LMFDB - Embedded newform 2352.4.a.ce.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2352,4,Mod(1,2352)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://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);

sage: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")

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:traces = [2,0,6,0,18,0,0,0,18,0,-58] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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(\sqrt{137}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 34 $ x^2 - x - 34
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 1176)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-5.35235$ 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)

$f(q)$	$=$	$q+3.00000 q^{3} +20.7047 q^{5} +9.00000 q^{9} +6.11410 q^{11} -22.8188 q^{13} +62.1141 q^{15} +8.70470 q^{17} +75.0470 q^{19} -178.570 q^{23} +303.685 q^{25} +27.0000 q^{27} -122.228 q^{29} -102.591 q^{31} +18.3423 q^{33} +182.228 q^{37} -68.4564 q^{39} +488.208 q^{41} +388.000 q^{43} +186.342 q^{45} -254.550 q^{47} +26.1141 q^{51} +610.228 q^{53} +126.591 q^{55} +225.141 q^{57} +31.2752 q^{59} -587.503 q^{61} -472.456 q^{65} +689.597 q^{67} -535.711 q^{69} +563.940 q^{71} +275.315 q^{73} +911.054 q^{75} +219.315 q^{79} +81.0000 q^{81} -1447.28 q^{83} +180.228 q^{85} -366.685 q^{87} +1018.57 q^{89} -307.772 q^{93} +1553.83 q^{95} +792.497 q^{97} +55.0269 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 6 q^{3} + 18 q^{5} + 18 q^{9} - 58 q^{11} + 48 q^{13} + 54 q^{15} - 6 q^{17} - 84 q^{19} - 6 q^{23} + 186 q^{25} + 54 q^{27} - 104 q^{29} - 252 q^{31} - 174 q^{33} + 224 q^{37} + 144 q^{39} + 438 q^{41}+ \cdots - 522 q^{99}+O(q^{100}) $ 2 * q + 6 * q^3 + 18 * q^5 + 18 * q^9 - 58 * q^11 + 48 * q^13 + 54 * q^15 - 6 * q^17 - 84 * q^19 - 6 * q^23 + 186 * q^25 + 54 * q^27 - 104 * q^29 - 252 * q^31 - 174 * q^33 + 224 * q^37 + 144 * q^39 + 438 * q^41 + 776 * q^43 + 162 * q^45 + 240 * q^47 - 18 * q^51 + 1080 * q^53 + 300 * q^55 - 252 * q^57 - 312 * q^59 - 660 * q^61 - 664 * q^65 + 396 * q^67 - 18 * q^69 - 66 * q^71 + 972 * q^73 + 558 * q^75 + 860 * q^79 + 162 * q^81 - 2520 * q^83 + 220 * q^85 - 312 * q^87 + 1686 * q^89 - 756 * q^93 + 1984 * q^95 + 2100 * q^97 - 522 * q^99

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
$5$	20.7047	1.85188	0.925942	−	0.377665i	$-0.123273\pi$
$5$	20.7047	1.85188	0.925942	+	0.377665i	$0.123273\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	6.11410	0.167588	0.0837941	−	0.996483i	$-0.473296\pi$
$11$	6.11410	0.167588	0.0837941	+	0.996483i	$0.473296\pi$
$12$	0	0
$13$	−22.8188	−0.486830	−0.243415	−	0.969922i	$-0.578268\pi$
$13$	−22.8188	−0.486830	−0.243415	+	0.969922i	$0.578268\pi$
$14$	0	0
$15$	62.1141	1.06919
$16$	0	0
$17$	8.70470	0.124188	0.0620941	−	0.998070i	$-0.480222\pi$
$17$	8.70470	0.124188	0.0620941	+	0.998070i	$0.480222\pi$
$18$	0	0
$19$	75.0470	0.906156	0.453078	−	0.891471i	$-0.350326\pi$
$19$	75.0470	0.906156	0.453078	+	0.891471i	$0.350326\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−178.570	−1.61889	−0.809446	−	0.587194i	$-0.800233\pi$
$23$	−178.570	−1.61889	−0.809446	+	0.587194i	$0.800233\pi$
$24$	0	0
$25$	303.685	2.42948
$26$	0	0
$27$	27.0000	0.192450
$28$	0	0
$29$	−122.228	−0.782662	−0.391331	−	0.920250i	$-0.627985\pi$
$29$	−122.228	−0.782662	−0.391331	+	0.920250i	$0.627985\pi$
$30$	0	0
$31$	−102.591	−0.594381	−0.297191	−	0.954818i	$-0.596050\pi$
$31$	−102.591	−0.594381	−0.297191	+	0.954818i	$0.596050\pi$
$32$	0	0
$33$	18.3423	0.0967571
$34$	0	0
$35$	0	0
$36$	0	0
$37$	182.228	0.809679	0.404840	−	0.914388i	$-0.367327\pi$
$37$	182.228	0.809679	0.404840	+	0.914388i	$0.367327\pi$
$38$	0	0
$39$	−68.4564	−0.281072
$40$	0	0
$41$	488.208	1.85964	0.929821	−	0.368013i	$-0.119962\pi$
$41$	488.208	1.85964	0.929821	+	0.368013i	$0.119962\pi$
$42$	0	0
$43$	388.000	1.37603	0.688017	−	0.725695i	$-0.258482\pi$
$43$	388.000	1.37603	0.688017	+	0.725695i	$0.258482\pi$
$44$	0	0
$45$	186.342	0.617295
$46$	0	0
$47$	−254.550	−0.790000	−0.395000	−	0.918681i	$-0.629255\pi$
$47$	−254.550	−0.790000	−0.395000	+	0.918681i	$0.629255\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	26.1141	0.0717001
$52$	0	0
$53$	610.228	1.58153	0.790767	−	0.612117i	$-0.209682\pi$
$53$	610.228	1.58153	0.790767	+	0.612117i	$0.209682\pi$
$54$	0	0
$55$	126.591	0.310354
$56$	0	0
$57$	225.141	0.523169
$58$	0	0
$59$	31.2752	0.0690116	0.0345058	−	0.999404i	$-0.489014\pi$
$59$	31.2752	0.0690116	0.0345058	+	0.999404i	$0.489014\pi$
$60$	0	0
$61$	−587.503	−1.23315	−0.616575	−	0.787296i	$-0.711480\pi$
$61$	−587.503	−1.23315	−0.616575	+	0.787296i	$0.711480\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−472.456	−0.901554
$66$	0	0
$67$	689.597	1.25743	0.628714	−	0.777636i	$-0.283581\pi$
$67$	689.597	1.25743	0.628714	+	0.777636i	$0.283581\pi$
$68$	0	0
$69$	−535.711	−0.934668
$70$	0	0
$71$	563.940	0.942638	0.471319	−	0.881963i	$-0.343778\pi$
$71$	563.940	0.942638	0.471319	+	0.881963i	$0.343778\pi$
$72$	0	0
$73$	275.315	0.441414	0.220707	−	0.975340i	$-0.429164\pi$
$73$	275.315	0.441414	0.220707	+	0.975340i	$0.429164\pi$
$74$	0	0
$75$	911.054	1.40266
$76$	0	0
$77$	0	0
$78$	0	0
$79$	219.315	0.312341	0.156170	−	0.987730i	$-0.450085\pi$
$79$	219.315	0.312341	0.156170	+	0.987730i	$0.450085\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	−1447.28	−1.91396	−0.956982	−	0.290146i	$-0.906296\pi$
$83$	−1447.28	−1.91396	−0.956982	+	0.290146i	$0.906296\pi$
$84$	0	0
$85$	180.228	0.229982
$86$	0	0
$87$	−366.685	−0.451870
$88$	0	0
$89$	1018.57	1.21313	0.606563	−	0.795035i	$-0.292548\pi$
$89$	1018.57	1.21313	0.606563	+	0.795035i	$0.292548\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−307.772	−0.343166
$94$	0	0
$95$	1553.83	1.67810
$96$	0	0
$97$	792.497	0.829545	0.414772	−	0.909925i	$-0.363861\pi$
$97$	792.497	0.829545	0.414772	+	0.909925i	$0.363861\pi$
$98$	0	0
$99$	55.0269	0.0558627
$100$	0	0
$101$	509.658	0.502107	0.251054	−	0.967973i	$-0.419223\pi$
$101$	509.658	0.502107	0.251054	+	0.967973i	$0.419223\pi$
$102$	0	0
$103$	217.490	0.208057	0.104029	−	0.994574i	$-0.466827\pi$
$103$	217.490	0.208057	0.104029	+	0.994574i	$0.466827\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	1897.31	1.71420	0.857102	−	0.515147i	$-0.172263\pi$
$107$	1897.31	1.71420	0.857102	+	0.515147i	$0.172263\pi$
$108$	0	0
$109$	850.456	0.747330	0.373665	−	0.927564i	$-0.378101\pi$
$109$	850.456	0.747330	0.373665	+	0.927564i	$0.378101\pi$
$110$	0	0
$111$	546.685	0.467469
$112$	0	0
$113$	1527.48	1.27162	0.635809	−	0.771846i	$-0.280666\pi$
$113$	1527.48	1.27162	0.635809	+	0.771846i	$0.280666\pi$
$114$	0	0
$115$	−3697.25	−2.99800
$116$	0	0
$117$	−205.369	−0.162277
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−1293.62	−0.971914
$122$	0	0
$123$	1464.62	1.07366
$124$	0	0
$125$	3699.61	2.64723
$126$	0	0
$127$	−1038.51	−0.725613	−0.362807	−	0.931864i	$-0.618181\pi$
$127$	−1038.51	−0.725613	−0.362807	+	0.931864i	$0.618181\pi$
$128$	0	0
$129$	1164.00	0.794453
$130$	0	0
$131$	961.181	0.641059	0.320530	−	0.947238i	$-0.396139\pi$
$131$	961.181	0.641059	0.320530	+	0.947238i	$0.396139\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	559.027	0.356395
$136$	0	0
$137$	−1690.00	−1.05392	−0.526958	−	0.849891i	$-0.676667\pi$
$137$	−1690.00	−1.05392	−0.526958	+	0.849891i	$0.676667\pi$
$138$	0	0
$139$	1732.30	1.05706	0.528530	−	0.848915i	$-0.322743\pi$
$139$	1732.30	1.05706	0.528530	+	0.848915i	$0.322743\pi$
$140$	0	0
$141$	−763.651	−0.456107
$142$	0	0
$143$	−139.516	−0.0815871
$144$	0	0
$145$	−2530.70	−1.44940
$146$	0	0
$147$	0	0
$148$	0	0
$149$	2725.88	1.49874	0.749372	−	0.662150i	$-0.230356\pi$
$149$	2725.88	1.49874	0.749372	+	0.662150i	$0.230356\pi$
$150$	0	0
$151$	−1129.83	−0.608900	−0.304450	−	0.952528i	$-0.598473\pi$
$151$	−1129.83	−0.608900	−0.304450	+	0.952528i	$0.598473\pi$
$152$	0	0
$153$	78.3423	0.0413961
$154$	0	0
$155$	−2124.11	−1.10073
$156$	0	0
$157$	−719.691	−0.365845	−0.182922	−	0.983127i	$-0.558556\pi$
$157$	−719.691	−0.365845	−0.182922	+	0.983127i	$0.558556\pi$
$158$	0	0
$159$	1830.68	0.913099
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−2469.25	−1.18654	−0.593271	−	0.805003i	$-0.702164\pi$
$163$	−2469.25	−1.18654	−0.593271	+	0.805003i	$0.702164\pi$
$164$	0	0
$165$	379.772	0.179183
$166$	0	0
$167$	−2419.03	−1.12090	−0.560451	−	0.828188i	$-0.689372\pi$
$167$	−2419.03	−1.12090	−0.560451	+	0.828188i	$0.689372\pi$
$168$	0	0
$169$	−1676.30	−0.762996
$170$	0	0
$171$	675.423	0.302052
$172$	0	0
$173$	3559.16	1.56415	0.782075	−	0.623184i	$-0.214161\pi$
$173$	3559.16	1.56415	0.782075	+	0.623184i	$0.214161\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	93.8256	0.0398439
$178$	0	0
$179$	−3864.15	−1.61352	−0.806761	−	0.590878i	$-0.798782\pi$
$179$	−3864.15	−1.61352	−0.806761	+	0.590878i	$0.798782\pi$
$180$	0	0
$181$	4100.46	1.68389	0.841946	−	0.539561i	$-0.181410\pi$
$181$	4100.46	1.68389	0.841946	+	0.539561i	$0.181410\pi$
$182$	0	0
$183$	−1762.51	−0.711959
$184$	0	0
$185$	3772.98	1.49943
$186$	0	0
$187$	53.2214	0.0208125
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−1120.17	−0.424359	−0.212179	−	0.977231i	$-0.568056\pi$
$191$	−1120.17	−0.424359	−0.212179	+	0.977231i	$0.568056\pi$
$192$	0	0
$193$	−1418.44	−0.529025	−0.264512	−	0.964382i	$-0.585211\pi$
$193$	−1418.44	−0.529025	−0.264512	+	0.964382i	$0.585211\pi$
$194$	0	0
$195$	−1417.37	−0.520512
$196$	0	0
$197$	−4611.01	−1.66762	−0.833809	−	0.552053i	$-0.813844\pi$
$197$	−4611.01	−1.66762	−0.833809	+	0.552053i	$0.813844\pi$
$198$	0	0
$199$	−5049.83	−1.79886	−0.899429	−	0.437067i	$-0.856017\pi$
$199$	−5049.83	−1.79886	−0.899429	+	0.437067i	$0.856017\pi$
$200$	0	0
$201$	2068.79	0.725977
$202$	0	0
$203$	0	0
$204$	0	0
$205$	10108.2	3.44384
$206$	0	0
$207$	−1607.13	−0.539631
$208$	0	0
$209$	458.845	0.151861
$210$	0	0
$211$	1712.11	0.558608	0.279304	−	0.960203i	$-0.409896\pi$
$211$	1712.11	0.558608	0.279304	+	0.960203i	$0.409896\pi$
$212$	0	0
$213$	1691.82	0.544233
$214$	0	0
$215$	8033.42	2.54826
$216$	0	0
$217$	0	0
$218$	0	0
$219$	825.946	0.254851
$220$	0	0
$221$	−198.631	−0.0604586
$222$	0	0
$223$	161.906	0.0486190	0.0243095	−	0.999704i	$-0.492261\pi$
$223$	161.906	0.0486190	0.0243095	+	0.999704i	$0.492261\pi$
$224$	0	0
$225$	2733.16	0.809826
$226$	0	0
$227$	−3947.76	−1.15428	−0.577141	−	0.816645i	$-0.695832\pi$
$227$	−3947.76	−1.15428	−0.577141	+	0.816645i	$0.695832\pi$
$228$	0	0
$229$	−3584.78	−1.03445	−0.517225	−	0.855850i	$-0.673035\pi$
$229$	−3584.78	−1.03445	−0.517225	+	0.855850i	$0.673035\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−3448.85	−0.969706	−0.484853	−	0.874596i	$-0.661127\pi$
$233$	−3448.85	−0.969706	−0.484853	+	0.874596i	$0.661127\pi$
$234$	0	0
$235$	−5270.39	−1.46299
$236$	0	0
$237$	657.946	0.180330
$238$	0	0
$239$	−3818.57	−1.03348	−0.516742	−	0.856141i	$-0.672855\pi$
$239$	−3818.57	−1.03348	−0.516742	+	0.856141i	$0.672855\pi$
$240$	0	0
$241$	−1314.54	−0.351356	−0.175678	−	0.984448i	$-0.556212\pi$
$241$	−1314.54	−0.351356	−0.175678	+	0.984448i	$0.556212\pi$
$242$	0	0
$243$	243.000	0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−1712.48	−0.441144
$248$	0	0
$249$	−4341.83	−1.10503
$250$	0	0
$251$	4829.66	1.21452	0.607262	−	0.794501i	$-0.292268\pi$
$251$	4829.66	1.21452	0.607262	+	0.794501i	$0.292268\pi$
$252$	0	0
$253$	−1091.80	−0.271307
$254$	0	0
$255$	540.685	0.132780
$256$	0	0
$257$	−1479.93	−0.359203	−0.179602	−	0.983739i	$-0.557481\pi$
$257$	−1479.93	−0.359203	−0.179602	+	0.983739i	$0.557481\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−1100.05	−0.260887
$262$	0	0
$263$	1191.01	0.279244	0.139622	−	0.990205i	$-0.455411\pi$
$263$	1191.01	0.279244	0.139622	+	0.990205i	$0.455411\pi$
$264$	0	0
$265$	12634.6	2.92882
$266$	0	0
$267$	3055.71	0.700399
$268$	0	0
$269$	558.772	0.126650	0.0633251	−	0.997993i	$-0.479830\pi$
$269$	558.772	0.126650	0.0633251	+	0.997993i	$0.479830\pi$
$270$	0	0
$271$	−5992.31	−1.34320	−0.671599	−	0.740914i	$-0.734392\pi$
$271$	−5992.31	−1.34320	−0.671599	+	0.740914i	$0.734392\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	1856.76	0.407152
$276$	0	0
$277$	7884.03	1.71013	0.855064	−	0.518523i	$-0.173518\pi$
$277$	7884.03	1.71013	0.855064	+	0.518523i	$0.173518\pi$
$278$	0	0
$279$	−923.315	−0.198127
$280$	0	0
$281$	−75.8256	−0.0160974	−0.00804871	−	0.999968i	$-0.502562\pi$
$281$	−75.8256	−0.0160974	−0.00804871	+	0.999968i	$0.502562\pi$
$282$	0	0
$283$	−803.369	−0.168747	−0.0843733	−	0.996434i	$-0.526889\pi$
$283$	−803.369	−0.168747	−0.0843733	+	0.996434i	$0.526889\pi$
$284$	0	0
$285$	4661.48	0.968849
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−4837.23	−0.984577
$290$	0	0
$291$	2377.49	0.478938
$292$	0	0
$293$	−2626.26	−0.523645	−0.261822	−	0.965116i	$-0.584323\pi$
$293$	−2626.26	−0.523645	−0.261822	+	0.965116i	$0.584323\pi$
$294$	0	0
$295$	647.544	0.127802
$296$	0	0
$297$	165.081	0.0322524
$298$	0	0
$299$	4074.76	0.788126
$300$	0	0
$301$	0	0
$302$	0	0
$303$	1528.97	0.289892
$304$	0	0
$305$	−12164.1	−2.28365
$306$	0	0
$307$	4865.84	0.904587	0.452293	−	0.891869i	$-0.350606\pi$
$307$	4865.84	0.904587	0.452293	+	0.891869i	$0.350606\pi$
$308$	0	0
$309$	652.469	0.120122
$310$	0	0
$311$	−6152.97	−1.12187	−0.560937	−	0.827859i	$-0.689559\pi$
$311$	−6152.97	−1.12187	−0.560937	+	0.827859i	$0.689559\pi$
$312$	0	0
$313$	7892.78	1.42532	0.712662	−	0.701508i	$-0.247489\pi$
$313$	7892.78	1.42532	0.712662	+	0.701508i	$0.247489\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−3688.75	−0.653568	−0.326784	−	0.945099i	$-0.605965\pi$
$317$	−3688.75	−0.653568	−0.326784	+	0.945099i	$0.605965\pi$
$318$	0	0
$319$	−747.315	−0.131165
$320$	0	0
$321$	5691.93	0.989696
$322$	0	0
$323$	653.262	0.112534
$324$	0	0
$325$	−6929.72	−1.18274
$326$	0	0
$327$	2551.37	0.431471
$328$	0	0
$329$	0	0
$330$	0	0
$331$	1981.80	0.329092	0.164546	−	0.986369i	$-0.447384\pi$
$331$	1981.80	0.329092	0.164546	+	0.986369i	$0.447384\pi$
$332$	0	0
$333$	1640.05	0.269893
$334$	0	0
$335$	14277.9	2.32861
$336$	0	0
$337$	1578.35	0.255128	0.127564	−	0.991830i	$-0.459284\pi$
$337$	1578.35	0.255128	0.127564	+	0.991830i	$0.459284\pi$
$338$	0	0
$339$	4582.43	0.734169
$340$	0	0
$341$	−627.249	−0.0996113
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−11091.7	−1.73090
$346$	0	0
$347$	3213.51	0.497148	0.248574	−	0.968613i	$-0.420038\pi$
$347$	3213.51	0.497148	0.248574	+	0.968613i	$0.420038\pi$
$348$	0	0
$349$	−725.920	−0.111340	−0.0556699	−	0.998449i	$-0.517729\pi$
$349$	−725.920	−0.111340	−0.0556699	+	0.998449i	$0.517729\pi$
$350$	0	0
$351$	−616.108	−0.0936906
$352$	0	0
$353$	−619.403	−0.0933923	−0.0466962	−	0.998909i	$-0.514869\pi$
$353$	−619.403	−0.0933923	−0.0466962	+	0.998909i	$0.514869\pi$
$354$	0	0
$355$	11676.2	1.74566
$356$	0	0
$357$	0	0
$358$	0	0
$359$	7912.05	1.16318	0.581590	−	0.813482i	$-0.302431\pi$
$359$	7912.05	1.16318	0.581590	+	0.813482i	$0.302431\pi$
$360$	0	0
$361$	−1226.95	−0.178881
$362$	0	0
$363$	−3880.85	−0.561135
$364$	0	0
$365$	5700.32	0.817448
$366$	0	0
$367$	−2916.08	−0.414764	−0.207382	−	0.978260i	$-0.566494\pi$
$367$	−2916.08	−0.414764	−0.207382	+	0.978260i	$0.566494\pi$
$368$	0	0
$369$	4393.87	0.619880
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−53.1688	−0.00738063	−0.00369031	−	0.999993i	$-0.501175\pi$
$373$	−53.1688	−0.00738063	−0.00369031	+	0.999993i	$0.501175\pi$
$374$	0	0
$375$	11098.8	1.52838
$376$	0	0
$377$	2789.10	0.381024
$378$	0	0
$379$	12650.6	1.71456	0.857281	−	0.514848i	$-0.172152\pi$
$379$	12650.6	1.71456	0.857281	+	0.514848i	$0.172152\pi$
$380$	0	0
$381$	−3115.53	−0.418933
$382$	0	0
$383$	10401.5	1.38770	0.693852	−	0.720118i	$-0.255912\pi$
$383$	10401.5	1.38770	0.693852	+	0.720118i	$0.255912\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	3492.00	0.458678
$388$	0	0
$389$	2665.16	0.347375	0.173687	−	0.984801i	$-0.444432\pi$
$389$	2665.16	0.347375	0.173687	+	0.984801i	$0.444432\pi$
$390$	0	0
$391$	−1554.40	−0.201047
$392$	0	0
$393$	2883.54	0.370116
$394$	0	0
$395$	4540.86	0.578419
$396$	0	0
$397$	10880.9	1.37556	0.687779	−	0.725920i	$-0.258586\pi$
$397$	10880.9	1.37556	0.687779	+	0.725920i	$0.258586\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−12599.0	−1.56899	−0.784493	−	0.620137i	$-0.787077\pi$
$401$	−12599.0	−1.56899	−0.784493	+	0.620137i	$0.787077\pi$
$402$	0	0
$403$	2340.99	0.289363
$404$	0	0
$405$	1677.08	0.205765
$406$	0	0
$407$	1114.16	0.135693
$408$	0	0
$409$	−5740.47	−0.694005	−0.347002	−	0.937864i	$-0.612800\pi$
$409$	−5740.47	−0.694005	−0.347002	+	0.937864i	$0.612800\pi$
$410$	0	0
$411$	−5070.00	−0.608478
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−29965.4	−3.54444
$416$	0	0
$417$	5196.89	0.610294
$418$	0	0
$419$	3433.66	0.400347	0.200174	−	0.979760i	$-0.435849\pi$
$419$	3433.66	0.400347	0.200174	+	0.979760i	$0.435849\pi$
$420$	0	0
$421$	−4793.88	−0.554963	−0.277481	−	0.960731i	$-0.589500\pi$
$421$	−4793.88	−0.554963	−0.277481	+	0.960731i	$0.589500\pi$
$422$	0	0
$423$	−2290.95	−0.263333
$424$	0	0
$425$	2643.48	0.301712
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−418.549	−0.0471043
$430$	0	0
$431$	−5350.29	−0.597945	−0.298973	−	0.954262i	$-0.596644\pi$
$431$	−5350.29	−0.597945	−0.298973	+	0.954262i	$0.596644\pi$
$432$	0	0
$433$	−4440.03	−0.492781	−0.246390	−	0.969171i	$-0.579245\pi$
$433$	−4440.03	−0.492781	−0.246390	+	0.969171i	$0.579245\pi$
$434$	0	0
$435$	−7592.09	−0.836812
$436$	0	0
$437$	−13401.2	−1.46697
$438$	0	0
$439$	−10376.0	−1.12806	−0.564030	−	0.825754i	$-0.690750\pi$
$439$	−10376.0	−1.12806	−0.564030	+	0.825754i	$0.690750\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	4186.32	0.448979	0.224490	−	0.974476i	$-0.427928\pi$
$443$	4186.32	0.448979	0.224490	+	0.974476i	$0.427928\pi$
$444$	0	0
$445$	21089.2	2.24657
$446$	0	0
$447$	8177.64	0.865300
$448$	0	0
$449$	5216.36	0.548275	0.274137	−	0.961691i	$-0.411608\pi$
$449$	5216.36	0.548275	0.274137	+	0.961691i	$0.411608\pi$
$450$	0	0
$451$	2984.95	0.311654
$452$	0	0
$453$	−3389.48	−0.351549
$454$	0	0
$455$	0	0
$456$	0	0
$457$	9469.65	0.969303	0.484652	−	0.874707i	$-0.338946\pi$
$457$	9469.65	0.969303	0.484652	+	0.874707i	$0.338946\pi$
$458$	0	0
$459$	235.027	0.0239000
$460$	0	0
$461$	−3455.07	−0.349064	−0.174532	−	0.984651i	$-0.555841\pi$
$461$	−3455.07	−0.349064	−0.174532	+	0.984651i	$0.555841\pi$
$462$	0	0
$463$	10881.9	1.09228	0.546139	−	0.837695i	$-0.316097\pi$
$463$	10881.9	1.09228	0.546139	+	0.837695i	$0.316097\pi$
$464$	0	0
$465$	−6372.32	−0.635504
$466$	0	0
$467$	−12053.6	−1.19437	−0.597186	−	0.802102i	$-0.703715\pi$
$467$	−12053.6	−1.19437	−0.597186	+	0.802102i	$0.703715\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−2159.07	−0.211221
$472$	0	0
$473$	2372.27	0.230607
$474$	0	0
$475$	22790.6	2.20148
$476$	0	0
$477$	5492.05	0.527178
$478$	0	0
$479$	7836.59	0.747522	0.373761	−	0.927525i	$-0.378068\pi$
$479$	7836.59	0.747522	0.373761	+	0.927525i	$0.378068\pi$
$480$	0	0
$481$	−4158.23	−0.394177
$482$	0	0
$483$	0	0
$484$	0	0
$485$	16408.4	1.53622
$486$	0	0
$487$	−8799.54	−0.818779	−0.409390	−	0.912360i	$-0.634258\pi$
$487$	−8799.54	−0.818779	−0.409390	+	0.912360i	$0.634258\pi$
$488$	0	0
$489$	−7407.75	−0.685051
$490$	0	0
$491$	11382.9	1.04623	0.523117	−	0.852261i	$-0.324769\pi$
$491$	11382.9	1.04623	0.523117	+	0.852261i	$0.324769\pi$
$492$	0	0
$493$	−1063.96	−0.0971974
$494$	0	0
$495$	1139.32	0.103451
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−17510.7	−1.57092	−0.785458	−	0.618915i	$-0.787573\pi$
$499$	−17510.7	−1.57092	−0.785458	+	0.618915i	$0.787573\pi$
$500$	0	0
$501$	−7257.10	−0.647153
$502$	0	0
$503$	14573.9	1.29188	0.645940	−	0.763388i	$-0.276465\pi$
$503$	14573.9	1.29188	0.645940	+	0.763388i	$0.276465\pi$
$504$	0	0
$505$	10552.3	0.929845
$506$	0	0
$507$	−5028.91	−0.440516
$508$	0	0
$509$	17451.3	1.51968	0.759839	−	0.650112i	$-0.225278\pi$
$509$	17451.3	1.51968	0.759839	+	0.650112i	$0.225278\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	2026.27	0.174390
$514$	0	0
$515$	4503.06	0.385298
$516$	0	0
$517$	−1556.35	−0.132395
$518$	0	0
$519$	10677.5	0.903063
$520$	0	0
$521$	−3210.52	−0.269972	−0.134986	−	0.990848i	$-0.543099\pi$
$521$	−3210.52	−0.269972	−0.134986	+	0.990848i	$0.543099\pi$
$522$	0	0
$523$	16647.8	1.39188	0.695942	−	0.718098i	$-0.254987\pi$
$523$	16647.8	1.39188	0.695942	+	0.718098i	$0.254987\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−893.020	−0.0738151
$528$	0	0
$529$	19720.4	1.62081
$530$	0	0
$531$	281.477	0.0230039
$532$	0	0
$533$	−11140.3	−0.905330
$534$	0	0
$535$	39283.2	3.17451
$536$	0	0
$537$	−11592.5	−0.931568
$538$	0	0
$539$	0	0
$540$	0	0
$541$	10709.9	0.851118	0.425559	−	0.904931i	$-0.360078\pi$
$541$	10709.9	0.851118	0.425559	+	0.904931i	$0.360078\pi$
$542$	0	0
$543$	12301.4	0.972196
$544$	0	0
$545$	17608.4	1.38397
$546$	0	0
$547$	−1263.05	−0.0987277	−0.0493638	−	0.998781i	$-0.515719\pi$
$547$	−1263.05	−0.0987277	−0.0493638	+	0.998781i	$0.515719\pi$
$548$	0	0
$549$	−5287.53	−0.411050
$550$	0	0
$551$	−9172.86	−0.709214
$552$	0	0
$553$	0	0
$554$	0	0
$555$	11318.9	0.865698
$556$	0	0
$557$	−24281.3	−1.84710	−0.923548	−	0.383483i	$-0.874725\pi$
$557$	−24281.3	−1.84710	−0.923548	+	0.383483i	$0.874725\pi$
$558$	0	0
$559$	−8853.69	−0.669895
$560$	0	0
$561$	159.664	0.0120161
$562$	0	0
$563$	5467.57	0.409291	0.204645	−	0.978836i	$-0.434396\pi$
$563$	5467.57	0.409291	0.204645	+	0.978836i	$0.434396\pi$
$564$	0	0
$565$	31625.9	2.35489
$566$	0	0
$567$	0	0
$568$	0	0
$569$	6474.08	0.476991	0.238495	−	0.971144i	$-0.423346\pi$
$569$	6474.08	0.476991	0.238495	+	0.971144i	$0.423346\pi$
$570$	0	0
$571$	−18112.8	−1.32749	−0.663747	−	0.747957i	$-0.731035\pi$
$571$	−18112.8	−1.32749	−0.663747	+	0.747957i	$0.731035\pi$
$572$	0	0
$573$	−3360.50	−0.245004
$574$	0	0
$575$	−54229.1	−3.93306
$576$	0	0
$577$	10175.8	0.734181	0.367090	−	0.930185i	$-0.380354\pi$
$577$	10175.8	0.734181	0.367090	+	0.930185i	$0.380354\pi$
$578$	0	0
$579$	−4255.33	−0.305433
$580$	0	0
$581$	0	0
$582$	0	0
$583$	3731.00	0.265046
$584$	0	0
$585$	−4252.11	−0.300518
$586$	0	0
$587$	−11566.7	−0.813304	−0.406652	−	0.913583i	$-0.633304\pi$
$587$	−11566.7	−0.813304	−0.406652	+	0.913583i	$0.633304\pi$
$588$	0	0
$589$	−7699.12	−0.538602
$590$	0	0
$591$	−13833.0	−0.962799
$592$	0	0
$593$	4736.73	0.328017	0.164009	−	0.986459i	$-0.447557\pi$
$593$	4736.73	0.328017	0.164009	+	0.986459i	$0.447557\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−15149.5	−1.03857
$598$	0	0
$599$	−7774.56	−0.530317	−0.265158	−	0.964205i	$-0.585424\pi$
$599$	−7774.56	−0.530317	−0.265158	+	0.964205i	$0.585424\pi$
$600$	0	0
$601$	−16662.3	−1.13089	−0.565447	−	0.824784i	$-0.691296\pi$
$601$	−16662.3	−1.13089	−0.565447	+	0.824784i	$0.691296\pi$
$602$	0	0
$603$	6206.38	0.419143
$604$	0	0
$605$	−26784.0	−1.79987
$606$	0	0
$607$	26523.4	1.77356	0.886781	−	0.462190i	$-0.152936\pi$
$607$	26523.4	1.77356	0.886781	+	0.462190i	$0.152936\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	5808.53	0.384596
$612$	0	0
$613$	−6963.91	−0.458841	−0.229421	−	0.973327i	$-0.573683\pi$
$613$	−6963.91	−0.458841	−0.229421	+	0.973327i	$0.573683\pi$
$614$	0	0
$615$	30324.6	1.98830
$616$	0	0
$617$	−22945.5	−1.49716	−0.748582	−	0.663042i	$-0.769265\pi$
$617$	−22945.5	−1.49716	−0.748582	+	0.663042i	$0.769265\pi$
$618$	0	0
$619$	−20783.6	−1.34954	−0.674769	−	0.738029i	$-0.735757\pi$
$619$	−20783.6	−1.34954	−0.674769	+	0.738029i	$0.735757\pi$
$620$	0	0
$621$	−4821.40	−0.311556
$622$	0	0
$623$	0	0
$624$	0	0
$625$	38638.8	2.47288
$626$	0	0
$627$	1376.53	0.0876770
$628$	0	0
$629$	1586.24	0.100553
$630$	0	0
$631$	−909.976	−0.0574098	−0.0287049	−	0.999588i	$-0.509138\pi$
$631$	−909.976	−0.0574098	−0.0287049	+	0.999588i	$0.509138\pi$
$632$	0	0
$633$	5136.32	0.322513
$634$	0	0
$635$	−21502.0	−1.34375
$636$	0	0
$637$	0	0
$638$	0	0
$639$	5075.46	0.314213
$640$	0	0
$641$	−25103.9	−1.54687	−0.773437	−	0.633874i	$-0.781464\pi$
$641$	−25103.9	−1.54687	−0.773437	+	0.633874i	$0.781464\pi$
$642$	0	0
$643$	23810.1	1.46031	0.730156	−	0.683281i	$-0.239448\pi$
$643$	23810.1	1.46031	0.730156	+	0.683281i	$0.239448\pi$
$644$	0	0
$645$	24100.3	1.47124
$646$	0	0
$647$	10985.6	0.667527	0.333763	−	0.942657i	$-0.391681\pi$
$647$	10985.6	0.667527	0.333763	+	0.942657i	$0.391681\pi$
$648$	0	0
$649$	191.220	0.0115655
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−24506.4	−1.46862	−0.734309	−	0.678815i	$-0.762494\pi$
$653$	−24506.4	−1.46862	−0.734309	+	0.678815i	$0.762494\pi$
$654$	0	0
$655$	19901.0	1.18717
$656$	0	0
$657$	2477.84	0.147138
$658$	0	0
$659$	−22113.2	−1.30714	−0.653571	−	0.756865i	$-0.726730\pi$
$659$	−22113.2	−1.30714	−0.653571	+	0.756865i	$0.726730\pi$
$660$	0	0
$661$	−23937.9	−1.40859	−0.704294	−	0.709909i	$-0.748736\pi$
$661$	−23937.9	−1.40859	−0.704294	+	0.709909i	$0.748736\pi$
$662$	0	0
$663$	−595.892	−0.0349058
$664$	0	0
$665$	0	0
$666$	0	0
$667$	21826.4	1.26705
$668$	0	0
$669$	485.718	0.0280702
$670$	0	0
$671$	−3592.05	−0.206661
$672$	0	0
$673$	−15407.1	−0.882468	−0.441234	−	0.897392i	$-0.645459\pi$
$673$	−15407.1	−0.882468	−0.441234	+	0.897392i	$0.645459\pi$
$674$	0	0
$675$	8199.48	0.467553
$676$	0	0
$677$	−5145.81	−0.292126	−0.146063	−	0.989275i	$-0.546660\pi$
$677$	−5145.81	−0.292126	−0.146063	+	0.989275i	$0.546660\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−11843.3	−0.666425
$682$	0	0
$683$	−18247.7	−1.02230	−0.511148	−	0.859492i	$-0.670780\pi$
$683$	−18247.7	−1.02230	−0.511148	+	0.859492i	$0.670780\pi$
$684$	0	0
$685$	−34990.9	−1.95173
$686$	0	0
$687$	−10754.3	−0.597240
$688$	0	0
$689$	−13924.7	−0.769939
$690$	0	0
$691$	10359.8	0.570338	0.285169	−	0.958477i	$-0.407950\pi$
$691$	10359.8	0.570338	0.285169	+	0.958477i	$0.407950\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	35866.7	1.95755
$696$	0	0
$697$	4249.70	0.230946
$698$	0	0
$699$	−10346.5	−0.559860
$700$	0	0
$701$	−369.772	−0.0199231	−0.00996155	−	0.999950i	$-0.503171\pi$
$701$	−369.772	−0.0199231	−0.00996155	+	0.999950i	$0.503171\pi$
$702$	0	0
$703$	13675.7	0.733696
$704$	0	0
$705$	−15811.2	−0.844657
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−16229.0	−0.859652	−0.429826	−	0.902912i	$-0.641425\pi$
$709$	−16229.0	−0.859652	−0.429826	+	0.902912i	$0.641425\pi$
$710$	0	0
$711$	1973.84	0.104114
$712$	0	0
$713$	18319.7	0.962239
$714$	0	0
$715$	−2888.65	−0.151090
$716$	0	0
$717$	−11455.7	−0.596683
$718$	0	0
$719$	−27233.9	−1.41259	−0.706294	−	0.707918i	$-0.749635\pi$
$719$	−27233.9	−1.41259	−0.706294	+	0.707918i	$0.749635\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−3943.61	−0.202855
$724$	0	0
$725$	−37118.8	−1.90146
$726$	0	0
$727$	−27653.4	−1.41074	−0.705370	−	0.708840i	$-0.749219\pi$
$727$	−27653.4	−1.41074	−0.705370	+	0.708840i	$0.749219\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	3377.42	0.170887
$732$	0	0
$733$	−20513.2	−1.03366	−0.516831	−	0.856088i	$-0.672888\pi$
$733$	−20513.2	−1.03366	−0.516831	+	0.856088i	$0.672888\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	4216.27	0.210730
$738$	0	0
$739$	5032.87	0.250524	0.125262	−	0.992124i	$-0.460023\pi$
$739$	5032.87	0.250524	0.125262	+	0.992124i	$0.460023\pi$
$740$	0	0
$741$	−5137.45	−0.254695
$742$	0	0
$743$	6528.88	0.322371	0.161185	−	0.986924i	$-0.448468\pi$
$743$	6528.88	0.322371	0.161185	+	0.986924i	$0.448468\pi$
$744$	0	0
$745$	56438.5	2.77550
$746$	0	0
$747$	−13025.5	−0.637988
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−11101.7	−0.539423	−0.269711	−	0.962941i	$-0.586928\pi$
$751$	−11101.7	−0.539423	−0.269711	+	0.962941i	$0.586928\pi$
$752$	0	0
$753$	14489.0	0.701206
$754$	0	0
$755$	−23392.7	−1.12761
$756$	0	0
$757$	−6375.85	−0.306122	−0.153061	−	0.988217i	$-0.548913\pi$
$757$	−6375.85	−0.306122	−0.153061	+	0.988217i	$0.548913\pi$
$758$	0	0
$759$	−3275.39	−0.156639
$760$	0	0
$761$	−4038.93	−0.192393	−0.0961965	−	0.995362i	$-0.530668\pi$
$761$	−4038.93	−0.192393	−0.0961965	+	0.995362i	$0.530668\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	1622.05	0.0766608
$766$	0	0
$767$	−713.662	−0.0335969
$768$	0	0
$769$	18864.4	0.884614	0.442307	−	0.896864i	$-0.354160\pi$
$769$	18864.4	0.884614	0.442307	+	0.896864i	$0.354160\pi$
$770$	0	0
$771$	−4439.78	−0.207386
$772$	0	0
$773$	10489.2	0.488060	0.244030	−	0.969768i	$-0.421530\pi$
$773$	10489.2	0.488060	0.244030	+	0.969768i	$0.421530\pi$
$774$	0	0
$775$	−31155.2	−1.44404
$776$	0	0
$777$	0	0
$778$	0	0
$779$	36638.6	1.68513
$780$	0	0
$781$	3447.98	0.157975
$782$	0	0
$783$	−3300.16	−0.150623
$784$	0	0
$785$	−14901.0	−0.677502
$786$	0	0
$787$	30948.6	1.40178	0.700888	−	0.713272i	$-0.252787\pi$
$787$	30948.6	1.40178	0.700888	+	0.713272i	$0.252787\pi$
$788$	0	0
$789$	3573.04	0.161221
$790$	0	0
$791$	0	0
$792$	0	0
$793$	13406.1	0.600335
$794$	0	0
$795$	37903.8	1.69095
$796$	0	0
$797$	−38324.5	−1.70329	−0.851646	−	0.524117i	$-0.824395\pi$
$797$	−38324.5	−1.70329	−0.851646	+	0.524117i	$0.824395\pi$
$798$	0	0
$799$	−2215.78	−0.0981087
$800$	0	0
$801$	9167.13	0.404375
$802$	0	0
$803$	1683.31	0.0739758
$804$	0	0
$805$	0	0
$806$	0	0
$807$	1676.31	0.0731215
$808$	0	0
$809$	19368.1	0.841714	0.420857	−	0.907127i	$-0.361729\pi$
$809$	19368.1	0.841714	0.420857	+	0.907127i	$0.361729\pi$
$810$	0	0
$811$	5621.93	0.243419	0.121710	−	0.992566i	$-0.461162\pi$
$811$	5621.93	0.243419	0.121710	+	0.992566i	$0.461162\pi$
$812$	0	0
$813$	−17976.9	−0.775496
$814$	0	0
$815$	−51125.1	−2.19734
$816$	0	0
$817$	29118.2	1.24690
$818$	0	0
$819$	0	0
$820$	0	0
$821$	12512.3	0.531891	0.265945	−	0.963988i	$-0.414316\pi$
$821$	12512.3	0.531891	0.265945	+	0.963988i	$0.414316\pi$
$822$	0	0
$823$	−26084.1	−1.10478	−0.552391	−	0.833585i	$-0.686284\pi$
$823$	−26084.1	−1.10478	−0.552391	+	0.833585i	$0.686284\pi$
$824$	0	0
$825$	5570.27	0.235069
$826$	0	0
$827$	22210.4	0.933895	0.466947	−	0.884285i	$-0.345354\pi$
$827$	22210.4	0.933895	0.466947	+	0.884285i	$0.345354\pi$
$828$	0	0
$829$	20945.2	0.877511	0.438755	−	0.898607i	$-0.355419\pi$
$829$	20945.2	0.877511	0.438755	+	0.898607i	$0.355419\pi$
$830$	0	0
$831$	23652.1	0.987343
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−50085.4	−2.07578
$836$	0	0
$837$	−2769.95	−0.114389
$838$	0	0
$839$	−1357.80	−0.0558718	−0.0279359	−	0.999610i	$-0.508893\pi$
$839$	−1357.80	−0.0558718	−0.0279359	+	0.999610i	$0.508893\pi$
$840$	0	0
$841$	−9449.27	−0.387440
$842$	0	0
$843$	−227.477	−0.00929385
$844$	0	0
$845$	−34707.3	−1.41298
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−2410.11	−0.0974259
$850$	0	0
$851$	−32540.6	−1.31078
$852$	0	0
$853$	−24726.4	−0.992516	−0.496258	−	0.868175i	$-0.665293\pi$
$853$	−24726.4	−0.992516	−0.496258	+	0.868175i	$0.665293\pi$
$854$	0	0
$855$	13984.4	0.559365
$856$	0	0
$857$	20843.2	0.830792	0.415396	−	0.909641i	$-0.363643\pi$
$857$	20843.2	0.830792	0.415396	+	0.909641i	$0.363643\pi$
$858$	0	0
$859$	46160.3	1.83349	0.916745	−	0.399473i	$-0.130807\pi$
$859$	46160.3	1.83349	0.916745	+	0.399473i	$0.130807\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−8635.84	−0.340634	−0.170317	−	0.985389i	$-0.554479\pi$
$863$	−8635.84	−0.340634	−0.170317	+	0.985389i	$0.554479\pi$
$864$	0	0
$865$	73691.4	2.89663
$866$	0	0
$867$	−14511.7	−0.568446
$868$	0	0
$869$	1340.92	0.0523446
$870$	0	0
$871$	−15735.8	−0.612155
$872$	0	0
$873$	7132.47	0.276515
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−23282.3	−0.896453	−0.448226	−	0.893920i	$-0.647944\pi$
$877$	−23282.3	−0.896453	−0.448226	+	0.893920i	$0.647944\pi$
$878$	0	0
$879$	−7878.79	−0.302326
$880$	0	0
$881$	7097.62	0.271424	0.135712	−	0.990748i	$-0.456668\pi$
$881$	7097.62	0.271424	0.135712	+	0.990748i	$0.456668\pi$
$882$	0	0
$883$	35486.1	1.35244	0.676219	−	0.736700i	$-0.263617\pi$
$883$	35486.1	1.35244	0.676219	+	0.736700i	$0.263617\pi$
$884$	0	0
$885$	1942.63	0.0737862
$886$	0	0
$887$	45062.5	1.70581	0.852903	−	0.522070i	$-0.174840\pi$
$887$	45062.5	1.70581	0.852903	+	0.522070i	$0.174840\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	495.242	0.0186209
$892$	0	0
$893$	−19103.2	−0.715863
$894$	0	0
$895$	−80006.2	−2.98806
$896$	0	0
$897$	12224.3	0.455025
$898$	0	0
$899$	12539.5	0.465200
$900$	0	0
$901$	5311.85	0.196408
$902$	0	0
$903$	0	0
$904$	0	0
$905$	84898.7	3.11837
$906$	0	0
$907$	−10899.4	−0.399018	−0.199509	−	0.979896i	$-0.563935\pi$
$907$	−10899.4	−0.399018	−0.199509	+	0.979896i	$0.563935\pi$
$908$	0	0
$909$	4586.92	0.167369
$910$	0	0
$911$	10217.7	0.371600	0.185800	−	0.982588i	$-0.440512\pi$
$911$	10217.7	0.371600	0.185800	+	0.982588i	$0.440512\pi$
$912$	0	0
$913$	−8848.78	−0.320758
$914$	0	0
$915$	−36492.2	−1.31847
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−4396.64	−0.157815	−0.0789074	−	0.996882i	$-0.525143\pi$
$919$	−4396.64	−0.157815	−0.0789074	+	0.996882i	$0.525143\pi$
$920$	0	0
$921$	14597.5	0.522263
$922$	0	0
$923$	−12868.4	−0.458905
$924$	0	0
$925$	55339.9	1.96710
$926$	0	0
$927$	1957.41	0.0693525
$928$	0	0
$929$	−42006.3	−1.48351	−0.741756	−	0.670670i	$-0.766007\pi$
$929$	−42006.3	−1.48351	−0.741756	+	0.670670i	$0.766007\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−18458.9	−0.647714
$934$	0	0
$935$	1101.93	0.0385423
$936$	0	0
$937$	39752.5	1.38597	0.692987	−	0.720950i	$-0.256294\pi$
$937$	39752.5	1.38597	0.692987	+	0.720950i	$0.256294\pi$
$938$	0	0
$939$	23678.3	0.822911
$940$	0	0
$941$	−13609.1	−0.471460	−0.235730	−	0.971819i	$-0.575748\pi$
$941$	−13609.1	−0.471460	−0.235730	+	0.971819i	$0.575748\pi$
$942$	0	0
$943$	−87179.6	−3.01056
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−30929.7	−1.06133	−0.530665	−	0.847581i	$-0.678058\pi$
$947$	−30929.7	−1.06133	−0.530665	+	0.847581i	$0.678058\pi$
$948$	0	0
$949$	−6282.37	−0.214894
$950$	0	0
$951$	−11066.3	−0.377337
$952$	0	0
$953$	−26166.5	−0.889418	−0.444709	−	0.895675i	$-0.646693\pi$
$953$	−26166.5	−0.889418	−0.444709	+	0.895675i	$0.646693\pi$
$954$	0	0
$955$	−23192.7	−0.785863
$956$	0	0
$957$	−2241.95	−0.0757281
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−19266.2	−0.646711
$962$	0	0
$963$	17075.8	0.571401
$964$	0	0
$965$	−29368.4	−0.979693
$966$	0	0
$967$	55883.6	1.85842	0.929212	−	0.369546i	$-0.120487\pi$
$967$	55883.6	1.85842	0.929212	+	0.369546i	$0.120487\pi$
$968$	0	0
$969$	1959.78	0.0649715
$970$	0	0
$971$	−10618.3	−0.350933	−0.175467	−	0.984485i	$-0.556143\pi$
$971$	−10618.3	−0.350933	−0.175467	+	0.984485i	$0.556143\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	−20789.2	−0.682857
$976$	0	0
$977$	11493.1	0.376352	0.188176	−	0.982135i	$-0.439743\pi$
$977$	11493.1	0.376352	0.188176	+	0.982135i	$0.439743\pi$
$978$	0	0
$979$	6227.64	0.203306
$980$	0	0
$981$	7654.11	0.249110
$982$	0	0
$983$	−25414.1	−0.824602	−0.412301	−	0.911048i	$-0.635275\pi$
$983$	−25414.1	−0.824602	−0.412301	+	0.911048i	$0.635275\pi$
$984$	0	0
$985$	−95469.5	−3.08824
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−69285.4	−2.22765
$990$	0	0
$991$	39966.1	1.28109	0.640547	−	0.767919i	$-0.278708\pi$
$991$	39966.1	1.28109	0.640547	+	0.767919i	$0.278708\pi$
$992$	0	0
$993$	5945.40	0.190002
$994$	0	0
$995$	−104555.	−3.33128
$996$	0	0
$997$	−16507.7	−0.524376	−0.262188	−	0.965017i	$-0.584444\pi$
$997$	−16507.7	−0.524376	−0.262188	+	0.965017i	$0.584444\pi$
$998$	0	0
$999$	4920.16	0.155823

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2352.4.a.ce.1.2		2
4.3	odd	2		1176.4.a.s.1.2	✓	2
7.6	odd	2		2352.4.a.bm.1.1		2
28.27	even	2		1176.4.a.t.1.1	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1176.4.a.s.1.2	✓	2	4.3	odd	2
1176.4.a.t.1.1	yes	2	28.27	even	2
2352.4.a.bm.1.1		2	7.6	odd	2
2352.4.a.ce.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 $\F_{q}$
Belyi maps

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

\(f(q)\)	\(=\)	\(q+3.00000 q^{3} +20.7047 q^{5} +9.00000 q^{9} +6.11410 q^{11} -22.8188 q^{13} +62.1141 q^{15} +8.70470 q^{17} +75.0470 q^{19} -178.570 q^{23} +303.685 q^{25} +27.0000 q^{27} -122.228 q^{29} -102.591 q^{31} +18.3423 q^{33} +182.228 q^{37} -68.4564 q^{39} +488.208 q^{41} +388.000 q^{43} +186.342 q^{45} -254.550 q^{47} +26.1141 q^{51} +610.228 q^{53} +126.591 q^{55} +225.141 q^{57} +31.2752 q^{59} -587.503 q^{61} -472.456 q^{65} +689.597 q^{67} -535.711 q^{69} +563.940 q^{71} +275.315 q^{73} +911.054 q^{75} +219.315 q^{79} +81.0000 q^{81} -1447.28 q^{83} +180.228 q^{85} -366.685 q^{87} +1018.57 q^{89} -307.772 q^{93} +1553.83 q^{95} +792.497 q^{97} +55.0269 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 6 q^{3} + 18 q^{5} + 18 q^{9} - 58 q^{11} + 48 q^{13} + 54 q^{15} - 6 q^{17} - 84 q^{19} - 6 q^{23} + 186 q^{25} + 54 q^{27} - 104 q^{29} - 252 q^{31} - 174 q^{33} + 224 q^{37} + 144 q^{39} + 438 q^{41}+ \cdots - 522 q^{99}+O(q^{100}) \) 2 * q + 6 * q^3 + 18 * q^5 + 18 * q^9 - 58 * q^11 + 48 * q^13 + 54 * q^15 - 6 * q^17 - 84 * q^19 - 6 * q^23 + 186 * q^25 + 54 * q^27 - 104 * q^29 - 252 * q^31 - 174 * q^33 + 224 * q^37 + 144 * q^39 + 438 * q^41 + 776 * q^43 + 162 * q^45 + 240 * q^47 - 18 * q^51 + 1080 * q^53 + 300 * q^55 - 252 * q^57 - 312 * q^59 - 660 * q^61 - 664 * q^65 + 396 * q^67 - 18 * q^69 - 66 * q^71 + 972 * q^73 + 558 * q^75 + 860 * q^79 + 162 * q^81 - 2520 * q^83 + 220 * q^85 - 312 * q^87 + 1686 * q^89 - 756 * q^93 + 1984 * q^95 + 2100 * q^97 - 522 * q^99

Introduction

L-functions

Modular forms

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Database

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