LMFDB - Embedded newform 3640.2.a.w.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3640,2,Mod(1,3640)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3640, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 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("3640.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3640 = 2^{3} \cdot 5 \cdot 7 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3640.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:	$29.0655463357$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.23724.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 7x^{2} + 4x + 10 $ x^4 - x^3 - 7x^2 + 4x + 10
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-2.13169$ of defining polynomial
Character	$\chi$	$=$	3640.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.544101 q^{3} -1.00000 q^{5} +1.00000 q^{7} -2.70395 q^{9} +O(q^{10})$	$q-0.544101 q^{3} -1.00000 q^{5} +1.00000 q^{7} -2.70395 q^{9} -4.26338 q^{11} +1.00000 q^{13} +0.544101 q^{15} -3.70395 q^{17} -3.64763 q^{19} -0.544101 q^{21} -3.35158 q^{23} +1.00000 q^{25} +3.10353 q^{27} +8.80748 q^{29} -2.54410 q^{31} +2.31971 q^{33} -1.00000 q^{35} -11.1272 q^{37} -0.544101 q^{39} +9.91101 q^{41} +9.35158 q^{43} +2.70395 q^{45} +7.40791 q^{47} +1.00000 q^{49} +2.01532 q^{51} -1.08820 q^{53} +4.26338 q^{55} +1.98468 q^{57} +6.86381 q^{59} +4.20705 q^{61} -2.70395 q^{63} -1.00000 q^{65} +2.61575 q^{67} +1.82360 q^{69} -8.43978 q^{71} -11.6150 q^{73} -0.544101 q^{75} -4.26338 q^{77} +1.52755 q^{79} +6.42323 q^{81} -1.08820 q^{83} +3.70395 q^{85} -4.79216 q^{87} -0.487772 q^{89} +1.00000 q^{91} +1.38425 q^{93} +3.64763 q^{95} +10.4961 q^{97} +11.5280 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + q^{3} - 4 q^{5} + 4 q^{7} + 3 q^{9}+O(q^{10}) $ 4 * q + q^3 - 4 * q^5 + 4 * q^7 + 3 * q^9	$ 4 q + q^{3} - 4 q^{5} + 4 q^{7} + 3 q^{9} + 2 q^{11} + 4 q^{13} - q^{15} - q^{17} - 3 q^{19} + q^{21} + 12 q^{23} + 4 q^{25} + 4 q^{27} + 13 q^{29} - 7 q^{31} - 12 q^{33} - 4 q^{35} - q^{37} + q^{39} + 9 q^{41} + 12 q^{43} - 3 q^{45} + 2 q^{47} + 4 q^{49} + 6 q^{51} + 2 q^{53} - 2 q^{55} + 10 q^{57} + 3 q^{59} + 3 q^{63} - 4 q^{65} + 3 q^{67} + 20 q^{69} - 2 q^{71} - 2 q^{73} + q^{75} + 2 q^{77} + 5 q^{79} - 4 q^{81} + 2 q^{83} + q^{85} + q^{87} - q^{89} + 4 q^{91} + 13 q^{93} + 3 q^{95} + 8 q^{97} + 8 q^{99}+O(q^{100}) $ 4 * q + q^3 - 4 * q^5 + 4 * q^7 + 3 * q^9 + 2 * q^11 + 4 * q^13 - q^15 - q^17 - 3 * q^19 + q^21 + 12 * q^23 + 4 * q^25 + 4 * q^27 + 13 * q^29 - 7 * q^31 - 12 * q^33 - 4 * q^35 - q^37 + q^39 + 9 * q^41 + 12 * q^43 - 3 * q^45 + 2 * q^47 + 4 * q^49 + 6 * q^51 + 2 * q^53 - 2 * q^55 + 10 * q^57 + 3 * q^59 + 3 * q^63 - 4 * q^65 + 3 * q^67 + 20 * q^69 - 2 * q^71 - 2 * q^73 + q^75 + 2 * q^77 + 5 * q^79 - 4 * q^81 + 2 * q^83 + q^85 + q^87 - q^89 + 4 * q^91 + 13 * q^93 + 3 * q^95 + 8 * q^97 + 8 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	−0.544101	−0.314137	−0.157068	−	0.987588i	$-0.550204\pi$
$3$	−0.544101	−0.314137	−0.157068	+	0.987588i	$0.550204\pi$
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	−2.70395	−0.901318
$10$	0	0
$11$	−4.26338	−1.28546	−0.642729	−	0.766094i	$-0.722198\pi$
$11$	−4.26338	−1.28546	−0.642729	+	0.766094i	$0.722198\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	0.544101	0.140486
$16$	0	0
$17$	−3.70395	−0.898341	−0.449170	−	0.893446i	$-0.648280\pi$
$17$	−3.70395	−0.898341	−0.449170	+	0.893446i	$0.648280\pi$
$18$	0	0
$19$	−3.64763	−0.836823	−0.418411	−	0.908258i	$-0.637413\pi$
$19$	−3.64763	−0.836823	−0.418411	+	0.908258i	$0.637413\pi$
$20$	0	0
$21$	−0.544101	−0.118733
$22$	0	0
$23$	−3.35158	−0.698853	−0.349426	−	0.936964i	$-0.613624\pi$
$23$	−3.35158	−0.698853	−0.349426	+	0.936964i	$0.613624\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	3.10353	0.597274
$28$	0	0
$29$	8.80748	1.63551	0.817754	−	0.575568i	$-0.195219\pi$
$29$	8.80748	1.63551	0.817754	+	0.575568i	$0.195219\pi$
$30$	0	0
$31$	−2.54410	−0.456934	−0.228467	−	0.973552i	$-0.573371\pi$
$31$	−2.54410	−0.456934	−0.228467	+	0.973552i	$0.573371\pi$
$32$	0	0
$33$	2.31971	0.403809
$34$	0	0
$35$	−1.00000	−0.169031
$36$	0	0
$37$	−11.1272	−1.82930	−0.914649	−	0.404249i	$-0.867533\pi$
$37$	−11.1272	−1.82930	−0.914649	+	0.404249i	$0.867533\pi$
$38$	0	0
$39$	−0.544101	−0.0871258
$40$	0	0
$41$	9.91101	1.54784	0.773920	−	0.633284i	$-0.218293\pi$
$41$	9.91101	1.54784	0.773920	+	0.633284i	$0.218293\pi$
$42$	0	0
$43$	9.35158	1.42610	0.713051	−	0.701112i	$-0.247313\pi$
$43$	9.35158	1.42610	0.713051	+	0.701112i	$0.247313\pi$
$44$	0	0
$45$	2.70395	0.403082
$46$	0	0
$47$	7.40791	1.08055	0.540277	−	0.841487i	$-0.318319\pi$
$47$	7.40791	1.08055	0.540277	+	0.841487i	$0.318319\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	2.01532	0.282202
$52$	0	0
$53$	−1.08820	−0.149476	−0.0747380	−	0.997203i	$-0.523812\pi$
$53$	−1.08820	−0.149476	−0.0747380	+	0.997203i	$0.523812\pi$
$54$	0	0
$55$	4.26338	0.574874
$56$	0	0
$57$	1.98468	0.262877
$58$	0	0
$59$	6.86381	0.893592	0.446796	−	0.894636i	$-0.352565\pi$
$59$	6.86381	0.893592	0.446796	+	0.894636i	$0.352565\pi$
$60$	0	0
$61$	4.20705	0.538658	0.269329	−	0.963048i	$-0.413198\pi$
$61$	4.20705	0.538658	0.269329	+	0.963048i	$0.413198\pi$
$62$	0	0
$63$	−2.70395	−0.340666
$64$	0	0
$65$	−1.00000	−0.124035
$66$	0	0
$67$	2.61575	0.319565	0.159783	−	0.987152i	$-0.448921\pi$
$67$	2.61575	0.319565	0.159783	+	0.987152i	$0.448921\pi$
$68$	0	0
$69$	1.82360	0.219535
$70$	0	0
$71$	−8.43978	−1.00162	−0.500809	−	0.865558i	$-0.666964\pi$
$71$	−8.43978	−1.00162	−0.500809	+	0.865558i	$0.666964\pi$
$72$	0	0
$73$	−11.6150	−1.35943	−0.679714	−	0.733477i	$-0.737896\pi$
$73$	−11.6150	−1.35943	−0.679714	+	0.733477i	$0.737896\pi$
$74$	0	0
$75$	−0.544101	−0.0628273
$76$	0	0
$77$	−4.26338	−0.485857
$78$	0	0
$79$	1.52755	0.171863	0.0859315	−	0.996301i	$-0.472613\pi$
$79$	1.52755	0.171863	0.0859315	+	0.996301i	$0.472613\pi$
$80$	0	0
$81$	6.42323	0.713693
$82$	0	0
$83$	−1.08820	−0.119446	−0.0597228	−	0.998215i	$-0.519022\pi$
$83$	−1.08820	−0.119446	−0.0597228	+	0.998215i	$0.519022\pi$
$84$	0	0
$85$	3.70395	0.401750
$86$	0	0
$87$	−4.79216	−0.513773
$88$	0	0
$89$	−0.487772	−0.0517038	−0.0258519	−	0.999666i	$-0.508230\pi$
$89$	−0.487772	−0.0517038	−0.0258519	+	0.999666i	$0.508230\pi$
$90$	0	0
$91$	1.00000	0.104828
$92$	0	0
$93$	1.38425	0.143540
$94$	0	0
$95$	3.64763	0.374239
$96$	0	0
$97$	10.4961	1.06572	0.532859	−	0.846204i	$-0.321118\pi$
$97$	10.4961	1.06572	0.532859	+	0.846204i	$0.321118\pi$
$98$	0	0
$99$	11.5280	1.15861
$100$	0	0
$101$	6.00000	0.597022	0.298511	−	0.954406i	$-0.403510\pi$
$101$	6.00000	0.597022	0.298511	+	0.954406i	$0.403510\pi$
$102$	0	0
$103$	−9.05554	−0.892268	−0.446134	−	0.894966i	$-0.647200\pi$
$103$	−9.05554	−0.892268	−0.446134	+	0.894966i	$0.647200\pi$
$104$	0	0
$105$	0.544101	0.0530988
$106$	0	0
$107$	17.3516	1.67744	0.838720	−	0.544562i	$-0.183304\pi$
$107$	17.3516	1.67744	0.838720	+	0.544562i	$0.183304\pi$
$108$	0	0
$109$	−0.526759	−0.0504543	−0.0252272	−	0.999682i	$-0.508031\pi$
$109$	−0.526759	−0.0504543	−0.0252272	+	0.999682i	$0.508031\pi$
$110$	0	0
$111$	6.05431	0.574650
$112$	0	0
$113$	11.5843	1.08976	0.544880	−	0.838514i	$-0.316575\pi$
$113$	11.5843	1.08976	0.544880	+	0.838514i	$0.316575\pi$
$114$	0	0
$115$	3.35158	0.312537
$116$	0	0
$117$	−2.70395	−0.249981
$118$	0	0
$119$	−3.70395	−0.339541
$120$	0	0
$121$	7.17640	0.652400
$122$	0	0
$123$	−5.39258	−0.486233
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−8.47043	−0.751629	−0.375815	−	0.926695i	$-0.622637\pi$
$127$	−8.47043	−0.751629	−0.375815	+	0.926695i	$0.622637\pi$
$128$	0	0
$129$	−5.08820	−0.447991
$130$	0	0
$131$	−14.7032	−1.28462	−0.642311	−	0.766444i	$-0.722024\pi$
$131$	−14.7032	−1.28462	−0.642311	+	0.766444i	$0.722024\pi$
$132$	0	0
$133$	−3.64763	−0.316289
$134$	0	0
$135$	−3.10353	−0.267109
$136$	0	0
$137$	17.3036	1.47835	0.739173	−	0.673516i	$-0.235217\pi$
$137$	17.3036	1.47835	0.739173	+	0.673516i	$0.235217\pi$
$138$	0	0
$139$	−6.31971	−0.536031	−0.268015	−	0.963415i	$-0.586368\pi$
$139$	−6.31971	−0.536031	−0.268015	+	0.963415i	$0.586368\pi$
$140$	0	0
$141$	−4.03065	−0.339442
$142$	0	0
$143$	−4.26338	−0.356522
$144$	0	0
$145$	−8.80748	−0.731421
$146$	0	0
$147$	−0.544101	−0.0448767
$148$	0	0
$149$	16.5268	1.35392	0.676962	−	0.736018i	$-0.263296\pi$
$149$	16.5268	1.35392	0.676962	+	0.736018i	$0.263296\pi$
$150$	0	0
$151$	−2.96813	−0.241543	−0.120771	−	0.992680i	$-0.538537\pi$
$151$	−2.96813	−0.241543	−0.120771	+	0.992680i	$0.538537\pi$
$152$	0	0
$153$	10.0153	0.809691
$154$	0	0
$155$	2.54410	0.204347
$156$	0	0
$157$	8.66418	0.691476	0.345738	−	0.938331i	$-0.387629\pi$
$157$	8.66418	0.691476	0.345738	+	0.938331i	$0.387629\pi$
$158$	0	0
$159$	0.592091	0.0469559
$160$	0	0
$161$	−3.35158	−0.264142
$162$	0	0
$163$	−18.1847	−1.42434	−0.712169	−	0.702008i	$-0.752287\pi$
$163$	−18.1847	−1.42434	−0.712169	+	0.702008i	$0.752287\pi$
$164$	0	0
$165$	−2.31971	−0.180589
$166$	0	0
$167$	−6.73381	−0.521078	−0.260539	−	0.965463i	$-0.583900\pi$
$167$	−6.73381	−0.521078	−0.260539	+	0.965463i	$0.583900\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	9.86302	0.754244
$172$	0	0
$173$	22.2791	1.69385	0.846926	−	0.531711i	$-0.178451\pi$
$173$	22.2791	1.69385	0.846926	+	0.531711i	$0.178451\pi$
$174$	0	0
$175$	1.00000	0.0755929
$176$	0	0
$177$	−3.73460	−0.280710
$178$	0	0
$179$	15.1119	1.12951	0.564757	−	0.825257i	$-0.308970\pi$
$179$	15.1119	1.12951	0.564757	+	0.825257i	$0.308970\pi$
$180$	0	0
$181$	−3.61496	−0.268698	−0.134349	−	0.990934i	$-0.542894\pi$
$181$	−3.61496	−0.268698	−0.134349	+	0.990934i	$0.542894\pi$
$182$	0	0
$183$	−2.28906	−0.169212
$184$	0	0
$185$	11.1272	0.818087
$186$	0	0
$187$	15.7914	1.15478
$188$	0	0
$189$	3.10353	0.225748
$190$	0	0
$191$	13.2398	0.958001	0.479001	−	0.877814i	$-0.340999\pi$
$191$	13.2398	0.958001	0.479001	+	0.877814i	$0.340999\pi$
$192$	0	0
$193$	−1.32050	−0.0950517	−0.0475259	−	0.998870i	$-0.515134\pi$
$193$	−1.32050	−0.0950517	−0.0475259	+	0.998870i	$0.515134\pi$
$194$	0	0
$195$	0.544101	0.0389639
$196$	0	0
$197$	1.96101	0.139716	0.0698582	−	0.997557i	$-0.477745\pi$
$197$	1.96101	0.139716	0.0698582	+	0.997557i	$0.477745\pi$
$198$	0	0
$199$	−14.3504	−1.01727	−0.508635	−	0.860982i	$-0.669850\pi$
$199$	−14.3504	−1.01727	−0.508635	+	0.860982i	$0.669850\pi$
$200$	0	0
$201$	−1.42323	−0.100387
$202$	0	0
$203$	8.80748	0.618164
$204$	0	0
$205$	−9.91101	−0.692215
$206$	0	0
$207$	9.06252	0.629889
$208$	0	0
$209$	15.5512	1.07570
$210$	0	0
$211$	−8.82280	−0.607387	−0.303693	−	0.952770i	$-0.598220\pi$
$211$	−8.82280	−0.607387	−0.303693	+	0.952770i	$0.598220\pi$
$212$	0	0
$213$	4.59209	0.314645
$214$	0	0
$215$	−9.35158	−0.637773
$216$	0	0
$217$	−2.54410	−0.172705
$218$	0	0
$219$	6.31971	0.427046
$220$	0	0
$221$	−3.70395	−0.249155
$222$	0	0
$223$	−6.06375	−0.406058	−0.203029	−	0.979173i	$-0.565079\pi$
$223$	−6.06375	−0.406058	−0.203029	+	0.979173i	$0.565079\pi$
$224$	0	0
$225$	−2.70395	−0.180264
$226$	0	0
$227$	15.6787	1.04063	0.520316	−	0.853974i	$-0.325814\pi$
$227$	15.6787	1.04063	0.520316	+	0.853974i	$0.325814\pi$
$228$	0	0
$229$	20.1670	1.33267	0.666335	−	0.745652i	$-0.267862\pi$
$229$	20.1670	1.33267	0.666335	+	0.745652i	$0.267862\pi$
$230$	0	0
$231$	2.31971	0.152626
$232$	0	0
$233$	11.6150	0.760921	0.380461	−	0.924797i	$-0.375765\pi$
$233$	11.6150	0.760921	0.380461	+	0.924797i	$0.375765\pi$
$234$	0	0
$235$	−7.40791	−0.483239
$236$	0	0
$237$	−0.831142	−0.0539885
$238$	0	0
$239$	2.85547	0.184705	0.0923525	−	0.995726i	$-0.470561\pi$
$239$	2.85547	0.184705	0.0923525	+	0.995726i	$0.470561\pi$
$240$	0	0
$241$	0.679499	0.0437704	0.0218852	−	0.999760i	$-0.493033\pi$
$241$	0.679499	0.0437704	0.0218852	+	0.999760i	$0.493033\pi$
$242$	0	0
$243$	−12.8055	−0.821471
$244$	0	0
$245$	−1.00000	−0.0638877
$246$	0	0
$247$	−3.64763	−0.232093
$248$	0	0
$249$	0.592091	0.0375222
$250$	0	0
$251$	−4.63942	−0.292837	−0.146419	−	0.989223i	$-0.546775\pi$
$251$	−4.63942	−0.292837	−0.146419	+	0.989223i	$0.546775\pi$
$252$	0	0
$253$	14.2891	0.898345
$254$	0	0
$255$	−2.01532	−0.126204
$256$	0	0
$257$	18.8465	1.17561	0.587805	−	0.809003i	$-0.299992\pi$
$257$	18.8465	1.17561	0.587805	+	0.809003i	$0.299992\pi$
$258$	0	0
$259$	−11.1272	−0.691410
$260$	0	0
$261$	−23.8150	−1.47411
$262$	0	0
$263$	7.35158	0.453318	0.226659	−	0.973974i	$-0.427220\pi$
$263$	7.35158	0.453318	0.226659	+	0.973974i	$0.427220\pi$
$264$	0	0
$265$	1.08820	0.0668477
$266$	0	0
$267$	0.265397	0.0162421
$268$	0	0
$269$	−9.37726	−0.571742	−0.285871	−	0.958268i	$-0.592283\pi$
$269$	−9.37726	−0.571742	−0.285871	+	0.958268i	$0.592283\pi$
$270$	0	0
$271$	1.84769	0.112239	0.0561196	−	0.998424i	$-0.482127\pi$
$271$	1.84769	0.112239	0.0561196	+	0.998424i	$0.482127\pi$
$272$	0	0
$273$	−0.544101	−0.0329305
$274$	0	0
$275$	−4.26338	−0.257091
$276$	0	0
$277$	14.2891	0.858546	0.429273	−	0.903175i	$-0.358770\pi$
$277$	14.2891	0.858546	0.429273	+	0.903175i	$0.358770\pi$
$278$	0	0
$279$	6.87913	0.411843
$280$	0	0
$281$	−10.6701	−0.636523	−0.318261	−	0.948003i	$-0.603099\pi$
$281$	−10.6701	−0.636523	−0.318261	+	0.948003i	$0.603099\pi$
$282$	0	0
$283$	−2.62476	−0.156026	−0.0780128	−	0.996952i	$-0.524858\pi$
$283$	−2.62476	−0.156026	−0.0780128	+	0.996952i	$0.524858\pi$
$284$	0	0
$285$	−1.98468	−0.117562
$286$	0	0
$287$	9.91101	0.585028
$288$	0	0
$289$	−3.28072	−0.192984
$290$	0	0
$291$	−5.71094	−0.334781
$292$	0	0
$293$	25.1993	1.47216	0.736079	−	0.676896i	$-0.236675\pi$
$293$	25.1993	1.47216	0.736079	+	0.676896i	$0.236675\pi$
$294$	0	0
$295$	−6.86381	−0.399626
$296$	0	0
$297$	−13.2315	−0.767770
$298$	0	0
$299$	−3.35158	−0.193827
$300$	0	0
$301$	9.35158	0.539016
$302$	0	0
$303$	−3.26460	−0.187547
$304$	0	0
$305$	−4.20705	−0.240895
$306$	0	0
$307$	1.58431	0.0904214	0.0452107	−	0.998977i	$-0.485604\pi$
$307$	1.58431	0.0904214	0.0452107	+	0.998977i	$0.485604\pi$
$308$	0	0
$309$	4.92712	0.280294
$310$	0	0
$311$	−28.0473	−1.59042	−0.795209	−	0.606336i	$-0.792639\pi$
$311$	−28.0473	−1.59042	−0.795209	+	0.606336i	$0.792639\pi$
$312$	0	0
$313$	−18.0721	−1.02149	−0.510747	−	0.859731i	$-0.670631\pi$
$313$	−18.0721	−1.02149	−0.510747	+	0.859731i	$0.670631\pi$
$314$	0	0
$315$	2.70395	0.152351
$316$	0	0
$317$	1.36058	0.0764181	0.0382090	−	0.999270i	$-0.487835\pi$
$317$	1.36058	0.0764181	0.0382090	+	0.999270i	$0.487835\pi$
$318$	0	0
$319$	−37.5496	−2.10238
$320$	0	0
$321$	−9.44101	−0.526946
$322$	0	0
$323$	13.5106	0.751752
$324$	0	0
$325$	1.00000	0.0554700
$326$	0	0
$327$	0.286610	0.0158496
$328$	0	0
$329$	7.40791	0.408411
$330$	0	0
$331$	12.0547	0.662589	0.331294	−	0.943527i	$-0.392515\pi$
$331$	12.0547	0.662589	0.331294	+	0.943527i	$0.392515\pi$
$332$	0	0
$333$	30.0874	1.64878
$334$	0	0
$335$	−2.61575	−0.142914
$336$	0	0
$337$	19.4063	1.05713	0.528565	−	0.848893i	$-0.322730\pi$
$337$	19.4063	1.05713	0.528565	+	0.848893i	$0.322730\pi$
$338$	0	0
$339$	−6.30303	−0.342334
$340$	0	0
$341$	10.8465	0.587369
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	−1.82360	−0.0981792
$346$	0	0
$347$	31.3193	1.68131	0.840655	−	0.541571i	$-0.182170\pi$
$347$	31.3193	1.68131	0.840655	+	0.541571i	$0.182170\pi$
$348$	0	0
$349$	28.7115	1.53689	0.768446	−	0.639915i	$-0.221030\pi$
$349$	28.7115	1.53689	0.768446	+	0.639915i	$0.221030\pi$
$350$	0	0
$351$	3.10353	0.165654
$352$	0	0
$353$	−16.5905	−0.883023	−0.441512	−	0.897255i	$-0.645558\pi$
$353$	−16.5905	−0.883023	−0.441512	+	0.897255i	$0.645558\pi$
$354$	0	0
$355$	8.43978	0.447937
$356$	0	0
$357$	2.01532	0.106662
$358$	0	0
$359$	−1.84928	−0.0976011	−0.0488006	−	0.998809i	$-0.515540\pi$
$359$	−1.84928	−0.0976011	−0.0488006	+	0.998809i	$0.515540\pi$
$360$	0	0
$361$	−5.69482	−0.299728
$362$	0	0
$363$	−3.90469	−0.204943
$364$	0	0
$365$	11.6150	0.607955
$366$	0	0
$367$	5.84769	0.305247	0.152623	−	0.988284i	$-0.451228\pi$
$367$	5.84769	0.305247	0.152623	+	0.988284i	$0.451228\pi$
$368$	0	0
$369$	−26.7989	−1.39510
$370$	0	0
$371$	−1.08820	−0.0564966
$372$	0	0
$373$	26.9591	1.39589	0.697945	−	0.716151i	$-0.254098\pi$
$373$	26.9591	1.39589	0.697945	+	0.716151i	$0.254098\pi$
$374$	0	0
$375$	0.544101	0.0280972
$376$	0	0
$377$	8.80748	0.453608
$378$	0	0
$379$	−21.8783	−1.12381	−0.561907	−	0.827200i	$-0.689932\pi$
$379$	−21.8783	−1.12381	−0.561907	+	0.827200i	$0.689932\pi$
$380$	0	0
$381$	4.60877	0.236114
$382$	0	0
$383$	−5.75826	−0.294234	−0.147117	−	0.989119i	$-0.546999\pi$
$383$	−5.75826	−0.294234	−0.147117	+	0.989119i	$0.546999\pi$
$384$	0	0
$385$	4.26338	0.217282
$386$	0	0
$387$	−25.2862	−1.28537
$388$	0	0
$389$	2.88814	0.146434	0.0732172	−	0.997316i	$-0.476673\pi$
$389$	2.88814	0.146434	0.0732172	+	0.997316i	$0.476673\pi$
$390$	0	0
$391$	12.4141	0.627808
$392$	0	0
$393$	8.00000	0.403547
$394$	0	0
$395$	−1.52755	−0.0768595
$396$	0	0
$397$	16.6685	0.836567	0.418283	−	0.908317i	$-0.362632\pi$
$397$	16.6685	0.836567	0.418283	+	0.908317i	$0.362632\pi$
$398$	0	0
$399$	1.98468	0.0993581
$400$	0	0
$401$	−22.6701	−1.13209	−0.566044	−	0.824375i	$-0.691527\pi$
$401$	−22.6701	−1.13209	−0.566044	+	0.824375i	$0.691527\pi$
$402$	0	0
$403$	−2.54410	−0.126731
$404$	0	0
$405$	−6.42323	−0.319173
$406$	0	0
$407$	47.4394	2.35148
$408$	0	0
$409$	3.74404	0.185131	0.0925654	−	0.995707i	$-0.470493\pi$
$409$	3.74404	0.185131	0.0925654	+	0.995707i	$0.470493\pi$
$410$	0	0
$411$	−9.41490	−0.464403
$412$	0	0
$413$	6.86381	0.337746
$414$	0	0
$415$	1.08820	0.0534177
$416$	0	0
$417$	3.43856	0.168387
$418$	0	0
$419$	−23.6954	−1.15760	−0.578798	−	0.815471i	$-0.696478\pi$
$419$	−23.6954	−1.15760	−0.578798	+	0.815471i	$0.696478\pi$
$420$	0	0
$421$	33.9637	1.65529	0.827645	−	0.561251i	$-0.189680\pi$
$421$	33.9637	1.65529	0.827645	+	0.561251i	$0.189680\pi$
$422$	0	0
$423$	−20.0306	−0.973923
$424$	0	0
$425$	−3.70395	−0.179668
$426$	0	0
$427$	4.20705	0.203593
$428$	0	0
$429$	2.31971	0.111997
$430$	0	0
$431$	22.1980	1.06924	0.534621	−	0.845092i	$-0.320454\pi$
$431$	22.1980	1.06924	0.534621	+	0.845092i	$0.320454\pi$
$432$	0	0
$433$	6.47245	0.311046	0.155523	−	0.987832i	$-0.450294\pi$
$433$	6.47245	0.311046	0.155523	+	0.987832i	$0.450294\pi$
$434$	0	0
$435$	4.79216	0.229766
$436$	0	0
$437$	12.2253	0.584816
$438$	0	0
$439$	31.7873	1.51713	0.758563	−	0.651600i	$-0.225902\pi$
$439$	31.7873	1.51713	0.758563	+	0.651600i	$0.225902\pi$
$440$	0	0
$441$	−2.70395	−0.128760
$442$	0	0
$443$	1.70352	0.0809368	0.0404684	−	0.999181i	$-0.487115\pi$
$443$	1.70352	0.0809368	0.0404684	+	0.999181i	$0.487115\pi$
$444$	0	0
$445$	0.487772	0.0231226
$446$	0	0
$447$	−8.99222	−0.425317
$448$	0	0
$449$	−1.40791	−0.0664433	−0.0332217	−	0.999448i	$-0.510577\pi$
$449$	−1.40791	−0.0664433	−0.0332217	+	0.999448i	$0.510577\pi$
$450$	0	0
$451$	−42.2544	−1.98968
$452$	0	0
$453$	1.61496	0.0758774
$454$	0	0
$455$	−1.00000	−0.0468807
$456$	0	0
$457$	−13.9263	−0.651446	−0.325723	−	0.945465i	$-0.605608\pi$
$457$	−13.9263	−0.651446	−0.325723	+	0.945465i	$0.605608\pi$
$458$	0	0
$459$	−11.4953	−0.536555
$460$	0	0
$461$	−29.1869	−1.35937	−0.679685	−	0.733504i	$-0.737883\pi$
$461$	−29.1869	−1.35937	−0.679685	+	0.733504i	$0.737883\pi$
$462$	0	0
$463$	15.0812	0.700884	0.350442	−	0.936585i	$-0.386031\pi$
$463$	15.0812	0.700884	0.350442	+	0.936585i	$0.386031\pi$
$464$	0	0
$465$	−1.38425	−0.0641929
$466$	0	0
$467$	7.02489	0.325073	0.162536	−	0.986703i	$-0.448032\pi$
$467$	7.02489	0.325073	0.162536	+	0.986703i	$0.448032\pi$
$468$	0	0
$469$	2.61575	0.120784
$470$	0	0
$471$	−4.71418	−0.217218
$472$	0	0
$473$	−39.8693	−1.83319
$474$	0	0
$475$	−3.64763	−0.167365
$476$	0	0
$477$	2.94245	0.134725
$478$	0	0
$479$	−0.0148925	−0.000680457 0	−0.000340229	−	1.00000i	$-0.500108\pi$
$479$	−0.0148925	−0.000680457 0	−0.000340229		1.00000i	$0.500108\pi$
$480$	0	0
$481$	−11.1272	−0.507356
$482$	0	0
$483$	1.82360	0.0829765
$484$	0	0
$485$	−10.4961	−0.476604
$486$	0	0
$487$	4.68244	0.212181	0.106091	−	0.994356i	$-0.466167\pi$
$487$	4.68244	0.212181	0.106091	+	0.994356i	$0.466167\pi$
$488$	0	0
$489$	9.89433	0.447437
$490$	0	0
$491$	−38.1070	−1.71975	−0.859873	−	0.510508i	$-0.829457\pi$
$491$	−38.1070	−1.71975	−0.859873	+	0.510508i	$0.829457\pi$
$492$	0	0
$493$	−32.6225	−1.46924
$494$	0	0
$495$	−11.5280	−0.518144
$496$	0	0
$497$	−8.43978	−0.378576
$498$	0	0
$499$	9.35158	0.418634	0.209317	−	0.977848i	$-0.432876\pi$
$499$	9.35158	0.418634	0.209317	+	0.977848i	$0.432876\pi$
$500$	0	0
$501$	3.66387	0.163690
$502$	0	0
$503$	−37.0776	−1.65321	−0.826604	−	0.562783i	$-0.809731\pi$
$503$	−37.0776	−1.65321	−0.826604	+	0.562783i	$0.809731\pi$
$504$	0	0
$505$	−6.00000	−0.266996
$506$	0	0
$507$	−0.544101	−0.0241644
$508$	0	0
$509$	−28.6946	−1.27187	−0.635933	−	0.771745i	$-0.719384\pi$
$509$	−28.6946	−1.27187	−0.635933	+	0.771745i	$0.719384\pi$
$510$	0	0
$511$	−11.6150	−0.513816
$512$	0	0
$513$	−11.3205	−0.499812
$514$	0	0
$515$	9.05554	0.399035
$516$	0	0
$517$	−31.5827	−1.38901
$518$	0	0
$519$	−12.1221	−0.532101
$520$	0	0
$521$	−27.7276	−1.21477	−0.607384	−	0.794408i	$-0.707781\pi$
$521$	−27.7276	−1.21477	−0.607384	+	0.794408i	$0.707781\pi$
$522$	0	0
$523$	−7.46424	−0.326388	−0.163194	−	0.986594i	$-0.552180\pi$
$523$	−7.46424	−0.326388	−0.163194	+	0.986594i	$0.552180\pi$
$524$	0	0
$525$	−0.544101	−0.0237465
$526$	0	0
$527$	9.42323	0.410482
$528$	0	0
$529$	−11.7669	−0.511605
$530$	0	0
$531$	−18.5594	−0.805410
$532$	0	0
$533$	9.91101	0.429293
$534$	0	0
$535$	−17.3516	−0.750174
$536$	0	0
$537$	−8.22238	−0.354822
$538$	0	0
$539$	−4.26338	−0.183637
$540$	0	0
$541$	15.3426	0.659629	0.329814	−	0.944046i	$-0.393014\pi$
$541$	15.3426	0.659629	0.329814	+	0.944046i	$0.393014\pi$
$542$	0	0
$543$	1.96690	0.0844078
$544$	0	0
$545$	0.526759	0.0225639
$546$	0	0
$547$	27.1752	1.16193	0.580963	−	0.813930i	$-0.302676\pi$
$547$	27.1752	1.16193	0.580963	+	0.813930i	$0.302676\pi$
$548$	0	0
$549$	−11.3757	−0.485502
$550$	0	0
$551$	−32.1264	−1.36863
$552$	0	0
$553$	1.52755	0.0649581
$554$	0	0
$555$	−6.05431	−0.256991
$556$	0	0
$557$	−26.3127	−1.11491	−0.557453	−	0.830209i	$-0.688221\pi$
$557$	−26.3127	−1.11491	−0.557453	+	0.830209i	$0.688221\pi$
$558$	0	0
$559$	9.35158	0.395530
$560$	0	0
$561$	−8.59209	−0.362758
$562$	0	0
$563$	37.7754	1.59204	0.796021	−	0.605269i	$-0.206934\pi$
$563$	37.7754	1.59204	0.796021	+	0.605269i	$0.206934\pi$
$564$	0	0
$565$	−11.5843	−0.487356
$566$	0	0
$567$	6.42323	0.269750
$568$	0	0
$569$	17.9570	0.752796	0.376398	−	0.926458i	$-0.377163\pi$
$569$	17.9570	0.752796	0.376398	+	0.926458i	$0.377163\pi$
$570$	0	0
$571$	−4.54907	−0.190373	−0.0951863	−	0.995459i	$-0.530345\pi$
$571$	−4.54907	−0.190373	−0.0951863	+	0.995459i	$0.530345\pi$
$572$	0	0
$573$	−7.20381	−0.300943
$574$	0	0
$575$	−3.35158	−0.139771
$576$	0	0
$577$	16.6685	0.693918	0.346959	−	0.937880i	$-0.387214\pi$
$577$	16.6685	0.693918	0.346959	+	0.937880i	$0.387214\pi$
$578$	0	0
$579$	0.718485	0.0298592
$580$	0	0
$581$	−1.08820	−0.0451462
$582$	0	0
$583$	4.63942	0.192145
$584$	0	0
$585$	2.70395	0.111795
$586$	0	0
$587$	−34.8473	−1.43830	−0.719152	−	0.694853i	$-0.755469\pi$
$587$	−34.8473	−1.43830	−0.719152	+	0.694853i	$0.755469\pi$
$588$	0	0
$589$	9.27993	0.382373
$590$	0	0
$591$	−1.06699	−0.0438901
$592$	0	0
$593$	−17.8734	−0.733971	−0.366986	−	0.930227i	$-0.619610\pi$
$593$	−17.8734	−0.733971	−0.366986	+	0.930227i	$0.619610\pi$
$594$	0	0
$595$	3.70395	0.151847
$596$	0	0
$597$	7.80804	0.319562
$598$	0	0
$599$	33.7591	1.37936	0.689680	−	0.724114i	$-0.257751\pi$
$599$	33.7591	1.37936	0.689680	+	0.724114i	$0.257751\pi$
$600$	0	0
$601$	−31.3757	−1.27984	−0.639920	−	0.768441i	$-0.721033\pi$
$601$	−31.3757	−1.27984	−0.639920	+	0.768441i	$0.721033\pi$
$602$	0	0
$603$	−7.07288	−0.288030
$604$	0	0
$605$	−7.17640	−0.291762
$606$	0	0
$607$	5.08728	0.206486	0.103243	−	0.994656i	$-0.467078\pi$
$607$	5.08728	0.206486	0.103243	+	0.994656i	$0.467078\pi$
$608$	0	0
$609$	−4.79216	−0.194188
$610$	0	0
$611$	7.40791	0.299692
$612$	0	0
$613$	28.3504	1.14506	0.572530	−	0.819884i	$-0.305962\pi$
$613$	28.3504	1.14506	0.572530	+	0.819884i	$0.305962\pi$
$614$	0	0
$615$	5.39258	0.217450
$616$	0	0
$617$	1.03389	0.0416229	0.0208114	−	0.999783i	$-0.493375\pi$
$617$	1.03389	0.0416229	0.0208114	+	0.999783i	$0.493375\pi$
$618$	0	0
$619$	−26.0622	−1.04753	−0.523764	−	0.851863i	$-0.675473\pi$
$619$	−26.0622	−1.04753	−0.523764	+	0.851863i	$0.675473\pi$
$620$	0	0
$621$	−10.4017	−0.417406
$622$	0	0
$623$	−0.487772	−0.0195422
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−8.46143	−0.337917
$628$	0	0
$629$	41.2146	1.64333
$630$	0	0
$631$	−20.4911	−0.815739	−0.407870	−	0.913040i	$-0.633728\pi$
$631$	−20.4911	−0.815739	−0.407870	+	0.913040i	$0.633728\pi$
$632$	0	0
$633$	4.80049	0.190802
$634$	0	0
$635$	8.47043	0.336139
$636$	0	0
$637$	1.00000	0.0396214
$638$	0	0
$639$	22.8208	0.902776
$640$	0	0
$641$	37.7650	1.49163	0.745814	−	0.666154i	$-0.232061\pi$
$641$	37.7650	1.49163	0.745814	+	0.666154i	$0.232061\pi$
$642$	0	0
$643$	29.9637	1.18165	0.590827	−	0.806798i	$-0.298802\pi$
$643$	29.9637	1.18165	0.590827	+	0.806798i	$0.298802\pi$
$644$	0	0
$645$	5.08820	0.200348
$646$	0	0
$647$	12.3024	0.483656	0.241828	−	0.970319i	$-0.422253\pi$
$647$	12.3024	0.483656	0.241828	+	0.970319i	$0.422253\pi$
$648$	0	0
$649$	−29.2630	−1.14867
$650$	0	0
$651$	1.38425	0.0542529
$652$	0	0
$653$	−0.240149	−0.00939775	−0.00469888	−	0.999989i	$-0.501496\pi$
$653$	−0.240149	−0.00939775	−0.00469888	+	0.999989i	$0.501496\pi$
$654$	0	0
$655$	14.7032	0.574500
$656$	0	0
$657$	31.4063	1.22528
$658$	0	0
$659$	31.0465	1.20940	0.604701	−	0.796453i	$-0.293293\pi$
$659$	31.0465	1.20940	0.604701	+	0.796453i	$0.293293\pi$
$660$	0	0
$661$	−42.9828	−1.67184	−0.835918	−	0.548854i	$-0.815064\pi$
$661$	−42.9828	−1.67184	−0.835918	+	0.548854i	$0.815064\pi$
$662$	0	0
$663$	2.01532	0.0782687
$664$	0	0
$665$	3.64763	0.141449
$666$	0	0
$667$	−29.5190	−1.14298
$668$	0	0
$669$	3.29929	0.127558
$670$	0	0
$671$	−17.9363	−0.692421
$672$	0	0
$673$	0.512532	0.0197567	0.00987834	−	0.999951i	$-0.496856\pi$
$673$	0.512532	0.0197567	0.00987834	+	0.999951i	$0.496856\pi$
$674$	0	0
$675$	3.10353	0.119455
$676$	0	0
$677$	−19.0535	−0.732286	−0.366143	−	0.930559i	$-0.619322\pi$
$677$	−19.0535	−0.732286	−0.366143	+	0.930559i	$0.619322\pi$
$678$	0	0
$679$	10.4961	0.402804
$680$	0	0
$681$	−8.53079	−0.326901
$682$	0	0
$683$	−38.0057	−1.45425	−0.727123	−	0.686507i	$-0.759143\pi$
$683$	−38.0057	−1.45425	−0.727123	+	0.686507i	$0.759143\pi$
$684$	0	0
$685$	−17.3036	−0.661136
$686$	0	0
$687$	−10.9729	−0.418641
$688$	0	0
$689$	−1.08820	−0.0414572
$690$	0	0
$691$	19.0693	0.725429	0.362715	−	0.931900i	$-0.381850\pi$
$691$	19.0693	0.725429	0.362715	+	0.931900i	$0.381850\pi$
$692$	0	0
$693$	11.5280	0.437912
$694$	0	0
$695$	6.31971	0.239720
$696$	0	0
$697$	−36.7099	−1.39049
$698$	0	0
$699$	−6.31971	−0.239033
$700$	0	0
$701$	−22.1181	−0.835387	−0.417694	−	0.908588i	$-0.637161\pi$
$701$	−22.1181	−0.835387	−0.417694	+	0.908588i	$0.637161\pi$
$702$	0	0
$703$	40.5878	1.53080
$704$	0	0
$705$	4.03065	0.151803
$706$	0	0
$707$	6.00000	0.225653
$708$	0	0
$709$	0.465462	0.0174808	0.00874040	−	0.999962i	$-0.497218\pi$
$709$	0.465462	0.0174808	0.00874040	+	0.999962i	$0.497218\pi$
$710$	0	0
$711$	−4.13043	−0.154903
$712$	0	0
$713$	8.52676	0.319330
$714$	0	0
$715$	4.26338	0.159441
$716$	0	0
$717$	−1.55366	−0.0580226
$718$	0	0
$719$	3.36058	0.125329	0.0626643	−	0.998035i	$-0.480040\pi$
$719$	3.36058	0.125329	0.0626643	+	0.998035i	$0.480040\pi$
$720$	0	0
$721$	−9.05554	−0.337246
$722$	0	0
$723$	−0.369716	−0.0137499
$724$	0	0
$725$	8.80748	0.327102
$726$	0	0
$727$	−36.5425	−1.35529	−0.677643	−	0.735391i	$-0.736999\pi$
$727$	−36.5425	−1.35529	−0.677643	+	0.735391i	$0.736999\pi$
$728$	0	0
$729$	−12.3022	−0.455638
$730$	0	0
$731$	−34.6378	−1.28113
$732$	0	0
$733$	−14.4448	−0.533529	−0.266765	−	0.963762i	$-0.585955\pi$
$733$	−14.4448	−0.533529	−0.266765	+	0.963762i	$0.585955\pi$
$734$	0	0
$735$	0.544101	0.0200695
$736$	0	0
$737$	−11.1519	−0.410787
$738$	0	0
$739$	22.6146	0.831891	0.415946	−	0.909389i	$-0.363451\pi$
$739$	22.6146	0.831891	0.415946	+	0.909389i	$0.363451\pi$
$740$	0	0
$741$	1.98468	0.0729089
$742$	0	0
$743$	−10.0237	−0.367732	−0.183866	−	0.982951i	$-0.558861\pi$
$743$	−10.0237	−0.367732	−0.183866	+	0.982951i	$0.558861\pi$
$744$	0	0
$745$	−16.5268	−0.605494
$746$	0	0
$747$	2.94245	0.107659
$748$	0	0
$749$	17.3516	0.634013
$750$	0	0
$751$	−19.6564	−0.717272	−0.358636	−	0.933477i	$-0.616758\pi$
$751$	−19.6564	−0.717272	−0.358636	+	0.933477i	$0.616758\pi$
$752$	0	0
$753$	2.52431	0.0919909
$754$	0	0
$755$	2.96813	0.108021
$756$	0	0
$757$	22.9424	0.833857	0.416929	−	0.908939i	$-0.363106\pi$
$757$	22.9424	0.833857	0.416929	+	0.908939i	$0.363106\pi$
$758$	0	0
$759$	−7.77469	−0.282203
$760$	0	0
$761$	−31.7177	−1.14977	−0.574883	−	0.818236i	$-0.694952\pi$
$761$	−31.7177	−1.14977	−0.574883	+	0.818236i	$0.694952\pi$
$762$	0	0
$763$	−0.526759	−0.0190699
$764$	0	0
$765$	−10.0153	−0.362105
$766$	0	0
$767$	6.86381	0.247838
$768$	0	0
$769$	12.3173	0.444172	0.222086	−	0.975027i	$-0.428713\pi$
$769$	12.3173	0.444172	0.222086	+	0.975027i	$0.428713\pi$
$770$	0	0
$771$	−10.2544	−0.369302
$772$	0	0
$773$	−21.7607	−0.782678	−0.391339	−	0.920246i	$-0.627988\pi$
$773$	−21.7607	−0.782678	−0.391339	+	0.920246i	$0.627988\pi$
$774$	0	0
$775$	−2.54410	−0.0913868
$776$	0	0
$777$	6.05431	0.217197
$778$	0	0
$779$	−36.1516	−1.29527
$780$	0	0
$781$	35.9820	1.28754
$782$	0	0
$783$	27.3342	0.976846
$784$	0	0
$785$	−8.66418	−0.309238
$786$	0	0
$787$	−36.2834	−1.29336	−0.646682	−	0.762759i	$-0.723844\pi$
$787$	−36.2834	−1.29336	−0.646682	+	0.762759i	$0.723844\pi$
$788$	0	0
$789$	−4.00000	−0.142404
$790$	0	0
$791$	11.5843	0.411891
$792$	0	0
$793$	4.20705	0.149397
$794$	0	0
$795$	−0.592091	−0.0209993
$796$	0	0
$797$	43.8328	1.55264	0.776319	−	0.630341i	$-0.217085\pi$
$797$	43.8328	1.55264	0.776319	+	0.630341i	$0.217085\pi$
$798$	0	0
$799$	−27.4386	−0.970706
$800$	0	0
$801$	1.31891	0.0466015
$802$	0	0
$803$	49.5190	1.74749
$804$	0	0
$805$	3.35158	0.118128
$806$	0	0
$807$	5.10217	0.179605
$808$	0	0
$809$	47.1576	1.65797	0.828986	−	0.559269i	$-0.188918\pi$
$809$	47.1576	1.65797	0.828986	+	0.559269i	$0.188918\pi$
$810$	0	0
$811$	−25.3491	−0.890128	−0.445064	−	0.895499i	$-0.646819\pi$
$811$	−25.3491	−0.890128	−0.445064	+	0.895499i	$0.646819\pi$
$812$	0	0
$813$	−1.00533	−0.0352585
$814$	0	0
$815$	18.1847	0.636984
$816$	0	0
$817$	−34.1111	−1.19340
$818$	0	0
$819$	−2.70395	−0.0944838
$820$	0	0
$821$	0.910212	0.0317666	0.0158833	−	0.999874i	$-0.494944\pi$
$821$	0.910212	0.0317666	0.0158833	+	0.999874i	$0.494944\pi$
$822$	0	0
$823$	−15.1736	−0.528918	−0.264459	−	0.964397i	$-0.585193\pi$
$823$	−15.1736	−0.528918	−0.264459	+	0.964397i	$0.585193\pi$
$824$	0	0
$825$	2.31971	0.0807619
$826$	0	0
$827$	23.4552	0.815618	0.407809	−	0.913067i	$-0.366293\pi$
$827$	23.4552	0.815618	0.407809	+	0.913067i	$0.366293\pi$
$828$	0	0
$829$	−18.7976	−0.652866	−0.326433	−	0.945220i	$-0.605847\pi$
$829$	−18.7976	−0.652866	−0.326433	+	0.945220i	$0.605847\pi$
$830$	0	0
$831$	−7.77469	−0.269701
$832$	0	0
$833$	−3.70395	−0.128334
$834$	0	0
$835$	6.73381	0.233033
$836$	0	0
$837$	−7.89568	−0.272915
$838$	0	0
$839$	−12.2161	−0.421745	−0.210873	−	0.977514i	$-0.567630\pi$
$839$	−12.2161	−0.421745	−0.210873	+	0.977514i	$0.567630\pi$
$840$	0	0
$841$	48.5717	1.67489
$842$	0	0
$843$	5.80559	0.199955
$844$	0	0
$845$	−1.00000	−0.0344010
$846$	0	0
$847$	7.17640	0.246584
$848$	0	0
$849$	1.42813	0.0490134
$850$	0	0
$851$	37.2937	1.27841
$852$	0	0
$853$	54.1748	1.85491	0.927455	−	0.373934i	$-0.121991\pi$
$853$	54.1748	1.85491	0.927455	+	0.373934i	$0.121991\pi$
$854$	0	0
$855$	−9.86302	−0.337308
$856$	0	0
$857$	5.73595	0.195936	0.0979682	−	0.995190i	$-0.468766\pi$
$857$	5.73595	0.195936	0.0979682	+	0.995190i	$0.468766\pi$
$858$	0	0
$859$	−16.7669	−0.572079	−0.286040	−	0.958218i	$-0.592339\pi$
$859$	−16.7669	−0.572079	−0.286040	+	0.958218i	$0.592339\pi$
$860$	0	0
$861$	−5.39258	−0.183779
$862$	0	0
$863$	−24.1347	−0.821556	−0.410778	−	0.911735i	$-0.634743\pi$
$863$	−24.1347	−0.821556	−0.410778	+	0.911735i	$0.634743\pi$
$864$	0	0
$865$	−22.2791	−0.757513
$866$	0	0
$867$	1.78504	0.0606232
$868$	0	0
$869$	−6.51253	−0.220923
$870$	0	0
$871$	2.61575	0.0886314
$872$	0	0
$873$	−28.3810	−0.960551
$874$	0	0
$875$	−1.00000	−0.0338062
$876$	0	0
$877$	21.4300	0.723639	0.361820	−	0.932248i	$-0.382156\pi$
$877$	21.4300	0.723639	0.361820	+	0.932248i	$0.382156\pi$
$878$	0	0
$879$	−13.7109	−0.462459
$880$	0	0
$881$	27.5023	0.926576	0.463288	−	0.886208i	$-0.346670\pi$
$881$	27.5023	0.926576	0.463288	+	0.886208i	$0.346670\pi$
$882$	0	0
$883$	40.5982	1.36624	0.683119	−	0.730307i	$-0.260623\pi$
$883$	40.5982	1.36624	0.683119	+	0.730307i	$0.260623\pi$
$884$	0	0
$885$	3.73460	0.125537
$886$	0	0
$887$	−5.56452	−0.186838	−0.0934191	−	0.995627i	$-0.529780\pi$
$887$	−5.56452	−0.186838	−0.0934191	+	0.995627i	$0.529780\pi$
$888$	0	0
$889$	−8.47043	−0.284089
$890$	0	0
$891$	−27.3847	−0.917421
$892$	0	0
$893$	−27.0213	−0.904233
$894$	0	0
$895$	−15.1119	−0.505134
$896$	0	0
$897$	1.82360	0.0608881
$898$	0	0
$899$	−22.4071	−0.747319
$900$	0	0
$901$	4.03065	0.134280
$902$	0	0
$903$	−5.08820	−0.169325
$904$	0	0
$905$	3.61496	0.120165
$906$	0	0
$907$	11.2405	0.373235	0.186618	−	0.982433i	$-0.440247\pi$
$907$	11.2405	0.373235	0.186618	+	0.982433i	$0.440247\pi$
$908$	0	0
$909$	−16.2237	−0.538107
$910$	0	0
$911$	47.8126	1.58410	0.792051	−	0.610455i	$-0.209014\pi$
$911$	47.8126	1.58410	0.792051	+	0.610455i	$0.209014\pi$
$912$	0	0
$913$	4.63942	0.153542
$914$	0	0
$915$	2.28906	0.0756740
$916$	0	0
$917$	−14.7032	−0.485541
$918$	0	0
$919$	32.4085	1.06906	0.534528	−	0.845151i	$-0.320489\pi$
$919$	32.4085	1.06906	0.534528	+	0.845151i	$0.320489\pi$
$920$	0	0
$921$	−0.862025	−0.0284047
$922$	0	0
$923$	−8.43978	−0.277799
$924$	0	0
$925$	−11.1272	−0.365860
$926$	0	0
$927$	24.4858	0.804218
$928$	0	0
$929$	0.343604	0.0112733	0.00563665	−	0.999984i	$-0.498206\pi$
$929$	0.343604	0.0112733	0.00563665	+	0.999984i	$0.498206\pi$
$930$	0	0
$931$	−3.64763	−0.119546
$932$	0	0
$933$	15.2606	0.499609
$934$	0	0
$935$	−15.7914	−0.516433
$936$	0	0
$937$	−45.3969	−1.48305	−0.741526	−	0.670924i	$-0.765898\pi$
$937$	−45.3969	−1.48305	−0.741526	+	0.670924i	$0.765898\pi$
$938$	0	0
$939$	9.83303	0.320889
$940$	0	0
$941$	55.3324	1.80379	0.901893	−	0.431960i	$-0.142178\pi$
$941$	55.3324	1.80379	0.901893	+	0.431960i	$0.142178\pi$
$942$	0	0
$943$	−33.2175	−1.08171
$944$	0	0
$945$	−3.10353	−0.100958
$946$	0	0
$947$	−36.9016	−1.19914	−0.599571	−	0.800322i	$-0.704662\pi$
$947$	−36.9016	−1.19914	−0.599571	+	0.800322i	$0.704662\pi$
$948$	0	0
$949$	−11.6150	−0.377038
$950$	0	0
$951$	−0.740295	−0.0240057
$952$	0	0
$953$	−40.5716	−1.31424	−0.657122	−	0.753784i	$-0.728226\pi$
$953$	−40.5716	−1.31424	−0.657122	+	0.753784i	$0.728226\pi$
$954$	0	0
$955$	−13.2398	−0.428431
$956$	0	0
$957$	20.4308	0.660433
$958$	0	0
$959$	17.3036	0.558762
$960$	0	0
$961$	−24.5276	−0.791211
$962$	0	0
$963$	−46.9179	−1.51191
$964$	0	0
$965$	1.32050	0.0425084
$966$	0	0
$967$	26.3587	0.847638	0.423819	−	0.905747i	$-0.360689\pi$
$967$	26.3587	0.847638	0.423819	+	0.905747i	$0.360689\pi$
$968$	0	0
$969$	−7.35115	−0.236153
$970$	0	0
$971$	40.0442	1.28508	0.642539	−	0.766253i	$-0.277881\pi$
$971$	40.0442	1.28508	0.642539	+	0.766253i	$0.277881\pi$
$972$	0	0
$973$	−6.31971	−0.202601
$974$	0	0
$975$	−0.544101	−0.0174252
$976$	0	0
$977$	40.3724	1.29163	0.645814	−	0.763495i	$-0.276518\pi$
$977$	40.3724	1.29163	0.645814	+	0.763495i	$0.276518\pi$
$978$	0	0
$979$	2.07956	0.0664630
$980$	0	0
$981$	1.42433	0.0454754
$982$	0	0
$983$	−57.6582	−1.83901	−0.919506	−	0.393075i	$-0.871411\pi$
$983$	−57.6582	−1.83901	−0.919506	+	0.393075i	$0.871411\pi$
$984$	0	0
$985$	−1.96101	−0.0624831
$986$	0	0
$987$	−4.03065	−0.128297
$988$	0	0
$989$	−31.3426	−0.996636
$990$	0	0
$991$	−46.4031	−1.47404	−0.737021	−	0.675870i	$-0.763768\pi$
$991$	−46.4031	−1.47404	−0.737021	+	0.675870i	$0.763768\pi$
$992$	0	0
$993$	−6.55899	−0.208143
$994$	0	0
$995$	14.3504	0.454937
$996$	0	0
$997$	15.7943	0.500211	0.250105	−	0.968219i	$-0.419535\pi$
$997$	15.7943	0.500211	0.250105	+	0.968219i	$0.419535\pi$
$998$	0	0
$999$	−34.5335	−1.09259

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3640.2.a.w.1.2	✓	4
4.3	odd	2		7280.2.a.bs.1.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3640.2.a.w.1.2	✓	4	1.1	even	1	trivial
7280.2.a.bs.1.3		4	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 3640 = 2^{3} \cdot 5 \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3640.a (trivial)

\(f(q)\)	\(=\)	\(q-0.544101 q^{3} -1.00000 q^{5} +1.00000 q^{7} -2.70395 q^{9} +O(q^{10})\)	\(q-0.544101 q^{3} -1.00000 q^{5} +1.00000 q^{7} -2.70395 q^{9} -4.26338 q^{11} +1.00000 q^{13} +0.544101 q^{15} -3.70395 q^{17} -3.64763 q^{19} -0.544101 q^{21} -3.35158 q^{23} +1.00000 q^{25} +3.10353 q^{27} +8.80748 q^{29} -2.54410 q^{31} +2.31971 q^{33} -1.00000 q^{35} -11.1272 q^{37} -0.544101 q^{39} +9.91101 q^{41} +9.35158 q^{43} +2.70395 q^{45} +7.40791 q^{47} +1.00000 q^{49} +2.01532 q^{51} -1.08820 q^{53} +4.26338 q^{55} +1.98468 q^{57} +6.86381 q^{59} +4.20705 q^{61} -2.70395 q^{63} -1.00000 q^{65} +2.61575 q^{67} +1.82360 q^{69} -8.43978 q^{71} -11.6150 q^{73} -0.544101 q^{75} -4.26338 q^{77} +1.52755 q^{79} +6.42323 q^{81} -1.08820 q^{83} +3.70395 q^{85} -4.79216 q^{87} -0.487772 q^{89} +1.00000 q^{91} +1.38425 q^{93} +3.64763 q^{95} +10.4961 q^{97} +11.5280 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + q^{3} - 4 q^{5} + 4 q^{7} + 3 q^{9}+O(q^{10}) \) 4 * q + q^3 - 4 * q^5 + 4 * q^7 + 3 * q^9	\( 4 q + q^{3} - 4 q^{5} + 4 q^{7} + 3 q^{9} + 2 q^{11} + 4 q^{13} - q^{15} - q^{17} - 3 q^{19} + q^{21} + 12 q^{23} + 4 q^{25} + 4 q^{27} + 13 q^{29} - 7 q^{31} - 12 q^{33} - 4 q^{35} - q^{37} + q^{39} + 9 q^{41} + 12 q^{43} - 3 q^{45} + 2 q^{47} + 4 q^{49} + 6 q^{51} + 2 q^{53} - 2 q^{55} + 10 q^{57} + 3 q^{59} + 3 q^{63} - 4 q^{65} + 3 q^{67} + 20 q^{69} - 2 q^{71} - 2 q^{73} + q^{75} + 2 q^{77} + 5 q^{79} - 4 q^{81} + 2 q^{83} + q^{85} + q^{87} - q^{89} + 4 q^{91} + 13 q^{93} + 3 q^{95} + 8 q^{97} + 8 q^{99}+O(q^{100}) \) 4 * q + q^3 - 4 * q^5 + 4 * q^7 + 3 * q^9 + 2 * q^11 + 4 * q^13 - q^15 - q^17 - 3 * q^19 + q^21 + 12 * q^23 + 4 * q^25 + 4 * q^27 + 13 * q^29 - 7 * q^31 - 12 * q^33 - 4 * q^35 - q^37 + q^39 + 9 * q^41 + 12 * q^43 - 3 * q^45 + 2 * q^47 + 4 * q^49 + 6 * q^51 + 2 * q^53 - 2 * q^55 + 10 * q^57 + 3 * q^59 + 3 * q^63 - 4 * q^65 + 3 * q^67 + 20 * q^69 - 2 * q^71 - 2 * q^73 + q^75 + 2 * q^77 + 5 * q^79 - 4 * q^81 + 2 * q^83 + q^85 + q^87 - q^89 + 4 * q^91 + 13 * q^93 + 3 * q^95 + 8 * q^97 + 8 * q^99

Introduction

L-functions

Modular forms

Varieties

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Representations

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Database

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