LMFDB - Embedded newform 2352.4.a.cd.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:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 7 $
Twist minimal:	no (minimal twist has level 294)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.41421$ 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)

$f(q)$	$=$	$q+3.00000 q^{3} +15.8995 q^{5} +9.00000 q^{9} +O(q^{10})$	$q+3.00000 q^{3} +15.8995 q^{5} +9.00000 q^{9} -57.3970 q^{11} -5.69848 q^{13} +47.6985 q^{15} +51.8995 q^{17} +16.2010 q^{19} +213.397 q^{23} +127.794 q^{25} +27.0000 q^{27} -218.191 q^{29} -251.397 q^{31} -172.191 q^{33} +386.794 q^{37} -17.0955 q^{39} +328.503 q^{41} +37.5879 q^{43} +143.095 q^{45} -254.995 q^{47} +155.698 q^{51} +211.588 q^{53} -912.583 q^{55} +48.6030 q^{57} -412.201 q^{59} +836.693 q^{61} -90.6030 q^{65} +165.588 q^{67} +640.191 q^{69} +465.015 q^{71} -449.658 q^{73} +383.382 q^{75} +343.558 q^{79} +81.0000 q^{81} +1502.33 q^{83} +825.176 q^{85} -654.573 q^{87} +341.085 q^{89} -754.191 q^{93} +257.588 q^{95} +865.437 q^{97} -516.573 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 6 q^{3} + 12 q^{5} + 18 q^{9}+O(q^{10}) $ 2 * q + 6 * q^3 + 12 * q^5 + 18 * q^9	$ 2 q + 6 q^{3} + 12 q^{5} + 18 q^{9} + 4 q^{11} + 48 q^{13} + 36 q^{15} + 84 q^{17} + 72 q^{19} + 308 q^{23} + 18 q^{25} + 54 q^{27} - 80 q^{29} - 384 q^{31} + 12 q^{33} + 536 q^{37} + 144 q^{39} + 756 q^{41} - 400 q^{43} + 108 q^{45} - 312 q^{47} + 252 q^{51} - 52 q^{53} - 1152 q^{55} + 216 q^{57} - 864 q^{59} + 1416 q^{61} - 300 q^{65} - 144 q^{67} + 924 q^{69} + 1524 q^{71} + 744 q^{73} + 54 q^{75} - 976 q^{79} + 162 q^{81} + 312 q^{83} + 700 q^{85} - 240 q^{87} + 108 q^{89} - 1152 q^{93} + 40 q^{95} - 744 q^{97} + 36 q^{99}+O(q^{100}) $ 2 * q + 6 * q^3 + 12 * q^5 + 18 * q^9 + 4 * q^11 + 48 * q^13 + 36 * q^15 + 84 * q^17 + 72 * q^19 + 308 * q^23 + 18 * q^25 + 54 * q^27 - 80 * q^29 - 384 * q^31 + 12 * q^33 + 536 * q^37 + 144 * q^39 + 756 * q^41 - 400 * q^43 + 108 * q^45 - 312 * q^47 + 252 * q^51 - 52 * q^53 - 1152 * q^55 + 216 * q^57 - 864 * q^59 + 1416 * q^61 - 300 * q^65 - 144 * q^67 + 924 * q^69 + 1524 * q^71 + 744 * q^73 + 54 * q^75 - 976 * q^79 + 162 * q^81 + 312 * q^83 + 700 * q^85 - 240 * q^87 + 108 * q^89 - 1152 * q^93 + 40 * q^95 - 744 * q^97 + 36 * 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
$5$	15.8995	1.42209	0.711047	−	0.703144i	$-0.248221\pi$
$5$	15.8995	1.42209	0.711047	+	0.703144i	$0.248221\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	−57.3970	−1.57326	−0.786629	−	0.617426i	$-0.788176\pi$
$11$	−57.3970	−1.57326	−0.786629	+	0.617426i	$0.788176\pi$
$12$	0	0
$13$	−5.69848	−0.121575	−0.0607875	−	0.998151i	$-0.519361\pi$
$13$	−5.69848	−0.121575	−0.0607875	+	0.998151i	$0.519361\pi$
$14$	0	0
$15$	47.6985	0.821046
$16$	0	0
$17$	51.8995	0.740440	0.370220	−	0.928944i	$-0.379282\pi$
$17$	51.8995	0.740440	0.370220	+	0.928944i	$0.379282\pi$
$18$	0	0
$19$	16.2010	0.195619	0.0978096	−	0.995205i	$-0.468816\pi$
$19$	16.2010	0.195619	0.0978096	+	0.995205i	$0.468816\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	213.397	1.93462	0.967312	−	0.253590i	$-0.0816114\pi$
$23$	213.397	1.93462	0.967312	+	0.253590i	$0.0816114\pi$
$24$	0	0
$25$	127.794	1.02235
$26$	0	0
$27$	27.0000	0.192450
$28$	0	0
$29$	−218.191	−1.39714	−0.698570	−	0.715542i	$-0.746180\pi$
$29$	−218.191	−1.39714	−0.698570	+	0.715542i	$0.746180\pi$
$30$	0	0
$31$	−251.397	−1.45652	−0.728262	−	0.685299i	$-0.759671\pi$
$31$	−251.397	−1.45652	−0.728262	+	0.685299i	$0.759671\pi$
$32$	0	0
$33$	−172.191	−0.908321
$34$	0	0
$35$	0	0
$36$	0	0
$37$	386.794	1.71861	0.859304	−	0.511464i	$-0.170897\pi$
$37$	386.794	1.71861	0.859304	+	0.511464i	$0.170897\pi$
$38$	0	0
$39$	−17.0955	−0.0701914
$40$	0	0
$41$	328.503	1.25130	0.625652	−	0.780102i	$-0.284833\pi$
$41$	328.503	1.25130	0.625652	+	0.780102i	$0.284833\pi$
$42$	0	0
$43$	37.5879	0.133305	0.0666523	−	0.997776i	$-0.478768\pi$
$43$	37.5879	0.133305	0.0666523	+	0.997776i	$0.478768\pi$
$44$	0	0
$45$	143.095	0.474031
$46$	0	0
$47$	−254.995	−0.791379	−0.395690	−	0.918384i	$-0.629494\pi$
$47$	−254.995	−0.791379	−0.395690	+	0.918384i	$0.629494\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	155.698	0.427493
$52$	0	0
$53$	211.588	0.548374	0.274187	−	0.961676i	$-0.411591\pi$
$53$	211.588	0.548374	0.274187	+	0.961676i	$0.411591\pi$
$54$	0	0
$55$	−912.583	−2.23732
$56$	0	0
$57$	48.6030	0.112941
$58$	0	0
$59$	−412.201	−0.909559	−0.454780	−	0.890604i	$-0.650282\pi$
$59$	−412.201	−0.909559	−0.454780	+	0.890604i	$0.650282\pi$
$60$	0	0
$61$	836.693	1.75619	0.878095	−	0.478486i	$-0.158814\pi$
$61$	836.693	1.75619	0.878095	+	0.478486i	$0.158814\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−90.6030	−0.172891
$66$	0	0
$67$	165.588	0.301937	0.150969	−	0.988539i	$-0.451761\pi$
$67$	165.588	0.301937	0.150969	+	0.988539i	$0.451761\pi$
$68$	0	0
$69$	640.191	1.11696
$70$	0	0
$71$	465.015	0.777284	0.388642	−	0.921389i	$-0.372944\pi$
$71$	465.015	0.777284	0.388642	+	0.921389i	$0.372944\pi$
$72$	0	0
$73$	−449.658	−0.720938	−0.360469	−	0.932771i	$-0.617383\pi$
$73$	−449.658	−0.720938	−0.360469	+	0.932771i	$0.617383\pi$
$74$	0	0
$75$	383.382	0.590255
$76$	0	0
$77$	0	0
$78$	0	0
$79$	343.558	0.489282	0.244641	−	0.969614i	$-0.421330\pi$
$79$	343.558	0.489282	0.244641	+	0.969614i	$0.421330\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	1502.33	1.98677	0.993387	−	0.114812i	$-0.0366265\pi$
$83$	1502.33	1.98677	0.993387	+	0.114812i	$0.0366265\pi$
$84$	0	0
$85$	825.176	1.05298
$86$	0	0
$87$	−654.573	−0.806639
$88$	0	0
$89$	341.085	0.406236	0.203118	−	0.979154i	$-0.434893\pi$
$89$	341.085	0.406236	0.203118	+	0.979154i	$0.434893\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−754.191	−0.840924
$94$	0	0
$95$	257.588	0.278189
$96$	0	0
$97$	865.437	0.905895	0.452947	−	0.891537i	$-0.350373\pi$
$97$	865.437	0.905895	0.452947	+	0.891537i	$0.350373\pi$
$98$	0	0
$99$	−516.573	−0.524419
$100$	0	0
$101$	243.256	0.239652	0.119826	−	0.992795i	$-0.461766\pi$
$101$	243.256	0.239652	0.119826	+	0.992795i	$0.461766\pi$
$102$	0	0
$103$	−953.346	−0.912000	−0.456000	−	0.889980i	$-0.650718\pi$
$103$	−953.346	−0.912000	−0.456000	+	0.889980i	$0.650718\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	1344.95	1.21516	0.607578	−	0.794260i	$-0.292141\pi$
$107$	1344.95	1.21516	0.607578	+	0.794260i	$0.292141\pi$
$108$	0	0
$109$	1734.35	1.52404	0.762022	−	0.647551i	$-0.224207\pi$
$109$	1734.35	1.52404	0.762022	+	0.647551i	$0.224207\pi$
$110$	0	0
$111$	1160.38	0.992239
$112$	0	0
$113$	1441.18	1.19977	0.599887	−	0.800085i	$-0.295212\pi$
$113$	1441.18	1.19977	0.599887	+	0.800085i	$0.295212\pi$
$114$	0	0
$115$	3392.90	2.75122
$116$	0	0
$117$	−51.2864	−0.0405250
$118$	0	0
$119$	0	0
$120$	0	0
$121$	1963.41	1.47514
$122$	0	0
$123$	985.508	0.722441
$124$	0	0
$125$	44.4222	0.0317860
$126$	0	0
$127$	−1184.70	−0.827759	−0.413880	−	0.910332i	$-0.635827\pi$
$127$	−1184.70	−0.827759	−0.413880	+	0.910332i	$0.635827\pi$
$128$	0	0
$129$	112.764	0.0769634
$130$	0	0
$131$	297.588	0.198476	0.0992381	−	0.995064i	$-0.468359\pi$
$131$	297.588	0.198476	0.0992381	+	0.995064i	$0.468359\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	429.286	0.273682
$136$	0	0
$137$	620.985	0.387258	0.193629	−	0.981075i	$-0.437974\pi$
$137$	620.985	0.387258	0.193629	+	0.981075i	$0.437974\pi$
$138$	0	0
$139$	−898.754	−0.548426	−0.274213	−	0.961669i	$-0.588417\pi$
$139$	−898.754	−0.548426	−0.274213	+	0.961669i	$0.588417\pi$
$140$	0	0
$141$	−764.985	−0.456903
$142$	0	0
$143$	327.076	0.191269
$144$	0	0
$145$	−3469.13	−1.98686
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−3054.70	−1.67954	−0.839769	−	0.542945i	$-0.817309\pi$
$149$	−3054.70	−1.67954	−0.839769	+	0.542945i	$0.817309\pi$
$150$	0	0
$151$	65.1455	0.0351090	0.0175545	−	0.999846i	$-0.494412\pi$
$151$	65.1455	0.0351090	0.0175545	+	0.999846i	$0.494412\pi$
$152$	0	0
$153$	467.095	0.246813
$154$	0	0
$155$	−3997.08	−2.07131
$156$	0	0
$157$	−1542.22	−0.783966	−0.391983	−	0.919973i	$-0.628211\pi$
$157$	−1542.22	−0.783966	−0.391983	+	0.919973i	$0.628211\pi$
$158$	0	0
$159$	634.764	0.316604
$160$	0	0
$161$	0	0
$162$	0	0
$163$	2514.73	1.20840	0.604200	−	0.796833i	$-0.293493\pi$
$163$	2514.73	1.20840	0.604200	+	0.796833i	$0.293493\pi$
$164$	0	0
$165$	−2737.75	−1.29172
$166$	0	0
$167$	528.643	0.244956	0.122478	−	0.992471i	$-0.460916\pi$
$167$	528.643	0.244956	0.122478	+	0.992471i	$0.460916\pi$
$168$	0	0
$169$	−2164.53	−0.985220
$170$	0	0
$171$	145.809	0.0652064
$172$	0	0
$173$	−96.8439	−0.0425602	−0.0212801	−	0.999774i	$-0.506774\pi$
$173$	−96.8439	−0.0425602	−0.0212801	+	0.999774i	$0.506774\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−1236.60	−0.525134
$178$	0	0
$179$	534.542	0.223204	0.111602	−	0.993753i	$-0.464402\pi$
$179$	534.542	0.223204	0.111602	+	0.993753i	$0.464402\pi$
$180$	0	0
$181$	−2087.00	−0.857049	−0.428524	−	0.903530i	$-0.640966\pi$
$181$	−2087.00	−0.857049	−0.428524	+	0.903530i	$0.640966\pi$
$182$	0	0
$183$	2510.08	1.01394
$184$	0	0
$185$	6149.83	2.44402
$186$	0	0
$187$	−2978.87	−1.16490
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−3387.69	−1.28337	−0.641687	−	0.766966i	$-0.721765\pi$
$191$	−3387.69	−1.28337	−0.641687	+	0.766966i	$0.721765\pi$
$192$	0	0
$193$	−1908.35	−0.711742	−0.355871	−	0.934535i	$-0.615816\pi$
$193$	−1908.35	−0.711742	−0.355871	+	0.934535i	$0.615816\pi$
$194$	0	0
$195$	−271.809	−0.0998187
$196$	0	0
$197$	−2061.88	−0.745699	−0.372850	−	0.927892i	$-0.621619\pi$
$197$	−2061.88	−0.745699	−0.372850	+	0.927892i	$0.621619\pi$
$198$	0	0
$199$	3171.50	1.12976	0.564878	−	0.825174i	$-0.308923\pi$
$199$	3171.50	1.12976	0.564878	+	0.825174i	$0.308923\pi$
$200$	0	0
$201$	496.764	0.174323
$202$	0	0
$203$	0	0
$204$	0	0
$205$	5223.02	1.77947
$206$	0	0
$207$	1920.57	0.644875
$208$	0	0
$209$	−929.889	−0.307760
$210$	0	0
$211$	−1349.97	−0.440454	−0.220227	−	0.975449i	$-0.570680\pi$
$211$	−1349.97	−0.440454	−0.220227	+	0.975449i	$0.570680\pi$
$212$	0	0
$213$	1395.05	0.448765
$214$	0	0
$215$	597.628	0.189572
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−1348.97	−0.416234
$220$	0	0
$221$	−295.748	−0.0900190
$222$	0	0
$223$	−1361.85	−0.408951	−0.204476	−	0.978872i	$-0.565549\pi$
$223$	−1361.85	−0.408951	−0.204476	+	0.978872i	$0.565549\pi$
$224$	0	0
$225$	1150.15	0.340784
$226$	0	0
$227$	1861.81	0.544373	0.272186	−	0.962245i	$-0.412253\pi$
$227$	1861.81	0.544373	0.272186	+	0.962245i	$0.412253\pi$
$228$	0	0
$229$	5358.78	1.54637	0.773184	−	0.634181i	$-0.218663\pi$
$229$	5358.78	1.54637	0.773184	+	0.634181i	$0.218663\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	5441.12	1.52987	0.764935	−	0.644107i	$-0.222771\pi$
$233$	5441.12	1.52987	0.764935	+	0.644107i	$0.222771\pi$
$234$	0	0
$235$	−4054.29	−1.12542
$236$	0	0
$237$	1030.67	0.282487
$238$	0	0
$239$	1157.28	0.313213	0.156607	−	0.987661i	$-0.449945\pi$
$239$	1157.28	0.313213	0.156607	+	0.987661i	$0.449945\pi$
$240$	0	0
$241$	3969.38	1.06095	0.530477	−	0.847699i	$-0.322013\pi$
$241$	3969.38	1.06095	0.530477	+	0.847699i	$0.322013\pi$
$242$	0	0
$243$	243.000	0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−92.3212	−0.0237824
$248$	0	0
$249$	4506.99	1.14706
$250$	0	0
$251$	5978.75	1.50349	0.751744	−	0.659455i	$-0.229213\pi$
$251$	5978.75	1.50349	0.751744	+	0.659455i	$0.229213\pi$
$252$	0	0
$253$	−12248.3	−3.04366
$254$	0	0
$255$	2475.53	0.607935
$256$	0	0
$257$	−4650.15	−1.12867	−0.564335	−	0.825546i	$-0.690868\pi$
$257$	−4650.15	−1.12867	−0.564335	+	0.825546i	$0.690868\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−1963.72	−0.465713
$262$	0	0
$263$	−3695.37	−0.866411	−0.433205	−	0.901295i	$-0.642618\pi$
$263$	−3695.37	−0.866411	−0.433205	+	0.901295i	$0.642618\pi$
$264$	0	0
$265$	3364.14	0.779840
$266$	0	0
$267$	1023.26	0.234540
$268$	0	0
$269$	7157.69	1.62235	0.811175	−	0.584804i	$-0.198829\pi$
$269$	7157.69	1.62235	0.811175	+	0.584804i	$0.198829\pi$
$270$	0	0
$271$	4038.37	0.905216	0.452608	−	0.891710i	$-0.350494\pi$
$271$	4038.37	0.905216	0.452608	+	0.891710i	$0.350494\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−7334.98	−1.60842
$276$	0	0
$277$	−2754.82	−0.597550	−0.298775	−	0.954324i	$-0.596578\pi$
$277$	−2754.82	−0.597550	−0.298775	+	0.954324i	$0.596578\pi$
$278$	0	0
$279$	−2262.57	−0.485508
$280$	0	0
$281$	−772.742	−0.164050	−0.0820248	−	0.996630i	$-0.526139\pi$
$281$	−772.742	−0.164050	−0.0820248	+	0.996630i	$0.526139\pi$
$282$	0	0
$283$	−6745.49	−1.41688	−0.708441	−	0.705770i	$-0.750601\pi$
$283$	−6745.49	−1.41688	−0.708441	+	0.705770i	$0.750601\pi$
$284$	0	0
$285$	772.764	0.160613
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−2219.44	−0.451749
$290$	0	0
$291$	2596.31	0.523019
$292$	0	0
$293$	−1922.69	−0.383362	−0.191681	−	0.981457i	$-0.561394\pi$
$293$	−1922.69	−0.383362	−0.191681	+	0.981457i	$0.561394\pi$
$294$	0	0
$295$	−6553.79	−1.29348
$296$	0	0
$297$	−1549.72	−0.302774
$298$	0	0
$299$	−1216.04	−0.235202
$300$	0	0
$301$	0	0
$302$	0	0
$303$	729.768	0.138363
$304$	0	0
$305$	13303.0	2.49747
$306$	0	0
$307$	2016.68	0.374913	0.187456	−	0.982273i	$-0.439976\pi$
$307$	2016.68	0.374913	0.187456	+	0.982273i	$0.439976\pi$
$308$	0	0
$309$	−2860.04	−0.526544
$310$	0	0
$311$	7149.99	1.30366	0.651831	−	0.758365i	$-0.274001\pi$
$311$	7149.99	1.30366	0.651831	+	0.758365i	$0.274001\pi$
$312$	0	0
$313$	−8596.49	−1.55240	−0.776202	−	0.630484i	$-0.782856\pi$
$313$	−8596.49	−1.55240	−0.776202	+	0.630484i	$0.782856\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−2853.24	−0.505532	−0.252766	−	0.967527i	$-0.581340\pi$
$317$	−2853.24	−0.505532	−0.252766	+	0.967527i	$0.581340\pi$
$318$	0	0
$319$	12523.5	2.19806
$320$	0	0
$321$	4034.86	0.701570
$322$	0	0
$323$	840.824	0.144844
$324$	0	0
$325$	−728.232	−0.124292
$326$	0	0
$327$	5203.05	0.879907
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−1619.12	−0.268866	−0.134433	−	0.990923i	$-0.542921\pi$
$331$	−1619.12	−0.268866	−0.134433	+	0.990923i	$0.542921\pi$
$332$	0	0
$333$	3481.15	0.572870
$334$	0	0
$335$	2632.76	0.429383
$336$	0	0
$337$	−3278.67	−0.529972	−0.264986	−	0.964252i	$-0.585367\pi$
$337$	−3278.67	−0.529972	−0.264986	+	0.964252i	$0.585367\pi$
$338$	0	0
$339$	4323.53	0.692690
$340$	0	0
$341$	14429.4	2.29149
$342$	0	0
$343$	0	0
$344$	0	0
$345$	10178.7	1.58842
$346$	0	0
$347$	2850.30	0.440957	0.220479	−	0.975392i	$-0.429238\pi$
$347$	2850.30	0.440957	0.220479	+	0.975392i	$0.429238\pi$
$348$	0	0
$349$	4725.32	0.724758	0.362379	−	0.932031i	$-0.381965\pi$
$349$	4725.32	0.724758	0.362379	+	0.932031i	$0.381965\pi$
$350$	0	0
$351$	−153.859	−0.0233971
$352$	0	0
$353$	−6727.44	−1.01435	−0.507175	−	0.861843i	$-0.669310\pi$
$353$	−6727.44	−1.01435	−0.507175	+	0.861843i	$0.669310\pi$
$354$	0	0
$355$	7393.51	1.10537
$356$	0	0
$357$	0	0
$358$	0	0
$359$	7331.89	1.07789	0.538945	−	0.842341i	$-0.318823\pi$
$359$	7331.89	1.07789	0.538945	+	0.842341i	$0.318823\pi$
$360$	0	0
$361$	−6596.53	−0.961733
$362$	0	0
$363$	5890.24	0.851673
$364$	0	0
$365$	−7149.34	−1.02524
$366$	0	0
$367$	2774.43	0.394616	0.197308	−	0.980342i	$-0.436780\pi$
$367$	2774.43	0.394616	0.197308	+	0.980342i	$0.436780\pi$
$368$	0	0
$369$	2956.52	0.417101
$370$	0	0
$371$	0	0
$372$	0	0
$373$	2527.35	0.350834	0.175417	−	0.984494i	$-0.443873\pi$
$373$	2527.35	0.350834	0.175417	+	0.984494i	$0.443873\pi$
$374$	0	0
$375$	133.267	0.0183516
$376$	0	0
$377$	1243.36	0.169857
$378$	0	0
$379$	−3116.40	−0.422371	−0.211186	−	0.977446i	$-0.567732\pi$
$379$	−3116.40	−0.422371	−0.211186	+	0.977446i	$0.567732\pi$
$380$	0	0
$381$	−3554.11	−0.477907
$382$	0	0
$383$	1518.07	0.202532	0.101266	−	0.994859i	$-0.467711\pi$
$383$	1518.07	0.202532	0.101266	+	0.994859i	$0.467711\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	338.291	0.0444349
$388$	0	0
$389$	−3246.77	−0.423182	−0.211591	−	0.977358i	$-0.567865\pi$
$389$	−3246.77	−0.423182	−0.211591	+	0.977358i	$0.567865\pi$
$390$	0	0
$391$	11075.2	1.43247
$392$	0	0
$393$	892.764	0.114590
$394$	0	0
$395$	5462.39	0.695804
$396$	0	0
$397$	1830.62	0.231427	0.115713	−	0.993283i	$-0.463085\pi$
$397$	1830.62	0.231427	0.115713	+	0.993283i	$0.463085\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−3385.81	−0.421644	−0.210822	−	0.977524i	$-0.567614\pi$
$401$	−3385.81	−0.421644	−0.210822	+	0.977524i	$0.567614\pi$
$402$	0	0
$403$	1432.58	0.177077
$404$	0	0
$405$	1287.86	0.158010
$406$	0	0
$407$	−22200.8	−2.70382
$408$	0	0
$409$	9253.17	1.11868	0.559340	−	0.828938i	$-0.311055\pi$
$409$	9253.17	1.11868	0.559340	+	0.828938i	$0.311055\pi$
$410$	0	0
$411$	1862.95	0.223583
$412$	0	0
$413$	0	0
$414$	0	0
$415$	23886.3	2.82538
$416$	0	0
$417$	−2696.26	−0.316634
$418$	0	0
$419$	−3547.52	−0.413622	−0.206811	−	0.978381i	$-0.566308\pi$
$419$	−3547.52	−0.413622	−0.206811	+	0.978381i	$0.566308\pi$
$420$	0	0
$421$	7848.87	0.908624	0.454312	−	0.890843i	$-0.349885\pi$
$421$	7848.87	0.908624	0.454312	+	0.890843i	$0.349885\pi$
$422$	0	0
$423$	−2294.95	−0.263793
$424$	0	0
$425$	6632.44	0.756990
$426$	0	0
$427$	0	0
$428$	0	0
$429$	981.227	0.110429
$430$	0	0
$431$	4447.32	0.497030	0.248515	−	0.968628i	$-0.420057\pi$
$431$	4447.32	0.497030	0.248515	+	0.968628i	$0.420057\pi$
$432$	0	0
$433$	−6994.82	−0.776327	−0.388164	−	0.921590i	$-0.626890\pi$
$433$	−6994.82	−0.776327	−0.388164	+	0.921590i	$0.626890\pi$
$434$	0	0
$435$	−10407.4	−1.14712
$436$	0	0
$437$	3457.25	0.378450
$438$	0	0
$439$	−636.182	−0.0691647	−0.0345823	−	0.999402i	$-0.511010\pi$
$439$	−636.182	−0.0691647	−0.0345823	+	0.999402i	$0.511010\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	4474.24	0.479859	0.239929	−	0.970790i	$-0.422876\pi$
$443$	4474.24	0.479859	0.239929	+	0.970790i	$0.422876\pi$
$444$	0	0
$445$	5423.08	0.577705
$446$	0	0
$447$	−9164.11	−0.969681
$448$	0	0
$449$	−2389.42	−0.251144	−0.125572	−	0.992085i	$-0.540077\pi$
$449$	−2389.42	−0.251144	−0.125572	+	0.992085i	$0.540077\pi$
$450$	0	0
$451$	−18855.0	−1.96862
$452$	0	0
$453$	195.436	0.0202702
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−4438.34	−0.454304	−0.227152	−	0.973859i	$-0.572941\pi$
$457$	−4438.34	−0.454304	−0.227152	+	0.973859i	$0.572941\pi$
$458$	0	0
$459$	1401.29	0.142498
$460$	0	0
$461$	−14079.8	−1.42248	−0.711240	−	0.702949i	$-0.751866\pi$
$461$	−14079.8	−1.42248	−0.711240	+	0.702949i	$0.751866\pi$
$462$	0	0
$463$	−4687.50	−0.470511	−0.235255	−	0.971934i	$-0.575593\pi$
$463$	−4687.50	−0.470511	−0.235255	+	0.971934i	$0.575593\pi$
$464$	0	0
$465$	−11991.3	−1.19587
$466$	0	0
$467$	−8447.26	−0.837029	−0.418514	−	0.908210i	$-0.637449\pi$
$467$	−8447.26	−0.837029	−0.418514	+	0.908210i	$0.637449\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−4626.66	−0.452623
$472$	0	0
$473$	−2157.43	−0.209723
$474$	0	0
$475$	2070.39	0.199992
$476$	0	0
$477$	1904.29	0.182791
$478$	0	0
$479$	4369.41	0.416792	0.208396	−	0.978045i	$-0.433176\pi$
$479$	4369.41	0.416792	0.208396	+	0.978045i	$0.433176\pi$
$480$	0	0
$481$	−2204.14	−0.208940
$482$	0	0
$483$	0	0
$484$	0	0
$485$	13760.0	1.28827
$486$	0	0
$487$	14477.7	1.34712	0.673561	−	0.739132i	$-0.264764\pi$
$487$	14477.7	1.34712	0.673561	+	0.739132i	$0.264764\pi$
$488$	0	0
$489$	7544.20	0.697670
$490$	0	0
$491$	−9306.12	−0.855355	−0.427677	−	0.903931i	$-0.640668\pi$
$491$	−9306.12	−0.855355	−0.427677	+	0.903931i	$0.640668\pi$
$492$	0	0
$493$	−11324.0	−1.03450
$494$	0	0
$495$	−8213.25	−0.745774
$496$	0	0
$497$	0	0
$498$	0	0
$499$	12237.5	1.09785	0.548926	−	0.835871i	$-0.315037\pi$
$499$	12237.5	1.09785	0.548926	+	0.835871i	$0.315037\pi$
$500$	0	0
$501$	1585.93	0.141425
$502$	0	0
$503$	−5524.30	−0.489694	−0.244847	−	0.969562i	$-0.578738\pi$
$503$	−5524.30	−0.489694	−0.244847	+	0.969562i	$0.578738\pi$
$504$	0	0
$505$	3867.65	0.340808
$506$	0	0
$507$	−6493.58	−0.568817
$508$	0	0
$509$	−10079.6	−0.877743	−0.438871	−	0.898550i	$-0.644622\pi$
$509$	−10079.6	−0.877743	−0.438871	+	0.898550i	$0.644622\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	437.427	0.0376470
$514$	0	0
$515$	−15157.7	−1.29695
$516$	0	0
$517$	14635.9	1.24504
$518$	0	0
$519$	−290.532	−0.0245721
$520$	0	0
$521$	5706.61	0.479868	0.239934	−	0.970789i	$-0.422874\pi$
$521$	5706.61	0.479868	0.239934	+	0.970789i	$0.422874\pi$
$522$	0	0
$523$	10657.3	0.891032	0.445516	−	0.895274i	$-0.353020\pi$
$523$	10657.3	0.891032	0.445516	+	0.895274i	$0.353020\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−13047.4	−1.07847
$528$	0	0
$529$	33371.3	2.74277
$530$	0	0
$531$	−3709.81	−0.303186
$532$	0	0
$533$	−1871.97	−0.152127
$534$	0	0
$535$	21384.1	1.72807
$536$	0	0
$537$	1603.63	0.128867
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−4010.48	−0.318714	−0.159357	−	0.987221i	$-0.550942\pi$
$541$	−4010.48	−0.318714	−0.159357	+	0.987221i	$0.550942\pi$
$542$	0	0
$543$	−6261.01	−0.494817
$544$	0	0
$545$	27575.3	2.16733
$546$	0	0
$547$	17619.8	1.37728	0.688638	−	0.725105i	$-0.258209\pi$
$547$	17619.8	1.37728	0.688638	+	0.725105i	$0.258209\pi$
$548$	0	0
$549$	7530.24	0.585397
$550$	0	0
$551$	−3534.91	−0.273307
$552$	0	0
$553$	0	0
$554$	0	0
$555$	18449.5	1.41106
$556$	0	0
$557$	−10337.7	−0.786395	−0.393198	−	0.919454i	$-0.628631\pi$
$557$	−10337.7	−0.786395	−0.393198	+	0.919454i	$0.628631\pi$
$558$	0	0
$559$	−214.194	−0.0162065
$560$	0	0
$561$	−8936.62	−0.672557
$562$	0	0
$563$	24023.7	1.79836	0.899180	−	0.437580i	$-0.144164\pi$
$563$	24023.7	1.79836	0.899180	+	0.437580i	$0.144164\pi$
$564$	0	0
$565$	22914.0	1.70619
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−13179.2	−0.971001	−0.485501	−	0.874236i	$-0.661363\pi$
$569$	−13179.2	−0.971001	−0.485501	+	0.874236i	$0.661363\pi$
$570$	0	0
$571$	−7776.26	−0.569924	−0.284962	−	0.958539i	$-0.591981\pi$
$571$	−7776.26	−0.569924	−0.284962	+	0.958539i	$0.591981\pi$
$572$	0	0
$573$	−10163.1	−0.740956
$574$	0	0
$575$	27270.8	1.97787
$576$	0	0
$577$	20167.0	1.45505	0.727525	−	0.686081i	$-0.240670\pi$
$577$	20167.0	1.45505	0.727525	+	0.686081i	$0.240670\pi$
$578$	0	0
$579$	−5725.05	−0.410924
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−12144.5	−0.862734
$584$	0	0
$585$	−815.427	−0.0576304
$586$	0	0
$587$	8365.08	0.588184	0.294092	−	0.955777i	$-0.404983\pi$
$587$	8365.08	0.588184	0.294092	+	0.955777i	$0.404983\pi$
$588$	0	0
$589$	−4072.88	−0.284924
$590$	0	0
$591$	−6185.64	−0.430530
$592$	0	0
$593$	−27621.9	−1.91281	−0.956403	−	0.292050i	$-0.905663\pi$
$593$	−27621.9	−1.91281	−0.956403	+	0.292050i	$0.905663\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	9514.49	0.652265
$598$	0	0
$599$	−538.318	−0.0367197	−0.0183598	−	0.999831i	$-0.505844\pi$
$599$	−538.318	−0.0367197	−0.0183598	+	0.999831i	$0.505844\pi$
$600$	0	0
$601$	6958.64	0.472294	0.236147	−	0.971717i	$-0.424115\pi$
$601$	6958.64	0.472294	0.236147	+	0.971717i	$0.424115\pi$
$602$	0	0
$603$	1490.29	0.100646
$604$	0	0
$605$	31217.3	2.09779
$606$	0	0
$607$	17297.6	1.15665	0.578326	−	0.815806i	$-0.303706\pi$
$607$	17297.6	1.15665	0.578326	+	0.815806i	$0.303706\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	1453.08	0.0962120
$612$	0	0
$613$	−839.158	−0.0552908	−0.0276454	−	0.999618i	$-0.508801\pi$
$613$	−839.158	−0.0552908	−0.0276454	+	0.999618i	$0.508801\pi$
$614$	0	0
$615$	15669.1	1.02738
$616$	0	0
$617$	−16040.0	−1.04659	−0.523295	−	0.852152i	$-0.675297\pi$
$617$	−16040.0	−1.04659	−0.523295	+	0.852152i	$0.675297\pi$
$618$	0	0
$619$	5429.28	0.352538	0.176269	−	0.984342i	$-0.443597\pi$
$619$	5429.28	0.352538	0.176269	+	0.984342i	$0.443597\pi$
$620$	0	0
$621$	5761.72	0.372318
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−15268.0	−0.977149
$626$	0	0
$627$	−2789.67	−0.177685
$628$	0	0
$629$	20074.4	1.27253
$630$	0	0
$631$	1807.86	0.114057	0.0570284	−	0.998373i	$-0.481837\pi$
$631$	1807.86	0.114057	0.0570284	+	0.998373i	$0.481837\pi$
$632$	0	0
$633$	−4049.91	−0.254296
$634$	0	0
$635$	−18836.2	−1.17715
$636$	0	0
$637$	0	0
$638$	0	0
$639$	4185.14	0.259095
$640$	0	0
$641$	−5904.56	−0.363832	−0.181916	−	0.983314i	$-0.558230\pi$
$641$	−5904.56	−0.363832	−0.181916	+	0.983314i	$0.558230\pi$
$642$	0	0
$643$	−8092.42	−0.496320	−0.248160	−	0.968719i	$-0.579826\pi$
$643$	−8092.42	−0.496320	−0.248160	+	0.968719i	$0.579826\pi$
$644$	0	0
$645$	1792.88	0.109449
$646$	0	0
$647$	−20192.7	−1.22698	−0.613490	−	0.789702i	$-0.710235\pi$
$647$	−20192.7	−1.22698	−0.613490	+	0.789702i	$0.710235\pi$
$648$	0	0
$649$	23659.1	1.43097
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−21180.8	−1.26933	−0.634664	−	0.772789i	$-0.718861\pi$
$653$	−21180.8	−1.26933	−0.634664	+	0.772789i	$0.718861\pi$
$654$	0	0
$655$	4731.50	0.282252
$656$	0	0
$657$	−4046.92	−0.240313
$658$	0	0
$659$	−28411.3	−1.67944	−0.839718	−	0.543023i	$-0.817280\pi$
$659$	−28411.3	−1.67944	−0.839718	+	0.543023i	$0.817280\pi$
$660$	0	0
$661$	16704.9	0.982975	0.491488	−	0.870885i	$-0.336453\pi$
$661$	16704.9	0.982975	0.491488	+	0.870885i	$0.336453\pi$
$662$	0	0
$663$	−887.245	−0.0519725
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−46561.3	−2.70294
$668$	0	0
$669$	−4085.55	−0.236108
$670$	0	0
$671$	−48023.7	−2.76294
$672$	0	0
$673$	−9047.09	−0.518187	−0.259093	−	0.965852i	$-0.583424\pi$
$673$	−9047.09	−0.518187	−0.259093	+	0.965852i	$0.583424\pi$
$674$	0	0
$675$	3450.44	0.196752
$676$	0	0
$677$	−7844.26	−0.445316	−0.222658	−	0.974897i	$-0.571473\pi$
$677$	−7844.26	−0.445316	−0.222658	+	0.974897i	$0.571473\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	5585.43	0.314294
$682$	0	0
$683$	−25766.1	−1.44350	−0.721751	−	0.692153i	$-0.756662\pi$
$683$	−25766.1	−1.44350	−0.721751	+	0.692153i	$0.756662\pi$
$684$	0	0
$685$	9873.35	0.550717
$686$	0	0
$687$	16076.3	0.892796
$688$	0	0
$689$	−1205.73	−0.0666686
$690$	0	0
$691$	−24674.1	−1.35839	−0.679195	−	0.733958i	$-0.737671\pi$
$691$	−24674.1	−1.35839	−0.679195	+	0.733958i	$0.737671\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−14289.7	−0.779914
$696$	0	0
$697$	17049.1	0.926515
$698$	0	0
$699$	16323.4	0.883271
$700$	0	0
$701$	29377.9	1.58286	0.791431	−	0.611258i	$-0.209336\pi$
$701$	29377.9	1.58286	0.791431	+	0.611258i	$0.209336\pi$
$702$	0	0
$703$	6266.45	0.336193
$704$	0	0
$705$	−12162.9	−0.649759
$706$	0	0
$707$	0	0
$708$	0	0
$709$	30594.3	1.62058	0.810291	−	0.586028i	$-0.199309\pi$
$709$	30594.3	1.62058	0.810291	+	0.586028i	$0.199309\pi$
$710$	0	0
$711$	3092.02	0.163094
$712$	0	0
$713$	−53647.4	−2.81782
$714$	0	0
$715$	5200.34	0.272002
$716$	0	0
$717$	3471.83	0.180834
$718$	0	0
$719$	−1946.94	−0.100985	−0.0504927	−	0.998724i	$-0.516079\pi$
$719$	−1946.94	−0.100985	−0.0504927	+	0.998724i	$0.516079\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	11908.1	0.612542
$724$	0	0
$725$	−27883.5	−1.42837
$726$	0	0
$727$	15750.6	0.803518	0.401759	−	0.915745i	$-0.368399\pi$
$727$	15750.6	0.803518	0.401759	+	0.915745i	$0.368399\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	1950.79	0.0987040
$732$	0	0
$733$	−14349.6	−0.723074	−0.361537	−	0.932358i	$-0.617748\pi$
$733$	−14349.6	−0.723074	−0.361537	+	0.932358i	$0.617748\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−9504.24	−0.475025
$738$	0	0
$739$	17758.1	0.883953	0.441976	−	0.897027i	$-0.354278\pi$
$739$	17758.1	0.883953	0.441976	+	0.897027i	$0.354278\pi$
$740$	0	0
$741$	−276.964	−0.0137308
$742$	0	0
$743$	29187.6	1.44117	0.720586	−	0.693366i	$-0.243873\pi$
$743$	29187.6	1.44117	0.720586	+	0.693366i	$0.243873\pi$
$744$	0	0
$745$	−48568.2	−2.38846
$746$	0	0
$747$	13521.0	0.662258
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−13781.9	−0.669654	−0.334827	−	0.942280i	$-0.608678\pi$
$751$	−13781.9	−0.669654	−0.334827	+	0.942280i	$0.608678\pi$
$752$	0	0
$753$	17936.3	0.868039
$754$	0	0
$755$	1035.78	0.0499283
$756$	0	0
$757$	36952.7	1.77420	0.887099	−	0.461579i	$-0.152717\pi$
$757$	36952.7	1.77420	0.887099	+	0.461579i	$0.152717\pi$
$758$	0	0
$759$	−36745.0	−1.75726
$760$	0	0
$761$	−28816.5	−1.37266	−0.686332	−	0.727288i	$-0.740780\pi$
$761$	−28816.5	−1.37266	−0.686332	+	0.727288i	$0.740780\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	7426.58	0.350992
$766$	0	0
$767$	2348.92	0.110580
$768$	0	0
$769$	−25285.2	−1.18571	−0.592854	−	0.805310i	$-0.701999\pi$
$769$	−25285.2	−1.18571	−0.592854	+	0.805310i	$0.701999\pi$
$770$	0	0
$771$	−13950.4	−0.651638
$772$	0	0
$773$	18418.2	0.856995	0.428497	−	0.903543i	$-0.359043\pi$
$773$	18418.2	0.856995	0.428497	+	0.903543i	$0.359043\pi$
$774$	0	0
$775$	−32127.0	−1.48908
$776$	0	0
$777$	0	0
$778$	0	0
$779$	5322.07	0.244779
$780$	0	0
$781$	−26690.5	−1.22287
$782$	0	0
$783$	−5891.15	−0.268880
$784$	0	0
$785$	−24520.5	−1.11487
$786$	0	0
$787$	−11075.8	−0.501664	−0.250832	−	0.968031i	$-0.580704\pi$
$787$	−11075.8	−0.501664	−0.250832	+	0.968031i	$0.580704\pi$
$788$	0	0
$789$	−11086.1	−0.500223
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−4767.88	−0.213509
$794$	0	0
$795$	10092.4	0.450241
$796$	0	0
$797$	4838.83	0.215057	0.107528	−	0.994202i	$-0.465706\pi$
$797$	4838.83	0.215057	0.107528	+	0.994202i	$0.465706\pi$
$798$	0	0
$799$	−13234.1	−0.585969
$800$	0	0
$801$	3069.77	0.135412
$802$	0	0
$803$	25809.0	1.13422
$804$	0	0
$805$	0	0
$806$	0	0
$807$	21473.1	0.936664
$808$	0	0
$809$	−31509.9	−1.36938	−0.684690	−	0.728834i	$-0.740062\pi$
$809$	−31509.9	−1.36938	−0.684690	+	0.728834i	$0.740062\pi$
$810$	0	0
$811$	29463.3	1.27570	0.637851	−	0.770160i	$-0.279823\pi$
$811$	29463.3	1.27570	0.637851	+	0.770160i	$0.279823\pi$
$812$	0	0
$813$	12115.1	0.522627
$814$	0	0
$815$	39983.0	1.71846
$816$	0	0
$817$	608.962	0.0260770
$818$	0	0
$819$	0	0
$820$	0	0
$821$	3502.22	0.148877	0.0744386	−	0.997226i	$-0.476284\pi$
$821$	3502.22	0.148877	0.0744386	+	0.997226i	$0.476284\pi$
$822$	0	0
$823$	−39993.0	−1.69389	−0.846943	−	0.531684i	$-0.821560\pi$
$823$	−39993.0	−1.69389	−0.846943	+	0.531684i	$0.821560\pi$
$824$	0	0
$825$	−22005.0	−0.928623
$826$	0	0
$827$	10733.6	0.451322	0.225661	−	0.974206i	$-0.427546\pi$
$827$	10733.6	0.451322	0.225661	+	0.974206i	$0.427546\pi$
$828$	0	0
$829$	−14537.5	−0.609056	−0.304528	−	0.952503i	$-0.598499\pi$
$829$	−14537.5	−0.609056	−0.304528	+	0.952503i	$0.598499\pi$
$830$	0	0
$831$	−8264.47	−0.344996
$832$	0	0
$833$	0	0
$834$	0	0
$835$	8405.16	0.348351
$836$	0	0
$837$	−6787.72	−0.280308
$838$	0	0
$839$	−7353.57	−0.302591	−0.151295	−	0.988489i	$-0.548344\pi$
$839$	−7353.57	−0.302591	−0.151295	+	0.988489i	$0.548344\pi$
$840$	0	0
$841$	23218.3	0.951998
$842$	0	0
$843$	−2318.23	−0.0947141
$844$	0	0
$845$	−34414.9	−1.40107
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−20236.5	−0.818037
$850$	0	0
$851$	82540.7	3.32486
$852$	0	0
$853$	19293.6	0.774442	0.387221	−	0.921987i	$-0.373435\pi$
$853$	19293.6	0.774442	0.387221	+	0.921987i	$0.373435\pi$
$854$	0	0
$855$	2318.29	0.0927297
$856$	0	0
$857$	−14161.6	−0.564468	−0.282234	−	0.959346i	$-0.591076\pi$
$857$	−14161.6	−0.564468	−0.282234	+	0.959346i	$0.591076\pi$
$858$	0	0
$859$	−8219.13	−0.326465	−0.163232	−	0.986588i	$-0.552192\pi$
$859$	−8219.13	−0.326465	−0.163232	+	0.986588i	$0.552192\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−2574.95	−0.101567	−0.0507835	−	0.998710i	$-0.516172\pi$
$863$	−2574.95	−0.101567	−0.0507835	+	0.998710i	$0.516172\pi$
$864$	0	0
$865$	−1539.77	−0.0605246
$866$	0	0
$867$	−6658.33	−0.260817
$868$	0	0
$869$	−19719.2	−0.769766
$870$	0	0
$871$	−943.600	−0.0367080
$872$	0	0
$873$	7788.93	0.301965
$874$	0	0
$875$	0	0
$876$	0	0
$877$	30981.1	1.19288	0.596442	−	0.802656i	$-0.296581\pi$
$877$	30981.1	1.19288	0.596442	+	0.802656i	$0.296581\pi$
$878$	0	0
$879$	−5768.08	−0.221334
$880$	0	0
$881$	−41781.8	−1.59780	−0.798902	−	0.601461i	$-0.794585\pi$
$881$	−41781.8	−1.59780	−0.798902	+	0.601461i	$0.794585\pi$
$882$	0	0
$883$	39289.6	1.49740	0.748699	−	0.662911i	$-0.230679\pi$
$883$	39289.6	1.49740	0.748699	+	0.662911i	$0.230679\pi$
$884$	0	0
$885$	−19661.4	−0.746790
$886$	0	0
$887$	−5832.44	−0.220783	−0.110391	−	0.993888i	$-0.535210\pi$
$887$	−5832.44	−0.220783	−0.110391	+	0.993888i	$0.535210\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−4649.15	−0.174806
$892$	0	0
$893$	−4131.18	−0.154809
$894$	0	0
$895$	8498.95	0.317418
$896$	0	0
$897$	−3648.12	−0.135794
$898$	0	0
$899$	54852.5	2.03497
$900$	0	0
$901$	10981.3	0.406038
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−33182.3	−1.21880
$906$	0	0
$907$	−42387.9	−1.55178	−0.775892	−	0.630866i	$-0.782700\pi$
$907$	−42387.9	−1.55178	−0.775892	+	0.630866i	$0.782700\pi$
$908$	0	0
$909$	2189.30	0.0798841
$910$	0	0
$911$	−2275.12	−0.0827423	−0.0413711	−	0.999144i	$-0.513173\pi$
$911$	−2275.12	−0.0827423	−0.0413711	+	0.999144i	$0.513173\pi$
$912$	0	0
$913$	−86229.3	−3.12571
$914$	0	0
$915$	39909.0	1.44191
$916$	0	0
$917$	0	0
$918$	0	0
$919$	31284.3	1.12293	0.561465	−	0.827500i	$-0.310238\pi$
$919$	31284.3	1.12293	0.561465	+	0.827500i	$0.310238\pi$
$920$	0	0
$921$	6050.05	0.216456
$922$	0	0
$923$	−2649.88	−0.0944983
$924$	0	0
$925$	49429.9	1.75702
$926$	0	0
$927$	−8580.12	−0.304000
$928$	0	0
$929$	32196.6	1.13707	0.568535	−	0.822659i	$-0.307511\pi$
$929$	32196.6	1.13707	0.568535	+	0.822659i	$0.307511\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	21450.0	0.752669
$934$	0	0
$935$	−47362.6	−1.65660
$936$	0	0
$937$	22293.6	0.777269	0.388635	−	0.921392i	$-0.372947\pi$
$937$	22293.6	0.777269	0.388635	+	0.921392i	$0.372947\pi$
$938$	0	0
$939$	−25789.5	−0.896281
$940$	0	0
$941$	−31809.7	−1.10198	−0.550991	−	0.834511i	$-0.685750\pi$
$941$	−31809.7	−1.10198	−0.550991	+	0.834511i	$0.685750\pi$
$942$	0	0
$943$	70101.4	2.42080
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−18982.6	−0.651376	−0.325688	−	0.945477i	$-0.605596\pi$
$947$	−18982.6	−0.651376	−0.325688	+	0.945477i	$0.605596\pi$
$948$	0	0
$949$	2562.37	0.0876481
$950$	0	0
$951$	−8559.71	−0.291869
$952$	0	0
$953$	9254.58	0.314570	0.157285	−	0.987553i	$-0.449726\pi$
$953$	9254.58	0.314570	0.157285	+	0.987553i	$0.449726\pi$
$954$	0	0
$955$	−53862.5	−1.82508
$956$	0	0
$957$	37570.5	1.26905
$958$	0	0
$959$	0	0
$960$	0	0
$961$	33409.4	1.12146
$962$	0	0
$963$	12104.6	0.405052
$964$	0	0
$965$	−30341.8	−1.01216
$966$	0	0
$967$	−15317.5	−0.509387	−0.254694	−	0.967022i	$-0.581975\pi$
$967$	−15317.5	−0.509387	−0.254694	+	0.967022i	$0.581975\pi$
$968$	0	0
$969$	2522.47	0.0836259
$970$	0	0
$971$	−22432.6	−0.741398	−0.370699	−	0.928753i	$-0.620882\pi$
$971$	−22432.6	−0.741398	−0.370699	+	0.928753i	$0.620882\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	−2184.70	−0.0717603
$976$	0	0
$977$	−627.864	−0.0205600	−0.0102800	−	0.999947i	$-0.503272\pi$
$977$	−627.864	−0.0205600	−0.0102800	+	0.999947i	$0.503272\pi$
$978$	0	0
$979$	−19577.3	−0.639114
$980$	0	0
$981$	15609.2	0.508015
$982$	0	0
$983$	−45032.6	−1.46116	−0.730579	−	0.682828i	$-0.760750\pi$
$983$	−45032.6	−1.46116	−0.730579	+	0.682828i	$0.760750\pi$
$984$	0	0
$985$	−32782.8	−1.06045
$986$	0	0
$987$	0	0
$988$	0	0
$989$	8021.14	0.257894
$990$	0	0
$991$	31981.0	1.02514	0.512569	−	0.858646i	$-0.328694\pi$
$991$	31981.0	1.02514	0.512569	+	0.858646i	$0.328694\pi$
$992$	0	0
$993$	−4857.35	−0.155230
$994$	0	0
$995$	50425.2	1.60662
$996$	0	0
$997$	12378.8	0.393221	0.196611	−	0.980482i	$-0.437007\pi$
$997$	12378.8	0.393221	0.196611	+	0.980482i	$0.437007\pi$
$998$	0	0
$999$	10443.4	0.330746

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2352.4.a.cd.1.2		2
4.3	odd	2		294.4.a.j.1.2	✓	2
7.6	odd	2		2352.4.a.bn.1.1		2
12.11	even	2		882.4.a.bc.1.1		2
28.3	even	6		294.4.e.n.79.2		4
28.11	odd	6		294.4.e.o.79.1		4
28.19	even	6		294.4.e.n.67.2		4
28.23	odd	6		294.4.e.o.67.1		4
28.27	even	2		294.4.a.k.1.1	yes	2
84.11	even	6		882.4.g.bd.667.2		4
84.23	even	6		882.4.g.bd.361.2		4
84.47	odd	6		882.4.g.y.361.1		4
84.59	odd	6		882.4.g.y.667.1		4
84.83	odd	2		882.4.a.bi.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
294.4.a.j.1.2	✓	2	4.3	odd	2
294.4.a.k.1.1	yes	2	28.27	even	2
294.4.e.n.67.2		4	28.19	even	6
294.4.e.n.79.2		4	28.3	even	6
294.4.e.o.67.1		4	28.23	odd	6
294.4.e.o.79.1		4	28.11	odd	6
882.4.a.bc.1.1		2	12.11	even	2
882.4.a.bi.1.2		2	84.83	odd	2
882.4.g.y.361.1		4	84.47	odd	6
882.4.g.y.667.1		4	84.59	odd	6
882.4.g.bd.361.2		4	84.23	even	6
882.4.g.bd.667.2		4	84.11	even	6
2352.4.a.bn.1.1		2	7.6	odd	2
2352.4.a.cd.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)

\(f(q)\)	\(=\)	\(q+3.00000 q^{3} +15.8995 q^{5} +9.00000 q^{9} +O(q^{10})\)	\(q+3.00000 q^{3} +15.8995 q^{5} +9.00000 q^{9} -57.3970 q^{11} -5.69848 q^{13} +47.6985 q^{15} +51.8995 q^{17} +16.2010 q^{19} +213.397 q^{23} +127.794 q^{25} +27.0000 q^{27} -218.191 q^{29} -251.397 q^{31} -172.191 q^{33} +386.794 q^{37} -17.0955 q^{39} +328.503 q^{41} +37.5879 q^{43} +143.095 q^{45} -254.995 q^{47} +155.698 q^{51} +211.588 q^{53} -912.583 q^{55} +48.6030 q^{57} -412.201 q^{59} +836.693 q^{61} -90.6030 q^{65} +165.588 q^{67} +640.191 q^{69} +465.015 q^{71} -449.658 q^{73} +383.382 q^{75} +343.558 q^{79} +81.0000 q^{81} +1502.33 q^{83} +825.176 q^{85} -654.573 q^{87} +341.085 q^{89} -754.191 q^{93} +257.588 q^{95} +865.437 q^{97} -516.573 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 6 q^{3} + 12 q^{5} + 18 q^{9}+O(q^{10}) \) 2 * q + 6 * q^3 + 12 * q^5 + 18 * q^9	\( 2 q + 6 q^{3} + 12 q^{5} + 18 q^{9} + 4 q^{11} + 48 q^{13} + 36 q^{15} + 84 q^{17} + 72 q^{19} + 308 q^{23} + 18 q^{25} + 54 q^{27} - 80 q^{29} - 384 q^{31} + 12 q^{33} + 536 q^{37} + 144 q^{39} + 756 q^{41} - 400 q^{43} + 108 q^{45} - 312 q^{47} + 252 q^{51} - 52 q^{53} - 1152 q^{55} + 216 q^{57} - 864 q^{59} + 1416 q^{61} - 300 q^{65} - 144 q^{67} + 924 q^{69} + 1524 q^{71} + 744 q^{73} + 54 q^{75} - 976 q^{79} + 162 q^{81} + 312 q^{83} + 700 q^{85} - 240 q^{87} + 108 q^{89} - 1152 q^{93} + 40 q^{95} - 744 q^{97} + 36 q^{99}+O(q^{100}) \) 2 * q + 6 * q^3 + 12 * q^5 + 18 * q^9 + 4 * q^11 + 48 * q^13 + 36 * q^15 + 84 * q^17 + 72 * q^19 + 308 * q^23 + 18 * q^25 + 54 * q^27 - 80 * q^29 - 384 * q^31 + 12 * q^33 + 536 * q^37 + 144 * q^39 + 756 * q^41 - 400 * q^43 + 108 * q^45 - 312 * q^47 + 252 * q^51 - 52 * q^53 - 1152 * q^55 + 216 * q^57 - 864 * q^59 + 1416 * q^61 - 300 * q^65 - 144 * q^67 + 924 * q^69 + 1524 * q^71 + 744 * q^73 + 54 * q^75 - 976 * q^79 + 162 * q^81 + 312 * q^83 + 700 * q^85 - 240 * q^87 + 108 * q^89 - 1152 * q^93 + 40 * q^95 - 744 * q^97 + 36 * q^99

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