LMFDB - Embedded newform 7168.2.a.bd.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7168,2,Mod(1,7168)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7168, 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("7168.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7168 = 2^{10} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7168.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:	$57.2367681689$
Analytic rank:	$1$
Dimension:	$8$
Coefficient field:	8.8.13747093504.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 12x^{6} + 38x^{4} - 20x^{2} + 1 $ x^8 - 12x^6 + 38x^4 - 20*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 112)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-2.10489$ of defining polynomial
Character	$\chi$	$=$	7168.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.01155 q^{3} +1.22714 q^{5} +1.00000 q^{7} -1.97676 q^{9} +O(q^{10})$	$q-1.01155 q^{3} +1.22714 q^{5} +1.00000 q^{7} -1.97676 q^{9} -4.20978 q^{11} -2.85695 q^{13} -1.24132 q^{15} -0.264559 q^{17} +6.41995 q^{19} -1.01155 q^{21} +1.54621 q^{23} -3.49412 q^{25} +5.03426 q^{27} +0.464044 q^{29} -6.04033 q^{31} +4.25841 q^{33} +1.22714 q^{35} +9.40259 q^{37} +2.88995 q^{39} +11.0327 q^{41} +4.78580 q^{43} -2.42577 q^{45} +3.12566 q^{47} +1.00000 q^{49} +0.267615 q^{51} -0.608892 q^{53} -5.16599 q^{55} -6.49412 q^{57} +6.54272 q^{59} -6.88400 q^{61} -1.97676 q^{63} -3.50588 q^{65} +4.72877 q^{67} -1.56407 q^{69} -9.03885 q^{71} -14.8146 q^{73} +3.53449 q^{75} -4.20978 q^{77} +12.5904 q^{79} +0.837864 q^{81} +1.01155 q^{83} -0.324651 q^{85} -0.469405 q^{87} -10.9924 q^{89} -2.85695 q^{91} +6.11011 q^{93} +7.87820 q^{95} -14.2452 q^{97} +8.32172 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 8 q^{7} + 8 q^{9}+O(q^{10}) $ 8 * q + 8 * q^7 + 8 * q^9	$ 8 q + 8 q^{7} + 8 q^{9} - 8 q^{15} - 24 q^{17} - 8 q^{25} - 16 q^{31} - 24 q^{33} - 16 q^{39} - 8 q^{41} + 24 q^{47} + 8 q^{49} - 8 q^{55} - 32 q^{57} + 8 q^{63} - 48 q^{65} - 56 q^{71} - 48 q^{73} - 24 q^{79} - 40 q^{81} + 24 q^{87} - 24 q^{89} - 16 q^{95} - 32 q^{97}+O(q^{100}) $ 8 * q + 8 * q^7 + 8 * q^9 - 8 * q^15 - 24 * q^17 - 8 * q^25 - 16 * q^31 - 24 * q^33 - 16 * q^39 - 8 * q^41 + 24 * q^47 + 8 * q^49 - 8 * q^55 - 32 * q^57 + 8 * q^63 - 48 * q^65 - 56 * q^71 - 48 * q^73 - 24 * q^79 - 40 * q^81 + 24 * q^87 - 24 * q^89 - 16 * q^95 - 32 * q^97

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
$3$	−1.01155	−0.584020	−0.292010	−	0.956415i	$-0.594324\pi$
$3$	−1.01155	−0.584020	−0.292010	+	0.956415i	$0.594324\pi$
$4$	0	0
$5$	1.22714	0.548795	0.274397	−	0.961616i	$-0.411522\pi$
$5$	1.22714	0.548795	0.274397	+	0.961616i	$0.411522\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	−1.97676	−0.658920
$10$	0	0
$11$	−4.20978	−1.26929	−0.634647	−	0.772802i	$-0.718855\pi$
$11$	−4.20978	−1.26929	−0.634647	+	0.772802i	$0.718855\pi$
$12$	0	0
$13$	−2.85695	−0.792374	−0.396187	−	0.918170i	$-0.629667\pi$
$13$	−2.85695	−0.792374	−0.396187	+	0.918170i	$0.629667\pi$
$14$	0	0
$15$	−1.24132	−0.320507
$16$	0	0
$17$	−0.264559	−0.0641649	−0.0320825	−	0.999485i	$-0.510214\pi$
$17$	−0.264559	−0.0641649	−0.0320825	+	0.999485i	$0.510214\pi$
$18$	0	0
$19$	6.41995	1.47284	0.736419	−	0.676526i	$-0.236515\pi$
$19$	6.41995	1.47284	0.736419	+	0.676526i	$0.236515\pi$
$20$	0	0
$21$	−1.01155	−0.220739
$22$	0	0
$23$	1.54621	0.322407	0.161203	−	0.986921i	$-0.448462\pi$
$23$	1.54621	0.322407	0.161203	+	0.986921i	$0.448462\pi$
$24$	0	0
$25$	−3.49412	−0.698824
$26$	0	0
$27$	5.03426	0.968843
$28$	0	0
$29$	0.464044	0.0861708	0.0430854	−	0.999071i	$-0.486281\pi$
$29$	0.464044	0.0861708	0.0430854	+	0.999071i	$0.486281\pi$
$30$	0	0
$31$	−6.04033	−1.08488	−0.542438	−	0.840096i	$-0.682499\pi$
$31$	−6.04033	−1.08488	−0.542438	+	0.840096i	$0.682499\pi$
$32$	0	0
$33$	4.25841	0.741294
$34$	0	0
$35$	1.22714	0.207425
$36$	0	0
$37$	9.40259	1.54578	0.772888	−	0.634543i	$-0.218812\pi$
$37$	9.40259	1.54578	0.772888	+	0.634543i	$0.218812\pi$
$38$	0	0
$39$	2.88995	0.462763
$40$	0	0
$41$	11.0327	1.72302	0.861510	−	0.507741i	$-0.169519\pi$
$41$	11.0327	1.72302	0.861510	+	0.507741i	$0.169519\pi$
$42$	0	0
$43$	4.78580	0.729828	0.364914	−	0.931041i	$-0.381098\pi$
$43$	4.78580	0.729828	0.364914	+	0.931041i	$0.381098\pi$
$44$	0	0
$45$	−2.42577	−0.361612
$46$	0	0
$47$	3.12566	0.455925	0.227962	−	0.973670i	$-0.426794\pi$
$47$	3.12566	0.455925	0.227962	+	0.973670i	$0.426794\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0.267615	0.0374736
$52$	0	0
$53$	−0.608892	−0.0836378	−0.0418189	−	0.999125i	$-0.513315\pi$
$53$	−0.608892	−0.0836378	−0.0418189	+	0.999125i	$0.513315\pi$
$54$	0	0
$55$	−5.16599	−0.696582
$56$	0	0
$57$	−6.49412	−0.860167
$58$	0	0
$59$	6.54272	0.851790	0.425895	−	0.904773i	$-0.359959\pi$
$59$	6.54272	0.851790	0.425895	+	0.904773i	$0.359959\pi$
$60$	0	0
$61$	−6.88400	−0.881405	−0.440703	−	0.897653i	$-0.645271\pi$
$61$	−6.88400	−0.881405	−0.440703	+	0.897653i	$0.645271\pi$
$62$	0	0
$63$	−1.97676	−0.249048
$64$	0	0
$65$	−3.50588	−0.434851
$66$	0	0
$67$	4.72877	0.577711	0.288855	−	0.957373i	$-0.406725\pi$
$67$	4.72877	0.577711	0.288855	+	0.957373i	$0.406725\pi$
$68$	0	0
$69$	−1.56407	−0.188292
$70$	0	0
$71$	−9.03885	−1.07271	−0.536357	−	0.843991i	$-0.680200\pi$
$71$	−9.03885	−1.07271	−0.536357	+	0.843991i	$0.680200\pi$
$72$	0	0
$73$	−14.8146	−1.73392	−0.866960	−	0.498377i	$-0.833930\pi$
$73$	−14.8146	−1.73392	−0.866960	+	0.498377i	$0.833930\pi$
$74$	0	0
$75$	3.53449	0.408128
$76$	0	0
$77$	−4.20978	−0.479748
$78$	0	0
$79$	12.5904	1.41653	0.708265	−	0.705947i	$-0.249478\pi$
$79$	12.5904	1.41653	0.708265	+	0.705947i	$0.249478\pi$
$80$	0	0
$81$	0.837864	0.0930960
$82$	0	0
$83$	1.01155	0.111032	0.0555162	−	0.998458i	$-0.482320\pi$
$83$	1.01155	0.111032	0.0555162	+	0.998458i	$0.482320\pi$
$84$	0	0
$85$	−0.324651	−0.0352134
$86$	0	0
$87$	−0.469405	−0.0503255
$88$	0	0
$89$	−10.9924	−1.16519	−0.582595	−	0.812763i	$-0.697962\pi$
$89$	−10.9924	−1.16519	−0.582595	+	0.812763i	$0.697962\pi$
$90$	0	0
$91$	−2.85695	−0.299489
$92$	0	0
$93$	6.11011	0.633589
$94$	0	0
$95$	7.87820	0.808286
$96$	0	0
$97$	−14.2452	−1.44638	−0.723189	−	0.690650i	$-0.757325\pi$
$97$	−14.2452	−1.44638	−0.723189	+	0.690650i	$0.757325\pi$
$98$	0	0
$99$	8.32172	0.836364
$100$	0	0
$101$	−14.8915	−1.48176	−0.740881	−	0.671636i	$-0.765592\pi$
$101$	−14.8915	−1.48176	−0.740881	+	0.671636i	$0.765592\pi$
$102$	0	0
$103$	−12.9862	−1.27957	−0.639785	−	0.768554i	$-0.720977\pi$
$103$	−12.9862	−1.27957	−0.639785	+	0.768554i	$0.720977\pi$
$104$	0	0
$105$	−1.24132	−0.121140
$106$	0	0
$107$	−1.75549	−0.169710	−0.0848548	−	0.996393i	$-0.527043\pi$
$107$	−1.75549	−0.169710	−0.0848548	+	0.996393i	$0.527043\pi$
$108$	0	0
$109$	−15.4906	−1.48373	−0.741866	−	0.670548i	$-0.766059\pi$
$109$	−15.4906	−1.48373	−0.741866	+	0.670548i	$0.766059\pi$
$110$	0	0
$111$	−9.51121	−0.902764
$112$	0	0
$113$	−7.63302	−0.718054	−0.359027	−	0.933327i	$-0.616891\pi$
$113$	−7.63302	−0.718054	−0.359027	+	0.933327i	$0.616891\pi$
$114$	0	0
$115$	1.89742	0.176935
$116$	0	0
$117$	5.64750	0.522111
$118$	0	0
$119$	−0.264559	−0.0242521
$120$	0	0
$121$	6.72221	0.611110
$122$	0	0
$123$	−11.1602	−1.00628
$124$	0	0
$125$	−10.4235	−0.932306
$126$	0	0
$127$	10.7393	0.952959	0.476479	−	0.879186i	$-0.341913\pi$
$127$	10.7393	0.952959	0.476479	+	0.879186i	$0.341913\pi$
$128$	0	0
$129$	−4.84109	−0.426234
$130$	0	0
$131$	−18.2248	−1.59231	−0.796154	−	0.605094i	$-0.793135\pi$
$131$	−18.2248	−1.59231	−0.796154	+	0.605094i	$0.793135\pi$
$132$	0	0
$133$	6.41995	0.556680
$134$	0	0
$135$	6.17775	0.531696
$136$	0	0
$137$	−7.34374	−0.627418	−0.313709	−	0.949519i	$-0.601572\pi$
$137$	−7.34374	−0.627418	−0.313709	+	0.949519i	$0.601572\pi$
$138$	0	0
$139$	11.8254	1.00302	0.501510	−	0.865152i	$-0.332778\pi$
$139$	11.8254	1.00302	0.501510	+	0.865152i	$0.332778\pi$
$140$	0	0
$141$	−3.16177	−0.266269
$142$	0	0
$143$	12.0271	1.00576
$144$	0	0
$145$	0.569448	0.0472901
$146$	0	0
$147$	−1.01155	−0.0834315
$148$	0	0
$149$	18.2125	1.49203	0.746014	−	0.665930i	$-0.231965\pi$
$149$	18.2125	1.49203	0.746014	+	0.665930i	$0.231965\pi$
$150$	0	0
$151$	−13.0171	−1.05932	−0.529658	−	0.848211i	$-0.677680\pi$
$151$	−13.0171	−1.05932	−0.529658	+	0.848211i	$0.677680\pi$
$152$	0	0
$153$	0.522969	0.0422796
$154$	0	0
$155$	−7.41234	−0.595374
$156$	0	0
$157$	−20.7282	−1.65429	−0.827145	−	0.561989i	$-0.810036\pi$
$157$	−20.7282	−1.65429	−0.827145	+	0.561989i	$0.810036\pi$
$158$	0	0
$159$	0.615927	0.0488462
$160$	0	0
$161$	1.54621	0.121858
$162$	0	0
$163$	6.28992	0.492664	0.246332	−	0.969185i	$-0.420775\pi$
$163$	6.28992	0.492664	0.246332	+	0.969185i	$0.420775\pi$
$164$	0	0
$165$	5.22568	0.406818
$166$	0	0
$167$	−5.45765	−0.422326	−0.211163	−	0.977451i	$-0.567725\pi$
$167$	−5.45765	−0.422326	−0.211163	+	0.977451i	$0.567725\pi$
$168$	0	0
$169$	−4.83786	−0.372143
$170$	0	0
$171$	−12.6907	−0.970483
$172$	0	0
$173$	16.4364	1.24964	0.624820	−	0.780769i	$-0.285172\pi$
$173$	16.4364	1.24964	0.624820	+	0.780769i	$0.285172\pi$
$174$	0	0
$175$	−3.49412	−0.264131
$176$	0	0
$177$	−6.61831	−0.497462
$178$	0	0
$179$	−18.3024	−1.36799	−0.683994	−	0.729488i	$-0.739758\pi$
$179$	−18.3024	−1.36799	−0.683994	+	0.729488i	$0.739758\pi$
$180$	0	0
$181$	−1.96673	−0.146186	−0.0730930	−	0.997325i	$-0.523287\pi$
$181$	−1.96673	−0.146186	−0.0730930	+	0.997325i	$0.523287\pi$
$182$	0	0
$183$	6.96353	0.514759
$184$	0	0
$185$	11.5383	0.848313
$186$	0	0
$187$	1.11373	0.0814442
$188$	0	0
$189$	5.03426	0.366188
$190$	0	0
$191$	3.47088	0.251144	0.125572	−	0.992084i	$-0.459923\pi$
$191$	3.47088	0.251144	0.125572	+	0.992084i	$0.459923\pi$
$192$	0	0
$193$	−7.58654	−0.546091	−0.273046	−	0.962001i	$-0.588031\pi$
$193$	−7.58654	−0.546091	−0.273046	+	0.962001i	$0.588031\pi$
$194$	0	0
$195$	3.54638	0.253962
$196$	0	0
$197$	11.0703	0.788724	0.394362	−	0.918955i	$-0.370966\pi$
$197$	11.0703	0.788724	0.394362	+	0.918955i	$0.370966\pi$
$198$	0	0
$199$	19.6696	1.39434	0.697170	−	0.716906i	$-0.254442\pi$
$199$	19.6696	1.39434	0.697170	+	0.716906i	$0.254442\pi$
$200$	0	0
$201$	−4.78340	−0.337395
$202$	0	0
$203$	0.464044	0.0325695
$204$	0	0
$205$	13.5387	0.945584
$206$	0	0
$207$	−3.05648	−0.212440
$208$	0	0
$209$	−27.0266	−1.86947
$210$	0	0
$211$	1.61064	0.110881	0.0554406	−	0.998462i	$-0.482344\pi$
$211$	1.61064	0.110881	0.0554406	+	0.998462i	$0.482344\pi$
$212$	0	0
$213$	9.14328	0.626487
$214$	0	0
$215$	5.87286	0.400526
$216$	0	0
$217$	−6.04033	−0.410044
$218$	0	0
$219$	14.9858	1.01265
$220$	0	0
$221$	0.755830	0.0508426
$222$	0	0
$223$	8.02710	0.537534	0.268767	−	0.963205i	$-0.413384\pi$
$223$	8.02710	0.537534	0.268767	+	0.963205i	$0.413384\pi$
$224$	0	0
$225$	6.90704	0.460469
$226$	0	0
$227$	−5.05484	−0.335502	−0.167751	−	0.985829i	$-0.553650\pi$
$227$	−5.05484	−0.335502	−0.167751	+	0.985829i	$0.553650\pi$
$228$	0	0
$229$	0.899570	0.0594453	0.0297226	−	0.999558i	$-0.490538\pi$
$229$	0.899570	0.0594453	0.0297226	+	0.999558i	$0.490538\pi$
$230$	0	0
$231$	4.25841	0.280183
$232$	0	0
$233$	−24.5633	−1.60920	−0.804598	−	0.593820i	$-0.797619\pi$
$233$	−24.5633	−1.60920	−0.804598	+	0.593820i	$0.797619\pi$
$234$	0	0
$235$	3.83563	0.250209
$236$	0	0
$237$	−12.7359	−0.827283
$238$	0	0
$239$	−29.3026	−1.89543	−0.947714	−	0.319122i	$-0.896612\pi$
$239$	−29.3026	−1.89543	−0.947714	+	0.319122i	$0.896612\pi$
$240$	0	0
$241$	−7.94029	−0.511479	−0.255739	−	0.966746i	$-0.582319\pi$
$241$	−7.94029	−0.511479	−0.255739	+	0.966746i	$0.582319\pi$
$242$	0	0
$243$	−15.9503	−1.02321
$244$	0	0
$245$	1.22714	0.0783992
$246$	0	0
$247$	−18.3415	−1.16704
$248$	0	0
$249$	−1.02324	−0.0648452
$250$	0	0
$251$	−9.80525	−0.618902	−0.309451	−	0.950915i	$-0.600145\pi$
$251$	−9.80525	−0.618902	−0.309451	+	0.950915i	$0.600145\pi$
$252$	0	0
$253$	−6.50919	−0.409229
$254$	0	0
$255$	0.328402	0.0205653
$256$	0	0
$257$	−8.30489	−0.518045	−0.259022	−	0.965871i	$-0.583400\pi$
$257$	−8.30489	−0.518045	−0.259022	+	0.965871i	$0.583400\pi$
$258$	0	0
$259$	9.40259	0.584248
$260$	0	0
$261$	−0.917304	−0.0567797
$262$	0	0
$263$	−13.8535	−0.854242	−0.427121	−	0.904194i	$-0.640472\pi$
$263$	−13.8535	−0.854242	−0.427121	+	0.904194i	$0.640472\pi$
$264$	0	0
$265$	−0.747198	−0.0459000
$266$	0	0
$267$	11.1194	0.680494
$268$	0	0
$269$	1.81896	0.110904	0.0554521	−	0.998461i	$-0.482340\pi$
$269$	1.81896	0.110904	0.0554521	+	0.998461i	$0.482340\pi$
$270$	0	0
$271$	−16.3703	−0.994425	−0.497212	−	0.867629i	$-0.665643\pi$
$271$	−16.3703	−0.994425	−0.497212	+	0.867629i	$0.665643\pi$
$272$	0	0
$273$	2.88995	0.174908
$274$	0	0
$275$	14.7095	0.887014
$276$	0	0
$277$	−11.6933	−0.702583	−0.351291	−	0.936266i	$-0.614257\pi$
$277$	−11.6933	−0.702583	−0.351291	+	0.936266i	$0.614257\pi$
$278$	0	0
$279$	11.9403	0.714846
$280$	0	0
$281$	−30.2126	−1.80233	−0.901165	−	0.433476i	$-0.857287\pi$
$281$	−30.2126	−1.80233	−0.901165	+	0.433476i	$0.857287\pi$
$282$	0	0
$283$	16.6986	0.992629	0.496314	−	0.868143i	$-0.334686\pi$
$283$	16.6986	0.992629	0.496314	+	0.868143i	$0.334686\pi$
$284$	0	0
$285$	−7.96921	−0.472055
$286$	0	0
$287$	11.0327	0.651240
$288$	0	0
$289$	−16.9300	−0.995883
$290$	0	0
$291$	14.4098	0.844715
$292$	0	0
$293$	21.7575	1.27109	0.635543	−	0.772066i	$-0.280776\pi$
$293$	21.7575	1.27109	0.635543	+	0.772066i	$0.280776\pi$
$294$	0	0
$295$	8.02885	0.467458
$296$	0	0
$297$	−21.1931	−1.22975
$298$	0	0
$299$	−4.41743	−0.255467
$300$	0	0
$301$	4.78580	0.275849
$302$	0	0
$303$	15.0636	0.865379
$304$	0	0
$305$	−8.44764	−0.483711
$306$	0	0
$307$	11.5300	0.658050	0.329025	−	0.944321i	$-0.393280\pi$
$307$	11.5300	0.658050	0.329025	+	0.944321i	$0.393280\pi$
$308$	0	0
$309$	13.1363	0.747295
$310$	0	0
$311$	31.1072	1.76393	0.881964	−	0.471316i	$-0.156221\pi$
$311$	31.1072	1.76393	0.881964	+	0.471316i	$0.156221\pi$
$312$	0	0
$313$	3.17986	0.179736	0.0898681	−	0.995954i	$-0.471355\pi$
$313$	3.17986	0.179736	0.0898681	+	0.995954i	$0.471355\pi$
$314$	0	0
$315$	−2.42577	−0.136676
$316$	0	0
$317$	−2.21953	−0.124661	−0.0623307	−	0.998056i	$-0.519853\pi$
$317$	−2.21953	−0.124661	−0.0623307	+	0.998056i	$0.519853\pi$
$318$	0	0
$319$	−1.95352	−0.109376
$320$	0	0
$321$	1.77577	0.0991139
$322$	0	0
$323$	−1.69845	−0.0945045
$324$	0	0
$325$	9.98251	0.553730
$326$	0	0
$327$	15.6696	0.866530
$328$	0	0
$329$	3.12566	0.172323
$330$	0	0
$331$	2.77139	0.152329	0.0761647	−	0.997095i	$-0.475733\pi$
$331$	2.77139	0.152329	0.0761647	+	0.997095i	$0.475733\pi$
$332$	0	0
$333$	−18.5867	−1.01854
$334$	0	0
$335$	5.80287	0.317045
$336$	0	0
$337$	28.7067	1.56375	0.781876	−	0.623434i	$-0.214263\pi$
$337$	28.7067	1.56375	0.781876	+	0.623434i	$0.214263\pi$
$338$	0	0
$339$	7.72120	0.419358
$340$	0	0
$341$	25.4284	1.37703
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	−1.91934	−0.103334
$346$	0	0
$347$	−6.24912	−0.335470	−0.167735	−	0.985832i	$-0.553645\pi$
$347$	−6.24912	−0.335470	−0.167735	+	0.985832i	$0.553645\pi$
$348$	0	0
$349$	10.5561	0.565053	0.282527	−	0.959259i	$-0.408827\pi$
$349$	10.5561	0.565053	0.282527	+	0.959259i	$0.408827\pi$
$350$	0	0
$351$	−14.3826	−0.767686
$352$	0	0
$353$	−6.70687	−0.356971	−0.178485	−	0.983943i	$-0.557120\pi$
$353$	−6.70687	−0.356971	−0.178485	+	0.983943i	$0.557120\pi$
$354$	0	0
$355$	−11.0920	−0.588700
$356$	0	0
$357$	0.267615	0.0141637
$358$	0	0
$359$	−11.7540	−0.620353	−0.310176	−	0.950679i	$-0.600388\pi$
$359$	−11.7540	−0.620353	−0.310176	+	0.950679i	$0.600388\pi$
$360$	0	0
$361$	22.2158	1.16925
$362$	0	0
$363$	−6.79987	−0.356901
$364$	0	0
$365$	−18.1797	−0.951567
$366$	0	0
$367$	34.4703	1.79934	0.899669	−	0.436573i	$-0.143808\pi$
$367$	34.4703	1.79934	0.899669	+	0.436573i	$0.143808\pi$
$368$	0	0
$369$	−21.8090	−1.13533
$370$	0	0
$371$	−0.608892	−0.0316121
$372$	0	0
$373$	12.4416	0.644201	0.322101	−	0.946705i	$-0.395611\pi$
$373$	12.4416	0.644201	0.322101	+	0.946705i	$0.395611\pi$
$374$	0	0
$375$	10.5439	0.544486
$376$	0	0
$377$	−1.32575	−0.0682795
$378$	0	0
$379$	0.242681	0.0124657	0.00623284	−	0.999981i	$-0.498016\pi$
$379$	0.242681	0.0124657	0.00623284	+	0.999981i	$0.498016\pi$
$380$	0	0
$381$	−10.8634	−0.556547
$382$	0	0
$383$	−21.7161	−1.10964	−0.554819	−	0.831971i	$-0.687213\pi$
$383$	−21.7161	−1.10964	−0.554819	+	0.831971i	$0.687213\pi$
$384$	0	0
$385$	−5.16599	−0.263283
$386$	0	0
$387$	−9.46039	−0.480898
$388$	0	0
$389$	16.0258	0.812543	0.406271	−	0.913752i	$-0.366829\pi$
$389$	16.0258	0.812543	0.406271	+	0.913752i	$0.366829\pi$
$390$	0	0
$391$	−0.409063	−0.0206872
$392$	0	0
$393$	18.4353	0.929940
$394$	0	0
$395$	15.4502	0.777384
$396$	0	0
$397$	−20.8563	−1.04675	−0.523375	−	0.852102i	$-0.675327\pi$
$397$	−20.8563	−1.04675	−0.523375	+	0.852102i	$0.675327\pi$
$398$	0	0
$399$	−6.49412	−0.325113
$400$	0	0
$401$	0.146885	0.00733506	0.00366753	−	0.999993i	$-0.498833\pi$
$401$	0.146885	0.00733506	0.00366753	+	0.999993i	$0.498833\pi$
$402$	0	0
$403$	17.2569	0.859627
$404$	0	0
$405$	1.02818	0.0510906
$406$	0	0
$407$	−39.5828	−1.96205
$408$	0	0
$409$	2.27164	0.112326	0.0561628	−	0.998422i	$-0.482113\pi$
$409$	2.27164	0.112326	0.0561628	+	0.998422i	$0.482113\pi$
$410$	0	0
$411$	7.42858	0.366425
$412$	0	0
$413$	6.54272	0.321946
$414$	0	0
$415$	1.24132	0.0609340
$416$	0	0
$417$	−11.9620	−0.585784
$418$	0	0
$419$	27.3868	1.33793	0.668966	−	0.743293i	$-0.266737\pi$
$419$	27.3868	1.33793	0.668966	+	0.743293i	$0.266737\pi$
$420$	0	0
$421$	11.4693	0.558981	0.279491	−	0.960148i	$-0.409834\pi$
$421$	11.4693	0.558981	0.279491	+	0.960148i	$0.409834\pi$
$422$	0	0
$423$	−6.17869	−0.300418
$424$	0	0
$425$	0.924401	0.0448400
$426$	0	0
$427$	−6.88400	−0.333140
$428$	0	0
$429$	−12.1660	−0.587382
$430$	0	0
$431$	−10.7658	−0.518569	−0.259284	−	0.965801i	$-0.583487\pi$
$431$	−10.7658	−0.518569	−0.259284	+	0.965801i	$0.583487\pi$
$432$	0	0
$433$	−4.99439	−0.240015	−0.120008	−	0.992773i	$-0.538292\pi$
$433$	−4.99439	−0.240015	−0.120008	+	0.992773i	$0.538292\pi$
$434$	0	0
$435$	−0.576027	−0.0276184
$436$	0	0
$437$	9.92659	0.474853
$438$	0	0
$439$	−1.69924	−0.0811004	−0.0405502	−	0.999178i	$-0.512911\pi$
$439$	−1.69924	−0.0811004	−0.0405502	+	0.999178i	$0.512911\pi$
$440$	0	0
$441$	−1.97676	−0.0941315
$442$	0	0
$443$	11.1764	0.531007	0.265504	−	0.964110i	$-0.414462\pi$
$443$	11.1764	0.531007	0.265504	+	0.964110i	$0.414462\pi$
$444$	0	0
$445$	−13.4892	−0.639450
$446$	0	0
$447$	−18.4229	−0.871375
$448$	0	0
$449$	4.05419	0.191329	0.0956646	−	0.995414i	$-0.469502\pi$
$449$	4.05419	0.191329	0.0956646	+	0.995414i	$0.469502\pi$
$450$	0	0
$451$	−46.4452	−2.18702
$452$	0	0
$453$	13.1675	0.618662
$454$	0	0
$455$	−3.50588	−0.164358
$456$	0	0
$457$	10.1498	0.474789	0.237395	−	0.971413i	$-0.423707\pi$
$457$	10.1498	0.474789	0.237395	+	0.971413i	$0.423707\pi$
$458$	0	0
$459$	−1.33186	−0.0621658
$460$	0	0
$461$	1.45111	0.0675848	0.0337924	−	0.999429i	$-0.489241\pi$
$461$	1.45111	0.0675848	0.0337924	+	0.999429i	$0.489241\pi$
$462$	0	0
$463$	10.5945	0.492369	0.246185	−	0.969223i	$-0.420823\pi$
$463$	10.5945	0.492369	0.246185	+	0.969223i	$0.420823\pi$
$464$	0	0
$465$	7.49798	0.347710
$466$	0	0
$467$	−13.1061	−0.606478	−0.303239	−	0.952915i	$-0.598068\pi$
$467$	−13.1061	−0.606478	−0.303239	+	0.952915i	$0.598068\pi$
$468$	0	0
$469$	4.72877	0.218354
$470$	0	0
$471$	20.9677	0.966139
$472$	0	0
$473$	−20.1472	−0.926367
$474$	0	0
$475$	−22.4321	−1.02926
$476$	0	0
$477$	1.20363	0.0551106
$478$	0	0
$479$	−33.4766	−1.52958	−0.764792	−	0.644277i	$-0.777158\pi$
$479$	−33.4766	−1.52958	−0.764792	+	0.644277i	$0.777158\pi$
$480$	0	0
$481$	−26.8627	−1.22483
$482$	0	0
$483$	−1.56407	−0.0711677
$484$	0	0
$485$	−17.4809	−0.793765
$486$	0	0
$487$	23.0248	1.04335	0.521677	−	0.853143i	$-0.325307\pi$
$487$	23.0248	1.04335	0.521677	+	0.853143i	$0.325307\pi$
$488$	0	0
$489$	−6.36258	−0.287726
$490$	0	0
$491$	−31.0196	−1.39989	−0.699947	−	0.714195i	$-0.746793\pi$
$491$	−31.0196	−1.39989	−0.699947	+	0.714195i	$0.746793\pi$
$492$	0	0
$493$	−0.122767	−0.00552914
$494$	0	0
$495$	10.2119	0.458992
$496$	0	0
$497$	−9.03885	−0.405448
$498$	0	0
$499$	−8.65754	−0.387565	−0.193782	−	0.981045i	$-0.562076\pi$
$499$	−8.65754	−0.387565	−0.193782	+	0.981045i	$0.562076\pi$
$500$	0	0
$501$	5.52070	0.246647
$502$	0	0
$503$	−11.5286	−0.514034	−0.257017	−	0.966407i	$-0.582740\pi$
$503$	−11.5286	−0.514034	−0.257017	+	0.966407i	$0.582740\pi$
$504$	0	0
$505$	−18.2740	−0.813183
$506$	0	0
$507$	4.89376	0.217339
$508$	0	0
$509$	25.3620	1.12415	0.562076	−	0.827085i	$-0.310003\pi$
$509$	25.3620	1.12415	0.562076	+	0.827085i	$0.310003\pi$
$510$	0	0
$511$	−14.8146	−0.655360
$512$	0	0
$513$	32.3197	1.42695
$514$	0	0
$515$	−15.9359	−0.702222
$516$	0	0
$517$	−13.1583	−0.578703
$518$	0	0
$519$	−16.6263	−0.729815
$520$	0	0
$521$	−0.287525	−0.0125967	−0.00629835	−	0.999980i	$-0.502005\pi$
$521$	−0.287525	−0.0125967	−0.00629835	+	0.999980i	$0.502005\pi$
$522$	0	0
$523$	−5.43483	−0.237648	−0.118824	−	0.992915i	$-0.537913\pi$
$523$	−5.43483	−0.237648	−0.118824	+	0.992915i	$0.537913\pi$
$524$	0	0
$525$	3.53449	0.154258
$526$	0	0
$527$	1.59802	0.0696109
$528$	0	0
$529$	−20.6092	−0.896054
$530$	0	0
$531$	−12.9334	−0.561261
$532$	0	0
$533$	−31.5198	−1.36528
$534$	0	0
$535$	−2.15424	−0.0931358
$536$	0	0
$537$	18.5139	0.798932
$538$	0	0
$539$	−4.20978	−0.181328
$540$	0	0
$541$	−37.2691	−1.60232	−0.801161	−	0.598449i	$-0.795784\pi$
$541$	−37.2691	−1.60232	−0.801161	+	0.598449i	$0.795784\pi$
$542$	0	0
$543$	1.98945	0.0853757
$544$	0	0
$545$	−19.0092	−0.814264
$546$	0	0
$547$	14.7369	0.630103	0.315052	−	0.949075i	$-0.397978\pi$
$547$	14.7369	0.630103	0.315052	+	0.949075i	$0.397978\pi$
$548$	0	0
$549$	13.6080	0.580776
$550$	0	0
$551$	2.97914	0.126916
$552$	0	0
$553$	12.5904	0.535398
$554$	0	0
$555$	−11.6716	−0.495432
$556$	0	0
$557$	10.9562	0.464229	0.232114	−	0.972688i	$-0.425436\pi$
$557$	10.9562	0.464229	0.232114	+	0.972688i	$0.425436\pi$
$558$	0	0
$559$	−13.6728	−0.578297
$560$	0	0
$561$	−1.12660	−0.0475651
$562$	0	0
$563$	−19.8795	−0.837822	−0.418911	−	0.908027i	$-0.637588\pi$
$563$	−19.8795	−0.837822	−0.418911	+	0.908027i	$0.637588\pi$
$564$	0	0
$565$	−9.36680	−0.394064
$566$	0	0
$567$	0.837864	0.0351870
$568$	0	0
$569$	−26.9645	−1.13041	−0.565205	−	0.824951i	$-0.691203\pi$
$569$	−26.9645	−1.13041	−0.565205	+	0.824951i	$0.691203\pi$
$570$	0	0
$571$	−30.8484	−1.29097	−0.645484	−	0.763774i	$-0.723344\pi$
$571$	−30.8484	−1.29097	−0.645484	+	0.763774i	$0.723344\pi$
$572$	0	0
$573$	−3.51098	−0.146673
$574$	0	0
$575$	−5.40264	−0.225306
$576$	0	0
$577$	8.16250	0.339809	0.169905	−	0.985461i	$-0.445654\pi$
$577$	8.16250	0.339809	0.169905	+	0.985461i	$0.445654\pi$
$578$	0	0
$579$	7.67419	0.318928
$580$	0	0
$581$	1.01155	0.0419663
$582$	0	0
$583$	2.56330	0.106161
$584$	0	0
$585$	6.93028	0.286532
$586$	0	0
$587$	−0.0235079	−0.000970273 0	−0.000485137	−	1.00000i	$-0.500154\pi$
$587$	−0.0235079	−0.000970273 0	−0.000485137		1.00000i	$0.500154\pi$
$588$	0	0
$589$	−38.7786	−1.59785
$590$	0	0
$591$	−11.1982	−0.460631
$592$	0	0
$593$	−16.7146	−0.686386	−0.343193	−	0.939265i	$-0.611508\pi$
$593$	−16.7146	−0.686386	−0.343193	+	0.939265i	$0.611508\pi$
$594$	0	0
$595$	−0.324651	−0.0133094
$596$	0	0
$597$	−19.8968	−0.814323
$598$	0	0
$599$	17.5716	0.717954	0.358977	−	0.933346i	$-0.383126\pi$
$599$	17.5716	0.717954	0.358977	+	0.933346i	$0.383126\pi$
$600$	0	0
$601$	−6.99237	−0.285225	−0.142612	−	0.989779i	$-0.545550\pi$
$601$	−6.99237	−0.285225	−0.142612	+	0.989779i	$0.545550\pi$
$602$	0	0
$603$	−9.34764	−0.380665
$604$	0	0
$605$	8.24911	0.335374
$606$	0	0
$607$	6.25546	0.253901	0.126951	−	0.991909i	$-0.459481\pi$
$607$	6.25546	0.253901	0.126951	+	0.991909i	$0.459481\pi$
$608$	0	0
$609$	−0.469405	−0.0190213
$610$	0	0
$611$	−8.92985	−0.361263
$612$	0	0
$613$	−8.05956	−0.325522	−0.162761	−	0.986665i	$-0.552040\pi$
$613$	−8.05956	−0.325522	−0.162761	+	0.986665i	$0.552040\pi$
$614$	0	0
$615$	−13.6951	−0.552240
$616$	0	0
$617$	−44.4895	−1.79108	−0.895541	−	0.444980i	$-0.853211\pi$
$617$	−44.4895	−1.79108	−0.895541	+	0.444980i	$0.853211\pi$
$618$	0	0
$619$	−19.6224	−0.788690	−0.394345	−	0.918962i	$-0.629029\pi$
$619$	−19.6224	−0.788690	−0.394345	+	0.918962i	$0.629029\pi$
$620$	0	0
$621$	7.78401	0.312362
$622$	0	0
$623$	−10.9924	−0.440400
$624$	0	0
$625$	4.67950	0.187180
$626$	0	0
$627$	27.3388	1.09181
$628$	0	0
$629$	−2.48754	−0.0991846
$630$	0	0
$631$	8.64101	0.343993	0.171997	−	0.985098i	$-0.444978\pi$
$631$	8.64101	0.343993	0.171997	+	0.985098i	$0.444978\pi$
$632$	0	0
$633$	−1.62925	−0.0647569
$634$	0	0
$635$	13.1786	0.522979
$636$	0	0
$637$	−2.85695	−0.113196
$638$	0	0
$639$	17.8676	0.706833
$640$	0	0
$641$	15.0886	0.595966	0.297983	−	0.954571i	$-0.403686\pi$
$641$	15.0886	0.595966	0.297983	+	0.954571i	$0.403686\pi$
$642$	0	0
$643$	−17.4736	−0.689091	−0.344546	−	0.938770i	$-0.611967\pi$
$643$	−17.4736	−0.689091	−0.344546	+	0.938770i	$0.611967\pi$
$644$	0	0
$645$	−5.94071	−0.233915
$646$	0	0
$647$	40.8085	1.60435	0.802173	−	0.597091i	$-0.203677\pi$
$647$	40.8085	1.60435	0.802173	+	0.597091i	$0.203677\pi$
$648$	0	0
$649$	−27.5434	−1.08117
$650$	0	0
$651$	6.11011	0.239474
$652$	0	0
$653$	−13.9778	−0.546994	−0.273497	−	0.961873i	$-0.588180\pi$
$653$	−13.9778	−0.546994	−0.273497	+	0.961873i	$0.588180\pi$
$654$	0	0
$655$	−22.3644	−0.873850
$656$	0	0
$657$	29.2850	1.14252
$658$	0	0
$659$	3.84433	0.149754	0.0748769	−	0.997193i	$-0.476144\pi$
$659$	3.84433	0.149754	0.0748769	+	0.997193i	$0.476144\pi$
$660$	0	0
$661$	13.8211	0.537577	0.268789	−	0.963199i	$-0.413377\pi$
$661$	13.8211	0.537577	0.268789	+	0.963199i	$0.413377\pi$
$662$	0	0
$663$	−0.764562	−0.0296931
$664$	0	0
$665$	7.87820	0.305503
$666$	0	0
$667$	0.717509	0.0277821
$668$	0	0
$669$	−8.11983	−0.313931
$670$	0	0
$671$	28.9801	1.11876
$672$	0	0
$673$	−19.1036	−0.736391	−0.368195	−	0.929748i	$-0.620024\pi$
$673$	−19.1036	−0.736391	−0.368195	+	0.929748i	$0.620024\pi$
$674$	0	0
$675$	−17.5903	−0.677051
$676$	0	0
$677$	14.7396	0.566488	0.283244	−	0.959048i	$-0.408589\pi$
$677$	14.7396	0.566488	0.283244	+	0.959048i	$0.408589\pi$
$678$	0	0
$679$	−14.2452	−0.546680
$680$	0	0
$681$	5.11324	0.195940
$682$	0	0
$683$	−49.5658	−1.89658	−0.948291	−	0.317401i	$-0.897190\pi$
$683$	−49.5658	−1.89658	−0.948291	+	0.317401i	$0.897190\pi$
$684$	0	0
$685$	−9.01182	−0.344324
$686$	0	0
$687$	−0.909963	−0.0347172
$688$	0	0
$689$	1.73957	0.0662724
$690$	0	0
$691$	−5.67744	−0.215980	−0.107990	−	0.994152i	$-0.534441\pi$
$691$	−5.67744	−0.215980	−0.107990	+	0.994152i	$0.534441\pi$
$692$	0	0
$693$	8.32172	0.316116
$694$	0	0
$695$	14.5115	0.550452
$696$	0	0
$697$	−2.91880	−0.110557
$698$	0	0
$699$	24.8471	0.939803
$700$	0	0
$701$	−9.77425	−0.369168	−0.184584	−	0.982817i	$-0.559094\pi$
$701$	−9.77425	−0.369168	−0.184584	+	0.982817i	$0.559094\pi$
$702$	0	0
$703$	60.3642	2.27668
$704$	0	0
$705$	−3.87995	−0.146127
$706$	0	0
$707$	−14.8915	−0.560054
$708$	0	0
$709$	26.1760	0.983059	0.491529	−	0.870861i	$-0.336438\pi$
$709$	26.1760	0.983059	0.491529	+	0.870861i	$0.336438\pi$
$710$	0	0
$711$	−24.8882	−0.933380
$712$	0	0
$713$	−9.33961	−0.349771
$714$	0	0
$715$	14.7590	0.551954
$716$	0	0
$717$	29.6411	1.10697
$718$	0	0
$719$	14.5342	0.542034	0.271017	−	0.962575i	$-0.412640\pi$
$719$	14.5342	0.542034	0.271017	+	0.962575i	$0.412640\pi$
$720$	0	0
$721$	−12.9862	−0.483632
$722$	0	0
$723$	8.03202	0.298714
$724$	0	0
$725$	−1.62143	−0.0602183
$726$	0	0
$727$	−35.2605	−1.30774	−0.653870	−	0.756607i	$-0.726856\pi$
$727$	−35.2605	−1.30774	−0.653870	+	0.756607i	$0.726856\pi$
$728$	0	0
$729$	13.6210	0.504481
$730$	0	0
$731$	−1.26613	−0.0468294
$732$	0	0
$733$	6.61975	0.244506	0.122253	−	0.992499i	$-0.460988\pi$
$733$	6.61975	0.244506	0.122253	+	0.992499i	$0.460988\pi$
$734$	0	0
$735$	−1.24132	−0.0457868
$736$	0	0
$737$	−19.9070	−0.733285
$738$	0	0
$739$	15.7033	0.577655	0.288827	−	0.957381i	$-0.406735\pi$
$739$	15.7033	0.577655	0.288827	+	0.957381i	$0.406735\pi$
$740$	0	0
$741$	18.5533	0.681574
$742$	0	0
$743$	9.88941	0.362807	0.181404	−	0.983409i	$-0.441936\pi$
$743$	9.88941	0.362807	0.181404	+	0.983409i	$0.441936\pi$
$744$	0	0
$745$	22.3494	0.818817
$746$	0	0
$747$	−1.99960	−0.0731615
$748$	0	0
$749$	−1.75549	−0.0641442
$750$	0	0
$751$	−3.91696	−0.142932	−0.0714659	−	0.997443i	$-0.522768\pi$
$751$	−3.91696	−0.142932	−0.0714659	+	0.997443i	$0.522768\pi$
$752$	0	0
$753$	9.91853	0.361451
$754$	0	0
$755$	−15.9738	−0.581347
$756$	0	0
$757$	−43.5939	−1.58445	−0.792224	−	0.610230i	$-0.791077\pi$
$757$	−43.5939	−1.58445	−0.792224	+	0.610230i	$0.791077\pi$
$758$	0	0
$759$	6.58439	0.238998
$760$	0	0
$761$	30.8862	1.11962	0.559812	−	0.828620i	$-0.310873\pi$
$761$	30.8862	1.11962	0.559812	+	0.828620i	$0.310873\pi$
$762$	0	0
$763$	−15.4906	−0.560798
$764$	0	0
$765$	0.641758	0.0232028
$766$	0	0
$767$	−18.6922	−0.674936
$768$	0	0
$769$	32.1016	1.15761	0.578807	−	0.815465i	$-0.303519\pi$
$769$	32.1016	1.15761	0.578807	+	0.815465i	$0.303519\pi$
$770$	0	0
$771$	8.40084	0.302549
$772$	0	0
$773$	−43.8031	−1.57549	−0.787744	−	0.616003i	$-0.788751\pi$
$773$	−43.8031	−1.57549	−0.787744	+	0.616003i	$0.788751\pi$
$774$	0	0
$775$	21.1057	0.758137
$776$	0	0
$777$	−9.51121	−0.341213
$778$	0	0
$779$	70.8294	2.53773
$780$	0	0
$781$	38.0515	1.36159
$782$	0	0
$783$	2.33612	0.0834860
$784$	0	0
$785$	−25.4364	−0.907865
$786$	0	0
$787$	−34.6494	−1.23512	−0.617559	−	0.786525i	$-0.711878\pi$
$787$	−34.6494	−1.23512	−0.617559	+	0.786525i	$0.711878\pi$
$788$	0	0
$789$	14.0135	0.498895
$790$	0	0
$791$	−7.63302	−0.271399
$792$	0	0
$793$	19.6672	0.698403
$794$	0	0
$795$	0.755830	0.0268065
$796$	0	0
$797$	15.3473	0.543628	0.271814	−	0.962350i	$-0.412376\pi$
$797$	15.3473	0.543628	0.271814	+	0.962350i	$0.412376\pi$
$798$	0	0
$799$	−0.826921	−0.0292544
$800$	0	0
$801$	21.7293	0.767767
$802$	0	0
$803$	62.3662	2.20086
$804$	0	0
$805$	1.89742	0.0668752
$806$	0	0
$807$	−1.83998	−0.0647703
$808$	0	0
$809$	−40.1576	−1.41186	−0.705932	−	0.708279i	$-0.749472\pi$
$809$	−40.1576	−1.41186	−0.705932	+	0.708279i	$0.749472\pi$
$810$	0	0
$811$	−22.2240	−0.780390	−0.390195	−	0.920732i	$-0.627592\pi$
$811$	−22.2240	−0.780390	−0.390195	+	0.920732i	$0.627592\pi$
$812$	0	0
$813$	16.5594	0.580764
$814$	0	0
$815$	7.71862	0.270372
$816$	0	0
$817$	30.7246	1.07492
$818$	0	0
$819$	5.64750	0.197340
$820$	0	0
$821$	3.23544	0.112917	0.0564587	−	0.998405i	$-0.482019\pi$
$821$	3.23544	0.112917	0.0564587	+	0.998405i	$0.482019\pi$
$822$	0	0
$823$	−10.5792	−0.368767	−0.184384	−	0.982854i	$-0.559029\pi$
$823$	−10.5792	−0.368767	−0.184384	+	0.982854i	$0.559029\pi$
$824$	0	0
$825$	−14.8794	−0.518034
$826$	0	0
$827$	−23.7870	−0.827156	−0.413578	−	0.910469i	$-0.635721\pi$
$827$	−23.7870	−0.827156	−0.413578	+	0.910469i	$0.635721\pi$
$828$	0	0
$829$	−7.73348	−0.268595	−0.134297	−	0.990941i	$-0.542878\pi$
$829$	−7.73348	−0.268595	−0.134297	+	0.990941i	$0.542878\pi$
$830$	0	0
$831$	11.8284	0.410323
$832$	0	0
$833$	−0.264559	−0.00916642
$834$	0	0
$835$	−6.69731	−0.231770
$836$	0	0
$837$	−30.4086	−1.05107
$838$	0	0
$839$	−6.99735	−0.241575	−0.120788	−	0.992678i	$-0.538542\pi$
$839$	−6.99735	−0.241575	−0.120788	+	0.992678i	$0.538542\pi$
$840$	0	0
$841$	−28.7847	−0.992575
$842$	0	0
$843$	30.5616	1.05260
$844$	0	0
$845$	−5.93675	−0.204230
$846$	0	0
$847$	6.72221	0.230978
$848$	0	0
$849$	−16.8915	−0.579715
$850$	0	0
$851$	14.5384	0.498369
$852$	0	0
$853$	46.1566	1.58037	0.790186	−	0.612867i	$-0.209984\pi$
$853$	46.1566	1.58037	0.790186	+	0.612867i	$0.209984\pi$
$854$	0	0
$855$	−15.5733	−0.532596
$856$	0	0
$857$	−43.6532	−1.49116	−0.745582	−	0.666414i	$-0.767828\pi$
$857$	−43.6532	−1.49116	−0.745582	+	0.666414i	$0.767828\pi$
$858$	0	0
$859$	−9.38409	−0.320181	−0.160090	−	0.987102i	$-0.551179\pi$
$859$	−9.38409	−0.320181	−0.160090	+	0.987102i	$0.551179\pi$
$860$	0	0
$861$	−11.1602	−0.380337
$862$	0	0
$863$	8.11592	0.276269	0.138135	−	0.990413i	$-0.455889\pi$
$863$	8.11592	0.276269	0.138135	+	0.990413i	$0.455889\pi$
$864$	0	0
$865$	20.1699	0.685795
$866$	0	0
$867$	17.1256	0.581616
$868$	0	0
$869$	−53.0027	−1.79799
$870$	0	0
$871$	−13.5098	−0.457763
$872$	0	0
$873$	28.1593	0.953048
$874$	0	0
$875$	−10.4235	−0.352378
$876$	0	0
$877$	32.8275	1.10851	0.554253	−	0.832348i	$-0.313004\pi$
$877$	32.8275	1.10851	0.554253	+	0.832348i	$0.313004\pi$
$878$	0	0
$879$	−22.0088	−0.742340
$880$	0	0
$881$	12.2614	0.413096	0.206548	−	0.978436i	$-0.433777\pi$
$881$	12.2614	0.413096	0.206548	+	0.978436i	$0.433777\pi$
$882$	0	0
$883$	−5.82911	−0.196165	−0.0980826	−	0.995178i	$-0.531271\pi$
$883$	−5.82911	−0.196165	−0.0980826	+	0.995178i	$0.531271\pi$
$884$	0	0
$885$	−8.12160	−0.273005
$886$	0	0
$887$	−23.8826	−0.801899	−0.400949	−	0.916100i	$-0.631320\pi$
$887$	−23.8826	−0.801899	−0.400949	+	0.916100i	$0.631320\pi$
$888$	0	0
$889$	10.7393	0.360185
$890$	0	0
$891$	−3.52722	−0.118166
$892$	0	0
$893$	20.0666	0.671503
$894$	0	0
$895$	−22.4597	−0.750744
$896$	0	0
$897$	4.46847	0.149198
$898$	0	0
$899$	−2.80298	−0.0934846
$900$	0	0
$901$	0.161088	0.00536661
$902$	0	0
$903$	−4.84109	−0.161101
$904$	0	0
$905$	−2.41346	−0.0802261
$906$	0	0
$907$	47.8587	1.58912	0.794560	−	0.607185i	$-0.207701\pi$
$907$	47.8587	1.58912	0.794560	+	0.607185i	$0.207701\pi$
$908$	0	0
$909$	29.4370	0.976363
$910$	0	0
$911$	0.866439	0.0287064	0.0143532	−	0.999897i	$-0.495431\pi$
$911$	0.866439	0.0287064	0.0143532	+	0.999897i	$0.495431\pi$
$912$	0	0
$913$	−4.25841	−0.140933
$914$	0	0
$915$	8.54524	0.282497
$916$	0	0
$917$	−18.2248	−0.601836
$918$	0	0
$919$	32.5204	1.07275	0.536375	−	0.843980i	$-0.319793\pi$
$919$	32.5204	1.07275	0.536375	+	0.843980i	$0.319793\pi$
$920$	0	0
$921$	−11.6632	−0.384314
$922$	0	0
$923$	25.8235	0.849991
$924$	0	0
$925$	−32.8538	−1.08023
$926$	0	0
$927$	25.6707	0.843135
$928$	0	0
$929$	10.2017	0.334705	0.167353	−	0.985897i	$-0.446478\pi$
$929$	10.2017	0.334705	0.167353	+	0.985897i	$0.446478\pi$
$930$	0	0
$931$	6.41995	0.210405
$932$	0	0
$933$	−31.4666	−1.03017
$934$	0	0
$935$	1.36671	0.0446962
$936$	0	0
$937$	−5.06532	−0.165477	−0.0827384	−	0.996571i	$-0.526367\pi$
$937$	−5.06532	−0.165477	−0.0827384	+	0.996571i	$0.526367\pi$
$938$	0	0
$939$	−3.21659	−0.104970
$940$	0	0
$941$	−41.6093	−1.35643	−0.678213	−	0.734866i	$-0.737245\pi$
$941$	−41.6093	−1.35643	−0.678213	+	0.734866i	$0.737245\pi$
$942$	0	0
$943$	17.0589	0.555513
$944$	0	0
$945$	6.17775	0.200962
$946$	0	0
$947$	−26.8755	−0.873337	−0.436668	−	0.899623i	$-0.643842\pi$
$947$	−26.8755	−0.873337	−0.436668	+	0.899623i	$0.643842\pi$
$948$	0	0
$949$	42.3246	1.37391
$950$	0	0
$951$	2.24518	0.0728049
$952$	0	0
$953$	39.1113	1.26694	0.633470	−	0.773767i	$-0.281630\pi$
$953$	39.1113	1.26694	0.633470	+	0.773767i	$0.281630\pi$
$954$	0	0
$955$	4.25927	0.137827
$956$	0	0
$957$	1.97609	0.0638779
$958$	0	0
$959$	−7.34374	−0.237142
$960$	0	0
$961$	5.48559	0.176955
$962$	0	0
$963$	3.47018	0.111825
$964$	0	0
$965$	−9.30976	−0.299692
$966$	0	0
$967$	−53.9124	−1.73371	−0.866853	−	0.498565i	$-0.833861\pi$
$967$	−53.9124	−1.73371	−0.866853	+	0.498565i	$0.833861\pi$
$968$	0	0
$969$	1.71808	0.0551926
$970$	0	0
$971$	36.9258	1.18501	0.592503	−	0.805568i	$-0.298140\pi$
$971$	36.9258	1.18501	0.592503	+	0.805568i	$0.298140\pi$
$972$	0	0
$973$	11.8254	0.379106
$974$	0	0
$975$	−10.0978	−0.323390
$976$	0	0
$977$	57.0755	1.82601	0.913004	−	0.407950i	$-0.133756\pi$
$977$	57.0755	1.82601	0.913004	+	0.407950i	$0.133756\pi$
$978$	0	0
$979$	46.2754	1.47897
$980$	0	0
$981$	30.6212	0.977661
$982$	0	0
$983$	46.5572	1.48494	0.742471	−	0.669878i	$-0.233654\pi$
$983$	46.5572	1.48494	0.742471	+	0.669878i	$0.233654\pi$
$984$	0	0
$985$	13.5848	0.432847
$986$	0	0
$987$	−3.16177	−0.100640
$988$	0	0
$989$	7.39985	0.235302
$990$	0	0
$991$	−52.9443	−1.68183	−0.840915	−	0.541167i	$-0.817983\pi$
$991$	−52.9443	−1.68183	−0.840915	+	0.541167i	$0.817983\pi$
$992$	0	0
$993$	−2.80341	−0.0889635
$994$	0	0
$995$	24.1374	0.765206
$996$	0	0
$997$	22.6910	0.718630	0.359315	−	0.933216i	$-0.383010\pi$
$997$	22.6910	0.718630	0.359315	+	0.933216i	$0.383010\pi$
$998$	0	0
$999$	47.3350	1.49761

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7168.2.a.bd.1.4		8
4.3	odd	2		7168.2.a.bc.1.5		8
8.3	odd	2		7168.2.a.bc.1.4		8
8.5	even	2	inner	7168.2.a.bd.1.5		8
32.3	odd	8		896.2.m.e.673.3		8
32.5	even	8		448.2.m.c.113.3		8
32.11	odd	8		896.2.m.e.225.3		8
32.13	even	8		448.2.m.c.337.3		8
32.19	odd	8		112.2.m.c.29.4	✓	8
32.21	even	8		896.2.m.f.225.2		8
32.27	odd	8		112.2.m.c.85.4	yes	8
32.29	even	8		896.2.m.f.673.2		8
224.19	even	24		784.2.x.j.557.2		16
224.27	even	8		784.2.m.g.197.4		8
224.51	odd	24		784.2.x.k.557.2		16
224.59	even	24		784.2.x.j.373.2		16
224.83	even	8		784.2.m.g.589.4		8
224.115	even	24		784.2.x.j.765.2		16
224.123	odd	24		784.2.x.k.373.2		16
224.179	odd	24		784.2.x.k.765.2		16
224.187	even	24		784.2.x.j.165.2		16
224.219	odd	24		784.2.x.k.165.2		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
112.2.m.c.29.4	✓	8	32.19	odd	8
112.2.m.c.85.4	yes	8	32.27	odd	8
448.2.m.c.113.3		8	32.5	even	8
448.2.m.c.337.3		8	32.13	even	8
784.2.m.g.197.4		8	224.27	even	8
784.2.m.g.589.4		8	224.83	even	8
784.2.x.j.165.2		16	224.187	even	24
784.2.x.j.373.2		16	224.59	even	24
784.2.x.j.557.2		16	224.19	even	24
784.2.x.j.765.2		16	224.115	even	24
784.2.x.k.165.2		16	224.219	odd	24
784.2.x.k.373.2		16	224.123	odd	24
784.2.x.k.557.2		16	224.51	odd	24
784.2.x.k.765.2		16	224.179	odd	24
896.2.m.e.225.3		8	32.11	odd	8
896.2.m.e.673.3		8	32.3	odd	8
896.2.m.f.225.2		8	32.21	even	8
896.2.m.f.673.2		8	32.29	even	8
7168.2.a.bc.1.4		8	8.3	odd	2
7168.2.a.bc.1.5		8	4.3	odd	2
7168.2.a.bd.1.4		8	1.1	even	1	trivial
7168.2.a.bd.1.5		8	8.5	even	2	inner

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:	\( N \)	\(=\)	\( 7168 = 2^{10} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7168.a (trivial)

\(f(q)\)	\(=\)	\(q-1.01155 q^{3} +1.22714 q^{5} +1.00000 q^{7} -1.97676 q^{9} +O(q^{10})\)	\(q-1.01155 q^{3} +1.22714 q^{5} +1.00000 q^{7} -1.97676 q^{9} -4.20978 q^{11} -2.85695 q^{13} -1.24132 q^{15} -0.264559 q^{17} +6.41995 q^{19} -1.01155 q^{21} +1.54621 q^{23} -3.49412 q^{25} +5.03426 q^{27} +0.464044 q^{29} -6.04033 q^{31} +4.25841 q^{33} +1.22714 q^{35} +9.40259 q^{37} +2.88995 q^{39} +11.0327 q^{41} +4.78580 q^{43} -2.42577 q^{45} +3.12566 q^{47} +1.00000 q^{49} +0.267615 q^{51} -0.608892 q^{53} -5.16599 q^{55} -6.49412 q^{57} +6.54272 q^{59} -6.88400 q^{61} -1.97676 q^{63} -3.50588 q^{65} +4.72877 q^{67} -1.56407 q^{69} -9.03885 q^{71} -14.8146 q^{73} +3.53449 q^{75} -4.20978 q^{77} +12.5904 q^{79} +0.837864 q^{81} +1.01155 q^{83} -0.324651 q^{85} -0.469405 q^{87} -10.9924 q^{89} -2.85695 q^{91} +6.11011 q^{93} +7.87820 q^{95} -14.2452 q^{97} +8.32172 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{7} + 8 q^{9}+O(q^{10}) \) 8 * q + 8 * q^7 + 8 * q^9	\( 8 q + 8 q^{7} + 8 q^{9} - 8 q^{15} - 24 q^{17} - 8 q^{25} - 16 q^{31} - 24 q^{33} - 16 q^{39} - 8 q^{41} + 24 q^{47} + 8 q^{49} - 8 q^{55} - 32 q^{57} + 8 q^{63} - 48 q^{65} - 56 q^{71} - 48 q^{73} - 24 q^{79} - 40 q^{81} + 24 q^{87} - 24 q^{89} - 16 q^{95} - 32 q^{97}+O(q^{100}) \) 8 * q + 8 * q^7 + 8 * q^9 - 8 * q^15 - 24 * q^17 - 8 * q^25 - 16 * q^31 - 24 * q^33 - 16 * q^39 - 8 * q^41 + 24 * q^47 + 8 * q^49 - 8 * q^55 - 32 * q^57 + 8 * q^63 - 48 * q^65 - 56 * q^71 - 48 * q^73 - 24 * q^79 - 40 * q^81 + 24 * q^87 - 24 * q^89 - 16 * q^95 - 32 * q^97

Introduction

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