LMFDB - Embedded newform 4012.2.a.h.1.1

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			1.1
Root			$1.28051$ 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)

$f(q)$	$=$	$q-3.40908 q^{3} +1.59062 q^{5} +2.94577 q^{7} +8.62180 q^{9} +O(q^{10})$	$q-3.40908 q^{3} +1.59062 q^{5} +2.94577 q^{7} +8.62180 q^{9} -4.17423 q^{11} +2.42836 q^{13} -5.42255 q^{15} +1.00000 q^{17} +2.92275 q^{19} -10.0424 q^{21} -5.49621 q^{23} -2.46993 q^{25} -19.1652 q^{27} +6.57198 q^{29} -9.00599 q^{31} +14.2303 q^{33} +4.68561 q^{35} -9.65198 q^{37} -8.27846 q^{39} -6.88635 q^{41} +8.59296 q^{43} +13.7140 q^{45} -9.92863 q^{47} +1.67758 q^{49} -3.40908 q^{51} +7.14948 q^{53} -6.63962 q^{55} -9.96389 q^{57} -1.00000 q^{59} +2.68573 q^{61} +25.3979 q^{63} +3.86260 q^{65} -12.4960 q^{67} +18.7370 q^{69} -2.57055 q^{71} -14.5249 q^{73} +8.42017 q^{75} -12.2963 q^{77} +5.62340 q^{79} +39.4701 q^{81} +11.7633 q^{83} +1.59062 q^{85} -22.4044 q^{87} +3.27169 q^{89} +7.15339 q^{91} +30.7021 q^{93} +4.64899 q^{95} +5.72240 q^{97} -35.9894 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−3.40908	−1.96823	−0.984116	−	0.177528i	$-0.943190\pi$
$3$	−3.40908	−1.96823	−0.984116	+	0.177528i	$0.943190\pi$
$4$	0	0
$5$	1.59062	0.711347	0.355674	−	0.934610i	$-0.384251\pi$
$5$	1.59062	0.711347	0.355674	+	0.934610i	$0.384251\pi$
$6$	0	0
$7$	2.94577	1.11340	0.556699	−	0.830714i	$-0.312068\pi$
$7$	2.94577	1.11340	0.556699	+	0.830714i	$0.312068\pi$
$8$	0	0
$9$	8.62180	2.87393
$10$	0	0
$11$	−4.17423	−1.25858	−0.629289	−	0.777171i	$-0.716654\pi$
$11$	−4.17423	−1.25858	−0.629289	+	0.777171i	$0.716654\pi$
$12$	0	0
$13$	2.42836	0.673505	0.336753	−	0.941593i	$-0.390671\pi$
$13$	2.42836	0.673505	0.336753	+	0.941593i	$0.390671\pi$
$14$	0	0
$15$	−5.42255	−1.40010
$16$	0	0
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	0	0
$19$	2.92275	0.670526	0.335263	−	0.942125i	$-0.391175\pi$
$19$	2.92275	0.670526	0.335263	+	0.942125i	$0.391175\pi$
$20$	0	0
$21$	−10.0424	−2.19142
$22$	0	0
$23$	−5.49621	−1.14604	−0.573020	−	0.819541i	$-0.694228\pi$
$23$	−5.49621	−1.14604	−0.573020	+	0.819541i	$0.694228\pi$
$24$	0	0
$25$	−2.46993	−0.493985
$26$	0	0
$27$	−19.1652	−3.68834
$28$	0	0
$29$	6.57198	1.22039	0.610193	−	0.792253i	$-0.291092\pi$
$29$	6.57198	1.22039	0.610193	+	0.792253i	$0.291092\pi$
$30$	0	0
$31$	−9.00599	−1.61752	−0.808762	−	0.588137i	$-0.799862\pi$
$31$	−9.00599	−1.61752	−0.808762	+	0.588137i	$0.799862\pi$
$32$	0	0
$33$	14.2303	2.47717
$34$	0	0
$35$	4.68561	0.792012
$36$	0	0
$37$	−9.65198	−1.58678	−0.793388	−	0.608717i	$-0.791685\pi$
$37$	−9.65198	−1.58678	−0.793388	+	0.608717i	$0.791685\pi$
$38$	0	0
$39$	−8.27846	−1.32561
$40$	0	0
$41$	−6.88635	−1.07547	−0.537734	−	0.843115i	$-0.680719\pi$
$41$	−6.88635	−1.07547	−0.537734	+	0.843115i	$0.680719\pi$
$42$	0	0
$43$	8.59296	1.31041	0.655207	−	0.755449i	$-0.272581\pi$
$43$	8.59296	1.31041	0.655207	+	0.755449i	$0.272581\pi$
$44$	0	0
$45$	13.7140	2.04437
$46$	0	0
$47$	−9.92863	−1.44824	−0.724120	−	0.689674i	$-0.757754\pi$
$47$	−9.92863	−1.44824	−0.724120	+	0.689674i	$0.757754\pi$
$48$	0	0
$49$	1.67758	0.239655
$50$	0	0
$51$	−3.40908	−0.477366
$52$	0	0
$53$	7.14948	0.982056	0.491028	−	0.871144i	$-0.336621\pi$
$53$	7.14948	0.982056	0.491028	+	0.871144i	$0.336621\pi$
$54$	0	0
$55$	−6.63962	−0.895286
$56$	0	0
$57$	−9.96389	−1.31975
$58$	0	0
$59$	−1.00000	−0.130189
$59$	−1.00000	−0.130189
$60$	0	0
$61$	2.68573	0.343873	0.171936	−	0.985108i	$-0.444998\pi$
$61$	2.68573	0.343873	0.171936	+	0.985108i	$0.444998\pi$
$62$	0	0
$63$	25.3979	3.19983
$64$	0	0
$65$	3.86260	0.479096
$66$	0	0
$67$	−12.4960	−1.52664	−0.763318	−	0.646023i	$-0.776431\pi$
$67$	−12.4960	−1.52664	−0.763318	+	0.646023i	$0.776431\pi$
$68$	0	0
$69$	18.7370	2.25567
$70$	0	0
$71$	−2.57055	−0.305068	−0.152534	−	0.988298i	$-0.548743\pi$
$71$	−2.57055	−0.305068	−0.152534	+	0.988298i	$0.548743\pi$
$72$	0	0
$73$	−14.5249	−1.70001	−0.850007	−	0.526772i	$-0.823402\pi$
$73$	−14.5249	−1.70001	−0.850007	+	0.526772i	$0.823402\pi$
$74$	0	0
$75$	8.42017	0.972277
$76$	0	0
$77$	−12.2963	−1.40130
$78$	0	0
$79$	5.62340	0.632681	0.316341	−	0.948646i	$-0.397546\pi$
$79$	5.62340	0.632681	0.316341	+	0.948646i	$0.397546\pi$
$80$	0	0
$81$	39.4701	4.38557
$82$	0	0
$83$	11.7633	1.29119	0.645595	−	0.763680i	$-0.276609\pi$
$83$	11.7633	1.29119	0.645595	+	0.763680i	$0.276609\pi$
$84$	0	0
$85$	1.59062	0.172527
$86$	0	0
$87$	−22.4044	−2.40200
$88$	0	0
$89$	3.27169	0.346799	0.173399	−	0.984852i	$-0.444525\pi$
$89$	3.27169	0.346799	0.173399	+	0.984852i	$0.444525\pi$
$90$	0	0
$91$	7.15339	0.749879
$92$	0	0
$93$	30.7021	3.18366
$94$	0	0
$95$	4.64899	0.476977
$96$	0	0
$97$	5.72240	0.581022	0.290511	−	0.956872i	$-0.406175\pi$
$97$	5.72240	0.581022	0.290511	+	0.956872i	$0.406175\pi$
$98$	0	0
$99$	−35.9894	−3.61707
$100$	0	0
$101$	−8.87600	−0.883195	−0.441597	−	0.897213i	$-0.645588\pi$
$101$	−8.87600	−0.883195	−0.441597	+	0.897213i	$0.645588\pi$
$102$	0	0
$103$	16.3111	1.60718	0.803590	−	0.595184i	$-0.202921\pi$
$103$	16.3111	1.60718	0.803590	+	0.595184i	$0.202921\pi$
$104$	0	0
$105$	−15.9736	−1.55886
$106$	0	0
$107$	6.78597	0.656024	0.328012	−	0.944674i	$-0.393621\pi$
$107$	6.78597	0.656024	0.328012	+	0.944674i	$0.393621\pi$
$108$	0	0
$109$	14.5931	1.39777	0.698883	−	0.715236i	$-0.253681\pi$
$109$	14.5931	1.39777	0.698883	+	0.715236i	$0.253681\pi$
$110$	0	0
$111$	32.9043	3.12314
$112$	0	0
$113$	−14.9549	−1.40684	−0.703419	−	0.710776i	$-0.748344\pi$
$113$	−14.9549	−1.40684	−0.703419	+	0.710776i	$0.748344\pi$
$114$	0	0
$115$	−8.74239	−0.815232
$116$	0	0
$117$	20.9368	1.93561
$118$	0	0
$119$	2.94577	0.270039
$120$	0	0
$121$	6.42422	0.584020
$122$	0	0
$123$	23.4761	2.11677
$124$	0	0
$125$	−11.8818	−1.06274
$126$	0	0
$127$	1.28605	0.114118	0.0570592	−	0.998371i	$-0.481828\pi$
$127$	1.28605	0.114118	0.0570592	+	0.998371i	$0.481828\pi$
$128$	0	0
$129$	−29.2941	−2.57920
$130$	0	0
$131$	−4.83652	−0.422569	−0.211284	−	0.977425i	$-0.567765\pi$
$131$	−4.83652	−0.422569	−0.211284	+	0.977425i	$0.567765\pi$
$132$	0	0
$133$	8.60977	0.746562
$134$	0	0
$135$	−30.4845	−2.62369
$136$	0	0
$137$	−14.8237	−1.26647	−0.633237	−	0.773958i	$-0.718274\pi$
$137$	−14.8237	−1.26647	−0.633237	+	0.773958i	$0.718274\pi$
$138$	0	0
$139$	5.91536	0.501734	0.250867	−	0.968022i	$-0.419284\pi$
$139$	5.91536	0.501734	0.250867	+	0.968022i	$0.419284\pi$
$140$	0	0
$141$	33.8475	2.85047
$142$	0	0
$143$	−10.1365	−0.847659
$144$	0	0
$145$	10.4535	0.868118
$146$	0	0
$147$	−5.71902	−0.471696
$148$	0	0
$149$	−12.1293	−0.993671	−0.496836	−	0.867845i	$-0.665505\pi$
$149$	−12.1293	−0.993671	−0.496836	+	0.867845i	$0.665505\pi$
$150$	0	0
$151$	−11.0316	−0.897740	−0.448870	−	0.893597i	$-0.648173\pi$
$151$	−11.0316	−0.897740	−0.448870	+	0.893597i	$0.648173\pi$
$152$	0	0
$153$	8.62180	0.697032
$154$	0	0
$155$	−14.3251	−1.15062
$156$	0	0
$157$	3.64832	0.291168	0.145584	−	0.989346i	$-0.453494\pi$
$157$	3.64832	0.291168	0.145584	+	0.989346i	$0.453494\pi$
$158$	0	0
$159$	−24.3731	−1.93291
$160$	0	0
$161$	−16.1906	−1.27600
$162$	0	0
$163$	−15.4775	−1.21229	−0.606146	−	0.795354i	$-0.707285\pi$
$163$	−15.4775	−1.21229	−0.606146	+	0.795354i	$0.707285\pi$
$164$	0	0
$165$	22.6350	1.76213
$166$	0	0
$167$	−2.02052	−0.156353	−0.0781763	−	0.996940i	$-0.524910\pi$
$167$	−2.02052	−0.156353	−0.0781763	+	0.996940i	$0.524910\pi$
$168$	0	0
$169$	−7.10308	−0.546391
$170$	0	0
$171$	25.1994	1.92705
$172$	0	0
$173$	−7.55138	−0.574120	−0.287060	−	0.957913i	$-0.592678\pi$
$173$	−7.55138	−0.574120	−0.287060	+	0.957913i	$0.592678\pi$
$174$	0	0
$175$	−7.27584	−0.550002
$176$	0	0
$177$	3.40908	0.256242
$178$	0	0
$179$	−5.50033	−0.411114	−0.205557	−	0.978645i	$-0.565901\pi$
$179$	−5.50033	−0.411114	−0.205557	+	0.978645i	$0.565901\pi$
$180$	0	0
$181$	−4.40123	−0.327141	−0.163570	−	0.986532i	$-0.552301\pi$
$181$	−4.40123	−0.327141	−0.163570	+	0.986532i	$0.552301\pi$
$182$	0	0
$183$	−9.15587	−0.676822
$184$	0	0
$185$	−15.3526	−1.12875
$186$	0	0
$187$	−4.17423	−0.305250
$188$	0	0
$189$	−56.4562	−4.10659
$190$	0	0
$191$	−23.6314	−1.70991	−0.854954	−	0.518703i	$-0.826415\pi$
$191$	−23.6314	−1.70991	−0.854954	+	0.518703i	$0.826415\pi$
$192$	0	0
$193$	25.2551	1.81790	0.908950	−	0.416905i	$-0.136885\pi$
$193$	25.2551	1.81790	0.908950	+	0.416905i	$0.136885\pi$
$194$	0	0
$195$	−13.1679	−0.942972
$196$	0	0
$197$	4.34112	0.309292	0.154646	−	0.987970i	$-0.450576\pi$
$197$	4.34112	0.309292	0.154646	+	0.987970i	$0.450576\pi$
$198$	0	0
$199$	20.0389	1.42052	0.710259	−	0.703940i	$-0.248578\pi$
$199$	20.0389	1.42052	0.710259	+	0.703940i	$0.248578\pi$
$200$	0	0
$201$	42.6000	3.00477
$202$	0	0
$203$	19.3596	1.35878
$204$	0	0
$205$	−10.9536	−0.765031
$206$	0	0
$207$	−47.3873	−3.29364
$208$	0	0
$209$	−12.2003	−0.843909
$210$	0	0
$211$	−8.53895	−0.587846	−0.293923	−	0.955829i	$-0.594961\pi$
$211$	−8.53895	−0.587846	−0.293923	+	0.955829i	$0.594961\pi$
$212$	0	0
$213$	8.76320	0.600445
$214$	0	0
$215$	13.6681	0.932159
$216$	0	0
$217$	−26.5296	−1.80095
$218$	0	0
$219$	49.5166	3.34602
$220$	0	0
$221$	2.42836	0.163349
$222$	0	0
$223$	−18.1338	−1.21433	−0.607164	−	0.794577i	$-0.707693\pi$
$223$	−18.1338	−1.21433	−0.607164	+	0.794577i	$0.707693\pi$
$224$	0	0
$225$	−21.2952	−1.41968
$226$	0	0
$227$	−17.7022	−1.17493	−0.587467	−	0.809248i	$-0.699875\pi$
$227$	−17.7022	−1.17493	−0.587467	+	0.809248i	$0.699875\pi$
$228$	0	0
$229$	20.1316	1.33034	0.665168	−	0.746694i	$-0.268360\pi$
$229$	20.1316	1.33034	0.665168	+	0.746694i	$0.268360\pi$
$230$	0	0
$231$	41.9192	2.75808
$232$	0	0
$233$	−16.8272	−1.10239	−0.551193	−	0.834378i	$-0.685827\pi$
$233$	−16.8272	−1.10239	−0.551193	+	0.834378i	$0.685827\pi$
$234$	0	0
$235$	−15.7927	−1.03020
$236$	0	0
$237$	−19.1706	−1.24526
$238$	0	0
$239$	16.1020	1.04155	0.520776	−	0.853693i	$-0.325643\pi$
$239$	16.1020	1.04155	0.520776	+	0.853693i	$0.325643\pi$
$240$	0	0
$241$	−16.9320	−1.09068	−0.545341	−	0.838214i	$-0.683600\pi$
$241$	−16.9320	−1.09068	−0.545341	+	0.838214i	$0.683600\pi$
$242$	0	0
$243$	−77.0611	−4.94347
$244$	0	0
$245$	2.66840	0.170478
$246$	0	0
$247$	7.09749	0.451603
$248$	0	0
$249$	−40.1020	−2.54136
$250$	0	0
$251$	7.27674	0.459304	0.229652	−	0.973273i	$-0.426241\pi$
$251$	7.27674	0.459304	0.229652	+	0.973273i	$0.426241\pi$
$252$	0	0
$253$	22.9425	1.44238
$254$	0	0
$255$	−5.42255	−0.339573
$256$	0	0
$257$	2.10165	0.131098	0.0655488	−	0.997849i	$-0.479120\pi$
$257$	2.10165	0.131098	0.0655488	+	0.997849i	$0.479120\pi$
$258$	0	0
$259$	−28.4325	−1.76671
$260$	0	0
$261$	56.6623	3.50731
$262$	0	0
$263$	3.69630	0.227924	0.113962	−	0.993485i	$-0.463646\pi$
$263$	3.69630	0.227924	0.113962	+	0.993485i	$0.463646\pi$
$264$	0	0
$265$	11.3721	0.698583
$266$	0	0
$267$	−11.1535	−0.682580
$268$	0	0
$269$	13.0959	0.798472	0.399236	−	0.916848i	$-0.369275\pi$
$269$	13.0959	0.798472	0.399236	+	0.916848i	$0.369275\pi$
$270$	0	0
$271$	−12.7015	−0.771559	−0.385779	−	0.922591i	$-0.626067\pi$
$271$	−12.7015	−0.771559	−0.385779	+	0.922591i	$0.626067\pi$
$272$	0	0
$273$	−24.3865	−1.47594
$274$	0	0
$275$	10.3100	0.621719
$276$	0	0
$277$	−3.18601	−0.191429	−0.0957143	−	0.995409i	$-0.530514\pi$
$277$	−3.18601	−0.191429	−0.0957143	+	0.995409i	$0.530514\pi$
$278$	0	0
$279$	−77.6479	−4.64866
$280$	0	0
$281$	−11.7913	−0.703412	−0.351706	−	0.936111i	$-0.614398\pi$
$281$	−11.7913	−0.703412	−0.351706	+	0.936111i	$0.614398\pi$
$282$	0	0
$283$	−21.1924	−1.25976	−0.629880	−	0.776693i	$-0.716896\pi$
$283$	−21.1924	−1.25976	−0.629880	+	0.776693i	$0.716896\pi$
$284$	0	0
$285$	−15.8488	−0.938801
$286$	0	0
$287$	−20.2856	−1.19742
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	−19.5081	−1.14359
$292$	0	0
$293$	17.1741	1.00332	0.501662	−	0.865064i	$-0.332722\pi$
$293$	17.1741	1.00332	0.501662	+	0.865064i	$0.332722\pi$
$294$	0	0
$295$	−1.59062	−0.0926095
$296$	0	0
$297$	79.9999	4.64206
$298$	0	0
$299$	−13.3468	−0.771864
$300$	0	0
$301$	25.3129	1.45901
$302$	0	0
$303$	30.2590	1.73833
$304$	0	0
$305$	4.27198	0.244613
$306$	0	0
$307$	23.8735	1.36253	0.681267	−	0.732035i	$-0.261429\pi$
$307$	23.8735	1.36253	0.681267	+	0.732035i	$0.261429\pi$
$308$	0	0
$309$	−55.6058	−3.16330
$310$	0	0
$311$	−28.2306	−1.60081	−0.800405	−	0.599459i	$-0.795382\pi$
$311$	−28.2306	−1.60081	−0.800405	+	0.599459i	$0.795382\pi$
$312$	0	0
$313$	13.2718	0.750169	0.375084	−	0.926991i	$-0.377614\pi$
$313$	13.2718	0.750169	0.375084	+	0.926991i	$0.377614\pi$
$314$	0	0
$315$	40.3984	2.27619
$316$	0	0
$317$	−18.6354	−1.04667	−0.523333	−	0.852128i	$-0.675312\pi$
$317$	−18.6354	−1.04667	−0.523333	+	0.852128i	$0.675312\pi$
$318$	0	0
$319$	−27.4330	−1.53595
$320$	0	0
$321$	−23.1339	−1.29121
$322$	0	0
$323$	2.92275	0.162626
$324$	0	0
$325$	−5.99786	−0.332702
$326$	0	0
$327$	−49.7490	−2.75113
$328$	0	0
$329$	−29.2475	−1.61247
$330$	0	0
$331$	15.7755	0.867101	0.433551	−	0.901129i	$-0.357261\pi$
$331$	15.7755	0.867101	0.433551	+	0.901129i	$0.357261\pi$
$332$	0	0
$333$	−83.2175	−4.56029
$334$	0	0
$335$	−19.8765	−1.08597
$336$	0	0
$337$	14.6987	0.800690	0.400345	−	0.916364i	$-0.368890\pi$
$337$	14.6987	0.800690	0.400345	+	0.916364i	$0.368890\pi$
$338$	0	0
$339$	50.9823	2.76898
$340$	0	0
$341$	37.5931	2.03578
$342$	0	0
$343$	−15.6786	−0.846567
$344$	0	0
$345$	29.8035	1.60457
$346$	0	0
$347$	8.03339	0.431255	0.215628	−	0.976476i	$-0.430820\pi$
$347$	8.03339	0.431255	0.215628	+	0.976476i	$0.430820\pi$
$348$	0	0
$349$	−13.0860	−0.700477	−0.350239	−	0.936660i	$-0.613900\pi$
$349$	−13.0860	−0.700477	−0.350239	+	0.936660i	$0.613900\pi$
$350$	0	0
$351$	−46.5399	−2.48411
$352$	0	0
$353$	−28.0347	−1.49214	−0.746068	−	0.665870i	$-0.768061\pi$
$353$	−28.0347	−1.49214	−0.746068	+	0.665870i	$0.768061\pi$
$354$	0	0
$355$	−4.08877	−0.217009
$356$	0	0
$357$	−10.0424	−0.531499
$358$	0	0
$359$	−13.7340	−0.724852	−0.362426	−	0.932013i	$-0.618051\pi$
$359$	−13.7340	−0.724852	−0.362426	+	0.932013i	$0.618051\pi$
$360$	0	0
$361$	−10.4575	−0.550395
$362$	0	0
$363$	−21.9007	−1.14949
$364$	0	0
$365$	−23.1036	−1.20930
$366$	0	0
$367$	0.538662	0.0281179	0.0140590	−	0.999901i	$-0.495525\pi$
$367$	0.538662	0.0281179	0.0140590	+	0.999901i	$0.495525\pi$
$368$	0	0
$369$	−59.3728	−3.09082
$370$	0	0
$371$	21.0608	1.09342
$372$	0	0
$373$	34.0974	1.76550	0.882748	−	0.469846i	$-0.155691\pi$
$373$	34.0974	1.76550	0.882748	+	0.469846i	$0.155691\pi$
$374$	0	0
$375$	40.5060	2.09172
$376$	0	0
$377$	15.9591	0.821936
$378$	0	0
$379$	20.5286	1.05448	0.527241	−	0.849716i	$-0.323227\pi$
$379$	20.5286	1.05448	0.527241	+	0.849716i	$0.323227\pi$
$380$	0	0
$381$	−4.38424	−0.224612
$382$	0	0
$383$	−28.1459	−1.43819	−0.719095	−	0.694912i	$-0.755443\pi$
$383$	−28.1459	−1.43819	−0.719095	+	0.694912i	$0.755443\pi$
$384$	0	0
$385$	−19.5588	−0.996810
$386$	0	0
$387$	74.0868	3.76604
$388$	0	0
$389$	−29.2411	−1.48258	−0.741292	−	0.671182i	$-0.765787\pi$
$389$	−29.2411	−1.48258	−0.741292	+	0.671182i	$0.765787\pi$
$390$	0	0
$391$	−5.49621	−0.277955
$392$	0	0
$393$	16.4881	0.831713
$394$	0	0
$395$	8.94469	0.450056
$396$	0	0
$397$	−13.1613	−0.660546	−0.330273	−	0.943885i	$-0.607141\pi$
$397$	−13.1613	−0.660546	−0.330273	+	0.943885i	$0.607141\pi$
$398$	0	0
$399$	−29.3514	−1.46941
$400$	0	0
$401$	14.9015	0.744147	0.372074	−	0.928203i	$-0.378647\pi$
$401$	14.9015	0.744147	0.372074	+	0.928203i	$0.378647\pi$
$402$	0	0
$403$	−21.8698	−1.08941
$404$	0	0
$405$	62.7820	3.11966
$406$	0	0
$407$	40.2896	1.99708
$408$	0	0
$409$	−12.1960	−0.603054	−0.301527	−	0.953458i	$-0.597496\pi$
$409$	−12.1960	−0.603054	−0.301527	+	0.953458i	$0.597496\pi$
$410$	0	0
$411$	50.5352	2.49272
$412$	0	0
$413$	−2.94577	−0.144952
$414$	0	0
$415$	18.7110	0.918485
$416$	0	0
$417$	−20.1659	−0.987529
$418$	0	0
$419$	22.9344	1.12042	0.560210	−	0.828351i	$-0.310720\pi$
$419$	22.9344	1.12042	0.560210	+	0.828351i	$0.310720\pi$
$420$	0	0
$421$	−15.0532	−0.733648	−0.366824	−	0.930290i	$-0.619555\pi$
$421$	−15.0532	−0.733648	−0.366824	+	0.930290i	$0.619555\pi$
$422$	0	0
$423$	−85.6027	−4.16215
$424$	0	0
$425$	−2.46993	−0.119809
$426$	0	0
$427$	7.91157	0.382867
$428$	0	0
$429$	34.5562	1.66839
$430$	0	0
$431$	23.2512	1.11997	0.559985	−	0.828503i	$-0.310807\pi$
$431$	23.2512	1.11997	0.559985	+	0.828503i	$0.310807\pi$
$432$	0	0
$433$	−17.4847	−0.840261	−0.420131	−	0.907464i	$-0.638016\pi$
$433$	−17.4847	−0.840261	−0.420131	+	0.907464i	$0.638016\pi$
$434$	0	0
$435$	−35.6369	−1.70866
$436$	0	0
$437$	−16.0641	−0.768449
$438$	0	0
$439$	33.0940	1.57949	0.789745	−	0.613435i	$-0.210213\pi$
$439$	33.0940	1.57949	0.789745	+	0.613435i	$0.210213\pi$
$440$	0	0
$441$	14.4638	0.688753
$442$	0	0
$443$	−8.02662	−0.381356	−0.190678	−	0.981653i	$-0.561069\pi$
$443$	−8.02662	−0.381356	−0.190678	+	0.981653i	$0.561069\pi$
$444$	0	0
$445$	5.20402	0.246694
$446$	0	0
$447$	41.3497	1.95578
$448$	0	0
$449$	9.79131	0.462080	0.231040	−	0.972944i	$-0.425787\pi$
$449$	9.79131	0.462080	0.231040	+	0.972944i	$0.425787\pi$
$450$	0	0
$451$	28.7452	1.35356
$452$	0	0
$453$	37.6076	1.76696
$454$	0	0
$455$	11.3783	0.533425
$456$	0	0
$457$	23.9268	1.11925	0.559624	−	0.828747i	$-0.310946\pi$
$457$	23.9268	1.11925	0.559624	+	0.828747i	$0.310946\pi$
$458$	0	0
$459$	−19.1652	−0.894553
$460$	0	0
$461$	−22.4132	−1.04389	−0.521943	−	0.852980i	$-0.674793\pi$
$461$	−22.4132	−1.04389	−0.521943	+	0.852980i	$0.674793\pi$
$462$	0	0
$463$	11.3848	0.529095	0.264548	−	0.964373i	$-0.414777\pi$
$463$	11.3848	0.529095	0.264548	+	0.964373i	$0.414777\pi$
$464$	0	0
$465$	48.8354	2.26469
$466$	0	0
$467$	−7.38533	−0.341752	−0.170876	−	0.985293i	$-0.554660\pi$
$467$	−7.38533	−0.341752	−0.170876	+	0.985293i	$0.554660\pi$
$468$	0	0
$469$	−36.8105	−1.69975
$470$	0	0
$471$	−12.4374	−0.573085
$472$	0	0
$473$	−35.8690	−1.64926
$474$	0	0
$475$	−7.21899	−0.331230
$476$	0	0
$477$	61.6414	2.82237
$478$	0	0
$479$	32.8937	1.50295	0.751475	−	0.659761i	$-0.229343\pi$
$479$	32.8937	1.50295	0.751475	+	0.659761i	$0.229343\pi$
$480$	0	0
$481$	−23.4385	−1.06870
$482$	0	0
$483$	55.1950	2.51146
$484$	0	0
$485$	9.10217	0.413308
$486$	0	0
$487$	−26.9696	−1.22211	−0.611055	−	0.791588i	$-0.709255\pi$
$487$	−26.9696	−1.22211	−0.611055	+	0.791588i	$0.709255\pi$
$488$	0	0
$489$	52.7640	2.38607
$490$	0	0
$491$	−6.84487	−0.308905	−0.154452	−	0.988000i	$-0.549361\pi$
$491$	−6.84487	−0.308905	−0.154452	+	0.988000i	$0.549361\pi$
$492$	0	0
$493$	6.57198	0.295987
$494$	0	0
$495$	−57.2455	−2.57299
$496$	0	0
$497$	−7.57226	−0.339662
$498$	0	0
$499$	30.5644	1.36825	0.684126	−	0.729364i	$-0.260184\pi$
$499$	30.5644	1.36825	0.684126	+	0.729364i	$0.260184\pi$
$500$	0	0
$501$	6.88811	0.307738
$502$	0	0
$503$	−13.5892	−0.605913	−0.302957	−	0.953004i	$-0.597974\pi$
$503$	−13.5892	−0.605913	−0.302957	+	0.953004i	$0.597974\pi$
$504$	0	0
$505$	−14.1183	−0.628258
$506$	0	0
$507$	24.2149	1.07542
$508$	0	0
$509$	4.30545	0.190836	0.0954179	−	0.995437i	$-0.469581\pi$
$509$	4.30545	0.190836	0.0954179	+	0.995437i	$0.469581\pi$
$510$	0	0
$511$	−42.7871	−1.89279
$512$	0	0
$513$	−56.0151	−2.47313
$514$	0	0
$515$	25.9448	1.14326
$516$	0	0
$517$	41.4444	1.82272
$518$	0	0
$519$	25.7432	1.13000
$520$	0	0
$521$	36.9996	1.62098	0.810491	−	0.585752i	$-0.199201\pi$
$521$	36.9996	1.62098	0.810491	+	0.585752i	$0.199201\pi$
$522$	0	0
$523$	5.96168	0.260686	0.130343	−	0.991469i	$-0.458392\pi$
$523$	5.96168	0.260686	0.130343	+	0.991469i	$0.458392\pi$
$524$	0	0
$525$	24.8039	1.08253
$526$	0	0
$527$	−9.00599	−0.392307
$528$	0	0
$529$	7.20837	0.313407
$530$	0	0
$531$	−8.62180	−0.374154
$532$	0	0
$533$	−16.7225	−0.724333
$534$	0	0
$535$	10.7939	0.466661
$536$	0	0
$537$	18.7510	0.809167
$538$	0	0
$539$	−7.00263	−0.301625
$540$	0	0
$541$	−2.12311	−0.0912798	−0.0456399	−	0.998958i	$-0.514533\pi$
$541$	−2.12311	−0.0912798	−0.0456399	+	0.998958i	$0.514533\pi$
$542$	0	0
$543$	15.0041	0.643889
$544$	0	0
$545$	23.2121	0.994297
$546$	0	0
$547$	−5.90947	−0.252671	−0.126335	−	0.991988i	$-0.540322\pi$
$547$	−5.90947	−0.252671	−0.126335	+	0.991988i	$0.540322\pi$
$548$	0	0
$549$	23.1559	0.988269
$550$	0	0
$551$	19.2083	0.818300
$552$	0	0
$553$	16.5653	0.704426
$554$	0	0
$555$	52.3383	2.22164
$556$	0	0
$557$	6.01480	0.254855	0.127428	−	0.991848i	$-0.459328\pi$
$557$	6.01480	0.254855	0.127428	+	0.991848i	$0.459328\pi$
$558$	0	0
$559$	20.8668	0.882570
$560$	0	0
$561$	14.2303	0.600803
$562$	0	0
$563$	−21.5624	−0.908745	−0.454373	−	0.890812i	$-0.650137\pi$
$563$	−21.5624	−0.908745	−0.454373	+	0.890812i	$0.650137\pi$
$564$	0	0
$565$	−23.7875	−1.00075
$566$	0	0
$567$	116.270	4.88288
$568$	0	0
$569$	30.5241	1.27964	0.639819	−	0.768526i	$-0.279009\pi$
$569$	30.5241	1.27964	0.639819	+	0.768526i	$0.279009\pi$
$570$	0	0
$571$	20.6758	0.865257	0.432629	−	0.901572i	$-0.357586\pi$
$571$	20.6758	0.865257	0.432629	+	0.901572i	$0.357586\pi$
$572$	0	0
$573$	80.5613	3.36550
$574$	0	0
$575$	13.5752	0.566127
$576$	0	0
$577$	44.5040	1.85273	0.926363	−	0.376631i	$-0.122918\pi$
$577$	44.5040	1.85273	0.926363	+	0.376631i	$0.122918\pi$
$578$	0	0
$579$	−86.0965	−3.57805
$580$	0	0
$581$	34.6520	1.43761
$582$	0	0
$583$	−29.8436	−1.23600
$584$	0	0
$585$	33.3025	1.37689
$586$	0	0
$587$	−42.0273	−1.73465	−0.867325	−	0.497741i	$-0.834163\pi$
$587$	−42.0273	−1.73465	−0.867325	+	0.497741i	$0.834163\pi$
$588$	0	0
$589$	−26.3223	−1.08459
$590$	0	0
$591$	−14.7992	−0.608758
$592$	0	0
$593$	−39.2967	−1.61372	−0.806861	−	0.590741i	$-0.798835\pi$
$593$	−39.2967	−1.61372	−0.806861	+	0.590741i	$0.798835\pi$
$594$	0	0
$595$	4.68561	0.192091
$596$	0	0
$597$	−68.3141	−2.79591
$598$	0	0
$599$	−9.84577	−0.402287	−0.201144	−	0.979562i	$-0.564466\pi$
$599$	−9.84577	−0.402287	−0.201144	+	0.979562i	$0.564466\pi$
$600$	0	0
$601$	−14.3910	−0.587023	−0.293511	−	0.955956i	$-0.594824\pi$
$601$	−14.3910	−0.587023	−0.293511	+	0.955956i	$0.594824\pi$
$602$	0	0
$603$	−107.738	−4.38745
$604$	0	0
$605$	10.2185	0.415441
$606$	0	0
$607$	−37.5219	−1.52297	−0.761483	−	0.648185i	$-0.775528\pi$
$607$	−37.5219	−1.52297	−0.761483	+	0.648185i	$0.775528\pi$
$608$	0	0
$609$	−65.9982	−2.67438
$610$	0	0
$611$	−24.1103	−0.975397
$612$	0	0
$613$	−6.02703	−0.243430	−0.121715	−	0.992565i	$-0.538839\pi$
$613$	−6.02703	−0.243430	−0.121715	+	0.992565i	$0.538839\pi$
$614$	0	0
$615$	37.3416	1.50576
$616$	0	0
$617$	−43.5592	−1.75363	−0.876813	−	0.480831i	$-0.840335\pi$
$617$	−43.5592	−1.75363	−0.876813	+	0.480831i	$0.840335\pi$
$618$	0	0
$619$	−36.6728	−1.47400	−0.737002	−	0.675891i	$-0.763759\pi$
$619$	−36.6728	−1.47400	−0.737002	+	0.675891i	$0.763759\pi$
$620$	0	0
$621$	105.336	4.22698
$622$	0	0
$623$	9.63767	0.386125
$624$	0	0
$625$	−6.54984	−0.261993
$626$	0	0
$627$	41.5916	1.66101
$628$	0	0
$629$	−9.65198	−0.384850
$630$	0	0
$631$	3.86792	0.153979	0.0769896	−	0.997032i	$-0.475469\pi$
$631$	3.86792	0.153979	0.0769896	+	0.997032i	$0.475469\pi$
$632$	0	0
$633$	29.1099	1.15702
$634$	0	0
$635$	2.04562	0.0811779
$636$	0	0
$637$	4.07378	0.161409
$638$	0	0
$639$	−22.1628	−0.876746
$640$	0	0
$641$	−4.35173	−0.171883	−0.0859415	−	0.996300i	$-0.527390\pi$
$641$	−4.35173	−0.171883	−0.0859415	+	0.996300i	$0.527390\pi$
$642$	0	0
$643$	−39.8526	−1.57163	−0.785816	−	0.618461i	$-0.787757\pi$
$643$	−39.8526	−1.57163	−0.785816	+	0.618461i	$0.787757\pi$
$644$	0	0
$645$	−46.5957	−1.83470
$646$	0	0
$647$	12.6138	0.495901	0.247950	−	0.968773i	$-0.420243\pi$
$647$	12.6138	0.495901	0.247950	+	0.968773i	$0.420243\pi$
$648$	0	0
$649$	4.17423	0.163853
$650$	0	0
$651$	90.4415	3.54468
$652$	0	0
$653$	7.84100	0.306842	0.153421	−	0.988161i	$-0.450971\pi$
$653$	7.84100	0.306842	0.153421	+	0.988161i	$0.450971\pi$
$654$	0	0
$655$	−7.69307	−0.300593
$656$	0	0
$657$	−125.231	−4.88573
$658$	0	0
$659$	20.4556	0.796836	0.398418	−	0.917204i	$-0.369559\pi$
$659$	20.4556	0.796836	0.398418	+	0.917204i	$0.369559\pi$
$660$	0	0
$661$	−49.5519	−1.92734	−0.963672	−	0.267088i	$-0.913939\pi$
$661$	−49.5519	−1.92734	−0.963672	+	0.267088i	$0.913939\pi$
$662$	0	0
$663$	−8.27846	−0.321509
$664$	0	0
$665$	13.6949	0.531065
$666$	0	0
$667$	−36.1210	−1.39861
$668$	0	0
$669$	61.8195	2.39008
$670$	0	0
$671$	−11.2109	−0.432791
$672$	0	0
$673$	−20.5054	−0.790426	−0.395213	−	0.918589i	$-0.629329\pi$
$673$	−20.5054	−0.790426	−0.395213	+	0.918589i	$0.629329\pi$
$674$	0	0
$675$	47.3365	1.82198
$676$	0	0
$677$	4.99787	0.192084	0.0960419	−	0.995377i	$-0.469382\pi$
$677$	4.99787	0.192084	0.0960419	+	0.995377i	$0.469382\pi$
$678$	0	0
$679$	16.8569	0.646908
$680$	0	0
$681$	60.3480	2.31254
$682$	0	0
$683$	0.642542	0.0245862	0.0122931	−	0.999924i	$-0.496087\pi$
$683$	0.642542	0.0245862	0.0122931	+	0.999924i	$0.496087\pi$
$684$	0	0
$685$	−23.5789	−0.900903
$686$	0	0
$687$	−68.6303	−2.61841
$688$	0	0
$689$	17.3615	0.661420
$690$	0	0
$691$	28.0564	1.06732	0.533658	−	0.845701i	$-0.320817\pi$
$691$	28.0564	1.06732	0.533658	+	0.845701i	$0.320817\pi$
$692$	0	0
$693$	−106.017	−4.02724
$694$	0	0
$695$	9.40910	0.356907
$696$	0	0
$697$	−6.88635	−0.260839
$698$	0	0
$699$	57.3652	2.16975
$700$	0	0
$701$	−9.65416	−0.364633	−0.182316	−	0.983240i	$-0.558359\pi$
$701$	−9.65416	−0.364633	−0.182316	+	0.983240i	$0.558359\pi$
$702$	0	0
$703$	−28.2104	−1.06397
$704$	0	0
$705$	53.8385	2.02767
$706$	0	0
$707$	−26.1467	−0.983347
$708$	0	0
$709$	35.5575	1.33539	0.667695	−	0.744435i	$-0.267281\pi$
$709$	35.5575	1.33539	0.667695	+	0.744435i	$0.267281\pi$
$710$	0	0
$711$	48.4838	1.81829
$712$	0	0
$713$	49.4988	1.85375
$714$	0	0
$715$	−16.1234	−0.602980
$716$	0	0
$717$	−54.8930	−2.05002
$718$	0	0
$719$	34.9436	1.30318	0.651589	−	0.758572i	$-0.274103\pi$
$719$	34.9436	1.30318	0.651589	+	0.758572i	$0.274103\pi$
$720$	0	0
$721$	48.0488	1.78943
$722$	0	0
$723$	57.7223	2.14672
$724$	0	0
$725$	−16.2323	−0.602853
$726$	0	0
$727$	−2.61532	−0.0969969	−0.0484984	−	0.998823i	$-0.515444\pi$
$727$	−2.61532	−0.0969969	−0.0484984	+	0.998823i	$0.515444\pi$
$728$	0	0
$729$	144.297	5.34433
$730$	0	0
$731$	8.59296	0.317822
$732$	0	0
$733$	−36.3602	−1.34299	−0.671497	−	0.741007i	$-0.734349\pi$
$733$	−36.3602	−1.34299	−0.671497	+	0.741007i	$0.734349\pi$
$734$	0	0
$735$	−9.09678	−0.335540
$736$	0	0
$737$	52.1614	1.92139
$738$	0	0
$739$	−7.19437	−0.264649	−0.132325	−	0.991206i	$-0.542244\pi$
$739$	−7.19437	−0.264649	−0.132325	+	0.991206i	$0.542244\pi$
$740$	0	0
$741$	−24.1959	−0.888859
$742$	0	0
$743$	23.1042	0.847610	0.423805	−	0.905753i	$-0.360694\pi$
$743$	23.1042	0.847610	0.423805	+	0.905753i	$0.360694\pi$
$744$	0	0
$745$	−19.2931	−0.706845
$746$	0	0
$747$	101.421	3.71080
$748$	0	0
$749$	19.9899	0.730416
$750$	0	0
$751$	27.1876	0.992091	0.496046	−	0.868296i	$-0.334785\pi$
$751$	27.1876	0.992091	0.496046	+	0.868296i	$0.334785\pi$
$752$	0	0
$753$	−24.8070	−0.904016
$754$	0	0
$755$	−17.5471	−0.638605
$756$	0	0
$757$	−14.2270	−0.517090	−0.258545	−	0.965999i	$-0.583243\pi$
$757$	−14.2270	−0.517090	−0.258545	+	0.965999i	$0.583243\pi$
$758$	0	0
$759$	−78.2127	−2.83894
$760$	0	0
$761$	44.2010	1.60228	0.801142	−	0.598474i	$-0.204226\pi$
$761$	44.2010	1.60228	0.801142	+	0.598474i	$0.204226\pi$
$762$	0	0
$763$	42.9880	1.55627
$764$	0	0
$765$	13.7140	0.495831
$766$	0	0
$767$	−2.42836	−0.0876829
$768$	0	0
$769$	53.0932	1.91459	0.957295	−	0.289112i	$-0.0933601\pi$
$769$	53.0932	1.91459	0.957295	+	0.289112i	$0.0933601\pi$
$770$	0	0
$771$	−7.16470	−0.258030
$772$	0	0
$773$	−1.81951	−0.0654431	−0.0327216	−	0.999465i	$-0.510417\pi$
$773$	−1.81951	−0.0654431	−0.0327216	+	0.999465i	$0.510417\pi$
$774$	0	0
$775$	22.2441	0.799032
$776$	0	0
$777$	96.9287	3.47730
$778$	0	0
$779$	−20.1271	−0.721129
$780$	0	0
$781$	10.7301	0.383952
$782$	0	0
$783$	−125.953	−4.50120
$784$	0	0
$785$	5.80309	0.207121
$786$	0	0
$787$	−15.6710	−0.558610	−0.279305	−	0.960202i	$-0.590104\pi$
$787$	−15.6710	−0.558610	−0.279305	+	0.960202i	$0.590104\pi$
$788$	0	0
$789$	−12.6010	−0.448607
$790$	0	0
$791$	−44.0537	−1.56637
$792$	0	0
$793$	6.52192	0.231600
$794$	0	0
$795$	−38.7684	−1.37497
$796$	0	0
$797$	−6.09659	−0.215952	−0.107976	−	0.994153i	$-0.534437\pi$
$797$	−6.09659	−0.215952	−0.107976	+	0.994153i	$0.534437\pi$
$798$	0	0
$799$	−9.92863	−0.351250
$800$	0	0
$801$	28.2079	0.996677
$802$	0	0
$803$	60.6304	2.13960
$804$	0	0
$805$	−25.7531	−0.907678
$806$	0	0
$807$	−44.6450	−1.57158
$808$	0	0
$809$	18.0683	0.635248	0.317624	−	0.948217i	$-0.397115\pi$
$809$	18.0683	0.635248	0.317624	+	0.948217i	$0.397115\pi$
$810$	0	0
$811$	−39.0436	−1.37101	−0.685504	−	0.728069i	$-0.740418\pi$
$811$	−39.0436	−1.37101	−0.685504	+	0.728069i	$0.740418\pi$
$812$	0	0
$813$	43.3002	1.51861
$814$	0	0
$815$	−24.6188	−0.862360
$816$	0	0
$817$	25.1151	0.878666
$818$	0	0
$819$	61.6752	2.15510
$820$	0	0
$821$	16.0672	0.560749	0.280375	−	0.959891i	$-0.409541\pi$
$821$	16.0672	0.560749	0.280375	+	0.959891i	$0.409541\pi$
$822$	0	0
$823$	−10.8890	−0.379566	−0.189783	−	0.981826i	$-0.560778\pi$
$823$	−10.8890	−0.379566	−0.189783	+	0.981826i	$0.560778\pi$
$824$	0	0
$825$	−35.1477	−1.22369
$826$	0	0
$827$	−39.8450	−1.38555	−0.692774	−	0.721155i	$-0.743612\pi$
$827$	−39.8450	−1.38555	−0.692774	+	0.721155i	$0.743612\pi$
$828$	0	0
$829$	28.1612	0.978079	0.489040	−	0.872262i	$-0.337347\pi$
$829$	28.1612	0.978079	0.489040	+	0.872262i	$0.337347\pi$
$830$	0	0
$831$	10.8613	0.376776
$832$	0	0
$833$	1.67758	0.0581249
$834$	0	0
$835$	−3.21388	−0.111221
$836$	0	0
$837$	172.601	5.96597
$838$	0	0
$839$	1.84632	0.0637420	0.0318710	−	0.999492i	$-0.489853\pi$
$839$	1.84632	0.0637420	0.0318710	+	0.999492i	$0.489853\pi$
$840$	0	0
$841$	14.1909	0.489342
$842$	0	0
$843$	40.1975	1.38448
$844$	0	0
$845$	−11.2983	−0.388674
$846$	0	0
$847$	18.9243	0.650247
$848$	0	0
$849$	72.2466	2.47950
$850$	0	0
$851$	53.0493	1.81851
$852$	0	0
$853$	13.0869	0.448089	0.224044	−	0.974579i	$-0.428074\pi$
$853$	13.0869	0.448089	0.224044	+	0.974579i	$0.428074\pi$
$854$	0	0
$855$	40.0827	1.37080
$856$	0	0
$857$	−19.4032	−0.662800	−0.331400	−	0.943490i	$-0.607521\pi$
$857$	−19.4032	−0.662800	−0.331400	+	0.943490i	$0.607521\pi$
$858$	0	0
$859$	14.3515	0.489667	0.244833	−	0.969565i	$-0.421267\pi$
$859$	14.3515	0.489667	0.244833	+	0.969565i	$0.421267\pi$
$860$	0	0
$861$	69.1553	2.35681
$862$	0	0
$863$	−13.9696	−0.475532	−0.237766	−	0.971323i	$-0.576415\pi$
$863$	−13.9696	−0.475532	−0.237766	+	0.971323i	$0.576415\pi$
$864$	0	0
$865$	−12.0114	−0.408399
$866$	0	0
$867$	−3.40908	−0.115778
$868$	0	0
$869$	−23.4734	−0.796279
$870$	0	0
$871$	−30.3449	−1.02820
$872$	0	0
$873$	49.3374	1.66982
$874$	0	0
$875$	−35.0012	−1.18325
$876$	0	0
$877$	−45.1153	−1.52344	−0.761718	−	0.647909i	$-0.775644\pi$
$877$	−45.1153	−1.52344	−0.761718	+	0.647909i	$0.775644\pi$
$878$	0	0
$879$	−58.5480	−1.97477
$880$	0	0
$881$	−55.6135	−1.87367	−0.936833	−	0.349776i	$-0.886258\pi$
$881$	−55.6135	−1.87367	−0.936833	+	0.349776i	$0.886258\pi$
$882$	0	0
$883$	−3.80160	−0.127934	−0.0639671	−	0.997952i	$-0.520375\pi$
$883$	−3.80160	−0.127934	−0.0639671	+	0.997952i	$0.520375\pi$
$884$	0	0
$885$	5.42255	0.182277
$886$	0	0
$887$	2.79464	0.0938347	0.0469173	−	0.998899i	$-0.485060\pi$
$887$	2.79464	0.0938347	0.0469173	+	0.998899i	$0.485060\pi$
$888$	0	0
$889$	3.78841	0.127059
$890$	0	0
$891$	−164.757	−5.51958
$892$	0	0
$893$	−29.0189	−0.971082
$894$	0	0
$895$	−8.74893	−0.292445
$896$	0	0
$897$	45.5002	1.51921
$898$	0	0
$899$	−59.1872	−1.97400
$900$	0	0
$901$	7.14948	0.238184
$902$	0	0
$903$	−86.2937	−2.87167
$904$	0	0
$905$	−7.00068	−0.232711
$906$	0	0
$907$	24.6132	0.817266	0.408633	−	0.912699i	$-0.366006\pi$
$907$	24.6132	0.817266	0.408633	+	0.912699i	$0.366006\pi$
$908$	0	0
$909$	−76.5271	−2.53824
$910$	0	0
$911$	18.5202	0.613602	0.306801	−	0.951774i	$-0.400741\pi$
$911$	18.5202	0.613602	0.306801	+	0.951774i	$0.400741\pi$
$912$	0	0
$913$	−49.1028	−1.62506
$914$	0	0
$915$	−14.5635	−0.481455
$916$	0	0
$917$	−14.2473	−0.470487
$918$	0	0
$919$	15.2854	0.504219	0.252109	−	0.967699i	$-0.418876\pi$
$919$	15.2854	0.504219	0.252109	+	0.967699i	$0.418876\pi$
$920$	0	0
$921$	−81.3867	−2.68178
$922$	0	0
$923$	−6.24221	−0.205465
$924$	0	0
$925$	23.8397	0.783843
$926$	0	0
$927$	140.631	4.61893
$928$	0	0
$929$	−29.9097	−0.981307	−0.490653	−	0.871355i	$-0.663242\pi$
$929$	−29.9097	−0.981307	−0.490653	+	0.871355i	$0.663242\pi$
$930$	0	0
$931$	4.90317	0.160695
$932$	0	0
$933$	96.2403	3.15077
$934$	0	0
$935$	−6.63962	−0.217139
$936$	0	0
$937$	−39.9851	−1.30625	−0.653127	−	0.757248i	$-0.726543\pi$
$937$	−39.9851	−1.30625	−0.653127	+	0.757248i	$0.726543\pi$
$938$	0	0
$939$	−45.2447	−1.47651
$940$	0	0
$941$	−44.4993	−1.45064	−0.725319	−	0.688413i	$-0.758308\pi$
$941$	−44.4993	−1.45064	−0.725319	+	0.688413i	$0.758308\pi$
$942$	0	0
$943$	37.8489	1.23253
$944$	0	0
$945$	−89.8005	−2.92121
$946$	0	0
$947$	20.3620	0.661677	0.330839	−	0.943687i	$-0.392668\pi$
$947$	20.3620	0.661677	0.330839	+	0.943687i	$0.392668\pi$
$948$	0	0
$949$	−35.2717	−1.14497
$950$	0	0
$951$	63.5294	2.06008
$952$	0	0
$953$	42.8579	1.38830	0.694152	−	0.719828i	$-0.255780\pi$
$953$	42.8579	1.38830	0.694152	+	0.719828i	$0.255780\pi$
$954$	0	0
$955$	−37.5886	−1.21634
$956$	0	0
$957$	93.5211	3.02311
$958$	0	0
$959$	−43.6673	−1.41009
$960$	0	0
$961$	50.1078	1.61638
$962$	0	0
$963$	58.5073	1.88537
$964$	0	0
$965$	40.1712	1.29316
$966$	0	0
$967$	−34.2690	−1.10202	−0.551009	−	0.834499i	$-0.685757\pi$
$967$	−34.2690	−1.10202	−0.551009	+	0.834499i	$0.685757\pi$
$968$	0	0
$969$	−9.96389	−0.320086
$970$	0	0
$971$	52.8803	1.69701	0.848505	−	0.529188i	$-0.177503\pi$
$971$	52.8803	1.69701	0.848505	+	0.529188i	$0.177503\pi$
$972$	0	0
$973$	17.4253	0.558630
$974$	0	0
$975$	20.4472	0.654834
$976$	0	0
$977$	−13.1858	−0.421850	−0.210925	−	0.977502i	$-0.567648\pi$
$977$	−13.1858	−0.421850	−0.210925	+	0.977502i	$0.567648\pi$
$978$	0	0
$979$	−13.6568	−0.436473
$980$	0	0
$981$	125.819	4.01709
$982$	0	0
$983$	−8.64908	−0.275863	−0.137931	−	0.990442i	$-0.544045\pi$
$983$	−8.64908	−0.275863	−0.137931	+	0.990442i	$0.544045\pi$
$984$	0	0
$985$	6.90507	0.220014
$986$	0	0
$987$	99.7070	3.17371
$988$	0	0
$989$	−47.2287	−1.50179
$990$	0	0
$991$	34.8233	1.10620	0.553099	−	0.833115i	$-0.313445\pi$
$991$	34.8233	1.10620	0.553099	+	0.833115i	$0.313445\pi$
$992$	0	0
$993$	−53.7800	−1.70666
$994$	0	0
$995$	31.8742	1.01048
$996$	0	0
$997$	−47.7664	−1.51278	−0.756388	−	0.654123i	$-0.773038\pi$
$997$	−47.7664	−1.51278	−0.756388	+	0.654123i	$0.773038\pi$
$998$	0	0
$999$	184.982	5.85256

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4012.2.a.h.1.1	✓	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)

\(f(q)\)	\(=\)	\(q-3.40908 q^{3} +1.59062 q^{5} +2.94577 q^{7} +8.62180 q^{9} +O(q^{10})\)	\(q-3.40908 q^{3} +1.59062 q^{5} +2.94577 q^{7} +8.62180 q^{9} -4.17423 q^{11} +2.42836 q^{13} -5.42255 q^{15} +1.00000 q^{17} +2.92275 q^{19} -10.0424 q^{21} -5.49621 q^{23} -2.46993 q^{25} -19.1652 q^{27} +6.57198 q^{29} -9.00599 q^{31} +14.2303 q^{33} +4.68561 q^{35} -9.65198 q^{37} -8.27846 q^{39} -6.88635 q^{41} +8.59296 q^{43} +13.7140 q^{45} -9.92863 q^{47} +1.67758 q^{49} -3.40908 q^{51} +7.14948 q^{53} -6.63962 q^{55} -9.96389 q^{57} -1.00000 q^{59} +2.68573 q^{61} +25.3979 q^{63} +3.86260 q^{65} -12.4960 q^{67} +18.7370 q^{69} -2.57055 q^{71} -14.5249 q^{73} +8.42017 q^{75} -12.2963 q^{77} +5.62340 q^{79} +39.4701 q^{81} +11.7633 q^{83} +1.59062 q^{85} -22.4044 q^{87} +3.27169 q^{89} +7.15339 q^{91} +30.7021 q^{93} +4.64899 q^{95} +5.72240 q^{97} -35.9894 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 15 q - q^{3} + q^{5} - 11 q^{7} + 16 q^{9}+O(q^{10}) \) 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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