LMFDB - Embedded newform 2800.2.a.bh.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2800,2,Mod(1,2800)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2800, 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("2800.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2800 = 2^{4} \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2800.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:	$22.3581125660$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 175)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.618034$ of defining polynomial
Character	$\chi$	$=$	2800.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.23607 q^{3} +1.00000 q^{7} -1.47214 q^{9} +O(q^{10})$	$q+1.23607 q^{3} +1.00000 q^{7} -1.47214 q^{9} -4.23607 q^{11} +3.23607 q^{13} +6.47214 q^{17} -4.47214 q^{19} +1.23607 q^{21} +1.76393 q^{23} -5.52786 q^{27} +5.00000 q^{29} +9.70820 q^{31} -5.23607 q^{33} -3.00000 q^{37} +4.00000 q^{39} +9.23607 q^{41} +6.23607 q^{43} -2.00000 q^{47} +1.00000 q^{49} +8.00000 q^{51} +0.472136 q^{53} -5.52786 q^{57} +1.70820 q^{59} +3.70820 q^{61} -1.47214 q^{63} +0.236068 q^{67} +2.18034 q^{69} +4.70820 q^{71} +13.2361 q^{73} -4.23607 q^{77} -11.1803 q^{79} -2.41641 q^{81} +5.70820 q^{83} +6.18034 q^{87} +12.7639 q^{89} +3.23607 q^{91} +12.0000 q^{93} -0.763932 q^{97} +6.23607 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} + 2 q^{7} + 6 q^{9}+O(q^{10}) $ 2 * q - 2 * q^3 + 2 * q^7 + 6 * q^9	$ 2 q - 2 q^{3} + 2 q^{7} + 6 q^{9} - 4 q^{11} + 2 q^{13} + 4 q^{17} - 2 q^{21} + 8 q^{23} - 20 q^{27} + 10 q^{29} + 6 q^{31} - 6 q^{33} - 6 q^{37} + 8 q^{39} + 14 q^{41} + 8 q^{43} - 4 q^{47} + 2 q^{49} + 16 q^{51} - 8 q^{53} - 20 q^{57} - 10 q^{59} - 6 q^{61} + 6 q^{63} - 4 q^{67} - 18 q^{69} - 4 q^{71} + 22 q^{73} - 4 q^{77} + 22 q^{81} - 2 q^{83} - 10 q^{87} + 30 q^{89} + 2 q^{91} + 24 q^{93} - 6 q^{97} + 8 q^{99}+O(q^{100}) $ 2 * q - 2 * q^3 + 2 * q^7 + 6 * q^9 - 4 * q^11 + 2 * q^13 + 4 * q^17 - 2 * q^21 + 8 * q^23 - 20 * q^27 + 10 * q^29 + 6 * q^31 - 6 * q^33 - 6 * q^37 + 8 * q^39 + 14 * q^41 + 8 * q^43 - 4 * q^47 + 2 * q^49 + 16 * q^51 - 8 * q^53 - 20 * q^57 - 10 * q^59 - 6 * q^61 + 6 * q^63 - 4 * q^67 - 18 * q^69 - 4 * q^71 + 22 * q^73 - 4 * q^77 + 22 * q^81 - 2 * q^83 - 10 * q^87 + 30 * q^89 + 2 * q^91 + 24 * q^93 - 6 * q^97 + 8 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	1.23607	0.713644	0.356822	−	0.934172i	$-0.383860\pi$
$3$	1.23607	0.713644	0.356822	+	0.934172i	$0.383860\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	−1.47214	−0.490712
$10$	0	0
$11$	−4.23607	−1.27722	−0.638611	−	0.769529i	$-0.720491\pi$
$11$	−4.23607	−1.27722	−0.638611	+	0.769529i	$0.720491\pi$
$12$	0	0
$13$	3.23607	0.897524	0.448762	−	0.893651i	$-0.351865\pi$
$13$	3.23607	0.897524	0.448762	+	0.893651i	$0.351865\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	6.47214	1.56972	0.784862	−	0.619671i	$-0.212734\pi$
$17$	6.47214	1.56972	0.784862	+	0.619671i	$0.212734\pi$
$18$	0	0
$19$	−4.47214	−1.02598	−0.512989	−	0.858395i	$-0.671462\pi$
$19$	−4.47214	−1.02598	−0.512989	+	0.858395i	$0.671462\pi$
$20$	0	0
$21$	1.23607	0.269732
$22$	0	0
$23$	1.76393	0.367805	0.183903	−	0.982944i	$-0.441127\pi$
$23$	1.76393	0.367805	0.183903	+	0.982944i	$0.441127\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−5.52786	−1.06384
$28$	0	0
$29$	5.00000	0.928477	0.464238	−	0.885710i	$-0.346328\pi$
$29$	5.00000	0.928477	0.464238	+	0.885710i	$0.346328\pi$
$30$	0	0
$31$	9.70820	1.74364	0.871822	−	0.489822i	$-0.162938\pi$
$31$	9.70820	1.74364	0.871822	+	0.489822i	$0.162938\pi$
$32$	0	0
$33$	−5.23607	−0.911482
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−3.00000	−0.493197	−0.246598	−	0.969118i	$-0.579313\pi$
$37$	−3.00000	−0.493197	−0.246598	+	0.969118i	$0.579313\pi$
$38$	0	0
$39$	4.00000	0.640513
$40$	0	0
$41$	9.23607	1.44243	0.721216	−	0.692711i	$-0.243584\pi$
$41$	9.23607	1.44243	0.721216	+	0.692711i	$0.243584\pi$
$42$	0	0
$43$	6.23607	0.950991	0.475496	−	0.879718i	$-0.342269\pi$
$43$	6.23607	0.950991	0.475496	+	0.879718i	$0.342269\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−2.00000	−0.291730	−0.145865	−	0.989305i	$-0.546597\pi$
$47$	−2.00000	−0.291730	−0.145865	+	0.989305i	$0.546597\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	8.00000	1.12022
$52$	0	0
$53$	0.472136	0.0648529	0.0324264	−	0.999474i	$-0.489677\pi$
$53$	0.472136	0.0648529	0.0324264	+	0.999474i	$0.489677\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−5.52786	−0.732183
$58$	0	0
$59$	1.70820	0.222389	0.111195	−	0.993799i	$-0.464532\pi$
$59$	1.70820	0.222389	0.111195	+	0.993799i	$0.464532\pi$
$60$	0	0
$61$	3.70820	0.474787	0.237393	−	0.971414i	$-0.423707\pi$
$61$	3.70820	0.474787	0.237393	+	0.971414i	$0.423707\pi$
$62$	0	0
$63$	−1.47214	−0.185472
$64$	0	0
$65$	0	0
$66$	0	0
$67$	0.236068	0.0288403	0.0144201	−	0.999896i	$-0.495410\pi$
$67$	0.236068	0.0288403	0.0144201	+	0.999896i	$0.495410\pi$
$68$	0	0
$69$	2.18034	0.262482
$70$	0	0
$71$	4.70820	0.558761	0.279381	−	0.960180i	$-0.409871\pi$
$71$	4.70820	0.558761	0.279381	+	0.960180i	$0.409871\pi$
$72$	0	0
$73$	13.2361	1.54916	0.774582	−	0.632473i	$-0.217960\pi$
$73$	13.2361	1.54916	0.774582	+	0.632473i	$0.217960\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−4.23607	−0.482745
$78$	0	0
$79$	−11.1803	−1.25789	−0.628943	−	0.777451i	$-0.716512\pi$
$79$	−11.1803	−1.25789	−0.628943	+	0.777451i	$0.716512\pi$
$80$	0	0
$81$	−2.41641	−0.268490
$82$	0	0
$83$	5.70820	0.626557	0.313278	−	0.949661i	$-0.398573\pi$
$83$	5.70820	0.626557	0.313278	+	0.949661i	$0.398573\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	6.18034	0.662602
$88$	0	0
$89$	12.7639	1.35297	0.676487	−	0.736455i	$-0.263501\pi$
$89$	12.7639	1.35297	0.676487	+	0.736455i	$0.263501\pi$
$90$	0	0
$91$	3.23607	0.339232
$92$	0	0
$93$	12.0000	1.24434
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−0.763932	−0.0775655	−0.0387828	−	0.999248i	$-0.512348\pi$
$97$	−0.763932	−0.0775655	−0.0387828	+	0.999248i	$0.512348\pi$
$98$	0	0
$99$	6.23607	0.626748
$100$	0	0
$101$	9.23607	0.919023	0.459512	−	0.888172i	$-0.348024\pi$
$101$	9.23607	0.919023	0.459512	+	0.888172i	$0.348024\pi$
$102$	0	0
$103$	−0.472136	−0.0465209	−0.0232605	−	0.999729i	$-0.507405\pi$
$103$	−0.472136	−0.0465209	−0.0232605	+	0.999729i	$0.507405\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	8.00000	0.773389	0.386695	−	0.922208i	$-0.373617\pi$
$107$	8.00000	0.773389	0.386695	+	0.922208i	$0.373617\pi$
$108$	0	0
$109$	−18.4164	−1.76397	−0.881986	−	0.471276i	$-0.843794\pi$
$109$	−18.4164	−1.76397	−0.881986	+	0.471276i	$0.843794\pi$
$110$	0	0
$111$	−3.70820	−0.351967
$112$	0	0
$113$	−12.4164	−1.16804	−0.584019	−	0.811740i	$-0.698521\pi$
$113$	−12.4164	−1.16804	−0.584019	+	0.811740i	$0.698521\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−4.76393	−0.440426
$118$	0	0
$119$	6.47214	0.593300
$120$	0	0
$121$	6.94427	0.631297
$122$	0	0
$123$	11.4164	1.02938
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−17.6525	−1.56640	−0.783202	−	0.621767i	$-0.786415\pi$
$127$	−17.6525	−1.56640	−0.783202	+	0.621767i	$0.786415\pi$
$128$	0	0
$129$	7.70820	0.678670
$130$	0	0
$131$	−0.944272	−0.0825014	−0.0412507	−	0.999149i	$-0.513134\pi$
$131$	−0.944272	−0.0825014	−0.0412507	+	0.999149i	$0.513134\pi$
$132$	0	0
$133$	−4.47214	−0.387783
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−6.94427	−0.593289	−0.296645	−	0.954988i	$-0.595868\pi$
$137$	−6.94427	−0.593289	−0.296645	+	0.954988i	$0.595868\pi$
$138$	0	0
$139$	20.6525	1.75172	0.875860	−	0.482565i	$-0.160295\pi$
$139$	20.6525	1.75172	0.875860	+	0.482565i	$0.160295\pi$
$140$	0	0
$141$	−2.47214	−0.208191
$142$	0	0
$143$	−13.7082	−1.14634
$144$	0	0
$145$	0	0
$146$	0	0
$147$	1.23607	0.101949
$148$	0	0
$149$	−13.9443	−1.14236	−0.571180	−	0.820825i	$-0.693514\pi$
$149$	−13.9443	−1.14236	−0.571180	+	0.820825i	$0.693514\pi$
$150$	0	0
$151$	15.7639	1.28285	0.641425	−	0.767185i	$-0.278343\pi$
$151$	15.7639	1.28285	0.641425	+	0.767185i	$0.278343\pi$
$152$	0	0
$153$	−9.52786	−0.770282
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−5.23607	−0.417884	−0.208942	−	0.977928i	$-0.567002\pi$
$157$	−5.23607	−0.417884	−0.208942	+	0.977928i	$0.567002\pi$
$158$	0	0
$159$	0.583592	0.0462819
$160$	0	0
$161$	1.76393	0.139017
$162$	0	0
$163$	−10.4721	−0.820241	−0.410120	−	0.912031i	$-0.634513\pi$
$163$	−10.4721	−0.820241	−0.410120	+	0.912031i	$0.634513\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	0.763932	0.0591148	0.0295574	−	0.999563i	$-0.490590\pi$
$167$	0.763932	0.0591148	0.0295574	+	0.999563i	$0.490590\pi$
$168$	0	0
$169$	−2.52786	−0.194451
$170$	0	0
$171$	6.58359	0.503460
$172$	0	0
$173$	20.4721	1.55647	0.778234	−	0.627975i	$-0.216116\pi$
$173$	20.4721	1.55647	0.778234	+	0.627975i	$0.216116\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	2.11146	0.158707
$178$	0	0
$179$	−3.41641	−0.255354	−0.127677	−	0.991816i	$-0.540752\pi$
$179$	−3.41641	−0.255354	−0.127677	+	0.991816i	$0.540752\pi$
$180$	0	0
$181$	−14.1803	−1.05402	−0.527008	−	0.849860i	$-0.676686\pi$
$181$	−14.1803	−1.05402	−0.527008	+	0.849860i	$0.676686\pi$
$182$	0	0
$183$	4.58359	0.338829
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−27.4164	−2.00489
$188$	0	0
$189$	−5.52786	−0.402093
$190$	0	0
$191$	2.47214	0.178877	0.0894387	−	0.995992i	$-0.471493\pi$
$191$	2.47214	0.178877	0.0894387	+	0.995992i	$0.471493\pi$
$192$	0	0
$193$	14.4164	1.03772	0.518858	−	0.854861i	$-0.326357\pi$
$193$	14.4164	1.03772	0.518858	+	0.854861i	$0.326357\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−7.47214	−0.532368	−0.266184	−	0.963922i	$-0.585763\pi$
$197$	−7.47214	−0.532368	−0.266184	+	0.963922i	$0.585763\pi$
$198$	0	0
$199$	−2.76393	−0.195930	−0.0979650	−	0.995190i	$-0.531233\pi$
$199$	−2.76393	−0.195930	−0.0979650	+	0.995190i	$0.531233\pi$
$200$	0	0
$201$	0.291796	0.0205817
$202$	0	0
$203$	5.00000	0.350931
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−2.59675	−0.180486
$208$	0	0
$209$	18.9443	1.31040
$210$	0	0
$211$	−12.0000	−0.826114	−0.413057	−	0.910705i	$-0.635539\pi$
$211$	−12.0000	−0.826114	−0.413057	+	0.910705i	$0.635539\pi$
$212$	0	0
$213$	5.81966	0.398757
$214$	0	0
$215$	0	0
$216$	0	0
$217$	9.70820	0.659036
$218$	0	0
$219$	16.3607	1.10555
$220$	0	0
$221$	20.9443	1.40886
$222$	0	0
$223$	−2.18034	−0.146006	−0.0730032	−	0.997332i	$-0.523258\pi$
$223$	−2.18034	−0.146006	−0.0730032	+	0.997332i	$0.523258\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−5.41641	−0.359500	−0.179750	−	0.983712i	$-0.557529\pi$
$227$	−5.41641	−0.359500	−0.179750	+	0.983712i	$0.557529\pi$
$228$	0	0
$229$	4.47214	0.295527	0.147764	−	0.989023i	$-0.452793\pi$
$229$	4.47214	0.295527	0.147764	+	0.989023i	$0.452793\pi$
$230$	0	0
$231$	−5.23607	−0.344508
$232$	0	0
$233$	9.94427	0.651471	0.325735	−	0.945461i	$-0.394388\pi$
$233$	9.94427	0.651471	0.325735	+	0.945461i	$0.394388\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−13.8197	−0.897683
$238$	0	0
$239$	14.4721	0.936125	0.468062	−	0.883695i	$-0.344952\pi$
$239$	14.4721	0.936125	0.468062	+	0.883695i	$0.344952\pi$
$240$	0	0
$241$	−12.4721	−0.803401	−0.401700	−	0.915771i	$-0.631581\pi$
$241$	−12.4721	−0.803401	−0.401700	+	0.915771i	$0.631581\pi$
$242$	0	0
$243$	13.5967	0.872232
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−14.4721	−0.920840
$248$	0	0
$249$	7.05573	0.447139
$250$	0	0
$251$	2.47214	0.156040	0.0780199	−	0.996952i	$-0.475140\pi$
$251$	2.47214	0.156040	0.0780199	+	0.996952i	$0.475140\pi$
$252$	0	0
$253$	−7.47214	−0.469769
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−18.6525	−1.16351	−0.581755	−	0.813364i	$-0.697634\pi$
$257$	−18.6525	−1.16351	−0.581755	+	0.813364i	$0.697634\pi$
$258$	0	0
$259$	−3.00000	−0.186411
$260$	0	0
$261$	−7.36068	−0.455615
$262$	0	0
$263$	11.7639	0.725395	0.362698	−	0.931907i	$-0.381856\pi$
$263$	11.7639	0.725395	0.362698	+	0.931907i	$0.381856\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	15.7771	0.965542
$268$	0	0
$269$	1.70820	0.104151	0.0520755	−	0.998643i	$-0.483416\pi$
$269$	1.70820	0.104151	0.0520755	+	0.998643i	$0.483416\pi$
$270$	0	0
$271$	−10.2918	−0.625182	−0.312591	−	0.949888i	$-0.601197\pi$
$271$	−10.2918	−0.625182	−0.312591	+	0.949888i	$0.601197\pi$
$272$	0	0
$273$	4.00000	0.242091
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−15.8885	−0.954650	−0.477325	−	0.878727i	$-0.658394\pi$
$277$	−15.8885	−0.954650	−0.477325	+	0.878727i	$0.658394\pi$
$278$	0	0
$279$	−14.2918	−0.855627
$280$	0	0
$281$	29.3607	1.75151	0.875756	−	0.482755i	$-0.160364\pi$
$281$	29.3607	1.75151	0.875756	+	0.482755i	$0.160364\pi$
$282$	0	0
$283$	−9.41641	−0.559747	−0.279874	−	0.960037i	$-0.590293\pi$
$283$	−9.41641	−0.559747	−0.279874	+	0.960037i	$0.590293\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	9.23607	0.545188
$288$	0	0
$289$	24.8885	1.46403
$290$	0	0
$291$	−0.944272	−0.0553542
$292$	0	0
$293$	−9.12461	−0.533066	−0.266533	−	0.963826i	$-0.585878\pi$
$293$	−9.12461	−0.533066	−0.266533	+	0.963826i	$0.585878\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	23.4164	1.35876
$298$	0	0
$299$	5.70820	0.330114
$300$	0	0
$301$	6.23607	0.359441
$302$	0	0
$303$	11.4164	0.655855
$304$	0	0
$305$	0	0
$306$	0	0
$307$	31.4164	1.79303	0.896515	−	0.443014i	$-0.146091\pi$
$307$	31.4164	1.79303	0.896515	+	0.443014i	$0.146091\pi$
$308$	0	0
$309$	−0.583592	−0.0331994
$310$	0	0
$311$	20.3607	1.15455	0.577274	−	0.816550i	$-0.304116\pi$
$311$	20.3607	1.15455	0.577274	+	0.816550i	$0.304116\pi$
$312$	0	0
$313$	−28.4721	−1.60934	−0.804670	−	0.593722i	$-0.797658\pi$
$313$	−28.4721	−1.60934	−0.804670	+	0.593722i	$0.797658\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	19.3607	1.08740	0.543702	−	0.839278i	$-0.317022\pi$
$317$	19.3607	1.08740	0.543702	+	0.839278i	$0.317022\pi$
$318$	0	0
$319$	−21.1803	−1.18587
$320$	0	0
$321$	9.88854	0.551925
$322$	0	0
$323$	−28.9443	−1.61050
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−22.7639	−1.25885
$328$	0	0
$329$	−2.00000	−0.110264
$330$	0	0
$331$	11.2918	0.620653	0.310327	−	0.950630i	$-0.399562\pi$
$331$	11.2918	0.620653	0.310327	+	0.950630i	$0.399562\pi$
$332$	0	0
$333$	4.41641	0.242018
$334$	0	0
$335$	0	0
$336$	0	0
$337$	7.52786	0.410069	0.205034	−	0.978755i	$-0.434269\pi$
$337$	7.52786	0.410069	0.205034	+	0.978755i	$0.434269\pi$
$338$	0	0
$339$	−15.3475	−0.833563
$340$	0	0
$341$	−41.1246	−2.22702
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	15.7639	0.846252	0.423126	−	0.906071i	$-0.360933\pi$
$347$	15.7639	0.846252	0.423126	+	0.906071i	$0.360933\pi$
$348$	0	0
$349$	−4.47214	−0.239388	−0.119694	−	0.992811i	$-0.538191\pi$
$349$	−4.47214	−0.239388	−0.119694	+	0.992811i	$0.538191\pi$
$350$	0	0
$351$	−17.8885	−0.954820
$352$	0	0
$353$	−20.1803	−1.07409	−0.537046	−	0.843553i	$-0.680460\pi$
$353$	−20.1803	−1.07409	−0.537046	+	0.843553i	$0.680460\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	8.00000	0.423405
$358$	0	0
$359$	−10.1246	−0.534357	−0.267178	−	0.963647i	$-0.586091\pi$
$359$	−10.1246	−0.534357	−0.267178	+	0.963647i	$0.586091\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	8.58359	0.450522
$364$	0	0
$365$	0	0
$366$	0	0
$367$	3.12461	0.163103	0.0815517	−	0.996669i	$-0.474012\pi$
$367$	3.12461	0.163103	0.0815517	+	0.996669i	$0.474012\pi$
$368$	0	0
$369$	−13.5967	−0.707818
$370$	0	0
$371$	0.472136	0.0245121
$372$	0	0
$373$	−15.8328	−0.819792	−0.409896	−	0.912132i	$-0.634435\pi$
$373$	−15.8328	−0.819792	−0.409896	+	0.912132i	$0.634435\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	16.1803	0.833330
$378$	0	0
$379$	−11.1803	−0.574295	−0.287148	−	0.957886i	$-0.592707\pi$
$379$	−11.1803	−0.574295	−0.287148	+	0.957886i	$0.592707\pi$
$380$	0	0
$381$	−21.8197	−1.11786
$382$	0	0
$383$	−28.7639	−1.46977	−0.734884	−	0.678193i	$-0.762763\pi$
$383$	−28.7639	−1.46977	−0.734884	+	0.678193i	$0.762763\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−9.18034	−0.466663
$388$	0	0
$389$	−32.8885	−1.66752	−0.833758	−	0.552131i	$-0.813815\pi$
$389$	−32.8885	−1.66752	−0.833758	+	0.552131i	$0.813815\pi$
$390$	0	0
$391$	11.4164	0.577353
$392$	0	0
$393$	−1.16718	−0.0588767
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−26.9443	−1.35229	−0.676147	−	0.736767i	$-0.736352\pi$
$397$	−26.9443	−1.35229	−0.676147	+	0.736767i	$0.736352\pi$
$398$	0	0
$399$	−5.52786	−0.276739
$400$	0	0
$401$	11.4721	0.572891	0.286446	−	0.958097i	$-0.407526\pi$
$401$	11.4721	0.572891	0.286446	+	0.958097i	$0.407526\pi$
$402$	0	0
$403$	31.4164	1.56496
$404$	0	0
$405$	0	0
$406$	0	0
$407$	12.7082	0.629922
$408$	0	0
$409$	15.5279	0.767803	0.383902	−	0.923374i	$-0.374580\pi$
$409$	15.5279	0.767803	0.383902	+	0.923374i	$0.374580\pi$
$410$	0	0
$411$	−8.58359	−0.423397
$412$	0	0
$413$	1.70820	0.0840552
$414$	0	0
$415$	0	0
$416$	0	0
$417$	25.5279	1.25010
$418$	0	0
$419$	3.81966	0.186603	0.0933013	−	0.995638i	$-0.470258\pi$
$419$	3.81966	0.186603	0.0933013	+	0.995638i	$0.470258\pi$
$420$	0	0
$421$	−13.0000	−0.633581	−0.316791	−	0.948495i	$-0.602605\pi$
$421$	−13.0000	−0.633581	−0.316791	+	0.948495i	$0.602605\pi$
$422$	0	0
$423$	2.94427	0.143155
$424$	0	0
$425$	0	0
$426$	0	0
$427$	3.70820	0.179453
$428$	0	0
$429$	−16.9443	−0.818077
$430$	0	0
$431$	−26.4721	−1.27512	−0.637559	−	0.770402i	$-0.720056\pi$
$431$	−26.4721	−1.27512	−0.637559	+	0.770402i	$0.720056\pi$
$432$	0	0
$433$	−16.3607	−0.786244	−0.393122	−	0.919486i	$-0.628605\pi$
$433$	−16.3607	−0.786244	−0.393122	+	0.919486i	$0.628605\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−7.88854	−0.377360
$438$	0	0
$439$	21.7082	1.03608	0.518038	−	0.855358i	$-0.326663\pi$
$439$	21.7082	1.03608	0.518038	+	0.855358i	$0.326663\pi$
$440$	0	0
$441$	−1.47214	−0.0701017
$442$	0	0
$443$	7.41641	0.352364	0.176182	−	0.984358i	$-0.443625\pi$
$443$	7.41641	0.352364	0.176182	+	0.984358i	$0.443625\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−17.2361	−0.815238
$448$	0	0
$449$	29.4721	1.39088	0.695438	−	0.718586i	$-0.255210\pi$
$449$	29.4721	1.39088	0.695438	+	0.718586i	$0.255210\pi$
$450$	0	0
$451$	−39.1246	−1.84231
$452$	0	0
$453$	19.4853	0.915499
$454$	0	0
$455$	0	0
$456$	0	0
$457$	21.4721	1.00442	0.502212	−	0.864744i	$-0.332520\pi$
$457$	21.4721	1.00442	0.502212	+	0.864744i	$0.332520\pi$
$458$	0	0
$459$	−35.7771	−1.66993
$460$	0	0
$461$	8.18034	0.380996	0.190498	−	0.981688i	$-0.438990\pi$
$461$	8.18034	0.380996	0.190498	+	0.981688i	$0.438990\pi$
$462$	0	0
$463$	21.8885	1.01725	0.508623	−	0.860989i	$-0.330155\pi$
$463$	21.8885	1.01725	0.508623	+	0.860989i	$0.330155\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−10.9443	−0.506441	−0.253220	−	0.967409i	$-0.581490\pi$
$467$	−10.9443	−0.506441	−0.253220	+	0.967409i	$0.581490\pi$
$468$	0	0
$469$	0.236068	0.0109006
$470$	0	0
$471$	−6.47214	−0.298220
$472$	0	0
$473$	−26.4164	−1.21463
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−0.695048	−0.0318241
$478$	0	0
$479$	3.81966	0.174525	0.0872624	−	0.996185i	$-0.472188\pi$
$479$	3.81966	0.174525	0.0872624	+	0.996185i	$0.472188\pi$
$480$	0	0
$481$	−9.70820	−0.442656
$482$	0	0
$483$	2.18034	0.0992089
$484$	0	0
$485$	0	0
$486$	0	0
$487$	10.2361	0.463841	0.231920	−	0.972735i	$-0.425499\pi$
$487$	10.2361	0.463841	0.231920	+	0.972735i	$0.425499\pi$
$488$	0	0
$489$	−12.9443	−0.585360
$490$	0	0
$491$	10.2361	0.461947	0.230974	−	0.972960i	$-0.425809\pi$
$491$	10.2361	0.461947	0.230974	+	0.972960i	$0.425809\pi$
$492$	0	0
$493$	32.3607	1.45745
$494$	0	0
$495$	0	0
$496$	0	0
$497$	4.70820	0.211192
$498$	0	0
$499$	−28.9443	−1.29572	−0.647862	−	0.761758i	$-0.724337\pi$
$499$	−28.9443	−1.29572	−0.647862	+	0.761758i	$0.724337\pi$
$500$	0	0
$501$	0.944272	0.0421870
$502$	0	0
$503$	−43.8885	−1.95689	−0.978447	−	0.206499i	$-0.933793\pi$
$503$	−43.8885	−1.95689	−0.978447	+	0.206499i	$0.933793\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−3.12461	−0.138769
$508$	0	0
$509$	9.34752	0.414322	0.207161	−	0.978307i	$-0.433578\pi$
$509$	9.34752	0.414322	0.207161	+	0.978307i	$0.433578\pi$
$510$	0	0
$511$	13.2361	0.585529
$512$	0	0
$513$	24.7214	1.09147
$514$	0	0
$515$	0	0
$516$	0	0
$517$	8.47214	0.372604
$518$	0	0
$519$	25.3050	1.11076
$520$	0	0
$521$	22.0000	0.963837	0.481919	−	0.876216i	$-0.339940\pi$
$521$	22.0000	0.963837	0.481919	+	0.876216i	$0.339940\pi$
$522$	0	0
$523$	−28.3607	−1.24013	−0.620063	−	0.784552i	$-0.712893\pi$
$523$	−28.3607	−1.24013	−0.620063	+	0.784552i	$0.712893\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	62.8328	2.73704
$528$	0	0
$529$	−19.8885	−0.864719
$530$	0	0
$531$	−2.51471	−0.109129
$532$	0	0
$533$	29.8885	1.29462
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−4.22291	−0.182232
$538$	0	0
$539$	−4.23607	−0.182460
$540$	0	0
$541$	−1.94427	−0.0835908	−0.0417954	−	0.999126i	$-0.513308\pi$
$541$	−1.94427	−0.0835908	−0.0417954	+	0.999126i	$0.513308\pi$
$542$	0	0
$543$	−17.5279	−0.752193
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−14.2361	−0.608690	−0.304345	−	0.952562i	$-0.598438\pi$
$547$	−14.2361	−0.608690	−0.304345	+	0.952562i	$0.598438\pi$
$548$	0	0
$549$	−5.45898	−0.232984
$550$	0	0
$551$	−22.3607	−0.952597
$552$	0	0
$553$	−11.1803	−0.475436
$554$	0	0
$555$	0	0
$556$	0	0
$557$	44.8885	1.90199	0.950994	−	0.309208i	$-0.100064\pi$
$557$	44.8885	1.90199	0.950994	+	0.309208i	$0.100064\pi$
$558$	0	0
$559$	20.1803	0.853537
$560$	0	0
$561$	−33.8885	−1.43078
$562$	0	0
$563$	−9.41641	−0.396854	−0.198427	−	0.980116i	$-0.563583\pi$
$563$	−9.41641	−0.396854	−0.198427	+	0.980116i	$0.563583\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−2.41641	−0.101480
$568$	0	0
$569$	13.9443	0.584574	0.292287	−	0.956331i	$-0.405584\pi$
$569$	13.9443	0.584574	0.292287	+	0.956331i	$0.405584\pi$
$570$	0	0
$571$	12.5967	0.527157	0.263579	−	0.964638i	$-0.415097\pi$
$571$	12.5967	0.527157	0.263579	+	0.964638i	$0.415097\pi$
$572$	0	0
$573$	3.05573	0.127655
$574$	0	0
$575$	0	0
$576$	0	0
$577$	2.00000	0.0832611	0.0416305	−	0.999133i	$-0.486745\pi$
$577$	2.00000	0.0832611	0.0416305	+	0.999133i	$0.486745\pi$
$578$	0	0
$579$	17.8197	0.740560
$580$	0	0
$581$	5.70820	0.236816
$582$	0	0
$583$	−2.00000	−0.0828315
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−29.2361	−1.20670	−0.603351	−	0.797476i	$-0.706168\pi$
$587$	−29.2361	−1.20670	−0.603351	+	0.797476i	$0.706168\pi$
$588$	0	0
$589$	−43.4164	−1.78894
$590$	0	0
$591$	−9.23607	−0.379921
$592$	0	0
$593$	−25.3050	−1.03915	−0.519575	−	0.854425i	$-0.673910\pi$
$593$	−25.3050	−1.03915	−0.519575	+	0.854425i	$0.673910\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−3.41641	−0.139824
$598$	0	0
$599$	−11.1803	−0.456816	−0.228408	−	0.973565i	$-0.573352\pi$
$599$	−11.1803	−0.456816	−0.228408	+	0.973565i	$0.573352\pi$
$600$	0	0
$601$	−19.0557	−0.777299	−0.388650	−	0.921386i	$-0.627058\pi$
$601$	−19.0557	−0.777299	−0.388650	+	0.921386i	$0.627058\pi$
$602$	0	0
$603$	−0.347524	−0.0141523
$604$	0	0
$605$	0	0
$606$	0	0
$607$	33.1246	1.34449	0.672243	−	0.740330i	$-0.265331\pi$
$607$	33.1246	1.34449	0.672243	+	0.740330i	$0.265331\pi$
$608$	0	0
$609$	6.18034	0.250440
$610$	0	0
$611$	−6.47214	−0.261835
$612$	0	0
$613$	17.5836	0.710195	0.355097	−	0.934829i	$-0.384448\pi$
$613$	17.5836	0.710195	0.355097	+	0.934829i	$0.384448\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−11.9443	−0.480858	−0.240429	−	0.970667i	$-0.577288\pi$
$617$	−11.9443	−0.480858	−0.240429	+	0.970667i	$0.577288\pi$
$618$	0	0
$619$	−1.70820	−0.0686585	−0.0343293	−	0.999411i	$-0.510929\pi$
$619$	−1.70820	−0.0686585	−0.0343293	+	0.999411i	$0.510929\pi$
$620$	0	0
$621$	−9.75078	−0.391285
$622$	0	0
$623$	12.7639	0.511376
$624$	0	0
$625$	0	0
$626$	0	0
$627$	23.4164	0.935161
$628$	0	0
$629$	−19.4164	−0.774183
$630$	0	0
$631$	3.65248	0.145403	0.0727014	−	0.997354i	$-0.476838\pi$
$631$	3.65248	0.145403	0.0727014	+	0.997354i	$0.476838\pi$
$632$	0	0
$633$	−14.8328	−0.589551
$634$	0	0
$635$	0	0
$636$	0	0
$637$	3.23607	0.128218
$638$	0	0
$639$	−6.93112	−0.274191
$640$	0	0
$641$	−9.83282	−0.388373	−0.194186	−	0.980965i	$-0.562207\pi$
$641$	−9.83282	−0.388373	−0.194186	+	0.980965i	$0.562207\pi$
$642$	0	0
$643$	9.52786	0.375742	0.187871	−	0.982194i	$-0.439841\pi$
$643$	9.52786	0.375742	0.187871	+	0.982194i	$0.439841\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	15.8885	0.624643	0.312322	−	0.949976i	$-0.398893\pi$
$647$	15.8885	0.624643	0.312322	+	0.949976i	$0.398893\pi$
$648$	0	0
$649$	−7.23607	−0.284041
$650$	0	0
$651$	12.0000	0.470317
$652$	0	0
$653$	−42.9443	−1.68054	−0.840270	−	0.542169i	$-0.817603\pi$
$653$	−42.9443	−1.68054	−0.840270	+	0.542169i	$0.817603\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−19.4853	−0.760194
$658$	0	0
$659$	17.8885	0.696839	0.348419	−	0.937339i	$-0.386719\pi$
$659$	17.8885	0.696839	0.348419	+	0.937339i	$0.386719\pi$
$660$	0	0
$661$	46.7214	1.81725	0.908625	−	0.417613i	$-0.137133\pi$
$661$	46.7214	1.81725	0.908625	+	0.417613i	$0.137133\pi$
$662$	0	0
$663$	25.8885	1.00543
$664$	0	0
$665$	0	0
$666$	0	0
$667$	8.81966	0.341499
$668$	0	0
$669$	−2.69505	−0.104197
$670$	0	0
$671$	−15.7082	−0.606408
$672$	0	0
$673$	−28.4721	−1.09752	−0.548760	−	0.835980i	$-0.684900\pi$
$673$	−28.4721	−1.09752	−0.548760	+	0.835980i	$0.684900\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−30.3607	−1.16686	−0.583428	−	0.812165i	$-0.698289\pi$
$677$	−30.3607	−1.16686	−0.583428	+	0.812165i	$0.698289\pi$
$678$	0	0
$679$	−0.763932	−0.0293170
$680$	0	0
$681$	−6.69505	−0.256555
$682$	0	0
$683$	−26.1246	−0.999630	−0.499815	−	0.866132i	$-0.666599\pi$
$683$	−26.1246	−0.999630	−0.499815	+	0.866132i	$0.666599\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	5.52786	0.210901
$688$	0	0
$689$	1.52786	0.0582070
$690$	0	0
$691$	−18.1803	−0.691613	−0.345806	−	0.938306i	$-0.612395\pi$
$691$	−18.1803	−0.691613	−0.345806	+	0.938306i	$0.612395\pi$
$692$	0	0
$693$	6.23607	0.236889
$694$	0	0
$695$	0	0
$696$	0	0
$697$	59.7771	2.26422
$698$	0	0
$699$	12.2918	0.464918
$700$	0	0
$701$	−46.9443	−1.77306	−0.886530	−	0.462670i	$-0.846891\pi$
$701$	−46.9443	−1.77306	−0.886530	+	0.462670i	$0.846891\pi$
$702$	0	0
$703$	13.4164	0.506009
$704$	0	0
$705$	0	0
$706$	0	0
$707$	9.23607	0.347358
$708$	0	0
$709$	−47.8885	−1.79849	−0.899246	−	0.437443i	$-0.855884\pi$
$709$	−47.8885	−1.79849	−0.899246	+	0.437443i	$0.855884\pi$
$710$	0	0
$711$	16.4590	0.617260
$712$	0	0
$713$	17.1246	0.641322
$714$	0	0
$715$	0	0
$716$	0	0
$717$	17.8885	0.668060
$718$	0	0
$719$	6.18034	0.230488	0.115244	−	0.993337i	$-0.463235\pi$
$719$	6.18034	0.230488	0.115244	+	0.993337i	$0.463235\pi$
$720$	0	0
$721$	−0.472136	−0.0175833
$722$	0	0
$723$	−15.4164	−0.573342
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−20.9443	−0.776780	−0.388390	−	0.921495i	$-0.626969\pi$
$727$	−20.9443	−0.776780	−0.388390	+	0.921495i	$0.626969\pi$
$728$	0	0
$729$	24.0557	0.890953
$730$	0	0
$731$	40.3607	1.49279
$732$	0	0
$733$	−4.00000	−0.147743	−0.0738717	−	0.997268i	$-0.523536\pi$
$733$	−4.00000	−0.147743	−0.0738717	+	0.997268i	$0.523536\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−1.00000	−0.0368355
$738$	0	0
$739$	5.65248	0.207930	0.103965	−	0.994581i	$-0.466847\pi$
$739$	5.65248	0.207930	0.103965	+	0.994581i	$0.466847\pi$
$740$	0	0
$741$	−17.8885	−0.657152
$742$	0	0
$743$	−1.52786	−0.0560519	−0.0280259	−	0.999607i	$-0.508922\pi$
$743$	−1.52786	−0.0560519	−0.0280259	+	0.999607i	$0.508922\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−8.40325	−0.307459
$748$	0	0
$749$	8.00000	0.292314
$750$	0	0
$751$	−20.9443	−0.764267	−0.382134	−	0.924107i	$-0.624811\pi$
$751$	−20.9443	−0.764267	−0.382134	+	0.924107i	$0.624811\pi$
$752$	0	0
$753$	3.05573	0.111357
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−46.4164	−1.68703	−0.843517	−	0.537103i	$-0.819519\pi$
$757$	−46.4164	−1.68703	−0.843517	+	0.537103i	$0.819519\pi$
$758$	0	0
$759$	−9.23607	−0.335248
$760$	0	0
$761$	−43.7771	−1.58692	−0.793459	−	0.608624i	$-0.791722\pi$
$761$	−43.7771	−1.58692	−0.793459	+	0.608624i	$0.791722\pi$
$762$	0	0
$763$	−18.4164	−0.666719
$764$	0	0
$765$	0	0
$766$	0	0
$767$	5.52786	0.199600
$768$	0	0
$769$	33.0132	1.19048	0.595242	−	0.803546i	$-0.297056\pi$
$769$	33.0132	1.19048	0.595242	+	0.803546i	$0.297056\pi$
$770$	0	0
$771$	−23.0557	−0.830332
$772$	0	0
$773$	−27.8197	−1.00060	−0.500302	−	0.865851i	$-0.666778\pi$
$773$	−27.8197	−1.00060	−0.500302	+	0.865851i	$0.666778\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−3.70820	−0.133031
$778$	0	0
$779$	−41.3050	−1.47990
$780$	0	0
$781$	−19.9443	−0.713662
$782$	0	0
$783$	−27.6393	−0.987749
$784$	0	0
$785$	0	0
$786$	0	0
$787$	45.2361	1.61249	0.806246	−	0.591581i	$-0.201496\pi$
$787$	45.2361	1.61249	0.806246	+	0.591581i	$0.201496\pi$
$788$	0	0
$789$	14.5410	0.517674
$790$	0	0
$791$	−12.4164	−0.441477
$792$	0	0
$793$	12.0000	0.426132
$794$	0	0
$795$	0	0
$796$	0	0
$797$	8.58359	0.304046	0.152023	−	0.988377i	$-0.451421\pi$
$797$	8.58359	0.304046	0.152023	+	0.988377i	$0.451421\pi$
$798$	0	0
$799$	−12.9443	−0.457935
$800$	0	0
$801$	−18.7902	−0.663921
$802$	0	0
$803$	−56.0689	−1.97863
$804$	0	0
$805$	0	0
$806$	0	0
$807$	2.11146	0.0743268
$808$	0	0
$809$	20.5279	0.721721	0.360861	−	0.932620i	$-0.382483\pi$
$809$	20.5279	0.721721	0.360861	+	0.932620i	$0.382483\pi$
$810$	0	0
$811$	−46.7214	−1.64061	−0.820304	−	0.571927i	$-0.806196\pi$
$811$	−46.7214	−1.64061	−0.820304	+	0.571927i	$0.806196\pi$
$812$	0	0
$813$	−12.7214	−0.446158
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−27.8885	−0.975697
$818$	0	0
$819$	−4.76393	−0.166465
$820$	0	0
$821$	−24.8328	−0.866671	−0.433336	−	0.901233i	$-0.642664\pi$
$821$	−24.8328	−0.866671	−0.433336	+	0.901233i	$0.642664\pi$
$822$	0	0
$823$	−0.347524	−0.0121139	−0.00605697	−	0.999982i	$-0.501928\pi$
$823$	−0.347524	−0.0121139	−0.00605697	+	0.999982i	$0.501928\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−25.5410	−0.888148	−0.444074	−	0.895990i	$-0.646467\pi$
$827$	−25.5410	−0.888148	−0.444074	+	0.895990i	$0.646467\pi$
$828$	0	0
$829$	−52.3607	−1.81856	−0.909281	−	0.416183i	$-0.863368\pi$
$829$	−52.3607	−1.81856	−0.909281	+	0.416183i	$0.863368\pi$
$830$	0	0
$831$	−19.6393	−0.681280
$832$	0	0
$833$	6.47214	0.224246
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−53.6656	−1.85496
$838$	0	0
$839$	−0.652476	−0.0225260	−0.0112630	−	0.999937i	$-0.503585\pi$
$839$	−0.652476	−0.0225260	−0.0112630	+	0.999937i	$0.503585\pi$
$840$	0	0
$841$	−4.00000	−0.137931
$842$	0	0
$843$	36.2918	1.24996
$844$	0	0
$845$	0	0
$846$	0	0
$847$	6.94427	0.238608
$848$	0	0
$849$	−11.6393	−0.399460
$850$	0	0
$851$	−5.29180	−0.181400
$852$	0	0
$853$	−0.583592	−0.0199818	−0.00999091	−	0.999950i	$-0.503180\pi$
$853$	−0.583592	−0.0199818	−0.00999091	+	0.999950i	$0.503180\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	38.1803	1.30422	0.652108	−	0.758126i	$-0.273885\pi$
$857$	38.1803	1.30422	0.652108	+	0.758126i	$0.273885\pi$
$858$	0	0
$859$	22.3607	0.762937	0.381468	−	0.924382i	$-0.375419\pi$
$859$	22.3607	0.762937	0.381468	+	0.924382i	$0.375419\pi$
$860$	0	0
$861$	11.4164	0.389070
$862$	0	0
$863$	49.6525	1.69019	0.845095	−	0.534616i	$-0.179544\pi$
$863$	49.6525	1.69019	0.845095	+	0.534616i	$0.179544\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	30.7639	1.04480
$868$	0	0
$869$	47.3607	1.60660
$870$	0	0
$871$	0.763932	0.0258848
$872$	0	0
$873$	1.12461	0.0380623
$874$	0	0
$875$	0	0
$876$	0	0
$877$	14.3607	0.484926	0.242463	−	0.970161i	$-0.422045\pi$
$877$	14.3607	0.484926	0.242463	+	0.970161i	$0.422045\pi$
$878$	0	0
$879$	−11.2786	−0.380419
$880$	0	0
$881$	28.1803	0.949420	0.474710	−	0.880142i	$-0.342553\pi$
$881$	28.1803	0.949420	0.474710	+	0.880142i	$0.342553\pi$
$882$	0	0
$883$	−50.5967	−1.70272	−0.851358	−	0.524585i	$-0.824220\pi$
$883$	−50.5967	−1.70272	−0.851358	+	0.524585i	$0.824220\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−52.6525	−1.76790	−0.883949	−	0.467584i	$-0.845124\pi$
$887$	−52.6525	−1.76790	−0.883949	+	0.467584i	$0.845124\pi$
$888$	0	0
$889$	−17.6525	−0.592045
$890$	0	0
$891$	10.2361	0.342921
$892$	0	0
$893$	8.94427	0.299309
$894$	0	0
$895$	0	0
$896$	0	0
$897$	7.05573	0.235584
$898$	0	0
$899$	48.5410	1.61893
$900$	0	0
$901$	3.05573	0.101801
$902$	0	0
$903$	7.70820	0.256513
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−18.8328	−0.625333	−0.312667	−	0.949863i	$-0.601222\pi$
$907$	−18.8328	−0.625333	−0.312667	+	0.949863i	$0.601222\pi$
$908$	0	0
$909$	−13.5967	−0.450976
$910$	0	0
$911$	−23.1803	−0.767999	−0.383999	−	0.923333i	$-0.625454\pi$
$911$	−23.1803	−0.767999	−0.383999	+	0.923333i	$0.625454\pi$
$912$	0	0
$913$	−24.1803	−0.800252
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−0.944272	−0.0311826
$918$	0	0
$919$	32.2361	1.06337	0.531685	−	0.846942i	$-0.321559\pi$
$919$	32.2361	1.06337	0.531685	+	0.846942i	$0.321559\pi$
$920$	0	0
$921$	38.8328	1.27958
$922$	0	0
$923$	15.2361	0.501501
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0.695048	0.0228284
$928$	0	0
$929$	51.7082	1.69649	0.848246	−	0.529603i	$-0.177659\pi$
$929$	51.7082	1.69649	0.848246	+	0.529603i	$0.177659\pi$
$930$	0	0
$931$	−4.47214	−0.146568
$932$	0	0
$933$	25.1672	0.823937
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−30.7639	−1.00501	−0.502507	−	0.864573i	$-0.667589\pi$
$937$	−30.7639	−1.00501	−0.502507	+	0.864573i	$0.667589\pi$
$938$	0	0
$939$	−35.1935	−1.14850
$940$	0	0
$941$	−0.763932	−0.0249035	−0.0124517	−	0.999922i	$-0.503964\pi$
$941$	−0.763932	−0.0249035	−0.0124517	+	0.999922i	$0.503964\pi$
$942$	0	0
$943$	16.2918	0.530534
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−18.8328	−0.611984	−0.305992	−	0.952034i	$-0.598988\pi$
$947$	−18.8328	−0.611984	−0.305992	+	0.952034i	$0.598988\pi$
$948$	0	0
$949$	42.8328	1.39041
$950$	0	0
$951$	23.9311	0.776020
$952$	0	0
$953$	5.47214	0.177260	0.0886299	−	0.996065i	$-0.471751\pi$
$953$	5.47214	0.177260	0.0886299	+	0.996065i	$0.471751\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−26.1803	−0.846290
$958$	0	0
$959$	−6.94427	−0.224242
$960$	0	0
$961$	63.2492	2.04030
$962$	0	0
$963$	−11.7771	−0.379511
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−49.8885	−1.60431	−0.802154	−	0.597118i	$-0.796313\pi$
$967$	−49.8885	−1.60431	−0.802154	+	0.597118i	$0.796313\pi$
$968$	0	0
$969$	−35.7771	−1.14933
$970$	0	0
$971$	18.0000	0.577647	0.288824	−	0.957382i	$-0.406736\pi$
$971$	18.0000	0.577647	0.288824	+	0.957382i	$0.406736\pi$
$972$	0	0
$973$	20.6525	0.662088
$974$	0	0
$975$	0	0
$976$	0	0
$977$	2.52786	0.0808735	0.0404368	−	0.999182i	$-0.487125\pi$
$977$	2.52786	0.0808735	0.0404368	+	0.999182i	$0.487125\pi$
$978$	0	0
$979$	−54.0689	−1.72805
$980$	0	0
$981$	27.1115	0.865602
$982$	0	0
$983$	32.5410	1.03790	0.518949	−	0.854805i	$-0.326324\pi$
$983$	32.5410	1.03790	0.518949	+	0.854805i	$0.326324\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−2.47214	−0.0786890
$988$	0	0
$989$	11.0000	0.349780
$990$	0	0
$991$	9.18034	0.291623	0.145812	−	0.989312i	$-0.453421\pi$
$991$	9.18034	0.291623	0.145812	+	0.989312i	$0.453421\pi$
$992$	0	0
$993$	13.9574	0.442926
$994$	0	0
$995$	0	0
$996$	0	0
$997$	18.5836	0.588548	0.294274	−	0.955721i	$-0.404922\pi$
$997$	18.5836	0.588548	0.294274	+	0.955721i	$0.404922\pi$
$998$	0	0
$999$	16.5836	0.524682

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2800.2.a.bh.1.2		2
4.3	odd	2		175.2.a.d.1.1	✓	2
5.2	odd	4		2800.2.g.s.449.2		4
5.3	odd	4		2800.2.g.s.449.3		4
5.4	even	2		2800.2.a.bp.1.1		2
12.11	even	2		1575.2.a.s.1.2		2
20.3	even	4		175.2.b.c.99.4		4
20.7	even	4		175.2.b.c.99.1		4
20.19	odd	2		175.2.a.e.1.2	yes	2
28.27	even	2		1225.2.a.n.1.1		2
60.23	odd	4		1575.2.d.k.1324.1		4
60.47	odd	4		1575.2.d.k.1324.4		4
60.59	even	2		1575.2.a.n.1.1		2
140.27	odd	4		1225.2.b.k.99.1		4
140.83	odd	4		1225.2.b.k.99.4		4
140.139	even	2		1225.2.a.u.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
175.2.a.d.1.1	✓	2	4.3	odd	2
175.2.a.e.1.2	yes	2	20.19	odd	2
175.2.b.c.99.1		4	20.7	even	4
175.2.b.c.99.4		4	20.3	even	4
1225.2.a.n.1.1		2	28.27	even	2
1225.2.a.u.1.2		2	140.139	even	2
1225.2.b.k.99.1		4	140.27	odd	4
1225.2.b.k.99.4		4	140.83	odd	4
1575.2.a.n.1.1		2	60.59	even	2
1575.2.a.s.1.2		2	12.11	even	2
1575.2.d.k.1324.1		4	60.23	odd	4
1575.2.d.k.1324.4		4	60.47	odd	4
2800.2.a.bh.1.2		2	1.1	even	1	trivial
2800.2.a.bp.1.1		2	5.4	even	2
2800.2.g.s.449.2		4	5.2	odd	4
2800.2.g.s.449.3		4	5.3	odd	4

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:	\( N \)	\(=\)	\( 2800 = 2^{4} \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2800.a (trivial)

\(f(q)\)	\(=\)	\(q+1.23607 q^{3} +1.00000 q^{7} -1.47214 q^{9} +O(q^{10})\)	\(q+1.23607 q^{3} +1.00000 q^{7} -1.47214 q^{9} -4.23607 q^{11} +3.23607 q^{13} +6.47214 q^{17} -4.47214 q^{19} +1.23607 q^{21} +1.76393 q^{23} -5.52786 q^{27} +5.00000 q^{29} +9.70820 q^{31} -5.23607 q^{33} -3.00000 q^{37} +4.00000 q^{39} +9.23607 q^{41} +6.23607 q^{43} -2.00000 q^{47} +1.00000 q^{49} +8.00000 q^{51} +0.472136 q^{53} -5.52786 q^{57} +1.70820 q^{59} +3.70820 q^{61} -1.47214 q^{63} +0.236068 q^{67} +2.18034 q^{69} +4.70820 q^{71} +13.2361 q^{73} -4.23607 q^{77} -11.1803 q^{79} -2.41641 q^{81} +5.70820 q^{83} +6.18034 q^{87} +12.7639 q^{89} +3.23607 q^{91} +12.0000 q^{93} -0.763932 q^{97} +6.23607 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} + 2 q^{7} + 6 q^{9}+O(q^{10}) \) 2 * q - 2 * q^3 + 2 * q^7 + 6 * q^9	\( 2 q - 2 q^{3} + 2 q^{7} + 6 q^{9} - 4 q^{11} + 2 q^{13} + 4 q^{17} - 2 q^{21} + 8 q^{23} - 20 q^{27} + 10 q^{29} + 6 q^{31} - 6 q^{33} - 6 q^{37} + 8 q^{39} + 14 q^{41} + 8 q^{43} - 4 q^{47} + 2 q^{49} + 16 q^{51} - 8 q^{53} - 20 q^{57} - 10 q^{59} - 6 q^{61} + 6 q^{63} - 4 q^{67} - 18 q^{69} - 4 q^{71} + 22 q^{73} - 4 q^{77} + 22 q^{81} - 2 q^{83} - 10 q^{87} + 30 q^{89} + 2 q^{91} + 24 q^{93} - 6 q^{97} + 8 q^{99}+O(q^{100}) \) 2 * q - 2 * q^3 + 2 * q^7 + 6 * q^9 - 4 * q^11 + 2 * q^13 + 4 * q^17 - 2 * q^21 + 8 * q^23 - 20 * q^27 + 10 * q^29 + 6 * q^31 - 6 * q^33 - 6 * q^37 + 8 * q^39 + 14 * q^41 + 8 * q^43 - 4 * q^47 + 2 * q^49 + 16 * q^51 - 8 * q^53 - 20 * q^57 - 10 * q^59 - 6 * q^61 + 6 * q^63 - 4 * q^67 - 18 * q^69 - 4 * q^71 + 22 * q^73 - 4 * q^77 + 22 * q^81 - 2 * q^83 - 10 * q^87 + 30 * q^89 + 2 * q^91 + 24 * q^93 - 6 * q^97 + 8 * q^99

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