LMFDB - Embedded newform 6160.2.a.bm.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6160,2,Mod(1,6160)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6160, 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("6160.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6160 = 2^{4} \cdot 5 \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6160.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:	$49.1878476451$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.148.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 3x + 1 $ x^3 - x^2 - 3*x + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 3080)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.17009$ of defining polynomial
Character	$\chi$	$=$	6160.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.70928 q^{3} -1.00000 q^{5} -1.00000 q^{7} -0.0783777 q^{9} +O(q^{10})$	$q-1.70928 q^{3} -1.00000 q^{5} -1.00000 q^{7} -0.0783777 q^{9} +1.00000 q^{11} -3.70928 q^{13} +1.70928 q^{15} +3.70928 q^{17} +5.26180 q^{19} +1.70928 q^{21} +2.34017 q^{23} +1.00000 q^{25} +5.26180 q^{27} -5.41855 q^{29} -8.04945 q^{31} -1.70928 q^{33} +1.00000 q^{35} -3.07838 q^{37} +6.34017 q^{39} -6.04945 q^{41} +8.49693 q^{43} +0.0783777 q^{45} -10.3896 q^{47} +1.00000 q^{49} -6.34017 q^{51} -7.75872 q^{53} -1.00000 q^{55} -8.99386 q^{57} +3.36910 q^{59} -7.89269 q^{61} +0.0783777 q^{63} +3.70928 q^{65} +3.81658 q^{67} -4.00000 q^{69} +3.41855 q^{71} -1.86603 q^{73} -1.70928 q^{75} -1.00000 q^{77} -13.1773 q^{79} -8.75872 q^{81} +2.73820 q^{83} -3.70928 q^{85} +9.26180 q^{87} +0.523590 q^{89} +3.70928 q^{91} +13.7587 q^{93} -5.26180 q^{95} -10.0989 q^{97} -0.0783777 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 2 q^{3} - 3 q^{5} - 3 q^{7} + 3 q^{9}+O(q^{10}) $ 3 * q + 2 * q^3 - 3 * q^5 - 3 * q^7 + 3 * q^9	$ 3 q + 2 q^{3} - 3 q^{5} - 3 q^{7} + 3 q^{9} + 3 q^{11} - 4 q^{13} - 2 q^{15} + 4 q^{17} + 8 q^{19} - 2 q^{21} - 4 q^{23} + 3 q^{25} + 8 q^{27} - 2 q^{29} - 6 q^{31} + 2 q^{33} + 3 q^{35} - 6 q^{37} + 8 q^{39} + 8 q^{43} - 3 q^{45} - 2 q^{47} + 3 q^{49} - 8 q^{51} + 2 q^{53} - 3 q^{55} + 8 q^{57} + 14 q^{59} - 12 q^{61} - 3 q^{63} + 4 q^{65} + 16 q^{67} - 12 q^{69} - 4 q^{71} + 8 q^{73} + 2 q^{75} - 3 q^{77} - q^{81} + 16 q^{83} - 4 q^{85} + 20 q^{87} - 14 q^{89} + 4 q^{91} + 16 q^{93} - 8 q^{95} + 6 q^{97} + 3 q^{99}+O(q^{100}) $ 3 * q + 2 * q^3 - 3 * q^5 - 3 * q^7 + 3 * q^9 + 3 * q^11 - 4 * q^13 - 2 * q^15 + 4 * q^17 + 8 * q^19 - 2 * q^21 - 4 * q^23 + 3 * q^25 + 8 * q^27 - 2 * q^29 - 6 * q^31 + 2 * q^33 + 3 * q^35 - 6 * q^37 + 8 * q^39 + 8 * q^43 - 3 * q^45 - 2 * q^47 + 3 * q^49 - 8 * q^51 + 2 * q^53 - 3 * q^55 + 8 * q^57 + 14 * q^59 - 12 * q^61 - 3 * q^63 + 4 * q^65 + 16 * q^67 - 12 * q^69 - 4 * q^71 + 8 * q^73 + 2 * q^75 - 3 * q^77 - q^81 + 16 * q^83 - 4 * q^85 + 20 * q^87 - 14 * q^89 + 4 * q^91 + 16 * q^93 - 8 * q^95 + 6 * q^97 + 3 * 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
$3$	−1.70928	−0.986851	−0.493425	−	0.869788i	$-0.664255\pi$
$3$	−1.70928	−0.986851	−0.493425	+	0.869788i	$0.664255\pi$
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	−0.0783777	−0.0261259
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	−3.70928	−1.02877	−0.514384	−	0.857560i	$-0.671979\pi$
$13$	−3.70928	−1.02877	−0.514384	+	0.857560i	$0.671979\pi$
$14$	0	0
$15$	1.70928	0.441333
$16$	0	0
$17$	3.70928	0.899631	0.449816	−	0.893121i	$-0.351490\pi$
$17$	3.70928	0.899631	0.449816	+	0.893121i	$0.351490\pi$
$18$	0	0
$19$	5.26180	1.20714	0.603569	−	0.797311i	$-0.293745\pi$
$19$	5.26180	1.20714	0.603569	+	0.797311i	$0.293745\pi$
$20$	0	0
$21$	1.70928	0.372994
$22$	0	0
$23$	2.34017	0.487960	0.243980	−	0.969780i	$-0.421547\pi$
$23$	2.34017	0.487960	0.243980	+	0.969780i	$0.421547\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	5.26180	1.01263
$28$	0	0
$29$	−5.41855	−1.00620	−0.503100	−	0.864228i	$-0.667807\pi$
$29$	−5.41855	−1.00620	−0.503100	+	0.864228i	$0.667807\pi$
$30$	0	0
$31$	−8.04945	−1.44572	−0.722862	−	0.690993i	$-0.757174\pi$
$31$	−8.04945	−1.44572	−0.722862	+	0.690993i	$0.757174\pi$
$32$	0	0
$33$	−1.70928	−0.297547
$34$	0	0
$35$	1.00000	0.169031
$36$	0	0
$37$	−3.07838	−0.506082	−0.253041	−	0.967456i	$-0.581431\pi$
$37$	−3.07838	−0.506082	−0.253041	+	0.967456i	$0.581431\pi$
$38$	0	0
$39$	6.34017	1.01524
$40$	0	0
$41$	−6.04945	−0.944765	−0.472383	−	0.881394i	$-0.656606\pi$
$41$	−6.04945	−0.944765	−0.472383	+	0.881394i	$0.656606\pi$
$42$	0	0
$43$	8.49693	1.29577	0.647885	−	0.761738i	$-0.275654\pi$
$43$	8.49693	1.29577	0.647885	+	0.761738i	$0.275654\pi$
$44$	0	0
$45$	0.0783777	0.0116839
$46$	0	0
$47$	−10.3896	−1.51548	−0.757741	−	0.652555i	$-0.773697\pi$
$47$	−10.3896	−1.51548	−0.757741	+	0.652555i	$0.773697\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−6.34017	−0.887802
$52$	0	0
$53$	−7.75872	−1.06574	−0.532871	−	0.846196i	$-0.678887\pi$
$53$	−7.75872	−1.06574	−0.532871	+	0.846196i	$0.678887\pi$
$54$	0	0
$55$	−1.00000	−0.134840
$56$	0	0
$57$	−8.99386	−1.19127
$58$	0	0
$59$	3.36910	0.438620	0.219310	−	0.975655i	$-0.429619\pi$
$59$	3.36910	0.438620	0.219310	+	0.975655i	$0.429619\pi$
$60$	0	0
$61$	−7.89269	−1.01056	−0.505278	−	0.862957i	$-0.668610\pi$
$61$	−7.89269	−1.01056	−0.505278	+	0.862957i	$0.668610\pi$
$62$	0	0
$63$	0.0783777	0.00987467
$64$	0	0
$65$	3.70928	0.460079
$66$	0	0
$67$	3.81658	0.466270	0.233135	−	0.972444i	$-0.425102\pi$
$67$	3.81658	0.466270	0.233135	+	0.972444i	$0.425102\pi$
$68$	0	0
$69$	−4.00000	−0.481543
$70$	0	0
$71$	3.41855	0.405707	0.202854	−	0.979209i	$-0.434978\pi$
$71$	3.41855	0.405707	0.202854	+	0.979209i	$0.434978\pi$
$72$	0	0
$73$	−1.86603	−0.218402	−0.109201	−	0.994020i	$-0.534829\pi$
$73$	−1.86603	−0.218402	−0.109201	+	0.994020i	$0.534829\pi$
$74$	0	0
$75$	−1.70928	−0.197370
$76$	0	0
$77$	−1.00000	−0.113961
$78$	0	0
$79$	−13.1773	−1.48256	−0.741280	−	0.671196i	$-0.765781\pi$
$79$	−13.1773	−1.48256	−0.741280	+	0.671196i	$0.765781\pi$
$80$	0	0
$81$	−8.75872	−0.973192
$82$	0	0
$83$	2.73820	0.300557	0.150279	−	0.988644i	$-0.451983\pi$
$83$	2.73820	0.300557	0.150279	+	0.988644i	$0.451983\pi$
$84$	0	0
$85$	−3.70928	−0.402327
$86$	0	0
$87$	9.26180	0.992969
$88$	0	0
$89$	0.523590	0.0555005	0.0277502	−	0.999615i	$-0.491166\pi$
$89$	0.523590	0.0555005	0.0277502	+	0.999615i	$0.491166\pi$
$90$	0	0
$91$	3.70928	0.388838
$92$	0	0
$93$	13.7587	1.42671
$94$	0	0
$95$	−5.26180	−0.539849
$96$	0	0
$97$	−10.0989	−1.02539	−0.512694	−	0.858572i	$-0.671352\pi$
$97$	−10.0989	−1.02539	−0.512694	+	0.858572i	$0.671352\pi$
$98$	0	0
$99$	−0.0783777	−0.00787726
$100$	0	0
$101$	−4.78765	−0.476389	−0.238195	−	0.971217i	$-0.576556\pi$
$101$	−4.78765	−0.476389	−0.238195	+	0.971217i	$0.576556\pi$
$102$	0	0
$103$	13.1278	1.29352	0.646762	−	0.762692i	$-0.276123\pi$
$103$	13.1278	1.29352	0.646762	+	0.762692i	$0.276123\pi$
$104$	0	0
$105$	−1.70928	−0.166808
$106$	0	0
$107$	4.36683	0.422158	0.211079	−	0.977469i	$-0.432302\pi$
$107$	4.36683	0.422158	0.211079	+	0.977469i	$0.432302\pi$
$108$	0	0
$109$	8.25565	0.790748	0.395374	−	0.918520i	$-0.370615\pi$
$109$	8.25565	0.790748	0.395374	+	0.918520i	$0.370615\pi$
$110$	0	0
$111$	5.26180	0.499428
$112$	0	0
$113$	14.6803	1.38101	0.690505	−	0.723327i	$-0.257388\pi$
$113$	14.6803	1.38101	0.690505	+	0.723327i	$0.257388\pi$
$114$	0	0
$115$	−2.34017	−0.218222
$116$	0	0
$117$	0.290725	0.0268775
$118$	0	0
$119$	−3.70928	−0.340029
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	10.3402	0.932342
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−6.83710	−0.606695	−0.303347	−	0.952880i	$-0.598104\pi$
$127$	−6.83710	−0.606695	−0.303347	+	0.952880i	$0.598104\pi$
$128$	0	0
$129$	−14.5236	−1.27873
$130$	0	0
$131$	8.00000	0.698963	0.349482	−	0.936943i	$-0.386358\pi$
$131$	8.00000	0.698963	0.349482	+	0.936943i	$0.386358\pi$
$132$	0	0
$133$	−5.26180	−0.456256
$134$	0	0
$135$	−5.26180	−0.452863
$136$	0	0
$137$	−1.60197	−0.136865	−0.0684327	−	0.997656i	$-0.521800\pi$
$137$	−1.60197	−0.136865	−0.0684327	+	0.997656i	$0.521800\pi$
$138$	0	0
$139$	−8.09890	−0.686939	−0.343470	−	0.939164i	$-0.611602\pi$
$139$	−8.09890	−0.686939	−0.343470	+	0.939164i	$0.611602\pi$
$140$	0	0
$141$	17.7587	1.49555
$142$	0	0
$143$	−3.70928	−0.310185
$144$	0	0
$145$	5.41855	0.449986
$146$	0	0
$147$	−1.70928	−0.140979
$148$	0	0
$149$	0.156755	0.0128419	0.00642096	−	0.999979i	$-0.497956\pi$
$149$	0.156755	0.0128419	0.00642096	+	0.999979i	$0.497956\pi$
$150$	0	0
$151$	3.81658	0.310589	0.155295	−	0.987868i	$-0.450367\pi$
$151$	3.81658	0.310589	0.155295	+	0.987868i	$0.450367\pi$
$152$	0	0
$153$	−0.290725	−0.0235037
$154$	0	0
$155$	8.04945	0.646547
$156$	0	0
$157$	9.51745	0.759575	0.379787	−	0.925074i	$-0.375997\pi$
$157$	9.51745	0.759575	0.379787	+	0.925074i	$0.375997\pi$
$158$	0	0
$159$	13.2618	1.05173
$160$	0	0
$161$	−2.34017	−0.184431
$162$	0	0
$163$	19.0205	1.48980	0.744901	−	0.667175i	$-0.232497\pi$
$163$	19.0205	1.48980	0.744901	+	0.667175i	$0.232497\pi$
$164$	0	0
$165$	1.70928	0.133067
$166$	0	0
$167$	4.36683	0.337916	0.168958	−	0.985623i	$-0.445960\pi$
$167$	4.36683	0.337916	0.168958	+	0.985623i	$0.445960\pi$
$168$	0	0
$169$	0.758724	0.0583634
$170$	0	0
$171$	−0.412408	−0.0315376
$172$	0	0
$173$	11.8082	0.897759	0.448879	−	0.893592i	$-0.351823\pi$
$173$	11.8082	0.897759	0.448879	+	0.893592i	$0.351823\pi$
$174$	0	0
$175$	−1.00000	−0.0755929
$176$	0	0
$177$	−5.75872	−0.432852
$178$	0	0
$179$	9.84324	0.735719	0.367859	−	0.929881i	$-0.380091\pi$
$179$	9.84324	0.735719	0.367859	+	0.929881i	$0.380091\pi$
$180$	0	0
$181$	7.26180	0.539765	0.269882	−	0.962893i	$-0.413015\pi$
$181$	7.26180	0.539765	0.269882	+	0.962893i	$0.413015\pi$
$182$	0	0
$183$	13.4908	0.997268
$184$	0	0
$185$	3.07838	0.226327
$186$	0	0
$187$	3.70928	0.271249
$188$	0	0
$189$	−5.26180	−0.382739
$190$	0	0
$191$	4.31351	0.312115	0.156057	−	0.987748i	$-0.450122\pi$
$191$	4.31351	0.312115	0.156057	+	0.987748i	$0.450122\pi$
$192$	0	0
$193$	0.921622	0.0663398	0.0331699	−	0.999450i	$-0.489440\pi$
$193$	0.921622	0.0663398	0.0331699	+	0.999450i	$0.489440\pi$
$194$	0	0
$195$	−6.34017	−0.454029
$196$	0	0
$197$	4.92162	0.350651	0.175326	−	0.984511i	$-0.443902\pi$
$197$	4.92162	0.350651	0.175326	+	0.984511i	$0.443902\pi$
$198$	0	0
$199$	−9.99159	−0.708285	−0.354143	−	0.935191i	$-0.615227\pi$
$199$	−9.99159	−0.708285	−0.354143	+	0.935191i	$0.615227\pi$
$200$	0	0
$201$	−6.52359	−0.460139
$202$	0	0
$203$	5.41855	0.380308
$204$	0	0
$205$	6.04945	0.422512
$206$	0	0
$207$	−0.183417	−0.0127484
$208$	0	0
$209$	5.26180	0.363966
$210$	0	0
$211$	16.6803	1.14832	0.574161	−	0.818742i	$-0.305328\pi$
$211$	16.6803	1.14832	0.574161	+	0.818742i	$0.305328\pi$
$212$	0	0
$213$	−5.84324	−0.400373
$214$	0	0
$215$	−8.49693	−0.579486
$216$	0	0
$217$	8.04945	0.546432
$218$	0	0
$219$	3.18956	0.215531
$220$	0	0
$221$	−13.7587	−0.925512
$222$	0	0
$223$	18.9711	1.27040	0.635198	−	0.772349i	$-0.280918\pi$
$223$	18.9711	1.27040	0.635198	+	0.772349i	$0.280918\pi$
$224$	0	0
$225$	−0.0783777	−0.00522518
$226$	0	0
$227$	−2.52359	−0.167497	−0.0837483	−	0.996487i	$-0.526689\pi$
$227$	−2.52359	−0.167497	−0.0837483	+	0.996487i	$0.526689\pi$
$228$	0	0
$229$	4.25565	0.281221	0.140611	−	0.990065i	$-0.455093\pi$
$229$	4.25565	0.281221	0.140611	+	0.990065i	$0.455093\pi$
$230$	0	0
$231$	1.70928	0.112462
$232$	0	0
$233$	9.81658	0.643106	0.321553	−	0.946892i	$-0.395795\pi$
$233$	9.81658	0.643106	0.321553	+	0.946892i	$0.395795\pi$
$234$	0	0
$235$	10.3896	0.677744
$236$	0	0
$237$	22.5236	1.46306
$238$	0	0
$239$	−23.2039	−1.50094	−0.750469	−	0.660906i	$-0.770172\pi$
$239$	−23.2039	−1.50094	−0.750469	+	0.660906i	$0.770172\pi$
$240$	0	0
$241$	−9.73594	−0.627147	−0.313573	−	0.949564i	$-0.601526\pi$
$241$	−9.73594	−0.627147	−0.313573	+	0.949564i	$0.601526\pi$
$242$	0	0
$243$	−0.814315	−0.0522383
$244$	0	0
$245$	−1.00000	−0.0638877
$246$	0	0
$247$	−19.5174	−1.24187
$248$	0	0
$249$	−4.68035	−0.296605
$250$	0	0
$251$	−25.4101	−1.60387	−0.801937	−	0.597409i	$-0.796197\pi$
$251$	−25.4101	−1.60387	−0.801937	+	0.597409i	$0.796197\pi$
$252$	0	0
$253$	2.34017	0.147125
$254$	0	0
$255$	6.34017	0.397037
$256$	0	0
$257$	−0.837101	−0.0522170	−0.0261085	−	0.999659i	$-0.508312\pi$
$257$	−0.837101	−0.0522170	−0.0261085	+	0.999659i	$0.508312\pi$
$258$	0	0
$259$	3.07838	0.191281
$260$	0	0
$261$	0.424694	0.0262879
$262$	0	0
$263$	8.00000	0.493301	0.246651	−	0.969104i	$-0.420670\pi$
$263$	8.00000	0.493301	0.246651	+	0.969104i	$0.420670\pi$
$264$	0	0
$265$	7.75872	0.476615
$266$	0	0
$267$	−0.894960	−0.0547707
$268$	0	0
$269$	6.09890	0.371856	0.185928	−	0.982563i	$-0.440471\pi$
$269$	6.09890	0.371856	0.185928	+	0.982563i	$0.440471\pi$
$270$	0	0
$271$	29.1461	1.77050	0.885249	−	0.465117i	$-0.153988\pi$
$271$	29.1461	1.77050	0.885249	+	0.465117i	$0.153988\pi$
$272$	0	0
$273$	−6.34017	−0.383725
$274$	0	0
$275$	1.00000	0.0603023
$276$	0	0
$277$	15.4452	0.928013	0.464006	−	0.885832i	$-0.346411\pi$
$277$	15.4452	0.928013	0.464006	+	0.885832i	$0.346411\pi$
$278$	0	0
$279$	0.630898	0.0377709
$280$	0	0
$281$	−7.26180	−0.433202	−0.216601	−	0.976260i	$-0.569497\pi$
$281$	−7.26180	−0.433202	−0.216601	+	0.976260i	$0.569497\pi$
$282$	0	0
$283$	14.8371	0.881974	0.440987	−	0.897513i	$-0.354628\pi$
$283$	14.8371	0.881974	0.440987	+	0.897513i	$0.354628\pi$
$284$	0	0
$285$	8.99386	0.532750
$286$	0	0
$287$	6.04945	0.357088
$288$	0	0
$289$	−3.24128	−0.190663
$290$	0	0
$291$	17.2618	1.01190
$292$	0	0
$293$	−12.1217	−0.708156	−0.354078	−	0.935216i	$-0.615205\pi$
$293$	−12.1217	−0.708156	−0.354078	+	0.935216i	$0.615205\pi$
$294$	0	0
$295$	−3.36910	−0.196157
$296$	0	0
$297$	5.26180	0.305320
$298$	0	0
$299$	−8.68035	−0.501997
$300$	0	0
$301$	−8.49693	−0.489755
$302$	0	0
$303$	8.18342	0.470125
$304$	0	0
$305$	7.89269	0.451934
$306$	0	0
$307$	1.47641	0.0842631	0.0421316	−	0.999112i	$-0.486585\pi$
$307$	1.47641	0.0842631	0.0421316	+	0.999112i	$0.486585\pi$
$308$	0	0
$309$	−22.4391	−1.27651
$310$	0	0
$311$	−12.1483	−0.688869	−0.344435	−	0.938810i	$-0.611929\pi$
$311$	−12.1483	−0.688869	−0.344435	+	0.938810i	$0.611929\pi$
$312$	0	0
$313$	−9.68649	−0.547513	−0.273756	−	0.961799i	$-0.588266\pi$
$313$	−9.68649	−0.547513	−0.273756	+	0.961799i	$0.588266\pi$
$314$	0	0
$315$	−0.0783777	−0.00441609
$316$	0	0
$317$	33.1506	1.86192	0.930962	−	0.365116i	$-0.118971\pi$
$317$	33.1506	1.86192	0.930962	+	0.365116i	$0.118971\pi$
$318$	0	0
$319$	−5.41855	−0.303381
$320$	0	0
$321$	−7.46412	−0.416607
$322$	0	0
$323$	19.5174	1.08598
$324$	0	0
$325$	−3.70928	−0.205754
$326$	0	0
$327$	−14.1112	−0.780350
$328$	0	0
$329$	10.3896	0.572798
$330$	0	0
$331$	−19.2039	−1.05554	−0.527772	−	0.849386i	$-0.676972\pi$
$331$	−19.2039	−1.05554	−0.527772	+	0.849386i	$0.676972\pi$
$332$	0	0
$333$	0.241276	0.0132219
$334$	0	0
$335$	−3.81658	−0.208522
$336$	0	0
$337$	−7.54411	−0.410954	−0.205477	−	0.978662i	$-0.565875\pi$
$337$	−7.54411	−0.410954	−0.205477	+	0.978662i	$0.565875\pi$
$338$	0	0
$339$	−25.0928	−1.36285
$340$	0	0
$341$	−8.04945	−0.435902
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	4.00000	0.215353
$346$	0	0
$347$	−5.17727	−0.277931	−0.138965	−	0.990297i	$-0.544378\pi$
$347$	−5.17727	−0.277931	−0.138965	+	0.990297i	$0.544378\pi$
$348$	0	0
$349$	−17.2534	−0.923553	−0.461776	−	0.886996i	$-0.652788\pi$
$349$	−17.2534	−0.923553	−0.461776	+	0.886996i	$0.652788\pi$
$350$	0	0
$351$	−19.5174	−1.04176
$352$	0	0
$353$	13.5174	0.719461	0.359731	−	0.933056i	$-0.382869\pi$
$353$	13.5174	0.719461	0.359731	+	0.933056i	$0.382869\pi$
$354$	0	0
$355$	−3.41855	−0.181438
$356$	0	0
$357$	6.34017	0.335558
$358$	0	0
$359$	8.86376	0.467812	0.233906	−	0.972259i	$-0.424849\pi$
$359$	8.86376	0.467812	0.233906	+	0.972259i	$0.424849\pi$
$360$	0	0
$361$	8.68649	0.457184
$362$	0	0
$363$	−1.70928	−0.0897137
$364$	0	0
$365$	1.86603	0.0976725
$366$	0	0
$367$	6.97107	0.363887	0.181943	−	0.983309i	$-0.441761\pi$
$367$	6.97107	0.363887	0.181943	+	0.983309i	$0.441761\pi$
$368$	0	0
$369$	0.474142	0.0246829
$370$	0	0
$371$	7.75872	0.402813
$372$	0	0
$373$	23.2762	1.20519	0.602597	−	0.798045i	$-0.294133\pi$
$373$	23.2762	1.20519	0.602597	+	0.798045i	$0.294133\pi$
$374$	0	0
$375$	1.70928	0.0882666
$376$	0	0
$377$	20.0989	1.03515
$378$	0	0
$379$	−0.581449	−0.0298670	−0.0149335	−	0.999888i	$-0.504754\pi$
$379$	−0.581449	−0.0298670	−0.0149335	+	0.999888i	$0.504754\pi$
$380$	0	0
$381$	11.6865	0.598717
$382$	0	0
$383$	−30.1217	−1.53915	−0.769573	−	0.638559i	$-0.779531\pi$
$383$	−30.1217	−1.53915	−0.769573	+	0.638559i	$0.779531\pi$
$384$	0	0
$385$	1.00000	0.0509647
$386$	0	0
$387$	−0.665970	−0.0338532
$388$	0	0
$389$	16.5236	0.837779	0.418890	−	0.908037i	$-0.362419\pi$
$389$	16.5236	0.837779	0.418890	+	0.908037i	$0.362419\pi$
$390$	0	0
$391$	8.68035	0.438984
$392$	0	0
$393$	−13.6742	−0.689772
$394$	0	0
$395$	13.1773	0.663021
$396$	0	0
$397$	24.3545	1.22232	0.611160	−	0.791507i	$-0.290703\pi$
$397$	24.3545	1.22232	0.611160	+	0.791507i	$0.290703\pi$
$398$	0	0
$399$	8.99386	0.450256
$400$	0	0
$401$	28.5236	1.42440	0.712200	−	0.701977i	$-0.247699\pi$
$401$	28.5236	1.42440	0.712200	+	0.701977i	$0.247699\pi$
$402$	0	0
$403$	29.8576	1.48731
$404$	0	0
$405$	8.75872	0.435224
$406$	0	0
$407$	−3.07838	−0.152590
$408$	0	0
$409$	−13.2702	−0.656169	−0.328085	−	0.944648i	$-0.606403\pi$
$409$	−13.2702	−0.656169	−0.328085	+	0.944648i	$0.606403\pi$
$410$	0	0
$411$	2.73820	0.135066
$412$	0	0
$413$	−3.36910	−0.165783
$414$	0	0
$415$	−2.73820	−0.134413
$416$	0	0
$417$	13.8432	0.677907
$418$	0	0
$419$	−16.6309	−0.812473	−0.406236	−	0.913768i	$-0.633159\pi$
$419$	−16.6309	−0.812473	−0.406236	+	0.913768i	$0.633159\pi$
$420$	0	0
$421$	29.4329	1.43447	0.717237	−	0.696830i	$-0.245407\pi$
$421$	29.4329	1.43447	0.717237	+	0.696830i	$0.245407\pi$
$422$	0	0
$423$	0.814315	0.0395934
$424$	0	0
$425$	3.70928	0.179926
$426$	0	0
$427$	7.89269	0.381954
$428$	0	0
$429$	6.34017	0.306106
$430$	0	0
$431$	40.6947	1.96020	0.980098	−	0.198515i	$-0.0636118\pi$
$431$	40.6947	1.96020	0.980098	+	0.198515i	$0.0636118\pi$
$432$	0	0
$433$	−13.1050	−0.629788	−0.314894	−	0.949127i	$-0.601969\pi$
$433$	−13.1050	−0.629788	−0.314894	+	0.949127i	$0.601969\pi$
$434$	0	0
$435$	−9.26180	−0.444069
$436$	0	0
$437$	12.3135	0.589035
$438$	0	0
$439$	30.5692	1.45899	0.729493	−	0.683988i	$-0.239756\pi$
$439$	30.5692	1.45899	0.729493	+	0.683988i	$0.239756\pi$
$440$	0	0
$441$	−0.0783777	−0.00373227
$442$	0	0
$443$	−13.4452	−0.638801	−0.319401	−	0.947620i	$-0.603482\pi$
$443$	−13.4452	−0.638801	−0.319401	+	0.947620i	$0.603482\pi$
$444$	0	0
$445$	−0.523590	−0.0248206
$446$	0	0
$447$	−0.267938	−0.0126730
$448$	0	0
$449$	20.3857	0.962063	0.481031	−	0.876703i	$-0.340262\pi$
$449$	20.3857	0.962063	0.481031	+	0.876703i	$0.340262\pi$
$450$	0	0
$451$	−6.04945	−0.284857
$452$	0	0
$453$	−6.52359	−0.306505
$454$	0	0
$455$	−3.70928	−0.173894
$456$	0	0
$457$	−8.43907	−0.394763	−0.197382	−	0.980327i	$-0.563244\pi$
$457$	−8.43907	−0.394763	−0.197382	+	0.980327i	$0.563244\pi$
$458$	0	0
$459$	19.5174	0.910996
$460$	0	0
$461$	−19.0966	−0.889419	−0.444709	−	0.895675i	$-0.646693\pi$
$461$	−19.0966	−0.889419	−0.444709	+	0.895675i	$0.646693\pi$
$462$	0	0
$463$	37.0616	1.72240	0.861198	−	0.508269i	$-0.169714\pi$
$463$	37.0616	1.72240	0.861198	+	0.508269i	$0.169714\pi$
$464$	0	0
$465$	−13.7587	−0.638046
$466$	0	0
$467$	23.0166	1.06508	0.532542	−	0.846404i	$-0.321237\pi$
$467$	23.0166	1.06508	0.532542	+	0.846404i	$0.321237\pi$
$468$	0	0
$469$	−3.81658	−0.176233
$470$	0	0
$471$	−16.2679	−0.749587
$472$	0	0
$473$	8.49693	0.390689
$474$	0	0
$475$	5.26180	0.241428
$476$	0	0
$477$	0.608111	0.0278435
$478$	0	0
$479$	28.4657	1.30063	0.650316	−	0.759664i	$-0.274636\pi$
$479$	28.4657	1.30063	0.650316	+	0.759664i	$0.274636\pi$
$480$	0	0
$481$	11.4186	0.520641
$482$	0	0
$483$	4.00000	0.182006
$484$	0	0
$485$	10.0989	0.458567
$486$	0	0
$487$	15.3874	0.697268	0.348634	−	0.937259i	$-0.386646\pi$
$487$	15.3874	0.697268	0.348634	+	0.937259i	$0.386646\pi$
$488$	0	0
$489$	−32.5113	−1.47021
$490$	0	0
$491$	25.1773	1.13623	0.568117	−	0.822948i	$-0.307672\pi$
$491$	25.1773	1.13623	0.568117	+	0.822948i	$0.307672\pi$
$492$	0	0
$493$	−20.0989	−0.905209
$494$	0	0
$495$	0.0783777	0.00352282
$496$	0	0
$497$	−3.41855	−0.153343
$498$	0	0
$499$	44.2967	1.98299	0.991496	−	0.130136i	$-0.0415415\pi$
$499$	44.2967	1.98299	0.991496	+	0.130136i	$0.0415415\pi$
$500$	0	0
$501$	−7.46412	−0.333472
$502$	0	0
$503$	−24.5814	−1.09603	−0.548016	−	0.836468i	$-0.684617\pi$
$503$	−24.5814	−1.09603	−0.548016	+	0.836468i	$0.684617\pi$
$504$	0	0
$505$	4.78765	0.213048
$506$	0	0
$507$	−1.29687	−0.0575959
$508$	0	0
$509$	14.7792	0.655078	0.327539	−	0.944838i	$-0.393781\pi$
$509$	14.7792	0.655078	0.327539	+	0.944838i	$0.393781\pi$
$510$	0	0
$511$	1.86603	0.0825483
$512$	0	0
$513$	27.6865	1.22239
$514$	0	0
$515$	−13.1278	−0.578481
$516$	0	0
$517$	−10.3896	−0.456935
$518$	0	0
$519$	−20.1834	−0.885954
$520$	0	0
$521$	37.9299	1.66174	0.830869	−	0.556469i	$-0.187844\pi$
$521$	37.9299	1.66174	0.830869	+	0.556469i	$0.187844\pi$
$522$	0	0
$523$	−2.04104	−0.0892484	−0.0446242	−	0.999004i	$-0.514209\pi$
$523$	−2.04104	−0.0892484	−0.0446242	+	0.999004i	$0.514209\pi$
$524$	0	0
$525$	1.70928	0.0745989
$526$	0	0
$527$	−29.8576	−1.30062
$528$	0	0
$529$	−17.5236	−0.761895
$530$	0	0
$531$	−0.264063	−0.0114593
$532$	0	0
$533$	22.4391	0.971944
$534$	0	0
$535$	−4.36683	−0.188795
$536$	0	0
$537$	−16.8248	−0.726044
$538$	0	0
$539$	1.00000	0.0430730
$540$	0	0
$541$	21.2039	0.911628	0.455814	−	0.890075i	$-0.349348\pi$
$541$	21.2039	0.911628	0.455814	+	0.890075i	$0.349348\pi$
$542$	0	0
$543$	−12.4124	−0.532667
$544$	0	0
$545$	−8.25565	−0.353633
$546$	0	0
$547$	−15.3197	−0.655021	−0.327511	−	0.944848i	$-0.606210\pi$
$547$	−15.3197	−0.655021	−0.327511	+	0.944848i	$0.606210\pi$
$548$	0	0
$549$	0.618611	0.0264017
$550$	0	0
$551$	−28.5113	−1.21462
$552$	0	0
$553$	13.1773	0.560355
$554$	0	0
$555$	−5.26180	−0.223351
$556$	0	0
$557$	−16.4924	−0.698805	−0.349403	−	0.936973i	$-0.613615\pi$
$557$	−16.4924	−0.698805	−0.349403	+	0.936973i	$0.613615\pi$
$558$	0	0
$559$	−31.5174	−1.33305
$560$	0	0
$561$	−6.34017	−0.267682
$562$	0	0
$563$	35.6163	1.50105	0.750525	−	0.660842i	$-0.229801\pi$
$563$	35.6163	1.50105	0.750525	+	0.660842i	$0.229801\pi$
$564$	0	0
$565$	−14.6803	−0.617607
$566$	0	0
$567$	8.75872	0.367832
$568$	0	0
$569$	14.1522	0.593292	0.296646	−	0.954988i	$-0.404132\pi$
$569$	14.1522	0.593292	0.296646	+	0.954988i	$0.404132\pi$
$570$	0	0
$571$	−2.04104	−0.0854148	−0.0427074	−	0.999088i	$-0.513598\pi$
$571$	−2.04104	−0.0854148	−0.0427074	+	0.999088i	$0.513598\pi$
$572$	0	0
$573$	−7.37298	−0.308011
$574$	0	0
$575$	2.34017	0.0975920
$576$	0	0
$577$	−20.6225	−0.858525	−0.429263	−	0.903180i	$-0.641227\pi$
$577$	−20.6225	−0.858525	−0.429263	+	0.903180i	$0.641227\pi$
$578$	0	0
$579$	−1.57531	−0.0654675
$580$	0	0
$581$	−2.73820	−0.113600
$582$	0	0
$583$	−7.75872	−0.321334
$584$	0	0
$585$	−0.290725	−0.0120200
$586$	0	0
$587$	−25.8082	−1.06522	−0.532609	−	0.846362i	$-0.678788\pi$
$587$	−25.8082	−1.06522	−0.532609	+	0.846362i	$0.678788\pi$
$588$	0	0
$589$	−42.3545	−1.74519
$590$	0	0
$591$	−8.41241	−0.346040
$592$	0	0
$593$	5.28458	0.217012	0.108506	−	0.994096i	$-0.465393\pi$
$593$	5.28458	0.217012	0.108506	+	0.994096i	$0.465393\pi$
$594$	0	0
$595$	3.70928	0.152065
$596$	0	0
$597$	17.0784	0.698971
$598$	0	0
$599$	9.88882	0.404046	0.202023	−	0.979381i	$-0.435248\pi$
$599$	9.88882	0.404046	0.202023	+	0.979381i	$0.435248\pi$
$600$	0	0
$601$	−27.7237	−1.13087	−0.565436	−	0.824792i	$-0.691292\pi$
$601$	−27.7237	−1.13087	−0.565436	+	0.824792i	$0.691292\pi$
$602$	0	0
$603$	−0.299135	−0.0121817
$604$	0	0
$605$	−1.00000	−0.0406558
$606$	0	0
$607$	−12.3668	−0.501954	−0.250977	−	0.967993i	$-0.580752\pi$
$607$	−12.3668	−0.501954	−0.250977	+	0.967993i	$0.580752\pi$
$608$	0	0
$609$	−9.26180	−0.375307
$610$	0	0
$611$	38.5380	1.55908
$612$	0	0
$613$	23.1773	0.936121	0.468061	−	0.883696i	$-0.344953\pi$
$613$	23.1773	0.936121	0.468061	+	0.883696i	$0.344953\pi$
$614$	0	0
$615$	−10.3402	−0.416956
$616$	0	0
$617$	−8.24128	−0.331781	−0.165891	−	0.986144i	$-0.553050\pi$
$617$	−8.24128	−0.331781	−0.165891	+	0.986144i	$0.553050\pi$
$618$	0	0
$619$	−5.04331	−0.202708	−0.101354	−	0.994850i	$-0.532317\pi$
$619$	−5.04331	−0.202708	−0.101354	+	0.994850i	$0.532317\pi$
$620$	0	0
$621$	12.3135	0.494124
$622$	0	0
$623$	−0.523590	−0.0209772
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−8.99386	−0.359180
$628$	0	0
$629$	−11.4186	−0.455287
$630$	0	0
$631$	−21.4764	−0.854962	−0.427481	−	0.904024i	$-0.640599\pi$
$631$	−21.4764	−0.854962	−0.427481	+	0.904024i	$0.640599\pi$
$632$	0	0
$633$	−28.5113	−1.13322
$634$	0	0
$635$	6.83710	0.271322
$636$	0	0
$637$	−3.70928	−0.146967
$638$	0	0
$639$	−0.267938	−0.0105995
$640$	0	0
$641$	13.9155	0.549628	0.274814	−	0.961497i	$-0.411384\pi$
$641$	13.9155	0.549628	0.274814	+	0.961497i	$0.411384\pi$
$642$	0	0
$643$	−0.500804	−0.0197498	−0.00987489	−	0.999951i	$-0.503143\pi$
$643$	−0.500804	−0.0197498	−0.00987489	+	0.999951i	$0.503143\pi$
$644$	0	0
$645$	14.5236	0.571866
$646$	0	0
$647$	−38.4352	−1.51104	−0.755522	−	0.655124i	$-0.772617\pi$
$647$	−38.4352	−1.51104	−0.755522	+	0.655124i	$0.772617\pi$
$648$	0	0
$649$	3.36910	0.132249
$650$	0	0
$651$	−13.7587	−0.539247
$652$	0	0
$653$	48.9816	1.91680	0.958398	−	0.285434i	$-0.0921376\pi$
$653$	48.9816	1.91680	0.958398	+	0.285434i	$0.0921376\pi$
$654$	0	0
$655$	−8.00000	−0.312586
$656$	0	0
$657$	0.146255	0.00570596
$658$	0	0
$659$	4.49693	0.175175	0.0875877	−	0.996157i	$-0.472084\pi$
$659$	4.49693	0.175175	0.0875877	+	0.996157i	$0.472084\pi$
$660$	0	0
$661$	−39.6742	−1.54315	−0.771574	−	0.636140i	$-0.780530\pi$
$661$	−39.6742	−1.54315	−0.771574	+	0.636140i	$0.780530\pi$
$662$	0	0
$663$	23.5174	0.913342
$664$	0	0
$665$	5.26180	0.204044
$666$	0	0
$667$	−12.6803	−0.490985
$668$	0	0
$669$	−32.4268	−1.25369
$670$	0	0
$671$	−7.89269	−0.304694
$672$	0	0
$673$	−16.7070	−0.644008	−0.322004	−	0.946738i	$-0.604356\pi$
$673$	−16.7070	−0.644008	−0.322004	+	0.946738i	$0.604356\pi$
$674$	0	0
$675$	5.26180	0.202527
$676$	0	0
$677$	−17.9194	−0.688697	−0.344348	−	0.938842i	$-0.611900\pi$
$677$	−17.9194	−0.688697	−0.344348	+	0.938842i	$0.611900\pi$
$678$	0	0
$679$	10.0989	0.387560
$680$	0	0
$681$	4.31351	0.165294
$682$	0	0
$683$	9.39189	0.359371	0.179685	−	0.983724i	$-0.442492\pi$
$683$	9.39189	0.359371	0.179685	+	0.983724i	$0.442492\pi$
$684$	0	0
$685$	1.60197	0.0612081
$686$	0	0
$687$	−7.27408	−0.277524
$688$	0	0
$689$	28.7792	1.09640
$690$	0	0
$691$	17.3568	0.660284	0.330142	−	0.943931i	$-0.392903\pi$
$691$	17.3568	0.660284	0.330142	+	0.943931i	$0.392903\pi$
$692$	0	0
$693$	0.0783777	0.00297732
$694$	0	0
$695$	8.09890	0.307209
$696$	0	0
$697$	−22.4391	−0.849940
$698$	0	0
$699$	−16.7792	−0.634649
$700$	0	0
$701$	25.2495	0.953661	0.476830	−	0.878995i	$-0.341786\pi$
$701$	25.2495	0.953661	0.476830	+	0.878995i	$0.341786\pi$
$702$	0	0
$703$	−16.1978	−0.610911
$704$	0	0
$705$	−17.7587	−0.668832
$706$	0	0
$707$	4.78765	0.180058
$708$	0	0
$709$	−31.2450	−1.17343	−0.586715	−	0.809794i	$-0.699579\pi$
$709$	−31.2450	−1.17343	−0.586715	+	0.809794i	$0.699579\pi$
$710$	0	0
$711$	1.03281	0.0387332
$712$	0	0
$713$	−18.8371	−0.705455
$714$	0	0
$715$	3.70928	0.138719
$716$	0	0
$717$	39.6619	1.48120
$718$	0	0
$719$	46.1361	1.72058	0.860292	−	0.509801i	$-0.170281\pi$
$719$	46.1361	1.72058	0.860292	+	0.509801i	$0.170281\pi$
$720$	0	0
$721$	−13.1278	−0.488906
$722$	0	0
$723$	16.6414	0.618900
$724$	0	0
$725$	−5.41855	−0.201240
$726$	0	0
$727$	−40.9588	−1.51908	−0.759539	−	0.650462i	$-0.774575\pi$
$727$	−40.9588	−1.51908	−0.759539	+	0.650462i	$0.774575\pi$
$728$	0	0
$729$	27.6681	1.02474
$730$	0	0
$731$	31.5174	1.16571
$732$	0	0
$733$	50.1171	1.85112	0.925560	−	0.378602i	$-0.123595\pi$
$733$	50.1171	1.85112	0.925560	+	0.378602i	$0.123595\pi$
$734$	0	0
$735$	1.70928	0.0630476
$736$	0	0
$737$	3.81658	0.140586
$738$	0	0
$739$	−43.0349	−1.58306	−0.791532	−	0.611128i	$-0.790716\pi$
$739$	−43.0349	−1.58306	−0.791532	+	0.611128i	$0.790716\pi$
$740$	0	0
$741$	33.3607	1.22554
$742$	0	0
$743$	36.1445	1.32601	0.663006	−	0.748614i	$-0.269280\pi$
$743$	36.1445	1.32601	0.663006	+	0.748614i	$0.269280\pi$
$744$	0	0
$745$	−0.156755	−0.00574308
$746$	0	0
$747$	−0.214614	−0.00785233
$748$	0	0
$749$	−4.36683	−0.159561
$750$	0	0
$751$	−7.53427	−0.274929	−0.137465	−	0.990507i	$-0.543895\pi$
$751$	−7.53427	−0.274929	−0.137465	+	0.990507i	$0.543895\pi$
$752$	0	0
$753$	43.4329	1.58278
$754$	0	0
$755$	−3.81658	−0.138900
$756$	0	0
$757$	45.7686	1.66349	0.831743	−	0.555160i	$-0.187343\pi$
$757$	45.7686	1.66349	0.831743	+	0.555160i	$0.187343\pi$
$758$	0	0
$759$	−4.00000	−0.145191
$760$	0	0
$761$	−51.7692	−1.87663	−0.938316	−	0.345778i	$-0.887615\pi$
$761$	−51.7692	−1.87663	−0.938316	+	0.345778i	$0.887615\pi$
$762$	0	0
$763$	−8.25565	−0.298875
$764$	0	0
$765$	0.290725	0.0105112
$766$	0	0
$767$	−12.4969	−0.451238
$768$	0	0
$769$	0.259528	0.00935881	0.00467941	−	0.999989i	$-0.498510\pi$
$769$	0.259528	0.00935881	0.00467941	+	0.999989i	$0.498510\pi$
$770$	0	0
$771$	1.43084	0.0515303
$772$	0	0
$773$	25.7854	0.927436	0.463718	−	0.885983i	$-0.346515\pi$
$773$	25.7854	0.927436	0.463718	+	0.885983i	$0.346515\pi$
$774$	0	0
$775$	−8.04945	−0.289145
$776$	0	0
$777$	−5.26180	−0.188766
$778$	0	0
$779$	−31.8310	−1.14046
$780$	0	0
$781$	3.41855	0.122325
$782$	0	0
$783$	−28.5113	−1.01891
$784$	0	0
$785$	−9.51745	−0.339692
$786$	0	0
$787$	−24.0989	−0.859033	−0.429516	−	0.903059i	$-0.641316\pi$
$787$	−24.0989	−0.859033	−0.429516	+	0.903059i	$0.641316\pi$
$788$	0	0
$789$	−13.6742	−0.486815
$790$	0	0
$791$	−14.6803	−0.521973
$792$	0	0
$793$	29.2762	1.03963
$794$	0	0
$795$	−13.2618	−0.470347
$796$	0	0
$797$	48.1568	1.70580	0.852900	−	0.522074i	$-0.174842\pi$
$797$	48.1568	1.70580	0.852900	+	0.522074i	$0.174842\pi$
$798$	0	0
$799$	−38.5380	−1.36338
$800$	0	0
$801$	−0.0410378	−0.00145000
$802$	0	0
$803$	−1.86603	−0.0658508
$804$	0	0
$805$	2.34017	0.0824803
$806$	0	0
$807$	−10.4247	−0.366967
$808$	0	0
$809$	7.84324	0.275754	0.137877	−	0.990449i	$-0.455972\pi$
$809$	7.84324	0.275754	0.137877	+	0.990449i	$0.455972\pi$
$810$	0	0
$811$	−1.10957	−0.0389624	−0.0194812	−	0.999810i	$-0.506201\pi$
$811$	−1.10957	−0.0389624	−0.0194812	+	0.999810i	$0.506201\pi$
$812$	0	0
$813$	−49.8187	−1.74722
$814$	0	0
$815$	−19.0205	−0.666260
$816$	0	0
$817$	44.7091	1.56417
$818$	0	0
$819$	−0.290725	−0.0101587
$820$	0	0
$821$	−5.88428	−0.205363	−0.102681	−	0.994714i	$-0.532742\pi$
$821$	−5.88428	−0.205363	−0.102681	+	0.994714i	$0.532742\pi$
$822$	0	0
$823$	0.711543	0.0248028	0.0124014	−	0.999923i	$-0.496052\pi$
$823$	0.711543	0.0248028	0.0124014	+	0.999923i	$0.496052\pi$
$824$	0	0
$825$	−1.70928	−0.0595093
$826$	0	0
$827$	−44.5790	−1.55016	−0.775082	−	0.631861i	$-0.782291\pi$
$827$	−44.5790	−1.55016	−0.775082	+	0.631861i	$0.782291\pi$
$828$	0	0
$829$	−54.8248	−1.90414	−0.952072	−	0.305873i	$-0.901052\pi$
$829$	−54.8248	−1.90414	−0.952072	+	0.305873i	$0.901052\pi$
$830$	0	0
$831$	−26.4001	−0.915810
$832$	0	0
$833$	3.70928	0.128519
$834$	0	0
$835$	−4.36683	−0.151121
$836$	0	0
$837$	−42.3545	−1.46399
$838$	0	0
$839$	−14.0084	−0.483624	−0.241812	−	0.970323i	$-0.577742\pi$
$839$	−14.0084	−0.483624	−0.241812	+	0.970323i	$0.577742\pi$
$840$	0	0
$841$	0.360692	0.0124377
$842$	0	0
$843$	12.4124	0.427506
$844$	0	0
$845$	−0.758724	−0.0261009
$846$	0	0
$847$	−1.00000	−0.0343604
$848$	0	0
$849$	−25.3607	−0.870377
$850$	0	0
$851$	−7.20394	−0.246948
$852$	0	0
$853$	28.8554	0.987988	0.493994	−	0.869465i	$-0.335536\pi$
$853$	28.8554	0.987988	0.493994	+	0.869465i	$0.335536\pi$
$854$	0	0
$855$	0.412408	0.0141040
$856$	0	0
$857$	24.8554	0.849043	0.424521	−	0.905418i	$-0.360442\pi$
$857$	24.8554	0.849043	0.424521	+	0.905418i	$0.360442\pi$
$858$	0	0
$859$	37.1955	1.26909	0.634547	−	0.772884i	$-0.281186\pi$
$859$	37.1955	1.26909	0.634547	+	0.772884i	$0.281186\pi$
$860$	0	0
$861$	−10.3402	−0.352392
$862$	0	0
$863$	−7.54864	−0.256959	−0.128479	−	0.991712i	$-0.541010\pi$
$863$	−7.54864	−0.256959	−0.128479	+	0.991712i	$0.541010\pi$
$864$	0	0
$865$	−11.8082	−0.401490
$866$	0	0
$867$	5.54023	0.188156
$868$	0	0
$869$	−13.1773	−0.447008
$870$	0	0
$871$	−14.1568	−0.479683
$872$	0	0
$873$	0.791529	0.0267892
$874$	0	0
$875$	1.00000	0.0338062
$876$	0	0
$877$	57.1650	1.93032	0.965162	−	0.261652i	$-0.0842673\pi$
$877$	57.1650	1.93032	0.965162	+	0.261652i	$0.0842673\pi$
$878$	0	0
$879$	20.7193	0.698844
$880$	0	0
$881$	−26.4657	−0.891653	−0.445827	−	0.895119i	$-0.647090\pi$
$881$	−26.4657	−0.891653	−0.445827	+	0.895119i	$0.647090\pi$
$882$	0	0
$883$	−38.6369	−1.30023	−0.650117	−	0.759834i	$-0.725280\pi$
$883$	−38.6369	−1.30023	−0.650117	+	0.759834i	$0.725280\pi$
$884$	0	0
$885$	5.75872	0.193577
$886$	0	0
$887$	54.9360	1.84457	0.922285	−	0.386510i	$-0.126320\pi$
$887$	54.9360	1.84457	0.922285	+	0.386510i	$0.126320\pi$
$888$	0	0
$889$	6.83710	0.229309
$890$	0	0
$891$	−8.75872	−0.293428
$892$	0	0
$893$	−54.6681	−1.82940
$894$	0	0
$895$	−9.84324	−0.329023
$896$	0	0
$897$	14.8371	0.495396
$898$	0	0
$899$	43.6163	1.45469
$900$	0	0
$901$	−28.7792	−0.958776
$902$	0	0
$903$	14.5236	0.483315
$904$	0	0
$905$	−7.26180	−0.241390
$906$	0	0
$907$	−31.3340	−1.04043	−0.520215	−	0.854036i	$-0.674148\pi$
$907$	−31.3340	−1.04043	−0.520215	+	0.854036i	$0.674148\pi$
$908$	0	0
$909$	0.375245	0.0124461
$910$	0	0
$911$	9.62863	0.319011	0.159505	−	0.987197i	$-0.449010\pi$
$911$	9.62863	0.319011	0.159505	+	0.987197i	$0.449010\pi$
$912$	0	0
$913$	2.73820	0.0906214
$914$	0	0
$915$	−13.4908	−0.445992
$916$	0	0
$917$	−8.00000	−0.264183
$918$	0	0
$919$	−33.7275	−1.11257	−0.556284	−	0.830992i	$-0.687773\pi$
$919$	−33.7275	−1.11257	−0.556284	+	0.830992i	$0.687773\pi$
$920$	0	0
$921$	−2.52359	−0.0831551
$922$	0	0
$923$	−12.6803	−0.417379
$924$	0	0
$925$	−3.07838	−0.101216
$926$	0	0
$927$	−1.02893	−0.0337945
$928$	0	0
$929$	2.41241	0.0791485	0.0395743	−	0.999217i	$-0.487400\pi$
$929$	2.41241	0.0791485	0.0395743	+	0.999217i	$0.487400\pi$
$930$	0	0
$931$	5.26180	0.172448
$932$	0	0
$933$	20.7649	0.679811
$934$	0	0
$935$	−3.70928	−0.121306
$936$	0	0
$937$	−34.7442	−1.13504	−0.567521	−	0.823359i	$-0.692098\pi$
$937$	−34.7442	−1.13504	−0.567521	+	0.823359i	$0.692098\pi$
$938$	0	0
$939$	16.5569	0.540313
$940$	0	0
$941$	12.1606	0.396425	0.198213	−	0.980159i	$-0.436486\pi$
$941$	12.1606	0.396425	0.198213	+	0.980159i	$0.436486\pi$
$942$	0	0
$943$	−14.1568	−0.461007
$944$	0	0
$945$	5.26180	0.171166
$946$	0	0
$947$	35.1194	1.14123	0.570614	−	0.821219i	$-0.306705\pi$
$947$	35.1194	1.14123	0.570614	+	0.821219i	$0.306705\pi$
$948$	0	0
$949$	6.92162	0.224685
$950$	0	0
$951$	−56.6635	−1.83744
$952$	0	0
$953$	−17.1727	−0.556280	−0.278140	−	0.960541i	$-0.589718\pi$
$953$	−17.1727	−0.556280	−0.278140	+	0.960541i	$0.589718\pi$
$954$	0	0
$955$	−4.31351	−0.139582
$956$	0	0
$957$	9.26180	0.299391
$958$	0	0
$959$	1.60197	0.0517303
$960$	0	0
$961$	33.7936	1.09012
$962$	0	0
$963$	−0.342263	−0.0110293
$964$	0	0
$965$	−0.921622	−0.0296681
$966$	0	0
$967$	31.7542	1.02115	0.510573	−	0.859834i	$-0.329433\pi$
$967$	31.7542	1.02115	0.510573	+	0.859834i	$0.329433\pi$
$968$	0	0
$969$	−33.3607	−1.07170
$970$	0	0
$971$	13.1135	0.420831	0.210415	−	0.977612i	$-0.432518\pi$
$971$	13.1135	0.420831	0.210415	+	0.977612i	$0.432518\pi$
$972$	0	0
$973$	8.09890	0.259639
$974$	0	0
$975$	6.34017	0.203048
$976$	0	0
$977$	37.5174	1.20029	0.600145	−	0.799891i	$-0.295110\pi$
$977$	37.5174	1.20029	0.600145	+	0.799891i	$0.295110\pi$
$978$	0	0
$979$	0.523590	0.0167340
$980$	0	0
$981$	−0.647059	−0.0206590
$982$	0	0
$983$	−46.9009	−1.49591	−0.747954	−	0.663751i	$-0.768963\pi$
$983$	−46.9009	−1.49591	−0.747954	+	0.663751i	$0.768963\pi$
$984$	0	0
$985$	−4.92162	−0.156816
$986$	0	0
$987$	−17.7587	−0.565266
$988$	0	0
$989$	19.8843	0.632283
$990$	0	0
$991$	−26.0866	−0.828668	−0.414334	−	0.910125i	$-0.635986\pi$
$991$	−26.0866	−0.828668	−0.414334	+	0.910125i	$0.635986\pi$
$992$	0	0
$993$	32.8248	1.04166
$994$	0	0
$995$	9.99159	0.316755
$996$	0	0
$997$	−21.5525	−0.682575	−0.341288	−	0.939959i	$-0.610863\pi$
$997$	−21.5525	−0.682575	−0.341288	+	0.939959i	$0.610863\pi$
$998$	0	0
$999$	−16.1978	−0.512476

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6160.2.a.bm.1.1		3
4.3	odd	2		3080.2.a.k.1.3	✓	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3080.2.a.k.1.3	✓	3	4.3	odd	2
6160.2.a.bm.1.1		3	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 6160 = 2^{4} \cdot 5 \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6160.a (trivial)

\(f(q)\)	\(=\)	\(q-1.70928 q^{3} -1.00000 q^{5} -1.00000 q^{7} -0.0783777 q^{9} +O(q^{10})\)	\(q-1.70928 q^{3} -1.00000 q^{5} -1.00000 q^{7} -0.0783777 q^{9} +1.00000 q^{11} -3.70928 q^{13} +1.70928 q^{15} +3.70928 q^{17} +5.26180 q^{19} +1.70928 q^{21} +2.34017 q^{23} +1.00000 q^{25} +5.26180 q^{27} -5.41855 q^{29} -8.04945 q^{31} -1.70928 q^{33} +1.00000 q^{35} -3.07838 q^{37} +6.34017 q^{39} -6.04945 q^{41} +8.49693 q^{43} +0.0783777 q^{45} -10.3896 q^{47} +1.00000 q^{49} -6.34017 q^{51} -7.75872 q^{53} -1.00000 q^{55} -8.99386 q^{57} +3.36910 q^{59} -7.89269 q^{61} +0.0783777 q^{63} +3.70928 q^{65} +3.81658 q^{67} -4.00000 q^{69} +3.41855 q^{71} -1.86603 q^{73} -1.70928 q^{75} -1.00000 q^{77} -13.1773 q^{79} -8.75872 q^{81} +2.73820 q^{83} -3.70928 q^{85} +9.26180 q^{87} +0.523590 q^{89} +3.70928 q^{91} +13.7587 q^{93} -5.26180 q^{95} -10.0989 q^{97} -0.0783777 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 2 q^{3} - 3 q^{5} - 3 q^{7} + 3 q^{9}+O(q^{10}) \) 3 * q + 2 * q^3 - 3 * q^5 - 3 * q^7 + 3 * q^9	\( 3 q + 2 q^{3} - 3 q^{5} - 3 q^{7} + 3 q^{9} + 3 q^{11} - 4 q^{13} - 2 q^{15} + 4 q^{17} + 8 q^{19} - 2 q^{21} - 4 q^{23} + 3 q^{25} + 8 q^{27} - 2 q^{29} - 6 q^{31} + 2 q^{33} + 3 q^{35} - 6 q^{37} + 8 q^{39} + 8 q^{43} - 3 q^{45} - 2 q^{47} + 3 q^{49} - 8 q^{51} + 2 q^{53} - 3 q^{55} + 8 q^{57} + 14 q^{59} - 12 q^{61} - 3 q^{63} + 4 q^{65} + 16 q^{67} - 12 q^{69} - 4 q^{71} + 8 q^{73} + 2 q^{75} - 3 q^{77} - q^{81} + 16 q^{83} - 4 q^{85} + 20 q^{87} - 14 q^{89} + 4 q^{91} + 16 q^{93} - 8 q^{95} + 6 q^{97} + 3 q^{99}+O(q^{100}) \) 3 * q + 2 * q^3 - 3 * q^5 - 3 * q^7 + 3 * q^9 + 3 * q^11 - 4 * q^13 - 2 * q^15 + 4 * q^17 + 8 * q^19 - 2 * q^21 - 4 * q^23 + 3 * q^25 + 8 * q^27 - 2 * q^29 - 6 * q^31 + 2 * q^33 + 3 * q^35 - 6 * q^37 + 8 * q^39 + 8 * q^43 - 3 * q^45 - 2 * q^47 + 3 * q^49 - 8 * q^51 + 2 * q^53 - 3 * q^55 + 8 * q^57 + 14 * q^59 - 12 * q^61 - 3 * q^63 + 4 * q^65 + 16 * q^67 - 12 * q^69 - 4 * q^71 + 8 * q^73 + 2 * q^75 - 3 * q^77 - q^81 + 16 * q^83 - 4 * q^85 + 20 * q^87 - 14 * q^89 + 4 * q^91 + 16 * q^93 - 8 * q^95 + 6 * q^97 + 3 * q^99

Introduction

L-functions

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