LMFDB - Embedded newform 8512.2.a.bk.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8512,2,Mod(1,8512)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8512, 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("8512.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8512 = 2^{6} \cdot 7 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8512.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:	$67.9686622005$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.1101.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 9x + 12 $ x^3 - x^2 - 9*x + 12
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1064)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.43163$ of defining polynomial
Character	$\chi$	$=$	8512.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.43163 q^{3} +2.51882 q^{5} -1.00000 q^{7} -0.950444 q^{9} -1.95044 q^{11} -5.38207 q^{13} -3.60601 q^{15} -3.95044 q^{17} -1.00000 q^{19} +1.43163 q^{21} +8.51882 q^{23} +1.34444 q^{25} +5.65556 q^{27} -3.95044 q^{29} -5.95044 q^{31} +2.79231 q^{33} -2.51882 q^{35} -7.55645 q^{37} +7.70512 q^{39} +6.81370 q^{41} +5.03763 q^{43} -2.39399 q^{45} -5.65556 q^{47} +1.00000 q^{49} +5.65556 q^{51} -11.4316 q^{53} -4.91281 q^{55} +1.43163 q^{57} +0.617929 q^{59} -5.48118 q^{61} +0.950444 q^{63} -13.5565 q^{65} -0.223935 q^{67} -12.1958 q^{69} +13.5565 q^{71} +7.43163 q^{73} -1.92473 q^{75} +1.95044 q^{77} +5.03763 q^{79} -5.24533 q^{81} -13.7804 q^{83} -9.95044 q^{85} +5.65556 q^{87} +10.9385 q^{89} +5.38207 q^{91} +8.51882 q^{93} -2.51882 q^{95} +4.69320 q^{97} +1.85379 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - q^{3} - 2 q^{5} - 3 q^{7} + 10 q^{9} + 7 q^{11} + 5 q^{15} + q^{17} - 3 q^{19} + q^{21} + 16 q^{23} + 7 q^{25} + 14 q^{27} + q^{29} - 5 q^{31} + 12 q^{33} + 2 q^{35} + 6 q^{37} + 33 q^{39} + q^{41}+ \cdots + 54 q^{99}+O(q^{100}) $ 3 * q - q^3 - 2 * q^5 - 3 * q^7 + 10 * q^9 + 7 * q^11 + 5 * q^15 + q^17 - 3 * q^19 + q^21 + 16 * q^23 + 7 * q^25 + 14 * q^27 + q^29 - 5 * q^31 + 12 * q^33 + 2 * q^35 + 6 * q^37 + 33 * q^39 + q^41 - 4 * q^43 - 23 * q^45 - 14 * q^47 + 3 * q^49 + 14 * q^51 - 31 * q^53 - 21 * q^55 + q^57 + 18 * q^59 - 26 * q^61 - 10 * q^63 - 12 * q^65 - q^67 - q^69 + 12 * q^71 + 19 * q^73 - 44 * q^75 - 7 * q^77 - 4 * q^79 + 7 * q^81 - 13 * q^83 - 17 * q^85 + 14 * q^87 - 12 * q^89 + 16 * q^93 + 2 * q^95 - 8 * q^97 + 54 * q^99

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
$3$	−1.43163	−0.826550	−0.413275	−	0.910606i	$-0.635615\pi$
$3$	−1.43163	−0.826550	−0.413275	+	0.910606i	$0.635615\pi$
$4$	0	0
$5$	2.51882	1.12645	0.563225	−	0.826304i	$-0.309561\pi$
$5$	2.51882	1.12645	0.563225	+	0.826304i	$0.309561\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	−0.950444	−0.316815
$10$	0	0
$11$	−1.95044	−0.588081	−0.294040	−	0.955793i	$-0.595000\pi$
$11$	−1.95044	−0.588081	−0.294040	+	0.955793i	$0.595000\pi$
$12$	0	0
$13$	−5.38207	−1.49272	−0.746359	−	0.665544i	$-0.768200\pi$
$13$	−5.38207	−1.49272	−0.746359	+	0.665544i	$0.768200\pi$
$14$	0	0
$15$	−3.60601	−0.931067
$16$	0	0
$17$	−3.95044	−0.958123	−0.479062	−	0.877781i	$-0.659023\pi$
$17$	−3.95044	−0.958123	−0.479062	+	0.877781i	$0.659023\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	1.43163	0.312407
$22$	0	0
$23$	8.51882	1.77630	0.888148	−	0.459557i	$-0.151992\pi$
$23$	8.51882	1.77630	0.888148	+	0.459557i	$0.151992\pi$
$24$	0	0
$25$	1.34444	0.268888
$26$	0	0
$27$	5.65556	1.08841
$28$	0	0
$29$	−3.95044	−0.733579	−0.366789	−	0.930304i	$-0.619543\pi$
$29$	−3.95044	−0.733579	−0.366789	+	0.930304i	$0.619543\pi$
$30$	0	0
$31$	−5.95044	−1.06873	−0.534366	−	0.845253i	$-0.679449\pi$
$31$	−5.95044	−1.06873	−0.534366	+	0.845253i	$0.679449\pi$
$32$	0	0
$33$	2.79231	0.486078
$34$	0	0
$35$	−2.51882	−0.425758
$36$	0	0
$37$	−7.55645	−1.24227	−0.621136	−	0.783703i	$-0.713329\pi$
$37$	−7.55645	−1.24227	−0.621136	+	0.783703i	$0.713329\pi$
$38$	0	0
$39$	7.70512	1.23381
$40$	0	0
$41$	6.81370	1.06412	0.532060	−	0.846706i	$-0.321418\pi$
$41$	6.81370	1.06412	0.532060	+	0.846706i	$0.321418\pi$
$42$	0	0
$43$	5.03763	0.768232	0.384116	−	0.923285i	$-0.374506\pi$
$43$	5.03763	0.768232	0.384116	+	0.923285i	$0.374506\pi$
$44$	0	0
$45$	−2.39399	−0.356876
$46$	0	0
$47$	−5.65556	−0.824949	−0.412474	−	0.910969i	$-0.635335\pi$
$47$	−5.65556	−0.824949	−0.412474	+	0.910969i	$0.635335\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	5.65556	0.791937
$52$	0	0
$53$	−11.4316	−1.57025	−0.785127	−	0.619334i	$-0.787403\pi$
$53$	−11.4316	−1.57025	−0.785127	+	0.619334i	$0.787403\pi$
$54$	0	0
$55$	−4.91281	−0.662443
$56$	0	0
$57$	1.43163	0.189624
$58$	0	0
$59$	0.617929	0.0804475	0.0402238	−	0.999191i	$-0.487193\pi$
$59$	0.617929	0.0804475	0.0402238	+	0.999191i	$0.487193\pi$
$60$	0	0
$61$	−5.48118	−0.701794	−0.350897	−	0.936414i	$-0.614123\pi$
$61$	−5.48118	−0.701794	−0.350897	+	0.936414i	$0.614123\pi$
$62$	0	0
$63$	0.950444	0.119745
$64$	0	0
$65$	−13.5565	−1.68147
$66$	0	0
$67$	−0.223935	−0.0273581	−0.0136790	−	0.999906i	$-0.504354\pi$
$67$	−0.223935	−0.0273581	−0.0136790	+	0.999906i	$0.504354\pi$
$68$	0	0
$69$	−12.1958	−1.46820
$70$	0	0
$71$	13.5565	1.60885	0.804427	−	0.594051i	$-0.202472\pi$
$71$	13.5565	1.60885	0.804427	+	0.594051i	$0.202472\pi$
$72$	0	0
$73$	7.43163	0.869806	0.434903	−	0.900477i	$-0.356783\pi$
$73$	7.43163	0.869806	0.434903	+	0.900477i	$0.356783\pi$
$74$	0	0
$75$	−1.92473	−0.222249
$76$	0	0
$77$	1.95044	0.222274
$78$	0	0
$79$	5.03763	0.566778	0.283389	−	0.959005i	$-0.408541\pi$
$79$	5.03763	0.566778	0.283389	+	0.959005i	$0.408541\pi$
$80$	0	0
$81$	−5.24533	−0.582814
$82$	0	0
$83$	−13.7804	−1.51259	−0.756297	−	0.654229i	$-0.772993\pi$
$83$	−13.7804	−1.51259	−0.756297	+	0.654229i	$0.772993\pi$
$84$	0	0
$85$	−9.95044	−1.07928
$86$	0	0
$87$	5.65556	0.606340
$88$	0	0
$89$	10.9385	1.15948	0.579740	−	0.814801i	$-0.303154\pi$
$89$	10.9385	1.15948	0.579740	+	0.814801i	$0.303154\pi$
$90$	0	0
$91$	5.38207	0.564194
$92$	0	0
$93$	8.51882	0.883360
$94$	0	0
$95$	−2.51882	−0.258425
$96$	0	0
$97$	4.69320	0.476522	0.238261	−	0.971201i	$-0.423423\pi$
$97$	4.69320	0.476522	0.238261	+	0.971201i	$0.423423\pi$
$98$	0	0
$99$	1.85379	0.186313
$100$	0	0
$101$	−17.3821	−1.72958	−0.864790	−	0.502133i	$-0.832549\pi$
$101$	−17.3821	−1.72958	−0.864790	+	0.502133i	$0.832549\pi$
$102$	0	0
$103$	2.34444	0.231004	0.115502	−	0.993307i	$-0.463152\pi$
$103$	2.34444	0.231004	0.115502	+	0.993307i	$0.463152\pi$
$104$	0	0
$105$	3.60601	0.351910
$106$	0	0
$107$	15.2830	1.47746	0.738730	−	0.674002i	$-0.235426\pi$
$107$	15.2830	1.47746	0.738730	+	0.674002i	$0.235426\pi$
$108$	0	0
$109$	13.3821	1.28177	0.640885	−	0.767637i	$-0.278568\pi$
$109$	13.3821	1.28177	0.640885	+	0.767637i	$0.278568\pi$
$110$	0	0
$111$	10.8180	1.02680
$112$	0	0
$113$	14.6436	1.37756	0.688779	−	0.724971i	$-0.258147\pi$
$113$	14.6436	1.37756	0.688779	+	0.724971i	$0.258147\pi$
$114$	0	0
$115$	21.4573	2.00091
$116$	0	0
$117$	5.11536	0.472915
$118$	0	0
$119$	3.95044	0.362137
$120$	0	0
$121$	−7.19577	−0.654161
$122$	0	0
$123$	−9.75467	−0.879549
$124$	0	0
$125$	−9.20769	−0.823561
$126$	0	0
$127$	18.1744	1.61272	0.806358	−	0.591428i	$-0.201436\pi$
$127$	18.1744	1.61272	0.806358	+	0.591428i	$0.201436\pi$
$128$	0	0
$129$	−7.21201	−0.634982
$130$	0	0
$131$	4.81370	0.420575	0.210287	−	0.977640i	$-0.432560\pi$
$131$	4.81370	0.420575	0.210287	+	0.977640i	$0.432560\pi$
$132$	0	0
$133$	1.00000	0.0867110
$134$	0	0
$135$	14.2453	1.22604
$136$	0	0
$137$	−11.7265	−1.00186	−0.500932	−	0.865487i	$-0.667009\pi$
$137$	−11.7265	−1.00186	−0.500932	+	0.865487i	$0.667009\pi$
$138$	0	0
$139$	9.65556	0.818974	0.409487	−	0.912316i	$-0.365708\pi$
$139$	9.65556	0.818974	0.409487	+	0.912316i	$0.365708\pi$
$140$	0	0
$141$	8.09666	0.681861
$142$	0	0
$143$	10.4974	0.877839
$144$	0	0
$145$	−9.95044	−0.826339
$146$	0	0
$147$	−1.43163	−0.118079
$148$	0	0
$149$	−4.69320	−0.384482	−0.192241	−	0.981348i	$-0.561575\pi$
$149$	−4.69320	−0.384482	−0.192241	+	0.981348i	$0.561575\pi$
$150$	0	0
$151$	−4.22394	−0.343739	−0.171869	−	0.985120i	$-0.554981\pi$
$151$	−4.22394	−0.343739	−0.171869	+	0.985120i	$0.554981\pi$
$152$	0	0
$153$	3.75467	0.303547
$154$	0	0
$155$	−14.9881	−1.20387
$156$	0	0
$157$	20.8890	1.66712	0.833560	−	0.552428i	$-0.186299\pi$
$157$	20.8890	1.66712	0.833560	+	0.552428i	$0.186299\pi$
$158$	0	0
$159$	16.3658	1.29789
$160$	0	0
$161$	−8.51882	−0.671377
$162$	0	0
$163$	15.1581	1.18728	0.593638	−	0.804732i	$-0.297691\pi$
$163$	15.1581	1.18728	0.593638	+	0.804732i	$0.297691\pi$
$164$	0	0
$165$	7.03331	0.547543
$166$	0	0
$167$	−6.24533	−0.483278	−0.241639	−	0.970366i	$-0.577685\pi$
$167$	−6.24533	−0.483278	−0.241639	+	0.970366i	$0.577685\pi$
$168$	0	0
$169$	15.9667	1.22821
$170$	0	0
$171$	0.950444	0.0726823
$172$	0	0
$173$	10.9385	0.831640	0.415820	−	0.909447i	$-0.363495\pi$
$173$	10.9385	0.831640	0.415820	+	0.909447i	$0.363495\pi$
$174$	0	0
$175$	−1.34444	−0.101630
$176$	0	0
$177$	−0.884644	−0.0664939
$178$	0	0
$179$	−15.4359	−1.15374	−0.576868	−	0.816837i	$-0.695725\pi$
$179$	−15.4359	−1.15374	−0.576868	+	0.816837i	$0.695725\pi$
$180$	0	0
$181$	2.12482	0.157937	0.0789684	−	0.996877i	$-0.474837\pi$
$181$	2.12482	0.157937	0.0789684	+	0.996877i	$0.474837\pi$
$182$	0	0
$183$	7.84701	0.580068
$184$	0	0
$185$	−19.0333	−1.39936
$186$	0	0
$187$	7.70512	0.563454
$188$	0	0
$189$	−5.65556	−0.411382
$190$	0	0
$191$	−7.50689	−0.543180	−0.271590	−	0.962413i	$-0.587549\pi$
$191$	−7.50689	−0.543180	−0.271590	+	0.962413i	$0.587549\pi$
$192$	0	0
$193$	−6.22394	−0.448009	−0.224004	−	0.974588i	$-0.571913\pi$
$193$	−6.22394	−0.448009	−0.224004	+	0.974588i	$0.571913\pi$
$194$	0	0
$195$	19.4078	1.38982
$196$	0	0
$197$	10.2949	0.733480	0.366740	−	0.930323i	$-0.380474\pi$
$197$	10.2949	0.733480	0.366740	+	0.930323i	$0.380474\pi$
$198$	0	0
$199$	−19.3821	−1.37396	−0.686979	−	0.726677i	$-0.741064\pi$
$199$	−19.3821	−1.37396	−0.686979	+	0.726677i	$0.741064\pi$
$200$	0	0
$201$	0.320592	0.0226128
$202$	0	0
$203$	3.95044	0.277267
$204$	0	0
$205$	17.1625	1.19868
$206$	0	0
$207$	−8.09666	−0.562757
$208$	0	0
$209$	1.95044	0.134915
$210$	0	0
$211$	−4.39399	−0.302495	−0.151248	−	0.988496i	$-0.548329\pi$
$211$	−4.39399	−0.302495	−0.151248	+	0.988496i	$0.548329\pi$
$212$	0	0
$213$	−19.4078	−1.32980
$214$	0	0
$215$	12.6889	0.865374
$216$	0	0
$217$	5.95044	0.403942
$218$	0	0
$219$	−10.6393	−0.718939
$220$	0	0
$221$	21.2616	1.43021
$222$	0	0
$223$	−11.8299	−0.792191	−0.396096	−	0.918209i	$-0.629635\pi$
$223$	−11.8299	−0.792191	−0.396096	+	0.918209i	$0.629635\pi$
$224$	0	0
$225$	−1.27781	−0.0851875
$226$	0	0
$227$	14.2992	0.949071	0.474536	−	0.880236i	$-0.342616\pi$
$227$	14.2992	0.949071	0.474536	+	0.880236i	$0.342616\pi$
$228$	0	0
$229$	14.3488	0.948193	0.474096	−	0.880473i	$-0.342775\pi$
$229$	14.3488	0.948193	0.474096	+	0.880473i	$0.342775\pi$
$230$	0	0
$231$	−2.79231	−0.183720
$232$	0	0
$233$	−15.8513	−1.03846	−0.519228	−	0.854636i	$-0.673780\pi$
$233$	−15.8513	−1.03846	−0.519228	+	0.854636i	$0.673780\pi$
$234$	0	0
$235$	−14.2453	−0.929263
$236$	0	0
$237$	−7.21201	−0.468471
$238$	0	0
$239$	7.28296	0.471095	0.235548	−	0.971863i	$-0.424312\pi$
$239$	7.28296	0.471095	0.235548	+	0.971863i	$0.424312\pi$
$240$	0	0
$241$	7.82562	0.504093	0.252046	−	0.967715i	$-0.418896\pi$
$241$	7.82562	0.504093	0.252046	+	0.967715i	$0.418896\pi$
$242$	0	0
$243$	−9.45734	−0.606688
$244$	0	0
$245$	2.51882	0.160921
$246$	0	0
$247$	5.38207	0.342453
$248$	0	0
$249$	19.7284	1.25023
$250$	0	0
$251$	30.1010	1.89996	0.949978	−	0.312316i	$-0.101105\pi$
$251$	30.1010	1.89996	0.949978	+	0.312316i	$0.101105\pi$
$252$	0	0
$253$	−16.6155	−1.04461
$254$	0	0
$255$	14.2453	0.892077
$256$	0	0
$257$	3.43163	0.214059	0.107030	−	0.994256i	$-0.465866\pi$
$257$	3.43163	0.214059	0.107030	+	0.994256i	$0.465866\pi$
$258$	0	0
$259$	7.55645	0.469535
$260$	0	0
$261$	3.75467	0.232409
$262$	0	0
$263$	7.70512	0.475118	0.237559	−	0.971373i	$-0.423653\pi$
$263$	7.70512	0.475118	0.237559	+	0.971373i	$0.423653\pi$
$264$	0	0
$265$	−28.7942	−1.76881
$266$	0	0
$267$	−15.6599	−0.958369
$268$	0	0
$269$	−28.8180	−1.75707	−0.878533	−	0.477682i	$-0.841477\pi$
$269$	−28.8180	−1.75707	−0.878533	+	0.477682i	$0.841477\pi$
$270$	0	0
$271$	19.3368	1.17463	0.587315	−	0.809359i	$-0.300185\pi$
$271$	19.3368	1.17463	0.587315	+	0.809359i	$0.300185\pi$
$272$	0	0
$273$	−7.70512	−0.466335
$274$	0	0
$275$	−2.62225	−0.158128
$276$	0	0
$277$	−19.9762	−1.20025	−0.600125	−	0.799906i	$-0.704883\pi$
$277$	−19.9762	−1.20025	−0.600125	+	0.799906i	$0.704883\pi$
$278$	0	0
$279$	5.65556	0.338590
$280$	0	0
$281$	28.1462	1.67906	0.839531	−	0.543312i	$-0.182830\pi$
$281$	28.1462	1.67906	0.839531	+	0.543312i	$0.182830\pi$
$282$	0	0
$283$	−6.56837	−0.390449	−0.195225	−	0.980759i	$-0.562544\pi$
$283$	−6.56837	−0.390449	−0.195225	+	0.980759i	$0.562544\pi$
$284$	0	0
$285$	3.60601	0.213601
$286$	0	0
$287$	−6.81370	−0.402200
$288$	0	0
$289$	−1.39399	−0.0819996
$290$	0	0
$291$	−6.71891	−0.393869
$292$	0	0
$293$	13.4812	0.787579	0.393790	−	0.919201i	$-0.371164\pi$
$293$	13.4812	0.787579	0.393790	+	0.919201i	$0.371164\pi$
$294$	0	0
$295$	1.55645	0.0906200
$296$	0	0
$297$	−11.0309	−0.640075
$298$	0	0
$299$	−45.8489	−2.65151
$300$	0	0
$301$	−5.03763	−0.290364
$302$	0	0
$303$	24.8846	1.42959
$304$	0	0
$305$	−13.8061	−0.790535
$306$	0	0
$307$	0.323048	0.0184373	0.00921865	−	0.999958i	$-0.497066\pi$
$307$	0.323048	0.0184373	0.00921865	+	0.999958i	$0.497066\pi$
$308$	0	0
$309$	−3.35636	−0.190937
$310$	0	0
$311$	−17.8513	−1.01226	−0.506128	−	0.862458i	$-0.668924\pi$
$311$	−17.8513	−1.01226	−0.506128	+	0.862458i	$0.668924\pi$
$312$	0	0
$313$	−12.1744	−0.688137	−0.344068	−	0.938945i	$-0.611805\pi$
$313$	−12.1744	−0.688137	−0.344068	+	0.938945i	$0.611805\pi$
$314$	0	0
$315$	2.39399	0.134886
$316$	0	0
$317$	7.72651	0.433964	0.216982	−	0.976176i	$-0.430379\pi$
$317$	7.72651	0.433964	0.216982	+	0.976176i	$0.430379\pi$
$318$	0	0
$319$	7.70512	0.431404
$320$	0	0
$321$	−21.8795	−1.22119
$322$	0	0
$323$	3.95044	0.219809
$324$	0	0
$325$	−7.23586	−0.401373
$326$	0	0
$327$	−19.1581	−1.05945
$328$	0	0
$329$	5.65556	0.311801
$330$	0	0
$331$	10.1205	0.556273	0.278137	−	0.960542i	$-0.410283\pi$
$331$	10.1205	0.556273	0.278137	+	0.960542i	$0.410283\pi$
$332$	0	0
$333$	7.18198	0.393570
$334$	0	0
$335$	−0.564052	−0.0308175
$336$	0	0
$337$	33.2377	1.81057	0.905287	−	0.424800i	$-0.139656\pi$
$337$	33.2377	1.81057	0.905287	+	0.424800i	$0.139656\pi$
$338$	0	0
$339$	−20.9642	−1.13862
$340$	0	0
$341$	11.6060	0.628500
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	−30.7189	−1.65385
$346$	0	0
$347$	−3.84701	−0.206518	−0.103259	−	0.994654i	$-0.532927\pi$
$347$	−3.84701	−0.206518	−0.103259	+	0.994654i	$0.532927\pi$
$348$	0	0
$349$	37.2096	1.99178	0.995891	−	0.0905605i	$-0.0288659\pi$
$349$	37.2096	1.99178	0.995891	+	0.0905605i	$0.0288659\pi$
$350$	0	0
$351$	−30.4386	−1.62469
$352$	0	0
$353$	−15.2616	−0.812291	−0.406146	−	0.913808i	$-0.633127\pi$
$353$	−15.2616	−0.812291	−0.406146	+	0.913808i	$0.633127\pi$
$354$	0	0
$355$	34.1462	1.81229
$356$	0	0
$357$	−5.65556	−0.299324
$358$	0	0
$359$	−30.0019	−1.58344	−0.791719	−	0.610886i	$-0.790814\pi$
$359$	−30.0019	−1.58344	−0.791719	+	0.610886i	$0.790814\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	10.3017	0.540697
$364$	0	0
$365$	18.7189	0.979792
$366$	0	0
$367$	0.717041	0.0374293	0.0187146	−	0.999825i	$-0.494043\pi$
$367$	0.717041	0.0374293	0.0187146	+	0.999825i	$0.494043\pi$
$368$	0	0
$369$	−6.47604	−0.337129
$370$	0	0
$371$	11.4316	0.593501
$372$	0	0
$373$	−32.1719	−1.66580	−0.832900	−	0.553424i	$-0.813321\pi$
$373$	−32.1719	−1.66580	−0.832900	+	0.553424i	$0.813321\pi$
$374$	0	0
$375$	13.1820	0.680715
$376$	0	0
$377$	21.2616	1.09503
$378$	0	0
$379$	−14.8351	−0.762027	−0.381014	−	0.924569i	$-0.624425\pi$
$379$	−14.8351	−0.762027	−0.381014	+	0.924569i	$0.624425\pi$
$380$	0	0
$381$	−26.0189	−1.33299
$382$	0	0
$383$	25.1086	1.28299	0.641494	−	0.767128i	$-0.278315\pi$
$383$	25.1086	1.28299	0.641494	+	0.767128i	$0.278315\pi$
$384$	0	0
$385$	4.91281	0.250380
$386$	0	0
$387$	−4.78799	−0.243387
$388$	0	0
$389$	1.77606	0.0900501	0.0450250	−	0.998986i	$-0.485663\pi$
$389$	1.77606	0.0900501	0.0450250	+	0.998986i	$0.485663\pi$
$390$	0	0
$391$	−33.6531	−1.70191
$392$	0	0
$393$	−6.89142	−0.347626
$394$	0	0
$395$	12.6889	0.638447
$396$	0	0
$397$	10.5898	0.531485	0.265742	−	0.964044i	$-0.414383\pi$
$397$	10.5898	0.531485	0.265742	+	0.964044i	$0.414383\pi$
$398$	0	0
$399$	−1.43163	−0.0716710
$400$	0	0
$401$	35.8513	1.79033	0.895165	−	0.445735i	$-0.147058\pi$
$401$	35.8513	1.79033	0.895165	+	0.445735i	$0.147058\pi$
$402$	0	0
$403$	32.0257	1.59531
$404$	0	0
$405$	−13.2120	−0.656510
$406$	0	0
$407$	14.7384	0.730557
$408$	0	0
$409$	−9.50689	−0.470086	−0.235043	−	0.971985i	$-0.575523\pi$
$409$	−9.50689	−0.470086	−0.235043	+	0.971985i	$0.575523\pi$
$410$	0	0
$411$	16.7880	0.828090
$412$	0	0
$413$	−0.617929	−0.0304063
$414$	0	0
$415$	−34.7103	−1.70386
$416$	0	0
$417$	−13.8232	−0.676923
$418$	0	0
$419$	−23.1838	−1.13261	−0.566303	−	0.824197i	$-0.691627\pi$
$419$	−23.1838	−1.13261	−0.566303	+	0.824197i	$0.691627\pi$
$420$	0	0
$421$	−27.0095	−1.31636	−0.658180	−	0.752860i	$-0.728674\pi$
$421$	−27.0095	−1.31636	−0.658180	+	0.752860i	$0.728674\pi$
$422$	0	0
$423$	5.37529	0.261356
$424$	0	0
$425$	−5.31112	−0.257627
$426$	0	0
$427$	5.48118	0.265253
$428$	0	0
$429$	−15.0284	−0.725578
$430$	0	0
$431$	−10.7641	−0.518490	−0.259245	−	0.965812i	$-0.583474\pi$
$431$	−10.7641	−0.518490	−0.259245	+	0.965812i	$0.583474\pi$
$432$	0	0
$433$	15.7547	0.757121	0.378561	−	0.925576i	$-0.376419\pi$
$433$	15.7547	0.757121	0.378561	+	0.925576i	$0.376419\pi$
$434$	0	0
$435$	14.2453	0.683011
$436$	0	0
$437$	−8.51882	−0.407510
$438$	0	0
$439$	3.45302	0.164804	0.0824018	−	0.996599i	$-0.473741\pi$
$439$	3.45302	0.164804	0.0824018	+	0.996599i	$0.473741\pi$
$440$	0	0
$441$	−0.950444	−0.0452592
$442$	0	0
$443$	−19.5069	−0.926800	−0.463400	−	0.886149i	$-0.653371\pi$
$443$	−19.5069	−0.926800	−0.463400	+	0.886149i	$0.653371\pi$
$444$	0	0
$445$	27.5521	1.30610
$446$	0	0
$447$	6.71891	0.317793
$448$	0	0
$449$	7.43163	0.350720	0.175360	−	0.984504i	$-0.443891\pi$
$449$	7.43163	0.350720	0.175360	+	0.984504i	$0.443891\pi$
$450$	0	0
$451$	−13.2897	−0.625789
$452$	0	0
$453$	6.04710	0.284118
$454$	0	0
$455$	13.5565	0.635536
$456$	0	0
$457$	1.50689	0.0704895	0.0352448	−	0.999379i	$-0.488779\pi$
$457$	1.50689	0.0704895	0.0352448	+	0.999379i	$0.488779\pi$
$458$	0	0
$459$	−22.3420	−1.04283
$460$	0	0
$461$	−12.9171	−0.601611	−0.300805	−	0.953686i	$-0.597255\pi$
$461$	−12.9171	−0.601611	−0.300805	+	0.953686i	$0.597255\pi$
$462$	0	0
$463$	9.30680	0.432524	0.216262	−	0.976335i	$-0.430613\pi$
$463$	9.30680	0.432524	0.216262	+	0.976335i	$0.430613\pi$
$464$	0	0
$465$	21.4573	0.995060
$466$	0	0
$467$	−4.29488	−0.198743	−0.0993717	−	0.995050i	$-0.531683\pi$
$467$	−4.29488	−0.198743	−0.0993717	+	0.995050i	$0.531683\pi$
$468$	0	0
$469$	0.223935	0.0103404
$470$	0	0
$471$	−29.9052	−1.37796
$472$	0	0
$473$	−9.82562	−0.451783
$474$	0	0
$475$	−1.34444	−0.0616870
$476$	0	0
$477$	10.8651	0.497480
$478$	0	0
$479$	−40.8651	−1.86717	−0.933587	−	0.358350i	$-0.883340\pi$
$479$	−40.8651	−1.86717	−0.933587	+	0.358350i	$0.883340\pi$
$480$	0	0
$481$	40.6694	1.85436
$482$	0	0
$483$	12.1958	0.554927
$484$	0	0
$485$	11.8213	0.536778
$486$	0	0
$487$	37.1600	1.68388	0.841940	−	0.539571i	$-0.181414\pi$
$487$	37.1600	1.68388	0.841940	+	0.539571i	$0.181414\pi$
$488$	0	0
$489$	−21.7008	−0.981344
$490$	0	0
$491$	37.8770	1.70937	0.854683	−	0.519149i	$-0.173751\pi$
$491$	37.8770	1.70937	0.854683	+	0.519149i	$0.173751\pi$
$492$	0	0
$493$	15.6060	0.702859
$494$	0	0
$495$	4.66935	0.209872
$496$	0	0
$497$	−13.5565	−0.608090
$498$	0	0
$499$	29.3325	1.31310	0.656552	−	0.754281i	$-0.272014\pi$
$499$	29.3325	1.31310	0.656552	+	0.754281i	$0.272014\pi$
$500$	0	0
$501$	8.94098	0.399453
$502$	0	0
$503$	32.0000	1.42681	0.713405	−	0.700752i	$-0.247152\pi$
$503$	32.0000	1.42681	0.713405	+	0.700752i	$0.247152\pi$
$504$	0	0
$505$	−43.7823	−1.94828
$506$	0	0
$507$	−22.8583	−1.01517
$508$	0	0
$509$	6.00000	0.265945	0.132973	−	0.991120i	$-0.457548\pi$
$509$	6.00000	0.265945	0.132973	+	0.991120i	$0.457548\pi$
$510$	0	0
$511$	−7.43163	−0.328756
$512$	0	0
$513$	−5.65556	−0.249699
$514$	0	0
$515$	5.90521	0.260215
$516$	0	0
$517$	11.0309	0.485137
$518$	0	0
$519$	−15.6599	−0.687393
$520$	0	0
$521$	11.9052	0.521577	0.260788	−	0.965396i	$-0.416018\pi$
$521$	11.9052	0.521577	0.260788	+	0.965396i	$0.416018\pi$
$522$	0	0
$523$	25.0095	1.09359	0.546794	−	0.837267i	$-0.315848\pi$
$523$	25.0095	1.09359	0.546794	+	0.837267i	$0.315848\pi$
$524$	0	0
$525$	1.92473	0.0840022
$526$	0	0
$527$	23.5069	1.02398
$528$	0	0
$529$	49.5702	2.15523
$530$	0	0
$531$	−0.587307	−0.0254869
$532$	0	0
$533$	−36.6718	−1.58843
$534$	0	0
$535$	38.4950	1.66428
$536$	0	0
$537$	22.0985	0.953622
$538$	0	0
$539$	−1.95044	−0.0840116
$540$	0	0
$541$	−8.27349	−0.355705	−0.177853	−	0.984057i	$-0.556915\pi$
$541$	−8.27349	−0.355705	−0.177853	+	0.984057i	$0.556915\pi$
$542$	0	0
$543$	−3.04195	−0.130543
$544$	0	0
$545$	33.7070	1.44385
$546$	0	0
$547$	31.0590	1.32799	0.663994	−	0.747738i	$-0.268860\pi$
$547$	31.0590	1.32799	0.663994	+	0.747738i	$0.268860\pi$
$548$	0	0
$549$	5.20956	0.222338
$550$	0	0
$551$	3.95044	0.168295
$552$	0	0
$553$	−5.03763	−0.214222
$554$	0	0
$555$	27.2486	1.15664
$556$	0	0
$557$	19.0633	0.807740	0.403870	−	0.914816i	$-0.367665\pi$
$557$	19.0633	0.807740	0.403870	+	0.914816i	$0.367665\pi$
$558$	0	0
$559$	−27.1129	−1.14675
$560$	0	0
$561$	−11.0309	−0.465723
$562$	0	0
$563$	33.4573	1.41006	0.705029	−	0.709178i	$-0.250934\pi$
$563$	33.4573	1.41006	0.705029	+	0.709178i	$0.250934\pi$
$564$	0	0
$565$	36.8846	1.55175
$566$	0	0
$567$	5.24533	0.220283
$568$	0	0
$569$	18.3206	0.768039	0.384020	−	0.923325i	$-0.374540\pi$
$569$	18.3206	0.768039	0.384020	+	0.923325i	$0.374540\pi$
$570$	0	0
$571$	12.9667	0.542639	0.271319	−	0.962489i	$-0.412540\pi$
$571$	12.9667	0.542639	0.271319	+	0.962489i	$0.412540\pi$
$572$	0	0
$573$	10.7471	0.448965
$574$	0	0
$575$	11.4530	0.477624
$576$	0	0
$577$	6.39399	0.266185	0.133093	−	0.991104i	$-0.457509\pi$
$577$	6.39399	0.266185	0.133093	+	0.991104i	$0.457509\pi$
$578$	0	0
$579$	8.91035	0.370302
$580$	0	0
$581$	13.7804	0.571707
$582$	0	0
$583$	22.2967	0.923437
$584$	0	0
$585$	12.8846	0.532714
$586$	0	0
$587$	−26.5659	−1.09649	−0.548246	−	0.836317i	$-0.684704\pi$
$587$	−26.5659	−1.09649	−0.548246	+	0.836317i	$0.684704\pi$
$588$	0	0
$589$	5.95044	0.245184
$590$	0	0
$591$	−14.7384	−0.606258
$592$	0	0
$593$	−17.3821	−0.713796	−0.356898	−	0.934143i	$-0.616166\pi$
$593$	−17.3821	−0.713796	−0.356898	+	0.934143i	$0.616166\pi$
$594$	0	0
$595$	9.95044	0.407928
$596$	0	0
$597$	27.7479	1.13565
$598$	0	0
$599$	24.8933	1.01711	0.508556	−	0.861029i	$-0.330179\pi$
$599$	24.8933	1.01711	0.508556	+	0.861029i	$0.330179\pi$
$600$	0	0
$601$	20.8608	0.850930	0.425465	−	0.904975i	$-0.360111\pi$
$601$	20.8608	0.850930	0.425465	+	0.904975i	$0.360111\pi$
$602$	0	0
$603$	0.212838	0.00866743
$604$	0	0
$605$	−18.1248	−0.736879
$606$	0	0
$607$	−40.7685	−1.65474	−0.827370	−	0.561657i	$-0.810164\pi$
$607$	−40.7685	−1.65474	−0.827370	+	0.561657i	$0.810164\pi$
$608$	0	0
$609$	−5.65556	−0.229175
$610$	0	0
$611$	30.4386	1.23142
$612$	0	0
$613$	43.3839	1.75226	0.876130	−	0.482074i	$-0.160116\pi$
$613$	43.3839	1.75226	0.876130	+	0.482074i	$0.160116\pi$
$614$	0	0
$615$	−24.5702	−0.990768
$616$	0	0
$617$	−5.43595	−0.218843	−0.109422	−	0.993995i	$-0.534900\pi$
$617$	−5.43595	−0.218843	−0.109422	+	0.993995i	$0.534900\pi$
$618$	0	0
$619$	−0.393994	−0.0158359	−0.00791797	−	0.999969i	$-0.502520\pi$
$619$	−0.393994	−0.0158359	−0.00791797	+	0.999969i	$0.502520\pi$
$620$	0	0
$621$	48.1787	1.93334
$622$	0	0
$623$	−10.9385	−0.438243
$624$	0	0
$625$	−29.9147	−1.19659
$626$	0	0
$627$	−2.79231	−0.111514
$628$	0	0
$629$	29.8513	1.19025
$630$	0	0
$631$	15.5326	0.618343	0.309172	−	0.951006i	$-0.399948\pi$
$631$	15.5326	0.618343	0.309172	+	0.951006i	$0.399948\pi$
$632$	0	0
$633$	6.29056	0.250027
$634$	0	0
$635$	45.7779	1.81664
$636$	0	0
$637$	−5.38207	−0.213245
$638$	0	0
$639$	−12.8846	−0.509709
$640$	0	0
$641$	−1.92660	−0.0760961	−0.0380480	−	0.999276i	$-0.512114\pi$
$641$	−1.92660	−0.0760961	−0.0380480	+	0.999276i	$0.512114\pi$
$642$	0	0
$643$	1.20769	0.0476267	0.0238134	−	0.999716i	$-0.492419\pi$
$643$	1.20769	0.0476267	0.0238134	+	0.999716i	$0.492419\pi$
$644$	0	0
$645$	−18.1657	−0.715275
$646$	0	0
$647$	26.3444	1.03571	0.517853	−	0.855469i	$-0.326731\pi$
$647$	26.3444	1.03571	0.517853	+	0.855469i	$0.326731\pi$
$648$	0	0
$649$	−1.20524	−0.0473096
$650$	0	0
$651$	−8.51882	−0.333879
$652$	0	0
$653$	2.85893	0.111879	0.0559394	−	0.998434i	$-0.482185\pi$
$653$	2.85893	0.111879	0.0559394	+	0.998434i	$0.482185\pi$
$654$	0	0
$655$	12.1248	0.473756
$656$	0	0
$657$	−7.06334	−0.275567
$658$	0	0
$659$	−31.3796	−1.22238	−0.611188	−	0.791485i	$-0.709308\pi$
$659$	−31.3796	−1.22238	−0.611188	+	0.791485i	$0.709308\pi$
$660$	0	0
$661$	−34.6975	−1.34958	−0.674788	−	0.738011i	$-0.735765\pi$
$661$	−34.6975	−1.34958	−0.674788	+	0.738011i	$0.735765\pi$
$662$	0	0
$663$	−30.4386	−1.18214
$664$	0	0
$665$	2.51882	0.0976755
$666$	0	0
$667$	−33.6531	−1.30305
$668$	0	0
$669$	16.9361	0.654786
$670$	0	0
$671$	10.6907	0.412711
$672$	0	0
$673$	−38.8846	−1.49889	−0.749446	−	0.662065i	$-0.769680\pi$
$673$	−38.8846	−1.49889	−0.749446	+	0.662065i	$0.769680\pi$
$674$	0	0
$675$	7.60355	0.292661
$676$	0	0
$677$	−7.28974	−0.280167	−0.140084	−	0.990140i	$-0.544737\pi$
$677$	−7.28974	−0.280167	−0.140084	+	0.990140i	$0.544737\pi$
$678$	0	0
$679$	−4.69320	−0.180108
$680$	0	0
$681$	−20.4711	−0.784455
$682$	0	0
$683$	0.320592	0.0122671	0.00613355	−	0.999981i	$-0.498048\pi$
$683$	0.320592	0.0122671	0.00613355	+	0.999981i	$0.498048\pi$
$684$	0	0
$685$	−29.5369	−1.12855
$686$	0	0
$687$	−20.5421	−0.783729
$688$	0	0
$689$	61.5258	2.34395
$690$	0	0
$691$	32.9385	1.25304	0.626520	−	0.779405i	$-0.284479\pi$
$691$	32.9385	1.25304	0.626520	+	0.779405i	$0.284479\pi$
$692$	0	0
$693$	−1.85379	−0.0704196
$694$	0	0
$695$	24.3206	0.922533
$696$	0	0
$697$	−26.9171	−1.01956
$698$	0	0
$699$	22.6932	0.858335
$700$	0	0
$701$	25.6599	0.969160	0.484580	−	0.874747i	$-0.338972\pi$
$701$	25.6599	0.969160	0.484580	+	0.874747i	$0.338972\pi$
$702$	0	0
$703$	7.55645	0.284997
$704$	0	0
$705$	20.3940	0.768082
$706$	0	0
$707$	17.3821	0.653720
$708$	0	0
$709$	3.55645	0.133565	0.0667826	−	0.997768i	$-0.478727\pi$
$709$	3.55645	0.133565	0.0667826	+	0.997768i	$0.478727\pi$
$710$	0	0
$711$	−4.78799	−0.179564
$712$	0	0
$713$	−50.6907	−1.89838
$714$	0	0
$715$	26.4411	0.988841
$716$	0	0
$717$	−10.4265	−0.389384
$718$	0	0
$719$	5.92473	0.220955	0.110478	−	0.993879i	$-0.464762\pi$
$719$	5.92473	0.220955	0.110478	+	0.993879i	$0.464762\pi$
$720$	0	0
$721$	−2.34444	−0.0873114
$722$	0	0
$723$	−11.2034	−0.416658
$724$	0	0
$725$	−5.31112	−0.197250
$726$	0	0
$727$	−19.8299	−0.735452	−0.367726	−	0.929934i	$-0.619864\pi$
$727$	−19.8299	−0.735452	−0.367726	+	0.929934i	$0.619864\pi$
$728$	0	0
$729$	29.2754	1.08427
$730$	0	0
$731$	−19.9009	−0.736061
$732$	0	0
$733$	−11.3864	−0.420566	−0.210283	−	0.977641i	$-0.567439\pi$
$733$	−11.3864	−0.420566	−0.210283	+	0.977641i	$0.567439\pi$
$734$	0	0
$735$	−3.60601	−0.133010
$736$	0	0
$737$	0.436773	0.0160888
$738$	0	0
$739$	29.2163	1.07474	0.537370	−	0.843347i	$-0.319418\pi$
$739$	29.2163	1.07474	0.537370	+	0.843347i	$0.319418\pi$
$740$	0	0
$741$	−7.70512	−0.283055
$742$	0	0
$743$	15.8299	0.580744	0.290372	−	0.956914i	$-0.406221\pi$
$743$	15.8299	0.580744	0.290372	+	0.956914i	$0.406221\pi$
$744$	0	0
$745$	−11.8213	−0.433099
$746$	0	0
$747$	13.0975	0.479212
$748$	0	0
$749$	−15.2830	−0.558427
$750$	0	0
$751$	−35.0590	−1.27932	−0.639661	−	0.768657i	$-0.720925\pi$
$751$	−35.0590	−1.27932	−0.639661	+	0.768657i	$0.720925\pi$
$752$	0	0
$753$	−43.0934	−1.57041
$754$	0	0
$755$	−10.6393	−0.387204
$756$	0	0
$757$	−47.8770	−1.74012	−0.870060	−	0.492945i	$-0.835920\pi$
$757$	−47.8770	−1.74012	−0.870060	+	0.492945i	$0.835920\pi$
$758$	0	0
$759$	23.7872	0.863419
$760$	0	0
$761$	31.7070	1.14938	0.574689	−	0.818372i	$-0.305123\pi$
$761$	31.7070	1.14938	0.574689	+	0.818372i	$0.305123\pi$
$762$	0	0
$763$	−13.3821	−0.484463
$764$	0	0
$765$	9.45734	0.341931
$766$	0	0
$767$	−3.32574	−0.120085
$768$	0	0
$769$	−36.7641	−1.32575	−0.662874	−	0.748731i	$-0.730664\pi$
$769$	−36.7641	−1.32575	−0.662874	+	0.748731i	$0.730664\pi$
$770$	0	0
$771$	−4.91281	−0.176931
$772$	0	0
$773$	33.5044	1.20507	0.602535	−	0.798092i	$-0.294157\pi$
$773$	33.5044	1.20507	0.602535	+	0.798092i	$0.294157\pi$
$774$	0	0
$775$	−8.00000	−0.287368
$776$	0	0
$777$	−10.8180	−0.388094
$778$	0	0
$779$	−6.81370	−0.244126
$780$	0	0
$781$	−26.4411	−0.946137
$782$	0	0
$783$	−22.3420	−0.798437
$784$	0	0
$785$	52.6155	1.87793
$786$	0	0
$787$	2.12296	0.0756753	0.0378376	−	0.999284i	$-0.487953\pi$
$787$	2.12296	0.0756753	0.0378376	+	0.999284i	$0.487953\pi$
$788$	0	0
$789$	−11.0309	−0.392709
$790$	0	0
$791$	−14.6436	−0.520668
$792$	0	0
$793$	29.5001	1.04758
$794$	0	0
$795$	41.2225	1.46201
$796$	0	0
$797$	−34.1676	−1.21028	−0.605139	−	0.796120i	$-0.706883\pi$
$797$	−34.1676	−1.21028	−0.605139	+	0.796120i	$0.706883\pi$
$798$	0	0
$799$	22.3420	0.790402
$800$	0	0
$801$	−10.3964	−0.367340
$802$	0	0
$803$	−14.4950	−0.511516
$804$	0	0
$805$	−21.4573	−0.756272
$806$	0	0
$807$	41.2567	1.45230
$808$	0	0
$809$	−1.63172	−0.0573681	−0.0286841	−	0.999589i	$-0.509132\pi$
$809$	−1.63172	−0.0573681	−0.0286841	+	0.999589i	$0.509132\pi$
$810$	0	0
$811$	19.4617	0.683391	0.341696	−	0.939811i	$-0.388999\pi$
$811$	19.4617	0.683391	0.341696	+	0.939811i	$0.388999\pi$
$812$	0	0
$813$	−27.6831	−0.970890
$814$	0	0
$815$	38.1806	1.33741
$816$	0	0
$817$	−5.03763	−0.176244
$818$	0	0
$819$	−5.11536	−0.178745
$820$	0	0
$821$	−39.0376	−1.36242	−0.681211	−	0.732087i	$-0.738547\pi$
$821$	−39.0376	−1.36242	−0.681211	+	0.732087i	$0.738547\pi$
$822$	0	0
$823$	1.48550	0.0517814	0.0258907	−	0.999665i	$-0.491758\pi$
$823$	1.48550	0.0517814	0.0258907	+	0.999665i	$0.491758\pi$
$824$	0	0
$825$	3.75408	0.130700
$826$	0	0
$827$	10.5941	0.368392	0.184196	−	0.982889i	$-0.441032\pi$
$827$	10.5941	0.368392	0.184196	+	0.982889i	$0.441032\pi$
$828$	0	0
$829$	38.8394	1.34895	0.674474	−	0.738298i	$-0.264370\pi$
$829$	38.8394	1.34895	0.674474	+	0.738298i	$0.264370\pi$
$830$	0	0
$831$	28.5984	0.992068
$832$	0	0
$833$	−3.95044	−0.136875
$834$	0	0
$835$	−15.7308	−0.544388
$836$	0	0
$837$	−33.6531	−1.16322
$838$	0	0
$839$	−40.4993	−1.39819	−0.699095	−	0.715028i	$-0.746414\pi$
$839$	−40.4993	−1.39819	−0.699095	+	0.715028i	$0.746414\pi$
$840$	0	0
$841$	−13.3940	−0.461862
$842$	0	0
$843$	−40.2949	−1.38783
$844$	0	0
$845$	40.2172	1.38351
$846$	0	0
$847$	7.19577	0.247250
$848$	0	0
$849$	9.40346	0.322726
$850$	0	0
$851$	−64.3720	−2.20664
$852$	0	0
$853$	−8.37015	−0.286588	−0.143294	−	0.989680i	$-0.545770\pi$
$853$	−8.37015	−0.286588	−0.143294	+	0.989680i	$0.545770\pi$
$854$	0	0
$855$	2.39399	0.0818729
$856$	0	0
$857$	−14.3230	−0.489266	−0.244633	−	0.969616i	$-0.578667\pi$
$857$	−14.3230	−0.489266	−0.244633	+	0.969616i	$0.578667\pi$
$858$	0	0
$859$	−0.912810	−0.0311447	−0.0155723	−	0.999879i	$-0.504957\pi$
$859$	−0.912810	−0.0311447	−0.0155723	+	0.999879i	$0.504957\pi$
$860$	0	0
$861$	9.75467	0.332438
$862$	0	0
$863$	24.1958	0.823634	0.411817	−	0.911267i	$-0.364894\pi$
$863$	24.1958	0.823634	0.411817	+	0.911267i	$0.364894\pi$
$864$	0	0
$865$	27.5521	0.936801
$866$	0	0
$867$	1.99568	0.0677768
$868$	0	0
$869$	−9.82562	−0.333311
$870$	0	0
$871$	1.20524	0.0408379
$872$	0	0
$873$	−4.46062	−0.150969
$874$	0	0
$875$	9.20769	0.311277
$876$	0	0
$877$	4.86325	0.164220	0.0821102	−	0.996623i	$-0.473834\pi$
$877$	4.86325	0.164220	0.0821102	+	0.996623i	$0.473834\pi$
$878$	0	0
$879$	−19.3000	−0.650974
$880$	0	0
$881$	−36.8608	−1.24187	−0.620936	−	0.783861i	$-0.713247\pi$
$881$	−36.8608	−1.24187	−0.620936	+	0.783861i	$0.713247\pi$
$882$	0	0
$883$	−17.2077	−0.579085	−0.289542	−	0.957165i	$-0.593503\pi$
$883$	−17.2077	−0.579085	−0.289542	+	0.957165i	$0.593503\pi$
$884$	0	0
$885$	−2.22826	−0.0749020
$886$	0	0
$887$	−18.3959	−0.617672	−0.308836	−	0.951115i	$-0.599940\pi$
$887$	−18.3959	−0.617672	−0.308836	+	0.951115i	$0.599940\pi$
$888$	0	0
$889$	−18.1744	−0.609549
$890$	0	0
$891$	10.2307	0.342742
$892$	0	0
$893$	5.65556	0.189256
$894$	0	0
$895$	−38.8803	−1.29963
$896$	0	0
$897$	65.6385	2.19161
$898$	0	0
$899$	23.5069	0.783999
$900$	0	0
$901$	45.1600	1.50450
$902$	0	0
$903$	7.21201	0.240001
$904$	0	0
$905$	5.35204	0.177908
$906$	0	0
$907$	−31.7308	−1.05360	−0.526802	−	0.849988i	$-0.676609\pi$
$907$	−31.7308	−1.05360	−0.526802	+	0.849988i	$0.676609\pi$
$908$	0	0
$909$	16.5207	0.547956
$910$	0	0
$911$	10.9624	0.363199	0.181600	−	0.983373i	$-0.441872\pi$
$911$	10.9624	0.363199	0.181600	+	0.983373i	$0.441872\pi$
$912$	0	0
$913$	26.8779	0.889528
$914$	0	0
$915$	19.7652	0.653417
$916$	0	0
$917$	−4.81370	−0.158962
$918$	0	0
$919$	33.1086	1.09215	0.546076	−	0.837736i	$-0.316121\pi$
$919$	33.1086	1.09215	0.546076	+	0.837736i	$0.316121\pi$
$920$	0	0
$921$	−0.462484	−0.0152394
$922$	0	0
$923$	−72.9618	−2.40157
$924$	0	0
$925$	−10.1592	−0.334032
$926$	0	0
$927$	−2.22826	−0.0731855
$928$	0	0
$929$	2.74275	0.0899868	0.0449934	−	0.998987i	$-0.485673\pi$
$929$	2.74275	0.0899868	0.0449934	+	0.998987i	$0.485673\pi$
$930$	0	0
$931$	−1.00000	−0.0327737
$932$	0	0
$933$	25.5565	0.836681
$934$	0	0
$935$	19.4078	0.634702
$936$	0	0
$937$	21.2291	0.693524	0.346762	−	0.937953i	$-0.387281\pi$
$937$	21.2291	0.693524	0.346762	+	0.937953i	$0.387281\pi$
$938$	0	0
$939$	17.4292	0.568780
$940$	0	0
$941$	−37.9523	−1.23721	−0.618605	−	0.785702i	$-0.712302\pi$
$941$	−37.9523	−1.23721	−0.618605	+	0.785702i	$0.712302\pi$
$942$	0	0
$943$	58.0446	1.89019
$944$	0	0
$945$	−14.2453	−0.463400
$946$	0	0
$947$	21.7241	0.705937	0.352968	−	0.935635i	$-0.385172\pi$
$947$	21.7241	0.705937	0.352968	+	0.935635i	$0.385172\pi$
$948$	0	0
$949$	−39.9975	−1.29838
$950$	0	0
$951$	−11.0615	−0.358693
$952$	0	0
$953$	−4.39831	−0.142475	−0.0712377	−	0.997459i	$-0.522695\pi$
$953$	−4.39831	−0.142475	−0.0712377	+	0.997459i	$0.522695\pi$
$954$	0	0
$955$	−18.9085	−0.611864
$956$	0	0
$957$	−11.0309	−0.356577
$958$	0	0
$959$	11.7265	0.378669
$960$	0	0
$961$	4.40778	0.142187
$962$	0	0
$963$	−14.5256	−0.468081
$964$	0	0
$965$	−15.6770	−0.504659
$966$	0	0
$967$	59.5865	1.91617	0.958086	−	0.286481i	$-0.0924854\pi$
$967$	59.5865	1.91617	0.958086	+	0.286481i	$0.0924854\pi$
$968$	0	0
$969$	−5.65556	−0.181683
$970$	0	0
$971$	−6.14621	−0.197241	−0.0986207	−	0.995125i	$-0.531443\pi$
$971$	−6.14621	−0.197241	−0.0986207	+	0.995125i	$0.531443\pi$
$972$	0	0
$973$	−9.65556	−0.309543
$974$	0	0
$975$	10.3591	0.331755
$976$	0	0
$977$	−13.3539	−0.427229	−0.213615	−	0.976918i	$-0.568524\pi$
$977$	−13.3539	−0.427229	−0.213615	+	0.976918i	$0.568524\pi$
$978$	0	0
$979$	−21.3350	−0.681869
$980$	0	0
$981$	−12.7189	−0.406083
$982$	0	0
$983$	29.7975	0.950391	0.475196	−	0.879880i	$-0.342377\pi$
$983$	29.7975	0.950391	0.475196	+	0.879880i	$0.342377\pi$
$984$	0	0
$985$	25.9309	0.826228
$986$	0	0
$987$	−8.09666	−0.257719
$988$	0	0
$989$	42.9147	1.36461
$990$	0	0
$991$	43.0138	1.36638	0.683189	−	0.730242i	$-0.260593\pi$
$991$	43.0138	1.36638	0.683189	+	0.730242i	$0.260593\pi$
$992$	0	0
$993$	−14.4888	−0.459788
$994$	0	0
$995$	−48.8199	−1.54769
$996$	0	0
$997$	39.0352	1.23626	0.618128	−	0.786077i	$-0.287891\pi$
$997$	39.0352	1.23626	0.618128	+	0.786077i	$0.287891\pi$
$998$	0	0
$999$	−42.7360	−1.35211

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8512.2.a.bk.1.2		3
4.3	odd	2		8512.2.a.bo.1.2		3
8.3	odd	2		1064.2.a.f.1.2	✓	3
8.5	even	2		2128.2.a.q.1.2		3
24.11	even	2		9576.2.a.ca.1.3		3
56.27	even	2		7448.2.a.bi.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1064.2.a.f.1.2	✓	3	8.3	odd	2
2128.2.a.q.1.2		3	8.5	even	2
7448.2.a.bi.1.2		3	56.27	even	2
8512.2.a.bk.1.2		3	1.1	even	1	trivial
8512.2.a.bo.1.2		3	4.3	odd	2
9576.2.a.ca.1.3		3	24.11	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:	\( N \)	\(=\)	\( 8512 = 2^{6} \cdot 7 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8512.a (trivial)

\(f(q)\)	\(=\)	\(q-1.43163 q^{3} +2.51882 q^{5} -1.00000 q^{7} -0.950444 q^{9} -1.95044 q^{11} -5.38207 q^{13} -3.60601 q^{15} -3.95044 q^{17} -1.00000 q^{19} +1.43163 q^{21} +8.51882 q^{23} +1.34444 q^{25} +5.65556 q^{27} -3.95044 q^{29} -5.95044 q^{31} +2.79231 q^{33} -2.51882 q^{35} -7.55645 q^{37} +7.70512 q^{39} +6.81370 q^{41} +5.03763 q^{43} -2.39399 q^{45} -5.65556 q^{47} +1.00000 q^{49} +5.65556 q^{51} -11.4316 q^{53} -4.91281 q^{55} +1.43163 q^{57} +0.617929 q^{59} -5.48118 q^{61} +0.950444 q^{63} -13.5565 q^{65} -0.223935 q^{67} -12.1958 q^{69} +13.5565 q^{71} +7.43163 q^{73} -1.92473 q^{75} +1.95044 q^{77} +5.03763 q^{79} -5.24533 q^{81} -13.7804 q^{83} -9.95044 q^{85} +5.65556 q^{87} +10.9385 q^{89} +5.38207 q^{91} +8.51882 q^{93} -2.51882 q^{95} +4.69320 q^{97} +1.85379 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - q^{3} - 2 q^{5} - 3 q^{7} + 10 q^{9} + 7 q^{11} + 5 q^{15} + q^{17} - 3 q^{19} + q^{21} + 16 q^{23} + 7 q^{25} + 14 q^{27} + q^{29} - 5 q^{31} + 12 q^{33} + 2 q^{35} + 6 q^{37} + 33 q^{39} + q^{41}+ \cdots + 54 q^{99}+O(q^{100}) \) 3 * q - q^3 - 2 * q^5 - 3 * q^7 + 10 * q^9 + 7 * q^11 + 5 * q^15 + q^17 - 3 * q^19 + q^21 + 16 * q^23 + 7 * q^25 + 14 * q^27 + q^29 - 5 * q^31 + 12 * q^33 + 2 * q^35 + 6 * q^37 + 33 * q^39 + q^41 - 4 * q^43 - 23 * q^45 - 14 * q^47 + 3 * q^49 + 14 * q^51 - 31 * q^53 - 21 * q^55 + q^57 + 18 * q^59 - 26 * q^61 - 10 * q^63 - 12 * q^65 - q^67 - q^69 + 12 * q^71 + 19 * q^73 - 44 * q^75 - 7 * q^77 - 4 * q^79 + 7 * q^81 - 13 * q^83 - 17 * q^85 + 14 * q^87 - 12 * q^89 + 16 * q^93 + 2 * q^95 - 8 * q^97 + 54 * q^99

Introduction

L-functions

Modular forms

Varieties

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Database

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