LMFDB - Embedded newform 6160.2.a.bi.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:	$1$
Dimension:	$3$
Coefficient field:	3.3.892.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 8x + 10 $ x^3 - x^2 - 8*x + 10
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 770)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.59774$ 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)

$f(q)$	$=$	$q-3.34596 q^{3} -1.00000 q^{5} +1.00000 q^{7} +8.19547 q^{9} +O(q^{10})$	$q-3.34596 q^{3} -1.00000 q^{5} +1.00000 q^{7} +8.19547 q^{9} -1.00000 q^{11} +6.69193 q^{13} +3.34596 q^{15} -7.19547 q^{17} +1.84951 q^{19} -3.34596 q^{21} -1.84951 q^{23} +1.00000 q^{25} -17.3839 q^{27} +6.84242 q^{29} -6.00000 q^{31} +3.34596 q^{33} -1.00000 q^{35} -6.54143 q^{37} -22.3909 q^{39} -9.34596 q^{41} -0.503544 q^{43} -8.19547 q^{45} +1.49646 q^{47} +1.00000 q^{49} +24.0758 q^{51} +6.84242 q^{53} +1.00000 q^{55} -6.18838 q^{57} +7.88740 q^{59} +4.50354 q^{61} +8.19547 q^{63} -6.69193 q^{65} -8.00000 q^{67} +6.18838 q^{69} -0.300986 q^{71} -10.1884 q^{73} -3.34596 q^{75} -1.00000 q^{77} +12.2404 q^{79} +33.5793 q^{81} +1.30807 q^{83} +7.19547 q^{85} -22.8945 q^{87} -8.69193 q^{89} +6.69193 q^{91} +20.0758 q^{93} -1.84951 q^{95} +3.84951 q^{97} -8.19547 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{5} + 3 q^{7} + 11 q^{9}+O(q^{10}) $ 3 * q - 3 * q^5 + 3 * q^7 + 11 * q^9	$ 3 q - 3 q^{5} + 3 q^{7} + 11 q^{9} - 3 q^{11} - 8 q^{17} + 2 q^{19} - 2 q^{23} + 3 q^{25} - 12 q^{27} + 4 q^{29} - 18 q^{31} - 3 q^{35} + 4 q^{37} - 40 q^{39} - 18 q^{41} - 8 q^{43} - 11 q^{45} - 2 q^{47} + 3 q^{49} + 12 q^{51} + 4 q^{53} + 3 q^{55} + 8 q^{57} - 10 q^{59} + 20 q^{61} + 11 q^{63} - 24 q^{67} - 8 q^{69} - 8 q^{71} - 4 q^{73} - 3 q^{77} + 6 q^{79} + 47 q^{81} + 24 q^{83} + 8 q^{85} - 48 q^{87} - 6 q^{89} - 2 q^{95} + 8 q^{97} - 11 q^{99}+O(q^{100}) $ 3 * q - 3 * q^5 + 3 * q^7 + 11 * q^9 - 3 * q^11 - 8 * q^17 + 2 * q^19 - 2 * q^23 + 3 * q^25 - 12 * q^27 + 4 * q^29 - 18 * q^31 - 3 * q^35 + 4 * q^37 - 40 * q^39 - 18 * q^41 - 8 * q^43 - 11 * q^45 - 2 * q^47 + 3 * q^49 + 12 * q^51 + 4 * q^53 + 3 * q^55 + 8 * q^57 - 10 * q^59 + 20 * q^61 + 11 * q^63 - 24 * q^67 - 8 * q^69 - 8 * q^71 - 4 * q^73 - 3 * q^77 + 6 * q^79 + 47 * q^81 + 24 * q^83 + 8 * q^85 - 48 * q^87 - 6 * q^89 - 2 * q^95 + 8 * q^97 - 11 * 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
$3$	−3.34596	−1.93179	−0.965896	−	0.258929i	$-0.916630\pi$
$3$	−3.34596	−1.93179	−0.965896	+	0.258929i	$0.916630\pi$
$4$	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
$9$	8.19547	2.73182
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	6.69193	1.85601	0.928003	−	0.372572i	$-0.121524\pi$
$13$	6.69193	1.85601	0.928003	+	0.372572i	$0.121524\pi$
$14$	0	0
$15$	3.34596	0.863924
$16$	0	0
$17$	−7.19547	−1.74516	−0.872579	−	0.488473i	$-0.837554\pi$
$17$	−7.19547	−1.74516	−0.872579	+	0.488473i	$0.837554\pi$
$18$	0	0
$19$	1.84951	0.424306	0.212153	−	0.977236i	$-0.431952\pi$
$19$	1.84951	0.424306	0.212153	+	0.977236i	$0.431952\pi$
$20$	0	0
$21$	−3.34596	−0.730149
$22$	0	0
$23$	−1.84951	−0.385649	−0.192824	−	0.981233i	$-0.561765\pi$
$23$	−1.84951	−0.385649	−0.192824	+	0.981233i	$0.561765\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−17.3839	−3.34552
$28$	0	0
$29$	6.84242	1.27061	0.635303	−	0.772263i	$-0.280875\pi$
$29$	6.84242	1.27061	0.635303	+	0.772263i	$0.280875\pi$
$30$	0	0
$31$	−6.00000	−1.07763	−0.538816	−	0.842424i	$-0.681128\pi$
$31$	−6.00000	−1.07763	−0.538816	+	0.842424i	$0.681128\pi$
$32$	0	0
$33$	3.34596	0.582457
$34$	0	0
$35$	−1.00000	−0.169031
$36$	0	0
$37$	−6.54143	−1.07541	−0.537703	−	0.843135i	$-0.680708\pi$
$37$	−6.54143	−1.07541	−0.537703	+	0.843135i	$0.680708\pi$
$38$	0	0
$39$	−22.3909	−3.58542
$40$	0	0
$41$	−9.34596	−1.45959	−0.729797	−	0.683664i	$-0.760385\pi$
$41$	−9.34596	−1.45959	−0.729797	+	0.683664i	$0.760385\pi$
$42$	0	0
$43$	−0.503544	−0.0767897	−0.0383949	−	0.999263i	$-0.512224\pi$
$43$	−0.503544	−0.0767897	−0.0383949	+	0.999263i	$0.512224\pi$
$44$	0	0
$45$	−8.19547	−1.22171
$46$	0	0
$47$	1.49646	0.218281	0.109140	−	0.994026i	$-0.465190\pi$
$47$	1.49646	0.218281	0.109140	+	0.994026i	$0.465190\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	24.0758	3.37128
$52$	0	0
$53$	6.84242	0.939879	0.469939	−	0.882699i	$-0.344276\pi$
$53$	6.84242	0.939879	0.469939	+	0.882699i	$0.344276\pi$
$54$	0	0
$55$	1.00000	0.134840
$56$	0	0
$57$	−6.18838	−0.819671
$58$	0	0
$59$	7.88740	1.02685	0.513426	−	0.858134i	$-0.328376\pi$
$59$	7.88740	1.02685	0.513426	+	0.858134i	$0.328376\pi$
$60$	0	0
$61$	4.50354	0.576620	0.288310	−	0.957537i	$-0.406907\pi$
$61$	4.50354	0.576620	0.288310	+	0.957537i	$0.406907\pi$
$62$	0	0
$63$	8.19547	1.03253
$64$	0	0
$65$	−6.69193	−0.830031
$66$	0	0
$67$	−8.00000	−0.977356	−0.488678	−	0.872464i	$-0.662521\pi$
$67$	−8.00000	−0.977356	−0.488678	+	0.872464i	$0.662521\pi$
$68$	0	0
$69$	6.18838	0.744994
$70$	0	0
$71$	−0.300986	−0.0357204	−0.0178602	−	0.999840i	$-0.505685\pi$
$71$	−0.300986	−0.0357204	−0.0178602	+	0.999840i	$0.505685\pi$
$72$	0	0
$73$	−10.1884	−1.19246	−0.596230	−	0.802814i	$-0.703335\pi$
$73$	−10.1884	−1.19246	−0.596230	+	0.802814i	$0.703335\pi$
$74$	0	0
$75$	−3.34596	−0.386359
$76$	0	0
$77$	−1.00000	−0.113961
$78$	0	0
$79$	12.2404	1.37716	0.688579	−	0.725161i	$-0.258235\pi$
$79$	12.2404	1.37716	0.688579	+	0.725161i	$0.258235\pi$
$80$	0	0
$81$	33.5793	3.73104
$82$	0	0
$83$	1.30807	0.143580	0.0717899	−	0.997420i	$-0.477129\pi$
$83$	1.30807	0.143580	0.0717899	+	0.997420i	$0.477129\pi$
$84$	0	0
$85$	7.19547	0.780458
$86$	0	0
$87$	−22.8945	−2.45455
$88$	0	0
$89$	−8.69193	−0.921342	−0.460671	−	0.887571i	$-0.652391\pi$
$89$	−8.69193	−0.921342	−0.460671	+	0.887571i	$0.652391\pi$
$90$	0	0
$91$	6.69193	0.701505
$92$	0	0
$93$	20.0758	2.08176
$94$	0	0
$95$	−1.84951	−0.189755
$96$	0	0
$97$	3.84951	0.390858	0.195429	−	0.980718i	$-0.437390\pi$
$97$	3.84951	0.390858	0.195429	+	0.980718i	$0.437390\pi$
$98$	0	0
$99$	−8.19547	−0.823676
$100$	0	0
$101$	6.89448	0.686027	0.343013	−	0.939331i	$-0.388552\pi$
$101$	6.89448	0.686027	0.343013	+	0.939331i	$0.388552\pi$
$102$	0	0
$103$	10.5793	1.04241	0.521206	−	0.853431i	$-0.325482\pi$
$103$	10.5793	1.04241	0.521206	+	0.853431i	$0.325482\pi$
$104$	0	0
$105$	3.34596	0.326533
$106$	0	0
$107$	7.49646	0.724710	0.362355	−	0.932040i	$-0.381973\pi$
$107$	7.49646	0.724710	0.362355	+	0.932040i	$0.381973\pi$
$108$	0	0
$109$	1.45857	0.139705	0.0698527	−	0.997557i	$-0.477747\pi$
$109$	1.45857	0.139705	0.0698527	+	0.997557i	$0.477747\pi$
$110$	0	0
$111$	21.8874	2.07746
$112$	0	0
$113$	−14.0000	−1.31701	−0.658505	−	0.752577i	$-0.728811\pi$
$113$	−14.0000	−1.31701	−0.658505	+	0.752577i	$0.728811\pi$
$114$	0	0
$115$	1.84951	0.172467
$116$	0	0
$117$	54.8435	5.07028
$118$	0	0
$119$	−7.19547	−0.659608
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	31.2713	2.81963
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	19.7748	1.75473	0.877365	−	0.479824i	$-0.159300\pi$
$127$	19.7748	1.75473	0.877365	+	0.479824i	$0.159300\pi$
$128$	0	0
$129$	1.68484	0.148342
$130$	0	0
$131$	−2.15049	−0.187889	−0.0939447	−	0.995577i	$-0.529948\pi$
$131$	−2.15049	−0.187889	−0.0939447	+	0.995577i	$0.529948\pi$
$132$	0	0
$133$	1.84951	0.160373
$134$	0	0
$135$	17.3839	1.49616
$136$	0	0
$137$	−8.39094	−0.716886	−0.358443	−	0.933552i	$-0.616692\pi$
$137$	−8.39094	−0.716886	−0.358443	+	0.933552i	$0.616692\pi$
$138$	0	0
$139$	−17.9253	−1.52040	−0.760201	−	0.649687i	$-0.774900\pi$
$139$	−17.9253	−1.52040	−0.760201	+	0.649687i	$0.774900\pi$
$140$	0	0
$141$	−5.00709	−0.421673
$142$	0	0
$143$	−6.69193	−0.559607
$144$	0	0
$145$	−6.84242	−0.568232
$146$	0	0
$147$	−3.34596	−0.275970
$148$	0	0
$149$	−0.451479	−0.0369866	−0.0184933	−	0.999829i	$-0.505887\pi$
$149$	−0.451479	−0.0369866	−0.0184933	+	0.999829i	$0.505887\pi$
$150$	0	0
$151$	−19.2334	−1.56519	−0.782594	−	0.622532i	$-0.786104\pi$
$151$	−19.2334	−1.56519	−0.782594	+	0.622532i	$0.786104\pi$
$152$	0	0
$153$	−58.9703	−4.76746
$154$	0	0
$155$	6.00000	0.481932
$156$	0	0
$157$	−9.69901	−0.774066	−0.387033	−	0.922066i	$-0.626500\pi$
$157$	−9.69901	−0.774066	−0.387033	+	0.922066i	$0.626500\pi$
$158$	0	0
$159$	−22.8945	−1.81565
$160$	0	0
$161$	−1.84951	−0.145762
$162$	0	0
$163$	8.00000	0.626608	0.313304	−	0.949653i	$-0.398564\pi$
$163$	8.00000	0.626608	0.313304	+	0.949653i	$0.398564\pi$
$164$	0	0
$165$	−3.34596	−0.260483
$166$	0	0
$167$	−8.00000	−0.619059	−0.309529	−	0.950890i	$-0.600171\pi$
$167$	−8.00000	−0.619059	−0.309529	+	0.950890i	$0.600171\pi$
$168$	0	0
$169$	31.7819	2.44476
$170$	0	0
$171$	15.1576	1.15913
$172$	0	0
$173$	−15.4965	−1.17817	−0.589087	−	0.808070i	$-0.700512\pi$
$173$	−15.4965	−1.17817	−0.589087	+	0.808070i	$0.700512\pi$
$174$	0	0
$175$	1.00000	0.0755929
$176$	0	0
$177$	−26.3909	−1.98366
$178$	0	0
$179$	13.8874	1.03799	0.518996	−	0.854776i	$-0.326306\pi$
$179$	13.8874	1.03799	0.518996	+	0.854776i	$0.326306\pi$
$180$	0	0
$181$	18.7819	1.39605	0.698023	−	0.716075i	$-0.254063\pi$
$181$	18.7819	1.39605	0.698023	+	0.716075i	$0.254063\pi$
$182$	0	0
$183$	−15.0687	−1.11391
$184$	0	0
$185$	6.54143	0.480936
$186$	0	0
$187$	7.19547	0.526185
$188$	0	0
$189$	−17.3839	−1.26449
$190$	0	0
$191$	6.39094	0.462432	0.231216	−	0.972902i	$-0.425730\pi$
$191$	6.39094	0.462432	0.231216	+	0.972902i	$0.425730\pi$
$192$	0	0
$193$	−13.1955	−0.949831	−0.474915	−	0.880031i	$-0.657521\pi$
$193$	−13.1955	−0.949831	−0.474915	+	0.880031i	$0.657521\pi$
$194$	0	0
$195$	22.3909	1.60345
$196$	0	0
$197$	−4.69193	−0.334286	−0.167143	−	0.985933i	$-0.553454\pi$
$197$	−4.69193	−0.334286	−0.167143	+	0.985933i	$0.553454\pi$
$198$	0	0
$199$	9.49646	0.673186	0.336593	−	0.941650i	$-0.390725\pi$
$199$	9.49646	0.673186	0.336593	+	0.941650i	$0.390725\pi$
$200$	0	0
$201$	26.7677	1.88805
$202$	0	0
$203$	6.84242	0.480244
$204$	0	0
$205$	9.34596	0.652750
$206$	0	0
$207$	−15.1576	−1.05352
$208$	0	0
$209$	−1.84951	−0.127933
$210$	0	0
$211$	2.39094	0.164599	0.0822996	−	0.996608i	$-0.473774\pi$
$211$	2.39094	0.164599	0.0822996	+	0.996608i	$0.473774\pi$
$212$	0	0
$213$	1.00709	0.0690045
$214$	0	0
$215$	0.503544	0.0343414
$216$	0	0
$217$	−6.00000	−0.407307
$218$	0	0
$219$	34.0900	2.30359
$220$	0	0
$221$	−48.1516	−3.23902
$222$	0	0
$223$	−0.188383	−0.0126150	−0.00630752	−	0.999980i	$-0.502008\pi$
$223$	−0.188383	−0.0126150	−0.00630752	+	0.999980i	$0.502008\pi$
$224$	0	0
$225$	8.19547	0.546365
$226$	0	0
$227$	6.99291	0.464136	0.232068	−	0.972700i	$-0.425451\pi$
$227$	6.99291	0.464136	0.232068	+	0.972700i	$0.425451\pi$
$228$	0	0
$229$	−9.19547	−0.607654	−0.303827	−	0.952727i	$-0.598264\pi$
$229$	−9.19547	−0.607654	−0.303827	+	0.952727i	$0.598264\pi$
$230$	0	0
$231$	3.34596	0.220148
$232$	0	0
$233$	−0.992912	−0.0650479	−0.0325239	−	0.999471i	$-0.510355\pi$
$233$	−0.992912	−0.0650479	−0.0325239	+	0.999471i	$0.510355\pi$
$234$	0	0
$235$	−1.49646	−0.0976180
$236$	0	0
$237$	−40.9561	−2.66038
$238$	0	0
$239$	1.14341	0.0739607	0.0369804	−	0.999316i	$-0.488226\pi$
$239$	1.14341	0.0739607	0.0369804	+	0.999316i	$0.488226\pi$
$240$	0	0
$241$	8.33888	0.537154	0.268577	−	0.963258i	$-0.413447\pi$
$241$	8.33888	0.537154	0.268577	+	0.963258i	$0.413447\pi$
$242$	0	0
$243$	−60.2036	−3.86206
$244$	0	0
$245$	−1.00000	−0.0638877
$246$	0	0
$247$	12.3768	0.787515
$248$	0	0
$249$	−4.37677	−0.277366
$250$	0	0
$251$	−23.8874	−1.50776	−0.753880	−	0.657013i	$-0.771820\pi$
$251$	−23.8874	−1.50776	−0.753880	+	0.657013i	$0.771820\pi$
$252$	0	0
$253$	1.84951	0.116278
$254$	0	0
$255$	−24.0758	−1.50768
$256$	0	0
$257$	−31.6243	−1.97267	−0.986335	−	0.164753i	$-0.947317\pi$
$257$	−31.6243	−1.97267	−0.986335	+	0.164753i	$0.947317\pi$
$258$	0	0
$259$	−6.54143	−0.406465
$260$	0	0
$261$	56.0768	3.47107
$262$	0	0
$263$	−13.3839	−0.825284	−0.412642	−	0.910893i	$-0.635394\pi$
$263$	−13.3839	−0.825284	−0.412642	+	0.910893i	$0.635394\pi$
$264$	0	0
$265$	−6.84242	−0.420326
$266$	0	0
$267$	29.0829	1.77984
$268$	0	0
$269$	5.79744	0.353476	0.176738	−	0.984258i	$-0.443445\pi$
$269$	5.79744	0.353476	0.176738	+	0.984258i	$0.443445\pi$
$270$	0	0
$271$	20.0758	1.21952	0.609758	−	0.792587i	$-0.291266\pi$
$271$	20.0758	1.21952	0.609758	+	0.792587i	$0.291266\pi$
$272$	0	0
$273$	−22.3909	−1.35516
$274$	0	0
$275$	−1.00000	−0.0603023
$276$	0	0
$277$	22.0000	1.32185	0.660926	−	0.750451i	$-0.270164\pi$
$277$	22.0000	1.32185	0.660926	+	0.750451i	$0.270164\pi$
$278$	0	0
$279$	−49.1728	−2.94390
$280$	0	0
$281$	−6.30099	−0.375885	−0.187943	−	0.982180i	$-0.560182\pi$
$281$	−6.30099	−0.375885	−0.187943	+	0.982180i	$0.560182\pi$
$282$	0	0
$283$	−22.3909	−1.33100	−0.665502	−	0.746396i	$-0.731782\pi$
$283$	−22.3909	−1.33100	−0.665502	+	0.746396i	$0.731782\pi$
$284$	0	0
$285$	6.18838	0.366568
$286$	0	0
$287$	−9.34596	−0.551675
$288$	0	0
$289$	34.7748	2.04558
$290$	0	0
$291$	−12.8803	−0.755057
$292$	0	0
$293$	−10.6919	−0.624629	−0.312315	−	0.949979i	$-0.601104\pi$
$293$	−10.6919	−0.624629	−0.312315	+	0.949979i	$0.601104\pi$
$294$	0	0
$295$	−7.88740	−0.459222
$296$	0	0
$297$	17.3839	1.00871
$298$	0	0
$299$	−12.3768	−0.715767
$300$	0	0
$301$	−0.503544	−0.0290238
$302$	0	0
$303$	−23.0687	−1.32526
$304$	0	0
$305$	−4.50354	−0.257872
$306$	0	0
$307$	9.08287	0.518387	0.259193	−	0.965825i	$-0.416543\pi$
$307$	9.08287	0.518387	0.259193	+	0.965825i	$0.416543\pi$
$308$	0	0
$309$	−35.3980	−2.01372
$310$	0	0
$311$	11.3839	0.645519	0.322760	−	0.946481i	$-0.395389\pi$
$311$	11.3839	0.645519	0.322760	+	0.946481i	$0.395389\pi$
$312$	0	0
$313$	−23.6243	−1.33532	−0.667662	−	0.744464i	$-0.732705\pi$
$313$	−23.6243	−1.33532	−0.667662	+	0.744464i	$0.732705\pi$
$314$	0	0
$315$	−8.19547	−0.461762
$316$	0	0
$317$	−9.53435	−0.535502	−0.267751	−	0.963488i	$-0.586280\pi$
$317$	−9.53435	−0.535502	−0.267751	+	0.963488i	$0.586280\pi$
$318$	0	0
$319$	−6.84242	−0.383102
$320$	0	0
$321$	−25.0829	−1.39999
$322$	0	0
$323$	−13.3081	−0.740481
$324$	0	0
$325$	6.69193	0.371201
$326$	0	0
$327$	−4.88031	−0.269882
$328$	0	0
$329$	1.49646	0.0825023
$330$	0	0
$331$	−0.503544	−0.0276773	−0.0138386	−	0.999904i	$-0.504405\pi$
$331$	−0.503544	−0.0276773	−0.0138386	+	0.999904i	$0.504405\pi$
$332$	0	0
$333$	−53.6101	−2.93782
$334$	0	0
$335$	8.00000	0.437087
$336$	0	0
$337$	12.3909	0.674978	0.337489	−	0.941330i	$-0.390422\pi$
$337$	12.3909	0.674978	0.337489	+	0.941330i	$0.390422\pi$
$338$	0	0
$339$	46.8435	2.54419
$340$	0	0
$341$	6.00000	0.324918
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	−6.18838	−0.333171
$346$	0	0
$347$	−20.8803	−1.12091	−0.560457	−	0.828184i	$-0.689374\pi$
$347$	−20.8803	−1.12091	−0.560457	+	0.828184i	$0.689374\pi$
$348$	0	0
$349$	−22.1884	−1.18772	−0.593858	−	0.804570i	$-0.702396\pi$
$349$	−22.1884	−1.18772	−0.593858	+	0.804570i	$0.702396\pi$
$350$	0	0
$351$	−116.331	−6.20931
$352$	0	0
$353$	−31.6243	−1.68319	−0.841596	−	0.540108i	$-0.818383\pi$
$353$	−31.6243	−1.68319	−0.841596	+	0.540108i	$0.818383\pi$
$354$	0	0
$355$	0.300986	0.0159747
$356$	0	0
$357$	24.0758	1.27423
$358$	0	0
$359$	−30.5273	−1.61117	−0.805584	−	0.592482i	$-0.798148\pi$
$359$	−30.5273	−1.61117	−0.805584	+	0.592482i	$0.798148\pi$
$360$	0	0
$361$	−15.5793	−0.819964
$362$	0	0
$363$	−3.34596	−0.175618
$364$	0	0
$365$	10.1884	0.533284
$366$	0	0
$367$	10.5793	0.552236	0.276118	−	0.961124i	$-0.410952\pi$
$367$	10.5793	0.552236	0.276118	+	0.961124i	$0.410952\pi$
$368$	0	0
$369$	−76.5946	−3.98735
$370$	0	0
$371$	6.84242	0.355241
$372$	0	0
$373$	36.4667	1.88818	0.944088	−	0.329695i	$-0.106946\pi$
$373$	36.4667	1.88818	0.944088	+	0.329695i	$0.106946\pi$
$374$	0	0
$375$	3.34596	0.172785
$376$	0	0
$377$	45.7890	2.35825
$378$	0	0
$379$	−14.9929	−0.770134	−0.385067	−	0.922889i	$-0.625822\pi$
$379$	−14.9929	−0.770134	−0.385067	+	0.922889i	$0.625822\pi$
$380$	0	0
$381$	−66.1657	−3.38977
$382$	0	0
$383$	−35.1813	−1.79768	−0.898840	−	0.438277i	$-0.855589\pi$
$383$	−35.1813	−1.79768	−0.898840	+	0.438277i	$0.855589\pi$
$384$	0	0
$385$	1.00000	0.0509647
$386$	0	0
$387$	−4.12678	−0.209776
$388$	0	0
$389$	2.30099	0.116665	0.0583323	−	0.998297i	$-0.481422\pi$
$389$	2.30099	0.116665	0.0583323	+	0.998297i	$0.481422\pi$
$390$	0	0
$391$	13.3081	0.673018
$392$	0	0
$393$	7.19547	0.362963
$394$	0	0
$395$	−12.2404	−0.615884
$396$	0	0
$397$	18.0758	0.907197	0.453599	−	0.891206i	$-0.350140\pi$
$397$	18.0758	0.907197	0.453599	+	0.891206i	$0.350140\pi$
$398$	0	0
$399$	−6.18838	−0.309807
$400$	0	0
$401$	22.5793	1.12756	0.563779	−	0.825926i	$-0.309347\pi$
$401$	22.5793	1.12756	0.563779	+	0.825926i	$0.309347\pi$
$402$	0	0
$403$	−40.1516	−2.00009
$404$	0	0
$405$	−33.5793	−1.66857
$406$	0	0
$407$	6.54143	0.324247
$408$	0	0
$409$	1.04498	0.0516708	0.0258354	−	0.999666i	$-0.491775\pi$
$409$	1.04498	0.0516708	0.0258354	+	0.999666i	$0.491775\pi$
$410$	0	0
$411$	28.0758	1.38488
$412$	0	0
$413$	7.88740	0.388113
$414$	0	0
$415$	−1.30807	−0.0642108
$416$	0	0
$417$	59.9774	2.93710
$418$	0	0
$419$	−5.49646	−0.268519	−0.134260	−	0.990946i	$-0.542866\pi$
$419$	−5.49646	−0.268519	−0.134260	+	0.990946i	$0.542866\pi$
$420$	0	0
$421$	2.60197	0.126812	0.0634062	−	0.997988i	$-0.479804\pi$
$421$	2.60197	0.126812	0.0634062	+	0.997988i	$0.479804\pi$
$422$	0	0
$423$	12.2642	0.596304
$424$	0	0
$425$	−7.19547	−0.349032
$426$	0	0
$427$	4.50354	0.217942
$428$	0	0
$429$	22.3909	1.08104
$430$	0	0
$431$	−19.2334	−0.926438	−0.463219	−	0.886244i	$-0.653306\pi$
$431$	−19.2334	−0.926438	−0.463219	+	0.886244i	$0.653306\pi$
$432$	0	0
$433$	−8.93237	−0.429263	−0.214631	−	0.976695i	$-0.568855\pi$
$433$	−8.93237	−0.429263	−0.214631	+	0.976695i	$0.568855\pi$
$434$	0	0
$435$	22.8945	1.09771
$436$	0	0
$437$	−3.42068	−0.163633
$438$	0	0
$439$	0.300986	0.0143653	0.00718264	−	0.999974i	$-0.497714\pi$
$439$	0.300986	0.0143653	0.00718264	+	0.999974i	$0.497714\pi$
$440$	0	0
$441$	8.19547	0.390260
$442$	0	0
$443$	−30.7677	−1.46182	−0.730909	−	0.682475i	$-0.760904\pi$
$443$	−30.7677	−1.46182	−0.730909	+	0.682475i	$0.760904\pi$
$444$	0	0
$445$	8.69193	0.412037
$446$	0	0
$447$	1.51063	0.0714504
$448$	0	0
$449$	−1.19547	−0.0564177	−0.0282089	−	0.999602i	$-0.508980\pi$
$449$	−1.19547	−0.0564177	−0.0282089	+	0.999602i	$0.508980\pi$
$450$	0	0
$451$	9.34596	0.440084
$452$	0	0
$453$	64.3541	3.02362
$454$	0	0
$455$	−6.69193	−0.313722
$456$	0	0
$457$	−25.7748	−1.20569	−0.602847	−	0.797857i	$-0.705967\pi$
$457$	−25.7748	−1.20569	−0.602847	+	0.797857i	$0.705967\pi$
$458$	0	0
$459$	125.085	5.83847
$460$	0	0
$461$	11.5722	0.538973	0.269486	−	0.963004i	$-0.413146\pi$
$461$	11.5722	0.538973	0.269486	+	0.963004i	$0.413146\pi$
$462$	0	0
$463$	32.9182	1.52984	0.764919	−	0.644126i	$-0.222779\pi$
$463$	32.9182	1.52984	0.764919	+	0.644126i	$0.222779\pi$
$464$	0	0
$465$	−20.0758	−0.930992
$466$	0	0
$467$	14.6399	0.677452	0.338726	−	0.940885i	$-0.390004\pi$
$467$	14.6399	0.677452	0.338726	+	0.940885i	$0.390004\pi$
$468$	0	0
$469$	−8.00000	−0.369406
$470$	0	0
$471$	32.4525	1.49533
$472$	0	0
$473$	0.503544	0.0231530
$474$	0	0
$475$	1.84951	0.0848612
$476$	0	0
$477$	56.0768	2.56758
$478$	0	0
$479$	−10.3909	−0.474774	−0.237387	−	0.971415i	$-0.576291\pi$
$479$	−10.3909	−0.474774	−0.237387	+	0.971415i	$0.576291\pi$
$480$	0	0
$481$	−43.7748	−1.99596
$482$	0	0
$483$	6.18838	0.281581
$484$	0	0
$485$	−3.84951	−0.174797
$486$	0	0
$487$	−39.3091	−1.78127	−0.890634	−	0.454722i	$-0.849739\pi$
$487$	−39.3091	−1.78127	−0.890634	+	0.454722i	$0.849739\pi$
$488$	0	0
$489$	−26.7677	−1.21048
$490$	0	0
$491$	−23.7748	−1.07294	−0.536471	−	0.843919i	$-0.680243\pi$
$491$	−23.7748	−1.07294	−0.536471	+	0.843919i	$0.680243\pi$
$492$	0	0
$493$	−49.2344	−2.21741
$494$	0	0
$495$	8.19547	0.368359
$496$	0	0
$497$	−0.300986	−0.0135011
$498$	0	0
$499$	6.11260	0.273638	0.136819	−	0.990596i	$-0.456312\pi$
$499$	6.11260	0.273638	0.136819	+	0.990596i	$0.456312\pi$
$500$	0	0
$501$	26.7677	1.19589
$502$	0	0
$503$	13.1586	0.586715	0.293358	−	0.956003i	$-0.405227\pi$
$503$	13.1586	0.586715	0.293358	+	0.956003i	$0.405227\pi$
$504$	0	0
$505$	−6.89448	−0.306801
$506$	0	0
$507$	−106.341	−4.72277
$508$	0	0
$509$	34.1799	1.51500	0.757499	−	0.652836i	$-0.226421\pi$
$509$	34.1799	1.51500	0.757499	+	0.652836i	$0.226421\pi$
$510$	0	0
$511$	−10.1884	−0.450708
$512$	0	0
$513$	−32.1516	−1.41953
$514$	0	0
$515$	−10.5793	−0.466181
$516$	0	0
$517$	−1.49646	−0.0658141
$518$	0	0
$519$	51.8506	2.27599
$520$	0	0
$521$	−26.6778	−1.16877	−0.584387	−	0.811475i	$-0.698665\pi$
$521$	−26.6778	−1.16877	−0.584387	+	0.811475i	$0.698665\pi$
$522$	0	0
$523$	−25.4880	−1.11451	−0.557256	−	0.830341i	$-0.688146\pi$
$523$	−25.4880	−1.11451	−0.557256	+	0.830341i	$0.688146\pi$
$524$	0	0
$525$	−3.34596	−0.146030
$526$	0	0
$527$	43.1728	1.88064
$528$	0	0
$529$	−19.5793	−0.851275
$530$	0	0
$531$	64.6409	2.80518
$532$	0	0
$533$	−62.5425	−2.70902
$534$	0	0
$535$	−7.49646	−0.324100
$536$	0	0
$537$	−46.4667	−2.00519
$538$	0	0
$539$	−1.00000	−0.0430730
$540$	0	0
$541$	4.45148	0.191384	0.0956920	−	0.995411i	$-0.469494\pi$
$541$	4.45148	0.191384	0.0956920	+	0.995411i	$0.469494\pi$
$542$	0	0
$543$	−62.8435	−2.69687
$544$	0	0
$545$	−1.45857	−0.0624781
$546$	0	0
$547$	1.38385	0.0591693	0.0295846	−	0.999562i	$-0.490582\pi$
$547$	1.38385	0.0591693	0.0295846	+	0.999562i	$0.490582\pi$
$548$	0	0
$549$	36.9087	1.57522
$550$	0	0
$551$	12.6551	0.539126
$552$	0	0
$553$	12.2404	0.520517
$554$	0	0
$555$	−21.8874	−0.929068
$556$	0	0
$557$	−6.70610	−0.284147	−0.142073	−	0.989856i	$-0.545377\pi$
$557$	−6.70610	−0.284147	−0.142073	+	0.989856i	$0.545377\pi$
$558$	0	0
$559$	−3.36968	−0.142522
$560$	0	0
$561$	−24.0758	−1.01648
$562$	0	0
$563$	−3.59488	−0.151506	−0.0757532	−	0.997127i	$-0.524136\pi$
$563$	−3.59488	−0.151506	−0.0757532	+	0.997127i	$0.524136\pi$
$564$	0	0
$565$	14.0000	0.588984
$566$	0	0
$567$	33.5793	1.41020
$568$	0	0
$569$	31.7606	1.33147	0.665737	−	0.746186i	$-0.268117\pi$
$569$	31.7606	1.33147	0.665737	+	0.746186i	$0.268117\pi$
$570$	0	0
$571$	5.68484	0.237903	0.118952	−	0.992900i	$-0.462047\pi$
$571$	5.68484	0.237903	0.118952	+	0.992900i	$0.462047\pi$
$572$	0	0
$573$	−21.3839	−0.893323
$574$	0	0
$575$	−1.84951	−0.0771298
$576$	0	0
$577$	13.5343	0.563442	0.281721	−	0.959496i	$-0.409095\pi$
$577$	13.5343	0.563442	0.281721	+	0.959496i	$0.409095\pi$
$578$	0	0
$579$	44.1516	1.83488
$580$	0	0
$581$	1.30807	0.0542680
$582$	0	0
$583$	−6.84242	−0.283384
$584$	0	0
$585$	−54.8435	−2.26750
$586$	0	0
$587$	0.0520650	0.00214895	0.00107448	−	0.999999i	$-0.499658\pi$
$587$	0.0520650	0.00214895	0.00107448	+	0.999999i	$0.499658\pi$
$588$	0	0
$589$	−11.0970	−0.457246
$590$	0	0
$591$	15.6990	0.645771
$592$	0	0
$593$	31.9774	1.31315	0.656576	−	0.754260i	$-0.272004\pi$
$593$	31.9774	1.31315	0.656576	+	0.754260i	$0.272004\pi$
$594$	0	0
$595$	7.19547	0.294986
$596$	0	0
$597$	−31.7748	−1.30046
$598$	0	0
$599$	−21.0829	−0.861423	−0.430711	−	0.902490i	$-0.641737\pi$
$599$	−21.0829	−0.861423	−0.430711	+	0.902490i	$0.641737\pi$
$600$	0	0
$601$	−21.6469	−0.882997	−0.441499	−	0.897262i	$-0.645553\pi$
$601$	−21.6469	−0.882997	−0.441499	+	0.897262i	$0.645553\pi$
$602$	0	0
$603$	−65.5638	−2.66996
$604$	0	0
$605$	−1.00000	−0.0406558
$606$	0	0
$607$	−21.6622	−0.879241	−0.439621	−	0.898184i	$-0.644887\pi$
$607$	−21.6622	−0.879241	−0.439621	+	0.898184i	$0.644887\pi$
$608$	0	0
$609$	−22.8945	−0.927731
$610$	0	0
$611$	10.0142	0.405130
$612$	0	0
$613$	8.31516	0.335846	0.167923	−	0.985800i	$-0.446294\pi$
$613$	8.31516	0.335846	0.167923	+	0.985800i	$0.446294\pi$
$614$	0	0
$615$	−31.2713	−1.26098
$616$	0	0
$617$	12.4667	0.501891	0.250946	−	0.968001i	$-0.419258\pi$
$617$	12.4667	0.501891	0.250946	+	0.968001i	$0.419258\pi$
$618$	0	0
$619$	−5.27125	−0.211869	−0.105935	−	0.994373i	$-0.533783\pi$
$619$	−5.27125	−0.211869	−0.105935	+	0.994373i	$0.533783\pi$
$620$	0	0
$621$	32.1516	1.29020
$622$	0	0
$623$	−8.69193	−0.348235
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	6.18838	0.247140
$628$	0	0
$629$	47.0687	1.87675
$630$	0	0
$631$	17.1586	0.683075	0.341537	−	0.939868i	$-0.389052\pi$
$631$	17.1586	0.683075	0.341537	+	0.939868i	$0.389052\pi$
$632$	0	0
$633$	−8.00000	−0.317971
$634$	0	0
$635$	−19.7748	−0.784739
$636$	0	0
$637$	6.69193	0.265144
$638$	0	0
$639$	−2.46672	−0.0975820
$640$	0	0
$641$	24.7677	0.978266	0.489133	−	0.872209i	$-0.337313\pi$
$641$	24.7677	0.978266	0.489133	+	0.872209i	$0.337313\pi$
$642$	0	0
$643$	−1.03080	−0.0406509	−0.0203254	−	0.999793i	$-0.506470\pi$
$643$	−1.03080	−0.0406509	−0.0203254	+	0.999793i	$0.506470\pi$
$644$	0	0
$645$	−1.68484	−0.0663405
$646$	0	0
$647$	−12.7904	−0.502841	−0.251420	−	0.967878i	$-0.580898\pi$
$647$	−12.7904	−0.502841	−0.251420	+	0.967878i	$0.580898\pi$
$648$	0	0
$649$	−7.88740	−0.309607
$650$	0	0
$651$	20.0758	0.786832
$652$	0	0
$653$	−22.3162	−0.873301	−0.436651	−	0.899631i	$-0.643835\pi$
$653$	−22.3162	−0.873301	−0.436651	+	0.899631i	$0.643835\pi$
$654$	0	0
$655$	2.15049	0.0840267
$656$	0	0
$657$	−83.4986	−3.25759
$658$	0	0
$659$	−22.6919	−0.883952	−0.441976	−	0.897027i	$-0.645722\pi$
$659$	−22.6919	−0.883952	−0.441976	+	0.897027i	$0.645722\pi$
$660$	0	0
$661$	43.5638	1.69443	0.847217	−	0.531247i	$-0.178276\pi$
$661$	43.5638	1.69443	0.847217	+	0.531247i	$0.178276\pi$
$662$	0	0
$663$	161.113	6.25712
$664$	0	0
$665$	−1.84951	−0.0717208
$666$	0	0
$667$	−12.6551	−0.490008
$668$	0	0
$669$	0.630322	0.0243697
$670$	0	0
$671$	−4.50354	−0.173857
$672$	0	0
$673$	38.0000	1.46479	0.732396	−	0.680879i	$-0.238402\pi$
$673$	38.0000	1.46479	0.732396	+	0.680879i	$0.238402\pi$
$674$	0	0
$675$	−17.3839	−0.669105
$676$	0	0
$677$	−16.0984	−0.618713	−0.309356	−	0.950946i	$-0.600114\pi$
$677$	−16.0984	−0.618713	−0.309356	+	0.950946i	$0.600114\pi$
$678$	0	0
$679$	3.84951	0.147731
$680$	0	0
$681$	−23.3980	−0.896614
$682$	0	0
$683$	−29.0829	−1.11282	−0.556412	−	0.830906i	$-0.687823\pi$
$683$	−29.0829	−1.11282	−0.556412	+	0.830906i	$0.687823\pi$
$684$	0	0
$685$	8.39094	0.320601
$686$	0	0
$687$	30.7677	1.17386
$688$	0	0
$689$	45.7890	1.74442
$690$	0	0
$691$	−7.58641	−0.288601	−0.144300	−	0.989534i	$-0.546093\pi$
$691$	−7.58641	−0.288601	−0.144300	+	0.989534i	$0.546093\pi$
$692$	0	0
$693$	−8.19547	−0.311320
$694$	0	0
$695$	17.9253	0.679945
$696$	0	0
$697$	67.2486	2.54722
$698$	0	0
$699$	3.32225	0.125659
$700$	0	0
$701$	4.07471	0.153900	0.0769499	−	0.997035i	$-0.475482\pi$
$701$	4.07471	0.153900	0.0769499	+	0.997035i	$0.475482\pi$
$702$	0	0
$703$	−12.0984	−0.456301
$704$	0	0
$705$	5.00709	0.188578
$706$	0	0
$707$	6.89448	0.259294
$708$	0	0
$709$	−46.4525	−1.74456	−0.872281	−	0.489005i	$-0.837360\pi$
$709$	−46.4525	−1.74456	−0.872281	+	0.489005i	$0.837360\pi$
$710$	0	0
$711$	100.316	3.76215
$712$	0	0
$713$	11.0970	0.415588
$714$	0	0
$715$	6.69193	0.250264
$716$	0	0
$717$	−3.82579	−0.142877
$718$	0	0
$719$	17.8732	0.666559	0.333279	−	0.942828i	$-0.391845\pi$
$719$	17.8732	0.666559	0.333279	+	0.942828i	$0.391845\pi$
$720$	0	0
$721$	10.5793	0.393995
$722$	0	0
$723$	−27.9016	−1.03767
$724$	0	0
$725$	6.84242	0.254121
$726$	0	0
$727$	11.8874	0.440879	0.220440	−	0.975401i	$-0.429251\pi$
$727$	11.8874	0.440879	0.220440	+	0.975401i	$0.429251\pi$
$728$	0	0
$729$	100.701	3.72967
$730$	0	0
$731$	3.62323	0.134010
$732$	0	0
$733$	−31.6764	−1.16999	−0.584997	−	0.811036i	$-0.698904\pi$
$733$	−31.6764	−1.16999	−0.584997	+	0.811036i	$0.698904\pi$
$734$	0	0
$735$	3.34596	0.123418
$736$	0	0
$737$	8.00000	0.294684
$738$	0	0
$739$	−28.5567	−1.05047	−0.525237	−	0.850956i	$-0.676023\pi$
$739$	−28.5567	−1.05047	−0.525237	+	0.850956i	$0.676023\pi$
$740$	0	0
$741$	−41.4122	−1.52132
$742$	0	0
$743$	−13.9858	−0.513090	−0.256545	−	0.966532i	$-0.582584\pi$
$743$	−13.9858	−0.513090	−0.256545	+	0.966532i	$0.582584\pi$
$744$	0	0
$745$	0.451479	0.0165409
$746$	0	0
$747$	10.7203	0.392234
$748$	0	0
$749$	7.49646	0.273915
$750$	0	0
$751$	−18.9171	−0.690296	−0.345148	−	0.938548i	$-0.612171\pi$
$751$	−18.9171	−0.690296	−0.345148	+	0.938548i	$0.612171\pi$
$752$	0	0
$753$	79.9264	2.91268
$754$	0	0
$755$	19.2334	0.699974
$756$	0	0
$757$	10.9182	0.396829	0.198414	−	0.980118i	$-0.436421\pi$
$757$	10.9182	0.396829	0.198414	+	0.980118i	$0.436421\pi$
$758$	0	0
$759$	−6.18838	−0.224624
$760$	0	0
$761$	−30.3531	−1.10030	−0.550149	−	0.835067i	$-0.685429\pi$
$761$	−30.3531	−1.10030	−0.550149	+	0.835067i	$0.685429\pi$
$762$	0	0
$763$	1.45857	0.0528036
$764$	0	0
$765$	58.9703	2.13207
$766$	0	0
$767$	52.7819	1.90584
$768$	0	0
$769$	−19.7369	−0.711731	−0.355865	−	0.934537i	$-0.615814\pi$
$769$	−19.7369	−0.711731	−0.355865	+	0.934537i	$0.615814\pi$
$770$	0	0
$771$	105.814	3.81079
$772$	0	0
$773$	−24.3909	−0.877281	−0.438641	−	0.898663i	$-0.644540\pi$
$773$	−24.3909	−0.877281	−0.438641	+	0.898663i	$0.644540\pi$
$774$	0	0
$775$	−6.00000	−0.215526
$776$	0	0
$777$	21.8874	0.785206
$778$	0	0
$779$	−17.2854	−0.619315
$780$	0	0
$781$	0.300986	0.0107701
$782$	0	0
$783$	−118.948	−4.25084
$784$	0	0
$785$	9.69901	0.346173
$786$	0	0
$787$	−33.7890	−1.20445	−0.602223	−	0.798328i	$-0.705718\pi$
$787$	−33.7890	−1.20445	−0.602223	+	0.798328i	$0.705718\pi$
$788$	0	0
$789$	44.7819	1.59428
$790$	0	0
$791$	−14.0000	−0.497783
$792$	0	0
$793$	30.1374	1.07021
$794$	0	0
$795$	22.8945	0.811984
$796$	0	0
$797$	36.8435	1.30506	0.652532	−	0.757761i	$-0.273707\pi$
$797$	36.8435	1.30506	0.652532	+	0.757761i	$0.273707\pi$
$798$	0	0
$799$	−10.7677	−0.380934
$800$	0	0
$801$	−71.2344	−2.51694
$802$	0	0
$803$	10.1884	0.359540
$804$	0	0
$805$	1.84951	0.0651866
$806$	0	0
$807$	−19.3980	−0.682843
$808$	0	0
$809$	37.6990	1.32543	0.662713	−	0.748873i	$-0.269405\pi$
$809$	37.6990	1.32543	0.662713	+	0.748873i	$0.269405\pi$
$810$	0	0
$811$	−38.3304	−1.34596	−0.672981	−	0.739660i	$-0.734987\pi$
$811$	−38.3304	−1.34596	−0.672981	+	0.739660i	$0.734987\pi$
$812$	0	0
$813$	−67.1728	−2.35585
$814$	0	0
$815$	−8.00000	−0.280228
$816$	0	0
$817$	−0.931308	−0.0325823
$818$	0	0
$819$	54.8435	1.91639
$820$	0	0
$821$	−18.1363	−0.632962	−0.316481	−	0.948599i	$-0.602501\pi$
$821$	−18.1363	−0.632962	−0.316481	+	0.948599i	$0.602501\pi$
$822$	0	0
$823$	7.86368	0.274111	0.137055	−	0.990563i	$-0.456236\pi$
$823$	7.86368	0.274111	0.137055	+	0.990563i	$0.456236\pi$
$824$	0	0
$825$	3.34596	0.116491
$826$	0	0
$827$	−12.0000	−0.417281	−0.208640	−	0.977992i	$-0.566904\pi$
$827$	−12.0000	−0.417281	−0.208640	+	0.977992i	$0.566904\pi$
$828$	0	0
$829$	2.80453	0.0974053	0.0487027	−	0.998813i	$-0.484491\pi$
$829$	2.80453	0.0974053	0.0487027	+	0.998813i	$0.484491\pi$
$830$	0	0
$831$	−73.6112	−2.55354
$832$	0	0
$833$	−7.19547	−0.249308
$834$	0	0
$835$	8.00000	0.276851
$836$	0	0
$837$	104.303	3.60524
$838$	0	0
$839$	−36.9929	−1.27714	−0.638569	−	0.769565i	$-0.720473\pi$
$839$	−36.9929	−1.27714	−0.638569	+	0.769565i	$0.720473\pi$
$840$	0	0
$841$	17.8187	0.614438
$842$	0	0
$843$	21.0829	0.726133
$844$	0	0
$845$	−31.7819	−1.09333
$846$	0	0
$847$	1.00000	0.0343604
$848$	0	0
$849$	74.9193	2.57122
$850$	0	0
$851$	12.0984	0.414729
$852$	0	0
$853$	−49.2571	−1.68653	−0.843265	−	0.537498i	$-0.819370\pi$
$853$	−49.2571	−1.68653	−0.843265	+	0.537498i	$0.819370\pi$
$854$	0	0
$855$	−15.1576	−0.518378
$856$	0	0
$857$	−20.3541	−0.695283	−0.347642	−	0.937627i	$-0.613017\pi$
$857$	−20.3541	−0.695283	−0.347642	+	0.937627i	$0.613017\pi$
$858$	0	0
$859$	34.2783	1.16956	0.584781	−	0.811191i	$-0.301180\pi$
$859$	34.2783	1.16956	0.584781	+	0.811191i	$0.301180\pi$
$860$	0	0
$861$	31.2713	1.06572
$862$	0	0
$863$	12.4373	0.423371	0.211685	−	0.977338i	$-0.432105\pi$
$863$	12.4373	0.423371	0.211685	+	0.977338i	$0.432105\pi$
$864$	0	0
$865$	15.4965	0.526895
$866$	0	0
$867$	−116.355	−3.95163
$868$	0	0
$869$	−12.2404	−0.415229
$870$	0	0
$871$	−53.5354	−1.81398
$872$	0	0
$873$	31.5485	1.06776
$874$	0	0
$875$	−1.00000	−0.0338062
$876$	0	0
$877$	34.4809	1.16434	0.582169	−	0.813068i	$-0.302204\pi$
$877$	34.4809	1.16434	0.582169	+	0.813068i	$0.302204\pi$
$878$	0	0
$879$	35.7748	1.20665
$880$	0	0
$881$	27.3839	0.922585	0.461293	−	0.887248i	$-0.347386\pi$
$881$	27.3839	0.922585	0.461293	+	0.887248i	$0.347386\pi$
$882$	0	0
$883$	−22.0900	−0.743386	−0.371693	−	0.928356i	$-0.621223\pi$
$883$	−22.0900	−0.743386	−0.371693	+	0.928356i	$0.621223\pi$
$884$	0	0
$885$	26.3909	0.887122
$886$	0	0
$887$	−25.7890	−0.865909	−0.432954	−	0.901416i	$-0.642529\pi$
$887$	−25.7890	−0.865909	−0.432954	+	0.901416i	$0.642529\pi$
$888$	0	0
$889$	19.7748	0.663225
$890$	0	0
$891$	−33.5793	−1.12495
$892$	0	0
$893$	2.76771	0.0926178
$894$	0	0
$895$	−13.8874	−0.464204
$896$	0	0
$897$	41.4122	1.38271
$898$	0	0
$899$	−41.0545	−1.36924
$900$	0	0
$901$	−49.2344	−1.64024
$902$	0	0
$903$	1.68484	0.0560679
$904$	0	0
$905$	−18.7819	−0.624331
$906$	0	0
$907$	45.4596	1.50946	0.754731	−	0.656034i	$-0.227767\pi$
$907$	45.4596	1.50946	0.754731	+	0.656034i	$0.227767\pi$
$908$	0	0
$909$	56.5035	1.87410
$910$	0	0
$911$	15.8506	0.525153	0.262576	−	0.964911i	$-0.415428\pi$
$911$	15.8506	0.525153	0.262576	+	0.964911i	$0.415428\pi$
$912$	0	0
$913$	−1.30807	−0.0432909
$914$	0	0
$915$	15.0687	0.498156
$916$	0	0
$917$	−2.15049	−0.0710155
$918$	0	0
$919$	−18.6314	−0.614593	−0.307296	−	0.951614i	$-0.599424\pi$
$919$	−18.6314	−0.614593	−0.307296	+	0.951614i	$0.599424\pi$
$920$	0	0
$921$	−30.3909	−1.00142
$922$	0	0
$923$	−2.01418	−0.0662974
$924$	0	0
$925$	−6.54143	−0.215081
$926$	0	0
$927$	86.7025	2.84768
$928$	0	0
$929$	−29.2486	−0.959616	−0.479808	−	0.877374i	$-0.659294\pi$
$929$	−29.2486	−0.959616	−0.479808	+	0.877374i	$0.659294\pi$
$930$	0	0
$931$	1.84951	0.0606151
$932$	0	0
$933$	−38.0900	−1.24701
$934$	0	0
$935$	−7.19547	−0.235317
$936$	0	0
$937$	−46.4441	−1.51726	−0.758631	−	0.651521i	$-0.774131\pi$
$937$	−46.4441	−1.51726	−0.758631	+	0.651521i	$0.774131\pi$
$938$	0	0
$939$	79.0460	2.57957
$940$	0	0
$941$	−39.2713	−1.28021	−0.640103	−	0.768289i	$-0.721108\pi$
$941$	−39.2713	−1.28021	−0.640103	+	0.768289i	$0.721108\pi$
$942$	0	0
$943$	17.2854	0.562891
$944$	0	0
$945$	17.3839	0.565497
$946$	0	0
$947$	−54.2415	−1.76261	−0.881306	−	0.472546i	$-0.843335\pi$
$947$	−54.2415	−1.76261	−0.881306	+	0.472546i	$0.843335\pi$
$948$	0	0
$949$	−68.1799	−2.21321
$950$	0	0
$951$	31.9016	1.03448
$952$	0	0
$953$	59.3612	1.92290	0.961449	−	0.274983i	$-0.0886723\pi$
$953$	59.3612	1.92290	0.961449	+	0.274983i	$0.0886723\pi$
$954$	0	0
$955$	−6.39094	−0.206806
$956$	0	0
$957$	22.8945	0.740074
$958$	0	0
$959$	−8.39094	−0.270958
$960$	0	0
$961$	5.00000	0.161290
$962$	0	0
$963$	61.4370	1.97978
$964$	0	0
$965$	13.1955	0.424777
$966$	0	0
$967$	18.7677	0.603529	0.301764	−	0.953383i	$-0.402424\pi$
$967$	18.7677	0.603529	0.301764	+	0.953383i	$0.402424\pi$
$968$	0	0
$969$	44.5283	1.43046
$970$	0	0
$971$	59.4370	1.90742	0.953712	−	0.300722i	$-0.0972277\pi$
$971$	59.4370	1.90742	0.953712	+	0.300722i	$0.0972277\pi$
$972$	0	0
$973$	−17.9253	−0.574658
$974$	0	0
$975$	−22.3909	−0.717084
$976$	0	0
$977$	−32.7677	−1.04833	−0.524166	−	0.851616i	$-0.675623\pi$
$977$	−32.7677	−1.04833	−0.524166	+	0.851616i	$0.675623\pi$
$978$	0	0
$979$	8.69193	0.277795
$980$	0	0
$981$	11.9536	0.381650
$982$	0	0
$983$	−13.4207	−0.428053	−0.214027	−	0.976828i	$-0.568658\pi$
$983$	−13.4207	−0.428053	−0.214027	+	0.976828i	$0.568658\pi$
$984$	0	0
$985$	4.69193	0.149497
$986$	0	0
$987$	−5.00709	−0.159377
$988$	0	0
$989$	0.931308	0.0296139
$990$	0	0
$991$	38.1657	1.21237	0.606187	−	0.795322i	$-0.292698\pi$
$991$	38.1657	1.21237	0.606187	+	0.795322i	$0.292698\pi$
$992$	0	0
$993$	1.68484	0.0534668
$994$	0	0
$995$	−9.49646	−0.301058
$996$	0	0
$997$	−33.6622	−1.06609	−0.533046	−	0.846086i	$-0.678953\pi$
$997$	−33.6622	−1.06609	−0.533046	+	0.846086i	$0.678953\pi$
$998$	0	0
$999$	113.715	3.59779

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6160.2.a.bi.1.1		3
4.3	odd	2		770.2.a.l.1.3	✓	3
12.11	even	2		6930.2.a.cl.1.3		3
20.3	even	4		3850.2.c.z.1849.6		6
20.7	even	4		3850.2.c.z.1849.1		6
20.19	odd	2		3850.2.a.bu.1.1		3
28.27	even	2		5390.2.a.bz.1.1		3
44.43	even	2		8470.2.a.cl.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
770.2.a.l.1.3	✓	3	4.3	odd	2
3850.2.a.bu.1.1		3	20.19	odd	2
3850.2.c.z.1849.1		6	20.7	even	4
3850.2.c.z.1849.6		6	20.3	even	4
5390.2.a.bz.1.1		3	28.27	even	2
6160.2.a.bi.1.1		3	1.1	even	1	trivial
6930.2.a.cl.1.3		3	12.11	even	2
8470.2.a.cl.1.3		3	44.43	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 $\F_{q}$

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

\(f(q)\)	\(=\)	\(q-3.34596 q^{3} -1.00000 q^{5} +1.00000 q^{7} +8.19547 q^{9} +O(q^{10})\)	\(q-3.34596 q^{3} -1.00000 q^{5} +1.00000 q^{7} +8.19547 q^{9} -1.00000 q^{11} +6.69193 q^{13} +3.34596 q^{15} -7.19547 q^{17} +1.84951 q^{19} -3.34596 q^{21} -1.84951 q^{23} +1.00000 q^{25} -17.3839 q^{27} +6.84242 q^{29} -6.00000 q^{31} +3.34596 q^{33} -1.00000 q^{35} -6.54143 q^{37} -22.3909 q^{39} -9.34596 q^{41} -0.503544 q^{43} -8.19547 q^{45} +1.49646 q^{47} +1.00000 q^{49} +24.0758 q^{51} +6.84242 q^{53} +1.00000 q^{55} -6.18838 q^{57} +7.88740 q^{59} +4.50354 q^{61} +8.19547 q^{63} -6.69193 q^{65} -8.00000 q^{67} +6.18838 q^{69} -0.300986 q^{71} -10.1884 q^{73} -3.34596 q^{75} -1.00000 q^{77} +12.2404 q^{79} +33.5793 q^{81} +1.30807 q^{83} +7.19547 q^{85} -22.8945 q^{87} -8.69193 q^{89} +6.69193 q^{91} +20.0758 q^{93} -1.84951 q^{95} +3.84951 q^{97} -8.19547 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{5} + 3 q^{7} + 11 q^{9}+O(q^{10}) \) 3 * q - 3 * q^5 + 3 * q^7 + 11 * q^9	\( 3 q - 3 q^{5} + 3 q^{7} + 11 q^{9} - 3 q^{11} - 8 q^{17} + 2 q^{19} - 2 q^{23} + 3 q^{25} - 12 q^{27} + 4 q^{29} - 18 q^{31} - 3 q^{35} + 4 q^{37} - 40 q^{39} - 18 q^{41} - 8 q^{43} - 11 q^{45} - 2 q^{47} + 3 q^{49} + 12 q^{51} + 4 q^{53} + 3 q^{55} + 8 q^{57} - 10 q^{59} + 20 q^{61} + 11 q^{63} - 24 q^{67} - 8 q^{69} - 8 q^{71} - 4 q^{73} - 3 q^{77} + 6 q^{79} + 47 q^{81} + 24 q^{83} + 8 q^{85} - 48 q^{87} - 6 q^{89} - 2 q^{95} + 8 q^{97} - 11 q^{99}+O(q^{100}) \) 3 * q - 3 * q^5 + 3 * q^7 + 11 * q^9 - 3 * q^11 - 8 * q^17 + 2 * q^19 - 2 * q^23 + 3 * q^25 - 12 * q^27 + 4 * q^29 - 18 * q^31 - 3 * q^35 + 4 * q^37 - 40 * q^39 - 18 * q^41 - 8 * q^43 - 11 * q^45 - 2 * q^47 + 3 * q^49 + 12 * q^51 + 4 * q^53 + 3 * q^55 + 8 * q^57 - 10 * q^59 + 20 * q^61 + 11 * q^63 - 24 * q^67 - 8 * q^69 - 8 * q^71 - 4 * q^73 - 3 * q^77 + 6 * q^79 + 47 * q^81 + 24 * q^83 + 8 * q^85 - 48 * q^87 - 6 * q^89 - 2 * q^95 + 8 * q^97 - 11 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more