	−0.559020	+	0.829154i	$0.688823\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	918.000	0.217712
$262$	0	0
$263$	3565.97	0.836074	0.418037	−	0.908430i	$-0.362718\pi$
$263$	3565.97	0.836074	0.418037	+	0.908430i	$0.362718\pi$
$264$	0	0
$265$	9107.63	2.11124
$266$	0	0
$267$	−1690.04	−0.387373
$268$	0	0
$269$	−926.084	−0.209905	−0.104952	−	0.994477i	$-0.533469\pi$
$269$	−926.084	−0.209905	−0.104952	+	0.994477i	$0.533469\pi$
$270$	0	0
$271$	−4720.24	−1.05806	−0.529030	−	0.848603i	$-0.677444\pi$
$271$	−4720.24	−1.05806	−0.529030	+	0.848603i	$0.677444\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−10383.9	−2.27699
$276$	0	0
$277$	−5273.15	−1.14380	−0.571900	−	0.820323i	$-0.693793\pi$
$277$	−5273.15	−1.14380	−0.571900	+	0.820323i	$0.693793\pi$
$278$	0	0
$279$	−2211.49	−0.474546
$280$	0	0
$281$	4130.22	0.876826	0.438413	−	0.898774i	$-0.355541\pi$
$281$	4130.22	0.876826	0.438413	+	0.898774i	$0.355541\pi$
$282$	0	0
$283$	3161.77	0.664126	0.332063	−	0.943257i	$-0.392255\pi$
$283$	3161.77	0.664126	0.332063	+	0.943257i	$0.392255\pi$
$284$	0	0
$285$	−1281.45	−0.266340
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−4654.67	−0.947419
$290$	0	0
$291$	4455.09	0.897463
$292$	0	0
$293$	−3546.23	−0.707075	−0.353537	−	0.935420i	$-0.615021\pi$
$293$	−3546.23	−0.707075	−0.353537	+	0.935420i	$0.615021\pi$
$294$	0	0
$295$	7752.92	1.53014
$296$	0	0
$297$	846.654	0.165414
$298$	0	0
$299$	−6843.95	−1.32373
$300$	0	0
$301$	0	0
$302$	0	0
$303$	4578.76	0.868128
$304$	0	0
$305$	7341.04	1.37819
$306$	0	0
$307$	−596.966	−0.110979	−0.0554896	−	0.998459i	$-0.517672\pi$
$307$	−596.966	−0.110979	−0.0554896	+	0.998459i	$0.517672\pi$
$308$	0	0
$309$	893.965	0.164582
$310$	0	0
$311$	−3234.94	−0.589828	−0.294914	−	0.955524i	$-0.595291\pi$
$311$	−3234.94	−0.589828	−0.294914	+	0.955524i	$0.595291\pi$
$312$	0	0
$313$	−2307.12	−0.416632	−0.208316	−	0.978062i	$-0.566798\pi$
$313$	−2307.12	−0.416632	−0.208316	+	0.978062i	$0.566798\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	5190.60	0.919663	0.459831	−	0.888006i	$-0.347910\pi$
$317$	5190.60	0.919663	0.459831	+	0.888006i	$0.347910\pi$
$318$	0	0
$319$	−3198.47	−0.561379
$320$	0	0
$321$	779.747	0.135580
$322$	0	0
$323$	321.454	0.0553751
$324$	0	0
$325$	−28053.0	−4.78800
$326$	0	0
$327$	4308.98	0.728706
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−1651.08	−0.274174	−0.137087	−	0.990559i	$-0.543774\pi$
$331$	−1651.08	−0.274174	−0.137087	+	0.990559i	$0.543774\pi$
$332$	0	0
$333$	1937.56	0.318852
$334$	0	0
$335$	−1479.76	−0.241336
$336$	0	0
$337$	−5633.73	−0.910650	−0.455325	−	0.890325i	$-0.650477\pi$
$337$	−5633.73	−0.910650	−0.455325	+	0.890325i	$0.650477\pi$
$338$	0	0
$339$	−363.384	−0.0582191
$340$	0	0
$341$	7705.21	1.22364
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−5176.29	−0.807774
$346$	0	0
$347$	2701.17	0.417886	0.208943	−	0.977928i	$-0.432998\pi$
$347$	2701.17	0.417886	0.208943	+	0.977928i	$0.432998\pi$
$348$	0	0
$349$	−5855.26	−0.898065	−0.449032	−	0.893515i	$-0.648231\pi$
$349$	−5855.26	−0.898065	−0.449032	+	0.893515i	$0.648231\pi$
$350$	0	0
$351$	2287.31	0.347828
$352$	0	0
$353$	−7167.35	−1.08068	−0.540339	−	0.841447i	$-0.681704\pi$
$353$	−7167.35	−1.08068	−0.540339	+	0.841447i	$0.681704\pi$
$354$	0	0
$355$	10015.3	1.49734
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−5899.16	−0.867258	−0.433629	−	0.901091i	$-0.642767\pi$
$359$	−5899.16	−0.867258	−0.433629	+	0.901091i	$0.642767\pi$
$360$	0	0
$361$	−6459.00	−0.941682
$362$	0	0
$363$	1043.11	0.150824
$364$	0	0
$365$	−15966.4	−2.28964
$366$	0	0
$367$	10663.5	1.51670	0.758349	−	0.651848i	$-0.226006\pi$
$367$	10663.5	1.51670	0.758349	+	0.651848i	$0.226006\pi$
$368$	0	0
$369$	−1358.40	−0.191641
$370$	0	0
$371$	0	0
$372$	0	0
$373$	3563.67	0.494692	0.247346	−	0.968927i	$-0.420442\pi$
$373$	3563.67	0.494692	0.247346	+	0.968927i	$0.420442\pi$
$374$	0	0
$375$	−13208.3	−1.81886
$376$	0	0
$377$	−8640.94	−1.18045
$378$	0	0
$379$	−2702.04	−0.366212	−0.183106	−	0.983093i	$-0.558615\pi$
$379$	−2702.04	−0.366212	−0.183106	+	0.983093i	$0.558615\pi$
$380$	0	0
$381$	1489.24	0.200252
$382$	0	0
$383$	6244.41	0.833092	0.416546	−	0.909115i	$-0.363240\pi$
$383$	6244.41	0.833092	0.416546	+	0.909115i	$0.363240\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−3975.49	−0.522185
$388$	0	0
$389$	3753.93	0.489285	0.244642	−	0.969613i	$-0.421329\pi$
$389$	3753.93	0.489285	0.244642	+	0.969613i	$0.421329\pi$
$390$	0	0
$391$	1298.48	0.167946
$392$	0	0
$393$	−8023.78	−1.02989
$394$	0	0
$395$	−27737.9	−3.53328
$396$	0	0
$397$	−5717.71	−0.722830	−0.361415	−	0.932405i	$-0.617706\pi$
$397$	−5717.71	−0.722830	−0.361415	+	0.932405i	$0.617706\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	622.680	0.0775440	0.0387720	−	0.999248i	$-0.487655\pi$
$401$	622.680	0.0775440	0.0387720	+	0.999248i	$0.487655\pi$
$402$	0	0
$403$	20816.3	2.57303
$404$	0	0
$405$	1729.96	0.212253
$406$	0	0
$407$	−6750.81	−0.822175
$408$	0	0
$409$	6462.06	0.781243	0.390622	−	0.920551i	$-0.372260\pi$
$409$	6462.06	0.781243	0.390622	+	0.920551i	$0.372260\pi$
$410$	0	0
$411$	−7111.38	−0.853475
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−27618.8	−3.26688
$416$	0	0
$417$	286.326	0.0336245
$418$	0	0
$419$	1418.38	0.165375	0.0826877	−	0.996576i	$-0.473650\pi$
$419$	1418.38	0.165375	0.0826877	+	0.996576i	$0.473650\pi$
$420$	0	0
$421$	−5773.61	−0.668381	−0.334191	−	0.942505i	$-0.608463\pi$
$421$	−5773.61	−0.668381	−0.334191	+	0.942505i	$0.608463\pi$
$422$	0	0
$423$	−1857.82	−0.213547
$424$	0	0
$425$	5322.39	0.607468
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−7969.38	−0.896889
$430$	0	0
$431$	14770.4	1.65073	0.825366	−	0.564598i	$-0.190969\pi$
$431$	14770.4	1.65073	0.825366	+	0.564598i	$0.190969\pi$
$432$	0	0
$433$	2353.06	0.261156	0.130578	−	0.991438i	$-0.458317\pi$
$433$	2353.06	0.261156	0.130578	+	0.991438i	$0.458317\pi$
$434$	0	0
$435$	−6535.41	−0.720343
$436$	0	0
$437$	1615.76	0.176870
$438$	0	0
$439$	−8437.95	−0.917361	−0.458681	−	0.888601i	$-0.651678\pi$
$439$	−8437.95	−0.917361	−0.458681	+	0.888601i	$0.651678\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−5577.32	−0.598163	−0.299082	−	0.954228i	$-0.596680\pi$
$443$	−5577.32	−0.598163	−0.299082	+	0.954228i	$0.596680\pi$
$444$	0	0
$445$	12031.7	1.28170
$446$	0	0
$447$	6784.69	0.717908
$448$	0	0
$449$	2837.09	0.298198	0.149099	−	0.988822i	$-0.452363\pi$
$449$	2837.09	0.298198	0.149099	+	0.988822i	$0.452363\pi$
$450$	0	0
$451$	4732.90	0.494154
$452$	0	0
$453$	380.651	0.0394803
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−4460.98	−0.456621	−0.228310	−	0.973588i	$-0.573320\pi$
$457$	−4460.98	−0.456621	−0.228310	+	0.973588i	$0.573320\pi$
$458$	0	0
$459$	−433.962	−0.0441299
$460$	0	0
$461$	5008.00	0.505956	0.252978	−	0.967472i	$-0.418590\pi$
$461$	5008.00	0.505956	0.252978	+	0.967472i	$0.418590\pi$
$462$	0	0
$463$	−348.867	−0.0350178	−0.0175089	−	0.999847i	$-0.505574\pi$
$463$	−348.867	−0.0350178	−0.0175089	+	0.999847i	$0.505574\pi$
$464$	0	0
$465$	15744.0	1.57013
$466$	0	0
$467$	9786.32	0.969715	0.484857	−	0.874593i	$-0.338871\pi$
$467$	9786.32	0.969715	0.484857	+	0.874593i	$0.338871\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	8933.20	0.873928
$472$	0	0
$473$	13851.3	1.34648
$474$	0	0
$475$	6622.91	0.639747
$476$	0	0
$477$	3837.92	0.368399
$478$	0	0
$479$	−13073.7	−1.24708	−0.623540	−	0.781792i	$-0.714306\pi$
$479$	−13073.7	−1.24708	−0.623540	+	0.781792i	$0.714306\pi$
$480$	0	0
$481$	−18237.9	−1.72885
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−31716.6	−2.96944
$486$	0	0
$487$	−1220.76	−0.113589	−0.0567944	−	0.998386i	$-0.518088\pi$
$487$	−1220.76	−0.113589	−0.0567944	+	0.998386i	$0.518088\pi$
$488$	0	0
$489$	−909.088	−0.0840703
$490$	0	0
$491$	6367.89	0.585293	0.292647	−	0.956221i	$-0.405464\pi$
$491$	6367.89	0.585293	0.292647	+	0.956221i	$0.405464\pi$
$492$	0	0
$493$	1639.41	0.149768
$494$	0	0
$495$	−6027.49	−0.547304
$496$	0	0
$497$	0	0
$498$	0	0
$499$	11715.5	1.05101	0.525507	−	0.850789i	$-0.323876\pi$
$499$	11715.5	1.05101	0.525507	+	0.850789i	$0.323876\pi$
$500$	0	0
$501$	−10643.7	−0.949153
$502$	0	0
$503$	−13412.1	−1.18890	−0.594450	−	0.804133i	$-0.702630\pi$
$503$	−13412.1	−1.18890	−0.594450	+	0.804133i	$0.702630\pi$
$504$	0	0
$505$	−32597.0	−2.87237
$506$	0	0
$507$	−14939.0	−1.30860
$508$	0	0
$509$	−10029.1	−0.873345	−0.436672	−	0.899621i	$-0.643843\pi$
$509$	−10029.1	−0.873345	−0.436672	+	0.899621i	$0.643843\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−540.000	−0.0464748
$514$	0	0
$515$	−6364.30	−0.544553
$516$	0	0
$517$	6472.97	0.550640
$518$	0	0
$519$	849.306	0.0718312
$520$	0	0
$521$	14668.9	1.23350	0.616752	−	0.787158i	$-0.288448\pi$
$521$	14668.9	1.23350	0.616752	+	0.787158i	$0.288448\pi$
$522$	0	0
$523$	4305.05	0.359936	0.179968	−	0.983672i	$-0.442401\pi$
$523$	4305.05	0.359936	0.179968	+	0.983672i	$0.442401\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−3949.39	−0.326448
$528$	0	0
$529$	−5640.33	−0.463576
$530$	0	0
$531$	3267.05	0.267002
$532$	0	0
$533$	12786.3	1.03909
$534$	0	0
$535$	−5551.16	−0.448594
$536$	0	0
$537$	4834.51	0.388500
$538$	0	0
$539$	0	0
$540$	0	0
$541$	1876.27	0.149108	0.0745539	−	0.997217i	$-0.476247\pi$
$541$	1876.27	0.149108	0.0745539	+	0.997217i	$0.476247\pi$
$542$	0	0
$543$	−8384.98	−0.662678
$544$	0	0
$545$	−30676.4	−2.41107
$546$	0	0
$547$	4753.99	0.371602	0.185801	−	0.982587i	$-0.440512\pi$
$547$	4753.99	0.371602	0.185801	+	0.982587i	$0.440512\pi$
$548$	0	0
$549$	3093.49	0.240486
$550$	0	0
$551$	2040.00	0.157726
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−13793.9	−1.05499
$556$	0	0
$557$	−18408.3	−1.40033	−0.700165	−	0.713981i	$-0.746890\pi$
$557$	−18408.3	−1.40033	−0.700165	+	0.713981i	$0.746890\pi$
$558$	0	0
$559$	37420.4	2.83133
$560$	0	0
$561$	1512.00	0.113791
$562$	0	0
$563$	−1220.61	−0.0913723	−0.0456861	−	0.998956i	$-0.514547\pi$
$563$	−1220.61	−0.0913723	−0.0456861	+	0.998956i	$0.514547\pi$
$564$	0	0
$565$	2587.00	0.192630
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−12709.3	−0.936381	−0.468191	−	0.883627i	$-0.655094\pi$
$569$	−12709.3	−0.936381	−0.468191	+	0.883627i	$0.655094\pi$
$570$	0	0
$571$	−6789.95	−0.497637	−0.248818	−	0.968550i	$-0.580042\pi$
$571$	−6789.95	−0.497637	−0.248818	+	0.968550i	$0.580042\pi$
$572$	0	0
$573$	−8360.97	−0.609572
$574$	0	0
$575$	26752.5	1.94027
$576$	0	0
$577$	4818.41	0.347648	0.173824	−	0.984777i	$-0.444388\pi$
$577$	4818.41	0.347648	0.173824	+	0.984777i	$0.444388\pi$
$578$	0	0
$579$	5255.60	0.377228
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−13372.0	−0.949934
$584$	0	0
$585$	−16283.8	−1.15086
$586$	0	0
$587$	2086.69	0.146724	0.0733620	−	0.997305i	$-0.476627\pi$
$587$	2086.69	0.146724	0.0733620	+	0.997305i	$0.476627\pi$
$588$	0	0
$589$	−4914.42	−0.343795
$590$	0	0
$591$	−2894.15	−0.201437
$592$	0	0
$593$	−22800.0	−1.57889	−0.789447	−	0.613818i	$-0.789633\pi$
$593$	−22800.0	−1.57889	−0.789447	+	0.613818i	$0.789633\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	15853.1	1.08681
$598$	0	0
$599$	−21433.3	−1.46200	−0.731001	−	0.682376i	$-0.760947\pi$
$599$	−21433.3	−1.46200	−0.731001	+	0.682376i	$0.760947\pi$
$600$	0	0
$601$	9749.85	0.661738	0.330869	−	0.943677i	$-0.392658\pi$
$601$	9749.85	0.661738	0.330869	+	0.943677i	$0.392658\pi$
$602$	0	0
$603$	−623.564	−0.0421119
$604$	0	0
$605$	−7426.10	−0.499031
$606$	0	0
$607$	22837.3	1.52708	0.763541	−	0.645760i	$-0.223459\pi$
$607$	22837.3	1.52708	0.763541	+	0.645760i	$0.223459\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	17487.3	1.15787
$612$	0	0
$613$	−280.302	−0.0184687	−0.00923434	−	0.999957i	$-0.502939\pi$
$613$	−280.302	−0.0184687	−0.00923434	+	0.999957i	$0.502939\pi$
$614$	0	0
$615$	9670.69	0.634081
$616$	0	0
$617$	−13271.3	−0.865934	−0.432967	−	0.901410i	$-0.642533\pi$
$617$	−13271.3	−0.865934	−0.432967	+	0.901410i	$0.642533\pi$
$618$	0	0
$619$	30464.9	1.97817	0.989085	−	0.147349i	$-0.0470741\pi$
$619$	30464.9	1.97817	0.989085	+	0.147349i	$0.0470741\pi$
$620$	0	0
$621$	−2181.27	−0.140952
$622$	0	0
$623$	0	0
$624$	0	0
$625$	52639.1	3.36890
$626$	0	0
$627$	1881.45	0.119837
$628$	0	0
$629$	3460.20	0.219344
$630$	0	0
$631$	8564.10	0.540303	0.270152	−	0.962818i	$-0.412926\pi$
$631$	8564.10	0.540303	0.270152	+	0.962818i	$0.412926\pi$
$632$	0	0
$633$	−5994.11	−0.376373
$634$	0	0
$635$	−10602.2	−0.662573
$636$	0	0
$637$	0	0
$638$	0	0
$639$	4220.40	0.261278
$640$	0	0
$641$	−3170.52	−0.195363	−0.0976817	−	0.995218i	$-0.531143\pi$
$641$	−3170.52	−0.195363	−0.0976817	+	0.995218i	$0.531143\pi$
$642$	0	0
$643$	−22499.6	−1.37993	−0.689966	−	0.723842i	$-0.742374\pi$
$643$	−22499.6	−1.37993	−0.689966	+	0.723842i	$0.742374\pi$
$644$	0	0
$645$	28302.2	1.72775
$646$	0	0
$647$	24475.4	1.48722	0.743608	−	0.668615i	$-0.233113\pi$
$647$	24475.4	1.48722	0.743608	+	0.668615i	$0.233113\pi$
$648$	0	0
$649$	−11383.0	−0.688476
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−6047.57	−0.362419	−0.181210	−	0.983445i	$-0.558001\pi$
$653$	−6047.57	−0.362419	−0.181210	+	0.983445i	$0.558001\pi$
$654$	0	0
$655$	57122.8	3.40759
$656$	0	0
$657$	−6728.18	−0.399530
$658$	0	0
$659$	−5415.60	−0.320124	−0.160062	−	0.987107i	$-0.551169\pi$
$659$	−5415.60	−0.320124	−0.160062	+	0.987107i	$0.551169\pi$
$660$	0	0
$661$	21652.6	1.27411	0.637056	−	0.770817i	$-0.280152\pi$
$661$	21652.6	1.27411	0.637056	+	0.770817i	$0.280152\pi$
$662$	0	0
$663$	4084.80	0.239277
$664$	0	0
$665$	0	0
$666$	0	0
$667$	8240.36	0.478363
$668$	0	0
$669$	11908.2	0.688189
$670$	0	0
$671$	−10778.3	−0.620104
$672$	0	0
$673$	525.611	0.0301052	0.0150526	−	0.999887i	$-0.495208\pi$
$673$	525.611	0.0301052	0.0150526	+	0.999887i	$0.495208\pi$
$674$	0	0
$675$	−8940.92	−0.509832
$676$	0	0
$677$	−1169.54	−0.0663947	−0.0331974	−	0.999449i	$-0.510569\pi$
$677$	−1169.54	−0.0663947	−0.0331974	+	0.999449i	$0.510569\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	6525.96	0.367218
$682$	0	0
$683$	11840.7	0.663356	0.331678	−	0.943393i	$-0.392385\pi$
$683$	11840.7	0.663356	0.331678	+	0.943393i	$0.392385\pi$
$684$	0	0
$685$	50627.2	2.82389
$686$	0	0
$687$	7701.99	0.427728
$688$	0	0
$689$	−36125.6	−1.99750
$690$	0	0
$691$	−267.476	−0.0147254	−0.00736271	−	0.999973i	$-0.502344\pi$
$691$	−267.476	−0.0147254	−0.00736271	+	0.999973i	$0.502344\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−2038.41	−0.111253
$696$	0	0
$697$	−2425.90	−0.131833
$698$	0	0
$699$	6344.04	0.343281
$700$	0	0
$701$	9700.72	0.522669	0.261335	−	0.965248i	$-0.415837\pi$
$701$	9700.72	0.522669	0.261335	+	0.965248i	$0.415837\pi$
$702$	0	0
$703$	4305.70	0.230999
$704$	0	0
$705$	13226.2	0.706562
$706$	0	0
$707$	0	0
$708$	0	0
$709$	33319.9	1.76496	0.882479	−	0.470353i	$-0.155873\pi$
$709$	33319.9	1.76496	0.882479	+	0.470353i	$0.155873\pi$
$710$	0	0
$711$	−11688.6	−0.616538
$712$	0	0
$713$	−19851.3	−1.04269
$714$	0	0
$715$	56735.5	2.96753
$716$	0	0
$717$	16254.4	0.846625
$718$	0	0
$719$	34321.8	1.78023	0.890117	−	0.455733i	$-0.150623\pi$
$719$	34321.8	1.78023	0.890117	+	0.455733i	$0.150623\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	1129.13	0.0580815
$724$	0	0
$725$	33776.8	1.73026
$726$	0	0
$727$	27768.3	1.41660	0.708301	−	0.705910i	$-0.249462\pi$
$727$	27768.3	1.41660	0.708301	+	0.705910i	$0.249462\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	−7099.64	−0.359220
$732$	0	0
$733$	4358.45	0.219622	0.109811	−	0.993952i	$-0.464975\pi$
$733$	4358.45	0.219622	0.109811	+	0.993952i	$0.464975\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	2172.60	0.108587
$738$	0	0
$739$	34023.0	1.69358	0.846789	−	0.531928i	$-0.178532\pi$
$739$	34023.0	1.69358	0.846789	+	0.531928i	$0.178532\pi$
$740$	0	0
$741$	5082.91	0.251991
$742$	0	0
$743$	−19494.5	−0.962563	−0.481281	−	0.876566i	$-0.659828\pi$
$743$	−19494.5	−0.962563	−0.481281	+	0.876566i	$0.659828\pi$
$744$	0	0
$745$	−48301.5	−2.37534
$746$	0	0
$747$	−11638.5	−0.570052
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−3936.95	−0.191294	−0.0956468	−	0.995415i	$-0.530492\pi$
$751$	−3936.95	−0.191294	−0.0956468	+	0.995415i	$0.530492\pi$
$752$	0	0
$753$	−5501.81	−0.266264
$754$	0	0
$755$	−2709.93	−0.130628
$756$	0	0
$757$	27339.3	1.31263	0.656316	−	0.754486i	$-0.272114\pi$
$757$	27339.3	1.31263	0.656316	+	0.754486i	$0.272114\pi$
$758$	0	0
$759$	7599.92	0.363452
$760$	0	0
$761$	−1917.52	−0.0913402	−0.0456701	−	0.998957i	$-0.514542\pi$
$761$	−1917.52	−0.0913402	−0.0456701	+	0.998957i	$0.514542\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	3089.46	0.146013
$766$	0	0
$767$	−30752.1	−1.44771
$768$	0	0
$769$	9055.78	0.424655	0.212328	−	0.977199i	$-0.431896\pi$
$769$	9055.78	0.424655	0.212328	+	0.977199i	$0.431896\pi$
$770$	0	0
$771$	13819.1	0.645501
$772$	0	0
$773$	29066.5	1.35246	0.676228	−	0.736693i	$-0.263614\pi$
$773$	29066.5	1.35246	0.676228	+	0.736693i	$0.263614\pi$
$774$	0	0
$775$	−81369.4	−3.77145
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−3018.66	−0.138838
$780$	0	0
$781$	−14704.6	−0.673716
$782$	0	0
$783$	−2754.00	−0.125696
$784$	0	0
$785$	−63597.1	−2.89156
$786$	0	0
$787$	15939.6	0.721962	0.360981	−	0.932573i	$-0.382442\pi$
$787$	15939.6	0.721962	0.360981	+	0.932573i	$0.382442\pi$
$788$	0	0
$789$	−10697.9	−0.482707
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−29118.4	−1.30394
$794$	0	0
$795$	−27322.9	−1.21892
$796$	0	0
$797$	−16655.0	−0.740212	−0.370106	−	0.928989i	$-0.620679\pi$
$797$	−16655.0	−0.740212	−0.370106	+	0.928989i	$0.620679\pi$
$798$	0	0
$799$	−3317.79	−0.146902
$800$	0	0
$801$	5070.11	0.223650
$802$	0	0
$803$	23442.1	1.03021
$804$	0	0
$805$	0	0
$806$	0	0
$807$	2778.25	0.121189
$808$	0	0
$809$	−3173.02	−0.137896	−0.0689478	−	0.997620i	$-0.521964\pi$
$809$	−3173.02	−0.137896	−0.0689478	+	0.997620i	$0.521964\pi$
$810$	0	0
$811$	2879.83	0.124691	0.0623455	−	0.998055i	$-0.480142\pi$
$811$	2879.83	0.124691	0.0623455	+	0.998055i	$0.480142\pi$
$812$	0	0
$813$	14160.7	0.610872
$814$	0	0
$815$	6471.96	0.278163
$816$	0	0
$817$	−8834.42	−0.378307
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−16714.5	−0.710522	−0.355261	−	0.934767i	$-0.615608\pi$
$821$	−16714.5	−0.710522	−0.355261	+	0.934767i	$0.615608\pi$
$822$	0	0
$823$	−37949.4	−1.60733	−0.803665	−	0.595082i	$-0.797119\pi$
$823$	−37949.4	−1.60733	−0.803665	+	0.595082i	$0.797119\pi$
$824$	0	0
$825$	31151.7	1.31462
$826$	0	0
$827$	−17817.1	−0.749165	−0.374583	−	0.927194i	$-0.622214\pi$
$827$	−17817.1	−0.749165	−0.374583	+	0.927194i	$0.622214\pi$
$828$	0	0
$829$	22637.8	0.948425	0.474213	−	0.880410i	$-0.342733\pi$
$829$	22637.8	0.948425	0.474213	+	0.880410i	$0.342733\pi$
$830$	0	0
$831$	15819.4	0.660373
$832$	0	0
$833$	0	0
$834$	0	0
$835$	75774.5	3.14046
$836$	0	0
$837$	6634.47	0.273979
$838$	0	0
$839$	−43907.0	−1.80672	−0.903359	−	0.428884i	$-0.858907\pi$
$839$	−43907.0	−1.80672	−0.903359	+	0.428884i	$0.858907\pi$
$840$	0	0
$841$	−13985.0	−0.573414
$842$	0	0
$843$	−12390.6	−0.506235
$844$	0	0
$845$	106353.	4.32978
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−9485.30	−0.383433
$850$	0	0
$851$	17392.4	0.700592
$852$	0	0
$853$	−41480.0	−1.66500	−0.832502	−	0.554022i	$-0.813092\pi$
$853$	−41480.0	−1.66500	−0.832502	+	0.554022i	$0.813092\pi$
$854$	0	0
$855$	3844.36	0.153771
$856$	0	0
$857$	34365.9	1.36980	0.684899	−	0.728638i	$-0.259846\pi$
$857$	34365.9	1.36980	0.684899	+	0.728638i	$0.259846\pi$
$858$	0	0
$859$	13207.6	0.524606	0.262303	−	0.964986i	$-0.415518\pi$
$859$	13207.6	0.524606	0.262303	+	0.964986i	$0.415518\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	5894.31	0.232497	0.116248	−	0.993220i	$-0.462913\pi$
$863$	5894.31	0.232497	0.116248	+	0.993220i	$0.462913\pi$
$864$	0	0
$865$	−6046.36	−0.237668
$866$	0	0
$867$	13964.0	0.546993
$868$	0	0
$869$	40725.3	1.58977
$870$	0	0
$871$	5869.48	0.228335
$872$	0	0
$873$	−13365.3	−0.518151
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−36195.4	−1.39365	−0.696825	−	0.717241i	$-0.745405\pi$
$877$	−36195.4	−1.39365	−0.696825	+	0.717241i	$0.745405\pi$
$878$	0	0
$879$	10638.7	0.408230
$880$	0	0
$881$	2389.46	0.0913766	0.0456883	−	0.998956i	$-0.485452\pi$
$881$	2389.46	0.0913766	0.0456883	+	0.998956i	$0.485452\pi$
$882$	0	0
$883$	−7146.69	−0.272373	−0.136186	−	0.990683i	$-0.543485\pi$
$883$	−7146.69	−0.272373	−0.136186	+	0.990683i	$0.543485\pi$
$884$	0	0
$885$	−23258.8	−0.883429
$886$	0	0
$887$	−13497.3	−0.510930	−0.255465	−	0.966818i	$-0.582229\pi$
$887$	−13497.3	−0.510930	−0.255465	+	0.966818i	$0.582229\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−2539.96	−0.0955016
$892$	0	0
$893$	−4128.49	−0.154708
$894$	0	0
$895$	−34417.8	−1.28543
$896$	0	0
$897$	20531.8	0.764257
$898$	0	0
$899$	−25063.5	−0.929828
$900$	0	0
$901$	6853.97	0.253428
$902$	0	0
$903$	0	0
$904$	0	0
$905$	59694.3	2.19260
$906$	0	0
$907$	−37635.1	−1.37779	−0.688894	−	0.724862i	$-0.741903\pi$
$907$	−37635.1	−1.37779	−0.688894	+	0.724862i	$0.741903\pi$
$908$	0	0
$909$	−13736.3	−0.501214
$910$	0	0
$911$	−53207.9	−1.93508	−0.967540	−	0.252718i	$-0.918675\pi$
$911$	−53207.9	−1.93508	−0.967540	+	0.252718i	$0.918675\pi$
$912$	0	0
$913$	40550.4	1.46991
$914$	0	0
$915$	−22023.1	−0.795696
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−45909.8	−1.64790	−0.823952	−	0.566659i	$-0.808236\pi$
$919$	−45909.8	−1.64790	−0.823952	+	0.566659i	$0.808236\pi$
$920$	0	0
$921$	1790.90	0.0640739
$922$	0	0
$923$	−39725.7	−1.41667
$924$	0	0
$925$	71290.6	2.53408
$926$	0	0
$927$	−2681.89	−0.0950216
$928$	0	0
$929$	−3855.67	−0.136168	−0.0680841	−	0.997680i	$-0.521689\pi$
$929$	−3855.67	−0.136168	−0.0680841	+	0.997680i	$0.521689\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	9704.81	0.340537
$934$	0	0
$935$	−10764.2	−0.376500
$936$	0	0
$937$	−5584.25	−0.194695	−0.0973476	−	0.995250i	$-0.531036\pi$
$937$	−5584.25	−0.194695	−0.0973476	+	0.995250i	$0.531036\pi$
$938$	0	0
$939$	6921.35	0.240543
$940$	0	0
$941$	51277.2	1.77640	0.888199	−	0.459460i	$-0.151957\pi$
$941$	51277.2	1.77640	0.888199	+	0.459460i	$0.151957\pi$
$942$	0	0
$943$	−12193.6	−0.421078
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−48182.1	−1.65333	−0.826667	−	0.562692i	$-0.809766\pi$
$947$	−48182.1	−1.65333	−0.826667	+	0.562692i	$0.809766\pi$
$948$	0	0
$949$	63331.0	2.16629
$950$	0	0
$951$	−15571.8	−0.530968
$952$	0	0
$953$	6599.75	0.224330	0.112165	−	0.993690i	$-0.464221\pi$
$953$	6599.75	0.224330	0.112165	+	0.993690i	$0.464221\pi$
$954$	0	0
$955$	59523.3	2.01689
$956$	0	0
$957$	9595.41	0.324112
$958$	0	0
$959$	0	0
$960$	0	0
$961$	30587.8	1.02675
$962$	0	0
$963$	−2339.24	−0.0782772
$964$	0	0
$965$	−37415.6	−1.24814
$966$	0	0
$967$	−36957.9	−1.22905	−0.614523	−	0.788899i	$-0.710651\pi$
$967$	−36957.9	−1.22905	−0.614523	+	0.788899i	$0.710651\pi$
$968$	0	0
$969$	−964.361	−0.0319708
$970$	0	0
$971$	−38023.4	−1.25667	−0.628336	−	0.777942i	$-0.716264\pi$
$971$	−38023.4	−1.25667	−0.628336	+	0.777942i	$0.716264\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	84159.1	2.76436
$976$	0	0
$977$	−34863.9	−1.14165	−0.570826	−	0.821071i	$-0.693377\pi$
$977$	−34863.9	−1.14165	−0.570826	+	0.821071i	$0.693377\pi$
$978$	0	0
$979$	−17665.2	−0.576691
$980$	0	0
$981$	−12926.9	−0.420719
$982$	0	0
$983$	3499.55	0.113549	0.0567743	−	0.998387i	$-0.481918\pi$
$983$	3499.55	0.113549	0.0567743	+	0.998387i	$0.481918\pi$
$984$	0	0
$985$	20604.0	0.666494
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−35685.7	−1.14736
$990$	0	0
$991$	10944.5	0.350821	0.175410	−	0.984495i	$-0.443875\pi$
$991$	10944.5	0.350821	0.175410	+	0.984495i	$0.443875\pi$
$992$	0	0
$993$	4953.24	0.158294
$994$	0	0
$995$	−112861.	−3.59592
$996$	0	0
$997$	−46149.3	−1.46596	−0.732980	−	0.680250i	$-0.761871\pi$
$997$	−46149.3	−1.46596	−0.732980	+	0.680250i	$0.761871\pi$
$998$	0	0
$999$	−5812.69	−0.184090

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2352.4.a.br.1.2		2
4.3	odd	2		1176.4.a.w.1.2		2
7.6	odd	2		336.4.a.o.1.1		2
21.20	even	2		1008.4.a.bg.1.2		2
28.27	even	2		168.4.a.g.1.1	✓	2
56.13	odd	2		1344.4.a.bi.1.2		2
56.27	even	2		1344.4.a.bq.1.2		2
84.83	odd	2		504.4.a.n.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
168.4.a.g.1.1	✓	2	28.27	even	2
336.4.a.o.1.1		2	7.6	odd	2
504.4.a.n.1.2		2	84.83	odd	2
1008.4.a.bg.1.2		2	21.20	even	2
1176.4.a.w.1.2		2	4.3	odd	2
1344.4.a.bi.1.2		2	56.13	odd	2
1344.4.a.bq.1.2		2	56.27	even	2
2352.4.a.br.1.2		2	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 2352 = 2^{4} \cdot 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2352.a (trivial)

\(f(q)\)	\(=\)	\(q-3.00000 q^{3} +21.3576 q^{5} +9.00000 q^{9} +O(q^{10})\)	\(q-3.00000 q^{3} +21.3576 q^{5} +9.00000 q^{9} -31.3576 q^{11} -84.7151 q^{13} -64.0727 q^{15} +16.0727 q^{17} +20.0000 q^{19} +80.7878 q^{23} +331.145 q^{25} -27.0000 q^{27} +102.000 q^{29} -245.721 q^{31} +94.0727 q^{33} +215.285 q^{37} +254.145 q^{39} -150.933 q^{41} -441.721 q^{43} +192.218 q^{45} -206.424 q^{47} -48.2180 q^{51} +426.436 q^{53} -669.721 q^{55} -60.0000 q^{57} +363.006 q^{59} +343.721 q^{61} -1809.31 q^{65} -69.2849 q^{67} -242.363 q^{69} +468.933 q^{71} -747.576 q^{73} -993.436 q^{75} -1298.74 q^{79} +81.0000 q^{81} -1293.16 q^{83} +343.273 q^{85} -306.000 q^{87} +563.346 q^{89} +737.163 q^{93} +427.151 q^{95} -1485.03 q^{97} -282.218 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 6 q^{3} + 6 q^{5} + 18 q^{9}+O(q^{10}) \) 2 * q - 6 * q^3 + 6 * q^5 + 18 * q^9	\( 2 q - 6 q^{3} + 6 q^{5} + 18 q^{9} - 26 q^{11} - 96 q^{13} - 18 q^{15} - 78 q^{17} + 40 q^{19} - 22 q^{23} + 442 q^{25} - 54 q^{27} + 204 q^{29} + 96 q^{31} + 78 q^{33} + 504 q^{37} + 288 q^{39} + 102 q^{41} - 296 q^{43} + 54 q^{45} - 780 q^{47} + 234 q^{51} + 192 q^{53} - 752 q^{55} - 120 q^{57} + 212 q^{59} + 100 q^{61} - 1636 q^{65} - 212 q^{67} + 66 q^{69} + 534 q^{71} - 1128 q^{73} - 1326 q^{75} - 468 q^{79} + 162 q^{81} - 824 q^{83} + 1788 q^{85} - 612 q^{87} + 2118 q^{89} - 288 q^{93} + 120 q^{95} - 400 q^{97} - 234 q^{99}+O(q^{100}) \) 2 * q - 6 * q^3 + 6 * q^5 + 18 * q^9 - 26 * q^11 - 96 * q^13 - 18 * q^15 - 78 * q^17 + 40 * q^19 - 22 * q^23 + 442 * q^25 - 54 * q^27 + 204 * q^29 + 96 * q^31 + 78 * q^33 + 504 * q^37 + 288 * q^39 + 102 * q^41 - 296 * q^43 + 54 * q^45 - 780 * q^47 + 234 * q^51 + 192 * q^53 - 752 * q^55 - 120 * q^57 + 212 * q^59 + 100 * q^61 - 1636 * q^65 - 212 * q^67 + 66 * q^69 + 534 * q^71 - 1128 * q^73 - 1326 * q^75 - 468 * q^79 + 162 * q^81 - 824 * q^83 + 1788 * q^85 - 612 * q^87 + 2118 * q^89 - 288 * q^93 + 120 * q^95 - 400 * q^97 - 234 * q^99

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