LMFDB - Embedded newform 3808.2.a.j.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3808,2,Mod(1,3808)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3808, 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("3808.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3808 = 2^{5} \cdot 7 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3808.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:	$30.4070330897$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	6.6.4022000.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 7x^{4} + 10x^{3} + 14x^{2} - 8x - 4 $ x^6 - 2x^5 - 7x^4 + 10x^3 + 14x^2 - 8*x - 4
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.32040$ of defining polynomial
Character	$\chi$	$=$	3808.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.49796 q^{3} +3.75449 q^{5} -1.00000 q^{7} +3.23981 q^{9} -2.17220 q^{11} -3.12612 q^{13} -9.37857 q^{15} +1.00000 q^{17} +5.42606 q^{19} +2.49796 q^{21} -2.65751 q^{23} +9.09622 q^{25} -0.599024 q^{27} -6.42444 q^{29} -1.93987 q^{31} +5.42606 q^{33} -3.75449 q^{35} +9.89820 q^{37} +7.80893 q^{39} -5.75781 q^{41} -4.87730 q^{43} +12.1638 q^{45} -12.3835 q^{47} +1.00000 q^{49} -2.49796 q^{51} +0.635847 q^{53} -8.15550 q^{55} -13.5541 q^{57} -15.1965 q^{59} -8.50901 q^{61} -3.23981 q^{63} -11.7370 q^{65} +8.06191 q^{67} +6.63835 q^{69} +5.65018 q^{71} +6.15385 q^{73} -22.7220 q^{75} +2.17220 q^{77} +0.358213 q^{79} -8.22308 q^{81} +1.64243 q^{83} +3.75449 q^{85} +16.0480 q^{87} +7.35413 q^{89} +3.12612 q^{91} +4.84571 q^{93} +20.3721 q^{95} -4.12785 q^{97} -7.03749 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 2 q^{3} + 2 q^{5} - 6 q^{7} + 4 q^{9} - 2 q^{11} - 4 q^{13} - 8 q^{15} + 6 q^{17} + 2 q^{19} + 2 q^{21} - 10 q^{23} - 2 q^{27} + 2 q^{29} - 12 q^{31} + 2 q^{33} - 2 q^{35} + 2 q^{37} - 10 q^{39} - 8 q^{43}+ \cdots - 2 q^{99}+O(q^{100}) $ 6 * q - 2 * q^3 + 2 * q^5 - 6 * q^7 + 4 * q^9 - 2 * q^11 - 4 * q^13 - 8 * q^15 + 6 * q^17 + 2 * q^19 + 2 * q^21 - 10 * q^23 - 2 * q^27 + 2 * q^29 - 12 * q^31 + 2 * q^33 - 2 * q^35 + 2 * q^37 - 10 * q^39 - 8 * q^43 + 14 * q^45 - 22 * q^47 + 6 * q^49 - 2 * q^51 - 6 * q^53 - 24 * q^55 - 8 * q^57 - 8 * q^59 - 4 * q^61 - 4 * q^63 - 2 * q^65 + 8 * q^67 + 4 * q^69 - 18 * q^71 + 8 * q^73 - 30 * q^75 + 2 * q^77 - 14 * q^79 - 26 * q^81 + 2 * q^85 - 22 * q^87 + 2 * q^89 + 4 * q^91 - 8 * q^93 - 2 * q^95 + 2 * q^97 - 2 * q^99

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	−2.49796	−1.44220	−0.721099	−	0.692832i	$-0.756363\pi$
$3$	−2.49796	−1.44220	−0.721099	+	0.692832i	$0.756363\pi$
$4$	0	0
$5$	3.75449	1.67906	0.839530	−	0.543313i	$-0.182830\pi$
$5$	3.75449	1.67906	0.839530	+	0.543313i	$0.182830\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	3.23981	1.07994
$10$	0	0
$11$	−2.17220	−0.654942	−0.327471	−	0.944861i	$-0.606196\pi$
$11$	−2.17220	−0.654942	−0.327471	+	0.944861i	$0.606196\pi$
$12$	0	0
$13$	−3.12612	−0.867030	−0.433515	−	0.901146i	$-0.642727\pi$
$13$	−3.12612	−0.867030	−0.433515	+	0.901146i	$0.642727\pi$
$14$	0	0
$15$	−9.37857	−2.42154
$16$	0	0
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	0	0
$19$	5.42606	1.24482	0.622412	−	0.782690i	$-0.286153\pi$
$19$	5.42606	1.24482	0.622412	+	0.782690i	$0.286153\pi$
$20$	0	0
$21$	2.49796	0.545100
$22$	0	0
$23$	−2.65751	−0.554129	−0.277064	−	0.960851i	$-0.589362\pi$
$23$	−2.65751	−0.554129	−0.277064	+	0.960851i	$0.589362\pi$
$24$	0	0
$25$	9.09622	1.81924
$26$	0	0
$27$	−0.599024	−0.115282
$28$	0	0
$29$	−6.42444	−1.19299	−0.596494	−	0.802617i	$-0.703440\pi$
$29$	−6.42444	−1.19299	−0.596494	+	0.802617i	$0.703440\pi$
$30$	0	0
$31$	−1.93987	−0.348410	−0.174205	−	0.984709i	$-0.555736\pi$
$31$	−1.93987	−0.348410	−0.174205	+	0.984709i	$0.555736\pi$
$32$	0	0
$33$	5.42606	0.944556
$34$	0	0
$35$	−3.75449	−0.634625
$36$	0	0
$37$	9.89820	1.62725	0.813627	−	0.581388i	$-0.197490\pi$
$37$	9.89820	1.62725	0.813627	+	0.581388i	$0.197490\pi$
$38$	0	0
$39$	7.80893	1.25043
$40$	0	0
$41$	−5.75781	−0.899218	−0.449609	−	0.893225i	$-0.648437\pi$
$41$	−5.75781	−0.899218	−0.449609	+	0.893225i	$0.648437\pi$
$42$	0	0
$43$	−4.87730	−0.743781	−0.371891	−	0.928277i	$-0.621290\pi$
$43$	−4.87730	−0.743781	−0.371891	+	0.928277i	$0.621290\pi$
$44$	0	0
$45$	12.1638	1.81328
$46$	0	0
$47$	−12.3835	−1.80632	−0.903160	−	0.429304i	$-0.858759\pi$
$47$	−12.3835	−1.80632	−0.903160	+	0.429304i	$0.858759\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−2.49796	−0.349784
$52$	0	0
$53$	0.635847	0.0873403	0.0436701	−	0.999046i	$-0.486095\pi$
$53$	0.635847	0.0873403	0.0436701	+	0.999046i	$0.486095\pi$
$54$	0	0
$55$	−8.15550	−1.09969
$56$	0	0
$57$	−13.5541	−1.79528
$58$	0	0
$59$	−15.1965	−1.97842	−0.989209	−	0.146513i	$-0.953195\pi$
$59$	−15.1965	−1.97842	−0.989209	+	0.146513i	$0.953195\pi$
$60$	0	0
$61$	−8.50901	−1.08947	−0.544734	−	0.838609i	$-0.683369\pi$
$61$	−8.50901	−1.08947	−0.544734	+	0.838609i	$0.683369\pi$
$62$	0	0
$63$	−3.23981	−0.408177
$64$	0	0
$65$	−11.7370	−1.45580
$66$	0	0
$67$	8.06191	0.984919	0.492459	−	0.870335i	$-0.336098\pi$
$67$	8.06191	0.984919	0.492459	+	0.870335i	$0.336098\pi$
$68$	0	0
$69$	6.63835	0.799164
$70$	0	0
$71$	5.65018	0.670554	0.335277	−	0.942120i	$-0.391170\pi$
$71$	5.65018	0.670554	0.335277	+	0.942120i	$0.391170\pi$
$72$	0	0
$73$	6.15385	0.720253	0.360127	−	0.932903i	$-0.382733\pi$
$73$	6.15385	0.720253	0.360127	+	0.932903i	$0.382733\pi$
$74$	0	0
$75$	−22.7220	−2.62371
$76$	0	0
$77$	2.17220	0.247545
$78$	0	0
$79$	0.358213	0.0403021	0.0201510	−	0.999797i	$-0.493585\pi$
$79$	0.358213	0.0403021	0.0201510	+	0.999797i	$0.493585\pi$
$80$	0	0
$81$	−8.22308	−0.913675
$82$	0	0
$83$	1.64243	0.180280	0.0901401	−	0.995929i	$-0.471269\pi$
$83$	1.64243	0.180280	0.0901401	+	0.995929i	$0.471269\pi$
$84$	0	0
$85$	3.75449	0.407232
$86$	0	0
$87$	16.0480	1.72053
$88$	0	0
$89$	7.35413	0.779537	0.389768	−	0.920913i	$-0.372555\pi$
$89$	7.35413	0.779537	0.389768	+	0.920913i	$0.372555\pi$
$90$	0	0
$91$	3.12612	0.327707
$92$	0	0
$93$	4.84571	0.502476
$94$	0	0
$95$	20.3721	2.09013
$96$	0	0
$97$	−4.12785	−0.419120	−0.209560	−	0.977796i	$-0.567203\pi$
$97$	−4.12785	−0.419120	−0.209560	+	0.977796i	$0.567203\pi$
$98$	0	0
$99$	−7.03749	−0.707295
$100$	0	0
$101$	10.5907	1.05381	0.526905	−	0.849924i	$-0.323352\pi$
$101$	10.5907	1.05381	0.526905	+	0.849924i	$0.323352\pi$
$102$	0	0
$103$	−0.360814	−0.0355520	−0.0177760	−	0.999842i	$-0.505659\pi$
$103$	−0.360814	−0.0355520	−0.0177760	+	0.999842i	$0.505659\pi$
$104$	0	0
$105$	9.37857	0.915255
$106$	0	0
$107$	8.00092	0.773478	0.386739	−	0.922189i	$-0.373601\pi$
$107$	8.00092	0.773478	0.386739	+	0.922189i	$0.373601\pi$
$108$	0	0
$109$	16.5377	1.58402	0.792011	−	0.610507i	$-0.209034\pi$
$109$	16.5377	1.58402	0.792011	+	0.610507i	$0.209034\pi$
$110$	0	0
$111$	−24.7253	−2.34682
$112$	0	0
$113$	−19.1675	−1.80313	−0.901563	−	0.432648i	$-0.857579\pi$
$113$	−19.1675	−1.80313	−0.901563	+	0.432648i	$0.857579\pi$
$114$	0	0
$115$	−9.97760	−0.930416
$116$	0	0
$117$	−10.1280	−0.936336
$118$	0	0
$119$	−1.00000	−0.0916698
$120$	0	0
$121$	−6.28156	−0.571051
$122$	0	0
$123$	14.3828	1.29685
$124$	0	0
$125$	15.3792	1.37556
$126$	0	0
$127$	−11.4384	−1.01500	−0.507499	−	0.861652i	$-0.669430\pi$
$127$	−11.4384	−1.01500	−0.507499	+	0.861652i	$0.669430\pi$
$128$	0	0
$129$	12.1833	1.07268
$130$	0	0
$131$	14.1746	1.23844	0.619221	−	0.785217i	$-0.287449\pi$
$131$	14.1746	1.23844	0.619221	+	0.785217i	$0.287449\pi$
$132$	0	0
$133$	−5.42606	−0.470499
$134$	0	0
$135$	−2.24903	−0.193566
$136$	0	0
$137$	−18.3485	−1.56761	−0.783807	−	0.621004i	$-0.786725\pi$
$137$	−18.3485	−1.56761	−0.783807	+	0.621004i	$0.786725\pi$
$138$	0	0
$139$	9.48487	0.804496	0.402248	−	0.915531i	$-0.368229\pi$
$139$	9.48487	0.804496	0.402248	+	0.915531i	$0.368229\pi$
$140$	0	0
$141$	30.9335	2.60507
$142$	0	0
$143$	6.79055	0.567854
$144$	0	0
$145$	−24.1205	−2.00310
$146$	0	0
$147$	−2.49796	−0.206028
$148$	0	0
$149$	−3.74520	−0.306819	−0.153409	−	0.988163i	$-0.549025\pi$
$149$	−3.74520	−0.306819	−0.153409	+	0.988163i	$0.549025\pi$
$150$	0	0
$151$	−13.6859	−1.11375	−0.556873	−	0.830598i	$-0.687999\pi$
$151$	−13.6859	−1.11375	−0.556873	+	0.830598i	$0.687999\pi$
$152$	0	0
$153$	3.23981	0.261923
$154$	0	0
$155$	−7.28321	−0.585002
$156$	0	0
$157$	−15.7238	−1.25490	−0.627449	−	0.778658i	$-0.715901\pi$
$157$	−15.7238	−1.25490	−0.627449	+	0.778658i	$0.715901\pi$
$158$	0	0
$159$	−1.58832	−0.125962
$160$	0	0
$161$	2.65751	0.209441
$162$	0	0
$163$	−11.0584	−0.866158	−0.433079	−	0.901356i	$-0.642573\pi$
$163$	−11.0584	−0.866158	−0.433079	+	0.901356i	$0.642573\pi$
$164$	0	0
$165$	20.3721	1.58597
$166$	0	0
$167$	16.3018	1.26147	0.630737	−	0.775997i	$-0.282753\pi$
$167$	16.3018	1.26147	0.630737	+	0.775997i	$0.282753\pi$
$168$	0	0
$169$	−3.22737	−0.248259
$170$	0	0
$171$	17.5794	1.34433
$172$	0	0
$173$	−19.2226	−1.46147	−0.730734	−	0.682662i	$-0.760822\pi$
$173$	−19.2226	−1.46147	−0.730734	+	0.682662i	$0.760822\pi$
$174$	0	0
$175$	−9.09622	−0.687609
$176$	0	0
$177$	37.9603	2.85327
$178$	0	0
$179$	6.54001	0.488823	0.244412	−	0.969672i	$-0.421405\pi$
$179$	6.54001	0.488823	0.244412	+	0.969672i	$0.421405\pi$
$180$	0	0
$181$	−11.8889	−0.883696	−0.441848	−	0.897090i	$-0.645677\pi$
$181$	−11.8889	−0.883696	−0.441848	+	0.897090i	$0.645677\pi$
$182$	0	0
$183$	21.2552	1.57123
$184$	0	0
$185$	37.1627	2.73226
$186$	0	0
$187$	−2.17220	−0.158847
$188$	0	0
$189$	0.599024	0.0435726
$190$	0	0
$191$	−22.2959	−1.61327	−0.806637	−	0.591047i	$-0.798715\pi$
$191$	−22.2959	−1.61327	−0.806637	+	0.591047i	$0.798715\pi$
$192$	0	0
$193$	−13.9716	−1.00570	−0.502849	−	0.864374i	$-0.667715\pi$
$193$	−13.9716	−1.00570	−0.502849	+	0.864374i	$0.667715\pi$
$194$	0	0
$195$	29.3186	2.09955
$196$	0	0
$197$	−20.1280	−1.43406	−0.717030	−	0.697043i	$-0.754499\pi$
$197$	−20.1280	−1.43406	−0.717030	+	0.697043i	$0.754499\pi$
$198$	0	0
$199$	−2.58221	−0.183048	−0.0915241	−	0.995803i	$-0.529174\pi$
$199$	−2.58221	−0.183048	−0.0915241	+	0.995803i	$0.529174\pi$
$200$	0	0
$201$	−20.1383	−1.42045
$202$	0	0
$203$	6.42444	0.450907
$204$	0	0
$205$	−21.6176	−1.50984
$206$	0	0
$207$	−8.60981	−0.598423
$208$	0	0
$209$	−11.7865	−0.815287
$210$	0	0
$211$	5.99915	0.412999	0.206499	−	0.978447i	$-0.433793\pi$
$211$	5.99915	0.412999	0.206499	+	0.978447i	$0.433793\pi$
$212$	0	0
$213$	−14.1139	−0.967071
$214$	0	0
$215$	−18.3118	−1.24885
$216$	0	0
$217$	1.93987	0.131687
$218$	0	0
$219$	−15.3721	−1.03875
$220$	0	0
$221$	−3.12612	−0.210286
$222$	0	0
$223$	−3.25568	−0.218016	−0.109008	−	0.994041i	$-0.534767\pi$
$223$	−3.25568	−0.218016	−0.109008	+	0.994041i	$0.534767\pi$
$224$	0	0
$225$	29.4700	1.96466
$226$	0	0
$227$	−18.8830	−1.25331	−0.626655	−	0.779297i	$-0.715577\pi$
$227$	−18.8830	−1.25331	−0.626655	+	0.779297i	$0.715577\pi$
$228$	0	0
$229$	18.3534	1.21283	0.606415	−	0.795149i	$-0.292607\pi$
$229$	18.3534	1.21283	0.606415	+	0.795149i	$0.292607\pi$
$230$	0	0
$231$	−5.42606	−0.357009
$232$	0	0
$233$	−23.3324	−1.52855	−0.764277	−	0.644888i	$-0.776904\pi$
$233$	−23.3324	−1.52855	−0.764277	+	0.644888i	$0.776904\pi$
$234$	0	0
$235$	−46.4938	−3.03292
$236$	0	0
$237$	−0.894801	−0.0581236
$238$	0	0
$239$	5.82115	0.376539	0.188269	−	0.982117i	$-0.439712\pi$
$239$	5.82115	0.376539	0.188269	+	0.982117i	$0.439712\pi$
$240$	0	0
$241$	−5.10553	−0.328876	−0.164438	−	0.986387i	$-0.552581\pi$
$241$	−5.10553	−0.328876	−0.164438	+	0.986387i	$0.552581\pi$
$242$	0	0
$243$	22.3380	1.43298
$244$	0	0
$245$	3.75449	0.239866
$246$	0	0
$247$	−16.9625	−1.07930
$248$	0	0
$249$	−4.10273	−0.260000
$250$	0	0
$251$	15.8392	0.999764	0.499882	−	0.866094i	$-0.333377\pi$
$251$	15.8392	0.999764	0.499882	+	0.866094i	$0.333377\pi$
$252$	0	0
$253$	5.77263	0.362922
$254$	0	0
$255$	−9.37857	−0.587309
$256$	0	0
$257$	−12.5576	−0.783323	−0.391661	−	0.920109i	$-0.628100\pi$
$257$	−12.5576	−0.783323	−0.391661	+	0.920109i	$0.628100\pi$
$258$	0	0
$259$	−9.89820	−0.615044
$260$	0	0
$261$	−20.8139	−1.28835
$262$	0	0
$263$	13.4492	0.829312	0.414656	−	0.909978i	$-0.363902\pi$
$263$	13.4492	0.829312	0.414656	+	0.909978i	$0.363902\pi$
$264$	0	0
$265$	2.38728	0.146650
$266$	0	0
$267$	−18.3703	−1.12425
$268$	0	0
$269$	11.5448	0.703901	0.351951	−	0.936019i	$-0.385519\pi$
$269$	11.5448	0.703901	0.351951	+	0.936019i	$0.385519\pi$
$270$	0	0
$271$	−20.9170	−1.27062	−0.635309	−	0.772258i	$-0.719127\pi$
$271$	−20.9170	−1.27062	−0.635309	+	0.772258i	$0.719127\pi$
$272$	0	0
$273$	−7.80893	−0.472618
$274$	0	0
$275$	−19.7588	−1.19150
$276$	0	0
$277$	−13.6005	−0.817177	−0.408589	−	0.912719i	$-0.633979\pi$
$277$	−13.6005	−0.817177	−0.408589	+	0.912719i	$0.633979\pi$
$278$	0	0
$279$	−6.28479	−0.376260
$280$	0	0
$281$	−18.9440	−1.13011	−0.565053	−	0.825055i	$-0.691144\pi$
$281$	−18.9440	−1.13011	−0.565053	+	0.825055i	$0.691144\pi$
$282$	0	0
$283$	−3.60723	−0.214427	−0.107214	−	0.994236i	$-0.534193\pi$
$283$	−3.60723	−0.214427	−0.107214	+	0.994236i	$0.534193\pi$
$284$	0	0
$285$	−50.8887	−3.01439
$286$	0	0
$287$	5.75781	0.339872
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	10.3112	0.604454
$292$	0	0
$293$	−17.8392	−1.04218	−0.521090	−	0.853502i	$-0.674474\pi$
$293$	−17.8392	−1.04218	−0.521090	+	0.853502i	$0.674474\pi$
$294$	0	0
$295$	−57.0552	−3.32188
$296$	0	0
$297$	1.30120	0.0755031
$298$	0	0
$299$	8.30769	0.480446
$300$	0	0
$301$	4.87730	0.281123
$302$	0	0
$303$	−26.4550	−1.51980
$304$	0	0
$305$	−31.9470	−1.82928
$306$	0	0
$307$	23.8837	1.36311	0.681557	−	0.731765i	$-0.261303\pi$
$307$	23.8837	1.36311	0.681557	+	0.731765i	$0.261303\pi$
$308$	0	0
$309$	0.901299	0.0512731
$310$	0	0
$311$	20.9315	1.18692	0.593458	−	0.804865i	$-0.297762\pi$
$311$	20.9315	1.18692	0.593458	+	0.804865i	$0.297762\pi$
$312$	0	0
$313$	−1.23562	−0.0698415	−0.0349208	−	0.999390i	$-0.511118\pi$
$313$	−1.23562	−0.0698415	−0.0349208	+	0.999390i	$0.511118\pi$
$314$	0	0
$315$	−12.1638	−0.685354
$316$	0	0
$317$	10.2042	0.573124	0.286562	−	0.958062i	$-0.407488\pi$
$317$	10.2042	0.573124	0.286562	+	0.958062i	$0.407488\pi$
$318$	0	0
$319$	13.9551	0.781338
$320$	0	0
$321$	−19.9860	−1.11551
$322$	0	0
$323$	5.42606	0.301914
$324$	0	0
$325$	−28.4359	−1.57734
$326$	0	0
$327$	−41.3105	−2.28447
$328$	0	0
$329$	12.3835	0.682725
$330$	0	0
$331$	33.1815	1.82382	0.911911	−	0.410388i	$-0.134607\pi$
$331$	33.1815	1.82382	0.911911	+	0.410388i	$0.134607\pi$
$332$	0	0
$333$	32.0682	1.75733
$334$	0	0
$335$	30.2684	1.65374
$336$	0	0
$337$	−7.82641	−0.426332	−0.213166	−	0.977016i	$-0.568377\pi$
$337$	−7.82641	−0.426332	−0.213166	+	0.977016i	$0.568377\pi$
$338$	0	0
$339$	47.8796	2.60046
$340$	0	0
$341$	4.21377	0.228188
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	24.9236	1.34184
$346$	0	0
$347$	−22.2668	−1.19535	−0.597673	−	0.801740i	$-0.703908\pi$
$347$	−22.2668	−1.19535	−0.597673	+	0.801740i	$0.703908\pi$
$348$	0	0
$349$	22.3528	1.19652	0.598260	−	0.801302i	$-0.295859\pi$
$349$	22.3528	1.19652	0.598260	+	0.801302i	$0.295859\pi$
$350$	0	0
$351$	1.87262	0.0999531
$352$	0	0
$353$	17.6036	0.936945	0.468473	−	0.883478i	$-0.344804\pi$
$353$	17.6036	0.936945	0.468473	+	0.883478i	$0.344804\pi$
$354$	0	0
$355$	21.2136	1.12590
$356$	0	0
$357$	2.49796	0.132206
$358$	0	0
$359$	−25.3868	−1.33987	−0.669933	−	0.742421i	$-0.733677\pi$
$359$	−25.3868	−1.33987	−0.669933	+	0.742421i	$0.733677\pi$
$360$	0	0
$361$	10.4421	0.549586
$362$	0	0
$363$	15.6911	0.823569
$364$	0	0
$365$	23.1046	1.20935
$366$	0	0
$367$	−15.4184	−0.804834	−0.402417	−	0.915457i	$-0.631830\pi$
$367$	−15.4184	−0.804834	−0.402417	+	0.915457i	$0.631830\pi$
$368$	0	0
$369$	−18.6542	−0.971097
$370$	0	0
$371$	−0.635847	−0.0330115
$372$	0	0
$373$	−10.1197	−0.523981	−0.261990	−	0.965070i	$-0.584379\pi$
$373$	−10.1197	−0.523981	−0.261990	+	0.965070i	$0.584379\pi$
$374$	0	0
$375$	−38.4167	−1.98383
$376$	0	0
$377$	20.0836	1.03436
$378$	0	0
$379$	−12.0607	−0.619519	−0.309759	−	0.950815i	$-0.600248\pi$
$379$	−12.0607	−0.619519	−0.309759	+	0.950815i	$0.600248\pi$
$380$	0	0
$381$	28.5728	1.46383
$382$	0	0
$383$	−17.6283	−0.900764	−0.450382	−	0.892836i	$-0.648712\pi$
$383$	−17.6283	−0.900764	−0.450382	+	0.892836i	$0.648712\pi$
$384$	0	0
$385$	8.15550	0.415643
$386$	0	0
$387$	−15.8015	−0.803236
$388$	0	0
$389$	15.2574	0.773579	0.386789	−	0.922168i	$-0.373584\pi$
$389$	15.2574	0.773579	0.386789	+	0.922168i	$0.373584\pi$
$390$	0	0
$391$	−2.65751	−0.134396
$392$	0	0
$393$	−35.4076	−1.78608
$394$	0	0
$395$	1.34491	0.0676696
$396$	0	0
$397$	−10.2174	−0.512796	−0.256398	−	0.966571i	$-0.582536\pi$
$397$	−10.2174	−0.512796	−0.256398	+	0.966571i	$0.582536\pi$
$398$	0	0
$399$	13.5541	0.678553
$400$	0	0
$401$	19.8570	0.991612	0.495806	−	0.868433i	$-0.334873\pi$
$401$	19.8570	0.991612	0.495806	+	0.868433i	$0.334873\pi$
$402$	0	0
$403$	6.06425	0.302082
$404$	0	0
$405$	−30.8735	−1.53412
$406$	0	0
$407$	−21.5008	−1.06576
$408$	0	0
$409$	35.0632	1.73376	0.866881	−	0.498515i	$-0.166121\pi$
$409$	35.0632	1.73376	0.866881	+	0.498515i	$0.166121\pi$
$410$	0	0
$411$	45.8337	2.26081
$412$	0	0
$413$	15.1965	0.747772
$414$	0	0
$415$	6.16649	0.302701
$416$	0	0
$417$	−23.6928	−1.16024
$418$	0	0
$419$	4.35320	0.212668	0.106334	−	0.994330i	$-0.466089\pi$
$419$	4.35320	0.212668	0.106334	+	0.994330i	$0.466089\pi$
$420$	0	0
$421$	−31.8257	−1.55109	−0.775545	−	0.631292i	$-0.782525\pi$
$421$	−31.8257	−1.55109	−0.775545	+	0.631292i	$0.782525\pi$
$422$	0	0
$423$	−40.1202	−1.95071
$424$	0	0
$425$	9.09622	0.441231
$426$	0	0
$427$	8.50901	0.411780
$428$	0	0
$429$	−16.9625	−0.818958
$430$	0	0
$431$	−6.61361	−0.318567	−0.159283	−	0.987233i	$-0.550918\pi$
$431$	−6.61361	−0.318567	−0.159283	+	0.987233i	$0.550918\pi$
$432$	0	0
$433$	−8.04250	−0.386498	−0.193249	−	0.981150i	$-0.561902\pi$
$433$	−8.04250	−0.386498	−0.193249	+	0.981150i	$0.561902\pi$
$434$	0	0
$435$	60.2521	2.88887
$436$	0	0
$437$	−14.4198	−0.689793
$438$	0	0
$439$	8.24142	0.393342	0.196671	−	0.980470i	$-0.436987\pi$
$439$	8.24142	0.393342	0.196671	+	0.980470i	$0.436987\pi$
$440$	0	0
$441$	3.23981	0.154276
$442$	0	0
$443$	−21.8146	−1.03644	−0.518221	−	0.855247i	$-0.673406\pi$
$443$	−21.8146	−1.03644	−0.518221	+	0.855247i	$0.673406\pi$
$444$	0	0
$445$	27.6110	1.30889
$446$	0	0
$447$	9.35536	0.442493
$448$	0	0
$449$	−2.33895	−0.110382	−0.0551910	−	0.998476i	$-0.517577\pi$
$449$	−2.33895	−0.110382	−0.0551910	+	0.998476i	$0.517577\pi$
$450$	0	0
$451$	12.5071	0.588936
$452$	0	0
$453$	34.1869	1.60624
$454$	0	0
$455$	11.7370	0.550239
$456$	0	0
$457$	11.5619	0.540844	0.270422	−	0.962742i	$-0.412837\pi$
$457$	11.5619	0.540844	0.270422	+	0.962742i	$0.412837\pi$
$458$	0	0
$459$	−0.599024	−0.0279600
$460$	0	0
$461$	12.3116	0.573410	0.286705	−	0.958019i	$-0.407440\pi$
$461$	12.3116	0.573410	0.286705	+	0.958019i	$0.407440\pi$
$462$	0	0
$463$	−33.8875	−1.57489	−0.787444	−	0.616387i	$-0.788596\pi$
$463$	−33.8875	−1.57489	−0.787444	+	0.616387i	$0.788596\pi$
$464$	0	0
$465$	18.1932	0.843688
$466$	0	0
$467$	−11.4345	−0.529124	−0.264562	−	0.964369i	$-0.585227\pi$
$467$	−11.4345	−0.529124	−0.264562	+	0.964369i	$0.585227\pi$
$468$	0	0
$469$	−8.06191	−0.372264
$470$	0	0
$471$	39.2775	1.80981
$472$	0	0
$473$	10.5945	0.487134
$474$	0	0
$475$	49.3566	2.26464
$476$	0	0
$477$	2.06002	0.0943218
$478$	0	0
$479$	−14.8696	−0.679410	−0.339705	−	0.940532i	$-0.610327\pi$
$479$	−14.8696	−0.679410	−0.339705	+	0.940532i	$0.610327\pi$
$480$	0	0
$481$	−30.9430	−1.41088
$482$	0	0
$483$	−6.63835	−0.302055
$484$	0	0
$485$	−15.4980	−0.703728
$486$	0	0
$487$	−21.3571	−0.967782	−0.483891	−	0.875128i	$-0.660777\pi$
$487$	−21.3571	−0.967782	−0.483891	+	0.875128i	$0.660777\pi$
$488$	0	0
$489$	27.6233	1.24917
$490$	0	0
$491$	−30.7492	−1.38769	−0.693845	−	0.720124i	$-0.744085\pi$
$491$	−30.7492	−1.38769	−0.693845	+	0.720124i	$0.744085\pi$
$492$	0	0
$493$	−6.42444	−0.289342
$494$	0	0
$495$	−26.4222	−1.18759
$496$	0	0
$497$	−5.65018	−0.253445
$498$	0	0
$499$	23.9607	1.07263	0.536314	−	0.844019i	$-0.319816\pi$
$499$	23.9607	1.07263	0.536314	+	0.844019i	$0.319816\pi$
$500$	0	0
$501$	−40.7213	−1.81929
$502$	0	0
$503$	20.0642	0.894617	0.447308	−	0.894380i	$-0.352383\pi$
$503$	20.0642	0.894617	0.447308	+	0.894380i	$0.352383\pi$
$504$	0	0
$505$	39.7625	1.76941
$506$	0	0
$507$	8.06184	0.358039
$508$	0	0
$509$	−21.8762	−0.969646	−0.484823	−	0.874612i	$-0.661116\pi$
$509$	−21.8762	−0.969646	−0.484823	+	0.874612i	$0.661116\pi$
$510$	0	0
$511$	−6.15385	−0.272230
$512$	0	0
$513$	−3.25034	−0.143506
$514$	0	0
$515$	−1.35467	−0.0596940
$516$	0	0
$517$	26.8994	1.18303
$518$	0	0
$519$	48.0173	2.10773
$520$	0	0
$521$	−16.6864	−0.731047	−0.365523	−	0.930802i	$-0.619110\pi$
$521$	−16.6864	−0.731047	−0.365523	+	0.930802i	$0.619110\pi$
$522$	0	0
$523$	−33.6620	−1.47194	−0.735969	−	0.677015i	$-0.763273\pi$
$523$	−33.6620	−1.47194	−0.735969	+	0.677015i	$0.763273\pi$
$524$	0	0
$525$	22.7220	0.991669
$526$	0	0
$527$	−1.93987	−0.0845019
$528$	0	0
$529$	−15.9376	−0.692941
$530$	0	0
$531$	−49.2337	−2.13656
$532$	0	0
$533$	17.9996	0.779649
$534$	0	0
$535$	30.0394	1.29872
$536$	0	0
$537$	−16.3367	−0.704980
$538$	0	0
$539$	−2.17220	−0.0935631
$540$	0	0
$541$	4.26889	0.183534	0.0917669	−	0.995781i	$-0.470749\pi$
$541$	4.26889	0.183534	0.0917669	+	0.995781i	$0.470749\pi$
$542$	0	0
$543$	29.6980	1.27446
$544$	0	0
$545$	62.0906	2.65967
$546$	0	0
$547$	23.9460	1.02386	0.511929	−	0.859028i	$-0.328931\pi$
$547$	23.9460	1.02386	0.511929	+	0.859028i	$0.328931\pi$
$548$	0	0
$549$	−27.5675	−1.17655
$550$	0	0
$551$	−34.8594	−1.48506
$552$	0	0
$553$	−0.358213	−0.0152328
$554$	0	0
$555$	−92.8310	−3.94046
$556$	0	0
$557$	−20.5598	−0.871147	−0.435573	−	0.900153i	$-0.643454\pi$
$557$	−20.5598	−0.871147	−0.435573	+	0.900153i	$0.643454\pi$
$558$	0	0
$559$	15.2470	0.644881
$560$	0	0
$561$	5.42606	0.229088
$562$	0	0
$563$	−31.7372	−1.33757	−0.668783	−	0.743458i	$-0.733184\pi$
$563$	−31.7372	−1.33757	−0.668783	+	0.743458i	$0.733184\pi$
$564$	0	0
$565$	−71.9642	−3.02756
$566$	0	0
$567$	8.22308	0.345337
$568$	0	0
$569$	9.23097	0.386982	0.193491	−	0.981102i	$-0.438019\pi$
$569$	9.23097	0.386982	0.193491	+	0.981102i	$0.438019\pi$
$570$	0	0
$571$	3.96659	0.165996	0.0829982	−	0.996550i	$-0.473550\pi$
$571$	3.96659	0.165996	0.0829982	+	0.996550i	$0.473550\pi$
$572$	0	0
$573$	55.6943	2.32666
$574$	0	0
$575$	−24.1733	−1.00810
$576$	0	0
$577$	−13.3603	−0.556198	−0.278099	−	0.960552i	$-0.589704\pi$
$577$	−13.3603	−0.556198	−0.278099	+	0.960552i	$0.589704\pi$
$578$	0	0
$579$	34.9005	1.45042
$580$	0	0
$581$	−1.64243	−0.0681395
$582$	0	0
$583$	−1.38118	−0.0572028
$584$	0	0
$585$	−38.0256	−1.57216
$586$	0	0
$587$	21.4953	0.887208	0.443604	−	0.896223i	$-0.353700\pi$
$587$	21.4953	0.887208	0.443604	+	0.896223i	$0.353700\pi$
$588$	0	0
$589$	−10.5258	−0.433709
$590$	0	0
$591$	50.2789	2.06820
$592$	0	0
$593$	11.2025	0.460031	0.230016	−	0.973187i	$-0.426122\pi$
$593$	11.2025	0.460031	0.230016	+	0.973187i	$0.426122\pi$
$594$	0	0
$595$	−3.75449	−0.153919
$596$	0	0
$597$	6.45027	0.263992
$598$	0	0
$599$	1.35862	0.0555118	0.0277559	−	0.999615i	$-0.491164\pi$
$599$	1.35862	0.0555118	0.0277559	+	0.999615i	$0.491164\pi$
$600$	0	0
$601$	19.6294	0.800701	0.400350	−	0.916362i	$-0.368888\pi$
$601$	19.6294	0.800701	0.400350	+	0.916362i	$0.368888\pi$
$602$	0	0
$603$	26.1190	1.06365
$604$	0	0
$605$	−23.5841	−0.958829
$606$	0	0
$607$	−6.71007	−0.272353	−0.136177	−	0.990685i	$-0.543482\pi$
$607$	−6.71007	−0.272353	−0.136177	+	0.990685i	$0.543482\pi$
$608$	0	0
$609$	−16.0480	−0.650297
$610$	0	0
$611$	38.7123	1.56613
$612$	0	0
$613$	−5.64980	−0.228193	−0.114097	−	0.993470i	$-0.536397\pi$
$613$	−5.64980	−0.228193	−0.114097	+	0.993470i	$0.536397\pi$
$614$	0	0
$615$	54.0000	2.17749
$616$	0	0
$617$	−9.36400	−0.376981	−0.188490	−	0.982075i	$-0.560359\pi$
$617$	−9.36400	−0.376981	−0.188490	+	0.982075i	$0.560359\pi$
$618$	0	0
$619$	12.2615	0.492833	0.246416	−	0.969164i	$-0.420747\pi$
$619$	12.2615	0.492833	0.246416	+	0.969164i	$0.420747\pi$
$620$	0	0
$621$	1.59191	0.0638812
$622$	0	0
$623$	−7.35413	−0.294637
$624$	0	0
$625$	12.2601	0.490403
$626$	0	0
$627$	29.4421	1.17581
$628$	0	0
$629$	9.89820	0.394667
$630$	0	0
$631$	−28.8988	−1.15044	−0.575222	−	0.817997i	$-0.695084\pi$
$631$	−28.8988	−1.15044	−0.575222	+	0.817997i	$0.695084\pi$
$632$	0	0
$633$	−14.9856	−0.595626
$634$	0	0
$635$	−42.9456	−1.70424
$636$	0	0
$637$	−3.12612	−0.123861
$638$	0	0
$639$	18.3055	0.724154
$640$	0	0
$641$	31.7913	1.25568	0.627841	−	0.778341i	$-0.283939\pi$
$641$	31.7913	1.25568	0.627841	+	0.778341i	$0.283939\pi$
$642$	0	0
$643$	5.19435	0.204845	0.102423	−	0.994741i	$-0.467341\pi$
$643$	5.19435	0.204845	0.102423	+	0.994741i	$0.467341\pi$
$644$	0	0
$645$	45.7421	1.80109
$646$	0	0
$647$	23.3503	0.917993	0.458997	−	0.888438i	$-0.348209\pi$
$647$	23.3503	0.917993	0.458997	+	0.888438i	$0.348209\pi$
$648$	0	0
$649$	33.0098	1.29575
$650$	0	0
$651$	−4.84571	−0.189918
$652$	0	0
$653$	24.1334	0.944414	0.472207	−	0.881488i	$-0.343458\pi$
$653$	24.1334	0.944414	0.472207	+	0.881488i	$0.343458\pi$
$654$	0	0
$655$	53.2185	2.07942
$656$	0	0
$657$	19.9373	0.777827
$658$	0	0
$659$	−34.6004	−1.34784	−0.673920	−	0.738804i	$-0.735391\pi$
$659$	−34.6004	−1.34784	−0.673920	+	0.738804i	$0.735391\pi$
$660$	0	0
$661$	−19.9502	−0.775973	−0.387986	−	0.921665i	$-0.626829\pi$
$661$	−19.9502	−0.775973	−0.387986	+	0.921665i	$0.626829\pi$
$662$	0	0
$663$	7.80893	0.303274
$664$	0	0
$665$	−20.3721	−0.789996
$666$	0	0
$667$	17.0730	0.661069
$668$	0	0
$669$	8.13255	0.314423
$670$	0	0
$671$	18.4832	0.713538
$672$	0	0
$673$	45.1276	1.73954	0.869771	−	0.493455i	$-0.164266\pi$
$673$	45.1276	1.73954	0.869771	+	0.493455i	$0.164266\pi$
$674$	0	0
$675$	−5.44885	−0.209726
$676$	0	0
$677$	−1.91883	−0.0737465	−0.0368732	−	0.999320i	$-0.511740\pi$
$677$	−1.91883	−0.0737465	−0.0368732	+	0.999320i	$0.511740\pi$
$678$	0	0
$679$	4.12785	0.158412
$680$	0	0
$681$	47.1691	1.80752
$682$	0	0
$683$	−6.73903	−0.257862	−0.128931	−	0.991654i	$-0.541155\pi$
$683$	−6.73903	−0.257862	−0.128931	+	0.991654i	$0.541155\pi$
$684$	0	0
$685$	−68.8892	−2.63212
$686$	0	0
$687$	−45.8461	−1.74914
$688$	0	0
$689$	−1.98773	−0.0757266
$690$	0	0
$691$	7.23784	0.275340	0.137670	−	0.990478i	$-0.456039\pi$
$691$	7.23784	0.275340	0.137670	+	0.990478i	$0.456039\pi$
$692$	0	0
$693$	7.03749	0.267332
$694$	0	0
$695$	35.6109	1.35080
$696$	0	0
$697$	−5.75781	−0.218092
$698$	0	0
$699$	58.2833	2.20448
$700$	0	0
$701$	14.3845	0.543296	0.271648	−	0.962397i	$-0.412431\pi$
$701$	14.3845	0.543296	0.271648	+	0.962397i	$0.412431\pi$
$702$	0	0
$703$	53.7082	2.02564
$704$	0	0
$705$	116.140	4.37407
$706$	0	0
$707$	−10.5907	−0.398303
$708$	0	0
$709$	−0.273628	−0.0102763	−0.00513816	−	0.999987i	$-0.501636\pi$
$709$	−0.273628	−0.0102763	−0.00513816	+	0.999987i	$0.501636\pi$
$710$	0	0
$711$	1.16054	0.0435236
$712$	0	0
$713$	5.15521	0.193064
$714$	0	0
$715$	25.4951	0.953461
$716$	0	0
$717$	−14.5410	−0.543044
$718$	0	0
$719$	47.8532	1.78462	0.892312	−	0.451420i	$-0.149082\pi$
$719$	47.8532	1.78462	0.892312	+	0.451420i	$0.149082\pi$
$720$	0	0
$721$	0.360814	0.0134374
$722$	0	0
$723$	12.7534	0.474304
$724$	0	0
$725$	−58.4381	−2.17034
$726$	0	0
$727$	−11.9347	−0.442633	−0.221317	−	0.975202i	$-0.571035\pi$
$727$	−11.9347	−0.442633	−0.221317	+	0.975202i	$0.571035\pi$
$728$	0	0
$729$	−31.1302	−1.15297
$730$	0	0
$731$	−4.87730	−0.180394
$732$	0	0
$733$	29.8523	1.10262	0.551309	−	0.834301i	$-0.314128\pi$
$733$	29.8523	1.10262	0.551309	+	0.834301i	$0.314128\pi$
$734$	0	0
$735$	−9.37857	−0.345934
$736$	0	0
$737$	−17.5120	−0.645064
$738$	0	0
$739$	37.7013	1.38686	0.693432	−	0.720522i	$-0.256098\pi$
$739$	37.7013	1.38686	0.693432	+	0.720522i	$0.256098\pi$
$740$	0	0
$741$	42.3717	1.55656
$742$	0	0
$743$	−51.3102	−1.88239	−0.941195	−	0.337865i	$-0.890295\pi$
$743$	−51.3102	−1.88239	−0.941195	+	0.337865i	$0.890295\pi$
$744$	0	0
$745$	−14.0613	−0.515167
$746$	0	0
$747$	5.32116	0.194691
$748$	0	0
$749$	−8.00092	−0.292347
$750$	0	0
$751$	21.1470	0.771665	0.385832	−	0.922569i	$-0.373914\pi$
$751$	21.1470	0.771665	0.385832	+	0.922569i	$0.373914\pi$
$752$	0	0
$753$	−39.5658	−1.44186
$754$	0	0
$755$	−51.3837	−1.87005
$756$	0	0
$757$	31.9921	1.16277	0.581386	−	0.813628i	$-0.302511\pi$
$757$	31.9921	1.16277	0.581386	+	0.813628i	$0.302511\pi$
$758$	0	0
$759$	−14.4198	−0.523406
$760$	0	0
$761$	43.8285	1.58878	0.794391	−	0.607407i	$-0.207790\pi$
$761$	43.8285	1.58878	0.794391	+	0.607407i	$0.207790\pi$
$762$	0	0
$763$	−16.5377	−0.598704
$764$	0	0
$765$	12.1638	0.439784
$766$	0	0
$767$	47.5061	1.71535
$768$	0	0
$769$	43.6455	1.57390	0.786949	−	0.617019i	$-0.211660\pi$
$769$	43.6455	1.57390	0.786949	+	0.617019i	$0.211660\pi$
$770$	0	0
$771$	31.3684	1.12971
$772$	0	0
$773$	15.5207	0.558241	0.279121	−	0.960256i	$-0.409957\pi$
$773$	15.5207	0.558241	0.279121	+	0.960256i	$0.409957\pi$
$774$	0	0
$775$	−17.6454	−0.633843
$776$	0	0
$777$	24.7253	0.887015
$778$	0	0
$779$	−31.2422	−1.11937
$780$	0	0
$781$	−12.2733	−0.439174
$782$	0	0
$783$	3.84839	0.137530
$784$	0	0
$785$	−59.0350	−2.10705
$786$	0	0
$787$	9.06022	0.322962	0.161481	−	0.986876i	$-0.448373\pi$
$787$	9.06022	0.322962	0.161481	+	0.986876i	$0.448373\pi$
$788$	0	0
$789$	−33.5955	−1.19603
$790$	0	0
$791$	19.1675	0.681517
$792$	0	0
$793$	26.6002	0.944601
$794$	0	0
$795$	−5.96333	−0.211498
$796$	0	0
$797$	−50.8427	−1.80094	−0.900471	−	0.434917i	$-0.856778\pi$
$797$	−50.8427	−1.80094	−0.900471	+	0.434917i	$0.856778\pi$
$798$	0	0
$799$	−12.3835	−0.438097
$800$	0	0
$801$	23.8260	0.841849
$802$	0	0
$803$	−13.3674	−0.471724
$804$	0	0
$805$	9.97760	0.351664
$806$	0	0
$807$	−28.8385	−1.01516
$808$	0	0
$809$	29.0616	1.02175	0.510877	−	0.859654i	$-0.329321\pi$
$809$	29.0616	1.02175	0.510877	+	0.859654i	$0.329321\pi$
$810$	0	0
$811$	−25.7311	−0.903541	−0.451771	−	0.892134i	$-0.649207\pi$
$811$	−25.7311	−0.903541	−0.451771	+	0.892134i	$0.649207\pi$
$812$	0	0
$813$	52.2499	1.83248
$814$	0	0
$815$	−41.5185	−1.45433
$816$	0	0
$817$	−26.4645	−0.925877
$818$	0	0
$819$	10.1280	0.353902
$820$	0	0
$821$	−33.6757	−1.17529	−0.587645	−	0.809119i	$-0.699945\pi$
$821$	−33.6757	−1.17529	−0.587645	+	0.809119i	$0.699945\pi$
$822$	0	0
$823$	39.5248	1.37775	0.688874	−	0.724881i	$-0.258105\pi$
$823$	39.5248	1.37775	0.688874	+	0.724881i	$0.258105\pi$
$824$	0	0
$825$	49.3566	1.71838
$826$	0	0
$827$	37.0422	1.28808	0.644042	−	0.764990i	$-0.277256\pi$
$827$	37.0422	1.28808	0.644042	+	0.764990i	$0.277256\pi$
$828$	0	0
$829$	22.0798	0.766865	0.383432	−	0.923569i	$-0.374742\pi$
$829$	22.0798	0.766865	0.383432	+	0.923569i	$0.374742\pi$
$830$	0	0
$831$	33.9736	1.17853
$832$	0	0
$833$	1.00000	0.0346479
$834$	0	0
$835$	61.2051	2.11809
$836$	0	0
$837$	1.16203	0.0401655
$838$	0	0
$839$	−44.5480	−1.53797	−0.768983	−	0.639269i	$-0.779237\pi$
$839$	−44.5480	−1.53797	−0.768983	+	0.639269i	$0.779237\pi$
$840$	0	0
$841$	12.2734	0.423221
$842$	0	0
$843$	47.3214	1.62984
$844$	0	0
$845$	−12.1171	−0.416842
$846$	0	0
$847$	6.28156	0.215837
$848$	0	0
$849$	9.01071	0.309247
$850$	0	0
$851$	−26.3045	−0.901708
$852$	0	0
$853$	26.5019	0.907407	0.453704	−	0.891153i	$-0.350103\pi$
$853$	26.5019	0.907407	0.453704	+	0.891153i	$0.350103\pi$
$854$	0	0
$855$	66.0017	2.25721
$856$	0	0
$857$	56.3112	1.92355	0.961776	−	0.273837i	$-0.0882928\pi$
$857$	56.3112	1.92355	0.961776	+	0.273837i	$0.0882928\pi$
$858$	0	0
$859$	−18.5821	−0.634013	−0.317006	−	0.948423i	$-0.602678\pi$
$859$	−18.5821	−0.634013	−0.317006	+	0.948423i	$0.602678\pi$
$860$	0	0
$861$	−14.3828	−0.490163
$862$	0	0
$863$	47.2009	1.60674	0.803369	−	0.595481i	$-0.203039\pi$
$863$	47.2009	1.60674	0.803369	+	0.595481i	$0.203039\pi$
$864$	0	0
$865$	−72.1712	−2.45389
$866$	0	0
$867$	−2.49796	−0.0848352
$868$	0	0
$869$	−0.778109	−0.0263955
$870$	0	0
$871$	−25.2025	−0.853954
$872$	0	0
$873$	−13.3734	−0.452622
$874$	0	0
$875$	−15.3792	−0.519912
$876$	0	0
$877$	6.76248	0.228353	0.114176	−	0.993460i	$-0.463577\pi$
$877$	6.76248	0.228353	0.114176	+	0.993460i	$0.463577\pi$
$878$	0	0
$879$	44.5617	1.50303
$880$	0	0
$881$	25.4892	0.858753	0.429377	−	0.903126i	$-0.358733\pi$
$881$	25.4892	0.858753	0.429377	+	0.903126i	$0.358733\pi$
$882$	0	0
$883$	−31.1313	−1.04765	−0.523826	−	0.851825i	$-0.675496\pi$
$883$	−31.1313	−1.04765	−0.523826	+	0.851825i	$0.675496\pi$
$884$	0	0
$885$	142.522	4.79081
$886$	0	0
$887$	57.6042	1.93416	0.967080	−	0.254474i	$-0.0819022\pi$
$887$	57.6042	1.93416	0.967080	+	0.254474i	$0.0819022\pi$
$888$	0	0
$889$	11.4384	0.383633
$890$	0	0
$891$	17.8621	0.598404
$892$	0	0
$893$	−67.1937	−2.24855
$894$	0	0
$895$	24.5544	0.820764
$896$	0	0
$897$	−20.7523	−0.692899
$898$	0	0
$899$	12.4625	0.415649
$900$	0	0
$901$	0.635847	0.0211831
$902$	0	0
$903$	−12.1833	−0.405435
$904$	0	0
$905$	−44.6369	−1.48378
$906$	0	0
$907$	46.8298	1.55496	0.777479	−	0.628909i	$-0.216498\pi$
$907$	46.8298	1.55496	0.777479	+	0.628909i	$0.216498\pi$
$908$	0	0
$909$	34.3117	1.13805
$910$	0	0
$911$	26.3016	0.871411	0.435705	−	0.900089i	$-0.356499\pi$
$911$	26.3016	0.871411	0.435705	+	0.900089i	$0.356499\pi$
$912$	0	0
$913$	−3.56768	−0.118073
$914$	0	0
$915$	79.8024	2.63819
$916$	0	0
$917$	−14.1746	−0.468087
$918$	0	0
$919$	0.441997	0.0145801	0.00729007	−	0.999973i	$-0.497679\pi$
$919$	0.441997	0.0145801	0.00729007	+	0.999973i	$0.497679\pi$
$920$	0	0
$921$	−59.6605	−1.96588
$922$	0	0
$923$	−17.6632	−0.581390
$924$	0	0
$925$	90.0361	2.96037
$926$	0	0
$927$	−1.16897	−0.0383939
$928$	0	0
$929$	−10.1004	−0.331383	−0.165691	−	0.986178i	$-0.552986\pi$
$929$	−10.1004	−0.331383	−0.165691	+	0.986178i	$0.552986\pi$
$930$	0	0
$931$	5.42606	0.177832
$932$	0	0
$933$	−52.2860	−1.71177
$934$	0	0
$935$	−8.15550	−0.266713
$936$	0	0
$937$	−41.5166	−1.35629	−0.678144	−	0.734929i	$-0.737215\pi$
$937$	−41.5166	−1.35629	−0.678144	+	0.734929i	$0.737215\pi$
$938$	0	0
$939$	3.08654	0.100725
$940$	0	0
$941$	14.7445	0.480657	0.240328	−	0.970692i	$-0.422745\pi$
$941$	14.7445	0.480657	0.240328	+	0.970692i	$0.422745\pi$
$942$	0	0
$943$	15.3014	0.498283
$944$	0	0
$945$	2.24903	0.0731610
$946$	0	0
$947$	−39.0836	−1.27005	−0.635024	−	0.772493i	$-0.719010\pi$
$947$	−39.0836	−1.27005	−0.635024	+	0.772493i	$0.719010\pi$
$948$	0	0
$949$	−19.2377	−0.624481
$950$	0	0
$951$	−25.4896	−0.826558
$952$	0	0
$953$	56.8507	1.84158	0.920788	−	0.390064i	$-0.127547\pi$
$953$	56.8507	1.84158	0.920788	+	0.390064i	$0.127547\pi$
$954$	0	0
$955$	−83.7098	−2.70879
$956$	0	0
$957$	−34.8594	−1.12684
$958$	0	0
$959$	18.3485	0.592503
$960$	0	0
$961$	−27.2369	−0.878610
$962$	0	0
$963$	25.9214	0.835306
$964$	0	0
$965$	−52.4563	−1.68863
$966$	0	0
$967$	41.6625	1.33978	0.669888	−	0.742463i	$-0.266342\pi$
$967$	41.6625	1.33978	0.669888	+	0.742463i	$0.266342\pi$
$968$	0	0
$969$	−13.5541	−0.435420
$970$	0	0
$971$	33.4402	1.07315	0.536573	−	0.843854i	$-0.319718\pi$
$971$	33.4402	1.07315	0.536573	+	0.843854i	$0.319718\pi$
$972$	0	0
$973$	−9.48487	−0.304071
$974$	0	0
$975$	71.0317	2.27483
$976$	0	0
$977$	−33.2659	−1.06427	−0.532136	−	0.846659i	$-0.678610\pi$
$977$	−33.2659	−1.06427	−0.532136	+	0.846659i	$0.678610\pi$
$978$	0	0
$979$	−15.9746	−0.510551
$980$	0	0
$981$	53.5789	1.71064
$982$	0	0
$983$	−51.5812	−1.64518	−0.822592	−	0.568632i	$-0.807473\pi$
$983$	−51.5812	−1.64518	−0.822592	+	0.568632i	$0.807473\pi$
$984$	0	0
$985$	−75.5704	−2.40787
$986$	0	0
$987$	−30.9335	−0.984625
$988$	0	0
$989$	12.9615	0.412151
$990$	0	0
$991$	52.1728	1.65732	0.828662	−	0.559750i	$-0.189103\pi$
$991$	52.1728	1.65732	0.828662	+	0.559750i	$0.189103\pi$
$992$	0	0
$993$	−82.8861	−2.63031
$994$	0	0
$995$	−9.69490	−0.307349
$996$	0	0
$997$	−32.6169	−1.03299	−0.516494	−	0.856291i	$-0.672763\pi$
$997$	−32.6169	−1.03299	−0.516494	+	0.856291i	$0.672763\pi$
$998$	0	0
$999$	−5.92926	−0.187593

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3808.2.a.j.1.1	✓	6
4.3	odd	2		3808.2.a.n.1.6	yes	6
8.3	odd	2		7616.2.a.bw.1.1		6
8.5	even	2		7616.2.a.ca.1.6		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3808.2.a.j.1.1	✓	6	1.1	even	1	trivial
3808.2.a.n.1.6	yes	6	4.3	odd	2
7616.2.a.bw.1.1		6	8.3	odd	2
7616.2.a.ca.1.6		6	8.5	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 3808 = 2^{5} \cdot 7 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3808.a (trivial)

\(f(q)\)	\(=\)	\(q-2.49796 q^{3} +3.75449 q^{5} -1.00000 q^{7} +3.23981 q^{9} -2.17220 q^{11} -3.12612 q^{13} -9.37857 q^{15} +1.00000 q^{17} +5.42606 q^{19} +2.49796 q^{21} -2.65751 q^{23} +9.09622 q^{25} -0.599024 q^{27} -6.42444 q^{29} -1.93987 q^{31} +5.42606 q^{33} -3.75449 q^{35} +9.89820 q^{37} +7.80893 q^{39} -5.75781 q^{41} -4.87730 q^{43} +12.1638 q^{45} -12.3835 q^{47} +1.00000 q^{49} -2.49796 q^{51} +0.635847 q^{53} -8.15550 q^{55} -13.5541 q^{57} -15.1965 q^{59} -8.50901 q^{61} -3.23981 q^{63} -11.7370 q^{65} +8.06191 q^{67} +6.63835 q^{69} +5.65018 q^{71} +6.15385 q^{73} -22.7220 q^{75} +2.17220 q^{77} +0.358213 q^{79} -8.22308 q^{81} +1.64243 q^{83} +3.75449 q^{85} +16.0480 q^{87} +7.35413 q^{89} +3.12612 q^{91} +4.84571 q^{93} +20.3721 q^{95} -4.12785 q^{97} -7.03749 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 2 q^{3} + 2 q^{5} - 6 q^{7} + 4 q^{9} - 2 q^{11} - 4 q^{13} - 8 q^{15} + 6 q^{17} + 2 q^{19} + 2 q^{21} - 10 q^{23} - 2 q^{27} + 2 q^{29} - 12 q^{31} + 2 q^{33} - 2 q^{35} + 2 q^{37} - 10 q^{39} - 8 q^{43}+ \cdots - 2 q^{99}+O(q^{100}) \) 6 * q - 2 * q^3 + 2 * q^5 - 6 * q^7 + 4 * q^9 - 2 * q^11 - 4 * q^13 - 8 * q^15 + 6 * q^17 + 2 * q^19 + 2 * q^21 - 10 * q^23 - 2 * q^27 + 2 * q^29 - 12 * q^31 + 2 * q^33 - 2 * q^35 + 2 * q^37 - 10 * q^39 - 8 * q^43 + 14 * q^45 - 22 * q^47 + 6 * q^49 - 2 * q^51 - 6 * q^53 - 24 * q^55 - 8 * q^57 - 8 * q^59 - 4 * q^61 - 4 * q^63 - 2 * q^65 + 8 * q^67 + 4 * q^69 - 18 * q^71 + 8 * q^73 - 30 * q^75 + 2 * q^77 - 14 * q^79 - 26 * q^81 + 2 * q^85 - 22 * q^87 + 2 * q^89 + 4 * q^91 - 8 * q^93 - 2 * q^95 + 2 * q^97 - 2 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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