LMFDB - Embedded newform 8624.2.a.bx.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8624,2,Mod(1,8624)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8624, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("8624.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8624 = 2^{4} \cdot 7^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8624.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:	$68.8629867032$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 2156)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.41421$ of defining polynomial
Character	$\chi$	$=$	8624.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.82843 q^{3} +1.41421 q^{5} +5.00000 q^{9} +O(q^{10})$	$q-2.82843 q^{3} +1.41421 q^{5} +5.00000 q^{9} +1.00000 q^{11} -1.41421 q^{13} -4.00000 q^{15} +7.07107 q^{17} -2.82843 q^{19} -4.00000 q^{23} -3.00000 q^{25} -5.65685 q^{27} +5.65685 q^{31} -2.82843 q^{33} -8.00000 q^{37} +4.00000 q^{39} +9.89949 q^{41} -4.00000 q^{43} +7.07107 q^{45} -20.0000 q^{51} -6.00000 q^{53} +1.41421 q^{55} +8.00000 q^{57} -8.48528 q^{59} -1.41421 q^{61} -2.00000 q^{65} +8.00000 q^{67} +11.3137 q^{69} -8.00000 q^{71} -1.41421 q^{73} +8.48528 q^{75} -16.0000 q^{79} +1.00000 q^{81} +2.82843 q^{83} +10.0000 q^{85} -15.5563 q^{89} -16.0000 q^{93} -4.00000 q^{95} +9.89949 q^{97} +5.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 10 q^{9}+O(q^{10}) $ 2 * q + 10 * q^9	$ 2 q + 10 q^{9} + 2 q^{11} - 8 q^{15} - 8 q^{23} - 6 q^{25} - 16 q^{37} + 8 q^{39} - 8 q^{43} - 40 q^{51} - 12 q^{53} + 16 q^{57} - 4 q^{65} + 16 q^{67} - 16 q^{71} - 32 q^{79} + 2 q^{81} + 20 q^{85} - 32 q^{93} - 8 q^{95} + 10 q^{99}+O(q^{100}) $ 2 * q + 10 * q^9 + 2 * q^11 - 8 * q^15 - 8 * q^23 - 6 * q^25 - 16 * q^37 + 8 * q^39 - 8 * q^43 - 40 * q^51 - 12 * q^53 + 16 * q^57 - 4 * q^65 + 16 * q^67 - 16 * q^71 - 32 * q^79 + 2 * q^81 + 20 * q^85 - 32 * q^93 - 8 * q^95 + 10 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	−2.82843	−1.63299	−0.816497	−	0.577350i	$-0.804087\pi$
$3$	−2.82843	−1.63299	−0.816497	+	0.577350i	$0.804087\pi$
$4$	0	0
$5$	1.41421	0.632456	0.316228	−	0.948683i	$-0.397584\pi$
$5$	1.41421	0.632456	0.316228	+	0.948683i	$0.397584\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	5.00000	1.66667
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	−1.41421	−0.392232	−0.196116	−	0.980581i	$-0.562833\pi$
$13$	−1.41421	−0.392232	−0.196116	+	0.980581i	$0.562833\pi$
$14$	0	0
$15$	−4.00000	−1.03280
$16$	0	0
$17$	7.07107	1.71499	0.857493	−	0.514496i	$-0.172021\pi$
$17$	7.07107	1.71499	0.857493	+	0.514496i	$0.172021\pi$
$18$	0	0
$19$	−2.82843	−0.648886	−0.324443	−	0.945905i	$-0.605177\pi$
$19$	−2.82843	−0.648886	−0.324443	+	0.945905i	$0.605177\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−4.00000	−0.834058	−0.417029	−	0.908893i	$-0.636929\pi$
$23$	−4.00000	−0.834058	−0.417029	+	0.908893i	$0.636929\pi$
$24$	0	0
$25$	−3.00000	−0.600000
$26$	0	0
$27$	−5.65685	−1.08866
$28$	0	0
$29$	0	0		−	1.00000i	$-0.5\pi$
$29$	0	0			1.00000i	$0.5\pi$
$30$	0	0
$31$	5.65685	1.01600	0.508001	−	0.861357i	$-0.330385\pi$
$31$	5.65685	1.01600	0.508001	+	0.861357i	$0.330385\pi$
$32$	0	0
$33$	−2.82843	−0.492366
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−8.00000	−1.31519	−0.657596	−	0.753371i	$-0.728427\pi$
$37$	−8.00000	−1.31519	−0.657596	+	0.753371i	$0.728427\pi$
$38$	0	0
$39$	4.00000	0.640513
$40$	0	0
$41$	9.89949	1.54604	0.773021	−	0.634381i	$-0.218745\pi$
$41$	9.89949	1.54604	0.773021	+	0.634381i	$0.218745\pi$
$42$	0	0
$43$	−4.00000	−0.609994	−0.304997	−	0.952353i	$-0.598656\pi$
$43$	−4.00000	−0.609994	−0.304997	+	0.952353i	$0.598656\pi$
$44$	0	0
$45$	7.07107	1.05409
$46$	0	0
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	−20.0000	−2.80056
$52$	0	0
$53$	−6.00000	−0.824163	−0.412082	−	0.911147i	$-0.635198\pi$
$53$	−6.00000	−0.824163	−0.412082	+	0.911147i	$0.635198\pi$
$54$	0	0
$55$	1.41421	0.190693
$56$	0	0
$57$	8.00000	1.05963
$58$	0	0
$59$	−8.48528	−1.10469	−0.552345	−	0.833616i	$-0.686267\pi$
$59$	−8.48528	−1.10469	−0.552345	+	0.833616i	$0.686267\pi$
$60$	0	0
$61$	−1.41421	−0.181071	−0.0905357	−	0.995893i	$-0.528858\pi$
$61$	−1.41421	−0.181071	−0.0905357	+	0.995893i	$0.528858\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−2.00000	−0.248069
$66$	0	0
$67$	8.00000	0.977356	0.488678	−	0.872464i	$-0.337479\pi$
$67$	8.00000	0.977356	0.488678	+	0.872464i	$0.337479\pi$
$68$	0	0
$69$	11.3137	1.36201
$70$	0	0
$71$	−8.00000	−0.949425	−0.474713	−	0.880141i	$-0.657448\pi$
$71$	−8.00000	−0.949425	−0.474713	+	0.880141i	$0.657448\pi$
$72$	0	0
$73$	−1.41421	−0.165521	−0.0827606	−	0.996569i	$-0.526374\pi$
$73$	−1.41421	−0.165521	−0.0827606	+	0.996569i	$0.526374\pi$
$74$	0	0
$75$	8.48528	0.979796
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−16.0000	−1.80014	−0.900070	−	0.435745i	$-0.856485\pi$
$79$	−16.0000	−1.80014	−0.900070	+	0.435745i	$0.856485\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	2.82843	0.310460	0.155230	−	0.987878i	$-0.450388\pi$
$83$	2.82843	0.310460	0.155230	+	0.987878i	$0.450388\pi$
$84$	0	0
$85$	10.0000	1.08465
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−15.5563	−1.64897	−0.824485	−	0.565884i	$-0.808535\pi$
$89$	−15.5563	−1.64897	−0.824485	+	0.565884i	$0.808535\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−16.0000	−1.65912
$94$	0	0
$95$	−4.00000	−0.410391
$96$	0	0
$97$	9.89949	1.00514	0.502571	−	0.864536i	$-0.332388\pi$
$97$	9.89949	1.00514	0.502571	+	0.864536i	$0.332388\pi$
$98$	0	0
$99$	5.00000	0.502519
$100$	0	0
$101$	−18.3848	−1.82935	−0.914677	−	0.404186i	$-0.867555\pi$
$101$	−18.3848	−1.82935	−0.914677	+	0.404186i	$0.867555\pi$
$102$	0	0
$103$	16.9706	1.67216	0.836080	−	0.548608i	$-0.184842\pi$
$103$	16.9706	1.67216	0.836080	+	0.548608i	$0.184842\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	8.00000	0.773389	0.386695	−	0.922208i	$-0.373617\pi$
$107$	8.00000	0.773389	0.386695	+	0.922208i	$0.373617\pi$
$108$	0	0
$109$	16.0000	1.53252	0.766261	−	0.642529i	$-0.222115\pi$
$109$	16.0000	1.53252	0.766261	+	0.642529i	$0.222115\pi$
$110$	0	0
$111$	22.6274	2.14770
$112$	0	0
$113$	6.00000	0.564433	0.282216	−	0.959351i	$-0.408930\pi$
$113$	6.00000	0.564433	0.282216	+	0.959351i	$0.408930\pi$
$114$	0	0
$115$	−5.65685	−0.527504
$116$	0	0
$117$	−7.07107	−0.653720
$118$	0	0
$119$	0	0
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−28.0000	−2.52467
$124$	0	0
$125$	−11.3137	−1.01193
$126$	0	0
$127$	−12.0000	−1.06483	−0.532414	−	0.846484i	$-0.678715\pi$
$127$	−12.0000	−1.06483	−0.532414	+	0.846484i	$0.678715\pi$
$128$	0	0
$129$	11.3137	0.996116
$130$	0	0
$131$	−2.82843	−0.247121	−0.123560	−	0.992337i	$-0.539431\pi$
$131$	−2.82843	−0.247121	−0.123560	+	0.992337i	$0.539431\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−8.00000	−0.688530
$136$	0	0
$137$	16.0000	1.36697	0.683486	−	0.729964i	$-0.260463\pi$
$137$	16.0000	1.36697	0.683486	+	0.729964i	$0.260463\pi$
$138$	0	0
$139$	19.7990	1.67933	0.839664	−	0.543106i	$-0.182752\pi$
$139$	19.7990	1.67933	0.839664	+	0.543106i	$0.182752\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−1.41421	−0.118262
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−22.0000	−1.80231	−0.901155	−	0.433497i	$-0.857280\pi$
$149$	−22.0000	−1.80231	−0.901155	+	0.433497i	$0.857280\pi$
$150$	0	0
$151$	−4.00000	−0.325515	−0.162758	−	0.986666i	$-0.552039\pi$
$151$	−4.00000	−0.325515	−0.162758	+	0.986666i	$0.552039\pi$
$152$	0	0
$153$	35.3553	2.85831
$154$	0	0
$155$	8.00000	0.642575
$156$	0	0
$157$	−15.5563	−1.24153	−0.620766	−	0.783996i	$-0.713178\pi$
$157$	−15.5563	−1.24153	−0.620766	+	0.783996i	$0.713178\pi$
$158$	0	0
$159$	16.9706	1.34585
$160$	0	0
$161$	0	0
$162$	0	0
$163$	4.00000	0.313304	0.156652	−	0.987654i	$-0.449930\pi$
$163$	4.00000	0.313304	0.156652	+	0.987654i	$0.449930\pi$
$164$	0	0
$165$	−4.00000	−0.311400
$166$	0	0
$167$	−5.65685	−0.437741	−0.218870	−	0.975754i	$-0.570237\pi$
$167$	−5.65685	−0.437741	−0.218870	+	0.975754i	$0.570237\pi$
$168$	0	0
$169$	−11.0000	−0.846154
$170$	0	0
$171$	−14.1421	−1.08148
$172$	0	0
$173$	1.41421	0.107521	0.0537603	−	0.998554i	$-0.482879\pi$
$173$	1.41421	0.107521	0.0537603	+	0.998554i	$0.482879\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	24.0000	1.80395
$178$	0	0
$179$	16.0000	1.19590	0.597948	−	0.801535i	$-0.295983\pi$
$179$	16.0000	1.19590	0.597948	+	0.801535i	$0.295983\pi$
$180$	0	0
$181$	12.7279	0.946059	0.473029	−	0.881047i	$-0.343160\pi$
$181$	12.7279	0.946059	0.473029	+	0.881047i	$0.343160\pi$
$182$	0	0
$183$	4.00000	0.295689
$184$	0	0
$185$	−11.3137	−0.831800
$186$	0	0
$187$	7.07107	0.517088
$188$	0	0
$189$	0	0
$190$	0	0
$191$	0	0		−	1.00000i	$-0.5\pi$
$191$	0	0			1.00000i	$0.5\pi$
$192$	0	0
$193$	6.00000	0.431889	0.215945	−	0.976406i	$-0.430717\pi$
$193$	6.00000	0.431889	0.215945	+	0.976406i	$0.430717\pi$
$194$	0	0
$195$	5.65685	0.405096
$196$	0	0
$197$	6.00000	0.427482	0.213741	−	0.976890i	$-0.431435\pi$
$197$	6.00000	0.427482	0.213741	+	0.976890i	$0.431435\pi$
$198$	0	0
$199$	16.9706	1.20301	0.601506	−	0.798869i	$-0.294568\pi$
$199$	16.9706	1.20301	0.601506	+	0.798869i	$0.294568\pi$
$200$	0	0
$201$	−22.6274	−1.59601
$202$	0	0
$203$	0	0
$204$	0	0
$205$	14.0000	0.977802
$206$	0	0
$207$	−20.0000	−1.39010
$208$	0	0
$209$	−2.82843	−0.195646
$210$	0	0
$211$	8.00000	0.550743	0.275371	−	0.961338i	$-0.411199\pi$
$211$	8.00000	0.550743	0.275371	+	0.961338i	$0.411199\pi$
$212$	0	0
$213$	22.6274	1.55041
$214$	0	0
$215$	−5.65685	−0.385794
$216$	0	0
$217$	0	0
$218$	0	0
$219$	4.00000	0.270295
$220$	0	0
$221$	−10.0000	−0.672673
$222$	0	0
$223$	11.3137	0.757622	0.378811	−	0.925474i	$-0.376333\pi$
$223$	11.3137	0.757622	0.378811	+	0.925474i	$0.376333\pi$
$224$	0	0
$225$	−15.0000	−1.00000
$226$	0	0
$227$	2.82843	0.187729	0.0938647	−	0.995585i	$-0.470078\pi$
$227$	2.82843	0.187729	0.0938647	+	0.995585i	$0.470078\pi$
$228$	0	0
$229$	12.7279	0.841085	0.420542	−	0.907273i	$-0.361840\pi$
$229$	12.7279	0.841085	0.420542	+	0.907273i	$0.361840\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−24.0000	−1.57229	−0.786146	−	0.618041i	$-0.787927\pi$
$233$	−24.0000	−1.57229	−0.786146	+	0.618041i	$0.787927\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	45.2548	2.93962
$238$	0	0
$239$	−20.0000	−1.29369	−0.646846	−	0.762620i	$-0.723912\pi$
$239$	−20.0000	−1.29369	−0.646846	+	0.762620i	$0.723912\pi$
$240$	0	0
$241$	−26.8701	−1.73085	−0.865426	−	0.501036i	$-0.832952\pi$
$241$	−26.8701	−1.73085	−0.865426	+	0.501036i	$0.832952\pi$
$242$	0	0
$243$	14.1421	0.907218
$244$	0	0
$245$	0	0
$246$	0	0
$247$	4.00000	0.254514
$248$	0	0
$249$	−8.00000	−0.506979
$250$	0	0
$251$	−14.1421	−0.892644	−0.446322	−	0.894873i	$-0.647266\pi$
$251$	−14.1421	−0.892644	−0.446322	+	0.894873i	$0.647266\pi$
$252$	0	0
$253$	−4.00000	−0.251478
$254$	0	0
$255$	−28.2843	−1.77123
$256$	0	0
$257$	−1.41421	−0.0882162	−0.0441081	−	0.999027i	$-0.514045\pi$
$257$	−1.41421	−0.0882162	−0.0441081	+	0.999027i	$0.514045\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	0	0
$262$	0	0
$263$	24.0000	1.47990	0.739952	−	0.672660i	$-0.234848\pi$
$263$	24.0000	1.47990	0.739952	+	0.672660i	$0.234848\pi$
$264$	0	0
$265$	−8.48528	−0.521247
$266$	0	0
$267$	44.0000	2.69276
$268$	0	0
$269$	−7.07107	−0.431131	−0.215565	−	0.976489i	$-0.569159\pi$
$269$	−7.07107	−0.431131	−0.215565	+	0.976489i	$0.569159\pi$
$270$	0	0
$271$	−16.9706	−1.03089	−0.515444	−	0.856923i	$-0.672373\pi$
$271$	−16.9706	−1.03089	−0.515444	+	0.856923i	$0.672373\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−3.00000	−0.180907
$276$	0	0
$277$	−10.0000	−0.600842	−0.300421	−	0.953807i	$-0.597127\pi$
$277$	−10.0000	−0.600842	−0.300421	+	0.953807i	$0.597127\pi$
$278$	0	0
$279$	28.2843	1.69334
$280$	0	0
$281$	24.0000	1.43172	0.715860	−	0.698244i	$-0.246035\pi$
$281$	24.0000	1.43172	0.715860	+	0.698244i	$0.246035\pi$
$282$	0	0
$283$	−8.48528	−0.504398	−0.252199	−	0.967675i	$-0.581154\pi$
$283$	−8.48528	−0.504398	−0.252199	+	0.967675i	$0.581154\pi$
$284$	0	0
$285$	11.3137	0.670166
$286$	0	0
$287$	0	0
$288$	0	0
$289$	33.0000	1.94118
$290$	0	0
$291$	−28.0000	−1.64139
$292$	0	0
$293$	−29.6985	−1.73500	−0.867502	−	0.497434i	$-0.834276\pi$
$293$	−29.6985	−1.73500	−0.867502	+	0.497434i	$0.834276\pi$
$294$	0	0
$295$	−12.0000	−0.698667
$296$	0	0
$297$	−5.65685	−0.328244
$298$	0	0
$299$	5.65685	0.327144
$300$	0	0
$301$	0	0
$302$	0	0
$303$	52.0000	2.98732
$304$	0	0
$305$	−2.00000	−0.114520
$306$	0	0
$307$	−14.1421	−0.807134	−0.403567	−	0.914950i	$-0.632230\pi$
$307$	−14.1421	−0.807134	−0.403567	+	0.914950i	$0.632230\pi$
$308$	0	0
$309$	−48.0000	−2.73062
$310$	0	0
$311$	5.65685	0.320771	0.160385	−	0.987054i	$-0.448726\pi$
$311$	5.65685	0.320771	0.160385	+	0.987054i	$0.448726\pi$
$312$	0	0
$313$	26.8701	1.51879	0.759393	−	0.650633i	$-0.225496\pi$
$313$	26.8701	1.51879	0.759393	+	0.650633i	$0.225496\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−2.00000	−0.112331	−0.0561656	−	0.998421i	$-0.517887\pi$
$317$	−2.00000	−0.112331	−0.0561656	+	0.998421i	$0.517887\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	−22.6274	−1.26294
$322$	0	0
$323$	−20.0000	−1.11283
$324$	0	0
$325$	4.24264	0.235339
$326$	0	0
$327$	−45.2548	−2.50260
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−28.0000	−1.53902	−0.769510	−	0.638635i	$-0.779499\pi$
$331$	−28.0000	−1.53902	−0.769510	+	0.638635i	$0.779499\pi$
$332$	0	0
$333$	−40.0000	−2.19199
$334$	0	0
$335$	11.3137	0.618134
$336$	0	0
$337$	16.0000	0.871576	0.435788	−	0.900049i	$-0.356470\pi$
$337$	16.0000	0.871576	0.435788	+	0.900049i	$0.356470\pi$
$338$	0	0
$339$	−16.9706	−0.921714
$340$	0	0
$341$	5.65685	0.306336
$342$	0	0
$343$	0	0
$344$	0	0
$345$	16.0000	0.861411
$346$	0	0
$347$	−12.0000	−0.644194	−0.322097	−	0.946707i	$-0.604388\pi$
$347$	−12.0000	−0.644194	−0.322097	+	0.946707i	$0.604388\pi$
$348$	0	0
$349$	−9.89949	−0.529908	−0.264954	−	0.964261i	$-0.585357\pi$
$349$	−9.89949	−0.529908	−0.264954	+	0.964261i	$0.585357\pi$
$350$	0	0
$351$	8.00000	0.427008
$352$	0	0
$353$	9.89949	0.526897	0.263448	−	0.964673i	$-0.415140\pi$
$353$	9.89949	0.526897	0.263448	+	0.964673i	$0.415140\pi$
$354$	0	0
$355$	−11.3137	−0.600469
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−20.0000	−1.05556	−0.527780	−	0.849381i	$-0.676975\pi$
$359$	−20.0000	−1.05556	−0.527780	+	0.849381i	$0.676975\pi$
$360$	0	0
$361$	−11.0000	−0.578947
$362$	0	0
$363$	−2.82843	−0.148454
$364$	0	0
$365$	−2.00000	−0.104685
$366$	0	0
$367$	33.9411	1.77171	0.885856	−	0.463960i	$-0.153572\pi$
$367$	33.9411	1.77171	0.885856	+	0.463960i	$0.153572\pi$
$368$	0	0
$369$	49.4975	2.57674
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−38.0000	−1.96757	−0.983783	−	0.179364i	$-0.942596\pi$
$373$	−38.0000	−1.96757	−0.983783	+	0.179364i	$0.942596\pi$
$374$	0	0
$375$	32.0000	1.65247
$376$	0	0
$377$	0	0
$378$	0	0
$379$	−36.0000	−1.84920	−0.924598	−	0.380945i	$-0.875599\pi$
$379$	−36.0000	−1.84920	−0.924598	+	0.380945i	$0.875599\pi$
$380$	0	0
$381$	33.9411	1.73886
$382$	0	0
$383$	−22.6274	−1.15621	−0.578103	−	0.815963i	$-0.696207\pi$
$383$	−22.6274	−1.15621	−0.578103	+	0.815963i	$0.696207\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−20.0000	−1.01666
$388$	0	0
$389$	8.00000	0.405616	0.202808	−	0.979219i	$-0.434993\pi$
$389$	8.00000	0.405616	0.202808	+	0.979219i	$0.434993\pi$
$390$	0	0
$391$	−28.2843	−1.43040
$392$	0	0
$393$	8.00000	0.403547
$394$	0	0
$395$	−22.6274	−1.13851
$396$	0	0
$397$	−7.07107	−0.354887	−0.177443	−	0.984131i	$-0.556783\pi$
$397$	−7.07107	−0.354887	−0.177443	+	0.984131i	$0.556783\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−24.0000	−1.19850	−0.599251	−	0.800561i	$-0.704535\pi$
$401$	−24.0000	−1.19850	−0.599251	+	0.800561i	$0.704535\pi$
$402$	0	0
$403$	−8.00000	−0.398508
$404$	0	0
$405$	1.41421	0.0702728
$406$	0	0
$407$	−8.00000	−0.396545
$408$	0	0
$409$	−24.0416	−1.18878	−0.594391	−	0.804176i	$-0.702607\pi$
$409$	−24.0416	−1.18878	−0.594391	+	0.804176i	$0.702607\pi$
$410$	0	0
$411$	−45.2548	−2.23226
$412$	0	0
$413$	0	0
$414$	0	0
$415$	4.00000	0.196352
$416$	0	0
$417$	−56.0000	−2.74233
$418$	0	0
$419$	25.4558	1.24360	0.621800	−	0.783176i	$-0.286402\pi$
$419$	25.4558	1.24360	0.621800	+	0.783176i	$0.286402\pi$
$420$	0	0
$421$	−10.0000	−0.487370	−0.243685	−	0.969854i	$-0.578356\pi$
$421$	−10.0000	−0.487370	−0.243685	+	0.969854i	$0.578356\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−21.2132	−1.02899
$426$	0	0
$427$	0	0
$428$	0	0
$429$	4.00000	0.193122
$430$	0	0
$431$	−4.00000	−0.192673	−0.0963366	−	0.995349i	$-0.530713\pi$
$431$	−4.00000	−0.192673	−0.0963366	+	0.995349i	$0.530713\pi$
$432$	0	0
$433$	12.7279	0.611665	0.305832	−	0.952085i	$-0.401065\pi$
$433$	12.7279	0.611665	0.305832	+	0.952085i	$0.401065\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	11.3137	0.541208
$438$	0	0
$439$	16.9706	0.809961	0.404980	−	0.914325i	$-0.367278\pi$
$439$	16.9706	0.809961	0.404980	+	0.914325i	$0.367278\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	0	0		−	1.00000i	$-0.5\pi$
$443$	0	0			1.00000i	$0.5\pi$
$444$	0	0
$445$	−22.0000	−1.04290
$446$	0	0
$447$	62.2254	2.94316
$448$	0	0
$449$	−18.0000	−0.849473	−0.424736	−	0.905317i	$-0.639633\pi$
$449$	−18.0000	−0.849473	−0.424736	+	0.905317i	$0.639633\pi$
$450$	0	0
$451$	9.89949	0.466149
$452$	0	0
$453$	11.3137	0.531564
$454$	0	0
$455$	0	0
$456$	0	0
$457$	14.0000	0.654892	0.327446	−	0.944870i	$-0.393812\pi$
$457$	14.0000	0.654892	0.327446	+	0.944870i	$0.393812\pi$
$458$	0	0
$459$	−40.0000	−1.86704
$460$	0	0
$461$	24.0416	1.11973	0.559865	−	0.828584i	$-0.310853\pi$
$461$	24.0416	1.11973	0.559865	+	0.828584i	$0.310853\pi$
$462$	0	0
$463$	−16.0000	−0.743583	−0.371792	−	0.928316i	$-0.621256\pi$
$463$	−16.0000	−0.743583	−0.371792	+	0.928316i	$0.621256\pi$
$464$	0	0
$465$	−22.6274	−1.04932
$466$	0	0
$467$	−8.48528	−0.392652	−0.196326	−	0.980539i	$-0.562901\pi$
$467$	−8.48528	−0.392652	−0.196326	+	0.980539i	$0.562901\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	44.0000	2.02741
$472$	0	0
$473$	−4.00000	−0.183920
$474$	0	0
$475$	8.48528	0.389331
$476$	0	0
$477$	−30.0000	−1.37361
$478$	0	0
$479$	33.9411	1.55081	0.775405	−	0.631464i	$-0.217546\pi$
$479$	33.9411	1.55081	0.775405	+	0.631464i	$0.217546\pi$
$480$	0	0
$481$	11.3137	0.515861
$482$	0	0
$483$	0	0
$484$	0	0
$485$	14.0000	0.635707
$486$	0	0
$487$	20.0000	0.906287	0.453143	−	0.891438i	$-0.350303\pi$
$487$	20.0000	0.906287	0.453143	+	0.891438i	$0.350303\pi$
$488$	0	0
$489$	−11.3137	−0.511624
$490$	0	0
$491$	32.0000	1.44414	0.722070	−	0.691820i	$-0.243191\pi$
$491$	32.0000	1.44414	0.722070	+	0.691820i	$0.243191\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	7.07107	0.317821
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−24.0000	−1.07439	−0.537194	−	0.843459i	$-0.680516\pi$
$499$	−24.0000	−1.07439	−0.537194	+	0.843459i	$0.680516\pi$
$500$	0	0
$501$	16.0000	0.714827
$502$	0	0
$503$	5.65685	0.252227	0.126113	−	0.992016i	$-0.459750\pi$
$503$	5.65685	0.252227	0.126113	+	0.992016i	$0.459750\pi$
$504$	0	0
$505$	−26.0000	−1.15698
$506$	0	0
$507$	31.1127	1.38176
$508$	0	0
$509$	−29.6985	−1.31636	−0.658181	−	0.752860i	$-0.728674\pi$
$509$	−29.6985	−1.31636	−0.658181	+	0.752860i	$0.728674\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	16.0000	0.706417
$514$	0	0
$515$	24.0000	1.05757
$516$	0	0
$517$	0	0
$518$	0	0
$519$	−4.00000	−0.175581
$520$	0	0
$521$	−7.07107	−0.309789	−0.154895	−	0.987931i	$-0.549504\pi$
$521$	−7.07107	−0.309789	−0.154895	+	0.987931i	$0.549504\pi$
$522$	0	0
$523$	−42.4264	−1.85518	−0.927589	−	0.373603i	$-0.878122\pi$
$523$	−42.4264	−1.85518	−0.927589	+	0.373603i	$0.878122\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	40.0000	1.74243
$528$	0	0
$529$	−7.00000	−0.304348
$530$	0	0
$531$	−42.4264	−1.84115
$532$	0	0
$533$	−14.0000	−0.606407
$534$	0	0
$535$	11.3137	0.489134
$536$	0	0
$537$	−45.2548	−1.95289
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−30.0000	−1.28980	−0.644900	−	0.764267i	$-0.723101\pi$
$541$	−30.0000	−1.28980	−0.644900	+	0.764267i	$0.723101\pi$
$542$	0	0
$543$	−36.0000	−1.54491
$544$	0	0
$545$	22.6274	0.969252
$546$	0	0
$547$	20.0000	0.855138	0.427569	−	0.903983i	$-0.359370\pi$
$547$	20.0000	0.855138	0.427569	+	0.903983i	$0.359370\pi$
$548$	0	0
$549$	−7.07107	−0.301786
$550$	0	0
$551$	0	0
$552$	0	0
$553$	0	0
$554$	0	0
$555$	32.0000	1.35832
$556$	0	0
$557$	−18.0000	−0.762684	−0.381342	−	0.924434i	$-0.624538\pi$
$557$	−18.0000	−0.762684	−0.381342	+	0.924434i	$0.624538\pi$
$558$	0	0
$559$	5.65685	0.239259
$560$	0	0
$561$	−20.0000	−0.844401
$562$	0	0
$563$	−36.7696	−1.54965	−0.774826	−	0.632175i	$-0.782163\pi$
$563$	−36.7696	−1.54965	−0.774826	+	0.632175i	$0.782163\pi$
$564$	0	0
$565$	8.48528	0.356978
$566$	0	0
$567$	0	0
$568$	0	0
$569$	0	0		−	1.00000i	$-0.5\pi$
$569$	0	0			1.00000i	$0.5\pi$
$570$	0	0
$571$	−44.0000	−1.84134	−0.920671	−	0.390339i	$-0.872358\pi$
$571$	−44.0000	−1.84134	−0.920671	+	0.390339i	$0.872358\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	12.0000	0.500435
$576$	0	0
$577$	−21.2132	−0.883117	−0.441559	−	0.897232i	$-0.645574\pi$
$577$	−21.2132	−0.883117	−0.441559	+	0.897232i	$0.645574\pi$
$578$	0	0
$579$	−16.9706	−0.705273
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−6.00000	−0.248495
$584$	0	0
$585$	−10.0000	−0.413449
$586$	0	0
$587$	14.1421	0.583708	0.291854	−	0.956463i	$-0.405728\pi$
$587$	14.1421	0.583708	0.291854	+	0.956463i	$0.405728\pi$
$588$	0	0
$589$	−16.0000	−0.659269
$590$	0	0
$591$	−16.9706	−0.698076
$592$	0	0
$593$	7.07107	0.290374	0.145187	−	0.989404i	$-0.453622\pi$
$593$	7.07107	0.290374	0.145187	+	0.989404i	$0.453622\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−48.0000	−1.96451
$598$	0	0
$599$	0	0		−	1.00000i	$-0.5\pi$
$599$	0	0			1.00000i	$0.5\pi$
$600$	0	0
$601$	−35.3553	−1.44217	−0.721087	−	0.692844i	$-0.756357\pi$
$601$	−35.3553	−1.44217	−0.721087	+	0.692844i	$0.756357\pi$
$602$	0	0
$603$	40.0000	1.62893
$604$	0	0
$605$	1.41421	0.0574960
$606$	0	0
$607$	0	0		−	1.00000i	$-0.5\pi$
$607$	0	0			1.00000i	$0.5\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	0	0
$612$	0	0
$613$	32.0000	1.29247	0.646234	−	0.763139i	$-0.276343\pi$
$613$	32.0000	1.29247	0.646234	+	0.763139i	$0.276343\pi$
$614$	0	0
$615$	−39.5980	−1.59674
$616$	0	0
$617$	−24.0000	−0.966204	−0.483102	−	0.875564i	$-0.660490\pi$
$617$	−24.0000	−0.966204	−0.483102	+	0.875564i	$0.660490\pi$
$618$	0	0
$619$	−8.48528	−0.341052	−0.170526	−	0.985353i	$-0.554547\pi$
$619$	−8.48528	−0.341052	−0.170526	+	0.985353i	$0.554547\pi$
$620$	0	0
$621$	22.6274	0.908007
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−1.00000	−0.0400000
$626$	0	0
$627$	8.00000	0.319489
$628$	0	0
$629$	−56.5685	−2.25554
$630$	0	0
$631$	−48.0000	−1.91085	−0.955425	−	0.295234i	$-0.904602\pi$
$631$	−48.0000	−1.91085	−0.955425	+	0.295234i	$0.904602\pi$
$632$	0	0
$633$	−22.6274	−0.899359
$634$	0	0
$635$	−16.9706	−0.673456
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−40.0000	−1.58238
$640$	0	0
$641$	−24.0000	−0.947943	−0.473972	−	0.880540i	$-0.657180\pi$
$641$	−24.0000	−0.947943	−0.473972	+	0.880540i	$0.657180\pi$
$642$	0	0
$643$	−42.4264	−1.67313	−0.836567	−	0.547865i	$-0.815441\pi$
$643$	−42.4264	−1.67313	−0.836567	+	0.547865i	$0.815441\pi$
$644$	0	0
$645$	16.0000	0.629999
$646$	0	0
$647$	−28.2843	−1.11197	−0.555985	−	0.831193i	$-0.687659\pi$
$647$	−28.2843	−1.11197	−0.555985	+	0.831193i	$0.687659\pi$
$648$	0	0
$649$	−8.48528	−0.333076
$650$	0	0
$651$	0	0
$652$	0	0
$653$	24.0000	0.939193	0.469596	−	0.882881i	$-0.344399\pi$
$653$	24.0000	0.939193	0.469596	+	0.882881i	$0.344399\pi$
$654$	0	0
$655$	−4.00000	−0.156293
$656$	0	0
$657$	−7.07107	−0.275869
$658$	0	0
$659$	28.0000	1.09073	0.545363	−	0.838200i	$-0.316392\pi$
$659$	28.0000	1.09073	0.545363	+	0.838200i	$0.316392\pi$
$660$	0	0
$661$	−21.2132	−0.825098	−0.412549	−	0.910935i	$-0.635361\pi$
$661$	−21.2132	−0.825098	−0.412549	+	0.910935i	$0.635361\pi$
$662$	0	0
$663$	28.2843	1.09847
$664$	0	0
$665$	0	0
$666$	0	0
$667$	0	0
$668$	0	0
$669$	−32.0000	−1.23719
$670$	0	0
$671$	−1.41421	−0.0545951
$672$	0	0
$673$	8.00000	0.308377	0.154189	−	0.988041i	$-0.450724\pi$
$673$	8.00000	0.308377	0.154189	+	0.988041i	$0.450724\pi$
$674$	0	0
$675$	16.9706	0.653197
$676$	0	0
$677$	−49.4975	−1.90234	−0.951171	−	0.308664i	$-0.900118\pi$
$677$	−49.4975	−1.90234	−0.951171	+	0.308664i	$0.900118\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−8.00000	−0.306561
$682$	0	0
$683$	−48.0000	−1.83667	−0.918334	−	0.395805i	$-0.870466\pi$
$683$	−48.0000	−1.83667	−0.918334	+	0.395805i	$0.870466\pi$
$684$	0	0
$685$	22.6274	0.864549
$686$	0	0
$687$	−36.0000	−1.37349
$688$	0	0
$689$	8.48528	0.323263
$690$	0	0
$691$	2.82843	0.107598	0.0537992	−	0.998552i	$-0.482867\pi$
$691$	2.82843	0.107598	0.0537992	+	0.998552i	$0.482867\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	28.0000	1.06210
$696$	0	0
$697$	70.0000	2.65144
$698$	0	0
$699$	67.8823	2.56754
$700$	0	0
$701$	32.0000	1.20862	0.604312	−	0.796748i	$-0.293448\pi$
$701$	32.0000	1.20862	0.604312	+	0.796748i	$0.293448\pi$
$702$	0	0
$703$	22.6274	0.853409
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	40.0000	1.50223	0.751116	−	0.660171i	$-0.229516\pi$
$709$	40.0000	1.50223	0.751116	+	0.660171i	$0.229516\pi$
$710$	0	0
$711$	−80.0000	−3.00023
$712$	0	0
$713$	−22.6274	−0.847403
$714$	0	0
$715$	−2.00000	−0.0747958
$716$	0	0
$717$	56.5685	2.11259
$718$	0	0
$719$	11.3137	0.421930	0.210965	−	0.977494i	$-0.432339\pi$
$719$	11.3137	0.421930	0.210965	+	0.977494i	$0.432339\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	76.0000	2.82647
$724$	0	0
$725$	0	0
$726$	0	0
$727$	11.3137	0.419602	0.209801	−	0.977744i	$-0.432718\pi$
$727$	11.3137	0.419602	0.209801	+	0.977744i	$0.432718\pi$
$728$	0	0
$729$	−43.0000	−1.59259
$730$	0	0
$731$	−28.2843	−1.04613
$732$	0	0
$733$	−21.2132	−0.783528	−0.391764	−	0.920066i	$-0.628135\pi$
$733$	−21.2132	−0.783528	−0.391764	+	0.920066i	$0.628135\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	8.00000	0.294684
$738$	0	0
$739$	−36.0000	−1.32428	−0.662141	−	0.749380i	$-0.730352\pi$
$739$	−36.0000	−1.32428	−0.662141	+	0.749380i	$0.730352\pi$
$740$	0	0
$741$	−11.3137	−0.415619
$742$	0	0
$743$	−12.0000	−0.440237	−0.220119	−	0.975473i	$-0.570644\pi$
$743$	−12.0000	−0.440237	−0.220119	+	0.975473i	$0.570644\pi$
$744$	0	0
$745$	−31.1127	−1.13988
$746$	0	0
$747$	14.1421	0.517434
$748$	0	0
$749$	0	0
$750$	0	0
$751$	28.0000	1.02173	0.510867	−	0.859660i	$-0.329324\pi$
$751$	28.0000	1.02173	0.510867	+	0.859660i	$0.329324\pi$
$752$	0	0
$753$	40.0000	1.45768
$754$	0	0
$755$	−5.65685	−0.205874
$756$	0	0
$757$	−8.00000	−0.290765	−0.145382	−	0.989376i	$-0.546441\pi$
$757$	−8.00000	−0.290765	−0.145382	+	0.989376i	$0.546441\pi$
$758$	0	0
$759$	11.3137	0.410662
$760$	0	0
$761$	12.7279	0.461387	0.230693	−	0.973026i	$-0.425901\pi$
$761$	12.7279	0.461387	0.230693	+	0.973026i	$0.425901\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	50.0000	1.80775
$766$	0	0
$767$	12.0000	0.433295
$768$	0	0
$769$	−4.24264	−0.152994	−0.0764968	−	0.997070i	$-0.524373\pi$
$769$	−4.24264	−0.152994	−0.0764968	+	0.997070i	$0.524373\pi$
$770$	0	0
$771$	4.00000	0.144056
$772$	0	0
$773$	−9.89949	−0.356060	−0.178030	−	0.984025i	$-0.556972\pi$
$773$	−9.89949	−0.356060	−0.178030	+	0.984025i	$0.556972\pi$
$774$	0	0
$775$	−16.9706	−0.609601
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−28.0000	−1.00320
$780$	0	0
$781$	−8.00000	−0.286263
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−22.0000	−0.785214
$786$	0	0
$787$	8.48528	0.302468	0.151234	−	0.988498i	$-0.451675\pi$
$787$	8.48528	0.302468	0.151234	+	0.988498i	$0.451675\pi$
$788$	0	0
$789$	−67.8823	−2.41667
$790$	0	0
$791$	0	0
$792$	0	0
$793$	2.00000	0.0710221
$794$	0	0
$795$	24.0000	0.851192
$796$	0	0
$797$	15.5563	0.551034	0.275517	−	0.961296i	$-0.411151\pi$
$797$	15.5563	0.551034	0.275517	+	0.961296i	$0.411151\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	−77.7817	−2.74828
$802$	0	0
$803$	−1.41421	−0.0499065
$804$	0	0
$805$	0	0
$806$	0	0
$807$	20.0000	0.704033
$808$	0	0
$809$	18.0000	0.632846	0.316423	−	0.948618i	$-0.397518\pi$
$809$	18.0000	0.632846	0.316423	+	0.948618i	$0.397518\pi$
$810$	0	0
$811$	−8.48528	−0.297959	−0.148979	−	0.988840i	$-0.547599\pi$
$811$	−8.48528	−0.297959	−0.148979	+	0.988840i	$0.547599\pi$
$812$	0	0
$813$	48.0000	1.68343
$814$	0	0
$815$	5.65685	0.198151
$816$	0	0
$817$	11.3137	0.395817
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−26.0000	−0.907406	−0.453703	−	0.891153i	$-0.649897\pi$
$821$	−26.0000	−0.907406	−0.453703	+	0.891153i	$0.649897\pi$
$822$	0	0
$823$	8.00000	0.278862	0.139431	−	0.990232i	$-0.455473\pi$
$823$	8.00000	0.278862	0.139431	+	0.990232i	$0.455473\pi$
$824$	0	0
$825$	8.48528	0.295420
$826$	0	0
$827$	8.00000	0.278187	0.139094	−	0.990279i	$-0.455581\pi$
$827$	8.00000	0.278187	0.139094	+	0.990279i	$0.455581\pi$
$828$	0	0
$829$	15.5563	0.540294	0.270147	−	0.962819i	$-0.412928\pi$
$829$	15.5563	0.540294	0.270147	+	0.962819i	$0.412928\pi$
$830$	0	0
$831$	28.2843	0.981170
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−8.00000	−0.276851
$836$	0	0
$837$	−32.0000	−1.10608
$838$	0	0
$839$	−28.2843	−0.976481	−0.488241	−	0.872709i	$-0.662361\pi$
$839$	−28.2843	−0.976481	−0.488241	+	0.872709i	$0.662361\pi$
$840$	0	0
$841$	−29.0000	−1.00000
$842$	0	0
$843$	−67.8823	−2.33799
$844$	0	0
$845$	−15.5563	−0.535155
$846$	0	0
$847$	0	0
$848$	0	0
$849$	24.0000	0.823678
$850$	0	0
$851$	32.0000	1.09695
$852$	0	0
$853$	18.3848	0.629483	0.314741	−	0.949177i	$-0.398082\pi$
$853$	18.3848	0.629483	0.314741	+	0.949177i	$0.398082\pi$
$854$	0	0
$855$	−20.0000	−0.683986
$856$	0	0
$857$	−21.2132	−0.724629	−0.362315	−	0.932056i	$-0.618013\pi$
$857$	−21.2132	−0.724629	−0.362315	+	0.932056i	$0.618013\pi$
$858$	0	0
$859$	−8.48528	−0.289514	−0.144757	−	0.989467i	$-0.546240\pi$
$859$	−8.48528	−0.289514	−0.144757	+	0.989467i	$0.546240\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	8.00000	0.272323	0.136162	−	0.990687i	$-0.456523\pi$
$863$	8.00000	0.272323	0.136162	+	0.990687i	$0.456523\pi$
$864$	0	0
$865$	2.00000	0.0680020
$866$	0	0
$867$	−93.3381	−3.16993
$868$	0	0
$869$	−16.0000	−0.542763
$870$	0	0
$871$	−11.3137	−0.383350
$872$	0	0
$873$	49.4975	1.67524
$874$	0	0
$875$	0	0
$876$	0	0
$877$	32.0000	1.08056	0.540282	−	0.841484i	$-0.318318\pi$
$877$	32.0000	1.08056	0.540282	+	0.841484i	$0.318318\pi$
$878$	0	0
$879$	84.0000	2.83325
$880$	0	0
$881$	12.7279	0.428815	0.214407	−	0.976744i	$-0.431218\pi$
$881$	12.7279	0.428815	0.214407	+	0.976744i	$0.431218\pi$
$882$	0	0
$883$	−24.0000	−0.807664	−0.403832	−	0.914833i	$-0.632322\pi$
$883$	−24.0000	−0.807664	−0.403832	+	0.914833i	$0.632322\pi$
$884$	0	0
$885$	33.9411	1.14092
$886$	0	0
$887$	−5.65685	−0.189939	−0.0949693	−	0.995480i	$-0.530275\pi$
$887$	−5.65685	−0.189939	−0.0949693	+	0.995480i	$0.530275\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	1.00000	0.0335013
$892$	0	0
$893$	0	0
$894$	0	0
$895$	22.6274	0.756351
$896$	0	0
$897$	−16.0000	−0.534224
$898$	0	0
$899$	0	0
$900$	0	0
$901$	−42.4264	−1.41343
$902$	0	0
$903$	0	0
$904$	0	0
$905$	18.0000	0.598340
$906$	0	0
$907$	32.0000	1.06254	0.531271	−	0.847202i	$-0.321714\pi$
$907$	32.0000	1.06254	0.531271	+	0.847202i	$0.321714\pi$
$908$	0	0
$909$	−91.9239	−3.04892
$910$	0	0
$911$	−12.0000	−0.397578	−0.198789	−	0.980042i	$-0.563701\pi$
$911$	−12.0000	−0.397578	−0.198789	+	0.980042i	$0.563701\pi$
$912$	0	0
$913$	2.82843	0.0936073
$914$	0	0
$915$	5.65685	0.187010
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−8.00000	−0.263896	−0.131948	−	0.991257i	$-0.542123\pi$
$919$	−8.00000	−0.263896	−0.131948	+	0.991257i	$0.542123\pi$
$920$	0	0
$921$	40.0000	1.31804
$922$	0	0
$923$	11.3137	0.372395
$924$	0	0
$925$	24.0000	0.789115
$926$	0	0
$927$	84.8528	2.78693
$928$	0	0
$929$	1.41421	0.0463988	0.0231994	−	0.999731i	$-0.492615\pi$
$929$	1.41421	0.0463988	0.0231994	+	0.999731i	$0.492615\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−16.0000	−0.523816
$934$	0	0
$935$	10.0000	0.327035
$936$	0	0
$937$	21.2132	0.693005	0.346503	−	0.938049i	$-0.387369\pi$
$937$	21.2132	0.693005	0.346503	+	0.938049i	$0.387369\pi$
$938$	0	0
$939$	−76.0000	−2.48017
$940$	0	0
$941$	43.8406	1.42916	0.714582	−	0.699552i	$-0.246617\pi$
$941$	43.8406	1.42916	0.714582	+	0.699552i	$0.246617\pi$
$942$	0	0
$943$	−39.5980	−1.28949
$944$	0	0
$945$	0	0
$946$	0	0
$947$	36.0000	1.16984	0.584921	−	0.811090i	$-0.301125\pi$
$947$	36.0000	1.16984	0.584921	+	0.811090i	$0.301125\pi$
$948$	0	0
$949$	2.00000	0.0649227
$950$	0	0
$951$	5.65685	0.183436
$952$	0	0
$953$	42.0000	1.36051	0.680257	−	0.732974i	$-0.261868\pi$
$953$	42.0000	1.36051	0.680257	+	0.732974i	$0.261868\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	1.00000	0.0322581
$962$	0	0
$963$	40.0000	1.28898
$964$	0	0
$965$	8.48528	0.273151
$966$	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
$969$	56.5685	1.81724
$970$	0	0
$971$	2.82843	0.0907685	0.0453843	−	0.998970i	$-0.485549\pi$
$971$	2.82843	0.0907685	0.0453843	+	0.998970i	$0.485549\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	−12.0000	−0.384308
$976$	0	0
$977$	−32.0000	−1.02377	−0.511885	−	0.859054i	$-0.671053\pi$
$977$	−32.0000	−1.02377	−0.511885	+	0.859054i	$0.671053\pi$
$978$	0	0
$979$	−15.5563	−0.497183
$980$	0	0
$981$	80.0000	2.55420
$982$	0	0
$983$	−28.2843	−0.902128	−0.451064	−	0.892492i	$-0.648955\pi$
$983$	−28.2843	−0.902128	−0.451064	+	0.892492i	$0.648955\pi$
$984$	0	0
$985$	8.48528	0.270364
$986$	0	0
$987$	0	0
$988$	0	0
$989$	16.0000	0.508770
$990$	0	0
$991$	16.0000	0.508257	0.254128	−	0.967170i	$-0.418211\pi$
$991$	16.0000	0.508257	0.254128	+	0.967170i	$0.418211\pi$
$992$	0	0
$993$	79.1960	2.51321
$994$	0	0
$995$	24.0000	0.760851
$996$	0	0
$997$	−38.1838	−1.20929	−0.604646	−	0.796494i	$-0.706685\pi$
$997$	−38.1838	−1.20929	−0.604646	+	0.796494i	$0.706685\pi$
$998$	0	0
$999$	45.2548	1.43180

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8624.2.a.bx.1.1		2
4.3	odd	2		2156.2.a.f.1.2	yes	2
7.6	odd	2	inner	8624.2.a.bx.1.2		2
28.3	even	6		2156.2.i.f.177.2		4
28.11	odd	6		2156.2.i.f.177.1		4
28.19	even	6		2156.2.i.f.1145.2		4
28.23	odd	6		2156.2.i.f.1145.1		4
28.27	even	2		2156.2.a.f.1.1	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2156.2.a.f.1.1	✓	2	28.27	even	2
2156.2.a.f.1.2	yes	2	4.3	odd	2
2156.2.i.f.177.1		4	28.11	odd	6
2156.2.i.f.177.2		4	28.3	even	6
2156.2.i.f.1145.1		4	28.23	odd	6
2156.2.i.f.1145.2		4	28.19	even	6
8624.2.a.bx.1.1		2	1.1	even	1	trivial
8624.2.a.bx.1.2		2	7.6	odd	2	inner

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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:	\( N \)	\(=\)	\( 8624 = 2^{4} \cdot 7^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8624.a (trivial)

\(f(q)\)	\(=\)	\(q-2.82843 q^{3} +1.41421 q^{5} +5.00000 q^{9} +O(q^{10})\)	\(q-2.82843 q^{3} +1.41421 q^{5} +5.00000 q^{9} +1.00000 q^{11} -1.41421 q^{13} -4.00000 q^{15} +7.07107 q^{17} -2.82843 q^{19} -4.00000 q^{23} -3.00000 q^{25} -5.65685 q^{27} +5.65685 q^{31} -2.82843 q^{33} -8.00000 q^{37} +4.00000 q^{39} +9.89949 q^{41} -4.00000 q^{43} +7.07107 q^{45} -20.0000 q^{51} -6.00000 q^{53} +1.41421 q^{55} +8.00000 q^{57} -8.48528 q^{59} -1.41421 q^{61} -2.00000 q^{65} +8.00000 q^{67} +11.3137 q^{69} -8.00000 q^{71} -1.41421 q^{73} +8.48528 q^{75} -16.0000 q^{79} +1.00000 q^{81} +2.82843 q^{83} +10.0000 q^{85} -15.5563 q^{89} -16.0000 q^{93} -4.00000 q^{95} +9.89949 q^{97} +5.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 10 q^{9}+O(q^{10}) \) 2 * q + 10 * q^9	\( 2 q + 10 q^{9} + 2 q^{11} - 8 q^{15} - 8 q^{23} - 6 q^{25} - 16 q^{37} + 8 q^{39} - 8 q^{43} - 40 q^{51} - 12 q^{53} + 16 q^{57} - 4 q^{65} + 16 q^{67} - 16 q^{71} - 32 q^{79} + 2 q^{81} + 20 q^{85} - 32 q^{93} - 8 q^{95} + 10 q^{99}+O(q^{100}) \) 2 * q + 10 * q^9 + 2 * q^11 - 8 * q^15 - 8 * q^23 - 6 * q^25 - 16 * q^37 + 8 * q^39 - 8 * q^43 - 40 * q^51 - 12 * q^53 + 16 * q^57 - 4 * q^65 + 16 * q^67 - 16 * q^71 - 32 * q^79 + 2 * q^81 + 20 * q^85 - 32 * q^93 - 8 * q^95 + 10 * q^99

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