LMFDB - Embedded newform 8024.2.a.w.1.8

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:	$1$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - 2 x^{19} - 29 x^{18} + 51 x^{17} + 341 x^{16} - 514 x^{15} - 2114 x^{14} + 2629 x^{13} + \cdots + 4 $ x^20 - 2x^19 - 29x^18 + 51x^17 + 341x^16 - 514x^15 - 2114x^14 + 2629x^13 + 7532x^12 - 7342x^11 - 15668x^10 + 11260x^9 + 18269x^8 - 9059x^7 - 10680x^6 + 3386x^5 + 2473x^4 - 427x^3 - 193x^2 + 18*x + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{8} $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Root			$0.609415$ of defining polynomial
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)

$f(q)$	$=$	$q-0.609415 q^{3} +0.676769 q^{5} -1.71345 q^{7} -2.62861 q^{9} +O(q^{10})$	$q-0.609415 q^{3} +0.676769 q^{5} -1.71345 q^{7} -2.62861 q^{9} +0.335917 q^{11} -0.910614 q^{13} -0.412433 q^{15} +1.00000 q^{17} -0.578329 q^{19} +1.04420 q^{21} +0.787611 q^{23} -4.54198 q^{25} +3.43016 q^{27} +5.28823 q^{29} +8.71631 q^{31} -0.204713 q^{33} -1.15961 q^{35} -3.88781 q^{37} +0.554942 q^{39} +9.42716 q^{41} -0.222142 q^{43} -1.77897 q^{45} -3.91585 q^{47} -4.06408 q^{49} -0.609415 q^{51} +2.46304 q^{53} +0.227338 q^{55} +0.352442 q^{57} -1.00000 q^{59} +6.35019 q^{61} +4.50400 q^{63} -0.616276 q^{65} -0.214838 q^{67} -0.479982 q^{69} -14.1852 q^{71} +7.06182 q^{73} +2.76795 q^{75} -0.575578 q^{77} +2.73359 q^{79} +5.79545 q^{81} +6.26931 q^{83} +0.676769 q^{85} -3.22272 q^{87} -8.55200 q^{89} +1.56029 q^{91} -5.31185 q^{93} -0.391396 q^{95} +2.43622 q^{97} -0.882996 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q - 2 q^{3} - 2 q^{5} + 5 q^{7} + 2 q^{9}+O(q^{10}) $ 20 * q - 2 * q^3 - 2 * q^5 + 5 * q^7 + 2 * q^9	$ 20 q - 2 q^{3} - 2 q^{5} + 5 q^{7} + 2 q^{9} + 8 q^{11} - 3 q^{13} - 2 q^{15} + 20 q^{17} - 9 q^{19} - 31 q^{21} - 8 q^{23} - 4 q^{25} - 17 q^{27} - 37 q^{29} - 9 q^{31} - 8 q^{33} - 8 q^{35} - 11 q^{37} - 12 q^{39} - 27 q^{41} + 17 q^{43} - 4 q^{45} + 7 q^{47} - 9 q^{49} - 2 q^{51} - q^{53} - 21 q^{55} - 57 q^{57} - 20 q^{59} - 12 q^{61} - 14 q^{63} - 59 q^{65} - 26 q^{67} + 14 q^{69} - 3 q^{71} - 45 q^{73} + 15 q^{75} + 6 q^{77} + 5 q^{79} + 12 q^{81} - 17 q^{83} - 2 q^{85} - 32 q^{87} - 40 q^{89} - 36 q^{91} - 4 q^{93} + 5 q^{95} - 77 q^{97} + 54 q^{99}+O(q^{100}) $ 20 * q - 2 * q^3 - 2 * q^5 + 5 * q^7 + 2 * q^9 + 8 * q^11 - 3 * q^13 - 2 * q^15 + 20 * q^17 - 9 * q^19 - 31 * q^21 - 8 * q^23 - 4 * q^25 - 17 * q^27 - 37 * q^29 - 9 * q^31 - 8 * q^33 - 8 * q^35 - 11 * q^37 - 12 * q^39 - 27 * q^41 + 17 * q^43 - 4 * q^45 + 7 * q^47 - 9 * q^49 - 2 * q^51 - q^53 - 21 * q^55 - 57 * q^57 - 20 * q^59 - 12 * q^61 - 14 * q^63 - 59 * q^65 - 26 * q^67 + 14 * q^69 - 3 * q^71 - 45 * q^73 + 15 * q^75 + 6 * q^77 + 5 * q^79 + 12 * q^81 - 17 * q^83 - 2 * q^85 - 32 * q^87 - 40 * q^89 - 36 * q^91 - 4 * q^93 + 5 * q^95 - 77 * q^97 + 54 * 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
$3$	−0.609415	−0.351846	−0.175923	−	0.984404i	$-0.556291\pi$
$3$	−0.609415	−0.351846	−0.175923	+	0.984404i	$0.556291\pi$
$4$	0	0
$5$	0.676769	0.302660	0.151330	−	0.988483i	$-0.451644\pi$
$5$	0.676769	0.302660	0.151330	+	0.988483i	$0.451644\pi$
$6$	0	0
$7$	−1.71345	−0.647624	−0.323812	−	0.946121i	$-0.604965\pi$
$7$	−1.71345	−0.647624	−0.323812	+	0.946121i	$0.604965\pi$
$8$	0	0
$9$	−2.62861	−0.876205
$10$	0	0
$11$	0.335917	0.101283	0.0506414	−	0.998717i	$-0.483873\pi$
$11$	0.335917	0.101283	0.0506414	+	0.998717i	$0.483873\pi$
$12$	0	0
$13$	−0.910614	−0.252559	−0.126279	−	0.991995i	$-0.540304\pi$
$13$	−0.910614	−0.252559	−0.126279	+	0.991995i	$0.540304\pi$
$14$	0	0
$15$	−0.412433	−0.106490
$16$	0	0
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	0	0
$19$	−0.578329	−0.132678	−0.0663389	−	0.997797i	$-0.521132\pi$
$19$	−0.578329	−0.132678	−0.0663389	+	0.997797i	$0.521132\pi$
$20$	0	0
$21$	1.04420	0.227864
$22$	0	0
$23$	0.787611	0.164228	0.0821141	−	0.996623i	$-0.473833\pi$
$23$	0.787611	0.164228	0.0821141	+	0.996623i	$0.473833\pi$
$24$	0	0
$25$	−4.54198	−0.908397
$26$	0	0
$27$	3.43016	0.660135
$28$	0	0
$29$	5.28823	0.981999	0.490999	−	0.871160i	$-0.336632\pi$
$29$	5.28823	0.981999	0.490999	+	0.871160i	$0.336632\pi$
$30$	0	0
$31$	8.71631	1.56550	0.782748	−	0.622339i	$-0.213817\pi$
$31$	8.71631	1.56550	0.782748	+	0.622339i	$0.213817\pi$
$32$	0	0
$33$	−0.204713	−0.0356359
$34$	0	0
$35$	−1.15961	−0.196010
$36$	0	0
$37$	−3.88781	−0.639152	−0.319576	−	0.947561i	$-0.603540\pi$
$37$	−3.88781	−0.639152	−0.319576	+	0.947561i	$0.603540\pi$
$38$	0	0
$39$	0.554942	0.0888618
$40$	0	0
$41$	9.42716	1.47228	0.736138	−	0.676832i	$-0.236647\pi$
$41$	9.42716	1.47228	0.736138	+	0.676832i	$0.236647\pi$
$42$	0	0
$43$	−0.222142	−0.0338763	−0.0169382	−	0.999857i	$-0.505392\pi$
$43$	−0.222142	−0.0338763	−0.0169382	+	0.999857i	$0.505392\pi$
$44$	0	0
$45$	−1.77897	−0.265192
$46$	0	0
$47$	−3.91585	−0.571186	−0.285593	−	0.958351i	$-0.592190\pi$
$47$	−3.91585	−0.571186	−0.285593	+	0.958351i	$0.592190\pi$
$48$	0	0
$49$	−4.06408	−0.580583
$50$	0	0
$51$	−0.609415	−0.0853351
$52$	0	0
$53$	2.46304	0.338324	0.169162	−	0.985588i	$-0.445894\pi$
$53$	2.46304	0.338324	0.169162	+	0.985588i	$0.445894\pi$
$54$	0	0
$55$	0.227338	0.0306543
$56$	0	0
$57$	0.352442	0.0466821
$58$	0	0
$59$	−1.00000	−0.130189
$59$	−1.00000	−0.130189
$60$	0	0
$61$	6.35019	0.813059	0.406529	−	0.913638i	$-0.366739\pi$
$61$	6.35019	0.813059	0.406529	+	0.913638i	$0.366739\pi$
$62$	0	0
$63$	4.50400	0.567451
$64$	0	0
$65$	−0.616276	−0.0764396
$66$	0	0
$67$	−0.214838	−0.0262467	−0.0131233	−	0.999914i	$-0.504177\pi$
$67$	−0.214838	−0.0262467	−0.0131233	+	0.999914i	$0.504177\pi$
$68$	0	0
$69$	−0.479982	−0.0577830
$70$	0	0
$71$	−14.1852	−1.68347	−0.841735	−	0.539891i	$-0.818465\pi$
$71$	−14.1852	−1.68347	−0.841735	+	0.539891i	$0.818465\pi$
$72$	0	0
$73$	7.06182	0.826524	0.413262	−	0.910612i	$-0.364389\pi$
$73$	7.06182	0.826524	0.413262	+	0.910612i	$0.364389\pi$
$74$	0	0
$75$	2.76795	0.319616
$76$	0	0
$77$	−0.575578	−0.0655932
$78$	0	0
$79$	2.73359	0.307553	0.153777	−	0.988106i	$-0.450856\pi$
$79$	2.73359	0.307553	0.153777	+	0.988106i	$0.450856\pi$
$80$	0	0
$81$	5.79545	0.643939
$82$	0	0
$83$	6.26931	0.688147	0.344073	−	0.938943i	$-0.388193\pi$
$83$	6.26931	0.688147	0.344073	+	0.938943i	$0.388193\pi$
$84$	0	0
$85$	0.676769	0.0734059
$86$	0	0
$87$	−3.22272	−0.345512
$88$	0	0
$89$	−8.55200	−0.906510	−0.453255	−	0.891381i	$-0.649737\pi$
$89$	−8.55200	−0.906510	−0.453255	+	0.891381i	$0.649737\pi$
$90$	0	0
$91$	1.56029	0.163563
$92$	0	0
$93$	−5.31185	−0.550813
$94$	0	0
$95$	−0.391396	−0.0401563
$96$	0	0
$97$	2.43622	0.247360	0.123680	−	0.992322i	$-0.460530\pi$
$97$	2.43622	0.247360	0.123680	+	0.992322i	$0.460530\pi$
$98$	0	0
$99$	−0.882996	−0.0887445
$100$	0	0
$101$	−14.6359	−1.45632	−0.728162	−	0.685405i	$-0.759625\pi$
$101$	−14.6359	−1.45632	−0.728162	+	0.685405i	$0.759625\pi$
$102$	0	0
$103$	15.2236	1.50002	0.750011	−	0.661425i	$-0.230048\pi$
$103$	15.2236	1.50002	0.750011	+	0.661425i	$0.230048\pi$
$104$	0	0
$105$	0.706685	0.0689654
$106$	0	0
$107$	−7.40877	−0.716233	−0.358117	−	0.933677i	$-0.616581\pi$
$107$	−7.40877	−0.716233	−0.358117	+	0.933677i	$0.616581\pi$
$108$	0	0
$109$	−18.4815	−1.77021	−0.885104	−	0.465393i	$-0.845913\pi$
$109$	−18.4815	−1.77021	−0.885104	+	0.465393i	$0.845913\pi$
$110$	0	0
$111$	2.36929	0.224883
$112$	0	0
$113$	0.702339	0.0660705	0.0330353	−	0.999454i	$-0.489483\pi$
$113$	0.702339	0.0660705	0.0330353	+	0.999454i	$0.489483\pi$
$114$	0	0
$115$	0.533031	0.0497054
$116$	0	0
$117$	2.39365	0.221293
$118$	0	0
$119$	−1.71345	−0.157072
$120$	0	0
$121$	−10.8872	−0.989742
$122$	0	0
$123$	−5.74505	−0.518014
$124$	0	0
$125$	−6.45772	−0.577596
$126$	0	0
$127$	0.209611	0.0185999	0.00929996	−	0.999957i	$-0.497040\pi$
$127$	0.209611	0.0185999	0.00929996	+	0.999957i	$0.497040\pi$
$128$	0	0
$129$	0.135377	0.0119192
$130$	0	0
$131$	5.21670	0.455786	0.227893	−	0.973686i	$-0.426816\pi$
$131$	5.21670	0.455786	0.227893	+	0.973686i	$0.426816\pi$
$132$	0	0
$133$	0.990940	0.0859254
$134$	0	0
$135$	2.32143	0.199797
$136$	0	0
$137$	−15.7874	−1.34881	−0.674403	−	0.738363i	$-0.735599\pi$
$137$	−15.7874	−1.34881	−0.674403	+	0.738363i	$0.735599\pi$
$138$	0	0
$139$	20.2670	1.71903	0.859513	−	0.511114i	$-0.170767\pi$
$139$	20.2670	1.71903	0.859513	+	0.511114i	$0.170767\pi$
$140$	0	0
$141$	2.38638	0.200969
$142$	0	0
$143$	−0.305891	−0.0255799
$144$	0	0
$145$	3.57891	0.297212
$146$	0	0
$147$	2.47671	0.204276
$148$	0	0
$149$	−2.19786	−0.180055	−0.0900277	−	0.995939i	$-0.528696\pi$
$149$	−2.19786	−0.180055	−0.0900277	+	0.995939i	$0.528696\pi$
$150$	0	0
$151$	−13.2483	−1.07813	−0.539064	−	0.842264i	$-0.681222\pi$
$151$	−13.2483	−1.07813	−0.539064	+	0.842264i	$0.681222\pi$
$152$	0	0
$153$	−2.62861	−0.212511
$154$	0	0
$155$	5.89893	0.473814
$156$	0	0
$157$	−15.4611	−1.23393	−0.616967	−	0.786989i	$-0.711639\pi$
$157$	−15.4611	−1.23393	−0.616967	+	0.786989i	$0.711639\pi$
$158$	0	0
$159$	−1.50101	−0.119038
$160$	0	0
$161$	−1.34953	−0.106358
$162$	0	0
$163$	−16.2563	−1.27329	−0.636647	−	0.771155i	$-0.719679\pi$
$163$	−16.2563	−1.27329	−0.636647	+	0.771155i	$0.719679\pi$
$164$	0	0
$165$	−0.138543	−0.0107856
$166$	0	0
$167$	−9.18304	−0.710605	−0.355302	−	0.934751i	$-0.615622\pi$
$167$	−9.18304	−0.710605	−0.355302	+	0.934751i	$0.615622\pi$
$168$	0	0
$169$	−12.1708	−0.936214
$170$	0	0
$171$	1.52020	0.116253
$172$	0	0
$173$	−16.0768	−1.22230	−0.611150	−	0.791515i	$-0.709293\pi$
$173$	−16.0768	−1.22230	−0.611150	+	0.791515i	$0.709293\pi$
$174$	0	0
$175$	7.78247	0.588300
$176$	0	0
$177$	0.609415	0.0458064
$178$	0	0
$179$	−10.5565	−0.789030	−0.394515	−	0.918890i	$-0.629087\pi$
$179$	−10.5565	−0.789030	−0.394515	+	0.918890i	$0.629087\pi$
$180$	0	0
$181$	−11.1880	−0.831594	−0.415797	−	0.909458i	$-0.636497\pi$
$181$	−11.1880	−0.831594	−0.415797	+	0.909458i	$0.636497\pi$
$182$	0	0
$183$	−3.86990	−0.286071
$184$	0	0
$185$	−2.63115	−0.193446
$186$	0	0
$187$	0.335917	0.0245647
$188$	0	0
$189$	−5.87742	−0.427519
$190$	0	0
$191$	−19.2773	−1.39486	−0.697430	−	0.716653i	$-0.745673\pi$
$191$	−19.2773	−1.39486	−0.697430	+	0.716653i	$0.745673\pi$
$192$	0	0
$193$	9.01918	0.649215	0.324607	−	0.945849i	$-0.394768\pi$
$193$	9.01918	0.649215	0.324607	+	0.945849i	$0.394768\pi$
$194$	0	0
$195$	0.375567	0.0268949
$196$	0	0
$197$	1.00385	0.0715213	0.0357607	−	0.999360i	$-0.488615\pi$
$197$	1.00385	0.0715213	0.0357607	+	0.999360i	$0.488615\pi$
$198$	0	0
$199$	26.0027	1.84328	0.921641	−	0.388044i	$-0.126849\pi$
$199$	26.0027	1.84328	0.921641	+	0.388044i	$0.126849\pi$
$200$	0	0
$201$	0.130926	0.00923478
$202$	0	0
$203$	−9.06112	−0.635966
$204$	0	0
$205$	6.38002	0.445600
$206$	0	0
$207$	−2.07032	−0.143897
$208$	0	0
$209$	−0.194271	−0.0134380
$210$	0	0
$211$	−2.61761	−0.180204	−0.0901018	−	0.995933i	$-0.528719\pi$
$211$	−2.61761	−0.180204	−0.0901018	+	0.995933i	$0.528719\pi$
$212$	0	0
$213$	8.64465	0.592322
$214$	0	0
$215$	−0.150339	−0.0102530
$216$	0	0
$217$	−14.9350	−1.01385
$218$	0	0
$219$	−4.30358	−0.290809
$220$	0	0
$221$	−0.910614	−0.0612545
$222$	0	0
$223$	17.3692	1.16313	0.581565	−	0.813500i	$-0.302441\pi$
$223$	17.3692	1.16313	0.581565	+	0.813500i	$0.302441\pi$
$224$	0	0
$225$	11.9391	0.795941
$226$	0	0
$227$	14.5293	0.964347	0.482173	−	0.876076i	$-0.339848\pi$
$227$	14.5293	0.964347	0.482173	+	0.876076i	$0.339848\pi$
$228$	0	0
$229$	9.30115	0.614637	0.307319	−	0.951607i	$-0.400568\pi$
$229$	9.30115	0.614637	0.307319	+	0.951607i	$0.400568\pi$
$230$	0	0
$231$	0.350766	0.0230787
$232$	0	0
$233$	23.8522	1.56261	0.781304	−	0.624150i	$-0.214555\pi$
$233$	23.8522	1.56261	0.781304	+	0.624150i	$0.214555\pi$
$234$	0	0
$235$	−2.65013	−0.172875
$236$	0	0
$237$	−1.66589	−0.108211
$238$	0	0
$239$	−15.6671	−1.01342	−0.506710	−	0.862116i	$-0.669139\pi$
$239$	−15.6671	−1.01342	−0.506710	+	0.862116i	$0.669139\pi$
$240$	0	0
$241$	−5.45074	−0.351113	−0.175556	−	0.984469i	$-0.556172\pi$
$241$	−5.45074	−0.351113	−0.175556	+	0.984469i	$0.556172\pi$
$242$	0	0
$243$	−13.8223	−0.886702
$244$	0	0
$245$	−2.75045	−0.175720
$246$	0	0
$247$	0.526635	0.0335090
$248$	0	0
$249$	−3.82061	−0.242122
$250$	0	0
$251$	9.24082	0.583275	0.291638	−	0.956529i	$-0.405800\pi$
$251$	9.24082	0.583275	0.291638	+	0.956529i	$0.405800\pi$
$252$	0	0
$253$	0.264572	0.0166335
$254$	0	0
$255$	−0.412433	−0.0258276
$256$	0	0
$257$	−19.4558	−1.21362	−0.606811	−	0.794846i	$-0.707551\pi$
$257$	−19.4558	−1.21362	−0.606811	+	0.794846i	$0.707551\pi$
$258$	0	0
$259$	6.66158	0.413930
$260$	0	0
$261$	−13.9007	−0.860432
$262$	0	0
$263$	19.8856	1.22620	0.613100	−	0.790005i	$-0.289922\pi$
$263$	19.8856	1.22620	0.613100	+	0.790005i	$0.289922\pi$
$264$	0	0
$265$	1.66691	0.102397
$266$	0	0
$267$	5.21171	0.318952
$268$	0	0
$269$	6.16717	0.376019	0.188010	−	0.982167i	$-0.439796\pi$
$269$	6.16717	0.376019	0.188010	+	0.982167i	$0.439796\pi$
$270$	0	0
$271$	0.525032	0.0318934	0.0159467	−	0.999873i	$-0.494924\pi$
$271$	0.525032	0.0318934	0.0159467	+	0.999873i	$0.494924\pi$
$272$	0	0
$273$	−0.950866	−0.0575490
$274$	0	0
$275$	−1.52573	−0.0920050
$276$	0	0
$277$	−19.8479	−1.19255	−0.596273	−	0.802782i	$-0.703352\pi$
$277$	−19.8479	−1.19255	−0.596273	+	0.802782i	$0.703352\pi$
$278$	0	0
$279$	−22.9118	−1.37169
$280$	0	0
$281$	−31.7353	−1.89317	−0.946584	−	0.322458i	$-0.895491\pi$
$281$	−31.7353	−1.89317	−0.946584	+	0.322458i	$0.895491\pi$
$282$	0	0
$283$	−15.2238	−0.904959	−0.452480	−	0.891775i	$-0.649461\pi$
$283$	−15.2238	−0.904959	−0.452480	+	0.891775i	$0.649461\pi$
$284$	0	0
$285$	0.238522	0.0141288
$286$	0	0
$287$	−16.1530	−0.953481
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	−1.48467	−0.0870328
$292$	0	0
$293$	16.8255	0.982955	0.491477	−	0.870890i	$-0.336457\pi$
$293$	16.8255	0.982955	0.491477	+	0.870890i	$0.336457\pi$
$294$	0	0
$295$	−0.676769	−0.0394030
$296$	0	0
$297$	1.15225	0.0668603
$298$	0	0
$299$	−0.717209	−0.0414773
$300$	0	0
$301$	0.380629	0.0219391
$302$	0	0
$303$	8.91932	0.512402
$304$	0	0
$305$	4.29761	0.246081
$306$	0	0
$307$	13.3960	0.764552	0.382276	−	0.924048i	$-0.375140\pi$
$307$	13.3960	0.764552	0.382276	+	0.924048i	$0.375140\pi$
$308$	0	0
$309$	−9.27747	−0.527777
$310$	0	0
$311$	−1.29051	−0.0731778	−0.0365889	−	0.999330i	$-0.511649\pi$
$311$	−1.29051	−0.0731778	−0.0365889	+	0.999330i	$0.511649\pi$
$312$	0	0
$313$	3.90116	0.220506	0.110253	−	0.993904i	$-0.464834\pi$
$313$	3.90116	0.220506	0.110253	+	0.993904i	$0.464834\pi$
$314$	0	0
$315$	3.04817	0.171745
$316$	0	0
$317$	−16.6566	−0.935528	−0.467764	−	0.883853i	$-0.654940\pi$
$317$	−16.6566	−0.935528	−0.467764	+	0.883853i	$0.654940\pi$
$318$	0	0
$319$	1.77641	0.0994596
$320$	0	0
$321$	4.51502	0.252004
$322$	0	0
$323$	−0.578329	−0.0321791
$324$	0	0
$325$	4.13599	0.229424
$326$	0	0
$327$	11.2629	0.622840
$328$	0	0
$329$	6.70963	0.369914
$330$	0	0
$331$	−27.8093	−1.52853	−0.764267	−	0.644900i	$-0.776899\pi$
$331$	−27.8093	−1.52853	−0.764267	+	0.644900i	$0.776899\pi$
$332$	0	0
$333$	10.2196	0.560028
$334$	0	0
$335$	−0.145396	−0.00794383
$336$	0	0
$337$	8.62671	0.469927	0.234963	−	0.972004i	$-0.424503\pi$
$337$	8.62671	0.469927	0.234963	+	0.972004i	$0.424503\pi$
$338$	0	0
$339$	−0.428016	−0.0232466
$340$	0	0
$341$	2.92796	0.158558
$342$	0	0
$343$	18.9578	1.02362
$344$	0	0
$345$	−0.324837	−0.0174886
$346$	0	0
$347$	2.26696	0.121697	0.0608484	−	0.998147i	$-0.480619\pi$
$347$	2.26696	0.121697	0.0608484	+	0.998147i	$0.480619\pi$
$348$	0	0
$349$	−22.9034	−1.22599	−0.612995	−	0.790087i	$-0.710036\pi$
$349$	−22.9034	−1.22599	−0.612995	+	0.790087i	$0.710036\pi$
$350$	0	0
$351$	−3.12355	−0.166723
$352$	0	0
$353$	−22.4055	−1.19253	−0.596263	−	0.802789i	$-0.703349\pi$
$353$	−22.4055	−1.19253	−0.596263	+	0.802789i	$0.703349\pi$
$354$	0	0
$355$	−9.60009	−0.509520
$356$	0	0
$357$	1.04420	0.0552651
$358$	0	0
$359$	6.46419	0.341167	0.170583	−	0.985343i	$-0.445435\pi$
$359$	6.46419	0.341167	0.170583	+	0.985343i	$0.445435\pi$
$360$	0	0
$361$	−18.6655	−0.982397
$362$	0	0
$363$	6.63480	0.348237
$364$	0	0
$365$	4.77922	0.250156
$366$	0	0
$367$	−14.6223	−0.763276	−0.381638	−	0.924312i	$-0.624640\pi$
$367$	−14.6223	−0.763276	−0.381638	+	0.924312i	$0.624640\pi$
$368$	0	0
$369$	−24.7804	−1.29001
$370$	0	0
$371$	−4.22030	−0.219107
$372$	0	0
$373$	−14.0424	−0.727089	−0.363545	−	0.931577i	$-0.618434\pi$
$373$	−14.0424	−0.727089	−0.363545	+	0.931577i	$0.618434\pi$
$374$	0	0
$375$	3.93543	0.203225
$376$	0	0
$377$	−4.81553	−0.248012
$378$	0	0
$379$	25.6541	1.31777	0.658883	−	0.752246i	$-0.271029\pi$
$379$	25.6541	1.31777	0.658883	+	0.752246i	$0.271029\pi$
$380$	0	0
$381$	−0.127740	−0.00654431
$382$	0	0
$383$	21.0771	1.07699	0.538496	−	0.842628i	$-0.318993\pi$
$383$	21.0771	1.07699	0.538496	+	0.842628i	$0.318993\pi$
$384$	0	0
$385$	−0.389534	−0.0198525
$386$	0	0
$387$	0.583925	0.0296826
$388$	0	0
$389$	−13.5914	−0.689110	−0.344555	−	0.938766i	$-0.611970\pi$
$389$	−13.5914	−0.689110	−0.344555	+	0.938766i	$0.611970\pi$
$390$	0	0
$391$	0.787611	0.0398312
$392$	0	0
$393$	−3.17914	−0.160366
$394$	0	0
$395$	1.85001	0.0930841
$396$	0	0
$397$	−3.12682	−0.156930	−0.0784652	−	0.996917i	$-0.525002\pi$
$397$	−3.12682	−0.156930	−0.0784652	+	0.996917i	$0.525002\pi$
$398$	0	0
$399$	−0.603893	−0.0302325
$400$	0	0
$401$	13.9417	0.696215	0.348107	−	0.937455i	$-0.386824\pi$
$401$	13.9417	0.696215	0.348107	+	0.937455i	$0.386824\pi$
$402$	0	0
$403$	−7.93719	−0.395380
$404$	0	0
$405$	3.92218	0.194895
$406$	0	0
$407$	−1.30598	−0.0647351
$408$	0	0
$409$	0.312940	0.0154739	0.00773695	−	0.999970i	$-0.497537\pi$
$409$	0.312940	0.0154739	0.00773695	+	0.999970i	$0.497537\pi$
$410$	0	0
$411$	9.62106	0.474572
$412$	0	0
$413$	1.71345	0.0843135
$414$	0	0
$415$	4.24288	0.208275
$416$	0	0
$417$	−12.3510	−0.604832
$418$	0	0
$419$	−7.26178	−0.354761	−0.177381	−	0.984142i	$-0.556762\pi$
$419$	−7.26178	−0.354761	−0.177381	+	0.984142i	$0.556762\pi$
$420$	0	0
$421$	7.27760	0.354688	0.177344	−	0.984149i	$-0.443249\pi$
$421$	7.27760	0.354688	0.177344	+	0.984149i	$0.443249\pi$
$422$	0	0
$423$	10.2933	0.500476
$424$	0	0
$425$	−4.54198	−0.220319
$426$	0	0
$427$	−10.8807	−0.526556
$428$	0	0
$429$	0.186414	0.00900017
$430$	0	0
$431$	−11.2828	−0.543472	−0.271736	−	0.962372i	$-0.587598\pi$
$431$	−11.2828	−0.543472	−0.271736	+	0.962372i	$0.587598\pi$
$432$	0	0
$433$	−27.4420	−1.31878	−0.659390	−	0.751801i	$-0.729185\pi$
$433$	−27.4420	−1.31878	−0.659390	+	0.751801i	$0.729185\pi$
$434$	0	0
$435$	−2.18104	−0.104573
$436$	0	0
$437$	−0.455498	−0.0217894
$438$	0	0
$439$	−30.3931	−1.45059	−0.725293	−	0.688441i	$-0.758295\pi$
$439$	−30.3931	−1.45059	−0.725293	+	0.688441i	$0.758295\pi$
$440$	0	0
$441$	10.6829	0.508709
$442$	0	0
$443$	−33.5441	−1.59373	−0.796865	−	0.604157i	$-0.793510\pi$
$443$	−33.5441	−1.59373	−0.796865	+	0.604157i	$0.793510\pi$
$444$	0	0
$445$	−5.78773	−0.274365
$446$	0	0
$447$	1.33941	0.0633518
$448$	0	0
$449$	31.1952	1.47219	0.736096	−	0.676878i	$-0.236667\pi$
$449$	31.1952	1.47219	0.736096	+	0.676878i	$0.236667\pi$
$450$	0	0
$451$	3.16675	0.149116
$452$	0	0
$453$	8.07369	0.379335
$454$	0	0
$455$	1.05596	0.0495041
$456$	0	0
$457$	−19.7145	−0.922205	−0.461103	−	0.887347i	$-0.652546\pi$
$457$	−19.7145	−0.922205	−0.461103	+	0.887347i	$0.652546\pi$
$458$	0	0
$459$	3.43016	0.160106
$460$	0	0
$461$	−12.6568	−0.589488	−0.294744	−	0.955576i	$-0.595234\pi$
$461$	−12.6568	−0.589488	−0.294744	+	0.955576i	$0.595234\pi$
$462$	0	0
$463$	−12.8207	−0.595827	−0.297913	−	0.954593i	$-0.596291\pi$
$463$	−12.8207	−0.595827	−0.297913	+	0.954593i	$0.596291\pi$
$464$	0	0
$465$	−3.59490	−0.166709
$466$	0	0
$467$	−0.0920969	−0.00426174	−0.00213087	−	0.999998i	$-0.500678\pi$
$467$	−0.0920969	−0.00426174	−0.00213087	+	0.999998i	$0.500678\pi$
$468$	0	0
$469$	0.368115	0.0169980
$470$	0	0
$471$	9.42225	0.434154
$472$	0	0
$473$	−0.0746212	−0.00343109
$474$	0	0
$475$	2.62676	0.120524
$476$	0	0
$477$	−6.47437	−0.296441
$478$	0	0
$479$	11.1928	0.511411	0.255705	−	0.966755i	$-0.417692\pi$
$479$	11.1928	0.511411	0.255705	+	0.966755i	$0.417692\pi$
$480$	0	0
$481$	3.54029	0.161423
$482$	0	0
$483$	0.822426	0.0374217
$484$	0	0
$485$	1.64876	0.0748662
$486$	0	0
$487$	31.4422	1.42478	0.712391	−	0.701783i	$-0.247612\pi$
$487$	31.4422	1.42478	0.712391	+	0.701783i	$0.247612\pi$
$488$	0	0
$489$	9.90685	0.448003
$490$	0	0
$491$	30.2935	1.36713	0.683564	−	0.729891i	$-0.260429\pi$
$491$	30.2935	1.36713	0.683564	+	0.729891i	$0.260429\pi$
$492$	0	0
$493$	5.28823	0.238170
$494$	0	0
$495$	−0.597585	−0.0268594
$496$	0	0
$497$	24.3056	1.09026
$498$	0	0
$499$	−12.2117	−0.546671	−0.273335	−	0.961919i	$-0.588127\pi$
$499$	−12.2117	−0.546671	−0.273335	+	0.961919i	$0.588127\pi$
$500$	0	0
$501$	5.59628	0.250023
$502$	0	0
$503$	13.8307	0.616679	0.308340	−	0.951276i	$-0.400227\pi$
$503$	13.8307	0.616679	0.308340	+	0.951276i	$0.400227\pi$
$504$	0	0
$505$	−9.90512	−0.440772
$506$	0	0
$507$	7.41706	0.329403
$508$	0	0
$509$	11.1673	0.494984	0.247492	−	0.968890i	$-0.420394\pi$
$509$	11.1673	0.494984	0.247492	+	0.968890i	$0.420394\pi$
$510$	0	0
$511$	−12.1001	−0.535277
$512$	0	0
$513$	−1.98376	−0.0875852
$514$	0	0
$515$	10.3028	0.453997
$516$	0	0
$517$	−1.31540	−0.0578513
$518$	0	0
$519$	9.79747	0.430061
$520$	0	0
$521$	−43.6409	−1.91194	−0.955971	−	0.293462i	$-0.905193\pi$
$521$	−43.6409	−1.91194	−0.955971	+	0.293462i	$0.905193\pi$
$522$	0	0
$523$	20.7731	0.908345	0.454173	−	0.890914i	$-0.349935\pi$
$523$	20.7731	0.908345	0.454173	+	0.890914i	$0.349935\pi$
$524$	0	0
$525$	−4.74275	−0.206991
$526$	0	0
$527$	8.71631	0.379689
$528$	0	0
$529$	−22.3797	−0.973029
$530$	0	0
$531$	2.62861	0.114072
$532$	0	0
$533$	−8.58451	−0.371836
$534$	0	0
$535$	−5.01403	−0.216775
$536$	0	0
$537$	6.43329	0.277617
$538$	0	0
$539$	−1.36519	−0.0588031
$540$	0	0
$541$	20.2257	0.869571	0.434786	−	0.900534i	$-0.356824\pi$
$541$	20.2257	0.869571	0.434786	+	0.900534i	$0.356824\pi$
$542$	0	0
$543$	6.81810	0.292593
$544$	0	0
$545$	−12.5077	−0.535772
$546$	0	0
$547$	20.1539	0.861720	0.430860	−	0.902419i	$-0.358210\pi$
$547$	20.1539	0.861720	0.430860	+	0.902419i	$0.358210\pi$
$548$	0	0
$549$	−16.6922	−0.712406
$550$	0	0
$551$	−3.05834	−0.130289
$552$	0	0
$553$	−4.68388	−0.199179
$554$	0	0
$555$	1.60346	0.0680632
$556$	0	0
$557$	2.69080	0.114013	0.0570064	−	0.998374i	$-0.481844\pi$
$557$	2.69080	0.114013	0.0570064	+	0.998374i	$0.481844\pi$
$558$	0	0
$559$	0.202285	0.00855576
$560$	0	0
$561$	−0.204713	−0.00864298
$562$	0	0
$563$	−32.6104	−1.37437	−0.687183	−	0.726484i	$-0.741153\pi$
$563$	−32.6104	−1.37437	−0.687183	+	0.726484i	$0.741153\pi$
$564$	0	0
$565$	0.475322	0.0199969
$566$	0	0
$567$	−9.93023	−0.417030
$568$	0	0
$569$	30.7936	1.29093	0.645466	−	0.763789i	$-0.276663\pi$
$569$	30.7936	1.29093	0.645466	+	0.763789i	$0.276663\pi$
$570$	0	0
$571$	−40.1143	−1.67873	−0.839366	−	0.543567i	$-0.817073\pi$
$571$	−40.1143	−1.67873	−0.839366	+	0.543567i	$0.817073\pi$
$572$	0	0
$573$	11.7479	0.490775
$574$	0	0
$575$	−3.57731	−0.149184
$576$	0	0
$577$	10.4209	0.433829	0.216915	−	0.976191i	$-0.430401\pi$
$577$	10.4209	0.433829	0.216915	+	0.976191i	$0.430401\pi$
$578$	0	0
$579$	−5.49642	−0.228424
$580$	0	0
$581$	−10.7422	−0.445660
$582$	0	0
$583$	0.827377	0.0342664
$584$	0	0
$585$	1.61995	0.0669767
$586$	0	0
$587$	35.2300	1.45410	0.727049	−	0.686585i	$-0.240891\pi$
$587$	35.2300	1.45410	0.727049	+	0.686585i	$0.240891\pi$
$588$	0	0
$589$	−5.04090	−0.207707
$590$	0	0
$591$	−0.611761	−0.0251645
$592$	0	0
$593$	9.32426	0.382902	0.191451	−	0.981502i	$-0.438681\pi$
$593$	9.32426	0.382902	0.191451	+	0.981502i	$0.438681\pi$
$594$	0	0
$595$	−1.15961	−0.0475395
$596$	0	0
$597$	−15.8464	−0.648551
$598$	0	0
$599$	36.1196	1.47581	0.737903	−	0.674907i	$-0.235816\pi$
$599$	36.1196	1.47581	0.737903	+	0.674907i	$0.235816\pi$
$600$	0	0
$601$	−26.7022	−1.08920	−0.544602	−	0.838695i	$-0.683319\pi$
$601$	−26.7022	−1.08920	−0.544602	+	0.838695i	$0.683319\pi$
$602$	0	0
$603$	0.564727	0.0229975
$604$	0	0
$605$	−7.36810	−0.299556
$606$	0	0
$607$	23.1001	0.937602	0.468801	−	0.883304i	$-0.344686\pi$
$607$	23.1001	0.937602	0.468801	+	0.883304i	$0.344686\pi$
$608$	0	0
$609$	5.52198	0.223762
$610$	0	0
$611$	3.56583	0.144258
$612$	0	0
$613$	7.96758	0.321807	0.160904	−	0.986970i	$-0.448559\pi$
$613$	7.96758	0.321807	0.160904	+	0.986970i	$0.448559\pi$
$614$	0	0
$615$	−3.88808	−0.156782
$616$	0	0
$617$	−4.39592	−0.176973	−0.0884866	−	0.996077i	$-0.528203\pi$
$617$	−4.39592	−0.176973	−0.0884866	+	0.996077i	$0.528203\pi$
$618$	0	0
$619$	−42.0596	−1.69052	−0.845259	−	0.534357i	$-0.820554\pi$
$619$	−42.0596	−1.69052	−0.845259	+	0.534357i	$0.820554\pi$
$620$	0	0
$621$	2.70163	0.108413
$622$	0	0
$623$	14.6534	0.587078
$624$	0	0
$625$	18.3395	0.733581
$626$	0	0
$627$	0.118391	0.00472810
$628$	0	0
$629$	−3.88781	−0.155017
$630$	0	0
$631$	−46.1695	−1.83798	−0.918990	−	0.394281i	$-0.870994\pi$
$631$	−46.1695	−1.83798	−0.918990	+	0.394281i	$0.870994\pi$
$632$	0	0
$633$	1.59521	0.0634039
$634$	0	0
$635$	0.141858	0.00562946
$636$	0	0
$637$	3.70081	0.146631
$638$	0	0
$639$	37.2873	1.47506
$640$	0	0
$641$	−48.7680	−1.92622	−0.963111	−	0.269105i	$-0.913272\pi$
$641$	−48.7680	−1.92622	−0.963111	+	0.269105i	$0.913272\pi$
$642$	0	0
$643$	22.9464	0.904919	0.452459	−	0.891785i	$-0.350547\pi$
$643$	22.9464	0.904919	0.452459	+	0.891785i	$0.350547\pi$
$644$	0	0
$645$	0.0916187	0.00360748
$646$	0	0
$647$	−12.7070	−0.499565	−0.249783	−	0.968302i	$-0.580359\pi$
$647$	−12.7070	−0.499565	−0.249783	+	0.968302i	$0.580359\pi$
$648$	0	0
$649$	−0.335917	−0.0131859
$650$	0	0
$651$	9.10160	0.356720
$652$	0	0
$653$	−20.7119	−0.810520	−0.405260	−	0.914202i	$-0.632819\pi$
$653$	−20.7119	−0.810520	−0.405260	+	0.914202i	$0.632819\pi$
$654$	0	0
$655$	3.53051	0.137948
$656$	0	0
$657$	−18.5628	−0.724204
$658$	0	0
$659$	−6.86475	−0.267413	−0.133706	−	0.991021i	$-0.542688\pi$
$659$	−6.86475	−0.267413	−0.133706	+	0.991021i	$0.542688\pi$
$660$	0	0
$661$	28.3858	1.10408	0.552041	−	0.833817i	$-0.313849\pi$
$661$	28.3858	1.10408	0.552041	+	0.833817i	$0.313849\pi$
$662$	0	0
$663$	0.554942	0.0215521
$664$	0	0
$665$	0.670638	0.0260062
$666$	0	0
$667$	4.16506	0.161272
$668$	0	0
$669$	−10.5851	−0.409242
$670$	0	0
$671$	2.13314	0.0823489
$672$	0	0
$673$	17.3372	0.668301	0.334151	−	0.942520i	$-0.391551\pi$
$673$	17.3372	0.668301	0.334151	+	0.942520i	$0.391551\pi$
$674$	0	0
$675$	−15.5797	−0.599664
$676$	0	0
$677$	2.34946	0.0902969	0.0451484	−	0.998980i	$-0.485624\pi$
$677$	2.34946	0.0902969	0.0451484	+	0.998980i	$0.485624\pi$
$678$	0	0
$679$	−4.17434	−0.160197
$680$	0	0
$681$	−8.85440	−0.339301
$682$	0	0
$683$	17.8119	0.681555	0.340778	−	0.940144i	$-0.389310\pi$
$683$	17.8119	0.681555	0.340778	+	0.940144i	$0.389310\pi$
$684$	0	0
$685$	−10.6844	−0.408230
$686$	0	0
$687$	−5.66826	−0.216257
$688$	0	0
$689$	−2.24288	−0.0854468
$690$	0	0
$691$	−42.5230	−1.61765	−0.808825	−	0.588050i	$-0.799896\pi$
$691$	−42.5230	−1.61765	−0.808825	+	0.588050i	$0.799896\pi$
$692$	0	0
$693$	1.51297	0.0574731
$694$	0	0
$695$	13.7161	0.520281
$696$	0	0
$697$	9.42716	0.357079
$698$	0	0
$699$	−14.5359	−0.549797
$700$	0	0
$701$	−16.6481	−0.628792	−0.314396	−	0.949292i	$-0.601802\pi$
$701$	−16.6481	−0.628792	−0.314396	+	0.949292i	$0.601802\pi$
$702$	0	0
$703$	2.24843	0.0848013
$704$	0	0
$705$	1.61503	0.0608255
$706$	0	0
$707$	25.0779	0.943151
$708$	0	0
$709$	−0.741683	−0.0278545	−0.0139272	−	0.999903i	$-0.504433\pi$
$709$	−0.741683	−0.0278545	−0.0139272	+	0.999903i	$0.504433\pi$
$710$	0	0
$711$	−7.18555	−0.269479
$712$	0	0
$713$	6.86506	0.257099
$714$	0	0
$715$	−0.207018	−0.00774202
$716$	0	0
$717$	9.54776	0.356568
$718$	0	0
$719$	0.869547	0.0324286	0.0162143	−	0.999869i	$-0.494839\pi$
$719$	0.869547	0.0324286	0.0162143	+	0.999869i	$0.494839\pi$
$720$	0	0
$721$	−26.0849	−0.971451
$722$	0	0
$723$	3.32176	0.123538
$724$	0	0
$725$	−24.0190	−0.892044
$726$	0	0
$727$	−5.86294	−0.217444	−0.108722	−	0.994072i	$-0.534676\pi$
$727$	−5.86294	−0.217444	−0.108722	+	0.994072i	$0.534676\pi$
$728$	0	0
$729$	−8.96283	−0.331956
$730$	0	0
$731$	−0.222142	−0.00821621
$732$	0	0
$733$	35.9995	1.32967	0.664836	−	0.746990i	$-0.268502\pi$
$733$	35.9995	1.32967	0.664836	+	0.746990i	$0.268502\pi$
$734$	0	0
$735$	1.67616	0.0618262
$736$	0	0
$737$	−0.0721679	−0.00265834
$738$	0	0
$739$	−12.1226	−0.445937	−0.222968	−	0.974826i	$-0.571575\pi$
$739$	−12.1226	−0.445937	−0.222968	+	0.974826i	$0.571575\pi$
$740$	0	0
$741$	−0.320939	−0.0117900
$742$	0	0
$743$	−15.7302	−0.577087	−0.288543	−	0.957467i	$-0.593171\pi$
$743$	−15.7302	−0.577087	−0.288543	+	0.957467i	$0.593171\pi$
$744$	0	0
$745$	−1.48744	−0.0544957
$746$	0	0
$747$	−16.4796	−0.602957
$748$	0	0
$749$	12.6946	0.463850
$750$	0	0
$751$	−18.3700	−0.670331	−0.335165	−	0.942159i	$-0.608792\pi$
$751$	−18.3700	−0.670331	−0.335165	+	0.942159i	$0.608792\pi$
$752$	0	0
$753$	−5.63149	−0.205223
$754$	0	0
$755$	−8.96602	−0.326307
$756$	0	0
$757$	−10.9888	−0.399394	−0.199697	−	0.979858i	$-0.563996\pi$
$757$	−10.9888	−0.399394	−0.199697	+	0.979858i	$0.563996\pi$
$758$	0	0
$759$	−0.161234	−0.00585242
$760$	0	0
$761$	−23.1522	−0.839268	−0.419634	−	0.907693i	$-0.637842\pi$
$761$	−23.1522	−0.839268	−0.419634	+	0.907693i	$0.637842\pi$
$762$	0	0
$763$	31.6672	1.14643
$764$	0	0
$765$	−1.77897	−0.0643186
$766$	0	0
$767$	0.910614	0.0328804
$768$	0	0
$769$	53.4827	1.92864	0.964318	−	0.264748i	$-0.0852889\pi$
$769$	53.4827	1.92864	0.964318	+	0.264748i	$0.0852889\pi$
$770$	0	0
$771$	11.8567	0.427008
$772$	0	0
$773$	−35.4163	−1.27383	−0.636917	−	0.770932i	$-0.719791\pi$
$773$	−35.4163	−1.27383	−0.636917	+	0.770932i	$0.719791\pi$
$774$	0	0
$775$	−39.5893	−1.42209
$776$	0	0
$777$	−4.05966	−0.145640
$778$	0	0
$779$	−5.45201	−0.195338
$780$	0	0
$781$	−4.76504	−0.170507
$782$	0	0
$783$	18.1395	0.648252
$784$	0	0
$785$	−10.4636	−0.373463
$786$	0	0
$787$	−40.3376	−1.43788	−0.718939	−	0.695073i	$-0.755372\pi$
$787$	−40.3376	−1.43788	−0.718939	+	0.695073i	$0.755372\pi$
$788$	0	0
$789$	−12.1186	−0.431434
$790$	0	0
$791$	−1.20343	−0.0427889
$792$	0	0
$793$	−5.78257	−0.205345
$794$	0	0
$795$	−1.01584	−0.0360281
$796$	0	0
$797$	22.5956	0.800377	0.400189	−	0.916433i	$-0.368945\pi$
$797$	22.5956	0.800377	0.400189	+	0.916433i	$0.368945\pi$
$798$	0	0
$799$	−3.91585	−0.138533
$800$	0	0
$801$	22.4799	0.794288
$802$	0	0
$803$	2.37219	0.0837126
$804$	0	0
$805$	−0.913323	−0.0321904
$806$	0	0
$807$	−3.75837	−0.132301
$808$	0	0
$809$	−36.2970	−1.27614	−0.638068	−	0.769980i	$-0.720266\pi$
$809$	−36.2970	−1.27614	−0.638068	+	0.769980i	$0.720266\pi$
$810$	0	0
$811$	50.8579	1.78586	0.892931	−	0.450194i	$-0.148645\pi$
$811$	50.8579	1.78586	0.892931	+	0.450194i	$0.148645\pi$
$812$	0	0
$813$	−0.319962	−0.0112216
$814$	0	0
$815$	−11.0018	−0.385376
$816$	0	0
$817$	0.128471	0.00449464
$818$	0	0
$819$	−4.10141	−0.143315
$820$	0	0
$821$	36.2540	1.26527	0.632636	−	0.774449i	$-0.281973\pi$
$821$	36.2540	1.26527	0.632636	+	0.774449i	$0.281973\pi$
$822$	0	0
$823$	−46.3859	−1.61691	−0.808456	−	0.588556i	$-0.799697\pi$
$823$	−46.3859	−1.61691	−0.808456	+	0.588556i	$0.799697\pi$
$824$	0	0
$825$	0.929803	0.0323716
$826$	0	0
$827$	−52.4212	−1.82286	−0.911432	−	0.411452i	$-0.865022\pi$
$827$	−52.4212	−1.82286	−0.911432	+	0.411452i	$0.865022\pi$
$828$	0	0
$829$	21.0084	0.729650	0.364825	−	0.931076i	$-0.381129\pi$
$829$	21.0084	0.729650	0.364825	+	0.931076i	$0.381129\pi$
$830$	0	0
$831$	12.0956	0.419592
$832$	0	0
$833$	−4.06408	−0.140812
$834$	0	0
$835$	−6.21480	−0.215072
$836$	0	0
$837$	29.8984	1.03344
$838$	0	0
$839$	−47.6965	−1.64667	−0.823333	−	0.567558i	$-0.807888\pi$
$839$	−47.6965	−1.64667	−0.823333	+	0.567558i	$0.807888\pi$
$840$	0	0
$841$	−1.03467	−0.0356784
$842$	0	0
$843$	19.3399	0.666103
$844$	0	0
$845$	−8.23681	−0.283355
$846$	0	0
$847$	18.6546	0.640981
$848$	0	0
$849$	9.27759	0.318406
$850$	0	0
$851$	−3.06208	−0.104967
$852$	0	0
$853$	−29.2818	−1.00259	−0.501294	−	0.865277i	$-0.667143\pi$
$853$	−29.2818	−1.00259	−0.501294	+	0.865277i	$0.667143\pi$
$854$	0	0
$855$	1.02883	0.0351852
$856$	0	0
$857$	41.6053	1.42121	0.710604	−	0.703592i	$-0.248422\pi$
$857$	41.6053	1.42121	0.710604	+	0.703592i	$0.248422\pi$
$858$	0	0
$859$	28.8573	0.984600	0.492300	−	0.870426i	$-0.336156\pi$
$859$	28.8573	0.984600	0.492300	+	0.870426i	$0.336156\pi$
$860$	0	0
$861$	9.84388	0.335478
$862$	0	0
$863$	−9.62096	−0.327501	−0.163751	−	0.986502i	$-0.552359\pi$
$863$	−9.62096	−0.327501	−0.163751	+	0.986502i	$0.552359\pi$
$864$	0	0
$865$	−10.8803	−0.369942
$866$	0	0
$867$	−0.609415	−0.0206968
$868$	0	0
$869$	0.918260	0.0311498
$870$	0	0
$871$	0.195635	0.00662883
$872$	0	0
$873$	−6.40388	−0.216738
$874$	0	0
$875$	11.0650	0.374065
$876$	0	0
$877$	30.1140	1.01688	0.508439	−	0.861098i	$-0.330223\pi$
$877$	30.1140	1.01688	0.508439	+	0.861098i	$0.330223\pi$
$878$	0	0
$879$	−10.2537	−0.345848
$880$	0	0
$881$	−24.4438	−0.823532	−0.411766	−	0.911290i	$-0.635088\pi$
$881$	−24.4438	−0.823532	−0.411766	+	0.911290i	$0.635088\pi$
$882$	0	0
$883$	−6.02784	−0.202853	−0.101427	−	0.994843i	$-0.532341\pi$
$883$	−6.02784	−0.202853	−0.101427	+	0.994843i	$0.532341\pi$
$884$	0	0
$885$	0.412433	0.0138638
$886$	0	0
$887$	−30.6215	−1.02817	−0.514084	−	0.857740i	$-0.671868\pi$
$887$	−30.6215	−1.02817	−0.514084	+	0.857740i	$0.671868\pi$
$888$	0	0
$889$	−0.359158	−0.0120458
$890$	0	0
$891$	1.94679	0.0652199
$892$	0	0
$893$	2.26465	0.0757837
$894$	0	0
$895$	−7.14431	−0.238808
$896$	0	0
$897$	0.437078	0.0145936
$898$	0	0
$899$	46.0938	1.53732
$900$	0	0
$901$	2.46304	0.0820557
$902$	0	0
$903$	−0.231961	−0.00771919
$904$	0	0
$905$	−7.57166	−0.251691
$906$	0	0
$907$	1.55679	0.0516923	0.0258462	−	0.999666i	$-0.491772\pi$
$907$	1.55679	0.0516923	0.0258462	+	0.999666i	$0.491772\pi$
$908$	0	0
$909$	38.4721	1.27604
$910$	0	0
$911$	45.0358	1.49210	0.746051	−	0.665889i	$-0.231948\pi$
$911$	45.0358	1.49210	0.746051	+	0.665889i	$0.231948\pi$
$912$	0	0
$913$	2.10597	0.0696974
$914$	0	0
$915$	−2.61903	−0.0865825
$916$	0	0
$917$	−8.93857	−0.295178
$918$	0	0
$919$	−27.8496	−0.918673	−0.459337	−	0.888262i	$-0.651913\pi$
$919$	−27.8496	−0.918673	−0.459337	+	0.888262i	$0.651913\pi$
$920$	0	0
$921$	−8.16375	−0.269005
$922$	0	0
$923$	12.9172	0.425175
$924$	0	0
$925$	17.6584	0.580604
$926$	0	0
$927$	−40.0169	−1.31433
$928$	0	0
$929$	12.2536	0.402028	0.201014	−	0.979588i	$-0.435576\pi$
$929$	12.2536	0.402028	0.201014	+	0.979588i	$0.435576\pi$
$930$	0	0
$931$	2.35038	0.0770305
$932$	0	0
$933$	0.786453	0.0257473
$934$	0	0
$935$	0.227338	0.00743476
$936$	0	0
$937$	−18.3010	−0.597868	−0.298934	−	0.954274i	$-0.596631\pi$
$937$	−18.3010	−0.597868	−0.298934	+	0.954274i	$0.596631\pi$
$938$	0	0
$939$	−2.37742	−0.0775843
$940$	0	0
$941$	0.588015	0.0191688	0.00958438	−	0.999954i	$-0.496949\pi$
$941$	0.588015	0.0191688	0.00958438	+	0.999954i	$0.496949\pi$
$942$	0	0
$943$	7.42493	0.241789
$944$	0	0
$945$	−3.97766	−0.129393
$946$	0	0
$947$	−1.46909	−0.0477391	−0.0238695	−	0.999715i	$-0.507599\pi$
$947$	−1.46909	−0.0477391	−0.0238695	+	0.999715i	$0.507599\pi$
$948$	0	0
$949$	−6.43059	−0.208746
$950$	0	0
$951$	10.1508	0.329162
$952$	0	0
$953$	−44.9347	−1.45558	−0.727789	−	0.685801i	$-0.759452\pi$
$953$	−44.9347	−1.45558	−0.727789	+	0.685801i	$0.759452\pi$
$954$	0	0
$955$	−13.0463	−0.422169
$956$	0	0
$957$	−1.08257	−0.0349944
$958$	0	0
$959$	27.0509	0.873519
$960$	0	0
$961$	44.9741	1.45078
$962$	0	0
$963$	19.4748	0.627567
$964$	0	0
$965$	6.10391	0.196492
$966$	0	0
$967$	−8.78435	−0.282486	−0.141243	−	0.989975i	$-0.545110\pi$
$967$	−8.78435	−0.282486	−0.141243	+	0.989975i	$0.545110\pi$
$968$	0	0
$969$	0.352442	0.0113221
$970$	0	0
$971$	20.0863	0.644600	0.322300	−	0.946638i	$-0.395544\pi$
$971$	20.0863	0.644600	0.322300	+	0.946638i	$0.395544\pi$
$972$	0	0
$973$	−34.7266	−1.11328
$974$	0	0
$975$	−2.52054	−0.0807217
$976$	0	0
$977$	−50.2080	−1.60630	−0.803148	−	0.595779i	$-0.796843\pi$
$977$	−50.2080	−1.60630	−0.803148	+	0.595779i	$0.796843\pi$
$978$	0	0
$979$	−2.87276	−0.0918139
$980$	0	0
$981$	48.5808	1.55106
$982$	0	0
$983$	−47.8130	−1.52500	−0.762499	−	0.646989i	$-0.776028\pi$
$983$	−47.8130	−1.52500	−0.762499	+	0.646989i	$0.776028\pi$
$984$	0	0
$985$	0.679375	0.0216467
$986$	0	0
$987$	−4.08895	−0.130153
$988$	0	0
$989$	−0.174961	−0.00556344
$990$	0	0
$991$	0.216755	0.00688544	0.00344272	−	0.999994i	$-0.498904\pi$
$991$	0.216755	0.00688544	0.00344272	+	0.999994i	$0.498904\pi$
$992$	0	0
$993$	16.9474	0.537809
$994$	0	0
$995$	17.5978	0.557888
$996$	0	0
$997$	−32.0778	−1.01592	−0.507958	−	0.861382i	$-0.669599\pi$
$997$	−32.0778	−1.01592	−0.507958	+	0.861382i	$0.669599\pi$
$998$	0	0
$999$	−13.3358	−0.421926

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8024.2.a.w.1.8	✓	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8024.2.a.w.1.8	✓	20	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)

\(f(q)\)	\(=\)	\(q-0.609415 q^{3} +0.676769 q^{5} -1.71345 q^{7} -2.62861 q^{9} +O(q^{10})\)	\(q-0.609415 q^{3} +0.676769 q^{5} -1.71345 q^{7} -2.62861 q^{9} +0.335917 q^{11} -0.910614 q^{13} -0.412433 q^{15} +1.00000 q^{17} -0.578329 q^{19} +1.04420 q^{21} +0.787611 q^{23} -4.54198 q^{25} +3.43016 q^{27} +5.28823 q^{29} +8.71631 q^{31} -0.204713 q^{33} -1.15961 q^{35} -3.88781 q^{37} +0.554942 q^{39} +9.42716 q^{41} -0.222142 q^{43} -1.77897 q^{45} -3.91585 q^{47} -4.06408 q^{49} -0.609415 q^{51} +2.46304 q^{53} +0.227338 q^{55} +0.352442 q^{57} -1.00000 q^{59} +6.35019 q^{61} +4.50400 q^{63} -0.616276 q^{65} -0.214838 q^{67} -0.479982 q^{69} -14.1852 q^{71} +7.06182 q^{73} +2.76795 q^{75} -0.575578 q^{77} +2.73359 q^{79} +5.79545 q^{81} +6.26931 q^{83} +0.676769 q^{85} -3.22272 q^{87} -8.55200 q^{89} +1.56029 q^{91} -5.31185 q^{93} -0.391396 q^{95} +2.43622 q^{97} -0.882996 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 2 q^{3} - 2 q^{5} + 5 q^{7} + 2 q^{9}+O(q^{10}) \) 20 * q - 2 * q^3 - 2 * q^5 + 5 * q^7 + 2 * q^9	\( 20 q - 2 q^{3} - 2 q^{5} + 5 q^{7} + 2 q^{9} + 8 q^{11} - 3 q^{13} - 2 q^{15} + 20 q^{17} - 9 q^{19} - 31 q^{21} - 8 q^{23} - 4 q^{25} - 17 q^{27} - 37 q^{29} - 9 q^{31} - 8 q^{33} - 8 q^{35} - 11 q^{37} - 12 q^{39} - 27 q^{41} + 17 q^{43} - 4 q^{45} + 7 q^{47} - 9 q^{49} - 2 q^{51} - q^{53} - 21 q^{55} - 57 q^{57} - 20 q^{59} - 12 q^{61} - 14 q^{63} - 59 q^{65} - 26 q^{67} + 14 q^{69} - 3 q^{71} - 45 q^{73} + 15 q^{75} + 6 q^{77} + 5 q^{79} + 12 q^{81} - 17 q^{83} - 2 q^{85} - 32 q^{87} - 40 q^{89} - 36 q^{91} - 4 q^{93} + 5 q^{95} - 77 q^{97} + 54 q^{99}+O(q^{100}) \) 20 * q - 2 * q^3 - 2 * q^5 + 5 * q^7 + 2 * q^9 + 8 * q^11 - 3 * q^13 - 2 * q^15 + 20 * q^17 - 9 * q^19 - 31 * q^21 - 8 * q^23 - 4 * q^25 - 17 * q^27 - 37 * q^29 - 9 * q^31 - 8 * q^33 - 8 * q^35 - 11 * q^37 - 12 * q^39 - 27 * q^41 + 17 * q^43 - 4 * q^45 + 7 * q^47 - 9 * q^49 - 2 * q^51 - q^53 - 21 * q^55 - 57 * q^57 - 20 * q^59 - 12 * q^61 - 14 * q^63 - 59 * q^65 - 26 * q^67 + 14 * q^69 - 3 * q^71 - 45 * q^73 + 15 * q^75 + 6 * q^77 + 5 * q^79 + 12 * q^81 - 17 * q^83 - 2 * q^85 - 32 * q^87 - 40 * q^89 - 36 * q^91 - 4 * q^93 + 5 * q^95 - 77 * q^97 + 54 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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