LMFDB - Embedded newform 6160.2.a.bv.1.5

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:	$5$
Coefficient field:	5.5.8892720.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 2x^{4} - 10x^{3} + 14x^{2} + 20x - 20 $ x^5 - 2x^4 - 10x^3 + 14x^2 + 20x - 20
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
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			1.5
Root			$-2.47436$ 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)

$f(q)$	$=$	$q+2.47436 q^{3} -1.00000 q^{5} -1.00000 q^{7} +3.12244 q^{9} +O(q^{10})$	$q+2.47436 q^{3} -1.00000 q^{5} -1.00000 q^{7} +3.12244 q^{9} -1.00000 q^{11} +5.50406 q^{13} -2.47436 q^{15} -3.50406 q^{17} -5.51815 q^{19} -2.47436 q^{21} -3.15214 q^{23} +1.00000 q^{25} +0.302965 q^{27} -10.2233 q^{29} -3.38162 q^{31} -2.47436 q^{33} +1.00000 q^{35} +1.15214 q^{37} +13.6190 q^{39} -4.99766 q^{41} +11.6190 q^{43} -3.12244 q^{45} +7.99250 q^{47} +1.00000 q^{49} -8.67029 q^{51} +0.907259 q^{53} +1.00000 q^{55} -13.6539 q^{57} -14.9761 q^{59} +0.618384 q^{61} -3.12244 q^{63} -5.50406 q^{65} -15.1587 q^{67} -7.79953 q^{69} -7.03629 q^{71} -1.59164 q^{73} +2.47436 q^{75} +1.00000 q^{77} -10.3444 q^{79} -8.61768 q^{81} +11.6539 q^{83} +3.50406 q^{85} -25.2961 q^{87} +4.48845 q^{89} -5.50406 q^{91} -8.36733 q^{93} +5.51815 q^{95} +0.383952 q^{97} -3.12244 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 2 q^{3} - 5 q^{5} - 5 q^{7} + 9 q^{9}+O(q^{10}) $ 5 * q - 2 * q^3 - 5 * q^5 - 5 * q^7 + 9 * q^9	$ 5 q - 2 q^{3} - 5 q^{5} - 5 q^{7} + 9 q^{9} - 5 q^{11} + 8 q^{13} + 2 q^{15} + 2 q^{17} - 4 q^{19} + 2 q^{21} - 4 q^{23} + 5 q^{25} - 14 q^{27} - 4 q^{29} - 4 q^{31} + 2 q^{33} + 5 q^{35} - 6 q^{37} + 4 q^{39} - 4 q^{41} - 6 q^{43} - 9 q^{45} + 2 q^{47} + 5 q^{49} - 8 q^{51} + 6 q^{53} + 5 q^{55} - 16 q^{57} - 20 q^{59} + 16 q^{61} - 9 q^{63} - 8 q^{65} - 22 q^{67} + 34 q^{69} + 12 q^{71} + 30 q^{73} - 2 q^{75} + 5 q^{77} - 6 q^{79} + 13 q^{81} + 6 q^{83} - 2 q^{85} + 4 q^{87} + 4 q^{89} - 8 q^{91} - 22 q^{93} + 4 q^{95} + 10 q^{97} - 9 q^{99}+O(q^{100}) $ 5 * q - 2 * q^3 - 5 * q^5 - 5 * q^7 + 9 * q^9 - 5 * q^11 + 8 * q^13 + 2 * q^15 + 2 * q^17 - 4 * q^19 + 2 * q^21 - 4 * q^23 + 5 * q^25 - 14 * q^27 - 4 * q^29 - 4 * q^31 + 2 * q^33 + 5 * q^35 - 6 * q^37 + 4 * q^39 - 4 * q^41 - 6 * q^43 - 9 * q^45 + 2 * q^47 + 5 * q^49 - 8 * q^51 + 6 * q^53 + 5 * q^55 - 16 * q^57 - 20 * q^59 + 16 * q^61 - 9 * q^63 - 8 * q^65 - 22 * q^67 + 34 * q^69 + 12 * q^71 + 30 * q^73 - 2 * q^75 + 5 * q^77 - 6 * q^79 + 13 * q^81 + 6 * q^83 - 2 * q^85 + 4 * q^87 + 4 * q^89 - 8 * q^91 - 22 * q^93 + 4 * q^95 + 10 * q^97 - 9 * 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
$3$	2.47436	1.42857	0.714285	−	0.699855i	$-0.246752\pi$
$3$	2.47436	1.42857	0.714285	+	0.699855i	$0.246752\pi$
$4$	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
$9$	3.12244	1.04081
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	5.50406	1.52655	0.763276	−	0.646073i	$-0.223590\pi$
$13$	5.50406	1.52655	0.763276	+	0.646073i	$0.223590\pi$
$14$	0	0
$15$	−2.47436	−0.638876
$16$	0	0
$17$	−3.50406	−0.849859	−0.424929	−	0.905226i	$-0.639701\pi$
$17$	−3.50406	−0.849859	−0.424929	+	0.905226i	$0.639701\pi$
$18$	0	0
$19$	−5.51815	−1.26595	−0.632975	−	0.774172i	$-0.718166\pi$
$19$	−5.51815	−1.26595	−0.632975	+	0.774172i	$0.718166\pi$
$20$	0	0
$21$	−2.47436	−0.539949
$22$	0	0
$23$	−3.15214	−0.657267	−0.328634	−	0.944457i	$-0.606588\pi$
$23$	−3.15214	−0.657267	−0.328634	+	0.944457i	$0.606588\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0.302965	0.0583056
$28$	0	0
$29$	−10.2233	−1.89842	−0.949209	−	0.314645i	$-0.898115\pi$
$29$	−10.2233	−1.89842	−0.949209	+	0.314645i	$0.898115\pi$
$30$	0	0
$31$	−3.38162	−0.607356	−0.303678	−	0.952775i	$-0.598215\pi$
$31$	−3.38162	−0.607356	−0.303678	+	0.952775i	$0.598215\pi$
$32$	0	0
$33$	−2.47436	−0.430730
$34$	0	0
$35$	1.00000	0.169031
$36$	0	0
$37$	1.15214	0.189411	0.0947056	−	0.995505i	$-0.469809\pi$
$37$	1.15214	0.189411	0.0947056	+	0.995505i	$0.469809\pi$
$38$	0	0
$39$	13.6190	2.18079
$40$	0	0
$41$	−4.99766	−0.780504	−0.390252	−	0.920708i	$-0.627612\pi$
$41$	−4.99766	−0.780504	−0.390252	+	0.920708i	$0.627612\pi$
$42$	0	0
$43$	11.6190	1.77188	0.885941	−	0.463798i	$-0.153514\pi$
$43$	11.6190	1.77188	0.885941	+	0.463798i	$0.153514\pi$
$44$	0	0
$45$	−3.12244	−0.465466
$46$	0	0
$47$	7.99250	1.16583	0.582913	−	0.812534i	$-0.301913\pi$
$47$	7.99250	1.16583	0.582913	+	0.812534i	$0.301913\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−8.67029	−1.21408
$52$	0	0
$53$	0.907259	0.124622	0.0623108	−	0.998057i	$-0.480153\pi$
$53$	0.907259	0.124622	0.0623108	+	0.998057i	$0.480153\pi$
$54$	0	0
$55$	1.00000	0.134840
$56$	0	0
$57$	−13.6539	−1.80850
$58$	0	0
$59$	−14.9761	−1.94972	−0.974860	−	0.222820i	$-0.928474\pi$
$59$	−14.9761	−1.94972	−0.974860	+	0.222820i	$0.928474\pi$
$60$	0	0
$61$	0.618384	0.0791759	0.0395880	−	0.999216i	$-0.487395\pi$
$61$	0.618384	0.0791759	0.0395880	+	0.999216i	$0.487395\pi$
$62$	0	0
$63$	−3.12244	−0.393391
$64$	0	0
$65$	−5.50406	−0.682694
$66$	0	0
$67$	−15.1587	−1.85193	−0.925967	−	0.377604i	$-0.876748\pi$
$67$	−15.1587	−1.85193	−0.925967	+	0.377604i	$0.876748\pi$
$68$	0	0
$69$	−7.79953	−0.938953
$70$	0	0
$71$	−7.03629	−0.835055	−0.417527	−	0.908664i	$-0.637103\pi$
$71$	−7.03629	−0.835055	−0.417527	+	0.908664i	$0.637103\pi$
$72$	0	0
$73$	−1.59164	−0.186287	−0.0931436	−	0.995653i	$-0.529692\pi$
$73$	−1.59164	−0.186287	−0.0931436	+	0.995653i	$0.529692\pi$
$74$	0	0
$75$	2.47436	0.285714
$76$	0	0
$77$	1.00000	0.113961
$78$	0	0
$79$	−10.3444	−1.16384	−0.581919	−	0.813247i	$-0.697698\pi$
$79$	−10.3444	−1.16384	−0.581919	+	0.813247i	$0.697698\pi$
$80$	0	0
$81$	−8.61768	−0.957520
$82$	0	0
$83$	11.6539	1.27918	0.639589	−	0.768717i	$-0.279105\pi$
$83$	11.6539	1.27918	0.639589	+	0.768717i	$0.279105\pi$
$84$	0	0
$85$	3.50406	0.380068
$86$	0	0
$87$	−25.2961	−2.71203
$88$	0	0
$89$	4.48845	0.475774	0.237887	−	0.971293i	$-0.423545\pi$
$89$	4.48845	0.475774	0.237887	+	0.971293i	$0.423545\pi$
$90$	0	0
$91$	−5.50406	−0.576982
$92$	0	0
$93$	−8.36733	−0.867651
$94$	0	0
$95$	5.51815	0.566150
$96$	0	0
$97$	0.383952	0.0389845	0.0194922	−	0.999810i	$-0.493795\pi$
$97$	0.383952	0.0389845	0.0194922	+	0.999810i	$0.493795\pi$
$98$	0	0
$99$	−3.12244	−0.313817
$100$	0	0
$101$	−0.921348	−0.0916776	−0.0458388	−	0.998949i	$-0.514596\pi$
$101$	−0.921348	−0.0916776	−0.0458388	+	0.998949i	$0.514596\pi$
$102$	0	0
$103$	−1.25917	−0.124070	−0.0620351	−	0.998074i	$-0.519759\pi$
$103$	−1.25917	−0.124070	−0.0620351	+	0.998074i	$0.519759\pi$
$104$	0	0
$105$	2.47436	0.241473
$106$	0	0
$107$	4.63543	0.448124	0.224062	−	0.974575i	$-0.428068\pi$
$107$	4.63543	0.448124	0.224062	+	0.974575i	$0.428068\pi$
$108$	0	0
$109$	9.88051	0.946382	0.473191	−	0.880960i	$-0.343102\pi$
$109$	9.88051	0.946382	0.473191	+	0.880960i	$0.343102\pi$
$110$	0	0
$111$	2.85081	0.270587
$112$	0	0
$113$	−9.80985	−0.922833	−0.461416	−	0.887184i	$-0.652659\pi$
$113$	−9.80985	−0.922833	−0.461416	+	0.887184i	$0.652659\pi$
$114$	0	0
$115$	3.15214	0.293939
$116$	0	0
$117$	17.1861	1.58886
$118$	0	0
$119$	3.50406	0.321216
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−12.3660	−1.11501
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	20.2299	1.79511	0.897556	−	0.440900i	$-0.145341\pi$
$127$	20.2299	1.79511	0.897556	+	0.440900i	$0.145341\pi$
$128$	0	0
$129$	28.7496	2.53126
$130$	0	0
$131$	−7.12760	−0.622741	−0.311371	−	0.950289i	$-0.600788\pi$
$131$	−7.12760	−0.622741	−0.311371	+	0.950289i	$0.600788\pi$
$132$	0	0
$133$	5.51815	0.478484
$134$	0	0
$135$	−0.302965	−0.0260751
$136$	0	0
$137$	−20.1669	−1.72297	−0.861485	−	0.507783i	$-0.830465\pi$
$137$	−20.1669	−1.72297	−0.861485	+	0.507783i	$0.830465\pi$
$138$	0	0
$139$	−23.1126	−1.96039	−0.980193	−	0.198045i	$-0.936541\pi$
$139$	−23.1126	−1.96039	−0.980193	+	0.198045i	$0.936541\pi$
$140$	0	0
$141$	19.7763	1.66547
$142$	0	0
$143$	−5.50406	−0.460272
$144$	0	0
$145$	10.2233	0.848999
$146$	0	0
$147$	2.47436	0.204082
$148$	0	0
$149$	4.86773	0.398780	0.199390	−	0.979920i	$-0.436104\pi$
$149$	4.86773	0.398780	0.199390	+	0.979920i	$0.436104\pi$
$150$	0	0
$151$	7.94488	0.646545	0.323272	−	0.946306i	$-0.395217\pi$
$151$	7.94488	0.646545	0.323272	+	0.946306i	$0.395217\pi$
$152$	0	0
$153$	−10.9412	−0.884545
$154$	0	0
$155$	3.38162	0.271618
$156$	0	0
$157$	19.9863	1.59508	0.797541	−	0.603264i	$-0.206134\pi$
$157$	19.9863	1.59508	0.797541	+	0.603264i	$0.206134\pi$
$158$	0	0
$159$	2.24488	0.178031
$160$	0	0
$161$	3.15214	0.248424
$162$	0	0
$163$	4.07927	0.319513	0.159757	−	0.987156i	$-0.448929\pi$
$163$	4.07927	0.319513	0.159757	+	0.987156i	$0.448929\pi$
$164$	0	0
$165$	2.47436	0.192628
$166$	0	0
$167$	−25.4534	−1.96964	−0.984821	−	0.173571i	$-0.944469\pi$
$167$	−25.4534	−1.96964	−0.984821	+	0.173571i	$0.944469\pi$
$168$	0	0
$169$	17.2947	1.33036
$170$	0	0
$171$	−17.2301	−1.31762
$172$	0	0
$173$	23.5870	1.79328	0.896642	−	0.442756i	$-0.145999\pi$
$173$	23.5870	1.79328	0.896642	+	0.442756i	$0.145999\pi$
$174$	0	0
$175$	−1.00000	−0.0755929
$176$	0	0
$177$	−37.0562	−2.78531
$178$	0	0
$179$	−21.5560	−1.61117	−0.805584	−	0.592481i	$-0.798148\pi$
$179$	−21.5560	−1.61117	−0.805584	+	0.592481i	$0.798148\pi$
$180$	0	0
$181$	7.78462	0.578626	0.289313	−	0.957235i	$-0.406573\pi$
$181$	7.78462	0.578626	0.289313	+	0.957235i	$0.406573\pi$
$182$	0	0
$183$	1.53010	0.113108
$184$	0	0
$185$	−1.15214	−0.0847072
$186$	0	0
$187$	3.50406	0.256242
$188$	0	0
$189$	−0.302965	−0.0220374
$190$	0	0
$191$	−11.9850	−0.867205	−0.433602	−	0.901104i	$-0.642758\pi$
$191$	−11.9850	−0.867205	−0.433602	+	0.901104i	$0.642758\pi$
$192$	0	0
$193$	15.6784	1.12856	0.564278	−	0.825585i	$-0.309155\pi$
$193$	15.6784	1.12856	0.564278	+	0.825585i	$0.309155\pi$
$194$	0	0
$195$	−13.6190	−0.975277
$196$	0	0
$197$	−19.2181	−1.36924	−0.684618	−	0.728902i	$-0.740031\pi$
$197$	−19.2181	−1.36924	−0.684618	+	0.728902i	$0.740031\pi$
$198$	0	0
$199$	−14.4282	−1.02279	−0.511395	−	0.859346i	$-0.670871\pi$
$199$	−14.4282	−1.02279	−0.511395	+	0.859346i	$0.670871\pi$
$200$	0	0
$201$	−37.5081	−2.64562
$202$	0	0
$203$	10.2233	0.717535
$204$	0	0
$205$	4.99766	0.349052
$206$	0	0
$207$	−9.84238	−0.684093
$208$	0	0
$209$	5.51815	0.381698
$210$	0	0
$211$	16.2299	1.11731	0.558656	−	0.829400i	$-0.311317\pi$
$211$	16.2299	1.11731	0.558656	+	0.829400i	$0.311317\pi$
$212$	0	0
$213$	−17.4103	−1.19293
$214$	0	0
$215$	−11.6190	−0.792409
$216$	0	0
$217$	3.38162	0.229559
$218$	0	0
$219$	−3.93828	−0.266124
$220$	0	0
$221$	−19.2865	−1.29735
$222$	0	0
$223$	4.26729	0.285759	0.142879	−	0.989740i	$-0.454364\pi$
$223$	4.26729	0.285759	0.142879	+	0.989740i	$0.454364\pi$
$224$	0	0
$225$	3.12244	0.208163
$226$	0	0
$227$	−13.8974	−0.922405	−0.461202	−	0.887295i	$-0.652582\pi$
$227$	−13.8974	−0.922405	−0.461202	+	0.887295i	$0.652582\pi$
$228$	0	0
$229$	6.31328	0.417194	0.208597	−	0.978002i	$-0.433110\pi$
$229$	6.31328	0.417194	0.208597	+	0.978002i	$0.433110\pi$
$230$	0	0
$231$	2.47436	0.162801
$232$	0	0
$233$	23.1109	1.51405	0.757023	−	0.653389i	$-0.226653\pi$
$233$	23.1109	1.51405	0.757023	+	0.653389i	$0.226653\pi$
$234$	0	0
$235$	−7.99250	−0.521373
$236$	0	0
$237$	−25.5958	−1.66262
$238$	0	0
$239$	−8.92713	−0.577448	−0.288724	−	0.957412i	$-0.593231\pi$
$239$	−8.92713	−0.577448	−0.288724	+	0.957412i	$0.593231\pi$
$240$	0	0
$241$	−25.3182	−1.63089	−0.815444	−	0.578836i	$-0.803507\pi$
$241$	−25.3182	−1.63089	−0.815444	+	0.578836i	$0.803507\pi$
$242$	0	0
$243$	−22.2321	−1.42619
$244$	0	0
$245$	−1.00000	−0.0638877
$246$	0	0
$247$	−30.3722	−1.93254
$248$	0	0
$249$	28.8358	1.82740
$250$	0	0
$251$	−2.43158	−0.153480	−0.0767400	−	0.997051i	$-0.524451\pi$
$251$	−2.43158	−0.153480	−0.0767400	+	0.997051i	$0.524451\pi$
$252$	0	0
$253$	3.15214	0.198174
$254$	0	0
$255$	8.67029	0.542955
$256$	0	0
$257$	7.47498	0.466276	0.233138	−	0.972444i	$-0.425101\pi$
$257$	7.47498	0.466276	0.233138	+	0.972444i	$0.425101\pi$
$258$	0	0
$259$	−1.15214	−0.0715907
$260$	0	0
$261$	−31.9217	−1.97590
$262$	0	0
$263$	17.0316	1.05021	0.525107	−	0.851036i	$-0.324025\pi$
$263$	17.0316	1.05021	0.525107	+	0.851036i	$0.324025\pi$
$264$	0	0
$265$	−0.907259	−0.0557325
$266$	0	0
$267$	11.1060	0.679677
$268$	0	0
$269$	25.6094	1.56143	0.780716	−	0.624887i	$-0.214855\pi$
$269$	25.6094	1.56143	0.780716	+	0.624887i	$0.214855\pi$
$270$	0	0
$271$	−4.94871	−0.300613	−0.150306	−	0.988639i	$-0.548026\pi$
$271$	−4.94871	−0.300613	−0.150306	+	0.988639i	$0.548026\pi$
$272$	0	0
$273$	−13.6190	−0.824260
$274$	0	0
$275$	−1.00000	−0.0603023
$276$	0	0
$277$	−24.0408	−1.44447	−0.722235	−	0.691648i	$-0.756885\pi$
$277$	−24.0408	−1.44447	−0.722235	+	0.691648i	$0.756885\pi$
$278$	0	0
$279$	−10.5589	−0.632145
$280$	0	0
$281$	5.10602	0.304599	0.152300	−	0.988334i	$-0.451332\pi$
$281$	5.10602	0.304599	0.152300	+	0.988334i	$0.451332\pi$
$282$	0	0
$283$	−17.2218	−1.02373	−0.511864	−	0.859066i	$-0.671045\pi$
$283$	−17.2218	−1.02373	−0.511864	+	0.859066i	$0.671045\pi$
$284$	0	0
$285$	13.6539	0.808785
$286$	0	0
$287$	4.99766	0.295003
$288$	0	0
$289$	−4.72158	−0.277740
$290$	0	0
$291$	0.950035	0.0556920
$292$	0	0
$293$	24.4391	1.42775	0.713874	−	0.700274i	$-0.246939\pi$
$293$	24.4391	1.42775	0.713874	+	0.700274i	$0.246939\pi$
$294$	0	0
$295$	14.9761	0.871941
$296$	0	0
$297$	−0.302965	−0.0175798
$298$	0	0
$299$	−17.3496	−1.00335
$300$	0	0
$301$	−11.6190	−0.669708
$302$	0	0
$303$	−2.27974	−0.130968
$304$	0	0
$305$	−0.618384	−0.0354085
$306$	0	0
$307$	−4.76455	−0.271927	−0.135964	−	0.990714i	$-0.543413\pi$
$307$	−4.76455	−0.271927	−0.135964	+	0.990714i	$0.543413\pi$
$308$	0	0
$309$	−3.11565	−0.177243
$310$	0	0
$311$	20.3495	1.15391	0.576957	−	0.816774i	$-0.304240\pi$
$311$	20.3495	1.15391	0.576957	+	0.816774i	$0.304240\pi$
$312$	0	0
$313$	−13.5476	−0.765758	−0.382879	−	0.923798i	$-0.625067\pi$
$313$	−13.5476	−0.765758	−0.382879	+	0.923798i	$0.625067\pi$
$314$	0	0
$315$	3.12244	0.175930
$316$	0	0
$317$	8.12073	0.456105	0.228053	−	0.973649i	$-0.426764\pi$
$317$	8.12073	0.456105	0.228053	+	0.973649i	$0.426764\pi$
$318$	0	0
$319$	10.2233	0.572395
$320$	0	0
$321$	11.4697	0.640177
$322$	0	0
$323$	19.3359	1.07588
$324$	0	0
$325$	5.50406	0.305310
$326$	0	0
$327$	24.4479	1.35197
$328$	0	0
$329$	−7.99250	−0.440641
$330$	0	0
$331$	−6.54785	−0.359902	−0.179951	−	0.983676i	$-0.557594\pi$
$331$	−6.54785	−0.359902	−0.179951	+	0.983676i	$0.557594\pi$
$332$	0	0
$333$	3.59750	0.197142
$334$	0	0
$335$	15.1587	0.828210
$336$	0	0
$337$	9.96047	0.542581	0.271291	−	0.962497i	$-0.412550\pi$
$337$	9.96047	0.542581	0.271291	+	0.962497i	$0.412550\pi$
$338$	0	0
$339$	−24.2731	−1.31833
$340$	0	0
$341$	3.38162	0.183125
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	7.79953	0.419912
$346$	0	0
$347$	−20.4698	−1.09888	−0.549439	−	0.835534i	$-0.685158\pi$
$347$	−20.4698	−1.09888	−0.549439	+	0.835534i	$0.685158\pi$
$348$	0	0
$349$	−15.7585	−0.843533	−0.421766	−	0.906705i	$-0.638590\pi$
$349$	−15.7585	−0.843533	−0.421766	+	0.906705i	$0.638590\pi$
$350$	0	0
$351$	1.66754	0.0890065
$352$	0	0
$353$	−17.4886	−0.930827	−0.465413	−	0.885093i	$-0.654094\pi$
$353$	−17.4886	−0.930827	−0.465413	+	0.885093i	$0.654094\pi$
$354$	0	0
$355$	7.03629	0.373448
$356$	0	0
$357$	8.67029	0.458880
$358$	0	0
$359$	33.4995	1.76804	0.884018	−	0.467452i	$-0.154828\pi$
$359$	33.4995	1.76804	0.884018	+	0.467452i	$0.154828\pi$
$360$	0	0
$361$	11.4499	0.602629
$362$	0	0
$363$	2.47436	0.129870
$364$	0	0
$365$	1.59164	0.0833102
$366$	0	0
$367$	13.0053	0.678871	0.339435	−	0.940629i	$-0.389764\pi$
$367$	13.0053	0.678871	0.339435	+	0.940629i	$0.389764\pi$
$368$	0	0
$369$	−15.6049	−0.812359
$370$	0	0
$371$	−0.907259	−0.0471026
$372$	0	0
$373$	−27.2879	−1.41291	−0.706456	−	0.707757i	$-0.749707\pi$
$373$	−27.2879	−1.41291	−0.706456	+	0.707757i	$0.749707\pi$
$374$	0	0
$375$	−2.47436	−0.127775
$376$	0	0
$377$	−56.2696	−2.89803
$378$	0	0
$379$	−31.4265	−1.61427	−0.807136	−	0.590365i	$-0.798984\pi$
$379$	−31.4265	−1.61427	−0.807136	+	0.590365i	$0.798984\pi$
$380$	0	0
$381$	50.0560	2.56445
$382$	0	0
$383$	−32.6873	−1.67024	−0.835122	−	0.550064i	$-0.814603\pi$
$383$	−32.6873	−1.67024	−0.835122	+	0.550064i	$0.814603\pi$
$384$	0	0
$385$	−1.00000	−0.0509647
$386$	0	0
$387$	36.2797	1.84420
$388$	0	0
$389$	−2.57775	−0.130697	−0.0653486	−	0.997862i	$-0.520816\pi$
$389$	−2.57775	−0.130697	−0.0653486	+	0.997862i	$0.520816\pi$
$390$	0	0
$391$	11.0453	0.558584
$392$	0	0
$393$	−17.6362	−0.889630
$394$	0	0
$395$	10.3444	0.520484
$396$	0	0
$397$	8.29102	0.416114	0.208057	−	0.978117i	$-0.433286\pi$
$397$	8.29102	0.416114	0.208057	+	0.978117i	$0.433286\pi$
$398$	0	0
$399$	13.6539	0.683548
$400$	0	0
$401$	−2.74832	−0.137245	−0.0686223	−	0.997643i	$-0.521860\pi$
$401$	−2.74832	−0.137245	−0.0686223	+	0.997643i	$0.521860\pi$
$402$	0	0
$403$	−18.6126	−0.927160
$404$	0	0
$405$	8.61768	0.428216
$406$	0	0
$407$	−1.15214	−0.0571096
$408$	0	0
$409$	−2.90108	−0.143449	−0.0717247	−	0.997424i	$-0.522850\pi$
$409$	−2.90108	−0.143449	−0.0717247	+	0.997424i	$0.522850\pi$
$410$	0	0
$411$	−49.9000	−2.46138
$412$	0	0
$413$	14.9761	0.736925
$414$	0	0
$415$	−11.6539	−0.572066
$416$	0	0
$417$	−57.1888	−2.80055
$418$	0	0
$419$	−10.4773	−0.511850	−0.255925	−	0.966697i	$-0.582380\pi$
$419$	−10.4773	−0.511850	−0.255925	+	0.966697i	$0.582380\pi$
$420$	0	0
$421$	29.5805	1.44167	0.720833	−	0.693109i	$-0.243760\pi$
$421$	29.5805	1.44167	0.720833	+	0.693109i	$0.243760\pi$
$422$	0	0
$423$	24.9561	1.21341
$424$	0	0
$425$	−3.50406	−0.169972
$426$	0	0
$427$	−0.618384	−0.0299257
$428$	0	0
$429$	−13.6190	−0.657532
$430$	0	0
$431$	−20.8936	−1.00641	−0.503204	−	0.864167i	$-0.667846\pi$
$431$	−20.8936	−1.00641	−0.503204	+	0.864167i	$0.667846\pi$
$432$	0	0
$433$	−27.8967	−1.34063	−0.670316	−	0.742076i	$-0.733841\pi$
$433$	−27.8967	−1.34063	−0.670316	+	0.742076i	$0.733841\pi$
$434$	0	0
$435$	25.2961	1.21285
$436$	0	0
$437$	17.3940	0.832067
$438$	0	0
$439$	13.0184	0.621336	0.310668	−	0.950518i	$-0.399447\pi$
$439$	13.0184	0.621336	0.310668	+	0.950518i	$0.399447\pi$
$440$	0	0
$441$	3.12244	0.148688
$442$	0	0
$443$	22.9686	1.09127	0.545635	−	0.838023i	$-0.316288\pi$
$443$	22.9686	1.09127	0.545635	+	0.838023i	$0.316288\pi$
$444$	0	0
$445$	−4.48845	−0.212773
$446$	0	0
$447$	12.0445	0.569685
$448$	0	0
$449$	−6.88567	−0.324955	−0.162478	−	0.986712i	$-0.551949\pi$
$449$	−6.88567	−0.324955	−0.162478	+	0.986712i	$0.551949\pi$
$450$	0	0
$451$	4.99766	0.235331
$452$	0	0
$453$	19.6585	0.923635
$454$	0	0
$455$	5.50406	0.258034
$456$	0	0
$457$	37.3173	1.74563	0.872814	−	0.488052i	$-0.162292\pi$
$457$	37.3173	1.74563	0.872814	+	0.488052i	$0.162292\pi$
$458$	0	0
$459$	−1.06161	−0.0495515
$460$	0	0
$461$	−14.6718	−0.683334	−0.341667	−	0.939821i	$-0.610991\pi$
$461$	−14.6718	−0.683334	−0.341667	+	0.939821i	$0.610991\pi$
$462$	0	0
$463$	1.70432	0.0792064	0.0396032	−	0.999215i	$-0.487391\pi$
$463$	1.70432	0.0792064	0.0396032	+	0.999215i	$0.487391\pi$
$464$	0	0
$465$	8.36733	0.388025
$466$	0	0
$467$	9.67827	0.447857	0.223929	−	0.974606i	$-0.428112\pi$
$467$	9.67827	0.447857	0.223929	+	0.974606i	$0.428112\pi$
$468$	0	0
$469$	15.1587	0.699965
$470$	0	0
$471$	49.4533	2.27869
$472$	0	0
$473$	−11.6190	−0.534242
$474$	0	0
$475$	−5.51815	−0.253190
$476$	0	0
$477$	2.83286	0.129708
$478$	0	0
$479$	−25.5097	−1.16557	−0.582785	−	0.812626i	$-0.698037\pi$
$479$	−25.5097	−1.16557	−0.582785	+	0.812626i	$0.698037\pi$
$480$	0	0
$481$	6.34146	0.289146
$482$	0	0
$483$	7.79953	0.354891
$484$	0	0
$485$	−0.383952	−0.0174344
$486$	0	0
$487$	12.9235	0.585619	0.292810	−	0.956171i	$-0.405410\pi$
$487$	12.9235	0.585619	0.292810	+	0.956171i	$0.405410\pi$
$488$	0	0
$489$	10.0936	0.456447
$490$	0	0
$491$	−10.7370	−0.484553	−0.242276	−	0.970207i	$-0.577894\pi$
$491$	−10.7370	−0.484553	−0.242276	+	0.970207i	$0.577894\pi$
$492$	0	0
$493$	35.8230	1.61339
$494$	0	0
$495$	3.12244	0.140343
$496$	0	0
$497$	7.03629	0.315621
$498$	0	0
$499$	−4.95004	−0.221594	−0.110797	−	0.993843i	$-0.535340\pi$
$499$	−4.95004	−0.221594	−0.110797	+	0.993843i	$0.535340\pi$
$500$	0	0
$501$	−62.9808	−2.81377
$502$	0	0
$503$	8.47813	0.378021	0.189010	−	0.981975i	$-0.439472\pi$
$503$	8.47813	0.378021	0.189010	+	0.981975i	$0.439472\pi$
$504$	0	0
$505$	0.921348	0.0409995
$506$	0	0
$507$	42.7932	1.90051
$508$	0	0
$509$	34.7097	1.53848	0.769241	−	0.638959i	$-0.220634\pi$
$509$	34.7097	1.53848	0.769241	+	0.638959i	$0.220634\pi$
$510$	0	0
$511$	1.59164	0.0704100
$512$	0	0
$513$	−1.67180	−0.0738120
$514$	0	0
$515$	1.25917	0.0554859
$516$	0	0
$517$	−7.99250	−0.351510
$518$	0	0
$519$	58.3626	2.56183
$520$	0	0
$521$	−6.64575	−0.291156	−0.145578	−	0.989347i	$-0.546504\pi$
$521$	−6.64575	−0.291156	−0.145578	+	0.989347i	$0.546504\pi$
$522$	0	0
$523$	33.5363	1.46644	0.733220	−	0.679991i	$-0.238017\pi$
$523$	33.5363	1.46644	0.733220	+	0.679991i	$0.238017\pi$
$524$	0	0
$525$	−2.47436	−0.107990
$526$	0	0
$527$	11.8494	0.516167
$528$	0	0
$529$	−13.0640	−0.568000
$530$	0	0
$531$	−46.7619	−2.02930
$532$	0	0
$533$	−27.5074	−1.19148
$534$	0	0
$535$	−4.63543	−0.200407
$536$	0	0
$537$	−53.3371	−2.30167
$538$	0	0
$539$	−1.00000	−0.0430730
$540$	0	0
$541$	17.1955	0.739293	0.369647	−	0.929172i	$-0.379479\pi$
$541$	17.1955	0.739293	0.369647	+	0.929172i	$0.379479\pi$
$542$	0	0
$543$	19.2619	0.826608
$544$	0	0
$545$	−9.88051	−0.423235
$546$	0	0
$547$	−15.8662	−0.678390	−0.339195	−	0.940716i	$-0.610155\pi$
$547$	−15.8662	−0.678390	−0.339195	+	0.940716i	$0.610155\pi$
$548$	0	0
$549$	1.93087	0.0824074
$550$	0	0
$551$	56.4137	2.40330
$552$	0	0
$553$	10.3444	0.439889
$554$	0	0
$555$	−2.85081	−0.121010
$556$	0	0
$557$	−10.1938	−0.431925	−0.215962	−	0.976402i	$-0.569289\pi$
$557$	−10.1938	−0.431925	−0.215962	+	0.976402i	$0.569289\pi$
$558$	0	0
$559$	63.9517	2.70487
$560$	0	0
$561$	8.67029	0.366060
$562$	0	0
$563$	6.52827	0.275134	0.137567	−	0.990492i	$-0.456072\pi$
$563$	6.52827	0.275134	0.137567	+	0.990492i	$0.456072\pi$
$564$	0	0
$565$	9.80985	0.412703
$566$	0	0
$567$	8.61768	0.361909
$568$	0	0
$569$	17.9850	0.753971	0.376985	−	0.926219i	$-0.376961\pi$
$569$	17.9850	0.753971	0.376985	+	0.926219i	$0.376961\pi$
$570$	0	0
$571$	8.99780	0.376546	0.188273	−	0.982117i	$-0.439711\pi$
$571$	8.99780	0.376546	0.188273	+	0.982117i	$0.439711\pi$
$572$	0	0
$573$	−29.6552	−1.23886
$574$	0	0
$575$	−3.15214	−0.131453
$576$	0	0
$577$	38.0523	1.58414	0.792070	−	0.610431i	$-0.209004\pi$
$577$	38.0523	1.58414	0.792070	+	0.610431i	$0.209004\pi$
$578$	0	0
$579$	38.7940	1.61222
$580$	0	0
$581$	−11.6539	−0.483484
$582$	0	0
$583$	−0.907259	−0.0375748
$584$	0	0
$585$	−17.1861	−0.710558
$586$	0	0
$587$	41.5316	1.71419	0.857095	−	0.515158i	$-0.172267\pi$
$587$	41.5316	1.71419	0.857095	+	0.515158i	$0.172267\pi$
$588$	0	0
$589$	18.6603	0.768882
$590$	0	0
$591$	−47.5525	−1.95605
$592$	0	0
$593$	22.9810	0.943715	0.471857	−	0.881675i	$-0.343584\pi$
$593$	22.9810	0.943715	0.471857	+	0.881675i	$0.343584\pi$
$594$	0	0
$595$	−3.50406	−0.143652
$596$	0	0
$597$	−35.7006	−1.46113
$598$	0	0
$599$	−21.4932	−0.878189	−0.439094	−	0.898441i	$-0.644701\pi$
$599$	−21.4932	−0.878189	−0.439094	+	0.898441i	$0.644701\pi$
$600$	0	0
$601$	−3.79172	−0.154667	−0.0773337	−	0.997005i	$-0.524641\pi$
$601$	−3.79172	−0.154667	−0.0773337	+	0.997005i	$0.524641\pi$
$602$	0	0
$603$	−47.3323	−1.92752
$604$	0	0
$605$	−1.00000	−0.0406558
$606$	0	0
$607$	0.953386	0.0386967	0.0193484	−	0.999813i	$-0.493841\pi$
$607$	0.953386	0.0386967	0.0193484	+	0.999813i	$0.493841\pi$
$608$	0	0
$609$	25.2961	1.02505
$610$	0	0
$611$	43.9912	1.77969
$612$	0	0
$613$	7.53437	0.304310	0.152155	−	0.988357i	$-0.451379\pi$
$613$	7.53437	0.304310	0.152155	+	0.988357i	$0.451379\pi$
$614$	0	0
$615$	12.3660	0.498645
$616$	0	0
$617$	12.3686	0.497940	0.248970	−	0.968511i	$-0.419908\pi$
$617$	12.3686	0.497940	0.248970	+	0.968511i	$0.419908\pi$
$618$	0	0
$619$	−6.17770	−0.248303	−0.124151	−	0.992263i	$-0.539621\pi$
$619$	−6.17770	−0.248303	−0.124151	+	0.992263i	$0.539621\pi$
$620$	0	0
$621$	−0.954988	−0.0383224
$622$	0	0
$623$	−4.48845	−0.179826
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	13.6539	0.545283
$628$	0	0
$629$	−4.03718	−0.160973
$630$	0	0
$631$	−22.3872	−0.891220	−0.445610	−	0.895227i	$-0.647013\pi$
$631$	−22.3872	−0.891220	−0.445610	+	0.895227i	$0.647013\pi$
$632$	0	0
$633$	40.1585	1.59616
$634$	0	0
$635$	−20.2299	−0.802799
$636$	0	0
$637$	5.50406	0.218079
$638$	0	0
$639$	−21.9704	−0.869136
$640$	0	0
$641$	−1.16773	−0.0461227	−0.0230614	−	0.999734i	$-0.507341\pi$
$641$	−1.16773	−0.0461227	−0.0230614	+	0.999734i	$0.507341\pi$
$642$	0	0
$643$	−6.03032	−0.237813	−0.118906	−	0.992905i	$-0.537939\pi$
$643$	−6.03032	−0.237813	−0.118906	+	0.992905i	$0.537939\pi$
$644$	0	0
$645$	−28.7496	−1.13201
$646$	0	0
$647$	−18.2404	−0.717105	−0.358553	−	0.933509i	$-0.616730\pi$
$647$	−18.2404	−0.717105	−0.358553	+	0.933509i	$0.616730\pi$
$648$	0	0
$649$	14.9761	0.587863
$650$	0	0
$651$	8.36733	0.327941
$652$	0	0
$653$	0.625485	0.0244771	0.0122386	−	0.999925i	$-0.496104\pi$
$653$	0.625485	0.0244771	0.0122386	+	0.999925i	$0.496104\pi$
$654$	0	0
$655$	7.12760	0.278498
$656$	0	0
$657$	−4.96980	−0.193890
$658$	0	0
$659$	45.6456	1.77810	0.889049	−	0.457812i	$-0.151367\pi$
$659$	45.6456	1.77810	0.889049	+	0.457812i	$0.151367\pi$
$660$	0	0
$661$	−27.9448	−1.08693	−0.543463	−	0.839433i	$-0.682887\pi$
$661$	−27.9448	−1.08693	−0.543463	+	0.839433i	$0.682887\pi$
$662$	0	0
$663$	−47.7218	−1.85336
$664$	0	0
$665$	−5.51815	−0.213985
$666$	0	0
$667$	32.2253	1.24777
$668$	0	0
$669$	10.5588	0.408227
$670$	0	0
$671$	−0.618384	−0.0238724
$672$	0	0
$673$	−13.0327	−0.502372	−0.251186	−	0.967939i	$-0.580821\pi$
$673$	−13.0327	−0.502372	−0.251186	+	0.967939i	$0.580821\pi$
$674$	0	0
$675$	0.302965	0.0116611
$676$	0	0
$677$	31.0938	1.19503	0.597517	−	0.801856i	$-0.296154\pi$
$677$	31.0938	1.19503	0.597517	+	0.801856i	$0.296154\pi$
$678$	0	0
$679$	−0.383952	−0.0147347
$680$	0	0
$681$	−34.3872	−1.31772
$682$	0	0
$683$	−40.3481	−1.54388	−0.771938	−	0.635698i	$-0.780712\pi$
$683$	−40.3481	−1.54388	−0.771938	+	0.635698i	$0.780712\pi$
$684$	0	0
$685$	20.1669	0.770536
$686$	0	0
$687$	15.6213	0.595991
$688$	0	0
$689$	4.99361	0.190241
$690$	0	0
$691$	−26.5902	−1.01154	−0.505770	−	0.862669i	$-0.668791\pi$
$691$	−26.5902	−1.01154	−0.505770	+	0.862669i	$0.668791\pi$
$692$	0	0
$693$	3.12244	0.118612
$694$	0	0
$695$	23.1126	0.876711
$696$	0	0
$697$	17.5121	0.663318
$698$	0	0
$699$	57.1846	2.16292
$700$	0	0
$701$	−27.6352	−1.04377	−0.521884	−	0.853016i	$-0.674771\pi$
$701$	−27.6352	−1.04377	−0.521884	+	0.853016i	$0.674771\pi$
$702$	0	0
$703$	−6.35770	−0.239785
$704$	0	0
$705$	−19.7763	−0.744819
$706$	0	0
$707$	0.921348	0.0346509
$708$	0	0
$709$	16.5465	0.621418	0.310709	−	0.950505i	$-0.399434\pi$
$709$	16.5465	0.621418	0.310709	+	0.950505i	$0.399434\pi$
$710$	0	0
$711$	−32.2998	−1.21134
$712$	0	0
$713$	10.6593	0.399195
$714$	0	0
$715$	5.50406	0.205840
$716$	0	0
$717$	−22.0889	−0.824925
$718$	0	0
$719$	−22.3569	−0.833771	−0.416886	−	0.908959i	$-0.636878\pi$
$719$	−22.3569	−0.833771	−0.416886	+	0.908959i	$0.636878\pi$
$720$	0	0
$721$	1.25917	0.0468941
$722$	0	0
$723$	−62.6462	−2.32984
$724$	0	0
$725$	−10.2233	−0.379684
$726$	0	0
$727$	28.1634	1.04452	0.522261	−	0.852786i	$-0.325089\pi$
$727$	28.1634	1.04452	0.522261	+	0.852786i	$0.325089\pi$
$728$	0	0
$729$	−29.1571	−1.07989
$730$	0	0
$731$	−40.7137	−1.50585
$732$	0	0
$733$	10.2314	0.377905	0.188953	−	0.981986i	$-0.439491\pi$
$733$	10.2314	0.377905	0.188953	+	0.981986i	$0.439491\pi$
$734$	0	0
$735$	−2.47436	−0.0912680
$736$	0	0
$737$	15.1587	0.558379
$738$	0	0
$739$	−16.6059	−0.610859	−0.305429	−	0.952215i	$-0.598800\pi$
$739$	−16.6059	−0.610859	−0.305429	+	0.952215i	$0.598800\pi$
$740$	0	0
$741$	−75.1517	−2.76077
$742$	0	0
$743$	52.7551	1.93540	0.967699	−	0.252108i	$-0.0811240\pi$
$743$	52.7551	1.93540	0.967699	+	0.252108i	$0.0811240\pi$
$744$	0	0
$745$	−4.86773	−0.178340
$746$	0	0
$747$	36.3885	1.33139
$748$	0	0
$749$	−4.63543	−0.169375
$750$	0	0
$751$	37.3197	1.36181	0.680907	−	0.732370i	$-0.261586\pi$
$751$	37.3197	1.36181	0.680907	+	0.732370i	$0.261586\pi$
$752$	0	0
$753$	−6.01660	−0.219257
$754$	0	0
$755$	−7.94488	−0.289144
$756$	0	0
$757$	−1.68144	−0.0611131	−0.0305566	−	0.999533i	$-0.509728\pi$
$757$	−1.68144	−0.0611131	−0.0305566	+	0.999533i	$0.509728\pi$
$758$	0	0
$759$	7.79953	0.283105
$760$	0	0
$761$	−6.34160	−0.229883	−0.114941	−	0.993372i	$-0.536668\pi$
$761$	−6.34160	−0.229883	−0.114941	+	0.993372i	$0.536668\pi$
$762$	0	0
$763$	−9.88051	−0.357699
$764$	0	0
$765$	10.9412	0.395581
$766$	0	0
$767$	−82.4292	−2.97635
$768$	0	0
$769$	−17.5912	−0.634356	−0.317178	−	0.948366i	$-0.602735\pi$
$769$	−17.5912	−0.634356	−0.317178	+	0.948366i	$0.602735\pi$
$770$	0	0
$771$	18.4958	0.666109
$772$	0	0
$773$	−30.3991	−1.09338	−0.546689	−	0.837336i	$-0.684112\pi$
$773$	−30.3991	−1.09338	−0.546689	+	0.837336i	$0.684112\pi$
$774$	0	0
$775$	−3.38162	−0.121471
$776$	0	0
$777$	−2.85081	−0.102272
$778$	0	0
$779$	27.5778	0.988079
$780$	0	0
$781$	7.03629	0.251778
$782$	0	0
$783$	−3.09730	−0.110688
$784$	0	0
$785$	−19.9863	−0.713343
$786$	0	0
$787$	53.6700	1.91313	0.956565	−	0.291520i	$-0.0941610\pi$
$787$	53.6700	1.91313	0.956565	+	0.291520i	$0.0941610\pi$
$788$	0	0
$789$	42.1423	1.50031
$790$	0	0
$791$	9.80985	0.348798
$792$	0	0
$793$	3.40362	0.120866
$794$	0	0
$795$	−2.24488	−0.0796178
$796$	0	0
$797$	−27.4102	−0.970920	−0.485460	−	0.874259i	$-0.661348\pi$
$797$	−27.4102	−0.970920	−0.485460	+	0.874259i	$0.661348\pi$
$798$	0	0
$799$	−28.0062	−0.990788
$800$	0	0
$801$	14.0149	0.495193
$802$	0	0
$803$	1.59164	0.0561677
$804$	0	0
$805$	−3.15214	−0.111098
$806$	0	0
$807$	63.3667	2.23061
$808$	0	0
$809$	12.9009	0.453570	0.226785	−	0.973945i	$-0.427178\pi$
$809$	12.9009	0.453570	0.226785	+	0.973945i	$0.427178\pi$
$810$	0	0
$811$	−33.7480	−1.18505	−0.592527	−	0.805551i	$-0.701870\pi$
$811$	−33.7480	−1.18505	−0.592527	+	0.805551i	$0.701870\pi$
$812$	0	0
$813$	−12.2449	−0.429447
$814$	0	0
$815$	−4.07927	−0.142891
$816$	0	0
$817$	−64.1154	−2.24311
$818$	0	0
$819$	−17.1861	−0.600531
$820$	0	0
$821$	3.17201	0.110704	0.0553520	−	0.998467i	$-0.482372\pi$
$821$	3.17201	0.110704	0.0553520	+	0.998467i	$0.482372\pi$
$822$	0	0
$823$	−22.3222	−0.778104	−0.389052	−	0.921216i	$-0.627197\pi$
$823$	−22.3222	−0.778104	−0.389052	+	0.921216i	$0.627197\pi$
$824$	0	0
$825$	−2.47436	−0.0861460
$826$	0	0
$827$	−36.1217	−1.25607	−0.628036	−	0.778184i	$-0.716141\pi$
$827$	−36.1217	−1.25607	−0.628036	+	0.778184i	$0.716141\pi$
$828$	0	0
$829$	2.74568	0.0953614	0.0476807	−	0.998863i	$-0.484817\pi$
$829$	2.74568	0.0953614	0.0476807	+	0.998863i	$0.484817\pi$
$830$	0	0
$831$	−59.4855	−2.06353
$832$	0	0
$833$	−3.50406	−0.121408
$834$	0	0
$835$	25.4534	0.880851
$836$	0	0
$837$	−1.02451	−0.0354123
$838$	0	0
$839$	4.67646	0.161449	0.0807247	−	0.996736i	$-0.474277\pi$
$839$	4.67646	0.161449	0.0807247	+	0.996736i	$0.474277\pi$
$840$	0	0
$841$	75.5158	2.60399
$842$	0	0
$843$	12.6341	0.435142
$844$	0	0
$845$	−17.2947	−0.594954
$846$	0	0
$847$	−1.00000	−0.0343604
$848$	0	0
$849$	−42.6128	−1.46247
$850$	0	0
$851$	−3.63172	−0.124494
$852$	0	0
$853$	30.6558	1.04963	0.524817	−	0.851215i	$-0.324134\pi$
$853$	30.6558	1.04963	0.524817	+	0.851215i	$0.324134\pi$
$854$	0	0
$855$	17.2301	0.589257
$856$	0	0
$857$	28.4139	0.970599	0.485300	−	0.874348i	$-0.338711\pi$
$857$	28.4139	0.970599	0.485300	+	0.874348i	$0.338711\pi$
$858$	0	0
$859$	10.8500	0.370197	0.185099	−	0.982720i	$-0.440740\pi$
$859$	10.8500	0.370197	0.185099	+	0.982720i	$0.440740\pi$
$860$	0	0
$861$	12.3660	0.421432
$862$	0	0
$863$	−14.4911	−0.493281	−0.246641	−	0.969107i	$-0.579327\pi$
$863$	−14.4911	−0.493281	−0.246641	+	0.969107i	$0.579327\pi$
$864$	0	0
$865$	−23.5870	−0.801981
$866$	0	0
$867$	−11.6829	−0.396771
$868$	0	0
$869$	10.3444	0.350910
$870$	0	0
$871$	−83.4346	−2.82707
$872$	0	0
$873$	1.19887	0.0405756
$874$	0	0
$875$	1.00000	0.0338062
$876$	0	0
$877$	−5.24672	−0.177169	−0.0885846	−	0.996069i	$-0.528234\pi$
$877$	−5.24672	−0.177169	−0.0885846	+	0.996069i	$0.528234\pi$
$878$	0	0
$879$	60.4711	2.03964
$880$	0	0
$881$	−25.5069	−0.859349	−0.429674	−	0.902984i	$-0.641372\pi$
$881$	−25.5069	−0.859349	−0.429674	+	0.902984i	$0.641372\pi$
$882$	0	0
$883$	52.4959	1.76663	0.883314	−	0.468783i	$-0.155307\pi$
$883$	52.4959	1.76663	0.883314	+	0.468783i	$0.155307\pi$
$884$	0	0
$885$	37.0562	1.24563
$886$	0	0
$887$	49.0807	1.64797	0.823984	−	0.566612i	$-0.191746\pi$
$887$	49.0807	1.64797	0.823984	+	0.566612i	$0.191746\pi$
$888$	0	0
$889$	−20.2299	−0.678489
$890$	0	0
$891$	8.61768	0.288703
$892$	0	0
$893$	−44.1038	−1.47588
$894$	0	0
$895$	21.5560	0.720536
$896$	0	0
$897$	−42.9290	−1.43336
$898$	0	0
$899$	34.5713	1.15302
$900$	0	0
$901$	−3.17909	−0.105911
$902$	0	0
$903$	−28.7496	−0.956726
$904$	0	0
$905$	−7.78462	−0.258769
$906$	0	0
$907$	−54.3002	−1.80301	−0.901505	−	0.432769i	$-0.857536\pi$
$907$	−54.3002	−1.80301	−0.901505	+	0.432769i	$0.857536\pi$
$908$	0	0
$909$	−2.87686	−0.0954193
$910$	0	0
$911$	11.1551	0.369585	0.184792	−	0.982778i	$-0.440839\pi$
$911$	11.1551	0.369585	0.184792	+	0.982778i	$0.440839\pi$
$912$	0	0
$913$	−11.6539	−0.385687
$914$	0	0
$915$	−1.53010	−0.0505836
$916$	0	0
$917$	7.12760	0.235374
$918$	0	0
$919$	−55.1118	−1.81797	−0.908986	−	0.416827i	$-0.863142\pi$
$919$	−55.1118	−1.81797	−0.908986	+	0.416827i	$0.863142\pi$
$920$	0	0
$921$	−11.7892	−0.388468
$922$	0	0
$923$	−38.7282	−1.27475
$924$	0	0
$925$	1.15214	0.0378822
$926$	0	0
$927$	−3.93170	−0.129134
$928$	0	0
$929$	46.5594	1.52756	0.763782	−	0.645474i	$-0.223340\pi$
$929$	46.5594	1.52756	0.763782	+	0.645474i	$0.223340\pi$
$930$	0	0
$931$	−5.51815	−0.180850
$932$	0	0
$933$	50.3519	1.64845
$934$	0	0
$935$	−3.50406	−0.114595
$936$	0	0
$937$	4.26597	0.139363	0.0696816	−	0.997569i	$-0.477802\pi$
$937$	4.26597	0.139363	0.0696816	+	0.997569i	$0.477802\pi$
$938$	0	0
$939$	−33.5217	−1.09394
$940$	0	0
$941$	18.4952	0.602925	0.301463	−	0.953478i	$-0.402525\pi$
$941$	18.4952	0.602925	0.301463	+	0.953478i	$0.402525\pi$
$942$	0	0
$943$	15.7534	0.513000
$944$	0	0
$945$	0.302965	0.00985544
$946$	0	0
$947$	4.79245	0.155734	0.0778668	−	0.996964i	$-0.475189\pi$
$947$	4.79245	0.155734	0.0778668	+	0.996964i	$0.475189\pi$
$948$	0	0
$949$	−8.76047	−0.284377
$950$	0	0
$951$	20.0936	0.651579
$952$	0	0
$953$	55.2697	1.79036	0.895181	−	0.445702i	$-0.147046\pi$
$953$	55.2697	1.79036	0.895181	+	0.445702i	$0.147046\pi$
$954$	0	0
$955$	11.9850	0.387826
$956$	0	0
$957$	25.2961	0.817706
$958$	0	0
$959$	20.1669	0.651222
$960$	0	0
$961$	−19.5647	−0.631118
$962$	0	0
$963$	14.4739	0.466414
$964$	0	0
$965$	−15.6784	−0.504706
$966$	0	0
$967$	2.68823	0.0864476	0.0432238	−	0.999065i	$-0.486237\pi$
$967$	2.68823	0.0864476	0.0432238	+	0.999065i	$0.486237\pi$
$968$	0	0
$969$	47.8439	1.53697
$970$	0	0
$971$	22.0970	0.709127	0.354563	−	0.935032i	$-0.384630\pi$
$971$	22.0970	0.709127	0.354563	+	0.935032i	$0.384630\pi$
$972$	0	0
$973$	23.1126	0.740956
$974$	0	0
$975$	13.6190	0.436157
$976$	0	0
$977$	22.7894	0.729097	0.364548	−	0.931184i	$-0.381223\pi$
$977$	22.7894	0.729097	0.364548	+	0.931184i	$0.381223\pi$
$978$	0	0
$979$	−4.48845	−0.143451
$980$	0	0
$981$	30.8513	0.985007
$982$	0	0
$983$	7.04944	0.224842	0.112421	−	0.993661i	$-0.464139\pi$
$983$	7.04944	0.224842	0.112421	+	0.993661i	$0.464139\pi$
$984$	0	0
$985$	19.2181	0.612341
$986$	0	0
$987$	−19.7763	−0.629487
$988$	0	0
$989$	−36.6248	−1.16460
$990$	0	0
$991$	6.15730	0.195593	0.0977966	−	0.995206i	$-0.468821\pi$
$991$	6.15730	0.195593	0.0977966	+	0.995206i	$0.468821\pi$
$992$	0	0
$993$	−16.2017	−0.514146
$994$	0	0
$995$	14.4282	0.457406
$996$	0	0
$997$	−1.97924	−0.0626831	−0.0313415	−	0.999509i	$-0.509978\pi$
$997$	−1.97924	−0.0626831	−0.0313415	+	0.999509i	$0.509978\pi$
$998$	0	0
$999$	0.349059	0.0110437

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6160.2.a.bv.1.5		5
4.3	odd	2		3080.2.a.t.1.1	✓	5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3080.2.a.t.1.1	✓	5	4.3	odd	2
6160.2.a.bv.1.5		5	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)

\(f(q)\)	\(=\)	\(q+2.47436 q^{3} -1.00000 q^{5} -1.00000 q^{7} +3.12244 q^{9} +O(q^{10})\)	\(q+2.47436 q^{3} -1.00000 q^{5} -1.00000 q^{7} +3.12244 q^{9} -1.00000 q^{11} +5.50406 q^{13} -2.47436 q^{15} -3.50406 q^{17} -5.51815 q^{19} -2.47436 q^{21} -3.15214 q^{23} +1.00000 q^{25} +0.302965 q^{27} -10.2233 q^{29} -3.38162 q^{31} -2.47436 q^{33} +1.00000 q^{35} +1.15214 q^{37} +13.6190 q^{39} -4.99766 q^{41} +11.6190 q^{43} -3.12244 q^{45} +7.99250 q^{47} +1.00000 q^{49} -8.67029 q^{51} +0.907259 q^{53} +1.00000 q^{55} -13.6539 q^{57} -14.9761 q^{59} +0.618384 q^{61} -3.12244 q^{63} -5.50406 q^{65} -15.1587 q^{67} -7.79953 q^{69} -7.03629 q^{71} -1.59164 q^{73} +2.47436 q^{75} +1.00000 q^{77} -10.3444 q^{79} -8.61768 q^{81} +11.6539 q^{83} +3.50406 q^{85} -25.2961 q^{87} +4.48845 q^{89} -5.50406 q^{91} -8.36733 q^{93} +5.51815 q^{95} +0.383952 q^{97} -3.12244 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 2 q^{3} - 5 q^{5} - 5 q^{7} + 9 q^{9}+O(q^{10}) \) 5 * q - 2 * q^3 - 5 * q^5 - 5 * q^7 + 9 * q^9	\( 5 q - 2 q^{3} - 5 q^{5} - 5 q^{7} + 9 q^{9} - 5 q^{11} + 8 q^{13} + 2 q^{15} + 2 q^{17} - 4 q^{19} + 2 q^{21} - 4 q^{23} + 5 q^{25} - 14 q^{27} - 4 q^{29} - 4 q^{31} + 2 q^{33} + 5 q^{35} - 6 q^{37} + 4 q^{39} - 4 q^{41} - 6 q^{43} - 9 q^{45} + 2 q^{47} + 5 q^{49} - 8 q^{51} + 6 q^{53} + 5 q^{55} - 16 q^{57} - 20 q^{59} + 16 q^{61} - 9 q^{63} - 8 q^{65} - 22 q^{67} + 34 q^{69} + 12 q^{71} + 30 q^{73} - 2 q^{75} + 5 q^{77} - 6 q^{79} + 13 q^{81} + 6 q^{83} - 2 q^{85} + 4 q^{87} + 4 q^{89} - 8 q^{91} - 22 q^{93} + 4 q^{95} + 10 q^{97} - 9 q^{99}+O(q^{100}) \) 5 * q - 2 * q^3 - 5 * q^5 - 5 * q^7 + 9 * q^9 - 5 * q^11 + 8 * q^13 + 2 * q^15 + 2 * q^17 - 4 * q^19 + 2 * q^21 - 4 * q^23 + 5 * q^25 - 14 * q^27 - 4 * q^29 - 4 * q^31 + 2 * q^33 + 5 * q^35 - 6 * q^37 + 4 * q^39 - 4 * q^41 - 6 * q^43 - 9 * q^45 + 2 * q^47 + 5 * q^49 - 8 * q^51 + 6 * q^53 + 5 * q^55 - 16 * q^57 - 20 * q^59 + 16 * q^61 - 9 * q^63 - 8 * q^65 - 22 * q^67 + 34 * q^69 + 12 * q^71 + 30 * q^73 - 2 * q^75 + 5 * q^77 - 6 * q^79 + 13 * q^81 + 6 * q^83 - 2 * q^85 + 4 * q^87 + 4 * q^89 - 8 * q^91 - 22 * q^93 + 4 * q^95 + 10 * q^97 - 9 * q^99

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