LMFDB - Embedded newform 7360.2.a.cd.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7360,2,Mod(1,7360)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7360, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("7360.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7360 = 2^{6} \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7360.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:	$58.7698958877$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.1573.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 7x + 2 $ x^3 - x^2 - 7*x + 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 3680)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$0.277754$ of defining polynomial
Character	$\chi$	$=$	7360.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.277754 q^{3} -1.00000 q^{5} +1.27775 q^{7} -2.92285 q^{9} +O(q^{10})$	$q+0.277754 q^{3} -1.00000 q^{5} +1.27775 q^{7} -2.92285 q^{9} +0.277754 q^{11} +3.92285 q^{13} -0.277754 q^{15} +5.47836 q^{17} +3.92285 q^{19} +0.354902 q^{21} +1.00000 q^{23} +1.00000 q^{25} -1.64510 q^{27} -0.799393 q^{29} +5.27775 q^{31} +0.0771475 q^{33} -1.27775 q^{35} +1.75612 q^{37} +1.08959 q^{39} +10.0339 q^{41} +4.00000 q^{43} +2.92285 q^{45} -8.95672 q^{47} -5.36734 q^{49} +1.52164 q^{51} -10.6451 q^{53} -0.277754 q^{55} +1.08959 q^{57} -8.08959 q^{59} -15.0124 q^{61} -3.73469 q^{63} -3.92285 q^{65} -12.6451 q^{67} +0.277754 q^{69} +9.32407 q^{71} +4.55551 q^{73} +0.277754 q^{75} +0.354902 q^{77} -8.00000 q^{79} +8.31162 q^{81} +2.64510 q^{83} -5.47836 q^{85} -0.222035 q^{87} +13.2902 q^{89} +5.01244 q^{91} +1.46592 q^{93} -3.92285 q^{95} +3.16674 q^{97} -0.811835 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + q^{3} - 3 q^{5} + 4 q^{7} + 6 q^{9}+O(q^{10}) $ 3 * q + q^3 - 3 * q^5 + 4 * q^7 + 6 * q^9	$ 3 q + q^{3} - 3 q^{5} + 4 q^{7} + 6 q^{9} + q^{11} - 3 q^{13} - q^{15} + 2 q^{17} - 3 q^{19} + 16 q^{21} + 3 q^{23} + 3 q^{25} + 10 q^{27} - 17 q^{29} + 16 q^{31} + 15 q^{33} - 4 q^{35} - 9 q^{37} - 12 q^{39} + 16 q^{41} + 12 q^{43} - 6 q^{45} + 2 q^{47} - q^{49} + 19 q^{51} - 17 q^{53} - q^{55} - 12 q^{57} - 9 q^{59} - 15 q^{61} + 19 q^{63} + 3 q^{65} - 23 q^{67} + q^{69} - 16 q^{71} + 14 q^{73} + q^{75} + 16 q^{77} - 24 q^{79} + 11 q^{81} - 7 q^{83} - 2 q^{85} - 2 q^{87} + 10 q^{89} - 15 q^{91} + 20 q^{93} + 3 q^{95} + 9 q^{97} + 13 q^{99}+O(q^{100}) $ 3 * q + q^3 - 3 * q^5 + 4 * q^7 + 6 * q^9 + q^11 - 3 * q^13 - q^15 + 2 * q^17 - 3 * q^19 + 16 * q^21 + 3 * q^23 + 3 * q^25 + 10 * q^27 - 17 * q^29 + 16 * q^31 + 15 * q^33 - 4 * q^35 - 9 * q^37 - 12 * q^39 + 16 * q^41 + 12 * q^43 - 6 * q^45 + 2 * q^47 - q^49 + 19 * q^51 - 17 * q^53 - q^55 - 12 * q^57 - 9 * q^59 - 15 * q^61 + 19 * q^63 + 3 * q^65 - 23 * q^67 + q^69 - 16 * q^71 + 14 * q^73 + q^75 + 16 * q^77 - 24 * q^79 + 11 * q^81 - 7 * q^83 - 2 * q^85 - 2 * q^87 + 10 * q^89 - 15 * q^91 + 20 * q^93 + 3 * q^95 + 9 * q^97 + 13 * 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
$3$	0.277754	0.160362	0.0801808	−	0.996780i	$-0.474450\pi$
$3$	0.277754	0.160362	0.0801808	+	0.996780i	$0.474450\pi$
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	1.27775	0.482946	0.241473	−	0.970408i	$-0.422370\pi$
$7$	1.27775	0.482946	0.241473	+	0.970408i	$0.422370\pi$
$8$	0	0
$9$	−2.92285	−0.974284
$10$	0	0
$11$	0.277754	0.0837461	0.0418730	−	0.999123i	$-0.486667\pi$
$11$	0.277754	0.0837461	0.0418730	+	0.999123i	$0.486667\pi$
$12$	0	0
$13$	3.92285	1.08800	0.544002	−	0.839084i	$-0.316908\pi$
$13$	3.92285	1.08800	0.544002	+	0.839084i	$0.316908\pi$
$14$	0	0
$15$	−0.277754	−0.0717159
$16$	0	0
$17$	5.47836	1.32870	0.664349	−	0.747423i	$-0.268709\pi$
$17$	5.47836	1.32870	0.664349	+	0.747423i	$0.268709\pi$
$18$	0	0
$19$	3.92285	0.899964	0.449982	−	0.893038i	$-0.351430\pi$
$19$	3.92285	0.899964	0.449982	+	0.893038i	$0.351430\pi$
$20$	0	0
$21$	0.354902	0.0774459
$22$	0	0
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−1.64510	−0.316599
$28$	0	0
$29$	−0.799393	−0.148444	−0.0742218	−	0.997242i	$-0.523647\pi$
$29$	−0.799393	−0.148444	−0.0742218	+	0.997242i	$0.523647\pi$
$30$	0	0
$31$	5.27775	0.947913	0.473956	−	0.880548i	$-0.342825\pi$
$31$	5.27775	0.947913	0.473956	+	0.880548i	$0.342825\pi$
$32$	0	0
$33$	0.0771475	0.0134297
$34$	0	0
$35$	−1.27775	−0.215980
$36$	0	0
$37$	1.75612	0.288704	0.144352	−	0.989526i	$-0.453890\pi$
$37$	1.75612	0.288704	0.144352	+	0.989526i	$0.453890\pi$
$38$	0	0
$39$	1.08959	0.174474
$40$	0	0
$41$	10.0339	1.56703	0.783514	−	0.621375i	$-0.213425\pi$
$41$	10.0339	1.56703	0.783514	+	0.621375i	$0.213425\pi$
$42$	0	0
$43$	4.00000	0.609994	0.304997	−	0.952353i	$-0.401344\pi$
$43$	4.00000	0.609994	0.304997	+	0.952353i	$0.401344\pi$
$44$	0	0
$45$	2.92285	0.435713
$46$	0	0
$47$	−8.95672	−1.30647	−0.653236	−	0.757154i	$-0.726589\pi$
$47$	−8.95672	−1.30647	−0.653236	+	0.757154i	$0.726589\pi$
$48$	0	0
$49$	−5.36734	−0.766763
$50$	0	0
$51$	1.52164	0.213072
$52$	0	0
$53$	−10.6451	−1.46222	−0.731108	−	0.682261i	$-0.760997\pi$
$53$	−10.6451	−1.46222	−0.731108	+	0.682261i	$0.760997\pi$
$54$	0	0
$55$	−0.277754	−0.0374524
$56$	0	0
$57$	1.08959	0.144320
$58$	0	0
$59$	−8.08959	−1.05317	−0.526587	−	0.850121i	$-0.676529\pi$
$59$	−8.08959	−1.05317	−0.526587	+	0.850121i	$0.676529\pi$
$60$	0	0
$61$	−15.0124	−1.92215	−0.961073	−	0.276294i	$-0.910894\pi$
$61$	−15.0124	−1.92215	−0.961073	+	0.276294i	$0.910894\pi$
$62$	0	0
$63$	−3.73469	−0.470526
$64$	0	0
$65$	−3.92285	−0.486570
$66$	0	0
$67$	−12.6451	−1.54484	−0.772422	−	0.635109i	$-0.780955\pi$
$67$	−12.6451	−1.54484	−0.772422	+	0.635109i	$0.780955\pi$
$68$	0	0
$69$	0.277754	0.0334377
$70$	0	0
$71$	9.32407	1.10656	0.553282	−	0.832994i	$-0.313375\pi$
$71$	9.32407	1.10656	0.553282	+	0.832994i	$0.313375\pi$
$72$	0	0
$73$	4.55551	0.533182	0.266591	−	0.963810i	$-0.414103\pi$
$73$	4.55551	0.533182	0.266591	+	0.963810i	$0.414103\pi$
$74$	0	0
$75$	0.277754	0.0320723
$76$	0	0
$77$	0.354902	0.0404448
$78$	0	0
$79$	−8.00000	−0.900070	−0.450035	−	0.893011i	$-0.648589\pi$
$79$	−8.00000	−0.900070	−0.450035	+	0.893011i	$0.648589\pi$
$80$	0	0
$81$	8.31162	0.923514
$82$	0	0
$83$	2.64510	0.290337	0.145169	−	0.989407i	$-0.453628\pi$
$83$	2.64510	0.290337	0.145169	+	0.989407i	$0.453628\pi$
$84$	0	0
$85$	−5.47836	−0.594212
$86$	0	0
$87$	−0.222035	−0.0238046
$88$	0	0
$89$	13.2902	1.40876	0.704379	−	0.709824i	$-0.251226\pi$
$89$	13.2902	1.40876	0.704379	+	0.709824i	$0.251226\pi$
$90$	0	0
$91$	5.01244	0.525447
$92$	0	0
$93$	1.46592	0.152009
$94$	0	0
$95$	−3.92285	−0.402476
$96$	0	0
$97$	3.16674	0.321533	0.160767	−	0.986992i	$-0.448603\pi$
$97$	3.16674	0.321533	0.160767	+	0.986992i	$0.448603\pi$
$98$	0	0
$99$	−0.811835	−0.0815925
$100$	0	0
$101$	−5.60182	−0.557402	−0.278701	−	0.960378i	$-0.589904\pi$
$101$	−5.60182	−0.557402	−0.278701	+	0.960378i	$0.589904\pi$
$102$	0	0
$103$	2.83326	0.279170	0.139585	−	0.990210i	$-0.455423\pi$
$103$	2.83326	0.279170	0.139585	+	0.990210i	$0.455423\pi$
$104$	0	0
$105$	−0.354902	−0.0346349
$106$	0	0
$107$	19.2006	1.85619	0.928096	−	0.372340i	$-0.121444\pi$
$107$	19.2006	1.85619	0.928096	+	0.372340i	$0.121444\pi$
$108$	0	0
$109$	4.07715	0.390520	0.195260	−	0.980752i	$-0.437445\pi$
$109$	4.07715	0.390520	0.195260	+	0.980752i	$0.437445\pi$
$110$	0	0
$111$	0.487769	0.0462970
$112$	0	0
$113$	11.7561	1.10592	0.552961	−	0.833207i	$-0.313498\pi$
$113$	11.7561	1.10592	0.552961	+	0.833207i	$0.313498\pi$
$114$	0	0
$115$	−1.00000	−0.0932505
$116$	0	0
$117$	−11.4659	−1.06002
$118$	0	0
$119$	7.00000	0.641689
$120$	0	0
$121$	−10.9229	−0.992987
$122$	0	0
$123$	2.78695	0.251291
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−17.8457	−1.58355	−0.791775	−	0.610813i	$-0.790843\pi$
$127$	−17.8457	−1.58355	−0.791775	+	0.610813i	$0.790843\pi$
$128$	0	0
$129$	1.11102	0.0978196
$130$	0	0
$131$	12.4012	1.08350	0.541750	−	0.840540i	$-0.317762\pi$
$131$	12.4012	1.08350	0.541750	+	0.840540i	$0.317762\pi$
$132$	0	0
$133$	5.01244	0.434634
$134$	0	0
$135$	1.64510	0.141588
$136$	0	0
$137$	15.0339	1.28443	0.642215	−	0.766524i	$-0.278016\pi$
$137$	15.0339	1.28443	0.642215	+	0.766524i	$0.278016\pi$
$138$	0	0
$139$	18.0030	1.52700	0.763499	−	0.645809i	$-0.223480\pi$
$139$	18.0030	1.52700	0.763499	+	0.645809i	$0.223480\pi$
$140$	0	0
$141$	−2.48777	−0.209508
$142$	0	0
$143$	1.08959	0.0911160
$144$	0	0
$145$	0.799393	0.0663860
$146$	0	0
$147$	−1.49080	−0.122959
$148$	0	0
$149$	14.3241	1.17347	0.586737	−	0.809778i	$-0.300412\pi$
$149$	14.3241	1.17347	0.586737	+	0.809778i	$0.300412\pi$
$150$	0	0
$151$	24.1912	1.96865	0.984326	−	0.176359i	$-0.0564319\pi$
$151$	24.1912	1.96865	0.984326	+	0.176359i	$0.0564319\pi$
$152$	0	0
$153$	−16.0124	−1.29453
$154$	0	0
$155$	−5.27775	−0.423919
$156$	0	0
$157$	−12.7994	−1.02150	−0.510751	−	0.859728i	$-0.670633\pi$
$157$	−12.7994	−1.02150	−0.510751	+	0.859728i	$0.670633\pi$
$158$	0	0
$159$	−2.95672	−0.234483
$160$	0	0
$161$	1.27775	0.100701
$162$	0	0
$163$	−19.4351	−1.52227	−0.761137	−	0.648592i	$-0.775358\pi$
$163$	−19.4351	−1.52227	−0.761137	+	0.648592i	$0.775358\pi$
$164$	0	0
$165$	−0.0771475	−0.00600592
$166$	0	0
$167$	24.8024	1.91927	0.959635	−	0.281249i	$-0.0907488\pi$
$167$	24.8024	1.91927	0.959635	+	0.281249i	$0.0907488\pi$
$168$	0	0
$169$	2.38877	0.183752
$170$	0	0
$171$	−11.4659	−0.876821
$172$	0	0
$173$	12.6790	0.963964	0.481982	−	0.876181i	$-0.339917\pi$
$173$	12.6790	0.963964	0.481982	+	0.876181i	$0.339917\pi$
$174$	0	0
$175$	1.27775	0.0965892
$176$	0	0
$177$	−2.24692	−0.168889
$178$	0	0
$179$	0.154295	0.0115325	0.00576627	−	0.999983i	$-0.498165\pi$
$179$	0.154295	0.0115325	0.00576627	+	0.999983i	$0.498165\pi$
$180$	0	0
$181$	−0.410621	−0.0305212	−0.0152606	−	0.999884i	$-0.504858\pi$
$181$	−0.410621	−0.0305212	−0.0152606	+	0.999884i	$0.504858\pi$
$182$	0	0
$183$	−4.16977	−0.308238
$184$	0	0
$185$	−1.75612	−0.129112
$186$	0	0
$187$	1.52164	0.111273
$188$	0	0
$189$	−2.10203	−0.152900
$190$	0	0
$191$	−7.44449	−0.538664	−0.269332	−	0.963047i	$-0.586803\pi$
$191$	−7.44449	−0.538664	−0.269332	+	0.963047i	$0.586803\pi$
$192$	0	0
$193$	−8.06774	−0.580729	−0.290364	−	0.956916i	$-0.593776\pi$
$193$	−8.06774	−0.580729	−0.290364	+	0.956916i	$0.593776\pi$
$194$	0	0
$195$	−1.08959	−0.0780271
$196$	0	0
$197$	−18.4569	−1.31500	−0.657501	−	0.753454i	$-0.728386\pi$
$197$	−18.4569	−1.31500	−0.657501	+	0.753454i	$0.728386\pi$
$198$	0	0
$199$	3.29020	0.233236	0.116618	−	0.993177i	$-0.462795\pi$
$199$	3.29020	0.233236	0.116618	+	0.993177i	$0.462795\pi$
$200$	0	0
$201$	−3.51223	−0.247734
$202$	0	0
$203$	−1.02143	−0.0716902
$204$	0	0
$205$	−10.0339	−0.700796
$206$	0	0
$207$	−2.92285	−0.203152
$208$	0	0
$209$	1.08959	0.0753685
$210$	0	0
$211$	26.3365	1.81308	0.906540	−	0.422120i	$-0.138714\pi$
$211$	26.3365	1.81308	0.906540	+	0.422120i	$0.138714\pi$
$212$	0	0
$213$	2.58980	0.177450
$214$	0	0
$215$	−4.00000	−0.272798
$216$	0	0
$217$	6.74367	0.457790
$218$	0	0
$219$	1.26531	0.0855019
$220$	0	0
$221$	21.4908	1.44563
$222$	0	0
$223$	25.9134	1.73529	0.867646	−	0.497182i	$-0.165632\pi$
$223$	25.9134	1.73529	0.867646	+	0.497182i	$0.165632\pi$
$224$	0	0
$225$	−2.92285	−0.194857
$226$	0	0
$227$	10.7347	0.712486	0.356243	−	0.934393i	$-0.384057\pi$
$227$	10.7347	0.712486	0.356243	+	0.934393i	$0.384057\pi$
$228$	0	0
$229$	−4.22203	−0.279000	−0.139500	−	0.990222i	$-0.544550\pi$
$229$	−4.22203	−0.279000	−0.139500	+	0.990222i	$0.544550\pi$
$230$	0	0
$231$	0.0985755	0.00648579
$232$	0	0
$233$	−4.55551	−0.298441	−0.149221	−	0.988804i	$-0.547676\pi$
$233$	−4.55551	−0.298441	−0.149221	+	0.988804i	$0.547676\pi$
$234$	0	0
$235$	8.95672	0.584272
$236$	0	0
$237$	−2.22203	−0.144337
$238$	0	0
$239$	24.3365	1.57420	0.787099	−	0.616827i	$-0.211582\pi$
$239$	24.3365	1.57420	0.787099	+	0.616827i	$0.211582\pi$
$240$	0	0
$241$	15.6914	1.01077	0.505386	−	0.862893i	$-0.331350\pi$
$241$	15.6914	1.01077	0.505386	+	0.862893i	$0.331350\pi$
$242$	0	0
$243$	7.24388	0.464695
$244$	0	0
$245$	5.36734	0.342907
$246$	0	0
$247$	15.3888	0.979164
$248$	0	0
$249$	0.734688	0.0465589
$250$	0	0
$251$	30.6575	1.93509	0.967543	−	0.252705i	$-0.0813202\pi$
$251$	30.6575	1.93509	0.967543	+	0.252705i	$0.0813202\pi$
$252$	0	0
$253$	0.277754	0.0174623
$254$	0	0
$255$	−1.52164	−0.0952887
$256$	0	0
$257$	4.24692	0.264916	0.132458	−	0.991189i	$-0.457713\pi$
$257$	4.24692	0.264916	0.132458	+	0.991189i	$0.457713\pi$
$258$	0	0
$259$	2.24388	0.139428
$260$	0	0
$261$	2.33651	0.144626
$262$	0	0
$263$	20.2130	1.24639	0.623195	−	0.782066i	$-0.285834\pi$
$263$	20.2130	1.24639	0.623195	+	0.782066i	$0.285834\pi$
$264$	0	0
$265$	10.6451	0.653923
$266$	0	0
$267$	3.69141	0.225911
$268$	0	0
$269$	−6.08959	−0.371289	−0.185644	−	0.982617i	$-0.559437\pi$
$269$	−6.08959	−0.371289	−0.185644	+	0.982617i	$0.559437\pi$
$270$	0	0
$271$	8.78999	0.533954	0.266977	−	0.963703i	$-0.413975\pi$
$271$	8.78999	0.533954	0.266977	+	0.963703i	$0.413975\pi$
$272$	0	0
$273$	1.39223	0.0842614
$274$	0	0
$275$	0.277754	0.0167492
$276$	0	0
$277$	−22.8024	−1.37007	−0.685033	−	0.728512i	$-0.740212\pi$
$277$	−22.8024	−1.37007	−0.685033	+	0.728512i	$0.740212\pi$
$278$	0	0
$279$	−15.4261	−0.923536
$280$	0	0
$281$	−23.3579	−1.39342	−0.696709	−	0.717354i	$-0.745353\pi$
$281$	−23.3579	−1.39342	−0.696709	+	0.717354i	$0.745353\pi$
$282$	0	0
$283$	12.2439	0.727823	0.363912	−	0.931433i	$-0.381441\pi$
$283$	12.2439	0.727823	0.363912	+	0.931433i	$0.381441\pi$
$284$	0	0
$285$	−1.08959	−0.0645417
$286$	0	0
$287$	12.8208	0.756789
$288$	0	0
$289$	13.0124	0.765438
$290$	0	0
$291$	0.879575	0.0515616
$292$	0	0
$293$	−26.1573	−1.52813	−0.764064	−	0.645141i	$-0.776799\pi$
$293$	−26.1573	−1.52813	−0.764064	+	0.645141i	$0.776799\pi$
$294$	0	0
$295$	8.08959	0.470994
$296$	0	0
$297$	−0.456933	−0.0265140
$298$	0	0
$299$	3.92285	0.226864
$300$	0	0
$301$	5.11102	0.294594
$302$	0	0
$303$	−1.55593	−0.0893859
$304$	0	0
$305$	15.0124	0.859610
$306$	0	0
$307$	−5.38877	−0.307553	−0.153777	−	0.988106i	$-0.549144\pi$
$307$	−5.38877	−0.307553	−0.153777	+	0.988106i	$0.549144\pi$
$308$	0	0
$309$	0.786951	0.0447681
$310$	0	0
$311$	26.5804	1.50724	0.753618	−	0.657313i	$-0.228307\pi$
$311$	26.5804	1.50724	0.753618	+	0.657313i	$0.228307\pi$
$312$	0	0
$313$	25.1235	1.42006	0.710031	−	0.704170i	$-0.248681\pi$
$313$	25.1235	1.42006	0.710031	+	0.704170i	$0.248681\pi$
$314$	0	0
$315$	3.73469	0.210426
$316$	0	0
$317$	−19.5894	−1.10025	−0.550125	−	0.835083i	$-0.685420\pi$
$317$	−19.5894	−1.10025	−0.550125	+	0.835083i	$0.685420\pi$
$318$	0	0
$319$	−0.222035	−0.0124316
$320$	0	0
$321$	5.33305	0.297662
$322$	0	0
$323$	21.4908	1.19578
$324$	0	0
$325$	3.92285	0.217601
$326$	0	0
$327$	1.13245	0.0626244
$328$	0	0
$329$	−11.4445	−0.630955
$330$	0	0
$331$	−31.1141	−1.71018	−0.855091	−	0.518477i	$-0.826499\pi$
$331$	−31.1141	−1.71018	−0.855091	+	0.518477i	$0.826499\pi$
$332$	0	0
$333$	−5.13287	−0.281279
$334$	0	0
$335$	12.6451	0.690876
$336$	0	0
$337$	17.9229	0.976320	0.488160	−	0.872754i	$-0.337668\pi$
$337$	17.9229	0.976320	0.488160	+	0.872754i	$0.337668\pi$
$338$	0	0
$339$	3.26531	0.177347
$340$	0	0
$341$	1.46592	0.0793840
$342$	0	0
$343$	−15.8024	−0.853251
$344$	0	0
$345$	−0.277754	−0.0149538
$346$	0	0
$347$	7.16674	0.384731	0.192365	−	0.981323i	$-0.438384\pi$
$347$	7.16674	0.384731	0.192365	+	0.981323i	$0.438384\pi$
$348$	0	0
$349$	−7.93529	−0.424767	−0.212383	−	0.977186i	$-0.568123\pi$
$349$	−7.93529	−0.424767	−0.212383	+	0.977186i	$0.568123\pi$
$350$	0	0
$351$	−6.45348	−0.344461
$352$	0	0
$353$	0	0		−	1.00000i	$-0.5\pi$
$353$	0	0			1.00000i	$0.5\pi$
$354$	0	0
$355$	−9.32407	−0.494870
$356$	0	0
$357$	1.94428	0.102902
$358$	0	0
$359$	−27.6049	−1.45693	−0.728464	−	0.685084i	$-0.759766\pi$
$359$	−27.6049	−1.45693	−0.728464	+	0.685084i	$0.759766\pi$
$360$	0	0
$361$	−3.61123	−0.190065
$362$	0	0
$363$	−3.03387	−0.159237
$364$	0	0
$365$	−4.55551	−0.238446
$366$	0	0
$367$	−11.5341	−0.602074	−0.301037	−	0.953612i	$-0.597333\pi$
$367$	−11.5341	−0.602074	−0.301037	+	0.953612i	$0.597333\pi$
$368$	0	0
$369$	−29.3275	−1.52673
$370$	0	0
$371$	−13.6018	−0.706171
$372$	0	0
$373$	11.6237	0.601851	0.300925	−	0.953648i	$-0.402704\pi$
$373$	11.6237	0.601851	0.300925	+	0.953648i	$0.402704\pi$
$374$	0	0
$375$	−0.277754	−0.0143432
$376$	0	0
$377$	−3.13590	−0.161507
$378$	0	0
$379$	−12.2778	−0.630666	−0.315333	−	0.948981i	$-0.602116\pi$
$379$	−12.2778	−0.630666	−0.315333	+	0.948981i	$0.602116\pi$
$380$	0	0
$381$	−4.95672	−0.253941
$382$	0	0
$383$	−4.64510	−0.237353	−0.118677	−	0.992933i	$-0.537865\pi$
$383$	−4.64510	−0.237353	−0.118677	+	0.992933i	$0.537865\pi$
$384$	0	0
$385$	−0.354902	−0.0180875
$386$	0	0
$387$	−11.6914	−0.594308
$388$	0	0
$389$	9.90142	0.502022	0.251011	−	0.967984i	$-0.419237\pi$
$389$	9.90142	0.502022	0.251011	+	0.967984i	$0.419237\pi$
$390$	0	0
$391$	5.47836	0.277053
$392$	0	0
$393$	3.44449	0.173752
$394$	0	0
$395$	8.00000	0.402524
$396$	0	0
$397$	6.34549	0.318471	0.159236	−	0.987241i	$-0.449097\pi$
$397$	6.34549	0.318471	0.159236	+	0.987241i	$0.449097\pi$
$398$	0	0
$399$	1.39223	0.0696986
$400$	0	0
$401$	−5.84571	−0.291921	−0.145960	−	0.989290i	$-0.546627\pi$
$401$	−5.84571	−0.291921	−0.145960	+	0.989290i	$0.546627\pi$
$402$	0	0
$403$	20.7039	1.03133
$404$	0	0
$405$	−8.31162	−0.413008
$406$	0	0
$407$	0.487769	0.0241778
$408$	0	0
$409$	7.52467	0.372071	0.186036	−	0.982543i	$-0.440436\pi$
$409$	7.52467	0.372071	0.186036	+	0.982543i	$0.440436\pi$
$410$	0	0
$411$	4.17572	0.205973
$412$	0	0
$413$	−10.3365	−0.508626
$414$	0	0
$415$	−2.64510	−0.129843
$416$	0	0
$417$	5.00042	0.244872
$418$	0	0
$419$	−14.4012	−0.703545	−0.351773	−	0.936085i	$-0.614421\pi$
$419$	−14.4012	−0.703545	−0.351773	+	0.936085i	$0.614421\pi$
$420$	0	0
$421$	18.4784	0.900580	0.450290	−	0.892882i	$-0.351321\pi$
$421$	18.4784	0.900580	0.450290	+	0.892882i	$0.351321\pi$
$422$	0	0
$423$	26.1792	1.27288
$424$	0	0
$425$	5.47836	0.265740
$426$	0	0
$427$	−19.1822	−0.928292
$428$	0	0
$429$	0.302638	0.0146115
$430$	0	0
$431$	−13.1359	−0.632734	−0.316367	−	0.948637i	$-0.602463\pi$
$431$	−13.1359	−0.632734	−0.316367	+	0.948637i	$0.602463\pi$
$432$	0	0
$433$	−36.1479	−1.73716	−0.868579	−	0.495550i	$-0.834966\pi$
$433$	−36.1479	−1.73716	−0.868579	+	0.495550i	$0.834966\pi$
$434$	0	0
$435$	0.222035	0.0106458
$436$	0	0
$437$	3.92285	0.187655
$438$	0	0
$439$	−8.16977	−0.389922	−0.194961	−	0.980811i	$-0.562458\pi$
$439$	−8.16977	−0.389922	−0.194961	+	0.980811i	$0.562458\pi$
$440$	0	0
$441$	15.6880	0.747045
$442$	0	0
$443$	−1.36734	−0.0649645	−0.0324822	−	0.999472i	$-0.510341\pi$
$443$	−1.36734	−0.0649645	−0.0324822	+	0.999472i	$0.510341\pi$
$444$	0	0
$445$	−13.2902	−0.630016
$446$	0	0
$447$	3.97857	0.188180
$448$	0	0
$449$	−0.944281	−0.0445634	−0.0222817	−	0.999752i	$-0.507093\pi$
$449$	−0.944281	−0.0445634	−0.0222817	+	0.999752i	$0.507093\pi$
$450$	0	0
$451$	2.78695	0.131232
$452$	0	0
$453$	6.71921	0.315696
$454$	0	0
$455$	−5.01244	−0.234987
$456$	0	0
$457$	−7.75612	−0.362816	−0.181408	−	0.983408i	$-0.558065\pi$
$457$	−7.75612	−0.362816	−0.181408	+	0.983408i	$0.558065\pi$
$458$	0	0
$459$	−9.01244	−0.420665
$460$	0	0
$461$	−9.84571	−0.458560	−0.229280	−	0.973360i	$-0.573637\pi$
$461$	−9.84571	−0.458560	−0.229280	+	0.973360i	$0.573637\pi$
$462$	0	0
$463$	7.29020	0.338804	0.169402	−	0.985547i	$-0.445816\pi$
$463$	7.29020	0.338804	0.169402	+	0.985547i	$0.445816\pi$
$464$	0	0
$465$	−1.46592	−0.0679804
$466$	0	0
$467$	37.5153	1.73600	0.868000	−	0.496565i	$-0.165405\pi$
$467$	37.5153	1.73600	0.868000	+	0.496565i	$0.165405\pi$
$468$	0	0
$469$	−16.1573	−0.746076
$470$	0	0
$471$	−3.55509	−0.163810
$472$	0	0
$473$	1.11102	0.0510846
$474$	0	0
$475$	3.92285	0.179993
$476$	0	0
$477$	31.1141	1.42461
$478$	0	0
$479$	−26.1792	−1.19616	−0.598079	−	0.801437i	$-0.704069\pi$
$479$	−26.1792	−1.19616	−0.598079	+	0.801437i	$0.704069\pi$
$480$	0	0
$481$	6.88898	0.314111
$482$	0	0
$483$	0.354902	0.0161486
$484$	0	0
$485$	−3.16674	−0.143794
$486$	0	0
$487$	−10.9567	−0.496496	−0.248248	−	0.968696i	$-0.579855\pi$
$487$	−10.9567	−0.496496	−0.248248	+	0.968696i	$0.579855\pi$
$488$	0	0
$489$	−5.39818	−0.244114
$490$	0	0
$491$	19.6884	0.888524	0.444262	−	0.895897i	$-0.353466\pi$
$491$	19.6884	0.888524	0.444262	+	0.895897i	$0.353466\pi$
$492$	0	0
$493$	−4.37936	−0.197237
$494$	0	0
$495$	0.811835	0.0364893
$496$	0	0
$497$	11.9139	0.534410
$498$	0	0
$499$	−32.8243	−1.46942	−0.734708	−	0.678383i	$-0.762681\pi$
$499$	−32.8243	−1.46942	−0.734708	+	0.678383i	$0.762681\pi$
$500$	0	0
$501$	6.88898	0.307777
$502$	0	0
$503$	17.9010	0.798166	0.399083	−	0.916915i	$-0.369328\pi$
$503$	17.9010	0.798166	0.399083	+	0.916915i	$0.369328\pi$
$504$	0	0
$505$	5.60182	0.249278
$506$	0	0
$507$	0.663492	0.0294667
$508$	0	0
$509$	43.3579	1.92181	0.960903	−	0.276884i	$-0.0893018\pi$
$509$	43.3579	1.92181	0.960903	+	0.276884i	$0.0893018\pi$
$510$	0	0
$511$	5.82082	0.257498
$512$	0	0
$513$	−6.45348	−0.284928
$514$	0	0
$515$	−2.83326	−0.124848
$516$	0	0
$517$	−2.48777	−0.109412
$518$	0	0
$519$	3.52164	0.154583
$520$	0	0
$521$	−25.5122	−1.11771	−0.558856	−	0.829265i	$-0.688759\pi$
$521$	−25.5122	−1.11771	−0.558856	+	0.829265i	$0.688759\pi$
$522$	0	0
$523$	24.4012	1.06699	0.533495	−	0.845803i	$-0.320878\pi$
$523$	24.4012	1.06699	0.533495	+	0.845803i	$0.320878\pi$
$524$	0	0
$525$	0.354902	0.0154892
$526$	0	0
$527$	28.9134	1.25949
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	23.6447	1.02609
$532$	0	0
$533$	39.3614	1.70493
$534$	0	0
$535$	−19.2006	−0.830115
$536$	0	0
$537$	0.0428561	0.00184938
$538$	0	0
$539$	−1.49080	−0.0642134
$540$	0	0
$541$	16.3147	0.701422	0.350711	−	0.936484i	$-0.385940\pi$
$541$	16.3147	0.701422	0.350711	+	0.936484i	$0.385940\pi$
$542$	0	0
$543$	−0.114052	−0.00489443
$544$	0	0
$545$	−4.07715	−0.174646
$546$	0	0
$547$	−11.1016	−0.474671	−0.237335	−	0.971428i	$-0.576274\pi$
$547$	−11.1016	−0.474671	−0.237335	+	0.971428i	$0.576274\pi$
$548$	0	0
$549$	43.8792	1.87272
$550$	0	0
$551$	−3.13590	−0.133594
$552$	0	0
$553$	−10.2220	−0.434685
$554$	0	0
$555$	−0.487769	−0.0207046
$556$	0	0
$557$	−39.8487	−1.68845	−0.844223	−	0.535993i	$-0.819937\pi$
$557$	−39.8487	−1.68845	−0.844223	+	0.535993i	$0.819937\pi$
$558$	0	0
$559$	15.6914	0.663676
$560$	0	0
$561$	0.422642	0.0178440
$562$	0	0
$563$	−11.0463	−0.465547	−0.232773	−	0.972531i	$-0.574780\pi$
$563$	−11.0463	−0.465547	−0.232773	+	0.972531i	$0.574780\pi$
$564$	0	0
$565$	−11.7561	−0.494584
$566$	0	0
$567$	10.6202	0.446007
$568$	0	0
$569$	−38.4938	−1.61375	−0.806873	−	0.590725i	$-0.798842\pi$
$569$	−38.4938	−1.61375	−0.806873	+	0.590725i	$0.798842\pi$
$570$	0	0
$571$	20.2778	0.848598	0.424299	−	0.905522i	$-0.360521\pi$
$571$	20.2778	0.848598	0.424299	+	0.905522i	$0.360521\pi$
$572$	0	0
$573$	−2.06774	−0.0863811
$574$	0	0
$575$	1.00000	0.0417029
$576$	0	0
$577$	−14.8024	−0.616233	−0.308117	−	0.951349i	$-0.599699\pi$
$577$	−14.8024	−0.616233	−0.308117	+	0.951349i	$0.599699\pi$
$578$	0	0
$579$	−2.24085	−0.0931265
$580$	0	0
$581$	3.37979	0.140217
$582$	0	0
$583$	−2.95672	−0.122455
$584$	0	0
$585$	11.4659	0.474057
$586$	0	0
$587$	36.7716	1.51773	0.758863	−	0.651250i	$-0.225755\pi$
$587$	36.7716	1.51773	0.758863	+	0.651250i	$0.225755\pi$
$588$	0	0
$589$	20.7039	0.853087
$590$	0	0
$591$	−5.12649	−0.210876
$592$	0	0
$593$	−8.73469	−0.358691	−0.179345	−	0.983786i	$-0.557398\pi$
$593$	−8.73469	−0.358691	−0.179345	+	0.983786i	$0.557398\pi$
$594$	0	0
$595$	−7.00000	−0.286972
$596$	0	0
$597$	0.913866	0.0374021
$598$	0	0
$599$	−20.8581	−0.852241	−0.426120	−	0.904666i	$-0.640120\pi$
$599$	−20.8581	−0.852241	−0.426120	+	0.904666i	$0.640120\pi$
$600$	0	0
$601$	42.7283	1.74292	0.871462	−	0.490463i	$-0.163172\pi$
$601$	42.7283	1.74292	0.871462	+	0.490463i	$0.163172\pi$
$602$	0	0
$603$	36.9598	1.50512
$604$	0	0
$605$	10.9229	0.444077
$606$	0	0
$607$	1.29020	0.0523675	0.0261837	−	0.999657i	$-0.491665\pi$
$607$	1.29020	0.0523675	0.0261837	+	0.999657i	$0.491665\pi$
$608$	0	0
$609$	−0.283706	−0.0114964
$610$	0	0
$611$	−35.1359	−1.42145
$612$	0	0
$613$	−42.4938	−1.71631	−0.858155	−	0.513391i	$-0.828389\pi$
$613$	−42.4938	−1.71631	−0.858155	+	0.513391i	$0.828389\pi$
$614$	0	0
$615$	−2.78695	−0.112381
$616$	0	0
$617$	−6.34246	−0.255338	−0.127669	−	0.991817i	$-0.540749\pi$
$617$	−6.34246	−0.255338	−0.127669	+	0.991817i	$0.540749\pi$
$618$	0	0
$619$	−22.3241	−0.897280	−0.448640	−	0.893713i	$-0.648091\pi$
$619$	−22.3241	−0.897280	−0.448640	+	0.893713i	$0.648091\pi$
$620$	0	0
$621$	−1.64510	−0.0660155
$622$	0	0
$623$	16.9816	0.680354
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0.302638	0.0120862
$628$	0	0
$629$	9.62064	0.383600
$630$	0	0
$631$	−30.6481	−1.22008	−0.610041	−	0.792370i	$-0.708847\pi$
$631$	−30.6481	−1.22008	−0.610041	+	0.792370i	$0.708847\pi$
$632$	0	0
$633$	7.31508	0.290748
$634$	0	0
$635$	17.8457	0.708185
$636$	0	0
$637$	−21.0553	−0.834241
$638$	0	0
$639$	−27.2529	−1.07811
$640$	0	0
$641$	−24.3147	−0.960371	−0.480186	−	0.877167i	$-0.659431\pi$
$641$	−24.3147	−0.960371	−0.480186	+	0.877167i	$0.659431\pi$
$642$	0	0
$643$	23.6944	0.934418	0.467209	−	0.884147i	$-0.345260\pi$
$643$	23.6944	0.934418	0.467209	+	0.884147i	$0.345260\pi$
$644$	0	0
$645$	−1.11102	−0.0437463
$646$	0	0
$647$	32.8951	1.29324	0.646619	−	0.762813i	$-0.276182\pi$
$647$	32.8951	1.29324	0.646619	+	0.762813i	$0.276182\pi$
$648$	0	0
$649$	−2.24692	−0.0881993
$650$	0	0
$651$	1.87308	0.0734120
$652$	0	0
$653$	48.9045	1.91378	0.956890	−	0.290452i	$-0.0938055\pi$
$653$	48.9045	1.91378	0.956890	+	0.290452i	$0.0938055\pi$
$654$	0	0
$655$	−12.4012	−0.484556
$656$	0	0
$657$	−13.3151	−0.519471
$658$	0	0
$659$	−21.9134	−0.853627	−0.426813	−	0.904340i	$-0.640364\pi$
$659$	−21.9134	−0.853627	−0.426813	+	0.904340i	$0.640364\pi$
$660$	0	0
$661$	−6.34549	−0.246811	−0.123406	−	0.992356i	$-0.539382\pi$
$661$	−6.34549	−0.246811	−0.123406	+	0.992356i	$0.539382\pi$
$662$	0	0
$663$	5.96916	0.231823
$664$	0	0
$665$	−5.01244	−0.194374
$666$	0	0
$667$	−0.799393	−0.0309526
$668$	0	0
$669$	7.19757	0.278274
$670$	0	0
$671$	−4.16977	−0.160972
$672$	0	0
$673$	22.9816	0.885876	0.442938	−	0.896552i	$-0.353936\pi$
$673$	22.9816	0.885876	0.442938	+	0.896552i	$0.353936\pi$
$674$	0	0
$675$	−1.64510	−0.0633199
$676$	0	0
$677$	9.97815	0.383491	0.191746	−	0.981445i	$-0.438585\pi$
$677$	9.97815	0.383491	0.191746	+	0.981445i	$0.438585\pi$
$678$	0	0
$679$	4.04631	0.155283
$680$	0	0
$681$	2.98161	0.114255
$682$	0	0
$683$	−15.1265	−0.578799	−0.289400	−	0.957208i	$-0.593456\pi$
$683$	−15.1265	−0.578799	−0.289400	+	0.957208i	$0.593456\pi$
$684$	0	0
$685$	−15.0339	−0.574415
$686$	0	0
$687$	−1.17269	−0.0447409
$688$	0	0
$689$	−41.7592	−1.59090
$690$	0	0
$691$	−35.8457	−1.36363	−0.681817	−	0.731522i	$-0.738810\pi$
$691$	−35.8457	−1.36363	−0.681817	+	0.731522i	$0.738810\pi$
$692$	0	0
$693$	−1.03733	−0.0394047
$694$	0	0
$695$	−18.0030	−0.682894
$696$	0	0
$697$	54.9692	2.08211
$698$	0	0
$699$	−1.26531	−0.0478585
$700$	0	0
$701$	13.0339	0.492282	0.246141	−	0.969234i	$-0.420837\pi$
$701$	13.0339	0.492282	0.246141	+	0.969234i	$0.420837\pi$
$702$	0	0
$703$	6.88898	0.259823
$704$	0	0
$705$	2.48777	0.0936948
$706$	0	0
$707$	−7.15775	−0.269195
$708$	0	0
$709$	2.21001	0.0829988	0.0414994	−	0.999139i	$-0.486787\pi$
$709$	2.21001	0.0829988	0.0414994	+	0.999139i	$0.486787\pi$
$710$	0	0
$711$	23.3828	0.876924
$712$	0	0
$713$	5.27775	0.197653
$714$	0	0
$715$	−1.08959	−0.0407483
$716$	0	0
$717$	6.75957	0.252441
$718$	0	0
$719$	37.9259	1.41440	0.707198	−	0.707015i	$-0.249959\pi$
$719$	37.9259	1.41440	0.707198	+	0.707015i	$0.249959\pi$
$720$	0	0
$721$	3.62021	0.134824
$722$	0	0
$723$	4.35836	0.162089
$724$	0	0
$725$	−0.799393	−0.0296887
$726$	0	0
$727$	26.5465	0.984556	0.492278	−	0.870438i	$-0.336164\pi$
$727$	26.5465	0.984556	0.492278	+	0.870438i	$0.336164\pi$
$728$	0	0
$729$	−22.9229	−0.848995
$730$	0	0
$731$	21.9134	0.810498
$732$	0	0
$733$	8.37936	0.309499	0.154749	−	0.987954i	$-0.450543\pi$
$733$	8.37936	0.309499	0.154749	+	0.987954i	$0.450543\pi$
$734$	0	0
$735$	1.49080	0.0549891
$736$	0	0
$737$	−3.51223	−0.129375
$738$	0	0
$739$	23.4904	0.864108	0.432054	−	0.901848i	$-0.357789\pi$
$739$	23.4904	0.864108	0.432054	+	0.901848i	$0.357789\pi$
$740$	0	0
$741$	4.27430	0.157020
$742$	0	0
$743$	28.7681	1.05540	0.527700	−	0.849431i	$-0.323054\pi$
$743$	28.7681	1.05540	0.527700	+	0.849431i	$0.323054\pi$
$744$	0	0
$745$	−14.3241	−0.524793
$746$	0	0
$747$	−7.73123	−0.282871
$748$	0	0
$749$	24.5337	0.896440
$750$	0	0
$751$	−3.13590	−0.114431	−0.0572153	−	0.998362i	$-0.518222\pi$
$751$	−3.13590	−0.114431	−0.0572153	+	0.998362i	$0.518222\pi$
$752$	0	0
$753$	8.51527	0.310314
$754$	0	0
$755$	−24.1912	−0.880408
$756$	0	0
$757$	0.0218495	0.000794132 0	0.000397066	−	1.00000i	$-0.499874\pi$
$757$	0.0218495	0.000794132 0	0.000397066		1.00000i	$0.499874\pi$
$758$	0	0
$759$	0.0771475	0.00280028
$760$	0	0
$761$	−8.83368	−0.320221	−0.160110	−	0.987099i	$-0.551185\pi$
$761$	−8.83368	−0.320221	−0.160110	+	0.987099i	$0.551185\pi$
$762$	0	0
$763$	5.20959	0.188600
$764$	0	0
$765$	16.0124	0.578931
$766$	0	0
$767$	−31.7343	−1.14586
$768$	0	0
$769$	−53.2036	−1.91857	−0.959286	−	0.282436i	$-0.908858\pi$
$769$	−53.2036	−1.91857	−0.959286	+	0.282436i	$0.908858\pi$
$770$	0	0
$771$	1.17960	0.0424823
$772$	0	0
$773$	15.3768	0.553063	0.276532	−	0.961005i	$-0.410815\pi$
$773$	15.3768	0.553063	0.276532	+	0.961005i	$0.410815\pi$
$774$	0	0
$775$	5.27775	0.189583
$776$	0	0
$777$	0.623249	0.0223589
$778$	0	0
$779$	39.3614	1.41027
$780$	0	0
$781$	2.58980	0.0926703
$782$	0	0
$783$	1.31508	0.0469971
$784$	0	0
$785$	12.7994	0.456830
$786$	0	0
$787$	36.0957	1.28667	0.643336	−	0.765584i	$-0.277550\pi$
$787$	36.0957	1.28667	0.643336	+	0.765584i	$0.277550\pi$
$788$	0	0
$789$	5.61426	0.199873
$790$	0	0
$791$	15.0214	0.534100
$792$	0	0
$793$	−58.8916	−2.09130
$794$	0	0
$795$	2.95672	0.104864
$796$	0	0
$797$	6.13245	0.217222	0.108611	−	0.994084i	$-0.465360\pi$
$797$	6.13245	0.217222	0.108611	+	0.994084i	$0.465360\pi$
$798$	0	0
$799$	−49.0682	−1.73591
$800$	0	0
$801$	−38.8453	−1.37253
$802$	0	0
$803$	1.26531	0.0446519
$804$	0	0
$805$	−1.27775	−0.0450349
$806$	0	0
$807$	−1.69141	−0.0595405
$808$	0	0
$809$	−30.7008	−1.07938	−0.539692	−	0.841863i	$-0.681459\pi$
$809$	−30.7008	−1.07938	−0.539692	+	0.841863i	$0.681459\pi$
$810$	0	0
$811$	−0.490803	−0.0172344	−0.00861722	−	0.999963i	$-0.502743\pi$
$811$	−0.490803	−0.0172344	−0.00861722	+	0.999963i	$0.502743\pi$
$812$	0	0
$813$	2.44146	0.0856256
$814$	0	0
$815$	19.4351	0.680781
$816$	0	0
$817$	15.6914	0.548973
$818$	0	0
$819$	−14.6506	−0.511934
$820$	0	0
$821$	18.8024	0.656209	0.328105	−	0.944641i	$-0.393590\pi$
$821$	18.8024	0.656209	0.328105	+	0.944641i	$0.393590\pi$
$822$	0	0
$823$	36.2718	1.26436	0.632178	−	0.774823i	$-0.282161\pi$
$823$	36.2718	1.26436	0.632178	+	0.774823i	$0.282161\pi$
$824$	0	0
$825$	0.0771475	0.00268593
$826$	0	0
$827$	−51.6018	−1.79437	−0.897186	−	0.441654i	$-0.854392\pi$
$827$	−51.6018	−1.79437	−0.897186	+	0.441654i	$0.854392\pi$
$828$	0	0
$829$	−17.2255	−0.598266	−0.299133	−	0.954211i	$-0.596697\pi$
$829$	−17.2255	−0.598266	−0.299133	+	0.954211i	$0.596697\pi$
$830$	0	0
$831$	−6.33347	−0.219706
$832$	0	0
$833$	−29.4042	−1.01880
$834$	0	0
$835$	−24.8024	−0.858323
$836$	0	0
$837$	−8.68242	−0.300108
$838$	0	0
$839$	13.7592	0.475019	0.237509	−	0.971385i	$-0.423669\pi$
$839$	13.7592	0.475019	0.237509	+	0.971385i	$0.423669\pi$
$840$	0	0
$841$	−28.3610	−0.977965
$842$	0	0
$843$	−6.48777	−0.223451
$844$	0	0
$845$	−2.38877	−0.0821762
$846$	0	0
$847$	−13.9567	−0.479559
$848$	0	0
$849$	3.40079	0.116715
$850$	0	0
$851$	1.75612	0.0601989
$852$	0	0
$853$	22.3490	0.765213	0.382607	−	0.923911i	$-0.375026\pi$
$853$	22.3490	0.765213	0.382607	+	0.923911i	$0.375026\pi$
$854$	0	0
$855$	11.4659	0.392126
$856$	0	0
$857$	35.9134	1.22678	0.613390	−	0.789780i	$-0.289805\pi$
$857$	35.9134	1.22678	0.613390	+	0.789780i	$0.289805\pi$
$858$	0	0
$859$	5.64552	0.192623	0.0963113	−	0.995351i	$-0.469296\pi$
$859$	5.64552	0.192623	0.0963113	+	0.995351i	$0.469296\pi$
$860$	0	0
$861$	3.56104	0.121360
$862$	0	0
$863$	−36.5616	−1.24457	−0.622285	−	0.782791i	$-0.713796\pi$
$863$	−36.5616	−1.24457	−0.622285	+	0.782791i	$0.713796\pi$
$864$	0	0
$865$	−12.6790	−0.431098
$866$	0	0
$867$	3.61426	0.122747
$868$	0	0
$869$	−2.22203	−0.0753774
$870$	0	0
$871$	−49.6049	−1.68080
$872$	0	0
$873$	−9.25590	−0.313265
$874$	0	0
$875$	−1.27775	−0.0431960
$876$	0	0
$877$	−6.45693	−0.218035	−0.109018	−	0.994040i	$-0.534770\pi$
$877$	−6.45693	−0.218035	−0.109018	+	0.994040i	$0.534770\pi$
$878$	0	0
$879$	−7.26531	−0.245053
$880$	0	0
$881$	12.3147	0.414891	0.207446	−	0.978247i	$-0.433485\pi$
$881$	12.3147	0.414891	0.207446	+	0.978247i	$0.433485\pi$
$882$	0	0
$883$	7.43508	0.250210	0.125105	−	0.992143i	$-0.460073\pi$
$883$	7.43508	0.250210	0.125105	+	0.992143i	$0.460073\pi$
$884$	0	0
$885$	2.24692	0.0755293
$886$	0	0
$887$	23.6914	0.795480	0.397740	−	0.917498i	$-0.369795\pi$
$887$	23.6914	0.795480	0.397740	+	0.917498i	$0.369795\pi$
$888$	0	0
$889$	−22.8024	−0.764769
$890$	0	0
$891$	2.30859	0.0773407
$892$	0	0
$893$	−35.1359	−1.17578
$894$	0	0
$895$	−0.154295	−0.00515751
$896$	0	0
$897$	1.08959	0.0363803
$898$	0	0
$899$	−4.21900	−0.140712
$900$	0	0
$901$	−58.3177	−1.94284
$902$	0	0
$903$	1.41961	0.0472416
$904$	0	0
$905$	0.410621	0.0136495
$906$	0	0
$907$	10.5337	0.349764	0.174882	−	0.984589i	$-0.444046\pi$
$907$	10.5337	0.349764	0.174882	+	0.984589i	$0.444046\pi$
$908$	0	0
$909$	16.3733	0.543068
$910$	0	0
$911$	37.9383	1.25695	0.628476	−	0.777829i	$-0.283679\pi$
$911$	37.9383	1.25695	0.628476	+	0.777829i	$0.283679\pi$
$912$	0	0
$913$	0.734688	0.0243146
$914$	0	0
$915$	4.16977	0.137848
$916$	0	0
$917$	15.8457	0.523271
$918$	0	0
$919$	27.2036	0.897365	0.448683	−	0.893691i	$-0.351893\pi$
$919$	27.2036	0.897365	0.448683	+	0.893691i	$0.351893\pi$
$920$	0	0
$921$	−1.49675	−0.0493198
$922$	0	0
$923$	36.5769	1.20394
$924$	0	0
$925$	1.75612	0.0577407
$926$	0	0
$927$	−8.28121	−0.271991
$928$	0	0
$929$	1.84874	0.0606552	0.0303276	−	0.999540i	$-0.490345\pi$
$929$	1.84874	0.0606552	0.0303276	+	0.999540i	$0.490345\pi$
$930$	0	0
$931$	−21.0553	−0.690060
$932$	0	0
$933$	7.38282	0.241703
$934$	0	0
$935$	−1.52164	−0.0497629
$936$	0	0
$937$	51.8363	1.69342	0.846709	−	0.532056i	$-0.178581\pi$
$937$	51.8363	1.69342	0.846709	+	0.532056i	$0.178581\pi$
$938$	0	0
$939$	6.97815	0.227723
$940$	0	0
$941$	30.4569	0.992868	0.496434	−	0.868075i	$-0.334643\pi$
$941$	30.4569	0.992868	0.496434	+	0.868075i	$0.334643\pi$
$942$	0	0
$943$	10.0339	0.326748
$944$	0	0
$945$	2.10203	0.0683791
$946$	0	0
$947$	38.8118	1.26122	0.630608	−	0.776102i	$-0.282806\pi$
$947$	38.8118	1.26122	0.630608	+	0.776102i	$0.282806\pi$
$948$	0	0
$949$	17.8706	0.580104
$950$	0	0
$951$	−5.44104	−0.176438
$952$	0	0
$953$	2.49979	0.0809761	0.0404881	−	0.999180i	$-0.487109\pi$
$953$	2.49979	0.0809761	0.0404881	+	0.999180i	$0.487109\pi$
$954$	0	0
$955$	7.44449	0.240898
$956$	0	0
$957$	−0.0616712	−0.00199355
$958$	0	0
$959$	19.2096	0.620310
$960$	0	0
$961$	−3.14531	−0.101462
$962$	0	0
$963$	−56.1205	−1.80846
$964$	0	0
$965$	8.06774	0.259710
$966$	0	0
$967$	1.35794	0.0436683	0.0218341	−	0.999762i	$-0.493049\pi$
$967$	1.35794	0.0436683	0.0218341	+	0.999762i	$0.493049\pi$
$968$	0	0
$969$	5.96916	0.191757
$970$	0	0
$971$	−4.59241	−0.147378	−0.0736888	−	0.997281i	$-0.523477\pi$
$971$	−4.59241	−0.147378	−0.0736888	+	0.997281i	$0.523477\pi$
$972$	0	0
$973$	23.0035	0.737457
$974$	0	0
$975$	1.08959	0.0348948
$976$	0	0
$977$	−9.45693	−0.302554	−0.151277	−	0.988491i	$-0.548339\pi$
$977$	−9.45693	−0.302554	−0.151277	+	0.988491i	$0.548339\pi$
$978$	0	0
$979$	3.69141	0.117978
$980$	0	0
$981$	−11.9169	−0.380477
$982$	0	0
$983$	−24.6571	−0.786440	−0.393220	−	0.919444i	$-0.628639\pi$
$983$	−24.6571	−0.786440	−0.393220	+	0.919444i	$0.628639\pi$
$984$	0	0
$985$	18.4569	0.588087
$986$	0	0
$987$	−3.17876	−0.101181
$988$	0	0
$989$	4.00000	0.127193
$990$	0	0
$991$	−18.7437	−0.595412	−0.297706	−	0.954658i	$-0.596222\pi$
$991$	−18.7437	−0.595412	−0.297706	+	0.954658i	$0.596222\pi$
$992$	0	0
$993$	−8.64206	−0.274248
$994$	0	0
$995$	−3.29020	−0.104306
$996$	0	0
$997$	−17.0682	−0.540554	−0.270277	−	0.962783i	$-0.587115\pi$
$997$	−17.0682	−0.540554	−0.270277	+	0.962783i	$0.587115\pi$
$998$	0	0
$999$	−2.88898	−0.0914034

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7360.2.a.cd.1.2		3
4.3	odd	2		7360.2.a.bx.1.2		3
8.3	odd	2		3680.2.a.r.1.2	yes	3
8.5	even	2		3680.2.a.q.1.2	✓	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3680.2.a.q.1.2	✓	3	8.5	even	2
3680.2.a.r.1.2	yes	3	8.3	odd	2
7360.2.a.bx.1.2		3	4.3	odd	2
7360.2.a.cd.1.2		3	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:	\( N \)	\(=\)	\( 7360 = 2^{6} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7360.a (trivial)

\(f(q)\)	\(=\)	\(q+0.277754 q^{3} -1.00000 q^{5} +1.27775 q^{7} -2.92285 q^{9} +O(q^{10})\)	\(q+0.277754 q^{3} -1.00000 q^{5} +1.27775 q^{7} -2.92285 q^{9} +0.277754 q^{11} +3.92285 q^{13} -0.277754 q^{15} +5.47836 q^{17} +3.92285 q^{19} +0.354902 q^{21} +1.00000 q^{23} +1.00000 q^{25} -1.64510 q^{27} -0.799393 q^{29} +5.27775 q^{31} +0.0771475 q^{33} -1.27775 q^{35} +1.75612 q^{37} +1.08959 q^{39} +10.0339 q^{41} +4.00000 q^{43} +2.92285 q^{45} -8.95672 q^{47} -5.36734 q^{49} +1.52164 q^{51} -10.6451 q^{53} -0.277754 q^{55} +1.08959 q^{57} -8.08959 q^{59} -15.0124 q^{61} -3.73469 q^{63} -3.92285 q^{65} -12.6451 q^{67} +0.277754 q^{69} +9.32407 q^{71} +4.55551 q^{73} +0.277754 q^{75} +0.354902 q^{77} -8.00000 q^{79} +8.31162 q^{81} +2.64510 q^{83} -5.47836 q^{85} -0.222035 q^{87} +13.2902 q^{89} +5.01244 q^{91} +1.46592 q^{93} -3.92285 q^{95} +3.16674 q^{97} -0.811835 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + q^{3} - 3 q^{5} + 4 q^{7} + 6 q^{9}+O(q^{10}) \) 3 * q + q^3 - 3 * q^5 + 4 * q^7 + 6 * q^9	\( 3 q + q^{3} - 3 q^{5} + 4 q^{7} + 6 q^{9} + q^{11} - 3 q^{13} - q^{15} + 2 q^{17} - 3 q^{19} + 16 q^{21} + 3 q^{23} + 3 q^{25} + 10 q^{27} - 17 q^{29} + 16 q^{31} + 15 q^{33} - 4 q^{35} - 9 q^{37} - 12 q^{39} + 16 q^{41} + 12 q^{43} - 6 q^{45} + 2 q^{47} - q^{49} + 19 q^{51} - 17 q^{53} - q^{55} - 12 q^{57} - 9 q^{59} - 15 q^{61} + 19 q^{63} + 3 q^{65} - 23 q^{67} + q^{69} - 16 q^{71} + 14 q^{73} + q^{75} + 16 q^{77} - 24 q^{79} + 11 q^{81} - 7 q^{83} - 2 q^{85} - 2 q^{87} + 10 q^{89} - 15 q^{91} + 20 q^{93} + 3 q^{95} + 9 q^{97} + 13 q^{99}+O(q^{100}) \) 3 * q + q^3 - 3 * q^5 + 4 * q^7 + 6 * q^9 + q^11 - 3 * q^13 - q^15 + 2 * q^17 - 3 * q^19 + 16 * q^21 + 3 * q^23 + 3 * q^25 + 10 * q^27 - 17 * q^29 + 16 * q^31 + 15 * q^33 - 4 * q^35 - 9 * q^37 - 12 * q^39 + 16 * q^41 + 12 * q^43 - 6 * q^45 + 2 * q^47 - q^49 + 19 * q^51 - 17 * q^53 - q^55 - 12 * q^57 - 9 * q^59 - 15 * q^61 + 19 * q^63 + 3 * q^65 - 23 * q^67 + q^69 - 16 * q^71 + 14 * q^73 + q^75 + 16 * q^77 - 24 * q^79 + 11 * q^81 - 7 * q^83 - 2 * q^85 - 2 * q^87 + 10 * q^89 - 15 * q^91 + 20 * q^93 + 3 * q^95 + 9 * q^97 + 13 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more