LMFDB - Embedded newform 7744.2.a.dq.1.3

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:	$4$
Coefficient field:	4.4.4400.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 7x^{2} + 11 $ x^4 - 7*x^2 + 11
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 352)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$1.54336$ 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)

$f(q)$	$=$	$q+1.54336 q^{3} +0.381966 q^{5} -1.54336 q^{7} -0.618034 q^{9} +1.61803 q^{13} +0.589512 q^{15} +0.854102 q^{17} -4.63009 q^{19} -2.38197 q^{21} +4.99442 q^{23} -4.85410 q^{25} -5.58394 q^{27} -0.145898 q^{29} +2.13287 q^{31} -0.589512 q^{35} -1.85410 q^{37} +2.49721 q^{39} -0.618034 q^{41} -1.90770 q^{43} -0.236068 q^{45} +9.62451 q^{47} -4.61803 q^{49} +1.31819 q^{51} +8.56231 q^{53} -7.14590 q^{57} +10.3532 q^{59} -12.5623 q^{61} +0.953850 q^{63} +0.618034 q^{65} -8.08115 q^{67} +7.70820 q^{69} -10.9427 q^{71} -9.85410 q^{73} -7.49164 q^{75} -15.9371 q^{79} -6.76393 q^{81} -10.9427 q^{83} +0.326238 q^{85} -0.225173 q^{87} +8.18034 q^{89} -2.49721 q^{91} +3.29180 q^{93} -1.76854 q^{95} -3.14590 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 6 q^{5} + 2 q^{9} + 2 q^{13} - 10 q^{17} - 14 q^{21} - 6 q^{25} - 14 q^{29} + 6 q^{37} + 2 q^{41} + 8 q^{45} - 14 q^{49} - 6 q^{53} - 42 q^{57} - 10 q^{61} - 2 q^{65} + 4 q^{69} - 26 q^{73} - 36 q^{81}+ \cdots - 26 q^{97}+O(q^{100}) $ 4 * q + 6 * q^5 + 2 * q^9 + 2 * q^13 - 10 * q^17 - 14 * q^21 - 6 * q^25 - 14 * q^29 + 6 * q^37 + 2 * q^41 + 8 * q^45 - 14 * q^49 - 6 * q^53 - 42 * q^57 - 10 * q^61 - 2 * q^65 + 4 * q^69 - 26 * q^73 - 36 * q^81 - 30 * q^85 - 12 * q^89 + 40 * q^93 - 26 * 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
$3$	1.54336	0.891060	0.445530	−	0.895267i	$-0.353015\pi$
$3$	1.54336	0.891060	0.445530	+	0.895267i	$0.353015\pi$
$4$	0	0
$5$	0.381966	0.170820	0.0854102	−	0.996346i	$-0.472780\pi$
$5$	0.381966	0.170820	0.0854102	+	0.996346i	$0.472780\pi$
$6$	0	0
$7$	−1.54336	−0.583336	−0.291668	−	0.956520i	$-0.594210\pi$
$7$	−1.54336	−0.583336	−0.291668	+	0.956520i	$0.594210\pi$
$8$	0	0
$9$	−0.618034	−0.206011
$10$	0	0
$11$	0	0
$11$	0	0
$12$	0	0
$13$	1.61803	0.448762	0.224381	−	0.974502i	$-0.427964\pi$
$13$	1.61803	0.448762	0.224381	+	0.974502i	$0.427964\pi$
$14$	0	0
$15$	0.589512	0.152211
$16$	0	0
$17$	0.854102	0.207150	0.103575	−	0.994622i	$-0.466972\pi$
$17$	0.854102	0.207150	0.103575	+	0.994622i	$0.466972\pi$
$18$	0	0
$19$	−4.63009	−1.06221	−0.531107	−	0.847305i	$-0.678224\pi$
$19$	−4.63009	−1.06221	−0.531107	+	0.847305i	$0.678224\pi$
$20$	0	0
$21$	−2.38197	−0.519788
$22$	0	0
$23$	4.99442	1.04141	0.520705	−	0.853737i	$-0.325669\pi$
$23$	4.99442	1.04141	0.520705	+	0.853737i	$0.325669\pi$
$24$	0	0
$25$	−4.85410	−0.970820
$26$	0	0
$27$	−5.58394	−1.07463
$28$	0	0
$29$	−0.145898	−0.0270926	−0.0135463	−	0.999908i	$-0.504312\pi$
$29$	−0.145898	−0.0270926	−0.0135463	+	0.999908i	$0.504312\pi$
$30$	0	0
$31$	2.13287	0.383075	0.191538	−	0.981485i	$-0.438653\pi$
$31$	2.13287	0.383075	0.191538	+	0.981485i	$0.438653\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−0.589512	−0.0996457
$36$	0	0
$37$	−1.85410	−0.304812	−0.152406	−	0.988318i	$-0.548702\pi$
$37$	−1.85410	−0.304812	−0.152406	+	0.988318i	$0.548702\pi$
$38$	0	0
$39$	2.49721	0.399874
$40$	0	0
$41$	−0.618034	−0.0965207	−0.0482603	−	0.998835i	$-0.515368\pi$
$41$	−0.618034	−0.0965207	−0.0482603	+	0.998835i	$0.515368\pi$
$42$	0	0
$43$	−1.90770	−0.290922	−0.145461	−	0.989364i	$-0.546466\pi$
$43$	−1.90770	−0.290922	−0.145461	+	0.989364i	$0.546466\pi$
$44$	0	0
$45$	−0.236068	−0.0351909
$46$	0	0
$47$	9.62451	1.40388	0.701940	−	0.712237i	$-0.252318\pi$
$47$	9.62451	1.40388	0.701940	+	0.712237i	$0.252318\pi$
$48$	0	0
$49$	−4.61803	−0.659719
$50$	0	0
$51$	1.31819	0.184583
$52$	0	0
$53$	8.56231	1.17612	0.588062	−	0.808816i	$-0.299891\pi$
$53$	8.56231	1.17612	0.588062	+	0.808816i	$0.299891\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−7.14590	−0.946497
$58$	0	0
$59$	10.3532	1.34787	0.673935	−	0.738791i	$-0.264603\pi$
$59$	10.3532	1.34787	0.673935	+	0.738791i	$0.264603\pi$
$60$	0	0
$61$	−12.5623	−1.60844	−0.804219	−	0.594333i	$-0.797416\pi$
$61$	−12.5623	−1.60844	−0.804219	+	0.594333i	$0.797416\pi$
$62$	0	0
$63$	0.953850	0.120174
$64$	0	0
$65$	0.618034	0.0766577
$66$	0	0
$67$	−8.08115	−0.987269	−0.493635	−	0.869669i	$-0.664332\pi$
$67$	−8.08115	−0.987269	−0.493635	+	0.869669i	$0.664332\pi$
$68$	0	0
$69$	7.70820	0.927959
$70$	0	0
$71$	−10.9427	−1.29866	−0.649330	−	0.760507i	$-0.724950\pi$
$71$	−10.9427	−1.29866	−0.649330	+	0.760507i	$0.724950\pi$
$72$	0	0
$73$	−9.85410	−1.15334	−0.576668	−	0.816979i	$-0.695647\pi$
$73$	−9.85410	−1.15334	−0.576668	+	0.816979i	$0.695647\pi$
$74$	0	0
$75$	−7.49164	−0.865060
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−15.9371	−1.79307	−0.896533	−	0.442977i	$-0.853922\pi$
$79$	−15.9371	−1.79307	−0.896533	+	0.442977i	$0.853922\pi$
$80$	0	0
$81$	−6.76393	−0.751548
$82$	0	0
$83$	−10.9427	−1.20112	−0.600559	−	0.799581i	$-0.705055\pi$
$83$	−10.9427	−1.20112	−0.600559	+	0.799581i	$0.705055\pi$
$84$	0	0
$85$	0.326238	0.0353855
$86$	0	0
$87$	−0.225173	−0.0241411
$88$	0	0
$89$	8.18034	0.867114	0.433557	−	0.901126i	$-0.357258\pi$
$89$	8.18034	0.867114	0.433557	+	0.901126i	$0.357258\pi$
$90$	0	0
$91$	−2.49721	−0.261779
$92$	0	0
$93$	3.29180	0.341343
$94$	0	0
$95$	−1.76854	−0.181448
$96$	0	0
$97$	−3.14590	−0.319418	−0.159709	−	0.987164i	$-0.551056\pi$
$97$	−3.14590	−0.319418	−0.159709	+	0.987164i	$0.551056\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	15.0344	1.49598	0.747991	−	0.663708i	$-0.231018\pi$
$101$	15.0344	1.49598	0.747991	+	0.663708i	$0.231018\pi$
$102$	0	0
$103$	7.26646	0.715986	0.357993	−	0.933724i	$-0.383461\pi$
$103$	7.26646	0.715986	0.357993	+	0.933724i	$0.383461\pi$
$104$	0	0
$105$	−0.909830	−0.0887903
$106$	0	0
$107$	6.53779	0.632032	0.316016	−	0.948754i	$-0.397655\pi$
$107$	6.53779	0.632032	0.316016	+	0.948754i	$0.397655\pi$
$108$	0	0
$109$	−14.4721	−1.38618	−0.693090	−	0.720851i	$-0.743751\pi$
$109$	−14.4721	−1.38618	−0.693090	+	0.720851i	$0.743751\pi$
$110$	0	0
$111$	−2.86155	−0.271606
$112$	0	0
$113$	−19.5623	−1.84027	−0.920133	−	0.391605i	$-0.871920\pi$
$113$	−19.5623	−1.84027	−0.920133	+	0.391605i	$0.871920\pi$
$114$	0	0
$115$	1.90770	0.177894
$116$	0	0
$117$	−1.00000	−0.0924500
$118$	0	0
$119$	−1.31819	−0.120838
$120$	0	0
$121$	0	0
$122$	0	0
$123$	−0.953850	−0.0860058
$124$	0	0
$125$	−3.76393	−0.336656
$126$	0	0
$127$	−17.1161	−1.51881	−0.759406	−	0.650617i	$-0.774510\pi$
$127$	−17.1161	−1.51881	−0.759406	+	0.650617i	$0.774510\pi$
$128$	0	0
$129$	−2.94427	−0.259229
$130$	0	0
$131$	8.80982	0.769718	0.384859	−	0.922975i	$-0.374250\pi$
$131$	8.80982	0.769718	0.384859	+	0.922975i	$0.374250\pi$
$132$	0	0
$133$	7.14590	0.619628
$134$	0	0
$135$	−2.13287	−0.183569
$136$	0	0
$137$	−10.0902	−0.862061	−0.431031	−	0.902337i	$-0.641850\pi$
$137$	−10.0902	−0.862061	−0.431031	+	0.902337i	$0.641850\pi$
$138$	0	0
$139$	−5.35876	−0.454524	−0.227262	−	0.973834i	$-0.572977\pi$
$139$	−5.35876	−0.454524	−0.227262	+	0.973834i	$0.572977\pi$
$140$	0	0
$141$	14.8541	1.25094
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−0.0557281	−0.00462797
$146$	0	0
$147$	−7.12730	−0.587850
$148$	0	0
$149$	−20.5623	−1.68453	−0.842265	−	0.539064i	$-0.818778\pi$
$149$	−20.5623	−1.68453	−0.842265	+	0.539064i	$0.818778\pi$
$150$	0	0
$151$	16.5266	1.34492	0.672459	−	0.740134i	$-0.265238\pi$
$151$	16.5266	1.34492	0.672459	+	0.740134i	$0.265238\pi$
$152$	0	0
$153$	−0.527864	−0.0426753
$154$	0	0
$155$	0.814685	0.0654371
$156$	0	0
$157$	13.3820	1.06800	0.533999	−	0.845485i	$-0.320689\pi$
$157$	13.3820	1.06800	0.533999	+	0.845485i	$0.320689\pi$
$158$	0	0
$159$	13.2147	1.04800
$160$	0	0
$161$	−7.70820	−0.607492
$162$	0	0
$163$	9.03500	0.707676	0.353838	−	0.935307i	$-0.384876\pi$
$163$	9.03500	0.707676	0.353838	+	0.935307i	$0.384876\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	5.94827	0.460291	0.230146	−	0.973156i	$-0.426080\pi$
$167$	5.94827	0.460291	0.230146	+	0.973156i	$0.426080\pi$
$168$	0	0
$169$	−10.3820	−0.798613
$170$	0	0
$171$	2.86155	0.218828
$172$	0	0
$173$	−1.67376	−0.127254	−0.0636269	−	0.997974i	$-0.520267\pi$
$173$	−1.67376	−0.127254	−0.0636269	+	0.997974i	$0.520267\pi$
$174$	0	0
$175$	7.49164	0.566314
$176$	0	0
$177$	15.9787	1.20103
$178$	0	0
$179$	1.54336	0.115356	0.0576781	−	0.998335i	$-0.481630\pi$
$179$	1.54336	0.115356	0.0576781	+	0.998335i	$0.481630\pi$
$180$	0	0
$181$	20.3820	1.51498	0.757490	−	0.652847i	$-0.226426\pi$
$181$	20.3820	1.51498	0.757490	+	0.652847i	$0.226426\pi$
$182$	0	0
$183$	−19.3882	−1.43322
$184$	0	0
$185$	−0.708204	−0.0520682
$186$	0	0
$187$	0	0
$188$	0	0
$189$	8.61803	0.626870
$190$	0	0
$191$	3.45106	0.249710	0.124855	−	0.992175i	$-0.460153\pi$
$191$	3.45106	0.249710	0.124855	+	0.992175i	$0.460153\pi$
$192$	0	0
$193$	−3.32624	−0.239428	−0.119714	−	0.992808i	$-0.538198\pi$
$193$	−3.32624	−0.239428	−0.119714	+	0.992808i	$0.538198\pi$
$194$	0	0
$195$	0.953850	0.0683066
$196$	0	0
$197$	−14.1803	−1.01031	−0.505154	−	0.863029i	$-0.668564\pi$
$197$	−14.1803	−1.01031	−0.505154	+	0.863029i	$0.668564\pi$
$198$	0	0
$199$	−18.0700	−1.28095	−0.640474	−	0.767980i	$-0.721262\pi$
$199$	−18.0700	−1.28095	−0.640474	+	0.767980i	$0.721262\pi$
$200$	0	0
$201$	−12.4721	−0.879717
$202$	0	0
$203$	0.225173	0.0158041
$204$	0	0
$205$	−0.236068	−0.0164877
$206$	0	0
$207$	−3.08672	−0.214542
$208$	0	0
$209$	0	0
$210$	0	0
$211$	−24.0183	−1.65349	−0.826743	−	0.562580i	$-0.809809\pi$
$211$	−24.0183	−1.65349	−0.826743	+	0.562580i	$0.809809\pi$
$212$	0	0
$213$	−16.8885	−1.15718
$214$	0	0
$215$	−0.728677	−0.0496953
$216$	0	0
$217$	−3.29180	−0.223462
$218$	0	0
$219$	−15.2084	−1.02769
$220$	0	0
$221$	1.38197	0.0929611
$222$	0	0
$223$	−21.5211	−1.44116	−0.720578	−	0.693374i	$-0.756124\pi$
$223$	−21.5211	−1.44116	−0.720578	+	0.693374i	$0.756124\pi$
$224$	0	0
$225$	3.00000	0.200000
$226$	0	0
$227$	−9.62451	−0.638801	−0.319401	−	0.947620i	$-0.603482\pi$
$227$	−9.62451	−0.638801	−0.319401	+	0.947620i	$0.603482\pi$
$228$	0	0
$229$	8.67376	0.573178	0.286589	−	0.958054i	$-0.407479\pi$
$229$	8.67376	0.573178	0.286589	+	0.958054i	$0.407479\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−16.0902	−1.05410	−0.527051	−	0.849834i	$-0.676702\pi$
$233$	−16.0902	−1.05410	−0.527051	+	0.849834i	$0.676702\pi$
$234$	0	0
$235$	3.67624	0.239811
$236$	0	0
$237$	−24.5967	−1.59773
$238$	0	0
$239$	−15.7980	−1.02188	−0.510942	−	0.859615i	$-0.670703\pi$
$239$	−15.7980	−1.02188	−0.510942	+	0.859615i	$0.670703\pi$
$240$	0	0
$241$	7.41641	0.477733	0.238866	−	0.971052i	$-0.423224\pi$
$241$	7.41641	0.477733	0.238866	+	0.971052i	$0.423224\pi$
$242$	0	0
$243$	6.31261	0.404954
$244$	0	0
$245$	−1.76393	−0.112693
$246$	0	0
$247$	−7.49164	−0.476681
$248$	0	0
$249$	−16.8885	−1.07027
$250$	0	0
$251$	14.4798	0.913955	0.456977	−	0.889478i	$-0.348932\pi$
$251$	14.4798	0.913955	0.456977	+	0.889478i	$0.348932\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	0.503503	0.0315306
$256$	0	0
$257$	−14.3262	−0.893646	−0.446823	−	0.894622i	$-0.647445\pi$
$257$	−14.3262	−0.893646	−0.446823	+	0.894622i	$0.647445\pi$
$258$	0	0
$259$	2.86155	0.177808
$260$	0	0
$261$	0.0901699	0.00558138
$262$	0	0
$263$	−15.7119	−0.968840	−0.484420	−	0.874835i	$-0.660969\pi$
$263$	−15.7119	−0.968840	−0.484420	+	0.874835i	$0.660969\pi$
$264$	0	0
$265$	3.27051	0.200906
$266$	0	0
$267$	12.6252	0.772651
$268$	0	0
$269$	−13.7984	−0.841302	−0.420651	−	0.907223i	$-0.638198\pi$
$269$	−13.7984	−0.841302	−0.420651	+	0.907223i	$0.638198\pi$
$270$	0	0
$271$	25.7868	1.56644	0.783218	−	0.621747i	$-0.213577\pi$
$271$	25.7868	1.56644	0.783218	+	0.621747i	$0.213577\pi$
$272$	0	0
$273$	−3.85410	−0.233261
$274$	0	0
$275$	0	0
$276$	0	0
$277$	11.3262	0.680528	0.340264	−	0.940330i	$-0.389484\pi$
$277$	11.3262	0.680528	0.340264	+	0.940330i	$0.389484\pi$
$278$	0	0
$279$	−1.31819	−0.0789179
$280$	0	0
$281$	−14.3820	−0.857956	−0.428978	−	0.903315i	$-0.641126\pi$
$281$	−14.3820	−0.857956	−0.428978	+	0.903315i	$0.641126\pi$
$282$	0	0
$283$	20.3420	1.20921	0.604604	−	0.796526i	$-0.293331\pi$
$283$	20.3420	1.20921	0.604604	+	0.796526i	$0.293331\pi$
$284$	0	0
$285$	−2.72949	−0.161681
$286$	0	0
$287$	0.953850	0.0563040
$288$	0	0
$289$	−16.2705	−0.957089
$290$	0	0
$291$	−4.85526	−0.284620
$292$	0	0
$293$	−7.56231	−0.441795	−0.220897	−	0.975297i	$-0.570899\pi$
$293$	−7.56231	−0.441795	−0.220897	+	0.975297i	$0.570899\pi$
$294$	0	0
$295$	3.95457	0.230244
$296$	0	0
$297$	0	0
$298$	0	0
$299$	8.08115	0.467345
$300$	0	0
$301$	2.94427	0.169705
$302$	0	0
$303$	23.2036	1.33301
$304$	0	0
$305$	−4.79837	−0.274754
$306$	0	0
$307$	8.80982	0.502803	0.251402	−	0.967883i	$-0.419109\pi$
$307$	8.80982	0.502803	0.251402	+	0.967883i	$0.419109\pi$
$308$	0	0
$309$	11.2148	0.637987
$310$	0	0
$311$	12.2609	0.695251	0.347626	−	0.937633i	$-0.386988\pi$
$311$	12.2609	0.695251	0.347626	+	0.937633i	$0.386988\pi$
$312$	0	0
$313$	−23.3262	−1.31848	−0.659238	−	0.751934i	$-0.729121\pi$
$313$	−23.3262	−1.31848	−0.659238	+	0.751934i	$0.729121\pi$
$314$	0	0
$315$	0.364338	0.0205281
$316$	0	0
$317$	13.0344	0.732087	0.366044	−	0.930598i	$-0.380712\pi$
$317$	13.0344	0.732087	0.366044	+	0.930598i	$0.380712\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	10.0902	0.563178
$322$	0	0
$323$	−3.95457	−0.220038
$324$	0	0
$325$	−7.85410	−0.435667
$326$	0	0
$327$	−22.3357	−1.23517
$328$	0	0
$329$	−14.8541	−0.818933
$330$	0	0
$331$	3.81540	0.209713	0.104857	−	0.994487i	$-0.466562\pi$
$331$	3.81540	0.209713	0.104857	+	0.994487i	$0.466562\pi$
$332$	0	0
$333$	1.14590	0.0627948
$334$	0	0
$335$	−3.08672	−0.168646
$336$	0	0
$337$	−14.0344	−0.764505	−0.382252	−	0.924058i	$-0.624852\pi$
$337$	−14.0344	−0.764505	−0.382252	+	0.924058i	$0.624852\pi$
$338$	0	0
$339$	−30.1917	−1.63979
$340$	0	0
$341$	0	0
$342$	0	0
$343$	17.9308	0.968174
$344$	0	0
$345$	2.94427	0.158514
$346$	0	0
$347$	17.5665	0.943019	0.471509	−	0.881861i	$-0.343709\pi$
$347$	17.5665	0.943019	0.471509	+	0.881861i	$0.343709\pi$
$348$	0	0
$349$	10.0344	0.537131	0.268566	−	0.963261i	$-0.413450\pi$
$349$	10.0344	0.537131	0.268566	+	0.963261i	$0.413450\pi$
$350$	0	0
$351$	−9.03500	−0.482253
$352$	0	0
$353$	19.7082	1.04896	0.524481	−	0.851422i	$-0.324259\pi$
$353$	19.7082	1.04896	0.524481	+	0.851422i	$0.324259\pi$
$354$	0	0
$355$	−4.17974	−0.221838
$356$	0	0
$357$	−2.03444	−0.107674
$358$	0	0
$359$	29.6022	1.56234	0.781172	−	0.624315i	$-0.214622\pi$
$359$	29.6022	1.56234	0.781172	+	0.624315i	$0.214622\pi$
$360$	0	0
$361$	2.43769	0.128300
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−3.76393	−0.197013
$366$	0	0
$367$	−26.9658	−1.40760	−0.703802	−	0.710396i	$-0.748516\pi$
$367$	−26.9658	−1.40760	−0.703802	+	0.710396i	$0.748516\pi$
$368$	0	0
$369$	0.381966	0.0198844
$370$	0	0
$371$	−13.2147	−0.686075
$372$	0	0
$373$	−6.29180	−0.325777	−0.162888	−	0.986644i	$-0.552081\pi$
$373$	−6.29180	−0.325777	−0.162888	+	0.986644i	$0.552081\pi$
$374$	0	0
$375$	−5.80911	−0.299981
$376$	0	0
$377$	−0.236068	−0.0121581
$378$	0	0
$379$	3.31190	0.170121	0.0850604	−	0.996376i	$-0.472892\pi$
$379$	3.31190	0.170121	0.0850604	+	0.996376i	$0.472892\pi$
$380$	0	0
$381$	−26.4164	−1.35335
$382$	0	0
$383$	−26.3763	−1.34777	−0.673883	−	0.738838i	$-0.735375\pi$
$383$	−26.3763	−1.34777	−0.673883	+	0.738838i	$0.735375\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	1.17902	0.0599331
$388$	0	0
$389$	−23.5623	−1.19466	−0.597328	−	0.801997i	$-0.703771\pi$
$389$	−23.5623	−1.19466	−0.597328	+	0.801997i	$0.703771\pi$
$390$	0	0
$391$	4.26575	0.215728
$392$	0	0
$393$	13.5967	0.685865
$394$	0	0
$395$	−6.08744	−0.306292
$396$	0	0
$397$	14.1803	0.711691	0.355845	−	0.934545i	$-0.384193\pi$
$397$	14.1803	0.711691	0.355845	+	0.934545i	$0.384193\pi$
$398$	0	0
$399$	11.0287	0.552126
$400$	0	0
$401$	14.6738	0.732773	0.366386	−	0.930463i	$-0.380595\pi$
$401$	14.6738	0.732773	0.366386	+	0.930463i	$0.380595\pi$
$402$	0	0
$403$	3.45106	0.171910
$404$	0	0
$405$	−2.58359	−0.128380
$406$	0	0
$407$	0	0
$408$	0	0
$409$	4.56231	0.225592	0.112796	−	0.993618i	$-0.464019\pi$
$409$	4.56231	0.225592	0.112796	+	0.993618i	$0.464019\pi$
$410$	0	0
$411$	−15.5728	−0.768149
$412$	0	0
$413$	−15.9787	−0.786261
$414$	0	0
$415$	−4.17974	−0.205175
$416$	0	0
$417$	−8.27051	−0.405009
$418$	0	0
$419$	27.3302	1.33517	0.667583	−	0.744535i	$-0.267329\pi$
$419$	27.3302	1.33517	0.667583	+	0.744535i	$0.267329\pi$
$420$	0	0
$421$	−8.32624	−0.405796	−0.202898	−	0.979200i	$-0.565036\pi$
$421$	−8.32624	−0.405796	−0.202898	+	0.979200i	$0.565036\pi$
$422$	0	0
$423$	−5.94827	−0.289215
$424$	0	0
$425$	−4.14590	−0.201106
$426$	0	0
$427$	19.3882	0.938260
$428$	0	0
$429$	0	0
$430$	0	0
$431$	9.48534	0.456893	0.228446	−	0.973556i	$-0.426635\pi$
$431$	9.48534	0.456893	0.228446	+	0.973556i	$0.426635\pi$
$432$	0	0
$433$	−33.0902	−1.59021	−0.795106	−	0.606470i	$-0.792585\pi$
$433$	−33.0902	−1.59021	−0.795106	+	0.606470i	$0.792585\pi$
$434$	0	0
$435$	−0.0860086	−0.00412380
$436$	0	0
$437$	−23.1246	−1.10620
$438$	0	0
$439$	19.6994	0.940199	0.470100	−	0.882613i	$-0.344218\pi$
$439$	19.6994	0.940199	0.470100	+	0.882613i	$0.344218\pi$
$440$	0	0
$441$	2.85410	0.135910
$442$	0	0
$443$	5.80911	0.275999	0.138000	−	0.990432i	$-0.455933\pi$
$443$	5.80911	0.275999	0.138000	+	0.990432i	$0.455933\pi$
$444$	0	0
$445$	3.12461	0.148121
$446$	0	0
$447$	−31.7351	−1.50102
$448$	0	0
$449$	25.7984	1.21750	0.608750	−	0.793362i	$-0.291671\pi$
$449$	25.7984	1.21750	0.608750	+	0.793362i	$0.291671\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	25.5066	1.19840
$454$	0	0
$455$	−0.953850	−0.0447172
$456$	0	0
$457$	−28.7426	−1.34452	−0.672262	−	0.740313i	$-0.734677\pi$
$457$	−28.7426	−1.34452	−0.672262	+	0.740313i	$0.734677\pi$
$458$	0	0
$459$	−4.76925	−0.222610
$460$	0	0
$461$	−2.76393	−0.128729	−0.0643646	−	0.997926i	$-0.520502\pi$
$461$	−2.76393	−0.128729	−0.0643646	+	0.997926i	$0.520502\pi$
$462$	0	0
$463$	31.8742	1.48132	0.740661	−	0.671879i	$-0.234513\pi$
$463$	31.8742	1.48132	0.740661	+	0.671879i	$0.234513\pi$
$464$	0	0
$465$	1.25735	0.0583084
$466$	0	0
$467$	4.04057	0.186975	0.0934877	−	0.995620i	$-0.470198\pi$
$467$	4.04057	0.186975	0.0934877	+	0.995620i	$0.470198\pi$
$468$	0	0
$469$	12.4721	0.575910
$470$	0	0
$471$	20.6532	0.951650
$472$	0	0
$473$	0	0
$474$	0	0
$475$	22.4749	1.03122
$476$	0	0
$477$	−5.29180	−0.242295
$478$	0	0
$479$	−19.0238	−0.869222	−0.434611	−	0.900618i	$-0.643114\pi$
$479$	−19.0238	−0.869222	−0.434611	+	0.900618i	$0.643114\pi$
$480$	0	0
$481$	−3.00000	−0.136788
$482$	0	0
$483$	−11.8965	−0.541312
$484$	0	0
$485$	−1.20163	−0.0545630
$486$	0	0
$487$	−9.62451	−0.436128	−0.218064	−	0.975934i	$-0.569974\pi$
$487$	−9.62451	−0.436128	−0.218064	+	0.975934i	$0.569974\pi$
$488$	0	0
$489$	13.9443	0.630582
$490$	0	0
$491$	33.1393	1.49555	0.747777	−	0.663950i	$-0.231121\pi$
$491$	33.1393	1.49555	0.747777	+	0.663950i	$0.231121\pi$
$492$	0	0
$493$	−0.124612	−0.00561223
$494$	0	0
$495$	0	0
$496$	0	0
$497$	16.8885	0.757555
$498$	0	0
$499$	−8.89583	−0.398232	−0.199116	−	0.979976i	$-0.563807\pi$
$499$	−8.89583	−0.398232	−0.199116	+	0.979976i	$0.563807\pi$
$500$	0	0
$501$	9.18034	0.410147
$502$	0	0
$503$	39.3127	1.75287	0.876434	−	0.481522i	$-0.159916\pi$
$503$	39.3127	1.75287	0.876434	+	0.481522i	$0.159916\pi$
$504$	0	0
$505$	5.74265	0.255544
$506$	0	0
$507$	−16.0231	−0.711612
$508$	0	0
$509$	−11.0902	−0.491563	−0.245782	−	0.969325i	$-0.579045\pi$
$509$	−11.0902	−0.491563	−0.245782	+	0.969325i	$0.579045\pi$
$510$	0	0
$511$	15.2084	0.672782
$512$	0	0
$513$	25.8541	1.14149
$514$	0	0
$515$	2.77554	0.122305
$516$	0	0
$517$	0	0
$518$	0	0
$519$	−2.58322	−0.113391
$520$	0	0
$521$	13.3820	0.586275	0.293137	−	0.956070i	$-0.405301\pi$
$521$	13.3820	0.586275	0.293137	+	0.956070i	$0.405301\pi$
$522$	0	0
$523$	−3.59023	−0.156990	−0.0784948	−	0.996915i	$-0.525011\pi$
$523$	−3.59023	−0.156990	−0.0784948	+	0.996915i	$0.525011\pi$
$524$	0	0
$525$	11.5623	0.504620
$526$	0	0
$527$	1.82169	0.0793541
$528$	0	0
$529$	1.94427	0.0845336
$530$	0	0
$531$	−6.39862	−0.277677
$532$	0	0
$533$	−1.00000	−0.0433148
$534$	0	0
$535$	2.49721	0.107964
$536$	0	0
$537$	2.38197	0.102789
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−37.0344	−1.59224	−0.796118	−	0.605142i	$-0.793116\pi$
$541$	−37.0344	−1.59224	−0.796118	+	0.605142i	$0.793116\pi$
$542$	0	0
$543$	31.4568	1.34994
$544$	0	0
$545$	−5.52786	−0.236788
$546$	0	0
$547$	−31.0596	−1.32801	−0.664005	−	0.747728i	$-0.731145\pi$
$547$	−31.0596	−1.32801	−0.664005	+	0.747728i	$0.731145\pi$
$548$	0	0
$549$	7.76393	0.331357
$550$	0	0
$551$	0.675520	0.0287781
$552$	0	0
$553$	24.5967	1.04596
$554$	0	0
$555$	−1.09301	−0.0463959
$556$	0	0
$557$	−19.8541	−0.841245	−0.420623	−	0.907236i	$-0.638188\pi$
$557$	−19.8541	−0.841245	−0.420623	+	0.907236i	$0.638188\pi$
$558$	0	0
$559$	−3.08672	−0.130555
$560$	0	0
$561$	0	0
$562$	0	0
$563$	36.3652	1.53261	0.766305	−	0.642478i	$-0.222093\pi$
$563$	36.3652	1.53261	0.766305	+	0.642478i	$0.222093\pi$
$564$	0	0
$565$	−7.47214	−0.314355
$566$	0	0
$567$	10.4392	0.438405
$568$	0	0
$569$	−6.43769	−0.269882	−0.134941	−	0.990854i	$-0.543085\pi$
$569$	−6.43769	−0.269882	−0.134941	+	0.990854i	$0.543085\pi$
$570$	0	0
$571$	39.6771	1.66043	0.830217	−	0.557441i	$-0.188217\pi$
$571$	39.6771	1.66043	0.830217	+	0.557441i	$0.188217\pi$
$572$	0	0
$573$	5.32624	0.222507
$574$	0	0
$575$	−24.2434	−1.01102
$576$	0	0
$577$	0.381966	0.0159015	0.00795073	−	0.999968i	$-0.497469\pi$
$577$	0.381966	0.0159015	0.00795073	+	0.999968i	$0.497469\pi$
$578$	0	0
$579$	−5.13359	−0.213345
$580$	0	0
$581$	16.8885	0.700655
$582$	0	0
$583$	0	0
$584$	0	0
$585$	−0.381966	−0.0157924
$586$	0	0
$587$	−1.54336	−0.0637014	−0.0318507	−	0.999493i	$-0.510140\pi$
$587$	−1.54336	−0.0637014	−0.0318507	+	0.999493i	$0.510140\pi$
$588$	0	0
$589$	−9.87539	−0.406908
$590$	0	0
$591$	−21.8854	−0.900245
$592$	0	0
$593$	−38.6525	−1.58727	−0.793633	−	0.608396i	$-0.791813\pi$
$593$	−38.6525	−1.58727	−0.793633	+	0.608396i	$0.791813\pi$
$594$	0	0
$595$	−0.503503	−0.0206416
$596$	0	0
$597$	−27.8885	−1.14140
$598$	0	0
$599$	26.3763	1.07771	0.538854	−	0.842399i	$-0.318858\pi$
$599$	26.3763	1.07771	0.538854	+	0.842399i	$0.318858\pi$
$600$	0	0
$601$	−26.9098	−1.09767	−0.548837	−	0.835929i	$-0.684929\pi$
$601$	−26.9098	−1.09767	−0.548837	+	0.835929i	$0.684929\pi$
$602$	0	0
$603$	4.99442	0.203389
$604$	0	0
$605$	0	0
$606$	0	0
$607$	0.953850	0.0387156	0.0193578	−	0.999813i	$-0.493838\pi$
$607$	0.953850	0.0387156	0.0193578	+	0.999813i	$0.493838\pi$
$608$	0	0
$609$	0.347524	0.0140824
$610$	0	0
$611$	15.5728	0.630007
$612$	0	0
$613$	17.8541	0.721120	0.360560	−	0.932736i	$-0.382586\pi$
$613$	17.8541	0.721120	0.360560	+	0.932736i	$0.382586\pi$
$614$	0	0
$615$	−0.364338	−0.0146915
$616$	0	0
$617$	−10.0689	−0.405358	−0.202679	−	0.979245i	$-0.564965\pi$
$617$	−10.0689	−0.405358	−0.202679	+	0.979245i	$0.564965\pi$
$618$	0	0
$619$	−36.5043	−1.46723	−0.733616	−	0.679564i	$-0.762169\pi$
$619$	−36.5043	−1.46723	−0.733616	+	0.679564i	$0.762169\pi$
$620$	0	0
$621$	−27.8885	−1.11913
$622$	0	0
$623$	−12.6252	−0.505819
$624$	0	0
$625$	22.8328	0.913313
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−1.58359	−0.0631420
$630$	0	0
$631$	−4.17974	−0.166393	−0.0831964	−	0.996533i	$-0.526513\pi$
$631$	−4.17974	−0.166393	−0.0831964	+	0.996533i	$0.526513\pi$
$632$	0	0
$633$	−37.0689	−1.47336
$634$	0	0
$635$	−6.53779	−0.259444
$636$	0	0
$637$	−7.47214	−0.296057
$638$	0	0
$639$	6.76296	0.267539
$640$	0	0
$641$	10.9787	0.433633	0.216817	−	0.976212i	$-0.430433\pi$
$641$	10.9787	0.433633	0.216817	+	0.976212i	$0.430433\pi$
$642$	0	0
$643$	44.4463	1.75279	0.876396	−	0.481592i	$-0.159941\pi$
$643$	44.4463	1.75279	0.876396	+	0.481592i	$0.159941\pi$
$644$	0	0
$645$	−1.12461	−0.0442815
$646$	0	0
$647$	46.3540	1.82236	0.911182	−	0.412004i	$-0.135171\pi$
$647$	46.3540	1.82236	0.911182	+	0.412004i	$0.135171\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	−5.08043	−0.199118
$652$	0	0
$653$	10.7984	0.422573	0.211287	−	0.977424i	$-0.432235\pi$
$653$	10.7984	0.422573	0.211287	+	0.977424i	$0.432235\pi$
$654$	0	0
$655$	3.36505	0.131484
$656$	0	0
$657$	6.09017	0.237600
$658$	0	0
$659$	−11.1679	−0.435039	−0.217519	−	0.976056i	$-0.569796\pi$
$659$	−11.1679	−0.435039	−0.217519	+	0.976056i	$0.569796\pi$
$660$	0	0
$661$	31.4164	1.22196	0.610978	−	0.791647i	$-0.290776\pi$
$661$	31.4164	1.22196	0.610978	+	0.791647i	$0.290776\pi$
$662$	0	0
$663$	2.13287	0.0828340
$664$	0	0
$665$	2.72949	0.105845
$666$	0	0
$667$	−0.728677	−0.0282145
$668$	0	0
$669$	−33.2148	−1.28416
$670$	0	0
$671$	0	0
$672$	0	0
$673$	44.3820	1.71080	0.855400	−	0.517969i	$-0.173312\pi$
$673$	44.3820	1.71080	0.855400	+	0.517969i	$0.173312\pi$
$674$	0	0
$675$	27.1050	1.04327
$676$	0	0
$677$	23.9787	0.921577	0.460788	−	0.887510i	$-0.347567\pi$
$677$	23.9787	0.921577	0.460788	+	0.887510i	$0.347567\pi$
$678$	0	0
$679$	4.85526	0.186328
$680$	0	0
$681$	−14.8541	−0.569210
$682$	0	0
$683$	−26.1511	−1.00065	−0.500323	−	0.865839i	$-0.666785\pi$
$683$	−26.1511	−1.00065	−0.500323	+	0.865839i	$0.666785\pi$
$684$	0	0
$685$	−3.85410	−0.147258
$686$	0	0
$687$	13.3868	0.510737
$688$	0	0
$689$	13.8541	0.527799
$690$	0	0
$691$	−20.2029	−0.768553	−0.384277	−	0.923218i	$-0.625549\pi$
$691$	−20.2029	−0.768553	−0.384277	+	0.923218i	$0.625549\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−2.04687	−0.0776420
$696$	0	0
$697$	−0.527864	−0.0199943
$698$	0	0
$699$	−24.8330	−0.939269
$700$	0	0
$701$	−23.7426	−0.896747	−0.448374	−	0.893846i	$-0.647997\pi$
$701$	−23.7426	−0.896747	−0.448374	+	0.893846i	$0.647997\pi$
$702$	0	0
$703$	8.58465	0.323776
$704$	0	0
$705$	5.67376	0.213686
$706$	0	0
$707$	−23.2036	−0.872661
$708$	0	0
$709$	11.6180	0.436324	0.218162	−	0.975913i	$-0.429994\pi$
$709$	11.6180	0.436324	0.218162	+	0.975913i	$0.429994\pi$
$710$	0	0
$711$	9.84968	0.369392
$712$	0	0
$713$	10.6525	0.398938
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−24.3820	−0.910561
$718$	0	0
$719$	−23.8791	−0.890540	−0.445270	−	0.895396i	$-0.646892\pi$
$719$	−23.8791	−0.890540	−0.445270	+	0.895396i	$0.646892\pi$
$720$	0	0
$721$	−11.2148	−0.417660
$722$	0	0
$723$	11.4462	0.425689
$724$	0	0
$725$	0.708204	0.0263020
$726$	0	0
$727$	11.4462	0.424516	0.212258	−	0.977214i	$-0.431918\pi$
$727$	11.4462	0.424516	0.212258	+	0.977214i	$0.431918\pi$
$728$	0	0
$729$	30.0344	1.11239
$730$	0	0
$731$	−1.62937	−0.0602644
$732$	0	0
$733$	−6.43769	−0.237782	−0.118891	−	0.992907i	$-0.537934\pi$
$733$	−6.43769	−0.237782	−0.118891	+	0.992907i	$0.537934\pi$
$734$	0	0
$735$	−2.72239	−0.100417
$736$	0	0
$737$	0	0
$738$	0	0
$739$	17.3945	0.639866	0.319933	−	0.947440i	$-0.396340\pi$
$739$	17.3945	0.639866	0.319933	+	0.947440i	$0.396340\pi$
$740$	0	0
$741$	−11.5623	−0.424752
$742$	0	0
$743$	12.4001	0.454914	0.227457	−	0.973788i	$-0.426959\pi$
$743$	12.4001	0.454914	0.227457	+	0.973788i	$0.426959\pi$
$744$	0	0
$745$	−7.85410	−0.287752
$746$	0	0
$747$	6.76296	0.247444
$748$	0	0
$749$	−10.0902	−0.368687
$750$	0	0
$751$	44.5855	1.62695	0.813474	−	0.581602i	$-0.197574\pi$
$751$	44.5855	1.62695	0.813474	+	0.581602i	$0.197574\pi$
$752$	0	0
$753$	22.3475	0.814389
$754$	0	0
$755$	6.31261	0.229739
$756$	0	0
$757$	−32.1591	−1.16884	−0.584420	−	0.811451i	$-0.698678\pi$
$757$	−32.1591	−1.16884	−0.584420	+	0.811451i	$0.698678\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−18.8541	−0.683461	−0.341730	−	0.939798i	$-0.611013\pi$
$761$	−18.8541	−0.683461	−0.341730	+	0.939798i	$0.611013\pi$
$762$	0	0
$763$	22.3357	0.808608
$764$	0	0
$765$	−0.201626	−0.00728981
$766$	0	0
$767$	16.7518	0.604873
$768$	0	0
$769$	25.1246	0.906017	0.453008	−	0.891506i	$-0.350351\pi$
$769$	25.1246	0.906017	0.453008	+	0.891506i	$0.350351\pi$
$770$	0	0
$771$	−22.1106	−0.796293
$772$	0	0
$773$	−54.2148	−1.94997	−0.974985	−	0.222270i	$-0.928653\pi$
$773$	−54.2148	−1.94997	−0.974985	+	0.222270i	$0.928653\pi$
$774$	0	0
$775$	−10.3532	−0.371897
$776$	0	0
$777$	4.41641	0.158438
$778$	0	0
$779$	2.86155	0.102526
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0.814685	0.0291145
$784$	0	0
$785$	5.11146	0.182436
$786$	0	0
$787$	−18.2952	−0.652152	−0.326076	−	0.945343i	$-0.605727\pi$
$787$	−18.2952	−0.652152	−0.326076	+	0.945343i	$0.605727\pi$
$788$	0	0
$789$	−24.2492	−0.863295
$790$	0	0
$791$	30.1917	1.07349
$792$	0	0
$793$	−20.3262	−0.721806
$794$	0	0
$795$	5.04758	0.179019
$796$	0	0
$797$	−37.9787	−1.34528	−0.672638	−	0.739972i	$-0.734839\pi$
$797$	−37.9787	−1.34528	−0.672638	+	0.739972i	$0.734839\pi$
$798$	0	0
$799$	8.22031	0.290814
$800$	0	0
$801$	−5.05573	−0.178635
$802$	0	0
$803$	0	0
$804$	0	0
$805$	−2.94427	−0.103772
$806$	0	0
$807$	−21.2959	−0.749651
$808$	0	0
$809$	−7.79837	−0.274176	−0.137088	−	0.990559i	$-0.543774\pi$
$809$	−7.79837	−0.274176	−0.137088	+	0.990559i	$0.543774\pi$
$810$	0	0
$811$	21.2427	0.745933	0.372967	−	0.927845i	$-0.378341\pi$
$811$	21.2427	0.745933	0.372967	+	0.927845i	$0.378341\pi$
$812$	0	0
$813$	39.7984	1.39579
$814$	0	0
$815$	3.45106	0.120885
$816$	0	0
$817$	8.83282	0.309021
$818$	0	0
$819$	1.54336	0.0539294
$820$	0	0
$821$	−15.0902	−0.526651	−0.263325	−	0.964707i	$-0.584819\pi$
$821$	−15.0902	−0.526651	−0.263325	+	0.964707i	$0.584819\pi$
$822$	0	0
$823$	−44.7246	−1.55900	−0.779502	−	0.626400i	$-0.784528\pi$
$823$	−44.7246	−1.55900	−0.779502	+	0.626400i	$0.784528\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−18.2952	−0.636185	−0.318093	−	0.948060i	$-0.603042\pi$
$827$	−18.2952	−0.636185	−0.318093	+	0.948060i	$0.603042\pi$
$828$	0	0
$829$	13.1591	0.457033	0.228516	−	0.973540i	$-0.426613\pi$
$829$	13.1591	0.457033	0.228516	+	0.973540i	$0.426613\pi$
$830$	0	0
$831$	17.4805	0.606391
$832$	0	0
$833$	−3.94427	−0.136661
$834$	0	0
$835$	2.27204	0.0786271
$836$	0	0
$837$	−11.9098	−0.411664
$838$	0	0
$839$	−16.9770	−0.586110	−0.293055	−	0.956096i	$-0.594672\pi$
$839$	−16.9770	−0.586110	−0.293055	+	0.956096i	$0.594672\pi$
$840$	0	0
$841$	−28.9787	−0.999266
$842$	0	0
$843$	−22.1966	−0.764491
$844$	0	0
$845$	−3.96556	−0.136419
$846$	0	0
$847$	0	0
$848$	0	0
$849$	31.3951	1.07748
$850$	0	0
$851$	−9.26017	−0.317435
$852$	0	0
$853$	34.7426	1.18957	0.594783	−	0.803886i	$-0.297238\pi$
$853$	34.7426	1.18957	0.594783	+	0.803886i	$0.297238\pi$
$854$	0	0
$855$	1.09301	0.0373803
$856$	0	0
$857$	−31.1246	−1.06320	−0.531598	−	0.846997i	$-0.678408\pi$
$857$	−31.1246	−1.06320	−0.531598	+	0.846997i	$0.678408\pi$
$858$	0	0
$859$	3.81540	0.130180	0.0650899	−	0.997879i	$-0.479267\pi$
$859$	3.81540	0.130180	0.0650899	+	0.997879i	$0.479267\pi$
$860$	0	0
$861$	1.47214	0.0501703
$862$	0	0
$863$	−11.3930	−0.387824	−0.193912	−	0.981019i	$-0.562118\pi$
$863$	−11.3930	−0.387824	−0.193912	+	0.981019i	$0.562118\pi$
$864$	0	0
$865$	−0.639320	−0.0217375
$866$	0	0
$867$	−25.1113	−0.852824
$868$	0	0
$869$	0	0
$870$	0	0
$871$	−13.0756	−0.443049
$872$	0	0
$873$	1.94427	0.0658036
$874$	0	0
$875$	5.80911	0.196384
$876$	0	0
$877$	24.6869	0.833618	0.416809	−	0.908994i	$-0.363148\pi$
$877$	24.6869	0.833618	0.416809	+	0.908994i	$0.363148\pi$
$878$	0	0
$879$	−11.6714	−0.393666
$880$	0	0
$881$	−1.23607	−0.0416442	−0.0208221	−	0.999783i	$-0.506628\pi$
$881$	−1.23607	−0.0416442	−0.0208221	+	0.999783i	$0.506628\pi$
$882$	0	0
$883$	20.7924	0.699719	0.349860	−	0.936802i	$-0.386229\pi$
$883$	20.7924	0.699719	0.349860	+	0.936802i	$0.386229\pi$
$884$	0	0
$885$	6.10333	0.205161
$886$	0	0
$887$	15.0693	0.505977	0.252988	−	0.967469i	$-0.418587\pi$
$887$	15.0693	0.505977	0.252988	+	0.967469i	$0.418587\pi$
$888$	0	0
$889$	26.4164	0.885978
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−44.5623	−1.49122
$894$	0	0
$895$	0.589512	0.0197052
$896$	0	0
$897$	12.4721	0.416432
$898$	0	0
$899$	−0.311182	−0.0103785
$900$	0	0
$901$	7.31308	0.243634
$902$	0	0
$903$	4.54408	0.151217
$904$	0	0
$905$	7.78522	0.258789
$906$	0	0
$907$	−43.7176	−1.45162	−0.725810	−	0.687895i	$-0.758535\pi$
$907$	−43.7176	−1.45162	−0.725810	+	0.687895i	$0.758535\pi$
$908$	0	0
$909$	−9.29180	−0.308189
$910$	0	0
$911$	2.41120	0.0798867	0.0399434	−	0.999202i	$-0.487282\pi$
$911$	2.41120	0.0798867	0.0399434	+	0.999202i	$0.487282\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	−7.40563	−0.244822
$916$	0	0
$917$	−13.5967	−0.449004
$918$	0	0
$919$	1.40420	0.0463202	0.0231601	−	0.999732i	$-0.492627\pi$
$919$	1.40420	0.0463202	0.0231601	+	0.999732i	$0.492627\pi$
$920$	0	0
$921$	13.5967	0.448028
$922$	0	0
$923$	−17.7057	−0.582789
$924$	0	0
$925$	9.00000	0.295918
$926$	0	0
$927$	−4.49092	−0.147501
$928$	0	0
$929$	50.6312	1.66116	0.830578	−	0.556903i	$-0.188010\pi$
$929$	50.6312	1.66116	0.830578	+	0.556903i	$0.188010\pi$
$930$	0	0
$931$	21.3819	0.700763
$932$	0	0
$933$	18.9230	0.619511
$934$	0	0
$935$	0	0
$936$	0	0
$937$	0.201626	0.00658684	0.00329342	−	0.999995i	$-0.498952\pi$
$937$	0.201626	0.00658684	0.00329342	+	0.999995i	$0.498952\pi$
$938$	0	0
$939$	−36.0008	−1.17484
$940$	0	0
$941$	23.5066	0.766293	0.383146	−	0.923688i	$-0.374841\pi$
$941$	23.5066	0.766293	0.383146	+	0.923688i	$0.374841\pi$
$942$	0	0
$943$	−3.08672	−0.100518
$944$	0	0
$945$	3.29180	0.107082
$946$	0	0
$947$	23.7931	0.773172	0.386586	−	0.922253i	$-0.373654\pi$
$947$	23.7931	0.773172	0.386586	+	0.922253i	$0.373654\pi$
$948$	0	0
$949$	−15.9443	−0.517573
$950$	0	0
$951$	20.1169	0.652334
$952$	0	0
$953$	44.8673	1.45339	0.726697	−	0.686959i	$-0.241055\pi$
$953$	44.8673	1.45339	0.726697	+	0.686959i	$0.241055\pi$
$954$	0	0
$955$	1.31819	0.0426556
$956$	0	0
$957$	0	0
$958$	0	0
$959$	15.5728	0.502871
$960$	0	0
$961$	−26.4508	−0.853253
$962$	0	0
$963$	−4.04057	−0.130206
$964$	0	0
$965$	−1.27051	−0.0408992
$966$	0	0
$967$	−60.3834	−1.94180	−0.970900	−	0.239484i	$-0.923022\pi$
$967$	−60.3834	−1.94180	−0.970900	+	0.239484i	$0.923022\pi$
$968$	0	0
$969$	−6.10333	−0.196067
$970$	0	0
$971$	18.4343	0.591586	0.295793	−	0.955252i	$-0.404416\pi$
$971$	18.4343	0.591586	0.295793	+	0.955252i	$0.404416\pi$
$972$	0	0
$973$	8.27051	0.265140
$974$	0	0
$975$	−12.1217	−0.388206
$976$	0	0
$977$	9.03444	0.289037	0.144519	−	0.989502i	$-0.453837\pi$
$977$	9.03444	0.289037	0.144519	+	0.989502i	$0.453837\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	8.94427	0.285569
$982$	0	0
$983$	−0.0860086	−0.00274325	−0.00137162	−	0.999999i	$-0.500437\pi$
$983$	−0.0860086	−0.00274325	−0.00137162	+	0.999999i	$0.500437\pi$
$984$	0	0
$985$	−5.41641	−0.172581
$986$	0	0
$987$	−22.9253	−0.729719
$988$	0	0
$989$	−9.52786	−0.302968
$990$	0	0
$991$	−15.2616	−0.484801	−0.242400	−	0.970176i	$-0.577935\pi$
$991$	−15.2616	−0.484801	−0.242400	+	0.970176i	$0.577935\pi$
$992$	0	0
$993$	5.88854	0.186867
$994$	0	0
$995$	−6.90212	−0.218812
$996$	0	0
$997$	27.2016	0.861484	0.430742	−	0.902475i	$-0.358252\pi$
$997$	27.2016	0.861484	0.430742	+	0.902475i	$0.358252\pi$
$998$	0	0
$999$	10.3532	0.327560

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7744.2.a.dq.1.3		4
4.3	odd	2	inner	7744.2.a.dq.1.2		4
8.3	odd	2		3872.2.a.bf.1.3		4
8.5	even	2		3872.2.a.bf.1.2		4
11.2	odd	10		704.2.m.k.257.2		8
11.6	odd	10		704.2.m.k.641.2		8
11.10	odd	2		7744.2.a.dp.1.3		4
44.35	even	10		704.2.m.k.257.1		8
44.39	even	10		704.2.m.k.641.1		8
44.43	even	2		7744.2.a.dp.1.2		4
88.13	odd	10		352.2.m.c.257.1	✓	8
88.21	odd	2		3872.2.a.bg.1.2		4
88.35	even	10		352.2.m.c.257.2	yes	8
88.43	even	2		3872.2.a.bg.1.3		4
88.61	odd	10		352.2.m.c.289.1	yes	8
88.83	even	10		352.2.m.c.289.2	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
352.2.m.c.257.1	✓	8	88.13	odd	10
352.2.m.c.257.2	yes	8	88.35	even	10
352.2.m.c.289.1	yes	8	88.61	odd	10
352.2.m.c.289.2	yes	8	88.83	even	10
704.2.m.k.257.1		8	44.35	even	10
704.2.m.k.257.2		8	11.2	odd	10
704.2.m.k.641.1		8	44.39	even	10
704.2.m.k.641.2		8	11.6	odd	10
3872.2.a.bf.1.2		4	8.5	even	2
3872.2.a.bf.1.3		4	8.3	odd	2
3872.2.a.bg.1.2		4	88.21	odd	2
3872.2.a.bg.1.3		4	88.43	even	2
7744.2.a.dp.1.2		4	44.43	even	2
7744.2.a.dp.1.3		4	11.10	odd	2
7744.2.a.dq.1.2		4	4.3	odd	2	inner
7744.2.a.dq.1.3		4	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+1.54336 q^{3} +0.381966 q^{5} -1.54336 q^{7} -0.618034 q^{9} +1.61803 q^{13} +0.589512 q^{15} +0.854102 q^{17} -4.63009 q^{19} -2.38197 q^{21} +4.99442 q^{23} -4.85410 q^{25} -5.58394 q^{27} -0.145898 q^{29} +2.13287 q^{31} -0.589512 q^{35} -1.85410 q^{37} +2.49721 q^{39} -0.618034 q^{41} -1.90770 q^{43} -0.236068 q^{45} +9.62451 q^{47} -4.61803 q^{49} +1.31819 q^{51} +8.56231 q^{53} -7.14590 q^{57} +10.3532 q^{59} -12.5623 q^{61} +0.953850 q^{63} +0.618034 q^{65} -8.08115 q^{67} +7.70820 q^{69} -10.9427 q^{71} -9.85410 q^{73} -7.49164 q^{75} -15.9371 q^{79} -6.76393 q^{81} -10.9427 q^{83} +0.326238 q^{85} -0.225173 q^{87} +8.18034 q^{89} -2.49721 q^{91} +3.29180 q^{93} -1.76854 q^{95} -3.14590 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 6 q^{5} + 2 q^{9} + 2 q^{13} - 10 q^{17} - 14 q^{21} - 6 q^{25} - 14 q^{29} + 6 q^{37} + 2 q^{41} + 8 q^{45} - 14 q^{49} - 6 q^{53} - 42 q^{57} - 10 q^{61} - 2 q^{65} + 4 q^{69} - 26 q^{73} - 36 q^{81}+ \cdots - 26 q^{97}+O(q^{100}) \) 4 * q + 6 * q^5 + 2 * q^9 + 2 * q^13 - 10 * q^17 - 14 * q^21 - 6 * q^25 - 14 * q^29 + 6 * q^37 + 2 * q^41 + 8 * q^45 - 14 * q^49 - 6 * q^53 - 42 * q^57 - 10 * q^61 - 2 * q^65 + 4 * q^69 - 26 * q^73 - 36 * q^81 - 30 * q^85 - 12 * q^89 + 40 * q^93 - 26 * q^97

Introduction

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