LMFDB - Embedded newform 1088.2.a.v.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1088,2,Mod(1,1088)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1088, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("1088.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1088 = 2^{6} \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1088.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:	$8.68772373992$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.148.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 3x + 1 $ x^3 - x^2 - 3*x + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 544)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.48119$ of defining polynomial
Character	$\chi$	$=$	1088.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.806063 q^{3} -2.96239 q^{5} -4.15633 q^{7} -2.35026 q^{9} +O(q^{10})$	$q+0.806063 q^{3} -2.96239 q^{5} -4.15633 q^{7} -2.35026 q^{9} +3.19394 q^{11} +5.35026 q^{13} -2.38787 q^{15} +1.00000 q^{17} +1.61213 q^{19} -3.35026 q^{21} +8.46898 q^{23} +3.77575 q^{25} -4.31265 q^{27} +6.96239 q^{29} -6.54420 q^{31} +2.57452 q^{33} +12.3127 q^{35} +6.96239 q^{37} +4.31265 q^{39} +8.70052 q^{41} +4.31265 q^{43} +6.96239 q^{45} -5.92478 q^{47} +10.2750 q^{49} +0.806063 q^{51} -4.70052 q^{53} -9.46168 q^{55} +1.29948 q^{57} -7.53690 q^{59} +10.1866 q^{61} +9.76845 q^{63} -15.8496 q^{65} -3.22425 q^{67} +6.82653 q^{69} -13.7685 q^{71} +5.22425 q^{73} +3.04349 q^{75} -13.2750 q^{77} +13.2447 q^{79} +3.57452 q^{81} -4.31265 q^{83} -2.96239 q^{85} +5.61213 q^{87} -7.27504 q^{89} -22.2374 q^{91} -5.27504 q^{93} -4.77575 q^{95} -2.77575 q^{97} -7.50659 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 2 q^{3} + 2 q^{5} - 2 q^{7} + 3 q^{9}+O(q^{10}) $ 3 * q + 2 * q^3 + 2 * q^5 - 2 * q^7 + 3 * q^9	$ 3 q + 2 q^{3} + 2 q^{5} - 2 q^{7} + 3 q^{9} + 10 q^{11} + 6 q^{13} - 8 q^{15} + 3 q^{17} + 4 q^{19} - 6 q^{23} + 13 q^{25} + 8 q^{27} + 10 q^{29} - 10 q^{31} - 4 q^{33} + 16 q^{35} + 10 q^{37} - 8 q^{39} + 6 q^{41} - 8 q^{43} + 10 q^{45} + 4 q^{47} - q^{49} + 2 q^{51} + 6 q^{53} + 16 q^{55} + 24 q^{57} + 18 q^{61} + 18 q^{63} - 4 q^{65} - 8 q^{67} - 8 q^{69} - 30 q^{71} + 14 q^{73} - 34 q^{75} - 8 q^{77} + 10 q^{79} - q^{81} + 8 q^{83} + 2 q^{85} + 16 q^{87} + 10 q^{89} - 24 q^{91} + 16 q^{93} - 16 q^{95} - 10 q^{97} - 2 q^{99}+O(q^{100}) $ 3 * q + 2 * q^3 + 2 * q^5 - 2 * q^7 + 3 * q^9 + 10 * q^11 + 6 * q^13 - 8 * q^15 + 3 * q^17 + 4 * q^19 - 6 * q^23 + 13 * q^25 + 8 * q^27 + 10 * q^29 - 10 * q^31 - 4 * q^33 + 16 * q^35 + 10 * q^37 - 8 * q^39 + 6 * q^41 - 8 * q^43 + 10 * q^45 + 4 * q^47 - q^49 + 2 * q^51 + 6 * q^53 + 16 * q^55 + 24 * q^57 + 18 * q^61 + 18 * q^63 - 4 * q^65 - 8 * q^67 - 8 * q^69 - 30 * q^71 + 14 * q^73 - 34 * q^75 - 8 * q^77 + 10 * q^79 - q^81 + 8 * q^83 + 2 * q^85 + 16 * q^87 + 10 * q^89 - 24 * q^91 + 16 * q^93 - 16 * q^95 - 10 * q^97 - 2 * 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
$3$	0.806063	0.465381	0.232690	−	0.972551i	$-0.425247\pi$
$3$	0.806063	0.465381	0.232690	+	0.972551i	$0.425247\pi$
$4$	0	0
$5$	−2.96239	−1.32482	−0.662410	−	0.749141i	$-0.730466\pi$
$5$	−2.96239	−1.32482	−0.662410	+	0.749141i	$0.730466\pi$
$6$	0	0
$7$	−4.15633	−1.57094	−0.785472	−	0.618898i	$-0.787580\pi$
$7$	−4.15633	−1.57094	−0.785472	+	0.618898i	$0.787580\pi$
$8$	0	0
$9$	−2.35026	−0.783421
$10$	0	0
$11$	3.19394	0.963008	0.481504	−	0.876444i	$-0.340091\pi$
$11$	3.19394	0.963008	0.481504	+	0.876444i	$0.340091\pi$
$12$	0	0
$13$	5.35026	1.48390	0.741948	−	0.670458i	$-0.233902\pi$
$13$	5.35026	1.48390	0.741948	+	0.670458i	$0.233902\pi$
$14$	0	0
$15$	−2.38787	−0.616546
$16$	0	0
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	0	0
$19$	1.61213	0.369847	0.184924	−	0.982753i	$-0.440796\pi$
$19$	1.61213	0.369847	0.184924	+	0.982753i	$0.440796\pi$
$20$	0	0
$21$	−3.35026	−0.731087
$22$	0	0
$23$	8.46898	1.76590	0.882952	−	0.469464i	$-0.155553\pi$
$23$	8.46898	1.76590	0.882952	+	0.469464i	$0.155553\pi$
$24$	0	0
$25$	3.77575	0.755149
$26$	0	0
$27$	−4.31265	−0.829970
$28$	0	0
$29$	6.96239	1.29288	0.646442	−	0.762964i	$-0.276256\pi$
$29$	6.96239	1.29288	0.646442	+	0.762964i	$0.276256\pi$
$30$	0	0
$31$	−6.54420	−1.17537	−0.587686	−	0.809089i	$-0.699961\pi$
$31$	−6.54420	−1.17537	−0.587686	+	0.809089i	$0.699961\pi$
$32$	0	0
$33$	2.57452	0.448166
$34$	0	0
$35$	12.3127	2.08122
$36$	0	0
$37$	6.96239	1.14461	0.572305	−	0.820041i	$-0.306049\pi$
$37$	6.96239	1.14461	0.572305	+	0.820041i	$0.306049\pi$
$38$	0	0
$39$	4.31265	0.690577
$40$	0	0
$41$	8.70052	1.35879	0.679397	−	0.733771i	$-0.262242\pi$
$41$	8.70052	1.35879	0.679397	+	0.733771i	$0.262242\pi$
$42$	0	0
$43$	4.31265	0.657673	0.328837	−	0.944387i	$-0.393343\pi$
$43$	4.31265	0.657673	0.328837	+	0.944387i	$0.393343\pi$
$44$	0	0
$45$	6.96239	1.03789
$46$	0	0
$47$	−5.92478	−0.864218	−0.432109	−	0.901821i	$-0.642230\pi$
$47$	−5.92478	−0.864218	−0.432109	+	0.901821i	$0.642230\pi$
$48$	0	0
$49$	10.2750	1.46786
$50$	0	0
$51$	0.806063	0.112871
$52$	0	0
$53$	−4.70052	−0.645667	−0.322833	−	0.946456i	$-0.604635\pi$
$53$	−4.70052	−0.645667	−0.322833	+	0.946456i	$0.604635\pi$
$54$	0	0
$55$	−9.46168	−1.27581
$56$	0	0
$57$	1.29948	0.172120
$58$	0	0
$59$	−7.53690	−0.981221	−0.490611	−	0.871379i	$-0.663226\pi$
$59$	−7.53690	−0.981221	−0.490611	+	0.871379i	$0.663226\pi$
$60$	0	0
$61$	10.1866	1.30427	0.652133	−	0.758105i	$-0.273874\pi$
$61$	10.1866	1.30427	0.652133	+	0.758105i	$0.273874\pi$
$62$	0	0
$63$	9.76845	1.23071
$64$	0	0
$65$	−15.8496	−1.96590
$66$	0	0
$67$	−3.22425	−0.393905	−0.196953	−	0.980413i	$-0.563105\pi$
$67$	−3.22425	−0.393905	−0.196953	+	0.980413i	$0.563105\pi$
$68$	0	0
$69$	6.82653	0.821818
$70$	0	0
$71$	−13.7685	−1.63401	−0.817007	−	0.576627i	$-0.804368\pi$
$71$	−13.7685	−1.63401	−0.817007	+	0.576627i	$0.804368\pi$
$72$	0	0
$73$	5.22425	0.611453	0.305726	−	0.952119i	$-0.401101\pi$
$73$	5.22425	0.611453	0.305726	+	0.952119i	$0.401101\pi$
$74$	0	0
$75$	3.04349	0.351432
$76$	0	0
$77$	−13.2750	−1.51283
$78$	0	0
$79$	13.2447	1.49015	0.745074	−	0.666982i	$-0.232414\pi$
$79$	13.2447	1.49015	0.745074	+	0.666982i	$0.232414\pi$
$80$	0	0
$81$	3.57452	0.397168
$82$	0	0
$83$	−4.31265	−0.473375	−0.236687	−	0.971586i	$-0.576062\pi$
$83$	−4.31265	−0.473375	−0.236687	+	0.971586i	$0.576062\pi$
$84$	0	0
$85$	−2.96239	−0.321316
$86$	0	0
$87$	5.61213	0.601683
$88$	0	0
$89$	−7.27504	−0.771153	−0.385576	−	0.922676i	$-0.625997\pi$
$89$	−7.27504	−0.771153	−0.385576	+	0.922676i	$0.625997\pi$
$90$	0	0
$91$	−22.2374	−2.33112
$92$	0	0
$93$	−5.27504	−0.546996
$94$	0	0
$95$	−4.77575	−0.489981
$96$	0	0
$97$	−2.77575	−0.281834	−0.140917	−	0.990021i	$-0.545005\pi$
$97$	−2.77575	−0.281834	−0.140917	+	0.990021i	$0.545005\pi$
$98$	0	0
$99$	−7.50659	−0.754440
$100$	0	0
$101$	−2.64974	−0.263659	−0.131829	−	0.991272i	$-0.542085\pi$
$101$	−2.64974	−0.263659	−0.131829	+	0.991272i	$0.542085\pi$
$102$	0	0
$103$	0	0		−	1.00000i	$-0.5\pi$
$103$	0	0			1.00000i	$0.5\pi$
$104$	0	0
$105$	9.92478	0.968559
$106$	0	0
$107$	−4.34297	−0.419851	−0.209925	−	0.977717i	$-0.567322\pi$
$107$	−4.34297	−0.419851	−0.209925	+	0.977717i	$0.567322\pi$
$108$	0	0
$109$	13.6629	1.30867	0.654335	−	0.756205i	$-0.272949\pi$
$109$	13.6629	1.30867	0.654335	+	0.756205i	$0.272949\pi$
$110$	0	0
$111$	5.61213	0.532680
$112$	0	0
$113$	5.47627	0.515164	0.257582	−	0.966256i	$-0.417074\pi$
$113$	5.47627	0.515164	0.257582	+	0.966256i	$0.417074\pi$
$114$	0	0
$115$	−25.0884	−2.33951
$116$	0	0
$117$	−12.5745	−1.16251
$118$	0	0
$119$	−4.15633	−0.381010
$120$	0	0
$121$	−0.798769	−0.0726154
$122$	0	0
$123$	7.01317	0.632357
$124$	0	0
$125$	3.62672	0.324383
$126$	0	0
$127$	16.3127	1.44751	0.723757	−	0.690055i	$-0.242414\pi$
$127$	16.3127	1.44751	0.723757	+	0.690055i	$0.242414\pi$
$128$	0	0
$129$	3.47627	0.306068
$130$	0	0
$131$	10.7308	0.937558	0.468779	−	0.883316i	$-0.344694\pi$
$131$	10.7308	0.937558	0.468779	+	0.883316i	$0.344694\pi$
$132$	0	0
$133$	−6.70052	−0.581009
$134$	0	0
$135$	12.7757	1.09956
$136$	0	0
$137$	7.79877	0.666294	0.333147	−	0.942875i	$-0.391889\pi$
$137$	7.79877	0.666294	0.333147	+	0.942875i	$0.391889\pi$
$138$	0	0
$139$	18.2071	1.54431	0.772153	−	0.635436i	$-0.219180\pi$
$139$	18.2071	1.54431	0.772153	+	0.635436i	$0.219180\pi$
$140$	0	0
$141$	−4.77575	−0.402190
$142$	0	0
$143$	17.0884	1.42900
$144$	0	0
$145$	−20.6253	−1.71284
$146$	0	0
$147$	8.28233	0.683115
$148$	0	0
$149$	5.84955	0.479214	0.239607	−	0.970870i	$-0.422981\pi$
$149$	5.84955	0.479214	0.239607	+	0.970870i	$0.422981\pi$
$150$	0	0
$151$	3.53690	0.287829	0.143915	−	0.989590i	$-0.454031\pi$
$151$	3.53690	0.287829	0.143915	+	0.989590i	$0.454031\pi$
$152$	0	0
$153$	−2.35026	−0.190007
$154$	0	0
$155$	19.3865	1.55716
$156$	0	0
$157$	−6.62530	−0.528757	−0.264378	−	0.964419i	$-0.585167\pi$
$157$	−6.62530	−0.528757	−0.264378	+	0.964419i	$0.585167\pi$
$158$	0	0
$159$	−3.78892	−0.300481
$160$	0	0
$161$	−35.1998	−2.77413
$162$	0	0
$163$	−3.50659	−0.274657	−0.137329	−	0.990526i	$-0.543852\pi$
$163$	−3.50659	−0.274657	−0.137329	+	0.990526i	$0.543852\pi$
$164$	0	0
$165$	−7.62672	−0.593739
$166$	0	0
$167$	−7.06793	−0.546933	−0.273466	−	0.961882i	$-0.588170\pi$
$167$	−7.06793	−0.546933	−0.273466	+	0.961882i	$0.588170\pi$
$168$	0	0
$169$	15.6253	1.20195
$170$	0	0
$171$	−3.78892	−0.289746
$172$	0	0
$173$	2.18664	0.166247	0.0831237	−	0.996539i	$-0.473510\pi$
$173$	2.18664	0.166247	0.0831237	+	0.996539i	$0.473510\pi$
$174$	0	0
$175$	−15.6932	−1.18630
$176$	0	0
$177$	−6.07522	−0.456642
$178$	0	0
$179$	4.83638	0.361488	0.180744	−	0.983530i	$-0.442149\pi$
$179$	4.83638	0.361488	0.180744	+	0.983530i	$0.442149\pi$
$180$	0	0
$181$	2.18664	0.162532	0.0812659	−	0.996692i	$-0.474104\pi$
$181$	2.18664	0.162532	0.0812659	+	0.996692i	$0.474104\pi$
$182$	0	0
$183$	8.21108	0.606980
$184$	0	0
$185$	−20.6253	−1.51640
$186$	0	0
$187$	3.19394	0.233564
$188$	0	0
$189$	17.9248	1.30384
$190$	0	0
$191$	−2.07522	−0.150158	−0.0750789	−	0.997178i	$-0.523921\pi$
$191$	−2.07522	−0.150158	−0.0750789	+	0.997178i	$0.523921\pi$
$192$	0	0
$193$	3.29948	0.237502	0.118751	−	0.992924i	$-0.462111\pi$
$193$	3.29948	0.237502	0.118751	+	0.992924i	$0.462111\pi$
$194$	0	0
$195$	−12.7757	−0.914890
$196$	0	0
$197$	8.88717	0.633184	0.316592	−	0.948562i	$-0.397461\pi$
$197$	8.88717	0.633184	0.316592	+	0.948562i	$0.397461\pi$
$198$	0	0
$199$	6.60483	0.468204	0.234102	−	0.972212i	$-0.424785\pi$
$199$	6.60483	0.468204	0.234102	+	0.972212i	$0.424785\pi$
$200$	0	0
$201$	−2.59895	−0.183316
$202$	0	0
$203$	−28.9380	−2.03105
$204$	0	0
$205$	−25.7743	−1.80016
$206$	0	0
$207$	−19.9043	−1.38345
$208$	0	0
$209$	5.14903	0.356166
$210$	0	0
$211$	−10.7308	−0.738742	−0.369371	−	0.929282i	$-0.620427\pi$
$211$	−10.7308	−0.738742	−0.369371	+	0.929282i	$0.620427\pi$
$212$	0	0
$213$	−11.0982	−0.760439
$214$	0	0
$215$	−12.7757	−0.871299
$216$	0	0
$217$	27.1998	1.84644
$218$	0	0
$219$	4.21108	0.284558
$220$	0	0
$221$	5.35026	0.359898
$222$	0	0
$223$	28.6859	1.92095	0.960476	−	0.278362i	$-0.0897916\pi$
$223$	28.6859	1.92095	0.960476	+	0.278362i	$0.0897916\pi$
$224$	0	0
$225$	−8.87399	−0.591599
$226$	0	0
$227$	18.5198	1.22920	0.614600	−	0.788839i	$-0.289317\pi$
$227$	18.5198	1.22920	0.614600	+	0.788839i	$0.289317\pi$
$228$	0	0
$229$	−11.2750	−0.745076	−0.372538	−	0.928017i	$-0.621512\pi$
$229$	−11.2750	−0.745076	−0.372538	+	0.928017i	$0.621512\pi$
$230$	0	0
$231$	−10.7005	−0.704043
$232$	0	0
$233$	12.5501	0.822183	0.411091	−	0.911594i	$-0.365148\pi$
$233$	12.5501	0.822183	0.411091	+	0.911594i	$0.365148\pi$
$234$	0	0
$235$	17.5515	1.14493
$236$	0	0
$237$	10.6761	0.693486
$238$	0	0
$239$	−26.7005	−1.72711	−0.863557	−	0.504252i	$-0.831768\pi$
$239$	−26.7005	−1.72711	−0.863557	+	0.504252i	$0.831768\pi$
$240$	0	0
$241$	−7.92478	−0.510480	−0.255240	−	0.966878i	$-0.582154\pi$
$241$	−7.92478	−0.510480	−0.255240	+	0.966878i	$0.582154\pi$
$242$	0	0
$243$	15.8192	1.01480
$244$	0	0
$245$	−30.4387	−1.94465
$246$	0	0
$247$	8.62530	0.548815
$248$	0	0
$249$	−3.47627	−0.220300
$250$	0	0
$251$	0	0		−	1.00000i	$-0.5\pi$
$251$	0	0			1.00000i	$0.5\pi$
$252$	0	0
$253$	27.0494	1.70058
$254$	0	0
$255$	−2.38787	−0.149534
$256$	0	0
$257$	2.64974	0.165286	0.0826431	−	0.996579i	$-0.473664\pi$
$257$	2.64974	0.165286	0.0826431	+	0.996579i	$0.473664\pi$
$258$	0	0
$259$	−28.9380	−1.79812
$260$	0	0
$261$	−16.3634	−1.01287
$262$	0	0
$263$	−11.5369	−0.711396	−0.355698	−	0.934601i	$-0.615757\pi$
$263$	−11.5369	−0.711396	−0.355698	+	0.934601i	$0.615757\pi$
$264$	0	0
$265$	13.9248	0.855392
$266$	0	0
$267$	−5.86414	−0.358880
$268$	0	0
$269$	−26.0362	−1.58745	−0.793727	−	0.608274i	$-0.791862\pi$
$269$	−26.0362	−1.58745	−0.793727	+	0.608274i	$0.791862\pi$
$270$	0	0
$271$	−17.1490	−1.04173	−0.520865	−	0.853639i	$-0.674390\pi$
$271$	−17.1490	−1.04173	−0.520865	+	0.853639i	$0.674390\pi$
$272$	0	0
$273$	−17.9248	−1.08486
$274$	0	0
$275$	12.0595	0.727215
$276$	0	0
$277$	21.9149	1.31674	0.658370	−	0.752694i	$-0.271246\pi$
$277$	21.9149	1.31674	0.658370	+	0.752694i	$0.271246\pi$
$278$	0	0
$279$	15.3806	0.920811
$280$	0	0
$281$	2.62530	0.156612	0.0783062	−	0.996929i	$-0.475049\pi$
$281$	2.62530	0.156612	0.0783062	+	0.996929i	$0.475049\pi$
$282$	0	0
$283$	7.04349	0.418692	0.209346	−	0.977842i	$-0.432866\pi$
$283$	7.04349	0.418692	0.209346	+	0.977842i	$0.432866\pi$
$284$	0	0
$285$	−3.84955	−0.228028
$286$	0	0
$287$	−36.1622	−2.13459
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	−2.23743	−0.131160
$292$	0	0
$293$	−0.598953	−0.0349912	−0.0174956	−	0.999847i	$-0.505569\pi$
$293$	−0.598953	−0.0349912	−0.0174956	+	0.999847i	$0.505569\pi$
$294$	0	0
$295$	22.3272	1.29994
$296$	0	0
$297$	−13.7743	−0.799268
$298$	0	0
$299$	45.3112	2.62042
$300$	0	0
$301$	−17.9248	−1.03317
$302$	0	0
$303$	−2.13586	−0.122702
$304$	0	0
$305$	−30.1768	−1.72792
$306$	0	0
$307$	26.7005	1.52388	0.761940	−	0.647648i	$-0.224247\pi$
$307$	26.7005	1.52388	0.761940	+	0.647648i	$0.224247\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−11.8437	−0.671593	−0.335797	−	0.941935i	$-0.609005\pi$
$311$	−11.8437	−0.671593	−0.335797	+	0.941935i	$0.609005\pi$
$312$	0	0
$313$	−33.1754	−1.87518	−0.937592	−	0.347738i	$-0.886950\pi$
$313$	−33.1754	−1.87518	−0.937592	+	0.347738i	$0.886950\pi$
$314$	0	0
$315$	−28.9380	−1.63047
$316$	0	0
$317$	13.0376	0.732265	0.366133	−	0.930563i	$-0.380682\pi$
$317$	13.0376	0.732265	0.366133	+	0.930563i	$0.380682\pi$
$318$	0	0
$319$	22.2374	1.24506
$320$	0	0
$321$	−3.50071	−0.195390
$322$	0	0
$323$	1.61213	0.0897011
$324$	0	0
$325$	20.2012	1.12056
$326$	0	0
$327$	11.0132	0.609030
$328$	0	0
$329$	24.6253	1.35764
$330$	0	0
$331$	9.71370	0.533913	0.266957	−	0.963709i	$-0.413982\pi$
$331$	9.71370	0.533913	0.266957	+	0.963709i	$0.413982\pi$
$332$	0	0
$333$	−16.3634	−0.896711
$334$	0	0
$335$	9.55149	0.521854
$336$	0	0
$337$	−6.62530	−0.360903	−0.180452	−	0.983584i	$-0.557756\pi$
$337$	−6.62530	−0.360903	−0.180452	+	0.983584i	$0.557756\pi$
$338$	0	0
$339$	4.41422	0.239748
$340$	0	0
$341$	−20.9018	−1.13189
$342$	0	0
$343$	−13.6121	−0.734986
$344$	0	0
$345$	−20.2228	−1.08876
$346$	0	0
$347$	−18.5804	−0.997448	−0.498724	−	0.866761i	$-0.666198\pi$
$347$	−18.5804	−0.997448	−0.498724	+	0.866761i	$0.666198\pi$
$348$	0	0
$349$	−12.7005	−0.679843	−0.339922	−	0.940454i	$-0.610401\pi$
$349$	−12.7005	−0.679843	−0.339922	+	0.940454i	$0.610401\pi$
$350$	0	0
$351$	−23.0738	−1.23159
$352$	0	0
$353$	6.77575	0.360637	0.180318	−	0.983608i	$-0.442287\pi$
$353$	6.77575	0.360637	0.180318	+	0.983608i	$0.442287\pi$
$354$	0	0
$355$	40.7875	2.16478
$356$	0	0
$357$	−3.35026	−0.177315
$358$	0	0
$359$	−32.4142	−1.71076	−0.855379	−	0.518003i	$-0.826675\pi$
$359$	−32.4142	−1.71076	−0.855379	+	0.518003i	$0.826675\pi$
$360$	0	0
$361$	−16.4010	−0.863213
$362$	0	0
$363$	−0.643859	−0.0337938
$364$	0	0
$365$	−15.4763	−0.810065
$366$	0	0
$367$	−10.2315	−0.534082	−0.267041	−	0.963685i	$-0.586046\pi$
$367$	−10.2315	−0.534082	−0.267041	+	0.963685i	$0.586046\pi$
$368$	0	0
$369$	−20.4485	−1.06451
$370$	0	0
$371$	19.5369	1.01431
$372$	0	0
$373$	−21.4518	−1.11073	−0.555367	−	0.831605i	$-0.687422\pi$
$373$	−21.4518	−1.11073	−0.555367	+	0.831605i	$0.687422\pi$
$374$	0	0
$375$	2.92336	0.150962
$376$	0	0
$377$	37.2506	1.91850
$378$	0	0
$379$	0.806063	0.0414047	0.0207023	−	0.999786i	$-0.493410\pi$
$379$	0.806063	0.0414047	0.0207023	+	0.999786i	$0.493410\pi$
$380$	0	0
$381$	13.1490	0.673645
$382$	0	0
$383$	10.9116	0.557557	0.278778	−	0.960355i	$-0.410070\pi$
$383$	10.9116	0.557557	0.278778	+	0.960355i	$0.410070\pi$
$384$	0	0
$385$	39.3258	2.00423
$386$	0	0
$387$	−10.1359	−0.515235
$388$	0	0
$389$	−23.1246	−1.17246	−0.586232	−	0.810143i	$-0.699389\pi$
$389$	−23.1246	−1.17246	−0.586232	+	0.810143i	$0.699389\pi$
$390$	0	0
$391$	8.46898	0.428295
$392$	0	0
$393$	8.64974	0.436322
$394$	0	0
$395$	−39.2360	−1.97418
$396$	0	0
$397$	34.8119	1.74716	0.873581	−	0.486679i	$-0.161792\pi$
$397$	34.8119	1.74716	0.873581	+	0.486679i	$0.161792\pi$
$398$	0	0
$399$	−5.40105	−0.270391
$400$	0	0
$401$	−17.4763	−0.872723	−0.436362	−	0.899771i	$-0.643733\pi$
$401$	−17.4763	−0.872723	−0.436362	+	0.899771i	$0.643733\pi$
$402$	0	0
$403$	−35.0132	−1.74413
$404$	0	0
$405$	−10.5891	−0.526177
$406$	0	0
$407$	22.2374	1.10227
$408$	0	0
$409$	10.0000	0.494468	0.247234	−	0.968956i	$-0.420478\pi$
$409$	10.0000	0.494468	0.247234	+	0.968956i	$0.420478\pi$
$410$	0	0
$411$	6.28630	0.310080
$412$	0	0
$413$	31.3258	1.54144
$414$	0	0
$415$	12.7757	0.627137
$416$	0	0
$417$	14.6761	0.718691
$418$	0	0
$419$	−26.0567	−1.27295	−0.636476	−	0.771297i	$-0.719608\pi$
$419$	−26.0567	−1.27295	−0.636476	+	0.771297i	$0.719608\pi$
$420$	0	0
$421$	37.3503	1.82034	0.910170	−	0.414235i	$-0.135951\pi$
$421$	37.3503	1.82034	0.910170	+	0.414235i	$0.135951\pi$
$422$	0	0
$423$	13.9248	0.677046
$424$	0	0
$425$	3.77575	0.183151
$426$	0	0
$427$	−42.3390	−2.04893
$428$	0	0
$429$	13.7743	0.665031
$430$	0	0
$431$	1.39517	0.0672028	0.0336014	−	0.999435i	$-0.489302\pi$
$431$	1.39517	0.0672028	0.0336014	+	0.999435i	$0.489302\pi$
$432$	0	0
$433$	31.1246	1.49575	0.747876	−	0.663838i	$-0.231074\pi$
$433$	31.1246	1.49575	0.747876	+	0.663838i	$0.231074\pi$
$434$	0	0
$435$	−16.6253	−0.797122
$436$	0	0
$437$	13.6531	0.653115
$438$	0	0
$439$	−31.0191	−1.48046	−0.740229	−	0.672354i	$-0.765283\pi$
$439$	−31.0191	−1.48046	−0.740229	+	0.672354i	$0.765283\pi$
$440$	0	0
$441$	−24.1490	−1.14995
$442$	0	0
$443$	16.6253	0.789892	0.394946	−	0.918704i	$-0.370763\pi$
$443$	16.6253	0.789892	0.394946	+	0.918704i	$0.370763\pi$
$444$	0	0
$445$	21.5515	1.02164
$446$	0	0
$447$	4.71511	0.223017
$448$	0	0
$449$	−7.55149	−0.356377	−0.178188	−	0.983996i	$-0.557024\pi$
$449$	−7.55149	−0.356377	−0.178188	+	0.983996i	$0.557024\pi$
$450$	0	0
$451$	27.7889	1.30853
$452$	0	0
$453$	2.85097	0.133950
$454$	0	0
$455$	65.8759	3.08831
$456$	0	0
$457$	−33.8251	−1.58227	−0.791136	−	0.611640i	$-0.790510\pi$
$457$	−33.8251	−1.58227	−0.791136	+	0.611640i	$0.790510\pi$
$458$	0	0
$459$	−4.31265	−0.201297
$460$	0	0
$461$	4.55008	0.211918	0.105959	−	0.994370i	$-0.466209\pi$
$461$	4.55008	0.211918	0.105959	+	0.994370i	$0.466209\pi$
$462$	0	0
$463$	31.0738	1.44412	0.722061	−	0.691829i	$-0.243195\pi$
$463$	31.0738	1.44412	0.722061	+	0.691829i	$0.243195\pi$
$464$	0	0
$465$	15.6267	0.724672
$466$	0	0
$467$	−24.5647	−1.13672	−0.568359	−	0.822781i	$-0.692421\pi$
$467$	−24.5647	−1.13672	−0.568359	+	0.822781i	$0.692421\pi$
$468$	0	0
$469$	13.4010	0.618803
$470$	0	0
$471$	−5.34041	−0.246073
$472$	0	0
$473$	13.7743	0.633344
$474$	0	0
$475$	6.08698	0.279290
$476$	0	0
$477$	11.0475	0.505828
$478$	0	0
$479$	−3.99412	−0.182496	−0.0912480	−	0.995828i	$-0.529086\pi$
$479$	−3.99412	−0.182496	−0.0912480	+	0.995828i	$0.529086\pi$
$480$	0	0
$481$	37.2506	1.69848
$482$	0	0
$483$	−28.3733	−1.29103
$484$	0	0
$485$	8.22284	0.373380
$486$	0	0
$487$	−43.4821	−1.97036	−0.985182	−	0.171511i	$-0.945135\pi$
$487$	−43.4821	−1.97036	−0.985182	+	0.171511i	$0.945135\pi$
$488$	0	0
$489$	−2.82653	−0.127820
$490$	0	0
$491$	42.8627	1.93437	0.967184	−	0.254077i	$-0.0817718\pi$
$491$	42.8627	1.93437	0.967184	+	0.254077i	$0.0817718\pi$
$492$	0	0
$493$	6.96239	0.313570
$494$	0	0
$495$	22.2374	0.999498
$496$	0	0
$497$	57.2262	2.56694
$498$	0	0
$499$	−17.0581	−0.763625	−0.381812	−	0.924240i	$-0.624700\pi$
$499$	−17.0581	−0.763625	−0.381812	+	0.924240i	$0.624700\pi$
$500$	0	0
$501$	−5.69720	−0.254532
$502$	0	0
$503$	1.29360	0.0576786	0.0288393	−	0.999584i	$-0.490819\pi$
$503$	1.29360	0.0576786	0.0288393	+	0.999584i	$0.490819\pi$
$504$	0	0
$505$	7.84955	0.349301
$506$	0	0
$507$	12.5950	0.559363
$508$	0	0
$509$	0.448507	0.0198797	0.00993987	−	0.999951i	$-0.496836\pi$
$509$	0.448507	0.0198797	0.00993987	+	0.999951i	$0.496836\pi$
$510$	0	0
$511$	−21.7137	−0.960557
$512$	0	0
$513$	−6.95254	−0.306962
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−18.9234	−0.832249
$518$	0	0
$519$	1.76257	0.0773683
$520$	0	0
$521$	−21.9511	−0.961696	−0.480848	−	0.876804i	$-0.659671\pi$
$521$	−21.9511	−0.961696	−0.480848	+	0.876804i	$0.659671\pi$
$522$	0	0
$523$	−29.4010	−1.28562	−0.642809	−	0.766026i	$-0.722231\pi$
$523$	−29.4010	−1.28562	−0.642809	+	0.766026i	$0.722231\pi$
$524$	0	0
$525$	−12.6497	−0.552080
$526$	0	0
$527$	−6.54420	−0.285070
$528$	0	0
$529$	48.7235	2.11842
$530$	0	0
$531$	17.7137	0.768709
$532$	0	0
$533$	46.5501	2.01631
$534$	0	0
$535$	12.8656	0.556227
$536$	0	0
$537$	3.89843	0.168230
$538$	0	0
$539$	32.8178	1.41356
$540$	0	0
$541$	1.13918	0.0489773	0.0244886	−	0.999700i	$-0.492204\pi$
$541$	1.13918	0.0489773	0.0244886	+	0.999700i	$0.492204\pi$
$542$	0	0
$543$	1.76257	0.0756392
$544$	0	0
$545$	−40.4749	−1.73375
$546$	0	0
$547$	−2.04491	−0.0874339	−0.0437169	−	0.999044i	$-0.513920\pi$
$547$	−2.04491	−0.0874339	−0.0437169	+	0.999044i	$0.513920\pi$
$548$	0	0
$549$	−23.9413	−1.02179
$550$	0	0
$551$	11.2243	0.478169
$552$	0	0
$553$	−55.0494	−2.34094
$554$	0	0
$555$	−16.6253	−0.705705
$556$	0	0
$557$	−29.4518	−1.24791	−0.623957	−	0.781459i	$-0.714476\pi$
$557$	−29.4518	−1.24791	−0.623957	+	0.781459i	$0.714476\pi$
$558$	0	0
$559$	23.0738	0.975918
$560$	0	0
$561$	2.57452	0.108696
$562$	0	0
$563$	−15.4156	−0.649692	−0.324846	−	0.945767i	$-0.605312\pi$
$563$	−15.4156	−0.649692	−0.324846	+	0.945767i	$0.605312\pi$
$564$	0	0
$565$	−16.2228	−0.682500
$566$	0	0
$567$	−14.8568	−0.623929
$568$	0	0
$569$	15.5223	0.650729	0.325365	−	0.945589i	$-0.394513\pi$
$569$	15.5223	0.650729	0.325365	+	0.945589i	$0.394513\pi$
$570$	0	0
$571$	43.0435	1.80131	0.900657	−	0.434531i	$-0.143086\pi$
$571$	43.0435	1.80131	0.900657	+	0.434531i	$0.143086\pi$
$572$	0	0
$573$	−1.67276	−0.0698806
$574$	0	0
$575$	31.9767	1.33352
$576$	0	0
$577$	30.7513	1.28019	0.640097	−	0.768294i	$-0.278894\pi$
$577$	30.7513	1.28019	0.640097	+	0.768294i	$0.278894\pi$
$578$	0	0
$579$	2.65959	0.110529
$580$	0	0
$581$	17.9248	0.743645
$582$	0	0
$583$	−15.0132	−0.621782
$584$	0	0
$585$	37.2506	1.54012
$586$	0	0
$587$	−25.6121	−1.05713	−0.528563	−	0.848894i	$-0.677269\pi$
$587$	−25.6121	−1.05713	−0.528563	+	0.848894i	$0.677269\pi$
$588$	0	0
$589$	−10.5501	−0.434708
$590$	0	0
$591$	7.16362	0.294672
$592$	0	0
$593$	24.0263	0.986644	0.493322	−	0.869847i	$-0.335782\pi$
$593$	24.0263	0.986644	0.493322	+	0.869847i	$0.335782\pi$
$594$	0	0
$595$	12.3127	0.504769
$596$	0	0
$597$	5.32391	0.217893
$598$	0	0
$599$	22.0263	0.899972	0.449986	−	0.893036i	$-0.351429\pi$
$599$	22.0263	0.899972	0.449986	+	0.893036i	$0.351429\pi$
$600$	0	0
$601$	43.2506	1.76423	0.882114	−	0.471035i	$-0.156120\pi$
$601$	43.2506	1.76423	0.882114	+	0.471035i	$0.156120\pi$
$602$	0	0
$603$	7.57784	0.308594
$604$	0	0
$605$	2.36626	0.0962023
$606$	0	0
$607$	−23.1695	−0.940421	−0.470210	−	0.882554i	$-0.655822\pi$
$607$	−23.1695	−0.940421	−0.470210	+	0.882554i	$0.655822\pi$
$608$	0	0
$609$	−23.3258	−0.945210
$610$	0	0
$611$	−31.6991	−1.28241
$612$	0	0
$613$	41.6991	1.68421	0.842106	−	0.539313i	$-0.181316\pi$
$613$	41.6991	1.68421	0.842106	+	0.539313i	$0.181316\pi$
$614$	0	0
$615$	−20.7757	−0.837759
$616$	0	0
$617$	8.70052	0.350270	0.175135	−	0.984544i	$-0.443964\pi$
$617$	8.70052	0.350270	0.175135	+	0.984544i	$0.443964\pi$
$618$	0	0
$619$	−16.2823	−0.654442	−0.327221	−	0.944948i	$-0.606112\pi$
$619$	−16.2823	−0.654442	−0.327221	+	0.944948i	$0.606112\pi$
$620$	0	0
$621$	−36.5237	−1.46565
$622$	0	0
$623$	30.2374	1.21144
$624$	0	0
$625$	−29.6225	−1.18490
$626$	0	0
$627$	4.15045	0.165753
$628$	0	0
$629$	6.96239	0.277609
$630$	0	0
$631$	19.6385	0.781795	0.390898	−	0.920434i	$-0.372165\pi$
$631$	19.6385	0.781795	0.390898	+	0.920434i	$0.372165\pi$
$632$	0	0
$633$	−8.64974	−0.343796
$634$	0	0
$635$	−48.3244	−1.91770
$636$	0	0
$637$	54.9741	2.17816
$638$	0	0
$639$	32.3595	1.28012
$640$	0	0
$641$	−46.3244	−1.82970	−0.914852	−	0.403789i	$-0.867693\pi$
$641$	−46.3244	−1.82970	−0.914852	+	0.403789i	$0.867693\pi$
$642$	0	0
$643$	5.70308	0.224907	0.112454	−	0.993657i	$-0.464129\pi$
$643$	5.70308	0.224907	0.112454	+	0.993657i	$0.464129\pi$
$644$	0	0
$645$	−10.2981	−0.405486
$646$	0	0
$647$	13.8035	0.542672	0.271336	−	0.962485i	$-0.412535\pi$
$647$	13.8035	0.542672	0.271336	+	0.962485i	$0.412535\pi$
$648$	0	0
$649$	−24.0724	−0.944924
$650$	0	0
$651$	21.9248	0.859300
$652$	0	0
$653$	−4.26187	−0.166780	−0.0833898	−	0.996517i	$-0.526575\pi$
$653$	−4.26187	−0.166780	−0.0833898	+	0.996517i	$0.526575\pi$
$654$	0	0
$655$	−31.7889	−1.24210
$656$	0	0
$657$	−12.2784	−0.479025
$658$	0	0
$659$	23.1754	0.902785	0.451392	−	0.892326i	$-0.350928\pi$
$659$	23.1754	0.902785	0.451392	+	0.892326i	$0.350928\pi$
$660$	0	0
$661$	−34.7269	−1.35072	−0.675359	−	0.737489i	$-0.736011\pi$
$661$	−34.7269	−1.35072	−0.675359	+	0.737489i	$0.736011\pi$
$662$	0	0
$663$	4.31265	0.167489
$664$	0	0
$665$	19.8496	0.769733
$666$	0	0
$667$	58.9643	2.28311
$668$	0	0
$669$	23.1227	0.893975
$670$	0	0
$671$	32.5355	1.25602
$672$	0	0
$673$	37.8496	1.45899	0.729497	−	0.683984i	$-0.239754\pi$
$673$	37.8496	1.45899	0.729497	+	0.683984i	$0.239754\pi$
$674$	0	0
$675$	−16.2835	−0.626751
$676$	0	0
$677$	23.9610	0.920895	0.460448	−	0.887687i	$-0.347689\pi$
$677$	23.9610	0.920895	0.460448	+	0.887687i	$0.347689\pi$
$678$	0	0
$679$	11.5369	0.442746
$680$	0	0
$681$	14.9281	0.572046
$682$	0	0
$683$	8.18076	0.313028	0.156514	−	0.987676i	$-0.449974\pi$
$683$	8.18076	0.313028	0.156514	+	0.987676i	$0.449974\pi$
$684$	0	0
$685$	−23.1030	−0.882720
$686$	0	0
$687$	−9.08840	−0.346744
$688$	0	0
$689$	−25.1490	−0.958102
$690$	0	0
$691$	2.56864	0.0977155	0.0488578	−	0.998806i	$-0.484442\pi$
$691$	2.56864	0.0977155	0.0488578	+	0.998806i	$0.484442\pi$
$692$	0	0
$693$	31.1998	1.18518
$694$	0	0
$695$	−53.9365	−2.04593
$696$	0	0
$697$	8.70052	0.329556
$698$	0	0
$699$	10.1162	0.382628
$700$	0	0
$701$	9.19982	0.347472	0.173736	−	0.984792i	$-0.444416\pi$
$701$	9.19982	0.347472	0.173736	+	0.984792i	$0.444416\pi$
$702$	0	0
$703$	11.2243	0.423331
$704$	0	0
$705$	14.1476	0.532830
$706$	0	0
$707$	11.0132	0.414193
$708$	0	0
$709$	−32.6155	−1.22490	−0.612449	−	0.790510i	$-0.709816\pi$
$709$	−32.6155	−1.22490	−0.612449	+	0.790510i	$0.709816\pi$
$710$	0	0
$711$	−31.1286	−1.16741
$712$	0	0
$713$	−55.4227	−2.07559
$714$	0	0
$715$	−50.6225	−1.89317
$716$	0	0
$717$	−21.5223	−0.803766
$718$	0	0
$719$	−16.8714	−0.629198	−0.314599	−	0.949225i	$-0.601870\pi$
$719$	−16.8714	−0.629198	−0.314599	+	0.949225i	$0.601870\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−6.38787	−0.237568
$724$	0	0
$725$	26.2882	0.976320
$726$	0	0
$727$	13.9248	0.516441	0.258221	−	0.966086i	$-0.416864\pi$
$727$	13.9248	0.516441	0.258221	+	0.966086i	$0.416864\pi$
$728$	0	0
$729$	2.02776	0.0751023
$730$	0	0
$731$	4.31265	0.159509
$732$	0	0
$733$	−19.7743	−0.730382	−0.365191	−	0.930933i	$-0.618996\pi$
$733$	−19.7743	−0.730382	−0.365191	+	0.930933i	$0.618996\pi$
$734$	0	0
$735$	−24.5355	−0.905005
$736$	0	0
$737$	−10.2981	−0.379334
$738$	0	0
$739$	16.9868	0.624871	0.312435	−	0.949939i	$-0.398855\pi$
$739$	16.9868	0.624871	0.312435	+	0.949939i	$0.398855\pi$
$740$	0	0
$741$	6.95254	0.255408
$742$	0	0
$743$	−10.3938	−0.381310	−0.190655	−	0.981657i	$-0.561061\pi$
$743$	−10.3938	−0.381310	−0.190655	+	0.981657i	$0.561061\pi$
$744$	0	0
$745$	−17.3287	−0.634873
$746$	0	0
$747$	10.1359	0.370852
$748$	0	0
$749$	18.0508	0.659561
$750$	0	0
$751$	−42.2433	−1.54148	−0.770740	−	0.637150i	$-0.780113\pi$
$751$	−42.2433	−1.54148	−0.770740	+	0.637150i	$0.780113\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−10.4777	−0.381322
$756$	0	0
$757$	23.0230	0.836786	0.418393	−	0.908266i	$-0.362593\pi$
$757$	23.0230	0.836786	0.418393	+	0.908266i	$0.362593\pi$
$758$	0	0
$759$	21.8035	0.791417
$760$	0	0
$761$	−10.8002	−0.391506	−0.195753	−	0.980653i	$-0.562715\pi$
$761$	−10.8002	−0.391506	−0.195753	+	0.980653i	$0.562715\pi$
$762$	0	0
$763$	−56.7875	−2.05585
$764$	0	0
$765$	6.96239	0.251726
$766$	0	0
$767$	−40.3244	−1.45603
$768$	0	0
$769$	−17.4518	−0.629329	−0.314665	−	0.949203i	$-0.601892\pi$
$769$	−17.4518	−0.629329	−0.314665	+	0.949203i	$0.601892\pi$
$770$	0	0
$771$	2.13586	0.0769210
$772$	0	0
$773$	−11.2750	−0.405535	−0.202767	−	0.979227i	$-0.564994\pi$
$773$	−11.2750	−0.405535	−0.202767	+	0.979227i	$0.564994\pi$
$774$	0	0
$775$	−24.7092	−0.887582
$776$	0	0
$777$	−23.3258	−0.836809
$778$	0	0
$779$	14.0263	0.502546
$780$	0	0
$781$	−43.9756	−1.57357
$782$	0	0
$783$	−30.0263	−1.07305
$784$	0	0
$785$	19.6267	0.700508
$786$	0	0
$787$	2.50800	0.0894006	0.0447003	−	0.999000i	$-0.485767\pi$
$787$	2.50800	0.0894006	0.0447003	+	0.999000i	$0.485767\pi$
$788$	0	0
$789$	−9.29948	−0.331070
$790$	0	0
$791$	−22.7612	−0.809294
$792$	0	0
$793$	54.5012	1.93539
$794$	0	0
$795$	11.2243	0.398083
$796$	0	0
$797$	−38.8773	−1.37711	−0.688553	−	0.725186i	$-0.741754\pi$
$797$	−38.8773	−1.37711	−0.688553	+	0.725186i	$0.741754\pi$
$798$	0	0
$799$	−5.92478	−0.209604
$800$	0	0
$801$	17.0982	0.604137
$802$	0	0
$803$	16.6859	0.588834
$804$	0	0
$805$	104.276	3.67523
$806$	0	0
$807$	−20.9868	−0.738771
$808$	0	0
$809$	44.1768	1.55317	0.776587	−	0.630010i	$-0.216949\pi$
$809$	44.1768	1.55317	0.776587	+	0.630010i	$0.216949\pi$
$810$	0	0
$811$	−20.0713	−0.704797	−0.352399	−	0.935850i	$-0.614634\pi$
$811$	−20.0713	−0.704797	−0.352399	+	0.935850i	$0.614634\pi$
$812$	0	0
$813$	−13.8232	−0.484801
$814$	0	0
$815$	10.3879	0.363871
$816$	0	0
$817$	6.95254	0.243239
$818$	0	0
$819$	52.2638	1.82624
$820$	0	0
$821$	−16.4847	−0.575320	−0.287660	−	0.957733i	$-0.592877\pi$
$821$	−16.4847	−0.575320	−0.287660	+	0.957733i	$0.592877\pi$
$822$	0	0
$823$	−22.6048	−0.787955	−0.393977	−	0.919120i	$-0.628901\pi$
$823$	−22.6048	−0.787955	−0.393977	+	0.919120i	$0.628901\pi$
$824$	0	0
$825$	9.72072	0.338432
$826$	0	0
$827$	30.5198	1.06128	0.530638	−	0.847599i	$-0.321952\pi$
$827$	30.5198	1.06128	0.530638	+	0.847599i	$0.321952\pi$
$828$	0	0
$829$	−6.62530	−0.230106	−0.115053	−	0.993359i	$-0.536704\pi$
$829$	−6.62530	−0.230106	−0.115053	+	0.993359i	$0.536704\pi$
$830$	0	0
$831$	17.6648	0.612786
$832$	0	0
$833$	10.2750	0.356009
$834$	0	0
$835$	20.9380	0.724588
$836$	0	0
$837$	28.2228	0.975524
$838$	0	0
$839$	10.9175	0.376913	0.188457	−	0.982082i	$-0.439651\pi$
$839$	10.9175	0.376913	0.188457	+	0.982082i	$0.439651\pi$
$840$	0	0
$841$	19.4749	0.671547
$842$	0	0
$843$	2.11616	0.0728844
$844$	0	0
$845$	−46.2882	−1.59236
$846$	0	0
$847$	3.31994	0.114075
$848$	0	0
$849$	5.67750	0.194851
$850$	0	0
$851$	58.9643	2.02127
$852$	0	0
$853$	−29.7645	−1.01912	−0.509558	−	0.860436i	$-0.670191\pi$
$853$	−29.7645	−1.01912	−0.509558	+	0.860436i	$0.670191\pi$
$854$	0	0
$855$	11.2243	0.383861
$856$	0	0
$857$	0.0752228	0.00256956	0.00128478	−	0.999999i	$-0.499591\pi$
$857$	0.0752228	0.00256956	0.00128478	+	0.999999i	$0.499591\pi$
$858$	0	0
$859$	−8.78751	−0.299826	−0.149913	−	0.988699i	$-0.547899\pi$
$859$	−8.78751	−0.299826	−0.149913	+	0.988699i	$0.547899\pi$
$860$	0	0
$861$	−29.1490	−0.993396
$862$	0	0
$863$	−20.4749	−0.696972	−0.348486	−	0.937314i	$-0.613304\pi$
$863$	−20.4749	−0.696972	−0.348486	+	0.937314i	$0.613304\pi$
$864$	0	0
$865$	−6.47768	−0.220248
$866$	0	0
$867$	0.806063	0.0273753
$868$	0	0
$869$	42.3028	1.43502
$870$	0	0
$871$	−17.2506	−0.584514
$872$	0	0
$873$	6.52373	0.220795
$874$	0	0
$875$	−15.0738	−0.509588
$876$	0	0
$877$	11.0640	0.373603	0.186802	−	0.982398i	$-0.440188\pi$
$877$	11.0640	0.373603	0.186802	+	0.982398i	$0.440188\pi$
$878$	0	0
$879$	−0.482794	−0.0162842
$880$	0	0
$881$	−25.7283	−0.866808	−0.433404	−	0.901200i	$-0.642688\pi$
$881$	−25.7283	−0.866808	−0.433404	+	0.901200i	$0.642688\pi$
$882$	0	0
$883$	5.82321	0.195967	0.0979833	−	0.995188i	$-0.468761\pi$
$883$	5.82321	0.195967	0.0979833	+	0.995188i	$0.468761\pi$
$884$	0	0
$885$	17.9972	0.604968
$886$	0	0
$887$	21.6786	0.727898	0.363949	−	0.931419i	$-0.381428\pi$
$887$	21.6786	0.727898	0.363949	+	0.931419i	$0.381428\pi$
$888$	0	0
$889$	−67.8007	−2.27396
$890$	0	0
$891$	11.4168	0.382476
$892$	0	0
$893$	−9.55149	−0.319629
$894$	0	0
$895$	−14.3272	−0.478907
$896$	0	0
$897$	36.5237	1.21949
$898$	0	0
$899$	−45.5633	−1.51962
$900$	0	0
$901$	−4.70052	−0.156597
$902$	0	0
$903$	−14.4485	−0.480816
$904$	0	0
$905$	−6.47768	−0.215326
$906$	0	0
$907$	31.5475	1.04752	0.523759	−	0.851866i	$-0.324529\pi$
$907$	31.5475	1.04752	0.523759	+	0.851866i	$0.324529\pi$
$908$	0	0
$909$	6.22758	0.206556
$910$	0	0
$911$	38.0205	1.25967	0.629837	−	0.776727i	$-0.283122\pi$
$911$	38.0205	1.25967	0.629837	+	0.776727i	$0.283122\pi$
$912$	0	0
$913$	−13.7743	−0.455864
$914$	0	0
$915$	−24.3244	−0.804140
$916$	0	0
$917$	−44.6009	−1.47285
$918$	0	0
$919$	−31.5975	−1.04231	−0.521153	−	0.853463i	$-0.674498\pi$
$919$	−31.5975	−1.04231	−0.521153	+	0.853463i	$0.674498\pi$
$920$	0	0
$921$	21.5223	0.709184
$922$	0	0
$923$	−73.6648	−2.42471
$924$	0	0
$925$	26.2882	0.864351
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−7.92478	−0.260004	−0.130002	−	0.991514i	$-0.541498\pi$
$929$	−7.92478	−0.260004	−0.130002	+	0.991514i	$0.541498\pi$
$930$	0	0
$931$	16.5647	0.542885
$932$	0	0
$933$	−9.54675	−0.312547
$934$	0	0
$935$	−9.46168	−0.309430
$936$	0	0
$937$	−35.4010	−1.15650	−0.578251	−	0.815859i	$-0.696265\pi$
$937$	−35.4010	−1.15650	−0.578251	+	0.815859i	$0.696265\pi$
$938$	0	0
$939$	−26.7415	−0.872675
$940$	0	0
$941$	−40.9887	−1.33619	−0.668097	−	0.744074i	$-0.732891\pi$
$941$	−40.9887	−1.33619	−0.668097	+	0.744074i	$0.732891\pi$
$942$	0	0
$943$	73.6845	2.39950
$944$	0	0
$945$	−53.1002	−1.72735
$946$	0	0
$947$	19.7586	0.642068	0.321034	−	0.947068i	$-0.395970\pi$
$947$	19.7586	0.642068	0.321034	+	0.947068i	$0.395970\pi$
$948$	0	0
$949$	27.9511	0.907332
$950$	0	0
$951$	10.5091	0.340782
$952$	0	0
$953$	−28.6761	−0.928910	−0.464455	−	0.885597i	$-0.653750\pi$
$953$	−28.6761	−0.928910	−0.464455	+	0.885597i	$0.653750\pi$
$954$	0	0
$955$	6.14762	0.198932
$956$	0	0
$957$	17.9248	0.579426
$958$	0	0
$959$	−32.4142	−1.04671
$960$	0	0
$961$	11.8265	0.381501
$962$	0	0
$963$	10.2071	0.328920
$964$	0	0
$965$	−9.77433	−0.314647
$966$	0	0
$967$	24.5355	0.789008	0.394504	−	0.918894i	$-0.370916\pi$
$967$	24.5355	0.789008	0.394504	+	0.918894i	$0.370916\pi$
$968$	0	0
$969$	1.29948	0.0417452
$970$	0	0
$971$	34.3390	1.10199	0.550995	−	0.834508i	$-0.314248\pi$
$971$	34.3390	1.10199	0.550995	+	0.834508i	$0.314248\pi$
$972$	0	0
$973$	−75.6747	−2.42602
$974$	0	0
$975$	16.2835	0.521489
$976$	0	0
$977$	−2.27171	−0.0726786	−0.0363393	−	0.999340i	$-0.511570\pi$
$977$	−2.27171	−0.0726786	−0.0363393	+	0.999340i	$0.511570\pi$
$978$	0	0
$979$	−23.2360	−0.742626
$980$	0	0
$981$	−32.1114	−1.02524
$982$	0	0
$983$	48.8832	1.55913	0.779566	−	0.626320i	$-0.215440\pi$
$983$	48.8832	1.55913	0.779566	+	0.626320i	$0.215440\pi$
$984$	0	0
$985$	−26.3272	−0.838856
$986$	0	0
$987$	19.8496	0.631818
$988$	0	0
$989$	36.5237	1.16139
$990$	0	0
$991$	−12.5179	−0.397643	−0.198821	−	0.980036i	$-0.563711\pi$
$991$	−12.5179	−0.397643	−0.198821	+	0.980036i	$0.563711\pi$
$992$	0	0
$993$	7.82986	0.248473
$994$	0	0
$995$	−19.5661	−0.620286
$996$	0	0
$997$	50.4387	1.59741	0.798704	−	0.601724i	$-0.205519\pi$
$997$	50.4387	1.59741	0.798704	+	0.601724i	$0.205519\pi$
$998$	0	0
$999$	−30.0263	−0.949992

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1088.2.a.v.1.2		3
3.2	odd	2		9792.2.a.dc.1.3		3
4.3	odd	2		1088.2.a.u.1.2		3
8.3	odd	2		544.2.a.j.1.2	yes	3
8.5	even	2		544.2.a.i.1.2	✓	3
12.11	even	2		9792.2.a.dd.1.3		3
24.5	odd	2		4896.2.a.be.1.1		3
24.11	even	2		4896.2.a.bf.1.1		3
136.67	odd	2		9248.2.a.t.1.2		3
136.101	even	2		9248.2.a.u.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
544.2.a.i.1.2	✓	3	8.5	even	2
544.2.a.j.1.2	yes	3	8.3	odd	2
1088.2.a.u.1.2		3	4.3	odd	2
1088.2.a.v.1.2		3	1.1	even	1	trivial
4896.2.a.be.1.1		3	24.5	odd	2
4896.2.a.bf.1.1		3	24.11	even	2
9248.2.a.t.1.2		3	136.67	odd	2
9248.2.a.u.1.2		3	136.101	even	2
9792.2.a.dc.1.3		3	3.2	odd	2
9792.2.a.dd.1.3		3	12.11	even	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 1088 = 2^{6} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1088.a (trivial)

\(f(q)\)	\(=\)	\(q+0.806063 q^{3} -2.96239 q^{5} -4.15633 q^{7} -2.35026 q^{9} +O(q^{10})\)	\(q+0.806063 q^{3} -2.96239 q^{5} -4.15633 q^{7} -2.35026 q^{9} +3.19394 q^{11} +5.35026 q^{13} -2.38787 q^{15} +1.00000 q^{17} +1.61213 q^{19} -3.35026 q^{21} +8.46898 q^{23} +3.77575 q^{25} -4.31265 q^{27} +6.96239 q^{29} -6.54420 q^{31} +2.57452 q^{33} +12.3127 q^{35} +6.96239 q^{37} +4.31265 q^{39} +8.70052 q^{41} +4.31265 q^{43} +6.96239 q^{45} -5.92478 q^{47} +10.2750 q^{49} +0.806063 q^{51} -4.70052 q^{53} -9.46168 q^{55} +1.29948 q^{57} -7.53690 q^{59} +10.1866 q^{61} +9.76845 q^{63} -15.8496 q^{65} -3.22425 q^{67} +6.82653 q^{69} -13.7685 q^{71} +5.22425 q^{73} +3.04349 q^{75} -13.2750 q^{77} +13.2447 q^{79} +3.57452 q^{81} -4.31265 q^{83} -2.96239 q^{85} +5.61213 q^{87} -7.27504 q^{89} -22.2374 q^{91} -5.27504 q^{93} -4.77575 q^{95} -2.77575 q^{97} -7.50659 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 2 q^{3} + 2 q^{5} - 2 q^{7} + 3 q^{9}+O(q^{10}) \) 3 * q + 2 * q^3 + 2 * q^5 - 2 * q^7 + 3 * q^9	\( 3 q + 2 q^{3} + 2 q^{5} - 2 q^{7} + 3 q^{9} + 10 q^{11} + 6 q^{13} - 8 q^{15} + 3 q^{17} + 4 q^{19} - 6 q^{23} + 13 q^{25} + 8 q^{27} + 10 q^{29} - 10 q^{31} - 4 q^{33} + 16 q^{35} + 10 q^{37} - 8 q^{39} + 6 q^{41} - 8 q^{43} + 10 q^{45} + 4 q^{47} - q^{49} + 2 q^{51} + 6 q^{53} + 16 q^{55} + 24 q^{57} + 18 q^{61} + 18 q^{63} - 4 q^{65} - 8 q^{67} - 8 q^{69} - 30 q^{71} + 14 q^{73} - 34 q^{75} - 8 q^{77} + 10 q^{79} - q^{81} + 8 q^{83} + 2 q^{85} + 16 q^{87} + 10 q^{89} - 24 q^{91} + 16 q^{93} - 16 q^{95} - 10 q^{97} - 2 q^{99}+O(q^{100}) \) 3 * q + 2 * q^3 + 2 * q^5 - 2 * q^7 + 3 * q^9 + 10 * q^11 + 6 * q^13 - 8 * q^15 + 3 * q^17 + 4 * q^19 - 6 * q^23 + 13 * q^25 + 8 * q^27 + 10 * q^29 - 10 * q^31 - 4 * q^33 + 16 * q^35 + 10 * q^37 - 8 * q^39 + 6 * q^41 - 8 * q^43 + 10 * q^45 + 4 * q^47 - q^49 + 2 * q^51 + 6 * q^53 + 16 * q^55 + 24 * q^57 + 18 * q^61 + 18 * q^63 - 4 * q^65 - 8 * q^67 - 8 * q^69 - 30 * q^71 + 14 * q^73 - 34 * q^75 - 8 * q^77 + 10 * q^79 - q^81 + 8 * q^83 + 2 * q^85 + 16 * q^87 + 10 * q^89 - 24 * q^91 + 16 * q^93 - 16 * q^95 - 10 * q^97 - 2 * q^99

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