LMFDB - Embedded newform 6240.2.a.ce.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:	$4$
Coefficient field:	4.4.15317.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 4x^{2} + 5x + 2 $ x^4 - 2x^3 - 4x^2 + 5*x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.69353$ 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)

$f(q)$	$=$	$q-1.00000 q^{3} +1.00000 q^{5} -2.74252 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} +1.00000 q^{5} -2.74252 q^{7} +1.00000 q^{9} +5.76763 q^{11} +1.00000 q^{13} -1.00000 q^{15} -0.742521 q^{17} -2.26394 q^{19} +2.74252 q^{21} -2.74252 q^{23} +1.00000 q^{25} -1.00000 q^{27} -0.263945 q^{29} -5.76763 q^{33} -2.74252 q^{35} +7.76763 q^{37} -1.00000 q^{39} -3.76763 q^{41} +5.28906 q^{43} +1.00000 q^{45} +0.521423 q^{49} +0.742521 q^{51} +13.0567 q^{53} +5.76763 q^{55} +2.26394 q^{57} -10.0502 q^{59} +13.2527 q^{61} -2.74252 q^{63} +1.00000 q^{65} -10.2462 q^{67} +2.74252 q^{69} +16.5417 q^{71} +9.02511 q^{73} -1.00000 q^{75} -15.8179 q^{77} -1.00647 q^{79} +1.00000 q^{81} -6.24621 q^{83} -0.742521 q^{85} +0.263945 q^{87} +0.993534 q^{89} -2.74252 q^{91} -2.26394 q^{95} -5.27041 q^{97} +5.76763 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{3} + 4 q^{5} - 3 q^{7} + 4 q^{9}+O(q^{10}) $ 4 * q - 4 * q^3 + 4 * q^5 - 3 * q^7 + 4 * q^9	$ 4 q - 4 q^{3} + 4 q^{5} - 3 q^{7} + 4 q^{9} - 3 q^{11} + 4 q^{13} - 4 q^{15} + 5 q^{17} - 8 q^{19} + 3 q^{21} - 3 q^{23} + 4 q^{25} - 4 q^{27} + 3 q^{33} - 3 q^{35} + 5 q^{37} - 4 q^{39} + 11 q^{41} + 2 q^{43} + 4 q^{45} + 9 q^{49} - 5 q^{51} + 7 q^{53} - 3 q^{55} + 8 q^{57} - 4 q^{59} + 11 q^{61} - 3 q^{63} + 4 q^{65} - 8 q^{67} + 3 q^{69} + 5 q^{71} + 18 q^{73} - 4 q^{75} - q^{77} + 5 q^{79} + 4 q^{81} + 8 q^{83} + 5 q^{85} + 13 q^{89} - 3 q^{91} - 8 q^{95} - 11 q^{97} - 3 q^{99}+O(q^{100}) $ 4 * q - 4 * q^3 + 4 * q^5 - 3 * q^7 + 4 * q^9 - 3 * q^11 + 4 * q^13 - 4 * q^15 + 5 * q^17 - 8 * q^19 + 3 * q^21 - 3 * q^23 + 4 * q^25 - 4 * q^27 + 3 * q^33 - 3 * q^35 + 5 * q^37 - 4 * q^39 + 11 * q^41 + 2 * q^43 + 4 * q^45 + 9 * q^49 - 5 * q^51 + 7 * q^53 - 3 * q^55 + 8 * q^57 - 4 * q^59 + 11 * q^61 - 3 * q^63 + 4 * q^65 - 8 * q^67 + 3 * q^69 + 5 * q^71 + 18 * q^73 - 4 * q^75 - q^77 + 5 * q^79 + 4 * q^81 + 8 * q^83 + 5 * q^85 + 13 * q^89 - 3 * q^91 - 8 * q^95 - 11 * q^97 - 3 * 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
$7$	−2.74252	−1.03658	−0.518288	−	0.855206i	$-0.673430\pi$
$7$	−2.74252	−1.03658	−0.518288	+	0.855206i	$0.673430\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	5.76763	1.73901	0.869504	−	0.493927i	$-0.164439\pi$
$11$	5.76763	1.73901	0.869504	+	0.493927i	$0.164439\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	−1.00000	−0.258199
$16$	0	0
$17$	−0.742521	−0.180088	−0.0900439	−	0.995938i	$-0.528701\pi$
$17$	−0.742521	−0.180088	−0.0900439	+	0.995938i	$0.528701\pi$
$18$	0	0
$19$	−2.26394	−0.519385	−0.259692	−	0.965691i	$-0.583621\pi$
$19$	−2.26394	−0.519385	−0.259692	+	0.965691i	$0.583621\pi$
$20$	0	0
$21$	2.74252	0.598467
$22$	0	0
$23$	−2.74252	−0.571855	−0.285928	−	0.958251i	$-0.592302\pi$
$23$	−2.74252	−0.571855	−0.285928	+	0.958251i	$0.592302\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−0.263945	−0.0490133	−0.0245067	−	0.999700i	$-0.507801\pi$
$29$	−0.263945	−0.0490133	−0.0245067	+	0.999700i	$0.507801\pi$
$30$	0	0
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	0	0
$33$	−5.76763	−1.00402
$34$	0	0
$35$	−2.74252	−0.463571
$36$	0	0
$37$	7.76763	1.27699	0.638496	−	0.769625i	$-0.279557\pi$
$37$	7.76763	1.27699	0.638496	+	0.769625i	$0.279557\pi$
$38$	0	0
$39$	−1.00000	−0.160128
$40$	0	0
$41$	−3.76763	−0.588406	−0.294203	−	0.955743i	$-0.595054\pi$
$41$	−3.76763	−0.588406	−0.294203	+	0.955743i	$0.595054\pi$
$42$	0	0
$43$	5.28906	0.806574	0.403287	−	0.915074i	$-0.367868\pi$
$43$	5.28906	0.806574	0.403287	+	0.915074i	$0.367868\pi$
$44$	0	0
$45$	1.00000	0.149071
$46$	0	0
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	0	0
$49$	0.521423	0.0744891
$50$	0	0
$51$	0.742521	0.103974
$52$	0	0
$53$	13.0567	1.79347	0.896737	−	0.442563i	$-0.145931\pi$
$53$	13.0567	1.79347	0.896737	+	0.442563i	$0.145931\pi$
$54$	0	0
$55$	5.76763	0.777708
$56$	0	0
$57$	2.26394	0.299867
$58$	0	0
$59$	−10.0502	−1.30843	−0.654214	−	0.756309i	$-0.727000\pi$
$59$	−10.0502	−1.30843	−0.654214	+	0.756309i	$0.727000\pi$
$60$	0	0
$61$	13.2527	1.69683	0.848416	−	0.529330i	$-0.177557\pi$
$61$	13.2527	1.69683	0.848416	+	0.529330i	$0.177557\pi$
$62$	0	0
$63$	−2.74252	−0.345525
$64$	0	0
$65$	1.00000	0.124035
$66$	0	0
$67$	−10.2462	−1.25177	−0.625887	−	0.779914i	$-0.715263\pi$
$67$	−10.2462	−1.25177	−0.625887	+	0.779914i	$0.715263\pi$
$68$	0	0
$69$	2.74252	0.330161
$70$	0	0
$71$	16.5417	1.96314	0.981571	−	0.191096i	$-0.0612041\pi$
$71$	16.5417	1.96314	0.981571	+	0.191096i	$0.0612041\pi$
$72$	0	0
$73$	9.02511	1.05631	0.528155	−	0.849148i	$-0.322884\pi$
$73$	9.02511	1.05631	0.528155	+	0.849148i	$0.322884\pi$
$74$	0	0
$75$	−1.00000	−0.115470
$76$	0	0
$77$	−15.8179	−1.80261
$78$	0	0
$79$	−1.00647	−0.113236	−0.0566181	−	0.998396i	$-0.518032\pi$
$79$	−1.00647	−0.113236	−0.0566181	+	0.998396i	$0.518032\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−6.24621	−0.685611	−0.342805	−	0.939406i	$-0.611377\pi$
$83$	−6.24621	−0.685611	−0.342805	+	0.939406i	$0.611377\pi$
$84$	0	0
$85$	−0.742521	−0.0805377
$86$	0	0
$87$	0.263945	0.0282979
$88$	0	0
$89$	0.993534	0.105314	0.0526572	−	0.998613i	$-0.483231\pi$
$89$	0.993534	0.105314	0.0526572	+	0.998613i	$0.483231\pi$
$90$	0	0
$91$	−2.74252	−0.287494
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−2.26394	−0.232276
$96$	0	0
$97$	−5.27041	−0.535129	−0.267565	−	0.963540i	$-0.586219\pi$
$97$	−5.27041	−0.535129	−0.267565	+	0.963540i	$0.586219\pi$
$98$	0	0
$99$	5.76763	0.579669
$100$	0	0
$101$	−17.2843	−1.71985	−0.859924	−	0.510422i	$-0.829489\pi$
$101$	−17.2843	−1.71985	−0.859924	+	0.510422i	$0.829489\pi$
$102$	0	0
$103$	−16.8243	−1.65775	−0.828875	−	0.559434i	$-0.811019\pi$
$103$	−16.8243	−1.65775	−0.828875	+	0.559434i	$0.811019\pi$
$104$	0	0
$105$	2.74252	0.267643
$106$	0	0
$107$	−10.2955	−0.995306	−0.497653	−	0.867376i	$-0.665805\pi$
$107$	−10.2955	−0.995306	−0.497653	+	0.867376i	$0.665805\pi$
$108$	0	0
$109$	10.5102	1.00669	0.503345	−	0.864085i	$-0.332102\pi$
$109$	10.5102	1.00669	0.503345	+	0.864085i	$0.332102\pi$
$110$	0	0
$111$	−7.76763	−0.737271
$112$	0	0
$113$	−12.5231	−1.17807	−0.589037	−	0.808106i	$-0.700493\pi$
$113$	−12.5231	−1.17807	−0.589037	+	0.808106i	$0.700493\pi$
$114$	0	0
$115$	−2.74252	−0.255741
$116$	0	0
$117$	1.00000	0.0924500
$118$	0	0
$119$	2.03638	0.186675
$120$	0	0
$121$	22.2656	2.02415
$122$	0	0
$123$	3.76763	0.339716
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−3.27613	−0.290709	−0.145355	−	0.989380i	$-0.546432\pi$
$127$	−3.27613	−0.290709	−0.145355	+	0.989380i	$0.546432\pi$
$128$	0	0
$129$	−5.28906	−0.465676
$130$	0	0
$131$	−3.02511	−0.264305	−0.132153	−	0.991229i	$-0.542189\pi$
$131$	−3.02511	−0.264305	−0.132153	+	0.991229i	$0.542189\pi$
$132$	0	0
$133$	6.20892	0.538381
$134$	0	0
$135$	−1.00000	−0.0860663
$136$	0	0
$137$	12.7741	1.09137	0.545683	−	0.837992i	$-0.316270\pi$
$137$	12.7741	1.09137	0.545683	+	0.837992i	$0.316270\pi$
$138$	0	0
$139$	7.78057	0.659939	0.329970	−	0.943992i	$-0.392962\pi$
$139$	7.78057	0.659939	0.329970	+	0.943992i	$0.392962\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	5.76763	0.482314
$144$	0	0
$145$	−0.263945	−0.0219194
$146$	0	0
$147$	−0.521423	−0.0430063
$148$	0	0
$149$	18.3458	1.50294	0.751471	−	0.659766i	$-0.229345\pi$
$149$	18.3458	1.50294	0.751471	+	0.659766i	$0.229345\pi$
$150$	0	0
$151$	10.9701	0.892733	0.446366	−	0.894850i	$-0.352718\pi$
$151$	10.9701	0.892733	0.446366	+	0.894850i	$0.352718\pi$
$152$	0	0
$153$	−0.742521	−0.0600293
$154$	0	0
$155$	0	0
$156$	0	0
$157$	2.19598	0.175259	0.0876293	−	0.996153i	$-0.472071\pi$
$157$	2.19598	0.175259	0.0876293	+	0.996153i	$0.472071\pi$
$158$	0	0
$159$	−13.0567	−1.03546
$160$	0	0
$161$	7.52142	0.592771
$162$	0	0
$163$	−10.4915	−0.821758	−0.410879	−	0.911690i	$-0.634778\pi$
$163$	−10.4915	−0.821758	−0.410879	+	0.911690i	$0.634778\pi$
$164$	0	0
$165$	−5.76763	−0.449010
$166$	0	0
$167$	−16.6283	−1.28674	−0.643370	−	0.765555i	$-0.722464\pi$
$167$	−16.6283	−1.28674	−0.643370	+	0.765555i	$0.722464\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−2.26394	−0.173128
$172$	0	0
$173$	−2.19598	−0.166958	−0.0834788	−	0.996510i	$-0.526603\pi$
$173$	−2.19598	−0.166958	−0.0834788	+	0.996510i	$0.526603\pi$
$174$	0	0
$175$	−2.74252	−0.207315
$176$	0	0
$177$	10.0502	0.755421
$178$	0	0
$179$	1.01218	0.0756540	0.0378270	−	0.999284i	$-0.487956\pi$
$179$	1.01218	0.0756540	0.0378270	+	0.999284i	$0.487956\pi$
$180$	0	0
$181$	16.3328	1.21401	0.607004	−	0.794698i	$-0.292371\pi$
$181$	16.3328	1.21401	0.607004	+	0.794698i	$0.292371\pi$
$182$	0	0
$183$	−13.2527	−0.979666
$184$	0	0
$185$	7.76763	0.571088
$186$	0	0
$187$	−4.28259	−0.313174
$188$	0	0
$189$	2.74252	0.199489
$190$	0	0
$191$	22.5054	1.62843	0.814215	−	0.580563i	$-0.197168\pi$
$191$	22.5054	1.62843	0.814215	+	0.580563i	$0.197168\pi$
$192$	0	0
$193$	−12.2778	−0.883775	−0.441887	−	0.897071i	$-0.645691\pi$
$193$	−12.2778	−0.883775	−0.441887	+	0.897071i	$0.645691\pi$
$194$	0	0
$195$	−1.00000	−0.0716115
$196$	0	0
$197$	1.80402	0.128531	0.0642654	−	0.997933i	$-0.479530\pi$
$197$	1.80402	0.128531	0.0642654	+	0.997933i	$0.479530\pi$
$198$	0	0
$199$	25.2163	1.78754	0.893768	−	0.448530i	$-0.148052\pi$
$199$	25.2163	1.78754	0.893768	+	0.448530i	$0.148052\pi$
$200$	0	0
$201$	10.2462	0.722712
$202$	0	0
$203$	0.723874	0.0508060
$204$	0	0
$205$	−3.76763	−0.263143
$206$	0	0
$207$	−2.74252	−0.190618
$208$	0	0
$209$	−13.0576	−0.903214
$210$	0	0
$211$	19.0705	1.31287	0.656435	−	0.754383i	$-0.272064\pi$
$211$	19.0705	1.31287	0.656435	+	0.754383i	$0.272064\pi$
$212$	0	0
$213$	−16.5417	−1.13342
$214$	0	0
$215$	5.28906	0.360711
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−9.02511	−0.609861
$220$	0	0
$221$	−0.742521	−0.0499474
$222$	0	0
$223$	17.6032	1.17880	0.589400	−	0.807842i	$-0.299364\pi$
$223$	17.6032	1.17880	0.589400	+	0.807842i	$0.299364\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	6.01293	0.399092	0.199546	−	0.979888i	$-0.436053\pi$
$227$	6.01293	0.399092	0.199546	+	0.979888i	$0.436053\pi$
$228$	0	0
$229$	−15.5305	−1.02628	−0.513141	−	0.858304i	$-0.671518\pi$
$229$	−15.5305	−1.02628	−0.513141	+	0.858304i	$0.671518\pi$
$230$	0	0
$231$	15.8179	1.04074
$232$	0	0
$233$	21.7628	1.42573	0.712865	−	0.701301i	$-0.247397\pi$
$233$	21.7628	1.42573	0.712865	+	0.701301i	$0.247397\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	1.00647	0.0653770
$238$	0	0
$239$	22.2955	1.44218	0.721089	−	0.692843i	$-0.243642\pi$
$239$	22.2955	1.44218	0.721089	+	0.692843i	$0.243642\pi$
$240$	0	0
$241$	10.9572	0.705812	0.352906	−	0.935659i	$-0.385193\pi$
$241$	10.9572	0.705812	0.352906	+	0.935659i	$0.385193\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	0.521423	0.0333125
$246$	0	0
$247$	−2.26394	−0.144051
$248$	0	0
$249$	6.24621	0.395838
$250$	0	0
$251$	13.9952	0.883369	0.441685	−	0.897170i	$-0.354381\pi$
$251$	13.9952	0.883369	0.441685	+	0.897170i	$0.354381\pi$
$252$	0	0
$253$	−15.8179	−0.994460
$254$	0	0
$255$	0.742521	0.0464985
$256$	0	0
$257$	19.6032	1.22282	0.611408	−	0.791316i	$-0.290603\pi$
$257$	19.6032	1.22282	0.611408	+	0.791316i	$0.290603\pi$
$258$	0	0
$259$	−21.3029	−1.32370
$260$	0	0
$261$	−0.263945	−0.0163378
$262$	0	0
$263$	4.05503	0.250044	0.125022	−	0.992154i	$-0.460100\pi$
$263$	4.05503	0.250044	0.125022	+	0.992154i	$0.460100\pi$
$264$	0	0
$265$	13.0567	0.802066
$266$	0	0
$267$	−0.993534	−0.0608033
$268$	0	0
$269$	−3.34409	−0.203893	−0.101946	−	0.994790i	$-0.532507\pi$
$269$	−3.34409	−0.203893	−0.101946	+	0.994790i	$0.532507\pi$
$270$	0	0
$271$	−0.565184	−0.0343325	−0.0171662	−	0.999853i	$-0.505464\pi$
$271$	−0.565184	−0.0343325	−0.0171662	+	0.999853i	$0.505464\pi$
$272$	0	0
$273$	2.74252	0.165985
$274$	0	0
$275$	5.76763	0.347801
$276$	0	0
$277$	−7.81695	−0.469675	−0.234837	−	0.972035i	$-0.575456\pi$
$277$	−7.81695	−0.469675	−0.234837	+	0.972035i	$0.575456\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	21.2761	1.26923	0.634614	−	0.772830i	$-0.281159\pi$
$281$	21.2761	1.26923	0.634614	+	0.772830i	$0.281159\pi$
$282$	0	0
$283$	−18.2721	−1.08616	−0.543081	−	0.839680i	$-0.682742\pi$
$283$	−18.2721	−1.08616	−0.543081	+	0.839680i	$0.682742\pi$
$284$	0	0
$285$	2.26394	0.134105
$286$	0	0
$287$	10.3328	0.609927
$288$	0	0
$289$	−16.4487	−0.967568
$290$	0	0
$291$	5.27041	0.308957
$292$	0	0
$293$	−7.81695	−0.456671	−0.228335	−	0.973583i	$-0.573328\pi$
$293$	−7.81695	−0.456671	−0.228335	+	0.973583i	$0.573328\pi$
$294$	0	0
$295$	−10.0502	−0.585147
$296$	0	0
$297$	−5.76763	−0.334672
$298$	0	0
$299$	−2.74252	−0.158604
$300$	0	0
$301$	−14.5054	−0.836075
$302$	0	0
$303$	17.2843	0.992955
$304$	0	0
$305$	13.2527	0.758846
$306$	0	0
$307$	9.49889	0.542130	0.271065	−	0.962561i	$-0.412624\pi$
$307$	9.49889	0.542130	0.271065	+	0.962561i	$0.412624\pi$
$308$	0	0
$309$	16.8243	0.957103
$310$	0	0
$311$	5.98707	0.339495	0.169748	−	0.985488i	$-0.445705\pi$
$311$	5.98707	0.339495	0.169748	+	0.985488i	$0.445705\pi$
$312$	0	0
$313$	16.4422	0.929368	0.464684	−	0.885477i	$-0.346168\pi$
$313$	16.4422	0.929368	0.464684	+	0.885477i	$0.346168\pi$
$314$	0	0
$315$	−2.74252	−0.154524
$316$	0	0
$317$	−12.5781	−0.706457	−0.353229	−	0.935537i	$-0.614916\pi$
$317$	−12.5781	−0.706457	−0.353229	+	0.935537i	$0.614916\pi$
$318$	0	0
$319$	−1.52234	−0.0852345
$320$	0	0
$321$	10.2955	0.574640
$322$	0	0
$323$	1.68103	0.0935349
$324$	0	0
$325$	1.00000	0.0554700
$326$	0	0
$327$	−10.5102	−0.581213
$328$	0	0
$329$	0	0
$330$	0	0
$331$	0.706141	0.0388130	0.0194065	−	0.999812i	$-0.493822\pi$
$331$	0.706141	0.0388130	0.0194065	+	0.999812i	$0.493822\pi$
$332$	0	0
$333$	7.76763	0.425664
$334$	0	0
$335$	−10.2462	−0.559810
$336$	0	0
$337$	13.5984	0.740754	0.370377	−	0.928882i	$-0.379229\pi$
$337$	13.5984	0.740754	0.370377	+	0.928882i	$0.379229\pi$
$338$	0	0
$339$	12.5231	0.680161
$340$	0	0
$341$	0	0
$342$	0	0
$343$	17.7676	0.959362
$344$	0	0
$345$	2.74252	0.147652
$346$	0	0
$347$	35.8810	1.92619	0.963097	−	0.269154i	$-0.0867442\pi$
$347$	35.8810	1.92619	0.963097	+	0.269154i	$0.0867442\pi$
$348$	0	0
$349$	19.0753	1.02108	0.510540	−	0.859854i	$-0.329446\pi$
$349$	19.0753	1.02108	0.510540	+	0.859854i	$0.329446\pi$
$350$	0	0
$351$	−1.00000	−0.0533761
$352$	0	0
$353$	11.8169	0.628953	0.314476	−	0.949265i	$-0.398171\pi$
$353$	11.8169	0.628953	0.314476	+	0.949265i	$0.398171\pi$
$354$	0	0
$355$	16.5417	0.877944
$356$	0	0
$357$	−2.03638	−0.107777
$358$	0	0
$359$	13.7538	0.725897	0.362949	−	0.931809i	$-0.381770\pi$
$359$	13.7538	0.725897	0.362949	+	0.931809i	$0.381770\pi$
$360$	0	0
$361$	−13.8746	−0.730240
$362$	0	0
$363$	−22.2656	−1.16864
$364$	0	0
$365$	9.02511	0.472396
$366$	0	0
$367$	6.81140	0.355552	0.177776	−	0.984071i	$-0.443110\pi$
$367$	6.81140	0.355552	0.177776	+	0.984071i	$0.443110\pi$
$368$	0	0
$369$	−3.76763	−0.196135
$370$	0	0
$371$	−35.8083	−1.85907
$372$	0	0
$373$	31.3522	1.62336	0.811678	−	0.584105i	$-0.198554\pi$
$373$	31.3522	1.62336	0.811678	+	0.584105i	$0.198554\pi$
$374$	0	0
$375$	−1.00000	−0.0516398
$376$	0	0
$377$	−0.263945	−0.0135938
$378$	0	0
$379$	−17.6633	−0.907302	−0.453651	−	0.891179i	$-0.649879\pi$
$379$	−17.6633	−0.907302	−0.453651	+	0.891179i	$0.649879\pi$
$380$	0	0
$381$	3.27613	0.167841
$382$	0	0
$383$	−6.54082	−0.334220	−0.167110	−	0.985938i	$-0.553444\pi$
$383$	−6.54082	−0.334220	−0.167110	+	0.985938i	$0.553444\pi$
$384$	0	0
$385$	−15.8179	−0.806153
$386$	0	0
$387$	5.28906	0.268858
$388$	0	0
$389$	−6.31417	−0.320141	−0.160071	−	0.987106i	$-0.551172\pi$
$389$	−6.31417	−0.320141	−0.160071	+	0.987106i	$0.551172\pi$
$390$	0	0
$391$	2.03638	0.102984
$392$	0	0
$393$	3.02511	0.152597
$394$	0	0
$395$	−1.00647	−0.0506408
$396$	0	0
$397$	−25.2172	−1.26562	−0.632808	−	0.774309i	$-0.718098\pi$
$397$	−25.2172	−1.26562	−0.632808	+	0.774309i	$0.718098\pi$
$398$	0	0
$399$	−6.20892	−0.310835
$400$	0	0
$401$	10.9199	0.545312	0.272656	−	0.962112i	$-0.412098\pi$
$401$	10.9199	0.545312	0.272656	+	0.962112i	$0.412098\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	1.00000	0.0496904
$406$	0	0
$407$	44.8009	2.22070
$408$	0	0
$409$	−13.0074	−0.643174	−0.321587	−	0.946880i	$-0.604216\pi$
$409$	−13.0074	−0.643174	−0.321587	+	0.946880i	$0.604216\pi$
$410$	0	0
$411$	−12.7741	−0.630100
$412$	0	0
$413$	27.5630	1.35628
$414$	0	0
$415$	−6.24621	−0.306614
$416$	0	0
$417$	−7.78057	−0.381016
$418$	0	0
$419$	38.0584	1.85927	0.929636	−	0.368479i	$-0.120121\pi$
$419$	38.0584	1.85927	0.929636	+	0.368479i	$0.120121\pi$
$420$	0	0
$421$	9.48984	0.462507	0.231253	−	0.972894i	$-0.425717\pi$
$421$	9.48984	0.462507	0.231253	+	0.972894i	$0.425717\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−0.742521	−0.0360176
$426$	0	0
$427$	−36.3458	−1.75889
$428$	0	0
$429$	−5.76763	−0.278464
$430$	0	0
$431$	−25.6487	−1.23545	−0.617726	−	0.786393i	$-0.711946\pi$
$431$	−25.6487	−1.23545	−0.617726	+	0.786393i	$0.711946\pi$
$432$	0	0
$433$	−12.0502	−0.579097	−0.289548	−	0.957163i	$-0.593505\pi$
$433$	−12.0502	−0.579097	−0.289548	+	0.957163i	$0.593505\pi$
$434$	0	0
$435$	0.263945	0.0126552
$436$	0	0
$437$	6.20892	0.297013
$438$	0	0
$439$	17.8049	0.849783	0.424891	−	0.905244i	$-0.360312\pi$
$439$	17.8049	0.849783	0.424891	+	0.905244i	$0.360312\pi$
$440$	0	0
$441$	0.521423	0.0248297
$442$	0	0
$443$	14.3328	0.680973	0.340486	−	0.940249i	$-0.389408\pi$
$443$	14.3328	0.680973	0.340486	+	0.940249i	$0.389408\pi$
$444$	0	0
$445$	0.993534	0.0470980
$446$	0	0
$447$	−18.3458	−0.867724
$448$	0	0
$449$	35.9913	1.69853	0.849267	−	0.527963i	$-0.177044\pi$
$449$	35.9913	1.69853	0.849267	+	0.527963i	$0.177044\pi$
$450$	0	0
$451$	−21.7303	−1.02324
$452$	0	0
$453$	−10.9701	−0.515420
$454$	0	0
$455$	−2.74252	−0.128571
$456$	0	0
$457$	−2.30033	−0.107605	−0.0538023	−	0.998552i	$-0.517134\pi$
$457$	−2.30033	−0.107605	−0.0538023	+	0.998552i	$0.517134\pi$
$458$	0	0
$459$	0.742521	0.0346579
$460$	0	0
$461$	−33.2804	−1.55002	−0.775011	−	0.631948i	$-0.782256\pi$
$461$	−33.2804	−1.55002	−0.775011	+	0.631948i	$0.782256\pi$
$462$	0	0
$463$	−0.792748	−0.0368421	−0.0184211	−	0.999830i	$-0.505864\pi$
$463$	−0.792748	−0.0368421	−0.0184211	+	0.999830i	$0.505864\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−32.3103	−1.49514	−0.747571	−	0.664182i	$-0.768780\pi$
$467$	−32.3103	−1.49514	−0.747571	+	0.664182i	$0.768780\pi$
$468$	0	0
$469$	28.1005	1.29756
$470$	0	0
$471$	−2.19598	−0.101186
$472$	0	0
$473$	30.5054	1.40264
$474$	0	0
$475$	−2.26394	−0.103877
$476$	0	0
$477$	13.0567	0.597825
$478$	0	0
$479$	20.2826	0.926735	0.463368	−	0.886166i	$-0.346641\pi$
$479$	20.2826	0.926735	0.463368	+	0.886166i	$0.346641\pi$
$480$	0	0
$481$	7.76763	0.354174
$482$	0	0
$483$	−7.52142	−0.342237
$484$	0	0
$485$	−5.27041	−0.239317
$486$	0	0
$487$	19.3709	0.877778	0.438889	−	0.898541i	$-0.355372\pi$
$487$	19.3709	0.877778	0.438889	+	0.898541i	$0.355372\pi$
$488$	0	0
$489$	10.4915	0.474442
$490$	0	0
$491$	−3.98227	−0.179717	−0.0898586	−	0.995955i	$-0.528642\pi$
$491$	−3.98227	−0.179717	−0.0898586	+	0.995955i	$0.528642\pi$
$492$	0	0
$493$	0.195985	0.00882670
$494$	0	0
$495$	5.76763	0.259236
$496$	0	0
$497$	−45.3661	−2.03495
$498$	0	0
$499$	−17.7619	−0.795133	−0.397566	−	0.917573i	$-0.630145\pi$
$499$	−17.7619	−0.795133	−0.397566	+	0.917573i	$0.630145\pi$
$500$	0	0
$501$	16.6283	0.742900
$502$	0	0
$503$	−16.0454	−0.715430	−0.357715	−	0.933831i	$-0.616444\pi$
$503$	−16.0454	−0.715430	−0.357715	+	0.933831i	$0.616444\pi$
$504$	0	0
$505$	−17.2843	−0.769139
$506$	0	0
$507$	−1.00000	−0.0444116
$508$	0	0
$509$	16.7879	0.744113	0.372056	−	0.928210i	$-0.378653\pi$
$509$	16.7879	0.744113	0.372056	+	0.928210i	$0.378653\pi$
$510$	0	0
$511$	−24.7516	−1.09494
$512$	0	0
$513$	2.26394	0.0999556
$514$	0	0
$515$	−16.8243	−0.741368
$516$	0	0
$517$	0	0
$518$	0	0
$519$	2.19598	0.0963930
$520$	0	0
$521$	30.5279	1.33745	0.668726	−	0.743509i	$-0.266840\pi$
$521$	30.5279	1.33745	0.668726	+	0.743509i	$0.266840\pi$
$522$	0	0
$523$	−36.0277	−1.57538	−0.787690	−	0.616071i	$-0.788723\pi$
$523$	−36.0277	−1.57538	−0.787690	+	0.616071i	$0.788723\pi$
$524$	0	0
$525$	2.74252	0.119693
$526$	0	0
$527$	0	0
$528$	0	0
$529$	−15.4786	−0.672982
$530$	0	0
$531$	−10.0502	−0.436143
$532$	0	0
$533$	−3.76763	−0.163194
$534$	0	0
$535$	−10.2955	−0.445114
$536$	0	0
$537$	−1.01218	−0.0436788
$538$	0	0
$539$	3.00738	0.129537
$540$	0	0
$541$	−2.04543	−0.0879397	−0.0439699	−	0.999033i	$-0.514001\pi$
$541$	−2.04543	−0.0879397	−0.0439699	+	0.999033i	$0.514001\pi$
$542$	0	0
$543$	−16.3328	−0.700908
$544$	0	0
$545$	10.5102	0.450206
$546$	0	0
$547$	−28.5183	−1.21935	−0.609677	−	0.792650i	$-0.708701\pi$
$547$	−28.5183	−1.21935	−0.609677	+	0.792650i	$0.708701\pi$
$548$	0	0
$549$	13.2527	0.565610
$550$	0	0
$551$	0.597556	0.0254568
$552$	0	0
$553$	2.76025	0.117378
$554$	0	0
$555$	−7.76763	−0.329718
$556$	0	0
$557$	23.0835	0.978078	0.489039	−	0.872262i	$-0.337348\pi$
$557$	23.0835	0.978078	0.489039	+	0.872262i	$0.337348\pi$
$558$	0	0
$559$	5.28906	0.223703
$560$	0	0
$561$	4.28259	0.180811
$562$	0	0
$563$	25.6949	1.08291	0.541455	−	0.840730i	$-0.317874\pi$
$563$	25.6949	1.08291	0.541455	+	0.840730i	$0.317874\pi$
$564$	0	0
$565$	−12.5231	−0.526850
$566$	0	0
$567$	−2.74252	−0.115175
$568$	0	0
$569$	−24.0761	−1.00932	−0.504661	−	0.863318i	$-0.668383\pi$
$569$	−24.0761	−1.00932	−0.504661	+	0.863318i	$0.668383\pi$
$570$	0	0
$571$	6.82341	0.285551	0.142775	−	0.989755i	$-0.454397\pi$
$571$	6.82341	0.285551	0.142775	+	0.989755i	$0.454397\pi$
$572$	0	0
$573$	−22.5054	−0.940175
$574$	0	0
$575$	−2.74252	−0.114371
$576$	0	0
$577$	−19.7126	−0.820647	−0.410323	−	0.911940i	$-0.634584\pi$
$577$	−19.7126	−0.820647	−0.410323	+	0.911940i	$0.634584\pi$
$578$	0	0
$579$	12.2778	0.510248
$580$	0	0
$581$	17.1304	0.710687
$582$	0	0
$583$	75.3062	3.11887
$584$	0	0
$585$	1.00000	0.0413449
$586$	0	0
$587$	3.50940	0.144849	0.0724243	−	0.997374i	$-0.476926\pi$
$587$	3.50940	0.144849	0.0724243	+	0.997374i	$0.476926\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	−1.80402	−0.0742073
$592$	0	0
$593$	15.2891	0.627846	0.313923	−	0.949448i	$-0.398357\pi$
$593$	15.2891	0.627846	0.313923	+	0.949448i	$0.398357\pi$
$594$	0	0
$595$	2.03638	0.0834835
$596$	0	0
$597$	−25.2163	−1.03203
$598$	0	0
$599$	−7.07054	−0.288894	−0.144447	−	0.989513i	$-0.546140\pi$
$599$	−7.07054	−0.288894	−0.144447	+	0.989513i	$0.546140\pi$
$600$	0	0
$601$	35.5620	1.45061	0.725303	−	0.688430i	$-0.241700\pi$
$601$	35.5620	1.45061	0.725303	+	0.688430i	$0.241700\pi$
$602$	0	0
$603$	−10.2462	−0.417258
$604$	0	0
$605$	22.2656	0.905226
$606$	0	0
$607$	−15.7409	−0.638902	−0.319451	−	0.947603i	$-0.603498\pi$
$607$	−15.7409	−0.638902	−0.319451	+	0.947603i	$0.603498\pi$
$608$	0	0
$609$	−0.723874	−0.0293329
$610$	0	0
$611$	0	0
$612$	0	0
$613$	−11.7676	−0.475290	−0.237645	−	0.971352i	$-0.576376\pi$
$613$	−11.7676	−0.475290	−0.237645	+	0.971352i	$0.576376\pi$
$614$	0	0
$615$	3.76763	0.151926
$616$	0	0
$617$	−24.7741	−0.997368	−0.498684	−	0.866784i	$-0.666183\pi$
$617$	−24.7741	−0.997368	−0.498684	+	0.866784i	$0.666183\pi$
$618$	0	0
$619$	−1.80882	−0.0727025	−0.0363512	−	0.999339i	$-0.511574\pi$
$619$	−1.80882	−0.0727025	−0.0363512	+	0.999339i	$0.511574\pi$
$620$	0	0
$621$	2.74252	0.110054
$622$	0	0
$623$	−2.72479	−0.109166
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	13.0576	0.521471
$628$	0	0
$629$	−5.76763	−0.229971
$630$	0	0
$631$	−38.5685	−1.53539	−0.767694	−	0.640816i	$-0.778596\pi$
$631$	−38.5685	−1.53539	−0.767694	+	0.640816i	$0.778596\pi$
$632$	0	0
$633$	−19.0705	−0.757986
$634$	0	0
$635$	−3.27613	−0.130009
$636$	0	0
$637$	0.521423	0.0206595
$638$	0	0
$639$	16.5417	0.654381
$640$	0	0
$641$	−22.1263	−0.873937	−0.436969	−	0.899477i	$-0.643948\pi$
$641$	−22.1263	−0.873937	−0.436969	+	0.899477i	$0.643948\pi$
$642$	0	0
$643$	28.0770	1.10725	0.553624	−	0.832766i	$-0.313244\pi$
$643$	28.0770	1.10725	0.553624	+	0.832766i	$0.313244\pi$
$644$	0	0
$645$	−5.28906	−0.208256
$646$	0	0
$647$	−23.2981	−0.915943	−0.457971	−	0.888967i	$-0.651424\pi$
$647$	−23.2981	−0.915943	−0.457971	+	0.888967i	$0.651424\pi$
$648$	0	0
$649$	−57.9660	−2.27537
$650$	0	0
$651$	0	0
$652$	0	0
$653$	6.82433	0.267057	0.133528	−	0.991045i	$-0.457369\pi$
$653$	6.82433	0.267057	0.133528	+	0.991045i	$0.457369\pi$
$654$	0	0
$655$	−3.02511	−0.118201
$656$	0	0
$657$	9.02511	0.352103
$658$	0	0
$659$	−24.5733	−0.957240	−0.478620	−	0.878022i	$-0.658863\pi$
$659$	−24.5733	−0.957240	−0.478620	+	0.878022i	$0.658863\pi$
$660$	0	0
$661$	34.9653	1.35999	0.679996	−	0.733216i	$-0.261982\pi$
$661$	34.9653	1.35999	0.679996	+	0.733216i	$0.261982\pi$
$662$	0	0
$663$	0.742521	0.0288371
$664$	0	0
$665$	6.20892	0.240771
$666$	0	0
$667$	0.723874	0.0280285
$668$	0	0
$669$	−17.6032	−0.680580
$670$	0	0
$671$	76.4366	2.95080
$672$	0	0
$673$	−27.0576	−1.04299	−0.521497	−	0.853253i	$-0.674626\pi$
$673$	−27.0576	−1.04299	−0.521497	+	0.853253i	$0.674626\pi$
$674$	0	0
$675$	−1.00000	−0.0384900
$676$	0	0
$677$	4.42835	0.170195	0.0850977	−	0.996373i	$-0.472880\pi$
$677$	4.42835	0.170195	0.0850977	+	0.996373i	$0.472880\pi$
$678$	0	0
$679$	14.4542	0.554702
$680$	0	0
$681$	−6.01293	−0.230416
$682$	0	0
$683$	10.8373	0.414676	0.207338	−	0.978269i	$-0.433520\pi$
$683$	10.8373	0.414676	0.207338	+	0.978269i	$0.433520\pi$
$684$	0	0
$685$	12.7741	0.488073
$686$	0	0
$687$	15.5305	0.592524
$688$	0	0
$689$	13.0567	0.497420
$690$	0	0
$691$	−19.7863	−0.752706	−0.376353	−	0.926476i	$-0.622822\pi$
$691$	−19.7863	−0.752706	−0.376353	+	0.926476i	$0.622822\pi$
$692$	0	0
$693$	−15.8179	−0.600871
$694$	0	0
$695$	7.78057	0.295134
$696$	0	0
$697$	2.79755	0.105965
$698$	0	0
$699$	−21.7628	−0.823146
$700$	0	0
$701$	18.2041	0.687560	0.343780	−	0.939050i	$-0.388293\pi$
$701$	18.2041	0.687560	0.343780	+	0.939050i	$0.388293\pi$
$702$	0	0
$703$	−17.5855	−0.663250
$704$	0	0
$705$	0	0
$706$	0	0
$707$	47.4024	1.78275
$708$	0	0
$709$	−15.5305	−0.583259	−0.291629	−	0.956531i	$-0.594197\pi$
$709$	−15.5305	−0.583259	−0.291629	+	0.956531i	$0.594197\pi$
$710$	0	0
$711$	−1.00647	−0.0377454
$712$	0	0
$713$	0	0
$714$	0	0
$715$	5.76763	0.215697
$716$	0	0
$717$	−22.2955	−0.832642
$718$	0	0
$719$	51.9904	1.93891	0.969457	−	0.245260i	$-0.0788733\pi$
$719$	51.9904	1.93891	0.969457	+	0.245260i	$0.0788733\pi$
$720$	0	0
$721$	46.1411	1.71838
$722$	0	0
$723$	−10.9572	−0.407501
$724$	0	0
$725$	−0.263945	−0.00980266
$726$	0	0
$727$	14.3094	0.530705	0.265353	−	0.964151i	$-0.414512\pi$
$727$	14.3094	0.530705	0.265353	+	0.964151i	$0.414512\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−3.92724	−0.145254
$732$	0	0
$733$	48.3605	1.78624	0.893118	−	0.449822i	$-0.148513\pi$
$733$	48.3605	1.78624	0.893118	+	0.449822i	$0.148513\pi$
$734$	0	0
$735$	−0.521423	−0.0192330
$736$	0	0
$737$	−59.0964	−2.17684
$738$	0	0
$739$	−37.1986	−1.36837	−0.684186	−	0.729308i	$-0.739842\pi$
$739$	−37.1986	−1.36837	−0.684186	+	0.729308i	$0.739842\pi$
$740$	0	0
$741$	2.26394	0.0831681
$742$	0	0
$743$	32.0632	1.17628	0.588142	−	0.808758i	$-0.299860\pi$
$743$	32.0632	1.17628	0.588142	+	0.808758i	$0.299860\pi$
$744$	0	0
$745$	18.3458	0.672136
$746$	0	0
$747$	−6.24621	−0.228537
$748$	0	0
$749$	28.2357	1.03171
$750$	0	0
$751$	−18.6875	−0.681916	−0.340958	−	0.940078i	$-0.610751\pi$
$751$	−18.6875	−0.681916	−0.340958	+	0.940078i	$0.610751\pi$
$752$	0	0
$753$	−13.9952	−0.510013
$754$	0	0
$755$	10.9701	0.399242
$756$	0	0
$757$	11.4496	0.416142	0.208071	−	0.978114i	$-0.433282\pi$
$757$	11.4496	0.416142	0.208071	+	0.978114i	$0.433282\pi$
$758$	0	0
$759$	15.8179	0.574152
$760$	0	0
$761$	26.9199	0.975844	0.487922	−	0.872887i	$-0.337755\pi$
$761$	26.9199	0.975844	0.487922	+	0.872887i	$0.337755\pi$
$762$	0	0
$763$	−28.8243	−1.04351
$764$	0	0
$765$	−0.742521	−0.0268459
$766$	0	0
$767$	−10.0502	−0.362893
$768$	0	0
$769$	−39.5223	−1.42521	−0.712606	−	0.701564i	$-0.752485\pi$
$769$	−39.5223	−1.42521	−0.712606	+	0.701564i	$0.752485\pi$
$770$	0	0
$771$	−19.6032	−0.705993
$772$	0	0
$773$	−30.2237	−1.08707	−0.543535	−	0.839386i	$-0.682915\pi$
$773$	−30.2237	−1.08707	−0.543535	+	0.839386i	$0.682915\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	21.3029	0.764237
$778$	0	0
$779$	8.52972	0.305609
$780$	0	0
$781$	95.4067	3.41392
$782$	0	0
$783$	0.263945	0.00943262
$784$	0	0
$785$	2.19598	0.0783781
$786$	0	0
$787$	−19.8040	−0.705937	−0.352968	−	0.935635i	$-0.614828\pi$
$787$	−19.8040	−0.705937	−0.352968	+	0.935635i	$0.614828\pi$
$788$	0	0
$789$	−4.05503	−0.144363
$790$	0	0
$791$	34.3448	1.22116
$792$	0	0
$793$	13.2527	0.470616
$794$	0	0
$795$	−13.0567	−0.463073
$796$	0	0
$797$	−33.4487	−1.18481	−0.592406	−	0.805639i	$-0.701822\pi$
$797$	−33.4487	−1.18481	−0.592406	+	0.805639i	$0.701822\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	0.993534	0.0351048
$802$	0	0
$803$	52.0536	1.83693
$804$	0	0
$805$	7.52142	0.265095
$806$	0	0
$807$	3.34409	0.117717
$808$	0	0
$809$	−46.9848	−1.65190	−0.825950	−	0.563744i	$-0.809360\pi$
$809$	−46.9848	−1.65190	−0.825950	+	0.563744i	$0.809360\pi$
$810$	0	0
$811$	−17.8720	−0.627570	−0.313785	−	0.949494i	$-0.601597\pi$
$811$	−17.8720	−0.627570	−0.313785	+	0.949494i	$0.601597\pi$
$812$	0	0
$813$	0.565184	0.0198219
$814$	0	0
$815$	−10.4915	−0.367502
$816$	0	0
$817$	−11.9741	−0.418922
$818$	0	0
$819$	−2.74252	−0.0958315
$820$	0	0
$821$	−5.82746	−0.203380	−0.101690	−	0.994816i	$-0.532425\pi$
$821$	−5.82746	−0.203380	−0.101690	+	0.994816i	$0.532425\pi$
$822$	0	0
$823$	34.7386	1.21091	0.605456	−	0.795879i	$-0.292991\pi$
$823$	34.7386	1.21091	0.605456	+	0.795879i	$0.292991\pi$
$824$	0	0
$825$	−5.76763	−0.200803
$826$	0	0
$827$	15.7685	0.548326	0.274163	−	0.961683i	$-0.411599\pi$
$827$	15.7685	0.548326	0.274163	+	0.961683i	$0.411599\pi$
$828$	0	0
$829$	−20.5781	−0.714708	−0.357354	−	0.933969i	$-0.616321\pi$
$829$	−20.5781	−0.714708	−0.357354	+	0.933969i	$0.616321\pi$
$830$	0	0
$831$	7.81695	0.271167
$832$	0	0
$833$	−0.387168	−0.0134146
$834$	0	0
$835$	−16.6283	−0.575448
$836$	0	0
$837$	0	0
$838$	0	0
$839$	32.6018	1.12554	0.562770	−	0.826614i	$-0.309736\pi$
$839$	32.6018	1.12554	0.562770	+	0.826614i	$0.309736\pi$
$840$	0	0
$841$	−28.9303	−0.997598
$842$	0	0
$843$	−21.2761	−0.732789
$844$	0	0
$845$	1.00000	0.0344010
$846$	0	0
$847$	−61.0639	−2.09818
$848$	0	0
$849$	18.2721	0.627096
$850$	0	0
$851$	−21.3029	−0.730254
$852$	0	0
$853$	7.59442	0.260028	0.130014	−	0.991512i	$-0.458498\pi$
$853$	7.59442	0.260028	0.130014	+	0.991512i	$0.458498\pi$
$854$	0	0
$855$	−2.26394	−0.0774253
$856$	0	0
$857$	−31.7499	−1.08456	−0.542278	−	0.840199i	$-0.682438\pi$
$857$	−31.7499	−1.08456	−0.542278	+	0.840199i	$0.682438\pi$
$858$	0	0
$859$	7.74510	0.264259	0.132130	−	0.991232i	$-0.457818\pi$
$859$	7.74510	0.264259	0.132130	+	0.991232i	$0.457818\pi$
$860$	0	0
$861$	−10.3328	−0.352142
$862$	0	0
$863$	44.9476	1.53003	0.765016	−	0.644011i	$-0.222731\pi$
$863$	44.9476	1.53003	0.765016	+	0.644011i	$0.222731\pi$
$864$	0	0
$865$	−2.19598	−0.0746657
$866$	0	0
$867$	16.4487	0.558626
$868$	0	0
$869$	−5.80493	−0.196919
$870$	0	0
$871$	−10.2462	−0.347180
$872$	0	0
$873$	−5.27041	−0.178376
$874$	0	0
$875$	−2.74252	−0.0927141
$876$	0	0
$877$	14.1263	0.477012	0.238506	−	0.971141i	$-0.423342\pi$
$877$	14.1263	0.477012	0.238506	+	0.971141i	$0.423342\pi$
$878$	0	0
$879$	7.81695	0.263659
$880$	0	0
$881$	56.0033	1.88680	0.943400	−	0.331657i	$-0.107608\pi$
$881$	56.0033	1.88680	0.943400	+	0.331657i	$0.107608\pi$
$882$	0	0
$883$	36.5183	1.22894	0.614469	−	0.788941i	$-0.289370\pi$
$883$	36.5183	1.22894	0.614469	+	0.788941i	$0.289370\pi$
$884$	0	0
$885$	10.0502	0.337835
$886$	0	0
$887$	6.21463	0.208667	0.104333	−	0.994542i	$-0.466729\pi$
$887$	6.21463	0.208667	0.104333	+	0.994542i	$0.466729\pi$
$888$	0	0
$889$	8.98485	0.301342
$890$	0	0
$891$	5.76763	0.193223
$892$	0	0
$893$	0	0
$894$	0	0
$895$	1.01218	0.0338335
$896$	0	0
$897$	2.74252	0.0915701
$898$	0	0
$899$	0	0
$900$	0	0
$901$	−9.69487	−0.322983
$902$	0	0
$903$	14.5054	0.482708
$904$	0	0
$905$	16.3328	0.542921
$906$	0	0
$907$	40.9230	1.35882	0.679412	−	0.733757i	$-0.262235\pi$
$907$	40.9230	1.35882	0.679412	+	0.733757i	$0.262235\pi$
$908$	0	0
$909$	−17.2843	−0.573283
$910$	0	0
$911$	7.00738	0.232165	0.116082	−	0.993240i	$-0.462966\pi$
$911$	7.00738	0.232165	0.116082	+	0.993240i	$0.462966\pi$
$912$	0	0
$913$	−36.0259	−1.19228
$914$	0	0
$915$	−13.2527	−0.438120
$916$	0	0
$917$	8.29644	0.273973
$918$	0	0
$919$	44.8027	1.47790	0.738952	−	0.673758i	$-0.235321\pi$
$919$	44.8027	1.47790	0.738952	+	0.673758i	$0.235321\pi$
$920$	0	0
$921$	−9.49889	−0.312999
$922$	0	0
$923$	16.5417	0.544478
$924$	0	0
$925$	7.76763	0.255398
$926$	0	0
$927$	−16.8243	−0.552583
$928$	0	0
$929$	9.11985	0.299213	0.149606	−	0.988746i	$-0.452199\pi$
$929$	9.11985	0.299213	0.149606	+	0.988746i	$0.452199\pi$
$930$	0	0
$931$	−1.18047	−0.0386885
$932$	0	0
$933$	−5.98707	−0.196008
$934$	0	0
$935$	−4.28259	−0.140056
$936$	0	0
$937$	−4.37904	−0.143057	−0.0715285	−	0.997439i	$-0.522788\pi$
$937$	−4.37904	−0.143057	−0.0715285	+	0.997439i	$0.522788\pi$
$938$	0	0
$939$	−16.4422	−0.536571
$940$	0	0
$941$	−39.8212	−1.29813	−0.649067	−	0.760731i	$-0.724840\pi$
$941$	−39.8212	−1.29813	−0.649067	+	0.760731i	$0.724840\pi$
$942$	0	0
$943$	10.3328	0.336483
$944$	0	0
$945$	2.74252	0.0892142
$946$	0	0
$947$	47.8949	1.55637	0.778187	−	0.628033i	$-0.216140\pi$
$947$	47.8949	1.55637	0.778187	+	0.628033i	$0.216140\pi$
$948$	0	0
$949$	9.02511	0.292968
$950$	0	0
$951$	12.5781	0.407873
$952$	0	0
$953$	−7.22018	−0.233885	−0.116942	−	0.993139i	$-0.537309\pi$
$953$	−7.22018	−0.233885	−0.116942	+	0.993139i	$0.537309\pi$
$954$	0	0
$955$	22.5054	0.728256
$956$	0	0
$957$	1.52234	0.0492102
$958$	0	0
$959$	−35.0332	−1.13128
$960$	0	0
$961$	−31.0000	−1.00000
$962$	0	0
$963$	−10.2955	−0.331769
$964$	0	0
$965$	−12.2778	−0.395236
$966$	0	0
$967$	3.16103	0.101652	0.0508260	−	0.998708i	$-0.483815\pi$
$967$	3.16103	0.101652	0.0508260	+	0.998708i	$0.483815\pi$
$968$	0	0
$969$	−1.68103	−0.0540024
$970$	0	0
$971$	−28.9280	−0.928343	−0.464172	−	0.885745i	$-0.653648\pi$
$971$	−28.9280	−0.928343	−0.464172	+	0.885745i	$0.653648\pi$
$972$	0	0
$973$	−21.3384	−0.684077
$974$	0	0
$975$	−1.00000	−0.0320256
$976$	0	0
$977$	5.79441	0.185380	0.0926898	−	0.995695i	$-0.470453\pi$
$977$	5.79441	0.185380	0.0926898	+	0.995695i	$0.470453\pi$
$978$	0	0
$979$	5.73034	0.183142
$980$	0	0
$981$	10.5102	0.335563
$982$	0	0
$983$	−45.2471	−1.44316	−0.721579	−	0.692332i	$-0.756583\pi$
$983$	−45.2471	−1.44316	−0.721579	+	0.692332i	$0.756583\pi$
$984$	0	0
$985$	1.80402	0.0574807
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−14.5054	−0.461243
$990$	0	0
$991$	−39.5118	−1.25513	−0.627567	−	0.778562i	$-0.715949\pi$
$991$	−39.5118	−1.25513	−0.627567	+	0.778562i	$0.715949\pi$
$992$	0	0
$993$	−0.706141	−0.0224087
$994$	0	0
$995$	25.2163	0.799410
$996$	0	0
$997$	38.2495	1.21138	0.605688	−	0.795703i	$-0.292898\pi$
$997$	38.2495	1.21138	0.605688	+	0.795703i	$0.292898\pi$
$998$	0	0
$999$	−7.76763	−0.245757

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6240.2.a.ce.1.2	✓	4
4.3	odd	2		6240.2.a.cf.1.3	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6240.2.a.ce.1.2	✓	4	1.1	even	1	trivial
6240.2.a.cf.1.3	yes	4	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 6240 = 2^{5} \cdot 3 \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6240.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +1.00000 q^{5} -2.74252 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +1.00000 q^{5} -2.74252 q^{7} +1.00000 q^{9} +5.76763 q^{11} +1.00000 q^{13} -1.00000 q^{15} -0.742521 q^{17} -2.26394 q^{19} +2.74252 q^{21} -2.74252 q^{23} +1.00000 q^{25} -1.00000 q^{27} -0.263945 q^{29} -5.76763 q^{33} -2.74252 q^{35} +7.76763 q^{37} -1.00000 q^{39} -3.76763 q^{41} +5.28906 q^{43} +1.00000 q^{45} +0.521423 q^{49} +0.742521 q^{51} +13.0567 q^{53} +5.76763 q^{55} +2.26394 q^{57} -10.0502 q^{59} +13.2527 q^{61} -2.74252 q^{63} +1.00000 q^{65} -10.2462 q^{67} +2.74252 q^{69} +16.5417 q^{71} +9.02511 q^{73} -1.00000 q^{75} -15.8179 q^{77} -1.00647 q^{79} +1.00000 q^{81} -6.24621 q^{83} -0.742521 q^{85} +0.263945 q^{87} +0.993534 q^{89} -2.74252 q^{91} -2.26394 q^{95} -5.27041 q^{97} +5.76763 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{3} + 4 q^{5} - 3 q^{7} + 4 q^{9}+O(q^{10}) \) 4 * q - 4 * q^3 + 4 * q^5 - 3 * q^7 + 4 * q^9	\( 4 q - 4 q^{3} + 4 q^{5} - 3 q^{7} + 4 q^{9} - 3 q^{11} + 4 q^{13} - 4 q^{15} + 5 q^{17} - 8 q^{19} + 3 q^{21} - 3 q^{23} + 4 q^{25} - 4 q^{27} + 3 q^{33} - 3 q^{35} + 5 q^{37} - 4 q^{39} + 11 q^{41} + 2 q^{43} + 4 q^{45} + 9 q^{49} - 5 q^{51} + 7 q^{53} - 3 q^{55} + 8 q^{57} - 4 q^{59} + 11 q^{61} - 3 q^{63} + 4 q^{65} - 8 q^{67} + 3 q^{69} + 5 q^{71} + 18 q^{73} - 4 q^{75} - q^{77} + 5 q^{79} + 4 q^{81} + 8 q^{83} + 5 q^{85} + 13 q^{89} - 3 q^{91} - 8 q^{95} - 11 q^{97} - 3 q^{99}+O(q^{100}) \) 4 * q - 4 * q^3 + 4 * q^5 - 3 * q^7 + 4 * q^9 - 3 * q^11 + 4 * q^13 - 4 * q^15 + 5 * q^17 - 8 * q^19 + 3 * q^21 - 3 * q^23 + 4 * q^25 - 4 * q^27 + 3 * q^33 - 3 * q^35 + 5 * q^37 - 4 * q^39 + 11 * q^41 + 2 * q^43 + 4 * q^45 + 9 * q^49 - 5 * q^51 + 7 * q^53 - 3 * q^55 + 8 * q^57 - 4 * q^59 + 11 * q^61 - 3 * q^63 + 4 * q^65 - 8 * q^67 + 3 * q^69 + 5 * q^71 + 18 * q^73 - 4 * q^75 - q^77 + 5 * q^79 + 4 * q^81 + 8 * q^83 + 5 * q^85 + 13 * q^89 - 3 * q^91 - 8 * q^95 - 11 * q^97 - 3 * q^99

Introduction

L-functions

Modular forms

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Database

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