LMFDB - Embedded newform 3800.2.a.n.1.2

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:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 760)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.41421$ 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)

$f(q)$	$=$	$q+1.41421 q^{3} +2.82843 q^{7} -1.00000 q^{9} +O(q^{10})$	$q+1.41421 q^{3} +2.82843 q^{7} -1.00000 q^{9} -0.828427 q^{11} -3.41421 q^{13} -4.82843 q^{17} -1.00000 q^{19} +4.00000 q^{21} -4.00000 q^{23} -5.65685 q^{27} -0.828427 q^{29} -1.17157 q^{33} -10.2426 q^{37} -4.82843 q^{39} -0.828427 q^{41} -2.82843 q^{43} +8.48528 q^{47} +1.00000 q^{49} -6.82843 q^{51} -13.0711 q^{53} -1.41421 q^{57} +2.82843 q^{59} -1.65685 q^{61} -2.82843 q^{63} +9.41421 q^{67} -5.65685 q^{69} +15.3137 q^{71} -12.1421 q^{73} -2.34315 q^{77} -9.17157 q^{79} -5.00000 q^{81} -8.00000 q^{83} -1.17157 q^{87} -3.17157 q^{89} -9.65685 q^{91} +2.24264 q^{97} +0.828427 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^9	$ 2 q - 2 q^{9} + 4 q^{11} - 4 q^{13} - 4 q^{17} - 2 q^{19} + 8 q^{21} - 8 q^{23} + 4 q^{29} - 8 q^{33} - 12 q^{37} - 4 q^{39} + 4 q^{41} + 2 q^{49} - 8 q^{51} - 12 q^{53} + 8 q^{61} + 16 q^{67} + 8 q^{71} + 4 q^{73} - 16 q^{77} - 24 q^{79} - 10 q^{81} - 16 q^{83} - 8 q^{87} - 12 q^{89} - 8 q^{91} - 4 q^{97} - 4 q^{99}+O(q^{100}) $ 2 * q - 2 * q^9 + 4 * q^11 - 4 * q^13 - 4 * q^17 - 2 * q^19 + 8 * q^21 - 8 * q^23 + 4 * q^29 - 8 * q^33 - 12 * q^37 - 4 * q^39 + 4 * q^41 + 2 * q^49 - 8 * q^51 - 12 * q^53 + 8 * q^61 + 16 * q^67 + 8 * q^71 + 4 * q^73 - 16 * q^77 - 24 * q^79 - 10 * q^81 - 16 * q^83 - 8 * q^87 - 12 * q^89 - 8 * q^91 - 4 * q^97 - 4 * 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
$3$	1.41421	0.816497	0.408248	−	0.912871i	$-0.366140\pi$
$3$	1.41421	0.816497	0.408248	+	0.912871i	$0.366140\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	2.82843	1.06904	0.534522	−	0.845154i	$-0.320491\pi$
$7$	2.82843	1.06904	0.534522	+	0.845154i	$0.320491\pi$
$8$	0	0
$9$	−1.00000	−0.333333
$10$	0	0
$11$	−0.828427	−0.249780	−0.124890	−	0.992171i	$-0.539858\pi$
$11$	−0.828427	−0.249780	−0.124890	+	0.992171i	$0.539858\pi$
$12$	0	0
$13$	−3.41421	−0.946932	−0.473466	−	0.880812i	$-0.656997\pi$
$13$	−3.41421	−0.946932	−0.473466	+	0.880812i	$0.656997\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−4.82843	−1.17107	−0.585533	−	0.810649i	$-0.699115\pi$
$17$	−4.82843	−1.17107	−0.585533	+	0.810649i	$0.699115\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	4.00000	0.872872
$22$	0	0
$23$	−4.00000	−0.834058	−0.417029	−	0.908893i	$-0.636929\pi$
$23$	−4.00000	−0.834058	−0.417029	+	0.908893i	$0.636929\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−5.65685	−1.08866
$28$	0	0
$29$	−0.828427	−0.153835	−0.0769175	−	0.997037i	$-0.524508\pi$
$29$	−0.828427	−0.153835	−0.0769175	+	0.997037i	$0.524508\pi$
$30$	0	0
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	0	0
$33$	−1.17157	−0.203945
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−10.2426	−1.68388	−0.841940	−	0.539571i	$-0.818586\pi$
$37$	−10.2426	−1.68388	−0.841940	+	0.539571i	$0.818586\pi$
$38$	0	0
$39$	−4.82843	−0.773167
$40$	0	0
$41$	−0.828427	−0.129379	−0.0646893	−	0.997905i	$-0.520606\pi$
$41$	−0.828427	−0.129379	−0.0646893	+	0.997905i	$0.520606\pi$
$42$	0	0
$43$	−2.82843	−0.431331	−0.215666	−	0.976467i	$-0.569192\pi$
$43$	−2.82843	−0.431331	−0.215666	+	0.976467i	$0.569192\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	8.48528	1.23771	0.618853	−	0.785507i	$-0.287598\pi$
$47$	8.48528	1.23771	0.618853	+	0.785507i	$0.287598\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−6.82843	−0.956171
$52$	0	0
$53$	−13.0711	−1.79545	−0.897725	−	0.440557i	$-0.854781\pi$
$53$	−13.0711	−1.79545	−0.897725	+	0.440557i	$0.854781\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−1.41421	−0.187317
$58$	0	0
$59$	2.82843	0.368230	0.184115	−	0.982905i	$-0.441058\pi$
$59$	2.82843	0.368230	0.184115	+	0.982905i	$0.441058\pi$
$60$	0	0
$61$	−1.65685	−0.212138	−0.106069	−	0.994359i	$-0.533827\pi$
$61$	−1.65685	−0.212138	−0.106069	+	0.994359i	$0.533827\pi$
$62$	0	0
$63$	−2.82843	−0.356348
$64$	0	0
$65$	0	0
$66$	0	0
$67$	9.41421	1.15013	0.575065	−	0.818108i	$-0.304977\pi$
$67$	9.41421	1.15013	0.575065	+	0.818108i	$0.304977\pi$
$68$	0	0
$69$	−5.65685	−0.681005
$70$	0	0
$71$	15.3137	1.81740	0.908701	−	0.417447i	$-0.137075\pi$
$71$	15.3137	1.81740	0.908701	+	0.417447i	$0.137075\pi$
$72$	0	0
$73$	−12.1421	−1.42113	−0.710565	−	0.703632i	$-0.751560\pi$
$73$	−12.1421	−1.42113	−0.710565	+	0.703632i	$0.751560\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−2.34315	−0.267026
$78$	0	0
$79$	−9.17157	−1.03188	−0.515941	−	0.856624i	$-0.672558\pi$
$79$	−9.17157	−1.03188	−0.515941	+	0.856624i	$0.672558\pi$
$80$	0	0
$81$	−5.00000	−0.555556
$82$	0	0
$83$	−8.00000	−0.878114	−0.439057	−	0.898459i	$-0.644687\pi$
$83$	−8.00000	−0.878114	−0.439057	+	0.898459i	$0.644687\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−1.17157	−0.125606
$88$	0	0
$89$	−3.17157	−0.336186	−0.168093	−	0.985771i	$-0.553761\pi$
$89$	−3.17157	−0.336186	−0.168093	+	0.985771i	$0.553761\pi$
$90$	0	0
$91$	−9.65685	−1.01231
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	2.24264	0.227706	0.113853	−	0.993498i	$-0.463681\pi$
$97$	2.24264	0.227706	0.113853	+	0.993498i	$0.463681\pi$
$98$	0	0
$99$	0.828427	0.0832601
$100$	0	0
$101$	17.6569	1.75692	0.878461	−	0.477813i	$-0.158571\pi$
$101$	17.6569	1.75692	0.878461	+	0.477813i	$0.158571\pi$
$102$	0	0
$103$	−8.24264	−0.812172	−0.406086	−	0.913835i	$-0.633107\pi$
$103$	−8.24264	−0.812172	−0.406086	+	0.913835i	$0.633107\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−1.41421	−0.136717	−0.0683586	−	0.997661i	$-0.521776\pi$
$107$	−1.41421	−0.136717	−0.0683586	+	0.997661i	$0.521776\pi$
$108$	0	0
$109$	3.65685	0.350263	0.175132	−	0.984545i	$-0.443965\pi$
$109$	3.65685	0.350263	0.175132	+	0.984545i	$0.443965\pi$
$110$	0	0
$111$	−14.4853	−1.37488
$112$	0	0
$113$	14.7279	1.38549	0.692743	−	0.721184i	$-0.256402\pi$
$113$	14.7279	1.38549	0.692743	+	0.721184i	$0.256402\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	3.41421	0.315644
$118$	0	0
$119$	−13.6569	−1.25192
$120$	0	0
$121$	−10.3137	−0.937610
$122$	0	0
$123$	−1.17157	−0.105637
$124$	0	0
$125$	0	0
$126$	0	0
$127$	13.4142	1.19032	0.595159	−	0.803608i	$-0.297089\pi$
$127$	13.4142	1.19032	0.595159	+	0.803608i	$0.297089\pi$
$128$	0	0
$129$	−4.00000	−0.352180
$130$	0	0
$131$	7.31371	0.639002	0.319501	−	0.947586i	$-0.396485\pi$
$131$	7.31371	0.639002	0.319501	+	0.947586i	$0.396485\pi$
$132$	0	0
$133$	−2.82843	−0.245256
$134$	0	0
$135$	0	0
$136$	0	0
$137$	7.17157	0.612709	0.306354	−	0.951918i	$-0.400891\pi$
$137$	7.17157	0.612709	0.306354	+	0.951918i	$0.400891\pi$
$138$	0	0
$139$	−3.17157	−0.269009	−0.134505	−	0.990913i	$-0.542944\pi$
$139$	−3.17157	−0.269009	−0.134505	+	0.990913i	$0.542944\pi$
$140$	0	0
$141$	12.0000	1.01058
$142$	0	0
$143$	2.82843	0.236525
$144$	0	0
$145$	0	0
$146$	0	0
$147$	1.41421	0.116642
$148$	0	0
$149$	1.65685	0.135735	0.0678674	−	0.997694i	$-0.478381\pi$
$149$	1.65685	0.135735	0.0678674	+	0.997694i	$0.478381\pi$
$150$	0	0
$151$	6.14214	0.499840	0.249920	−	0.968267i	$-0.419596\pi$
$151$	6.14214	0.499840	0.249920	+	0.968267i	$0.419596\pi$
$152$	0	0
$153$	4.82843	0.390355
$154$	0	0
$155$	0	0
$156$	0	0
$157$	10.4853	0.836817	0.418408	−	0.908259i	$-0.362588\pi$
$157$	10.4853	0.836817	0.418408	+	0.908259i	$0.362588\pi$
$158$	0	0
$159$	−18.4853	−1.46598
$160$	0	0
$161$	−11.3137	−0.891645
$162$	0	0
$163$	−14.8284	−1.16145	−0.580726	−	0.814099i	$-0.697231\pi$
$163$	−14.8284	−1.16145	−0.580726	+	0.814099i	$0.697231\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−2.10051	−0.162542	−0.0812710	−	0.996692i	$-0.525898\pi$
$167$	−2.10051	−0.162542	−0.0812710	+	0.996692i	$0.525898\pi$
$168$	0	0
$169$	−1.34315	−0.103319
$170$	0	0
$171$	1.00000	0.0764719
$172$	0	0
$173$	10.7279	0.815629	0.407814	−	0.913065i	$-0.366291\pi$
$173$	10.7279	0.815629	0.407814	+	0.913065i	$0.366291\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	4.00000	0.300658
$178$	0	0
$179$	8.48528	0.634220	0.317110	−	0.948389i	$-0.397288\pi$
$179$	8.48528	0.634220	0.317110	+	0.948389i	$0.397288\pi$
$180$	0	0
$181$	−5.31371	−0.394965	−0.197482	−	0.980306i	$-0.563277\pi$
$181$	−5.31371	−0.394965	−0.197482	+	0.980306i	$0.563277\pi$
$182$	0	0
$183$	−2.34315	−0.173210
$184$	0	0
$185$	0	0
$186$	0	0
$187$	4.00000	0.292509
$188$	0	0
$189$	−16.0000	−1.16383
$190$	0	0
$191$	3.31371	0.239772	0.119886	−	0.992788i	$-0.461747\pi$
$191$	3.31371	0.239772	0.119886	+	0.992788i	$0.461747\pi$
$192$	0	0
$193$	6.92893	0.498755	0.249378	−	0.968406i	$-0.419774\pi$
$193$	6.92893	0.498755	0.249378	+	0.968406i	$0.419774\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−25.3137	−1.80353	−0.901764	−	0.432230i	$-0.857727\pi$
$197$	−25.3137	−1.80353	−0.901764	+	0.432230i	$0.857727\pi$
$198$	0	0
$199$	−5.65685	−0.401004	−0.200502	−	0.979693i	$-0.564257\pi$
$199$	−5.65685	−0.401004	−0.200502	+	0.979693i	$0.564257\pi$
$200$	0	0
$201$	13.3137	0.939077
$202$	0	0
$203$	−2.34315	−0.164457
$204$	0	0
$205$	0	0
$206$	0	0
$207$	4.00000	0.278019
$208$	0	0
$209$	0.828427	0.0573035
$210$	0	0
$211$	−16.4853	−1.13489	−0.567447	−	0.823410i	$-0.692069\pi$
$211$	−16.4853	−1.13489	−0.567447	+	0.823410i	$0.692069\pi$
$212$	0	0
$213$	21.6569	1.48390
$214$	0	0
$215$	0	0
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−17.1716	−1.16035
$220$	0	0
$221$	16.4853	1.10892
$222$	0	0
$223$	−13.4142	−0.898282	−0.449141	−	0.893461i	$-0.648270\pi$
$223$	−13.4142	−0.898282	−0.449141	+	0.893461i	$0.648270\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−24.0416	−1.59570	−0.797850	−	0.602857i	$-0.794029\pi$
$227$	−24.0416	−1.59570	−0.797850	+	0.602857i	$0.794029\pi$
$228$	0	0
$229$	11.3137	0.747631	0.373815	−	0.927503i	$-0.378049\pi$
$229$	11.3137	0.747631	0.373815	+	0.927503i	$0.378049\pi$
$230$	0	0
$231$	−3.31371	−0.218026
$232$	0	0
$233$	10.0000	0.655122	0.327561	−	0.944830i	$-0.393773\pi$
$233$	10.0000	0.655122	0.327561	+	0.944830i	$0.393773\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−12.9706	−0.842529
$238$	0	0
$239$	8.00000	0.517477	0.258738	−	0.965947i	$-0.416693\pi$
$239$	8.00000	0.517477	0.258738	+	0.965947i	$0.416693\pi$
$240$	0	0
$241$	8.14214	0.524481	0.262241	−	0.965003i	$-0.415539\pi$
$241$	8.14214	0.524481	0.262241	+	0.965003i	$0.415539\pi$
$242$	0	0
$243$	9.89949	0.635053
$244$	0	0
$245$	0	0
$246$	0	0
$247$	3.41421	0.217241
$248$	0	0
$249$	−11.3137	−0.716977
$250$	0	0
$251$	28.9706	1.82861	0.914303	−	0.405031i	$-0.132739\pi$
$251$	28.9706	1.82861	0.914303	+	0.405031i	$0.132739\pi$
$252$	0	0
$253$	3.31371	0.208331
$254$	0	0
$255$	0	0
$256$	0	0
$257$	4.10051	0.255782	0.127891	−	0.991788i	$-0.459179\pi$
$257$	4.10051	0.255782	0.127891	+	0.991788i	$0.459179\pi$
$258$	0	0
$259$	−28.9706	−1.80014
$260$	0	0
$261$	0.828427	0.0512784
$262$	0	0
$263$	−27.3137	−1.68424	−0.842118	−	0.539294i	$-0.818691\pi$
$263$	−27.3137	−1.68424	−0.842118	+	0.539294i	$0.818691\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−4.48528	−0.274495
$268$	0	0
$269$	−29.3137	−1.78729	−0.893644	−	0.448776i	$-0.851860\pi$
$269$	−29.3137	−1.78729	−0.893644	+	0.448776i	$0.851860\pi$
$270$	0	0
$271$	−9.51472	−0.577978	−0.288989	−	0.957332i	$-0.593319\pi$
$271$	−9.51472	−0.577978	−0.288989	+	0.957332i	$0.593319\pi$
$272$	0	0
$273$	−13.6569	−0.826550
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−19.1716	−1.15191	−0.575954	−	0.817482i	$-0.695369\pi$
$277$	−19.1716	−1.15191	−0.575954	+	0.817482i	$0.695369\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	11.1716	0.666440	0.333220	−	0.942849i	$-0.391865\pi$
$281$	11.1716	0.666440	0.333220	+	0.942849i	$0.391865\pi$
$282$	0	0
$283$	−6.34315	−0.377061	−0.188530	−	0.982067i	$-0.560372\pi$
$283$	−6.34315	−0.377061	−0.188530	+	0.982067i	$0.560372\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−2.34315	−0.138312
$288$	0	0
$289$	6.31371	0.371395
$290$	0	0
$291$	3.17157	0.185921
$292$	0	0
$293$	−32.3848	−1.89194	−0.945969	−	0.324256i	$-0.894886\pi$
$293$	−32.3848	−1.89194	−0.945969	+	0.324256i	$0.894886\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	4.68629	0.271926
$298$	0	0
$299$	13.6569	0.789796
$300$	0	0
$301$	−8.00000	−0.461112
$302$	0	0
$303$	24.9706	1.43452
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−1.41421	−0.0807134	−0.0403567	−	0.999185i	$-0.512849\pi$
$307$	−1.41421	−0.0807134	−0.0403567	+	0.999185i	$0.512849\pi$
$308$	0	0
$309$	−11.6569	−0.663135
$310$	0	0
$311$	19.1716	1.08712	0.543560	−	0.839370i	$-0.317076\pi$
$311$	19.1716	1.08712	0.543560	+	0.839370i	$0.317076\pi$
$312$	0	0
$313$	5.31371	0.300349	0.150174	−	0.988660i	$-0.452017\pi$
$313$	5.31371	0.300349	0.150174	+	0.988660i	$0.452017\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−0.100505	−0.00564493	−0.00282246	−	0.999996i	$-0.500898\pi$
$317$	−0.100505	−0.00564493	−0.00282246	+	0.999996i	$0.500898\pi$
$318$	0	0
$319$	0.686292	0.0384249
$320$	0	0
$321$	−2.00000	−0.111629
$322$	0	0
$323$	4.82843	0.268661
$324$	0	0
$325$	0	0
$326$	0	0
$327$	5.17157	0.285989
$328$	0	0
$329$	24.0000	1.32316
$330$	0	0
$331$	−17.4558	−0.959460	−0.479730	−	0.877416i	$-0.659265\pi$
$331$	−17.4558	−0.959460	−0.479730	+	0.877416i	$0.659265\pi$
$332$	0	0
$333$	10.2426	0.561293
$334$	0	0
$335$	0	0
$336$	0	0
$337$	20.3848	1.11043	0.555215	−	0.831707i	$-0.312636\pi$
$337$	20.3848	1.11043	0.555215	+	0.831707i	$0.312636\pi$
$338$	0	0
$339$	20.8284	1.13124
$340$	0	0
$341$	0	0
$342$	0	0
$343$	−16.9706	−0.916324
$344$	0	0
$345$	0	0
$346$	0	0
$347$	20.2843	1.08892	0.544458	−	0.838788i	$-0.316735\pi$
$347$	20.2843	1.08892	0.544458	+	0.838788i	$0.316735\pi$
$348$	0	0
$349$	12.3431	0.660713	0.330357	−	0.943856i	$-0.392831\pi$
$349$	12.3431	0.660713	0.330357	+	0.943856i	$0.392831\pi$
$350$	0	0
$351$	19.3137	1.03089
$352$	0	0
$353$	12.1421	0.646261	0.323130	−	0.946354i	$-0.395265\pi$
$353$	12.1421	0.646261	0.323130	+	0.946354i	$0.395265\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−19.3137	−1.02219
$358$	0	0
$359$	−19.4558	−1.02684	−0.513420	−	0.858137i	$-0.671622\pi$
$359$	−19.4558	−1.02684	−0.513420	+	0.858137i	$0.671622\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	−14.5858	−0.765555
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−31.1127	−1.62407	−0.812035	−	0.583609i	$-0.801640\pi$
$367$	−31.1127	−1.62407	−0.812035	+	0.583609i	$0.801640\pi$
$368$	0	0
$369$	0.828427	0.0431262
$370$	0	0
$371$	−36.9706	−1.91942
$372$	0	0
$373$	33.5563	1.73748	0.868741	−	0.495267i	$-0.164930\pi$
$373$	33.5563	1.73748	0.868741	+	0.495267i	$0.164930\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	2.82843	0.145671
$378$	0	0
$379$	9.65685	0.496039	0.248020	−	0.968755i	$-0.420220\pi$
$379$	9.65685	0.496039	0.248020	+	0.968755i	$0.420220\pi$
$380$	0	0
$381$	18.9706	0.971891
$382$	0	0
$383$	−32.7279	−1.67232	−0.836159	−	0.548487i	$-0.815204\pi$
$383$	−32.7279	−1.67232	−0.836159	+	0.548487i	$0.815204\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	2.82843	0.143777
$388$	0	0
$389$	9.31371	0.472224	0.236112	−	0.971726i	$-0.424127\pi$
$389$	9.31371	0.472224	0.236112	+	0.971726i	$0.424127\pi$
$390$	0	0
$391$	19.3137	0.976736
$392$	0	0
$393$	10.3431	0.521743
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−36.8284	−1.84837	−0.924183	−	0.381950i	$-0.875253\pi$
$397$	−36.8284	−1.84837	−0.924183	+	0.381950i	$0.875253\pi$
$398$	0	0
$399$	−4.00000	−0.200250
$400$	0	0
$401$	14.9706	0.747594	0.373797	−	0.927510i	$-0.378056\pi$
$401$	14.9706	0.747594	0.373797	+	0.927510i	$0.378056\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	0	0
$406$	0	0
$407$	8.48528	0.420600
$408$	0	0
$409$	−30.2843	−1.49746	−0.748730	−	0.662875i	$-0.769336\pi$
$409$	−30.2843	−1.49746	−0.748730	+	0.662875i	$0.769336\pi$
$410$	0	0
$411$	10.1421	0.500275
$412$	0	0
$413$	8.00000	0.393654
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−4.48528	−0.219645
$418$	0	0
$419$	1.65685	0.0809426	0.0404713	−	0.999181i	$-0.487114\pi$
$419$	1.65685	0.0809426	0.0404713	+	0.999181i	$0.487114\pi$
$420$	0	0
$421$	−20.8284	−1.01512	−0.507558	−	0.861618i	$-0.669452\pi$
$421$	−20.8284	−1.01512	−0.507558	+	0.861618i	$0.669452\pi$
$422$	0	0
$423$	−8.48528	−0.412568
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−4.68629	−0.226786
$428$	0	0
$429$	4.00000	0.193122
$430$	0	0
$431$	−14.8284	−0.714260	−0.357130	−	0.934055i	$-0.616245\pi$
$431$	−14.8284	−0.714260	−0.357130	+	0.934055i	$0.616245\pi$
$432$	0	0
$433$	18.0416	0.867025	0.433513	−	0.901147i	$-0.357274\pi$
$433$	18.0416	0.867025	0.433513	+	0.901147i	$0.357274\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	4.00000	0.191346
$438$	0	0
$439$	14.1421	0.674967	0.337484	−	0.941331i	$-0.390424\pi$
$439$	14.1421	0.674967	0.337484	+	0.941331i	$0.390424\pi$
$440$	0	0
$441$	−1.00000	−0.0476190
$442$	0	0
$443$	−0.686292	−0.0326067	−0.0163033	−	0.999867i	$-0.505190\pi$
$443$	−0.686292	−0.0326067	−0.0163033	+	0.999867i	$0.505190\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	2.34315	0.110827
$448$	0	0
$449$	26.2843	1.24043	0.620216	−	0.784431i	$-0.287045\pi$
$449$	26.2843	1.24043	0.620216	+	0.784431i	$0.287045\pi$
$450$	0	0
$451$	0.686292	0.0323162
$452$	0	0
$453$	8.68629	0.408118
$454$	0	0
$455$	0	0
$456$	0	0
$457$	34.2843	1.60375	0.801875	−	0.597491i	$-0.203836\pi$
$457$	34.2843	1.60375	0.801875	+	0.597491i	$0.203836\pi$
$458$	0	0
$459$	27.3137	1.27489
$460$	0	0
$461$	−8.62742	−0.401819	−0.200909	−	0.979610i	$-0.564390\pi$
$461$	−8.62742	−0.401819	−0.200909	+	0.979610i	$0.564390\pi$
$462$	0	0
$463$	−26.6274	−1.23748	−0.618741	−	0.785595i	$-0.712357\pi$
$463$	−26.6274	−1.23748	−0.618741	+	0.785595i	$0.712357\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−24.0000	−1.11059	−0.555294	−	0.831654i	$-0.687394\pi$
$467$	−24.0000	−1.11059	−0.555294	+	0.831654i	$0.687394\pi$
$468$	0	0
$469$	26.6274	1.22954
$470$	0	0
$471$	14.8284	0.683258
$472$	0	0
$473$	2.34315	0.107738
$474$	0	0
$475$	0	0
$476$	0	0
$477$	13.0711	0.598483
$478$	0	0
$479$	23.4558	1.07172	0.535862	−	0.844305i	$-0.319987\pi$
$479$	23.4558	1.07172	0.535862	+	0.844305i	$0.319987\pi$
$480$	0	0
$481$	34.9706	1.59452
$482$	0	0
$483$	−16.0000	−0.728025
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−27.5563	−1.24870	−0.624349	−	0.781146i	$-0.714636\pi$
$487$	−27.5563	−1.24870	−0.624349	+	0.781146i	$0.714636\pi$
$488$	0	0
$489$	−20.9706	−0.948322
$490$	0	0
$491$	−12.9706	−0.585353	−0.292677	−	0.956211i	$-0.594546\pi$
$491$	−12.9706	−0.585353	−0.292677	+	0.956211i	$0.594546\pi$
$492$	0	0
$493$	4.00000	0.180151
$494$	0	0
$495$	0	0
$496$	0	0
$497$	43.3137	1.94289
$498$	0	0
$499$	4.14214	0.185427	0.0927137	−	0.995693i	$-0.470446\pi$
$499$	4.14214	0.185427	0.0927137	+	0.995693i	$0.470446\pi$
$500$	0	0
$501$	−2.97056	−0.132715
$502$	0	0
$503$	1.65685	0.0738755	0.0369377	−	0.999318i	$-0.488240\pi$
$503$	1.65685	0.0738755	0.0369377	+	0.999318i	$0.488240\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−1.89949	−0.0843595
$508$	0	0
$509$	−15.4558	−0.685068	−0.342534	−	0.939505i	$-0.611285\pi$
$509$	−15.4558	−0.685068	−0.342534	+	0.939505i	$0.611285\pi$
$510$	0	0
$511$	−34.3431	−1.51925
$512$	0	0
$513$	5.65685	0.249756
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−7.02944	−0.309154
$518$	0	0
$519$	15.1716	0.665958
$520$	0	0
$521$	14.6863	0.643418	0.321709	−	0.946839i	$-0.395743\pi$
$521$	14.6863	0.643418	0.321709	+	0.946839i	$0.395743\pi$
$522$	0	0
$523$	36.2426	1.58478	0.792390	−	0.610015i	$-0.208837\pi$
$523$	36.2426	1.58478	0.792390	+	0.610015i	$0.208837\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	0	0
$528$	0	0
$529$	−7.00000	−0.304348
$530$	0	0
$531$	−2.82843	−0.122743
$532$	0	0
$533$	2.82843	0.122513
$534$	0	0
$535$	0	0
$536$	0	0
$537$	12.0000	0.517838
$538$	0	0
$539$	−0.828427	−0.0356829
$540$	0	0
$541$	36.2843	1.55998	0.779991	−	0.625790i	$-0.215223\pi$
$541$	36.2843	1.55998	0.779991	+	0.625790i	$0.215223\pi$
$542$	0	0
$543$	−7.51472	−0.322487
$544$	0	0
$545$	0	0
$546$	0	0
$547$	39.0711	1.67056	0.835279	−	0.549826i	$-0.185306\pi$
$547$	39.0711	1.67056	0.835279	+	0.549826i	$0.185306\pi$
$548$	0	0
$549$	1.65685	0.0707128
$550$	0	0
$551$	0.828427	0.0352922
$552$	0	0
$553$	−25.9411	−1.10313
$554$	0	0
$555$	0	0
$556$	0	0
$557$	5.79899	0.245711	0.122856	−	0.992425i	$-0.460795\pi$
$557$	5.79899	0.245711	0.122856	+	0.992425i	$0.460795\pi$
$558$	0	0
$559$	9.65685	0.408441
$560$	0	0
$561$	5.65685	0.238833
$562$	0	0
$563$	4.24264	0.178806	0.0894030	−	0.995996i	$-0.471504\pi$
$563$	4.24264	0.178806	0.0894030	+	0.995996i	$0.471504\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−14.1421	−0.593914
$568$	0	0
$569$	−31.9411	−1.33904	−0.669521	−	0.742793i	$-0.733501\pi$
$569$	−31.9411	−1.33904	−0.669521	+	0.742793i	$0.733501\pi$
$570$	0	0
$571$	−30.4853	−1.27577	−0.637885	−	0.770132i	$-0.720190\pi$
$571$	−30.4853	−1.27577	−0.637885	+	0.770132i	$0.720190\pi$
$572$	0	0
$573$	4.68629	0.195773
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−44.6274	−1.85786	−0.928932	−	0.370251i	$-0.879272\pi$
$577$	−44.6274	−1.85786	−0.928932	+	0.370251i	$0.879272\pi$
$578$	0	0
$579$	9.79899	0.407232
$580$	0	0
$581$	−22.6274	−0.938743
$582$	0	0
$583$	10.8284	0.448468
$584$	0	0
$585$	0	0
$586$	0	0
$587$	5.65685	0.233483	0.116742	−	0.993162i	$-0.462755\pi$
$587$	5.65685	0.233483	0.116742	+	0.993162i	$0.462755\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	−35.7990	−1.47257
$592$	0	0
$593$	−16.6274	−0.682806	−0.341403	−	0.939917i	$-0.610902\pi$
$593$	−16.6274	−0.682806	−0.341403	+	0.939917i	$0.610902\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−8.00000	−0.327418
$598$	0	0
$599$	6.14214	0.250961	0.125480	−	0.992096i	$-0.459953\pi$
$599$	6.14214	0.250961	0.125480	+	0.992096i	$0.459953\pi$
$600$	0	0
$601$	−14.4853	−0.590867	−0.295433	−	0.955363i	$-0.595464\pi$
$601$	−14.4853	−0.590867	−0.295433	+	0.955363i	$0.595464\pi$
$602$	0	0
$603$	−9.41421	−0.383376
$604$	0	0
$605$	0	0
$606$	0	0
$607$	7.75736	0.314862	0.157431	−	0.987530i	$-0.449679\pi$
$607$	7.75736	0.314862	0.157431	+	0.987530i	$0.449679\pi$
$608$	0	0
$609$	−3.31371	−0.134278
$610$	0	0
$611$	−28.9706	−1.17202
$612$	0	0
$613$	−17.7990	−0.718894	−0.359447	−	0.933165i	$-0.617035\pi$
$613$	−17.7990	−0.718894	−0.359447	+	0.933165i	$0.617035\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	37.7990	1.52173	0.760865	−	0.648910i	$-0.224775\pi$
$617$	37.7990	1.52173	0.760865	+	0.648910i	$0.224775\pi$
$618$	0	0
$619$	40.1421	1.61345	0.806724	−	0.590928i	$-0.201238\pi$
$619$	40.1421	1.61345	0.806724	+	0.590928i	$0.201238\pi$
$620$	0	0
$621$	22.6274	0.908007
$622$	0	0
$623$	−8.97056	−0.359398
$624$	0	0
$625$	0	0
$626$	0	0
$627$	1.17157	0.0467881
$628$	0	0
$629$	49.4558	1.97193
$630$	0	0
$631$	−34.4853	−1.37284	−0.686419	−	0.727207i	$-0.740818\pi$
$631$	−34.4853	−1.37284	−0.686419	+	0.727207i	$0.740818\pi$
$632$	0	0
$633$	−23.3137	−0.926637
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−3.41421	−0.135276
$638$	0	0
$639$	−15.3137	−0.605801
$640$	0	0
$641$	27.4558	1.08444	0.542220	−	0.840236i	$-0.317584\pi$
$641$	27.4558	1.08444	0.542220	+	0.840236i	$0.317584\pi$
$642$	0	0
$643$	35.3137	1.39264	0.696318	−	0.717733i	$-0.254820\pi$
$643$	35.3137	1.39264	0.696318	+	0.717733i	$0.254820\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	16.6863	0.656006	0.328003	−	0.944677i	$-0.393624\pi$
$647$	16.6863	0.656006	0.328003	+	0.944677i	$0.393624\pi$
$648$	0	0
$649$	−2.34315	−0.0919765
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−12.1421	−0.475158	−0.237579	−	0.971368i	$-0.576354\pi$
$653$	−12.1421	−0.475158	−0.237579	+	0.971368i	$0.576354\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	12.1421	0.473710
$658$	0	0
$659$	−28.4853	−1.10963	−0.554815	−	0.831974i	$-0.687211\pi$
$659$	−28.4853	−1.10963	−0.554815	+	0.831974i	$0.687211\pi$
$660$	0	0
$661$	8.82843	0.343386	0.171693	−	0.985151i	$-0.445076\pi$
$661$	8.82843	0.343386	0.171693	+	0.985151i	$0.445076\pi$
$662$	0	0
$663$	23.3137	0.905429
$664$	0	0
$665$	0	0
$666$	0	0
$667$	3.31371	0.128307
$668$	0	0
$669$	−18.9706	−0.733444
$670$	0	0
$671$	1.37258	0.0529880
$672$	0	0
$673$	−8.38478	−0.323209	−0.161605	−	0.986856i	$-0.551667\pi$
$673$	−8.38478	−0.323209	−0.161605	+	0.986856i	$0.551667\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	7.61522	0.292677	0.146338	−	0.989235i	$-0.453251\pi$
$677$	7.61522	0.292677	0.146338	+	0.989235i	$0.453251\pi$
$678$	0	0
$679$	6.34315	0.243428
$680$	0	0
$681$	−34.0000	−1.30288
$682$	0	0
$683$	9.89949	0.378794	0.189397	−	0.981901i	$-0.439347\pi$
$683$	9.89949	0.378794	0.189397	+	0.981901i	$0.439347\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	16.0000	0.610438
$688$	0	0
$689$	44.6274	1.70017
$690$	0	0
$691$	−35.1716	−1.33799	−0.668995	−	0.743267i	$-0.733275\pi$
$691$	−35.1716	−1.33799	−0.668995	+	0.743267i	$0.733275\pi$
$692$	0	0
$693$	2.34315	0.0890087
$694$	0	0
$695$	0	0
$696$	0	0
$697$	4.00000	0.151511
$698$	0	0
$699$	14.1421	0.534905
$700$	0	0
$701$	−24.0000	−0.906467	−0.453234	−	0.891392i	$-0.649730\pi$
$701$	−24.0000	−0.906467	−0.453234	+	0.891392i	$0.649730\pi$
$702$	0	0
$703$	10.2426	0.386309
$704$	0	0
$705$	0	0
$706$	0	0
$707$	49.9411	1.87823
$708$	0	0
$709$	−18.2843	−0.686680	−0.343340	−	0.939211i	$-0.611558\pi$
$709$	−18.2843	−0.686680	−0.343340	+	0.939211i	$0.611558\pi$
$710$	0	0
$711$	9.17157	0.343961
$712$	0	0
$713$	0	0
$714$	0	0
$715$	0	0
$716$	0	0
$717$	11.3137	0.422518
$718$	0	0
$719$	−31.4558	−1.17311	−0.586553	−	0.809911i	$-0.699515\pi$
$719$	−31.4558	−1.17311	−0.586553	+	0.809911i	$0.699515\pi$
$720$	0	0
$721$	−23.3137	−0.868248
$722$	0	0
$723$	11.5147	0.428237
$724$	0	0
$725$	0	0
$726$	0	0
$727$	41.4558	1.53751	0.768756	−	0.639542i	$-0.220876\pi$
$727$	41.4558	1.53751	0.768756	+	0.639542i	$0.220876\pi$
$728$	0	0
$729$	29.0000	1.07407
$730$	0	0
$731$	13.6569	0.505117
$732$	0	0
$733$	−4.34315	−0.160418	−0.0802089	−	0.996778i	$-0.525559\pi$
$733$	−4.34315	−0.160418	−0.0802089	+	0.996778i	$0.525559\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−7.79899	−0.287279
$738$	0	0
$739$	0.686292	0.0252456	0.0126228	−	0.999920i	$-0.495982\pi$
$739$	0.686292	0.0252456	0.0126228	+	0.999920i	$0.495982\pi$
$740$	0	0
$741$	4.82843	0.177377
$742$	0	0
$743$	15.7574	0.578081	0.289041	−	0.957317i	$-0.406664\pi$
$743$	15.7574	0.578081	0.289041	+	0.957317i	$0.406664\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	8.00000	0.292705
$748$	0	0
$749$	−4.00000	−0.146157
$750$	0	0
$751$	−44.4853	−1.62329	−0.811645	−	0.584150i	$-0.801428\pi$
$751$	−44.4853	−1.62329	−0.811645	+	0.584150i	$0.801428\pi$
$752$	0	0
$753$	40.9706	1.49305
$754$	0	0
$755$	0	0
$756$	0	0
$757$	22.2843	0.809936	0.404968	−	0.914331i	$-0.367283\pi$
$757$	22.2843	0.809936	0.404968	+	0.914331i	$0.367283\pi$
$758$	0	0
$759$	4.68629	0.170102
$760$	0	0
$761$	1.37258	0.0497561	0.0248780	−	0.999690i	$-0.492080\pi$
$761$	1.37258	0.0497561	0.0248780	+	0.999690i	$0.492080\pi$
$762$	0	0
$763$	10.3431	0.374447
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−9.65685	−0.348689
$768$	0	0
$769$	40.0000	1.44244	0.721218	−	0.692708i	$-0.243582\pi$
$769$	40.0000	1.44244	0.721218	+	0.692708i	$0.243582\pi$
$770$	0	0
$771$	5.79899	0.208846
$772$	0	0
$773$	48.8701	1.75773	0.878867	−	0.477067i	$-0.158300\pi$
$773$	48.8701	1.75773	0.878867	+	0.477067i	$0.158300\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−40.9706	−1.46981
$778$	0	0
$779$	0.828427	0.0296815
$780$	0	0
$781$	−12.6863	−0.453951
$782$	0	0
$783$	4.68629	0.167474
$784$	0	0
$785$	0	0
$786$	0	0
$787$	1.41421	0.0504113	0.0252056	−	0.999682i	$-0.491976\pi$
$787$	1.41421	0.0504113	0.0252056	+	0.999682i	$0.491976\pi$
$788$	0	0
$789$	−38.6274	−1.37517
$790$	0	0
$791$	41.6569	1.48115
$792$	0	0
$793$	5.65685	0.200881
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−27.6985	−0.981131	−0.490565	−	0.871404i	$-0.663210\pi$
$797$	−27.6985	−0.981131	−0.490565	+	0.871404i	$0.663210\pi$
$798$	0	0
$799$	−40.9706	−1.44943
$800$	0	0
$801$	3.17157	0.112062
$802$	0	0
$803$	10.0589	0.354970
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−41.4558	−1.45931
$808$	0	0
$809$	38.0000	1.33601	0.668004	−	0.744157i	$-0.267149\pi$
$809$	38.0000	1.33601	0.668004	+	0.744157i	$0.267149\pi$
$810$	0	0
$811$	−0.686292	−0.0240990	−0.0120495	−	0.999927i	$-0.503836\pi$
$811$	−0.686292	−0.0240990	−0.0120495	+	0.999927i	$0.503836\pi$
$812$	0	0
$813$	−13.4558	−0.471917
$814$	0	0
$815$	0	0
$816$	0	0
$817$	2.82843	0.0989541
$818$	0	0
$819$	9.65685	0.337438
$820$	0	0
$821$	−49.5980	−1.73098	−0.865491	−	0.500925i	$-0.832993\pi$
$821$	−49.5980	−1.73098	−0.865491	+	0.500925i	$0.832993\pi$
$822$	0	0
$823$	16.4853	0.574641	0.287320	−	0.957835i	$-0.407236\pi$
$823$	16.4853	0.574641	0.287320	+	0.957835i	$0.407236\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−13.6985	−0.476343	−0.238171	−	0.971223i	$-0.576548\pi$
$827$	−13.6985	−0.476343	−0.238171	+	0.971223i	$0.576548\pi$
$828$	0	0
$829$	−49.5980	−1.72261	−0.861305	−	0.508089i	$-0.830352\pi$
$829$	−49.5980	−1.72261	−0.861305	+	0.508089i	$0.830352\pi$
$830$	0	0
$831$	−27.1127	−0.940529
$832$	0	0
$833$	−4.82843	−0.167295
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	0	0
$839$	0.201010	0.00693964	0.00346982	−	0.999994i	$-0.498896\pi$
$839$	0.201010	0.00693964	0.00346982	+	0.999994i	$0.498896\pi$
$840$	0	0
$841$	−28.3137	−0.976335
$842$	0	0
$843$	15.7990	0.544146
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−29.1716	−1.00235
$848$	0	0
$849$	−8.97056	−0.307869
$850$	0	0
$851$	40.9706	1.40445
$852$	0	0
$853$	27.9411	0.956686	0.478343	−	0.878173i	$-0.341238\pi$
$853$	27.9411	0.956686	0.478343	+	0.878173i	$0.341238\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−31.2132	−1.06622	−0.533111	−	0.846045i	$-0.678977\pi$
$857$	−31.2132	−1.06622	−0.533111	+	0.846045i	$0.678977\pi$
$858$	0	0
$859$	6.34315	0.216425	0.108213	−	0.994128i	$-0.465487\pi$
$859$	6.34315	0.216425	0.108213	+	0.994128i	$0.465487\pi$
$860$	0	0
$861$	−3.31371	−0.112931
$862$	0	0
$863$	44.0416	1.49919	0.749597	−	0.661894i	$-0.230247\pi$
$863$	44.0416	1.49919	0.749597	+	0.661894i	$0.230247\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	8.92893	0.303242
$868$	0	0
$869$	7.59798	0.257744
$870$	0	0
$871$	−32.1421	−1.08909
$872$	0	0
$873$	−2.24264	−0.0759019
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−18.7279	−0.632397	−0.316198	−	0.948693i	$-0.602407\pi$
$877$	−18.7279	−0.632397	−0.316198	+	0.948693i	$0.602407\pi$
$878$	0	0
$879$	−45.7990	−1.54476
$880$	0	0
$881$	−1.65685	−0.0558208	−0.0279104	−	0.999610i	$-0.508885\pi$
$881$	−1.65685	−0.0558208	−0.0279104	+	0.999610i	$0.508885\pi$
$882$	0	0
$883$	10.8284	0.364406	0.182203	−	0.983261i	$-0.441677\pi$
$883$	10.8284	0.364406	0.182203	+	0.983261i	$0.441677\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	15.2721	0.512786	0.256393	−	0.966573i	$-0.417466\pi$
$887$	15.2721	0.512786	0.256393	+	0.966573i	$0.417466\pi$
$888$	0	0
$889$	37.9411	1.27250
$890$	0	0
$891$	4.14214	0.138767
$892$	0	0
$893$	−8.48528	−0.283949
$894$	0	0
$895$	0	0
$896$	0	0
$897$	19.3137	0.644866
$898$	0	0
$899$	0	0
$900$	0	0
$901$	63.1127	2.10259
$902$	0	0
$903$	−11.3137	−0.376497
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−48.0416	−1.59520	−0.797598	−	0.603189i	$-0.793896\pi$
$907$	−48.0416	−1.59520	−0.797598	+	0.603189i	$0.793896\pi$
$908$	0	0
$909$	−17.6569	−0.585641
$910$	0	0
$911$	−9.17157	−0.303868	−0.151934	−	0.988391i	$-0.548550\pi$
$911$	−9.17157	−0.303868	−0.151934	+	0.988391i	$0.548550\pi$
$912$	0	0
$913$	6.62742	0.219335
$914$	0	0
$915$	0	0
$916$	0	0
$917$	20.6863	0.683122
$918$	0	0
$919$	−17.9411	−0.591823	−0.295912	−	0.955215i	$-0.595623\pi$
$919$	−17.9411	−0.591823	−0.295912	+	0.955215i	$0.595623\pi$
$920$	0	0
$921$	−2.00000	−0.0659022
$922$	0	0
$923$	−52.2843	−1.72096
$924$	0	0
$925$	0	0
$926$	0	0
$927$	8.24264	0.270724
$928$	0	0
$929$	41.3137	1.35546	0.677729	−	0.735311i	$-0.262964\pi$
$929$	41.3137	1.35546	0.677729	+	0.735311i	$0.262964\pi$
$930$	0	0
$931$	−1.00000	−0.0327737
$932$	0	0
$933$	27.1127	0.887630
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−55.4558	−1.81166	−0.905832	−	0.423638i	$-0.860753\pi$
$937$	−55.4558	−1.81166	−0.905832	+	0.423638i	$0.860753\pi$
$938$	0	0
$939$	7.51472	0.245234
$940$	0	0
$941$	21.5980	0.704074	0.352037	−	0.935986i	$-0.385489\pi$
$941$	21.5980	0.704074	0.352037	+	0.935986i	$0.385489\pi$
$942$	0	0
$943$	3.31371	0.107909
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−47.3137	−1.53749	−0.768744	−	0.639556i	$-0.779118\pi$
$947$	−47.3137	−1.53749	−0.768744	+	0.639556i	$0.779118\pi$
$948$	0	0
$949$	41.4558	1.34571
$950$	0	0
$951$	−0.142136	−0.00460906
$952$	0	0
$953$	34.7279	1.12495	0.562474	−	0.826815i	$-0.309850\pi$
$953$	34.7279	1.12495	0.562474	+	0.826815i	$0.309850\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0.970563	0.0313738
$958$	0	0
$959$	20.2843	0.655013
$960$	0	0
$961$	−31.0000	−1.00000
$962$	0	0
$963$	1.41421	0.0455724
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−37.6569	−1.21096	−0.605481	−	0.795859i	$-0.707019\pi$
$967$	−37.6569	−1.21096	−0.605481	+	0.795859i	$0.707019\pi$
$968$	0	0
$969$	6.82843	0.219361
$970$	0	0
$971$	25.9411	0.832490	0.416245	−	0.909252i	$-0.363346\pi$
$971$	25.9411	0.832490	0.416245	+	0.909252i	$0.363346\pi$
$972$	0	0
$973$	−8.97056	−0.287583
$974$	0	0
$975$	0	0
$976$	0	0
$977$	53.8406	1.72251	0.861257	−	0.508170i	$-0.169678\pi$
$977$	53.8406	1.72251	0.861257	+	0.508170i	$0.169678\pi$
$978$	0	0
$979$	2.62742	0.0839726
$980$	0	0
$981$	−3.65685	−0.116754
$982$	0	0
$983$	−4.04163	−0.128908	−0.0644540	−	0.997921i	$-0.520531\pi$
$983$	−4.04163	−0.128908	−0.0644540	+	0.997921i	$0.520531\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	33.9411	1.08036
$988$	0	0
$989$	11.3137	0.359755
$990$	0	0
$991$	44.7696	1.42215	0.711076	−	0.703115i	$-0.248208\pi$
$991$	44.7696	1.42215	0.711076	+	0.703115i	$0.248208\pi$
$992$	0	0
$993$	−24.6863	−0.783396
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−12.8284	−0.406280	−0.203140	−	0.979150i	$-0.565115\pi$
$997$	−12.8284	−0.406280	−0.203140	+	0.979150i	$0.565115\pi$
$998$	0	0
$999$	57.9411	1.83318

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3800.2.a.n.1.2		2
4.3	odd	2		7600.2.a.ba.1.1		2
5.2	odd	4		3800.2.d.i.3649.1		4
5.3	odd	4		3800.2.d.i.3649.3		4
5.4	even	2		760.2.a.f.1.1	✓	2
15.14	odd	2		6840.2.a.z.1.1		2
20.19	odd	2		1520.2.a.m.1.2		2
40.19	odd	2		6080.2.a.bg.1.1		2
40.29	even	2		6080.2.a.bf.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
760.2.a.f.1.1	✓	2	5.4	even	2
1520.2.a.m.1.2		2	20.19	odd	2
3800.2.a.n.1.2		2	1.1	even	1	trivial
3800.2.d.i.3649.1		4	5.2	odd	4
3800.2.d.i.3649.3		4	5.3	odd	4
6080.2.a.bf.1.2		2	40.29	even	2
6080.2.a.bg.1.1		2	40.19	odd	2
6840.2.a.z.1.1		2	15.14	odd	2
7600.2.a.ba.1.1		2	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)

\(f(q)\)	\(=\)	\(q+1.41421 q^{3} +2.82843 q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q+1.41421 q^{3} +2.82843 q^{7} -1.00000 q^{9} -0.828427 q^{11} -3.41421 q^{13} -4.82843 q^{17} -1.00000 q^{19} +4.00000 q^{21} -4.00000 q^{23} -5.65685 q^{27} -0.828427 q^{29} -1.17157 q^{33} -10.2426 q^{37} -4.82843 q^{39} -0.828427 q^{41} -2.82843 q^{43} +8.48528 q^{47} +1.00000 q^{49} -6.82843 q^{51} -13.0711 q^{53} -1.41421 q^{57} +2.82843 q^{59} -1.65685 q^{61} -2.82843 q^{63} +9.41421 q^{67} -5.65685 q^{69} +15.3137 q^{71} -12.1421 q^{73} -2.34315 q^{77} -9.17157 q^{79} -5.00000 q^{81} -8.00000 q^{83} -1.17157 q^{87} -3.17157 q^{89} -9.65685 q^{91} +2.24264 q^{97} +0.828427 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^9	\( 2 q - 2 q^{9} + 4 q^{11} - 4 q^{13} - 4 q^{17} - 2 q^{19} + 8 q^{21} - 8 q^{23} + 4 q^{29} - 8 q^{33} - 12 q^{37} - 4 q^{39} + 4 q^{41} + 2 q^{49} - 8 q^{51} - 12 q^{53} + 8 q^{61} + 16 q^{67} + 8 q^{71} + 4 q^{73} - 16 q^{77} - 24 q^{79} - 10 q^{81} - 16 q^{83} - 8 q^{87} - 12 q^{89} - 8 q^{91} - 4 q^{97} - 4 q^{99}+O(q^{100}) \) 2 * q - 2 * q^9 + 4 * q^11 - 4 * q^13 - 4 * q^17 - 2 * q^19 + 8 * q^21 - 8 * q^23 + 4 * q^29 - 8 * q^33 - 12 * q^37 - 4 * q^39 + 4 * q^41 + 2 * q^49 - 8 * q^51 - 12 * q^53 + 8 * q^61 + 16 * q^67 + 8 * q^71 + 4 * q^73 - 16 * q^77 - 24 * q^79 - 10 * q^81 - 16 * q^83 - 8 * q^87 - 12 * q^89 - 8 * q^91 - 4 * q^97 - 4 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more