LMFDB - Embedded newform 6240.2.a.bv.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6240,2,Mod(1,6240)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6240, 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("6240.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6240 = 2^{5} \cdot 3 \cdot 5 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6240.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:	$49.8266508613$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.61803$ of defining polynomial
Character	$\chi$	$=$	6240.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} +1.00000 q^{5} +5.23607 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +1.00000 q^{5} +5.23607 q^{7} +1.00000 q^{9} -2.47214 q^{11} -1.00000 q^{13} +1.00000 q^{15} +0.763932 q^{17} +5.23607 q^{19} +5.23607 q^{21} +2.76393 q^{23} +1.00000 q^{25} +1.00000 q^{27} +4.76393 q^{29} -8.94427 q^{31} -2.47214 q^{33} +5.23607 q^{35} -8.47214 q^{37} -1.00000 q^{39} -3.52786 q^{41} +4.94427 q^{43} +1.00000 q^{45} +12.9443 q^{47} +20.4164 q^{49} +0.763932 q^{51} -8.47214 q^{53} -2.47214 q^{55} +5.23607 q^{57} +2.47214 q^{59} +10.9443 q^{61} +5.23607 q^{63} -1.00000 q^{65} -8.00000 q^{67} +2.76393 q^{69} -4.00000 q^{71} +9.70820 q^{73} +1.00000 q^{75} -12.9443 q^{77} +4.94427 q^{79} +1.00000 q^{81} +8.00000 q^{83} +0.763932 q^{85} +4.76393 q^{87} -14.9443 q^{89} -5.23607 q^{91} -8.94427 q^{93} +5.23607 q^{95} +14.6525 q^{97} -2.47214 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} + 2 q^{5} + 6 q^{7} + 2 q^{9}+O(q^{10}) $ 2 * q + 2 * q^3 + 2 * q^5 + 6 * q^7 + 2 * q^9	$ 2 q + 2 q^{3} + 2 q^{5} + 6 q^{7} + 2 q^{9} + 4 q^{11} - 2 q^{13} + 2 q^{15} + 6 q^{17} + 6 q^{19} + 6 q^{21} + 10 q^{23} + 2 q^{25} + 2 q^{27} + 14 q^{29} + 4 q^{33} + 6 q^{35} - 8 q^{37} - 2 q^{39} - 16 q^{41} - 8 q^{43} + 2 q^{45} + 8 q^{47} + 14 q^{49} + 6 q^{51} - 8 q^{53} + 4 q^{55} + 6 q^{57} - 4 q^{59} + 4 q^{61} + 6 q^{63} - 2 q^{65} - 16 q^{67} + 10 q^{69} - 8 q^{71} + 6 q^{73} + 2 q^{75} - 8 q^{77} - 8 q^{79} + 2 q^{81} + 16 q^{83} + 6 q^{85} + 14 q^{87} - 12 q^{89} - 6 q^{91} + 6 q^{95} - 2 q^{97} + 4 q^{99}+O(q^{100}) $ 2 * q + 2 * q^3 + 2 * q^5 + 6 * q^7 + 2 * q^9 + 4 * q^11 - 2 * q^13 + 2 * q^15 + 6 * q^17 + 6 * q^19 + 6 * q^21 + 10 * q^23 + 2 * q^25 + 2 * q^27 + 14 * q^29 + 4 * q^33 + 6 * q^35 - 8 * q^37 - 2 * q^39 - 16 * q^41 - 8 * q^43 + 2 * q^45 + 8 * q^47 + 14 * q^49 + 6 * q^51 - 8 * q^53 + 4 * q^55 + 6 * q^57 - 4 * q^59 + 4 * q^61 + 6 * q^63 - 2 * q^65 - 16 * q^67 + 10 * q^69 - 8 * q^71 + 6 * q^73 + 2 * q^75 - 8 * q^77 - 8 * q^79 + 2 * q^81 + 16 * q^83 + 6 * q^85 + 14 * q^87 - 12 * q^89 - 6 * q^91 + 6 * q^95 - 2 * q^97 + 4 * 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
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	5.23607	1.97905	0.989524	−	0.144370i	$-0.0461154\pi$
$7$	5.23607	1.97905	0.989524	+	0.144370i	$0.0461154\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−2.47214	−0.745377	−0.372689	−	0.927957i	$-0.621564\pi$
$11$	−2.47214	−0.745377	−0.372689	+	0.927957i	$0.621564\pi$
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	1.00000	0.258199
$16$	0	0
$17$	0.763932	0.185281	0.0926404	−	0.995700i	$-0.470469\pi$
$17$	0.763932	0.185281	0.0926404	+	0.995700i	$0.470469\pi$
$18$	0	0
$19$	5.23607	1.20124	0.600618	−	0.799536i	$-0.294921\pi$
$19$	5.23607	1.20124	0.600618	+	0.799536i	$0.294921\pi$
$20$	0	0
$21$	5.23607	1.14260
$22$	0	0
$23$	2.76393	0.576320	0.288160	−	0.957582i	$-0.406957\pi$
$23$	2.76393	0.576320	0.288160	+	0.957582i	$0.406957\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	4.76393	0.884640	0.442320	−	0.896857i	$-0.354156\pi$
$29$	4.76393	0.884640	0.442320	+	0.896857i	$0.354156\pi$
$30$	0	0
$31$	−8.94427	−1.60644	−0.803219	−	0.595683i	$-0.796881\pi$
$31$	−8.94427	−1.60644	−0.803219	+	0.595683i	$0.796881\pi$
$32$	0	0
$33$	−2.47214	−0.430344
$34$	0	0
$35$	5.23607	0.885057
$36$	0	0
$37$	−8.47214	−1.39281	−0.696405	−	0.717649i	$-0.745218\pi$
$37$	−8.47214	−1.39281	−0.696405	+	0.717649i	$0.745218\pi$
$38$	0	0
$39$	−1.00000	−0.160128
$40$	0	0
$41$	−3.52786	−0.550960	−0.275480	−	0.961307i	$-0.588837\pi$
$41$	−3.52786	−0.550960	−0.275480	+	0.961307i	$0.588837\pi$
$42$	0	0
$43$	4.94427	0.753994	0.376997	−	0.926214i	$-0.376957\pi$
$43$	4.94427	0.753994	0.376997	+	0.926214i	$0.376957\pi$
$44$	0	0
$45$	1.00000	0.149071
$46$	0	0
$47$	12.9443	1.88812	0.944058	−	0.329779i	$-0.106974\pi$
$47$	12.9443	1.88812	0.944058	+	0.329779i	$0.106974\pi$
$48$	0	0
$49$	20.4164	2.91663
$50$	0	0
$51$	0.763932	0.106972
$52$	0	0
$53$	−8.47214	−1.16374	−0.581869	−	0.813283i	$-0.697678\pi$
$53$	−8.47214	−1.16374	−0.581869	+	0.813283i	$0.697678\pi$
$54$	0	0
$55$	−2.47214	−0.333343
$56$	0	0
$57$	5.23607	0.693534
$58$	0	0
$59$	2.47214	0.321845	0.160922	−	0.986967i	$-0.448553\pi$
$59$	2.47214	0.321845	0.160922	+	0.986967i	$0.448553\pi$
$60$	0	0
$61$	10.9443	1.40127	0.700635	−	0.713520i	$-0.252900\pi$
$61$	10.9443	1.40127	0.700635	+	0.713520i	$0.252900\pi$
$62$	0	0
$63$	5.23607	0.659683
$64$	0	0
$65$	−1.00000	−0.124035
$66$	0	0
$67$	−8.00000	−0.977356	−0.488678	−	0.872464i	$-0.662521\pi$
$67$	−8.00000	−0.977356	−0.488678	+	0.872464i	$0.662521\pi$
$68$	0	0
$69$	2.76393	0.332738
$70$	0	0
$71$	−4.00000	−0.474713	−0.237356	−	0.971423i	$-0.576281\pi$
$71$	−4.00000	−0.474713	−0.237356	+	0.971423i	$0.576281\pi$
$72$	0	0
$73$	9.70820	1.13626	0.568130	−	0.822939i	$-0.307667\pi$
$73$	9.70820	1.13626	0.568130	+	0.822939i	$0.307667\pi$
$74$	0	0
$75$	1.00000	0.115470
$76$	0	0
$77$	−12.9443	−1.47514
$78$	0	0
$79$	4.94427	0.556274	0.278137	−	0.960541i	$-0.410283\pi$
$79$	4.94427	0.556274	0.278137	+	0.960541i	$0.410283\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	8.00000	0.878114	0.439057	−	0.898459i	$-0.355313\pi$
$83$	8.00000	0.878114	0.439057	+	0.898459i	$0.355313\pi$
$84$	0	0
$85$	0.763932	0.0828601
$86$	0	0
$87$	4.76393	0.510747
$88$	0	0
$89$	−14.9443	−1.58409	−0.792045	−	0.610463i	$-0.790983\pi$
$89$	−14.9443	−1.58409	−0.792045	+	0.610463i	$0.790983\pi$
$90$	0	0
$91$	−5.23607	−0.548889
$92$	0	0
$93$	−8.94427	−0.927478
$94$	0	0
$95$	5.23607	0.537209
$96$	0	0
$97$	14.6525	1.48773	0.743867	−	0.668328i	$-0.232990\pi$
$97$	14.6525	1.48773	0.743867	+	0.668328i	$0.232990\pi$
$98$	0	0
$99$	−2.47214	−0.248459
$100$	0	0
$101$	2.29180	0.228042	0.114021	−	0.993478i	$-0.463627\pi$
$101$	2.29180	0.228042	0.114021	+	0.993478i	$0.463627\pi$
$102$	0	0
$103$	4.00000	0.394132	0.197066	−	0.980390i	$-0.436859\pi$
$103$	4.00000	0.394132	0.197066	+	0.980390i	$0.436859\pi$
$104$	0	0
$105$	5.23607	0.510988
$106$	0	0
$107$	−12.0000	−1.16008	−0.580042	−	0.814587i	$-0.696964\pi$
$107$	−12.0000	−1.16008	−0.580042	+	0.814587i	$0.696964\pi$
$108$	0	0
$109$	−12.1803	−1.16666	−0.583332	−	0.812233i	$-0.698252\pi$
$109$	−12.1803	−1.16666	−0.583332	+	0.812233i	$0.698252\pi$
$110$	0	0
$111$	−8.47214	−0.804140
$112$	0	0
$113$	11.2361	1.05700	0.528500	−	0.848933i	$-0.322755\pi$
$113$	11.2361	1.05700	0.528500	+	0.848933i	$0.322755\pi$
$114$	0	0
$115$	2.76393	0.257738
$116$	0	0
$117$	−1.00000	−0.0924500
$118$	0	0
$119$	4.00000	0.366679
$120$	0	0
$121$	−4.88854	−0.444413
$122$	0	0
$123$	−3.52786	−0.318097
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	4.00000	0.354943	0.177471	−	0.984126i	$-0.443208\pi$
$127$	4.00000	0.354943	0.177471	+	0.984126i	$0.443208\pi$
$128$	0	0
$129$	4.94427	0.435319
$130$	0	0
$131$	−20.6525	−1.80442	−0.902208	−	0.431302i	$-0.858054\pi$
$131$	−20.6525	−1.80442	−0.902208	+	0.431302i	$0.858054\pi$
$132$	0	0
$133$	27.4164	2.37730
$134$	0	0
$135$	1.00000	0.0860663
$136$	0	0
$137$	−0.472136	−0.0403373	−0.0201686	−	0.999797i	$-0.506420\pi$
$137$	−0.472136	−0.0403373	−0.0201686	+	0.999797i	$0.506420\pi$
$138$	0	0
$139$	6.47214	0.548959	0.274480	−	0.961593i	$-0.411494\pi$
$139$	6.47214	0.548959	0.274480	+	0.961593i	$0.411494\pi$
$140$	0	0
$141$	12.9443	1.09010
$142$	0	0
$143$	2.47214	0.206730
$144$	0	0
$145$	4.76393	0.395623
$146$	0	0
$147$	20.4164	1.68392
$148$	0	0
$149$	−4.47214	−0.366372	−0.183186	−	0.983078i	$-0.558641\pi$
$149$	−4.47214	−0.366372	−0.183186	+	0.983078i	$0.558641\pi$
$150$	0	0
$151$	16.9443	1.37891	0.689453	−	0.724331i	$-0.257851\pi$
$151$	16.9443	1.37891	0.689453	+	0.724331i	$0.257851\pi$
$152$	0	0
$153$	0.763932	0.0617602
$154$	0	0
$155$	−8.94427	−0.718421
$156$	0	0
$157$	13.4164	1.07075	0.535373	−	0.844616i	$-0.320171\pi$
$157$	13.4164	1.07075	0.535373	+	0.844616i	$0.320171\pi$
$158$	0	0
$159$	−8.47214	−0.671884
$160$	0	0
$161$	14.4721	1.14056
$162$	0	0
$163$	−10.4721	−0.820241	−0.410120	−	0.912031i	$-0.634513\pi$
$163$	−10.4721	−0.820241	−0.410120	+	0.912031i	$0.634513\pi$
$164$	0	0
$165$	−2.47214	−0.192456
$166$	0	0
$167$	−23.4164	−1.81202	−0.906008	−	0.423261i	$-0.860886\pi$
$167$	−23.4164	−1.81202	−0.906008	+	0.423261i	$0.860886\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	5.23607	0.400412
$172$	0	0
$173$	4.47214	0.340010	0.170005	−	0.985443i	$-0.445622\pi$
$173$	4.47214	0.340010	0.170005	+	0.985443i	$0.445622\pi$
$174$	0	0
$175$	5.23607	0.395810
$176$	0	0
$177$	2.47214	0.185817
$178$	0	0
$179$	−21.2361	−1.58726	−0.793629	−	0.608402i	$-0.791811\pi$
$179$	−21.2361	−1.58726	−0.793629	+	0.608402i	$0.791811\pi$
$180$	0	0
$181$	11.5279	0.856859	0.428430	−	0.903575i	$-0.359067\pi$
$181$	11.5279	0.856859	0.428430	+	0.903575i	$0.359067\pi$
$182$	0	0
$183$	10.9443	0.809024
$184$	0	0
$185$	−8.47214	−0.622884
$186$	0	0
$187$	−1.88854	−0.138104
$188$	0	0
$189$	5.23607	0.380868
$190$	0	0
$191$	−3.05573	−0.221105	−0.110552	−	0.993870i	$-0.535262\pi$
$191$	−3.05573	−0.221105	−0.110552	+	0.993870i	$0.535262\pi$
$192$	0	0
$193$	20.1803	1.45261	0.726306	−	0.687371i	$-0.241235\pi$
$193$	20.1803	1.45261	0.726306	+	0.687371i	$0.241235\pi$
$194$	0	0
$195$	−1.00000	−0.0716115
$196$	0	0
$197$	17.4164	1.24087	0.620434	−	0.784259i	$-0.286957\pi$
$197$	17.4164	1.24087	0.620434	+	0.784259i	$0.286957\pi$
$198$	0	0
$199$	25.8885	1.83519	0.917595	−	0.397516i	$-0.130128\pi$
$199$	25.8885	1.83519	0.917595	+	0.397516i	$0.130128\pi$
$200$	0	0
$201$	−8.00000	−0.564276
$202$	0	0
$203$	24.9443	1.75074
$204$	0	0
$205$	−3.52786	−0.246397
$206$	0	0
$207$	2.76393	0.192107
$208$	0	0
$209$	−12.9443	−0.895374
$210$	0	0
$211$	−16.9443	−1.16649	−0.583246	−	0.812296i	$-0.698218\pi$
$211$	−16.9443	−1.16649	−0.583246	+	0.812296i	$0.698218\pi$
$212$	0	0
$213$	−4.00000	−0.274075
$214$	0	0
$215$	4.94427	0.337197
$216$	0	0
$217$	−46.8328	−3.17922
$218$	0	0
$219$	9.70820	0.656020
$220$	0	0
$221$	−0.763932	−0.0513876
$222$	0	0
$223$	−10.1803	−0.681726	−0.340863	−	0.940113i	$-0.610719\pi$
$223$	−10.1803	−0.681726	−0.340863	+	0.940113i	$0.610719\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	13.8885	0.921815	0.460908	−	0.887448i	$-0.347524\pi$
$227$	13.8885	0.921815	0.460908	+	0.887448i	$0.347524\pi$
$228$	0	0
$229$	−7.23607	−0.478173	−0.239086	−	0.970998i	$-0.576848\pi$
$229$	−7.23607	−0.478173	−0.239086	+	0.970998i	$0.576848\pi$
$230$	0	0
$231$	−12.9443	−0.851671
$232$	0	0
$233$	−15.2361	−0.998148	−0.499074	−	0.866559i	$-0.666326\pi$
$233$	−15.2361	−0.998148	−0.499074	+	0.866559i	$0.666326\pi$
$234$	0	0
$235$	12.9443	0.844391
$236$	0	0
$237$	4.94427	0.321165
$238$	0	0
$239$	−28.9443	−1.87225	−0.936125	−	0.351668i	$-0.885614\pi$
$239$	−28.9443	−1.87225	−0.936125	+	0.351668i	$0.885614\pi$
$240$	0	0
$241$	10.0000	0.644157	0.322078	−	0.946713i	$-0.395619\pi$
$241$	10.0000	0.644157	0.322078	+	0.946713i	$0.395619\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	20.4164	1.30436
$246$	0	0
$247$	−5.23607	−0.333163
$248$	0	0
$249$	8.00000	0.506979
$250$	0	0
$251$	−0.291796	−0.0184180	−0.00920900	−	0.999958i	$-0.502931\pi$
$251$	−0.291796	−0.0184180	−0.00920900	+	0.999958i	$0.502931\pi$
$252$	0	0
$253$	−6.83282	−0.429575
$254$	0	0
$255$	0.763932	0.0478393
$256$	0	0
$257$	6.29180	0.392471	0.196236	−	0.980557i	$-0.437128\pi$
$257$	6.29180	0.392471	0.196236	+	0.980557i	$0.437128\pi$
$258$	0	0
$259$	−44.3607	−2.75644
$260$	0	0
$261$	4.76393	0.294880
$262$	0	0
$263$	−12.6525	−0.780185	−0.390093	−	0.920776i	$-0.627557\pi$
$263$	−12.6525	−0.780185	−0.390093	+	0.920776i	$0.627557\pi$
$264$	0	0
$265$	−8.47214	−0.520439
$266$	0	0
$267$	−14.9443	−0.914575
$268$	0	0
$269$	20.1803	1.23042	0.615209	−	0.788364i	$-0.289072\pi$
$269$	20.1803	1.23042	0.615209	+	0.788364i	$0.289072\pi$
$270$	0	0
$271$	−13.8885	−0.843669	−0.421834	−	0.906673i	$-0.638614\pi$
$271$	−13.8885	−0.843669	−0.421834	+	0.906673i	$0.638614\pi$
$272$	0	0
$273$	−5.23607	−0.316901
$274$	0	0
$275$	−2.47214	−0.149075
$276$	0	0
$277$	−12.4721	−0.749378	−0.374689	−	0.927151i	$-0.622251\pi$
$277$	−12.4721	−0.749378	−0.374689	+	0.927151i	$0.622251\pi$
$278$	0	0
$279$	−8.94427	−0.535480
$280$	0	0
$281$	−27.8885	−1.66369	−0.831846	−	0.555007i	$-0.812715\pi$
$281$	−27.8885	−1.66369	−0.831846	+	0.555007i	$0.812715\pi$
$282$	0	0
$283$	−16.0000	−0.951101	−0.475551	−	0.879688i	$-0.657751\pi$
$283$	−16.0000	−0.951101	−0.475551	+	0.879688i	$0.657751\pi$
$284$	0	0
$285$	5.23607	0.310158
$286$	0	0
$287$	−18.4721	−1.09038
$288$	0	0
$289$	−16.4164	−0.965671
$290$	0	0
$291$	14.6525	0.858943
$292$	0	0
$293$	−3.52786	−0.206100	−0.103050	−	0.994676i	$-0.532860\pi$
$293$	−3.52786	−0.206100	−0.103050	+	0.994676i	$0.532860\pi$
$294$	0	0
$295$	2.47214	0.143933
$296$	0	0
$297$	−2.47214	−0.143448
$298$	0	0
$299$	−2.76393	−0.159842
$300$	0	0
$301$	25.8885	1.49219
$302$	0	0
$303$	2.29180	0.131660
$304$	0	0
$305$	10.9443	0.626667
$306$	0	0
$307$	−20.9443	−1.19535	−0.597676	−	0.801737i	$-0.703909\pi$
$307$	−20.9443	−1.19535	−0.597676	+	0.801737i	$0.703909\pi$
$308$	0	0
$309$	4.00000	0.227552
$310$	0	0
$311$	3.05573	0.173274	0.0866372	−	0.996240i	$-0.472388\pi$
$311$	3.05573	0.173274	0.0866372	+	0.996240i	$0.472388\pi$
$312$	0	0
$313$	−5.41641	−0.306153	−0.153077	−	0.988214i	$-0.548918\pi$
$313$	−5.41641	−0.306153	−0.153077	+	0.988214i	$0.548918\pi$
$314$	0	0
$315$	5.23607	0.295019
$316$	0	0
$317$	−2.00000	−0.112331	−0.0561656	−	0.998421i	$-0.517887\pi$
$317$	−2.00000	−0.112331	−0.0561656	+	0.998421i	$0.517887\pi$
$318$	0	0
$319$	−11.7771	−0.659390
$320$	0	0
$321$	−12.0000	−0.669775
$322$	0	0
$323$	4.00000	0.222566
$324$	0	0
$325$	−1.00000	−0.0554700
$326$	0	0
$327$	−12.1803	−0.673574
$328$	0	0
$329$	67.7771	3.73667
$330$	0	0
$331$	0.291796	0.0160386	0.00801928	−	0.999968i	$-0.497447\pi$
$331$	0.291796	0.0160386	0.00801928	+	0.999968i	$0.497447\pi$
$332$	0	0
$333$	−8.47214	−0.464270
$334$	0	0
$335$	−8.00000	−0.437087
$336$	0	0
$337$	23.5279	1.28164	0.640822	−	0.767689i	$-0.278594\pi$
$337$	23.5279	1.28164	0.640822	+	0.767689i	$0.278594\pi$
$338$	0	0
$339$	11.2361	0.610259
$340$	0	0
$341$	22.1115	1.19740
$342$	0	0
$343$	70.2492	3.79310
$344$	0	0
$345$	2.76393	0.148805
$346$	0	0
$347$	−11.4164	−0.612865	−0.306432	−	0.951892i	$-0.599135\pi$
$347$	−11.4164	−0.612865	−0.306432	+	0.951892i	$0.599135\pi$
$348$	0	0
$349$	−10.2918	−0.550907	−0.275454	−	0.961314i	$-0.588828\pi$
$349$	−10.2918	−0.550907	−0.275454	+	0.961314i	$0.588828\pi$
$350$	0	0
$351$	−1.00000	−0.0533761
$352$	0	0
$353$	4.47214	0.238028	0.119014	−	0.992893i	$-0.462027\pi$
$353$	4.47214	0.238028	0.119014	+	0.992893i	$0.462027\pi$
$354$	0	0
$355$	−4.00000	−0.212298
$356$	0	0
$357$	4.00000	0.211702
$358$	0	0
$359$	7.05573	0.372387	0.186194	−	0.982513i	$-0.440385\pi$
$359$	7.05573	0.372387	0.186194	+	0.982513i	$0.440385\pi$
$360$	0	0
$361$	8.41641	0.442969
$362$	0	0
$363$	−4.88854	−0.256582
$364$	0	0
$365$	9.70820	0.508151
$366$	0	0
$367$	18.8328	0.983065	0.491532	−	0.870859i	$-0.336437\pi$
$367$	18.8328	0.983065	0.491532	+	0.870859i	$0.336437\pi$
$368$	0	0
$369$	−3.52786	−0.183653
$370$	0	0
$371$	−44.3607	−2.30309
$372$	0	0
$373$	21.4164	1.10890	0.554450	−	0.832217i	$-0.312929\pi$
$373$	21.4164	1.10890	0.554450	+	0.832217i	$0.312929\pi$
$374$	0	0
$375$	1.00000	0.0516398
$376$	0	0
$377$	−4.76393	−0.245355
$378$	0	0
$379$	−37.2361	−1.91269	−0.956344	−	0.292243i	$-0.905598\pi$
$379$	−37.2361	−1.91269	−0.956344	+	0.292243i	$0.905598\pi$
$380$	0	0
$381$	4.00000	0.204926
$382$	0	0
$383$	25.3050	1.29302	0.646511	−	0.762904i	$-0.276227\pi$
$383$	25.3050	1.29302	0.646511	+	0.762904i	$0.276227\pi$
$384$	0	0
$385$	−12.9443	−0.659701
$386$	0	0
$387$	4.94427	0.251331
$388$	0	0
$389$	7.23607	0.366883	0.183442	−	0.983031i	$-0.441276\pi$
$389$	7.23607	0.366883	0.183442	+	0.983031i	$0.441276\pi$
$390$	0	0
$391$	2.11146	0.106781
$392$	0	0
$393$	−20.6525	−1.04178
$394$	0	0
$395$	4.94427	0.248773
$396$	0	0
$397$	4.47214	0.224450	0.112225	−	0.993683i	$-0.464202\pi$
$397$	4.47214	0.224450	0.112225	+	0.993683i	$0.464202\pi$
$398$	0	0
$399$	27.4164	1.37254
$400$	0	0
$401$	−21.4164	−1.06948	−0.534742	−	0.845015i	$-0.679591\pi$
$401$	−21.4164	−1.06948	−0.534742	+	0.845015i	$0.679591\pi$
$402$	0	0
$403$	8.94427	0.445546
$404$	0	0
$405$	1.00000	0.0496904
$406$	0	0
$407$	20.9443	1.03817
$408$	0	0
$409$	7.52786	0.372229	0.186114	−	0.982528i	$-0.440410\pi$
$409$	7.52786	0.372229	0.186114	+	0.982528i	$0.440410\pi$
$410$	0	0
$411$	−0.472136	−0.0232887
$412$	0	0
$413$	12.9443	0.636946
$414$	0	0
$415$	8.00000	0.392705
$416$	0	0
$417$	6.47214	0.316942
$418$	0	0
$419$	15.1246	0.738886	0.369443	−	0.929253i	$-0.379549\pi$
$419$	15.1246	0.738886	0.369443	+	0.929253i	$0.379549\pi$
$420$	0	0
$421$	14.2918	0.696540	0.348270	−	0.937394i	$-0.386769\pi$
$421$	14.2918	0.696540	0.348270	+	0.937394i	$0.386769\pi$
$422$	0	0
$423$	12.9443	0.629372
$424$	0	0
$425$	0.763932	0.0370561
$426$	0	0
$427$	57.3050	2.77318
$428$	0	0
$429$	2.47214	0.119356
$430$	0	0
$431$	9.88854	0.476314	0.238157	−	0.971227i	$-0.423457\pi$
$431$	9.88854	0.476314	0.238157	+	0.971227i	$0.423457\pi$
$432$	0	0
$433$	−6.58359	−0.316387	−0.158194	−	0.987408i	$-0.550567\pi$
$433$	−6.58359	−0.316387	−0.158194	+	0.987408i	$0.550567\pi$
$434$	0	0
$435$	4.76393	0.228413
$436$	0	0
$437$	14.4721	0.692296
$438$	0	0
$439$	−21.8885	−1.04468	−0.522342	−	0.852736i	$-0.674941\pi$
$439$	−21.8885	−1.04468	−0.522342	+	0.852736i	$0.674941\pi$
$440$	0	0
$441$	20.4164	0.972210
$442$	0	0
$443$	−30.4721	−1.44777	−0.723887	−	0.689918i	$-0.757647\pi$
$443$	−30.4721	−1.44777	−0.723887	+	0.689918i	$0.757647\pi$
$444$	0	0
$445$	−14.9443	−0.708426
$446$	0	0
$447$	−4.47214	−0.211525
$448$	0	0
$449$	28.8328	1.36070	0.680352	−	0.732885i	$-0.261827\pi$
$449$	28.8328	1.36070	0.680352	+	0.732885i	$0.261827\pi$
$450$	0	0
$451$	8.72136	0.410673
$452$	0	0
$453$	16.9443	0.796111
$454$	0	0
$455$	−5.23607	−0.245471
$456$	0	0
$457$	33.7082	1.57680	0.788402	−	0.615161i	$-0.210909\pi$
$457$	33.7082	1.57680	0.788402	+	0.615161i	$0.210909\pi$
$458$	0	0
$459$	0.763932	0.0356573
$460$	0	0
$461$	30.0000	1.39724	0.698620	−	0.715493i	$-0.253798\pi$
$461$	30.0000	1.39724	0.698620	+	0.715493i	$0.253798\pi$
$462$	0	0
$463$	7.70820	0.358231	0.179115	−	0.983828i	$-0.442676\pi$
$463$	7.70820	0.358231	0.179115	+	0.983828i	$0.442676\pi$
$464$	0	0
$465$	−8.94427	−0.414781
$466$	0	0
$467$	8.94427	0.413892	0.206946	−	0.978352i	$-0.433648\pi$
$467$	8.94427	0.413892	0.206946	+	0.978352i	$0.433648\pi$
$468$	0	0
$469$	−41.8885	−1.93423
$470$	0	0
$471$	13.4164	0.618195
$472$	0	0
$473$	−12.2229	−0.562010
$474$	0	0
$475$	5.23607	0.240247
$476$	0	0
$477$	−8.47214	−0.387912
$478$	0	0
$479$	22.8328	1.04326	0.521629	−	0.853172i	$-0.325325\pi$
$479$	22.8328	1.04326	0.521629	+	0.853172i	$0.325325\pi$
$480$	0	0
$481$	8.47214	0.386296
$482$	0	0
$483$	14.4721	0.658505
$484$	0	0
$485$	14.6525	0.665335
$486$	0	0
$487$	26.1803	1.18634	0.593172	−	0.805076i	$-0.297875\pi$
$487$	26.1803	1.18634	0.593172	+	0.805076i	$0.297875\pi$
$488$	0	0
$489$	−10.4721	−0.473566
$490$	0	0
$491$	−31.7082	−1.43097	−0.715486	−	0.698627i	$-0.753795\pi$
$491$	−31.7082	−1.43097	−0.715486	+	0.698627i	$0.753795\pi$
$492$	0	0
$493$	3.63932	0.163907
$494$	0	0
$495$	−2.47214	−0.111114
$496$	0	0
$497$	−20.9443	−0.939479
$498$	0	0
$499$	10.7639	0.481860	0.240930	−	0.970543i	$-0.422548\pi$
$499$	10.7639	0.481860	0.240930	+	0.970543i	$0.422548\pi$
$500$	0	0
$501$	−23.4164	−1.04617
$502$	0	0
$503$	−33.5967	−1.49800	−0.749002	−	0.662567i	$-0.769467\pi$
$503$	−33.5967	−1.49800	−0.749002	+	0.662567i	$0.769467\pi$
$504$	0	0
$505$	2.29180	0.101984
$506$	0	0
$507$	1.00000	0.0444116
$508$	0	0
$509$	−3.88854	−0.172357	−0.0861783	−	0.996280i	$-0.527465\pi$
$509$	−3.88854	−0.172357	−0.0861783	+	0.996280i	$0.527465\pi$
$510$	0	0
$511$	50.8328	2.24871
$512$	0	0
$513$	5.23607	0.231178
$514$	0	0
$515$	4.00000	0.176261
$516$	0	0
$517$	−32.0000	−1.40736
$518$	0	0
$519$	4.47214	0.196305
$520$	0	0
$521$	−18.3607	−0.804396	−0.402198	−	0.915553i	$-0.631754\pi$
$521$	−18.3607	−0.804396	−0.402198	+	0.915553i	$0.631754\pi$
$522$	0	0
$523$	−42.8328	−1.87295	−0.936474	−	0.350737i	$-0.885931\pi$
$523$	−42.8328	−1.87295	−0.936474	+	0.350737i	$0.885931\pi$
$524$	0	0
$525$	5.23607	0.228521
$526$	0	0
$527$	−6.83282	−0.297642
$528$	0	0
$529$	−15.3607	−0.667856
$530$	0	0
$531$	2.47214	0.107282
$532$	0	0
$533$	3.52786	0.152809
$534$	0	0
$535$	−12.0000	−0.518805
$536$	0	0
$537$	−21.2361	−0.916404
$538$	0	0
$539$	−50.4721	−2.17399
$540$	0	0
$541$	29.1246	1.25216	0.626082	−	0.779757i	$-0.284657\pi$
$541$	29.1246	1.25216	0.626082	+	0.779757i	$0.284657\pi$
$542$	0	0
$543$	11.5279	0.494708
$544$	0	0
$545$	−12.1803	−0.521748
$546$	0	0
$547$	−5.88854	−0.251776	−0.125888	−	0.992044i	$-0.540178\pi$
$547$	−5.88854	−0.251776	−0.125888	+	0.992044i	$0.540178\pi$
$548$	0	0
$549$	10.9443	0.467090
$550$	0	0
$551$	24.9443	1.06266
$552$	0	0
$553$	25.8885	1.10089
$554$	0	0
$555$	−8.47214	−0.359622
$556$	0	0
$557$	−3.88854	−0.164763	−0.0823814	−	0.996601i	$-0.526253\pi$
$557$	−3.88854	−0.164763	−0.0823814	+	0.996601i	$0.526253\pi$
$558$	0	0
$559$	−4.94427	−0.209120
$560$	0	0
$561$	−1.88854	−0.0797344
$562$	0	0
$563$	37.3050	1.57222	0.786108	−	0.618089i	$-0.212093\pi$
$563$	37.3050	1.57222	0.786108	+	0.618089i	$0.212093\pi$
$564$	0	0
$565$	11.2361	0.472705
$566$	0	0
$567$	5.23607	0.219894
$568$	0	0
$569$	−26.3607	−1.10510	−0.552549	−	0.833481i	$-0.686345\pi$
$569$	−26.3607	−1.10510	−0.552549	+	0.833481i	$0.686345\pi$
$570$	0	0
$571$	37.3050	1.56116	0.780582	−	0.625054i	$-0.214923\pi$
$571$	37.3050	1.56116	0.780582	+	0.625054i	$0.214923\pi$
$572$	0	0
$573$	−3.05573	−0.127655
$574$	0	0
$575$	2.76393	0.115264
$576$	0	0
$577$	−7.59675	−0.316257	−0.158128	−	0.987419i	$-0.550546\pi$
$577$	−7.59675	−0.316257	−0.158128	+	0.987419i	$0.550546\pi$
$578$	0	0
$579$	20.1803	0.838666
$580$	0	0
$581$	41.8885	1.73783
$582$	0	0
$583$	20.9443	0.867423
$584$	0	0
$585$	−1.00000	−0.0413449
$586$	0	0
$587$	−16.9443	−0.699365	−0.349682	−	0.936868i	$-0.613711\pi$
$587$	−16.9443	−0.699365	−0.349682	+	0.936868i	$0.613711\pi$
$588$	0	0
$589$	−46.8328	−1.92971
$590$	0	0
$591$	17.4164	0.716415
$592$	0	0
$593$	−6.00000	−0.246390	−0.123195	−	0.992382i	$-0.539314\pi$
$593$	−6.00000	−0.246390	−0.123195	+	0.992382i	$0.539314\pi$
$594$	0	0
$595$	4.00000	0.163984
$596$	0	0
$597$	25.8885	1.05955
$598$	0	0
$599$	−16.0000	−0.653742	−0.326871	−	0.945069i	$-0.605994\pi$
$599$	−16.0000	−0.653742	−0.326871	+	0.945069i	$0.605994\pi$
$600$	0	0
$601$	42.3607	1.72793	0.863964	−	0.503553i	$-0.167974\pi$
$601$	42.3607	1.72793	0.863964	+	0.503553i	$0.167974\pi$
$602$	0	0
$603$	−8.00000	−0.325785
$604$	0	0
$605$	−4.88854	−0.198748
$606$	0	0
$607$	16.9443	0.687747	0.343873	−	0.939016i	$-0.388261\pi$
$607$	16.9443	0.687747	0.343873	+	0.939016i	$0.388261\pi$
$608$	0	0
$609$	24.9443	1.01079
$610$	0	0
$611$	−12.9443	−0.523669
$612$	0	0
$613$	−29.4164	−1.18812	−0.594059	−	0.804422i	$-0.702475\pi$
$613$	−29.4164	−1.18812	−0.594059	+	0.804422i	$0.702475\pi$
$614$	0	0
$615$	−3.52786	−0.142257
$616$	0	0
$617$	17.4164	0.701158	0.350579	−	0.936533i	$-0.385985\pi$
$617$	17.4164	0.701158	0.350579	+	0.936533i	$0.385985\pi$
$618$	0	0
$619$	−3.34752	−0.134548	−0.0672742	−	0.997735i	$-0.521430\pi$
$619$	−3.34752	−0.134548	−0.0672742	+	0.997735i	$0.521430\pi$
$620$	0	0
$621$	2.76393	0.110913
$622$	0	0
$623$	−78.2492	−3.13499
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−12.9443	−0.516944
$628$	0	0
$629$	−6.47214	−0.258061
$630$	0	0
$631$	−3.41641	−0.136005	−0.0680025	−	0.997685i	$-0.521663\pi$
$631$	−3.41641	−0.136005	−0.0680025	+	0.997685i	$0.521663\pi$
$632$	0	0
$633$	−16.9443	−0.673474
$634$	0	0
$635$	4.00000	0.158735
$636$	0	0
$637$	−20.4164	−0.808928
$638$	0	0
$639$	−4.00000	−0.158238
$640$	0	0
$641$	26.0000	1.02694	0.513469	−	0.858108i	$-0.328360\pi$
$641$	26.0000	1.02694	0.513469	+	0.858108i	$0.328360\pi$
$642$	0	0
$643$	−8.00000	−0.315489	−0.157745	−	0.987480i	$-0.550422\pi$
$643$	−8.00000	−0.315489	−0.157745	+	0.987480i	$0.550422\pi$
$644$	0	0
$645$	4.94427	0.194681
$646$	0	0
$647$	16.2918	0.640497	0.320248	−	0.947334i	$-0.396234\pi$
$647$	16.2918	0.640497	0.320248	+	0.947334i	$0.396234\pi$
$648$	0	0
$649$	−6.11146	−0.239896
$650$	0	0
$651$	−46.8328	−1.83552
$652$	0	0
$653$	26.0000	1.01746	0.508729	−	0.860927i	$-0.330115\pi$
$653$	26.0000	1.01746	0.508729	+	0.860927i	$0.330115\pi$
$654$	0	0
$655$	−20.6525	−0.806959
$656$	0	0
$657$	9.70820	0.378753
$658$	0	0
$659$	−39.7082	−1.54681	−0.773406	−	0.633911i	$-0.781449\pi$
$659$	−39.7082	−1.54681	−0.773406	+	0.633911i	$0.781449\pi$
$660$	0	0
$661$	13.1246	0.510488	0.255244	−	0.966877i	$-0.417844\pi$
$661$	13.1246	0.510488	0.255244	+	0.966877i	$0.417844\pi$
$662$	0	0
$663$	−0.763932	−0.0296687
$664$	0	0
$665$	27.4164	1.06316
$666$	0	0
$667$	13.1672	0.509835
$668$	0	0
$669$	−10.1803	−0.393595
$670$	0	0
$671$	−27.0557	−1.04447
$672$	0	0
$673$	22.9443	0.884437	0.442218	−	0.896907i	$-0.354192\pi$
$673$	22.9443	0.884437	0.442218	+	0.896907i	$0.354192\pi$
$674$	0	0
$675$	1.00000	0.0384900
$676$	0	0
$677$	−30.0000	−1.15299	−0.576497	−	0.817099i	$-0.695581\pi$
$677$	−30.0000	−1.15299	−0.576497	+	0.817099i	$0.695581\pi$
$678$	0	0
$679$	76.7214	2.94430
$680$	0	0
$681$	13.8885	0.532210
$682$	0	0
$683$	−14.1115	−0.539960	−0.269980	−	0.962866i	$-0.587017\pi$
$683$	−14.1115	−0.539960	−0.269980	+	0.962866i	$0.587017\pi$
$684$	0	0
$685$	−0.472136	−0.0180394
$686$	0	0
$687$	−7.23607	−0.276073
$688$	0	0
$689$	8.47214	0.322763
$690$	0	0
$691$	8.29180	0.315435	0.157717	−	0.987484i	$-0.449586\pi$
$691$	8.29180	0.315435	0.157717	+	0.987484i	$0.449586\pi$
$692$	0	0
$693$	−12.9443	−0.491712
$694$	0	0
$695$	6.47214	0.245502
$696$	0	0
$697$	−2.69505	−0.102082
$698$	0	0
$699$	−15.2361	−0.576281
$700$	0	0
$701$	23.2361	0.877614	0.438807	−	0.898581i	$-0.355401\pi$
$701$	23.2361	0.877614	0.438807	+	0.898581i	$0.355401\pi$
$702$	0	0
$703$	−44.3607	−1.67309
$704$	0	0
$705$	12.9443	0.487509
$706$	0	0
$707$	12.0000	0.451306
$708$	0	0
$709$	−13.3475	−0.501277	−0.250638	−	0.968081i	$-0.580641\pi$
$709$	−13.3475	−0.501277	−0.250638	+	0.968081i	$0.580641\pi$
$710$	0	0
$711$	4.94427	0.185425
$712$	0	0
$713$	−24.7214	−0.925822
$714$	0	0
$715$	2.47214	0.0924526
$716$	0	0
$717$	−28.9443	−1.08094
$718$	0	0
$719$	34.4721	1.28559	0.642797	−	0.766037i	$-0.277774\pi$
$719$	34.4721	1.28559	0.642797	+	0.766037i	$0.277774\pi$
$720$	0	0
$721$	20.9443	0.780005
$722$	0	0
$723$	10.0000	0.371904
$724$	0	0
$725$	4.76393	0.176928
$726$	0	0
$727$	−14.4721	−0.536742	−0.268371	−	0.963316i	$-0.586485\pi$
$727$	−14.4721	−0.536742	−0.268371	+	0.963316i	$0.586485\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	3.77709	0.139701
$732$	0	0
$733$	−16.4721	−0.608412	−0.304206	−	0.952606i	$-0.598391\pi$
$733$	−16.4721	−0.608412	−0.304206	+	0.952606i	$0.598391\pi$
$734$	0	0
$735$	20.4164	0.753071
$736$	0	0
$737$	19.7771	0.728498
$738$	0	0
$739$	−2.18034	−0.0802051	−0.0401025	−	0.999196i	$-0.512768\pi$
$739$	−2.18034	−0.0802051	−0.0401025	+	0.999196i	$0.512768\pi$
$740$	0	0
$741$	−5.23607	−0.192352
$742$	0	0
$743$	−47.4164	−1.73954	−0.869770	−	0.493458i	$-0.835733\pi$
$743$	−47.4164	−1.73954	−0.869770	+	0.493458i	$0.835733\pi$
$744$	0	0
$745$	−4.47214	−0.163846
$746$	0	0
$747$	8.00000	0.292705
$748$	0	0
$749$	−62.8328	−2.29586
$750$	0	0
$751$	20.0000	0.729810	0.364905	−	0.931045i	$-0.381101\pi$
$751$	20.0000	0.729810	0.364905	+	0.931045i	$0.381101\pi$
$752$	0	0
$753$	−0.291796	−0.0106336
$754$	0	0
$755$	16.9443	0.616665
$756$	0	0
$757$	−9.05573	−0.329136	−0.164568	−	0.986366i	$-0.552623\pi$
$757$	−9.05573	−0.329136	−0.164568	+	0.986366i	$0.552623\pi$
$758$	0	0
$759$	−6.83282	−0.248015
$760$	0	0
$761$	−29.4164	−1.06634	−0.533172	−	0.846007i	$-0.679000\pi$
$761$	−29.4164	−1.06634	−0.533172	+	0.846007i	$0.679000\pi$
$762$	0	0
$763$	−63.7771	−2.30889
$764$	0	0
$765$	0.763932	0.0276200
$766$	0	0
$767$	−2.47214	−0.0892637
$768$	0	0
$769$	−20.8328	−0.751251	−0.375625	−	0.926772i	$-0.622572\pi$
$769$	−20.8328	−0.751251	−0.375625	+	0.926772i	$0.622572\pi$
$770$	0	0
$771$	6.29180	0.226594
$772$	0	0
$773$	−1.63932	−0.0589623	−0.0294811	−	0.999565i	$-0.509385\pi$
$773$	−1.63932	−0.0589623	−0.0294811	+	0.999565i	$0.509385\pi$
$774$	0	0
$775$	−8.94427	−0.321288
$776$	0	0
$777$	−44.3607	−1.59143
$778$	0	0
$779$	−18.4721	−0.661833
$780$	0	0
$781$	9.88854	0.353840
$782$	0	0
$783$	4.76393	0.170249
$784$	0	0
$785$	13.4164	0.478852
$786$	0	0
$787$	−12.3607	−0.440611	−0.220305	−	0.975431i	$-0.570705\pi$
$787$	−12.3607	−0.440611	−0.220305	+	0.975431i	$0.570705\pi$
$788$	0	0
$789$	−12.6525	−0.450440
$790$	0	0
$791$	58.8328	2.09185
$792$	0	0
$793$	−10.9443	−0.388642
$794$	0	0
$795$	−8.47214	−0.300476
$796$	0	0
$797$	−45.4164	−1.60873	−0.804366	−	0.594134i	$-0.797495\pi$
$797$	−45.4164	−1.60873	−0.804366	+	0.594134i	$0.797495\pi$
$798$	0	0
$799$	9.88854	0.349832
$800$	0	0
$801$	−14.9443	−0.528030
$802$	0	0
$803$	−24.0000	−0.846942
$804$	0	0
$805$	14.4721	0.510076
$806$	0	0
$807$	20.1803	0.710382
$808$	0	0
$809$	42.7214	1.50200	0.751002	−	0.660300i	$-0.229571\pi$
$809$	42.7214	1.50200	0.751002	+	0.660300i	$0.229571\pi$
$810$	0	0
$811$	−7.70820	−0.270672	−0.135336	−	0.990800i	$-0.543211\pi$
$811$	−7.70820	−0.270672	−0.135336	+	0.990800i	$0.543211\pi$
$812$	0	0
$813$	−13.8885	−0.487092
$814$	0	0
$815$	−10.4721	−0.366823
$816$	0	0
$817$	25.8885	0.905725
$818$	0	0
$819$	−5.23607	−0.182963
$820$	0	0
$821$	−11.3050	−0.394546	−0.197273	−	0.980349i	$-0.563208\pi$
$821$	−11.3050	−0.394546	−0.197273	+	0.980349i	$0.563208\pi$
$822$	0	0
$823$	−23.7771	−0.828817	−0.414409	−	0.910091i	$-0.636012\pi$
$823$	−23.7771	−0.828817	−0.414409	+	0.910091i	$0.636012\pi$
$824$	0	0
$825$	−2.47214	−0.0860687
$826$	0	0
$827$	20.9443	0.728304	0.364152	−	0.931340i	$-0.381359\pi$
$827$	20.9443	0.728304	0.364152	+	0.931340i	$0.381359\pi$
$828$	0	0
$829$	−8.11146	−0.281723	−0.140861	−	0.990029i	$-0.544987\pi$
$829$	−8.11146	−0.281723	−0.140861	+	0.990029i	$0.544987\pi$
$830$	0	0
$831$	−12.4721	−0.432654
$832$	0	0
$833$	15.5967	0.540395
$834$	0	0
$835$	−23.4164	−0.810358
$836$	0	0
$837$	−8.94427	−0.309159
$838$	0	0
$839$	17.8885	0.617581	0.308791	−	0.951130i	$-0.400076\pi$
$839$	17.8885	0.617581	0.308791	+	0.951130i	$0.400076\pi$
$840$	0	0
$841$	−6.30495	−0.217412
$842$	0	0
$843$	−27.8885	−0.960532
$844$	0	0
$845$	1.00000	0.0344010
$846$	0	0
$847$	−25.5967	−0.879515
$848$	0	0
$849$	−16.0000	−0.549119
$850$	0	0
$851$	−23.4164	−0.802704
$852$	0	0
$853$	9.41641	0.322412	0.161206	−	0.986921i	$-0.448462\pi$
$853$	9.41641	0.322412	0.161206	+	0.986921i	$0.448462\pi$
$854$	0	0
$855$	5.23607	0.179070
$856$	0	0
$857$	21.1246	0.721603	0.360801	−	0.932643i	$-0.382503\pi$
$857$	21.1246	0.721603	0.360801	+	0.932643i	$0.382503\pi$
$858$	0	0
$859$	−0.944272	−0.0322181	−0.0161091	−	0.999870i	$-0.505128\pi$
$859$	−0.944272	−0.0322181	−0.0161091	+	0.999870i	$0.505128\pi$
$860$	0	0
$861$	−18.4721	−0.629529
$862$	0	0
$863$	0.583592	0.0198657	0.00993285	−	0.999951i	$-0.496838\pi$
$863$	0.583592	0.0198657	0.00993285	+	0.999951i	$0.496838\pi$
$864$	0	0
$865$	4.47214	0.152057
$866$	0	0
$867$	−16.4164	−0.557530
$868$	0	0
$869$	−12.2229	−0.414634
$870$	0	0
$871$	8.00000	0.271070
$872$	0	0
$873$	14.6525	0.495911
$874$	0	0
$875$	5.23607	0.177011
$876$	0	0
$877$	−38.0000	−1.28317	−0.641584	−	0.767052i	$-0.721723\pi$
$877$	−38.0000	−1.28317	−0.641584	+	0.767052i	$0.721723\pi$
$878$	0	0
$879$	−3.52786	−0.118992
$880$	0	0
$881$	33.4164	1.12583	0.562914	−	0.826516i	$-0.309680\pi$
$881$	33.4164	1.12583	0.562914	+	0.826516i	$0.309680\pi$
$882$	0	0
$883$	−13.8885	−0.467387	−0.233693	−	0.972310i	$-0.575081\pi$
$883$	−13.8885	−0.467387	−0.233693	+	0.972310i	$0.575081\pi$
$884$	0	0
$885$	2.47214	0.0830999
$886$	0	0
$887$	−44.0689	−1.47969	−0.739844	−	0.672778i	$-0.765101\pi$
$887$	−44.0689	−1.47969	−0.739844	+	0.672778i	$0.765101\pi$
$888$	0	0
$889$	20.9443	0.702448
$890$	0	0
$891$	−2.47214	−0.0828197
$892$	0	0
$893$	67.7771	2.26807
$894$	0	0
$895$	−21.2361	−0.709843
$896$	0	0
$897$	−2.76393	−0.0922850
$898$	0	0
$899$	−42.6099	−1.42112
$900$	0	0
$901$	−6.47214	−0.215618
$902$	0	0
$903$	25.8885	0.861517
$904$	0	0
$905$	11.5279	0.383199
$906$	0	0
$907$	24.0000	0.796907	0.398453	−	0.917189i	$-0.369547\pi$
$907$	24.0000	0.796907	0.398453	+	0.917189i	$0.369547\pi$
$908$	0	0
$909$	2.29180	0.0760141
$910$	0	0
$911$	45.5279	1.50841	0.754203	−	0.656642i	$-0.228024\pi$
$911$	45.5279	1.50841	0.754203	+	0.656642i	$0.228024\pi$
$912$	0	0
$913$	−19.7771	−0.654526
$914$	0	0
$915$	10.9443	0.361806
$916$	0	0
$917$	−108.138	−3.57102
$918$	0	0
$919$	−55.7771	−1.83992	−0.919958	−	0.392017i	$-0.871778\pi$
$919$	−55.7771	−1.83992	−0.919958	+	0.392017i	$0.871778\pi$
$920$	0	0
$921$	−20.9443	−0.690137
$922$	0	0
$923$	4.00000	0.131662
$924$	0	0
$925$	−8.47214	−0.278562
$926$	0	0
$927$	4.00000	0.131377
$928$	0	0
$929$	12.1115	0.397364	0.198682	−	0.980064i	$-0.436334\pi$
$929$	12.1115	0.397364	0.198682	+	0.980064i	$0.436334\pi$
$930$	0	0
$931$	106.902	3.50356
$932$	0	0
$933$	3.05573	0.100040
$934$	0	0
$935$	−1.88854	−0.0617620
$936$	0	0
$937$	−54.0000	−1.76410	−0.882052	−	0.471153i	$-0.843838\pi$
$937$	−54.0000	−1.76410	−0.882052	+	0.471153i	$0.843838\pi$
$938$	0	0
$939$	−5.41641	−0.176758
$940$	0	0
$941$	24.4721	0.797769	0.398884	−	0.917001i	$-0.369397\pi$
$941$	24.4721	0.797769	0.398884	+	0.917001i	$0.369397\pi$
$942$	0	0
$943$	−9.75078	−0.317529
$944$	0	0
$945$	5.23607	0.170329
$946$	0	0
$947$	−33.8885	−1.10123	−0.550615	−	0.834759i	$-0.685607\pi$
$947$	−33.8885	−1.10123	−0.550615	+	0.834759i	$0.685607\pi$
$948$	0	0
$949$	−9.70820	−0.315142
$950$	0	0
$951$	−2.00000	−0.0648544
$952$	0	0
$953$	19.8197	0.642022	0.321011	−	0.947076i	$-0.395977\pi$
$953$	19.8197	0.642022	0.321011	+	0.947076i	$0.395977\pi$
$954$	0	0
$955$	−3.05573	−0.0988810
$956$	0	0
$957$	−11.7771	−0.380699
$958$	0	0
$959$	−2.47214	−0.0798294
$960$	0	0
$961$	49.0000	1.58065
$962$	0	0
$963$	−12.0000	−0.386695
$964$	0	0
$965$	20.1803	0.649628
$966$	0	0
$967$	5.23607	0.168381	0.0841903	−	0.996450i	$-0.473170\pi$
$967$	5.23607	0.168381	0.0841903	+	0.996450i	$0.473170\pi$
$968$	0	0
$969$	4.00000	0.128499
$970$	0	0
$971$	6.54102	0.209911	0.104956	−	0.994477i	$-0.466530\pi$
$971$	6.54102	0.209911	0.104956	+	0.994477i	$0.466530\pi$
$972$	0	0
$973$	33.8885	1.08642
$974$	0	0
$975$	−1.00000	−0.0320256
$976$	0	0
$977$	−57.7771	−1.84845	−0.924226	−	0.381845i	$-0.875289\pi$
$977$	−57.7771	−1.84845	−0.924226	+	0.381845i	$0.875289\pi$
$978$	0	0
$979$	36.9443	1.18074
$980$	0	0
$981$	−12.1803	−0.388888
$982$	0	0
$983$	−7.41641	−0.236547	−0.118273	−	0.992981i	$-0.537736\pi$
$983$	−7.41641	−0.236547	−0.118273	+	0.992981i	$0.537736\pi$
$984$	0	0
$985$	17.4164	0.554933
$986$	0	0
$987$	67.7771	2.15737
$988$	0	0
$989$	13.6656	0.434542
$990$	0	0
$991$	−44.9443	−1.42770	−0.713851	−	0.700298i	$-0.753051\pi$
$991$	−44.9443	−1.42770	−0.713851	+	0.700298i	$0.753051\pi$
$992$	0	0
$993$	0.291796	0.00925987
$994$	0	0
$995$	25.8885	0.820722
$996$	0	0
$997$	−7.52786	−0.238410	−0.119205	−	0.992870i	$-0.538035\pi$
$997$	−7.52786	−0.238410	−0.119205	+	0.992870i	$0.538035\pi$
$998$	0	0
$999$	−8.47214	−0.268047

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6240.2.a.bv.1.2	yes	2
4.3	odd	2		6240.2.a.bl.1.1	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6240.2.a.bl.1.1	✓	2	4.3	odd	2
6240.2.a.bv.1.2	yes	2	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:	\( N \)	\(=\)	\( 6240 = 2^{5} \cdot 3 \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6240.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +1.00000 q^{5} +5.23607 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +1.00000 q^{5} +5.23607 q^{7} +1.00000 q^{9} -2.47214 q^{11} -1.00000 q^{13} +1.00000 q^{15} +0.763932 q^{17} +5.23607 q^{19} +5.23607 q^{21} +2.76393 q^{23} +1.00000 q^{25} +1.00000 q^{27} +4.76393 q^{29} -8.94427 q^{31} -2.47214 q^{33} +5.23607 q^{35} -8.47214 q^{37} -1.00000 q^{39} -3.52786 q^{41} +4.94427 q^{43} +1.00000 q^{45} +12.9443 q^{47} +20.4164 q^{49} +0.763932 q^{51} -8.47214 q^{53} -2.47214 q^{55} +5.23607 q^{57} +2.47214 q^{59} +10.9443 q^{61} +5.23607 q^{63} -1.00000 q^{65} -8.00000 q^{67} +2.76393 q^{69} -4.00000 q^{71} +9.70820 q^{73} +1.00000 q^{75} -12.9443 q^{77} +4.94427 q^{79} +1.00000 q^{81} +8.00000 q^{83} +0.763932 q^{85} +4.76393 q^{87} -14.9443 q^{89} -5.23607 q^{91} -8.94427 q^{93} +5.23607 q^{95} +14.6525 q^{97} -2.47214 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} + 2 q^{5} + 6 q^{7} + 2 q^{9}+O(q^{10}) \) 2 * q + 2 * q^3 + 2 * q^5 + 6 * q^7 + 2 * q^9	\( 2 q + 2 q^{3} + 2 q^{5} + 6 q^{7} + 2 q^{9} + 4 q^{11} - 2 q^{13} + 2 q^{15} + 6 q^{17} + 6 q^{19} + 6 q^{21} + 10 q^{23} + 2 q^{25} + 2 q^{27} + 14 q^{29} + 4 q^{33} + 6 q^{35} - 8 q^{37} - 2 q^{39} - 16 q^{41} - 8 q^{43} + 2 q^{45} + 8 q^{47} + 14 q^{49} + 6 q^{51} - 8 q^{53} + 4 q^{55} + 6 q^{57} - 4 q^{59} + 4 q^{61} + 6 q^{63} - 2 q^{65} - 16 q^{67} + 10 q^{69} - 8 q^{71} + 6 q^{73} + 2 q^{75} - 8 q^{77} - 8 q^{79} + 2 q^{81} + 16 q^{83} + 6 q^{85} + 14 q^{87} - 12 q^{89} - 6 q^{91} + 6 q^{95} - 2 q^{97} + 4 q^{99}+O(q^{100}) \) 2 * q + 2 * q^3 + 2 * q^5 + 6 * q^7 + 2 * q^9 + 4 * q^11 - 2 * q^13 + 2 * q^15 + 6 * q^17 + 6 * q^19 + 6 * q^21 + 10 * q^23 + 2 * q^25 + 2 * q^27 + 14 * q^29 + 4 * q^33 + 6 * q^35 - 8 * q^37 - 2 * q^39 - 16 * q^41 - 8 * q^43 + 2 * q^45 + 8 * q^47 + 14 * q^49 + 6 * q^51 - 8 * q^53 + 4 * q^55 + 6 * q^57 - 4 * q^59 + 4 * q^61 + 6 * q^63 - 2 * q^65 - 16 * q^67 + 10 * q^69 - 8 * q^71 + 6 * q^73 + 2 * q^75 - 8 * q^77 - 8 * q^79 + 2 * q^81 + 16 * q^83 + 6 * q^85 + 14 * q^87 - 12 * q^89 - 6 * q^91 + 6 * q^95 - 2 * q^97 + 4 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more