LMFDB - Embedded newform 2352.4.a.cl.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:	$4$
Coefficient field:	4.4.136768.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 23x^{2} + 18x + 119 $ x^4 - 2x^3 - 23x^2 + 18*x + 119
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{4}\cdot 7 $
Twist minimal:	no (minimal twist has level 588)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-2.51732$ 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)

$f(q)$	$=$	$q-3.00000 q^{3} -10.6550 q^{5} +9.00000 q^{9} +O(q^{10})$	$q-3.00000 q^{3} -10.6550 q^{5} +9.00000 q^{9} +6.65399 q^{11} +75.9335 q^{13} +31.9651 q^{15} -104.287 q^{17} -85.4943 q^{19} +68.6733 q^{23} -11.4700 q^{25} -27.0000 q^{27} +87.7843 q^{29} -62.7683 q^{31} -19.9620 q^{33} +42.2093 q^{37} -227.800 q^{39} +313.904 q^{41} -306.591 q^{43} -95.8954 q^{45} -215.081 q^{47} +312.861 q^{51} +525.024 q^{53} -70.8986 q^{55} +256.483 q^{57} -360.491 q^{59} +800.726 q^{61} -809.075 q^{65} +40.2286 q^{67} -206.020 q^{69} +298.781 q^{71} +517.126 q^{73} +34.4099 q^{75} +1222.47 q^{79} +81.0000 q^{81} -1328.55 q^{83} +1111.18 q^{85} -263.353 q^{87} -639.938 q^{89} +188.305 q^{93} +910.946 q^{95} +1425.65 q^{97} +59.8859 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 12 q^{3} + 36 q^{9}+O(q^{10}) $ 4 * q - 12 * q^3 + 36 * q^9	$ 4 q - 12 q^{3} + 36 q^{9} + 48 q^{17} - 192 q^{19} - 192 q^{23} + 324 q^{25} - 108 q^{27} + 96 q^{29} - 48 q^{31} + 256 q^{37} + 1008 q^{41} + 112 q^{43} - 864 q^{47} - 144 q^{51} - 648 q^{53} - 2352 q^{55} + 576 q^{57} - 336 q^{59} + 960 q^{61} - 360 q^{65} - 720 q^{67} + 576 q^{69} + 1344 q^{71} + 672 q^{73} - 972 q^{75} + 1984 q^{79} + 324 q^{81} - 3120 q^{83} + 680 q^{85} - 288 q^{87} + 2160 q^{89} + 144 q^{93} + 3744 q^{95} + 2016 q^{97}+O(q^{100}) $ 4 * q - 12 * q^3 + 36 * q^9 + 48 * q^17 - 192 * q^19 - 192 * q^23 + 324 * q^25 - 108 * q^27 + 96 * q^29 - 48 * q^31 + 256 * q^37 + 1008 * q^41 + 112 * q^43 - 864 * q^47 - 144 * q^51 - 648 * q^53 - 2352 * q^55 + 576 * q^57 - 336 * q^59 + 960 * q^61 - 360 * q^65 - 720 * q^67 + 576 * q^69 + 1344 * q^71 + 672 * q^73 - 972 * q^75 + 1984 * q^79 + 324 * q^81 - 3120 * q^83 + 680 * q^85 - 288 * q^87 + 2160 * q^89 + 144 * q^93 + 3744 * q^95 + 2016 * q^97

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
$5$	−10.6550	−0.953016	−0.476508	−	0.879170i	$-0.658098\pi$
$5$	−10.6550	−0.953016	−0.476508	+	0.879170i	$0.658098\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	6.65399	0.182387	0.0911933	−	0.995833i	$-0.470932\pi$
$11$	6.65399	0.182387	0.0911933	+	0.995833i	$0.470932\pi$
$12$	0	0
$13$	75.9335	1.62001	0.810006	−	0.586422i	$-0.199464\pi$
$13$	75.9335	1.62001	0.810006	+	0.586422i	$0.199464\pi$
$14$	0	0
$15$	31.9651	0.550224
$16$	0	0
$17$	−104.287	−1.48784	−0.743921	−	0.668268i	$-0.767036\pi$
$17$	−104.287	−1.48784	−0.743921	+	0.668268i	$0.767036\pi$
$18$	0	0
$19$	−85.4943	−1.03230	−0.516151	−	0.856498i	$-0.672636\pi$
$19$	−85.4943	−1.03230	−0.516151	+	0.856498i	$0.672636\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	68.6733	0.622581	0.311291	−	0.950315i	$-0.399239\pi$
$23$	68.6733	0.622581	0.311291	+	0.950315i	$0.399239\pi$
$24$	0	0
$25$	−11.4700	−0.0917596
$26$	0	0
$27$	−27.0000	−0.192450
$28$	0	0
$29$	87.7843	0.562108	0.281054	−	0.959692i	$-0.409316\pi$
$29$	87.7843	0.562108	0.281054	+	0.959692i	$0.409316\pi$
$30$	0	0
$31$	−62.7683	−0.363662	−0.181831	−	0.983330i	$-0.558202\pi$
$31$	−62.7683	−0.363662	−0.181831	+	0.983330i	$0.558202\pi$
$32$	0	0
$33$	−19.9620	−0.105301
$34$	0	0
$35$	0	0
$36$	0	0
$37$	42.2093	0.187545	0.0937726	−	0.995594i	$-0.470107\pi$
$37$	42.2093	0.187545	0.0937726	+	0.995594i	$0.470107\pi$
$38$	0	0
$39$	−227.800	−0.935314
$40$	0	0
$41$	313.904	1.19570	0.597848	−	0.801609i	$-0.296023\pi$
$41$	313.904	1.19570	0.597848	+	0.801609i	$0.296023\pi$
$42$	0	0
$43$	−306.591	−1.08732	−0.543659	−	0.839306i	$-0.682962\pi$
$43$	−306.591	−1.08732	−0.543659	+	0.839306i	$0.682962\pi$
$44$	0	0
$45$	−95.8954	−0.317672
$46$	0	0
$47$	−215.081	−0.667508	−0.333754	−	0.942660i	$-0.608315\pi$
$47$	−215.081	−0.667508	−0.333754	+	0.942660i	$0.608315\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	312.861	0.859006
$52$	0	0
$53$	525.024	1.36071	0.680354	−	0.732884i	$-0.261826\pi$
$53$	525.024	1.36071	0.680354	+	0.732884i	$0.261826\pi$
$54$	0	0
$55$	−70.8986	−0.173818
$56$	0	0
$57$	256.483	0.596000
$58$	0	0
$59$	−360.491	−0.795457	−0.397729	−	0.917503i	$-0.630201\pi$
$59$	−360.491	−0.795457	−0.397729	+	0.917503i	$0.630201\pi$
$60$	0	0
$61$	800.726	1.68070	0.840348	−	0.542048i	$-0.182351\pi$
$61$	800.726	1.68070	0.840348	+	0.542048i	$0.182351\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−809.075	−1.54390
$66$	0	0
$67$	40.2286	0.0733538	0.0366769	−	0.999327i	$-0.488323\pi$
$67$	40.2286	0.0733538	0.0366769	+	0.999327i	$0.488323\pi$
$68$	0	0
$69$	−206.020	−0.359447
$70$	0	0
$71$	298.781	0.499419	0.249709	−	0.968321i	$-0.419665\pi$
$71$	298.781	0.499419	0.249709	+	0.968321i	$0.419665\pi$
$72$	0	0
$73$	517.126	0.829110	0.414555	−	0.910024i	$-0.363937\pi$
$73$	517.126	0.829110	0.414555	+	0.910024i	$0.363937\pi$
$74$	0	0
$75$	34.4099	0.0529774
$76$	0	0
$77$	0	0
$78$	0	0
$79$	1222.47	1.74099	0.870494	−	0.492178i	$-0.163799\pi$
$79$	1222.47	1.74099	0.870494	+	0.492178i	$0.163799\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	−1328.55	−1.75696	−0.878479	−	0.477782i	$-0.841441\pi$
$83$	−1328.55	−1.75696	−0.878479	+	0.477782i	$0.841441\pi$
$84$	0	0
$85$	1111.18	1.41794
$86$	0	0
$87$	−263.353	−0.324533
$88$	0	0
$89$	−639.938	−0.762172	−0.381086	−	0.924540i	$-0.624450\pi$
$89$	−639.938	−0.762172	−0.381086	+	0.924540i	$0.624450\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	188.305	0.209960
$94$	0	0
$95$	910.946	0.983801
$96$	0	0
$97$	1425.65	1.49230	0.746149	−	0.665779i	$-0.231901\pi$
$97$	1425.65	1.49230	0.746149	+	0.665779i	$0.231901\pi$
$98$	0	0
$99$	59.8859	0.0607956
$100$	0	0
$101$	−992.170	−0.977471	−0.488736	−	0.872432i	$-0.662542\pi$
$101$	−992.170	−0.977471	−0.488736	+	0.872432i	$0.662542\pi$
$102$	0	0
$103$	−267.572	−0.255967	−0.127984	−	0.991776i	$-0.540851\pi$
$103$	−267.572	−0.255967	−0.127984	+	0.991776i	$0.540851\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	1536.62	1.38833	0.694164	−	0.719817i	$-0.255774\pi$
$107$	1536.62	1.38833	0.694164	+	0.719817i	$0.255774\pi$
$108$	0	0
$109$	998.820	0.877703	0.438852	−	0.898560i	$-0.355385\pi$
$109$	998.820	0.877703	0.438852	+	0.898560i	$0.355385\pi$
$110$	0	0
$111$	−126.628	−0.108279
$112$	0	0
$113$	939.006	0.781719	0.390860	−	0.920450i	$-0.372178\pi$
$113$	939.006	0.781719	0.390860	+	0.920450i	$0.372178\pi$
$114$	0	0
$115$	−731.717	−0.593330
$116$	0	0
$117$	683.401	0.540004
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−1286.72	−0.966735
$122$	0	0
$123$	−941.712	−0.690336
$124$	0	0
$125$	1454.09	1.04046
$126$	0	0
$127$	1621.27	1.13279	0.566397	−	0.824133i	$-0.308337\pi$
$127$	1621.27	1.13279	0.566397	+	0.824133i	$0.308337\pi$
$128$	0	0
$129$	919.773	0.627764
$130$	0	0
$131$	−1518.43	−1.01271	−0.506357	−	0.862324i	$-0.669008\pi$
$131$	−1518.43	−1.01271	−0.506357	+	0.862324i	$0.669008\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	287.686	0.183408
$136$	0	0
$137$	−2484.99	−1.54969	−0.774844	−	0.632152i	$-0.782172\pi$
$137$	−2484.99	−1.54969	−0.774844	+	0.632152i	$0.782172\pi$
$138$	0	0
$139$	1655.36	1.01011	0.505057	−	0.863086i	$-0.331471\pi$
$139$	1655.36	1.01011	0.505057	+	0.863086i	$0.331471\pi$
$140$	0	0
$141$	645.244	0.385386
$142$	0	0
$143$	505.260	0.295469
$144$	0	0
$145$	−935.346	−0.535698
$146$	0	0
$147$	0	0
$148$	0	0
$149$	350.191	0.192542	0.0962709	−	0.995355i	$-0.469308\pi$
$149$	350.191	0.192542	0.0962709	+	0.995355i	$0.469308\pi$
$150$	0	0
$151$	−3338.14	−1.79903	−0.899516	−	0.436888i	$-0.856081\pi$
$151$	−3338.14	−1.79903	−0.899516	+	0.436888i	$0.856081\pi$
$152$	0	0
$153$	−938.583	−0.495947
$154$	0	0
$155$	668.799	0.346576
$156$	0	0
$157$	−1743.67	−0.886371	−0.443185	−	0.896430i	$-0.646152\pi$
$157$	−1743.67	−0.886371	−0.443185	+	0.896430i	$0.646152\pi$
$158$	0	0
$159$	−1575.07	−0.785605
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−3431.56	−1.64896	−0.824480	−	0.565891i	$-0.808532\pi$
$163$	−3431.56	−1.64896	−0.824480	+	0.565891i	$0.808532\pi$
$164$	0	0
$165$	212.696	0.100354
$166$	0	0
$167$	−3381.05	−1.56667	−0.783335	−	0.621600i	$-0.786483\pi$
$167$	−3381.05	−1.56667	−0.783335	+	0.621600i	$0.786483\pi$
$168$	0	0
$169$	3568.89	1.62444
$170$	0	0
$171$	−769.449	−0.344101
$172$	0	0
$173$	−1341.00	−0.589332	−0.294666	−	0.955600i	$-0.595208\pi$
$173$	−1341.00	−0.589332	−0.294666	+	0.955600i	$0.595208\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	1081.47	0.459257
$178$	0	0
$179$	−1125.15	−0.469818	−0.234909	−	0.972017i	$-0.575479\pi$
$179$	−1125.15	−0.469818	−0.234909	+	0.972017i	$0.575479\pi$
$180$	0	0
$181$	−3535.04	−1.45170	−0.725848	−	0.687855i	$-0.758553\pi$
$181$	−3535.04	−1.45170	−0.725848	+	0.687855i	$0.758553\pi$
$182$	0	0
$183$	−2402.18	−0.970350
$184$	0	0
$185$	−449.742	−0.178734
$186$	0	0
$187$	−693.925	−0.271363
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−2639.89	−1.00008	−0.500041	−	0.866001i	$-0.666682\pi$
$191$	−2639.89	−1.00008	−0.500041	+	0.866001i	$0.666682\pi$
$192$	0	0
$193$	1047.05	0.390510	0.195255	−	0.980752i	$-0.437447\pi$
$193$	1047.05	0.390510	0.195255	+	0.980752i	$0.437447\pi$
$194$	0	0
$195$	2427.22	0.891370
$196$	0	0
$197$	−4585.45	−1.65837	−0.829187	−	0.558972i	$-0.811196\pi$
$197$	−4585.45	−1.65837	−0.829187	+	0.558972i	$0.811196\pi$
$198$	0	0
$199$	−830.742	−0.295928	−0.147964	−	0.988993i	$-0.547272\pi$
$199$	−830.742	−0.295928	−0.147964	+	0.988993i	$0.547272\pi$
$200$	0	0
$201$	−120.686	−0.0423508
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−3344.66	−1.13952
$206$	0	0
$207$	618.059	0.207527
$208$	0	0
$209$	−568.878	−0.188278
$210$	0	0
$211$	2630.08	0.858114	0.429057	−	0.903277i	$-0.358846\pi$
$211$	2630.08	0.858114	0.429057	+	0.903277i	$0.358846\pi$
$212$	0	0
$213$	−896.342	−0.288340
$214$	0	0
$215$	3266.74	1.03623
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−1551.38	−0.478687
$220$	0	0
$221$	−7918.87	−2.41032
$222$	0	0
$223$	863.988	0.259448	0.129724	−	0.991550i	$-0.458591\pi$
$223$	863.988	0.259448	0.129724	+	0.991550i	$0.458591\pi$
$224$	0	0
$225$	−103.230	−0.0305865
$226$	0	0
$227$	4160.36	1.21644	0.608221	−	0.793767i	$-0.291883\pi$
$227$	4160.36	1.21644	0.608221	+	0.793767i	$0.291883\pi$
$228$	0	0
$229$	−181.210	−0.0522914	−0.0261457	−	0.999658i	$-0.508323\pi$
$229$	−181.210	−0.0522914	−0.0261457	+	0.999658i	$0.508323\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	2150.45	0.604637	0.302319	−	0.953207i	$-0.402239\pi$
$233$	2150.45	0.604637	0.302319	+	0.953207i	$0.402239\pi$
$234$	0	0
$235$	2291.70	0.636146
$236$	0	0
$237$	−3667.40	−1.00516
$238$	0	0
$239$	6939.47	1.87815	0.939073	−	0.343719i	$-0.111687\pi$
$239$	6939.47	1.87815	0.939073	+	0.343719i	$0.111687\pi$
$240$	0	0
$241$	−206.170	−0.0551060	−0.0275530	−	0.999620i	$-0.508772\pi$
$241$	−206.170	−0.0551060	−0.0275530	+	0.999620i	$0.508772\pi$
$242$	0	0
$243$	−243.000	−0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−6491.88	−1.67234
$248$	0	0
$249$	3985.65	1.01438
$250$	0	0
$251$	−5011.93	−1.26036	−0.630180	−	0.776449i	$-0.717019\pi$
$251$	−5011.93	−1.26036	−0.630180	+	0.776449i	$0.717019\pi$
$252$	0	0
$253$	456.951	0.113551
$254$	0	0
$255$	−3333.55	−0.818647
$256$	0	0
$257$	−5863.08	−1.42307	−0.711535	−	0.702651i	$-0.752000\pi$
$257$	−5863.08	−1.42307	−0.711535	+	0.702651i	$0.752000\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	790.059	0.187369
$262$	0	0
$263$	5639.56	1.32224	0.661122	−	0.750279i	$-0.270081\pi$
$263$	5639.56	1.32224	0.661122	+	0.750279i	$0.270081\pi$
$264$	0	0
$265$	−5594.15	−1.29678
$266$	0	0
$267$	1919.81	0.440040
$268$	0	0
$269$	−2976.63	−0.674679	−0.337339	−	0.941383i	$-0.609527\pi$
$269$	−2976.63	−0.674679	−0.337339	+	0.941383i	$0.609527\pi$
$270$	0	0
$271$	2807.33	0.629275	0.314637	−	0.949212i	$-0.398117\pi$
$271$	2807.33	0.629275	0.314637	+	0.949212i	$0.398117\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−76.3210	−0.0167357
$276$	0	0
$277$	−1918.39	−0.416118	−0.208059	−	0.978116i	$-0.566715\pi$
$277$	−1918.39	−0.416118	−0.208059	+	0.978116i	$0.566715\pi$
$278$	0	0
$279$	−564.915	−0.121221
$280$	0	0
$281$	5209.50	1.10595	0.552977	−	0.833197i	$-0.313492\pi$
$281$	5209.50	1.10595	0.552977	+	0.833197i	$0.313492\pi$
$282$	0	0
$283$	−7496.70	−1.57467	−0.787337	−	0.616523i	$-0.788541\pi$
$283$	−7496.70	−1.57467	−0.787337	+	0.616523i	$0.788541\pi$
$284$	0	0
$285$	−2732.84	−0.567998
$286$	0	0
$287$	0	0
$288$	0	0
$289$	5962.78	1.21367
$290$	0	0
$291$	−4276.95	−0.861579
$292$	0	0
$293$	−3358.94	−0.669732	−0.334866	−	0.942266i	$-0.608691\pi$
$293$	−3358.94	−0.669732	−0.334866	+	0.942266i	$0.608691\pi$
$294$	0	0
$295$	3841.05	0.758084
$296$	0	0
$297$	−179.658	−0.0351003
$298$	0	0
$299$	5214.60	1.00859
$300$	0	0
$301$	0	0
$302$	0	0
$303$	2976.51	0.564343
$304$	0	0
$305$	−8531.77	−1.60173
$306$	0	0
$307$	7330.92	1.36286	0.681429	−	0.731884i	$-0.261359\pi$
$307$	7330.92	1.36286	0.681429	+	0.731884i	$0.261359\pi$
$308$	0	0
$309$	802.716	0.147783
$310$	0	0
$311$	2038.53	0.371687	0.185843	−	0.982579i	$-0.440498\pi$
$311$	2038.53	0.371687	0.185843	+	0.982579i	$0.440498\pi$
$312$	0	0
$313$	4138.87	0.747421	0.373711	−	0.927545i	$-0.378085\pi$
$313$	4138.87	0.747421	0.373711	+	0.927545i	$0.378085\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−7552.25	−1.33810	−0.669049	−	0.743219i	$-0.733298\pi$
$317$	−7552.25	−1.33810	−0.669049	+	0.743219i	$0.733298\pi$
$318$	0	0
$319$	584.116	0.102521
$320$	0	0
$321$	−4609.87	−0.801552
$322$	0	0
$323$	8915.94	1.53590
$324$	0	0
$325$	−870.953	−0.148652
$326$	0	0
$327$	−2996.46	−0.506742
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−5577.70	−0.926218	−0.463109	−	0.886301i	$-0.653266\pi$
$331$	−5577.70	−0.926218	−0.463109	+	0.886301i	$0.653266\pi$
$332$	0	0
$333$	379.884	0.0625151
$334$	0	0
$335$	−428.637	−0.0699073
$336$	0	0
$337$	4467.16	0.722083	0.361041	−	0.932550i	$-0.382421\pi$
$337$	4467.16	0.722083	0.361041	+	0.932550i	$0.382421\pi$
$338$	0	0
$339$	−2817.02	−0.451326
$340$	0	0
$341$	−417.660	−0.0663271
$342$	0	0
$343$	0	0
$344$	0	0
$345$	2195.15	0.342559
$346$	0	0
$347$	−9788.41	−1.51432	−0.757161	−	0.653229i	$-0.773414\pi$
$347$	−9788.41	−1.51432	−0.757161	+	0.653229i	$0.773414\pi$
$348$	0	0
$349$	−4746.95	−0.728075	−0.364038	−	0.931384i	$-0.618602\pi$
$349$	−4746.95	−0.728075	−0.364038	+	0.931384i	$0.618602\pi$
$350$	0	0
$351$	−2050.20	−0.311771
$352$	0	0
$353$	−9434.66	−1.42254	−0.711269	−	0.702920i	$-0.751879\pi$
$353$	−9434.66	−1.42254	−0.711269	+	0.702920i	$0.751879\pi$
$354$	0	0
$355$	−3183.52	−0.475954
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−11307.5	−1.66237	−0.831183	−	0.555999i	$-0.812336\pi$
$359$	−11307.5	−1.66237	−0.831183	+	0.555999i	$0.812336\pi$
$360$	0	0
$361$	450.275	0.0656474
$362$	0	0
$363$	3860.17	0.558145
$364$	0	0
$365$	−5510.00	−0.790155
$366$	0	0
$367$	−6089.61	−0.866144	−0.433072	−	0.901359i	$-0.642570\pi$
$367$	−6089.61	−0.866144	−0.433072	+	0.901359i	$0.642570\pi$
$368$	0	0
$369$	2825.14	0.398565
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−2601.84	−0.361175	−0.180588	−	0.983559i	$-0.557800\pi$
$373$	−2601.84	−0.361175	−0.180588	+	0.983559i	$0.557800\pi$
$374$	0	0
$375$	−4362.28	−0.600713
$376$	0	0
$377$	6665.77	0.910622
$378$	0	0
$379$	−10416.2	−1.41173	−0.705865	−	0.708347i	$-0.749441\pi$
$379$	−10416.2	−1.41173	−0.705865	+	0.708347i	$0.749441\pi$
$380$	0	0
$381$	−4863.82	−0.654018
$382$	0	0
$383$	1664.46	0.222063	0.111031	−	0.993817i	$-0.464585\pi$
$383$	1664.46	0.222063	0.111031	+	0.993817i	$0.464585\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−2759.32	−0.362440
$388$	0	0
$389$	−1334.16	−0.173893	−0.0869465	−	0.996213i	$-0.527711\pi$
$389$	−1334.16	−0.173893	−0.0869465	+	0.996213i	$0.527711\pi$
$390$	0	0
$391$	−7161.73	−0.926302
$392$	0	0
$393$	4555.28	0.584691
$394$	0	0
$395$	−13025.4	−1.65919
$396$	0	0
$397$	4444.88	0.561920	0.280960	−	0.959720i	$-0.409347\pi$
$397$	4444.88	0.561920	0.280960	+	0.959720i	$0.409347\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−2649.82	−0.329989	−0.164995	−	0.986294i	$-0.552761\pi$
$401$	−2649.82	−0.329989	−0.164995	+	0.986294i	$0.552761\pi$
$402$	0	0
$403$	−4766.21	−0.589137
$404$	0	0
$405$	−863.059	−0.105891
$406$	0	0
$407$	280.860	0.0342057
$408$	0	0
$409$	−9592.66	−1.15972	−0.579862	−	0.814715i	$-0.696893\pi$
$409$	−9592.66	−1.15972	−0.579862	+	0.814715i	$0.696893\pi$
$410$	0	0
$411$	7454.98	0.894713
$412$	0	0
$413$	0	0
$414$	0	0
$415$	14155.8	1.67441
$416$	0	0
$417$	−4966.09	−0.583190
$418$	0	0
$419$	−16850.1	−1.96463	−0.982317	−	0.187224i	$-0.940051\pi$
$419$	−16850.1	−1.96463	−0.982317	+	0.187224i	$0.940051\pi$
$420$	0	0
$421$	1691.10	0.195770	0.0978849	−	0.995198i	$-0.468792\pi$
$421$	1691.10	0.195770	0.0978849	+	0.995198i	$0.468792\pi$
$422$	0	0
$423$	−1935.73	−0.222503
$424$	0	0
$425$	1196.17	0.136524
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−1515.78	−0.170589
$430$	0	0
$431$	6789.34	0.758773	0.379386	−	0.925238i	$-0.376135\pi$
$431$	6789.34	0.758773	0.379386	+	0.925238i	$0.376135\pi$
$432$	0	0
$433$	10386.6	1.15276	0.576382	−	0.817180i	$-0.304464\pi$
$433$	10386.6	1.15276	0.576382	+	0.817180i	$0.304464\pi$
$434$	0	0
$435$	2806.04	0.309286
$436$	0	0
$437$	−5871.17	−0.642692
$438$	0	0
$439$	4212.26	0.457950	0.228975	−	0.973432i	$-0.426463\pi$
$439$	4212.26	0.457950	0.228975	+	0.973432i	$0.426463\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	1809.96	0.194117	0.0970585	−	0.995279i	$-0.469057\pi$
$443$	1809.96	0.194117	0.0970585	+	0.995279i	$0.469057\pi$
$444$	0	0
$445$	6818.57	0.726362
$446$	0	0
$447$	−1050.57	−0.111164
$448$	0	0
$449$	1768.29	0.185859	0.0929297	−	0.995673i	$-0.470377\pi$
$449$	1768.29	0.185859	0.0929297	+	0.995673i	$0.470377\pi$
$450$	0	0
$451$	2088.71	0.218079
$452$	0	0
$453$	10014.4	1.03867
$454$	0	0
$455$	0	0
$456$	0	0
$457$	4372.61	0.447575	0.223788	−	0.974638i	$-0.428158\pi$
$457$	4372.61	0.447575	0.223788	+	0.974638i	$0.428158\pi$
$458$	0	0
$459$	2815.75	0.286335
$460$	0	0
$461$	−1164.34	−0.117633	−0.0588165	−	0.998269i	$-0.518733\pi$
$461$	−1164.34	−0.117633	−0.0588165	+	0.998269i	$0.518733\pi$
$462$	0	0
$463$	14893.9	1.49498	0.747491	−	0.664272i	$-0.231258\pi$
$463$	14893.9	1.49498	0.747491	+	0.664272i	$0.231258\pi$
$464$	0	0
$465$	−2006.40	−0.200096
$466$	0	0
$467$	−19160.3	−1.89857	−0.949285	−	0.314418i	$-0.898191\pi$
$467$	−19160.3	−1.89857	−0.949285	+	0.314418i	$0.898191\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	5231.02	0.511746
$472$	0	0
$473$	−2040.05	−0.198312
$474$	0	0
$475$	980.616	0.0947237
$476$	0	0
$477$	4725.21	0.453569
$478$	0	0
$479$	9406.28	0.897253	0.448626	−	0.893719i	$-0.351913\pi$
$479$	9406.28	0.897253	0.448626	+	0.893719i	$0.351913\pi$
$480$	0	0
$481$	3205.10	0.303825
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−15190.4	−1.42218
$486$	0	0
$487$	−12460.8	−1.15945	−0.579725	−	0.814812i	$-0.696840\pi$
$487$	−12460.8	−1.15945	−0.579725	+	0.814812i	$0.696840\pi$
$488$	0	0
$489$	10294.7	0.952028
$490$	0	0
$491$	−12868.0	−1.18274	−0.591369	−	0.806401i	$-0.701412\pi$
$491$	−12868.0	−1.18274	−0.591369	+	0.806401i	$0.701412\pi$
$492$	0	0
$493$	−9154.76	−0.836328
$494$	0	0
$495$	−638.087	−0.0579392
$496$	0	0
$497$	0	0
$498$	0	0
$499$	13705.0	1.22950	0.614750	−	0.788722i	$-0.289257\pi$
$499$	13705.0	1.22950	0.614750	+	0.788722i	$0.289257\pi$
$500$	0	0
$501$	10143.2	0.904517
$502$	0	0
$503$	−10126.1	−0.897616	−0.448808	−	0.893628i	$-0.648151\pi$
$503$	−10126.1	−0.897616	−0.448808	+	0.893628i	$0.648151\pi$
$504$	0	0
$505$	10571.6	0.931546
$506$	0	0
$507$	−10706.7	−0.937870
$508$	0	0
$509$	−6236.06	−0.543042	−0.271521	−	0.962432i	$-0.587527\pi$
$509$	−6236.06	−0.543042	−0.271521	+	0.962432i	$0.587527\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	2308.35	0.198667
$514$	0	0
$515$	2850.99	0.243941
$516$	0	0
$517$	−1431.15	−0.121744
$518$	0	0
$519$	4023.00	0.340251
$520$	0	0
$521$	17016.8	1.43094	0.715468	−	0.698645i	$-0.246213\pi$
$521$	17016.8	1.43094	0.715468	+	0.698645i	$0.246213\pi$
$522$	0	0
$523$	5814.62	0.486148	0.243074	−	0.970008i	$-0.421844\pi$
$523$	5814.62	0.486148	0.243074	+	0.970008i	$0.421844\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	6545.92	0.541071
$528$	0	0
$529$	−7450.98	−0.612393
$530$	0	0
$531$	−3244.42	−0.265152
$532$	0	0
$533$	23835.8	1.93704
$534$	0	0
$535$	−16372.8	−1.32310
$536$	0	0
$537$	3375.44	0.271250
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−10069.2	−0.800199	−0.400100	−	0.916472i	$-0.631025\pi$
$541$	−10069.2	−0.800199	−0.400100	+	0.916472i	$0.631025\pi$
$542$	0	0
$543$	10605.1	0.838137
$544$	0	0
$545$	−10642.5	−0.836466
$546$	0	0
$547$	7437.51	0.581362	0.290681	−	0.956820i	$-0.406118\pi$
$547$	7437.51	0.581362	0.290681	+	0.956820i	$0.406118\pi$
$548$	0	0
$549$	7206.53	0.560232
$550$	0	0
$551$	−7505.06	−0.580265
$552$	0	0
$553$	0	0
$554$	0	0
$555$	1349.23	0.103192
$556$	0	0
$557$	21787.1	1.65736	0.828680	−	0.559723i	$-0.189092\pi$
$557$	21787.1	1.65736	0.828680	+	0.559723i	$0.189092\pi$
$558$	0	0
$559$	−23280.5	−1.76147
$560$	0	0
$561$	2081.77	0.156671
$562$	0	0
$563$	−2572.48	−0.192570	−0.0962851	−	0.995354i	$-0.530696\pi$
$563$	−2572.48	−0.192570	−0.0962851	+	0.995354i	$0.530696\pi$
$564$	0	0
$565$	−10005.2	−0.744991
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−17580.6	−1.29528	−0.647642	−	0.761945i	$-0.724245\pi$
$569$	−17580.6	−1.29528	−0.647642	+	0.761945i	$0.724245\pi$
$570$	0	0
$571$	−7220.75	−0.529210	−0.264605	−	0.964357i	$-0.585242\pi$
$571$	−7220.75	−0.529210	−0.264605	+	0.964357i	$0.585242\pi$
$572$	0	0
$573$	7919.67	0.577398
$574$	0	0
$575$	−787.679	−0.0571278
$576$	0	0
$577$	11155.1	0.804839	0.402419	−	0.915455i	$-0.368169\pi$
$577$	11155.1	0.804839	0.402419	+	0.915455i	$0.368169\pi$
$578$	0	0
$579$	−3141.16	−0.225461
$580$	0	0
$581$	0	0
$582$	0	0
$583$	3493.50	0.248175
$584$	0	0
$585$	−7281.67	−0.514633
$586$	0	0
$587$	15216.2	1.06992	0.534958	−	0.844879i	$-0.320327\pi$
$587$	15216.2	1.06992	0.534958	+	0.844879i	$0.320327\pi$
$588$	0	0
$589$	5366.33	0.375409
$590$	0	0
$591$	13756.3	0.957462
$592$	0	0
$593$	3843.89	0.266188	0.133094	−	0.991103i	$-0.457509\pi$
$593$	3843.89	0.266188	0.133094	+	0.991103i	$0.457509\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	2492.23	0.170854
$598$	0	0
$599$	−18747.0	−1.27876	−0.639382	−	0.768889i	$-0.720810\pi$
$599$	−18747.0	−1.27876	−0.639382	+	0.768889i	$0.720810\pi$
$600$	0	0
$601$	−9864.63	−0.669529	−0.334764	−	0.942302i	$-0.608657\pi$
$601$	−9864.63	−0.669529	−0.334764	+	0.942302i	$0.608657\pi$
$602$	0	0
$603$	362.057	0.0244513
$604$	0	0
$605$	13710.1	0.921314
$606$	0	0
$607$	−22292.8	−1.49067	−0.745336	−	0.666689i	$-0.767711\pi$
$607$	−22292.8	−1.49067	−0.745336	+	0.666689i	$0.767711\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−16331.9	−1.08137
$612$	0	0
$613$	6046.78	0.398413	0.199206	−	0.979958i	$-0.436164\pi$
$613$	6046.78	0.398413	0.199206	+	0.979958i	$0.436164\pi$
$614$	0	0
$615$	10034.0	0.657901
$616$	0	0
$617$	−9383.68	−0.612273	−0.306137	−	0.951988i	$-0.599036\pi$
$617$	−9383.68	−0.612273	−0.306137	+	0.951988i	$0.599036\pi$
$618$	0	0
$619$	3989.50	0.259049	0.129525	−	0.991576i	$-0.458655\pi$
$619$	3989.50	0.259049	0.129525	+	0.991576i	$0.458655\pi$
$620$	0	0
$621$	−1854.18	−0.119816
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−14059.7	−0.899821
$626$	0	0
$627$	1706.63	0.108702
$628$	0	0
$629$	−4401.88	−0.279038
$630$	0	0
$631$	−11434.0	−0.721363	−0.360681	−	0.932689i	$-0.617456\pi$
$631$	−11434.0	−0.721363	−0.360681	+	0.932689i	$0.617456\pi$
$632$	0	0
$633$	−7890.24	−0.495432
$634$	0	0
$635$	−17274.7	−1.07957
$636$	0	0
$637$	0	0
$638$	0	0
$639$	2689.03	0.166473
$640$	0	0
$641$	−15097.3	−0.930274	−0.465137	−	0.885239i	$-0.653995\pi$
$641$	−15097.3	−0.930274	−0.465137	+	0.885239i	$0.653995\pi$
$642$	0	0
$643$	−4170.19	−0.255764	−0.127882	−	0.991789i	$-0.540818\pi$
$643$	−4170.19	−0.255764	−0.127882	+	0.991789i	$0.540818\pi$
$644$	0	0
$645$	−9800.23	−0.598269
$646$	0	0
$647$	4563.88	0.277318	0.138659	−	0.990340i	$-0.455721\pi$
$647$	4563.88	0.277318	0.138659	+	0.990340i	$0.455721\pi$
$648$	0	0
$649$	−2398.71	−0.145081
$650$	0	0
$651$	0	0
$652$	0	0
$653$	3819.04	0.228868	0.114434	−	0.993431i	$-0.463495\pi$
$653$	3819.04	0.228868	0.114434	+	0.993431i	$0.463495\pi$
$654$	0	0
$655$	16178.9	0.965134
$656$	0	0
$657$	4654.13	0.276370
$658$	0	0
$659$	4326.02	0.255717	0.127859	−	0.991792i	$-0.459190\pi$
$659$	4326.02	0.255717	0.127859	+	0.991792i	$0.459190\pi$
$660$	0	0
$661$	−29317.8	−1.72516	−0.862579	−	0.505923i	$-0.831152\pi$
$661$	−29317.8	−1.72516	−0.862579	+	0.505923i	$0.831152\pi$
$662$	0	0
$663$	23756.6	1.39160
$664$	0	0
$665$	0	0
$666$	0	0
$667$	6028.43	0.349958
$668$	0	0
$669$	−2591.96	−0.149792
$670$	0	0
$671$	5328.02	0.306536
$672$	0	0
$673$	−5483.56	−0.314080	−0.157040	−	0.987592i	$-0.550195\pi$
$673$	−5483.56	−0.314080	−0.157040	+	0.987592i	$0.550195\pi$
$674$	0	0
$675$	309.689	0.0176591
$676$	0	0
$677$	−12069.3	−0.685174	−0.342587	−	0.939486i	$-0.611303\pi$
$677$	−12069.3	−0.685174	−0.342587	+	0.939486i	$0.611303\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−12481.1	−0.702314
$682$	0	0
$683$	−30796.7	−1.72533	−0.862667	−	0.505772i	$-0.831208\pi$
$683$	−30796.7	−1.72533	−0.862667	+	0.505772i	$0.831208\pi$
$684$	0	0
$685$	26477.7	1.47688
$686$	0	0
$687$	543.631	0.0301904
$688$	0	0
$689$	39866.9	2.20436
$690$	0	0
$691$	6501.96	0.357954	0.178977	−	0.983853i	$-0.442721\pi$
$691$	6501.96	0.357954	0.178977	+	0.983853i	$0.442721\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−17638.0	−0.962656
$696$	0	0
$697$	−32736.1	−1.77901
$698$	0	0
$699$	−6451.34	−0.349088
$700$	0	0
$701$	−2235.98	−0.120473	−0.0602367	−	0.998184i	$-0.519186\pi$
$701$	−2235.98	−0.120473	−0.0602367	+	0.998184i	$0.519186\pi$
$702$	0	0
$703$	−3608.66	−0.193603
$704$	0	0
$705$	−6875.11	−0.367279
$706$	0	0
$707$	0	0
$708$	0	0
$709$	35564.5	1.88385	0.941926	−	0.335820i	$-0.109014\pi$
$709$	35564.5	1.88385	0.941926	+	0.335820i	$0.109014\pi$
$710$	0	0
$711$	11002.2	0.580330
$712$	0	0
$713$	−4310.50	−0.226409
$714$	0	0
$715$	−5383.57	−0.281586
$716$	0	0
$717$	−20818.4	−1.08435
$718$	0	0
$719$	9924.15	0.514754	0.257377	−	0.966311i	$-0.417142\pi$
$719$	9924.15	0.514754	0.257377	+	0.966311i	$0.417142\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	618.509	0.0318155
$724$	0	0
$725$	−1006.88	−0.0515788
$726$	0	0
$727$	−880.081	−0.0448974	−0.0224487	−	0.999748i	$-0.507146\pi$
$727$	−880.081	−0.0448974	−0.0224487	+	0.999748i	$0.507146\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	31973.5	1.61776
$732$	0	0
$733$	−17336.2	−0.873568	−0.436784	−	0.899566i	$-0.643883\pi$
$733$	−17336.2	−0.873568	−0.436784	+	0.899566i	$0.643883\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	267.681	0.0133787
$738$	0	0
$739$	24586.4	1.22385	0.611924	−	0.790917i	$-0.290396\pi$
$739$	24586.4	1.22385	0.611924	+	0.790917i	$0.290396\pi$
$740$	0	0
$741$	19475.6	0.965527
$742$	0	0
$743$	−14579.3	−0.719870	−0.359935	−	0.932977i	$-0.617201\pi$
$743$	−14579.3	−0.719870	−0.359935	+	0.932977i	$0.617201\pi$
$744$	0	0
$745$	−3731.30	−0.183496
$746$	0	0
$747$	−11957.0	−0.585652
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−16086.0	−0.781604	−0.390802	−	0.920475i	$-0.627802\pi$
$751$	−16086.0	−0.781604	−0.390802	+	0.920475i	$0.627802\pi$
$752$	0	0
$753$	15035.8	0.727669
$754$	0	0
$755$	35568.0	1.71451
$756$	0	0
$757$	−37017.3	−1.77730	−0.888651	−	0.458584i	$-0.848357\pi$
$757$	−37017.3	−1.77730	−0.888651	+	0.458584i	$0.848357\pi$
$758$	0	0
$759$	−1370.85	−0.0655584
$760$	0	0
$761$	24606.5	1.17212	0.586060	−	0.810267i	$-0.300678\pi$
$761$	24606.5	1.17212	0.586060	+	0.810267i	$0.300678\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	10000.6	0.472646
$766$	0	0
$767$	−27373.4	−1.28865
$768$	0	0
$769$	−12715.6	−0.596275	−0.298137	−	0.954523i	$-0.596365\pi$
$769$	−12715.6	−0.596275	−0.298137	+	0.954523i	$0.596365\pi$
$770$	0	0
$771$	17589.2	0.821610
$772$	0	0
$773$	3174.23	0.147696	0.0738480	−	0.997270i	$-0.476472\pi$
$773$	3174.23	0.147696	0.0738480	+	0.997270i	$0.476472\pi$
$774$	0	0
$775$	719.949	0.0333695
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−26837.0	−1.23432
$780$	0	0
$781$	1988.08	0.0910874
$782$	0	0
$783$	−2370.18	−0.108178
$784$	0	0
$785$	18578.9	0.844726
$786$	0	0
$787$	15569.9	0.705218	0.352609	−	0.935771i	$-0.385295\pi$
$787$	15569.9	0.705218	0.352609	+	0.935771i	$0.385295\pi$
$788$	0	0
$789$	−16918.7	−0.763398
$790$	0	0
$791$	0	0
$792$	0	0
$793$	60801.9	2.72275
$794$	0	0
$795$	16782.5	0.748695
$796$	0	0
$797$	−31514.5	−1.40063	−0.700314	−	0.713835i	$-0.746957\pi$
$797$	−31514.5	−1.40063	−0.700314	+	0.713835i	$0.746957\pi$
$798$	0	0
$799$	22430.2	0.993146
$800$	0	0
$801$	−5759.44	−0.254057
$802$	0	0
$803$	3440.95	0.151219
$804$	0	0
$805$	0	0
$806$	0	0
$807$	8929.90	0.389526
$808$	0	0
$809$	11029.1	0.479310	0.239655	−	0.970858i	$-0.422966\pi$
$809$	11029.1	0.479310	0.239655	+	0.970858i	$0.422966\pi$
$810$	0	0
$811$	20830.5	0.901920	0.450960	−	0.892544i	$-0.351082\pi$
$811$	20830.5	0.901920	0.450960	+	0.892544i	$0.351082\pi$
$812$	0	0
$813$	−8422.00	−0.363312
$814$	0	0
$815$	36563.4	1.57149
$816$	0	0
$817$	26211.8	1.12244
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−9349.06	−0.397423	−0.198712	−	0.980058i	$-0.563676\pi$
$821$	−9349.06	−0.397423	−0.198712	+	0.980058i	$0.563676\pi$
$822$	0	0
$823$	32890.6	1.39307	0.696533	−	0.717525i	$-0.254725\pi$
$823$	32890.6	1.39307	0.696533	+	0.717525i	$0.254725\pi$
$824$	0	0
$825$	228.963	0.00966238
$826$	0	0
$827$	13081.2	0.550033	0.275016	−	0.961440i	$-0.411317\pi$
$827$	13081.2	0.550033	0.275016	+	0.961440i	$0.411317\pi$
$828$	0	0
$829$	28791.2	1.20622	0.603112	−	0.797657i	$-0.293927\pi$
$829$	28791.2	1.20622	0.603112	+	0.797657i	$0.293927\pi$
$830$	0	0
$831$	5755.17	0.240246
$832$	0	0
$833$	0	0
$834$	0	0
$835$	36025.3	1.49306
$836$	0	0
$837$	1694.74	0.0699868
$838$	0	0
$839$	36766.8	1.51291	0.756455	−	0.654046i	$-0.226930\pi$
$839$	36766.8	1.51291	0.756455	+	0.654046i	$0.226930\pi$
$840$	0	0
$841$	−16682.9	−0.684034
$842$	0	0
$843$	−15628.5	−0.638523
$844$	0	0
$845$	−38026.7	−1.54812
$846$	0	0
$847$	0	0
$848$	0	0
$849$	22490.1	0.909138
$850$	0	0
$851$	2898.65	0.116762
$852$	0	0
$853$	5899.55	0.236808	0.118404	−	0.992966i	$-0.462222\pi$
$853$	5899.55	0.236808	0.118404	+	0.992966i	$0.462222\pi$
$854$	0	0
$855$	8198.51	0.327934
$856$	0	0
$857$	42277.0	1.68513	0.842564	−	0.538596i	$-0.181045\pi$
$857$	42277.0	1.68513	0.842564	+	0.538596i	$0.181045\pi$
$858$	0	0
$859$	6343.46	0.251963	0.125981	−	0.992033i	$-0.459792\pi$
$859$	6343.46	0.251963	0.125981	+	0.992033i	$0.459792\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−6929.78	−0.273340	−0.136670	−	0.990617i	$-0.543640\pi$
$863$	−6929.78	−0.273340	−0.136670	+	0.990617i	$0.543640\pi$
$864$	0	0
$865$	14288.4	0.561643
$866$	0	0
$867$	−17888.3	−0.700715
$868$	0	0
$869$	8134.27	0.317533
$870$	0	0
$871$	3054.69	0.118834
$872$	0	0
$873$	12830.9	0.497433
$874$	0	0
$875$	0	0
$876$	0	0
$877$	3287.88	0.126595	0.0632976	−	0.997995i	$-0.479838\pi$
$877$	3287.88	0.126595	0.0632976	+	0.997995i	$0.479838\pi$
$878$	0	0
$879$	10076.8	0.386670
$880$	0	0
$881$	−46875.9	−1.79261	−0.896304	−	0.443439i	$-0.853758\pi$
$881$	−46875.9	−1.79261	−0.896304	+	0.443439i	$0.853758\pi$
$882$	0	0
$883$	−42479.4	−1.61897	−0.809483	−	0.587144i	$-0.800252\pi$
$883$	−42479.4	−1.61897	−0.809483	+	0.587144i	$0.800252\pi$
$884$	0	0
$885$	−11523.2	−0.437680
$886$	0	0
$887$	−3680.15	−0.139309	−0.0696547	−	0.997571i	$-0.522190\pi$
$887$	−3680.15	−0.139309	−0.0696547	+	0.997571i	$0.522190\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	538.973	0.0202652
$892$	0	0
$893$	18388.2	0.689069
$894$	0	0
$895$	11988.5	0.447745
$896$	0	0
$897$	−15643.8	−0.582309
$898$	0	0
$899$	−5510.07	−0.204417
$900$	0	0
$901$	−54753.1	−2.02452
$902$	0	0
$903$	0	0
$904$	0	0
$905$	37666.0	1.38349
$906$	0	0
$907$	7102.18	0.260005	0.130002	−	0.991514i	$-0.458502\pi$
$907$	7102.18	0.260005	0.130002	+	0.991514i	$0.458502\pi$
$908$	0	0
$909$	−8929.53	−0.325824
$910$	0	0
$911$	−38119.0	−1.38632	−0.693161	−	0.720783i	$-0.743782\pi$
$911$	−38119.0	−1.38632	−0.693161	+	0.720783i	$0.743782\pi$
$912$	0	0
$913$	−8840.17	−0.320446
$914$	0	0
$915$	25595.3	0.924760
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−17190.6	−0.617045	−0.308522	−	0.951217i	$-0.599834\pi$
$919$	−17190.6	−0.617045	−0.308522	+	0.951217i	$0.599834\pi$
$920$	0	0
$921$	−21992.8	−0.786847
$922$	0	0
$923$	22687.5	0.809065
$924$	0	0
$925$	−484.139	−0.0172091
$926$	0	0
$927$	−2408.15	−0.0853225
$928$	0	0
$929$	32256.5	1.13918	0.569591	−	0.821928i	$-0.307101\pi$
$929$	32256.5	1.13918	0.569591	+	0.821928i	$0.307101\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−6115.59	−0.214593
$934$	0	0
$935$	7393.80	0.258613
$936$	0	0
$937$	26213.6	0.913940	0.456970	−	0.889482i	$-0.348935\pi$
$937$	26213.6	0.913940	0.456970	+	0.889482i	$0.348935\pi$
$938$	0	0
$939$	−12416.6	−0.431524
$940$	0	0
$941$	34240.1	1.18618	0.593089	−	0.805137i	$-0.297908\pi$
$941$	34240.1	1.18618	0.593089	+	0.805137i	$0.297908\pi$
$942$	0	0
$943$	21556.8	0.744418
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−36004.5	−1.23547	−0.617735	−	0.786386i	$-0.711950\pi$
$947$	−36004.5	−1.23547	−0.617735	+	0.786386i	$0.711950\pi$
$948$	0	0
$949$	39267.2	1.34317
$950$	0	0
$951$	22656.8	0.772551
$952$	0	0
$953$	−29488.1	−1.00232	−0.501161	−	0.865354i	$-0.667093\pi$
$953$	−29488.1	−1.00232	−0.501161	+	0.865354i	$0.667093\pi$
$954$	0	0
$955$	28128.2	0.953095
$956$	0	0
$957$	−1752.35	−0.0591905
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−25851.1	−0.867750
$962$	0	0
$963$	13829.6	0.462776
$964$	0	0
$965$	−11156.4	−0.372163
$966$	0	0
$967$	56009.0	1.86260	0.931298	−	0.364259i	$-0.118678\pi$
$967$	56009.0	1.86260	0.931298	+	0.364259i	$0.118678\pi$
$968$	0	0
$969$	−26747.8	−0.886753
$970$	0	0
$971$	−13576.9	−0.448715	−0.224358	−	0.974507i	$-0.572028\pi$
$971$	−13576.9	−0.448715	−0.224358	+	0.974507i	$0.572028\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	2612.86	0.0858241
$976$	0	0
$977$	−31775.1	−1.04051	−0.520254	−	0.854011i	$-0.674163\pi$
$977$	−31775.1	−1.04051	−0.520254	+	0.854011i	$0.674163\pi$
$978$	0	0
$979$	−4258.14	−0.139010
$980$	0	0
$981$	8989.38	0.292568
$982$	0	0
$983$	−32755.4	−1.06280	−0.531401	−	0.847120i	$-0.678334\pi$
$983$	−32755.4	−1.06280	−0.531401	+	0.847120i	$0.678334\pi$
$984$	0	0
$985$	48858.2	1.58046
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−21054.6	−0.676944
$990$	0	0
$991$	40797.0	1.30773	0.653865	−	0.756611i	$-0.273146\pi$
$991$	40797.0	1.30773	0.653865	+	0.756611i	$0.273146\pi$
$992$	0	0
$993$	16733.1	0.534752
$994$	0	0
$995$	8851.60	0.282025
$996$	0	0
$997$	−17370.5	−0.551784	−0.275892	−	0.961189i	$-0.588973\pi$
$997$	−17370.5	−0.551784	−0.275892	+	0.961189i	$0.588973\pi$
$998$	0	0
$999$	−1139.65	−0.0360931

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2352.4.a.cl.1.2		4
4.3	odd	2		588.4.a.k.1.2	yes	4
7.6	odd	2		2352.4.a.cq.1.3		4
12.11	even	2		1764.4.a.ba.1.3		4
28.3	even	6		588.4.i.l.373.2		8
28.11	odd	6		588.4.i.k.373.3		8
28.19	even	6		588.4.i.l.361.2		8
28.23	odd	6		588.4.i.k.361.3		8
28.27	even	2		588.4.a.j.1.3	✓	4
84.11	even	6		1764.4.k.bd.1549.2		8
84.23	even	6		1764.4.k.bd.361.2		8
84.47	odd	6		1764.4.k.bb.361.3		8
84.59	odd	6		1764.4.k.bb.1549.3		8
84.83	odd	2		1764.4.a.bc.1.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
588.4.a.j.1.3	✓	4	28.27	even	2
588.4.a.k.1.2	yes	4	4.3	odd	2
588.4.i.k.361.3		8	28.23	odd	6
588.4.i.k.373.3		8	28.11	odd	6
588.4.i.l.361.2		8	28.19	even	6
588.4.i.l.373.2		8	28.3	even	6
1764.4.a.ba.1.3		4	12.11	even	2
1764.4.a.bc.1.2		4	84.83	odd	2
1764.4.k.bb.361.3		8	84.47	odd	6
1764.4.k.bb.1549.3		8	84.59	odd	6
1764.4.k.bd.361.2		8	84.23	even	6
1764.4.k.bd.1549.2		8	84.11	even	6
2352.4.a.cl.1.2		4	1.1	even	1	trivial
2352.4.a.cq.1.3		4	7.6	odd	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:	\( N \)	\(=\)	\( 2352 = 2^{4} \cdot 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2352.a (trivial)

\(f(q)\)	\(=\)	\(q-3.00000 q^{3} -10.6550 q^{5} +9.00000 q^{9} +O(q^{10})\)	\(q-3.00000 q^{3} -10.6550 q^{5} +9.00000 q^{9} +6.65399 q^{11} +75.9335 q^{13} +31.9651 q^{15} -104.287 q^{17} -85.4943 q^{19} +68.6733 q^{23} -11.4700 q^{25} -27.0000 q^{27} +87.7843 q^{29} -62.7683 q^{31} -19.9620 q^{33} +42.2093 q^{37} -227.800 q^{39} +313.904 q^{41} -306.591 q^{43} -95.8954 q^{45} -215.081 q^{47} +312.861 q^{51} +525.024 q^{53} -70.8986 q^{55} +256.483 q^{57} -360.491 q^{59} +800.726 q^{61} -809.075 q^{65} +40.2286 q^{67} -206.020 q^{69} +298.781 q^{71} +517.126 q^{73} +34.4099 q^{75} +1222.47 q^{79} +81.0000 q^{81} -1328.55 q^{83} +1111.18 q^{85} -263.353 q^{87} -639.938 q^{89} +188.305 q^{93} +910.946 q^{95} +1425.65 q^{97} +59.8859 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 12 q^{3} + 36 q^{9}+O(q^{10}) \) 4 * q - 12 * q^3 + 36 * q^9	\( 4 q - 12 q^{3} + 36 q^{9} + 48 q^{17} - 192 q^{19} - 192 q^{23} + 324 q^{25} - 108 q^{27} + 96 q^{29} - 48 q^{31} + 256 q^{37} + 1008 q^{41} + 112 q^{43} - 864 q^{47} - 144 q^{51} - 648 q^{53} - 2352 q^{55} + 576 q^{57} - 336 q^{59} + 960 q^{61} - 360 q^{65} - 720 q^{67} + 576 q^{69} + 1344 q^{71} + 672 q^{73} - 972 q^{75} + 1984 q^{79} + 324 q^{81} - 3120 q^{83} + 680 q^{85} - 288 q^{87} + 2160 q^{89} + 144 q^{93} + 3744 q^{95} + 2016 q^{97}+O(q^{100}) \) 4 * q - 12 * q^3 + 36 * q^9 + 48 * q^17 - 192 * q^19 - 192 * q^23 + 324 * q^25 - 108 * q^27 + 96 * q^29 - 48 * q^31 + 256 * q^37 + 1008 * q^41 + 112 * q^43 - 864 * q^47 - 144 * q^51 - 648 * q^53 - 2352 * q^55 + 576 * q^57 - 336 * q^59 + 960 * q^61 - 360 * q^65 - 720 * q^67 + 576 * q^69 + 1344 * q^71 + 672 * q^73 - 972 * q^75 + 1984 * q^79 + 324 * q^81 - 3120 * q^83 + 680 * q^85 - 288 * q^87 + 2160 * q^89 + 144 * q^93 + 3744 * q^95 + 2016 * q^97

Introduction

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