LMFDB - Embedded newform 4012.2.a.i.1.12

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			1.12
Root			$1.33821$ 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)

$f(q)$	$=$	$q+1.33821 q^{3} +3.45044 q^{5} -1.09754 q^{7} -1.20921 q^{9} +O(q^{10})$	$q+1.33821 q^{3} +3.45044 q^{5} -1.09754 q^{7} -1.20921 q^{9} -2.07064 q^{11} -0.787376 q^{13} +4.61740 q^{15} -1.00000 q^{17} +5.10502 q^{19} -1.46874 q^{21} +5.70643 q^{23} +6.90555 q^{25} -5.63278 q^{27} +0.628146 q^{29} +11.0042 q^{31} -2.77094 q^{33} -3.78700 q^{35} +1.84027 q^{37} -1.05367 q^{39} +3.21649 q^{41} -11.9573 q^{43} -4.17230 q^{45} +7.01036 q^{47} -5.79540 q^{49} -1.33821 q^{51} +7.97710 q^{53} -7.14462 q^{55} +6.83157 q^{57} -1.00000 q^{59} +12.0168 q^{61} +1.32715 q^{63} -2.71680 q^{65} -0.837540 q^{67} +7.63637 q^{69} +12.4992 q^{71} +12.8344 q^{73} +9.24105 q^{75} +2.27261 q^{77} +7.79793 q^{79} -3.91020 q^{81} -1.69226 q^{83} -3.45044 q^{85} +0.840589 q^{87} +3.60405 q^{89} +0.864178 q^{91} +14.7259 q^{93} +17.6146 q^{95} +9.09780 q^{97} +2.50383 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	1.33821	0.772613	0.386307	−	0.922370i	$-0.373751\pi$
$3$	1.33821	0.772613	0.386307	+	0.922370i	$0.373751\pi$
$4$	0	0
$5$	3.45044	1.54308	0.771542	−	0.636178i	$-0.219486\pi$
$5$	3.45044	1.54308	0.771542	+	0.636178i	$0.219486\pi$
$6$	0	0
$7$	−1.09754	−0.414832	−0.207416	−	0.978253i	$-0.566505\pi$
$7$	−1.09754	−0.414832	−0.207416	+	0.978253i	$0.566505\pi$
$8$	0	0
$9$	−1.20921	−0.403069
$10$	0	0
$11$	−2.07064	−0.624321	−0.312160	−	0.950029i	$-0.601053\pi$
$11$	−2.07064	−0.624321	−0.312160	+	0.950029i	$0.601053\pi$
$12$	0	0
$13$	−0.787376	−0.218379	−0.109189	−	0.994021i	$-0.534826\pi$
$13$	−0.787376	−0.218379	−0.109189	+	0.994021i	$0.534826\pi$
$14$	0	0
$15$	4.61740	1.19221
$16$	0	0
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	0	0
$19$	5.10502	1.17117	0.585586	−	0.810610i	$-0.300864\pi$
$19$	5.10502	1.17117	0.585586	+	0.810610i	$0.300864\pi$
$20$	0	0
$21$	−1.46874	−0.320504
$22$	0	0
$23$	5.70643	1.18987	0.594936	−	0.803773i	$-0.297177\pi$
$23$	5.70643	1.18987	0.594936	+	0.803773i	$0.297177\pi$
$24$	0	0
$25$	6.90555	1.38111
$26$	0	0
$27$	−5.63278	−1.08403
$28$	0	0
$29$	0.628146	0.116644	0.0583219	−	0.998298i	$-0.481425\pi$
$29$	0.628146	0.116644	0.0583219	+	0.998298i	$0.481425\pi$
$30$	0	0
$31$	11.0042	1.97641	0.988205	−	0.153137i	$-0.0489376\pi$
$31$	11.0042	1.97641	0.988205	+	0.153137i	$0.0489376\pi$
$32$	0	0
$33$	−2.77094	−0.482359
$34$	0	0
$35$	−3.78700	−0.640121
$36$	0	0
$37$	1.84027	0.302539	0.151269	−	0.988493i	$-0.451664\pi$
$37$	1.84027	0.302539	0.151269	+	0.988493i	$0.451664\pi$
$38$	0	0
$39$	−1.05367	−0.168722
$40$	0	0
$41$	3.21649	0.502331	0.251166	−	0.967944i	$-0.419186\pi$
$41$	3.21649	0.502331	0.251166	+	0.967944i	$0.419186\pi$
$42$	0	0
$43$	−11.9573	−1.82347	−0.911736	−	0.410777i	$-0.865258\pi$
$43$	−11.9573	−1.82347	−0.911736	+	0.410777i	$0.865258\pi$
$44$	0	0
$45$	−4.17230	−0.621970
$46$	0	0
$47$	7.01036	1.02257	0.511283	−	0.859413i	$-0.329171\pi$
$47$	7.01036	1.02257	0.511283	+	0.859413i	$0.329171\pi$
$48$	0	0
$49$	−5.79540	−0.827915
$50$	0	0
$51$	−1.33821	−0.187386
$52$	0	0
$53$	7.97710	1.09574	0.547869	−	0.836564i	$-0.315439\pi$
$53$	7.97710	1.09574	0.547869	+	0.836564i	$0.315439\pi$
$54$	0	0
$55$	−7.14462	−0.963380
$56$	0	0
$57$	6.83157	0.904863
$58$	0	0
$59$	−1.00000	−0.130189
$59$	−1.00000	−0.130189
$60$	0	0
$61$	12.0168	1.53859	0.769296	−	0.638892i	$-0.220607\pi$
$61$	12.0168	1.53859	0.769296	+	0.638892i	$0.220607\pi$
$62$	0	0
$63$	1.32715	0.167206
$64$	0	0
$65$	−2.71680	−0.336977
$66$	0	0
$67$	−0.837540	−0.102322	−0.0511609	−	0.998690i	$-0.516292\pi$
$67$	−0.837540	−0.102322	−0.0511609	+	0.998690i	$0.516292\pi$
$68$	0	0
$69$	7.63637	0.919311
$70$	0	0
$71$	12.4992	1.48338	0.741691	−	0.670741i	$-0.234024\pi$
$71$	12.4992	1.48338	0.741691	+	0.670741i	$0.234024\pi$
$72$	0	0
$73$	12.8344	1.50215	0.751076	−	0.660216i	$-0.229535\pi$
$73$	12.8344	1.50215	0.751076	+	0.660216i	$0.229535\pi$
$74$	0	0
$75$	9.24105	1.06706
$76$	0	0
$77$	2.27261	0.258988
$78$	0	0
$79$	7.79793	0.877335	0.438668	−	0.898649i	$-0.355451\pi$
$79$	7.79793	0.877335	0.438668	+	0.898649i	$0.355451\pi$
$80$	0	0
$81$	−3.91020	−0.434467
$82$	0	0
$83$	−1.69226	−0.185750	−0.0928751	−	0.995678i	$-0.529606\pi$
$83$	−1.69226	−0.185750	−0.0928751	+	0.995678i	$0.529606\pi$
$84$	0	0
$85$	−3.45044	−0.374253
$86$	0	0
$87$	0.840589	0.0901206
$88$	0	0
$89$	3.60405	0.382029	0.191014	−	0.981587i	$-0.438822\pi$
$89$	3.60405	0.382029	0.191014	+	0.981587i	$0.438822\pi$
$90$	0	0
$91$	0.864178	0.0905905
$92$	0	0
$93$	14.7259	1.52700
$94$	0	0
$95$	17.6146	1.80722
$96$	0	0
$97$	9.09780	0.923742	0.461871	−	0.886947i	$-0.347178\pi$
$97$	9.09780	0.923742	0.461871	+	0.886947i	$0.347178\pi$
$98$	0	0
$99$	2.50383	0.251644
$100$	0	0
$101$	−11.7376	−1.16794	−0.583969	−	0.811776i	$-0.698501\pi$
$101$	−11.7376	−1.16794	−0.583969	+	0.811776i	$0.698501\pi$
$102$	0	0
$103$	−2.53431	−0.249713	−0.124857	−	0.992175i	$-0.539847\pi$
$103$	−2.53431	−0.249713	−0.124857	+	0.992175i	$0.539847\pi$
$104$	0	0
$105$	−5.06779	−0.494566
$106$	0	0
$107$	6.46546	0.625040	0.312520	−	0.949911i	$-0.398827\pi$
$107$	6.46546	0.625040	0.312520	+	0.949911i	$0.398827\pi$
$108$	0	0
$109$	−13.2228	−1.26652	−0.633259	−	0.773940i	$-0.718283\pi$
$109$	−13.2228	−1.26652	−0.633259	+	0.773940i	$0.718283\pi$
$110$	0	0
$111$	2.46266	0.233746
$112$	0	0
$113$	8.42935	0.792967	0.396484	−	0.918042i	$-0.370230\pi$
$113$	8.42935	0.792967	0.396484	+	0.918042i	$0.370230\pi$
$114$	0	0
$115$	19.6897	1.83607
$116$	0	0
$117$	0.952101	0.0880217
$118$	0	0
$119$	1.09754	0.100611
$120$	0	0
$121$	−6.71246	−0.610223
$122$	0	0
$123$	4.30432	0.388108
$124$	0	0
$125$	6.57500	0.588086
$126$	0	0
$127$	7.07659	0.627946	0.313973	−	0.949432i	$-0.398340\pi$
$127$	7.07659	0.627946	0.313973	+	0.949432i	$0.398340\pi$
$128$	0	0
$129$	−16.0013	−1.40884
$130$	0	0
$131$	−16.9630	−1.48207	−0.741033	−	0.671469i	$-0.765664\pi$
$131$	−16.9630	−1.48207	−0.741033	+	0.671469i	$0.765664\pi$
$132$	0	0
$133$	−5.60297	−0.485839
$134$	0	0
$135$	−19.4356	−1.67275
$136$	0	0
$137$	−6.14420	−0.524935	−0.262467	−	0.964941i	$-0.584536\pi$
$137$	−6.14420	−0.524935	−0.262467	+	0.964941i	$0.584536\pi$
$138$	0	0
$139$	−4.55271	−0.386156	−0.193078	−	0.981183i	$-0.561847\pi$
$139$	−4.55271	−0.386156	−0.193078	+	0.981183i	$0.561847\pi$
$140$	0	0
$141$	9.38129	0.790048
$142$	0	0
$143$	1.63037	0.136338
$144$	0	0
$145$	2.16738	0.179991
$146$	0	0
$147$	−7.75544	−0.639658
$148$	0	0
$149$	−22.8193	−1.86943	−0.934714	−	0.355401i	$-0.884344\pi$
$149$	−22.8193	−1.86943	−0.934714	+	0.355401i	$0.884344\pi$
$150$	0	0
$151$	21.6297	1.76020	0.880102	−	0.474785i	$-0.157474\pi$
$151$	21.6297	1.76020	0.880102	+	0.474785i	$0.157474\pi$
$152$	0	0
$153$	1.20921	0.0977586
$154$	0	0
$155$	37.9693	3.04977
$156$	0	0
$157$	2.46100	0.196409	0.0982045	−	0.995166i	$-0.468690\pi$
$157$	2.46100	0.196409	0.0982045	+	0.995166i	$0.468690\pi$
$158$	0	0
$159$	10.6750	0.846582
$160$	0	0
$161$	−6.26304	−0.493597
$162$	0	0
$163$	−24.5869	−1.92579	−0.962896	−	0.269874i	$-0.913018\pi$
$163$	−24.5869	−1.92579	−0.962896	+	0.269874i	$0.913018\pi$
$164$	0	0
$165$	−9.56097	−0.744320
$166$	0	0
$167$	6.73578	0.521231	0.260615	−	0.965443i	$-0.416075\pi$
$167$	6.73578	0.521231	0.260615	+	0.965443i	$0.416075\pi$
$168$	0	0
$169$	−12.3800	−0.952311
$170$	0	0
$171$	−6.17302	−0.472063
$172$	0	0
$173$	−12.4714	−0.948183	−0.474092	−	0.880475i	$-0.657224\pi$
$173$	−12.4714	−0.948183	−0.474092	+	0.880475i	$0.657224\pi$
$174$	0	0
$175$	−7.57913	−0.572929
$176$	0	0
$177$	−1.33821	−0.100586
$178$	0	0
$179$	1.54154	0.115220	0.0576100	−	0.998339i	$-0.481652\pi$
$179$	1.54154	0.115220	0.0576100	+	0.998339i	$0.481652\pi$
$180$	0	0
$181$	−11.5931	−0.861710	−0.430855	−	0.902421i	$-0.641788\pi$
$181$	−11.5931	−0.861710	−0.430855	+	0.902421i	$0.641788\pi$
$182$	0	0
$183$	16.0809	1.18874
$184$	0	0
$185$	6.34975	0.466843
$186$	0	0
$187$	2.07064	0.151420
$188$	0	0
$189$	6.18221	0.449690
$190$	0	0
$191$	7.48069	0.541284	0.270642	−	0.962680i	$-0.412764\pi$
$191$	7.48069	0.541284	0.270642	+	0.962680i	$0.412764\pi$
$192$	0	0
$193$	−4.86586	−0.350252	−0.175126	−	0.984546i	$-0.556033\pi$
$193$	−4.86586	−0.350252	−0.175126	+	0.984546i	$0.556033\pi$
$194$	0	0
$195$	−3.63563	−0.260353
$196$	0	0
$197$	4.88082	0.347744	0.173872	−	0.984768i	$-0.444372\pi$
$197$	4.88082	0.347744	0.173872	+	0.984768i	$0.444372\pi$
$198$	0	0
$199$	22.1249	1.56839	0.784196	−	0.620513i	$-0.213076\pi$
$199$	22.1249	1.56839	0.784196	+	0.620513i	$0.213076\pi$
$200$	0	0
$201$	−1.12080	−0.0790552
$202$	0	0
$203$	−0.689417	−0.0483876
$204$	0	0
$205$	11.0983	0.775140
$206$	0	0
$207$	−6.90025	−0.479601
$208$	0	0
$209$	−10.5706	−0.731187
$210$	0	0
$211$	−14.5125	−0.999079	−0.499540	−	0.866291i	$-0.666497\pi$
$211$	−14.5125	−0.999079	−0.499540	+	0.866291i	$0.666497\pi$
$212$	0	0
$213$	16.7265	1.14608
$214$	0	0
$215$	−41.2580	−2.81377
$216$	0	0
$217$	−12.0775	−0.819877
$218$	0	0
$219$	17.1750	1.16058
$220$	0	0
$221$	0.787376	0.0529647
$222$	0	0
$223$	−5.74898	−0.384980	−0.192490	−	0.981299i	$-0.561656\pi$
$223$	−5.74898	−0.384980	−0.192490	+	0.981299i	$0.561656\pi$
$224$	0	0
$225$	−8.35024	−0.556683
$226$	0	0
$227$	4.74165	0.314714	0.157357	−	0.987542i	$-0.449703\pi$
$227$	4.74165	0.314714	0.157357	+	0.987542i	$0.449703\pi$
$228$	0	0
$229$	−16.7329	−1.10574	−0.552869	−	0.833268i	$-0.686467\pi$
$229$	−16.7329	−1.10574	−0.552869	+	0.833268i	$0.686467\pi$
$230$	0	0
$231$	3.04122	0.200098
$232$	0	0
$233$	−9.41847	−0.617024	−0.308512	−	0.951220i	$-0.599831\pi$
$233$	−9.41847	−0.617024	−0.308512	+	0.951220i	$0.599831\pi$
$234$	0	0
$235$	24.1888	1.57791
$236$	0	0
$237$	10.4352	0.677841
$238$	0	0
$239$	−14.4178	−0.932609	−0.466305	−	0.884624i	$-0.654415\pi$
$239$	−14.4178	−0.932609	−0.466305	+	0.884624i	$0.654415\pi$
$240$	0	0
$241$	−12.0994	−0.779388	−0.389694	−	0.920944i	$-0.627419\pi$
$241$	−12.0994	−0.779388	−0.389694	+	0.920944i	$0.627419\pi$
$242$	0	0
$243$	11.6657	0.748355
$244$	0	0
$245$	−19.9967	−1.27754
$246$	0	0
$247$	−4.01957	−0.255759
$248$	0	0
$249$	−2.26460	−0.143513
$250$	0	0
$251$	−16.3313	−1.03082	−0.515412	−	0.856942i	$-0.672361\pi$
$251$	−16.3313	−1.03082	−0.515412	+	0.856942i	$0.672361\pi$
$252$	0	0
$253$	−11.8159	−0.742862
$254$	0	0
$255$	−4.61740	−0.289153
$256$	0	0
$257$	25.2771	1.57674	0.788370	−	0.615201i	$-0.210925\pi$
$257$	25.2771	1.57674	0.788370	+	0.615201i	$0.210925\pi$
$258$	0	0
$259$	−2.01978	−0.125503
$260$	0	0
$261$	−0.759559	−0.0470155
$262$	0	0
$263$	12.7316	0.785065	0.392533	−	0.919738i	$-0.371599\pi$
$263$	12.7316	0.785065	0.392533	+	0.919738i	$0.371599\pi$
$264$	0	0
$265$	27.5245	1.69082
$266$	0	0
$267$	4.82296	0.295160
$268$	0	0
$269$	−0.361688	−0.0220525	−0.0110263	−	0.999939i	$-0.503510\pi$
$269$	−0.361688	−0.0220525	−0.0110263	+	0.999939i	$0.503510\pi$
$270$	0	0
$271$	13.5712	0.824390	0.412195	−	0.911096i	$-0.364762\pi$
$271$	13.5712	0.824390	0.412195	+	0.911096i	$0.364762\pi$
$272$	0	0
$273$	1.15645	0.0699914
$274$	0	0
$275$	−14.2989	−0.862256
$276$	0	0
$277$	−4.73968	−0.284780	−0.142390	−	0.989811i	$-0.545479\pi$
$277$	−4.73968	−0.284780	−0.142390	+	0.989811i	$0.545479\pi$
$278$	0	0
$279$	−13.3063	−0.796629
$280$	0	0
$281$	−20.5630	−1.22668	−0.613342	−	0.789818i	$-0.710175\pi$
$281$	−20.5630	−1.22668	−0.613342	+	0.789818i	$0.710175\pi$
$282$	0	0
$283$	29.0461	1.72661	0.863306	−	0.504681i	$-0.168390\pi$
$283$	29.0461	1.72661	0.863306	+	0.504681i	$0.168390\pi$
$284$	0	0
$285$	23.5719	1.39628
$286$	0	0
$287$	−3.53023	−0.208383
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	12.1747	0.713695
$292$	0	0
$293$	−26.1984	−1.53052	−0.765262	−	0.643719i	$-0.777391\pi$
$293$	−26.1984	−1.53052	−0.765262	+	0.643719i	$0.777391\pi$
$294$	0	0
$295$	−3.45044	−0.200893
$296$	0	0
$297$	11.6635	0.676782
$298$	0	0
$299$	−4.49311	−0.259843
$300$	0	0
$301$	13.1236	0.756434
$302$	0	0
$303$	−15.7073	−0.902364
$304$	0	0
$305$	41.4632	2.37418
$306$	0	0
$307$	−23.3961	−1.33529	−0.667644	−	0.744481i	$-0.732697\pi$
$307$	−23.3961	−1.33529	−0.667644	+	0.744481i	$0.732697\pi$
$308$	0	0
$309$	−3.39143	−0.192932
$310$	0	0
$311$	6.38678	0.362161	0.181081	−	0.983468i	$-0.442041\pi$
$311$	6.38678	0.362161	0.181081	+	0.983468i	$0.442041\pi$
$312$	0	0
$313$	−23.5653	−1.33199	−0.665996	−	0.745955i	$-0.731993\pi$
$313$	−23.5653	−1.33199	−0.665996	+	0.745955i	$0.731993\pi$
$314$	0	0
$315$	4.57927	0.258013
$316$	0	0
$317$	4.19780	0.235772	0.117886	−	0.993027i	$-0.462388\pi$
$317$	4.19780	0.235772	0.117886	+	0.993027i	$0.462388\pi$
$318$	0	0
$319$	−1.30066	−0.0728232
$320$	0	0
$321$	8.65211	0.482914
$322$	0	0
$323$	−5.10502	−0.284051
$324$	0	0
$325$	−5.43727	−0.301605
$326$	0	0
$327$	−17.6949	−0.978528
$328$	0	0
$329$	−7.69416	−0.424193
$330$	0	0
$331$	0.648543	0.0356471	0.0178236	−	0.999841i	$-0.494326\pi$
$331$	0.648543	0.0356471	0.0178236	+	0.999841i	$0.494326\pi$
$332$	0	0
$333$	−2.22527	−0.121944
$334$	0	0
$335$	−2.88989	−0.157891
$336$	0	0
$337$	−3.41537	−0.186047	−0.0930236	−	0.995664i	$-0.529653\pi$
$337$	−3.41537	−0.186047	−0.0930236	+	0.995664i	$0.529653\pi$
$338$	0	0
$339$	11.2802	0.612657
$340$	0	0
$341$	−22.7857	−1.23391
$342$	0	0
$343$	14.0435	0.758277
$344$	0	0
$345$	26.3489	1.41857
$346$	0	0
$347$	20.9236	1.12324	0.561618	−	0.827397i	$-0.310179\pi$
$347$	20.9236	1.12324	0.561618	+	0.827397i	$0.310179\pi$
$348$	0	0
$349$	28.3081	1.51530	0.757648	−	0.652663i	$-0.226348\pi$
$349$	28.3081	1.51530	0.757648	+	0.652663i	$0.226348\pi$
$350$	0	0
$351$	4.43512	0.236729
$352$	0	0
$353$	−35.9684	−1.91440	−0.957201	−	0.289423i	$-0.906537\pi$
$353$	−35.9684	−1.91440	−0.957201	+	0.289423i	$0.906537\pi$
$354$	0	0
$355$	43.1278	2.28899
$356$	0	0
$357$	1.46874	0.0777337
$358$	0	0
$359$	−22.1526	−1.16917	−0.584585	−	0.811333i	$-0.698743\pi$
$359$	−22.1526	−1.16917	−0.584585	+	0.811333i	$0.698743\pi$
$360$	0	0
$361$	7.06123	0.371644
$362$	0	0
$363$	−8.98265	−0.471467
$364$	0	0
$365$	44.2843	2.31795
$366$	0	0
$367$	20.9634	1.09428	0.547139	−	0.837042i	$-0.315717\pi$
$367$	20.9634	1.09428	0.547139	+	0.837042i	$0.315717\pi$
$368$	0	0
$369$	−3.88940	−0.202474
$370$	0	0
$371$	−8.75520	−0.454547
$372$	0	0
$373$	8.95736	0.463795	0.231897	−	0.972740i	$-0.425507\pi$
$373$	8.95736	0.463795	0.231897	+	0.972740i	$0.425507\pi$
$374$	0	0
$375$	8.79871	0.454363
$376$	0	0
$377$	−0.494587	−0.0254725
$378$	0	0
$379$	30.1505	1.54873	0.774363	−	0.632742i	$-0.218071\pi$
$379$	30.1505	1.54873	0.774363	+	0.632742i	$0.218071\pi$
$380$	0	0
$381$	9.46993	0.485159
$382$	0	0
$383$	−18.2171	−0.930850	−0.465425	−	0.885087i	$-0.654099\pi$
$383$	−18.2171	−0.930850	−0.465425	+	0.885087i	$0.654099\pi$
$384$	0	0
$385$	7.84152	0.399641
$386$	0	0
$387$	14.4589	0.734985
$388$	0	0
$389$	−29.3137	−1.48626	−0.743132	−	0.669144i	$-0.766661\pi$
$389$	−29.3137	−1.48626	−0.743132	+	0.669144i	$0.766661\pi$
$390$	0	0
$391$	−5.70643	−0.288586
$392$	0	0
$393$	−22.7000	−1.14506
$394$	0	0
$395$	26.9063	1.35380
$396$	0	0
$397$	8.84057	0.443696	0.221848	−	0.975081i	$-0.428791\pi$
$397$	8.84057	0.443696	0.221848	+	0.975081i	$0.428791\pi$
$398$	0	0
$399$	−7.49793	−0.375366
$400$	0	0
$401$	−4.43564	−0.221505	−0.110753	−	0.993848i	$-0.535326\pi$
$401$	−4.43564	−0.221505	−0.110753	+	0.993848i	$0.535326\pi$
$402$	0	0
$403$	−8.66443	−0.431606
$404$	0	0
$405$	−13.4919	−0.670419
$406$	0	0
$407$	−3.81054	−0.188881
$408$	0	0
$409$	−11.2641	−0.556974	−0.278487	−	0.960440i	$-0.589833\pi$
$409$	−11.2641	−0.556974	−0.278487	+	0.960440i	$0.589833\pi$
$410$	0	0
$411$	−8.22221	−0.405571
$412$	0	0
$413$	1.09754	0.0540065
$414$	0	0
$415$	−5.83906	−0.286628
$416$	0	0
$417$	−6.09246	−0.298349
$418$	0	0
$419$	32.4001	1.58285	0.791424	−	0.611268i	$-0.209340\pi$
$419$	32.4001	1.58285	0.791424	+	0.611268i	$0.209340\pi$
$420$	0	0
$421$	−4.64255	−0.226264	−0.113132	−	0.993580i	$-0.536088\pi$
$421$	−4.64255	−0.226264	−0.113132	+	0.993580i	$0.536088\pi$
$422$	0	0
$423$	−8.47697	−0.412164
$424$	0	0
$425$	−6.90555	−0.334969
$426$	0	0
$427$	−13.1889	−0.638257
$428$	0	0
$429$	2.18177	0.105337
$430$	0	0
$431$	11.7830	0.567569	0.283785	−	0.958888i	$-0.408410\pi$
$431$	11.7830	0.567569	0.283785	+	0.958888i	$0.408410\pi$
$432$	0	0
$433$	25.6059	1.23054	0.615271	−	0.788316i	$-0.289047\pi$
$433$	25.6059	1.23054	0.615271	+	0.788316i	$0.289047\pi$
$434$	0	0
$435$	2.90040	0.139064
$436$	0	0
$437$	29.1314	1.39355
$438$	0	0
$439$	−4.61337	−0.220184	−0.110092	−	0.993921i	$-0.535115\pi$
$439$	−4.61337	−0.220184	−0.110092	+	0.993921i	$0.535115\pi$
$440$	0	0
$441$	7.00784	0.333707
$442$	0	0
$443$	−24.9391	−1.18489	−0.592447	−	0.805610i	$-0.701838\pi$
$443$	−24.9391	−1.18489	−0.592447	+	0.805610i	$0.701838\pi$
$444$	0	0
$445$	12.4356	0.589503
$446$	0	0
$447$	−30.5369	−1.44434
$448$	0	0
$449$	29.2549	1.38062	0.690312	−	0.723511i	$-0.257473\pi$
$449$	29.2549	1.38062	0.690312	+	0.723511i	$0.257473\pi$
$450$	0	0
$451$	−6.66019	−0.313616
$452$	0	0
$453$	28.9450	1.35996
$454$	0	0
$455$	2.98180	0.139789
$456$	0	0
$457$	−31.7472	−1.48507	−0.742536	−	0.669806i	$-0.766377\pi$
$457$	−31.7472	−1.48507	−0.742536	+	0.669806i	$0.766377\pi$
$458$	0	0
$459$	5.63278	0.262916
$460$	0	0
$461$	−17.4281	−0.811708	−0.405854	−	0.913938i	$-0.633026\pi$
$461$	−17.4281	−0.811708	−0.405854	+	0.913938i	$0.633026\pi$
$462$	0	0
$463$	12.5828	0.584771	0.292386	−	0.956301i	$-0.405551\pi$
$463$	12.5828	0.584771	0.292386	+	0.956301i	$0.405551\pi$
$464$	0	0
$465$	50.8107	2.35629
$466$	0	0
$467$	5.57783	0.258111	0.129056	−	0.991637i	$-0.458805\pi$
$467$	5.57783	0.258111	0.129056	+	0.991637i	$0.458805\pi$
$468$	0	0
$469$	0.919235	0.0424463
$470$	0	0
$471$	3.29332	0.151748
$472$	0	0
$473$	24.7593	1.13843
$474$	0	0
$475$	35.2530	1.61752
$476$	0	0
$477$	−9.64596	−0.441658
$478$	0	0
$479$	−40.1527	−1.83462	−0.917311	−	0.398171i	$-0.869645\pi$
$479$	−40.1527	−1.83462	−0.917311	+	0.398171i	$0.869645\pi$
$480$	0	0
$481$	−1.44899	−0.0660681
$482$	0	0
$483$	−8.38123	−0.381359
$484$	0	0
$485$	31.3914	1.42541
$486$	0	0
$487$	11.9789	0.542816	0.271408	−	0.962464i	$-0.412511\pi$
$487$	11.9789	0.542816	0.271408	+	0.962464i	$0.412511\pi$
$488$	0	0
$489$	−32.9023	−1.48789
$490$	0	0
$491$	5.29324	0.238881	0.119440	−	0.992841i	$-0.461890\pi$
$491$	5.29324	0.238881	0.119440	+	0.992841i	$0.461890\pi$
$492$	0	0
$493$	−0.628146	−0.0282903
$494$	0	0
$495$	8.63932	0.388309
$496$	0	0
$497$	−13.7184	−0.615354
$498$	0	0
$499$	0.343017	0.0153556	0.00767778	−	0.999971i	$-0.497556\pi$
$499$	0.343017	0.0153556	0.00767778	+	0.999971i	$0.497556\pi$
$500$	0	0
$501$	9.01386	0.402710
$502$	0	0
$503$	−14.6507	−0.653244	−0.326622	−	0.945155i	$-0.605910\pi$
$503$	−14.6507	−0.653244	−0.326622	+	0.945155i	$0.605910\pi$
$504$	0	0
$505$	−40.5000	−1.80223
$506$	0	0
$507$	−16.5670	−0.735768
$508$	0	0
$509$	27.1132	1.20177	0.600885	−	0.799335i	$-0.294815\pi$
$509$	27.1132	1.20177	0.600885	+	0.799335i	$0.294815\pi$
$510$	0	0
$511$	−14.0863	−0.623140
$512$	0	0
$513$	−28.7555	−1.26958
$514$	0	0
$515$	−8.74449	−0.385328
$516$	0	0
$517$	−14.5159	−0.638409
$518$	0	0
$519$	−16.6893	−0.732579
$520$	0	0
$521$	3.28791	0.144046	0.0720229	−	0.997403i	$-0.477055\pi$
$521$	3.28791	0.144046	0.0720229	+	0.997403i	$0.477055\pi$
$522$	0	0
$523$	−24.4050	−1.06716	−0.533578	−	0.845751i	$-0.679153\pi$
$523$	−24.4050	−1.06716	−0.533578	+	0.845751i	$0.679153\pi$
$524$	0	0
$525$	−10.1424	−0.442652
$526$	0	0
$527$	−11.0042	−0.479350
$528$	0	0
$529$	9.56332	0.415796
$530$	0	0
$531$	1.20921	0.0524751
$532$	0	0
$533$	−2.53259	−0.109699
$534$	0	0
$535$	22.3087	0.964489
$536$	0	0
$537$	2.06290	0.0890205
$538$	0	0
$539$	12.0002	0.516884
$540$	0	0
$541$	10.4509	0.449318	0.224659	−	0.974437i	$-0.427873\pi$
$541$	10.4509	0.449318	0.224659	+	0.974437i	$0.427873\pi$
$542$	0	0
$543$	−15.5140	−0.665768
$544$	0	0
$545$	−45.6246	−1.95434
$546$	0	0
$547$	−20.7777	−0.888389	−0.444194	−	0.895930i	$-0.646510\pi$
$547$	−20.7777	−0.888389	−0.444194	+	0.895930i	$0.646510\pi$
$548$	0	0
$549$	−14.5308	−0.620159
$550$	0	0
$551$	3.20670	0.136610
$552$	0	0
$553$	−8.55855	−0.363947
$554$	0	0
$555$	8.49728	0.360689
$556$	0	0
$557$	18.9722	0.803876	0.401938	−	0.915667i	$-0.368337\pi$
$557$	18.9722	0.803876	0.401938	+	0.915667i	$0.368337\pi$
$558$	0	0
$559$	9.41490	0.398208
$560$	0	0
$561$	2.77094	0.116989
$562$	0	0
$563$	−28.6144	−1.20595	−0.602976	−	0.797759i	$-0.706019\pi$
$563$	−28.6144	−1.20595	−0.602976	+	0.797759i	$0.706019\pi$
$564$	0	0
$565$	29.0850	1.22362
$566$	0	0
$567$	4.29161	0.180231
$568$	0	0
$569$	31.6348	1.32620	0.663100	−	0.748531i	$-0.269240\pi$
$569$	31.6348	1.32620	0.663100	+	0.748531i	$0.269240\pi$
$570$	0	0
$571$	43.3282	1.81323	0.906614	−	0.421961i	$-0.138658\pi$
$571$	43.3282	1.81323	0.906614	+	0.421961i	$0.138658\pi$
$572$	0	0
$573$	10.0107	0.418203
$574$	0	0
$575$	39.4060	1.64335
$576$	0	0
$577$	−8.20420	−0.341545	−0.170773	−	0.985310i	$-0.554626\pi$
$577$	−8.20420	−0.341545	−0.170773	+	0.985310i	$0.554626\pi$
$578$	0	0
$579$	−6.51152	−0.270610
$580$	0	0
$581$	1.85733	0.0770551
$582$	0	0
$583$	−16.5177	−0.684092
$584$	0	0
$585$	3.28517	0.135825
$586$	0	0
$587$	29.4613	1.21600	0.607999	−	0.793938i	$-0.291972\pi$
$587$	29.4613	1.21600	0.607999	+	0.793938i	$0.291972\pi$
$588$	0	0
$589$	56.1766	2.31472
$590$	0	0
$591$	6.53154	0.268671
$592$	0	0
$593$	−5.53173	−0.227161	−0.113580	−	0.993529i	$-0.536232\pi$
$593$	−5.53173	−0.227161	−0.113580	+	0.993529i	$0.536232\pi$
$594$	0	0
$595$	3.78700	0.155252
$596$	0	0
$597$	29.6076	1.21176
$598$	0	0
$599$	−35.1152	−1.43477	−0.717384	−	0.696677i	$-0.754661\pi$
$599$	−35.1152	−1.43477	−0.717384	+	0.696677i	$0.754661\pi$
$600$	0	0
$601$	−7.28852	−0.297305	−0.148653	−	0.988889i	$-0.547494\pi$
$601$	−7.28852	−0.297305	−0.148653	+	0.988889i	$0.547494\pi$
$602$	0	0
$603$	1.01276	0.0412428
$604$	0	0
$605$	−23.1609	−0.941627
$606$	0	0
$607$	44.3345	1.79948	0.899742	−	0.436423i	$-0.143755\pi$
$607$	44.3345	1.79948	0.899742	+	0.436423i	$0.143755\pi$
$608$	0	0
$609$	−0.922581	−0.0373849
$610$	0	0
$611$	−5.51979	−0.223307
$612$	0	0
$613$	−41.5756	−1.67922	−0.839612	−	0.543187i	$-0.817218\pi$
$613$	−41.5756	−1.67922	−0.839612	+	0.543187i	$0.817218\pi$
$614$	0	0
$615$	14.8518	0.598883
$616$	0	0
$617$	1.67672	0.0675020	0.0337510	−	0.999430i	$-0.489255\pi$
$617$	1.67672	0.0675020	0.0337510	+	0.999430i	$0.489255\pi$
$618$	0	0
$619$	−33.9475	−1.36447	−0.682233	−	0.731135i	$-0.738991\pi$
$619$	−33.9475	−1.36447	−0.682233	+	0.731135i	$0.738991\pi$
$620$	0	0
$621$	−32.1431	−1.28986
$622$	0	0
$623$	−3.95560	−0.158478
$624$	0	0
$625$	−11.8411	−0.473644
$626$	0	0
$627$	−14.1457	−0.564925
$628$	0	0
$629$	−1.84027	−0.0733765
$630$	0	0
$631$	−20.9561	−0.834248	−0.417124	−	0.908850i	$-0.636962\pi$
$631$	−20.9561	−0.834248	−0.417124	+	0.908850i	$0.636962\pi$
$632$	0	0
$633$	−19.4207	−0.771902
$634$	0	0
$635$	24.4174	0.968974
$636$	0	0
$637$	4.56316	0.180799
$638$	0	0
$639$	−15.1141	−0.597906
$640$	0	0
$641$	13.6452	0.538955	0.269477	−	0.963007i	$-0.413149\pi$
$641$	13.6452	0.538955	0.269477	+	0.963007i	$0.413149\pi$
$642$	0	0
$643$	−34.0759	−1.34382	−0.671911	−	0.740632i	$-0.734526\pi$
$643$	−34.0759	−1.34382	−0.671911	+	0.740632i	$0.734526\pi$
$644$	0	0
$645$	−55.2117	−2.17396
$646$	0	0
$647$	33.7796	1.32801	0.664007	−	0.747727i	$-0.268855\pi$
$647$	33.7796	1.32801	0.664007	+	0.747727i	$0.268855\pi$
$648$	0	0
$649$	2.07064	0.0812797
$650$	0	0
$651$	−16.1622	−0.633448
$652$	0	0
$653$	−36.0290	−1.40992	−0.704961	−	0.709246i	$-0.749036\pi$
$653$	−36.0290	−1.40992	−0.704961	+	0.709246i	$0.749036\pi$
$654$	0	0
$655$	−58.5299	−2.28695
$656$	0	0
$657$	−15.5194	−0.605470
$658$	0	0
$659$	48.2334	1.87891	0.939453	−	0.342679i	$-0.111334\pi$
$659$	48.2334	1.87891	0.939453	+	0.342679i	$0.111334\pi$
$660$	0	0
$661$	30.3568	1.18074	0.590372	−	0.807132i	$-0.298981\pi$
$661$	30.3568	1.18074	0.590372	+	0.807132i	$0.298981\pi$
$662$	0	0
$663$	1.05367	0.0409212
$664$	0	0
$665$	−19.3327	−0.749691
$666$	0	0
$667$	3.58447	0.138791
$668$	0	0
$669$	−7.69331	−0.297441
$670$	0	0
$671$	−24.8824	−0.960575
$672$	0	0
$673$	−9.02294	−0.347809	−0.173904	−	0.984763i	$-0.555638\pi$
$673$	−9.02294	−0.347809	−0.173904	+	0.984763i	$0.555638\pi$
$674$	0	0
$675$	−38.8975	−1.49716
$676$	0	0
$677$	16.8670	0.648252	0.324126	−	0.946014i	$-0.394930\pi$
$677$	16.8670	0.648252	0.324126	+	0.946014i	$0.394930\pi$
$678$	0	0
$679$	−9.98522	−0.383197
$680$	0	0
$681$	6.34530	0.243152
$682$	0	0
$683$	−37.7775	−1.44551	−0.722757	−	0.691102i	$-0.757126\pi$
$683$	−37.7775	−1.44551	−0.722757	+	0.691102i	$0.757126\pi$
$684$	0	0
$685$	−21.2002	−0.810019
$686$	0	0
$687$	−22.3920	−0.854308
$688$	0	0
$689$	−6.28098	−0.239286
$690$	0	0
$691$	4.75206	0.180777	0.0903885	−	0.995907i	$-0.471189\pi$
$691$	4.75206	0.180777	0.0903885	+	0.995907i	$0.471189\pi$
$692$	0	0
$693$	−2.74806	−0.104390
$694$	0	0
$695$	−15.7089	−0.595871
$696$	0	0
$697$	−3.21649	−0.121833
$698$	0	0
$699$	−12.6038	−0.476721
$700$	0	0
$701$	36.3011	1.37107	0.685537	−	0.728038i	$-0.259568\pi$
$701$	36.3011	1.37107	0.685537	+	0.728038i	$0.259568\pi$
$702$	0	0
$703$	9.39463	0.354325
$704$	0	0
$705$	32.3696	1.21911
$706$	0	0
$707$	12.8825	0.484497
$708$	0	0
$709$	−22.4300	−0.842376	−0.421188	−	0.906973i	$-0.638387\pi$
$709$	−22.4300	−0.842376	−0.421188	+	0.906973i	$0.638387\pi$
$710$	0	0
$711$	−9.42931	−0.353627
$712$	0	0
$713$	62.7946	2.35168
$714$	0	0
$715$	5.62550	0.210382
$716$	0	0
$717$	−19.2940	−0.720546
$718$	0	0
$719$	27.3206	1.01889	0.509444	−	0.860504i	$-0.329851\pi$
$719$	27.3206	1.01889	0.509444	+	0.860504i	$0.329851\pi$
$720$	0	0
$721$	2.78151	0.103589
$722$	0	0
$723$	−16.1914	−0.602165
$724$	0	0
$725$	4.33770	0.161098
$726$	0	0
$727$	−10.9614	−0.406534	−0.203267	−	0.979123i	$-0.565156\pi$
$727$	−10.9614	−0.406534	−0.203267	+	0.979123i	$0.565156\pi$
$728$	0	0
$729$	27.3417	1.01266
$730$	0	0
$731$	11.9573	0.442257
$732$	0	0
$733$	10.8231	0.399762	0.199881	−	0.979820i	$-0.435944\pi$
$733$	10.8231	0.399762	0.199881	+	0.979820i	$0.435944\pi$
$734$	0	0
$735$	−26.7597	−0.987046
$736$	0	0
$737$	1.73424	0.0638817
$738$	0	0
$739$	−31.6309	−1.16356	−0.581780	−	0.813346i	$-0.697643\pi$
$739$	−31.6309	−1.16356	−0.581780	+	0.813346i	$0.697643\pi$
$740$	0	0
$741$	−5.37901	−0.197603
$742$	0	0
$743$	30.4253	1.11620	0.558098	−	0.829775i	$-0.311531\pi$
$743$	30.4253	1.11620	0.558098	+	0.829775i	$0.311531\pi$
$744$	0	0
$745$	−78.7366	−2.88469
$746$	0	0
$747$	2.04630	0.0748701
$748$	0	0
$749$	−7.09611	−0.259286
$750$	0	0
$751$	−26.8840	−0.981010	−0.490505	−	0.871438i	$-0.663188\pi$
$751$	−26.8840	−0.981010	−0.490505	+	0.871438i	$0.663188\pi$
$752$	0	0
$753$	−21.8547	−0.796429
$754$	0	0
$755$	74.6322	2.71614
$756$	0	0
$757$	−24.1369	−0.877271	−0.438635	−	0.898665i	$-0.644538\pi$
$757$	−24.1369	−0.877271	−0.438635	+	0.898665i	$0.644538\pi$
$758$	0	0
$759$	−15.8122	−0.573945
$760$	0	0
$761$	−14.6936	−0.532642	−0.266321	−	0.963884i	$-0.585808\pi$
$761$	−14.6936	−0.532642	−0.266321	+	0.963884i	$0.585808\pi$
$762$	0	0
$763$	14.5126	0.525392
$764$	0	0
$765$	4.17230	0.150850
$766$	0	0
$767$	0.787376	0.0284305
$768$	0	0
$769$	−0.509185	−0.0183617	−0.00918084	−	0.999958i	$-0.502922\pi$
$769$	−0.509185	−0.0183617	−0.00918084	+	0.999958i	$0.502922\pi$
$770$	0	0
$771$	33.8259	1.21821
$772$	0	0
$773$	48.6312	1.74914	0.874571	−	0.484898i	$-0.161143\pi$
$773$	48.6312	1.74914	0.874571	+	0.484898i	$0.161143\pi$
$774$	0	0
$775$	75.9900	2.72964
$776$	0	0
$777$	−2.70287	−0.0969651
$778$	0	0
$779$	16.4203	0.588317
$780$	0	0
$781$	−25.8813	−0.926107
$782$	0	0
$783$	−3.53821	−0.126445
$784$	0	0
$785$	8.49153	0.303076
$786$	0	0
$787$	−10.2060	−0.363806	−0.181903	−	0.983316i	$-0.558226\pi$
$787$	−10.2060	−0.363806	−0.181903	+	0.983316i	$0.558226\pi$
$788$	0	0
$789$	17.0375	0.606552
$790$	0	0
$791$	−9.25157	−0.328948
$792$	0	0
$793$	−9.46173	−0.335996
$794$	0	0
$795$	36.8335	1.30635
$796$	0	0
$797$	−11.5797	−0.410173	−0.205086	−	0.978744i	$-0.565748\pi$
$797$	−11.5797	−0.410173	−0.205086	+	0.978744i	$0.565748\pi$
$798$	0	0
$799$	−7.01036	−0.248009
$800$	0	0
$801$	−4.35804	−0.153984
$802$	0	0
$803$	−26.5754	−0.937824
$804$	0	0
$805$	−21.6103	−0.761662
$806$	0	0
$807$	−0.484013	−0.0170381
$808$	0	0
$809$	13.5391	0.476009	0.238004	−	0.971264i	$-0.423507\pi$
$809$	13.5391	0.476009	0.238004	+	0.971264i	$0.423507\pi$
$810$	0	0
$811$	−11.7928	−0.414100	−0.207050	−	0.978330i	$-0.566386\pi$
$811$	−11.7928	−0.414100	−0.207050	+	0.978330i	$0.566386\pi$
$812$	0	0
$813$	18.1610	0.636934
$814$	0	0
$815$	−84.8355	−2.97166
$816$	0	0
$817$	−61.0423	−2.13560
$818$	0	0
$819$	−1.04497	−0.0365142
$820$	0	0
$821$	2.91970	0.101898	0.0509491	−	0.998701i	$-0.483775\pi$
$821$	2.91970	0.101898	0.0509491	+	0.998701i	$0.483775\pi$
$822$	0	0
$823$	29.2265	1.01877	0.509387	−	0.860538i	$-0.329872\pi$
$823$	29.2265	1.01877	0.509387	+	0.860538i	$0.329872\pi$
$824$	0	0
$825$	−19.1349	−0.666191
$826$	0	0
$827$	31.0284	1.07896	0.539481	−	0.841998i	$-0.318621\pi$
$827$	31.0284	1.07896	0.539481	+	0.841998i	$0.318621\pi$
$828$	0	0
$829$	−13.1474	−0.456630	−0.228315	−	0.973587i	$-0.573322\pi$
$829$	−13.1474	−0.456630	−0.228315	+	0.973587i	$0.573322\pi$
$830$	0	0
$831$	−6.34267	−0.220025
$832$	0	0
$833$	5.79540	0.200799
$834$	0	0
$835$	23.2414	0.804303
$836$	0	0
$837$	−61.9842	−2.14249
$838$	0	0
$839$	20.2455	0.698954	0.349477	−	0.936945i	$-0.386359\pi$
$839$	20.2455	0.698954	0.349477	+	0.936945i	$0.386359\pi$
$840$	0	0
$841$	−28.6054	−0.986394
$842$	0	0
$843$	−27.5175	−0.947752
$844$	0	0
$845$	−42.7166	−1.46950
$846$	0	0
$847$	7.36720	0.253140
$848$	0	0
$849$	38.8696	1.33400
$850$	0	0
$851$	10.5014	0.359983
$852$	0	0
$853$	8.64303	0.295932	0.147966	−	0.988992i	$-0.452727\pi$
$853$	8.64303	0.295932	0.147966	+	0.988992i	$0.452727\pi$
$854$	0	0
$855$	−21.2997	−0.728433
$856$	0	0
$857$	6.58079	0.224795	0.112398	−	0.993663i	$-0.464147\pi$
$857$	6.58079	0.224795	0.112398	+	0.993663i	$0.464147\pi$
$858$	0	0
$859$	−44.8319	−1.52965	−0.764823	−	0.644241i	$-0.777173\pi$
$859$	−44.8319	−1.52965	−0.764823	+	0.644241i	$0.777173\pi$
$860$	0	0
$861$	−4.72418	−0.160999
$862$	0	0
$863$	−14.2066	−0.483599	−0.241800	−	0.970326i	$-0.577738\pi$
$863$	−14.2066	−0.483599	−0.241800	+	0.970326i	$0.577738\pi$
$864$	0	0
$865$	−43.0319	−1.46313
$866$	0	0
$867$	1.33821	0.0454478
$868$	0	0
$869$	−16.1467	−0.547739
$870$	0	0
$871$	0.659459	0.0223449
$872$	0	0
$873$	−11.0011	−0.372332
$874$	0	0
$875$	−7.21634	−0.243957
$876$	0	0
$877$	0.306807	0.0103601	0.00518007	−	0.999987i	$-0.498351\pi$
$877$	0.306807	0.0103601	0.00518007	+	0.999987i	$0.498351\pi$
$878$	0	0
$879$	−35.0588	−1.18250
$880$	0	0
$881$	−28.3604	−0.955487	−0.477744	−	0.878499i	$-0.658545\pi$
$881$	−28.3604	−0.955487	−0.477744	+	0.878499i	$0.658545\pi$
$882$	0	0
$883$	56.9954	1.91805	0.959024	−	0.283323i	$-0.0914370\pi$
$883$	56.9954	1.91805	0.959024	+	0.283323i	$0.0914370\pi$
$884$	0	0
$885$	−4.61740	−0.155212
$886$	0	0
$887$	44.5897	1.49718	0.748588	−	0.663036i	$-0.230732\pi$
$887$	44.5897	1.49718	0.748588	+	0.663036i	$0.230732\pi$
$888$	0	0
$889$	−7.76685	−0.260492
$890$	0	0
$891$	8.09661	0.271247
$892$	0	0
$893$	35.7880	1.19760
$894$	0	0
$895$	5.31899	0.177794
$896$	0	0
$897$	−6.01270	−0.200758
$898$	0	0
$899$	6.91224	0.230536
$900$	0	0
$901$	−7.97710	−0.265756
$902$	0	0
$903$	17.5621	0.584431
$904$	0	0
$905$	−40.0014	−1.32969
$906$	0	0
$907$	37.8660	1.25732	0.628660	−	0.777680i	$-0.283604\pi$
$907$	37.8660	1.25732	0.628660	+	0.777680i	$0.283604\pi$
$908$	0	0
$909$	14.1932	0.470759
$910$	0	0
$911$	31.3203	1.03769	0.518843	−	0.854869i	$-0.326363\pi$
$911$	31.3203	1.03769	0.518843	+	0.854869i	$0.326363\pi$
$912$	0	0
$913$	3.50407	0.115968
$914$	0	0
$915$	55.4863	1.83432
$916$	0	0
$917$	18.6176	0.614808
$918$	0	0
$919$	−42.1603	−1.39074	−0.695370	−	0.718652i	$-0.744759\pi$
$919$	−42.1603	−1.39074	−0.695370	+	0.718652i	$0.744759\pi$
$920$	0	0
$921$	−31.3088	−1.03166
$922$	0	0
$923$	−9.84158	−0.323939
$924$	0	0
$925$	12.7081	0.417840
$926$	0	0
$927$	3.06451	0.100652
$928$	0	0
$929$	−23.7991	−0.780824	−0.390412	−	0.920640i	$-0.627667\pi$
$929$	−23.7991	−0.780824	−0.390412	+	0.920640i	$0.627667\pi$
$930$	0	0
$931$	−29.5856	−0.969630
$932$	0	0
$933$	8.54682	0.279810
$934$	0	0
$935$	7.14462	0.233654
$936$	0	0
$937$	−47.8887	−1.56446	−0.782228	−	0.622992i	$-0.785917\pi$
$937$	−47.8887	−1.56446	−0.782228	+	0.622992i	$0.785917\pi$
$938$	0	0
$939$	−31.5353	−1.02911
$940$	0	0
$941$	−3.12917	−0.102008	−0.0510039	−	0.998698i	$-0.516242\pi$
$941$	−3.12917	−0.102008	−0.0510039	+	0.998698i	$0.516242\pi$
$942$	0	0
$943$	18.3547	0.597710
$944$	0	0
$945$	21.3314	0.693910
$946$	0	0
$947$	−51.9946	−1.68960	−0.844798	−	0.535085i	$-0.820279\pi$
$947$	−51.9946	−1.68960	−0.844798	+	0.535085i	$0.820279\pi$
$948$	0	0
$949$	−10.1055	−0.328038
$950$	0	0
$951$	5.61751	0.182160
$952$	0	0
$953$	41.0007	1.32814	0.664071	−	0.747669i	$-0.268827\pi$
$953$	41.0007	1.32814	0.664071	+	0.747669i	$0.268827\pi$
$954$	0	0
$955$	25.8117	0.835247
$956$	0	0
$957$	−1.74055	−0.0562641
$958$	0	0
$959$	6.74352	0.217760
$960$	0	0
$961$	90.0921	2.90620
$962$	0	0
$963$	−7.81808	−0.251934
$964$	0	0
$965$	−16.7894	−0.540469
$966$	0	0
$967$	−1.44502	−0.0464688	−0.0232344	−	0.999730i	$-0.507396\pi$
$967$	−1.44502	−0.0464688	−0.0232344	+	0.999730i	$0.507396\pi$
$968$	0	0
$969$	−6.83157	−0.219461
$970$	0	0
$971$	−9.25699	−0.297071	−0.148536	−	0.988907i	$-0.547456\pi$
$971$	−9.25699	−0.297071	−0.148536	+	0.988907i	$0.547456\pi$
$972$	0	0
$973$	4.99679	0.160190
$974$	0	0
$975$	−7.27618	−0.233024
$976$	0	0
$977$	32.9873	1.05536	0.527678	−	0.849444i	$-0.323063\pi$
$977$	32.9873	1.05536	0.527678	+	0.849444i	$0.323063\pi$
$978$	0	0
$979$	−7.46269	−0.238509
$980$	0	0
$981$	15.9891	0.510494
$982$	0	0
$983$	−14.2460	−0.454377	−0.227189	−	0.973851i	$-0.572953\pi$
$983$	−14.2460	−0.454377	−0.227189	+	0.973851i	$0.572953\pi$
$984$	0	0
$985$	16.8410	0.536598
$986$	0	0
$987$	−10.2964	−0.327737
$988$	0	0
$989$	−68.2335	−2.16970
$990$	0	0
$991$	30.7804	0.977773	0.488887	−	0.872347i	$-0.337403\pi$
$991$	30.7804	0.977773	0.488887	+	0.872347i	$0.337403\pi$
$992$	0	0
$993$	0.867883	0.0275414
$994$	0	0
$995$	76.3407	2.42016
$996$	0	0
$997$	−59.3522	−1.87970	−0.939852	−	0.341582i	$-0.889037\pi$
$997$	−59.3522	−1.87970	−0.939852	+	0.341582i	$0.889037\pi$
$998$	0	0
$999$	−10.3659	−0.327961

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4012.2.a.i.1.12	✓	18

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4012.2.a.i.1.12	✓	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)

\(f(q)\)	\(=\)	\(q+1.33821 q^{3} +3.45044 q^{5} -1.09754 q^{7} -1.20921 q^{9} +O(q^{10})\)	\(q+1.33821 q^{3} +3.45044 q^{5} -1.09754 q^{7} -1.20921 q^{9} -2.07064 q^{11} -0.787376 q^{13} +4.61740 q^{15} -1.00000 q^{17} +5.10502 q^{19} -1.46874 q^{21} +5.70643 q^{23} +6.90555 q^{25} -5.63278 q^{27} +0.628146 q^{29} +11.0042 q^{31} -2.77094 q^{33} -3.78700 q^{35} +1.84027 q^{37} -1.05367 q^{39} +3.21649 q^{41} -11.9573 q^{43} -4.17230 q^{45} +7.01036 q^{47} -5.79540 q^{49} -1.33821 q^{51} +7.97710 q^{53} -7.14462 q^{55} +6.83157 q^{57} -1.00000 q^{59} +12.0168 q^{61} +1.32715 q^{63} -2.71680 q^{65} -0.837540 q^{67} +7.63637 q^{69} +12.4992 q^{71} +12.8344 q^{73} +9.24105 q^{75} +2.27261 q^{77} +7.79793 q^{79} -3.91020 q^{81} -1.69226 q^{83} -3.45044 q^{85} +0.840589 q^{87} +3.60405 q^{89} +0.864178 q^{91} +14.7259 q^{93} +17.6146 q^{95} +9.09780 q^{97} +2.50383 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q + 8 q^{3} + 4 q^{5} + 2 q^{7} + 18 q^{9}+O(q^{10}) \) 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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