LMFDB - Embedded newform 7744.2.a.bo.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7744,2,Mod(1,7744)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7744, 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("7744.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7744 = 2^{6} \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7744.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:	$61.8361513253$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 44)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$1.61803$ of defining polynomial
Character	$\chi$	$=$	7744.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.61803 q^{3} +1.61803 q^{5} -3.85410 q^{7} +3.85410 q^{9} +2.38197 q^{13} -4.23607 q^{15} -2.38197 q^{17} -3.85410 q^{19} +10.0902 q^{21} +2.47214 q^{23} -2.38197 q^{25} -2.23607 q^{27} +8.61803 q^{29} -0.854102 q^{31} -6.23607 q^{35} +1.85410 q^{37} -6.23607 q^{39} -8.61803 q^{41} +6.23607 q^{45} -1.38197 q^{47} +7.85410 q^{49} +6.23607 q^{51} +4.09017 q^{53} +10.0902 q^{57} +1.09017 q^{59} +2.38197 q^{61} -14.8541 q^{63} +3.85410 q^{65} +12.9443 q^{67} -6.47214 q^{69} +6.38197 q^{71} +0.909830 q^{73} +6.23607 q^{75} -7.14590 q^{79} -5.70820 q^{81} -13.0344 q^{83} -3.85410 q^{85} -22.5623 q^{87} +0.472136 q^{89} -9.18034 q^{91} +2.23607 q^{93} -6.23607 q^{95} -14.5623 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 3 q^{3} + q^{5} - q^{7} + q^{9} + 7 q^{13} - 4 q^{15} - 7 q^{17} - q^{19} + 9 q^{21} - 4 q^{23} - 7 q^{25} + 15 q^{29} + 5 q^{31} - 8 q^{35} - 3 q^{37} - 8 q^{39} - 15 q^{41} + 8 q^{45} - 5 q^{47}+ \cdots - 9 q^{97}+O(q^{100}) $ 2 * q - 3 * q^3 + q^5 - q^7 + q^9 + 7 * q^13 - 4 * q^15 - 7 * q^17 - q^19 + 9 * q^21 - 4 * q^23 - 7 * q^25 + 15 * q^29 + 5 * q^31 - 8 * q^35 - 3 * q^37 - 8 * q^39 - 15 * q^41 + 8 * q^45 - 5 * q^47 + 9 * q^49 + 8 * q^51 - 3 * q^53 + 9 * q^57 - 9 * q^59 + 7 * q^61 - 23 * q^63 + q^65 + 8 * q^67 - 4 * q^69 + 15 * q^71 + 13 * q^73 + 8 * q^75 - 21 * q^79 + 2 * q^81 + 3 * q^83 - q^85 - 25 * q^87 - 8 * q^89 + 4 * q^91 - 8 * q^95 - 9 * q^97

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
$3$	−2.61803	−1.51152	−0.755761	−	0.654847i	$-0.772733\pi$
$3$	−2.61803	−1.51152	−0.755761	+	0.654847i	$0.772733\pi$
$4$	0	0
$5$	1.61803	0.723607	0.361803	−	0.932254i	$-0.382161\pi$
$5$	1.61803	0.723607	0.361803	+	0.932254i	$0.382161\pi$
$6$	0	0
$7$	−3.85410	−1.45671	−0.728357	−	0.685198i	$-0.759716\pi$
$7$	−3.85410	−1.45671	−0.728357	+	0.685198i	$0.759716\pi$
$8$	0	0
$9$	3.85410	1.28470
$10$	0	0
$11$	0	0
$11$	0	0
$12$	0	0
$13$	2.38197	0.660639	0.330319	−	0.943869i	$-0.392844\pi$
$13$	2.38197	0.660639	0.330319	+	0.943869i	$0.392844\pi$
$14$	0	0
$15$	−4.23607	−1.09375
$16$	0	0
$17$	−2.38197	−0.577712	−0.288856	−	0.957373i	$-0.593275\pi$
$17$	−2.38197	−0.577712	−0.288856	+	0.957373i	$0.593275\pi$
$18$	0	0
$19$	−3.85410	−0.884192	−0.442096	−	0.896968i	$-0.645765\pi$
$19$	−3.85410	−0.884192	−0.442096	+	0.896968i	$0.645765\pi$
$20$	0	0
$21$	10.0902	2.20186
$22$	0	0
$23$	2.47214	0.515476	0.257738	−	0.966215i	$-0.417023\pi$
$23$	2.47214	0.515476	0.257738	+	0.966215i	$0.417023\pi$
$24$	0	0
$25$	−2.38197	−0.476393
$26$	0	0
$27$	−2.23607	−0.430331
$28$	0	0
$29$	8.61803	1.60033	0.800164	−	0.599781i	$-0.204746\pi$
$29$	8.61803	1.60033	0.800164	+	0.599781i	$0.204746\pi$
$30$	0	0
$31$	−0.854102	−0.153401	−0.0767006	−	0.997054i	$-0.524439\pi$
$31$	−0.854102	−0.153401	−0.0767006	+	0.997054i	$0.524439\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−6.23607	−1.05409
$36$	0	0
$37$	1.85410	0.304812	0.152406	−	0.988318i	$-0.451298\pi$
$37$	1.85410	0.304812	0.152406	+	0.988318i	$0.451298\pi$
$38$	0	0
$39$	−6.23607	−0.998570
$40$	0	0
$41$	−8.61803	−1.34591	−0.672955	−	0.739683i	$-0.734975\pi$
$41$	−8.61803	−1.34591	−0.672955	+	0.739683i	$0.734975\pi$
$42$	0	0
$43$	0	0		−	1.00000i	$-0.5\pi$
$43$	0	0			1.00000i	$0.5\pi$
$44$	0	0
$45$	6.23607	0.929618
$46$	0	0
$47$	−1.38197	−0.201580	−0.100790	−	0.994908i	$-0.532137\pi$
$47$	−1.38197	−0.201580	−0.100790	+	0.994908i	$0.532137\pi$
$48$	0	0
$49$	7.85410	1.12201
$50$	0	0
$51$	6.23607	0.873224
$52$	0	0
$53$	4.09017	0.561828	0.280914	−	0.959733i	$-0.409362\pi$
$53$	4.09017	0.561828	0.280914	+	0.959733i	$0.409362\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	10.0902	1.33648
$58$	0	0
$59$	1.09017	0.141928	0.0709640	−	0.997479i	$-0.477392\pi$
$59$	1.09017	0.141928	0.0709640	+	0.997479i	$0.477392\pi$
$60$	0	0
$61$	2.38197	0.304979	0.152490	−	0.988305i	$-0.451271\pi$
$61$	2.38197	0.304979	0.152490	+	0.988305i	$0.451271\pi$
$62$	0	0
$63$	−14.8541	−1.87144
$64$	0	0
$65$	3.85410	0.478043
$66$	0	0
$67$	12.9443	1.58139	0.790697	−	0.612207i	$-0.209718\pi$
$67$	12.9443	1.58139	0.790697	+	0.612207i	$0.209718\pi$
$68$	0	0
$69$	−6.47214	−0.779154
$70$	0	0
$71$	6.38197	0.757400	0.378700	−	0.925519i	$-0.376371\pi$
$71$	6.38197	0.757400	0.378700	+	0.925519i	$0.376371\pi$
$72$	0	0
$73$	0.909830	0.106488	0.0532438	−	0.998582i	$-0.483044\pi$
$73$	0.909830	0.106488	0.0532438	+	0.998582i	$0.483044\pi$
$74$	0	0
$75$	6.23607	0.720079
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−7.14590	−0.803976	−0.401988	−	0.915645i	$-0.631681\pi$
$79$	−7.14590	−0.803976	−0.401988	+	0.915645i	$0.631681\pi$
$80$	0	0
$81$	−5.70820	−0.634245
$82$	0	0
$83$	−13.0344	−1.43072	−0.715358	−	0.698758i	$-0.753736\pi$
$83$	−13.0344	−1.43072	−0.715358	+	0.698758i	$0.753736\pi$
$84$	0	0
$85$	−3.85410	−0.418036
$86$	0	0
$87$	−22.5623	−2.41893
$88$	0	0
$89$	0.472136	0.0500463	0.0250232	−	0.999687i	$-0.492034\pi$
$89$	0.472136	0.0500463	0.0250232	+	0.999687i	$0.492034\pi$
$90$	0	0
$91$	−9.18034	−0.962361
$92$	0	0
$93$	2.23607	0.231869
$94$	0	0
$95$	−6.23607	−0.639807
$96$	0	0
$97$	−14.5623	−1.47858	−0.739289	−	0.673388i	$-0.764838\pi$
$97$	−14.5623	−1.47858	−0.739289	+	0.673388i	$0.764838\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	17.7984	1.77100	0.885502	−	0.464635i	$-0.153815\pi$
$101$	17.7984	1.77100	0.885502	+	0.464635i	$0.153815\pi$
$102$	0	0
$103$	7.85410	0.773888	0.386944	−	0.922103i	$-0.373531\pi$
$103$	7.85410	0.773888	0.386944	+	0.922103i	$0.373531\pi$
$104$	0	0
$105$	16.3262	1.59328
$106$	0	0
$107$	11.5623	1.11777	0.558885	−	0.829245i	$-0.311229\pi$
$107$	11.5623	1.11777	0.558885	+	0.829245i	$0.311229\pi$
$108$	0	0
$109$	−12.4721	−1.19461	−0.597307	−	0.802013i	$-0.703763\pi$
$109$	−12.4721	−1.19461	−0.597307	+	0.802013i	$0.703763\pi$
$110$	0	0
$111$	−4.85410	−0.460731
$112$	0	0
$113$	17.5623	1.65212	0.826061	−	0.563580i	$-0.190576\pi$
$113$	17.5623	1.65212	0.826061	+	0.563580i	$0.190576\pi$
$114$	0	0
$115$	4.00000	0.373002
$116$	0	0
$117$	9.18034	0.848723
$118$	0	0
$119$	9.18034	0.841560
$120$	0	0
$121$	0	0
$122$	0	0
$123$	22.5623	2.03437
$124$	0	0
$125$	−11.9443	−1.06833
$126$	0	0
$127$	−2.38197	−0.211365	−0.105683	−	0.994400i	$-0.533703\pi$
$127$	−2.38197	−0.211365	−0.105683	+	0.994400i	$0.533703\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−9.52786	−0.832453	−0.416227	−	0.909261i	$-0.636648\pi$
$131$	−9.52786	−0.832453	−0.416227	+	0.909261i	$0.636648\pi$
$132$	0	0
$133$	14.8541	1.28801
$134$	0	0
$135$	−3.61803	−0.311391
$136$	0	0
$137$	13.7984	1.17887	0.589437	−	0.807814i	$-0.299350\pi$
$137$	13.7984	1.17887	0.589437	+	0.807814i	$0.299350\pi$
$138$	0	0
$139$	−11.5623	−0.980702	−0.490351	−	0.871525i	$-0.663131\pi$
$139$	−11.5623	−0.980702	−0.490351	+	0.871525i	$0.663131\pi$
$140$	0	0
$141$	3.61803	0.304693
$142$	0	0
$143$	0	0
$144$	0	0
$145$	13.9443	1.15801
$146$	0	0
$147$	−20.5623	−1.69595
$148$	0	0
$149$	2.38197	0.195138	0.0975691	−	0.995229i	$-0.468893\pi$
$149$	2.38197	0.195138	0.0975691	+	0.995229i	$0.468893\pi$
$150$	0	0
$151$	19.2705	1.56821	0.784106	−	0.620627i	$-0.213122\pi$
$151$	19.2705	1.56821	0.784106	+	0.620627i	$0.213122\pi$
$152$	0	0
$153$	−9.18034	−0.742186
$154$	0	0
$155$	−1.38197	−0.111002
$156$	0	0
$157$	−17.5623	−1.40162	−0.700812	−	0.713346i	$-0.747179\pi$
$157$	−17.5623	−1.40162	−0.700812	+	0.713346i	$0.747179\pi$
$158$	0	0
$159$	−10.7082	−0.849216
$160$	0	0
$161$	−9.52786	−0.750901
$162$	0	0
$163$	11.3262	0.887139	0.443570	−	0.896240i	$-0.353712\pi$
$163$	11.3262	0.887139	0.443570	+	0.896240i	$0.353712\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	17.7984	1.37728	0.688640	−	0.725104i	$-0.258208\pi$
$167$	17.7984	1.37728	0.688640	+	0.725104i	$0.258208\pi$
$168$	0	0
$169$	−7.32624	−0.563557
$170$	0	0
$171$	−14.8541	−1.13592
$172$	0	0
$173$	−6.79837	−0.516871	−0.258435	−	0.966029i	$-0.583207\pi$
$173$	−6.79837	−0.516871	−0.258435	+	0.966029i	$0.583207\pi$
$174$	0	0
$175$	9.18034	0.693968
$176$	0	0
$177$	−2.85410	−0.214527
$178$	0	0
$179$	−2.61803	−0.195681	−0.0978405	−	0.995202i	$-0.531194\pi$
$179$	−2.61803	−0.195681	−0.0978405	+	0.995202i	$0.531194\pi$
$180$	0	0
$181$	1.61803	0.120268	0.0601338	−	0.998190i	$-0.480847\pi$
$181$	1.61803	0.120268	0.0601338	+	0.998190i	$0.480847\pi$
$182$	0	0
$183$	−6.23607	−0.460983
$184$	0	0
$185$	3.00000	0.220564
$186$	0	0
$187$	0	0
$188$	0	0
$189$	8.61803	0.626870
$190$	0	0
$191$	−26.0344	−1.88379	−0.941893	−	0.335913i	$-0.890955\pi$
$191$	−26.0344	−1.88379	−0.941893	+	0.335913i	$0.890955\pi$
$192$	0	0
$193$	22.5623	1.62407	0.812035	−	0.583609i	$-0.198360\pi$
$193$	22.5623	1.62407	0.812035	+	0.583609i	$0.198360\pi$
$194$	0	0
$195$	−10.0902	−0.722572
$196$	0	0
$197$	−2.94427	−0.209771	−0.104885	−	0.994484i	$-0.533448\pi$
$197$	−2.94427	−0.209771	−0.104885	+	0.994484i	$0.533448\pi$
$198$	0	0
$199$	0.944272	0.0669377	0.0334688	−	0.999440i	$-0.489345\pi$
$199$	0.944272	0.0669377	0.0334688	+	0.999440i	$0.489345\pi$
$200$	0	0
$201$	−33.8885	−2.39031
$202$	0	0
$203$	−33.2148	−2.33122
$204$	0	0
$205$	−13.9443	−0.973910
$206$	0	0
$207$	9.52786	0.662232
$208$	0	0
$209$	0	0
$210$	0	0
$211$	−11.9098	−0.819907	−0.409953	−	0.912107i	$-0.634455\pi$
$211$	−11.9098	−0.819907	−0.409953	+	0.912107i	$0.634455\pi$
$212$	0	0
$213$	−16.7082	−1.14483
$214$	0	0
$215$	0	0
$216$	0	0
$217$	3.29180	0.223462
$218$	0	0
$219$	−2.38197	−0.160958
$220$	0	0
$221$	−5.67376	−0.381659
$222$	0	0
$223$	12.1459	0.813349	0.406675	−	0.913573i	$-0.366688\pi$
$223$	12.1459	0.813349	0.406675	+	0.913573i	$0.366688\pi$
$224$	0	0
$225$	−9.18034	−0.612023
$226$	0	0
$227$	−5.67376	−0.376581	−0.188290	−	0.982113i	$-0.560295\pi$
$227$	−5.67376	−0.376581	−0.188290	+	0.982113i	$0.560295\pi$
$228$	0	0
$229$	−16.2705	−1.07519	−0.537593	−	0.843205i	$-0.680666\pi$
$229$	−16.2705	−1.07519	−0.537593	+	0.843205i	$0.680666\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−2.38197	−0.156048	−0.0780239	−	0.996951i	$-0.524861\pi$
$233$	−2.38197	−0.156048	−0.0780239	+	0.996951i	$0.524861\pi$
$234$	0	0
$235$	−2.23607	−0.145865
$236$	0	0
$237$	18.7082	1.21523
$238$	0	0
$239$	−11.5623	−0.747903	−0.373952	−	0.927448i	$-0.621997\pi$
$239$	−11.5623	−0.747903	−0.373952	+	0.927448i	$0.621997\pi$
$240$	0	0
$241$	−12.4721	−0.803401	−0.401700	−	0.915771i	$-0.631581\pi$
$241$	−12.4721	−0.803401	−0.401700	+	0.915771i	$0.631581\pi$
$242$	0	0
$243$	21.6525	1.38901
$244$	0	0
$245$	12.7082	0.811897
$246$	0	0
$247$	−9.18034	−0.584131
$248$	0	0
$249$	34.1246	2.16256
$250$	0	0
$251$	25.7984	1.62838	0.814189	−	0.580599i	$-0.197182\pi$
$251$	25.7984	1.62838	0.814189	+	0.580599i	$0.197182\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	10.0902	0.631871
$256$	0	0
$257$	−4.90983	−0.306267	−0.153133	−	0.988206i	$-0.548936\pi$
$257$	−4.90983	−0.306267	−0.153133	+	0.988206i	$0.548936\pi$
$258$	0	0
$259$	−7.14590	−0.444024
$260$	0	0
$261$	33.2148	2.05594
$262$	0	0
$263$	5.88854	0.363103	0.181552	−	0.983381i	$-0.441888\pi$
$263$	5.88854	0.363103	0.181552	+	0.983381i	$0.441888\pi$
$264$	0	0
$265$	6.61803	0.406543
$266$	0	0
$267$	−1.23607	−0.0756461
$268$	0	0
$269$	24.0902	1.46880	0.734402	−	0.678715i	$-0.237463\pi$
$269$	24.0902	1.46880	0.734402	+	0.678715i	$0.237463\pi$
$270$	0	0
$271$	−19.2705	−1.17060	−0.585300	−	0.810817i	$-0.699023\pi$
$271$	−19.2705	−1.17060	−0.585300	+	0.810817i	$0.699023\pi$
$272$	0	0
$273$	24.0344	1.45463
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−22.5623	−1.35564	−0.677819	−	0.735229i	$-0.737075\pi$
$277$	−22.5623	−1.35564	−0.677819	+	0.735229i	$0.737075\pi$
$278$	0	0
$279$	−3.29180	−0.197075
$280$	0	0
$281$	−27.3262	−1.63015	−0.815073	−	0.579358i	$-0.803303\pi$
$281$	−27.3262	−1.63015	−0.815073	+	0.579358i	$0.803303\pi$
$282$	0	0
$283$	11.5623	0.687308	0.343654	−	0.939096i	$-0.388335\pi$
$283$	11.5623	0.687308	0.343654	+	0.939096i	$0.388335\pi$
$284$	0	0
$285$	16.3262	0.967083
$286$	0	0
$287$	33.2148	1.96061
$288$	0	0
$289$	−11.3262	−0.666249
$290$	0	0
$291$	38.1246	2.23490
$292$	0	0
$293$	8.61803	0.503471	0.251735	−	0.967796i	$-0.418999\pi$
$293$	8.61803	0.503471	0.251735	+	0.967796i	$0.418999\pi$
$294$	0	0
$295$	1.76393	0.102700
$296$	0	0
$297$	0	0
$298$	0	0
$299$	5.88854	0.340543
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−46.5967	−2.67691
$304$	0	0
$305$	3.85410	0.220685
$306$	0	0
$307$	−9.52786	−0.543784	−0.271892	−	0.962328i	$-0.587649\pi$
$307$	−9.52786	−0.543784	−0.271892	+	0.962328i	$0.587649\pi$
$308$	0	0
$309$	−20.5623	−1.16975
$310$	0	0
$311$	−16.7984	−0.952548	−0.476274	−	0.879297i	$-0.658013\pi$
$311$	−16.7984	−0.952548	−0.476274	+	0.879297i	$0.658013\pi$
$312$	0	0
$313$	−7.50658	−0.424297	−0.212148	−	0.977237i	$-0.568046\pi$
$313$	−7.50658	−0.424297	−0.212148	+	0.977237i	$0.568046\pi$
$314$	0	0
$315$	−24.0344	−1.35419
$316$	0	0
$317$	14.5623	0.817901	0.408950	−	0.912557i	$-0.365895\pi$
$317$	14.5623	0.817901	0.408950	+	0.912557i	$0.365895\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	−30.2705	−1.68954
$322$	0	0
$323$	9.18034	0.510808
$324$	0	0
$325$	−5.67376	−0.314724
$326$	0	0
$327$	32.6525	1.80569
$328$	0	0
$329$	5.32624	0.293645
$330$	0	0
$331$	−12.0000	−0.659580	−0.329790	−	0.944054i	$-0.606978\pi$
$331$	−12.0000	−0.659580	−0.329790	+	0.944054i	$0.606978\pi$
$332$	0	0
$333$	7.14590	0.391593
$334$	0	0
$335$	20.9443	1.14431
$336$	0	0
$337$	−24.0344	−1.30924	−0.654620	−	0.755958i	$-0.727171\pi$
$337$	−24.0344	−1.30924	−0.654620	+	0.755958i	$0.727171\pi$
$338$	0	0
$339$	−45.9787	−2.49722
$340$	0	0
$341$	0	0
$342$	0	0
$343$	−3.29180	−0.177740
$344$	0	0
$345$	−10.4721	−0.563801
$346$	0	0
$347$	17.7984	0.955467	0.477733	−	0.878505i	$-0.341458\pi$
$347$	17.7984	0.955467	0.477733	+	0.878505i	$0.341458\pi$
$348$	0	0
$349$	−16.3262	−0.873923	−0.436962	−	0.899480i	$-0.643945\pi$
$349$	−16.3262	−0.873923	−0.436962	+	0.899480i	$0.643945\pi$
$350$	0	0
$351$	−5.32624	−0.284294
$352$	0	0
$353$	−14.9443	−0.795403	−0.397702	−	0.917515i	$-0.630192\pi$
$353$	−14.9443	−0.795403	−0.397702	+	0.917515i	$0.630192\pi$
$354$	0	0
$355$	10.3262	0.548060
$356$	0	0
$357$	−24.0344	−1.27204
$358$	0	0
$359$	11.5623	0.610235	0.305118	−	0.952315i	$-0.401304\pi$
$359$	11.5623	0.610235	0.305118	+	0.952315i	$0.401304\pi$
$360$	0	0
$361$	−4.14590	−0.218205
$362$	0	0
$363$	0	0
$364$	0	0
$365$	1.47214	0.0770551
$366$	0	0
$367$	−20.1459	−1.05161	−0.525804	−	0.850606i	$-0.676235\pi$
$367$	−20.1459	−1.05161	−0.525804	+	0.850606i	$0.676235\pi$
$368$	0	0
$369$	−33.2148	−1.72909
$370$	0	0
$371$	−15.7639	−0.818423
$372$	0	0
$373$	−6.58359	−0.340885	−0.170443	−	0.985368i	$-0.554520\pi$
$373$	−6.58359	−0.340885	−0.170443	+	0.985368i	$0.554520\pi$
$374$	0	0
$375$	31.2705	1.61480
$376$	0	0
$377$	20.5279	1.05724
$378$	0	0
$379$	3.14590	0.161594	0.0807970	−	0.996731i	$-0.474253\pi$
$379$	3.14590	0.161594	0.0807970	+	0.996731i	$0.474253\pi$
$380$	0	0
$381$	6.23607	0.319483
$382$	0	0
$383$	−19.9098	−1.01734	−0.508672	−	0.860960i	$-0.669864\pi$
$383$	−19.9098	−1.01734	−0.508672	+	0.860960i	$0.669864\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	26.7984	1.35873	0.679366	−	0.733800i	$-0.262255\pi$
$389$	26.7984	1.35873	0.679366	+	0.733800i	$0.262255\pi$
$390$	0	0
$391$	−5.88854	−0.297796
$392$	0	0
$393$	24.9443	1.25827
$394$	0	0
$395$	−11.5623	−0.581763
$396$	0	0
$397$	−32.8328	−1.64783	−0.823916	−	0.566712i	$-0.808215\pi$
$397$	−32.8328	−1.64783	−0.823916	+	0.566712i	$0.808215\pi$
$398$	0	0
$399$	−38.8885	−1.94686
$400$	0	0
$401$	6.74265	0.336712	0.168356	−	0.985726i	$-0.446154\pi$
$401$	6.74265	0.336712	0.168356	+	0.985726i	$0.446154\pi$
$402$	0	0
$403$	−2.03444	−0.101343
$404$	0	0
$405$	−9.23607	−0.458944
$406$	0	0
$407$	0	0
$408$	0	0
$409$	7.14590	0.353342	0.176671	−	0.984270i	$-0.443467\pi$
$409$	7.14590	0.353342	0.176671	+	0.984270i	$0.443467\pi$
$410$	0	0
$411$	−36.1246	−1.78190
$412$	0	0
$413$	−4.20163	−0.206749
$414$	0	0
$415$	−21.0902	−1.03528
$416$	0	0
$417$	30.2705	1.48235
$418$	0	0
$419$	−4.58359	−0.223923	−0.111962	−	0.993713i	$-0.535713\pi$
$419$	−4.58359	−0.223923	−0.111962	+	0.993713i	$0.535713\pi$
$420$	0	0
$421$	9.27051	0.451817	0.225909	−	0.974149i	$-0.427465\pi$
$421$	9.27051	0.451817	0.225909	+	0.974149i	$0.427465\pi$
$422$	0	0
$423$	−5.32624	−0.258971
$424$	0	0
$425$	5.67376	0.275218
$426$	0	0
$427$	−9.18034	−0.444268
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−16.6738	−0.803147	−0.401573	−	0.915827i	$-0.631537\pi$
$431$	−16.6738	−0.803147	−0.401573	+	0.915827i	$0.631537\pi$
$432$	0	0
$433$	19.4508	0.934748	0.467374	−	0.884060i	$-0.345200\pi$
$433$	19.4508	0.934748	0.467374	+	0.884060i	$0.345200\pi$
$434$	0	0
$435$	−36.5066	−1.75036
$436$	0	0
$437$	−9.52786	−0.455780
$438$	0	0
$439$	−15.4164	−0.735785	−0.367893	−	0.929868i	$-0.619921\pi$
$439$	−15.4164	−0.735785	−0.367893	+	0.929868i	$0.619921\pi$
$440$	0	0
$441$	30.2705	1.44145
$442$	0	0
$443$	3.27051	0.155387	0.0776933	−	0.996977i	$-0.475245\pi$
$443$	3.27051	0.155387	0.0776933	+	0.996977i	$0.475245\pi$
$444$	0	0
$445$	0.763932	0.0362139
$446$	0	0
$447$	−6.23607	−0.294956
$448$	0	0
$449$	−14.5623	−0.687238	−0.343619	−	0.939109i	$-0.611653\pi$
$449$	−14.5623	−0.687238	−0.343619	+	0.939109i	$0.611653\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	−50.4508	−2.37039
$454$	0	0
$455$	−14.8541	−0.696371
$456$	0	0
$457$	7.14590	0.334271	0.167136	−	0.985934i	$-0.446548\pi$
$457$	7.14590	0.334271	0.167136	+	0.985934i	$0.446548\pi$
$458$	0	0
$459$	5.32624	0.248607
$460$	0	0
$461$	−33.7771	−1.57316	−0.786578	−	0.617491i	$-0.788149\pi$
$461$	−33.7771	−1.57316	−0.786578	+	0.617491i	$0.788149\pi$
$462$	0	0
$463$	12.0000	0.557687	0.278844	−	0.960337i	$-0.410049\pi$
$463$	12.0000	0.557687	0.278844	+	0.960337i	$0.410049\pi$
$464$	0	0
$465$	3.61803	0.167782
$466$	0	0
$467$	12.6738	0.586472	0.293236	−	0.956040i	$-0.405268\pi$
$467$	12.6738	0.586472	0.293236	+	0.956040i	$0.405268\pi$
$468$	0	0
$469$	−49.8885	−2.30364
$470$	0	0
$471$	45.9787	2.11859
$472$	0	0
$473$	0	0
$474$	0	0
$475$	9.18034	0.421223
$476$	0	0
$477$	15.7639	0.721781
$478$	0	0
$479$	−27.3262	−1.24857	−0.624284	−	0.781198i	$-0.714609\pi$
$479$	−27.3262	−1.24857	−0.624284	+	0.781198i	$0.714609\pi$
$480$	0	0
$481$	4.41641	0.201371
$482$	0	0
$483$	24.9443	1.13500
$484$	0	0
$485$	−23.5623	−1.06991
$486$	0	0
$487$	18.0344	0.817219	0.408609	−	0.912709i	$-0.366014\pi$
$487$	18.0344	0.817219	0.408609	+	0.912709i	$0.366014\pi$
$488$	0	0
$489$	−29.6525	−1.34093
$490$	0	0
$491$	−11.5623	−0.521800	−0.260900	−	0.965366i	$-0.584019\pi$
$491$	−11.5623	−0.521800	−0.260900	+	0.965366i	$0.584019\pi$
$492$	0	0
$493$	−20.5279	−0.924528
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−24.5967	−1.10331
$498$	0	0
$499$	−23.2705	−1.04173	−0.520866	−	0.853639i	$-0.674391\pi$
$499$	−23.2705	−1.04173	−0.520866	+	0.853639i	$0.674391\pi$
$500$	0	0
$501$	−46.5967	−2.08179
$502$	0	0
$503$	−36.5066	−1.62775	−0.813874	−	0.581042i	$-0.802645\pi$
$503$	−36.5066	−1.62775	−0.813874	+	0.581042i	$0.802645\pi$
$504$	0	0
$505$	28.7984	1.28151
$506$	0	0
$507$	19.1803	0.851829
$508$	0	0
$509$	28.6869	1.27153	0.635763	−	0.771885i	$-0.280686\pi$
$509$	28.6869	1.27153	0.635763	+	0.771885i	$0.280686\pi$
$510$	0	0
$511$	−3.50658	−0.155122
$512$	0	0
$513$	8.61803	0.380495
$514$	0	0
$515$	12.7082	0.559990
$516$	0	0
$517$	0	0
$518$	0	0
$519$	17.7984	0.781262
$520$	0	0
$521$	7.67376	0.336194	0.168097	−	0.985770i	$-0.446238\pi$
$521$	7.67376	0.336194	0.168097	+	0.985770i	$0.446238\pi$
$522$	0	0
$523$	−28.4508	−1.24407	−0.622034	−	0.782990i	$-0.713694\pi$
$523$	−28.4508	−1.24407	−0.622034	+	0.782990i	$0.713694\pi$
$524$	0	0
$525$	−24.0344	−1.04895
$526$	0	0
$527$	2.03444	0.0886217
$528$	0	0
$529$	−16.8885	−0.734285
$530$	0	0
$531$	4.20163	0.182335
$532$	0	0
$533$	−20.5279	−0.889160
$534$	0	0
$535$	18.7082	0.808826
$536$	0	0
$537$	6.85410	0.295776
$538$	0	0
$539$	0	0
$540$	0	0
$541$	8.27051	0.355577	0.177788	−	0.984069i	$-0.443106\pi$
$541$	8.27051	0.355577	0.177788	+	0.984069i	$0.443106\pi$
$542$	0	0
$543$	−4.23607	−0.181787
$544$	0	0
$545$	−20.1803	−0.864431
$546$	0	0
$547$	−38.3262	−1.63871	−0.819356	−	0.573285i	$-0.805669\pi$
$547$	−38.3262	−1.63871	−0.819356	+	0.573285i	$0.805669\pi$
$548$	0	0
$549$	9.18034	0.391807
$550$	0	0
$551$	−33.2148	−1.41500
$552$	0	0
$553$	27.5410	1.17116
$554$	0	0
$555$	−7.85410	−0.333388
$556$	0	0
$557$	−31.7426	−1.34498	−0.672490	−	0.740107i	$-0.734775\pi$
$557$	−31.7426	−1.34498	−0.672490	+	0.740107i	$0.734775\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0	0
$562$	0	0
$563$	13.0344	0.549336	0.274668	−	0.961539i	$-0.411432\pi$
$563$	13.0344	0.549336	0.274668	+	0.961539i	$0.411432\pi$
$564$	0	0
$565$	28.4164	1.19549
$566$	0	0
$567$	22.0000	0.923913
$568$	0	0
$569$	−29.9230	−1.25444	−0.627218	−	0.778843i	$-0.715807\pi$
$569$	−29.9230	−1.25444	−0.627218	+	0.778843i	$0.715807\pi$
$570$	0	0
$571$	−9.52786	−0.398729	−0.199364	−	0.979925i	$-0.563888\pi$
$571$	−9.52786	−0.398729	−0.199364	+	0.979925i	$0.563888\pi$
$572$	0	0
$573$	68.1591	2.84739
$574$	0	0
$575$	−5.88854	−0.245569
$576$	0	0
$577$	−36.0902	−1.50245	−0.751227	−	0.660044i	$-0.770538\pi$
$577$	−36.0902	−1.50245	−0.751227	+	0.660044i	$0.770538\pi$
$578$	0	0
$579$	−59.0689	−2.45482
$580$	0	0
$581$	50.2361	2.08414
$582$	0	0
$583$	0	0
$584$	0	0
$585$	14.8541	0.614142
$586$	0	0
$587$	−7.85410	−0.324173	−0.162087	−	0.986777i	$-0.551822\pi$
$587$	−7.85410	−0.324173	−0.162087	+	0.986777i	$0.551822\pi$
$588$	0	0
$589$	3.29180	0.135636
$590$	0	0
$591$	7.70820	0.317073
$592$	0	0
$593$	−37.4164	−1.53651	−0.768254	−	0.640145i	$-0.778874\pi$
$593$	−37.4164	−1.53651	−0.768254	+	0.640145i	$0.778874\pi$
$594$	0	0
$595$	14.8541	0.608959
$596$	0	0
$597$	−2.47214	−0.101178
$598$	0	0
$599$	−45.9787	−1.87864	−0.939320	−	0.343043i	$-0.888542\pi$
$599$	−45.9787	−1.87864	−0.939320	+	0.343043i	$0.888542\pi$
$600$	0	0
$601$	−33.5623	−1.36904	−0.684518	−	0.728996i	$-0.739987\pi$
$601$	−33.5623	−1.36904	−0.684518	+	0.728996i	$0.739987\pi$
$602$	0	0
$603$	49.8885	2.03162
$604$	0	0
$605$	0	0
$606$	0	0
$607$	21.4377	0.870129	0.435065	−	0.900399i	$-0.356726\pi$
$607$	21.4377	0.870129	0.435065	+	0.900399i	$0.356726\pi$
$608$	0	0
$609$	86.9574	3.52369
$610$	0	0
$611$	−3.29180	−0.133172
$612$	0	0
$613$	−10.4377	−0.421574	−0.210787	−	0.977532i	$-0.567603\pi$
$613$	−10.4377	−0.421574	−0.210787	+	0.977532i	$0.567603\pi$
$614$	0	0
$615$	36.5066	1.47209
$616$	0	0
$617$	−7.52786	−0.303060	−0.151530	−	0.988453i	$-0.548420\pi$
$617$	−7.52786	−0.303060	−0.151530	+	0.988453i	$0.548420\pi$
$618$	0	0
$619$	22.3262	0.897367	0.448684	−	0.893691i	$-0.351893\pi$
$619$	22.3262	0.897367	0.448684	+	0.893691i	$0.351893\pi$
$620$	0	0
$621$	−5.52786	−0.221826
$622$	0	0
$623$	−1.81966	−0.0729031
$624$	0	0
$625$	−7.41641	−0.296656
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−4.41641	−0.176094
$630$	0	0
$631$	23.8541	0.949617	0.474808	−	0.880089i	$-0.342517\pi$
$631$	23.8541	0.949617	0.474808	+	0.880089i	$0.342517\pi$
$632$	0	0
$633$	31.1803	1.23931
$634$	0	0
$635$	−3.85410	−0.152945
$636$	0	0
$637$	18.7082	0.741246
$638$	0	0
$639$	24.5967	0.973032
$640$	0	0
$641$	27.0902	1.07000	0.534999	−	0.844853i	$-0.320312\pi$
$641$	27.0902	1.07000	0.534999	+	0.844853i	$0.320312\pi$
$642$	0	0
$643$	−12.2705	−0.483902	−0.241951	−	0.970289i	$-0.577787\pi$
$643$	−12.2705	−0.483902	−0.241951	+	0.970289i	$0.577787\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−6.74265	−0.265081	−0.132540	−	0.991178i	$-0.542313\pi$
$647$	−6.74265	−0.265081	−0.132540	+	0.991178i	$0.542313\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	−8.61803	−0.337767
$652$	0	0
$653$	−7.67376	−0.300298	−0.150149	−	0.988663i	$-0.547975\pi$
$653$	−7.67376	−0.300298	−0.150149	+	0.988663i	$0.547975\pi$
$654$	0	0
$655$	−15.4164	−0.602369
$656$	0	0
$657$	3.50658	0.136805
$658$	0	0
$659$	−40.3607	−1.57223	−0.786114	−	0.618081i	$-0.787910\pi$
$659$	−40.3607	−1.57223	−0.786114	+	0.618081i	$0.787910\pi$
$660$	0	0
$661$	30.3607	1.18089	0.590447	−	0.807077i	$-0.298952\pi$
$661$	30.3607	1.18089	0.590447	+	0.807077i	$0.298952\pi$
$662$	0	0
$663$	14.8541	0.576886
$664$	0	0
$665$	24.0344	0.932016
$666$	0	0
$667$	21.3050	0.824931
$668$	0	0
$669$	−31.7984	−1.22940
$670$	0	0
$671$	0	0
$672$	0	0
$673$	−36.8541	−1.42062	−0.710311	−	0.703888i	$-0.751446\pi$
$673$	−36.8541	−1.42062	−0.710311	+	0.703888i	$0.751446\pi$
$674$	0	0
$675$	5.32624	0.205007
$676$	0	0
$677$	−3.50658	−0.134769	−0.0673844	−	0.997727i	$-0.521465\pi$
$677$	−3.50658	−0.134769	−0.0673844	+	0.997727i	$0.521465\pi$
$678$	0	0
$679$	56.1246	2.15386
$680$	0	0
$681$	14.8541	0.569210
$682$	0	0
$683$	−0.944272	−0.0361316	−0.0180658	−	0.999837i	$-0.505751\pi$
$683$	−0.944272	−0.0361316	−0.0180658	+	0.999837i	$0.505751\pi$
$684$	0	0
$685$	22.3262	0.853042
$686$	0	0
$687$	42.5967	1.62517
$688$	0	0
$689$	9.74265	0.371165
$690$	0	0
$691$	−24.0902	−0.916433	−0.458217	−	0.888841i	$-0.651512\pi$
$691$	−24.0902	−0.916433	−0.458217	+	0.888841i	$0.651512\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−18.7082	−0.709643
$696$	0	0
$697$	20.5279	0.777548
$698$	0	0
$699$	6.23607	0.235870
$700$	0	0
$701$	48.9787	1.84990	0.924950	−	0.380088i	$-0.124106\pi$
$701$	48.9787	1.84990	0.924950	+	0.380088i	$0.124106\pi$
$702$	0	0
$703$	−7.14590	−0.269513
$704$	0	0
$705$	5.85410	0.220478
$706$	0	0
$707$	−68.5967	−2.57985
$708$	0	0
$709$	−23.3262	−0.876035	−0.438018	−	0.898966i	$-0.644319\pi$
$709$	−23.3262	−0.876035	−0.438018	+	0.898966i	$0.644319\pi$
$710$	0	0
$711$	−27.5410	−1.03287
$712$	0	0
$713$	−2.11146	−0.0790747
$714$	0	0
$715$	0	0
$716$	0	0
$717$	30.2705	1.13047
$718$	0	0
$719$	39.2705	1.46454	0.732271	−	0.681013i	$-0.238460\pi$
$719$	39.2705	1.46454	0.732271	+	0.681013i	$0.238460\pi$
$720$	0	0
$721$	−30.2705	−1.12733
$722$	0	0
$723$	32.6525	1.21436
$724$	0	0
$725$	−20.5279	−0.762386
$726$	0	0
$727$	−32.0000	−1.18681	−0.593407	−	0.804902i	$-0.702218\pi$
$727$	−32.0000	−1.18681	−0.593407	+	0.804902i	$0.702218\pi$
$728$	0	0
$729$	−39.5623	−1.46527
$730$	0	0
$731$	0	0
$732$	0	0
$733$	33.5623	1.23965	0.619826	−	0.784739i	$-0.287203\pi$
$733$	33.5623	1.23965	0.619826	+	0.784739i	$0.287203\pi$
$734$	0	0
$735$	−33.2705	−1.22720
$736$	0	0
$737$	0	0
$738$	0	0
$739$	−23.6869	−0.871337	−0.435669	−	0.900107i	$-0.643488\pi$
$739$	−23.6869	−0.871337	−0.435669	+	0.900107i	$0.643488\pi$
$740$	0	0
$741$	24.0344	0.882927
$742$	0	0
$743$	3.50658	0.128644	0.0643219	−	0.997929i	$-0.479512\pi$
$743$	3.50658	0.128644	0.0643219	+	0.997929i	$0.479512\pi$
$744$	0	0
$745$	3.85410	0.141203
$746$	0	0
$747$	−50.2361	−1.83804
$748$	0	0
$749$	−44.5623	−1.62827
$750$	0	0
$751$	−6.90983	−0.252143	−0.126072	−	0.992021i	$-0.540237\pi$
$751$	−6.90983	−0.252143	−0.126072	+	0.992021i	$0.540237\pi$
$752$	0	0
$753$	−67.5410	−2.46133
$754$	0	0
$755$	31.1803	1.13477
$756$	0	0
$757$	−26.7426	−0.971978	−0.485989	−	0.873965i	$-0.661540\pi$
$757$	−26.7426	−0.971978	−0.485989	+	0.873965i	$0.661540\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	18.9230	0.685958	0.342979	−	0.939343i	$-0.388564\pi$
$761$	18.9230	0.685958	0.342979	+	0.939343i	$0.388564\pi$
$762$	0	0
$763$	48.0689	1.74021
$764$	0	0
$765$	−14.8541	−0.537051
$766$	0	0
$767$	2.59675	0.0937631
$768$	0	0
$769$	−37.4164	−1.34927	−0.674635	−	0.738151i	$-0.735699\pi$
$769$	−37.4164	−1.34927	−0.674635	+	0.738151i	$0.735699\pi$
$770$	0	0
$771$	12.8541	0.462929
$772$	0	0
$773$	28.3262	1.01882	0.509412	−	0.860523i	$-0.329863\pi$
$773$	28.3262	1.01882	0.509412	+	0.860523i	$0.329863\pi$
$774$	0	0
$775$	2.03444	0.0730793
$776$	0	0
$777$	18.7082	0.671153
$778$	0	0
$779$	33.2148	1.19004
$780$	0	0
$781$	0	0
$782$	0	0
$783$	−19.2705	−0.688672
$784$	0	0
$785$	−28.4164	−1.01423
$786$	0	0
$787$	−16.6738	−0.594355	−0.297178	−	0.954822i	$-0.596045\pi$
$787$	−16.6738	−0.594355	−0.297178	+	0.954822i	$0.596045\pi$
$788$	0	0
$789$	−15.4164	−0.548839
$790$	0	0
$791$	−67.6869	−2.40667
$792$	0	0
$793$	5.67376	0.201481
$794$	0	0
$795$	−17.3262	−0.614498
$796$	0	0
$797$	−13.7984	−0.488763	−0.244382	−	0.969679i	$-0.578585\pi$
$797$	−13.7984	−0.488763	−0.244382	+	0.969679i	$0.578585\pi$
$798$	0	0
$799$	3.29180	0.116455
$800$	0	0
$801$	1.81966	0.0642945
$802$	0	0
$803$	0	0
$804$	0	0
$805$	−15.4164	−0.543357
$806$	0	0
$807$	−63.0689	−2.22013
$808$	0	0
$809$	18.9230	0.665297	0.332648	−	0.943051i	$-0.392058\pi$
$809$	18.9230	0.665297	0.332648	+	0.943051i	$0.392058\pi$
$810$	0	0
$811$	30.6180	1.07514	0.537572	−	0.843218i	$-0.319342\pi$
$811$	30.6180	1.07514	0.537572	+	0.843218i	$0.319342\pi$
$812$	0	0
$813$	50.4508	1.76939
$814$	0	0
$815$	18.3262	0.641940
$816$	0	0
$817$	0	0
$818$	0	0
$819$	−35.3820	−1.23635
$820$	0	0
$821$	−25.8541	−0.902314	−0.451157	−	0.892445i	$-0.648989\pi$
$821$	−25.8541	−0.902314	−0.451157	+	0.892445i	$0.648989\pi$
$822$	0	0
$823$	39.5066	1.37711	0.688556	−	0.725183i	$-0.258245\pi$
$823$	39.5066	1.37711	0.688556	+	0.725183i	$0.258245\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	13.0344	0.453252	0.226626	−	0.973982i	$-0.427231\pi$
$827$	13.0344	0.453252	0.226626	+	0.973982i	$0.427231\pi$
$828$	0	0
$829$	−34.8673	−1.21099	−0.605495	−	0.795849i	$-0.707025\pi$
$829$	−34.8673	−1.21099	−0.605495	+	0.795849i	$0.707025\pi$
$830$	0	0
$831$	59.0689	2.04908
$832$	0	0
$833$	−18.7082	−0.648201
$834$	0	0
$835$	28.7984	0.996609
$836$	0	0
$837$	1.90983	0.0660134
$838$	0	0
$839$	2.61803	0.0903846	0.0451923	−	0.998978i	$-0.485610\pi$
$839$	2.61803	0.0903846	0.0451923	+	0.998978i	$0.485610\pi$
$840$	0	0
$841$	45.2705	1.56105
$842$	0	0
$843$	71.5410	2.46400
$844$	0	0
$845$	−11.8541	−0.407794
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−30.2705	−1.03888
$850$	0	0
$851$	4.58359	0.157124
$852$	0	0
$853$	17.7984	0.609405	0.304702	−	0.952448i	$-0.401443\pi$
$853$	17.7984	0.609405	0.304702	+	0.952448i	$0.401443\pi$
$854$	0	0
$855$	−24.0344	−0.821961
$856$	0	0
$857$	−2.94427	−0.100574	−0.0502872	−	0.998735i	$-0.516014\pi$
$857$	−2.94427	−0.100574	−0.0502872	+	0.998735i	$0.516014\pi$
$858$	0	0
$859$	−31.0557	−1.05961	−0.529804	−	0.848120i	$-0.677734\pi$
$859$	−31.0557	−1.05961	−0.529804	+	0.848120i	$0.677734\pi$
$860$	0	0
$861$	−86.9574	−2.96350
$862$	0	0
$863$	25.2148	0.858321	0.429161	−	0.903228i	$-0.358809\pi$
$863$	25.2148	0.858321	0.429161	+	0.903228i	$0.358809\pi$
$864$	0	0
$865$	−11.0000	−0.374011
$866$	0	0
$867$	29.6525	1.00705
$868$	0	0
$869$	0	0
$870$	0	0
$871$	30.8328	1.04473
$872$	0	0
$873$	−56.1246	−1.89953
$874$	0	0
$875$	46.0344	1.55625
$876$	0	0
$877$	27.6738	0.934477	0.467238	−	0.884131i	$-0.345249\pi$
$877$	27.6738	0.934477	0.467238	+	0.884131i	$0.345249\pi$
$878$	0	0
$879$	−22.5623	−0.761008
$880$	0	0
$881$	15.8885	0.535299	0.267649	−	0.963516i	$-0.413753\pi$
$881$	15.8885	0.535299	0.267649	+	0.963516i	$0.413753\pi$
$882$	0	0
$883$	5.02129	0.168980	0.0844899	−	0.996424i	$-0.473074\pi$
$883$	5.02129	0.168980	0.0844899	+	0.996424i	$0.473074\pi$
$884$	0	0
$885$	−4.61803	−0.155234
$886$	0	0
$887$	2.03444	0.0683099	0.0341549	−	0.999417i	$-0.489126\pi$
$887$	2.03444	0.0683099	0.0341549	+	0.999417i	$0.489126\pi$
$888$	0	0
$889$	9.18034	0.307899
$890$	0	0
$891$	0	0
$892$	0	0
$893$	5.32624	0.178236
$894$	0	0
$895$	−4.23607	−0.141596
$896$	0	0
$897$	−15.4164	−0.514739
$898$	0	0
$899$	−7.36068	−0.245492
$900$	0	0
$901$	−9.74265	−0.324575
$902$	0	0
$903$	0	0
$904$	0	0
$905$	2.61803	0.0870264
$906$	0	0
$907$	1.97871	0.0657021	0.0328511	−	0.999460i	$-0.489541\pi$
$907$	1.97871	0.0657021	0.0328511	+	0.999460i	$0.489541\pi$
$908$	0	0
$909$	68.5967	2.27521
$910$	0	0
$911$	19.5066	0.646282	0.323141	−	0.946351i	$-0.395261\pi$
$911$	19.5066	0.646282	0.323141	+	0.946351i	$0.395261\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	−10.0902	−0.333571
$916$	0	0
$917$	36.7214	1.21265
$918$	0	0
$919$	−21.4377	−0.707164	−0.353582	−	0.935403i	$-0.615037\pi$
$919$	−21.4377	−0.707164	−0.353582	+	0.935403i	$0.615037\pi$
$920$	0	0
$921$	24.9443	0.821942
$922$	0	0
$923$	15.2016	0.500368
$924$	0	0
$925$	−4.41641	−0.145211
$926$	0	0
$927$	30.2705	0.994214
$928$	0	0
$929$	19.9098	0.653220	0.326610	−	0.945159i	$-0.394094\pi$
$929$	19.9098	0.653220	0.326610	+	0.945159i	$0.394094\pi$
$930$	0	0
$931$	−30.2705	−0.992076
$932$	0	0
$933$	43.9787	1.43980
$934$	0	0
$935$	0	0
$936$	0	0
$937$	7.14590	0.233446	0.116723	−	0.993164i	$-0.462761\pi$
$937$	7.14590	0.233446	0.116723	+	0.993164i	$0.462761\pi$
$938$	0	0
$939$	19.6525	0.641334
$940$	0	0
$941$	11.9098	0.388249	0.194125	−	0.980977i	$-0.437813\pi$
$941$	11.9098	0.388249	0.194125	+	0.980977i	$0.437813\pi$
$942$	0	0
$943$	−21.3050	−0.693785
$944$	0	0
$945$	13.9443	0.453607
$946$	0	0
$947$	−23.7771	−0.772652	−0.386326	−	0.922362i	$-0.626256\pi$
$947$	−23.7771	−0.772652	−0.386326	+	0.922362i	$0.626256\pi$
$948$	0	0
$949$	2.16718	0.0703498
$950$	0	0
$951$	−38.1246	−1.23628
$952$	0	0
$953$	−45.3394	−1.46869	−0.734344	−	0.678778i	$-0.762510\pi$
$953$	−45.3394	−1.46869	−0.734344	+	0.678778i	$0.762510\pi$
$954$	0	0
$955$	−42.1246	−1.36312
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−53.1803	−1.71728
$960$	0	0
$961$	−30.2705	−0.976468
$962$	0	0
$963$	44.5623	1.43600
$964$	0	0
$965$	36.5066	1.17519
$966$	0	0
$967$	44.0000	1.41494	0.707472	−	0.706741i	$-0.249835\pi$
$967$	44.0000	1.41494	0.707472	+	0.706741i	$0.249835\pi$
$968$	0	0
$969$	−24.0344	−0.772098
$970$	0	0
$971$	6.90983	0.221747	0.110873	−	0.993835i	$-0.464635\pi$
$971$	6.90983	0.221747	0.110873	+	0.993835i	$0.464635\pi$
$972$	0	0
$973$	44.5623	1.42860
$974$	0	0
$975$	14.8541	0.475712
$976$	0	0
$977$	−20.4508	−0.654281	−0.327140	−	0.944976i	$-0.606085\pi$
$977$	−20.4508	−0.654281	−0.327140	+	0.944976i	$0.606085\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	−48.0689	−1.53472
$982$	0	0
$983$	39.2705	1.25253	0.626267	−	0.779608i	$-0.284582\pi$
$983$	39.2705	1.25253	0.626267	+	0.779608i	$0.284582\pi$
$984$	0	0
$985$	−4.76393	−0.151791
$986$	0	0
$987$	−13.9443	−0.443851
$988$	0	0
$989$	0	0
$990$	0	0
$991$	−24.0000	−0.762385	−0.381193	−	0.924496i	$-0.624487\pi$
$991$	−24.0000	−0.762385	−0.381193	+	0.924496i	$0.624487\pi$
$992$	0	0
$993$	31.4164	0.996970
$994$	0	0
$995$	1.52786	0.0484365
$996$	0	0
$997$	29.9230	0.947670	0.473835	−	0.880614i	$-0.342869\pi$
$997$	29.9230	0.947670	0.473835	+	0.880614i	$0.342869\pi$
$998$	0	0
$999$	−4.14590	−0.131170

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7744.2.a.bo.1.1		2
4.3	odd	2		7744.2.a.db.1.2		2
8.3	odd	2		484.2.a.c.1.1		2
8.5	even	2		1936.2.a.z.1.2		2
11.2	odd	10		704.2.m.d.257.1		4
11.6	odd	10		704.2.m.d.641.1		4
11.10	odd	2		7744.2.a.bp.1.1		2
24.11	even	2		4356.2.a.u.1.2		2
44.35	even	10		704.2.m.e.257.1		4
44.39	even	10		704.2.m.e.641.1		4
44.43	even	2		7744.2.a.da.1.2		2
88.3	odd	10		484.2.e.d.9.1		4
88.13	odd	10		176.2.m.b.81.1		4
88.19	even	10		484.2.e.e.9.1		4
88.21	odd	2		1936.2.a.ba.1.2		2
88.27	odd	10		484.2.e.c.245.1		4
88.35	even	10		44.2.e.a.37.1	yes	4
88.43	even	2		484.2.a.b.1.1		2
88.51	even	10		484.2.e.e.269.1		4
88.59	odd	10		484.2.e.d.269.1		4
88.61	odd	10		176.2.m.b.113.1		4
88.75	odd	10		484.2.e.c.81.1		4
88.83	even	10		44.2.e.a.25.1	✓	4
264.35	odd	10		396.2.j.a.37.1		4
264.83	odd	10		396.2.j.a.289.1		4
264.131	odd	2		4356.2.a.t.1.2		2
440.83	odd	20		1100.2.cb.a.949.2		8
440.123	odd	20		1100.2.cb.a.1049.1		8
440.259	even	10		1100.2.n.a.201.1		4
440.299	even	10		1100.2.n.a.301.1		4
440.347	odd	20		1100.2.cb.a.949.1		8
440.387	odd	20		1100.2.cb.a.1049.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
44.2.e.a.25.1	✓	4	88.83	even	10
44.2.e.a.37.1	yes	4	88.35	even	10
176.2.m.b.81.1		4	88.13	odd	10
176.2.m.b.113.1		4	88.61	odd	10
396.2.j.a.37.1		4	264.35	odd	10
396.2.j.a.289.1		4	264.83	odd	10
484.2.a.b.1.1		2	88.43	even	2
484.2.a.c.1.1		2	8.3	odd	2
484.2.e.c.81.1		4	88.75	odd	10
484.2.e.c.245.1		4	88.27	odd	10
484.2.e.d.9.1		4	88.3	odd	10
484.2.e.d.269.1		4	88.59	odd	10
484.2.e.e.9.1		4	88.19	even	10
484.2.e.e.269.1		4	88.51	even	10
704.2.m.d.257.1		4	11.2	odd	10
704.2.m.d.641.1		4	11.6	odd	10
704.2.m.e.257.1		4	44.35	even	10
704.2.m.e.641.1		4	44.39	even	10
1100.2.n.a.201.1		4	440.259	even	10
1100.2.n.a.301.1		4	440.299	even	10
1100.2.cb.a.949.1		8	440.347	odd	20
1100.2.cb.a.949.2		8	440.83	odd	20
1100.2.cb.a.1049.1		8	440.123	odd	20
1100.2.cb.a.1049.2		8	440.387	odd	20
1936.2.a.z.1.2		2	8.5	even	2
1936.2.a.ba.1.2		2	88.21	odd	2
4356.2.a.t.1.2		2	264.131	odd	2
4356.2.a.u.1.2		2	24.11	even	2
7744.2.a.bo.1.1		2	1.1	even	1	trivial
7744.2.a.bp.1.1		2	11.10	odd	2
7744.2.a.da.1.2		2	44.43	even	2
7744.2.a.db.1.2		2	4.3	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 7744 = 2^{6} \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7744.a (trivial)

\(f(q)\)	\(=\)	\(q-2.61803 q^{3} +1.61803 q^{5} -3.85410 q^{7} +3.85410 q^{9} +2.38197 q^{13} -4.23607 q^{15} -2.38197 q^{17} -3.85410 q^{19} +10.0902 q^{21} +2.47214 q^{23} -2.38197 q^{25} -2.23607 q^{27} +8.61803 q^{29} -0.854102 q^{31} -6.23607 q^{35} +1.85410 q^{37} -6.23607 q^{39} -8.61803 q^{41} +6.23607 q^{45} -1.38197 q^{47} +7.85410 q^{49} +6.23607 q^{51} +4.09017 q^{53} +10.0902 q^{57} +1.09017 q^{59} +2.38197 q^{61} -14.8541 q^{63} +3.85410 q^{65} +12.9443 q^{67} -6.47214 q^{69} +6.38197 q^{71} +0.909830 q^{73} +6.23607 q^{75} -7.14590 q^{79} -5.70820 q^{81} -13.0344 q^{83} -3.85410 q^{85} -22.5623 q^{87} +0.472136 q^{89} -9.18034 q^{91} +2.23607 q^{93} -6.23607 q^{95} -14.5623 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 3 q^{3} + q^{5} - q^{7} + q^{9} + 7 q^{13} - 4 q^{15} - 7 q^{17} - q^{19} + 9 q^{21} - 4 q^{23} - 7 q^{25} + 15 q^{29} + 5 q^{31} - 8 q^{35} - 3 q^{37} - 8 q^{39} - 15 q^{41} + 8 q^{45} - 5 q^{47}+ \cdots - 9 q^{97}+O(q^{100}) \) 2 * q - 3 * q^3 + q^5 - q^7 + q^9 + 7 * q^13 - 4 * q^15 - 7 * q^17 - q^19 + 9 * q^21 - 4 * q^23 - 7 * q^25 + 15 * q^29 + 5 * q^31 - 8 * q^35 - 3 * q^37 - 8 * q^39 - 15 * q^41 + 8 * q^45 - 5 * q^47 + 9 * q^49 + 8 * q^51 - 3 * q^53 + 9 * q^57 - 9 * q^59 + 7 * q^61 - 23 * q^63 + q^65 + 8 * q^67 - 4 * q^69 + 15 * q^71 + 13 * q^73 + 8 * q^75 - 21 * q^79 + 2 * q^81 + 3 * q^83 - q^85 - 25 * q^87 - 8 * q^89 + 4 * q^91 - 8 * q^95 - 9 * q^97

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