LMFDB - Embedded newform 4016.2.a.h.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4016,2,Mod(1,4016)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4016, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("4016.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4016 = 2^{4} \cdot 251 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4016.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:	$32.0679214517$
Analytic rank:	$1$
Dimension:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - x^{8} - 11x^{7} + 7x^{6} + 40x^{5} - 11x^{4} - 53x^{3} - 2x^{2} + 13x + 1 $ x^9 - x^8 - 11x^7 + 7x^6 + 40x^5 - 11x^4 - 53x^3 - 2x^2 + 13*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 2008)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-0.600027$ of defining polynomial
Character	$\chi$	$=$	4016.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.600027 q^{3} +2.23756 q^{5} -0.532333 q^{7} -2.63997 q^{9} +O(q^{10})$	$q-0.600027 q^{3} +2.23756 q^{5} -0.532333 q^{7} -2.63997 q^{9} +5.09276 q^{11} -3.68266 q^{13} -1.34260 q^{15} -0.993973 q^{17} -2.48968 q^{19} +0.319414 q^{21} +2.14850 q^{23} +0.00667253 q^{25} +3.38413 q^{27} -9.53291 q^{29} +0.838484 q^{31} -3.05580 q^{33} -1.19113 q^{35} -1.17933 q^{37} +2.20970 q^{39} -3.98579 q^{41} -3.86444 q^{43} -5.90708 q^{45} +4.30822 q^{47} -6.71662 q^{49} +0.596411 q^{51} -0.989226 q^{53} +11.3954 q^{55} +1.49388 q^{57} -6.11779 q^{59} +10.0825 q^{61} +1.40534 q^{63} -8.24017 q^{65} +0.877968 q^{67} -1.28916 q^{69} -2.80233 q^{71} -1.96868 q^{73} -0.00400370 q^{75} -2.71104 q^{77} -4.66529 q^{79} +5.88933 q^{81} -11.4865 q^{83} -2.22407 q^{85} +5.72001 q^{87} +1.60788 q^{89} +1.96040 q^{91} -0.503113 q^{93} -5.57081 q^{95} +8.01260 q^{97} -13.4447 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q + q^{3} - 5 q^{5} - 4 q^{9}+O(q^{10}) $ 9 * q + q^3 - 5 * q^5 - 4 * q^9	$ 9 q + q^{3} - 5 q^{5} - 4 q^{9} + 3 q^{11} - 3 q^{13} + q^{15} - 11 q^{17} - 4 q^{19} - 3 q^{21} + 9 q^{23} - 12 q^{25} + 7 q^{27} - 9 q^{29} - 3 q^{31} - 14 q^{33} + 8 q^{35} - 10 q^{37} + q^{39} - 23 q^{41} - 10 q^{45} + 11 q^{47} - 21 q^{49} + 3 q^{51} - 21 q^{53} + 4 q^{55} - 21 q^{57} + 4 q^{59} - 11 q^{61} + 2 q^{63} - 29 q^{65} + 4 q^{67} - 14 q^{69} + 19 q^{71} - 31 q^{73} - 16 q^{75} - 26 q^{77} - 4 q^{79} - 27 q^{81} + 22 q^{83} + 4 q^{85} + 6 q^{87} - 36 q^{89} - 14 q^{91} - 32 q^{93} + 3 q^{95} - 38 q^{97}+O(q^{100}) $ 9 * q + q^3 - 5 * q^5 - 4 * q^9 + 3 * q^11 - 3 * q^13 + q^15 - 11 * q^17 - 4 * q^19 - 3 * q^21 + 9 * q^23 - 12 * q^25 + 7 * q^27 - 9 * q^29 - 3 * q^31 - 14 * q^33 + 8 * q^35 - 10 * q^37 + q^39 - 23 * q^41 - 10 * q^45 + 11 * q^47 - 21 * q^49 + 3 * q^51 - 21 * q^53 + 4 * q^55 - 21 * q^57 + 4 * q^59 - 11 * q^61 + 2 * q^63 - 29 * q^65 + 4 * q^67 - 14 * q^69 + 19 * q^71 - 31 * q^73 - 16 * q^75 - 26 * q^77 - 4 * q^79 - 27 * q^81 + 22 * q^83 + 4 * q^85 + 6 * q^87 - 36 * q^89 - 14 * q^91 - 32 * q^93 + 3 * q^95 - 38 * q^97

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
$3$	−0.600027	−0.346426	−0.173213	−	0.984884i	$-0.555415\pi$
$3$	−0.600027	−0.346426	−0.173213	+	0.984884i	$0.555415\pi$
$4$	0	0
$5$	2.23756	1.00067	0.500334	−	0.865833i	$-0.333211\pi$
$5$	2.23756	1.00067	0.500334	+	0.865833i	$0.333211\pi$
$6$	0	0
$7$	−0.532333	−0.201203	−0.100601	−	0.994927i	$-0.532077\pi$
$7$	−0.532333	−0.201203	−0.100601	+	0.994927i	$0.532077\pi$
$8$	0	0
$9$	−2.63997	−0.879989
$10$	0	0
$11$	5.09276	1.53553	0.767763	−	0.640734i	$-0.221370\pi$
$11$	5.09276	1.53553	0.767763	+	0.640734i	$0.221370\pi$
$12$	0	0
$13$	−3.68266	−1.02139	−0.510693	−	0.859763i	$-0.670611\pi$
$13$	−3.68266	−1.02139	−0.510693	+	0.859763i	$0.670611\pi$
$14$	0	0
$15$	−1.34260	−0.346657
$16$	0	0
$17$	−0.993973	−0.241074	−0.120537	−	0.992709i	$-0.538462\pi$
$17$	−0.993973	−0.241074	−0.120537	+	0.992709i	$0.538462\pi$
$18$	0	0
$19$	−2.48968	−0.571172	−0.285586	−	0.958353i	$-0.592188\pi$
$19$	−2.48968	−0.571172	−0.285586	+	0.958353i	$0.592188\pi$
$20$	0	0
$21$	0.319414	0.0697019
$22$	0	0
$23$	2.14850	0.447992	0.223996	−	0.974590i	$-0.428090\pi$
$23$	2.14850	0.447992	0.223996	+	0.974590i	$0.428090\pi$
$24$	0	0
$25$	0.00667253	0.00133451
$26$	0	0
$27$	3.38413	0.651277
$28$	0	0
$29$	−9.53291	−1.77022	−0.885108	−	0.465385i	$-0.845916\pi$
$29$	−9.53291	−1.77022	−0.885108	+	0.465385i	$0.845916\pi$
$30$	0	0
$31$	0.838484	0.150596	0.0752981	−	0.997161i	$-0.476009\pi$
$31$	0.838484	0.150596	0.0752981	+	0.997161i	$0.476009\pi$
$32$	0	0
$33$	−3.05580	−0.531946
$34$	0	0
$35$	−1.19113	−0.201337
$36$	0	0
$37$	−1.17933	−0.193881	−0.0969405	−	0.995290i	$-0.530906\pi$
$37$	−1.17933	−0.193881	−0.0969405	+	0.995290i	$0.530906\pi$
$38$	0	0
$39$	2.20970	0.353835
$40$	0	0
$41$	−3.98579	−0.622475	−0.311238	−	0.950332i	$-0.600743\pi$
$41$	−3.98579	−0.622475	−0.311238	+	0.950332i	$0.600743\pi$
$42$	0	0
$43$	−3.86444	−0.589322	−0.294661	−	0.955602i	$-0.595207\pi$
$43$	−3.86444	−0.589322	−0.294661	+	0.955602i	$0.595207\pi$
$44$	0	0
$45$	−5.90708	−0.880576
$46$	0	0
$47$	4.30822	0.628418	0.314209	−	0.949354i	$-0.398261\pi$
$47$	4.30822	0.628418	0.314209	+	0.949354i	$0.398261\pi$
$48$	0	0
$49$	−6.71662	−0.959517
$50$	0	0
$51$	0.596411	0.0835143
$52$	0	0
$53$	−0.989226	−0.135881	−0.0679403	−	0.997689i	$-0.521643\pi$
$53$	−0.989226	−0.135881	−0.0679403	+	0.997689i	$0.521643\pi$
$54$	0	0
$55$	11.3954	1.53655
$56$	0	0
$57$	1.49388	0.197869
$58$	0	0
$59$	−6.11779	−0.796468	−0.398234	−	0.917284i	$-0.630377\pi$
$59$	−6.11779	−0.796468	−0.398234	+	0.917284i	$0.630377\pi$
$60$	0	0
$61$	10.0825	1.29094	0.645468	−	0.763787i	$-0.276662\pi$
$61$	10.0825	1.29094	0.645468	+	0.763787i	$0.276662\pi$
$62$	0	0
$63$	1.40534	0.177056
$64$	0	0
$65$	−8.24017	−1.02207
$66$	0	0
$67$	0.877968	0.107261	0.0536304	−	0.998561i	$-0.482921\pi$
$67$	0.877968	0.107261	0.0536304	+	0.998561i	$0.482921\pi$
$68$	0	0
$69$	−1.28916	−0.155196
$70$	0	0
$71$	−2.80233	−0.332575	−0.166288	−	0.986077i	$-0.553178\pi$
$71$	−2.80233	−0.332575	−0.166288	+	0.986077i	$0.553178\pi$
$72$	0	0
$73$	−1.96868	−0.230417	−0.115208	−	0.993341i	$-0.536754\pi$
$73$	−1.96868	−0.230417	−0.115208	+	0.993341i	$0.536754\pi$
$74$	0	0
$75$	−0.00400370	−0.000462307 0
$76$	0	0
$77$	−2.71104	−0.308952
$78$	0	0
$79$	−4.66529	−0.524886	−0.262443	−	0.964948i	$-0.584528\pi$
$79$	−4.66529	−0.524886	−0.262443	+	0.964948i	$0.584528\pi$
$80$	0	0
$81$	5.88933	0.654370
$82$	0	0
$83$	−11.4865	−1.26080	−0.630401	−	0.776270i	$-0.717110\pi$
$83$	−11.4865	−1.26080	−0.630401	+	0.776270i	$0.717110\pi$
$84$	0	0
$85$	−2.22407	−0.241235
$86$	0	0
$87$	5.72001	0.613249
$88$	0	0
$89$	1.60788	0.170435	0.0852176	−	0.996362i	$-0.472841\pi$
$89$	1.60788	0.170435	0.0852176	+	0.996362i	$0.472841\pi$
$90$	0	0
$91$	1.96040	0.205506
$92$	0	0
$93$	−0.503113	−0.0521704
$94$	0	0
$95$	−5.57081	−0.571553
$96$	0	0
$97$	8.01260	0.813556	0.406778	−	0.913527i	$-0.366652\pi$
$97$	8.01260	0.813556	0.406778	+	0.913527i	$0.366652\pi$
$98$	0	0
$99$	−13.4447	−1.35125
$100$	0	0
$101$	−8.32986	−0.828852	−0.414426	−	0.910083i	$-0.636018\pi$
$101$	−8.32986	−0.828852	−0.414426	+	0.910083i	$0.636018\pi$
$102$	0	0
$103$	−9.77547	−0.963206	−0.481603	−	0.876390i	$-0.659945\pi$
$103$	−9.77547	−0.963206	−0.481603	+	0.876390i	$0.659945\pi$
$104$	0	0
$105$	0.714708	0.0697484
$106$	0	0
$107$	−4.49903	−0.434938	−0.217469	−	0.976067i	$-0.569780\pi$
$107$	−4.49903	−0.434938	−0.217469	+	0.976067i	$0.569780\pi$
$108$	0	0
$109$	−4.92852	−0.472067	−0.236033	−	0.971745i	$-0.575847\pi$
$109$	−4.92852	−0.472067	−0.236033	+	0.971745i	$0.575847\pi$
$110$	0	0
$111$	0.707632	0.0671654
$112$	0	0
$113$	−4.70947	−0.443030	−0.221515	−	0.975157i	$-0.571100\pi$
$113$	−4.70947	−0.443030	−0.221515	+	0.975157i	$0.571100\pi$
$114$	0	0
$115$	4.80739	0.448291
$116$	0	0
$117$	9.72211	0.898809
$118$	0	0
$119$	0.529125	0.0485048
$120$	0	0
$121$	14.9362	1.35784
$122$	0	0
$123$	2.39158	0.215642
$124$	0	0
$125$	−11.1729	−0.999332
$126$	0	0
$127$	−5.31995	−0.472069	−0.236035	−	0.971745i	$-0.575848\pi$
$127$	−5.31995	−0.472069	−0.236035	+	0.971745i	$0.575848\pi$
$128$	0	0
$129$	2.31877	0.204156
$130$	0	0
$131$	2.04223	0.178431	0.0892153	−	0.996012i	$-0.471564\pi$
$131$	2.04223	0.178431	0.0892153	+	0.996012i	$0.471564\pi$
$132$	0	0
$133$	1.32534	0.114922
$134$	0	0
$135$	7.57220	0.651711
$136$	0	0
$137$	−1.23607	−0.105605	−0.0528023	−	0.998605i	$-0.516815\pi$
$137$	−1.23607	−0.105605	−0.0528023	+	0.998605i	$0.516815\pi$
$138$	0	0
$139$	14.5986	1.23824	0.619120	−	0.785297i	$-0.287490\pi$
$139$	14.5986	1.23824	0.619120	+	0.785297i	$0.287490\pi$
$140$	0	0
$141$	−2.58505	−0.217700
$142$	0	0
$143$	−18.7549	−1.56836
$144$	0	0
$145$	−21.3304	−1.77140
$146$	0	0
$147$	4.03016	0.332402
$148$	0	0
$149$	1.11114	0.0910281	0.0455141	−	0.998964i	$-0.485507\pi$
$149$	1.11114	0.0910281	0.0455141	+	0.998964i	$0.485507\pi$
$150$	0	0
$151$	22.7530	1.85161	0.925807	−	0.377996i	$-0.123387\pi$
$151$	22.7530	1.85161	0.925807	+	0.377996i	$0.123387\pi$
$152$	0	0
$153$	2.62406	0.212142
$154$	0	0
$155$	1.87616	0.150697
$156$	0	0
$157$	−13.5336	−1.08010	−0.540048	−	0.841634i	$-0.681594\pi$
$157$	−13.5336	−1.08010	−0.540048	+	0.841634i	$0.681594\pi$
$158$	0	0
$159$	0.593563	0.0470726
$160$	0	0
$161$	−1.14371	−0.0901374
$162$	0	0
$163$	9.61026	0.752734	0.376367	−	0.926471i	$-0.377173\pi$
$163$	9.61026	0.752734	0.376367	+	0.926471i	$0.377173\pi$
$164$	0	0
$165$	−6.83753	−0.532301
$166$	0	0
$167$	9.97516	0.771901	0.385950	−	0.922520i	$-0.373874\pi$
$167$	9.97516	0.771901	0.385950	+	0.922520i	$0.373874\pi$
$168$	0	0
$169$	0.561994	0.0432303
$170$	0	0
$171$	6.57268	0.502625
$172$	0	0
$173$	−11.8604	−0.901730	−0.450865	−	0.892592i	$-0.648884\pi$
$173$	−11.8604	−0.901730	−0.450865	+	0.892592i	$0.648884\pi$
$174$	0	0
$175$	−0.00355200	−0.000268506 0
$176$	0	0
$177$	3.67084	0.275917
$178$	0	0
$179$	−3.52673	−0.263600	−0.131800	−	0.991276i	$-0.542076\pi$
$179$	−3.52673	−0.263600	−0.131800	+	0.991276i	$0.542076\pi$
$180$	0	0
$181$	−10.3749	−0.771162	−0.385581	−	0.922674i	$-0.625999\pi$
$181$	−10.3749	−0.771162	−0.385581	+	0.922674i	$0.625999\pi$
$182$	0	0
$183$	−6.04980	−0.447214
$184$	0	0
$185$	−2.63883	−0.194010
$186$	0	0
$187$	−5.06207	−0.370175
$188$	0	0
$189$	−1.80149	−0.131039
$190$	0	0
$191$	0.635023	0.0459486	0.0229743	−	0.999736i	$-0.492686\pi$
$191$	0.635023	0.0459486	0.0229743	+	0.999736i	$0.492686\pi$
$192$	0	0
$193$	7.55542	0.543851	0.271925	−	0.962318i	$-0.412340\pi$
$193$	7.55542	0.543851	0.271925	+	0.962318i	$0.412340\pi$
$194$	0	0
$195$	4.94433	0.354071
$196$	0	0
$197$	12.2589	0.873408	0.436704	−	0.899605i	$-0.356146\pi$
$197$	12.2589	0.873408	0.436704	+	0.899605i	$0.356146\pi$
$198$	0	0
$199$	−1.71633	−0.121668	−0.0608338	−	0.998148i	$-0.519376\pi$
$199$	−1.71633	−0.121668	−0.0608338	+	0.998148i	$0.519376\pi$
$200$	0	0
$201$	−0.526805	−0.0371579
$202$	0	0
$203$	5.07468	0.356173
$204$	0	0
$205$	−8.91843	−0.622890
$206$	0	0
$207$	−5.67196	−0.394228
$208$	0	0
$209$	−12.6794	−0.877049
$210$	0	0
$211$	−10.0333	−0.690717	−0.345359	−	0.938471i	$-0.612243\pi$
$211$	−10.0333	−0.690717	−0.345359	+	0.938471i	$0.612243\pi$
$212$	0	0
$213$	1.68147	0.115213
$214$	0	0
$215$	−8.64692	−0.589715
$216$	0	0
$217$	−0.446353	−0.0303004
$218$	0	0
$219$	1.18126	0.0798224
$220$	0	0
$221$	3.66047	0.246230
$222$	0	0
$223$	−22.8840	−1.53242	−0.766212	−	0.642587i	$-0.777861\pi$
$223$	−22.8840	−1.53242	−0.766212	+	0.642587i	$0.777861\pi$
$224$	0	0
$225$	−0.0176152	−0.00117435
$226$	0	0
$227$	−3.92121	−0.260260	−0.130130	−	0.991497i	$-0.541539\pi$
$227$	−3.92121	−0.260260	−0.130130	+	0.991497i	$0.541539\pi$
$228$	0	0
$229$	15.0372	0.993684	0.496842	−	0.867841i	$-0.334493\pi$
$229$	15.0372	0.993684	0.496842	+	0.867841i	$0.334493\pi$
$230$	0	0
$231$	1.62670	0.107029
$232$	0	0
$233$	−15.1671	−0.993630	−0.496815	−	0.867857i	$-0.665497\pi$
$233$	−15.1671	−0.993630	−0.496815	+	0.867857i	$0.665497\pi$
$234$	0	0
$235$	9.63989	0.628837
$236$	0	0
$237$	2.79930	0.181834
$238$	0	0
$239$	−13.4573	−0.870479	−0.435239	−	0.900315i	$-0.643336\pi$
$239$	−13.4573	−0.870479	−0.435239	+	0.900315i	$0.643336\pi$
$240$	0	0
$241$	−20.2308	−1.30318	−0.651591	−	0.758570i	$-0.725898\pi$
$241$	−20.2308	−1.30318	−0.651591	+	0.758570i	$0.725898\pi$
$242$	0	0
$243$	−13.6862	−0.877968
$244$	0	0
$245$	−15.0288	−0.960157
$246$	0	0
$247$	9.16866	0.583388
$248$	0	0
$249$	6.89219	0.436774
$250$	0	0
$251$	−1.00000	−0.0631194
$251$	−1.00000	−0.0631194
$252$	0	0
$253$	10.9418	0.687904
$254$	0	0
$255$	1.33451	0.0835700
$256$	0	0
$257$	−14.2469	−0.888697	−0.444348	−	0.895854i	$-0.646565\pi$
$257$	−14.2469	−0.888697	−0.444348	+	0.895854i	$0.646565\pi$
$258$	0	0
$259$	0.627797	0.0390094
$260$	0	0
$261$	25.1666	1.55777
$262$	0	0
$263$	8.34401	0.514514	0.257257	−	0.966343i	$-0.417181\pi$
$263$	8.34401	0.514514	0.257257	+	0.966343i	$0.417181\pi$
$264$	0	0
$265$	−2.21345	−0.135971
$266$	0	0
$267$	−0.964773	−0.0590432
$268$	0	0
$269$	3.56186	0.217170	0.108585	−	0.994087i	$-0.465368\pi$
$269$	3.56186	0.217170	0.108585	+	0.994087i	$0.465368\pi$
$270$	0	0
$271$	−15.1885	−0.922635	−0.461317	−	0.887235i	$-0.652623\pi$
$271$	−15.1885	−0.922635	−0.461317	+	0.887235i	$0.652623\pi$
$272$	0	0
$273$	−1.17629	−0.0711926
$274$	0	0
$275$	0.0339816	0.00204917
$276$	0	0
$277$	18.8454	1.13231	0.566155	−	0.824299i	$-0.308430\pi$
$277$	18.8454	1.13231	0.566155	+	0.824299i	$0.308430\pi$
$278$	0	0
$279$	−2.21357	−0.132523
$280$	0	0
$281$	−15.6312	−0.932479	−0.466240	−	0.884659i	$-0.654392\pi$
$281$	−15.6312	−0.932479	−0.466240	+	0.884659i	$0.654392\pi$
$282$	0	0
$283$	2.75640	0.163851	0.0819255	−	0.996638i	$-0.473893\pi$
$283$	2.75640	0.163851	0.0819255	+	0.996638i	$0.473893\pi$
$284$	0	0
$285$	3.34264	0.198001
$286$	0	0
$287$	2.12176	0.125244
$288$	0	0
$289$	−16.0120	−0.941883
$290$	0	0
$291$	−4.80778	−0.281837
$292$	0	0
$293$	−5.26559	−0.307619	−0.153810	−	0.988100i	$-0.549154\pi$
$293$	−5.26559	−0.307619	−0.153810	+	0.988100i	$0.549154\pi$
$294$	0	0
$295$	−13.6889	−0.796999
$296$	0	0
$297$	17.2346	1.00005
$298$	0	0
$299$	−7.91218	−0.457573
$300$	0	0
$301$	2.05717	0.118573
$302$	0	0
$303$	4.99815	0.287136
$304$	0	0
$305$	22.5603	1.29180
$306$	0	0
$307$	13.4123	0.765478	0.382739	−	0.923856i	$-0.374981\pi$
$307$	13.4123	0.765478	0.382739	+	0.923856i	$0.374981\pi$
$308$	0	0
$309$	5.86555	0.333679
$310$	0	0
$311$	−3.10776	−0.176225	−0.0881125	−	0.996111i	$-0.528083\pi$
$311$	−3.10776	−0.176225	−0.0881125	+	0.996111i	$0.528083\pi$
$312$	0	0
$313$	18.0012	1.01749	0.508744	−	0.860918i	$-0.330110\pi$
$313$	18.0012	1.01749	0.508744	+	0.860918i	$0.330110\pi$
$314$	0	0
$315$	3.14453	0.177174
$316$	0	0
$317$	0.340055	0.0190994	0.00954969	−	0.999954i	$-0.496960\pi$
$317$	0.340055	0.0190994	0.00954969	+	0.999954i	$0.496960\pi$
$318$	0	0
$319$	−48.5488	−2.71821
$320$	0	0
$321$	2.69954	0.150674
$322$	0	0
$323$	2.47468	0.137695
$324$	0	0
$325$	−0.0245727	−0.00136305
$326$	0	0
$327$	2.95725	0.163536
$328$	0	0
$329$	−2.29341	−0.126440
$330$	0	0
$331$	−7.83103	−0.430433	−0.215216	−	0.976566i	$-0.569046\pi$
$331$	−7.83103	−0.430433	−0.215216	+	0.976566i	$0.569046\pi$
$332$	0	0
$333$	3.11340	0.170613
$334$	0	0
$335$	1.96451	0.107332
$336$	0	0
$337$	−18.4094	−1.00282	−0.501412	−	0.865209i	$-0.667186\pi$
$337$	−18.4094	−1.00282	−0.501412	+	0.865209i	$0.667186\pi$
$338$	0	0
$339$	2.82581	0.153477
$340$	0	0
$341$	4.27020	0.231244
$342$	0	0
$343$	7.30181	0.394261
$344$	0	0
$345$	−2.88456	−0.155300
$346$	0	0
$347$	−23.1401	−1.24223	−0.621113	−	0.783721i	$-0.713319\pi$
$347$	−23.1401	−1.24223	−0.621113	+	0.783721i	$0.713319\pi$
$348$	0	0
$349$	−26.6755	−1.42791	−0.713953	−	0.700194i	$-0.753097\pi$
$349$	−26.6755	−1.42791	−0.713953	+	0.700194i	$0.753097\pi$
$350$	0	0
$351$	−12.4626	−0.665205
$352$	0	0
$353$	−22.2028	−1.18174	−0.590869	−	0.806768i	$-0.701215\pi$
$353$	−22.2028	−1.18174	−0.590869	+	0.806768i	$0.701215\pi$
$354$	0	0
$355$	−6.27038	−0.332797
$356$	0	0
$357$	−0.317489	−0.0168033
$358$	0	0
$359$	−5.66930	−0.299214	−0.149607	−	0.988746i	$-0.547801\pi$
$359$	−5.66930	−0.299214	−0.149607	+	0.988746i	$0.547801\pi$
$360$	0	0
$361$	−12.8015	−0.673762
$362$	0	0
$363$	−8.96214	−0.470390
$364$	0	0
$365$	−4.40504	−0.230571
$366$	0	0
$367$	23.9290	1.24908	0.624542	−	0.780991i	$-0.285286\pi$
$367$	23.9290	1.24908	0.624542	+	0.780991i	$0.285286\pi$
$368$	0	0
$369$	10.5223	0.547771
$370$	0	0
$371$	0.526598	0.0273396
$372$	0	0
$373$	34.1176	1.76654	0.883271	−	0.468862i	$-0.155336\pi$
$373$	34.1176	1.76654	0.883271	+	0.468862i	$0.155336\pi$
$374$	0	0
$375$	6.70403	0.346194
$376$	0	0
$377$	35.1065	1.80808
$378$	0	0
$379$	5.40285	0.277526	0.138763	−	0.990326i	$-0.455687\pi$
$379$	5.40285	0.277526	0.138763	+	0.990326i	$0.455687\pi$
$380$	0	0
$381$	3.19211	0.163537
$382$	0	0
$383$	1.71883	0.0878279	0.0439140	−	0.999035i	$-0.486017\pi$
$383$	1.71883	0.0878279	0.0439140	+	0.999035i	$0.486017\pi$
$384$	0	0
$385$	−6.06612	−0.309158
$386$	0	0
$387$	10.2020	0.518597
$388$	0	0
$389$	−9.38408	−0.475792	−0.237896	−	0.971291i	$-0.576458\pi$
$389$	−9.38408	−0.475792	−0.237896	+	0.971291i	$0.576458\pi$
$390$	0	0
$391$	−2.13555	−0.107999
$392$	0	0
$393$	−1.22539	−0.0618130
$394$	0	0
$395$	−10.4389	−0.525236
$396$	0	0
$397$	29.6592	1.48855	0.744277	−	0.667871i	$-0.232794\pi$
$397$	29.6592	1.48855	0.744277	+	0.667871i	$0.232794\pi$
$398$	0	0
$399$	−0.795240	−0.0398118
$400$	0	0
$401$	6.60653	0.329914	0.164957	−	0.986301i	$-0.447251\pi$
$401$	6.60653	0.329914	0.164957	+	0.986301i	$0.447251\pi$
$402$	0	0
$403$	−3.08785	−0.153817
$404$	0	0
$405$	13.1777	0.654806
$406$	0	0
$407$	−6.00606	−0.297709
$408$	0	0
$409$	−7.99048	−0.395104	−0.197552	−	0.980292i	$-0.563299\pi$
$409$	−7.99048	−0.395104	−0.197552	+	0.980292i	$0.563299\pi$
$410$	0	0
$411$	0.741676	0.0365842
$412$	0	0
$413$	3.25670	0.160252
$414$	0	0
$415$	−25.7016	−1.26164
$416$	0	0
$417$	−8.75957	−0.428958
$418$	0	0
$419$	31.1997	1.52420	0.762102	−	0.647457i	$-0.224167\pi$
$419$	31.1997	1.52420	0.762102	+	0.647457i	$0.224167\pi$
$420$	0	0
$421$	7.48196	0.364648	0.182324	−	0.983238i	$-0.441638\pi$
$421$	7.48196	0.364648	0.182324	+	0.983238i	$0.441638\pi$
$422$	0	0
$423$	−11.3736	−0.553001
$424$	0	0
$425$	−0.00663231	−0.000321714 0
$426$	0	0
$427$	−5.36726	−0.259740
$428$	0	0
$429$	11.2535	0.543322
$430$	0	0
$431$	14.6863	0.707413	0.353706	−	0.935357i	$-0.384921\pi$
$431$	14.6863	0.707413	0.353706	+	0.935357i	$0.384921\pi$
$432$	0	0
$433$	−17.9301	−0.861664	−0.430832	−	0.902432i	$-0.641780\pi$
$433$	−17.9301	−0.861664	−0.430832	+	0.902432i	$0.641780\pi$
$434$	0	0
$435$	12.7989	0.613658
$436$	0	0
$437$	−5.34907	−0.255881
$438$	0	0
$439$	−12.7398	−0.608039	−0.304020	−	0.952666i	$-0.598329\pi$
$439$	−12.7398	−0.608039	−0.304020	+	0.952666i	$0.598329\pi$
$440$	0	0
$441$	17.7317	0.844365
$442$	0	0
$443$	10.9473	0.520120	0.260060	−	0.965592i	$-0.416258\pi$
$443$	10.9473	0.520120	0.260060	+	0.965592i	$0.416258\pi$
$444$	0	0
$445$	3.59773	0.170549
$446$	0	0
$447$	−0.666714	−0.0315345
$448$	0	0
$449$	−10.8664	−0.512819	−0.256409	−	0.966568i	$-0.582540\pi$
$449$	−10.8664	−0.512819	−0.256409	+	0.966568i	$0.582540\pi$
$450$	0	0
$451$	−20.2987	−0.955826
$452$	0	0
$453$	−13.6524	−0.641447
$454$	0	0
$455$	4.38651	0.205643
$456$	0	0
$457$	−11.6316	−0.544105	−0.272052	−	0.962282i	$-0.587702\pi$
$457$	−11.6316	−0.544105	−0.272052	+	0.962282i	$0.587702\pi$
$458$	0	0
$459$	−3.36374	−0.157006
$460$	0	0
$461$	−10.9455	−0.509782	−0.254891	−	0.966970i	$-0.582040\pi$
$461$	−10.9455	−0.509782	−0.254891	+	0.966970i	$0.582040\pi$
$462$	0	0
$463$	−37.0847	−1.72347	−0.861737	−	0.507356i	$-0.830623\pi$
$463$	−37.0847	−1.72347	−0.861737	+	0.507356i	$0.830623\pi$
$464$	0	0
$465$	−1.12575	−0.0522052
$466$	0	0
$467$	36.7770	1.70184	0.850919	−	0.525297i	$-0.176046\pi$
$467$	36.7770	1.70184	0.850919	+	0.525297i	$0.176046\pi$
$468$	0	0
$469$	−0.467371	−0.0215812
$470$	0	0
$471$	8.12051	0.374173
$472$	0	0
$473$	−19.6807	−0.904919
$474$	0	0
$475$	−0.0166125	−0.000762232 0
$476$	0	0
$477$	2.61152	0.119574
$478$	0	0
$479$	−4.17414	−0.190721	−0.0953607	−	0.995443i	$-0.530400\pi$
$479$	−4.17414	−0.190721	−0.0953607	+	0.995443i	$0.530400\pi$
$480$	0	0
$481$	4.34308	0.198028
$482$	0	0
$483$	0.686260	0.0312259
$484$	0	0
$485$	17.9287	0.814099
$486$	0	0
$487$	−25.9015	−1.17371	−0.586856	−	0.809692i	$-0.699634\pi$
$487$	−25.9015	−1.17371	−0.586856	+	0.809692i	$0.699634\pi$
$488$	0	0
$489$	−5.76642	−0.260766
$490$	0	0
$491$	−36.0089	−1.62506	−0.812528	−	0.582922i	$-0.801909\pi$
$491$	−36.0089	−1.62506	−0.812528	+	0.582922i	$0.801909\pi$
$492$	0	0
$493$	9.47546	0.426753
$494$	0	0
$495$	−30.0834	−1.35215
$496$	0	0
$497$	1.49177	0.0669151
$498$	0	0
$499$	−4.09812	−0.183457	−0.0917284	−	0.995784i	$-0.529239\pi$
$499$	−4.09812	−0.183457	−0.0917284	+	0.995784i	$0.529239\pi$
$500$	0	0
$501$	−5.98537	−0.267407
$502$	0	0
$503$	−0.653440	−0.0291355	−0.0145677	−	0.999894i	$-0.504637\pi$
$503$	−0.653440	−0.0291355	−0.0145677	+	0.999894i	$0.504637\pi$
$504$	0	0
$505$	−18.6386	−0.829405
$506$	0	0
$507$	−0.337212	−0.0149761
$508$	0	0
$509$	28.3462	1.25642	0.628211	−	0.778043i	$-0.283788\pi$
$509$	28.3462	1.25642	0.628211	+	0.778043i	$0.283788\pi$
$510$	0	0
$511$	1.04799	0.0463605
$512$	0	0
$513$	−8.42542	−0.371991
$514$	0	0
$515$	−21.8732	−0.963848
$516$	0	0
$517$	21.9407	0.964952
$518$	0	0
$519$	7.11656	0.312383
$520$	0	0
$521$	15.2168	0.666658	0.333329	−	0.942811i	$-0.391828\pi$
$521$	15.2168	0.666658	0.333329	+	0.942811i	$0.391828\pi$
$522$	0	0
$523$	16.2441	0.710303	0.355151	−	0.934809i	$-0.384429\pi$
$523$	16.2441	0.710303	0.355151	+	0.934809i	$0.384429\pi$
$524$	0	0
$525$	0.00213130	9.30175e−5 0
$526$	0	0
$527$	−0.833431	−0.0363048
$528$	0	0
$529$	−18.3840	−0.799303
$530$	0	0
$531$	16.1508	0.700883
$532$	0	0
$533$	14.6783	0.635788
$534$	0	0
$535$	−10.0668	−0.435228
$536$	0	0
$537$	2.11613	0.0913179
$538$	0	0
$539$	−34.2062	−1.47336
$540$	0	0
$541$	35.5803	1.52972	0.764858	−	0.644199i	$-0.222809\pi$
$541$	35.5803	1.52972	0.764858	+	0.644199i	$0.222809\pi$
$542$	0	0
$543$	6.22524	0.267151
$544$	0	0
$545$	−11.0279	−0.472382
$546$	0	0
$547$	−6.95502	−0.297375	−0.148688	−	0.988884i	$-0.547505\pi$
$547$	−6.95502	−0.297375	−0.148688	+	0.988884i	$0.547505\pi$
$548$	0	0
$549$	−26.6176	−1.13601
$550$	0	0
$551$	23.7339	1.01110
$552$	0	0
$553$	2.48348	0.105608
$554$	0	0
$555$	1.58337	0.0672102
$556$	0	0
$557$	−15.2640	−0.646755	−0.323378	−	0.946270i	$-0.604818\pi$
$557$	−15.2640	−0.646755	−0.323378	+	0.946270i	$0.604818\pi$
$558$	0	0
$559$	14.2314	0.601925
$560$	0	0
$561$	3.03738	0.128238
$562$	0	0
$563$	−3.86646	−0.162952	−0.0814759	−	0.996675i	$-0.525963\pi$
$563$	−3.86646	−0.162952	−0.0814759	+	0.996675i	$0.525963\pi$
$564$	0	0
$565$	−10.5377	−0.443325
$566$	0	0
$567$	−3.13508	−0.131661
$568$	0	0
$569$	−38.9015	−1.63084	−0.815418	−	0.578872i	$-0.803493\pi$
$569$	−38.9015	−1.63084	−0.815418	+	0.578872i	$0.803493\pi$
$570$	0	0
$571$	−27.9257	−1.16865	−0.584327	−	0.811519i	$-0.698641\pi$
$571$	−27.9257	−1.16865	−0.584327	+	0.811519i	$0.698641\pi$
$572$	0	0
$573$	−0.381031	−0.0159178
$574$	0	0
$575$	0.0143359	0.000597848 0
$576$	0	0
$577$	28.4216	1.18321	0.591603	−	0.806229i	$-0.298495\pi$
$577$	28.4216	1.18321	0.591603	+	0.806229i	$0.298495\pi$
$578$	0	0
$579$	−4.53346	−0.188404
$580$	0	0
$581$	6.11462	0.253677
$582$	0	0
$583$	−5.03789	−0.208648
$584$	0	0
$585$	21.7538	0.899408
$586$	0	0
$587$	29.1182	1.20184	0.600918	−	0.799311i	$-0.294802\pi$
$587$	29.1182	1.20184	0.600918	+	0.799311i	$0.294802\pi$
$588$	0	0
$589$	−2.08756	−0.0860164
$590$	0	0
$591$	−7.35565	−0.302571
$592$	0	0
$593$	16.3827	0.672758	0.336379	−	0.941727i	$-0.390798\pi$
$593$	16.3827	0.672758	0.336379	+	0.941727i	$0.390798\pi$
$594$	0	0
$595$	1.18395	0.0485371
$596$	0	0
$597$	1.02985	0.0421488
$598$	0	0
$599$	15.0044	0.613062	0.306531	−	0.951861i	$-0.400832\pi$
$599$	15.0044	0.613062	0.306531	+	0.951861i	$0.400832\pi$
$600$	0	0
$601$	−47.1637	−1.92385	−0.961923	−	0.273319i	$-0.911878\pi$
$601$	−47.1637	−1.92385	−0.961923	+	0.273319i	$0.911878\pi$
$602$	0	0
$603$	−2.31781	−0.0943884
$604$	0	0
$605$	33.4207	1.35874
$606$	0	0
$607$	25.0030	1.01484	0.507421	−	0.861698i	$-0.330599\pi$
$607$	25.0030	1.01484	0.507421	+	0.861698i	$0.330599\pi$
$608$	0	0
$609$	−3.04495	−0.123387
$610$	0	0
$611$	−15.8657	−0.641858
$612$	0	0
$613$	31.1706	1.25897	0.629485	−	0.777013i	$-0.283266\pi$
$613$	31.1706	1.25897	0.629485	+	0.777013i	$0.283266\pi$
$614$	0	0
$615$	5.35130	0.215785
$616$	0	0
$617$	−3.64051	−0.146561	−0.0732806	−	0.997311i	$-0.523347\pi$
$617$	−3.64051	−0.146561	−0.0732806	+	0.997311i	$0.523347\pi$
$618$	0	0
$619$	37.8291	1.52048	0.760240	−	0.649642i	$-0.225081\pi$
$619$	37.8291	1.52048	0.760240	+	0.649642i	$0.225081\pi$
$620$	0	0
$621$	7.27080	0.291767
$622$	0	0
$623$	−0.855928	−0.0342920
$624$	0	0
$625$	−25.0333	−1.00133
$626$	0	0
$627$	7.60796	0.303833
$628$	0	0
$629$	1.17223	0.0467397
$630$	0	0
$631$	18.5244	0.737446	0.368723	−	0.929539i	$-0.379795\pi$
$631$	18.5244	0.737446	0.368723	+	0.929539i	$0.379795\pi$
$632$	0	0
$633$	6.02022	0.239282
$634$	0	0
$635$	−11.9037	−0.472384
$636$	0	0
$637$	24.7350	0.980038
$638$	0	0
$639$	7.39806	0.292663
$640$	0	0
$641$	37.7859	1.49245	0.746226	−	0.665692i	$-0.231864\pi$
$641$	37.7859	1.49245	0.746226	+	0.665692i	$0.231864\pi$
$642$	0	0
$643$	−0.540217	−0.0213041	−0.0106520	−	0.999943i	$-0.503391\pi$
$643$	−0.540217	−0.0213041	−0.0106520	+	0.999943i	$0.503391\pi$
$644$	0	0
$645$	5.18839	0.204293
$646$	0	0
$647$	11.9248	0.468812	0.234406	−	0.972139i	$-0.424685\pi$
$647$	11.9248	0.468812	0.234406	+	0.972139i	$0.424685\pi$
$648$	0	0
$649$	−31.1564	−1.22300
$650$	0	0
$651$	0.267824	0.0104968
$652$	0	0
$653$	−7.76559	−0.303891	−0.151945	−	0.988389i	$-0.548554\pi$
$653$	−7.76559	−0.303891	−0.151945	+	0.988389i	$0.548554\pi$
$654$	0	0
$655$	4.56961	0.178550
$656$	0	0
$657$	5.19726	0.202764
$658$	0	0
$659$	30.1347	1.17388	0.586941	−	0.809630i	$-0.300332\pi$
$659$	30.1347	1.17388	0.586941	+	0.809630i	$0.300332\pi$
$660$	0	0
$661$	9.67594	0.376350	0.188175	−	0.982135i	$-0.439743\pi$
$661$	9.67594	0.376350	0.188175	+	0.982135i	$0.439743\pi$
$662$	0	0
$663$	−2.19638	−0.0853003
$664$	0	0
$665$	2.96553	0.114998
$666$	0	0
$667$	−20.4814	−0.793044
$668$	0	0
$669$	13.7310	0.530872
$670$	0	0
$671$	51.3480	1.98227
$672$	0	0
$673$	13.3665	0.515242	0.257621	−	0.966246i	$-0.417061\pi$
$673$	13.3665	0.515242	0.257621	+	0.966246i	$0.417061\pi$
$674$	0	0
$675$	0.0225807	0.000869132 0
$676$	0	0
$677$	19.2961	0.741610	0.370805	−	0.928711i	$-0.379082\pi$
$677$	19.2961	0.741610	0.370805	+	0.928711i	$0.379082\pi$
$678$	0	0
$679$	−4.26537	−0.163690
$680$	0	0
$681$	2.35283	0.0901607
$682$	0	0
$683$	37.2603	1.42573	0.712863	−	0.701303i	$-0.247398\pi$
$683$	37.2603	1.42573	0.712863	+	0.701303i	$0.247398\pi$
$684$	0	0
$685$	−2.76578	−0.105675
$686$	0	0
$687$	−9.02272	−0.344238
$688$	0	0
$689$	3.64299	0.138787
$690$	0	0
$691$	−36.2662	−1.37963	−0.689815	−	0.723985i	$-0.742308\pi$
$691$	−36.2662	−1.37963	−0.689815	+	0.723985i	$0.742308\pi$
$692$	0	0
$693$	7.15707	0.271875
$694$	0	0
$695$	32.6653	1.23907
$696$	0	0
$697$	3.96176	0.150063
$698$	0	0
$699$	9.10067	0.344219
$700$	0	0
$701$	23.4035	0.883937	0.441968	−	0.897031i	$-0.354280\pi$
$701$	23.4035	0.883937	0.441968	+	0.897031i	$0.354280\pi$
$702$	0	0
$703$	2.93616	0.110739
$704$	0	0
$705$	−5.78420	−0.217845
$706$	0	0
$707$	4.43426	0.166768
$708$	0	0
$709$	14.0459	0.527506	0.263753	−	0.964590i	$-0.415040\pi$
$709$	14.0459	0.527506	0.263753	+	0.964590i	$0.415040\pi$
$710$	0	0
$711$	12.3162	0.461894
$712$	0	0
$713$	1.80148	0.0674659
$714$	0	0
$715$	−41.9652	−1.56941
$716$	0	0
$717$	8.07473	0.301556
$718$	0	0
$719$	18.1906	0.678396	0.339198	−	0.940715i	$-0.389844\pi$
$719$	18.1906	0.678396	0.339198	+	0.940715i	$0.389844\pi$
$720$	0	0
$721$	5.20380	0.193800
$722$	0	0
$723$	12.1391	0.451456
$724$	0	0
$725$	−0.0636086	−0.00236236
$726$	0	0
$727$	24.5672	0.911147	0.455574	−	0.890198i	$-0.349434\pi$
$727$	24.5672	0.911147	0.455574	+	0.890198i	$0.349434\pi$
$728$	0	0
$729$	−9.45592	−0.350219
$730$	0	0
$731$	3.84115	0.142070
$732$	0	0
$733$	41.9650	1.55001	0.775006	−	0.631954i	$-0.217747\pi$
$733$	41.9650	1.55001	0.775006	+	0.631954i	$0.217747\pi$
$734$	0	0
$735$	9.01771	0.332623
$736$	0	0
$737$	4.47128	0.164702
$738$	0	0
$739$	−51.5106	−1.89485	−0.947425	−	0.319978i	$-0.896324\pi$
$739$	−51.5106	−1.89485	−0.947425	+	0.319978i	$0.896324\pi$
$740$	0	0
$741$	−5.50144	−0.202101
$742$	0	0
$743$	17.2811	0.633981	0.316991	−	0.948429i	$-0.397328\pi$
$743$	17.2811	0.633981	0.316991	+	0.948429i	$0.397328\pi$
$744$	0	0
$745$	2.48624	0.0910889
$746$	0	0
$747$	30.3239	1.10949
$748$	0	0
$749$	2.39498	0.0875107
$750$	0	0
$751$	4.16570	0.152009	0.0760043	−	0.997107i	$-0.475784\pi$
$751$	4.16570	0.152009	0.0760043	+	0.997107i	$0.475784\pi$
$752$	0	0
$753$	0.600027	0.0218662
$754$	0	0
$755$	50.9112	1.85285
$756$	0	0
$757$	4.25135	0.154518	0.0772589	−	0.997011i	$-0.475383\pi$
$757$	4.25135	0.154518	0.0772589	+	0.997011i	$0.475383\pi$
$758$	0	0
$759$	−6.56537	−0.238308
$760$	0	0
$761$	10.3895	0.376620	0.188310	−	0.982110i	$-0.439699\pi$
$761$	10.3895	0.376620	0.188310	+	0.982110i	$0.439699\pi$
$762$	0	0
$763$	2.62361	0.0949812
$764$	0	0
$765$	5.87148	0.212284
$766$	0	0
$767$	22.5297	0.813502
$768$	0	0
$769$	−9.29363	−0.335137	−0.167569	−	0.985860i	$-0.553592\pi$
$769$	−9.29363	−0.335137	−0.167569	+	0.985860i	$0.553592\pi$
$770$	0	0
$771$	8.54852	0.307868
$772$	0	0
$773$	14.2469	0.512424	0.256212	−	0.966621i	$-0.417525\pi$
$773$	14.2469	0.512424	0.256212	+	0.966621i	$0.417525\pi$
$774$	0	0
$775$	0.00559481	0.000200971 0
$776$	0	0
$777$	−0.376696	−0.0135139
$778$	0	0
$779$	9.92334	0.355540
$780$	0	0
$781$	−14.2716	−0.510678
$782$	0	0
$783$	−32.2606	−1.15290
$784$	0	0
$785$	−30.2822	−1.08082
$786$	0	0
$787$	−9.30156	−0.331565	−0.165782	−	0.986162i	$-0.553015\pi$
$787$	−9.30156	−0.331565	−0.165782	+	0.986162i	$0.553015\pi$
$788$	0	0
$789$	−5.00664	−0.178241
$790$	0	0
$791$	2.50701	0.0891388
$792$	0	0
$793$	−37.1306	−1.31855
$794$	0	0
$795$	1.32813	0.0471040
$796$	0	0
$797$	−13.3144	−0.471622	−0.235811	−	0.971799i	$-0.575775\pi$
$797$	−13.3144	−0.471622	−0.235811	+	0.971799i	$0.575775\pi$
$798$	0	0
$799$	−4.28225	−0.151495
$800$	0	0
$801$	−4.24476	−0.149981
$802$	0	0
$803$	−10.0260	−0.353811
$804$	0	0
$805$	−2.55913	−0.0901975
$806$	0	0
$807$	−2.13721	−0.0752334
$808$	0	0
$809$	22.5699	0.793516	0.396758	−	0.917923i	$-0.370135\pi$
$809$	22.5699	0.793516	0.396758	+	0.917923i	$0.370135\pi$
$810$	0	0
$811$	−45.6794	−1.60402	−0.802010	−	0.597310i	$-0.796236\pi$
$811$	−45.6794	−1.60402	−0.802010	+	0.597310i	$0.796236\pi$
$812$	0	0
$813$	9.11351	0.319625
$814$	0	0
$815$	21.5035	0.753236
$816$	0	0
$817$	9.62123	0.336604
$818$	0	0
$819$	−5.17540	−0.180843
$820$	0	0
$821$	−2.87513	−0.100343	−0.0501714	−	0.998741i	$-0.515977\pi$
$821$	−2.87513	−0.100343	−0.0501714	+	0.998741i	$0.515977\pi$
$822$	0	0
$823$	34.0085	1.18546	0.592731	−	0.805400i	$-0.298050\pi$
$823$	34.0085	1.18546	0.592731	+	0.805400i	$0.298050\pi$
$824$	0	0
$825$	−0.0203899	−0.000709884 0
$826$	0	0
$827$	18.8922	0.656947	0.328474	−	0.944513i	$-0.393466\pi$
$827$	18.8922	0.656947	0.328474	+	0.944513i	$0.393466\pi$
$828$	0	0
$829$	3.75186	0.130307	0.0651537	−	0.997875i	$-0.479246\pi$
$829$	3.75186	0.130307	0.0651537	+	0.997875i	$0.479246\pi$
$830$	0	0
$831$	−11.3078	−0.392262
$832$	0	0
$833$	6.67614	0.231315
$834$	0	0
$835$	22.3200	0.772416
$836$	0	0
$837$	2.83754	0.0980798
$838$	0	0
$839$	20.3264	0.701744	0.350872	−	0.936424i	$-0.385885\pi$
$839$	20.3264	0.701744	0.350872	+	0.936424i	$0.385885\pi$
$840$	0	0
$841$	61.8763	2.13367
$842$	0	0
$843$	9.37915	0.323035
$844$	0	0
$845$	1.25750	0.0432592
$846$	0	0
$847$	−7.95104	−0.273201
$848$	0	0
$849$	−1.65392	−0.0567622
$850$	0	0
$851$	−2.53379	−0.0868573
$852$	0	0
$853$	20.0385	0.686106	0.343053	−	0.939316i	$-0.388539\pi$
$853$	20.0385	0.686106	0.343053	+	0.939316i	$0.388539\pi$
$854$	0	0
$855$	14.7068	0.502961
$856$	0	0
$857$	12.0900	0.412986	0.206493	−	0.978448i	$-0.433795\pi$
$857$	12.0900	0.412986	0.206493	+	0.978448i	$0.433795\pi$
$858$	0	0
$859$	44.5445	1.51984	0.759920	−	0.650017i	$-0.225238\pi$
$859$	44.5445	1.51984	0.759920	+	0.650017i	$0.225238\pi$
$860$	0	0
$861$	−1.27312	−0.0433877
$862$	0	0
$863$	20.8453	0.709584	0.354792	−	0.934945i	$-0.384552\pi$
$863$	20.8453	0.709584	0.354792	+	0.934945i	$0.384552\pi$
$864$	0	0
$865$	−26.5384	−0.902331
$866$	0	0
$867$	9.60765	0.326293
$868$	0	0
$869$	−23.7592	−0.805975
$870$	0	0
$871$	−3.23326	−0.109555
$872$	0	0
$873$	−21.1530	−0.715920
$874$	0	0
$875$	5.94768	0.201068
$876$	0	0
$877$	58.3453	1.97018	0.985091	−	0.172035i	$-0.0550341\pi$
$877$	58.3453	1.97018	0.985091	+	0.172035i	$0.0550341\pi$
$878$	0	0
$879$	3.15950	0.106567
$880$	0	0
$881$	37.7244	1.27097	0.635484	−	0.772114i	$-0.280801\pi$
$881$	37.7244	1.27097	0.635484	+	0.772114i	$0.280801\pi$
$882$	0	0
$883$	−20.1922	−0.679521	−0.339760	−	0.940512i	$-0.610346\pi$
$883$	−20.1922	−0.679521	−0.339760	+	0.940512i	$0.610346\pi$
$884$	0	0
$885$	8.21372	0.276101
$886$	0	0
$887$	21.8723	0.734401	0.367200	−	0.930142i	$-0.380316\pi$
$887$	21.8723	0.734401	0.367200	+	0.930142i	$0.380316\pi$
$888$	0	0
$889$	2.83198	0.0949817
$890$	0	0
$891$	29.9929	1.00480
$892$	0	0
$893$	−10.7261	−0.358935
$894$	0	0
$895$	−7.89127	−0.263776
$896$	0	0
$897$	4.74753	0.158515
$898$	0	0
$899$	−7.99319	−0.266588
$900$	0	0
$901$	0.983265	0.0327573
$902$	0	0
$903$	−1.23436	−0.0410769
$904$	0	0
$905$	−23.2145	−0.771676
$906$	0	0
$907$	−46.8301	−1.55497	−0.777484	−	0.628903i	$-0.783504\pi$
$907$	−46.8301	−1.55497	−0.777484	+	0.628903i	$0.783504\pi$
$908$	0	0
$909$	21.9906	0.729381
$910$	0	0
$911$	−16.3686	−0.542316	−0.271158	−	0.962535i	$-0.587407\pi$
$911$	−16.3686	−0.542316	−0.271158	+	0.962535i	$0.587407\pi$
$912$	0	0
$913$	−58.4978	−1.93599
$914$	0	0
$915$	−13.5368	−0.447512
$916$	0	0
$917$	−1.08715	−0.0359008
$918$	0	0
$919$	32.3562	1.06733	0.533666	−	0.845695i	$-0.320814\pi$
$919$	32.3562	1.06733	0.533666	+	0.845695i	$0.320814\pi$
$920$	0	0
$921$	−8.04772	−0.265182
$922$	0	0
$923$	10.3200	0.339688
$924$	0	0
$925$	−0.00786913	−0.000258735 0
$926$	0	0
$927$	25.8069	0.847610
$928$	0	0
$929$	14.8594	0.487522	0.243761	−	0.969835i	$-0.421619\pi$
$929$	14.8594	0.487522	0.243761	+	0.969835i	$0.421619\pi$
$930$	0	0
$931$	16.7223	0.548050
$932$	0	0
$933$	1.86474	0.0610489
$934$	0	0
$935$	−11.3267	−0.370422
$936$	0	0
$937$	−50.6989	−1.65626	−0.828131	−	0.560535i	$-0.810596\pi$
$937$	−50.6989	−1.65626	−0.828131	+	0.560535i	$0.810596\pi$
$938$	0	0
$939$	−10.8012	−0.352484
$940$	0	0
$941$	15.3955	0.501880	0.250940	−	0.968003i	$-0.419260\pi$
$941$	15.3955	0.501880	0.250940	+	0.968003i	$0.419260\pi$
$942$	0	0
$943$	−8.56344	−0.278864
$944$	0	0
$945$	−4.03093	−0.131126
$946$	0	0
$947$	39.5927	1.28659	0.643295	−	0.765619i	$-0.277567\pi$
$947$	39.5927	1.28659	0.643295	+	0.765619i	$0.277567\pi$
$948$	0	0
$949$	7.24999	0.235345
$950$	0	0
$951$	−0.204042	−0.00661652
$952$	0	0
$953$	−3.25185	−0.105338	−0.0526688	−	0.998612i	$-0.516773\pi$
$953$	−3.25185	−0.105338	−0.0526688	+	0.998612i	$0.516773\pi$
$954$	0	0
$955$	1.42090	0.0459793
$956$	0	0
$957$	29.1306	0.941659
$958$	0	0
$959$	0.658001	0.0212480
$960$	0	0
$961$	−30.2969	−0.977321
$962$	0	0
$963$	11.8773	0.382740
$964$	0	0
$965$	16.9057	0.544214
$966$	0	0
$967$	−0.601688	−0.0193490	−0.00967448	−	0.999953i	$-0.503080\pi$
$967$	−0.601688	−0.0193490	−0.00967448	+	0.999953i	$0.503080\pi$
$968$	0	0
$969$	−1.48487	−0.0477010
$970$	0	0
$971$	−39.2864	−1.26076	−0.630380	−	0.776287i	$-0.717101\pi$
$971$	−39.2864	−1.26076	−0.630380	+	0.776287i	$0.717101\pi$
$972$	0	0
$973$	−7.77133	−0.249137
$974$	0	0
$975$	0.0147443	0.000472194 0
$976$	0	0
$977$	−20.7438	−0.663652	−0.331826	−	0.943341i	$-0.607665\pi$
$977$	−20.7438	−0.663652	−0.331826	+	0.943341i	$0.607665\pi$
$978$	0	0
$979$	8.18856	0.261707
$980$	0	0
$981$	13.0111	0.415413
$982$	0	0
$983$	29.7828	0.949923	0.474962	−	0.880007i	$-0.342462\pi$
$983$	29.7828	0.949923	0.474962	+	0.880007i	$0.342462\pi$
$984$	0	0
$985$	27.4299	0.873990
$986$	0	0
$987$	1.37611	0.0438019
$988$	0	0
$989$	−8.30274	−0.264012
$990$	0	0
$991$	−54.2518	−1.72337	−0.861683	−	0.507448i	$-0.830589\pi$
$991$	−54.2518	−1.72337	−0.861683	+	0.507448i	$0.830589\pi$
$992$	0	0
$993$	4.69883	0.149113
$994$	0	0
$995$	−3.84040	−0.121749
$996$	0	0
$997$	−4.47307	−0.141664	−0.0708318	−	0.997488i	$-0.522565\pi$
$997$	−4.47307	−0.141664	−0.0708318	+	0.997488i	$0.522565\pi$
$998$	0	0
$999$	−3.99102	−0.126270

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4016.2.a.h.1.4		9
4.3	odd	2		2008.2.a.a.1.6	✓	9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2008.2.a.a.1.6	✓	9	4.3	odd	2
4016.2.a.h.1.4		9	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:	\( N \)	\(=\)	\( 4016 = 2^{4} \cdot 251 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4016.a (trivial)

\(f(q)\)	\(=\)	\(q-0.600027 q^{3} +2.23756 q^{5} -0.532333 q^{7} -2.63997 q^{9} +O(q^{10})\)	\(q-0.600027 q^{3} +2.23756 q^{5} -0.532333 q^{7} -2.63997 q^{9} +5.09276 q^{11} -3.68266 q^{13} -1.34260 q^{15} -0.993973 q^{17} -2.48968 q^{19} +0.319414 q^{21} +2.14850 q^{23} +0.00667253 q^{25} +3.38413 q^{27} -9.53291 q^{29} +0.838484 q^{31} -3.05580 q^{33} -1.19113 q^{35} -1.17933 q^{37} +2.20970 q^{39} -3.98579 q^{41} -3.86444 q^{43} -5.90708 q^{45} +4.30822 q^{47} -6.71662 q^{49} +0.596411 q^{51} -0.989226 q^{53} +11.3954 q^{55} +1.49388 q^{57} -6.11779 q^{59} +10.0825 q^{61} +1.40534 q^{63} -8.24017 q^{65} +0.877968 q^{67} -1.28916 q^{69} -2.80233 q^{71} -1.96868 q^{73} -0.00400370 q^{75} -2.71104 q^{77} -4.66529 q^{79} +5.88933 q^{81} -11.4865 q^{83} -2.22407 q^{85} +5.72001 q^{87} +1.60788 q^{89} +1.96040 q^{91} -0.503113 q^{93} -5.57081 q^{95} +8.01260 q^{97} -13.4447 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q + q^{3} - 5 q^{5} - 4 q^{9}+O(q^{10}) \) 9 * q + q^3 - 5 * q^5 - 4 * q^9	\( 9 q + q^{3} - 5 q^{5} - 4 q^{9} + 3 q^{11} - 3 q^{13} + q^{15} - 11 q^{17} - 4 q^{19} - 3 q^{21} + 9 q^{23} - 12 q^{25} + 7 q^{27} - 9 q^{29} - 3 q^{31} - 14 q^{33} + 8 q^{35} - 10 q^{37} + q^{39} - 23 q^{41} - 10 q^{45} + 11 q^{47} - 21 q^{49} + 3 q^{51} - 21 q^{53} + 4 q^{55} - 21 q^{57} + 4 q^{59} - 11 q^{61} + 2 q^{63} - 29 q^{65} + 4 q^{67} - 14 q^{69} + 19 q^{71} - 31 q^{73} - 16 q^{75} - 26 q^{77} - 4 q^{79} - 27 q^{81} + 22 q^{83} + 4 q^{85} + 6 q^{87} - 36 q^{89} - 14 q^{91} - 32 q^{93} + 3 q^{95} - 38 q^{97}+O(q^{100}) \) 9 * q + q^3 - 5 * q^5 - 4 * q^9 + 3 * q^11 - 3 * q^13 + q^15 - 11 * q^17 - 4 * q^19 - 3 * q^21 + 9 * q^23 - 12 * q^25 + 7 * q^27 - 9 * q^29 - 3 * q^31 - 14 * q^33 + 8 * q^35 - 10 * q^37 + q^39 - 23 * q^41 - 10 * q^45 + 11 * q^47 - 21 * q^49 + 3 * q^51 - 21 * q^53 + 4 * q^55 - 21 * q^57 + 4 * q^59 - 11 * q^61 + 2 * q^63 - 29 * q^65 + 4 * q^67 - 14 * q^69 + 19 * q^71 - 31 * q^73 - 16 * q^75 - 26 * q^77 - 4 * q^79 - 27 * q^81 + 22 * q^83 + 4 * q^85 + 6 * q^87 - 36 * q^89 - 14 * q^91 - 32 * q^93 + 3 * q^95 - 38 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more