LMFDB - Embedded newform 4016.2.a.f.1.1

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:	$6$
Coefficient field:	6.6.60853001.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 3x^{5} - 8x^{4} + 15x^{3} + 20x^{2} - 12x - 5 $ x^6 - 3x^5 - 8x^4 + 15x^3 + 20x^2 - 12*x - 5
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 502)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$0.730357$ 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)

$f(q)$	$=$	$q-3.20334 q^{3} +1.15583 q^{5} +3.99175 q^{7} +7.26139 q^{9} +O(q^{10})$	$q-3.20334 q^{3} +1.15583 q^{5} +3.99175 q^{7} +7.26139 q^{9} +3.92729 q^{11} -1.69512 q^{13} -3.70252 q^{15} +0.837764 q^{17} -1.86789 q^{19} -12.7869 q^{21} -1.87301 q^{23} -3.66406 q^{25} -13.6507 q^{27} -10.1028 q^{29} -9.22393 q^{31} -12.5805 q^{33} +4.61379 q^{35} -4.58130 q^{37} +5.43003 q^{39} +2.89716 q^{41} +7.59775 q^{43} +8.39294 q^{45} -7.88848 q^{47} +8.93407 q^{49} -2.68364 q^{51} -11.2188 q^{53} +4.53929 q^{55} +5.98350 q^{57} +4.32990 q^{59} -6.07271 q^{61} +28.9857 q^{63} -1.95927 q^{65} -13.5799 q^{67} +5.99987 q^{69} -8.56565 q^{71} +8.77732 q^{73} +11.7372 q^{75} +15.6768 q^{77} -8.00100 q^{79} +21.9436 q^{81} -7.50634 q^{83} +0.968313 q^{85} +32.3627 q^{87} +5.34791 q^{89} -6.76648 q^{91} +29.5474 q^{93} -2.15897 q^{95} +12.6479 q^{97} +28.5176 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - q^{3} - q^{5} - 6 q^{7} + 15 q^{9}+O(q^{10}) $ 6 * q - q^3 - q^5 - 6 * q^7 + 15 * q^9	$ 6 q - q^{3} - q^{5} - 6 q^{7} + 15 q^{9} - q^{11} - 5 q^{13} - 2 q^{15} + 8 q^{17} - 3 q^{19} - 14 q^{21} - 18 q^{23} - q^{25} - 16 q^{27} + q^{29} - 6 q^{31} - 16 q^{33} - 6 q^{35} - 13 q^{37} + 6 q^{39} + 4 q^{41} + 5 q^{43} - 23 q^{45} - 8 q^{47} + 8 q^{49} + 16 q^{51} - 3 q^{53} + 30 q^{55} - 24 q^{57} + 5 q^{59} - 61 q^{61} + 27 q^{63} - q^{65} + 13 q^{67} - 21 q^{69} - 22 q^{71} + 6 q^{73} + 30 q^{75} + 4 q^{77} - 28 q^{79} + 2 q^{81} - 14 q^{83} - 16 q^{85} + 24 q^{87} + 18 q^{89} + 16 q^{91} + 27 q^{93} - 20 q^{95} + 16 q^{97} - 5 q^{99}+O(q^{100}) $ 6 * q - q^3 - q^5 - 6 * q^7 + 15 * q^9 - q^11 - 5 * q^13 - 2 * q^15 + 8 * q^17 - 3 * q^19 - 14 * q^21 - 18 * q^23 - q^25 - 16 * q^27 + q^29 - 6 * q^31 - 16 * q^33 - 6 * q^35 - 13 * q^37 + 6 * q^39 + 4 * q^41 + 5 * q^43 - 23 * q^45 - 8 * q^47 + 8 * q^49 + 16 * q^51 - 3 * q^53 + 30 * q^55 - 24 * q^57 + 5 * q^59 - 61 * q^61 + 27 * q^63 - q^65 + 13 * q^67 - 21 * q^69 - 22 * q^71 + 6 * q^73 + 30 * q^75 + 4 * q^77 - 28 * q^79 + 2 * q^81 - 14 * q^83 - 16 * q^85 + 24 * q^87 + 18 * q^89 + 16 * q^91 + 27 * q^93 - 20 * q^95 + 16 * q^97 - 5 * 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
$3$	−3.20334	−1.84945	−0.924725	−	0.380636i	$-0.875705\pi$
$3$	−3.20334	−1.84945	−0.924725	+	0.380636i	$0.875705\pi$
$4$	0	0
$5$	1.15583	0.516903	0.258452	−	0.966024i	$-0.416788\pi$
$5$	1.15583	0.516903	0.258452	+	0.966024i	$0.416788\pi$
$6$	0	0
$7$	3.99175	1.50874	0.754370	−	0.656450i	$-0.227943\pi$
$7$	3.99175	1.50874	0.754370	+	0.656450i	$0.227943\pi$
$8$	0	0
$9$	7.26139	2.42046
$10$	0	0
$11$	3.92729	1.18412	0.592062	−	0.805893i	$-0.298314\pi$
$11$	3.92729	1.18412	0.592062	+	0.805893i	$0.298314\pi$
$12$	0	0
$13$	−1.69512	−0.470141	−0.235070	−	0.971978i	$-0.575532\pi$
$13$	−1.69512	−0.470141	−0.235070	+	0.971978i	$0.575532\pi$
$14$	0	0
$15$	−3.70252	−0.955986
$16$	0	0
$17$	0.837764	0.203188	0.101594	−	0.994826i	$-0.467606\pi$
$17$	0.837764	0.203188	0.101594	+	0.994826i	$0.467606\pi$
$18$	0	0
$19$	−1.86789	−0.428524	−0.214262	−	0.976776i	$-0.568735\pi$
$19$	−1.86789	−0.428524	−0.214262	+	0.976776i	$0.568735\pi$
$20$	0	0
$21$	−12.7869	−2.79034
$22$	0	0
$23$	−1.87301	−0.390549	−0.195274	−	0.980749i	$-0.562560\pi$
$23$	−1.87301	−0.390549	−0.195274	+	0.980749i	$0.562560\pi$
$24$	0	0
$25$	−3.66406	−0.732811
$26$	0	0
$27$	−13.6507	−2.62708
$28$	0	0
$29$	−10.1028	−1.87604	−0.938021	−	0.346578i	$-0.887344\pi$
$29$	−10.1028	−1.87604	−0.938021	+	0.346578i	$0.887344\pi$
$30$	0	0
$31$	−9.22393	−1.65667	−0.828333	−	0.560236i	$-0.810710\pi$
$31$	−9.22393	−1.65667	−0.828333	+	0.560236i	$0.810710\pi$
$32$	0	0
$33$	−12.5805	−2.18998
$34$	0	0
$35$	4.61379	0.779872
$36$	0	0
$37$	−4.58130	−0.753162	−0.376581	−	0.926384i	$-0.622900\pi$
$37$	−4.58130	−0.753162	−0.376581	+	0.926384i	$0.622900\pi$
$38$	0	0
$39$	5.43003	0.869501
$40$	0	0
$41$	2.89716	0.452461	0.226230	−	0.974074i	$-0.427360\pi$
$41$	2.89716	0.452461	0.226230	+	0.974074i	$0.427360\pi$
$42$	0	0
$43$	7.59775	1.15865	0.579323	−	0.815098i	$-0.303317\pi$
$43$	7.59775	1.15865	0.579323	+	0.815098i	$0.303317\pi$
$44$	0	0
$45$	8.39294	1.25115
$46$	0	0
$47$	−7.88848	−1.15065	−0.575327	−	0.817924i	$-0.695125\pi$
$47$	−7.88848	−1.15065	−0.575327	+	0.817924i	$0.695125\pi$
$48$	0	0
$49$	8.93407	1.27630
$50$	0	0
$51$	−2.68364	−0.375785
$52$	0	0
$53$	−11.2188	−1.54102	−0.770511	−	0.637426i	$-0.779999\pi$
$53$	−11.2188	−1.54102	−0.770511	+	0.637426i	$0.779999\pi$
$54$	0	0
$55$	4.53929	0.612077
$56$	0	0
$57$	5.98350	0.792534
$58$	0	0
$59$	4.32990	0.563705	0.281853	−	0.959458i	$-0.409051\pi$
$59$	4.32990	0.563705	0.281853	+	0.959458i	$0.409051\pi$
$60$	0	0
$61$	−6.07271	−0.777530	−0.388765	−	0.921337i	$-0.627098\pi$
$61$	−6.07271	−0.777530	−0.388765	+	0.921337i	$0.627098\pi$
$62$	0	0
$63$	28.9857	3.65185
$64$	0	0
$65$	−1.95927	−0.243017
$66$	0	0
$67$	−13.5799	−1.65905	−0.829525	−	0.558470i	$-0.811389\pi$
$67$	−13.5799	−1.65905	−0.829525	+	0.558470i	$0.811389\pi$
$68$	0	0
$69$	5.99987	0.722300
$70$	0	0
$71$	−8.56565	−1.01656	−0.508278	−	0.861193i	$-0.669718\pi$
$71$	−8.56565	−1.01656	−0.508278	+	0.861193i	$0.669718\pi$
$72$	0	0
$73$	8.77732	1.02731	0.513654	−	0.857998i	$-0.328292\pi$
$73$	8.77732	1.02731	0.513654	+	0.857998i	$0.328292\pi$
$74$	0	0
$75$	11.7372	1.35530
$76$	0	0
$77$	15.6768	1.78653
$78$	0	0
$79$	−8.00100	−0.900183	−0.450091	−	0.892983i	$-0.648609\pi$
$79$	−8.00100	−0.900183	−0.450091	+	0.892983i	$0.648609\pi$
$80$	0	0
$81$	21.9436	2.43818
$82$	0	0
$83$	−7.50634	−0.823927	−0.411964	−	0.911200i	$-0.635157\pi$
$83$	−7.50634	−0.823927	−0.411964	+	0.911200i	$0.635157\pi$
$84$	0	0
$85$	0.968313	0.105028
$86$	0	0
$87$	32.3627	3.46965
$88$	0	0
$89$	5.34791	0.566877	0.283439	−	0.958990i	$-0.408525\pi$
$89$	5.34791	0.566877	0.283439	+	0.958990i	$0.408525\pi$
$90$	0	0
$91$	−6.76648	−0.709320
$92$	0	0
$93$	29.5474	3.06392
$94$	0	0
$95$	−2.15897	−0.221506
$96$	0	0
$97$	12.6479	1.28420	0.642098	−	0.766623i	$-0.278064\pi$
$97$	12.6479	1.28420	0.642098	+	0.766623i	$0.278064\pi$
$98$	0	0
$99$	28.5176	2.86613
$100$	0	0
$101$	−9.24820	−0.920230	−0.460115	−	0.887859i	$-0.652192\pi$
$101$	−9.24820	−0.920230	−0.460115	+	0.887859i	$0.652192\pi$
$102$	0	0
$103$	14.1913	1.39831	0.699154	−	0.714971i	$-0.253560\pi$
$103$	14.1913	1.39831	0.699154	+	0.714971i	$0.253560\pi$
$104$	0	0
$105$	−14.7795	−1.44233
$106$	0	0
$107$	−6.21432	−0.600761	−0.300380	−	0.953820i	$-0.597114\pi$
$107$	−6.21432	−0.600761	−0.300380	+	0.953820i	$0.597114\pi$
$108$	0	0
$109$	−20.3134	−1.94567	−0.972836	−	0.231497i	$-0.925638\pi$
$109$	−20.3134	−1.94567	−0.972836	+	0.231497i	$0.925638\pi$
$110$	0	0
$111$	14.6755	1.39293
$112$	0	0
$113$	7.01510	0.659925	0.329963	−	0.943994i	$-0.392964\pi$
$113$	7.01510	0.659925	0.329963	+	0.943994i	$0.392964\pi$
$114$	0	0
$115$	−2.16488	−0.201876
$116$	0	0
$117$	−12.3089	−1.13796
$118$	0	0
$119$	3.34414	0.306557
$120$	0	0
$121$	4.42363	0.402148
$122$	0	0
$123$	−9.28060	−0.836804
$124$	0	0
$125$	−10.0142	−0.895696
$126$	0	0
$127$	−8.15528	−0.723664	−0.361832	−	0.932243i	$-0.617849\pi$
$127$	−8.15528	−0.723664	−0.361832	+	0.932243i	$0.617849\pi$
$128$	0	0
$129$	−24.3382	−2.14286
$130$	0	0
$131$	−4.04337	−0.353270	−0.176635	−	0.984276i	$-0.556521\pi$
$131$	−4.04337	−0.353270	−0.176635	+	0.984276i	$0.556521\pi$
$132$	0	0
$133$	−7.45617	−0.646532
$134$	0	0
$135$	−15.7779	−1.35794
$136$	0	0
$137$	3.61786	0.309094	0.154547	−	0.987985i	$-0.450608\pi$
$137$	3.61786	0.309094	0.154547	+	0.987985i	$0.450608\pi$
$138$	0	0
$139$	−11.9159	−1.01069	−0.505345	−	0.862917i	$-0.668635\pi$
$139$	−11.9159	−1.01069	−0.505345	+	0.862917i	$0.668635\pi$
$140$	0	0
$141$	25.2695	2.12807
$142$	0	0
$143$	−6.65722	−0.556704
$144$	0	0
$145$	−11.6771	−0.969732
$146$	0	0
$147$	−28.6189	−2.36044
$148$	0	0
$149$	−2.87060	−0.235168	−0.117584	−	0.993063i	$-0.537515\pi$
$149$	−2.87060	−0.235168	−0.117584	+	0.993063i	$0.537515\pi$
$150$	0	0
$151$	13.3885	1.08954	0.544772	−	0.838584i	$-0.316616\pi$
$151$	13.3885	1.08954	0.544772	+	0.838584i	$0.316616\pi$
$152$	0	0
$153$	6.08333	0.491808
$154$	0	0
$155$	−10.6613	−0.856336
$156$	0	0
$157$	−10.1119	−0.807016	−0.403508	−	0.914976i	$-0.632209\pi$
$157$	−10.1119	−0.807016	−0.403508	+	0.914976i	$0.632209\pi$
$158$	0	0
$159$	35.9377	2.85004
$160$	0	0
$161$	−7.47657	−0.589236
$162$	0	0
$163$	21.4079	1.67679	0.838397	−	0.545060i	$-0.183493\pi$
$163$	21.4079	1.67679	0.838397	+	0.545060i	$0.183493\pi$
$164$	0	0
$165$	−14.5409	−1.13201
$166$	0	0
$167$	−17.4442	−1.34987	−0.674937	−	0.737875i	$-0.735829\pi$
$167$	−17.4442	−1.34987	−0.674937	+	0.737875i	$0.735829\pi$
$168$	0	0
$169$	−10.1266	−0.778968
$170$	0	0
$171$	−13.5635	−1.03723
$172$	0	0
$173$	18.0412	1.37164	0.685822	−	0.727769i	$-0.259443\pi$
$173$	18.0412	1.37164	0.685822	+	0.727769i	$0.259443\pi$
$174$	0	0
$175$	−14.6260	−1.10562
$176$	0	0
$177$	−13.8702	−1.04254
$178$	0	0
$179$	16.1744	1.20893	0.604466	−	0.796631i	$-0.293386\pi$
$179$	16.1744	1.20893	0.604466	+	0.796631i	$0.293386\pi$
$180$	0	0
$181$	4.56478	0.339298	0.169649	−	0.985505i	$-0.445737\pi$
$181$	4.56478	0.339298	0.169649	+	0.985505i	$0.445737\pi$
$182$	0	0
$183$	19.4530	1.43800
$184$	0	0
$185$	−5.29521	−0.389312
$186$	0	0
$187$	3.29014	0.240599
$188$	0	0
$189$	−54.4902	−3.96358
$190$	0	0
$191$	−23.5977	−1.70747	−0.853734	−	0.520710i	$-0.825667\pi$
$191$	−23.5977	−1.70747	−0.853734	+	0.520710i	$0.825667\pi$
$192$	0	0
$193$	−4.82917	−0.347611	−0.173805	−	0.984780i	$-0.555606\pi$
$193$	−4.82917	−0.347611	−0.173805	+	0.984780i	$0.555606\pi$
$194$	0	0
$195$	6.27620	0.449448
$196$	0	0
$197$	−0.767410	−0.0546757	−0.0273379	−	0.999626i	$-0.508703\pi$
$197$	−0.767410	−0.0546757	−0.0273379	+	0.999626i	$0.508703\pi$
$198$	0	0
$199$	21.8253	1.54715	0.773576	−	0.633703i	$-0.218466\pi$
$199$	21.8253	1.54715	0.773576	+	0.633703i	$0.218466\pi$
$200$	0	0
$201$	43.5011	3.06833
$202$	0	0
$203$	−40.3278	−2.83046
$204$	0	0
$205$	3.34863	0.233878
$206$	0	0
$207$	−13.6006	−0.945309
$208$	0	0
$209$	−7.33577	−0.507426
$210$	0	0
$211$	−1.80664	−0.124374	−0.0621870	−	0.998065i	$-0.519808\pi$
$211$	−1.80664	−0.124374	−0.0621870	+	0.998065i	$0.519808\pi$
$212$	0	0
$213$	27.4387	1.88007
$214$	0	0
$215$	8.78172	0.598908
$216$	0	0
$217$	−36.8196	−2.49948
$218$	0	0
$219$	−28.1167	−1.89995
$220$	0	0
$221$	−1.42011	−0.0955267
$222$	0	0
$223$	10.3162	0.690827	0.345413	−	0.938451i	$-0.387739\pi$
$223$	10.3162	0.690827	0.345413	+	0.938451i	$0.387739\pi$
$224$	0	0
$225$	−26.6061	−1.77374
$226$	0	0
$227$	13.8985	0.922474	0.461237	−	0.887277i	$-0.347406\pi$
$227$	13.8985	0.922474	0.461237	+	0.887277i	$0.347406\pi$
$228$	0	0
$229$	12.1492	0.802839	0.401419	−	0.915894i	$-0.368517\pi$
$229$	12.1492	0.802839	0.401419	+	0.915894i	$0.368517\pi$
$230$	0	0
$231$	−50.2180	−3.30410
$232$	0	0
$233$	4.39115	0.287674	0.143837	−	0.989601i	$-0.454056\pi$
$233$	4.39115	0.287674	0.143837	+	0.989601i	$0.454056\pi$
$234$	0	0
$235$	−9.11775	−0.594776
$236$	0	0
$237$	25.6299	1.66484
$238$	0	0
$239$	−13.7533	−0.889626	−0.444813	−	0.895623i	$-0.646730\pi$
$239$	−13.7533	−0.889626	−0.444813	+	0.895623i	$0.646730\pi$
$240$	0	0
$241$	−1.31579	−0.0847574	−0.0423787	−	0.999102i	$-0.513494\pi$
$241$	−1.31579	−0.0847574	−0.0423787	+	0.999102i	$0.513494\pi$
$242$	0	0
$243$	−29.3409	−1.88222
$244$	0	0
$245$	10.3263	0.659721
$246$	0	0
$247$	3.16630	0.201467
$248$	0	0
$249$	24.0454	1.52381
$250$	0	0
$251$	1.00000	0.0631194
$251$	1.00000	0.0631194
$252$	0	0
$253$	−7.35584	−0.462458
$254$	0	0
$255$	−3.10184	−0.194245
$256$	0	0
$257$	15.2840	0.953389	0.476695	−	0.879069i	$-0.341835\pi$
$257$	15.2840	0.953389	0.476695	+	0.879069i	$0.341835\pi$
$258$	0	0
$259$	−18.2874	−1.13633
$260$	0	0
$261$	−73.3604	−4.54089
$262$	0	0
$263$	6.85516	0.422707	0.211354	−	0.977410i	$-0.432213\pi$
$263$	6.85516	0.422707	0.211354	+	0.977410i	$0.432213\pi$
$264$	0	0
$265$	−12.9671	−0.796560
$266$	0	0
$267$	−17.1312	−1.04841
$268$	0	0
$269$	27.7491	1.69189	0.845946	−	0.533268i	$-0.179036\pi$
$269$	27.7491	1.69189	0.845946	+	0.533268i	$0.179036\pi$
$270$	0	0
$271$	−10.3613	−0.629406	−0.314703	−	0.949190i	$-0.601905\pi$
$271$	−10.3613	−0.629406	−0.314703	+	0.949190i	$0.601905\pi$
$272$	0	0
$273$	21.6753	1.31185
$274$	0	0
$275$	−14.3898	−0.867739
$276$	0	0
$277$	15.9887	0.960667	0.480334	−	0.877086i	$-0.340516\pi$
$277$	15.9887	0.960667	0.480334	+	0.877086i	$0.340516\pi$
$278$	0	0
$279$	−66.9786	−4.00990
$280$	0	0
$281$	3.22781	0.192555	0.0962775	−	0.995355i	$-0.469306\pi$
$281$	3.22781	0.192555	0.0962775	+	0.995355i	$0.469306\pi$
$282$	0	0
$283$	−28.4815	−1.69305	−0.846525	−	0.532350i	$-0.821309\pi$
$283$	−28.4815	−1.69305	−0.846525	+	0.532350i	$0.821309\pi$
$284$	0	0
$285$	6.91591	0.409663
$286$	0	0
$287$	11.5648	0.682646
$288$	0	0
$289$	−16.2982	−0.958715
$290$	0	0
$291$	−40.5154	−2.37506
$292$	0	0
$293$	−0.732169	−0.0427738	−0.0213869	−	0.999771i	$-0.506808\pi$
$293$	−0.732169	−0.0427738	−0.0213869	+	0.999771i	$0.506808\pi$
$294$	0	0
$295$	5.00463	0.291381
$296$	0	0
$297$	−53.6103	−3.11078
$298$	0	0
$299$	3.17496	0.183613
$300$	0	0
$301$	30.3283	1.74810
$302$	0	0
$303$	29.6251	1.70192
$304$	0	0
$305$	−7.01902	−0.401908
$306$	0	0
$307$	−27.1281	−1.54828	−0.774142	−	0.633011i	$-0.781819\pi$
$307$	−27.1281	−1.54828	−0.774142	+	0.633011i	$0.781819\pi$
$308$	0	0
$309$	−45.4595	−2.58610
$310$	0	0
$311$	13.9547	0.791300	0.395650	−	0.918401i	$-0.370519\pi$
$311$	13.9547	0.791300	0.395650	+	0.918401i	$0.370519\pi$
$312$	0	0
$313$	−5.90475	−0.333756	−0.166878	−	0.985978i	$-0.553369\pi$
$313$	−5.90475	−0.333756	−0.166878	+	0.985978i	$0.553369\pi$
$314$	0	0
$315$	33.5025	1.88765
$316$	0	0
$317$	12.8044	0.719165	0.359582	−	0.933113i	$-0.382919\pi$
$317$	12.8044	0.719165	0.359582	+	0.933113i	$0.382919\pi$
$318$	0	0
$319$	−39.6766	−2.22147
$320$	0	0
$321$	19.9066	1.11108
$322$	0	0
$323$	−1.56485	−0.0870708
$324$	0	0
$325$	6.21100	0.344524
$326$	0	0
$327$	65.0708	3.59842
$328$	0	0
$329$	−31.4888	−1.73604
$330$	0	0
$331$	9.22701	0.507162	0.253581	−	0.967314i	$-0.418391\pi$
$331$	9.22701	0.507162	0.253581	+	0.967314i	$0.418391\pi$
$332$	0	0
$333$	−33.2666	−1.82300
$334$	0	0
$335$	−15.6961	−0.857568
$336$	0	0
$337$	−32.2226	−1.75528	−0.877639	−	0.479323i	$-0.840882\pi$
$337$	−32.2226	−1.75528	−0.877639	+	0.479323i	$0.840882\pi$
$338$	0	0
$339$	−22.4718	−1.22050
$340$	0	0
$341$	−36.2251	−1.96170
$342$	0	0
$343$	7.72032	0.416858
$344$	0	0
$345$	6.93484	0.373359
$346$	0	0
$347$	33.7002	1.80912	0.904561	−	0.426344i	$-0.140199\pi$
$347$	33.7002	1.80912	0.904561	+	0.426344i	$0.140199\pi$
$348$	0	0
$349$	3.20195	0.171397	0.0856983	−	0.996321i	$-0.472688\pi$
$349$	3.20195	0.171397	0.0856983	+	0.996321i	$0.472688\pi$
$350$	0	0
$351$	23.1395	1.23510
$352$	0	0
$353$	16.6946	0.888562	0.444281	−	0.895887i	$-0.353459\pi$
$353$	16.6946	0.888562	0.444281	+	0.895887i	$0.353459\pi$
$354$	0	0
$355$	−9.90044	−0.525461
$356$	0	0
$357$	−10.7124	−0.566962
$358$	0	0
$359$	−19.7956	−1.04477	−0.522385	−	0.852710i	$-0.674958\pi$
$359$	−19.7956	−1.04477	−0.522385	+	0.852710i	$0.674958\pi$
$360$	0	0
$361$	−15.5110	−0.816367
$362$	0	0
$363$	−14.1704	−0.743753
$364$	0	0
$365$	10.1451	0.531018
$366$	0	0
$367$	20.8134	1.08645	0.543224	−	0.839588i	$-0.317203\pi$
$367$	20.8134	1.08645	0.543224	+	0.839588i	$0.317203\pi$
$368$	0	0
$369$	21.0374	1.09517
$370$	0	0
$371$	−44.7827	−2.32500
$372$	0	0
$373$	4.36136	0.225823	0.112911	−	0.993605i	$-0.463982\pi$
$373$	4.36136	0.225823	0.112911	+	0.993605i	$0.463982\pi$
$374$	0	0
$375$	32.0788	1.65654
$376$	0	0
$377$	17.1254	0.882004
$378$	0	0
$379$	−0.888279	−0.0456278	−0.0228139	−	0.999740i	$-0.507263\pi$
$379$	−0.888279	−0.0456278	−0.0228139	+	0.999740i	$0.507263\pi$
$380$	0	0
$381$	26.1241	1.33838
$382$	0	0
$383$	38.6469	1.97476	0.987382	−	0.158354i	$-0.0506186\pi$
$383$	38.6469	1.97476	0.987382	+	0.158354i	$0.0506186\pi$
$384$	0	0
$385$	18.1197	0.923465
$386$	0	0
$387$	55.1703	2.80446
$388$	0	0
$389$	0.995864	0.0504923	0.0252461	−	0.999681i	$-0.491963\pi$
$389$	0.995864	0.0504923	0.0252461	+	0.999681i	$0.491963\pi$
$390$	0	0
$391$	−1.56914	−0.0793546
$392$	0	0
$393$	12.9523	0.653356
$394$	0	0
$395$	−9.24780	−0.465307
$396$	0	0
$397$	−34.0416	−1.70850	−0.854249	−	0.519865i	$-0.825982\pi$
$397$	−34.0416	−1.70850	−0.854249	+	0.519865i	$0.825982\pi$
$398$	0	0
$399$	23.8846	1.19573
$400$	0	0
$401$	20.1680	1.00714	0.503571	−	0.863954i	$-0.332019\pi$
$401$	20.1680	1.00714	0.503571	+	0.863954i	$0.332019\pi$
$402$	0	0
$403$	15.6356	0.778866
$404$	0	0
$405$	25.3631	1.26030
$406$	0	0
$407$	−17.9921	−0.891836
$408$	0	0
$409$	−9.04023	−0.447010	−0.223505	−	0.974703i	$-0.571750\pi$
$409$	−9.04023	−0.447010	−0.223505	+	0.974703i	$0.571750\pi$
$410$	0	0
$411$	−11.5892	−0.571654
$412$	0	0
$413$	17.2839	0.850485
$414$	0	0
$415$	−8.67605	−0.425891
$416$	0	0
$417$	38.1706	1.86922
$418$	0	0
$419$	30.2372	1.47718	0.738591	−	0.674154i	$-0.235492\pi$
$419$	30.2372	1.47718	0.738591	+	0.674154i	$0.235492\pi$
$420$	0	0
$421$	−27.5186	−1.34118	−0.670588	−	0.741830i	$-0.733958\pi$
$421$	−27.5186	−1.34118	−0.670588	+	0.741830i	$0.733958\pi$
$422$	0	0
$423$	−57.2814	−2.78511
$424$	0	0
$425$	−3.06961	−0.148898
$426$	0	0
$427$	−24.2407	−1.17309
$428$	0	0
$429$	21.3253	1.02960
$430$	0	0
$431$	−10.4596	−0.503822	−0.251911	−	0.967750i	$-0.581059\pi$
$431$	−10.4596	−0.503822	−0.251911	+	0.967750i	$0.581059\pi$
$432$	0	0
$433$	−0.409610	−0.0196846	−0.00984231	−	0.999952i	$-0.503133\pi$
$433$	−0.409610	−0.0196846	−0.00984231	+	0.999952i	$0.503133\pi$
$434$	0	0
$435$	37.4058	1.79347
$436$	0	0
$437$	3.49857	0.167360
$438$	0	0
$439$	20.4964	0.978240	0.489120	−	0.872217i	$-0.337318\pi$
$439$	20.4964	0.978240	0.489120	+	0.872217i	$0.337318\pi$
$440$	0	0
$441$	64.8738	3.08923
$442$	0	0
$443$	−27.3575	−1.29979	−0.649896	−	0.760023i	$-0.725188\pi$
$443$	−27.3575	−1.29979	−0.649896	+	0.760023i	$0.725188\pi$
$444$	0	0
$445$	6.18128	0.293021
$446$	0	0
$447$	9.19550	0.434932
$448$	0	0
$449$	−2.65943	−0.125506	−0.0627532	−	0.998029i	$-0.519988\pi$
$449$	−2.65943	−0.125506	−0.0627532	+	0.998029i	$0.519988\pi$
$450$	0	0
$451$	11.3780	0.535769
$452$	0	0
$453$	−42.8881	−2.01506
$454$	0	0
$455$	−7.82090	−0.366650
$456$	0	0
$457$	−3.56782	−0.166896	−0.0834478	−	0.996512i	$-0.526593\pi$
$457$	−3.56782	−0.166896	−0.0834478	+	0.996512i	$0.526593\pi$
$458$	0	0
$459$	−11.4361	−0.533790
$460$	0	0
$461$	0.340886	0.0158766	0.00793831	−	0.999968i	$-0.497473\pi$
$461$	0.340886	0.0158766	0.00793831	+	0.999968i	$0.497473\pi$
$462$	0	0
$463$	8.74942	0.406620	0.203310	−	0.979114i	$-0.434830\pi$
$463$	8.74942	0.406620	0.203310	+	0.979114i	$0.434830\pi$
$464$	0	0
$465$	34.1518	1.58375
$466$	0	0
$467$	−25.0085	−1.15725	−0.578627	−	0.815592i	$-0.696411\pi$
$467$	−25.0085	−1.15725	−0.578627	+	0.815592i	$0.696411\pi$
$468$	0	0
$469$	−54.2076	−2.50307
$470$	0	0
$471$	32.3918	1.49254
$472$	0	0
$473$	29.8386	1.37198
$474$	0	0
$475$	6.84407	0.314027
$476$	0	0
$477$	−81.4642	−3.72999
$478$	0	0
$479$	−28.1827	−1.28770	−0.643850	−	0.765151i	$-0.722664\pi$
$479$	−28.1827	−1.28770	−0.643850	+	0.765151i	$0.722664\pi$
$480$	0	0
$481$	7.76584	0.354092
$482$	0	0
$483$	23.9500	1.08976
$484$	0	0
$485$	14.6188	0.663805
$486$	0	0
$487$	−9.07287	−0.411131	−0.205565	−	0.978643i	$-0.565903\pi$
$487$	−9.07287	−0.411131	−0.205565	+	0.978643i	$0.565903\pi$
$488$	0	0
$489$	−68.5767	−3.10115
$490$	0	0
$491$	38.7115	1.74702	0.873512	−	0.486802i	$-0.161837\pi$
$491$	38.7115	1.74702	0.873512	+	0.486802i	$0.161837\pi$
$492$	0	0
$493$	−8.46376	−0.381189
$494$	0	0
$495$	32.9615	1.48151
$496$	0	0
$497$	−34.1919	−1.53372
$498$	0	0
$499$	10.3222	0.462083	0.231042	−	0.972944i	$-0.425787\pi$
$499$	10.3222	0.462083	0.231042	+	0.972944i	$0.425787\pi$
$500$	0	0
$501$	55.8798	2.49652
$502$	0	0
$503$	12.5744	0.560667	0.280333	−	0.959903i	$-0.409555\pi$
$503$	12.5744	0.560667	0.280333	+	0.959903i	$0.409555\pi$
$504$	0	0
$505$	−10.6894	−0.475670
$506$	0	0
$507$	32.4389	1.44066
$508$	0	0
$509$	4.54879	0.201621	0.100811	−	0.994906i	$-0.467856\pi$
$509$	4.54879	0.201621	0.100811	+	0.994906i	$0.467856\pi$
$510$	0	0
$511$	35.0369	1.54994
$512$	0	0
$513$	25.4980	1.12577
$514$	0	0
$515$	16.4027	0.722790
$516$	0	0
$517$	−30.9804	−1.36252
$518$	0	0
$519$	−57.7920	−2.53679
$520$	0	0
$521$	0.341732	0.0149715	0.00748577	−	0.999972i	$-0.497617\pi$
$521$	0.341732	0.0149715	0.00748577	+	0.999972i	$0.497617\pi$
$522$	0	0
$523$	12.2809	0.537007	0.268504	−	0.963279i	$-0.413471\pi$
$523$	12.2809	0.537007	0.268504	+	0.963279i	$0.413471\pi$
$524$	0	0
$525$	46.8520	2.04479
$526$	0	0
$527$	−7.72747	−0.336614
$528$	0	0
$529$	−19.4919	−0.847472
$530$	0	0
$531$	31.4411	1.36443
$532$	0	0
$533$	−4.91103	−0.212720
$534$	0	0
$535$	−7.18270	−0.310535
$536$	0	0
$537$	−51.8122	−2.23586
$538$	0	0
$539$	35.0867	1.51129
$540$	0	0
$541$	−7.99736	−0.343833	−0.171917	−	0.985111i	$-0.554996\pi$
$541$	−7.99736	−0.343833	−0.171917	+	0.985111i	$0.554996\pi$
$542$	0	0
$543$	−14.6226	−0.627514
$544$	0	0
$545$	−23.4789	−1.00572
$546$	0	0
$547$	4.93724	0.211101	0.105551	−	0.994414i	$-0.466340\pi$
$547$	4.93724	0.211101	0.105551	+	0.994414i	$0.466340\pi$
$548$	0	0
$549$	−44.0963	−1.88198
$550$	0	0
$551$	18.8709	0.803929
$552$	0	0
$553$	−31.9380	−1.35814
$554$	0	0
$555$	16.9624	0.720012
$556$	0	0
$557$	36.7910	1.55888	0.779441	−	0.626475i	$-0.215503\pi$
$557$	36.7910	1.55888	0.779441	+	0.626475i	$0.215503\pi$
$558$	0	0
$559$	−12.8791	−0.544727
$560$	0	0
$561$	−10.5395	−0.444976
$562$	0	0
$563$	39.1386	1.64950	0.824748	−	0.565501i	$-0.191317\pi$
$563$	39.1386	1.64950	0.824748	+	0.565501i	$0.191317\pi$
$564$	0	0
$565$	8.10827	0.341118
$566$	0	0
$567$	87.5936	3.67858
$568$	0	0
$569$	14.0631	0.589556	0.294778	−	0.955566i	$-0.404754\pi$
$569$	14.0631	0.589556	0.294778	+	0.955566i	$0.404754\pi$
$570$	0	0
$571$	−13.1651	−0.550941	−0.275470	−	0.961310i	$-0.588834\pi$
$571$	−13.1651	−0.550941	−0.275470	+	0.961310i	$0.588834\pi$
$572$	0	0
$573$	75.5914	3.15787
$574$	0	0
$575$	6.86280	0.286198
$576$	0	0
$577$	36.1773	1.50608	0.753041	−	0.657973i	$-0.228586\pi$
$577$	36.1773	1.50608	0.753041	+	0.657973i	$0.228586\pi$
$578$	0	0
$579$	15.4695	0.642889
$580$	0	0
$581$	−29.9634	−1.24309
$582$	0	0
$583$	−44.0596	−1.82476
$584$	0	0
$585$	−14.2270	−0.588214
$586$	0	0
$587$	37.6309	1.55320	0.776598	−	0.629997i	$-0.216944\pi$
$587$	37.6309	1.55320	0.776598	+	0.629997i	$0.216944\pi$
$588$	0	0
$589$	17.2293	0.709922
$590$	0	0
$591$	2.45828	0.101120
$592$	0	0
$593$	15.6040	0.640780	0.320390	−	0.947286i	$-0.396186\pi$
$593$	15.6040	0.640780	0.320390	+	0.947286i	$0.396186\pi$
$594$	0	0
$595$	3.86526	0.158460
$596$	0	0
$597$	−69.9138	−2.86138
$598$	0	0
$599$	−25.3810	−1.03704	−0.518520	−	0.855066i	$-0.673517\pi$
$599$	−25.3810	−1.03704	−0.518520	+	0.855066i	$0.673517\pi$
$600$	0	0
$601$	−3.29867	−0.134555	−0.0672777	−	0.997734i	$-0.521431\pi$
$601$	−3.29867	−0.134555	−0.0672777	+	0.997734i	$0.521431\pi$
$602$	0	0
$603$	−98.6090	−4.01567
$604$	0	0
$605$	5.11297	0.207872
$606$	0	0
$607$	−9.19344	−0.373150	−0.186575	−	0.982441i	$-0.559739\pi$
$607$	−9.19344	−0.373150	−0.186575	+	0.982441i	$0.559739\pi$
$608$	0	0
$609$	129.184	5.23479
$610$	0	0
$611$	13.3719	0.540969
$612$	0	0
$613$	39.8121	1.60800	0.803998	−	0.594631i	$-0.202702\pi$
$613$	39.8121	1.60800	0.803998	+	0.594631i	$0.202702\pi$
$614$	0	0
$615$	−10.7268	−0.432546
$616$	0	0
$617$	−30.6552	−1.23413	−0.617066	−	0.786911i	$-0.711679\pi$
$617$	−30.6552	−1.23413	−0.617066	+	0.786911i	$0.711679\pi$
$618$	0	0
$619$	27.9655	1.12403	0.562014	−	0.827128i	$-0.310027\pi$
$619$	27.9655	1.12403	0.562014	+	0.827128i	$0.310027\pi$
$620$	0	0
$621$	25.5678	1.02600
$622$	0	0
$623$	21.3475	0.855270
$624$	0	0
$625$	6.74558	0.269823
$626$	0	0
$627$	23.4990	0.938458
$628$	0	0
$629$	−3.83805	−0.153033
$630$	0	0
$631$	−4.30466	−0.171366	−0.0856829	−	0.996322i	$-0.527307\pi$
$631$	−4.30466	−0.171366	−0.0856829	+	0.996322i	$0.527307\pi$
$632$	0	0
$633$	5.78727	0.230023
$634$	0	0
$635$	−9.42612	−0.374064
$636$	0	0
$637$	−15.1443	−0.600038
$638$	0	0
$639$	−62.1986	−2.46054
$640$	0	0
$641$	15.0331	0.593771	0.296886	−	0.954913i	$-0.404052\pi$
$641$	15.0331	0.593771	0.296886	+	0.954913i	$0.404052\pi$
$642$	0	0
$643$	25.2358	0.995204	0.497602	−	0.867405i	$-0.334214\pi$
$643$	25.2358	0.995204	0.497602	+	0.867405i	$0.334214\pi$
$644$	0	0
$645$	−28.1308	−1.10765
$646$	0	0
$647$	−20.7607	−0.816186	−0.408093	−	0.912940i	$-0.633806\pi$
$647$	−20.7607	−0.816186	−0.408093	+	0.912940i	$0.633806\pi$
$648$	0	0
$649$	17.0048	0.667497
$650$	0	0
$651$	117.946	4.62266
$652$	0	0
$653$	−27.5527	−1.07822	−0.539110	−	0.842235i	$-0.681239\pi$
$653$	−27.5527	−1.07822	−0.539110	+	0.842235i	$0.681239\pi$
$654$	0	0
$655$	−4.67345	−0.182607
$656$	0	0
$657$	63.7355	2.48656
$658$	0	0
$659$	−41.7388	−1.62591	−0.812956	−	0.582326i	$-0.802143\pi$
$659$	−41.7388	−1.62591	−0.812956	+	0.582326i	$0.802143\pi$
$660$	0	0
$661$	−10.1367	−0.394272	−0.197136	−	0.980376i	$-0.563164\pi$
$661$	−10.1367	−0.394272	−0.197136	+	0.980376i	$0.563164\pi$
$662$	0	0
$663$	4.54909	0.176672
$664$	0	0
$665$	−8.61806	−0.334194
$666$	0	0
$667$	18.9226	0.732686
$668$	0	0
$669$	−33.0465	−1.27765
$670$	0	0
$671$	−23.8493	−0.920692
$672$	0	0
$673$	−40.9232	−1.57747	−0.788737	−	0.614730i	$-0.789265\pi$
$673$	−40.9232	−1.57747	−0.788737	+	0.614730i	$0.789265\pi$
$674$	0	0
$675$	50.0169	1.92515
$676$	0	0
$677$	−21.6307	−0.831335	−0.415668	−	0.909517i	$-0.636452\pi$
$677$	−21.6307	−0.831335	−0.415668	+	0.909517i	$0.636452\pi$
$678$	0	0
$679$	50.4871	1.93752
$680$	0	0
$681$	−44.5215	−1.70607
$682$	0	0
$683$	−3.79398	−0.145172	−0.0725862	−	0.997362i	$-0.523125\pi$
$683$	−3.79398	−0.145172	−0.0725862	+	0.997362i	$0.523125\pi$
$684$	0	0
$685$	4.18163	0.159772
$686$	0	0
$687$	−38.9179	−1.48481
$688$	0	0
$689$	19.0172	0.724497
$690$	0	0
$691$	9.81627	0.373428	0.186714	−	0.982414i	$-0.440216\pi$
$691$	9.81627	0.373428	0.186714	+	0.982414i	$0.440216\pi$
$692$	0	0
$693$	113.835	4.32424
$694$	0	0
$695$	−13.7727	−0.522429
$696$	0	0
$697$	2.42714	0.0919344
$698$	0	0
$699$	−14.0663	−0.532038
$700$	0	0
$701$	10.1003	0.381483	0.190741	−	0.981640i	$-0.438911\pi$
$701$	10.1003	0.381483	0.190741	+	0.981640i	$0.438911\pi$
$702$	0	0
$703$	8.55739	0.322748
$704$	0	0
$705$	29.2073	1.10001
$706$	0	0
$707$	−36.9165	−1.38839
$708$	0	0
$709$	−37.1164	−1.39394	−0.696969	−	0.717102i	$-0.745468\pi$
$709$	−37.1164	−1.39394	−0.696969	+	0.717102i	$0.745468\pi$
$710$	0	0
$711$	−58.0984	−2.17886
$712$	0	0
$713$	17.2765	0.647009
$714$	0	0
$715$	−7.69462	−0.287762
$716$	0	0
$717$	44.0565	1.64532
$718$	0	0
$719$	−9.74662	−0.363488	−0.181744	−	0.983346i	$-0.558174\pi$
$719$	−9.74662	−0.363488	−0.181744	+	0.983346i	$0.558174\pi$
$720$	0	0
$721$	56.6480	2.10968
$722$	0	0
$723$	4.21492	0.156755
$724$	0	0
$725$	37.0172	1.37478
$726$	0	0
$727$	−46.3912	−1.72055	−0.860277	−	0.509827i	$-0.829710\pi$
$727$	−46.3912	−1.72055	−0.860277	+	0.509827i	$0.829710\pi$
$728$	0	0
$729$	28.1580	1.04289
$730$	0	0
$731$	6.36512	0.235423
$732$	0	0
$733$	18.0764	0.667668	0.333834	−	0.942632i	$-0.391657\pi$
$733$	18.0764	0.667668	0.333834	+	0.942632i	$0.391657\pi$
$734$	0	0
$735$	−33.0786	−1.22012
$736$	0	0
$737$	−53.3323	−1.96452
$738$	0	0
$739$	−7.37931	−0.271452	−0.135726	−	0.990746i	$-0.543337\pi$
$739$	−7.37931	−0.271452	−0.135726	+	0.990746i	$0.543337\pi$
$740$	0	0
$741$	−10.1427	−0.372602
$742$	0	0
$743$	22.5896	0.828730	0.414365	−	0.910111i	$-0.364004\pi$
$743$	22.5896	0.828730	0.414365	+	0.910111i	$0.364004\pi$
$744$	0	0
$745$	−3.31792	−0.121559
$746$	0	0
$747$	−54.5065	−1.99429
$748$	0	0
$749$	−24.8060	−0.906391
$750$	0	0
$751$	−3.29152	−0.120109	−0.0600546	−	0.998195i	$-0.519127\pi$
$751$	−3.29152	−0.120109	−0.0600546	+	0.998195i	$0.519127\pi$
$752$	0	0
$753$	−3.20334	−0.116736
$754$	0	0
$755$	15.4749	0.563189
$756$	0	0
$757$	26.2462	0.953935	0.476968	−	0.878921i	$-0.341736\pi$
$757$	26.2462	0.953935	0.476968	+	0.878921i	$0.341736\pi$
$758$	0	0
$759$	23.5633	0.855292
$760$	0	0
$761$	−13.3451	−0.483760	−0.241880	−	0.970306i	$-0.577764\pi$
$761$	−13.3451	−0.483760	−0.241880	+	0.970306i	$0.577764\pi$
$762$	0	0
$763$	−81.0860	−2.93551
$764$	0	0
$765$	7.03130	0.254217
$766$	0	0
$767$	−7.33969	−0.265021
$768$	0	0
$769$	−7.61171	−0.274485	−0.137243	−	0.990537i	$-0.543824\pi$
$769$	−7.61171	−0.274485	−0.137243	+	0.990537i	$0.543824\pi$
$770$	0	0
$771$	−48.9598	−1.76325
$772$	0	0
$773$	28.9398	1.04089	0.520446	−	0.853895i	$-0.325766\pi$
$773$	28.9398	1.04089	0.520446	+	0.853895i	$0.325766\pi$
$774$	0	0
$775$	33.7970	1.21402
$776$	0	0
$777$	58.5808	2.10158
$778$	0	0
$779$	−5.41159	−0.193890
$780$	0	0
$781$	−33.6398	−1.20373
$782$	0	0
$783$	137.910	4.92851
$784$	0	0
$785$	−11.6876	−0.417149
$786$	0	0
$787$	−16.0168	−0.570937	−0.285469	−	0.958388i	$-0.592149\pi$
$787$	−16.0168	−0.570937	−0.285469	+	0.958388i	$0.592149\pi$
$788$	0	0
$789$	−21.9594	−0.781776
$790$	0	0
$791$	28.0025	0.995656
$792$	0	0
$793$	10.2939	0.365549
$794$	0	0
$795$	41.5379	1.47320
$796$	0	0
$797$	11.6001	0.410897	0.205448	−	0.978668i	$-0.434135\pi$
$797$	11.6001	0.410897	0.205448	+	0.978668i	$0.434135\pi$
$798$	0	0
$799$	−6.60868	−0.233798
$800$	0	0
$801$	38.8333	1.37211
$802$	0	0
$803$	34.4711	1.21646
$804$	0	0
$805$	−8.64165	−0.304578
$806$	0	0
$807$	−88.8899	−3.12907
$808$	0	0
$809$	6.61791	0.232673	0.116337	−	0.993210i	$-0.462885\pi$
$809$	6.61791	0.232673	0.116337	+	0.993210i	$0.462885\pi$
$810$	0	0
$811$	46.7646	1.64213	0.821064	−	0.570837i	$-0.193381\pi$
$811$	46.7646	1.64213	0.821064	+	0.570837i	$0.193381\pi$
$812$	0	0
$813$	33.1909	1.16405
$814$	0	0
$815$	24.7439	0.866740
$816$	0	0
$817$	−14.1918	−0.496508
$818$	0	0
$819$	−49.1341	−1.71688
$820$	0	0
$821$	−35.6596	−1.24453	−0.622264	−	0.782808i	$-0.713787\pi$
$821$	−35.6596	−1.24453	−0.622264	+	0.782808i	$0.713787\pi$
$822$	0	0
$823$	50.0419	1.74435	0.872175	−	0.489194i	$-0.162709\pi$
$823$	50.0419	1.74435	0.872175	+	0.489194i	$0.162709\pi$
$824$	0	0
$825$	46.0955	1.60484
$826$	0	0
$827$	56.1772	1.95347	0.976737	−	0.214440i	$-0.0687928\pi$
$827$	56.1772	1.95347	0.976737	+	0.214440i	$0.0687928\pi$
$828$	0	0
$829$	−18.5774	−0.645219	−0.322609	−	0.946532i	$-0.604560\pi$
$829$	−18.5774	−0.645219	−0.322609	+	0.946532i	$0.604560\pi$
$830$	0	0
$831$	−51.2172	−1.77671
$832$	0	0
$833$	7.48464	0.259327
$834$	0	0
$835$	−20.1626	−0.697754
$836$	0	0
$837$	125.913	4.35219
$838$	0	0
$839$	2.66473	0.0919968	0.0459984	−	0.998942i	$-0.485353\pi$
$839$	2.66473	0.0919968	0.0459984	+	0.998942i	$0.485353\pi$
$840$	0	0
$841$	73.0665	2.51953
$842$	0	0
$843$	−10.3398	−0.356121
$844$	0	0
$845$	−11.7046	−0.402651
$846$	0	0
$847$	17.6580	0.606737
$848$	0	0
$849$	91.2359	3.13121
$850$	0	0
$851$	8.58081	0.294146
$852$	0	0
$853$	−40.7600	−1.39560	−0.697798	−	0.716294i	$-0.745837\pi$
$853$	−40.7600	−1.39560	−0.697798	+	0.716294i	$0.745837\pi$
$854$	0	0
$855$	−15.6771	−0.536146
$856$	0	0
$857$	−13.7445	−0.469504	−0.234752	−	0.972055i	$-0.575428\pi$
$857$	−13.7445	−0.469504	−0.234752	+	0.972055i	$0.575428\pi$
$858$	0	0
$859$	−56.2082	−1.91780	−0.958900	−	0.283746i	$-0.908423\pi$
$859$	−56.2082	−1.91780	−0.958900	+	0.283746i	$0.908423\pi$
$860$	0	0
$861$	−37.0458	−1.26252
$862$	0	0
$863$	−44.4420	−1.51282	−0.756412	−	0.654095i	$-0.773050\pi$
$863$	−44.4420	−1.51282	−0.756412	+	0.654095i	$0.773050\pi$
$864$	0	0
$865$	20.8525	0.709008
$866$	0	0
$867$	52.2085	1.77309
$868$	0	0
$869$	−31.4223	−1.06593
$870$	0	0
$871$	23.0195	0.779987
$872$	0	0
$873$	91.8411	3.10835
$874$	0	0
$875$	−39.9741	−1.35137
$876$	0	0
$877$	0.972692	0.0328455	0.0164227	−	0.999865i	$-0.494772\pi$
$877$	0.972692	0.0328455	0.0164227	+	0.999865i	$0.494772\pi$
$878$	0	0
$879$	2.34539	0.0791079
$880$	0	0
$881$	−47.5755	−1.60286	−0.801430	−	0.598089i	$-0.795927\pi$
$881$	−47.5755	−1.60286	−0.801430	+	0.598089i	$0.795927\pi$
$882$	0	0
$883$	−21.1429	−0.711516	−0.355758	−	0.934578i	$-0.615777\pi$
$883$	−21.1429	−0.711516	−0.355758	+	0.934578i	$0.615777\pi$
$884$	0	0
$885$	−16.0315	−0.538895
$886$	0	0
$887$	1.00200	0.0336439	0.0168219	−	0.999859i	$-0.494645\pi$
$887$	1.00200	0.0336439	0.0168219	+	0.999859i	$0.494645\pi$
$888$	0	0
$889$	−32.5538	−1.09182
$890$	0	0
$891$	86.1791	2.88711
$892$	0	0
$893$	14.7348	0.493083
$894$	0	0
$895$	18.6949	0.624901
$896$	0	0
$897$	−10.1705	−0.339583
$898$	0	0
$899$	93.1875	3.10798
$900$	0	0
$901$	−9.39872	−0.313117
$902$	0	0
$903$	−97.1520	−3.23302
$904$	0	0
$905$	5.27611	0.175384
$906$	0	0
$907$	−52.5280	−1.74416	−0.872081	−	0.489362i	$-0.837230\pi$
$907$	−52.5280	−1.74416	−0.872081	+	0.489362i	$0.837230\pi$
$908$	0	0
$909$	−67.1548	−2.22739
$910$	0	0
$911$	16.9564	0.561791	0.280896	−	0.959738i	$-0.409368\pi$
$911$	16.9564	0.561791	0.280896	+	0.959738i	$0.409368\pi$
$912$	0	0
$913$	−29.4796	−0.975632
$914$	0	0
$915$	22.4843	0.743309
$916$	0	0
$917$	−16.1401	−0.532993
$918$	0	0
$919$	35.0215	1.15525	0.577627	−	0.816301i	$-0.303979\pi$
$919$	35.0215	1.15525	0.577627	+	0.816301i	$0.303979\pi$
$920$	0	0
$921$	86.9007	2.86348
$922$	0	0
$923$	14.5198	0.477924
$924$	0	0
$925$	16.7862	0.551925
$926$	0	0
$927$	103.048	3.38455
$928$	0	0
$929$	−44.0005	−1.44361	−0.721805	−	0.692096i	$-0.756687\pi$
$929$	−44.0005	−1.44361	−0.721805	+	0.692096i	$0.756687\pi$
$930$	0	0
$931$	−16.6879	−0.546924
$932$	0	0
$933$	−44.7018	−1.46347
$934$	0	0
$935$	3.80285	0.124366
$936$	0	0
$937$	41.4406	1.35380	0.676902	−	0.736073i	$-0.263322\pi$
$937$	41.4406	1.35380	0.676902	+	0.736073i	$0.263322\pi$
$938$	0	0
$939$	18.9149	0.617265
$940$	0	0
$941$	38.6854	1.26111	0.630553	−	0.776146i	$-0.282828\pi$
$941$	38.6854	1.26111	0.630553	+	0.776146i	$0.282828\pi$
$942$	0	0
$943$	−5.42640	−0.176708
$944$	0	0
$945$	−62.9814	−2.04879
$946$	0	0
$947$	−19.4690	−0.632658	−0.316329	−	0.948649i	$-0.602450\pi$
$947$	−19.4690	−0.632658	−0.316329	+	0.948649i	$0.602450\pi$
$948$	0	0
$949$	−14.8786	−0.482979
$950$	0	0
$951$	−41.0168	−1.33006
$952$	0	0
$953$	16.4595	0.533177	0.266588	−	0.963810i	$-0.414104\pi$
$953$	16.4595	0.533177	0.266588	+	0.963810i	$0.414104\pi$
$954$	0	0
$955$	−27.2749	−0.882595
$956$	0	0
$957$	127.098	4.10849
$958$	0	0
$959$	14.4416	0.466343
$960$	0	0
$961$	54.0808	1.74454
$962$	0	0
$963$	−45.1246	−1.45412
$964$	0	0
$965$	−5.58170	−0.179681
$966$	0	0
$967$	−56.4418	−1.81505	−0.907523	−	0.420003i	$-0.862029\pi$
$967$	−56.4418	−1.81505	−0.907523	+	0.420003i	$0.862029\pi$
$968$	0	0
$969$	5.01276	0.161033
$970$	0	0
$971$	−30.9420	−0.992975	−0.496487	−	0.868044i	$-0.665377\pi$
$971$	−30.9420	−0.992975	−0.496487	+	0.868044i	$0.665377\pi$
$972$	0	0
$973$	−47.5651	−1.52487
$974$	0	0
$975$	−19.8959	−0.637180
$976$	0	0
$977$	−9.67723	−0.309602	−0.154801	−	0.987946i	$-0.549474\pi$
$977$	−9.67723	−0.309602	−0.154801	+	0.987946i	$0.549474\pi$
$978$	0	0
$979$	21.0028	0.671253
$980$	0	0
$981$	−147.504	−4.70943
$982$	0	0
$983$	3.67901	0.117342	0.0586711	−	0.998277i	$-0.481314\pi$
$983$	3.67901	0.117342	0.0586711	+	0.998277i	$0.481314\pi$
$984$	0	0
$985$	−0.886996	−0.0282621
$986$	0	0
$987$	100.869	3.21071
$988$	0	0
$989$	−14.2306	−0.452508
$990$	0	0
$991$	−19.7539	−0.627502	−0.313751	−	0.949505i	$-0.601586\pi$
$991$	−19.7539	−0.627502	−0.313751	+	0.949505i	$0.601586\pi$
$992$	0	0
$993$	−29.5573	−0.937971
$994$	0	0
$995$	25.2263	0.799728
$996$	0	0
$997$	8.02113	0.254032	0.127016	−	0.991901i	$-0.459460\pi$
$997$	8.02113	0.254032	0.127016	+	0.991901i	$0.459460\pi$
$998$	0	0
$999$	62.5380	1.97861

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4016.2.a.f.1.1		6
4.3	odd	2		502.2.a.e.1.6	✓	6
12.11	even	2		4518.2.a.x.1.2		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
502.2.a.e.1.6	✓	6	4.3	odd	2
4016.2.a.f.1.1		6	1.1	even	1	trivial
4518.2.a.x.1.2		6	12.11	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:	\( N \)	\(=\)	\( 4016 = 2^{4} \cdot 251 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4016.a (trivial)

\(f(q)\)	\(=\)	\(q-3.20334 q^{3} +1.15583 q^{5} +3.99175 q^{7} +7.26139 q^{9} +O(q^{10})\)	\(q-3.20334 q^{3} +1.15583 q^{5} +3.99175 q^{7} +7.26139 q^{9} +3.92729 q^{11} -1.69512 q^{13} -3.70252 q^{15} +0.837764 q^{17} -1.86789 q^{19} -12.7869 q^{21} -1.87301 q^{23} -3.66406 q^{25} -13.6507 q^{27} -10.1028 q^{29} -9.22393 q^{31} -12.5805 q^{33} +4.61379 q^{35} -4.58130 q^{37} +5.43003 q^{39} +2.89716 q^{41} +7.59775 q^{43} +8.39294 q^{45} -7.88848 q^{47} +8.93407 q^{49} -2.68364 q^{51} -11.2188 q^{53} +4.53929 q^{55} +5.98350 q^{57} +4.32990 q^{59} -6.07271 q^{61} +28.9857 q^{63} -1.95927 q^{65} -13.5799 q^{67} +5.99987 q^{69} -8.56565 q^{71} +8.77732 q^{73} +11.7372 q^{75} +15.6768 q^{77} -8.00100 q^{79} +21.9436 q^{81} -7.50634 q^{83} +0.968313 q^{85} +32.3627 q^{87} +5.34791 q^{89} -6.76648 q^{91} +29.5474 q^{93} -2.15897 q^{95} +12.6479 q^{97} +28.5176 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - q^{3} - q^{5} - 6 q^{7} + 15 q^{9}+O(q^{10}) \) 6 * q - q^3 - q^5 - 6 * q^7 + 15 * q^9	\( 6 q - q^{3} - q^{5} - 6 q^{7} + 15 q^{9} - q^{11} - 5 q^{13} - 2 q^{15} + 8 q^{17} - 3 q^{19} - 14 q^{21} - 18 q^{23} - q^{25} - 16 q^{27} + q^{29} - 6 q^{31} - 16 q^{33} - 6 q^{35} - 13 q^{37} + 6 q^{39} + 4 q^{41} + 5 q^{43} - 23 q^{45} - 8 q^{47} + 8 q^{49} + 16 q^{51} - 3 q^{53} + 30 q^{55} - 24 q^{57} + 5 q^{59} - 61 q^{61} + 27 q^{63} - q^{65} + 13 q^{67} - 21 q^{69} - 22 q^{71} + 6 q^{73} + 30 q^{75} + 4 q^{77} - 28 q^{79} + 2 q^{81} - 14 q^{83} - 16 q^{85} + 24 q^{87} + 18 q^{89} + 16 q^{91} + 27 q^{93} - 20 q^{95} + 16 q^{97} - 5 q^{99}+O(q^{100}) \) 6 * q - q^3 - q^5 - 6 * q^7 + 15 * q^9 - q^11 - 5 * q^13 - 2 * q^15 + 8 * q^17 - 3 * q^19 - 14 * q^21 - 18 * q^23 - q^25 - 16 * q^27 + q^29 - 6 * q^31 - 16 * q^33 - 6 * q^35 - 13 * q^37 + 6 * q^39 + 4 * q^41 + 5 * q^43 - 23 * q^45 - 8 * q^47 + 8 * q^49 + 16 * q^51 - 3 * q^53 + 30 * q^55 - 24 * q^57 + 5 * q^59 - 61 * q^61 + 27 * q^63 - q^65 + 13 * q^67 - 21 * q^69 - 22 * q^71 + 6 * q^73 + 30 * q^75 + 4 * q^77 - 28 * q^79 + 2 * q^81 - 14 * q^83 - 16 * q^85 + 24 * q^87 + 18 * q^89 + 16 * q^91 + 27 * q^93 - 20 * q^95 + 16 * q^97 - 5 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more