LMFDB - Embedded newform 4923.2.a.l.1.16

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			1.16
Root			$2.35947$ 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)

$f(q)$	$=$	$q+2.35947 q^{2} +3.56710 q^{4} +4.18174 q^{5} -2.82979 q^{7} +3.69754 q^{8} +O(q^{10})$	\(q+2.35947 q^{2} +3.56710 q^{4} +4.18174 q^{5} -2.82979 q^{7} +3.69754 q^{8} +9.86669 q^{10} -2.48715 q^{11} +4.60177 q^{13} -6.67680 q^{14} +1.59003 q^{16} +5.70168 q^{17} +0.542873 q^{19} +14.9167 q^{20} -5.86836 q^{22} -6.30681 q^{23} +12.4869 q^{25} +10.8577 q^{26} -10.0941 q^{28} +4.16997 q^{29} +5.03942 q^{31} -3.64346 q^{32} +13.4529 q^{34} -11.8334 q^{35} -4.02373 q^{37} +1.28089 q^{38} +15.4621 q^{40} +2.04830 q^{41} +7.11938 q^{43} -8.87193 q^{44} -14.8807 q^{46} +4.23012 q^{47} +1.00770 q^{49} +29.4625 q^{50} +16.4150 q^{52} +8.60588 q^{53} -10.4006 q^{55} -10.4632 q^{56} +9.83893 q^{58} -2.15472 q^{59} -8.53383 q^{61} +11.8904 q^{62} -11.7767 q^{64} +19.2434 q^{65} -3.63078 q^{67} +20.3385 q^{68} -27.9206 q^{70} -7.43976 q^{71} +10.8830 q^{73} -9.49386 q^{74} +1.93648 q^{76} +7.03811 q^{77} +4.82054 q^{79} +6.64907 q^{80} +4.83290 q^{82} -11.0018 q^{83} +23.8429 q^{85} +16.7980 q^{86} -9.19633 q^{88} -2.44024 q^{89} -13.0220 q^{91} -22.4970 q^{92} +9.98085 q^{94} +2.27015 q^{95} +5.35684 q^{97} +2.37763 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	2.35947	1.66840	0.834199	−	0.551463i	$-0.185930\pi$
$2$	2.35947	1.66840	0.834199	+	0.551463i	$0.185930\pi$
$3$	0	0
$3$	0	0
$4$	3.56710	1.78355
$5$	4.18174	1.87013	0.935065	−	0.354477i	$-0.115341\pi$
$5$	4.18174	1.87013	0.935065	+	0.354477i	$0.115341\pi$
$6$	0	0
$7$	−2.82979	−1.06956	−0.534779	−	0.844992i	$-0.679605\pi$
$7$	−2.82979	−1.06956	−0.534779	+	0.844992i	$0.679605\pi$
$8$	3.69754	1.30728
$9$	0	0
$10$	9.86669	3.12012
$11$	−2.48715	−0.749904	−0.374952	−	0.927044i	$-0.622341\pi$
$11$	−2.48715	−0.749904	−0.374952	+	0.927044i	$0.622341\pi$
$12$	0	0
$13$	4.60177	1.27630	0.638150	−	0.769912i	$-0.279700\pi$
$13$	4.60177	1.27630	0.638150	+	0.769912i	$0.279700\pi$
$14$	−6.67680	−1.78445
$15$	0	0
$16$	1.59003	0.397506
$17$	5.70168	1.38286	0.691430	−	0.722444i	$-0.256981\pi$
$17$	5.70168	1.38286	0.691430	+	0.722444i	$0.256981\pi$
$18$	0	0
$19$	0.542873	0.124544	0.0622718	−	0.998059i	$-0.480165\pi$
$19$	0.542873	0.124544	0.0622718	+	0.998059i	$0.480165\pi$
$20$	14.9167	3.33547
$21$	0	0
$22$	−5.86836	−1.25114
$23$	−6.30681	−1.31506	−0.657530	−	0.753428i	$-0.728399\pi$
$23$	−6.30681	−1.31506	−0.657530	+	0.753428i	$0.728399\pi$
$24$	0	0
$25$	12.4869	2.49738
$26$	10.8577	2.12938
$27$	0	0
$28$	−10.0941	−1.90761
$29$	4.16997	0.774344	0.387172	−	0.922007i	$-0.373452\pi$
$29$	4.16997	0.774344	0.387172	+	0.922007i	$0.373452\pi$
$30$	0	0
$31$	5.03942	0.905106	0.452553	−	0.891737i	$-0.350513\pi$
$31$	5.03942	0.905106	0.452553	+	0.891737i	$0.350513\pi$
$32$	−3.64346	−0.644078
$33$	0	0
$34$	13.4529	2.30716
$35$	−11.8334	−2.00021
$36$	0	0
$37$	−4.02373	−0.661496	−0.330748	−	0.943719i	$-0.607301\pi$
$37$	−4.02373	−0.661496	−0.330748	+	0.943719i	$0.607301\pi$
$38$	1.28089	0.207788
$39$	0	0
$40$	15.4621	2.44478
$41$	2.04830	0.319890	0.159945	−	0.987126i	$-0.448868\pi$
$41$	2.04830	0.319890	0.159945	+	0.987126i	$0.448868\pi$
$42$	0	0
$43$	7.11938	1.08570	0.542848	−	0.839831i	$-0.317346\pi$
$43$	7.11938	1.08570	0.542848	+	0.839831i	$0.317346\pi$
$44$	−8.87193	−1.33749
$45$	0	0
$46$	−14.8807	−2.19404
$47$	4.23012	0.617027	0.308513	−	0.951220i	$-0.400169\pi$
$47$	4.23012	0.617027	0.308513	+	0.951220i	$0.400169\pi$
$48$	0	0
$49$	1.00770	0.143956
$50$	29.4625	4.16663
$51$	0	0
$52$	16.4150	2.27635
$53$	8.60588	1.18211	0.591054	−	0.806632i	$-0.298712\pi$
$53$	8.60588	1.18211	0.591054	+	0.806632i	$0.298712\pi$
$54$	0	0
$55$	−10.4006	−1.40242
$56$	−10.4632	−1.39821
$57$	0	0
$58$	9.83893	1.29191
$59$	−2.15472	−0.280520	−0.140260	−	0.990115i	$-0.544794\pi$
$59$	−2.15472	−0.280520	−0.140260	+	0.990115i	$0.544794\pi$
$60$	0	0
$61$	−8.53383	−1.09264	−0.546322	−	0.837575i	$-0.683973\pi$
$61$	−8.53383	−1.09264	−0.546322	+	0.837575i	$0.683973\pi$
$62$	11.8904	1.51008
$63$	0	0
$64$	−11.7767	−1.47209
$65$	19.2434	2.38685
$66$	0	0
$67$	−3.63078	−0.443570	−0.221785	−	0.975096i	$-0.571188\pi$
$67$	−3.63078	−0.443570	−0.221785	+	0.975096i	$0.571188\pi$
$68$	20.3385	2.46640
$69$	0	0
$70$	−27.9206	−3.33715
$71$	−7.43976	−0.882938	−0.441469	−	0.897277i	$-0.645542\pi$
$71$	−7.43976	−0.882938	−0.441469	+	0.897277i	$0.645542\pi$
$72$	0	0
$73$	10.8830	1.27376	0.636881	−	0.770962i	$-0.280224\pi$
$73$	10.8830	1.27376	0.636881	+	0.770962i	$0.280224\pi$
$74$	−9.49386	−1.10364
$75$	0	0
$76$	1.93648	0.222130
$77$	7.03811	0.802067
$78$	0	0
$79$	4.82054	0.542353	0.271177	−	0.962530i	$-0.412587\pi$
$79$	4.82054	0.542353	0.271177	+	0.962530i	$0.412587\pi$
$80$	6.64907	0.743388
$81$	0	0
$82$	4.83290	0.533704
$83$	−11.0018	−1.20760	−0.603801	−	0.797135i	$-0.706348\pi$
$83$	−11.0018	−1.20760	−0.603801	+	0.797135i	$0.706348\pi$
$84$	0	0
$85$	23.8429	2.58613
$86$	16.7980	1.81137
$87$	0	0
$88$	−9.19633	−0.980333
$89$	−2.44024	−0.258665	−0.129332	−	0.991601i	$-0.541283\pi$
$89$	−2.44024	−0.258665	−0.129332	+	0.991601i	$0.541283\pi$
$90$	0	0
$91$	−13.0220	−1.36508
$92$	−22.4970	−2.34548
$93$	0	0
$94$	9.98085	1.02945
$95$	2.27015	0.232913
$96$	0	0
$97$	5.35684	0.543904	0.271952	−	0.962311i	$-0.412331\pi$
$97$	5.35684	0.543904	0.271952	+	0.962311i	$0.412331\pi$
$98$	2.37763	0.240177
$99$	0	0
$100$	44.5421	4.45421
$101$	−16.3478	−1.62666	−0.813332	−	0.581800i	$-0.802349\pi$
$101$	−16.3478	−1.62666	−0.813332	+	0.581800i	$0.802349\pi$
$102$	0	0
$103$	−11.8061	−1.16329	−0.581645	−	0.813443i	$-0.697591\pi$
$103$	−11.8061	−1.16329	−0.581645	+	0.813443i	$0.697591\pi$
$104$	17.0152	1.66848
$105$	0	0
$106$	20.3053	1.97223
$107$	12.0397	1.16392	0.581959	−	0.813218i	$-0.302286\pi$
$107$	12.0397	1.16392	0.581959	+	0.813218i	$0.302286\pi$
$108$	0	0
$109$	1.39459	0.133577	0.0667886	−	0.997767i	$-0.478725\pi$
$109$	1.39459	0.133577	0.0667886	+	0.997767i	$0.478725\pi$
$110$	−24.5399	−2.33979
$111$	0	0
$112$	−4.49943	−0.425156
$113$	−4.16686	−0.391985	−0.195992	−	0.980605i	$-0.562793\pi$
$113$	−4.16686	−0.391985	−0.195992	+	0.980605i	$0.562793\pi$
$114$	0	0
$115$	−26.3734	−2.45933
$116$	14.8747	1.38108
$117$	0	0
$118$	−5.08399	−0.468019
$119$	−16.1345	−1.47905
$120$	0	0
$121$	−4.81408	−0.437644
$122$	−20.1353	−1.82297
$123$	0	0
$124$	17.9761	1.61430
$125$	31.3083	2.80030
$126$	0	0
$127$	8.57110	0.760562	0.380281	−	0.924871i	$-0.375827\pi$
$127$	8.57110	0.760562	0.380281	+	0.924871i	$0.375827\pi$
$128$	−20.4998	−1.81195
$129$	0	0
$130$	45.4042	3.98221
$131$	19.4849	1.70241	0.851204	−	0.524836i	$-0.175873\pi$
$131$	19.4849	1.70241	0.851204	+	0.524836i	$0.175873\pi$
$132$	0	0
$133$	−1.53621	−0.133207
$134$	−8.56672	−0.740052
$135$	0	0
$136$	21.0822	1.80778
$137$	−16.6524	−1.42271	−0.711357	−	0.702830i	$-0.751919\pi$
$137$	−16.6524	−1.42271	−0.711357	+	0.702830i	$0.751919\pi$
$138$	0	0
$139$	−7.41300	−0.628763	−0.314381	−	0.949297i	$-0.601797\pi$
$139$	−7.41300	−0.628763	−0.314381	+	0.949297i	$0.601797\pi$
$140$	−42.2111	−3.56749
$141$	0	0
$142$	−17.5539	−1.47309
$143$	−11.4453	−0.957103
$144$	0	0
$145$	17.4377	1.44812
$146$	25.6782	2.12514
$147$	0	0
$148$	−14.3530	−1.17981
$149$	20.4476	1.67513	0.837566	−	0.546335i	$-0.183978\pi$
$149$	20.4476	1.67513	0.837566	+	0.546335i	$0.183978\pi$
$150$	0	0
$151$	15.4794	1.25970	0.629849	−	0.776717i	$-0.283117\pi$
$151$	15.4794	1.25970	0.629849	+	0.776717i	$0.283117\pi$
$152$	2.00729	0.162813
$153$	0	0
$154$	16.6062	1.33817
$155$	21.0735	1.69267
$156$	0	0
$157$	−16.5044	−1.31719	−0.658596	−	0.752497i	$-0.728849\pi$
$157$	−16.5044	−1.31719	−0.658596	+	0.752497i	$0.728849\pi$
$158$	11.3739	0.904861
$159$	0	0
$160$	−15.2360	−1.20451
$161$	17.8469	1.40653
$162$	0	0
$163$	2.84622	0.222933	0.111467	−	0.993768i	$-0.464445\pi$
$163$	2.84622	0.222933	0.111467	+	0.993768i	$0.464445\pi$
$164$	7.30649	0.570541
$165$	0	0
$166$	−25.9584	−2.01476
$167$	−11.3203	−0.875995	−0.437997	−	0.898976i	$-0.644312\pi$
$167$	−11.3203	−0.875995	−0.437997	+	0.898976i	$0.644312\pi$
$168$	0	0
$169$	8.17626	0.628943
$170$	56.2567	4.31469
$171$	0	0
$172$	25.3956	1.93640
$173$	14.3600	1.09177	0.545884	−	0.837861i	$-0.316194\pi$
$173$	14.3600	1.09177	0.545884	+	0.837861i	$0.316194\pi$
$174$	0	0
$175$	−35.3353	−2.67110
$176$	−3.95463	−0.298092
$177$	0	0
$178$	−5.75768	−0.431556
$179$	−9.00995	−0.673435	−0.336718	−	0.941606i	$-0.609317\pi$
$179$	−9.00995	−0.673435	−0.336718	+	0.941606i	$0.609317\pi$
$180$	0	0
$181$	−24.6614	−1.83306	−0.916532	−	0.399961i	$-0.869024\pi$
$181$	−24.6614	−1.83306	−0.916532	+	0.399961i	$0.869024\pi$
$182$	−30.7251	−2.27749
$183$	0	0
$184$	−23.3197	−1.71915
$185$	−16.8262	−1.23708
$186$	0	0
$187$	−14.1809	−1.03701
$188$	15.0893	1.10050
$189$	0	0
$190$	5.35636	0.388591
$191$	−0.990564	−0.0716747	−0.0358374	−	0.999358i	$-0.511410\pi$
$191$	−0.990564	−0.0716747	−0.0358374	+	0.999358i	$0.511410\pi$
$192$	0	0
$193$	−22.7042	−1.63428	−0.817141	−	0.576438i	$-0.804442\pi$
$193$	−22.7042	−1.63428	−0.817141	+	0.576438i	$0.804442\pi$
$194$	12.6393	0.907449
$195$	0	0
$196$	3.59455	0.256754
$197$	−10.6866	−0.761389	−0.380694	−	0.924701i	$-0.624315\pi$
$197$	−10.6866	−0.761389	−0.380694	+	0.924701i	$0.624315\pi$
$198$	0	0
$199$	−3.22512	−0.228623	−0.114311	−	0.993445i	$-0.536466\pi$
$199$	−3.22512	−0.228623	−0.114311	+	0.993445i	$0.536466\pi$
$200$	46.1709	3.26477
$201$	0	0
$202$	−38.5721	−2.71392
$203$	−11.8001	−0.828207
$204$	0	0
$205$	8.56544	0.598236
$206$	−27.8562	−1.94083
$207$	0	0
$208$	7.31693	0.507337
$209$	−1.35021	−0.0933958
$210$	0	0
$211$	−5.91177	−0.406983	−0.203491	−	0.979077i	$-0.565229\pi$
$211$	−5.91177	−0.406983	−0.203491	+	0.979077i	$0.565229\pi$
$212$	30.6981	2.10835
$213$	0	0
$214$	28.4072	1.94188
$215$	29.7714	2.03039
$216$	0	0
$217$	−14.2605	−0.968065
$218$	3.29049	0.222860
$219$	0	0
$220$	−37.1001	−2.50129
$221$	26.2378	1.76494
$222$	0	0
$223$	−12.2358	−0.819367	−0.409684	−	0.912228i	$-0.634361\pi$
$223$	−12.2358	−0.819367	−0.409684	+	0.912228i	$0.634361\pi$
$224$	10.3102	0.688880
$225$	0	0
$226$	−9.83157	−0.653987
$227$	−0.447253	−0.0296852	−0.0148426	−	0.999890i	$-0.504725\pi$
$227$	−0.447253	−0.0296852	−0.0148426	+	0.999890i	$0.504725\pi$
$228$	0	0
$229$	−11.3140	−0.747652	−0.373826	−	0.927499i	$-0.621954\pi$
$229$	−11.3140	−0.747652	−0.373826	+	0.927499i	$0.621954\pi$
$230$	−62.2273	−4.10315
$231$	0	0
$232$	15.4186	1.01228
$233$	−19.0486	−1.24792	−0.623958	−	0.781458i	$-0.714476\pi$
$233$	−19.0486	−1.24792	−0.623958	+	0.781458i	$0.714476\pi$
$234$	0	0
$235$	17.6893	1.15392
$236$	−7.68609	−0.500322
$237$	0	0
$238$	−38.0690	−2.46764
$239$	−4.17495	−0.270055	−0.135027	−	0.990842i	$-0.543112\pi$
$239$	−4.17495	−0.270055	−0.135027	+	0.990842i	$0.543112\pi$
$240$	0	0
$241$	4.77764	0.307755	0.153878	−	0.988090i	$-0.450824\pi$
$241$	4.77764	0.307755	0.153878	+	0.988090i	$0.450824\pi$
$242$	−11.3587	−0.730164
$243$	0	0
$244$	−30.4411	−1.94879
$245$	4.21392	0.269217
$246$	0	0
$247$	2.49817	0.158955
$248$	18.6334	1.18322
$249$	0	0
$250$	73.8711	4.67202
$251$	−4.20802	−0.265608	−0.132804	−	0.991142i	$-0.542398\pi$
$251$	−4.20802	−0.265608	−0.132804	+	0.991142i	$0.542398\pi$
$252$	0	0
$253$	15.6860	0.986169
$254$	20.2233	1.26892
$255$	0	0
$256$	−24.8154	−1.55096
$257$	−19.5268	−1.21805	−0.609023	−	0.793152i	$-0.708438\pi$
$257$	−19.5268	−1.21805	−0.609023	+	0.793152i	$0.708438\pi$
$258$	0	0
$259$	11.3863	0.707509
$260$	68.6431	4.25707
$261$	0	0
$262$	45.9742	2.84029
$263$	−5.05834	−0.311910	−0.155955	−	0.987764i	$-0.549846\pi$
$263$	−5.05834	−0.311910	−0.155955	+	0.987764i	$0.549846\pi$
$264$	0	0
$265$	35.9875	2.21070
$266$	−3.62465	−0.222242
$267$	0	0
$268$	−12.9514	−0.791131
$269$	16.9375	1.03270	0.516349	−	0.856378i	$-0.327291\pi$
$269$	16.9375	1.03270	0.516349	+	0.856378i	$0.327291\pi$
$270$	0	0
$271$	−11.5325	−0.700547	−0.350274	−	0.936647i	$-0.613911\pi$
$271$	−11.5325	−0.700547	−0.350274	+	0.936647i	$0.613911\pi$
$272$	9.06581	0.549695
$273$	0	0
$274$	−39.2910	−2.37365
$275$	−31.0569	−1.87280
$276$	0	0
$277$	−14.9882	−0.900556	−0.450278	−	0.892889i	$-0.648675\pi$
$277$	−14.9882	−0.900556	−0.450278	+	0.892889i	$0.648675\pi$
$278$	−17.4908	−1.04903
$279$	0	0
$280$	−43.7545	−2.61483
$281$	25.9673	1.54908	0.774541	−	0.632524i	$-0.217981\pi$
$281$	25.9673	1.54908	0.774541	+	0.632524i	$0.217981\pi$
$282$	0	0
$283$	−27.6595	−1.64419	−0.822094	−	0.569351i	$-0.807194\pi$
$283$	−27.6595	−1.64419	−0.822094	+	0.569351i	$0.807194\pi$
$284$	−26.5384	−1.57477
$285$	0	0
$286$	−27.0048	−1.59683
$287$	−5.79625	−0.342142
$288$	0	0
$289$	15.5091	0.912300
$290$	41.1438	2.41605
$291$	0	0
$292$	38.8209	2.27182
$293$	−0.415058	−0.0242480	−0.0121240	−	0.999927i	$-0.503859\pi$
$293$	−0.415058	−0.0242480	−0.0121240	+	0.999927i	$0.503859\pi$
$294$	0	0
$295$	−9.01045	−0.524609
$296$	−14.8779	−0.864759
$297$	0	0
$298$	48.2455	2.79479
$299$	−29.0225	−1.67841
$300$	0	0
$301$	−20.1463	−1.16122
$302$	36.5233	2.10168
$303$	0	0
$304$	0.863182	0.0495069
$305$	−35.6862	−2.04339
$306$	0	0
$307$	−5.03206	−0.287195	−0.143597	−	0.989636i	$-0.545867\pi$
$307$	−5.03206	−0.287195	−0.143597	+	0.989636i	$0.545867\pi$
$308$	25.1057	1.43053
$309$	0	0
$310$	49.7224	2.82404
$311$	7.79527	0.442029	0.221015	−	0.975270i	$-0.429063\pi$
$311$	7.79527	0.442029	0.221015	+	0.975270i	$0.429063\pi$
$312$	0	0
$313$	−13.4305	−0.759134	−0.379567	−	0.925164i	$-0.623927\pi$
$313$	−13.4305	−0.759134	−0.379567	+	0.925164i	$0.623927\pi$
$314$	−38.9416	−2.19760
$315$	0	0
$316$	17.1954	0.967315
$317$	9.15464	0.514176	0.257088	−	0.966388i	$-0.417237\pi$
$317$	9.15464	0.514176	0.257088	+	0.966388i	$0.417237\pi$
$318$	0	0
$319$	−10.3713	−0.580684
$320$	−49.2470	−2.75299
$321$	0	0
$322$	42.1093	2.34666
$323$	3.09529	0.172226
$324$	0	0
$325$	57.4619	3.18741
$326$	6.71557	0.371941
$327$	0	0
$328$	7.57366	0.418185
$329$	−11.9703	−0.659947
$330$	0	0
$331$	9.02794	0.496221	0.248110	−	0.968732i	$-0.420190\pi$
$331$	9.02794	0.496221	0.248110	+	0.968732i	$0.420190\pi$
$332$	−39.2445	−2.15382
$333$	0	0
$334$	−26.7100	−1.46151
$335$	−15.1830	−0.829534
$336$	0	0
$337$	−29.1258	−1.58658	−0.793292	−	0.608841i	$-0.791635\pi$
$337$	−29.1258	−1.58658	−0.793292	+	0.608841i	$0.791635\pi$
$338$	19.2917	1.04933
$339$	0	0
$340$	85.0501	4.61249
$341$	−12.5338	−0.678743
$342$	0	0
$343$	16.9569	0.915589
$344$	26.3242	1.41931
$345$	0	0
$346$	33.8819	1.82150
$347$	2.77201	0.148810	0.0744048	−	0.997228i	$-0.476294\pi$
$347$	2.77201	0.148810	0.0744048	+	0.997228i	$0.476294\pi$
$348$	0	0
$349$	1.73834	0.0930509	0.0465255	−	0.998917i	$-0.485185\pi$
$349$	1.73834	0.0930509	0.0465255	+	0.998917i	$0.485185\pi$
$350$	−83.3727	−4.45646
$351$	0	0
$352$	9.06183	0.482997
$353$	25.4703	1.35565	0.677825	−	0.735224i	$-0.262923\pi$
$353$	25.4703	1.35565	0.677825	+	0.735224i	$0.262923\pi$
$354$	0	0
$355$	−31.1111	−1.65121
$356$	−8.70459	−0.461342
$357$	0	0
$358$	−21.2587	−1.12356
$359$	32.6283	1.72205	0.861027	−	0.508559i	$-0.169822\pi$
$359$	32.6283	1.72205	0.861027	+	0.508559i	$0.169822\pi$
$360$	0	0
$361$	−18.7053	−0.984489
$362$	−58.1878	−3.05828
$363$	0	0
$364$	−46.4509	−2.43469
$365$	45.5099	2.38210
$366$	0	0
$367$	6.81294	0.355632	0.177816	−	0.984064i	$-0.443097\pi$
$367$	6.81294	0.355632	0.177816	+	0.984064i	$0.443097\pi$
$368$	−10.0280	−0.522745
$369$	0	0
$370$	−39.7008	−2.06395
$371$	−24.3528	−1.26433
$372$	0	0
$373$	10.6694	0.552440	0.276220	−	0.961095i	$-0.410918\pi$
$373$	10.6694	0.552440	0.276220	+	0.961095i	$0.410918\pi$
$374$	−33.4595	−1.73015
$375$	0	0
$376$	15.6410	0.806625
$377$	19.1892	0.988296
$378$	0	0
$379$	22.2741	1.14415	0.572073	−	0.820203i	$-0.306139\pi$
$379$	22.2741	1.14415	0.572073	+	0.820203i	$0.306139\pi$
$380$	8.09787	0.415412
$381$	0	0
$382$	−2.33721	−0.119582
$383$	2.85301	0.145782	0.0728911	−	0.997340i	$-0.476777\pi$
$383$	2.85301	0.145782	0.0728911	+	0.997340i	$0.476777\pi$
$384$	0	0
$385$	29.4315	1.49997
$386$	−53.5698	−2.72663
$387$	0	0
$388$	19.1084	0.970082
$389$	12.7913	0.648545	0.324273	−	0.945964i	$-0.394881\pi$
$389$	12.7913	0.648545	0.324273	+	0.945964i	$0.394881\pi$
$390$	0	0
$391$	−35.9594	−1.81854
$392$	3.72599	0.188191
$393$	0	0
$394$	−25.2147	−1.27030
$395$	20.1582	1.01427
$396$	0	0
$397$	0.584024	0.0293113	0.0146557	−	0.999893i	$-0.495335\pi$
$397$	0.584024	0.0293113	0.0146557	+	0.999893i	$0.495335\pi$
$398$	−7.60958	−0.381434
$399$	0	0
$400$	19.8545	0.992726
$401$	−1.23150	−0.0614983	−0.0307492	−	0.999527i	$-0.509789\pi$
$401$	−1.23150	−0.0614983	−0.0307492	+	0.999527i	$0.509789\pi$
$402$	0	0
$403$	23.1902	1.15519
$404$	−58.3142	−2.90124
$405$	0	0
$406$	−27.8421	−1.38178
$407$	10.0076	0.496059
$408$	0	0
$409$	−1.18813	−0.0587492	−0.0293746	−	0.999568i	$-0.509352\pi$
$409$	−1.18813	−0.0587492	−0.0293746	+	0.999568i	$0.509352\pi$
$410$	20.2099	0.998096
$411$	0	0
$412$	−42.1136	−2.07479
$413$	6.09739	0.300033
$414$	0	0
$415$	−46.0065	−2.25837
$416$	−16.7663	−0.822038
$417$	0	0
$418$	−3.18577	−0.155821
$419$	2.74974	0.134333	0.0671667	−	0.997742i	$-0.478604\pi$
$419$	2.74974	0.134333	0.0671667	+	0.997742i	$0.478604\pi$
$420$	0	0
$421$	26.4727	1.29020	0.645100	−	0.764098i	$-0.276816\pi$
$421$	26.4727	1.29020	0.645100	+	0.764098i	$0.276816\pi$
$422$	−13.9486	−0.679009
$423$	0	0
$424$	31.8206	1.54534
$425$	71.1964	3.45353
$426$	0	0
$427$	24.1489	1.16865
$428$	42.9467	2.07591
$429$	0	0
$430$	70.2447	3.38750
$431$	14.7022	0.708179	0.354090	−	0.935212i	$-0.384791\pi$
$431$	14.7022	0.708179	0.354090	+	0.935212i	$0.384791\pi$
$432$	0	0
$433$	34.4192	1.65408	0.827040	−	0.562143i	$-0.190023\pi$
$433$	34.4192	1.65408	0.827040	+	0.562143i	$0.190023\pi$
$434$	−33.6472	−1.61512
$435$	0	0
$436$	4.97464	0.238242
$437$	−3.42380	−0.163782
$438$	0	0
$439$	17.3151	0.826403	0.413202	−	0.910640i	$-0.364411\pi$
$439$	17.3151	0.826403	0.413202	+	0.910640i	$0.364411\pi$
$440$	−38.4566	−1.83335
$441$	0	0
$442$	61.9073	2.94463
$443$	−19.9059	−0.945757	−0.472878	−	0.881128i	$-0.656785\pi$
$443$	−19.9059	−0.945757	−0.472878	+	0.881128i	$0.656785\pi$
$444$	0	0
$445$	−10.2044	−0.483737
$446$	−28.8699	−1.36703
$447$	0	0
$448$	33.3255	1.57448
$449$	9.26847	0.437406	0.218703	−	0.975791i	$-0.429817\pi$
$449$	9.26847	0.437406	0.218703	+	0.975791i	$0.429817\pi$
$450$	0	0
$451$	−5.09443	−0.239887
$452$	−14.8636	−0.699125
$453$	0	0
$454$	−1.05528	−0.0495268
$455$	−54.4547	−2.55287
$456$	0	0
$457$	−9.70114	−0.453800	−0.226900	−	0.973918i	$-0.572859\pi$
$457$	−9.70114	−0.453800	−0.226900	+	0.973918i	$0.572859\pi$
$458$	−26.6951	−1.24738
$459$	0	0
$460$	−94.0767	−4.38635
$461$	−25.5277	−1.18895	−0.594473	−	0.804116i	$-0.702639\pi$
$461$	−25.5277	−1.18895	−0.594473	+	0.804116i	$0.702639\pi$
$462$	0	0
$463$	−38.9438	−1.80987	−0.904937	−	0.425546i	$-0.860082\pi$
$463$	−38.9438	−1.80987	−0.904937	+	0.425546i	$0.860082\pi$
$464$	6.63036	0.307807
$465$	0	0
$466$	−44.9447	−2.08202
$467$	14.9435	0.691501	0.345751	−	0.938326i	$-0.387624\pi$
$467$	14.9435	0.691501	0.345751	+	0.938326i	$0.387624\pi$
$468$	0	0
$469$	10.2743	0.474425
$470$	41.7373	1.92520
$471$	0	0
$472$	−7.96714	−0.366717
$473$	−17.7070	−0.814168
$474$	0	0
$475$	6.77881	0.311033
$476$	−57.5535	−2.63796
$477$	0	0
$478$	−9.85067	−0.450559
$479$	−3.22823	−0.147502	−0.0737508	−	0.997277i	$-0.523497\pi$
$479$	−3.22823	−0.147502	−0.0737508	+	0.997277i	$0.523497\pi$
$480$	0	0
$481$	−18.5162	−0.844268
$482$	11.2727	0.513458
$483$	0	0
$484$	−17.1723	−0.780560
$485$	22.4009	1.01717
$486$	0	0
$487$	−7.84073	−0.355297	−0.177649	−	0.984094i	$-0.556849\pi$
$487$	−7.84073	−0.355297	−0.177649	+	0.984094i	$0.556849\pi$
$488$	−31.5541	−1.42839
$489$	0	0
$490$	9.94261	0.449161
$491$	−18.3532	−0.828269	−0.414134	−	0.910216i	$-0.635916\pi$
$491$	−18.3532	−0.828269	−0.414134	+	0.910216i	$0.635916\pi$
$492$	0	0
$493$	23.7758	1.07081
$494$	5.89437	0.265200
$495$	0	0
$496$	8.01280	0.359785
$497$	21.0529	0.944354
$498$	0	0
$499$	26.3880	1.18129	0.590644	−	0.806932i	$-0.298874\pi$
$499$	26.3880	1.18129	0.590644	+	0.806932i	$0.298874\pi$
$500$	111.680	4.99448
$501$	0	0
$502$	−9.92869	−0.443139
$503$	5.59404	0.249426	0.124713	−	0.992193i	$-0.460199\pi$
$503$	5.59404	0.249426	0.124713	+	0.992193i	$0.460199\pi$
$504$	0	0
$505$	−68.3620	−3.04207
$506$	37.0106	1.64532
$507$	0	0
$508$	30.5740	1.35650
$509$	−38.7081	−1.71571	−0.857853	−	0.513895i	$-0.828202\pi$
$509$	−38.7081	−1.71571	−0.857853	+	0.513895i	$0.828202\pi$
$510$	0	0
$511$	−30.7966	−1.36236
$512$	−17.5515	−0.775676
$513$	0	0
$514$	−46.0729	−2.03219
$515$	−49.3700	−2.17550
$516$	0	0
$517$	−10.5210	−0.462711
$518$	26.8656	1.18041
$519$	0	0
$520$	71.1531	3.12027
$521$	−5.16331	−0.226209	−0.113104	−	0.993583i	$-0.536079\pi$
$521$	−5.16331	−0.226209	−0.113104	+	0.993583i	$0.536079\pi$
$522$	0	0
$523$	32.8624	1.43697	0.718487	−	0.695541i	$-0.244835\pi$
$523$	32.8624	1.43697	0.718487	+	0.695541i	$0.244835\pi$
$524$	69.5048	3.03633
$525$	0	0
$526$	−11.9350	−0.520391
$527$	28.7331	1.25163
$528$	0	0
$529$	16.7758	0.729384
$530$	84.9115	3.68832
$531$	0	0
$532$	−5.47984	−0.237581
$533$	9.42579	0.408276
$534$	0	0
$535$	50.3467	2.17668
$536$	−13.4249	−0.579869
$537$	0	0
$538$	39.9636	1.72295
$539$	−2.50629	−0.107954
$540$	0	0
$541$	2.08381	0.0895898	0.0447949	−	0.998996i	$-0.485737\pi$
$541$	2.08381	0.0895898	0.0447949	+	0.998996i	$0.485737\pi$
$542$	−27.2105	−1.16879
$543$	0	0
$544$	−20.7738	−0.890670
$545$	5.83180	0.249807
$546$	0	0
$547$	−1.00000	−0.0427569
$547$	−1.00000	−0.0427569
$548$	−59.4010	−2.53749
$549$	0	0
$550$	−73.2778	−3.12457
$551$	2.26376	0.0964396
$552$	0	0
$553$	−13.6411	−0.580079
$554$	−35.3643	−1.50249
$555$	0	0
$556$	−26.4430	−1.12143
$557$	29.8940	1.26665	0.633324	−	0.773887i	$-0.281690\pi$
$557$	29.8940	1.26665	0.633324	+	0.773887i	$0.281690\pi$
$558$	0	0
$559$	32.7617	1.38567
$560$	−18.8154	−0.795098
$561$	0	0
$562$	61.2692	2.58449
$563$	−41.3819	−1.74404	−0.872020	−	0.489470i	$-0.837190\pi$
$563$	−41.3819	−1.74404	−0.872020	+	0.489470i	$0.837190\pi$
$564$	0	0
$565$	−17.4247	−0.733062
$566$	−65.2619	−2.74316
$567$	0	0
$568$	−27.5088	−1.15424
$569$	−34.2607	−1.43628	−0.718141	−	0.695898i	$-0.755007\pi$
$569$	−34.2607	−1.43628	−0.718141	+	0.695898i	$0.755007\pi$
$570$	0	0
$571$	−22.9912	−0.962151	−0.481075	−	0.876679i	$-0.659754\pi$
$571$	−22.9912	−0.962151	−0.481075	+	0.876679i	$0.659754\pi$
$572$	−40.8265	−1.70704
$573$	0	0
$574$	−13.6761	−0.570828
$575$	−78.7526	−3.28421
$576$	0	0
$577$	13.6464	0.568109	0.284054	−	0.958808i	$-0.408320\pi$
$577$	13.6464	0.568109	0.284054	+	0.958808i	$0.408320\pi$
$578$	36.5933	1.52208
$579$	0	0
$580$	62.2022	2.58280
$581$	31.1327	1.29160
$582$	0	0
$583$	−21.4041	−0.886468
$584$	40.2404	1.66516
$585$	0	0
$586$	−0.979318	−0.0404553
$587$	7.33098	0.302582	0.151291	−	0.988489i	$-0.451657\pi$
$587$	7.33098	0.302582	0.151291	+	0.988489i	$0.451657\pi$
$588$	0	0
$589$	2.73576	0.112725
$590$	−21.2599	−0.875256
$591$	0	0
$592$	−6.39782	−0.262949
$593$	−15.0502	−0.618037	−0.309018	−	0.951056i	$-0.600000\pi$
$593$	−15.0502	−0.618037	−0.309018	+	0.951056i	$0.600000\pi$
$594$	0	0
$595$	−67.4703	−2.76601
$596$	72.9387	2.98769
$597$	0	0
$598$	−68.4777	−2.80026
$599$	−26.1936	−1.07024	−0.535122	−	0.844775i	$-0.679734\pi$
$599$	−26.1936	−1.07024	−0.535122	+	0.844775i	$0.679734\pi$
$600$	0	0
$601$	−11.7023	−0.477345	−0.238673	−	0.971100i	$-0.576712\pi$
$601$	−11.7023	−0.477345	−0.238673	+	0.971100i	$0.576712\pi$
$602$	−47.5347	−1.93737
$603$	0	0
$604$	55.2168	2.24674
$605$	−20.1312	−0.818450
$606$	0	0
$607$	5.53127	0.224507	0.112254	−	0.993680i	$-0.464193\pi$
$607$	5.53127	0.224507	0.112254	+	0.993680i	$0.464193\pi$
$608$	−1.97793	−0.0802158
$609$	0	0
$610$	−84.2006	−3.40918
$611$	19.4660	0.787512
$612$	0	0
$613$	23.0113	0.929416	0.464708	−	0.885464i	$-0.346159\pi$
$613$	23.0113	0.929416	0.464708	+	0.885464i	$0.346159\pi$
$614$	−11.8730	−0.479155
$615$	0	0
$616$	26.0237	1.04852
$617$	18.8683	0.759610	0.379805	−	0.925067i	$-0.375991\pi$
$617$	18.8683	0.759610	0.379805	+	0.925067i	$0.375991\pi$
$618$	0	0
$619$	−6.14818	−0.247116	−0.123558	−	0.992337i	$-0.539431\pi$
$619$	−6.14818	−0.247116	−0.123558	+	0.992337i	$0.539431\pi$
$620$	75.1714	3.01896
$621$	0	0
$622$	18.3927	0.737481
$623$	6.90536	0.276657
$624$	0	0
$625$	68.4886	2.73954
$626$	−31.6888	−1.26654
$627$	0	0
$628$	−58.8728	−2.34928
$629$	−22.9420	−0.914757
$630$	0	0
$631$	−32.0198	−1.27469	−0.637344	−	0.770580i	$-0.719967\pi$
$631$	−32.0198	−1.27469	−0.637344	+	0.770580i	$0.719967\pi$
$632$	17.8241	0.709006
$633$	0	0
$634$	21.6001	0.857850
$635$	35.8421	1.42235
$636$	0	0
$637$	4.63718	0.183732
$638$	−24.4709	−0.968812
$639$	0	0
$640$	−85.7249	−3.38857
$641$	10.9848	0.433874	0.216937	−	0.976186i	$-0.430393\pi$
$641$	10.9848	0.433874	0.216937	+	0.976186i	$0.430393\pi$
$642$	0	0
$643$	30.7503	1.21267	0.606336	−	0.795209i	$-0.292639\pi$
$643$	30.7503	1.21267	0.606336	+	0.795209i	$0.292639\pi$
$644$	63.6618	2.50863
$645$	0	0
$646$	7.30324	0.287342
$647$	26.5562	1.04403	0.522016	−	0.852936i	$-0.325180\pi$
$647$	26.5562	1.04403	0.522016	+	0.852936i	$0.325180\pi$
$648$	0	0
$649$	5.35910	0.210363
$650$	135.580	5.31787
$651$	0	0
$652$	10.1528	0.397613
$653$	7.83426	0.306578	0.153289	−	0.988181i	$-0.451013\pi$
$653$	7.83426	0.306578	0.153289	+	0.988181i	$0.451013\pi$
$654$	0	0
$655$	81.4809	3.18372
$656$	3.25684	0.127158
$657$	0	0
$658$	−28.2437	−1.10105
$659$	36.0459	1.40415	0.702074	−	0.712104i	$-0.252258\pi$
$659$	36.0459	1.40415	0.702074	+	0.712104i	$0.252258\pi$
$660$	0	0
$661$	−6.74525	−0.262360	−0.131180	−	0.991359i	$-0.541877\pi$
$661$	−6.74525	−0.262360	−0.131180	+	0.991359i	$0.541877\pi$
$662$	21.3012	0.827893
$663$	0	0
$664$	−40.6795	−1.57867
$665$	−6.42405	−0.249114
$666$	0	0
$667$	−26.2992	−1.01831
$668$	−40.3808	−1.56238
$669$	0	0
$670$	−35.8238	−1.38399
$671$	21.2249	0.819379
$672$	0	0
$673$	32.6652	1.25915	0.629576	−	0.776939i	$-0.283229\pi$
$673$	32.6652	1.25915	0.629576	+	0.776939i	$0.283229\pi$
$674$	−68.7216	−2.64706
$675$	0	0
$676$	29.1656	1.12175
$677$	28.5071	1.09562	0.547809	−	0.836603i	$-0.315462\pi$
$677$	28.5071	1.09562	0.547809	+	0.836603i	$0.315462\pi$
$678$	0	0
$679$	−15.1587	−0.581738
$680$	88.1600	3.38078
$681$	0	0
$682$	−29.5731	−1.13241
$683$	27.5926	1.05580	0.527900	−	0.849306i	$-0.322980\pi$
$683$	27.5926	1.05580	0.527900	+	0.849306i	$0.322980\pi$
$684$	0	0
$685$	−69.6362	−2.66066
$686$	40.0094	1.52757
$687$	0	0
$688$	11.3200	0.431571
$689$	39.6023	1.50873
$690$	0	0
$691$	−4.93442	−0.187714	−0.0938571	−	0.995586i	$-0.529920\pi$
$691$	−4.93442	−0.187714	−0.0938571	+	0.995586i	$0.529920\pi$
$692$	51.2235	1.94723
$693$	0	0
$694$	6.54049	0.248273
$695$	−30.9992	−1.17587
$696$	0	0
$697$	11.6787	0.442363
$698$	4.10155	0.155246
$699$	0	0
$700$	−126.045	−4.76405
$701$	16.8337	0.635799	0.317900	−	0.948124i	$-0.397023\pi$
$701$	16.8337	0.635799	0.317900	+	0.948124i	$0.397023\pi$
$702$	0	0
$703$	−2.18437	−0.0823851
$704$	29.2904	1.10392
$705$	0	0
$706$	60.0965	2.26176
$707$	46.2607	1.73981
$708$	0	0
$709$	−1.51372	−0.0568489	−0.0284244	−	0.999596i	$-0.509049\pi$
$709$	−1.51372	−0.0568489	−0.0284244	+	0.999596i	$0.509049\pi$
$710$	−73.4058	−2.75487
$711$	0	0
$712$	−9.02288	−0.338147
$713$	−31.7826	−1.19027
$714$	0	0
$715$	−47.8612	−1.78991
$716$	−32.1394	−1.20111
$717$	0	0
$718$	76.9855	2.87307
$719$	48.0138	1.79061	0.895306	−	0.445453i	$-0.146957\pi$
$719$	48.0138	1.79061	0.895306	+	0.445453i	$0.146957\pi$
$720$	0	0
$721$	33.4087	1.24421
$722$	−44.1346	−1.64252
$723$	0	0
$724$	−87.9697	−3.26937
$725$	52.0701	1.93383
$726$	0	0
$727$	−52.7641	−1.95691	−0.978456	−	0.206456i	$-0.933807\pi$
$727$	−52.7641	−1.95691	−0.978456	+	0.206456i	$0.933807\pi$
$728$	−48.1494	−1.78454
$729$	0	0
$730$	107.379	3.97429
$731$	40.5924	1.50136
$732$	0	0
$733$	−17.8293	−0.658541	−0.329271	−	0.944236i	$-0.606803\pi$
$733$	−17.8293	−0.658541	−0.329271	+	0.944236i	$0.606803\pi$
$734$	16.0749	0.593336
$735$	0	0
$736$	22.9786	0.847002
$737$	9.03030	0.332635
$738$	0	0
$739$	−36.1098	−1.32832	−0.664160	−	0.747591i	$-0.731211\pi$
$739$	−36.1098	−1.32832	−0.664160	+	0.747591i	$0.731211\pi$
$740$	−60.0207	−2.20640
$741$	0	0
$742$	−57.4598	−2.10941
$743$	0.881568	0.0323416	0.0161708	−	0.999869i	$-0.494852\pi$
$743$	0.881568	0.0323416	0.0161708	+	0.999869i	$0.494852\pi$
$744$	0	0
$745$	85.5065	3.13272
$746$	25.1741	0.921689
$747$	0	0
$748$	−50.5849	−1.84957
$749$	−34.0697	−1.24488
$750$	0	0
$751$	−13.6390	−0.497694	−0.248847	−	0.968543i	$-0.580052\pi$
$751$	−13.6390	−0.497694	−0.248847	+	0.968543i	$0.580052\pi$
$752$	6.72600	0.245272
$753$	0	0
$754$	45.2764	1.64887
$755$	64.7309	2.35580
$756$	0	0
$757$	−19.9724	−0.725910	−0.362955	−	0.931807i	$-0.618232\pi$
$757$	−19.9724	−0.725910	−0.362955	+	0.931807i	$0.618232\pi$
$758$	52.5552	1.90889
$759$	0	0
$760$	8.39397	0.304481
$761$	−44.3290	−1.60693	−0.803463	−	0.595355i	$-0.797011\pi$
$761$	−44.3290	−1.60693	−0.803463	+	0.595355i	$0.797011\pi$
$762$	0	0
$763$	−3.94639	−0.142869
$764$	−3.53345	−0.127836
$765$	0	0
$766$	6.73161	0.243223
$767$	−9.91550	−0.358028
$768$	0	0
$769$	−35.4396	−1.27799	−0.638993	−	0.769213i	$-0.720649\pi$
$769$	−35.4396	−1.27799	−0.638993	+	0.769213i	$0.720649\pi$
$770$	69.4428	2.50255
$771$	0	0
$772$	−80.9882	−2.91483
$773$	35.9274	1.29222	0.646110	−	0.763244i	$-0.276395\pi$
$773$	35.9274	1.29222	0.646110	+	0.763244i	$0.276395\pi$
$774$	0	0
$775$	62.9268	2.26040
$776$	19.8071	0.711033
$777$	0	0
$778$	30.1807	1.08203
$779$	1.11197	0.0398403
$780$	0	0
$781$	18.5038	0.662119
$782$	−84.8451	−3.03405
$783$	0	0
$784$	1.60226	0.0572236
$785$	−69.0169	−2.46332
$786$	0	0
$787$	4.68054	0.166843	0.0834216	−	0.996514i	$-0.473415\pi$
$787$	4.68054	0.166843	0.0834216	+	0.996514i	$0.473415\pi$
$788$	−38.1202	−1.35798
$789$	0	0
$790$	47.5628	1.69221
$791$	11.7913	0.419251
$792$	0	0
$793$	−39.2707	−1.39454
$794$	1.37799	0.0489030
$795$	0	0
$796$	−11.5043	−0.407761
$797$	−51.2468	−1.81526	−0.907628	−	0.419776i	$-0.862109\pi$
$797$	−51.2468	−1.81526	−0.907628	+	0.419776i	$0.862109\pi$
$798$	0	0
$799$	24.1188	0.853262
$800$	−45.4956	−1.60851
$801$	0	0
$802$	−2.90570	−0.102604
$803$	−27.0677	−0.955199
$804$	0	0
$805$	74.6311	2.63040
$806$	54.7167	1.92731
$807$	0	0
$808$	−60.4465	−2.12650
$809$	4.96718	0.174637	0.0873183	−	0.996180i	$-0.472170\pi$
$809$	4.96718	0.174637	0.0873183	+	0.996180i	$0.472170\pi$
$810$	0	0
$811$	5.27601	0.185266	0.0926328	−	0.995700i	$-0.470472\pi$
$811$	5.27601	0.185266	0.0926328	+	0.995700i	$0.470472\pi$
$812$	−42.0923	−1.47715
$813$	0	0
$814$	23.6127	0.827624
$815$	11.9021	0.416914
$816$	0	0
$817$	3.86492	0.135216
$818$	−2.80336	−0.0980170
$819$	0	0
$820$	30.5538	1.06699
$821$	−32.6165	−1.13832	−0.569162	−	0.822225i	$-0.692732\pi$
$821$	−32.6165	−1.13832	−0.569162	+	0.822225i	$0.692732\pi$
$822$	0	0
$823$	−41.0761	−1.43182	−0.715912	−	0.698191i	$-0.753989\pi$
$823$	−41.0761	−1.43182	−0.715912	+	0.698191i	$0.753989\pi$
$824$	−43.6535	−1.52074
$825$	0	0
$826$	14.3866	0.500574
$827$	17.2551	0.600019	0.300010	−	0.953936i	$-0.403010\pi$
$827$	17.2551	0.600019	0.300010	+	0.953936i	$0.403010\pi$
$828$	0	0
$829$	−7.35445	−0.255430	−0.127715	−	0.991811i	$-0.540764\pi$
$829$	−7.35445	−0.255430	−0.127715	+	0.991811i	$0.540764\pi$
$830$	−108.551	−3.76786
$831$	0	0
$832$	−54.1936	−1.87882
$833$	5.74555	0.199072
$834$	0	0
$835$	−47.3387	−1.63822
$836$	−4.81633	−0.166576
$837$	0	0
$838$	6.48793	0.224122
$839$	7.34393	0.253541	0.126770	−	0.991932i	$-0.459539\pi$
$839$	7.34393	0.253541	0.126770	+	0.991932i	$0.459539\pi$
$840$	0	0
$841$	−11.6113	−0.400391
$842$	62.4615	2.15257
$843$	0	0
$844$	−21.0879	−0.725875
$845$	34.1910	1.17621
$846$	0	0
$847$	13.6228	0.468086
$848$	13.6836	0.469896
$849$	0	0
$850$	167.986	5.76187
$851$	25.3769	0.869908
$852$	0	0
$853$	19.4141	0.664725	0.332362	−	0.943152i	$-0.392154\pi$
$853$	19.4141	0.664725	0.332362	+	0.943152i	$0.392154\pi$
$854$	56.9787	1.94977
$855$	0	0
$856$	44.5171	1.52156
$857$	−11.7234	−0.400462	−0.200231	−	0.979749i	$-0.564169\pi$
$857$	−11.7234	−0.400462	−0.200231	+	0.979749i	$0.564169\pi$
$858$	0	0
$859$	33.3903	1.13926	0.569632	−	0.821900i	$-0.307086\pi$
$859$	33.3903	1.13926	0.569632	+	0.821900i	$0.307086\pi$
$860$	106.198	3.62131
$861$	0	0
$862$	34.6894	1.18152
$863$	34.6843	1.18067	0.590334	−	0.807159i	$-0.298996\pi$
$863$	34.6843	1.18067	0.590334	+	0.807159i	$0.298996\pi$
$864$	0	0
$865$	60.0496	2.04175
$866$	81.2110	2.75966
$867$	0	0
$868$	−50.8686	−1.72659
$869$	−11.9894	−0.406713
$870$	0	0
$871$	−16.7080	−0.566129
$872$	5.15654	0.174623
$873$	0	0
$874$	−8.07835	−0.273254
$875$	−88.5959	−2.99509
$876$	0	0
$877$	11.7682	0.397384	0.198692	−	0.980062i	$-0.436331\pi$
$877$	11.7682	0.397384	0.198692	+	0.980062i	$0.436331\pi$
$878$	40.8544	1.37877
$879$	0	0
$880$	−16.5372	−0.557470
$881$	38.4286	1.29469	0.647346	−	0.762197i	$-0.275879\pi$
$881$	38.4286	1.29469	0.647346	+	0.762197i	$0.275879\pi$
$882$	0	0
$883$	−43.7693	−1.47296	−0.736478	−	0.676462i	$-0.763512\pi$
$883$	−43.7693	−1.47296	−0.736478	+	0.676462i	$0.763512\pi$
$884$	93.5929	3.14787
$885$	0	0
$886$	−46.9673	−1.57790
$887$	20.7665	0.697271	0.348635	−	0.937258i	$-0.386645\pi$
$887$	20.7665	0.697271	0.348635	+	0.937258i	$0.386645\pi$
$888$	0	0
$889$	−24.2544	−0.813466
$890$	−24.0771	−0.807066
$891$	0	0
$892$	−43.6462	−1.46138
$893$	2.29642	0.0768467
$894$	0	0
$895$	−37.6772	−1.25941
$896$	58.0102	1.93798
$897$	0	0
$898$	21.8687	0.729767
$899$	21.0142	0.700864
$900$	0	0
$901$	49.0679	1.63469
$902$	−12.0202	−0.400227
$903$	0	0
$904$	−15.4071	−0.512433
$905$	−103.127	−3.42807
$906$	0	0
$907$	12.9518	0.430056	0.215028	−	0.976608i	$-0.431016\pi$
$907$	12.9518	0.430056	0.215028	+	0.976608i	$0.431016\pi$
$908$	−1.59540	−0.0529451
$909$	0	0
$910$	−128.484	−4.25921
$911$	−5.57453	−0.184692	−0.0923462	−	0.995727i	$-0.529437\pi$
$911$	−5.57453	−0.184692	−0.0923462	+	0.995727i	$0.529437\pi$
$912$	0	0
$913$	27.3631	0.905585
$914$	−22.8896	−0.757119
$915$	0	0
$916$	−40.3583	−1.33348
$917$	−55.1382	−1.82082
$918$	0	0
$919$	−24.5634	−0.810273	−0.405137	−	0.914256i	$-0.632776\pi$
$919$	−24.5634	−0.810273	−0.405137	+	0.914256i	$0.632776\pi$
$920$	−97.5167	−3.21503
$921$	0	0
$922$	−60.2320	−1.98363
$923$	−34.2361	−1.12689
$924$	0	0
$925$	−50.2439	−1.65201
$926$	−91.8869	−3.01959
$927$	0	0
$928$	−15.1931	−0.498738
$929$	−28.8268	−0.945775	−0.472888	−	0.881123i	$-0.656788\pi$
$929$	−28.8268	−0.945775	−0.472888	+	0.881123i	$0.656788\pi$
$930$	0	0
$931$	0.547050	0.0179289
$932$	−67.9484	−2.22572
$933$	0	0
$934$	35.2587	1.15370
$935$	−59.3009	−1.93935
$936$	0	0
$937$	−15.5358	−0.507534	−0.253767	−	0.967265i	$-0.581670\pi$
$937$	−15.5358	−0.507534	−0.253767	+	0.967265i	$0.581670\pi$
$938$	24.2420	0.791529
$939$	0	0
$940$	63.0994	2.05808
$941$	53.4497	1.74241	0.871205	−	0.490919i	$-0.163339\pi$
$941$	53.4497	1.74241	0.871205	+	0.490919i	$0.163339\pi$
$942$	0	0
$943$	−12.9182	−0.420675
$944$	−3.42605	−0.111508
$945$	0	0
$946$	−41.7791	−1.35836
$947$	−26.4923	−0.860885	−0.430443	−	0.902618i	$-0.641643\pi$
$947$	−26.4923	−0.860885	−0.430443	+	0.902618i	$0.641643\pi$
$948$	0	0
$949$	50.0811	1.62570
$950$	15.9944	0.518927
$951$	0	0
$952$	−59.6580	−1.93353
$953$	26.3995	0.855162	0.427581	−	0.903977i	$-0.359366\pi$
$953$	26.3995	0.855162	0.427581	+	0.903977i	$0.359366\pi$
$954$	0	0
$955$	−4.14228	−0.134041
$956$	−14.8925	−0.481657
$957$	0	0
$958$	−7.61691	−0.246091
$959$	47.1229	1.52168
$960$	0	0
$961$	−5.60426	−0.180783
$962$	−43.6886	−1.40858
$963$	0	0
$964$	17.0424	0.548897
$965$	−94.9429	−3.05632
$966$	0	0
$967$	23.1774	0.745334	0.372667	−	0.927965i	$-0.378443\pi$
$967$	23.1774	0.745334	0.372667	+	0.927965i	$0.378443\pi$
$968$	−17.8002	−0.572121
$969$	0	0
$970$	52.8542	1.69705
$971$	−9.57093	−0.307146	−0.153573	−	0.988137i	$-0.549078\pi$
$971$	−9.57093	−0.307146	−0.153573	+	0.988137i	$0.549078\pi$
$972$	0	0
$973$	20.9772	0.672499
$974$	−18.5000	−0.592777
$975$	0	0
$976$	−13.5690	−0.434333
$977$	−15.5814	−0.498493	−0.249247	−	0.968440i	$-0.580183\pi$
$977$	−15.5814	−0.498493	−0.249247	+	0.968440i	$0.580183\pi$
$978$	0	0
$979$	6.06925	0.193974
$980$	15.0315	0.480163
$981$	0	0
$982$	−43.3039	−1.38188
$983$	45.6013	1.45445	0.727227	−	0.686397i	$-0.240809\pi$
$983$	45.6013	1.45445	0.727227	+	0.686397i	$0.240809\pi$
$984$	0	0
$985$	−44.6885	−1.42390
$986$	56.0984	1.78654
$987$	0	0
$988$	8.91125	0.283505
$989$	−44.9006	−1.42776
$990$	0	0
$991$	26.6593	0.846860	0.423430	−	0.905929i	$-0.360826\pi$
$991$	26.6593	0.846860	0.423430	+	0.905929i	$0.360826\pi$
$992$	−18.3609	−0.582959
$993$	0	0
$994$	49.6738	1.57556
$995$	−13.4866	−0.427554
$996$	0	0
$997$	−21.8151	−0.690891	−0.345446	−	0.938439i	$-0.612272\pi$
$997$	−21.8151	−0.690891	−0.345446	+	0.938439i	$0.612272\pi$
$998$	62.2616	1.97086
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4923.2.a.l.1.16		18
3.2	odd	2		547.2.a.b.1.3	✓	18
12.11	even	2		8752.2.a.s.1.18		18

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
547.2.a.b.1.3	✓	18	3.2	odd	2
4923.2.a.l.1.16		18	1.1	even	1	trivial
8752.2.a.s.1.18		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)

\(f(q)\)	\(=\)	\(q+2.35947 q^{2} +3.56710 q^{4} +4.18174 q^{5} -2.82979 q^{7} +3.69754 q^{8} +O(q^{10})\)	\(q+2.35947 q^{2} +3.56710 q^{4} +4.18174 q^{5} -2.82979 q^{7} +3.69754 q^{8} +9.86669 q^{10} -2.48715 q^{11} +4.60177 q^{13} -6.67680 q^{14} +1.59003 q^{16} +5.70168 q^{17} +0.542873 q^{19} +14.9167 q^{20} -5.86836 q^{22} -6.30681 q^{23} +12.4869 q^{25} +10.8577 q^{26} -10.0941 q^{28} +4.16997 q^{29} +5.03942 q^{31} -3.64346 q^{32} +13.4529 q^{34} -11.8334 q^{35} -4.02373 q^{37} +1.28089 q^{38} +15.4621 q^{40} +2.04830 q^{41} +7.11938 q^{43} -8.87193 q^{44} -14.8807 q^{46} +4.23012 q^{47} +1.00770 q^{49} +29.4625 q^{50} +16.4150 q^{52} +8.60588 q^{53} -10.4006 q^{55} -10.4632 q^{56} +9.83893 q^{58} -2.15472 q^{59} -8.53383 q^{61} +11.8904 q^{62} -11.7767 q^{64} +19.2434 q^{65} -3.63078 q^{67} +20.3385 q^{68} -27.9206 q^{70} -7.43976 q^{71} +10.8830 q^{73} -9.49386 q^{74} +1.93648 q^{76} +7.03811 q^{77} +4.82054 q^{79} +6.64907 q^{80} +4.83290 q^{82} -11.0018 q^{83} +23.8429 q^{85} +16.7980 q^{86} -9.19633 q^{88} -2.44024 q^{89} -13.0220 q^{91} -22.4970 q^{92} +9.98085 q^{94} +2.27015 q^{95} +5.35684 q^{97} +2.37763 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q + 4 q^{2} + 16 q^{4} + 27 q^{5} - 11 q^{7} + 12 q^{8}+O(q^{10}) \) 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

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