LMFDB - Embedded newform 3483.2.a.s.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3483,2,Mod(1,3483)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3483, 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("3483.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3483 = 3^{4} \cdot 43 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3483.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:	$27.8118950240$
Analytic rank:	$0$
Dimension:	$19$
Coefficient field:	$\mathbb{Q}[x]/(x^{19} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{19} - 4 x^{18} - 22 x^{17} + 100 x^{16} + 181 x^{15} - 1020 x^{14} - 619 x^{13} + 5458 x^{12} + \cdots + 9 $ x^19 - 4x^18 - 22x^17 + 100x^16 + 181x^15 - 1020x^14 - 619x^13 + 5458x^12 + 145x^11 - 16363x^10 + 4943x^9 + 27069x^8 - 13490x^7 - 22501x^6 + 13478x^5 + 7567x^4 - 4310x^3 - 855x^2 + 225x + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 387)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.55346$ of defining polynomial
Character	$\chi$	$=$	3483.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.55346 q^{2} +4.52016 q^{4} +2.79948 q^{5} -0.153982 q^{7} -6.43513 q^{8} +O(q^{10})$	\(q-2.55346 q^{2} +4.52016 q^{4} +2.79948 q^{5} -0.153982 q^{7} -6.43513 q^{8} -7.14837 q^{10} +3.71589 q^{11} -1.28451 q^{13} +0.393187 q^{14} +7.39154 q^{16} +0.563748 q^{17} -6.62305 q^{19} +12.6541 q^{20} -9.48839 q^{22} +6.04515 q^{23} +2.83710 q^{25} +3.27995 q^{26} -0.696024 q^{28} -8.68393 q^{29} -1.69358 q^{31} -6.00374 q^{32} -1.43951 q^{34} -0.431070 q^{35} +4.07664 q^{37} +16.9117 q^{38} -18.0150 q^{40} +11.1145 q^{41} +1.00000 q^{43} +16.7964 q^{44} -15.4361 q^{46} -7.38495 q^{47} -6.97629 q^{49} -7.24443 q^{50} -5.80620 q^{52} +2.30102 q^{53} +10.4026 q^{55} +0.990895 q^{56} +22.1741 q^{58} +5.89963 q^{59} -4.15013 q^{61} +4.32448 q^{62} +0.547229 q^{64} -3.59597 q^{65} +10.6879 q^{67} +2.54823 q^{68} +1.10072 q^{70} +12.6183 q^{71} +1.47133 q^{73} -10.4095 q^{74} -29.9373 q^{76} -0.572181 q^{77} +8.32840 q^{79} +20.6925 q^{80} -28.3805 q^{82} +6.97738 q^{83} +1.57820 q^{85} -2.55346 q^{86} -23.9123 q^{88} +15.8842 q^{89} +0.197792 q^{91} +27.3251 q^{92} +18.8572 q^{94} -18.5411 q^{95} +16.5463 q^{97} +17.8137 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 19 q + 4 q^{2} + 22 q^{4} + 9 q^{5} + 7 q^{7} + 12 q^{8}+O(q^{10}) $ 19 * q + 4 * q^2 + 22 * q^4 + 9 * q^5 + 7 * q^7 + 12 * q^8	$ 19 q + 4 q^{2} + 22 q^{4} + 9 q^{5} + 7 q^{7} + 12 q^{8} - 7 q^{10} + 5 q^{11} - 5 q^{13} + 17 q^{14} + 24 q^{16} + 21 q^{17} - 4 q^{19} + 21 q^{20} - 20 q^{22} + 22 q^{23} + 10 q^{25} + 17 q^{26} - q^{28} + 30 q^{29} - 5 q^{31} + 48 q^{32} - 6 q^{34} + 53 q^{35} - q^{37} + 21 q^{38} + 16 q^{40} + 29 q^{41} + 19 q^{43} + 29 q^{44} + 32 q^{47} - 10 q^{49} - 11 q^{50} + q^{52} + 38 q^{53} + 2 q^{55} + 46 q^{56} + 30 q^{58} + 30 q^{59} - 10 q^{61} + 25 q^{62} + 14 q^{64} + 8 q^{65} + 3 q^{67} + 47 q^{68} + 56 q^{70} + 21 q^{71} + 8 q^{73} + 28 q^{74} - 36 q^{76} + 49 q^{77} + 4 q^{79} + 70 q^{80} - 4 q^{82} + 29 q^{83} - 4 q^{85} + 4 q^{86} - 47 q^{88} + 54 q^{89} + 4 q^{91} + 12 q^{92} - 23 q^{94} + 33 q^{95} - 4 q^{97} + 49 q^{98}+O(q^{100}) $ 19 * q + 4 * q^2 + 22 * q^4 + 9 * q^5 + 7 * q^7 + 12 * q^8 - 7 * q^10 + 5 * q^11 - 5 * q^13 + 17 * q^14 + 24 * q^16 + 21 * q^17 - 4 * q^19 + 21 * q^20 - 20 * q^22 + 22 * q^23 + 10 * q^25 + 17 * q^26 - q^28 + 30 * q^29 - 5 * q^31 + 48 * q^32 - 6 * q^34 + 53 * q^35 - q^37 + 21 * q^38 + 16 * q^40 + 29 * q^41 + 19 * q^43 + 29 * q^44 + 32 * q^47 - 10 * q^49 - 11 * q^50 + q^52 + 38 * q^53 + 2 * q^55 + 46 * q^56 + 30 * q^58 + 30 * q^59 - 10 * q^61 + 25 * q^62 + 14 * q^64 + 8 * q^65 + 3 * q^67 + 47 * q^68 + 56 * q^70 + 21 * q^71 + 8 * q^73 + 28 * q^74 - 36 * q^76 + 49 * q^77 + 4 * q^79 + 70 * q^80 - 4 * q^82 + 29 * q^83 - 4 * q^85 + 4 * q^86 - 47 * q^88 + 54 * q^89 + 4 * q^91 + 12 * q^92 - 23 * q^94 + 33 * q^95 - 4 * q^97 + 49 * 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 $

$2$	−2.55346	−1.80557	−0.902785	−	0.430093i	$-0.858481\pi$
$2$	−2.55346	−1.80557	−0.902785	+	0.430093i	$0.858481\pi$
$3$	0	0
$3$	0	0
$4$	4.52016	2.26008
$5$	2.79948	1.25197	0.625983	−	0.779836i	$-0.284698\pi$
$5$	2.79948	1.25197	0.625983	+	0.779836i	$0.284698\pi$
$6$	0	0
$7$	−0.153982	−0.0581997	−0.0290999	−	0.999577i	$-0.509264\pi$
$7$	−0.153982	−0.0581997	−0.0290999	+	0.999577i	$0.509264\pi$
$8$	−6.43513	−2.27516
$9$	0	0
$10$	−7.14837	−2.26051
$11$	3.71589	1.12038	0.560192	−	0.828363i	$-0.310727\pi$
$11$	3.71589	1.12038	0.560192	+	0.828363i	$0.310727\pi$
$12$	0	0
$13$	−1.28451	−0.356259	−0.178130	−	0.984007i	$-0.557005\pi$
$13$	−1.28451	−0.356259	−0.178130	+	0.984007i	$0.557005\pi$
$14$	0.393187	0.105084
$15$	0	0
$16$	7.39154	1.84788
$17$	0.563748	0.136729	0.0683645	−	0.997660i	$-0.478222\pi$
$17$	0.563748	0.136729	0.0683645	+	0.997660i	$0.478222\pi$
$18$	0	0
$19$	−6.62305	−1.51943	−0.759716	−	0.650255i	$-0.774662\pi$
$19$	−6.62305	−1.51943	−0.759716	+	0.650255i	$0.774662\pi$
$20$	12.6541	2.82955
$21$	0	0
$22$	−9.48839	−2.02293
$23$	6.04515	1.26050	0.630251	−	0.776392i	$-0.282952\pi$
$23$	6.04515	1.26050	0.630251	+	0.776392i	$0.282952\pi$
$24$	0	0
$25$	2.83710	0.567421
$26$	3.27995	0.643251
$27$	0	0
$28$	−0.696024	−0.131536
$29$	−8.68393	−1.61257	−0.806283	−	0.591530i	$-0.798524\pi$
$29$	−8.68393	−1.61257	−0.806283	+	0.591530i	$0.798524\pi$
$30$	0	0
$31$	−1.69358	−0.304175	−0.152088	−	0.988367i	$-0.548600\pi$
$31$	−1.69358	−0.304175	−0.152088	+	0.988367i	$0.548600\pi$
$32$	−6.00374	−1.06132
$33$	0	0
$34$	−1.43951	−0.246874
$35$	−0.431070	−0.0728641
$36$	0	0
$37$	4.07664	0.670195	0.335097	−	0.942183i	$-0.391231\pi$
$37$	4.07664	0.670195	0.335097	+	0.942183i	$0.391231\pi$
$38$	16.9117	2.74344
$39$	0	0
$40$	−18.0150	−2.84843
$41$	11.1145	1.73580	0.867899	−	0.496741i	$-0.165470\pi$
$41$	11.1145	1.73580	0.867899	+	0.496741i	$0.165470\pi$
$42$	0	0
$43$	1.00000	0.152499
$43$	1.00000	0.152499
$44$	16.7964	2.53216
$45$	0	0
$46$	−15.4361	−2.27592
$47$	−7.38495	−1.07721	−0.538603	−	0.842560i	$-0.681048\pi$
$47$	−7.38495	−1.07721	−0.538603	+	0.842560i	$0.681048\pi$
$48$	0	0
$49$	−6.97629	−0.996613
$50$	−7.24443	−1.02452
$51$	0	0
$52$	−5.80620	−0.805175
$53$	2.30102	0.316069	0.158034	−	0.987434i	$-0.449484\pi$
$53$	2.30102	0.316069	0.158034	+	0.987434i	$0.449484\pi$
$54$	0	0
$55$	10.4026	1.40268
$56$	0.990895	0.132414
$57$	0	0
$58$	22.1741	2.91160
$59$	5.89963	0.768067	0.384033	−	0.923319i	$-0.374535\pi$
$59$	5.89963	0.768067	0.384033	+	0.923319i	$0.374535\pi$
$60$	0	0
$61$	−4.15013	−0.531370	−0.265685	−	0.964060i	$-0.585598\pi$
$61$	−4.15013	−0.531370	−0.265685	+	0.964060i	$0.585598\pi$
$62$	4.32448	0.549210
$63$	0	0
$64$	0.547229	0.0684037
$65$	−3.59597	−0.446025
$66$	0	0
$67$	10.6879	1.30573	0.652865	−	0.757474i	$-0.273567\pi$
$67$	10.6879	1.30573	0.652865	+	0.757474i	$0.273567\pi$
$68$	2.54823	0.309019
$69$	0	0
$70$	1.10072	0.131561
$71$	12.6183	1.49752	0.748758	−	0.662843i	$-0.230650\pi$
$71$	12.6183	1.49752	0.748758	+	0.662843i	$0.230650\pi$
$72$	0	0
$73$	1.47133	0.172206	0.0861030	−	0.996286i	$-0.472559\pi$
$73$	1.47133	0.172206	0.0861030	+	0.996286i	$0.472559\pi$
$74$	−10.4095	−1.21008
$75$	0	0
$76$	−29.9373	−3.43404
$77$	−0.572181	−0.0652061
$78$	0	0
$79$	8.32840	0.937018	0.468509	−	0.883459i	$-0.344791\pi$
$79$	8.32840	0.937018	0.468509	+	0.883459i	$0.344791\pi$
$80$	20.6925	2.31349
$81$	0	0
$82$	−28.3805	−3.13410
$83$	6.97738	0.765867	0.382933	−	0.923776i	$-0.374914\pi$
$83$	6.97738	0.765867	0.382933	+	0.923776i	$0.374914\pi$
$84$	0	0
$85$	1.57820	0.171180
$86$	−2.55346	−0.275347
$87$	0	0
$88$	−23.9123	−2.54906
$89$	15.8842	1.68373	0.841864	−	0.539690i	$-0.181459\pi$
$89$	15.8842	1.68373	0.841864	+	0.539690i	$0.181459\pi$
$90$	0	0
$91$	0.197792	0.0207342
$92$	27.3251	2.84884
$93$	0	0
$94$	18.8572	1.94497
$95$	−18.5411	−1.90228
$96$	0	0
$97$	16.5463	1.68003	0.840013	−	0.542566i	$-0.182547\pi$
$97$	16.5463	1.68003	0.840013	+	0.542566i	$0.182547\pi$
$98$	17.8137	1.79945
$99$	0	0
$100$	12.8242	1.28242
$101$	11.1988	1.11432	0.557162	−	0.830404i	$-0.311890\pi$
$101$	11.1988	1.11432	0.557162	+	0.830404i	$0.311890\pi$
$102$	0	0
$103$	14.3840	1.41729	0.708647	−	0.705563i	$-0.249306\pi$
$103$	14.3840	1.41729	0.708647	+	0.705563i	$0.249306\pi$
$104$	8.26601	0.810549
$105$	0	0
$106$	−5.87556	−0.570684
$107$	−10.3317	−0.998801	−0.499400	−	0.866371i	$-0.666446\pi$
$107$	−10.3317	−0.998801	−0.499400	+	0.866371i	$0.666446\pi$
$108$	0	0
$109$	8.69073	0.832421	0.416211	−	0.909268i	$-0.363358\pi$
$109$	8.69073	0.832421	0.416211	+	0.909268i	$0.363358\pi$
$110$	−26.5626	−2.53264
$111$	0	0
$112$	−1.13816	−0.107546
$113$	5.39658	0.507667	0.253834	−	0.967248i	$-0.418308\pi$
$113$	5.39658	0.507667	0.253834	+	0.967248i	$0.418308\pi$
$114$	0	0
$115$	16.9233	1.57811
$116$	−39.2528	−3.64453
$117$	0	0
$118$	−15.0645	−1.38680
$119$	−0.0868071	−0.00795759
$120$	0	0
$121$	2.80787	0.255261
$122$	10.5972	0.959425
$123$	0	0
$124$	−7.65524	−0.687461
$125$	−6.05499	−0.541575
$126$	0	0
$127$	8.93444	0.792803	0.396402	−	0.918077i	$-0.370259\pi$
$127$	8.93444	0.792803	0.396402	+	0.918077i	$0.370259\pi$
$128$	10.6101	0.937813
$129$	0	0
$130$	9.18216	0.805329
$131$	−9.00722	−0.786964	−0.393482	−	0.919332i	$-0.628730\pi$
$131$	−9.00722	−0.786964	−0.393482	+	0.919332i	$0.628730\pi$
$132$	0	0
$133$	1.01983	0.0884305
$134$	−27.2910	−2.35759
$135$	0	0
$136$	−3.62780	−0.311081
$137$	−14.6070	−1.24796	−0.623982	−	0.781439i	$-0.714486\pi$
$137$	−14.6070	−1.24796	−0.623982	+	0.781439i	$0.714486\pi$
$138$	0	0
$139$	−12.4481	−1.05583	−0.527917	−	0.849296i	$-0.677027\pi$
$139$	−12.4481	−1.05583	−0.527917	+	0.849296i	$0.677027\pi$
$140$	−1.94851	−0.164679
$141$	0	0
$142$	−32.2203	−2.70387
$143$	−4.77311	−0.399148
$144$	0	0
$145$	−24.3105	−2.01888
$146$	−3.75698	−0.310930
$147$	0	0
$148$	18.4271	1.51469
$149$	16.3496	1.33941	0.669706	−	0.742626i	$-0.266420\pi$
$149$	16.3496	1.33941	0.669706	+	0.742626i	$0.266420\pi$
$150$	0	0
$151$	−16.0457	−1.30578	−0.652891	−	0.757452i	$-0.726444\pi$
$151$	−16.0457	−1.30578	−0.652891	+	0.757452i	$0.726444\pi$
$152$	42.6202	3.45696
$153$	0	0
$154$	1.46104	0.117734
$155$	−4.74114	−0.380817
$156$	0	0
$157$	1.51581	0.120975	0.0604875	−	0.998169i	$-0.480734\pi$
$157$	1.51581	0.120975	0.0604875	+	0.998169i	$0.480734\pi$
$158$	−21.2662	−1.69185
$159$	0	0
$160$	−16.8074	−1.32874
$161$	−0.930845	−0.0733608
$162$	0	0
$163$	18.1419	1.42098	0.710490	−	0.703707i	$-0.248473\pi$
$163$	18.1419	1.42098	0.710490	+	0.703707i	$0.248473\pi$
$164$	50.2395	3.92304
$165$	0	0
$166$	−17.8165	−1.38283
$167$	−21.0440	−1.62844	−0.814218	−	0.580559i	$-0.802834\pi$
$167$	−21.0440	−1.62844	−0.814218	+	0.580559i	$0.802834\pi$
$168$	0	0
$169$	−11.3500	−0.873079
$170$	−4.02988	−0.309078
$171$	0	0
$172$	4.52016	0.344659
$173$	15.7058	1.19409	0.597043	−	0.802209i	$-0.296342\pi$
$173$	15.7058	1.19409	0.597043	+	0.802209i	$0.296342\pi$
$174$	0	0
$175$	−0.436863	−0.0330237
$176$	27.4662	2.07034
$177$	0	0
$178$	−40.5598	−3.04009
$179$	15.3843	1.14988	0.574939	−	0.818196i	$-0.305026\pi$
$179$	15.3843	1.14988	0.574939	+	0.818196i	$0.305026\pi$
$180$	0	0
$181$	−4.19026	−0.311460	−0.155730	−	0.987800i	$-0.549773\pi$
$181$	−4.19026	−0.311460	−0.155730	+	0.987800i	$0.549773\pi$
$182$	−0.505053	−0.0374370
$183$	0	0
$184$	−38.9014	−2.86785
$185$	11.4125	0.839062
$186$	0	0
$187$	2.09483	0.153189
$188$	−33.3812	−2.43457
$189$	0	0
$190$	47.3440	3.43470
$191$	−16.5234	−1.19559	−0.597795	−	0.801649i	$-0.703956\pi$
$191$	−16.5234	−1.19559	−0.597795	+	0.801649i	$0.703956\pi$
$192$	0	0
$193$	−0.445795	−0.0320890	−0.0160445	−	0.999871i	$-0.505107\pi$
$193$	−0.445795	−0.0320890	−0.0160445	+	0.999871i	$0.505107\pi$
$194$	−42.2504	−3.03341
$195$	0	0
$196$	−31.5340	−2.25243
$197$	2.28962	0.163128	0.0815642	−	0.996668i	$-0.474008\pi$
$197$	2.28962	0.163128	0.0815642	+	0.996668i	$0.474008\pi$
$198$	0	0
$199$	−3.92413	−0.278174	−0.139087	−	0.990280i	$-0.544417\pi$
$199$	−3.92413	−0.278174	−0.139087	+	0.990280i	$0.544417\pi$
$200$	−18.2571	−1.29097
$201$	0	0
$202$	−28.5957	−2.01199
$203$	1.33717	0.0938509
$204$	0	0
$205$	31.1149	2.17316
$206$	−36.7289	−2.55902
$207$	0	0
$208$	−9.49452	−0.658327
$209$	−24.6106	−1.70235
$210$	0	0
$211$	−18.0578	−1.24315	−0.621574	−	0.783356i	$-0.713506\pi$
$211$	−18.0578	−1.24315	−0.621574	+	0.783356i	$0.713506\pi$
$212$	10.4010	0.714341
$213$	0	0
$214$	26.3815	1.80340
$215$	2.79948	0.190923
$216$	0	0
$217$	0.260780	0.0177029
$218$	−22.1914	−1.50299
$219$	0	0
$220$	47.0214	3.17018
$221$	−0.724141	−0.0487110
$222$	0	0
$223$	−8.31822	−0.557030	−0.278515	−	0.960432i	$-0.589842\pi$
$223$	−8.31822	−0.557030	−0.278515	+	0.960432i	$0.589842\pi$
$224$	0.924468	0.0617686
$225$	0	0
$226$	−13.7799	−0.916628
$227$	−13.9353	−0.924920	−0.462460	−	0.886640i	$-0.653033\pi$
$227$	−13.9353	−0.924920	−0.462460	+	0.886640i	$0.653033\pi$
$228$	0	0
$229$	−1.66826	−0.110242	−0.0551209	−	0.998480i	$-0.517554\pi$
$229$	−1.66826	−0.110242	−0.0551209	+	0.998480i	$0.517554\pi$
$230$	−43.2130	−2.84938
$231$	0	0
$232$	55.8823	3.66885
$233$	0.0999337	0.00654687	0.00327344	−	0.999995i	$-0.498958\pi$
$233$	0.0999337	0.00654687	0.00327344	+	0.999995i	$0.498958\pi$
$234$	0	0
$235$	−20.6740	−1.34863
$236$	26.6673	1.73589
$237$	0	0
$238$	0.221659	0.0143680
$239$	11.6816	0.755619	0.377810	−	0.925883i	$-0.376677\pi$
$239$	11.6816	0.755619	0.377810	+	0.925883i	$0.376677\pi$
$240$	0	0
$241$	−5.01008	−0.322727	−0.161364	−	0.986895i	$-0.551589\pi$
$241$	−5.01008	−0.322727	−0.161364	+	0.986895i	$0.551589\pi$
$242$	−7.16979	−0.460892
$243$	0	0
$244$	−18.7593	−1.20094
$245$	−19.5300	−1.24773
$246$	0	0
$247$	8.50739	0.541312
$248$	10.8984	0.692049
$249$	0	0
$250$	15.4612	0.977851
$251$	−4.73634	−0.298955	−0.149478	−	0.988765i	$-0.547759\pi$
$251$	−4.73634	−0.298955	−0.149478	+	0.988765i	$0.547759\pi$
$252$	0	0
$253$	22.4631	1.41225
$254$	−22.8137	−1.43146
$255$	0	0
$256$	−28.1871	−1.76169
$257$	16.3913	1.02246	0.511232	−	0.859443i	$-0.329189\pi$
$257$	16.3913	1.02246	0.511232	+	0.859443i	$0.329189\pi$
$258$	0	0
$259$	−0.627729	−0.0390052
$260$	−16.2544	−1.00805
$261$	0	0
$262$	22.9996	1.42092
$263$	16.0440	0.989316	0.494658	−	0.869088i	$-0.335293\pi$
$263$	16.0440	0.989316	0.494658	+	0.869088i	$0.335293\pi$
$264$	0	0
$265$	6.44166	0.395708
$266$	−2.60410	−0.159667
$267$	0	0
$268$	48.3109	2.95106
$269$	15.7847	0.962412	0.481206	−	0.876607i	$-0.340199\pi$
$269$	15.7847	0.962412	0.481206	+	0.876607i	$0.340199\pi$
$270$	0	0
$271$	30.4843	1.85179	0.925894	−	0.377784i	$-0.123314\pi$
$271$	30.4843	1.85179	0.925894	+	0.377784i	$0.123314\pi$
$272$	4.16697	0.252660
$273$	0	0
$274$	37.2985	2.25329
$275$	10.5424	0.635729
$276$	0	0
$277$	19.3537	1.16285	0.581424	−	0.813601i	$-0.302496\pi$
$277$	19.3537	1.16285	0.581424	+	0.813601i	$0.302496\pi$
$278$	31.7857	1.90638
$279$	0	0
$280$	2.77399	0.165778
$281$	−18.3103	−1.09230	−0.546152	−	0.837686i	$-0.683908\pi$
$281$	−18.3103	−1.09230	−0.546152	+	0.837686i	$0.683908\pi$
$282$	0	0
$283$	12.0194	0.714480	0.357240	−	0.934013i	$-0.383718\pi$
$283$	12.0194	0.714480	0.357240	+	0.934013i	$0.383718\pi$
$284$	57.0368	3.38451
$285$	0	0
$286$	12.1880	0.720689
$287$	−1.71144	−0.101023
$288$	0	0
$289$	−16.6822	−0.981305
$290$	62.0759	3.64522
$291$	0	0
$292$	6.65064	0.389199
$293$	17.5836	1.02725	0.513623	−	0.858016i	$-0.328303\pi$
$293$	17.5836	1.02725	0.513623	+	0.858016i	$0.328303\pi$
$294$	0	0
$295$	16.5159	0.961594
$296$	−26.2337	−1.52480
$297$	0	0
$298$	−41.7481	−2.41840
$299$	−7.76507	−0.449066
$300$	0	0
$301$	−0.153982	−0.00887538
$302$	40.9721	2.35768
$303$	0	0
$304$	−48.9545	−2.80774
$305$	−11.6182	−0.665257
$306$	0	0
$307$	−9.91335	−0.565785	−0.282892	−	0.959152i	$-0.591294\pi$
$307$	−9.91335	−0.565785	−0.282892	+	0.959152i	$0.591294\pi$
$308$	−2.58635	−0.147371
$309$	0	0
$310$	12.1063	0.687592
$311$	−26.0941	−1.47966	−0.739831	−	0.672792i	$-0.765095\pi$
$311$	−26.0941	−1.47966	−0.739831	+	0.672792i	$0.765095\pi$
$312$	0	0
$313$	4.77476	0.269886	0.134943	−	0.990853i	$-0.456915\pi$
$313$	4.77476	0.269886	0.134943	+	0.990853i	$0.456915\pi$
$314$	−3.87057	−0.218429
$315$	0	0
$316$	37.6457	2.11774
$317$	21.3069	1.19671	0.598357	−	0.801229i	$-0.295820\pi$
$317$	21.3069	1.19671	0.598357	+	0.801229i	$0.295820\pi$
$318$	0	0
$319$	−32.2686	−1.80669
$320$	1.53196	0.0856391
$321$	0	0
$322$	2.37688	0.132458
$323$	−3.73373	−0.207750
$324$	0	0
$325$	−3.64429	−0.202149
$326$	−46.3245	−2.56568
$327$	0	0
$328$	−71.5235	−3.94922
$329$	1.13715	0.0626931
$330$	0	0
$331$	4.07663	0.224072	0.112036	−	0.993704i	$-0.464263\pi$
$331$	4.07663	0.224072	0.112036	+	0.993704i	$0.464263\pi$
$332$	31.5389	1.73092
$333$	0	0
$334$	53.7351	2.94026
$335$	29.9205	1.63473
$336$	0	0
$337$	4.60476	0.250837	0.125419	−	0.992104i	$-0.459973\pi$
$337$	4.60476	0.250837	0.125419	+	0.992104i	$0.459973\pi$
$338$	28.9819	1.57641
$339$	0	0
$340$	7.13374	0.386881
$341$	−6.29315	−0.340793
$342$	0	0
$343$	2.15210	0.116202
$344$	−6.43513	−0.346959
$345$	0	0
$346$	−40.1040	−2.15601
$347$	13.2518	0.711392	0.355696	−	0.934602i	$-0.384244\pi$
$347$	13.2518	0.711392	0.355696	+	0.934602i	$0.384244\pi$
$348$	0	0
$349$	2.93184	0.156938	0.0784690	−	0.996917i	$-0.474997\pi$
$349$	2.93184	0.156938	0.0784690	+	0.996917i	$0.474997\pi$
$350$	1.11551	0.0596266
$351$	0	0
$352$	−22.3093	−1.18909
$353$	29.4408	1.56697	0.783487	−	0.621408i	$-0.213439\pi$
$353$	29.4408	1.56697	0.783487	+	0.621408i	$0.213439\pi$
$354$	0	0
$355$	35.3247	1.87484
$356$	71.7994	3.80536
$357$	0	0
$358$	−39.2832	−2.07618
$359$	−35.4312	−1.86999	−0.934994	−	0.354664i	$-0.884595\pi$
$359$	−35.4312	−1.86999	−0.934994	+	0.354664i	$0.884595\pi$
$360$	0	0
$361$	24.8648	1.30867
$362$	10.6997	0.562362
$363$	0	0
$364$	0.894051	0.0468610
$365$	4.11896	0.215596
$366$	0	0
$367$	−7.91098	−0.412950	−0.206475	−	0.978452i	$-0.566199\pi$
$367$	−7.91098	−0.412950	−0.206475	+	0.978452i	$0.566199\pi$
$368$	44.6830	2.32926
$369$	0	0
$370$	−29.1413	−1.51498
$371$	−0.354315	−0.0183951
$372$	0	0
$373$	1.94397	0.100655	0.0503274	−	0.998733i	$-0.483974\pi$
$373$	1.94397	0.100655	0.0503274	+	0.998733i	$0.483974\pi$
$374$	−5.34906	−0.276594
$375$	0	0
$376$	47.5232	2.45082
$377$	11.1546	0.574492
$378$	0	0
$379$	8.52784	0.438046	0.219023	−	0.975720i	$-0.429713\pi$
$379$	8.52784	0.438046	0.219023	+	0.975720i	$0.429713\pi$
$380$	−83.8088	−4.29930
$381$	0	0
$382$	42.1918	2.15872
$383$	10.0894	0.515543	0.257772	−	0.966206i	$-0.417012\pi$
$383$	10.0894	0.515543	0.257772	+	0.966206i	$0.417012\pi$
$384$	0	0
$385$	−1.60181	−0.0816358
$386$	1.13832	0.0579390
$387$	0	0
$388$	74.7922	3.79700
$389$	1.58439	0.0803320	0.0401660	−	0.999193i	$-0.487211\pi$
$389$	1.58439	0.0803320	0.0401660	+	0.999193i	$0.487211\pi$
$390$	0	0
$391$	3.40794	0.172347
$392$	44.8934	2.26746
$393$	0	0
$394$	−5.84645	−0.294540
$395$	23.3152	1.17312
$396$	0	0
$397$	−5.37215	−0.269620	−0.134810	−	0.990871i	$-0.543042\pi$
$397$	−5.37215	−0.269620	−0.134810	+	0.990871i	$0.543042\pi$
$398$	10.0201	0.502262
$399$	0	0
$400$	20.9706	1.04853
$401$	−13.5019	−0.674251	−0.337126	−	0.941460i	$-0.609455\pi$
$401$	−13.5019	−0.674251	−0.337126	+	0.941460i	$0.609455\pi$
$402$	0	0
$403$	2.17542	0.108365
$404$	50.6205	2.51846
$405$	0	0
$406$	−3.41441	−0.169454
$407$	15.1483	0.750876
$408$	0	0
$409$	18.7344	0.926355	0.463177	−	0.886266i	$-0.346709\pi$
$409$	18.7344	0.926355	0.463177	+	0.886266i	$0.346709\pi$
$410$	−79.4508	−3.92379
$411$	0	0
$412$	65.0178	3.20320
$413$	−0.908437	−0.0447013
$414$	0	0
$415$	19.5331	0.958840
$416$	7.71187	0.378106
$417$	0	0
$418$	62.8421	3.07371
$419$	9.19588	0.449248	0.224624	−	0.974446i	$-0.427885\pi$
$419$	9.19588	0.449248	0.224624	+	0.974446i	$0.427885\pi$
$420$	0	0
$421$	−4.97712	−0.242570	−0.121285	−	0.992618i	$-0.538702\pi$
$421$	−4.97712	−0.242570	−0.121285	+	0.992618i	$0.538702\pi$
$422$	46.1098	2.24459
$423$	0	0
$424$	−14.8074	−0.719109
$425$	1.59941	0.0775829
$426$	0	0
$427$	0.639045	0.0309256
$428$	−46.7009	−2.25737
$429$	0	0
$430$	−7.14837	−0.344725
$431$	−2.97671	−0.143383	−0.0716916	−	0.997427i	$-0.522840\pi$
$431$	−2.97671	−0.143383	−0.0716916	+	0.997427i	$0.522840\pi$
$432$	0	0
$433$	−2.07208	−0.0995780	−0.0497890	−	0.998760i	$-0.515855\pi$
$433$	−2.07208	−0.0995780	−0.0497890	+	0.998760i	$0.515855\pi$
$434$	−0.665892	−0.0319639
$435$	0	0
$436$	39.2835	1.88134
$437$	−40.0373	−1.91525
$438$	0	0
$439$	−21.6777	−1.03462	−0.517310	−	0.855798i	$-0.673067\pi$
$439$	−21.6777	−1.03462	−0.517310	+	0.855798i	$0.673067\pi$
$440$	−66.9420	−3.19134
$441$	0	0
$442$	1.84907	0.0879511
$443$	17.8155	0.846438	0.423219	−	0.906027i	$-0.360900\pi$
$443$	17.8155	0.846438	0.423219	+	0.906027i	$0.360900\pi$
$444$	0	0
$445$	44.4677	2.10797
$446$	21.2403	1.00576
$447$	0	0
$448$	−0.0842635	−0.00398107
$449$	−39.1785	−1.84895	−0.924475	−	0.381243i	$-0.875496\pi$
$449$	−39.1785	−1.84895	−0.924475	+	0.381243i	$0.875496\pi$
$450$	0	0
$451$	41.3004	1.94476
$452$	24.3934	1.14737
$453$	0	0
$454$	35.5833	1.67001
$455$	0.553714	0.0259585
$456$	0	0
$457$	3.07612	0.143895	0.0719474	−	0.997408i	$-0.477079\pi$
$457$	3.07612	0.143895	0.0719474	+	0.997408i	$0.477079\pi$
$458$	4.25984	0.199049
$459$	0	0
$460$	76.4960	3.56665
$461$	−4.15304	−0.193426	−0.0967131	−	0.995312i	$-0.530833\pi$
$461$	−4.15304	−0.193426	−0.0967131	+	0.995312i	$0.530833\pi$
$462$	0	0
$463$	7.09324	0.329651	0.164825	−	0.986323i	$-0.447294\pi$
$463$	7.09324	0.329651	0.164825	+	0.986323i	$0.447294\pi$
$464$	−64.1876	−2.97984
$465$	0	0
$466$	−0.255177	−0.0118208
$467$	−10.7211	−0.496112	−0.248056	−	0.968746i	$-0.579792\pi$
$467$	−10.7211	−0.496112	−0.248056	+	0.968746i	$0.579792\pi$
$468$	0	0
$469$	−1.64574	−0.0759932
$470$	52.7904	2.43504
$471$	0	0
$472$	−37.9649	−1.74748
$473$	3.71589	0.170857
$474$	0	0
$475$	−18.7903	−0.862157
$476$	−0.392382	−0.0179848
$477$	0	0
$478$	−29.8285	−1.36432
$479$	−6.81573	−0.311419	−0.155709	−	0.987803i	$-0.549766\pi$
$479$	−6.81573	−0.311419	−0.155709	+	0.987803i	$0.549766\pi$
$480$	0	0
$481$	−5.23649	−0.238763
$482$	12.7930	0.582707
$483$	0	0
$484$	12.6920	0.576911
$485$	46.3212	2.10334
$486$	0	0
$487$	−17.3377	−0.785645	−0.392823	−	0.919614i	$-0.628501\pi$
$487$	−17.3377	−0.785645	−0.392823	+	0.919614i	$0.628501\pi$
$488$	26.7066	1.20895
$489$	0	0
$490$	49.8691	2.25286
$491$	15.9023	0.717662	0.358831	−	0.933403i	$-0.383175\pi$
$491$	15.9023	0.717662	0.358831	+	0.933403i	$0.383175\pi$
$492$	0	0
$493$	−4.89555	−0.220485
$494$	−21.7233	−0.977376
$495$	0	0
$496$	−12.5181	−0.562081
$497$	−1.94299	−0.0871551
$498$	0	0
$499$	−4.52496	−0.202565	−0.101282	−	0.994858i	$-0.532295\pi$
$499$	−4.52496	−0.202565	−0.101282	+	0.994858i	$0.532295\pi$
$500$	−27.3695	−1.22400
$501$	0	0
$502$	12.0941	0.539784
$503$	−19.5945	−0.873674	−0.436837	−	0.899541i	$-0.643901\pi$
$503$	−19.5945	−0.873674	−0.436837	+	0.899541i	$0.643901\pi$
$504$	0	0
$505$	31.3509	1.39510
$506$	−57.3588	−2.54991
$507$	0	0
$508$	40.3851	1.79180
$509$	−28.8325	−1.27798	−0.638988	−	0.769217i	$-0.720647\pi$
$509$	−28.8325	−1.27798	−0.638988	+	0.769217i	$0.720647\pi$
$510$	0	0
$511$	−0.226558	−0.0100223
$512$	50.7542	2.24304
$513$	0	0
$514$	−41.8546	−1.84613
$515$	40.2676	1.77440
$516$	0	0
$517$	−27.4417	−1.20688
$518$	1.60288	0.0704265
$519$	0	0
$520$	23.1405	1.01478
$521$	39.0982	1.71292	0.856461	−	0.516211i	$-0.172658\pi$
$521$	39.0982	1.71292	0.856461	+	0.516211i	$0.172658\pi$
$522$	0	0
$523$	−20.7298	−0.906453	−0.453226	−	0.891395i	$-0.649727\pi$
$523$	−20.7298	−0.906453	−0.453226	+	0.891395i	$0.649727\pi$
$524$	−40.7141	−1.77860
$525$	0	0
$526$	−40.9677	−1.78628
$527$	−0.954751	−0.0415896
$528$	0	0
$529$	13.5439	0.588864
$530$	−16.4485	−0.714478
$531$	0	0
$532$	4.60980	0.199860
$533$	−14.2767	−0.618394
$534$	0	0
$535$	−28.9234	−1.25047
$536$	−68.7778	−2.97075
$537$	0	0
$538$	−40.3057	−1.73770
$539$	−25.9232	−1.11659
$540$	0	0
$541$	42.2499	1.81647	0.908233	−	0.418466i	$-0.137432\pi$
$541$	42.2499	1.81647	0.908233	+	0.418466i	$0.137432\pi$
$542$	−77.8404	−3.34353
$543$	0	0
$544$	−3.38460	−0.145113
$545$	24.3296	1.04216
$546$	0	0
$547$	−13.6870	−0.585215	−0.292608	−	0.956233i	$-0.594523\pi$
$547$	−13.6870	−0.585215	−0.292608	+	0.956233i	$0.594523\pi$
$548$	−66.0262	−2.82050
$549$	0	0
$550$	−26.9195	−1.14785
$551$	57.5141	2.45018
$552$	0	0
$553$	−1.28242	−0.0545342
$554$	−49.4188	−2.09960
$555$	0	0
$556$	−56.2674	−2.38627
$557$	−19.8203	−0.839813	−0.419907	−	0.907567i	$-0.637937\pi$
$557$	−19.8203	−0.839813	−0.419907	+	0.907567i	$0.637937\pi$
$558$	0	0
$559$	−1.28451	−0.0543291
$560$	−3.18627	−0.134645
$561$	0	0
$562$	46.7547	1.97223
$563$	41.3186	1.74137	0.870686	−	0.491839i	$-0.163675\pi$
$563$	41.3186	1.74137	0.870686	+	0.491839i	$0.163675\pi$
$564$	0	0
$565$	15.1076	0.635582
$566$	−30.6911	−1.29004
$567$	0	0
$568$	−81.2005	−3.40710
$569$	5.55964	0.233072	0.116536	−	0.993186i	$-0.462821\pi$
$569$	5.55964	0.233072	0.116536	+	0.993186i	$0.462821\pi$
$570$	0	0
$571$	−42.3313	−1.77151	−0.885756	−	0.464152i	$-0.846359\pi$
$571$	−42.3313	−1.77151	−0.885756	+	0.464152i	$0.846359\pi$
$572$	−21.5752	−0.902106
$573$	0	0
$574$	4.37009	0.182404
$575$	17.1507	0.715234
$576$	0	0
$577$	18.7514	0.780631	0.390316	−	0.920681i	$-0.372366\pi$
$577$	18.7514	0.780631	0.390316	+	0.920681i	$0.372366\pi$
$578$	42.5973	1.77181
$579$	0	0
$580$	−109.887	−4.56283
$581$	−1.07439	−0.0445732
$582$	0	0
$583$	8.55034	0.354119
$584$	−9.46820	−0.391797
$585$	0	0
$586$	−44.8990	−1.85476
$587$	−12.0355	−0.496757	−0.248379	−	0.968663i	$-0.579898\pi$
$587$	−12.0355	−0.496757	−0.248379	+	0.968663i	$0.579898\pi$
$588$	0	0
$589$	11.2166	0.462174
$590$	−42.1727	−1.73622
$591$	0	0
$592$	30.1326	1.23844
$593$	15.9739	0.655970	0.327985	−	0.944683i	$-0.393630\pi$
$593$	15.9739	0.655970	0.327985	+	0.944683i	$0.393630\pi$
$594$	0	0
$595$	−0.243015	−0.00996264
$596$	73.9029	3.02718
$597$	0	0
$598$	19.8278	0.810819
$599$	2.24550	0.0917485	0.0458743	−	0.998947i	$-0.485393\pi$
$599$	2.24550	0.0917485	0.0458743	+	0.998947i	$0.485393\pi$
$600$	0	0
$601$	4.78769	0.195294	0.0976470	−	0.995221i	$-0.468868\pi$
$601$	4.78769	0.195294	0.0976470	+	0.995221i	$0.468868\pi$
$602$	0.393187	0.0160251
$603$	0	0
$604$	−72.5292	−2.95117
$605$	7.86059	0.319578
$606$	0	0
$607$	29.0005	1.17709	0.588547	−	0.808463i	$-0.299700\pi$
$607$	29.0005	1.17709	0.588547	+	0.808463i	$0.299700\pi$
$608$	39.7631	1.61260
$609$	0	0
$610$	29.6667	1.20117
$611$	9.48606	0.383765
$612$	0	0
$613$	−36.6015	−1.47832	−0.739160	−	0.673530i	$-0.764777\pi$
$613$	−36.6015	−1.47832	−0.739160	+	0.673530i	$0.764777\pi$
$614$	25.3133	1.02156
$615$	0	0
$616$	3.68206	0.148354
$617$	−35.8483	−1.44320	−0.721599	−	0.692312i	$-0.756592\pi$
$617$	−35.8483	−1.44320	−0.721599	+	0.692312i	$0.756592\pi$
$618$	0	0
$619$	−32.9537	−1.32452	−0.662261	−	0.749273i	$-0.730403\pi$
$619$	−32.9537	−1.32452	−0.662261	+	0.749273i	$0.730403\pi$
$620$	−21.4307	−0.860678
$621$	0	0
$622$	66.6303	2.67163
$623$	−2.44589	−0.0979925
$624$	0	0
$625$	−31.1364	−1.24545
$626$	−12.1922	−0.487297
$627$	0	0
$628$	6.85172	0.273413
$629$	2.29820	0.0916351
$630$	0	0
$631$	10.8066	0.430203	0.215101	−	0.976592i	$-0.430992\pi$
$631$	10.8066	0.430203	0.215101	+	0.976592i	$0.430992\pi$
$632$	−53.5944	−2.13187
$633$	0	0
$634$	−54.4063	−2.16075
$635$	25.0118	0.992563
$636$	0	0
$637$	8.96113	0.355053
$638$	82.3965	3.26211
$639$	0	0
$640$	29.7029	1.17411
$641$	−15.9647	−0.630567	−0.315284	−	0.948998i	$-0.602100\pi$
$641$	−15.9647	−0.630567	−0.315284	+	0.948998i	$0.602100\pi$
$642$	0	0
$643$	35.6262	1.40496	0.702481	−	0.711703i	$-0.252076\pi$
$643$	35.6262	1.40496	0.702481	+	0.711703i	$0.252076\pi$
$644$	−4.20757	−0.165801
$645$	0	0
$646$	9.53394	0.375108
$647$	16.4474	0.646612	0.323306	−	0.946294i	$-0.395206\pi$
$647$	16.4474	0.646612	0.323306	+	0.946294i	$0.395206\pi$
$648$	0	0
$649$	21.9224	0.860530
$650$	9.30556	0.364994
$651$	0	0
$652$	82.0042	3.21153
$653$	−5.83888	−0.228493	−0.114246	−	0.993452i	$-0.536445\pi$
$653$	−5.83888	−0.228493	−0.114246	+	0.993452i	$0.536445\pi$
$654$	0	0
$655$	−25.2155	−0.985253
$656$	82.1535	3.20755
$657$	0	0
$658$	−2.90367	−0.113197
$659$	0.758647	0.0295527	0.0147763	−	0.999891i	$-0.495296\pi$
$659$	0.758647	0.0295527	0.0147763	+	0.999891i	$0.495296\pi$
$660$	0	0
$661$	−45.0061	−1.75053	−0.875266	−	0.483641i	$-0.839314\pi$
$661$	−45.0061	−1.75053	−0.875266	+	0.483641i	$0.839314\pi$
$662$	−10.4095	−0.404577
$663$	0	0
$664$	−44.9004	−1.74247
$665$	2.85500	0.110712
$666$	0	0
$667$	−52.4957	−2.03264
$668$	−95.1225	−3.68040
$669$	0	0
$670$	−76.4008	−2.95162
$671$	−15.4214	−0.595338
$672$	0	0
$673$	−16.8848	−0.650863	−0.325431	−	0.945566i	$-0.605509\pi$
$673$	−16.8848	−0.650863	−0.325431	+	0.945566i	$0.605509\pi$
$674$	−11.7581	−0.452904
$675$	0	0
$676$	−51.3040	−1.97323
$677$	35.6991	1.37203	0.686014	−	0.727589i	$-0.259359\pi$
$677$	35.6991	1.37203	0.686014	+	0.727589i	$0.259359\pi$
$678$	0	0
$679$	−2.54784	−0.0977771
$680$	−10.1560	−0.389463
$681$	0	0
$682$	16.0693	0.615326
$683$	13.0146	0.497989	0.248995	−	0.968505i	$-0.419900\pi$
$683$	13.0146	0.497989	0.248995	+	0.968505i	$0.419900\pi$
$684$	0	0
$685$	−40.8922	−1.56241
$686$	−5.49530	−0.209811
$687$	0	0
$688$	7.39154	0.281800
$689$	−2.95568	−0.112603
$690$	0	0
$691$	−0.835726	−0.0317925	−0.0158963	−	0.999874i	$-0.505060\pi$
$691$	−0.835726	−0.0317925	−0.0158963	+	0.999874i	$0.505060\pi$
$692$	70.9926	2.69873
$693$	0	0
$694$	−33.8379	−1.28447
$695$	−34.8482	−1.32187
$696$	0	0
$697$	6.26580	0.237334
$698$	−7.48635	−0.283362
$699$	0	0
$700$	−1.97469	−0.0746363
$701$	45.3676	1.71351	0.856755	−	0.515723i	$-0.172477\pi$
$701$	45.3676	1.71351	0.856755	+	0.515723i	$0.172477\pi$
$702$	0	0
$703$	−26.9998	−1.01832
$704$	2.03345	0.0766384
$705$	0	0
$706$	−75.1758	−2.82928
$707$	−1.72442	−0.0648534
$708$	0	0
$709$	0.203131	0.00762875	0.00381438	−	0.999993i	$-0.498786\pi$
$709$	0.203131	0.00762875	0.00381438	+	0.999993i	$0.498786\pi$
$710$	−90.2003	−3.38516
$711$	0	0
$712$	−102.217	−3.83075
$713$	−10.2379	−0.383413
$714$	0	0
$715$	−13.3622	−0.499719
$716$	69.5396	2.59882
$717$	0	0
$718$	90.4722	3.37639
$719$	−15.9357	−0.594302	−0.297151	−	0.954830i	$-0.596037\pi$
$719$	−15.9357	−0.594302	−0.297151	+	0.954830i	$0.596037\pi$
$720$	0	0
$721$	−2.21487	−0.0824861
$722$	−63.4913	−2.36290
$723$	0	0
$724$	−18.9407	−0.703924
$725$	−24.6372	−0.915003
$726$	0	0
$727$	−45.8449	−1.70029	−0.850146	−	0.526546i	$-0.823487\pi$
$727$	−45.8449	−1.70029	−0.850146	+	0.526546i	$0.823487\pi$
$728$	−1.27282	−0.0471737
$729$	0	0
$730$	−10.5176	−0.389274
$731$	0.563748	0.0208510
$732$	0	0
$733$	6.34339	0.234299	0.117149	−	0.993114i	$-0.462624\pi$
$733$	6.34339	0.234299	0.117149	+	0.993114i	$0.462624\pi$
$734$	20.2004	0.745610
$735$	0	0
$736$	−36.2935	−1.33780
$737$	39.7150	1.46292
$738$	0	0
$739$	32.6732	1.20190	0.600951	−	0.799286i	$-0.294789\pi$
$739$	32.6732	1.20190	0.600951	+	0.799286i	$0.294789\pi$
$740$	51.5862	1.89635
$741$	0	0
$742$	0.904730	0.0332137
$743$	−20.3284	−0.745776	−0.372888	−	0.927876i	$-0.621632\pi$
$743$	−20.3284	−0.745776	−0.372888	+	0.927876i	$0.621632\pi$
$744$	0	0
$745$	45.7705	1.67690
$746$	−4.96385	−0.181739
$747$	0	0
$748$	9.46897	0.346220
$749$	1.59089	0.0581299
$750$	0	0
$751$	−13.3906	−0.488631	−0.244316	−	0.969696i	$-0.578563\pi$
$751$	−13.3906	−0.488631	−0.244316	+	0.969696i	$0.578563\pi$
$752$	−54.5862	−1.99055
$753$	0	0
$754$	−28.4829	−1.03728
$755$	−44.9197	−1.63479
$756$	0	0
$757$	33.6454	1.22286	0.611432	−	0.791297i	$-0.290594\pi$
$757$	33.6454	1.22286	0.611432	+	0.791297i	$0.290594\pi$
$758$	−21.7755	−0.790922
$759$	0	0
$760$	119.315	4.32799
$761$	14.0630	0.509782	0.254891	−	0.966970i	$-0.417960\pi$
$761$	14.0630	0.509782	0.254891	+	0.966970i	$0.417960\pi$
$762$	0	0
$763$	−1.33822	−0.0484467
$764$	−74.6883	−2.70213
$765$	0	0
$766$	−25.7628	−0.930849
$767$	−7.57815	−0.273631
$768$	0	0
$769$	−3.66927	−0.132317	−0.0661587	−	0.997809i	$-0.521074\pi$
$769$	−3.66927	−0.132317	−0.0661587	+	0.997809i	$0.521074\pi$
$770$	4.09016	0.147399
$771$	0	0
$772$	−2.01507	−0.0725238
$773$	27.7108	0.996686	0.498343	−	0.866980i	$-0.333942\pi$
$773$	27.7108	0.996686	0.498343	+	0.866980i	$0.333942\pi$
$774$	0	0
$775$	−4.80485	−0.172595
$776$	−106.478	−3.82234
$777$	0	0
$778$	−4.04569	−0.145045
$779$	−73.6121	−2.63743
$780$	0	0
$781$	46.8883	1.67779
$782$	−8.70205	−0.311185
$783$	0	0
$784$	−51.5655	−1.84163
$785$	4.24349	0.151457
$786$	0	0
$787$	−27.3666	−0.975515	−0.487757	−	0.872979i	$-0.662185\pi$
$787$	−27.3666	−0.975515	−0.487757	+	0.872979i	$0.662185\pi$
$788$	10.3494	0.368683
$789$	0	0
$790$	−59.5345	−2.11814
$791$	−0.830975	−0.0295461
$792$	0	0
$793$	5.33089	0.189306
$794$	13.7176	0.486819
$795$	0	0
$796$	−17.7377	−0.628696
$797$	38.4034	1.36032	0.680159	−	0.733065i	$-0.261911\pi$
$797$	38.4034	1.36032	0.680159	+	0.733065i	$0.261911\pi$
$798$	0	0
$799$	−4.16325	−0.147285
$800$	−17.0332	−0.602215
$801$	0	0
$802$	34.4765	1.21741
$803$	5.46730	0.192937
$804$	0	0
$805$	−2.60588	−0.0918453
$806$	−5.55485	−0.195661
$807$	0	0
$808$	−72.0659	−2.53527
$809$	38.1968	1.34293	0.671464	−	0.741037i	$-0.265666\pi$
$809$	38.1968	1.34293	0.671464	+	0.741037i	$0.265666\pi$
$810$	0	0
$811$	−7.50867	−0.263665	−0.131833	−	0.991272i	$-0.542086\pi$
$811$	−7.50867	−0.263665	−0.131833	+	0.991272i	$0.542086\pi$
$812$	6.04422	0.212111
$813$	0	0
$814$	−38.6807	−1.35576
$815$	50.7878	1.77902
$816$	0	0
$817$	−6.62305	−0.231711
$818$	−47.8375	−1.67260
$819$	0	0
$820$	140.645	4.91152
$821$	7.21215	0.251706	0.125853	−	0.992049i	$-0.459833\pi$
$821$	7.21215	0.251706	0.125853	+	0.992049i	$0.459833\pi$
$822$	0	0
$823$	3.90142	0.135995	0.0679975	−	0.997685i	$-0.478339\pi$
$823$	3.90142	0.135995	0.0679975	+	0.997685i	$0.478339\pi$
$824$	−92.5627	−3.22457
$825$	0	0
$826$	2.31966	0.0807112
$827$	−56.2324	−1.95539	−0.977696	−	0.210023i	$-0.932646\pi$
$827$	−56.2324	−1.95539	−0.977696	+	0.210023i	$0.932646\pi$
$828$	0	0
$829$	−21.3784	−0.742502	−0.371251	−	0.928533i	$-0.621071\pi$
$829$	−21.3784	−0.742502	−0.371251	+	0.928533i	$0.621071\pi$
$830$	−49.8769	−1.73125
$831$	0	0
$832$	−0.702922	−0.0243694
$833$	−3.93287	−0.136266
$834$	0	0
$835$	−58.9124	−2.03875
$836$	−111.244	−3.84744
$837$	0	0
$838$	−23.4813	−0.811149
$839$	32.4668	1.12088	0.560439	−	0.828196i	$-0.310632\pi$
$839$	32.4668	1.12088	0.560439	+	0.828196i	$0.310632\pi$
$840$	0	0
$841$	46.4107	1.60037
$842$	12.7089	0.437977
$843$	0	0
$844$	−81.6240	−2.80961
$845$	−31.7742	−1.09307
$846$	0	0
$847$	−0.432362	−0.0148561
$848$	17.0081	0.584059
$849$	0	0
$850$	−4.08404	−0.140081
$851$	24.6439	0.844781
$852$	0	0
$853$	−8.76506	−0.300110	−0.150055	−	0.988678i	$-0.547945\pi$
$853$	−8.76506	−0.300110	−0.150055	+	0.988678i	$0.547945\pi$
$854$	−1.63178	−0.0558383
$855$	0	0
$856$	66.4857	2.27244
$857$	39.2894	1.34210	0.671049	−	0.741413i	$-0.265844\pi$
$857$	39.2894	1.34210	0.671049	+	0.741413i	$0.265844\pi$
$858$	0	0
$859$	−37.5081	−1.27976	−0.639880	−	0.768475i	$-0.721016\pi$
$859$	−37.5081	−1.27976	−0.639880	+	0.768475i	$0.721016\pi$
$860$	12.6541	0.431502
$861$	0	0
$862$	7.60092	0.258888
$863$	35.2574	1.20018	0.600088	−	0.799934i	$-0.295132\pi$
$863$	35.2574	1.20018	0.600088	+	0.799934i	$0.295132\pi$
$864$	0	0
$865$	43.9680	1.49496
$866$	5.29099	0.179795
$867$	0	0
$868$	1.17877	0.0400100
$869$	30.9475	1.04982
$870$	0	0
$871$	−13.7287	−0.465179
$872$	−55.9260	−1.89389
$873$	0	0
$874$	102.234	3.45811
$875$	0.932360	0.0315195
$876$	0	0
$877$	−8.80708	−0.297394	−0.148697	−	0.988883i	$-0.547508\pi$
$877$	−8.80708	−0.297394	−0.148697	+	0.988883i	$0.547508\pi$
$878$	55.3531	1.86808
$879$	0	0
$880$	76.8911	2.59200
$881$	−22.3061	−0.751512	−0.375756	−	0.926719i	$-0.622617\pi$
$881$	−22.3061	−0.751512	−0.375756	+	0.926719i	$0.622617\pi$
$882$	0	0
$883$	−12.9751	−0.436648	−0.218324	−	0.975876i	$-0.570059\pi$
$883$	−12.9751	−0.436648	−0.218324	+	0.975876i	$0.570059\pi$
$884$	−3.27324	−0.110091
$885$	0	0
$886$	−45.4911	−1.52830
$887$	−36.9094	−1.23930	−0.619648	−	0.784880i	$-0.712725\pi$
$887$	−36.9094	−1.23930	−0.619648	+	0.784880i	$0.712725\pi$
$888$	0	0
$889$	−1.37574	−0.0461409
$890$	−113.546	−3.80609
$891$	0	0
$892$	−37.5997	−1.25893
$893$	48.9109	1.63674
$894$	0	0
$895$	43.0681	1.43961
$896$	−1.63377	−0.0545805
$897$	0	0
$898$	100.041	3.33841
$899$	14.7069	0.490503
$900$	0	0
$901$	1.29719	0.0432158
$902$	−105.459	−3.51140
$903$	0	0
$904$	−34.7277	−1.15503
$905$	−11.7306	−0.389937
$906$	0	0
$907$	47.8042	1.58731	0.793656	−	0.608367i	$-0.208175\pi$
$907$	47.8042	1.58731	0.793656	+	0.608367i	$0.208175\pi$
$908$	−62.9899	−2.09039
$909$	0	0
$910$	−1.41389	−0.0468699
$911$	16.2435	0.538172	0.269086	−	0.963116i	$-0.413278\pi$
$911$	16.2435	0.538172	0.269086	+	0.963116i	$0.413278\pi$
$912$	0	0
$913$	25.9272	0.858065
$914$	−7.85475	−0.259812
$915$	0	0
$916$	−7.54081	−0.249155
$917$	1.38695	0.0458011
$918$	0	0
$919$	−16.1366	−0.532297	−0.266149	−	0.963932i	$-0.585751\pi$
$919$	−16.1366	−0.532297	−0.266149	+	0.963932i	$0.585751\pi$
$920$	−108.904	−3.59045
$921$	0	0
$922$	10.6046	0.349245
$923$	−16.2084	−0.533505
$924$	0	0
$925$	11.5658	0.380282
$926$	−18.1123	−0.595207
$927$	0	0
$928$	52.1360	1.71145
$929$	14.2652	0.468026	0.234013	−	0.972233i	$-0.424814\pi$
$929$	14.2652	0.468026	0.234013	+	0.972233i	$0.424814\pi$
$930$	0	0
$931$	46.2043	1.51429
$932$	0.451716	0.0147965
$933$	0	0
$934$	27.3758	0.895764
$935$	5.86444	0.191788
$936$	0	0
$937$	28.3390	0.925795	0.462897	−	0.886412i	$-0.346810\pi$
$937$	28.3390	0.925795	0.462897	+	0.886412i	$0.346810\pi$
$938$	4.20233	0.137211
$939$	0	0
$940$	−93.4500	−3.04800
$941$	48.4004	1.57781	0.788904	−	0.614517i	$-0.210649\pi$
$941$	48.4004	1.57781	0.788904	+	0.614517i	$0.210649\pi$
$942$	0	0
$943$	67.1890	2.18798
$944$	43.6074	1.41930
$945$	0	0
$946$	−9.48839	−0.308494
$947$	−34.7556	−1.12941	−0.564703	−	0.825294i	$-0.691009\pi$
$947$	−34.7556	−1.12941	−0.564703	+	0.825294i	$0.691009\pi$
$948$	0	0
$949$	−1.88994	−0.0613500
$950$	47.9802	1.55668
$951$	0	0
$952$	0.558615	0.0181048
$953$	−39.1055	−1.26675	−0.633375	−	0.773845i	$-0.718331\pi$
$953$	−39.1055	−1.26675	−0.633375	+	0.773845i	$0.718331\pi$
$954$	0	0
$955$	−46.2569	−1.49684
$956$	52.8027	1.70776
$957$	0	0
$958$	17.4037	0.562288
$959$	2.24922	0.0726312
$960$	0	0
$961$	−28.1318	−0.907477
$962$	13.3712	0.431104
$963$	0	0
$964$	−22.6464	−0.729390
$965$	−1.24800	−0.0401744
$966$	0	0
$967$	−5.69699	−0.183203	−0.0916014	−	0.995796i	$-0.529199\pi$
$967$	−5.69699	−0.183203	−0.0916014	+	0.995796i	$0.529199\pi$
$968$	−18.0690	−0.580761
$969$	0	0
$970$	−118.279	−3.79772
$971$	24.9838	0.801768	0.400884	−	0.916129i	$-0.368703\pi$
$971$	24.9838	0.801768	0.400884	+	0.916129i	$0.368703\pi$
$972$	0	0
$973$	1.91678	0.0614493
$974$	44.2711	1.41854
$975$	0	0
$976$	−30.6759	−0.981910
$977$	−13.4725	−0.431024	−0.215512	−	0.976501i	$-0.569142\pi$
$977$	−13.4725	−0.431024	−0.215512	+	0.976501i	$0.569142\pi$
$978$	0	0
$979$	59.0242	1.88642
$980$	−88.2788	−2.81996
$981$	0	0
$982$	−40.6060	−1.29579
$983$	−11.2269	−0.358082	−0.179041	−	0.983842i	$-0.557299\pi$
$983$	−11.2269	−0.358082	−0.179041	+	0.983842i	$0.557299\pi$
$984$	0	0
$985$	6.40974	0.204231
$986$	12.5006	0.398100
$987$	0	0
$988$	38.4548	1.22341
$989$	6.04515	0.192225
$990$	0	0
$991$	−55.8592	−1.77443	−0.887213	−	0.461360i	$-0.847362\pi$
$991$	−55.8592	−1.77443	−0.887213	+	0.461360i	$0.847362\pi$
$992$	10.1678	0.322828
$993$	0	0
$994$	4.96135	0.157365
$995$	−10.9855	−0.348265
$996$	0	0
$997$	−53.7266	−1.70154	−0.850769	−	0.525539i	$-0.823864\pi$
$997$	−53.7266	−1.70154	−0.850769	+	0.525539i	$0.823864\pi$
$998$	11.5543	0.365745
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3483.2.a.s.1.1		19
3.2	odd	2		3483.2.a.r.1.19		19
9.2	odd	6		1161.2.f.c.388.1		38
9.4	even	3		387.2.f.c.259.19	yes	38
9.5	odd	6		1161.2.f.c.775.1		38
9.7	even	3		387.2.f.c.130.19	✓	38

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
387.2.f.c.130.19	✓	38	9.7	even	3
387.2.f.c.259.19	yes	38	9.4	even	3
1161.2.f.c.388.1		38	9.2	odd	6
1161.2.f.c.775.1		38	9.5	odd	6
3483.2.a.r.1.19		19	3.2	odd	2
3483.2.a.s.1.1		19	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 3483 = 3^{4} \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3483.a (trivial)

\(f(q)\)	\(=\)	\(q-2.55346 q^{2} +4.52016 q^{4} +2.79948 q^{5} -0.153982 q^{7} -6.43513 q^{8} +O(q^{10})\)	\(q-2.55346 q^{2} +4.52016 q^{4} +2.79948 q^{5} -0.153982 q^{7} -6.43513 q^{8} -7.14837 q^{10} +3.71589 q^{11} -1.28451 q^{13} +0.393187 q^{14} +7.39154 q^{16} +0.563748 q^{17} -6.62305 q^{19} +12.6541 q^{20} -9.48839 q^{22} +6.04515 q^{23} +2.83710 q^{25} +3.27995 q^{26} -0.696024 q^{28} -8.68393 q^{29} -1.69358 q^{31} -6.00374 q^{32} -1.43951 q^{34} -0.431070 q^{35} +4.07664 q^{37} +16.9117 q^{38} -18.0150 q^{40} +11.1145 q^{41} +1.00000 q^{43} +16.7964 q^{44} -15.4361 q^{46} -7.38495 q^{47} -6.97629 q^{49} -7.24443 q^{50} -5.80620 q^{52} +2.30102 q^{53} +10.4026 q^{55} +0.990895 q^{56} +22.1741 q^{58} +5.89963 q^{59} -4.15013 q^{61} +4.32448 q^{62} +0.547229 q^{64} -3.59597 q^{65} +10.6879 q^{67} +2.54823 q^{68} +1.10072 q^{70} +12.6183 q^{71} +1.47133 q^{73} -10.4095 q^{74} -29.9373 q^{76} -0.572181 q^{77} +8.32840 q^{79} +20.6925 q^{80} -28.3805 q^{82} +6.97738 q^{83} +1.57820 q^{85} -2.55346 q^{86} -23.9123 q^{88} +15.8842 q^{89} +0.197792 q^{91} +27.3251 q^{92} +18.8572 q^{94} -18.5411 q^{95} +16.5463 q^{97} +17.8137 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 19 q + 4 q^{2} + 22 q^{4} + 9 q^{5} + 7 q^{7} + 12 q^{8}+O(q^{10}) \) 19 * q + 4 * q^2 + 22 * q^4 + 9 * q^5 + 7 * q^7 + 12 * q^8	\( 19 q + 4 q^{2} + 22 q^{4} + 9 q^{5} + 7 q^{7} + 12 q^{8} - 7 q^{10} + 5 q^{11} - 5 q^{13} + 17 q^{14} + 24 q^{16} + 21 q^{17} - 4 q^{19} + 21 q^{20} - 20 q^{22} + 22 q^{23} + 10 q^{25} + 17 q^{26} - q^{28} + 30 q^{29} - 5 q^{31} + 48 q^{32} - 6 q^{34} + 53 q^{35} - q^{37} + 21 q^{38} + 16 q^{40} + 29 q^{41} + 19 q^{43} + 29 q^{44} + 32 q^{47} - 10 q^{49} - 11 q^{50} + q^{52} + 38 q^{53} + 2 q^{55} + 46 q^{56} + 30 q^{58} + 30 q^{59} - 10 q^{61} + 25 q^{62} + 14 q^{64} + 8 q^{65} + 3 q^{67} + 47 q^{68} + 56 q^{70} + 21 q^{71} + 8 q^{73} + 28 q^{74} - 36 q^{76} + 49 q^{77} + 4 q^{79} + 70 q^{80} - 4 q^{82} + 29 q^{83} - 4 q^{85} + 4 q^{86} - 47 q^{88} + 54 q^{89} + 4 q^{91} + 12 q^{92} - 23 q^{94} + 33 q^{95} - 4 q^{97} + 49 q^{98}+O(q^{100}) \) 19 * q + 4 * q^2 + 22 * q^4 + 9 * q^5 + 7 * q^7 + 12 * q^8 - 7 * q^10 + 5 * q^11 - 5 * q^13 + 17 * q^14 + 24 * q^16 + 21 * q^17 - 4 * q^19 + 21 * q^20 - 20 * q^22 + 22 * q^23 + 10 * q^25 + 17 * q^26 - q^28 + 30 * q^29 - 5 * q^31 + 48 * q^32 - 6 * q^34 + 53 * q^35 - q^37 + 21 * q^38 + 16 * q^40 + 29 * q^41 + 19 * q^43 + 29 * q^44 + 32 * q^47 - 10 * q^49 - 11 * q^50 + q^52 + 38 * q^53 + 2 * q^55 + 46 * q^56 + 30 * q^58 + 30 * q^59 - 10 * q^61 + 25 * q^62 + 14 * q^64 + 8 * q^65 + 3 * q^67 + 47 * q^68 + 56 * q^70 + 21 * q^71 + 8 * q^73 + 28 * q^74 - 36 * q^76 + 49 * q^77 + 4 * q^79 + 70 * q^80 - 4 * q^82 + 29 * q^83 - 4 * q^85 + 4 * q^86 - 47 * q^88 + 54 * q^89 + 4 * q^91 + 12 * q^92 - 23 * q^94 + 33 * q^95 - 4 * q^97 + 49 * q^98

Introduction

L-functions

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Database

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