LMFDB - Embedded newform 4842.2.a.n.1.2

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:	$1$
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} - 15x^{5} + 16x^{4} + 49x^{3} - 53x^{2} - 44x + 48 $ x^7 - x^6 - 15x^5 + 16x^4 + 49x^3 - 53x^2 - 44*x + 48
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
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			1.2
Root			$1.19866$ 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)

$f(q)$	$=$	\(q-1.00000 q^{2} +1.00000 q^{4} -3.84433 q^{5} -4.95403 q^{7} -1.00000 q^{8} +3.84433 q^{10} +0.453515 q^{11} -0.0945931 q^{13} +4.95403 q^{14} +1.00000 q^{16} -1.41349 q^{17} -2.49192 q^{19} -3.84433 q^{20} -0.453515 q^{22} +2.25654 q^{23} +9.77888 q^{25} +0.0945931 q^{26} -4.95403 q^{28} +6.64864 q^{29} +5.36736 q^{31} -1.00000 q^{32} +1.41349 q^{34} +19.0449 q^{35} +7.31117 q^{37} +2.49192 q^{38} +3.84433 q^{40} +10.6653 q^{41} -12.2765 q^{43} +0.453515 q^{44} -2.25654 q^{46} +1.43212 q^{47} +17.5424 q^{49} -9.77888 q^{50} -0.0945931 q^{52} -9.96897 q^{53} -1.74346 q^{55} +4.95403 q^{56} -6.64864 q^{58} -7.89955 q^{59} +0.950317 q^{61} -5.36736 q^{62} +1.00000 q^{64} +0.363647 q^{65} -8.06276 q^{67} -1.41349 q^{68} -19.0449 q^{70} +3.06085 q^{71} +12.8264 q^{73} -7.31117 q^{74} -2.49192 q^{76} -2.24672 q^{77} +13.7673 q^{79} -3.84433 q^{80} -10.6653 q^{82} -10.1171 q^{83} +5.43392 q^{85} +12.2765 q^{86} -0.453515 q^{88} +1.77803 q^{89} +0.468616 q^{91} +2.25654 q^{92} -1.43212 q^{94} +9.57975 q^{95} -12.9137 q^{97} -17.5424 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q - 7 q^{2} + 7 q^{4} - 7 q^{5} + 6 q^{7} - 7 q^{8} + 7 q^{10} + 3 q^{11} - 9 q^{13} - 6 q^{14} + 7 q^{16} - 8 q^{17} - 11 q^{19} - 7 q^{20} - 3 q^{22} - 12 q^{23} + 22 q^{25} + 9 q^{26} + 6 q^{28} + 5 q^{29}+ \cdots - 15 q^{98}+O(q^{100}) $ 7 * q - 7 * q^2 + 7 * q^4 - 7 * q^5 + 6 * q^7 - 7 * q^8 + 7 * q^10 + 3 * q^11 - 9 * q^13 - 6 * q^14 + 7 * q^16 - 8 * q^17 - 11 * q^19 - 7 * q^20 - 3 * q^22 - 12 * q^23 + 22 * q^25 + 9 * q^26 + 6 * q^28 + 5 * q^29 + 14 * q^31 - 7 * q^32 + 8 * q^34 + 4 * q^35 + 13 * q^37 + 11 * q^38 + 7 * q^40 - 12 * q^41 - 11 * q^43 + 3 * q^44 + 12 * q^46 - 2 * q^47 + 15 * q^49 - 22 * q^50 - 9 * q^52 - 19 * q^53 - 40 * q^55 - 6 * q^56 - 5 * q^58 + 9 * q^59 - 3 * q^61 - 14 * q^62 + 7 * q^64 + 10 * q^65 - 33 * q^67 - 8 * q^68 - 4 * q^70 - 28 * q^71 - 14 * q^73 - 13 * q^74 - 11 * q^76 - 10 * q^77 + 2 * q^79 - 7 * q^80 + 12 * q^82 + 7 * q^83 - 16 * q^85 + 11 * q^86 - 3 * q^88 - 18 * q^89 - 26 * q^91 - 12 * q^92 + 2 * q^94 + 34 * q^95 - 4 * q^97 - 15 * 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
$5$	−3.84433	−1.71924	−0.859619	−	0.510936i	$-0.829299\pi$
$5$	−3.84433	−1.71924	−0.859619	+	0.510936i	$0.829299\pi$
$6$	0	0
$7$	−4.95403	−1.87245	−0.936223	−	0.351407i	$-0.885703\pi$
$7$	−4.95403	−1.87245	−0.936223	+	0.351407i	$0.885703\pi$
$8$	−1.00000	−0.353553
$9$	0	0
$10$	3.84433	1.21568
$11$	0.453515	0.136740	0.0683699	−	0.997660i	$-0.478220\pi$
$11$	0.453515	0.136740	0.0683699	+	0.997660i	$0.478220\pi$
$12$	0	0
$13$	−0.0945931	−0.0262354	−0.0131177	−	0.999914i	$-0.504176\pi$
$13$	−0.0945931	−0.0262354	−0.0131177	+	0.999914i	$0.504176\pi$
$14$	4.95403	1.32402
$15$	0	0
$16$	1.00000	0.250000
$17$	−1.41349	−0.342822	−0.171411	−	0.985200i	$-0.554833\pi$
$17$	−1.41349	−0.342822	−0.171411	+	0.985200i	$0.554833\pi$
$18$	0	0
$19$	−2.49192	−0.571685	−0.285842	−	0.958277i	$-0.592273\pi$
$19$	−2.49192	−0.571685	−0.285842	+	0.958277i	$0.592273\pi$
$20$	−3.84433	−0.859619
$21$	0	0
$22$	−0.453515	−0.0966896
$23$	2.25654	0.470521	0.235261	−	0.971932i	$-0.424406\pi$
$23$	2.25654	0.470521	0.235261	+	0.971932i	$0.424406\pi$
$24$	0	0
$25$	9.77888	1.95578
$26$	0.0945931	0.0185512
$27$	0	0
$28$	−4.95403	−0.936223
$29$	6.64864	1.23462	0.617311	−	0.786720i	$-0.288222\pi$
$29$	6.64864	1.23462	0.617311	+	0.786720i	$0.288222\pi$
$30$	0	0
$31$	5.36736	0.964006	0.482003	−	0.876170i	$-0.339909\pi$
$31$	5.36736	0.964006	0.482003	+	0.876170i	$0.339909\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	1.41349	0.242412
$35$	19.0449	3.21918
$36$	0	0
$37$	7.31117	1.20195	0.600974	−	0.799268i	$-0.294779\pi$
$37$	7.31117	1.20195	0.600974	+	0.799268i	$0.294779\pi$
$38$	2.49192	0.404242
$39$	0	0
$40$	3.84433	0.607842
$41$	10.6653	1.66563	0.832817	−	0.553548i	$-0.186726\pi$
$41$	10.6653	1.66563	0.832817	+	0.553548i	$0.186726\pi$
$42$	0	0
$43$	−12.2765	−1.87215	−0.936076	−	0.351798i	$-0.885570\pi$
$43$	−12.2765	−1.87215	−0.936076	+	0.351798i	$0.885570\pi$
$44$	0.453515	0.0683699
$45$	0	0
$46$	−2.25654	−0.332709
$47$	1.43212	0.208897	0.104448	−	0.994530i	$-0.466692\pi$
$47$	1.43212	0.208897	0.104448	+	0.994530i	$0.466692\pi$
$48$	0	0
$49$	17.5424	2.50605
$50$	−9.77888	−1.38294
$51$	0	0
$52$	−0.0945931	−0.0131177
$53$	−9.96897	−1.36934	−0.684672	−	0.728852i	$-0.740054\pi$
$53$	−9.96897	−1.36934	−0.684672	+	0.728852i	$0.740054\pi$
$54$	0	0
$55$	−1.74346	−0.235088
$56$	4.95403	0.662010
$57$	0	0
$58$	−6.64864	−0.873009
$59$	−7.89955	−1.02843	−0.514217	−	0.857660i	$-0.671917\pi$
$59$	−7.89955	−1.02843	−0.514217	+	0.857660i	$0.671917\pi$
$60$	0	0
$61$	0.950317	0.121676	0.0608378	−	0.998148i	$-0.480623\pi$
$61$	0.950317	0.121676	0.0608378	+	0.998148i	$0.480623\pi$
$62$	−5.36736	−0.681655
$63$	0	0
$64$	1.00000	0.125000
$65$	0.363647	0.0451049
$66$	0	0
$67$	−8.06276	−0.985023	−0.492512	−	0.870306i	$-0.663921\pi$
$67$	−8.06276	−0.985023	−0.492512	+	0.870306i	$0.663921\pi$
$68$	−1.41349	−0.171411
$69$	0	0
$70$	−19.0449	−2.27630
$71$	3.06085	0.363256	0.181628	−	0.983367i	$-0.441863\pi$
$71$	3.06085	0.363256	0.181628	+	0.983367i	$0.441863\pi$
$72$	0	0
$73$	12.8264	1.50121	0.750607	−	0.660749i	$-0.229761\pi$
$73$	12.8264	1.50121	0.750607	+	0.660749i	$0.229761\pi$
$74$	−7.31117	−0.849906
$75$	0	0
$76$	−2.49192	−0.285842
$77$	−2.24672	−0.256038
$78$	0	0
$79$	13.7673	1.54894	0.774469	−	0.632611i	$-0.218017\pi$
$79$	13.7673	1.54894	0.774469	+	0.632611i	$0.218017\pi$
$80$	−3.84433	−0.429809
$81$	0	0
$82$	−10.6653	−1.17778
$83$	−10.1171	−1.11049	−0.555246	−	0.831686i	$-0.687376\pi$
$83$	−10.1171	−1.11049	−0.555246	+	0.831686i	$0.687376\pi$
$84$	0	0
$85$	5.43392	0.589392
$86$	12.2765	1.32381
$87$	0	0
$88$	−0.453515	−0.0483448
$89$	1.77803	0.188471	0.0942354	−	0.995550i	$-0.469959\pi$
$89$	1.77803	0.188471	0.0942354	+	0.995550i	$0.469959\pi$
$90$	0	0
$91$	0.468616	0.0491244
$92$	2.25654	0.235261
$93$	0	0
$94$	−1.43212	−0.147712
$95$	9.57975	0.982862
$96$	0	0
$97$	−12.9137	−1.31119	−0.655593	−	0.755115i	$-0.727581\pi$
$97$	−12.9137	−1.31119	−0.655593	+	0.755115i	$0.727581\pi$
$98$	−17.5424	−1.77205
$99$	0	0
$100$	9.77888	0.977888
$101$	6.03448	0.600453	0.300226	−	0.953868i	$-0.402938\pi$
$101$	6.03448	0.600453	0.300226	+	0.953868i	$0.402938\pi$
$102$	0	0
$103$	−1.09297	−0.107694	−0.0538468	−	0.998549i	$-0.517148\pi$
$103$	−1.09297	−0.107694	−0.0538468	+	0.998549i	$0.517148\pi$
$104$	0.0945931	0.00927561
$105$	0	0
$106$	9.96897	0.968272
$107$	−5.58489	−0.539911	−0.269956	−	0.962873i	$-0.587009\pi$
$107$	−5.58489	−0.539911	−0.269956	+	0.962873i	$0.587009\pi$
$108$	0	0
$109$	−6.76759	−0.648218	−0.324109	−	0.946020i	$-0.605064\pi$
$109$	−6.76759	−0.648218	−0.324109	+	0.946020i	$0.605064\pi$
$110$	1.74346	0.166232
$111$	0	0
$112$	−4.95403	−0.468111
$113$	0.653864	0.0615103	0.0307552	−	0.999527i	$-0.490209\pi$
$113$	0.653864	0.0615103	0.0307552	+	0.999527i	$0.490209\pi$
$114$	0	0
$115$	−8.67489	−0.808937
$116$	6.64864	0.617311
$117$	0	0
$118$	7.89955	0.727213
$119$	7.00247	0.641915
$120$	0	0
$121$	−10.7943	−0.981302
$122$	−0.950317	−0.0860376
$123$	0	0
$124$	5.36736	0.482003
$125$	−18.3716	−1.64321
$126$	0	0
$127$	8.29619	0.736168	0.368084	−	0.929792i	$-0.380014\pi$
$127$	8.29619	0.736168	0.368084	+	0.929792i	$0.380014\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	−0.363647	−0.0318940
$131$	15.8736	1.38688	0.693440	−	0.720514i	$-0.256094\pi$
$131$	15.8736	1.38688	0.693440	+	0.720514i	$0.256094\pi$
$132$	0	0
$133$	12.3450	1.07045
$134$	8.06276	0.696517
$135$	0	0
$136$	1.41349	0.121206
$137$	5.16032	0.440876	0.220438	−	0.975401i	$-0.429251\pi$
$137$	5.16032	0.440876	0.220438	+	0.975401i	$0.429251\pi$
$138$	0	0
$139$	15.2369	1.29237	0.646187	−	0.763179i	$-0.276363\pi$
$139$	15.2369	1.29237	0.646187	+	0.763179i	$0.276363\pi$
$140$	19.0449	1.60959
$141$	0	0
$142$	−3.06085	−0.256861
$143$	−0.0428993	−0.00358742
$144$	0	0
$145$	−25.5596	−2.12261
$146$	−12.8264	−1.06152
$147$	0	0
$148$	7.31117	0.600974
$149$	−16.0979	−1.31879	−0.659397	−	0.751795i	$-0.729188\pi$
$149$	−16.0979	−1.31879	−0.659397	+	0.751795i	$0.729188\pi$
$150$	0	0
$151$	−7.57215	−0.616212	−0.308106	−	0.951352i	$-0.599695\pi$
$151$	−7.57215	−0.616212	−0.308106	+	0.951352i	$0.599695\pi$
$152$	2.49192	0.202121
$153$	0	0
$154$	2.24672	0.181046
$155$	−20.6339	−1.65735
$156$	0	0
$157$	21.8948	1.74739	0.873696	−	0.486472i	$-0.161717\pi$
$157$	21.8948	1.74739	0.873696	+	0.486472i	$0.161717\pi$
$158$	−13.7673	−1.09527
$159$	0	0
$160$	3.84433	0.303921
$161$	−11.1790	−0.881025
$162$	0	0
$163$	−6.02116	−0.471614	−0.235807	−	0.971800i	$-0.575773\pi$
$163$	−6.02116	−0.471614	−0.235807	+	0.971800i	$0.575773\pi$
$164$	10.6653	0.832817
$165$	0	0
$166$	10.1171	0.785237
$167$	−19.6784	−1.52276	−0.761382	−	0.648304i	$-0.775479\pi$
$167$	−19.6784	−1.52276	−0.761382	+	0.648304i	$0.775479\pi$
$168$	0	0
$169$	−12.9911	−0.999312
$170$	−5.43392	−0.416763
$171$	0	0
$172$	−12.2765	−0.936076
$173$	−12.4812	−0.948931	−0.474466	−	0.880274i	$-0.657359\pi$
$173$	−12.4812	−0.948931	−0.474466	+	0.880274i	$0.657359\pi$
$174$	0	0
$175$	−48.4448	−3.66209
$176$	0.453515	0.0341849
$177$	0	0
$178$	−1.77803	−0.133269
$179$	−7.84995	−0.586733	−0.293366	−	0.956000i	$-0.594776\pi$
$179$	−7.84995	−0.586733	−0.293366	+	0.956000i	$0.594776\pi$
$180$	0	0
$181$	−17.9797	−1.33642	−0.668209	−	0.743973i	$-0.732939\pi$
$181$	−17.9797	−1.33642	−0.668209	+	0.743973i	$0.732939\pi$
$182$	−0.468616	−0.0347362
$183$	0	0
$184$	−2.25654	−0.166354
$185$	−28.1065	−2.06643
$186$	0	0
$187$	−0.641038	−0.0468774
$188$	1.43212	0.104448
$189$	0	0
$190$	−9.57975	−0.694988
$191$	12.3054	0.890386	0.445193	−	0.895435i	$-0.353135\pi$
$191$	12.3054	0.890386	0.445193	+	0.895435i	$0.353135\pi$
$192$	0	0
$193$	5.82207	0.419081	0.209541	−	0.977800i	$-0.432803\pi$
$193$	5.82207	0.419081	0.209541	+	0.977800i	$0.432803\pi$
$194$	12.9137	0.927148
$195$	0	0
$196$	17.5424	1.25303
$197$	21.6619	1.54334	0.771671	−	0.636021i	$-0.219421\pi$
$197$	21.6619	1.54334	0.771671	+	0.636021i	$0.219421\pi$
$198$	0	0
$199$	5.71843	0.405369	0.202684	−	0.979244i	$-0.435033\pi$
$199$	5.71843	0.405369	0.202684	+	0.979244i	$0.435033\pi$
$200$	−9.77888	−0.691471
$201$	0	0
$202$	−6.03448	−0.424584
$203$	−32.9375	−2.31176
$204$	0	0
$205$	−41.0008	−2.86362
$206$	1.09297	0.0761509
$207$	0	0
$208$	−0.0945931	−0.00655885
$209$	−1.13012	−0.0781721
$210$	0	0
$211$	−20.0839	−1.38264	−0.691318	−	0.722551i	$-0.742970\pi$
$211$	−20.0839	−1.38264	−0.691318	+	0.722551i	$0.742970\pi$
$212$	−9.96897	−0.684672
$213$	0	0
$214$	5.58489	0.381775
$215$	47.1950	3.21867
$216$	0	0
$217$	−26.5900	−1.80505
$218$	6.76759	0.458359
$219$	0	0
$220$	−1.74346	−0.117544
$221$	0.133706	0.00899406
$222$	0	0
$223$	−5.45166	−0.365070	−0.182535	−	0.983199i	$-0.558430\pi$
$223$	−5.45166	−0.365070	−0.182535	+	0.983199i	$0.558430\pi$
$224$	4.95403	0.331005
$225$	0	0
$226$	−0.653864	−0.0434944
$227$	−14.8158	−0.983360	−0.491680	−	0.870776i	$-0.663617\pi$
$227$	−14.8158	−0.983360	−0.491680	+	0.870776i	$0.663617\pi$
$228$	0	0
$229$	21.9863	1.45289	0.726447	−	0.687223i	$-0.241170\pi$
$229$	21.9863	1.45289	0.726447	+	0.687223i	$0.241170\pi$
$230$	8.67489	0.572005
$231$	0	0
$232$	−6.64864	−0.436504
$233$	−17.7722	−1.16429	−0.582147	−	0.813084i	$-0.697787\pi$
$233$	−17.7722	−1.16429	−0.582147	+	0.813084i	$0.697787\pi$
$234$	0	0
$235$	−5.50555	−0.359143
$236$	−7.89955	−0.514217
$237$	0	0
$238$	−7.00247	−0.453902
$239$	12.0820	0.781517	0.390759	−	0.920493i	$-0.372213\pi$
$239$	12.0820	0.781517	0.390759	+	0.920493i	$0.372213\pi$
$240$	0	0
$241$	11.3215	0.729285	0.364642	−	0.931148i	$-0.381191\pi$
$241$	11.3215	0.729285	0.364642	+	0.931148i	$0.381191\pi$
$242$	10.7943	0.693885
$243$	0	0
$244$	0.950317	0.0608378
$245$	−67.4387	−4.30850
$246$	0	0
$247$	0.235718	0.0149984
$248$	−5.36736	−0.340827
$249$	0	0
$250$	18.3716	1.16192
$251$	−8.60802	−0.543333	−0.271667	−	0.962391i	$-0.587575\pi$
$251$	−8.60802	−0.543333	−0.271667	+	0.962391i	$0.587575\pi$
$252$	0	0
$253$	1.02337	0.0643390
$254$	−8.29619	−0.520550
$255$	0	0
$256$	1.00000	0.0625000
$257$	14.3976	0.898096	0.449048	−	0.893508i	$-0.351763\pi$
$257$	14.3976	0.898096	0.449048	+	0.893508i	$0.351763\pi$
$258$	0	0
$259$	−36.2197	−2.25058
$260$	0.363647	0.0225524
$261$	0	0
$262$	−15.8736	−0.980673
$263$	22.3410	1.37761	0.688803	−	0.724949i	$-0.258137\pi$
$263$	22.3410	1.37761	0.688803	+	0.724949i	$0.258137\pi$
$264$	0	0
$265$	38.3240	2.35423
$266$	−12.3450	−0.756922
$267$	0	0
$268$	−8.06276	−0.492512
$269$	1.00000	0.0609711
$269$	1.00000	0.0609711
$270$	0	0
$271$	16.7534	1.01770	0.508848	−	0.860856i	$-0.330071\pi$
$271$	16.7534	1.01770	0.508848	+	0.860856i	$0.330071\pi$
$272$	−1.41349	−0.0857054
$273$	0	0
$274$	−5.16032	−0.311747
$275$	4.43487	0.267432
$276$	0	0
$277$	14.7850	0.888345	0.444172	−	0.895941i	$-0.353498\pi$
$277$	14.7850	0.888345	0.444172	+	0.895941i	$0.353498\pi$
$278$	−15.2369	−0.913846
$279$	0	0
$280$	−19.0449	−1.13815
$281$	−25.0439	−1.49400	−0.746998	−	0.664827i	$-0.768505\pi$
$281$	−25.0439	−1.49400	−0.746998	+	0.664827i	$0.768505\pi$
$282$	0	0
$283$	24.6030	1.46249	0.731247	−	0.682113i	$-0.238939\pi$
$283$	24.6030	1.46249	0.731247	+	0.682113i	$0.238939\pi$
$284$	3.06085	0.181628
$285$	0	0
$286$	0.0428993	0.00253669
$287$	−52.8360	−3.11881
$288$	0	0
$289$	−15.0020	−0.882473
$290$	25.5596	1.50091
$291$	0	0
$292$	12.8264	0.750607
$293$	11.4618	0.669607	0.334804	−	0.942288i	$-0.391330\pi$
$293$	11.4618	0.669607	0.334804	+	0.942288i	$0.391330\pi$
$294$	0	0
$295$	30.3685	1.76812
$296$	−7.31117	−0.424953
$297$	0	0
$298$	16.0979	0.932528
$299$	−0.213453	−0.0123443
$300$	0	0
$301$	60.8182	3.50550
$302$	7.57215	0.435728
$303$	0	0
$304$	−2.49192	−0.142921
$305$	−3.65333	−0.209189
$306$	0	0
$307$	−17.7679	−1.01407	−0.507033	−	0.861927i	$-0.669258\pi$
$307$	−17.7679	−1.01407	−0.507033	+	0.861927i	$0.669258\pi$
$308$	−2.24672	−0.128019
$309$	0	0
$310$	20.6339	1.17193
$311$	−7.62146	−0.432173	−0.216087	−	0.976374i	$-0.569329\pi$
$311$	−7.62146	−0.432173	−0.216087	+	0.976374i	$0.569329\pi$
$312$	0	0
$313$	7.97031	0.450509	0.225254	−	0.974300i	$-0.427679\pi$
$313$	7.97031	0.450509	0.225254	+	0.974300i	$0.427679\pi$
$314$	−21.8948	−1.23559
$315$	0	0
$316$	13.7673	0.774469
$317$	−1.71640	−0.0964028	−0.0482014	−	0.998838i	$-0.515349\pi$
$317$	−1.71640	−0.0964028	−0.0482014	+	0.998838i	$0.515349\pi$
$318$	0	0
$319$	3.01525	0.168822
$320$	−3.84433	−0.214905
$321$	0	0
$322$	11.1790	0.622979
$323$	3.52230	0.195986
$324$	0	0
$325$	−0.925014	−0.0513106
$326$	6.02116	0.333481
$327$	0	0
$328$	−10.6653	−0.588891
$329$	−7.09477	−0.391147
$330$	0	0
$331$	−13.7537	−0.755970	−0.377985	−	0.925812i	$-0.623383\pi$
$331$	−13.7537	−0.755970	−0.377985	+	0.925812i	$0.623383\pi$
$332$	−10.1171	−0.555246
$333$	0	0
$334$	19.6784	1.07676
$335$	30.9959	1.69349
$336$	0	0
$337$	18.7032	1.01883	0.509413	−	0.860522i	$-0.329863\pi$
$337$	18.7032	1.01883	0.509413	+	0.860522i	$0.329863\pi$
$338$	12.9911	0.706620
$339$	0	0
$340$	5.43392	0.294696
$341$	2.43417	0.131818
$342$	0	0
$343$	−52.2272	−2.82000
$344$	12.2765	0.661906
$345$	0	0
$346$	12.4812	0.670996
$347$	−12.9029	−0.692662	−0.346331	−	0.938112i	$-0.612573\pi$
$347$	−12.9029	−0.692662	−0.346331	+	0.938112i	$0.612573\pi$
$348$	0	0
$349$	−4.29547	−0.229931	−0.114966	−	0.993369i	$-0.536676\pi$
$349$	−4.29547	−0.229931	−0.114966	+	0.993369i	$0.536676\pi$
$350$	48.4448	2.58949
$351$	0	0
$352$	−0.453515	−0.0241724
$353$	22.3707	1.19067	0.595337	−	0.803476i	$-0.297018\pi$
$353$	22.3707	1.19067	0.595337	+	0.803476i	$0.297018\pi$
$354$	0	0
$355$	−11.7669	−0.624523
$356$	1.77803	0.0942354
$357$	0	0
$358$	7.84995	0.414883
$359$	−32.0628	−1.69221	−0.846105	−	0.533016i	$-0.821058\pi$
$359$	−32.0628	−1.69221	−0.846105	+	0.533016i	$0.821058\pi$
$360$	0	0
$361$	−12.7903	−0.673176
$362$	17.9797	0.944991
$363$	0	0
$364$	0.468616	0.0245622
$365$	−49.3088	−2.58094
$366$	0	0
$367$	−4.79576	−0.250337	−0.125168	−	0.992136i	$-0.539947\pi$
$367$	−4.79576	−0.250337	−0.125168	+	0.992136i	$0.539947\pi$
$368$	2.25654	0.117630
$369$	0	0
$370$	28.1065	1.46119
$371$	49.3865	2.56402
$372$	0	0
$373$	32.9108	1.70406	0.852028	−	0.523497i	$-0.175373\pi$
$373$	32.9108	1.70406	0.852028	+	0.523497i	$0.175373\pi$
$374$	0.641038	0.0331473
$375$	0	0
$376$	−1.43212	−0.0738561
$377$	−0.628915	−0.0323908
$378$	0	0
$379$	−8.98570	−0.461565	−0.230782	−	0.973005i	$-0.574129\pi$
$379$	−8.98570	−0.461565	−0.230782	+	0.973005i	$0.574129\pi$
$380$	9.57975	0.491431
$381$	0	0
$382$	−12.3054	−0.629598
$383$	−0.787688	−0.0402490	−0.0201245	−	0.999797i	$-0.506406\pi$
$383$	−0.787688	−0.0402490	−0.0201245	+	0.999797i	$0.506406\pi$
$384$	0	0
$385$	8.63715	0.440190
$386$	−5.82207	−0.296335
$387$	0	0
$388$	−12.9137	−0.655593
$389$	4.15609	0.210722	0.105361	−	0.994434i	$-0.466400\pi$
$389$	4.15609	0.210722	0.105361	+	0.994434i	$0.466400\pi$
$390$	0	0
$391$	−3.18960	−0.161305
$392$	−17.5424	−0.886024
$393$	0	0
$394$	−21.6619	−1.09131
$395$	−52.9259	−2.66299
$396$	0	0
$397$	1.90587	0.0956527	0.0478263	−	0.998856i	$-0.484771\pi$
$397$	1.90587	0.0956527	0.0478263	+	0.998856i	$0.484771\pi$
$398$	−5.71843	−0.286639
$399$	0	0
$400$	9.77888	0.488944
$401$	11.0163	0.550128	0.275064	−	0.961426i	$-0.411301\pi$
$401$	11.0163	0.550128	0.275064	+	0.961426i	$0.411301\pi$
$402$	0	0
$403$	−0.507715	−0.0252911
$404$	6.03448	0.300226
$405$	0	0
$406$	32.9375	1.63466
$407$	3.31572	0.164354
$408$	0	0
$409$	−5.30078	−0.262107	−0.131053	−	0.991375i	$-0.541836\pi$
$409$	−5.30078	−0.262107	−0.131053	+	0.991375i	$0.541836\pi$
$410$	41.0008	2.02489
$411$	0	0
$412$	−1.09297	−0.0538468
$413$	39.1346	1.92569
$414$	0	0
$415$	38.8934	1.90920
$416$	0.0945931	0.00463781
$417$	0	0
$418$	1.13012	0.0552760
$419$	25.5590	1.24864	0.624319	−	0.781169i	$-0.285376\pi$
$419$	25.5590	1.24864	0.624319	+	0.781169i	$0.285376\pi$
$420$	0	0
$421$	−24.8795	−1.21255	−0.606276	−	0.795254i	$-0.707337\pi$
$421$	−24.8795	−1.21255	−0.606276	+	0.795254i	$0.707337\pi$
$422$	20.0839	0.977671
$423$	0	0
$424$	9.96897	0.484136
$425$	−13.8224	−0.670483
$426$	0	0
$427$	−4.70789	−0.227831
$428$	−5.58489	−0.269956
$429$	0	0
$430$	−47.1950	−2.27595
$431$	−19.4524	−0.936988	−0.468494	−	0.883467i	$-0.655203\pi$
$431$	−19.4524	−0.936988	−0.468494	+	0.883467i	$0.655203\pi$
$432$	0	0
$433$	−16.7217	−0.803593	−0.401797	−	0.915729i	$-0.631614\pi$
$433$	−16.7217	−0.803593	−0.401797	+	0.915729i	$0.631614\pi$
$434$	26.5900	1.27636
$435$	0	0
$436$	−6.76759	−0.324109
$437$	−5.62311	−0.268990
$438$	0	0
$439$	−14.8215	−0.707390	−0.353695	−	0.935361i	$-0.615075\pi$
$439$	−14.8215	−0.707390	−0.353695	+	0.935361i	$0.615075\pi$
$440$	1.74346	0.0831162
$441$	0	0
$442$	−0.133706	−0.00635976
$443$	24.8635	1.18130	0.590651	−	0.806927i	$-0.298871\pi$
$443$	24.8635	1.18130	0.590651	+	0.806927i	$0.298871\pi$
$444$	0	0
$445$	−6.83533	−0.324026
$446$	5.45166	0.258144
$447$	0	0
$448$	−4.95403	−0.234056
$449$	−22.0294	−1.03963	−0.519817	−	0.854278i	$-0.674000\pi$
$449$	−22.0294	−1.03963	−0.519817	+	0.854278i	$0.674000\pi$
$450$	0	0
$451$	4.83685	0.227759
$452$	0.653864	0.0307552
$453$	0	0
$454$	14.8158	0.695340
$455$	−1.80152	−0.0844564
$456$	0	0
$457$	−27.1457	−1.26982	−0.634911	−	0.772585i	$-0.718963\pi$
$457$	−27.1457	−1.26982	−0.634911	+	0.772585i	$0.718963\pi$
$458$	−21.9863	−1.02735
$459$	0	0
$460$	−8.67489	−0.404469
$461$	−27.6585	−1.28819	−0.644093	−	0.764947i	$-0.722765\pi$
$461$	−27.6585	−1.28819	−0.644093	+	0.764947i	$0.722765\pi$
$462$	0	0
$463$	31.2336	1.45155	0.725774	−	0.687933i	$-0.241482\pi$
$463$	31.2336	1.45155	0.725774	+	0.687933i	$0.241482\pi$
$464$	6.64864	0.308655
$465$	0	0
$466$	17.7722	0.823280
$467$	13.4254	0.621255	0.310628	−	0.950532i	$-0.399461\pi$
$467$	13.4254	0.621255	0.310628	+	0.950532i	$0.399461\pi$
$468$	0	0
$469$	39.9431	1.84440
$470$	5.50555	0.253952
$471$	0	0
$472$	7.89955	0.363606
$473$	−5.56758	−0.255998
$474$	0	0
$475$	−24.3682	−1.11809
$476$	7.00247	0.320958
$477$	0	0
$478$	−12.0820	−0.552616
$479$	−2.09566	−0.0957533	−0.0478767	−	0.998853i	$-0.515245\pi$
$479$	−2.09566	−0.0957533	−0.0478767	+	0.998853i	$0.515245\pi$
$480$	0	0
$481$	−0.691586	−0.0315336
$482$	−11.3215	−0.515682
$483$	0	0
$484$	−10.7943	−0.490651
$485$	49.6445	2.25424
$486$	0	0
$487$	−38.1661	−1.72947	−0.864736	−	0.502226i	$-0.832515\pi$
$487$	−38.1661	−1.72947	−0.864736	+	0.502226i	$0.832515\pi$
$488$	−0.950317	−0.0430188
$489$	0	0
$490$	67.4387	3.04657
$491$	−6.36732	−0.287353	−0.143676	−	0.989625i	$-0.545892\pi$
$491$	−6.36732	−0.287353	−0.143676	+	0.989625i	$0.545892\pi$
$492$	0	0
$493$	−9.39778	−0.423255
$494$	−0.235718	−0.0106055
$495$	0	0
$496$	5.36736	0.241001
$497$	−15.1635	−0.680176
$498$	0	0
$499$	33.6939	1.50835	0.754173	−	0.656676i	$-0.228038\pi$
$499$	33.6939	1.50835	0.754173	+	0.656676i	$0.228038\pi$
$500$	−18.3716	−0.821603
$501$	0	0
$502$	8.60802	0.384195
$503$	29.6505	1.32205	0.661026	−	0.750363i	$-0.270121\pi$
$503$	29.6505	1.32205	0.661026	+	0.750363i	$0.270121\pi$
$504$	0	0
$505$	−23.1985	−1.03232
$506$	−1.02337	−0.0454945
$507$	0	0
$508$	8.29619	0.368084
$509$	3.41765	0.151485	0.0757423	−	0.997127i	$-0.475867\pi$
$509$	3.41765	0.151485	0.0757423	+	0.997127i	$0.475867\pi$
$510$	0	0
$511$	−63.5422	−2.81094
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	−14.3976	−0.635050
$515$	4.20174	0.185151
$516$	0	0
$517$	0.649488	0.0285645
$518$	36.2197	1.59140
$519$	0	0
$520$	−0.363647	−0.0159470
$521$	−28.2730	−1.23866	−0.619330	−	0.785130i	$-0.712596\pi$
$521$	−28.2730	−1.23866	−0.619330	+	0.785130i	$0.712596\pi$
$522$	0	0
$523$	−17.2424	−0.753956	−0.376978	−	0.926222i	$-0.623037\pi$
$523$	−17.2424	−0.753956	−0.376978	+	0.926222i	$0.623037\pi$
$524$	15.8736	0.693440
$525$	0	0
$526$	−22.3410	−0.974114
$527$	−7.58670	−0.330482
$528$	0	0
$529$	−17.9080	−0.778610
$530$	−38.3240	−1.66469
$531$	0	0
$532$	12.3450	0.535225
$533$	−1.00886	−0.0436986
$534$	0	0
$535$	21.4702	0.928236
$536$	8.06276	0.348258
$537$	0	0
$538$	−1.00000	−0.0431131
$539$	7.95572	0.342677
$540$	0	0
$541$	−2.32570	−0.0999896	−0.0499948	−	0.998749i	$-0.515920\pi$
$541$	−2.32570	−0.0999896	−0.0499948	+	0.998749i	$0.515920\pi$
$542$	−16.7534	−0.719620
$543$	0	0
$544$	1.41349	0.0606029
$545$	26.0169	1.11444
$546$	0	0
$547$	26.5955	1.13714	0.568570	−	0.822635i	$-0.307497\pi$
$547$	26.5955	1.13714	0.568570	+	0.822635i	$0.307497\pi$
$548$	5.16032	0.220438
$549$	0	0
$550$	−4.43487	−0.189103
$551$	−16.5679	−0.705814
$552$	0	0
$553$	−68.2034	−2.90030
$554$	−14.7850	−0.628155
$555$	0	0
$556$	15.2369	0.646187
$557$	−19.7222	−0.835658	−0.417829	−	0.908526i	$-0.637209\pi$
$557$	−19.7222	−0.835658	−0.417829	+	0.908526i	$0.637209\pi$
$558$	0	0
$559$	1.16127	0.0491166
$560$	19.0449	0.804795
$561$	0	0
$562$	25.0439	1.05641
$563$	25.9388	1.09319	0.546595	−	0.837397i	$-0.315924\pi$
$563$	25.9388	1.09319	0.546595	+	0.837397i	$0.315924\pi$
$564$	0	0
$565$	−2.51367	−0.105751
$566$	−24.6030	−1.03414
$567$	0	0
$568$	−3.06085	−0.128430
$569$	44.9748	1.88544	0.942720	−	0.333584i	$-0.108258\pi$
$569$	44.9748	1.88544	0.942720	+	0.333584i	$0.108258\pi$
$570$	0	0
$571$	−44.4599	−1.86059	−0.930294	−	0.366814i	$-0.880448\pi$
$571$	−44.4599	−1.86059	−0.930294	+	0.366814i	$0.880448\pi$
$572$	−0.0428993	−0.00179371
$573$	0	0
$574$	52.8360	2.20533
$575$	22.0664	0.920234
$576$	0	0
$577$	−36.6191	−1.52447	−0.762236	−	0.647299i	$-0.775898\pi$
$577$	−36.6191	−1.52447	−0.762236	+	0.647299i	$0.775898\pi$
$578$	15.0020	0.624003
$579$	0	0
$580$	−25.5596	−1.06130
$581$	50.1202	2.07934
$582$	0	0
$583$	−4.52107	−0.187244
$584$	−12.8264	−0.530759
$585$	0	0
$586$	−11.4618	−0.473484
$587$	−20.5256	−0.847184	−0.423592	−	0.905853i	$-0.639231\pi$
$587$	−20.5256	−0.847184	−0.423592	+	0.905853i	$0.639231\pi$
$588$	0	0
$589$	−13.3750	−0.551108
$590$	−30.3685	−1.25025
$591$	0	0
$592$	7.31117	0.300487
$593$	1.99866	0.0820753	0.0410377	−	0.999158i	$-0.486934\pi$
$593$	1.99866	0.0820753	0.0410377	+	0.999158i	$0.486934\pi$
$594$	0	0
$595$	−26.9198	−1.10360
$596$	−16.0979	−0.659397
$597$	0	0
$598$	0.213453	0.00872874
$599$	−21.2473	−0.868141	−0.434070	−	0.900879i	$-0.642923\pi$
$599$	−21.2473	−0.868141	−0.434070	+	0.900879i	$0.642923\pi$
$600$	0	0
$601$	−22.2574	−0.907899	−0.453949	−	0.891028i	$-0.649985\pi$
$601$	−22.2574	−0.907899	−0.453949	+	0.891028i	$0.649985\pi$
$602$	−60.8182	−2.47876
$603$	0	0
$604$	−7.57215	−0.308106
$605$	41.4970	1.68709
$606$	0	0
$607$	7.20720	0.292531	0.146266	−	0.989245i	$-0.453275\pi$
$607$	7.20720	0.292531	0.146266	+	0.989245i	$0.453275\pi$
$608$	2.49192	0.101061
$609$	0	0
$610$	3.65333	0.147919
$611$	−0.135469	−0.00548048
$612$	0	0
$613$	15.9436	0.643956	0.321978	−	0.946747i	$-0.395652\pi$
$613$	15.9436	0.643956	0.321978	+	0.946747i	$0.395652\pi$
$614$	17.7679	0.717052
$615$	0	0
$616$	2.24672	0.0905231
$617$	−18.0019	−0.724729	−0.362364	−	0.932037i	$-0.618030\pi$
$617$	−18.0019	−0.724729	−0.362364	+	0.932037i	$0.618030\pi$
$618$	0	0
$619$	−17.9006	−0.719487	−0.359744	−	0.933051i	$-0.617136\pi$
$619$	−17.9006	−0.719487	−0.359744	+	0.933051i	$0.617136\pi$
$620$	−20.6339	−0.828677
$621$	0	0
$622$	7.62146	0.305593
$623$	−8.80840	−0.352901
$624$	0	0
$625$	21.7321	0.869285
$626$	−7.97031	−0.318558
$627$	0	0
$628$	21.8948	0.873696
$629$	−10.3343	−0.412054
$630$	0	0
$631$	25.6888	1.02265	0.511327	−	0.859386i	$-0.329154\pi$
$631$	25.6888	1.02265	0.511327	+	0.859386i	$0.329154\pi$
$632$	−13.7673	−0.547633
$633$	0	0
$634$	1.71640	0.0681671
$635$	−31.8933	−1.26565
$636$	0	0
$637$	−1.65939	−0.0657473
$638$	−3.01525	−0.119375
$639$	0	0
$640$	3.84433	0.151961
$641$	−25.4553	−1.00542	−0.502712	−	0.864454i	$-0.667664\pi$
$641$	−25.4553	−1.00542	−0.502712	+	0.864454i	$0.667664\pi$
$642$	0	0
$643$	22.7411	0.896823	0.448411	−	0.893827i	$-0.351990\pi$
$643$	22.7411	0.896823	0.448411	+	0.893827i	$0.351990\pi$
$644$	−11.1790	−0.440513
$645$	0	0
$646$	−3.52230	−0.138583
$647$	−28.5975	−1.12428	−0.562142	−	0.827041i	$-0.690023\pi$
$647$	−28.5975	−1.12428	−0.562142	+	0.827041i	$0.690023\pi$
$648$	0	0
$649$	−3.58256	−0.140628
$650$	0.925014	0.0362820
$651$	0	0
$652$	−6.02116	−0.235807
$653$	5.54791	0.217107	0.108553	−	0.994091i	$-0.465378\pi$
$653$	5.54791	0.217107	0.108553	+	0.994091i	$0.465378\pi$
$654$	0	0
$655$	−61.0233	−2.38438
$656$	10.6653	0.416409
$657$	0	0
$658$	7.09477	0.276583
$659$	27.7914	1.08260	0.541299	−	0.840830i	$-0.317933\pi$
$659$	27.7914	1.08260	0.541299	+	0.840830i	$0.317933\pi$
$660$	0	0
$661$	1.92881	0.0750221	0.0375111	−	0.999296i	$-0.488057\pi$
$661$	1.92881	0.0750221	0.0375111	+	0.999296i	$0.488057\pi$
$662$	13.7537	0.534552
$663$	0	0
$664$	10.1171	0.392618
$665$	−47.4584	−1.84036
$666$	0	0
$667$	15.0029	0.580915
$668$	−19.6784	−0.761382
$669$	0	0
$670$	−30.9959	−1.19748
$671$	0.430982	0.0166379
$672$	0	0
$673$	19.2283	0.741195	0.370597	−	0.928794i	$-0.379153\pi$
$673$	19.2283	0.741195	0.370597	+	0.928794i	$0.379153\pi$
$674$	−18.7032	−0.720419
$675$	0	0
$676$	−12.9911	−0.499656
$677$	−3.56493	−0.137011	−0.0685057	−	0.997651i	$-0.521823\pi$
$677$	−3.56493	−0.137011	−0.0685057	+	0.997651i	$0.521823\pi$
$678$	0	0
$679$	63.9747	2.45512
$680$	−5.43392	−0.208381
$681$	0	0
$682$	−2.43417	−0.0932094
$683$	−36.0055	−1.37771	−0.688856	−	0.724899i	$-0.741887\pi$
$683$	−36.0055	−1.37771	−0.688856	+	0.724899i	$0.741887\pi$
$684$	0	0
$685$	−19.8380	−0.757971
$686$	52.2272	1.99404
$687$	0	0
$688$	−12.2765	−0.468038
$689$	0.942995	0.0359253
$690$	0	0
$691$	6.11836	0.232754	0.116377	−	0.993205i	$-0.462872\pi$
$691$	6.11836	0.232754	0.116377	+	0.993205i	$0.462872\pi$
$692$	−12.4812	−0.474466
$693$	0	0
$694$	12.9029	0.489786
$695$	−58.5755	−2.22190
$696$	0	0
$697$	−15.0752	−0.571016
$698$	4.29547	0.162586
$699$	0	0
$700$	−48.4448	−1.83104
$701$	27.4930	1.03840	0.519198	−	0.854654i	$-0.326231\pi$
$701$	27.4930	1.03840	0.519198	+	0.854654i	$0.326231\pi$
$702$	0	0
$703$	−18.2188	−0.687136
$704$	0.453515	0.0170925
$705$	0	0
$706$	−22.3707	−0.841934
$707$	−29.8949	−1.12432
$708$	0	0
$709$	−41.8276	−1.57087	−0.785433	−	0.618946i	$-0.787560\pi$
$709$	−41.8276	−1.57087	−0.785433	+	0.618946i	$0.787560\pi$
$710$	11.7669	0.441604
$711$	0	0
$712$	−1.77803	−0.0666345
$713$	12.1117	0.453585
$714$	0	0
$715$	0.164919	0.00616763
$716$	−7.84995	−0.293366
$717$	0	0
$718$	32.0628	1.19657
$719$	36.4204	1.35825	0.679126	−	0.734022i	$-0.262359\pi$
$719$	36.4204	1.35825	0.679126	+	0.734022i	$0.262359\pi$
$720$	0	0
$721$	5.41461	0.201650
$722$	12.7903	0.476008
$723$	0	0
$724$	−17.9797	−0.668209
$725$	65.0162	2.41464
$726$	0	0
$727$	1.23081	0.0456481	0.0228241	−	0.999739i	$-0.492734\pi$
$727$	1.23081	0.0456481	0.0228241	+	0.999739i	$0.492734\pi$
$728$	−0.468616	−0.0173681
$729$	0	0
$730$	49.3088	1.82500
$731$	17.3527	0.641814
$732$	0	0
$733$	27.5812	1.01873	0.509367	−	0.860550i	$-0.329880\pi$
$733$	27.5812	1.01873	0.509367	+	0.860550i	$0.329880\pi$
$734$	4.79576	0.177015
$735$	0	0
$736$	−2.25654	−0.0831772
$737$	−3.65658	−0.134692
$738$	0	0
$739$	19.3896	0.713257	0.356629	−	0.934246i	$-0.383926\pi$
$739$	19.3896	0.713257	0.356629	+	0.934246i	$0.383926\pi$
$740$	−28.1065	−1.03322
$741$	0	0
$742$	−49.3865	−1.81304
$743$	3.02696	0.111048	0.0555242	−	0.998457i	$-0.482317\pi$
$743$	3.02696	0.111048	0.0555242	+	0.998457i	$0.482317\pi$
$744$	0	0
$745$	61.8858	2.26732
$746$	−32.9108	−1.20495
$747$	0	0
$748$	−0.641038	−0.0234387
$749$	27.6677	1.01096
$750$	0	0
$751$	5.10910	0.186434	0.0932168	−	0.995646i	$-0.470285\pi$
$751$	5.10910	0.186434	0.0932168	+	0.995646i	$0.470285\pi$
$752$	1.43212	0.0522241
$753$	0	0
$754$	0.628915	0.0229037
$755$	29.1098	1.05942
$756$	0	0
$757$	−27.3805	−0.995161	−0.497580	−	0.867418i	$-0.665778\pi$
$757$	−27.3805	−0.995161	−0.497580	+	0.867418i	$0.665778\pi$
$758$	8.98570	0.326375
$759$	0	0
$760$	−9.57975	−0.347494
$761$	49.3210	1.78789	0.893943	−	0.448181i	$-0.147928\pi$
$761$	49.3210	1.78789	0.893943	+	0.448181i	$0.147928\pi$
$762$	0	0
$763$	33.5268	1.21375
$764$	12.3054	0.445193
$765$	0	0
$766$	0.787688	0.0284603
$767$	0.747243	0.0269814
$768$	0	0
$769$	11.8488	0.427277	0.213639	−	0.976913i	$-0.431468\pi$
$769$	11.8488	0.427277	0.213639	+	0.976913i	$0.431468\pi$
$770$	−8.63715	−0.311261
$771$	0	0
$772$	5.82207	0.209541
$773$	−14.0516	−0.505401	−0.252700	−	0.967545i	$-0.581319\pi$
$773$	−14.0516	−0.505401	−0.252700	+	0.967545i	$0.581319\pi$
$774$	0	0
$775$	52.4867	1.88538
$776$	12.9137	0.463574
$777$	0	0
$778$	−4.15609	−0.149003
$779$	−26.5770	−0.952218
$780$	0	0
$781$	1.38814	0.0496715
$782$	3.18960	0.114060
$783$	0	0
$784$	17.5424	0.626513
$785$	−84.1707	−3.00418
$786$	0	0
$787$	−22.7651	−0.811489	−0.405744	−	0.913987i	$-0.632988\pi$
$787$	−22.7651	−0.811489	−0.405744	+	0.913987i	$0.632988\pi$
$788$	21.6619	0.771671
$789$	0	0
$790$	52.9259	1.88302
$791$	−3.23926	−0.115175
$792$	0	0
$793$	−0.0898934	−0.00319221
$794$	−1.90587	−0.0676367
$795$	0	0
$796$	5.71843	0.202684
$797$	−32.9141	−1.16588	−0.582938	−	0.812517i	$-0.698097\pi$
$797$	−32.9141	−1.16588	−0.582938	+	0.812517i	$0.698097\pi$
$798$	0	0
$799$	−2.02429	−0.0716143
$800$	−9.77888	−0.345736
$801$	0	0
$802$	−11.0163	−0.388999
$803$	5.81695	0.205276
$804$	0	0
$805$	42.9756	1.51469
$806$	0.507715	0.0178835
$807$	0	0
$808$	−6.03448	−0.212292
$809$	35.8700	1.26112	0.630560	−	0.776140i	$-0.282825\pi$
$809$	35.8700	1.26112	0.630560	+	0.776140i	$0.282825\pi$
$810$	0	0
$811$	−16.6343	−0.584109	−0.292055	−	0.956402i	$-0.594339\pi$
$811$	−16.6343	−0.584109	−0.292055	+	0.956402i	$0.594339\pi$
$812$	−32.9375	−1.15588
$813$	0	0
$814$	−3.31572	−0.116216
$815$	23.1473	0.810816
$816$	0	0
$817$	30.5921	1.07028
$818$	5.30078	0.185337
$819$	0	0
$820$	−41.0008	−1.43181
$821$	−32.4924	−1.13399	−0.566997	−	0.823720i	$-0.691895\pi$
$821$	−32.4924	−1.13399	−0.566997	+	0.823720i	$0.691895\pi$
$822$	0	0
$823$	−29.9885	−1.04533	−0.522666	−	0.852537i	$-0.675063\pi$
$823$	−29.9885	−1.04533	−0.522666	+	0.852537i	$0.675063\pi$
$824$	1.09297	0.0380754
$825$	0	0
$826$	−39.1346	−1.36167
$827$	−31.1200	−1.08215	−0.541075	−	0.840975i	$-0.681982\pi$
$827$	−31.1200	−1.08215	−0.541075	+	0.840975i	$0.681982\pi$
$828$	0	0
$829$	−15.8365	−0.550025	−0.275013	−	0.961441i	$-0.588682\pi$
$829$	−15.8365	−0.550025	−0.275013	+	0.961441i	$0.588682\pi$
$830$	−38.8934	−1.35001
$831$	0	0
$832$	−0.0945931	−0.00327942
$833$	−24.7960	−0.859129
$834$	0	0
$835$	75.6504	2.61799
$836$	−1.13012	−0.0390860
$837$	0	0
$838$	−25.5590	−0.882921
$839$	12.2519	0.422981	0.211491	−	0.977380i	$-0.432168\pi$
$839$	12.2519	0.422981	0.211491	+	0.977380i	$0.432168\pi$
$840$	0	0
$841$	15.2044	0.524289
$842$	24.8795	0.857403
$843$	0	0
$844$	−20.0839	−0.691318
$845$	49.9419	1.71805
$846$	0	0
$847$	53.4754	1.83744
$848$	−9.96897	−0.342336
$849$	0	0
$850$	13.8224	0.474103
$851$	16.4979	0.565542
$852$	0	0
$853$	−9.41699	−0.322431	−0.161216	−	0.986919i	$-0.551541\pi$
$853$	−9.41699	−0.322431	−0.161216	+	0.986919i	$0.551541\pi$
$854$	4.70789	0.161101
$855$	0	0
$856$	5.58489	0.190888
$857$	−19.2781	−0.658529	−0.329264	−	0.944238i	$-0.606801\pi$
$857$	−19.2781	−0.658529	−0.329264	+	0.944238i	$0.606801\pi$
$858$	0	0
$859$	−48.7953	−1.66487	−0.832437	−	0.554120i	$-0.813055\pi$
$859$	−48.7953	−1.66487	−0.832437	+	0.554120i	$0.813055\pi$
$860$	47.1950	1.60934
$861$	0	0
$862$	19.4524	0.662551
$863$	−27.3052	−0.929481	−0.464741	−	0.885447i	$-0.653852\pi$
$863$	−27.3052	−0.929481	−0.464741	+	0.885447i	$0.653852\pi$
$864$	0	0
$865$	47.9820	1.63144
$866$	16.7217	0.568226
$867$	0	0
$868$	−26.5900	−0.902524
$869$	6.24366	0.211802
$870$	0	0
$871$	0.762681	0.0258425
$872$	6.76759	0.229180
$873$	0	0
$874$	5.62311	0.190205
$875$	91.0134	3.07681
$876$	0	0
$877$	−14.3551	−0.484737	−0.242368	−	0.970184i	$-0.577924\pi$
$877$	−14.3551	−0.484737	−0.242368	+	0.970184i	$0.577924\pi$
$878$	14.8215	0.500201
$879$	0	0
$880$	−1.74346	−0.0587720
$881$	20.4566	0.689201	0.344600	−	0.938749i	$-0.388014\pi$
$881$	20.4566	0.689201	0.344600	+	0.938749i	$0.388014\pi$
$882$	0	0
$883$	13.2088	0.444510	0.222255	−	0.974989i	$-0.428658\pi$
$883$	13.2088	0.444510	0.222255	+	0.974989i	$0.428658\pi$
$884$	0.133706	0.00449703
$885$	0	0
$886$	−24.8635	−0.835307
$887$	21.4469	0.720115	0.360058	−	0.932930i	$-0.382757\pi$
$887$	21.4469	0.720115	0.360058	+	0.932930i	$0.382757\pi$
$888$	0	0
$889$	−41.0996	−1.37844
$890$	6.83533	0.229121
$891$	0	0
$892$	−5.45166	−0.182535
$893$	−3.56873	−0.119423
$894$	0	0
$895$	30.1778	1.00873
$896$	4.95403	0.165502
$897$	0	0
$898$	22.0294	0.735132
$899$	35.6856	1.19018
$900$	0	0
$901$	14.0910	0.469441
$902$	−4.83685	−0.161050
$903$	0	0
$904$	−0.653864	−0.0217472
$905$	69.1198	2.29762
$906$	0	0
$907$	−53.7295	−1.78406	−0.892029	−	0.451977i	$-0.850719\pi$
$907$	−53.7295	−1.78406	−0.892029	+	0.451977i	$0.850719\pi$
$908$	−14.8158	−0.491680
$909$	0	0
$910$	1.80152	0.0597197
$911$	17.7091	0.586728	0.293364	−	0.956001i	$-0.405225\pi$
$911$	17.7091	0.586728	0.293364	+	0.956001i	$0.405225\pi$
$912$	0	0
$913$	−4.58824	−0.151848
$914$	27.1457	0.897900
$915$	0	0
$916$	21.9863	0.726447
$917$	−78.6381	−2.59686
$918$	0	0
$919$	59.7641	1.97144	0.985718	−	0.168406i	$-0.0538619\pi$
$919$	59.7641	1.97144	0.985718	+	0.168406i	$0.0538619\pi$
$920$	8.67489	0.286003
$921$	0	0
$922$	27.6585	0.910885
$923$	−0.289535	−0.00953015
$924$	0	0
$925$	71.4950	2.35074
$926$	−31.2336	−1.02640
$927$	0	0
$928$	−6.64864	−0.218252
$929$	53.9545	1.77019	0.885095	−	0.465410i	$-0.154093\pi$
$929$	53.9545	1.77019	0.885095	+	0.465410i	$0.154093\pi$
$930$	0	0
$931$	−43.7141	−1.43267
$932$	−17.7722	−0.582147
$933$	0	0
$934$	−13.4254	−0.439294
$935$	2.46436	0.0805933
$936$	0	0
$937$	−23.7929	−0.777280	−0.388640	−	0.921390i	$-0.627055\pi$
$937$	−23.7929	−0.777280	−0.388640	+	0.921390i	$0.627055\pi$
$938$	−39.9431	−1.30419
$939$	0	0
$940$	−5.50555	−0.179571
$941$	30.7434	1.00221	0.501103	−	0.865388i	$-0.332928\pi$
$941$	30.7434	1.00221	0.501103	+	0.865388i	$0.332928\pi$
$942$	0	0
$943$	24.0666	0.783716
$944$	−7.89955	−0.257108
$945$	0	0
$946$	5.56758	0.181018
$947$	−44.2821	−1.43897	−0.719487	−	0.694506i	$-0.755623\pi$
$947$	−44.2821	−1.43897	−0.719487	+	0.694506i	$0.755623\pi$
$948$	0	0
$949$	−1.21329	−0.0393849
$950$	24.3682	0.790608
$951$	0	0
$952$	−7.00247	−0.226951
$953$	9.40636	0.304702	0.152351	−	0.988326i	$-0.451316\pi$
$953$	9.40636	0.304702	0.152351	+	0.988326i	$0.451316\pi$
$954$	0	0
$955$	−47.3059	−1.53078
$956$	12.0820	0.390759
$957$	0	0
$958$	2.09566	0.0677078
$959$	−25.5644	−0.825517
$960$	0	0
$961$	−2.19148	−0.0706930
$962$	0.691586	0.0222976
$963$	0	0
$964$	11.3215	0.364642
$965$	−22.3819	−0.720500
$966$	0	0
$967$	−56.8679	−1.82875	−0.914373	−	0.404872i	$-0.867316\pi$
$967$	−56.8679	−1.82875	−0.914373	+	0.404872i	$0.867316\pi$
$968$	10.7943	0.346943
$969$	0	0
$970$	−49.6445	−1.59399
$971$	21.1723	0.679452	0.339726	−	0.940524i	$-0.389666\pi$
$971$	21.1723	0.679452	0.339726	+	0.940524i	$0.389666\pi$
$972$	0	0
$973$	−75.4838	−2.41990
$974$	38.1661	1.22292
$975$	0	0
$976$	0.950317	0.0304189
$977$	−6.11162	−0.195528	−0.0977641	−	0.995210i	$-0.531169\pi$
$977$	−6.11162	−0.195528	−0.0977641	+	0.995210i	$0.531169\pi$
$978$	0	0
$979$	0.806362	0.0257714
$980$	−67.4387	−2.15425
$981$	0	0
$982$	6.36732	0.203189
$983$	−42.4508	−1.35397	−0.676985	−	0.735997i	$-0.736714\pi$
$983$	−42.4508	−1.35397	−0.676985	+	0.735997i	$0.736714\pi$
$984$	0	0
$985$	−83.2753	−2.65337
$986$	9.39778	0.299286
$987$	0	0
$988$	0.235718	0.00749919
$989$	−27.7025	−0.880887
$990$	0	0
$991$	−52.9269	−1.68128	−0.840640	−	0.541594i	$-0.817821\pi$
$991$	−52.9269	−1.68128	−0.840640	+	0.541594i	$0.817821\pi$
$992$	−5.36736	−0.170414
$993$	0	0
$994$	15.1635	0.480957
$995$	−21.9835	−0.696925
$996$	0	0
$997$	−43.4208	−1.37515	−0.687575	−	0.726113i	$-0.741325\pi$
$997$	−43.4208	−1.37515	−0.687575	+	0.726113i	$0.741325\pi$
$998$	−33.6939	−1.06656
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4842.2.a.n.1.2		7
3.2	odd	2		538.2.a.e.1.6	✓	7
12.11	even	2		4304.2.a.h.1.2		7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
538.2.a.e.1.6	✓	7	3.2	odd	2
4304.2.a.h.1.2		7	12.11	even	2
4842.2.a.n.1.2		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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} -3.84433 q^{5} -4.95403 q^{7} -1.00000 q^{8} +3.84433 q^{10} +0.453515 q^{11} -0.0945931 q^{13} +4.95403 q^{14} +1.00000 q^{16} -1.41349 q^{17} -2.49192 q^{19} -3.84433 q^{20} -0.453515 q^{22} +2.25654 q^{23} +9.77888 q^{25} +0.0945931 q^{26} -4.95403 q^{28} +6.64864 q^{29} +5.36736 q^{31} -1.00000 q^{32} +1.41349 q^{34} +19.0449 q^{35} +7.31117 q^{37} +2.49192 q^{38} +3.84433 q^{40} +10.6653 q^{41} -12.2765 q^{43} +0.453515 q^{44} -2.25654 q^{46} +1.43212 q^{47} +17.5424 q^{49} -9.77888 q^{50} -0.0945931 q^{52} -9.96897 q^{53} -1.74346 q^{55} +4.95403 q^{56} -6.64864 q^{58} -7.89955 q^{59} +0.950317 q^{61} -5.36736 q^{62} +1.00000 q^{64} +0.363647 q^{65} -8.06276 q^{67} -1.41349 q^{68} -19.0449 q^{70} +3.06085 q^{71} +12.8264 q^{73} -7.31117 q^{74} -2.49192 q^{76} -2.24672 q^{77} +13.7673 q^{79} -3.84433 q^{80} -10.6653 q^{82} -10.1171 q^{83} +5.43392 q^{85} +12.2765 q^{86} -0.453515 q^{88} +1.77803 q^{89} +0.468616 q^{91} +2.25654 q^{92} -1.43212 q^{94} +9.57975 q^{95} -12.9137 q^{97} -17.5424 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q - 7 q^{2} + 7 q^{4} - 7 q^{5} + 6 q^{7} - 7 q^{8} + 7 q^{10} + 3 q^{11} - 9 q^{13} - 6 q^{14} + 7 q^{16} - 8 q^{17} - 11 q^{19} - 7 q^{20} - 3 q^{22} - 12 q^{23} + 22 q^{25} + 9 q^{26} + 6 q^{28} + 5 q^{29}+ \cdots - 15 q^{98}+O(q^{100}) \) 7 * q - 7 * q^2 + 7 * q^4 - 7 * q^5 + 6 * q^7 - 7 * q^8 + 7 * q^10 + 3 * q^11 - 9 * q^13 - 6 * q^14 + 7 * q^16 - 8 * q^17 - 11 * q^19 - 7 * q^20 - 3 * q^22 - 12 * q^23 + 22 * q^25 + 9 * q^26 + 6 * q^28 + 5 * q^29 + 14 * q^31 - 7 * q^32 + 8 * q^34 + 4 * q^35 + 13 * q^37 + 11 * q^38 + 7 * q^40 - 12 * q^41 - 11 * q^43 + 3 * q^44 + 12 * q^46 - 2 * q^47 + 15 * q^49 - 22 * q^50 - 9 * q^52 - 19 * q^53 - 40 * q^55 - 6 * q^56 - 5 * q^58 + 9 * q^59 - 3 * q^61 - 14 * q^62 + 7 * q^64 + 10 * q^65 - 33 * q^67 - 8 * q^68 - 4 * q^70 - 28 * q^71 - 14 * q^73 - 13 * q^74 - 11 * q^76 - 10 * q^77 + 2 * q^79 - 7 * q^80 + 12 * q^82 + 7 * q^83 - 16 * q^85 + 11 * q^86 - 3 * q^88 - 18 * q^89 - 26 * q^91 - 12 * q^92 + 2 * q^94 + 34 * q^95 - 4 * q^97 - 15 * q^98

Introduction

L-functions

Modular forms

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