LMFDB - Embedded newform 4012.2.a.h.1.14

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			1.14
Root			$0.186961$ 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)

$f(q)$	$=$	$q+2.17219 q^{3} -1.55586 q^{5} +3.67456 q^{7} +1.71840 q^{9} +O(q^{10})$	$q+2.17219 q^{3} -1.55586 q^{5} +3.67456 q^{7} +1.71840 q^{9} -6.34872 q^{11} -4.15200 q^{13} -3.37962 q^{15} +1.00000 q^{17} +4.98305 q^{19} +7.98184 q^{21} -3.19399 q^{23} -2.57930 q^{25} -2.78388 q^{27} -5.70145 q^{29} -0.739366 q^{31} -13.7906 q^{33} -5.71711 q^{35} +3.82932 q^{37} -9.01893 q^{39} -10.3229 q^{41} +2.00486 q^{43} -2.67359 q^{45} -12.9026 q^{47} +6.50243 q^{49} +2.17219 q^{51} +1.42170 q^{53} +9.87773 q^{55} +10.8241 q^{57} -1.00000 q^{59} +7.05667 q^{61} +6.31436 q^{63} +6.45994 q^{65} +13.7911 q^{67} -6.93794 q^{69} +12.3952 q^{71} -9.68558 q^{73} -5.60271 q^{75} -23.3288 q^{77} -3.90718 q^{79} -11.2023 q^{81} -11.8048 q^{83} -1.55586 q^{85} -12.3846 q^{87} +12.5399 q^{89} -15.2568 q^{91} -1.60604 q^{93} -7.75293 q^{95} -11.7550 q^{97} -10.9096 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	2.17219	1.25411	0.627056	−	0.778974i	$-0.284260\pi$
$3$	2.17219	1.25411	0.627056	+	0.778974i	$0.284260\pi$
$4$	0	0
$5$	−1.55586	−0.695802	−0.347901	−	0.937531i	$-0.613106\pi$
$5$	−1.55586	−0.695802	−0.347901	+	0.937531i	$0.613106\pi$
$6$	0	0
$7$	3.67456	1.38885	0.694427	−	0.719563i	$-0.255658\pi$
$7$	3.67456	1.38885	0.694427	+	0.719563i	$0.255658\pi$
$8$	0	0
$9$	1.71840	0.572799
$10$	0	0
$11$	−6.34872	−1.91421	−0.957105	−	0.289740i	$-0.906431\pi$
$11$	−6.34872	−1.91421	−0.957105	+	0.289740i	$0.906431\pi$
$12$	0	0
$13$	−4.15200	−1.15156	−0.575779	−	0.817605i	$-0.695301\pi$
$13$	−4.15200	−1.15156	−0.575779	+	0.817605i	$0.695301\pi$
$14$	0	0
$15$	−3.37962	−0.872615
$16$	0	0
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	0	0
$19$	4.98305	1.14319	0.571595	−	0.820536i	$-0.306325\pi$
$19$	4.98305	1.14319	0.571595	+	0.820536i	$0.306325\pi$
$20$	0	0
$21$	7.98184	1.74178
$22$	0	0
$23$	−3.19399	−0.665993	−0.332996	−	0.942928i	$-0.608060\pi$
$23$	−3.19399	−0.665993	−0.332996	+	0.942928i	$0.608060\pi$
$24$	0	0
$25$	−2.57930	−0.515859
$26$	0	0
$27$	−2.78388	−0.535759
$28$	0	0
$29$	−5.70145	−1.05873	−0.529366	−	0.848393i	$-0.677570\pi$
$29$	−5.70145	−1.05873	−0.529366	+	0.848393i	$0.677570\pi$
$30$	0	0
$31$	−0.739366	−0.132794	−0.0663970	−	0.997793i	$-0.521150\pi$
$31$	−0.739366	−0.132794	−0.0663970	+	0.997793i	$0.521150\pi$
$32$	0	0
$33$	−13.7906	−2.40064
$34$	0	0
$35$	−5.71711	−0.966368
$36$	0	0
$37$	3.82932	0.629537	0.314768	−	0.949168i	$-0.398073\pi$
$37$	3.82932	0.629537	0.314768	+	0.949168i	$0.398073\pi$
$38$	0	0
$39$	−9.01893	−1.44418
$40$	0	0
$41$	−10.3229	−1.61217	−0.806084	−	0.591801i	$-0.798417\pi$
$41$	−10.3229	−1.61217	−0.806084	+	0.591801i	$0.798417\pi$
$42$	0	0
$43$	2.00486	0.305738	0.152869	−	0.988246i	$-0.451149\pi$
$43$	2.00486	0.305738	0.152869	+	0.988246i	$0.451149\pi$
$44$	0	0
$45$	−2.67359	−0.398555
$46$	0	0
$47$	−12.9026	−1.88203	−0.941017	−	0.338360i	$-0.890128\pi$
$47$	−12.9026	−1.88203	−0.941017	+	0.338360i	$0.890128\pi$
$48$	0	0
$49$	6.50243	0.928918
$50$	0	0
$51$	2.17219	0.304167
$52$	0	0
$53$	1.42170	0.195285	0.0976427	−	0.995222i	$-0.468870\pi$
$53$	1.42170	0.195285	0.0976427	+	0.995222i	$0.468870\pi$
$54$	0	0
$55$	9.87773	1.33191
$56$	0	0
$57$	10.8241	1.43369
$58$	0	0
$59$	−1.00000	−0.130189
$59$	−1.00000	−0.130189
$60$	0	0
$61$	7.05667	0.903514	0.451757	−	0.892141i	$-0.350797\pi$
$61$	7.05667	0.903514	0.451757	+	0.892141i	$0.350797\pi$
$62$	0	0
$63$	6.31436	0.795534
$64$	0	0
$65$	6.45994	0.801257
$66$	0	0
$67$	13.7911	1.68485	0.842423	−	0.538816i	$-0.181128\pi$
$67$	13.7911	1.68485	0.842423	+	0.538816i	$0.181128\pi$
$68$	0	0
$69$	−6.93794	−0.835230
$70$	0	0
$71$	12.3952	1.47104	0.735520	−	0.677503i	$-0.236938\pi$
$71$	12.3952	1.47104	0.735520	+	0.677503i	$0.236938\pi$
$72$	0	0
$73$	−9.68558	−1.13361	−0.566806	−	0.823852i	$-0.691821\pi$
$73$	−9.68558	−1.13361	−0.566806	+	0.823852i	$0.691821\pi$
$74$	0	0
$75$	−5.60271	−0.646945
$76$	0	0
$77$	−23.3288	−2.65856
$78$	0	0
$79$	−3.90718	−0.439592	−0.219796	−	0.975546i	$-0.570539\pi$
$79$	−3.90718	−0.439592	−0.219796	+	0.975546i	$0.570539\pi$
$80$	0	0
$81$	−11.2023	−1.24470
$82$	0	0
$83$	−11.8048	−1.29574	−0.647870	−	0.761751i	$-0.724340\pi$
$83$	−11.8048	−1.29574	−0.647870	+	0.761751i	$0.724340\pi$
$84$	0	0
$85$	−1.55586	−0.168757
$86$	0	0
$87$	−12.3846	−1.32777
$88$	0	0
$89$	12.5399	1.32923	0.664613	−	0.747187i	$-0.268596\pi$
$89$	12.5399	1.32923	0.664613	+	0.747187i	$0.268596\pi$
$90$	0	0
$91$	−15.2568	−1.59935
$92$	0	0
$93$	−1.60604	−0.166539
$94$	0	0
$95$	−7.75293	−0.795434
$96$	0	0
$97$	−11.7550	−1.19354	−0.596768	−	0.802414i	$-0.703549\pi$
$97$	−11.7550	−1.19354	−0.596768	+	0.802414i	$0.703549\pi$
$98$	0	0
$99$	−10.9096	−1.09646
$100$	0	0
$101$	−5.85059	−0.582155	−0.291078	−	0.956699i	$-0.594014\pi$
$101$	−5.85059	−0.582155	−0.291078	+	0.956699i	$0.594014\pi$
$102$	0	0
$103$	−15.8038	−1.55719	−0.778597	−	0.627524i	$-0.784068\pi$
$103$	−15.8038	−1.55719	−0.778597	+	0.627524i	$0.784068\pi$
$104$	0	0
$105$	−12.4186	−1.21194
$106$	0	0
$107$	1.33817	0.129366	0.0646831	−	0.997906i	$-0.479396\pi$
$107$	1.33817	0.129366	0.0646831	+	0.997906i	$0.479396\pi$
$108$	0	0
$109$	−11.1650	−1.06941	−0.534707	−	0.845037i	$-0.679578\pi$
$109$	−11.1650	−1.06941	−0.534707	+	0.845037i	$0.679578\pi$
$110$	0	0
$111$	8.31801	0.789510
$112$	0	0
$113$	−19.5289	−1.83712	−0.918560	−	0.395281i	$-0.870647\pi$
$113$	−19.5289	−1.83712	−0.918560	+	0.395281i	$0.870647\pi$
$114$	0	0
$115$	4.96940	0.463399
$116$	0	0
$117$	−7.13479	−0.659611
$118$	0	0
$119$	3.67456	0.336847
$120$	0	0
$121$	29.3062	2.66420
$122$	0	0
$123$	−22.4233	−2.02184
$124$	0	0
$125$	11.7923	1.05474
$126$	0	0
$127$	15.5792	1.38243	0.691216	−	0.722648i	$-0.257075\pi$
$127$	15.5792	1.38243	0.691216	+	0.722648i	$0.257075\pi$
$128$	0	0
$129$	4.35492	0.383430
$130$	0	0
$131$	10.0616	0.879086	0.439543	−	0.898221i	$-0.355140\pi$
$131$	10.0616	0.879086	0.439543	+	0.898221i	$0.355140\pi$
$132$	0	0
$133$	18.3105	1.58772
$134$	0	0
$135$	4.33134	0.372782
$136$	0	0
$137$	−7.95481	−0.679625	−0.339813	−	0.940493i	$-0.610364\pi$
$137$	−7.95481	−0.679625	−0.339813	+	0.940493i	$0.610364\pi$
$138$	0	0
$139$	−1.21005	−0.102635	−0.0513174	−	0.998682i	$-0.516342\pi$
$139$	−1.21005	−0.102635	−0.0513174	+	0.998682i	$0.516342\pi$
$140$	0	0
$141$	−28.0268	−2.36028
$142$	0	0
$143$	26.3599	2.20433
$144$	0	0
$145$	8.87066	0.736669
$146$	0	0
$147$	14.1245	1.16497
$148$	0	0
$149$	−11.8543	−0.971140	−0.485570	−	0.874198i	$-0.661388\pi$
$149$	−11.8543	−0.971140	−0.485570	+	0.874198i	$0.661388\pi$
$150$	0	0
$151$	8.98649	0.731311	0.365655	−	0.930750i	$-0.380845\pi$
$151$	8.98649	0.731311	0.365655	+	0.930750i	$0.380845\pi$
$152$	0	0
$153$	1.71840	0.138924
$154$	0	0
$155$	1.15035	0.0923984
$156$	0	0
$157$	16.5557	1.32129	0.660646	−	0.750698i	$-0.270282\pi$
$157$	16.5557	1.32129	0.660646	+	0.750698i	$0.270282\pi$
$158$	0	0
$159$	3.08820	0.244910
$160$	0	0
$161$	−11.7365	−0.924967
$162$	0	0
$163$	−5.01082	−0.392478	−0.196239	−	0.980556i	$-0.562873\pi$
$163$	−5.01082	−0.392478	−0.196239	+	0.980556i	$0.562873\pi$
$164$	0	0
$165$	21.4563	1.67037
$166$	0	0
$167$	8.14710	0.630441	0.315221	−	0.949018i	$-0.397921\pi$
$167$	8.14710	0.630441	0.315221	+	0.949018i	$0.397921\pi$
$168$	0	0
$169$	4.23914	0.326087
$170$	0	0
$171$	8.56285	0.654818
$172$	0	0
$173$	−4.10700	−0.312250	−0.156125	−	0.987737i	$-0.549900\pi$
$173$	−4.10700	−0.312250	−0.156125	+	0.987737i	$0.549900\pi$
$174$	0	0
$175$	−9.47779	−0.716453
$176$	0	0
$177$	−2.17219	−0.163272
$178$	0	0
$179$	21.4122	1.60043	0.800213	−	0.599716i	$-0.204720\pi$
$179$	21.4122	1.60043	0.800213	+	0.599716i	$0.204720\pi$
$180$	0	0
$181$	−22.1815	−1.64874	−0.824368	−	0.566055i	$-0.808469\pi$
$181$	−22.1815	−1.64874	−0.824368	+	0.566055i	$0.808469\pi$
$182$	0	0
$183$	15.3284	1.13311
$184$	0	0
$185$	−5.95790	−0.438033
$186$	0	0
$187$	−6.34872	−0.464264
$188$	0	0
$189$	−10.2296	−0.744091
$190$	0	0
$191$	14.5280	1.05121	0.525606	−	0.850728i	$-0.323839\pi$
$191$	14.5280	1.05121	0.525606	+	0.850728i	$0.323839\pi$
$192$	0	0
$193$	−3.63201	−0.261438	−0.130719	−	0.991419i	$-0.541729\pi$
$193$	−3.63201	−0.261438	−0.130719	+	0.991419i	$0.541729\pi$
$194$	0	0
$195$	14.0322	1.00487
$196$	0	0
$197$	6.89639	0.491348	0.245674	−	0.969353i	$-0.420991\pi$
$197$	6.89639	0.491348	0.245674	+	0.969353i	$0.420991\pi$
$198$	0	0
$199$	−21.8775	−1.55085	−0.775426	−	0.631438i	$-0.782465\pi$
$199$	−21.8775	−1.55085	−0.775426	+	0.631438i	$0.782465\pi$
$200$	0	0
$201$	29.9568	2.11299
$202$	0	0
$203$	−20.9503	−1.47043
$204$	0	0
$205$	16.0610	1.12175
$206$	0	0
$207$	−5.48854	−0.381480
$208$	0	0
$209$	−31.6360	−2.18831
$210$	0	0
$211$	4.57911	0.315239	0.157619	−	0.987500i	$-0.449618\pi$
$211$	4.57911	0.315239	0.157619	+	0.987500i	$0.449618\pi$
$212$	0	0
$213$	26.9247	1.84485
$214$	0	0
$215$	−3.11928	−0.212733
$216$	0	0
$217$	−2.71685	−0.184432
$218$	0	0
$219$	−21.0389	−1.42168
$220$	0	0
$221$	−4.15200	−0.279294
$222$	0	0
$223$	−24.3347	−1.62957	−0.814785	−	0.579763i	$-0.803145\pi$
$223$	−24.3347	−1.62957	−0.814785	+	0.579763i	$0.803145\pi$
$224$	0	0
$225$	−4.43225	−0.295483
$226$	0	0
$227$	15.8973	1.05514	0.527569	−	0.849512i	$-0.323104\pi$
$227$	15.8973	1.05514	0.527569	+	0.849512i	$0.323104\pi$
$228$	0	0
$229$	5.36933	0.354815	0.177408	−	0.984137i	$-0.443229\pi$
$229$	5.36933	0.354815	0.177408	+	0.984137i	$0.443229\pi$
$230$	0	0
$231$	−50.6745	−3.33414
$232$	0	0
$233$	18.7838	1.23057	0.615284	−	0.788306i	$-0.289041\pi$
$233$	18.7838	1.23057	0.615284	+	0.788306i	$0.289041\pi$
$234$	0	0
$235$	20.0746	1.30952
$236$	0	0
$237$	−8.48713	−0.551298
$238$	0	0
$239$	22.1621	1.43355	0.716774	−	0.697306i	$-0.245618\pi$
$239$	22.1621	1.43355	0.716774	+	0.697306i	$0.245618\pi$
$240$	0	0
$241$	3.32558	0.214219	0.107110	−	0.994247i	$-0.465840\pi$
$241$	3.32558	0.214219	0.107110	+	0.994247i	$0.465840\pi$
$242$	0	0
$243$	−15.9818	−1.02524
$244$	0	0
$245$	−10.1169	−0.646343
$246$	0	0
$247$	−20.6896	−1.31645
$248$	0	0
$249$	−25.6421	−1.62500
$250$	0	0
$251$	11.8881	0.750369	0.375185	−	0.926950i	$-0.377579\pi$
$251$	11.8881	0.750369	0.375185	+	0.926950i	$0.377579\pi$
$252$	0	0
$253$	20.2777	1.27485
$254$	0	0
$255$	−3.37962	−0.211640
$256$	0	0
$257$	−1.28171	−0.0799511	−0.0399755	−	0.999201i	$-0.512728\pi$
$257$	−1.28171	−0.0799511	−0.0399755	+	0.999201i	$0.512728\pi$
$258$	0	0
$259$	14.0711	0.874335
$260$	0	0
$261$	−9.79735	−0.606441
$262$	0	0
$263$	14.2974	0.881613	0.440806	−	0.897602i	$-0.354693\pi$
$263$	14.2974	0.881613	0.440806	+	0.897602i	$0.354693\pi$
$264$	0	0
$265$	−2.21197	−0.135880
$266$	0	0
$267$	27.2390	1.66700
$268$	0	0
$269$	−11.6736	−0.711751	−0.355875	−	0.934533i	$-0.615817\pi$
$269$	−11.6736	−0.711751	−0.355875	+	0.934533i	$0.615817\pi$
$270$	0	0
$271$	9.26307	0.562692	0.281346	−	0.959606i	$-0.409219\pi$
$271$	9.26307	0.562692	0.281346	+	0.959606i	$0.409219\pi$
$272$	0	0
$273$	−33.1406	−2.00576
$274$	0	0
$275$	16.3752	0.987463
$276$	0	0
$277$	−5.13477	−0.308518	−0.154259	−	0.988030i	$-0.549299\pi$
$277$	−5.13477	−0.308518	−0.154259	+	0.988030i	$0.549299\pi$
$278$	0	0
$279$	−1.27052	−0.0760643
$280$	0	0
$281$	−13.9831	−0.834163	−0.417082	−	0.908869i	$-0.636947\pi$
$281$	−13.9831	−0.834163	−0.417082	+	0.908869i	$0.636947\pi$
$282$	0	0
$283$	−1.14870	−0.0682832	−0.0341416	−	0.999417i	$-0.510870\pi$
$283$	−1.14870	−0.0682832	−0.0341416	+	0.999417i	$0.510870\pi$
$284$	0	0
$285$	−16.8408	−0.997564
$286$	0	0
$287$	−37.9322	−2.23907
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	−25.5340	−1.49683
$292$	0	0
$293$	−20.4455	−1.19444	−0.597219	−	0.802078i	$-0.703728\pi$
$293$	−20.4455	−1.19444	−0.597219	+	0.802078i	$0.703728\pi$
$294$	0	0
$295$	1.55586	0.0905857
$296$	0	0
$297$	17.6741	1.02555
$298$	0	0
$299$	13.2615	0.766930
$300$	0	0
$301$	7.36697	0.424625
$302$	0	0
$303$	−12.7086	−0.730088
$304$	0	0
$305$	−10.9792	−0.628667
$306$	0	0
$307$	10.4102	0.594143	0.297072	−	0.954855i	$-0.403990\pi$
$307$	10.4102	0.594143	0.297072	+	0.954855i	$0.403990\pi$
$308$	0	0
$309$	−34.3288	−1.95290
$310$	0	0
$311$	−0.411745	−0.0233479	−0.0116740	−	0.999932i	$-0.503716\pi$
$311$	−0.411745	−0.0233479	−0.0116740	+	0.999932i	$0.503716\pi$
$312$	0	0
$313$	−33.9391	−1.91835	−0.959175	−	0.282814i	$-0.908732\pi$
$313$	−33.9391	−1.91835	−0.959175	+	0.282814i	$0.908732\pi$
$314$	0	0
$315$	−9.82427	−0.553535
$316$	0	0
$317$	35.2112	1.97766	0.988828	−	0.149060i	$-0.0476248\pi$
$317$	35.2112	1.97766	0.988828	+	0.149060i	$0.0476248\pi$
$318$	0	0
$319$	36.1969	2.02664
$320$	0	0
$321$	2.90676	0.162240
$322$	0	0
$323$	4.98305	0.277264
$324$	0	0
$325$	10.7092	0.594042
$326$	0	0
$327$	−24.2525	−1.34117
$328$	0	0
$329$	−47.4113	−2.61387
$330$	0	0
$331$	9.12087	0.501328	0.250664	−	0.968074i	$-0.419351\pi$
$331$	9.12087	0.501328	0.250664	+	0.968074i	$0.419351\pi$
$332$	0	0
$333$	6.58030	0.360598
$334$	0	0
$335$	−21.4570	−1.17232
$336$	0	0
$337$	−22.4333	−1.22202	−0.611009	−	0.791624i	$-0.709236\pi$
$337$	−22.4333	−1.22202	−0.611009	+	0.791624i	$0.709236\pi$
$338$	0	0
$339$	−42.4203	−2.30396
$340$	0	0
$341$	4.69403	0.254196
$342$	0	0
$343$	−1.82837	−0.0987227
$344$	0	0
$345$	10.7945	0.581155
$346$	0	0
$347$	−28.9023	−1.55156	−0.775780	−	0.631004i	$-0.782643\pi$
$347$	−28.9023	−1.55156	−0.775780	+	0.631004i	$0.782643\pi$
$348$	0	0
$349$	3.32482	0.177973	0.0889867	−	0.996033i	$-0.471637\pi$
$349$	3.32482	0.177973	0.0889867	+	0.996033i	$0.471637\pi$
$350$	0	0
$351$	11.5587	0.616957
$352$	0	0
$353$	−31.2964	−1.66574	−0.832871	−	0.553468i	$-0.813304\pi$
$353$	−31.2964	−1.66574	−0.832871	+	0.553468i	$0.813304\pi$
$354$	0	0
$355$	−19.2852	−1.02355
$356$	0	0
$357$	7.98184	0.422444
$358$	0	0
$359$	−21.1650	−1.11704	−0.558522	−	0.829490i	$-0.688631\pi$
$359$	−21.1650	−1.11704	−0.558522	+	0.829490i	$0.688631\pi$
$360$	0	0
$361$	5.83078	0.306883
$362$	0	0
$363$	63.6586	3.34121
$364$	0	0
$365$	15.0694	0.788769
$366$	0	0
$367$	24.1767	1.26201	0.631007	−	0.775777i	$-0.282642\pi$
$367$	24.1767	1.26201	0.631007	+	0.775777i	$0.282642\pi$
$368$	0	0
$369$	−17.7389	−0.923448
$370$	0	0
$371$	5.22412	0.271223
$372$	0	0
$373$	8.52147	0.441225	0.220613	−	0.975362i	$-0.429194\pi$
$373$	8.52147	0.441225	0.220613	+	0.975362i	$0.429194\pi$
$374$	0	0
$375$	25.6152	1.32276
$376$	0	0
$377$	23.6724	1.21919
$378$	0	0
$379$	−8.65777	−0.444720	−0.222360	−	0.974965i	$-0.571376\pi$
$379$	−8.65777	−0.444720	−0.222360	+	0.974965i	$0.571376\pi$
$380$	0	0
$381$	33.8410	1.73373
$382$	0	0
$383$	13.9393	0.712266	0.356133	−	0.934435i	$-0.384095\pi$
$383$	13.9393	0.712266	0.356133	+	0.934435i	$0.384095\pi$
$384$	0	0
$385$	36.2963	1.84983
$386$	0	0
$387$	3.44514	0.175126
$388$	0	0
$389$	−6.27194	−0.318000	−0.159000	−	0.987279i	$-0.550827\pi$
$389$	−6.27194	−0.318000	−0.159000	+	0.987279i	$0.550827\pi$
$390$	0	0
$391$	−3.19399	−0.161527
$392$	0	0
$393$	21.8557	1.10247
$394$	0	0
$395$	6.07903	0.305869
$396$	0	0
$397$	35.1419	1.76372	0.881861	−	0.471509i	$-0.156291\pi$
$397$	35.1419	1.76372	0.881861	+	0.471509i	$0.156291\pi$
$398$	0	0
$399$	39.7739	1.99119
$400$	0	0
$401$	−14.5893	−0.728555	−0.364277	−	0.931291i	$-0.618684\pi$
$401$	−14.5893	−0.728555	−0.364277	+	0.931291i	$0.618684\pi$
$402$	0	0
$403$	3.06985	0.152920
$404$	0	0
$405$	17.4292	0.866065
$406$	0	0
$407$	−24.3113	−1.20507
$408$	0	0
$409$	−1.91669	−0.0947744	−0.0473872	−	0.998877i	$-0.515089\pi$
$409$	−1.91669	−0.0947744	−0.0473872	+	0.998877i	$0.515089\pi$
$410$	0	0
$411$	−17.2793	−0.852327
$412$	0	0
$413$	−3.67456	−0.180814
$414$	0	0
$415$	18.3666	0.901579
$416$	0	0
$417$	−2.62845	−0.128716
$418$	0	0
$419$	−15.7408	−0.768989	−0.384495	−	0.923127i	$-0.625624\pi$
$419$	−15.7408	−0.768989	−0.384495	+	0.923127i	$0.625624\pi$
$420$	0	0
$421$	3.54380	0.172714	0.0863570	−	0.996264i	$-0.472477\pi$
$421$	3.54380	0.172714	0.0863570	+	0.996264i	$0.472477\pi$
$422$	0	0
$423$	−22.1717	−1.07803
$424$	0	0
$425$	−2.57930	−0.125114
$426$	0	0
$427$	25.9302	1.25485
$428$	0	0
$429$	57.2586	2.76447
$430$	0	0
$431$	−27.5323	−1.32619	−0.663093	−	0.748537i	$-0.730757\pi$
$431$	−27.5323	−1.32619	−0.663093	+	0.748537i	$0.730757\pi$
$432$	0	0
$433$	22.6816	1.09001	0.545003	−	0.838434i	$-0.316528\pi$
$433$	22.6816	1.09001	0.545003	+	0.838434i	$0.316528\pi$
$434$	0	0
$435$	19.2687	0.923866
$436$	0	0
$437$	−15.9158	−0.761356
$438$	0	0
$439$	6.23916	0.297779	0.148890	−	0.988854i	$-0.452430\pi$
$439$	6.23916	0.297779	0.148890	+	0.988854i	$0.452430\pi$
$440$	0	0
$441$	11.1737	0.532083
$442$	0	0
$443$	−11.3248	−0.538055	−0.269028	−	0.963132i	$-0.586702\pi$
$443$	−11.3248	−0.538055	−0.269028	+	0.963132i	$0.586702\pi$
$444$	0	0
$445$	−19.5103	−0.924879
$446$	0	0
$447$	−25.7497	−1.21792
$448$	0	0
$449$	22.1388	1.04479	0.522397	−	0.852702i	$-0.325038\pi$
$449$	22.1388	1.04479	0.522397	+	0.852702i	$0.325038\pi$
$450$	0	0
$451$	65.5373	3.08603
$452$	0	0
$453$	19.5203	0.917146
$454$	0	0
$455$	23.7375	1.11283
$456$	0	0
$457$	−26.9089	−1.25875	−0.629373	−	0.777103i	$-0.716688\pi$
$457$	−26.9089	−1.25875	−0.629373	+	0.777103i	$0.716688\pi$
$458$	0	0
$459$	−2.78388	−0.129941
$460$	0	0
$461$	−8.27862	−0.385574	−0.192787	−	0.981241i	$-0.561753\pi$
$461$	−8.27862	−0.385574	−0.192787	+	0.981241i	$0.561753\pi$
$462$	0	0
$463$	−22.4858	−1.04500	−0.522501	−	0.852639i	$-0.675001\pi$
$463$	−22.4858	−1.04500	−0.522501	+	0.852639i	$0.675001\pi$
$464$	0	0
$465$	2.49878	0.115878
$466$	0	0
$467$	−37.1288	−1.71812	−0.859058	−	0.511877i	$-0.828950\pi$
$467$	−37.1288	−1.71812	−0.859058	+	0.511877i	$0.828950\pi$
$468$	0	0
$469$	50.6762	2.34001
$470$	0	0
$471$	35.9622	1.65705
$472$	0	0
$473$	−12.7283	−0.585246
$474$	0	0
$475$	−12.8528	−0.589725
$476$	0	0
$477$	2.44304	0.111859
$478$	0	0
$479$	13.9484	0.637319	0.318659	−	0.947869i	$-0.396767\pi$
$479$	13.9484	0.637319	0.318659	+	0.947869i	$0.396767\pi$
$480$	0	0
$481$	−15.8994	−0.724949
$482$	0	0
$483$	−25.4939	−1.16001
$484$	0	0
$485$	18.2891	0.830465
$486$	0	0
$487$	−9.86606	−0.447074	−0.223537	−	0.974695i	$-0.571760\pi$
$487$	−9.86606	−0.447074	−0.223537	+	0.974695i	$0.571760\pi$
$488$	0	0
$489$	−10.8844	−0.492211
$490$	0	0
$491$	0.722035	0.0325850	0.0162925	−	0.999867i	$-0.494814\pi$
$491$	0.722035	0.0325850	0.0162925	+	0.999867i	$0.494814\pi$
$492$	0	0
$493$	−5.70145	−0.256780
$494$	0	0
$495$	16.9738	0.762918
$496$	0	0
$497$	45.5470	2.04306
$498$	0	0
$499$	−12.3555	−0.553109	−0.276555	−	0.960998i	$-0.589193\pi$
$499$	−12.3555	−0.553109	−0.276555	+	0.960998i	$0.589193\pi$
$500$	0	0
$501$	17.6970	0.790645
$502$	0	0
$503$	12.4241	0.553963	0.276982	−	0.960875i	$-0.410666\pi$
$503$	12.4241	0.553963	0.276982	+	0.960875i	$0.410666\pi$
$504$	0	0
$505$	9.10270	0.405065
$506$	0	0
$507$	9.20820	0.408950
$508$	0	0
$509$	11.0920	0.491645	0.245822	−	0.969315i	$-0.420942\pi$
$509$	11.0920	0.491645	0.245822	+	0.969315i	$0.420942\pi$
$510$	0	0
$511$	−35.5903	−1.57442
$512$	0	0
$513$	−13.8722	−0.612474
$514$	0	0
$515$	24.5885	1.08350
$516$	0	0
$517$	81.9148	3.60261
$518$	0	0
$519$	−8.92118	−0.391596
$520$	0	0
$521$	18.5403	0.812264	0.406132	−	0.913814i	$-0.366877\pi$
$521$	18.5403	0.812264	0.406132	+	0.913814i	$0.366877\pi$
$522$	0	0
$523$	0.716287	0.0313210	0.0156605	−	0.999877i	$-0.495015\pi$
$523$	0.716287	0.0313210	0.0156605	+	0.999877i	$0.495015\pi$
$524$	0	0
$525$	−20.5875	−0.898513
$526$	0	0
$527$	−0.739366	−0.0322073
$528$	0	0
$529$	−12.7984	−0.556454
$530$	0	0
$531$	−1.71840	−0.0745720
$532$	0	0
$533$	42.8608	1.85651
$534$	0	0
$535$	−2.08201	−0.0900133
$536$	0	0
$537$	46.5114	2.00711
$538$	0	0
$539$	−41.2821	−1.77814
$540$	0	0
$541$	20.8644	0.897030	0.448515	−	0.893775i	$-0.351953\pi$
$541$	20.8644	0.897030	0.448515	+	0.893775i	$0.351953\pi$
$542$	0	0
$543$	−48.1823	−2.06770
$544$	0	0
$545$	17.3712	0.744101
$546$	0	0
$547$	−17.6879	−0.756282	−0.378141	−	0.925748i	$-0.623437\pi$
$547$	−17.6879	−0.756282	−0.378141	+	0.925748i	$0.623437\pi$
$548$	0	0
$549$	12.1262	0.517531
$550$	0	0
$551$	−28.4106	−1.21033
$552$	0	0
$553$	−14.3572	−0.610530
$554$	0	0
$555$	−12.9417	−0.549343
$556$	0	0
$557$	22.5750	0.956534	0.478267	−	0.878214i	$-0.341265\pi$
$557$	22.5750	0.956534	0.478267	+	0.878214i	$0.341265\pi$
$558$	0	0
$559$	−8.32417	−0.352075
$560$	0	0
$561$	−13.7906	−0.582240
$562$	0	0
$563$	36.4613	1.53666	0.768331	−	0.640053i	$-0.221088\pi$
$563$	36.4613	1.53666	0.768331	+	0.640053i	$0.221088\pi$
$564$	0	0
$565$	30.3842	1.27827
$566$	0	0
$567$	−41.1636	−1.72871
$568$	0	0
$569$	−31.8765	−1.33633	−0.668167	−	0.744012i	$-0.732921\pi$
$569$	−31.8765	−1.33633	−0.668167	+	0.744012i	$0.732921\pi$
$570$	0	0
$571$	−9.98426	−0.417828	−0.208914	−	0.977934i	$-0.566993\pi$
$571$	−9.98426	−0.417828	−0.208914	+	0.977934i	$0.566993\pi$
$572$	0	0
$573$	31.5576	1.31834
$574$	0	0
$575$	8.23824	0.343558
$576$	0	0
$577$	−5.00971	−0.208557	−0.104278	−	0.994548i	$-0.533253\pi$
$577$	−5.00971	−0.208557	−0.104278	+	0.994548i	$0.533253\pi$
$578$	0	0
$579$	−7.88941	−0.327873
$580$	0	0
$581$	−43.3773	−1.79959
$582$	0	0
$583$	−9.02597	−0.373817
$584$	0	0
$585$	11.1007	0.458959
$586$	0	0
$587$	−0.694150	−0.0286506	−0.0143253	−	0.999897i	$-0.504560\pi$
$587$	−0.694150	−0.0286506	−0.0143253	+	0.999897i	$0.504560\pi$
$588$	0	0
$589$	−3.68430	−0.151809
$590$	0	0
$591$	14.9803	0.616205
$592$	0	0
$593$	−37.0937	−1.52325	−0.761627	−	0.648015i	$-0.775599\pi$
$593$	−37.0937	−1.52325	−0.761627	+	0.648015i	$0.775599\pi$
$594$	0	0
$595$	−5.71711	−0.234379
$596$	0	0
$597$	−47.5219	−1.94494
$598$	0	0
$599$	11.8762	0.485247	0.242623	−	0.970121i	$-0.421992\pi$
$599$	11.8762	0.485247	0.242623	+	0.970121i	$0.421992\pi$
$600$	0	0
$601$	30.3058	1.23620	0.618100	−	0.786100i	$-0.287903\pi$
$601$	30.3058	1.23620	0.618100	+	0.786100i	$0.287903\pi$
$602$	0	0
$603$	23.6985	0.965078
$604$	0	0
$605$	−45.5964	−1.85376
$606$	0	0
$607$	−29.9695	−1.21642	−0.608212	−	0.793775i	$-0.708113\pi$
$607$	−29.9695	−1.21642	−0.608212	+	0.793775i	$0.708113\pi$
$608$	0	0
$609$	−45.5081	−1.84408
$610$	0	0
$611$	53.5715	2.16727
$612$	0	0
$613$	−35.9928	−1.45373	−0.726867	−	0.686778i	$-0.759024\pi$
$613$	−35.9928	−1.45373	−0.726867	+	0.686778i	$0.759024\pi$
$614$	0	0
$615$	34.8875	1.40680
$616$	0	0
$617$	29.0716	1.17038	0.585190	−	0.810896i	$-0.301020\pi$
$617$	29.0716	1.17038	0.585190	+	0.810896i	$0.301020\pi$
$618$	0	0
$619$	−9.14834	−0.367703	−0.183851	−	0.982954i	$-0.558857\pi$
$619$	−9.14834	−0.367703	−0.183851	+	0.982954i	$0.558857\pi$
$620$	0	0
$621$	8.89169	0.356811
$622$	0	0
$623$	46.0787	1.84610
$624$	0	0
$625$	−5.45076	−0.218030
$626$	0	0
$627$	−68.7193	−2.74438
$628$	0	0
$629$	3.82932	0.152685
$630$	0	0
$631$	−2.28915	−0.0911295	−0.0455647	−	0.998961i	$-0.514509\pi$
$631$	−2.28915	−0.0911295	−0.0455647	+	0.998961i	$0.514509\pi$
$632$	0	0
$633$	9.94668	0.395345
$634$	0	0
$635$	−24.2391	−0.961899
$636$	0	0
$637$	−26.9981	−1.06970
$638$	0	0
$639$	21.2999	0.842610
$640$	0	0
$641$	18.5173	0.731390	0.365695	−	0.930735i	$-0.380831\pi$
$641$	18.5173	0.731390	0.365695	+	0.930735i	$0.380831\pi$
$642$	0	0
$643$	10.9655	0.432435	0.216218	−	0.976345i	$-0.430628\pi$
$643$	10.9655	0.432435	0.216218	+	0.976345i	$0.430628\pi$
$644$	0	0
$645$	−6.77566	−0.266791
$646$	0	0
$647$	−49.3691	−1.94090	−0.970450	−	0.241302i	$-0.922425\pi$
$647$	−49.3691	−1.94090	−0.970450	+	0.241302i	$0.922425\pi$
$648$	0	0
$649$	6.34872	0.249209
$650$	0	0
$651$	−5.90150	−0.231298
$652$	0	0
$653$	41.7378	1.63333	0.816663	−	0.577115i	$-0.195822\pi$
$653$	41.7378	1.63333	0.816663	+	0.577115i	$0.195822\pi$
$654$	0	0
$655$	−15.6545	−0.611670
$656$	0	0
$657$	−16.6437	−0.649331
$658$	0	0
$659$	7.44303	0.289939	0.144970	−	0.989436i	$-0.453692\pi$
$659$	7.44303	0.289939	0.144970	+	0.989436i	$0.453692\pi$
$660$	0	0
$661$	34.4324	1.33926	0.669632	−	0.742693i	$-0.266452\pi$
$661$	34.4324	1.33926	0.669632	+	0.742693i	$0.266452\pi$
$662$	0	0
$663$	−9.01893	−0.350266
$664$	0	0
$665$	−28.4887	−1.10474
$666$	0	0
$667$	18.2104	0.705108
$668$	0	0
$669$	−52.8595	−2.04366
$670$	0	0
$671$	−44.8008	−1.72952
$672$	0	0
$673$	−44.0615	−1.69845	−0.849224	−	0.528033i	$-0.822930\pi$
$673$	−44.0615	−1.69845	−0.849224	+	0.528033i	$0.822930\pi$
$674$	0	0
$675$	7.18046	0.276376
$676$	0	0
$677$	36.0848	1.38685	0.693425	−	0.720529i	$-0.256101\pi$
$677$	36.0848	1.38685	0.693425	+	0.720529i	$0.256101\pi$
$678$	0	0
$679$	−43.1944	−1.65765
$680$	0	0
$681$	34.5318	1.32326
$682$	0	0
$683$	−12.1107	−0.463404	−0.231702	−	0.972787i	$-0.574429\pi$
$683$	−12.1107	−0.463404	−0.231702	+	0.972787i	$0.574429\pi$
$684$	0	0
$685$	12.3766	0.472885
$686$	0	0
$687$	11.6632	0.444978
$688$	0	0
$689$	−5.90290	−0.224883
$690$	0	0
$691$	40.4179	1.53757	0.768784	−	0.639508i	$-0.220862\pi$
$691$	40.4179	1.53757	0.768784	+	0.639508i	$0.220862\pi$
$692$	0	0
$693$	−40.0881	−1.52282
$694$	0	0
$695$	1.88266	0.0714135
$696$	0	0
$697$	−10.3229	−0.391008
$698$	0	0
$699$	40.8019	1.54327
$700$	0	0
$701$	−14.8532	−0.560998	−0.280499	−	0.959854i	$-0.590500\pi$
$701$	−14.8532	−0.560998	−0.280499	+	0.959854i	$0.590500\pi$
$702$	0	0
$703$	19.0817	0.719680
$704$	0	0
$705$	43.6058	1.64229
$706$	0	0
$707$	−21.4984	−0.808529
$708$	0	0
$709$	4.99579	0.187621	0.0938104	−	0.995590i	$-0.470095\pi$
$709$	4.99579	0.187621	0.0938104	+	0.995590i	$0.470095\pi$
$710$	0	0
$711$	−6.71409	−0.251798
$712$	0	0
$713$	2.36153	0.0884399
$714$	0	0
$715$	−41.0124	−1.53378
$716$	0	0
$717$	48.1403	1.79783
$718$	0	0
$719$	44.3096	1.65247	0.826235	−	0.563325i	$-0.190478\pi$
$719$	44.3096	1.65247	0.826235	+	0.563325i	$0.190478\pi$
$720$	0	0
$721$	−58.0721	−2.16272
$722$	0	0
$723$	7.22377	0.268655
$724$	0	0
$725$	14.7057	0.546157
$726$	0	0
$727$	19.5577	0.725356	0.362678	−	0.931915i	$-0.381863\pi$
$727$	19.5577	0.725356	0.362678	+	0.931915i	$0.381863\pi$
$728$	0	0
$729$	−1.10865	−0.0410613
$730$	0	0
$731$	2.00486	0.0741523
$732$	0	0
$733$	40.9675	1.51317	0.756585	−	0.653896i	$-0.226867\pi$
$733$	40.9675	1.51317	0.756585	+	0.653896i	$0.226867\pi$
$734$	0	0
$735$	−21.9757	−0.810587
$736$	0	0
$737$	−87.5556	−3.22515
$738$	0	0
$739$	−39.2411	−1.44351	−0.721753	−	0.692150i	$-0.756663\pi$
$739$	−39.2411	−1.44351	−0.721753	+	0.692150i	$0.756663\pi$
$740$	0	0
$741$	−44.9418	−1.65098
$742$	0	0
$743$	43.5571	1.59796	0.798978	−	0.601360i	$-0.205374\pi$
$743$	43.5571	1.59796	0.798978	+	0.601360i	$0.205374\pi$
$744$	0	0
$745$	18.4436	0.675721
$746$	0	0
$747$	−20.2852	−0.742198
$748$	0	0
$749$	4.91721	0.179671
$750$	0	0
$751$	−1.53090	−0.0558635	−0.0279317	−	0.999610i	$-0.508892\pi$
$751$	−1.53090	−0.0558635	−0.0279317	+	0.999610i	$0.508892\pi$
$752$	0	0
$753$	25.8231	0.941048
$754$	0	0
$755$	−13.9817	−0.508848
$756$	0	0
$757$	48.6451	1.76804	0.884019	−	0.467451i	$-0.154828\pi$
$757$	48.6451	1.76804	0.884019	+	0.467451i	$0.154828\pi$
$758$	0	0
$759$	44.0470	1.59881
$760$	0	0
$761$	−7.11155	−0.257794	−0.128897	−	0.991658i	$-0.541144\pi$
$761$	−7.11155	−0.257794	−0.128897	+	0.991658i	$0.541144\pi$
$762$	0	0
$763$	−41.0266	−1.48526
$764$	0	0
$765$	−2.67359	−0.0966637
$766$	0	0
$767$	4.15200	0.149920
$768$	0	0
$769$	−3.72771	−0.134425	−0.0672123	−	0.997739i	$-0.521410\pi$
$769$	−3.72771	−0.134425	−0.0672123	+	0.997739i	$0.521410\pi$
$770$	0	0
$771$	−2.78412	−0.100268
$772$	0	0
$773$	22.4179	0.806316	0.403158	−	0.915130i	$-0.367912\pi$
$773$	22.4179	0.806316	0.403158	+	0.915130i	$0.367912\pi$
$774$	0	0
$775$	1.90704	0.0685030
$776$	0	0
$777$	30.5651	1.09652
$778$	0	0
$779$	−51.4396	−1.84301
$780$	0	0
$781$	−78.6937	−2.81588
$782$	0	0
$783$	15.8722	0.567225
$784$	0	0
$785$	−25.7584	−0.919358
$786$	0	0
$787$	−36.3330	−1.29513	−0.647565	−	0.762010i	$-0.724213\pi$
$787$	−36.3330	−1.29513	−0.647565	+	0.762010i	$0.724213\pi$
$788$	0	0
$789$	31.0565	1.10564
$790$	0	0
$791$	−71.7601	−2.55149
$792$	0	0
$793$	−29.2993	−1.04045
$794$	0	0
$795$	−4.80480	−0.170409
$796$	0	0
$797$	6.75510	0.239278	0.119639	−	0.992817i	$-0.461826\pi$
$797$	6.75510	0.239278	0.119639	+	0.992817i	$0.461826\pi$
$798$	0	0
$799$	−12.9026	−0.456460
$800$	0	0
$801$	21.5485	0.761380
$802$	0	0
$803$	61.4910	2.16997
$804$	0	0
$805$	18.2604	0.643594
$806$	0	0
$807$	−25.3572	−0.892616
$808$	0	0
$809$	34.0945	1.19870	0.599350	−	0.800487i	$-0.295426\pi$
$809$	34.0945	1.19870	0.599350	+	0.800487i	$0.295426\pi$
$810$	0	0
$811$	−20.6278	−0.724342	−0.362171	−	0.932112i	$-0.617964\pi$
$811$	−20.6278	−0.724342	−0.362171	+	0.932112i	$0.617964\pi$
$812$	0	0
$813$	20.1211	0.705679
$814$	0	0
$815$	7.79614	0.273087
$816$	0	0
$817$	9.99030	0.349516
$818$	0	0
$819$	−26.2172	−0.916105
$820$	0	0
$821$	−22.1821	−0.774161	−0.387081	−	0.922046i	$-0.626517\pi$
$821$	−22.1821	−0.774161	−0.387081	+	0.922046i	$0.626517\pi$
$822$	0	0
$823$	12.9803	0.452464	0.226232	−	0.974073i	$-0.427359\pi$
$823$	12.9803	0.452464	0.226232	+	0.974073i	$0.427359\pi$
$824$	0	0
$825$	35.5700	1.23839
$826$	0	0
$827$	−34.2437	−1.19077	−0.595385	−	0.803440i	$-0.703001\pi$
$827$	−34.2437	−1.19077	−0.595385	+	0.803440i	$0.703001\pi$
$828$	0	0
$829$	23.0777	0.801521	0.400761	−	0.916183i	$-0.368746\pi$
$829$	23.0777	0.801521	0.400761	+	0.916183i	$0.368746\pi$
$830$	0	0
$831$	−11.1537	−0.386917
$832$	0	0
$833$	6.50243	0.225296
$834$	0	0
$835$	−12.6758	−0.438663
$836$	0	0
$837$	2.05831	0.0711455
$838$	0	0
$839$	−34.3385	−1.18550	−0.592749	−	0.805388i	$-0.701957\pi$
$839$	−34.3385	−1.18550	−0.592749	+	0.805388i	$0.701957\pi$
$840$	0	0
$841$	3.50653	0.120915
$842$	0	0
$843$	−30.3740	−1.04613
$844$	0	0
$845$	−6.59551	−0.226892
$846$	0	0
$847$	107.688	3.70019
$848$	0	0
$849$	−2.49520	−0.0856349
$850$	0	0
$851$	−12.2308	−0.419267
$852$	0	0
$853$	6.78494	0.232312	0.116156	−	0.993231i	$-0.462943\pi$
$853$	6.78494	0.232312	0.116156	+	0.993231i	$0.462943\pi$
$854$	0	0
$855$	−13.3226	−0.455624
$856$	0	0
$857$	18.0374	0.616144	0.308072	−	0.951363i	$-0.400316\pi$
$857$	18.0374	0.616144	0.308072	+	0.951363i	$0.400316\pi$
$858$	0	0
$859$	12.2942	0.419474	0.209737	−	0.977758i	$-0.432739\pi$
$859$	12.2942	0.419474	0.209737	+	0.977758i	$0.432739\pi$
$860$	0	0
$861$	−82.3959	−2.80804
$862$	0	0
$863$	46.0910	1.56895	0.784477	−	0.620157i	$-0.212931\pi$
$863$	46.0910	1.56895	0.784477	+	0.620157i	$0.212931\pi$
$864$	0	0
$865$	6.38993	0.217264
$866$	0	0
$867$	2.17219	0.0737713
$868$	0	0
$869$	24.8056	0.841472
$870$	0	0
$871$	−57.2606	−1.94020
$872$	0	0
$873$	−20.1997	−0.683656
$874$	0	0
$875$	43.3317	1.46488
$876$	0	0
$877$	13.4575	0.454427	0.227213	−	0.973845i	$-0.427039\pi$
$877$	13.4575	0.454427	0.227213	+	0.973845i	$0.427039\pi$
$878$	0	0
$879$	−44.4114	−1.49796
$880$	0	0
$881$	14.8807	0.501345	0.250672	−	0.968072i	$-0.419348\pi$
$881$	14.8807	0.501345	0.250672	+	0.968072i	$0.419348\pi$
$882$	0	0
$883$	44.3490	1.49246	0.746231	−	0.665687i	$-0.231861\pi$
$883$	44.3490	1.49246	0.746231	+	0.665687i	$0.231861\pi$
$884$	0	0
$885$	3.37962	0.113605
$886$	0	0
$887$	−41.8672	−1.40576	−0.702881	−	0.711307i	$-0.748103\pi$
$887$	−41.8672	−1.40576	−0.702881	+	0.711307i	$0.748103\pi$
$888$	0	0
$889$	57.2468	1.92000
$890$	0	0
$891$	71.1203	2.38262
$892$	0	0
$893$	−64.2941	−2.15152
$894$	0	0
$895$	−33.3145	−1.11358
$896$	0	0
$897$	28.8064	0.961816
$898$	0	0
$899$	4.21546	0.140593
$900$	0	0
$901$	1.42170	0.0473637
$902$	0	0
$903$	16.0024	0.532528
$904$	0	0
$905$	34.5113	1.14719
$906$	0	0
$907$	−17.6391	−0.585698	−0.292849	−	0.956159i	$-0.594603\pi$
$907$	−17.6391	−0.585698	−0.292849	+	0.956159i	$0.594603\pi$
$908$	0	0
$909$	−10.0536	−0.333458
$910$	0	0
$911$	3.03791	0.100651	0.0503253	−	0.998733i	$-0.483974\pi$
$911$	3.03791	0.100651	0.0503253	+	0.998733i	$0.483974\pi$
$912$	0	0
$913$	74.9451	2.48032
$914$	0	0
$915$	−23.8489	−0.788419
$916$	0	0
$917$	36.9720	1.22092
$918$	0	0
$919$	−23.6673	−0.780714	−0.390357	−	0.920664i	$-0.627648\pi$
$919$	−23.6673	−0.780714	−0.390357	+	0.920664i	$0.627648\pi$
$920$	0	0
$921$	22.6130	0.745122
$922$	0	0
$923$	−51.4650	−1.69399
$924$	0	0
$925$	−9.87696	−0.324752
$926$	0	0
$927$	−27.1572	−0.891959
$928$	0	0
$929$	9.26938	0.304119	0.152059	−	0.988371i	$-0.451410\pi$
$929$	9.26938	0.304119	0.152059	+	0.988371i	$0.451410\pi$
$930$	0	0
$931$	32.4019	1.06193
$932$	0	0
$933$	−0.894387	−0.0292809
$934$	0	0
$935$	9.87773	0.323036
$936$	0	0
$937$	47.0278	1.53633	0.768166	−	0.640251i	$-0.221170\pi$
$937$	47.0278	1.53633	0.768166	+	0.640251i	$0.221170\pi$
$938$	0	0
$939$	−73.7220	−2.40583
$940$	0	0
$941$	−19.8894	−0.648377	−0.324189	−	0.945992i	$-0.605091\pi$
$941$	−19.8894	−0.648377	−0.324189	+	0.945992i	$0.605091\pi$
$942$	0	0
$943$	32.9713	1.07369
$944$	0	0
$945$	15.9158	0.517740
$946$	0	0
$947$	24.9644	0.811234	0.405617	−	0.914043i	$-0.367057\pi$
$947$	24.9644	0.811234	0.405617	+	0.914043i	$0.367057\pi$
$948$	0	0
$949$	40.2146	1.30542
$950$	0	0
$951$	76.4852	2.48020
$952$	0	0
$953$	5.09494	0.165041	0.0825207	−	0.996589i	$-0.473703\pi$
$953$	5.09494	0.165041	0.0825207	+	0.996589i	$0.473703\pi$
$954$	0	0
$955$	−22.6036	−0.731435
$956$	0	0
$957$	78.6264	2.54163
$958$	0	0
$959$	−29.2305	−0.943901
$960$	0	0
$961$	−30.4533	−0.982366
$962$	0	0
$963$	2.29951	0.0741008
$964$	0	0
$965$	5.65090	0.181909
$966$	0	0
$967$	7.96939	0.256278	0.128139	−	0.991756i	$-0.459100\pi$
$967$	7.96939	0.256278	0.128139	+	0.991756i	$0.459100\pi$
$968$	0	0
$969$	10.8241	0.347721
$970$	0	0
$971$	41.7000	1.33822	0.669109	−	0.743165i	$-0.266676\pi$
$971$	41.7000	1.33822	0.669109	+	0.743165i	$0.266676\pi$
$972$	0	0
$973$	−4.44639	−0.142545
$974$	0	0
$975$	23.2625	0.744996
$976$	0	0
$977$	5.24777	0.167891	0.0839455	−	0.996470i	$-0.473248\pi$
$977$	5.24777	0.167891	0.0839455	+	0.996470i	$0.473248\pi$
$978$	0	0
$979$	−79.6123	−2.54442
$980$	0	0
$981$	−19.1859	−0.612559
$982$	0	0
$983$	48.0403	1.53225	0.766124	−	0.642693i	$-0.222183\pi$
$983$	48.0403	1.53225	0.766124	+	0.642693i	$0.222183\pi$
$984$	0	0
$985$	−10.7298	−0.341881
$986$	0	0
$987$	−102.986	−3.27809
$988$	0	0
$989$	−6.40349	−0.203619
$990$	0	0
$991$	−46.2170	−1.46813	−0.734066	−	0.679078i	$-0.762380\pi$
$991$	−46.2170	−1.46813	−0.734066	+	0.679078i	$0.762380\pi$
$992$	0	0
$993$	19.8122	0.628722
$994$	0	0
$995$	34.0383	1.07909
$996$	0	0
$997$	26.7402	0.846872	0.423436	−	0.905926i	$-0.360824\pi$
$997$	26.7402	0.846872	0.423436	+	0.905926i	$0.360824\pi$
$998$	0	0
$999$	−10.6604	−0.337280

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4012.2.a.h.1.14	✓	15

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4012.2.a.h.1.14	✓	15	1.1	even	1	trivial

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

\(f(q)\)	\(=\)	\(q+2.17219 q^{3} -1.55586 q^{5} +3.67456 q^{7} +1.71840 q^{9} +O(q^{10})\)	\(q+2.17219 q^{3} -1.55586 q^{5} +3.67456 q^{7} +1.71840 q^{9} -6.34872 q^{11} -4.15200 q^{13} -3.37962 q^{15} +1.00000 q^{17} +4.98305 q^{19} +7.98184 q^{21} -3.19399 q^{23} -2.57930 q^{25} -2.78388 q^{27} -5.70145 q^{29} -0.739366 q^{31} -13.7906 q^{33} -5.71711 q^{35} +3.82932 q^{37} -9.01893 q^{39} -10.3229 q^{41} +2.00486 q^{43} -2.67359 q^{45} -12.9026 q^{47} +6.50243 q^{49} +2.17219 q^{51} +1.42170 q^{53} +9.87773 q^{55} +10.8241 q^{57} -1.00000 q^{59} +7.05667 q^{61} +6.31436 q^{63} +6.45994 q^{65} +13.7911 q^{67} -6.93794 q^{69} +12.3952 q^{71} -9.68558 q^{73} -5.60271 q^{75} -23.3288 q^{77} -3.90718 q^{79} -11.2023 q^{81} -11.8048 q^{83} -1.55586 q^{85} -12.3846 q^{87} +12.5399 q^{89} -15.2568 q^{91} -1.60604 q^{93} -7.75293 q^{95} -11.7550 q^{97} -10.9096 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 15 q - q^{3} + q^{5} - 11 q^{7} + 16 q^{9}+O(q^{10}) \) 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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