LMFDB - Embedded newform 7203.2.a.j.1.10

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [7203,2,Mod(1,7203)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(7203, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("7203.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 7203 = 3 \cdot 7^{4} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7203.a (trivial)

Newform invariants

comment:select newform

sage:traces = [24,-6,-24,32] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(4)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$57.5162445759$
Analytic rank:	$1$
Dimension:	$24$
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.10
Character	$\chi$	$=$	7203.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-0.796145 q^{2} -1.00000 q^{3} -1.36615 q^{4} -2.25441 q^{5} +0.796145 q^{6} +2.67995 q^{8} +1.00000 q^{9} +1.79483 q^{10} +1.03004 q^{11} +1.36615 q^{12} -3.01079 q^{13} +2.25441 q^{15} +0.598684 q^{16} +6.43512 q^{17} -0.796145 q^{18} -5.89578 q^{19} +3.07987 q^{20} -0.820061 q^{22} +2.43884 q^{23} -2.67995 q^{24} +0.0823483 q^{25} +2.39702 q^{26} -1.00000 q^{27} -5.31054 q^{29} -1.79483 q^{30} -5.96355 q^{31} -5.83653 q^{32} -1.03004 q^{33} -5.12328 q^{34} -1.36615 q^{36} -0.364356 q^{37} +4.69390 q^{38} +3.01079 q^{39} -6.04169 q^{40} +7.24996 q^{41} +8.97255 q^{43} -1.40719 q^{44} -2.25441 q^{45} -1.94167 q^{46} -2.92073 q^{47} -0.598684 q^{48} -0.0655611 q^{50} -6.43512 q^{51} +4.11320 q^{52} +1.55667 q^{53} +0.796145 q^{54} -2.32213 q^{55} +5.89578 q^{57} +4.22796 q^{58} -5.85015 q^{59} -3.07987 q^{60} -14.1918 q^{61} +4.74785 q^{62} +3.44935 q^{64} +6.78754 q^{65} +0.820061 q^{66} +9.44651 q^{67} -8.79136 q^{68} -2.43884 q^{69} +11.4001 q^{71} +2.67995 q^{72} +12.4955 q^{73} +0.290080 q^{74} -0.0823483 q^{75} +8.05455 q^{76} -2.39702 q^{78} -12.8912 q^{79} -1.34968 q^{80} +1.00000 q^{81} -5.77202 q^{82} +2.64870 q^{83} -14.5074 q^{85} -7.14344 q^{86} +5.31054 q^{87} +2.76045 q^{88} -1.97536 q^{89} +1.79483 q^{90} -3.33183 q^{92} +5.96355 q^{93} +2.32533 q^{94} +13.2915 q^{95} +5.83653 q^{96} +10.7738 q^{97} +1.03004 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q - 6 q^{2} - 24 q^{3} + 32 q^{4} - 14 q^{5} + 6 q^{6} - 18 q^{8} + 24 q^{9} - 16 q^{10} + 2 q^{11} - 32 q^{12} - 28 q^{13} + 14 q^{15} + 48 q^{16} - 30 q^{17} - 6 q^{18} - 16 q^{19} - 34 q^{20} + 18 q^{22}+ \cdots + 2 q^{99}+O(q^{100}) $ 24 * q - 6 * q^2 - 24 * q^3 + 32 * q^4 - 14 * q^5 + 6 * q^6 - 18 * q^8 + 24 * q^9 - 16 * q^10 + 2 * q^11 - 32 * q^12 - 28 * q^13 + 14 * q^15 + 48 * q^16 - 30 * q^17 - 6 * q^18 - 16 * q^19 - 34 * q^20 + 18 * q^22 + 4 * q^23 + 18 * q^24 + 42 * q^25 + 48 * q^26 - 24 * q^27 + 6 * q^29 + 16 * q^30 - 20 * q^31 - 42 * q^32 - 2 * q^33 - 24 * q^34 + 32 * q^36 + 6 * q^37 + 46 * q^38 + 28 * q^39 - 56 * q^40 - 32 * q^41 + 34 * q^43 + 12 * q^44 - 14 * q^45 + 40 * q^46 - 34 * q^47 - 48 * q^48 - 20 * q^50 + 30 * q^51 - 64 * q^52 + 14 * q^53 + 6 * q^54 + 72 * q^55 + 16 * q^57 + 32 * q^58 - 22 * q^59 + 34 * q^60 - 60 * q^61 + 62 * q^62 + 92 * q^64 + 14 * q^65 - 18 * q^66 + 8 * q^67 - 92 * q^68 - 4 * q^69 + 10 * q^71 - 18 * q^72 + 48 * q^73 - 92 * q^74 - 42 * q^75 - 56 * q^76 - 48 * q^78 + 50 * q^79 - 90 * q^80 + 24 * q^81 - 56 * q^82 - 56 * q^83 + 20 * q^85 - 170 * q^86 - 6 * q^87 - 52 * q^88 - 46 * q^89 - 16 * q^90 - 28 * q^92 + 20 * q^93 - 48 * q^94 + 18 * q^95 + 42 * q^96 - 36 * q^97 + 2 * 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 $

$2$	−0.796145	−0.562959	−0.281480	−	0.959567i	$-0.590825\pi$
$2$	−0.796145	−0.562959	−0.281480	+	0.959567i	$0.590825\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	−1.36615	−0.683077
$5$	−2.25441	−1.00820	−0.504101	−	0.863645i	$-0.668176\pi$
$5$	−2.25441	−1.00820	−0.504101	+	0.863645i	$0.668176\pi$
$6$	0.796145	0.325025
$7$	0	0
$7$	0	0
$8$	2.67995	0.947504
$9$	1.00000	0.333333
$10$	1.79483	0.567576
$11$	1.03004	0.310569	0.155284	−	0.987870i	$-0.450371\pi$
$11$	1.03004	0.310569	0.155284	+	0.987870i	$0.450371\pi$
$12$	1.36615	0.394375
$13$	−3.01079	−0.835042	−0.417521	−	0.908667i	$-0.637101\pi$
$13$	−3.01079	−0.835042	−0.417521	+	0.908667i	$0.637101\pi$
$14$	0	0
$15$	2.25441	0.582085
$16$	0.598684	0.149671
$17$	6.43512	1.56075	0.780373	−	0.625315i	$-0.215029\pi$
$17$	6.43512	1.56075	0.780373	+	0.625315i	$0.215029\pi$
$18$	−0.796145	−0.187653
$19$	−5.89578	−1.35259	−0.676293	−	0.736633i	$-0.736415\pi$
$19$	−5.89578	−1.35259	−0.676293	+	0.736633i	$0.736415\pi$
$20$	3.07987	0.688679
$21$	0	0
$22$	−0.820061	−0.174838
$23$	2.43884	0.508534	0.254267	−	0.967134i	$-0.418166\pi$
$23$	2.43884	0.508534	0.254267	+	0.967134i	$0.418166\pi$
$24$	−2.67995	−0.547042
$25$	0.0823483	0.0164697
$26$	2.39702	0.470095
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−5.31054	−0.986143	−0.493071	−	0.869989i	$-0.664126\pi$
$29$	−5.31054	−0.986143	−0.493071	+	0.869989i	$0.664126\pi$
$30$	−1.79483	−0.327690
$31$	−5.96355	−1.07109	−0.535543	−	0.844508i	$-0.679893\pi$
$31$	−5.96355	−1.07109	−0.535543	+	0.844508i	$0.679893\pi$
$32$	−5.83653	−1.03176
$33$	−1.03004	−0.179307
$34$	−5.12328	−0.878636
$35$	0	0
$36$	−1.36615	−0.227692
$37$	−0.364356	−0.0598997	−0.0299499	−	0.999551i	$-0.509535\pi$
$37$	−0.364356	−0.0598997	−0.0299499	+	0.999551i	$0.509535\pi$
$38$	4.69390	0.761451
$39$	3.01079	0.482112
$40$	−6.04169	−0.955274
$41$	7.24996	1.13225	0.566127	−	0.824318i	$-0.308441\pi$
$41$	7.24996	1.13225	0.566127	+	0.824318i	$0.308441\pi$
$42$	0	0
$43$	8.97255	1.36830	0.684150	−	0.729341i	$-0.260173\pi$
$43$	8.97255	1.36830	0.684150	+	0.729341i	$0.260173\pi$
$44$	−1.40719	−0.212142
$45$	−2.25441	−0.336067
$46$	−1.94167	−0.286284
$47$	−2.92073	−0.426033	−0.213016	−	0.977049i	$-0.568329\pi$
$47$	−2.92073	−0.426033	−0.213016	+	0.977049i	$0.568329\pi$
$48$	−0.598684	−0.0864125
$49$	0	0
$50$	−0.0655611	−0.00927175
$51$	−6.43512	−0.901097
$52$	4.11320	0.570398
$53$	1.55667	0.213825	0.106912	−	0.994268i	$-0.465904\pi$
$53$	1.55667	0.213825	0.106912	+	0.994268i	$0.465904\pi$
$54$	0.796145	0.108342
$55$	−2.32213	−0.313116
$56$	0	0
$57$	5.89578	0.780916
$58$	4.22796	0.555158
$59$	−5.85015	−0.761624	−0.380812	−	0.924652i	$-0.624356\pi$
$59$	−5.85015	−0.761624	−0.380812	+	0.924652i	$0.624356\pi$
$60$	−3.07987	−0.397609
$61$	−14.1918	−1.81707	−0.908535	−	0.417809i	$-0.862798\pi$
$61$	−14.1918	−1.81707	−0.908535	+	0.417809i	$0.862798\pi$
$62$	4.74785	0.602978
$63$	0	0
$64$	3.44935	0.431169
$65$	6.78754	0.841890
$66$	0.820061	0.100942
$67$	9.44651	1.15408	0.577038	−	0.816718i	$-0.304209\pi$
$67$	9.44651	1.15408	0.577038	+	0.816718i	$0.304209\pi$
$68$	−8.79136	−1.06611
$69$	−2.43884	−0.293602
$70$	0	0
$71$	11.4001	1.35295	0.676474	−	0.736467i	$-0.263507\pi$
$71$	11.4001	1.35295	0.676474	+	0.736467i	$0.263507\pi$
$72$	2.67995	0.315835
$73$	12.4955	1.46248	0.731242	−	0.682119i	$-0.238941\pi$
$73$	12.4955	1.46248	0.731242	+	0.682119i	$0.238941\pi$
$74$	0.290080	0.0337211
$75$	−0.0823483	−0.00950876
$76$	8.05455	0.923920
$77$	0	0
$78$	−2.39702	−0.271409
$79$	−12.8912	−1.45037	−0.725184	−	0.688555i	$-0.758245\pi$
$79$	−12.8912	−1.45037	−0.725184	+	0.688555i	$0.758245\pi$
$80$	−1.34968	−0.150898
$81$	1.00000	0.111111
$82$	−5.77202	−0.637413
$83$	2.64870	0.290733	0.145366	−	0.989378i	$-0.453564\pi$
$83$	2.64870	0.290733	0.145366	+	0.989378i	$0.453564\pi$
$84$	0	0
$85$	−14.5074	−1.57355
$86$	−7.14344	−0.770297
$87$	5.31054	0.569350
$88$	2.76045	0.294265
$89$	−1.97536	−0.209388	−0.104694	−	0.994504i	$-0.533386\pi$
$89$	−1.97536	−0.209388	−0.104694	+	0.994504i	$0.533386\pi$
$90$	1.79483	0.189192
$91$	0	0
$92$	−3.33183	−0.347368
$93$	5.96355	0.618392
$94$	2.32533	0.239839
$95$	13.2915	1.36368
$96$	5.83653	0.595688
$97$	10.7738	1.09391	0.546957	−	0.837161i	$-0.315786\pi$
$97$	10.7738	1.09391	0.546957	+	0.837161i	$0.315786\pi$
$98$	0	0
$99$	1.03004	0.103523
$100$	−0.112500	−0.0112500
$101$	−2.88285	−0.286855	−0.143427	−	0.989661i	$-0.545812\pi$
$101$	−2.88285	−0.286855	−0.143427	+	0.989661i	$0.545812\pi$
$102$	5.12328	0.507281
$103$	3.18333	0.313663	0.156831	−	0.987625i	$-0.449872\pi$
$103$	3.18333	0.313663	0.156831	+	0.987625i	$0.449872\pi$
$104$	−8.06874	−0.791205
$105$	0	0
$106$	−1.23933	−0.120374
$107$	11.1987	1.08262	0.541308	−	0.840825i	$-0.317929\pi$
$107$	11.1987	1.08262	0.541308	+	0.840825i	$0.317929\pi$
$108$	1.36615	0.131458
$109$	13.3668	1.28030	0.640152	−	0.768248i	$-0.278871\pi$
$109$	13.3668	1.28030	0.640152	+	0.768248i	$0.278871\pi$
$110$	1.84875	0.176271
$111$	0.364356	0.0345831
$112$	0	0
$113$	9.69666	0.912185	0.456093	−	0.889932i	$-0.349249\pi$
$113$	9.69666	0.912185	0.456093	+	0.889932i	$0.349249\pi$
$114$	−4.69390	−0.439624
$115$	−5.49814	−0.512704
$116$	7.25502	0.673611
$117$	−3.01079	−0.278347
$118$	4.65756	0.428763
$119$	0	0
$120$	6.04169	0.551528
$121$	−9.93902	−0.903547
$122$	11.2987	1.02294
$123$	−7.24996	−0.653707
$124$	8.14713	0.731634
$125$	11.0864	0.991596
$126$	0	0
$127$	−4.37273	−0.388017	−0.194008	−	0.981000i	$-0.562149\pi$
$127$	−4.37273	−0.388017	−0.194008	+	0.981000i	$0.562149\pi$
$128$	8.92687	0.789032
$129$	−8.97255	−0.789989
$130$	−5.40386	−0.473950
$131$	4.82488	0.421551	0.210776	−	0.977534i	$-0.432401\pi$
$131$	4.82488	0.421551	0.210776	+	0.977534i	$0.432401\pi$
$132$	1.40719	0.122480
$133$	0	0
$134$	−7.52079	−0.649697
$135$	2.25441	0.194028
$136$	17.2458	1.47881
$137$	−0.925604	−0.0790797	−0.0395399	−	0.999218i	$-0.512589\pi$
$137$	−0.925604	−0.0790797	−0.0395399	+	0.999218i	$0.512589\pi$
$138$	1.94167	0.165286
$139$	−0.767472	−0.0650961	−0.0325481	−	0.999470i	$-0.510362\pi$
$139$	−0.767472	−0.0650961	−0.0325481	+	0.999470i	$0.510362\pi$
$140$	0	0
$141$	2.92073	0.245970
$142$	−9.07616	−0.761654
$143$	−3.10123	−0.259338
$144$	0.598684	0.0498903
$145$	11.9721	0.994230
$146$	−9.94819	−0.823318
$147$	0	0
$148$	0.497766	0.0409161
$149$	−17.8730	−1.46421	−0.732107	−	0.681190i	$-0.761463\pi$
$149$	−17.8730	−1.46421	−0.732107	+	0.681190i	$0.761463\pi$
$150$	0.0655611	0.00535304
$151$	19.0466	1.54999	0.774993	−	0.631969i	$-0.217753\pi$
$151$	19.0466	1.54999	0.774993	+	0.631969i	$0.217753\pi$
$152$	−15.8004	−1.28158
$153$	6.43512	0.520248
$154$	0	0
$155$	13.4443	1.07987
$156$	−4.11320	−0.329319
$157$	14.7953	1.18079	0.590395	−	0.807114i	$-0.298972\pi$
$157$	14.7953	1.18079	0.590395	+	0.807114i	$0.298972\pi$
$158$	10.2632	0.816498
$159$	−1.55667	−0.123452
$160$	13.1579	1.04022
$161$	0	0
$162$	−0.796145	−0.0625510
$163$	9.53602	0.746919	0.373459	−	0.927647i	$-0.378172\pi$
$163$	9.53602	0.746919	0.373459	+	0.927647i	$0.378172\pi$
$164$	−9.90456	−0.773416
$165$	2.32213	0.180777
$166$	−2.10875	−0.163671
$167$	3.27141	0.253149	0.126575	−	0.991957i	$-0.459602\pi$
$167$	3.27141	0.253149	0.126575	+	0.991957i	$0.459602\pi$
$168$	0	0
$169$	−3.93516	−0.302705
$170$	11.5500	0.885842
$171$	−5.89578	−0.450862
$172$	−12.2579	−0.934654
$173$	−24.6552	−1.87450	−0.937252	−	0.348653i	$-0.886639\pi$
$173$	−24.6552	−1.87450	−0.937252	+	0.348653i	$0.886639\pi$
$174$	−4.22796	−0.320521
$175$	0	0
$176$	0.616668	0.0464831
$177$	5.85015	0.439724
$178$	1.57267	0.117877
$179$	−18.2211	−1.36191	−0.680953	−	0.732327i	$-0.738434\pi$
$179$	−18.2211	−1.36191	−0.680953	+	0.732327i	$0.738434\pi$
$180$	3.07987	0.229560
$181$	−11.6879	−0.868752	−0.434376	−	0.900732i	$-0.643031\pi$
$181$	−11.6879	−0.868752	−0.434376	+	0.900732i	$0.643031\pi$
$182$	0	0
$183$	14.1918	1.04909
$184$	6.53596	0.481838
$185$	0.821406	0.0603910
$186$	−4.74785	−0.348129
$187$	6.62843	0.484719
$188$	3.99017	0.291013
$189$	0	0
$190$	−10.5819	−0.767695
$191$	1.43231	0.103639	0.0518193	−	0.998656i	$-0.483498\pi$
$191$	1.43231	0.103639	0.0518193	+	0.998656i	$0.483498\pi$
$192$	−3.44935	−0.248936
$193$	25.2004	1.81397	0.906983	−	0.421168i	$-0.138380\pi$
$193$	25.2004	1.81397	0.906983	+	0.421168i	$0.138380\pi$
$194$	−8.57750	−0.615829
$195$	−6.78754	−0.486066
$196$	0	0
$197$	−19.7659	−1.40826	−0.704131	−	0.710070i	$-0.748663\pi$
$197$	−19.7659	−1.40826	−0.704131	+	0.710070i	$0.748663\pi$
$198$	−0.820061	−0.0582792
$199$	15.1743	1.07568	0.537840	−	0.843047i	$-0.319240\pi$
$199$	15.1743	1.07568	0.537840	+	0.843047i	$0.319240\pi$
$200$	0.220689	0.0156051
$201$	−9.44651	−0.666306
$202$	2.29517	0.161487
$203$	0	0
$204$	8.79136	0.615518
$205$	−16.3444	−1.14154
$206$	−2.53439	−0.176579
$207$	2.43884	0.169511
$208$	−1.80251	−0.124981
$209$	−6.07289	−0.420071
$210$	0	0
$211$	25.5923	1.76185	0.880924	−	0.473259i	$-0.156922\pi$
$211$	25.5923	1.76185	0.880924	+	0.473259i	$0.156922\pi$
$212$	−2.12664	−0.146059
$213$	−11.4001	−0.781125
$214$	−8.91575	−0.609468
$215$	−20.2278	−1.37952
$216$	−2.67995	−0.182347
$217$	0	0
$218$	−10.6419	−0.720759
$219$	−12.4955	−0.844365
$220$	3.17238	0.213882
$221$	−19.3748	−1.30329
$222$	−0.290080	−0.0194689
$223$	−2.16406	−0.144916	−0.0724580	−	0.997371i	$-0.523084\pi$
$223$	−2.16406	−0.144916	−0.0724580	+	0.997371i	$0.523084\pi$
$224$	0	0
$225$	0.0823483	0.00548989
$226$	−7.71994	−0.513523
$227$	−24.9814	−1.65807	−0.829036	−	0.559194i	$-0.811110\pi$
$227$	−24.9814	−1.65807	−0.829036	+	0.559194i	$0.811110\pi$
$228$	−8.05455	−0.533425
$229$	−6.74951	−0.446020	−0.223010	−	0.974816i	$-0.571588\pi$
$229$	−6.74951	−0.446020	−0.223010	+	0.974816i	$0.571588\pi$
$230$	4.37732	0.288632
$231$	0	0
$232$	−14.2320	−0.934374
$233$	22.0954	1.44752	0.723758	−	0.690054i	$-0.242413\pi$
$233$	22.0954	1.44752	0.723758	+	0.690054i	$0.242413\pi$
$234$	2.39702	0.156698
$235$	6.58452	0.429527
$236$	7.99220	0.520248
$237$	12.8912	0.837371
$238$	0	0
$239$	−14.1469	−0.915086	−0.457543	−	0.889188i	$-0.651270\pi$
$239$	−14.1469	−0.915086	−0.457543	+	0.889188i	$0.651270\pi$
$240$	1.34968	0.0871212
$241$	−11.2165	−0.722517	−0.361259	−	0.932466i	$-0.617653\pi$
$241$	−11.2165	−0.722517	−0.361259	+	0.932466i	$0.617653\pi$
$242$	7.91290	0.508660
$243$	−1.00000	−0.0641500
$244$	19.3881	1.24120
$245$	0	0
$246$	5.77202	0.368010
$247$	17.7509	1.12947
$248$	−15.9820	−1.01486
$249$	−2.64870	−0.167855
$250$	−8.82637	−0.558228
$251$	25.8994	1.63476	0.817379	−	0.576100i	$-0.195426\pi$
$251$	25.8994	1.63476	0.817379	+	0.576100i	$0.195426\pi$
$252$	0	0
$253$	2.51210	0.157935
$254$	3.48132	0.218438
$255$	14.5074	0.908487
$256$	−14.0058	−0.875362
$257$	8.33587	0.519977	0.259989	−	0.965612i	$-0.416281\pi$
$257$	8.33587	0.519977	0.259989	+	0.965612i	$0.416281\pi$
$258$	7.14344	0.444731
$259$	0	0
$260$	−9.27282	−0.575076
$261$	−5.31054	−0.328714
$262$	−3.84130	−0.237316
$263$	−24.5466	−1.51361	−0.756804	−	0.653642i	$-0.773240\pi$
$263$	−24.5466	−1.51361	−0.756804	+	0.653642i	$0.773240\pi$
$264$	−2.76045	−0.169894
$265$	−3.50936	−0.215578
$266$	0	0
$267$	1.97536	0.120890
$268$	−12.9054	−0.788322
$269$	30.1130	1.83602	0.918011	−	0.396555i	$-0.129794\pi$
$269$	30.1130	1.83602	0.918011	+	0.396555i	$0.129794\pi$
$270$	−1.79483	−0.109230
$271$	−16.0901	−0.977405	−0.488702	−	0.872450i	$-0.662530\pi$
$271$	−16.0901	−0.977405	−0.488702	+	0.872450i	$0.662530\pi$
$272$	3.85260	0.233598
$273$	0	0
$274$	0.736915	0.0445187
$275$	0.0848220	0.00511496
$276$	3.33183	0.200553
$277$	−19.9575	−1.19913	−0.599565	−	0.800326i	$-0.704660\pi$
$277$	−19.9575	−1.19913	−0.599565	+	0.800326i	$0.704660\pi$
$278$	0.611019	0.0366465
$279$	−5.96355	−0.357029
$280$	0	0
$281$	−14.6649	−0.874836	−0.437418	−	0.899258i	$-0.644107\pi$
$281$	−14.6649	−0.874836	−0.437418	+	0.899258i	$0.644107\pi$
$282$	−2.32533	−0.138471
$283$	0.883867	0.0525404	0.0262702	−	0.999655i	$-0.491637\pi$
$283$	0.883867	0.0525404	0.0262702	+	0.999655i	$0.491637\pi$
$284$	−15.5743	−0.924167
$285$	−13.2915	−0.787320
$286$	2.46903	0.145997
$287$	0	0
$288$	−5.83653	−0.343921
$289$	24.4107	1.43593
$290$	−9.53154	−0.559711
$291$	−10.7738	−0.631571
$292$	−17.0707	−0.998988
$293$	−11.2204	−0.655503	−0.327751	−	0.944764i	$-0.606291\pi$
$293$	−11.2204	−0.655503	−0.327751	+	0.944764i	$0.606291\pi$
$294$	0	0
$295$	13.1886	0.767871
$296$	−0.976454	−0.0567552
$297$	−1.03004	−0.0597690
$298$	14.2295	0.824293
$299$	−7.34283	−0.424647
$300$	0.112500	0.00649521
$301$	0	0
$302$	−15.1638	−0.872579
$303$	2.88285	0.165616
$304$	−3.52971	−0.202443
$305$	31.9940	1.83197
$306$	−5.12328	−0.292879
$307$	−15.8876	−0.906756	−0.453378	−	0.891318i	$-0.649781\pi$
$307$	−15.8876	−0.906756	−0.453378	+	0.891318i	$0.649781\pi$
$308$	0	0
$309$	−3.18333	−0.181093
$310$	−10.7036	−0.607923
$311$	−23.7475	−1.34660	−0.673299	−	0.739371i	$-0.735123\pi$
$311$	−23.7475	−1.34660	−0.673299	+	0.739371i	$0.735123\pi$
$312$	8.06874	0.456803
$313$	5.80782	0.328278	0.164139	−	0.986437i	$-0.447515\pi$
$313$	5.80782	0.328278	0.164139	+	0.986437i	$0.447515\pi$
$314$	−11.7792	−0.664737
$315$	0	0
$316$	17.6113	0.990713
$317$	12.7048	0.713574	0.356787	−	0.934186i	$-0.383872\pi$
$317$	12.7048	0.713574	0.356787	+	0.934186i	$0.383872\pi$
$318$	1.23933	0.0694982
$319$	−5.47007	−0.306265
$320$	−7.77625	−0.434705
$321$	−11.1987	−0.625048
$322$	0	0
$323$	−37.9401	−2.11104
$324$	−1.36615	−0.0758974
$325$	−0.247933	−0.0137529
$326$	−7.59205	−0.420485
$327$	−13.3668	−0.739184
$328$	19.4295	1.07281
$329$	0	0
$330$	−1.84875	−0.101770
$331$	21.6094	1.18776	0.593880	−	0.804554i	$-0.297596\pi$
$331$	21.6094	1.18776	0.593880	+	0.804554i	$0.297596\pi$
$332$	−3.61853	−0.198593
$333$	−0.364356	−0.0199666
$334$	−2.60452	−0.142513
$335$	−21.2963	−1.16354
$336$	0	0
$337$	−14.5829	−0.794383	−0.397192	−	0.917736i	$-0.630015\pi$
$337$	−14.5829	−0.794383	−0.397192	+	0.917736i	$0.630015\pi$
$338$	3.13296	0.170410
$339$	−9.69666	−0.526650
$340$	19.8193	1.07485
$341$	−6.14270	−0.332646
$342$	4.69390	0.253817
$343$	0	0
$344$	24.0459	1.29647
$345$	5.49814	0.296010
$346$	19.6291	1.05527
$347$	7.19196	0.386085	0.193042	−	0.981190i	$-0.438165\pi$
$347$	7.19196	0.386085	0.193042	+	0.981190i	$0.438165\pi$
$348$	−7.25502	−0.388910
$349$	−8.24964	−0.441593	−0.220797	−	0.975320i	$-0.570866\pi$
$349$	−8.24964	−0.441593	−0.220797	+	0.975320i	$0.570866\pi$
$350$	0	0
$351$	3.01079	0.160704
$352$	−6.01186	−0.320433
$353$	−26.6180	−1.41673	−0.708365	−	0.705846i	$-0.750567\pi$
$353$	−26.6180	−1.41673	−0.708365	+	0.705846i	$0.750567\pi$
$354$	−4.65756	−0.247547
$355$	−25.7005	−1.36404
$356$	2.69865	0.143028
$357$	0	0
$358$	14.5066	0.766698
$359$	35.3254	1.86440	0.932201	−	0.361940i	$-0.117886\pi$
$359$	35.3254	1.86440	0.932201	+	0.361940i	$0.117886\pi$
$360$	−6.04169	−0.318425
$361$	15.7603	0.829488
$362$	9.30522	0.489072
$363$	9.93902	0.521663
$364$	0	0
$365$	−28.1698	−1.47448
$366$	−11.2987	−0.590592
$367$	−12.5622	−0.655742	−0.327871	−	0.944722i	$-0.606331\pi$
$367$	−12.5622	−0.655742	−0.327871	+	0.944722i	$0.606331\pi$
$368$	1.46009	0.0761127
$369$	7.24996	0.377418
$370$	−0.653958	−0.0339977
$371$	0	0
$372$	−8.14713	−0.422409
$373$	18.6294	0.964595	0.482298	−	0.876007i	$-0.339802\pi$
$373$	18.6294	0.964595	0.482298	+	0.876007i	$0.339802\pi$
$374$	−5.27719	−0.272877
$375$	−11.0864	−0.572498
$376$	−7.82740	−0.403668
$377$	15.9889	0.823471
$378$	0	0
$379$	−34.2625	−1.75995	−0.879974	−	0.475021i	$-0.842440\pi$
$379$	−34.2625	−1.75995	−0.879974	+	0.475021i	$0.842440\pi$
$380$	−18.1582	−0.931497
$381$	4.37273	0.224022
$382$	−1.14033	−0.0583443
$383$	7.35822	0.375988	0.187994	−	0.982170i	$-0.439802\pi$
$383$	7.35822	0.375988	0.187994	+	0.982170i	$0.439802\pi$
$384$	−8.92687	−0.455548
$385$	0	0
$386$	−20.0632	−1.02119
$387$	8.97255	0.456100
$388$	−14.7187	−0.747227
$389$	−20.2773	−1.02810	−0.514049	−	0.857761i	$-0.671855\pi$
$389$	−20.2773	−1.02810	−0.514049	+	0.857761i	$0.671855\pi$
$390$	5.40386	0.273635
$391$	15.6942	0.793692
$392$	0	0
$393$	−4.82488	−0.243383
$394$	15.7365	0.792794
$395$	29.0619	1.46226
$396$	−1.40719	−0.0707141
$397$	23.3209	1.17044	0.585221	−	0.810874i	$-0.301008\pi$
$397$	23.3209	1.17044	0.585221	+	0.810874i	$0.301008\pi$
$398$	−12.0810	−0.605564
$399$	0	0
$400$	0.0493006	0.00246503
$401$	−4.33508	−0.216484	−0.108242	−	0.994125i	$-0.534522\pi$
$401$	−4.33508	−0.216484	−0.108242	+	0.994125i	$0.534522\pi$
$402$	7.52079	0.375103
$403$	17.9550	0.894402
$404$	3.93842	0.195944
$405$	−2.25441	−0.112022
$406$	0	0
$407$	−0.375301	−0.0186030
$408$	−17.2458	−0.853793
$409$	−12.1023	−0.598420	−0.299210	−	0.954187i	$-0.596723\pi$
$409$	−12.1023	−0.598420	−0.299210	+	0.954187i	$0.596723\pi$
$410$	13.0125	0.642640
$411$	0.925604	0.0456567
$412$	−4.34892	−0.214256
$413$	0	0
$414$	−1.94167	−0.0954279
$415$	−5.97125	−0.293117
$416$	17.5725	0.861565
$417$	0.767472	0.0375833
$418$	4.83490	0.236483
$419$	−17.0997	−0.835374	−0.417687	−	0.908591i	$-0.637159\pi$
$419$	−17.0997	−0.835374	−0.417687	+	0.908591i	$0.637159\pi$
$420$	0	0
$421$	−16.3056	−0.794687	−0.397344	−	0.917670i	$-0.630068\pi$
$421$	−16.3056	−0.794687	−0.397344	+	0.917670i	$0.630068\pi$
$422$	−20.3752	−0.991848
$423$	−2.92073	−0.142011
$424$	4.17178	0.202600
$425$	0.529921	0.0257049
$426$	9.07616	0.439741
$427$	0	0
$428$	−15.2991	−0.739510
$429$	3.10123	0.149729
$430$	16.1042	0.776615
$431$	10.2779	0.495068	0.247534	−	0.968879i	$-0.420380\pi$
$431$	10.2779	0.495068	0.247534	+	0.968879i	$0.420380\pi$
$432$	−0.598684	−0.0288042
$433$	−19.8322	−0.953076	−0.476538	−	0.879154i	$-0.658109\pi$
$433$	−19.8322	−0.953076	−0.476538	+	0.879154i	$0.658109\pi$
$434$	0	0
$435$	−11.9721	−0.574019
$436$	−18.2611	−0.874546
$437$	−14.3789	−0.687835
$438$	9.94819	0.475343
$439$	−32.6262	−1.55716	−0.778582	−	0.627543i	$-0.784061\pi$
$439$	−32.6262	−1.55716	−0.778582	+	0.627543i	$0.784061\pi$
$440$	−6.22318	−0.296678
$441$	0	0
$442$	15.4251	0.733698
$443$	5.47971	0.260349	0.130174	−	0.991491i	$-0.458446\pi$
$443$	5.47971	0.260349	0.130174	+	0.991491i	$0.458446\pi$
$444$	−0.497766	−0.0236229
$445$	4.45327	0.211105
$446$	1.72290	0.0815819
$447$	17.8730	0.845364
$448$	0	0
$449$	17.3854	0.820469	0.410234	−	0.911980i	$-0.365447\pi$
$449$	17.3854	0.820469	0.410234	+	0.911980i	$0.365447\pi$
$450$	−0.0655611	−0.00309058
$451$	7.46775	0.351643
$452$	−13.2471	−0.623093
$453$	−19.0466	−0.894885
$454$	19.8888	0.933428
$455$	0	0
$456$	15.8004	0.739920
$457$	−8.18572	−0.382912	−0.191456	−	0.981501i	$-0.561321\pi$
$457$	−8.18572	−0.382912	−0.191456	+	0.981501i	$0.561321\pi$
$458$	5.37358	0.251091
$459$	−6.43512	−0.300366
$460$	7.51131	0.350216
$461$	12.7675	0.594642	0.297321	−	0.954778i	$-0.403907\pi$
$461$	12.7675	0.594642	0.297321	+	0.954778i	$0.403907\pi$
$462$	0	0
$463$	26.0124	1.20890	0.604449	−	0.796644i	$-0.293393\pi$
$463$	26.0124	1.20890	0.604449	+	0.796644i	$0.293393\pi$
$464$	−3.17933	−0.147597
$465$	−13.4443	−0.623463
$466$	−17.5911	−0.814892
$467$	−38.4256	−1.77812	−0.889062	−	0.457788i	$-0.848642\pi$
$467$	−38.4256	−1.77812	−0.889062	+	0.457788i	$0.848642\pi$
$468$	4.11320	0.190133
$469$	0	0
$470$	−5.24223	−0.241806
$471$	−14.7953	−0.681730
$472$	−15.6781	−0.721642
$473$	9.24208	0.424951
$474$	−10.2632	−0.471406
$475$	−0.485508	−0.0222766
$476$	0	0
$477$	1.55667	0.0712748
$478$	11.2630	0.515156
$479$	3.22090	0.147167	0.0735834	−	0.997289i	$-0.476556\pi$
$479$	3.22090	0.147167	0.0735834	+	0.997289i	$0.476556\pi$
$480$	−13.1579	−0.600574
$481$	1.09700	0.0500188
$482$	8.92994	0.406748
$483$	0	0
$484$	13.5782	0.617192
$485$	−24.2885	−1.10288
$486$	0.796145	0.0361139
$487$	−40.5598	−1.83794	−0.918970	−	0.394327i	$-0.870978\pi$
$487$	−40.5598	−1.83794	−0.918970	+	0.394327i	$0.870978\pi$
$488$	−38.0332	−1.72168
$489$	−9.53602	−0.431234
$490$	0	0
$491$	−12.8273	−0.578888	−0.289444	−	0.957195i	$-0.593470\pi$
$491$	−12.8273	−0.578888	−0.289444	+	0.957195i	$0.593470\pi$
$492$	9.90456	0.446532
$493$	−34.1740	−1.53912
$494$	−14.1323	−0.635843
$495$	−2.32213	−0.104372
$496$	−3.57028	−0.160310
$497$	0	0
$498$	2.10875	0.0944954
$499$	−30.5986	−1.36978	−0.684891	−	0.728646i	$-0.740150\pi$
$499$	−30.5986	−1.36978	−0.684891	+	0.728646i	$0.740150\pi$
$500$	−15.1457	−0.677337
$501$	−3.27141	−0.146156
$502$	−20.6197	−0.920302
$503$	−13.9565	−0.622291	−0.311145	−	0.950362i	$-0.600713\pi$
$503$	−13.9565	−0.622291	−0.311145	+	0.950362i	$0.600713\pi$
$504$	0	0
$505$	6.49912	0.289207
$506$	−2.00000	−0.0889108
$507$	3.93516	0.174767
$508$	5.97382	0.265045
$509$	−18.1952	−0.806488	−0.403244	−	0.915093i	$-0.632117\pi$
$509$	−18.1952	−0.806488	−0.403244	+	0.915093i	$0.632117\pi$
$510$	−11.5500	−0.511441
$511$	0	0
$512$	−6.70311	−0.296239
$513$	5.89578	0.260305
$514$	−6.63656	−0.292726
$515$	−7.17651	−0.316235
$516$	12.2579	0.539623
$517$	−3.00847	−0.132312
$518$	0	0
$519$	24.6552	1.08225
$520$	18.1902	0.797694
$521$	−24.7653	−1.08499	−0.542493	−	0.840060i	$-0.682520\pi$
$521$	−24.7653	−1.08499	−0.542493	+	0.840060i	$0.682520\pi$
$522$	4.22796	0.185053
$523$	−16.0281	−0.700862	−0.350431	−	0.936589i	$-0.613965\pi$
$523$	−16.0281	−0.700862	−0.350431	+	0.936589i	$0.613965\pi$
$524$	−6.59152	−0.287952
$525$	0	0
$526$	19.5426	0.852100
$527$	−38.3762	−1.67169
$528$	−0.616668	−0.0268370
$529$	−17.0520	−0.741393
$530$	2.79396	0.121362
$531$	−5.85015	−0.253875
$532$	0	0
$533$	−21.8281	−0.945480
$534$	−1.57267	−0.0680562
$535$	−25.2463	−1.09149
$536$	25.3161	1.09349
$537$	18.2211	0.786297
$538$	−23.9743	−1.03361
$539$	0	0
$540$	−3.07987	−0.132536
$541$	−31.4594	−1.35254	−0.676272	−	0.736652i	$-0.736406\pi$
$541$	−31.4594	−1.35254	−0.676272	+	0.736652i	$0.736406\pi$
$542$	12.8101	0.550239
$543$	11.6879	0.501574
$544$	−37.5588	−1.61032
$545$	−30.1341	−1.29080
$546$	0	0
$547$	34.3727	1.46967	0.734836	−	0.678245i	$-0.237259\pi$
$547$	34.3727	1.46967	0.734836	+	0.678245i	$0.237259\pi$
$548$	1.26452	0.0540175
$549$	−14.1918	−0.605690
$550$	−0.0675306	−0.00287951
$551$	31.3098	1.33384
$552$	−6.53596	−0.278189
$553$	0	0
$554$	15.8891	0.675062
$555$	−0.821406	−0.0348668
$556$	1.04848	0.0444656
$557$	−36.3491	−1.54016	−0.770080	−	0.637947i	$-0.779784\pi$
$557$	−36.3491	−1.54016	−0.770080	+	0.637947i	$0.779784\pi$
$558$	4.74785	0.200993
$559$	−27.0144	−1.14259
$560$	0	0
$561$	−6.62843	−0.279852
$562$	11.6754	0.492497
$563$	30.4008	1.28124	0.640620	−	0.767858i	$-0.278677\pi$
$563$	30.4008	1.28124	0.640620	+	0.767858i	$0.278677\pi$
$564$	−3.99017	−0.168016
$565$	−21.8602	−0.919666
$566$	−0.703686	−0.0295781
$567$	0	0
$568$	30.5517	1.28192
$569$	22.2601	0.933191	0.466596	−	0.884471i	$-0.345480\pi$
$569$	22.2601	0.933191	0.466596	+	0.884471i	$0.345480\pi$
$570$	10.5819	0.443229
$571$	5.07223	0.212266	0.106133	−	0.994352i	$-0.466153\pi$
$571$	5.07223	0.212266	0.106133	+	0.994352i	$0.466153\pi$
$572$	4.23676	0.177148
$573$	−1.43231	−0.0598358
$574$	0	0
$575$	0.200834	0.00837537
$576$	3.44935	0.143723
$577$	28.9860	1.20670	0.603352	−	0.797475i	$-0.293832\pi$
$577$	28.9860	1.20670	0.603352	+	0.797475i	$0.293832\pi$
$578$	−19.4345	−0.808368
$579$	−25.2004	−1.04729
$580$	−16.3558	−0.679136
$581$	0	0
$582$	8.57750	0.355549
$583$	1.60343	0.0664072
$584$	33.4871	1.38571
$585$	6.78754	0.280630
$586$	8.93306	0.369021
$587$	−9.06939	−0.374334	−0.187167	−	0.982328i	$-0.559931\pi$
$587$	−9.06939	−0.374334	−0.187167	+	0.982328i	$0.559931\pi$
$588$	0	0
$589$	35.1598	1.44874
$590$	−10.5000	−0.432280
$591$	19.7659	0.813061
$592$	−0.218134	−0.00896525
$593$	5.68143	0.233308	0.116654	−	0.993173i	$-0.462783\pi$
$593$	5.68143	0.233308	0.116654	+	0.993173i	$0.462783\pi$
$594$	0.820061	0.0336475
$595$	0	0
$596$	24.4173	1.00017
$597$	−15.1743	−0.621045
$598$	5.84596	0.239059
$599$	−25.0703	−1.02434	−0.512172	−	0.858883i	$-0.671159\pi$
$599$	−25.0703	−1.02434	−0.512172	+	0.858883i	$0.671159\pi$
$600$	−0.220689	−0.00900959
$601$	−1.25109	−0.0510329	−0.0255165	−	0.999674i	$-0.508123\pi$
$601$	−1.25109	−0.0510329	−0.0255165	+	0.999674i	$0.508123\pi$
$602$	0	0
$603$	9.44651	0.384692
$604$	−26.0205	−1.05876
$605$	22.4066	0.910957
$606$	−2.29517	−0.0932348
$607$	20.8842	0.847664	0.423832	−	0.905741i	$-0.360685\pi$
$607$	20.8842	0.847664	0.423832	+	0.905741i	$0.360685\pi$
$608$	34.4109	1.39555
$609$	0	0
$610$	−25.4719	−1.03133
$611$	8.79370	0.355755
$612$	−8.79136	−0.355370
$613$	14.5959	0.589524	0.294762	−	0.955571i	$-0.404760\pi$
$613$	14.5959	0.589524	0.294762	+	0.955571i	$0.404760\pi$
$614$	12.6489	0.510467
$615$	16.3444	0.659068
$616$	0	0
$617$	−1.51555	−0.0610136	−0.0305068	−	0.999535i	$-0.509712\pi$
$617$	−1.51555	−0.0610136	−0.0305068	+	0.999535i	$0.509712\pi$
$618$	2.53439	0.101948
$619$	−2.70353	−0.108664	−0.0543321	−	0.998523i	$-0.517303\pi$
$619$	−2.70353	−0.108664	−0.0543321	+	0.998523i	$0.517303\pi$
$620$	−18.3669	−0.737634
$621$	−2.43884	−0.0978674
$622$	18.9064	0.758079
$623$	0	0
$624$	1.80251	0.0721581
$625$	−25.4050	−1.01620
$626$	−4.62387	−0.184807
$627$	6.07289	0.242528
$628$	−20.2126	−0.806571
$629$	−2.34467	−0.0934883
$630$	0	0
$631$	23.2082	0.923904	0.461952	−	0.886905i	$-0.347149\pi$
$631$	23.2082	0.923904	0.461952	+	0.886905i	$0.347149\pi$
$632$	−34.5476	−1.37423
$633$	−25.5923	−1.01720
$634$	−10.1149	−0.401713
$635$	9.85791	0.391199
$636$	2.12664	0.0843270
$637$	0	0
$638$	4.35497	0.172415
$639$	11.4001	0.450983
$640$	−20.1248	−0.795503
$641$	19.4809	0.769450	0.384725	−	0.923031i	$-0.374296\pi$
$641$	19.4809	0.769450	0.384725	+	0.923031i	$0.374296\pi$
$642$	8.91575	0.351877
$643$	6.33940	0.250001	0.125001	−	0.992157i	$-0.460107\pi$
$643$	6.33940	0.250001	0.125001	+	0.992157i	$0.460107\pi$
$644$	0	0
$645$	20.2278	0.796467
$646$	30.2058	1.18843
$647$	1.87238	0.0736107	0.0368054	−	0.999322i	$-0.488282\pi$
$647$	1.87238	0.0736107	0.0368054	+	0.999322i	$0.488282\pi$
$648$	2.67995	0.105278
$649$	−6.02589	−0.236537
$650$	0.197391	0.00774230
$651$	0	0
$652$	−13.0277	−0.510203
$653$	47.6133	1.86325	0.931626	−	0.363419i	$-0.118391\pi$
$653$	47.6133	1.86325	0.931626	+	0.363419i	$0.118391\pi$
$654$	10.6419	0.416131
$655$	−10.8772	−0.425009
$656$	4.34043	0.169465
$657$	12.4955	0.487494
$658$	0	0
$659$	−1.85049	−0.0720847	−0.0360423	−	0.999350i	$-0.511475\pi$
$659$	−1.85049	−0.0720847	−0.0360423	+	0.999350i	$0.511475\pi$
$660$	−3.17238	−0.123485
$661$	−10.1613	−0.395228	−0.197614	−	0.980280i	$-0.563319\pi$
$661$	−10.1613	−0.395228	−0.197614	+	0.980280i	$0.563319\pi$
$662$	−17.2042	−0.668660
$663$	19.3748	0.752454
$664$	7.09838	0.275470
$665$	0	0
$666$	0.290080	0.0112404
$667$	−12.9516	−0.501487
$668$	−4.46925	−0.172921
$669$	2.16406	0.0836673
$670$	16.9549	0.655026
$671$	−14.6181	−0.564325
$672$	0	0
$673$	26.8581	1.03530	0.517652	−	0.855591i	$-0.326806\pi$
$673$	26.8581	1.03530	0.517652	+	0.855591i	$0.326806\pi$
$674$	11.6101	0.447205
$675$	−0.0823483	−0.00316959
$676$	5.37603	0.206771
$677$	−46.1013	−1.77182	−0.885908	−	0.463861i	$-0.846464\pi$
$677$	−46.1013	−1.77182	−0.885908	+	0.463861i	$0.846464\pi$
$678$	7.71994	0.296483
$679$	0	0
$680$	−38.8790	−1.49094
$681$	24.9814	0.957289
$682$	4.89048	0.187266
$683$	−12.0857	−0.462447	−0.231224	−	0.972901i	$-0.574273\pi$
$683$	−12.0857	−0.462447	−0.231224	+	0.972901i	$0.574273\pi$
$684$	8.05455	0.307973
$685$	2.08669	0.0797283
$686$	0	0
$687$	6.74951	0.257510
$688$	5.37172	0.204795
$689$	−4.68679	−0.178552
$690$	−4.37732	−0.166642
$691$	31.1572	1.18528	0.592638	−	0.805469i	$-0.298086\pi$
$691$	31.1572	1.18528	0.592638	+	0.805469i	$0.298086\pi$
$692$	33.6829	1.28043
$693$	0	0
$694$	−5.72584	−0.217350
$695$	1.73019	0.0656300
$696$	14.2320	0.539461
$697$	46.6544	1.76716
$698$	6.56791	0.248599
$699$	−22.0954	−0.835723
$700$	0	0
$701$	−26.4543	−0.999163	−0.499582	−	0.866267i	$-0.666513\pi$
$701$	−26.4543	−0.999163	−0.499582	+	0.866267i	$0.666513\pi$
$702$	−2.39702	−0.0904698
$703$	2.14816	0.0810195
$704$	3.55297	0.133908
$705$	−6.58452	−0.247987
$706$	21.1917	0.797562
$707$	0	0
$708$	−7.99220	−0.300365
$709$	2.60534	0.0978458	0.0489229	−	0.998803i	$-0.484421\pi$
$709$	2.60534	0.0978458	0.0489229	+	0.998803i	$0.484421\pi$
$710$	20.4614	0.767901
$711$	−12.8912	−0.483456
$712$	−5.29386	−0.198396
$713$	−14.5442	−0.544683
$714$	0	0
$715$	6.99143	0.261465
$716$	24.8928	0.930287
$717$	14.1469	0.528325
$718$	−28.1241	−1.04958
$719$	21.5568	0.803934	0.401967	−	0.915654i	$-0.368327\pi$
$719$	21.5568	0.803934	0.401967	+	0.915654i	$0.368327\pi$
$720$	−1.34968	−0.0502995
$721$	0	0
$722$	−12.5474	−0.466968
$723$	11.2165	0.417145
$724$	15.9674	0.593424
$725$	−0.437314	−0.0162414
$726$	−7.91290	−0.293675
$727$	2.63775	0.0978288	0.0489144	−	0.998803i	$-0.484424\pi$
$727$	2.63775	0.0978288	0.0489144	+	0.998803i	$0.484424\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	22.4273	0.830071
$731$	57.7394	2.13557
$732$	−19.3881	−0.716606
$733$	5.54876	0.204948	0.102474	−	0.994736i	$-0.467324\pi$
$733$	5.54876	0.204948	0.102474	+	0.994736i	$0.467324\pi$
$734$	10.0013	0.369156
$735$	0	0
$736$	−14.2344	−0.524686
$737$	9.73029	0.358420
$738$	−5.77202	−0.212471
$739$	37.5222	1.38028	0.690139	−	0.723677i	$-0.257549\pi$
$739$	37.5222	1.38028	0.690139	+	0.723677i	$0.257549\pi$
$740$	−1.12217	−0.0412517
$741$	−17.7509	−0.652097
$742$	0	0
$743$	−20.5930	−0.755485	−0.377743	−	0.925911i	$-0.623300\pi$
$743$	−20.5930	−0.755485	−0.377743	+	0.925911i	$0.623300\pi$
$744$	15.9820	0.585928
$745$	40.2930	1.47622
$746$	−14.8317	−0.543028
$747$	2.64870	0.0969110
$748$	−9.05545	−0.331100
$749$	0	0
$750$	8.82637	0.322293
$751$	−12.8974	−0.470632	−0.235316	−	0.971919i	$-0.575613\pi$
$751$	−12.8974	−0.470632	−0.235316	+	0.971919i	$0.575613\pi$
$752$	−1.74859	−0.0637647
$753$	−25.8994	−0.943828
$754$	−12.7295	−0.463581
$755$	−42.9387	−1.56270
$756$	0	0
$757$	41.4852	1.50781	0.753903	−	0.656986i	$-0.228169\pi$
$757$	41.4852	1.50781	0.753903	+	0.656986i	$0.228169\pi$
$758$	27.2779	0.990779
$759$	−2.51210	−0.0911836
$760$	35.6205	1.29209
$761$	2.76709	0.100307	0.0501534	−	0.998742i	$-0.484029\pi$
$761$	2.76709	0.100307	0.0501534	+	0.998742i	$0.484029\pi$
$762$	−3.48132	−0.126115
$763$	0	0
$764$	−1.95676	−0.0707931
$765$	−14.5074	−0.524515
$766$	−5.85821	−0.211666
$767$	17.6135	0.635988
$768$	14.0058	0.505390
$769$	−10.4907	−0.378306	−0.189153	−	0.981948i	$-0.560574\pi$
$769$	−10.4907	−0.378306	−0.189153	+	0.981948i	$0.560574\pi$
$770$	0	0
$771$	−8.33587	−0.300209
$772$	−34.4276	−1.23908
$773$	−31.6774	−1.13936	−0.569678	−	0.821868i	$-0.692932\pi$
$773$	−31.6774	−1.13936	−0.569678	+	0.821868i	$0.692932\pi$
$774$	−7.14344	−0.256766
$775$	−0.491088	−0.0176404
$776$	28.8732	1.03649
$777$	0	0
$778$	16.1436	0.578778
$779$	−42.7442	−1.53147
$780$	9.27282	0.332020
$781$	11.7426	0.420183
$782$	−12.4949	−0.446816
$783$	5.31054	0.189783
$784$	0	0
$785$	−33.3545	−1.19047
$786$	3.84130	0.137015
$787$	−43.4596	−1.54917	−0.774583	−	0.632472i	$-0.782040\pi$
$787$	−43.4596	−1.54917	−0.774583	+	0.632472i	$0.782040\pi$
$788$	27.0033	0.961951
$789$	24.5466	0.873882
$790$	−23.1375	−0.823195
$791$	0	0
$792$	2.76045	0.0980883
$793$	42.7284	1.51733
$794$	−18.5668	−0.658911
$795$	3.50936	0.124464
$796$	−20.7305	−0.734773
$797$	−3.67444	−0.130155	−0.0650777	−	0.997880i	$-0.520730\pi$
$797$	−3.67444	−0.130155	−0.0650777	+	0.997880i	$0.520730\pi$
$798$	0	0
$799$	−18.7953	−0.664929
$800$	−0.480628	−0.0169928
$801$	−1.97536	−0.0697960
$802$	3.45135	0.121871
$803$	12.8708	0.454201
$804$	12.9054	0.455138
$805$	0	0
$806$	−14.2948	−0.503512
$807$	−30.1130	−1.06003
$808$	−7.72589	−0.271796
$809$	9.28813	0.326553	0.163277	−	0.986580i	$-0.447794\pi$
$809$	9.28813	0.326553	0.163277	+	0.986580i	$0.447794\pi$
$810$	1.79483	0.0630640
$811$	−49.0017	−1.72068	−0.860341	−	0.509718i	$-0.829750\pi$
$811$	−49.0017	−1.72068	−0.860341	+	0.509718i	$0.829750\pi$
$812$	0	0
$813$	16.0901	0.564305
$814$	0.298794	0.0104727
$815$	−21.4981	−0.753044
$816$	−3.85260	−0.134868
$817$	−52.9002	−1.85074
$818$	9.63517	0.336886
$819$	0	0
$820$	22.3289	0.779759
$821$	−46.0214	−1.60616	−0.803079	−	0.595873i	$-0.796806\pi$
$821$	−46.0214	−1.60616	−0.803079	+	0.595873i	$0.796806\pi$
$822$	−0.736915	−0.0257029
$823$	−16.1078	−0.561484	−0.280742	−	0.959783i	$-0.590580\pi$
$823$	−16.1078	−0.561484	−0.280742	+	0.959783i	$0.590580\pi$
$824$	8.53114	0.297196
$825$	−0.0848220	−0.00295312
$826$	0	0
$827$	19.0531	0.662542	0.331271	−	0.943536i	$-0.392522\pi$
$827$	19.0531	0.662542	0.331271	+	0.943536i	$0.392522\pi$
$828$	−3.33183	−0.115789
$829$	−15.6696	−0.544228	−0.272114	−	0.962265i	$-0.587723\pi$
$829$	−15.6696	−0.544228	−0.272114	+	0.962265i	$0.587723\pi$
$830$	4.75398	0.165013
$831$	19.9575	0.692318
$832$	−10.3853	−0.360044
$833$	0	0
$834$	−0.611019	−0.0211578
$835$	−7.37509	−0.255226
$836$	8.29650	0.286941
$837$	5.96355	0.206131
$838$	13.6138	0.470282
$839$	−15.3768	−0.530866	−0.265433	−	0.964129i	$-0.585515\pi$
$839$	−15.3768	−0.530866	−0.265433	+	0.964129i	$0.585515\pi$
$840$	0	0
$841$	−0.798146	−0.0275223
$842$	12.9816	0.447376
$843$	14.6649	0.505087
$844$	−34.9630	−1.20348
$845$	8.87145	0.305187
$846$	2.32533	0.0799463
$847$	0	0
$848$	0.931950	0.0320033
$849$	−0.883867	−0.0303342
$850$	−0.421894	−0.0144708
$851$	−0.888607	−0.0304610
$852$	15.5743	0.533568
$853$	−35.5627	−1.21764	−0.608822	−	0.793307i	$-0.708357\pi$
$853$	−35.5627	−1.21764	−0.608822	+	0.793307i	$0.708357\pi$
$854$	0	0
$855$	13.2915	0.454559
$856$	30.0118	1.02578
$857$	12.2667	0.419023	0.209512	−	0.977806i	$-0.432813\pi$
$857$	12.2667	0.419023	0.209512	+	0.977806i	$0.432813\pi$
$858$	−2.46903	−0.0842912
$859$	39.9500	1.36308	0.681539	−	0.731782i	$-0.261311\pi$
$859$	39.9500	1.36308	0.681539	+	0.731782i	$0.261311\pi$
$860$	27.6342	0.942320
$861$	0	0
$862$	−8.18268	−0.278703
$863$	−24.7316	−0.841872	−0.420936	−	0.907090i	$-0.638298\pi$
$863$	−24.7316	−0.841872	−0.420936	+	0.907090i	$0.638298\pi$
$864$	5.83653	0.198563
$865$	55.5829	1.88988
$866$	15.7893	0.536543
$867$	−24.4107	−0.829032
$868$	0	0
$869$	−13.2784	−0.450439
$870$	9.53154	0.323149
$871$	−28.4414	−0.963701
$872$	35.8222	1.21309
$873$	10.7738	0.364638
$874$	11.4477	0.387223
$875$	0	0
$876$	17.0707	0.576766
$877$	24.4944	0.827117	0.413558	−	0.910478i	$-0.364286\pi$
$877$	24.4944	0.827117	0.413558	+	0.910478i	$0.364286\pi$
$878$	25.9752	0.876620
$879$	11.2204	0.378455
$880$	−1.39022	−0.0468643
$881$	56.3191	1.89744	0.948720	−	0.316118i	$-0.102379\pi$
$881$	56.3191	1.89744	0.948720	+	0.316118i	$0.102379\pi$
$882$	0	0
$883$	−22.5309	−0.758224	−0.379112	−	0.925351i	$-0.623770\pi$
$883$	−22.5309	−0.758224	−0.379112	+	0.925351i	$0.623770\pi$
$884$	26.4689	0.890246
$885$	−13.1886	−0.443330
$886$	−4.36264	−0.146566
$887$	6.83879	0.229624	0.114812	−	0.993387i	$-0.463373\pi$
$887$	6.83879	0.229624	0.114812	+	0.993387i	$0.463373\pi$
$888$	0.976454	0.0327676
$889$	0	0
$890$	−3.54544	−0.118844
$891$	1.03004	0.0345076
$892$	2.95644	0.0989888
$893$	17.2200	0.576246
$894$	−14.2295	−0.475906
$895$	41.0777	1.37308
$896$	0	0
$897$	7.34283	0.245170
$898$	−13.8413	−0.461890
$899$	31.6697	1.05624
$900$	−0.112500	−0.00375001
$901$	10.0173	0.333726
$902$	−5.94541	−0.197960
$903$	0	0
$904$	25.9865	0.864299
$905$	26.3492	0.875876
$906$	15.1638	0.503784
$907$	−28.3167	−0.940239	−0.470120	−	0.882603i	$-0.655789\pi$
$907$	−28.3167	−0.940239	−0.470120	+	0.882603i	$0.655789\pi$
$908$	34.1284	1.13259
$909$	−2.88285	−0.0956182
$910$	0	0
$911$	14.6206	0.484402	0.242201	−	0.970226i	$-0.422131\pi$
$911$	14.6206	0.484402	0.242201	+	0.970226i	$0.422131\pi$
$912$	3.52971	0.116880
$913$	2.72827	0.0902925
$914$	6.51702	0.215564
$915$	−31.9940	−1.05769
$916$	9.22087	0.304666
$917$	0	0
$918$	5.12328	0.169094
$919$	27.0213	0.891349	0.445674	−	0.895195i	$-0.352964\pi$
$919$	27.0213	0.891349	0.445674	+	0.895195i	$0.352964\pi$
$920$	−14.7347	−0.485789
$921$	15.8876	0.523516
$922$	−10.1648	−0.334759
$923$	−34.3234	−1.12977
$924$	0	0
$925$	−0.0300041	−0.000986528 0
$926$	−20.7096	−0.680560
$927$	3.18333	0.104554
$928$	30.9951	1.01747
$929$	−32.8412	−1.07749	−0.538743	−	0.842470i	$-0.681101\pi$
$929$	−32.8412	−1.07749	−0.538743	+	0.842470i	$0.681101\pi$
$930$	10.7036	0.350984
$931$	0	0
$932$	−30.1857	−0.988764
$933$	23.7475	0.777458
$934$	30.5923	1.00101
$935$	−14.9432	−0.488694
$936$	−8.06874	−0.263735
$937$	−56.2631	−1.83803	−0.919017	−	0.394218i	$-0.871015\pi$
$937$	−56.2631	−1.83803	−0.919017	+	0.394218i	$0.871015\pi$
$938$	0	0
$939$	−5.80782	−0.189531
$940$	−8.99546	−0.293400
$941$	−27.9290	−0.910460	−0.455230	−	0.890374i	$-0.650443\pi$
$941$	−27.9290	−0.910460	−0.455230	+	0.890374i	$0.650443\pi$
$942$	11.7792	0.383786
$943$	17.6815	0.575789
$944$	−3.50239	−0.113993
$945$	0	0
$946$	−7.35803	−0.239230
$947$	−36.0960	−1.17296	−0.586481	−	0.809963i	$-0.699487\pi$
$947$	−36.0960	−1.17296	−0.586481	+	0.809963i	$0.699487\pi$
$948$	−17.6113	−0.571989
$949$	−37.6212	−1.22123
$950$	0.386534	0.0125408
$951$	−12.7048	−0.411982
$952$	0	0
$953$	−5.49666	−0.178054	−0.0890271	−	0.996029i	$-0.528376\pi$
$953$	−5.49666	−0.178054	−0.0890271	+	0.996029i	$0.528376\pi$
$954$	−1.23933	−0.0401248
$955$	−3.22902	−0.104489
$956$	19.3268	0.625074
$957$	5.47007	0.176822
$958$	−2.56431	−0.0828490
$959$	0	0
$960$	7.77625	0.250977
$961$	4.56397	0.147225
$962$	−0.873369	−0.0281586
$963$	11.1987	0.360872
$964$	15.3234	0.493535
$965$	−56.8120	−1.82884
$966$	0	0
$967$	2.15529	0.0693093	0.0346547	−	0.999399i	$-0.488967\pi$
$967$	2.15529	0.0693093	0.0346547	+	0.999399i	$0.488967\pi$
$968$	−26.6360	−0.856114
$969$	37.9401	1.21881
$970$	19.3372	0.620879
$971$	49.0664	1.57462	0.787308	−	0.616560i	$-0.211474\pi$
$971$	49.0664	1.57462	0.787308	+	0.616560i	$0.211474\pi$
$972$	1.36615	0.0438194
$973$	0	0
$974$	32.2915	1.03469
$975$	0.247933	0.00794022
$976$	−8.49638	−0.271962
$977$	−45.8282	−1.46618	−0.733088	−	0.680134i	$-0.761922\pi$
$977$	−45.8282	−1.46618	−0.733088	+	0.680134i	$0.761922\pi$
$978$	7.59205	0.242767
$979$	−2.03470	−0.0650293
$980$	0	0
$981$	13.3668	0.426768
$982$	10.2124	0.325890
$983$	9.93572	0.316900	0.158450	−	0.987367i	$-0.449350\pi$
$983$	9.93572	0.316900	0.158450	+	0.987367i	$0.449350\pi$
$984$	−19.4295	−0.619390
$985$	44.5604	1.41981
$986$	27.2074	0.866461
$987$	0	0
$988$	−24.2505	−0.771512
$989$	21.8826	0.695827
$990$	1.84875	0.0587571
$991$	15.4631	0.491202	0.245601	−	0.969371i	$-0.421015\pi$
$991$	15.4631	0.491202	0.245601	+	0.969371i	$0.421015\pi$
$992$	34.8065	1.10511
$993$	−21.6094	−0.685753
$994$	0	0
$995$	−34.2091	−1.08450
$996$	3.61853	0.114658
$997$	9.13783	0.289398	0.144699	−	0.989476i	$-0.453779\pi$
$997$	9.13783	0.289398	0.144699	+	0.989476i	$0.453779\pi$
$998$	24.3609	0.771131
$999$	0.364356	0.0115277

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7203.2.a.j.1.10	✓	24
7.6	odd	2		7203.2.a.l.1.10	yes	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
7203.2.a.j.1.10	✓	24	1.1	even	1	trivial
7203.2.a.l.1.10	yes	24	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 7203 = 3 \cdot 7^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7203.a (trivial)

\(f(q)\)	\(=\)	\(q-0.796145 q^{2} -1.00000 q^{3} -1.36615 q^{4} -2.25441 q^{5} +0.796145 q^{6} +2.67995 q^{8} +1.00000 q^{9} +1.79483 q^{10} +1.03004 q^{11} +1.36615 q^{12} -3.01079 q^{13} +2.25441 q^{15} +0.598684 q^{16} +6.43512 q^{17} -0.796145 q^{18} -5.89578 q^{19} +3.07987 q^{20} -0.820061 q^{22} +2.43884 q^{23} -2.67995 q^{24} +0.0823483 q^{25} +2.39702 q^{26} -1.00000 q^{27} -5.31054 q^{29} -1.79483 q^{30} -5.96355 q^{31} -5.83653 q^{32} -1.03004 q^{33} -5.12328 q^{34} -1.36615 q^{36} -0.364356 q^{37} +4.69390 q^{38} +3.01079 q^{39} -6.04169 q^{40} +7.24996 q^{41} +8.97255 q^{43} -1.40719 q^{44} -2.25441 q^{45} -1.94167 q^{46} -2.92073 q^{47} -0.598684 q^{48} -0.0655611 q^{50} -6.43512 q^{51} +4.11320 q^{52} +1.55667 q^{53} +0.796145 q^{54} -2.32213 q^{55} +5.89578 q^{57} +4.22796 q^{58} -5.85015 q^{59} -3.07987 q^{60} -14.1918 q^{61} +4.74785 q^{62} +3.44935 q^{64} +6.78754 q^{65} +0.820061 q^{66} +9.44651 q^{67} -8.79136 q^{68} -2.43884 q^{69} +11.4001 q^{71} +2.67995 q^{72} +12.4955 q^{73} +0.290080 q^{74} -0.0823483 q^{75} +8.05455 q^{76} -2.39702 q^{78} -12.8912 q^{79} -1.34968 q^{80} +1.00000 q^{81} -5.77202 q^{82} +2.64870 q^{83} -14.5074 q^{85} -7.14344 q^{86} +5.31054 q^{87} +2.76045 q^{88} -1.97536 q^{89} +1.79483 q^{90} -3.33183 q^{92} +5.96355 q^{93} +2.32533 q^{94} +13.2915 q^{95} +5.83653 q^{96} +10.7738 q^{97} +1.03004 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 6 q^{2} - 24 q^{3} + 32 q^{4} - 14 q^{5} + 6 q^{6} - 18 q^{8} + 24 q^{9} - 16 q^{10} + 2 q^{11} - 32 q^{12} - 28 q^{13} + 14 q^{15} + 48 q^{16} - 30 q^{17} - 6 q^{18} - 16 q^{19} - 34 q^{20} + 18 q^{22}+ \cdots + 2 q^{99}+O(q^{100}) \) 24 * q - 6 * q^2 - 24 * q^3 + 32 * q^4 - 14 * q^5 + 6 * q^6 - 18 * q^8 + 24 * q^9 - 16 * q^10 + 2 * q^11 - 32 * q^12 - 28 * q^13 + 14 * q^15 + 48 * q^16 - 30 * q^17 - 6 * q^18 - 16 * q^19 - 34 * q^20 + 18 * q^22 + 4 * q^23 + 18 * q^24 + 42 * q^25 + 48 * q^26 - 24 * q^27 + 6 * q^29 + 16 * q^30 - 20 * q^31 - 42 * q^32 - 2 * q^33 - 24 * q^34 + 32 * q^36 + 6 * q^37 + 46 * q^38 + 28 * q^39 - 56 * q^40 - 32 * q^41 + 34 * q^43 + 12 * q^44 - 14 * q^45 + 40 * q^46 - 34 * q^47 - 48 * q^48 - 20 * q^50 + 30 * q^51 - 64 * q^52 + 14 * q^53 + 6 * q^54 + 72 * q^55 + 16 * q^57 + 32 * q^58 - 22 * q^59 + 34 * q^60 - 60 * q^61 + 62 * q^62 + 92 * q^64 + 14 * q^65 - 18 * q^66 + 8 * q^67 - 92 * q^68 - 4 * q^69 + 10 * q^71 - 18 * q^72 + 48 * q^73 - 92 * q^74 - 42 * q^75 - 56 * q^76 - 48 * q^78 + 50 * q^79 - 90 * q^80 + 24 * q^81 - 56 * q^82 - 56 * q^83 + 20 * q^85 - 170 * q^86 - 6 * q^87 - 52 * q^88 - 46 * q^89 - 16 * q^90 - 28 * q^92 + 20 * q^93 - 48 * q^94 + 18 * q^95 + 42 * q^96 - 36 * q^97 + 2 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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