LMFDB - Embedded newform 4012.2.a.h.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4012,2,Mod(1,4012)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4012, 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("4012.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4012 = 2^{2} \cdot 17 \cdot 59 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4012.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:	$32.0359812909$
Analytic rank:	$1$
Dimension:	$15$
Coefficient field:	$\mathbb{Q}[x]/(x^{15} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{15} - 5 x^{14} - 16 x^{13} + 123 x^{12} - 76 x^{11} - 630 x^{10} + 937 x^{9} + 1083 x^{8} - 2602 x^{7} + \cdots - 2 $ x^15 - 5x^14 - 16x^13 + 123x^12 - 76x^11 - 630x^10 + 937x^9 + 1083x^8 - 2602x^7 - 249x^6 + 2736x^5 - 801x^4 - 900x^3 + 429x^2 - 36x - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$2.23095$ of defining polynomial
Character	$\chi$	$=$	4012.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.598056 q^{3} -0.722695 q^{5} -0.897746 q^{7} -2.64233 q^{9} +O(q^{10})$	$q-0.598056 q^{3} -0.722695 q^{5} -0.897746 q^{7} -2.64233 q^{9} +3.15421 q^{11} -5.02384 q^{13} +0.432212 q^{15} +1.00000 q^{17} +5.60899 q^{19} +0.536902 q^{21} +6.04958 q^{23} -4.47771 q^{25} +3.37443 q^{27} +1.54292 q^{29} -0.336375 q^{31} -1.88639 q^{33} +0.648796 q^{35} +4.94861 q^{37} +3.00454 q^{39} +7.50124 q^{41} -7.93860 q^{43} +1.90960 q^{45} +5.10041 q^{47} -6.19405 q^{49} -0.598056 q^{51} -9.86232 q^{53} -2.27953 q^{55} -3.35449 q^{57} -1.00000 q^{59} -2.08150 q^{61} +2.37214 q^{63} +3.63070 q^{65} +1.50739 q^{67} -3.61799 q^{69} -0.378963 q^{71} -15.4256 q^{73} +2.67792 q^{75} -2.83168 q^{77} -2.25294 q^{79} +5.90889 q^{81} -15.2628 q^{83} -0.722695 q^{85} -0.922750 q^{87} +6.09271 q^{89} +4.51014 q^{91} +0.201171 q^{93} -4.05359 q^{95} -2.30013 q^{97} -8.33446 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 15 q - q^{3} + q^{5} - 11 q^{7} + 16 q^{9}+O(q^{10}) $ 15 * q - q^3 + q^5 - 11 * q^7 + 16 * q^9	$ 15 q - q^{3} + q^{5} - 11 q^{7} + 16 q^{9} - 12 q^{11} - 10 q^{13} - 13 q^{15} + 15 q^{17} - 4 q^{21} - 21 q^{23} + 4 q^{25} - 37 q^{27} + 23 q^{29} - 31 q^{31} - 11 q^{33} - 35 q^{35} - 10 q^{37} - 2 q^{39} - 15 q^{41} - 3 q^{43} + 15 q^{45} - 47 q^{47} + 18 q^{49} - q^{51} - 7 q^{53} - 20 q^{55} - 30 q^{57} - 15 q^{59} + q^{61} + 19 q^{63} + 11 q^{65} - 20 q^{67} - 24 q^{69} - 13 q^{71} - 24 q^{73} - 10 q^{75} + 21 q^{77} - 34 q^{79} - 21 q^{81} - 40 q^{83} + q^{85} - 60 q^{87} - 46 q^{89} - 19 q^{91} + 33 q^{93} - 4 q^{95} - 5 q^{97} - 74 q^{99}+O(q^{100}) $ 15 * q - q^3 + q^5 - 11 * q^7 + 16 * q^9 - 12 * q^11 - 10 * q^13 - 13 * q^15 + 15 * q^17 - 4 * q^21 - 21 * q^23 + 4 * q^25 - 37 * q^27 + 23 * q^29 - 31 * q^31 - 11 * q^33 - 35 * q^35 - 10 * q^37 - 2 * q^39 - 15 * q^41 - 3 * q^43 + 15 * q^45 - 47 * q^47 + 18 * q^49 - q^51 - 7 * q^53 - 20 * q^55 - 30 * q^57 - 15 * q^59 + q^61 + 19 * q^63 + 11 * q^65 - 20 * q^67 - 24 * q^69 - 13 * q^71 - 24 * q^73 - 10 * q^75 + 21 * q^77 - 34 * q^79 - 21 * q^81 - 40 * q^83 + q^85 - 60 * q^87 - 46 * q^89 - 19 * q^91 + 33 * q^93 - 4 * q^95 - 5 * q^97 - 74 * 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
$3$	−0.598056	−0.345288	−0.172644	−	0.984984i	$-0.555231\pi$
$3$	−0.598056	−0.345288	−0.172644	+	0.984984i	$0.555231\pi$
$4$	0	0
$5$	−0.722695	−0.323199	−0.161599	−	0.986856i	$-0.551665\pi$
$5$	−0.722695	−0.323199	−0.161599	+	0.986856i	$0.551665\pi$
$6$	0	0
$7$	−0.897746	−0.339316	−0.169658	−	0.985503i	$-0.554266\pi$
$7$	−0.897746	−0.339316	−0.169658	+	0.985503i	$0.554266\pi$
$8$	0	0
$9$	−2.64233	−0.880776
$10$	0	0
$11$	3.15421	0.951030	0.475515	−	0.879708i	$-0.342262\pi$
$11$	3.15421	0.951030	0.475515	+	0.879708i	$0.342262\pi$
$12$	0	0
$13$	−5.02384	−1.39336	−0.696682	−	0.717381i	$-0.745341\pi$
$13$	−5.02384	−1.39336	−0.696682	+	0.717381i	$0.745341\pi$
$14$	0	0
$15$	0.432212	0.111597
$16$	0	0
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	0	0
$19$	5.60899	1.28679	0.643396	−	0.765534i	$-0.277525\pi$
$19$	5.60899	1.28679	0.643396	+	0.765534i	$0.277525\pi$
$20$	0	0
$21$	0.536902	0.117162
$22$	0	0
$23$	6.04958	1.26142	0.630712	−	0.776017i	$-0.282763\pi$
$23$	6.04958	1.26142	0.630712	+	0.776017i	$0.282763\pi$
$24$	0	0
$25$	−4.47771	−0.895543
$26$	0	0
$27$	3.37443	0.649409
$28$	0	0
$29$	1.54292	0.286512	0.143256	−	0.989686i	$-0.454243\pi$
$29$	1.54292	0.286512	0.143256	+	0.989686i	$0.454243\pi$
$30$	0	0
$31$	−0.336375	−0.0604148	−0.0302074	−	0.999544i	$-0.509617\pi$
$31$	−0.336375	−0.0604148	−0.0302074	+	0.999544i	$0.509617\pi$
$32$	0	0
$33$	−1.88639	−0.328379
$34$	0	0
$35$	0.648796	0.109667
$36$	0	0
$37$	4.94861	0.813546	0.406773	−	0.913529i	$-0.366654\pi$
$37$	4.94861	0.813546	0.406773	+	0.913529i	$0.366654\pi$
$38$	0	0
$39$	3.00454	0.481111
$40$	0	0
$41$	7.50124	1.17150	0.585749	−	0.810493i	$-0.300801\pi$
$41$	7.50124	1.17150	0.585749	+	0.810493i	$0.300801\pi$
$42$	0	0
$43$	−7.93860	−1.21063	−0.605313	−	0.795988i	$-0.706952\pi$
$43$	−7.93860	−1.21063	−0.605313	+	0.795988i	$0.706952\pi$
$44$	0	0
$45$	1.90960	0.284666
$46$	0	0
$47$	5.10041	0.743971	0.371985	−	0.928239i	$-0.378677\pi$
$47$	5.10041	0.743971	0.371985	+	0.928239i	$0.378677\pi$
$48$	0	0
$49$	−6.19405	−0.884865
$50$	0	0
$51$	−0.598056	−0.0837446
$52$	0	0
$53$	−9.86232	−1.35469	−0.677347	−	0.735664i	$-0.736870\pi$
$53$	−9.86232	−1.35469	−0.677347	+	0.735664i	$0.736870\pi$
$54$	0	0
$55$	−2.27953	−0.307372
$56$	0	0
$57$	−3.35449	−0.444313
$58$	0	0
$59$	−1.00000	−0.130189
$59$	−1.00000	−0.130189
$60$	0	0
$61$	−2.08150	−0.266508	−0.133254	−	0.991082i	$-0.542543\pi$
$61$	−2.08150	−0.266508	−0.133254	+	0.991082i	$0.542543\pi$
$62$	0	0
$63$	2.37214	0.298862
$64$	0	0
$65$	3.63070	0.450333
$66$	0	0
$67$	1.50739	0.184158	0.0920788	−	0.995752i	$-0.470649\pi$
$67$	1.50739	0.184158	0.0920788	+	0.995752i	$0.470649\pi$
$68$	0	0
$69$	−3.61799	−0.435554
$70$	0	0
$71$	−0.378963	−0.0449747	−0.0224873	−	0.999747i	$-0.507159\pi$
$71$	−0.378963	−0.0449747	−0.0224873	+	0.999747i	$0.507159\pi$
$72$	0	0
$73$	−15.4256	−1.80543	−0.902714	−	0.430241i	$-0.858429\pi$
$73$	−15.4256	−1.80543	−0.902714	+	0.430241i	$0.858429\pi$
$74$	0	0
$75$	2.67792	0.309220
$76$	0	0
$77$	−2.83168	−0.322700
$78$	0	0
$79$	−2.25294	−0.253476	−0.126738	−	0.991936i	$-0.540451\pi$
$79$	−2.25294	−0.253476	−0.126738	+	0.991936i	$0.540451\pi$
$80$	0	0
$81$	5.90889	0.656544
$82$	0	0
$83$	−15.2628	−1.67531	−0.837654	−	0.546201i	$-0.816073\pi$
$83$	−15.2628	−1.67531	−0.837654	+	0.546201i	$0.816073\pi$
$84$	0	0
$85$	−0.722695	−0.0783872
$86$	0	0
$87$	−0.922750	−0.0989292
$88$	0	0
$89$	6.09271	0.645826	0.322913	−	0.946429i	$-0.395338\pi$
$89$	6.09271	0.645826	0.322913	+	0.946429i	$0.395338\pi$
$90$	0	0
$91$	4.51014	0.472791
$92$	0	0
$93$	0.201171	0.0208605
$94$	0	0
$95$	−4.05359	−0.415889
$96$	0	0
$97$	−2.30013	−0.233543	−0.116771	−	0.993159i	$-0.537254\pi$
$97$	−2.30013	−0.233543	−0.116771	+	0.993159i	$0.537254\pi$
$98$	0	0
$99$	−8.33446	−0.837645
$100$	0	0
$101$	−16.1156	−1.60356	−0.801781	−	0.597618i	$-0.796114\pi$
$101$	−16.1156	−1.60356	−0.801781	+	0.597618i	$0.796114\pi$
$102$	0	0
$103$	−4.51354	−0.444732	−0.222366	−	0.974963i	$-0.571378\pi$
$103$	−4.51354	−0.444732	−0.222366	+	0.974963i	$0.571378\pi$
$104$	0	0
$105$	−0.388016	−0.0378665
$106$	0	0
$107$	15.2391	1.47322	0.736611	−	0.676317i	$-0.236425\pi$
$107$	15.2391	1.47322	0.736611	+	0.676317i	$0.236425\pi$
$108$	0	0
$109$	−3.35569	−0.321417	−0.160709	−	0.987002i	$-0.551378\pi$
$109$	−3.35569	−0.321417	−0.160709	+	0.987002i	$0.551378\pi$
$110$	0	0
$111$	−2.95954	−0.280908
$112$	0	0
$113$	−11.6430	−1.09529	−0.547643	−	0.836712i	$-0.684475\pi$
$113$	−11.6430	−1.09529	−0.547643	+	0.836712i	$0.684475\pi$
$114$	0	0
$115$	−4.37200	−0.407691
$116$	0	0
$117$	13.2746	1.22724
$118$	0	0
$119$	−0.897746	−0.0822963
$120$	0	0
$121$	−1.05097	−0.0955423
$122$	0	0
$123$	−4.48616	−0.404504
$124$	0	0
$125$	6.84949	0.612637
$126$	0	0
$127$	−11.8325	−1.04996	−0.524982	−	0.851113i	$-0.675928\pi$
$127$	−11.8325	−1.04996	−0.524982	+	0.851113i	$0.675928\pi$
$128$	0	0
$129$	4.74773	0.418014
$130$	0	0
$131$	13.3645	1.16766	0.583832	−	0.811874i	$-0.301552\pi$
$131$	13.3645	1.16766	0.583832	+	0.811874i	$0.301552\pi$
$132$	0	0
$133$	−5.03545	−0.436629
$134$	0	0
$135$	−2.43868	−0.209888
$136$	0	0
$137$	−12.7282	−1.08744	−0.543720	−	0.839267i	$-0.682985\pi$
$137$	−12.7282	−1.08744	−0.543720	+	0.839267i	$0.682985\pi$
$138$	0	0
$139$	−6.83692	−0.579900	−0.289950	−	0.957042i	$-0.593639\pi$
$139$	−6.83692	−0.579900	−0.289950	+	0.957042i	$0.593639\pi$
$140$	0	0
$141$	−3.05033	−0.256884
$142$	0	0
$143$	−15.8462	−1.32513
$144$	0	0
$145$	−1.11506	−0.0926005
$146$	0	0
$147$	3.70439	0.305533
$148$	0	0
$149$	4.68976	0.384200	0.192100	−	0.981375i	$-0.438470\pi$
$149$	4.68976	0.384200	0.192100	+	0.981375i	$0.438470\pi$
$150$	0	0
$151$	−5.89287	−0.479555	−0.239777	−	0.970828i	$-0.577074\pi$
$151$	−5.89287	−0.479555	−0.239777	+	0.970828i	$0.577074\pi$
$152$	0	0
$153$	−2.64233	−0.213620
$154$	0	0
$155$	0.243096	0.0195260
$156$	0	0
$157$	14.0943	1.12485	0.562423	−	0.826850i	$-0.309869\pi$
$157$	14.0943	1.12485	0.562423	+	0.826850i	$0.309869\pi$
$158$	0	0
$159$	5.89821	0.467759
$160$	0	0
$161$	−5.43099	−0.428022
$162$	0	0
$163$	−14.2020	−1.11238	−0.556192	−	0.831054i	$-0.687738\pi$
$163$	−14.2020	−1.11238	−0.556192	+	0.831054i	$0.687738\pi$
$164$	0	0
$165$	1.36329	0.106132
$166$	0	0
$167$	−21.6885	−1.67831	−0.839155	−	0.543893i	$-0.816950\pi$
$167$	−21.6885	−1.67831	−0.839155	+	0.543893i	$0.816950\pi$
$168$	0	0
$169$	12.2390	0.941461
$170$	0	0
$171$	−14.8208	−1.13338
$172$	0	0
$173$	21.5395	1.63762	0.818811	−	0.574064i	$-0.194634\pi$
$173$	21.5395	1.63762	0.818811	+	0.574064i	$0.194634\pi$
$174$	0	0
$175$	4.01985	0.303872
$176$	0	0
$177$	0.598056	0.0449526
$178$	0	0
$179$	−21.8090	−1.63008	−0.815041	−	0.579404i	$-0.803285\pi$
$179$	−21.8090	−1.63008	−0.815041	+	0.579404i	$0.803285\pi$
$180$	0	0
$181$	3.41728	0.254004	0.127002	−	0.991902i	$-0.459464\pi$
$181$	3.41728	0.254004	0.127002	+	0.991902i	$0.459464\pi$
$182$	0	0
$183$	1.24485	0.0920221
$184$	0	0
$185$	−3.57633	−0.262937
$186$	0	0
$187$	3.15421	0.230659
$188$	0	0
$189$	−3.02938	−0.220355
$190$	0	0
$191$	9.03103	0.653462	0.326731	−	0.945117i	$-0.394053\pi$
$191$	9.03103	0.653462	0.326731	+	0.945117i	$0.394053\pi$
$192$	0	0
$193$	8.25513	0.594218	0.297109	−	0.954844i	$-0.403978\pi$
$193$	8.25513	0.594218	0.297109	+	0.954844i	$0.403978\pi$
$194$	0	0
$195$	−2.17136	−0.155495
$196$	0	0
$197$	−6.24870	−0.445202	−0.222601	−	0.974910i	$-0.571455\pi$
$197$	−6.24870	−0.445202	−0.222601	+	0.974910i	$0.571455\pi$
$198$	0	0
$199$	10.9968	0.779546	0.389773	−	0.920911i	$-0.372554\pi$
$199$	10.9968	0.779546	0.389773	+	0.920911i	$0.372554\pi$
$200$	0	0
$201$	−0.901506	−0.0635873
$202$	0	0
$203$	−1.38515	−0.0972183
$204$	0	0
$205$	−5.42111	−0.378627
$206$	0	0
$207$	−15.9850	−1.11103
$208$	0	0
$209$	17.6919	1.22378
$210$	0	0
$211$	−14.1110	−0.971439	−0.485720	−	0.874115i	$-0.661442\pi$
$211$	−14.1110	−0.971439	−0.485720	+	0.874115i	$0.661442\pi$
$212$	0	0
$213$	0.226641	0.0155292
$214$	0	0
$215$	5.73719	0.391273
$216$	0	0
$217$	0.301980	0.0204997
$218$	0	0
$219$	9.22536	0.623392
$220$	0	0
$221$	−5.02384	−0.337940
$222$	0	0
$223$	26.5434	1.77748	0.888738	−	0.458415i	$-0.151583\pi$
$223$	26.5434	1.77748	0.888738	+	0.458415i	$0.151583\pi$
$224$	0	0
$225$	11.8316	0.788773
$226$	0	0
$227$	−3.94263	−0.261681	−0.130841	−	0.991403i	$-0.541768\pi$
$227$	−3.94263	−0.261681	−0.130841	+	0.991403i	$0.541768\pi$
$228$	0	0
$229$	−25.5148	−1.68606	−0.843032	−	0.537864i	$-0.819231\pi$
$229$	−25.5148	−1.68606	−0.843032	+	0.537864i	$0.819231\pi$
$230$	0	0
$231$	1.69350	0.111424
$232$	0	0
$233$	10.3170	0.675889	0.337944	−	0.941166i	$-0.390268\pi$
$233$	10.3170	0.675889	0.337944	+	0.941166i	$0.390268\pi$
$234$	0	0
$235$	−3.68604	−0.240450
$236$	0	0
$237$	1.34739	0.0875221
$238$	0	0
$239$	7.16562	0.463505	0.231753	−	0.972775i	$-0.425554\pi$
$239$	7.16562	0.463505	0.231753	+	0.972775i	$0.425554\pi$
$240$	0	0
$241$	1.68517	0.108551	0.0542756	−	0.998526i	$-0.482715\pi$
$241$	1.68517	0.108551	0.0542756	+	0.998526i	$0.482715\pi$
$242$	0	0
$243$	−13.6571	−0.876105
$244$	0	0
$245$	4.47641	0.285987
$246$	0	0
$247$	−28.1787	−1.79297
$248$	0	0
$249$	9.12799	0.578463
$250$	0	0
$251$	−29.3850	−1.85476	−0.927382	−	0.374116i	$-0.877946\pi$
$251$	−29.3850	−1.85476	−0.927382	+	0.374116i	$0.877946\pi$
$252$	0	0
$253$	19.0816	1.19965
$254$	0	0
$255$	0.432212	0.0270661
$256$	0	0
$257$	−21.4030	−1.33508	−0.667541	−	0.744573i	$-0.732653\pi$
$257$	−21.4030	−1.33508	−0.667541	+	0.744573i	$0.732653\pi$
$258$	0	0
$259$	−4.44260	−0.276049
$260$	0	0
$261$	−4.07689	−0.252353
$262$	0	0
$263$	0.904976	0.0558032	0.0279016	−	0.999611i	$-0.491117\pi$
$263$	0.904976	0.0558032	0.0279016	+	0.999611i	$0.491117\pi$
$264$	0	0
$265$	7.12744	0.437835
$266$	0	0
$267$	−3.64378	−0.222996
$268$	0	0
$269$	−10.2670	−0.625988	−0.312994	−	0.949755i	$-0.601332\pi$
$269$	−10.2670	−0.625988	−0.312994	+	0.949755i	$0.601332\pi$
$270$	0	0
$271$	−0.995326	−0.0604617	−0.0302309	−	0.999543i	$-0.509624\pi$
$271$	−0.995326	−0.0604617	−0.0302309	+	0.999543i	$0.509624\pi$
$272$	0	0
$273$	−2.69731	−0.163249
$274$	0	0
$275$	−14.1236	−0.851688
$276$	0	0
$277$	29.1390	1.75079	0.875397	−	0.483405i	$-0.160600\pi$
$277$	29.1390	1.75079	0.875397	+	0.483405i	$0.160600\pi$
$278$	0	0
$279$	0.888814	0.0532119
$280$	0	0
$281$	20.1193	1.20022	0.600108	−	0.799919i	$-0.295124\pi$
$281$	20.1193	1.20022	0.600108	+	0.799919i	$0.295124\pi$
$282$	0	0
$283$	−33.2593	−1.97706	−0.988531	−	0.151017i	$-0.951745\pi$
$283$	−33.2593	−1.97706	−0.988531	+	0.151017i	$0.951745\pi$
$284$	0	0
$285$	2.42427	0.143601
$286$	0	0
$287$	−6.73421	−0.397508
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	1.37560	0.0806394
$292$	0	0
$293$	0.0149832	0.000875328 0	0.000437664	−	1.00000i	$-0.499861\pi$
$293$	0.0149832	0.000875328 0	0.000437664		1.00000i	$0.499861\pi$
$294$	0	0
$295$	0.722695	0.0420769
$296$	0	0
$297$	10.6436	0.617607
$298$	0	0
$299$	−30.3921	−1.75762
$300$	0	0
$301$	7.12685	0.410785
$302$	0	0
$303$	9.63803	0.553690
$304$	0	0
$305$	1.50429	0.0861352
$306$	0	0
$307$	−14.6161	−0.834187	−0.417094	−	0.908863i	$-0.636951\pi$
$307$	−14.6161	−0.834187	−0.417094	+	0.908863i	$0.636951\pi$
$308$	0	0
$309$	2.69935	0.153561
$310$	0	0
$311$	−12.5646	−0.712472	−0.356236	−	0.934396i	$-0.615940\pi$
$311$	−12.5646	−0.712472	−0.356236	+	0.934396i	$0.615940\pi$
$312$	0	0
$313$	−32.4422	−1.83374	−0.916872	−	0.399182i	$-0.869294\pi$
$313$	−32.4422	−1.83374	−0.916872	+	0.399182i	$0.869294\pi$
$314$	0	0
$315$	−1.71433	−0.0965918
$316$	0	0
$317$	−30.3264	−1.70330	−0.851650	−	0.524110i	$-0.824398\pi$
$317$	−30.3264	−1.70330	−0.851650	+	0.524110i	$0.824398\pi$
$318$	0	0
$319$	4.86668	0.272482
$320$	0	0
$321$	−9.11384	−0.508685
$322$	0	0
$323$	5.60899	0.312093
$324$	0	0
$325$	22.4953	1.24782
$326$	0	0
$327$	2.00689	0.110981
$328$	0	0
$329$	−4.57887	−0.252441
$330$	0	0
$331$	6.68987	0.367708	0.183854	−	0.982954i	$-0.441143\pi$
$331$	6.68987	0.367708	0.183854	+	0.982954i	$0.441143\pi$
$332$	0	0
$333$	−13.0759	−0.716552
$334$	0	0
$335$	−1.08939	−0.0595195
$336$	0	0
$337$	−23.2738	−1.26781	−0.633903	−	0.773413i	$-0.718548\pi$
$337$	−23.2738	−1.26781	−0.633903	+	0.773413i	$0.718548\pi$
$338$	0	0
$339$	6.96319	0.378189
$340$	0	0
$341$	−1.06100	−0.0574562
$342$	0	0
$343$	11.8449	0.639565
$344$	0	0
$345$	2.61470	0.140771
$346$	0	0
$347$	13.8232	0.742069	0.371034	−	0.928619i	$-0.379003\pi$
$347$	13.8232	0.742069	0.371034	+	0.928619i	$0.379003\pi$
$348$	0	0
$349$	6.61687	0.354193	0.177096	−	0.984194i	$-0.443330\pi$
$349$	6.61687	0.354193	0.177096	+	0.984194i	$0.443330\pi$
$350$	0	0
$351$	−16.9526	−0.904862
$352$	0	0
$353$	4.58581	0.244078	0.122039	−	0.992525i	$-0.461057\pi$
$353$	4.58581	0.244078	0.122039	+	0.992525i	$0.461057\pi$
$354$	0	0
$355$	0.273875	0.0145358
$356$	0	0
$357$	0.536902	0.0284159
$358$	0	0
$359$	−1.10562	−0.0583524	−0.0291762	−	0.999574i	$-0.509288\pi$
$359$	−1.10562	−0.0583524	−0.0291762	+	0.999574i	$0.509288\pi$
$360$	0	0
$361$	12.4608	0.655832
$362$	0	0
$363$	0.628536	0.0329896
$364$	0	0
$365$	11.1480	0.583512
$366$	0	0
$367$	4.70175	0.245429	0.122715	−	0.992442i	$-0.460840\pi$
$367$	4.70175	0.245429	0.122715	+	0.992442i	$0.460840\pi$
$368$	0	0
$369$	−19.8208	−1.03183
$370$	0	0
$371$	8.85386	0.459669
$372$	0	0
$373$	0.392564	0.0203262	0.0101631	−	0.999948i	$-0.496765\pi$
$373$	0.392564	0.0203262	0.0101631	+	0.999948i	$0.496765\pi$
$374$	0	0
$375$	−4.09638	−0.211536
$376$	0	0
$377$	−7.75137	−0.399216
$378$	0	0
$379$	14.7192	0.756074	0.378037	−	0.925791i	$-0.376599\pi$
$379$	14.7192	0.756074	0.378037	+	0.925791i	$0.376599\pi$
$380$	0	0
$381$	7.07649	0.362539
$382$	0	0
$383$	8.97944	0.458828	0.229414	−	0.973329i	$-0.426319\pi$
$383$	8.97944	0.458828	0.229414	+	0.973329i	$0.426319\pi$
$384$	0	0
$385$	2.04644	0.104296
$386$	0	0
$387$	20.9764	1.06629
$388$	0	0
$389$	11.9291	0.604828	0.302414	−	0.953177i	$-0.402207\pi$
$389$	11.9291	0.604828	0.302414	+	0.953177i	$0.402207\pi$
$390$	0	0
$391$	6.04958	0.305940
$392$	0	0
$393$	−7.99274	−0.403180
$394$	0	0
$395$	1.62819	0.0819231
$396$	0	0
$397$	23.2894	1.16886	0.584432	−	0.811443i	$-0.301318\pi$
$397$	23.2894	1.16886	0.584432	+	0.811443i	$0.301318\pi$
$398$	0	0
$399$	3.01148	0.150763
$400$	0	0
$401$	−16.2351	−0.810741	−0.405370	−	0.914152i	$-0.632857\pi$
$401$	−16.2351	−0.810741	−0.405370	+	0.914152i	$0.632857\pi$
$402$	0	0
$403$	1.68990	0.0841797
$404$	0	0
$405$	−4.27032	−0.212194
$406$	0	0
$407$	15.6089	0.773707
$408$	0	0
$409$	−2.18391	−0.107987	−0.0539937	−	0.998541i	$-0.517195\pi$
$409$	−2.18391	−0.107987	−0.0539937	+	0.998541i	$0.517195\pi$
$410$	0	0
$411$	7.61215	0.375479
$412$	0	0
$413$	0.897746	0.0441752
$414$	0	0
$415$	11.0303	0.541458
$416$	0	0
$417$	4.08886	0.200232
$418$	0	0
$419$	23.2873	1.13766	0.568830	−	0.822455i	$-0.307396\pi$
$419$	23.2873	1.13766	0.568830	+	0.822455i	$0.307396\pi$
$420$	0	0
$421$	9.48703	0.462370	0.231185	−	0.972910i	$-0.425740\pi$
$421$	9.48703	0.462370	0.231185	+	0.972910i	$0.425740\pi$
$422$	0	0
$423$	−13.4770	−0.655272
$424$	0	0
$425$	−4.47771	−0.217201
$426$	0	0
$427$	1.86866	0.0904307
$428$	0	0
$429$	9.47694	0.457551
$430$	0	0
$431$	−2.08352	−0.100360	−0.0501799	−	0.998740i	$-0.515979\pi$
$431$	−2.08352	−0.100360	−0.0501799	+	0.998740i	$0.515979\pi$
$432$	0	0
$433$	−22.8924	−1.10014	−0.550069	−	0.835119i	$-0.685399\pi$
$433$	−22.8924	−1.10014	−0.550069	+	0.835119i	$0.685399\pi$
$434$	0	0
$435$	0.666866	0.0319738
$436$	0	0
$437$	33.9320	1.62319
$438$	0	0
$439$	−19.3223	−0.922203	−0.461101	−	0.887347i	$-0.652546\pi$
$439$	−19.3223	−0.922203	−0.461101	+	0.887347i	$0.652546\pi$
$440$	0	0
$441$	16.3667	0.779368
$442$	0	0
$443$	−21.9646	−1.04357	−0.521784	−	0.853078i	$-0.674733\pi$
$443$	−21.9646	−1.04357	−0.521784	+	0.853078i	$0.674733\pi$
$444$	0	0
$445$	−4.40317	−0.208730
$446$	0	0
$447$	−2.80474	−0.132659
$448$	0	0
$449$	14.1319	0.666926	0.333463	−	0.942763i	$-0.391783\pi$
$449$	14.1319	0.666926	0.333463	+	0.942763i	$0.391783\pi$
$450$	0	0
$451$	23.6605	1.11413
$452$	0	0
$453$	3.52426	0.165584
$454$	0	0
$455$	−3.25945	−0.152805
$456$	0	0
$457$	−5.41960	−0.253518	−0.126759	−	0.991934i	$-0.540457\pi$
$457$	−5.41960	−0.253518	−0.126759	+	0.991934i	$0.540457\pi$
$458$	0	0
$459$	3.37443	0.157505
$460$	0	0
$461$	−8.59496	−0.400307	−0.200154	−	0.979765i	$-0.564144\pi$
$461$	−8.59496	−0.400307	−0.200154	+	0.979765i	$0.564144\pi$
$462$	0	0
$463$	−20.2968	−0.943271	−0.471635	−	0.881794i	$-0.656336\pi$
$463$	−20.2968	−0.943271	−0.471635	+	0.881794i	$0.656336\pi$
$464$	0	0
$465$	−0.145385	−0.00674208
$466$	0	0
$467$	−35.8507	−1.65897	−0.829486	−	0.558528i	$-0.811366\pi$
$467$	−35.8507	−1.65897	−0.829486	+	0.558528i	$0.811366\pi$
$468$	0	0
$469$	−1.35326	−0.0624876
$470$	0	0
$471$	−8.42917	−0.388396
$472$	0	0
$473$	−25.0400	−1.15134
$474$	0	0
$475$	−25.1155	−1.15238
$476$	0	0
$477$	26.0595	1.19318
$478$	0	0
$479$	6.12414	0.279819	0.139910	−	0.990164i	$-0.455319\pi$
$479$	6.12414	0.279819	0.139910	+	0.990164i	$0.455319\pi$
$480$	0	0
$481$	−24.8610	−1.13357
$482$	0	0
$483$	3.24803	0.147791
$484$	0	0
$485$	1.66229	0.0754807
$486$	0	0
$487$	−20.2382	−0.917080	−0.458540	−	0.888674i	$-0.651627\pi$
$487$	−20.2382	−0.917080	−0.458540	+	0.888674i	$0.651627\pi$
$488$	0	0
$489$	8.49358	0.384093
$490$	0	0
$491$	−25.1806	−1.13639	−0.568193	−	0.822895i	$-0.692357\pi$
$491$	−25.1806	−1.13639	−0.568193	+	0.822895i	$0.692357\pi$
$492$	0	0
$493$	1.54292	0.0694895
$494$	0	0
$495$	6.02327	0.270726
$496$	0	0
$497$	0.340213	0.0152606
$498$	0	0
$499$	19.8528	0.888733	0.444367	−	0.895845i	$-0.353429\pi$
$499$	19.8528	0.888733	0.444367	+	0.895845i	$0.353429\pi$
$500$	0	0
$501$	12.9710	0.579499
$502$	0	0
$503$	30.5370	1.36158	0.680789	−	0.732480i	$-0.261637\pi$
$503$	30.5370	1.36158	0.680789	+	0.732480i	$0.261637\pi$
$504$	0	0
$505$	11.6467	0.518269
$506$	0	0
$507$	−7.31960	−0.325075
$508$	0	0
$509$	5.05281	0.223962	0.111981	−	0.993710i	$-0.464280\pi$
$509$	5.05281	0.223962	0.111981	+	0.993710i	$0.464280\pi$
$510$	0	0
$511$	13.8483	0.612611
$512$	0	0
$513$	18.9271	0.835654
$514$	0	0
$515$	3.26191	0.143737
$516$	0	0
$517$	16.0877	0.707538
$518$	0	0
$519$	−12.8818	−0.565450
$520$	0	0
$521$	−7.29891	−0.319771	−0.159885	−	0.987136i	$-0.551112\pi$
$521$	−7.29891	−0.319771	−0.159885	+	0.987136i	$0.551112\pi$
$522$	0	0
$523$	43.8726	1.91842	0.959208	−	0.282700i	$-0.0912302\pi$
$523$	43.8726	1.91842	0.959208	+	0.282700i	$0.0912302\pi$
$524$	0	0
$525$	−2.40409	−0.104923
$526$	0	0
$527$	−0.336375	−0.0146527
$528$	0	0
$529$	13.5974	0.591191
$530$	0	0
$531$	2.64233	0.114667
$532$	0	0
$533$	−37.6851	−1.63232
$534$	0	0
$535$	−11.0132	−0.476143
$536$	0	0
$537$	13.0430	0.562847
$538$	0	0
$539$	−19.5373	−0.841533
$540$	0	0
$541$	−23.9446	−1.02946	−0.514730	−	0.857352i	$-0.672108\pi$
$541$	−23.9446	−1.02946	−0.514730	+	0.857352i	$0.672108\pi$
$542$	0	0
$543$	−2.04372	−0.0877046
$544$	0	0
$545$	2.42514	0.103882
$546$	0	0
$547$	−22.8629	−0.977549	−0.488775	−	0.872410i	$-0.662556\pi$
$547$	−22.8629	−0.977549	−0.488775	+	0.872410i	$0.662556\pi$
$548$	0	0
$549$	5.50000	0.234734
$550$	0	0
$551$	8.65421	0.368682
$552$	0	0
$553$	2.02257	0.0860085
$554$	0	0
$555$	2.13885	0.0907890
$556$	0	0
$557$	11.4575	0.485469	0.242735	−	0.970093i	$-0.421956\pi$
$557$	11.4575	0.485469	0.242735	+	0.970093i	$0.421956\pi$
$558$	0	0
$559$	39.8823	1.68684
$560$	0	0
$561$	−1.88639	−0.0796436
$562$	0	0
$563$	−8.02319	−0.338137	−0.169069	−	0.985604i	$-0.554076\pi$
$563$	−8.02319	−0.338137	−0.169069	+	0.985604i	$0.554076\pi$
$564$	0	0
$565$	8.41437	0.353995
$566$	0	0
$567$	−5.30469	−0.222776
$568$	0	0
$569$	39.6808	1.66350	0.831752	−	0.555147i	$-0.187338\pi$
$569$	39.6808	1.66350	0.831752	+	0.555147i	$0.187338\pi$
$570$	0	0
$571$	34.9980	1.46462	0.732309	−	0.680972i	$-0.238442\pi$
$571$	34.9980	1.46462	0.732309	+	0.680972i	$0.238442\pi$
$572$	0	0
$573$	−5.40106	−0.225633
$574$	0	0
$575$	−27.0883	−1.12966
$576$	0	0
$577$	28.3839	1.18164	0.590819	−	0.806804i	$-0.298805\pi$
$577$	28.3839	1.18164	0.590819	+	0.806804i	$0.298805\pi$
$578$	0	0
$579$	−4.93703	−0.205176
$580$	0	0
$581$	13.7021	0.568459
$582$	0	0
$583$	−31.1078	−1.28835
$584$	0	0
$585$	−9.59351	−0.396643
$586$	0	0
$587$	12.9320	0.533761	0.266880	−	0.963730i	$-0.414007\pi$
$587$	12.9320	0.533761	0.266880	+	0.963730i	$0.414007\pi$
$588$	0	0
$589$	−1.88673	−0.0777412
$590$	0	0
$591$	3.73707	0.153723
$592$	0	0
$593$	−19.1158	−0.784990	−0.392495	−	0.919754i	$-0.628388\pi$
$593$	−19.1158	−0.784990	−0.392495	+	0.919754i	$0.628388\pi$
$594$	0	0
$595$	0.648796	0.0265981
$596$	0	0
$597$	−6.57673	−0.269168
$598$	0	0
$599$	−38.6586	−1.57955	−0.789773	−	0.613399i	$-0.789802\pi$
$599$	−38.6586	−1.57955	−0.789773	+	0.613399i	$0.789802\pi$
$600$	0	0
$601$	−21.9805	−0.896604	−0.448302	−	0.893882i	$-0.647971\pi$
$601$	−21.9805	−0.896604	−0.448302	+	0.893882i	$0.647971\pi$
$602$	0	0
$603$	−3.98303	−0.162202
$604$	0	0
$605$	0.759527	0.0308792
$606$	0	0
$607$	−37.7991	−1.53422	−0.767108	−	0.641518i	$-0.778305\pi$
$607$	−37.7991	−1.53422	−0.767108	+	0.641518i	$0.778305\pi$
$608$	0	0
$609$	0.828395	0.0335683
$610$	0	0
$611$	−25.6236	−1.03662
$612$	0	0
$613$	47.3135	1.91097	0.955486	−	0.295035i	$-0.0953315\pi$
$613$	47.3135	1.91097	0.955486	+	0.295035i	$0.0953315\pi$
$614$	0	0
$615$	3.24212	0.130735
$616$	0	0
$617$	−7.51762	−0.302648	−0.151324	−	0.988484i	$-0.548354\pi$
$617$	−7.51762	−0.302648	−0.151324	+	0.988484i	$0.548354\pi$
$618$	0	0
$619$	16.1529	0.649239	0.324620	−	0.945845i	$-0.394764\pi$
$619$	16.1529	0.649239	0.324620	+	0.945845i	$0.394764\pi$
$620$	0	0
$621$	20.4139	0.819180
$622$	0	0
$623$	−5.46971	−0.219139
$624$	0	0
$625$	17.4385	0.697539
$626$	0	0
$627$	−10.5808	−0.422555
$628$	0	0
$629$	4.94861	0.197314
$630$	0	0
$631$	39.8976	1.58830	0.794149	−	0.607723i	$-0.207917\pi$
$631$	39.8976	1.58830	0.794149	+	0.607723i	$0.207917\pi$
$632$	0	0
$633$	8.43915	0.335426
$634$	0	0
$635$	8.55127	0.339347
$636$	0	0
$637$	31.1179	1.23294
$638$	0	0
$639$	1.00135	0.0396126
$640$	0	0
$641$	−31.2119	−1.23280	−0.616398	−	0.787435i	$-0.711409\pi$
$641$	−31.2119	−1.23280	−0.616398	+	0.787435i	$0.711409\pi$
$642$	0	0
$643$	15.6614	0.617626	0.308813	−	0.951123i	$-0.400068\pi$
$643$	15.6614	0.617626	0.308813	+	0.951123i	$0.400068\pi$
$644$	0	0
$645$	−3.43116	−0.135102
$646$	0	0
$647$	−33.3617	−1.31158	−0.655792	−	0.754942i	$-0.727665\pi$
$647$	−33.3617	−1.31158	−0.655792	+	0.754942i	$0.727665\pi$
$648$	0	0
$649$	−3.15421	−0.123814
$650$	0	0
$651$	−0.180601	−0.00707830
$652$	0	0
$653$	21.6079	0.845583	0.422791	−	0.906227i	$-0.361050\pi$
$653$	21.6079	0.845583	0.422791	+	0.906227i	$0.361050\pi$
$654$	0	0
$655$	−9.65848	−0.377388
$656$	0	0
$657$	40.7595	1.59018
$658$	0	0
$659$	3.40802	0.132757	0.0663787	−	0.997795i	$-0.478855\pi$
$659$	3.40802	0.132757	0.0663787	+	0.997795i	$0.478855\pi$
$660$	0	0
$661$	−42.0718	−1.63641	−0.818203	−	0.574930i	$-0.805029\pi$
$661$	−42.0718	−1.63641	−0.818203	+	0.574930i	$0.805029\pi$
$662$	0	0
$663$	3.00454	0.116687
$664$	0	0
$665$	3.63909	0.141118
$666$	0	0
$667$	9.33399	0.361414
$668$	0	0
$669$	−15.8744	−0.613741
$670$	0	0
$671$	−6.56548	−0.253458
$672$	0	0
$673$	−24.0914	−0.928657	−0.464328	−	0.885663i	$-0.653704\pi$
$673$	−24.0914	−0.928657	−0.464328	+	0.885663i	$0.653704\pi$
$674$	0	0
$675$	−15.1097	−0.581573
$676$	0	0
$677$	−28.6952	−1.10284	−0.551422	−	0.834226i	$-0.685915\pi$
$677$	−28.6952	−1.10284	−0.551422	+	0.834226i	$0.685915\pi$
$678$	0	0
$679$	2.06493	0.0792448
$680$	0	0
$681$	2.35791	0.0903553
$682$	0	0
$683$	−42.8746	−1.64055	−0.820275	−	0.571969i	$-0.806180\pi$
$683$	−42.8746	−1.64055	−0.820275	+	0.571969i	$0.806180\pi$
$684$	0	0
$685$	9.19857	0.351459
$686$	0	0
$687$	15.2593	0.582177
$688$	0	0
$689$	49.5467	1.88758
$690$	0	0
$691$	31.8288	1.21083	0.605413	−	0.795912i	$-0.293008\pi$
$691$	31.8288	1.21083	0.605413	+	0.795912i	$0.293008\pi$
$692$	0	0
$693$	7.48223	0.284226
$694$	0	0
$695$	4.94101	0.187423
$696$	0	0
$697$	7.50124	0.284130
$698$	0	0
$699$	−6.17014	−0.233376
$700$	0	0
$701$	−32.8338	−1.24011	−0.620057	−	0.784557i	$-0.712891\pi$
$701$	−32.8338	−1.24011	−0.620057	+	0.784557i	$0.712891\pi$
$702$	0	0
$703$	27.7567	1.04686
$704$	0	0
$705$	2.20446	0.0830246
$706$	0	0
$707$	14.4677	0.544115
$708$	0	0
$709$	39.9877	1.50177	0.750885	−	0.660433i	$-0.229627\pi$
$709$	39.9877	1.50177	0.750885	+	0.660433i	$0.229627\pi$
$710$	0	0
$711$	5.95302	0.223256
$712$	0	0
$713$	−2.03493	−0.0762086
$714$	0	0
$715$	11.4520	0.428280
$716$	0	0
$717$	−4.28544	−0.160043
$718$	0	0
$719$	18.0501	0.673156	0.336578	−	0.941656i	$-0.390730\pi$
$719$	18.0501	0.673156	0.336578	+	0.941656i	$0.390730\pi$
$720$	0	0
$721$	4.05201	0.150905
$722$	0	0
$723$	−1.00782	−0.0374814
$724$	0	0
$725$	−6.90874	−0.256584
$726$	0	0
$727$	−43.4084	−1.60993	−0.804964	−	0.593324i	$-0.797815\pi$
$727$	−43.4084	−1.60993	−0.804964	+	0.593324i	$0.797815\pi$
$728$	0	0
$729$	−9.55895	−0.354035
$730$	0	0
$731$	−7.93860	−0.293620
$732$	0	0
$733$	−17.8872	−0.660679	−0.330339	−	0.943862i	$-0.607163\pi$
$733$	−17.8872	−0.660679	−0.330339	+	0.943862i	$0.607163\pi$
$734$	0	0
$735$	−2.67714	−0.0987478
$736$	0	0
$737$	4.75464	0.175139
$738$	0	0
$739$	14.3358	0.527353	0.263676	−	0.964611i	$-0.415065\pi$
$739$	14.3358	0.527353	0.263676	+	0.964611i	$0.415065\pi$
$740$	0	0
$741$	16.8524	0.619089
$742$	0	0
$743$	21.2710	0.780358	0.390179	−	0.920739i	$-0.372413\pi$
$743$	21.2710	0.780358	0.390179	+	0.920739i	$0.372413\pi$
$744$	0	0
$745$	−3.38926	−0.124173
$746$	0	0
$747$	40.3293	1.47557
$748$	0	0
$749$	−13.6809	−0.499888
$750$	0	0
$751$	19.1906	0.700273	0.350137	−	0.936699i	$-0.386135\pi$
$751$	19.1906	0.700273	0.350137	+	0.936699i	$0.386135\pi$
$752$	0	0
$753$	17.5739	0.640427
$754$	0	0
$755$	4.25874	0.154992
$756$	0	0
$757$	18.5624	0.674663	0.337331	−	0.941386i	$-0.390476\pi$
$757$	18.5624	0.674663	0.337331	+	0.941386i	$0.390476\pi$
$758$	0	0
$759$	−11.4119	−0.414225
$760$	0	0
$761$	47.9917	1.73970	0.869850	−	0.493317i	$-0.164216\pi$
$761$	47.9917	1.73970	0.869850	+	0.493317i	$0.164216\pi$
$762$	0	0
$763$	3.01256	0.109062
$764$	0	0
$765$	1.90960	0.0690416
$766$	0	0
$767$	5.02384	0.181400
$768$	0	0
$769$	22.8870	0.825325	0.412663	−	0.910884i	$-0.364599\pi$
$769$	22.8870	0.825325	0.412663	+	0.910884i	$0.364599\pi$
$770$	0	0
$771$	12.8002	0.460987
$772$	0	0
$773$	−20.7966	−0.748002	−0.374001	−	0.927428i	$-0.622014\pi$
$773$	−20.7966	−0.748002	−0.374001	+	0.927428i	$0.622014\pi$
$774$	0	0
$775$	1.50619	0.0541040
$776$	0	0
$777$	2.65692	0.0953165
$778$	0	0
$779$	42.0744	1.50747
$780$	0	0
$781$	−1.19533	−0.0427723
$782$	0	0
$783$	5.20646	0.186064
$784$	0	0
$785$	−10.1859	−0.363549
$786$	0	0
$787$	2.55544	0.0910915	0.0455457	−	0.998962i	$-0.485497\pi$
$787$	2.55544	0.0910915	0.0455457	+	0.998962i	$0.485497\pi$
$788$	0	0
$789$	−0.541226	−0.0192682
$790$	0	0
$791$	10.4525	0.371648
$792$	0	0
$793$	10.4571	0.371343
$794$	0	0
$795$	−4.26261	−0.151179
$796$	0	0
$797$	29.5465	1.04659	0.523296	−	0.852151i	$-0.324702\pi$
$797$	29.5465	1.04659	0.523296	+	0.852151i	$0.324702\pi$
$798$	0	0
$799$	5.10041	0.180439
$800$	0	0
$801$	−16.0989	−0.568828
$802$	0	0
$803$	−48.6555	−1.71702
$804$	0	0
$805$	3.92494	0.138336
$806$	0	0
$807$	6.14022	0.216146
$808$	0	0
$809$	−7.52215	−0.264465	−0.132232	−	0.991219i	$-0.542215\pi$
$809$	−7.52215	−0.264465	−0.132232	+	0.991219i	$0.542215\pi$
$810$	0	0
$811$	−15.1712	−0.532735	−0.266367	−	0.963872i	$-0.585823\pi$
$811$	−15.1712	−0.532735	−0.266367	+	0.963872i	$0.585823\pi$
$812$	0	0
$813$	0.595260	0.0208767
$814$	0	0
$815$	10.2637	0.359521
$816$	0	0
$817$	−44.5276	−1.55782
$818$	0	0
$819$	−11.9173	−0.416423
$820$	0	0
$821$	2.95245	0.103041	0.0515206	−	0.998672i	$-0.483593\pi$
$821$	2.95245	0.103041	0.0515206	+	0.998672i	$0.483593\pi$
$822$	0	0
$823$	36.6442	1.27734	0.638669	−	0.769482i	$-0.279485\pi$
$823$	36.6442	1.27734	0.638669	+	0.769482i	$0.279485\pi$
$824$	0	0
$825$	8.44672	0.294077
$826$	0	0
$827$	37.4921	1.30373	0.651863	−	0.758336i	$-0.273988\pi$
$827$	37.4921	1.30373	0.651863	+	0.758336i	$0.273988\pi$
$828$	0	0
$829$	47.1918	1.63904	0.819520	−	0.573050i	$-0.194240\pi$
$829$	47.1918	1.63904	0.819520	+	0.573050i	$0.194240\pi$
$830$	0	0
$831$	−17.4268	−0.604528
$832$	0	0
$833$	−6.19405	−0.214611
$834$	0	0
$835$	15.6742	0.542428
$836$	0	0
$837$	−1.13507	−0.0392339
$838$	0	0
$839$	21.4573	0.740789	0.370395	−	0.928874i	$-0.379222\pi$
$839$	21.4573	0.740789	0.370395	+	0.928874i	$0.379222\pi$
$840$	0	0
$841$	−26.6194	−0.917911
$842$	0	0
$843$	−12.0325	−0.414420
$844$	0	0
$845$	−8.84505	−0.304279
$846$	0	0
$847$	0.943500	0.0324191
$848$	0	0
$849$	19.8909	0.682655
$850$	0	0
$851$	29.9370	1.02623
$852$	0	0
$853$	−10.2314	−0.350318	−0.175159	−	0.984540i	$-0.556044\pi$
$853$	−10.2314	−0.350318	−0.175159	+	0.984540i	$0.556044\pi$
$854$	0	0
$855$	10.7109	0.366306
$856$	0	0
$857$	−47.6356	−1.62720	−0.813601	−	0.581424i	$-0.802496\pi$
$857$	−47.6356	−1.62720	−0.813601	+	0.581424i	$0.802496\pi$
$858$	0	0
$859$	−19.8896	−0.678625	−0.339312	−	0.940674i	$-0.610194\pi$
$859$	−19.8896	−0.678625	−0.339312	+	0.940674i	$0.610194\pi$
$860$	0	0
$861$	4.02744	0.137255
$862$	0	0
$863$	42.0187	1.43033	0.715166	−	0.698954i	$-0.246351\pi$
$863$	42.0187	1.43033	0.715166	+	0.698954i	$0.246351\pi$
$864$	0	0
$865$	−15.5665	−0.529277
$866$	0	0
$867$	−0.598056	−0.0203110
$868$	0	0
$869$	−7.10625	−0.241063
$870$	0	0
$871$	−7.57291	−0.256598
$872$	0	0
$873$	6.07769	0.205699
$874$	0	0
$875$	−6.14911	−0.207878
$876$	0	0
$877$	14.0085	0.473034	0.236517	−	0.971627i	$-0.423994\pi$
$877$	14.0085	0.473034	0.236517	+	0.971627i	$0.423994\pi$
$878$	0	0
$879$	−0.00896079	−0.000302240 0
$880$	0	0
$881$	18.4449	0.621423	0.310712	−	0.950504i	$-0.399433\pi$
$881$	18.4449	0.621423	0.310712	+	0.950504i	$0.399433\pi$
$882$	0	0
$883$	49.1476	1.65395	0.826974	−	0.562240i	$-0.190060\pi$
$883$	49.1476	1.65395	0.826974	+	0.562240i	$0.190060\pi$
$884$	0	0
$885$	−0.432212	−0.0145286
$886$	0	0
$887$	−14.8476	−0.498535	−0.249268	−	0.968435i	$-0.580190\pi$
$887$	−14.8476	−0.498535	−0.249268	+	0.968435i	$0.580190\pi$
$888$	0	0
$889$	10.6226	0.356270
$890$	0	0
$891$	18.6379	0.624393
$892$	0	0
$893$	28.6081	0.957335
$894$	0	0
$895$	15.7612	0.526840
$896$	0	0
$897$	18.1762	0.606885
$898$	0	0
$899$	−0.518999	−0.0173096
$900$	0	0
$901$	−9.86232	−0.328561
$902$	0	0
$903$	−4.26225	−0.141839
$904$	0	0
$905$	−2.46965	−0.0820939
$906$	0	0
$907$	36.7497	1.22025	0.610127	−	0.792303i	$-0.291118\pi$
$907$	36.7497	1.22025	0.610127	+	0.792303i	$0.291118\pi$
$908$	0	0
$909$	42.5827	1.41238
$910$	0	0
$911$	−50.0989	−1.65985	−0.829926	−	0.557874i	$-0.811617\pi$
$911$	−50.0989	−1.65985	−0.829926	+	0.557874i	$0.811617\pi$
$912$	0	0
$913$	−48.1420	−1.59327
$914$	0	0
$915$	−0.899648	−0.0297414
$916$	0	0
$917$	−11.9980	−0.396208
$918$	0	0
$919$	−40.8477	−1.34744	−0.673721	−	0.738986i	$-0.735305\pi$
$919$	−40.8477	−1.34744	−0.673721	+	0.738986i	$0.735305\pi$
$920$	0	0
$921$	8.74127	0.288035
$922$	0	0
$923$	1.90385	0.0626660
$924$	0	0
$925$	−22.1585	−0.728565
$926$	0	0
$927$	11.9263	0.391710
$928$	0	0
$929$	19.8271	0.650507	0.325253	−	0.945627i	$-0.394550\pi$
$929$	19.8271	0.650507	0.325253	+	0.945627i	$0.394550\pi$
$930$	0	0
$931$	−34.7424	−1.13864
$932$	0	0
$933$	7.51432	0.246008
$934$	0	0
$935$	−2.27953	−0.0745486
$936$	0	0
$937$	36.8080	1.20246	0.601232	−	0.799074i	$-0.294677\pi$
$937$	36.8080	1.20246	0.601232	+	0.799074i	$0.294677\pi$
$938$	0	0
$939$	19.4023	0.633169
$940$	0	0
$941$	29.9981	0.977909	0.488955	−	0.872309i	$-0.337378\pi$
$941$	29.9981	0.977909	0.488955	+	0.872309i	$0.337378\pi$
$942$	0	0
$943$	45.3794	1.47776
$944$	0	0
$945$	2.18932	0.0712185
$946$	0	0
$947$	10.7725	0.350060	0.175030	−	0.984563i	$-0.443998\pi$
$947$	10.7725	0.350060	0.175030	+	0.984563i	$0.443998\pi$
$948$	0	0
$949$	77.4957	2.51562
$950$	0	0
$951$	18.1369	0.588129
$952$	0	0
$953$	−27.5821	−0.893471	−0.446735	−	0.894666i	$-0.647413\pi$
$953$	−27.5821	−0.893471	−0.446735	+	0.894666i	$0.647413\pi$
$954$	0	0
$955$	−6.52668	−0.211198
$956$	0	0
$957$	−2.91055	−0.0940846
$958$	0	0
$959$	11.4267	0.368986
$960$	0	0
$961$	−30.8869	−0.996350
$962$	0	0
$963$	−40.2668	−1.29758
$964$	0	0
$965$	−5.96594	−0.192050
$966$	0	0
$967$	−16.4127	−0.527798	−0.263899	−	0.964550i	$-0.585009\pi$
$967$	−16.4127	−0.527798	−0.263899	+	0.964550i	$0.585009\pi$
$968$	0	0
$969$	−3.35449	−0.107762
$970$	0	0
$971$	15.5512	0.499062	0.249531	−	0.968367i	$-0.419724\pi$
$971$	15.5512	0.499062	0.249531	+	0.968367i	$0.419724\pi$
$972$	0	0
$973$	6.13782	0.196770
$974$	0	0
$975$	−13.4535	−0.430855
$976$	0	0
$977$	−0.269895	−0.00863471	−0.00431736	−	0.999991i	$-0.501374\pi$
$977$	−0.269895	−0.00863471	−0.00431736	+	0.999991i	$0.501374\pi$
$978$	0	0
$979$	19.2177	0.614200
$980$	0	0
$981$	8.86685	0.283097
$982$	0	0
$983$	−57.1626	−1.82320	−0.911602	−	0.411074i	$-0.865154\pi$
$983$	−57.1626	−1.82320	−0.911602	+	0.411074i	$0.865154\pi$
$984$	0	0
$985$	4.51590	0.143889
$986$	0	0
$987$	2.73842	0.0871649
$988$	0	0
$989$	−48.0252	−1.52711
$990$	0	0
$991$	−7.54039	−0.239528	−0.119764	−	0.992802i	$-0.538214\pi$
$991$	−7.54039	−0.239528	−0.119764	+	0.992802i	$0.538214\pi$
$992$	0	0
$993$	−4.00091	−0.126965
$994$	0	0
$995$	−7.94736	−0.251948
$996$	0	0
$997$	21.8260	0.691237	0.345619	−	0.938375i	$-0.387669\pi$
$997$	21.8260	0.691237	0.345619	+	0.938375i	$0.387669\pi$
$998$	0	0
$999$	16.6987	0.528324

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4012.2.a.h.1.6	✓	15

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4012.2.a.h.1.6	✓	15	1.1	even	1	trivial

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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:	\( N \)	\(=\)	\( 4012 = 2^{2} \cdot 17 \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4012.a (trivial)

\(f(q)\)	\(=\)	\(q-0.598056 q^{3} -0.722695 q^{5} -0.897746 q^{7} -2.64233 q^{9} +O(q^{10})\)	\(q-0.598056 q^{3} -0.722695 q^{5} -0.897746 q^{7} -2.64233 q^{9} +3.15421 q^{11} -5.02384 q^{13} +0.432212 q^{15} +1.00000 q^{17} +5.60899 q^{19} +0.536902 q^{21} +6.04958 q^{23} -4.47771 q^{25} +3.37443 q^{27} +1.54292 q^{29} -0.336375 q^{31} -1.88639 q^{33} +0.648796 q^{35} +4.94861 q^{37} +3.00454 q^{39} +7.50124 q^{41} -7.93860 q^{43} +1.90960 q^{45} +5.10041 q^{47} -6.19405 q^{49} -0.598056 q^{51} -9.86232 q^{53} -2.27953 q^{55} -3.35449 q^{57} -1.00000 q^{59} -2.08150 q^{61} +2.37214 q^{63} +3.63070 q^{65} +1.50739 q^{67} -3.61799 q^{69} -0.378963 q^{71} -15.4256 q^{73} +2.67792 q^{75} -2.83168 q^{77} -2.25294 q^{79} +5.90889 q^{81} -15.2628 q^{83} -0.722695 q^{85} -0.922750 q^{87} +6.09271 q^{89} +4.51014 q^{91} +0.201171 q^{93} -4.05359 q^{95} -2.30013 q^{97} -8.33446 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 15 q - q^{3} + q^{5} - 11 q^{7} + 16 q^{9}+O(q^{10}) \) 15 * q - q^3 + q^5 - 11 * q^7 + 16 * q^9	\( 15 q - q^{3} + q^{5} - 11 q^{7} + 16 q^{9} - 12 q^{11} - 10 q^{13} - 13 q^{15} + 15 q^{17} - 4 q^{21} - 21 q^{23} + 4 q^{25} - 37 q^{27} + 23 q^{29} - 31 q^{31} - 11 q^{33} - 35 q^{35} - 10 q^{37} - 2 q^{39} - 15 q^{41} - 3 q^{43} + 15 q^{45} - 47 q^{47} + 18 q^{49} - q^{51} - 7 q^{53} - 20 q^{55} - 30 q^{57} - 15 q^{59} + q^{61} + 19 q^{63} + 11 q^{65} - 20 q^{67} - 24 q^{69} - 13 q^{71} - 24 q^{73} - 10 q^{75} + 21 q^{77} - 34 q^{79} - 21 q^{81} - 40 q^{83} + q^{85} - 60 q^{87} - 46 q^{89} - 19 q^{91} + 33 q^{93} - 4 q^{95} - 5 q^{97} - 74 q^{99}+O(q^{100}) \) 15 * q - q^3 + q^5 - 11 * q^7 + 16 * q^9 - 12 * q^11 - 10 * q^13 - 13 * q^15 + 15 * q^17 - 4 * q^21 - 21 * q^23 + 4 * q^25 - 37 * q^27 + 23 * q^29 - 31 * q^31 - 11 * q^33 - 35 * q^35 - 10 * q^37 - 2 * q^39 - 15 * q^41 - 3 * q^43 + 15 * q^45 - 47 * q^47 + 18 * q^49 - q^51 - 7 * q^53 - 20 * q^55 - 30 * q^57 - 15 * q^59 + q^61 + 19 * q^63 + 11 * q^65 - 20 * q^67 - 24 * q^69 - 13 * q^71 - 24 * q^73 - 10 * q^75 + 21 * q^77 - 34 * q^79 - 21 * q^81 - 40 * q^83 + q^85 - 60 * q^87 - 46 * q^89 - 19 * q^91 + 33 * q^93 - 4 * q^95 - 5 * q^97 - 74 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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