LMFDB - Embedded newform 8024.2.a.ba.1.18

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8024,2,Mod(1,8024)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8024, 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("8024.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8024 = 2^{3} \cdot 17 \cdot 59 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8024.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:	$64.0719625819$
Analytic rank:	$0$
Dimension:	$30$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.18
Character	$\chi$	$=$	8024.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.597652 q^{3} -3.19982 q^{5} +4.21450 q^{7} -2.64281 q^{9} +O(q^{10})$	$q+0.597652 q^{3} -3.19982 q^{5} +4.21450 q^{7} -2.64281 q^{9} +5.94939 q^{11} -0.565788 q^{13} -1.91238 q^{15} -1.00000 q^{17} +5.00504 q^{19} +2.51880 q^{21} +5.77337 q^{23} +5.23884 q^{25} -3.37244 q^{27} -0.0765582 q^{29} -10.9331 q^{31} +3.55566 q^{33} -13.4856 q^{35} +6.62492 q^{37} -0.338145 q^{39} +2.55967 q^{41} +1.61320 q^{43} +8.45652 q^{45} +7.42334 q^{47} +10.7620 q^{49} -0.597652 q^{51} -11.6959 q^{53} -19.0370 q^{55} +2.99127 q^{57} +1.00000 q^{59} +11.6150 q^{61} -11.1381 q^{63} +1.81042 q^{65} -7.00194 q^{67} +3.45047 q^{69} +8.35620 q^{71} +0.574110 q^{73} +3.13100 q^{75} +25.0737 q^{77} +4.78437 q^{79} +5.91289 q^{81} -3.23362 q^{83} +3.19982 q^{85} -0.0457552 q^{87} +14.4811 q^{89} -2.38451 q^{91} -6.53419 q^{93} -16.0152 q^{95} -18.3927 q^{97} -15.7231 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 30 q + 4 q^{3} + 2 q^{5} + 3 q^{7} + 34 q^{9}+O(q^{10}) $ 30 * q + 4 * q^3 + 2 * q^5 + 3 * q^7 + 34 * q^9	$ 30 q + 4 q^{3} + 2 q^{5} + 3 q^{7} + 34 q^{9} + 3 q^{11} + 9 q^{13} + 14 q^{15} - 30 q^{17} + 24 q^{19} + 7 q^{21} + 9 q^{23} + 40 q^{25} + 19 q^{27} + 9 q^{29} + 11 q^{31} - 14 q^{33} + 30 q^{35} - 13 q^{37} + 16 q^{39} - 13 q^{41} + 23 q^{43} + 12 q^{45} + 43 q^{47} + 35 q^{49} - 4 q^{51} - 4 q^{53} + 43 q^{55} + 3 q^{57} + 30 q^{59} + 43 q^{61} + 38 q^{63} + 3 q^{65} + 50 q^{67} + 34 q^{69} + 3 q^{71} - 16 q^{73} + 21 q^{75} + 18 q^{77} + 45 q^{79} + 6 q^{81} + 63 q^{83} - 2 q^{85} + 42 q^{87} + 6 q^{89} + 22 q^{91} - 2 q^{93} + 19 q^{95} - 28 q^{97} + 51 q^{99}+O(q^{100}) $ 30 * q + 4 * q^3 + 2 * q^5 + 3 * q^7 + 34 * q^9 + 3 * q^11 + 9 * q^13 + 14 * q^15 - 30 * q^17 + 24 * q^19 + 7 * q^21 + 9 * q^23 + 40 * q^25 + 19 * q^27 + 9 * q^29 + 11 * q^31 - 14 * q^33 + 30 * q^35 - 13 * q^37 + 16 * q^39 - 13 * q^41 + 23 * q^43 + 12 * q^45 + 43 * q^47 + 35 * q^49 - 4 * q^51 - 4 * q^53 + 43 * q^55 + 3 * q^57 + 30 * q^59 + 43 * q^61 + 38 * q^63 + 3 * q^65 + 50 * q^67 + 34 * q^69 + 3 * q^71 - 16 * q^73 + 21 * q^75 + 18 * q^77 + 45 * q^79 + 6 * q^81 + 63 * q^83 - 2 * q^85 + 42 * q^87 + 6 * q^89 + 22 * q^91 - 2 * q^93 + 19 * q^95 - 28 * q^97 + 51 * 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
$3$	0.597652	0.345055	0.172527	−	0.985005i	$-0.444807\pi$
$3$	0.597652	0.345055	0.172527	+	0.985005i	$0.444807\pi$
$4$	0	0
$5$	−3.19982	−1.43100	−0.715501	−	0.698611i	$-0.753802\pi$
$5$	−3.19982	−1.43100	−0.715501	+	0.698611i	$0.753802\pi$
$6$	0	0
$7$	4.21450	1.59293	0.796465	−	0.604684i	$-0.206701\pi$
$7$	4.21450	1.59293	0.796465	+	0.604684i	$0.206701\pi$
$8$	0	0
$9$	−2.64281	−0.880937
$10$	0	0
$11$	5.94939	1.79381	0.896904	−	0.442225i	$-0.145811\pi$
$11$	5.94939	1.79381	0.896904	+	0.442225i	$0.145811\pi$
$12$	0	0
$13$	−0.565788	−0.156921	−0.0784607	−	0.996917i	$-0.525001\pi$
$13$	−0.565788	−0.156921	−0.0784607	+	0.996917i	$0.525001\pi$
$14$	0	0
$15$	−1.91238	−0.493774
$16$	0	0
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	0	0
$19$	5.00504	1.14824	0.574118	−	0.818773i	$-0.305345\pi$
$19$	5.00504	1.14824	0.574118	+	0.818773i	$0.305345\pi$
$20$	0	0
$21$	2.51880	0.549648
$22$	0	0
$23$	5.77337	1.20383	0.601915	−	0.798560i	$-0.294404\pi$
$23$	5.77337	1.20383	0.601915	+	0.798560i	$0.294404\pi$
$24$	0	0
$25$	5.23884	1.04777
$26$	0	0
$27$	−3.37244	−0.649026
$28$	0	0
$29$	−0.0765582	−0.0142165	−0.00710825	−	0.999975i	$-0.502263\pi$
$29$	−0.0765582	−0.0142165	−0.00710825	+	0.999975i	$0.502263\pi$
$30$	0	0
$31$	−10.9331	−1.96364	−0.981822	−	0.189807i	$-0.939214\pi$
$31$	−10.9331	−1.96364	−0.981822	+	0.189807i	$0.939214\pi$
$32$	0	0
$33$	3.55566	0.618962
$34$	0	0
$35$	−13.4856	−2.27949
$36$	0	0
$37$	6.62492	1.08913	0.544565	−	0.838718i	$-0.316695\pi$
$37$	6.62492	1.08913	0.544565	+	0.838718i	$0.316695\pi$
$38$	0	0
$39$	−0.338145	−0.0541465
$40$	0	0
$41$	2.55967	0.399753	0.199876	−	0.979821i	$-0.435946\pi$
$41$	2.55967	0.399753	0.199876	+	0.979821i	$0.435946\pi$
$42$	0	0
$43$	1.61320	0.246010	0.123005	−	0.992406i	$-0.460747\pi$
$43$	1.61320	0.246010	0.123005	+	0.992406i	$0.460747\pi$
$44$	0	0
$45$	8.45652	1.26062
$46$	0	0
$47$	7.42334	1.08281	0.541403	−	0.840763i	$-0.317893\pi$
$47$	7.42334	1.08281	0.541403	+	0.840763i	$0.317893\pi$
$48$	0	0
$49$	10.7620	1.53743
$50$	0	0
$51$	−0.597652	−0.0836880
$52$	0	0
$53$	−11.6959	−1.60656	−0.803281	−	0.595601i	$-0.796914\pi$
$53$	−11.6959	−1.60656	−0.803281	+	0.595601i	$0.796914\pi$
$54$	0	0
$55$	−19.0370	−2.56694
$56$	0	0
$57$	2.99127	0.396204
$58$	0	0
$59$	1.00000	0.130189
$59$	1.00000	0.130189
$60$	0	0
$61$	11.6150	1.48715	0.743576	−	0.668651i	$-0.233128\pi$
$61$	11.6150	1.48715	0.743576	+	0.668651i	$0.233128\pi$
$62$	0	0
$63$	−11.1381	−1.40327
$64$	0	0
$65$	1.81042	0.224555
$66$	0	0
$67$	−7.00194	−0.855423	−0.427712	−	0.903915i	$-0.640680\pi$
$67$	−7.00194	−0.855423	−0.427712	+	0.903915i	$0.640680\pi$
$68$	0	0
$69$	3.45047	0.415387
$70$	0	0
$71$	8.35620	0.991698	0.495849	−	0.868409i	$-0.334857\pi$
$71$	8.35620	0.991698	0.495849	+	0.868409i	$0.334857\pi$
$72$	0	0
$73$	0.574110	0.0671945	0.0335973	−	0.999435i	$-0.489304\pi$
$73$	0.574110	0.0671945	0.0335973	+	0.999435i	$0.489304\pi$
$74$	0	0
$75$	3.13100	0.361537
$76$	0	0
$77$	25.0737	2.85741
$78$	0	0
$79$	4.78437	0.538284	0.269142	−	0.963101i	$-0.413260\pi$
$79$	4.78437	0.538284	0.269142	+	0.963101i	$0.413260\pi$
$80$	0	0
$81$	5.91289	0.656988
$82$	0	0
$83$	−3.23362	−0.354936	−0.177468	−	0.984127i	$-0.556791\pi$
$83$	−3.23362	−0.354936	−0.177468	+	0.984127i	$0.556791\pi$
$84$	0	0
$85$	3.19982	0.347069
$86$	0	0
$87$	−0.0457552	−0.00490547
$88$	0	0
$89$	14.4811	1.53499	0.767496	−	0.641054i	$-0.221503\pi$
$89$	14.4811	1.53499	0.767496	+	0.641054i	$0.221503\pi$
$90$	0	0
$91$	−2.38451	−0.249965
$92$	0	0
$93$	−6.53419	−0.677564
$94$	0	0
$95$	−16.0152	−1.64313
$96$	0	0
$97$	−18.3927	−1.86749	−0.933746	−	0.357936i	$-0.883480\pi$
$97$	−18.3927	−1.86749	−0.933746	+	0.357936i	$0.883480\pi$
$98$	0	0
$99$	−15.7231	−1.58023
$100$	0	0
$101$	−8.20964	−0.816890	−0.408445	−	0.912783i	$-0.633929\pi$
$101$	−8.20964	−0.816890	−0.408445	+	0.912783i	$0.633929\pi$
$102$	0	0
$103$	2.72652	0.268652	0.134326	−	0.990937i	$-0.457113\pi$
$103$	2.72652	0.268652	0.134326	+	0.990937i	$0.457113\pi$
$104$	0	0
$105$	−8.05972	−0.786548
$106$	0	0
$107$	−11.2340	−1.08603	−0.543017	−	0.839722i	$-0.682718\pi$
$107$	−11.2340	−1.08603	−0.543017	+	0.839722i	$0.682718\pi$
$108$	0	0
$109$	10.9200	1.04595	0.522973	−	0.852349i	$-0.324823\pi$
$109$	10.9200	1.04595	0.522973	+	0.852349i	$0.324823\pi$
$110$	0	0
$111$	3.95940	0.375810
$112$	0	0
$113$	−18.5822	−1.74807	−0.874034	−	0.485865i	$-0.838505\pi$
$113$	−18.5822	−1.74807	−0.874034	+	0.485865i	$0.838505\pi$
$114$	0	0
$115$	−18.4737	−1.72268
$116$	0	0
$117$	1.49527	0.138238
$118$	0	0
$119$	−4.21450	−0.386342
$120$	0	0
$121$	24.3952	2.21775
$122$	0	0
$123$	1.52979	0.137937
$124$	0	0
$125$	−0.764245	−0.0683562
$126$	0	0
$127$	−1.16408	−0.103296	−0.0516479	−	0.998665i	$-0.516447\pi$
$127$	−1.16408	−0.103296	−0.0516479	+	0.998665i	$0.516447\pi$
$128$	0	0
$129$	0.964130	0.0848869
$130$	0	0
$131$	−13.7999	−1.20570	−0.602851	−	0.797853i	$-0.705969\pi$
$131$	−13.7999	−1.20570	−0.602851	+	0.797853i	$0.705969\pi$
$132$	0	0
$133$	21.0937	1.82906
$134$	0	0
$135$	10.7912	0.928758
$136$	0	0
$137$	−6.52942	−0.557846	−0.278923	−	0.960313i	$-0.589977\pi$
$137$	−6.52942	−0.557846	−0.278923	+	0.960313i	$0.589977\pi$
$138$	0	0
$139$	−6.77205	−0.574397	−0.287199	−	0.957871i	$-0.592724\pi$
$139$	−6.77205	−0.574397	−0.287199	+	0.957871i	$0.592724\pi$
$140$	0	0
$141$	4.43658	0.373627
$142$	0	0
$143$	−3.36610	−0.281487
$144$	0	0
$145$	0.244972	0.0203439
$146$	0	0
$147$	6.43193	0.530497
$148$	0	0
$149$	−10.9988	−0.901057	−0.450528	−	0.892762i	$-0.648764\pi$
$149$	−10.9988	−0.901057	−0.450528	+	0.892762i	$0.648764\pi$
$150$	0	0
$151$	8.09867	0.659060	0.329530	−	0.944145i	$-0.393110\pi$
$151$	8.09867	0.659060	0.329530	+	0.944145i	$0.393110\pi$
$152$	0	0
$153$	2.64281	0.213659
$154$	0	0
$155$	34.9839	2.80998
$156$	0	0
$157$	8.13666	0.649376	0.324688	−	0.945821i	$-0.394741\pi$
$157$	8.13666	0.649376	0.324688	+	0.945821i	$0.394741\pi$
$158$	0	0
$159$	−6.99011	−0.554351
$160$	0	0
$161$	24.3319	1.91762
$162$	0	0
$163$	7.60121	0.595373	0.297686	−	0.954664i	$-0.403785\pi$
$163$	7.60121	0.595373	0.297686	+	0.954664i	$0.403785\pi$
$164$	0	0
$165$	−11.3775	−0.885736
$166$	0	0
$167$	16.4158	1.27029	0.635147	−	0.772392i	$-0.280940\pi$
$167$	16.4158	1.27029	0.635147	+	0.772392i	$0.280940\pi$
$168$	0	0
$169$	−12.6799	−0.975376
$170$	0	0
$171$	−13.2274	−1.01152
$172$	0	0
$173$	15.7257	1.19561	0.597803	−	0.801643i	$-0.296040\pi$
$173$	15.7257	1.19561	0.597803	+	0.801643i	$0.296040\pi$
$174$	0	0
$175$	22.0791	1.66902
$176$	0	0
$177$	0.597652	0.0449223
$178$	0	0
$179$	−7.39362	−0.552625	−0.276312	−	0.961068i	$-0.589112\pi$
$179$	−7.39362	−0.552625	−0.276312	+	0.961068i	$0.589112\pi$
$180$	0	0
$181$	23.5660	1.75164	0.875822	−	0.482634i	$-0.160320\pi$
$181$	23.5660	1.75164	0.875822	+	0.482634i	$0.160320\pi$
$182$	0	0
$183$	6.94175	0.513149
$184$	0	0
$185$	−21.1986	−1.55855
$186$	0	0
$187$	−5.94939	−0.435062
$188$	0	0
$189$	−14.2131	−1.03385
$190$	0	0
$191$	1.20456	0.0871592	0.0435796	−	0.999050i	$-0.486124\pi$
$191$	1.20456	0.0871592	0.0435796	+	0.999050i	$0.486124\pi$
$192$	0	0
$193$	4.89051	0.352027	0.176013	−	0.984388i	$-0.443680\pi$
$193$	4.89051	0.352027	0.176013	+	0.984388i	$0.443680\pi$
$194$	0	0
$195$	1.08200	0.0774837
$196$	0	0
$197$	−14.6629	−1.04469	−0.522345	−	0.852734i	$-0.674943\pi$
$197$	−14.6629	−1.04469	−0.522345	+	0.852734i	$0.674943\pi$
$198$	0	0
$199$	−14.1723	−1.00465	−0.502323	−	0.864680i	$-0.667521\pi$
$199$	−14.1723	−1.00465	−0.502323	+	0.864680i	$0.667521\pi$
$200$	0	0
$201$	−4.18473	−0.295168
$202$	0	0
$203$	−0.322655	−0.0226459
$204$	0	0
$205$	−8.19047	−0.572047
$206$	0	0
$207$	−15.2579	−1.06050
$208$	0	0
$209$	29.7769	2.05971
$210$	0	0
$211$	23.7374	1.63415	0.817075	−	0.576532i	$-0.195594\pi$
$211$	23.7374	1.63415	0.817075	+	0.576532i	$0.195594\pi$
$212$	0	0
$213$	4.99410	0.342190
$214$	0	0
$215$	−5.16193	−0.352041
$216$	0	0
$217$	−46.0775	−3.12795
$218$	0	0
$219$	0.343118	0.0231858
$220$	0	0
$221$	0.565788	0.0380590
$222$	0	0
$223$	22.2121	1.48743	0.743715	−	0.668496i	$-0.233062\pi$
$223$	22.2121	1.48743	0.743715	+	0.668496i	$0.233062\pi$
$224$	0	0
$225$	−13.8453	−0.923018
$226$	0	0
$227$	26.3144	1.74655	0.873274	−	0.487229i	$-0.161992\pi$
$227$	26.3144	1.74655	0.873274	+	0.487229i	$0.161992\pi$
$228$	0	0
$229$	−15.5730	−1.02909	−0.514547	−	0.857462i	$-0.672040\pi$
$229$	−15.5730	−1.02909	−0.514547	+	0.857462i	$0.672040\pi$
$230$	0	0
$231$	14.9853	0.985963
$232$	0	0
$233$	9.05035	0.592908	0.296454	−	0.955047i	$-0.404196\pi$
$233$	9.05035	0.592908	0.296454	+	0.955047i	$0.404196\pi$
$234$	0	0
$235$	−23.7534	−1.54950
$236$	0	0
$237$	2.85939	0.185737
$238$	0	0
$239$	−7.45554	−0.482259	−0.241129	−	0.970493i	$-0.577518\pi$
$239$	−7.45554	−0.482259	−0.241129	+	0.970493i	$0.577518\pi$
$240$	0	0
$241$	29.4558	1.89742	0.948708	−	0.316153i	$-0.102391\pi$
$241$	29.4558	1.89742	0.948708	+	0.316153i	$0.102391\pi$
$242$	0	0
$243$	13.6512	0.875723
$244$	0	0
$245$	−34.4365	−2.20006
$246$	0	0
$247$	−2.83179	−0.180183
$248$	0	0
$249$	−1.93258	−0.122472
$250$	0	0
$251$	10.0506	0.634385	0.317193	−	0.948361i	$-0.397260\pi$
$251$	10.0506	0.634385	0.317193	+	0.948361i	$0.397260\pi$
$252$	0	0
$253$	34.3480	2.15944
$254$	0	0
$255$	1.91238	0.119758
$256$	0	0
$257$	−22.1018	−1.37867	−0.689336	−	0.724442i	$-0.742097\pi$
$257$	−22.1018	−1.37867	−0.689336	+	0.724442i	$0.742097\pi$
$258$	0	0
$259$	27.9207	1.73491
$260$	0	0
$261$	0.202329	0.0125238
$262$	0	0
$263$	−12.6540	−0.780281	−0.390141	−	0.920755i	$-0.627574\pi$
$263$	−12.6540	−0.780281	−0.390141	+	0.920755i	$0.627574\pi$
$264$	0	0
$265$	37.4249	2.29899
$266$	0	0
$267$	8.65465	0.529656
$268$	0	0
$269$	19.8532	1.21047	0.605236	−	0.796046i	$-0.293079\pi$
$269$	19.8532	1.21047	0.605236	+	0.796046i	$0.293079\pi$
$270$	0	0
$271$	−11.2296	−0.682150	−0.341075	−	0.940036i	$-0.610791\pi$
$271$	−11.2296	−0.682150	−0.341075	+	0.940036i	$0.610791\pi$
$272$	0	0
$273$	−1.42511	−0.0862516
$274$	0	0
$275$	31.1679	1.87949
$276$	0	0
$277$	5.76667	0.346485	0.173243	−	0.984879i	$-0.444576\pi$
$277$	5.76667	0.346485	0.173243	+	0.984879i	$0.444576\pi$
$278$	0	0
$279$	28.8941	1.72985
$280$	0	0
$281$	−24.6309	−1.46936	−0.734678	−	0.678416i	$-0.762667\pi$
$281$	−24.6309	−1.46936	−0.734678	+	0.678416i	$0.762667\pi$
$282$	0	0
$283$	1.97137	0.117186	0.0585930	−	0.998282i	$-0.481339\pi$
$283$	1.97137	0.117186	0.0585930	+	0.998282i	$0.481339\pi$
$284$	0	0
$285$	−9.57153	−0.566969
$286$	0	0
$287$	10.7877	0.636779
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	−10.9924	−0.644387
$292$	0	0
$293$	23.2534	1.35848	0.679239	−	0.733917i	$-0.262310\pi$
$293$	23.2534	1.35848	0.679239	+	0.733917i	$0.262310\pi$
$294$	0	0
$295$	−3.19982	−0.186301
$296$	0	0
$297$	−20.0639	−1.16423
$298$	0	0
$299$	−3.26651	−0.188907
$300$	0	0
$301$	6.79881	0.391877
$302$	0	0
$303$	−4.90651	−0.281872
$304$	0	0
$305$	−37.1660	−2.12812
$306$	0	0
$307$	12.5601	0.716844	0.358422	−	0.933560i	$-0.383315\pi$
$307$	12.5601	0.716844	0.358422	+	0.933560i	$0.383315\pi$
$308$	0	0
$309$	1.62951	0.0926998
$310$	0	0
$311$	16.8956	0.958059	0.479030	−	0.877799i	$-0.340989\pi$
$311$	16.8956	0.958059	0.479030	+	0.877799i	$0.340989\pi$
$312$	0	0
$313$	30.8543	1.74399	0.871995	−	0.489514i	$-0.162826\pi$
$313$	30.8543	1.74399	0.871995	+	0.489514i	$0.162826\pi$
$314$	0	0
$315$	35.6400	2.00809
$316$	0	0
$317$	−7.72827	−0.434063	−0.217031	−	0.976165i	$-0.569637\pi$
$317$	−7.72827	−0.434063	−0.217031	+	0.976165i	$0.569637\pi$
$318$	0	0
$319$	−0.455475	−0.0255017
$320$	0	0
$321$	−6.71403	−0.374741
$322$	0	0
$323$	−5.00504	−0.278488
$324$	0	0
$325$	−2.96408	−0.164417
$326$	0	0
$327$	6.52636	0.360909
$328$	0	0
$329$	31.2857	1.72483
$330$	0	0
$331$	32.9846	1.81300	0.906498	−	0.422210i	$-0.138745\pi$
$331$	32.9846	1.81300	0.906498	+	0.422210i	$0.138745\pi$
$332$	0	0
$333$	−17.5084	−0.959456
$334$	0	0
$335$	22.4049	1.22411
$336$	0	0
$337$	23.4156	1.27553	0.637764	−	0.770232i	$-0.279859\pi$
$337$	23.4156	1.27553	0.637764	+	0.770232i	$0.279859\pi$
$338$	0	0
$339$	−11.1057	−0.603179
$340$	0	0
$341$	−65.0453	−3.52240
$342$	0	0
$343$	15.8550	0.856087
$344$	0	0
$345$	−11.0409	−0.594420
$346$	0	0
$347$	8.34801	0.448144	0.224072	−	0.974573i	$-0.428065\pi$
$347$	8.34801	0.448144	0.224072	+	0.974573i	$0.428065\pi$
$348$	0	0
$349$	25.6675	1.37395	0.686974	−	0.726682i	$-0.258938\pi$
$349$	25.6675	1.37395	0.686974	+	0.726682i	$0.258938\pi$
$350$	0	0
$351$	1.90809	0.101846
$352$	0	0
$353$	2.74983	0.146359	0.0731794	−	0.997319i	$-0.476685\pi$
$353$	2.74983	0.146359	0.0731794	+	0.997319i	$0.476685\pi$
$354$	0	0
$355$	−26.7383	−1.41912
$356$	0	0
$357$	−2.51880	−0.133309
$358$	0	0
$359$	8.93294	0.471462	0.235731	−	0.971818i	$-0.424252\pi$
$359$	8.93294	0.471462	0.235731	+	0.971818i	$0.424252\pi$
$360$	0	0
$361$	6.05044	0.318444
$362$	0	0
$363$	14.5799	0.765244
$364$	0	0
$365$	−1.83705	−0.0961555
$366$	0	0
$367$	−1.19550	−0.0624045	−0.0312023	−	0.999513i	$-0.509934\pi$
$367$	−1.19550	−0.0624045	−0.0312023	+	0.999513i	$0.509934\pi$
$368$	0	0
$369$	−6.76472	−0.352157
$370$	0	0
$371$	−49.2925	−2.55914
$372$	0	0
$373$	−3.31164	−0.171470	−0.0857352	−	0.996318i	$-0.527324\pi$
$373$	−3.31164	−0.171470	−0.0857352	+	0.996318i	$0.527324\pi$
$374$	0	0
$375$	−0.456753	−0.0235866
$376$	0	0
$377$	0.0433158	0.00223087
$378$	0	0
$379$	−32.4774	−1.66825	−0.834126	−	0.551573i	$-0.814028\pi$
$379$	−32.4774	−1.66825	−0.834126	+	0.551573i	$0.814028\pi$
$380$	0	0
$381$	−0.695717	−0.0356427
$382$	0	0
$383$	−20.8129	−1.06349	−0.531744	−	0.846905i	$-0.678463\pi$
$383$	−20.8129	−1.06349	−0.531744	+	0.846905i	$0.678463\pi$
$384$	0	0
$385$	−80.2313	−4.08896
$386$	0	0
$387$	−4.26337	−0.216719
$388$	0	0
$389$	−23.2018	−1.17638	−0.588188	−	0.808724i	$-0.700159\pi$
$389$	−23.2018	−1.17638	−0.588188	+	0.808724i	$0.700159\pi$
$390$	0	0
$391$	−5.77337	−0.291972
$392$	0	0
$393$	−8.24754	−0.416033
$394$	0	0
$395$	−15.3091	−0.770286
$396$	0	0
$397$	11.5211	0.578226	0.289113	−	0.957295i	$-0.406640\pi$
$397$	11.5211	0.578226	0.289113	+	0.957295i	$0.406640\pi$
$398$	0	0
$399$	12.6067	0.631125
$400$	0	0
$401$	9.30535	0.464687	0.232344	−	0.972634i	$-0.425361\pi$
$401$	9.30535	0.464687	0.232344	+	0.972634i	$0.425361\pi$
$402$	0	0
$403$	6.18582	0.308138
$404$	0	0
$405$	−18.9202	−0.940151
$406$	0	0
$407$	39.4142	1.95369
$408$	0	0
$409$	13.1571	0.650577	0.325289	−	0.945615i	$-0.394539\pi$
$409$	13.1571	0.650577	0.325289	+	0.945615i	$0.394539\pi$
$410$	0	0
$411$	−3.90232	−0.192487
$412$	0	0
$413$	4.21450	0.207382
$414$	0	0
$415$	10.3470	0.507914
$416$	0	0
$417$	−4.04733	−0.198199
$418$	0	0
$419$	−16.7397	−0.817790	−0.408895	−	0.912581i	$-0.634086\pi$
$419$	−16.7397	−0.817790	−0.408895	+	0.912581i	$0.634086\pi$
$420$	0	0
$421$	3.14716	0.153383	0.0766917	−	0.997055i	$-0.475564\pi$
$421$	3.14716	0.153383	0.0766917	+	0.997055i	$0.475564\pi$
$422$	0	0
$423$	−19.6185	−0.953884
$424$	0	0
$425$	−5.23884	−0.254121
$426$	0	0
$427$	48.9515	2.36893
$428$	0	0
$429$	−2.01175	−0.0971284
$430$	0	0
$431$	4.51298	0.217383	0.108691	−	0.994076i	$-0.465334\pi$
$431$	4.51298	0.217383	0.108691	+	0.994076i	$0.465334\pi$
$432$	0	0
$433$	15.1810	0.729555	0.364777	−	0.931095i	$-0.381145\pi$
$433$	15.1810	0.729555	0.364777	+	0.931095i	$0.381145\pi$
$434$	0	0
$435$	0.146408	0.00701974
$436$	0	0
$437$	28.8959	1.38228
$438$	0	0
$439$	27.4003	1.30774	0.653872	−	0.756606i	$-0.273144\pi$
$439$	27.4003	1.30774	0.653872	+	0.756606i	$0.273144\pi$
$440$	0	0
$441$	−28.4419	−1.35438
$442$	0	0
$443$	−11.1402	−0.529286	−0.264643	−	0.964346i	$-0.585254\pi$
$443$	−11.1402	−0.529286	−0.264643	+	0.964346i	$0.585254\pi$
$444$	0	0
$445$	−46.3368	−2.19658
$446$	0	0
$447$	−6.57346	−0.310914
$448$	0	0
$449$	2.03036	0.0958187	0.0479093	−	0.998852i	$-0.484744\pi$
$449$	2.03036	0.0958187	0.0479093	+	0.998852i	$0.484744\pi$
$450$	0	0
$451$	15.2285	0.717080
$452$	0	0
$453$	4.84019	0.227412
$454$	0	0
$455$	7.63002	0.357701
$456$	0	0
$457$	−14.2997	−0.668911	−0.334456	−	0.942411i	$-0.608552\pi$
$457$	−14.2997	−0.668911	−0.334456	+	0.942411i	$0.608552\pi$
$458$	0	0
$459$	3.37244	0.157412
$460$	0	0
$461$	26.0168	1.21172	0.605861	−	0.795571i	$-0.292829\pi$
$461$	26.0168	1.21172	0.605861	+	0.795571i	$0.292829\pi$
$462$	0	0
$463$	−18.7655	−0.872109	−0.436054	−	0.899920i	$-0.643624\pi$
$463$	−18.7655	−0.872109	−0.436054	+	0.899920i	$0.643624\pi$
$464$	0	0
$465$	20.9082	0.969596
$466$	0	0
$467$	6.25153	0.289286	0.144643	−	0.989484i	$-0.453797\pi$
$467$	6.25153	0.289286	0.144643	+	0.989484i	$0.453797\pi$
$468$	0	0
$469$	−29.5097	−1.36263
$470$	0	0
$471$	4.86289	0.224070
$472$	0	0
$473$	9.59753	0.441295
$474$	0	0
$475$	26.2206	1.20308
$476$	0	0
$477$	30.9102	1.41528
$478$	0	0
$479$	14.3534	0.655823	0.327912	−	0.944708i	$-0.393655\pi$
$479$	14.3534	0.655823	0.327912	+	0.944708i	$0.393655\pi$
$480$	0	0
$481$	−3.74830	−0.170908
$482$	0	0
$483$	14.5420	0.661683
$484$	0	0
$485$	58.8532	2.67239
$486$	0	0
$487$	−31.6741	−1.43529	−0.717645	−	0.696410i	$-0.754780\pi$
$487$	−31.6741	−1.43529	−0.717645	+	0.696410i	$0.754780\pi$
$488$	0	0
$489$	4.54288	0.205436
$490$	0	0
$491$	7.40786	0.334312	0.167156	−	0.985930i	$-0.446542\pi$
$491$	7.40786	0.334312	0.167156	+	0.985930i	$0.446542\pi$
$492$	0	0
$493$	0.0765582	0.00344801
$494$	0	0
$495$	50.3111	2.26132
$496$	0	0
$497$	35.2172	1.57971
$498$	0	0
$499$	39.5865	1.77213	0.886067	−	0.463557i	$-0.153427\pi$
$499$	39.5865	1.77213	0.886067	+	0.463557i	$0.153427\pi$
$500$	0	0
$501$	9.81094	0.438321
$502$	0	0
$503$	−14.7654	−0.658358	−0.329179	−	0.944268i	$-0.606772\pi$
$503$	−14.7654	−0.658358	−0.329179	+	0.944268i	$0.606772\pi$
$504$	0	0
$505$	26.2694	1.16897
$506$	0	0
$507$	−7.57816	−0.336558
$508$	0	0
$509$	36.1760	1.60347	0.801736	−	0.597678i	$-0.203910\pi$
$509$	36.1760	1.60347	0.801736	+	0.597678i	$0.203910\pi$
$510$	0	0
$511$	2.41959	0.107036
$512$	0	0
$513$	−16.8792	−0.745235
$514$	0	0
$515$	−8.72438	−0.384442
$516$	0	0
$517$	44.1644	1.94235
$518$	0	0
$519$	9.39853	0.412550
$520$	0	0
$521$	33.3919	1.46293	0.731463	−	0.681881i	$-0.238838\pi$
$521$	33.3919	1.46293	0.731463	+	0.681881i	$0.238838\pi$
$522$	0	0
$523$	−21.0560	−0.920715	−0.460358	−	0.887734i	$-0.652279\pi$
$523$	−21.0560	−0.920715	−0.460358	+	0.887734i	$0.652279\pi$
$524$	0	0
$525$	13.1956	0.575904
$526$	0	0
$527$	10.9331	0.476253
$528$	0	0
$529$	10.3318	0.449208
$530$	0	0
$531$	−2.64281	−0.114688
$532$	0	0
$533$	−1.44823	−0.0627298
$534$	0	0
$535$	35.9468	1.55412
$536$	0	0
$537$	−4.41881	−0.190686
$538$	0	0
$539$	64.0273	2.75785
$540$	0	0
$541$	2.95451	0.127024	0.0635122	−	0.997981i	$-0.479770\pi$
$541$	2.95451	0.127024	0.0635122	+	0.997981i	$0.479770\pi$
$542$	0	0
$543$	14.0842	0.604413
$544$	0	0
$545$	−34.9420	−1.49675
$546$	0	0
$547$	33.0752	1.41420	0.707098	−	0.707116i	$-0.250004\pi$
$547$	33.0752	1.41420	0.707098	+	0.707116i	$0.250004\pi$
$548$	0	0
$549$	−30.6963	−1.31009
$550$	0	0
$551$	−0.383177	−0.0163239
$552$	0	0
$553$	20.1637	0.857449
$554$	0	0
$555$	−12.6694	−0.537784
$556$	0	0
$557$	44.7331	1.89540	0.947700	−	0.319162i	$-0.103401\pi$
$557$	44.7331	1.89540	0.947700	+	0.319162i	$0.103401\pi$
$558$	0	0
$559$	−0.912727	−0.0386043
$560$	0	0
$561$	−3.55566	−0.150120
$562$	0	0
$563$	−25.4392	−1.07214	−0.536068	−	0.844175i	$-0.680091\pi$
$563$	−25.4392	−1.07214	−0.536068	+	0.844175i	$0.680091\pi$
$564$	0	0
$565$	59.4597	2.50149
$566$	0	0
$567$	24.9199	1.04654
$568$	0	0
$569$	−15.9911	−0.670380	−0.335190	−	0.942151i	$-0.608800\pi$
$569$	−15.9911	−0.670380	−0.335190	+	0.942151i	$0.608800\pi$
$570$	0	0
$571$	9.37682	0.392408	0.196204	−	0.980563i	$-0.437139\pi$
$571$	9.37682	0.392408	0.196204	+	0.980563i	$0.437139\pi$
$572$	0	0
$573$	0.719911	0.0300747
$574$	0	0
$575$	30.2458	1.26134
$576$	0	0
$577$	−33.9035	−1.41142	−0.705710	−	0.708501i	$-0.749372\pi$
$577$	−33.9035	−1.41142	−0.705710	+	0.708501i	$0.749372\pi$
$578$	0	0
$579$	2.92283	0.121468
$580$	0	0
$581$	−13.6281	−0.565388
$582$	0	0
$583$	−69.5837	−2.88186
$584$	0	0
$585$	−4.78460	−0.197819
$586$	0	0
$587$	−9.99500	−0.412538	−0.206269	−	0.978495i	$-0.566132\pi$
$587$	−9.99500	−0.412538	−0.206269	+	0.978495i	$0.566132\pi$
$588$	0	0
$589$	−54.7206	−2.25472
$590$	0	0
$591$	−8.76333	−0.360475
$592$	0	0
$593$	−40.6254	−1.66828	−0.834142	−	0.551549i	$-0.814037\pi$
$593$	−40.6254	−1.66828	−0.834142	+	0.551549i	$0.814037\pi$
$594$	0	0
$595$	13.4856	0.552857
$596$	0	0
$597$	−8.47009	−0.346658
$598$	0	0
$599$	−20.2046	−0.825536	−0.412768	−	0.910836i	$-0.635438\pi$
$599$	−20.2046	−0.825536	−0.412768	+	0.910836i	$0.635438\pi$
$600$	0	0
$601$	−28.1337	−1.14760	−0.573799	−	0.818996i	$-0.694531\pi$
$601$	−28.1337	−1.14760	−0.573799	+	0.818996i	$0.694531\pi$
$602$	0	0
$603$	18.5048	0.753574
$604$	0	0
$605$	−78.0603	−3.17360
$606$	0	0
$607$	5.24335	0.212821	0.106410	−	0.994322i	$-0.466064\pi$
$607$	5.24335	0.212821	0.106410	+	0.994322i	$0.466064\pi$
$608$	0	0
$609$	−0.192835	−0.00781408
$610$	0	0
$611$	−4.20004	−0.169915
$612$	0	0
$613$	16.7547	0.676717	0.338358	−	0.941017i	$-0.390128\pi$
$613$	16.7547	0.676717	0.338358	+	0.941017i	$0.390128\pi$
$614$	0	0
$615$	−4.89505	−0.197388
$616$	0	0
$617$	36.4112	1.46586	0.732931	−	0.680303i	$-0.238152\pi$
$617$	36.4112	1.46586	0.732931	+	0.680303i	$0.238152\pi$
$618$	0	0
$619$	17.1605	0.689741	0.344870	−	0.938650i	$-0.387923\pi$
$619$	17.1605	0.689741	0.344870	+	0.938650i	$0.387923\pi$
$620$	0	0
$621$	−19.4703	−0.781318
$622$	0	0
$623$	61.0305	2.44514
$624$	0	0
$625$	−23.7488	−0.949950
$626$	0	0
$627$	17.7962	0.710714
$628$	0	0
$629$	−6.62492	−0.264153
$630$	0	0
$631$	4.14095	0.164849	0.0824243	−	0.996597i	$-0.473734\pi$
$631$	4.14095	0.164849	0.0824243	+	0.996597i	$0.473734\pi$
$632$	0	0
$633$	14.1867	0.563871
$634$	0	0
$635$	3.72486	0.147816
$636$	0	0
$637$	−6.08902	−0.241256
$638$	0	0
$639$	−22.0839	−0.873624
$640$	0	0
$641$	−23.7685	−0.938799	−0.469399	−	0.882986i	$-0.655530\pi$
$641$	−23.7685	−0.938799	−0.469399	+	0.882986i	$0.655530\pi$
$642$	0	0
$643$	−6.02287	−0.237519	−0.118759	−	0.992923i	$-0.537892\pi$
$643$	−6.02287	−0.237519	−0.118759	+	0.992923i	$0.537892\pi$
$644$	0	0
$645$	−3.08504	−0.121473
$646$	0	0
$647$	43.7351	1.71940	0.859701	−	0.510797i	$-0.170650\pi$
$647$	43.7351	1.71940	0.859701	+	0.510797i	$0.170650\pi$
$648$	0	0
$649$	5.94939	0.233534
$650$	0	0
$651$	−27.5383	−1.07931
$652$	0	0
$653$	−0.420392	−0.0164512	−0.00822560	−	0.999966i	$-0.502618\pi$
$653$	−0.420392	−0.0164512	−0.00822560	+	0.999966i	$0.502618\pi$
$654$	0	0
$655$	44.1572	1.72536
$656$	0	0
$657$	−1.51727	−0.0591942
$658$	0	0
$659$	−38.6131	−1.50415	−0.752077	−	0.659075i	$-0.770948\pi$
$659$	−38.6131	−1.50415	−0.752077	+	0.659075i	$0.770948\pi$
$660$	0	0
$661$	7.73738	0.300949	0.150475	−	0.988614i	$-0.451920\pi$
$661$	7.73738	0.300949	0.150475	+	0.988614i	$0.451920\pi$
$662$	0	0
$663$	0.338145	0.0131325
$664$	0	0
$665$	−67.4961	−2.61739
$666$	0	0
$667$	−0.441999	−0.0171143
$668$	0	0
$669$	13.2751	0.513245
$670$	0	0
$671$	69.1023	2.66767
$672$	0	0
$673$	−41.2488	−1.59003	−0.795013	−	0.606593i	$-0.792536\pi$
$673$	−41.2488	−1.59003	−0.795013	+	0.606593i	$0.792536\pi$
$674$	0	0
$675$	−17.6677	−0.680029
$676$	0	0
$677$	12.8553	0.494071	0.247035	−	0.969006i	$-0.420544\pi$
$677$	12.8553	0.494071	0.247035	+	0.969006i	$0.420544\pi$
$678$	0	0
$679$	−77.5159	−2.97479
$680$	0	0
$681$	15.7269	0.602655
$682$	0	0
$683$	−14.3216	−0.547999	−0.274000	−	0.961730i	$-0.588347\pi$
$683$	−14.3216	−0.547999	−0.274000	+	0.961730i	$0.588347\pi$
$684$	0	0
$685$	20.8930	0.798279
$686$	0	0
$687$	−9.30726	−0.355094
$688$	0	0
$689$	6.61743	0.252104
$690$	0	0
$691$	9.45437	0.359661	0.179831	−	0.983698i	$-0.442445\pi$
$691$	9.45437	0.359661	0.179831	+	0.983698i	$0.442445\pi$
$692$	0	0
$693$	−66.2651	−2.51720
$694$	0	0
$695$	21.6693	0.821964
$696$	0	0
$697$	−2.55967	−0.0969543
$698$	0	0
$699$	5.40896	0.204586
$700$	0	0
$701$	−14.0128	−0.529255	−0.264628	−	0.964351i	$-0.585249\pi$
$701$	−14.0128	−0.529255	−0.264628	+	0.964351i	$0.585249\pi$
$702$	0	0
$703$	33.1580	1.25058
$704$	0	0
$705$	−14.1962	−0.534661
$706$	0	0
$707$	−34.5995	−1.30125
$708$	0	0
$709$	16.4708	0.618574	0.309287	−	0.950969i	$-0.399910\pi$
$709$	16.4708	0.618574	0.309287	+	0.950969i	$0.399910\pi$
$710$	0	0
$711$	−12.6442	−0.474194
$712$	0	0
$713$	−63.1208	−2.36389
$714$	0	0
$715$	10.7709	0.402809
$716$	0	0
$717$	−4.45582	−0.166406
$718$	0	0
$719$	−36.6670	−1.36745	−0.683725	−	0.729740i	$-0.739641\pi$
$719$	−36.6670	−1.36745	−0.683725	+	0.729740i	$0.739641\pi$
$720$	0	0
$721$	11.4909	0.427945
$722$	0	0
$723$	17.6043	0.654712
$724$	0	0
$725$	−0.401076	−0.0148956
$726$	0	0
$727$	−16.1623	−0.599427	−0.299714	−	0.954029i	$-0.596891\pi$
$727$	−16.1623	−0.599427	−0.299714	+	0.954029i	$0.596891\pi$
$728$	0	0
$729$	−9.58002	−0.354816
$730$	0	0
$731$	−1.61320	−0.0596662
$732$	0	0
$733$	16.8519	0.622441	0.311220	−	0.950338i	$-0.399262\pi$
$733$	16.8519	0.622441	0.311220	+	0.950338i	$0.399262\pi$
$734$	0	0
$735$	−20.5810	−0.759142
$736$	0	0
$737$	−41.6573	−1.53447
$738$	0	0
$739$	16.1029	0.592354	0.296177	−	0.955133i	$-0.404288\pi$
$739$	16.1029	0.592354	0.296177	+	0.955133i	$0.404288\pi$
$740$	0	0
$741$	−1.69243	−0.0621729
$742$	0	0
$743$	−15.3876	−0.564516	−0.282258	−	0.959339i	$-0.591083\pi$
$743$	−15.3876	−0.564516	−0.282258	+	0.959339i	$0.591083\pi$
$744$	0	0
$745$	35.1942	1.28941
$746$	0	0
$747$	8.54585	0.312676
$748$	0	0
$749$	−47.3457	−1.72998
$750$	0	0
$751$	−24.4956	−0.893858	−0.446929	−	0.894569i	$-0.647482\pi$
$751$	−24.4956	−0.893858	−0.446929	+	0.894569i	$0.647482\pi$
$752$	0	0
$753$	6.00674	0.218898
$754$	0	0
$755$	−25.9143	−0.943117
$756$	0	0
$757$	17.9677	0.653047	0.326524	−	0.945189i	$-0.394123\pi$
$757$	17.9677	0.653047	0.326524	+	0.945189i	$0.394123\pi$
$758$	0	0
$759$	20.5282	0.745125
$760$	0	0
$761$	1.50089	0.0544072	0.0272036	−	0.999630i	$-0.491340\pi$
$761$	1.50089	0.0544072	0.0272036	+	0.999630i	$0.491340\pi$
$762$	0	0
$763$	46.0223	1.66612
$764$	0	0
$765$	−8.45652	−0.305746
$766$	0	0
$767$	−0.565788	−0.0204294
$768$	0	0
$769$	−28.6086	−1.03165	−0.515827	−	0.856693i	$-0.672515\pi$
$769$	−28.6086	−1.03165	−0.515827	+	0.856693i	$0.672515\pi$
$770$	0	0
$771$	−13.2092	−0.475717
$772$	0	0
$773$	−14.3685	−0.516798	−0.258399	−	0.966038i	$-0.583195\pi$
$773$	−14.3685	−0.516798	−0.258399	+	0.966038i	$0.583195\pi$
$774$	0	0
$775$	−57.2768	−2.05744
$776$	0	0
$777$	16.6869	0.598639
$778$	0	0
$779$	12.8112	0.459010
$780$	0	0
$781$	49.7143	1.77892
$782$	0	0
$783$	0.258188	0.00922688
$784$	0	0
$785$	−26.0358	−0.929259
$786$	0	0
$787$	42.2800	1.50712	0.753560	−	0.657379i	$-0.228335\pi$
$787$	42.2800	1.50712	0.753560	+	0.657379i	$0.228335\pi$
$788$	0	0
$789$	−7.56271	−0.269240
$790$	0	0
$791$	−78.3147	−2.78455
$792$	0	0
$793$	−6.57165	−0.233366
$794$	0	0
$795$	22.3671	0.793278
$796$	0	0
$797$	8.27593	0.293149	0.146574	−	0.989200i	$-0.453175\pi$
$797$	8.27593	0.293149	0.146574	+	0.989200i	$0.453175\pi$
$798$	0	0
$799$	−7.42334	−0.262619
$800$	0	0
$801$	−38.2708	−1.35223
$802$	0	0
$803$	3.41560	0.120534
$804$	0	0
$805$	−77.8575	−2.74412
$806$	0	0
$807$	11.8653	0.417679
$808$	0	0
$809$	4.04426	0.142189	0.0710943	−	0.997470i	$-0.477351\pi$
$809$	4.04426	0.142189	0.0710943	+	0.997470i	$0.477351\pi$
$810$	0	0
$811$	15.2668	0.536091	0.268046	−	0.963406i	$-0.413622\pi$
$811$	15.2668	0.536091	0.268046	+	0.963406i	$0.413622\pi$
$812$	0	0
$813$	−6.71140	−0.235379
$814$	0	0
$815$	−24.3225	−0.851980
$816$	0	0
$817$	8.07411	0.282477
$818$	0	0
$819$	6.30182	0.220204
$820$	0	0
$821$	0.611222	0.0213318	0.0106659	−	0.999943i	$-0.496605\pi$
$821$	0.611222	0.0213318	0.0106659	+	0.999943i	$0.496605\pi$
$822$	0	0
$823$	−1.83022	−0.0637975	−0.0318987	−	0.999491i	$-0.510155\pi$
$823$	−1.83022	−0.0637975	−0.0318987	+	0.999491i	$0.510155\pi$
$824$	0	0
$825$	18.6276	0.648528
$826$	0	0
$827$	−42.3702	−1.47336	−0.736678	−	0.676244i	$-0.763607\pi$
$827$	−42.3702	−1.47336	−0.736678	+	0.676244i	$0.763607\pi$
$828$	0	0
$829$	−19.3543	−0.672202	−0.336101	−	0.941826i	$-0.609108\pi$
$829$	−19.3543	−0.672202	−0.336101	+	0.941826i	$0.609108\pi$
$830$	0	0
$831$	3.44646	0.119556
$832$	0	0
$833$	−10.7620	−0.372881
$834$	0	0
$835$	−52.5276	−1.81779
$836$	0	0
$837$	36.8712	1.27446
$838$	0	0
$839$	−43.7160	−1.50924	−0.754622	−	0.656159i	$-0.772180\pi$
$839$	−43.7160	−1.50924	−0.754622	+	0.656159i	$0.772180\pi$
$840$	0	0
$841$	−28.9941	−0.999798
$842$	0	0
$843$	−14.7207	−0.507008
$844$	0	0
$845$	40.5733	1.39576
$846$	0	0
$847$	102.814	3.53272
$848$	0	0
$849$	1.17820	0.0404356
$850$	0	0
$851$	38.2481	1.31113
$852$	0	0
$853$	−10.3214	−0.353399	−0.176700	−	0.984265i	$-0.556542\pi$
$853$	−10.3214	−0.353399	−0.176700	+	0.984265i	$0.556542\pi$
$854$	0	0
$855$	42.3252	1.44749
$856$	0	0
$857$	6.35964	0.217241	0.108621	−	0.994083i	$-0.465357\pi$
$857$	6.35964	0.217241	0.108621	+	0.994083i	$0.465357\pi$
$858$	0	0
$859$	35.7303	1.21910	0.609552	−	0.792746i	$-0.291349\pi$
$859$	35.7303	1.21910	0.609552	+	0.792746i	$0.291349\pi$
$860$	0	0
$861$	6.44730	0.219723
$862$	0	0
$863$	−33.9573	−1.15592	−0.577961	−	0.816065i	$-0.696151\pi$
$863$	−33.9573	−1.15592	−0.577961	+	0.816065i	$0.696151\pi$
$864$	0	0
$865$	−50.3195	−1.71092
$866$	0	0
$867$	0.597652	0.0202973
$868$	0	0
$869$	28.4641	0.965578
$870$	0	0
$871$	3.96162	0.134234
$872$	0	0
$873$	48.6083	1.64514
$874$	0	0
$875$	−3.22091	−0.108887
$876$	0	0
$877$	−3.84113	−0.129706	−0.0648528	−	0.997895i	$-0.520658\pi$
$877$	−3.84113	−0.129706	−0.0648528	+	0.997895i	$0.520658\pi$
$878$	0	0
$879$	13.8975	0.468749
$880$	0	0
$881$	−45.9151	−1.54692	−0.773459	−	0.633846i	$-0.781475\pi$
$881$	−45.9151	−1.54692	−0.773459	+	0.633846i	$0.781475\pi$
$882$	0	0
$883$	−4.66898	−0.157124	−0.0785618	−	0.996909i	$-0.525033\pi$
$883$	−4.66898	−0.157124	−0.0785618	+	0.996909i	$0.525033\pi$
$884$	0	0
$885$	−1.91238	−0.0642839
$886$	0	0
$887$	−13.5663	−0.455513	−0.227757	−	0.973718i	$-0.573139\pi$
$887$	−13.5663	−0.455513	−0.227757	+	0.973718i	$0.573139\pi$
$888$	0	0
$889$	−4.90603	−0.164543
$890$	0	0
$891$	35.1781	1.17851
$892$	0	0
$893$	37.1541	1.24332
$894$	0	0
$895$	23.6582	0.790807
$896$	0	0
$897$	−1.95223	−0.0651832
$898$	0	0
$899$	0.837019	0.0279161
$900$	0	0
$901$	11.6959	0.389648
$902$	0	0
$903$	4.06332	0.135219
$904$	0	0
$905$	−75.4068	−2.50661
$906$	0	0
$907$	−25.6370	−0.851263	−0.425632	−	0.904897i	$-0.639948\pi$
$907$	−25.6370	−0.851263	−0.425632	+	0.904897i	$0.639948\pi$
$908$	0	0
$909$	21.6965	0.719629
$910$	0	0
$911$	−46.8718	−1.55293	−0.776466	−	0.630159i	$-0.782990\pi$
$911$	−46.8718	−1.55293	−0.776466	+	0.630159i	$0.782990\pi$
$912$	0	0
$913$	−19.2381	−0.636687
$914$	0	0
$915$	−22.2123	−0.734317
$916$	0	0
$917$	−58.1597	−1.92060
$918$	0	0
$919$	−29.0728	−0.959023	−0.479511	−	0.877536i	$-0.659186\pi$
$919$	−29.0728	−0.959023	−0.479511	+	0.877536i	$0.659186\pi$
$920$	0	0
$921$	7.50658	0.247350
$922$	0	0
$923$	−4.72784	−0.155619
$924$	0	0
$925$	34.7069	1.14116
$926$	0	0
$927$	−7.20569	−0.236666
$928$	0	0
$929$	−46.0096	−1.50953	−0.754763	−	0.655997i	$-0.772248\pi$
$929$	−46.0096	−1.50953	−0.754763	+	0.655997i	$0.772248\pi$
$930$	0	0
$931$	53.8643	1.76533
$932$	0	0
$933$	10.0977	0.330583
$934$	0	0
$935$	19.0370	0.622575
$936$	0	0
$937$	−50.4196	−1.64714	−0.823568	−	0.567217i	$-0.808020\pi$
$937$	−50.4196	−1.64714	−0.823568	+	0.567217i	$0.808020\pi$
$938$	0	0
$939$	18.4402	0.601772
$940$	0	0
$941$	39.0753	1.27382	0.636910	−	0.770939i	$-0.280212\pi$
$941$	39.0753	1.27382	0.636910	+	0.770939i	$0.280212\pi$
$942$	0	0
$943$	14.7779	0.481235
$944$	0	0
$945$	45.4795	1.47945
$946$	0	0
$947$	−27.2608	−0.885858	−0.442929	−	0.896557i	$-0.646061\pi$
$947$	−27.2608	−0.885858	−0.442929	+	0.896557i	$0.646061\pi$
$948$	0	0
$949$	−0.324825	−0.0105443
$950$	0	0
$951$	−4.61882	−0.149775
$952$	0	0
$953$	6.99212	0.226497	0.113249	−	0.993567i	$-0.463874\pi$
$953$	6.99212	0.226497	0.113249	+	0.993567i	$0.463874\pi$
$954$	0	0
$955$	−3.85439	−0.124725
$956$	0	0
$957$	−0.272215	−0.00879947
$958$	0	0
$959$	−27.5182	−0.888610
$960$	0	0
$961$	88.5327	2.85589
$962$	0	0
$963$	29.6894	0.956727
$964$	0	0
$965$	−15.6488	−0.503751
$966$	0	0
$967$	1.44148	0.0463549	0.0231775	−	0.999731i	$-0.492622\pi$
$967$	1.44148	0.0463549	0.0231775	+	0.999731i	$0.492622\pi$
$968$	0	0
$969$	−2.99127	−0.0960936
$970$	0	0
$971$	−42.8698	−1.37576	−0.687878	−	0.725826i	$-0.741458\pi$
$971$	−42.8698	−1.37576	−0.687878	+	0.725826i	$0.741458\pi$
$972$	0	0
$973$	−28.5408	−0.914976
$974$	0	0
$975$	−1.77149	−0.0567330
$976$	0	0
$977$	28.4194	0.909216	0.454608	−	0.890692i	$-0.349779\pi$
$977$	28.4194	0.909216	0.454608	+	0.890692i	$0.349779\pi$
$978$	0	0
$979$	86.1536	2.75348
$980$	0	0
$981$	−28.8595	−0.921413
$982$	0	0
$983$	−1.99196	−0.0635335	−0.0317668	−	0.999495i	$-0.510113\pi$
$983$	−1.99196	−0.0635335	−0.0317668	+	0.999495i	$0.510113\pi$
$984$	0	0
$985$	46.9187	1.49495
$986$	0	0
$987$	18.6979	0.595162
$988$	0	0
$989$	9.31357	0.296154
$990$	0	0
$991$	−35.2507	−1.11978	−0.559888	−	0.828569i	$-0.689156\pi$
$991$	−35.2507	−1.11978	−0.559888	+	0.828569i	$0.689156\pi$
$992$	0	0
$993$	19.7133	0.625583
$994$	0	0
$995$	45.3487	1.43765
$996$	0	0
$997$	47.9165	1.51753	0.758765	−	0.651364i	$-0.225803\pi$
$997$	47.9165	1.51753	0.758765	+	0.651364i	$0.225803\pi$
$998$	0	0
$999$	−22.3421	−0.706874

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8024.2.a.ba.1.18	✓	30

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8024.2.a.ba.1.18	✓	30	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:	\( N \)	\(=\)	\( 8024 = 2^{3} \cdot 17 \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8024.a (trivial)

\(f(q)\)	\(=\)	\(q+0.597652 q^{3} -3.19982 q^{5} +4.21450 q^{7} -2.64281 q^{9} +O(q^{10})\)	\(q+0.597652 q^{3} -3.19982 q^{5} +4.21450 q^{7} -2.64281 q^{9} +5.94939 q^{11} -0.565788 q^{13} -1.91238 q^{15} -1.00000 q^{17} +5.00504 q^{19} +2.51880 q^{21} +5.77337 q^{23} +5.23884 q^{25} -3.37244 q^{27} -0.0765582 q^{29} -10.9331 q^{31} +3.55566 q^{33} -13.4856 q^{35} +6.62492 q^{37} -0.338145 q^{39} +2.55967 q^{41} +1.61320 q^{43} +8.45652 q^{45} +7.42334 q^{47} +10.7620 q^{49} -0.597652 q^{51} -11.6959 q^{53} -19.0370 q^{55} +2.99127 q^{57} +1.00000 q^{59} +11.6150 q^{61} -11.1381 q^{63} +1.81042 q^{65} -7.00194 q^{67} +3.45047 q^{69} +8.35620 q^{71} +0.574110 q^{73} +3.13100 q^{75} +25.0737 q^{77} +4.78437 q^{79} +5.91289 q^{81} -3.23362 q^{83} +3.19982 q^{85} -0.0457552 q^{87} +14.4811 q^{89} -2.38451 q^{91} -6.53419 q^{93} -16.0152 q^{95} -18.3927 q^{97} -15.7231 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 30 q + 4 q^{3} + 2 q^{5} + 3 q^{7} + 34 q^{9}+O(q^{10}) \) 30 * q + 4 * q^3 + 2 * q^5 + 3 * q^7 + 34 * q^9	\( 30 q + 4 q^{3} + 2 q^{5} + 3 q^{7} + 34 q^{9} + 3 q^{11} + 9 q^{13} + 14 q^{15} - 30 q^{17} + 24 q^{19} + 7 q^{21} + 9 q^{23} + 40 q^{25} + 19 q^{27} + 9 q^{29} + 11 q^{31} - 14 q^{33} + 30 q^{35} - 13 q^{37} + 16 q^{39} - 13 q^{41} + 23 q^{43} + 12 q^{45} + 43 q^{47} + 35 q^{49} - 4 q^{51} - 4 q^{53} + 43 q^{55} + 3 q^{57} + 30 q^{59} + 43 q^{61} + 38 q^{63} + 3 q^{65} + 50 q^{67} + 34 q^{69} + 3 q^{71} - 16 q^{73} + 21 q^{75} + 18 q^{77} + 45 q^{79} + 6 q^{81} + 63 q^{83} - 2 q^{85} + 42 q^{87} + 6 q^{89} + 22 q^{91} - 2 q^{93} + 19 q^{95} - 28 q^{97} + 51 q^{99}+O(q^{100}) \) 30 * q + 4 * q^3 + 2 * q^5 + 3 * q^7 + 34 * q^9 + 3 * q^11 + 9 * q^13 + 14 * q^15 - 30 * q^17 + 24 * q^19 + 7 * q^21 + 9 * q^23 + 40 * q^25 + 19 * q^27 + 9 * q^29 + 11 * q^31 - 14 * q^33 + 30 * q^35 - 13 * q^37 + 16 * q^39 - 13 * q^41 + 23 * q^43 + 12 * q^45 + 43 * q^47 + 35 * q^49 - 4 * q^51 - 4 * q^53 + 43 * q^55 + 3 * q^57 + 30 * q^59 + 43 * q^61 + 38 * q^63 + 3 * q^65 + 50 * q^67 + 34 * q^69 + 3 * q^71 - 16 * q^73 + 21 * q^75 + 18 * q^77 + 45 * q^79 + 6 * q^81 + 63 * q^83 - 2 * q^85 + 42 * q^87 + 6 * q^89 + 22 * q^91 - 2 * q^93 + 19 * q^95 - 28 * q^97 + 51 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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