LMFDB - Embedded newform 8496.2.a.z.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8496,2,Mod(1,8496)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8496, 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("8496.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8496 = 2^{4} \cdot 3^{2} \cdot 59 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8496.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:	$67.8409015573$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{11}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 11 $ x^2 - 11
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 354)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-3.31662$ of defining polynomial
Character	$\chi$	$=$	8496.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-4.31662 q^{5} -4.00000 q^{7} +O(q^{10})$	$q-4.31662 q^{5} -4.00000 q^{7} -2.00000 q^{11} -4.31662 q^{13} +6.63325 q^{17} +2.00000 q^{19} -4.63325 q^{23} +13.6332 q^{25} +8.31662 q^{29} -2.31662 q^{31} +17.2665 q^{35} -4.31662 q^{37} +2.63325 q^{41} -4.00000 q^{43} +8.63325 q^{47} +9.00000 q^{49} +3.68338 q^{53} +8.63325 q^{55} -1.00000 q^{59} -4.31662 q^{61} +18.6332 q^{65} -4.63325 q^{67} +6.31662 q^{71} +11.2665 q^{73} +8.00000 q^{77} -4.00000 q^{83} -28.6332 q^{85} -8.63325 q^{89} +17.2665 q^{91} -8.63325 q^{95} -5.36675 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{5} - 8 q^{7}+O(q^{10}) $ 2 * q - 2 * q^5 - 8 * q^7	$ 2 q - 2 q^{5} - 8 q^{7} - 4 q^{11} - 2 q^{13} + 4 q^{19} + 4 q^{23} + 14 q^{25} + 10 q^{29} + 2 q^{31} + 8 q^{35} - 2 q^{37} - 8 q^{41} - 8 q^{43} + 4 q^{47} + 18 q^{49} + 14 q^{53} + 4 q^{55} - 2 q^{59} - 2 q^{61} + 24 q^{65} + 4 q^{67} + 6 q^{71} - 4 q^{73} + 16 q^{77} - 8 q^{83} - 44 q^{85} - 4 q^{89} + 8 q^{91} - 4 q^{95} - 24 q^{97}+O(q^{100}) $ 2 * q - 2 * q^5 - 8 * q^7 - 4 * q^11 - 2 * q^13 + 4 * q^19 + 4 * q^23 + 14 * q^25 + 10 * q^29 + 2 * q^31 + 8 * q^35 - 2 * q^37 - 8 * q^41 - 8 * q^43 + 4 * q^47 + 18 * q^49 + 14 * q^53 + 4 * q^55 - 2 * q^59 - 2 * q^61 + 24 * q^65 + 4 * q^67 + 6 * q^71 - 4 * q^73 + 16 * q^77 - 8 * q^83 - 44 * q^85 - 4 * q^89 + 8 * q^91 - 4 * q^95 - 24 * 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
$3$	0	0
$3$	0	0
$4$	0	0
$5$	−4.31662	−1.93045	−0.965227	−	0.261414i	$-0.915811\pi$
$5$	−4.31662	−1.93045	−0.965227	+	0.261414i	$0.915811\pi$
$6$	0	0
$7$	−4.00000	−1.51186	−0.755929	−	0.654654i	$-0.772814\pi$
$7$	−4.00000	−1.51186	−0.755929	+	0.654654i	$0.772814\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−2.00000	−0.603023	−0.301511	−	0.953463i	$-0.597491\pi$
$11$	−2.00000	−0.603023	−0.301511	+	0.953463i	$0.597491\pi$
$12$	0	0
$13$	−4.31662	−1.19722	−0.598608	−	0.801042i	$-0.704279\pi$
$13$	−4.31662	−1.19722	−0.598608	+	0.801042i	$0.704279\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	6.63325	1.60880	0.804400	−	0.594089i	$-0.202487\pi$
$17$	6.63325	1.60880	0.804400	+	0.594089i	$0.202487\pi$
$18$	0	0
$19$	2.00000	0.458831	0.229416	−	0.973329i	$-0.426318\pi$
$19$	2.00000	0.458831	0.229416	+	0.973329i	$0.426318\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−4.63325	−0.966099	−0.483050	−	0.875593i	$-0.660471\pi$
$23$	−4.63325	−0.966099	−0.483050	+	0.875593i	$0.660471\pi$
$24$	0	0
$25$	13.6332	2.72665
$26$	0	0
$27$	0	0
$28$	0	0
$29$	8.31662	1.54436	0.772179	−	0.635405i	$-0.219167\pi$
$29$	8.31662	1.54436	0.772179	+	0.635405i	$0.219167\pi$
$30$	0	0
$31$	−2.31662	−0.416078	−0.208039	−	0.978121i	$-0.566708\pi$
$31$	−2.31662	−0.416078	−0.208039	+	0.978121i	$0.566708\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	17.2665	2.91857
$36$	0	0
$37$	−4.31662	−0.709649	−0.354824	−	0.934933i	$-0.615459\pi$
$37$	−4.31662	−0.709649	−0.354824	+	0.934933i	$0.615459\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	2.63325	0.411244	0.205622	−	0.978631i	$-0.434078\pi$
$41$	2.63325	0.411244	0.205622	+	0.978631i	$0.434078\pi$
$42$	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
$45$	0	0
$46$	0	0
$47$	8.63325	1.25929	0.629644	−	0.776883i	$-0.283201\pi$
$47$	8.63325	1.25929	0.629644	+	0.776883i	$0.283201\pi$
$48$	0	0
$49$	9.00000	1.28571
$50$	0	0
$51$	0	0
$52$	0	0
$53$	3.68338	0.505950	0.252975	−	0.967473i	$-0.418591\pi$
$53$	3.68338	0.505950	0.252975	+	0.967473i	$0.418591\pi$
$54$	0	0
$55$	8.63325	1.16411
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−1.00000	−0.130189
$59$	−1.00000	−0.130189
$60$	0	0
$61$	−4.31662	−0.552687	−0.276344	−	0.961059i	$-0.589123\pi$
$61$	−4.31662	−0.552687	−0.276344	+	0.961059i	$0.589123\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	18.6332	2.31117
$66$	0	0
$67$	−4.63325	−0.566042	−0.283021	−	0.959114i	$-0.591337\pi$
$67$	−4.63325	−0.566042	−0.283021	+	0.959114i	$0.591337\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	6.31662	0.749645	0.374823	−	0.927097i	$-0.377704\pi$
$71$	6.31662	0.749645	0.374823	+	0.927097i	$0.377704\pi$
$72$	0	0
$73$	11.2665	1.31864	0.659322	−	0.751861i	$-0.270843\pi$
$73$	11.2665	1.31864	0.659322	+	0.751861i	$0.270843\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	8.00000	0.911685
$78$	0	0
$79$	0	0		−	1.00000i	$-0.5\pi$
$79$	0	0			1.00000i	$0.5\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−4.00000	−0.439057	−0.219529	−	0.975606i	$-0.570452\pi$
$83$	−4.00000	−0.439057	−0.219529	+	0.975606i	$0.570452\pi$
$84$	0	0
$85$	−28.6332	−3.10571
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−8.63325	−0.915123	−0.457561	−	0.889178i	$-0.651277\pi$
$89$	−8.63325	−0.915123	−0.457561	+	0.889178i	$0.651277\pi$
$90$	0	0
$91$	17.2665	1.81002
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−8.63325	−0.885753
$96$	0	0
$97$	−5.36675	−0.544911	−0.272455	−	0.962168i	$-0.587836\pi$
$97$	−5.36675	−0.544911	−0.272455	+	0.962168i	$0.587836\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	1.36675	0.135997	0.0679984	−	0.997685i	$-0.478339\pi$
$101$	1.36675	0.135997	0.0679984	+	0.997685i	$0.478339\pi$
$102$	0	0
$103$	−1.68338	−0.165868	−0.0829339	−	0.996555i	$-0.526429\pi$
$103$	−1.68338	−0.165868	−0.0829339	+	0.996555i	$0.526429\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	12.6332	1.22130	0.610651	−	0.791900i	$-0.290908\pi$
$107$	12.6332	1.22130	0.610651	+	0.791900i	$0.290908\pi$
$108$	0	0
$109$	12.9499	1.24037	0.620187	−	0.784454i	$-0.287057\pi$
$109$	12.9499	1.24037	0.620187	+	0.784454i	$0.287057\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	14.0000	1.31701	0.658505	−	0.752577i	$-0.271189\pi$
$113$	14.0000	1.31701	0.658505	+	0.752577i	$0.271189\pi$
$114$	0	0
$115$	20.0000	1.86501
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−26.5330	−2.43228
$120$	0	0
$121$	−7.00000	−0.636364
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−37.2665	−3.33322
$126$	0	0
$127$	17.8997	1.58835	0.794173	−	0.607692i	$-0.207904\pi$
$127$	17.8997	1.58835	0.794173	+	0.607692i	$0.207904\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	19.2665	1.68332	0.841661	−	0.540006i	$-0.181578\pi$
$131$	19.2665	1.68332	0.841661	+	0.540006i	$0.181578\pi$
$132$	0	0
$133$	−8.00000	−0.693688
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−14.6332	−1.25020	−0.625101	−	0.780544i	$-0.714942\pi$
$137$	−14.6332	−1.25020	−0.625101	+	0.780544i	$0.714942\pi$
$138$	0	0
$139$	−10.0000	−0.848189	−0.424094	−	0.905618i	$-0.639408\pi$
$139$	−10.0000	−0.848189	−0.424094	+	0.905618i	$0.639408\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	8.63325	0.721949
$144$	0	0
$145$	−35.8997	−2.98131
$146$	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
$151$	−14.3166	−1.16507	−0.582535	−	0.812805i	$-0.697939\pi$
$151$	−14.3166	−1.16507	−0.582535	+	0.812805i	$0.697939\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	10.0000	0.803219
$156$	0	0
$157$	−13.5831	−1.08405	−0.542026	−	0.840362i	$-0.682342\pi$
$157$	−13.5831	−1.08405	−0.542026	+	0.840362i	$0.682342\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	18.5330	1.46060
$162$	0	0
$163$	10.0000	0.783260	0.391630	−	0.920123i	$-0.371911\pi$
$163$	10.0000	0.783260	0.391630	+	0.920123i	$0.371911\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	18.3166	1.41738	0.708691	−	0.705519i	$-0.249286\pi$
$167$	18.3166	1.41738	0.708691	+	0.705519i	$0.249286\pi$
$168$	0	0
$169$	5.63325	0.433327
$170$	0	0
$171$	0	0
$172$	0	0
$173$	22.6332	1.72077	0.860387	−	0.509641i	$-0.170222\pi$
$173$	22.6332	1.72077	0.860387	+	0.509641i	$0.170222\pi$
$174$	0	0
$175$	−54.5330	−4.12231
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−4.00000	−0.298974	−0.149487	−	0.988764i	$-0.547762\pi$
$179$	−4.00000	−0.298974	−0.149487	+	0.988764i	$0.547762\pi$
$180$	0	0
$181$	3.26650	0.242797	0.121398	−	0.992604i	$-0.461262\pi$
$181$	3.26650	0.242797	0.121398	+	0.992604i	$0.461262\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	18.6332	1.36994
$186$	0	0
$187$	−13.2665	−0.970143
$188$	0	0
$189$	0	0
$190$	0	0
$191$	0.633250	0.0458203	0.0229102	−	0.999738i	$-0.492707\pi$
$191$	0.633250	0.0458203	0.0229102	+	0.999738i	$0.492707\pi$
$192$	0	0
$193$	−10.0000	−0.719816	−0.359908	−	0.932988i	$-0.617192\pi$
$193$	−10.0000	−0.719816	−0.359908	+	0.932988i	$0.617192\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−15.6834	−1.11739	−0.558697	−	0.829372i	$-0.688699\pi$
$197$	−15.6834	−1.11739	−0.558697	+	0.829372i	$0.688699\pi$
$198$	0	0
$199$	−24.6332	−1.74620	−0.873102	−	0.487537i	$-0.837895\pi$
$199$	−24.6332	−1.74620	−0.873102	+	0.487537i	$0.837895\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−33.2665	−2.33485
$204$	0	0
$205$	−11.3668	−0.793888
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−4.00000	−0.276686
$210$	0	0
$211$	−16.6332	−1.14508	−0.572540	−	0.819877i	$-0.694042\pi$
$211$	−16.6332	−1.14508	−0.572540	+	0.819877i	$0.694042\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	17.2665	1.17757
$216$	0	0
$217$	9.26650	0.629051
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−28.6332	−1.92608
$222$	0	0
$223$	13.2665	0.888390	0.444195	−	0.895930i	$-0.353490\pi$
$223$	13.2665	0.888390	0.444195	+	0.895930i	$0.353490\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−21.2665	−1.41151	−0.705754	−	0.708457i	$-0.749391\pi$
$227$	−21.2665	−1.41151	−0.705754	+	0.708457i	$0.749391\pi$
$228$	0	0
$229$	−5.58312	−0.368943	−0.184472	−	0.982838i	$-0.559057\pi$
$229$	−5.58312	−0.368943	−0.184472	+	0.982838i	$0.559057\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	20.6332	1.35173	0.675865	−	0.737026i	$-0.263770\pi$
$233$	20.6332	1.35173	0.675865	+	0.737026i	$0.263770\pi$
$234$	0	0
$235$	−37.2665	−2.43100
$236$	0	0
$237$	0	0
$238$	0	0
$239$	2.94987	0.190812	0.0954058	−	0.995438i	$-0.469585\pi$
$239$	2.94987	0.190812	0.0954058	+	0.995438i	$0.469585\pi$
$240$	0	0
$241$	−11.3668	−0.732197	−0.366098	−	0.930576i	$-0.619307\pi$
$241$	−11.3668	−0.732197	−0.366098	+	0.930576i	$0.619307\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−38.8496	−2.48201
$246$	0	0
$247$	−8.63325	−0.549321
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−22.5330	−1.42227	−0.711135	−	0.703055i	$-0.751819\pi$
$251$	−22.5330	−1.42227	−0.711135	+	0.703055i	$0.751819\pi$
$252$	0	0
$253$	9.26650	0.582580
$254$	0	0
$255$	0	0
$256$	0	0
$257$	7.26650	0.453272	0.226636	−	0.973980i	$-0.427227\pi$
$257$	7.26650	0.453272	0.226636	+	0.973980i	$0.427227\pi$
$258$	0	0
$259$	17.2665	1.07289
$260$	0	0
$261$	0	0
$262$	0	0
$263$	7.58312	0.467595	0.233798	−	0.972285i	$-0.424885\pi$
$263$	7.58312	0.467595	0.233798	+	0.972285i	$0.424885\pi$
$264$	0	0
$265$	−15.8997	−0.976714
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−2.00000	−0.121942	−0.0609711	−	0.998140i	$-0.519420\pi$
$269$	−2.00000	−0.121942	−0.0609711	+	0.998140i	$0.519420\pi$
$270$	0	0
$271$	26.5330	1.61176	0.805882	−	0.592076i	$-0.201691\pi$
$271$	26.5330	1.61176	0.805882	+	0.592076i	$0.201691\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−27.2665	−1.64423
$276$	0	0
$277$	−18.6332	−1.11956	−0.559782	−	0.828640i	$-0.689115\pi$
$277$	−18.6332	−1.11956	−0.559782	+	0.828640i	$0.689115\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−15.2665	−0.910723	−0.455361	−	0.890307i	$-0.650490\pi$
$281$	−15.2665	−0.910723	−0.455361	+	0.890307i	$0.650490\pi$
$282$	0	0
$283$	24.0000	1.42665	0.713326	−	0.700832i	$-0.247188\pi$
$283$	24.0000	1.42665	0.713326	+	0.700832i	$0.247188\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−10.5330	−0.621743
$288$	0	0
$289$	27.0000	1.58824
$290$	0	0
$291$	0	0
$292$	0	0
$293$	20.9499	1.22390	0.611952	−	0.790895i	$-0.290384\pi$
$293$	20.9499	1.22390	0.611952	+	0.790895i	$0.290384\pi$
$294$	0	0
$295$	4.31662	0.251324
$296$	0	0
$297$	0	0
$298$	0	0
$299$	20.0000	1.15663
$300$	0	0
$301$	16.0000	0.922225
$302$	0	0
$303$	0	0
$304$	0	0
$305$	18.6332	1.06694
$306$	0	0
$307$	−19.2665	−1.09960	−0.549799	−	0.835297i	$-0.685296\pi$
$307$	−19.2665	−1.09960	−0.549799	+	0.835297i	$0.685296\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	2.31662	0.131364	0.0656819	−	0.997841i	$-0.479078\pi$
$311$	2.31662	0.131364	0.0656819	+	0.997841i	$0.479078\pi$
$312$	0	0
$313$	−18.6332	−1.05321	−0.526607	−	0.850109i	$-0.676536\pi$
$313$	−18.6332	−1.05321	−0.526607	+	0.850109i	$0.676536\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	11.6834	0.656204	0.328102	−	0.944642i	$-0.393591\pi$
$317$	11.6834	0.656204	0.328102	+	0.944642i	$0.393591\pi$
$318$	0	0
$319$	−16.6332	−0.931283
$320$	0	0
$321$	0	0
$322$	0	0
$323$	13.2665	0.738168
$324$	0	0
$325$	−58.8496	−3.26439
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−34.5330	−1.90387
$330$	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
$333$	0	0
$334$	0	0
$335$	20.0000	1.09272
$336$	0	0
$337$	19.2665	1.04951	0.524757	−	0.851252i	$-0.324156\pi$
$337$	19.2665	1.04951	0.524757	+	0.851252i	$0.324156\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	4.63325	0.250905
$342$	0	0
$343$	−8.00000	−0.431959
$344$	0	0
$345$	0	0
$346$	0	0
$347$	2.00000	0.107366	0.0536828	−	0.998558i	$-0.482904\pi$
$347$	2.00000	0.107366	0.0536828	+	0.998558i	$0.482904\pi$
$348$	0	0
$349$	28.9499	1.54965	0.774826	−	0.632175i	$-0.217838\pi$
$349$	28.9499	1.54965	0.774826	+	0.632175i	$0.217838\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	14.0000	0.745145	0.372572	−	0.928003i	$-0.378476\pi$
$353$	14.0000	0.745145	0.372572	+	0.928003i	$0.378476\pi$
$354$	0	0
$355$	−27.2665	−1.44716
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−18.3166	−0.966714	−0.483357	−	0.875423i	$-0.660583\pi$
$359$	−18.3166	−0.966714	−0.483357	+	0.875423i	$0.660583\pi$
$360$	0	0
$361$	−15.0000	−0.789474
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−48.6332	−2.54558
$366$	0	0
$367$	−22.3166	−1.16492	−0.582459	−	0.812860i	$-0.697909\pi$
$367$	−22.3166	−1.16492	−0.582459	+	0.812860i	$0.697909\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−14.7335	−0.764925
$372$	0	0
$373$	−3.26650	−0.169133	−0.0845665	−	0.996418i	$-0.526951\pi$
$373$	−3.26650	−0.169133	−0.0845665	+	0.996418i	$0.526951\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−35.8997	−1.84893
$378$	0	0
$379$	−20.0000	−1.02733	−0.513665	−	0.857991i	$-0.671713\pi$
$379$	−20.0000	−1.02733	−0.513665	+	0.857991i	$0.671713\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	32.2164	1.64618	0.823090	−	0.567911i	$-0.192248\pi$
$383$	32.2164	1.64618	0.823090	+	0.567911i	$0.192248\pi$
$384$	0	0
$385$	−34.5330	−1.75996
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−12.3166	−0.624478	−0.312239	−	0.950004i	$-0.601079\pi$
$389$	−12.3166	−0.624478	−0.312239	+	0.950004i	$0.601079\pi$
$390$	0	0
$391$	−30.7335	−1.55426
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	0.316625	0.0158909	0.00794547	−	0.999968i	$-0.497471\pi$
$397$	0.316625	0.0158909	0.00794547	+	0.999968i	$0.497471\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−21.8997	−1.09362	−0.546811	−	0.837256i	$-0.684158\pi$
$401$	−21.8997	−1.09362	−0.546811	+	0.837256i	$0.684158\pi$
$402$	0	0
$403$	10.0000	0.498135
$404$	0	0
$405$	0	0
$406$	0	0
$407$	8.63325	0.427934
$408$	0	0
$409$	−6.63325	−0.327993	−0.163997	−	0.986461i	$-0.552439\pi$
$409$	−6.63325	−0.327993	−0.163997	+	0.986461i	$0.552439\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	4.00000	0.196827
$414$	0	0
$415$	17.2665	0.847579
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−21.2665	−1.03894	−0.519468	−	0.854490i	$-0.673870\pi$
$419$	−21.2665	−1.03894	−0.519468	+	0.854490i	$0.673870\pi$
$420$	0	0
$421$	−15.6834	−0.764361	−0.382180	−	0.924088i	$-0.624827\pi$
$421$	−15.6834	−0.764361	−0.382180	+	0.924088i	$0.624827\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	90.4327	4.38663
$426$	0	0
$427$	17.2665	0.835584
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−8.63325	−0.415849	−0.207924	−	0.978145i	$-0.566671\pi$
$431$	−8.63325	−0.415849	−0.207924	+	0.978145i	$0.566671\pi$
$432$	0	0
$433$	20.6332	0.991571	0.495785	−	0.868445i	$-0.334880\pi$
$433$	20.6332	0.991571	0.495785	+	0.868445i	$0.334880\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−9.26650	−0.443277
$438$	0	0
$439$	−1.89975	−0.0906701	−0.0453350	−	0.998972i	$-0.514436\pi$
$439$	−1.89975	−0.0906701	−0.0453350	+	0.998972i	$0.514436\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−21.2665	−1.01040	−0.505201	−	0.863002i	$-0.668582\pi$
$443$	−21.2665	−1.01040	−0.505201	+	0.863002i	$0.668582\pi$
$444$	0	0
$445$	37.2665	1.76660
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−14.6332	−0.690586	−0.345293	−	0.938495i	$-0.612220\pi$
$449$	−14.6332	−0.690586	−0.345293	+	0.938495i	$0.612220\pi$
$450$	0	0
$451$	−5.26650	−0.247990
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−74.5330	−3.49416
$456$	0	0
$457$	33.1662	1.55145	0.775726	−	0.631070i	$-0.217384\pi$
$457$	33.1662	1.55145	0.775726	+	0.631070i	$0.217384\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	8.31662	0.387344	0.193672	−	0.981066i	$-0.437960\pi$
$461$	8.31662	0.387344	0.193672	+	0.981066i	$0.437960\pi$
$462$	0	0
$463$	−18.9499	−0.880675	−0.440338	−	0.897832i	$-0.645141\pi$
$463$	−18.9499	−0.880675	−0.440338	+	0.897832i	$0.645141\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−15.2665	−0.706449	−0.353225	−	0.935539i	$-0.614915\pi$
$467$	−15.2665	−0.706449	−0.353225	+	0.935539i	$0.614915\pi$
$468$	0	0
$469$	18.5330	0.855774
$470$	0	0
$471$	0	0
$472$	0	0
$473$	8.00000	0.367840
$474$	0	0
$475$	27.2665	1.25107
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−8.21637	−0.375416	−0.187708	−	0.982225i	$-0.560106\pi$
$479$	−8.21637	−0.375416	−0.187708	+	0.982225i	$0.560106\pi$
$480$	0	0
$481$	18.6332	0.849603
$482$	0	0
$483$	0	0
$484$	0	0
$485$	23.1662	1.05193
$486$	0	0
$487$	−0.633250	−0.0286953	−0.0143476	−	0.999897i	$-0.504567\pi$
$487$	−0.633250	−0.0286953	−0.0143476	+	0.999897i	$0.504567\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	1.89975	0.0857345	0.0428672	−	0.999081i	$-0.486351\pi$
$491$	1.89975	0.0857345	0.0428672	+	0.999081i	$0.486351\pi$
$492$	0	0
$493$	55.1662	2.48456
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−25.2665	−1.13336
$498$	0	0
$499$	14.5330	0.650586	0.325293	−	0.945613i	$-0.394537\pi$
$499$	14.5330	0.650586	0.325293	+	0.945613i	$0.394537\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	39.1662	1.74634	0.873168	−	0.487419i	$-0.162061\pi$
$503$	39.1662	1.74634	0.873168	+	0.487419i	$0.162061\pi$
$504$	0	0
$505$	−5.89975	−0.262535
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−11.8997	−0.527447	−0.263724	−	0.964598i	$-0.584951\pi$
$509$	−11.8997	−0.527447	−0.263724	+	0.964598i	$0.584951\pi$
$510$	0	0
$511$	−45.0660	−1.99360
$512$	0	0
$513$	0	0
$514$	0	0
$515$	7.26650	0.320200
$516$	0	0
$517$	−17.2665	−0.759380
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−21.3668	−0.936094	−0.468047	−	0.883703i	$-0.655042\pi$
$521$	−21.3668	−0.936094	−0.468047	+	0.883703i	$0.655042\pi$
$522$	0	0
$523$	14.5330	0.635484	0.317742	−	0.948177i	$-0.397075\pi$
$523$	14.5330	0.635484	0.317742	+	0.948177i	$0.397075\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−15.3668	−0.669386
$528$	0	0
$529$	−1.53300	−0.0666521
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−11.3668	−0.492349
$534$	0	0
$535$	−54.5330	−2.35767
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−18.0000	−0.775315
$540$	0	0
$541$	−3.05013	−0.131135	−0.0655676	−	0.997848i	$-0.520886\pi$
$541$	−3.05013	−0.131135	−0.0655676	+	0.997848i	$0.520886\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−55.8997	−2.39448
$546$	0	0
$547$	30.5330	1.30550	0.652748	−	0.757575i	$-0.273616\pi$
$547$	30.5330	1.30550	0.652748	+	0.757575i	$0.273616\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	16.6332	0.708600
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−12.3166	−0.521872	−0.260936	−	0.965356i	$-0.584031\pi$
$557$	−12.3166	−0.521872	−0.260936	+	0.965356i	$0.584031\pi$
$558$	0	0
$559$	17.2665	0.730295
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−14.0000	−0.590030	−0.295015	−	0.955493i	$-0.595325\pi$
$563$	−14.0000	−0.590030	−0.295015	+	0.955493i	$0.595325\pi$
$564$	0	0
$565$	−60.4327	−2.54242
$566$	0	0
$567$	0	0
$568$	0	0
$569$	19.2665	0.807694	0.403847	−	0.914827i	$-0.367673\pi$
$569$	19.2665	0.807694	0.403847	+	0.914827i	$0.367673\pi$
$570$	0	0
$571$	17.2665	0.722581	0.361290	−	0.932453i	$-0.382336\pi$
$571$	17.2665	0.722581	0.361290	+	0.932453i	$0.382336\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−63.1662	−2.63421
$576$	0	0
$577$	−3.36675	−0.140160	−0.0700798	−	0.997541i	$-0.522325\pi$
$577$	−3.36675	−0.140160	−0.0700798	+	0.997541i	$0.522325\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	16.0000	0.663792
$582$	0	0
$583$	−7.36675	−0.305100
$584$	0	0
$585$	0	0
$586$	0	0
$587$	20.7335	0.855763	0.427882	−	0.903835i	$-0.359260\pi$
$587$	20.7335	0.855763	0.427882	+	0.903835i	$0.359260\pi$
$588$	0	0
$589$	−4.63325	−0.190910
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−19.2665	−0.791180	−0.395590	−	0.918427i	$-0.629460\pi$
$593$	−19.2665	−0.791180	−0.395590	+	0.918427i	$0.629460\pi$
$594$	0	0
$595$	114.533	4.69540
$596$	0	0
$597$	0	0
$598$	0	0
$599$	1.05013	0.0429070	0.0214535	−	0.999770i	$-0.493171\pi$
$599$	1.05013	0.0429070	0.0214535	+	0.999770i	$0.493171\pi$
$600$	0	0
$601$	20.7335	0.845737	0.422869	−	0.906191i	$-0.361023\pi$
$601$	20.7335	0.845737	0.422869	+	0.906191i	$0.361023\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	30.2164	1.22847
$606$	0	0
$607$	−5.89975	−0.239463	−0.119732	−	0.992806i	$-0.538203\pi$
$607$	−5.89975	−0.239463	−0.119732	+	0.992806i	$0.538203\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−37.2665	−1.50764
$612$	0	0
$613$	20.9499	0.846157	0.423079	−	0.906093i	$-0.360949\pi$
$613$	20.9499	0.846157	0.423079	+	0.906093i	$0.360949\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	14.0000	0.563619	0.281809	−	0.959470i	$-0.409065\pi$
$617$	14.0000	0.563619	0.281809	+	0.959470i	$0.409065\pi$
$618$	0	0
$619$	18.7335	0.752963	0.376481	−	0.926424i	$-0.377134\pi$
$619$	18.7335	0.752963	0.376481	+	0.926424i	$0.377134\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	34.5330	1.38354
$624$	0	0
$625$	92.6992	3.70797
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−28.6332	−1.14168
$630$	0	0
$631$	34.5330	1.37474	0.687368	−	0.726309i	$-0.258766\pi$
$631$	34.5330	1.37474	0.687368	+	0.726309i	$0.258766\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−77.2665	−3.06623
$636$	0	0
$637$	−38.8496	−1.53928
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−26.6332	−1.05195	−0.525975	−	0.850500i	$-0.676299\pi$
$641$	−26.6332	−1.05195	−0.525975	+	0.850500i	$0.676299\pi$
$642$	0	0
$643$	5.26650	0.207690	0.103845	−	0.994593i	$-0.466885\pi$
$643$	5.26650	0.207690	0.103845	+	0.994593i	$0.466885\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	0.416876	0.0163891	0.00819454	−	0.999966i	$-0.497392\pi$
$647$	0.416876	0.0163891	0.00819454	+	0.999966i	$0.497392\pi$
$648$	0	0
$649$	2.00000	0.0785069
$650$	0	0
$651$	0	0
$652$	0	0
$653$	11.6834	0.457206	0.228603	−	0.973520i	$-0.426584\pi$
$653$	11.6834	0.457206	0.228603	+	0.973520i	$0.426584\pi$
$654$	0	0
$655$	−83.1662	−3.24957
$656$	0	0
$657$	0	0
$658$	0	0
$659$	5.26650	0.205154	0.102577	−	0.994725i	$-0.467291\pi$
$659$	5.26650	0.205154	0.102577	+	0.994725i	$0.467291\pi$
$660$	0	0
$661$	15.2665	0.593798	0.296899	−	0.954909i	$-0.404048\pi$
$661$	15.2665	0.593798	0.296899	+	0.954909i	$0.404048\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	34.5330	1.33913
$666$	0	0
$667$	−38.5330	−1.49200
$668$	0	0
$669$	0	0
$670$	0	0
$671$	8.63325	0.333283
$672$	0	0
$673$	−33.1662	−1.27846	−0.639232	−	0.769014i	$-0.720748\pi$
$673$	−33.1662	−1.27846	−0.639232	+	0.769014i	$0.720748\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	19.6834	0.756494	0.378247	−	0.925705i	$-0.376527\pi$
$677$	19.6834	0.756494	0.378247	+	0.925705i	$0.376527\pi$
$678$	0	0
$679$	21.4670	0.823828
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−40.5330	−1.55095	−0.775476	−	0.631377i	$-0.782490\pi$
$683$	−40.5330	−1.55095	−0.775476	+	0.631377i	$0.782490\pi$
$684$	0	0
$685$	63.1662	2.41346
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−15.8997	−0.605732
$690$	0	0
$691$	48.4327	1.84247	0.921234	−	0.389008i	$-0.127182\pi$
$691$	48.4327	1.84247	0.921234	+	0.389008i	$0.127182\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	43.1662	1.63739
$696$	0	0
$697$	17.4670	0.661610
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−37.1662	−1.40375	−0.701875	−	0.712300i	$-0.747653\pi$
$701$	−37.1662	−1.40375	−0.701875	+	0.712300i	$0.747653\pi$
$702$	0	0
$703$	−8.63325	−0.325609
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−5.46700	−0.205608
$708$	0	0
$709$	−31.2665	−1.17424	−0.587119	−	0.809501i	$-0.699738\pi$
$709$	−31.2665	−1.17424	−0.587119	+	0.809501i	$0.699738\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	10.7335	0.401973
$714$	0	0
$715$	−37.2665	−1.39369
$716$	0	0
$717$	0	0
$718$	0	0
$719$	41.8997	1.56260	0.781298	−	0.624158i	$-0.214558\pi$
$719$	41.8997	1.56260	0.781298	+	0.624158i	$0.214558\pi$
$720$	0	0
$721$	6.73350	0.250769
$722$	0	0
$723$	0	0
$724$	0	0
$725$	113.383	4.21092
$726$	0	0
$727$	35.1662	1.30424	0.652122	−	0.758114i	$-0.273879\pi$
$727$	35.1662	1.30424	0.652122	+	0.758114i	$0.273879\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−26.5330	−0.981358
$732$	0	0
$733$	−47.8997	−1.76922	−0.884609	−	0.466334i	$-0.845575\pi$
$733$	−47.8997	−1.76922	−0.884609	+	0.466334i	$0.845575\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	9.26650	0.341336
$738$	0	0
$739$	−18.5330	−0.681747	−0.340874	−	0.940109i	$-0.610723\pi$
$739$	−18.5330	−0.681747	−0.340874	+	0.940109i	$0.610723\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−20.2164	−0.741667	−0.370833	−	0.928699i	$-0.620928\pi$
$743$	−20.2164	−0.741667	−0.370833	+	0.928699i	$0.620928\pi$
$744$	0	0
$745$	94.9657	3.47928
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−50.5330	−1.84644
$750$	0	0
$751$	20.8496	0.760814	0.380407	−	0.924819i	$-0.375784\pi$
$751$	20.8496	0.760814	0.380407	+	0.924819i	$0.375784\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	61.7995	2.24911
$756$	0	0
$757$	11.2665	0.409488	0.204744	−	0.978816i	$-0.434364\pi$
$757$	11.2665	0.409488	0.204744	+	0.978816i	$0.434364\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	33.1662	1.20228	0.601138	−	0.799145i	$-0.294714\pi$
$761$	33.1662	1.20228	0.601138	+	0.799145i	$0.294714\pi$
$762$	0	0
$763$	−51.7995	−1.87527
$764$	0	0
$765$	0	0
$766$	0	0
$767$	4.31662	0.155864
$768$	0	0
$769$	−18.6332	−0.671932	−0.335966	−	0.941874i	$-0.609063\pi$
$769$	−18.6332	−0.671932	−0.335966	+	0.941874i	$0.609063\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−47.8997	−1.72283	−0.861417	−	0.507898i	$-0.830423\pi$
$773$	−47.8997	−1.72283	−0.861417	+	0.507898i	$0.830423\pi$
$774$	0	0
$775$	−31.5831	−1.13450
$776$	0	0
$777$	0	0
$778$	0	0
$779$	5.26650	0.188692
$780$	0	0
$781$	−12.6332	−0.452053
$782$	0	0
$783$	0	0
$784$	0	0
$785$	58.6332	2.09271
$786$	0	0
$787$	16.5330	0.589338	0.294669	−	0.955599i	$-0.404791\pi$
$787$	16.5330	0.589338	0.294669	+	0.955599i	$0.404791\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−56.0000	−1.99113
$792$	0	0
$793$	18.6332	0.661686
$794$	0	0
$795$	0	0
$796$	0	0
$797$	12.7335	0.451044	0.225522	−	0.974238i	$-0.427591\pi$
$797$	12.7335	0.451044	0.225522	+	0.974238i	$0.427591\pi$
$798$	0	0
$799$	57.2665	2.02594
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−22.5330	−0.795172
$804$	0	0
$805$	−80.0000	−2.81963
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−9.89975	−0.348057	−0.174028	−	0.984741i	$-0.555678\pi$
$809$	−9.89975	−0.348057	−0.174028	+	0.984741i	$0.555678\pi$
$810$	0	0
$811$	−25.8997	−0.909463	−0.454732	−	0.890629i	$-0.650265\pi$
$811$	−25.8997	−0.909463	−0.454732	+	0.890629i	$0.650265\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−43.1662	−1.51205
$816$	0	0
$817$	−8.00000	−0.279885
$818$	0	0
$819$	0	0
$820$	0	0
$821$	51.0660	1.78222	0.891108	−	0.453792i	$-0.149929\pi$
$821$	51.0660	1.78222	0.891108	+	0.453792i	$0.149929\pi$
$822$	0	0
$823$	12.8496	0.447910	0.223955	−	0.974600i	$-0.428103\pi$
$823$	12.8496	0.447910	0.223955	+	0.974600i	$0.428103\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−10.7335	−0.373240	−0.186620	−	0.982432i	$-0.559753\pi$
$827$	−10.7335	−0.373240	−0.186620	+	0.982432i	$0.559753\pi$
$828$	0	0
$829$	−14.0000	−0.486240	−0.243120	−	0.969996i	$-0.578171\pi$
$829$	−14.0000	−0.486240	−0.243120	+	0.969996i	$0.578171\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	59.6992	2.06846
$834$	0	0
$835$	−79.0660	−2.73619
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−1.89975	−0.0655866	−0.0327933	−	0.999462i	$-0.510440\pi$
$839$	−1.89975	−0.0655866	−0.0327933	+	0.999462i	$0.510440\pi$
$840$	0	0
$841$	40.1662	1.38504
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−24.3166	−0.836517
$846$	0	0
$847$	28.0000	0.962091
$848$	0	0
$849$	0	0
$850$	0	0
$851$	20.0000	0.685591
$852$	0	0
$853$	17.1662	0.587761	0.293881	−	0.955842i	$-0.405053\pi$
$853$	17.1662	0.587761	0.293881	+	0.955842i	$0.405053\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−13.8997	−0.474806	−0.237403	−	0.971411i	$-0.576296\pi$
$857$	−13.8997	−0.474806	−0.237403	+	0.971411i	$0.576296\pi$
$858$	0	0
$859$	−25.8997	−0.883688	−0.441844	−	0.897092i	$-0.645676\pi$
$859$	−25.8997	−0.883688	−0.441844	+	0.897092i	$0.645676\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−14.1003	−0.479978	−0.239989	−	0.970776i	$-0.577144\pi$
$863$	−14.1003	−0.479978	−0.239989	+	0.970776i	$0.577144\pi$
$864$	0	0
$865$	−97.6992	−3.32187
$866$	0	0
$867$	0	0
$868$	0	0
$869$	0	0
$870$	0	0
$871$	20.0000	0.677674
$872$	0	0
$873$	0	0
$874$	0	0
$875$	149.066	5.03935
$876$	0	0
$877$	−18.6332	−0.629200	−0.314600	−	0.949224i	$-0.601870\pi$
$877$	−18.6332	−0.629200	−0.314600	+	0.949224i	$0.601870\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−37.8997	−1.27687	−0.638437	−	0.769674i	$-0.720419\pi$
$881$	−37.8997	−1.27687	−0.638437	+	0.769674i	$0.720419\pi$
$882$	0	0
$883$	−13.2665	−0.446453	−0.223227	−	0.974767i	$-0.571659\pi$
$883$	−13.2665	−0.446453	−0.223227	+	0.974767i	$0.571659\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	14.7335	0.494703	0.247351	−	0.968926i	$-0.420440\pi$
$887$	14.7335	0.494703	0.247351	+	0.968926i	$0.420440\pi$
$888$	0	0
$889$	−71.5990	−2.40135
$890$	0	0
$891$	0	0
$892$	0	0
$893$	17.2665	0.577801
$894$	0	0
$895$	17.2665	0.577155
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−19.2665	−0.642574
$900$	0	0
$901$	24.4327	0.813973
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−14.1003	−0.468708
$906$	0	0
$907$	31.2665	1.03819	0.519094	−	0.854717i	$-0.326270\pi$
$907$	31.2665	1.03819	0.519094	+	0.854717i	$0.326270\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	47.5831	1.57650	0.788250	−	0.615356i	$-0.210988\pi$
$911$	47.5831	1.57650	0.788250	+	0.615356i	$0.210988\pi$
$912$	0	0
$913$	8.00000	0.264761
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−77.0660	−2.54494
$918$	0	0
$919$	0.416876	0.0137515	0.00687574	−	0.999976i	$-0.497811\pi$
$919$	0.416876	0.0137515	0.00687574	+	0.999976i	$0.497811\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−27.2665	−0.897488
$924$	0	0
$925$	−58.8496	−1.93496
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−31.3668	−1.02911	−0.514555	−	0.857457i	$-0.672043\pi$
$929$	−31.3668	−1.02911	−0.514555	+	0.857457i	$0.672043\pi$
$930$	0	0
$931$	18.0000	0.589926
$932$	0	0
$933$	0	0
$934$	0	0
$935$	57.2665	1.87281
$936$	0	0
$937$	12.7335	0.415985	0.207993	−	0.978130i	$-0.433307\pi$
$937$	12.7335	0.415985	0.207993	+	0.978130i	$0.433307\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	6.63325	0.216238	0.108119	−	0.994138i	$-0.465517\pi$
$941$	6.63325	0.216238	0.108119	+	0.994138i	$0.465517\pi$
$942$	0	0
$943$	−12.2005	−0.397303
$944$	0	0
$945$	0	0
$946$	0	0
$947$	31.1662	1.01277	0.506383	−	0.862308i	$-0.330982\pi$
$947$	31.1662	1.01277	0.506383	+	0.862308i	$0.330982\pi$
$948$	0	0
$949$	−48.6332	−1.57870
$950$	0	0
$951$	0	0
$952$	0	0
$953$	14.6332	0.474017	0.237009	−	0.971508i	$-0.423833\pi$
$953$	14.6332	0.474017	0.237009	+	0.971508i	$0.423833\pi$
$954$	0	0
$955$	−2.73350	−0.0884540
$956$	0	0
$957$	0	0
$958$	0	0
$959$	58.5330	1.89013
$960$	0	0
$961$	−25.6332	−0.826879
$962$	0	0
$963$	0	0
$964$	0	0
$965$	43.1662	1.38957
$966$	0	0
$967$	18.3166	0.589023	0.294511	−	0.955648i	$-0.404843\pi$
$967$	18.3166	0.589023	0.294511	+	0.955648i	$0.404843\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	51.1662	1.64200	0.821002	−	0.570926i	$-0.193416\pi$
$971$	51.1662	1.64200	0.821002	+	0.570926i	$0.193416\pi$
$972$	0	0
$973$	40.0000	1.28234
$974$	0	0
$975$	0	0
$976$	0	0
$977$	20.5330	0.656909	0.328454	−	0.944520i	$-0.393472\pi$
$977$	20.5330	0.656909	0.328454	+	0.944520i	$0.393472\pi$
$978$	0	0
$979$	17.2665	0.551840
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−33.8997	−1.08123	−0.540617	−	0.841269i	$-0.681809\pi$
$983$	−33.8997	−1.08123	−0.540617	+	0.841269i	$0.681809\pi$
$984$	0	0
$985$	67.6992	2.15708
$986$	0	0
$987$	0	0
$988$	0	0
$989$	18.5330	0.589315
$990$	0	0
$991$	2.94987	0.0937058	0.0468529	−	0.998902i	$-0.485081\pi$
$991$	2.94987	0.0937058	0.0468529	+	0.998902i	$0.485081\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	106.332	3.37097
$996$	0	0
$997$	30.6332	0.970165	0.485082	−	0.874468i	$-0.338790\pi$
$997$	30.6332	0.970165	0.485082	+	0.874468i	$0.338790\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8496.2.a.z.1.1		2
3.2	odd	2		2832.2.a.l.1.2		2
4.3	odd	2		1062.2.a.m.1.1		2
12.11	even	2		354.2.a.g.1.2	✓	2
60.59	even	2		8850.2.a.bm.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
354.2.a.g.1.2	✓	2	12.11	even	2
1062.2.a.m.1.1		2	4.3	odd	2
2832.2.a.l.1.2		2	3.2	odd	2
8496.2.a.z.1.1		2	1.1	even	1	trivial
8850.2.a.bm.1.2		2	60.59	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:	\( N \)	\(=\)	\( 8496 = 2^{4} \cdot 3^{2} \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8496.a (trivial)

\(f(q)\)	\(=\)	\(q-4.31662 q^{5} -4.00000 q^{7} +O(q^{10})\)	\(q-4.31662 q^{5} -4.00000 q^{7} -2.00000 q^{11} -4.31662 q^{13} +6.63325 q^{17} +2.00000 q^{19} -4.63325 q^{23} +13.6332 q^{25} +8.31662 q^{29} -2.31662 q^{31} +17.2665 q^{35} -4.31662 q^{37} +2.63325 q^{41} -4.00000 q^{43} +8.63325 q^{47} +9.00000 q^{49} +3.68338 q^{53} +8.63325 q^{55} -1.00000 q^{59} -4.31662 q^{61} +18.6332 q^{65} -4.63325 q^{67} +6.31662 q^{71} +11.2665 q^{73} +8.00000 q^{77} -4.00000 q^{83} -28.6332 q^{85} -8.63325 q^{89} +17.2665 q^{91} -8.63325 q^{95} -5.36675 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{5} - 8 q^{7}+O(q^{10}) \) 2 * q - 2 * q^5 - 8 * q^7	\( 2 q - 2 q^{5} - 8 q^{7} - 4 q^{11} - 2 q^{13} + 4 q^{19} + 4 q^{23} + 14 q^{25} + 10 q^{29} + 2 q^{31} + 8 q^{35} - 2 q^{37} - 8 q^{41} - 8 q^{43} + 4 q^{47} + 18 q^{49} + 14 q^{53} + 4 q^{55} - 2 q^{59} - 2 q^{61} + 24 q^{65} + 4 q^{67} + 6 q^{71} - 4 q^{73} + 16 q^{77} - 8 q^{83} - 44 q^{85} - 4 q^{89} + 8 q^{91} - 4 q^{95} - 24 q^{97}+O(q^{100}) \) 2 * q - 2 * q^5 - 8 * q^7 - 4 * q^11 - 2 * q^13 + 4 * q^19 + 4 * q^23 + 14 * q^25 + 10 * q^29 + 2 * q^31 + 8 * q^35 - 2 * q^37 - 8 * q^41 - 8 * q^43 + 4 * q^47 + 18 * q^49 + 14 * q^53 + 4 * q^55 - 2 * q^59 - 2 * q^61 + 24 * q^65 + 4 * q^67 + 6 * q^71 - 4 * q^73 + 16 * q^77 - 8 * q^83 - 44 * q^85 - 4 * q^89 + 8 * q^91 - 4 * q^95 - 24 * q^97

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