LMFDB - Embedded newform 4368.2.a.br.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4368,2,Mod(1,4368)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4368, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("4368.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4368 = 2^{4} \cdot 3 \cdot 7 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4368.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:	$34.8786556029$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.17428.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 6x^{2} + 4x + 6 $ x^4 - x^3 - 6x^2 + 4x + 6
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 273)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.787711$ of defining polynomial
Character	$\chi$	$=$	4368.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} -2.66208 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} -2.66208 q^{5} -1.00000 q^{7} +1.00000 q^{9} -1.57542 q^{11} +1.00000 q^{13} +2.66208 q^{15} +4.75902 q^{17} +2.23750 q^{19} +1.00000 q^{21} -5.84568 q^{23} +2.08666 q^{25} -1.00000 q^{27} +4.23750 q^{29} -7.28055 q^{31} +1.57542 q^{33} +2.66208 q^{35} +10.4750 q^{37} -1.00000 q^{39} -2.25127 q^{41} -0.913344 q^{43} -2.66208 q^{45} +2.09695 q^{47} +1.00000 q^{49} -4.75902 q^{51} +1.08666 q^{53} +4.19389 q^{55} -2.23750 q^{57} +12.3344 q^{59} -7.51805 q^{61} -1.00000 q^{63} -2.66208 q^{65} +15.6914 q^{67} +5.84568 q^{69} -10.0504 q^{71} +15.0797 q^{73} -2.08666 q^{75} +1.57542 q^{77} +11.7555 q^{79} +1.00000 q^{81} -7.42110 q^{83} -12.6689 q^{85} -4.23750 q^{87} -11.1371 q^{89} -1.00000 q^{91} +7.28055 q^{93} -5.95639 q^{95} -14.6047 q^{97} -1.57542 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{3} - 3 q^{5} - 4 q^{7} + 4 q^{9}+O(q^{10}) $ 4 * q - 4 * q^3 - 3 * q^5 - 4 * q^7 + 4 * q^9	$ 4 q - 4 q^{3} - 3 q^{5} - 4 q^{7} + 4 q^{9} + 2 q^{11} + 4 q^{13} + 3 q^{15} - 2 q^{17} - 7 q^{19} + 4 q^{21} - 3 q^{23} + 9 q^{25} - 4 q^{27} + q^{29} - 3 q^{31} - 2 q^{33} + 3 q^{35} + 10 q^{37} - 4 q^{39} - 16 q^{41} - 3 q^{43} - 3 q^{45} - 5 q^{47} + 4 q^{49} + 2 q^{51} + 5 q^{53} - 10 q^{55} + 7 q^{57} + 20 q^{59} + 12 q^{61} - 4 q^{63} - 3 q^{65} + 22 q^{67} + 3 q^{69} - 13 q^{73} - 9 q^{75} - 2 q^{77} - 11 q^{79} + 4 q^{81} - q^{83} + 8 q^{85} - q^{87} - 5 q^{89} - 4 q^{91} + 3 q^{93} - 13 q^{95} - 17 q^{97} + 2 q^{99}+O(q^{100}) $ 4 * q - 4 * q^3 - 3 * q^5 - 4 * q^7 + 4 * q^9 + 2 * q^11 + 4 * q^13 + 3 * q^15 - 2 * q^17 - 7 * q^19 + 4 * q^21 - 3 * q^23 + 9 * q^25 - 4 * q^27 + q^29 - 3 * q^31 - 2 * q^33 + 3 * q^35 + 10 * q^37 - 4 * q^39 - 16 * q^41 - 3 * q^43 - 3 * q^45 - 5 * q^47 + 4 * q^49 + 2 * q^51 + 5 * q^53 - 10 * q^55 + 7 * q^57 + 20 * q^59 + 12 * q^61 - 4 * q^63 - 3 * q^65 + 22 * q^67 + 3 * q^69 - 13 * q^73 - 9 * q^75 - 2 * q^77 - 11 * q^79 + 4 * q^81 - q^83 + 8 * q^85 - q^87 - 5 * q^89 - 4 * q^91 + 3 * q^93 - 13 * q^95 - 17 * q^97 + 2 * 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
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	−2.66208	−1.19052	−0.595259	−	0.803534i	$-0.702950\pi$
$5$	−2.66208	−1.19052	−0.595259	+	0.803534i	$0.702950\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−1.57542	−0.475007	−0.237504	−	0.971387i	$-0.576329\pi$
$11$	−1.57542	−0.475007	−0.237504	+	0.971387i	$0.576329\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	2.66208	0.687345
$16$	0	0
$17$	4.75902	1.15423	0.577116	−	0.816662i	$-0.304178\pi$
$17$	4.75902	1.15423	0.577116	+	0.816662i	$0.304178\pi$
$18$	0	0
$19$	2.23750	0.513317	0.256659	−	0.966502i	$-0.417378\pi$
$19$	2.23750	0.513317	0.256659	+	0.966502i	$0.417378\pi$
$20$	0	0
$21$	1.00000	0.218218
$22$	0	0
$23$	−5.84568	−1.21891	−0.609454	−	0.792821i	$-0.708611\pi$
$23$	−5.84568	−1.21891	−0.609454	+	0.792821i	$0.708611\pi$
$24$	0	0
$25$	2.08666	0.417331
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	4.23750	0.786884	0.393442	−	0.919349i	$-0.371284\pi$
$29$	4.23750	0.786884	0.393442	+	0.919349i	$0.371284\pi$
$30$	0	0
$31$	−7.28055	−1.30763	−0.653813	−	0.756656i	$-0.726832\pi$
$31$	−7.28055	−1.30763	−0.653813	+	0.756656i	$0.726832\pi$
$32$	0	0
$33$	1.57542	0.274246
$34$	0	0
$35$	2.66208	0.449973
$36$	0	0
$37$	10.4750	1.72208	0.861039	−	0.508538i	$-0.169814\pi$
$37$	10.4750	1.72208	0.861039	+	0.508538i	$0.169814\pi$
$38$	0	0
$39$	−1.00000	−0.160128
$40$	0	0
$41$	−2.25127	−0.351589	−0.175794	−	0.984427i	$-0.556249\pi$
$41$	−2.25127	−0.351589	−0.175794	+	0.984427i	$0.556249\pi$
$42$	0	0
$43$	−0.913344	−0.139284	−0.0696418	−	0.997572i	$-0.522186\pi$
$43$	−0.913344	−0.139284	−0.0696418	+	0.997572i	$0.522186\pi$
$44$	0	0
$45$	−2.66208	−0.396839
$46$	0	0
$47$	2.09695	0.305871	0.152936	−	0.988236i	$-0.451127\pi$
$47$	2.09695	0.305871	0.152936	+	0.988236i	$0.451127\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−4.75902	−0.666397
$52$	0	0
$53$	1.08666	0.149264	0.0746319	−	0.997211i	$-0.476222\pi$
$53$	1.08666	0.149264	0.0746319	+	0.997211i	$0.476222\pi$
$54$	0	0
$55$	4.19389	0.565504
$56$	0	0
$57$	−2.23750	−0.296364
$58$	0	0
$59$	12.3344	1.60581	0.802904	−	0.596108i	$-0.203287\pi$
$59$	12.3344	1.60581	0.802904	+	0.596108i	$0.203287\pi$
$60$	0	0
$61$	−7.51805	−0.962587	−0.481294	−	0.876559i	$-0.659833\pi$
$61$	−7.51805	−0.962587	−0.481294	+	0.876559i	$0.659833\pi$
$62$	0	0
$63$	−1.00000	−0.125988
$64$	0	0
$65$	−2.66208	−0.330190
$66$	0	0
$67$	15.6914	1.91700	0.958502	−	0.285084i	$-0.0920217\pi$
$67$	15.6914	1.91700	0.958502	+	0.285084i	$0.0920217\pi$
$68$	0	0
$69$	5.84568	0.703737
$70$	0	0
$71$	−10.0504	−1.19277	−0.596383	−	0.802700i	$-0.703396\pi$
$71$	−10.0504	−1.19277	−0.596383	+	0.802700i	$0.703396\pi$
$72$	0	0
$73$	15.0797	1.76495	0.882473	−	0.470363i	$-0.155877\pi$
$73$	15.0797	1.76495	0.882473	+	0.470363i	$0.155877\pi$
$74$	0	0
$75$	−2.08666	−0.240946
$76$	0	0
$77$	1.57542	0.179536
$78$	0	0
$79$	11.7555	1.32260	0.661301	−	0.750121i	$-0.270005\pi$
$79$	11.7555	1.32260	0.661301	+	0.750121i	$0.270005\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−7.42110	−0.814572	−0.407286	−	0.913301i	$-0.633525\pi$
$83$	−7.42110	−0.814572	−0.407286	+	0.913301i	$0.633525\pi$
$84$	0	0
$85$	−12.6689	−1.37413
$86$	0	0
$87$	−4.23750	−0.454308
$88$	0	0
$89$	−11.1371	−1.18053	−0.590264	−	0.807210i	$-0.700976\pi$
$89$	−11.1371	−1.18053	−0.590264	+	0.807210i	$0.700976\pi$
$90$	0	0
$91$	−1.00000	−0.104828
$92$	0	0
$93$	7.28055	0.754958
$94$	0	0
$95$	−5.95639	−0.611113
$96$	0	0
$97$	−14.6047	−1.48288	−0.741442	−	0.671017i	$-0.765858\pi$
$97$	−14.6047	−1.48288	−0.741442	+	0.671017i	$0.765858\pi$
$98$	0	0
$99$	−1.57542	−0.158336
$100$	0	0
$101$	−11.4349	−1.13781	−0.568906	−	0.822403i	$-0.692633\pi$
$101$	−11.4349	−1.13781	−0.568906	+	0.822403i	$0.692633\pi$
$102$	0	0
$103$	−11.1508	−1.09873	−0.549363	−	0.835584i	$-0.685129\pi$
$103$	−11.1508	−1.09873	−0.549363	+	0.835584i	$0.685129\pi$
$104$	0	0
$105$	−2.66208	−0.259792
$106$	0	0
$107$	−2.28403	−0.220805	−0.110403	−	0.993887i	$-0.535214\pi$
$107$	−2.28403	−0.220805	−0.110403	+	0.993887i	$0.535214\pi$
$108$	0	0
$109$	14.6689	1.40502	0.702512	−	0.711671i	$-0.252062\pi$
$109$	14.6689	1.40502	0.702512	+	0.711671i	$0.252062\pi$
$110$	0	0
$111$	−10.4750	−0.994243
$112$	0	0
$113$	−4.23750	−0.398630	−0.199315	−	0.979935i	$-0.563872\pi$
$113$	−4.23750	−0.398630	−0.199315	+	0.979935i	$0.563872\pi$
$114$	0	0
$115$	15.5617	1.45113
$116$	0	0
$117$	1.00000	0.0924500
$118$	0	0
$119$	−4.75902	−0.436259
$120$	0	0
$121$	−8.51805	−0.774368
$122$	0	0
$123$	2.25127	0.202990
$124$	0	0
$125$	7.75555	0.693677
$126$	0	0
$127$	−4.84916	−0.430293	−0.215147	−	0.976582i	$-0.569023\pi$
$127$	−4.84916	−0.430293	−0.215147	+	0.976582i	$0.569023\pi$
$128$	0	0
$129$	0.913344	0.0804154
$130$	0	0
$131$	6.36721	0.556305	0.278153	−	0.960537i	$-0.410278\pi$
$131$	6.36721	0.556305	0.278153	+	0.960537i	$0.410278\pi$
$132$	0	0
$133$	−2.23750	−0.194016
$134$	0	0
$135$	2.66208	0.229115
$136$	0	0
$137$	−18.0258	−1.54005	−0.770024	−	0.638015i	$-0.779756\pi$
$137$	−18.0258	−1.54005	−0.770024	+	0.638015i	$0.779756\pi$
$138$	0	0
$139$	−2.67585	−0.226962	−0.113481	−	0.993540i	$-0.536200\pi$
$139$	−2.67585	−0.226962	−0.113481	+	0.993540i	$0.536200\pi$
$140$	0	0
$141$	−2.09695	−0.176595
$142$	0	0
$143$	−1.57542	−0.131743
$144$	0	0
$145$	−11.2805	−0.936799
$146$	0	0
$147$	−1.00000	−0.0824786
$148$	0	0
$149$	−7.46471	−0.611533	−0.305766	−	0.952107i	$-0.598913\pi$
$149$	−7.46471	−0.611533	−0.305766	+	0.952107i	$0.598913\pi$
$150$	0	0
$151$	−13.0155	−1.05919	−0.529594	−	0.848251i	$-0.677656\pi$
$151$	−13.0155	−1.05919	−0.529594	+	0.848251i	$0.677656\pi$
$152$	0	0
$153$	4.75902	0.384744
$154$	0	0
$155$	19.3814	1.55675
$156$	0	0
$157$	−17.1233	−1.36659	−0.683294	−	0.730143i	$-0.739453\pi$
$157$	−17.1233	−1.36659	−0.683294	+	0.730143i	$0.739453\pi$
$158$	0	0
$159$	−1.08666	−0.0861774
$160$	0	0
$161$	5.84568	0.460704
$162$	0	0
$163$	0.849158	0.0665112	0.0332556	−	0.999447i	$-0.489412\pi$
$163$	0.849158	0.0665112	0.0332556	+	0.999447i	$0.489412\pi$
$164$	0	0
$165$	−4.19389	−0.326494
$166$	0	0
$167$	−18.3781	−1.42214	−0.711068	−	0.703123i	$-0.751788\pi$
$167$	−18.3781	−1.42214	−0.711068	+	0.703123i	$0.751788\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	2.23750	0.171106
$172$	0	0
$173$	−12.2565	−0.931844	−0.465922	−	0.884826i	$-0.654277\pi$
$173$	−12.2565	−0.931844	−0.465922	+	0.884826i	$0.654277\pi$
$174$	0	0
$175$	−2.08666	−0.157736
$176$	0	0
$177$	−12.3344	−0.927114
$178$	0	0
$179$	−18.5146	−1.38384	−0.691922	−	0.721972i	$-0.743236\pi$
$179$	−18.5146	−1.38384	−0.691922	+	0.721972i	$0.743236\pi$
$180$	0	0
$181$	−18.1664	−1.35029	−0.675147	−	0.737683i	$-0.735920\pi$
$181$	−18.1664	−1.35029	−0.675147	+	0.737683i	$0.735920\pi$
$182$	0	0
$183$	7.51805	0.555750
$184$	0	0
$185$	−27.8853	−2.05016
$186$	0	0
$187$	−7.49747	−0.548269
$188$	0	0
$189$	1.00000	0.0727393
$190$	0	0
$191$	−23.5151	−1.70149	−0.850747	−	0.525575i	$-0.823850\pi$
$191$	−23.5151	−1.70149	−0.850747	+	0.525575i	$0.823850\pi$
$192$	0	0
$193$	14.7344	1.06061	0.530303	−	0.847808i	$-0.322078\pi$
$193$	14.7344	1.06061	0.530303	+	0.847808i	$0.322078\pi$
$194$	0	0
$195$	2.66208	0.190635
$196$	0	0
$197$	14.5008	1.03314	0.516570	−	0.856245i	$-0.327209\pi$
$197$	14.5008	1.03314	0.516570	+	0.856245i	$0.327209\pi$
$198$	0	0
$199$	13.5180	0.958269	0.479135	−	0.877741i	$-0.340951\pi$
$199$	13.5180	0.958269	0.479135	+	0.877741i	$0.340951\pi$
$200$	0	0
$201$	−15.6914	−1.10678
$202$	0	0
$203$	−4.23750	−0.297414
$204$	0	0
$205$	5.99305	0.418572
$206$	0	0
$207$	−5.84568	−0.406303
$208$	0	0
$209$	−3.52500	−0.243830
$210$	0	0
$211$	−4.06419	−0.279790	−0.139895	−	0.990166i	$-0.544677\pi$
$211$	−4.06419	−0.279790	−0.139895	+	0.990166i	$0.544677\pi$
$212$	0	0
$213$	10.0504	0.688643
$214$	0	0
$215$	2.43139	0.165820
$216$	0	0
$217$	7.28055	0.494236
$218$	0	0
$219$	−15.0797	−1.01899
$220$	0	0
$221$	4.75902	0.320127
$222$	0	0
$223$	0.0641862	0.00429822	0.00214911	−	0.999998i	$-0.499316\pi$
$223$	0.0641862	0.00429822	0.00214911	+	0.999998i	$0.499316\pi$
$224$	0	0
$225$	2.08666	0.139110
$226$	0	0
$227$	1.68614	0.111913	0.0559564	−	0.998433i	$-0.482179\pi$
$227$	1.68614	0.111913	0.0559564	+	0.998433i	$0.482179\pi$
$228$	0	0
$229$	0.648310	0.0428415	0.0214208	−	0.999771i	$-0.493181\pi$
$229$	0.648310	0.0428415	0.0214208	+	0.999771i	$0.493181\pi$
$230$	0	0
$231$	−1.57542	−0.103655
$232$	0	0
$233$	3.10724	0.203562	0.101781	−	0.994807i	$-0.467546\pi$
$233$	3.10724	0.203562	0.101781	+	0.994807i	$0.467546\pi$
$234$	0	0
$235$	−5.58223	−0.364145
$236$	0	0
$237$	−11.7555	−0.763605
$238$	0	0
$239$	15.0659	0.974534	0.487267	−	0.873253i	$-0.337994\pi$
$239$	15.0659	0.974534	0.487267	+	0.873253i	$0.337994\pi$
$240$	0	0
$241$	25.7280	1.65729	0.828643	−	0.559777i	$-0.189113\pi$
$241$	25.7280	1.65729	0.828643	+	0.559777i	$0.189113\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	−2.66208	−0.170074
$246$	0	0
$247$	2.23750	0.142369
$248$	0	0
$249$	7.42110	0.470293
$250$	0	0
$251$	11.6258	0.733816	0.366908	−	0.930257i	$-0.380416\pi$
$251$	11.6258	0.733816	0.366908	+	0.930257i	$0.380416\pi$
$252$	0	0
$253$	9.20941	0.578991
$254$	0	0
$255$	12.6689	0.793357
$256$	0	0
$257$	−12.5651	−0.783791	−0.391896	−	0.920010i	$-0.628181\pi$
$257$	−12.5651	−0.783791	−0.391896	+	0.920010i	$0.628181\pi$
$258$	0	0
$259$	−10.4750	−0.650885
$260$	0	0
$261$	4.23750	0.262295
$262$	0	0
$263$	−9.37068	−0.577821	−0.288911	−	0.957356i	$-0.593293\pi$
$263$	−9.37068	−0.577821	−0.288911	+	0.957356i	$0.593293\pi$
$264$	0	0
$265$	−2.89276	−0.177701
$266$	0	0
$267$	11.1371	0.681578
$268$	0	0
$269$	−10.3918	−0.633600	−0.316800	−	0.948492i	$-0.602608\pi$
$269$	−10.3918	−0.633600	−0.316800	+	0.948492i	$0.602608\pi$
$270$	0	0
$271$	5.13026	0.311641	0.155821	−	0.987785i	$-0.450198\pi$
$271$	5.13026	0.311641	0.155821	+	0.987785i	$0.450198\pi$
$272$	0	0
$273$	1.00000	0.0605228
$274$	0	0
$275$	−3.28736	−0.198235
$276$	0	0
$277$	8.04361	0.483293	0.241647	−	0.970364i	$-0.422312\pi$
$277$	8.04361	0.483293	0.241647	+	0.970364i	$0.422312\pi$
$278$	0	0
$279$	−7.28055	−0.435875
$280$	0	0
$281$	−27.8525	−1.66154	−0.830770	−	0.556616i	$-0.812100\pi$
$281$	−27.8525	−1.66154	−0.830770	+	0.556616i	$0.812100\pi$
$282$	0	0
$283$	−23.8197	−1.41594	−0.707968	−	0.706244i	$-0.750388\pi$
$283$	−23.8197	−1.41594	−0.707968	+	0.706244i	$0.750388\pi$
$284$	0	0
$285$	5.95639	0.352826
$286$	0	0
$287$	2.25127	0.132888
$288$	0	0
$289$	5.64831	0.332254
$290$	0	0
$291$	14.6047	0.856143
$292$	0	0
$293$	−16.4612	−0.961675	−0.480838	−	0.876810i	$-0.659667\pi$
$293$	−16.4612	−0.961675	−0.480838	+	0.876810i	$0.659667\pi$
$294$	0	0
$295$	−32.8352	−1.91174
$296$	0	0
$297$	1.57542	0.0914152
$298$	0	0
$299$	−5.84568	−0.338064
$300$	0	0
$301$	0.913344	0.0526443
$302$	0	0
$303$	11.4349	0.656916
$304$	0	0
$305$	20.0136	1.14598
$306$	0	0
$307$	−1.95639	−0.111657	−0.0558287	−	0.998440i	$-0.517780\pi$
$307$	−1.95639	−0.111657	−0.0558287	+	0.998440i	$0.517780\pi$
$308$	0	0
$309$	11.1508	0.634349
$310$	0	0
$311$	6.67585	0.378552	0.189276	−	0.981924i	$-0.439386\pi$
$311$	6.67585	0.378552	0.189276	+	0.981924i	$0.439386\pi$
$312$	0	0
$313$	−6.78364	−0.383434	−0.191717	−	0.981450i	$-0.561406\pi$
$313$	−6.78364	−0.383434	−0.191717	+	0.981450i	$0.561406\pi$
$314$	0	0
$315$	2.66208	0.149991
$316$	0	0
$317$	−30.6947	−1.72399	−0.861993	−	0.506920i	$-0.830784\pi$
$317$	−30.6947	−1.72399	−0.861993	+	0.506920i	$0.830784\pi$
$318$	0	0
$319$	−6.67585	−0.373776
$320$	0	0
$321$	2.28403	0.127482
$322$	0	0
$323$	10.6483	0.592488
$324$	0	0
$325$	2.08666	0.115747
$326$	0	0
$327$	−14.6689	−0.811191
$328$	0	0
$329$	−2.09695	−0.115608
$330$	0	0
$331$	13.8267	0.759983	0.379992	−	0.924990i	$-0.375927\pi$
$331$	13.8267	0.759983	0.379992	+	0.924990i	$0.375927\pi$
$332$	0	0
$333$	10.4750	0.574026
$334$	0	0
$335$	−41.7716	−2.28223
$336$	0	0
$337$	23.8633	1.29992	0.649959	−	0.759969i	$-0.274786\pi$
$337$	23.8633	1.29992	0.649959	+	0.759969i	$0.274786\pi$
$338$	0	0
$339$	4.23750	0.230149
$340$	0	0
$341$	11.4699	0.621132
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	−15.5617	−0.837811
$346$	0	0
$347$	13.6012	0.730152	0.365076	−	0.930978i	$-0.381043\pi$
$347$	13.6012	0.730152	0.365076	+	0.930978i	$0.381043\pi$
$348$	0	0
$349$	−16.4039	−0.878078	−0.439039	−	0.898468i	$-0.644681\pi$
$349$	−16.4039	−0.878078	−0.439039	+	0.898468i	$0.644681\pi$
$350$	0	0
$351$	−1.00000	−0.0533761
$352$	0	0
$353$	−4.73322	−0.251924	−0.125962	−	0.992035i	$-0.540202\pi$
$353$	−4.73322	−0.251924	−0.125962	+	0.992035i	$0.540202\pi$
$354$	0	0
$355$	26.7550	1.42001
$356$	0	0
$357$	4.75902	0.251874
$358$	0	0
$359$	17.5479	0.926142	0.463071	−	0.886321i	$-0.346747\pi$
$359$	17.5479	0.926142	0.463071	+	0.886321i	$0.346747\pi$
$360$	0	0
$361$	−13.9936	−0.736505
$362$	0	0
$363$	8.51805	0.447082
$364$	0	0
$365$	−40.1433	−2.10120
$366$	0	0
$367$	11.1888	0.584052	0.292026	−	0.956410i	$-0.405671\pi$
$367$	11.1888	0.584052	0.292026	+	0.956410i	$0.405671\pi$
$368$	0	0
$369$	−2.25127	−0.117196
$370$	0	0
$371$	−1.08666	−0.0564164
$372$	0	0
$373$	−5.43195	−0.281256	−0.140628	−	0.990063i	$-0.544912\pi$
$373$	−5.43195	−0.281256	−0.140628	+	0.990063i	$0.544912\pi$
$374$	0	0
$375$	−7.75555	−0.400495
$376$	0	0
$377$	4.23750	0.218242
$378$	0	0
$379$	−10.8422	−0.556927	−0.278463	−	0.960447i	$-0.589825\pi$
$379$	−10.8422	−0.556927	−0.278463	+	0.960447i	$0.589825\pi$
$380$	0	0
$381$	4.84916	0.248430
$382$	0	0
$383$	−10.5353	−0.538328	−0.269164	−	0.963094i	$-0.586747\pi$
$383$	−10.5353	−0.538328	−0.269164	+	0.963094i	$0.586747\pi$
$384$	0	0
$385$	−4.19389	−0.213741
$386$	0	0
$387$	−0.913344	−0.0464279
$388$	0	0
$389$	6.00000	0.304212	0.152106	−	0.988364i	$-0.451394\pi$
$389$	6.00000	0.304212	0.152106	+	0.988364i	$0.451394\pi$
$390$	0	0
$391$	−27.8197	−1.40690
$392$	0	0
$393$	−6.36721	−0.321183
$394$	0	0
$395$	−31.2942	−1.57458
$396$	0	0
$397$	−35.4400	−1.77868	−0.889340	−	0.457246i	$-0.848836\pi$
$397$	−35.4400	−1.77868	−0.889340	+	0.457246i	$0.848836\pi$
$398$	0	0
$399$	2.23750	0.112015
$400$	0	0
$401$	23.7241	1.18473	0.592363	−	0.805671i	$-0.298195\pi$
$401$	23.7241	1.18473	0.592363	+	0.805671i	$0.298195\pi$
$402$	0	0
$403$	−7.28055	−0.362670
$404$	0	0
$405$	−2.66208	−0.132280
$406$	0	0
$407$	−16.5025	−0.818000
$408$	0	0
$409$	−3.95639	−0.195631	−0.0978156	−	0.995205i	$-0.531186\pi$
$409$	−3.95639	−0.195631	−0.0978156	+	0.995205i	$0.531186\pi$
$410$	0	0
$411$	18.0258	0.889147
$412$	0	0
$413$	−12.3344	−0.606938
$414$	0	0
$415$	19.7555	0.969762
$416$	0	0
$417$	2.67585	0.131037
$418$	0	0
$419$	18.0586	0.882219	0.441109	−	0.897453i	$-0.354585\pi$
$419$	18.0586	0.882219	0.441109	+	0.897453i	$0.354585\pi$
$420$	0	0
$421$	24.4956	1.19384	0.596921	−	0.802300i	$-0.296391\pi$
$421$	24.4956	1.19384	0.596921	+	0.802300i	$0.296391\pi$
$422$	0	0
$423$	2.09695	0.101957
$424$	0	0
$425$	9.93045	0.481697
$426$	0	0
$427$	7.51805	0.363824
$428$	0	0
$429$	1.57542	0.0760621
$430$	0	0
$431$	40.7779	1.96420	0.982101	−	0.188357i	$-0.0603163\pi$
$431$	40.7779	1.96420	0.982101	+	0.188357i	$0.0603163\pi$
$432$	0	0
$433$	−18.0791	−0.868828	−0.434414	−	0.900713i	$-0.643045\pi$
$433$	−18.0791	−0.868828	−0.434414	+	0.900713i	$0.643045\pi$
$434$	0	0
$435$	11.2805	0.540861
$436$	0	0
$437$	−13.0797	−0.625687
$438$	0	0
$439$	−20.7561	−0.990635	−0.495317	−	0.868712i	$-0.664948\pi$
$439$	−20.7561	−0.990635	−0.495317	+	0.868712i	$0.664948\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	8.24042	0.391514	0.195757	−	0.980652i	$-0.437284\pi$
$443$	8.24042	0.391514	0.195757	+	0.980652i	$0.437284\pi$
$444$	0	0
$445$	29.6478	1.40544
$446$	0	0
$447$	7.46471	0.353069
$448$	0	0
$449$	36.1061	1.70395	0.851975	−	0.523582i	$-0.175405\pi$
$449$	36.1061	1.70395	0.851975	+	0.523582i	$0.175405\pi$
$450$	0	0
$451$	3.54669	0.167007
$452$	0	0
$453$	13.0155	0.611522
$454$	0	0
$455$	2.66208	0.124800
$456$	0	0
$457$	−6.54052	−0.305953	−0.152976	−	0.988230i	$-0.548886\pi$
$457$	−6.54052	−0.305953	−0.152976	+	0.988230i	$0.548886\pi$
$458$	0	0
$459$	−4.75902	−0.222132
$460$	0	0
$461$	−14.6987	−0.684588	−0.342294	−	0.939593i	$-0.611204\pi$
$461$	−14.6987	−0.684588	−0.342294	+	0.939593i	$0.611204\pi$
$462$	0	0
$463$	−21.7775	−1.01208	−0.506042	−	0.862509i	$-0.668892\pi$
$463$	−21.7775	−1.01208	−0.506042	+	0.862509i	$0.668892\pi$
$464$	0	0
$465$	−19.3814	−0.898790
$466$	0	0
$467$	−35.0086	−1.62000	−0.810001	−	0.586428i	$-0.800534\pi$
$467$	−35.0086	−1.62000	−0.810001	+	0.586428i	$0.800534\pi$
$468$	0	0
$469$	−15.6914	−0.724560
$470$	0	0
$471$	17.1233	0.789000
$472$	0	0
$473$	1.43890	0.0661607
$474$	0	0
$475$	4.66889	0.214223
$476$	0	0
$477$	1.08666	0.0497546
$478$	0	0
$479$	−31.7814	−1.45213	−0.726064	−	0.687628i	$-0.758652\pi$
$479$	−31.7814	−1.45213	−0.726064	+	0.687628i	$0.758652\pi$
$480$	0	0
$481$	10.4750	0.477619
$482$	0	0
$483$	−5.84568	−0.265988
$484$	0	0
$485$	38.8789	1.76540
$486$	0	0
$487$	28.5817	1.29516	0.647580	−	0.761998i	$-0.275781\pi$
$487$	28.5817	1.29516	0.647580	+	0.761998i	$0.275781\pi$
$488$	0	0
$489$	−0.849158	−0.0384002
$490$	0	0
$491$	−21.7160	−0.980028	−0.490014	−	0.871715i	$-0.663008\pi$
$491$	−21.7160	−0.980028	−0.490014	+	0.871715i	$0.663008\pi$
$492$	0	0
$493$	20.1664	0.908247
$494$	0	0
$495$	4.19389	0.188501
$496$	0	0
$497$	10.0504	0.450823
$498$	0	0
$499$	42.6895	1.91104	0.955522	−	0.294921i	$-0.0952934\pi$
$499$	42.6895	1.91104	0.955522	+	0.294921i	$0.0952934\pi$
$500$	0	0
$501$	18.3781	0.821071
$502$	0	0
$503$	−21.8922	−0.976125	−0.488063	−	0.872809i	$-0.662296\pi$
$503$	−21.8922	−0.976125	−0.488063	+	0.872809i	$0.662296\pi$
$504$	0	0
$505$	30.4405	1.35458
$506$	0	0
$507$	−1.00000	−0.0444116
$508$	0	0
$509$	−5.36546	−0.237820	−0.118910	−	0.992905i	$-0.537940\pi$
$509$	−5.36546	−0.237820	−0.118910	+	0.992905i	$0.537940\pi$
$510$	0	0
$511$	−15.0797	−0.667087
$512$	0	0
$513$	−2.23750	−0.0987880
$514$	0	0
$515$	29.6844	1.30805
$516$	0	0
$517$	−3.30357	−0.145291
$518$	0	0
$519$	12.2565	0.538000
$520$	0	0
$521$	−11.5221	−0.504791	−0.252396	−	0.967624i	$-0.581218\pi$
$521$	−11.5221	−0.504791	−0.252396	+	0.967624i	$0.581218\pi$
$522$	0	0
$523$	3.62584	0.158547	0.0792734	−	0.996853i	$-0.474740\pi$
$523$	3.62584	0.158547	0.0792734	+	0.996853i	$0.474740\pi$
$524$	0	0
$525$	2.08666	0.0910691
$526$	0	0
$527$	−34.6483	−1.50930
$528$	0	0
$529$	11.1720	0.485738
$530$	0	0
$531$	12.3344	0.535269
$532$	0	0
$533$	−2.25127	−0.0975132
$534$	0	0
$535$	6.08026	0.262872
$536$	0	0
$537$	18.5146	0.798963
$538$	0	0
$539$	−1.57542	−0.0678582
$540$	0	0
$541$	37.2300	1.60064	0.800321	−	0.599572i	$-0.204662\pi$
$541$	37.2300	1.60064	0.800321	+	0.599572i	$0.204662\pi$
$542$	0	0
$543$	18.1664	0.779593
$544$	0	0
$545$	−39.0497	−1.67271
$546$	0	0
$547$	4.34529	0.185791	0.0928956	−	0.995676i	$-0.470388\pi$
$547$	4.34529	0.185791	0.0928956	+	0.995676i	$0.470388\pi$
$548$	0	0
$549$	−7.51805	−0.320862
$550$	0	0
$551$	9.48140	0.403921
$552$	0	0
$553$	−11.7555	−0.499897
$554$	0	0
$555$	27.8853	1.18366
$556$	0	0
$557$	28.0052	1.18662	0.593310	−	0.804974i	$-0.297821\pi$
$557$	28.0052	1.18662	0.593310	+	0.804974i	$0.297821\pi$
$558$	0	0
$559$	−0.913344	−0.0386303
$560$	0	0
$561$	7.49747	0.316543
$562$	0	0
$563$	22.2525	0.937829	0.468915	−	0.883243i	$-0.344645\pi$
$563$	22.2525	0.937829	0.468915	+	0.883243i	$0.344645\pi$
$564$	0	0
$565$	11.2805	0.474576
$566$	0	0
$567$	−1.00000	−0.0419961
$568$	0	0
$569$	3.10724	0.130262	0.0651311	−	0.997877i	$-0.479253\pi$
$569$	3.10724	0.130262	0.0651311	+	0.997877i	$0.479253\pi$
$570$	0	0
$571$	−16.5772	−0.693733	−0.346866	−	0.937915i	$-0.612754\pi$
$571$	−16.5772	−0.693733	−0.346866	+	0.937915i	$0.612754\pi$
$572$	0	0
$573$	23.5151	0.982358
$574$	0	0
$575$	−12.1979	−0.508689
$576$	0	0
$577$	−7.45253	−0.310253	−0.155126	−	0.987895i	$-0.549578\pi$
$577$	−7.45253	−0.310253	−0.155126	+	0.987895i	$0.549578\pi$
$578$	0	0
$579$	−14.7344	−0.612341
$580$	0	0
$581$	7.42110	0.307879
$582$	0	0
$583$	−1.71194	−0.0709014
$584$	0	0
$585$	−2.66208	−0.110063
$586$	0	0
$587$	−11.9892	−0.494845	−0.247423	−	0.968908i	$-0.579584\pi$
$587$	−11.9892	−0.494845	−0.247423	+	0.968908i	$0.579584\pi$
$588$	0	0
$589$	−16.2902	−0.671227
$590$	0	0
$591$	−14.5008	−0.596483
$592$	0	0
$593$	−33.4767	−1.37473	−0.687363	−	0.726314i	$-0.741232\pi$
$593$	−33.4767	−1.37473	−0.687363	+	0.726314i	$0.741232\pi$
$594$	0	0
$595$	12.6689	0.519374
$596$	0	0
$597$	−13.5180	−0.553257
$598$	0	0
$599$	13.5863	0.555120	0.277560	−	0.960708i	$-0.410474\pi$
$599$	13.5863	0.555120	0.277560	+	0.960708i	$0.410474\pi$
$600$	0	0
$601$	−6.78364	−0.276710	−0.138355	−	0.990383i	$-0.544182\pi$
$601$	−6.78364	−0.276710	−0.138355	+	0.990383i	$0.544182\pi$
$602$	0	0
$603$	15.6914	0.639002
$604$	0	0
$605$	22.6757	0.921898
$606$	0	0
$607$	17.4905	0.709918	0.354959	−	0.934882i	$-0.384495\pi$
$607$	17.4905	0.709918	0.354959	+	0.934882i	$0.384495\pi$
$608$	0	0
$609$	4.23750	0.171712
$610$	0	0
$611$	2.09695	0.0848334
$612$	0	0
$613$	−38.2455	−1.54472	−0.772361	−	0.635184i	$-0.780924\pi$
$613$	−38.2455	−1.54472	−0.772361	+	0.635184i	$0.780924\pi$
$614$	0	0
$615$	−5.99305	−0.241663
$616$	0	0
$617$	−34.1542	−1.37500	−0.687498	−	0.726187i	$-0.741291\pi$
$617$	−34.1542	−1.37500	−0.687498	+	0.726187i	$0.741291\pi$
$618$	0	0
$619$	8.56805	0.344379	0.172190	−	0.985064i	$-0.444916\pi$
$619$	8.56805	0.344379	0.172190	+	0.985064i	$0.444916\pi$
$620$	0	0
$621$	5.84568	0.234579
$622$	0	0
$623$	11.1371	0.446197
$624$	0	0
$625$	−31.0791	−1.24317
$626$	0	0
$627$	3.52500	0.140775
$628$	0	0
$629$	49.8508	1.98768
$630$	0	0
$631$	17.7119	0.705101	0.352551	−	0.935793i	$-0.385314\pi$
$631$	17.7119	0.705101	0.352551	+	0.935793i	$0.385314\pi$
$632$	0	0
$633$	4.06419	0.161537
$634$	0	0
$635$	12.9088	0.512271
$636$	0	0
$637$	1.00000	0.0396214
$638$	0	0
$639$	−10.0504	−0.397588
$640$	0	0
$641$	16.6117	0.656121	0.328061	−	0.944657i	$-0.393605\pi$
$641$	16.6117	0.656121	0.328061	+	0.944657i	$0.393605\pi$
$642$	0	0
$643$	−12.9500	−0.510698	−0.255349	−	0.966849i	$-0.582190\pi$
$643$	−12.9500	−0.510698	−0.255349	+	0.966849i	$0.582190\pi$
$644$	0	0
$645$	−2.43139	−0.0957360
$646$	0	0
$647$	23.6775	0.930857	0.465428	−	0.885086i	$-0.345900\pi$
$647$	23.6775	0.930857	0.465428	+	0.885086i	$0.345900\pi$
$648$	0	0
$649$	−19.4319	−0.762771
$650$	0	0
$651$	−7.28055	−0.285347
$652$	0	0
$653$	22.2811	0.871927	0.435963	−	0.899964i	$-0.356408\pi$
$653$	22.2811	0.871927	0.435963	+	0.899964i	$0.356408\pi$
$654$	0	0
$655$	−16.9500	−0.662291
$656$	0	0
$657$	15.0797	0.588315
$658$	0	0
$659$	−15.5165	−0.604435	−0.302218	−	0.953239i	$-0.597727\pi$
$659$	−15.5165	−0.604435	−0.302218	+	0.953239i	$0.597727\pi$
$660$	0	0
$661$	28.1444	1.09469	0.547346	−	0.836906i	$-0.315638\pi$
$661$	28.1444	1.09469	0.547346	+	0.836906i	$0.315638\pi$
$662$	0	0
$663$	−4.75902	−0.184825
$664$	0	0
$665$	5.95639	0.230979
$666$	0	0
$667$	−24.7711	−0.959139
$668$	0	0
$669$	−0.0641862	−0.00248158
$670$	0	0
$671$	11.8441	0.457236
$672$	0	0
$673$	7.13477	0.275025	0.137513	−	0.990500i	$-0.456089\pi$
$673$	7.13477	0.275025	0.137513	+	0.990500i	$0.456089\pi$
$674$	0	0
$675$	−2.08666	−0.0803154
$676$	0	0
$677$	−10.7660	−0.413770	−0.206885	−	0.978365i	$-0.566333\pi$
$677$	−10.7660	−0.413770	−0.206885	+	0.978365i	$0.566333\pi$
$678$	0	0
$679$	14.6047	0.560477
$680$	0	0
$681$	−1.68614	−0.0646129
$682$	0	0
$683$	12.2020	0.466898	0.233449	−	0.972369i	$-0.424999\pi$
$683$	12.2020	0.466898	0.233449	+	0.972369i	$0.424999\pi$
$684$	0	0
$685$	47.9861	1.83345
$686$	0	0
$687$	−0.648310	−0.0247346
$688$	0	0
$689$	1.08666	0.0413983
$690$	0	0
$691$	−32.0367	−1.21873	−0.609366	−	0.792889i	$-0.708576\pi$
$691$	−32.0367	−1.21873	−0.609366	+	0.792889i	$0.708576\pi$
$692$	0	0
$693$	1.57542	0.0598453
$694$	0	0
$695$	7.12331	0.270202
$696$	0	0
$697$	−10.7138	−0.405815
$698$	0	0
$699$	−3.10724	−0.117526
$700$	0	0
$701$	−14.9513	−0.564704	−0.282352	−	0.959311i	$-0.591115\pi$
$701$	−14.9513	−0.564704	−0.282352	+	0.959311i	$0.591115\pi$
$702$	0	0
$703$	23.4378	0.883973
$704$	0	0
$705$	5.58223	0.210239
$706$	0	0
$707$	11.4349	0.430053
$708$	0	0
$709$	−8.58279	−0.322333	−0.161167	−	0.986927i	$-0.551526\pi$
$709$	−8.58279	−0.322333	−0.161167	+	0.986927i	$0.551526\pi$
$710$	0	0
$711$	11.7555	0.440867
$712$	0	0
$713$	42.5598	1.59388
$714$	0	0
$715$	4.19389	0.156843
$716$	0	0
$717$	−15.0659	−0.562648
$718$	0	0
$719$	−35.9644	−1.34125	−0.670623	−	0.741798i	$-0.733973\pi$
$719$	−35.9644	−1.34125	−0.670623	+	0.741798i	$0.733973\pi$
$720$	0	0
$721$	11.1508	0.415279
$722$	0	0
$723$	−25.7280	−0.956835
$724$	0	0
$725$	8.84220	0.328391
$726$	0	0
$727$	31.5111	1.16868	0.584341	−	0.811508i	$-0.301353\pi$
$727$	31.5111	1.16868	0.584341	+	0.811508i	$0.301353\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−4.34662	−0.160766
$732$	0	0
$733$	−26.6047	−0.982667	−0.491334	−	0.870971i	$-0.663490\pi$
$733$	−26.6047	−0.982667	−0.491334	+	0.870971i	$0.663490\pi$
$734$	0	0
$735$	2.66208	0.0981922
$736$	0	0
$737$	−24.7205	−0.910591
$738$	0	0
$739$	50.3397	1.85177	0.925887	−	0.377800i	$-0.123319\pi$
$739$	50.3397	1.85177	0.925887	+	0.377800i	$0.123319\pi$
$740$	0	0
$741$	−2.23750	−0.0821966
$742$	0	0
$743$	−23.4021	−0.858540	−0.429270	−	0.903176i	$-0.641229\pi$
$743$	−23.4021	−0.858540	−0.429270	+	0.903176i	$0.641229\pi$
$744$	0	0
$745$	19.8716	0.728040
$746$	0	0
$747$	−7.42110	−0.271524
$748$	0	0
$749$	2.28403	0.0834565
$750$	0	0
$751$	−49.0305	−1.78915	−0.894574	−	0.446920i	$-0.852521\pi$
$751$	−49.0305	−1.78915	−0.894574	+	0.446920i	$0.852521\pi$
$752$	0	0
$753$	−11.6258	−0.423669
$754$	0	0
$755$	34.6483	1.26098
$756$	0	0
$757$	−35.3883	−1.28621	−0.643106	−	0.765778i	$-0.722354\pi$
$757$	−35.3883	−1.28621	−0.643106	+	0.765778i	$0.722354\pi$
$758$	0	0
$759$	−9.20941	−0.334280
$760$	0	0
$761$	27.7051	1.00431	0.502155	−	0.864778i	$-0.332541\pi$
$761$	27.7051	1.00431	0.502155	+	0.864778i	$0.332541\pi$
$762$	0	0
$763$	−14.6689	−0.531049
$764$	0	0
$765$	−12.6689	−0.458045
$766$	0	0
$767$	12.3344	0.445371
$768$	0	0
$769$	21.1383	0.762265	0.381133	−	0.924520i	$-0.375534\pi$
$769$	21.1383	0.762265	0.381133	+	0.924520i	$0.375534\pi$
$770$	0	0
$771$	12.5651	0.452522
$772$	0	0
$773$	−41.4537	−1.49099	−0.745493	−	0.666513i	$-0.767786\pi$
$773$	−41.4537	−1.49099	−0.745493	+	0.666513i	$0.767786\pi$
$774$	0	0
$775$	−15.1920	−0.545713
$776$	0	0
$777$	10.4750	0.375788
$778$	0	0
$779$	−5.03721	−0.180477
$780$	0	0
$781$	15.8336	0.566572
$782$	0	0
$783$	−4.23750	−0.151436
$784$	0	0
$785$	45.5836	1.62695
$786$	0	0
$787$	28.7711	1.02558	0.512789	−	0.858515i	$-0.328612\pi$
$787$	28.7711	1.02558	0.512789	+	0.858515i	$0.328612\pi$
$788$	0	0
$789$	9.37068	0.333605
$790$	0	0
$791$	4.23750	0.150668
$792$	0	0
$793$	−7.51805	−0.266974
$794$	0	0
$795$	2.89276	0.102596
$796$	0	0
$797$	19.5357	0.691990	0.345995	−	0.938236i	$-0.387541\pi$
$797$	19.5357	0.691990	0.345995	+	0.938236i	$0.387541\pi$
$798$	0	0
$799$	9.97942	0.353046
$800$	0	0
$801$	−11.1371	−0.393509
$802$	0	0
$803$	−23.7569	−0.838362
$804$	0	0
$805$	−15.5617	−0.548476
$806$	0	0
$807$	10.3918	0.365809
$808$	0	0
$809$	−36.1824	−1.27211	−0.636053	−	0.771645i	$-0.719434\pi$
$809$	−36.1824	−1.27211	−0.636053	+	0.771645i	$0.719434\pi$
$810$	0	0
$811$	−0.794087	−0.0278842	−0.0139421	−	0.999903i	$-0.504438\pi$
$811$	−0.794087	−0.0278842	−0.0139421	+	0.999903i	$0.504438\pi$
$812$	0	0
$813$	−5.13026	−0.179926
$814$	0	0
$815$	−2.26052	−0.0791827
$816$	0	0
$817$	−2.04361	−0.0714967
$818$	0	0
$819$	−1.00000	−0.0349428
$820$	0	0
$821$	−15.8525	−0.553256	−0.276628	−	0.960977i	$-0.589217\pi$
$821$	−15.8525	−0.553256	−0.276628	+	0.960977i	$0.589217\pi$
$822$	0	0
$823$	20.0412	0.698591	0.349295	−	0.937013i	$-0.386421\pi$
$823$	20.0412	0.698591	0.349295	+	0.937013i	$0.386421\pi$
$824$	0	0
$825$	3.28736	0.114451
$826$	0	0
$827$	−34.7848	−1.20959	−0.604794	−	0.796382i	$-0.706744\pi$
$827$	−34.7848	−1.20959	−0.604794	+	0.796382i	$0.706744\pi$
$828$	0	0
$829$	−30.6793	−1.06554	−0.532769	−	0.846261i	$-0.678848\pi$
$829$	−30.6793	−1.06554	−0.532769	+	0.846261i	$0.678848\pi$
$830$	0	0
$831$	−8.04361	−0.279030
$832$	0	0
$833$	4.75902	0.164890
$834$	0	0
$835$	48.9238	1.69308
$836$	0	0
$837$	7.28055	0.251653
$838$	0	0
$839$	−39.8238	−1.37487	−0.687436	−	0.726245i	$-0.741264\pi$
$839$	−39.8238	−1.37487	−0.687436	+	0.726245i	$0.741264\pi$
$840$	0	0
$841$	−11.0436	−0.380814
$842$	0	0
$843$	27.8525	0.959291
$844$	0	0
$845$	−2.66208	−0.0915782
$846$	0	0
$847$	8.51805	0.292684
$848$	0	0
$849$	23.8197	0.817491
$850$	0	0
$851$	−61.2335	−2.09906
$852$	0	0
$853$	24.7505	0.847440	0.423720	−	0.905793i	$-0.360724\pi$
$853$	24.7505	0.847440	0.423720	+	0.905793i	$0.360724\pi$
$854$	0	0
$855$	−5.95639	−0.203704
$856$	0	0
$857$	6.47097	0.221044	0.110522	−	0.993874i	$-0.464748\pi$
$857$	6.47097	0.221044	0.110522	+	0.993874i	$0.464748\pi$
$858$	0	0
$859$	−31.0741	−1.06023	−0.530117	−	0.847925i	$-0.677852\pi$
$859$	−31.0741	−1.06023	−0.530117	+	0.847925i	$0.677852\pi$
$860$	0	0
$861$	−2.25127	−0.0767230
$862$	0	0
$863$	−28.6391	−0.974885	−0.487442	−	0.873155i	$-0.662070\pi$
$863$	−28.6391	−0.974885	−0.487442	+	0.873155i	$0.662070\pi$
$864$	0	0
$865$	32.6277	1.10938
$866$	0	0
$867$	−5.64831	−0.191827
$868$	0	0
$869$	−18.5199	−0.628246
$870$	0	0
$871$	15.6914	0.531681
$872$	0	0
$873$	−14.6047	−0.494294
$874$	0	0
$875$	−7.75555	−0.262185
$876$	0	0
$877$	0.00110921	3.74555e−5 0	1.87278e−5	−	1.00000i	$-0.499994\pi$
$877$	0.00110921	3.74555e−5 0	1.87278e−5		1.00000i	$0.499994\pi$
$878$	0	0
$879$	16.4612	0.555223
$880$	0	0
$881$	23.9479	0.806824	0.403412	−	0.915019i	$-0.367824\pi$
$881$	23.9479	0.806824	0.403412	+	0.915019i	$0.367824\pi$
$882$	0	0
$883$	−51.9965	−1.74982	−0.874911	−	0.484283i	$-0.839081\pi$
$883$	−51.9965	−1.74982	−0.874911	+	0.484283i	$0.839081\pi$
$884$	0	0
$885$	32.8352	1.10374
$886$	0	0
$887$	−10.8697	−0.364970	−0.182485	−	0.983209i	$-0.558414\pi$
$887$	−10.8697	−0.364970	−0.182485	+	0.983209i	$0.558414\pi$
$888$	0	0
$889$	4.84916	0.162636
$890$	0	0
$891$	−1.57542	−0.0527786
$892$	0	0
$893$	4.69192	0.157009
$894$	0	0
$895$	49.2872	1.64749
$896$	0	0
$897$	5.84568	0.195182
$898$	0	0
$899$	−30.8513	−1.02895
$900$	0	0
$901$	5.17142	0.172285
$902$	0	0
$903$	−0.913344	−0.0303942
$904$	0	0
$905$	48.3603	1.60755
$906$	0	0
$907$	25.4892	0.846354	0.423177	−	0.906047i	$-0.360915\pi$
$907$	25.4892	0.846354	0.423177	+	0.906047i	$0.360915\pi$
$908$	0	0
$909$	−11.4349	−0.379271
$910$	0	0
$911$	−42.2059	−1.39834	−0.699172	−	0.714953i	$-0.746448\pi$
$911$	−42.2059	−1.39834	−0.699172	+	0.714953i	$0.746448\pi$
$912$	0	0
$913$	11.6914	0.386928
$914$	0	0
$915$	−20.0136	−0.661630
$916$	0	0
$917$	−6.36721	−0.210264
$918$	0	0
$919$	29.4033	0.969925	0.484963	−	0.874535i	$-0.338833\pi$
$919$	29.4033	0.969925	0.484963	+	0.874535i	$0.338833\pi$
$920$	0	0
$921$	1.95639	0.0644654
$922$	0	0
$923$	−10.0504	−0.330814
$924$	0	0
$925$	21.8577	0.718677
$926$	0	0
$927$	−11.1508	−0.366242
$928$	0	0
$929$	−13.3759	−0.438849	−0.219425	−	0.975629i	$-0.570418\pi$
$929$	−13.3759	−0.438849	−0.219425	+	0.975629i	$0.570418\pi$
$930$	0	0
$931$	2.23750	0.0733311
$932$	0	0
$933$	−6.67585	−0.218557
$934$	0	0
$935$	19.9588	0.652724
$936$	0	0
$937$	−24.9981	−0.816653	−0.408326	−	0.912836i	$-0.633887\pi$
$937$	−24.9981	−0.816653	−0.408326	+	0.912836i	$0.633887\pi$
$938$	0	0
$939$	6.78364	0.221376
$940$	0	0
$941$	10.5369	0.343493	0.171746	−	0.985141i	$-0.445059\pi$
$941$	10.5369	0.343493	0.171746	+	0.985141i	$0.445059\pi$
$942$	0	0
$943$	13.1602	0.428555
$944$	0	0
$945$	−2.66208	−0.0865974
$946$	0	0
$947$	52.7779	1.71505	0.857525	−	0.514442i	$-0.172001\pi$
$947$	52.7779	1.71505	0.857525	+	0.514442i	$0.172001\pi$
$948$	0	0
$949$	15.0797	0.489508
$950$	0	0
$951$	30.6947	0.995344
$952$	0	0
$953$	21.8486	0.707746	0.353873	−	0.935294i	$-0.384865\pi$
$953$	21.8486	0.707746	0.353873	+	0.935294i	$0.384865\pi$
$954$	0	0
$955$	62.5991	2.02566
$956$	0	0
$957$	6.67585	0.215799
$958$	0	0
$959$	18.0258	0.582084
$960$	0	0
$961$	22.0064	0.709884
$962$	0	0
$963$	−2.28403	−0.0736017
$964$	0	0
$965$	−39.2241	−1.26267
$966$	0	0
$967$	−1.94143	−0.0624323	−0.0312162	−	0.999513i	$-0.509938\pi$
$967$	−1.94143	−0.0624323	−0.0312162	+	0.999513i	$0.509938\pi$
$968$	0	0
$969$	−10.6483	−0.342073
$970$	0	0
$971$	14.9077	0.478412	0.239206	−	0.970969i	$-0.423113\pi$
$971$	14.9077	0.478412	0.239206	+	0.970969i	$0.423113\pi$
$972$	0	0
$973$	2.67585	0.0857837
$974$	0	0
$975$	−2.08666	−0.0668265
$976$	0	0
$977$	−16.5353	−0.529011	−0.264505	−	0.964384i	$-0.585209\pi$
$977$	−16.5353	−0.529011	−0.264505	+	0.964384i	$0.585209\pi$
$978$	0	0
$979$	17.5456	0.560759
$980$	0	0
$981$	14.6689	0.468342
$982$	0	0
$983$	30.0694	0.959065	0.479533	−	0.877524i	$-0.340806\pi$
$983$	30.0694	0.959065	0.479533	+	0.877524i	$0.340806\pi$
$984$	0	0
$985$	−38.6023	−1.22997
$986$	0	0
$987$	2.09695	0.0667465
$988$	0	0
$989$	5.33912	0.169774
$990$	0	0
$991$	−43.5902	−1.38469	−0.692345	−	0.721567i	$-0.743422\pi$
$991$	−43.5902	−1.38469	−0.692345	+	0.721567i	$0.743422\pi$
$992$	0	0
$993$	−13.8267	−0.438777
$994$	0	0
$995$	−35.9861	−1.14084
$996$	0	0
$997$	57.8577	1.83237	0.916186	−	0.400753i	$-0.131251\pi$
$997$	57.8577	1.83237	0.916186	+	0.400753i	$0.131251\pi$
$998$	0	0
$999$	−10.4750	−0.331414

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4368.2.a.br.1.2		4
4.3	odd	2		273.2.a.e.1.1	✓	4
12.11	even	2		819.2.a.k.1.4		4
20.19	odd	2		6825.2.a.bg.1.4		4
28.27	even	2		1911.2.a.s.1.1		4
52.51	odd	2		3549.2.a.w.1.4		4
84.83	odd	2		5733.2.a.bf.1.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
273.2.a.e.1.1	✓	4	4.3	odd	2
819.2.a.k.1.4		4	12.11	even	2
1911.2.a.s.1.1		4	28.27	even	2
3549.2.a.w.1.4		4	52.51	odd	2
4368.2.a.br.1.2		4	1.1	even	1	trivial
5733.2.a.bf.1.4		4	84.83	odd	2
6825.2.a.bg.1.4		4	20.19	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:	\( N \)	\(=\)	\( 4368 = 2^{4} \cdot 3 \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4368.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -2.66208 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} -2.66208 q^{5} -1.00000 q^{7} +1.00000 q^{9} -1.57542 q^{11} +1.00000 q^{13} +2.66208 q^{15} +4.75902 q^{17} +2.23750 q^{19} +1.00000 q^{21} -5.84568 q^{23} +2.08666 q^{25} -1.00000 q^{27} +4.23750 q^{29} -7.28055 q^{31} +1.57542 q^{33} +2.66208 q^{35} +10.4750 q^{37} -1.00000 q^{39} -2.25127 q^{41} -0.913344 q^{43} -2.66208 q^{45} +2.09695 q^{47} +1.00000 q^{49} -4.75902 q^{51} +1.08666 q^{53} +4.19389 q^{55} -2.23750 q^{57} +12.3344 q^{59} -7.51805 q^{61} -1.00000 q^{63} -2.66208 q^{65} +15.6914 q^{67} +5.84568 q^{69} -10.0504 q^{71} +15.0797 q^{73} -2.08666 q^{75} +1.57542 q^{77} +11.7555 q^{79} +1.00000 q^{81} -7.42110 q^{83} -12.6689 q^{85} -4.23750 q^{87} -11.1371 q^{89} -1.00000 q^{91} +7.28055 q^{93} -5.95639 q^{95} -14.6047 q^{97} -1.57542 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{3} - 3 q^{5} - 4 q^{7} + 4 q^{9}+O(q^{10}) \) 4 * q - 4 * q^3 - 3 * q^5 - 4 * q^7 + 4 * q^9	\( 4 q - 4 q^{3} - 3 q^{5} - 4 q^{7} + 4 q^{9} + 2 q^{11} + 4 q^{13} + 3 q^{15} - 2 q^{17} - 7 q^{19} + 4 q^{21} - 3 q^{23} + 9 q^{25} - 4 q^{27} + q^{29} - 3 q^{31} - 2 q^{33} + 3 q^{35} + 10 q^{37} - 4 q^{39} - 16 q^{41} - 3 q^{43} - 3 q^{45} - 5 q^{47} + 4 q^{49} + 2 q^{51} + 5 q^{53} - 10 q^{55} + 7 q^{57} + 20 q^{59} + 12 q^{61} - 4 q^{63} - 3 q^{65} + 22 q^{67} + 3 q^{69} - 13 q^{73} - 9 q^{75} - 2 q^{77} - 11 q^{79} + 4 q^{81} - q^{83} + 8 q^{85} - q^{87} - 5 q^{89} - 4 q^{91} + 3 q^{93} - 13 q^{95} - 17 q^{97} + 2 q^{99}+O(q^{100}) \) 4 * q - 4 * q^3 - 3 * q^5 - 4 * q^7 + 4 * q^9 + 2 * q^11 + 4 * q^13 + 3 * q^15 - 2 * q^17 - 7 * q^19 + 4 * q^21 - 3 * q^23 + 9 * q^25 - 4 * q^27 + q^29 - 3 * q^31 - 2 * q^33 + 3 * q^35 + 10 * q^37 - 4 * q^39 - 16 * q^41 - 3 * q^43 - 3 * q^45 - 5 * q^47 + 4 * q^49 + 2 * q^51 + 5 * q^53 - 10 * q^55 + 7 * q^57 + 20 * q^59 + 12 * q^61 - 4 * q^63 - 3 * q^65 + 22 * q^67 + 3 * q^69 - 13 * q^73 - 9 * q^75 - 2 * q^77 - 11 * q^79 + 4 * q^81 - q^83 + 8 * q^85 - q^87 - 5 * q^89 - 4 * q^91 + 3 * q^93 - 13 * q^95 - 17 * q^97 + 2 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more