LMFDB - Embedded newform 6024.2.a.o.1.13

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6024,2,Mod(1,6024)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6024, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("6024.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6024 = 2^{3} \cdot 3 \cdot 251 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6024.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:	$48.1018821776$
Analytic rank:	$1$
Dimension:	$14$
Coefficient field:	$\mathbb{Q}[x]/(x^{14} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{14} - 3 x^{13} - 43 x^{12} + 119 x^{11} + 679 x^{10} - 1667 x^{9} - 4890 x^{8} + 9662 x^{7} + \cdots - 416 $ x^14 - 3x^13 - 43x^12 + 119x^11 + 679x^10 - 1667x^9 - 4890x^8 + 9662x^7 + 16575x^6 - 20277x^5 - 25196x^4 + 8040x^3 + 10776x^2 + 912*x - 416
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.13
Root			$-3.58563$ of defining polynomial
Character	$\chi$	$=$	6024.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} +3.58563 q^{5} -1.47988 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} +3.58563 q^{5} -1.47988 q^{7} +1.00000 q^{9} -2.22283 q^{11} +3.79222 q^{13} -3.58563 q^{15} -4.21231 q^{17} -2.92795 q^{19} +1.47988 q^{21} -1.45540 q^{23} +7.85673 q^{25} -1.00000 q^{27} -8.55655 q^{29} +7.83747 q^{31} +2.22283 q^{33} -5.30631 q^{35} -2.57717 q^{37} -3.79222 q^{39} -4.27997 q^{41} +4.13688 q^{43} +3.58563 q^{45} -3.07190 q^{47} -4.80994 q^{49} +4.21231 q^{51} +2.67835 q^{53} -7.97025 q^{55} +2.92795 q^{57} -12.0794 q^{59} +3.15891 q^{61} -1.47988 q^{63} +13.5975 q^{65} -13.4806 q^{67} +1.45540 q^{69} -13.3683 q^{71} +10.4710 q^{73} -7.85673 q^{75} +3.28953 q^{77} -6.01002 q^{79} +1.00000 q^{81} +3.45375 q^{83} -15.1038 q^{85} +8.55655 q^{87} +15.3843 q^{89} -5.61205 q^{91} -7.83747 q^{93} -10.4985 q^{95} -9.36985 q^{97} -2.22283 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 14 q - 14 q^{3} - 3 q^{5} + 7 q^{7} + 14 q^{9}+O(q^{10}) $ 14 * q - 14 * q^3 - 3 * q^5 + 7 * q^7 + 14 * q^9	$ 14 q - 14 q^{3} - 3 q^{5} + 7 q^{7} + 14 q^{9} - 22 q^{11} + 9 q^{13} + 3 q^{15} - 7 q^{21} - 17 q^{23} + 25 q^{25} - 14 q^{27} - 18 q^{29} - 7 q^{31} + 22 q^{33} - 27 q^{35} + 9 q^{37} - 9 q^{39} - 6 q^{41} - 14 q^{43} - 3 q^{45} - 15 q^{47} + 15 q^{49} - 11 q^{53} + 2 q^{55} - 36 q^{59} + 2 q^{61} + 7 q^{63} + 8 q^{65} + 3 q^{67} + 17 q^{69} - 29 q^{71} + 2 q^{73} - 25 q^{75} + 8 q^{77} + 23 q^{79} + 14 q^{81} - 55 q^{83} + 7 q^{85} + 18 q^{87} + 9 q^{89} - 22 q^{91} + 7 q^{93} - 27 q^{95} + 17 q^{97} - 22 q^{99}+O(q^{100}) $ 14 * q - 14 * q^3 - 3 * q^5 + 7 * q^7 + 14 * q^9 - 22 * q^11 + 9 * q^13 + 3 * q^15 - 7 * q^21 - 17 * q^23 + 25 * q^25 - 14 * q^27 - 18 * q^29 - 7 * q^31 + 22 * q^33 - 27 * q^35 + 9 * q^37 - 9 * q^39 - 6 * q^41 - 14 * q^43 - 3 * q^45 - 15 * q^47 + 15 * q^49 - 11 * q^53 + 2 * q^55 - 36 * q^59 + 2 * q^61 + 7 * q^63 + 8 * q^65 + 3 * q^67 + 17 * q^69 - 29 * q^71 + 2 * q^73 - 25 * q^75 + 8 * q^77 + 23 * q^79 + 14 * q^81 - 55 * q^83 + 7 * q^85 + 18 * q^87 + 9 * q^89 - 22 * q^91 + 7 * q^93 - 27 * q^95 + 17 * q^97 - 22 * 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
$5$	3.58563	1.60354	0.801771	−	0.597631i	$-0.203891\pi$
$5$	3.58563	1.60354	0.801771	+	0.597631i	$0.203891\pi$
$6$	0	0
$7$	−1.47988	−0.559344	−0.279672	−	0.960096i	$-0.590226\pi$
$7$	−1.47988	−0.559344	−0.279672	+	0.960096i	$0.590226\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−2.22283	−0.670209	−0.335104	−	0.942181i	$-0.608772\pi$
$11$	−2.22283	−0.670209	−0.335104	+	0.942181i	$0.608772\pi$
$12$	0	0
$13$	3.79222	1.05177	0.525887	−	0.850555i	$-0.323734\pi$
$13$	3.79222	1.05177	0.525887	+	0.850555i	$0.323734\pi$
$14$	0	0
$15$	−3.58563	−0.925805
$16$	0	0
$17$	−4.21231	−1.02164	−0.510818	−	0.859689i	$-0.670657\pi$
$17$	−4.21231	−1.02164	−0.510818	+	0.859689i	$0.670657\pi$
$18$	0	0
$19$	−2.92795	−0.671717	−0.335859	−	0.941912i	$-0.609026\pi$
$19$	−2.92795	−0.671717	−0.335859	+	0.941912i	$0.609026\pi$
$20$	0	0
$21$	1.47988	0.322937
$22$	0	0
$23$	−1.45540	−0.303471	−0.151736	−	0.988421i	$-0.548486\pi$
$23$	−1.45540	−0.303471	−0.151736	+	0.988421i	$0.548486\pi$
$24$	0	0
$25$	7.85673	1.57135
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−8.55655	−1.58891	−0.794455	−	0.607323i	$-0.792244\pi$
$29$	−8.55655	−1.58891	−0.794455	+	0.607323i	$0.792244\pi$
$30$	0	0
$31$	7.83747	1.40765	0.703826	−	0.710372i	$-0.251473\pi$
$31$	7.83747	1.40765	0.703826	+	0.710372i	$0.251473\pi$
$32$	0	0
$33$	2.22283	0.386945
$34$	0	0
$35$	−5.30631	−0.896931
$36$	0	0
$37$	−2.57717	−0.423684	−0.211842	−	0.977304i	$-0.567946\pi$
$37$	−2.57717	−0.423684	−0.211842	+	0.977304i	$0.567946\pi$
$38$	0	0
$39$	−3.79222	−0.607242
$40$	0	0
$41$	−4.27997	−0.668419	−0.334209	−	0.942499i	$-0.608469\pi$
$41$	−4.27997	−0.668419	−0.334209	+	0.942499i	$0.608469\pi$
$42$	0	0
$43$	4.13688	0.630868	0.315434	−	0.948948i	$-0.397850\pi$
$43$	4.13688	0.630868	0.315434	+	0.948948i	$0.397850\pi$
$44$	0	0
$45$	3.58563	0.534514
$46$	0	0
$47$	−3.07190	−0.448082	−0.224041	−	0.974580i	$-0.571925\pi$
$47$	−3.07190	−0.448082	−0.224041	+	0.974580i	$0.571925\pi$
$48$	0	0
$49$	−4.80994	−0.687135
$50$	0	0
$51$	4.21231	0.589842
$52$	0	0
$53$	2.67835	0.367900	0.183950	−	0.982936i	$-0.441112\pi$
$53$	2.67835	0.367900	0.183950	+	0.982936i	$0.441112\pi$
$54$	0	0
$55$	−7.97025	−1.07471
$56$	0	0
$57$	2.92795	0.387816
$58$	0	0
$59$	−12.0794	−1.57260	−0.786301	−	0.617843i	$-0.788007\pi$
$59$	−12.0794	−1.57260	−0.786301	+	0.617843i	$0.788007\pi$
$60$	0	0
$61$	3.15891	0.404457	0.202229	−	0.979338i	$-0.435182\pi$
$61$	3.15891	0.404457	0.202229	+	0.979338i	$0.435182\pi$
$62$	0	0
$63$	−1.47988	−0.186448
$64$	0	0
$65$	13.5975	1.68656
$66$	0	0
$67$	−13.4806	−1.64691	−0.823457	−	0.567379i	$-0.807958\pi$
$67$	−13.4806	−1.64691	−0.823457	+	0.567379i	$0.807958\pi$
$68$	0	0
$69$	1.45540	0.175209
$70$	0	0
$71$	−13.3683	−1.58652	−0.793261	−	0.608882i	$-0.791618\pi$
$71$	−13.3683	−1.58652	−0.793261	+	0.608882i	$0.791618\pi$
$72$	0	0
$73$	10.4710	1.22554	0.612770	−	0.790261i	$-0.290055\pi$
$73$	10.4710	1.22554	0.612770	+	0.790261i	$0.290055\pi$
$74$	0	0
$75$	−7.85673	−0.907218
$76$	0	0
$77$	3.28953	0.374877
$78$	0	0
$79$	−6.01002	−0.676181	−0.338090	−	0.941114i	$-0.609781\pi$
$79$	−6.01002	−0.676181	−0.338090	+	0.941114i	$0.609781\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	3.45375	0.379098	0.189549	−	0.981871i	$-0.439297\pi$
$83$	3.45375	0.379098	0.189549	+	0.981871i	$0.439297\pi$
$84$	0	0
$85$	−15.1038	−1.63824
$86$	0	0
$87$	8.55655	0.917358
$88$	0	0
$89$	15.3843	1.63073	0.815365	−	0.578947i	$-0.196536\pi$
$89$	15.3843	1.63073	0.815365	+	0.578947i	$0.196536\pi$
$90$	0	0
$91$	−5.61205	−0.588303
$92$	0	0
$93$	−7.83747	−0.812708
$94$	0	0
$95$	−10.4985	−1.07713
$96$	0	0
$97$	−9.36985	−0.951364	−0.475682	−	0.879617i	$-0.657799\pi$
$97$	−9.36985	−0.951364	−0.475682	+	0.879617i	$0.657799\pi$
$98$	0	0
$99$	−2.22283	−0.223403
$100$	0	0
$101$	8.02785	0.798801	0.399400	−	0.916777i	$-0.369218\pi$
$101$	8.02785	0.798801	0.399400	+	0.916777i	$0.369218\pi$
$102$	0	0
$103$	−4.16682	−0.410569	−0.205284	−	0.978702i	$-0.565812\pi$
$103$	−4.16682	−0.410569	−0.205284	+	0.978702i	$0.565812\pi$
$104$	0	0
$105$	5.30631	0.517843
$106$	0	0
$107$	−19.6558	−1.90020	−0.950098	−	0.311951i	$-0.899017\pi$
$107$	−19.6558	−1.90020	−0.950098	+	0.311951i	$0.899017\pi$
$108$	0	0
$109$	2.67433	0.256154	0.128077	−	0.991764i	$-0.459120\pi$
$109$	2.67433	0.256154	0.128077	+	0.991764i	$0.459120\pi$
$110$	0	0
$111$	2.57717	0.244614
$112$	0	0
$113$	5.26485	0.495276	0.247638	−	0.968853i	$-0.420346\pi$
$113$	5.26485	0.495276	0.247638	+	0.968853i	$0.420346\pi$
$114$	0	0
$115$	−5.21851	−0.486629
$116$	0	0
$117$	3.79222	0.350591
$118$	0	0
$119$	6.23374	0.571446
$120$	0	0
$121$	−6.05902	−0.550820
$122$	0	0
$123$	4.27997	0.385912
$124$	0	0
$125$	10.2432	0.916179
$126$	0	0
$127$	−9.02213	−0.800585	−0.400292	−	0.916387i	$-0.631091\pi$
$127$	−9.02213	−0.800585	−0.400292	+	0.916387i	$0.631091\pi$
$128$	0	0
$129$	−4.13688	−0.364232
$130$	0	0
$131$	8.18602	0.715216	0.357608	−	0.933872i	$-0.383592\pi$
$131$	8.18602	0.715216	0.357608	+	0.933872i	$0.383592\pi$
$132$	0	0
$133$	4.33302	0.375721
$134$	0	0
$135$	−3.58563	−0.308602
$136$	0	0
$137$	−9.63996	−0.823597	−0.411799	−	0.911275i	$-0.635099\pi$
$137$	−9.63996	−0.823597	−0.411799	+	0.911275i	$0.635099\pi$
$138$	0	0
$139$	10.2077	0.865807	0.432904	−	0.901440i	$-0.357489\pi$
$139$	10.2077	0.865807	0.432904	+	0.901440i	$0.357489\pi$
$140$	0	0
$141$	3.07190	0.258700
$142$	0	0
$143$	−8.42947	−0.704908
$144$	0	0
$145$	−30.6806	−2.54788
$146$	0	0
$147$	4.80994	0.396717
$148$	0	0
$149$	0.113074	0.00926339	0.00463170	−	0.999989i	$-0.498526\pi$
$149$	0.113074	0.00926339	0.00463170	+	0.999989i	$0.498526\pi$
$150$	0	0
$151$	−12.8768	−1.04790	−0.523951	−	0.851749i	$-0.675542\pi$
$151$	−12.8768	−1.04790	−0.523951	+	0.851749i	$0.675542\pi$
$152$	0	0
$153$	−4.21231	−0.340545
$154$	0	0
$155$	28.1023	2.25723
$156$	0	0
$157$	−8.74045	−0.697564	−0.348782	−	0.937204i	$-0.613405\pi$
$157$	−8.74045	−0.697564	−0.348782	+	0.937204i	$0.613405\pi$
$158$	0	0
$159$	−2.67835	−0.212407
$160$	0	0
$161$	2.15382	0.169745
$162$	0	0
$163$	3.07679	0.240993	0.120497	−	0.992714i	$-0.461551\pi$
$163$	3.07679	0.240993	0.120497	+	0.992714i	$0.461551\pi$
$164$	0	0
$165$	7.97025	0.620483
$166$	0	0
$167$	−2.92117	−0.226047	−0.113023	−	0.993592i	$-0.536054\pi$
$167$	−2.92117	−0.226047	−0.113023	+	0.993592i	$0.536054\pi$
$168$	0	0
$169$	1.38095	0.106227
$170$	0	0
$171$	−2.92795	−0.223906
$172$	0	0
$173$	−9.49890	−0.722188	−0.361094	−	0.932529i	$-0.617597\pi$
$173$	−9.49890	−0.722188	−0.361094	+	0.932529i	$0.617597\pi$
$174$	0	0
$175$	−11.6271	−0.878923
$176$	0	0
$177$	12.0794	0.907943
$178$	0	0
$179$	17.7360	1.32565	0.662825	−	0.748774i	$-0.269357\pi$
$179$	17.7360	1.32565	0.662825	+	0.748774i	$0.269357\pi$
$180$	0	0
$181$	25.6798	1.90877	0.954384	−	0.298582i	$-0.0965137\pi$
$181$	25.6798	1.90877	0.954384	+	0.298582i	$0.0965137\pi$
$182$	0	0
$183$	−3.15891	−0.233513
$184$	0	0
$185$	−9.24078	−0.679395
$186$	0	0
$187$	9.36326	0.684709
$188$	0	0
$189$	1.47988	0.107646
$190$	0	0
$191$	−3.10368	−0.224574	−0.112287	−	0.993676i	$-0.535818\pi$
$191$	−3.10368	−0.224574	−0.112287	+	0.993676i	$0.535818\pi$
$192$	0	0
$193$	−1.83074	−0.131780	−0.0658898	−	0.997827i	$-0.520989\pi$
$193$	−1.83074	−0.131780	−0.0658898	+	0.997827i	$0.520989\pi$
$194$	0	0
$195$	−13.5975	−0.973737
$196$	0	0
$197$	−6.73467	−0.479826	−0.239913	−	0.970794i	$-0.577119\pi$
$197$	−6.73467	−0.479826	−0.239913	+	0.970794i	$0.577119\pi$
$198$	0	0
$199$	−14.4839	−1.02673	−0.513367	−	0.858169i	$-0.671602\pi$
$199$	−14.4839	−1.02673	−0.513367	+	0.858169i	$0.671602\pi$
$200$	0	0
$201$	13.4806	0.950846
$202$	0	0
$203$	12.6627	0.888747
$204$	0	0
$205$	−15.3464	−1.07184
$206$	0	0
$207$	−1.45540	−0.101157
$208$	0	0
$209$	6.50833	0.450191
$210$	0	0
$211$	25.5704	1.76034	0.880169	−	0.474661i	$-0.157429\pi$
$211$	25.5704	1.76034	0.880169	+	0.474661i	$0.157429\pi$
$212$	0	0
$213$	13.3683	0.915979
$214$	0	0
$215$	14.8333	1.01162
$216$	0	0
$217$	−11.5986	−0.787361
$218$	0	0
$219$	−10.4710	−0.707566
$220$	0	0
$221$	−15.9740	−1.07453
$222$	0	0
$223$	14.9853	1.00349	0.501744	−	0.865016i	$-0.332692\pi$
$223$	14.9853	1.00349	0.501744	+	0.865016i	$0.332692\pi$
$224$	0	0
$225$	7.85673	0.523782
$226$	0	0
$227$	−16.3969	−1.08830	−0.544149	−	0.838989i	$-0.683147\pi$
$227$	−16.3969	−1.08830	−0.544149	+	0.838989i	$0.683147\pi$
$228$	0	0
$229$	−13.0986	−0.865578	−0.432789	−	0.901495i	$-0.642471\pi$
$229$	−13.0986	−0.865578	−0.432789	+	0.901495i	$0.642471\pi$
$230$	0	0
$231$	−3.28953	−0.216435
$232$	0	0
$233$	22.3541	1.46446	0.732232	−	0.681055i	$-0.238478\pi$
$233$	22.3541	1.46446	0.732232	+	0.681055i	$0.238478\pi$
$234$	0	0
$235$	−11.0147	−0.718518
$236$	0	0
$237$	6.01002	0.390393
$238$	0	0
$239$	−3.20508	−0.207319	−0.103660	−	0.994613i	$-0.533055\pi$
$239$	−3.20508	−0.207319	−0.103660	+	0.994613i	$0.533055\pi$
$240$	0	0
$241$	12.8916	0.830420	0.415210	−	0.909726i	$-0.363708\pi$
$241$	12.8916	0.830420	0.415210	+	0.909726i	$0.363708\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	−17.2467	−1.10185
$246$	0	0
$247$	−11.1034	−0.706494
$248$	0	0
$249$	−3.45375	−0.218872
$250$	0	0
$251$	1.00000	0.0631194
$251$	1.00000	0.0631194
$252$	0	0
$253$	3.23510	0.203389
$254$	0	0
$255$	15.1038	0.945836
$256$	0	0
$257$	−16.9368	−1.05649	−0.528244	−	0.849093i	$-0.677149\pi$
$257$	−16.9368	−1.05649	−0.528244	+	0.849093i	$0.677149\pi$
$258$	0	0
$259$	3.81391	0.236985
$260$	0	0
$261$	−8.55655	−0.529637
$262$	0	0
$263$	−4.57193	−0.281917	−0.140959	−	0.990015i	$-0.545018\pi$
$263$	−4.57193	−0.281917	−0.140959	+	0.990015i	$0.545018\pi$
$264$	0	0
$265$	9.60357	0.589943
$266$	0	0
$267$	−15.3843	−0.941502
$268$	0	0
$269$	5.13089	0.312836	0.156418	−	0.987691i	$-0.450005\pi$
$269$	5.13089	0.312836	0.156418	+	0.987691i	$0.450005\pi$
$270$	0	0
$271$	1.50403	0.0913633	0.0456817	−	0.998956i	$-0.485454\pi$
$271$	1.50403	0.0913633	0.0456817	+	0.998956i	$0.485454\pi$
$272$	0	0
$273$	5.61205	0.339657
$274$	0	0
$275$	−17.4642	−1.05313
$276$	0	0
$277$	23.4856	1.41111	0.705555	−	0.708655i	$-0.250698\pi$
$277$	23.4856	1.41111	0.705555	+	0.708655i	$0.250698\pi$
$278$	0	0
$279$	7.83747	0.469217
$280$	0	0
$281$	−0.487050	−0.0290550	−0.0145275	−	0.999894i	$-0.504624\pi$
$281$	−0.487050	−0.0290550	−0.0145275	+	0.999894i	$0.504624\pi$
$282$	0	0
$283$	−16.4694	−0.979006	−0.489503	−	0.872002i	$-0.662822\pi$
$283$	−16.4694	−0.979006	−0.489503	+	0.872002i	$0.662822\pi$
$284$	0	0
$285$	10.4985	0.621879
$286$	0	0
$287$	6.33386	0.373876
$288$	0	0
$289$	0.743586	0.0437404
$290$	0	0
$291$	9.36985	0.549270
$292$	0	0
$293$	−6.04404	−0.353097	−0.176548	−	0.984292i	$-0.556493\pi$
$293$	−6.04404	−0.353097	−0.176548	+	0.984292i	$0.556493\pi$
$294$	0	0
$295$	−43.3122	−2.52173
$296$	0	0
$297$	2.22283	0.128982
$298$	0	0
$299$	−5.51919	−0.319183
$300$	0	0
$301$	−6.12210	−0.352872
$302$	0	0
$303$	−8.02785	−0.461188
$304$	0	0
$305$	11.3267	0.648564
$306$	0	0
$307$	11.6886	0.667104	0.333552	−	0.942732i	$-0.391753\pi$
$307$	11.6886	0.667104	0.333552	+	0.942732i	$0.391753\pi$
$308$	0	0
$309$	4.16682	0.237042
$310$	0	0
$311$	−7.42886	−0.421252	−0.210626	−	0.977567i	$-0.567550\pi$
$311$	−7.42886	−0.421252	−0.210626	+	0.977567i	$0.567550\pi$
$312$	0	0
$313$	17.3066	0.978229	0.489115	−	0.872220i	$-0.337320\pi$
$313$	17.3066	0.978229	0.489115	+	0.872220i	$0.337320\pi$
$314$	0	0
$315$	−5.30631	−0.298977
$316$	0	0
$317$	−29.5794	−1.66134	−0.830672	−	0.556762i	$-0.812043\pi$
$317$	−29.5794	−1.66134	−0.830672	+	0.556762i	$0.812043\pi$
$318$	0	0
$319$	19.0198	1.06490
$320$	0	0
$321$	19.6558	1.09708
$322$	0	0
$323$	12.3334	0.686250
$324$	0	0
$325$	29.7945	1.65270
$326$	0	0
$327$	−2.67433	−0.147891
$328$	0	0
$329$	4.54605	0.250632
$330$	0	0
$331$	−19.8501	−1.09106	−0.545530	−	0.838092i	$-0.683671\pi$
$331$	−19.8501	−1.09106	−0.545530	+	0.838092i	$0.683671\pi$
$332$	0	0
$333$	−2.57717	−0.141228
$334$	0	0
$335$	−48.3363	−2.64090
$336$	0	0
$337$	−32.1437	−1.75098	−0.875489	−	0.483238i	$-0.839461\pi$
$337$	−32.1437	−1.75098	−0.875489	+	0.483238i	$0.839461\pi$
$338$	0	0
$339$	−5.26485	−0.285948
$340$	0	0
$341$	−17.4214	−0.943421
$342$	0	0
$343$	17.4773	0.943688
$344$	0	0
$345$	5.21851	0.280955
$346$	0	0
$347$	−2.13236	−0.114471	−0.0572357	−	0.998361i	$-0.518229\pi$
$347$	−2.13236	−0.114471	−0.0572357	+	0.998361i	$0.518229\pi$
$348$	0	0
$349$	−19.2319	−1.02946	−0.514729	−	0.857353i	$-0.672108\pi$
$349$	−19.2319	−1.02946	−0.514729	+	0.857353i	$0.672108\pi$
$350$	0	0
$351$	−3.79222	−0.202414
$352$	0	0
$353$	0.983095	0.0523248	0.0261624	−	0.999658i	$-0.491671\pi$
$353$	0.983095	0.0523248	0.0261624	+	0.999658i	$0.491671\pi$
$354$	0	0
$355$	−47.9337	−2.54405
$356$	0	0
$357$	−6.23374	−0.329924
$358$	0	0
$359$	2.71646	0.143369	0.0716845	−	0.997427i	$-0.477163\pi$
$359$	2.71646	0.143369	0.0716845	+	0.997427i	$0.477163\pi$
$360$	0	0
$361$	−10.4271	−0.548796
$362$	0	0
$363$	6.05902	0.318016
$364$	0	0
$365$	37.5452	1.96521
$366$	0	0
$367$	30.3435	1.58392	0.791960	−	0.610573i	$-0.209061\pi$
$367$	30.3435	1.58392	0.791960	+	0.610573i	$0.209061\pi$
$368$	0	0
$369$	−4.27997	−0.222806
$370$	0	0
$371$	−3.96365	−0.205782
$372$	0	0
$373$	−26.8477	−1.39012	−0.695061	−	0.718951i	$-0.744623\pi$
$373$	−26.8477	−1.39012	−0.695061	+	0.718951i	$0.744623\pi$
$374$	0	0
$375$	−10.2432	−0.528956
$376$	0	0
$377$	−32.4483	−1.67117
$378$	0	0
$379$	0.636725	0.0327064	0.0163532	−	0.999866i	$-0.494794\pi$
$379$	0.636725	0.0327064	0.0163532	+	0.999866i	$0.494794\pi$
$380$	0	0
$381$	9.02213	0.462218
$382$	0	0
$383$	1.26060	0.0644139	0.0322069	−	0.999481i	$-0.489746\pi$
$383$	1.26060	0.0644139	0.0322069	+	0.999481i	$0.489746\pi$
$384$	0	0
$385$	11.7950	0.601131
$386$	0	0
$387$	4.13688	0.210289
$388$	0	0
$389$	−26.4181	−1.33945	−0.669725	−	0.742609i	$-0.733588\pi$
$389$	−26.4181	−1.33945	−0.669725	+	0.742609i	$0.733588\pi$
$390$	0	0
$391$	6.13059	0.310037
$392$	0	0
$393$	−8.18602	−0.412930
$394$	0	0
$395$	−21.5497	−1.08428
$396$	0	0
$397$	−15.4034	−0.773076	−0.386538	−	0.922273i	$-0.626329\pi$
$397$	−15.4034	−0.773076	−0.386538	+	0.922273i	$0.626329\pi$
$398$	0	0
$399$	−4.33302	−0.216922
$400$	0	0
$401$	−32.2717	−1.61157	−0.805785	−	0.592208i	$-0.798256\pi$
$401$	−32.2717	−1.61157	−0.805785	+	0.592208i	$0.798256\pi$
$402$	0	0
$403$	29.7214	1.48053
$404$	0	0
$405$	3.58563	0.178171
$406$	0	0
$407$	5.72861	0.283957
$408$	0	0
$409$	16.3230	0.807118	0.403559	−	0.914954i	$-0.367773\pi$
$409$	16.3230	0.807118	0.403559	+	0.914954i	$0.367773\pi$
$410$	0	0
$411$	9.63996	0.475504
$412$	0	0
$413$	17.8761	0.879625
$414$	0	0
$415$	12.3839	0.607899
$416$	0	0
$417$	−10.2077	−0.499874
$418$	0	0
$419$	32.0484	1.56566	0.782832	−	0.622233i	$-0.213774\pi$
$419$	32.0484	1.56566	0.782832	+	0.622233i	$0.213774\pi$
$420$	0	0
$421$	−16.9770	−0.827409	−0.413704	−	0.910411i	$-0.635765\pi$
$421$	−16.9770	−0.827409	−0.413704	+	0.910411i	$0.635765\pi$
$422$	0	0
$423$	−3.07190	−0.149361
$424$	0	0
$425$	−33.0950	−1.60534
$426$	0	0
$427$	−4.67482	−0.226231
$428$	0	0
$429$	8.42947	0.406979
$430$	0	0
$431$	4.23195	0.203846	0.101923	−	0.994792i	$-0.467500\pi$
$431$	4.23195	0.203846	0.101923	+	0.994792i	$0.467500\pi$
$432$	0	0
$433$	−21.4394	−1.03031	−0.515155	−	0.857097i	$-0.672266\pi$
$433$	−21.4394	−1.03031	−0.515155	+	0.857097i	$0.672266\pi$
$434$	0	0
$435$	30.6806	1.47102
$436$	0	0
$437$	4.26132	0.203847
$438$	0	0
$439$	−27.6843	−1.32130	−0.660650	−	0.750694i	$-0.729719\pi$
$439$	−27.6843	−1.32130	−0.660650	+	0.750694i	$0.729719\pi$
$440$	0	0
$441$	−4.80994	−0.229045
$442$	0	0
$443$	19.6407	0.933157	0.466578	−	0.884480i	$-0.345487\pi$
$443$	19.6407	0.933157	0.466578	+	0.884480i	$0.345487\pi$
$444$	0	0
$445$	55.1623	2.61494
$446$	0	0
$447$	−0.113074	−0.00534822
$448$	0	0
$449$	−7.10924	−0.335506	−0.167753	−	0.985829i	$-0.553651\pi$
$449$	−7.10924	−0.335506	−0.167753	+	0.985829i	$0.553651\pi$
$450$	0	0
$451$	9.51365	0.447980
$452$	0	0
$453$	12.8768	0.605006
$454$	0	0
$455$	−20.1227	−0.943368
$456$	0	0
$457$	27.5155	1.28712	0.643560	−	0.765396i	$-0.277457\pi$
$457$	27.5155	1.28712	0.643560	+	0.765396i	$0.277457\pi$
$458$	0	0
$459$	4.21231	0.196614
$460$	0	0
$461$	2.75453	0.128291	0.0641457	−	0.997941i	$-0.479568\pi$
$461$	2.75453	0.128291	0.0641457	+	0.997941i	$0.479568\pi$
$462$	0	0
$463$	−31.4962	−1.46375	−0.731877	−	0.681437i	$-0.761355\pi$
$463$	−31.4962	−1.46375	−0.731877	+	0.681437i	$0.761355\pi$
$464$	0	0
$465$	−28.1023	−1.30321
$466$	0	0
$467$	16.7704	0.776043	0.388022	−	0.921650i	$-0.373159\pi$
$467$	16.7704	0.776043	0.388022	+	0.921650i	$0.373159\pi$
$468$	0	0
$469$	19.9497	0.921191
$470$	0	0
$471$	8.74045	0.402739
$472$	0	0
$473$	−9.19558	−0.422813
$474$	0	0
$475$	−23.0041	−1.05550
$476$	0	0
$477$	2.67835	0.122633
$478$	0	0
$479$	−16.6079	−0.758836	−0.379418	−	0.925225i	$-0.623876\pi$
$479$	−16.6079	−0.758836	−0.379418	+	0.925225i	$0.623876\pi$
$480$	0	0
$481$	−9.77320	−0.445620
$482$	0	0
$483$	−2.15382	−0.0980021
$484$	0	0
$485$	−33.5968	−1.52555
$486$	0	0
$487$	26.7427	1.21183	0.605915	−	0.795530i	$-0.292807\pi$
$487$	26.7427	1.21183	0.605915	+	0.795530i	$0.292807\pi$
$488$	0	0
$489$	−3.07679	−0.139137
$490$	0	0
$491$	−22.1183	−0.998183	−0.499091	−	0.866549i	$-0.666333\pi$
$491$	−22.1183	−0.998183	−0.499091	+	0.866549i	$0.666333\pi$
$492$	0	0
$493$	36.0429	1.62329
$494$	0	0
$495$	−7.97025	−0.358236
$496$	0	0
$497$	19.7835	0.887411
$498$	0	0
$499$	−12.2450	−0.548161	−0.274080	−	0.961707i	$-0.588373\pi$
$499$	−12.2450	−0.548161	−0.274080	+	0.961707i	$0.588373\pi$
$500$	0	0
$501$	2.92117	0.130508
$502$	0	0
$503$	20.3872	0.909021	0.454510	−	0.890741i	$-0.349814\pi$
$503$	20.3872	0.909021	0.454510	+	0.890741i	$0.349814\pi$
$504$	0	0
$505$	28.7849	1.28091
$506$	0	0
$507$	−1.38095	−0.0613302
$508$	0	0
$509$	−41.8755	−1.85610	−0.928049	−	0.372457i	$-0.878515\pi$
$509$	−41.8755	−1.85610	−0.928049	+	0.372457i	$0.878515\pi$
$510$	0	0
$511$	−15.4959	−0.685498
$512$	0	0
$513$	2.92795	0.129272
$514$	0	0
$515$	−14.9407	−0.658364
$516$	0	0
$517$	6.82830	0.300308
$518$	0	0
$519$	9.49890	0.416956
$520$	0	0
$521$	38.1373	1.67083	0.835413	−	0.549622i	$-0.185228\pi$
$521$	38.1373	1.67083	0.835413	+	0.549622i	$0.185228\pi$
$522$	0	0
$523$	19.1038	0.835352	0.417676	−	0.908596i	$-0.362845\pi$
$523$	19.1038	0.835352	0.417676	+	0.908596i	$0.362845\pi$
$524$	0	0
$525$	11.6271	0.507446
$526$	0	0
$527$	−33.0139	−1.43811
$528$	0	0
$529$	−20.8818	−0.907905
$530$	0	0
$531$	−12.0794	−0.524201
$532$	0	0
$533$	−16.2306	−0.703025
$534$	0	0
$535$	−70.4783	−3.04704
$536$	0	0
$537$	−17.7360	−0.765364
$538$	0	0
$539$	10.6917	0.460524
$540$	0	0
$541$	7.34668	0.315859	0.157929	−	0.987450i	$-0.449518\pi$
$541$	7.34668	0.315859	0.157929	+	0.987450i	$0.449518\pi$
$542$	0	0
$543$	−25.6798	−1.10203
$544$	0	0
$545$	9.58915	0.410754
$546$	0	0
$547$	29.1781	1.24756	0.623782	−	0.781598i	$-0.285595\pi$
$547$	29.1781	1.24756	0.623782	+	0.781598i	$0.285595\pi$
$548$	0	0
$549$	3.15891	0.134819
$550$	0	0
$551$	25.0531	1.06730
$552$	0	0
$553$	8.89414	0.378217
$554$	0	0
$555$	9.24078	0.392249
$556$	0	0
$557$	4.95909	0.210124	0.105062	−	0.994466i	$-0.466496\pi$
$557$	4.95909	0.210124	0.105062	+	0.994466i	$0.466496\pi$
$558$	0	0
$559$	15.6880	0.663530
$560$	0	0
$561$	−9.36326	−0.395317
$562$	0	0
$563$	−27.3434	−1.15239	−0.576194	−	0.817313i	$-0.695463\pi$
$563$	−27.3434	−1.15239	−0.576194	+	0.817313i	$0.695463\pi$
$564$	0	0
$565$	18.8778	0.794196
$566$	0	0
$567$	−1.47988	−0.0621493
$568$	0	0
$569$	32.3113	1.35456	0.677279	−	0.735726i	$-0.263159\pi$
$569$	32.3113	1.35456	0.677279	+	0.735726i	$0.263159\pi$
$570$	0	0
$571$	16.4391	0.687953	0.343977	−	0.938978i	$-0.388226\pi$
$571$	16.4391	0.687953	0.343977	+	0.938978i	$0.388226\pi$
$572$	0	0
$573$	3.10368	0.129658
$574$	0	0
$575$	−11.4347	−0.476858
$576$	0	0
$577$	−11.5196	−0.479567	−0.239783	−	0.970826i	$-0.577076\pi$
$577$	−11.5196	−0.479567	−0.239783	+	0.970826i	$0.577076\pi$
$578$	0	0
$579$	1.83074	0.0760830
$580$	0	0
$581$	−5.11114	−0.212046
$582$	0	0
$583$	−5.95352	−0.246570
$584$	0	0
$585$	13.5975	0.562188
$586$	0	0
$587$	24.6706	1.01827	0.509133	−	0.860688i	$-0.329966\pi$
$587$	24.6706	1.01827	0.509133	+	0.860688i	$0.329966\pi$
$588$	0	0
$589$	−22.9477	−0.945544
$590$	0	0
$591$	6.73467	0.277027
$592$	0	0
$593$	25.5434	1.04894	0.524471	−	0.851428i	$-0.324263\pi$
$593$	25.5434	1.04894	0.524471	+	0.851428i	$0.324263\pi$
$594$	0	0
$595$	22.3519	0.916337
$596$	0	0
$597$	14.4839	0.592785
$598$	0	0
$599$	−26.6243	−1.08784	−0.543920	−	0.839137i	$-0.683061\pi$
$599$	−26.6243	−1.08784	−0.543920	+	0.839137i	$0.683061\pi$
$600$	0	0
$601$	−40.0970	−1.63559	−0.817796	−	0.575509i	$-0.804804\pi$
$601$	−40.0970	−1.63559	−0.817796	+	0.575509i	$0.804804\pi$
$602$	0	0
$603$	−13.4806	−0.548971
$604$	0	0
$605$	−21.7254	−0.883263
$606$	0	0
$607$	11.1596	0.452956	0.226478	−	0.974016i	$-0.427279\pi$
$607$	11.1596	0.452956	0.226478	+	0.974016i	$0.427279\pi$
$608$	0	0
$609$	−12.6627	−0.513118
$610$	0	0
$611$	−11.6493	−0.471281
$612$	0	0
$613$	−12.7214	−0.513812	−0.256906	−	0.966436i	$-0.582703\pi$
$613$	−12.7214	−0.513812	−0.256906	+	0.966436i	$0.582703\pi$
$614$	0	0
$615$	15.3464	0.618826
$616$	0	0
$617$	6.81438	0.274337	0.137168	−	0.990548i	$-0.456200\pi$
$617$	6.81438	0.274337	0.137168	+	0.990548i	$0.456200\pi$
$618$	0	0
$619$	−26.6101	−1.06955	−0.534776	−	0.844994i	$-0.679604\pi$
$619$	−26.6101	−1.06955	−0.534776	+	0.844994i	$0.679604\pi$
$620$	0	0
$621$	1.45540	0.0584031
$622$	0	0
$623$	−22.7669	−0.912138
$624$	0	0
$625$	−2.55539	−0.102216
$626$	0	0
$627$	−6.50833	−0.259918
$628$	0	0
$629$	10.8559	0.432851
$630$	0	0
$631$	−15.3400	−0.610678	−0.305339	−	0.952244i	$-0.598770\pi$
$631$	−15.3400	−0.610678	−0.305339	+	0.952244i	$0.598770\pi$
$632$	0	0
$633$	−25.5704	−1.01633
$634$	0	0
$635$	−32.3500	−1.28377
$636$	0	0
$637$	−18.2404	−0.722710
$638$	0	0
$639$	−13.3683	−0.528841
$640$	0	0
$641$	−37.1432	−1.46707	−0.733533	−	0.679654i	$-0.762130\pi$
$641$	−37.1432	−1.46707	−0.733533	+	0.679654i	$0.762130\pi$
$642$	0	0
$643$	39.7337	1.56694	0.783472	−	0.621428i	$-0.213447\pi$
$643$	39.7337	1.56694	0.783472	+	0.621428i	$0.213447\pi$
$644$	0	0
$645$	−14.8333	−0.584061
$646$	0	0
$647$	42.6637	1.67728	0.838641	−	0.544685i	$-0.183351\pi$
$647$	42.6637	1.67728	0.838641	+	0.544685i	$0.183351\pi$
$648$	0	0
$649$	26.8504	1.05397
$650$	0	0
$651$	11.5986	0.454583
$652$	0	0
$653$	−42.0476	−1.64545	−0.822725	−	0.568439i	$-0.807547\pi$
$653$	−42.0476	−1.64545	−0.822725	+	0.568439i	$0.807547\pi$
$654$	0	0
$655$	29.3520	1.14688
$656$	0	0
$657$	10.4710	0.408513
$658$	0	0
$659$	−39.3247	−1.53187	−0.765936	−	0.642917i	$-0.777724\pi$
$659$	−39.3247	−1.53187	−0.765936	+	0.642917i	$0.777724\pi$
$660$	0	0
$661$	7.45882	0.290114	0.145057	−	0.989423i	$-0.453663\pi$
$661$	7.45882	0.290114	0.145057	+	0.989423i	$0.453663\pi$
$662$	0	0
$663$	15.9740	0.620380
$664$	0	0
$665$	15.5366	0.602484
$666$	0	0
$667$	12.4532	0.482189
$668$	0	0
$669$	−14.9853	−0.579364
$670$	0	0
$671$	−7.02173	−0.271071
$672$	0	0
$673$	12.9696	0.499940	0.249970	−	0.968254i	$-0.419579\pi$
$673$	12.9696	0.499940	0.249970	+	0.968254i	$0.419579\pi$
$674$	0	0
$675$	−7.85673	−0.302406
$676$	0	0
$677$	−34.7510	−1.33559	−0.667794	−	0.744346i	$-0.732761\pi$
$677$	−34.7510	−1.33559	−0.667794	+	0.744346i	$0.732761\pi$
$678$	0	0
$679$	13.8663	0.532139
$680$	0	0
$681$	16.3969	0.628329
$682$	0	0
$683$	−7.44060	−0.284707	−0.142353	−	0.989816i	$-0.545467\pi$
$683$	−7.44060	−0.284707	−0.142353	+	0.989816i	$0.545467\pi$
$684$	0	0
$685$	−34.5653	−1.32067
$686$	0	0
$687$	13.0986	0.499742
$688$	0	0
$689$	10.1569	0.386947
$690$	0	0
$691$	−33.2418	−1.26458	−0.632290	−	0.774732i	$-0.717885\pi$
$691$	−33.2418	−1.26458	−0.632290	+	0.774732i	$0.717885\pi$
$692$	0	0
$693$	3.28953	0.124959
$694$	0	0
$695$	36.6011	1.38836
$696$	0	0
$697$	18.0286	0.682881
$698$	0	0
$699$	−22.3541	−0.845509
$700$	0	0
$701$	13.4904	0.509527	0.254764	−	0.967003i	$-0.418002\pi$
$701$	13.4904	0.509527	0.254764	+	0.967003i	$0.418002\pi$
$702$	0	0
$703$	7.54582	0.284596
$704$	0	0
$705$	11.0147	0.414837
$706$	0	0
$707$	−11.8803	−0.446804
$708$	0	0
$709$	−47.5321	−1.78511	−0.892553	−	0.450943i	$-0.851088\pi$
$709$	−47.5321	−1.78511	−0.892553	+	0.450943i	$0.851088\pi$
$710$	0	0
$711$	−6.01002	−0.225394
$712$	0	0
$713$	−11.4066	−0.427182
$714$	0	0
$715$	−30.2249	−1.13035
$716$	0	0
$717$	3.20508	0.119696
$718$	0	0
$719$	−22.7869	−0.849809	−0.424904	−	0.905238i	$-0.639692\pi$
$719$	−22.7869	−0.849809	−0.424904	+	0.905238i	$0.639692\pi$
$720$	0	0
$721$	6.16640	0.229649
$722$	0	0
$723$	−12.8916	−0.479443
$724$	0	0
$725$	−67.2265	−2.49673
$726$	0	0
$727$	48.2854	1.79080	0.895402	−	0.445258i	$-0.146888\pi$
$727$	48.2854	1.79080	0.895402	+	0.445258i	$0.146888\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−17.4258	−0.644518
$732$	0	0
$733$	7.52850	0.278072	0.139036	−	0.990287i	$-0.455600\pi$
$733$	7.52850	0.278072	0.139036	+	0.990287i	$0.455600\pi$
$734$	0	0
$735$	17.2467	0.636153
$736$	0	0
$737$	29.9650	1.10378
$738$	0	0
$739$	−27.6158	−1.01586	−0.507931	−	0.861398i	$-0.669589\pi$
$739$	−27.6158	−1.01586	−0.507931	+	0.861398i	$0.669589\pi$
$740$	0	0
$741$	11.1034	0.407894
$742$	0	0
$743$	−40.4237	−1.48300	−0.741501	−	0.670952i	$-0.765886\pi$
$743$	−40.4237	−1.48300	−0.741501	+	0.670952i	$0.765886\pi$
$744$	0	0
$745$	0.405442	0.0148542
$746$	0	0
$747$	3.45375	0.126366
$748$	0	0
$749$	29.0883	1.06286
$750$	0	0
$751$	50.0086	1.82484	0.912420	−	0.409255i	$-0.134211\pi$
$751$	50.0086	1.82484	0.912420	+	0.409255i	$0.134211\pi$
$752$	0	0
$753$	−1.00000	−0.0364420
$754$	0	0
$755$	−46.1715	−1.68035
$756$	0	0
$757$	42.5639	1.54701	0.773505	−	0.633790i	$-0.218502\pi$
$757$	42.5639	1.54701	0.773505	+	0.633790i	$0.218502\pi$
$758$	0	0
$759$	−3.23510	−0.117427
$760$	0	0
$761$	−52.0623	−1.88726	−0.943628	−	0.331007i	$-0.892612\pi$
$761$	−52.0623	−1.88726	−0.943628	+	0.331007i	$0.892612\pi$
$762$	0	0
$763$	−3.95770	−0.143278
$764$	0	0
$765$	−15.1038	−0.546079
$766$	0	0
$767$	−45.8077	−1.65402
$768$	0	0
$769$	5.23366	0.188731	0.0943653	−	0.995538i	$-0.469918\pi$
$769$	5.23366	0.188731	0.0943653	+	0.995538i	$0.469918\pi$
$770$	0	0
$771$	16.9368	0.609964
$772$	0	0
$773$	−1.52952	−0.0550131	−0.0275066	−	0.999622i	$-0.508757\pi$
$773$	−1.52952	−0.0550131	−0.0275066	+	0.999622i	$0.508757\pi$
$774$	0	0
$775$	61.5770	2.21191
$776$	0	0
$777$	−3.81391	−0.136823
$778$	0	0
$779$	12.5315	0.448988
$780$	0	0
$781$	29.7154	1.06330
$782$	0	0
$783$	8.55655	0.305786
$784$	0	0
$785$	−31.3400	−1.11857
$786$	0	0
$787$	−0.377405	−0.0134530	−0.00672651	−	0.999977i	$-0.502141\pi$
$787$	−0.377405	−0.0134530	−0.00672651	+	0.999977i	$0.502141\pi$
$788$	0	0
$789$	4.57193	0.162765
$790$	0	0
$791$	−7.79137	−0.277029
$792$	0	0
$793$	11.9793	0.425397
$794$	0	0
$795$	−9.60357	−0.340604
$796$	0	0
$797$	18.9070	0.669721	0.334861	−	0.942268i	$-0.391311\pi$
$797$	18.9070	0.669721	0.334861	+	0.942268i	$0.391311\pi$
$798$	0	0
$799$	12.9398	0.457777
$800$	0	0
$801$	15.3843	0.543577
$802$	0	0
$803$	−23.2753	−0.821368
$804$	0	0
$805$	7.72279	0.272193
$806$	0	0
$807$	−5.13089	−0.180616
$808$	0	0
$809$	−34.5929	−1.21622	−0.608111	−	0.793852i	$-0.708072\pi$
$809$	−34.5929	−1.21622	−0.608111	+	0.793852i	$0.708072\pi$
$810$	0	0
$811$	36.3807	1.27750	0.638750	−	0.769414i	$-0.279452\pi$
$811$	36.3807	1.27750	0.638750	+	0.769414i	$0.279452\pi$
$812$	0	0
$813$	−1.50403	−0.0527486
$814$	0	0
$815$	11.0322	0.386443
$816$	0	0
$817$	−12.1126	−0.423765
$818$	0	0
$819$	−5.61205	−0.196101
$820$	0	0
$821$	26.4992	0.924828	0.462414	−	0.886664i	$-0.346983\pi$
$821$	26.4992	0.924828	0.462414	+	0.886664i	$0.346983\pi$
$822$	0	0
$823$	20.6018	0.718133	0.359067	−	0.933312i	$-0.383095\pi$
$823$	20.6018	0.718133	0.359067	+	0.933312i	$0.383095\pi$
$824$	0	0
$825$	17.4642	0.608025
$826$	0	0
$827$	6.55484	0.227934	0.113967	−	0.993485i	$-0.463644\pi$
$827$	6.55484	0.227934	0.113967	+	0.993485i	$0.463644\pi$
$828$	0	0
$829$	−15.5037	−0.538467	−0.269234	−	0.963075i	$-0.586770\pi$
$829$	−15.5037	−0.538467	−0.269234	+	0.963075i	$0.586770\pi$
$830$	0	0
$831$	−23.4856	−0.814705
$832$	0	0
$833$	20.2610	0.702002
$834$	0	0
$835$	−10.4742	−0.362476
$836$	0	0
$837$	−7.83747	−0.270903
$838$	0	0
$839$	8.45593	0.291931	0.145966	−	0.989290i	$-0.453371\pi$
$839$	8.45593	0.291931	0.145966	+	0.989290i	$0.453371\pi$
$840$	0	0
$841$	44.2145	1.52464
$842$	0	0
$843$	0.487050	0.0167749
$844$	0	0
$845$	4.95158	0.170339
$846$	0	0
$847$	8.96665	0.308098
$848$	0	0
$849$	16.4694	0.565229
$850$	0	0
$851$	3.75081	0.128576
$852$	0	0
$853$	−0.141217	−0.00483518	−0.00241759	−	0.999997i	$-0.500770\pi$
$853$	−0.141217	−0.00483518	−0.00241759	+	0.999997i	$0.500770\pi$
$854$	0	0
$855$	−10.4985	−0.359042
$856$	0	0
$857$	4.06741	0.138940	0.0694700	−	0.997584i	$-0.477869\pi$
$857$	4.06741	0.138940	0.0694700	+	0.997584i	$0.477869\pi$
$858$	0	0
$859$	−17.8551	−0.609209	−0.304605	−	0.952479i	$-0.598524\pi$
$859$	−17.8551	−0.609209	−0.304605	+	0.952479i	$0.598524\pi$
$860$	0	0
$861$	−6.33386	−0.215857
$862$	0	0
$863$	3.64596	0.124110	0.0620550	−	0.998073i	$-0.480235\pi$
$863$	3.64596	0.124110	0.0620550	+	0.998073i	$0.480235\pi$
$864$	0	0
$865$	−34.0595	−1.15806
$866$	0	0
$867$	−0.743586	−0.0252535
$868$	0	0
$869$	13.3593	0.453182
$870$	0	0
$871$	−51.1213	−1.73218
$872$	0	0
$873$	−9.36985	−0.317121
$874$	0	0
$875$	−15.1587	−0.512459
$876$	0	0
$877$	46.9039	1.58383	0.791916	−	0.610630i	$-0.209084\pi$
$877$	46.9039	1.58383	0.791916	+	0.610630i	$0.209084\pi$
$878$	0	0
$879$	6.04404	0.203860
$880$	0	0
$881$	−14.0384	−0.472967	−0.236484	−	0.971636i	$-0.575995\pi$
$881$	−14.0384	−0.472967	−0.236484	+	0.971636i	$0.575995\pi$
$882$	0	0
$883$	31.6537	1.06523	0.532615	−	0.846357i	$-0.321209\pi$
$883$	31.6537	1.06523	0.532615	+	0.846357i	$0.321209\pi$
$884$	0	0
$885$	43.3122	1.45592
$886$	0	0
$887$	0.523524	0.0175782	0.00878911	−	0.999961i	$-0.497202\pi$
$887$	0.523524	0.0175782	0.00878911	+	0.999961i	$0.497202\pi$
$888$	0	0
$889$	13.3517	0.447802
$890$	0	0
$891$	−2.22283	−0.0744676
$892$	0	0
$893$	8.99435	0.300984
$894$	0	0
$895$	63.5947	2.12574
$896$	0	0
$897$	5.51919	0.184280
$898$	0	0
$899$	−67.0617	−2.23663
$900$	0	0
$901$	−11.2821	−0.375860
$902$	0	0
$903$	6.12210	0.203731
$904$	0	0
$905$	92.0784	3.06079
$906$	0	0
$907$	−11.6138	−0.385629	−0.192815	−	0.981235i	$-0.561762\pi$
$907$	−11.6138	−0.385629	−0.192815	+	0.981235i	$0.561762\pi$
$908$	0	0
$909$	8.02785	0.266267
$910$	0	0
$911$	−19.9411	−0.660679	−0.330340	−	0.943862i	$-0.607163\pi$
$911$	−19.9411	−0.660679	−0.330340	+	0.943862i	$0.607163\pi$
$912$	0	0
$913$	−7.67710	−0.254075
$914$	0	0
$915$	−11.3267	−0.374449
$916$	0	0
$917$	−12.1144	−0.400052
$918$	0	0
$919$	46.5296	1.53487	0.767435	−	0.641126i	$-0.221533\pi$
$919$	46.5296	1.53487	0.767435	+	0.641126i	$0.221533\pi$
$920$	0	0
$921$	−11.6886	−0.385153
$922$	0	0
$923$	−50.6954	−1.66866
$924$	0	0
$925$	−20.2481	−0.665755
$926$	0	0
$927$	−4.16682	−0.136856
$928$	0	0
$929$	40.9768	1.34441	0.672203	−	0.740367i	$-0.265348\pi$
$929$	40.9768	1.34441	0.672203	+	0.740367i	$0.265348\pi$
$930$	0	0
$931$	14.0833	0.461560
$932$	0	0
$933$	7.42886	0.243210
$934$	0	0
$935$	33.5732	1.09796
$936$	0	0
$937$	1.87910	0.0613875	0.0306937	−	0.999529i	$-0.490228\pi$
$937$	1.87910	0.0613875	0.0306937	+	0.999529i	$0.490228\pi$
$938$	0	0
$939$	−17.3066	−0.564781
$940$	0	0
$941$	−41.6260	−1.35697	−0.678485	−	0.734615i	$-0.737363\pi$
$941$	−41.6260	−1.35697	−0.678485	+	0.734615i	$0.737363\pi$
$942$	0	0
$943$	6.22905	0.202846
$944$	0	0
$945$	5.30631	0.172614
$946$	0	0
$947$	30.6746	0.996792	0.498396	−	0.866950i	$-0.333923\pi$
$947$	30.6746	0.996792	0.498396	+	0.866950i	$0.333923\pi$
$948$	0	0
$949$	39.7084	1.28899
$950$	0	0
$951$	29.5794	0.959177
$952$	0	0
$953$	48.8677	1.58298	0.791489	−	0.611183i	$-0.209306\pi$
$953$	48.8677	1.58298	0.791489	+	0.611183i	$0.209306\pi$
$954$	0	0
$955$	−11.1286	−0.360114
$956$	0	0
$957$	−19.0198	−0.614821
$958$	0	0
$959$	14.2660	0.460674
$960$	0	0
$961$	30.4260	0.981484
$962$	0	0
$963$	−19.6558	−0.633399
$964$	0	0
$965$	−6.56436	−0.211314
$966$	0	0
$967$	44.5556	1.43281	0.716406	−	0.697684i	$-0.245786\pi$
$967$	44.5556	1.43281	0.716406	+	0.697684i	$0.245786\pi$
$968$	0	0
$969$	−12.3334	−0.396207
$970$	0	0
$971$	30.7823	0.987851	0.493926	−	0.869504i	$-0.335562\pi$
$971$	30.7823	0.987851	0.493926	+	0.869504i	$0.335562\pi$
$972$	0	0
$973$	−15.1062	−0.484284
$974$	0	0
$975$	−29.7945	−0.954187
$976$	0	0
$977$	10.8410	0.346835	0.173417	−	0.984848i	$-0.444519\pi$
$977$	10.8410	0.346835	0.173417	+	0.984848i	$0.444519\pi$
$978$	0	0
$979$	−34.1966	−1.09293
$980$	0	0
$981$	2.67433	0.0853847
$982$	0	0
$983$	−28.1195	−0.896872	−0.448436	−	0.893815i	$-0.648019\pi$
$983$	−28.1195	−0.896872	−0.448436	+	0.893815i	$0.648019\pi$
$984$	0	0
$985$	−24.1480	−0.769420
$986$	0	0
$987$	−4.54605	−0.144702
$988$	0	0
$989$	−6.02080	−0.191450
$990$	0	0
$991$	46.5126	1.47752	0.738761	−	0.673968i	$-0.235411\pi$
$991$	46.5126	1.47752	0.738761	+	0.673968i	$0.235411\pi$
$992$	0	0
$993$	19.8501	0.629923
$994$	0	0
$995$	−51.9337	−1.64641
$996$	0	0
$997$	−30.7497	−0.973852	−0.486926	−	0.873443i	$-0.661882\pi$
$997$	−30.7497	−0.973852	−0.486926	+	0.873443i	$0.661882\pi$
$998$	0	0
$999$	2.57717	0.0815381

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6024.2.a.o.1.13	✓	14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6024.2.a.o.1.13	✓	14	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:	\( N \)	\(=\)	\( 6024 = 2^{3} \cdot 3 \cdot 251 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6024.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +3.58563 q^{5} -1.47988 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +3.58563 q^{5} -1.47988 q^{7} +1.00000 q^{9} -2.22283 q^{11} +3.79222 q^{13} -3.58563 q^{15} -4.21231 q^{17} -2.92795 q^{19} +1.47988 q^{21} -1.45540 q^{23} +7.85673 q^{25} -1.00000 q^{27} -8.55655 q^{29} +7.83747 q^{31} +2.22283 q^{33} -5.30631 q^{35} -2.57717 q^{37} -3.79222 q^{39} -4.27997 q^{41} +4.13688 q^{43} +3.58563 q^{45} -3.07190 q^{47} -4.80994 q^{49} +4.21231 q^{51} +2.67835 q^{53} -7.97025 q^{55} +2.92795 q^{57} -12.0794 q^{59} +3.15891 q^{61} -1.47988 q^{63} +13.5975 q^{65} -13.4806 q^{67} +1.45540 q^{69} -13.3683 q^{71} +10.4710 q^{73} -7.85673 q^{75} +3.28953 q^{77} -6.01002 q^{79} +1.00000 q^{81} +3.45375 q^{83} -15.1038 q^{85} +8.55655 q^{87} +15.3843 q^{89} -5.61205 q^{91} -7.83747 q^{93} -10.4985 q^{95} -9.36985 q^{97} -2.22283 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q - 14 q^{3} - 3 q^{5} + 7 q^{7} + 14 q^{9}+O(q^{10}) \) 14 * q - 14 * q^3 - 3 * q^5 + 7 * q^7 + 14 * q^9	\( 14 q - 14 q^{3} - 3 q^{5} + 7 q^{7} + 14 q^{9} - 22 q^{11} + 9 q^{13} + 3 q^{15} - 7 q^{21} - 17 q^{23} + 25 q^{25} - 14 q^{27} - 18 q^{29} - 7 q^{31} + 22 q^{33} - 27 q^{35} + 9 q^{37} - 9 q^{39} - 6 q^{41} - 14 q^{43} - 3 q^{45} - 15 q^{47} + 15 q^{49} - 11 q^{53} + 2 q^{55} - 36 q^{59} + 2 q^{61} + 7 q^{63} + 8 q^{65} + 3 q^{67} + 17 q^{69} - 29 q^{71} + 2 q^{73} - 25 q^{75} + 8 q^{77} + 23 q^{79} + 14 q^{81} - 55 q^{83} + 7 q^{85} + 18 q^{87} + 9 q^{89} - 22 q^{91} + 7 q^{93} - 27 q^{95} + 17 q^{97} - 22 q^{99}+O(q^{100}) \) 14 * q - 14 * q^3 - 3 * q^5 + 7 * q^7 + 14 * q^9 - 22 * q^11 + 9 * q^13 + 3 * q^15 - 7 * q^21 - 17 * q^23 + 25 * q^25 - 14 * q^27 - 18 * q^29 - 7 * q^31 + 22 * q^33 - 27 * q^35 + 9 * q^37 - 9 * q^39 - 6 * q^41 - 14 * q^43 - 3 * q^45 - 15 * q^47 + 15 * q^49 - 11 * q^53 + 2 * q^55 - 36 * q^59 + 2 * q^61 + 7 * q^63 + 8 * q^65 + 3 * q^67 + 17 * q^69 - 29 * q^71 + 2 * q^73 - 25 * q^75 + 8 * q^77 + 23 * q^79 + 14 * q^81 - 55 * q^83 + 7 * q^85 + 18 * q^87 + 9 * q^89 - 22 * q^91 + 7 * q^93 - 27 * q^95 + 17 * q^97 - 22 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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