LMFDB - Embedded newform 4012.2.a.i.1.16

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:	$0$
Dimension:	$18$
Coefficient field:	$\mathbb{Q}[x]/(x^{18} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{18} - 8 x^{17} - 4 x^{16} + 178 x^{15} - 265 x^{14} - 1405 x^{13} + 3503 x^{12} + 4295 x^{11} + \cdots - 16 $ x^18 - 8x^17 - 4x^16 + 178x^15 - 265x^14 - 1405x^13 + 3503x^12 + 4295x^11 - 17374x^10 - 893x^9 + 38112x^8 - 18700x^7 - 32137x^6 + 26381x^5 + 3964x^4 - 5788x^3 - 108x^2 + 232*x - 16
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.16
Root			$2.97166$ 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)

$f(q)$	$=$	$q+2.97166 q^{3} -4.20066 q^{5} +0.274921 q^{7} +5.83079 q^{9} +O(q^{10})$	$q+2.97166 q^{3} -4.20066 q^{5} +0.274921 q^{7} +5.83079 q^{9} -2.12930 q^{11} -6.94852 q^{13} -12.4829 q^{15} -1.00000 q^{17} +3.01007 q^{19} +0.816974 q^{21} +6.46848 q^{23} +12.6455 q^{25} +8.41216 q^{27} +3.91330 q^{29} +4.63145 q^{31} -6.32758 q^{33} -1.15485 q^{35} +1.57964 q^{37} -20.6487 q^{39} +10.7910 q^{41} +5.33316 q^{43} -24.4932 q^{45} +5.38581 q^{47} -6.92442 q^{49} -2.97166 q^{51} -6.66518 q^{53} +8.94447 q^{55} +8.94490 q^{57} -1.00000 q^{59} -6.84464 q^{61} +1.60301 q^{63} +29.1884 q^{65} -4.54387 q^{67} +19.2222 q^{69} +13.4629 q^{71} +11.7678 q^{73} +37.5782 q^{75} -0.585391 q^{77} +3.83930 q^{79} +7.50576 q^{81} +10.4787 q^{83} +4.20066 q^{85} +11.6290 q^{87} +14.9721 q^{89} -1.91030 q^{91} +13.7631 q^{93} -12.6443 q^{95} +3.68326 q^{97} -12.4155 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 18 q + 8 q^{3} + 4 q^{5} + 2 q^{7} + 18 q^{9}+O(q^{10}) $ 18 * q + 8 * q^3 + 4 * q^5 + 2 * q^7 + 18 * q^9	$ 18 q + 8 q^{3} + 4 q^{5} + 2 q^{7} + 18 q^{9} + 12 q^{11} + 2 q^{13} - 18 q^{17} + 5 q^{19} - 3 q^{21} + 21 q^{23} + 16 q^{25} + 26 q^{27} + 14 q^{29} + 15 q^{31} + 19 q^{33} + 20 q^{35} + 2 q^{37} - 14 q^{39} + 34 q^{41} + 21 q^{43} + 49 q^{45} + 69 q^{47} + 28 q^{49} - 8 q^{51} - 4 q^{53} + 18 q^{55} + 5 q^{57} - 18 q^{59} + 11 q^{61} + 35 q^{63} + 27 q^{65} + 34 q^{67} - 4 q^{69} + 37 q^{71} + 18 q^{73} + 72 q^{75} + 11 q^{77} + 11 q^{79} + 30 q^{81} + 28 q^{83} - 4 q^{85} + 7 q^{87} + 44 q^{89} - 23 q^{91} - 3 q^{93} - 11 q^{95} + 11 q^{97} + 56 q^{99}+O(q^{100}) $ 18 * q + 8 * q^3 + 4 * q^5 + 2 * q^7 + 18 * q^9 + 12 * q^11 + 2 * q^13 - 18 * q^17 + 5 * q^19 - 3 * q^21 + 21 * q^23 + 16 * q^25 + 26 * q^27 + 14 * q^29 + 15 * q^31 + 19 * q^33 + 20 * q^35 + 2 * q^37 - 14 * q^39 + 34 * q^41 + 21 * q^43 + 49 * q^45 + 69 * q^47 + 28 * q^49 - 8 * q^51 - 4 * q^53 + 18 * q^55 + 5 * q^57 - 18 * q^59 + 11 * q^61 + 35 * q^63 + 27 * q^65 + 34 * q^67 - 4 * q^69 + 37 * q^71 + 18 * q^73 + 72 * q^75 + 11 * q^77 + 11 * q^79 + 30 * q^81 + 28 * q^83 - 4 * q^85 + 7 * q^87 + 44 * q^89 - 23 * q^91 - 3 * q^93 - 11 * q^95 + 11 * q^97 + 56 * 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
$3$	2.97166	1.71569	0.857846	−	0.513907i	$-0.171802\pi$
$3$	2.97166	1.71569	0.857846	+	0.513907i	$0.171802\pi$
$4$	0	0
$5$	−4.20066	−1.87859	−0.939295	−	0.343109i	$-0.888520\pi$
$5$	−4.20066	−1.87859	−0.939295	+	0.343109i	$0.888520\pi$
$6$	0	0
$7$	0.274921	0.103911	0.0519553	−	0.998649i	$-0.483455\pi$
$7$	0.274921	0.103911	0.0519553	+	0.998649i	$0.483455\pi$
$8$	0	0
$9$	5.83079	1.94360
$10$	0	0
$11$	−2.12930	−0.642009	−0.321005	−	0.947078i	$-0.604020\pi$
$11$	−2.12930	−0.642009	−0.321005	+	0.947078i	$0.604020\pi$
$12$	0	0
$13$	−6.94852	−1.92717	−0.963586	−	0.267397i	$-0.913836\pi$
$13$	−6.94852	−1.92717	−0.963586	+	0.267397i	$0.913836\pi$
$14$	0	0
$15$	−12.4829	−3.22308
$16$	0	0
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	0	0
$19$	3.01007	0.690556	0.345278	−	0.938500i	$-0.387785\pi$
$19$	3.01007	0.690556	0.345278	+	0.938500i	$0.387785\pi$
$20$	0	0
$21$	0.816974	0.178278
$22$	0	0
$23$	6.46848	1.34877	0.674386	−	0.738379i	$-0.264409\pi$
$23$	6.46848	1.34877	0.674386	+	0.738379i	$0.264409\pi$
$24$	0	0
$25$	12.6455	2.52910
$26$	0	0
$27$	8.41216	1.61892
$28$	0	0
$29$	3.91330	0.726682	0.363341	−	0.931656i	$-0.381636\pi$
$29$	3.91330	0.726682	0.363341	+	0.931656i	$0.381636\pi$
$30$	0	0
$31$	4.63145	0.831832	0.415916	−	0.909403i	$-0.363461\pi$
$31$	4.63145	0.831832	0.415916	+	0.909403i	$0.363461\pi$
$32$	0	0
$33$	−6.32758	−1.10149
$34$	0	0
$35$	−1.15485	−0.195205
$36$	0	0
$37$	1.57964	0.259691	0.129846	−	0.991534i	$-0.458552\pi$
$37$	1.57964	0.259691	0.129846	+	0.991534i	$0.458552\pi$
$38$	0	0
$39$	−20.6487	−3.30643
$40$	0	0
$41$	10.7910	1.68528	0.842638	−	0.538480i	$-0.181001\pi$
$41$	10.7910	1.68528	0.842638	+	0.538480i	$0.181001\pi$
$42$	0	0
$43$	5.33316	0.813299	0.406649	−	0.913584i	$-0.366697\pi$
$43$	5.33316	0.813299	0.406649	+	0.913584i	$0.366697\pi$
$44$	0	0
$45$	−24.4932	−3.65122
$46$	0	0
$47$	5.38581	0.785601	0.392801	−	0.919624i	$-0.371506\pi$
$47$	5.38581	0.785601	0.392801	+	0.919624i	$0.371506\pi$
$48$	0	0
$49$	−6.92442	−0.989203
$50$	0	0
$51$	−2.97166	−0.416116
$52$	0	0
$53$	−6.66518	−0.915532	−0.457766	−	0.889073i	$-0.651350\pi$
$53$	−6.66518	−0.915532	−0.457766	+	0.889073i	$0.651350\pi$
$54$	0	0
$55$	8.94447	1.20607
$56$	0	0
$57$	8.94490	1.18478
$58$	0	0
$59$	−1.00000	−0.130189
$59$	−1.00000	−0.130189
$60$	0	0
$61$	−6.84464	−0.876367	−0.438183	−	0.898886i	$-0.644378\pi$
$61$	−6.84464	−0.876367	−0.438183	+	0.898886i	$0.644378\pi$
$62$	0	0
$63$	1.60301	0.201960
$64$	0	0
$65$	29.1884	3.62037
$66$	0	0
$67$	−4.54387	−0.555122	−0.277561	−	0.960708i	$-0.589526\pi$
$67$	−4.54387	−0.555122	−0.277561	+	0.960708i	$0.589526\pi$
$68$	0	0
$69$	19.2222	2.31408
$70$	0	0
$71$	13.4629	1.59775	0.798874	−	0.601499i	$-0.205429\pi$
$71$	13.4629	1.59775	0.798874	+	0.601499i	$0.205429\pi$
$72$	0	0
$73$	11.7678	1.37732	0.688660	−	0.725084i	$-0.258199\pi$
$73$	11.7678	1.37732	0.688660	+	0.725084i	$0.258199\pi$
$74$	0	0
$75$	37.5782	4.33916
$76$	0	0
$77$	−0.585391	−0.0667115
$78$	0	0
$79$	3.83930	0.431955	0.215978	−	0.976398i	$-0.430706\pi$
$79$	3.83930	0.431955	0.215978	+	0.976398i	$0.430706\pi$
$80$	0	0
$81$	7.50576	0.833973
$82$	0	0
$83$	10.4787	1.15019	0.575095	−	0.818086i	$-0.304965\pi$
$83$	10.4787	1.15019	0.575095	+	0.818086i	$0.304965\pi$
$84$	0	0
$85$	4.20066	0.455625
$86$	0	0
$87$	11.6290	1.24676
$88$	0	0
$89$	14.9721	1.58704	0.793518	−	0.608546i	$-0.208247\pi$
$89$	14.9721	1.58704	0.793518	+	0.608546i	$0.208247\pi$
$90$	0	0
$91$	−1.91030	−0.200254
$92$	0	0
$93$	13.7631	1.42717
$94$	0	0
$95$	−12.6443	−1.29727
$96$	0	0
$97$	3.68326	0.373979	0.186989	−	0.982362i	$-0.440127\pi$
$97$	3.68326	0.373979	0.186989	+	0.982362i	$0.440127\pi$
$98$	0	0
$99$	−12.4155	−1.24781
$100$	0	0
$101$	1.80434	0.179539	0.0897693	−	0.995963i	$-0.471387\pi$
$101$	1.80434	0.179539	0.0897693	+	0.995963i	$0.471387\pi$
$102$	0	0
$103$	−2.95238	−0.290906	−0.145453	−	0.989365i	$-0.546464\pi$
$103$	−2.95238	−0.290906	−0.145453	+	0.989365i	$0.546464\pi$
$104$	0	0
$105$	−3.43183	−0.334912
$106$	0	0
$107$	1.13313	0.109544	0.0547718	−	0.998499i	$-0.482557\pi$
$107$	1.13313	0.109544	0.0547718	+	0.998499i	$0.482557\pi$
$108$	0	0
$109$	1.48080	0.141835	0.0709174	−	0.997482i	$-0.477407\pi$
$109$	1.48080	0.141835	0.0709174	+	0.997482i	$0.477407\pi$
$110$	0	0
$111$	4.69417	0.445550
$112$	0	0
$113$	−6.75218	−0.635192	−0.317596	−	0.948226i	$-0.602876\pi$
$113$	−6.75218	−0.635192	−0.317596	+	0.948226i	$0.602876\pi$
$114$	0	0
$115$	−27.1719	−2.53379
$116$	0	0
$117$	−40.5154	−3.74565
$118$	0	0
$119$	−0.274921	−0.0252020
$120$	0	0
$121$	−6.46607	−0.587824
$122$	0	0
$123$	32.0673	2.89141
$124$	0	0
$125$	−32.1162	−2.87256
$126$	0	0
$127$	−16.5078	−1.46483	−0.732415	−	0.680859i	$-0.761607\pi$
$127$	−16.5078	−1.46483	−0.732415	+	0.680859i	$0.761607\pi$
$128$	0	0
$129$	15.8484	1.39537
$130$	0	0
$131$	0.513119	0.0448314	0.0224157	−	0.999749i	$-0.492864\pi$
$131$	0.513119	0.0448314	0.0224157	+	0.999749i	$0.492864\pi$
$132$	0	0
$133$	0.827531	0.0717561
$134$	0	0
$135$	−35.3366	−3.04129
$136$	0	0
$137$	16.5118	1.41070	0.705351	−	0.708859i	$-0.250790\pi$
$137$	16.5118	1.41070	0.705351	+	0.708859i	$0.250790\pi$
$138$	0	0
$139$	7.73056	0.655698	0.327849	−	0.944730i	$-0.393676\pi$
$139$	7.73056	0.655698	0.327849	+	0.944730i	$0.393676\pi$
$140$	0	0
$141$	16.0048	1.34785
$142$	0	0
$143$	14.7955	1.23726
$144$	0	0
$145$	−16.4385	−1.36514
$146$	0	0
$147$	−20.5770	−1.69717
$148$	0	0
$149$	−0.818945	−0.0670906	−0.0335453	−	0.999437i	$-0.510680\pi$
$149$	−0.818945	−0.0670906	−0.0335453	+	0.999437i	$0.510680\pi$
$150$	0	0
$151$	−4.42913	−0.360437	−0.180219	−	0.983627i	$-0.557681\pi$
$151$	−4.42913	−0.360437	−0.180219	+	0.983627i	$0.557681\pi$
$152$	0	0
$153$	−5.83079	−0.471392
$154$	0	0
$155$	−19.4551	−1.56267
$156$	0	0
$157$	−2.20139	−0.175690	−0.0878451	−	0.996134i	$-0.527998\pi$
$157$	−2.20139	−0.175690	−0.0878451	+	0.996134i	$0.527998\pi$
$158$	0	0
$159$	−19.8067	−1.57077
$160$	0	0
$161$	1.77832	0.140152
$162$	0	0
$163$	−18.7405	−1.46787	−0.733935	−	0.679219i	$-0.762318\pi$
$163$	−18.7405	−1.46787	−0.733935	+	0.679219i	$0.762318\pi$
$164$	0	0
$165$	26.5800	2.06925
$166$	0	0
$167$	−0.338629	−0.0262039	−0.0131020	−	0.999914i	$-0.504171\pi$
$167$	−0.338629	−0.0262039	−0.0131020	+	0.999914i	$0.504171\pi$
$168$	0	0
$169$	35.2819	2.71399
$170$	0	0
$171$	17.5511	1.34216
$172$	0	0
$173$	13.4103	1.01956	0.509782	−	0.860304i	$-0.329726\pi$
$173$	13.4103	1.01956	0.509782	+	0.860304i	$0.329726\pi$
$174$	0	0
$175$	3.47652	0.262801
$176$	0	0
$177$	−2.97166	−0.223364
$178$	0	0
$179$	5.52678	0.413091	0.206545	−	0.978437i	$-0.433778\pi$
$179$	5.52678	0.413091	0.206545	+	0.978437i	$0.433778\pi$
$180$	0	0
$181$	−5.45460	−0.405437	−0.202719	−	0.979237i	$-0.564978\pi$
$181$	−5.45460	−0.405437	−0.202719	+	0.979237i	$0.564978\pi$
$182$	0	0
$183$	−20.3400	−1.50357
$184$	0	0
$185$	−6.63553	−0.487854
$186$	0	0
$187$	2.12930	0.155710
$188$	0	0
$189$	2.31268	0.168223
$190$	0	0
$191$	−19.3930	−1.40323	−0.701615	−	0.712556i	$-0.747537\pi$
$191$	−19.3930	−1.40323	−0.701615	+	0.712556i	$0.747537\pi$
$192$	0	0
$193$	−9.52540	−0.685653	−0.342827	−	0.939399i	$-0.611384\pi$
$193$	−9.52540	−0.685653	−0.342827	+	0.939399i	$0.611384\pi$
$194$	0	0
$195$	86.7380	6.21144
$196$	0	0
$197$	18.1406	1.29247	0.646233	−	0.763140i	$-0.276344\pi$
$197$	18.1406	1.29247	0.646233	+	0.763140i	$0.276344\pi$
$198$	0	0
$199$	−21.1962	−1.50256	−0.751279	−	0.659984i	$-0.770563\pi$
$199$	−21.1962	−1.50256	−0.751279	+	0.659984i	$0.770563\pi$
$200$	0	0
$201$	−13.5029	−0.952418
$202$	0	0
$203$	1.07585	0.0755100
$204$	0	0
$205$	−45.3294	−3.16595
$206$	0	0
$207$	37.7164	2.62147
$208$	0	0
$209$	−6.40934	−0.443343
$210$	0	0
$211$	−12.4209	−0.855090	−0.427545	−	0.903994i	$-0.640621\pi$
$211$	−12.4209	−0.855090	−0.427545	+	0.903994i	$0.640621\pi$
$212$	0	0
$213$	40.0071	2.74124
$214$	0	0
$215$	−22.4028	−1.52786
$216$	0	0
$217$	1.27328	0.0864361
$218$	0	0
$219$	34.9700	2.36306
$220$	0	0
$221$	6.94852	0.467408
$222$	0	0
$223$	19.7619	1.32335	0.661676	−	0.749790i	$-0.269845\pi$
$223$	19.7619	1.32335	0.661676	+	0.749790i	$0.269845\pi$
$224$	0	0
$225$	73.7334	4.91556
$226$	0	0
$227$	−21.1509	−1.40383	−0.701916	−	0.712260i	$-0.747672\pi$
$227$	−21.1509	−1.40383	−0.701916	+	0.712260i	$0.747672\pi$
$228$	0	0
$229$	12.9314	0.854533	0.427266	−	0.904126i	$-0.359477\pi$
$229$	12.9314	0.854533	0.427266	+	0.904126i	$0.359477\pi$
$230$	0	0
$231$	−1.73959	−0.114456
$232$	0	0
$233$	28.8024	1.88691	0.943453	−	0.331506i	$-0.107557\pi$
$233$	28.8024	1.88691	0.943453	+	0.331506i	$0.107557\pi$
$234$	0	0
$235$	−22.6240	−1.47582
$236$	0	0
$237$	11.4091	0.741102
$238$	0	0
$239$	−14.6613	−0.948360	−0.474180	−	0.880428i	$-0.657255\pi$
$239$	−14.6613	−0.948360	−0.474180	+	0.880428i	$0.657255\pi$
$240$	0	0
$241$	5.04726	0.325122	0.162561	−	0.986698i	$-0.448025\pi$
$241$	5.04726	0.325122	0.162561	+	0.986698i	$0.448025\pi$
$242$	0	0
$243$	−2.93190	−0.188081
$244$	0	0
$245$	29.0871	1.85831
$246$	0	0
$247$	−20.9155	−1.33082
$248$	0	0
$249$	31.1393	1.97337
$250$	0	0
$251$	3.10386	0.195914	0.0979569	−	0.995191i	$-0.468769\pi$
$251$	3.10386	0.195914	0.0979569	+	0.995191i	$0.468769\pi$
$252$	0	0
$253$	−13.7734	−0.865924
$254$	0	0
$255$	12.4829	0.781712
$256$	0	0
$257$	−21.0933	−1.31576	−0.657882	−	0.753121i	$-0.728547\pi$
$257$	−21.0933	−1.31576	−0.657882	+	0.753121i	$0.728547\pi$
$258$	0	0
$259$	0.434277	0.0269847
$260$	0	0
$261$	22.8177	1.41238
$262$	0	0
$263$	1.95973	0.120842	0.0604210	−	0.998173i	$-0.480756\pi$
$263$	1.95973	0.120842	0.0604210	+	0.998173i	$0.480756\pi$
$264$	0	0
$265$	27.9981	1.71991
$266$	0	0
$267$	44.4920	2.72287
$268$	0	0
$269$	−14.2992	−0.871839	−0.435920	−	0.899986i	$-0.643577\pi$
$269$	−14.2992	−0.871839	−0.435920	+	0.899986i	$0.643577\pi$
$270$	0	0
$271$	3.90018	0.236919	0.118460	−	0.992959i	$-0.462204\pi$
$271$	3.90018	0.236919	0.118460	+	0.992959i	$0.462204\pi$
$272$	0	0
$273$	−5.67676	−0.343573
$274$	0	0
$275$	−26.9261	−1.62371
$276$	0	0
$277$	−1.23405	−0.0741469	−0.0370734	−	0.999313i	$-0.511804\pi$
$277$	−1.23405	−0.0741469	−0.0370734	+	0.999313i	$0.511804\pi$
$278$	0	0
$279$	27.0050	1.61675
$280$	0	0
$281$	−16.0133	−0.955272	−0.477636	−	0.878558i	$-0.658506\pi$
$281$	−16.0133	−0.955272	−0.477636	+	0.878558i	$0.658506\pi$
$282$	0	0
$283$	15.7015	0.933356	0.466678	−	0.884427i	$-0.345451\pi$
$283$	15.7015	0.933356	0.466678	+	0.884427i	$0.345451\pi$
$284$	0	0
$285$	−37.5745	−2.22572
$286$	0	0
$287$	2.96669	0.175118
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	10.9454	0.641632
$292$	0	0
$293$	−16.5596	−0.967422	−0.483711	−	0.875228i	$-0.660711\pi$
$293$	−16.5596	−0.967422	−0.483711	+	0.875228i	$0.660711\pi$
$294$	0	0
$295$	4.20066	0.244572
$296$	0	0
$297$	−17.9121	−1.03936
$298$	0	0
$299$	−44.9464	−2.59932
$300$	0	0
$301$	1.46620	0.0845103
$302$	0	0
$303$	5.36189	0.308033
$304$	0	0
$305$	28.7520	1.64633
$306$	0	0
$307$	27.7070	1.58132	0.790662	−	0.612254i	$-0.209737\pi$
$307$	27.7070	1.58132	0.790662	+	0.612254i	$0.209737\pi$
$308$	0	0
$309$	−8.77348	−0.499106
$310$	0	0
$311$	16.5823	0.940295	0.470148	−	0.882588i	$-0.344201\pi$
$311$	16.5823	0.940295	0.470148	+	0.882588i	$0.344201\pi$
$312$	0	0
$313$	12.5533	0.709555	0.354778	−	0.934951i	$-0.384557\pi$
$313$	12.5533	0.709555	0.354778	+	0.934951i	$0.384557\pi$
$314$	0	0
$315$	−6.73369	−0.379401
$316$	0	0
$317$	17.2735	0.970175	0.485087	−	0.874466i	$-0.338788\pi$
$317$	17.2735	0.970175	0.485087	+	0.874466i	$0.338788\pi$
$318$	0	0
$319$	−8.33261	−0.466537
$320$	0	0
$321$	3.36728	0.187943
$322$	0	0
$323$	−3.01007	−0.167485
$324$	0	0
$325$	−87.8676	−4.87402
$326$	0	0
$327$	4.40044	0.243345
$328$	0	0
$329$	1.48068	0.0816323
$330$	0	0
$331$	−13.2008	−0.725579	−0.362789	−	0.931871i	$-0.618176\pi$
$331$	−13.2008	−0.725579	−0.362789	+	0.931871i	$0.618176\pi$
$332$	0	0
$333$	9.21056	0.504736
$334$	0	0
$335$	19.0872	1.04285
$336$	0	0
$337$	0.401601	0.0218766	0.0109383	−	0.999940i	$-0.496518\pi$
$337$	0.401601	0.0218766	0.0109383	+	0.999940i	$0.496518\pi$
$338$	0	0
$339$	−20.0652	−1.08979
$340$	0	0
$341$	−9.86175	−0.534044
$342$	0	0
$343$	−3.82812	−0.206699
$344$	0	0
$345$	−80.7457	−4.34720
$346$	0	0
$347$	8.43850	0.453002	0.226501	−	0.974011i	$-0.427271\pi$
$347$	8.43850	0.453002	0.226501	+	0.974011i	$0.427271\pi$
$348$	0	0
$349$	15.6780	0.839223	0.419612	−	0.907704i	$-0.362166\pi$
$349$	15.6780	0.839223	0.419612	+	0.907704i	$0.362166\pi$
$350$	0	0
$351$	−58.4521	−3.11994
$352$	0	0
$353$	9.57012	0.509366	0.254683	−	0.967025i	$-0.418029\pi$
$353$	9.57012	0.509366	0.254683	+	0.967025i	$0.418029\pi$
$354$	0	0
$355$	−56.5529	−3.00151
$356$	0	0
$357$	−0.816974	−0.0432389
$358$	0	0
$359$	21.9372	1.15780	0.578900	−	0.815399i	$-0.303482\pi$
$359$	21.9372	1.15780	0.578900	+	0.815399i	$0.303482\pi$
$360$	0	0
$361$	−9.93951	−0.523132
$362$	0	0
$363$	−19.2150	−1.00853
$364$	0	0
$365$	−49.4326	−2.58742
$366$	0	0
$367$	36.9370	1.92810	0.964049	−	0.265725i	$-0.0856112\pi$
$367$	36.9370	1.92810	0.964049	+	0.265725i	$0.0856112\pi$
$368$	0	0
$369$	62.9203	3.27550
$370$	0	0
$371$	−1.83240	−0.0951335
$372$	0	0
$373$	−15.1105	−0.782391	−0.391196	−	0.920308i	$-0.627938\pi$
$373$	−15.1105	−0.782391	−0.391196	+	0.920308i	$0.627938\pi$
$374$	0	0
$375$	−95.4386	−4.92843
$376$	0	0
$377$	−27.1917	−1.40044
$378$	0	0
$379$	−31.3326	−1.60945	−0.804724	−	0.593649i	$-0.797687\pi$
$379$	−31.3326	−1.60945	−0.804724	+	0.593649i	$0.797687\pi$
$380$	0	0
$381$	−49.0556	−2.51320
$382$	0	0
$383$	−24.1505	−1.23404	−0.617018	−	0.786949i	$-0.711659\pi$
$383$	−24.1505	−1.23404	−0.617018	+	0.786949i	$0.711659\pi$
$384$	0	0
$385$	2.45903	0.125324
$386$	0	0
$387$	31.0965	1.58073
$388$	0	0
$389$	23.4494	1.18893	0.594467	−	0.804120i	$-0.297363\pi$
$389$	23.4494	1.18893	0.594467	+	0.804120i	$0.297363\pi$
$390$	0	0
$391$	−6.46848	−0.327125
$392$	0	0
$393$	1.52482	0.0769169
$394$	0	0
$395$	−16.1276	−0.811467
$396$	0	0
$397$	26.3297	1.32145	0.660726	−	0.750627i	$-0.270249\pi$
$397$	26.3297	1.32145	0.660726	+	0.750627i	$0.270249\pi$
$398$	0	0
$399$	2.45915	0.123111
$400$	0	0
$401$	39.3057	1.96283	0.981416	−	0.191892i	$-0.0614625\pi$
$401$	39.3057	1.96283	0.981416	+	0.191892i	$0.0614625\pi$
$402$	0	0
$403$	−32.1817	−1.60308
$404$	0	0
$405$	−31.5291	−1.56669
$406$	0	0
$407$	−3.36354	−0.166724
$408$	0	0
$409$	27.3619	1.35296	0.676479	−	0.736462i	$-0.263505\pi$
$409$	27.3619	1.35296	0.676479	+	0.736462i	$0.263505\pi$
$410$	0	0
$411$	49.0676	2.42033
$412$	0	0
$413$	−0.274921	−0.0135280
$414$	0	0
$415$	−44.0176	−2.16074
$416$	0	0
$417$	22.9726	1.12498
$418$	0	0
$419$	1.61363	0.0788311	0.0394155	−	0.999223i	$-0.487450\pi$
$419$	1.61363	0.0788311	0.0394155	+	0.999223i	$0.487450\pi$
$420$	0	0
$421$	1.77177	0.0863507	0.0431754	−	0.999068i	$-0.486253\pi$
$421$	1.77177	0.0863507	0.0431754	+	0.999068i	$0.486253\pi$
$422$	0	0
$423$	31.4036	1.52689
$424$	0	0
$425$	−12.6455	−0.613398
$426$	0	0
$427$	−1.88174	−0.0910637
$428$	0	0
$429$	43.9673	2.12276
$430$	0	0
$431$	−37.6737	−1.81468	−0.907338	−	0.420401i	$-0.861889\pi$
$431$	−37.6737	−1.81468	−0.907338	+	0.420401i	$0.861889\pi$
$432$	0	0
$433$	−3.65428	−0.175613	−0.0878067	−	0.996138i	$-0.527986\pi$
$433$	−3.65428	−0.175613	−0.0878067	+	0.996138i	$0.527986\pi$
$434$	0	0
$435$	−48.8496	−2.34216
$436$	0	0
$437$	19.4705	0.931403
$438$	0	0
$439$	−34.7465	−1.65836	−0.829181	−	0.558981i	$-0.811193\pi$
$439$	−34.7465	−1.65836	−0.829181	+	0.558981i	$0.811193\pi$
$440$	0	0
$441$	−40.3748	−1.92261
$442$	0	0
$443$	21.6561	1.02891	0.514456	−	0.857517i	$-0.327994\pi$
$443$	21.6561	1.02891	0.514456	+	0.857517i	$0.327994\pi$
$444$	0	0
$445$	−62.8926	−2.98139
$446$	0	0
$447$	−2.43363	−0.115107
$448$	0	0
$449$	−12.0779	−0.569993	−0.284996	−	0.958529i	$-0.591992\pi$
$449$	−12.0779	−0.569993	−0.284996	+	0.958529i	$0.591992\pi$
$450$	0	0
$451$	−22.9774	−1.08196
$452$	0	0
$453$	−13.1619	−0.618399
$454$	0	0
$455$	8.02450	0.376195
$456$	0	0
$457$	29.8952	1.39844	0.699220	−	0.714906i	$-0.253531\pi$
$457$	29.8952	1.39844	0.699220	+	0.714906i	$0.253531\pi$
$458$	0	0
$459$	−8.41216	−0.392646
$460$	0	0
$461$	34.3647	1.60052	0.800261	−	0.599651i	$-0.204694\pi$
$461$	34.3647	1.60052	0.800261	+	0.599651i	$0.204694\pi$
$462$	0	0
$463$	4.69621	0.218251	0.109126	−	0.994028i	$-0.465195\pi$
$463$	4.69621	0.218251	0.109126	+	0.994028i	$0.465195\pi$
$464$	0	0
$465$	−57.8141	−2.68106
$466$	0	0
$467$	42.7600	1.97870	0.989348	−	0.145569i	$-0.0465014\pi$
$467$	42.7600	1.97870	0.989348	+	0.145569i	$0.0465014\pi$
$468$	0	0
$469$	−1.24921	−0.0576830
$470$	0	0
$471$	−6.54180	−0.301430
$472$	0	0
$473$	−11.3559	−0.522145
$474$	0	0
$475$	38.0638	1.74649
$476$	0	0
$477$	−38.8633	−1.77943
$478$	0	0
$479$	−21.2450	−0.970707	−0.485354	−	0.874318i	$-0.661309\pi$
$479$	−21.2450	−0.970707	−0.485354	+	0.874318i	$0.661309\pi$
$480$	0	0
$481$	−10.9762	−0.500470
$482$	0	0
$483$	5.28458	0.240457
$484$	0	0
$485$	−15.4721	−0.702553
$486$	0	0
$487$	4.56549	0.206882	0.103441	−	0.994636i	$-0.467015\pi$
$487$	4.56549	0.206882	0.103441	+	0.994636i	$0.467015\pi$
$488$	0	0
$489$	−55.6905	−2.51841
$490$	0	0
$491$	−26.2309	−1.18379	−0.591893	−	0.806017i	$-0.701619\pi$
$491$	−26.2309	−1.18379	−0.591893	+	0.806017i	$0.701619\pi$
$492$	0	0
$493$	−3.91330	−0.176246
$494$	0	0
$495$	52.1534	2.34412
$496$	0	0
$497$	3.70123	0.166023
$498$	0	0
$499$	−18.8708	−0.844772	−0.422386	−	0.906416i	$-0.638807\pi$
$499$	−18.8708	−0.844772	−0.422386	+	0.906416i	$0.638807\pi$
$500$	0	0
$501$	−1.00629	−0.0449578
$502$	0	0
$503$	17.4202	0.776726	0.388363	−	0.921506i	$-0.373041\pi$
$503$	17.4202	0.776726	0.388363	+	0.921506i	$0.373041\pi$
$504$	0	0
$505$	−7.57941	−0.337279
$506$	0	0
$507$	104.846	4.65638
$508$	0	0
$509$	−11.7446	−0.520569	−0.260285	−	0.965532i	$-0.583816\pi$
$509$	−11.7446	−0.520569	−0.260285	+	0.965532i	$0.583816\pi$
$510$	0	0
$511$	3.23523	0.143118
$512$	0	0
$513$	25.3212	1.11796
$514$	0	0
$515$	12.4019	0.546494
$516$	0	0
$517$	−11.4680	−0.504363
$518$	0	0
$519$	39.8508	1.74926
$520$	0	0
$521$	27.6682	1.21216	0.606082	−	0.795402i	$-0.292740\pi$
$521$	27.6682	1.21216	0.606082	+	0.795402i	$0.292740\pi$
$522$	0	0
$523$	24.4250	1.06803	0.534015	−	0.845475i	$-0.320682\pi$
$523$	24.4250	1.06803	0.534015	+	0.845475i	$0.320682\pi$
$524$	0	0
$525$	10.3311	0.450885
$526$	0	0
$527$	−4.63145	−0.201749
$528$	0	0
$529$	18.8412	0.819184
$530$	0	0
$531$	−5.83079	−0.253035
$532$	0	0
$533$	−74.9817	−3.24782
$534$	0	0
$535$	−4.75988	−0.205788
$536$	0	0
$537$	16.4237	0.708736
$538$	0	0
$539$	14.7442	0.635077
$540$	0	0
$541$	5.41576	0.232842	0.116421	−	0.993200i	$-0.462858\pi$
$541$	5.41576	0.232842	0.116421	+	0.993200i	$0.462858\pi$
$542$	0	0
$543$	−16.2092	−0.695605
$544$	0	0
$545$	−6.22033	−0.266450
$546$	0	0
$547$	−5.31106	−0.227084	−0.113542	−	0.993533i	$-0.536220\pi$
$547$	−5.31106	−0.227084	−0.113542	+	0.993533i	$0.536220\pi$
$548$	0	0
$549$	−39.9097	−1.70330
$550$	0	0
$551$	11.7793	0.501815
$552$	0	0
$553$	1.05551	0.0448847
$554$	0	0
$555$	−19.7186	−0.837007
$556$	0	0
$557$	36.7605	1.55759	0.778795	−	0.627278i	$-0.215831\pi$
$557$	36.7605	1.55759	0.778795	+	0.627278i	$0.215831\pi$
$558$	0	0
$559$	−37.0576	−1.56737
$560$	0	0
$561$	6.32758	0.267150
$562$	0	0
$563$	8.80193	0.370957	0.185479	−	0.982648i	$-0.440616\pi$
$563$	8.80193	0.370957	0.185479	+	0.982648i	$0.440616\pi$
$564$	0	0
$565$	28.3636	1.19327
$566$	0	0
$567$	2.06349	0.0866586
$568$	0	0
$569$	−0.884606	−0.0370846	−0.0185423	−	0.999828i	$-0.505903\pi$
$569$	−0.884606	−0.0370846	−0.0185423	+	0.999828i	$0.505903\pi$
$570$	0	0
$571$	−30.5416	−1.27813	−0.639063	−	0.769154i	$-0.720678\pi$
$571$	−30.5416	−1.27813	−0.639063	+	0.769154i	$0.720678\pi$
$572$	0	0
$573$	−57.6296	−2.40751
$574$	0	0
$575$	81.7973	3.41118
$576$	0	0
$577$	−31.5415	−1.31309	−0.656545	−	0.754287i	$-0.727983\pi$
$577$	−31.5415	−1.31309	−0.656545	+	0.754287i	$0.727983\pi$
$578$	0	0
$579$	−28.3063	−1.17637
$580$	0	0
$581$	2.88083	0.119517
$582$	0	0
$583$	14.1922	0.587780
$584$	0	0
$585$	170.191	7.03654
$586$	0	0
$587$	44.1521	1.82235	0.911177	−	0.412015i	$-0.135175\pi$
$587$	44.1521	1.82235	0.911177	+	0.412015i	$0.135175\pi$
$588$	0	0
$589$	13.9410	0.574427
$590$	0	0
$591$	53.9078	2.21747
$592$	0	0
$593$	20.0572	0.823651	0.411826	−	0.911263i	$-0.364891\pi$
$593$	20.0572	0.823651	0.411826	+	0.911263i	$0.364891\pi$
$594$	0	0
$595$	1.15485	0.0473443
$596$	0	0
$597$	−62.9880	−2.57793
$598$	0	0
$599$	47.0812	1.92369	0.961844	−	0.273600i	$-0.0882144\pi$
$599$	47.0812	1.92369	0.961844	+	0.273600i	$0.0882144\pi$
$600$	0	0
$601$	−12.8400	−0.523754	−0.261877	−	0.965101i	$-0.584341\pi$
$601$	−12.8400	−0.523754	−0.261877	+	0.965101i	$0.584341\pi$
$602$	0	0
$603$	−26.4943	−1.07893
$604$	0	0
$605$	27.1617	1.10428
$606$	0	0
$607$	−34.0810	−1.38331	−0.691653	−	0.722230i	$-0.743117\pi$
$607$	−34.0810	−1.38331	−0.691653	+	0.722230i	$0.743117\pi$
$608$	0	0
$609$	3.19707	0.129552
$610$	0	0
$611$	−37.4234	−1.51399
$612$	0	0
$613$	−20.4484	−0.825902	−0.412951	−	0.910753i	$-0.635502\pi$
$613$	−20.4484	−0.825902	−0.412951	+	0.910753i	$0.635502\pi$
$614$	0	0
$615$	−134.704	−5.43178
$616$	0	0
$617$	−32.7584	−1.31880	−0.659402	−	0.751790i	$-0.729191\pi$
$617$	−32.7584	−1.31880	−0.659402	+	0.751790i	$0.729191\pi$
$618$	0	0
$619$	−28.5466	−1.14738	−0.573692	−	0.819071i	$-0.694489\pi$
$619$	−28.5466	−1.14738	−0.573692	+	0.819071i	$0.694489\pi$
$620$	0	0
$621$	54.4139	2.18356
$622$	0	0
$623$	4.11615	0.164910
$624$	0	0
$625$	71.6816	2.86726
$626$	0	0
$627$	−19.0464	−0.760641
$628$	0	0
$629$	−1.57964	−0.0629844
$630$	0	0
$631$	33.7323	1.34286	0.671432	−	0.741066i	$-0.265680\pi$
$631$	33.7323	1.34286	0.671432	+	0.741066i	$0.265680\pi$
$632$	0	0
$633$	−36.9108	−1.46707
$634$	0	0
$635$	69.3436	2.75182
$636$	0	0
$637$	48.1145	1.90636
$638$	0	0
$639$	78.4991	3.10538
$640$	0	0
$641$	−18.5102	−0.731108	−0.365554	−	0.930790i	$-0.619120\pi$
$641$	−18.5102	−0.731108	−0.365554	+	0.930790i	$0.619120\pi$
$642$	0	0
$643$	−13.9906	−0.551734	−0.275867	−	0.961196i	$-0.588965\pi$
$643$	−13.9906	−0.551734	−0.275867	+	0.961196i	$0.588965\pi$
$644$	0	0
$645$	−66.5735	−2.62133
$646$	0	0
$647$	22.8287	0.897487	0.448744	−	0.893660i	$-0.351872\pi$
$647$	22.8287	0.897487	0.448744	+	0.893660i	$0.351872\pi$
$648$	0	0
$649$	2.12930	0.0835825
$650$	0	0
$651$	3.78377	0.148298
$652$	0	0
$653$	19.5467	0.764922	0.382461	−	0.923972i	$-0.375077\pi$
$653$	19.5467	0.764922	0.382461	+	0.923972i	$0.375077\pi$
$654$	0	0
$655$	−2.15544	−0.0842199
$656$	0	0
$657$	68.6157	2.67696
$658$	0	0
$659$	−8.96583	−0.349259	−0.174630	−	0.984634i	$-0.555873\pi$
$659$	−8.96583	−0.349259	−0.174630	+	0.984634i	$0.555873\pi$
$660$	0	0
$661$	36.6471	1.42541	0.712703	−	0.701466i	$-0.247471\pi$
$661$	36.6471	1.42541	0.712703	+	0.701466i	$0.247471\pi$
$662$	0	0
$663$	20.6487	0.801928
$664$	0	0
$665$	−3.47618	−0.134800
$666$	0	0
$667$	25.3131	0.980128
$668$	0	0
$669$	58.7256	2.27046
$670$	0	0
$671$	14.5743	0.562635
$672$	0	0
$673$	−51.3518	−1.97947	−0.989733	−	0.142926i	$-0.954349\pi$
$673$	−51.3518	−1.97947	−0.989733	+	0.142926i	$0.954349\pi$
$674$	0	0
$675$	106.376	4.09442
$676$	0	0
$677$	−17.5438	−0.674263	−0.337132	−	0.941458i	$-0.609457\pi$
$677$	−17.5438	−0.674263	−0.337132	+	0.941458i	$0.609457\pi$
$678$	0	0
$679$	1.01261	0.0388603
$680$	0	0
$681$	−62.8533	−2.40854
$682$	0	0
$683$	36.9035	1.41207	0.706036	−	0.708176i	$-0.250482\pi$
$683$	36.9035	1.41207	0.706036	+	0.708176i	$0.250482\pi$
$684$	0	0
$685$	−69.3606	−2.65013
$686$	0	0
$687$	38.4279	1.46611
$688$	0	0
$689$	46.3131	1.76439
$690$	0	0
$691$	18.1486	0.690406	0.345203	−	0.938528i	$-0.387810\pi$
$691$	18.1486	0.690406	0.345203	+	0.938528i	$0.387810\pi$
$692$	0	0
$693$	−3.41329	−0.129660
$694$	0	0
$695$	−32.4735	−1.23179
$696$	0	0
$697$	−10.7910	−0.408740
$698$	0	0
$699$	85.5910	3.23735
$700$	0	0
$701$	−44.8193	−1.69280	−0.846401	−	0.532546i	$-0.821235\pi$
$701$	−44.8193	−1.69280	−0.846401	+	0.532546i	$0.821235\pi$
$702$	0	0
$703$	4.75482	0.179332
$704$	0	0
$705$	−67.2308	−2.53206
$706$	0	0
$707$	0.496052	0.0186559
$708$	0	0
$709$	−16.6682	−0.625988	−0.312994	−	0.949755i	$-0.601332\pi$
$709$	−16.6682	−0.625988	−0.312994	+	0.949755i	$0.601332\pi$
$710$	0	0
$711$	22.3862	0.839547
$712$	0	0
$713$	29.9584	1.12195
$714$	0	0
$715$	−62.1509	−2.32431
$716$	0	0
$717$	−43.5684	−1.62709
$718$	0	0
$719$	3.22826	0.120394	0.0601969	−	0.998187i	$-0.480827\pi$
$719$	3.22826	0.120394	0.0601969	+	0.998187i	$0.480827\pi$
$720$	0	0
$721$	−0.811672	−0.0302282
$722$	0	0
$723$	14.9988	0.557810
$724$	0	0
$725$	49.4858	1.83786
$726$	0	0
$727$	0.500824	0.0185745	0.00928726	−	0.999957i	$-0.497044\pi$
$727$	0.500824	0.0185745	0.00928726	+	0.999957i	$0.497044\pi$
$728$	0	0
$729$	−31.2299	−1.15666
$730$	0	0
$731$	−5.33316	−0.197254
$732$	0	0
$733$	−33.0391	−1.22033	−0.610163	−	0.792276i	$-0.708896\pi$
$733$	−33.0391	−1.22033	−0.610163	+	0.792276i	$0.708896\pi$
$734$	0	0
$735$	86.4371	3.18828
$736$	0	0
$737$	9.67527	0.356393
$738$	0	0
$739$	37.5267	1.38044	0.690222	−	0.723598i	$-0.257513\pi$
$739$	37.5267	1.38044	0.690222	+	0.723598i	$0.257513\pi$
$740$	0	0
$741$	−62.1539	−2.28328
$742$	0	0
$743$	3.38307	0.124113	0.0620564	−	0.998073i	$-0.480234\pi$
$743$	3.38307	0.124113	0.0620564	+	0.998073i	$0.480234\pi$
$744$	0	0
$745$	3.44011	0.126036
$746$	0	0
$747$	61.0993	2.23551
$748$	0	0
$749$	0.311521	0.0113827
$750$	0	0
$751$	19.9719	0.728785	0.364392	−	0.931245i	$-0.381277\pi$
$751$	19.9719	0.728785	0.364392	+	0.931245i	$0.381277\pi$
$752$	0	0
$753$	9.22362	0.336128
$754$	0	0
$755$	18.6052	0.677114
$756$	0	0
$757$	6.94229	0.252322	0.126161	−	0.992010i	$-0.459734\pi$
$757$	6.94229	0.252322	0.126161	+	0.992010i	$0.459734\pi$
$758$	0	0
$759$	−40.9298	−1.48566
$760$	0	0
$761$	−49.1640	−1.78220	−0.891098	−	0.453812i	$-0.850064\pi$
$761$	−49.1640	−1.78220	−0.891098	+	0.453812i	$0.850064\pi$
$762$	0	0
$763$	0.407103	0.0147381
$764$	0	0
$765$	24.4932	0.885552
$766$	0	0
$767$	6.94852	0.250897
$768$	0	0
$769$	34.5469	1.24579	0.622896	−	0.782304i	$-0.285956\pi$
$769$	34.5469	1.24579	0.622896	+	0.782304i	$0.285956\pi$
$770$	0	0
$771$	−62.6822	−2.25744
$772$	0	0
$773$	−39.5823	−1.42368	−0.711839	−	0.702343i	$-0.752137\pi$
$773$	−39.5823	−1.42368	−0.711839	+	0.702343i	$0.752137\pi$
$774$	0	0
$775$	58.5670	2.10379
$776$	0	0
$777$	1.29053	0.0462974
$778$	0	0
$779$	32.4817	1.16378
$780$	0	0
$781$	−28.6665	−1.02577
$782$	0	0
$783$	32.9194	1.17644
$784$	0	0
$785$	9.24729	0.330050
$786$	0	0
$787$	0.546122	0.0194671	0.00973357	−	0.999953i	$-0.496902\pi$
$787$	0.546122	0.0194671	0.00973357	+	0.999953i	$0.496902\pi$
$788$	0	0
$789$	5.82365	0.207327
$790$	0	0
$791$	−1.85632	−0.0660031
$792$	0	0
$793$	47.5601	1.68891
$794$	0	0
$795$	83.2010	2.95084
$796$	0	0
$797$	−13.8888	−0.491965	−0.245982	−	0.969274i	$-0.579111\pi$
$797$	−13.8888	−0.491965	−0.245982	+	0.969274i	$0.579111\pi$
$798$	0	0
$799$	−5.38581	−0.190536
$800$	0	0
$801$	87.2991	3.08456
$802$	0	0
$803$	−25.0573	−0.884252
$804$	0	0
$805$	−7.47013	−0.263287
$806$	0	0
$807$	−42.4925	−1.49581
$808$	0	0
$809$	−40.5209	−1.42464	−0.712320	−	0.701855i	$-0.752355\pi$
$809$	−40.5209	−1.42464	−0.712320	+	0.701855i	$0.752355\pi$
$810$	0	0
$811$	−17.2409	−0.605410	−0.302705	−	0.953084i	$-0.597890\pi$
$811$	−17.2409	−0.605410	−0.302705	+	0.953084i	$0.597890\pi$
$812$	0	0
$813$	11.5900	0.406480
$814$	0	0
$815$	78.7225	2.75753
$816$	0	0
$817$	16.0531	0.561629
$818$	0	0
$819$	−11.1385	−0.389212
$820$	0	0
$821$	−40.7034	−1.42056	−0.710279	−	0.703920i	$-0.751431\pi$
$821$	−40.7034	−1.42056	−0.710279	+	0.703920i	$0.751431\pi$
$822$	0	0
$823$	25.9775	0.905520	0.452760	−	0.891632i	$-0.350439\pi$
$823$	25.9775	0.905520	0.452760	+	0.891632i	$0.350439\pi$
$824$	0	0
$825$	−80.0155	−2.78578
$826$	0	0
$827$	−0.316315	−0.0109993	−0.00549967	−	0.999985i	$-0.501751\pi$
$827$	−0.316315	−0.0109993	−0.00549967	+	0.999985i	$0.501751\pi$
$828$	0	0
$829$	−9.24189	−0.320984	−0.160492	−	0.987037i	$-0.551308\pi$
$829$	−9.24189	−0.320984	−0.160492	+	0.987037i	$0.551308\pi$
$830$	0	0
$831$	−3.66718	−0.127213
$832$	0	0
$833$	6.92442	0.239917
$834$	0	0
$835$	1.42246	0.0492264
$836$	0	0
$837$	38.9605	1.34667
$838$	0	0
$839$	−51.4447	−1.77607	−0.888034	−	0.459777i	$-0.847929\pi$
$839$	−51.4447	−1.77607	−0.888034	+	0.459777i	$0.847929\pi$
$840$	0	0
$841$	−13.6860	−0.471933
$842$	0	0
$843$	−47.5861	−1.63895
$844$	0	0
$845$	−148.207	−5.09849
$846$	0	0
$847$	−1.77766	−0.0610811
$848$	0	0
$849$	46.6596	1.60135
$850$	0	0
$851$	10.2179	0.350264
$852$	0	0
$853$	−7.19210	−0.246253	−0.123126	−	0.992391i	$-0.539292\pi$
$853$	−7.19210	−0.246253	−0.123126	+	0.992391i	$0.539292\pi$
$854$	0	0
$855$	−73.7260	−2.52138
$856$	0	0
$857$	−12.3177	−0.420764	−0.210382	−	0.977619i	$-0.567471\pi$
$857$	−12.3177	−0.420764	−0.210382	+	0.977619i	$0.567471\pi$
$858$	0	0
$859$	−13.6564	−0.465950	−0.232975	−	0.972483i	$-0.574846\pi$
$859$	−13.6564	−0.465950	−0.232975	+	0.972483i	$0.574846\pi$
$860$	0	0
$861$	8.81600	0.300448
$862$	0	0
$863$	−7.13432	−0.242855	−0.121428	−	0.992600i	$-0.538747\pi$
$863$	−7.13432	−0.242855	−0.121428	+	0.992600i	$0.538747\pi$
$864$	0	0
$865$	−56.3319	−1.91534
$866$	0	0
$867$	2.97166	0.100923
$868$	0	0
$869$	−8.17504	−0.277319
$870$	0	0
$871$	31.5732	1.06982
$872$	0	0
$873$	21.4764	0.726864
$874$	0	0
$875$	−8.82943	−0.298489
$876$	0	0
$877$	−25.8328	−0.872312	−0.436156	−	0.899871i	$-0.643660\pi$
$877$	−25.8328	−0.872312	−0.436156	+	0.899871i	$0.643660\pi$
$878$	0	0
$879$	−49.2096	−1.65980
$880$	0	0
$881$	27.4871	0.926064	0.463032	−	0.886342i	$-0.346762\pi$
$881$	27.4871	0.926064	0.463032	+	0.886342i	$0.346762\pi$
$882$	0	0
$883$	−48.1207	−1.61939	−0.809696	−	0.586850i	$-0.800368\pi$
$883$	−48.1207	−1.61939	−0.809696	+	0.586850i	$0.800368\pi$
$884$	0	0
$885$	12.4829	0.419610
$886$	0	0
$887$	5.91374	0.198564	0.0992819	−	0.995059i	$-0.468345\pi$
$887$	5.91374	0.198564	0.0992819	+	0.995059i	$0.468345\pi$
$888$	0	0
$889$	−4.53835	−0.152211
$890$	0	0
$891$	−15.9820	−0.535418
$892$	0	0
$893$	16.2116	0.542502
$894$	0	0
$895$	−23.2161	−0.776029
$896$	0	0
$897$	−133.566	−4.45962
$898$	0	0
$899$	18.1243	0.604478
$900$	0	0
$901$	6.66518	0.222049
$902$	0	0
$903$	4.35705	0.144994
$904$	0	0
$905$	22.9129	0.761651
$906$	0	0
$907$	−7.36389	−0.244514	−0.122257	−	0.992498i	$-0.539013\pi$
$907$	−7.36389	−0.244514	−0.122257	+	0.992498i	$0.539013\pi$
$908$	0	0
$909$	10.5207	0.348951
$910$	0	0
$911$	−31.7282	−1.05120	−0.525601	−	0.850731i	$-0.676160\pi$
$911$	−31.7282	−1.05120	−0.525601	+	0.850731i	$0.676160\pi$
$912$	0	0
$913$	−22.3124	−0.738433
$914$	0	0
$915$	85.4413	2.82460
$916$	0	0
$917$	0.141067	0.00465846
$918$	0	0
$919$	−41.6140	−1.37272	−0.686359	−	0.727263i	$-0.740792\pi$
$919$	−41.6140	−1.37272	−0.686359	+	0.727263i	$0.740792\pi$
$920$	0	0
$921$	82.3360	2.71306
$922$	0	0
$923$	−93.5470	−3.07914
$924$	0	0
$925$	19.9754	0.656787
$926$	0	0
$927$	−17.2147	−0.565405
$928$	0	0
$929$	3.68178	0.120795	0.0603977	−	0.998174i	$-0.480763\pi$
$929$	3.68178	0.120795	0.0603977	+	0.998174i	$0.480763\pi$
$930$	0	0
$931$	−20.8429	−0.683100
$932$	0	0
$933$	49.2770	1.61326
$934$	0	0
$935$	−8.94447	−0.292516
$936$	0	0
$937$	−36.9742	−1.20789	−0.603946	−	0.797025i	$-0.706406\pi$
$937$	−36.9742	−1.20789	−0.603946	+	0.797025i	$0.706406\pi$
$938$	0	0
$939$	37.3042	1.21738
$940$	0	0
$941$	−45.1130	−1.47064	−0.735321	−	0.677719i	$-0.762969\pi$
$941$	−45.1130	−1.47064	−0.735321	+	0.677719i	$0.762969\pi$
$942$	0	0
$943$	69.8016	2.27305
$944$	0	0
$945$	−9.71479	−0.316022
$946$	0	0
$947$	−4.32362	−0.140499	−0.0702494	−	0.997529i	$-0.522380\pi$
$947$	−4.32362	−0.140499	−0.0702494	+	0.997529i	$0.522380\pi$
$948$	0	0
$949$	−81.7690	−2.65433
$950$	0	0
$951$	51.3310	1.66452
$952$	0	0
$953$	−42.2456	−1.36847	−0.684235	−	0.729262i	$-0.739864\pi$
$953$	−42.2456	−1.36847	−0.684235	+	0.729262i	$0.739864\pi$
$954$	0	0
$955$	81.4635	2.63610
$956$	0	0
$957$	−24.7617	−0.800433
$958$	0	0
$959$	4.53946	0.146587
$960$	0	0
$961$	−9.54970	−0.308055
$962$	0	0
$963$	6.60704	0.212909
$964$	0	0
$965$	40.0129	1.28806
$966$	0	0
$967$	−24.5676	−0.790041	−0.395021	−	0.918672i	$-0.629263\pi$
$967$	−24.5676	−0.790041	−0.395021	+	0.918672i	$0.629263\pi$
$968$	0	0
$969$	−8.94490	−0.287352
$970$	0	0
$971$	38.7477	1.24347	0.621736	−	0.783227i	$-0.286428\pi$
$971$	38.7477	1.24347	0.621736	+	0.783227i	$0.286428\pi$
$972$	0	0
$973$	2.12530	0.0681339
$974$	0	0
$975$	−261.113	−8.36231
$976$	0	0
$977$	6.20563	0.198536	0.0992678	−	0.995061i	$-0.468350\pi$
$977$	6.20563	0.198536	0.0992678	+	0.995061i	$0.468350\pi$
$978$	0	0
$979$	−31.8801	−1.01889
$980$	0	0
$981$	8.63423	0.275670
$982$	0	0
$983$	24.1434	0.770055	0.385028	−	0.922905i	$-0.374192\pi$
$983$	24.1434	0.770055	0.385028	+	0.922905i	$0.374192\pi$
$984$	0	0
$985$	−76.2025	−2.42801
$986$	0	0
$987$	4.40007	0.140056
$988$	0	0
$989$	34.4974	1.09695
$990$	0	0
$991$	−8.74721	−0.277864	−0.138932	−	0.990302i	$-0.544367\pi$
$991$	−8.74721	−0.277864	−0.138932	+	0.990302i	$0.544367\pi$
$992$	0	0
$993$	−39.2282	−1.24487
$994$	0	0
$995$	89.0379	2.82269
$996$	0	0
$997$	35.3343	1.11905	0.559524	−	0.828814i	$-0.310984\pi$
$997$	35.3343	1.11905	0.559524	+	0.828814i	$0.310984\pi$
$998$	0	0
$999$	13.2882	0.420420

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4012.2.a.i.1.16	✓	18

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4012.2.a.i.1.16	✓	18	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

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

\(f(q)\)	\(=\)	\(q+2.97166 q^{3} -4.20066 q^{5} +0.274921 q^{7} +5.83079 q^{9} +O(q^{10})\)	\(q+2.97166 q^{3} -4.20066 q^{5} +0.274921 q^{7} +5.83079 q^{9} -2.12930 q^{11} -6.94852 q^{13} -12.4829 q^{15} -1.00000 q^{17} +3.01007 q^{19} +0.816974 q^{21} +6.46848 q^{23} +12.6455 q^{25} +8.41216 q^{27} +3.91330 q^{29} +4.63145 q^{31} -6.32758 q^{33} -1.15485 q^{35} +1.57964 q^{37} -20.6487 q^{39} +10.7910 q^{41} +5.33316 q^{43} -24.4932 q^{45} +5.38581 q^{47} -6.92442 q^{49} -2.97166 q^{51} -6.66518 q^{53} +8.94447 q^{55} +8.94490 q^{57} -1.00000 q^{59} -6.84464 q^{61} +1.60301 q^{63} +29.1884 q^{65} -4.54387 q^{67} +19.2222 q^{69} +13.4629 q^{71} +11.7678 q^{73} +37.5782 q^{75} -0.585391 q^{77} +3.83930 q^{79} +7.50576 q^{81} +10.4787 q^{83} +4.20066 q^{85} +11.6290 q^{87} +14.9721 q^{89} -1.91030 q^{91} +13.7631 q^{93} -12.6443 q^{95} +3.68326 q^{97} -12.4155 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q + 8 q^{3} + 4 q^{5} + 2 q^{7} + 18 q^{9}+O(q^{10}) \) 18 * q + 8 * q^3 + 4 * q^5 + 2 * q^7 + 18 * q^9	\( 18 q + 8 q^{3} + 4 q^{5} + 2 q^{7} + 18 q^{9} + 12 q^{11} + 2 q^{13} - 18 q^{17} + 5 q^{19} - 3 q^{21} + 21 q^{23} + 16 q^{25} + 26 q^{27} + 14 q^{29} + 15 q^{31} + 19 q^{33} + 20 q^{35} + 2 q^{37} - 14 q^{39} + 34 q^{41} + 21 q^{43} + 49 q^{45} + 69 q^{47} + 28 q^{49} - 8 q^{51} - 4 q^{53} + 18 q^{55} + 5 q^{57} - 18 q^{59} + 11 q^{61} + 35 q^{63} + 27 q^{65} + 34 q^{67} - 4 q^{69} + 37 q^{71} + 18 q^{73} + 72 q^{75} + 11 q^{77} + 11 q^{79} + 30 q^{81} + 28 q^{83} - 4 q^{85} + 7 q^{87} + 44 q^{89} - 23 q^{91} - 3 q^{93} - 11 q^{95} + 11 q^{97} + 56 q^{99}+O(q^{100}) \) 18 * q + 8 * q^3 + 4 * q^5 + 2 * q^7 + 18 * q^9 + 12 * q^11 + 2 * q^13 - 18 * q^17 + 5 * q^19 - 3 * q^21 + 21 * q^23 + 16 * q^25 + 26 * q^27 + 14 * q^29 + 15 * q^31 + 19 * q^33 + 20 * q^35 + 2 * q^37 - 14 * q^39 + 34 * q^41 + 21 * q^43 + 49 * q^45 + 69 * q^47 + 28 * q^49 - 8 * q^51 - 4 * q^53 + 18 * q^55 + 5 * q^57 - 18 * q^59 + 11 * q^61 + 35 * q^63 + 27 * q^65 + 34 * q^67 - 4 * q^69 + 37 * q^71 + 18 * q^73 + 72 * q^75 + 11 * q^77 + 11 * q^79 + 30 * q^81 + 28 * q^83 - 4 * q^85 + 7 * q^87 + 44 * q^89 - 23 * q^91 - 3 * q^93 - 11 * q^95 + 11 * q^97 + 56 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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