LMFDB - Embedded newform 3800.2.a.ba.1.5

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} - 10x^{4} + 16x^{3} + 15x^{2} - 14x - 3 $ x^6 - 2x^5 - 10x^4 + 16x^3 + 15x^2 - 14*x - 3
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			1.5
Root			$-1.08999$ 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)

$f(q)$	$=$	$q+1.08999 q^{3} -4.19727 q^{7} -1.81192 q^{9} +O(q^{10})$	$q+1.08999 q^{3} -4.19727 q^{7} -1.81192 q^{9} -6.43052 q^{11} -2.24614 q^{13} +7.84744 q^{17} -1.00000 q^{19} -4.57497 q^{21} +0.859601 q^{23} -5.24494 q^{27} +8.38284 q^{29} +1.24541 q^{31} -7.00919 q^{33} +6.79977 q^{37} -2.44827 q^{39} +5.92480 q^{41} -6.81073 q^{43} +6.00919 q^{47} +10.6171 q^{49} +8.55363 q^{51} -13.7594 q^{53} -1.08999 q^{57} +6.88976 q^{59} -1.31884 q^{61} +7.60513 q^{63} -4.73266 q^{67} +0.936955 q^{69} +10.2546 q^{71} +9.86150 q^{73} +26.9906 q^{77} +6.47056 q^{79} -0.281158 q^{81} -5.42133 q^{83} +9.13720 q^{87} +3.10288 q^{89} +9.42766 q^{91} +1.35748 q^{93} -6.76032 q^{97} +11.6516 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 2 q^{3} - 2 q^{7} + 6 q^{9}+O(q^{10}) $ 6 * q - 2 * q^3 - 2 * q^7 + 6 * q^9	$ 6 q - 2 q^{3} - 2 q^{7} + 6 q^{9} + 3 q^{11} + q^{13} + 14 q^{17} - 6 q^{19} + 15 q^{21} - 12 q^{23} - 8 q^{27} + 9 q^{29} + 5 q^{31} - 2 q^{33} + 8 q^{37} + 12 q^{39} + 3 q^{41} - 15 q^{43} - 4 q^{47} + 22 q^{49} + 33 q^{51} - 13 q^{53} + 2 q^{57} + 9 q^{61} - 21 q^{63} + 3 q^{67} - 11 q^{69} + 19 q^{71} - 3 q^{73} + 36 q^{77} - 16 q^{79} + 26 q^{81} - 31 q^{83} + 25 q^{87} + 14 q^{89} + 42 q^{91} - 39 q^{93} + 11 q^{97} - 6 q^{99}+O(q^{100}) $ 6 * q - 2 * q^3 - 2 * q^7 + 6 * q^9 + 3 * q^11 + q^13 + 14 * q^17 - 6 * q^19 + 15 * q^21 - 12 * q^23 - 8 * q^27 + 9 * q^29 + 5 * q^31 - 2 * q^33 + 8 * q^37 + 12 * q^39 + 3 * q^41 - 15 * q^43 - 4 * q^47 + 22 * q^49 + 33 * q^51 - 13 * q^53 + 2 * q^57 + 9 * q^61 - 21 * q^63 + 3 * q^67 - 11 * q^69 + 19 * q^71 - 3 * q^73 + 36 * q^77 - 16 * q^79 + 26 * q^81 - 31 * q^83 + 25 * q^87 + 14 * q^89 + 42 * q^91 - 39 * q^93 + 11 * q^97 - 6 * 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
$3$	1.08999	0.629305	0.314653	−	0.949207i	$-0.398112\pi$
$3$	1.08999	0.629305	0.314653	+	0.949207i	$0.398112\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−4.19727	−1.58642	−0.793209	−	0.608949i	$-0.791591\pi$
$7$	−4.19727	−1.58642	−0.793209	+	0.608949i	$0.791591\pi$
$8$	0	0
$9$	−1.81192	−0.603975
$10$	0	0
$11$	−6.43052	−1.93887	−0.969437	−	0.245341i	$-0.921100\pi$
$11$	−6.43052	−1.93887	−0.969437	+	0.245341i	$0.921100\pi$
$12$	0	0
$13$	−2.24614	−0.622967	−0.311484	−	0.950251i	$-0.600826\pi$
$13$	−2.24614	−0.622967	−0.311484	+	0.950251i	$0.600826\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	7.84744	1.90328	0.951642	−	0.307208i	$-0.0993947\pi$
$17$	7.84744	1.90328	0.951642	+	0.307208i	$0.0993947\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	−4.57497	−0.998341
$22$	0	0
$23$	0.859601	0.179239	0.0896195	−	0.995976i	$-0.471435\pi$
$23$	0.859601	0.179239	0.0896195	+	0.995976i	$0.471435\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−5.24494	−1.00939
$28$	0	0
$29$	8.38284	1.55665	0.778327	−	0.627859i	$-0.216068\pi$
$29$	8.38284	1.55665	0.778327	+	0.627859i	$0.216068\pi$
$30$	0	0
$31$	1.24541	0.223681	0.111841	−	0.993726i	$-0.464325\pi$
$31$	1.24541	0.223681	0.111841	+	0.993726i	$0.464325\pi$
$32$	0	0
$33$	−7.00919	−1.22014
$34$	0	0
$35$	0	0
$36$	0	0
$37$	6.79977	1.11787	0.558937	−	0.829210i	$-0.311209\pi$
$37$	6.79977	1.11787	0.558937	+	0.829210i	$0.311209\pi$
$38$	0	0
$39$	−2.44827	−0.392037
$40$	0	0
$41$	5.92480	0.925298	0.462649	−	0.886542i	$-0.346899\pi$
$41$	5.92480	0.925298	0.462649	+	0.886542i	$0.346899\pi$
$42$	0	0
$43$	−6.81073	−1.03863	−0.519313	−	0.854584i	$-0.673812\pi$
$43$	−6.81073	−1.03863	−0.519313	+	0.854584i	$0.673812\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	6.00919	0.876531	0.438265	−	0.898846i	$-0.355593\pi$
$47$	6.00919	0.876531	0.438265	+	0.898846i	$0.355593\pi$
$48$	0	0
$49$	10.6171	1.51672
$50$	0	0
$51$	8.55363	1.19775
$52$	0	0
$53$	−13.7594	−1.88999	−0.944996	−	0.327082i	$-0.893935\pi$
$53$	−13.7594	−1.88999	−0.944996	+	0.327082i	$0.893935\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−1.08999	−0.144373
$58$	0	0
$59$	6.88976	0.896970	0.448485	−	0.893790i	$-0.351964\pi$
$59$	6.88976	0.896970	0.448485	+	0.893790i	$0.351964\pi$
$60$	0	0
$61$	−1.31884	−0.168860	−0.0844301	−	0.996429i	$-0.526907\pi$
$61$	−1.31884	−0.168860	−0.0844301	+	0.996429i	$0.526907\pi$
$62$	0	0
$63$	7.60513	0.958156
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−4.73266	−0.578187	−0.289093	−	0.957301i	$-0.593354\pi$
$67$	−4.73266	−0.578187	−0.289093	+	0.957301i	$0.593354\pi$
$68$	0	0
$69$	0.936955	0.112796
$70$	0	0
$71$	10.2546	1.21700	0.608498	−	0.793555i	$-0.291772\pi$
$71$	10.2546	1.21700	0.608498	+	0.793555i	$0.291772\pi$
$72$	0	0
$73$	9.86150	1.15420	0.577100	−	0.816673i	$-0.304184\pi$
$73$	9.86150	1.15420	0.577100	+	0.816673i	$0.304184\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	26.9906	3.07586
$78$	0	0
$79$	6.47056	0.727995	0.363997	−	0.931400i	$-0.381412\pi$
$79$	6.47056	0.727995	0.363997	+	0.931400i	$0.381412\pi$
$80$	0	0
$81$	−0.281158	−0.0312397
$82$	0	0
$83$	−5.42133	−0.595068	−0.297534	−	0.954711i	$-0.596164\pi$
$83$	−5.42133	−0.595068	−0.297534	+	0.954711i	$0.596164\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	9.13720	0.979611
$88$	0	0
$89$	3.10288	0.328905	0.164452	−	0.986385i	$-0.447414\pi$
$89$	3.10288	0.328905	0.164452	+	0.986385i	$0.447414\pi$
$90$	0	0
$91$	9.42766	0.988287
$92$	0	0
$93$	1.35748	0.140764
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−6.76032	−0.686406	−0.343203	−	0.939261i	$-0.611512\pi$
$97$	−6.76032	−0.686406	−0.343203	+	0.939261i	$0.611512\pi$
$98$	0	0
$99$	11.6516	1.17103
$100$	0	0
$101$	−3.05247	−0.303732	−0.151866	−	0.988401i	$-0.548528\pi$
$101$	−3.05247	−0.303732	−0.151866	+	0.988401i	$0.548528\pi$
$102$	0	0
$103$	−7.06366	−0.696003	−0.348002	−	0.937494i	$-0.613140\pi$
$103$	−7.06366	−0.696003	−0.348002	+	0.937494i	$0.613140\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	5.05281	0.488474	0.244237	−	0.969716i	$-0.421463\pi$
$107$	5.05281	0.488474	0.244237	+	0.969716i	$0.421463\pi$
$108$	0	0
$109$	−6.39334	−0.612371	−0.306185	−	0.951972i	$-0.599053\pi$
$109$	−6.39334	−0.612371	−0.306185	+	0.951972i	$0.599053\pi$
$110$	0	0
$111$	7.41167	0.703485
$112$	0	0
$113$	19.5837	1.84228	0.921140	−	0.389231i	$-0.127259\pi$
$113$	19.5837	1.84228	0.921140	+	0.389231i	$0.127259\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	4.06984	0.376257
$118$	0	0
$119$	−32.9378	−3.01941
$120$	0	0
$121$	30.3515	2.75923
$122$	0	0
$123$	6.45796	0.582295
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−6.40323	−0.568195	−0.284098	−	0.958795i	$-0.591694\pi$
$127$	−6.40323	−0.568195	−0.284098	+	0.958795i	$0.591694\pi$
$128$	0	0
$129$	−7.42362	−0.653613
$130$	0	0
$131$	12.1913	1.06516	0.532581	−	0.846379i	$-0.321222\pi$
$131$	12.1913	1.06516	0.532581	+	0.846379i	$0.321222\pi$
$132$	0	0
$133$	4.19727	0.363949
$134$	0	0
$135$	0	0
$136$	0	0
$137$	9.71228	0.829776	0.414888	−	0.909872i	$-0.363821\pi$
$137$	9.71228	0.829776	0.414888	+	0.909872i	$0.363821\pi$
$138$	0	0
$139$	20.0687	1.70220	0.851102	−	0.525000i	$-0.175935\pi$
$139$	20.0687	1.70220	0.851102	+	0.525000i	$0.175935\pi$
$140$	0	0
$141$	6.54995	0.551605
$142$	0	0
$143$	14.4438	1.20786
$144$	0	0
$145$	0	0
$146$	0	0
$147$	11.5725	0.954481
$148$	0	0
$149$	−14.8593	−1.21732	−0.608659	−	0.793432i	$-0.708292\pi$
$149$	−14.8593	−1.21732	−0.608659	+	0.793432i	$0.708292\pi$
$150$	0	0
$151$	−1.29571	−0.105444	−0.0527218	−	0.998609i	$-0.516790\pi$
$151$	−1.29571	−0.105444	−0.0527218	+	0.998609i	$0.516790\pi$
$152$	0	0
$153$	−14.2190	−1.14954
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−4.05355	−0.323508	−0.161754	−	0.986831i	$-0.551715\pi$
$157$	−4.05355	−0.323508	−0.161754	+	0.986831i	$0.551715\pi$
$158$	0	0
$159$	−14.9975	−1.18938
$160$	0	0
$161$	−3.60797	−0.284348
$162$	0	0
$163$	−8.09764	−0.634256	−0.317128	−	0.948383i	$-0.602719\pi$
$163$	−8.09764	−0.634256	−0.317128	+	0.948383i	$0.602719\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	18.6059	1.43976	0.719882	−	0.694096i	$-0.244196\pi$
$167$	18.6059	1.43976	0.719882	+	0.694096i	$0.244196\pi$
$168$	0	0
$169$	−7.95485	−0.611911
$170$	0	0
$171$	1.81192	0.138561
$172$	0	0
$173$	−8.35972	−0.635578	−0.317789	−	0.948161i	$-0.602940\pi$
$173$	−8.35972	−0.635578	−0.317789	+	0.948161i	$0.602940\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	7.50976	0.564468
$178$	0	0
$179$	14.5971	1.09104	0.545521	−	0.838097i	$-0.316332\pi$
$179$	14.5971	1.09104	0.545521	+	0.838097i	$0.316332\pi$
$180$	0	0
$181$	0.131566	0.00977923	0.00488961	−	0.999988i	$-0.498444\pi$
$181$	0.131566	0.00977923	0.00488961	+	0.999988i	$0.498444\pi$
$182$	0	0
$183$	−1.43752	−0.106265
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−50.4631	−3.69023
$188$	0	0
$189$	22.0144	1.60131
$190$	0	0
$191$	−22.5544	−1.63198	−0.815989	−	0.578067i	$-0.803807\pi$
$191$	−22.5544	−1.63198	−0.815989	+	0.578067i	$0.803807\pi$
$192$	0	0
$193$	−4.93863	−0.355490	−0.177745	−	0.984077i	$-0.556880\pi$
$193$	−4.93863	−0.355490	−0.177745	+	0.984077i	$0.556880\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−26.2422	−1.86968	−0.934840	−	0.355069i	$-0.884457\pi$
$197$	−26.2422	−1.86968	−0.934840	+	0.355069i	$0.884457\pi$
$198$	0	0
$199$	15.1880	1.07665	0.538323	−	0.842738i	$-0.319058\pi$
$199$	15.1880	1.07665	0.538323	+	0.842738i	$0.319058\pi$
$200$	0	0
$201$	−5.15855	−0.363856
$202$	0	0
$203$	−35.1850	−2.46950
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−1.55753	−0.108256
$208$	0	0
$209$	6.43052	0.444808
$210$	0	0
$211$	20.3632	1.40186	0.700929	−	0.713231i	$-0.252769\pi$
$211$	20.3632	1.40186	0.700929	+	0.713231i	$0.252769\pi$
$212$	0	0
$213$	11.1774	0.765863
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−5.22730	−0.354852
$218$	0	0
$219$	10.7489	0.726345
$220$	0	0
$221$	−17.6265	−1.18568
$222$	0	0
$223$	−15.3764	−1.02968	−0.514841	−	0.857285i	$-0.672149\pi$
$223$	−15.3764	−1.02968	−0.514841	+	0.857285i	$0.672149\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	20.0187	1.32868	0.664342	−	0.747428i	$-0.268712\pi$
$227$	20.0187	1.32868	0.664342	+	0.747428i	$0.268712\pi$
$228$	0	0
$229$	−12.6175	−0.833789	−0.416895	−	0.908955i	$-0.636882\pi$
$229$	−12.6175	−0.833789	−0.416895	+	0.908955i	$0.636882\pi$
$230$	0	0
$231$	29.4195	1.93566
$232$	0	0
$233$	−21.9230	−1.43622	−0.718112	−	0.695927i	$-0.754993\pi$
$233$	−21.9230	−1.43622	−0.718112	+	0.695927i	$0.754993\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	7.05284	0.458131
$238$	0	0
$239$	3.66866	0.237306	0.118653	−	0.992936i	$-0.462142\pi$
$239$	3.66866	0.237306	0.118653	+	0.992936i	$0.462142\pi$
$240$	0	0
$241$	−6.03281	−0.388608	−0.194304	−	0.980941i	$-0.562245\pi$
$241$	−6.03281	−0.388608	−0.194304	+	0.980941i	$0.562245\pi$
$242$	0	0
$243$	15.4284	0.989731
$244$	0	0
$245$	0	0
$246$	0	0
$247$	2.24614	0.142919
$248$	0	0
$249$	−5.90918	−0.374479
$250$	0	0
$251$	0.145763	0.00920048	0.00460024	−	0.999989i	$-0.498536\pi$
$251$	0.145763	0.00920048	0.00460024	+	0.999989i	$0.498536\pi$
$252$	0	0
$253$	−5.52768	−0.347522
$254$	0	0
$255$	0	0
$256$	0	0
$257$	14.7514	0.920164	0.460082	−	0.887876i	$-0.347820\pi$
$257$	14.7514	0.920164	0.460082	+	0.887876i	$0.347820\pi$
$258$	0	0
$259$	−28.5404	−1.77342
$260$	0	0
$261$	−15.1891	−0.940180
$262$	0	0
$263$	−30.2967	−1.86817	−0.934087	−	0.357046i	$-0.883784\pi$
$263$	−30.2967	−1.86817	−0.934087	+	0.357046i	$0.883784\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	3.38211	0.206981
$268$	0	0
$269$	20.3460	1.24052	0.620259	−	0.784397i	$-0.287027\pi$
$269$	20.3460	1.24052	0.620259	+	0.784397i	$0.287027\pi$
$270$	0	0
$271$	1.01349	0.0615651	0.0307825	−	0.999526i	$-0.490200\pi$
$271$	1.01349	0.0615651	0.0307825	+	0.999526i	$0.490200\pi$
$272$	0	0
$273$	10.2760	0.621934
$274$	0	0
$275$	0	0
$276$	0	0
$277$	10.4346	0.626953	0.313476	−	0.949596i	$-0.398506\pi$
$277$	10.4346	0.626953	0.313476	+	0.949596i	$0.398506\pi$
$278$	0	0
$279$	−2.25658	−0.135098
$280$	0	0
$281$	−8.12732	−0.484835	−0.242418	−	0.970172i	$-0.577940\pi$
$281$	−8.12732	−0.484835	−0.242418	+	0.970172i	$0.577940\pi$
$282$	0	0
$283$	−11.6397	−0.691907	−0.345953	−	0.938252i	$-0.612444\pi$
$283$	−11.6397	−0.691907	−0.345953	+	0.938252i	$0.612444\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−24.8680	−1.46791
$288$	0	0
$289$	44.5824	2.62249
$290$	0	0
$291$	−7.36867	−0.431959
$292$	0	0
$293$	2.21896	0.129633	0.0648164	−	0.997897i	$-0.479354\pi$
$293$	2.21896	0.129633	0.0648164	+	0.997897i	$0.479354\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	33.7277	1.95708
$298$	0	0
$299$	−1.93078	−0.111660
$300$	0	0
$301$	28.5864	1.64770
$302$	0	0
$303$	−3.32716	−0.191140
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−17.6712	−1.00855	−0.504274	−	0.863544i	$-0.668240\pi$
$307$	−17.6712	−1.00855	−0.504274	+	0.863544i	$0.668240\pi$
$308$	0	0
$309$	−7.69931	−0.437999
$310$	0	0
$311$	12.3628	0.701031	0.350516	−	0.936557i	$-0.386006\pi$
$311$	12.3628	0.701031	0.350516	+	0.936557i	$0.386006\pi$
$312$	0	0
$313$	−8.35545	−0.472278	−0.236139	−	0.971719i	$-0.575882\pi$
$313$	−8.35545	−0.472278	−0.236139	+	0.971719i	$0.575882\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	19.3526	1.08695	0.543475	−	0.839425i	$-0.317108\pi$
$317$	19.3526	1.08695	0.543475	+	0.839425i	$0.317108\pi$
$318$	0	0
$319$	−53.9060	−3.01816
$320$	0	0
$321$	5.50751	0.307399
$322$	0	0
$323$	−7.84744	−0.436643
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−6.96867	−0.385368
$328$	0	0
$329$	−25.2222	−1.39054
$330$	0	0
$331$	33.8929	1.86292	0.931461	−	0.363840i	$-0.118535\pi$
$331$	33.8929	1.86292	0.931461	+	0.363840i	$0.118535\pi$
$332$	0	0
$333$	−12.3207	−0.675168
$334$	0	0
$335$	0	0
$336$	0	0
$337$	28.6728	1.56191	0.780954	−	0.624588i	$-0.214733\pi$
$337$	28.6728	1.56191	0.780954	+	0.624588i	$0.214733\pi$
$338$	0	0
$339$	21.3460	1.15936
$340$	0	0
$341$	−8.00860	−0.433690
$342$	0	0
$343$	−15.1818	−0.819738
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−15.9590	−0.856725	−0.428362	−	0.903607i	$-0.640909\pi$
$347$	−15.9590	−0.856725	−0.428362	+	0.903607i	$0.640909\pi$
$348$	0	0
$349$	29.4645	1.57720	0.788599	−	0.614907i	$-0.210807\pi$
$349$	29.4645	1.57720	0.788599	+	0.614907i	$0.210807\pi$
$350$	0	0
$351$	11.7809	0.628817
$352$	0	0
$353$	28.3655	1.50974	0.754871	−	0.655873i	$-0.227699\pi$
$353$	28.3655	1.50974	0.754871	+	0.655873i	$0.227699\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−35.9019	−1.90013
$358$	0	0
$359$	0.0301128	0.00158929	0.000794647	−	1.00000i	$-0.499747\pi$
$359$	0.0301128	0.00158929	0.000794647		1.00000i	$0.499747\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	33.0829	1.73640
$364$	0	0
$365$	0	0
$366$	0	0
$367$	1.31478	0.0686310	0.0343155	−	0.999411i	$-0.489075\pi$
$367$	1.31478	0.0686310	0.0343155	+	0.999411i	$0.489075\pi$
$368$	0	0
$369$	−10.7353	−0.558857
$370$	0	0
$371$	57.7517	2.99832
$372$	0	0
$373$	3.63316	0.188118	0.0940589	−	0.995567i	$-0.470016\pi$
$373$	3.63316	0.188118	0.0940589	+	0.995567i	$0.470016\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−18.8290	−0.969745
$378$	0	0
$379$	24.4908	1.25801	0.629004	−	0.777402i	$-0.283463\pi$
$379$	24.4908	1.25801	0.629004	+	0.777402i	$0.283463\pi$
$380$	0	0
$381$	−6.97945	−0.357568
$382$	0	0
$383$	−3.23421	−0.165261	−0.0826303	−	0.996580i	$-0.526332\pi$
$383$	−3.23421	−0.165261	−0.0826303	+	0.996580i	$0.526332\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	12.3405	0.627304
$388$	0	0
$389$	22.9240	1.16229	0.581147	−	0.813799i	$-0.302604\pi$
$389$	22.9240	1.16229	0.581147	+	0.813799i	$0.302604\pi$
$390$	0	0
$391$	6.74567	0.341143
$392$	0	0
$393$	13.2884	0.670312
$394$	0	0
$395$	0	0
$396$	0	0
$397$	14.1567	0.710507	0.355253	−	0.934770i	$-0.384395\pi$
$397$	14.1567	0.710507	0.355253	+	0.934770i	$0.384395\pi$
$398$	0	0
$399$	4.57497	0.229035
$400$	0	0
$401$	−6.06391	−0.302817	−0.151409	−	0.988471i	$-0.548381\pi$
$401$	−6.06391	−0.302817	−0.151409	+	0.988471i	$0.548381\pi$
$402$	0	0
$403$	−2.79736	−0.139346
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−43.7260	−2.16742
$408$	0	0
$409$	3.95218	0.195423	0.0977113	−	0.995215i	$-0.468848\pi$
$409$	3.95218	0.195423	0.0977113	+	0.995215i	$0.468848\pi$
$410$	0	0
$411$	10.5863	0.522183
$412$	0	0
$413$	−28.9182	−1.42297
$414$	0	0
$415$	0	0
$416$	0	0
$417$	21.8746	1.07121
$418$	0	0
$419$	−12.6137	−0.616221	−0.308111	−	0.951351i	$-0.599697\pi$
$419$	−12.6137	−0.616221	−0.308111	+	0.951351i	$0.599697\pi$
$420$	0	0
$421$	−9.15060	−0.445973	−0.222986	−	0.974822i	$-0.571581\pi$
$421$	−9.15060	−0.445973	−0.222986	+	0.974822i	$0.571581\pi$
$422$	0	0
$423$	−10.8882	−0.529402
$424$	0	0
$425$	0	0
$426$	0	0
$427$	5.53553	0.267883
$428$	0	0
$429$	15.7436	0.760110
$430$	0	0
$431$	−27.2560	−1.31288	−0.656439	−	0.754379i	$-0.727938\pi$
$431$	−27.2560	−1.31288	−0.656439	+	0.754379i	$0.727938\pi$
$432$	0	0
$433$	−2.33860	−0.112386	−0.0561929	−	0.998420i	$-0.517896\pi$
$433$	−2.33860	−0.112386	−0.0561929	+	0.998420i	$0.517896\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−0.859601	−0.0411203
$438$	0	0
$439$	37.6773	1.79824	0.899119	−	0.437703i	$-0.144208\pi$
$439$	37.6773	1.79824	0.899119	+	0.437703i	$0.144208\pi$
$440$	0	0
$441$	−19.2373	−0.916062
$442$	0	0
$443$	−6.39870	−0.304011	−0.152006	−	0.988380i	$-0.548573\pi$
$443$	−6.39870	−0.304011	−0.152006	+	0.988380i	$0.548573\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−16.1964	−0.766065
$448$	0	0
$449$	−30.1784	−1.42421	−0.712103	−	0.702075i	$-0.752257\pi$
$449$	−30.1784	−1.42421	−0.712103	+	0.702075i	$0.752257\pi$
$450$	0	0
$451$	−38.0995	−1.79404
$452$	0	0
$453$	−1.41231	−0.0663562
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−7.21464	−0.337487	−0.168743	−	0.985660i	$-0.553971\pi$
$457$	−7.21464	−0.337487	−0.168743	+	0.985660i	$0.553971\pi$
$458$	0	0
$459$	−41.1594	−1.92116
$460$	0	0
$461$	−2.06925	−0.0963748	−0.0481874	−	0.998838i	$-0.515344\pi$
$461$	−2.06925	−0.0963748	−0.0481874	+	0.998838i	$0.515344\pi$
$462$	0	0
$463$	4.01966	0.186809	0.0934047	−	0.995628i	$-0.470225\pi$
$463$	4.01966	0.186809	0.0934047	+	0.995628i	$0.470225\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	21.5515	0.997285	0.498643	−	0.866808i	$-0.333832\pi$
$467$	21.5515	0.997285	0.498643	+	0.866808i	$0.333832\pi$
$468$	0	0
$469$	19.8643	0.917246
$470$	0	0
$471$	−4.41832	−0.203585
$472$	0	0
$473$	43.7965	2.01377
$474$	0	0
$475$	0	0
$476$	0	0
$477$	24.9309	1.14151
$478$	0	0
$479$	−31.8315	−1.45442	−0.727209	−	0.686416i	$-0.759183\pi$
$479$	−31.8315	−1.45442	−0.727209	+	0.686416i	$0.759183\pi$
$480$	0	0
$481$	−15.2732	−0.696400
$482$	0	0
$483$	−3.93265	−0.178942
$484$	0	0
$485$	0	0
$486$	0	0
$487$	3.54933	0.160835	0.0804177	−	0.996761i	$-0.474375\pi$
$487$	3.54933	0.160835	0.0804177	+	0.996761i	$0.474375\pi$
$488$	0	0
$489$	−8.82634	−0.399141
$490$	0	0
$491$	−16.5336	−0.746152	−0.373076	−	0.927801i	$-0.621697\pi$
$491$	−16.5336	−0.746152	−0.373076	+	0.927801i	$0.621697\pi$
$492$	0	0
$493$	65.7839	2.96276
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−43.0413	−1.93067
$498$	0	0
$499$	−32.8826	−1.47203	−0.736013	−	0.676968i	$-0.763294\pi$
$499$	−32.8826	−1.47203	−0.736013	+	0.676968i	$0.763294\pi$
$500$	0	0
$501$	20.2802	0.906051
$502$	0	0
$503$	17.2150	0.767580	0.383790	−	0.923420i	$-0.374619\pi$
$503$	17.2150	0.767580	0.383790	+	0.923420i	$0.374619\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−8.67070	−0.385079
$508$	0	0
$509$	28.4368	1.26044	0.630221	−	0.776416i	$-0.282964\pi$
$509$	28.4368	1.26044	0.630221	+	0.776416i	$0.282964\pi$
$510$	0	0
$511$	−41.3913	−1.83105
$512$	0	0
$513$	5.24494	0.231570
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−38.6422	−1.69948
$518$	0	0
$519$	−9.11201	−0.399973
$520$	0	0
$521$	25.1924	1.10370	0.551850	−	0.833944i	$-0.313922\pi$
$521$	25.1924	1.10370	0.551850	+	0.833944i	$0.313922\pi$
$522$	0	0
$523$	17.2647	0.754934	0.377467	−	0.926023i	$-0.376795\pi$
$523$	17.2647	0.754934	0.377467	+	0.926023i	$0.376795\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	9.77325	0.425730
$528$	0	0
$529$	−22.2611	−0.967873
$530$	0	0
$531$	−12.4837	−0.541747
$532$	0	0
$533$	−13.3079	−0.576431
$534$	0	0
$535$	0	0
$536$	0	0
$537$	15.9107	0.686598
$538$	0	0
$539$	−68.2732	−2.94073
$540$	0	0
$541$	−9.71009	−0.417469	−0.208735	−	0.977972i	$-0.566935\pi$
$541$	−9.71009	−0.417469	−0.208735	+	0.977972i	$0.566935\pi$
$542$	0	0
$543$	0.143406	0.00615412
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−5.60962	−0.239850	−0.119925	−	0.992783i	$-0.538265\pi$
$547$	−5.60962	−0.239850	−0.119925	+	0.992783i	$0.538265\pi$
$548$	0	0
$549$	2.38964	0.101987
$550$	0	0
$551$	−8.38284	−0.357121
$552$	0	0
$553$	−27.1587	−1.15490
$554$	0	0
$555$	0	0
$556$	0	0
$557$	17.5323	0.742868	0.371434	−	0.928459i	$-0.378866\pi$
$557$	17.5323	0.742868	0.371434	+	0.928459i	$0.378866\pi$
$558$	0	0
$559$	15.2979	0.647030
$560$	0	0
$561$	−55.0042	−2.32228
$562$	0	0
$563$	8.85461	0.373177	0.186589	−	0.982438i	$-0.440257\pi$
$563$	8.85461	0.373177	0.186589	+	0.982438i	$0.440257\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	1.18009	0.0495593
$568$	0	0
$569$	33.0220	1.38435	0.692177	−	0.721727i	$-0.256652\pi$
$569$	33.0220	1.38435	0.692177	+	0.721727i	$0.256652\pi$
$570$	0	0
$571$	12.9266	0.540963	0.270482	−	0.962725i	$-0.412817\pi$
$571$	12.9266	0.540963	0.270482	+	0.962725i	$0.412817\pi$
$572$	0	0
$573$	−24.5840	−1.02701
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−3.22539	−0.134275	−0.0671374	−	0.997744i	$-0.521387\pi$
$577$	−3.22539	−0.134275	−0.0671374	+	0.997744i	$0.521387\pi$
$578$	0	0
$579$	−5.38305	−0.223712
$580$	0	0
$581$	22.7548	0.944026
$582$	0	0
$583$	88.4797	3.66446
$584$	0	0
$585$	0	0
$586$	0	0
$587$	7.17382	0.296095	0.148048	−	0.988980i	$-0.452701\pi$
$587$	7.17382	0.296095	0.148048	+	0.988980i	$0.452701\pi$
$588$	0	0
$589$	−1.24541	−0.0513161
$590$	0	0
$591$	−28.6037	−1.17660
$592$	0	0
$593$	27.2576	1.11933	0.559667	−	0.828718i	$-0.310929\pi$
$593$	27.2576	1.11933	0.559667	+	0.828718i	$0.310929\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	16.5547	0.677540
$598$	0	0
$599$	−3.53091	−0.144269	−0.0721344	−	0.997395i	$-0.522981\pi$
$599$	−3.53091	−0.144269	−0.0721344	+	0.997395i	$0.522981\pi$
$600$	0	0
$601$	23.6488	0.964655	0.482327	−	0.875991i	$-0.339792\pi$
$601$	23.6488	0.964655	0.482327	+	0.875991i	$0.339792\pi$
$602$	0	0
$603$	8.57523	0.349210
$604$	0	0
$605$	0	0
$606$	0	0
$607$	28.6598	1.16326	0.581632	−	0.813452i	$-0.302414\pi$
$607$	28.6598	1.16326	0.581632	+	0.813452i	$0.302414\pi$
$608$	0	0
$609$	−38.3513	−1.55407
$610$	0	0
$611$	−13.4975	−0.546050
$612$	0	0
$613$	−4.64372	−0.187558	−0.0937790	−	0.995593i	$-0.529895\pi$
$613$	−4.64372	−0.187558	−0.0937790	+	0.995593i	$0.529895\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−4.49990	−0.181159	−0.0905796	−	0.995889i	$-0.528872\pi$
$617$	−4.49990	−0.181159	−0.0905796	+	0.995889i	$0.528872\pi$
$618$	0	0
$619$	20.5767	0.827049	0.413525	−	0.910493i	$-0.364298\pi$
$619$	20.5767	0.827049	0.413525	+	0.910493i	$0.364298\pi$
$620$	0	0
$621$	−4.50856	−0.180922
$622$	0	0
$623$	−13.0236	−0.521780
$624$	0	0
$625$	0	0
$626$	0	0
$627$	7.00919	0.279920
$628$	0	0
$629$	53.3608	2.12763
$630$	0	0
$631$	−32.5361	−1.29524	−0.647620	−	0.761963i	$-0.724236\pi$
$631$	−32.5361	−1.29524	−0.647620	+	0.761963i	$0.724236\pi$
$632$	0	0
$633$	22.1956	0.882196
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−23.8474	−0.944869
$638$	0	0
$639$	−18.5806	−0.735035
$640$	0	0
$641$	13.0922	0.517113	0.258556	−	0.965996i	$-0.416753\pi$
$641$	13.0922	0.517113	0.258556	+	0.965996i	$0.416753\pi$
$642$	0	0
$643$	16.8450	0.664304	0.332152	−	0.943226i	$-0.392225\pi$
$643$	16.8450	0.664304	0.332152	+	0.943226i	$0.392225\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−46.9300	−1.84501	−0.922504	−	0.385987i	$-0.873861\pi$
$647$	−46.9300	−1.84501	−0.922504	+	0.385987i	$0.873861\pi$
$648$	0	0
$649$	−44.3047	−1.73911
$650$	0	0
$651$	−5.69770	−0.223310
$652$	0	0
$653$	14.1069	0.552046	0.276023	−	0.961151i	$-0.410983\pi$
$653$	14.1069	0.552046	0.276023	+	0.961151i	$0.410983\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−17.8683	−0.697108
$658$	0	0
$659$	30.5841	1.19139	0.595693	−	0.803212i	$-0.296878\pi$
$659$	30.5841	1.19139	0.595693	+	0.803212i	$0.296878\pi$
$660$	0	0
$661$	−42.8953	−1.66843	−0.834216	−	0.551437i	$-0.814080\pi$
$661$	−42.8953	−1.66843	−0.834216	+	0.551437i	$0.814080\pi$
$662$	0	0
$663$	−19.2127	−0.746158
$664$	0	0
$665$	0	0
$666$	0	0
$667$	7.20589	0.279013
$668$	0	0
$669$	−16.7602	−0.647985
$670$	0	0
$671$	8.48082	0.327399
$672$	0	0
$673$	19.0609	0.734744	0.367372	−	0.930074i	$-0.380258\pi$
$673$	19.0609	0.734744	0.367372	+	0.930074i	$0.380258\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	48.2121	1.85294	0.926471	−	0.376366i	$-0.122827\pi$
$677$	48.2121	1.85294	0.926471	+	0.376366i	$0.122827\pi$
$678$	0	0
$679$	28.3749	1.08893
$680$	0	0
$681$	21.8201	0.836149
$682$	0	0
$683$	2.59451	0.0992763	0.0496381	−	0.998767i	$-0.484193\pi$
$683$	2.59451	0.0992763	0.0496381	+	0.998767i	$0.484193\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−13.7530	−0.524708
$688$	0	0
$689$	30.9054	1.17740
$690$	0	0
$691$	−15.0799	−0.573667	−0.286834	−	0.957980i	$-0.592603\pi$
$691$	−15.0799	−0.573667	−0.286834	+	0.957980i	$0.592603\pi$
$692$	0	0
$693$	−48.9049	−1.85774
$694$	0	0
$695$	0	0
$696$	0	0
$697$	46.4945	1.76111
$698$	0	0
$699$	−23.8958	−0.903824
$700$	0	0
$701$	27.2335	1.02860	0.514298	−	0.857612i	$-0.328053\pi$
$701$	27.2335	1.02860	0.514298	+	0.857612i	$0.328053\pi$
$702$	0	0
$703$	−6.79977	−0.256458
$704$	0	0
$705$	0	0
$706$	0	0
$707$	12.8120	0.481846
$708$	0	0
$709$	−29.9175	−1.12357	−0.561787	−	0.827282i	$-0.689886\pi$
$709$	−29.9175	−1.12357	−0.561787	+	0.827282i	$0.689886\pi$
$710$	0	0
$711$	−11.7242	−0.439690
$712$	0	0
$713$	1.07055	0.0400925
$714$	0	0
$715$	0	0
$716$	0	0
$717$	3.99880	0.149338
$718$	0	0
$719$	31.7693	1.18479	0.592397	−	0.805646i	$-0.298182\pi$
$719$	31.7693	1.18479	0.592397	+	0.805646i	$0.298182\pi$
$720$	0	0
$721$	29.6481	1.10415
$722$	0	0
$723$	−6.57570	−0.244553
$724$	0	0
$725$	0	0
$726$	0	0
$727$	23.7272	0.879993	0.439996	−	0.898000i	$-0.354980\pi$
$727$	23.7272	0.879993	0.439996	+	0.898000i	$0.354980\pi$
$728$	0	0
$729$	17.6602	0.654082
$730$	0	0
$731$	−53.4468	−1.97680
$732$	0	0
$733$	−37.4907	−1.38475	−0.692374	−	0.721538i	$-0.743435\pi$
$733$	−37.4907	−1.38475	−0.692374	+	0.721538i	$0.743435\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	30.4335	1.12103
$738$	0	0
$739$	7.10315	0.261294	0.130647	−	0.991429i	$-0.458295\pi$
$739$	7.10315	0.261294	0.130647	+	0.991429i	$0.458295\pi$
$740$	0	0
$741$	2.44827	0.0899394
$742$	0	0
$743$	10.6047	0.389049	0.194525	−	0.980898i	$-0.437684\pi$
$743$	10.6047	0.389049	0.194525	+	0.980898i	$0.437684\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	9.82303	0.359406
$748$	0	0
$749$	−21.2080	−0.774923
$750$	0	0
$751$	32.1380	1.17273	0.586365	−	0.810047i	$-0.300558\pi$
$751$	32.1380	1.17273	0.586365	+	0.810047i	$0.300558\pi$
$752$	0	0
$753$	0.158880	0.00578991
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−36.2721	−1.31833	−0.659167	−	0.751997i	$-0.729091\pi$
$757$	−36.2721	−1.31833	−0.659167	+	0.751997i	$0.729091\pi$
$758$	0	0
$759$	−6.02510	−0.218697
$760$	0	0
$761$	31.2638	1.13331	0.566656	−	0.823954i	$-0.308237\pi$
$761$	31.2638	1.13331	0.566656	+	0.823954i	$0.308237\pi$
$762$	0	0
$763$	26.8346	0.971476
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−15.4754	−0.558783
$768$	0	0
$769$	48.5188	1.74963	0.874816	−	0.484456i	$-0.160982\pi$
$769$	48.5188	1.74963	0.874816	+	0.484456i	$0.160982\pi$
$770$	0	0
$771$	16.0788	0.579064
$772$	0	0
$773$	−26.4968	−0.953023	−0.476512	−	0.879168i	$-0.658099\pi$
$773$	−26.4968	−0.953023	−0.476512	+	0.879168i	$0.658099\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−31.1088	−1.11602
$778$	0	0
$779$	−5.92480	−0.212278
$780$	0	0
$781$	−65.9424	−2.35960
$782$	0	0
$783$	−43.9675	−1.57127
$784$	0	0
$785$	0	0
$786$	0	0
$787$	16.9510	0.604238	0.302119	−	0.953270i	$-0.402306\pi$
$787$	16.9510	0.604238	0.302119	+	0.953270i	$0.402306\pi$
$788$	0	0
$789$	−33.0230	−1.17565
$790$	0	0
$791$	−82.1981	−2.92263
$792$	0	0
$793$	2.96230	0.105194
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−53.4354	−1.89278	−0.946390	−	0.323025i	$-0.895300\pi$
$797$	−53.4354	−1.89278	−0.946390	+	0.323025i	$0.895300\pi$
$798$	0	0
$799$	47.1568	1.66829
$800$	0	0
$801$	−5.62218	−0.198650
$802$	0	0
$803$	−63.4145	−2.23785
$804$	0	0
$805$	0	0
$806$	0	0
$807$	22.1769	0.780665
$808$	0	0
$809$	−3.88632	−0.136636	−0.0683179	−	0.997664i	$-0.521763\pi$
$809$	−3.88632	−0.136636	−0.0683179	+	0.997664i	$0.521763\pi$
$810$	0	0
$811$	10.8136	0.379716	0.189858	−	0.981812i	$-0.439197\pi$
$811$	10.8136	0.379716	0.189858	+	0.981812i	$0.439197\pi$
$812$	0	0
$813$	1.10469	0.0387432
$814$	0	0
$815$	0	0
$816$	0	0
$817$	6.81073	0.238277
$818$	0	0
$819$	−17.0822	−0.596900
$820$	0	0
$821$	−43.7227	−1.52593	−0.762966	−	0.646438i	$-0.776258\pi$
$821$	−43.7227	−1.52593	−0.762966	+	0.646438i	$0.776258\pi$
$822$	0	0
$823$	4.51329	0.157323	0.0786617	−	0.996901i	$-0.474935\pi$
$823$	4.51329	0.157323	0.0786617	+	0.996901i	$0.474935\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−9.56467	−0.332596	−0.166298	−	0.986076i	$-0.553181\pi$
$827$	−9.56467	−0.332596	−0.166298	+	0.986076i	$0.553181\pi$
$828$	0	0
$829$	20.0719	0.697126	0.348563	−	0.937285i	$-0.386670\pi$
$829$	20.0719	0.697126	0.348563	+	0.937285i	$0.386670\pi$
$830$	0	0
$831$	11.3736	0.394545
$832$	0	0
$833$	83.3168	2.88675
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−6.53208	−0.225782
$838$	0	0
$839$	12.4591	0.430135	0.215067	−	0.976599i	$-0.431003\pi$
$839$	12.4591	0.430135	0.215067	+	0.976599i	$0.431003\pi$
$840$	0	0
$841$	41.2720	1.42317
$842$	0	0
$843$	−8.85869	−0.305110
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−127.394	−4.37730
$848$	0	0
$849$	−12.6871	−0.435421
$850$	0	0
$851$	5.84508	0.200367
$852$	0	0
$853$	22.0112	0.753649	0.376824	−	0.926285i	$-0.377016\pi$
$853$	22.0112	0.753649	0.376824	+	0.926285i	$0.377016\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−15.8261	−0.540608	−0.270304	−	0.962775i	$-0.587124\pi$
$857$	−15.8261	−0.540608	−0.270304	+	0.962775i	$0.587124\pi$
$858$	0	0
$859$	−23.8728	−0.814530	−0.407265	−	0.913310i	$-0.633517\pi$
$859$	−23.8728	−0.814530	−0.407265	+	0.913310i	$0.633517\pi$
$860$	0	0
$861$	−27.1058	−0.923763
$862$	0	0
$863$	−29.6871	−1.01056	−0.505280	−	0.862956i	$-0.668611\pi$
$863$	−29.6871	−1.01056	−0.505280	+	0.862956i	$0.668611\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	48.5943	1.65035
$868$	0	0
$869$	−41.6090	−1.41149
$870$	0	0
$871$	10.6302	0.360192
$872$	0	0
$873$	12.2492	0.414572
$874$	0	0
$875$	0	0
$876$	0	0
$877$	20.3979	0.688790	0.344395	−	0.938825i	$-0.388084\pi$
$877$	20.3979	0.688790	0.344395	+	0.938825i	$0.388084\pi$
$878$	0	0
$879$	2.41864	0.0815786
$880$	0	0
$881$	−43.2919	−1.45854	−0.729270	−	0.684226i	$-0.760140\pi$
$881$	−43.2919	−1.45854	−0.729270	+	0.684226i	$0.760140\pi$
$882$	0	0
$883$	−9.74875	−0.328071	−0.164036	−	0.986454i	$-0.552451\pi$
$883$	−9.74875	−0.328071	−0.164036	+	0.986454i	$0.552451\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−10.5431	−0.354002	−0.177001	−	0.984211i	$-0.556640\pi$
$887$	−10.5431	−0.354002	−0.177001	+	0.984211i	$0.556640\pi$
$888$	0	0
$889$	26.8761	0.901395
$890$	0	0
$891$	1.80799	0.0605699
$892$	0	0
$893$	−6.00919	−0.201090
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−2.10453	−0.0702683
$898$	0	0
$899$	10.4400	0.348195
$900$	0	0
$901$	−107.976	−3.59719
$902$	0	0
$903$	31.1589	1.03690
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−48.2385	−1.60173	−0.800866	−	0.598844i	$-0.795627\pi$
$907$	−48.2385	−1.60173	−0.800866	+	0.598844i	$0.795627\pi$
$908$	0	0
$909$	5.53084	0.183447
$910$	0	0
$911$	−9.86236	−0.326755	−0.163377	−	0.986564i	$-0.552239\pi$
$911$	−9.86236	−0.326755	−0.163377	+	0.986564i	$0.552239\pi$
$912$	0	0
$913$	34.8619	1.15376
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−51.1703	−1.68979
$918$	0	0
$919$	38.0526	1.25524	0.627620	−	0.778520i	$-0.284029\pi$
$919$	38.0526	1.25524	0.627620	+	0.778520i	$0.284029\pi$
$920$	0	0
$921$	−19.2614	−0.634684
$922$	0	0
$923$	−23.0333	−0.758149
$924$	0	0
$925$	0	0
$926$	0	0
$927$	12.7988	0.420368
$928$	0	0
$929$	47.3243	1.55266	0.776330	−	0.630327i	$-0.217079\pi$
$929$	47.3243	1.55266	0.776330	+	0.630327i	$0.217079\pi$
$930$	0	0
$931$	−10.6171	−0.347960
$932$	0	0
$933$	13.4753	0.441163
$934$	0	0
$935$	0	0
$936$	0	0
$937$	19.0466	0.622224	0.311112	−	0.950373i	$-0.399299\pi$
$937$	19.0466	0.622224	0.311112	+	0.950373i	$0.399299\pi$
$938$	0	0
$939$	−9.10735	−0.297207
$940$	0	0
$941$	37.7416	1.23034	0.615171	−	0.788393i	$-0.289087\pi$
$941$	37.7416	1.23034	0.615171	+	0.788393i	$0.289087\pi$
$942$	0	0
$943$	5.09296	0.165850
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−4.72833	−0.153650	−0.0768251	−	0.997045i	$-0.524478\pi$
$947$	−4.72833	−0.153650	−0.0768251	+	0.997045i	$0.524478\pi$
$948$	0	0
$949$	−22.1503	−0.719030
$950$	0	0
$951$	21.0941	0.684024
$952$	0	0
$953$	−32.9799	−1.06832	−0.534162	−	0.845382i	$-0.679373\pi$
$953$	−32.9799	−1.06832	−0.534162	+	0.845382i	$0.679373\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−58.7569	−1.89934
$958$	0	0
$959$	−40.7650	−1.31637
$960$	0	0
$961$	−29.4490	−0.949967
$962$	0	0
$963$	−9.15531	−0.295026
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−3.71736	−0.119542	−0.0597712	−	0.998212i	$-0.519037\pi$
$967$	−3.71736	−0.119542	−0.0597712	+	0.998212i	$0.519037\pi$
$968$	0	0
$969$	−8.55363	−0.274782
$970$	0	0
$971$	18.6887	0.599749	0.299875	−	0.953979i	$-0.403055\pi$
$971$	18.6887	0.599749	0.299875	+	0.953979i	$0.403055\pi$
$972$	0	0
$973$	−84.2337	−2.70041
$974$	0	0
$975$	0	0
$976$	0	0
$977$	7.44705	0.238252	0.119126	−	0.992879i	$-0.461991\pi$
$977$	7.44705	0.238252	0.119126	+	0.992879i	$0.461991\pi$
$978$	0	0
$979$	−19.9531	−0.637705
$980$	0	0
$981$	11.5842	0.369856
$982$	0	0
$983$	13.7703	0.439204	0.219602	−	0.975590i	$-0.429524\pi$
$983$	13.7703	0.439204	0.219602	+	0.975590i	$0.429524\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−27.4919	−0.875077
$988$	0	0
$989$	−5.85450	−0.186162
$990$	0	0
$991$	32.0927	1.01946	0.509728	−	0.860335i	$-0.329746\pi$
$991$	32.0927	1.01946	0.509728	+	0.860335i	$0.329746\pi$
$992$	0	0
$993$	36.9429	1.17235
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−22.8068	−0.722299	−0.361149	−	0.932508i	$-0.617616\pi$
$997$	−22.8068	−0.722299	−0.361149	+	0.932508i	$0.617616\pi$
$998$	0	0
$999$	−35.6644	−1.12837

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3800.2.a.ba.1.5	✓	6
4.3	odd	2		7600.2.a.cl.1.2		6
5.2	odd	4		3800.2.d.q.3649.4		12
5.3	odd	4		3800.2.d.q.3649.9		12
5.4	even	2		3800.2.a.bc.1.2	yes	6
20.19	odd	2		7600.2.a.ch.1.5		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3800.2.a.ba.1.5	✓	6	1.1	even	1	trivial
3800.2.a.bc.1.2	yes	6	5.4	even	2
3800.2.d.q.3649.4		12	5.2	odd	4
3800.2.d.q.3649.9		12	5.3	odd	4
7600.2.a.ch.1.5		6	20.19	odd	2
7600.2.a.cl.1.2		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)

\(f(q)\)	\(=\)	\(q+1.08999 q^{3} -4.19727 q^{7} -1.81192 q^{9} +O(q^{10})\)	\(q+1.08999 q^{3} -4.19727 q^{7} -1.81192 q^{9} -6.43052 q^{11} -2.24614 q^{13} +7.84744 q^{17} -1.00000 q^{19} -4.57497 q^{21} +0.859601 q^{23} -5.24494 q^{27} +8.38284 q^{29} +1.24541 q^{31} -7.00919 q^{33} +6.79977 q^{37} -2.44827 q^{39} +5.92480 q^{41} -6.81073 q^{43} +6.00919 q^{47} +10.6171 q^{49} +8.55363 q^{51} -13.7594 q^{53} -1.08999 q^{57} +6.88976 q^{59} -1.31884 q^{61} +7.60513 q^{63} -4.73266 q^{67} +0.936955 q^{69} +10.2546 q^{71} +9.86150 q^{73} +26.9906 q^{77} +6.47056 q^{79} -0.281158 q^{81} -5.42133 q^{83} +9.13720 q^{87} +3.10288 q^{89} +9.42766 q^{91} +1.35748 q^{93} -6.76032 q^{97} +11.6516 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 2 q^{3} - 2 q^{7} + 6 q^{9}+O(q^{10}) \) 6 * q - 2 * q^3 - 2 * q^7 + 6 * q^9	\( 6 q - 2 q^{3} - 2 q^{7} + 6 q^{9} + 3 q^{11} + q^{13} + 14 q^{17} - 6 q^{19} + 15 q^{21} - 12 q^{23} - 8 q^{27} + 9 q^{29} + 5 q^{31} - 2 q^{33} + 8 q^{37} + 12 q^{39} + 3 q^{41} - 15 q^{43} - 4 q^{47} + 22 q^{49} + 33 q^{51} - 13 q^{53} + 2 q^{57} + 9 q^{61} - 21 q^{63} + 3 q^{67} - 11 q^{69} + 19 q^{71} - 3 q^{73} + 36 q^{77} - 16 q^{79} + 26 q^{81} - 31 q^{83} + 25 q^{87} + 14 q^{89} + 42 q^{91} - 39 q^{93} + 11 q^{97} - 6 q^{99}+O(q^{100}) \) 6 * q - 2 * q^3 - 2 * q^7 + 6 * q^9 + 3 * q^11 + q^13 + 14 * q^17 - 6 * q^19 + 15 * q^21 - 12 * q^23 - 8 * q^27 + 9 * q^29 + 5 * q^31 - 2 * q^33 + 8 * q^37 + 12 * q^39 + 3 * q^41 - 15 * q^43 - 4 * q^47 + 22 * q^49 + 33 * q^51 - 13 * q^53 + 2 * q^57 + 9 * q^61 - 21 * q^63 + 3 * q^67 - 11 * q^69 + 19 * q^71 - 3 * q^73 + 36 * q^77 - 16 * q^79 + 26 * q^81 - 31 * q^83 + 25 * q^87 + 14 * q^89 + 42 * q^91 - 39 * q^93 + 11 * q^97 - 6 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more