LMFDB - Embedded newform 6024.2.a.p.1.2

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:	$0$
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} - 6 x^{13} - 29 x^{12} + 210 x^{11} + 280 x^{10} - 2796 x^{9} - 863 x^{8} + 17652 x^{7} + \cdots - 3364 $ x^14 - 6x^13 - 29x^12 + 210x^11 + 280x^10 - 2796x^9 - 863x^8 + 17652x^7 - 1300x^6 - 53458x^5 + 10892x^4 + 67298x^3 - 13761x^2 - 22140*x - 3364
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-3.23770$ 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)

$f(q)$	$=$	$q-1.00000 q^{3} -3.23770 q^{5} +2.16912 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} -3.23770 q^{5} +2.16912 q^{7} +1.00000 q^{9} -1.14688 q^{11} -2.10679 q^{13} +3.23770 q^{15} +7.69596 q^{17} +6.58513 q^{19} -2.16912 q^{21} +7.25438 q^{23} +5.48268 q^{25} -1.00000 q^{27} -3.25115 q^{29} -6.11884 q^{31} +1.14688 q^{33} -7.02294 q^{35} -5.01961 q^{37} +2.10679 q^{39} -4.15988 q^{41} -6.96040 q^{43} -3.23770 q^{45} +8.84769 q^{47} -2.29494 q^{49} -7.69596 q^{51} +4.17888 q^{53} +3.71326 q^{55} -6.58513 q^{57} +1.07256 q^{59} -2.11883 q^{61} +2.16912 q^{63} +6.82114 q^{65} -8.68247 q^{67} -7.25438 q^{69} +8.82001 q^{71} +0.732215 q^{73} -5.48268 q^{75} -2.48772 q^{77} -1.27601 q^{79} +1.00000 q^{81} -10.0933 q^{83} -24.9172 q^{85} +3.25115 q^{87} -7.36931 q^{89} -4.56987 q^{91} +6.11884 q^{93} -21.3207 q^{95} -4.11616 q^{97} -1.14688 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 14 q - 14 q^{3} + 6 q^{5} + 3 q^{7} + 14 q^{9}+O(q^{10}) $ 14 * q - 14 * q^3 + 6 * q^5 + 3 * q^7 + 14 * q^9	$ 14 q - 14 q^{3} + 6 q^{5} + 3 q^{7} + 14 q^{9} + 7 q^{11} + 5 q^{13} - 6 q^{15} + 28 q^{17} - q^{19} - 3 q^{21} - 3 q^{23} + 24 q^{25} - 14 q^{27} + 15 q^{29} + 9 q^{31} - 7 q^{33} - 14 q^{35} + 4 q^{37} - 5 q^{39} + 32 q^{41} - 3 q^{43} + 6 q^{45} + 15 q^{47} + 27 q^{49} - 28 q^{51} + 8 q^{53} + q^{57} - 11 q^{59} + 25 q^{61} + 3 q^{63} + 18 q^{65} - 20 q^{67} + 3 q^{69} + 33 q^{71} + 8 q^{73} - 24 q^{75} + 14 q^{77} - 17 q^{79} + 14 q^{81} - 4 q^{83} + 16 q^{85} - 15 q^{87} + 14 q^{89} - 28 q^{91} - 9 q^{93} + 3 q^{95} + 9 q^{97} + 7 q^{99}+O(q^{100}) $ 14 * q - 14 * q^3 + 6 * q^5 + 3 * q^7 + 14 * q^9 + 7 * q^11 + 5 * q^13 - 6 * q^15 + 28 * q^17 - q^19 - 3 * q^21 - 3 * q^23 + 24 * q^25 - 14 * q^27 + 15 * q^29 + 9 * q^31 - 7 * q^33 - 14 * q^35 + 4 * q^37 - 5 * q^39 + 32 * q^41 - 3 * q^43 + 6 * q^45 + 15 * q^47 + 27 * q^49 - 28 * q^51 + 8 * q^53 + q^57 - 11 * q^59 + 25 * q^61 + 3 * q^63 + 18 * q^65 - 20 * q^67 + 3 * q^69 + 33 * q^71 + 8 * q^73 - 24 * q^75 + 14 * q^77 - 17 * q^79 + 14 * q^81 - 4 * q^83 + 16 * q^85 - 15 * q^87 + 14 * q^89 - 28 * q^91 - 9 * q^93 + 3 * q^95 + 9 * q^97 + 7 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	−3.23770	−1.44794	−0.723971	−	0.689830i	$-0.757685\pi$
$5$	−3.23770	−1.44794	−0.723971	+	0.689830i	$0.757685\pi$
$6$	0	0
$7$	2.16912	0.819849	0.409924	−	0.912120i	$-0.365555\pi$
$7$	2.16912	0.819849	0.409924	+	0.912120i	$0.365555\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−1.14688	−0.345798	−0.172899	−	0.984940i	$-0.555313\pi$
$11$	−1.14688	−0.345798	−0.172899	+	0.984940i	$0.555313\pi$
$12$	0	0
$13$	−2.10679	−0.584318	−0.292159	−	0.956370i	$-0.594374\pi$
$13$	−2.10679	−0.584318	−0.292159	+	0.956370i	$0.594374\pi$
$14$	0	0
$15$	3.23770	0.835970
$16$	0	0
$17$	7.69596	1.86654	0.933272	−	0.359171i	$-0.116940\pi$
$17$	7.69596	1.86654	0.933272	+	0.359171i	$0.116940\pi$
$18$	0	0
$19$	6.58513	1.51073	0.755367	−	0.655302i	$-0.227459\pi$
$19$	6.58513	1.51073	0.755367	+	0.655302i	$0.227459\pi$
$20$	0	0
$21$	−2.16912	−0.473340
$22$	0	0
$23$	7.25438	1.51264	0.756322	−	0.654200i	$-0.226994\pi$
$23$	7.25438	1.51264	0.756322	+	0.654200i	$0.226994\pi$
$24$	0	0
$25$	5.48268	1.09654
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−3.25115	−0.603723	−0.301861	−	0.953352i	$-0.597608\pi$
$29$	−3.25115	−0.603723	−0.301861	+	0.953352i	$0.597608\pi$
$30$	0	0
$31$	−6.11884	−1.09898	−0.549488	−	0.835502i	$-0.685177\pi$
$31$	−6.11884	−1.09898	−0.549488	+	0.835502i	$0.685177\pi$
$32$	0	0
$33$	1.14688	0.199647
$34$	0	0
$35$	−7.02294	−1.18709
$36$	0	0
$37$	−5.01961	−0.825219	−0.412610	−	0.910908i	$-0.635383\pi$
$37$	−5.01961	−0.825219	−0.412610	+	0.910908i	$0.635383\pi$
$38$	0	0
$39$	2.10679	0.337356
$40$	0	0
$41$	−4.15988	−0.649664	−0.324832	−	0.945772i	$-0.605308\pi$
$41$	−4.15988	−0.649664	−0.324832	+	0.945772i	$0.605308\pi$
$42$	0	0
$43$	−6.96040	−1.06145	−0.530725	−	0.847544i	$-0.678080\pi$
$43$	−6.96040	−1.06145	−0.530725	+	0.847544i	$0.678080\pi$
$44$	0	0
$45$	−3.23770	−0.482647
$46$	0	0
$47$	8.84769	1.29057	0.645284	−	0.763942i	$-0.276739\pi$
$47$	8.84769	1.29057	0.645284	+	0.763942i	$0.276739\pi$
$48$	0	0
$49$	−2.29494	−0.327848
$50$	0	0
$51$	−7.69596	−1.07765
$52$	0	0
$53$	4.17888	0.574013	0.287006	−	0.957929i	$-0.407340\pi$
$53$	4.17888	0.574013	0.287006	+	0.957929i	$0.407340\pi$
$54$	0	0
$55$	3.71326	0.500696
$56$	0	0
$57$	−6.58513	−0.872222
$58$	0	0
$59$	1.07256	0.139635	0.0698175	−	0.997560i	$-0.477758\pi$
$59$	1.07256	0.139635	0.0698175	+	0.997560i	$0.477758\pi$
$60$	0	0
$61$	−2.11883	−0.271288	−0.135644	−	0.990758i	$-0.543310\pi$
$61$	−2.11883	−0.271288	−0.135644	+	0.990758i	$0.543310\pi$
$62$	0	0
$63$	2.16912	0.273283
$64$	0	0
$65$	6.82114	0.846058
$66$	0	0
$67$	−8.68247	−1.06073	−0.530366	−	0.847769i	$-0.677946\pi$
$67$	−8.68247	−1.06073	−0.530366	+	0.847769i	$0.677946\pi$
$68$	0	0
$69$	−7.25438	−0.873325
$70$	0	0
$71$	8.82001	1.04674	0.523371	−	0.852105i	$-0.324674\pi$
$71$	8.82001	1.04674	0.523371	+	0.852105i	$0.324674\pi$
$72$	0	0
$73$	0.732215	0.0856993	0.0428497	−	0.999082i	$-0.486356\pi$
$73$	0.732215	0.0856993	0.0428497	+	0.999082i	$0.486356\pi$
$74$	0	0
$75$	−5.48268	−0.633086
$76$	0	0
$77$	−2.48772	−0.283502
$78$	0	0
$79$	−1.27601	−0.143562	−0.0717811	−	0.997420i	$-0.522868\pi$
$79$	−1.27601	−0.143562	−0.0717811	+	0.997420i	$0.522868\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−10.0933	−1.10788	−0.553940	−	0.832557i	$-0.686876\pi$
$83$	−10.0933	−1.10788	−0.553940	+	0.832557i	$0.686876\pi$
$84$	0	0
$85$	−24.9172	−2.70265
$86$	0	0
$87$	3.25115	0.348560
$88$	0	0
$89$	−7.36931	−0.781145	−0.390573	−	0.920572i	$-0.627723\pi$
$89$	−7.36931	−0.781145	−0.390573	+	0.920572i	$0.627723\pi$
$90$	0	0
$91$	−4.56987	−0.479052
$92$	0	0
$93$	6.11884	0.634494
$94$	0	0
$95$	−21.3207	−2.18745
$96$	0	0
$97$	−4.11616	−0.417933	−0.208966	−	0.977923i	$-0.567010\pi$
$97$	−4.11616	−0.417933	−0.208966	+	0.977923i	$0.567010\pi$
$98$	0	0
$99$	−1.14688	−0.115266
$100$	0	0
$101$	−0.837083	−0.0832929	−0.0416465	−	0.999132i	$-0.513260\pi$
$101$	−0.837083	−0.0832929	−0.0416465	+	0.999132i	$0.513260\pi$
$102$	0	0
$103$	8.89112	0.876068	0.438034	−	0.898958i	$-0.355675\pi$
$103$	8.89112	0.876068	0.438034	+	0.898958i	$0.355675\pi$
$104$	0	0
$105$	7.02294	0.685369
$106$	0	0
$107$	11.5664	1.11817	0.559084	−	0.829111i	$-0.311153\pi$
$107$	11.5664	1.11817	0.559084	+	0.829111i	$0.311153\pi$
$108$	0	0
$109$	11.6612	1.11694	0.558470	−	0.829525i	$-0.311389\pi$
$109$	11.6612	1.11694	0.558470	+	0.829525i	$0.311389\pi$
$110$	0	0
$111$	5.01961	0.476440
$112$	0	0
$113$	9.27477	0.872497	0.436249	−	0.899826i	$-0.356307\pi$
$113$	9.27477	0.872497	0.436249	+	0.899826i	$0.356307\pi$
$114$	0	0
$115$	−23.4875	−2.19022
$116$	0	0
$117$	−2.10679	−0.194773
$118$	0	0
$119$	16.6934	1.53028
$120$	0	0
$121$	−9.68466	−0.880423
$122$	0	0
$123$	4.15988	0.375084
$124$	0	0
$125$	−1.56278	−0.139779
$126$	0	0
$127$	1.94565	0.172649	0.0863245	−	0.996267i	$-0.472488\pi$
$127$	1.94565	0.172649	0.0863245	+	0.996267i	$0.472488\pi$
$128$	0	0
$129$	6.96040	0.612829
$130$	0	0
$131$	12.2278	1.06835	0.534173	−	0.845375i	$-0.320623\pi$
$131$	12.2278	1.06835	0.534173	+	0.845375i	$0.320623\pi$
$132$	0	0
$133$	14.2839	1.23857
$134$	0	0
$135$	3.23770	0.278657
$136$	0	0
$137$	−2.83704	−0.242384	−0.121192	−	0.992629i	$-0.538672\pi$
$137$	−2.83704	−0.242384	−0.121192	+	0.992629i	$0.538672\pi$
$138$	0	0
$139$	−14.1158	−1.19729	−0.598645	−	0.801014i	$-0.704294\pi$
$139$	−14.1158	−1.19729	−0.598645	+	0.801014i	$0.704294\pi$
$140$	0	0
$141$	−8.84769	−0.745110
$142$	0	0
$143$	2.41624	0.202056
$144$	0	0
$145$	10.5262	0.874156
$146$	0	0
$147$	2.29494	0.189283
$148$	0	0
$149$	24.1441	1.97796	0.988981	−	0.148039i	$-0.0472963\pi$
$149$	24.1441	1.97796	0.988981	+	0.148039i	$0.0472963\pi$
$150$	0	0
$151$	23.1803	1.88638	0.943192	−	0.332249i	$-0.107807\pi$
$151$	23.1803	1.88638	0.943192	+	0.332249i	$0.107807\pi$
$152$	0	0
$153$	7.69596	0.622181
$154$	0	0
$155$	19.8109	1.59125
$156$	0	0
$157$	3.44729	0.275124	0.137562	−	0.990493i	$-0.456073\pi$
$157$	3.44729	0.275124	0.137562	+	0.990493i	$0.456073\pi$
$158$	0	0
$159$	−4.17888	−0.331407
$160$	0	0
$161$	15.7356	1.24014
$162$	0	0
$163$	8.67246	0.679280	0.339640	−	0.940556i	$-0.389695\pi$
$163$	8.67246	0.679280	0.339640	+	0.940556i	$0.389695\pi$
$164$	0	0
$165$	−3.71326	−0.289077
$166$	0	0
$167$	−0.211060	−0.0163323	−0.00816617	−	0.999967i	$-0.502599\pi$
$167$	−0.211060	−0.0163323	−0.00816617	+	0.999967i	$0.502599\pi$
$168$	0	0
$169$	−8.56145	−0.658573
$170$	0	0
$171$	6.58513	0.503578
$172$	0	0
$173$	14.4807	1.10095	0.550473	−	0.834853i	$-0.314447\pi$
$173$	14.4807	1.10095	0.550473	+	0.834853i	$0.314447\pi$
$174$	0	0
$175$	11.8926	0.898994
$176$	0	0
$177$	−1.07256	−0.0806183
$178$	0	0
$179$	−10.7716	−0.805106	−0.402553	−	0.915397i	$-0.631877\pi$
$179$	−10.7716	−0.805106	−0.402553	+	0.915397i	$0.631877\pi$
$180$	0	0
$181$	22.0777	1.64102	0.820511	−	0.571631i	$-0.193689\pi$
$181$	22.0777	1.64102	0.820511	+	0.571631i	$0.193689\pi$
$182$	0	0
$183$	2.11883	0.156628
$184$	0	0
$185$	16.2520	1.19487
$186$	0	0
$187$	−8.82637	−0.645448
$188$	0	0
$189$	−2.16912	−0.157780
$190$	0	0
$191$	−12.9029	−0.933624	−0.466812	−	0.884357i	$-0.654598\pi$
$191$	−12.9029	−0.933624	−0.466812	+	0.884357i	$0.654598\pi$
$192$	0	0
$193$	7.56182	0.544312	0.272156	−	0.962253i	$-0.412263\pi$
$193$	7.56182	0.544312	0.272156	+	0.962253i	$0.412263\pi$
$194$	0	0
$195$	−6.82114	−0.488472
$196$	0	0
$197$	6.21970	0.443135	0.221568	−	0.975145i	$-0.428883\pi$
$197$	6.21970	0.443135	0.221568	+	0.975145i	$0.428883\pi$
$198$	0	0
$199$	−15.1130	−1.07133	−0.535667	−	0.844429i	$-0.679940\pi$
$199$	−15.1130	−1.07133	−0.535667	+	0.844429i	$0.679940\pi$
$200$	0	0
$201$	8.68247	0.612414
$202$	0	0
$203$	−7.05212	−0.494962
$204$	0	0
$205$	13.4684	0.940676
$206$	0	0
$207$	7.25438	0.504215
$208$	0	0
$209$	−7.55238	−0.522409
$210$	0	0
$211$	−23.2022	−1.59730	−0.798652	−	0.601794i	$-0.794453\pi$
$211$	−23.2022	−1.59730	−0.798652	+	0.601794i	$0.794453\pi$
$212$	0	0
$213$	−8.82001	−0.604337
$214$	0	0
$215$	22.5357	1.53692
$216$	0	0
$217$	−13.2725	−0.900994
$218$	0	0
$219$	−0.732215	−0.0494785
$220$	0	0
$221$	−16.2137	−1.09065
$222$	0	0
$223$	17.9988	1.20529	0.602644	−	0.798011i	$-0.294114\pi$
$223$	17.9988	1.20529	0.602644	+	0.798011i	$0.294114\pi$
$224$	0	0
$225$	5.48268	0.365512
$226$	0	0
$227$	10.9142	0.724404	0.362202	−	0.932100i	$-0.382025\pi$
$227$	10.9142	0.724404	0.362202	+	0.932100i	$0.382025\pi$
$228$	0	0
$229$	11.2380	0.742629	0.371315	−	0.928507i	$-0.378907\pi$
$229$	11.2380	0.742629	0.371315	+	0.928507i	$0.378907\pi$
$230$	0	0
$231$	2.48772	0.163680
$232$	0	0
$233$	−5.09586	−0.333841	−0.166921	−	0.985970i	$-0.553382\pi$
$233$	−5.09586	−0.333841	−0.166921	+	0.985970i	$0.553382\pi$
$234$	0	0
$235$	−28.6461	−1.86867
$236$	0	0
$237$	1.27601	0.0828857
$238$	0	0
$239$	−9.56268	−0.618559	−0.309279	−	0.950971i	$-0.600088\pi$
$239$	−9.56268	−0.618559	−0.309279	+	0.950971i	$0.600088\pi$
$240$	0	0
$241$	−14.8842	−0.958774	−0.479387	−	0.877604i	$-0.659141\pi$
$241$	−14.8842	−0.958774	−0.479387	+	0.877604i	$0.659141\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	7.43031	0.474705
$246$	0	0
$247$	−13.8735	−0.882748
$248$	0	0
$249$	10.0933	0.639634
$250$	0	0
$251$	1.00000	0.0631194
$251$	1.00000	0.0631194
$252$	0	0
$253$	−8.31993	−0.523070
$254$	0	0
$255$	24.9172	1.56037
$256$	0	0
$257$	4.32022	0.269488	0.134744	−	0.990880i	$-0.456979\pi$
$257$	4.32022	0.269488	0.134744	+	0.990880i	$0.456979\pi$
$258$	0	0
$259$	−10.8881	−0.676555
$260$	0	0
$261$	−3.25115	−0.201241
$262$	0	0
$263$	20.7538	1.27973	0.639866	−	0.768487i	$-0.278990\pi$
$263$	20.7538	1.27973	0.639866	+	0.768487i	$0.278990\pi$
$264$	0	0
$265$	−13.5299	−0.831138
$266$	0	0
$267$	7.36931	0.450994
$268$	0	0
$269$	14.3706	0.876191	0.438096	−	0.898928i	$-0.355653\pi$
$269$	14.3706	0.876191	0.438096	+	0.898928i	$0.355653\pi$
$270$	0	0
$271$	14.5103	0.881436	0.440718	−	0.897645i	$-0.354724\pi$
$271$	14.5103	0.881436	0.440718	+	0.897645i	$0.354724\pi$
$272$	0	0
$273$	4.56987	0.276581
$274$	0	0
$275$	−6.28800	−0.379181
$276$	0	0
$277$	−22.8166	−1.37092	−0.685458	−	0.728112i	$-0.740398\pi$
$277$	−22.8166	−1.37092	−0.685458	+	0.728112i	$0.740398\pi$
$278$	0	0
$279$	−6.11884	−0.366325
$280$	0	0
$281$	9.32076	0.556030	0.278015	−	0.960577i	$-0.410324\pi$
$281$	9.32076	0.556030	0.278015	+	0.960577i	$0.410324\pi$
$282$	0	0
$283$	8.33245	0.495313	0.247656	−	0.968848i	$-0.420340\pi$
$283$	8.33245	0.495313	0.247656	+	0.968848i	$0.420340\pi$
$284$	0	0
$285$	21.3207	1.26293
$286$	0	0
$287$	−9.02326	−0.532626
$288$	0	0
$289$	42.2277	2.48399
$290$	0	0
$291$	4.11616	0.241294
$292$	0	0
$293$	22.6015	1.32039	0.660197	−	0.751093i	$-0.270473\pi$
$293$	22.6015	1.32039	0.660197	+	0.751093i	$0.270473\pi$
$294$	0	0
$295$	−3.47261	−0.202183
$296$	0	0
$297$	1.14688	0.0665489
$298$	0	0
$299$	−15.2834	−0.883864
$300$	0	0
$301$	−15.0979	−0.870229
$302$	0	0
$303$	0.837083	0.0480892
$304$	0	0
$305$	6.86011	0.392809
$306$	0	0
$307$	−20.9003	−1.19284	−0.596422	−	0.802671i	$-0.703411\pi$
$307$	−20.9003	−1.19284	−0.596422	+	0.802671i	$0.703411\pi$
$308$	0	0
$309$	−8.89112	−0.505798
$310$	0	0
$311$	−24.2221	−1.37351	−0.686756	−	0.726888i	$-0.740966\pi$
$311$	−24.2221	−1.37351	−0.686756	+	0.726888i	$0.740966\pi$
$312$	0	0
$313$	−2.39631	−0.135447	−0.0677237	−	0.997704i	$-0.521574\pi$
$313$	−2.39631	−0.135447	−0.0677237	+	0.997704i	$0.521574\pi$
$314$	0	0
$315$	−7.02294	−0.395698
$316$	0	0
$317$	4.84370	0.272049	0.136025	−	0.990705i	$-0.456567\pi$
$317$	4.84370	0.272049	0.136025	+	0.990705i	$0.456567\pi$
$318$	0	0
$319$	3.72869	0.208766
$320$	0	0
$321$	−11.5664	−0.645575
$322$	0	0
$323$	50.6789	2.81985
$324$	0	0
$325$	−11.5508	−0.640726
$326$	0	0
$327$	−11.6612	−0.644866
$328$	0	0
$329$	19.1917	1.05807
$330$	0	0
$331$	−0.728998	−0.0400694	−0.0200347	−	0.999799i	$-0.506378\pi$
$331$	−0.728998	−0.0400694	−0.0200347	+	0.999799i	$0.506378\pi$
$332$	0	0
$333$	−5.01961	−0.275073
$334$	0	0
$335$	28.1112	1.53588
$336$	0	0
$337$	8.32092	0.453270	0.226635	−	0.973980i	$-0.427228\pi$
$337$	8.32092	0.453270	0.226635	+	0.973980i	$0.427228\pi$
$338$	0	0
$339$	−9.27477	−0.503737
$340$	0	0
$341$	7.01760	0.380024
$342$	0	0
$343$	−20.1618	−1.08863
$344$	0	0
$345$	23.4875	1.26452
$346$	0	0
$347$	−5.62750	−0.302100	−0.151050	−	0.988526i	$-0.548265\pi$
$347$	−5.62750	−0.302100	−0.151050	+	0.988526i	$0.548265\pi$
$348$	0	0
$349$	−20.4174	−1.09292	−0.546459	−	0.837486i	$-0.684024\pi$
$349$	−20.4174	−1.09292	−0.546459	+	0.837486i	$0.684024\pi$
$350$	0	0
$351$	2.10679	0.112452
$352$	0	0
$353$	−1.27995	−0.0681246	−0.0340623	−	0.999420i	$-0.510844\pi$
$353$	−1.27995	−0.0681246	−0.0340623	+	0.999420i	$0.510844\pi$
$354$	0	0
$355$	−28.5565	−1.51562
$356$	0	0
$357$	−16.6934	−0.883510
$358$	0	0
$359$	−30.8150	−1.62635	−0.813177	−	0.582016i	$-0.802264\pi$
$359$	−30.8150	−1.62635	−0.813177	+	0.582016i	$0.802264\pi$
$360$	0	0
$361$	24.3640	1.28232
$362$	0	0
$363$	9.68466	0.508313
$364$	0	0
$365$	−2.37069	−0.124088
$366$	0	0
$367$	24.4638	1.27700	0.638500	−	0.769622i	$-0.279555\pi$
$367$	24.4638	1.27700	0.638500	+	0.769622i	$0.279555\pi$
$368$	0	0
$369$	−4.15988	−0.216555
$370$	0	0
$371$	9.06447	0.470604
$372$	0	0
$373$	15.3331	0.793917	0.396958	−	0.917837i	$-0.370066\pi$
$373$	15.3331	0.793917	0.396958	+	0.917837i	$0.370066\pi$
$374$	0	0
$375$	1.56278	0.0807015
$376$	0	0
$377$	6.84948	0.352766
$378$	0	0
$379$	30.5853	1.57106	0.785530	−	0.618824i	$-0.212391\pi$
$379$	30.5853	1.57106	0.785530	+	0.618824i	$0.212391\pi$
$380$	0	0
$381$	−1.94565	−0.0996789
$382$	0	0
$383$	−1.84349	−0.0941977	−0.0470989	−	0.998890i	$-0.514998\pi$
$383$	−1.84349	−0.0941977	−0.0470989	+	0.998890i	$0.514998\pi$
$384$	0	0
$385$	8.05450	0.410495
$386$	0	0
$387$	−6.96040	−0.353817
$388$	0	0
$389$	24.6766	1.25115	0.625576	−	0.780163i	$-0.284864\pi$
$389$	24.6766	1.25115	0.625576	+	0.780163i	$0.284864\pi$
$390$	0	0
$391$	55.8294	2.82342
$392$	0	0
$393$	−12.2278	−0.616809
$394$	0	0
$395$	4.13133	0.207870
$396$	0	0
$397$	34.0054	1.70668	0.853341	−	0.521353i	$-0.174573\pi$
$397$	34.0054	1.70668	0.853341	+	0.521353i	$0.174573\pi$
$398$	0	0
$399$	−14.2839	−0.715090
$400$	0	0
$401$	14.4226	0.720228	0.360114	−	0.932908i	$-0.382738\pi$
$401$	14.4226	0.720228	0.360114	+	0.932908i	$0.382738\pi$
$402$	0	0
$403$	12.8911	0.642151
$404$	0	0
$405$	−3.23770	−0.160882
$406$	0	0
$407$	5.75691	0.285359
$408$	0	0
$409$	−13.7883	−0.681787	−0.340894	−	0.940102i	$-0.610730\pi$
$409$	−13.7883	−0.681787	−0.340894	+	0.940102i	$0.610730\pi$
$410$	0	0
$411$	2.83704	0.139941
$412$	0	0
$413$	2.32650	0.114480
$414$	0	0
$415$	32.6789	1.60415
$416$	0	0
$417$	14.1158	0.691256
$418$	0	0
$419$	−3.88734	−0.189909	−0.0949545	−	0.995482i	$-0.530271\pi$
$419$	−3.88734	−0.189909	−0.0949545	+	0.995482i	$0.530271\pi$
$420$	0	0
$421$	−19.9943	−0.974463	−0.487231	−	0.873273i	$-0.661993\pi$
$421$	−19.9943	−0.974463	−0.487231	+	0.873273i	$0.661993\pi$
$422$	0	0
$423$	8.84769	0.430190
$424$	0	0
$425$	42.1945	2.04673
$426$	0	0
$427$	−4.59598	−0.222415
$428$	0	0
$429$	−2.41624	−0.116657
$430$	0	0
$431$	−20.2269	−0.974294	−0.487147	−	0.873320i	$-0.661962\pi$
$431$	−20.2269	−0.974294	−0.487147	+	0.873320i	$0.661962\pi$
$432$	0	0
$433$	30.5797	1.46957	0.734784	−	0.678301i	$-0.237283\pi$
$433$	30.5797	1.46957	0.734784	+	0.678301i	$0.237283\pi$
$434$	0	0
$435$	−10.5262	−0.504694
$436$	0	0
$437$	47.7711	2.28520
$438$	0	0
$439$	−4.59334	−0.219228	−0.109614	−	0.993974i	$-0.534962\pi$
$439$	−4.59334	−0.219228	−0.109614	+	0.993974i	$0.534962\pi$
$440$	0	0
$441$	−2.29494	−0.109283
$442$	0	0
$443$	−23.7216	−1.12705	−0.563524	−	0.826100i	$-0.690555\pi$
$443$	−23.7216	−1.12705	−0.563524	+	0.826100i	$0.690555\pi$
$444$	0	0
$445$	23.8596	1.13105
$446$	0	0
$447$	−24.1441	−1.14198
$448$	0	0
$449$	−9.70483	−0.457999	−0.229000	−	0.973427i	$-0.573545\pi$
$449$	−9.70483	−0.457999	−0.229000	+	0.973427i	$0.573545\pi$
$450$	0	0
$451$	4.77090	0.224653
$452$	0	0
$453$	−23.1803	−1.08910
$454$	0	0
$455$	14.7958	0.693640
$456$	0	0
$457$	1.13693	0.0531835	0.0265917	−	0.999646i	$-0.491535\pi$
$457$	1.13693	0.0531835	0.0265917	+	0.999646i	$0.491535\pi$
$458$	0	0
$459$	−7.69596	−0.359216
$460$	0	0
$461$	29.8000	1.38792	0.693962	−	0.720012i	$-0.255864\pi$
$461$	29.8000	1.38792	0.693962	+	0.720012i	$0.255864\pi$
$462$	0	0
$463$	29.2772	1.36062	0.680312	−	0.732922i	$-0.261844\pi$
$463$	29.2772	1.36062	0.680312	+	0.732922i	$0.261844\pi$
$464$	0	0
$465$	−19.8109	−0.918711
$466$	0	0
$467$	−13.3299	−0.616835	−0.308417	−	0.951251i	$-0.599799\pi$
$467$	−13.3299	−0.616835	−0.308417	+	0.951251i	$0.599799\pi$
$468$	0	0
$469$	−18.8333	−0.869640
$470$	0	0
$471$	−3.44729	−0.158843
$472$	0	0
$473$	7.98277	0.367048
$474$	0	0
$475$	36.1042	1.65657
$476$	0	0
$477$	4.17888	0.191338
$478$	0	0
$479$	−3.02000	−0.137987	−0.0689936	−	0.997617i	$-0.521979\pi$
$479$	−3.02000	−0.137987	−0.0689936	+	0.997617i	$0.521979\pi$
$480$	0	0
$481$	10.5753	0.482190
$482$	0	0
$483$	−15.7356	−0.715995
$484$	0	0
$485$	13.3269	0.605142
$486$	0	0
$487$	−43.1607	−1.95580	−0.977900	−	0.209075i	$-0.932955\pi$
$487$	−43.1607	−1.95580	−0.977900	+	0.209075i	$0.932955\pi$
$488$	0	0
$489$	−8.67246	−0.392182
$490$	0	0
$491$	−18.9478	−0.855104	−0.427552	−	0.903991i	$-0.640624\pi$
$491$	−18.9478	−0.855104	−0.427552	+	0.903991i	$0.640624\pi$
$492$	0	0
$493$	−25.0207	−1.12688
$494$	0	0
$495$	3.71326	0.166899
$496$	0	0
$497$	19.1316	0.858171
$498$	0	0
$499$	18.9022	0.846177	0.423088	−	0.906088i	$-0.360946\pi$
$499$	18.9022	0.846177	0.423088	+	0.906088i	$0.360946\pi$
$500$	0	0
$501$	0.211060	0.00942948
$502$	0	0
$503$	3.51320	0.156646	0.0783230	−	0.996928i	$-0.475043\pi$
$503$	3.51320	0.156646	0.0783230	+	0.996928i	$0.475043\pi$
$504$	0	0
$505$	2.71022	0.120603
$506$	0	0
$507$	8.56145	0.380227
$508$	0	0
$509$	33.0985	1.46707	0.733534	−	0.679653i	$-0.237870\pi$
$509$	33.0985	1.46707	0.733534	+	0.679653i	$0.237870\pi$
$510$	0	0
$511$	1.58826	0.0702605
$512$	0	0
$513$	−6.58513	−0.290741
$514$	0	0
$515$	−28.7867	−1.26850
$516$	0	0
$517$	−10.1473	−0.446277
$518$	0	0
$519$	−14.4807	−0.635631
$520$	0	0
$521$	34.5532	1.51380	0.756901	−	0.653530i	$-0.226712\pi$
$521$	34.5532	1.51380	0.756901	+	0.653530i	$0.226712\pi$
$522$	0	0
$523$	12.1346	0.530609	0.265305	−	0.964165i	$-0.414528\pi$
$523$	12.1346	0.530609	0.265305	+	0.964165i	$0.414528\pi$
$524$	0	0
$525$	−11.8926	−0.519034
$526$	0	0
$527$	−47.0903	−2.05129
$528$	0	0
$529$	29.6261	1.28809
$530$	0	0
$531$	1.07256	0.0465450
$532$	0	0
$533$	8.76399	0.379610
$534$	0	0
$535$	−37.4486	−1.61904
$536$	0	0
$537$	10.7716	0.464828
$538$	0	0
$539$	2.63203	0.113369
$540$	0	0
$541$	17.2086	0.739857	0.369929	−	0.929060i	$-0.379382\pi$
$541$	17.2086	0.739857	0.369929	+	0.929060i	$0.379382\pi$
$542$	0	0
$543$	−22.0777	−0.947444
$544$	0	0
$545$	−37.7554	−1.61726
$546$	0	0
$547$	25.3187	1.08255	0.541274	−	0.840846i	$-0.317942\pi$
$547$	25.3187	1.08255	0.541274	+	0.840846i	$0.317942\pi$
$548$	0	0
$549$	−2.11883	−0.0904293
$550$	0	0
$551$	−21.4092	−0.912065
$552$	0	0
$553$	−2.76781	−0.117699
$554$	0	0
$555$	−16.2520	−0.689858
$556$	0	0
$557$	−36.1981	−1.53376	−0.766882	−	0.641789i	$-0.778193\pi$
$557$	−36.1981	−1.53376	−0.766882	+	0.641789i	$0.778193\pi$
$558$	0	0
$559$	14.6641	0.620225
$560$	0	0
$561$	8.82637	0.372650
$562$	0	0
$563$	6.81756	0.287326	0.143663	−	0.989627i	$-0.454112\pi$
$563$	6.81756	0.287326	0.143663	+	0.989627i	$0.454112\pi$
$564$	0	0
$565$	−30.0289	−1.26333
$566$	0	0
$567$	2.16912	0.0910943
$568$	0	0
$569$	11.7112	0.490960	0.245480	−	0.969402i	$-0.421054\pi$
$569$	11.7112	0.490960	0.245480	+	0.969402i	$0.421054\pi$
$570$	0	0
$571$	47.1031	1.97120	0.985601	−	0.169087i	$-0.0540819\pi$
$571$	47.1031	1.97120	0.985601	+	0.169087i	$0.0540819\pi$
$572$	0	0
$573$	12.9029	0.539028
$574$	0	0
$575$	39.7735	1.65867
$576$	0	0
$577$	26.4499	1.10112	0.550561	−	0.834795i	$-0.314414\pi$
$577$	26.4499	1.10112	0.550561	+	0.834795i	$0.314414\pi$
$578$	0	0
$579$	−7.56182	−0.314258
$580$	0	0
$581$	−21.8935	−0.908293
$582$	0	0
$583$	−4.79269	−0.198493
$584$	0	0
$585$	6.82114	0.282019
$586$	0	0
$587$	−6.13410	−0.253181	−0.126591	−	0.991955i	$-0.540403\pi$
$587$	−6.13410	−0.253181	−0.126591	+	0.991955i	$0.540403\pi$
$588$	0	0
$589$	−40.2934	−1.66026
$590$	0	0
$591$	−6.21970	−0.255844
$592$	0	0
$593$	31.7356	1.30323	0.651613	−	0.758551i	$-0.274093\pi$
$593$	31.7356	1.30323	0.651613	+	0.758551i	$0.274093\pi$
$594$	0	0
$595$	−54.0482	−2.21576
$596$	0	0
$597$	15.1130	0.618535
$598$	0	0
$599$	11.5277	0.471009	0.235504	−	0.971873i	$-0.424326\pi$
$599$	11.5277	0.471009	0.235504	+	0.971873i	$0.424326\pi$
$600$	0	0
$601$	41.6949	1.70077	0.850386	−	0.526159i	$-0.176368\pi$
$601$	41.6949	1.70077	0.850386	+	0.526159i	$0.176368\pi$
$602$	0	0
$603$	−8.68247	−0.353577
$604$	0	0
$605$	31.3560	1.27480
$606$	0	0
$607$	−12.2247	−0.496187	−0.248093	−	0.968736i	$-0.579804\pi$
$607$	−12.2247	−0.496187	−0.248093	+	0.968736i	$0.579804\pi$
$608$	0	0
$609$	7.05212	0.285766
$610$	0	0
$611$	−18.6402	−0.754102
$612$	0	0
$613$	40.6542	1.64201	0.821004	−	0.570923i	$-0.193415\pi$
$613$	40.6542	1.64201	0.821004	+	0.570923i	$0.193415\pi$
$614$	0	0
$615$	−13.4684	−0.543100
$616$	0	0
$617$	16.6964	0.672174	0.336087	−	0.941831i	$-0.390896\pi$
$617$	16.6964	0.672174	0.336087	+	0.941831i	$0.390896\pi$
$618$	0	0
$619$	−23.1817	−0.931753	−0.465876	−	0.884850i	$-0.654261\pi$
$619$	−23.1817	−0.931753	−0.465876	+	0.884850i	$0.654261\pi$
$620$	0	0
$621$	−7.25438	−0.291108
$622$	0	0
$623$	−15.9849	−0.640421
$624$	0	0
$625$	−22.3536	−0.894144
$626$	0	0
$627$	7.55238	0.301613
$628$	0	0
$629$	−38.6307	−1.54031
$630$	0	0
$631$	4.28808	0.170706	0.0853529	−	0.996351i	$-0.472798\pi$
$631$	4.28808	0.170706	0.0853529	+	0.996351i	$0.472798\pi$
$632$	0	0
$633$	23.2022	0.922203
$634$	0	0
$635$	−6.29944	−0.249986
$636$	0	0
$637$	4.83494	0.191567
$638$	0	0
$639$	8.82001	0.348914
$640$	0	0
$641$	−13.6880	−0.540645	−0.270322	−	0.962770i	$-0.587130\pi$
$641$	−13.6880	−0.540645	−0.270322	+	0.962770i	$0.587130\pi$
$642$	0	0
$643$	−21.3280	−0.841095	−0.420547	−	0.907271i	$-0.638162\pi$
$643$	−21.3280	−0.841095	−0.420547	+	0.907271i	$0.638162\pi$
$644$	0	0
$645$	−22.5357	−0.887341
$646$	0	0
$647$	−17.8142	−0.700348	−0.350174	−	0.936685i	$-0.613878\pi$
$647$	−17.8142	−0.700348	−0.350174	+	0.936685i	$0.613878\pi$
$648$	0	0
$649$	−1.23010	−0.0482856
$650$	0	0
$651$	13.2725	0.520189
$652$	0	0
$653$	14.1615	0.554182	0.277091	−	0.960844i	$-0.410630\pi$
$653$	14.1615	0.554182	0.277091	+	0.960844i	$0.410630\pi$
$654$	0	0
$655$	−39.5898	−1.54690
$656$	0	0
$657$	0.732215	0.0285664
$658$	0	0
$659$	−31.4531	−1.22524	−0.612619	−	0.790378i	$-0.709884\pi$
$659$	−31.4531	−1.22524	−0.612619	+	0.790378i	$0.709884\pi$
$660$	0	0
$661$	−1.02349	−0.0398092	−0.0199046	−	0.999802i	$-0.506336\pi$
$661$	−1.02349	−0.0398092	−0.0199046	+	0.999802i	$0.506336\pi$
$662$	0	0
$663$	16.2137	0.629690
$664$	0	0
$665$	−46.2470	−1.79338
$666$	0	0
$667$	−23.5851	−0.913218
$668$	0	0
$669$	−17.9988	−0.695873
$670$	0	0
$671$	2.43005	0.0938109
$672$	0	0
$673$	−32.3354	−1.24644	−0.623219	−	0.782048i	$-0.714175\pi$
$673$	−32.3354	−1.24644	−0.623219	+	0.782048i	$0.714175\pi$
$674$	0	0
$675$	−5.48268	−0.211029
$676$	0	0
$677$	15.0772	0.579464	0.289732	−	0.957108i	$-0.406434\pi$
$677$	15.0772	0.579464	0.289732	+	0.957108i	$0.406434\pi$
$678$	0	0
$679$	−8.92843	−0.342642
$680$	0	0
$681$	−10.9142	−0.418235
$682$	0	0
$683$	12.7043	0.486115	0.243058	−	0.970012i	$-0.421850\pi$
$683$	12.7043	0.486115	0.243058	+	0.970012i	$0.421850\pi$
$684$	0	0
$685$	9.18547	0.350959
$686$	0	0
$687$	−11.2380	−0.428757
$688$	0	0
$689$	−8.80401	−0.335406
$690$	0	0
$691$	37.0227	1.40841	0.704205	−	0.709997i	$-0.251304\pi$
$691$	37.0227	1.40841	0.704205	+	0.709997i	$0.251304\pi$
$692$	0	0
$693$	−2.48772	−0.0945008
$694$	0	0
$695$	45.7028	1.73361
$696$	0	0
$697$	−32.0143	−1.21263
$698$	0	0
$699$	5.09586	0.192743
$700$	0	0
$701$	−20.9139	−0.789909	−0.394954	−	0.918701i	$-0.629240\pi$
$701$	−20.9139	−0.789909	−0.394954	+	0.918701i	$0.629240\pi$
$702$	0	0
$703$	−33.0548	−1.24669
$704$	0	0
$705$	28.6461	1.07888
$706$	0	0
$707$	−1.81573	−0.0682876
$708$	0	0
$709$	36.8767	1.38493	0.692466	−	0.721450i	$-0.256524\pi$
$709$	36.8767	1.38493	0.692466	+	0.721450i	$0.256524\pi$
$710$	0	0
$711$	−1.27601	−0.0478541
$712$	0	0
$713$	−44.3884	−1.66236
$714$	0	0
$715$	−7.82305	−0.292566
$716$	0	0
$717$	9.56268	0.357125
$718$	0	0
$719$	−33.9896	−1.26760	−0.633800	−	0.773497i	$-0.718506\pi$
$719$	−33.9896	−1.26760	−0.633800	+	0.773497i	$0.718506\pi$
$720$	0	0
$721$	19.2859	0.718243
$722$	0	0
$723$	14.8842	0.553548
$724$	0	0
$725$	−17.8250	−0.662004
$726$	0	0
$727$	37.1131	1.37645	0.688224	−	0.725498i	$-0.258391\pi$
$727$	37.1131	1.37645	0.688224	+	0.725498i	$0.258391\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−53.5669	−1.98124
$732$	0	0
$733$	−35.1682	−1.29897	−0.649484	−	0.760375i	$-0.725015\pi$
$733$	−35.1682	−1.29897	−0.649484	+	0.760375i	$0.725015\pi$
$734$	0	0
$735$	−7.43031	−0.274071
$736$	0	0
$737$	9.95778	0.366800
$738$	0	0
$739$	44.0252	1.61949	0.809746	−	0.586780i	$-0.199605\pi$
$739$	44.0252	1.61949	0.809746	+	0.586780i	$0.199605\pi$
$740$	0	0
$741$	13.8735	0.509655
$742$	0	0
$743$	−7.91900	−0.290520	−0.145260	−	0.989394i	$-0.546402\pi$
$743$	−7.91900	−0.290520	−0.145260	+	0.989394i	$0.546402\pi$
$744$	0	0
$745$	−78.1713	−2.86398
$746$	0	0
$747$	−10.0933	−0.369293
$748$	0	0
$749$	25.0889	0.916729
$750$	0	0
$751$	16.0908	0.587160	0.293580	−	0.955934i	$-0.405153\pi$
$751$	16.0908	0.587160	0.293580	+	0.955934i	$0.405153\pi$
$752$	0	0
$753$	−1.00000	−0.0364420
$754$	0	0
$755$	−75.0507	−2.73137
$756$	0	0
$757$	−8.02466	−0.291661	−0.145831	−	0.989310i	$-0.546585\pi$
$757$	−8.02466	−0.291661	−0.145831	+	0.989310i	$0.546585\pi$
$758$	0	0
$759$	8.31993	0.301994
$760$	0	0
$761$	−19.9691	−0.723879	−0.361940	−	0.932202i	$-0.617885\pi$
$761$	−19.9691	−0.723879	−0.361940	+	0.932202i	$0.617885\pi$
$762$	0	0
$763$	25.2945	0.915722
$764$	0	0
$765$	−24.9172	−0.900882
$766$	0	0
$767$	−2.25965	−0.0815912
$768$	0	0
$769$	1.72601	0.0622417	0.0311208	−	0.999516i	$-0.490092\pi$
$769$	1.72601	0.0622417	0.0311208	+	0.999516i	$0.490092\pi$
$770$	0	0
$771$	−4.32022	−0.155589
$772$	0	0
$773$	20.0590	0.721471	0.360735	−	0.932668i	$-0.382526\pi$
$773$	20.0590	0.721471	0.360735	+	0.932668i	$0.382526\pi$
$774$	0	0
$775$	−33.5477	−1.20507
$776$	0	0
$777$	10.8881	0.390609
$778$	0	0
$779$	−27.3934	−0.981470
$780$	0	0
$781$	−10.1155	−0.361962
$782$	0	0
$783$	3.25115	0.116187
$784$	0	0
$785$	−11.1613	−0.398363
$786$	0	0
$787$	−21.4588	−0.764923	−0.382462	−	0.923971i	$-0.624924\pi$
$787$	−21.4588	−0.764923	−0.382462	+	0.923971i	$0.624924\pi$
$788$	0	0
$789$	−20.7538	−0.738853
$790$	0	0
$791$	20.1181	0.715316
$792$	0	0
$793$	4.46391	0.158518
$794$	0	0
$795$	13.5299	0.479857
$796$	0	0
$797$	2.69549	0.0954793	0.0477396	−	0.998860i	$-0.484798\pi$
$797$	2.69549	0.0954793	0.0477396	+	0.998860i	$0.484798\pi$
$798$	0	0
$799$	68.0915	2.40890
$800$	0	0
$801$	−7.36931	−0.260382
$802$	0	0
$803$	−0.839766	−0.0296347
$804$	0	0
$805$	−50.9471	−1.79565
$806$	0	0
$807$	−14.3706	−0.505869
$808$	0	0
$809$	20.9519	0.736628	0.368314	−	0.929701i	$-0.379935\pi$
$809$	20.9519	0.736628	0.368314	+	0.929701i	$0.379935\pi$
$810$	0	0
$811$	47.5349	1.66918	0.834588	−	0.550875i	$-0.185706\pi$
$811$	47.5349	1.66918	0.834588	+	0.550875i	$0.185706\pi$
$812$	0	0
$813$	−14.5103	−0.508898
$814$	0	0
$815$	−28.0788	−0.983558
$816$	0	0
$817$	−45.8352	−1.60357
$818$	0	0
$819$	−4.56987	−0.159684
$820$	0	0
$821$	−32.8606	−1.14684	−0.573421	−	0.819261i	$-0.694384\pi$
$821$	−32.8606	−1.14684	−0.573421	+	0.819261i	$0.694384\pi$
$822$	0	0
$823$	32.8069	1.14358	0.571788	−	0.820401i	$-0.306250\pi$
$823$	32.8069	1.14358	0.571788	+	0.820401i	$0.306250\pi$
$824$	0	0
$825$	6.28800	0.218920
$826$	0	0
$827$	37.2008	1.29360	0.646799	−	0.762660i	$-0.276107\pi$
$827$	37.2008	1.29360	0.646799	+	0.762660i	$0.276107\pi$
$828$	0	0
$829$	23.3132	0.809700	0.404850	−	0.914383i	$-0.367324\pi$
$829$	23.3132	0.809700	0.404850	+	0.914383i	$0.367324\pi$
$830$	0	0
$831$	22.8166	0.791499
$832$	0	0
$833$	−17.6617	−0.611943
$834$	0	0
$835$	0.683349	0.0236483
$836$	0	0
$837$	6.11884	0.211498
$838$	0	0
$839$	33.6457	1.16158	0.580789	−	0.814054i	$-0.302744\pi$
$839$	33.6457	1.16158	0.580789	+	0.814054i	$0.302744\pi$
$840$	0	0
$841$	−18.4300	−0.635519
$842$	0	0
$843$	−9.32076	−0.321024
$844$	0	0
$845$	27.7194	0.953575
$846$	0	0
$847$	−21.0071	−0.721814
$848$	0	0
$849$	−8.33245	−0.285969
$850$	0	0
$851$	−36.4142	−1.24826
$852$	0	0
$853$	−17.8627	−0.611609	−0.305804	−	0.952094i	$-0.598925\pi$
$853$	−17.8627	−0.611609	−0.305804	+	0.952094i	$0.598925\pi$
$854$	0	0
$855$	−21.3207	−0.729152
$856$	0	0
$857$	39.0408	1.33361	0.666805	−	0.745232i	$-0.267661\pi$
$857$	39.0408	1.33361	0.666805	+	0.745232i	$0.267661\pi$
$858$	0	0
$859$	−33.9381	−1.15795	−0.578977	−	0.815344i	$-0.696548\pi$
$859$	−33.9381	−1.15795	−0.578977	+	0.815344i	$0.696548\pi$
$860$	0	0
$861$	9.02326	0.307512
$862$	0	0
$863$	47.4235	1.61432	0.807158	−	0.590335i	$-0.201004\pi$
$863$	47.4235	1.61432	0.807158	+	0.590335i	$0.201004\pi$
$864$	0	0
$865$	−46.8841	−1.59411
$866$	0	0
$867$	−42.2277	−1.43413
$868$	0	0
$869$	1.46343	0.0496436
$870$	0	0
$871$	18.2921	0.619805
$872$	0	0
$873$	−4.11616	−0.139311
$874$	0	0
$875$	−3.38985	−0.114598
$876$	0	0
$877$	−16.3935	−0.553571	−0.276785	−	0.960932i	$-0.589269\pi$
$877$	−16.3935	−0.553571	−0.276785	+	0.960932i	$0.589269\pi$
$878$	0	0
$879$	−22.6015	−0.762330
$880$	0	0
$881$	32.8248	1.10590	0.552949	−	0.833215i	$-0.313503\pi$
$881$	32.8248	1.10590	0.552949	+	0.833215i	$0.313503\pi$
$882$	0	0
$883$	−52.8506	−1.77856	−0.889282	−	0.457360i	$-0.848795\pi$
$883$	−52.8506	−1.77856	−0.889282	+	0.457360i	$0.848795\pi$
$884$	0	0
$885$	3.47261	0.116731
$886$	0	0
$887$	−54.3107	−1.82358	−0.911788	−	0.410661i	$-0.865298\pi$
$887$	−54.3107	−1.82358	−0.911788	+	0.410661i	$0.865298\pi$
$888$	0	0
$889$	4.22035	0.141546
$890$	0	0
$891$	−1.14688	−0.0384220
$892$	0	0
$893$	58.2632	1.94971
$894$	0	0
$895$	34.8751	1.16575
$896$	0	0
$897$	15.2834	0.510299
$898$	0	0
$899$	19.8933	0.663477
$900$	0	0
$901$	32.1605	1.07142
$902$	0	0
$903$	15.0979	0.502427
$904$	0	0
$905$	−71.4808	−2.37610
$906$	0	0
$907$	−38.1370	−1.26632	−0.633159	−	0.774022i	$-0.718242\pi$
$907$	−38.1370	−1.26632	−0.633159	+	0.774022i	$0.718242\pi$
$908$	0	0
$909$	−0.837083	−0.0277643
$910$	0	0
$911$	−12.5958	−0.417316	−0.208658	−	0.977989i	$-0.566910\pi$
$911$	−12.5958	−0.417316	−0.208658	+	0.977989i	$0.566910\pi$
$912$	0	0
$913$	11.5758	0.383103
$914$	0	0
$915$	−6.86011	−0.226788
$916$	0	0
$917$	26.5234	0.875881
$918$	0	0
$919$	−26.8208	−0.884736	−0.442368	−	0.896834i	$-0.645862\pi$
$919$	−26.8208	−0.884736	−0.442368	+	0.896834i	$0.645862\pi$
$920$	0	0
$921$	20.9003	0.688688
$922$	0	0
$923$	−18.5819	−0.611630
$924$	0	0
$925$	−27.5209	−0.904883
$926$	0	0
$927$	8.89112	0.292023
$928$	0	0
$929$	30.1440	0.988994	0.494497	−	0.869179i	$-0.335352\pi$
$929$	30.1440	0.988994	0.494497	+	0.869179i	$0.335352\pi$
$930$	0	0
$931$	−15.1125	−0.495291
$932$	0	0
$933$	24.2221	0.792997
$934$	0	0
$935$	28.5771	0.934571
$936$	0	0
$937$	−39.8009	−1.30024	−0.650120	−	0.759831i	$-0.725281\pi$
$937$	−39.8009	−1.30024	−0.650120	+	0.759831i	$0.725281\pi$
$938$	0	0
$939$	2.39631	0.0782005
$940$	0	0
$941$	−15.7300	−0.512784	−0.256392	−	0.966573i	$-0.582534\pi$
$941$	−15.7300	−0.512784	−0.256392	+	0.966573i	$0.582534\pi$
$942$	0	0
$943$	−30.1774	−0.982711
$944$	0	0
$945$	7.02294	0.228456
$946$	0	0
$947$	6.51325	0.211652	0.105826	−	0.994385i	$-0.466251\pi$
$947$	6.51325	0.211652	0.105826	+	0.994385i	$0.466251\pi$
$948$	0	0
$949$	−1.54262	−0.0500756
$950$	0	0
$951$	−4.84370	−0.157068
$952$	0	0
$953$	−32.7505	−1.06089	−0.530447	−	0.847718i	$-0.677976\pi$
$953$	−32.7505	−1.06089	−0.530447	+	0.847718i	$0.677976\pi$
$954$	0	0
$955$	41.7758	1.35183
$956$	0	0
$957$	−3.72869	−0.120531
$958$	0	0
$959$	−6.15386	−0.198719
$960$	0	0
$961$	6.44019	0.207748
$962$	0	0
$963$	11.5664	0.372723
$964$	0	0
$965$	−24.4829	−0.788132
$966$	0	0
$967$	24.0579	0.773651	0.386826	−	0.922153i	$-0.373572\pi$
$967$	24.0579	0.773651	0.386826	+	0.922153i	$0.373572\pi$
$968$	0	0
$969$	−50.6789	−1.62804
$970$	0	0
$971$	40.5912	1.30263	0.651317	−	0.758806i	$-0.274217\pi$
$971$	40.5912	1.30263	0.651317	+	0.758806i	$0.274217\pi$
$972$	0	0
$973$	−30.6189	−0.981597
$974$	0	0
$975$	11.5508	0.369923
$976$	0	0
$977$	−28.0046	−0.895946	−0.447973	−	0.894047i	$-0.647854\pi$
$977$	−28.0046	−0.895946	−0.447973	+	0.894047i	$0.647854\pi$
$978$	0	0
$979$	8.45174	0.270119
$980$	0	0
$981$	11.6612	0.372313
$982$	0	0
$983$	32.7849	1.04568	0.522838	−	0.852432i	$-0.324873\pi$
$983$	32.7849	1.04568	0.522838	+	0.852432i	$0.324873\pi$
$984$	0	0
$985$	−20.1375	−0.641634
$986$	0	0
$987$	−19.1917	−0.610878
$988$	0	0
$989$	−50.4934	−1.60560
$990$	0	0
$991$	9.00632	0.286095	0.143048	−	0.989716i	$-0.454310\pi$
$991$	9.00632	0.286095	0.143048	+	0.989716i	$0.454310\pi$
$992$	0	0
$993$	0.728998	0.0231341
$994$	0	0
$995$	48.9314	1.55123
$996$	0	0
$997$	−7.79488	−0.246866	−0.123433	−	0.992353i	$-0.539390\pi$
$997$	−7.79488	−0.246866	−0.123433	+	0.992353i	$0.539390\pi$
$998$	0	0
$999$	5.01961	0.158813

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6024.2.a.p.1.2	✓	14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6024.2.a.p.1.2	✓	14	1.1	even	1	trivial

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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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -3.23770 q^{5} +2.16912 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} -3.23770 q^{5} +2.16912 q^{7} +1.00000 q^{9} -1.14688 q^{11} -2.10679 q^{13} +3.23770 q^{15} +7.69596 q^{17} +6.58513 q^{19} -2.16912 q^{21} +7.25438 q^{23} +5.48268 q^{25} -1.00000 q^{27} -3.25115 q^{29} -6.11884 q^{31} +1.14688 q^{33} -7.02294 q^{35} -5.01961 q^{37} +2.10679 q^{39} -4.15988 q^{41} -6.96040 q^{43} -3.23770 q^{45} +8.84769 q^{47} -2.29494 q^{49} -7.69596 q^{51} +4.17888 q^{53} +3.71326 q^{55} -6.58513 q^{57} +1.07256 q^{59} -2.11883 q^{61} +2.16912 q^{63} +6.82114 q^{65} -8.68247 q^{67} -7.25438 q^{69} +8.82001 q^{71} +0.732215 q^{73} -5.48268 q^{75} -2.48772 q^{77} -1.27601 q^{79} +1.00000 q^{81} -10.0933 q^{83} -24.9172 q^{85} +3.25115 q^{87} -7.36931 q^{89} -4.56987 q^{91} +6.11884 q^{93} -21.3207 q^{95} -4.11616 q^{97} -1.14688 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q - 14 q^{3} + 6 q^{5} + 3 q^{7} + 14 q^{9}+O(q^{10}) \) 14 * q - 14 * q^3 + 6 * q^5 + 3 * q^7 + 14 * q^9	\( 14 q - 14 q^{3} + 6 q^{5} + 3 q^{7} + 14 q^{9} + 7 q^{11} + 5 q^{13} - 6 q^{15} + 28 q^{17} - q^{19} - 3 q^{21} - 3 q^{23} + 24 q^{25} - 14 q^{27} + 15 q^{29} + 9 q^{31} - 7 q^{33} - 14 q^{35} + 4 q^{37} - 5 q^{39} + 32 q^{41} - 3 q^{43} + 6 q^{45} + 15 q^{47} + 27 q^{49} - 28 q^{51} + 8 q^{53} + q^{57} - 11 q^{59} + 25 q^{61} + 3 q^{63} + 18 q^{65} - 20 q^{67} + 3 q^{69} + 33 q^{71} + 8 q^{73} - 24 q^{75} + 14 q^{77} - 17 q^{79} + 14 q^{81} - 4 q^{83} + 16 q^{85} - 15 q^{87} + 14 q^{89} - 28 q^{91} - 9 q^{93} + 3 q^{95} + 9 q^{97} + 7 q^{99}+O(q^{100}) \) 14 * q - 14 * q^3 + 6 * q^5 + 3 * q^7 + 14 * q^9 + 7 * q^11 + 5 * q^13 - 6 * q^15 + 28 * q^17 - q^19 - 3 * q^21 - 3 * q^23 + 24 * q^25 - 14 * q^27 + 15 * q^29 + 9 * q^31 - 7 * q^33 - 14 * q^35 + 4 * q^37 - 5 * q^39 + 32 * q^41 - 3 * q^43 + 6 * q^45 + 15 * q^47 + 27 * q^49 - 28 * q^51 + 8 * q^53 + q^57 - 11 * q^59 + 25 * q^61 + 3 * q^63 + 18 * q^65 - 20 * q^67 + 3 * q^69 + 33 * q^71 + 8 * q^73 - 24 * q^75 + 14 * q^77 - 17 * q^79 + 14 * q^81 - 4 * q^83 + 16 * q^85 - 15 * q^87 + 14 * q^89 - 28 * q^91 - 9 * q^93 + 3 * q^95 + 9 * q^97 + 7 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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