LMFDB - Embedded newform 1520.2.a.p.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1520,2,Mod(1,1520)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1520, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("1520.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1520 = 2^{4} \cdot 5 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1520.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:	$12.1372611072$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.148.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 3x + 1 $ x^3 - x^2 - 3*x + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 95)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$2.17009$ of defining polynomial
Character	$\chi$	$=$	1520.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.70928 q^{3} +1.00000 q^{5} -1.07838 q^{7} -0.0783777 q^{9} +O(q^{10})$	$q+1.70928 q^{3} +1.00000 q^{5} -1.07838 q^{7} -0.0783777 q^{9} +6.34017 q^{11} +1.36910 q^{13} +1.70928 q^{15} +3.26180 q^{17} +1.00000 q^{19} -1.84324 q^{21} -2.34017 q^{23} +1.00000 q^{25} -5.26180 q^{27} +1.41855 q^{29} -8.68035 q^{31} +10.8371 q^{33} -1.07838 q^{35} +5.36910 q^{37} +2.34017 q^{39} -3.26180 q^{41} +11.9155 q^{43} -0.0783777 q^{45} -1.07838 q^{47} -5.83710 q^{49} +5.57531 q^{51} +6.63090 q^{53} +6.34017 q^{55} +1.70928 q^{57} +11.4186 q^{59} +5.60197 q^{61} +0.0845208 q^{63} +1.36910 q^{65} -10.3896 q^{67} -4.00000 q^{69} +10.8371 q^{71} +5.41855 q^{73} +1.70928 q^{75} -6.83710 q^{77} -14.2557 q^{79} -8.75872 q^{81} +14.3402 q^{83} +3.26180 q^{85} +2.42469 q^{87} +7.57531 q^{89} -1.47641 q^{91} -14.8371 q^{93} +1.00000 q^{95} -8.88655 q^{97} -0.496928 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 2 q^{3} + 3 q^{5} + 3 q^{9}+O(q^{10}) $ 3 * q - 2 * q^3 + 3 * q^5 + 3 * q^9	$ 3 q - 2 q^{3} + 3 q^{5} + 3 q^{9} + 8 q^{11} + 8 q^{13} - 2 q^{15} + 2 q^{17} + 3 q^{19} - 12 q^{21} + 4 q^{23} + 3 q^{25} - 8 q^{27} - 10 q^{29} - 4 q^{31} + 4 q^{33} + 20 q^{37} - 4 q^{39} - 2 q^{41} + 4 q^{43} + 3 q^{45} + 11 q^{49} - 4 q^{51} + 16 q^{53} + 8 q^{55} - 2 q^{57} + 20 q^{59} - 2 q^{61} + 32 q^{63} + 8 q^{65} - 2 q^{67} - 12 q^{69} + 4 q^{71} + 2 q^{73} - 2 q^{75} + 8 q^{77} - q^{81} + 32 q^{83} + 2 q^{85} + 28 q^{87} + 2 q^{89} - 20 q^{91} - 16 q^{93} + 3 q^{95} + 20 q^{97} + 16 q^{99}+O(q^{100}) $ 3 * q - 2 * q^3 + 3 * q^5 + 3 * q^9 + 8 * q^11 + 8 * q^13 - 2 * q^15 + 2 * q^17 + 3 * q^19 - 12 * q^21 + 4 * q^23 + 3 * q^25 - 8 * q^27 - 10 * q^29 - 4 * q^31 + 4 * q^33 + 20 * q^37 - 4 * q^39 - 2 * q^41 + 4 * q^43 + 3 * q^45 + 11 * q^49 - 4 * q^51 + 16 * q^53 + 8 * q^55 - 2 * q^57 + 20 * q^59 - 2 * q^61 + 32 * q^63 + 8 * q^65 - 2 * q^67 - 12 * q^69 + 4 * q^71 + 2 * q^73 - 2 * q^75 + 8 * q^77 - q^81 + 32 * q^83 + 2 * q^85 + 28 * q^87 + 2 * q^89 - 20 * q^91 - 16 * q^93 + 3 * q^95 + 20 * q^97 + 16 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	1.70928	0.986851	0.493425	−	0.869788i	$-0.335745\pi$
$3$	1.70928	0.986851	0.493425	+	0.869788i	$0.335745\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−1.07838	−0.407588	−0.203794	−	0.979014i	$-0.565327\pi$
$7$	−1.07838	−0.407588	−0.203794	+	0.979014i	$0.565327\pi$
$8$	0	0
$9$	−0.0783777	−0.0261259
$10$	0	0
$11$	6.34017	1.91163	0.955817	−	0.293962i	$-0.0949740\pi$
$11$	6.34017	1.91163	0.955817	+	0.293962i	$0.0949740\pi$
$12$	0	0
$13$	1.36910	0.379721	0.189860	−	0.981811i	$-0.439196\pi$
$13$	1.36910	0.379721	0.189860	+	0.981811i	$0.439196\pi$
$14$	0	0
$15$	1.70928	0.441333
$16$	0	0
$17$	3.26180	0.791102	0.395551	−	0.918444i	$-0.370554\pi$
$17$	3.26180	0.791102	0.395551	+	0.918444i	$0.370554\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	−1.84324	−0.402229
$22$	0	0
$23$	−2.34017	−0.487960	−0.243980	−	0.969780i	$-0.578453\pi$
$23$	−2.34017	−0.487960	−0.243980	+	0.969780i	$0.578453\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−5.26180	−1.01263
$28$	0	0
$29$	1.41855	0.263418	0.131709	−	0.991288i	$-0.457954\pi$
$29$	1.41855	0.263418	0.131709	+	0.991288i	$0.457954\pi$
$30$	0	0
$31$	−8.68035	−1.55904	−0.779518	−	0.626380i	$-0.784536\pi$
$31$	−8.68035	−1.55904	−0.779518	+	0.626380i	$0.784536\pi$
$32$	0	0
$33$	10.8371	1.88650
$34$	0	0
$35$	−1.07838	−0.182279
$36$	0	0
$37$	5.36910	0.882675	0.441337	−	0.897341i	$-0.354504\pi$
$37$	5.36910	0.882675	0.441337	+	0.897341i	$0.354504\pi$
$38$	0	0
$39$	2.34017	0.374728
$40$	0	0
$41$	−3.26180	−0.509407	−0.254703	−	0.967019i	$-0.581978\pi$
$41$	−3.26180	−0.509407	−0.254703	+	0.967019i	$0.581978\pi$
$42$	0	0
$43$	11.9155	1.81709	0.908547	−	0.417783i	$-0.137193\pi$
$43$	11.9155	1.81709	0.908547	+	0.417783i	$0.137193\pi$
$44$	0	0
$45$	−0.0783777	−0.0116839
$46$	0	0
$47$	−1.07838	−0.157298	−0.0786488	−	0.996902i	$-0.525061\pi$
$47$	−1.07838	−0.157298	−0.0786488	+	0.996902i	$0.525061\pi$
$48$	0	0
$49$	−5.83710	−0.833872
$50$	0	0
$51$	5.57531	0.780699
$52$	0	0
$53$	6.63090	0.910824	0.455412	−	0.890281i	$-0.349492\pi$
$53$	6.63090	0.910824	0.455412	+	0.890281i	$0.349492\pi$
$54$	0	0
$55$	6.34017	0.854909
$56$	0	0
$57$	1.70928	0.226399
$58$	0	0
$59$	11.4186	1.48657	0.743284	−	0.668976i	$-0.233267\pi$
$59$	11.4186	1.48657	0.743284	+	0.668976i	$0.233267\pi$
$60$	0	0
$61$	5.60197	0.717259	0.358629	−	0.933480i	$-0.383244\pi$
$61$	5.60197	0.717259	0.358629	+	0.933480i	$0.383244\pi$
$62$	0	0
$63$	0.0845208	0.0106486
$64$	0	0
$65$	1.36910	0.169816
$66$	0	0
$67$	−10.3896	−1.26929	−0.634647	−	0.772802i	$-0.718855\pi$
$67$	−10.3896	−1.26929	−0.634647	+	0.772802i	$0.718855\pi$
$68$	0	0
$69$	−4.00000	−0.481543
$70$	0	0
$71$	10.8371	1.28613	0.643064	−	0.765813i	$-0.277663\pi$
$71$	10.8371	1.28613	0.643064	+	0.765813i	$0.277663\pi$
$72$	0	0
$73$	5.41855	0.634193	0.317097	−	0.948393i	$-0.397292\pi$
$73$	5.41855	0.634193	0.317097	+	0.948393i	$0.397292\pi$
$74$	0	0
$75$	1.70928	0.197370
$76$	0	0
$77$	−6.83710	−0.779160
$78$	0	0
$79$	−14.2557	−1.60389	−0.801943	−	0.597400i	$-0.796200\pi$
$79$	−14.2557	−1.60389	−0.801943	+	0.597400i	$0.796200\pi$
$80$	0	0
$81$	−8.75872	−0.973192
$82$	0	0
$83$	14.3402	1.57404	0.787019	−	0.616928i	$-0.211623\pi$
$83$	14.3402	1.57404	0.787019	+	0.616928i	$0.211623\pi$
$84$	0	0
$85$	3.26180	0.353791
$86$	0	0
$87$	2.42469	0.259954
$88$	0	0
$89$	7.57531	0.802981	0.401490	−	0.915863i	$-0.368492\pi$
$89$	7.57531	0.802981	0.401490	+	0.915863i	$0.368492\pi$
$90$	0	0
$91$	−1.47641	−0.154770
$92$	0	0
$93$	−14.8371	−1.53854
$94$	0	0
$95$	1.00000	0.102598
$96$	0	0
$97$	−8.88655	−0.902292	−0.451146	−	0.892450i	$-0.648985\pi$
$97$	−8.88655	−0.902292	−0.451146	+	0.892450i	$0.648985\pi$
$98$	0	0
$99$	−0.496928	−0.0499432
$100$	0	0
$101$	−4.92162	−0.489720	−0.244860	−	0.969558i	$-0.578742\pi$
$101$	−4.92162	−0.489720	−0.244860	+	0.969558i	$0.578742\pi$
$102$	0	0
$103$	−6.38962	−0.629588	−0.314794	−	0.949160i	$-0.601935\pi$
$103$	−6.38962	−0.629588	−0.314794	+	0.949160i	$0.601935\pi$
$104$	0	0
$105$	−1.84324	−0.179882
$106$	0	0
$107$	−2.29072	−0.221453	−0.110726	−	0.993851i	$-0.535318\pi$
$107$	−2.29072	−0.221453	−0.110726	+	0.993851i	$0.535318\pi$
$108$	0	0
$109$	−12.8371	−1.22957	−0.614786	−	0.788694i	$-0.710757\pi$
$109$	−12.8371	−1.22957	−0.614786	+	0.788694i	$0.710757\pi$
$110$	0	0
$111$	9.17727	0.871068
$112$	0	0
$113$	−12.8865	−1.21226	−0.606132	−	0.795364i	$-0.707280\pi$
$113$	−12.8865	−1.21226	−0.606132	+	0.795364i	$0.707280\pi$
$114$	0	0
$115$	−2.34017	−0.218222
$116$	0	0
$117$	−0.107307	−0.00992055
$118$	0	0
$119$	−3.51745	−0.322444
$120$	0	0
$121$	29.1978	2.65434
$122$	0	0
$123$	−5.57531	−0.502708
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	8.23287	0.730549	0.365274	−	0.930900i	$-0.380975\pi$
$127$	8.23287	0.730549	0.365274	+	0.930900i	$0.380975\pi$
$128$	0	0
$129$	20.3668	1.79320
$130$	0	0
$131$	−1.47641	−0.128995	−0.0644973	−	0.997918i	$-0.520544\pi$
$131$	−1.47641	−0.128995	−0.0644973	+	0.997918i	$0.520544\pi$
$132$	0	0
$133$	−1.07838	−0.0935072
$134$	0	0
$135$	−5.26180	−0.452863
$136$	0	0
$137$	−3.94214	−0.336800	−0.168400	−	0.985719i	$-0.553860\pi$
$137$	−3.94214	−0.336800	−0.168400	+	0.985719i	$0.553860\pi$
$138$	0	0
$139$	−8.86376	−0.751815	−0.375907	−	0.926657i	$-0.622669\pi$
$139$	−8.86376	−0.751815	−0.375907	+	0.926657i	$0.622669\pi$
$140$	0	0
$141$	−1.84324	−0.155229
$142$	0	0
$143$	8.68035	0.725887
$144$	0	0
$145$	1.41855	0.117804
$146$	0	0
$147$	−9.97721	−0.822907
$148$	0	0
$149$	−19.7587	−1.61870	−0.809349	−	0.587328i	$-0.800180\pi$
$149$	−19.7587	−1.61870	−0.809349	+	0.587328i	$0.800180\pi$
$150$	0	0
$151$	−3.41855	−0.278198	−0.139099	−	0.990279i	$-0.544421\pi$
$151$	−3.41855	−0.278198	−0.139099	+	0.990279i	$0.544421\pi$
$152$	0	0
$153$	−0.255652	−0.0206683
$154$	0	0
$155$	−8.68035	−0.697222
$156$	0	0
$157$	9.41855	0.751682	0.375841	−	0.926684i	$-0.377354\pi$
$157$	9.41855	0.751682	0.375841	+	0.926684i	$0.377354\pi$
$158$	0	0
$159$	11.3340	0.898847
$160$	0	0
$161$	2.52359	0.198887
$162$	0	0
$163$	2.92162	0.228839	0.114420	−	0.993433i	$-0.463499\pi$
$163$	2.92162	0.228839	0.114420	+	0.993433i	$0.463499\pi$
$164$	0	0
$165$	10.8371	0.843667
$166$	0	0
$167$	−20.9132	−1.61831	−0.809156	−	0.587593i	$-0.800076\pi$
$167$	−20.9132	−1.61831	−0.809156	+	0.587593i	$0.800076\pi$
$168$	0	0
$169$	−11.1256	−0.855812
$170$	0	0
$171$	−0.0783777	−0.00599370
$172$	0	0
$173$	−1.05559	−0.0802551	−0.0401276	−	0.999195i	$-0.512776\pi$
$173$	−1.05559	−0.0802551	−0.0401276	+	0.999195i	$0.512776\pi$
$174$	0	0
$175$	−1.07838	−0.0815177
$176$	0	0
$177$	19.5174	1.46702
$178$	0	0
$179$	−0.894960	−0.0668925	−0.0334462	−	0.999441i	$-0.510648\pi$
$179$	−0.894960	−0.0668925	−0.0334462	+	0.999441i	$0.510648\pi$
$180$	0	0
$181$	−0.837101	−0.0622213	−0.0311106	−	0.999516i	$-0.509904\pi$
$181$	−0.837101	−0.0622213	−0.0311106	+	0.999516i	$0.509904\pi$
$182$	0	0
$183$	9.57531	0.707827
$184$	0	0
$185$	5.36910	0.394744
$186$	0	0
$187$	20.6803	1.51230
$188$	0	0
$189$	5.67420	0.412738
$190$	0	0
$191$	−22.0410	−1.59483	−0.797417	−	0.603429i	$-0.793801\pi$
$191$	−22.0410	−1.59483	−0.797417	+	0.603429i	$0.793801\pi$
$192$	0	0
$193$	12.7877	0.920475	0.460238	−	0.887796i	$-0.347764\pi$
$193$	12.7877	0.920475	0.460238	+	0.887796i	$0.347764\pi$
$194$	0	0
$195$	2.34017	0.167583
$196$	0	0
$197$	−9.20394	−0.655753	−0.327877	−	0.944721i	$-0.606333\pi$
$197$	−9.20394	−0.655753	−0.327877	+	0.944721i	$0.606333\pi$
$198$	0	0
$199$	16.1978	1.14823	0.574116	−	0.818774i	$-0.305346\pi$
$199$	16.1978	1.14823	0.574116	+	0.818774i	$0.305346\pi$
$200$	0	0
$201$	−17.7587	−1.25260
$202$	0	0
$203$	−1.52973	−0.107366
$204$	0	0
$205$	−3.26180	−0.227814
$206$	0	0
$207$	0.183417	0.0127484
$208$	0	0
$209$	6.34017	0.438559
$210$	0	0
$211$	−7.78539	−0.535968	−0.267984	−	0.963423i	$-0.586357\pi$
$211$	−7.78539	−0.535968	−0.267984	+	0.963423i	$0.586357\pi$
$212$	0	0
$213$	18.5236	1.26922
$214$	0	0
$215$	11.9155	0.812629
$216$	0	0
$217$	9.36069	0.635445
$218$	0	0
$219$	9.26180	0.625854
$220$	0	0
$221$	4.46573	0.300398
$222$	0	0
$223$	−12.5464	−0.840168	−0.420084	−	0.907485i	$-0.637999\pi$
$223$	−12.5464	−0.840168	−0.420084	+	0.907485i	$0.637999\pi$
$224$	0	0
$225$	−0.0783777	−0.00522518
$226$	0	0
$227$	−2.29072	−0.152041	−0.0760204	−	0.997106i	$-0.524221\pi$
$227$	−2.29072	−0.152041	−0.0760204	+	0.997106i	$0.524221\pi$
$228$	0	0
$229$	−5.91548	−0.390906	−0.195453	−	0.980713i	$-0.562618\pi$
$229$	−5.91548	−0.390906	−0.195453	+	0.980713i	$0.562618\pi$
$230$	0	0
$231$	−11.6865	−0.768915
$232$	0	0
$233$	13.5174	0.885557	0.442779	−	0.896631i	$-0.353993\pi$
$233$	13.5174	0.885557	0.442779	+	0.896631i	$0.353993\pi$
$234$	0	0
$235$	−1.07838	−0.0703456
$236$	0	0
$237$	−24.3668	−1.58280
$238$	0	0
$239$	−13.8432	−0.895445	−0.447723	−	0.894173i	$-0.647765\pi$
$239$	−13.8432	−0.895445	−0.447723	+	0.894173i	$0.647765\pi$
$240$	0	0
$241$	7.26180	0.467773	0.233887	−	0.972264i	$-0.424856\pi$
$241$	7.26180	0.467773	0.233887	+	0.972264i	$0.424856\pi$
$242$	0	0
$243$	0.814315	0.0522383
$244$	0	0
$245$	−5.83710	−0.372919
$246$	0	0
$247$	1.36910	0.0871139
$248$	0	0
$249$	24.5113	1.55334
$250$	0	0
$251$	−10.4703	−0.660877	−0.330439	−	0.943827i	$-0.607197\pi$
$251$	−10.4703	−0.660877	−0.330439	+	0.943827i	$0.607197\pi$
$252$	0	0
$253$	−14.8371	−0.932801
$254$	0	0
$255$	5.57531	0.349139
$256$	0	0
$257$	−23.6248	−1.47367	−0.736836	−	0.676072i	$-0.763681\pi$
$257$	−23.6248	−1.47367	−0.736836	+	0.676072i	$0.763681\pi$
$258$	0	0
$259$	−5.78992	−0.359768
$260$	0	0
$261$	−0.111183	−0.00688204
$262$	0	0
$263$	−5.65983	−0.349000	−0.174500	−	0.984657i	$-0.555831\pi$
$263$	−5.65983	−0.349000	−0.174500	+	0.984657i	$0.555831\pi$
$264$	0	0
$265$	6.63090	0.407333
$266$	0	0
$267$	12.9483	0.792422
$268$	0	0
$269$	−2.31351	−0.141057	−0.0705286	−	0.997510i	$-0.522469\pi$
$269$	−2.31351	−0.141057	−0.0705286	+	0.997510i	$0.522469\pi$
$270$	0	0
$271$	19.7009	1.19674	0.598371	−	0.801219i	$-0.295815\pi$
$271$	19.7009	1.19674	0.598371	+	0.801219i	$0.295815\pi$
$272$	0	0
$273$	−2.52359	−0.152735
$274$	0	0
$275$	6.34017	0.382327
$276$	0	0
$277$	−25.7321	−1.54609	−0.773045	−	0.634351i	$-0.781267\pi$
$277$	−25.7321	−1.54609	−0.773045	+	0.634351i	$0.781267\pi$
$278$	0	0
$279$	0.680346	0.0407312
$280$	0	0
$281$	−6.58145	−0.392616	−0.196308	−	0.980542i	$-0.562895\pi$
$281$	−6.58145	−0.392616	−0.196308	+	0.980542i	$0.562895\pi$
$282$	0	0
$283$	0.496928	0.0295393	0.0147697	−	0.999891i	$-0.495298\pi$
$283$	0.496928	0.0295393	0.0147697	+	0.999891i	$0.495298\pi$
$284$	0	0
$285$	1.70928	0.101249
$286$	0	0
$287$	3.51745	0.207628
$288$	0	0
$289$	−6.36069	−0.374158
$290$	0	0
$291$	−15.1896	−0.890428
$292$	0	0
$293$	6.63090	0.387381	0.193691	−	0.981063i	$-0.437954\pi$
$293$	6.63090	0.387381	0.193691	+	0.981063i	$0.437954\pi$
$294$	0	0
$295$	11.4186	0.664814
$296$	0	0
$297$	−33.3607	−1.93578
$298$	0	0
$299$	−3.20394	−0.185288
$300$	0	0
$301$	−12.8494	−0.740626
$302$	0	0
$303$	−8.41241	−0.483280
$304$	0	0
$305$	5.60197	0.320768
$306$	0	0
$307$	−14.6042	−0.833508	−0.416754	−	0.909019i	$-0.636832\pi$
$307$	−14.6042	−0.833508	−0.416754	+	0.909019i	$0.636832\pi$
$308$	0	0
$309$	−10.9216	−0.621309
$310$	0	0
$311$	−19.3340	−1.09633	−0.548166	−	0.836369i	$-0.684674\pi$
$311$	−19.3340	−1.09633	−0.548166	+	0.836369i	$0.684674\pi$
$312$	0	0
$313$	30.6803	1.73416	0.867078	−	0.498173i	$-0.165995\pi$
$313$	30.6803	1.73416	0.867078	+	0.498173i	$0.165995\pi$
$314$	0	0
$315$	0.0845208	0.00476221
$316$	0	0
$317$	18.7298	1.05197	0.525985	−	0.850494i	$-0.323697\pi$
$317$	18.7298	1.05197	0.525985	+	0.850494i	$0.323697\pi$
$318$	0	0
$319$	8.99386	0.503559
$320$	0	0
$321$	−3.91548	−0.218541
$322$	0	0
$323$	3.26180	0.181491
$324$	0	0
$325$	1.36910	0.0759441
$326$	0	0
$327$	−21.9421	−1.21340
$328$	0	0
$329$	1.16290	0.0641127
$330$	0	0
$331$	2.73820	0.150505	0.0752527	−	0.997164i	$-0.476024\pi$
$331$	2.73820	0.150505	0.0752527	+	0.997164i	$0.476024\pi$
$332$	0	0
$333$	−0.420818	−0.0230607
$334$	0	0
$335$	−10.3896	−0.567646
$336$	0	0
$337$	−6.04945	−0.329534	−0.164767	−	0.986332i	$-0.552687\pi$
$337$	−6.04945	−0.329534	−0.164767	+	0.986332i	$0.552687\pi$
$338$	0	0
$339$	−22.0267	−1.19632
$340$	0	0
$341$	−55.0349	−2.98031
$342$	0	0
$343$	13.8432	0.747465
$344$	0	0
$345$	−4.00000	−0.215353
$346$	0	0
$347$	5.97334	0.320666	0.160333	−	0.987063i	$-0.448743\pi$
$347$	5.97334	0.320666	0.160333	+	0.987063i	$0.448743\pi$
$348$	0	0
$349$	−10.0000	−0.535288	−0.267644	−	0.963518i	$-0.586245\pi$
$349$	−10.0000	−0.535288	−0.267644	+	0.963518i	$0.586245\pi$
$350$	0	0
$351$	−7.20394	−0.384518
$352$	0	0
$353$	−14.0989	−0.750409	−0.375204	−	0.926942i	$-0.622427\pi$
$353$	−14.0989	−0.750409	−0.375204	+	0.926942i	$0.622427\pi$
$354$	0	0
$355$	10.8371	0.575174
$356$	0	0
$357$	−6.01229	−0.318204
$358$	0	0
$359$	−6.02666	−0.318075	−0.159038	−	0.987273i	$-0.550839\pi$
$359$	−6.02666	−0.318075	−0.159038	+	0.987273i	$0.550839\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	49.9071	2.61944
$364$	0	0
$365$	5.41855	0.283620
$366$	0	0
$367$	−1.07838	−0.0562909	−0.0281454	−	0.999604i	$-0.508960\pi$
$367$	−1.07838	−0.0562909	−0.0281454	+	0.999604i	$0.508960\pi$
$368$	0	0
$369$	0.255652	0.0133087
$370$	0	0
$371$	−7.15061	−0.371241
$372$	0	0
$373$	12.3051	0.637134	0.318567	−	0.947900i	$-0.396798\pi$
$373$	12.3051	0.637134	0.318567	+	0.947900i	$0.396798\pi$
$374$	0	0
$375$	1.70928	0.0882666
$376$	0	0
$377$	1.94214	0.100025
$378$	0	0
$379$	−1.04718	−0.0537901	−0.0268950	−	0.999638i	$-0.508562\pi$
$379$	−1.04718	−0.0537901	−0.0268950	+	0.999638i	$0.508562\pi$
$380$	0	0
$381$	14.0722	0.720942
$382$	0	0
$383$	−0.0806452	−0.00412078	−0.00206039	−	0.999998i	$-0.500656\pi$
$383$	−0.0806452	−0.00412078	−0.00206039	+	0.999998i	$0.500656\pi$
$384$	0	0
$385$	−6.83710	−0.348451
$386$	0	0
$387$	−0.933908	−0.0474732
$388$	0	0
$389$	−20.5236	−1.04059	−0.520294	−	0.853987i	$-0.674178\pi$
$389$	−20.5236	−1.04059	−0.520294	+	0.853987i	$0.674178\pi$
$390$	0	0
$391$	−7.63317	−0.386026
$392$	0	0
$393$	−2.52359	−0.127298
$394$	0	0
$395$	−14.2557	−0.717280
$396$	0	0
$397$	39.4596	1.98042	0.990210	−	0.139586i	$-0.0445771\pi$
$397$	39.4596	1.98042	0.990210	+	0.139586i	$0.0445771\pi$
$398$	0	0
$399$	−1.84324	−0.0922776
$400$	0	0
$401$	0.470266	0.0234840	0.0117420	−	0.999931i	$-0.496262\pi$
$401$	0.470266	0.0234840	0.0117420	+	0.999931i	$0.496262\pi$
$402$	0	0
$403$	−11.8843	−0.591998
$404$	0	0
$405$	−8.75872	−0.435224
$406$	0	0
$407$	34.0410	1.68735
$408$	0	0
$409$	10.4826	0.518329	0.259164	−	0.965833i	$-0.416553\pi$
$409$	10.4826	0.518329	0.259164	+	0.965833i	$0.416553\pi$
$410$	0	0
$411$	−6.73820	−0.332371
$412$	0	0
$413$	−12.3135	−0.605908
$414$	0	0
$415$	14.3402	0.703931
$416$	0	0
$417$	−15.1506	−0.741929
$418$	0	0
$419$	−34.6681	−1.69365	−0.846823	−	0.531875i	$-0.821488\pi$
$419$	−34.6681	−1.69365	−0.846823	+	0.531875i	$0.821488\pi$
$420$	0	0
$421$	−34.6102	−1.68680	−0.843399	−	0.537288i	$-0.819449\pi$
$421$	−34.6102	−1.68680	−0.843399	+	0.537288i	$0.819449\pi$
$422$	0	0
$423$	0.0845208	0.00410954
$424$	0	0
$425$	3.26180	0.158220
$426$	0	0
$427$	−6.04104	−0.292346
$428$	0	0
$429$	14.8371	0.716342
$430$	0	0
$431$	−6.73820	−0.324568	−0.162284	−	0.986744i	$-0.551886\pi$
$431$	−6.73820	−0.324568	−0.162284	+	0.986744i	$0.551886\pi$
$432$	0	0
$433$	20.4741	0.983924	0.491962	−	0.870617i	$-0.336280\pi$
$433$	20.4741	0.983924	0.491962	+	0.870617i	$0.336280\pi$
$434$	0	0
$435$	2.42469	0.116255
$436$	0	0
$437$	−2.34017	−0.111946
$438$	0	0
$439$	21.4596	1.02421	0.512105	−	0.858923i	$-0.328866\pi$
$439$	21.4596	1.02421	0.512105	+	0.858923i	$0.328866\pi$
$440$	0	0
$441$	0.457499	0.0217857
$442$	0	0
$443$	21.5441	1.02359	0.511796	−	0.859107i	$-0.328980\pi$
$443$	21.5441	1.02359	0.511796	+	0.859107i	$0.328980\pi$
$444$	0	0
$445$	7.57531	0.359104
$446$	0	0
$447$	−33.7731	−1.59741
$448$	0	0
$449$	−8.47027	−0.399737	−0.199868	−	0.979823i	$-0.564051\pi$
$449$	−8.47027	−0.399737	−0.199868	+	0.979823i	$0.564051\pi$
$450$	0	0
$451$	−20.6803	−0.973799
$452$	0	0
$453$	−5.84324	−0.274540
$454$	0	0
$455$	−1.47641	−0.0692151
$456$	0	0
$457$	11.3607	0.531431	0.265715	−	0.964052i	$-0.414392\pi$
$457$	11.3607	0.531431	0.265715	+	0.964052i	$0.414392\pi$
$458$	0	0
$459$	−17.1629	−0.801096
$460$	0	0
$461$	3.04718	0.141921	0.0709607	−	0.997479i	$-0.477394\pi$
$461$	3.04718	0.141921	0.0709607	+	0.997479i	$0.477394\pi$
$462$	0	0
$463$	9.97334	0.463500	0.231750	−	0.972775i	$-0.425555\pi$
$463$	9.97334	0.463500	0.231750	+	0.972775i	$0.425555\pi$
$464$	0	0
$465$	−14.8371	−0.688054
$466$	0	0
$467$	−1.49079	−0.0689853	−0.0344927	−	0.999405i	$-0.510982\pi$
$467$	−1.49079	−0.0689853	−0.0344927	+	0.999405i	$0.510982\pi$
$468$	0	0
$469$	11.2039	0.517350
$470$	0	0
$471$	16.0989	0.741798
$472$	0	0
$473$	75.5462	3.47362
$474$	0	0
$475$	1.00000	0.0458831
$476$	0	0
$477$	−0.519715	−0.0237961
$478$	0	0
$479$	10.1711	0.464731	0.232365	−	0.972629i	$-0.425353\pi$
$479$	10.1711	0.464731	0.232365	+	0.972629i	$0.425353\pi$
$480$	0	0
$481$	7.35085	0.335170
$482$	0	0
$483$	4.31351	0.196272
$484$	0	0
$485$	−8.88655	−0.403517
$486$	0	0
$487$	24.2784	1.10016	0.550081	−	0.835112i	$-0.314597\pi$
$487$	24.2784	1.10016	0.550081	+	0.835112i	$0.314597\pi$
$488$	0	0
$489$	4.99386	0.225830
$490$	0	0
$491$	−19.2039	−0.866662	−0.433331	−	0.901235i	$-0.642662\pi$
$491$	−19.2039	−0.866662	−0.433331	+	0.901235i	$0.642662\pi$
$492$	0	0
$493$	4.62702	0.208391
$494$	0	0
$495$	−0.496928	−0.0223353
$496$	0	0
$497$	−11.6865	−0.524211
$498$	0	0
$499$	−7.33403	−0.328316	−0.164158	−	0.986434i	$-0.552491\pi$
$499$	−7.33403	−0.328316	−0.164158	+	0.986434i	$0.552491\pi$
$500$	0	0
$501$	−35.7464	−1.59703
$502$	0	0
$503$	−29.0616	−1.29579	−0.647895	−	0.761729i	$-0.724351\pi$
$503$	−29.0616	−1.29579	−0.647895	+	0.761729i	$0.724351\pi$
$504$	0	0
$505$	−4.92162	−0.219009
$506$	0	0
$507$	−19.0166	−0.844559
$508$	0	0
$509$	26.0456	1.15445	0.577225	−	0.816585i	$-0.304136\pi$
$509$	26.0456	1.15445	0.577225	+	0.816585i	$0.304136\pi$
$510$	0	0
$511$	−5.84324	−0.258490
$512$	0	0
$513$	−5.26180	−0.232314
$514$	0	0
$515$	−6.38962	−0.281560
$516$	0	0
$517$	−6.83710	−0.300695
$518$	0	0
$519$	−1.80430	−0.0791998
$520$	0	0
$521$	−38.8248	−1.70095	−0.850473	−	0.526019i	$-0.823684\pi$
$521$	−38.8248	−1.70095	−0.850473	+	0.526019i	$0.823684\pi$
$522$	0	0
$523$	−4.59970	−0.201131	−0.100565	−	0.994930i	$-0.532065\pi$
$523$	−4.59970	−0.201131	−0.100565	+	0.994930i	$0.532065\pi$
$524$	0	0
$525$	−1.84324	−0.0804458
$526$	0	0
$527$	−28.3135	−1.23336
$528$	0	0
$529$	−17.5236	−0.761895
$530$	0	0
$531$	−0.894960	−0.0388380
$532$	0	0
$533$	−4.46573	−0.193432
$534$	0	0
$535$	−2.29072	−0.0990367
$536$	0	0
$537$	−1.52973	−0.0660129
$538$	0	0
$539$	−37.0082	−1.59406
$540$	0	0
$541$	−12.1256	−0.521318	−0.260659	−	0.965431i	$-0.583940\pi$
$541$	−12.1256	−0.521318	−0.260659	+	0.965431i	$0.583940\pi$
$542$	0	0
$543$	−1.43084	−0.0614031
$544$	0	0
$545$	−12.8371	−0.549881
$546$	0	0
$547$	9.54023	0.407911	0.203955	−	0.978980i	$-0.434620\pi$
$547$	9.54023	0.407911	0.203955	+	0.978980i	$0.434620\pi$
$548$	0	0
$549$	−0.439070	−0.0187390
$550$	0	0
$551$	1.41855	0.0604323
$552$	0	0
$553$	15.3730	0.653726
$554$	0	0
$555$	9.17727	0.389554
$556$	0	0
$557$	19.9421	0.844976	0.422488	−	0.906369i	$-0.361157\pi$
$557$	19.9421	0.844976	0.422488	+	0.906369i	$0.361157\pi$
$558$	0	0
$559$	16.3135	0.689988
$560$	0	0
$561$	35.3484	1.49241
$562$	0	0
$563$	−23.5525	−0.992620	−0.496310	−	0.868145i	$-0.665312\pi$
$563$	−23.5525	−0.992620	−0.496310	+	0.868145i	$0.665312\pi$
$564$	0	0
$565$	−12.8865	−0.542141
$566$	0	0
$567$	9.44521	0.396662
$568$	0	0
$569$	−10.0000	−0.419222	−0.209611	−	0.977785i	$-0.567220\pi$
$569$	−10.0000	−0.419222	−0.209611	+	0.977785i	$0.567220\pi$
$570$	0	0
$571$	23.5031	0.983573	0.491786	−	0.870716i	$-0.336344\pi$
$571$	23.5031	0.983573	0.491786	+	0.870716i	$0.336344\pi$
$572$	0	0
$573$	−37.6742	−1.57386
$574$	0	0
$575$	−2.34017	−0.0975920
$576$	0	0
$577$	16.4703	0.685666	0.342833	−	0.939396i	$-0.388613\pi$
$577$	16.4703	0.685666	0.342833	+	0.939396i	$0.388613\pi$
$578$	0	0
$579$	21.8576	0.908372
$580$	0	0
$581$	−15.4641	−0.641560
$582$	0	0
$583$	42.0410	1.74116
$584$	0	0
$585$	−0.107307	−0.00443660
$586$	0	0
$587$	28.8104	1.18913	0.594567	−	0.804046i	$-0.297323\pi$
$587$	28.8104	1.18913	0.594567	+	0.804046i	$0.297323\pi$
$588$	0	0
$589$	−8.68035	−0.357667
$590$	0	0
$591$	−15.7321	−0.647131
$592$	0	0
$593$	−43.2450	−1.77586	−0.887929	−	0.459980i	$-0.847856\pi$
$593$	−43.2450	−1.77586	−0.887929	+	0.459980i	$0.847856\pi$
$594$	0	0
$595$	−3.51745	−0.144201
$596$	0	0
$597$	27.6865	1.13313
$598$	0	0
$599$	44.2967	1.80991	0.904957	−	0.425503i	$-0.139903\pi$
$599$	44.2967	1.80991	0.904957	+	0.425503i	$0.139903\pi$
$600$	0	0
$601$	−24.3090	−0.991584	−0.495792	−	0.868441i	$-0.665122\pi$
$601$	−24.3090	−0.991584	−0.495792	+	0.868441i	$0.665122\pi$
$602$	0	0
$603$	0.814315	0.0331615
$604$	0	0
$605$	29.1978	1.18706
$606$	0	0
$607$	6.29072	0.255333	0.127666	−	0.991817i	$-0.459251\pi$
$607$	6.29072	0.255333	0.127666	+	0.991817i	$0.459251\pi$
$608$	0	0
$609$	−2.61474	−0.105954
$610$	0	0
$611$	−1.47641	−0.0597291
$612$	0	0
$613$	−12.7915	−0.516645	−0.258322	−	0.966059i	$-0.583170\pi$
$613$	−12.7915	−0.516645	−0.258322	+	0.966059i	$0.583170\pi$
$614$	0	0
$615$	−5.57531	−0.224818
$616$	0	0
$617$	17.9299	0.721829	0.360914	−	0.932599i	$-0.382465\pi$
$617$	17.9299	0.721829	0.360914	+	0.932599i	$0.382465\pi$
$618$	0	0
$619$	−26.8515	−1.07925	−0.539626	−	0.841905i	$-0.681434\pi$
$619$	−26.8515	−1.07925	−0.539626	+	0.841905i	$0.681434\pi$
$620$	0	0
$621$	12.3135	0.494124
$622$	0	0
$623$	−8.16904	−0.327286
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	10.8371	0.432792
$628$	0	0
$629$	17.5129	0.698286
$630$	0	0
$631$	−35.5318	−1.41450	−0.707250	−	0.706964i	$-0.750064\pi$
$631$	−35.5318	−1.41450	−0.707250	+	0.706964i	$0.750064\pi$
$632$	0	0
$633$	−13.3074	−0.528920
$634$	0	0
$635$	8.23287	0.326711
$636$	0	0
$637$	−7.99159	−0.316638
$638$	0	0
$639$	−0.849388	−0.0336013
$640$	0	0
$641$	12.9360	0.510941	0.255471	−	0.966817i	$-0.417770\pi$
$641$	12.9360	0.510941	0.255471	+	0.966817i	$0.417770\pi$
$642$	0	0
$643$	−8.49693	−0.335086	−0.167543	−	0.985865i	$-0.553583\pi$
$643$	−8.49693	−0.335086	−0.167543	+	0.985865i	$0.553583\pi$
$644$	0	0
$645$	20.3668	0.801943
$646$	0	0
$647$	45.4908	1.78843	0.894214	−	0.447640i	$-0.147736\pi$
$647$	45.4908	1.78843	0.894214	+	0.447640i	$0.147736\pi$
$648$	0	0
$649$	72.3956	2.84178
$650$	0	0
$651$	16.0000	0.627089
$652$	0	0
$653$	35.9421	1.40652	0.703262	−	0.710930i	$-0.251726\pi$
$653$	35.9421	1.40652	0.703262	+	0.710930i	$0.251726\pi$
$654$	0	0
$655$	−1.47641	−0.0576881
$656$	0	0
$657$	−0.424694	−0.0165689
$658$	0	0
$659$	−30.9360	−1.20510	−0.602548	−	0.798083i	$-0.705848\pi$
$659$	−30.9360	−1.20510	−0.602548	+	0.798083i	$0.705848\pi$
$660$	0	0
$661$	10.0989	0.392802	0.196401	−	0.980524i	$-0.437075\pi$
$661$	10.0989	0.392802	0.196401	+	0.980524i	$0.437075\pi$
$662$	0	0
$663$	7.63317	0.296448
$664$	0	0
$665$	−1.07838	−0.0418177
$666$	0	0
$667$	−3.31965	−0.128538
$668$	0	0
$669$	−21.4452	−0.829120
$670$	0	0
$671$	35.5174	1.37114
$672$	0	0
$673$	−8.67194	−0.334279	−0.167139	−	0.985933i	$-0.553453\pi$
$673$	−8.67194	−0.334279	−0.167139	+	0.985933i	$0.553453\pi$
$674$	0	0
$675$	−5.26180	−0.202527
$676$	0	0
$677$	41.9793	1.61340	0.806698	−	0.590964i	$-0.201253\pi$
$677$	41.9793	1.61340	0.806698	+	0.590964i	$0.201253\pi$
$678$	0	0
$679$	9.58306	0.367764
$680$	0	0
$681$	−3.91548	−0.150041
$682$	0	0
$683$	−46.3896	−1.77505	−0.887525	−	0.460760i	$-0.847577\pi$
$683$	−46.3896	−1.77505	−0.887525	+	0.460760i	$0.847577\pi$
$684$	0	0
$685$	−3.94214	−0.150621
$686$	0	0
$687$	−10.1112	−0.385766
$688$	0	0
$689$	9.07838	0.345859
$690$	0	0
$691$	34.8515	1.32581	0.662906	−	0.748702i	$-0.269323\pi$
$691$	34.8515	1.32581	0.662906	+	0.748702i	$0.269323\pi$
$692$	0	0
$693$	0.535877	0.0203563
$694$	0	0
$695$	−8.86376	−0.336222
$696$	0	0
$697$	−10.6393	−0.402993
$698$	0	0
$699$	23.1050	0.873913
$700$	0	0
$701$	35.6430	1.34622	0.673109	−	0.739543i	$-0.264959\pi$
$701$	35.6430	1.34622	0.673109	+	0.739543i	$0.264959\pi$
$702$	0	0
$703$	5.36910	0.202500
$704$	0	0
$705$	−1.84324	−0.0694206
$706$	0	0
$707$	5.30737	0.199604
$708$	0	0
$709$	16.7214	0.627985	0.313992	−	0.949426i	$-0.398333\pi$
$709$	16.7214	0.627985	0.313992	+	0.949426i	$0.398333\pi$
$710$	0	0
$711$	1.11733	0.0419030
$712$	0	0
$713$	20.3135	0.760747
$714$	0	0
$715$	8.68035	0.324627
$716$	0	0
$717$	−23.6619	−0.883670
$718$	0	0
$719$	6.85148	0.255517	0.127758	−	0.991805i	$-0.459222\pi$
$719$	6.85148	0.255517	0.127758	+	0.991805i	$0.459222\pi$
$720$	0	0
$721$	6.89043	0.256613
$722$	0	0
$723$	12.4124	0.461622
$724$	0	0
$725$	1.41855	0.0526837
$726$	0	0
$727$	−34.4391	−1.27727	−0.638637	−	0.769508i	$-0.720502\pi$
$727$	−34.4391	−1.27727	−0.638637	+	0.769508i	$0.720502\pi$
$728$	0	0
$729$	27.6681	1.02474
$730$	0	0
$731$	38.8659	1.43751
$732$	0	0
$733$	−19.3607	−0.715103	−0.357552	−	0.933893i	$-0.616388\pi$
$733$	−19.3607	−0.715103	−0.357552	+	0.933893i	$0.616388\pi$
$734$	0	0
$735$	−9.97721	−0.368015
$736$	0	0
$737$	−65.8720	−2.42643
$738$	0	0
$739$	20.0000	0.735712	0.367856	−	0.929883i	$-0.380092\pi$
$739$	20.0000	0.735712	0.367856	+	0.929883i	$0.380092\pi$
$740$	0	0
$741$	2.34017	0.0859684
$742$	0	0
$743$	12.7154	0.466483	0.233242	−	0.972419i	$-0.425067\pi$
$743$	12.7154	0.466483	0.233242	+	0.972419i	$0.425067\pi$
$744$	0	0
$745$	−19.7587	−0.723904
$746$	0	0
$747$	−1.12395	−0.0411232
$748$	0	0
$749$	2.47027	0.0902616
$750$	0	0
$751$	−3.15836	−0.115250	−0.0576252	−	0.998338i	$-0.518353\pi$
$751$	−3.15836	−0.115250	−0.0576252	+	0.998338i	$0.518353\pi$
$752$	0	0
$753$	−17.8966	−0.652187
$754$	0	0
$755$	−3.41855	−0.124414
$756$	0	0
$757$	28.0410	1.01917	0.509584	−	0.860421i	$-0.329799\pi$
$757$	28.0410	1.01917	0.509584	+	0.860421i	$0.329799\pi$
$758$	0	0
$759$	−25.3607	−0.920535
$760$	0	0
$761$	−16.4924	−0.597849	−0.298924	−	0.954277i	$-0.596628\pi$
$761$	−16.4924	−0.597849	−0.298924	+	0.954277i	$0.596628\pi$
$762$	0	0
$763$	13.8432	0.501159
$764$	0	0
$765$	−0.255652	−0.00924312
$766$	0	0
$767$	15.6332	0.564481
$768$	0	0
$769$	26.9627	0.972298	0.486149	−	0.873876i	$-0.338401\pi$
$769$	26.9627	0.972298	0.486149	+	0.873876i	$0.338401\pi$
$770$	0	0
$771$	−40.3812	−1.45429
$772$	0	0
$773$	42.1939	1.51761	0.758805	−	0.651318i	$-0.225784\pi$
$773$	42.1939	1.51761	0.758805	+	0.651318i	$0.225784\pi$
$774$	0	0
$775$	−8.68035	−0.311807
$776$	0	0
$777$	−9.89657	−0.355037
$778$	0	0
$779$	−3.26180	−0.116866
$780$	0	0
$781$	68.7091	2.45860
$782$	0	0
$783$	−7.46412	−0.266746
$784$	0	0
$785$	9.41855	0.336162
$786$	0	0
$787$	35.4368	1.26319	0.631593	−	0.775300i	$-0.282401\pi$
$787$	35.4368	1.26319	0.631593	+	0.775300i	$0.282401\pi$
$788$	0	0
$789$	−9.67420	−0.344411
$790$	0	0
$791$	13.8966	0.494105
$792$	0	0
$793$	7.66967	0.272358
$794$	0	0
$795$	11.3340	0.401977
$796$	0	0
$797$	15.2579	0.540463	0.270232	−	0.962795i	$-0.412900\pi$
$797$	15.2579	0.540463	0.270232	+	0.962795i	$0.412900\pi$
$798$	0	0
$799$	−3.51745	−0.124438
$800$	0	0
$801$	−0.593735	−0.0209786
$802$	0	0
$803$	34.3545	1.21235
$804$	0	0
$805$	2.52359	0.0889449
$806$	0	0
$807$	−3.95443	−0.139202
$808$	0	0
$809$	−7.16290	−0.251834	−0.125917	−	0.992041i	$-0.540187\pi$
$809$	−7.16290	−0.251834	−0.125917	+	0.992041i	$0.540187\pi$
$810$	0	0
$811$	50.3545	1.76819	0.884094	−	0.467310i	$-0.154777\pi$
$811$	50.3545	1.76819	0.884094	+	0.467310i	$0.154777\pi$
$812$	0	0
$813$	33.6742	1.18101
$814$	0	0
$815$	2.92162	0.102340
$816$	0	0
$817$	11.9155	0.416870
$818$	0	0
$819$	0.115718	0.00404350
$820$	0	0
$821$	0.952819	0.0332536	0.0166268	−	0.999862i	$-0.494707\pi$
$821$	0.952819	0.0332536	0.0166268	+	0.999862i	$0.494707\pi$
$822$	0	0
$823$	44.2290	1.54173	0.770863	−	0.637001i	$-0.219825\pi$
$823$	44.2290	1.54173	0.770863	+	0.637001i	$0.219825\pi$
$824$	0	0
$825$	10.8371	0.377299
$826$	0	0
$827$	37.8615	1.31657	0.658287	−	0.752767i	$-0.271281\pi$
$827$	37.8615	1.31657	0.658287	+	0.752767i	$0.271281\pi$
$828$	0	0
$829$	−56.7214	−1.97002	−0.985008	−	0.172511i	$-0.944812\pi$
$829$	−56.7214	−1.97002	−0.985008	+	0.172511i	$0.944812\pi$
$830$	0	0
$831$	−43.9832	−1.52576
$832$	0	0
$833$	−19.0394	−0.659677
$834$	0	0
$835$	−20.9132	−0.723732
$836$	0	0
$837$	45.6742	1.57873
$838$	0	0
$839$	28.3591	0.979064	0.489532	−	0.871985i	$-0.337168\pi$
$839$	28.3591	0.979064	0.489532	+	0.871985i	$0.337168\pi$
$840$	0	0
$841$	−26.9877	−0.930611
$842$	0	0
$843$	−11.2495	−0.387454
$844$	0	0
$845$	−11.1256	−0.382731
$846$	0	0
$847$	−31.4863	−1.08188
$848$	0	0
$849$	0.849388	0.0291509
$850$	0	0
$851$	−12.5646	−0.430710
$852$	0	0
$853$	−17.0061	−0.582279	−0.291140	−	0.956681i	$-0.594034\pi$
$853$	−17.0061	−0.582279	−0.291140	+	0.956681i	$0.594034\pi$
$854$	0	0
$855$	−0.0783777	−0.00268046
$856$	0	0
$857$	15.4101	0.526400	0.263200	−	0.964741i	$-0.415222\pi$
$857$	15.4101	0.526400	0.263200	+	0.964741i	$0.415222\pi$
$858$	0	0
$859$	37.7275	1.28725	0.643623	−	0.765342i	$-0.277430\pi$
$859$	37.7275	1.28725	0.643623	+	0.765342i	$0.277430\pi$
$860$	0	0
$861$	6.01229	0.204898
$862$	0	0
$863$	−32.1340	−1.09385	−0.546927	−	0.837181i	$-0.684202\pi$
$863$	−32.1340	−1.09385	−0.546927	+	0.837181i	$0.684202\pi$
$864$	0	0
$865$	−1.05559	−0.0358912
$866$	0	0
$867$	−10.8722	−0.369238
$868$	0	0
$869$	−90.3833	−3.06604
$870$	0	0
$871$	−14.2245	−0.481977
$872$	0	0
$873$	0.696508	0.0235732
$874$	0	0
$875$	−1.07838	−0.0364558
$876$	0	0
$877$	−19.8927	−0.671729	−0.335864	−	0.941910i	$-0.609028\pi$
$877$	−19.8927	−0.671729	−0.335864	+	0.941910i	$0.609028\pi$
$878$	0	0
$879$	11.3340	0.382287
$880$	0	0
$881$	24.0722	0.811014	0.405507	−	0.914092i	$-0.367095\pi$
$881$	24.0722	0.811014	0.405507	+	0.914092i	$0.367095\pi$
$882$	0	0
$883$	31.1727	1.04905	0.524523	−	0.851396i	$-0.324244\pi$
$883$	31.1727	1.04905	0.524523	+	0.851396i	$0.324244\pi$
$884$	0	0
$885$	19.5174	0.656072
$886$	0	0
$887$	−4.86764	−0.163439	−0.0817197	−	0.996655i	$-0.526041\pi$
$887$	−4.86764	−0.163439	−0.0817197	+	0.996655i	$0.526041\pi$
$888$	0	0
$889$	−8.87814	−0.297763
$890$	0	0
$891$	−55.5318	−1.86039
$892$	0	0
$893$	−1.07838	−0.0360865
$894$	0	0
$895$	−0.894960	−0.0299152
$896$	0	0
$897$	−5.47641	−0.182852
$898$	0	0
$899$	−12.3135	−0.410679
$900$	0	0
$901$	21.6286	0.720554
$902$	0	0
$903$	−21.9631	−0.730888
$904$	0	0
$905$	−0.837101	−0.0278262
$906$	0	0
$907$	45.4778	1.51007	0.755033	−	0.655686i	$-0.227621\pi$
$907$	45.4778	1.51007	0.755033	+	0.655686i	$0.227621\pi$
$908$	0	0
$909$	0.385746	0.0127944
$910$	0	0
$911$	20.9483	0.694048	0.347024	−	0.937856i	$-0.387192\pi$
$911$	20.9483	0.694048	0.347024	+	0.937856i	$0.387192\pi$
$912$	0	0
$913$	90.9192	3.00899
$914$	0	0
$915$	9.57531	0.316550
$916$	0	0
$917$	1.59213	0.0525767
$918$	0	0
$919$	59.5174	1.96330	0.981650	−	0.190693i	$-0.0610735\pi$
$919$	59.5174	1.96330	0.981650	+	0.190693i	$0.0610735\pi$
$920$	0	0
$921$	−24.9627	−0.822548
$922$	0	0
$923$	14.8371	0.488369
$924$	0	0
$925$	5.36910	0.176535
$926$	0	0
$927$	0.500804	0.0164486
$928$	0	0
$929$	−16.7214	−0.548611	−0.274305	−	0.961643i	$-0.588448\pi$
$929$	−16.7214	−0.548611	−0.274305	+	0.961643i	$0.588448\pi$
$930$	0	0
$931$	−5.83710	−0.191303
$932$	0	0
$933$	−33.0472	−1.08192
$934$	0	0
$935$	20.6803	0.676320
$936$	0	0
$937$	−32.4534	−1.06021	−0.530104	−	0.847933i	$-0.677847\pi$
$937$	−32.4534	−1.06021	−0.530104	+	0.847933i	$0.677847\pi$
$938$	0	0
$939$	52.4412	1.71135
$940$	0	0
$941$	−23.6742	−0.771757	−0.385878	−	0.922550i	$-0.626102\pi$
$941$	−23.6742	−0.771757	−0.385878	+	0.922550i	$0.626102\pi$
$942$	0	0
$943$	7.63317	0.248570
$944$	0	0
$945$	5.67420	0.184582
$946$	0	0
$947$	−21.9733	−0.714038	−0.357019	−	0.934097i	$-0.616207\pi$
$947$	−21.9733	−0.714038	−0.357019	+	0.934097i	$0.616207\pi$
$948$	0	0
$949$	7.41855	0.240816
$950$	0	0
$951$	32.0144	1.03814
$952$	0	0
$953$	−53.2990	−1.72652	−0.863261	−	0.504757i	$-0.831582\pi$
$953$	−53.2990	−1.72652	−0.863261	+	0.504757i	$0.831582\pi$
$954$	0	0
$955$	−22.0410	−0.713231
$956$	0	0
$957$	15.3730	0.496938
$958$	0	0
$959$	4.25112	0.137276
$960$	0	0
$961$	44.3484	1.43059
$962$	0	0
$963$	0.179542	0.00578565
$964$	0	0
$965$	12.7877	0.411649
$966$	0	0
$967$	−15.8166	−0.508627	−0.254314	−	0.967122i	$-0.581850\pi$
$967$	−15.8166	−0.508627	−0.254314	+	0.967122i	$0.581850\pi$
$968$	0	0
$969$	5.57531	0.179105
$970$	0	0
$971$	43.1506	1.38477	0.692385	−	0.721529i	$-0.256560\pi$
$971$	43.1506	1.38477	0.692385	+	0.721529i	$0.256560\pi$
$972$	0	0
$973$	9.55849	0.306431
$974$	0	0
$975$	2.34017	0.0749455
$976$	0	0
$977$	−8.47414	−0.271112	−0.135556	−	0.990770i	$-0.543282\pi$
$977$	−8.47414	−0.271112	−0.135556	+	0.990770i	$0.543282\pi$
$978$	0	0
$979$	48.0288	1.53501
$980$	0	0
$981$	1.00614	0.0321237
$982$	0	0
$983$	5.59356	0.178407	0.0892034	−	0.996013i	$-0.471568\pi$
$983$	5.59356	0.178407	0.0892034	+	0.996013i	$0.471568\pi$
$984$	0	0
$985$	−9.20394	−0.293262
$986$	0	0
$987$	1.98771	0.0632696
$988$	0	0
$989$	−27.8843	−0.886669
$990$	0	0
$991$	−32.8950	−1.04494	−0.522471	−	0.852657i	$-0.674990\pi$
$991$	−32.8950	−1.04494	−0.522471	+	0.852657i	$0.674990\pi$
$992$	0	0
$993$	4.68035	0.148526
$994$	0	0
$995$	16.1978	0.513505
$996$	0	0
$997$	23.2618	0.736708	0.368354	−	0.929686i	$-0.379921\pi$
$997$	23.2618	0.736708	0.368354	+	0.929686i	$0.379921\pi$
$998$	0	0
$999$	−28.2511	−0.893826

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1520.2.a.p.1.3		3
4.3	odd	2		95.2.a.a.1.3	✓	3
5.4	even	2		7600.2.a.bx.1.1		3
8.3	odd	2		6080.2.a.bo.1.3		3
8.5	even	2		6080.2.a.by.1.1		3
12.11	even	2		855.2.a.i.1.1		3
20.3	even	4		475.2.b.d.324.1		6
20.7	even	4		475.2.b.d.324.6		6
20.19	odd	2		475.2.a.f.1.1		3
28.27	even	2		4655.2.a.u.1.3		3
60.59	even	2		4275.2.a.bk.1.3		3
76.75	even	2		1805.2.a.f.1.1		3
380.379	even	2		9025.2.a.bb.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
95.2.a.a.1.3	✓	3	4.3	odd	2
475.2.a.f.1.1		3	20.19	odd	2
475.2.b.d.324.1		6	20.3	even	4
475.2.b.d.324.6		6	20.7	even	4
855.2.a.i.1.1		3	12.11	even	2
1520.2.a.p.1.3		3	1.1	even	1	trivial
1805.2.a.f.1.1		3	76.75	even	2
4275.2.a.bk.1.3		3	60.59	even	2
4655.2.a.u.1.3		3	28.27	even	2
6080.2.a.bo.1.3		3	8.3	odd	2
6080.2.a.by.1.1		3	8.5	even	2
7600.2.a.bx.1.1		3	5.4	even	2
9025.2.a.bb.1.3		3	380.379	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:	\( N \)	\(=\)	\( 1520 = 2^{4} \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1520.a (trivial)

\(f(q)\)	\(=\)	\(q+1.70928 q^{3} +1.00000 q^{5} -1.07838 q^{7} -0.0783777 q^{9} +O(q^{10})\)	\(q+1.70928 q^{3} +1.00000 q^{5} -1.07838 q^{7} -0.0783777 q^{9} +6.34017 q^{11} +1.36910 q^{13} +1.70928 q^{15} +3.26180 q^{17} +1.00000 q^{19} -1.84324 q^{21} -2.34017 q^{23} +1.00000 q^{25} -5.26180 q^{27} +1.41855 q^{29} -8.68035 q^{31} +10.8371 q^{33} -1.07838 q^{35} +5.36910 q^{37} +2.34017 q^{39} -3.26180 q^{41} +11.9155 q^{43} -0.0783777 q^{45} -1.07838 q^{47} -5.83710 q^{49} +5.57531 q^{51} +6.63090 q^{53} +6.34017 q^{55} +1.70928 q^{57} +11.4186 q^{59} +5.60197 q^{61} +0.0845208 q^{63} +1.36910 q^{65} -10.3896 q^{67} -4.00000 q^{69} +10.8371 q^{71} +5.41855 q^{73} +1.70928 q^{75} -6.83710 q^{77} -14.2557 q^{79} -8.75872 q^{81} +14.3402 q^{83} +3.26180 q^{85} +2.42469 q^{87} +7.57531 q^{89} -1.47641 q^{91} -14.8371 q^{93} +1.00000 q^{95} -8.88655 q^{97} -0.496928 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 2 q^{3} + 3 q^{5} + 3 q^{9}+O(q^{10}) \) 3 * q - 2 * q^3 + 3 * q^5 + 3 * q^9	\( 3 q - 2 q^{3} + 3 q^{5} + 3 q^{9} + 8 q^{11} + 8 q^{13} - 2 q^{15} + 2 q^{17} + 3 q^{19} - 12 q^{21} + 4 q^{23} + 3 q^{25} - 8 q^{27} - 10 q^{29} - 4 q^{31} + 4 q^{33} + 20 q^{37} - 4 q^{39} - 2 q^{41} + 4 q^{43} + 3 q^{45} + 11 q^{49} - 4 q^{51} + 16 q^{53} + 8 q^{55} - 2 q^{57} + 20 q^{59} - 2 q^{61} + 32 q^{63} + 8 q^{65} - 2 q^{67} - 12 q^{69} + 4 q^{71} + 2 q^{73} - 2 q^{75} + 8 q^{77} - q^{81} + 32 q^{83} + 2 q^{85} + 28 q^{87} + 2 q^{89} - 20 q^{91} - 16 q^{93} + 3 q^{95} + 20 q^{97} + 16 q^{99}+O(q^{100}) \) 3 * q - 2 * q^3 + 3 * q^5 + 3 * q^9 + 8 * q^11 + 8 * q^13 - 2 * q^15 + 2 * q^17 + 3 * q^19 - 12 * q^21 + 4 * q^23 + 3 * q^25 - 8 * q^27 - 10 * q^29 - 4 * q^31 + 4 * q^33 + 20 * q^37 - 4 * q^39 - 2 * q^41 + 4 * q^43 + 3 * q^45 + 11 * q^49 - 4 * q^51 + 16 * q^53 + 8 * q^55 - 2 * q^57 + 20 * q^59 - 2 * q^61 + 32 * q^63 + 8 * q^65 - 2 * q^67 - 12 * q^69 + 4 * q^71 + 2 * q^73 - 2 * q^75 + 8 * q^77 - q^81 + 32 * q^83 + 2 * q^85 + 28 * q^87 + 2 * q^89 - 20 * q^91 - 16 * q^93 + 3 * q^95 + 20 * q^97 + 16 * q^99

Introduction

L-functions

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Database

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