LMFDB - Embedded newform 6240.2.a.bu.1.1

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:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{17}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 4 $ x^2 - x - 4
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.56155$ 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)

$f(q)$	$=$	$q+1.00000 q^{3} +1.00000 q^{5} -1.56155 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +1.00000 q^{5} -1.56155 q^{7} +1.00000 q^{9} -2.43845 q^{11} +1.00000 q^{13} +1.00000 q^{15} -6.68466 q^{17} +3.12311 q^{19} -1.56155 q^{21} +4.68466 q^{23} +1.00000 q^{25} +1.00000 q^{27} +1.12311 q^{29} -8.00000 q^{31} -2.43845 q^{33} -1.56155 q^{35} -7.56155 q^{37} +1.00000 q^{39} +7.56155 q^{41} -0.876894 q^{43} +1.00000 q^{45} -10.2462 q^{47} -4.56155 q^{49} -6.68466 q^{51} +9.80776 q^{53} -2.43845 q^{55} +3.12311 q^{57} -14.2462 q^{59} +3.56155 q^{61} -1.56155 q^{63} +1.00000 q^{65} +2.24621 q^{67} +4.68466 q^{69} -0.684658 q^{71} +8.24621 q^{73} +1.00000 q^{75} +3.80776 q^{77} -8.68466 q^{79} +1.00000 q^{81} +4.00000 q^{83} -6.68466 q^{85} +1.12311 q^{87} -9.80776 q^{89} -1.56155 q^{91} -8.00000 q^{93} +3.12311 q^{95} -14.6847 q^{97} -2.43845 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} + 2 q^{5} + q^{7} + 2 q^{9}+O(q^{10}) $ 2 * q + 2 * q^3 + 2 * q^5 + q^7 + 2 * q^9	$ 2 q + 2 q^{3} + 2 q^{5} + q^{7} + 2 q^{9} - 9 q^{11} + 2 q^{13} + 2 q^{15} - q^{17} - 2 q^{19} + q^{21} - 3 q^{23} + 2 q^{25} + 2 q^{27} - 6 q^{29} - 16 q^{31} - 9 q^{33} + q^{35} - 11 q^{37} + 2 q^{39} + 11 q^{41} - 10 q^{43} + 2 q^{45} - 4 q^{47} - 5 q^{49} - q^{51} - q^{53} - 9 q^{55} - 2 q^{57} - 12 q^{59} + 3 q^{61} + q^{63} + 2 q^{65} - 12 q^{67} - 3 q^{69} + 11 q^{71} + 2 q^{75} - 13 q^{77} - 5 q^{79} + 2 q^{81} + 8 q^{83} - q^{85} - 6 q^{87} + q^{89} + q^{91} - 16 q^{93} - 2 q^{95} - 17 q^{97} - 9 q^{99}+O(q^{100}) $ 2 * q + 2 * q^3 + 2 * q^5 + q^7 + 2 * q^9 - 9 * q^11 + 2 * q^13 + 2 * q^15 - q^17 - 2 * q^19 + q^21 - 3 * q^23 + 2 * q^25 + 2 * q^27 - 6 * q^29 - 16 * q^31 - 9 * q^33 + q^35 - 11 * q^37 + 2 * q^39 + 11 * q^41 - 10 * q^43 + 2 * q^45 - 4 * q^47 - 5 * q^49 - q^51 - q^53 - 9 * q^55 - 2 * q^57 - 12 * q^59 + 3 * q^61 + q^63 + 2 * q^65 - 12 * q^67 - 3 * q^69 + 11 * q^71 + 2 * q^75 - 13 * q^77 - 5 * q^79 + 2 * q^81 + 8 * q^83 - q^85 - 6 * q^87 + q^89 + q^91 - 16 * q^93 - 2 * q^95 - 17 * q^97 - 9 * 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
$7$	−1.56155	−0.590211	−0.295106	−	0.955465i	$-0.595355\pi$
$7$	−1.56155	−0.590211	−0.295106	+	0.955465i	$0.595355\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−2.43845	−0.735219	−0.367610	−	0.929980i	$-0.619824\pi$
$11$	−2.43845	−0.735219	−0.367610	+	0.929980i	$0.619824\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	1.00000	0.258199
$16$	0	0
$17$	−6.68466	−1.62127	−0.810634	−	0.585553i	$-0.800877\pi$
$17$	−6.68466	−1.62127	−0.810634	+	0.585553i	$0.800877\pi$
$18$	0	0
$19$	3.12311	0.716490	0.358245	−	0.933628i	$-0.383375\pi$
$19$	3.12311	0.716490	0.358245	+	0.933628i	$0.383375\pi$
$20$	0	0
$21$	−1.56155	−0.340759
$22$	0	0
$23$	4.68466	0.976819	0.488409	−	0.872615i	$-0.337577\pi$
$23$	4.68466	0.976819	0.488409	+	0.872615i	$0.337577\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	1.12311	0.208555	0.104278	−	0.994548i	$-0.466747\pi$
$29$	1.12311	0.208555	0.104278	+	0.994548i	$0.466747\pi$
$30$	0	0
$31$	−8.00000	−1.43684	−0.718421	−	0.695608i	$-0.755135\pi$
$31$	−8.00000	−1.43684	−0.718421	+	0.695608i	$0.755135\pi$
$32$	0	0
$33$	−2.43845	−0.424479
$34$	0	0
$35$	−1.56155	−0.263951
$36$	0	0
$37$	−7.56155	−1.24311	−0.621556	−	0.783370i	$-0.713499\pi$
$37$	−7.56155	−1.24311	−0.621556	+	0.783370i	$0.713499\pi$
$38$	0	0
$39$	1.00000	0.160128
$40$	0	0
$41$	7.56155	1.18092	0.590458	−	0.807068i	$-0.298947\pi$
$41$	7.56155	1.18092	0.590458	+	0.807068i	$0.298947\pi$
$42$	0	0
$43$	−0.876894	−0.133725	−0.0668626	−	0.997762i	$-0.521299\pi$
$43$	−0.876894	−0.133725	−0.0668626	+	0.997762i	$0.521299\pi$
$44$	0	0
$45$	1.00000	0.149071
$46$	0	0
$47$	−10.2462	−1.49456	−0.747282	−	0.664507i	$-0.768641\pi$
$47$	−10.2462	−1.49456	−0.747282	+	0.664507i	$0.768641\pi$
$48$	0	0
$49$	−4.56155	−0.651650
$50$	0	0
$51$	−6.68466	−0.936039
$52$	0	0
$53$	9.80776	1.34720	0.673600	−	0.739096i	$-0.264747\pi$
$53$	9.80776	1.34720	0.673600	+	0.739096i	$0.264747\pi$
$54$	0	0
$55$	−2.43845	−0.328800
$56$	0	0
$57$	3.12311	0.413665
$58$	0	0
$59$	−14.2462	−1.85470	−0.927349	−	0.374197i	$-0.877918\pi$
$59$	−14.2462	−1.85470	−0.927349	+	0.374197i	$0.877918\pi$
$60$	0	0
$61$	3.56155	0.456010	0.228005	−	0.973660i	$-0.426780\pi$
$61$	3.56155	0.456010	0.228005	+	0.973660i	$0.426780\pi$
$62$	0	0
$63$	−1.56155	−0.196737
$64$	0	0
$65$	1.00000	0.124035
$66$	0	0
$67$	2.24621	0.274418	0.137209	−	0.990542i	$-0.456187\pi$
$67$	2.24621	0.274418	0.137209	+	0.990542i	$0.456187\pi$
$68$	0	0
$69$	4.68466	0.563967
$70$	0	0
$71$	−0.684658	−0.0812540	−0.0406270	−	0.999174i	$-0.512936\pi$
$71$	−0.684658	−0.0812540	−0.0406270	+	0.999174i	$0.512936\pi$
$72$	0	0
$73$	8.24621	0.965146	0.482573	−	0.875856i	$-0.339702\pi$
$73$	8.24621	0.965146	0.482573	+	0.875856i	$0.339702\pi$
$74$	0	0
$75$	1.00000	0.115470
$76$	0	0
$77$	3.80776	0.433935
$78$	0	0
$79$	−8.68466	−0.977100	−0.488550	−	0.872536i	$-0.662474\pi$
$79$	−8.68466	−0.977100	−0.488550	+	0.872536i	$0.662474\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	4.00000	0.439057	0.219529	−	0.975606i	$-0.429548\pi$
$83$	4.00000	0.439057	0.219529	+	0.975606i	$0.429548\pi$
$84$	0	0
$85$	−6.68466	−0.725053
$86$	0	0
$87$	1.12311	0.120410
$88$	0	0
$89$	−9.80776	−1.03962	−0.519810	−	0.854282i	$-0.673997\pi$
$89$	−9.80776	−1.03962	−0.519810	+	0.854282i	$0.673997\pi$
$90$	0	0
$91$	−1.56155	−0.163695
$92$	0	0
$93$	−8.00000	−0.829561
$94$	0	0
$95$	3.12311	0.320424
$96$	0	0
$97$	−14.6847	−1.49100	−0.745501	−	0.666505i	$-0.767790\pi$
$97$	−14.6847	−1.49100	−0.745501	+	0.666505i	$0.767790\pi$
$98$	0	0
$99$	−2.43845	−0.245073
$100$	0	0
$101$	2.87689	0.286262	0.143131	−	0.989704i	$-0.454283\pi$
$101$	2.87689	0.286262	0.143131	+	0.989704i	$0.454283\pi$
$102$	0	0
$103$	2.24621	0.221326	0.110663	−	0.993858i	$-0.464703\pi$
$103$	2.24621	0.221326	0.110663	+	0.993858i	$0.464703\pi$
$104$	0	0
$105$	−1.56155	−0.152392
$106$	0	0
$107$	−7.80776	−0.754805	−0.377403	−	0.926049i	$-0.623183\pi$
$107$	−7.80776	−0.754805	−0.377403	+	0.926049i	$0.623183\pi$
$108$	0	0
$109$	−6.87689	−0.658687	−0.329344	−	0.944210i	$-0.606827\pi$
$109$	−6.87689	−0.658687	−0.329344	+	0.944210i	$0.606827\pi$
$110$	0	0
$111$	−7.56155	−0.717711
$112$	0	0
$113$	8.24621	0.775738	0.387869	−	0.921714i	$-0.373211\pi$
$113$	8.24621	0.775738	0.387869	+	0.921714i	$0.373211\pi$
$114$	0	0
$115$	4.68466	0.436847
$116$	0	0
$117$	1.00000	0.0924500
$118$	0	0
$119$	10.4384	0.956891
$120$	0	0
$121$	−5.05398	−0.459452
$122$	0	0
$123$	7.56155	0.681802
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−2.24621	−0.199319	−0.0996595	−	0.995022i	$-0.531775\pi$
$127$	−2.24621	−0.199319	−0.0996595	+	0.995022i	$0.531775\pi$
$128$	0	0
$129$	−0.876894	−0.0772062
$130$	0	0
$131$	−8.00000	−0.698963	−0.349482	−	0.936943i	$-0.613642\pi$
$131$	−8.00000	−0.698963	−0.349482	+	0.936943i	$0.613642\pi$
$132$	0	0
$133$	−4.87689	−0.422880
$134$	0	0
$135$	1.00000	0.0860663
$136$	0	0
$137$	−14.0000	−1.19610	−0.598050	−	0.801459i	$-0.704058\pi$
$137$	−14.0000	−1.19610	−0.598050	+	0.801459i	$0.704058\pi$
$138$	0	0
$139$	−19.8078	−1.68007	−0.840036	−	0.542530i	$-0.817466\pi$
$139$	−19.8078	−1.68007	−0.840036	+	0.542530i	$0.817466\pi$
$140$	0	0
$141$	−10.2462	−0.862887
$142$	0	0
$143$	−2.43845	−0.203913
$144$	0	0
$145$	1.12311	0.0932688
$146$	0	0
$147$	−4.56155	−0.376231
$148$	0	0
$149$	2.19224	0.179595	0.0897975	−	0.995960i	$-0.471378\pi$
$149$	2.19224	0.179595	0.0897975	+	0.995960i	$0.471378\pi$
$150$	0	0
$151$	−1.75379	−0.142721	−0.0713607	−	0.997451i	$-0.522734\pi$
$151$	−1.75379	−0.142721	−0.0713607	+	0.997451i	$0.522734\pi$
$152$	0	0
$153$	−6.68466	−0.540423
$154$	0	0
$155$	−8.00000	−0.642575
$156$	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
$159$	9.80776	0.777806
$160$	0	0
$161$	−7.31534	−0.576530
$162$	0	0
$163$	−15.8078	−1.23816	−0.619080	−	0.785328i	$-0.712494\pi$
$163$	−15.8078	−1.23816	−0.619080	+	0.785328i	$0.712494\pi$
$164$	0	0
$165$	−2.43845	−0.189833
$166$	0	0
$167$	−4.00000	−0.309529	−0.154765	−	0.987951i	$-0.549462\pi$
$167$	−4.00000	−0.309529	−0.154765	+	0.987951i	$0.549462\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	3.12311	0.238830
$172$	0	0
$173$	7.75379	0.589510	0.294755	−	0.955573i	$-0.404762\pi$
$173$	7.75379	0.589510	0.294755	+	0.955573i	$0.404762\pi$
$174$	0	0
$175$	−1.56155	−0.118042
$176$	0	0
$177$	−14.2462	−1.07081
$178$	0	0
$179$	−4.87689	−0.364516	−0.182258	−	0.983251i	$-0.558341\pi$
$179$	−4.87689	−0.364516	−0.182258	+	0.983251i	$0.558341\pi$
$180$	0	0
$181$	5.31534	0.395086	0.197543	−	0.980294i	$-0.436704\pi$
$181$	5.31534	0.395086	0.197543	+	0.980294i	$0.436704\pi$
$182$	0	0
$183$	3.56155	0.263278
$184$	0	0
$185$	−7.56155	−0.555936
$186$	0	0
$187$	16.3002	1.19199
$188$	0	0
$189$	−1.56155	−0.113586
$190$	0	0
$191$	25.3693	1.83566	0.917830	−	0.396974i	$-0.129940\pi$
$191$	25.3693	1.83566	0.917830	+	0.396974i	$0.129940\pi$
$192$	0	0
$193$	2.68466	0.193246	0.0966230	−	0.995321i	$-0.469196\pi$
$193$	2.68466	0.193246	0.0966230	+	0.995321i	$0.469196\pi$
$194$	0	0
$195$	1.00000	0.0716115
$196$	0	0
$197$	6.00000	0.427482	0.213741	−	0.976890i	$-0.431435\pi$
$197$	6.00000	0.427482	0.213741	+	0.976890i	$0.431435\pi$
$198$	0	0
$199$	0	0		−	1.00000i	$-0.5\pi$
$199$	0	0			1.00000i	$0.5\pi$
$200$	0	0
$201$	2.24621	0.158436
$202$	0	0
$203$	−1.75379	−0.123092
$204$	0	0
$205$	7.56155	0.528122
$206$	0	0
$207$	4.68466	0.325606
$208$	0	0
$209$	−7.61553	−0.526777
$210$	0	0
$211$	−20.4924	−1.41076	−0.705378	−	0.708831i	$-0.749223\pi$
$211$	−20.4924	−1.41076	−0.705378	+	0.708831i	$0.749223\pi$
$212$	0	0
$213$	−0.684658	−0.0469120
$214$	0	0
$215$	−0.876894	−0.0598037
$216$	0	0
$217$	12.4924	0.848041
$218$	0	0
$219$	8.24621	0.557227
$220$	0	0
$221$	−6.68466	−0.449659
$222$	0	0
$223$	−2.24621	−0.150417	−0.0752087	−	0.997168i	$-0.523962\pi$
$223$	−2.24621	−0.150417	−0.0752087	+	0.997168i	$0.523962\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	0.876894	0.0582015	0.0291008	−	0.999576i	$-0.490736\pi$
$227$	0.876894	0.0582015	0.0291008	+	0.999576i	$0.490736\pi$
$228$	0	0
$229$	29.6155	1.95705	0.978525	−	0.206130i	$-0.0660870\pi$
$229$	29.6155	1.95705	0.978525	+	0.206130i	$0.0660870\pi$
$230$	0	0
$231$	3.80776	0.250532
$232$	0	0
$233$	−19.5616	−1.28152	−0.640760	−	0.767741i	$-0.721381\pi$
$233$	−19.5616	−1.28152	−0.640760	+	0.767741i	$0.721381\pi$
$234$	0	0
$235$	−10.2462	−0.668389
$236$	0	0
$237$	−8.68466	−0.564129
$238$	0	0
$239$	3.80776	0.246304	0.123152	−	0.992388i	$-0.460700\pi$
$239$	3.80776	0.246304	0.123152	+	0.992388i	$0.460700\pi$
$240$	0	0
$241$	−18.8769	−1.21597	−0.607984	−	0.793949i	$-0.708021\pi$
$241$	−18.8769	−1.21597	−0.607984	+	0.793949i	$0.708021\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−4.56155	−0.291427
$246$	0	0
$247$	3.12311	0.198718
$248$	0	0
$249$	4.00000	0.253490
$250$	0	0
$251$	−30.2462	−1.90912	−0.954562	−	0.298013i	$-0.903676\pi$
$251$	−30.2462	−1.90912	−0.954562	+	0.298013i	$0.903676\pi$
$252$	0	0
$253$	−11.4233	−0.718176
$254$	0	0
$255$	−6.68466	−0.418610
$256$	0	0
$257$	14.4924	0.904012	0.452006	−	0.892015i	$-0.350708\pi$
$257$	14.4924	0.904012	0.452006	+	0.892015i	$0.350708\pi$
$258$	0	0
$259$	11.8078	0.733699
$260$	0	0
$261$	1.12311	0.0695185
$262$	0	0
$263$	−22.7386	−1.40212	−0.701062	−	0.713100i	$-0.747290\pi$
$263$	−22.7386	−1.40212	−0.701062	+	0.713100i	$0.747290\pi$
$264$	0	0
$265$	9.80776	0.602486
$266$	0	0
$267$	−9.80776	−0.600225
$268$	0	0
$269$	−5.12311	−0.312361	−0.156181	−	0.987729i	$-0.549918\pi$
$269$	−5.12311	−0.312361	−0.156181	+	0.987729i	$0.549918\pi$
$270$	0	0
$271$	19.1231	1.16165	0.580823	−	0.814030i	$-0.302731\pi$
$271$	19.1231	1.16165	0.580823	+	0.814030i	$0.302731\pi$
$272$	0	0
$273$	−1.56155	−0.0945095
$274$	0	0
$275$	−2.43845	−0.147044
$276$	0	0
$277$	10.8769	0.653529	0.326765	−	0.945106i	$-0.394042\pi$
$277$	10.8769	0.653529	0.326765	+	0.945106i	$0.394042\pi$
$278$	0	0
$279$	−8.00000	−0.478947
$280$	0	0
$281$	−2.49242	−0.148685	−0.0743427	−	0.997233i	$-0.523686\pi$
$281$	−2.49242	−0.148685	−0.0743427	+	0.997233i	$0.523686\pi$
$282$	0	0
$283$	−2.24621	−0.133523	−0.0667617	−	0.997769i	$-0.521267\pi$
$283$	−2.24621	−0.133523	−0.0667617	+	0.997769i	$0.521267\pi$
$284$	0	0
$285$	3.12311	0.184997
$286$	0	0
$287$	−11.8078	−0.696990
$288$	0	0
$289$	27.6847	1.62851
$290$	0	0
$291$	−14.6847	−0.860830
$292$	0	0
$293$	23.3693	1.36525	0.682625	−	0.730769i	$-0.260838\pi$
$293$	23.3693	1.36525	0.682625	+	0.730769i	$0.260838\pi$
$294$	0	0
$295$	−14.2462	−0.829446
$296$	0	0
$297$	−2.43845	−0.141493
$298$	0	0
$299$	4.68466	0.270921
$300$	0	0
$301$	1.36932	0.0789261
$302$	0	0
$303$	2.87689	0.165273
$304$	0	0
$305$	3.56155	0.203934
$306$	0	0
$307$	4.68466	0.267368	0.133684	−	0.991024i	$-0.457319\pi$
$307$	4.68466	0.267368	0.133684	+	0.991024i	$0.457319\pi$
$308$	0	0
$309$	2.24621	0.127782
$310$	0	0
$311$	−9.36932	−0.531285	−0.265643	−	0.964072i	$-0.585584\pi$
$311$	−9.36932	−0.531285	−0.265643	+	0.964072i	$0.585584\pi$
$312$	0	0
$313$	26.0000	1.46961	0.734803	−	0.678280i	$-0.237274\pi$
$313$	26.0000	1.46961	0.734803	+	0.678280i	$0.237274\pi$
$314$	0	0
$315$	−1.56155	−0.0879835
$316$	0	0
$317$	7.75379	0.435496	0.217748	−	0.976005i	$-0.430129\pi$
$317$	7.75379	0.435496	0.217748	+	0.976005i	$0.430129\pi$
$318$	0	0
$319$	−2.73863	−0.153334
$320$	0	0
$321$	−7.80776	−0.435787
$322$	0	0
$323$	−20.8769	−1.16162
$324$	0	0
$325$	1.00000	0.0554700
$326$	0	0
$327$	−6.87689	−0.380293
$328$	0	0
$329$	16.0000	0.882109
$330$	0	0
$331$	1.36932	0.0752645	0.0376322	−	0.999292i	$-0.488018\pi$
$331$	1.36932	0.0752645	0.0376322	+	0.999292i	$0.488018\pi$
$332$	0	0
$333$	−7.56155	−0.414371
$334$	0	0
$335$	2.24621	0.122724
$336$	0	0
$337$	−14.0000	−0.762629	−0.381314	−	0.924445i	$-0.624528\pi$
$337$	−14.0000	−0.762629	−0.381314	+	0.924445i	$0.624528\pi$
$338$	0	0
$339$	8.24621	0.447873
$340$	0	0
$341$	19.5076	1.05639
$342$	0	0
$343$	18.0540	0.974823
$344$	0	0
$345$	4.68466	0.252214
$346$	0	0
$347$	14.4384	0.775096	0.387548	−	0.921849i	$-0.373322\pi$
$347$	14.4384	0.775096	0.387548	+	0.921849i	$0.373322\pi$
$348$	0	0
$349$	−24.2462	−1.29787	−0.648935	−	0.760844i	$-0.724785\pi$
$349$	−24.2462	−1.29787	−0.648935	+	0.760844i	$0.724785\pi$
$350$	0	0
$351$	1.00000	0.0533761
$352$	0	0
$353$	23.8617	1.27003	0.635016	−	0.772499i	$-0.280993\pi$
$353$	23.8617	1.27003	0.635016	+	0.772499i	$0.280993\pi$
$354$	0	0
$355$	−0.684658	−0.0363379
$356$	0	0
$357$	10.4384	0.552461
$358$	0	0
$359$	−28.4924	−1.50377	−0.751886	−	0.659293i	$-0.770856\pi$
$359$	−28.4924	−1.50377	−0.751886	+	0.659293i	$0.770856\pi$
$360$	0	0
$361$	−9.24621	−0.486643
$362$	0	0
$363$	−5.05398	−0.265265
$364$	0	0
$365$	8.24621	0.431626
$366$	0	0
$367$	8.87689	0.463370	0.231685	−	0.972791i	$-0.425576\pi$
$367$	8.87689	0.463370	0.231685	+	0.972791i	$0.425576\pi$
$368$	0	0
$369$	7.56155	0.393639
$370$	0	0
$371$	−15.3153	−0.795133
$372$	0	0
$373$	2.49242	0.129053	0.0645264	−	0.997916i	$-0.479446\pi$
$373$	2.49242	0.129053	0.0645264	+	0.997916i	$0.479446\pi$
$374$	0	0
$375$	1.00000	0.0516398
$376$	0	0
$377$	1.12311	0.0578429
$378$	0	0
$379$	−14.2462	−0.731779	−0.365889	−	0.930658i	$-0.619235\pi$
$379$	−14.2462	−0.731779	−0.365889	+	0.930658i	$0.619235\pi$
$380$	0	0
$381$	−2.24621	−0.115077
$382$	0	0
$383$	−29.3693	−1.50070	−0.750351	−	0.661040i	$-0.770116\pi$
$383$	−29.3693	−1.50070	−0.750351	+	0.661040i	$0.770116\pi$
$384$	0	0
$385$	3.80776	0.194062
$386$	0	0
$387$	−0.876894	−0.0445750
$388$	0	0
$389$	−19.3693	−0.982063	−0.491032	−	0.871142i	$-0.663380\pi$
$389$	−19.3693	−0.982063	−0.491032	+	0.871142i	$0.663380\pi$
$390$	0	0
$391$	−31.3153	−1.58368
$392$	0	0
$393$	−8.00000	−0.403547
$394$	0	0
$395$	−8.68466	−0.436973
$396$	0	0
$397$	−31.5616	−1.58403	−0.792014	−	0.610502i	$-0.790968\pi$
$397$	−31.5616	−1.58403	−0.792014	+	0.610502i	$0.790968\pi$
$398$	0	0
$399$	−4.87689	−0.244150
$400$	0	0
$401$	30.4924	1.52272	0.761359	−	0.648330i	$-0.224532\pi$
$401$	30.4924	1.52272	0.761359	+	0.648330i	$0.224532\pi$
$402$	0	0
$403$	−8.00000	−0.398508
$404$	0	0
$405$	1.00000	0.0496904
$406$	0	0
$407$	18.4384	0.913960
$408$	0	0
$409$	−1.12311	−0.0555340	−0.0277670	−	0.999614i	$-0.508840\pi$
$409$	−1.12311	−0.0555340	−0.0277670	+	0.999614i	$0.508840\pi$
$410$	0	0
$411$	−14.0000	−0.690569
$412$	0	0
$413$	22.2462	1.09466
$414$	0	0
$415$	4.00000	0.196352
$416$	0	0
$417$	−19.8078	−0.969990
$418$	0	0
$419$	7.61553	0.372043	0.186021	−	0.982546i	$-0.440441\pi$
$419$	7.61553	0.372043	0.186021	+	0.982546i	$0.440441\pi$
$420$	0	0
$421$	−5.12311	−0.249685	−0.124842	−	0.992177i	$-0.539843\pi$
$421$	−5.12311	−0.249685	−0.124842	+	0.992177i	$0.539843\pi$
$422$	0	0
$423$	−10.2462	−0.498188
$424$	0	0
$425$	−6.68466	−0.324254
$426$	0	0
$427$	−5.56155	−0.269142
$428$	0	0
$429$	−2.43845	−0.117729
$430$	0	0
$431$	−8.00000	−0.385346	−0.192673	−	0.981263i	$-0.561716\pi$
$431$	−8.00000	−0.385346	−0.192673	+	0.981263i	$0.561716\pi$
$432$	0	0
$433$	−10.4924	−0.504234	−0.252117	−	0.967697i	$-0.581127\pi$
$433$	−10.4924	−0.504234	−0.252117	+	0.967697i	$0.581127\pi$
$434$	0	0
$435$	1.12311	0.0538488
$436$	0	0
$437$	14.6307	0.699880
$438$	0	0
$439$	24.6847	1.17813	0.589067	−	0.808084i	$-0.299495\pi$
$439$	24.6847	1.17813	0.589067	+	0.808084i	$0.299495\pi$
$440$	0	0
$441$	−4.56155	−0.217217
$442$	0	0
$443$	−25.5616	−1.21447	−0.607233	−	0.794524i	$-0.707721\pi$
$443$	−25.5616	−1.21447	−0.607233	+	0.794524i	$0.707721\pi$
$444$	0	0
$445$	−9.80776	−0.464933
$446$	0	0
$447$	2.19224	0.103689
$448$	0	0
$449$	34.3002	1.61873	0.809363	−	0.587309i	$-0.199813\pi$
$449$	34.3002	1.61873	0.809363	+	0.587309i	$0.199813\pi$
$450$	0	0
$451$	−18.4384	−0.868233
$452$	0	0
$453$	−1.75379	−0.0824002
$454$	0	0
$455$	−1.56155	−0.0732067
$456$	0	0
$457$	−25.4233	−1.18925	−0.594626	−	0.804003i	$-0.702700\pi$
$457$	−25.4233	−1.18925	−0.594626	+	0.804003i	$0.702700\pi$
$458$	0	0
$459$	−6.68466	−0.312013
$460$	0	0
$461$	1.80776	0.0841960	0.0420980	−	0.999113i	$-0.486596\pi$
$461$	1.80776	0.0841960	0.0420980	+	0.999113i	$0.486596\pi$
$462$	0	0
$463$	34.9309	1.62338	0.811688	−	0.584092i	$-0.198549\pi$
$463$	34.9309	1.62338	0.811688	+	0.584092i	$0.198549\pi$
$464$	0	0
$465$	−8.00000	−0.370991
$466$	0	0
$467$	26.9309	1.24621	0.623106	−	0.782137i	$-0.285871\pi$
$467$	26.9309	1.24621	0.623106	+	0.782137i	$0.285871\pi$
$468$	0	0
$469$	−3.50758	−0.161965
$470$	0	0
$471$	−2.00000	−0.0921551
$472$	0	0
$473$	2.13826	0.0983173
$474$	0	0
$475$	3.12311	0.143298
$476$	0	0
$477$	9.80776	0.449067
$478$	0	0
$479$	43.4233	1.98406	0.992030	−	0.125999i	$-0.0402135\pi$
$479$	43.4233	1.98406	0.992030	+	0.125999i	$0.0402135\pi$
$480$	0	0
$481$	−7.56155	−0.344777
$482$	0	0
$483$	−7.31534	−0.332860
$484$	0	0
$485$	−14.6847	−0.666796
$486$	0	0
$487$	3.31534	0.150232	0.0751162	−	0.997175i	$-0.476067\pi$
$487$	3.31534	0.150232	0.0751162	+	0.997175i	$0.476067\pi$
$488$	0	0
$489$	−15.8078	−0.714852
$490$	0	0
$491$	15.6155	0.704719	0.352359	−	0.935865i	$-0.385379\pi$
$491$	15.6155	0.704719	0.352359	+	0.935865i	$0.385379\pi$
$492$	0	0
$493$	−7.50758	−0.338124
$494$	0	0
$495$	−2.43845	−0.109600
$496$	0	0
$497$	1.06913	0.0479570
$498$	0	0
$499$	−37.8617	−1.69492	−0.847462	−	0.530856i	$-0.821871\pi$
$499$	−37.8617	−1.69492	−0.847462	+	0.530856i	$0.821871\pi$
$500$	0	0
$501$	−4.00000	−0.178707
$502$	0	0
$503$	14.7386	0.657163	0.328582	−	0.944476i	$-0.393429\pi$
$503$	14.7386	0.657163	0.328582	+	0.944476i	$0.393429\pi$
$504$	0	0
$505$	2.87689	0.128020
$506$	0	0
$507$	1.00000	0.0444116
$508$	0	0
$509$	6.68466	0.296292	0.148146	−	0.988965i	$-0.452669\pi$
$509$	6.68466	0.296292	0.148146	+	0.988965i	$0.452669\pi$
$510$	0	0
$511$	−12.8769	−0.569640
$512$	0	0
$513$	3.12311	0.137888
$514$	0	0
$515$	2.24621	0.0989799
$516$	0	0
$517$	24.9848	1.09883
$518$	0	0
$519$	7.75379	0.340354
$520$	0	0
$521$	12.7386	0.558090	0.279045	−	0.960278i	$-0.409982\pi$
$521$	12.7386	0.558090	0.279045	+	0.960278i	$0.409982\pi$
$522$	0	0
$523$	21.3693	0.934415	0.467207	−	0.884148i	$-0.345260\pi$
$523$	21.3693	0.934415	0.467207	+	0.884148i	$0.345260\pi$
$524$	0	0
$525$	−1.56155	−0.0681518
$526$	0	0
$527$	53.4773	2.32951
$528$	0	0
$529$	−1.05398	−0.0458250
$530$	0	0
$531$	−14.2462	−0.618233
$532$	0	0
$533$	7.56155	0.327527
$534$	0	0
$535$	−7.80776	−0.337559
$536$	0	0
$537$	−4.87689	−0.210454
$538$	0	0
$539$	11.1231	0.479106
$540$	0	0
$541$	−18.0000	−0.773880	−0.386940	−	0.922105i	$-0.626468\pi$
$541$	−18.0000	−0.773880	−0.386940	+	0.922105i	$0.626468\pi$
$542$	0	0
$543$	5.31534	0.228103
$544$	0	0
$545$	−6.87689	−0.294574
$546$	0	0
$547$	−13.7538	−0.588070	−0.294035	−	0.955795i	$-0.594998\pi$
$547$	−13.7538	−0.588070	−0.294035	+	0.955795i	$0.594998\pi$
$548$	0	0
$549$	3.56155	0.152003
$550$	0	0
$551$	3.50758	0.149428
$552$	0	0
$553$	13.5616	0.576696
$554$	0	0
$555$	−7.56155	−0.320970
$556$	0	0
$557$	−0.630683	−0.0267229	−0.0133615	−	0.999911i	$-0.504253\pi$
$557$	−0.630683	−0.0267229	−0.0133615	+	0.999911i	$0.504253\pi$
$558$	0	0
$559$	−0.876894	−0.0370887
$560$	0	0
$561$	16.3002	0.688194
$562$	0	0
$563$	−15.8078	−0.666218	−0.333109	−	0.942888i	$-0.608098\pi$
$563$	−15.8078	−0.666218	−0.333109	+	0.942888i	$0.608098\pi$
$564$	0	0
$565$	8.24621	0.346921
$566$	0	0
$567$	−1.56155	−0.0655791
$568$	0	0
$569$	32.2462	1.35183	0.675916	−	0.736979i	$-0.263748\pi$
$569$	32.2462	1.35183	0.675916	+	0.736979i	$0.263748\pi$
$570$	0	0
$571$	−16.6847	−0.698231	−0.349116	−	0.937080i	$-0.613518\pi$
$571$	−16.6847	−0.698231	−0.349116	+	0.937080i	$0.613518\pi$
$572$	0	0
$573$	25.3693	1.05982
$574$	0	0
$575$	4.68466	0.195364
$576$	0	0
$577$	−2.19224	−0.0912640	−0.0456320	−	0.998958i	$-0.514530\pi$
$577$	−2.19224	−0.0912640	−0.0456320	+	0.998958i	$0.514530\pi$
$578$	0	0
$579$	2.68466	0.111571
$580$	0	0
$581$	−6.24621	−0.259137
$582$	0	0
$583$	−23.9157	−0.990488
$584$	0	0
$585$	1.00000	0.0413449
$586$	0	0
$587$	16.8769	0.696584	0.348292	−	0.937386i	$-0.386762\pi$
$587$	16.8769	0.696584	0.348292	+	0.937386i	$0.386762\pi$
$588$	0	0
$589$	−24.9848	−1.02948
$590$	0	0
$591$	6.00000	0.246807
$592$	0	0
$593$	−39.3693	−1.61670	−0.808352	−	0.588699i	$-0.799640\pi$
$593$	−39.3693	−1.61670	−0.808352	+	0.588699i	$0.799640\pi$
$594$	0	0
$595$	10.4384	0.427935
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−6.24621	−0.255213	−0.127607	−	0.991825i	$-0.540730\pi$
$599$	−6.24621	−0.255213	−0.127607	+	0.991825i	$0.540730\pi$
$600$	0	0
$601$	37.4233	1.52653	0.763264	−	0.646087i	$-0.223596\pi$
$601$	37.4233	1.52653	0.763264	+	0.646087i	$0.223596\pi$
$602$	0	0
$603$	2.24621	0.0914728
$604$	0	0
$605$	−5.05398	−0.205473
$606$	0	0
$607$	21.3693	0.867354	0.433677	−	0.901068i	$-0.357216\pi$
$607$	21.3693	0.867354	0.433677	+	0.901068i	$0.357216\pi$
$608$	0	0
$609$	−1.75379	−0.0710671
$610$	0	0
$611$	−10.2462	−0.414517
$612$	0	0
$613$	−37.4233	−1.51151	−0.755756	−	0.654853i	$-0.772731\pi$
$613$	−37.4233	−1.51151	−0.755756	+	0.654853i	$0.772731\pi$
$614$	0	0
$615$	7.56155	0.304911
$616$	0	0
$617$	42.9848	1.73050	0.865252	−	0.501337i	$-0.167158\pi$
$617$	42.9848	1.73050	0.865252	+	0.501337i	$0.167158\pi$
$618$	0	0
$619$	−36.4924	−1.46675	−0.733377	−	0.679822i	$-0.762057\pi$
$619$	−36.4924	−1.46675	−0.733377	+	0.679822i	$0.762057\pi$
$620$	0	0
$621$	4.68466	0.187989
$622$	0	0
$623$	15.3153	0.613596
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−7.61553	−0.304135
$628$	0	0
$629$	50.5464	2.01542
$630$	0	0
$631$	10.7386	0.427498	0.213749	−	0.976889i	$-0.431432\pi$
$631$	10.7386	0.427498	0.213749	+	0.976889i	$0.431432\pi$
$632$	0	0
$633$	−20.4924	−0.814501
$634$	0	0
$635$	−2.24621	−0.0891382
$636$	0	0
$637$	−4.56155	−0.180735
$638$	0	0
$639$	−0.684658	−0.0270847
$640$	0	0
$641$	36.7386	1.45109	0.725544	−	0.688175i	$-0.241588\pi$
$641$	36.7386	1.45109	0.725544	+	0.688175i	$0.241588\pi$
$642$	0	0
$643$	2.93087	0.115582	0.0577911	−	0.998329i	$-0.481594\pi$
$643$	2.93087	0.115582	0.0577911	+	0.998329i	$0.481594\pi$
$644$	0	0
$645$	−0.876894	−0.0345277
$646$	0	0
$647$	−36.6847	−1.44222	−0.721111	−	0.692819i	$-0.756368\pi$
$647$	−36.6847	−1.44222	−0.721111	+	0.692819i	$0.756368\pi$
$648$	0	0
$649$	34.7386	1.36361
$650$	0	0
$651$	12.4924	0.489617
$652$	0	0
$653$	38.9848	1.52559	0.762797	−	0.646638i	$-0.223825\pi$
$653$	38.9848	1.52559	0.762797	+	0.646638i	$0.223825\pi$
$654$	0	0
$655$	−8.00000	−0.312586
$656$	0	0
$657$	8.24621	0.321715
$658$	0	0
$659$	−24.9848	−0.973271	−0.486636	−	0.873605i	$-0.661776\pi$
$659$	−24.9848	−0.973271	−0.486636	+	0.873605i	$0.661776\pi$
$660$	0	0
$661$	−44.7386	−1.74013	−0.870066	−	0.492936i	$-0.835924\pi$
$661$	−44.7386	−1.74013	−0.870066	+	0.492936i	$0.835924\pi$
$662$	0	0
$663$	−6.68466	−0.259611
$664$	0	0
$665$	−4.87689	−0.189118
$666$	0	0
$667$	5.26137	0.203721
$668$	0	0
$669$	−2.24621	−0.0868435
$670$	0	0
$671$	−8.68466	−0.335268
$672$	0	0
$673$	6.87689	0.265085	0.132542	−	0.991177i	$-0.457686\pi$
$673$	6.87689	0.265085	0.132542	+	0.991177i	$0.457686\pi$
$674$	0	0
$675$	1.00000	0.0384900
$676$	0	0
$677$	−37.4233	−1.43829	−0.719147	−	0.694858i	$-0.755467\pi$
$677$	−37.4233	−1.43829	−0.719147	+	0.694858i	$0.755467\pi$
$678$	0	0
$679$	22.9309	0.880006
$680$	0	0
$681$	0.876894	0.0336027
$682$	0	0
$683$	−4.38447	−0.167767	−0.0838836	−	0.996476i	$-0.526732\pi$
$683$	−4.38447	−0.167767	−0.0838836	+	0.996476i	$0.526732\pi$
$684$	0	0
$685$	−14.0000	−0.534913
$686$	0	0
$687$	29.6155	1.12990
$688$	0	0
$689$	9.80776	0.373646
$690$	0	0
$691$	29.8617	1.13599	0.567997	−	0.823031i	$-0.307718\pi$
$691$	29.8617	1.13599	0.567997	+	0.823031i	$0.307718\pi$
$692$	0	0
$693$	3.80776	0.144645
$694$	0	0
$695$	−19.8078	−0.751351
$696$	0	0
$697$	−50.5464	−1.91458
$698$	0	0
$699$	−19.5616	−0.739886
$700$	0	0
$701$	10.8769	0.410815	0.205407	−	0.978677i	$-0.434148\pi$
$701$	10.8769	0.410815	0.205407	+	0.978677i	$0.434148\pi$
$702$	0	0
$703$	−23.6155	−0.890677
$704$	0	0
$705$	−10.2462	−0.385895
$706$	0	0
$707$	−4.49242	−0.168955
$708$	0	0
$709$	−39.8617	−1.49704	−0.748520	−	0.663113i	$-0.769235\pi$
$709$	−39.8617	−1.49704	−0.748520	+	0.663113i	$0.769235\pi$
$710$	0	0
$711$	−8.68466	−0.325700
$712$	0	0
$713$	−37.4773	−1.40353
$714$	0	0
$715$	−2.43845	−0.0911928
$716$	0	0
$717$	3.80776	0.142204
$718$	0	0
$719$	44.4924	1.65929	0.829644	−	0.558293i	$-0.188544\pi$
$719$	44.4924	1.65929	0.829644	+	0.558293i	$0.188544\pi$
$720$	0	0
$721$	−3.50758	−0.130629
$722$	0	0
$723$	−18.8769	−0.702039
$724$	0	0
$725$	1.12311	0.0417111
$726$	0	0
$727$	5.36932	0.199137	0.0995685	−	0.995031i	$-0.468254\pi$
$727$	5.36932	0.199137	0.0995685	+	0.995031i	$0.468254\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	5.86174	0.216804
$732$	0	0
$733$	16.4384	0.607168	0.303584	−	0.952805i	$-0.401817\pi$
$733$	16.4384	0.607168	0.303584	+	0.952805i	$0.401817\pi$
$734$	0	0
$735$	−4.56155	−0.168255
$736$	0	0
$737$	−5.47727	−0.201758
$738$	0	0
$739$	−41.3693	−1.52179	−0.760897	−	0.648872i	$-0.775241\pi$
$739$	−41.3693	−1.52179	−0.760897	+	0.648872i	$0.775241\pi$
$740$	0	0
$741$	3.12311	0.114730
$742$	0	0
$743$	40.1080	1.47142	0.735709	−	0.677298i	$-0.236849\pi$
$743$	40.1080	1.47142	0.735709	+	0.677298i	$0.236849\pi$
$744$	0	0
$745$	2.19224	0.0803173
$746$	0	0
$747$	4.00000	0.146352
$748$	0	0
$749$	12.1922	0.445495
$750$	0	0
$751$	−52.7926	−1.92643	−0.963215	−	0.268733i	$-0.913395\pi$
$751$	−52.7926	−1.92643	−0.963215	+	0.268733i	$0.913395\pi$
$752$	0	0
$753$	−30.2462	−1.10223
$754$	0	0
$755$	−1.75379	−0.0638269
$756$	0	0
$757$	1.12311	0.0408200	0.0204100	−	0.999792i	$-0.493503\pi$
$757$	1.12311	0.0408200	0.0204100	+	0.999792i	$0.493503\pi$
$758$	0	0
$759$	−11.4233	−0.414639
$760$	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
$763$	10.7386	0.388765
$764$	0	0
$765$	−6.68466	−0.241684
$766$	0	0
$767$	−14.2462	−0.514401
$768$	0	0
$769$	20.7386	0.747854	0.373927	−	0.927458i	$-0.378011\pi$
$769$	20.7386	0.747854	0.373927	+	0.927458i	$0.378011\pi$
$770$	0	0
$771$	14.4924	0.521932
$772$	0	0
$773$	−23.8617	−0.858247	−0.429124	−	0.903246i	$-0.641178\pi$
$773$	−23.8617	−0.858247	−0.429124	+	0.903246i	$0.641178\pi$
$774$	0	0
$775$	−8.00000	−0.287368
$776$	0	0
$777$	11.8078	0.423601
$778$	0	0
$779$	23.6155	0.846114
$780$	0	0
$781$	1.66950	0.0597395
$782$	0	0
$783$	1.12311	0.0401365
$784$	0	0
$785$	−2.00000	−0.0713831
$786$	0	0
$787$	−35.2311	−1.25585	−0.627926	−	0.778273i	$-0.716096\pi$
$787$	−35.2311	−1.25585	−0.627926	+	0.778273i	$0.716096\pi$
$788$	0	0
$789$	−22.7386	−0.809517
$790$	0	0
$791$	−12.8769	−0.457850
$792$	0	0
$793$	3.56155	0.126474
$794$	0	0
$795$	9.80776	0.347846
$796$	0	0
$797$	22.6847	0.803532	0.401766	−	0.915742i	$-0.368397\pi$
$797$	22.6847	0.803532	0.401766	+	0.915742i	$0.368397\pi$
$798$	0	0
$799$	68.4924	2.42309
$800$	0	0
$801$	−9.80776	−0.346540
$802$	0	0
$803$	−20.1080	−0.709594
$804$	0	0
$805$	−7.31534	−0.257832
$806$	0	0
$807$	−5.12311	−0.180342
$808$	0	0
$809$	−2.49242	−0.0876289	−0.0438145	−	0.999040i	$-0.513951\pi$
$809$	−2.49242	−0.0876289	−0.0438145	+	0.999040i	$0.513951\pi$
$810$	0	0
$811$	−33.3693	−1.17176	−0.585878	−	0.810400i	$-0.699250\pi$
$811$	−33.3693	−1.17176	−0.585878	+	0.810400i	$0.699250\pi$
$812$	0	0
$813$	19.1231	0.670677
$814$	0	0
$815$	−15.8078	−0.553722
$816$	0	0
$817$	−2.73863	−0.0958127
$818$	0	0
$819$	−1.56155	−0.0545651
$820$	0	0
$821$	48.0540	1.67710	0.838548	−	0.544828i	$-0.183405\pi$
$821$	48.0540	1.67710	0.838548	+	0.544828i	$0.183405\pi$
$822$	0	0
$823$	24.4924	0.853752	0.426876	−	0.904310i	$-0.359614\pi$
$823$	24.4924	0.853752	0.426876	+	0.904310i	$0.359614\pi$
$824$	0	0
$825$	−2.43845	−0.0848958
$826$	0	0
$827$	−38.7386	−1.34707	−0.673537	−	0.739153i	$-0.735226\pi$
$827$	−38.7386	−1.34707	−0.673537	+	0.739153i	$0.735226\pi$
$828$	0	0
$829$	−0.246211	−0.00855127	−0.00427564	−	0.999991i	$-0.501361\pi$
$829$	−0.246211	−0.00855127	−0.00427564	+	0.999991i	$0.501361\pi$
$830$	0	0
$831$	10.8769	0.377315
$832$	0	0
$833$	30.4924	1.05650
$834$	0	0
$835$	−4.00000	−0.138426
$836$	0	0
$837$	−8.00000	−0.276520
$838$	0	0
$839$	29.5616	1.02058	0.510289	−	0.860003i	$-0.329538\pi$
$839$	29.5616	1.02058	0.510289	+	0.860003i	$0.329538\pi$
$840$	0	0
$841$	−27.7386	−0.956505
$842$	0	0
$843$	−2.49242	−0.0858436
$844$	0	0
$845$	1.00000	0.0344010
$846$	0	0
$847$	7.89205	0.271174
$848$	0	0
$849$	−2.24621	−0.0770898
$850$	0	0
$851$	−35.4233	−1.21429
$852$	0	0
$853$	25.8078	0.883641	0.441821	−	0.897103i	$-0.354333\pi$
$853$	25.8078	0.883641	0.441821	+	0.897103i	$0.354333\pi$
$854$	0	0
$855$	3.12311	0.106808
$856$	0	0
$857$	18.6847	0.638256	0.319128	−	0.947712i	$-0.396610\pi$
$857$	18.6847	0.638256	0.319128	+	0.947712i	$0.396610\pi$
$858$	0	0
$859$	30.9309	1.05535	0.527674	−	0.849447i	$-0.323064\pi$
$859$	30.9309	1.05535	0.527674	+	0.849447i	$0.323064\pi$
$860$	0	0
$861$	−11.8078	−0.402408
$862$	0	0
$863$	27.6155	0.940044	0.470022	−	0.882655i	$-0.344246\pi$
$863$	27.6155	0.940044	0.470022	+	0.882655i	$0.344246\pi$
$864$	0	0
$865$	7.75379	0.263637
$866$	0	0
$867$	27.6847	0.940220
$868$	0	0
$869$	21.1771	0.718383
$870$	0	0
$871$	2.24621	0.0761100
$872$	0	0
$873$	−14.6847	−0.497000
$874$	0	0
$875$	−1.56155	−0.0527901
$876$	0	0
$877$	−52.7386	−1.78086	−0.890429	−	0.455123i	$-0.849595\pi$
$877$	−52.7386	−1.78086	−0.890429	+	0.455123i	$0.849595\pi$
$878$	0	0
$879$	23.3693	0.788227
$880$	0	0
$881$	−27.8617	−0.938686	−0.469343	−	0.883016i	$-0.655509\pi$
$881$	−27.8617	−0.938686	−0.469343	+	0.883016i	$0.655509\pi$
$882$	0	0
$883$	−34.2462	−1.15248	−0.576238	−	0.817282i	$-0.695480\pi$
$883$	−34.2462	−1.15248	−0.576238	+	0.817282i	$0.695480\pi$
$884$	0	0
$885$	−14.2462	−0.478881
$886$	0	0
$887$	46.0540	1.54634	0.773171	−	0.634198i	$-0.218670\pi$
$887$	46.0540	1.54634	0.773171	+	0.634198i	$0.218670\pi$
$888$	0	0
$889$	3.50758	0.117640
$890$	0	0
$891$	−2.43845	−0.0816911
$892$	0	0
$893$	−32.0000	−1.07084
$894$	0	0
$895$	−4.87689	−0.163017
$896$	0	0
$897$	4.68466	0.156416
$898$	0	0
$899$	−8.98485	−0.299661
$900$	0	0
$901$	−65.5616	−2.18417
$902$	0	0
$903$	1.36932	0.0455680
$904$	0	0
$905$	5.31534	0.176688
$906$	0	0
$907$	29.3693	0.975192	0.487596	−	0.873069i	$-0.337874\pi$
$907$	29.3693	0.975192	0.487596	+	0.873069i	$0.337874\pi$
$908$	0	0
$909$	2.87689	0.0954206
$910$	0	0
$911$	−28.1080	−0.931258	−0.465629	−	0.884980i	$-0.654172\pi$
$911$	−28.1080	−0.931258	−0.465629	+	0.884980i	$0.654172\pi$
$912$	0	0
$913$	−9.75379	−0.322803
$914$	0	0
$915$	3.56155	0.117741
$916$	0	0
$917$	12.4924	0.412536
$918$	0	0
$919$	−27.4233	−0.904611	−0.452305	−	0.891863i	$-0.649398\pi$
$919$	−27.4233	−0.904611	−0.452305	+	0.891863i	$0.649398\pi$
$920$	0	0
$921$	4.68466	0.154365
$922$	0	0
$923$	−0.684658	−0.0225358
$924$	0	0
$925$	−7.56155	−0.248622
$926$	0	0
$927$	2.24621	0.0737753
$928$	0	0
$929$	−52.5464	−1.72399	−0.861996	−	0.506916i	$-0.830786\pi$
$929$	−52.5464	−1.72399	−0.861996	+	0.506916i	$0.830786\pi$
$930$	0	0
$931$	−14.2462	−0.466901
$932$	0	0
$933$	−9.36932	−0.306738
$934$	0	0
$935$	16.3002	0.533073
$936$	0	0
$937$	21.1231	0.690062	0.345031	−	0.938591i	$-0.387868\pi$
$937$	21.1231	0.690062	0.345031	+	0.938591i	$0.387868\pi$
$938$	0	0
$939$	26.0000	0.848478
$940$	0	0
$941$	32.4384	1.05746	0.528732	−	0.848789i	$-0.322668\pi$
$941$	32.4384	1.05746	0.528732	+	0.848789i	$0.322668\pi$
$942$	0	0
$943$	35.4233	1.15354
$944$	0	0
$945$	−1.56155	−0.0507973
$946$	0	0
$947$	36.9848	1.20185	0.600923	−	0.799307i	$-0.294800\pi$
$947$	36.9848	1.20185	0.600923	+	0.799307i	$0.294800\pi$
$948$	0	0
$949$	8.24621	0.267683
$950$	0	0
$951$	7.75379	0.251434
$952$	0	0
$953$	36.0540	1.16790	0.583951	−	0.811789i	$-0.301506\pi$
$953$	36.0540	1.16790	0.583951	+	0.811789i	$0.301506\pi$
$954$	0	0
$955$	25.3693	0.820932
$956$	0	0
$957$	−2.73863	−0.0885275
$958$	0	0
$959$	21.8617	0.705952
$960$	0	0
$961$	33.0000	1.06452
$962$	0	0
$963$	−7.80776	−0.251602
$964$	0	0
$965$	2.68466	0.0864222
$966$	0	0
$967$	−29.7538	−0.956817	−0.478409	−	0.878137i	$-0.658786\pi$
$967$	−29.7538	−0.956817	−0.478409	+	0.878137i	$0.658786\pi$
$968$	0	0
$969$	−20.8769	−0.670662
$970$	0	0
$971$	3.12311	0.100225	0.0501126	−	0.998744i	$-0.484042\pi$
$971$	3.12311	0.100225	0.0501126	+	0.998744i	$0.484042\pi$
$972$	0	0
$973$	30.9309	0.991598
$974$	0	0
$975$	1.00000	0.0320256
$976$	0	0
$977$	7.26137	0.232312	0.116156	−	0.993231i	$-0.462943\pi$
$977$	7.26137	0.232312	0.116156	+	0.993231i	$0.462943\pi$
$978$	0	0
$979$	23.9157	0.764350
$980$	0	0
$981$	−6.87689	−0.219562
$982$	0	0
$983$	26.2462	0.837124	0.418562	−	0.908188i	$-0.362534\pi$
$983$	26.2462	0.837124	0.418562	+	0.908188i	$0.362534\pi$
$984$	0	0
$985$	6.00000	0.191176
$986$	0	0
$987$	16.0000	0.509286
$988$	0	0
$989$	−4.10795	−0.130625
$990$	0	0
$991$	6.93087	0.220166	0.110083	−	0.993922i	$-0.464888\pi$
$991$	6.93087	0.220166	0.110083	+	0.993922i	$0.464888\pi$
$992$	0	0
$993$	1.36932	0.0434540
$994$	0	0
$995$	0	0
$996$	0	0
$997$	26.1080	0.826847	0.413424	−	0.910539i	$-0.364333\pi$
$997$	26.1080	0.826847	0.413424	+	0.910539i	$0.364333\pi$
$998$	0	0
$999$	−7.56155	−0.239237

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6240.2.a.bm.1.2	✓	2	4.3	odd	2
6240.2.a.bu.1.1	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +1.00000 q^{5} -1.56155 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +1.00000 q^{5} -1.56155 q^{7} +1.00000 q^{9} -2.43845 q^{11} +1.00000 q^{13} +1.00000 q^{15} -6.68466 q^{17} +3.12311 q^{19} -1.56155 q^{21} +4.68466 q^{23} +1.00000 q^{25} +1.00000 q^{27} +1.12311 q^{29} -8.00000 q^{31} -2.43845 q^{33} -1.56155 q^{35} -7.56155 q^{37} +1.00000 q^{39} +7.56155 q^{41} -0.876894 q^{43} +1.00000 q^{45} -10.2462 q^{47} -4.56155 q^{49} -6.68466 q^{51} +9.80776 q^{53} -2.43845 q^{55} +3.12311 q^{57} -14.2462 q^{59} +3.56155 q^{61} -1.56155 q^{63} +1.00000 q^{65} +2.24621 q^{67} +4.68466 q^{69} -0.684658 q^{71} +8.24621 q^{73} +1.00000 q^{75} +3.80776 q^{77} -8.68466 q^{79} +1.00000 q^{81} +4.00000 q^{83} -6.68466 q^{85} +1.12311 q^{87} -9.80776 q^{89} -1.56155 q^{91} -8.00000 q^{93} +3.12311 q^{95} -14.6847 q^{97} -2.43845 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} + 2 q^{5} + q^{7} + 2 q^{9}+O(q^{10}) \) 2 * q + 2 * q^3 + 2 * q^5 + q^7 + 2 * q^9	\( 2 q + 2 q^{3} + 2 q^{5} + q^{7} + 2 q^{9} - 9 q^{11} + 2 q^{13} + 2 q^{15} - q^{17} - 2 q^{19} + q^{21} - 3 q^{23} + 2 q^{25} + 2 q^{27} - 6 q^{29} - 16 q^{31} - 9 q^{33} + q^{35} - 11 q^{37} + 2 q^{39} + 11 q^{41} - 10 q^{43} + 2 q^{45} - 4 q^{47} - 5 q^{49} - q^{51} - q^{53} - 9 q^{55} - 2 q^{57} - 12 q^{59} + 3 q^{61} + q^{63} + 2 q^{65} - 12 q^{67} - 3 q^{69} + 11 q^{71} + 2 q^{75} - 13 q^{77} - 5 q^{79} + 2 q^{81} + 8 q^{83} - q^{85} - 6 q^{87} + q^{89} + q^{91} - 16 q^{93} - 2 q^{95} - 17 q^{97} - 9 q^{99}+O(q^{100}) \) 2 * q + 2 * q^3 + 2 * q^5 + q^7 + 2 * q^9 - 9 * q^11 + 2 * q^13 + 2 * q^15 - q^17 - 2 * q^19 + q^21 - 3 * q^23 + 2 * q^25 + 2 * q^27 - 6 * q^29 - 16 * q^31 - 9 * q^33 + q^35 - 11 * q^37 + 2 * q^39 + 11 * q^41 - 10 * q^43 + 2 * q^45 - 4 * q^47 - 5 * q^49 - q^51 - q^53 - 9 * q^55 - 2 * q^57 - 12 * q^59 + 3 * q^61 + q^63 + 2 * q^65 - 12 * q^67 - 3 * q^69 + 11 * q^71 + 2 * q^75 - 13 * q^77 - 5 * q^79 + 2 * q^81 + 8 * q^83 - q^85 - 6 * q^87 + q^89 + q^91 - 16 * q^93 - 2 * q^95 - 17 * q^97 - 9 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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