LMFDB - Embedded newform 8752.2.a.v.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8752,2,Mod(1,8752)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8752, 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("8752.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8752 = 2^{4} \cdot 547 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8752.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:	$69.8850718490$
Analytic rank:	$0$
Dimension:	$25$
Twist minimal:	no (minimal twist has level 547)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Character	$\chi$	$=$	8752.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.22215 q^{3} +0.712381 q^{5} -4.39659 q^{7} +1.93796 q^{9} +O(q^{10})$	$q-2.22215 q^{3} +0.712381 q^{5} -4.39659 q^{7} +1.93796 q^{9} -3.35537 q^{11} -2.53115 q^{13} -1.58302 q^{15} +4.59046 q^{17} +3.55428 q^{19} +9.76989 q^{21} +2.55608 q^{23} -4.49251 q^{25} +2.36001 q^{27} +5.14817 q^{29} -3.73387 q^{31} +7.45614 q^{33} -3.13205 q^{35} -10.4553 q^{37} +5.62460 q^{39} -6.42906 q^{41} +2.39294 q^{43} +1.38057 q^{45} -0.655082 q^{47} +12.3300 q^{49} -10.2007 q^{51} -1.90404 q^{53} -2.39030 q^{55} -7.89815 q^{57} -11.5584 q^{59} -4.56217 q^{61} -8.52042 q^{63} -1.80315 q^{65} -13.5959 q^{67} -5.67999 q^{69} -9.44418 q^{71} -8.94150 q^{73} +9.98305 q^{75} +14.7522 q^{77} -6.83371 q^{79} -11.0582 q^{81} -13.3811 q^{83} +3.27016 q^{85} -11.4400 q^{87} -15.4078 q^{89} +11.1284 q^{91} +8.29724 q^{93} +2.53200 q^{95} +6.59022 q^{97} -6.50257 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 25 q - 8 q^{3} + 29 q^{5} - 5 q^{7} + 29 q^{9}+O(q^{10}) $ 25 * q - 8 * q^3 + 29 * q^5 - 5 * q^7 + 29 * q^9	$ 25 q - 8 q^{3} + 29 q^{5} - 5 q^{7} + 29 q^{9} - 10 q^{11} + 19 q^{13} - 5 q^{15} + 40 q^{17} - 8 q^{21} - 26 q^{23} + 36 q^{25} - 11 q^{27} + 30 q^{29} + 5 q^{31} + 10 q^{33} - 11 q^{35} + 26 q^{37} + 17 q^{39} + 9 q^{41} + 10 q^{43} + 64 q^{45} - 28 q^{47} + 20 q^{49} + 9 q^{51} + 80 q^{53} + q^{55} - 8 q^{57} + 2 q^{59} + 22 q^{61} + 9 q^{63} + 30 q^{65} + 16 q^{67} + 22 q^{69} + q^{71} + 2 q^{73} + 31 q^{75} + 67 q^{77} + 34 q^{79} - 11 q^{81} - 15 q^{83} + 15 q^{85} + 29 q^{87} + 38 q^{89} + 41 q^{91} - 4 q^{93} + 46 q^{95} - 2 q^{97} + 41 q^{99}+O(q^{100}) $ 25 * q - 8 * q^3 + 29 * q^5 - 5 * q^7 + 29 * q^9 - 10 * q^11 + 19 * q^13 - 5 * q^15 + 40 * q^17 - 8 * q^21 - 26 * q^23 + 36 * q^25 - 11 * q^27 + 30 * q^29 + 5 * q^31 + 10 * q^33 - 11 * q^35 + 26 * q^37 + 17 * q^39 + 9 * q^41 + 10 * q^43 + 64 * q^45 - 28 * q^47 + 20 * q^49 + 9 * q^51 + 80 * q^53 + q^55 - 8 * q^57 + 2 * q^59 + 22 * q^61 + 9 * q^63 + 30 * q^65 + 16 * q^67 + 22 * q^69 + q^71 + 2 * q^73 + 31 * q^75 + 67 * q^77 + 34 * q^79 - 11 * q^81 - 15 * q^83 + 15 * q^85 + 29 * q^87 + 38 * q^89 + 41 * q^91 - 4 * q^93 + 46 * q^95 - 2 * q^97 + 41 * 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
$3$	−2.22215	−1.28296	−0.641480	−	0.767140i	$-0.721679\pi$
$3$	−2.22215	−1.28296	−0.641480	+	0.767140i	$0.721679\pi$
$4$	0	0
$5$	0.712381	0.318587	0.159293	−	0.987231i	$-0.449078\pi$
$5$	0.712381	0.318587	0.159293	+	0.987231i	$0.449078\pi$
$6$	0	0
$7$	−4.39659	−1.66176	−0.830878	−	0.556455i	$-0.812161\pi$
$7$	−4.39659	−1.66176	−0.830878	+	0.556455i	$0.812161\pi$
$8$	0	0
$9$	1.93796	0.645987
$10$	0	0
$11$	−3.35537	−1.01168	−0.505841	−	0.862627i	$-0.668818\pi$
$11$	−3.35537	−1.01168	−0.505841	+	0.862627i	$0.668818\pi$
$12$	0	0
$13$	−2.53115	−0.702015	−0.351008	−	0.936373i	$-0.614161\pi$
$13$	−2.53115	−0.702015	−0.351008	+	0.936373i	$0.614161\pi$
$14$	0	0
$15$	−1.58302	−0.408734
$16$	0	0
$17$	4.59046	1.11335	0.556675	−	0.830731i	$-0.312077\pi$
$17$	4.59046	1.11335	0.556675	+	0.830731i	$0.312077\pi$
$18$	0	0
$19$	3.55428	0.815407	0.407704	−	0.913114i	$-0.366330\pi$
$19$	3.55428	0.815407	0.407704	+	0.913114i	$0.366330\pi$
$20$	0	0
$21$	9.76989	2.13197
$22$	0	0
$23$	2.55608	0.532979	0.266489	−	0.963838i	$-0.414136\pi$
$23$	2.55608	0.532979	0.266489	+	0.963838i	$0.414136\pi$
$24$	0	0
$25$	−4.49251	−0.898503
$26$	0	0
$27$	2.36001	0.454185
$28$	0	0
$29$	5.14817	0.955990	0.477995	−	0.878362i	$-0.341364\pi$
$29$	5.14817	0.955990	0.477995	+	0.878362i	$0.341364\pi$
$30$	0	0
$31$	−3.73387	−0.670624	−0.335312	−	0.942107i	$-0.608842\pi$
$31$	−3.73387	−0.670624	−0.335312	+	0.942107i	$0.608842\pi$
$32$	0	0
$33$	7.45614	1.29795
$34$	0	0
$35$	−3.13205	−0.529413
$36$	0	0
$37$	−10.4553	−1.71883	−0.859417	−	0.511276i	$-0.829173\pi$
$37$	−10.4553	−1.71883	−0.859417	+	0.511276i	$0.829173\pi$
$38$	0	0
$39$	5.62460	0.900657
$40$	0	0
$41$	−6.42906	−1.00405	−0.502025	−	0.864853i	$-0.667411\pi$
$41$	−6.42906	−1.00405	−0.502025	+	0.864853i	$0.667411\pi$
$42$	0	0
$43$	2.39294	0.364920	0.182460	−	0.983213i	$-0.441594\pi$
$43$	2.39294	0.364920	0.182460	+	0.983213i	$0.441594\pi$
$44$	0	0
$45$	1.38057	0.205803
$46$	0	0
$47$	−0.655082	−0.0955535	−0.0477768	−	0.998858i	$-0.515214\pi$
$47$	−0.655082	−0.0955535	−0.0477768	+	0.998858i	$0.515214\pi$
$48$	0	0
$49$	12.3300	1.76143
$50$	0	0
$51$	−10.2007	−1.42838
$52$	0	0
$53$	−1.90404	−0.261540	−0.130770	−	0.991413i	$-0.541745\pi$
$53$	−1.90404	−0.261540	−0.130770	+	0.991413i	$0.541745\pi$
$54$	0	0
$55$	−2.39030	−0.322308
$56$	0	0
$57$	−7.89815	−1.04614
$58$	0	0
$59$	−11.5584	−1.50478	−0.752391	−	0.658717i	$-0.771099\pi$
$59$	−11.5584	−1.50478	−0.752391	+	0.658717i	$0.771099\pi$
$60$	0	0
$61$	−4.56217	−0.584126	−0.292063	−	0.956399i	$-0.594342\pi$
$61$	−4.56217	−0.584126	−0.292063	+	0.956399i	$0.594342\pi$
$62$	0	0
$63$	−8.52042	−1.07347
$64$	0	0
$65$	−1.80315	−0.223653
$66$	0	0
$67$	−13.5959	−1.66100	−0.830499	−	0.557020i	$-0.811944\pi$
$67$	−13.5959	−1.66100	−0.830499	+	0.557020i	$0.811944\pi$
$68$	0	0
$69$	−5.67999	−0.683790
$70$	0	0
$71$	−9.44418	−1.12082	−0.560409	−	0.828216i	$-0.689356\pi$
$71$	−9.44418	−1.12082	−0.560409	+	0.828216i	$0.689356\pi$
$72$	0	0
$73$	−8.94150	−1.04652	−0.523262	−	0.852172i	$-0.675285\pi$
$73$	−8.94150	−1.04652	−0.523262	+	0.852172i	$0.675285\pi$
$74$	0	0
$75$	9.98305	1.15274
$76$	0	0
$77$	14.7522	1.68117
$78$	0	0
$79$	−6.83371	−0.768852	−0.384426	−	0.923156i	$-0.625601\pi$
$79$	−6.83371	−0.768852	−0.384426	+	0.923156i	$0.625601\pi$
$80$	0	0
$81$	−11.0582	−1.22869
$82$	0	0
$83$	−13.3811	−1.46877	−0.734385	−	0.678733i	$-0.762530\pi$
$83$	−13.3811	−1.46877	−0.734385	+	0.678733i	$0.762530\pi$
$84$	0	0
$85$	3.27016	0.354698
$86$	0	0
$87$	−11.4400	−1.22650
$88$	0	0
$89$	−15.4078	−1.63323	−0.816614	−	0.577185i	$-0.804151\pi$
$89$	−15.4078	−1.63323	−0.816614	+	0.577185i	$0.804151\pi$
$90$	0	0
$91$	11.1284	1.16658
$92$	0	0
$93$	8.29724	0.860384
$94$	0	0
$95$	2.53200	0.259778
$96$	0	0
$97$	6.59022	0.669135	0.334568	−	0.942372i	$-0.391410\pi$
$97$	6.59022	0.669135	0.334568	+	0.942372i	$0.391410\pi$
$98$	0	0
$99$	−6.50257	−0.653533
$100$	0	0
$101$	−8.18518	−0.814455	−0.407228	−	0.913327i	$-0.633504\pi$
$101$	−8.18518	−0.814455	−0.407228	+	0.913327i	$0.633504\pi$
$102$	0	0
$103$	0.986016	0.0971550	0.0485775	−	0.998819i	$-0.484531\pi$
$103$	0.986016	0.0971550	0.0485775	+	0.998819i	$0.484531\pi$
$104$	0	0
$105$	6.95989	0.679216
$106$	0	0
$107$	13.5258	1.30758	0.653792	−	0.756674i	$-0.273177\pi$
$107$	13.5258	1.30758	0.653792	+	0.756674i	$0.273177\pi$
$108$	0	0
$109$	12.4417	1.19169	0.595847	−	0.803098i	$-0.296816\pi$
$109$	12.4417	1.19169	0.595847	+	0.803098i	$0.296816\pi$
$110$	0	0
$111$	23.2332	2.20520
$112$	0	0
$113$	−6.83528	−0.643009	−0.321505	−	0.946908i	$-0.604189\pi$
$113$	−6.83528	−0.643009	−0.321505	+	0.946908i	$0.604189\pi$
$114$	0	0
$115$	1.82090	0.169800
$116$	0	0
$117$	−4.90527	−0.453493
$118$	0	0
$119$	−20.1824	−1.85011
$120$	0	0
$121$	0.258495	0.0234995
$122$	0	0
$123$	14.2863	1.28816
$124$	0	0
$125$	−6.76229	−0.604838
$126$	0	0
$127$	8.02668	0.712253	0.356126	−	0.934438i	$-0.384097\pi$
$127$	8.02668	0.712253	0.356126	+	0.934438i	$0.384097\pi$
$128$	0	0
$129$	−5.31748	−0.468178
$130$	0	0
$131$	7.45795	0.651604	0.325802	−	0.945438i	$-0.394366\pi$
$131$	7.45795	0.651604	0.325802	+	0.945438i	$0.394366\pi$
$132$	0	0
$133$	−15.6267	−1.35501
$134$	0	0
$135$	1.68123	0.144697
$136$	0	0
$137$	5.26530	0.449845	0.224922	−	0.974377i	$-0.427787\pi$
$137$	5.26530	0.449845	0.224922	+	0.974377i	$0.427787\pi$
$138$	0	0
$139$	16.6957	1.41611	0.708056	−	0.706157i	$-0.249573\pi$
$139$	16.6957	1.41611	0.708056	+	0.706157i	$0.249573\pi$
$140$	0	0
$141$	1.45569	0.122591
$142$	0	0
$143$	8.49294	0.710216
$144$	0	0
$145$	3.66746	0.304566
$146$	0	0
$147$	−27.3992	−2.25984
$148$	0	0
$149$	−7.73867	−0.633976	−0.316988	−	0.948430i	$-0.602672\pi$
$149$	−7.73867	−0.633976	−0.316988	+	0.948430i	$0.602672\pi$
$150$	0	0
$151$	20.1661	1.64109	0.820547	−	0.571579i	$-0.193669\pi$
$151$	20.1661	1.64109	0.820547	+	0.571579i	$0.193669\pi$
$152$	0	0
$153$	8.89612	0.719209
$154$	0	0
$155$	−2.65994	−0.213652
$156$	0	0
$157$	23.0377	1.83861	0.919304	−	0.393548i	$-0.128752\pi$
$157$	23.0377	1.83861	0.919304	+	0.393548i	$0.128752\pi$
$158$	0	0
$159$	4.23106	0.335545
$160$	0	0
$161$	−11.2380	−0.885680
$162$	0	0
$163$	4.87315	0.381694	0.190847	−	0.981620i	$-0.438876\pi$
$163$	4.87315	0.381694	0.190847	+	0.981620i	$0.438876\pi$
$164$	0	0
$165$	5.31161	0.413509
$166$	0	0
$167$	−4.67914	−0.362083	−0.181041	−	0.983475i	$-0.557947\pi$
$167$	−4.67914	−0.362083	−0.181041	+	0.983475i	$0.557947\pi$
$168$	0	0
$169$	−6.59327	−0.507175
$170$	0	0
$171$	6.88805	0.526742
$172$	0	0
$173$	−8.50196	−0.646392	−0.323196	−	0.946332i	$-0.604757\pi$
$173$	−8.50196	−0.646392	−0.323196	+	0.946332i	$0.604757\pi$
$174$	0	0
$175$	19.7517	1.49309
$176$	0	0
$177$	25.6846	1.93057
$178$	0	0
$179$	−8.80118	−0.657831	−0.328916	−	0.944359i	$-0.606683\pi$
$179$	−8.80118	−0.657831	−0.328916	+	0.944359i	$0.606683\pi$
$180$	0	0
$181$	5.64971	0.419940	0.209970	−	0.977708i	$-0.432663\pi$
$181$	5.64971	0.419940	0.209970	+	0.977708i	$0.432663\pi$
$182$	0	0
$183$	10.1378	0.749410
$184$	0	0
$185$	−7.44813	−0.547597
$186$	0	0
$187$	−15.4027	−1.12635
$188$	0	0
$189$	−10.3760	−0.754744
$190$	0	0
$191$	−2.05404	−0.148625	−0.0743124	−	0.997235i	$-0.523676\pi$
$191$	−2.05404	−0.148625	−0.0743124	+	0.997235i	$0.523676\pi$
$192$	0	0
$193$	−18.3204	−1.31873	−0.659366	−	0.751822i	$-0.729175\pi$
$193$	−18.3204	−1.31873	−0.659366	+	0.751822i	$0.729175\pi$
$194$	0	0
$195$	4.00686	0.286937
$196$	0	0
$197$	22.2367	1.58430	0.792149	−	0.610328i	$-0.208962\pi$
$197$	22.2367	1.58430	0.792149	+	0.610328i	$0.208962\pi$
$198$	0	0
$199$	−3.69903	−0.262217	−0.131109	−	0.991368i	$-0.541854\pi$
$199$	−3.69903	−0.262217	−0.131109	+	0.991368i	$0.541854\pi$
$200$	0	0
$201$	30.2120	2.13099
$202$	0	0
$203$	−22.6344	−1.58862
$204$	0	0
$205$	−4.57994	−0.319877
$206$	0	0
$207$	4.95357	0.344297
$208$	0	0
$209$	−11.9259	−0.824933
$210$	0	0
$211$	−17.8259	−1.22719	−0.613593	−	0.789623i	$-0.710276\pi$
$211$	−17.8259	−1.22719	−0.613593	+	0.789623i	$0.710276\pi$
$212$	0	0
$213$	20.9864	1.43797
$214$	0	0
$215$	1.70469	0.116259
$216$	0	0
$217$	16.4163	1.11441
$218$	0	0
$219$	19.8694	1.34265
$220$	0	0
$221$	−11.6191	−0.781588
$222$	0	0
$223$	18.5405	1.24157	0.620783	−	0.783982i	$-0.286815\pi$
$223$	18.5405	1.24157	0.620783	+	0.783982i	$0.286815\pi$
$224$	0	0
$225$	−8.70631	−0.580421
$226$	0	0
$227$	1.75864	0.116725	0.0583625	−	0.998295i	$-0.481412\pi$
$227$	1.75864	0.116725	0.0583625	+	0.998295i	$0.481412\pi$
$228$	0	0
$229$	−7.73042	−0.510840	−0.255420	−	0.966830i	$-0.582214\pi$
$229$	−7.73042	−0.510840	−0.255420	+	0.966830i	$0.582214\pi$
$230$	0	0
$231$	−32.7816	−2.15687
$232$	0	0
$233$	22.8019	1.49380	0.746902	−	0.664934i	$-0.231540\pi$
$233$	22.8019	1.49380	0.746902	+	0.664934i	$0.231540\pi$
$234$	0	0
$235$	−0.466668	−0.0304421
$236$	0	0
$237$	15.1855	0.986407
$238$	0	0
$239$	14.1169	0.913149	0.456575	−	0.889685i	$-0.349076\pi$
$239$	14.1169	0.913149	0.456575	+	0.889685i	$0.349076\pi$
$240$	0	0
$241$	−4.96966	−0.320124	−0.160062	−	0.987107i	$-0.551169\pi$
$241$	−4.96966	−0.320124	−0.160062	+	0.987107i	$0.551169\pi$
$242$	0	0
$243$	17.4929	1.12217
$244$	0	0
$245$	8.78367	0.561168
$246$	0	0
$247$	−8.99642	−0.572428
$248$	0	0
$249$	29.7349	1.88437
$250$	0	0
$251$	−6.31919	−0.398864	−0.199432	−	0.979912i	$-0.563910\pi$
$251$	−6.31919	−0.398864	−0.199432	+	0.979912i	$0.563910\pi$
$252$	0	0
$253$	−8.57657	−0.539205
$254$	0	0
$255$	−7.26678	−0.455064
$256$	0	0
$257$	16.1829	1.00946	0.504730	−	0.863277i	$-0.331592\pi$
$257$	16.1829	1.00946	0.504730	+	0.863277i	$0.331592\pi$
$258$	0	0
$259$	45.9675	2.85628
$260$	0	0
$261$	9.97694	0.617557
$262$	0	0
$263$	17.0403	1.05075	0.525375	−	0.850871i	$-0.323925\pi$
$263$	17.0403	1.05075	0.525375	+	0.850871i	$0.323925\pi$
$264$	0	0
$265$	−1.35640	−0.0833230
$266$	0	0
$267$	34.2386	2.09537
$268$	0	0
$269$	14.8919	0.907975	0.453988	−	0.891008i	$-0.350001\pi$
$269$	14.8919	0.907975	0.453988	+	0.891008i	$0.350001\pi$
$270$	0	0
$271$	0.717096	0.0435605	0.0217802	−	0.999763i	$-0.493067\pi$
$271$	0.717096	0.0435605	0.0217802	+	0.999763i	$0.493067\pi$
$272$	0	0
$273$	−24.7291	−1.49667
$274$	0	0
$275$	15.0740	0.908998
$276$	0	0
$277$	−9.12809	−0.548454	−0.274227	−	0.961665i	$-0.588422\pi$
$277$	−9.12809	−0.548454	−0.274227	+	0.961665i	$0.588422\pi$
$278$	0	0
$279$	−7.23610	−0.433214
$280$	0	0
$281$	−26.4024	−1.57504	−0.787519	−	0.616291i	$-0.788635\pi$
$281$	−26.4024	−1.57504	−0.787519	+	0.616291i	$0.788635\pi$
$282$	0	0
$283$	−3.99826	−0.237672	−0.118836	−	0.992914i	$-0.537916\pi$
$283$	−3.99826	−0.237672	−0.118836	+	0.992914i	$0.537916\pi$
$284$	0	0
$285$	−5.62649	−0.333285
$286$	0	0
$287$	28.2659	1.66849
$288$	0	0
$289$	4.07228	0.239546
$290$	0	0
$291$	−14.6445	−0.858474
$292$	0	0
$293$	9.63049	0.562619	0.281309	−	0.959617i	$-0.409231\pi$
$293$	9.63049	0.562619	0.281309	+	0.959617i	$0.409231\pi$
$294$	0	0
$295$	−8.23402	−0.479403
$296$	0	0
$297$	−7.91871	−0.459490
$298$	0	0
$299$	−6.46981	−0.374159
$300$	0	0
$301$	−10.5208	−0.606408
$302$	0	0
$303$	18.1887	1.04491
$304$	0	0
$305$	−3.25000	−0.186095
$306$	0	0
$307$	33.0818	1.88808	0.944040	−	0.329832i	$-0.106992\pi$
$307$	33.0818	1.88808	0.944040	+	0.329832i	$0.106992\pi$
$308$	0	0
$309$	−2.19108	−0.124646
$310$	0	0
$311$	−18.0798	−1.02521	−0.512605	−	0.858624i	$-0.671320\pi$
$311$	−18.0798	−1.02521	−0.512605	+	0.858624i	$0.671320\pi$
$312$	0	0
$313$	10.3661	0.585927	0.292963	−	0.956124i	$-0.405359\pi$
$313$	10.3661	0.585927	0.292963	+	0.956124i	$0.405359\pi$
$314$	0	0
$315$	−6.06979	−0.341994
$316$	0	0
$317$	3.76302	0.211352	0.105676	−	0.994401i	$-0.466299\pi$
$317$	3.76302	0.211352	0.105676	+	0.994401i	$0.466299\pi$
$318$	0	0
$319$	−17.2740	−0.967158
$320$	0	0
$321$	−30.0563	−1.67758
$322$	0	0
$323$	16.3158	0.907833
$324$	0	0
$325$	11.3712	0.630762
$326$	0	0
$327$	−27.6472	−1.52890
$328$	0	0
$329$	2.88013	0.158787
$330$	0	0
$331$	13.0196	0.715623	0.357811	−	0.933794i	$-0.383523\pi$
$331$	13.0196	0.715623	0.357811	+	0.933794i	$0.383523\pi$
$332$	0	0
$333$	−20.2619	−1.11034
$334$	0	0
$335$	−9.68543	−0.529172
$336$	0	0
$337$	27.1502	1.47897	0.739483	−	0.673175i	$-0.235070\pi$
$337$	27.1502	1.47897	0.739483	+	0.673175i	$0.235070\pi$
$338$	0	0
$339$	15.1890	0.824955
$340$	0	0
$341$	12.5285	0.678458
$342$	0	0
$343$	−23.4339	−1.26531
$344$	0	0
$345$	−4.04632	−0.217846
$346$	0	0
$347$	−36.7795	−1.97443	−0.987213	−	0.159405i	$-0.949042\pi$
$347$	−36.7795	−1.97443	−0.987213	+	0.159405i	$0.949042\pi$
$348$	0	0
$349$	28.0449	1.50121	0.750606	−	0.660750i	$-0.229762\pi$
$349$	28.0449	1.50121	0.750606	+	0.660750i	$0.229762\pi$
$350$	0	0
$351$	−5.97355	−0.318845
$352$	0	0
$353$	−14.9368	−0.795005	−0.397502	−	0.917601i	$-0.630123\pi$
$353$	−14.9368	−0.795005	−0.397502	+	0.917601i	$0.630123\pi$
$354$	0	0
$355$	−6.72786	−0.357078
$356$	0	0
$357$	44.8483	2.37362
$358$	0	0
$359$	19.5526	1.03195	0.515973	−	0.856605i	$-0.327430\pi$
$359$	19.5526	1.03195	0.515973	+	0.856605i	$0.327430\pi$
$360$	0	0
$361$	−6.36711	−0.335111
$362$	0	0
$363$	−0.574415	−0.0301490
$364$	0	0
$365$	−6.36976	−0.333408
$366$	0	0
$367$	−9.76632	−0.509798	−0.254899	−	0.966968i	$-0.582042\pi$
$367$	−9.76632	−0.509798	−0.254899	+	0.966968i	$0.582042\pi$
$368$	0	0
$369$	−12.4593	−0.648603
$370$	0	0
$371$	8.37127	0.434615
$372$	0	0
$373$	−14.7414	−0.763280	−0.381640	−	0.924311i	$-0.624641\pi$
$373$	−14.7414	−0.763280	−0.381640	+	0.924311i	$0.624641\pi$
$374$	0	0
$375$	15.0268	0.775983
$376$	0	0
$377$	−13.0308	−0.671120
$378$	0	0
$379$	−11.0837	−0.569329	−0.284665	−	0.958627i	$-0.591882\pi$
$379$	−11.0837	−0.569329	−0.284665	+	0.958627i	$0.591882\pi$
$380$	0	0
$381$	−17.8365	−0.913792
$382$	0	0
$383$	−31.2011	−1.59430	−0.797151	−	0.603780i	$-0.793660\pi$
$383$	−31.2011	−1.59430	−0.797151	+	0.603780i	$0.793660\pi$
$384$	0	0
$385$	10.5092	0.535597
$386$	0	0
$387$	4.63743	0.235734
$388$	0	0
$389$	18.7044	0.948350	0.474175	−	0.880431i	$-0.342746\pi$
$389$	18.7044	0.948350	0.474175	+	0.880431i	$0.342746\pi$
$390$	0	0
$391$	11.7336	0.593391
$392$	0	0
$393$	−16.5727	−0.835983
$394$	0	0
$395$	−4.86821	−0.244946
$396$	0	0
$397$	9.10573	0.457003	0.228502	−	0.973544i	$-0.426617\pi$
$397$	9.10573	0.457003	0.228502	+	0.973544i	$0.426617\pi$
$398$	0	0
$399$	34.7249	1.73842
$400$	0	0
$401$	−26.3060	−1.31366	−0.656831	−	0.754038i	$-0.728103\pi$
$401$	−26.3060	−1.31366	−0.656831	+	0.754038i	$0.728103\pi$
$402$	0	0
$403$	9.45100	0.470788
$404$	0	0
$405$	−7.87765	−0.391444
$406$	0	0
$407$	35.0812	1.73891
$408$	0	0
$409$	−30.1487	−1.49076	−0.745378	−	0.666642i	$-0.767731\pi$
$409$	−30.1487	−1.49076	−0.745378	+	0.666642i	$0.767731\pi$
$410$	0	0
$411$	−11.7003	−0.577133
$412$	0	0
$413$	50.8178	2.50058
$414$	0	0
$415$	−9.53248	−0.467931
$416$	0	0
$417$	−37.1004	−1.81681
$418$	0	0
$419$	19.2797	0.941875	0.470937	−	0.882167i	$-0.343916\pi$
$419$	19.2797	0.941875	0.470937	+	0.882167i	$0.343916\pi$
$420$	0	0
$421$	−22.0329	−1.07382	−0.536910	−	0.843639i	$-0.680409\pi$
$421$	−22.0329	−1.07382	−0.536910	+	0.843639i	$0.680409\pi$
$422$	0	0
$423$	−1.26952	−0.0617263
$424$	0	0
$425$	−20.6227	−1.00035
$426$	0	0
$427$	20.0580	0.970674
$428$	0	0
$429$	−18.8726	−0.911178
$430$	0	0
$431$	14.5238	0.699585	0.349793	−	0.936827i	$-0.386252\pi$
$431$	14.5238	0.699585	0.349793	+	0.936827i	$0.386252\pi$
$432$	0	0
$433$	−14.0023	−0.672908	−0.336454	−	0.941700i	$-0.609228\pi$
$433$	−14.0023	−0.672908	−0.336454	+	0.941700i	$0.609228\pi$
$434$	0	0
$435$	−8.14965	−0.390746
$436$	0	0
$437$	9.08500	0.434595
$438$	0	0
$439$	−36.3096	−1.73296	−0.866482	−	0.499208i	$-0.833624\pi$
$439$	−36.3096	−1.73296	−0.866482	+	0.499208i	$0.833624\pi$
$440$	0	0
$441$	23.8951	1.13786
$442$	0	0
$443$	14.0802	0.668972	0.334486	−	0.942401i	$-0.391437\pi$
$443$	14.0802	0.668972	0.334486	+	0.942401i	$0.391437\pi$
$444$	0	0
$445$	−10.9763	−0.520324
$446$	0	0
$447$	17.1965	0.813366
$448$	0	0
$449$	0.723191	0.0341295	0.0170647	−	0.999854i	$-0.494568\pi$
$449$	0.723191	0.0341295	0.0170647	+	0.999854i	$0.494568\pi$
$450$	0	0
$451$	21.5719	1.01578
$452$	0	0
$453$	−44.8121	−2.10546
$454$	0	0
$455$	7.92769	0.371656
$456$	0	0
$457$	25.8037	1.20705	0.603523	−	0.797346i	$-0.293763\pi$
$457$	25.8037	1.20705	0.603523	+	0.797346i	$0.293763\pi$
$458$	0	0
$459$	10.8335	0.505666
$460$	0	0
$461$	14.5026	0.675453	0.337726	−	0.941244i	$-0.390342\pi$
$461$	14.5026	0.675453	0.337726	+	0.941244i	$0.390342\pi$
$462$	0	0
$463$	−36.0906	−1.67727	−0.838636	−	0.544693i	$-0.816646\pi$
$463$	−36.0906	−1.67727	−0.838636	+	0.544693i	$0.816646\pi$
$464$	0	0
$465$	5.91080	0.274107
$466$	0	0
$467$	−35.9415	−1.66318	−0.831588	−	0.555393i	$-0.812568\pi$
$467$	−35.9415	−1.66318	−0.831588	+	0.555393i	$0.812568\pi$
$468$	0	0
$469$	59.7754	2.76017
$470$	0	0
$471$	−51.1933	−2.35886
$472$	0	0
$473$	−8.02920	−0.369183
$474$	0	0
$475$	−15.9676	−0.732646
$476$	0	0
$477$	−3.68995	−0.168951
$478$	0	0
$479$	31.8677	1.45607	0.728037	−	0.685538i	$-0.240433\pi$
$479$	31.8677	1.45607	0.728037	+	0.685538i	$0.240433\pi$
$480$	0	0
$481$	26.4638	1.20665
$482$	0	0
$483$	24.9726	1.13629
$484$	0	0
$485$	4.69475	0.213177
$486$	0	0
$487$	−12.6821	−0.574681	−0.287341	−	0.957828i	$-0.592771\pi$
$487$	−12.6821	−0.574681	−0.287341	+	0.957828i	$0.592771\pi$
$488$	0	0
$489$	−10.8289	−0.489699
$490$	0	0
$491$	−33.7881	−1.52483	−0.762417	−	0.647086i	$-0.775987\pi$
$491$	−33.7881	−1.52483	−0.762417	+	0.647086i	$0.775987\pi$
$492$	0	0
$493$	23.6324	1.06435
$494$	0	0
$495$	−4.63231	−0.208207
$496$	0	0
$497$	41.5222	1.86253
$498$	0	0
$499$	15.2141	0.681077	0.340538	−	0.940231i	$-0.389391\pi$
$499$	15.2141	0.681077	0.340538	+	0.940231i	$0.389391\pi$
$500$	0	0
$501$	10.3978	0.464538
$502$	0	0
$503$	−18.6115	−0.829845	−0.414923	−	0.909857i	$-0.636191\pi$
$503$	−18.6115	−0.829845	−0.414923	+	0.909857i	$0.636191\pi$
$504$	0	0
$505$	−5.83097	−0.259475
$506$	0	0
$507$	14.6513	0.650685
$508$	0	0
$509$	−4.75744	−0.210870	−0.105435	−	0.994426i	$-0.533623\pi$
$509$	−4.75744	−0.210870	−0.105435	+	0.994426i	$0.533623\pi$
$510$	0	0
$511$	39.3121	1.73907
$512$	0	0
$513$	8.38814	0.370346
$514$	0	0
$515$	0.702419	0.0309523
$516$	0	0
$517$	2.19804	0.0966697
$518$	0	0
$519$	18.8926	0.829295
$520$	0	0
$521$	−37.2463	−1.63179	−0.815894	−	0.578202i	$-0.803755\pi$
$521$	−37.2463	−1.63179	−0.815894	+	0.578202i	$0.803755\pi$
$522$	0	0
$523$	31.6588	1.38434	0.692171	−	0.721734i	$-0.256655\pi$
$523$	31.6588	1.38434	0.692171	+	0.721734i	$0.256655\pi$
$524$	0	0
$525$	−43.8914	−1.91558
$526$	0	0
$527$	−17.1402	−0.746638
$528$	0	0
$529$	−16.4665	−0.715934
$530$	0	0
$531$	−22.3998	−0.972069
$532$	0	0
$533$	16.2729	0.704858
$534$	0	0
$535$	9.63549	0.416579
$536$	0	0
$537$	19.5576	0.843971
$538$	0	0
$539$	−41.3717	−1.78201
$540$	0	0
$541$	36.6462	1.57554	0.787771	−	0.615969i	$-0.211235\pi$
$541$	36.6462	1.57554	0.787771	+	0.615969i	$0.211235\pi$
$542$	0	0
$543$	−12.5545	−0.538766
$544$	0	0
$545$	8.86320	0.379658
$546$	0	0
$547$	−1.00000	−0.0427569
$547$	−1.00000	−0.0427569
$548$	0	0
$549$	−8.84130	−0.377337
$550$	0	0
$551$	18.2980	0.779522
$552$	0	0
$553$	30.0450	1.27764
$554$	0	0
$555$	16.5509	0.702546
$556$	0	0
$557$	−20.3359	−0.861660	−0.430830	−	0.902433i	$-0.641779\pi$
$557$	−20.3359	−0.861660	−0.430830	+	0.902433i	$0.641779\pi$
$558$	0	0
$559$	−6.05690	−0.256179
$560$	0	0
$561$	34.2271	1.44507
$562$	0	0
$563$	4.03557	0.170079	0.0850394	−	0.996378i	$-0.472898\pi$
$563$	4.03557	0.170079	0.0850394	+	0.996378i	$0.472898\pi$
$564$	0	0
$565$	−4.86933	−0.204854
$566$	0	0
$567$	48.6183	2.04178
$568$	0	0
$569$	1.98791	0.0833376	0.0416688	−	0.999131i	$-0.486733\pi$
$569$	1.98791	0.0833376	0.0416688	+	0.999131i	$0.486733\pi$
$570$	0	0
$571$	30.1856	1.26323	0.631614	−	0.775283i	$-0.282393\pi$
$571$	30.1856	1.26323	0.631614	+	0.775283i	$0.282393\pi$
$572$	0	0
$573$	4.56438	0.190680
$574$	0	0
$575$	−11.4832	−0.478883
$576$	0	0
$577$	31.1795	1.29802	0.649010	−	0.760780i	$-0.275183\pi$
$577$	31.1795	1.29802	0.649010	+	0.760780i	$0.275183\pi$
$578$	0	0
$579$	40.7108	1.69188
$580$	0	0
$581$	58.8314	2.44074
$582$	0	0
$583$	6.38875	0.264595
$584$	0	0
$585$	−3.49442	−0.144477
$586$	0	0
$587$	29.3436	1.21114	0.605571	−	0.795792i	$-0.292945\pi$
$587$	29.3436	1.21114	0.605571	+	0.795792i	$0.292945\pi$
$588$	0	0
$589$	−13.2712	−0.546832
$590$	0	0
$591$	−49.4133	−2.03259
$592$	0	0
$593$	−5.23773	−0.215088	−0.107544	−	0.994200i	$-0.534299\pi$
$593$	−5.23773	−0.215088	−0.107544	+	0.994200i	$0.534299\pi$
$594$	0	0
$595$	−14.3775	−0.589421
$596$	0	0
$597$	8.21981	0.336414
$598$	0	0
$599$	13.9395	0.569554	0.284777	−	0.958594i	$-0.408080\pi$
$599$	13.9395	0.569554	0.284777	+	0.958594i	$0.408080\pi$
$600$	0	0
$601$	−17.3985	−0.709701	−0.354851	−	0.934923i	$-0.615468\pi$
$601$	−17.3985	−0.709701	−0.354851	+	0.934923i	$0.615468\pi$
$602$	0	0
$603$	−26.3482	−1.07298
$604$	0	0
$605$	0.184147	0.00748664
$606$	0	0
$607$	−5.13339	−0.208358	−0.104179	−	0.994559i	$-0.533221\pi$
$607$	−5.13339	−0.208358	−0.104179	+	0.994559i	$0.533221\pi$
$608$	0	0
$609$	50.2970	2.03814
$610$	0	0
$611$	1.65811	0.0670800
$612$	0	0
$613$	8.90668	0.359737	0.179869	−	0.983691i	$-0.442433\pi$
$613$	8.90668	0.359737	0.179869	+	0.983691i	$0.442433\pi$
$614$	0	0
$615$	10.1773	0.410389
$616$	0	0
$617$	22.7075	0.914171	0.457085	−	0.889423i	$-0.348893\pi$
$617$	22.7075	0.914171	0.457085	+	0.889423i	$0.348893\pi$
$618$	0	0
$619$	18.2517	0.733600	0.366800	−	0.930300i	$-0.380453\pi$
$619$	18.2517	0.733600	0.366800	+	0.930300i	$0.380453\pi$
$620$	0	0
$621$	6.03237	0.242071
$622$	0	0
$623$	67.7419	2.71402
$624$	0	0
$625$	17.6452	0.705809
$626$	0	0
$627$	26.5012	1.05836
$628$	0	0
$629$	−47.9944	−1.91366
$630$	0	0
$631$	−23.0859	−0.919037	−0.459518	−	0.888168i	$-0.651978\pi$
$631$	−23.0859	−0.919037	−0.459518	+	0.888168i	$0.651978\pi$
$632$	0	0
$633$	39.6119	1.57443
$634$	0	0
$635$	5.71806	0.226914
$636$	0	0
$637$	−31.2091	−1.23655
$638$	0	0
$639$	−18.3025	−0.724034
$640$	0	0
$641$	−5.14884	−0.203367	−0.101683	−	0.994817i	$-0.532423\pi$
$641$	−5.14884	−0.203367	−0.101683	+	0.994817i	$0.532423\pi$
$642$	0	0
$643$	−25.7206	−1.01432	−0.507160	−	0.861852i	$-0.669305\pi$
$643$	−25.7206	−1.01432	−0.507160	+	0.861852i	$0.669305\pi$
$644$	0	0
$645$	−3.78807	−0.149155
$646$	0	0
$647$	−5.81471	−0.228600	−0.114300	−	0.993446i	$-0.536463\pi$
$647$	−5.81471	−0.228600	−0.114300	+	0.993446i	$0.536463\pi$
$648$	0	0
$649$	38.7828	1.52236
$650$	0	0
$651$	−36.4796	−1.42975
$652$	0	0
$653$	15.9057	0.622436	0.311218	−	0.950339i	$-0.399263\pi$
$653$	15.9057	0.622436	0.311218	+	0.950339i	$0.399263\pi$
$654$	0	0
$655$	5.31291	0.207592
$656$	0	0
$657$	−17.3283	−0.676040
$658$	0	0
$659$	30.7546	1.19803	0.599014	−	0.800739i	$-0.295560\pi$
$659$	30.7546	1.19803	0.599014	+	0.800739i	$0.295560\pi$
$660$	0	0
$661$	−31.9625	−1.24320	−0.621598	−	0.783336i	$-0.713516\pi$
$661$	−31.9625	−1.24320	−0.621598	+	0.783336i	$0.713516\pi$
$662$	0	0
$663$	25.8195	1.00275
$664$	0	0
$665$	−11.1322	−0.431687
$666$	0	0
$667$	13.1591	0.509522
$668$	0	0
$669$	−41.1999	−1.59288
$670$	0	0
$671$	15.3078	0.590949
$672$	0	0
$673$	5.89825	0.227361	0.113680	−	0.993517i	$-0.463736\pi$
$673$	5.89825	0.227361	0.113680	+	0.993517i	$0.463736\pi$
$674$	0	0
$675$	−10.6024	−0.408086
$676$	0	0
$677$	−32.3269	−1.24243	−0.621213	−	0.783642i	$-0.713360\pi$
$677$	−32.3269	−1.24243	−0.621213	+	0.783642i	$0.713360\pi$
$678$	0	0
$679$	−28.9745	−1.11194
$680$	0	0
$681$	−3.90796	−0.149753
$682$	0	0
$683$	45.4717	1.73993	0.869963	−	0.493117i	$-0.164143\pi$
$683$	45.4717	1.73993	0.869963	+	0.493117i	$0.164143\pi$
$684$	0	0
$685$	3.75090	0.143314
$686$	0	0
$687$	17.1782	0.655388
$688$	0	0
$689$	4.81941	0.183605
$690$	0	0
$691$	17.8493	0.679021	0.339510	−	0.940602i	$-0.389739\pi$
$691$	17.8493	0.679021	0.339510	+	0.940602i	$0.389739\pi$
$692$	0	0
$693$	28.5891	1.08601
$694$	0	0
$695$	11.8937	0.451154
$696$	0	0
$697$	−29.5123	−1.11786
$698$	0	0
$699$	−50.6694	−1.91649
$700$	0	0
$701$	22.0209	0.831717	0.415858	−	0.909429i	$-0.363481\pi$
$701$	22.0209	0.831717	0.415858	+	0.909429i	$0.363481\pi$
$702$	0	0
$703$	−37.1609	−1.40155
$704$	0	0
$705$	1.03701	0.0390560
$706$	0	0
$707$	35.9869	1.35343
$708$	0	0
$709$	47.4912	1.78357	0.891785	−	0.452459i	$-0.149453\pi$
$709$	47.4912	1.78357	0.891785	+	0.452459i	$0.149453\pi$
$710$	0	0
$711$	−13.2435	−0.496668
$712$	0	0
$713$	−9.54407	−0.357428
$714$	0	0
$715$	6.05022	0.226265
$716$	0	0
$717$	−31.3700	−1.17153
$718$	0	0
$719$	−17.4092	−0.649253	−0.324626	−	0.945842i	$-0.605239\pi$
$719$	−17.4092	−0.649253	−0.324626	+	0.945842i	$0.605239\pi$
$720$	0	0
$721$	−4.33511	−0.161448
$722$	0	0
$723$	11.0433	0.410707
$724$	0	0
$725$	−23.1282	−0.858960
$726$	0	0
$727$	−13.0030	−0.482254	−0.241127	−	0.970494i	$-0.577517\pi$
$727$	−13.0030	−0.482254	−0.241127	+	0.970494i	$0.577517\pi$
$728$	0	0
$729$	−5.69741	−0.211015
$730$	0	0
$731$	10.9847	0.406283
$732$	0	0
$733$	−6.25826	−0.231154	−0.115577	−	0.993299i	$-0.536872\pi$
$733$	−6.25826	−0.231154	−0.115577	+	0.993299i	$0.536872\pi$
$734$	0	0
$735$	−19.5187	−0.719956
$736$	0	0
$737$	45.6191	1.68040
$738$	0	0
$739$	−44.0177	−1.61922	−0.809608	−	0.586971i	$-0.800320\pi$
$739$	−44.0177	−1.61922	−0.809608	+	0.586971i	$0.800320\pi$
$740$	0	0
$741$	19.9914	0.734403
$742$	0	0
$743$	38.5408	1.41392	0.706962	−	0.707251i	$-0.250065\pi$
$743$	38.5408	1.41392	0.706962	+	0.707251i	$0.250065\pi$
$744$	0	0
$745$	−5.51288	−0.201976
$746$	0	0
$747$	−25.9321	−0.948807
$748$	0	0
$749$	−59.4672	−2.17288
$750$	0	0
$751$	32.4423	1.18383	0.591917	−	0.805999i	$-0.298371\pi$
$751$	32.4423	1.18383	0.591917	+	0.805999i	$0.298371\pi$
$752$	0	0
$753$	14.0422	0.511726
$754$	0	0
$755$	14.3660	0.522831
$756$	0	0
$757$	12.2641	0.445745	0.222873	−	0.974848i	$-0.428457\pi$
$757$	12.2641	0.445745	0.222873	+	0.974848i	$0.428457\pi$
$758$	0	0
$759$	19.0585	0.691778
$760$	0	0
$761$	21.1292	0.765932	0.382966	−	0.923762i	$-0.374903\pi$
$761$	21.1292	0.765932	0.382966	+	0.923762i	$0.374903\pi$
$762$	0	0
$763$	−54.7009	−1.98030
$764$	0	0
$765$	6.33743	0.229130
$766$	0	0
$767$	29.2562	1.05638
$768$	0	0
$769$	12.5981	0.454300	0.227150	−	0.973860i	$-0.427059\pi$
$769$	12.5981	0.454300	0.227150	+	0.973860i	$0.427059\pi$
$770$	0	0
$771$	−35.9608	−1.29510
$772$	0	0
$773$	17.2688	0.621116	0.310558	−	0.950554i	$-0.399484\pi$
$773$	17.2688	0.621116	0.310558	+	0.950554i	$0.399484\pi$
$774$	0	0
$775$	16.7745	0.602557
$776$	0	0
$777$	−102.147	−3.66449
$778$	0	0
$779$	−22.8507	−0.818710
$780$	0	0
$781$	31.6887	1.13391
$782$	0	0
$783$	12.1497	0.434196
$784$	0	0
$785$	16.4116	0.585756
$786$	0	0
$787$	44.8720	1.59951	0.799756	−	0.600325i	$-0.204962\pi$
$787$	44.8720	1.59951	0.799756	+	0.600325i	$0.204962\pi$
$788$	0	0
$789$	−37.8662	−1.34807
$790$	0	0
$791$	30.0519	1.06852
$792$	0	0
$793$	11.5475	0.410065
$794$	0	0
$795$	3.01413	0.106900
$796$	0	0
$797$	30.5278	1.08135	0.540675	−	0.841232i	$-0.318169\pi$
$797$	30.5278	1.08135	0.540675	+	0.841232i	$0.318169\pi$
$798$	0	0
$799$	−3.00713	−0.106384
$800$	0	0
$801$	−29.8598	−1.05504
$802$	0	0
$803$	30.0020	1.05875
$804$	0	0
$805$	−8.00575	−0.282166
$806$	0	0
$807$	−33.0921	−1.16490
$808$	0	0
$809$	−33.9180	−1.19249	−0.596247	−	0.802801i	$-0.703342\pi$
$809$	−33.9180	−1.19249	−0.596247	+	0.802801i	$0.703342\pi$
$810$	0	0
$811$	43.9741	1.54414	0.772069	−	0.635539i	$-0.219222\pi$
$811$	43.9741	1.54414	0.772069	+	0.635539i	$0.219222\pi$
$812$	0	0
$813$	−1.59350	−0.0558864
$814$	0	0
$815$	3.47154	0.121603
$816$	0	0
$817$	8.50518	0.297559
$818$	0	0
$819$	21.5665	0.753594
$820$	0	0
$821$	50.2569	1.75398	0.876989	−	0.480510i	$-0.159548\pi$
$821$	50.2569	1.75398	0.876989	+	0.480510i	$0.159548\pi$
$822$	0	0
$823$	−45.2402	−1.57697	−0.788487	−	0.615051i	$-0.789135\pi$
$823$	−45.2402	−1.57697	−0.788487	+	0.615051i	$0.789135\pi$
$824$	0	0
$825$	−33.4968	−1.16621
$826$	0	0
$827$	1.56540	0.0544343	0.0272172	−	0.999630i	$-0.491335\pi$
$827$	1.56540	0.0544343	0.0272172	+	0.999630i	$0.491335\pi$
$828$	0	0
$829$	2.28756	0.0794504	0.0397252	−	0.999211i	$-0.487352\pi$
$829$	2.28756	0.0794504	0.0397252	+	0.999211i	$0.487352\pi$
$830$	0	0
$831$	20.2840	0.703644
$832$	0	0
$833$	56.6004	1.96109
$834$	0	0
$835$	−3.33333	−0.115355
$836$	0	0
$837$	−8.81199	−0.304587
$838$	0	0
$839$	−19.8628	−0.685740	−0.342870	−	0.939383i	$-0.611399\pi$
$839$	−19.8628	−0.685740	−0.342870	+	0.939383i	$0.611399\pi$
$840$	0	0
$841$	−2.49638	−0.0860822
$842$	0	0
$843$	58.6702	2.02071
$844$	0	0
$845$	−4.69692	−0.161579
$846$	0	0
$847$	−1.13650	−0.0390505
$848$	0	0
$849$	8.88473	0.304923
$850$	0	0
$851$	−26.7244	−0.916101
$852$	0	0
$853$	−9.92703	−0.339895	−0.169948	−	0.985453i	$-0.554360\pi$
$853$	−9.92703	−0.339895	−0.169948	+	0.985453i	$0.554360\pi$
$854$	0	0
$855$	4.90692	0.167813
$856$	0	0
$857$	17.2232	0.588334	0.294167	−	0.955754i	$-0.404958\pi$
$857$	17.2232	0.588334	0.294167	+	0.955754i	$0.404958\pi$
$858$	0	0
$859$	−37.5632	−1.28164	−0.640820	−	0.767692i	$-0.721405\pi$
$859$	−37.5632	−1.28164	−0.640820	+	0.767692i	$0.721405\pi$
$860$	0	0
$861$	−62.8112	−2.14060
$862$	0	0
$863$	−10.5951	−0.360662	−0.180331	−	0.983606i	$-0.557717\pi$
$863$	−10.5951	−0.360662	−0.180331	+	0.983606i	$0.557717\pi$
$864$	0	0
$865$	−6.05664	−0.205932
$866$	0	0
$867$	−9.04923	−0.307328
$868$	0	0
$869$	22.9296	0.777834
$870$	0	0
$871$	34.4132	1.16605
$872$	0	0
$873$	12.7716	0.432252
$874$	0	0
$875$	29.7310	1.00509
$876$	0	0
$877$	15.5414	0.524797	0.262398	−	0.964960i	$-0.415487\pi$
$877$	15.5414	0.524797	0.262398	+	0.964960i	$0.415487\pi$
$878$	0	0
$879$	−21.4004	−0.721818
$880$	0	0
$881$	17.9521	0.604823	0.302411	−	0.953177i	$-0.402208\pi$
$881$	17.9521	0.604823	0.302411	+	0.953177i	$0.402208\pi$
$882$	0	0
$883$	45.2199	1.52177	0.760885	−	0.648887i	$-0.224765\pi$
$883$	45.2199	1.52177	0.760885	+	0.648887i	$0.224765\pi$
$884$	0	0
$885$	18.2972	0.615055
$886$	0	0
$887$	−16.0659	−0.539439	−0.269720	−	0.962939i	$-0.586931\pi$
$887$	−16.0659	−0.539439	−0.269720	+	0.962939i	$0.586931\pi$
$888$	0	0
$889$	−35.2900	−1.18359
$890$	0	0
$891$	37.1043	1.24304
$892$	0	0
$893$	−2.32834	−0.0779151
$894$	0	0
$895$	−6.26980	−0.209576
$896$	0	0
$897$	14.3769	0.480031
$898$	0	0
$899$	−19.2226	−0.641110
$900$	0	0
$901$	−8.74040	−0.291185
$902$	0	0
$903$	23.3788	0.777997
$904$	0	0
$905$	4.02475	0.133787
$906$	0	0
$907$	8.12301	0.269720	0.134860	−	0.990865i	$-0.456941\pi$
$907$	8.12301	0.269720	0.134860	+	0.990865i	$0.456941\pi$
$908$	0	0
$909$	−15.8625	−0.526127
$910$	0	0
$911$	−10.5301	−0.348876	−0.174438	−	0.984668i	$-0.555811\pi$
$911$	−10.5301	−0.348876	−0.174438	+	0.984668i	$0.555811\pi$
$912$	0	0
$913$	44.8987	1.48593
$914$	0	0
$915$	7.22200	0.238752
$916$	0	0
$917$	−32.7896	−1.08281
$918$	0	0
$919$	31.0906	1.02559	0.512793	−	0.858513i	$-0.328611\pi$
$919$	31.0906	1.02559	0.512793	+	0.858513i	$0.328611\pi$
$920$	0	0
$921$	−73.5129	−2.42233
$922$	0	0
$923$	23.9047	0.786831
$924$	0	0
$925$	46.9704	1.54438
$926$	0	0
$927$	1.91086	0.0627609
$928$	0	0
$929$	−29.4503	−0.966232	−0.483116	−	0.875556i	$-0.660495\pi$
$929$	−29.4503	−0.966232	−0.483116	+	0.875556i	$0.660495\pi$
$930$	0	0
$931$	43.8243	1.43628
$932$	0	0
$933$	40.1760	1.31530
$934$	0	0
$935$	−10.9726	−0.358842
$936$	0	0
$937$	31.6441	1.03377	0.516884	−	0.856055i	$-0.327092\pi$
$937$	31.6441	1.03377	0.516884	+	0.856055i	$0.327092\pi$
$938$	0	0
$939$	−23.0351	−0.751721
$940$	0	0
$941$	−30.6375	−0.998753	−0.499377	−	0.866385i	$-0.666438\pi$
$941$	−30.6375	−0.998753	−0.499377	+	0.866385i	$0.666438\pi$
$942$	0	0
$943$	−16.4332	−0.535137
$944$	0	0
$945$	−7.39168	−0.240451
$946$	0	0
$947$	−41.1947	−1.33865	−0.669323	−	0.742971i	$-0.733416\pi$
$947$	−41.1947	−1.33865	−0.669323	+	0.742971i	$0.733416\pi$
$948$	0	0
$949$	22.6323	0.734675
$950$	0	0
$951$	−8.36200	−0.271156
$952$	0	0
$953$	47.8644	1.55048	0.775240	−	0.631666i	$-0.217629\pi$
$953$	47.8644	1.55048	0.775240	+	0.631666i	$0.217629\pi$
$954$	0	0
$955$	−1.46326	−0.0473499
$956$	0	0
$957$	38.3854	1.24083
$958$	0	0
$959$	−23.1493	−0.747531
$960$	0	0
$961$	−17.0582	−0.550264
$962$	0	0
$963$	26.2124	0.844682
$964$	0	0
$965$	−13.0511	−0.420131
$966$	0	0
$967$	−15.1929	−0.488570	−0.244285	−	0.969703i	$-0.578553\pi$
$967$	−15.1929	−0.488570	−0.244285	+	0.969703i	$0.578553\pi$
$968$	0	0
$969$	−36.2561	−1.16471
$970$	0	0
$971$	−29.2489	−0.938641	−0.469321	−	0.883028i	$-0.655501\pi$
$971$	−29.2489	−0.938641	−0.469321	+	0.883028i	$0.655501\pi$
$972$	0	0
$973$	−73.4042	−2.35323
$974$	0	0
$975$	−25.2686	−0.809243
$976$	0	0
$977$	14.1107	0.451441	0.225720	−	0.974192i	$-0.427526\pi$
$977$	14.1107	0.451441	0.225720	+	0.974192i	$0.427526\pi$
$978$	0	0
$979$	51.6990	1.65231
$980$	0	0
$981$	24.1114	0.769819
$982$	0	0
$983$	−6.41828	−0.204711	−0.102356	−	0.994748i	$-0.532638\pi$
$983$	−6.41828	−0.204711	−0.102356	+	0.994748i	$0.532638\pi$
$984$	0	0
$985$	15.8410	0.504736
$986$	0	0
$987$	−6.40008	−0.203717
$988$	0	0
$989$	6.11654	0.194495
$990$	0	0
$991$	−34.3307	−1.09055	−0.545275	−	0.838257i	$-0.683575\pi$
$991$	−34.3307	−1.09055	−0.545275	+	0.838257i	$0.683575\pi$
$992$	0	0
$993$	−28.9316	−0.918115
$994$	0	0
$995$	−2.63512	−0.0835389
$996$	0	0
$997$	−23.9065	−0.757127	−0.378564	−	0.925575i	$-0.623582\pi$
$997$	−23.9065	−0.757127	−0.378564	+	0.925575i	$0.623582\pi$
$998$	0	0
$999$	−24.6745	−0.780668

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8752.2.a.v.1.6		25
4.3	odd	2		547.2.a.c.1.5	✓	25
12.11	even	2		4923.2.a.n.1.21		25

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
547.2.a.c.1.5	✓	25	4.3	odd	2
4923.2.a.n.1.21		25	12.11	even	2
8752.2.a.v.1.6		25	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 8752 = 2^{4} \cdot 547 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8752.a (trivial)

\(f(q)\)	\(=\)	\(q-2.22215 q^{3} +0.712381 q^{5} -4.39659 q^{7} +1.93796 q^{9} +O(q^{10})\)	\(q-2.22215 q^{3} +0.712381 q^{5} -4.39659 q^{7} +1.93796 q^{9} -3.35537 q^{11} -2.53115 q^{13} -1.58302 q^{15} +4.59046 q^{17} +3.55428 q^{19} +9.76989 q^{21} +2.55608 q^{23} -4.49251 q^{25} +2.36001 q^{27} +5.14817 q^{29} -3.73387 q^{31} +7.45614 q^{33} -3.13205 q^{35} -10.4553 q^{37} +5.62460 q^{39} -6.42906 q^{41} +2.39294 q^{43} +1.38057 q^{45} -0.655082 q^{47} +12.3300 q^{49} -10.2007 q^{51} -1.90404 q^{53} -2.39030 q^{55} -7.89815 q^{57} -11.5584 q^{59} -4.56217 q^{61} -8.52042 q^{63} -1.80315 q^{65} -13.5959 q^{67} -5.67999 q^{69} -9.44418 q^{71} -8.94150 q^{73} +9.98305 q^{75} +14.7522 q^{77} -6.83371 q^{79} -11.0582 q^{81} -13.3811 q^{83} +3.27016 q^{85} -11.4400 q^{87} -15.4078 q^{89} +11.1284 q^{91} +8.29724 q^{93} +2.53200 q^{95} +6.59022 q^{97} -6.50257 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 25 q - 8 q^{3} + 29 q^{5} - 5 q^{7} + 29 q^{9}+O(q^{10}) \) 25 * q - 8 * q^3 + 29 * q^5 - 5 * q^7 + 29 * q^9	\( 25 q - 8 q^{3} + 29 q^{5} - 5 q^{7} + 29 q^{9} - 10 q^{11} + 19 q^{13} - 5 q^{15} + 40 q^{17} - 8 q^{21} - 26 q^{23} + 36 q^{25} - 11 q^{27} + 30 q^{29} + 5 q^{31} + 10 q^{33} - 11 q^{35} + 26 q^{37} + 17 q^{39} + 9 q^{41} + 10 q^{43} + 64 q^{45} - 28 q^{47} + 20 q^{49} + 9 q^{51} + 80 q^{53} + q^{55} - 8 q^{57} + 2 q^{59} + 22 q^{61} + 9 q^{63} + 30 q^{65} + 16 q^{67} + 22 q^{69} + q^{71} + 2 q^{73} + 31 q^{75} + 67 q^{77} + 34 q^{79} - 11 q^{81} - 15 q^{83} + 15 q^{85} + 29 q^{87} + 38 q^{89} + 41 q^{91} - 4 q^{93} + 46 q^{95} - 2 q^{97} + 41 q^{99}+O(q^{100}) \) 25 * q - 8 * q^3 + 29 * q^5 - 5 * q^7 + 29 * q^9 - 10 * q^11 + 19 * q^13 - 5 * q^15 + 40 * q^17 - 8 * q^21 - 26 * q^23 + 36 * q^25 - 11 * q^27 + 30 * q^29 + 5 * q^31 + 10 * q^33 - 11 * q^35 + 26 * q^37 + 17 * q^39 + 9 * q^41 + 10 * q^43 + 64 * q^45 - 28 * q^47 + 20 * q^49 + 9 * q^51 + 80 * q^53 + q^55 - 8 * q^57 + 2 * q^59 + 22 * q^61 + 9 * q^63 + 30 * q^65 + 16 * q^67 + 22 * q^69 + q^71 + 2 * q^73 + 31 * q^75 + 67 * q^77 + 34 * q^79 - 11 * q^81 - 15 * q^83 + 15 * q^85 + 29 * q^87 + 38 * q^89 + 41 * q^91 - 4 * q^93 + 46 * q^95 - 2 * q^97 + 41 * q^99

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