LMFDB - Embedded newform 5776.2.a.ca.1.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [5776,2,Mod(1,5776)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("5776.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(5776, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 5776 = 2^{4} \cdot 19^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5776.a (trivial)

Newform invariants

comment:select newform

sage:traces = [8,0,-6,0,-2,0,2,0,2,0,-12,0,6,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(15)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$46.1215922075$
Analytic rank:	$1$
Dimension:	$8$
Coefficient field:	8.8.10564000000.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2x^{7} - 9x^{6} + 14x^{5} + 24x^{4} - 28x^{3} - 21x^{2} + 16x + 1 $ x^8 - 2x^7 - 9x^6 + 14x^5 + 24x^4 - 28x^3 - 21x^2 + 16*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 2888)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-0.0583903$ of defining polynomial
Character	$\chi$	$=$	5776.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.05839 q^{3} +3.24714 q^{5} +2.81724 q^{7} -1.87981 q^{9} -5.74166 q^{11} +2.81591 q^{13} -3.43674 q^{15} -5.80894 q^{17} -2.98174 q^{21} -4.37392 q^{23} +5.54391 q^{25} +5.16474 q^{27} +9.09159 q^{29} -5.54320 q^{31} +6.07691 q^{33} +9.14797 q^{35} +4.34678 q^{37} -2.98033 q^{39} -8.07410 q^{41} +0.0508452 q^{43} -6.10400 q^{45} +0.0700146 q^{47} +0.936845 q^{49} +6.14813 q^{51} +4.11253 q^{53} -18.6440 q^{55} -5.73018 q^{59} -3.98633 q^{61} -5.29588 q^{63} +9.14364 q^{65} -8.45190 q^{67} +4.62932 q^{69} -9.37179 q^{71} -3.12158 q^{73} -5.86762 q^{75} -16.1756 q^{77} +9.24692 q^{79} +0.173115 q^{81} -13.7604 q^{83} -18.8624 q^{85} -9.62245 q^{87} +10.4506 q^{89} +7.93309 q^{91} +5.86687 q^{93} -13.7369 q^{97} +10.7932 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 6 q^{3} - 2 q^{5} + 2 q^{7} + 2 q^{9} - 12 q^{11} + 6 q^{13} - 10 q^{17} - 4 q^{21} + 12 q^{23} + 2 q^{25} - 12 q^{27} + 18 q^{29} - 14 q^{31} + 40 q^{33} - 18 q^{35} + 16 q^{37} - 28 q^{39} + 12 q^{41}+ \cdots - 32 q^{99}+O(q^{100}) $ 8 * q - 6 * q^3 - 2 * q^5 + 2 * q^7 + 2 * q^9 - 12 * q^11 + 6 * q^13 - 10 * q^17 - 4 * q^21 + 12 * q^23 + 2 * q^25 - 12 * q^27 + 18 * q^29 - 14 * q^31 + 40 * q^33 - 18 * q^35 + 16 * q^37 - 28 * q^39 + 12 * q^41 - 8 * q^43 - 2 * q^45 - 24 * q^47 + 12 * q^49 + 22 * q^51 + 12 * q^53 + 2 * q^55 + 16 * q^59 - 12 * q^63 + 12 * q^65 - 42 * q^67 - 12 * q^69 - 36 * q^71 - 10 * q^73 - 32 * q^75 - 38 * q^77 - 6 * q^79 + 4 * q^81 - 12 * q^83 - 42 * q^85 + 12 * q^87 + 8 * q^89 - 16 * q^91 - 14 * q^93 + 28 * q^97 - 32 * q^99

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
$3$	−1.05839	−0.611062	−0.305531	−	0.952182i	$-0.598834\pi$
$3$	−1.05839	−0.611062	−0.305531	+	0.952182i	$0.598834\pi$
$4$	0	0
$5$	3.24714	1.45216	0.726082	−	0.687608i	$-0.241339\pi$
$5$	3.24714	1.45216	0.726082	+	0.687608i	$0.241339\pi$
$6$	0	0
$7$	2.81724	1.06482	0.532408	−	0.846488i	$-0.321287\pi$
$7$	2.81724	1.06482	0.532408	+	0.846488i	$0.321287\pi$
$8$	0	0
$9$	−1.87981	−0.626603
$10$	0	0
$11$	−5.74166	−1.73117	−0.865587	−	0.500758i	$-0.833055\pi$
$11$	−5.74166	−1.73117	−0.865587	+	0.500758i	$0.833055\pi$
$12$	0	0
$13$	2.81591	0.780992	0.390496	−	0.920605i	$-0.372303\pi$
$13$	2.81591	0.780992	0.390496	+	0.920605i	$0.372303\pi$
$14$	0	0
$15$	−3.43674	−0.887363
$16$	0	0
$17$	−5.80894	−1.40887	−0.704437	−	0.709766i	$-0.748801\pi$
$17$	−5.80894	−1.40887	−0.704437	+	0.709766i	$0.748801\pi$
$18$	0	0
$19$	0	0
$19$	0	0
$20$	0	0
$21$	−2.98174	−0.650669
$22$	0	0
$23$	−4.37392	−0.912026	−0.456013	−	0.889973i	$-0.650723\pi$
$23$	−4.37392	−0.912026	−0.456013	+	0.889973i	$0.650723\pi$
$24$	0	0
$25$	5.54391	1.10878
$26$	0	0
$27$	5.16474	0.993955
$28$	0	0
$29$	9.09159	1.68827	0.844133	−	0.536134i	$-0.180116\pi$
$29$	9.09159	1.68827	0.844133	+	0.536134i	$0.180116\pi$
$30$	0	0
$31$	−5.54320	−0.995588	−0.497794	−	0.867295i	$-0.665856\pi$
$31$	−5.54320	−0.995588	−0.497794	+	0.867295i	$0.665856\pi$
$32$	0	0
$33$	6.07691	1.05785
$34$	0	0
$35$	9.14797	1.54629
$36$	0	0
$37$	4.34678	0.714606	0.357303	−	0.933989i	$-0.383696\pi$
$37$	4.34678	0.714606	0.357303	+	0.933989i	$0.383696\pi$
$38$	0	0
$39$	−2.98033	−0.477235
$40$	0	0
$41$	−8.07410	−1.26096	−0.630482	−	0.776204i	$-0.717143\pi$
$41$	−8.07410	−1.26096	−0.630482	+	0.776204i	$0.717143\pi$
$42$	0	0
$43$	0.0508452	0.00775382	0.00387691	−	0.999992i	$-0.498766\pi$
$43$	0.0508452	0.00775382	0.00387691	+	0.999992i	$0.498766\pi$
$44$	0	0
$45$	−6.10400	−0.909931
$46$	0	0
$47$	0.0700146	0.0102127	0.00510634	−	0.999987i	$-0.498375\pi$
$47$	0.0700146	0.0102127	0.00510634	+	0.999987i	$0.498375\pi$
$48$	0	0
$49$	0.936845	0.133835
$50$	0	0
$51$	6.14813	0.860910
$52$	0	0
$53$	4.11253	0.564899	0.282449	−	0.959282i	$-0.408853\pi$
$53$	4.11253	0.564899	0.282449	+	0.959282i	$0.408853\pi$
$54$	0	0
$55$	−18.6440	−2.51395
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−5.73018	−0.746006	−0.373003	−	0.927830i	$-0.621672\pi$
$59$	−5.73018	−0.746006	−0.373003	+	0.927830i	$0.621672\pi$
$60$	0	0
$61$	−3.98633	−0.510398	−0.255199	−	0.966889i	$-0.582141\pi$
$61$	−3.98633	−0.510398	−0.255199	+	0.966889i	$0.582141\pi$
$62$	0	0
$63$	−5.29588	−0.667218
$64$	0	0
$65$	9.14364	1.13413
$66$	0	0
$67$	−8.45190	−1.03256	−0.516282	−	0.856418i	$-0.672684\pi$
$67$	−8.45190	−1.03256	−0.516282	+	0.856418i	$0.672684\pi$
$68$	0	0
$69$	4.62932	0.557305
$70$	0	0
$71$	−9.37179	−1.11223	−0.556114	−	0.831106i	$-0.687708\pi$
$71$	−9.37179	−1.11223	−0.556114	+	0.831106i	$0.687708\pi$
$72$	0	0
$73$	−3.12158	−0.365353	−0.182677	−	0.983173i	$-0.558476\pi$
$73$	−3.12158	−0.365353	−0.182677	+	0.983173i	$0.558476\pi$
$74$	0	0
$75$	−5.86762	−0.677535
$76$	0	0
$77$	−16.1756	−1.84338
$78$	0	0
$79$	9.24692	1.04036	0.520180	−	0.854057i	$-0.325865\pi$
$79$	9.24692	1.04036	0.520180	+	0.854057i	$0.325865\pi$
$80$	0	0
$81$	0.173115	0.0192350
$82$	0	0
$83$	−13.7604	−1.51040	−0.755199	−	0.655496i	$-0.772460\pi$
$83$	−13.7604	−1.51040	−0.755199	+	0.655496i	$0.772460\pi$
$84$	0	0
$85$	−18.8624	−2.04592
$86$	0	0
$87$	−9.62245	−1.03163
$88$	0	0
$89$	10.4506	1.10776	0.553879	−	0.832597i	$-0.313147\pi$
$89$	10.4506	1.10776	0.553879	+	0.832597i	$0.313147\pi$
$90$	0	0
$91$	7.93309	0.831614
$92$	0	0
$93$	5.86687	0.608366
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−13.7369	−1.39477	−0.697384	−	0.716698i	$-0.745653\pi$
$97$	−13.7369	−1.39477	−0.697384	+	0.716698i	$0.745653\pi$
$98$	0	0
$99$	10.7932	1.08476
$100$	0	0
$101$	−16.7163	−1.66333	−0.831665	−	0.555277i	$-0.812612\pi$
$101$	−16.7163	−1.66333	−0.831665	+	0.555277i	$0.812612\pi$
$102$	0	0
$103$	−6.13989	−0.604981	−0.302490	−	0.953152i	$-0.597818\pi$
$103$	−6.13989	−0.604981	−0.302490	+	0.953152i	$0.597818\pi$
$104$	0	0
$105$	−9.68212	−0.944879
$106$	0	0
$107$	−3.73031	−0.360623	−0.180311	−	0.983610i	$-0.557711\pi$
$107$	−3.73031	−0.360623	−0.180311	+	0.983610i	$0.557711\pi$
$108$	0	0
$109$	−4.69729	−0.449919	−0.224959	−	0.974368i	$-0.572225\pi$
$109$	−4.69729	−0.449919	−0.224959	+	0.974368i	$0.572225\pi$
$110$	0	0
$111$	−4.60059	−0.436668
$112$	0	0
$113$	15.8309	1.48924	0.744622	−	0.667487i	$-0.232630\pi$
$113$	15.8309	1.48924	0.744622	+	0.667487i	$0.232630\pi$
$114$	0	0
$115$	−14.2027	−1.32441
$116$	0	0
$117$	−5.29337	−0.489372
$118$	0	0
$119$	−16.3652	−1.50019
$120$	0	0
$121$	21.9666	1.99697
$122$	0	0
$123$	8.54555	0.770527
$124$	0	0
$125$	1.76615	0.157970
$126$	0	0
$127$	−2.85657	−0.253480	−0.126740	−	0.991936i	$-0.540451\pi$
$127$	−2.85657	−0.253480	−0.126740	+	0.991936i	$0.540451\pi$
$128$	0	0
$129$	−0.0538141	−0.00473807
$130$	0	0
$131$	−18.4249	−1.60979	−0.804894	−	0.593418i	$-0.797778\pi$
$131$	−18.4249	−1.60979	−0.804894	+	0.593418i	$0.797778\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	16.7706	1.44339
$136$	0	0
$137$	7.91284	0.676039	0.338020	−	0.941139i	$-0.390243\pi$
$137$	7.91284	0.676039	0.338020	+	0.941139i	$0.390243\pi$
$138$	0	0
$139$	10.2508	0.869458	0.434729	−	0.900561i	$-0.356844\pi$
$139$	10.2508	0.869458	0.434729	+	0.900561i	$0.356844\pi$
$140$	0	0
$141$	−0.0741028	−0.00624058
$142$	0	0
$143$	−16.1680	−1.35203
$144$	0	0
$145$	29.5217	2.45164
$146$	0	0
$147$	−0.991548	−0.0817815
$148$	0	0
$149$	−14.0529	−1.15126	−0.575631	−	0.817709i	$-0.695244\pi$
$149$	−14.0529	−1.15126	−0.575631	+	0.817709i	$0.695244\pi$
$150$	0	0
$151$	19.9660	1.62481	0.812406	−	0.583092i	$-0.198157\pi$
$151$	19.9660	1.62481	0.812406	+	0.583092i	$0.198157\pi$
$152$	0	0
$153$	10.9197	0.882806
$154$	0	0
$155$	−17.9995	−1.44576
$156$	0	0
$157$	−3.68869	−0.294390	−0.147195	−	0.989108i	$-0.547024\pi$
$157$	−3.68869	−0.294390	−0.147195	+	0.989108i	$0.547024\pi$
$158$	0	0
$159$	−4.35266	−0.345188
$160$	0	0
$161$	−12.3224	−0.971141
$162$	0	0
$163$	21.1527	1.65681	0.828405	−	0.560130i	$-0.189249\pi$
$163$	21.1527	1.65681	0.828405	+	0.560130i	$0.189249\pi$
$164$	0	0
$165$	19.7326	1.53618
$166$	0	0
$167$	−11.9102	−0.921642	−0.460821	−	0.887493i	$-0.652445\pi$
$167$	−11.9102	−0.921642	−0.460821	+	0.887493i	$0.652445\pi$
$168$	0	0
$169$	−5.07066	−0.390051
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−1.19805	−0.0910861	−0.0455431	−	0.998962i	$-0.514502\pi$
$173$	−1.19805	−0.0910861	−0.0455431	+	0.998962i	$0.514502\pi$
$174$	0	0
$175$	15.6185	1.18065
$176$	0	0
$177$	6.06476	0.455856
$178$	0	0
$179$	−1.35444	−0.101235	−0.0506177	−	0.998718i	$-0.516119\pi$
$179$	−1.35444	−0.101235	−0.0506177	+	0.998718i	$0.516119\pi$
$180$	0	0
$181$	−14.5947	−1.08482	−0.542408	−	0.840115i	$-0.682487\pi$
$181$	−14.5947	−1.08482	−0.542408	+	0.840115i	$0.682487\pi$
$182$	0	0
$183$	4.21910	0.311885
$184$	0	0
$185$	14.1146	1.03772
$186$	0	0
$187$	33.3529	2.43901
$188$	0	0
$189$	14.5503	1.05838
$190$	0	0
$191$	19.0267	1.37672	0.688361	−	0.725369i	$-0.258331\pi$
$191$	19.0267	1.37672	0.688361	+	0.725369i	$0.258331\pi$
$192$	0	0
$193$	4.66243	0.335609	0.167805	−	0.985820i	$-0.446332\pi$
$193$	4.66243	0.335609	0.167805	+	0.985820i	$0.446332\pi$
$194$	0	0
$195$	−9.67754	−0.693023
$196$	0	0
$197$	−15.2905	−1.08940	−0.544701	−	0.838631i	$-0.683357\pi$
$197$	−15.2905	−1.08940	−0.544701	+	0.838631i	$0.683357\pi$
$198$	0	0
$199$	−0.143336	−0.0101608	−0.00508042	−	0.999987i	$-0.501617\pi$
$199$	−0.143336	−0.0101608	−0.00508042	+	0.999987i	$0.501617\pi$
$200$	0	0
$201$	8.94541	0.630961
$202$	0	0
$203$	25.6132	1.79769
$204$	0	0
$205$	−26.2177	−1.83113
$206$	0	0
$207$	8.22215	0.571479
$208$	0	0
$209$	0	0
$210$	0	0
$211$	20.3475	1.40078	0.700392	−	0.713759i	$-0.253009\pi$
$211$	20.3475	1.40078	0.700392	+	0.713759i	$0.253009\pi$
$212$	0	0
$213$	9.91902	0.679640
$214$	0	0
$215$	0.165102	0.0112598
$216$	0	0
$217$	−15.6165	−1.06012
$218$	0	0
$219$	3.30385	0.223253
$220$	0	0
$221$	−16.3574	−1.10032
$222$	0	0
$223$	−23.9147	−1.60145	−0.800724	−	0.599034i	$-0.795552\pi$
$223$	−23.9147	−1.60145	−0.800724	+	0.599034i	$0.795552\pi$
$224$	0	0
$225$	−10.4215	−0.694767
$226$	0	0
$227$	−23.0675	−1.53104	−0.765521	−	0.643411i	$-0.777518\pi$
$227$	−23.0675	−1.53104	−0.765521	+	0.643411i	$0.777518\pi$
$228$	0	0
$229$	4.60612	0.304381	0.152191	−	0.988351i	$-0.451367\pi$
$229$	4.60612	0.304381	0.152191	+	0.988351i	$0.451367\pi$
$230$	0	0
$231$	17.1201	1.12642
$232$	0	0
$233$	−7.54056	−0.493998	−0.246999	−	0.969016i	$-0.579445\pi$
$233$	−7.54056	−0.493998	−0.246999	+	0.969016i	$0.579445\pi$
$234$	0	0
$235$	0.227347	0.0148305
$236$	0	0
$237$	−9.78686	−0.635725
$238$	0	0
$239$	−7.76510	−0.502283	−0.251141	−	0.967950i	$-0.580806\pi$
$239$	−7.76510	−0.502283	−0.251141	+	0.967950i	$0.580806\pi$
$240$	0	0
$241$	−1.05804	−0.0681544	−0.0340772	−	0.999419i	$-0.510849\pi$
$241$	−1.05804	−0.0681544	−0.0340772	+	0.999419i	$0.510849\pi$
$242$	0	0
$243$	−15.6775	−1.00571
$244$	0	0
$245$	3.04207	0.194350
$246$	0	0
$247$	0	0
$248$	0	0
$249$	14.5638	0.922946
$250$	0	0
$251$	10.5226	0.664179	0.332089	−	0.943248i	$-0.392246\pi$
$251$	10.5226	0.664179	0.332089	+	0.943248i	$0.392246\pi$
$252$	0	0
$253$	25.1136	1.57888
$254$	0	0
$255$	19.9638	1.25018
$256$	0	0
$257$	20.0181	1.24869	0.624347	−	0.781147i	$-0.285365\pi$
$257$	20.0181	1.24869	0.624347	+	0.781147i	$0.285365\pi$
$258$	0	0
$259$	12.2459	0.760924
$260$	0	0
$261$	−17.0905	−1.05787
$262$	0	0
$263$	−3.13462	−0.193289	−0.0966445	−	0.995319i	$-0.530811\pi$
$263$	−3.13462	−0.193289	−0.0966445	+	0.995319i	$0.530811\pi$
$264$	0	0
$265$	13.3539	0.820326
$266$	0	0
$267$	−11.0608	−0.676908
$268$	0	0
$269$	−4.54258	−0.276966	−0.138483	−	0.990365i	$-0.544223\pi$
$269$	−4.54258	−0.276966	−0.138483	+	0.990365i	$0.544223\pi$
$270$	0	0
$271$	−10.7856	−0.655178	−0.327589	−	0.944820i	$-0.606236\pi$
$271$	−10.7856	−0.655178	−0.327589	+	0.944820i	$0.606236\pi$
$272$	0	0
$273$	−8.39631	−0.508168
$274$	0	0
$275$	−31.8312	−1.91950
$276$	0	0
$277$	7.79802	0.468537	0.234269	−	0.972172i	$-0.424730\pi$
$277$	7.79802	0.468537	0.234269	+	0.972172i	$0.424730\pi$
$278$	0	0
$279$	10.4202	0.623839
$280$	0	0
$281$	−26.0265	−1.55261	−0.776305	−	0.630358i	$-0.782908\pi$
$281$	−26.0265	−1.55261	−0.776305	+	0.630358i	$0.782908\pi$
$282$	0	0
$283$	3.11584	0.185217	0.0926087	−	0.995703i	$-0.470479\pi$
$283$	3.11584	0.185217	0.0926087	+	0.995703i	$0.470479\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−22.7467	−1.34269
$288$	0	0
$289$	16.7438	0.984928
$290$	0	0
$291$	14.5390	0.852289
$292$	0	0
$293$	−5.04928	−0.294982	−0.147491	−	0.989063i	$-0.547120\pi$
$293$	−5.04928	−0.294982	−0.147491	+	0.989063i	$0.547120\pi$
$294$	0	0
$295$	−18.6067	−1.08332
$296$	0	0
$297$	−29.6542	−1.72071
$298$	0	0
$299$	−12.3166	−0.712285
$300$	0	0
$301$	0.143243	0.00825640
$302$	0	0
$303$	17.6923	1.01640
$304$	0	0
$305$	−12.9442	−0.741181
$306$	0	0
$307$	−20.3573	−1.16185	−0.580925	−	0.813957i	$-0.697309\pi$
$307$	−20.3573	−1.16185	−0.580925	+	0.813957i	$0.697309\pi$
$308$	0	0
$309$	6.49840	0.369681
$310$	0	0
$311$	12.2373	0.693912	0.346956	−	0.937881i	$-0.387215\pi$
$311$	12.2373	0.693912	0.346956	+	0.937881i	$0.387215\pi$
$312$	0	0
$313$	−5.04571	−0.285201	−0.142600	−	0.989780i	$-0.545546\pi$
$313$	−5.04571	−0.285201	−0.142600	+	0.989780i	$0.545546\pi$
$314$	0	0
$315$	−17.1964	−0.968910
$316$	0	0
$317$	2.05365	0.115345	0.0576723	−	0.998336i	$-0.481632\pi$
$317$	2.05365	0.115345	0.0576723	+	0.998336i	$0.481632\pi$
$318$	0	0
$319$	−52.2008	−2.92268
$320$	0	0
$321$	3.94812	0.220363
$322$	0	0
$323$	0	0
$324$	0	0
$325$	15.6111	0.865950
$326$	0	0
$327$	4.97157	0.274928
$328$	0	0
$329$	0.197248	0.0108746
$330$	0	0
$331$	2.42144	0.133094	0.0665472	−	0.997783i	$-0.478802\pi$
$331$	2.42144	0.133094	0.0665472	+	0.997783i	$0.478802\pi$
$332$	0	0
$333$	−8.17111	−0.447774
$334$	0	0
$335$	−27.4445	−1.49945
$336$	0	0
$337$	−3.98246	−0.216938	−0.108469	−	0.994100i	$-0.534595\pi$
$337$	−3.98246	−0.216938	−0.108469	+	0.994100i	$0.534595\pi$
$338$	0	0
$339$	−16.7552	−0.910020
$340$	0	0
$341$	31.8271	1.72354
$342$	0	0
$343$	−17.0814	−0.922307
$344$	0	0
$345$	15.0320	0.809298
$346$	0	0
$347$	0.721652	0.0387403	0.0193701	−	0.999812i	$-0.493834\pi$
$347$	0.721652	0.0387403	0.0193701	+	0.999812i	$0.493834\pi$
$348$	0	0
$349$	8.88227	0.475457	0.237729	−	0.971332i	$-0.423597\pi$
$349$	8.88227	0.475457	0.237729	+	0.971332i	$0.423597\pi$
$350$	0	0
$351$	14.5434	0.776271
$352$	0	0
$353$	−12.9565	−0.689604	−0.344802	−	0.938675i	$-0.612054\pi$
$353$	−12.9565	−0.689604	−0.344802	+	0.938675i	$0.612054\pi$
$354$	0	0
$355$	−30.4315	−1.61514
$356$	0	0
$357$	17.3207	0.916711
$358$	0	0
$359$	28.8612	1.52323	0.761617	−	0.648028i	$-0.224406\pi$
$359$	28.8612	1.52323	0.761617	+	0.648028i	$0.224406\pi$
$360$	0	0
$361$	0	0
$362$	0	0
$363$	−23.2493	−1.22027
$364$	0	0
$365$	−10.1362	−0.530553
$366$	0	0
$367$	4.93816	0.257770	0.128885	−	0.991660i	$-0.458860\pi$
$367$	4.93816	0.257770	0.128885	+	0.991660i	$0.458860\pi$
$368$	0	0
$369$	15.1778	0.790124
$370$	0	0
$371$	11.5860	0.601514
$372$	0	0
$373$	15.4596	0.800469	0.400234	−	0.916413i	$-0.368929\pi$
$373$	15.4596	0.800469	0.400234	+	0.916413i	$0.368929\pi$
$374$	0	0
$375$	−1.86928	−0.0965292
$376$	0	0
$377$	25.6011	1.31852
$378$	0	0
$379$	19.0983	0.981015	0.490507	−	0.871437i	$-0.336811\pi$
$379$	19.0983	0.981015	0.490507	+	0.871437i	$0.336811\pi$
$380$	0	0
$381$	3.02337	0.154892
$382$	0	0
$383$	10.7956	0.551632	0.275816	−	0.961210i	$-0.411052\pi$
$383$	10.7956	0.551632	0.275816	+	0.961210i	$0.411052\pi$
$384$	0	0
$385$	−52.5245	−2.67690
$386$	0	0
$387$	−0.0955794	−0.00485857
$388$	0	0
$389$	19.4666	0.986994	0.493497	−	0.869747i	$-0.335718\pi$
$389$	19.4666	0.986994	0.493497	+	0.869747i	$0.335718\pi$
$390$	0	0
$391$	25.4079	1.28493
$392$	0	0
$393$	19.5007	0.983681
$394$	0	0
$395$	30.0260	1.51077
$396$	0	0
$397$	−34.9087	−1.75202	−0.876009	−	0.482294i	$-0.839804\pi$
$397$	−34.9087	−1.75202	−0.876009	+	0.482294i	$0.839804\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−15.7225	−0.785145	−0.392572	−	0.919721i	$-0.628415\pi$
$401$	−15.7225	−0.785145	−0.392572	+	0.919721i	$0.628415\pi$
$402$	0	0
$403$	−15.6091	−0.777546
$404$	0	0
$405$	0.562129	0.0279324
$406$	0	0
$407$	−24.9577	−1.23711
$408$	0	0
$409$	16.5160	0.816665	0.408332	−	0.912833i	$-0.366110\pi$
$409$	16.5160	0.816665	0.408332	+	0.912833i	$0.366110\pi$
$410$	0	0
$411$	−8.37487	−0.413102
$412$	0	0
$413$	−16.1433	−0.794359
$414$	0	0
$415$	−44.6818	−2.19335
$416$	0	0
$417$	−10.8493	−0.531293
$418$	0	0
$419$	−32.2406	−1.57506	−0.787528	−	0.616279i	$-0.788639\pi$
$419$	−32.2406	−1.57506	−0.787528	+	0.616279i	$0.788639\pi$
$420$	0	0
$421$	−26.7306	−1.30277	−0.651384	−	0.758748i	$-0.725811\pi$
$421$	−26.7306	−1.30277	−0.651384	+	0.758748i	$0.725811\pi$
$422$	0	0
$423$	−0.131614	−0.00639930
$424$	0	0
$425$	−32.2042	−1.56214
$426$	0	0
$427$	−11.2305	−0.543480
$428$	0	0
$429$	17.1120	0.826177
$430$	0	0
$431$	28.4531	1.37054	0.685268	−	0.728291i	$-0.259685\pi$
$431$	28.4531	1.37054	0.685268	+	0.728291i	$0.259685\pi$
$432$	0	0
$433$	14.5323	0.698377	0.349189	−	0.937052i	$-0.386457\pi$
$433$	14.5323	0.698377	0.349189	+	0.937052i	$0.386457\pi$
$434$	0	0
$435$	−31.2454	−1.49810
$436$	0	0
$437$	0	0
$438$	0	0
$439$	22.6348	1.08030	0.540149	−	0.841569i	$-0.318368\pi$
$439$	22.6348	1.08030	0.540149	+	0.841569i	$0.318368\pi$
$440$	0	0
$441$	−1.76109	−0.0838615
$442$	0	0
$443$	11.9914	0.569728	0.284864	−	0.958568i	$-0.408052\pi$
$443$	11.9914	0.569728	0.284864	+	0.958568i	$0.408052\pi$
$444$	0	0
$445$	33.9344	1.60865
$446$	0	0
$447$	14.8735	0.703493
$448$	0	0
$449$	5.07083	0.239307	0.119654	−	0.992816i	$-0.461822\pi$
$449$	5.07083	0.239307	0.119654	+	0.992816i	$0.461822\pi$
$450$	0	0
$451$	46.3587	2.18295
$452$	0	0
$453$	−21.1319	−0.992861
$454$	0	0
$455$	25.7598	1.20764
$456$	0	0
$457$	−24.4719	−1.14475	−0.572375	−	0.819992i	$-0.693978\pi$
$457$	−24.4719	−1.14475	−0.572375	+	0.819992i	$0.693978\pi$
$458$	0	0
$459$	−30.0017	−1.40036
$460$	0	0
$461$	15.3965	0.717088	0.358544	−	0.933513i	$-0.383273\pi$
$461$	15.3965	0.717088	0.358544	+	0.933513i	$0.383273\pi$
$462$	0	0
$463$	−22.9681	−1.06742	−0.533708	−	0.845669i	$-0.679202\pi$
$463$	−22.9681	−1.06742	−0.533708	+	0.845669i	$0.679202\pi$
$464$	0	0
$465$	19.0505	0.883447
$466$	0	0
$467$	−34.7397	−1.60756	−0.803780	−	0.594926i	$-0.797181\pi$
$467$	−34.7397	−1.60756	−0.803780	+	0.594926i	$0.797181\pi$
$468$	0	0
$469$	−23.8110	−1.09949
$470$	0	0
$471$	3.90408	0.179890
$472$	0	0
$473$	−0.291936	−0.0134232
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−7.73077	−0.353967
$478$	0	0
$479$	9.76460	0.446156	0.223078	−	0.974801i	$-0.428390\pi$
$479$	9.76460	0.446156	0.223078	+	0.974801i	$0.428390\pi$
$480$	0	0
$481$	12.2401	0.558101
$482$	0	0
$483$	13.0419	0.593427
$484$	0	0
$485$	−44.6055	−2.02543
$486$	0	0
$487$	41.5764	1.88401	0.942003	−	0.335604i	$-0.108940\pi$
$487$	41.5764	1.88401	0.942003	+	0.335604i	$0.108940\pi$
$488$	0	0
$489$	−22.3878	−1.01241
$490$	0	0
$491$	−18.8922	−0.852591	−0.426295	−	0.904584i	$-0.640182\pi$
$491$	−18.8922	−0.852591	−0.426295	+	0.904584i	$0.640182\pi$
$492$	0	0
$493$	−52.8125	−2.37855
$494$	0	0
$495$	35.0471	1.57525
$496$	0	0
$497$	−26.4026	−1.18432
$498$	0	0
$499$	12.3404	0.552434	0.276217	−	0.961095i	$-0.410919\pi$
$499$	12.3404	0.552434	0.276217	+	0.961095i	$0.410919\pi$
$500$	0	0
$501$	12.6057	0.563180
$502$	0	0
$503$	−3.05058	−0.136019	−0.0680094	−	0.997685i	$-0.521665\pi$
$503$	−3.05058	−0.136019	−0.0680094	+	0.997685i	$0.521665\pi$
$504$	0	0
$505$	−54.2800	−2.41543
$506$	0	0
$507$	5.36674	0.238345
$508$	0	0
$509$	−10.7585	−0.476862	−0.238431	−	0.971159i	$-0.576633\pi$
$509$	−10.7585	−0.476862	−0.238431	+	0.971159i	$0.576633\pi$
$510$	0	0
$511$	−8.79424	−0.389034
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−19.9371	−0.878532
$516$	0	0
$517$	−0.402000	−0.0176799
$518$	0	0
$519$	1.26801	0.0556593
$520$	0	0
$521$	−13.5847	−0.595157	−0.297578	−	0.954697i	$-0.596179\pi$
$521$	−13.5847	−0.595157	−0.297578	+	0.954697i	$0.596179\pi$
$522$	0	0
$523$	−15.1776	−0.663672	−0.331836	−	0.943337i	$-0.607668\pi$
$523$	−15.1776	−0.663672	−0.331836	+	0.943337i	$0.607668\pi$
$524$	0	0
$525$	−16.5305	−0.721450
$526$	0	0
$527$	32.2001	1.40266
$528$	0	0
$529$	−3.86879	−0.168208
$530$	0	0
$531$	10.7716	0.467450
$532$	0	0
$533$	−22.7359	−0.984802
$534$	0	0
$535$	−12.1128	−0.523683
$536$	0	0
$537$	1.43352	0.0618610
$538$	0	0
$539$	−5.37904	−0.231692
$540$	0	0
$541$	10.6054	0.455963	0.227982	−	0.973665i	$-0.426787\pi$
$541$	10.6054	0.455963	0.227982	+	0.973665i	$0.426787\pi$
$542$	0	0
$543$	15.4469	0.662890
$544$	0	0
$545$	−15.2528	−0.653356
$546$	0	0
$547$	31.0842	1.32907	0.664533	−	0.747259i	$-0.268631\pi$
$547$	31.0842	1.32907	0.664533	+	0.747259i	$0.268631\pi$
$548$	0	0
$549$	7.49355	0.319817
$550$	0	0
$551$	0	0
$552$	0	0
$553$	26.0508	1.10779
$554$	0	0
$555$	−14.9387	−0.634114
$556$	0	0
$557$	20.0205	0.848298	0.424149	−	0.905592i	$-0.360573\pi$
$557$	20.0205	0.848298	0.424149	+	0.905592i	$0.360573\pi$
$558$	0	0
$559$	0.143175	0.00605568
$560$	0	0
$561$	−35.3004	−1.49039
$562$	0	0
$563$	−1.49920	−0.0631838	−0.0315919	−	0.999501i	$-0.510058\pi$
$563$	−1.49920	−0.0631838	−0.0315919	+	0.999501i	$0.510058\pi$
$564$	0	0
$565$	51.4050	2.16263
$566$	0	0
$567$	0.487707	0.0204818
$568$	0	0
$569$	−42.4569	−1.77989	−0.889944	−	0.456071i	$-0.849256\pi$
$569$	−42.4569	−1.77989	−0.889944	+	0.456071i	$0.849256\pi$
$570$	0	0
$571$	−5.09981	−0.213420	−0.106710	−	0.994290i	$-0.534032\pi$
$571$	−5.09981	−0.213420	−0.106710	+	0.994290i	$0.534032\pi$
$572$	0	0
$573$	−20.1376	−0.841262
$574$	0	0
$575$	−24.2486	−1.01124
$576$	0	0
$577$	−1.87153	−0.0779126	−0.0389563	−	0.999241i	$-0.512403\pi$
$577$	−1.87153	−0.0779126	−0.0389563	+	0.999241i	$0.512403\pi$
$578$	0	0
$579$	−4.93467	−0.205078
$580$	0	0
$581$	−38.7663	−1.60830
$582$	0	0
$583$	−23.6127	−0.977939
$584$	0	0
$585$	−17.1883	−0.710649
$586$	0	0
$587$	−17.9946	−0.742715	−0.371358	−	0.928490i	$-0.621108\pi$
$587$	−17.9946	−0.742715	−0.371358	+	0.928490i	$0.621108\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	16.1833	0.665692
$592$	0	0
$593$	11.4611	0.470652	0.235326	−	0.971917i	$-0.424384\pi$
$593$	11.4611	0.470652	0.235326	+	0.971917i	$0.424384\pi$
$594$	0	0
$595$	−53.1400	−2.17853
$596$	0	0
$597$	0.151706	0.00620890
$598$	0	0
$599$	29.0269	1.18601	0.593003	−	0.805200i	$-0.297942\pi$
$599$	29.0269	1.18601	0.593003	+	0.805200i	$0.297942\pi$
$600$	0	0
$601$	3.04702	0.124290	0.0621452	−	0.998067i	$-0.480206\pi$
$601$	3.04702	0.124290	0.0621452	+	0.998067i	$0.480206\pi$
$602$	0	0
$603$	15.8880	0.647008
$604$	0	0
$605$	71.3287	2.89992
$606$	0	0
$607$	6.52654	0.264904	0.132452	−	0.991189i	$-0.457715\pi$
$607$	6.52654	0.264904	0.132452	+	0.991189i	$0.457715\pi$
$608$	0	0
$609$	−27.1088	−1.09850
$610$	0	0
$611$	0.197155	0.00797603
$612$	0	0
$613$	−34.8632	−1.40811	−0.704055	−	0.710145i	$-0.748629\pi$
$613$	−34.8632	−1.40811	−0.704055	+	0.710145i	$0.748629\pi$
$614$	0	0
$615$	27.7486	1.11893
$616$	0	0
$617$	−27.0172	−1.08767	−0.543836	−	0.839192i	$-0.683029\pi$
$617$	−27.0172	−1.08767	−0.543836	+	0.839192i	$0.683029\pi$
$618$	0	0
$619$	−8.72746	−0.350786	−0.175393	−	0.984498i	$-0.556120\pi$
$619$	−8.72746	−0.350786	−0.175393	+	0.984498i	$0.556120\pi$
$620$	0	0
$621$	−22.5902	−0.906513
$622$	0	0
$623$	29.4417	1.17956
$624$	0	0
$625$	−21.9846	−0.879384
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−25.2502	−1.00679
$630$	0	0
$631$	−25.4746	−1.01413	−0.507063	−	0.861909i	$-0.669269\pi$
$631$	−25.4746	−1.01413	−0.507063	+	0.861909i	$0.669269\pi$
$632$	0	0
$633$	−21.5357	−0.855965
$634$	0	0
$635$	−9.27568	−0.368094
$636$	0	0
$637$	2.63807	0.104524
$638$	0	0
$639$	17.6172	0.696925
$640$	0	0
$641$	−21.6634	−0.855651	−0.427826	−	0.903861i	$-0.640720\pi$
$641$	−21.6634	−0.855651	−0.427826	+	0.903861i	$0.640720\pi$
$642$	0	0
$643$	5.26580	0.207663	0.103832	−	0.994595i	$-0.466890\pi$
$643$	5.26580	0.207663	0.103832	+	0.994595i	$0.466890\pi$
$644$	0	0
$645$	−0.174742	−0.00688045
$646$	0	0
$647$	0.104316	0.00410108	0.00205054	−	0.999998i	$-0.499347\pi$
$647$	0.104316	0.00410108	0.00205054	+	0.999998i	$0.499347\pi$
$648$	0	0
$649$	32.9007	1.29147
$650$	0	0
$651$	16.5284	0.647798
$652$	0	0
$653$	6.87399	0.269000	0.134500	−	0.990914i	$-0.457057\pi$
$653$	6.87399	0.269000	0.134500	+	0.990914i	$0.457057\pi$
$654$	0	0
$655$	−59.8281	−2.33768
$656$	0	0
$657$	5.86798	0.228932
$658$	0	0
$659$	21.9883	0.856541	0.428271	−	0.903650i	$-0.359123\pi$
$659$	21.9883	0.856541	0.428271	+	0.903650i	$0.359123\pi$
$660$	0	0
$661$	−5.66744	−0.220438	−0.110219	−	0.993907i	$-0.535155\pi$
$661$	−5.66744	−0.220438	−0.110219	+	0.993907i	$0.535155\pi$
$662$	0	0
$663$	17.3126	0.672364
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−39.7659	−1.53974
$668$	0	0
$669$	25.3111	0.978584
$670$	0	0
$671$	22.8882	0.883587
$672$	0	0
$673$	−50.4534	−1.94483	−0.972417	−	0.233248i	$-0.925065\pi$
$673$	−50.4534	−1.94483	−0.972417	+	0.233248i	$0.925065\pi$
$674$	0	0
$675$	28.6329	1.10208
$676$	0	0
$677$	−31.1360	−1.19665	−0.598327	−	0.801252i	$-0.704168\pi$
$677$	−31.1360	−1.19665	−0.598327	+	0.801252i	$0.704168\pi$
$678$	0	0
$679$	−38.7000	−1.48517
$680$	0	0
$681$	24.4144	0.935561
$682$	0	0
$683$	28.2048	1.07923	0.539613	−	0.841913i	$-0.318570\pi$
$683$	28.2048	1.07923	0.539613	+	0.841913i	$0.318570\pi$
$684$	0	0
$685$	25.6941	0.981720
$686$	0	0
$687$	−4.87508	−0.185996
$688$	0	0
$689$	11.5805	0.441182
$690$	0	0
$691$	21.1350	0.804015	0.402007	−	0.915636i	$-0.368313\pi$
$691$	21.1350	0.804015	0.402007	+	0.915636i	$0.368313\pi$
$692$	0	0
$693$	30.4071	1.15507
$694$	0	0
$695$	33.2856	1.26260
$696$	0	0
$697$	46.9020	1.77654
$698$	0	0
$699$	7.98085	0.301864
$700$	0	0
$701$	−12.8076	−0.483737	−0.241868	−	0.970309i	$-0.577760\pi$
$701$	−12.8076	−0.483737	−0.241868	+	0.970309i	$0.577760\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	−0.240622	−0.00906235
$706$	0	0
$707$	−47.0937	−1.77114
$708$	0	0
$709$	−35.8039	−1.34464	−0.672321	−	0.740259i	$-0.734703\pi$
$709$	−35.8039	−1.34464	−0.672321	+	0.740259i	$0.734703\pi$
$710$	0	0
$711$	−17.3825	−0.651893
$712$	0	0
$713$	24.2455	0.908002
$714$	0	0
$715$	−52.4997	−1.96338
$716$	0	0
$717$	8.21851	0.306926
$718$	0	0
$719$	29.7841	1.11076	0.555380	−	0.831597i	$-0.312573\pi$
$719$	29.7841	1.11076	0.555380	+	0.831597i	$0.312573\pi$
$720$	0	0
$721$	−17.2975	−0.644194
$722$	0	0
$723$	1.11982	0.0416466
$724$	0	0
$725$	50.4030	1.87192
$726$	0	0
$727$	41.3688	1.53429	0.767143	−	0.641477i	$-0.221678\pi$
$727$	41.3688	1.53429	0.767143	+	0.641477i	$0.221678\pi$
$728$	0	0
$729$	16.0735	0.595316
$730$	0	0
$731$	−0.295357	−0.0109242
$732$	0	0
$733$	21.2570	0.785144	0.392572	−	0.919721i	$-0.371585\pi$
$733$	21.2570	0.785144	0.392572	+	0.919721i	$0.371585\pi$
$734$	0	0
$735$	−3.21969	−0.118760
$736$	0	0
$737$	48.5279	1.78755
$738$	0	0
$739$	32.3222	1.18899	0.594496	−	0.804099i	$-0.297352\pi$
$739$	32.3222	1.18899	0.594496	+	0.804099i	$0.297352\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	11.7281	0.430262	0.215131	−	0.976585i	$-0.430982\pi$
$743$	11.7281	0.430262	0.215131	+	0.976585i	$0.430982\pi$
$744$	0	0
$745$	−45.6319	−1.67182
$746$	0	0
$747$	25.8669	0.946420
$748$	0	0
$749$	−10.5092	−0.383997
$750$	0	0
$751$	−26.6003	−0.970659	−0.485330	−	0.874331i	$-0.661300\pi$
$751$	−26.6003	−0.970659	−0.485330	+	0.874331i	$0.661300\pi$
$752$	0	0
$753$	−11.1370	−0.405854
$754$	0	0
$755$	64.8325	2.35950
$756$	0	0
$757$	10.7141	0.389411	0.194705	−	0.980862i	$-0.437625\pi$
$757$	10.7141	0.389411	0.194705	+	0.980862i	$0.437625\pi$
$758$	0	0
$759$	−26.5800	−0.964791
$760$	0	0
$761$	39.7224	1.43994	0.719968	−	0.694007i	$-0.244156\pi$
$761$	39.7224	1.43994	0.719968	+	0.694007i	$0.244156\pi$
$762$	0	0
$763$	−13.2334	−0.479081
$764$	0	0
$765$	35.4578	1.28198
$766$	0	0
$767$	−16.1357	−0.582625
$768$	0	0
$769$	20.0629	0.723485	0.361743	−	0.932278i	$-0.382182\pi$
$769$	20.0629	0.723485	0.361743	+	0.932278i	$0.382182\pi$
$770$	0	0
$771$	−21.1870	−0.763030
$772$	0	0
$773$	−22.4609	−0.807861	−0.403931	−	0.914790i	$-0.632356\pi$
$773$	−22.4609	−0.807861	−0.403931	+	0.914790i	$0.632356\pi$
$774$	0	0
$775$	−30.7310	−1.10389
$776$	0	0
$777$	−12.9610	−0.464972
$778$	0	0
$779$	0	0
$780$	0	0
$781$	53.8096	1.92546
$782$	0	0
$783$	46.9557	1.67806
$784$	0	0
$785$	−11.9777	−0.427503
$786$	0	0
$787$	−18.4292	−0.656929	−0.328465	−	0.944516i	$-0.606531\pi$
$787$	−18.4292	−0.656929	−0.328465	+	0.944516i	$0.606531\pi$
$788$	0	0
$789$	3.31765	0.118112
$790$	0	0
$791$	44.5994	1.58577
$792$	0	0
$793$	−11.2251	−0.398617
$794$	0	0
$795$	−14.1337	−0.501270
$796$	0	0
$797$	27.7727	0.983758	0.491879	−	0.870664i	$-0.336310\pi$
$797$	27.7727	0.983758	0.491879	+	0.870664i	$0.336310\pi$
$798$	0	0
$799$	−0.406711	−0.0143884
$800$	0	0
$801$	−19.6451	−0.694124
$802$	0	0
$803$	17.9230	0.632490
$804$	0	0
$805$	−40.0125	−1.41026
$806$	0	0
$807$	4.80783	0.169243
$808$	0	0
$809$	21.0668	0.740668	0.370334	−	0.928899i	$-0.379243\pi$
$809$	21.0668	0.740668	0.370334	+	0.928899i	$0.379243\pi$
$810$	0	0
$811$	34.9594	1.22759	0.613796	−	0.789465i	$-0.289642\pi$
$811$	34.9594	1.22759	0.613796	+	0.789465i	$0.289642\pi$
$812$	0	0
$813$	11.4154	0.400354
$814$	0	0
$815$	68.6858	2.40596
$816$	0	0
$817$	0	0
$818$	0	0
$819$	−14.9127	−0.521092
$820$	0	0
$821$	−29.8357	−1.04127	−0.520636	−	0.853779i	$-0.674305\pi$
$821$	−29.8357	−1.04127	−0.520636	+	0.853779i	$0.674305\pi$
$822$	0	0
$823$	4.47073	0.155840	0.0779200	−	0.996960i	$-0.475172\pi$
$823$	4.47073	0.155840	0.0779200	+	0.996960i	$0.475172\pi$
$824$	0	0
$825$	33.6899	1.17293
$826$	0	0
$827$	33.8135	1.17581	0.587905	−	0.808930i	$-0.299953\pi$
$827$	33.8135	1.17581	0.587905	+	0.808930i	$0.299953\pi$
$828$	0	0
$829$	0.0262614	0.000912096 0	0.000456048	−	1.00000i	$-0.499855\pi$
$829$	0.0262614	0.000912096 0	0.000456048		1.00000i	$0.499855\pi$
$830$	0	0
$831$	−8.25334	−0.286305
$832$	0	0
$833$	−5.44208	−0.188557
$834$	0	0
$835$	−38.6742	−1.33838
$836$	0	0
$837$	−28.6292	−0.989570
$838$	0	0
$839$	27.3456	0.944074	0.472037	−	0.881579i	$-0.343519\pi$
$839$	27.3456	0.944074	0.472037	+	0.881579i	$0.343519\pi$
$840$	0	0
$841$	53.6570	1.85024
$842$	0	0
$843$	27.5462	0.948740
$844$	0	0
$845$	−16.4652	−0.566418
$846$	0	0
$847$	61.8853	2.12640
$848$	0	0
$849$	−3.29777	−0.113179
$850$	0	0
$851$	−19.0125	−0.651739
$852$	0	0
$853$	−21.2406	−0.727264	−0.363632	−	0.931543i	$-0.618463\pi$
$853$	−21.2406	−0.727264	−0.363632	+	0.931543i	$0.618463\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	3.12677	0.106809	0.0534043	−	0.998573i	$-0.482993\pi$
$857$	3.12677	0.106809	0.0534043	+	0.998573i	$0.482993\pi$
$858$	0	0
$859$	−37.6878	−1.28589	−0.642946	−	0.765912i	$-0.722288\pi$
$859$	−37.6878	−1.28589	−0.642946	+	0.765912i	$0.722288\pi$
$860$	0	0
$861$	24.0749	0.820470
$862$	0	0
$863$	18.2687	0.621873	0.310937	−	0.950431i	$-0.399357\pi$
$863$	18.2687	0.621873	0.310937	+	0.950431i	$0.399357\pi$
$864$	0	0
$865$	−3.89024	−0.132272
$866$	0	0
$867$	−17.7215	−0.601852
$868$	0	0
$869$	−53.0927	−1.80105
$870$	0	0
$871$	−23.7998	−0.806425
$872$	0	0
$873$	25.8227	0.873966
$874$	0	0
$875$	4.97568	0.168209
$876$	0	0
$877$	36.8810	1.24538	0.622691	−	0.782468i	$-0.286039\pi$
$877$	36.8810	1.24538	0.622691	+	0.782468i	$0.286039\pi$
$878$	0	0
$879$	5.34411	0.180252
$880$	0	0
$881$	−30.7506	−1.03601	−0.518007	−	0.855376i	$-0.673326\pi$
$881$	−30.7506	−1.03601	−0.518007	+	0.855376i	$0.673326\pi$
$882$	0	0
$883$	−7.04420	−0.237056	−0.118528	−	0.992951i	$-0.537818\pi$
$883$	−7.04420	−0.237056	−0.118528	+	0.992951i	$0.537818\pi$
$884$	0	0
$885$	19.6931	0.661977
$886$	0	0
$887$	−39.4527	−1.32469	−0.662346	−	0.749198i	$-0.730439\pi$
$887$	−39.4527	−1.32469	−0.662346	+	0.749198i	$0.730439\pi$
$888$	0	0
$889$	−8.04765	−0.269909
$890$	0	0
$891$	−0.993967	−0.0332992
$892$	0	0
$893$	0	0
$894$	0	0
$895$	−4.39804	−0.147010
$896$	0	0
$897$	13.0357	0.435251
$898$	0	0
$899$	−50.3965	−1.68082
$900$	0	0
$901$	−23.8894	−0.795872
$902$	0	0
$903$	−0.151607	−0.00504517
$904$	0	0
$905$	−47.3910	−1.57533
$906$	0	0
$907$	5.30246	0.176065	0.0880326	−	0.996118i	$-0.471942\pi$
$907$	5.30246	0.176065	0.0880326	+	0.996118i	$0.471942\pi$
$908$	0	0
$909$	31.4234	1.04225
$910$	0	0
$911$	−44.6890	−1.48061	−0.740306	−	0.672270i	$-0.765319\pi$
$911$	−44.6890	−1.48061	−0.740306	+	0.672270i	$0.765319\pi$
$912$	0	0
$913$	79.0073	2.61476
$914$	0	0
$915$	13.7000	0.452908
$916$	0	0
$917$	−51.9073	−1.71413
$918$	0	0
$919$	−2.44156	−0.0805395	−0.0402698	−	0.999189i	$-0.512822\pi$
$919$	−2.44156	−0.0805395	−0.0402698	+	0.999189i	$0.512822\pi$
$920$	0	0
$921$	21.5459	0.709962
$922$	0	0
$923$	−26.3901	−0.868641
$924$	0	0
$925$	24.0981	0.792342
$926$	0	0
$927$	11.5418	0.379083
$928$	0	0
$929$	54.5996	1.79135	0.895677	−	0.444704i	$-0.146691\pi$
$929$	54.5996	1.79135	0.895677	+	0.444704i	$0.146691\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−12.9518	−0.424023
$934$	0	0
$935$	108.302	3.54184
$936$	0	0
$937$	12.1403	0.396606	0.198303	−	0.980141i	$-0.436457\pi$
$937$	12.1403	0.396606	0.198303	+	0.980141i	$0.436457\pi$
$938$	0	0
$939$	5.34034	0.174275
$940$	0	0
$941$	25.0301	0.815958	0.407979	−	0.912991i	$-0.366234\pi$
$941$	25.0301	0.815958	0.407979	+	0.912991i	$0.366234\pi$
$942$	0	0
$943$	35.3155	1.15003
$944$	0	0
$945$	47.2469	1.53694
$946$	0	0
$947$	56.2921	1.82925	0.914623	−	0.404307i	$-0.132487\pi$
$947$	56.2921	1.82925	0.914623	+	0.404307i	$0.132487\pi$
$948$	0	0
$949$	−8.79008	−0.285338
$950$	0	0
$951$	−2.17356	−0.0704826
$952$	0	0
$953$	26.0244	0.843014	0.421507	−	0.906825i	$-0.361501\pi$
$953$	26.0244	0.843014	0.421507	+	0.906825i	$0.361501\pi$
$954$	0	0
$955$	61.7822	1.99923
$956$	0	0
$957$	55.2488	1.78594
$958$	0	0
$959$	22.2924	0.719858
$960$	0	0
$961$	−0.272956	−0.00880504
$962$	0	0
$963$	7.01227	0.225967
$964$	0	0
$965$	15.1396	0.487360
$966$	0	0
$967$	24.2836	0.780907	0.390454	−	0.920623i	$-0.372318\pi$
$967$	24.2836	0.780907	0.390454	+	0.920623i	$0.372318\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−7.23140	−0.232067	−0.116033	−	0.993245i	$-0.537018\pi$
$971$	−7.23140	−0.232067	−0.116033	+	0.993245i	$0.537018\pi$
$972$	0	0
$973$	28.8788	0.925813
$974$	0	0
$975$	−16.5227	−0.529149
$976$	0	0
$977$	47.0692	1.50588	0.752939	−	0.658091i	$-0.228636\pi$
$977$	47.0692	1.50588	0.752939	+	0.658091i	$0.228636\pi$
$978$	0	0
$979$	−60.0035	−1.91772
$980$	0	0
$981$	8.83002	0.281921
$982$	0	0
$983$	−53.7899	−1.71563	−0.857816	−	0.513957i	$-0.828179\pi$
$983$	−53.7899	−1.71563	−0.857816	+	0.513957i	$0.828179\pi$
$984$	0	0
$985$	−49.6503	−1.58199
$986$	0	0
$987$	−0.208765	−0.00664508
$988$	0	0
$989$	−0.222393	−0.00707169
$990$	0	0
$991$	−17.5681	−0.558069	−0.279035	−	0.960281i	$-0.590014\pi$
$991$	−17.5681	−0.558069	−0.279035	+	0.960281i	$0.590014\pi$
$992$	0	0
$993$	−2.56283	−0.0813289
$994$	0	0
$995$	−0.465432	−0.0147552
$996$	0	0
$997$	25.4571	0.806236	0.403118	−	0.915148i	$-0.367926\pi$
$997$	25.4571	0.806236	0.403118	+	0.915148i	$0.367926\pi$
$998$	0	0
$999$	22.4500	0.710286

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5776.2.a.ca.1.4		8
4.3	odd	2		2888.2.a.w.1.5	yes	8
19.18	odd	2		5776.2.a.cc.1.5		8
76.75	even	2		2888.2.a.v.1.4	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2888.2.a.v.1.4	✓	8	76.75	even	2
2888.2.a.w.1.5	yes	8	4.3	odd	2
5776.2.a.ca.1.4		8	1.1	even	1	trivial
5776.2.a.cc.1.5		8	19.18	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 5776 = 2^{4} \cdot 19^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5776.a (trivial)

\(f(q)\)	\(=\)	\(q-1.05839 q^{3} +3.24714 q^{5} +2.81724 q^{7} -1.87981 q^{9} -5.74166 q^{11} +2.81591 q^{13} -3.43674 q^{15} -5.80894 q^{17} -2.98174 q^{21} -4.37392 q^{23} +5.54391 q^{25} +5.16474 q^{27} +9.09159 q^{29} -5.54320 q^{31} +6.07691 q^{33} +9.14797 q^{35} +4.34678 q^{37} -2.98033 q^{39} -8.07410 q^{41} +0.0508452 q^{43} -6.10400 q^{45} +0.0700146 q^{47} +0.936845 q^{49} +6.14813 q^{51} +4.11253 q^{53} -18.6440 q^{55} -5.73018 q^{59} -3.98633 q^{61} -5.29588 q^{63} +9.14364 q^{65} -8.45190 q^{67} +4.62932 q^{69} -9.37179 q^{71} -3.12158 q^{73} -5.86762 q^{75} -16.1756 q^{77} +9.24692 q^{79} +0.173115 q^{81} -13.7604 q^{83} -18.8624 q^{85} -9.62245 q^{87} +10.4506 q^{89} +7.93309 q^{91} +5.86687 q^{93} -13.7369 q^{97} +10.7932 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 6 q^{3} - 2 q^{5} + 2 q^{7} + 2 q^{9} - 12 q^{11} + 6 q^{13} - 10 q^{17} - 4 q^{21} + 12 q^{23} + 2 q^{25} - 12 q^{27} + 18 q^{29} - 14 q^{31} + 40 q^{33} - 18 q^{35} + 16 q^{37} - 28 q^{39} + 12 q^{41}+ \cdots - 32 q^{99}+O(q^{100}) \) 8 * q - 6 * q^3 - 2 * q^5 + 2 * q^7 + 2 * q^9 - 12 * q^11 + 6 * q^13 - 10 * q^17 - 4 * q^21 + 12 * q^23 + 2 * q^25 - 12 * q^27 + 18 * q^29 - 14 * q^31 + 40 * q^33 - 18 * q^35 + 16 * q^37 - 28 * q^39 + 12 * q^41 - 8 * q^43 - 2 * q^45 - 24 * q^47 + 12 * q^49 + 22 * q^51 + 12 * q^53 + 2 * q^55 + 16 * q^59 - 12 * q^63 + 12 * q^65 - 42 * q^67 - 12 * q^69 - 36 * q^71 - 10 * q^73 - 32 * q^75 - 38 * q^77 - 6 * q^79 + 4 * q^81 - 12 * q^83 - 42 * q^85 + 12 * q^87 + 8 * q^89 - 16 * q^91 - 14 * q^93 + 28 * q^97 - 32 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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