LMFDB - Embedded newform 6160.2.a.bx.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:	$0$
Dimension:	$5$
Coefficient field:	5.5.15785648.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 2x^{4} - 12x^{3} + 22x^{2} + 24x - 28 $ x^5 - 2x^4 - 12x^3 + 22x^2 + 24x - 28
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			$-3.08104$ 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)

$f(q)$	$=$	$q+3.08104 q^{3} +1.00000 q^{5} -1.00000 q^{7} +6.49280 q^{9} +O(q^{10})$	$q+3.08104 q^{3} +1.00000 q^{5} -1.00000 q^{7} +6.49280 q^{9} -1.00000 q^{11} +5.59262 q^{13} +3.08104 q^{15} +3.59262 q^{17} -6.08779 q^{19} -3.08104 q^{21} -4.98122 q^{23} +1.00000 q^{25} +10.7614 q^{27} +0.511583 q^{29} +7.08542 q^{31} -3.08104 q^{33} -1.00000 q^{35} -0.00438163 q^{37} +17.2311 q^{39} +3.97447 q^{41} +0.893428 q^{43} +6.49280 q^{45} -9.99235 q^{47} +1.00000 q^{49} +11.0690 q^{51} +6.98122 q^{53} -1.00000 q^{55} -18.7567 q^{57} +12.6616 q^{59} -7.08542 q^{61} -6.49280 q^{63} +5.59262 q^{65} +9.65488 q^{67} -15.3473 q^{69} +13.9624 q^{71} +12.7315 q^{73} +3.08104 q^{75} +1.00000 q^{77} -5.58059 q^{79} +13.6780 q^{81} +6.58587 q^{83} +3.59262 q^{85} +1.57621 q^{87} -5.78461 q^{89} -5.59262 q^{91} +21.8305 q^{93} -6.08779 q^{95} +12.2962 q^{97} -6.49280 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 2 q^{3} + 5 q^{5} - 5 q^{7} + 13 q^{9}+O(q^{10}) $ 5 * q - 2 * q^3 + 5 * q^5 - 5 * q^7 + 13 * q^9	$ 5 q - 2 q^{3} + 5 q^{5} - 5 q^{7} + 13 q^{9} - 5 q^{11} + 12 q^{13} - 2 q^{15} + 2 q^{17} - 8 q^{19} + 2 q^{21} - 4 q^{23} + 5 q^{25} - 2 q^{27} + 4 q^{29} + 2 q^{33} - 5 q^{35} + 18 q^{37} + 8 q^{39} + 4 q^{41} + 6 q^{43} + 13 q^{45} - 6 q^{47} + 5 q^{49} + 12 q^{51} + 14 q^{53} - 5 q^{55} - 12 q^{57} + 4 q^{59} - 13 q^{63} + 12 q^{65} - 6 q^{67} - 10 q^{69} + 28 q^{71} + 10 q^{73} - 2 q^{75} + 5 q^{77} + 14 q^{79} + 17 q^{81} + 22 q^{83} + 2 q^{85} - 16 q^{87} + 24 q^{89} - 12 q^{91} + 10 q^{93} - 8 q^{95} + 10 q^{97} - 13 q^{99}+O(q^{100}) $ 5 * q - 2 * q^3 + 5 * q^5 - 5 * q^7 + 13 * q^9 - 5 * q^11 + 12 * q^13 - 2 * q^15 + 2 * q^17 - 8 * q^19 + 2 * q^21 - 4 * q^23 + 5 * q^25 - 2 * q^27 + 4 * q^29 + 2 * q^33 - 5 * q^35 + 18 * q^37 + 8 * q^39 + 4 * q^41 + 6 * q^43 + 13 * q^45 - 6 * q^47 + 5 * q^49 + 12 * q^51 + 14 * q^53 - 5 * q^55 - 12 * q^57 + 4 * q^59 - 13 * q^63 + 12 * q^65 - 6 * q^67 - 10 * q^69 + 28 * q^71 + 10 * q^73 - 2 * q^75 + 5 * q^77 + 14 * q^79 + 17 * q^81 + 22 * q^83 + 2 * q^85 - 16 * q^87 + 24 * q^89 - 12 * q^91 + 10 * q^93 - 8 * q^95 + 10 * q^97 - 13 * 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
$3$	3.08104	1.77884	0.889419	−	0.457092i	$-0.151109\pi$
$3$	3.08104	1.77884	0.889419	+	0.457092i	$0.151109\pi$
$4$	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
$9$	6.49280	2.16427
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	5.59262	1.55111	0.775557	−	0.631278i	$-0.217469\pi$
$13$	5.59262	1.55111	0.775557	+	0.631278i	$0.217469\pi$
$14$	0	0
$15$	3.08104	0.795521
$16$	0	0
$17$	3.59262	0.871339	0.435669	−	0.900107i	$-0.356512\pi$
$17$	3.59262	0.871339	0.435669	+	0.900107i	$0.356512\pi$
$18$	0	0
$19$	−6.08779	−1.39663	−0.698317	−	0.715788i	$-0.746068\pi$
$19$	−6.08779	−1.39663	−0.698317	+	0.715788i	$0.746068\pi$
$20$	0	0
$21$	−3.08104	−0.672338
$22$	0	0
$23$	−4.98122	−1.03866	−0.519328	−	0.854575i	$-0.673818\pi$
$23$	−4.98122	−1.03866	−0.519328	+	0.854575i	$0.673818\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	10.7614	2.07104
$28$	0	0
$29$	0.511583	0.0949985	0.0474993	−	0.998871i	$-0.484875\pi$
$29$	0.511583	0.0949985	0.0474993	+	0.998871i	$0.484875\pi$
$30$	0	0
$31$	7.08542	1.27258	0.636290	−	0.771450i	$-0.280468\pi$
$31$	7.08542	1.27258	0.636290	+	0.771450i	$0.280468\pi$
$32$	0	0
$33$	−3.08104	−0.536340
$34$	0	0
$35$	−1.00000	−0.169031
$36$	0	0
$37$	−0.00438163	−0.000720336 0	−0.000360168	−	1.00000i	$-0.500115\pi$
$37$	−0.00438163	−0.000720336 0	−0.000360168		1.00000i	$0.500115\pi$
$38$	0	0
$39$	17.2311	2.75918
$40$	0	0
$41$	3.97447	0.620707	0.310354	−	0.950621i	$-0.399553\pi$
$41$	3.97447	0.620707	0.310354	+	0.950621i	$0.399553\pi$
$42$	0	0
$43$	0.893428	0.136246	0.0681232	−	0.997677i	$-0.478299\pi$
$43$	0.893428	0.136246	0.0681232	+	0.997677i	$0.478299\pi$
$44$	0	0
$45$	6.49280	0.967889
$46$	0	0
$47$	−9.99235	−1.45753	−0.728767	−	0.684762i	$-0.759906\pi$
$47$	−9.99235	−1.45753	−0.728767	+	0.684762i	$0.759906\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	11.0690	1.54997
$52$	0	0
$53$	6.98122	0.958944	0.479472	−	0.877557i	$-0.340828\pi$
$53$	6.98122	0.958944	0.479472	+	0.877557i	$0.340828\pi$
$54$	0	0
$55$	−1.00000	−0.134840
$56$	0	0
$57$	−18.7567	−2.48439
$58$	0	0
$59$	12.6616	1.64840	0.824202	−	0.566296i	$-0.191624\pi$
$59$	12.6616	1.64840	0.824202	+	0.566296i	$0.191624\pi$
$60$	0	0
$61$	−7.08542	−0.907195	−0.453598	−	0.891207i	$-0.649860\pi$
$61$	−7.08542	−0.907195	−0.453598	+	0.891207i	$0.649860\pi$
$62$	0	0
$63$	−6.49280	−0.818016
$64$	0	0
$65$	5.59262	0.693679
$66$	0	0
$67$	9.65488	1.17953	0.589765	−	0.807575i	$-0.299220\pi$
$67$	9.65488	1.17953	0.589765	+	0.807575i	$0.299220\pi$
$68$	0	0
$69$	−15.3473	−1.84760
$70$	0	0
$71$	13.9624	1.65704	0.828518	−	0.559963i	$-0.189184\pi$
$71$	13.9624	1.65704	0.828518	+	0.559963i	$0.189184\pi$
$72$	0	0
$73$	12.7315	1.49011	0.745057	−	0.667001i	$-0.232422\pi$
$73$	12.7315	1.49011	0.745057	+	0.667001i	$0.232422\pi$
$74$	0	0
$75$	3.08104	0.355768
$76$	0	0
$77$	1.00000	0.113961
$78$	0	0
$79$	−5.58059	−0.627865	−0.313933	−	0.949445i	$-0.601647\pi$
$79$	−5.58059	−0.627865	−0.313933	+	0.949445i	$0.601647\pi$
$80$	0	0
$81$	13.6780	1.51978
$82$	0	0
$83$	6.58587	0.722893	0.361447	−	0.932393i	$-0.382283\pi$
$83$	6.58587	0.722893	0.361447	+	0.932393i	$0.382283\pi$
$84$	0	0
$85$	3.59262	0.389674
$86$	0	0
$87$	1.57621	0.168987
$88$	0	0
$89$	−5.78461	−0.613168	−0.306584	−	0.951844i	$-0.599186\pi$
$89$	−5.78461	−0.613168	−0.306584	+	0.951844i	$0.599186\pi$
$90$	0	0
$91$	−5.59262	−0.586266
$92$	0	0
$93$	21.8305	2.26371
$94$	0	0
$95$	−6.08779	−0.624594
$96$	0	0
$97$	12.2962	1.24849	0.624245	−	0.781229i	$-0.285407\pi$
$97$	12.2962	1.24849	0.624245	+	0.781229i	$0.285407\pi$
$98$	0	0
$99$	−6.49280	−0.652551
$100$	0	0
$101$	10.8613	1.08074	0.540368	−	0.841429i	$-0.318285\pi$
$101$	10.8613	1.08074	0.540368	+	0.841429i	$0.318285\pi$
$102$	0	0
$103$	−0.194234	−0.0191385	−0.00956923	−	0.999954i	$-0.503046\pi$
$103$	−0.194234	−0.0191385	−0.00956923	+	0.999954i	$0.503046\pi$
$104$	0	0
$105$	−3.08104	−0.300679
$106$	0	0
$107$	−1.77585	−0.171678	−0.0858390	−	0.996309i	$-0.527357\pi$
$107$	−1.77585	−0.171678	−0.0858390	+	0.996309i	$0.527357\pi$
$108$	0	0
$109$	−15.6129	−1.49545	−0.747724	−	0.664010i	$-0.768853\pi$
$109$	−15.6129	−1.49545	−0.747724	+	0.664010i	$0.768853\pi$
$110$	0	0
$111$	−0.0135000	−0.00128136
$112$	0	0
$113$	−10.9991	−1.03471	−0.517354	−	0.855771i	$-0.673083\pi$
$113$	−10.9991	−1.03471	−0.517354	+	0.855771i	$0.673083\pi$
$114$	0	0
$115$	−4.98122	−0.464501
$116$	0	0
$117$	36.3118	3.35702
$118$	0	0
$119$	−3.59262	−0.329335
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	12.2455	1.10414
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−5.98650	−0.531216	−0.265608	−	0.964081i	$-0.585573\pi$
$127$	−5.98650	−0.531216	−0.265608	+	0.964081i	$0.585573\pi$
$128$	0	0
$129$	2.75269	0.242360
$130$	0	0
$131$	−3.85014	−0.336388	−0.168194	−	0.985754i	$-0.553794\pi$
$131$	−3.85014	−0.336388	−0.168194	+	0.985754i	$0.553794\pi$
$132$	0	0
$133$	6.08779	0.527878
$134$	0	0
$135$	10.7614	0.926198
$136$	0	0
$137$	−13.5016	−1.15352	−0.576758	−	0.816915i	$-0.695683\pi$
$137$	−13.5016	−1.15352	−0.576758	+	0.816915i	$0.695683\pi$
$138$	0	0
$139$	−16.6824	−1.41498	−0.707492	−	0.706721i	$-0.750174\pi$
$139$	−16.6824	−1.41498	−0.707492	+	0.706721i	$0.750174\pi$
$140$	0	0
$141$	−30.7868	−2.59272
$142$	0	0
$143$	−5.59262	−0.467678
$144$	0	0
$145$	0.511583	0.0424846
$146$	0	0
$147$	3.08104	0.254120
$148$	0	0
$149$	−12.6361	−1.03519	−0.517595	−	0.855626i	$-0.673172\pi$
$149$	−12.6361	−1.03519	−0.517595	+	0.855626i	$0.673172\pi$
$150$	0	0
$151$	−23.7427	−1.93215	−0.966075	−	0.258260i	$-0.916851\pi$
$151$	−23.7427	−1.93215	−0.966075	+	0.258260i	$0.916851\pi$
$152$	0	0
$153$	23.3262	1.88581
$154$	0	0
$155$	7.08542	0.569115
$156$	0	0
$157$	−9.58497	−0.764964	−0.382482	−	0.923963i	$-0.624931\pi$
$157$	−9.58497	−0.764964	−0.382482	+	0.923963i	$0.624931\pi$
$158$	0	0
$159$	21.5094	1.70581
$160$	0	0
$161$	4.98122	0.392575
$162$	0	0
$163$	−0.456134	−0.0357271	−0.0178636	−	0.999840i	$-0.505686\pi$
$163$	−0.456134	−0.0357271	−0.0178636	+	0.999840i	$0.505686\pi$
$164$	0	0
$165$	−3.08104	−0.239859
$166$	0	0
$167$	−9.36306	−0.724535	−0.362268	−	0.932074i	$-0.617997\pi$
$167$	−9.36306	−0.724535	−0.362268	+	0.932074i	$0.617997\pi$
$168$	0	0
$169$	18.2774	1.40595
$170$	0	0
$171$	−39.5268	−3.02269
$172$	0	0
$173$	2.76910	0.210531	0.105265	−	0.994444i	$-0.466431\pi$
$173$	2.76910	0.210531	0.105265	+	0.994444i	$0.466431\pi$
$174$	0	0
$175$	−1.00000	−0.0755929
$176$	0	0
$177$	39.0110	2.93224
$178$	0	0
$179$	22.1000	1.65183	0.825916	−	0.563793i	$-0.190659\pi$
$179$	22.1000	1.65183	0.825916	+	0.563793i	$0.190659\pi$
$180$	0	0
$181$	3.36306	0.249974	0.124987	−	0.992158i	$-0.460111\pi$
$181$	3.36306	0.249974	0.124987	+	0.992158i	$0.460111\pi$
$182$	0	0
$183$	−21.8305	−1.61375
$184$	0	0
$185$	−0.00438163	−0.000322144 0
$186$	0	0
$187$	−3.59262	−0.262718
$188$	0	0
$189$	−10.7614	−0.782780
$190$	0	0
$191$	−26.4950	−1.91711	−0.958555	−	0.284907i	$-0.908037\pi$
$191$	−26.4950	−1.91711	−0.958555	+	0.284907i	$0.908037\pi$
$192$	0	0
$193$	3.20882	0.230976	0.115488	−	0.993309i	$-0.463157\pi$
$193$	3.20882	0.230976	0.115488	+	0.993309i	$0.463157\pi$
$194$	0	0
$195$	17.2311	1.23394
$196$	0	0
$197$	−11.0310	−0.785926	−0.392963	−	0.919554i	$-0.628550\pi$
$197$	−11.0310	−0.785926	−0.392963	+	0.919554i	$0.628550\pi$
$198$	0	0
$199$	−8.07102	−0.572139	−0.286070	−	0.958209i	$-0.592349\pi$
$199$	−8.07102	−0.572139	−0.286070	+	0.958209i	$0.592349\pi$
$200$	0	0
$201$	29.7470	2.09819
$202$	0	0
$203$	−0.511583	−0.0359061
$204$	0	0
$205$	3.97447	0.277589
$206$	0	0
$207$	−32.3420	−2.24793
$208$	0	0
$209$	6.08779	0.421101
$210$	0	0
$211$	−1.13891	−0.0784059	−0.0392030	−	0.999231i	$-0.512482\pi$
$211$	−1.13891	−0.0784059	−0.0392030	+	0.999231i	$0.512482\pi$
$212$	0	0
$213$	43.0188	2.94760
$214$	0	0
$215$	0.893428	0.0609313
$216$	0	0
$217$	−7.08542	−0.480990
$218$	0	0
$219$	39.2263	2.65067
$220$	0	0
$221$	20.0922	1.35155
$222$	0	0
$223$	22.8560	1.53055	0.765277	−	0.643701i	$-0.222602\pi$
$223$	22.8560	1.53055	0.765277	+	0.643701i	$0.222602\pi$
$224$	0	0
$225$	6.49280	0.432853
$226$	0	0
$227$	−18.2778	−1.21314	−0.606571	−	0.795029i	$-0.707455\pi$
$227$	−18.2778	−1.21314	−0.606571	+	0.795029i	$0.707455\pi$
$228$	0	0
$229$	−15.9091	−1.05130	−0.525652	−	0.850700i	$-0.676179\pi$
$229$	−15.9091	−1.05130	−0.525652	+	0.850700i	$0.676179\pi$
$230$	0	0
$231$	3.08104	0.202717
$232$	0	0
$233$	12.6270	0.827221	0.413610	−	0.910454i	$-0.364268\pi$
$233$	12.6270	0.827221	0.413610	+	0.910454i	$0.364268\pi$
$234$	0	0
$235$	−9.99235	−0.651829
$236$	0	0
$237$	−17.1940	−1.11687
$238$	0	0
$239$	19.6593	1.27165	0.635826	−	0.771833i	$-0.280660\pi$
$239$	19.6593	1.27165	0.635826	+	0.771833i	$0.280660\pi$
$240$	0	0
$241$	9.16445	0.590334	0.295167	−	0.955446i	$-0.404625\pi$
$241$	9.16445	0.590334	0.295167	+	0.955446i	$0.404625\pi$
$242$	0	0
$243$	9.85823	0.632406
$244$	0	0
$245$	1.00000	0.0638877
$246$	0	0
$247$	−34.0467	−2.16634
$248$	0	0
$249$	20.2913	1.28591
$250$	0	0
$251$	16.4620	1.03907	0.519536	−	0.854449i	$-0.326105\pi$
$251$	16.4620	1.03907	0.519536	+	0.854449i	$0.326105\pi$
$252$	0	0
$253$	4.98122	0.313166
$254$	0	0
$255$	11.0690	0.693168
$256$	0	0
$257$	25.4263	1.58605	0.793026	−	0.609188i	$-0.208504\pi$
$257$	25.4263	1.58605	0.793026	+	0.609188i	$0.208504\pi$
$258$	0	0
$259$	0.00438163	0.000272261 0
$260$	0	0
$261$	3.32160	0.205602
$262$	0	0
$263$	25.5557	1.57583	0.787917	−	0.615782i	$-0.211160\pi$
$263$	25.5557	1.57583	0.787917	+	0.615782i	$0.211160\pi$
$264$	0	0
$265$	6.98122	0.428853
$266$	0	0
$267$	−17.8226	−1.09073
$268$	0	0
$269$	0.208408	0.0127069	0.00635343	−	0.999980i	$-0.497978\pi$
$269$	0.208408	0.0127069	0.00635343	+	0.999980i	$0.497978\pi$
$270$	0	0
$271$	−9.28749	−0.564175	−0.282087	−	0.959389i	$-0.591027\pi$
$271$	−9.28749	−0.564175	−0.282087	+	0.959389i	$0.591027\pi$
$272$	0	0
$273$	−17.2311	−1.04287
$274$	0	0
$275$	−1.00000	−0.0603023
$276$	0	0
$277$	13.9009	0.835221	0.417611	−	0.908626i	$-0.362868\pi$
$277$	13.9009	0.835221	0.417611	+	0.908626i	$0.362868\pi$
$278$	0	0
$279$	46.0042	2.75420
$280$	0	0
$281$	−2.93050	−0.174819	−0.0874096	−	0.996172i	$-0.527859\pi$
$281$	−2.93050	−0.174819	−0.0874096	+	0.996172i	$0.527859\pi$
$282$	0	0
$283$	−25.6627	−1.52549	−0.762745	−	0.646699i	$-0.776149\pi$
$283$	−25.6627	−1.52549	−0.762745	+	0.646699i	$0.776149\pi$
$284$	0	0
$285$	−18.7567	−1.11105
$286$	0	0
$287$	−3.97447	−0.234605
$288$	0	0
$289$	−4.09307	−0.240769
$290$	0	0
$291$	37.8851	2.22086
$292$	0	0
$293$	10.5672	0.617343	0.308672	−	0.951169i	$-0.400116\pi$
$293$	10.5672	0.617343	0.308672	+	0.951169i	$0.400116\pi$
$294$	0	0
$295$	12.6616	0.737188
$296$	0	0
$297$	−10.7614	−0.624443
$298$	0	0
$299$	−27.8581	−1.61107
$300$	0	0
$301$	−0.893428	−0.0514963
$302$	0	0
$303$	33.4640	1.92246
$304$	0	0
$305$	−7.08542	−0.405710
$306$	0	0
$307$	15.7248	0.897461	0.448730	−	0.893667i	$-0.351876\pi$
$307$	15.7248	0.897461	0.448730	+	0.893667i	$0.351876\pi$
$308$	0	0
$309$	−0.598443	−0.0340442
$310$	0	0
$311$	−27.6337	−1.56696	−0.783482	−	0.621414i	$-0.786559\pi$
$311$	−27.6337	−1.56696	−0.783482	+	0.621414i	$0.786559\pi$
$312$	0	0
$313$	30.0345	1.69765	0.848825	−	0.528675i	$-0.177311\pi$
$313$	30.0345	1.69765	0.848825	+	0.528675i	$0.177311\pi$
$314$	0	0
$315$	−6.49280	−0.365828
$316$	0	0
$317$	−23.9699	−1.34628	−0.673142	−	0.739513i	$-0.735056\pi$
$317$	−23.9699	−1.34628	−0.673142	+	0.739513i	$0.735056\pi$
$318$	0	0
$319$	−0.511583	−0.0286431
$320$	0	0
$321$	−5.47146	−0.305387
$322$	0	0
$323$	−21.8711	−1.21694
$324$	0	0
$325$	5.59262	0.310223
$326$	0	0
$327$	−48.1040	−2.66016
$328$	0	0
$329$	9.99235	0.550896
$330$	0	0
$331$	20.8213	1.14444	0.572221	−	0.820100i	$-0.306082\pi$
$331$	20.8213	1.14444	0.572221	+	0.820100i	$0.306082\pi$
$332$	0	0
$333$	−0.0284490	−0.00155900
$334$	0	0
$335$	9.65488	0.527502
$336$	0	0
$337$	−21.1778	−1.15363	−0.576813	−	0.816876i	$-0.695704\pi$
$337$	−21.1778	−1.15363	−0.576813	+	0.816876i	$0.695704\pi$
$338$	0	0
$339$	−33.8886	−1.84058
$340$	0	0
$341$	−7.08542	−0.383697
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	−15.3473	−0.826272
$346$	0	0
$347$	−29.4019	−1.57838	−0.789189	−	0.614150i	$-0.789499\pi$
$347$	−29.4019	−1.57838	−0.789189	+	0.614150i	$0.789499\pi$
$348$	0	0
$349$	−8.27540	−0.442972	−0.221486	−	0.975164i	$-0.571091\pi$
$349$	−8.27540	−0.442972	−0.221486	+	0.975164i	$0.571091\pi$
$350$	0	0
$351$	60.1847	3.21242
$352$	0	0
$353$	−4.36301	−0.232219	−0.116110	−	0.993236i	$-0.537042\pi$
$353$	−4.36301	−0.232219	−0.116110	+	0.993236i	$0.537042\pi$
$354$	0	0
$355$	13.9624	0.741049
$356$	0	0
$357$	−11.0690	−0.585834
$358$	0	0
$359$	15.4783	0.816916	0.408458	−	0.912777i	$-0.366067\pi$
$359$	15.4783	0.816916	0.408458	+	0.912777i	$0.366067\pi$
$360$	0	0
$361$	18.0612	0.950588
$362$	0	0
$363$	3.08104	0.161713
$364$	0	0
$365$	12.7315	0.666399
$366$	0	0
$367$	13.9413	0.727729	0.363864	−	0.931452i	$-0.381457\pi$
$367$	13.9413	0.727729	0.363864	+	0.931452i	$0.381457\pi$
$368$	0	0
$369$	25.8054	1.34338
$370$	0	0
$371$	−6.98122	−0.362447
$372$	0	0
$373$	−22.0166	−1.13998	−0.569988	−	0.821653i	$-0.693052\pi$
$373$	−22.0166	−1.13998	−0.569988	+	0.821653i	$0.693052\pi$
$374$	0	0
$375$	3.08104	0.159104
$376$	0	0
$377$	2.86109	0.147354
$378$	0	0
$379$	29.6802	1.52457	0.762286	−	0.647241i	$-0.224077\pi$
$379$	29.6802	1.52457	0.762286	+	0.647241i	$0.224077\pi$
$380$	0	0
$381$	−18.4446	−0.944947
$382$	0	0
$383$	−0.420879	−0.0215059	−0.0107529	−	0.999942i	$-0.503423\pi$
$383$	−0.420879	−0.0215059	−0.0107529	+	0.999942i	$0.503423\pi$
$384$	0	0
$385$	1.00000	0.0509647
$386$	0	0
$387$	5.80085	0.294874
$388$	0	0
$389$	−8.74571	−0.443425	−0.221712	−	0.975112i	$-0.571165\pi$
$389$	−8.74571	−0.443425	−0.221712	+	0.975112i	$0.571165\pi$
$390$	0	0
$391$	−17.8956	−0.905021
$392$	0	0
$393$	−11.8624	−0.598380
$394$	0	0
$395$	−5.58059	−0.280790
$396$	0	0
$397$	9.40063	0.471804	0.235902	−	0.971777i	$-0.424196\pi$
$397$	9.40063	0.471804	0.235902	+	0.971777i	$0.424196\pi$
$398$	0	0
$399$	18.7567	0.939010
$400$	0	0
$401$	−3.46046	−0.172807	−0.0864035	−	0.996260i	$-0.527537\pi$
$401$	−3.46046	−0.172807	−0.0864035	+	0.996260i	$0.527537\pi$
$402$	0	0
$403$	39.6261	1.97392
$404$	0	0
$405$	13.6780	0.679667
$406$	0	0
$407$	0.00438163	0.000217189 0
$408$	0	0
$409$	−4.30951	−0.213092	−0.106546	−	0.994308i	$-0.533979\pi$
$409$	−4.30951	−0.213092	−0.106546	+	0.994308i	$0.533979\pi$
$410$	0	0
$411$	−41.5988	−2.05192
$412$	0	0
$413$	−12.6616	−0.623038
$414$	0	0
$415$	6.58587	0.323288
$416$	0	0
$417$	−51.3992	−2.51703
$418$	0	0
$419$	26.4568	1.29250	0.646250	−	0.763126i	$-0.276336\pi$
$419$	26.4568	1.29250	0.646250	+	0.763126i	$0.276336\pi$
$420$	0	0
$421$	−16.3430	−0.796509	−0.398254	−	0.917275i	$-0.630384\pi$
$421$	−16.3430	−0.796509	−0.398254	+	0.917275i	$0.630384\pi$
$422$	0	0
$423$	−64.8783	−3.15449
$424$	0	0
$425$	3.59262	0.174268
$426$	0	0
$427$	7.08542	0.342888
$428$	0	0
$429$	−17.2311	−0.831924
$430$	0	0
$431$	−12.9742	−0.624947	−0.312474	−	0.949926i	$-0.601158\pi$
$431$	−12.9742	−0.624947	−0.312474	+	0.949926i	$0.601158\pi$
$432$	0	0
$433$	9.56660	0.459741	0.229871	−	0.973221i	$-0.426170\pi$
$433$	9.56660	0.459741	0.229871	+	0.973221i	$0.426170\pi$
$434$	0	0
$435$	1.57621	0.0755733
$436$	0	0
$437$	30.3246	1.45062
$438$	0	0
$439$	−14.3194	−0.683429	−0.341714	−	0.939804i	$-0.611008\pi$
$439$	−14.3194	−0.683429	−0.341714	+	0.939804i	$0.611008\pi$
$440$	0	0
$441$	6.49280	0.309181
$442$	0	0
$443$	14.2102	0.675145	0.337572	−	0.941300i	$-0.390394\pi$
$443$	14.2102	0.675145	0.337572	+	0.941300i	$0.390394\pi$
$444$	0	0
$445$	−5.78461	−0.274217
$446$	0	0
$447$	−38.9323	−1.84143
$448$	0	0
$449$	6.52070	0.307731	0.153865	−	0.988092i	$-0.450828\pi$
$449$	6.52070	0.307731	0.153865	+	0.988092i	$0.450828\pi$
$450$	0	0
$451$	−3.97447	−0.187150
$452$	0	0
$453$	−73.1521	−3.43698
$454$	0	0
$455$	−5.59262	−0.262186
$456$	0	0
$457$	−32.3783	−1.51459	−0.757297	−	0.653071i	$-0.773480\pi$
$457$	−32.3783	−1.51459	−0.757297	+	0.653071i	$0.773480\pi$
$458$	0	0
$459$	38.6618	1.80458
$460$	0	0
$461$	−11.7001	−0.544928	−0.272464	−	0.962166i	$-0.587839\pi$
$461$	−11.7001	−0.544928	−0.272464	+	0.962166i	$0.587839\pi$
$462$	0	0
$463$	−34.8611	−1.62013	−0.810066	−	0.586338i	$-0.800569\pi$
$463$	−34.8611	−1.62013	−0.810066	+	0.586338i	$0.800569\pi$
$464$	0	0
$465$	21.8305	1.01236
$466$	0	0
$467$	5.09454	0.235747	0.117874	−	0.993029i	$-0.462392\pi$
$467$	5.09454	0.235747	0.117874	+	0.993029i	$0.462392\pi$
$468$	0	0
$469$	−9.65488	−0.445821
$470$	0	0
$471$	−29.5317	−1.36075
$472$	0	0
$473$	−0.893428	−0.0410798
$474$	0	0
$475$	−6.08779	−0.279327
$476$	0	0
$477$	45.3276	2.07541
$478$	0	0
$479$	−0.450794	−0.0205973	−0.0102987	−	0.999947i	$-0.503278\pi$
$479$	−0.450794	−0.0205973	−0.0102987	+	0.999947i	$0.503278\pi$
$480$	0	0
$481$	−0.0245048	−0.00111732
$482$	0	0
$483$	15.3473	0.698327
$484$	0	0
$485$	12.2962	0.558342
$486$	0	0
$487$	−7.63390	−0.345925	−0.172962	−	0.984928i	$-0.555334\pi$
$487$	−7.63390	−0.345925	−0.172962	+	0.984928i	$0.555334\pi$
$488$	0	0
$489$	−1.40537	−0.0635528
$490$	0	0
$491$	−25.2158	−1.13797	−0.568986	−	0.822347i	$-0.692664\pi$
$491$	−25.2158	−1.13797	−0.568986	+	0.822347i	$0.692664\pi$
$492$	0	0
$493$	1.83792	0.0827759
$494$	0	0
$495$	−6.49280	−0.291830
$496$	0	0
$497$	−13.9624	−0.626301
$498$	0	0
$499$	−40.2555	−1.80209	−0.901043	−	0.433730i	$-0.857197\pi$
$499$	−40.2555	−1.80209	−0.901043	+	0.433730i	$0.857197\pi$
$500$	0	0
$501$	−28.8480	−1.28883
$502$	0	0
$503$	36.2843	1.61784	0.808920	−	0.587919i	$-0.200053\pi$
$503$	36.2843	1.61784	0.808920	+	0.587919i	$0.200053\pi$
$504$	0	0
$505$	10.8613	0.483320
$506$	0	0
$507$	56.3134	2.50097
$508$	0	0
$509$	−27.4631	−1.21728	−0.608640	−	0.793447i	$-0.708284\pi$
$509$	−27.4631	−1.21728	−0.608640	+	0.793447i	$0.708284\pi$
$510$	0	0
$511$	−12.7315	−0.563210
$512$	0	0
$513$	−65.5134	−2.89249
$514$	0	0
$515$	−0.194234	−0.00855898
$516$	0	0
$517$	9.99235	0.439463
$518$	0	0
$519$	8.53171	0.374500
$520$	0	0
$521$	−35.1079	−1.53810	−0.769052	−	0.639186i	$-0.779271\pi$
$521$	−35.1079	−1.53810	−0.769052	+	0.639186i	$0.779271\pi$
$522$	0	0
$523$	0.998661	0.0436684	0.0218342	−	0.999762i	$-0.493049\pi$
$523$	0.998661	0.0436684	0.0218342	+	0.999762i	$0.493049\pi$
$524$	0	0
$525$	−3.08104	−0.134468
$526$	0	0
$527$	25.4552	1.10885
$528$	0	0
$529$	1.81252	0.0788050
$530$	0	0
$531$	82.2094	3.56758
$532$	0	0
$533$	22.2277	0.962788
$534$	0	0
$535$	−1.77585	−0.0767767
$536$	0	0
$537$	68.0910	2.93834
$538$	0	0
$539$	−1.00000	−0.0430730
$540$	0	0
$541$	23.0066	0.989130	0.494565	−	0.869141i	$-0.335327\pi$
$541$	23.0066	0.989130	0.494565	+	0.869141i	$0.335327\pi$
$542$	0	0
$543$	10.3617	0.444664
$544$	0	0
$545$	−15.6129	−0.668784
$546$	0	0
$547$	−26.9433	−1.15201	−0.576006	−	0.817445i	$-0.695390\pi$
$547$	−26.9433	−1.15201	−0.576006	+	0.817445i	$0.695390\pi$
$548$	0	0
$549$	−46.0042	−1.96341
$550$	0	0
$551$	−3.11441	−0.132678
$552$	0	0
$553$	5.58059	0.237311
$554$	0	0
$555$	−0.0135000	−0.000573042 0
$556$	0	0
$557$	−28.4033	−1.20348	−0.601742	−	0.798690i	$-0.705527\pi$
$557$	−28.4033	−1.20348	−0.601742	+	0.798690i	$0.705527\pi$
$558$	0	0
$559$	4.99660	0.211334
$560$	0	0
$561$	−11.0690	−0.467334
$562$	0	0
$563$	−26.4950	−1.11663	−0.558315	−	0.829629i	$-0.688552\pi$
$563$	−26.4950	−1.11663	−0.558315	+	0.829629i	$0.688552\pi$
$564$	0	0
$565$	−10.9991	−0.462736
$566$	0	0
$567$	−13.6780	−0.574424
$568$	0	0
$569$	−4.01350	−0.168255	−0.0841273	−	0.996455i	$-0.526810\pi$
$569$	−4.01350	−0.168255	−0.0841273	+	0.996455i	$0.526810\pi$
$570$	0	0
$571$	−8.68371	−0.363402	−0.181701	−	0.983354i	$-0.558160\pi$
$571$	−8.68371	−0.363402	−0.181701	+	0.983354i	$0.558160\pi$
$572$	0	0
$573$	−81.6321	−3.41023
$574$	0	0
$575$	−4.98122	−0.207731
$576$	0	0
$577$	36.2123	1.50754	0.753769	−	0.657140i	$-0.228234\pi$
$577$	36.2123	1.50754	0.753769	+	0.657140i	$0.228234\pi$
$578$	0	0
$579$	9.88649	0.410869
$580$	0	0
$581$	−6.58587	−0.273228
$582$	0	0
$583$	−6.98122	−0.289132
$584$	0	0
$585$	36.3118	1.50131
$586$	0	0
$587$	35.6318	1.47068	0.735342	−	0.677697i	$-0.237022\pi$
$587$	35.6318	1.47068	0.735342	+	0.677697i	$0.237022\pi$
$588$	0	0
$589$	−43.1345	−1.77733
$590$	0	0
$591$	−33.9869	−1.39804
$592$	0	0
$593$	24.3253	0.998919	0.499459	−	0.866337i	$-0.333532\pi$
$593$	24.3253	0.998919	0.499459	+	0.866337i	$0.333532\pi$
$594$	0	0
$595$	−3.59262	−0.147283
$596$	0	0
$597$	−24.8671	−1.01774
$598$	0	0
$599$	16.6239	0.679233	0.339617	−	0.940564i	$-0.389703\pi$
$599$	16.6239	0.679233	0.339617	+	0.940564i	$0.389703\pi$
$600$	0	0
$601$	11.1086	0.453131	0.226565	−	0.973996i	$-0.427250\pi$
$601$	11.1086	0.453131	0.226565	+	0.973996i	$0.427250\pi$
$602$	0	0
$603$	62.6872	2.55282
$604$	0	0
$605$	1.00000	0.0406558
$606$	0	0
$607$	48.0642	1.95087	0.975433	−	0.220295i	$-0.0707020\pi$
$607$	48.0642	1.95087	0.975433	+	0.220295i	$0.0707020\pi$
$608$	0	0
$609$	−1.57621	−0.0638711
$610$	0	0
$611$	−55.8834	−2.26080
$612$	0	0
$613$	−8.64521	−0.349177	−0.174588	−	0.984642i	$-0.555859\pi$
$613$	−8.64521	−0.349177	−0.174588	+	0.984642i	$0.555859\pi$
$614$	0	0
$615$	12.2455	0.493786
$616$	0	0
$617$	−49.0997	−1.97668	−0.988340	−	0.152264i	$-0.951344\pi$
$617$	−49.0997	−1.97668	−0.988340	+	0.152264i	$0.951344\pi$
$618$	0	0
$619$	29.9932	1.20553	0.602764	−	0.797919i	$-0.294066\pi$
$619$	29.9932	1.20553	0.602764	+	0.797919i	$0.294066\pi$
$620$	0	0
$621$	−53.6051	−2.15110
$622$	0	0
$623$	5.78461	0.231756
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	18.7567	0.749071
$628$	0	0
$629$	−0.0157415	−0.000627656 0
$630$	0	0
$631$	20.0175	0.796885	0.398443	−	0.917193i	$-0.369551\pi$
$631$	20.0175	0.796885	0.398443	+	0.917193i	$0.369551\pi$
$632$	0	0
$633$	−3.50903	−0.139471
$634$	0	0
$635$	−5.98650	−0.237567
$636$	0	0
$637$	5.59262	0.221588
$638$	0	0
$639$	90.6553	3.58627
$640$	0	0
$641$	−5.85969	−0.231444	−0.115722	−	0.993282i	$-0.536918\pi$
$641$	−5.85969	−0.231444	−0.115722	+	0.993282i	$0.536918\pi$
$642$	0	0
$643$	−29.3362	−1.15691	−0.578454	−	0.815715i	$-0.696344\pi$
$643$	−29.3362	−1.15691	−0.578454	+	0.815715i	$0.696344\pi$
$644$	0	0
$645$	2.75269	0.108387
$646$	0	0
$647$	22.7736	0.895321	0.447661	−	0.894204i	$-0.352257\pi$
$647$	22.7736	0.895321	0.447661	+	0.894204i	$0.352257\pi$
$648$	0	0
$649$	−12.6616	−0.497012
$650$	0	0
$651$	−21.8305	−0.855603
$652$	0	0
$653$	10.0673	0.393964	0.196982	−	0.980407i	$-0.436886\pi$
$653$	10.0673	0.393964	0.196982	+	0.980407i	$0.436886\pi$
$654$	0	0
$655$	−3.85014	−0.150437
$656$	0	0
$657$	82.6633	3.22500
$658$	0	0
$659$	−43.3684	−1.68939	−0.844696	−	0.535246i	$-0.820219\pi$
$659$	−43.3684	−1.68939	−0.844696	+	0.535246i	$0.820219\pi$
$660$	0	0
$661$	19.7470	0.768071	0.384036	−	0.923318i	$-0.374534\pi$
$661$	19.7470	0.768071	0.384036	+	0.923318i	$0.374534\pi$
$662$	0	0
$663$	61.9048	2.40418
$664$	0	0
$665$	6.08779	0.236074
$666$	0	0
$667$	−2.54830	−0.0986707
$668$	0	0
$669$	70.4204	2.72261
$670$	0	0
$671$	7.08542	0.273530
$672$	0	0
$673$	20.6117	0.794521	0.397261	−	0.917706i	$-0.369961\pi$
$673$	20.6117	0.794521	0.397261	+	0.917706i	$0.369961\pi$
$674$	0	0
$675$	10.7614	0.414208
$676$	0	0
$677$	12.0193	0.461941	0.230971	−	0.972961i	$-0.425810\pi$
$677$	12.0193	0.461941	0.230971	+	0.972961i	$0.425810\pi$
$678$	0	0
$679$	−12.2962	−0.471885
$680$	0	0
$681$	−56.3147	−2.15798
$682$	0	0
$683$	17.6461	0.675210	0.337605	−	0.941288i	$-0.390383\pi$
$683$	17.6461	0.675210	0.337605	+	0.941288i	$0.390383\pi$
$684$	0	0
$685$	−13.5016	−0.515868
$686$	0	0
$687$	−49.0166	−1.87010
$688$	0	0
$689$	39.0433	1.48743
$690$	0	0
$691$	8.96967	0.341222	0.170611	−	0.985338i	$-0.445426\pi$
$691$	8.96967	0.341222	0.170611	+	0.985338i	$0.445426\pi$
$692$	0	0
$693$	6.49280	0.246641
$694$	0	0
$695$	−16.6824	−0.632800
$696$	0	0
$697$	14.2788	0.540846
$698$	0	0
$699$	38.9042	1.47149
$700$	0	0
$701$	38.7741	1.46448	0.732239	−	0.681048i	$-0.238475\pi$
$701$	38.7741	1.46448	0.732239	+	0.681048i	$0.238475\pi$
$702$	0	0
$703$	0.0266744	0.00100605
$704$	0	0
$705$	−30.7868	−1.15950
$706$	0	0
$707$	−10.8613	−0.408480
$708$	0	0
$709$	14.8990	0.559544	0.279772	−	0.960066i	$-0.409741\pi$
$709$	14.8990	0.559544	0.279772	+	0.960066i	$0.409741\pi$
$710$	0	0
$711$	−36.2336	−1.35887
$712$	0	0
$713$	−35.2940	−1.32177
$714$	0	0
$715$	−5.59262	−0.209152
$716$	0	0
$717$	60.5709	2.26206
$718$	0	0
$719$	18.2666	0.681231	0.340615	−	0.940203i	$-0.389365\pi$
$719$	18.2666	0.681231	0.340615	+	0.940203i	$0.389365\pi$
$720$	0	0
$721$	0.194234	0.00723366
$722$	0	0
$723$	28.2360	1.05011
$724$	0	0
$725$	0.511583	0.0189997
$726$	0	0
$727$	−42.0678	−1.56021	−0.780105	−	0.625649i	$-0.784834\pi$
$727$	−42.0678	−1.56021	−0.780105	+	0.625649i	$0.784834\pi$
$728$	0	0
$729$	−10.6605	−0.394835
$730$	0	0
$731$	3.20975	0.118717
$732$	0	0
$733$	−54.0964	−1.99810	−0.999048	−	0.0436317i	$-0.986107\pi$
$733$	−54.0964	−1.99810	−0.999048	+	0.0436317i	$0.986107\pi$
$734$	0	0
$735$	3.08104	0.113646
$736$	0	0
$737$	−9.65488	−0.355642
$738$	0	0
$739$	−38.1712	−1.40415	−0.702075	−	0.712103i	$-0.747743\pi$
$739$	−38.1712	−1.40415	−0.702075	+	0.712103i	$0.747743\pi$
$740$	0	0
$741$	−104.899	−3.85357
$742$	0	0
$743$	−22.4352	−0.823066	−0.411533	−	0.911395i	$-0.635007\pi$
$743$	−22.4352	−0.823066	−0.411533	+	0.911395i	$0.635007\pi$
$744$	0	0
$745$	−12.6361	−0.462951
$746$	0	0
$747$	42.7607	1.56453
$748$	0	0
$749$	1.77585	0.0648882
$750$	0	0
$751$	−5.78979	−0.211272	−0.105636	−	0.994405i	$-0.533688\pi$
$751$	−5.78979	−0.211272	−0.105636	+	0.994405i	$0.533688\pi$
$752$	0	0
$753$	50.7200	1.84834
$754$	0	0
$755$	−23.7427	−0.864084
$756$	0	0
$757$	8.53565	0.310233	0.155117	−	0.987896i	$-0.450425\pi$
$757$	8.53565	0.310233	0.155117	+	0.987896i	$0.450425\pi$
$758$	0	0
$759$	15.3473	0.557072
$760$	0	0
$761$	−39.0688	−1.41624	−0.708122	−	0.706090i	$-0.750457\pi$
$761$	−39.0688	−1.41624	−0.708122	+	0.706090i	$0.750457\pi$
$762$	0	0
$763$	15.6129	0.565226
$764$	0	0
$765$	23.3262	0.843359
$766$	0	0
$767$	70.8117	2.55686
$768$	0	0
$769$	−3.23778	−0.116757	−0.0583786	−	0.998295i	$-0.518593\pi$
$769$	−3.23778	−0.116757	−0.0583786	+	0.998295i	$0.518593\pi$
$770$	0	0
$771$	78.3396	2.82133
$772$	0	0
$773$	−20.2223	−0.727347	−0.363674	−	0.931526i	$-0.618478\pi$
$773$	−20.2223	−0.727347	−0.363674	+	0.931526i	$0.618478\pi$
$774$	0	0
$775$	7.08542	0.254516
$776$	0	0
$777$	0.0135000	0.000484309 0
$778$	0	0
$779$	−24.1957	−0.866901
$780$	0	0
$781$	−13.9624	−0.499615
$782$	0	0
$783$	5.50537	0.196746
$784$	0	0
$785$	−9.58497	−0.342102
$786$	0	0
$787$	−25.6955	−0.915947	−0.457974	−	0.888966i	$-0.651425\pi$
$787$	−25.6955	−0.915947	−0.457974	+	0.888966i	$0.651425\pi$
$788$	0	0
$789$	78.7382	2.80315
$790$	0	0
$791$	10.9991	0.391083
$792$	0	0
$793$	−39.6261	−1.40716
$794$	0	0
$795$	21.5094	0.762860
$796$	0	0
$797$	−10.4596	−0.370497	−0.185248	−	0.982692i	$-0.559309\pi$
$797$	−10.4596	−0.370497	−0.185248	+	0.982692i	$0.559309\pi$
$798$	0	0
$799$	−35.8987	−1.27001
$800$	0	0
$801$	−37.5583	−1.32706
$802$	0	0
$803$	−12.7315	−0.449286
$804$	0	0
$805$	4.98122	0.175565
$806$	0	0
$807$	0.642113	0.0226034
$808$	0	0
$809$	29.1612	1.02525	0.512626	−	0.858612i	$-0.328673\pi$
$809$	29.1612	1.02525	0.512626	+	0.858612i	$0.328673\pi$
$810$	0	0
$811$	−17.9633	−0.630776	−0.315388	−	0.948963i	$-0.602135\pi$
$811$	−17.9633	−0.630776	−0.315388	+	0.948963i	$0.602135\pi$
$812$	0	0
$813$	−28.6151	−1.00358
$814$	0	0
$815$	−0.456134	−0.0159777
$816$	0	0
$817$	−5.43900	−0.190286
$818$	0	0
$819$	−36.3118	−1.26884
$820$	0	0
$821$	−28.3207	−0.988399	−0.494200	−	0.869348i	$-0.664539\pi$
$821$	−28.3207	−0.988399	−0.494200	+	0.869348i	$0.664539\pi$
$822$	0	0
$823$	−53.5313	−1.86598	−0.932992	−	0.359897i	$-0.882812\pi$
$823$	−53.5313	−1.86598	−0.932992	+	0.359897i	$0.882812\pi$
$824$	0	0
$825$	−3.08104	−0.107268
$826$	0	0
$827$	38.2700	1.33078	0.665389	−	0.746497i	$-0.268266\pi$
$827$	38.2700	1.33078	0.665389	+	0.746497i	$0.268266\pi$
$828$	0	0
$829$	−39.3268	−1.36588	−0.682939	−	0.730476i	$-0.739298\pi$
$829$	−39.3268	−1.36588	−0.682939	+	0.730476i	$0.739298\pi$
$830$	0	0
$831$	42.8291	1.48572
$832$	0	0
$833$	3.59262	0.124477
$834$	0	0
$835$	−9.36306	−0.324022
$836$	0	0
$837$	76.2494	2.63556
$838$	0	0
$839$	−4.47548	−0.154511	−0.0772554	−	0.997011i	$-0.524616\pi$
$839$	−4.47548	−0.154511	−0.0772554	+	0.997011i	$0.524616\pi$
$840$	0	0
$841$	−28.7383	−0.990975
$842$	0	0
$843$	−9.02900	−0.310975
$844$	0	0
$845$	18.2774	0.628762
$846$	0	0
$847$	−1.00000	−0.0343604
$848$	0	0
$849$	−79.0678	−2.71360
$850$	0	0
$851$	0.0218259	0.000748181 0
$852$	0	0
$853$	7.85253	0.268866	0.134433	−	0.990923i	$-0.457079\pi$
$853$	7.85253	0.268866	0.134433	+	0.990923i	$0.457079\pi$
$854$	0	0
$855$	−39.5268	−1.35179
$856$	0	0
$857$	23.4105	0.799688	0.399844	−	0.916583i	$-0.369064\pi$
$857$	23.4105	0.799688	0.399844	+	0.916583i	$0.369064\pi$
$858$	0	0
$859$	−30.1104	−1.02735	−0.513676	−	0.857984i	$-0.671717\pi$
$859$	−30.1104	−1.02735	−0.513676	+	0.857984i	$0.671717\pi$
$860$	0	0
$861$	−12.2455	−0.417325
$862$	0	0
$863$	−30.5174	−1.03882	−0.519412	−	0.854524i	$-0.673849\pi$
$863$	−30.5174	−1.03882	−0.519412	+	0.854524i	$0.673849\pi$
$864$	0	0
$865$	2.76910	0.0941522
$866$	0	0
$867$	−12.6109	−0.428289
$868$	0	0
$869$	5.58059	0.189308
$870$	0	0
$871$	53.9961	1.82959
$872$	0	0
$873$	79.8367	2.70206
$874$	0	0
$875$	−1.00000	−0.0338062
$876$	0	0
$877$	45.2288	1.52727	0.763635	−	0.645649i	$-0.223413\pi$
$877$	45.2288	1.52727	0.763635	+	0.645649i	$0.223413\pi$
$878$	0	0
$879$	32.5580	1.09815
$880$	0	0
$881$	−4.31278	−0.145301	−0.0726507	−	0.997357i	$-0.523146\pi$
$881$	−4.31278	−0.145301	−0.0726507	+	0.997357i	$0.523146\pi$
$882$	0	0
$883$	−55.0939	−1.85406	−0.927028	−	0.374991i	$-0.877646\pi$
$883$	−55.0939	−1.85406	−0.927028	+	0.374991i	$0.877646\pi$
$884$	0	0
$885$	39.0110	1.31134
$886$	0	0
$887$	−39.0611	−1.31154	−0.655772	−	0.754959i	$-0.727657\pi$
$887$	−39.0611	−1.31154	−0.655772	+	0.754959i	$0.727657\pi$
$888$	0	0
$889$	5.98650	0.200781
$890$	0	0
$891$	−13.6780	−0.458232
$892$	0	0
$893$	60.8313	2.03564
$894$	0	0
$895$	22.1000	0.738722
$896$	0	0
$897$	−85.8317	−2.86584
$898$	0	0
$899$	3.62478	0.120893
$900$	0	0
$901$	25.0809	0.835565
$902$	0	0
$903$	−2.75269	−0.0916036
$904$	0	0
$905$	3.36306	0.111792
$906$	0	0
$907$	−1.70594	−0.0566449	−0.0283225	−	0.999599i	$-0.509017\pi$
$907$	−1.70594	−0.0566449	−0.0283225	+	0.999599i	$0.509017\pi$
$908$	0	0
$909$	70.5200	2.33900
$910$	0	0
$911$	−5.96243	−0.197544	−0.0987721	−	0.995110i	$-0.531491\pi$
$911$	−5.96243	−0.197544	−0.0987721	+	0.995110i	$0.531491\pi$
$912$	0	0
$913$	−6.58587	−0.217961
$914$	0	0
$915$	−21.8305	−0.721693
$916$	0	0
$917$	3.85014	0.127143
$918$	0	0
$919$	−52.0839	−1.71809	−0.859044	−	0.511902i	$-0.828941\pi$
$919$	−52.0839	−1.71809	−0.859044	+	0.511902i	$0.828941\pi$
$920$	0	0
$921$	48.4487	1.59644
$922$	0	0
$923$	78.0866	2.57025
$924$	0	0
$925$	−0.00438163	−0.000144067 0
$926$	0	0
$927$	−1.26112	−0.0414207
$928$	0	0
$929$	5.16815	0.169562	0.0847808	−	0.996400i	$-0.472981\pi$
$929$	5.16815	0.169562	0.0847808	+	0.996400i	$0.472981\pi$
$930$	0	0
$931$	−6.08779	−0.199519
$932$	0	0
$933$	−85.1406	−2.78738
$934$	0	0
$935$	−3.59262	−0.117491
$936$	0	0
$937$	−44.4410	−1.45182	−0.725912	−	0.687787i	$-0.758582\pi$
$937$	−44.4410	−1.45182	−0.725912	+	0.687787i	$0.758582\pi$
$938$	0	0
$939$	92.5374	3.01984
$940$	0	0
$941$	−18.6371	−0.607553	−0.303776	−	0.952743i	$-0.598248\pi$
$941$	−18.6371	−0.607553	−0.303776	+	0.952743i	$0.598248\pi$
$942$	0	0
$943$	−19.7977	−0.644701
$944$	0	0
$945$	−10.7614	−0.350070
$946$	0	0
$947$	−28.3254	−0.920453	−0.460227	−	0.887801i	$-0.652232\pi$
$947$	−28.3254	−0.920453	−0.460227	+	0.887801i	$0.652232\pi$
$948$	0	0
$949$	71.2026	2.31134
$950$	0	0
$951$	−73.8522	−2.39482
$952$	0	0
$953$	39.5352	1.28067	0.640335	−	0.768095i	$-0.278795\pi$
$953$	39.5352	1.28067	0.640335	+	0.768095i	$0.278795\pi$
$954$	0	0
$955$	−26.4950	−0.857358
$956$	0	0
$957$	−1.57621	−0.0509515
$958$	0	0
$959$	13.5016	0.435988
$960$	0	0
$961$	19.2032	0.619457
$962$	0	0
$963$	−11.5302	−0.371557
$964$	0	0
$965$	3.20882	0.103296
$966$	0	0
$967$	−44.0995	−1.41814	−0.709072	−	0.705136i	$-0.750886\pi$
$967$	−44.0995	−1.41814	−0.709072	+	0.705136i	$0.750886\pi$
$968$	0	0
$969$	−67.3858	−2.16474
$970$	0	0
$971$	25.1124	0.805896	0.402948	−	0.915223i	$-0.367986\pi$
$971$	25.1124	0.805896	0.402948	+	0.915223i	$0.367986\pi$
$972$	0	0
$973$	16.6824	0.534814
$974$	0	0
$975$	17.2311	0.551836
$976$	0	0
$977$	−34.0179	−1.08833	−0.544164	−	0.838979i	$-0.683153\pi$
$977$	−34.0179	−1.08833	−0.544164	+	0.838979i	$0.683153\pi$
$978$	0	0
$979$	5.78461	0.184877
$980$	0	0
$981$	−101.372	−3.23655
$982$	0	0
$983$	8.45158	0.269564	0.134782	−	0.990875i	$-0.456967\pi$
$983$	8.45158	0.269564	0.134782	+	0.990875i	$0.456967\pi$
$984$	0	0
$985$	−11.0310	−0.351477
$986$	0	0
$987$	30.7868	0.979955
$988$	0	0
$989$	−4.45036	−0.141513
$990$	0	0
$991$	−18.9433	−0.601754	−0.300877	−	0.953663i	$-0.597279\pi$
$991$	−18.9433	−0.601754	−0.300877	+	0.953663i	$0.597279\pi$
$992$	0	0
$993$	64.1512	2.03578
$994$	0	0
$995$	−8.07102	−0.255868
$996$	0	0
$997$	38.8783	1.23129	0.615644	−	0.788024i	$-0.288896\pi$
$997$	38.8783	1.23129	0.615644	+	0.788024i	$0.288896\pi$
$998$	0	0
$999$	−0.0471527	−0.00149185

Twists

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

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

\(f(q)\)	\(=\)	\(q+3.08104 q^{3} +1.00000 q^{5} -1.00000 q^{7} +6.49280 q^{9} +O(q^{10})\)	\(q+3.08104 q^{3} +1.00000 q^{5} -1.00000 q^{7} +6.49280 q^{9} -1.00000 q^{11} +5.59262 q^{13} +3.08104 q^{15} +3.59262 q^{17} -6.08779 q^{19} -3.08104 q^{21} -4.98122 q^{23} +1.00000 q^{25} +10.7614 q^{27} +0.511583 q^{29} +7.08542 q^{31} -3.08104 q^{33} -1.00000 q^{35} -0.00438163 q^{37} +17.2311 q^{39} +3.97447 q^{41} +0.893428 q^{43} +6.49280 q^{45} -9.99235 q^{47} +1.00000 q^{49} +11.0690 q^{51} +6.98122 q^{53} -1.00000 q^{55} -18.7567 q^{57} +12.6616 q^{59} -7.08542 q^{61} -6.49280 q^{63} +5.59262 q^{65} +9.65488 q^{67} -15.3473 q^{69} +13.9624 q^{71} +12.7315 q^{73} +3.08104 q^{75} +1.00000 q^{77} -5.58059 q^{79} +13.6780 q^{81} +6.58587 q^{83} +3.59262 q^{85} +1.57621 q^{87} -5.78461 q^{89} -5.59262 q^{91} +21.8305 q^{93} -6.08779 q^{95} +12.2962 q^{97} -6.49280 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 2 q^{3} + 5 q^{5} - 5 q^{7} + 13 q^{9}+O(q^{10}) \) 5 * q - 2 * q^3 + 5 * q^5 - 5 * q^7 + 13 * q^9	\( 5 q - 2 q^{3} + 5 q^{5} - 5 q^{7} + 13 q^{9} - 5 q^{11} + 12 q^{13} - 2 q^{15} + 2 q^{17} - 8 q^{19} + 2 q^{21} - 4 q^{23} + 5 q^{25} - 2 q^{27} + 4 q^{29} + 2 q^{33} - 5 q^{35} + 18 q^{37} + 8 q^{39} + 4 q^{41} + 6 q^{43} + 13 q^{45} - 6 q^{47} + 5 q^{49} + 12 q^{51} + 14 q^{53} - 5 q^{55} - 12 q^{57} + 4 q^{59} - 13 q^{63} + 12 q^{65} - 6 q^{67} - 10 q^{69} + 28 q^{71} + 10 q^{73} - 2 q^{75} + 5 q^{77} + 14 q^{79} + 17 q^{81} + 22 q^{83} + 2 q^{85} - 16 q^{87} + 24 q^{89} - 12 q^{91} + 10 q^{93} - 8 q^{95} + 10 q^{97} - 13 q^{99}+O(q^{100}) \) 5 * q - 2 * q^3 + 5 * q^5 - 5 * q^7 + 13 * q^9 - 5 * q^11 + 12 * q^13 - 2 * q^15 + 2 * q^17 - 8 * q^19 + 2 * q^21 - 4 * q^23 + 5 * q^25 - 2 * q^27 + 4 * q^29 + 2 * q^33 - 5 * q^35 + 18 * q^37 + 8 * q^39 + 4 * q^41 + 6 * q^43 + 13 * q^45 - 6 * q^47 + 5 * q^49 + 12 * q^51 + 14 * q^53 - 5 * q^55 - 12 * q^57 + 4 * q^59 - 13 * q^63 + 12 * q^65 - 6 * q^67 - 10 * q^69 + 28 * q^71 + 10 * q^73 - 2 * q^75 + 5 * q^77 + 14 * q^79 + 17 * q^81 + 22 * q^83 + 2 * q^85 - 16 * q^87 + 24 * q^89 - 12 * q^91 + 10 * q^93 - 8 * q^95 + 10 * q^97 - 13 * q^99

Introduction

L-functions

Modular forms

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Database

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