LMFDB - Embedded newform 4923.2.a.l.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4923,2,Mod(1,4923)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4923, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("4923.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4923 = 3^{2} \cdot 547 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4923.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:	$39.3103529151$
Analytic rank:	$0$
Dimension:	$18$
Coefficient field:	$\mathbb{Q}[x]/(x^{18} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{18} - 4 x^{17} - 18 x^{16} + 84 x^{15} + 116 x^{14} - 708 x^{13} - 282 x^{12} + 3104 x^{11} + \cdots + 328 $ x^18 - 4x^17 - 18x^16 + 84x^15 + 116x^14 - 708x^13 - 282x^12 + 3104x^11 - 137x^10 - 7703x^9 + 2068x^8 + 11068x^7 - 4274x^6 - 9021x^5 + 4048x^4 + 3834x^3 - 1851x^2 - 654*x + 328
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 547)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$-0.957552$ of defining polynomial
Character	$\chi$	$=$	4923.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.957552 q^{2} -1.08309 q^{4} +4.10274 q^{5} +4.97202 q^{7} +2.95222 q^{8} +O(q^{10})$	\(q-0.957552 q^{2} -1.08309 q^{4} +4.10274 q^{5} +4.97202 q^{7} +2.95222 q^{8} -3.92859 q^{10} -0.0769657 q^{11} -3.33082 q^{13} -4.76097 q^{14} -0.660720 q^{16} +5.96711 q^{17} -7.28632 q^{19} -4.44365 q^{20} +0.0736987 q^{22} -2.10873 q^{23} +11.8325 q^{25} +3.18943 q^{26} -5.38516 q^{28} -3.24142 q^{29} -6.20019 q^{31} -5.27177 q^{32} -5.71382 q^{34} +20.3989 q^{35} +2.16514 q^{37} +6.97703 q^{38} +12.1122 q^{40} +5.00098 q^{41} -1.06278 q^{43} +0.0833611 q^{44} +2.01922 q^{46} -5.86219 q^{47} +17.7210 q^{49} -11.3302 q^{50} +3.60759 q^{52} +9.01187 q^{53} -0.315771 q^{55} +14.6785 q^{56} +3.10383 q^{58} +4.47532 q^{59} +4.58346 q^{61} +5.93701 q^{62} +6.36944 q^{64} -13.6655 q^{65} +10.6474 q^{67} -6.46294 q^{68} -19.5330 q^{70} -3.35886 q^{71} +14.7261 q^{73} -2.07323 q^{74} +7.89177 q^{76} -0.382675 q^{77} -2.00500 q^{79} -2.71076 q^{80} -4.78870 q^{82} +8.56498 q^{83} +24.4815 q^{85} +1.01767 q^{86} -0.227220 q^{88} +8.84846 q^{89} -16.5609 q^{91} +2.28395 q^{92} +5.61336 q^{94} -29.8939 q^{95} -0.394709 q^{97} -16.9688 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 18 q + 4 q^{2} + 16 q^{4} + 27 q^{5} - 11 q^{7} + 12 q^{8}+O(q^{10}) $ 18 * q + 4 * q^2 + 16 * q^4 + 27 * q^5 - 11 * q^7 + 12 * q^8	$ 18 q + 4 q^{2} + 16 q^{4} + 27 q^{5} - 11 q^{7} + 12 q^{8} - 5 q^{10} - 2 q^{11} - 25 q^{13} + 7 q^{14} + 8 q^{16} + 30 q^{17} + 4 q^{19} + 41 q^{20} - 24 q^{22} + 26 q^{23} + 31 q^{25} + 18 q^{26} - 16 q^{28} + 18 q^{29} - 5 q^{31} + 28 q^{32} + 5 q^{34} + 9 q^{35} - 18 q^{37} + 45 q^{38} + 7 q^{40} + 17 q^{41} + 8 q^{43} - 12 q^{44} + 30 q^{46} + 52 q^{47} + 29 q^{49} - 13 q^{50} - 14 q^{52} + 60 q^{53} + 11 q^{55} - 7 q^{56} + 14 q^{58} + 8 q^{59} - 26 q^{61} - 4 q^{62} + 44 q^{64} + 6 q^{65} + 12 q^{67} + 61 q^{68} + 35 q^{70} + q^{71} - 2 q^{73} - 16 q^{74} + 66 q^{76} + 73 q^{77} + 18 q^{79} + 32 q^{80} + 44 q^{82} + 43 q^{83} + 51 q^{85} - 4 q^{86} - 17 q^{88} + 28 q^{89} - q^{91} + 68 q^{92} + 78 q^{94} + 18 q^{95} - 34 q^{97} - 34 q^{98}+O(q^{100}) $ 18 * q + 4 * q^2 + 16 * q^4 + 27 * q^5 - 11 * q^7 + 12 * q^8 - 5 * q^10 - 2 * q^11 - 25 * q^13 + 7 * q^14 + 8 * q^16 + 30 * q^17 + 4 * q^19 + 41 * q^20 - 24 * q^22 + 26 * q^23 + 31 * q^25 + 18 * q^26 - 16 * q^28 + 18 * q^29 - 5 * q^31 + 28 * q^32 + 5 * q^34 + 9 * q^35 - 18 * q^37 + 45 * q^38 + 7 * q^40 + 17 * q^41 + 8 * q^43 - 12 * q^44 + 30 * q^46 + 52 * q^47 + 29 * q^49 - 13 * q^50 - 14 * q^52 + 60 * q^53 + 11 * q^55 - 7 * q^56 + 14 * q^58 + 8 * q^59 - 26 * q^61 - 4 * q^62 + 44 * q^64 + 6 * q^65 + 12 * q^67 + 61 * q^68 + 35 * q^70 + q^71 - 2 * q^73 - 16 * q^74 + 66 * q^76 + 73 * q^77 + 18 * q^79 + 32 * q^80 + 44 * q^82 + 43 * q^83 + 51 * q^85 - 4 * q^86 - 17 * q^88 + 28 * q^89 - q^91 + 68 * q^92 + 78 * q^94 + 18 * q^95 - 34 * q^97 - 34 * q^98

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 $

$2$	−0.957552	−0.677092	−0.338546	−	0.940950i	$-0.609935\pi$
$2$	−0.957552	−0.677092	−0.338546	+	0.940950i	$0.609935\pi$
$3$	0	0
$3$	0	0
$4$	−1.08309	−0.541547
$5$	4.10274	1.83480	0.917401	−	0.397964i	$-0.130283\pi$
$5$	4.10274	1.83480	0.917401	+	0.397964i	$0.130283\pi$
$6$	0	0
$7$	4.97202	1.87925	0.939623	−	0.342210i	$-0.111176\pi$
$7$	4.97202	1.87925	0.939623	+	0.342210i	$0.111176\pi$
$8$	2.95222	1.04377
$9$	0	0
$10$	−3.92859	−1.24233
$11$	−0.0769657	−0.0232060	−0.0116030	−	0.999933i	$-0.503693\pi$
$11$	−0.0769657	−0.0232060	−0.0116030	+	0.999933i	$0.503693\pi$
$12$	0	0
$13$	−3.33082	−0.923803	−0.461901	−	0.886931i	$-0.652833\pi$
$13$	−3.33082	−0.923803	−0.461901	+	0.886931i	$0.652833\pi$
$14$	−4.76097	−1.27242
$15$	0	0
$16$	−0.660720	−0.165180
$17$	5.96711	1.44724	0.723618	−	0.690200i	$-0.242478\pi$
$17$	5.96711	1.44724	0.723618	+	0.690200i	$0.242478\pi$
$18$	0	0
$19$	−7.28632	−1.67160	−0.835798	−	0.549037i	$-0.814994\pi$
$19$	−7.28632	−1.67160	−0.835798	+	0.549037i	$0.814994\pi$
$20$	−4.44365	−0.993631
$21$	0	0
$22$	0.0736987	0.0157126
$23$	−2.10873	−0.439700	−0.219850	−	0.975534i	$-0.570557\pi$
$23$	−2.10873	−0.439700	−0.219850	+	0.975534i	$0.570557\pi$
$24$	0	0
$25$	11.8325	2.36650
$26$	3.18943	0.625499
$27$	0	0
$28$	−5.38516	−1.01770
$29$	−3.24142	−0.601916	−0.300958	−	0.953637i	$-0.597306\pi$
$29$	−3.24142	−0.601916	−0.300958	+	0.953637i	$0.597306\pi$
$30$	0	0
$31$	−6.20019	−1.11359	−0.556794	−	0.830651i	$-0.687969\pi$
$31$	−6.20019	−1.11359	−0.556794	+	0.830651i	$0.687969\pi$
$32$	−5.27177	−0.931927
$33$	0	0
$34$	−5.71382	−0.979912
$35$	20.3989	3.44805
$36$	0	0
$37$	2.16514	0.355946	0.177973	−	0.984035i	$-0.443046\pi$
$37$	2.16514	0.355946	0.177973	+	0.984035i	$0.443046\pi$
$38$	6.97703	1.13182
$39$	0	0
$40$	12.1122	1.91511
$41$	5.00098	0.781021	0.390511	−	0.920598i	$-0.372298\pi$
$41$	5.00098	0.781021	0.390511	+	0.920598i	$0.372298\pi$
$42$	0	0
$43$	−1.06278	−0.162072	−0.0810362	−	0.996711i	$-0.525823\pi$
$43$	−1.06278	−0.162072	−0.0810362	+	0.996711i	$0.525823\pi$
$44$	0.0833611	0.0125672
$45$	0	0
$46$	2.01922	0.297717
$47$	−5.86219	−0.855089	−0.427544	−	0.903994i	$-0.640621\pi$
$47$	−5.86219	−0.855089	−0.427544	+	0.903994i	$0.640621\pi$
$48$	0	0
$49$	17.7210	2.53157
$50$	−11.3302	−1.60234
$51$	0	0
$52$	3.60759	0.500283
$53$	9.01187	1.23788	0.618938	−	0.785440i	$-0.287563\pi$
$53$	9.01187	1.23788	0.618938	+	0.785440i	$0.287563\pi$
$54$	0	0
$55$	−0.315771	−0.0425785
$56$	14.6785	1.96150
$57$	0	0
$58$	3.10383	0.407552
$59$	4.47532	0.582637	0.291319	−	0.956626i	$-0.405906\pi$
$59$	4.47532	0.582637	0.291319	+	0.956626i	$0.405906\pi$
$60$	0	0
$61$	4.58346	0.586851	0.293426	−	0.955982i	$-0.405205\pi$
$61$	4.58346	0.586851	0.293426	+	0.955982i	$0.405205\pi$
$62$	5.93701	0.754001
$63$	0	0
$64$	6.36944	0.796180
$65$	−13.6655	−1.69500
$66$	0	0
$67$	10.6474	1.30078	0.650391	−	0.759599i	$-0.274605\pi$
$67$	10.6474	1.30078	0.650391	+	0.759599i	$0.274605\pi$
$68$	−6.46294	−0.783747
$69$	0	0
$70$	−19.5330	−2.33464
$71$	−3.35886	−0.398624	−0.199312	−	0.979936i	$-0.563871\pi$
$71$	−3.35886	−0.398624	−0.199312	+	0.979936i	$0.563871\pi$
$72$	0	0
$73$	14.7261	1.72356	0.861779	−	0.507284i	$-0.169351\pi$
$73$	14.7261	1.72356	0.861779	+	0.507284i	$0.169351\pi$
$74$	−2.07323	−0.241008
$75$	0	0
$76$	7.89177	0.905248
$77$	−0.382675	−0.0436099
$78$	0	0
$79$	−2.00500	−0.225580	−0.112790	−	0.993619i	$-0.535979\pi$
$79$	−2.00500	−0.225580	−0.112790	+	0.993619i	$0.535979\pi$
$80$	−2.71076	−0.303072
$81$	0	0
$82$	−4.78870	−0.528823
$83$	8.56498	0.940129	0.470065	−	0.882632i	$-0.344231\pi$
$83$	8.56498	0.940129	0.470065	+	0.882632i	$0.344231\pi$
$84$	0	0
$85$	24.4815	2.65539
$86$	1.01767	0.109738
$87$	0	0
$88$	−0.227220	−0.0242217
$89$	8.84846	0.937935	0.468967	−	0.883215i	$-0.344626\pi$
$89$	8.84846	0.937935	0.468967	+	0.883215i	$0.344626\pi$
$90$	0	0
$91$	−16.5609	−1.73605
$92$	2.28395	0.238118
$93$	0	0
$94$	5.61336	0.578973
$95$	−29.8939	−3.06705
$96$	0	0
$97$	−0.394709	−0.0400766	−0.0200383	−	0.999799i	$-0.506379\pi$
$97$	−0.394709	−0.0400766	−0.0200383	+	0.999799i	$0.506379\pi$
$98$	−16.9688	−1.71410
$99$	0	0
$100$	−12.8157	−1.28157
$101$	5.67033	0.564219	0.282110	−	0.959382i	$-0.408966\pi$
$101$	5.67033	0.564219	0.282110	+	0.959382i	$0.408966\pi$
$102$	0	0
$103$	−12.2307	−1.20513	−0.602565	−	0.798069i	$-0.705855\pi$
$103$	−12.2307	−1.20513	−0.602565	+	0.798069i	$0.705855\pi$
$104$	−9.83332	−0.964236
$105$	0	0
$106$	−8.62934	−0.838155
$107$	8.31411	0.803755	0.401878	−	0.915693i	$-0.368358\pi$
$107$	8.31411	0.803755	0.401878	+	0.915693i	$0.368358\pi$
$108$	0	0
$109$	0.677082	0.0648527	0.0324263	−	0.999474i	$-0.489677\pi$
$109$	0.677082	0.0648527	0.0324263	+	0.999474i	$0.489677\pi$
$110$	0.302367	0.0288295
$111$	0	0
$112$	−3.28511	−0.310414
$113$	8.32143	0.782814	0.391407	−	0.920218i	$-0.371988\pi$
$113$	8.32143	0.782814	0.391407	+	0.920218i	$0.371988\pi$
$114$	0	0
$115$	−8.65156	−0.806763
$116$	3.51076	0.325966
$117$	0	0
$118$	−4.28535	−0.394499
$119$	29.6686	2.71972
$120$	0	0
$121$	−10.9941	−0.999461
$122$	−4.38890	−0.397352
$123$	0	0
$124$	6.71539	0.603060
$125$	28.0319	2.50725
$126$	0	0
$127$	9.98242	0.885796	0.442898	−	0.896572i	$-0.353950\pi$
$127$	9.98242	0.885796	0.442898	+	0.896572i	$0.353950\pi$
$128$	4.44448	0.392840
$129$	0	0
$130$	13.0854	1.14767
$131$	−1.15417	−0.100840	−0.0504201	−	0.998728i	$-0.516056\pi$
$131$	−1.15417	−0.100840	−0.0504201	+	0.998728i	$0.516056\pi$
$132$	0	0
$133$	−36.2277	−3.14134
$134$	−10.1954	−0.880749
$135$	0	0
$136$	17.6162	1.51058
$137$	−5.05259	−0.431672	−0.215836	−	0.976430i	$-0.569248\pi$
$137$	−5.05259	−0.431672	−0.215836	+	0.976430i	$0.569248\pi$
$138$	0	0
$139$	−3.30416	−0.280256	−0.140128	−	0.990133i	$-0.544751\pi$
$139$	−3.30416	−0.280256	−0.140128	+	0.990133i	$0.544751\pi$
$140$	−22.0939	−1.86728
$141$	0	0
$142$	3.21629	0.269905
$143$	0.256359	0.0214378
$144$	0	0
$145$	−13.2987	−1.10440
$146$	−14.1010	−1.16701
$147$	0	0
$148$	−2.34505	−0.192762
$149$	23.7638	1.94680	0.973402	−	0.229103i	$-0.0735792\pi$
$149$	23.7638	1.94680	0.973402	+	0.229103i	$0.0735792\pi$
$150$	0	0
$151$	−5.72927	−0.466242	−0.233121	−	0.972448i	$-0.574894\pi$
$151$	−5.72927	−0.466242	−0.233121	+	0.972448i	$0.574894\pi$
$152$	−21.5108	−1.74476
$153$	0	0
$154$	0.366431	0.0295279
$155$	−25.4378	−2.04321
$156$	0	0
$157$	5.46006	0.435760	0.217880	−	0.975976i	$-0.430086\pi$
$157$	5.46006	0.435760	0.217880	+	0.975976i	$0.430086\pi$
$158$	1.91989	0.152738
$159$	0	0
$160$	−21.6287	−1.70990
$161$	−10.4846	−0.826305
$162$	0	0
$163$	−4.79414	−0.375506	−0.187753	−	0.982216i	$-0.560120\pi$
$163$	−4.79414	−0.375506	−0.187753	+	0.982216i	$0.560120\pi$
$164$	−5.41653	−0.422960
$165$	0	0
$166$	−8.20142	−0.636554
$167$	−8.88965	−0.687902	−0.343951	−	0.938988i	$-0.611765\pi$
$167$	−8.88965	−0.687902	−0.343951	+	0.938988i	$0.611765\pi$
$168$	0	0
$169$	−1.90565	−0.146588
$170$	−23.4423	−1.79794
$171$	0	0
$172$	1.15109	0.0877698
$173$	−18.2314	−1.38611	−0.693055	−	0.720885i	$-0.743736\pi$
$173$	−18.2314	−1.38611	−0.693055	+	0.720885i	$0.743736\pi$
$174$	0	0
$175$	58.8314	4.44723
$176$	0.0508528	0.00383317
$177$	0	0
$178$	−8.47286	−0.635068
$179$	10.4779	0.783152	0.391576	−	0.920146i	$-0.371930\pi$
$179$	10.4779	0.783152	0.391576	+	0.920146i	$0.371930\pi$
$180$	0	0
$181$	−11.8241	−0.878881	−0.439441	−	0.898272i	$-0.644823\pi$
$181$	−11.8241	−0.878881	−0.439441	+	0.898272i	$0.644823\pi$
$182$	15.8579	1.17547
$183$	0	0
$184$	−6.22543	−0.458945
$185$	8.88299	0.653091
$186$	0	0
$187$	−0.459263	−0.0335846
$188$	6.34931	0.463071
$189$	0	0
$190$	28.6249	2.07667
$191$	19.6548	1.42217	0.711087	−	0.703104i	$-0.248203\pi$
$191$	19.6548	1.42217	0.711087	+	0.703104i	$0.248203\pi$
$192$	0	0
$193$	7.35743	0.529600	0.264800	−	0.964303i	$-0.414694\pi$
$193$	7.35743	0.529600	0.264800	+	0.964303i	$0.414694\pi$
$194$	0.377955	0.0271356
$195$	0	0
$196$	−19.1935	−1.37096
$197$	−0.768399	−0.0547462	−0.0273731	−	0.999625i	$-0.508714\pi$
$197$	−0.768399	−0.0547462	−0.0273731	+	0.999625i	$0.508714\pi$
$198$	0	0
$199$	21.6917	1.53768	0.768841	−	0.639440i	$-0.220834\pi$
$199$	21.6917	1.53768	0.768841	+	0.639440i	$0.220834\pi$
$200$	34.9321	2.47008
$201$	0	0
$202$	−5.42964	−0.382028
$203$	−16.1164	−1.13115
$204$	0	0
$205$	20.5177	1.43302
$206$	11.7116	0.815984
$207$	0	0
$208$	2.20074	0.152594
$209$	0.560797	0.0387911
$210$	0	0
$211$	−10.4907	−0.722208	−0.361104	−	0.932526i	$-0.617600\pi$
$211$	−10.4907	−0.722208	−0.361104	+	0.932526i	$0.617600\pi$
$212$	−9.76070	−0.670368
$213$	0	0
$214$	−7.96119	−0.544216
$215$	−4.36031	−0.297371
$216$	0	0
$217$	−30.8275	−2.09271
$218$	−0.648341	−0.0439112
$219$	0	0
$220$	0.342009	0.0230583
$221$	−19.8754	−1.33696
$222$	0	0
$223$	−18.6471	−1.24870	−0.624352	−	0.781143i	$-0.714637\pi$
$223$	−18.6471	−1.24870	−0.624352	+	0.781143i	$0.714637\pi$
$224$	−26.2114	−1.75132
$225$	0	0
$226$	−7.96820	−0.530037
$227$	−27.9582	−1.85565	−0.927826	−	0.373014i	$-0.878324\pi$
$227$	−27.9582	−1.85565	−0.927826	+	0.373014i	$0.878324\pi$
$228$	0	0
$229$	−9.73578	−0.643358	−0.321679	−	0.946849i	$-0.604247\pi$
$229$	−9.73578	−0.643358	−0.321679	+	0.946849i	$0.604247\pi$
$230$	8.28432	0.546252
$231$	0	0
$232$	−9.56939	−0.628261
$233$	−19.0910	−1.25069	−0.625347	−	0.780347i	$-0.715043\pi$
$233$	−19.0910	−1.25069	−0.625347	+	0.780347i	$0.715043\pi$
$234$	0	0
$235$	−24.0511	−1.56892
$236$	−4.84719	−0.315525
$237$	0	0
$238$	−28.4092	−1.84150
$239$	26.9662	1.74430	0.872149	−	0.489241i	$-0.162726\pi$
$239$	26.9662	1.74430	0.872149	+	0.489241i	$0.162726\pi$
$240$	0	0
$241$	−6.54182	−0.421396	−0.210698	−	0.977551i	$-0.567574\pi$
$241$	−6.54182	−0.421396	−0.210698	+	0.977551i	$0.567574\pi$
$242$	10.5274	0.676727
$243$	0	0
$244$	−4.96431	−0.317808
$245$	72.7046	4.64493
$246$	0	0
$247$	24.2694	1.54422
$248$	−18.3044	−1.16233
$249$	0	0
$250$	−26.8420	−1.69764
$251$	22.2725	1.40583	0.702914	−	0.711274i	$-0.251882\pi$
$251$	22.2725	1.40583	0.702914	+	0.711274i	$0.251882\pi$
$252$	0	0
$253$	0.162300	0.0102037
$254$	−9.55868	−0.599765
$255$	0	0
$256$	−16.9947	−1.06217
$257$	10.9352	0.682119	0.341060	−	0.940042i	$-0.389214\pi$
$257$	10.9352	0.682119	0.341060	+	0.940042i	$0.389214\pi$
$258$	0	0
$259$	10.7651	0.668911
$260$	14.8010	0.917920
$261$	0	0
$262$	1.10518	0.0682781
$263$	−10.3267	−0.636774	−0.318387	−	0.947961i	$-0.603141\pi$
$263$	−10.3267	−0.636774	−0.318387	+	0.947961i	$0.603141\pi$
$264$	0	0
$265$	36.9734	2.27126
$266$	34.6899	2.12698
$267$	0	0
$268$	−11.5321	−0.704435
$269$	−6.36422	−0.388033	−0.194017	−	0.980998i	$-0.562152\pi$
$269$	−6.36422	−0.388033	−0.194017	+	0.980998i	$0.562152\pi$
$270$	0	0
$271$	−29.8982	−1.81619	−0.908095	−	0.418765i	$-0.862463\pi$
$271$	−29.8982	−1.81619	−0.908095	+	0.418765i	$0.862463\pi$
$272$	−3.94259	−0.239054
$273$	0	0
$274$	4.83812	0.292281
$275$	−0.910696	−0.0549171
$276$	0	0
$277$	7.02599	0.422151	0.211076	−	0.977470i	$-0.432303\pi$
$277$	7.02599	0.422151	0.211076	+	0.977470i	$0.432303\pi$
$278$	3.16391	0.189759
$279$	0	0
$280$	60.2221	3.59896
$281$	−19.7180	−1.17628	−0.588139	−	0.808760i	$-0.700139\pi$
$281$	−19.7180	−1.17628	−0.588139	+	0.808760i	$0.700139\pi$
$282$	0	0
$283$	−3.99962	−0.237753	−0.118876	−	0.992909i	$-0.537929\pi$
$283$	−3.99962	−0.237753	−0.118876	+	0.992909i	$0.537929\pi$
$284$	3.63796	0.215873
$285$	0	0
$286$	−0.245477	−0.0145154
$287$	24.8650	1.46773
$288$	0	0
$289$	18.6064	1.09449
$290$	12.7342	0.747778
$291$	0	0
$292$	−15.9497	−0.933387
$293$	24.6884	1.44231	0.721156	−	0.692772i	$-0.243611\pi$
$293$	24.6884	1.44231	0.721156	+	0.692772i	$0.243611\pi$
$294$	0	0
$295$	18.3611	1.06902
$296$	6.39196	0.371525
$297$	0	0
$298$	−22.7551	−1.31816
$299$	7.02379	0.406196
$300$	0	0
$301$	−5.28416	−0.304574
$302$	5.48608	0.315688
$303$	0	0
$304$	4.81421	0.276114
$305$	18.8047	1.07676
$306$	0	0
$307$	5.06946	0.289329	0.144665	−	0.989481i	$-0.453790\pi$
$307$	5.06946	0.289329	0.144665	+	0.989481i	$0.453790\pi$
$308$	0.414473	0.0236168
$309$	0	0
$310$	24.3580	1.38344
$311$	−3.96686	−0.224940	−0.112470	−	0.993655i	$-0.535876\pi$
$311$	−3.96686	−0.224940	−0.112470	+	0.993655i	$0.535876\pi$
$312$	0	0
$313$	19.7218	1.11474	0.557371	−	0.830263i	$-0.311810\pi$
$313$	19.7218	1.11474	0.557371	+	0.830263i	$0.311810\pi$
$314$	−5.22829	−0.295049
$315$	0	0
$316$	2.17160	0.122162
$317$	3.35701	0.188548	0.0942742	−	0.995546i	$-0.469947\pi$
$317$	3.35701	0.188548	0.0942742	+	0.995546i	$0.469947\pi$
$318$	0	0
$319$	0.249478	0.0139681
$320$	26.1322	1.46083
$321$	0	0
$322$	10.0396	0.559484
$323$	−43.4783	−2.41919
$324$	0	0
$325$	−39.4119	−2.18618
$326$	4.59064	0.254252
$327$	0	0
$328$	14.7640	0.815205
$329$	−29.1469	−1.60692
$330$	0	0
$331$	20.3450	1.11826	0.559130	−	0.829080i	$-0.311135\pi$
$331$	20.3450	1.11826	0.559130	+	0.829080i	$0.311135\pi$
$332$	−9.27668	−0.509124
$333$	0	0
$334$	8.51231	0.465773
$335$	43.6834	2.38668
$336$	0	0
$337$	−36.6342	−1.99559	−0.997795	−	0.0663680i	$-0.978859\pi$
$337$	−36.6342	−1.99559	−0.997795	+	0.0663680i	$0.978859\pi$
$338$	1.82476	0.0992537
$339$	0	0
$340$	−26.5158	−1.43802
$341$	0.477202	0.0258420
$342$	0	0
$343$	53.3049	2.87820
$344$	−3.13756	−0.169166
$345$	0	0
$346$	17.4575	0.938523
$347$	11.6505	0.625434	0.312717	−	0.949846i	$-0.398761\pi$
$347$	11.6505	0.625434	0.312717	+	0.949846i	$0.398761\pi$
$348$	0	0
$349$	34.6972	1.85730	0.928650	−	0.370958i	$-0.120970\pi$
$349$	34.6972	1.85730	0.928650	+	0.370958i	$0.120970\pi$
$350$	−56.3341	−3.01118
$351$	0	0
$352$	0.405746	0.0216263
$353$	−8.18793	−0.435800	−0.217900	−	0.975971i	$-0.569921\pi$
$353$	−8.18793	−0.435800	−0.217900	+	0.975971i	$0.569921\pi$
$354$	0	0
$355$	−13.7805	−0.731395
$356$	−9.58371	−0.507936
$357$	0	0
$358$	−10.0331	−0.530266
$359$	23.7939	1.25579	0.627896	−	0.778297i	$-0.283916\pi$
$359$	23.7939	1.25579	0.627896	+	0.778297i	$0.283916\pi$
$360$	0	0
$361$	34.0904	1.79423
$362$	11.3222	0.595083
$363$	0	0
$364$	17.9370	0.940155
$365$	60.4173	3.16239
$366$	0	0
$367$	19.8468	1.03600	0.517998	−	0.855382i	$-0.326678\pi$
$367$	19.8468	1.03600	0.517998	+	0.855382i	$0.326678\pi$
$368$	1.39328	0.0726296
$369$	0	0
$370$	−8.50593	−0.442202
$371$	44.8072	2.32627
$372$	0	0
$373$	−26.5360	−1.37398	−0.686991	−	0.726666i	$-0.741069\pi$
$373$	−26.5360	−1.37398	−0.686991	+	0.726666i	$0.741069\pi$
$374$	0.439768	0.0227399
$375$	0	0
$376$	−17.3065	−0.892515
$377$	10.7966	0.556052
$378$	0	0
$379$	1.81485	0.0932226	0.0466113	−	0.998913i	$-0.485158\pi$
$379$	1.81485	0.0932226	0.0466113	+	0.998913i	$0.485158\pi$
$380$	32.3779	1.66095
$381$	0	0
$382$	−18.8205	−0.962942
$383$	−20.8012	−1.06289	−0.531447	−	0.847092i	$-0.678351\pi$
$383$	−20.8012	−1.06289	−0.531447	+	0.847092i	$0.678351\pi$
$384$	0	0
$385$	−1.57002	−0.0800155
$386$	−7.04513	−0.358588
$387$	0	0
$388$	0.427507	0.0217034
$389$	9.93221	0.503583	0.251792	−	0.967782i	$-0.418980\pi$
$389$	9.93221	0.503583	0.251792	+	0.967782i	$0.418980\pi$
$390$	0	0
$391$	−12.5830	−0.636350
$392$	52.3163	2.64237
$393$	0	0
$394$	0.735782	0.0370682
$395$	−8.22598	−0.413894
$396$	0	0
$397$	−27.7376	−1.39211	−0.696055	−	0.717989i	$-0.745063\pi$
$397$	−27.7376	−1.39211	−0.696055	+	0.717989i	$0.745063\pi$
$398$	−20.7709	−1.04115
$399$	0	0
$400$	−7.81796	−0.390898
$401$	−37.8599	−1.89063	−0.945316	−	0.326157i	$-0.894246\pi$
$401$	−37.8599	−1.89063	−0.945316	+	0.326157i	$0.894246\pi$
$402$	0	0
$403$	20.6517	1.02874
$404$	−6.14150	−0.305551
$405$	0	0
$406$	15.4323	0.765891
$407$	−0.166641	−0.00826010
$408$	0	0
$409$	3.63895	0.179934	0.0899672	−	0.995945i	$-0.471324\pi$
$409$	3.63895	0.179934	0.0899672	+	0.995945i	$0.471324\pi$
$410$	−19.6468	−0.970285
$411$	0	0
$412$	13.2470	0.652635
$413$	22.2514	1.09492
$414$	0	0
$415$	35.1399	1.72495
$416$	17.5593	0.860916
$417$	0	0
$418$	−0.536992	−0.0262651
$419$	−7.47377	−0.365118	−0.182559	−	0.983195i	$-0.558438\pi$
$419$	−7.47377	−0.365118	−0.182559	+	0.983195i	$0.558438\pi$
$420$	0	0
$421$	23.1339	1.12748	0.563740	−	0.825952i	$-0.309362\pi$
$421$	23.1339	1.12748	0.563740	+	0.825952i	$0.309362\pi$
$422$	10.0454	0.489001
$423$	0	0
$424$	26.6051	1.29206
$425$	70.6058	3.42488
$426$	0	0
$427$	22.7890	1.10284
$428$	−9.00496	−0.435271
$429$	0	0
$430$	4.17523	0.201347
$431$	−6.19805	−0.298549	−0.149275	−	0.988796i	$-0.547694\pi$
$431$	−6.19805	−0.298549	−0.149275	+	0.988796i	$0.547694\pi$
$432$	0	0
$433$	2.84409	0.136678	0.0683390	−	0.997662i	$-0.478230\pi$
$433$	2.84409	0.136678	0.0683390	+	0.997662i	$0.478230\pi$
$434$	29.5189	1.41695
$435$	0	0
$436$	−0.733343	−0.0351208
$437$	15.3649	0.735001
$438$	0	0
$439$	5.09779	0.243304	0.121652	−	0.992573i	$-0.461181\pi$
$439$	5.09779	0.243304	0.121652	+	0.992573i	$0.461181\pi$
$440$	−0.932225	−0.0444421
$441$	0	0
$442$	19.0317	0.905245
$443$	−13.7065	−0.651217	−0.325609	−	0.945505i	$-0.605569\pi$
$443$	−13.7065	−0.651217	−0.325609	+	0.945505i	$0.605569\pi$
$444$	0	0
$445$	36.3029	1.72092
$446$	17.8556	0.845487
$447$	0	0
$448$	31.6690	1.49622
$449$	−30.5992	−1.44407	−0.722034	−	0.691858i	$-0.756793\pi$
$449$	−30.5992	−1.44407	−0.722034	+	0.691858i	$0.756793\pi$
$450$	0	0
$451$	−0.384904	−0.0181244
$452$	−9.01289	−0.423931
$453$	0	0
$454$	26.7714	1.25645
$455$	−67.9451	−3.18531
$456$	0	0
$457$	17.9572	0.840001	0.420001	−	0.907524i	$-0.362030\pi$
$457$	17.9572	0.840001	0.420001	+	0.907524i	$0.362030\pi$
$458$	9.32251	0.435612
$459$	0	0
$460$	9.37046	0.436900
$461$	18.3934	0.856668	0.428334	−	0.903620i	$-0.359101\pi$
$461$	18.3934	0.856668	0.428334	+	0.903620i	$0.359101\pi$
$462$	0	0
$463$	−34.4958	−1.60316	−0.801579	−	0.597889i	$-0.796006\pi$
$463$	−34.4958	−1.60316	−0.801579	+	0.597889i	$0.796006\pi$
$464$	2.14167	0.0994244
$465$	0	0
$466$	18.2806	0.846835
$467$	17.7648	0.822056	0.411028	−	0.911623i	$-0.365170\pi$
$467$	17.7648	0.822056	0.411028	+	0.911623i	$0.365170\pi$
$468$	0	0
$469$	52.9389	2.44449
$470$	23.0301	1.06230
$471$	0	0
$472$	13.2121	0.608138
$473$	0.0817977	0.00376106
$474$	0	0
$475$	−86.2153	−3.95583
$476$	−32.1339	−1.47285
$477$	0	0
$478$	−25.8215	−1.18105
$479$	−24.5625	−1.12229	−0.561145	−	0.827717i	$-0.689639\pi$
$479$	−24.5625	−1.12229	−0.561145	+	0.827717i	$0.689639\pi$
$480$	0	0
$481$	−7.21167	−0.328824
$482$	6.26414	0.285324
$483$	0	0
$484$	11.9076	0.541255
$485$	−1.61939	−0.0735327
$486$	0	0
$487$	−35.4671	−1.60717	−0.803584	−	0.595191i	$-0.797076\pi$
$487$	−35.4671	−1.60717	−0.803584	+	0.595191i	$0.797076\pi$
$488$	13.5314	0.612537
$489$	0	0
$490$	−69.6184	−3.14504
$491$	−27.8265	−1.25579	−0.627897	−	0.778296i	$-0.716084\pi$
$491$	−27.8265	−1.25579	−0.627897	+	0.778296i	$0.716084\pi$
$492$	0	0
$493$	−19.3419	−0.871115
$494$	−23.2392	−1.04558
$495$	0	0
$496$	4.09659	0.183942
$497$	−16.7003	−0.749112
$498$	0	0
$499$	−32.7456	−1.46590	−0.732948	−	0.680285i	$-0.761856\pi$
$499$	−32.7456	−1.46590	−0.732948	+	0.680285i	$0.761856\pi$
$500$	−30.3612	−1.35780
$501$	0	0
$502$	−21.3271	−0.951875
$503$	8.43515	0.376105	0.188052	−	0.982159i	$-0.439783\pi$
$503$	8.43515	0.376105	0.188052	+	0.982159i	$0.439783\pi$
$504$	0	0
$505$	23.2639	1.03523
$506$	−0.155411	−0.00690884
$507$	0	0
$508$	−10.8119	−0.479700
$509$	11.6347	0.515698	0.257849	−	0.966185i	$-0.416986\pi$
$509$	11.6347	0.515698	0.257849	+	0.966185i	$0.416986\pi$
$510$	0	0
$511$	73.2184	3.23899
$512$	7.38435	0.326345
$513$	0	0
$514$	−10.4710	−0.461857
$515$	−50.1796	−2.21118
$516$	0	0
$517$	0.451188	0.0198432
$518$	−10.3081	−0.452914
$519$	0	0
$520$	−40.3436	−1.76918
$521$	6.18104	0.270796	0.135398	−	0.990791i	$-0.456769\pi$
$521$	6.18104	0.270796	0.135398	+	0.990791i	$0.456769\pi$
$522$	0	0
$523$	−4.44108	−0.194195	−0.0970974	−	0.995275i	$-0.530956\pi$
$523$	−4.44108	−0.194195	−0.0970974	+	0.995275i	$0.530956\pi$
$524$	1.25007	0.0546098
$525$	0	0
$526$	9.88839	0.431154
$527$	−36.9972	−1.61162
$528$	0	0
$529$	−18.5533	−0.806664
$530$	−35.4039	−1.53785
$531$	0	0
$532$	39.2380	1.70118
$533$	−16.6573	−0.721510
$534$	0	0
$535$	34.1106	1.47473
$536$	31.4334	1.35772
$537$	0	0
$538$	6.09407	0.262734
$539$	−1.36391	−0.0587477
$540$	0	0
$541$	5.89672	0.253520	0.126760	−	0.991933i	$-0.459542\pi$
$541$	5.89672	0.253520	0.126760	+	0.991933i	$0.459542\pi$
$542$	28.6291	1.22973
$543$	0	0
$544$	−31.4573	−1.34872
$545$	2.77789	0.118992
$546$	0	0
$547$	−1.00000	−0.0427569
$547$	−1.00000	−0.0427569
$548$	5.47243	0.233771
$549$	0	0
$550$	0.872039	0.0371839
$551$	23.6180	1.00616
$552$	0	0
$553$	−9.96888	−0.423920
$554$	−6.72776	−0.285835
$555$	0	0
$556$	3.57872	0.151772
$557$	1.66388	0.0705009	0.0352504	−	0.999379i	$-0.488777\pi$
$557$	1.66388	0.0705009	0.0352504	+	0.999379i	$0.488777\pi$
$558$	0	0
$559$	3.53993	0.149723
$560$	−13.4780	−0.569548
$561$	0	0
$562$	18.8810	0.796448
$563$	5.94258	0.250450	0.125225	−	0.992128i	$-0.460035\pi$
$563$	5.94258	0.250450	0.125225	+	0.992128i	$0.460035\pi$
$564$	0	0
$565$	34.1407	1.43631
$566$	3.82985	0.160980
$567$	0	0
$568$	−9.91611	−0.416071
$569$	−16.7090	−0.700476	−0.350238	−	0.936661i	$-0.613899\pi$
$569$	−16.7090	−0.700476	−0.350238	+	0.936661i	$0.613899\pi$
$570$	0	0
$571$	−12.6323	−0.528644	−0.264322	−	0.964434i	$-0.585148\pi$
$571$	−12.6323	−0.528644	−0.264322	+	0.964434i	$0.585148\pi$
$572$	−0.277661	−0.0116096
$573$	0	0
$574$	−23.8095	−0.993789
$575$	−24.9515	−1.04055
$576$	0	0
$577$	−28.5904	−1.19023	−0.595116	−	0.803640i	$-0.702894\pi$
$577$	−28.5904	−1.19023	−0.595116	+	0.803640i	$0.702894\pi$
$578$	−17.8166	−0.741073
$579$	0	0
$580$	14.4037	0.598083
$581$	42.5853	1.76673
$582$	0	0
$583$	−0.693605	−0.0287262
$584$	43.4747	1.79900
$585$	0	0
$586$	−23.6404	−0.976578
$587$	22.0036	0.908187	0.454094	−	0.890954i	$-0.349963\pi$
$587$	22.0036	0.908187	0.454094	+	0.890954i	$0.349963\pi$
$588$	0	0
$589$	45.1766	1.86147
$590$	−17.5817	−0.723827
$591$	0	0
$592$	−1.43055	−0.0587951
$593$	−21.0528	−0.864537	−0.432268	−	0.901745i	$-0.642287\pi$
$593$	−21.0528	−0.864537	−0.432268	+	0.901745i	$0.642287\pi$
$594$	0	0
$595$	121.723	4.99014
$596$	−25.7384	−1.05429
$597$	0	0
$598$	−6.72564	−0.275032
$599$	−10.0460	−0.410468	−0.205234	−	0.978713i	$-0.565796\pi$
$599$	−10.0460	−0.410468	−0.205234	+	0.978713i	$0.565796\pi$
$600$	0	0
$601$	−24.1270	−0.984159	−0.492079	−	0.870550i	$-0.663763\pi$
$601$	−24.1270	−0.984159	−0.492079	+	0.870550i	$0.663763\pi$
$602$	5.05986	0.206225
$603$	0	0
$604$	6.20534	0.252492
$605$	−45.1059	−1.83381
$606$	0	0
$607$	−2.94817	−0.119662	−0.0598312	−	0.998209i	$-0.519056\pi$
$607$	−2.94817	−0.119662	−0.0598312	+	0.998209i	$0.519056\pi$
$608$	38.4118	1.55780
$609$	0	0
$610$	−18.0065	−0.729062
$611$	19.5259	0.789933
$612$	0	0
$613$	−6.60126	−0.266622	−0.133311	−	0.991074i	$-0.542561\pi$
$613$	−6.60126	−0.266622	−0.133311	+	0.991074i	$0.542561\pi$
$614$	−4.85427	−0.195902
$615$	0	0
$616$	−1.12974	−0.0455186
$617$	−17.3067	−0.696742	−0.348371	−	0.937357i	$-0.613265\pi$
$617$	−17.3067	−0.696742	−0.348371	+	0.937357i	$0.613265\pi$
$618$	0	0
$619$	15.8845	0.638451	0.319225	−	0.947679i	$-0.396577\pi$
$619$	15.8845	0.638451	0.319225	+	0.947679i	$0.396577\pi$
$620$	27.5515	1.10650
$621$	0	0
$622$	3.79847	0.152305
$623$	43.9947	1.76261
$624$	0	0
$625$	55.8453	2.23381
$626$	−18.8847	−0.754783
$627$	0	0
$628$	−5.91375	−0.235984
$629$	12.9196	0.515138
$630$	0	0
$631$	−17.6658	−0.703263	−0.351631	−	0.936139i	$-0.614373\pi$
$631$	−17.6658	−0.703263	−0.351631	+	0.936139i	$0.614373\pi$
$632$	−5.91920	−0.235453
$633$	0	0
$634$	−3.21451	−0.127665
$635$	40.9553	1.62526
$636$	0	0
$637$	−59.0254	−2.33867
$638$	−0.238888	−0.00945768
$639$	0	0
$640$	18.2345	0.720784
$641$	−49.6629	−1.96157	−0.980783	−	0.195102i	$-0.937496\pi$
$641$	−49.6629	−1.96157	−0.980783	+	0.195102i	$0.937496\pi$
$642$	0	0
$643$	−18.7162	−0.738096	−0.369048	−	0.929410i	$-0.620316\pi$
$643$	−18.7162	−0.738096	−0.369048	+	0.929410i	$0.620316\pi$
$644$	11.3558	0.447483
$645$	0	0
$646$	41.6327	1.63802
$647$	0.632028	0.0248476	0.0124238	−	0.999923i	$-0.496045\pi$
$647$	0.632028	0.0248476	0.0124238	+	0.999923i	$0.496045\pi$
$648$	0	0
$649$	−0.344446	−0.0135207
$650$	37.7389	1.48024
$651$	0	0
$652$	5.19251	0.203354
$653$	−12.2114	−0.477869	−0.238934	−	0.971036i	$-0.576798\pi$
$653$	−12.2114	−0.477869	−0.238934	+	0.971036i	$0.576798\pi$
$654$	0	0
$655$	−4.73526	−0.185022
$656$	−3.30424	−0.129009
$657$	0	0
$658$	27.9097	1.08803
$659$	8.51707	0.331778	0.165889	−	0.986144i	$-0.446951\pi$
$659$	8.51707	0.331778	0.165889	+	0.986144i	$0.446951\pi$
$660$	0	0
$661$	−5.76560	−0.224256	−0.112128	−	0.993694i	$-0.535767\pi$
$661$	−5.76560	−0.224256	−0.112128	+	0.993694i	$0.535767\pi$
$662$	−19.4814	−0.757165
$663$	0	0
$664$	25.2857	0.981277
$665$	−148.633	−5.76374
$666$	0	0
$667$	6.83527	0.264663
$668$	9.62833	0.372531
$669$	0	0
$670$	−41.8291	−1.61600
$671$	−0.352769	−0.0136185
$672$	0	0
$673$	−18.9482	−0.730400	−0.365200	−	0.930929i	$-0.618999\pi$
$673$	−18.9482	−0.730400	−0.365200	+	0.930929i	$0.618999\pi$
$674$	35.0791	1.35120
$675$	0	0
$676$	2.06400	0.0793845
$677$	−11.6427	−0.447466	−0.223733	−	0.974650i	$-0.571824\pi$
$677$	−11.6427	−0.447466	−0.223733	+	0.974650i	$0.571824\pi$
$678$	0	0
$679$	−1.96250	−0.0753139
$680$	72.2749	2.77162
$681$	0	0
$682$	−0.456946	−0.0174974
$683$	−32.4010	−1.23979	−0.619895	−	0.784685i	$-0.712825\pi$
$683$	−32.4010	−1.23979	−0.619895	+	0.784685i	$0.712825\pi$
$684$	0	0
$685$	−20.7295	−0.792032
$686$	−51.0422	−1.94880
$687$	0	0
$688$	0.702199	0.0267711
$689$	−30.0169	−1.14355
$690$	0	0
$691$	33.5061	1.27463	0.637317	−	0.770602i	$-0.280044\pi$
$691$	33.5061	1.27463	0.637317	+	0.770602i	$0.280044\pi$
$692$	19.7463	0.750643
$693$	0	0
$694$	−11.1560	−0.423476
$695$	−13.5561	−0.514213
$696$	0	0
$697$	29.8414	1.13032
$698$	−33.2244	−1.25756
$699$	0	0
$700$	−63.7199	−2.40839
$701$	18.1131	0.684122	0.342061	−	0.939678i	$-0.388875\pi$
$701$	18.1131	0.684122	0.342061	+	0.939678i	$0.388875\pi$
$702$	0	0
$703$	−15.7759	−0.594998
$704$	−0.490228	−0.0184762
$705$	0	0
$706$	7.84037	0.295076
$707$	28.1930	1.06031
$708$	0	0
$709$	−25.2354	−0.947736	−0.473868	−	0.880596i	$-0.657143\pi$
$709$	−25.2354	−0.947736	−0.473868	+	0.880596i	$0.657143\pi$
$710$	13.1956	0.495222
$711$	0	0
$712$	26.1226	0.978987
$713$	13.0745	0.489645
$714$	0	0
$715$	1.05177	0.0393341
$716$	−11.3485	−0.424114
$717$	0	0
$718$	−22.7839	−0.850287
$719$	−1.53980	−0.0574250	−0.0287125	−	0.999588i	$-0.509141\pi$
$719$	−1.53980	−0.0574250	−0.0287125	+	0.999588i	$0.509141\pi$
$720$	0	0
$721$	−60.8115	−2.26474
$722$	−32.6433	−1.21486
$723$	0	0
$724$	12.8067	0.475955
$725$	−38.3540	−1.42443
$726$	0	0
$727$	−29.6777	−1.10068	−0.550342	−	0.834939i	$-0.685503\pi$
$727$	−29.6777	−1.10068	−0.550342	+	0.834939i	$0.685503\pi$
$728$	−48.8915	−1.81204
$729$	0	0
$730$	−57.8527	−2.14123
$731$	−6.34173	−0.234557
$732$	0	0
$733$	−34.6887	−1.28126	−0.640629	−	0.767851i	$-0.721326\pi$
$733$	−34.6887	−1.28126	−0.640629	+	0.767851i	$0.721326\pi$
$734$	−19.0044	−0.701464
$735$	0	0
$736$	11.1167	0.409768
$737$	−0.819482	−0.0301860
$738$	0	0
$739$	−3.96845	−0.145982	−0.0729909	−	0.997333i	$-0.523254\pi$
$739$	−3.96845	−0.145982	−0.0729909	+	0.997333i	$0.523254\pi$
$740$	−9.62112	−0.353679
$741$	0	0
$742$	−42.9052	−1.57510
$743$	−47.6574	−1.74838	−0.874190	−	0.485584i	$-0.838607\pi$
$743$	−47.6574	−1.74838	−0.874190	+	0.485584i	$0.838607\pi$
$744$	0	0
$745$	97.4966	3.57200
$746$	25.4096	0.930312
$747$	0	0
$748$	0.497425	0.0181877
$749$	41.3379	1.51045
$750$	0	0
$751$	11.3760	0.415117	0.207558	−	0.978223i	$-0.433448\pi$
$751$	11.3760	0.415117	0.207558	+	0.978223i	$0.433448\pi$
$752$	3.87327	0.141243
$753$	0	0
$754$	−10.3383	−0.376498
$755$	−23.5057	−0.855461
$756$	0	0
$757$	35.7312	1.29867	0.649337	−	0.760501i	$-0.275047\pi$
$757$	35.7312	1.29867	0.649337	+	0.760501i	$0.275047\pi$
$758$	−1.73781	−0.0631202
$759$	0	0
$760$	−88.2534	−3.20129
$761$	−45.2992	−1.64209	−0.821047	−	0.570861i	$-0.806610\pi$
$761$	−45.2992	−1.64209	−0.821047	+	0.570861i	$0.806610\pi$
$762$	0	0
$763$	3.36646	0.121874
$764$	−21.2880	−0.770174
$765$	0	0
$766$	19.9183	0.719676
$767$	−14.9065	−0.538242
$768$	0	0
$769$	20.9580	0.755766	0.377883	−	0.925853i	$-0.376652\pi$
$769$	20.9580	0.755766	0.377883	+	0.925853i	$0.376652\pi$
$770$	1.50337	0.0541778
$771$	0	0
$772$	−7.96879	−0.286803
$773$	−9.94311	−0.357629	−0.178814	−	0.983883i	$-0.557226\pi$
$773$	−9.94311	−0.357629	−0.178814	+	0.983883i	$0.557226\pi$
$774$	0	0
$775$	−73.3637	−2.63530
$776$	−1.16527	−0.0418307
$777$	0	0
$778$	−9.51061	−0.340972
$779$	−36.4387	−1.30555
$780$	0	0
$781$	0.258517	0.00925048
$782$	12.0489	0.430867
$783$	0	0
$784$	−11.7086	−0.418164
$785$	22.4012	0.799533
$786$	0	0
$787$	43.0795	1.53562	0.767808	−	0.640680i	$-0.221347\pi$
$787$	43.0795	1.53562	0.767808	+	0.640680i	$0.221347\pi$
$788$	0.832248	0.0296476
$789$	0	0
$790$	7.87680	0.280244
$791$	41.3743	1.47110
$792$	0	0
$793$	−15.2667	−0.542135
$794$	26.5602	0.942586
$795$	0	0
$796$	−23.4941	−0.832727
$797$	−19.8313	−0.702460	−0.351230	−	0.936289i	$-0.614237\pi$
$797$	−19.8313	−0.702460	−0.351230	+	0.936289i	$0.614237\pi$
$798$	0	0
$799$	−34.9804	−1.23752
$800$	−62.3782	−2.20540
$801$	0	0
$802$	36.2528	1.28013
$803$	−1.13340	−0.0399970
$804$	0	0
$805$	−43.0157	−1.51611
$806$	−19.7751	−0.696548
$807$	0	0
$808$	16.7401	0.588914
$809$	−1.62413	−0.0571014	−0.0285507	−	0.999592i	$-0.509089\pi$
$809$	−1.62413	−0.0571014	−0.0285507	+	0.999592i	$0.509089\pi$
$810$	0	0
$811$	41.7383	1.46563	0.732814	−	0.680428i	$-0.238206\pi$
$811$	41.7383	1.46563	0.732814	+	0.680428i	$0.238206\pi$
$812$	17.4556	0.612570
$813$	0	0
$814$	0.159568	0.00559285
$815$	−19.6691	−0.688979
$816$	0	0
$817$	7.74375	0.270920
$818$	−3.48448	−0.121832
$819$	0	0
$820$	−22.2226	−0.776047
$821$	34.5850	1.20703	0.603513	−	0.797353i	$-0.293767\pi$
$821$	34.5850	1.20703	0.603513	+	0.797353i	$0.293767\pi$
$822$	0	0
$823$	−22.0897	−0.769999	−0.384999	−	0.922917i	$-0.625798\pi$
$823$	−22.0897	−0.769999	−0.384999	+	0.922917i	$0.625798\pi$
$824$	−36.1079	−1.25788
$825$	0	0
$826$	−21.3069	−0.741361
$827$	−4.29638	−0.149400	−0.0746999	−	0.997206i	$-0.523800\pi$
$827$	−4.29638	−0.149400	−0.0746999	+	0.997206i	$0.523800\pi$
$828$	0	0
$829$	9.89112	0.343533	0.171766	−	0.985138i	$-0.445053\pi$
$829$	9.89112	0.343533	0.171766	+	0.985138i	$0.445053\pi$
$830$	−33.6483	−1.16795
$831$	0	0
$832$	−21.2154	−0.735513
$833$	105.743	3.66378
$834$	0	0
$835$	−36.4720	−1.26216
$836$	−0.607396	−0.0210072
$837$	0	0
$838$	7.15653	0.247218
$839$	45.2914	1.56363	0.781817	−	0.623509i	$-0.214293\pi$
$839$	45.2914	1.56363	0.781817	+	0.623509i	$0.214293\pi$
$840$	0	0
$841$	−18.4932	−0.637697
$842$	−22.1520	−0.763407
$843$	0	0
$844$	11.3624	0.391109
$845$	−7.81838	−0.268960
$846$	0	0
$847$	−54.6628	−1.87823
$848$	−5.95432	−0.204472
$849$	0	0
$850$	−67.6087	−2.31896
$851$	−4.56568	−0.156510
$852$	0	0
$853$	−54.9342	−1.88091	−0.940456	−	0.339916i	$-0.889601\pi$
$853$	−54.9342	−1.88091	−0.940456	+	0.339916i	$0.889601\pi$
$854$	−21.8217	−0.746723
$855$	0	0
$856$	24.5451	0.838934
$857$	0.342745	0.0117079	0.00585397	−	0.999983i	$-0.498137\pi$
$857$	0.342745	0.0117079	0.00585397	+	0.999983i	$0.498137\pi$
$858$	0	0
$859$	13.8090	0.471157	0.235578	−	0.971855i	$-0.424302\pi$
$859$	13.8090	0.471157	0.235578	+	0.971855i	$0.424302\pi$
$860$	4.72263	0.161040
$861$	0	0
$862$	5.93495	0.202145
$863$	27.0862	0.922025	0.461013	−	0.887394i	$-0.347486\pi$
$863$	27.0862	0.922025	0.461013	+	0.887394i	$0.347486\pi$
$864$	0	0
$865$	−74.7988	−2.54324
$866$	−2.72336	−0.0925435
$867$	0	0
$868$	33.3891	1.13330
$869$	0.154316	0.00523481
$870$	0	0
$871$	−35.4644	−1.20167
$872$	1.99890	0.0676912
$873$	0	0
$874$	−14.7127	−0.497663
$875$	139.375	4.71175
$876$	0	0
$877$	37.0243	1.25022	0.625111	−	0.780536i	$-0.285054\pi$
$877$	37.0243	1.25022	0.625111	+	0.780536i	$0.285054\pi$
$878$	−4.88140	−0.164739
$879$	0	0
$880$	0.208636	0.00703311
$881$	−6.89398	−0.232264	−0.116132	−	0.993234i	$-0.537050\pi$
$881$	−6.89398	−0.232264	−0.116132	+	0.993234i	$0.537050\pi$
$882$	0	0
$883$	9.37818	0.315601	0.157801	−	0.987471i	$-0.449560\pi$
$883$	9.37818	0.315601	0.157801	+	0.987471i	$0.449560\pi$
$884$	21.5269	0.724027
$885$	0	0
$886$	13.1247	0.440934
$887$	−33.7032	−1.13164	−0.565822	−	0.824528i	$-0.691441\pi$
$887$	−33.7032	−1.13164	−0.565822	+	0.824528i	$0.691441\pi$
$888$	0	0
$889$	49.6328	1.66463
$890$	−34.7620	−1.16522
$891$	0	0
$892$	20.1966	0.676232
$893$	42.7138	1.42936
$894$	0	0
$895$	42.9879	1.43693
$896$	22.0980	0.738244
$897$	0	0
$898$	29.3004	0.977766
$899$	20.0974	0.670286
$900$	0	0
$901$	53.7748	1.79150
$902$	0.368566	0.0122719
$903$	0	0
$904$	24.5667	0.817076
$905$	−48.5114	−1.61257
$906$	0	0
$907$	−25.0104	−0.830455	−0.415228	−	0.909718i	$-0.636298\pi$
$907$	−25.0104	−0.830455	−0.415228	+	0.909718i	$0.636298\pi$
$908$	30.2814	1.00492
$909$	0	0
$910$	65.0609	2.15675
$911$	40.4013	1.33855	0.669277	−	0.743013i	$-0.266604\pi$
$911$	40.4013	1.33855	0.669277	+	0.743013i	$0.266604\pi$
$912$	0	0
$913$	−0.659210	−0.0218167
$914$	−17.1949	−0.568758
$915$	0	0
$916$	10.5448	0.348409
$917$	−5.73856	−0.189504
$918$	0	0
$919$	8.97289	0.295988	0.147994	−	0.988988i	$-0.452718\pi$
$919$	8.97289	0.295988	0.147994	+	0.988988i	$0.452718\pi$
$920$	−25.5413	−0.842073
$921$	0	0
$922$	−17.6127	−0.580043
$923$	11.1878	0.368250
$924$	0	0
$925$	25.6189	0.842346
$926$	33.0316	1.08548
$927$	0	0
$928$	17.0880	0.560942
$929$	1.61228	0.0528973	0.0264487	−	0.999650i	$-0.491580\pi$
$929$	1.61228	0.0528973	0.0264487	+	0.999650i	$0.491580\pi$
$930$	0	0
$931$	−129.121	−4.23176
$932$	20.6774	0.677310
$933$	0	0
$934$	−17.0107	−0.556607
$935$	−1.88424	−0.0616212
$936$	0	0
$937$	−12.9284	−0.422351	−0.211176	−	0.977448i	$-0.567729\pi$
$937$	−12.9284	−0.422351	−0.211176	+	0.977448i	$0.567729\pi$
$938$	−50.6918	−1.65514
$939$	0	0
$940$	26.0496	0.849643
$941$	18.4517	0.601510	0.300755	−	0.953701i	$-0.402761\pi$
$941$	18.4517	0.601510	0.300755	+	0.953701i	$0.402761\pi$
$942$	0	0
$943$	−10.5457	−0.343415
$944$	−2.95693	−0.0962399
$945$	0	0
$946$	−0.0783255	−0.00254658
$947$	−12.2973	−0.399607	−0.199803	−	0.979836i	$-0.564030\pi$
$947$	−12.2973	−0.399607	−0.199803	+	0.979836i	$0.564030\pi$
$948$	0	0
$949$	−49.0499	−1.59223
$950$	82.5556	2.67846
$951$	0	0
$952$	87.5883	2.83875
$953$	2.51896	0.0815970	0.0407985	−	0.999167i	$-0.487010\pi$
$953$	2.51896	0.0815970	0.0407985	+	0.999167i	$0.487010\pi$
$954$	0	0
$955$	80.6387	2.60941
$956$	−29.2069	−0.944619
$957$	0	0
$958$	23.5199	0.759894
$959$	−25.1216	−0.811218
$960$	0	0
$961$	7.44239	0.240077
$962$	6.90555	0.222644
$963$	0	0
$964$	7.08541	0.228206
$965$	30.1857	0.971711
$966$	0	0
$967$	−45.0973	−1.45023	−0.725116	−	0.688627i	$-0.758214\pi$
$967$	−45.0973	−1.45023	−0.725116	+	0.688627i	$0.758214\pi$
$968$	−32.4570	−1.04321
$969$	0	0
$970$	1.55065	0.0497884
$971$	−35.4672	−1.13820	−0.569098	−	0.822270i	$-0.692707\pi$
$971$	−35.4672	−1.13820	−0.569098	+	0.822270i	$0.692707\pi$
$972$	0	0
$973$	−16.4284	−0.526669
$974$	33.9616	1.08820
$975$	0	0
$976$	−3.02838	−0.0969360
$977$	−8.98700	−0.287520	−0.143760	−	0.989613i	$-0.545919\pi$
$977$	−8.98700	−0.287520	−0.143760	+	0.989613i	$0.545919\pi$
$978$	0	0
$979$	−0.681028	−0.0217658
$980$	−78.7459	−2.51545
$981$	0	0
$982$	26.6454	0.850287
$983$	8.91145	0.284231	0.142116	−	0.989850i	$-0.454610\pi$
$983$	8.91145	0.284231	0.142116	+	0.989850i	$0.454610\pi$
$984$	0	0
$985$	−3.15254	−0.100448
$986$	18.5209	0.589825
$987$	0	0
$988$	−26.2860	−0.836270
$989$	2.24111	0.0712633
$990$	0	0
$991$	24.7481	0.786149	0.393074	−	0.919507i	$-0.371411\pi$
$991$	24.7481	0.786149	0.393074	+	0.919507i	$0.371411\pi$
$992$	32.6860	1.03778
$993$	0	0
$994$	15.9914	0.507218
$995$	88.9954	2.82134
$996$	0	0
$997$	−39.8404	−1.26176	−0.630878	−	0.775882i	$-0.717305\pi$
$997$	−39.8404	−1.26176	−0.630878	+	0.775882i	$0.717305\pi$
$998$	31.3557	0.992546
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4923.2.a.l.1.6		18
3.2	odd	2		547.2.a.b.1.13	✓	18
12.11	even	2		8752.2.a.s.1.5		18

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
547.2.a.b.1.13	✓	18	3.2	odd	2
4923.2.a.l.1.6		18	1.1	even	1	trivial
8752.2.a.s.1.5		18	12.11	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 4923 = 3^{2} \cdot 547 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4923.a (trivial)

\(f(q)\)	\(=\)	\(q-0.957552 q^{2} -1.08309 q^{4} +4.10274 q^{5} +4.97202 q^{7} +2.95222 q^{8} +O(q^{10})\)	\(q-0.957552 q^{2} -1.08309 q^{4} +4.10274 q^{5} +4.97202 q^{7} +2.95222 q^{8} -3.92859 q^{10} -0.0769657 q^{11} -3.33082 q^{13} -4.76097 q^{14} -0.660720 q^{16} +5.96711 q^{17} -7.28632 q^{19} -4.44365 q^{20} +0.0736987 q^{22} -2.10873 q^{23} +11.8325 q^{25} +3.18943 q^{26} -5.38516 q^{28} -3.24142 q^{29} -6.20019 q^{31} -5.27177 q^{32} -5.71382 q^{34} +20.3989 q^{35} +2.16514 q^{37} +6.97703 q^{38} +12.1122 q^{40} +5.00098 q^{41} -1.06278 q^{43} +0.0833611 q^{44} +2.01922 q^{46} -5.86219 q^{47} +17.7210 q^{49} -11.3302 q^{50} +3.60759 q^{52} +9.01187 q^{53} -0.315771 q^{55} +14.6785 q^{56} +3.10383 q^{58} +4.47532 q^{59} +4.58346 q^{61} +5.93701 q^{62} +6.36944 q^{64} -13.6655 q^{65} +10.6474 q^{67} -6.46294 q^{68} -19.5330 q^{70} -3.35886 q^{71} +14.7261 q^{73} -2.07323 q^{74} +7.89177 q^{76} -0.382675 q^{77} -2.00500 q^{79} -2.71076 q^{80} -4.78870 q^{82} +8.56498 q^{83} +24.4815 q^{85} +1.01767 q^{86} -0.227220 q^{88} +8.84846 q^{89} -16.5609 q^{91} +2.28395 q^{92} +5.61336 q^{94} -29.8939 q^{95} -0.394709 q^{97} -16.9688 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q + 4 q^{2} + 16 q^{4} + 27 q^{5} - 11 q^{7} + 12 q^{8}+O(q^{10}) \) 18 * q + 4 * q^2 + 16 * q^4 + 27 * q^5 - 11 * q^7 + 12 * q^8	\( 18 q + 4 q^{2} + 16 q^{4} + 27 q^{5} - 11 q^{7} + 12 q^{8} - 5 q^{10} - 2 q^{11} - 25 q^{13} + 7 q^{14} + 8 q^{16} + 30 q^{17} + 4 q^{19} + 41 q^{20} - 24 q^{22} + 26 q^{23} + 31 q^{25} + 18 q^{26} - 16 q^{28} + 18 q^{29} - 5 q^{31} + 28 q^{32} + 5 q^{34} + 9 q^{35} - 18 q^{37} + 45 q^{38} + 7 q^{40} + 17 q^{41} + 8 q^{43} - 12 q^{44} + 30 q^{46} + 52 q^{47} + 29 q^{49} - 13 q^{50} - 14 q^{52} + 60 q^{53} + 11 q^{55} - 7 q^{56} + 14 q^{58} + 8 q^{59} - 26 q^{61} - 4 q^{62} + 44 q^{64} + 6 q^{65} + 12 q^{67} + 61 q^{68} + 35 q^{70} + q^{71} - 2 q^{73} - 16 q^{74} + 66 q^{76} + 73 q^{77} + 18 q^{79} + 32 q^{80} + 44 q^{82} + 43 q^{83} + 51 q^{85} - 4 q^{86} - 17 q^{88} + 28 q^{89} - q^{91} + 68 q^{92} + 78 q^{94} + 18 q^{95} - 34 q^{97} - 34 q^{98}+O(q^{100}) \) 18 * q + 4 * q^2 + 16 * q^4 + 27 * q^5 - 11 * q^7 + 12 * q^8 - 5 * q^10 - 2 * q^11 - 25 * q^13 + 7 * q^14 + 8 * q^16 + 30 * q^17 + 4 * q^19 + 41 * q^20 - 24 * q^22 + 26 * q^23 + 31 * q^25 + 18 * q^26 - 16 * q^28 + 18 * q^29 - 5 * q^31 + 28 * q^32 + 5 * q^34 + 9 * q^35 - 18 * q^37 + 45 * q^38 + 7 * q^40 + 17 * q^41 + 8 * q^43 - 12 * q^44 + 30 * q^46 + 52 * q^47 + 29 * q^49 - 13 * q^50 - 14 * q^52 + 60 * q^53 + 11 * q^55 - 7 * q^56 + 14 * q^58 + 8 * q^59 - 26 * q^61 - 4 * q^62 + 44 * q^64 + 6 * q^65 + 12 * q^67 + 61 * q^68 + 35 * q^70 + q^71 - 2 * q^73 - 16 * q^74 + 66 * q^76 + 73 * q^77 + 18 * q^79 + 32 * q^80 + 44 * q^82 + 43 * q^83 + 51 * q^85 - 4 * q^86 - 17 * q^88 + 28 * q^89 - q^91 + 68 * q^92 + 78 * q^94 + 18 * q^95 - 34 * q^97 - 34 * q^98

Introduction

L-functions

Modular forms

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Database

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