LMFDB - Embedded newform 3800.2.a.bb.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3800,2,Mod(1,3800)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3800, 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("3800.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3800 = 2^{3} \cdot 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3800.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:	$30.3431527681$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 12x^{4} + 16x^{3} + 33x^{2} - 4x - 7 $ x^6 - 2x^5 - 12x^4 + 16x^3 + 33x^2 - 4*x - 7
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-0.471016$ of defining polynomial
Character	$\chi$	$=$	3800.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.471016 q^{3} +0.567324 q^{7} -2.77814 q^{9} +O(q^{10})$	$q+0.471016 q^{3} +0.567324 q^{7} -2.77814 q^{9} +4.37804 q^{11} -0.165457 q^{13} -7.94693 q^{17} +1.00000 q^{19} +0.267219 q^{21} +3.87445 q^{23} -2.72160 q^{27} +3.53231 q^{29} +3.20380 q^{31} +2.06213 q^{33} +10.1779 q^{37} -0.0779330 q^{39} -5.97409 q^{41} +12.0904 q^{43} -5.46140 q^{47} -6.67814 q^{49} -3.74313 q^{51} -2.00333 q^{53} +0.471016 q^{57} +8.32164 q^{59} +11.8181 q^{61} -1.57611 q^{63} +8.79599 q^{67} +1.82493 q^{69} +0.720031 q^{71} +4.54017 q^{73} +2.48377 q^{77} +11.7123 q^{79} +7.05251 q^{81} -6.72351 q^{83} +1.66378 q^{87} +8.11941 q^{89} -0.0938678 q^{91} +1.50904 q^{93} -13.6321 q^{97} -12.1628 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 2 q^{3} - 2 q^{7} + 10 q^{9}+O(q^{10}) $ 6 * q - 2 * q^3 - 2 * q^7 + 10 * q^9	$ 6 q - 2 q^{3} - 2 q^{7} + 10 q^{9} + 3 q^{11} + 3 q^{13} + 2 q^{17} + 6 q^{19} + 11 q^{21} - 4 q^{23} - 20 q^{27} + 7 q^{29} + 5 q^{31} + 16 q^{33} + 8 q^{39} + 11 q^{41} + 7 q^{43} - 20 q^{47} - 2 q^{49} + 13 q^{51} + 7 q^{53} - 2 q^{57} - 4 q^{59} + 13 q^{61} + q^{63} - 25 q^{67} + 7 q^{69} + 29 q^{71} + 19 q^{73} - 24 q^{77} + 28 q^{79} + 38 q^{81} + 15 q^{83} - 57 q^{87} - 12 q^{89} + 27 q^{93} + 13 q^{97} + 10 q^{99}+O(q^{100}) $ 6 * q - 2 * q^3 - 2 * q^7 + 10 * q^9 + 3 * q^11 + 3 * q^13 + 2 * q^17 + 6 * q^19 + 11 * q^21 - 4 * q^23 - 20 * q^27 + 7 * q^29 + 5 * q^31 + 16 * q^33 + 8 * q^39 + 11 * q^41 + 7 * q^43 - 20 * q^47 - 2 * q^49 + 13 * q^51 + 7 * q^53 - 2 * q^57 - 4 * q^59 + 13 * q^61 + q^63 - 25 * q^67 + 7 * q^69 + 29 * q^71 + 19 * q^73 - 24 * q^77 + 28 * q^79 + 38 * q^81 + 15 * q^83 - 57 * q^87 - 12 * q^89 + 27 * q^93 + 13 * q^97 + 10 * 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
$3$	0.471016	0.271941	0.135971	−	0.990713i	$-0.456585\pi$
$3$	0.471016	0.271941	0.135971	+	0.990713i	$0.456585\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	0.567324	0.214428	0.107214	−	0.994236i	$-0.465807\pi$
$7$	0.567324	0.214428	0.107214	+	0.994236i	$0.465807\pi$
$8$	0	0
$9$	−2.77814	−0.926048
$10$	0	0
$11$	4.37804	1.32003	0.660014	−	0.751253i	$-0.270550\pi$
$11$	4.37804	1.32003	0.660014	+	0.751253i	$0.270550\pi$
$12$	0	0
$13$	−0.165457	−0.0458895	−0.0229448	−	0.999737i	$-0.507304\pi$
$13$	−0.165457	−0.0458895	−0.0229448	+	0.999737i	$0.507304\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−7.94693	−1.92741	−0.963707	−	0.266963i	$-0.913980\pi$
$17$	−7.94693	−1.92741	−0.963707	+	0.266963i	$0.913980\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	0.267219	0.0583119
$22$	0	0
$23$	3.87445	0.807879	0.403939	−	0.914786i	$-0.367641\pi$
$23$	3.87445	0.807879	0.403939	+	0.914786i	$0.367641\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−2.72160	−0.523772
$28$	0	0
$29$	3.53231	0.655934	0.327967	−	0.944689i	$-0.393636\pi$
$29$	3.53231	0.655934	0.327967	+	0.944689i	$0.393636\pi$
$30$	0	0
$31$	3.20380	0.575419	0.287709	−	0.957718i	$-0.407106\pi$
$31$	3.20380	0.575419	0.287709	+	0.957718i	$0.407106\pi$
$32$	0	0
$33$	2.06213	0.358970
$34$	0	0
$35$	0	0
$36$	0	0
$37$	10.1779	1.67323	0.836617	−	0.547788i	$-0.184530\pi$
$37$	10.1779	1.67323	0.836617	+	0.547788i	$0.184530\pi$
$38$	0	0
$39$	−0.0779330	−0.0124793
$40$	0	0
$41$	−5.97409	−0.932996	−0.466498	−	0.884522i	$-0.654485\pi$
$41$	−5.97409	−0.932996	−0.466498	+	0.884522i	$0.654485\pi$
$42$	0	0
$43$	12.0904	1.84376	0.921881	−	0.387472i	$-0.126652\pi$
$43$	12.0904	1.84376	0.921881	+	0.387472i	$0.126652\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−5.46140	−0.796627	−0.398314	−	0.917249i	$-0.630404\pi$
$47$	−5.46140	−0.796627	−0.398314	+	0.917249i	$0.630404\pi$
$48$	0	0
$49$	−6.67814	−0.954020
$50$	0	0
$51$	−3.74313	−0.524143
$52$	0	0
$53$	−2.00333	−0.275179	−0.137589	−	0.990489i	$-0.543935\pi$
$53$	−2.00333	−0.275179	−0.137589	+	0.990489i	$0.543935\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0.471016	0.0623876
$58$	0	0
$59$	8.32164	1.08339	0.541693	−	0.840577i	$-0.317784\pi$
$59$	8.32164	1.08339	0.541693	+	0.840577i	$0.317784\pi$
$60$	0	0
$61$	11.8181	1.51315	0.756573	−	0.653909i	$-0.226872\pi$
$61$	11.8181	1.51315	0.756573	+	0.653909i	$0.226872\pi$
$62$	0	0
$63$	−1.57611	−0.198571
$64$	0	0
$65$	0	0
$66$	0	0
$67$	8.79599	1.07460	0.537300	−	0.843391i	$-0.319444\pi$
$67$	8.79599	1.07460	0.537300	+	0.843391i	$0.319444\pi$
$68$	0	0
$69$	1.82493	0.219696
$70$	0	0
$71$	0.720031	0.0854520	0.0427260	−	0.999087i	$-0.486396\pi$
$71$	0.720031	0.0854520	0.0427260	+	0.999087i	$0.486396\pi$
$72$	0	0
$73$	4.54017	0.531386	0.265693	−	0.964058i	$-0.414399\pi$
$73$	4.54017	0.531386	0.265693	+	0.964058i	$0.414399\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	2.48377	0.283051
$78$	0	0
$79$	11.7123	1.31774	0.658870	−	0.752257i	$-0.271035\pi$
$79$	11.7123	1.31774	0.658870	+	0.752257i	$0.271035\pi$
$80$	0	0
$81$	7.05251	0.783613
$82$	0	0
$83$	−6.72351	−0.738001	−0.369000	−	0.929429i	$-0.620300\pi$
$83$	−6.72351	−0.738001	−0.369000	+	0.929429i	$0.620300\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	1.66378	0.178376
$88$	0	0
$89$	8.11941	0.860656	0.430328	−	0.902673i	$-0.358398\pi$
$89$	8.11941	0.860656	0.430328	+	0.902673i	$0.358398\pi$
$90$	0	0
$91$	−0.0938678	−0.00984002
$92$	0	0
$93$	1.50904	0.156480
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−13.6321	−1.38413	−0.692065	−	0.721835i	$-0.743299\pi$
$97$	−13.6321	−1.38413	−0.692065	+	0.721835i	$0.743299\pi$
$98$	0	0
$99$	−12.1628	−1.22241
$100$	0	0
$101$	19.2686	1.91730	0.958649	−	0.284590i	$-0.0918575\pi$
$101$	19.2686	1.91730	0.958649	+	0.284590i	$0.0918575\pi$
$102$	0	0
$103$	−19.2069	−1.89251	−0.946257	−	0.323417i	$-0.895168\pi$
$103$	−19.2069	−1.89251	−0.946257	+	0.323417i	$0.895168\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−12.0091	−1.16096	−0.580480	−	0.814275i	$-0.697135\pi$
$107$	−12.0091	−1.16096	−0.580480	+	0.814275i	$0.697135\pi$
$108$	0	0
$109$	−15.0032	−1.43704	−0.718521	−	0.695505i	$-0.755180\pi$
$109$	−15.0032	−1.43704	−0.718521	+	0.695505i	$0.755180\pi$
$110$	0	0
$111$	4.79395	0.455022
$112$	0	0
$113$	8.13049	0.764852	0.382426	−	0.923986i	$-0.375089\pi$
$113$	8.13049	0.764852	0.382426	+	0.923986i	$0.375089\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0.459664	0.0424959
$118$	0	0
$119$	−4.50848	−0.413292
$120$	0	0
$121$	8.16722	0.742474
$122$	0	0
$123$	−2.81389	−0.253720
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−8.37051	−0.742762	−0.371381	−	0.928480i	$-0.621116\pi$
$127$	−8.37051	−0.742762	−0.371381	+	0.928480i	$0.621116\pi$
$128$	0	0
$129$	5.69476	0.501395
$130$	0	0
$131$	−5.03206	−0.439653	−0.219826	−	0.975539i	$-0.570549\pi$
$131$	−5.03206	−0.439653	−0.219826	+	0.975539i	$0.570549\pi$
$132$	0	0
$133$	0.567324	0.0491932
$134$	0	0
$135$	0	0
$136$	0	0
$137$	20.6498	1.76423	0.882115	−	0.471033i	$-0.156119\pi$
$137$	20.6498	1.76423	0.882115	+	0.471033i	$0.156119\pi$
$138$	0	0
$139$	11.4726	0.973092	0.486546	−	0.873655i	$-0.338257\pi$
$139$	11.4726	0.973092	0.486546	+	0.873655i	$0.338257\pi$
$140$	0	0
$141$	−2.57241	−0.216636
$142$	0	0
$143$	−0.724378	−0.0605755
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−3.14551	−0.259438
$148$	0	0
$149$	13.0611	1.07001	0.535003	−	0.844850i	$-0.320311\pi$
$149$	13.0611	1.07001	0.535003	+	0.844850i	$0.320311\pi$
$150$	0	0
$151$	17.8177	1.44998	0.724992	−	0.688758i	$-0.241844\pi$
$151$	17.8177	1.44998	0.724992	+	0.688758i	$0.241844\pi$
$152$	0	0
$153$	22.0777	1.78488
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−11.3959	−0.909490	−0.454745	−	0.890622i	$-0.650270\pi$
$157$	−11.3959	−0.909490	−0.454745	+	0.890622i	$0.650270\pi$
$158$	0	0
$159$	−0.943601	−0.0748324
$160$	0	0
$161$	2.19807	0.173232
$162$	0	0
$163$	16.5741	1.29819	0.649093	−	0.760709i	$-0.275149\pi$
$163$	16.5741	1.29819	0.649093	+	0.760709i	$0.275149\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−19.3378	−1.49640	−0.748200	−	0.663473i	$-0.769082\pi$
$167$	−19.3378	−1.49640	−0.748200	+	0.663473i	$0.769082\pi$
$168$	0	0
$169$	−12.9726	−0.997894
$170$	0	0
$171$	−2.77814	−0.212450
$172$	0	0
$173$	9.31290	0.708046	0.354023	−	0.935237i	$-0.384813\pi$
$173$	9.31290	0.708046	0.354023	+	0.935237i	$0.384813\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	3.91963	0.294617
$178$	0	0
$179$	24.1703	1.80658	0.903288	−	0.429035i	$-0.141146\pi$
$179$	24.1703	1.80658	0.903288	+	0.429035i	$0.141146\pi$
$180$	0	0
$181$	18.6981	1.38982	0.694910	−	0.719097i	$-0.255444\pi$
$181$	18.6981	1.38982	0.694910	+	0.719097i	$0.255444\pi$
$182$	0	0
$183$	5.56649	0.411487
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−34.7920	−2.54424
$188$	0	0
$189$	−1.54403	−0.112312
$190$	0	0
$191$	6.27452	0.454008	0.227004	−	0.973894i	$-0.427107\pi$
$191$	6.27452	0.454008	0.227004	+	0.973894i	$0.427107\pi$
$192$	0	0
$193$	8.49096	0.611193	0.305596	−	0.952161i	$-0.401144\pi$
$193$	8.49096	0.611193	0.305596	+	0.952161i	$0.401144\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	5.97358	0.425600	0.212800	−	0.977096i	$-0.431742\pi$
$197$	5.97358	0.425600	0.212800	+	0.977096i	$0.431742\pi$
$198$	0	0
$199$	19.8763	1.40899	0.704497	−	0.709707i	$-0.251173\pi$
$199$	19.8763	1.40899	0.704497	+	0.709707i	$0.251173\pi$
$200$	0	0
$201$	4.14305	0.292228
$202$	0	0
$203$	2.00397	0.140651
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−10.7638	−0.748135
$208$	0	0
$209$	4.37804	0.302835
$210$	0	0
$211$	−2.84905	−0.196137	−0.0980685	−	0.995180i	$-0.531266\pi$
$211$	−2.84905	−0.196137	−0.0980685	+	0.995180i	$0.531266\pi$
$212$	0	0
$213$	0.339146	0.0232379
$214$	0	0
$215$	0	0
$216$	0	0
$217$	1.81759	0.123386
$218$	0	0
$219$	2.13849	0.144506
$220$	0	0
$221$	1.31488	0.0884481
$222$	0	0
$223$	−0.883914	−0.0591912	−0.0295956	−	0.999562i	$-0.509422\pi$
$223$	−0.883914	−0.0591912	−0.0295956	+	0.999562i	$0.509422\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	1.02836	0.0682549	0.0341275	−	0.999417i	$-0.489135\pi$
$227$	1.02836	0.0682549	0.0341275	+	0.999417i	$0.489135\pi$
$228$	0	0
$229$	−24.5878	−1.62481	−0.812404	−	0.583095i	$-0.801842\pi$
$229$	−24.5878	−1.62481	−0.812404	+	0.583095i	$0.801842\pi$
$230$	0	0
$231$	1.16989	0.0769734
$232$	0	0
$233$	−5.23585	−0.343012	−0.171506	−	0.985183i	$-0.554863\pi$
$233$	−5.23585	−0.343012	−0.171506	+	0.985183i	$0.554863\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	5.51669	0.358348
$238$	0	0
$239$	−12.8124	−0.828767	−0.414384	−	0.910102i	$-0.636003\pi$
$239$	−12.8124	−0.828767	−0.414384	+	0.910102i	$0.636003\pi$
$240$	0	0
$241$	−11.9996	−0.772962	−0.386481	−	0.922297i	$-0.626309\pi$
$241$	−11.9996	−0.772962	−0.386481	+	0.922297i	$0.626309\pi$
$242$	0	0
$243$	11.4866	0.736869
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−0.165457	−0.0105278
$248$	0	0
$249$	−3.16688	−0.200693
$250$	0	0
$251$	23.7189	1.49713	0.748563	−	0.663064i	$-0.230744\pi$
$251$	23.7189	1.49713	0.748563	+	0.663064i	$0.230744\pi$
$252$	0	0
$253$	16.9625	1.06642
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−2.11678	−0.132041	−0.0660206	−	0.997818i	$-0.521030\pi$
$257$	−2.11678	−0.132041	−0.0660206	+	0.997818i	$0.521030\pi$
$258$	0	0
$259$	5.77416	0.358789
$260$	0	0
$261$	−9.81327	−0.607426
$262$	0	0
$263$	−1.95674	−0.120658	−0.0603288	−	0.998179i	$-0.519215\pi$
$263$	−1.95674	−0.120658	−0.0603288	+	0.998179i	$0.519215\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	3.82437	0.234048
$268$	0	0
$269$	−22.8278	−1.39184	−0.695919	−	0.718120i	$-0.745003\pi$
$269$	−22.8278	−1.39184	−0.695919	+	0.718120i	$0.745003\pi$
$270$	0	0
$271$	−15.8203	−0.961014	−0.480507	−	0.876991i	$-0.659547\pi$
$271$	−15.8203	−0.961014	−0.480507	+	0.876991i	$0.659547\pi$
$272$	0	0
$273$	−0.0442133	−0.00267591
$274$	0	0
$275$	0	0
$276$	0	0
$277$	9.47924	0.569553	0.284776	−	0.958594i	$-0.408081\pi$
$277$	9.47924	0.569553	0.284776	+	0.958594i	$0.408081\pi$
$278$	0	0
$279$	−8.90061	−0.532866
$280$	0	0
$281$	13.3160	0.794367	0.397183	−	0.917739i	$-0.369988\pi$
$281$	13.3160	0.794367	0.397183	+	0.917739i	$0.369988\pi$
$282$	0	0
$283$	−9.20676	−0.547285	−0.273643	−	0.961831i	$-0.588229\pi$
$283$	−9.20676	−0.547285	−0.273643	+	0.961831i	$0.588229\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−3.38924	−0.200061
$288$	0	0
$289$	46.1537	2.71492
$290$	0	0
$291$	−6.42094	−0.376402
$292$	0	0
$293$	10.3435	0.604273	0.302137	−	0.953265i	$-0.402300\pi$
$293$	10.3435	0.604273	0.302137	+	0.953265i	$0.402300\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−11.9153	−0.691394
$298$	0	0
$299$	−0.641056	−0.0370732
$300$	0	0
$301$	6.85915	0.395355
$302$	0	0
$303$	9.07583	0.521393
$304$	0	0
$305$	0	0
$306$	0	0
$307$	22.0642	1.25927	0.629636	−	0.776891i	$-0.283204\pi$
$307$	22.0642	1.25927	0.629636	+	0.776891i	$0.283204\pi$
$308$	0	0
$309$	−9.04677	−0.514653
$310$	0	0
$311$	−31.4064	−1.78089	−0.890447	−	0.455087i	$-0.849608\pi$
$311$	−31.4064	−1.78089	−0.890447	+	0.455087i	$0.849608\pi$
$312$	0	0
$313$	−15.1428	−0.855924	−0.427962	−	0.903797i	$-0.640768\pi$
$313$	−15.1428	−0.855924	−0.427962	+	0.903797i	$0.640768\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−18.5791	−1.04350	−0.521752	−	0.853097i	$-0.674721\pi$
$317$	−18.5791	−1.04350	−0.521752	+	0.853097i	$0.674721\pi$
$318$	0	0
$319$	15.4646	0.865852
$320$	0	0
$321$	−5.65646	−0.315713
$322$	0	0
$323$	−7.94693	−0.442179
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−7.06673	−0.390791
$328$	0	0
$329$	−3.09839	−0.170820
$330$	0	0
$331$	22.8001	1.25320	0.626602	−	0.779339i	$-0.284445\pi$
$331$	22.8001	1.25320	0.626602	+	0.779339i	$0.284445\pi$
$332$	0	0
$333$	−28.2756	−1.54950
$334$	0	0
$335$	0	0
$336$	0	0
$337$	28.9306	1.57595	0.787976	−	0.615706i	$-0.211129\pi$
$337$	28.9306	1.57595	0.787976	+	0.615706i	$0.211129\pi$
$338$	0	0
$339$	3.82959	0.207995
$340$	0	0
$341$	14.0263	0.759569
$342$	0	0
$343$	−7.75994	−0.418997
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−20.2803	−1.08870	−0.544352	−	0.838857i	$-0.683224\pi$
$347$	−20.2803	−1.08870	−0.544352	+	0.838857i	$0.683224\pi$
$348$	0	0
$349$	20.4355	1.09389	0.546943	−	0.837170i	$-0.315792\pi$
$349$	20.4355	1.09389	0.546943	+	0.837170i	$0.315792\pi$
$350$	0	0
$351$	0.450308	0.0240357
$352$	0	0
$353$	7.50588	0.399498	0.199749	−	0.979847i	$-0.435987\pi$
$353$	7.50588	0.399498	0.199749	+	0.979847i	$0.435987\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−2.12357	−0.112391
$358$	0	0
$359$	−18.4383	−0.973135	−0.486567	−	0.873643i	$-0.661751\pi$
$359$	−18.4383	−0.973135	−0.486567	+	0.873643i	$0.661751\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	3.84689	0.201909
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−12.0105	−0.626943	−0.313471	−	0.949598i	$-0.601492\pi$
$367$	−12.0105	−0.626943	−0.313471	+	0.949598i	$0.601492\pi$
$368$	0	0
$369$	16.5969	0.863999
$370$	0	0
$371$	−1.13654	−0.0590061
$372$	0	0
$373$	−15.0719	−0.780395	−0.390197	−	0.920731i	$-0.627593\pi$
$373$	−15.0719	−0.780395	−0.390197	+	0.920731i	$0.627593\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−0.584446	−0.0301005
$378$	0	0
$379$	−8.50216	−0.436727	−0.218363	−	0.975868i	$-0.570072\pi$
$379$	−8.50216	−0.436727	−0.218363	+	0.975868i	$0.570072\pi$
$380$	0	0
$381$	−3.94264	−0.201988
$382$	0	0
$383$	−34.9232	−1.78449	−0.892247	−	0.451547i	$-0.850872\pi$
$383$	−34.9232	−1.78449	−0.892247	+	0.451547i	$0.850872\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−33.5888	−1.70741
$388$	0	0
$389$	−10.7455	−0.544820	−0.272410	−	0.962181i	$-0.587821\pi$
$389$	−10.7455	−0.544820	−0.272410	+	0.962181i	$0.587821\pi$
$390$	0	0
$391$	−30.7900	−1.55712
$392$	0	0
$393$	−2.37018	−0.119560
$394$	0	0
$395$	0	0
$396$	0	0
$397$	1.45462	0.0730051	0.0365025	−	0.999334i	$-0.488378\pi$
$397$	1.45462	0.0730051	0.0365025	+	0.999334i	$0.488378\pi$
$398$	0	0
$399$	0.267219	0.0133777
$400$	0	0
$401$	11.5379	0.576176	0.288088	−	0.957604i	$-0.406980\pi$
$401$	11.5379	0.576176	0.288088	+	0.957604i	$0.406980\pi$
$402$	0	0
$403$	−0.530091	−0.0264057
$404$	0	0
$405$	0	0
$406$	0	0
$407$	44.5592	2.20872
$408$	0	0
$409$	−20.5596	−1.01661	−0.508304	−	0.861178i	$-0.669727\pi$
$409$	−20.5596	−1.01661	−0.508304	+	0.861178i	$0.669727\pi$
$410$	0	0
$411$	9.72639	0.479767
$412$	0	0
$413$	4.72107	0.232308
$414$	0	0
$415$	0	0
$416$	0	0
$417$	5.40377	0.264624
$418$	0	0
$419$	−11.0760	−0.541096	−0.270548	−	0.962706i	$-0.587205\pi$
$419$	−11.0760	−0.541096	−0.270548	+	0.962706i	$0.587205\pi$
$420$	0	0
$421$	18.9178	0.921999	0.461000	−	0.887400i	$-0.347491\pi$
$421$	18.9178	0.921999	0.461000	+	0.887400i	$0.347491\pi$
$422$	0	0
$423$	15.1726	0.737715
$424$	0	0
$425$	0	0
$426$	0	0
$427$	6.70467	0.324461
$428$	0	0
$429$	−0.341194	−0.0164730
$430$	0	0
$431$	5.23923	0.252365	0.126182	−	0.992007i	$-0.459728\pi$
$431$	5.23923	0.252365	0.126182	+	0.992007i	$0.459728\pi$
$432$	0	0
$433$	−8.26963	−0.397413	−0.198707	−	0.980059i	$-0.563674\pi$
$433$	−8.26963	−0.397413	−0.198707	+	0.980059i	$0.563674\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	3.87445	0.185340
$438$	0	0
$439$	26.2786	1.25421	0.627104	−	0.778936i	$-0.284240\pi$
$439$	26.2786	1.25421	0.627104	+	0.778936i	$0.284240\pi$
$440$	0	0
$441$	18.5528	0.883469
$442$	0	0
$443$	−9.03372	−0.429205	−0.214602	−	0.976701i	$-0.568846\pi$
$443$	−9.03372	−0.429205	−0.214602	+	0.976701i	$0.568846\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	6.15198	0.290979
$448$	0	0
$449$	−10.5553	−0.498135	−0.249067	−	0.968486i	$-0.580124\pi$
$449$	−10.5553	−0.498135	−0.249067	+	0.968486i	$0.580124\pi$
$450$	0	0
$451$	−26.1548	−1.23158
$452$	0	0
$453$	8.39242	0.394310
$454$	0	0
$455$	0	0
$456$	0	0
$457$	13.7694	0.644105	0.322053	−	0.946722i	$-0.395627\pi$
$457$	13.7694	0.644105	0.322053	+	0.946722i	$0.395627\pi$
$458$	0	0
$459$	21.6284	1.00953
$460$	0	0
$461$	−0.0771895	−0.00359507	−0.00179754	−	0.999998i	$-0.500572\pi$
$461$	−0.0771895	−0.00359507	−0.00179754	+	0.999998i	$0.500572\pi$
$462$	0	0
$463$	7.73598	0.359521	0.179761	−	0.983710i	$-0.442468\pi$
$463$	7.73598	0.359521	0.179761	+	0.983710i	$0.442468\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−9.11290	−0.421695	−0.210847	−	0.977519i	$-0.567622\pi$
$467$	−9.11290	−0.421695	−0.210847	+	0.977519i	$0.567622\pi$
$468$	0	0
$469$	4.99017	0.230425
$470$	0	0
$471$	−5.36764	−0.247328
$472$	0	0
$473$	52.9321	2.43382
$474$	0	0
$475$	0	0
$476$	0	0
$477$	5.56554	0.254828
$478$	0	0
$479$	14.8868	0.680197	0.340099	−	0.940390i	$-0.389540\pi$
$479$	14.8868	0.680197	0.340099	+	0.940390i	$0.389540\pi$
$480$	0	0
$481$	−1.68400	−0.0767840
$482$	0	0
$483$	1.03533	0.0471090
$484$	0	0
$485$	0	0
$486$	0	0
$487$	37.1917	1.68532	0.842658	−	0.538449i	$-0.180989\pi$
$487$	37.1917	1.68532	0.842658	+	0.538449i	$0.180989\pi$
$488$	0	0
$489$	7.80668	0.353030
$490$	0	0
$491$	−6.94872	−0.313591	−0.156796	−	0.987631i	$-0.550116\pi$
$491$	−6.94872	−0.313591	−0.156796	+	0.987631i	$0.550116\pi$
$492$	0	0
$493$	−28.0711	−1.26426
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0.408491	0.0183233
$498$	0	0
$499$	12.2261	0.547314	0.273657	−	0.961827i	$-0.411767\pi$
$499$	12.2261	0.547314	0.273657	+	0.961827i	$0.411767\pi$
$500$	0	0
$501$	−9.10840	−0.406933
$502$	0	0
$503$	28.0625	1.25125	0.625623	−	0.780126i	$-0.284845\pi$
$503$	28.0625	1.25125	0.625623	+	0.780126i	$0.284845\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−6.11032	−0.271369
$508$	0	0
$509$	3.23221	0.143265	0.0716326	−	0.997431i	$-0.477179\pi$
$509$	3.23221	0.143265	0.0716326	+	0.997431i	$0.477179\pi$
$510$	0	0
$511$	2.57575	0.113944
$512$	0	0
$513$	−2.72160	−0.120162
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−23.9102	−1.05157
$518$	0	0
$519$	4.38653	0.192547
$520$	0	0
$521$	−27.8354	−1.21949	−0.609745	−	0.792598i	$-0.708728\pi$
$521$	−27.8354	−1.21949	−0.609745	+	0.792598i	$0.708728\pi$
$522$	0	0
$523$	10.1695	0.444682	0.222341	−	0.974969i	$-0.428630\pi$
$523$	10.1695	0.444682	0.222341	+	0.974969i	$0.428630\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−25.4604	−1.10907
$528$	0	0
$529$	−7.98863	−0.347332
$530$	0	0
$531$	−23.1187	−1.00327
$532$	0	0
$533$	0.988456	0.0428148
$534$	0	0
$535$	0	0
$536$	0	0
$537$	11.3846	0.491283
$538$	0	0
$539$	−29.2372	−1.25933
$540$	0	0
$541$	27.4801	1.18146	0.590731	−	0.806869i	$-0.298840\pi$
$541$	27.4801	1.18146	0.590731	+	0.806869i	$0.298840\pi$
$542$	0	0
$543$	8.80711	0.377949
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−5.84680	−0.249991	−0.124995	−	0.992157i	$-0.539892\pi$
$547$	−5.84680	−0.249991	−0.124995	+	0.992157i	$0.539892\pi$
$548$	0	0
$549$	−32.8322	−1.40125
$550$	0	0
$551$	3.53231	0.150482
$552$	0	0
$553$	6.64468	0.282561
$554$	0	0
$555$	0	0
$556$	0	0
$557$	12.2691	0.519860	0.259930	−	0.965628i	$-0.416301\pi$
$557$	12.2691	0.519860	0.259930	+	0.965628i	$0.416301\pi$
$558$	0	0
$559$	−2.00044	−0.0846095
$560$	0	0
$561$	−16.3876	−0.691884
$562$	0	0
$563$	−17.3234	−0.730096	−0.365048	−	0.930989i	$-0.618947\pi$
$563$	−17.3234	−0.730096	−0.365048	+	0.930989i	$0.618947\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	4.00106	0.168029
$568$	0	0
$569$	−10.3824	−0.435252	−0.217626	−	0.976032i	$-0.569831\pi$
$569$	−10.3824	−0.435252	−0.217626	+	0.976032i	$0.569831\pi$
$570$	0	0
$571$	10.3852	0.434607	0.217304	−	0.976104i	$-0.430274\pi$
$571$	10.3852	0.434607	0.217304	+	0.976104i	$0.430274\pi$
$572$	0	0
$573$	2.95540	0.123464
$574$	0	0
$575$	0	0
$576$	0	0
$577$	16.6524	0.693248	0.346624	−	0.938004i	$-0.387328\pi$
$577$	16.6524	0.693248	0.346624	+	0.938004i	$0.387328\pi$
$578$	0	0
$579$	3.99938	0.166209
$580$	0	0
$581$	−3.81441	−0.158248
$582$	0	0
$583$	−8.77065	−0.363243
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−13.4077	−0.553393	−0.276697	−	0.960957i	$-0.589240\pi$
$587$	−13.4077	−0.553393	−0.276697	+	0.960957i	$0.589240\pi$
$588$	0	0
$589$	3.20380	0.132010
$590$	0	0
$591$	2.81365	0.115738
$592$	0	0
$593$	20.8050	0.854361	0.427180	−	0.904166i	$-0.359507\pi$
$593$	20.8050	0.854361	0.427180	+	0.904166i	$0.359507\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	9.36207	0.383164
$598$	0	0
$599$	9.75788	0.398696	0.199348	−	0.979929i	$-0.436118\pi$
$599$	9.75788	0.398696	0.199348	+	0.979929i	$0.436118\pi$
$600$	0	0
$601$	3.27717	0.133679	0.0668393	−	0.997764i	$-0.478709\pi$
$601$	3.27717	0.133679	0.0668393	+	0.997764i	$0.478709\pi$
$602$	0	0
$603$	−24.4365	−0.995132
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−31.1984	−1.26630	−0.633151	−	0.774028i	$-0.718239\pi$
$607$	−31.1984	−1.26630	−0.633151	+	0.774028i	$0.718239\pi$
$608$	0	0
$609$	0.943901	0.0382488
$610$	0	0
$611$	0.903628	0.0365569
$612$	0	0
$613$	−30.3455	−1.22564	−0.612821	−	0.790222i	$-0.709965\pi$
$613$	−30.3455	−1.22564	−0.612821	+	0.790222i	$0.709965\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	40.7447	1.64032	0.820159	−	0.572135i	$-0.193885\pi$
$617$	40.7447	1.64032	0.820159	+	0.572135i	$0.193885\pi$
$618$	0	0
$619$	−13.7115	−0.551113	−0.275557	−	0.961285i	$-0.588862\pi$
$619$	−13.7115	−0.551113	−0.275557	+	0.961285i	$0.588862\pi$
$620$	0	0
$621$	−10.5447	−0.423144
$622$	0	0
$623$	4.60634	0.184549
$624$	0	0
$625$	0	0
$626$	0	0
$627$	2.06213	0.0823534
$628$	0	0
$629$	−80.8830	−3.22501
$630$	0	0
$631$	−32.3748	−1.28882	−0.644411	−	0.764680i	$-0.722897\pi$
$631$	−32.3748	−1.28882	−0.644411	+	0.764680i	$0.722897\pi$
$632$	0	0
$633$	−1.34195	−0.0533378
$634$	0	0
$635$	0	0
$636$	0	0
$637$	1.10495	0.0437796
$638$	0	0
$639$	−2.00035	−0.0791326
$640$	0	0
$641$	0.701397	0.0277035	0.0138518	−	0.999904i	$-0.495591\pi$
$641$	0.701397	0.0277035	0.0138518	+	0.999904i	$0.495591\pi$
$642$	0	0
$643$	−12.5736	−0.495856	−0.247928	−	0.968778i	$-0.579750\pi$
$643$	−12.5736	−0.495856	−0.247928	+	0.968778i	$0.579750\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−36.4494	−1.43297	−0.716487	−	0.697600i	$-0.754251\pi$
$647$	−36.4494	−1.43297	−0.716487	+	0.697600i	$0.754251\pi$
$648$	0	0
$649$	36.4325	1.43010
$650$	0	0
$651$	0.856115	0.0335538
$652$	0	0
$653$	−44.2535	−1.73177	−0.865887	−	0.500240i	$-0.833245\pi$
$653$	−44.2535	−1.73177	−0.865887	+	0.500240i	$0.833245\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−12.6132	−0.492089
$658$	0	0
$659$	−18.4958	−0.720493	−0.360247	−	0.932857i	$-0.617307\pi$
$659$	−18.4958	−0.720493	−0.360247	+	0.932857i	$0.617307\pi$
$660$	0	0
$661$	15.3505	0.597067	0.298533	−	0.954399i	$-0.403503\pi$
$661$	15.3505	0.597067	0.298533	+	0.954399i	$0.403503\pi$
$662$	0	0
$663$	0.619328	0.0240527
$664$	0	0
$665$	0	0
$666$	0	0
$667$	13.6858	0.529915
$668$	0	0
$669$	−0.416338	−0.0160965
$670$	0	0
$671$	51.7399	1.99740
$672$	0	0
$673$	−34.1219	−1.31530	−0.657651	−	0.753322i	$-0.728450\pi$
$673$	−34.1219	−1.31530	−0.657651	+	0.753322i	$0.728450\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	13.7808	0.529639	0.264819	−	0.964298i	$-0.414688\pi$
$677$	13.7808	0.529639	0.264819	+	0.964298i	$0.414688\pi$
$678$	0	0
$679$	−7.73382	−0.296797
$680$	0	0
$681$	0.484376	0.0185613
$682$	0	0
$683$	−35.8693	−1.37250	−0.686250	−	0.727366i	$-0.740744\pi$
$683$	−35.8693	−1.37250	−0.686250	+	0.727366i	$0.740744\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−11.5813	−0.441853
$688$	0	0
$689$	0.331465	0.0126278
$690$	0	0
$691$	17.5888	0.669111	0.334555	−	0.942376i	$-0.391414\pi$
$691$	17.5888	0.669111	0.334555	+	0.942376i	$0.391414\pi$
$692$	0	0
$693$	−6.90026	−0.262119
$694$	0	0
$695$	0	0
$696$	0	0
$697$	47.4757	1.79827
$698$	0	0
$699$	−2.46617	−0.0932792
$700$	0	0
$701$	−40.3585	−1.52432	−0.762159	−	0.647389i	$-0.775861\pi$
$701$	−40.3585	−1.52432	−0.762159	+	0.647389i	$0.775861\pi$
$702$	0	0
$703$	10.1779	0.383866
$704$	0	0
$705$	0	0
$706$	0	0
$707$	10.9315	0.411123
$708$	0	0
$709$	20.5258	0.770862	0.385431	−	0.922737i	$-0.374053\pi$
$709$	20.5258	0.770862	0.385431	+	0.922737i	$0.374053\pi$
$710$	0	0
$711$	−32.5385	−1.22029
$712$	0	0
$713$	12.4130	0.464869
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−6.03486	−0.225376
$718$	0	0
$719$	−29.8673	−1.11386	−0.556930	−	0.830559i	$-0.688021\pi$
$719$	−29.8673	−1.11386	−0.556930	+	0.830559i	$0.688021\pi$
$720$	0	0
$721$	−10.8965	−0.405808
$722$	0	0
$723$	−5.65200	−0.210200
$724$	0	0
$725$	0	0
$726$	0	0
$727$	26.4630	0.981458	0.490729	−	0.871312i	$-0.336731\pi$
$727$	26.4630	0.981458	0.490729	+	0.871312i	$0.336731\pi$
$728$	0	0
$729$	−15.7471	−0.583228
$730$	0	0
$731$	−96.0813	−3.55369
$732$	0	0
$733$	−30.5488	−1.12834	−0.564172	−	0.825657i	$-0.690804\pi$
$733$	−30.5488	−1.12834	−0.564172	+	0.825657i	$0.690804\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	38.5092	1.41850
$738$	0	0
$739$	−34.4001	−1.26543	−0.632714	−	0.774386i	$-0.718059\pi$
$739$	−34.4001	−1.26543	−0.632714	+	0.774386i	$0.718059\pi$
$740$	0	0
$741$	−0.0779330	−0.00286294
$742$	0	0
$743$	21.9370	0.804790	0.402395	−	0.915466i	$-0.368178\pi$
$743$	21.9370	0.804790	0.402395	+	0.915466i	$0.368178\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	18.6789	0.683424
$748$	0	0
$749$	−6.81303	−0.248943
$750$	0	0
$751$	−45.4750	−1.65941	−0.829703	−	0.558206i	$-0.811490\pi$
$751$	−45.4750	−1.65941	−0.829703	+	0.558206i	$0.811490\pi$
$752$	0	0
$753$	11.1720	0.407130
$754$	0	0
$755$	0	0
$756$	0	0
$757$	8.91122	0.323884	0.161942	−	0.986800i	$-0.448224\pi$
$757$	8.91122	0.323884	0.161942	+	0.986800i	$0.448224\pi$
$758$	0	0
$759$	7.98961	0.290005
$760$	0	0
$761$	−48.1068	−1.74387	−0.871934	−	0.489623i	$-0.837134\pi$
$761$	−48.1068	−1.74387	−0.871934	+	0.489623i	$0.837134\pi$
$762$	0	0
$763$	−8.51165	−0.308142
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−1.37687	−0.0497161
$768$	0	0
$769$	−11.2615	−0.406099	−0.203049	−	0.979169i	$-0.565085\pi$
$769$	−11.2615	−0.406099	−0.203049	+	0.979169i	$0.565085\pi$
$770$	0	0
$771$	−0.997039	−0.0359075
$772$	0	0
$773$	20.4897	0.736964	0.368482	−	0.929635i	$-0.379878\pi$
$773$	20.4897	0.736964	0.368482	+	0.929635i	$0.379878\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	2.71972	0.0975695
$778$	0	0
$779$	−5.97409	−0.214044
$780$	0	0
$781$	3.15232	0.112799
$782$	0	0
$783$	−9.61354	−0.343560
$784$	0	0
$785$	0	0
$786$	0	0
$787$	5.45356	0.194398	0.0971992	−	0.995265i	$-0.469012\pi$
$787$	5.45356	0.194398	0.0971992	+	0.995265i	$0.469012\pi$
$788$	0	0
$789$	−0.921655	−0.0328118
$790$	0	0
$791$	4.61262	0.164006
$792$	0	0
$793$	−1.95538	−0.0694376
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−4.96228	−0.175773	−0.0878864	−	0.996131i	$-0.528011\pi$
$797$	−4.96228	−0.175773	−0.0878864	+	0.996131i	$0.528011\pi$
$798$	0	0
$799$	43.4014	1.53543
$800$	0	0
$801$	−22.5569	−0.797008
$802$	0	0
$803$	19.8770	0.701445
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−10.7523	−0.378498
$808$	0	0
$809$	−18.4521	−0.648741	−0.324370	−	0.945930i	$-0.605152\pi$
$809$	−18.4521	−0.648741	−0.324370	+	0.945930i	$0.605152\pi$
$810$	0	0
$811$	31.3590	1.10116	0.550582	−	0.834781i	$-0.314406\pi$
$811$	31.3590	1.10116	0.550582	+	0.834781i	$0.314406\pi$
$812$	0	0
$813$	−7.45161	−0.261339
$814$	0	0
$815$	0	0
$816$	0	0
$817$	12.0904	0.422988
$818$	0	0
$819$	0.260778	0.00911233
$820$	0	0
$821$	−52.9504	−1.84798	−0.923991	−	0.382413i	$-0.875093\pi$
$821$	−52.9504	−1.84798	−0.923991	+	0.382413i	$0.875093\pi$
$822$	0	0
$823$	−46.0815	−1.60630	−0.803150	−	0.595776i	$-0.796845\pi$
$823$	−46.0815	−1.60630	−0.803150	+	0.595776i	$0.796845\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−10.9170	−0.379622	−0.189811	−	0.981821i	$-0.560788\pi$
$827$	−10.9170	−0.379622	−0.189811	+	0.981821i	$0.560788\pi$
$828$	0	0
$829$	48.6141	1.68844	0.844219	−	0.535998i	$-0.180064\pi$
$829$	48.6141	1.68844	0.844219	+	0.535998i	$0.180064\pi$
$830$	0	0
$831$	4.46488	0.154885
$832$	0	0
$833$	53.0707	1.83879
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−8.71945	−0.301388
$838$	0	0
$839$	−35.2171	−1.21583	−0.607915	−	0.794002i	$-0.707994\pi$
$839$	−35.2171	−1.21583	−0.607915	+	0.794002i	$0.707994\pi$
$840$	0	0
$841$	−16.5228	−0.569750
$842$	0	0
$843$	6.27206	0.216021
$844$	0	0
$845$	0	0
$846$	0	0
$847$	4.63346	0.159208
$848$	0	0
$849$	−4.33653	−0.148829
$850$	0	0
$851$	39.4337	1.35177
$852$	0	0
$853$	7.77006	0.266042	0.133021	−	0.991113i	$-0.457532\pi$
$853$	7.77006	0.266042	0.133021	+	0.991113i	$0.457532\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−20.8792	−0.713219	−0.356610	−	0.934253i	$-0.616067\pi$
$857$	−20.8792	−0.713219	−0.356610	+	0.934253i	$0.616067\pi$
$858$	0	0
$859$	5.48065	0.186997	0.0934987	−	0.995619i	$-0.470195\pi$
$859$	5.48065	0.186997	0.0934987	+	0.995619i	$0.470195\pi$
$860$	0	0
$861$	−1.59639	−0.0544048
$862$	0	0
$863$	−23.9562	−0.815479	−0.407740	−	0.913098i	$-0.633683\pi$
$863$	−23.9562	−0.815479	−0.407740	+	0.913098i	$0.633683\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	21.7391	0.738300
$868$	0	0
$869$	51.2770	1.73945
$870$	0	0
$871$	−1.45536	−0.0493129
$872$	0	0
$873$	37.8719	1.28177
$874$	0	0
$875$	0	0
$876$	0	0
$877$	18.0925	0.610941	0.305471	−	0.952201i	$-0.401186\pi$
$877$	18.0925	0.610941	0.305471	+	0.952201i	$0.401186\pi$
$878$	0	0
$879$	4.87195	0.164327
$880$	0	0
$881$	6.10173	0.205572	0.102786	−	0.994703i	$-0.467224\pi$
$881$	6.10173	0.205572	0.102786	+	0.994703i	$0.467224\pi$
$882$	0	0
$883$	−11.8538	−0.398914	−0.199457	−	0.979907i	$-0.563918\pi$
$883$	−11.8538	−0.398914	−0.199457	+	0.979907i	$0.563918\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	46.7513	1.56976	0.784878	−	0.619651i	$-0.212726\pi$
$887$	46.7513	1.56976	0.784878	+	0.619651i	$0.212726\pi$
$888$	0	0
$889$	−4.74879	−0.159269
$890$	0	0
$891$	30.8762	1.03439
$892$	0	0
$893$	−5.46140	−0.182759
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−0.301948	−0.0100817
$898$	0	0
$899$	11.3168	0.377437
$900$	0	0
$901$	15.9203	0.530383
$902$	0	0
$903$	3.23077	0.107513
$904$	0	0
$905$	0	0
$906$	0	0
$907$	36.8753	1.22442	0.612212	−	0.790693i	$-0.290280\pi$
$907$	36.8753	1.22442	0.612212	+	0.790693i	$0.290280\pi$
$908$	0	0
$909$	−53.5310	−1.77551
$910$	0	0
$911$	−24.9642	−0.827099	−0.413550	−	0.910482i	$-0.635711\pi$
$911$	−24.9642	−0.827099	−0.413550	+	0.910482i	$0.635711\pi$
$912$	0	0
$913$	−29.4358	−0.974182
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−2.85481	−0.0942740
$918$	0	0
$919$	−8.45204	−0.278807	−0.139404	−	0.990236i	$-0.544519\pi$
$919$	−8.45204	−0.278807	−0.139404	+	0.990236i	$0.544519\pi$
$920$	0	0
$921$	10.3926	0.342448
$922$	0	0
$923$	−0.119134	−0.00392135
$924$	0	0
$925$	0	0
$926$	0	0
$927$	53.3596	1.75256
$928$	0	0
$929$	34.3786	1.12793	0.563963	−	0.825800i	$-0.309276\pi$
$929$	34.3786	1.12793	0.563963	+	0.825800i	$0.309276\pi$
$930$	0	0
$931$	−6.67814	−0.218867
$932$	0	0
$933$	−14.7929	−0.484299
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−5.23838	−0.171130	−0.0855652	−	0.996333i	$-0.527270\pi$
$937$	−5.23838	−0.171130	−0.0855652	+	0.996333i	$0.527270\pi$
$938$	0	0
$939$	−7.13252	−0.232761
$940$	0	0
$941$	14.3720	0.468514	0.234257	−	0.972175i	$-0.424734\pi$
$941$	14.3720	0.468514	0.234257	+	0.972175i	$0.424734\pi$
$942$	0	0
$943$	−23.1463	−0.753748
$944$	0	0
$945$	0	0
$946$	0	0
$947$	27.8865	0.906190	0.453095	−	0.891462i	$-0.350320\pi$
$947$	27.8865	0.906190	0.453095	+	0.891462i	$0.350320\pi$
$948$	0	0
$949$	−0.751203	−0.0243851
$950$	0	0
$951$	−8.75104	−0.283772
$952$	0	0
$953$	−18.8741	−0.611392	−0.305696	−	0.952129i	$-0.598889\pi$
$953$	−18.8741	−0.611392	−0.305696	+	0.952129i	$0.598889\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	7.28408	0.235461
$958$	0	0
$959$	11.7151	0.378301
$960$	0	0
$961$	−20.7357	−0.668893
$962$	0	0
$963$	33.3629	1.07510
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−2.87426	−0.0924301	−0.0462150	−	0.998932i	$-0.514716\pi$
$967$	−2.87426	−0.0924301	−0.0462150	+	0.998932i	$0.514716\pi$
$968$	0	0
$969$	−3.74313	−0.120247
$970$	0	0
$971$	−50.4156	−1.61791	−0.808956	−	0.587869i	$-0.799967\pi$
$971$	−50.4156	−1.61791	−0.808956	+	0.587869i	$0.799967\pi$
$972$	0	0
$973$	6.50867	0.208658
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−48.5411	−1.55297	−0.776484	−	0.630137i	$-0.782999\pi$
$977$	−48.5411	−1.55297	−0.776484	+	0.630137i	$0.782999\pi$
$978$	0	0
$979$	35.5471	1.13609
$980$	0	0
$981$	41.6809	1.33077
$982$	0	0
$983$	5.00431	0.159613	0.0798063	−	0.996810i	$-0.474570\pi$
$983$	5.00431	0.159613	0.0798063	+	0.996810i	$0.474570\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−1.45939	−0.0464529
$988$	0	0
$989$	46.8435	1.48954
$990$	0	0
$991$	−43.9761	−1.39695	−0.698474	−	0.715635i	$-0.746137\pi$
$991$	−43.9761	−1.39695	−0.698474	+	0.715635i	$0.746137\pi$
$992$	0	0
$993$	10.7392	0.340798
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−11.4138	−0.361478	−0.180739	−	0.983531i	$-0.557849\pi$
$997$	−11.4138	−0.361478	−0.180739	+	0.983531i	$0.557849\pi$
$998$	0	0
$999$	−27.7001	−0.876393

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3800.2.a.bb.1.4	✓	6
4.3	odd	2		7600.2.a.cm.1.3		6
5.2	odd	4		3800.2.d.p.3649.6		12
5.3	odd	4		3800.2.d.p.3649.7		12
5.4	even	2		3800.2.a.bd.1.3	yes	6
20.19	odd	2		7600.2.a.ci.1.4		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3800.2.a.bb.1.4	✓	6	1.1	even	1	trivial
3800.2.a.bd.1.3	yes	6	5.4	even	2
3800.2.d.p.3649.6		12	5.2	odd	4
3800.2.d.p.3649.7		12	5.3	odd	4
7600.2.a.ci.1.4		6	20.19	odd	2
7600.2.a.cm.1.3		6	4.3	odd	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:	\( N \)	\(=\)	\( 3800 = 2^{3} \cdot 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3800.a (trivial)

\(f(q)\)	\(=\)	\(q+0.471016 q^{3} +0.567324 q^{7} -2.77814 q^{9} +O(q^{10})\)	\(q+0.471016 q^{3} +0.567324 q^{7} -2.77814 q^{9} +4.37804 q^{11} -0.165457 q^{13} -7.94693 q^{17} +1.00000 q^{19} +0.267219 q^{21} +3.87445 q^{23} -2.72160 q^{27} +3.53231 q^{29} +3.20380 q^{31} +2.06213 q^{33} +10.1779 q^{37} -0.0779330 q^{39} -5.97409 q^{41} +12.0904 q^{43} -5.46140 q^{47} -6.67814 q^{49} -3.74313 q^{51} -2.00333 q^{53} +0.471016 q^{57} +8.32164 q^{59} +11.8181 q^{61} -1.57611 q^{63} +8.79599 q^{67} +1.82493 q^{69} +0.720031 q^{71} +4.54017 q^{73} +2.48377 q^{77} +11.7123 q^{79} +7.05251 q^{81} -6.72351 q^{83} +1.66378 q^{87} +8.11941 q^{89} -0.0938678 q^{91} +1.50904 q^{93} -13.6321 q^{97} -12.1628 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 2 q^{3} - 2 q^{7} + 10 q^{9}+O(q^{10}) \) 6 * q - 2 * q^3 - 2 * q^7 + 10 * q^9	\( 6 q - 2 q^{3} - 2 q^{7} + 10 q^{9} + 3 q^{11} + 3 q^{13} + 2 q^{17} + 6 q^{19} + 11 q^{21} - 4 q^{23} - 20 q^{27} + 7 q^{29} + 5 q^{31} + 16 q^{33} + 8 q^{39} + 11 q^{41} + 7 q^{43} - 20 q^{47} - 2 q^{49} + 13 q^{51} + 7 q^{53} - 2 q^{57} - 4 q^{59} + 13 q^{61} + q^{63} - 25 q^{67} + 7 q^{69} + 29 q^{71} + 19 q^{73} - 24 q^{77} + 28 q^{79} + 38 q^{81} + 15 q^{83} - 57 q^{87} - 12 q^{89} + 27 q^{93} + 13 q^{97} + 10 q^{99}+O(q^{100}) \) 6 * q - 2 * q^3 - 2 * q^7 + 10 * q^9 + 3 * q^11 + 3 * q^13 + 2 * q^17 + 6 * q^19 + 11 * q^21 - 4 * q^23 - 20 * q^27 + 7 * q^29 + 5 * q^31 + 16 * q^33 + 8 * q^39 + 11 * q^41 + 7 * q^43 - 20 * q^47 - 2 * q^49 + 13 * q^51 + 7 * q^53 - 2 * q^57 - 4 * q^59 + 13 * q^61 + q^63 - 25 * q^67 + 7 * q^69 + 29 * q^71 + 19 * q^73 - 24 * q^77 + 28 * q^79 + 38 * q^81 + 15 * q^83 - 57 * q^87 - 12 * q^89 + 27 * q^93 + 13 * q^97 + 10 * q^99

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