LMFDB - Embedded newform 784.2.a.k.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [784,2,Mod(1,784)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(784, 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("784.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 784 = 2^{4} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	784.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:	$6.26027151847$
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:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 392)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.41421$ of defining polynomial
Character	$\chi$	$=$	784.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.41421 q^{3} +2.82843 q^{5} -1.00000 q^{9} +O(q^{10})$	$q-1.41421 q^{3} +2.82843 q^{5} -1.00000 q^{9} -6.00000 q^{11} -5.65685 q^{13} -4.00000 q^{15} +1.41421 q^{17} +4.24264 q^{19} -4.00000 q^{23} +3.00000 q^{25} +5.65685 q^{27} -6.00000 q^{29} -2.82843 q^{31} +8.48528 q^{33} +2.00000 q^{37} +8.00000 q^{39} -1.41421 q^{41} -10.0000 q^{43} -2.82843 q^{45} +2.82843 q^{47} -2.00000 q^{51} -2.00000 q^{53} -16.9706 q^{55} -6.00000 q^{57} -1.41421 q^{59} -8.48528 q^{61} -16.0000 q^{65} -4.00000 q^{67} +5.65685 q^{69} +12.0000 q^{71} -9.89949 q^{73} -4.24264 q^{75} +4.00000 q^{79} -5.00000 q^{81} -1.41421 q^{83} +4.00000 q^{85} +8.48528 q^{87} -4.24264 q^{89} +4.00000 q^{93} +12.0000 q^{95} +12.7279 q^{97} +6.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^9	$ 2 q - 2 q^{9} - 12 q^{11} - 8 q^{15} - 8 q^{23} + 6 q^{25} - 12 q^{29} + 4 q^{37} + 16 q^{39} - 20 q^{43} - 4 q^{51} - 4 q^{53} - 12 q^{57} - 32 q^{65} - 8 q^{67} + 24 q^{71} + 8 q^{79} - 10 q^{81} + 8 q^{85} + 8 q^{93} + 24 q^{95} + 12 q^{99}+O(q^{100}) $ 2 * q - 2 * q^9 - 12 * q^11 - 8 * q^15 - 8 * q^23 + 6 * q^25 - 12 * q^29 + 4 * q^37 + 16 * q^39 - 20 * q^43 - 4 * q^51 - 4 * q^53 - 12 * q^57 - 32 * q^65 - 8 * q^67 + 24 * q^71 + 8 * q^79 - 10 * q^81 + 8 * q^85 + 8 * q^93 + 24 * q^95 + 12 * 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
$3$	−1.41421	−0.816497	−0.408248	−	0.912871i	$-0.633860\pi$
$3$	−1.41421	−0.816497	−0.408248	+	0.912871i	$0.633860\pi$
$4$	0	0
$5$	2.82843	1.26491	0.632456	−	0.774597i	$-0.282047\pi$
$5$	2.82843	1.26491	0.632456	+	0.774597i	$0.282047\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	−1.00000	−0.333333
$10$	0	0
$11$	−6.00000	−1.80907	−0.904534	−	0.426401i	$-0.859781\pi$
$11$	−6.00000	−1.80907	−0.904534	+	0.426401i	$0.859781\pi$
$12$	0	0
$13$	−5.65685	−1.56893	−0.784465	−	0.620174i	$-0.787062\pi$
$13$	−5.65685	−1.56893	−0.784465	+	0.620174i	$0.787062\pi$
$14$	0	0
$15$	−4.00000	−1.03280
$16$	0	0
$17$	1.41421	0.342997	0.171499	−	0.985184i	$-0.445139\pi$
$17$	1.41421	0.342997	0.171499	+	0.985184i	$0.445139\pi$
$18$	0	0
$19$	4.24264	0.973329	0.486664	−	0.873589i	$-0.338214\pi$
$19$	4.24264	0.973329	0.486664	+	0.873589i	$0.338214\pi$
$20$	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
$29$	−6.00000	−1.11417	−0.557086	−	0.830455i	$-0.688081\pi$
$29$	−6.00000	−1.11417	−0.557086	+	0.830455i	$0.688081\pi$
$30$	0	0
$31$	−2.82843	−0.508001	−0.254000	−	0.967204i	$-0.581746\pi$
$31$	−2.82843	−0.508001	−0.254000	+	0.967204i	$0.581746\pi$
$32$	0	0
$33$	8.48528	1.47710
$34$	0	0
$35$	0	0
$36$	0	0
$37$	2.00000	0.328798	0.164399	−	0.986394i	$-0.447432\pi$
$37$	2.00000	0.328798	0.164399	+	0.986394i	$0.447432\pi$
$38$	0	0
$39$	8.00000	1.28103
$40$	0	0
$41$	−1.41421	−0.220863	−0.110432	−	0.993884i	$-0.535223\pi$
$41$	−1.41421	−0.220863	−0.110432	+	0.993884i	$0.535223\pi$
$42$	0	0
$43$	−10.0000	−1.52499	−0.762493	−	0.646997i	$-0.776025\pi$
$43$	−10.0000	−1.52499	−0.762493	+	0.646997i	$0.776025\pi$
$44$	0	0
$45$	−2.82843	−0.421637
$46$	0	0
$47$	2.82843	0.412568	0.206284	−	0.978492i	$-0.433863\pi$
$47$	2.82843	0.412568	0.206284	+	0.978492i	$0.433863\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	−2.00000	−0.280056
$52$	0	0
$53$	−2.00000	−0.274721	−0.137361	−	0.990521i	$-0.543862\pi$
$53$	−2.00000	−0.274721	−0.137361	+	0.990521i	$0.543862\pi$
$54$	0	0
$55$	−16.9706	−2.28831
$56$	0	0
$57$	−6.00000	−0.794719
$58$	0	0
$59$	−1.41421	−0.184115	−0.0920575	−	0.995754i	$-0.529344\pi$
$59$	−1.41421	−0.184115	−0.0920575	+	0.995754i	$0.529344\pi$
$60$	0	0
$61$	−8.48528	−1.08643	−0.543214	−	0.839594i	$-0.682793\pi$
$61$	−8.48528	−1.08643	−0.543214	+	0.839594i	$0.682793\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−16.0000	−1.98456
$66$	0	0
$67$	−4.00000	−0.488678	−0.244339	−	0.969690i	$-0.578571\pi$
$67$	−4.00000	−0.488678	−0.244339	+	0.969690i	$0.578571\pi$
$68$	0	0
$69$	5.65685	0.681005
$70$	0	0
$71$	12.0000	1.42414	0.712069	−	0.702109i	$-0.247758\pi$
$71$	12.0000	1.42414	0.712069	+	0.702109i	$0.247758\pi$
$72$	0	0
$73$	−9.89949	−1.15865	−0.579324	−	0.815097i	$-0.696683\pi$
$73$	−9.89949	−1.15865	−0.579324	+	0.815097i	$0.696683\pi$
$74$	0	0
$75$	−4.24264	−0.489898
$76$	0	0
$77$	0	0
$78$	0	0
$79$	4.00000	0.450035	0.225018	−	0.974355i	$-0.427756\pi$
$79$	4.00000	0.450035	0.225018	+	0.974355i	$0.427756\pi$
$80$	0	0
$81$	−5.00000	−0.555556
$82$	0	0
$83$	−1.41421	−0.155230	−0.0776151	−	0.996983i	$-0.524731\pi$
$83$	−1.41421	−0.155230	−0.0776151	+	0.996983i	$0.524731\pi$
$84$	0	0
$85$	4.00000	0.433861
$86$	0	0
$87$	8.48528	0.909718
$88$	0	0
$89$	−4.24264	−0.449719	−0.224860	−	0.974391i	$-0.572192\pi$
$89$	−4.24264	−0.449719	−0.224860	+	0.974391i	$0.572192\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	4.00000	0.414781
$94$	0	0
$95$	12.0000	1.23117
$96$	0	0
$97$	12.7279	1.29232	0.646162	−	0.763200i	$-0.276373\pi$
$97$	12.7279	1.29232	0.646162	+	0.763200i	$0.276373\pi$
$98$	0	0
$99$	6.00000	0.603023
$100$	0	0
$101$	−2.82843	−0.281439	−0.140720	−	0.990050i	$-0.544942\pi$
$101$	−2.82843	−0.281439	−0.140720	+	0.990050i	$0.544942\pi$
$102$	0	0
$103$	14.1421	1.39347	0.696733	−	0.717331i	$-0.254636\pi$
$103$	14.1421	1.39347	0.696733	+	0.717331i	$0.254636\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−4.00000	−0.386695	−0.193347	−	0.981130i	$-0.561934\pi$
$107$	−4.00000	−0.386695	−0.193347	+	0.981130i	$0.561934\pi$
$108$	0	0
$109$	−10.0000	−0.957826	−0.478913	−	0.877862i	$-0.658969\pi$
$109$	−10.0000	−0.957826	−0.478913	+	0.877862i	$0.658969\pi$
$110$	0	0
$111$	−2.82843	−0.268462
$112$	0	0
$113$	4.00000	0.376288	0.188144	−	0.982141i	$-0.439753\pi$
$113$	4.00000	0.376288	0.188144	+	0.982141i	$0.439753\pi$
$114$	0	0
$115$	−11.3137	−1.05501
$116$	0	0
$117$	5.65685	0.522976
$118$	0	0
$119$	0	0
$120$	0	0
$121$	25.0000	2.27273
$122$	0	0
$123$	2.00000	0.180334
$124$	0	0
$125$	−5.65685	−0.505964
$126$	0	0
$127$	8.00000	0.709885	0.354943	−	0.934888i	$-0.384500\pi$
$127$	8.00000	0.709885	0.354943	+	0.934888i	$0.384500\pi$
$128$	0	0
$129$	14.1421	1.24515
$130$	0	0
$131$	−21.2132	−1.85341	−0.926703	−	0.375794i	$-0.877370\pi$
$131$	−21.2132	−1.85341	−0.926703	+	0.375794i	$0.877370\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	16.0000	1.37706
$136$	0	0
$137$	−4.00000	−0.341743	−0.170872	−	0.985293i	$-0.554658\pi$
$137$	−4.00000	−0.341743	−0.170872	+	0.985293i	$0.554658\pi$
$138$	0	0
$139$	21.2132	1.79928	0.899640	−	0.436632i	$-0.143829\pi$
$139$	21.2132	1.79928	0.899640	+	0.436632i	$0.143829\pi$
$140$	0	0
$141$	−4.00000	−0.336861
$142$	0	0
$143$	33.9411	2.83830
$144$	0	0
$145$	−16.9706	−1.40933
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−6.00000	−0.491539	−0.245770	−	0.969328i	$-0.579041\pi$
$149$	−6.00000	−0.491539	−0.245770	+	0.969328i	$0.579041\pi$
$150$	0	0
$151$	−8.00000	−0.651031	−0.325515	−	0.945537i	$-0.605538\pi$
$151$	−8.00000	−0.651031	−0.325515	+	0.945537i	$0.605538\pi$
$152$	0	0
$153$	−1.41421	−0.114332
$154$	0	0
$155$	−8.00000	−0.642575
$156$	0	0
$157$	16.9706	1.35440	0.677199	−	0.735800i	$-0.263194\pi$
$157$	16.9706	1.35440	0.677199	+	0.735800i	$0.263194\pi$
$158$	0	0
$159$	2.82843	0.224309
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−2.00000	−0.156652	−0.0783260	−	0.996928i	$-0.524958\pi$
$163$	−2.00000	−0.156652	−0.0783260	+	0.996928i	$0.524958\pi$
$164$	0	0
$165$	24.0000	1.86840
$166$	0	0
$167$	2.82843	0.218870	0.109435	−	0.993994i	$-0.465096\pi$
$167$	2.82843	0.218870	0.109435	+	0.993994i	$0.465096\pi$
$168$	0	0
$169$	19.0000	1.46154
$170$	0	0
$171$	−4.24264	−0.324443
$172$	0	0
$173$	0	0		−	1.00000i	$-0.5\pi$
$173$	0	0			1.00000i	$0.5\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	2.00000	0.150329
$178$	0	0
$179$	−20.0000	−1.49487	−0.747435	−	0.664335i	$-0.768715\pi$
$179$	−20.0000	−1.49487	−0.747435	+	0.664335i	$0.768715\pi$
$180$	0	0
$181$	16.9706	1.26141	0.630706	−	0.776022i	$-0.282765\pi$
$181$	16.9706	1.26141	0.630706	+	0.776022i	$0.282765\pi$
$182$	0	0
$183$	12.0000	0.887066
$184$	0	0
$185$	5.65685	0.415900
$186$	0	0
$187$	−8.48528	−0.620505
$188$	0	0
$189$	0	0
$190$	0	0
$191$	4.00000	0.289430	0.144715	−	0.989473i	$-0.453773\pi$
$191$	4.00000	0.289430	0.144715	+	0.989473i	$0.453773\pi$
$192$	0	0
$193$	16.0000	1.15171	0.575853	−	0.817554i	$-0.304670\pi$
$193$	16.0000	1.15171	0.575853	+	0.817554i	$0.304670\pi$
$194$	0	0
$195$	22.6274	1.62038
$196$	0	0
$197$	18.0000	1.28245	0.641223	−	0.767354i	$-0.278427\pi$
$197$	18.0000	1.28245	0.641223	+	0.767354i	$0.278427\pi$
$198$	0	0
$199$	−25.4558	−1.80452	−0.902258	−	0.431196i	$-0.858092\pi$
$199$	−25.4558	−1.80452	−0.902258	+	0.431196i	$0.858092\pi$
$200$	0	0
$201$	5.65685	0.399004
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−4.00000	−0.279372
$206$	0	0
$207$	4.00000	0.278019
$208$	0	0
$209$	−25.4558	−1.76082
$210$	0	0
$211$	−12.0000	−0.826114	−0.413057	−	0.910705i	$-0.635539\pi$
$211$	−12.0000	−0.826114	−0.413057	+	0.910705i	$0.635539\pi$
$212$	0	0
$213$	−16.9706	−1.16280
$214$	0	0
$215$	−28.2843	−1.92897
$216$	0	0
$217$	0	0
$218$	0	0
$219$	14.0000	0.946032
$220$	0	0
$221$	−8.00000	−0.538138
$222$	0	0
$223$	−11.3137	−0.757622	−0.378811	−	0.925474i	$-0.623667\pi$
$223$	−11.3137	−0.757622	−0.378811	+	0.925474i	$0.623667\pi$
$224$	0	0
$225$	−3.00000	−0.200000
$226$	0	0
$227$	−9.89949	−0.657053	−0.328526	−	0.944495i	$-0.606552\pi$
$227$	−9.89949	−0.657053	−0.328526	+	0.944495i	$0.606552\pi$
$228$	0	0
$229$	0	0		−	1.00000i	$-0.5\pi$
$229$	0	0			1.00000i	$0.5\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	8.00000	0.524097	0.262049	−	0.965055i	$-0.415602\pi$
$233$	8.00000	0.524097	0.262049	+	0.965055i	$0.415602\pi$
$234$	0	0
$235$	8.00000	0.521862
$236$	0	0
$237$	−5.65685	−0.367452
$238$	0	0
$239$	4.00000	0.258738	0.129369	−	0.991596i	$-0.458705\pi$
$239$	4.00000	0.258738	0.129369	+	0.991596i	$0.458705\pi$
$240$	0	0
$241$	−1.41421	−0.0910975	−0.0455488	−	0.998962i	$-0.514504\pi$
$241$	−1.41421	−0.0910975	−0.0455488	+	0.998962i	$0.514504\pi$
$242$	0	0
$243$	−9.89949	−0.635053
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−24.0000	−1.52708
$248$	0	0
$249$	2.00000	0.126745
$250$	0	0
$251$	−1.41421	−0.0892644	−0.0446322	−	0.999003i	$-0.514212\pi$
$251$	−1.41421	−0.0892644	−0.0446322	+	0.999003i	$0.514212\pi$
$252$	0	0
$253$	24.0000	1.50887
$254$	0	0
$255$	−5.65685	−0.354246
$256$	0	0
$257$	−12.7279	−0.793946	−0.396973	−	0.917830i	$-0.629939\pi$
$257$	−12.7279	−0.793946	−0.396973	+	0.917830i	$0.629939\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	6.00000	0.371391
$262$	0	0
$263$	4.00000	0.246651	0.123325	−	0.992366i	$-0.460644\pi$
$263$	4.00000	0.246651	0.123325	+	0.992366i	$0.460644\pi$
$264$	0	0
$265$	−5.65685	−0.347498
$266$	0	0
$267$	6.00000	0.367194
$268$	0	0
$269$	28.2843	1.72452	0.862261	−	0.506464i	$-0.169048\pi$
$269$	28.2843	1.72452	0.862261	+	0.506464i	$0.169048\pi$
$270$	0	0
$271$	11.3137	0.687259	0.343629	−	0.939105i	$-0.388344\pi$
$271$	11.3137	0.687259	0.343629	+	0.939105i	$0.388344\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−18.0000	−1.08544
$276$	0	0
$277$	14.0000	0.841178	0.420589	−	0.907251i	$-0.361823\pi$
$277$	14.0000	0.841178	0.420589	+	0.907251i	$0.361823\pi$
$278$	0	0
$279$	2.82843	0.169334
$280$	0	0
$281$	−32.0000	−1.90896	−0.954480	−	0.298275i	$-0.903589\pi$
$281$	−32.0000	−1.90896	−0.954480	+	0.298275i	$0.903589\pi$
$282$	0	0
$283$	21.2132	1.26099	0.630497	−	0.776192i	$-0.282851\pi$
$283$	21.2132	1.26099	0.630497	+	0.776192i	$0.282851\pi$
$284$	0	0
$285$	−16.9706	−1.00525
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−15.0000	−0.882353
$290$	0	0
$291$	−18.0000	−1.05518
$292$	0	0
$293$	2.82843	0.165238	0.0826192	−	0.996581i	$-0.473671\pi$
$293$	2.82843	0.165238	0.0826192	+	0.996581i	$0.473671\pi$
$294$	0	0
$295$	−4.00000	−0.232889
$296$	0	0
$297$	−33.9411	−1.96946
$298$	0	0
$299$	22.6274	1.30858
$300$	0	0
$301$	0	0
$302$	0	0
$303$	4.00000	0.229794
$304$	0	0
$305$	−24.0000	−1.37424
$306$	0	0
$307$	1.41421	0.0807134	0.0403567	−	0.999185i	$-0.487151\pi$
$307$	1.41421	0.0807134	0.0403567	+	0.999185i	$0.487151\pi$
$308$	0	0
$309$	−20.0000	−1.13776
$310$	0	0
$311$	11.3137	0.641542	0.320771	−	0.947157i	$-0.396058\pi$
$311$	11.3137	0.641542	0.320771	+	0.947157i	$0.396058\pi$
$312$	0	0
$313$	21.2132	1.19904	0.599521	−	0.800359i	$-0.295358\pi$
$313$	21.2132	1.19904	0.599521	+	0.800359i	$0.295358\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−22.0000	−1.23564	−0.617822	−	0.786318i	$-0.711985\pi$
$317$	−22.0000	−1.23564	−0.617822	+	0.786318i	$0.711985\pi$
$318$	0	0
$319$	36.0000	2.01561
$320$	0	0
$321$	5.65685	0.315735
$322$	0	0
$323$	6.00000	0.333849
$324$	0	0
$325$	−16.9706	−0.941357
$326$	0	0
$327$	14.1421	0.782062
$328$	0	0
$329$	0	0
$330$	0	0
$331$	14.0000	0.769510	0.384755	−	0.923019i	$-0.374286\pi$
$331$	14.0000	0.769510	0.384755	+	0.923019i	$0.374286\pi$
$332$	0	0
$333$	−2.00000	−0.109599
$334$	0	0
$335$	−11.3137	−0.618134
$336$	0	0
$337$	−14.0000	−0.762629	−0.381314	−	0.924445i	$-0.624528\pi$
$337$	−14.0000	−0.762629	−0.381314	+	0.924445i	$0.624528\pi$
$338$	0	0
$339$	−5.65685	−0.307238
$340$	0	0
$341$	16.9706	0.919007
$342$	0	0
$343$	0	0
$344$	0	0
$345$	16.0000	0.861411
$346$	0	0
$347$	−10.0000	−0.536828	−0.268414	−	0.963304i	$-0.586500\pi$
$347$	−10.0000	−0.536828	−0.268414	+	0.963304i	$0.586500\pi$
$348$	0	0
$349$	16.9706	0.908413	0.454207	−	0.890896i	$-0.349923\pi$
$349$	16.9706	0.908413	0.454207	+	0.890896i	$0.349923\pi$
$350$	0	0
$351$	−32.0000	−1.70803
$352$	0	0
$353$	24.0416	1.27961	0.639803	−	0.768539i	$-0.279016\pi$
$353$	24.0416	1.27961	0.639803	+	0.768539i	$0.279016\pi$
$354$	0	0
$355$	33.9411	1.80141
$356$	0	0
$357$	0	0
$358$	0	0
$359$	8.00000	0.422224	0.211112	−	0.977462i	$-0.432292\pi$
$359$	8.00000	0.422224	0.211112	+	0.977462i	$0.432292\pi$
$360$	0	0
$361$	−1.00000	−0.0526316
$362$	0	0
$363$	−35.3553	−1.85567
$364$	0	0
$365$	−28.0000	−1.46559
$366$	0	0
$367$	−5.65685	−0.295285	−0.147643	−	0.989041i	$-0.547169\pi$
$367$	−5.65685	−0.295285	−0.147643	+	0.989041i	$0.547169\pi$
$368$	0	0
$369$	1.41421	0.0736210
$370$	0	0
$371$	0	0
$372$	0	0
$373$	26.0000	1.34623	0.673114	−	0.739538i	$-0.264956\pi$
$373$	26.0000	1.34623	0.673114	+	0.739538i	$0.264956\pi$
$374$	0	0
$375$	8.00000	0.413118
$376$	0	0
$377$	33.9411	1.74806
$378$	0	0
$379$	2.00000	0.102733	0.0513665	−	0.998680i	$-0.483642\pi$
$379$	2.00000	0.102733	0.0513665	+	0.998680i	$0.483642\pi$
$380$	0	0
$381$	−11.3137	−0.579619
$382$	0	0
$383$	−14.1421	−0.722629	−0.361315	−	0.932444i	$-0.617672\pi$
$383$	−14.1421	−0.722629	−0.361315	+	0.932444i	$0.617672\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	10.0000	0.508329
$388$	0	0
$389$	−30.0000	−1.52106	−0.760530	−	0.649303i	$-0.775061\pi$
$389$	−30.0000	−1.52106	−0.760530	+	0.649303i	$0.775061\pi$
$390$	0	0
$391$	−5.65685	−0.286079
$392$	0	0
$393$	30.0000	1.51330
$394$	0	0
$395$	11.3137	0.569254
$396$	0	0
$397$	28.2843	1.41955	0.709773	−	0.704430i	$-0.248797\pi$
$397$	28.2843	1.41955	0.709773	+	0.704430i	$0.248797\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−18.0000	−0.898877	−0.449439	−	0.893311i	$-0.648376\pi$
$401$	−18.0000	−0.898877	−0.449439	+	0.893311i	$0.648376\pi$
$402$	0	0
$403$	16.0000	0.797017
$404$	0	0
$405$	−14.1421	−0.702728
$406$	0	0
$407$	−12.0000	−0.594818
$408$	0	0
$409$	−4.24264	−0.209785	−0.104893	−	0.994484i	$-0.533450\pi$
$409$	−4.24264	−0.209785	−0.104893	+	0.994484i	$0.533450\pi$
$410$	0	0
$411$	5.65685	0.279032
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−4.00000	−0.196352
$416$	0	0
$417$	−30.0000	−1.46911
$418$	0	0
$419$	−32.5269	−1.58904	−0.794522	−	0.607236i	$-0.792278\pi$
$419$	−32.5269	−1.58904	−0.794522	+	0.607236i	$0.792278\pi$
$420$	0	0
$421$	−18.0000	−0.877266	−0.438633	−	0.898666i	$-0.644537\pi$
$421$	−18.0000	−0.877266	−0.438633	+	0.898666i	$0.644537\pi$
$422$	0	0
$423$	−2.82843	−0.137523
$424$	0	0
$425$	4.24264	0.205798
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−48.0000	−2.31746
$430$	0	0
$431$	−36.0000	−1.73406	−0.867029	−	0.498257i	$-0.833974\pi$
$431$	−36.0000	−1.73406	−0.867029	+	0.498257i	$0.833974\pi$
$432$	0	0
$433$	−15.5563	−0.747590	−0.373795	−	0.927511i	$-0.621944\pi$
$433$	−15.5563	−0.747590	−0.373795	+	0.927511i	$0.621944\pi$
$434$	0	0
$435$	24.0000	1.15071
$436$	0	0
$437$	−16.9706	−0.811812
$438$	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
$443$	−36.0000	−1.71041	−0.855206	−	0.518289i	$-0.826569\pi$
$443$	−36.0000	−1.71041	−0.855206	+	0.518289i	$0.826569\pi$
$444$	0	0
$445$	−12.0000	−0.568855
$446$	0	0
$447$	8.48528	0.401340
$448$	0	0
$449$	−2.00000	−0.0943858	−0.0471929	−	0.998886i	$-0.515028\pi$
$449$	−2.00000	−0.0943858	−0.0471929	+	0.998886i	$0.515028\pi$
$450$	0	0
$451$	8.48528	0.399556
$452$	0	0
$453$	11.3137	0.531564
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−24.0000	−1.12267	−0.561336	−	0.827588i	$-0.689713\pi$
$457$	−24.0000	−1.12267	−0.561336	+	0.827588i	$0.689713\pi$
$458$	0	0
$459$	8.00000	0.373408
$460$	0	0
$461$	−22.6274	−1.05386	−0.526932	−	0.849907i	$-0.676658\pi$
$461$	−22.6274	−1.05386	−0.526932	+	0.849907i	$0.676658\pi$
$462$	0	0
$463$	16.0000	0.743583	0.371792	−	0.928316i	$-0.378744\pi$
$463$	16.0000	0.743583	0.371792	+	0.928316i	$0.378744\pi$
$464$	0	0
$465$	11.3137	0.524661
$466$	0	0
$467$	21.2132	0.981630	0.490815	−	0.871264i	$-0.336699\pi$
$467$	21.2132	0.981630	0.490815	+	0.871264i	$0.336699\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−24.0000	−1.10586
$472$	0	0
$473$	60.0000	2.75880
$474$	0	0
$475$	12.7279	0.583997
$476$	0	0
$477$	2.00000	0.0915737
$478$	0	0
$479$	−19.7990	−0.904639	−0.452319	−	0.891856i	$-0.649403\pi$
$479$	−19.7990	−0.904639	−0.452319	+	0.891856i	$0.649403\pi$
$480$	0	0
$481$	−11.3137	−0.515861
$482$	0	0
$483$	0	0
$484$	0	0
$485$	36.0000	1.63468
$486$	0	0
$487$	−36.0000	−1.63132	−0.815658	−	0.578535i	$-0.803625\pi$
$487$	−36.0000	−1.63132	−0.815658	+	0.578535i	$0.803625\pi$
$488$	0	0
$489$	2.82843	0.127906
$490$	0	0
$491$	−28.0000	−1.26362	−0.631811	−	0.775122i	$-0.717688\pi$
$491$	−28.0000	−1.26362	−0.631811	+	0.775122i	$0.717688\pi$
$492$	0	0
$493$	−8.48528	−0.382158
$494$	0	0
$495$	16.9706	0.762770
$496$	0	0
$497$	0	0
$498$	0	0
$499$	28.0000	1.25345	0.626726	−	0.779240i	$-0.284395\pi$
$499$	28.0000	1.25345	0.626726	+	0.779240i	$0.284395\pi$
$500$	0	0
$501$	−4.00000	−0.178707
$502$	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
$505$	−8.00000	−0.355995
$506$	0	0
$507$	−26.8701	−1.19334
$508$	0	0
$509$	−39.5980	−1.75515	−0.877575	−	0.479440i	$-0.840840\pi$
$509$	−39.5980	−1.75515	−0.877575	+	0.479440i	$0.840840\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	24.0000	1.05963
$514$	0	0
$515$	40.0000	1.76261
$516$	0	0
$517$	−16.9706	−0.746364
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−32.5269	−1.42503	−0.712515	−	0.701657i	$-0.752444\pi$
$521$	−32.5269	−1.42503	−0.712515	+	0.701657i	$0.752444\pi$
$522$	0	0
$523$	−32.5269	−1.42230	−0.711151	−	0.703039i	$-0.751826\pi$
$523$	−32.5269	−1.42230	−0.711151	+	0.703039i	$0.751826\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−4.00000	−0.174243
$528$	0	0
$529$	−7.00000	−0.304348
$530$	0	0
$531$	1.41421	0.0613716
$532$	0	0
$533$	8.00000	0.346518
$534$	0	0
$535$	−11.3137	−0.489134
$536$	0	0
$537$	28.2843	1.22056
$538$	0	0
$539$	0	0
$540$	0	0
$541$	10.0000	0.429934	0.214967	−	0.976621i	$-0.431036\pi$
$541$	10.0000	0.429934	0.214967	+	0.976621i	$0.431036\pi$
$542$	0	0
$543$	−24.0000	−1.02994
$544$	0	0
$545$	−28.2843	−1.21157
$546$	0	0
$547$	−14.0000	−0.598597	−0.299298	−	0.954160i	$-0.596753\pi$
$547$	−14.0000	−0.598597	−0.299298	+	0.954160i	$0.596753\pi$
$548$	0	0
$549$	8.48528	0.362143
$550$	0	0
$551$	−25.4558	−1.08446
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−8.00000	−0.339581
$556$	0	0
$557$	18.0000	0.762684	0.381342	−	0.924434i	$-0.375462\pi$
$557$	18.0000	0.762684	0.381342	+	0.924434i	$0.375462\pi$
$558$	0	0
$559$	56.5685	2.39259
$560$	0	0
$561$	12.0000	0.506640
$562$	0	0
$563$	21.2132	0.894030	0.447015	−	0.894526i	$-0.352487\pi$
$563$	21.2132	0.894030	0.447015	+	0.894526i	$0.352487\pi$
$564$	0	0
$565$	11.3137	0.475971
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−22.0000	−0.922288	−0.461144	−	0.887325i	$-0.652561\pi$
$569$	−22.0000	−0.922288	−0.461144	+	0.887325i	$0.652561\pi$
$570$	0	0
$571$	10.0000	0.418487	0.209243	−	0.977864i	$-0.432900\pi$
$571$	10.0000	0.418487	0.209243	+	0.977864i	$0.432900\pi$
$572$	0	0
$573$	−5.65685	−0.236318
$574$	0	0
$575$	−12.0000	−0.500435
$576$	0	0
$577$	21.2132	0.883117	0.441559	−	0.897232i	$-0.354426\pi$
$577$	21.2132	0.883117	0.441559	+	0.897232i	$0.354426\pi$
$578$	0	0
$579$	−22.6274	−0.940363
$580$	0	0
$581$	0	0
$582$	0	0
$583$	12.0000	0.496989
$584$	0	0
$585$	16.0000	0.661519
$586$	0	0
$587$	4.24264	0.175113	0.0875563	−	0.996160i	$-0.472094\pi$
$587$	4.24264	0.175113	0.0875563	+	0.996160i	$0.472094\pi$
$588$	0	0
$589$	−12.0000	−0.494451
$590$	0	0
$591$	−25.4558	−1.04711
$592$	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
$597$	36.0000	1.47338
$598$	0	0
$599$	0	0		−	1.00000i	$-0.5\pi$
$599$	0	0			1.00000i	$0.5\pi$
$600$	0	0
$601$	26.8701	1.09605	0.548026	−	0.836461i	$-0.315379\pi$
$601$	26.8701	1.09605	0.548026	+	0.836461i	$0.315379\pi$
$602$	0	0
$603$	4.00000	0.162893
$604$	0	0
$605$	70.7107	2.87480
$606$	0	0
$607$	−39.5980	−1.60723	−0.803616	−	0.595148i	$-0.797093\pi$
$607$	−39.5980	−1.60723	−0.803616	+	0.595148i	$0.797093\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−16.0000	−0.647291
$612$	0	0
$613$	42.0000	1.69636	0.848182	−	0.529705i	$-0.177697\pi$
$613$	42.0000	1.69636	0.848182	+	0.529705i	$0.177697\pi$
$614$	0	0
$615$	5.65685	0.228106
$616$	0	0
$617$	22.0000	0.885687	0.442843	−	0.896599i	$-0.353970\pi$
$617$	22.0000	0.885687	0.442843	+	0.896599i	$0.353970\pi$
$618$	0	0
$619$	−4.24264	−0.170526	−0.0852631	−	0.996358i	$-0.527173\pi$
$619$	−4.24264	−0.170526	−0.0852631	+	0.996358i	$0.527173\pi$
$620$	0	0
$621$	−22.6274	−0.908007
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−31.0000	−1.24000
$626$	0	0
$627$	36.0000	1.43770
$628$	0	0
$629$	2.82843	0.112777
$630$	0	0
$631$	−28.0000	−1.11466	−0.557331	−	0.830290i	$-0.688175\pi$
$631$	−28.0000	−1.11466	−0.557331	+	0.830290i	$0.688175\pi$
$632$	0	0
$633$	16.9706	0.674519
$634$	0	0
$635$	22.6274	0.897942
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−12.0000	−0.474713
$640$	0	0
$641$	−6.00000	−0.236986	−0.118493	−	0.992955i	$-0.537806\pi$
$641$	−6.00000	−0.236986	−0.118493	+	0.992955i	$0.537806\pi$
$642$	0	0
$643$	−35.3553	−1.39428	−0.697139	−	0.716936i	$-0.745544\pi$
$643$	−35.3553	−1.39428	−0.697139	+	0.716936i	$0.745544\pi$
$644$	0	0
$645$	40.0000	1.57500
$646$	0	0
$647$	42.4264	1.66795	0.833977	−	0.551799i	$-0.186058\pi$
$647$	42.4264	1.66795	0.833977	+	0.551799i	$0.186058\pi$
$648$	0	0
$649$	8.48528	0.333076
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−26.0000	−1.01746	−0.508729	−	0.860927i	$-0.669885\pi$
$653$	−26.0000	−1.01746	−0.508729	+	0.860927i	$0.669885\pi$
$654$	0	0
$655$	−60.0000	−2.34439
$656$	0	0
$657$	9.89949	0.386216
$658$	0	0
$659$	−6.00000	−0.233727	−0.116863	−	0.993148i	$-0.537284\pi$
$659$	−6.00000	−0.233727	−0.116863	+	0.993148i	$0.537284\pi$
$660$	0	0
$661$	−36.7696	−1.43017	−0.715085	−	0.699038i	$-0.753612\pi$
$661$	−36.7696	−1.43017	−0.715085	+	0.699038i	$0.753612\pi$
$662$	0	0
$663$	11.3137	0.439388
$664$	0	0
$665$	0	0
$666$	0	0
$667$	24.0000	0.929284
$668$	0	0
$669$	16.0000	0.618596
$670$	0	0
$671$	50.9117	1.96542
$672$	0	0
$673$	−44.0000	−1.69608	−0.848038	−	0.529936i	$-0.822216\pi$
$673$	−44.0000	−1.69608	−0.848038	+	0.529936i	$0.822216\pi$
$674$	0	0
$675$	16.9706	0.653197
$676$	0	0
$677$	−33.9411	−1.30446	−0.652232	−	0.758020i	$-0.726167\pi$
$677$	−33.9411	−1.30446	−0.652232	+	0.758020i	$0.726167\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	14.0000	0.536481
$682$	0	0
$683$	−4.00000	−0.153056	−0.0765279	−	0.997067i	$-0.524383\pi$
$683$	−4.00000	−0.153056	−0.0765279	+	0.997067i	$0.524383\pi$
$684$	0	0
$685$	−11.3137	−0.432275
$686$	0	0
$687$	0	0
$688$	0	0
$689$	11.3137	0.431018
$690$	0	0
$691$	−43.8406	−1.66778	−0.833888	−	0.551934i	$-0.813890\pi$
$691$	−43.8406	−1.66778	−0.833888	+	0.551934i	$0.813890\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	60.0000	2.27593
$696$	0	0
$697$	−2.00000	−0.0757554
$698$	0	0
$699$	−11.3137	−0.427924
$700$	0	0
$701$	38.0000	1.43524	0.717620	−	0.696435i	$-0.245231\pi$
$701$	38.0000	1.43524	0.717620	+	0.696435i	$0.245231\pi$
$702$	0	0
$703$	8.48528	0.320028
$704$	0	0
$705$	−11.3137	−0.426099
$706$	0	0
$707$	0	0
$708$	0	0
$709$	34.0000	1.27690	0.638448	−	0.769665i	$-0.279577\pi$
$709$	34.0000	1.27690	0.638448	+	0.769665i	$0.279577\pi$
$710$	0	0
$711$	−4.00000	−0.150012
$712$	0	0
$713$	11.3137	0.423702
$714$	0	0
$715$	96.0000	3.59020
$716$	0	0
$717$	−5.65685	−0.211259
$718$	0	0
$719$	2.82843	0.105483	0.0527413	−	0.998608i	$-0.483204\pi$
$719$	2.82843	0.105483	0.0527413	+	0.998608i	$0.483204\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	2.00000	0.0743808
$724$	0	0
$725$	−18.0000	−0.668503
$726$	0	0
$727$	25.4558	0.944105	0.472052	−	0.881570i	$-0.343513\pi$
$727$	25.4558	0.944105	0.472052	+	0.881570i	$0.343513\pi$
$728$	0	0
$729$	29.0000	1.07407
$730$	0	0
$731$	−14.1421	−0.523066
$732$	0	0
$733$	8.48528	0.313411	0.156706	−	0.987645i	$-0.449913\pi$
$733$	8.48528	0.313411	0.156706	+	0.987645i	$0.449913\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	24.0000	0.884051
$738$	0	0
$739$	6.00000	0.220714	0.110357	−	0.993892i	$-0.464801\pi$
$739$	6.00000	0.220714	0.110357	+	0.993892i	$0.464801\pi$
$740$	0	0
$741$	33.9411	1.24686
$742$	0	0
$743$	24.0000	0.880475	0.440237	−	0.897881i	$-0.354894\pi$
$743$	24.0000	0.880475	0.440237	+	0.897881i	$0.354894\pi$
$744$	0	0
$745$	−16.9706	−0.621753
$746$	0	0
$747$	1.41421	0.0517434
$748$	0	0
$749$	0	0
$750$	0	0
$751$	12.0000	0.437886	0.218943	−	0.975738i	$-0.429739\pi$
$751$	12.0000	0.437886	0.218943	+	0.975738i	$0.429739\pi$
$752$	0	0
$753$	2.00000	0.0728841
$754$	0	0
$755$	−22.6274	−0.823496
$756$	0	0
$757$	−22.0000	−0.799604	−0.399802	−	0.916602i	$-0.630921\pi$
$757$	−22.0000	−0.799604	−0.399802	+	0.916602i	$0.630921\pi$
$758$	0	0
$759$	−33.9411	−1.23198
$760$	0	0
$761$	−26.8701	−0.974039	−0.487019	−	0.873391i	$-0.661916\pi$
$761$	−26.8701	−0.974039	−0.487019	+	0.873391i	$0.661916\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−4.00000	−0.144620
$766$	0	0
$767$	8.00000	0.288863
$768$	0	0
$769$	−29.6985	−1.07095	−0.535477	−	0.844550i	$-0.679868\pi$
$769$	−29.6985	−1.07095	−0.535477	+	0.844550i	$0.679868\pi$
$770$	0	0
$771$	18.0000	0.648254
$772$	0	0
$773$	−19.7990	−0.712120	−0.356060	−	0.934463i	$-0.615880\pi$
$773$	−19.7990	−0.712120	−0.356060	+	0.934463i	$0.615880\pi$
$774$	0	0
$775$	−8.48528	−0.304800
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−6.00000	−0.214972
$780$	0	0
$781$	−72.0000	−2.57636
$782$	0	0
$783$	−33.9411	−1.21296
$784$	0	0
$785$	48.0000	1.71319
$786$	0	0
$787$	32.5269	1.15946	0.579730	−	0.814809i	$-0.303158\pi$
$787$	32.5269	1.15946	0.579730	+	0.814809i	$0.303158\pi$
$788$	0	0
$789$	−5.65685	−0.201389
$790$	0	0
$791$	0	0
$792$	0	0
$793$	48.0000	1.70453
$794$	0	0
$795$	8.00000	0.283731
$796$	0	0
$797$	39.5980	1.40263	0.701316	−	0.712850i	$-0.252596\pi$
$797$	39.5980	1.40263	0.701316	+	0.712850i	$0.252596\pi$
$798$	0	0
$799$	4.00000	0.141510
$800$	0	0
$801$	4.24264	0.149906
$802$	0	0
$803$	59.3970	2.09607
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−40.0000	−1.40807
$808$	0	0
$809$	−32.0000	−1.12506	−0.562530	−	0.826777i	$-0.690172\pi$
$809$	−32.0000	−1.12506	−0.562530	+	0.826777i	$0.690172\pi$
$810$	0	0
$811$	−7.07107	−0.248299	−0.124149	−	0.992264i	$-0.539620\pi$
$811$	−7.07107	−0.248299	−0.124149	+	0.992264i	$0.539620\pi$
$812$	0	0
$813$	−16.0000	−0.561144
$814$	0	0
$815$	−5.65685	−0.198151
$816$	0	0
$817$	−42.4264	−1.48431
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−18.0000	−0.628204	−0.314102	−	0.949389i	$-0.601703\pi$
$821$	−18.0000	−0.628204	−0.314102	+	0.949389i	$0.601703\pi$
$822$	0	0
$823$	56.0000	1.95204	0.976019	−	0.217687i	$-0.0698512\pi$
$823$	56.0000	1.95204	0.976019	+	0.217687i	$0.0698512\pi$
$824$	0	0
$825$	25.4558	0.886259
$826$	0	0
$827$	4.00000	0.139094	0.0695468	−	0.997579i	$-0.477845\pi$
$827$	4.00000	0.139094	0.0695468	+	0.997579i	$0.477845\pi$
$828$	0	0
$829$	25.4558	0.884118	0.442059	−	0.896986i	$-0.354248\pi$
$829$	25.4558	0.884118	0.442059	+	0.896986i	$0.354248\pi$
$830$	0	0
$831$	−19.7990	−0.686819
$832$	0	0
$833$	0	0
$834$	0	0
$835$	8.00000	0.276851
$836$	0	0
$837$	−16.0000	−0.553041
$838$	0	0
$839$	31.1127	1.07413	0.537065	−	0.843541i	$-0.319533\pi$
$839$	31.1127	1.07413	0.537065	+	0.843541i	$0.319533\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	0	0
$843$	45.2548	1.55866
$844$	0	0
$845$	53.7401	1.84872
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−30.0000	−1.02960
$850$	0	0
$851$	−8.00000	−0.274236
$852$	0	0
$853$	−16.9706	−0.581061	−0.290531	−	0.956866i	$-0.593832\pi$
$853$	−16.9706	−0.581061	−0.290531	+	0.956866i	$0.593832\pi$
$854$	0	0
$855$	−12.0000	−0.410391
$856$	0	0
$857$	−7.07107	−0.241543	−0.120772	−	0.992680i	$-0.538537\pi$
$857$	−7.07107	−0.241543	−0.120772	+	0.992680i	$0.538537\pi$
$858$	0	0
$859$	−15.5563	−0.530776	−0.265388	−	0.964142i	$-0.585500\pi$
$859$	−15.5563	−0.530776	−0.265388	+	0.964142i	$0.585500\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−44.0000	−1.49778	−0.748889	−	0.662696i	$-0.769412\pi$
$863$	−44.0000	−1.49778	−0.748889	+	0.662696i	$0.769412\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	21.2132	0.720438
$868$	0	0
$869$	−24.0000	−0.814144
$870$	0	0
$871$	22.6274	0.766701
$872$	0	0
$873$	−12.7279	−0.430775
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−38.0000	−1.28317	−0.641584	−	0.767052i	$-0.721723\pi$
$877$	−38.0000	−1.28317	−0.641584	+	0.767052i	$0.721723\pi$
$878$	0	0
$879$	−4.00000	−0.134917
$880$	0	0
$881$	38.1838	1.28644	0.643222	−	0.765680i	$-0.277597\pi$
$881$	38.1838	1.28644	0.643222	+	0.765680i	$0.277597\pi$
$882$	0	0
$883$	28.0000	0.942275	0.471138	−	0.882060i	$-0.343844\pi$
$883$	28.0000	0.942275	0.471138	+	0.882060i	$0.343844\pi$
$884$	0	0
$885$	5.65685	0.190153
$886$	0	0
$887$	−36.7696	−1.23460	−0.617300	−	0.786728i	$-0.711774\pi$
$887$	−36.7696	−1.23460	−0.617300	+	0.786728i	$0.711774\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	30.0000	1.00504
$892$	0	0
$893$	12.0000	0.401565
$894$	0	0
$895$	−56.5685	−1.89088
$896$	0	0
$897$	−32.0000	−1.06845
$898$	0	0
$899$	16.9706	0.566000
$900$	0	0
$901$	−2.82843	−0.0942286
$902$	0	0
$903$	0	0
$904$	0	0
$905$	48.0000	1.59557
$906$	0	0
$907$	20.0000	0.664089	0.332045	−	0.943264i	$-0.392262\pi$
$907$	20.0000	0.664089	0.332045	+	0.943264i	$0.392262\pi$
$908$	0	0
$909$	2.82843	0.0938130
$910$	0	0
$911$	48.0000	1.59031	0.795155	−	0.606406i	$-0.207389\pi$
$911$	48.0000	1.59031	0.795155	+	0.606406i	$0.207389\pi$
$912$	0	0
$913$	8.48528	0.280822
$914$	0	0
$915$	33.9411	1.12206
$916$	0	0
$917$	0	0
$918$	0	0
$919$	32.0000	1.05558	0.527791	−	0.849374i	$-0.323020\pi$
$919$	32.0000	1.05558	0.527791	+	0.849374i	$0.323020\pi$
$920$	0	0
$921$	−2.00000	−0.0659022
$922$	0	0
$923$	−67.8823	−2.23437
$924$	0	0
$925$	6.00000	0.197279
$926$	0	0
$927$	−14.1421	−0.464489
$928$	0	0
$929$	−32.5269	−1.06717	−0.533587	−	0.845745i	$-0.679156\pi$
$929$	−32.5269	−1.06717	−0.533587	+	0.845745i	$0.679156\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−16.0000	−0.523816
$934$	0	0
$935$	−24.0000	−0.784884
$936$	0	0
$937$	−21.2132	−0.693005	−0.346503	−	0.938049i	$-0.612631\pi$
$937$	−21.2132	−0.693005	−0.346503	+	0.938049i	$0.612631\pi$
$938$	0	0
$939$	−30.0000	−0.979013
$940$	0	0
$941$	−31.1127	−1.01424	−0.507122	−	0.861874i	$-0.669291\pi$
$941$	−31.1127	−1.01424	−0.507122	+	0.861874i	$0.669291\pi$
$942$	0	0
$943$	5.65685	0.184213
$944$	0	0
$945$	0	0
$946$	0	0
$947$	42.0000	1.36482	0.682408	−	0.730971i	$-0.260933\pi$
$947$	42.0000	1.36482	0.682408	+	0.730971i	$0.260933\pi$
$948$	0	0
$949$	56.0000	1.81784
$950$	0	0
$951$	31.1127	1.00890
$952$	0	0
$953$	−26.0000	−0.842223	−0.421111	−	0.907009i	$-0.638360\pi$
$953$	−26.0000	−0.842223	−0.421111	+	0.907009i	$0.638360\pi$
$954$	0	0
$955$	11.3137	0.366103
$956$	0	0
$957$	−50.9117	−1.64574
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−23.0000	−0.741935
$962$	0	0
$963$	4.00000	0.128898
$964$	0	0
$965$	45.2548	1.45680
$966$	0	0
$967$	−60.0000	−1.92947	−0.964735	−	0.263223i	$-0.915214\pi$
$967$	−60.0000	−1.92947	−0.964735	+	0.263223i	$0.915214\pi$
$968$	0	0
$969$	−8.48528	−0.272587
$970$	0	0
$971$	32.5269	1.04384	0.521919	−	0.852995i	$-0.325216\pi$
$971$	32.5269	1.04384	0.521919	+	0.852995i	$0.325216\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	24.0000	0.768615
$976$	0	0
$977$	12.0000	0.383914	0.191957	−	0.981403i	$-0.438517\pi$
$977$	12.0000	0.383914	0.191957	+	0.981403i	$0.438517\pi$
$978$	0	0
$979$	25.4558	0.813572
$980$	0	0
$981$	10.0000	0.319275
$982$	0	0
$983$	−53.7401	−1.71404	−0.857022	−	0.515280i	$-0.827688\pi$
$983$	−53.7401	−1.71404	−0.857022	+	0.515280i	$0.827688\pi$
$984$	0	0
$985$	50.9117	1.62218
$986$	0	0
$987$	0	0
$988$	0	0
$989$	40.0000	1.27193
$990$	0	0
$991$	32.0000	1.01651	0.508257	−	0.861206i	$-0.330290\pi$
$991$	32.0000	1.01651	0.508257	+	0.861206i	$0.330290\pi$
$992$	0	0
$993$	−19.7990	−0.628302
$994$	0	0
$995$	−72.0000	−2.28255
$996$	0	0
$997$	−8.48528	−0.268732	−0.134366	−	0.990932i	$-0.542900\pi$
$997$	−8.48528	−0.268732	−0.134366	+	0.990932i	$0.542900\pi$
$998$	0	0
$999$	11.3137	0.357950

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	784.2.a.k.1.1		2
3.2	odd	2		7056.2.a.ct.1.1		2
4.3	odd	2		392.2.a.g.1.2	yes	2
7.2	even	3		784.2.i.n.753.2		4
7.3	odd	6		784.2.i.n.177.1		4
7.4	even	3		784.2.i.n.177.2		4
7.5	odd	6		784.2.i.n.753.1		4
7.6	odd	2	inner	784.2.a.k.1.2		2
8.3	odd	2		3136.2.a.bk.1.1		2
8.5	even	2		3136.2.a.bp.1.2		2
12.11	even	2		3528.2.a.be.1.1		2
20.19	odd	2		9800.2.a.bv.1.1		2
21.20	even	2		7056.2.a.ct.1.2		2
28.3	even	6		392.2.i.h.177.2		4
28.11	odd	6		392.2.i.h.177.1		4
28.19	even	6		392.2.i.h.361.2		4
28.23	odd	6		392.2.i.h.361.1		4
28.27	even	2		392.2.a.g.1.1	✓	2
56.13	odd	2		3136.2.a.bp.1.1		2
56.27	even	2		3136.2.a.bk.1.2		2
84.11	even	6		3528.2.s.bj.3313.2		4
84.23	even	6		3528.2.s.bj.361.2		4
84.47	odd	6		3528.2.s.bj.361.1		4
84.59	odd	6		3528.2.s.bj.3313.1		4
84.83	odd	2		3528.2.a.be.1.2		2
140.139	even	2		9800.2.a.bv.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
392.2.a.g.1.1	✓	2	28.27	even	2
392.2.a.g.1.2	yes	2	4.3	odd	2
392.2.i.h.177.1		4	28.11	odd	6
392.2.i.h.177.2		4	28.3	even	6
392.2.i.h.361.1		4	28.23	odd	6
392.2.i.h.361.2		4	28.19	even	6
784.2.a.k.1.1		2	1.1	even	1	trivial
784.2.a.k.1.2		2	7.6	odd	2	inner
784.2.i.n.177.1		4	7.3	odd	6
784.2.i.n.177.2		4	7.4	even	3
784.2.i.n.753.1		4	7.5	odd	6
784.2.i.n.753.2		4	7.2	even	3
3136.2.a.bk.1.1		2	8.3	odd	2
3136.2.a.bk.1.2		2	56.27	even	2
3136.2.a.bp.1.1		2	56.13	odd	2
3136.2.a.bp.1.2		2	8.5	even	2
3528.2.a.be.1.1		2	12.11	even	2
3528.2.a.be.1.2		2	84.83	odd	2
3528.2.s.bj.361.1		4	84.47	odd	6
3528.2.s.bj.361.2		4	84.23	even	6
3528.2.s.bj.3313.1		4	84.59	odd	6
3528.2.s.bj.3313.2		4	84.11	even	6
7056.2.a.ct.1.1		2	3.2	odd	2
7056.2.a.ct.1.2		2	21.20	even	2
9800.2.a.bv.1.1		2	20.19	odd	2
9800.2.a.bv.1.2		2	140.139	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 784 = 2^{4} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	784.a (trivial)

\(f(q)\)	\(=\)	\(q-1.41421 q^{3} +2.82843 q^{5} -1.00000 q^{9} +O(q^{10})\)	\(q-1.41421 q^{3} +2.82843 q^{5} -1.00000 q^{9} -6.00000 q^{11} -5.65685 q^{13} -4.00000 q^{15} +1.41421 q^{17} +4.24264 q^{19} -4.00000 q^{23} +3.00000 q^{25} +5.65685 q^{27} -6.00000 q^{29} -2.82843 q^{31} +8.48528 q^{33} +2.00000 q^{37} +8.00000 q^{39} -1.41421 q^{41} -10.0000 q^{43} -2.82843 q^{45} +2.82843 q^{47} -2.00000 q^{51} -2.00000 q^{53} -16.9706 q^{55} -6.00000 q^{57} -1.41421 q^{59} -8.48528 q^{61} -16.0000 q^{65} -4.00000 q^{67} +5.65685 q^{69} +12.0000 q^{71} -9.89949 q^{73} -4.24264 q^{75} +4.00000 q^{79} -5.00000 q^{81} -1.41421 q^{83} +4.00000 q^{85} +8.48528 q^{87} -4.24264 q^{89} +4.00000 q^{93} +12.0000 q^{95} +12.7279 q^{97} +6.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^9	\( 2 q - 2 q^{9} - 12 q^{11} - 8 q^{15} - 8 q^{23} + 6 q^{25} - 12 q^{29} + 4 q^{37} + 16 q^{39} - 20 q^{43} - 4 q^{51} - 4 q^{53} - 12 q^{57} - 32 q^{65} - 8 q^{67} + 24 q^{71} + 8 q^{79} - 10 q^{81} + 8 q^{85} + 8 q^{93} + 24 q^{95} + 12 q^{99}+O(q^{100}) \) 2 * q - 2 * q^9 - 12 * q^11 - 8 * q^15 - 8 * q^23 + 6 * q^25 - 12 * q^29 + 4 * q^37 + 16 * q^39 - 20 * q^43 - 4 * q^51 - 4 * q^53 - 12 * q^57 - 32 * q^65 - 8 * q^67 + 24 * q^71 + 8 * q^79 - 10 * q^81 + 8 * q^85 + 8 * q^93 + 24 * q^95 + 12 * q^99

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