LMFDB - Embedded newform 6039.2.a.i.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6039,2,Mod(1,6039)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6039, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("6039.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6039 = 3^{2} \cdot 11 \cdot 61 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6039.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:	$48.2216577807$
Analytic rank:	$0$
Dimension:	$13$
Coefficient field:	$\mathbb{Q}[x]/(x^{13} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{13} - 2 x^{12} - 19 x^{11} + 35 x^{10} + 136 x^{9} - 220 x^{8} - 469 x^{7} + 610 x^{6} + 841 x^{5} + \cdots - 47 $ x^13 - 2x^12 - 19x^11 + 35x^10 + 136x^9 - 220x^8 - 469x^7 + 610x^6 + 841x^5 - 760x^4 - 742x^3 + 366x^2 + 236x - 47
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 2013)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.53773$ of defining polynomial
Character	$\chi$	$=$	6039.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.53773 q^{2} +4.44007 q^{4} -0.994065 q^{5} +4.88646 q^{7} -6.19224 q^{8} +O(q^{10})$	\(q-2.53773 q^{2} +4.44007 q^{4} -0.994065 q^{5} +4.88646 q^{7} -6.19224 q^{8} +2.52267 q^{10} -1.00000 q^{11} +4.29008 q^{13} -12.4005 q^{14} +6.83409 q^{16} +3.82389 q^{17} -2.93110 q^{19} -4.41372 q^{20} +2.53773 q^{22} +5.41371 q^{23} -4.01184 q^{25} -10.8871 q^{26} +21.6962 q^{28} +2.31016 q^{29} -6.92787 q^{31} -4.95859 q^{32} -9.70401 q^{34} -4.85746 q^{35} +0.949760 q^{37} +7.43834 q^{38} +6.15549 q^{40} +4.66091 q^{41} +4.23337 q^{43} -4.44007 q^{44} -13.7385 q^{46} -8.27575 q^{47} +16.8775 q^{49} +10.1810 q^{50} +19.0483 q^{52} +11.5324 q^{53} +0.994065 q^{55} -30.2582 q^{56} -5.86257 q^{58} -5.12033 q^{59} -1.00000 q^{61} +17.5811 q^{62} -1.08461 q^{64} -4.26462 q^{65} -6.83412 q^{67} +16.9784 q^{68} +12.3269 q^{70} -4.72943 q^{71} +15.9491 q^{73} -2.41023 q^{74} -13.0143 q^{76} -4.88646 q^{77} -6.64878 q^{79} -6.79353 q^{80} -11.8281 q^{82} +8.78161 q^{83} -3.80120 q^{85} -10.7431 q^{86} +6.19224 q^{88} -0.0660596 q^{89} +20.9633 q^{91} +24.0372 q^{92} +21.0016 q^{94} +2.91370 q^{95} +11.3016 q^{97} -42.8306 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 13 q - 2 q^{2} + 16 q^{4} - 3 q^{5} + 11 q^{7} - 9 q^{8}+O(q^{10}) $ 13 * q - 2 * q^2 + 16 * q^4 - 3 * q^5 + 11 * q^7 - 9 * q^8	$ 13 q - 2 q^{2} + 16 q^{4} - 3 q^{5} + 11 q^{7} - 9 q^{8} + 6 q^{10} - 13 q^{11} + 13 q^{13} - q^{14} + 18 q^{16} - 17 q^{17} + 14 q^{19} + 7 q^{20} + 2 q^{22} - 7 q^{23} + 18 q^{25} + 10 q^{26} + 19 q^{28} + 6 q^{29} + 27 q^{31} - 5 q^{32} + 6 q^{34} - 14 q^{35} + 10 q^{37} - 2 q^{38} + 8 q^{40} - 3 q^{41} + 29 q^{43} - 16 q^{44} - 24 q^{46} - 8 q^{47} + 8 q^{49} + 27 q^{50} + 37 q^{52} + 24 q^{53} + 3 q^{55} - 24 q^{56} - 5 q^{58} - 13 q^{59} - 13 q^{61} - 39 q^{62} + 47 q^{64} + 11 q^{65} + 44 q^{67} + 8 q^{68} - 12 q^{70} - 3 q^{71} + 48 q^{73} + 22 q^{74} + 47 q^{76} - 11 q^{77} - 17 q^{79} + 26 q^{80} + 56 q^{82} - 50 q^{83} + 8 q^{85} - 18 q^{86} + 9 q^{88} + 15 q^{89} + 47 q^{91} - 14 q^{92} + 45 q^{94} + q^{95} + 27 q^{97} - 47 q^{98}+O(q^{100}) $ 13 * q - 2 * q^2 + 16 * q^4 - 3 * q^5 + 11 * q^7 - 9 * q^8 + 6 * q^10 - 13 * q^11 + 13 * q^13 - q^14 + 18 * q^16 - 17 * q^17 + 14 * q^19 + 7 * q^20 + 2 * q^22 - 7 * q^23 + 18 * q^25 + 10 * q^26 + 19 * q^28 + 6 * q^29 + 27 * q^31 - 5 * q^32 + 6 * q^34 - 14 * q^35 + 10 * q^37 - 2 * q^38 + 8 * q^40 - 3 * q^41 + 29 * q^43 - 16 * q^44 - 24 * q^46 - 8 * q^47 + 8 * q^49 + 27 * q^50 + 37 * q^52 + 24 * q^53 + 3 * q^55 - 24 * q^56 - 5 * q^58 - 13 * q^59 - 13 * q^61 - 39 * q^62 + 47 * q^64 + 11 * q^65 + 44 * q^67 + 8 * q^68 - 12 * q^70 - 3 * q^71 + 48 * q^73 + 22 * q^74 + 47 * q^76 - 11 * q^77 - 17 * q^79 + 26 * q^80 + 56 * q^82 - 50 * q^83 + 8 * q^85 - 18 * q^86 + 9 * q^88 + 15 * q^89 + 47 * q^91 - 14 * q^92 + 45 * q^94 + q^95 + 27 * q^97 - 47 * 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 $

$2$	−2.53773	−1.79445	−0.897223	−	0.441578i	$-0.854419\pi$
$2$	−2.53773	−1.79445	−0.897223	+	0.441578i	$0.854419\pi$
$3$	0	0
$3$	0	0
$4$	4.44007	2.22004
$5$	−0.994065	−0.444559	−0.222280	−	0.974983i	$-0.571350\pi$
$5$	−0.994065	−0.444559	−0.222280	+	0.974983i	$0.571350\pi$
$6$	0	0
$7$	4.88646	1.84691	0.923455	−	0.383707i	$-0.125353\pi$
$7$	4.88646	1.84691	0.923455	+	0.383707i	$0.125353\pi$
$8$	−6.19224	−2.18929
$9$	0	0
$10$	2.52267	0.797737
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	4.29008	1.18985	0.594927	−	0.803779i	$-0.297181\pi$
$13$	4.29008	1.18985	0.594927	+	0.803779i	$0.297181\pi$
$14$	−12.4005	−3.31418
$15$	0	0
$16$	6.83409	1.70852
$17$	3.82389	0.927431	0.463715	−	0.885984i	$-0.346516\pi$
$17$	3.82389	0.927431	0.463715	+	0.885984i	$0.346516\pi$
$18$	0	0
$19$	−2.93110	−0.672441	−0.336220	−	0.941783i	$-0.609149\pi$
$19$	−2.93110	−0.672441	−0.336220	+	0.941783i	$0.609149\pi$
$20$	−4.41372	−0.986937
$21$	0	0
$22$	2.53773	0.541046
$23$	5.41371	1.12884	0.564418	−	0.825489i	$-0.309101\pi$
$23$	5.41371	1.12884	0.564418	+	0.825489i	$0.309101\pi$
$24$	0	0
$25$	−4.01184	−0.802367
$26$	−10.8871	−2.13513
$27$	0	0
$28$	21.6962	4.10021
$29$	2.31016	0.428986	0.214493	−	0.976725i	$-0.431190\pi$
$29$	2.31016	0.428986	0.214493	+	0.976725i	$0.431190\pi$
$30$	0	0
$31$	−6.92787	−1.24428	−0.622141	−	0.782905i	$-0.713737\pi$
$31$	−6.92787	−1.24428	−0.622141	+	0.782905i	$0.713737\pi$
$32$	−4.95859	−0.876564
$33$	0	0
$34$	−9.70401	−1.66422
$35$	−4.85746	−0.821061
$36$	0	0
$37$	0.949760	0.156140	0.0780698	−	0.996948i	$-0.475124\pi$
$37$	0.949760	0.156140	0.0780698	+	0.996948i	$0.475124\pi$
$38$	7.43834	1.20666
$39$	0	0
$40$	6.15549	0.973268
$41$	4.66091	0.727912	0.363956	−	0.931416i	$-0.381426\pi$
$41$	4.66091	0.727912	0.363956	+	0.931416i	$0.381426\pi$
$42$	0	0
$43$	4.23337	0.645583	0.322791	−	0.946470i	$-0.395379\pi$
$43$	4.23337	0.645583	0.322791	+	0.946470i	$0.395379\pi$
$44$	−4.44007	−0.669366
$45$	0	0
$46$	−13.7385	−2.02563
$47$	−8.27575	−1.20714	−0.603571	−	0.797309i	$-0.706256\pi$
$47$	−8.27575	−1.20714	−0.603571	+	0.797309i	$0.706256\pi$
$48$	0	0
$49$	16.8775	2.41108
$50$	10.1810	1.43980
$51$	0	0
$52$	19.0483	2.64152
$53$	11.5324	1.58410	0.792051	−	0.610455i	$-0.209013\pi$
$53$	11.5324	1.58410	0.792051	+	0.610455i	$0.209013\pi$
$54$	0	0
$55$	0.994065	0.134040
$56$	−30.2582	−4.04342
$57$	0	0
$58$	−5.86257	−0.769793
$59$	−5.12033	−0.666610	−0.333305	−	0.942819i	$-0.608164\pi$
$59$	−5.12033	−0.666610	−0.333305	+	0.942819i	$0.608164\pi$
$60$	0	0
$61$	−1.00000	−0.128037
$61$	−1.00000	−0.128037
$62$	17.5811	2.23280
$63$	0	0
$64$	−1.08461	−0.135577
$65$	−4.26462	−0.528961
$66$	0	0
$67$	−6.83412	−0.834920	−0.417460	−	0.908695i	$-0.637080\pi$
$67$	−6.83412	−0.834920	−0.417460	+	0.908695i	$0.637080\pi$
$68$	16.9784	2.05893
$69$	0	0
$70$	12.3269	1.47335
$71$	−4.72943	−0.561280	−0.280640	−	0.959813i	$-0.590547\pi$
$71$	−4.72943	−0.561280	−0.280640	+	0.959813i	$0.590547\pi$
$72$	0	0
$73$	15.9491	1.86670	0.933348	−	0.358972i	$-0.116873\pi$
$73$	15.9491	1.86670	0.933348	+	0.358972i	$0.116873\pi$
$74$	−2.41023	−0.280184
$75$	0	0
$76$	−13.0143	−1.49284
$77$	−4.88646	−0.556864
$78$	0	0
$79$	−6.64878	−0.748046	−0.374023	−	0.927419i	$-0.622022\pi$
$79$	−6.64878	−0.748046	−0.374023	+	0.927419i	$0.622022\pi$
$80$	−6.79353	−0.759540
$81$	0	0
$82$	−11.8281	−1.30620
$83$	8.78161	0.963906	0.481953	−	0.876197i	$-0.339928\pi$
$83$	8.78161	0.963906	0.481953	+	0.876197i	$0.339928\pi$
$84$	0	0
$85$	−3.80120	−0.412298
$86$	−10.7431	−1.15846
$87$	0	0
$88$	6.19224	0.660095
$89$	−0.0660596	−0.00700230	−0.00350115	−	0.999994i	$-0.501114\pi$
$89$	−0.0660596	−0.00700230	−0.00350115	+	0.999994i	$0.501114\pi$
$90$	0	0
$91$	20.9633	2.19755
$92$	24.0372	2.50606
$93$	0	0
$94$	21.0016	2.16615
$95$	2.91370	0.298940
$96$	0	0
$97$	11.3016	1.14751	0.573753	−	0.819028i	$-0.305487\pi$
$97$	11.3016	1.14751	0.573753	+	0.819028i	$0.305487\pi$
$98$	−42.8306	−4.32654
$99$	0	0
$100$	−17.8128	−1.78128
$101$	6.79683	0.676310	0.338155	−	0.941091i	$-0.390197\pi$
$101$	6.79683	0.676310	0.338155	+	0.941091i	$0.390197\pi$
$102$	0	0
$103$	12.1574	1.19790	0.598952	−	0.800785i	$-0.295584\pi$
$103$	12.1574	1.19790	0.598952	+	0.800785i	$0.295584\pi$
$104$	−26.5652	−2.60493
$105$	0	0
$106$	−29.2662	−2.84258
$107$	−3.63314	−0.351229	−0.175614	−	0.984459i	$-0.556191\pi$
$107$	−3.63314	−0.351229	−0.175614	+	0.984459i	$0.556191\pi$
$108$	0	0
$109$	4.81079	0.460790	0.230395	−	0.973097i	$-0.425998\pi$
$109$	4.81079	0.460790	0.230395	+	0.973097i	$0.425998\pi$
$110$	−2.52267	−0.240527
$111$	0	0
$112$	33.3945	3.15549
$113$	4.35919	0.410078	0.205039	−	0.978754i	$-0.434268\pi$
$113$	4.35919	0.410078	0.205039	+	0.978754i	$0.434268\pi$
$114$	0	0
$115$	−5.38157	−0.501834
$116$	10.2573	0.952365
$117$	0	0
$118$	12.9940	1.19620
$119$	18.6853	1.71288
$120$	0	0
$121$	1.00000	0.0909091
$122$	2.53773	0.229755
$123$	0	0
$124$	−30.7602	−2.76235
$125$	8.95835	0.801259
$126$	0	0
$127$	−20.5780	−1.82600	−0.913001	−	0.407957i	$-0.866241\pi$
$127$	−20.5780	−1.82600	−0.913001	+	0.407957i	$0.866241\pi$
$128$	12.6696	1.11985
$129$	0	0
$130$	10.8225	0.949192
$131$	−8.91370	−0.778794	−0.389397	−	0.921070i	$-0.627317\pi$
$131$	−8.91370	−0.778794	−0.389397	+	0.921070i	$0.627317\pi$
$132$	0	0
$133$	−14.3227	−1.24194
$134$	17.3431	1.49822
$135$	0	0
$136$	−23.6785	−2.03041
$137$	11.8906	1.01589	0.507943	−	0.861391i	$-0.330406\pi$
$137$	11.8906	1.01589	0.507943	+	0.861391i	$0.330406\pi$
$138$	0	0
$139$	3.99386	0.338755	0.169377	−	0.985551i	$-0.445824\pi$
$139$	3.99386	0.338755	0.169377	+	0.985551i	$0.445824\pi$
$140$	−21.5675	−1.82278
$141$	0	0
$142$	12.0020	1.00719
$143$	−4.29008	−0.358755
$144$	0	0
$145$	−2.29645	−0.190710
$146$	−40.4744	−3.34969
$147$	0	0
$148$	4.21700	0.346636
$149$	−18.2887	−1.49827	−0.749134	−	0.662418i	$-0.769530\pi$
$149$	−18.2887	−1.49827	−0.749134	+	0.662418i	$0.769530\pi$
$150$	0	0
$151$	11.2751	0.917554	0.458777	−	0.888551i	$-0.348288\pi$
$151$	11.2751	0.917554	0.458777	+	0.888551i	$0.348288\pi$
$152$	18.1501	1.47217
$153$	0	0
$154$	12.4005	0.999263
$155$	6.88675	0.553157
$156$	0	0
$157$	−16.6823	−1.33139	−0.665695	−	0.746224i	$-0.731865\pi$
$157$	−16.6823	−1.33139	−0.665695	+	0.746224i	$0.731865\pi$
$158$	16.8728	1.34233
$159$	0	0
$160$	4.92916	0.389685
$161$	26.4539	2.08486
$162$	0	0
$163$	10.4676	0.819882	0.409941	−	0.912112i	$-0.365549\pi$
$163$	10.4676	0.819882	0.409941	+	0.912112i	$0.365549\pi$
$164$	20.6948	1.61599
$165$	0	0
$166$	−22.2853	−1.72968
$167$	21.0144	1.62614	0.813070	−	0.582166i	$-0.197795\pi$
$167$	21.0144	1.62614	0.813070	+	0.582166i	$0.197795\pi$
$168$	0	0
$169$	5.40481	0.415754
$170$	9.64641	0.739846
$171$	0	0
$172$	18.7965	1.43322
$173$	−3.21998	−0.244811	−0.122405	−	0.992480i	$-0.539061\pi$
$173$	−3.21998	−0.244811	−0.122405	+	0.992480i	$0.539061\pi$
$174$	0	0
$175$	−19.6037	−1.48190
$176$	−6.83409	−0.515139
$177$	0	0
$178$	0.167641	0.0125652
$179$	7.27099	0.543460	0.271730	−	0.962374i	$-0.412404\pi$
$179$	7.27099	0.543460	0.271730	+	0.962374i	$0.412404\pi$
$180$	0	0
$181$	13.4180	0.997355	0.498678	−	0.866788i	$-0.333819\pi$
$181$	13.4180	0.997355	0.498678	+	0.866788i	$0.333819\pi$
$182$	−53.1993	−3.94339
$183$	0	0
$184$	−33.5230	−2.47135
$185$	−0.944123	−0.0694133
$186$	0	0
$187$	−3.82389	−0.279631
$188$	−36.7449	−2.67990
$189$	0	0
$190$	−7.39419	−0.536431
$191$	−16.2153	−1.17329	−0.586647	−	0.809843i	$-0.699552\pi$
$191$	−16.2153	−1.17329	−0.586647	+	0.809843i	$0.699552\pi$
$192$	0	0
$193$	15.4786	1.11417	0.557086	−	0.830455i	$-0.311919\pi$
$193$	15.4786	1.11417	0.557086	+	0.830455i	$0.311919\pi$
$194$	−28.6805	−2.05914
$195$	0	0
$196$	74.9374	5.35267
$197$	17.7360	1.26364	0.631818	−	0.775117i	$-0.282309\pi$
$197$	17.7360	1.26364	0.631818	+	0.775117i	$0.282309\pi$
$198$	0	0
$199$	−2.42918	−0.172200	−0.0861000	−	0.996286i	$-0.527440\pi$
$199$	−2.42918	−0.172200	−0.0861000	+	0.996286i	$0.527440\pi$
$200$	24.8423	1.75661
$201$	0	0
$202$	−17.2485	−1.21360
$203$	11.2885	0.792299
$204$	0	0
$205$	−4.63325	−0.323600
$206$	−30.8522	−2.14957
$207$	0	0
$208$	29.3188	2.03289
$209$	2.93110	0.202748
$210$	0	0
$211$	−23.4818	−1.61655	−0.808276	−	0.588804i	$-0.799599\pi$
$211$	−23.4818	−1.61655	−0.808276	+	0.588804i	$0.799599\pi$
$212$	51.2048	3.51676
$213$	0	0
$214$	9.21992	0.630261
$215$	−4.20824	−0.287000
$216$	0	0
$217$	−33.8528	−2.29808
$218$	−12.2085	−0.826863
$219$	0	0
$220$	4.41372	0.297573
$221$	16.4048	1.10351
$222$	0	0
$223$	19.8001	1.32591	0.662957	−	0.748657i	$-0.269301\pi$
$223$	19.8001	1.32591	0.662957	+	0.748657i	$0.269301\pi$
$224$	−24.2300	−1.61893
$225$	0	0
$226$	−11.0624	−0.735862
$227$	−22.2706	−1.47815	−0.739077	−	0.673621i	$-0.764738\pi$
$227$	−22.2706	−1.47815	−0.739077	+	0.673621i	$0.764738\pi$
$228$	0	0
$229$	4.54201	0.300144	0.150072	−	0.988675i	$-0.452049\pi$
$229$	4.54201	0.300144	0.150072	+	0.988675i	$0.452049\pi$
$230$	13.6570	0.900515
$231$	0	0
$232$	−14.3051	−0.939175
$233$	9.10518	0.596500	0.298250	−	0.954488i	$-0.403597\pi$
$233$	9.10518	0.596500	0.298250	+	0.954488i	$0.403597\pi$
$234$	0	0
$235$	8.22663	0.536646
$236$	−22.7346	−1.47990
$237$	0	0
$238$	−47.4183	−3.07367
$239$	7.66236	0.495637	0.247818	−	0.968807i	$-0.420286\pi$
$239$	7.66236	0.495637	0.247818	+	0.968807i	$0.420286\pi$
$240$	0	0
$241$	−3.33162	−0.214608	−0.107304	−	0.994226i	$-0.534222\pi$
$241$	−3.33162	−0.214608	−0.107304	+	0.994226i	$0.534222\pi$
$242$	−2.53773	−0.163131
$243$	0	0
$244$	−4.44007	−0.284246
$245$	−16.7774	−1.07187
$246$	0	0
$247$	−12.5747	−0.800107
$248$	42.8991	2.72409
$249$	0	0
$250$	−22.7339	−1.43782
$251$	−29.0747	−1.83518	−0.917589	−	0.397529i	$-0.869868\pi$
$251$	−29.0747	−1.83518	−0.917589	+	0.397529i	$0.869868\pi$
$252$	0	0
$253$	−5.41371	−0.340357
$254$	52.2214	3.27666
$255$	0	0
$256$	−29.9829	−1.87393
$257$	20.2898	1.26564	0.632822	−	0.774297i	$-0.281896\pi$
$257$	20.2898	1.26564	0.632822	+	0.774297i	$0.281896\pi$
$258$	0	0
$259$	4.64097	0.288376
$260$	−18.9352	−1.17431
$261$	0	0
$262$	22.6206	1.39750
$263$	−32.3400	−1.99417	−0.997085	−	0.0762942i	$-0.975691\pi$
$263$	−32.3400	−1.99417	−0.997085	+	0.0762942i	$0.975691\pi$
$264$	0	0
$265$	−11.4640	−0.704227
$266$	36.3472	2.22859
$267$	0	0
$268$	−30.3440	−1.85355
$269$	−17.3768	−1.05948	−0.529742	−	0.848159i	$-0.677711\pi$
$269$	−17.3768	−1.05948	−0.529742	+	0.848159i	$0.677711\pi$
$270$	0	0
$271$	−11.5350	−0.700701	−0.350350	−	0.936619i	$-0.613937\pi$
$271$	−11.5350	−0.700701	−0.350350	+	0.936619i	$0.613937\pi$
$272$	26.1328	1.58454
$273$	0	0
$274$	−30.1752	−1.82295
$275$	4.01184	0.241923
$276$	0	0
$277$	−11.7386	−0.705307	−0.352653	−	0.935754i	$-0.614721\pi$
$277$	−11.7386	−0.705307	−0.352653	+	0.935754i	$0.614721\pi$
$278$	−10.1353	−0.607877
$279$	0	0
$280$	30.0786	1.79754
$281$	−7.78483	−0.464404	−0.232202	−	0.972668i	$-0.574593\pi$
$281$	−7.78483	−0.464404	−0.232202	+	0.972668i	$0.574593\pi$
$282$	0	0
$283$	−24.6554	−1.46561	−0.732806	−	0.680438i	$-0.761790\pi$
$283$	−24.6554	−1.46561	−0.732806	+	0.680438i	$0.761790\pi$
$284$	−20.9990	−1.24606
$285$	0	0
$286$	10.8871	0.643766
$287$	22.7754	1.34439
$288$	0	0
$289$	−2.37783	−0.139873
$290$	5.82777	0.342219
$291$	0	0
$292$	70.8150	4.14413
$293$	13.4592	0.786297	0.393149	−	0.919475i	$-0.371386\pi$
$293$	13.4592	0.786297	0.393149	+	0.919475i	$0.371386\pi$
$294$	0	0
$295$	5.08994	0.296348
$296$	−5.88114	−0.341835
$297$	0	0
$298$	46.4118	2.68856
$299$	23.2252	1.34315
$300$	0	0
$301$	20.6862	1.19233
$302$	−28.6131	−1.64650
$303$	0	0
$304$	−20.0314	−1.14888
$305$	0.994065	0.0569200
$306$	0	0
$307$	15.7969	0.901579	0.450789	−	0.892630i	$-0.351143\pi$
$307$	15.7969	0.901579	0.450789	+	0.892630i	$0.351143\pi$
$308$	−21.6962	−1.23626
$309$	0	0
$310$	−17.4767	−0.992611
$311$	−20.8528	−1.18246	−0.591228	−	0.806504i	$-0.701357\pi$
$311$	−20.8528	−1.18246	−0.591228	+	0.806504i	$0.701357\pi$
$312$	0	0
$313$	7.74049	0.437518	0.218759	−	0.975779i	$-0.429799\pi$
$313$	7.74049	0.437518	0.218759	+	0.975779i	$0.429799\pi$
$314$	42.3351	2.38911
$315$	0	0
$316$	−29.5211	−1.66069
$317$	−3.37084	−0.189325	−0.0946627	−	0.995509i	$-0.530177\pi$
$317$	−3.37084	−0.189325	−0.0946627	+	0.995509i	$0.530177\pi$
$318$	0	0
$319$	−2.31016	−0.129344
$320$	1.07818	0.0602719
$321$	0	0
$322$	−67.1328	−3.74116
$323$	−11.2082	−0.623642
$324$	0	0
$325$	−17.2111	−0.954700
$326$	−26.5638	−1.47123
$327$	0	0
$328$	−28.8615	−1.59361
$329$	−40.4391	−2.22948
$330$	0	0
$331$	11.3276	0.622622	0.311311	−	0.950308i	$-0.399232\pi$
$331$	11.3276	0.622622	0.311311	+	0.950308i	$0.399232\pi$
$332$	38.9910	2.13991
$333$	0	0
$334$	−53.3288	−2.91802
$335$	6.79355	0.371172
$336$	0	0
$337$	27.5129	1.49872	0.749362	−	0.662161i	$-0.230360\pi$
$337$	27.5129	1.49872	0.749362	+	0.662161i	$0.230360\pi$
$338$	−13.7159	−0.746049
$339$	0	0
$340$	−16.8776	−0.915316
$341$	6.92787	0.375165
$342$	0	0
$343$	48.2662	2.60613
$344$	−26.2140	−1.41337
$345$	0	0
$346$	8.17145	0.439300
$347$	7.46395	0.400686	0.200343	−	0.979726i	$-0.435794\pi$
$347$	7.46395	0.400686	0.200343	+	0.979726i	$0.435794\pi$
$348$	0	0
$349$	−8.64667	−0.462845	−0.231423	−	0.972853i	$-0.574338\pi$
$349$	−8.64667	−0.462845	−0.231423	+	0.972853i	$0.574338\pi$
$350$	49.7489	2.65919
$351$	0	0
$352$	4.95859	0.264294
$353$	−33.8131	−1.79969	−0.899845	−	0.436210i	$-0.856320\pi$
$353$	−33.8131	−1.79969	−0.899845	+	0.436210i	$0.856320\pi$
$354$	0	0
$355$	4.70136	0.249522
$356$	−0.293309	−0.0155454
$357$	0	0
$358$	−18.4518	−0.975209
$359$	−4.96562	−0.262076	−0.131038	−	0.991377i	$-0.541831\pi$
$359$	−4.96562	−0.262076	−0.131038	+	0.991377i	$0.541831\pi$
$360$	0	0
$361$	−10.4087	−0.547824
$362$	−34.0514	−1.78970
$363$	0	0
$364$	93.0787	4.87865
$365$	−15.8544	−0.829857
$366$	0	0
$367$	28.2934	1.47690	0.738451	−	0.674307i	$-0.235558\pi$
$367$	28.2934	1.47690	0.738451	+	0.674307i	$0.235558\pi$
$368$	36.9978	1.92864
$369$	0	0
$370$	2.39593	0.124558
$371$	56.3528	2.92569
$372$	0	0
$373$	−18.5022	−0.958007	−0.479003	−	0.877813i	$-0.659002\pi$
$373$	−18.5022	−0.958007	−0.479003	+	0.877813i	$0.659002\pi$
$374$	9.70401	0.501782
$375$	0	0
$376$	51.2454	2.64278
$377$	9.91079	0.510432
$378$	0	0
$379$	10.1763	0.522721	0.261361	−	0.965241i	$-0.415829\pi$
$379$	10.1763	0.522721	0.261361	+	0.965241i	$0.415829\pi$
$380$	12.9371	0.663657
$381$	0	0
$382$	41.1499	2.10541
$383$	18.0563	0.922634	0.461317	−	0.887235i	$-0.347377\pi$
$383$	18.0563	0.922634	0.461317	+	0.887235i	$0.347377\pi$
$384$	0	0
$385$	4.85746	0.247559
$386$	−39.2804	−1.99932
$387$	0	0
$388$	50.1801	2.54751
$389$	−12.9240	−0.655271	−0.327636	−	0.944804i	$-0.606252\pi$
$389$	−12.9240	−0.655271	−0.327636	+	0.944804i	$0.606252\pi$
$390$	0	0
$391$	20.7014	1.04692
$392$	−104.510	−5.27854
$393$	0	0
$394$	−45.0091	−2.26753
$395$	6.60932	0.332551
$396$	0	0
$397$	25.2642	1.26798	0.633988	−	0.773343i	$-0.281417\pi$
$397$	25.2642	1.26798	0.633988	+	0.773343i	$0.281417\pi$
$398$	6.16460	0.309004
$399$	0	0
$400$	−27.4173	−1.37086
$401$	33.4265	1.66924	0.834620	−	0.550826i	$-0.185687\pi$
$401$	33.4265	1.66924	0.834620	+	0.550826i	$0.185687\pi$
$402$	0	0
$403$	−29.7211	−1.48052
$404$	30.1784	1.50143
$405$	0	0
$406$	−28.6472	−1.42174
$407$	−0.949760	−0.0470779
$408$	0	0
$409$	−3.25363	−0.160882	−0.0804408	−	0.996759i	$-0.525633\pi$
$409$	−3.25363	−0.160882	−0.0804408	+	0.996759i	$0.525633\pi$
$410$	11.7579	0.580683
$411$	0	0
$412$	53.9797	2.65939
$413$	−25.0203	−1.23117
$414$	0	0
$415$	−8.72948	−0.428513
$416$	−21.2728	−1.04298
$417$	0	0
$418$	−7.43834	−0.363821
$419$	35.2785	1.72347	0.861735	−	0.507359i	$-0.169378\pi$
$419$	35.2785	1.72347	0.861735	+	0.507359i	$0.169378\pi$
$420$	0	0
$421$	7.26689	0.354167	0.177083	−	0.984196i	$-0.443334\pi$
$421$	7.26689	0.354167	0.177083	+	0.984196i	$0.443334\pi$
$422$	59.5904	2.90081
$423$	0	0
$424$	−71.4116	−3.46806
$425$	−15.3408	−0.744140
$426$	0	0
$427$	−4.88646	−0.236473
$428$	−16.1314	−0.779740
$429$	0	0
$430$	10.6794	0.515005
$431$	7.77524	0.374520	0.187260	−	0.982310i	$-0.440039\pi$
$431$	7.77524	0.374520	0.187260	+	0.982310i	$0.440039\pi$
$432$	0	0
$433$	−35.7180	−1.71650	−0.858250	−	0.513232i	$-0.828448\pi$
$433$	−35.7180	−1.71650	−0.858250	+	0.513232i	$0.828448\pi$
$434$	85.9092	4.12378
$435$	0	0
$436$	21.3603	1.02297
$437$	−15.8681	−0.759075
$438$	0	0
$439$	15.8024	0.754205	0.377103	−	0.926172i	$-0.376920\pi$
$439$	15.8024	0.754205	0.377103	+	0.926172i	$0.376920\pi$
$440$	−6.15549	−0.293451
$441$	0	0
$442$	−41.6310	−1.98018
$443$	2.20489	0.104757	0.0523787	−	0.998627i	$-0.483320\pi$
$443$	2.20489	0.104757	0.0523787	+	0.998627i	$0.483320\pi$
$444$	0	0
$445$	0.0656675	0.00311294
$446$	−50.2473	−2.37928
$447$	0	0
$448$	−5.29993	−0.250398
$449$	11.0478	0.521379	0.260690	−	0.965423i	$-0.416050\pi$
$449$	11.0478	0.521379	0.260690	+	0.965423i	$0.416050\pi$
$450$	0	0
$451$	−4.66091	−0.219474
$452$	19.3551	0.910387
$453$	0	0
$454$	56.5169	2.65247
$455$	−20.8389	−0.976943
$456$	0	0
$457$	−32.3367	−1.51265	−0.756324	−	0.654197i	$-0.773007\pi$
$457$	−32.3367	−1.51265	−0.756324	+	0.654197i	$0.773007\pi$
$458$	−11.5264	−0.538593
$459$	0	0
$460$	−23.8946	−1.11409
$461$	15.6990	0.731175	0.365587	−	0.930777i	$-0.380868\pi$
$461$	15.6990	0.731175	0.365587	+	0.930777i	$0.380868\pi$
$462$	0	0
$463$	−5.21953	−0.242572	−0.121286	−	0.992618i	$-0.538702\pi$
$463$	−5.21953	−0.242572	−0.121286	+	0.992618i	$0.538702\pi$
$464$	15.7879	0.732933
$465$	0	0
$466$	−23.1065	−1.07039
$467$	−38.1044	−1.76326	−0.881630	−	0.471941i	$-0.843554\pi$
$467$	−38.1044	−1.76326	−0.881630	+	0.471941i	$0.843554\pi$
$468$	0	0
$469$	−33.3947	−1.54202
$470$	−20.8770	−0.962982
$471$	0	0
$472$	31.7063	1.45940
$473$	−4.23337	−0.194650
$474$	0	0
$475$	11.7591	0.539544
$476$	82.9642	3.80266
$477$	0	0
$478$	−19.4450	−0.889393
$479$	13.9545	0.637597	0.318799	−	0.947822i	$-0.396721\pi$
$479$	13.9545	0.637597	0.318799	+	0.947822i	$0.396721\pi$
$480$	0	0
$481$	4.07455	0.185783
$482$	8.45475	0.385103
$483$	0	0
$484$	4.44007	0.201821
$485$	−11.2346	−0.510135
$486$	0	0
$487$	−28.5606	−1.29421	−0.647103	−	0.762403i	$-0.724020\pi$
$487$	−28.5606	−1.29421	−0.647103	+	0.762403i	$0.724020\pi$
$488$	6.19224	0.280310
$489$	0	0
$490$	42.5764	1.92341
$491$	−3.97165	−0.179238	−0.0896189	−	0.995976i	$-0.528565\pi$
$491$	−3.97165	−0.179238	−0.0896189	+	0.995976i	$0.528565\pi$
$492$	0	0
$493$	8.83382	0.397855
$494$	31.9111	1.43575
$495$	0	0
$496$	−47.3457	−2.12589
$497$	−23.1102	−1.03663
$498$	0	0
$499$	−24.1339	−1.08038	−0.540192	−	0.841542i	$-0.681648\pi$
$499$	−24.1339	−1.08038	−0.540192	+	0.841542i	$0.681648\pi$
$500$	39.7757	1.77882
$501$	0	0
$502$	73.7837	3.29313
$503$	13.5112	0.602433	0.301217	−	0.953556i	$-0.402607\pi$
$503$	13.5112	0.602433	0.301217	+	0.953556i	$0.402607\pi$
$504$	0	0
$505$	−6.75649	−0.300660
$506$	13.7385	0.610752
$507$	0	0
$508$	−91.3678	−4.05379
$509$	25.6211	1.13563	0.567817	−	0.823155i	$-0.307788\pi$
$509$	25.6211	1.13563	0.567817	+	0.823155i	$0.307788\pi$
$510$	0	0
$511$	77.9345	3.44762
$512$	50.7492	2.24282
$513$	0	0
$514$	−51.4901	−2.27113
$515$	−12.0852	−0.532539
$516$	0	0
$517$	8.27575	0.363967
$518$	−11.7775	−0.517475
$519$	0	0
$520$	26.4076	1.15805
$521$	−1.69899	−0.0744343	−0.0372172	−	0.999307i	$-0.511849\pi$
$521$	−1.69899	−0.0744343	−0.0372172	+	0.999307i	$0.511849\pi$
$522$	0	0
$523$	−6.21946	−0.271958	−0.135979	−	0.990712i	$-0.543418\pi$
$523$	−6.21946	−0.271958	−0.135979	+	0.990712i	$0.543418\pi$
$524$	−39.5775	−1.72895
$525$	0	0
$526$	82.0702	3.57843
$527$	−26.4914	−1.15399
$528$	0	0
$529$	6.30822	0.274271
$530$	29.0925	1.26370
$531$	0	0
$532$	−63.5939	−2.75714
$533$	19.9957	0.866109
$534$	0	0
$535$	3.61157	0.156142
$536$	42.3185	1.82788
$537$	0	0
$538$	44.0977	1.90119
$539$	−16.8775	−0.726967
$540$	0	0
$541$	29.7950	1.28099	0.640493	−	0.767964i	$-0.278730\pi$
$541$	29.7950	1.28099	0.640493	+	0.767964i	$0.278730\pi$
$542$	29.2727	1.25737
$543$	0	0
$544$	−18.9611	−0.812952
$545$	−4.78224	−0.204849
$546$	0	0
$547$	−7.33226	−0.313505	−0.156752	−	0.987638i	$-0.550102\pi$
$547$	−7.33226	−0.313505	−0.156752	+	0.987638i	$0.550102\pi$
$548$	52.7953	2.25530
$549$	0	0
$550$	−10.1810	−0.434117
$551$	−6.77132	−0.288468
$552$	0	0
$553$	−32.4890	−1.38157
$554$	29.7895	1.26563
$555$	0	0
$556$	17.7330	0.752048
$557$	−28.4514	−1.20552	−0.602762	−	0.797921i	$-0.705933\pi$
$557$	−28.4514	−1.20552	−0.602762	+	0.797921i	$0.705933\pi$
$558$	0	0
$559$	18.1615	0.768149
$560$	−33.1963	−1.40280
$561$	0	0
$562$	19.7558	0.833348
$563$	13.4623	0.567369	0.283684	−	0.958918i	$-0.408443\pi$
$563$	13.4623	0.567369	0.283684	+	0.958918i	$0.408443\pi$
$564$	0	0
$565$	−4.33331	−0.182304
$566$	62.5687	2.62996
$567$	0	0
$568$	29.2858	1.22880
$569$	9.09616	0.381331	0.190666	−	0.981655i	$-0.438935\pi$
$569$	9.09616	0.381331	0.190666	+	0.981655i	$0.438935\pi$
$570$	0	0
$571$	46.0280	1.92621	0.963107	−	0.269120i	$-0.0867329\pi$
$571$	46.0280	1.92621	0.963107	+	0.269120i	$0.0867329\pi$
$572$	−19.0483	−0.796448
$573$	0	0
$574$	−57.7977	−2.41243
$575$	−21.7189	−0.905741
$576$	0	0
$577$	−7.77431	−0.323649	−0.161824	−	0.986820i	$-0.551738\pi$
$577$	−7.77431	−0.323649	−0.161824	+	0.986820i	$0.551738\pi$
$578$	6.03430	0.250994
$579$	0	0
$580$	−10.1964	−0.423383
$581$	42.9110	1.78025
$582$	0	0
$583$	−11.5324	−0.477625
$584$	−98.7605	−4.08674
$585$	0	0
$586$	−34.1559	−1.41097
$587$	−37.6111	−1.55238	−0.776188	−	0.630501i	$-0.782849\pi$
$587$	−37.6111	−1.55238	−0.776188	+	0.630501i	$0.782849\pi$
$588$	0	0
$589$	20.3063	0.836706
$590$	−12.9169	−0.531780
$591$	0	0
$592$	6.49075	0.266768
$593$	−24.6250	−1.01123	−0.505614	−	0.862760i	$-0.668734\pi$
$593$	−24.6250	−1.01123	−0.505614	+	0.862760i	$0.668734\pi$
$594$	0	0
$595$	−18.5744	−0.761477
$596$	−81.2031	−3.32621
$597$	0	0
$598$	−58.9394	−2.41021
$599$	22.6718	0.926343	0.463171	−	0.886269i	$-0.346711\pi$
$599$	22.6718	0.926343	0.463171	+	0.886269i	$0.346711\pi$
$600$	0	0
$601$	−29.1465	−1.18891	−0.594455	−	0.804129i	$-0.702632\pi$
$601$	−29.1465	−1.18891	−0.594455	+	0.804129i	$0.702632\pi$
$602$	−52.4960	−2.13958
$603$	0	0
$604$	50.0622	2.03700
$605$	−0.994065	−0.0404145
$606$	0	0
$607$	8.63031	0.350293	0.175147	−	0.984542i	$-0.443960\pi$
$607$	8.63031	0.350293	0.175147	+	0.984542i	$0.443960\pi$
$608$	14.5341	0.589437
$609$	0	0
$610$	−2.52267	−0.102140
$611$	−35.5036	−1.43632
$612$	0	0
$613$	−17.1543	−0.692854	−0.346427	−	0.938077i	$-0.612605\pi$
$613$	−17.1543	−0.692854	−0.346427	+	0.938077i	$0.612605\pi$
$614$	−40.0884	−1.61783
$615$	0	0
$616$	30.2582	1.21914
$617$	43.1267	1.73622	0.868108	−	0.496376i	$-0.165336\pi$
$617$	43.1267	1.73622	0.868108	+	0.496376i	$0.165336\pi$
$618$	0	0
$619$	−16.8162	−0.675902	−0.337951	−	0.941164i	$-0.609734\pi$
$619$	−16.8162	−0.675902	−0.337951	+	0.941164i	$0.609734\pi$
$620$	30.5777	1.22803
$621$	0	0
$622$	52.9189	2.12185
$623$	−0.322798	−0.0129326
$624$	0	0
$625$	11.1540	0.446160
$626$	−19.6433	−0.785103
$627$	0	0
$628$	−74.0705	−2.95574
$629$	3.63178	0.144809
$630$	0	0
$631$	37.2478	1.48281	0.741406	−	0.671057i	$-0.234160\pi$
$631$	37.2478	1.48281	0.741406	+	0.671057i	$0.234160\pi$
$632$	41.1709	1.63769
$633$	0	0
$634$	8.55429	0.339734
$635$	20.4559	0.811766
$636$	0	0
$637$	72.4060	2.86883
$638$	5.86257	0.232101
$639$	0	0
$640$	−12.5944	−0.497839
$641$	15.9557	0.630213	0.315107	−	0.949056i	$-0.397960\pi$
$641$	15.9557	0.630213	0.315107	+	0.949056i	$0.397960\pi$
$642$	0	0
$643$	35.2795	1.39129	0.695644	−	0.718387i	$-0.255119\pi$
$643$	35.2795	1.39129	0.695644	+	0.718387i	$0.255119\pi$
$644$	117.457	4.62846
$645$	0	0
$646$	28.4434	1.11909
$647$	25.5200	1.00329	0.501647	−	0.865072i	$-0.332727\pi$
$647$	25.5200	1.00329	0.501647	+	0.865072i	$0.332727\pi$
$648$	0	0
$649$	5.12033	0.200990
$650$	43.6771	1.71316
$651$	0	0
$652$	46.4767	1.82017
$653$	3.73660	0.146225	0.0731123	−	0.997324i	$-0.476707\pi$
$653$	3.73660	0.146225	0.0731123	+	0.997324i	$0.476707\pi$
$654$	0	0
$655$	8.86080	0.346220
$656$	31.8531	1.24365
$657$	0	0
$658$	102.624	4.00069
$659$	13.9968	0.545237	0.272618	−	0.962122i	$-0.412110\pi$
$659$	13.9968	0.545237	0.272618	+	0.962122i	$0.412110\pi$
$660$	0	0
$661$	16.3657	0.636551	0.318275	−	0.947998i	$-0.396896\pi$
$661$	16.3657	0.636551	0.318275	+	0.947998i	$0.396896\pi$
$662$	−28.7464	−1.11726
$663$	0	0
$664$	−54.3778	−2.11027
$665$	14.2377	0.552115
$666$	0	0
$667$	12.5065	0.484255
$668$	93.3053	3.61009
$669$	0	0
$670$	−17.2402	−0.666047
$671$	1.00000	0.0386046
$672$	0	0
$673$	27.6477	1.06574	0.532871	−	0.846196i	$-0.321113\pi$
$673$	27.6477	1.06574	0.532871	+	0.846196i	$0.321113\pi$
$674$	−69.8203	−2.68938
$675$	0	0
$676$	23.9977	0.922989
$677$	−29.5915	−1.13729	−0.568647	−	0.822581i	$-0.692533\pi$
$677$	−29.5915	−1.13729	−0.568647	+	0.822581i	$0.692533\pi$
$678$	0	0
$679$	55.2250	2.11934
$680$	23.5379	0.902639
$681$	0	0
$682$	−17.5811	−0.673214
$683$	4.11331	0.157391	0.0786957	−	0.996899i	$-0.474924\pi$
$683$	4.11331	0.157391	0.0786957	+	0.996899i	$0.474924\pi$
$684$	0	0
$685$	−11.8201	−0.451622
$686$	−122.487	−4.67656
$687$	0	0
$688$	28.9312	1.10299
$689$	49.4751	1.88485
$690$	0	0
$691$	34.3281	1.30590	0.652951	−	0.757400i	$-0.273531\pi$
$691$	34.3281	1.30590	0.652951	+	0.757400i	$0.273531\pi$
$692$	−14.2970	−0.543489
$693$	0	0
$694$	−18.9415	−0.719009
$695$	−3.97016	−0.150597
$696$	0	0
$697$	17.8228	0.675088
$698$	21.9429	0.830551
$699$	0	0
$700$	−87.0418	−3.28987
$701$	3.38074	0.127689	0.0638443	−	0.997960i	$-0.479664\pi$
$701$	3.38074	0.127689	0.0638443	+	0.997960i	$0.479664\pi$
$702$	0	0
$703$	−2.78384	−0.104995
$704$	1.08461	0.0408779
$705$	0	0
$706$	85.8085	3.22945
$707$	33.2124	1.24908
$708$	0	0
$709$	28.0573	1.05371	0.526856	−	0.849954i	$-0.323371\pi$
$709$	28.0573	1.05371	0.526856	+	0.849954i	$0.323371\pi$
$710$	−11.9308	−0.447754
$711$	0	0
$712$	0.409057	0.0153301
$713$	−37.5055	−1.40459
$714$	0	0
$715$	4.26462	0.159488
$716$	32.2837	1.20650
$717$	0	0
$718$	12.6014	0.470280
$719$	−0.0861753	−0.00321380	−0.00160690	−	0.999999i	$-0.500511\pi$
$719$	−0.0861753	−0.00321380	−0.00160690	+	0.999999i	$0.500511\pi$
$720$	0	0
$721$	59.4067	2.21242
$722$	26.4143	0.983040
$723$	0	0
$724$	59.5771	2.21416
$725$	−9.26799	−0.344205
$726$	0	0
$727$	−12.6578	−0.469453	−0.234727	−	0.972061i	$-0.575419\pi$
$727$	−12.6578	−0.469453	−0.234727	+	0.972061i	$0.575419\pi$
$728$	−129.810	−4.81108
$729$	0	0
$730$	40.2342	1.48913
$731$	16.1880	0.598733
$732$	0	0
$733$	25.0980	0.927016	0.463508	−	0.886093i	$-0.346591\pi$
$733$	25.0980	0.927016	0.463508	+	0.886093i	$0.346591\pi$
$734$	−71.8009	−2.65022
$735$	0	0
$736$	−26.8444	−0.989497
$737$	6.83412	0.251738
$738$	0	0
$739$	31.4481	1.15684	0.578419	−	0.815740i	$-0.303670\pi$
$739$	31.4481	1.15684	0.578419	+	0.815740i	$0.303670\pi$
$740$	−4.19197	−0.154100
$741$	0	0
$742$	−143.008	−5.25000
$743$	−2.21453	−0.0812431	−0.0406215	−	0.999175i	$-0.512934\pi$
$743$	−2.21453	−0.0812431	−0.0406215	+	0.999175i	$0.512934\pi$
$744$	0	0
$745$	18.1802	0.666069
$746$	46.9535	1.71909
$747$	0	0
$748$	−16.9784	−0.620790
$749$	−17.7532	−0.648688
$750$	0	0
$751$	34.7061	1.26644	0.633222	−	0.773970i	$-0.281732\pi$
$751$	34.7061	1.26644	0.633222	+	0.773970i	$0.281732\pi$
$752$	−56.5572	−2.06243
$753$	0	0
$754$	−25.1509	−0.915942
$755$	−11.2082	−0.407907
$756$	0	0
$757$	−23.4680	−0.852960	−0.426480	−	0.904497i	$-0.640247\pi$
$757$	−23.4680	−0.852960	−0.426480	+	0.904497i	$0.640247\pi$
$758$	−25.8247	−0.937995
$759$	0	0
$760$	−18.0424	−0.654465
$761$	5.93914	0.215294	0.107647	−	0.994189i	$-0.465668\pi$
$761$	5.93914	0.215294	0.107647	+	0.994189i	$0.465668\pi$
$762$	0	0
$763$	23.5078	0.851038
$764$	−71.9969	−2.60476
$765$	0	0
$766$	−45.8220	−1.65562
$767$	−21.9666	−0.793169
$768$	0	0
$769$	−4.29907	−0.155028	−0.0775142	−	0.996991i	$-0.524698\pi$
$769$	−4.29907	−0.155028	−0.0775142	+	0.996991i	$0.524698\pi$
$770$	−12.3269	−0.444231
$771$	0	0
$772$	68.7260	2.47350
$773$	19.6190	0.705646	0.352823	−	0.935690i	$-0.385222\pi$
$773$	19.6190	0.705646	0.352823	+	0.935690i	$0.385222\pi$
$774$	0	0
$775$	27.7935	0.998371
$776$	−69.9824	−2.51222
$777$	0	0
$778$	32.7975	1.17585
$779$	−13.6616	−0.489477
$780$	0	0
$781$	4.72943	0.169232
$782$	−52.5347	−1.87864
$783$	0	0
$784$	115.343	4.11938
$785$	16.5833	0.591882
$786$	0	0
$787$	−19.9085	−0.709662	−0.354831	−	0.934931i	$-0.615462\pi$
$787$	−19.9085	−0.709662	−0.354831	+	0.934931i	$0.615462\pi$
$788$	78.7490	2.80532
$789$	0	0
$790$	−16.7727	−0.596744
$791$	21.3010	0.757376
$792$	0	0
$793$	−4.29008	−0.152345
$794$	−64.1138	−2.27531
$795$	0	0
$796$	−10.7857	−0.382290
$797$	31.1383	1.10298	0.551488	−	0.834183i	$-0.314060\pi$
$797$	31.1383	1.10298	0.551488	+	0.834183i	$0.314060\pi$
$798$	0	0
$799$	−31.6456	−1.11954
$800$	19.8931	0.703326
$801$	0	0
$802$	−84.8274	−2.99536
$803$	−15.9491	−0.562830
$804$	0	0
$805$	−26.2969	−0.926843
$806$	75.4242	2.65670
$807$	0	0
$808$	−42.0876	−1.48064
$809$	−28.9916	−1.01929	−0.509645	−	0.860385i	$-0.670223\pi$
$809$	−28.9916	−1.01929	−0.509645	+	0.860385i	$0.670223\pi$
$810$	0	0
$811$	52.5821	1.84641	0.923204	−	0.384309i	$-0.125560\pi$
$811$	52.5821	1.84641	0.923204	+	0.384309i	$0.125560\pi$
$812$	50.1219	1.75893
$813$	0	0
$814$	2.41023	0.0844787
$815$	−10.4054	−0.364486
$816$	0	0
$817$	−12.4084	−0.434116
$818$	8.25684	0.288693
$819$	0	0
$820$	−20.5719	−0.718403
$821$	3.62268	0.126432	0.0632162	−	0.998000i	$-0.479864\pi$
$821$	3.62268	0.126432	0.0632162	+	0.998000i	$0.479864\pi$
$822$	0	0
$823$	52.2870	1.82261	0.911306	−	0.411730i	$-0.135075\pi$
$823$	52.2870	1.82261	0.911306	+	0.411730i	$0.135075\pi$
$824$	−75.2815	−2.62256
$825$	0	0
$826$	63.4947	2.20926
$827$	21.6441	0.752637	0.376319	−	0.926490i	$-0.377190\pi$
$827$	21.6441	0.752637	0.376319	+	0.926490i	$0.377190\pi$
$828$	0	0
$829$	23.1504	0.804048	0.402024	−	0.915629i	$-0.368307\pi$
$829$	23.1504	0.804048	0.402024	+	0.915629i	$0.368307\pi$
$830$	22.1531	0.768944
$831$	0	0
$832$	−4.65308	−0.161317
$833$	64.5379	2.23611
$834$	0	0
$835$	−20.8896	−0.722916
$836$	13.0143	0.450109
$837$	0	0
$838$	−89.5274	−3.09267
$839$	−45.0573	−1.55555	−0.777776	−	0.628541i	$-0.783652\pi$
$839$	−45.0573	−1.55555	−0.777776	+	0.628541i	$0.783652\pi$
$840$	0	0
$841$	−23.6631	−0.815971
$842$	−18.4414	−0.635533
$843$	0	0
$844$	−104.261	−3.58880
$845$	−5.37273	−0.184827
$846$	0	0
$847$	4.88646	0.167901
$848$	78.8137	2.70647
$849$	0	0
$850$	38.9309	1.33532
$851$	5.14172	0.176256
$852$	0	0
$853$	−25.4763	−0.872291	−0.436146	−	0.899876i	$-0.643657\pi$
$853$	−25.4763	−0.872291	−0.436146	+	0.899876i	$0.643657\pi$
$854$	12.4005	0.424337
$855$	0	0
$856$	22.4973	0.768941
$857$	19.0519	0.650802	0.325401	−	0.945576i	$-0.394501\pi$
$857$	19.0519	0.650802	0.325401	+	0.945576i	$0.394501\pi$
$858$	0	0
$859$	−0.566950	−0.0193441	−0.00967204	−	0.999953i	$-0.503079\pi$
$859$	−0.566950	−0.0193441	−0.00967204	+	0.999953i	$0.503079\pi$
$860$	−18.6849	−0.637150
$861$	0	0
$862$	−19.7315	−0.672056
$863$	−5.04043	−0.171578	−0.0857891	−	0.996313i	$-0.527341\pi$
$863$	−5.04043	−0.171578	−0.0857891	+	0.996313i	$0.527341\pi$
$864$	0	0
$865$	3.20087	0.108833
$866$	90.6427	3.08017
$867$	0	0
$868$	−150.309	−5.10181
$869$	6.64878	0.225544
$870$	0	0
$871$	−29.3189	−0.993434
$872$	−29.7896	−1.00880
$873$	0	0
$874$	40.2690	1.36212
$875$	43.7746	1.47985
$876$	0	0
$877$	21.1269	0.713406	0.356703	−	0.934218i	$-0.383901\pi$
$877$	21.1269	0.713406	0.356703	+	0.934218i	$0.383901\pi$
$878$	−40.1021	−1.35338
$879$	0	0
$880$	6.79353	0.229010
$881$	43.4917	1.46527	0.732635	−	0.680621i	$-0.238290\pi$
$881$	43.4917	1.46527	0.732635	+	0.680621i	$0.238290\pi$
$882$	0	0
$883$	−11.2005	−0.376925	−0.188463	−	0.982080i	$-0.560350\pi$
$883$	−11.2005	−0.376925	−0.188463	+	0.982080i	$0.560350\pi$
$884$	72.8386	2.44983
$885$	0	0
$886$	−5.59541	−0.187981
$887$	36.9615	1.24105	0.620523	−	0.784188i	$-0.286920\pi$
$887$	36.9615	1.24105	0.620523	+	0.784188i	$0.286920\pi$
$888$	0	0
$889$	−100.554	−3.37246
$890$	−0.166646	−0.00558600
$891$	0	0
$892$	87.9139	2.94358
$893$	24.2571	0.811731
$894$	0	0
$895$	−7.22784	−0.241600
$896$	61.9098	2.06826
$897$	0	0
$898$	−28.0364	−0.935587
$899$	−16.0045	−0.533780
$900$	0	0
$901$	44.0988	1.46914
$902$	11.8281	0.393834
$903$	0	0
$904$	−26.9931	−0.897778
$905$	−13.3384	−0.443384
$906$	0	0
$907$	52.8559	1.75505	0.877526	−	0.479529i	$-0.159193\pi$
$907$	52.8559	1.75505	0.877526	+	0.479529i	$0.159193\pi$
$908$	−98.8832	−3.28155
$909$	0	0
$910$	52.8835	1.75307
$911$	6.97550	0.231109	0.115554	−	0.993301i	$-0.463136\pi$
$911$	6.97550	0.231109	0.115554	+	0.993301i	$0.463136\pi$
$912$	0	0
$913$	−8.78161	−0.290629
$914$	82.0619	2.71437
$915$	0	0
$916$	20.1668	0.666331
$917$	−43.5565	−1.43836
$918$	0	0
$919$	−33.6287	−1.10931	−0.554655	−	0.832081i	$-0.687150\pi$
$919$	−33.6287	−1.10931	−0.554655	+	0.832081i	$0.687150\pi$
$920$	33.3240	1.09866
$921$	0	0
$922$	−39.8398	−1.31205
$923$	−20.2897	−0.667842
$924$	0	0
$925$	−3.81028	−0.125281
$926$	13.2458	0.435282
$927$	0	0
$928$	−11.4552	−0.376034
$929$	6.54384	0.214696	0.107348	−	0.994221i	$-0.465764\pi$
$929$	6.54384	0.214696	0.107348	+	0.994221i	$0.465764\pi$
$930$	0	0
$931$	−49.4697	−1.62131
$932$	40.4276	1.32425
$933$	0	0
$934$	96.6986	3.16408
$935$	3.80120	0.124312
$936$	0	0
$937$	−35.2517	−1.15162	−0.575811	−	0.817583i	$-0.695314\pi$
$937$	−35.2517	−1.15162	−0.575811	+	0.817583i	$0.695314\pi$
$938$	84.7466	2.76708
$939$	0	0
$940$	36.5268	1.19137
$941$	−2.92372	−0.0953107	−0.0476553	−	0.998864i	$-0.515175\pi$
$941$	−2.92372	−0.0953107	−0.0476553	+	0.998864i	$0.515175\pi$
$942$	0	0
$943$	25.2328	0.821693
$944$	−34.9928	−1.13892
$945$	0	0
$946$	10.7431	0.349290
$947$	23.8069	0.773621	0.386811	−	0.922159i	$-0.373577\pi$
$947$	23.8069	0.773621	0.386811	+	0.922159i	$0.373577\pi$
$948$	0	0
$949$	68.4228	2.22110
$950$	−29.8414	−0.968183
$951$	0	0
$952$	−115.704	−3.74999
$953$	43.2272	1.40027	0.700133	−	0.714012i	$-0.253124\pi$
$953$	43.2272	1.40027	0.700133	+	0.714012i	$0.253124\pi$
$954$	0	0
$955$	16.1190	0.521599
$956$	34.0214	1.10033
$957$	0	0
$958$	−35.4127	−1.14413
$959$	58.1032	1.87625
$960$	0	0
$961$	16.9954	0.548239
$962$	−10.3401	−0.333378
$963$	0	0
$964$	−14.7926	−0.476438
$965$	−15.3867	−0.495315
$966$	0	0
$967$	−4.95474	−0.159334	−0.0796669	−	0.996822i	$-0.525386\pi$
$967$	−4.95474	−0.159334	−0.0796669	+	0.996822i	$0.525386\pi$
$968$	−6.19224	−0.199026
$969$	0	0
$970$	28.5103	0.915409
$971$	−59.7469	−1.91737	−0.958685	−	0.284471i	$-0.908182\pi$
$971$	−59.7469	−1.91737	−0.958685	+	0.284471i	$0.908182\pi$
$972$	0	0
$973$	19.5159	0.625650
$974$	72.4791	2.32238
$975$	0	0
$976$	−6.83409	−0.218754
$977$	34.4068	1.10077	0.550386	−	0.834911i	$-0.314481\pi$
$977$	34.4068	1.10077	0.550386	+	0.834911i	$0.314481\pi$
$978$	0	0
$979$	0.0660596	0.00211127
$980$	−74.4927	−2.37958
$981$	0	0
$982$	10.0790	0.321633
$983$	25.9260	0.826912	0.413456	−	0.910524i	$-0.364321\pi$
$983$	25.9260	0.826912	0.413456	+	0.910524i	$0.364321\pi$
$984$	0	0
$985$	−17.6307	−0.561761
$986$	−22.4178	−0.713929
$987$	0	0
$988$	−55.8324	−1.77627
$989$	22.9182	0.728757
$990$	0	0
$991$	−43.2573	−1.37411	−0.687057	−	0.726604i	$-0.741098\pi$
$991$	−43.2573	−1.37411	−0.687057	+	0.726604i	$0.741098\pi$
$992$	34.3525	1.09069
$993$	0	0
$994$	58.6474	1.86018
$995$	2.41476	0.0765531
$996$	0	0
$997$	4.53605	0.143658	0.0718291	−	0.997417i	$-0.477116\pi$
$997$	4.53605	0.143658	0.0718291	+	0.997417i	$0.477116\pi$
$998$	61.2454	1.93869
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6039.2.a.i.1.2		13
3.2	odd	2		2013.2.a.e.1.12	✓	13

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2013.2.a.e.1.12	✓	13	3.2	odd	2
6039.2.a.i.1.2		13	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:	\( N \)	\(=\)	\( 6039 = 3^{2} \cdot 11 \cdot 61 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6039.a (trivial)

\(f(q)\)	\(=\)	\(q-2.53773 q^{2} +4.44007 q^{4} -0.994065 q^{5} +4.88646 q^{7} -6.19224 q^{8} +O(q^{10})\)	\(q-2.53773 q^{2} +4.44007 q^{4} -0.994065 q^{5} +4.88646 q^{7} -6.19224 q^{8} +2.52267 q^{10} -1.00000 q^{11} +4.29008 q^{13} -12.4005 q^{14} +6.83409 q^{16} +3.82389 q^{17} -2.93110 q^{19} -4.41372 q^{20} +2.53773 q^{22} +5.41371 q^{23} -4.01184 q^{25} -10.8871 q^{26} +21.6962 q^{28} +2.31016 q^{29} -6.92787 q^{31} -4.95859 q^{32} -9.70401 q^{34} -4.85746 q^{35} +0.949760 q^{37} +7.43834 q^{38} +6.15549 q^{40} +4.66091 q^{41} +4.23337 q^{43} -4.44007 q^{44} -13.7385 q^{46} -8.27575 q^{47} +16.8775 q^{49} +10.1810 q^{50} +19.0483 q^{52} +11.5324 q^{53} +0.994065 q^{55} -30.2582 q^{56} -5.86257 q^{58} -5.12033 q^{59} -1.00000 q^{61} +17.5811 q^{62} -1.08461 q^{64} -4.26462 q^{65} -6.83412 q^{67} +16.9784 q^{68} +12.3269 q^{70} -4.72943 q^{71} +15.9491 q^{73} -2.41023 q^{74} -13.0143 q^{76} -4.88646 q^{77} -6.64878 q^{79} -6.79353 q^{80} -11.8281 q^{82} +8.78161 q^{83} -3.80120 q^{85} -10.7431 q^{86} +6.19224 q^{88} -0.0660596 q^{89} +20.9633 q^{91} +24.0372 q^{92} +21.0016 q^{94} +2.91370 q^{95} +11.3016 q^{97} -42.8306 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 13 q - 2 q^{2} + 16 q^{4} - 3 q^{5} + 11 q^{7} - 9 q^{8}+O(q^{10}) \) 13 * q - 2 * q^2 + 16 * q^4 - 3 * q^5 + 11 * q^7 - 9 * q^8	\( 13 q - 2 q^{2} + 16 q^{4} - 3 q^{5} + 11 q^{7} - 9 q^{8} + 6 q^{10} - 13 q^{11} + 13 q^{13} - q^{14} + 18 q^{16} - 17 q^{17} + 14 q^{19} + 7 q^{20} + 2 q^{22} - 7 q^{23} + 18 q^{25} + 10 q^{26} + 19 q^{28} + 6 q^{29} + 27 q^{31} - 5 q^{32} + 6 q^{34} - 14 q^{35} + 10 q^{37} - 2 q^{38} + 8 q^{40} - 3 q^{41} + 29 q^{43} - 16 q^{44} - 24 q^{46} - 8 q^{47} + 8 q^{49} + 27 q^{50} + 37 q^{52} + 24 q^{53} + 3 q^{55} - 24 q^{56} - 5 q^{58} - 13 q^{59} - 13 q^{61} - 39 q^{62} + 47 q^{64} + 11 q^{65} + 44 q^{67} + 8 q^{68} - 12 q^{70} - 3 q^{71} + 48 q^{73} + 22 q^{74} + 47 q^{76} - 11 q^{77} - 17 q^{79} + 26 q^{80} + 56 q^{82} - 50 q^{83} + 8 q^{85} - 18 q^{86} + 9 q^{88} + 15 q^{89} + 47 q^{91} - 14 q^{92} + 45 q^{94} + q^{95} + 27 q^{97} - 47 q^{98}+O(q^{100}) \) 13 * q - 2 * q^2 + 16 * q^4 - 3 * q^5 + 11 * q^7 - 9 * q^8 + 6 * q^10 - 13 * q^11 + 13 * q^13 - q^14 + 18 * q^16 - 17 * q^17 + 14 * q^19 + 7 * q^20 + 2 * q^22 - 7 * q^23 + 18 * q^25 + 10 * q^26 + 19 * q^28 + 6 * q^29 + 27 * q^31 - 5 * q^32 + 6 * q^34 - 14 * q^35 + 10 * q^37 - 2 * q^38 + 8 * q^40 - 3 * q^41 + 29 * q^43 - 16 * q^44 - 24 * q^46 - 8 * q^47 + 8 * q^49 + 27 * q^50 + 37 * q^52 + 24 * q^53 + 3 * q^55 - 24 * q^56 - 5 * q^58 - 13 * q^59 - 13 * q^61 - 39 * q^62 + 47 * q^64 + 11 * q^65 + 44 * q^67 + 8 * q^68 - 12 * q^70 - 3 * q^71 + 48 * q^73 + 22 * q^74 + 47 * q^76 - 11 * q^77 - 17 * q^79 + 26 * q^80 + 56 * q^82 - 50 * q^83 + 8 * q^85 - 18 * q^86 + 9 * q^88 + 15 * q^89 + 47 * q^91 - 14 * q^92 + 45 * q^94 + q^95 + 27 * q^97 - 47 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more