LMFDB - Embedded newform 4842.2.a.t.1.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4842,2,Mod(1,4842)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4842, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("4842.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4842 = 2 \cdot 3^{2} \cdot 269 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4842.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:	$38.6635646587$
Analytic rank:	$0$
Dimension:	$13$
Coefficient field:	$\mathbb{Q}[x]/(x^{13} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{13} - 5 x^{12} - 33 x^{11} + 200 x^{10} + 236 x^{9} - 2569 x^{8} + 1311 x^{7} + 11583 x^{6} + \cdots + 2160 $ x^13 - 5x^12 - 33x^11 + 200x^10 + 236x^9 - 2569x^8 + 1311x^7 + 11583x^6 - 15582x^5 - 10013x^4 + 24596x^3 - 6260x^2 - 5472x + 2160
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.9
Root			$-0.515478$ of defining polynomial
Character	$\chi$	$=$	4842.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.00000 q^{2} +1.00000 q^{4} +0.515478 q^{5} -4.54762 q^{7} -1.00000 q^{8} -0.515478 q^{10} -0.646622 q^{11} -5.04122 q^{13} +4.54762 q^{14} +1.00000 q^{16} -0.197175 q^{17} -6.64024 q^{19} +0.515478 q^{20} +0.646622 q^{22} -6.07059 q^{23} -4.73428 q^{25} +5.04122 q^{26} -4.54762 q^{28} -2.46368 q^{29} +4.70580 q^{31} -1.00000 q^{32} +0.197175 q^{34} -2.34420 q^{35} +6.75434 q^{37} +6.64024 q^{38} -0.515478 q^{40} -12.4131 q^{41} +5.56005 q^{43} -0.646622 q^{44} +6.07059 q^{46} +10.8909 q^{47} +13.6808 q^{49} +4.73428 q^{50} -5.04122 q^{52} -10.5761 q^{53} -0.333319 q^{55} +4.54762 q^{56} +2.46368 q^{58} -12.4961 q^{59} +8.74339 q^{61} -4.70580 q^{62} +1.00000 q^{64} -2.59864 q^{65} +12.5544 q^{67} -0.197175 q^{68} +2.34420 q^{70} +2.61034 q^{71} +4.04118 q^{73} -6.75434 q^{74} -6.64024 q^{76} +2.94059 q^{77} -13.0886 q^{79} +0.515478 q^{80} +12.4131 q^{82} +10.7430 q^{83} -0.101639 q^{85} -5.56005 q^{86} +0.646622 q^{88} +12.1522 q^{89} +22.9255 q^{91} -6.07059 q^{92} -10.8909 q^{94} -3.42290 q^{95} -9.57744 q^{97} -13.6808 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 13 q - 13 q^{2} + 13 q^{4} - 5 q^{5} + 6 q^{7} - 13 q^{8} + 5 q^{10} - 5 q^{11} + 8 q^{13} - 6 q^{14} + 13 q^{16} - 7 q^{17} + 13 q^{19} - 5 q^{20} + 5 q^{22} + 4 q^{23} + 26 q^{25} - 8 q^{26} + 6 q^{28}+ \cdots - 39 q^{98}+O(q^{100}) $ 13 * q - 13 * q^2 + 13 * q^4 - 5 * q^5 + 6 * q^7 - 13 * q^8 + 5 * q^10 - 5 * q^11 + 8 * q^13 - 6 * q^14 + 13 * q^16 - 7 * q^17 + 13 * q^19 - 5 * q^20 + 5 * q^22 + 4 * q^23 + 26 * q^25 - 8 * q^26 + 6 * q^28 - 22 * q^29 + 8 * q^31 - 13 * q^32 + 7 * q^34 - 8 * q^35 + 36 * q^37 - 13 * q^38 + 5 * q^40 - 16 * q^41 + 14 * q^43 - 5 * q^44 - 4 * q^46 - 4 * q^47 + 39 * q^49 - 26 * q^50 + 8 * q^52 - 21 * q^53 - 6 * q^56 + 22 * q^58 + 4 * q^59 + 8 * q^61 - 8 * q^62 + 13 * q^64 - 24 * q^65 + 40 * q^67 - 7 * q^68 + 8 * q^70 - 7 * q^71 + 31 * q^73 - 36 * q^74 + 13 * q^76 - 10 * q^77 + 43 * q^79 - 5 * q^80 + 16 * q^82 + 10 * q^83 + 30 * q^85 - 14 * q^86 + 5 * q^88 + 2 * q^89 + 24 * q^91 + 4 * q^92 + 4 * q^94 - 2 * q^95 + 27 * q^97 - 39 * q^98

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 $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	0.515478	0.230529	0.115264	−	0.993335i	$-0.463228\pi$
$5$	0.515478	0.230529	0.115264	+	0.993335i	$0.463228\pi$
$6$	0	0
$7$	−4.54762	−1.71884	−0.859419	−	0.511273i	$-0.829174\pi$
$7$	−4.54762	−1.71884	−0.859419	+	0.511273i	$0.829174\pi$
$8$	−1.00000	−0.353553
$9$	0	0
$10$	−0.515478	−0.163008
$11$	−0.646622	−0.194964	−0.0974820	−	0.995237i	$-0.531079\pi$
$11$	−0.646622	−0.194964	−0.0974820	+	0.995237i	$0.531079\pi$
$12$	0	0
$13$	−5.04122	−1.39818	−0.699092	−	0.715032i	$-0.746412\pi$
$13$	−5.04122	−1.39818	−0.699092	+	0.715032i	$0.746412\pi$
$14$	4.54762	1.21540
$15$	0	0
$16$	1.00000	0.250000
$17$	−0.197175	−0.0478219	−0.0239110	−	0.999714i	$-0.507612\pi$
$17$	−0.197175	−0.0478219	−0.0239110	+	0.999714i	$0.507612\pi$
$18$	0	0
$19$	−6.64024	−1.52338	−0.761688	−	0.647944i	$-0.775629\pi$
$19$	−6.64024	−1.52338	−0.761688	+	0.647944i	$0.775629\pi$
$20$	0.515478	0.115264
$21$	0	0
$22$	0.646622	0.137860
$23$	−6.07059	−1.26581	−0.632903	−	0.774231i	$-0.718137\pi$
$23$	−6.07059	−1.26581	−0.632903	+	0.774231i	$0.718137\pi$
$24$	0	0
$25$	−4.73428	−0.946857
$26$	5.04122	0.988665
$27$	0	0
$28$	−4.54762	−0.859419
$29$	−2.46368	−0.457494	−0.228747	−	0.973486i	$-0.573463\pi$
$29$	−2.46368	−0.457494	−0.228747	+	0.973486i	$0.573463\pi$
$30$	0	0
$31$	4.70580	0.845187	0.422593	−	0.906319i	$-0.361120\pi$
$31$	4.70580	0.845187	0.422593	+	0.906319i	$0.361120\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	0.197175	0.0338152
$35$	−2.34420	−0.396241
$36$	0	0
$37$	6.75434	1.11041	0.555203	−	0.831715i	$-0.312641\pi$
$37$	6.75434	1.11041	0.555203	+	0.831715i	$0.312641\pi$
$38$	6.64024	1.07719
$39$	0	0
$40$	−0.515478	−0.0815042
$41$	−12.4131	−1.93861	−0.969303	−	0.245868i	$-0.920927\pi$
$41$	−12.4131	−1.93861	−0.969303	+	0.245868i	$0.920927\pi$
$42$	0	0
$43$	5.56005	0.847899	0.423950	−	0.905686i	$-0.360643\pi$
$43$	5.56005	0.847899	0.423950	+	0.905686i	$0.360643\pi$
$44$	−0.646622	−0.0974820
$45$	0	0
$46$	6.07059	0.895059
$47$	10.8909	1.58860	0.794299	−	0.607527i	$-0.207838\pi$
$47$	10.8909	1.58860	0.794299	+	0.607527i	$0.207838\pi$
$48$	0	0
$49$	13.6808	1.95440
$50$	4.73428	0.669529
$51$	0	0
$52$	−5.04122	−0.699092
$53$	−10.5761	−1.45274	−0.726368	−	0.687306i	$-0.758793\pi$
$53$	−10.5761	−1.45274	−0.726368	+	0.687306i	$0.758793\pi$
$54$	0	0
$55$	−0.333319	−0.0449448
$56$	4.54762	0.607701
$57$	0	0
$58$	2.46368	0.323497
$59$	−12.4961	−1.62686	−0.813428	−	0.581665i	$-0.802401\pi$
$59$	−12.4961	−1.62686	−0.813428	+	0.581665i	$0.802401\pi$
$60$	0	0
$61$	8.74339	1.11948	0.559738	−	0.828670i	$-0.310902\pi$
$61$	8.74339	1.11948	0.559738	+	0.828670i	$0.310902\pi$
$62$	−4.70580	−0.597637
$63$	0	0
$64$	1.00000	0.125000
$65$	−2.59864	−0.322321
$66$	0	0
$67$	12.5544	1.53377	0.766884	−	0.641785i	$-0.221806\pi$
$67$	12.5544	1.53377	0.766884	+	0.641785i	$0.221806\pi$
$68$	−0.197175	−0.0239110
$69$	0	0
$70$	2.34420	0.280185
$71$	2.61034	0.309790	0.154895	−	0.987931i	$-0.450496\pi$
$71$	2.61034	0.309790	0.154895	+	0.987931i	$0.450496\pi$
$72$	0	0
$73$	4.04118	0.472985	0.236492	−	0.971633i	$-0.424002\pi$
$73$	4.04118	0.472985	0.236492	+	0.971633i	$0.424002\pi$
$74$	−6.75434	−0.785176
$75$	0	0
$76$	−6.64024	−0.761688
$77$	2.94059	0.335111
$78$	0	0
$79$	−13.0886	−1.47258	−0.736292	−	0.676664i	$-0.763425\pi$
$79$	−13.0886	−1.47258	−0.736292	+	0.676664i	$0.763425\pi$
$80$	0.515478	0.0576322
$81$	0	0
$82$	12.4131	1.37080
$83$	10.7430	1.17920	0.589599	−	0.807696i	$-0.299286\pi$
$83$	10.7430	1.17920	0.589599	+	0.807696i	$0.299286\pi$
$84$	0	0
$85$	−0.101639	−0.0110243
$86$	−5.56005	−0.599555
$87$	0	0
$88$	0.646622	0.0689302
$89$	12.1522	1.28813	0.644065	−	0.764971i	$-0.277247\pi$
$89$	12.1522	1.28813	0.644065	+	0.764971i	$0.277247\pi$
$90$	0	0
$91$	22.9255	2.40325
$92$	−6.07059	−0.632903
$93$	0	0
$94$	−10.8909	−1.12331
$95$	−3.42290	−0.351182
$96$	0	0
$97$	−9.57744	−0.972442	−0.486221	−	0.873836i	$-0.661625\pi$
$97$	−9.57744	−0.972442	−0.486221	+	0.873836i	$0.661625\pi$
$98$	−13.6808	−1.38197
$99$	0	0
$100$	−4.73428	−0.473428
$101$	−15.2726	−1.51968	−0.759838	−	0.650112i	$-0.774722\pi$
$101$	−15.2726	−1.51968	−0.759838	+	0.650112i	$0.774722\pi$
$102$	0	0
$103$	19.0993	1.88191	0.940954	−	0.338535i	$-0.109931\pi$
$103$	19.0993	1.88191	0.940954	+	0.338535i	$0.109931\pi$
$104$	5.04122	0.494333
$105$	0	0
$106$	10.5761	1.02724
$107$	−1.09762	−0.106111	−0.0530556	−	0.998592i	$-0.516896\pi$
$107$	−1.09762	−0.106111	−0.0530556	+	0.998592i	$0.516896\pi$
$108$	0	0
$109$	−9.93839	−0.951925	−0.475963	−	0.879465i	$-0.657900\pi$
$109$	−9.93839	−0.951925	−0.475963	+	0.879465i	$0.657900\pi$
$110$	0.333319	0.0317808
$111$	0	0
$112$	−4.54762	−0.429709
$113$	13.2480	1.24626	0.623131	−	0.782117i	$-0.285860\pi$
$113$	13.2480	1.24626	0.623131	+	0.782117i	$0.285860\pi$
$114$	0	0
$115$	−3.12925	−0.291804
$116$	−2.46368	−0.228747
$117$	0	0
$118$	12.4961	1.15036
$119$	0.896676	0.0821981
$120$	0	0
$121$	−10.5819	−0.961989
$122$	−8.74339	−0.791589
$123$	0	0
$124$	4.70580	0.422593
$125$	−5.01781	−0.448806
$126$	0	0
$127$	−0.269090	−0.0238778	−0.0119389	−	0.999929i	$-0.503800\pi$
$127$	−0.269090	−0.0238778	−0.0119389	+	0.999929i	$0.503800\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	2.59864	0.227916
$131$	17.4788	1.52713	0.763565	−	0.645731i	$-0.223447\pi$
$131$	17.4788	1.52713	0.763565	+	0.645731i	$0.223447\pi$
$132$	0	0
$133$	30.1973	2.61844
$134$	−12.5544	−1.08454
$135$	0	0
$136$	0.197175	0.0169076
$137$	−19.0418	−1.62685	−0.813425	−	0.581670i	$-0.802400\pi$
$137$	−19.0418	−1.62685	−0.813425	+	0.581670i	$0.802400\pi$
$138$	0	0
$139$	0.925657	0.0785132	0.0392566	−	0.999229i	$-0.487501\pi$
$139$	0.925657	0.0785132	0.0392566	+	0.999229i	$0.487501\pi$
$140$	−2.34420	−0.198121
$141$	0	0
$142$	−2.61034	−0.219055
$143$	3.25977	0.272595
$144$	0	0
$145$	−1.26997	−0.105466
$146$	−4.04118	−0.334451
$147$	0	0
$148$	6.75434	0.555203
$149$	9.09980	0.745485	0.372742	−	0.927935i	$-0.378418\pi$
$149$	9.09980	0.745485	0.372742	+	0.927935i	$0.378418\pi$
$150$	0	0
$151$	2.98407	0.242840	0.121420	−	0.992601i	$-0.461255\pi$
$151$	2.98407	0.242840	0.121420	+	0.992601i	$0.461255\pi$
$152$	6.64024	0.538595
$153$	0	0
$154$	−2.94059	−0.236959
$155$	2.42574	0.194840
$156$	0	0
$157$	1.57112	0.125389	0.0626947	−	0.998033i	$-0.480031\pi$
$157$	1.57112	0.125389	0.0626947	+	0.998033i	$0.480031\pi$
$158$	13.0886	1.04127
$159$	0	0
$160$	−0.515478	−0.0407521
$161$	27.6067	2.17571
$162$	0	0
$163$	5.81182	0.455217	0.227608	−	0.973753i	$-0.426909\pi$
$163$	5.81182	0.455217	0.227608	+	0.973753i	$0.426909\pi$
$164$	−12.4131	−0.969303
$165$	0	0
$166$	−10.7430	−0.833818
$167$	1.78569	0.138181	0.0690903	−	0.997610i	$-0.477990\pi$
$167$	1.78569	0.138181	0.0690903	+	0.997610i	$0.477990\pi$
$168$	0	0
$169$	12.4139	0.954917
$170$	0.101639	0.00779538
$171$	0	0
$172$	5.56005	0.423950
$173$	−5.50216	−0.418321	−0.209161	−	0.977881i	$-0.567073\pi$
$173$	−5.50216	−0.418321	−0.209161	+	0.977881i	$0.567073\pi$
$174$	0	0
$175$	21.5297	1.62749
$176$	−0.646622	−0.0487410
$177$	0	0
$178$	−12.1522	−0.910846
$179$	13.3214	0.995686	0.497843	−	0.867267i	$-0.334126\pi$
$179$	13.3214	0.995686	0.497843	+	0.867267i	$0.334126\pi$
$180$	0	0
$181$	−4.70187	−0.349487	−0.174744	−	0.984614i	$-0.555910\pi$
$181$	−4.70187	−0.349487	−0.174744	+	0.984614i	$0.555910\pi$
$182$	−22.9255	−1.69935
$183$	0	0
$184$	6.07059	0.447530
$185$	3.48171	0.255981
$186$	0	0
$187$	0.127498	0.00932356
$188$	10.8909	0.794299
$189$	0	0
$190$	3.42290	0.248323
$191$	−13.2513	−0.958833	−0.479417	−	0.877587i	$-0.659152\pi$
$191$	−13.2513	−0.958833	−0.479417	+	0.877587i	$0.659152\pi$
$192$	0	0
$193$	20.5170	1.47685	0.738423	−	0.674338i	$-0.235571\pi$
$193$	20.5170	1.47685	0.738423	+	0.674338i	$0.235571\pi$
$194$	9.57744	0.687620
$195$	0	0
$196$	13.6808	0.977201
$197$	−8.86861	−0.631862	−0.315931	−	0.948782i	$-0.602317\pi$
$197$	−8.86861	−0.631862	−0.315931	+	0.948782i	$0.602317\pi$
$198$	0	0
$199$	6.26292	0.443967	0.221983	−	0.975050i	$-0.428747\pi$
$199$	6.26292	0.443967	0.221983	+	0.975050i	$0.428747\pi$
$200$	4.73428	0.334764
$201$	0	0
$202$	15.2726	1.07457
$203$	11.2039	0.786358
$204$	0	0
$205$	−6.39870	−0.446905
$206$	−19.0993	−1.33071
$207$	0	0
$208$	−5.04122	−0.349546
$209$	4.29373	0.297003
$210$	0	0
$211$	−17.3450	−1.19408	−0.597038	−	0.802213i	$-0.703656\pi$
$211$	−17.3450	−1.19408	−0.597038	+	0.802213i	$0.703656\pi$
$212$	−10.5761	−0.726368
$213$	0	0
$214$	1.09762	0.0750320
$215$	2.86608	0.195465
$216$	0	0
$217$	−21.4002	−1.45274
$218$	9.93839	0.673113
$219$	0	0
$220$	−0.333319	−0.0224724
$221$	0.994003	0.0668639
$222$	0	0
$223$	17.6322	1.18074	0.590369	−	0.807133i	$-0.298982\pi$
$223$	17.6322	1.18074	0.590369	+	0.807133i	$0.298982\pi$
$224$	4.54762	0.303850
$225$	0	0
$226$	−13.2480	−0.881241
$227$	−8.20255	−0.544422	−0.272211	−	0.962238i	$-0.587755\pi$
$227$	−8.20255	−0.544422	−0.272211	+	0.962238i	$0.587755\pi$
$228$	0	0
$229$	−7.30547	−0.482759	−0.241380	−	0.970431i	$-0.577600\pi$
$229$	−7.30547	−0.482759	−0.241380	+	0.970431i	$0.577600\pi$
$230$	3.12925	0.206337
$231$	0	0
$232$	2.46368	0.161749
$233$	−5.33860	−0.349743	−0.174872	−	0.984591i	$-0.555951\pi$
$233$	−5.33860	−0.349743	−0.174872	+	0.984591i	$0.555951\pi$
$234$	0	0
$235$	5.61401	0.366218
$236$	−12.4961	−0.813428
$237$	0	0
$238$	−0.896676	−0.0581229
$239$	0.0672071	0.00434727	0.00217363	−	0.999998i	$-0.499308\pi$
$239$	0.0672071	0.00434727	0.00217363	+	0.999998i	$0.499308\pi$
$240$	0	0
$241$	−11.4205	−0.735656	−0.367828	−	0.929894i	$-0.619898\pi$
$241$	−11.4205	−0.735656	−0.367828	+	0.929894i	$0.619898\pi$
$242$	10.5819	0.680229
$243$	0	0
$244$	8.74339	0.559738
$245$	7.05216	0.450546
$246$	0	0
$247$	33.4749	2.12996
$248$	−4.70580	−0.298819
$249$	0	0
$250$	5.01781	0.317354
$251$	−22.3307	−1.40950	−0.704752	−	0.709454i	$-0.748942\pi$
$251$	−22.3307	−1.40950	−0.704752	+	0.709454i	$0.748942\pi$
$252$	0	0
$253$	3.92538	0.246786
$254$	0.269090	0.0168842
$255$	0	0
$256$	1.00000	0.0625000
$257$	8.56194	0.534079	0.267040	−	0.963686i	$-0.413955\pi$
$257$	8.56194	0.534079	0.267040	+	0.963686i	$0.413955\pi$
$258$	0	0
$259$	−30.7161	−1.90861
$260$	−2.59864	−0.161161
$261$	0	0
$262$	−17.4788	−1.07984
$263$	−25.3781	−1.56488	−0.782441	−	0.622725i	$-0.786026\pi$
$263$	−25.3781	−1.56488	−0.782441	+	0.622725i	$0.786026\pi$
$264$	0	0
$265$	−5.45173	−0.334897
$266$	−30.1973	−1.85151
$267$	0	0
$268$	12.5544	0.766884
$269$	1.00000	0.0609711
$269$	1.00000	0.0609711
$270$	0	0
$271$	−8.76637	−0.532519	−0.266259	−	0.963901i	$-0.585788\pi$
$271$	−8.76637	−0.532519	−0.266259	+	0.963901i	$0.585788\pi$
$272$	−0.197175	−0.0119555
$273$	0	0
$274$	19.0418	1.15036
$275$	3.06129	0.184603
$276$	0	0
$277$	14.7263	0.884816	0.442408	−	0.896814i	$-0.354124\pi$
$277$	14.7263	0.884816	0.442408	+	0.896814i	$0.354124\pi$
$278$	−0.925657	−0.0555172
$279$	0	0
$280$	2.34420	0.140092
$281$	−20.6994	−1.23482	−0.617410	−	0.786641i	$-0.711818\pi$
$281$	−20.6994	−1.23482	−0.617410	+	0.786641i	$0.711818\pi$
$282$	0	0
$283$	7.14383	0.424657	0.212328	−	0.977198i	$-0.431895\pi$
$283$	7.14383	0.424657	0.212328	+	0.977198i	$0.431895\pi$
$284$	2.61034	0.154895
$285$	0	0
$286$	−3.25977	−0.192754
$287$	56.4502	3.33215
$288$	0	0
$289$	−16.9611	−0.997713
$290$	1.26997	0.0745754
$291$	0	0
$292$	4.04118	0.236492
$293$	15.8364	0.925170	0.462585	−	0.886575i	$-0.346922\pi$
$293$	15.8364	0.925170	0.462585	+	0.886575i	$0.346922\pi$
$294$	0	0
$295$	−6.44148	−0.375037
$296$	−6.75434	−0.392588
$297$	0	0
$298$	−9.09980	−0.527137
$299$	30.6032	1.76983
$300$	0	0
$301$	−25.2850	−1.45740
$302$	−2.98407	−0.171714
$303$	0	0
$304$	−6.64024	−0.380844
$305$	4.50702	0.258071
$306$	0	0
$307$	−1.78858	−0.102080	−0.0510398	−	0.998697i	$-0.516254\pi$
$307$	−1.78858	−0.102080	−0.0510398	+	0.998697i	$0.516254\pi$
$308$	2.94059	0.167556
$309$	0	0
$310$	−2.42574	−0.137773
$311$	32.7903	1.85937	0.929684	−	0.368358i	$-0.120080\pi$
$311$	32.7903	1.85937	0.929684	+	0.368358i	$0.120080\pi$
$312$	0	0
$313$	−8.23738	−0.465604	−0.232802	−	0.972524i	$-0.574789\pi$
$313$	−8.23738	−0.465604	−0.232802	+	0.972524i	$0.574789\pi$
$314$	−1.57112	−0.0886636
$315$	0	0
$316$	−13.0886	−0.736292
$317$	−20.6827	−1.16166	−0.580829	−	0.814026i	$-0.697271\pi$
$317$	−20.6827	−1.16166	−0.580829	+	0.814026i	$0.697271\pi$
$318$	0	0
$319$	1.59307	0.0891949
$320$	0.515478	0.0288161
$321$	0	0
$322$	−27.6067	−1.53846
$323$	1.30929	0.0728508
$324$	0	0
$325$	23.8666	1.32388
$326$	−5.81182	−0.321887
$327$	0	0
$328$	12.4131	0.685401
$329$	−49.5275	−2.73054
$330$	0	0
$331$	21.1281	1.16130	0.580652	−	0.814152i	$-0.302798\pi$
$331$	21.1281	1.16130	0.580652	+	0.814152i	$0.302798\pi$
$332$	10.7430	0.589599
$333$	0	0
$334$	−1.78569	−0.0977085
$335$	6.47154	0.353578
$336$	0	0
$337$	25.5481	1.39169	0.695846	−	0.718191i	$-0.255030\pi$
$337$	25.5481	1.39169	0.695846	+	0.718191i	$0.255030\pi$
$338$	−12.4139	−0.675229
$339$	0	0
$340$	−0.101639	−0.00551217
$341$	−3.04288	−0.164781
$342$	0	0
$343$	−30.3818	−1.64046
$344$	−5.56005	−0.299778
$345$	0	0
$346$	5.50216	0.295798
$347$	23.2178	1.24640	0.623198	−	0.782064i	$-0.285833\pi$
$347$	23.2178	1.24640	0.623198	+	0.782064i	$0.285833\pi$
$348$	0	0
$349$	−8.33926	−0.446390	−0.223195	−	0.974774i	$-0.571649\pi$
$349$	−8.33926	−0.446390	−0.223195	+	0.974774i	$0.571649\pi$
$350$	−21.5297	−1.15081
$351$	0	0
$352$	0.646622	0.0344651
$353$	−19.3248	−1.02856	−0.514278	−	0.857624i	$-0.671940\pi$
$353$	−19.3248	−1.02856	−0.514278	+	0.857624i	$0.671940\pi$
$354$	0	0
$355$	1.34557	0.0714156
$356$	12.1522	0.644065
$357$	0	0
$358$	−13.3214	−0.704057
$359$	−26.4164	−1.39420	−0.697102	−	0.716972i	$-0.745528\pi$
$359$	−26.4164	−1.39420	−0.697102	+	0.716972i	$0.745528\pi$
$360$	0	0
$361$	25.0928	1.32067
$362$	4.70187	0.247125
$363$	0	0
$364$	22.9255	1.20163
$365$	2.08314	0.109037
$366$	0	0
$367$	−11.4430	−0.597322	−0.298661	−	0.954359i	$-0.596540\pi$
$367$	−11.4430	−0.597322	−0.298661	+	0.954359i	$0.596540\pi$
$368$	−6.07059	−0.316451
$369$	0	0
$370$	−3.48171	−0.181006
$371$	48.0959	2.49702
$372$	0	0
$373$	29.1222	1.50789	0.753944	−	0.656938i	$-0.228149\pi$
$373$	29.1222	1.50789	0.753944	+	0.656938i	$0.228149\pi$
$374$	−0.127498	−0.00659275
$375$	0	0
$376$	−10.8909	−0.561654
$377$	12.4200	0.639661
$378$	0	0
$379$	−0.256508	−0.0131759	−0.00658797	−	0.999978i	$-0.502097\pi$
$379$	−0.256508	−0.0131759	−0.00658797	+	0.999978i	$0.502097\pi$
$380$	−3.42290	−0.175591
$381$	0	0
$382$	13.2513	0.677997
$383$	−17.7921	−0.909132	−0.454566	−	0.890713i	$-0.650206\pi$
$383$	−17.7921	−0.909132	−0.454566	+	0.890713i	$0.650206\pi$
$384$	0	0
$385$	1.51581	0.0772528
$386$	−20.5170	−1.04429
$387$	0	0
$388$	−9.57744	−0.486221
$389$	−24.3756	−1.23589	−0.617945	−	0.786221i	$-0.712035\pi$
$389$	−24.3756	−1.23589	−0.617945	+	0.786221i	$0.712035\pi$
$390$	0	0
$391$	1.19697	0.0605333
$392$	−13.6808	−0.690985
$393$	0	0
$394$	8.86861	0.446794
$395$	−6.74689	−0.339473
$396$	0	0
$397$	6.54642	0.328555	0.164278	−	0.986414i	$-0.447471\pi$
$397$	6.54642	0.328555	0.164278	+	0.986414i	$0.447471\pi$
$398$	−6.26292	−0.313932
$399$	0	0
$400$	−4.73428	−0.236714
$401$	−35.6532	−1.78044	−0.890218	−	0.455534i	$-0.849448\pi$
$401$	−35.6532	−1.78044	−0.890218	+	0.455534i	$0.849448\pi$
$402$	0	0
$403$	−23.7230	−1.18173
$404$	−15.2726	−0.759838
$405$	0	0
$406$	−11.2039	−0.556039
$407$	−4.36750	−0.216489
$408$	0	0
$409$	−7.24483	−0.358234	−0.179117	−	0.983828i	$-0.557324\pi$
$409$	−7.24483	−0.358234	−0.179117	+	0.983828i	$0.557324\pi$
$410$	6.39870	0.316009
$411$	0	0
$412$	19.0993	0.940954
$413$	56.8276	2.79630
$414$	0	0
$415$	5.53778	0.271839
$416$	5.04122	0.247166
$417$	0	0
$418$	−4.29373	−0.210013
$419$	31.0240	1.51562	0.757810	−	0.652475i	$-0.226269\pi$
$419$	31.0240	1.51562	0.757810	+	0.652475i	$0.226269\pi$
$420$	0	0
$421$	5.11144	0.249116	0.124558	−	0.992212i	$-0.460249\pi$
$421$	5.11144	0.249116	0.124558	+	0.992212i	$0.460249\pi$
$422$	17.3450	0.844339
$423$	0	0
$424$	10.5761	0.513620
$425$	0.933482	0.0452805
$426$	0	0
$427$	−39.7616	−1.92420
$428$	−1.09762	−0.0530556
$429$	0	0
$430$	−2.86608	−0.138215
$431$	25.5427	1.23035	0.615173	−	0.788392i	$-0.289086\pi$
$431$	25.5427	1.23035	0.615173	+	0.788392i	$0.289086\pi$
$432$	0	0
$433$	20.6432	0.992047	0.496024	−	0.868309i	$-0.334793\pi$
$433$	20.6432	0.992047	0.496024	+	0.868309i	$0.334793\pi$
$434$	21.4002	1.02724
$435$	0	0
$436$	−9.93839	−0.475963
$437$	40.3102	1.92830
$438$	0	0
$439$	−34.9387	−1.66753	−0.833767	−	0.552117i	$-0.813820\pi$
$439$	−34.9387	−1.66753	−0.833767	+	0.552117i	$0.813820\pi$
$440$	0.333319	0.0158904
$441$	0	0
$442$	−0.994003	−0.0472799
$443$	18.3591	0.872267	0.436133	−	0.899882i	$-0.356348\pi$
$443$	18.3591	0.872267	0.436133	+	0.899882i	$0.356348\pi$
$444$	0	0
$445$	6.26419	0.296951
$446$	−17.6322	−0.834908
$447$	0	0
$448$	−4.54762	−0.214855
$449$	7.17048	0.338396	0.169198	−	0.985582i	$-0.445882\pi$
$449$	7.17048	0.338396	0.169198	+	0.985582i	$0.445882\pi$
$450$	0	0
$451$	8.02661	0.377958
$452$	13.2480	0.623131
$453$	0	0
$454$	8.20255	0.384965
$455$	11.8176	0.554018
$456$	0	0
$457$	15.6440	0.731796	0.365898	−	0.930655i	$-0.380762\pi$
$457$	15.6440	0.731796	0.365898	+	0.930655i	$0.380762\pi$
$458$	7.30547	0.341362
$459$	0	0
$460$	−3.12925	−0.145902
$461$	−8.75585	−0.407800	−0.203900	−	0.978992i	$-0.565362\pi$
$461$	−8.75585	−0.407800	−0.203900	+	0.978992i	$0.565362\pi$
$462$	0	0
$463$	19.5927	0.910552	0.455276	−	0.890350i	$-0.349541\pi$
$463$	19.5927	0.910552	0.455276	+	0.890350i	$0.349541\pi$
$464$	−2.46368	−0.114374
$465$	0	0
$466$	5.33860	0.247306
$467$	25.7091	1.18967	0.594837	−	0.803847i	$-0.297217\pi$
$467$	25.7091	1.18967	0.594837	+	0.803847i	$0.297217\pi$
$468$	0	0
$469$	−57.0928	−2.63630
$470$	−5.61401	−0.258955
$471$	0	0
$472$	12.4961	0.575181
$473$	−3.59525	−0.165310
$474$	0	0
$475$	31.4368	1.44242
$476$	0.896676	0.0410991
$477$	0	0
$478$	−0.0672071	−0.00307398
$479$	−9.00827	−0.411598	−0.205799	−	0.978594i	$-0.565979\pi$
$479$	−9.00827	−0.411598	−0.205799	+	0.978594i	$0.565979\pi$
$480$	0	0
$481$	−34.0501	−1.55255
$482$	11.4205	0.520187
$483$	0	0
$484$	−10.5819	−0.480995
$485$	−4.93696	−0.224176
$486$	0	0
$487$	33.1458	1.50198	0.750989	−	0.660315i	$-0.229577\pi$
$487$	33.1458	1.50198	0.750989	+	0.660315i	$0.229577\pi$
$488$	−8.74339	−0.395795
$489$	0	0
$490$	−7.05216	−0.318584
$491$	38.0044	1.71511	0.857557	−	0.514390i	$-0.171981\pi$
$491$	38.0044	1.71511	0.857557	+	0.514390i	$0.171981\pi$
$492$	0	0
$493$	0.485776	0.0218783
$494$	−33.4749	−1.50611
$495$	0	0
$496$	4.70580	0.211297
$497$	−11.8708	−0.532479
$498$	0	0
$499$	11.6632	0.522117	0.261059	−	0.965323i	$-0.415928\pi$
$499$	11.6632	0.522117	0.261059	+	0.965323i	$0.415928\pi$
$500$	−5.01781	−0.224403
$501$	0	0
$502$	22.3307	0.996670
$503$	17.6725	0.787977	0.393989	−	0.919115i	$-0.371095\pi$
$503$	17.6725	0.787977	0.393989	+	0.919115i	$0.371095\pi$
$504$	0	0
$505$	−7.87267	−0.350329
$506$	−3.92538	−0.174504
$507$	0	0
$508$	−0.269090	−0.0119389
$509$	−22.3438	−0.990370	−0.495185	−	0.868788i	$-0.664900\pi$
$509$	−22.3438	−0.990370	−0.495185	+	0.868788i	$0.664900\pi$
$510$	0	0
$511$	−18.3778	−0.812984
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	−8.56194	−0.377651
$515$	9.84526	0.433834
$516$	0	0
$517$	−7.04229	−0.309719
$518$	30.7161	1.34959
$519$	0	0
$520$	2.59864	0.113958
$521$	15.1174	0.662307	0.331153	−	0.943577i	$-0.392562\pi$
$521$	15.1174	0.662307	0.331153	+	0.943577i	$0.392562\pi$
$522$	0	0
$523$	−38.2974	−1.67463	−0.837313	−	0.546723i	$-0.815875\pi$
$523$	−38.2974	−1.67463	−0.837313	+	0.546723i	$0.815875\pi$
$524$	17.4788	0.763565
$525$	0	0
$526$	25.3781	1.10654
$527$	−0.927866	−0.0404185
$528$	0	0
$529$	13.8520	0.602263
$530$	5.45173	0.236808
$531$	0	0
$532$	30.1973	1.30922
$533$	62.5774	2.71053
$534$	0	0
$535$	−0.565800	−0.0244617
$536$	−12.5544	−0.542269
$537$	0	0
$538$	−1.00000	−0.0431131
$539$	−8.84632	−0.381038
$540$	0	0
$541$	25.1246	1.08019	0.540095	−	0.841604i	$-0.318388\pi$
$541$	25.1246	1.08019	0.540095	+	0.841604i	$0.318388\pi$
$542$	8.76637	0.376548
$543$	0	0
$544$	0.197175	0.00845381
$545$	−5.12302	−0.219446
$546$	0	0
$547$	−10.7226	−0.458466	−0.229233	−	0.973372i	$-0.573622\pi$
$547$	−10.7226	−0.458466	−0.229233	+	0.973372i	$0.573622\pi$
$548$	−19.0418	−0.813425
$549$	0	0
$550$	−3.06129	−0.130534
$551$	16.3594	0.696936
$552$	0	0
$553$	59.5220	2.53113
$554$	−14.7263	−0.625659
$555$	0	0
$556$	0.925657	0.0392566
$557$	33.4302	1.41648	0.708241	−	0.705971i	$-0.249489\pi$
$557$	33.4302	1.41648	0.708241	+	0.705971i	$0.249489\pi$
$558$	0	0
$559$	−28.0294	−1.18552
$560$	−2.34420	−0.0990603
$561$	0	0
$562$	20.6994	0.873150
$563$	−39.7267	−1.67428	−0.837141	−	0.546987i	$-0.815775\pi$
$563$	−39.7267	−1.67428	−0.837141	+	0.546987i	$0.815775\pi$
$564$	0	0
$565$	6.82903	0.287299
$566$	−7.14383	−0.300278
$567$	0	0
$568$	−2.61034	−0.109527
$569$	28.6505	1.20109	0.600545	−	0.799591i	$-0.294950\pi$
$569$	28.6505	1.20109	0.600545	+	0.799591i	$0.294950\pi$
$570$	0	0
$571$	23.8663	0.998773	0.499386	−	0.866379i	$-0.333559\pi$
$571$	23.8663	0.998773	0.499386	+	0.866379i	$0.333559\pi$
$572$	3.25977	0.136298
$573$	0	0
$574$	−56.4502	−2.35619
$575$	28.7399	1.19854
$576$	0	0
$577$	44.3037	1.84439	0.922194	−	0.386727i	$-0.126395\pi$
$577$	44.3037	1.84439	0.922194	+	0.386727i	$0.126395\pi$
$578$	16.9611	0.705490
$579$	0	0
$580$	−1.26997	−0.0527328
$581$	−48.8550	−2.02685
$582$	0	0
$583$	6.83873	0.283231
$584$	−4.04118	−0.167225
$585$	0	0
$586$	−15.8364	−0.654194
$587$	−29.0162	−1.19763	−0.598814	−	0.800888i	$-0.704361\pi$
$587$	−29.0162	−1.19763	−0.598814	+	0.800888i	$0.704361\pi$
$588$	0	0
$589$	−31.2477	−1.28754
$590$	6.44148	0.265191
$591$	0	0
$592$	6.75434	0.277602
$593$	40.8507	1.67754	0.838768	−	0.544489i	$-0.183276\pi$
$593$	40.8507	1.67754	0.838768	+	0.544489i	$0.183276\pi$
$594$	0	0
$595$	0.462217	0.0189490
$596$	9.09980	0.372742
$597$	0	0
$598$	−30.6032	−1.25146
$599$	15.5327	0.634648	0.317324	−	0.948317i	$-0.397216\pi$
$599$	15.5327	0.634648	0.317324	+	0.948317i	$0.397216\pi$
$600$	0	0
$601$	40.8195	1.66506	0.832532	−	0.553977i	$-0.186890\pi$
$601$	40.8195	1.66506	0.832532	+	0.553977i	$0.186890\pi$
$602$	25.2850	1.03054
$603$	0	0
$604$	2.98407	0.121420
$605$	−5.45473	−0.221766
$606$	0	0
$607$	−17.1079	−0.694389	−0.347194	−	0.937793i	$-0.612866\pi$
$607$	−17.1079	−0.694389	−0.347194	+	0.937793i	$0.612866\pi$
$608$	6.64024	0.269297
$609$	0	0
$610$	−4.50702	−0.182484
$611$	−54.9034	−2.22115
$612$	0	0
$613$	−38.1661	−1.54151	−0.770756	−	0.637130i	$-0.780121\pi$
$613$	−38.1661	−1.54151	−0.770756	+	0.637130i	$0.780121\pi$
$614$	1.78858	0.0721812
$615$	0	0
$616$	−2.94059	−0.118480
$617$	−15.7366	−0.633531	−0.316766	−	0.948504i	$-0.602597\pi$
$617$	−15.7366	−0.633531	−0.316766	+	0.948504i	$0.602597\pi$
$618$	0	0
$619$	−1.66495	−0.0669201	−0.0334601	−	0.999440i	$-0.510653\pi$
$619$	−1.66495	−0.0669201	−0.0334601	+	0.999440i	$0.510653\pi$
$620$	2.42574	0.0974199
$621$	0	0
$622$	−32.7903	−1.31477
$623$	−55.2635	−2.21409
$624$	0	0
$625$	21.0848	0.843394
$626$	8.23738	0.329232
$627$	0	0
$628$	1.57112	0.0626947
$629$	−1.33179	−0.0531018
$630$	0	0
$631$	31.1359	1.23950	0.619751	−	0.784798i	$-0.287233\pi$
$631$	31.1359	1.23950	0.619751	+	0.784798i	$0.287233\pi$
$632$	13.0886	0.520637
$633$	0	0
$634$	20.6827	0.821416
$635$	−0.138710	−0.00550453
$636$	0	0
$637$	−68.9680	−2.73261
$638$	−1.59307	−0.0630703
$639$	0	0
$640$	−0.515478	−0.0203761
$641$	6.63337	0.262002	0.131001	−	0.991382i	$-0.458181\pi$
$641$	6.63337	0.262002	0.131001	+	0.991382i	$0.458181\pi$
$642$	0	0
$643$	−11.8643	−0.467884	−0.233942	−	0.972251i	$-0.575163\pi$
$643$	−11.8643	−0.467884	−0.233942	+	0.972251i	$0.575163\pi$
$644$	27.6067	1.08786
$645$	0	0
$646$	−1.30929	−0.0515133
$647$	3.22521	0.126796	0.0633981	−	0.997988i	$-0.479806\pi$
$647$	3.22521	0.126796	0.0633981	+	0.997988i	$0.479806\pi$
$648$	0	0
$649$	8.08027	0.317178
$650$	−23.8666	−0.936124
$651$	0	0
$652$	5.81182	0.227608
$653$	−32.6756	−1.27870	−0.639348	−	0.768918i	$-0.720795\pi$
$653$	−32.6756	−1.27870	−0.639348	+	0.768918i	$0.720795\pi$
$654$	0	0
$655$	9.00993	0.352047
$656$	−12.4131	−0.484652
$657$	0	0
$658$	49.5275	1.93078
$659$	21.9028	0.853213	0.426606	−	0.904437i	$-0.359709\pi$
$659$	21.9028	0.853213	0.426606	+	0.904437i	$0.359709\pi$
$660$	0	0
$661$	23.0216	0.895436	0.447718	−	0.894175i	$-0.352237\pi$
$661$	23.0216	0.895436	0.447718	+	0.894175i	$0.352237\pi$
$662$	−21.1281	−0.821167
$663$	0	0
$664$	−10.7430	−0.416909
$665$	15.5660	0.603624
$666$	0	0
$667$	14.9560	0.579099
$668$	1.78569	0.0690903
$669$	0	0
$670$	−6.47154	−0.250017
$671$	−5.65367	−0.218257
$672$	0	0
$673$	−22.8348	−0.880216	−0.440108	−	0.897945i	$-0.645060\pi$
$673$	−22.8348	−0.880216	−0.440108	+	0.897945i	$0.645060\pi$
$674$	−25.5481	−0.984075
$675$	0	0
$676$	12.4139	0.477459
$677$	−36.3791	−1.39816	−0.699081	−	0.715042i	$-0.746408\pi$
$677$	−36.3791	−1.39816	−0.699081	+	0.715042i	$0.746408\pi$
$678$	0	0
$679$	43.5545	1.67147
$680$	0.101639	0.00389769
$681$	0	0
$682$	3.04288	0.116518
$683$	−5.64183	−0.215879	−0.107939	−	0.994157i	$-0.534425\pi$
$683$	−5.64183	−0.215879	−0.107939	+	0.994157i	$0.534425\pi$
$684$	0	0
$685$	−9.81562	−0.375036
$686$	30.3818	1.15998
$687$	0	0
$688$	5.56005	0.211975
$689$	53.3164	2.03119
$690$	0	0
$691$	0.339381	0.0129107	0.00645533	−	0.999979i	$-0.497945\pi$
$691$	0.339381	0.0129107	0.00645533	+	0.999979i	$0.497945\pi$
$692$	−5.50216	−0.209161
$693$	0	0
$694$	−23.2178	−0.881336
$695$	0.477156	0.0180995
$696$	0	0
$697$	2.44756	0.0927079
$698$	8.33926	0.315646
$699$	0	0
$700$	21.5297	0.813746
$701$	−35.5085	−1.34114	−0.670569	−	0.741847i	$-0.733950\pi$
$701$	−35.5085	−1.34114	−0.670569	+	0.741847i	$0.733950\pi$
$702$	0	0
$703$	−44.8504	−1.69157
$704$	−0.646622	−0.0243705
$705$	0	0
$706$	19.3248	0.727299
$707$	69.4537	2.61208
$708$	0	0
$709$	−28.8982	−1.08529	−0.542647	−	0.839961i	$-0.682578\pi$
$709$	−28.8982	−1.08529	−0.542647	+	0.839961i	$0.682578\pi$
$710$	−1.34557	−0.0504984
$711$	0	0
$712$	−12.1522	−0.455423
$713$	−28.5670	−1.06984
$714$	0	0
$715$	1.68034	0.0628411
$716$	13.3214	0.497843
$717$	0	0
$718$	26.4164	0.985851
$719$	−29.9219	−1.11590	−0.557950	−	0.829875i	$-0.688412\pi$
$719$	−29.9219	−1.11590	−0.557950	+	0.829875i	$0.688412\pi$
$720$	0	0
$721$	−86.8562	−3.23469
$722$	−25.0928	−0.933857
$723$	0	0
$724$	−4.70187	−0.174744
$725$	11.6638	0.433181
$726$	0	0
$727$	−4.86802	−0.180545	−0.0902724	−	0.995917i	$-0.528774\pi$
$727$	−4.86802	−0.180545	−0.0902724	+	0.995917i	$0.528774\pi$
$728$	−22.9255	−0.849677
$729$	0	0
$730$	−2.08314	−0.0771005
$731$	−1.09630	−0.0405482
$732$	0	0
$733$	44.8714	1.65736	0.828681	−	0.559721i	$-0.189092\pi$
$733$	44.8714	1.65736	0.828681	+	0.559721i	$0.189092\pi$
$734$	11.4430	0.422371
$735$	0	0
$736$	6.07059	0.223765
$737$	−8.11798	−0.299030
$738$	0	0
$739$	24.9297	0.917053	0.458526	−	0.888681i	$-0.348377\pi$
$739$	24.9297	0.917053	0.458526	+	0.888681i	$0.348377\pi$
$740$	3.48171	0.127990
$741$	0	0
$742$	−48.0959	−1.76566
$743$	10.1479	0.372289	0.186144	−	0.982522i	$-0.440401\pi$
$743$	10.1479	0.372289	0.186144	+	0.982522i	$0.440401\pi$
$744$	0	0
$745$	4.69075	0.171856
$746$	−29.1222	−1.06624
$747$	0	0
$748$	0.127498	0.00466178
$749$	4.99157	0.182388
$750$	0	0
$751$	49.0715	1.79065	0.895323	−	0.445418i	$-0.146945\pi$
$751$	49.0715	1.79065	0.895323	+	0.445418i	$0.146945\pi$
$752$	10.8909	0.397150
$753$	0	0
$754$	−12.4200	−0.452309
$755$	1.53822	0.0559817
$756$	0	0
$757$	0.698550	0.0253892	0.0126946	−	0.999919i	$-0.495959\pi$
$757$	0.698550	0.0253892	0.0126946	+	0.999919i	$0.495959\pi$
$758$	0.256508	0.00931680
$759$	0	0
$760$	3.42290	0.124162
$761$	7.36467	0.266969	0.133485	−	0.991051i	$-0.457383\pi$
$761$	7.36467	0.266969	0.133485	+	0.991051i	$0.457383\pi$
$762$	0	0
$763$	45.1960	1.63621
$764$	−13.2513	−0.479417
$765$	0	0
$766$	17.7921	0.642853
$767$	62.9957	2.27464
$768$	0	0
$769$	13.0544	0.470755	0.235377	−	0.971904i	$-0.424367\pi$
$769$	13.0544	0.470755	0.235377	+	0.971904i	$0.424367\pi$
$770$	−1.51581	−0.0546260
$771$	0	0
$772$	20.5170	0.738423
$773$	−17.6944	−0.636423	−0.318211	−	0.948020i	$-0.603082\pi$
$773$	−17.6944	−0.636423	−0.318211	+	0.948020i	$0.603082\pi$
$774$	0	0
$775$	−22.2786	−0.800271
$776$	9.57744	0.343810
$777$	0	0
$778$	24.3756	0.873906
$779$	82.4262	2.95323
$780$	0	0
$781$	−1.68790	−0.0603980
$782$	−1.19697	−0.0428035
$783$	0	0
$784$	13.6808	0.488600
$785$	0.809880	0.0289058
$786$	0	0
$787$	27.0542	0.964379	0.482190	−	0.876067i	$-0.339842\pi$
$787$	27.0542	0.964379	0.482190	+	0.876067i	$0.339842\pi$
$788$	−8.86861	−0.315931
$789$	0	0
$790$	6.74689	0.240044
$791$	−60.2466	−2.14212
$792$	0	0
$793$	−44.0774	−1.56523
$794$	−6.54642	−0.232324
$795$	0	0
$796$	6.26292	0.221983
$797$	−8.09022	−0.286570	−0.143285	−	0.989681i	$-0.545767\pi$
$797$	−8.09022	−0.286570	−0.143285	+	0.989681i	$0.545767\pi$
$798$	0	0
$799$	−2.14741	−0.0759699
$800$	4.73428	0.167382
$801$	0	0
$802$	35.6532	1.25896
$803$	−2.61312	−0.0922150
$804$	0	0
$805$	14.2306	0.501564
$806$	23.7230	0.835607
$807$	0	0
$808$	15.2726	0.537287
$809$	−42.3580	−1.48923	−0.744615	−	0.667495i	$-0.767367\pi$
$809$	−42.3580	−1.48923	−0.744615	+	0.667495i	$0.767367\pi$
$810$	0	0
$811$	34.4740	1.21055	0.605273	−	0.796018i	$-0.293064\pi$
$811$	34.4740	1.21055	0.605273	+	0.796018i	$0.293064\pi$
$812$	11.2039	0.393179
$813$	0	0
$814$	4.36750	0.153081
$815$	2.99586	0.104940
$816$	0	0
$817$	−36.9200	−1.29167
$818$	7.24483	0.253309
$819$	0	0
$820$	−6.39870	−0.223452
$821$	0.0792741	0.00276669	0.00138334	−	0.999999i	$-0.499560\pi$
$821$	0.0792741	0.00276669	0.00138334	+	0.999999i	$0.499560\pi$
$822$	0	0
$823$	−53.2780	−1.85716	−0.928578	−	0.371138i	$-0.878968\pi$
$823$	−53.2780	−1.85716	−0.928578	+	0.371138i	$0.878968\pi$
$824$	−19.0993	−0.665355
$825$	0	0
$826$	−56.8276	−1.97728
$827$	−32.4912	−1.12983	−0.564915	−	0.825149i	$-0.691091\pi$
$827$	−32.4912	−1.12983	−0.564915	+	0.825149i	$0.691091\pi$
$828$	0	0
$829$	24.2734	0.843049	0.421525	−	0.906817i	$-0.361495\pi$
$829$	24.2734	0.843049	0.421525	+	0.906817i	$0.361495\pi$
$830$	−5.53778	−0.192219
$831$	0	0
$832$	−5.04122	−0.174773
$833$	−2.69751	−0.0934633
$834$	0	0
$835$	0.920482	0.0318546
$836$	4.29373	0.148502
$837$	0	0
$838$	−31.0240	−1.07171
$839$	−39.4787	−1.36296	−0.681478	−	0.731838i	$-0.738663\pi$
$839$	−39.4787	−1.36296	−0.681478	+	0.731838i	$0.738663\pi$
$840$	0	0
$841$	−22.9303	−0.790699
$842$	−5.11144	−0.176152
$843$	0	0
$844$	−17.3450	−0.597038
$845$	6.39910	0.220136
$846$	0	0
$847$	48.1223	1.65350
$848$	−10.5761	−0.363184
$849$	0	0
$850$	−0.933482	−0.0320182
$851$	−41.0028	−1.40556
$852$	0	0
$853$	−11.8518	−0.405799	−0.202900	−	0.979200i	$-0.565036\pi$
$853$	−11.8518	−0.405799	−0.202900	+	0.979200i	$0.565036\pi$
$854$	39.7616	1.36061
$855$	0	0
$856$	1.09762	0.0375160
$857$	14.8008	0.505586	0.252793	−	0.967520i	$-0.418651\pi$
$857$	14.8008	0.505586	0.252793	+	0.967520i	$0.418651\pi$
$858$	0	0
$859$	−40.1831	−1.37103	−0.685514	−	0.728059i	$-0.740423\pi$
$859$	−40.1831	−1.37103	−0.685514	+	0.728059i	$0.740423\pi$
$860$	2.86608	0.0977325
$861$	0	0
$862$	−25.5427	−0.869986
$863$	−0.337706	−0.0114957	−0.00574783	−	0.999983i	$-0.501830\pi$
$863$	−0.337706	−0.0114957	−0.00574783	+	0.999983i	$0.501830\pi$
$864$	0	0
$865$	−2.83624	−0.0964351
$866$	−20.6432	−0.701483
$867$	0	0
$868$	−21.4002	−0.726369
$869$	8.46339	0.287101
$870$	0	0
$871$	−63.2897	−2.14449
$872$	9.93839	0.336556
$873$	0	0
$874$	−40.3102	−1.36351
$875$	22.8191	0.771425
$876$	0	0
$877$	33.4043	1.12798	0.563992	−	0.825780i	$-0.309265\pi$
$877$	33.4043	1.12798	0.563992	+	0.825780i	$0.309265\pi$
$878$	34.9387	1.17912
$879$	0	0
$880$	−0.333319	−0.0112362
$881$	33.0905	1.11485	0.557424	−	0.830228i	$-0.311790\pi$
$881$	33.0905	1.11485	0.557424	+	0.830228i	$0.311790\pi$
$882$	0	0
$883$	−51.6307	−1.73751	−0.868756	−	0.495241i	$-0.835080\pi$
$883$	−51.6307	−1.73751	−0.868756	+	0.495241i	$0.835080\pi$
$884$	0.994003	0.0334319
$885$	0	0
$886$	−18.3591	−0.616786
$887$	−25.4802	−0.855542	−0.427771	−	0.903887i	$-0.640701\pi$
$887$	−25.4802	−0.855542	−0.427771	+	0.903887i	$0.640701\pi$
$888$	0	0
$889$	1.22372	0.0410421
$890$	−6.26419	−0.209976
$891$	0	0
$892$	17.6322	0.590369
$893$	−72.3181	−2.42003
$894$	0	0
$895$	6.86687	0.229534
$896$	4.54762	0.151925
$897$	0	0
$898$	−7.17048	−0.239282
$899$	−11.5936	−0.386668
$900$	0	0
$901$	2.08534	0.0694727
$902$	−8.02661	−0.267257
$903$	0	0
$904$	−13.2480	−0.440620
$905$	−2.42371	−0.0805669
$906$	0	0
$907$	46.7342	1.55178	0.775892	−	0.630866i	$-0.217300\pi$
$907$	46.7342	1.55178	0.775892	+	0.630866i	$0.217300\pi$
$908$	−8.20255	−0.272211
$909$	0	0
$910$	−11.8176	−0.391750
$911$	15.3274	0.507821	0.253910	−	0.967228i	$-0.418283\pi$
$911$	15.3274	0.507821	0.253910	+	0.967228i	$0.418283\pi$
$912$	0	0
$913$	−6.94666	−0.229901
$914$	−15.6440	−0.517458
$915$	0	0
$916$	−7.30547	−0.241380
$917$	−79.4868	−2.62489
$918$	0	0
$919$	−56.6646	−1.86919	−0.934597	−	0.355709i	$-0.884239\pi$
$919$	−56.6646	−1.86919	−0.934597	+	0.355709i	$0.884239\pi$
$920$	3.12925	0.103168
$921$	0	0
$922$	8.75585	0.288358
$923$	−13.1593	−0.433144
$924$	0	0
$925$	−31.9769	−1.05140
$926$	−19.5927	−0.643858
$927$	0	0
$928$	2.46368	0.0808743
$929$	−27.7882	−0.911703	−0.455851	−	0.890056i	$-0.650665\pi$
$929$	−27.7882	−0.911703	−0.455851	+	0.890056i	$0.650665\pi$
$930$	0	0
$931$	−90.8439	−2.97729
$932$	−5.33860	−0.174872
$933$	0	0
$934$	−25.7091	−0.841226
$935$	0.0657222	0.00214935
$936$	0	0
$937$	−17.6446	−0.576425	−0.288213	−	0.957566i	$-0.593061\pi$
$937$	−17.6446	−0.576425	−0.288213	+	0.957566i	$0.593061\pi$
$938$	57.0928	1.86414
$939$	0	0
$940$	5.61401	0.183109
$941$	−10.5607	−0.344269	−0.172134	−	0.985073i	$-0.555066\pi$
$941$	−10.5607	−0.344269	−0.172134	+	0.985073i	$0.555066\pi$
$942$	0	0
$943$	75.3551	2.45390
$944$	−12.4961	−0.406714
$945$	0	0
$946$	3.59525	0.116892
$947$	3.84850	0.125059	0.0625297	−	0.998043i	$-0.480083\pi$
$947$	3.84850	0.125059	0.0625297	+	0.998043i	$0.480083\pi$
$948$	0	0
$949$	−20.3725	−0.661320
$950$	−31.4368	−1.01994
$951$	0	0
$952$	−0.896676	−0.0290614
$953$	18.3569	0.594639	0.297320	−	0.954778i	$-0.403907\pi$
$953$	18.3569	0.594639	0.297320	+	0.954778i	$0.403907\pi$
$954$	0	0
$955$	−6.83077	−0.221039
$956$	0.0672071	0.00217363
$957$	0	0
$958$	9.00827	0.291044
$959$	86.5948	2.79629
$960$	0	0
$961$	−8.85543	−0.285659
$962$	34.0501	1.09782
$963$	0	0
$964$	−11.4205	−0.367828
$965$	10.5761	0.340456
$966$	0	0
$967$	−22.7305	−0.730964	−0.365482	−	0.930818i	$-0.619096\pi$
$967$	−22.7305	−0.730964	−0.365482	+	0.930818i	$0.619096\pi$
$968$	10.5819	0.340114
$969$	0	0
$970$	4.93696	0.158516
$971$	58.1610	1.86648	0.933238	−	0.359259i	$-0.116971\pi$
$971$	58.1610	1.86648	0.933238	+	0.359259i	$0.116971\pi$
$972$	0	0
$973$	−4.20953	−0.134951
$974$	−33.1458	−1.06206
$975$	0	0
$976$	8.74339	0.279869
$977$	48.2740	1.54442	0.772210	−	0.635367i	$-0.219151\pi$
$977$	48.2740	1.54442	0.772210	+	0.635367i	$0.219151\pi$
$978$	0	0
$979$	−7.85788	−0.251139
$980$	7.05216	0.225273
$981$	0	0
$982$	−38.0044	−1.21277
$983$	−7.77950	−0.248128	−0.124064	−	0.992274i	$-0.539593\pi$
$983$	−7.77950	−0.248128	−0.124064	+	0.992274i	$0.539593\pi$
$984$	0	0
$985$	−4.57157	−0.145662
$986$	−0.485776	−0.0154703
$987$	0	0
$988$	33.4749	1.06498
$989$	−33.7527	−1.07327
$990$	0	0
$991$	26.9972	0.857595	0.428798	−	0.903401i	$-0.358937\pi$
$991$	26.9972	0.857595	0.428798	+	0.903401i	$0.358937\pi$
$992$	−4.70580	−0.149409
$993$	0	0
$994$	11.8708	0.376520
$995$	3.22840	0.102347
$996$	0	0
$997$	−27.5982	−0.874045	−0.437023	−	0.899451i	$-0.643967\pi$
$997$	−27.5982	−0.874045	−0.437023	+	0.899451i	$0.643967\pi$
$998$	−11.6632	−0.369193
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4842.2.a.t.1.9	✓	13
3.2	odd	2		4842.2.a.u.1.5	yes	13

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4842.2.a.t.1.9	✓	13	1.1	even	1	trivial
4842.2.a.u.1.5	yes	13	3.2	odd	2

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:	\( N \)	\(=\)	\( 4842 = 2 \cdot 3^{2} \cdot 269 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4842.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} +0.515478 q^{5} -4.54762 q^{7} -1.00000 q^{8} -0.515478 q^{10} -0.646622 q^{11} -5.04122 q^{13} +4.54762 q^{14} +1.00000 q^{16} -0.197175 q^{17} -6.64024 q^{19} +0.515478 q^{20} +0.646622 q^{22} -6.07059 q^{23} -4.73428 q^{25} +5.04122 q^{26} -4.54762 q^{28} -2.46368 q^{29} +4.70580 q^{31} -1.00000 q^{32} +0.197175 q^{34} -2.34420 q^{35} +6.75434 q^{37} +6.64024 q^{38} -0.515478 q^{40} -12.4131 q^{41} +5.56005 q^{43} -0.646622 q^{44} +6.07059 q^{46} +10.8909 q^{47} +13.6808 q^{49} +4.73428 q^{50} -5.04122 q^{52} -10.5761 q^{53} -0.333319 q^{55} +4.54762 q^{56} +2.46368 q^{58} -12.4961 q^{59} +8.74339 q^{61} -4.70580 q^{62} +1.00000 q^{64} -2.59864 q^{65} +12.5544 q^{67} -0.197175 q^{68} +2.34420 q^{70} +2.61034 q^{71} +4.04118 q^{73} -6.75434 q^{74} -6.64024 q^{76} +2.94059 q^{77} -13.0886 q^{79} +0.515478 q^{80} +12.4131 q^{82} +10.7430 q^{83} -0.101639 q^{85} -5.56005 q^{86} +0.646622 q^{88} +12.1522 q^{89} +22.9255 q^{91} -6.07059 q^{92} -10.8909 q^{94} -3.42290 q^{95} -9.57744 q^{97} -13.6808 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 13 q - 13 q^{2} + 13 q^{4} - 5 q^{5} + 6 q^{7} - 13 q^{8} + 5 q^{10} - 5 q^{11} + 8 q^{13} - 6 q^{14} + 13 q^{16} - 7 q^{17} + 13 q^{19} - 5 q^{20} + 5 q^{22} + 4 q^{23} + 26 q^{25} - 8 q^{26} + 6 q^{28}+ \cdots - 39 q^{98}+O(q^{100}) \) 13 * q - 13 * q^2 + 13 * q^4 - 5 * q^5 + 6 * q^7 - 13 * q^8 + 5 * q^10 - 5 * q^11 + 8 * q^13 - 6 * q^14 + 13 * q^16 - 7 * q^17 + 13 * q^19 - 5 * q^20 + 5 * q^22 + 4 * q^23 + 26 * q^25 - 8 * q^26 + 6 * q^28 - 22 * q^29 + 8 * q^31 - 13 * q^32 + 7 * q^34 - 8 * q^35 + 36 * q^37 - 13 * q^38 + 5 * q^40 - 16 * q^41 + 14 * q^43 - 5 * q^44 - 4 * q^46 - 4 * q^47 + 39 * q^49 - 26 * q^50 + 8 * q^52 - 21 * q^53 - 6 * q^56 + 22 * q^58 + 4 * q^59 + 8 * q^61 - 8 * q^62 + 13 * q^64 - 24 * q^65 + 40 * q^67 - 7 * q^68 + 8 * q^70 - 7 * q^71 + 31 * q^73 - 36 * q^74 + 13 * q^76 - 10 * q^77 + 43 * q^79 - 5 * q^80 + 16 * q^82 + 10 * q^83 + 30 * q^85 - 14 * q^86 + 5 * q^88 + 2 * q^89 + 24 * q^91 + 4 * q^92 + 4 * q^94 - 2 * q^95 + 27 * q^97 - 39 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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