LMFDB - Embedded newform 4012.2.a.i.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4012, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("4012.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$32.0359812909$
Analytic rank:	$0$
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			1.6
Root			$-0.397638$ of defining polynomial
Character	$\chi$	$=$	4012.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.397638 q^{3} +3.51710 q^{5} +2.09256 q^{7} -2.84188 q^{9} +O(q^{10})$	$q-0.397638 q^{3} +3.51710 q^{5} +2.09256 q^{7} -2.84188 q^{9} +2.13166 q^{11} +5.28609 q^{13} -1.39853 q^{15} -1.00000 q^{17} -0.313338 q^{19} -0.832081 q^{21} +1.96600 q^{23} +7.36999 q^{25} +2.32295 q^{27} -3.78015 q^{29} +1.00501 q^{31} -0.847627 q^{33} +7.35974 q^{35} -1.71657 q^{37} -2.10195 q^{39} +8.45487 q^{41} +6.98844 q^{43} -9.99519 q^{45} -2.81326 q^{47} -2.62119 q^{49} +0.397638 q^{51} -1.10259 q^{53} +7.49725 q^{55} +0.124595 q^{57} -1.00000 q^{59} +3.70140 q^{61} -5.94682 q^{63} +18.5917 q^{65} -1.96713 q^{67} -0.781757 q^{69} +5.84738 q^{71} -9.52409 q^{73} -2.93058 q^{75} +4.46062 q^{77} -3.16268 q^{79} +7.60196 q^{81} +10.2945 q^{83} -3.51710 q^{85} +1.50313 q^{87} -6.74642 q^{89} +11.0615 q^{91} -0.399630 q^{93} -1.10204 q^{95} -0.816018 q^{97} -6.05792 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−0.397638	−0.229576	−0.114788	−	0.993390i	$-0.536619\pi$
$3$	−0.397638	−0.229576	−0.114788	+	0.993390i	$0.536619\pi$
$4$	0	0
$5$	3.51710	1.57289	0.786447	−	0.617657i	$-0.211918\pi$
$5$	3.51710	1.57289	0.786447	+	0.617657i	$0.211918\pi$
$6$	0	0
$7$	2.09256	0.790914	0.395457	−	0.918485i	$-0.370586\pi$
$7$	2.09256	0.790914	0.395457	+	0.918485i	$0.370586\pi$
$8$	0	0
$9$	−2.84188	−0.947295
$10$	0	0
$11$	2.13166	0.642719	0.321359	−	0.946957i	$-0.395860\pi$
$11$	2.13166	0.642719	0.321359	+	0.946957i	$0.395860\pi$
$12$	0	0
$13$	5.28609	1.46610	0.733049	−	0.680176i	$-0.238097\pi$
$13$	5.28609	1.46610	0.733049	+	0.680176i	$0.238097\pi$
$14$	0	0
$15$	−1.39853	−0.361099
$16$	0	0
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	0	0
$19$	−0.313338	−0.0718846	−0.0359423	−	0.999354i	$-0.511443\pi$
$19$	−0.313338	−0.0718846	−0.0359423	+	0.999354i	$0.511443\pi$
$20$	0	0
$21$	−0.832081	−0.181575
$22$	0	0
$23$	1.96600	0.409940	0.204970	−	0.978768i	$-0.434290\pi$
$23$	1.96600	0.409940	0.204970	+	0.978768i	$0.434290\pi$
$24$	0	0
$25$	7.36999	1.47400
$26$	0	0
$27$	2.32295	0.447052
$28$	0	0
$29$	−3.78015	−0.701957	−0.350978	−	0.936384i	$-0.614151\pi$
$29$	−3.78015	−0.701957	−0.350978	+	0.936384i	$0.614151\pi$
$30$	0	0
$31$	1.00501	0.180505	0.0902527	−	0.995919i	$-0.471233\pi$
$31$	1.00501	0.180505	0.0902527	+	0.995919i	$0.471233\pi$
$32$	0	0
$33$	−0.847627	−0.147553
$34$	0	0
$35$	7.35974	1.24402
$36$	0	0
$37$	−1.71657	−0.282203	−0.141101	−	0.989995i	$-0.545064\pi$
$37$	−1.71657	−0.282203	−0.141101	+	0.989995i	$0.545064\pi$
$38$	0	0
$39$	−2.10195	−0.336581
$40$	0	0
$41$	8.45487	1.32043	0.660214	−	0.751077i	$-0.270465\pi$
$41$	8.45487	1.32043	0.660214	+	0.751077i	$0.270465\pi$
$42$	0	0
$43$	6.98844	1.06573	0.532864	−	0.846201i	$-0.321116\pi$
$43$	6.98844	1.06573	0.532864	+	0.846201i	$0.321116\pi$
$44$	0	0
$45$	−9.99519	−1.48999
$46$	0	0
$47$	−2.81326	−0.410356	−0.205178	−	0.978725i	$-0.565777\pi$
$47$	−2.81326	−0.410356	−0.205178	+	0.978725i	$0.565777\pi$
$48$	0	0
$49$	−2.62119	−0.374456
$50$	0	0
$51$	0.397638	0.0556804
$52$	0	0
$53$	−1.10259	−0.151452	−0.0757261	−	0.997129i	$-0.524127\pi$
$53$	−1.10259	−0.151452	−0.0757261	+	0.997129i	$0.524127\pi$
$54$	0	0
$55$	7.49725	1.01093
$56$	0	0
$57$	0.124595	0.0165030
$58$	0	0
$59$	−1.00000	−0.130189
$59$	−1.00000	−0.130189
$60$	0	0
$61$	3.70140	0.473916	0.236958	−	0.971520i	$-0.423850\pi$
$61$	3.70140	0.473916	0.236958	+	0.971520i	$0.423850\pi$
$62$	0	0
$63$	−5.94682	−0.749228
$64$	0	0
$65$	18.5917	2.30602
$66$	0	0
$67$	−1.96713	−0.240323	−0.120162	−	0.992754i	$-0.538341\pi$
$67$	−1.96713	−0.240323	−0.120162	+	0.992754i	$0.538341\pi$
$68$	0	0
$69$	−0.781757	−0.0941125
$70$	0	0
$71$	5.84738	0.693957	0.346978	−	0.937873i	$-0.387208\pi$
$71$	5.84738	0.693957	0.346978	+	0.937873i	$0.387208\pi$
$72$	0	0
$73$	−9.52409	−1.11471	−0.557355	−	0.830274i	$-0.688184\pi$
$73$	−9.52409	−1.11471	−0.557355	+	0.830274i	$0.688184\pi$
$74$	0	0
$75$	−2.93058	−0.338395
$76$	0	0
$77$	4.46062	0.508335
$78$	0	0
$79$	−3.16268	−0.355829	−0.177914	−	0.984046i	$-0.556935\pi$
$79$	−3.16268	−0.355829	−0.177914	+	0.984046i	$0.556935\pi$
$80$	0	0
$81$	7.60196	0.844662
$82$	0	0
$83$	10.2945	1.12997	0.564983	−	0.825102i	$-0.308883\pi$
$83$	10.2945	1.12997	0.564983	+	0.825102i	$0.308883\pi$
$84$	0	0
$85$	−3.51710	−0.381483
$86$	0	0
$87$	1.50313	0.161153
$88$	0	0
$89$	−6.74642	−0.715119	−0.357559	−	0.933890i	$-0.616391\pi$
$89$	−6.74642	−0.715119	−0.357559	+	0.933890i	$0.616391\pi$
$90$	0	0
$91$	11.0615	1.15956
$92$	0	0
$93$	−0.399630	−0.0414397
$94$	0	0
$95$	−1.10204	−0.113067
$96$	0	0
$97$	−0.816018	−0.0828540	−0.0414270	−	0.999142i	$-0.513190\pi$
$97$	−0.816018	−0.0828540	−0.0414270	+	0.999142i	$0.513190\pi$
$98$	0	0
$99$	−6.05792	−0.608844
$100$	0	0
$101$	9.91621	0.986700	0.493350	−	0.869831i	$-0.335772\pi$
$101$	9.91621	0.986700	0.493350	+	0.869831i	$0.335772\pi$
$102$	0	0
$103$	1.13269	0.111607	0.0558037	−	0.998442i	$-0.482228\pi$
$103$	1.13269	0.111607	0.0558037	+	0.998442i	$0.482228\pi$
$104$	0	0
$105$	−2.92651	−0.285598
$106$	0	0
$107$	0.842959	0.0814919	0.0407460	−	0.999170i	$-0.487027\pi$
$107$	0.842959	0.0814919	0.0407460	+	0.999170i	$0.487027\pi$
$108$	0	0
$109$	−12.7518	−1.22140	−0.610702	−	0.791860i	$-0.709113\pi$
$109$	−12.7518	−1.22140	−0.610702	+	0.791860i	$0.709113\pi$
$110$	0	0
$111$	0.682574	0.0647870
$112$	0	0
$113$	−6.64115	−0.624746	−0.312373	−	0.949959i	$-0.601124\pi$
$113$	−6.64115	−0.624746	−0.312373	+	0.949959i	$0.601124\pi$
$114$	0	0
$115$	6.91463	0.644793
$116$	0	0
$117$	−15.0225	−1.38883
$118$	0	0
$119$	−2.09256	−0.191825
$120$	0	0
$121$	−6.45604	−0.586912
$122$	0	0
$123$	−3.36197	−0.303139
$124$	0	0
$125$	8.33548	0.745548
$126$	0	0
$127$	12.3325	1.09434	0.547168	−	0.837023i	$-0.315706\pi$
$127$	12.3325	1.09434	0.547168	+	0.837023i	$0.315706\pi$
$128$	0	0
$129$	−2.77887	−0.244666
$130$	0	0
$131$	11.0860	0.968591	0.484296	−	0.874904i	$-0.339076\pi$
$131$	11.0860	0.968591	0.484296	+	0.874904i	$0.339076\pi$
$132$	0	0
$133$	−0.655678	−0.0568545
$134$	0	0
$135$	8.17005	0.703166
$136$	0	0
$137$	−16.1316	−1.37821	−0.689107	−	0.724660i	$-0.741997\pi$
$137$	−16.1316	−1.37821	−0.689107	+	0.724660i	$0.741997\pi$
$138$	0	0
$139$	−1.49987	−0.127217	−0.0636085	−	0.997975i	$-0.520261\pi$
$139$	−1.49987	−0.127217	−0.0636085	+	0.997975i	$0.520261\pi$
$140$	0	0
$141$	1.11866	0.0942079
$142$	0	0
$143$	11.2681	0.942289
$144$	0	0
$145$	−13.2952	−1.10410
$146$	0	0
$147$	1.04228	0.0859661
$148$	0	0
$149$	20.1848	1.65360	0.826801	−	0.562494i	$-0.190158\pi$
$149$	20.1848	1.65360	0.826801	+	0.562494i	$0.190158\pi$
$150$	0	0
$151$	−11.2181	−0.912919	−0.456459	−	0.889744i	$-0.650883\pi$
$151$	−11.2181	−0.912919	−0.456459	+	0.889744i	$0.650883\pi$
$152$	0	0
$153$	2.84188	0.229753
$154$	0	0
$155$	3.53472	0.283916
$156$	0	0
$157$	−17.1974	−1.37250	−0.686252	−	0.727364i	$-0.740745\pi$
$157$	−17.1974	−1.37250	−0.686252	+	0.727364i	$0.740745\pi$
$158$	0	0
$159$	0.438431	0.0347698
$160$	0	0
$161$	4.11398	0.324227
$162$	0	0
$163$	8.78568	0.688147	0.344074	−	0.938943i	$-0.388193\pi$
$163$	8.78568	0.688147	0.344074	+	0.938943i	$0.388193\pi$
$164$	0	0
$165$	−2.98119	−0.232085
$166$	0	0
$167$	−11.3720	−0.879989	−0.439995	−	0.898000i	$-0.645020\pi$
$167$	−11.3720	−0.879989	−0.439995	+	0.898000i	$0.645020\pi$
$168$	0	0
$169$	14.9428	1.14944
$170$	0	0
$171$	0.890470	0.0680959
$172$	0	0
$173$	−5.56171	−0.422849	−0.211424	−	0.977394i	$-0.567810\pi$
$173$	−5.56171	−0.422849	−0.211424	+	0.977394i	$0.567810\pi$
$174$	0	0
$175$	15.4221	1.16580
$176$	0	0
$177$	0.397638	0.0298883
$178$	0	0
$179$	−20.1429	−1.50555	−0.752775	−	0.658278i	$-0.771285\pi$
$179$	−20.1429	−1.50555	−0.752775	+	0.658278i	$0.771285\pi$
$180$	0	0
$181$	20.5930	1.53066	0.765332	−	0.643636i	$-0.222575\pi$
$181$	20.5930	1.53066	0.765332	+	0.643636i	$0.222575\pi$
$182$	0	0
$183$	−1.47182	−0.108800
$184$	0	0
$185$	−6.03736	−0.443875
$186$	0	0
$187$	−2.13166	−0.155882
$188$	0	0
$189$	4.86092	0.353580
$190$	0	0
$191$	−18.8386	−1.36311	−0.681557	−	0.731765i	$-0.738697\pi$
$191$	−18.8386	−1.36311	−0.681557	+	0.731765i	$0.738697\pi$
$192$	0	0
$193$	5.09645	0.366850	0.183425	−	0.983034i	$-0.441281\pi$
$193$	5.09645	0.366850	0.183425	+	0.983034i	$0.441281\pi$
$194$	0	0
$195$	−7.39276	−0.529406
$196$	0	0
$197$	14.5200	1.03451	0.517253	−	0.855832i	$-0.326955\pi$
$197$	14.5200	1.03451	0.517253	+	0.855832i	$0.326955\pi$
$198$	0	0
$199$	11.5348	0.817683	0.408841	−	0.912605i	$-0.365933\pi$
$199$	11.5348	0.817683	0.408841	+	0.912605i	$0.365933\pi$
$200$	0	0
$201$	0.782206	0.0551725
$202$	0	0
$203$	−7.91020	−0.555187
$204$	0	0
$205$	29.7366	2.07689
$206$	0	0
$207$	−5.58716	−0.388334
$208$	0	0
$209$	−0.667929	−0.0462016
$210$	0	0
$211$	−3.58232	−0.246617	−0.123309	−	0.992368i	$-0.539350\pi$
$211$	−3.58232	−0.246617	−0.123309	+	0.992368i	$0.539350\pi$
$212$	0	0
$213$	−2.32514	−0.159316
$214$	0	0
$215$	24.5790	1.67628
$216$	0	0
$217$	2.10305	0.142764
$218$	0	0
$219$	3.78713	0.255911
$220$	0	0
$221$	−5.28609	−0.355581
$222$	0	0
$223$	−9.09404	−0.608982	−0.304491	−	0.952515i	$-0.598486\pi$
$223$	−9.09404	−0.608982	−0.304491	+	0.952515i	$0.598486\pi$
$224$	0	0
$225$	−20.9446	−1.39631
$226$	0	0
$227$	20.1161	1.33515	0.667577	−	0.744541i	$-0.267332\pi$
$227$	20.1161	1.33515	0.667577	+	0.744541i	$0.267332\pi$
$228$	0	0
$229$	6.20267	0.409884	0.204942	−	0.978774i	$-0.434299\pi$
$229$	6.20267	0.409884	0.204942	+	0.978774i	$0.434299\pi$
$230$	0	0
$231$	−1.77371	−0.116702
$232$	0	0
$233$	−10.2091	−0.668822	−0.334411	−	0.942427i	$-0.608537\pi$
$233$	−10.2091	−0.668822	−0.334411	+	0.942427i	$0.608537\pi$
$234$	0	0
$235$	−9.89451	−0.645447
$236$	0	0
$237$	1.25760	0.0816898
$238$	0	0
$239$	26.7172	1.72819	0.864096	−	0.503327i	$-0.167891\pi$
$239$	26.7172	1.72819	0.864096	+	0.503327i	$0.167891\pi$
$240$	0	0
$241$	4.55950	0.293703	0.146852	−	0.989159i	$-0.453086\pi$
$241$	4.55950	0.293703	0.146852	+	0.989159i	$0.453086\pi$
$242$	0	0
$243$	−9.99168	−0.640967
$244$	0	0
$245$	−9.21898	−0.588979
$246$	0	0
$247$	−1.65633	−0.105390
$248$	0	0
$249$	−4.09347	−0.259413
$250$	0	0
$251$	12.7559	0.805147	0.402573	−	0.915388i	$-0.368116\pi$
$251$	12.7559	0.805147	0.402573	+	0.915388i	$0.368116\pi$
$252$	0	0
$253$	4.19085	0.263476
$254$	0	0
$255$	1.39853	0.0875794
$256$	0	0
$257$	−25.4383	−1.58680	−0.793400	−	0.608701i	$-0.791691\pi$
$257$	−25.4383	−1.58680	−0.793400	+	0.608701i	$0.791691\pi$
$258$	0	0
$259$	−3.59203	−0.223198
$260$	0	0
$261$	10.7428	0.664960
$262$	0	0
$263$	10.8246	0.667472	0.333736	−	0.942666i	$-0.391691\pi$
$263$	10.8246	0.667472	0.333736	+	0.942666i	$0.391691\pi$
$264$	0	0
$265$	−3.87791	−0.238218
$266$	0	0
$267$	2.68263	0.164174
$268$	0	0
$269$	−8.94585	−0.545438	−0.272719	−	0.962094i	$-0.587923\pi$
$269$	−8.94585	−0.545438	−0.272719	+	0.962094i	$0.587923\pi$
$270$	0	0
$271$	−11.5841	−0.703684	−0.351842	−	0.936059i	$-0.614445\pi$
$271$	−11.5841	−0.703684	−0.351842	+	0.936059i	$0.614445\pi$
$272$	0	0
$273$	−4.39845	−0.266207
$274$	0	0
$275$	15.7103	0.947366
$276$	0	0
$277$	11.6556	0.700318	0.350159	−	0.936690i	$-0.386128\pi$
$277$	11.6556	0.700318	0.350159	+	0.936690i	$0.386128\pi$
$278$	0	0
$279$	−2.85613	−0.170992
$280$	0	0
$281$	−21.5175	−1.28363	−0.641815	−	0.766860i	$-0.721818\pi$
$281$	−21.5175	−1.28363	−0.641815	+	0.766860i	$0.721818\pi$
$282$	0	0
$283$	6.24899	0.371464	0.185732	−	0.982600i	$-0.440534\pi$
$283$	6.24899	0.371464	0.185732	+	0.982600i	$0.440534\pi$
$284$	0	0
$285$	0.438213	0.0259575
$286$	0	0
$287$	17.6923	1.04434
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	0.324479	0.0190213
$292$	0	0
$293$	−16.7780	−0.980179	−0.490090	−	0.871672i	$-0.663036\pi$
$293$	−16.7780	−0.980179	−0.490090	+	0.871672i	$0.663036\pi$
$294$	0	0
$295$	−3.51710	−0.204773
$296$	0	0
$297$	4.95174	0.287329
$298$	0	0
$299$	10.3925	0.601013
$300$	0	0
$301$	14.6237	0.842898
$302$	0	0
$303$	−3.94306	−0.226523
$304$	0	0
$305$	13.0182	0.745420
$306$	0	0
$307$	4.05035	0.231166	0.115583	−	0.993298i	$-0.463126\pi$
$307$	4.05035	0.231166	0.115583	+	0.993298i	$0.463126\pi$
$308$	0	0
$309$	−0.450401	−0.0256224
$310$	0	0
$311$	23.1035	1.31008	0.655040	−	0.755594i	$-0.272652\pi$
$311$	23.1035	1.31008	0.655040	+	0.755594i	$0.272652\pi$
$312$	0	0
$313$	−17.1017	−0.966646	−0.483323	−	0.875442i	$-0.660570\pi$
$313$	−17.1017	−0.966646	−0.483323	+	0.875442i	$0.660570\pi$
$314$	0	0
$315$	−20.9155	−1.17846
$316$	0	0
$317$	19.9023	1.11783	0.558913	−	0.829226i	$-0.311219\pi$
$317$	19.9023	1.11783	0.558913	+	0.829226i	$0.311219\pi$
$318$	0	0
$319$	−8.05799	−0.451161
$320$	0	0
$321$	−0.335192	−0.0187086
$322$	0	0
$323$	0.313338	0.0174346
$324$	0	0
$325$	38.9584	2.16102
$326$	0	0
$327$	5.07061	0.280405
$328$	0	0
$329$	−5.88692	−0.324556
$330$	0	0
$331$	22.2544	1.22321	0.611606	−	0.791162i	$-0.290524\pi$
$331$	22.2544	1.22321	0.611606	+	0.791162i	$0.290524\pi$
$332$	0	0
$333$	4.87830	0.267329
$334$	0	0
$335$	−6.91860	−0.378003
$336$	0	0
$337$	−10.7248	−0.584220	−0.292110	−	0.956385i	$-0.594357\pi$
$337$	−10.7248	−0.584220	−0.292110	+	0.956385i	$0.594357\pi$
$338$	0	0
$339$	2.64077	0.143427
$340$	0	0
$341$	2.14234	0.116014
$342$	0	0
$343$	−20.1329	−1.08708
$344$	0	0
$345$	−2.74952	−0.148029
$346$	0	0
$347$	0.288624	0.0154942	0.00774709	−	0.999970i	$-0.497534\pi$
$347$	0.288624	0.0154942	0.00774709	+	0.999970i	$0.497534\pi$
$348$	0	0
$349$	2.48677	0.133114	0.0665568	−	0.997783i	$-0.478799\pi$
$349$	2.48677	0.133114	0.0665568	+	0.997783i	$0.478799\pi$
$350$	0	0
$351$	12.2793	0.655423
$352$	0	0
$353$	1.11908	0.0595627	0.0297814	−	0.999556i	$-0.490519\pi$
$353$	1.11908	0.0595627	0.0297814	+	0.999556i	$0.490519\pi$
$354$	0	0
$355$	20.5658	1.09152
$356$	0	0
$357$	0.832081	0.0440384
$358$	0	0
$359$	24.2819	1.28155	0.640775	−	0.767729i	$-0.278613\pi$
$359$	24.2819	1.28155	0.640775	+	0.767729i	$0.278613\pi$
$360$	0	0
$361$	−18.9018	−0.994833
$362$	0	0
$363$	2.56716	0.134741
$364$	0	0
$365$	−33.4972	−1.75332
$366$	0	0
$367$	24.0299	1.25435	0.627176	−	0.778878i	$-0.284211\pi$
$367$	24.0299	1.25435	0.627176	+	0.778878i	$0.284211\pi$
$368$	0	0
$369$	−24.0278	−1.25083
$370$	0	0
$371$	−2.30723	−0.119786
$372$	0	0
$373$	6.84116	0.354222	0.177111	−	0.984191i	$-0.443325\pi$
$373$	6.84116	0.354222	0.177111	+	0.984191i	$0.443325\pi$
$374$	0	0
$375$	−3.31450	−0.171160
$376$	0	0
$377$	−19.9822	−1.02914
$378$	0	0
$379$	13.8801	0.712972	0.356486	−	0.934301i	$-0.383975\pi$
$379$	13.8801	0.712972	0.356486	+	0.934301i	$0.383975\pi$
$380$	0	0
$381$	−4.90388	−0.251233
$382$	0	0
$383$	4.72146	0.241255	0.120628	−	0.992698i	$-0.461509\pi$
$383$	4.72146	0.241255	0.120628	+	0.992698i	$0.461509\pi$
$384$	0	0
$385$	15.6885	0.799558
$386$	0	0
$387$	−19.8603	−1.00956
$388$	0	0
$389$	−2.83082	−0.143528	−0.0717640	−	0.997422i	$-0.522863\pi$
$389$	−2.83082	−0.143528	−0.0717640	+	0.997422i	$0.522863\pi$
$390$	0	0
$391$	−1.96600	−0.0994251
$392$	0	0
$393$	−4.40822	−0.222365
$394$	0	0
$395$	−11.1234	−0.559681
$396$	0	0
$397$	7.20101	0.361408	0.180704	−	0.983537i	$-0.442162\pi$
$397$	7.20101	0.361408	0.180704	+	0.983537i	$0.442162\pi$
$398$	0	0
$399$	0.260722	0.0130524
$400$	0	0
$401$	−6.15793	−0.307512	−0.153756	−	0.988109i	$-0.549137\pi$
$401$	−6.15793	−0.307512	−0.153756	+	0.988109i	$0.549137\pi$
$402$	0	0
$403$	5.31258	0.264638
$404$	0	0
$405$	26.7368	1.32856
$406$	0	0
$407$	−3.65914	−0.181377
$408$	0	0
$409$	−23.5045	−1.16222	−0.581110	−	0.813825i	$-0.697382\pi$
$409$	−23.5045	−1.16222	−0.581110	+	0.813825i	$0.697382\pi$
$410$	0	0
$411$	6.41452	0.316405
$412$	0	0
$413$	−2.09256	−0.102968
$414$	0	0
$415$	36.2067	1.77732
$416$	0	0
$417$	0.596403	0.0292060
$418$	0	0
$419$	−6.50507	−0.317793	−0.158897	−	0.987295i	$-0.550794\pi$
$419$	−6.50507	−0.317793	−0.158897	+	0.987295i	$0.550794\pi$
$420$	0	0
$421$	18.1579	0.884960	0.442480	−	0.896778i	$-0.354099\pi$
$421$	18.1579	0.884960	0.442480	+	0.896778i	$0.354099\pi$
$422$	0	0
$423$	7.99496	0.388728
$424$	0	0
$425$	−7.36999	−0.357497
$426$	0	0
$427$	7.74541	0.374826
$428$	0	0
$429$	−4.48063	−0.216327
$430$	0	0
$431$	27.4204	1.32079	0.660396	−	0.750917i	$-0.270388\pi$
$431$	27.4204	1.32079	0.660396	+	0.750917i	$0.270388\pi$
$432$	0	0
$433$	−4.71932	−0.226796	−0.113398	−	0.993550i	$-0.536174\pi$
$433$	−4.71932	−0.226796	−0.113398	+	0.993550i	$0.536174\pi$
$434$	0	0
$435$	5.28666	0.253476
$436$	0	0
$437$	−0.616024	−0.0294684
$438$	0	0
$439$	6.56575	0.313366	0.156683	−	0.987649i	$-0.449920\pi$
$439$	6.56575	0.313366	0.156683	+	0.987649i	$0.449920\pi$
$440$	0	0
$441$	7.44912	0.354720
$442$	0	0
$443$	−0.143634	−0.00682424	−0.00341212	−	0.999994i	$-0.501086\pi$
$443$	−0.143634	−0.00682424	−0.00341212	+	0.999994i	$0.501086\pi$
$444$	0	0
$445$	−23.7278	−1.12481
$446$	0	0
$447$	−8.02623	−0.379628
$448$	0	0
$449$	15.9059	0.750645	0.375323	−	0.926894i	$-0.377532\pi$
$449$	15.9059	0.750645	0.375323	+	0.926894i	$0.377532\pi$
$450$	0	0
$451$	18.0229	0.848664
$452$	0	0
$453$	4.46075	0.209584
$454$	0	0
$455$	38.9043	1.82386
$456$	0	0
$457$	−2.39293	−0.111936	−0.0559682	−	0.998433i	$-0.517825\pi$
$457$	−2.39293	−0.111936	−0.0559682	+	0.998433i	$0.517825\pi$
$458$	0	0
$459$	−2.32295	−0.108426
$460$	0	0
$461$	−0.296763	−0.0138216	−0.00691082	−	0.999976i	$-0.502200\pi$
$461$	−0.296763	−0.0138216	−0.00691082	+	0.999976i	$0.502200\pi$
$462$	0	0
$463$	15.8608	0.737115	0.368558	−	0.929605i	$-0.379852\pi$
$463$	15.8608	0.737115	0.368558	+	0.929605i	$0.379852\pi$
$464$	0	0
$465$	−1.40554	−0.0651803
$466$	0	0
$467$	−11.6855	−0.540740	−0.270370	−	0.962757i	$-0.587146\pi$
$467$	−11.6855	−0.540740	−0.270370	+	0.962757i	$0.587146\pi$
$468$	0	0
$469$	−4.11634	−0.190075
$470$	0	0
$471$	6.83834	0.315094
$472$	0	0
$473$	14.8970	0.684963
$474$	0	0
$475$	−2.30930	−0.105958
$476$	0	0
$477$	3.13343	0.143470
$478$	0	0
$479$	13.1532	0.600985	0.300492	−	0.953784i	$-0.402849\pi$
$479$	13.1532	0.600985	0.300492	+	0.953784i	$0.402849\pi$
$480$	0	0
$481$	−9.07396	−0.413737
$482$	0	0
$483$	−1.63587	−0.0744349
$484$	0	0
$485$	−2.87002	−0.130321
$486$	0	0
$487$	−15.2900	−0.692857	−0.346428	−	0.938076i	$-0.612606\pi$
$487$	−15.2900	−0.692857	−0.346428	+	0.938076i	$0.612606\pi$
$488$	0	0
$489$	−3.49352	−0.157982
$490$	0	0
$491$	16.4240	0.741204	0.370602	−	0.928792i	$-0.379151\pi$
$491$	16.4240	0.741204	0.370602	+	0.928792i	$0.379151\pi$
$492$	0	0
$493$	3.78015	0.170250
$494$	0	0
$495$	−21.3063	−0.957648
$496$	0	0
$497$	12.2360	0.548860
$498$	0	0
$499$	3.92260	0.175599	0.0877997	−	0.996138i	$-0.472016\pi$
$499$	3.92260	0.175599	0.0877997	+	0.996138i	$0.472016\pi$
$500$	0	0
$501$	4.52192	0.202025
$502$	0	0
$503$	−4.97604	−0.221870	−0.110935	−	0.993828i	$-0.535385\pi$
$503$	−4.97604	−0.221870	−0.110935	+	0.993828i	$0.535385\pi$
$504$	0	0
$505$	34.8763	1.55198
$506$	0	0
$507$	−5.94180	−0.263885
$508$	0	0
$509$	22.5969	1.00159	0.500795	−	0.865566i	$-0.333041\pi$
$509$	22.5969	1.00159	0.500795	+	0.865566i	$0.333041\pi$
$510$	0	0
$511$	−19.9297	−0.881639
$512$	0	0
$513$	−0.727869	−0.0321362
$514$	0	0
$515$	3.98379	0.175547
$516$	0	0
$517$	−5.99690	−0.263744
$518$	0	0
$519$	2.21154	0.0970759
$520$	0	0
$521$	27.6030	1.20931	0.604655	−	0.796487i	$-0.293311\pi$
$521$	27.6030	1.20931	0.604655	+	0.796487i	$0.293311\pi$
$522$	0	0
$523$	17.8807	0.781870	0.390935	−	0.920418i	$-0.372152\pi$
$523$	17.8807	0.781870	0.390935	+	0.920418i	$0.372152\pi$
$524$	0	0
$525$	−6.13242	−0.267641
$526$	0	0
$527$	−1.00501	−0.0437790
$528$	0	0
$529$	−19.1348	−0.831949
$530$	0	0
$531$	2.84188	0.123327
$532$	0	0
$533$	44.6932	1.93588
$534$	0	0
$535$	2.96477	0.128178
$536$	0	0
$537$	8.00957	0.345638
$538$	0	0
$539$	−5.58748	−0.240670
$540$	0	0
$541$	−4.46207	−0.191839	−0.0959196	−	0.995389i	$-0.530579\pi$
$541$	−4.46207	−0.191839	−0.0959196	+	0.995389i	$0.530579\pi$
$542$	0	0
$543$	−8.18854	−0.351404
$544$	0	0
$545$	−44.8495	−1.92114
$546$	0	0
$547$	−17.0679	−0.729769	−0.364884	−	0.931053i	$-0.618891\pi$
$547$	−17.0679	−0.729769	−0.364884	+	0.931053i	$0.618891\pi$
$548$	0	0
$549$	−10.5190	−0.448938
$550$	0	0
$551$	1.18447	0.0504599
$552$	0	0
$553$	−6.61809	−0.281430
$554$	0	0
$555$	2.40068	0.101903
$556$	0	0
$557$	−14.6275	−0.619786	−0.309893	−	0.950771i	$-0.600293\pi$
$557$	−14.6275	−0.619786	−0.309893	+	0.950771i	$0.600293\pi$
$558$	0	0
$559$	36.9415	1.56246
$560$	0	0
$561$	0.847627	0.0357868
$562$	0	0
$563$	−13.7293	−0.578621	−0.289310	−	0.957235i	$-0.593426\pi$
$563$	−13.7293	−0.578621	−0.289310	+	0.957235i	$0.593426\pi$
$564$	0	0
$565$	−23.3576	−0.982660
$566$	0	0
$567$	15.9076	0.668055
$568$	0	0
$569$	−33.3820	−1.39945	−0.699723	−	0.714414i	$-0.746693\pi$
$569$	−33.3820	−1.39945	−0.699723	+	0.714414i	$0.746693\pi$
$570$	0	0
$571$	−6.55546	−0.274337	−0.137169	−	0.990548i	$-0.543800\pi$
$571$	−6.55546	−0.274337	−0.137169	+	0.990548i	$0.543800\pi$
$572$	0	0
$573$	7.49094	0.312939
$574$	0	0
$575$	14.4894	0.604251
$576$	0	0
$577$	−22.3206	−0.929217	−0.464609	−	0.885516i	$-0.653805\pi$
$577$	−22.3206	−0.929217	−0.464609	+	0.885516i	$0.653805\pi$
$578$	0	0
$579$	−2.02654	−0.0842201
$580$	0	0
$581$	21.5418	0.893706
$582$	0	0
$583$	−2.35034	−0.0973412
$584$	0	0
$585$	−52.8355	−2.18448
$586$	0	0
$587$	−38.0865	−1.57200	−0.786000	−	0.618227i	$-0.787851\pi$
$587$	−38.0865	−1.57200	−0.786000	+	0.618227i	$0.787851\pi$
$588$	0	0
$589$	−0.314908	−0.0129756
$590$	0	0
$591$	−5.77369	−0.237498
$592$	0	0
$593$	5.13305	0.210789	0.105395	−	0.994430i	$-0.466389\pi$
$593$	5.13305	0.210789	0.105395	+	0.994430i	$0.466389\pi$
$594$	0	0
$595$	−7.35974	−0.301720
$596$	0	0
$597$	−4.58668	−0.187720
$598$	0	0
$599$	−39.3268	−1.60685	−0.803424	−	0.595407i	$-0.796991\pi$
$599$	−39.3268	−1.60685	−0.803424	+	0.595407i	$0.796991\pi$
$600$	0	0
$601$	6.55602	0.267425	0.133713	−	0.991020i	$-0.457310\pi$
$601$	6.55602	0.267425	0.133713	+	0.991020i	$0.457310\pi$
$602$	0	0
$603$	5.59036	0.227657
$604$	0	0
$605$	−22.7065	−0.923151
$606$	0	0
$607$	−21.5350	−0.874078	−0.437039	−	0.899443i	$-0.643973\pi$
$607$	−21.5350	−0.874078	−0.437039	+	0.899443i	$0.643973\pi$
$608$	0	0
$609$	3.14539	0.127458
$610$	0	0
$611$	−14.8711	−0.601622
$612$	0	0
$613$	3.00257	0.121273	0.0606363	−	0.998160i	$-0.480687\pi$
$613$	3.00257	0.121273	0.0606363	+	0.998160i	$0.480687\pi$
$614$	0	0
$615$	−11.8244	−0.476805
$616$	0	0
$617$	−4.82687	−0.194323	−0.0971613	−	0.995269i	$-0.530976\pi$
$617$	−4.82687	−0.194323	−0.0971613	+	0.995269i	$0.530976\pi$
$618$	0	0
$619$	−40.3444	−1.62158	−0.810789	−	0.585339i	$-0.800962\pi$
$619$	−40.3444	−1.62158	−0.810789	+	0.585339i	$0.800962\pi$
$620$	0	0
$621$	4.56694	0.183265
$622$	0	0
$623$	−14.1173	−0.565597
$624$	0	0
$625$	−7.53324	−0.301330
$626$	0	0
$627$	0.265594	0.0106068
$628$	0	0
$629$	1.71657	0.0684442
$630$	0	0
$631$	−41.4560	−1.65034	−0.825168	−	0.564887i	$-0.808920\pi$
$631$	−41.4560	−1.65034	−0.825168	+	0.564887i	$0.808920\pi$
$632$	0	0
$633$	1.42447	0.0566174
$634$	0	0
$635$	43.3747	1.72127
$636$	0	0
$637$	−13.8558	−0.548988
$638$	0	0
$639$	−16.6176	−0.657381
$640$	0	0
$641$	−7.69665	−0.303999	−0.152000	−	0.988381i	$-0.548571\pi$
$641$	−7.69665	−0.303999	−0.152000	+	0.988381i	$0.548571\pi$
$642$	0	0
$643$	−43.7692	−1.72609	−0.863045	−	0.505127i	$-0.831446\pi$
$643$	−43.7692	−1.72609	−0.863045	+	0.505127i	$0.831446\pi$
$644$	0	0
$645$	−9.77355	−0.384833
$646$	0	0
$647$	−34.1682	−1.34329	−0.671646	−	0.740872i	$-0.734412\pi$
$647$	−34.1682	−1.34329	−0.671646	+	0.740872i	$0.734412\pi$
$648$	0	0
$649$	−2.13166	−0.0836749
$650$	0	0
$651$	−0.836250	−0.0327752
$652$	0	0
$653$	3.64712	0.142723	0.0713614	−	0.997451i	$-0.477266\pi$
$653$	3.64712	0.142723	0.0713614	+	0.997451i	$0.477266\pi$
$654$	0	0
$655$	38.9907	1.52349
$656$	0	0
$657$	27.0663	1.05596
$658$	0	0
$659$	−15.9833	−0.622621	−0.311310	−	0.950308i	$-0.600768\pi$
$659$	−15.9833	−0.622621	−0.311310	+	0.950308i	$0.600768\pi$
$660$	0	0
$661$	−51.2178	−1.99214	−0.996070	−	0.0885641i	$-0.971772\pi$
$661$	−51.2178	−1.99214	−0.996070	+	0.0885641i	$0.971772\pi$
$662$	0	0
$663$	2.10195	0.0816329
$664$	0	0
$665$	−2.30609	−0.0894262
$666$	0	0
$667$	−7.43180	−0.287760
$668$	0	0
$669$	3.61613	0.139808
$670$	0	0
$671$	7.89012	0.304595
$672$	0	0
$673$	17.9920	0.693541	0.346770	−	0.937950i	$-0.387278\pi$
$673$	17.9920	0.693541	0.346770	+	0.937950i	$0.387278\pi$
$674$	0	0
$675$	17.1201	0.658954
$676$	0	0
$677$	−47.2320	−1.81527	−0.907636	−	0.419758i	$-0.862115\pi$
$677$	−47.2320	−1.81527	−0.907636	+	0.419758i	$0.862115\pi$
$678$	0	0
$679$	−1.70757	−0.0655304
$680$	0	0
$681$	−7.99892	−0.306519
$682$	0	0
$683$	11.1699	0.427406	0.213703	−	0.976899i	$-0.431448\pi$
$683$	11.1699	0.427406	0.213703	+	0.976899i	$0.431448\pi$
$684$	0	0
$685$	−56.7364	−2.16778
$686$	0	0
$687$	−2.46641	−0.0940996
$688$	0	0
$689$	−5.82838	−0.222044
$690$	0	0
$691$	−30.8006	−1.17171	−0.585854	−	0.810416i	$-0.699241\pi$
$691$	−30.8006	−1.17171	−0.585854	+	0.810416i	$0.699241\pi$
$692$	0	0
$693$	−12.6766	−0.481543
$694$	0	0
$695$	−5.27518	−0.200099
$696$	0	0
$697$	−8.45487	−0.320251
$698$	0	0
$699$	4.05953	0.153546
$700$	0	0
$701$	−20.3970	−0.770385	−0.385193	−	0.922836i	$-0.625865\pi$
$701$	−20.3970	−0.770385	−0.385193	+	0.922836i	$0.625865\pi$
$702$	0	0
$703$	0.537867	0.0202860
$704$	0	0
$705$	3.93443	0.148179
$706$	0	0
$707$	20.7503	0.780395
$708$	0	0
$709$	−21.3756	−0.802777	−0.401388	−	0.915908i	$-0.631472\pi$
$709$	−21.3756	−0.802777	−0.401388	+	0.915908i	$0.631472\pi$
$710$	0	0
$711$	8.98796	0.337075
$712$	0	0
$713$	1.97586	0.0739964
$714$	0	0
$715$	39.6311	1.48212
$716$	0	0
$717$	−10.6238	−0.396752
$718$	0	0
$719$	−26.8240	−1.00037	−0.500183	−	0.865920i	$-0.666734\pi$
$719$	−26.8240	−1.00037	−0.500183	+	0.865920i	$0.666734\pi$
$720$	0	0
$721$	2.37023	0.0882718
$722$	0	0
$723$	−1.81303	−0.0674272
$724$	0	0
$725$	−27.8597	−1.03468
$726$	0	0
$727$	−19.2656	−0.714522	−0.357261	−	0.934005i	$-0.616289\pi$
$727$	−19.2656	−0.714522	−0.357261	+	0.934005i	$0.616289\pi$
$728$	0	0
$729$	−18.8328	−0.697512
$730$	0	0
$731$	−6.98844	−0.258477
$732$	0	0
$733$	−3.82684	−0.141347	−0.0706737	−	0.997499i	$-0.522515\pi$
$733$	−3.82684	−0.141347	−0.0706737	+	0.997499i	$0.522515\pi$
$734$	0	0
$735$	3.66581	0.135216
$736$	0	0
$737$	−4.19325	−0.154460
$738$	0	0
$739$	38.4046	1.41273	0.706367	−	0.707845i	$-0.250333\pi$
$739$	38.4046	1.41273	0.706367	+	0.707845i	$0.250333\pi$
$740$	0	0
$741$	0.658620	0.0241950
$742$	0	0
$743$	−15.7849	−0.579092	−0.289546	−	0.957164i	$-0.593504\pi$
$743$	−15.7849	−0.579092	−0.289546	+	0.957164i	$0.593504\pi$
$744$	0	0
$745$	70.9919	2.60094
$746$	0	0
$747$	−29.2557	−1.07041
$748$	0	0
$749$	1.76394	0.0644531
$750$	0	0
$751$	−10.5279	−0.384169	−0.192084	−	0.981378i	$-0.561525\pi$
$751$	−10.5279	−0.384169	−0.192084	+	0.981378i	$0.561525\pi$
$752$	0	0
$753$	−5.07224	−0.184843
$754$	0	0
$755$	−39.4553	−1.43592
$756$	0	0
$757$	46.7275	1.69834	0.849171	−	0.528119i	$-0.177102\pi$
$757$	46.7275	1.69834	0.849171	+	0.528119i	$0.177102\pi$
$758$	0	0
$759$	−1.66644	−0.0604879
$760$	0	0
$761$	−37.1530	−1.34680	−0.673398	−	0.739280i	$-0.735166\pi$
$761$	−37.1530	−1.34680	−0.673398	+	0.739280i	$0.735166\pi$
$762$	0	0
$763$	−26.6840	−0.966026
$764$	0	0
$765$	9.99519	0.361377
$766$	0	0
$767$	−5.28609	−0.190870
$768$	0	0
$769$	−6.64383	−0.239583	−0.119791	−	0.992799i	$-0.538223\pi$
$769$	−6.64383	−0.239583	−0.119791	+	0.992799i	$0.538223\pi$
$770$	0	0
$771$	10.1152	0.364291
$772$	0	0
$773$	14.0230	0.504372	0.252186	−	0.967679i	$-0.418851\pi$
$773$	14.0230	0.504372	0.252186	+	0.967679i	$0.418851\pi$
$774$	0	0
$775$	7.40692	0.266064
$776$	0	0
$777$	1.42833	0.0512409
$778$	0	0
$779$	−2.64923	−0.0949185
$780$	0	0
$781$	12.4646	0.446019
$782$	0	0
$783$	−8.78112	−0.313812
$784$	0	0
$785$	−60.4850	−2.15880
$786$	0	0
$787$	−35.5409	−1.26690	−0.633448	−	0.773785i	$-0.718361\pi$
$787$	−35.5409	−1.26690	−0.633448	+	0.773785i	$0.718361\pi$
$788$	0	0
$789$	−4.30426	−0.153236
$790$	0	0
$791$	−13.8970	−0.494121
$792$	0	0
$793$	19.5659	0.694807
$794$	0	0
$795$	1.54200	0.0546892
$796$	0	0
$797$	−6.17954	−0.218891	−0.109445	−	0.993993i	$-0.534907\pi$
$797$	−6.17954	−0.218891	−0.109445	+	0.993993i	$0.534907\pi$
$798$	0	0
$799$	2.81326	0.0995259
$800$	0	0
$801$	19.1725	0.677428
$802$	0	0
$803$	−20.3021	−0.716445
$804$	0	0
$805$	14.4693	0.509976
$806$	0	0
$807$	3.55721	0.125220
$808$	0	0
$809$	−13.4030	−0.471223	−0.235612	−	0.971847i	$-0.575709\pi$
$809$	−13.4030	−0.471223	−0.235612	+	0.971847i	$0.575709\pi$
$810$	0	0
$811$	−0.0498140	−0.00174921	−0.000874603	−	1.00000i	$-0.500278\pi$
$811$	−0.0498140	−0.00174921	−0.000874603		1.00000i	$0.500278\pi$
$812$	0	0
$813$	4.60627	0.161549
$814$	0	0
$815$	30.9001	1.08238
$816$	0	0
$817$	−2.18974	−0.0766094
$818$	0	0
$819$	−31.4354	−1.09844
$820$	0	0
$821$	24.4767	0.854244	0.427122	−	0.904194i	$-0.359528\pi$
$821$	24.4767	0.854244	0.427122	+	0.904194i	$0.359528\pi$
$822$	0	0
$823$	39.2416	1.36788	0.683938	−	0.729540i	$-0.260266\pi$
$823$	39.2416	1.36788	0.683938	+	0.729540i	$0.260266\pi$
$824$	0	0
$825$	−6.24700	−0.217493
$826$	0	0
$827$	−3.13905	−0.109155	−0.0545777	−	0.998510i	$-0.517381\pi$
$827$	−3.13905	−0.109155	−0.0545777	+	0.998510i	$0.517381\pi$
$828$	0	0
$829$	−25.5649	−0.887905	−0.443953	−	0.896050i	$-0.646424\pi$
$829$	−25.5649	−0.887905	−0.443953	+	0.896050i	$0.646424\pi$
$830$	0	0
$831$	−4.63471	−0.160776
$832$	0	0
$833$	2.62119	0.0908188
$834$	0	0
$835$	−39.9963	−1.38413
$836$	0	0
$837$	2.33459	0.0806954
$838$	0	0
$839$	38.7802	1.33884	0.669420	−	0.742884i	$-0.266543\pi$
$839$	38.7802	1.33884	0.669420	+	0.742884i	$0.266543\pi$
$840$	0	0
$841$	−14.7104	−0.507256
$842$	0	0
$843$	8.55618	0.294691
$844$	0	0
$845$	52.5551	1.80795
$846$	0	0
$847$	−13.5097	−0.464197
$848$	0	0
$849$	−2.48483	−0.0852793
$850$	0	0
$851$	−3.37479	−0.115686
$852$	0	0
$853$	−51.8220	−1.77435	−0.887175	−	0.461432i	$-0.847336\pi$
$853$	−51.8220	−1.77435	−0.887175	+	0.461432i	$0.847336\pi$
$854$	0	0
$855$	3.13187	0.107108
$856$	0	0
$857$	−6.33207	−0.216299	−0.108150	−	0.994135i	$-0.534493\pi$
$857$	−6.33207	−0.216299	−0.108150	+	0.994135i	$0.534493\pi$
$858$	0	0
$859$	54.9511	1.87491	0.937454	−	0.348108i	$-0.113176\pi$
$859$	54.9511	1.87491	0.937454	+	0.348108i	$0.113176\pi$
$860$	0	0
$861$	−7.03513	−0.239757
$862$	0	0
$863$	47.5882	1.61992	0.809960	−	0.586485i	$-0.199489\pi$
$863$	47.5882	1.61992	0.809960	+	0.586485i	$0.199489\pi$
$864$	0	0
$865$	−19.5611	−0.665096
$866$	0	0
$867$	−0.397638	−0.0135045
$868$	0	0
$869$	−6.74174	−0.228698
$870$	0	0
$871$	−10.3984	−0.352338
$872$	0	0
$873$	2.31903	0.0784872
$874$	0	0
$875$	17.4425	0.589664
$876$	0	0
$877$	−1.76816	−0.0597066	−0.0298533	−	0.999554i	$-0.509504\pi$
$877$	−1.76816	−0.0597066	−0.0298533	+	0.999554i	$0.509504\pi$
$878$	0	0
$879$	6.67155	0.225026
$880$	0	0
$881$	26.8980	0.906216	0.453108	−	0.891456i	$-0.350315\pi$
$881$	26.8980	0.906216	0.453108	+	0.891456i	$0.350315\pi$
$882$	0	0
$883$	3.94184	0.132654	0.0663268	−	0.997798i	$-0.478872\pi$
$883$	3.94184	0.132654	0.0663268	+	0.997798i	$0.478872\pi$
$884$	0	0
$885$	1.39853	0.0470111
$886$	0	0
$887$	13.0720	0.438914	0.219457	−	0.975622i	$-0.429571\pi$
$887$	13.0720	0.438914	0.219457	+	0.975622i	$0.429571\pi$
$888$	0	0
$889$	25.8066	0.865525
$890$	0	0
$891$	16.2048	0.542880
$892$	0	0
$893$	0.881500	0.0294983
$894$	0	0
$895$	−70.8445	−2.36807
$896$	0	0
$897$	−4.13244	−0.137978
$898$	0	0
$899$	−3.79910	−0.126707
$900$	0	0
$901$	1.10259	0.0367325
$902$	0	0
$903$	−5.81495	−0.193509
$904$	0	0
$905$	72.4275	2.40757
$906$	0	0
$907$	0.785027	0.0260664	0.0130332	−	0.999915i	$-0.495851\pi$
$907$	0.785027	0.0260664	0.0130332	+	0.999915i	$0.495851\pi$
$908$	0	0
$909$	−28.1807	−0.934696
$910$	0	0
$911$	−49.5108	−1.64037	−0.820184	−	0.572100i	$-0.806129\pi$
$911$	−49.5108	−1.64037	−0.820184	+	0.572100i	$0.806129\pi$
$912$	0	0
$913$	21.9443	0.726251
$914$	0	0
$915$	−5.17652	−0.171131
$916$	0	0
$917$	23.1982	0.766072
$918$	0	0
$919$	−41.8467	−1.38040	−0.690198	−	0.723620i	$-0.742477\pi$
$919$	−41.8467	−1.38040	−0.690198	+	0.723620i	$0.742477\pi$
$920$	0	0
$921$	−1.61057	−0.0530701
$922$	0	0
$923$	30.9098	1.01741
$924$	0	0
$925$	−12.6511	−0.415966
$926$	0	0
$927$	−3.21898	−0.105725
$928$	0	0
$929$	−28.7091	−0.941914	−0.470957	−	0.882156i	$-0.656091\pi$
$929$	−28.7091	−0.941914	−0.470957	+	0.882156i	$0.656091\pi$
$930$	0	0
$931$	0.821318	0.0269176
$932$	0	0
$933$	−9.18682	−0.300763
$934$	0	0
$935$	−7.49725	−0.245186
$936$	0	0
$937$	10.8346	0.353951	0.176975	−	0.984215i	$-0.443369\pi$
$937$	10.8346	0.353951	0.176975	+	0.984215i	$0.443369\pi$
$938$	0	0
$939$	6.80028	0.221919
$940$	0	0
$941$	15.2863	0.498318	0.249159	−	0.968463i	$-0.419846\pi$
$941$	15.2863	0.498318	0.249159	+	0.968463i	$0.419846\pi$
$942$	0	0
$943$	16.6223	0.541297
$944$	0	0
$945$	17.0963	0.556144
$946$	0	0
$947$	18.1774	0.590687	0.295343	−	0.955391i	$-0.404566\pi$
$947$	18.1774	0.590687	0.295343	+	0.955391i	$0.404566\pi$
$948$	0	0
$949$	−50.3452	−1.63427
$950$	0	0
$951$	−7.91391	−0.256626
$952$	0	0
$953$	5.31873	0.172290	0.0861452	−	0.996283i	$-0.472545\pi$
$953$	5.31873	0.172290	0.0861452	+	0.996283i	$0.472545\pi$
$954$	0	0
$955$	−66.2573	−2.14404
$956$	0	0
$957$	3.20416	0.103576
$958$	0	0
$959$	−33.7563	−1.09005
$960$	0	0
$961$	−29.9900	−0.967418
$962$	0	0
$963$	−2.39559	−0.0771969
$964$	0	0
$965$	17.9247	0.577017
$966$	0	0
$967$	8.43090	0.271119	0.135560	−	0.990769i	$-0.456717\pi$
$967$	8.43090	0.271119	0.135560	+	0.990769i	$0.456717\pi$
$968$	0	0
$969$	−0.124595	−0.00400256
$970$	0	0
$971$	−27.2209	−0.873561	−0.436781	−	0.899568i	$-0.643881\pi$
$971$	−27.2209	−0.873561	−0.436781	+	0.899568i	$0.643881\pi$
$972$	0	0
$973$	−3.13856	−0.100618
$974$	0	0
$975$	−15.4913	−0.496120
$976$	0	0
$977$	57.8187	1.84978	0.924892	−	0.380229i	$-0.124155\pi$
$977$	57.8187	1.84978	0.924892	+	0.380229i	$0.124155\pi$
$978$	0	0
$979$	−14.3811	−0.459620
$980$	0	0
$981$	36.2393	1.15703
$982$	0	0
$983$	−29.1220	−0.928846	−0.464423	−	0.885613i	$-0.653738\pi$
$983$	−29.1220	−0.928846	−0.464423	+	0.885613i	$0.653738\pi$
$984$	0	0
$985$	51.0683	1.62717
$986$	0	0
$987$	2.34086	0.0745103
$988$	0	0
$989$	13.7393	0.436885
$990$	0	0
$991$	−31.9458	−1.01479	−0.507396	−	0.861713i	$-0.669392\pi$
$991$	−31.9458	−1.01479	−0.507396	+	0.861713i	$0.669392\pi$
$992$	0	0
$993$	−8.84919	−0.280820
$994$	0	0
$995$	40.5692	1.28613
$996$	0	0
$997$	6.32071	0.200179	0.100089	−	0.994978i	$-0.468087\pi$
$997$	6.32071	0.200179	0.100089	+	0.994978i	$0.468087\pi$
$998$	0	0
$999$	−3.98752	−0.126159

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4012.2.a.i.1.6	✓	18

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4012.2.a.i.1.6	✓	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)

\(f(q)\)	\(=\)	\(q-0.397638 q^{3} +3.51710 q^{5} +2.09256 q^{7} -2.84188 q^{9} +O(q^{10})\)	\(q-0.397638 q^{3} +3.51710 q^{5} +2.09256 q^{7} -2.84188 q^{9} +2.13166 q^{11} +5.28609 q^{13} -1.39853 q^{15} -1.00000 q^{17} -0.313338 q^{19} -0.832081 q^{21} +1.96600 q^{23} +7.36999 q^{25} +2.32295 q^{27} -3.78015 q^{29} +1.00501 q^{31} -0.847627 q^{33} +7.35974 q^{35} -1.71657 q^{37} -2.10195 q^{39} +8.45487 q^{41} +6.98844 q^{43} -9.99519 q^{45} -2.81326 q^{47} -2.62119 q^{49} +0.397638 q^{51} -1.10259 q^{53} +7.49725 q^{55} +0.124595 q^{57} -1.00000 q^{59} +3.70140 q^{61} -5.94682 q^{63} +18.5917 q^{65} -1.96713 q^{67} -0.781757 q^{69} +5.84738 q^{71} -9.52409 q^{73} -2.93058 q^{75} +4.46062 q^{77} -3.16268 q^{79} +7.60196 q^{81} +10.2945 q^{83} -3.51710 q^{85} +1.50313 q^{87} -6.74642 q^{89} +11.0615 q^{91} -0.399630 q^{93} -1.10204 q^{95} -0.816018 q^{97} -6.05792 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q + 8 q^{3} + 4 q^{5} + 2 q^{7} + 18 q^{9}+O(q^{10}) \) 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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