LMFDB - Embedded newform 4842.2.a.o.1.1

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:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - x^{6} - 10x^{5} + 7x^{4} + 27x^{3} - 15x^{2} - 20x + 12 $ x^7 - x^6 - 10x^5 + 7x^4 + 27x^3 - 15x^2 - 20*x + 12
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 538)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.49844$ 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)

$f(q)$	$=$	\(q+1.00000 q^{2} +1.00000 q^{4} -2.98038 q^{5} -2.56529 q^{7} +1.00000 q^{8} -2.98038 q^{10} +5.77573 q^{11} -0.953732 q^{13} -2.56529 q^{14} +1.00000 q^{16} -2.51668 q^{17} -7.19922 q^{19} -2.98038 q^{20} +5.77573 q^{22} +5.17399 q^{23} +3.88265 q^{25} -0.953732 q^{26} -2.56529 q^{28} -4.63236 q^{29} +8.16921 q^{31} +1.00000 q^{32} -2.51668 q^{34} +7.64552 q^{35} -9.43775 q^{37} -7.19922 q^{38} -2.98038 q^{40} -2.46810 q^{41} +5.86496 q^{43} +5.77573 q^{44} +5.17399 q^{46} +9.50969 q^{47} -0.419304 q^{49} +3.88265 q^{50} -0.953732 q^{52} +7.51226 q^{53} -17.2139 q^{55} -2.56529 q^{56} -4.63236 q^{58} +1.01485 q^{59} -5.86716 q^{61} +8.16921 q^{62} +1.00000 q^{64} +2.84248 q^{65} +12.1288 q^{67} -2.51668 q^{68} +7.64552 q^{70} +12.3537 q^{71} +0.650878 q^{73} -9.43775 q^{74} -7.19922 q^{76} -14.8164 q^{77} +15.5158 q^{79} -2.98038 q^{80} -2.46810 q^{82} +8.27171 q^{83} +7.50067 q^{85} +5.86496 q^{86} +5.77573 q^{88} -2.89852 q^{89} +2.44660 q^{91} +5.17399 q^{92} +9.50969 q^{94} +21.4564 q^{95} +0.956247 q^{97} -0.419304 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q + 7 q^{2} + 7 q^{4} + 6 q^{5} - 3 q^{7} + 7 q^{8} + 6 q^{10} + 12 q^{11} + 3 q^{13} - 3 q^{14} + 7 q^{16} + 8 q^{17} - 7 q^{19} + 6 q^{20} + 12 q^{22} + 22 q^{23} + 9 q^{25} + 3 q^{26} - 3 q^{28} + 7 q^{29}+ \cdots + 6 q^{98}+O(q^{100}) $ 7 * q + 7 * q^2 + 7 * q^4 + 6 * q^5 - 3 * q^7 + 7 * q^8 + 6 * q^10 + 12 * q^11 + 3 * q^13 - 3 * q^14 + 7 * q^16 + 8 * q^17 - 7 * q^19 + 6 * q^20 + 12 * q^22 + 22 * q^23 + 9 * q^25 + 3 * q^26 - 3 * q^28 + 7 * q^29 - 3 * q^31 + 7 * q^32 + 8 * q^34 + 25 * q^35 - 13 * q^37 - 7 * q^38 + 6 * q^40 + 11 * q^41 - 14 * q^43 + 12 * q^44 + 22 * q^46 + 23 * q^47 + 6 * q^49 + 9 * q^50 + 3 * q^52 + 21 * q^53 + 15 * q^55 - 3 * q^56 + 7 * q^58 + 20 * q^59 + 13 * q^61 - 3 * q^62 + 7 * q^64 + 29 * q^65 - 30 * q^67 + 8 * q^68 + 25 * q^70 + 23 * q^71 + 9 * q^73 - 13 * q^74 - 7 * q^76 - 8 * q^77 + 11 * q^79 + 6 * q^80 + 11 * q^82 + 3 * q^83 + 21 * q^85 - 14 * q^86 + 12 * q^88 - 16 * q^89 - 10 * q^91 + 22 * q^92 + 23 * q^94 + 21 * q^97 + 6 * 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
$5$	−2.98038	−1.33287	−0.666433	−	0.745565i	$-0.732180\pi$
$5$	−2.98038	−1.33287	−0.666433	+	0.745565i	$0.732180\pi$
$6$	0	0
$7$	−2.56529	−0.969587	−0.484794	−	0.874629i	$-0.661105\pi$
$7$	−2.56529	−0.969587	−0.484794	+	0.874629i	$0.661105\pi$
$8$	1.00000	0.353553
$9$	0	0
$10$	−2.98038	−0.942478
$11$	5.77573	1.74145	0.870724	−	0.491772i	$-0.163651\pi$
$11$	5.77573	1.74145	0.870724	+	0.491772i	$0.163651\pi$
$12$	0	0
$13$	−0.953732	−0.264518	−0.132259	−	0.991215i	$-0.542223\pi$
$13$	−0.953732	−0.264518	−0.132259	+	0.991215i	$0.542223\pi$
$14$	−2.56529	−0.685602
$15$	0	0
$16$	1.00000	0.250000
$17$	−2.51668	−0.610385	−0.305193	−	0.952291i	$-0.598721\pi$
$17$	−2.51668	−0.610385	−0.305193	+	0.952291i	$0.598721\pi$
$18$	0	0
$19$	−7.19922	−1.65161	−0.825807	−	0.563953i	$-0.809280\pi$
$19$	−7.19922	−1.65161	−0.825807	+	0.563953i	$0.809280\pi$
$20$	−2.98038	−0.666433
$21$	0	0
$22$	5.77573	1.23139
$23$	5.17399	1.07885	0.539426	−	0.842033i	$-0.318641\pi$
$23$	5.17399	1.07885	0.539426	+	0.842033i	$0.318641\pi$
$24$	0	0
$25$	3.88265	0.776530
$26$	−0.953732	−0.187042
$27$	0	0
$28$	−2.56529	−0.484794
$29$	−4.63236	−0.860208	−0.430104	−	0.902779i	$-0.641523\pi$
$29$	−4.63236	−0.860208	−0.430104	+	0.902779i	$0.641523\pi$
$30$	0	0
$31$	8.16921	1.46723	0.733617	−	0.679563i	$-0.237831\pi$
$31$	8.16921	1.46723	0.733617	+	0.679563i	$0.237831\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	−2.51668	−0.431608
$35$	7.64552	1.29233
$36$	0	0
$37$	−9.43775	−1.55156	−0.775778	−	0.631006i	$-0.782642\pi$
$37$	−9.43775	−1.55156	−0.775778	+	0.631006i	$0.782642\pi$
$38$	−7.19922	−1.16787
$39$	0	0
$40$	−2.98038	−0.471239
$41$	−2.46810	−0.385453	−0.192726	−	0.981253i	$-0.561733\pi$
$41$	−2.46810	−0.385453	−0.192726	+	0.981253i	$0.561733\pi$
$42$	0	0
$43$	5.86496	0.894398	0.447199	−	0.894435i	$-0.352422\pi$
$43$	5.86496	0.894398	0.447199	+	0.894435i	$0.352422\pi$
$44$	5.77573	0.870724
$45$	0	0
$46$	5.17399	0.762864
$47$	9.50969	1.38713	0.693565	−	0.720394i	$-0.256039\pi$
$47$	9.50969	1.38713	0.693565	+	0.720394i	$0.256039\pi$
$48$	0	0
$49$	−0.419304	−0.0599006
$50$	3.88265	0.549089
$51$	0	0
$52$	−0.953732	−0.132259
$53$	7.51226	1.03189	0.515944	−	0.856622i	$-0.327441\pi$
$53$	7.51226	1.03189	0.515944	+	0.856622i	$0.327441\pi$
$54$	0	0
$55$	−17.2139	−2.32112
$56$	−2.56529	−0.342801
$57$	0	0
$58$	−4.63236	−0.608259
$59$	1.01485	0.132122	0.0660610	−	0.997816i	$-0.478957\pi$
$59$	1.01485	0.132122	0.0660610	+	0.997816i	$0.478957\pi$
$60$	0	0
$61$	−5.86716	−0.751212	−0.375606	−	0.926779i	$-0.622565\pi$
$61$	−5.86716	−0.751212	−0.375606	+	0.926779i	$0.622565\pi$
$62$	8.16921	1.03749
$63$	0	0
$64$	1.00000	0.125000
$65$	2.84248	0.352566
$66$	0	0
$67$	12.1288	1.48177	0.740883	−	0.671634i	$-0.234407\pi$
$67$	12.1288	1.48177	0.740883	+	0.671634i	$0.234407\pi$
$68$	−2.51668	−0.305193
$69$	0	0
$70$	7.64552	0.913815
$71$	12.3537	1.46611	0.733057	−	0.680167i	$-0.238093\pi$
$71$	12.3537	1.46611	0.733057	+	0.680167i	$0.238093\pi$
$72$	0	0
$73$	0.650878	0.0761795	0.0380897	−	0.999274i	$-0.487873\pi$
$73$	0.650878	0.0761795	0.0380897	+	0.999274i	$0.487873\pi$
$74$	−9.43775	−1.09712
$75$	0	0
$76$	−7.19922	−0.825807
$77$	−14.8164	−1.68849
$78$	0	0
$79$	15.5158	1.74567	0.872834	−	0.488018i	$-0.162280\pi$
$79$	15.5158	1.74567	0.872834	+	0.488018i	$0.162280\pi$
$80$	−2.98038	−0.333216
$81$	0	0
$82$	−2.46810	−0.272556
$83$	8.27171	0.907939	0.453969	−	0.891017i	$-0.350008\pi$
$83$	8.27171	0.907939	0.453969	+	0.891017i	$0.350008\pi$
$84$	0	0
$85$	7.50067	0.813562
$86$	5.86496	0.632435
$87$	0	0
$88$	5.77573	0.615695
$89$	−2.89852	−0.307242	−0.153621	−	0.988130i	$-0.549093\pi$
$89$	−2.89852	−0.307242	−0.153621	+	0.988130i	$0.549093\pi$
$90$	0	0
$91$	2.44660	0.256473
$92$	5.17399	0.539426
$93$	0	0
$94$	9.50969	0.980850
$95$	21.4564	2.20138
$96$	0	0
$97$	0.956247	0.0970922	0.0485461	−	0.998821i	$-0.484541\pi$
$97$	0.956247	0.0970922	0.0485461	+	0.998821i	$0.484541\pi$
$98$	−0.419304	−0.0423561
$99$	0	0
$100$	3.88265	0.388265
$101$	−5.92549	−0.589608	−0.294804	−	0.955558i	$-0.595254\pi$
$101$	−5.92549	−0.589608	−0.294804	+	0.955558i	$0.595254\pi$
$102$	0	0
$103$	−7.57015	−0.745909	−0.372955	−	0.927850i	$-0.621655\pi$
$103$	−7.57015	−0.745909	−0.372955	+	0.927850i	$0.621655\pi$
$104$	−0.953732	−0.0935211
$105$	0	0
$106$	7.51226	0.729655
$107$	−16.0455	−1.55118	−0.775589	−	0.631238i	$-0.782547\pi$
$107$	−16.0455	−1.55118	−0.775589	+	0.631238i	$0.782547\pi$
$108$	0	0
$109$	2.50942	0.240359	0.120179	−	0.992752i	$-0.461653\pi$
$109$	2.50942	0.240359	0.120179	+	0.992752i	$0.461653\pi$
$110$	−17.2139	−1.64128
$111$	0	0
$112$	−2.56529	−0.242397
$113$	13.4589	1.26611	0.633055	−	0.774107i	$-0.281801\pi$
$113$	13.4589	1.26611	0.633055	+	0.774107i	$0.281801\pi$
$114$	0	0
$115$	−15.4205	−1.43796
$116$	−4.63236	−0.430104
$117$	0	0
$118$	1.01485	0.0934244
$119$	6.45602	0.591822
$120$	0	0
$121$	22.3591	2.03264
$122$	−5.86716	−0.531187
$123$	0	0
$124$	8.16921	0.733617
$125$	3.33013	0.297856
$126$	0	0
$127$	2.35435	0.208915	0.104458	−	0.994529i	$-0.466689\pi$
$127$	2.35435	0.208915	0.104458	+	0.994529i	$0.466689\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	2.84248	0.249302
$131$	−8.14252	−0.711415	−0.355708	−	0.934597i	$-0.615760\pi$
$131$	−8.14252	−0.711415	−0.355708	+	0.934597i	$0.615760\pi$
$132$	0	0
$133$	18.4681	1.60138
$134$	12.1288	1.04777
$135$	0	0
$136$	−2.51668	−0.215804
$137$	13.5630	1.15876	0.579382	−	0.815056i	$-0.303294\pi$
$137$	13.5630	1.15876	0.579382	+	0.815056i	$0.303294\pi$
$138$	0	0
$139$	−17.6145	−1.49405	−0.747023	−	0.664798i	$-0.768518\pi$
$139$	−17.6145	−1.49405	−0.747023	+	0.664798i	$0.768518\pi$
$140$	7.64552	0.646165
$141$	0	0
$142$	12.3537	1.03670
$143$	−5.50850	−0.460644
$144$	0	0
$145$	13.8062	1.14654
$146$	0.650878	0.0538670
$147$	0	0
$148$	−9.43775	−0.775778
$149$	−4.73354	−0.387787	−0.193893	−	0.981023i	$-0.562112\pi$
$149$	−4.73354	−0.387787	−0.193893	+	0.981023i	$0.562112\pi$
$150$	0	0
$151$	18.5426	1.50897	0.754486	−	0.656316i	$-0.227886\pi$
$151$	18.5426	1.50897	0.754486	+	0.656316i	$0.227886\pi$
$152$	−7.19922	−0.583934
$153$	0	0
$154$	−14.8164	−1.19394
$155$	−24.3473	−1.95562
$156$	0	0
$157$	6.62583	0.528799	0.264399	−	0.964413i	$-0.414826\pi$
$157$	6.62583	0.528799	0.264399	+	0.964413i	$0.414826\pi$
$158$	15.5158	1.23437
$159$	0	0
$160$	−2.98038	−0.235620
$161$	−13.2728	−1.04604
$162$	0	0
$163$	−7.35217	−0.575867	−0.287933	−	0.957650i	$-0.592968\pi$
$163$	−7.35217	−0.575867	−0.287933	+	0.957650i	$0.592968\pi$
$164$	−2.46810	−0.192726
$165$	0	0
$166$	8.27171	0.642010
$167$	18.4854	1.43045	0.715223	−	0.698896i	$-0.246325\pi$
$167$	18.4854	1.43045	0.715223	+	0.698896i	$0.246325\pi$
$168$	0	0
$169$	−12.0904	−0.930030
$170$	7.50067	0.575275
$171$	0	0
$172$	5.86496	0.447199
$173$	15.2035	1.15590	0.577951	−	0.816071i	$-0.303852\pi$
$173$	15.2035	1.15590	0.577951	+	0.816071i	$0.303852\pi$
$174$	0	0
$175$	−9.96011	−0.752913
$176$	5.77573	0.435362
$177$	0	0
$178$	−2.89852	−0.217253
$179$	8.92937	0.667412	0.333706	−	0.942677i	$-0.391701\pi$
$179$	8.92937	0.667412	0.333706	+	0.942677i	$0.391701\pi$
$180$	0	0
$181$	8.96265	0.666189	0.333094	−	0.942894i	$-0.391907\pi$
$181$	8.96265	0.666189	0.333094	+	0.942894i	$0.391907\pi$
$182$	2.44660	0.181354
$183$	0	0
$184$	5.17399	0.381432
$185$	28.1280	2.06802
$186$	0	0
$187$	−14.5357	−1.06295
$188$	9.50969	0.693565
$189$	0	0
$190$	21.4564	1.55661
$191$	2.71179	0.196218	0.0981091	−	0.995176i	$-0.468721\pi$
$191$	2.71179	0.196218	0.0981091	+	0.995176i	$0.468721\pi$
$192$	0	0
$193$	−10.4616	−0.753042	−0.376521	−	0.926408i	$-0.622880\pi$
$193$	−10.4616	−0.753042	−0.376521	+	0.926408i	$0.622880\pi$
$194$	0.956247	0.0686545
$195$	0	0
$196$	−0.419304	−0.0299503
$197$	8.17270	0.582280	0.291140	−	0.956680i	$-0.405965\pi$
$197$	8.17270	0.582280	0.291140	+	0.956680i	$0.405965\pi$
$198$	0	0
$199$	23.4576	1.66287	0.831433	−	0.555625i	$-0.187521\pi$
$199$	23.4576	1.66287	0.831433	+	0.555625i	$0.187521\pi$
$200$	3.88265	0.274545
$201$	0	0
$202$	−5.92549	−0.416916
$203$	11.8833	0.834047
$204$	0	0
$205$	7.35587	0.513757
$206$	−7.57015	−0.527437
$207$	0	0
$208$	−0.953732	−0.0661294
$209$	−41.5807	−2.87620
$210$	0	0
$211$	23.7236	1.63320	0.816600	−	0.577203i	$-0.195856\pi$
$211$	23.7236	1.63320	0.816600	+	0.577203i	$0.195856\pi$
$212$	7.51226	0.515944
$213$	0	0
$214$	−16.0455	−1.09685
$215$	−17.4798	−1.19211
$216$	0	0
$217$	−20.9564	−1.42261
$218$	2.50942	0.169959
$219$	0	0
$220$	−17.2139	−1.16056
$221$	2.40024	0.161458
$222$	0	0
$223$	−4.63802	−0.310585	−0.155292	−	0.987869i	$-0.549632\pi$
$223$	−4.63802	−0.310585	−0.155292	+	0.987869i	$0.549632\pi$
$224$	−2.56529	−0.171400
$225$	0	0
$226$	13.4589	0.895274
$227$	−2.30295	−0.152852	−0.0764261	−	0.997075i	$-0.524351\pi$
$227$	−2.30295	−0.152852	−0.0764261	+	0.997075i	$0.524351\pi$
$228$	0	0
$229$	1.98890	0.131430	0.0657150	−	0.997838i	$-0.479067\pi$
$229$	1.98890	0.131430	0.0657150	+	0.997838i	$0.479067\pi$
$230$	−15.4205	−1.01679
$231$	0	0
$232$	−4.63236	−0.304129
$233$	−5.58928	−0.366166	−0.183083	−	0.983097i	$-0.558608\pi$
$233$	−5.58928	−0.366166	−0.183083	+	0.983097i	$0.558608\pi$
$234$	0	0
$235$	−28.3425	−1.84886
$236$	1.01485	0.0660610
$237$	0	0
$238$	6.45602	0.418481
$239$	3.34553	0.216404	0.108202	−	0.994129i	$-0.465491\pi$
$239$	3.34553	0.216404	0.108202	+	0.994129i	$0.465491\pi$
$240$	0	0
$241$	26.7139	1.72079	0.860395	−	0.509627i	$-0.170217\pi$
$241$	26.7139	1.72079	0.860395	+	0.509627i	$0.170217\pi$
$242$	22.3591	1.43729
$243$	0	0
$244$	−5.86716	−0.375606
$245$	1.24968	0.0798394
$246$	0	0
$247$	6.86613	0.436881
$248$	8.16921	0.518745
$249$	0	0
$250$	3.33013	0.210616
$251$	29.3076	1.84988	0.924940	−	0.380112i	$-0.124115\pi$
$251$	29.3076	1.84988	0.924940	+	0.380112i	$0.124115\pi$
$252$	0	0
$253$	29.8836	1.87877
$254$	2.35435	0.147725
$255$	0	0
$256$	1.00000	0.0625000
$257$	−10.3803	−0.647504	−0.323752	−	0.946142i	$-0.604944\pi$
$257$	−10.3803	−0.647504	−0.323752	+	0.946142i	$0.604944\pi$
$258$	0	0
$259$	24.2105	1.50437
$260$	2.84248	0.176283
$261$	0	0
$262$	−8.14252	−0.503046
$263$	8.83916	0.545046	0.272523	−	0.962149i	$-0.412142\pi$
$263$	8.83916	0.545046	0.272523	+	0.962149i	$0.412142\pi$
$264$	0	0
$265$	−22.3894	−1.37537
$266$	18.4681	1.13235
$267$	0	0
$268$	12.1288	0.740883
$269$	1.00000	0.0609711
$269$	1.00000	0.0609711
$270$	0	0
$271$	7.39604	0.449277	0.224639	−	0.974442i	$-0.427880\pi$
$271$	7.39604	0.449277	0.224639	+	0.974442i	$0.427880\pi$
$272$	−2.51668	−0.152596
$273$	0	0
$274$	13.5630	0.819370
$275$	22.4251	1.35229
$276$	0	0
$277$	13.5773	0.815779	0.407890	−	0.913031i	$-0.366265\pi$
$277$	13.5773	0.815779	0.407890	+	0.913031i	$0.366265\pi$
$278$	−17.6145	−1.05645
$279$	0	0
$280$	7.64552	0.456907
$281$	−13.5723	−0.809653	−0.404826	−	0.914394i	$-0.632668\pi$
$281$	−13.5723	−0.809653	−0.404826	+	0.914394i	$0.632668\pi$
$282$	0	0
$283$	20.5026	1.21875	0.609377	−	0.792880i	$-0.291419\pi$
$283$	20.5026	1.21875	0.609377	+	0.792880i	$0.291419\pi$
$284$	12.3537	0.733057
$285$	0	0
$286$	−5.50850	−0.325724
$287$	6.33139	0.373730
$288$	0	0
$289$	−10.6663	−0.627430
$290$	13.8062	0.810727
$291$	0	0
$292$	0.650878	0.0380897
$293$	26.0133	1.51971	0.759857	−	0.650090i	$-0.225269\pi$
$293$	26.0133	1.51971	0.759857	+	0.650090i	$0.225269\pi$
$294$	0	0
$295$	−3.02463	−0.176101
$296$	−9.43775	−0.548558
$297$	0	0
$298$	−4.73354	−0.274207
$299$	−4.93460	−0.285376
$300$	0	0
$301$	−15.0453	−0.867197
$302$	18.5426	1.06700
$303$	0	0
$304$	−7.19922	−0.412903
$305$	17.4863	1.00126
$306$	0	0
$307$	−17.3993	−0.993031	−0.496516	−	0.868028i	$-0.665387\pi$
$307$	−17.3993	−0.993031	−0.496516	+	0.868028i	$0.665387\pi$
$308$	−14.8164	−0.844243
$309$	0	0
$310$	−24.3473	−1.38284
$311$	13.4740	0.764038	0.382019	−	0.924155i	$-0.375229\pi$
$311$	13.4740	0.764038	0.382019	+	0.924155i	$0.375229\pi$
$312$	0	0
$313$	−9.89195	−0.559126	−0.279563	−	0.960127i	$-0.590190\pi$
$313$	−9.89195	−0.559126	−0.279563	+	0.960127i	$0.590190\pi$
$314$	6.62583	0.373917
$315$	0	0
$316$	15.5158	0.872834
$317$	−19.2200	−1.07950	−0.539751	−	0.841825i	$-0.681482\pi$
$317$	−19.2200	−1.07950	−0.539751	+	0.841825i	$0.681482\pi$
$318$	0	0
$319$	−26.7553	−1.49801
$320$	−2.98038	−0.166608
$321$	0	0
$322$	−13.2728	−0.739663
$323$	18.1182	1.00812
$324$	0	0
$325$	−3.70301	−0.205406
$326$	−7.35217	−0.407199
$327$	0	0
$328$	−2.46810	−0.136278
$329$	−24.3951	−1.34494
$330$	0	0
$331$	−10.6168	−0.583550	−0.291775	−	0.956487i	$-0.594246\pi$
$331$	−10.6168	−0.583550	−0.291775	+	0.956487i	$0.594246\pi$
$332$	8.27171	0.453969
$333$	0	0
$334$	18.4854	1.01148
$335$	−36.1483	−1.97499
$336$	0	0
$337$	−4.73621	−0.257998	−0.128999	−	0.991645i	$-0.541176\pi$
$337$	−4.73621	−0.257998	−0.128999	+	0.991645i	$0.541176\pi$
$338$	−12.0904	−0.657631
$339$	0	0
$340$	7.50067	0.406781
$341$	47.1832	2.55511
$342$	0	0
$343$	19.0326	1.02767
$344$	5.86496	0.316217
$345$	0	0
$346$	15.2035	0.817346
$347$	9.70333	0.520902	0.260451	−	0.965487i	$-0.416129\pi$
$347$	9.70333	0.520902	0.260451	+	0.965487i	$0.416129\pi$
$348$	0	0
$349$	−34.9874	−1.87283	−0.936416	−	0.350893i	$-0.885878\pi$
$349$	−34.9874	−1.87283	−0.936416	+	0.350893i	$0.885878\pi$
$350$	−9.96011	−0.532390
$351$	0	0
$352$	5.77573	0.307847
$353$	−17.9950	−0.957777	−0.478888	−	0.877876i	$-0.658960\pi$
$353$	−17.9950	−0.957777	−0.478888	+	0.877876i	$0.658960\pi$
$354$	0	0
$355$	−36.8187	−1.95413
$356$	−2.89852	−0.153621
$357$	0	0
$358$	8.92937	0.471932
$359$	8.64349	0.456186	0.228093	−	0.973639i	$-0.426751\pi$
$359$	8.64349	0.456186	0.228093	+	0.973639i	$0.426751\pi$
$360$	0	0
$361$	32.8287	1.72783
$362$	8.96265	0.471067
$363$	0	0
$364$	2.44660	0.128236
$365$	−1.93986	−0.101537
$366$	0	0
$367$	−25.4599	−1.32899	−0.664497	−	0.747291i	$-0.731354\pi$
$367$	−25.4599	−1.32899	−0.664497	+	0.747291i	$0.731354\pi$
$368$	5.17399	0.269713
$369$	0	0
$370$	28.1280	1.46231
$371$	−19.2711	−1.00051
$372$	0	0
$373$	−5.63673	−0.291859	−0.145929	−	0.989295i	$-0.546617\pi$
$373$	−5.63673	−0.291859	−0.145929	+	0.989295i	$0.546617\pi$
$374$	−14.5357	−0.751622
$375$	0	0
$376$	9.50969	0.490425
$377$	4.41803	0.227540
$378$	0	0
$379$	−16.2741	−0.835944	−0.417972	−	0.908460i	$-0.637259\pi$
$379$	−16.2741	−0.835944	−0.417972	+	0.908460i	$0.637259\pi$
$380$	21.4564	1.10069
$381$	0	0
$382$	2.71179	0.138747
$383$	−17.3292	−0.885483	−0.442742	−	0.896649i	$-0.645994\pi$
$383$	−17.3292	−0.885483	−0.442742	+	0.896649i	$0.645994\pi$
$384$	0	0
$385$	44.1585	2.25052
$386$	−10.4616	−0.532481
$387$	0	0
$388$	0.956247	0.0485461
$389$	−32.7406	−1.66002	−0.830008	−	0.557752i	$-0.811664\pi$
$389$	−32.7406	−1.66002	−0.830008	+	0.557752i	$0.811664\pi$
$390$	0	0
$391$	−13.0213	−0.658516
$392$	−0.419304	−0.0211781
$393$	0	0
$394$	8.17270	0.411734
$395$	−46.2430	−2.32674
$396$	0	0
$397$	−10.1894	−0.511392	−0.255696	−	0.966757i	$-0.582305\pi$
$397$	−10.1894	−0.511392	−0.255696	+	0.966757i	$0.582305\pi$
$398$	23.4576	1.17582
$399$	0	0
$400$	3.88265	0.194132
$401$	−27.6214	−1.37935	−0.689673	−	0.724121i	$-0.742246\pi$
$401$	−27.6214	−1.37935	−0.689673	+	0.724121i	$0.742246\pi$
$402$	0	0
$403$	−7.79124	−0.388109
$404$	−5.92549	−0.294804
$405$	0	0
$406$	11.8833	0.589760
$407$	−54.5099	−2.70195
$408$	0	0
$409$	−32.3617	−1.60018	−0.800092	−	0.599877i	$-0.795216\pi$
$409$	−32.3617	−1.60018	−0.800092	+	0.599877i	$0.795216\pi$
$410$	7.35587	0.363281
$411$	0	0
$412$	−7.57015	−0.372955
$413$	−2.60338	−0.128104
$414$	0	0
$415$	−24.6528	−1.21016
$416$	−0.953732	−0.0467606
$417$	0	0
$418$	−41.5807	−2.03378
$419$	20.7391	1.01317	0.506586	−	0.862189i	$-0.330907\pi$
$419$	20.7391	1.01317	0.506586	+	0.862189i	$0.330907\pi$
$420$	0	0
$421$	−2.74188	−0.133631	−0.0668155	−	0.997765i	$-0.521284\pi$
$421$	−2.74188	−0.133631	−0.0668155	+	0.997765i	$0.521284\pi$
$422$	23.7236	1.15485
$423$	0	0
$424$	7.51226	0.364828
$425$	−9.77140	−0.473983
$426$	0	0
$427$	15.0509	0.728366
$428$	−16.0455	−0.775589
$429$	0	0
$430$	−17.4798	−0.842950
$431$	1.93294	0.0931065	0.0465533	−	0.998916i	$-0.485176\pi$
$431$	1.93294	0.0931065	0.0465533	+	0.998916i	$0.485176\pi$
$432$	0	0
$433$	4.35794	0.209429	0.104715	−	0.994502i	$-0.466607\pi$
$433$	4.35794	0.209429	0.104715	+	0.994502i	$0.466607\pi$
$434$	−20.9564	−1.00594
$435$	0	0
$436$	2.50942	0.120179
$437$	−37.2487	−1.78185
$438$	0	0
$439$	8.19465	0.391109	0.195555	−	0.980693i	$-0.437349\pi$
$439$	8.19465	0.391109	0.195555	+	0.980693i	$0.437349\pi$
$440$	−17.2139	−0.820638
$441$	0	0
$442$	2.40024	0.114168
$443$	5.51425	0.261990	0.130995	−	0.991383i	$-0.458183\pi$
$443$	5.51425	0.261990	0.130995	+	0.991383i	$0.458183\pi$
$444$	0	0
$445$	8.63867	0.409512
$446$	−4.63802	−0.219616
$447$	0	0
$448$	−2.56529	−0.121198
$449$	19.1545	0.903956	0.451978	−	0.892029i	$-0.350719\pi$
$449$	19.1545	0.903956	0.451978	+	0.892029i	$0.350719\pi$
$450$	0	0
$451$	−14.2551	−0.671246
$452$	13.4589	0.633055
$453$	0	0
$454$	−2.30295	−0.108083
$455$	−7.29178	−0.341844
$456$	0	0
$457$	−41.6610	−1.94882	−0.974409	−	0.224783i	$-0.927833\pi$
$457$	−41.6610	−1.94882	−0.974409	+	0.224783i	$0.927833\pi$
$458$	1.98890	0.0929350
$459$	0	0
$460$	−15.4205	−0.718982
$461$	−28.7942	−1.34108	−0.670539	−	0.741874i	$-0.733937\pi$
$461$	−28.7942	−1.34108	−0.670539	+	0.741874i	$0.733937\pi$
$462$	0	0
$463$	1.89638	0.0881321	0.0440660	−	0.999029i	$-0.485969\pi$
$463$	1.89638	0.0881321	0.0440660	+	0.999029i	$0.485969\pi$
$464$	−4.63236	−0.215052
$465$	0	0
$466$	−5.58928	−0.258918
$467$	9.53499	0.441227	0.220613	−	0.975361i	$-0.429194\pi$
$467$	9.53499	0.441227	0.220613	+	0.975361i	$0.429194\pi$
$468$	0	0
$469$	−31.1138	−1.43670
$470$	−28.3425	−1.30734
$471$	0	0
$472$	1.01485	0.0467122
$473$	33.8744	1.55755
$474$	0	0
$475$	−27.9520	−1.28253
$476$	6.45602	0.295911
$477$	0	0
$478$	3.34553	0.153021
$479$	−11.2845	−0.515602	−0.257801	−	0.966198i	$-0.582998\pi$
$479$	−11.2845	−0.515602	−0.257801	+	0.966198i	$0.582998\pi$
$480$	0	0
$481$	9.00108	0.410414
$482$	26.7139	1.21678
$483$	0	0
$484$	22.3591	1.01632
$485$	−2.84998	−0.129411
$486$	0	0
$487$	13.3515	0.605014	0.302507	−	0.953147i	$-0.402176\pi$
$487$	13.3515	0.605014	0.302507	+	0.953147i	$0.402176\pi$
$488$	−5.86716	−0.265594
$489$	0	0
$490$	1.24968	0.0564550
$491$	−37.9516	−1.71273	−0.856365	−	0.516370i	$-0.827283\pi$
$491$	−37.9516	−1.71273	−0.856365	+	0.516370i	$0.827283\pi$
$492$	0	0
$493$	11.6582	0.525058
$494$	6.86613	0.308922
$495$	0	0
$496$	8.16921	0.366808
$497$	−31.6908	−1.42153
$498$	0	0
$499$	−27.6365	−1.23718	−0.618590	−	0.785714i	$-0.712296\pi$
$499$	−27.6365	−1.23718	−0.618590	+	0.785714i	$0.712296\pi$
$500$	3.33013	0.148928
$501$	0	0
$502$	29.3076	1.30806
$503$	−20.9442	−0.933855	−0.466927	−	0.884296i	$-0.654639\pi$
$503$	−20.9442	−0.933855	−0.466927	+	0.884296i	$0.654639\pi$
$504$	0	0
$505$	17.6602	0.785868
$506$	29.8836	1.32849
$507$	0	0
$508$	2.35435	0.104458
$509$	25.5377	1.13194	0.565969	−	0.824426i	$-0.308502\pi$
$509$	25.5377	1.13194	0.565969	+	0.824426i	$0.308502\pi$
$510$	0	0
$511$	−1.66969	−0.0738627
$512$	1.00000	0.0441942
$513$	0	0
$514$	−10.3803	−0.457854
$515$	22.5619	0.994197
$516$	0	0
$517$	54.9254	2.41562
$518$	24.2105	1.06375
$519$	0	0
$520$	2.84248	0.124651
$521$	38.8849	1.70358	0.851788	−	0.523886i	$-0.175518\pi$
$521$	38.8849	1.70358	0.851788	+	0.523886i	$0.175518\pi$
$522$	0	0
$523$	−25.0299	−1.09448	−0.547240	−	0.836976i	$-0.684321\pi$
$523$	−25.0299	−1.09448	−0.547240	+	0.836976i	$0.684321\pi$
$524$	−8.14252	−0.355708
$525$	0	0
$526$	8.83916	0.385406
$527$	−20.5593	−0.895578
$528$	0	0
$529$	3.77022	0.163922
$530$	−22.3894	−0.972532
$531$	0	0
$532$	18.4681	0.800692
$533$	2.35391	0.101959
$534$	0	0
$535$	47.8217	2.06751
$536$	12.1288	0.523883
$537$	0	0
$538$	1.00000	0.0431131
$539$	−2.42179	−0.104314
$540$	0	0
$541$	0.997803	0.0428989	0.0214494	−	0.999770i	$-0.493172\pi$
$541$	0.997803	0.0428989	0.0214494	+	0.999770i	$0.493172\pi$
$542$	7.39604	0.317687
$543$	0	0
$544$	−2.51668	−0.107902
$545$	−7.47902	−0.320366
$546$	0	0
$547$	−15.8011	−0.675608	−0.337804	−	0.941217i	$-0.609684\pi$
$547$	−15.8011	−0.675608	−0.337804	+	0.941217i	$0.609684\pi$
$548$	13.5630	0.579382
$549$	0	0
$550$	22.4251	0.956211
$551$	33.3494	1.42073
$552$	0	0
$553$	−39.8026	−1.69258
$554$	13.5773	0.576843
$555$	0	0
$556$	−17.6145	−0.747023
$557$	36.3525	1.54031	0.770153	−	0.637860i	$-0.220180\pi$
$557$	36.3525	1.54031	0.770153	+	0.637860i	$0.220180\pi$
$558$	0	0
$559$	−5.59360	−0.236584
$560$	7.64552	0.323082
$561$	0	0
$562$	−13.5723	−0.572511
$563$	20.1622	0.849736	0.424868	−	0.905255i	$-0.360320\pi$
$563$	20.1622	0.849736	0.424868	+	0.905255i	$0.360320\pi$
$564$	0	0
$565$	−40.1127	−1.68755
$566$	20.5026	0.861790
$567$	0	0
$568$	12.3537	0.518349
$569$	16.0424	0.672530	0.336265	−	0.941767i	$-0.390836\pi$
$569$	16.0424	0.672530	0.336265	+	0.941767i	$0.390836\pi$
$570$	0	0
$571$	−32.2187	−1.34831	−0.674156	−	0.738589i	$-0.735493\pi$
$571$	−32.2187	−1.34831	−0.674156	+	0.738589i	$0.735493\pi$
$572$	−5.50850	−0.230322
$573$	0	0
$574$	6.33139	0.264267
$575$	20.0888	0.837761
$576$	0	0
$577$	42.6582	1.77589	0.887943	−	0.459954i	$-0.152134\pi$
$577$	42.6582	1.77589	0.887943	+	0.459954i	$0.152134\pi$
$578$	−10.6663	−0.443660
$579$	0	0
$580$	13.8062	0.573271
$581$	−21.2193	−0.880326
$582$	0	0
$583$	43.3888	1.79698
$584$	0.650878	0.0269335
$585$	0	0
$586$	26.0133	1.07460
$587$	13.8741	0.572646	0.286323	−	0.958133i	$-0.407567\pi$
$587$	13.8741	0.572646	0.286323	+	0.958133i	$0.407567\pi$
$588$	0	0
$589$	−58.8119	−2.42330
$590$	−3.02463	−0.124522
$591$	0	0
$592$	−9.43775	−0.387889
$593$	−33.6435	−1.38157	−0.690786	−	0.723059i	$-0.742735\pi$
$593$	−33.6435	−1.38157	−0.690786	+	0.723059i	$0.742735\pi$
$594$	0	0
$595$	−19.2414	−0.788819
$596$	−4.73354	−0.193893
$597$	0	0
$598$	−4.93460	−0.201791
$599$	−12.8677	−0.525761	−0.262881	−	0.964828i	$-0.584673\pi$
$599$	−12.8677	−0.525761	−0.262881	+	0.964828i	$0.584673\pi$
$600$	0	0
$601$	17.2785	0.704804	0.352402	−	0.935849i	$-0.385365\pi$
$601$	17.2785	0.704804	0.352402	+	0.935849i	$0.385365\pi$
$602$	−15.0453	−0.613201
$603$	0	0
$604$	18.5426	0.754486
$605$	−66.6384	−2.70924
$606$	0	0
$607$	−36.7366	−1.49109	−0.745547	−	0.666453i	$-0.767812\pi$
$607$	−36.7366	−1.49109	−0.745547	+	0.666453i	$0.767812\pi$
$608$	−7.19922	−0.291967
$609$	0	0
$610$	17.4863	0.708001
$611$	−9.06970	−0.366921
$612$	0	0
$613$	−35.5029	−1.43395	−0.716975	−	0.697099i	$-0.754474\pi$
$613$	−35.5029	−1.43395	−0.716975	+	0.697099i	$0.754474\pi$
$614$	−17.3993	−0.702179
$615$	0	0
$616$	−14.8164	−0.596970
$617$	−5.97403	−0.240505	−0.120253	−	0.992743i	$-0.538370\pi$
$617$	−5.97403	−0.240505	−0.120253	+	0.992743i	$0.538370\pi$
$618$	0	0
$619$	1.54631	0.0621513	0.0310757	−	0.999517i	$-0.490107\pi$
$619$	1.54631	0.0621513	0.0310757	+	0.999517i	$0.490107\pi$
$620$	−24.3473	−0.977812
$621$	0	0
$622$	13.4740	0.540256
$623$	7.43552	0.297898
$624$	0	0
$625$	−29.3383	−1.17353
$626$	−9.89195	−0.395362
$627$	0	0
$628$	6.62583	0.264399
$629$	23.7518	0.947047
$630$	0	0
$631$	33.5953	1.33741	0.668705	−	0.743528i	$-0.266849\pi$
$631$	33.5953	1.33741	0.668705	+	0.743528i	$0.266849\pi$
$632$	15.5158	0.617187
$633$	0	0
$634$	−19.2200	−0.763324
$635$	−7.01686	−0.278456
$636$	0	0
$637$	0.399904	0.0158448
$638$	−26.7553	−1.05925
$639$	0	0
$640$	−2.98038	−0.117810
$641$	−22.6820	−0.895884	−0.447942	−	0.894063i	$-0.647843\pi$
$641$	−22.6820	−0.895884	−0.447942	+	0.894063i	$0.647843\pi$
$642$	0	0
$643$	35.9742	1.41868	0.709341	−	0.704865i	$-0.248993\pi$
$643$	35.9742	1.41868	0.709341	+	0.704865i	$0.248993\pi$
$644$	−13.2728	−0.523021
$645$	0	0
$646$	18.1182	0.712849
$647$	29.8868	1.17497	0.587485	−	0.809235i	$-0.300118\pi$
$647$	29.8868	1.17497	0.587485	+	0.809235i	$0.300118\pi$
$648$	0	0
$649$	5.86149	0.230084
$650$	−3.70301	−0.145244
$651$	0	0
$652$	−7.35217	−0.287933
$653$	48.1187	1.88303	0.941516	−	0.336968i	$-0.109402\pi$
$653$	48.1187	1.88303	0.941516	+	0.336968i	$0.109402\pi$
$654$	0	0
$655$	24.2678	0.948220
$656$	−2.46810	−0.0963632
$657$	0	0
$658$	−24.3951	−0.951019
$659$	27.3372	1.06491	0.532454	−	0.846459i	$-0.321270\pi$
$659$	27.3372	1.06491	0.532454	+	0.846459i	$0.321270\pi$
$660$	0	0
$661$	−20.6868	−0.804623	−0.402311	−	0.915503i	$-0.631793\pi$
$661$	−20.6868	−0.804623	−0.402311	+	0.915503i	$0.631793\pi$
$662$	−10.6168	−0.412632
$663$	0	0
$664$	8.27171	0.321005
$665$	−55.0418	−2.13443
$666$	0	0
$667$	−23.9678	−0.928037
$668$	18.4854	0.715223
$669$	0	0
$670$	−36.1483	−1.39653
$671$	−33.8871	−1.30820
$672$	0	0
$673$	−3.68048	−0.141872	−0.0709360	−	0.997481i	$-0.522599\pi$
$673$	−3.68048	−0.141872	−0.0709360	+	0.997481i	$0.522599\pi$
$674$	−4.73621	−0.182432
$675$	0	0
$676$	−12.0904	−0.465015
$677$	17.3358	0.666268	0.333134	−	0.942880i	$-0.391894\pi$
$677$	17.3358	0.666268	0.333134	+	0.942880i	$0.391894\pi$
$678$	0	0
$679$	−2.45305	−0.0941393
$680$	7.50067	0.287637
$681$	0	0
$682$	47.1832	1.80674
$683$	−2.26489	−0.0866636	−0.0433318	−	0.999061i	$-0.513797\pi$
$683$	−2.26489	−0.0866636	−0.0433318	+	0.999061i	$0.513797\pi$
$684$	0	0
$685$	−40.4228	−1.54448
$686$	19.0326	0.726670
$687$	0	0
$688$	5.86496	0.223599
$689$	−7.16468	−0.272953
$690$	0	0
$691$	−34.9262	−1.32865	−0.664327	−	0.747442i	$-0.731282\pi$
$691$	−34.9262	−1.32865	−0.664327	+	0.747442i	$0.731282\pi$
$692$	15.2035	0.577951
$693$	0	0
$694$	9.70333	0.368333
$695$	52.4980	1.99136
$696$	0	0
$697$	6.21143	0.235275
$698$	−34.9874	−1.32429
$699$	0	0
$700$	−9.96011	−0.376457
$701$	8.67601	0.327688	0.163844	−	0.986486i	$-0.447611\pi$
$701$	8.67601	0.327688	0.163844	+	0.986486i	$0.447611\pi$
$702$	0	0
$703$	67.9444	2.56257
$704$	5.77573	0.217681
$705$	0	0
$706$	−17.9950	−0.677251
$707$	15.2006	0.571676
$708$	0	0
$709$	−52.4850	−1.97111	−0.985557	−	0.169342i	$-0.945836\pi$
$709$	−52.4850	−1.97111	−0.985557	+	0.169342i	$0.945836\pi$
$710$	−36.8187	−1.38178
$711$	0	0
$712$	−2.89852	−0.108626
$713$	42.2675	1.58293
$714$	0	0
$715$	16.4174	0.613976
$716$	8.92937	0.333706
$717$	0	0
$718$	8.64349	0.322572
$719$	−12.2225	−0.455822	−0.227911	−	0.973682i	$-0.573190\pi$
$719$	−12.2225	−0.455822	−0.227911	+	0.973682i	$0.573190\pi$
$720$	0	0
$721$	19.4196	0.723224
$722$	32.8287	1.22176
$723$	0	0
$724$	8.96265	0.333094
$725$	−17.9858	−0.667977
$726$	0	0
$727$	32.8117	1.21692	0.608460	−	0.793585i	$-0.291788\pi$
$727$	32.8117	1.21692	0.608460	+	0.793585i	$0.291788\pi$
$728$	2.44660	0.0906769
$729$	0	0
$730$	−1.93986	−0.0717975
$731$	−14.7602	−0.545927
$732$	0	0
$733$	−11.8567	−0.437937	−0.218968	−	0.975732i	$-0.570269\pi$
$733$	−11.8567	−0.437937	−0.218968	+	0.975732i	$0.570269\pi$
$734$	−25.4599	−0.939741
$735$	0	0
$736$	5.17399	0.190716
$737$	70.0525	2.58042
$738$	0	0
$739$	23.7096	0.872173	0.436086	−	0.899905i	$-0.356364\pi$
$739$	23.7096	0.872173	0.436086	+	0.899905i	$0.356364\pi$
$740$	28.1280	1.03401
$741$	0	0
$742$	−19.2711	−0.707464
$743$	−16.3166	−0.598600	−0.299300	−	0.954159i	$-0.596753\pi$
$743$	−16.3166	−0.598600	−0.299300	+	0.954159i	$0.596753\pi$
$744$	0	0
$745$	14.1077	0.516867
$746$	−5.63673	−0.206375
$747$	0	0
$748$	−14.5357	−0.531477
$749$	41.1613	1.50400
$750$	0	0
$751$	30.5188	1.11365	0.556824	−	0.830630i	$-0.312020\pi$
$751$	30.5188	1.11365	0.556824	+	0.830630i	$0.312020\pi$
$752$	9.50969	0.346783
$753$	0	0
$754$	4.41803	0.160895
$755$	−55.2639	−2.01126
$756$	0	0
$757$	21.0756	0.766007	0.383003	−	0.923747i	$-0.374890\pi$
$757$	21.0756	0.766007	0.383003	+	0.923747i	$0.374890\pi$
$758$	−16.2741	−0.591101
$759$	0	0
$760$	21.4564	0.778305
$761$	9.37156	0.339719	0.169859	−	0.985468i	$-0.445669\pi$
$761$	9.37156	0.339719	0.169859	+	0.985468i	$0.445669\pi$
$762$	0	0
$763$	−6.43738	−0.233049
$764$	2.71179	0.0981091
$765$	0	0
$766$	−17.3292	−0.626131
$767$	−0.967894	−0.0349486
$768$	0	0
$769$	26.9367	0.971363	0.485681	−	0.874136i	$-0.338572\pi$
$769$	26.9367	0.971363	0.485681	+	0.874136i	$0.338572\pi$
$770$	44.1585	1.59136
$771$	0	0
$772$	−10.4616	−0.376521
$773$	14.7111	0.529121	0.264560	−	0.964369i	$-0.414773\pi$
$773$	14.7111	0.529121	0.264560	+	0.964369i	$0.414773\pi$
$774$	0	0
$775$	31.7182	1.13935
$776$	0.956247	0.0343273
$777$	0	0
$778$	−32.7406	−1.17381
$779$	17.7684	0.636619
$780$	0	0
$781$	71.3516	2.55316
$782$	−13.0213	−0.465641
$783$	0	0
$784$	−0.419304	−0.0149751
$785$	−19.7475	−0.704818
$786$	0	0
$787$	3.73905	0.133283	0.0666414	−	0.997777i	$-0.478772\pi$
$787$	3.73905	0.133283	0.0666414	+	0.997777i	$0.478772\pi$
$788$	8.17270	0.291140
$789$	0	0
$790$	−46.2430	−1.64525
$791$	−34.5260	−1.22760
$792$	0	0
$793$	5.59569	0.198709
$794$	−10.1894	−0.361609
$795$	0	0
$796$	23.4576	0.831433
$797$	−7.11196	−0.251919	−0.125959	−	0.992035i	$-0.540201\pi$
$797$	−7.11196	−0.251919	−0.125959	+	0.992035i	$0.540201\pi$
$798$	0	0
$799$	−23.9329	−0.846685
$800$	3.88265	0.137272
$801$	0	0
$802$	−27.6214	−0.975345
$803$	3.75929	0.132663
$804$	0	0
$805$	39.5579	1.39423
$806$	−7.79124	−0.274435
$807$	0	0
$808$	−5.92549	−0.208458
$809$	10.2508	0.360398	0.180199	−	0.983630i	$-0.442326\pi$
$809$	10.2508	0.360398	0.180199	+	0.983630i	$0.442326\pi$
$810$	0	0
$811$	−17.6750	−0.620655	−0.310327	−	0.950630i	$-0.600439\pi$
$811$	−17.6750	−0.620655	−0.310327	+	0.950630i	$0.600439\pi$
$812$	11.8833	0.417023
$813$	0	0
$814$	−54.5099	−1.91057
$815$	21.9123	0.767553
$816$	0	0
$817$	−42.2231	−1.47720
$818$	−32.3617	−1.13150
$819$	0	0
$820$	7.35587	0.256878
$821$	−14.4456	−0.504153	−0.252077	−	0.967707i	$-0.581114\pi$
$821$	−14.4456	−0.504153	−0.252077	+	0.967707i	$0.581114\pi$
$822$	0	0
$823$	−5.39437	−0.188036	−0.0940179	−	0.995571i	$-0.529971\pi$
$823$	−5.39437	−0.188036	−0.0940179	+	0.995571i	$0.529971\pi$
$824$	−7.57015	−0.263719
$825$	0	0
$826$	−2.60338	−0.0905831
$827$	40.7336	1.41645	0.708223	−	0.705989i	$-0.249497\pi$
$827$	40.7336	1.41645	0.708223	+	0.705989i	$0.249497\pi$
$828$	0	0
$829$	−15.9379	−0.553547	−0.276773	−	0.960935i	$-0.589265\pi$
$829$	−15.9379	−0.553547	−0.276773	+	0.960935i	$0.589265\pi$
$830$	−24.6528	−0.855712
$831$	0	0
$832$	−0.953732	−0.0330647
$833$	1.05526	0.0365625
$834$	0	0
$835$	−55.0936	−1.90659
$836$	−41.5807	−1.43810
$837$	0	0
$838$	20.7391	0.716421
$839$	−47.0290	−1.62362	−0.811811	−	0.583921i	$-0.801518\pi$
$839$	−47.0290	−1.62362	−0.811811	+	0.583921i	$0.801518\pi$
$840$	0	0
$841$	−7.54122	−0.260042
$842$	−2.74188	−0.0944914
$843$	0	0
$844$	23.7236	0.816600
$845$	36.0339	1.23961
$846$	0	0
$847$	−57.3574	−1.97082
$848$	7.51226	0.257972
$849$	0	0
$850$	−9.77140	−0.335156
$851$	−48.8308	−1.67390
$852$	0	0
$853$	5.50121	0.188358	0.0941790	−	0.995555i	$-0.469977\pi$
$853$	5.50121	0.188358	0.0941790	+	0.995555i	$0.469977\pi$
$854$	15.0509	0.515032
$855$	0	0
$856$	−16.0455	−0.548424
$857$	6.74343	0.230351	0.115176	−	0.993345i	$-0.463257\pi$
$857$	6.74343	0.230351	0.115176	+	0.993345i	$0.463257\pi$
$858$	0	0
$859$	38.3733	1.30928	0.654641	−	0.755940i	$-0.272820\pi$
$859$	38.3733	1.30928	0.654641	+	0.755940i	$0.272820\pi$
$860$	−17.4798	−0.596056
$861$	0	0
$862$	1.93294	0.0658363
$863$	−27.5099	−0.936449	−0.468224	−	0.883610i	$-0.655106\pi$
$863$	−27.5099	−0.936449	−0.468224	+	0.883610i	$0.655106\pi$
$864$	0	0
$865$	−45.3122	−1.54066
$866$	4.35794	0.148089
$867$	0	0
$868$	−20.9564	−0.711306
$869$	89.6153	3.03999
$870$	0	0
$871$	−11.5676	−0.391953
$872$	2.50942	0.0849797
$873$	0	0
$874$	−37.2487	−1.25996
$875$	−8.54273	−0.288797
$876$	0	0
$877$	40.4307	1.36525	0.682625	−	0.730769i	$-0.260838\pi$
$877$	40.4307	1.36525	0.682625	+	0.730769i	$0.260838\pi$
$878$	8.19465	0.276556
$879$	0	0
$880$	−17.2139	−0.580279
$881$	−14.9852	−0.504866	−0.252433	−	0.967614i	$-0.581231\pi$
$881$	−14.9852	−0.504866	−0.252433	+	0.967614i	$0.581231\pi$
$882$	0	0
$883$	45.8998	1.54465	0.772325	−	0.635227i	$-0.219094\pi$
$883$	45.8998	1.54465	0.772325	+	0.635227i	$0.219094\pi$
$884$	2.40024	0.0807289
$885$	0	0
$886$	5.51425	0.185255
$887$	−38.4897	−1.29236	−0.646179	−	0.763186i	$-0.723634\pi$
$887$	−38.4897	−1.29236	−0.646179	+	0.763186i	$0.723634\pi$
$888$	0	0
$889$	−6.03959	−0.202561
$890$	8.63867	0.289569
$891$	0	0
$892$	−4.63802	−0.155292
$893$	−68.4623	−2.29100
$894$	0	0
$895$	−26.6129	−0.889570
$896$	−2.56529	−0.0857002
$897$	0	0
$898$	19.1545	0.639193
$899$	−37.8427	−1.26213
$900$	0	0
$901$	−18.9060	−0.629849
$902$	−14.2551	−0.474643
$903$	0	0
$904$	13.4589	0.447637
$905$	−26.7121	−0.887940
$906$	0	0
$907$	−3.30242	−0.109655	−0.0548275	−	0.998496i	$-0.517461\pi$
$907$	−3.30242	−0.109655	−0.0548275	+	0.998496i	$0.517461\pi$
$908$	−2.30295	−0.0764261
$909$	0	0
$910$	−7.29178	−0.241720
$911$	−0.593599	−0.0196668	−0.00983341	−	0.999952i	$-0.503130\pi$
$911$	−0.593599	−0.0196668	−0.00983341	+	0.999952i	$0.503130\pi$
$912$	0	0
$913$	47.7752	1.58113
$914$	−41.6610	−1.37802
$915$	0	0
$916$	1.98890	0.0657150
$917$	20.8879	0.689779
$918$	0	0
$919$	7.37035	0.243125	0.121563	−	0.992584i	$-0.461209\pi$
$919$	7.37035	0.243125	0.121563	+	0.992584i	$0.461209\pi$
$920$	−15.4205	−0.508397
$921$	0	0
$922$	−28.7942	−0.948285
$923$	−11.7821	−0.387813
$924$	0	0
$925$	−36.6435	−1.20483
$926$	1.89638	0.0623188
$927$	0	0
$928$	−4.63236	−0.152065
$929$	−0.386422	−0.0126781	−0.00633905	−	0.999980i	$-0.502018\pi$
$929$	−0.386422	−0.0126781	−0.00633905	+	0.999980i	$0.502018\pi$
$930$	0	0
$931$	3.01866	0.0989327
$932$	−5.58928	−0.183083
$933$	0	0
$934$	9.53499	0.311994
$935$	43.3218	1.41678
$936$	0	0
$937$	−29.0617	−0.949405	−0.474702	−	0.880146i	$-0.657444\pi$
$937$	−29.0617	−0.949405	−0.474702	+	0.880146i	$0.657444\pi$
$938$	−31.1138	−1.01590
$939$	0	0
$940$	−28.3425	−0.924429
$941$	22.0276	0.718081	0.359040	−	0.933322i	$-0.383104\pi$
$941$	22.0276	0.718081	0.359040	+	0.933322i	$0.383104\pi$
$942$	0	0
$943$	−12.7699	−0.415847
$944$	1.01485	0.0330305
$945$	0	0
$946$	33.8744	1.10135
$947$	6.64851	0.216047	0.108024	−	0.994148i	$-0.465548\pi$
$947$	6.64851	0.216047	0.108024	+	0.994148i	$0.465548\pi$
$948$	0	0
$949$	−0.620763	−0.0201508
$950$	−27.9520	−0.906884
$951$	0	0
$952$	6.45602	0.209241
$953$	57.4933	1.86239	0.931195	−	0.364521i	$-0.118767\pi$
$953$	57.4933	1.86239	0.931195	+	0.364521i	$0.118767\pi$
$954$	0	0
$955$	−8.08216	−0.261532
$956$	3.34553	0.108202
$957$	0	0
$958$	−11.2845	−0.364586
$959$	−34.7929	−1.12352
$960$	0	0
$961$	35.7360	1.15277
$962$	9.00108	0.290207
$963$	0	0
$964$	26.7139	0.860395
$965$	31.1795	1.00370
$966$	0	0
$967$	44.0592	1.41685	0.708424	−	0.705788i	$-0.249407\pi$
$967$	44.0592	1.41685	0.708424	+	0.705788i	$0.249407\pi$
$968$	22.3591	0.718647
$969$	0	0
$970$	−2.84998	−0.0915072
$971$	−25.8900	−0.830850	−0.415425	−	0.909627i	$-0.636367\pi$
$971$	−25.8900	−0.830850	−0.415425	+	0.909627i	$0.636367\pi$
$972$	0	0
$973$	45.1864	1.44861
$974$	13.3515	0.427810
$975$	0	0
$976$	−5.86716	−0.187803
$977$	−16.1799	−0.517642	−0.258821	−	0.965925i	$-0.583334\pi$
$977$	−16.1799	−0.517642	−0.258821	+	0.965925i	$0.583334\pi$
$978$	0	0
$979$	−16.7410	−0.535046
$980$	1.24968	0.0399197
$981$	0	0
$982$	−37.9516	−1.21108
$983$	27.7004	0.883504	0.441752	−	0.897137i	$-0.354357\pi$
$983$	27.7004	0.883504	0.441752	+	0.897137i	$0.354357\pi$
$984$	0	0
$985$	−24.3577	−0.776101
$986$	11.6582	0.371272
$987$	0	0
$988$	6.86613	0.218441
$989$	30.3453	0.964923
$990$	0	0
$991$	−15.6306	−0.496523	−0.248262	−	0.968693i	$-0.579859\pi$
$991$	−15.6306	−0.496523	−0.248262	+	0.968693i	$0.579859\pi$
$992$	8.16921	0.259373
$993$	0	0
$994$	−31.6908	−1.00517
$995$	−69.9126	−2.21638
$996$	0	0
$997$	59.0197	1.86917	0.934586	−	0.355737i	$-0.115770\pi$
$997$	59.0197	1.86917	0.934586	+	0.355737i	$0.115770\pi$
$998$	−27.6365	−0.874819
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4842.2.a.o.1.1		7
3.2	odd	2		538.2.a.d.1.4	✓	7
12.11	even	2		4304.2.a.i.1.4		7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
538.2.a.d.1.4	✓	7	3.2	odd	2
4304.2.a.i.1.4		7	12.11	even	2
4842.2.a.o.1.1		7	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:	\( N \)	\(=\)	\( 4842 = 2 \cdot 3^{2} \cdot 269 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4842.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} -2.98038 q^{5} -2.56529 q^{7} +1.00000 q^{8} -2.98038 q^{10} +5.77573 q^{11} -0.953732 q^{13} -2.56529 q^{14} +1.00000 q^{16} -2.51668 q^{17} -7.19922 q^{19} -2.98038 q^{20} +5.77573 q^{22} +5.17399 q^{23} +3.88265 q^{25} -0.953732 q^{26} -2.56529 q^{28} -4.63236 q^{29} +8.16921 q^{31} +1.00000 q^{32} -2.51668 q^{34} +7.64552 q^{35} -9.43775 q^{37} -7.19922 q^{38} -2.98038 q^{40} -2.46810 q^{41} +5.86496 q^{43} +5.77573 q^{44} +5.17399 q^{46} +9.50969 q^{47} -0.419304 q^{49} +3.88265 q^{50} -0.953732 q^{52} +7.51226 q^{53} -17.2139 q^{55} -2.56529 q^{56} -4.63236 q^{58} +1.01485 q^{59} -5.86716 q^{61} +8.16921 q^{62} +1.00000 q^{64} +2.84248 q^{65} +12.1288 q^{67} -2.51668 q^{68} +7.64552 q^{70} +12.3537 q^{71} +0.650878 q^{73} -9.43775 q^{74} -7.19922 q^{76} -14.8164 q^{77} +15.5158 q^{79} -2.98038 q^{80} -2.46810 q^{82} +8.27171 q^{83} +7.50067 q^{85} +5.86496 q^{86} +5.77573 q^{88} -2.89852 q^{89} +2.44660 q^{91} +5.17399 q^{92} +9.50969 q^{94} +21.4564 q^{95} +0.956247 q^{97} -0.419304 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q + 7 q^{2} + 7 q^{4} + 6 q^{5} - 3 q^{7} + 7 q^{8} + 6 q^{10} + 12 q^{11} + 3 q^{13} - 3 q^{14} + 7 q^{16} + 8 q^{17} - 7 q^{19} + 6 q^{20} + 12 q^{22} + 22 q^{23} + 9 q^{25} + 3 q^{26} - 3 q^{28} + 7 q^{29}+ \cdots + 6 q^{98}+O(q^{100}) \) 7 * q + 7 * q^2 + 7 * q^4 + 6 * q^5 - 3 * q^7 + 7 * q^8 + 6 * q^10 + 12 * q^11 + 3 * q^13 - 3 * q^14 + 7 * q^16 + 8 * q^17 - 7 * q^19 + 6 * q^20 + 12 * q^22 + 22 * q^23 + 9 * q^25 + 3 * q^26 - 3 * q^28 + 7 * q^29 - 3 * q^31 + 7 * q^32 + 8 * q^34 + 25 * q^35 - 13 * q^37 - 7 * q^38 + 6 * q^40 + 11 * q^41 - 14 * q^43 + 12 * q^44 + 22 * q^46 + 23 * q^47 + 6 * q^49 + 9 * q^50 + 3 * q^52 + 21 * q^53 + 15 * q^55 - 3 * q^56 + 7 * q^58 + 20 * q^59 + 13 * q^61 - 3 * q^62 + 7 * q^64 + 29 * q^65 - 30 * q^67 + 8 * q^68 + 25 * q^70 + 23 * q^71 + 9 * q^73 - 13 * q^74 - 7 * q^76 - 8 * q^77 + 11 * q^79 + 6 * q^80 + 11 * q^82 + 3 * q^83 + 21 * q^85 - 14 * q^86 + 12 * q^88 - 16 * q^89 - 10 * q^91 + 22 * q^92 + 23 * q^94 + 21 * q^97 + 6 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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