LMFDB - Embedded newform 6240.2.a.cd.1.3

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:	$3$
Coefficient field:	3.3.568.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 6x - 2 $ x^3 - x^2 - 6*x - 2
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			1.3
Root			$-1.76156$ 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)

$f(q)$	$=$	$q+1.00000 q^{3} +1.00000 q^{5} +4.62620 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +1.00000 q^{5} +4.62620 q^{7} +1.00000 q^{9} -0.626198 q^{11} +1.00000 q^{13} +1.00000 q^{15} -0.896916 q^{17} +3.52311 q^{19} +4.62620 q^{21} -8.14931 q^{23} +1.00000 q^{25} +1.00000 q^{27} +3.79383 q^{29} +5.72928 q^{31} -0.626198 q^{33} +4.62620 q^{35} +0.896916 q^{37} +1.00000 q^{39} -0.896916 q^{41} +5.25240 q^{43} +1.00000 q^{45} -8.77551 q^{47} +14.4017 q^{49} -0.896916 q^{51} +3.10308 q^{53} -0.626198 q^{55} +3.52311 q^{57} +2.27072 q^{59} -7.40171 q^{61} +4.62620 q^{63} +1.00000 q^{65} -1.72928 q^{67} -8.14931 q^{69} +6.42003 q^{71} +12.5048 q^{73} +1.00000 q^{75} -2.89692 q^{77} +12.1493 q^{79} +1.00000 q^{81} -5.31695 q^{83} -0.896916 q^{85} +3.79383 q^{87} +7.40171 q^{89} +4.62620 q^{91} +5.72928 q^{93} +3.52311 q^{95} -11.4017 q^{97} -0.626198 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{3} + 3 q^{5} + 5 q^{7} + 3 q^{9}+O(q^{10}) $ 3 * q + 3 * q^3 + 3 * q^5 + 5 * q^7 + 3 * q^9	$ 3 q + 3 q^{3} + 3 q^{5} + 5 q^{7} + 3 q^{9} + 7 q^{11} + 3 q^{13} + 3 q^{15} + q^{17} - 2 q^{19} + 5 q^{21} - 3 q^{23} + 3 q^{25} + 3 q^{27} + 4 q^{29} + 12 q^{31} + 7 q^{33} + 5 q^{35} - q^{37} + 3 q^{39} + q^{41} - 2 q^{43} + 3 q^{45} + 4 q^{47} + 4 q^{49} + q^{51} + 13 q^{53} + 7 q^{55} - 2 q^{57} + 12 q^{59} + 17 q^{61} + 5 q^{63} + 3 q^{65} - 3 q^{69} + 3 q^{71} + 2 q^{73} + 3 q^{75} - 5 q^{77} + 15 q^{79} + 3 q^{81} + 4 q^{83} + q^{85} + 4 q^{87} - 17 q^{89} + 5 q^{91} + 12 q^{93} - 2 q^{95} + 5 q^{97} + 7 q^{99}+O(q^{100}) $ 3 * q + 3 * q^3 + 3 * q^5 + 5 * q^7 + 3 * q^9 + 7 * q^11 + 3 * q^13 + 3 * q^15 + q^17 - 2 * q^19 + 5 * q^21 - 3 * q^23 + 3 * q^25 + 3 * q^27 + 4 * q^29 + 12 * q^31 + 7 * q^33 + 5 * q^35 - q^37 + 3 * q^39 + q^41 - 2 * q^43 + 3 * q^45 + 4 * q^47 + 4 * q^49 + q^51 + 13 * q^53 + 7 * q^55 - 2 * q^57 + 12 * q^59 + 17 * q^61 + 5 * q^63 + 3 * q^65 - 3 * q^69 + 3 * q^71 + 2 * q^73 + 3 * q^75 - 5 * q^77 + 15 * q^79 + 3 * q^81 + 4 * q^83 + q^85 + 4 * q^87 - 17 * q^89 + 5 * q^91 + 12 * q^93 - 2 * q^95 + 5 * q^97 + 7 * 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
$7$	4.62620	1.74854	0.874269	−	0.485441i	$-0.161341\pi$
$7$	4.62620	1.74854	0.874269	+	0.485441i	$0.161341\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−0.626198	−0.188806	−0.0944029	−	0.995534i	$-0.530094\pi$
$11$	−0.626198	−0.188806	−0.0944029	+	0.995534i	$0.530094\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	1.00000	0.258199
$16$	0	0
$17$	−0.896916	−0.217534	−0.108767	−	0.994067i	$-0.534690\pi$
$17$	−0.896916	−0.217534	−0.108767	+	0.994067i	$0.534690\pi$
$18$	0	0
$19$	3.52311	0.808258	0.404129	−	0.914702i	$-0.367575\pi$
$19$	3.52311	0.808258	0.404129	+	0.914702i	$0.367575\pi$
$20$	0	0
$21$	4.62620	1.00952
$22$	0	0
$23$	−8.14931	−1.69925	−0.849625	−	0.527388i	$-0.823171\pi$
$23$	−8.14931	−1.69925	−0.849625	+	0.527388i	$0.823171\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	3.79383	0.704497	0.352249	−	0.935907i	$-0.385417\pi$
$29$	3.79383	0.704497	0.352249	+	0.935907i	$0.385417\pi$
$30$	0	0
$31$	5.72928	1.02901	0.514505	−	0.857488i	$-0.327976\pi$
$31$	5.72928	1.02901	0.514505	+	0.857488i	$0.327976\pi$
$32$	0	0
$33$	−0.626198	−0.109007
$34$	0	0
$35$	4.62620	0.781970
$36$	0	0
$37$	0.896916	0.147452	0.0737261	−	0.997279i	$-0.476511\pi$
$37$	0.896916	0.147452	0.0737261	+	0.997279i	$0.476511\pi$
$38$	0	0
$39$	1.00000	0.160128
$40$	0	0
$41$	−0.896916	−0.140075	−0.0700374	−	0.997544i	$-0.522312\pi$
$41$	−0.896916	−0.140075	−0.0700374	+	0.997544i	$0.522312\pi$
$42$	0	0
$43$	5.25240	0.800983	0.400491	−	0.916300i	$-0.368839\pi$
$43$	5.25240	0.800983	0.400491	+	0.916300i	$0.368839\pi$
$44$	0	0
$45$	1.00000	0.149071
$46$	0	0
$47$	−8.77551	−1.28004	−0.640020	−	0.768358i	$-0.721074\pi$
$47$	−8.77551	−1.28004	−0.640020	+	0.768358i	$0.721074\pi$
$48$	0	0
$49$	14.4017	2.05739
$50$	0	0
$51$	−0.896916	−0.125593
$52$	0	0
$53$	3.10308	0.426241	0.213121	−	0.977026i	$-0.431637\pi$
$53$	3.10308	0.426241	0.213121	+	0.977026i	$0.431637\pi$
$54$	0	0
$55$	−0.626198	−0.0844365
$56$	0	0
$57$	3.52311	0.466648
$58$	0	0
$59$	2.27072	0.295622	0.147811	−	0.989016i	$-0.452777\pi$
$59$	2.27072	0.295622	0.147811	+	0.989016i	$0.452777\pi$
$60$	0	0
$61$	−7.40171	−0.947692	−0.473846	−	0.880608i	$-0.657135\pi$
$61$	−7.40171	−0.947692	−0.473846	+	0.880608i	$0.657135\pi$
$62$	0	0
$63$	4.62620	0.582846
$64$	0	0
$65$	1.00000	0.124035
$66$	0	0
$67$	−1.72928	−0.211265	−0.105633	−	0.994405i	$-0.533687\pi$
$67$	−1.72928	−0.211265	−0.105633	+	0.994405i	$0.533687\pi$
$68$	0	0
$69$	−8.14931	−0.981062
$70$	0	0
$71$	6.42003	0.761917	0.380959	−	0.924592i	$-0.375594\pi$
$71$	6.42003	0.761917	0.380959	+	0.924592i	$0.375594\pi$
$72$	0	0
$73$	12.5048	1.46358	0.731788	−	0.681533i	$-0.238686\pi$
$73$	12.5048	1.46358	0.731788	+	0.681533i	$0.238686\pi$
$74$	0	0
$75$	1.00000	0.115470
$76$	0	0
$77$	−2.89692	−0.330134
$78$	0	0
$79$	12.1493	1.36690	0.683452	−	0.729995i	$-0.260478\pi$
$79$	12.1493	1.36690	0.683452	+	0.729995i	$0.260478\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−5.31695	−0.583611	−0.291805	−	0.956478i	$-0.594256\pi$
$83$	−5.31695	−0.583611	−0.291805	+	0.956478i	$0.594256\pi$
$84$	0	0
$85$	−0.896916	−0.0972842
$86$	0	0
$87$	3.79383	0.406742
$88$	0	0
$89$	7.40171	0.784580	0.392290	−	0.919842i	$-0.371683\pi$
$89$	7.40171	0.784580	0.392290	+	0.919842i	$0.371683\pi$
$90$	0	0
$91$	4.62620	0.484957
$92$	0	0
$93$	5.72928	0.594099
$94$	0	0
$95$	3.52311	0.361464
$96$	0	0
$97$	−11.4017	−1.15767	−0.578834	−	0.815445i	$-0.696492\pi$
$97$	−11.4017	−1.15767	−0.578834	+	0.815445i	$0.696492\pi$
$98$	0	0
$99$	−0.626198	−0.0629353
$100$	0	0
$101$	3.79383	0.377500	0.188750	−	0.982025i	$-0.439556\pi$
$101$	3.79383	0.377500	0.188750	+	0.982025i	$0.439556\pi$
$102$	0	0
$103$	11.0462	1.08842	0.544209	−	0.838950i	$-0.316830\pi$
$103$	11.0462	1.08842	0.544209	+	0.838950i	$0.316830\pi$
$104$	0	0
$105$	4.62620	0.451471
$106$	0	0
$107$	−11.6079	−1.12218	−0.561088	−	0.827756i	$-0.689617\pi$
$107$	−11.6079	−1.12218	−0.561088	+	0.827756i	$0.689617\pi$
$108$	0	0
$109$	−7.79383	−0.746514	−0.373257	−	0.927728i	$-0.621759\pi$
$109$	−7.79383	−0.746514	−0.373257	+	0.927728i	$0.621759\pi$
$110$	0	0
$111$	0.896916	0.0851315
$112$	0	0
$113$	12.5048	1.17635	0.588176	−	0.808733i	$-0.299846\pi$
$113$	12.5048	1.17635	0.588176	+	0.808733i	$0.299846\pi$
$114$	0	0
$115$	−8.14931	−0.759927
$116$	0	0
$117$	1.00000	0.0924500
$118$	0	0
$119$	−4.14931	−0.380367
$120$	0	0
$121$	−10.6079	−0.964352
$122$	0	0
$123$	−0.896916	−0.0808722
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	7.45856	0.661840	0.330920	−	0.943659i	$-0.392641\pi$
$127$	7.45856	0.661840	0.330920	+	0.943659i	$0.392641\pi$
$128$	0	0
$129$	5.25240	0.462448
$130$	0	0
$131$	15.0462	1.31460	0.657298	−	0.753631i	$-0.271699\pi$
$131$	15.0462	1.31460	0.657298	+	0.753631i	$0.271699\pi$
$132$	0	0
$133$	16.2986	1.41327
$134$	0	0
$135$	1.00000	0.0860663
$136$	0	0
$137$	−6.00000	−0.512615	−0.256307	−	0.966595i	$-0.582506\pi$
$137$	−6.00000	−0.512615	−0.256307	+	0.966595i	$0.582506\pi$
$138$	0	0
$139$	−12.1493	−1.03049	−0.515246	−	0.857043i	$-0.672299\pi$
$139$	−12.1493	−1.03049	−0.515246	+	0.857043i	$0.672299\pi$
$140$	0	0
$141$	−8.77551	−0.739031
$142$	0	0
$143$	−0.626198	−0.0523653
$144$	0	0
$145$	3.79383	0.315061
$146$	0	0
$147$	14.4017	1.18783
$148$	0	0
$149$	−17.6079	−1.44249	−0.721247	−	0.692678i	$-0.756431\pi$
$149$	−17.6079	−1.44249	−0.721247	+	0.692678i	$0.756431\pi$
$150$	0	0
$151$	10.2707	0.835819	0.417910	−	0.908489i	$-0.362763\pi$
$151$	10.2707	0.835819	0.417910	+	0.908489i	$0.362763\pi$
$152$	0	0
$153$	−0.896916	−0.0725114
$154$	0	0
$155$	5.72928	0.460187
$156$	0	0
$157$	−2.00000	−0.159617	−0.0798087	−	0.996810i	$-0.525431\pi$
$157$	−2.00000	−0.159617	−0.0798087	+	0.996810i	$0.525431\pi$
$158$	0	0
$159$	3.10308	0.246091
$160$	0	0
$161$	−37.7003	−2.97120
$162$	0	0
$163$	−5.87859	−0.460447	−0.230224	−	0.973138i	$-0.573946\pi$
$163$	−5.87859	−0.460447	−0.230224	+	0.973138i	$0.573946\pi$
$164$	0	0
$165$	−0.626198	−0.0487495
$166$	0	0
$167$	12.2341	0.946701	0.473351	−	0.880874i	$-0.343044\pi$
$167$	12.2341	0.946701	0.473351	+	0.880874i	$0.343044\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	3.52311	0.269419
$172$	0	0
$173$	−12.5048	−0.950722	−0.475361	−	0.879791i	$-0.657682\pi$
$173$	−12.5048	−0.950722	−0.475361	+	0.879791i	$0.657682\pi$
$174$	0	0
$175$	4.62620	0.349708
$176$	0	0
$177$	2.27072	0.170678
$178$	0	0
$179$	−5.79383	−0.433051	−0.216526	−	0.976277i	$-0.569473\pi$
$179$	−5.79383	−0.433051	−0.216526	+	0.976277i	$0.569473\pi$
$180$	0	0
$181$	9.19554	0.683499	0.341750	−	0.939791i	$-0.388981\pi$
$181$	9.19554	0.683499	0.341750	+	0.939791i	$0.388981\pi$
$182$	0	0
$183$	−7.40171	−0.547150
$184$	0	0
$185$	0.896916	0.0659426
$186$	0	0
$187$	0.561647	0.0410717
$188$	0	0
$189$	4.62620	0.336506
$190$	0	0
$191$	6.74760	0.488239	0.244120	−	0.969745i	$-0.421501\pi$
$191$	6.74760	0.488239	0.244120	+	0.969745i	$0.421501\pi$
$192$	0	0
$193$	−21.6079	−1.55537	−0.777684	−	0.628655i	$-0.783606\pi$
$193$	−21.6079	−1.55537	−0.777684	+	0.628655i	$0.783606\pi$
$194$	0	0
$195$	1.00000	0.0716115
$196$	0	0
$197$	−5.58767	−0.398105	−0.199052	−	0.979989i	$-0.563786\pi$
$197$	−5.58767	−0.398105	−0.199052	+	0.979989i	$0.563786\pi$
$198$	0	0
$199$	−14.0925	−0.998988	−0.499494	−	0.866317i	$-0.666481\pi$
$199$	−14.0925	−0.998988	−0.499494	+	0.866317i	$0.666481\pi$
$200$	0	0
$201$	−1.72928	−0.121974
$202$	0	0
$203$	17.5510	1.23184
$204$	0	0
$205$	−0.896916	−0.0626434
$206$	0	0
$207$	−8.14931	−0.566416
$208$	0	0
$209$	−2.20617	−0.152604
$210$	0	0
$211$	17.5510	1.20826	0.604131	−	0.796885i	$-0.293520\pi$
$211$	17.5510	1.20826	0.604131	+	0.796885i	$0.293520\pi$
$212$	0	0
$213$	6.42003	0.439893
$214$	0	0
$215$	5.25240	0.358210
$216$	0	0
$217$	26.5048	1.79926
$218$	0	0
$219$	12.5048	0.844996
$220$	0	0
$221$	−0.896916	−0.0603331
$222$	0	0
$223$	13.1878	0.883123	0.441562	−	0.897231i	$-0.354425\pi$
$223$	13.1878	0.883123	0.441562	+	0.897231i	$0.354425\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	−14.4402	−0.958432	−0.479216	−	0.877697i	$-0.659079\pi$
$227$	−14.4402	−0.958432	−0.479216	+	0.877697i	$0.659079\pi$
$228$	0	0
$229$	−18.2986	−1.20921	−0.604604	−	0.796527i	$-0.706668\pi$
$229$	−18.2986	−1.20921	−0.604604	+	0.796527i	$0.706668\pi$
$230$	0	0
$231$	−2.89692	−0.190603
$232$	0	0
$233$	7.40171	0.484902	0.242451	−	0.970164i	$-0.422049\pi$
$233$	7.40171	0.484902	0.242451	+	0.970164i	$0.422049\pi$
$234$	0	0
$235$	−8.77551	−0.572451
$236$	0	0
$237$	12.1493	0.789183
$238$	0	0
$239$	14.4200	0.932754	0.466377	−	0.884586i	$-0.345559\pi$
$239$	14.4200	0.932754	0.466377	+	0.884586i	$0.345559\pi$
$240$	0	0
$241$	−22.2986	−1.43638	−0.718190	−	0.695847i	$-0.755029\pi$
$241$	−22.2986	−1.43638	−0.718190	+	0.695847i	$0.755029\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	14.4017	0.920091
$246$	0	0
$247$	3.52311	0.224170
$248$	0	0
$249$	−5.31695	−0.336948
$250$	0	0
$251$	−24.5972	−1.55256	−0.776282	−	0.630385i	$-0.782897\pi$
$251$	−24.5972	−1.55256	−0.776282	+	0.630385i	$0.782897\pi$
$252$	0	0
$253$	5.10308	0.320828
$254$	0	0
$255$	−0.896916	−0.0561671
$256$	0	0
$257$	2.00000	0.124757	0.0623783	−	0.998053i	$-0.480131\pi$
$257$	2.00000	0.124757	0.0623783	+	0.998053i	$0.480131\pi$
$258$	0	0
$259$	4.14931	0.257826
$260$	0	0
$261$	3.79383	0.234832
$262$	0	0
$263$	−0.412335	−0.0254257	−0.0127128	−	0.999919i	$-0.504047\pi$
$263$	−0.412335	−0.0254257	−0.0127128	+	0.999919i	$0.504047\pi$
$264$	0	0
$265$	3.10308	0.190621
$266$	0	0
$267$	7.40171	0.452977
$268$	0	0
$269$	3.79383	0.231314	0.115657	−	0.993289i	$-0.463103\pi$
$269$	3.79383	0.231314	0.115657	+	0.993289i	$0.463103\pi$
$270$	0	0
$271$	1.97209	0.119796	0.0598981	−	0.998204i	$-0.480922\pi$
$271$	1.97209	0.119796	0.0598981	+	0.998204i	$0.480922\pi$
$272$	0	0
$273$	4.62620	0.279990
$274$	0	0
$275$	−0.626198	−0.0377612
$276$	0	0
$277$	−14.7110	−0.883896	−0.441948	−	0.897041i	$-0.645712\pi$
$277$	−14.7110	−0.883896	−0.441948	+	0.897041i	$0.645712\pi$
$278$	0	0
$279$	5.72928	0.343003
$280$	0	0
$281$	−12.9171	−0.770571	−0.385286	−	0.922797i	$-0.625897\pi$
$281$	−12.9171	−0.770571	−0.385286	+	0.922797i	$0.625897\pi$
$282$	0	0
$283$	−9.96336	−0.592260	−0.296130	−	0.955148i	$-0.595696\pi$
$283$	−9.96336	−0.592260	−0.296130	+	0.955148i	$0.595696\pi$
$284$	0	0
$285$	3.52311	0.208691
$286$	0	0
$287$	−4.14931	−0.244926
$288$	0	0
$289$	−16.1955	−0.952679
$290$	0	0
$291$	−11.4017	−0.668380
$292$	0	0
$293$	18.7110	1.09311	0.546553	−	0.837425i	$-0.315940\pi$
$293$	18.7110	1.09311	0.546553	+	0.837425i	$0.315940\pi$
$294$	0	0
$295$	2.27072	0.132206
$296$	0	0
$297$	−0.626198	−0.0363357
$298$	0	0
$299$	−8.14931	−0.471287
$300$	0	0
$301$	24.2986	1.40055
$302$	0	0
$303$	3.79383	0.217950
$304$	0	0
$305$	−7.40171	−0.423821
$306$	0	0
$307$	2.12141	0.121075	0.0605375	−	0.998166i	$-0.480719\pi$
$307$	2.12141	0.121075	0.0605375	+	0.998166i	$0.480719\pi$
$308$	0	0
$309$	11.0462	0.628398
$310$	0	0
$311$	13.7938	0.782176	0.391088	−	0.920353i	$-0.372099\pi$
$311$	13.7938	0.782176	0.391088	+	0.920353i	$0.372099\pi$
$312$	0	0
$313$	8.91713	0.504026	0.252013	−	0.967724i	$-0.418907\pi$
$313$	8.91713	0.504026	0.252013	+	0.967724i	$0.418907\pi$
$314$	0	0
$315$	4.62620	0.260657
$316$	0	0
$317$	12.0925	0.679180	0.339590	−	0.940574i	$-0.389712\pi$
$317$	12.0925	0.679180	0.339590	+	0.940574i	$0.389712\pi$
$318$	0	0
$319$	−2.37569	−0.133013
$320$	0	0
$321$	−11.6079	−0.647888
$322$	0	0
$323$	−3.15994	−0.175824
$324$	0	0
$325$	1.00000	0.0554700
$326$	0	0
$327$	−7.79383	−0.431000
$328$	0	0
$329$	−40.5972	−2.23820
$330$	0	0
$331$	−2.56934	−0.141224	−0.0706119	−	0.997504i	$-0.522495\pi$
$331$	−2.56934	−0.141224	−0.0706119	+	0.997504i	$0.522495\pi$
$332$	0	0
$333$	0.896916	0.0491507
$334$	0	0
$335$	−1.72928	−0.0944808
$336$	0	0
$337$	28.5048	1.55275	0.776377	−	0.630268i	$-0.217055\pi$
$337$	28.5048	1.55275	0.776377	+	0.630268i	$0.217055\pi$
$338$	0	0
$339$	12.5048	0.679167
$340$	0	0
$341$	−3.58767	−0.194283
$342$	0	0
$343$	34.2418	1.84888
$344$	0	0
$345$	−8.14931	−0.438744
$346$	0	0
$347$	27.9065	1.49810	0.749050	−	0.662514i	$-0.230510\pi$
$347$	27.9065	1.49810	0.749050	+	0.662514i	$0.230510\pi$
$348$	0	0
$349$	−16.0925	−0.861409	−0.430705	−	0.902493i	$-0.641735\pi$
$349$	−16.0925	−0.861409	−0.430705	+	0.902493i	$0.641735\pi$
$350$	0	0
$351$	1.00000	0.0533761
$352$	0	0
$353$	−26.7110	−1.42168	−0.710840	−	0.703353i	$-0.751685\pi$
$353$	−26.7110	−1.42168	−0.710840	+	0.703353i	$0.751685\pi$
$354$	0	0
$355$	6.42003	0.340740
$356$	0	0
$357$	−4.14931	−0.219605
$358$	0	0
$359$	27.6926	1.46156	0.730781	−	0.682612i	$-0.239156\pi$
$359$	27.6926	1.46156	0.730781	+	0.682612i	$0.239156\pi$
$360$	0	0
$361$	−6.58767	−0.346719
$362$	0	0
$363$	−10.6079	−0.556769
$364$	0	0
$365$	12.5048	0.654531
$366$	0	0
$367$	20.2986	1.05958	0.529790	−	0.848129i	$-0.322271\pi$
$367$	20.2986	1.05958	0.529790	+	0.848129i	$0.322271\pi$
$368$	0	0
$369$	−0.896916	−0.0466916
$370$	0	0
$371$	14.3555	0.745299
$372$	0	0
$373$	−26.5972	−1.37715	−0.688577	−	0.725164i	$-0.741764\pi$
$373$	−26.5972	−1.37715	−0.688577	+	0.725164i	$0.741764\pi$
$374$	0	0
$375$	1.00000	0.0516398
$376$	0	0
$377$	3.79383	0.195392
$378$	0	0
$379$	−6.68305	−0.343285	−0.171643	−	0.985159i	$-0.554907\pi$
$379$	−6.68305	−0.343285	−0.171643	+	0.985159i	$0.554907\pi$
$380$	0	0
$381$	7.45856	0.382114
$382$	0	0
$383$	−11.1108	−0.567734	−0.283867	−	0.958864i	$-0.591617\pi$
$383$	−11.1108	−0.567734	−0.283867	+	0.958864i	$0.591617\pi$
$384$	0	0
$385$	−2.89692	−0.147641
$386$	0	0
$387$	5.25240	0.266994
$388$	0	0
$389$	−15.7938	−0.800779	−0.400390	−	0.916345i	$-0.631125\pi$
$389$	−15.7938	−0.800779	−0.400390	+	0.916345i	$0.631125\pi$
$390$	0	0
$391$	7.30925	0.369645
$392$	0	0
$393$	15.0462	0.758982
$394$	0	0
$395$	12.1493	0.611298
$396$	0	0
$397$	29.9065	1.50096	0.750482	−	0.660891i	$-0.229821\pi$
$397$	29.9065	1.50096	0.750482	+	0.660891i	$0.229821\pi$
$398$	0	0
$399$	16.2986	0.815952
$400$	0	0
$401$	−30.0000	−1.49813	−0.749064	−	0.662497i	$-0.769497\pi$
$401$	−30.0000	−1.49813	−0.749064	+	0.662497i	$0.769497\pi$
$402$	0	0
$403$	5.72928	0.285396
$404$	0	0
$405$	1.00000	0.0496904
$406$	0	0
$407$	−0.561647	−0.0278398
$408$	0	0
$409$	−2.71096	−0.134048	−0.0670242	−	0.997751i	$-0.521350\pi$
$409$	−2.71096	−0.134048	−0.0670242	+	0.997751i	$0.521350\pi$
$410$	0	0
$411$	−6.00000	−0.295958
$412$	0	0
$413$	10.5048	0.516907
$414$	0	0
$415$	−5.31695	−0.260999
$416$	0	0
$417$	−12.1493	−0.594954
$418$	0	0
$419$	−10.3353	−0.504911	−0.252455	−	0.967609i	$-0.581238\pi$
$419$	−10.3353	−0.504911	−0.252455	+	0.967609i	$0.581238\pi$
$420$	0	0
$421$	16.8034	0.818948	0.409474	−	0.912322i	$-0.365712\pi$
$421$	16.8034	0.818948	0.409474	+	0.912322i	$0.365712\pi$
$422$	0	0
$423$	−8.77551	−0.426680
$424$	0	0
$425$	−0.896916	−0.0435068
$426$	0	0
$427$	−34.2418	−1.65708
$428$	0	0
$429$	−0.626198	−0.0302331
$430$	0	0
$431$	24.2341	1.16731	0.583657	−	0.812000i	$-0.301621\pi$
$431$	24.2341	1.16731	0.583657	+	0.812000i	$0.301621\pi$
$432$	0	0
$433$	−23.4219	−1.12559	−0.562793	−	0.826598i	$-0.690273\pi$
$433$	−23.4219	−1.12559	−0.562793	+	0.826598i	$0.690273\pi$
$434$	0	0
$435$	3.79383	0.181900
$436$	0	0
$437$	−28.7110	−1.37343
$438$	0	0
$439$	−0.690749	−0.0329676	−0.0164838	−	0.999864i	$-0.505247\pi$
$439$	−0.690749	−0.0329676	−0.0164838	+	0.999864i	$0.505247\pi$
$440$	0	0
$441$	14.4017	0.685796
$442$	0	0
$443$	28.9894	1.37733	0.688663	−	0.725081i	$-0.258198\pi$
$443$	28.9894	1.37733	0.688663	+	0.725081i	$0.258198\pi$
$444$	0	0
$445$	7.40171	0.350875
$446$	0	0
$447$	−17.6079	−0.832824
$448$	0	0
$449$	−13.3093	−0.628102	−0.314051	−	0.949406i	$-0.601686\pi$
$449$	−13.3093	−0.628102	−0.314051	+	0.949406i	$0.601686\pi$
$450$	0	0
$451$	0.561647	0.0264469
$452$	0	0
$453$	10.2707	0.482560
$454$	0	0
$455$	4.62620	0.216880
$456$	0	0
$457$	29.1955	1.36571	0.682855	−	0.730554i	$-0.260738\pi$
$457$	29.1955	1.36571	0.682855	+	0.730554i	$0.260738\pi$
$458$	0	0
$459$	−0.896916	−0.0418645
$460$	0	0
$461$	35.7003	1.66273	0.831365	−	0.555727i	$-0.187560\pi$
$461$	35.7003	1.66273	0.831365	+	0.555727i	$0.187560\pi$
$462$	0	0
$463$	28.6262	1.33037	0.665186	−	0.746678i	$-0.268352\pi$
$463$	28.6262	1.33037	0.665186	+	0.746678i	$0.268352\pi$
$464$	0	0
$465$	5.72928	0.265689
$466$	0	0
$467$	−27.7370	−1.28351	−0.641757	−	0.766908i	$-0.721794\pi$
$467$	−27.7370	−1.28351	−0.641757	+	0.766908i	$0.721794\pi$
$468$	0	0
$469$	−8.00000	−0.369406
$470$	0	0
$471$	−2.00000	−0.0921551
$472$	0	0
$473$	−3.28904	−0.151230
$474$	0	0
$475$	3.52311	0.161652
$476$	0	0
$477$	3.10308	0.142080
$478$	0	0
$479$	29.1676	1.33270	0.666352	−	0.745638i	$-0.267855\pi$
$479$	29.1676	1.33270	0.666352	+	0.745638i	$0.267855\pi$
$480$	0	0
$481$	0.896916	0.0408959
$482$	0	0
$483$	−37.7003	−1.71542
$484$	0	0
$485$	−11.4017	−0.517725
$486$	0	0
$487$	−31.1310	−1.41068	−0.705340	−	0.708869i	$-0.749206\pi$
$487$	−31.1310	−1.41068	−0.705340	+	0.708869i	$0.749206\pi$
$488$	0	0
$489$	−5.87859	−0.265839
$490$	0	0
$491$	13.7938	0.622507	0.311253	−	0.950327i	$-0.399251\pi$
$491$	13.7938	0.622507	0.311253	+	0.950327i	$0.399251\pi$
$492$	0	0
$493$	−3.40275	−0.153252
$494$	0	0
$495$	−0.626198	−0.0281455
$496$	0	0
$497$	29.7003	1.33224
$498$	0	0
$499$	−19.3940	−0.868195	−0.434098	−	0.900866i	$-0.642933\pi$
$499$	−19.3940	−0.868195	−0.434098	+	0.900866i	$0.642933\pi$
$500$	0	0
$501$	12.2341	0.546578
$502$	0	0
$503$	−21.6801	−0.966669	−0.483334	−	0.875436i	$-0.660574\pi$
$503$	−21.6801	−0.966669	−0.483334	+	0.875436i	$0.660574\pi$
$504$	0	0
$505$	3.79383	0.168823
$506$	0	0
$507$	1.00000	0.0444116
$508$	0	0
$509$	18.3188	0.811968	0.405984	−	0.913880i	$-0.366929\pi$
$509$	18.3188	0.811968	0.405984	+	0.913880i	$0.366929\pi$
$510$	0	0
$511$	57.8496	2.55912
$512$	0	0
$513$	3.52311	0.155549
$514$	0	0
$515$	11.0462	0.486755
$516$	0	0
$517$	5.49521	0.241679
$518$	0	0
$519$	−12.5048	−0.548899
$520$	0	0
$521$	39.6068	1.73521	0.867603	−	0.497257i	$-0.165659\pi$
$521$	39.6068	1.73521	0.867603	+	0.497257i	$0.165659\pi$
$522$	0	0
$523$	−1.79383	−0.0784388	−0.0392194	−	0.999231i	$-0.512487\pi$
$523$	−1.79383	−0.0784388	−0.0392194	+	0.999231i	$0.512487\pi$
$524$	0	0
$525$	4.62620	0.201904
$526$	0	0
$527$	−5.13869	−0.223845
$528$	0	0
$529$	43.4113	1.88745
$530$	0	0
$531$	2.27072	0.0985408
$532$	0	0
$533$	−0.896916	−0.0388498
$534$	0	0
$535$	−11.6079	−0.501852
$536$	0	0
$537$	−5.79383	−0.250022
$538$	0	0
$539$	−9.01832	−0.388447
$540$	0	0
$541$	−37.1020	−1.59514	−0.797571	−	0.603225i	$-0.793882\pi$
$541$	−37.1020	−1.59514	−0.797571	+	0.603225i	$0.793882\pi$
$542$	0	0
$543$	9.19554	0.394618
$544$	0	0
$545$	−7.79383	−0.333851
$546$	0	0
$547$	−6.50479	−0.278125	−0.139062	−	0.990284i	$-0.544409\pi$
$547$	−6.50479	−0.278125	−0.139062	+	0.990284i	$0.544409\pi$
$548$	0	0
$549$	−7.40171	−0.315897
$550$	0	0
$551$	13.3661	0.569415
$552$	0	0
$553$	56.2051	2.39009
$554$	0	0
$555$	0.896916	0.0380720
$556$	0	0
$557$	−35.7205	−1.51353	−0.756764	−	0.653688i	$-0.773221\pi$
$557$	−35.7205	−1.51353	−0.756764	+	0.653688i	$0.773221\pi$
$558$	0	0
$559$	5.25240	0.222153
$560$	0	0
$561$	0.561647	0.0237128
$562$	0	0
$563$	−24.1493	−1.01777	−0.508886	−	0.860834i	$-0.669943\pi$
$563$	−24.1493	−1.01777	−0.508886	+	0.860834i	$0.669943\pi$
$564$	0	0
$565$	12.5048	0.526081
$566$	0	0
$567$	4.62620	0.194282
$568$	0	0
$569$	15.0096	0.629235	0.314617	−	0.949219i	$-0.398124\pi$
$569$	15.0096	0.629235	0.314617	+	0.949219i	$0.398124\pi$
$570$	0	0
$571$	13.7003	0.573341	0.286671	−	0.958029i	$-0.407451\pi$
$571$	13.7003	0.573341	0.286671	+	0.958029i	$0.407451\pi$
$572$	0	0
$573$	6.74760	0.281885
$574$	0	0
$575$	−8.14931	−0.339850
$576$	0	0
$577$	−25.4942	−1.06134	−0.530668	−	0.847580i	$-0.678059\pi$
$577$	−25.4942	−1.06134	−0.530668	+	0.847580i	$0.678059\pi$
$578$	0	0
$579$	−21.6079	−0.897993
$580$	0	0
$581$	−24.5972	−1.02047
$582$	0	0
$583$	−1.94315	−0.0804768
$584$	0	0
$585$	1.00000	0.0413449
$586$	0	0
$587$	13.4865	0.556646	0.278323	−	0.960488i	$-0.410221\pi$
$587$	13.4865	0.556646	0.278323	+	0.960488i	$0.410221\pi$
$588$	0	0
$589$	20.1849	0.831705
$590$	0	0
$591$	−5.58767	−0.229846
$592$	0	0
$593$	−19.7938	−0.812835	−0.406418	−	0.913687i	$-0.633222\pi$
$593$	−19.7938	−0.812835	−0.406418	+	0.913687i	$0.633222\pi$
$594$	0	0
$595$	−4.14931	−0.170105
$596$	0	0
$597$	−14.0925	−0.576766
$598$	0	0
$599$	37.1387	1.51745	0.758723	−	0.651414i	$-0.225824\pi$
$599$	37.1387	1.51745	0.758723	+	0.651414i	$0.225824\pi$
$600$	0	0
$601$	47.4017	1.93356	0.966778	−	0.255617i	$-0.0822787\pi$
$601$	47.4017	1.93356	0.966778	+	0.255617i	$0.0822787\pi$
$602$	0	0
$603$	−1.72928	−0.0704218
$604$	0	0
$605$	−10.6079	−0.431271
$606$	0	0
$607$	18.3911	0.746471	0.373236	−	0.927737i	$-0.378248\pi$
$607$	18.3911	0.746471	0.373236	+	0.927737i	$0.378248\pi$
$608$	0	0
$609$	17.5510	0.711203
$610$	0	0
$611$	−8.77551	−0.355019
$612$	0	0
$613$	4.18596	0.169069	0.0845346	−	0.996421i	$-0.473060\pi$
$613$	4.18596	0.169069	0.0845346	+	0.996421i	$0.473060\pi$
$614$	0	0
$615$	−0.896916	−0.0361672
$616$	0	0
$617$	−34.1849	−1.37623	−0.688116	−	0.725600i	$-0.741562\pi$
$617$	−34.1849	−1.37623	−0.688116	+	0.725600i	$0.741562\pi$
$618$	0	0
$619$	−18.7389	−0.753179	−0.376589	−	0.926380i	$-0.622903\pi$
$619$	−18.7389	−0.753179	−0.376589	+	0.926380i	$0.622903\pi$
$620$	0	0
$621$	−8.14931	−0.327021
$622$	0	0
$623$	34.2418	1.37187
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−2.20617	−0.0881058
$628$	0	0
$629$	−0.804459	−0.0320759
$630$	0	0
$631$	10.2707	0.408871	0.204435	−	0.978880i	$-0.434464\pi$
$631$	10.2707	0.408871	0.204435	+	0.978880i	$0.434464\pi$
$632$	0	0
$633$	17.5510	0.697590
$634$	0	0
$635$	7.45856	0.295984
$636$	0	0
$637$	14.4017	0.570616
$638$	0	0
$639$	6.42003	0.253972
$640$	0	0
$641$	−38.0000	−1.50091	−0.750455	−	0.660922i	$-0.770166\pi$
$641$	−38.0000	−1.50091	−0.750455	+	0.660922i	$0.770166\pi$
$642$	0	0
$643$	43.1868	1.70312	0.851561	−	0.524256i	$-0.175657\pi$
$643$	43.1868	1.70312	0.851561	+	0.524256i	$0.175657\pi$
$644$	0	0
$645$	5.25240	0.206813
$646$	0	0
$647$	6.24177	0.245389	0.122695	−	0.992444i	$-0.460846\pi$
$647$	6.24177	0.245389	0.122695	+	0.992444i	$0.460846\pi$
$648$	0	0
$649$	−1.42192	−0.0558152
$650$	0	0
$651$	26.5048	1.03880
$652$	0	0
$653$	−18.0000	−0.704394	−0.352197	−	0.935926i	$-0.614565\pi$
$653$	−18.0000	−0.704394	−0.352197	+	0.935926i	$0.614565\pi$
$654$	0	0
$655$	15.0462	0.587905
$656$	0	0
$657$	12.5048	0.487858
$658$	0	0
$659$	−39.6435	−1.54429	−0.772145	−	0.635446i	$-0.780816\pi$
$659$	−39.6435	−1.54429	−0.772145	+	0.635446i	$0.780816\pi$
$660$	0	0
$661$	27.0096	1.05055	0.525276	−	0.850932i	$-0.323962\pi$
$661$	27.0096	1.05055	0.525276	+	0.850932i	$0.323962\pi$
$662$	0	0
$663$	−0.896916	−0.0348333
$664$	0	0
$665$	16.2986	0.632034
$666$	0	0
$667$	−30.9171	−1.19712
$668$	0	0
$669$	13.1878	0.509872
$670$	0	0
$671$	4.63494	0.178930
$672$	0	0
$673$	−18.7110	−0.721254	−0.360627	−	0.932710i	$-0.617437\pi$
$673$	−18.7110	−0.721254	−0.360627	+	0.932710i	$0.617437\pi$
$674$	0	0
$675$	1.00000	0.0384900
$676$	0	0
$677$	−29.4942	−1.13355	−0.566776	−	0.823872i	$-0.691810\pi$
$677$	−29.4942	−1.13355	−0.566776	+	0.823872i	$0.691810\pi$
$678$	0	0
$679$	−52.7466	−2.02423
$680$	0	0
$681$	−14.4402	−0.553351
$682$	0	0
$683$	32.9450	1.26061	0.630303	−	0.776349i	$-0.282931\pi$
$683$	32.9450	1.26061	0.630303	+	0.776349i	$0.282931\pi$
$684$	0	0
$685$	−6.00000	−0.229248
$686$	0	0
$687$	−18.2986	−0.698136
$688$	0	0
$689$	3.10308	0.118218
$690$	0	0
$691$	−21.0741	−0.801698	−0.400849	−	0.916144i	$-0.631285\pi$
$691$	−21.0741	−0.801698	−0.400849	+	0.916144i	$0.631285\pi$
$692$	0	0
$693$	−2.89692	−0.110045
$694$	0	0
$695$	−12.1493	−0.460850
$696$	0	0
$697$	0.804459	0.0304711
$698$	0	0
$699$	7.40171	0.279958
$700$	0	0
$701$	−3.38150	−0.127717	−0.0638587	−	0.997959i	$-0.520341\pi$
$701$	−3.38150	−0.127717	−0.0638587	+	0.997959i	$0.520341\pi$
$702$	0	0
$703$	3.15994	0.119179
$704$	0	0
$705$	−8.77551	−0.330505
$706$	0	0
$707$	17.5510	0.660074
$708$	0	0
$709$	−27.7205	−1.04107	−0.520533	−	0.853841i	$-0.674267\pi$
$709$	−27.7205	−1.04107	−0.520533	+	0.853841i	$0.674267\pi$
$710$	0	0
$711$	12.1493	0.455635
$712$	0	0
$713$	−46.6897	−1.74854
$714$	0	0
$715$	−0.626198	−0.0234185
$716$	0	0
$717$	14.4200	0.538526
$718$	0	0
$719$	−6.91713	−0.257965	−0.128983	−	0.991647i	$-0.541171\pi$
$719$	−6.91713	−0.257965	−0.128983	+	0.991647i	$0.541171\pi$
$720$	0	0
$721$	51.1020	1.90314
$722$	0	0
$723$	−22.2986	−0.829295
$724$	0	0
$725$	3.79383	0.140899
$726$	0	0
$727$	−34.7476	−1.28872	−0.644359	−	0.764723i	$-0.722876\pi$
$727$	−34.7476	−1.28872	−0.644359	+	0.764723i	$0.722876\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−4.71096	−0.174241
$732$	0	0
$733$	−37.1955	−1.37385	−0.686924	−	0.726729i	$-0.741040\pi$
$733$	−37.1955	−1.37385	−0.686924	+	0.726729i	$0.741040\pi$
$734$	0	0
$735$	14.4017	0.531215
$736$	0	0
$737$	1.08287	0.0398881
$738$	0	0
$739$	−54.0279	−1.98745	−0.993724	−	0.111857i	$-0.964320\pi$
$739$	−54.0279	−1.98745	−0.993724	+	0.111857i	$0.964320\pi$
$740$	0	0
$741$	3.52311	0.129425
$742$	0	0
$743$	−19.9354	−0.731361	−0.365680	−	0.930741i	$-0.619164\pi$
$743$	−19.9354	−0.731361	−0.365680	+	0.930741i	$0.619164\pi$
$744$	0	0
$745$	−17.6079	−0.645103
$746$	0	0
$747$	−5.31695	−0.194537
$748$	0	0
$749$	−53.7003	−1.96217
$750$	0	0
$751$	30.4846	1.11240	0.556199	−	0.831049i	$-0.312259\pi$
$751$	30.4846	1.11240	0.556199	+	0.831049i	$0.312259\pi$
$752$	0	0
$753$	−24.5972	−0.896374
$754$	0	0
$755$	10.2707	0.373790
$756$	0	0
$757$	−26.8959	−0.977547	−0.488774	−	0.872411i	$-0.662556\pi$
$757$	−26.8959	−0.977547	−0.488774	+	0.872411i	$0.662556\pi$
$758$	0	0
$759$	5.10308	0.185230
$760$	0	0
$761$	−22.0000	−0.797499	−0.398750	−	0.917060i	$-0.630556\pi$
$761$	−22.0000	−0.797499	−0.398750	+	0.917060i	$0.630556\pi$
$762$	0	0
$763$	−36.0558	−1.30531
$764$	0	0
$765$	−0.896916	−0.0324281
$766$	0	0
$767$	2.27072	0.0819909
$768$	0	0
$769$	18.0000	0.649097	0.324548	−	0.945869i	$-0.394788\pi$
$769$	18.0000	0.649097	0.324548	+	0.945869i	$0.394788\pi$
$770$	0	0
$771$	2.00000	0.0720282
$772$	0	0
$773$	9.28904	0.334104	0.167052	−	0.985948i	$-0.446575\pi$
$773$	9.28904	0.334104	0.167052	+	0.985948i	$0.446575\pi$
$774$	0	0
$775$	5.72928	0.205802
$776$	0	0
$777$	4.14931	0.148856
$778$	0	0
$779$	−3.15994	−0.113217
$780$	0	0
$781$	−4.02021	−0.143854
$782$	0	0
$783$	3.79383	0.135581
$784$	0	0
$785$	−2.00000	−0.0713831
$786$	0	0
$787$	0.646409	0.0230420	0.0115210	−	0.999934i	$-0.496333\pi$
$787$	0.646409	0.0230420	0.0115210	+	0.999934i	$0.496333\pi$
$788$	0	0
$789$	−0.412335	−0.0146795
$790$	0	0
$791$	57.8496	2.05690
$792$	0	0
$793$	−7.40171	−0.262842
$794$	0	0
$795$	3.10308	0.110055
$796$	0	0
$797$	−9.01063	−0.319173	−0.159586	−	0.987184i	$-0.551016\pi$
$797$	−9.01063	−0.319173	−0.159586	+	0.987184i	$0.551016\pi$
$798$	0	0
$799$	7.87090	0.278452
$800$	0	0
$801$	7.40171	0.261527
$802$	0	0
$803$	−7.83048	−0.276332
$804$	0	0
$805$	−37.7003	−1.32876
$806$	0	0
$807$	3.79383	0.133549
$808$	0	0
$809$	−42.1849	−1.48314	−0.741571	−	0.670874i	$-0.765919\pi$
$809$	−42.1849	−1.48314	−0.741571	+	0.670874i	$0.765919\pi$
$810$	0	0
$811$	7.11078	0.249693	0.124847	−	0.992176i	$-0.460156\pi$
$811$	7.11078	0.249693	0.124847	+	0.992176i	$0.460156\pi$
$812$	0	0
$813$	1.97209	0.0691643
$814$	0	0
$815$	−5.87859	−0.205918
$816$	0	0
$817$	18.5048	0.647401
$818$	0	0
$819$	4.62620	0.161652
$820$	0	0
$821$	30.2051	1.05417	0.527083	−	0.849814i	$-0.323286\pi$
$821$	30.2051	1.05417	0.527083	+	0.849814i	$0.323286\pi$
$822$	0	0
$823$	15.4586	0.538852	0.269426	−	0.963021i	$-0.413166\pi$
$823$	15.4586	0.538852	0.269426	+	0.963021i	$0.413166\pi$
$824$	0	0
$825$	−0.626198	−0.0218014
$826$	0	0
$827$	−24.7755	−0.861529	−0.430764	−	0.902464i	$-0.641756\pi$
$827$	−24.7755	−0.861529	−0.430764	+	0.902464i	$0.641756\pi$
$828$	0	0
$829$	43.6068	1.51453	0.757264	−	0.653109i	$-0.226536\pi$
$829$	43.6068	1.51453	0.757264	+	0.653109i	$0.226536\pi$
$830$	0	0
$831$	−14.7110	−0.510318
$832$	0	0
$833$	−12.9171	−0.447552
$834$	0	0
$835$	12.2341	0.423378
$836$	0	0
$837$	5.72928	0.198033
$838$	0	0
$839$	49.3930	1.70523	0.852617	−	0.522536i	$-0.175014\pi$
$839$	49.3930	1.70523	0.852617	+	0.522536i	$0.175014\pi$
$840$	0	0
$841$	−14.6068	−0.503684
$842$	0	0
$843$	−12.9171	−0.444889
$844$	0	0
$845$	1.00000	0.0344010
$846$	0	0
$847$	−49.0741	−1.68621
$848$	0	0
$849$	−9.96336	−0.341941
$850$	0	0
$851$	−7.30925	−0.250558
$852$	0	0
$853$	−32.4846	−1.11225	−0.556125	−	0.831098i	$-0.687713\pi$
$853$	−32.4846	−1.11225	−0.556125	+	0.831098i	$0.687713\pi$
$854$	0	0
$855$	3.52311	0.120488
$856$	0	0
$857$	13.9798	0.477541	0.238770	−	0.971076i	$-0.423256\pi$
$857$	13.9798	0.477541	0.238770	+	0.971076i	$0.423256\pi$
$858$	0	0
$859$	−37.8294	−1.29072	−0.645362	−	0.763877i	$-0.723293\pi$
$859$	−37.8294	−1.29072	−0.645362	+	0.763877i	$0.723293\pi$
$860$	0	0
$861$	−4.14931	−0.141408
$862$	0	0
$863$	−23.0375	−0.784205	−0.392103	−	0.919921i	$-0.628252\pi$
$863$	−23.0375	−0.784205	−0.392103	+	0.919921i	$0.628252\pi$
$864$	0	0
$865$	−12.5048	−0.425176
$866$	0	0
$867$	−16.1955	−0.550029
$868$	0	0
$869$	−7.60788	−0.258080
$870$	0	0
$871$	−1.72928	−0.0585945
$872$	0	0
$873$	−11.4017	−0.385889
$874$	0	0
$875$	4.62620	0.156394
$876$	0	0
$877$	−18.5972	−0.627985	−0.313992	−	0.949426i	$-0.601667\pi$
$877$	−18.5972	−0.627985	−0.313992	+	0.949426i	$0.601667\pi$
$878$	0	0
$879$	18.7110	0.631105
$880$	0	0
$881$	−8.20617	−0.276473	−0.138236	−	0.990399i	$-0.544143\pi$
$881$	−8.20617	−0.276473	−0.138236	+	0.990399i	$0.544143\pi$
$882$	0	0
$883$	−20.9538	−0.705151	−0.352575	−	0.935783i	$-0.614694\pi$
$883$	−20.9538	−0.705151	−0.352575	+	0.935783i	$0.614694\pi$
$884$	0	0
$885$	2.27072	0.0763294
$886$	0	0
$887$	−30.8969	−1.03742	−0.518708	−	0.854951i	$-0.673587\pi$
$887$	−30.8969	−1.03742	−0.518708	+	0.854951i	$0.673587\pi$
$888$	0	0
$889$	34.5048	1.15725
$890$	0	0
$891$	−0.626198	−0.0209784
$892$	0	0
$893$	−30.9171	−1.03460
$894$	0	0
$895$	−5.79383	−0.193666
$896$	0	0
$897$	−8.14931	−0.272098
$898$	0	0
$899$	21.7359	0.724934
$900$	0	0
$901$	−2.78321	−0.0927220
$902$	0	0
$903$	24.2986	0.808608
$904$	0	0
$905$	9.19554	0.305670
$906$	0	0
$907$	22.3353	0.741630	0.370815	−	0.928707i	$-0.379078\pi$
$907$	22.3353	0.741630	0.370815	+	0.928707i	$0.379078\pi$
$908$	0	0
$909$	3.79383	0.125833
$910$	0	0
$911$	13.6647	0.452733	0.226366	−	0.974042i	$-0.427315\pi$
$911$	13.6647	0.452733	0.226366	+	0.974042i	$0.427315\pi$
$912$	0	0
$913$	3.32946	0.110189
$914$	0	0
$915$	−7.40171	−0.244693
$916$	0	0
$917$	69.6068	2.29862
$918$	0	0
$919$	−8.26302	−0.272572	−0.136286	−	0.990670i	$-0.543517\pi$
$919$	−8.26302	−0.272572	−0.136286	+	0.990670i	$0.543517\pi$
$920$	0	0
$921$	2.12141	0.0699027
$922$	0	0
$923$	6.42003	0.211318
$924$	0	0
$925$	0.896916	0.0294904
$926$	0	0
$927$	11.0462	0.362806
$928$	0	0
$929$	33.9065	1.11244	0.556218	−	0.831036i	$-0.312252\pi$
$929$	33.9065	1.11244	0.556218	+	0.831036i	$0.312252\pi$
$930$	0	0
$931$	50.7389	1.66290
$932$	0	0
$933$	13.7938	0.451590
$934$	0	0
$935$	0.561647	0.0183678
$936$	0	0
$937$	15.7938	0.515962	0.257981	−	0.966150i	$-0.416943\pi$
$937$	15.7938	0.515962	0.257981	+	0.966150i	$0.416943\pi$
$938$	0	0
$939$	8.91713	0.290999
$940$	0	0
$941$	−39.1031	−1.27472	−0.637362	−	0.770564i	$-0.719974\pi$
$941$	−39.1031	−1.27472	−0.637362	+	0.770564i	$0.719974\pi$
$942$	0	0
$943$	7.30925	0.238022
$944$	0	0
$945$	4.62620	0.150490
$946$	0	0
$947$	10.3265	0.335567	0.167784	−	0.985824i	$-0.446339\pi$
$947$	10.3265	0.335567	0.167784	+	0.985824i	$0.446339\pi$
$948$	0	0
$949$	12.5048	0.405923
$950$	0	0
$951$	12.0925	0.392125
$952$	0	0
$953$	−19.4017	−0.628483	−0.314241	−	0.949343i	$-0.601750\pi$
$953$	−19.4017	−0.628483	−0.314241	+	0.949343i	$0.601750\pi$
$954$	0	0
$955$	6.74760	0.218347
$956$	0	0
$957$	−2.37569	−0.0767952
$958$	0	0
$959$	−27.7572	−0.896326
$960$	0	0
$961$	1.82467	0.0588603
$962$	0	0
$963$	−11.6079	−0.374059
$964$	0	0
$965$	−21.6079	−0.695582
$966$	0	0
$967$	−47.8217	−1.53784	−0.768922	−	0.639343i	$-0.779206\pi$
$967$	−47.8217	−1.53784	−0.768922	+	0.639343i	$0.779206\pi$
$968$	0	0
$969$	−3.15994	−0.101512
$970$	0	0
$971$	2.33527	0.0749424	0.0374712	−	0.999298i	$-0.488070\pi$
$971$	2.33527	0.0749424	0.0374712	+	0.999298i	$0.488070\pi$
$972$	0	0
$973$	−56.2051	−1.80185
$974$	0	0
$975$	1.00000	0.0320256
$976$	0	0
$977$	21.1020	0.675114	0.337557	−	0.941305i	$-0.390399\pi$
$977$	21.1020	0.675114	0.337557	+	0.941305i	$0.390399\pi$
$978$	0	0
$979$	−4.63494	−0.148133
$980$	0	0
$981$	−7.79383	−0.248838
$982$	0	0
$983$	10.8122	0.344854	0.172427	−	0.985022i	$-0.444839\pi$
$983$	10.8122	0.344854	0.172427	+	0.985022i	$0.444839\pi$
$984$	0	0
$985$	−5.58767	−0.178038
$986$	0	0
$987$	−40.5972	−1.29222
$988$	0	0
$989$	−42.8034	−1.36107
$990$	0	0
$991$	−36.7466	−1.16729	−0.583647	−	0.812008i	$-0.698375\pi$
$991$	−36.7466	−1.16729	−0.583647	+	0.812008i	$0.698375\pi$
$992$	0	0
$993$	−2.56934	−0.0815356
$994$	0	0
$995$	−14.0925	−0.446761
$996$	0	0
$997$	22.2986	0.706205	0.353102	−	0.935585i	$-0.385127\pi$
$997$	22.2986	0.706205	0.353102	+	0.935585i	$0.385127\pi$
$998$	0	0
$999$	0.896916	0.0283772

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6240.2.a.cd.1.3	yes	3
4.3	odd	2		6240.2.a.by.1.1	✓	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6240.2.a.by.1.1	✓	3	4.3	odd	2
6240.2.a.cd.1.3	yes	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:	\( N \)	\(=\)	\( 6240 = 2^{5} \cdot 3 \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6240.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +1.00000 q^{5} +4.62620 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +1.00000 q^{5} +4.62620 q^{7} +1.00000 q^{9} -0.626198 q^{11} +1.00000 q^{13} +1.00000 q^{15} -0.896916 q^{17} +3.52311 q^{19} +4.62620 q^{21} -8.14931 q^{23} +1.00000 q^{25} +1.00000 q^{27} +3.79383 q^{29} +5.72928 q^{31} -0.626198 q^{33} +4.62620 q^{35} +0.896916 q^{37} +1.00000 q^{39} -0.896916 q^{41} +5.25240 q^{43} +1.00000 q^{45} -8.77551 q^{47} +14.4017 q^{49} -0.896916 q^{51} +3.10308 q^{53} -0.626198 q^{55} +3.52311 q^{57} +2.27072 q^{59} -7.40171 q^{61} +4.62620 q^{63} +1.00000 q^{65} -1.72928 q^{67} -8.14931 q^{69} +6.42003 q^{71} +12.5048 q^{73} +1.00000 q^{75} -2.89692 q^{77} +12.1493 q^{79} +1.00000 q^{81} -5.31695 q^{83} -0.896916 q^{85} +3.79383 q^{87} +7.40171 q^{89} +4.62620 q^{91} +5.72928 q^{93} +3.52311 q^{95} -11.4017 q^{97} -0.626198 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{3} + 3 q^{5} + 5 q^{7} + 3 q^{9}+O(q^{10}) \) 3 * q + 3 * q^3 + 3 * q^5 + 5 * q^7 + 3 * q^9	\( 3 q + 3 q^{3} + 3 q^{5} + 5 q^{7} + 3 q^{9} + 7 q^{11} + 3 q^{13} + 3 q^{15} + q^{17} - 2 q^{19} + 5 q^{21} - 3 q^{23} + 3 q^{25} + 3 q^{27} + 4 q^{29} + 12 q^{31} + 7 q^{33} + 5 q^{35} - q^{37} + 3 q^{39} + q^{41} - 2 q^{43} + 3 q^{45} + 4 q^{47} + 4 q^{49} + q^{51} + 13 q^{53} + 7 q^{55} - 2 q^{57} + 12 q^{59} + 17 q^{61} + 5 q^{63} + 3 q^{65} - 3 q^{69} + 3 q^{71} + 2 q^{73} + 3 q^{75} - 5 q^{77} + 15 q^{79} + 3 q^{81} + 4 q^{83} + q^{85} + 4 q^{87} - 17 q^{89} + 5 q^{91} + 12 q^{93} - 2 q^{95} + 5 q^{97} + 7 q^{99}+O(q^{100}) \) 3 * q + 3 * q^3 + 3 * q^5 + 5 * q^7 + 3 * q^9 + 7 * q^11 + 3 * q^13 + 3 * q^15 + q^17 - 2 * q^19 + 5 * q^21 - 3 * q^23 + 3 * q^25 + 3 * q^27 + 4 * q^29 + 12 * q^31 + 7 * q^33 + 5 * q^35 - q^37 + 3 * q^39 + q^41 - 2 * q^43 + 3 * q^45 + 4 * q^47 + 4 * q^49 + q^51 + 13 * q^53 + 7 * q^55 - 2 * q^57 + 12 * q^59 + 17 * q^61 + 5 * q^63 + 3 * q^65 - 3 * q^69 + 3 * q^71 + 2 * q^73 + 3 * q^75 - 5 * q^77 + 15 * q^79 + 3 * q^81 + 4 * q^83 + q^85 + 4 * q^87 - 17 * q^89 + 5 * q^91 + 12 * q^93 - 2 * q^95 + 5 * q^97 + 7 * q^99

Introduction

L-functions

Modular forms

Varieties

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Database

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