LMFDB - Embedded newform 1456.2.a.u.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1456,2,Mod(1,1456)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1456, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1456.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1456 = 2^{4} \cdot 7 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1456.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:	$11.6262185343$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.183064.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 10x^{2} + 6x + 8 $ x^4 - x^3 - 10x^2 + 6x + 8
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 728)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$3.26869$ of defining polynomial
Character	$\chi$	$=$	1456.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.26869 q^{3} -1.04496 q^{5} -1.00000 q^{7} +7.68433 q^{9} +O(q^{10})$	$q-3.26869 q^{3} -1.04496 q^{5} -1.00000 q^{7} +7.68433 q^{9} -3.26869 q^{11} +1.00000 q^{13} +3.41564 q^{15} -7.95302 q^{17} -3.04496 q^{19} +3.26869 q^{21} +6.31365 q^{23} -3.90806 q^{25} -15.3116 q^{27} -4.46060 q^{29} -3.19191 q^{31} +10.6843 q^{33} +1.04496 q^{35} -1.26869 q^{37} -3.26869 q^{39} +10.7742 q^{41} -6.90806 q^{43} -8.02980 q^{45} -3.19191 q^{47} +1.00000 q^{49} +25.9960 q^{51} +8.46060 q^{53} +3.41564 q^{55} +9.95302 q^{57} +8.62729 q^{59} -9.31163 q^{61} -7.68433 q^{63} -1.04496 q^{65} +0.146952 q^{67} -20.6374 q^{69} +16.4904 q^{71} -1.63937 q^{73} +12.7742 q^{75} +3.26869 q^{77} +14.7611 q^{79} +26.9960 q^{81} -1.58234 q^{83} +8.31057 q^{85} +14.5803 q^{87} +2.66114 q^{89} -1.00000 q^{91} +10.4334 q^{93} +3.18185 q^{95} -1.34547 q^{97} -25.1177 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - q^{3} - 4 q^{7} + 9 q^{9}+O(q^{10}) $ 4 * q - q^3 - 4 * q^7 + 9 * q^9	$ 4 q - q^{3} - 4 q^{7} + 9 q^{9} - q^{11} + 4 q^{13} + 4 q^{15} + 2 q^{17} - 8 q^{19} + q^{21} + 9 q^{23} + 14 q^{25} - 7 q^{27} - 4 q^{29} - 11 q^{31} + 21 q^{33} + 7 q^{37} - q^{39} + 13 q^{41} + 2 q^{43} + 12 q^{45} - 11 q^{47} + 4 q^{49} + 28 q^{51} + 20 q^{53} + 4 q^{55} + 6 q^{57} + 2 q^{59} + 17 q^{61} - 9 q^{63} + 3 q^{67} - 27 q^{69} + 8 q^{71} + 11 q^{73} + 21 q^{75} + q^{77} + 27 q^{79} + 32 q^{81} + 22 q^{83} - 8 q^{85} - 8 q^{87} + 10 q^{89} - 4 q^{91} - 27 q^{93} + 34 q^{95} + 17 q^{97} - 10 q^{99}+O(q^{100}) $ 4 * q - q^3 - 4 * q^7 + 9 * q^9 - q^11 + 4 * q^13 + 4 * q^15 + 2 * q^17 - 8 * q^19 + q^21 + 9 * q^23 + 14 * q^25 - 7 * q^27 - 4 * q^29 - 11 * q^31 + 21 * q^33 + 7 * q^37 - q^39 + 13 * q^41 + 2 * q^43 + 12 * q^45 - 11 * q^47 + 4 * q^49 + 28 * q^51 + 20 * q^53 + 4 * q^55 + 6 * q^57 + 2 * q^59 + 17 * q^61 - 9 * q^63 + 3 * q^67 - 27 * q^69 + 8 * q^71 + 11 * q^73 + 21 * q^75 + q^77 + 27 * q^79 + 32 * q^81 + 22 * q^83 - 8 * q^85 - 8 * q^87 + 10 * q^89 - 4 * q^91 - 27 * q^93 + 34 * q^95 + 17 * q^97 - 10 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	−3.26869	−1.88718	−0.943589	−	0.331118i	$-0.892574\pi$
$3$	−3.26869	−1.88718	−0.943589	+	0.331118i	$0.892574\pi$
$4$	0	0
$5$	−1.04496	−0.467319	−0.233660	−	0.972318i	$-0.575070\pi$
$5$	−1.04496	−0.467319	−0.233660	+	0.972318i	$0.575070\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	7.68433	2.56144
$10$	0	0
$11$	−3.26869	−0.985547	−0.492773	−	0.870158i	$-0.664017\pi$
$11$	−3.26869	−0.985547	−0.492773	+	0.870158i	$0.664017\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	3.41564	0.881915
$16$	0	0
$17$	−7.95302	−1.92889	−0.964445	−	0.264282i	$-0.914865\pi$
$17$	−7.95302	−1.92889	−0.964445	+	0.264282i	$0.914865\pi$
$18$	0	0
$19$	−3.04496	−0.698561	−0.349281	−	0.937018i	$-0.613574\pi$
$19$	−3.04496	−0.698561	−0.349281	+	0.937018i	$0.613574\pi$
$20$	0	0
$21$	3.26869	0.713287
$22$	0	0
$23$	6.31365	1.31649	0.658243	−	0.752805i	$-0.271300\pi$
$23$	6.31365	1.31649	0.658243	+	0.752805i	$0.271300\pi$
$24$	0	0
$25$	−3.90806	−0.781613
$26$	0	0
$27$	−15.3116	−2.94672
$28$	0	0
$29$	−4.46060	−0.828313	−0.414156	−	0.910206i	$-0.635923\pi$
$29$	−4.46060	−0.828313	−0.414156	+	0.910206i	$0.635923\pi$
$30$	0	0
$31$	−3.19191	−0.573284	−0.286642	−	0.958038i	$-0.592539\pi$
$31$	−3.19191	−0.573284	−0.286642	+	0.958038i	$0.592539\pi$
$32$	0	0
$33$	10.6843	1.85990
$34$	0	0
$35$	1.04496	0.176630
$36$	0	0
$37$	−1.26869	−0.208571	−0.104286	−	0.994547i	$-0.533256\pi$
$37$	−1.26869	−0.208571	−0.104286	+	0.994547i	$0.533256\pi$
$38$	0	0
$39$	−3.26869	−0.523409
$40$	0	0
$41$	10.7742	1.68265	0.841327	−	0.540526i	$-0.181775\pi$
$41$	10.7742	1.68265	0.841327	+	0.540526i	$0.181775\pi$
$42$	0	0
$43$	−6.90806	−1.05347	−0.526735	−	0.850030i	$-0.676584\pi$
$43$	−6.90806	−1.05347	−0.526735	+	0.850030i	$0.676584\pi$
$44$	0	0
$45$	−8.02980	−1.19701
$46$	0	0
$47$	−3.19191	−0.465588	−0.232794	−	0.972526i	$-0.574787\pi$
$47$	−3.19191	−0.465588	−0.232794	+	0.972526i	$0.574787\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	25.9960	3.64016
$52$	0	0
$53$	8.46060	1.16215	0.581076	−	0.813849i	$-0.302632\pi$
$53$	8.46060	1.16215	0.581076	+	0.813849i	$0.302632\pi$
$54$	0	0
$55$	3.41564	0.460565
$56$	0	0
$57$	9.95302	1.31831
$58$	0	0
$59$	8.62729	1.12318	0.561589	−	0.827416i	$-0.310190\pi$
$59$	8.62729	1.12318	0.561589	+	0.827416i	$0.310190\pi$
$60$	0	0
$61$	−9.31163	−1.19223	−0.596116	−	0.802898i	$-0.703290\pi$
$61$	−9.31163	−1.19223	−0.596116	+	0.802898i	$0.703290\pi$
$62$	0	0
$63$	−7.68433	−0.968135
$64$	0	0
$65$	−1.04496	−0.129611
$66$	0	0
$67$	0.146952	0.0179531	0.00897655	−	0.999960i	$-0.497143\pi$
$67$	0.146952	0.0179531	0.00897655	+	0.999960i	$0.497143\pi$
$68$	0	0
$69$	−20.6374	−2.48445
$70$	0	0
$71$	16.4904	1.95705	0.978525	−	0.206127i	$-0.0660860\pi$
$71$	16.4904	1.95705	0.978525	+	0.206127i	$0.0660860\pi$
$72$	0	0
$73$	−1.63937	−0.191874	−0.0959371	−	0.995387i	$-0.530585\pi$
$73$	−1.63937	−0.191874	−0.0959371	+	0.995387i	$0.530585\pi$
$74$	0	0
$75$	12.7742	1.47504
$76$	0	0
$77$	3.26869	0.372502
$78$	0	0
$79$	14.7611	1.66075	0.830377	−	0.557201i	$-0.188125\pi$
$79$	14.7611	1.66075	0.830377	+	0.557201i	$0.188125\pi$
$80$	0	0
$81$	26.9960	2.99955
$82$	0	0
$83$	−1.58234	−0.173684	−0.0868420	−	0.996222i	$-0.527678\pi$
$83$	−1.58234	−0.173684	−0.0868420	+	0.996222i	$0.527678\pi$
$84$	0	0
$85$	8.31057	0.901408
$86$	0	0
$87$	14.5803	1.56317
$88$	0	0
$89$	2.66114	0.282080	0.141040	−	0.990004i	$-0.454955\pi$
$89$	2.66114	0.282080	0.141040	+	0.990004i	$0.454955\pi$
$90$	0	0
$91$	−1.00000	−0.104828
$92$	0	0
$93$	10.4334	1.08189
$94$	0	0
$95$	3.18185	0.326451
$96$	0	0
$97$	−1.34547	−0.136612	−0.0683058	−	0.997664i	$-0.521759\pi$
$97$	−1.34547	−0.136612	−0.0683058	+	0.997664i	$0.521759\pi$
$98$	0	0
$99$	−25.1177	−2.52442
$100$	0	0
$101$	−2.43741	−0.242531	−0.121265	−	0.992620i	$-0.538695\pi$
$101$	−2.43741	−0.242531	−0.121265	+	0.992620i	$0.538695\pi$
$102$	0	0
$103$	−0.293905	−0.0289593	−0.0144797	−	0.999895i	$-0.504609\pi$
$103$	−0.293905	−0.0289593	−0.0144797	+	0.999895i	$0.504609\pi$
$104$	0	0
$105$	−3.41564	−0.333333
$106$	0	0
$107$	10.0899	0.975429	0.487714	−	0.873003i	$-0.337831\pi$
$107$	10.0899	0.975429	0.487714	+	0.873003i	$0.337831\pi$
$108$	0	0
$109$	7.65912	0.733610	0.366805	−	0.930298i	$-0.380452\pi$
$109$	7.65912	0.733610	0.366805	+	0.930298i	$0.380452\pi$
$110$	0	0
$111$	4.14695	0.393611
$112$	0	0
$113$	8.31365	0.782082	0.391041	−	0.920373i	$-0.372115\pi$
$113$	8.31365	0.782082	0.391041	+	0.920373i	$0.372115\pi$
$114$	0	0
$115$	−6.59749	−0.615219
$116$	0	0
$117$	7.68433	0.710417
$118$	0	0
$119$	7.95302	0.729052
$120$	0	0
$121$	−0.315668	−0.0286971
$122$	0	0
$123$	−35.2177	−3.17547
$124$	0	0
$125$	9.30855	0.832582
$126$	0	0
$127$	−1.78940	−0.158784	−0.0793920	−	0.996843i	$-0.525298\pi$
$127$	−1.78940	−0.158784	−0.0793920	+	0.996843i	$0.525298\pi$
$128$	0	0
$129$	22.5803	1.98809
$130$	0	0
$131$	19.7262	1.72349	0.861744	−	0.507344i	$-0.169373\pi$
$131$	19.7262	1.72349	0.861744	+	0.507344i	$0.169373\pi$
$132$	0	0
$133$	3.04496	0.264031
$134$	0	0
$135$	16.0000	1.37706
$136$	0	0
$137$	−19.4116	−1.65844	−0.829222	−	0.558919i	$-0.811216\pi$
$137$	−19.4116	−1.65844	−0.829222	+	0.558919i	$0.811216\pi$
$138$	0	0
$139$	6.24693	0.529857	0.264929	−	0.964268i	$-0.414652\pi$
$139$	6.24693	0.529857	0.264929	+	0.964268i	$0.414652\pi$
$140$	0	0
$141$	10.4334	0.878648
$142$	0	0
$143$	−3.26869	−0.273342
$144$	0	0
$145$	4.66114	0.387086
$146$	0	0
$147$	−3.26869	−0.269597
$148$	0	0
$149$	−23.0848	−1.89118	−0.945591	−	0.325358i	$-0.894515\pi$
$149$	−23.0848	−1.89118	−0.945591	+	0.325358i	$0.894515\pi$
$150$	0	0
$151$	3.41564	0.277961	0.138981	−	0.990295i	$-0.455617\pi$
$151$	3.41564	0.277961	0.138981	+	0.990295i	$0.455617\pi$
$152$	0	0
$153$	−61.1137	−4.94075
$154$	0	0
$155$	3.33541	0.267907
$156$	0	0
$157$	6.01006	0.479655	0.239827	−	0.970816i	$-0.422909\pi$
$157$	6.01006	0.479655	0.239827	+	0.970816i	$0.422909\pi$
$158$	0	0
$159$	−27.6551	−2.19319
$160$	0	0
$161$	−6.31365	−0.497585
$162$	0	0
$163$	−15.0111	−1.17576	−0.587881	−	0.808948i	$-0.700038\pi$
$163$	−15.0111	−1.17576	−0.587881	+	0.808948i	$0.700038\pi$
$164$	0	0
$165$	−11.1647	−0.869169
$166$	0	0
$167$	−8.23379	−0.637150	−0.318575	−	0.947898i	$-0.603204\pi$
$167$	−8.23379	−0.637150	−0.318575	+	0.947898i	$0.603204\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−23.3985	−1.78933
$172$	0	0
$173$	5.59547	0.425416	0.212708	−	0.977116i	$-0.431772\pi$
$173$	5.59547	0.425416	0.212708	+	0.977116i	$0.431772\pi$
$174$	0	0
$175$	3.90806	0.295422
$176$	0	0
$177$	−28.1999	−2.11964
$178$	0	0
$179$	19.5354	1.46014	0.730071	−	0.683372i	$-0.239487\pi$
$179$	19.5354	1.46014	0.730071	+	0.683372i	$0.239487\pi$
$180$	0	0
$181$	10.6374	0.790668	0.395334	−	0.918537i	$-0.370629\pi$
$181$	10.6374	0.790668	0.395334	+	0.918537i	$0.370629\pi$
$182$	0	0
$183$	30.4368	2.24995
$184$	0	0
$185$	1.32573	0.0974694
$186$	0	0
$187$	25.9960	1.90101
$188$	0	0
$189$	15.3116	1.11376
$190$	0	0
$191$	−16.8272	−1.21758	−0.608788	−	0.793333i	$-0.708344\pi$
$191$	−16.8272	−1.21758	−0.608788	+	0.793333i	$0.708344\pi$
$192$	0	0
$193$	16.5374	1.19039	0.595193	−	0.803583i	$-0.297075\pi$
$193$	16.5374	1.19039	0.595193	+	0.803583i	$0.297075\pi$
$194$	0	0
$195$	3.41564	0.244599
$196$	0	0
$197$	4.41670	0.314677	0.157338	−	0.987545i	$-0.449709\pi$
$197$	4.41670	0.314677	0.157338	+	0.987545i	$0.449709\pi$
$198$	0	0
$199$	−11.5692	−0.820119	−0.410059	−	0.912059i	$-0.634492\pi$
$199$	−11.5692	−0.820119	−0.410059	+	0.912059i	$0.634492\pi$
$200$	0	0
$201$	−0.480342	−0.0338807
$202$	0	0
$203$	4.46060	0.313073
$204$	0	0
$205$	−11.2586	−0.786337
$206$	0	0
$207$	48.5162	3.37211
$208$	0	0
$209$	9.95302	0.688465
$210$	0	0
$211$	−2.28077	−0.157015	−0.0785073	−	0.996914i	$-0.525015\pi$
$211$	−2.28077	−0.157015	−0.0785073	+	0.996914i	$0.525015\pi$
$212$	0	0
$213$	−53.9020	−3.69330
$214$	0	0
$215$	7.21863	0.492307
$216$	0	0
$217$	3.19191	0.216681
$218$	0	0
$219$	5.35860	0.362101
$220$	0	0
$221$	−7.95302	−0.534978
$222$	0	0
$223$	−27.3656	−1.83254	−0.916268	−	0.400567i	$-0.868813\pi$
$223$	−27.3656	−1.83254	−0.916268	+	0.400567i	$0.868813\pi$
$224$	0	0
$225$	−30.0309	−2.00206
$226$	0	0
$227$	7.90604	0.524742	0.262371	−	0.964967i	$-0.415496\pi$
$227$	7.90604	0.524742	0.262371	+	0.964967i	$0.415496\pi$
$228$	0	0
$229$	13.2787	0.877484	0.438742	−	0.898613i	$-0.355424\pi$
$229$	13.2787	0.877484	0.438742	+	0.898613i	$0.355424\pi$
$230$	0	0
$231$	−10.6843	−0.702977
$232$	0	0
$233$	21.6823	1.42046	0.710228	−	0.703972i	$-0.248592\pi$
$233$	21.6823	1.42046	0.710228	+	0.703972i	$0.248592\pi$
$234$	0	0
$235$	3.33541	0.217578
$236$	0	0
$237$	−48.2495	−3.13414
$238$	0	0
$239$	18.3575	1.18745	0.593726	−	0.804668i	$-0.297656\pi$
$239$	18.3575	1.18745	0.593726	+	0.804668i	$0.297656\pi$
$240$	0	0
$241$	−3.42878	−0.220867	−0.110433	−	0.993884i	$-0.535224\pi$
$241$	−3.42878	−0.220867	−0.110433	+	0.993884i	$0.535224\pi$
$242$	0	0
$243$	−42.3065	−2.71397
$244$	0	0
$245$	−1.04496	−0.0667599
$246$	0	0
$247$	−3.04496	−0.193746
$248$	0	0
$249$	5.17217	0.327773
$250$	0	0
$251$	−6.43336	−0.406070	−0.203035	−	0.979171i	$-0.565081\pi$
$251$	−6.43336	−0.406070	−0.203035	+	0.979171i	$0.565081\pi$
$252$	0	0
$253$	−20.6374	−1.29746
$254$	0	0
$255$	−27.1647	−1.70112
$256$	0	0
$257$	16.8742	1.05258	0.526292	−	0.850304i	$-0.323582\pi$
$257$	16.8742	1.05258	0.526292	+	0.850304i	$0.323582\pi$
$258$	0	0
$259$	1.26869	0.0788325
$260$	0	0
$261$	−34.2767	−2.12168
$262$	0	0
$263$	−24.1364	−1.48831	−0.744157	−	0.668005i	$-0.767148\pi$
$263$	−24.1364	−1.48831	−0.744157	+	0.668005i	$0.767148\pi$
$264$	0	0
$265$	−8.84097	−0.543096
$266$	0	0
$267$	−8.69843	−0.532335
$268$	0	0
$269$	14.0329	0.855600	0.427800	−	0.903873i	$-0.359289\pi$
$269$	14.0329	0.855600	0.427800	+	0.903873i	$0.359289\pi$
$270$	0	0
$271$	15.6055	0.947968	0.473984	−	0.880533i	$-0.342815\pi$
$271$	15.6055	0.947968	0.473984	+	0.880533i	$0.342815\pi$
$272$	0	0
$273$	3.26869	0.197830
$274$	0	0
$275$	12.7742	0.770316
$276$	0	0
$277$	12.9343	0.777149	0.388574	−	0.921417i	$-0.372968\pi$
$277$	12.9343	0.777149	0.388574	+	0.921417i	$0.372968\pi$
$278$	0	0
$279$	−24.5277	−1.46843
$280$	0	0
$281$	−16.8479	−1.00506	−0.502532	−	0.864558i	$-0.667598\pi$
$281$	−16.8479	−1.00506	−0.502532	+	0.864558i	$0.667598\pi$
$282$	0	0
$283$	5.96308	0.354468	0.177234	−	0.984169i	$-0.443285\pi$
$283$	5.96308	0.354468	0.177234	+	0.984169i	$0.443285\pi$
$284$	0	0
$285$	−10.4005	−0.616072
$286$	0	0
$287$	−10.7742	−0.635984
$288$	0	0
$289$	46.2505	2.72062
$290$	0	0
$291$	4.39792	0.257811
$292$	0	0
$293$	−12.1439	−0.709453	−0.354726	−	0.934970i	$-0.615426\pi$
$293$	−12.1439	−0.709453	−0.354726	+	0.934970i	$0.615426\pi$
$294$	0	0
$295$	−9.01516	−0.524883
$296$	0	0
$297$	50.0490	2.90413
$298$	0	0
$299$	6.31365	0.365128
$300$	0	0
$301$	6.90806	0.398174
$302$	0	0
$303$	7.96712	0.457699
$304$	0	0
$305$	9.73025	0.557153
$306$	0	0
$307$	−20.5076	−1.17043	−0.585215	−	0.810878i	$-0.698990\pi$
$307$	−20.5076	−1.17043	−0.585215	+	0.810878i	$0.698990\pi$
$308$	0	0
$309$	0.960684	0.0546514
$310$	0	0
$311$	−12.0167	−0.681403	−0.340701	−	0.940172i	$-0.610664\pi$
$311$	−12.0167	−0.681403	−0.340701	+	0.940172i	$0.610664\pi$
$312$	0	0
$313$	−19.4116	−1.09721	−0.548604	−	0.836082i	$-0.684841\pi$
$313$	−19.4116	−1.09721	−0.548604	+	0.836082i	$0.684841\pi$
$314$	0	0
$315$	8.02980	0.452428
$316$	0	0
$317$	−7.60957	−0.427396	−0.213698	−	0.976900i	$-0.568551\pi$
$317$	−7.60957	−0.427396	−0.213698	+	0.976900i	$0.568551\pi$
$318$	0	0
$319$	14.5803	0.816341
$320$	0	0
$321$	−32.9808	−1.84081
$322$	0	0
$323$	24.2166	1.34745
$324$	0	0
$325$	−3.90806	−0.216780
$326$	0	0
$327$	−25.0353	−1.38445
$328$	0	0
$329$	3.19191	0.175976
$330$	0	0
$331$	11.1581	0.613303	0.306651	−	0.951822i	$-0.400791\pi$
$331$	11.1581	0.613303	0.306651	+	0.951822i	$0.400791\pi$
$332$	0	0
$333$	−9.74903	−0.534244
$334$	0	0
$335$	−0.153559	−0.00838983
$336$	0	0
$337$	−5.34892	−0.291374	−0.145687	−	0.989331i	$-0.546539\pi$
$337$	−5.34892	−0.291374	−0.145687	+	0.989331i	$0.546539\pi$
$338$	0	0
$339$	−27.1747	−1.47593
$340$	0	0
$341$	10.4334	0.564998
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	21.5652	1.16103
$346$	0	0
$347$	−13.2546	−0.711544	−0.355772	−	0.934573i	$-0.615782\pi$
$347$	−13.2546	−0.711544	−0.355772	+	0.934573i	$0.615782\pi$
$348$	0	0
$349$	11.7359	0.628208	0.314104	−	0.949389i	$-0.398296\pi$
$349$	11.7359	0.628208	0.314104	+	0.949389i	$0.398296\pi$
$350$	0	0
$351$	−15.3116	−0.817274
$352$	0	0
$353$	24.0026	1.27753	0.638764	−	0.769403i	$-0.279446\pi$
$353$	24.0026	1.27753	0.638764	+	0.769403i	$0.279446\pi$
$354$	0	0
$355$	−17.2318	−0.914567
$356$	0	0
$357$	−25.9960	−1.37585
$358$	0	0
$359$	−3.41564	−0.180271	−0.0901353	−	0.995930i	$-0.528730\pi$
$359$	−3.41564	−0.180271	−0.0901353	+	0.995930i	$0.528730\pi$
$360$	0	0
$361$	−9.72823	−0.512012
$362$	0	0
$363$	1.03182	0.0541566
$364$	0	0
$365$	1.71308	0.0896665
$366$	0	0
$367$	−0.678316	−0.0354078	−0.0177039	−	0.999843i	$-0.505636\pi$
$367$	−0.678316	−0.0354078	−0.0177039	+	0.999843i	$0.505636\pi$
$368$	0	0
$369$	82.7929	4.31003
$370$	0	0
$371$	−8.46060	−0.439252
$372$	0	0
$373$	−22.7132	−1.17604	−0.588022	−	0.808845i	$-0.700093\pi$
$373$	−22.7132	−1.17604	−0.588022	+	0.808845i	$0.700093\pi$
$374$	0	0
$375$	−30.4268	−1.57123
$376$	0	0
$377$	−4.46060	−0.229733
$378$	0	0
$379$	−10.4944	−0.539063	−0.269532	−	0.962992i	$-0.586869\pi$
$379$	−10.4944	−0.539063	−0.269532	+	0.962992i	$0.586869\pi$
$380$	0	0
$381$	5.84901	0.299654
$382$	0	0
$383$	−29.5551	−1.51020	−0.755098	−	0.655612i	$-0.772411\pi$
$383$	−29.5551	−1.51020	−0.755098	+	0.655612i	$0.772411\pi$
$384$	0	0
$385$	−3.41564	−0.174077
$386$	0	0
$387$	−53.0839	−2.69840
$388$	0	0
$389$	22.4475	1.13813	0.569066	−	0.822292i	$-0.307305\pi$
$389$	22.4475	1.13813	0.569066	+	0.822292i	$0.307305\pi$
$390$	0	0
$391$	−50.2126	−2.53936
$392$	0	0
$393$	−64.4789	−3.25253
$394$	0	0
$395$	−15.4247	−0.776103
$396$	0	0
$397$	35.2207	1.76768	0.883839	−	0.467791i	$-0.154950\pi$
$397$	35.2207	1.76768	0.883839	+	0.467791i	$0.154950\pi$
$398$	0	0
$399$	−9.95302	−0.498274
$400$	0	0
$401$	6.26763	0.312991	0.156495	−	0.987679i	$-0.449980\pi$
$401$	6.26763	0.312991	0.156495	+	0.987679i	$0.449980\pi$
$402$	0	0
$403$	−3.19191	−0.159000
$404$	0	0
$405$	−28.2096	−1.40175
$406$	0	0
$407$	4.14695	0.205557
$408$	0	0
$409$	33.2186	1.64256	0.821278	−	0.570528i	$-0.193262\pi$
$409$	33.2186	1.64256	0.821278	+	0.570528i	$0.193262\pi$
$410$	0	0
$411$	63.4505	3.12978
$412$	0	0
$413$	−8.62729	−0.424521
$414$	0	0
$415$	1.65347	0.0811659
$416$	0	0
$417$	−20.4193	−0.999936
$418$	0	0
$419$	5.77980	0.282362	0.141181	−	0.989984i	$-0.454910\pi$
$419$	5.77980	0.282362	0.141181	+	0.989984i	$0.454910\pi$
$420$	0	0
$421$	−16.7010	−0.813957	−0.406978	−	0.913438i	$-0.633418\pi$
$421$	−16.7010	−0.813957	−0.406978	+	0.913438i	$0.633418\pi$
$422$	0	0
$423$	−24.5277	−1.19258
$424$	0	0
$425$	31.0809	1.50765
$426$	0	0
$427$	9.31163	0.450621
$428$	0	0
$429$	10.6843	0.515844
$430$	0	0
$431$	16.4475	0.792246	0.396123	−	0.918197i	$-0.370355\pi$
$431$	16.4475	0.792246	0.396123	+	0.918197i	$0.370355\pi$
$432$	0	0
$433$	−6.74137	−0.323970	−0.161985	−	0.986793i	$-0.551790\pi$
$433$	−6.74137	−0.323970	−0.161985	+	0.986793i	$0.551790\pi$
$434$	0	0
$435$	−15.2358	−0.730501
$436$	0	0
$437$	−19.2248	−0.919646
$438$	0	0
$439$	−16.1737	−0.771927	−0.385964	−	0.922514i	$-0.626131\pi$
$439$	−16.1737	−0.771927	−0.385964	+	0.922514i	$0.626131\pi$
$440$	0	0
$441$	7.68433	0.365921
$442$	0	0
$443$	23.2677	1.10548	0.552741	−	0.833353i	$-0.313582\pi$
$443$	23.2677	1.10548	0.552741	+	0.833353i	$0.313582\pi$
$444$	0	0
$445$	−2.78078	−0.131821
$446$	0	0
$447$	75.4571	3.56900
$448$	0	0
$449$	18.4676	0.871539	0.435770	−	0.900058i	$-0.356476\pi$
$449$	18.4676	0.871539	0.435770	+	0.900058i	$0.356476\pi$
$450$	0	0
$451$	−35.2177	−1.65834
$452$	0	0
$453$	−11.1647	−0.524562
$454$	0	0
$455$	1.04496	0.0489884
$456$	0	0
$457$	−31.5914	−1.47778	−0.738892	−	0.673823i	$-0.764651\pi$
$457$	−31.5914	−1.47778	−0.738892	+	0.673823i	$0.764651\pi$
$458$	0	0
$459$	121.774	5.68391
$460$	0	0
$461$	22.3798	1.04233	0.521165	−	0.853456i	$-0.325498\pi$
$461$	22.3798	1.04233	0.521165	+	0.853456i	$0.325498\pi$
$462$	0	0
$463$	−7.38937	−0.343413	−0.171707	−	0.985148i	$-0.554928\pi$
$463$	−7.38937	−0.343413	−0.171707	+	0.985148i	$0.554928\pi$
$464$	0	0
$465$	−10.9024	−0.505588
$466$	0	0
$467$	19.5955	0.906770	0.453385	−	0.891315i	$-0.350216\pi$
$467$	19.5955	0.906770	0.453385	+	0.891315i	$0.350216\pi$
$468$	0	0
$469$	−0.146952	−0.00678563
$470$	0	0
$471$	−19.6450	−0.905195
$472$	0	0
$473$	22.5803	1.03824
$474$	0	0
$475$	11.8999	0.546004
$476$	0	0
$477$	65.0141	2.97679
$478$	0	0
$479$	19.2449	0.879322	0.439661	−	0.898164i	$-0.355099\pi$
$479$	19.2449	0.879322	0.439661	+	0.898164i	$0.355099\pi$
$480$	0	0
$481$	−1.26869	−0.0578473
$482$	0	0
$483$	20.6374	0.939032
$484$	0	0
$485$	1.40596	0.0638413
$486$	0	0
$487$	−26.8708	−1.21763	−0.608815	−	0.793312i	$-0.708355\pi$
$487$	−26.8708	−1.21763	−0.608815	+	0.793312i	$0.708355\pi$
$488$	0	0
$489$	49.0667	2.21887
$490$	0	0
$491$	20.1536	0.909517	0.454759	−	0.890615i	$-0.349726\pi$
$491$	20.1536	0.909517	0.454759	+	0.890615i	$0.349726\pi$
$492$	0	0
$493$	35.4752	1.59772
$494$	0	0
$495$	26.2469	1.17971
$496$	0	0
$497$	−16.4904	−0.739696
$498$	0	0
$499$	6.70715	0.300253	0.150127	−	0.988667i	$-0.452032\pi$
$499$	6.70715	0.300253	0.150127	+	0.988667i	$0.452032\pi$
$500$	0	0
$501$	26.9137	1.20242
$502$	0	0
$503$	40.5535	1.80819	0.904095	−	0.427332i	$-0.140547\pi$
$503$	40.5535	1.80819	0.904095	+	0.427332i	$0.140547\pi$
$504$	0	0
$505$	2.54699	0.113339
$506$	0	0
$507$	−3.26869	−0.145168
$508$	0	0
$509$	−32.9974	−1.46258	−0.731292	−	0.682065i	$-0.761082\pi$
$509$	−32.9974	−1.46258	−0.731292	+	0.682065i	$0.761082\pi$
$510$	0	0
$511$	1.63937	0.0725216
$512$	0	0
$513$	46.6233	2.05847
$514$	0	0
$515$	0.307118	0.0135332
$516$	0	0
$517$	10.4334	0.458859
$518$	0	0
$519$	−18.2899	−0.802836
$520$	0	0
$521$	−7.10103	−0.311102	−0.155551	−	0.987828i	$-0.549715\pi$
$521$	−7.10103	−0.311102	−0.155551	+	0.987828i	$0.549715\pi$
$522$	0	0
$523$	23.2854	1.01820	0.509099	−	0.860708i	$-0.329979\pi$
$523$	23.2854	1.01820	0.509099	+	0.860708i	$0.329979\pi$
$524$	0	0
$525$	−12.7742	−0.557514
$526$	0	0
$527$	25.3853	1.10580
$528$	0	0
$529$	16.8621	0.733137
$530$	0	0
$531$	66.2950	2.87696
$532$	0	0
$533$	10.7742	0.466684
$534$	0	0
$535$	−10.5435	−0.455837
$536$	0	0
$537$	−63.8550	−2.75555
$538$	0	0
$539$	−3.26869	−0.140792
$540$	0	0
$541$	−12.0865	−0.519638	−0.259819	−	0.965657i	$-0.583663\pi$
$541$	−12.0865	−0.519638	−0.259819	+	0.965657i	$0.583663\pi$
$542$	0	0
$543$	−34.7702	−1.49213
$544$	0	0
$545$	−8.00345	−0.342830
$546$	0	0
$547$	26.4324	1.13017	0.565084	−	0.825033i	$-0.308844\pi$
$547$	26.4324	1.13017	0.565084	+	0.825033i	$0.308844\pi$
$548$	0	0
$549$	−71.5536	−3.05383
$550$	0	0
$551$	13.5823	0.578627
$552$	0	0
$553$	−14.7611	−0.627706
$554$	0	0
$555$	−4.33339	−0.183942
$556$	0	0
$557$	11.3157	0.479460	0.239730	−	0.970840i	$-0.422941\pi$
$557$	11.3157	0.479460	0.239730	+	0.970840i	$0.422941\pi$
$558$	0	0
$559$	−6.90806	−0.292180
$560$	0	0
$561$	−84.9727	−3.58755
$562$	0	0
$563$	9.16211	0.386137	0.193068	−	0.981185i	$-0.438156\pi$
$563$	9.16211	0.386137	0.193068	+	0.981185i	$0.438156\pi$
$564$	0	0
$565$	−8.68741	−0.365482
$566$	0	0
$567$	−26.9960	−1.13372
$568$	0	0
$569$	−27.9521	−1.17181	−0.585906	−	0.810379i	$-0.699261\pi$
$569$	−27.9521	−1.17181	−0.585906	+	0.810379i	$0.699261\pi$
$570$	0	0
$571$	18.7081	0.782910	0.391455	−	0.920197i	$-0.371972\pi$
$571$	18.7081	0.782910	0.391455	+	0.920197i	$0.371972\pi$
$572$	0	0
$573$	55.0030	2.29778
$574$	0	0
$575$	−24.6741	−1.02898
$576$	0	0
$577$	−9.82017	−0.408819	−0.204410	−	0.978885i	$-0.565527\pi$
$577$	−9.82017	−0.408819	−0.204410	+	0.978885i	$0.565527\pi$
$578$	0	0
$579$	−54.0556	−2.24647
$580$	0	0
$581$	1.58234	0.0656464
$582$	0	0
$583$	−27.6551	−1.14536
$584$	0	0
$585$	−8.02980	−0.331991
$586$	0	0
$587$	−11.0450	−0.455874	−0.227937	−	0.973676i	$-0.573198\pi$
$587$	−11.0450	−0.455874	−0.227937	+	0.973676i	$0.573198\pi$
$588$	0	0
$589$	9.71923	0.400474
$590$	0	0
$591$	−14.4368	−0.593851
$592$	0	0
$593$	14.2601	0.585591	0.292795	−	0.956175i	$-0.405415\pi$
$593$	14.2601	0.585591	0.292795	+	0.956175i	$0.405415\pi$
$594$	0	0
$595$	−8.31057	−0.340700
$596$	0	0
$597$	37.8161	1.54771
$598$	0	0
$599$	9.09854	0.371756	0.185878	−	0.982573i	$-0.440487\pi$
$599$	9.09854	0.371756	0.185878	+	0.982573i	$0.440487\pi$
$600$	0	0
$601$	41.2768	1.68372	0.841858	−	0.539699i	$-0.181462\pi$
$601$	41.2768	1.68372	0.841858	+	0.539699i	$0.181462\pi$
$602$	0	0
$603$	1.12923	0.0459859
$604$	0	0
$605$	0.329860	0.0134107
$606$	0	0
$607$	6.13285	0.248925	0.124462	−	0.992224i	$-0.460279\pi$
$607$	6.13285	0.248925	0.124462	+	0.992224i	$0.460279\pi$
$608$	0	0
$609$	−14.5803	−0.590824
$610$	0	0
$611$	−3.19191	−0.129131
$612$	0	0
$613$	19.4217	0.784433	0.392217	−	0.919873i	$-0.371708\pi$
$613$	19.4217	0.784433	0.392217	+	0.919873i	$0.371708\pi$
$614$	0	0
$615$	36.8010	1.48396
$616$	0	0
$617$	22.1798	0.892926	0.446463	−	0.894802i	$-0.352683\pi$
$617$	22.1798	0.892926	0.446463	+	0.894802i	$0.352683\pi$
$618$	0	0
$619$	−49.1807	−1.97674	−0.988371	−	0.152064i	$-0.951408\pi$
$619$	−49.1807	−1.97674	−0.988371	+	0.152064i	$0.951408\pi$
$620$	0	0
$621$	−96.6722	−3.87932
$622$	0	0
$623$	−2.66114	−0.106616
$624$	0	0
$625$	9.81328	0.392531
$626$	0	0
$627$	−32.5333	−1.29926
$628$	0	0
$629$	10.0899	0.402311
$630$	0	0
$631$	−12.2676	−0.488367	−0.244183	−	0.969729i	$-0.578520\pi$
$631$	−12.2676	−0.488367	−0.244183	+	0.969729i	$0.578520\pi$
$632$	0	0
$633$	7.45513	0.296315
$634$	0	0
$635$	1.86985	0.0742028
$636$	0	0
$637$	1.00000	0.0396214
$638$	0	0
$639$	126.718	5.01287
$640$	0	0
$641$	−27.2308	−1.07555	−0.537776	−	0.843088i	$-0.680735\pi$
$641$	−27.2308	−1.07555	−0.537776	+	0.843088i	$0.680735\pi$
$642$	0	0
$643$	−6.06364	−0.239127	−0.119563	−	0.992827i	$-0.538149\pi$
$643$	−6.06364	−0.239127	−0.119563	+	0.992827i	$0.538149\pi$
$644$	0	0
$645$	−23.5955	−0.929071
$646$	0	0
$647$	−32.4904	−1.27733	−0.638665	−	0.769485i	$-0.720513\pi$
$647$	−32.4904	−1.27733	−0.638665	+	0.769485i	$0.720513\pi$
$648$	0	0
$649$	−28.1999	−1.10694
$650$	0	0
$651$	−10.4334	−0.408916
$652$	0	0
$653$	−16.6233	−0.650518	−0.325259	−	0.945625i	$-0.605452\pi$
$653$	−16.6233	−0.650518	−0.325259	+	0.945625i	$0.605452\pi$
$654$	0	0
$655$	−20.6131	−0.805419
$656$	0	0
$657$	−12.5975	−0.491475
$658$	0	0
$659$	7.04841	0.274567	0.137284	−	0.990532i	$-0.456163\pi$
$659$	7.04841	0.274567	0.137284	+	0.990532i	$0.456163\pi$
$660$	0	0
$661$	−45.1904	−1.75770	−0.878852	−	0.477094i	$-0.841690\pi$
$661$	−45.1904	−1.75770	−0.878852	+	0.477094i	$0.841690\pi$
$662$	0	0
$663$	25.9960	1.00960
$664$	0	0
$665$	−3.18185	−0.123387
$666$	0	0
$667$	−28.1627	−1.09046
$668$	0	0
$669$	89.4496	3.45832
$670$	0	0
$671$	30.4368	1.17500
$672$	0	0
$673$	46.7490	1.80204	0.901020	−	0.433777i	$-0.142819\pi$
$673$	46.7490	1.80204	0.901020	+	0.433777i	$0.142819\pi$
$674$	0	0
$675$	59.8388	2.30320
$676$	0	0
$677$	10.3302	0.397023	0.198512	−	0.980099i	$-0.436389\pi$
$677$	10.3302	0.397023	0.198512	+	0.980099i	$0.436389\pi$
$678$	0	0
$679$	1.34547	0.0516344
$680$	0	0
$681$	−25.8424	−0.990283
$682$	0	0
$683$	−27.6389	−1.05757	−0.528786	−	0.848755i	$-0.677353\pi$
$683$	−27.6389	−1.05757	−0.528786	+	0.848755i	$0.677353\pi$
$684$	0	0
$685$	20.2843	0.775023
$686$	0	0
$687$	−43.4041	−1.65597
$688$	0	0
$689$	8.46060	0.322323
$690$	0	0
$691$	−29.3348	−1.11595	−0.557975	−	0.829858i	$-0.688421\pi$
$691$	−29.3348	−1.11595	−0.557975	+	0.829858i	$0.688421\pi$
$692$	0	0
$693$	25.1177	0.954142
$694$	0	0
$695$	−6.52777	−0.247613
$696$	0	0
$697$	−85.6878	−3.24566
$698$	0	0
$699$	−70.8727	−2.68065
$700$	0	0
$701$	4.58789	0.173282	0.0866411	−	0.996240i	$-0.472387\pi$
$701$	4.58789	0.173282	0.0866411	+	0.996240i	$0.472387\pi$
$702$	0	0
$703$	3.86311	0.145700
$704$	0	0
$705$	−10.9024	−0.410609
$706$	0	0
$707$	2.43741	0.0916681
$708$	0	0
$709$	−43.0566	−1.61703	−0.808513	−	0.588479i	$-0.799727\pi$
$709$	−43.0566	−1.61703	−0.808513	+	0.588479i	$0.799727\pi$
$710$	0	0
$711$	113.429	4.25393
$712$	0	0
$713$	−20.1526	−0.754721
$714$	0	0
$715$	3.41564	0.127738
$716$	0	0
$717$	−60.0051	−2.24093
$718$	0	0
$719$	−26.0389	−0.971087	−0.485543	−	0.874213i	$-0.661378\pi$
$719$	−26.0389	−0.971087	−0.485543	+	0.874213i	$0.661378\pi$
$720$	0	0
$721$	0.293905	0.0109456
$722$	0	0
$723$	11.2076	0.416815
$724$	0	0
$725$	17.4323	0.647420
$726$	0	0
$727$	−31.6717	−1.17464	−0.587320	−	0.809355i	$-0.699817\pi$
$727$	−31.6717	−1.17464	−0.587320	+	0.809355i	$0.699817\pi$
$728$	0	0
$729$	57.2990	2.12219
$730$	0	0
$731$	54.9400	2.03203
$732$	0	0
$733$	27.8762	1.02963	0.514816	−	0.857301i	$-0.327860\pi$
$733$	27.8762	1.02963	0.514816	+	0.857301i	$0.327860\pi$
$734$	0	0
$735$	3.41564	0.125988
$736$	0	0
$737$	−0.480342	−0.0176936
$738$	0	0
$739$	8.15701	0.300060	0.150030	−	0.988681i	$-0.452063\pi$
$739$	8.15701	0.300060	0.150030	+	0.988681i	$0.452063\pi$
$740$	0	0
$741$	9.95302	0.365633
$742$	0	0
$743$	−52.3516	−1.92059	−0.960297	−	0.278981i	$-0.910003\pi$
$743$	−52.3516	−1.92059	−0.960297	+	0.278981i	$0.910003\pi$
$744$	0	0
$745$	24.1227	0.883786
$746$	0	0
$747$	−12.1592	−0.444882
$748$	0	0
$749$	−10.0899	−0.368677
$750$	0	0
$751$	9.08533	0.331528	0.165764	−	0.986165i	$-0.446991\pi$
$751$	9.08533	0.331528	0.165764	+	0.986165i	$0.446991\pi$
$752$	0	0
$753$	21.0287	0.766327
$754$	0	0
$755$	−3.56920	−0.129897
$756$	0	0
$757$	−34.3425	−1.24820	−0.624099	−	0.781345i	$-0.714534\pi$
$757$	−34.3425	−1.24820	−0.624099	+	0.781345i	$0.714534\pi$
$758$	0	0
$759$	67.4571	2.44854
$760$	0	0
$761$	−3.97681	−0.144159	−0.0720796	−	0.997399i	$-0.522964\pi$
$761$	−3.97681	−0.144159	−0.0720796	+	0.997399i	$0.522964\pi$
$762$	0	0
$763$	−7.65912	−0.277279
$764$	0	0
$765$	63.8612	2.30891
$766$	0	0
$767$	8.62729	0.311514
$768$	0	0
$769$	43.1616	1.55645	0.778223	−	0.627987i	$-0.216121\pi$
$769$	43.1616	1.55645	0.778223	+	0.627987i	$0.216121\pi$
$770$	0	0
$771$	−55.1566	−1.98642
$772$	0	0
$773$	−26.0859	−0.938244	−0.469122	−	0.883133i	$-0.655429\pi$
$773$	−26.0859	−0.938244	−0.469122	+	0.883133i	$0.655429\pi$
$774$	0	0
$775$	12.4742	0.448086
$776$	0	0
$777$	−4.14695	−0.148771
$778$	0	0
$779$	−32.8071	−1.17544
$780$	0	0
$781$	−53.9020	−1.92877
$782$	0	0
$783$	68.2990	2.44081
$784$	0	0
$785$	−6.28026	−0.224152
$786$	0	0
$787$	51.5106	1.83615	0.918077	−	0.396401i	$-0.129741\pi$
$787$	51.5106	1.83615	0.918077	+	0.396401i	$0.129741\pi$
$788$	0	0
$789$	78.8943	2.80871
$790$	0	0
$791$	−8.31365	−0.295599
$792$	0	0
$793$	−9.31163	−0.330666
$794$	0	0
$795$	28.8984	1.02492
$796$	0	0
$797$	36.1762	1.28143	0.640714	−	0.767780i	$-0.278638\pi$
$797$	36.1762	1.28143	0.640714	+	0.767780i	$0.278638\pi$
$798$	0	0
$799$	25.3853	0.898068
$800$	0	0
$801$	20.4491	0.722532
$802$	0	0
$803$	5.35860	0.189101
$804$	0	0
$805$	6.59749	0.232531
$806$	0	0
$807$	−45.8691	−1.61467
$808$	0	0
$809$	10.9121	0.383649	0.191825	−	0.981429i	$-0.438560\pi$
$809$	10.9121	0.383649	0.191825	+	0.981429i	$0.438560\pi$
$810$	0	0
$811$	46.5092	1.63316	0.816579	−	0.577234i	$-0.195868\pi$
$811$	46.5092	1.63316	0.816579	+	0.577234i	$0.195868\pi$
$812$	0	0
$813$	−51.0096	−1.78899
$814$	0	0
$815$	15.6860	0.549456
$816$	0	0
$817$	21.0348	0.735913
$818$	0	0
$819$	−7.68433	−0.268512
$820$	0	0
$821$	52.1081	1.81859	0.909293	−	0.416157i	$-0.136623\pi$
$821$	52.1081	1.81859	0.909293	+	0.416157i	$0.136623\pi$
$822$	0	0
$823$	−14.5904	−0.508588	−0.254294	−	0.967127i	$-0.581843\pi$
$823$	−14.5904	−0.508588	−0.254294	+	0.967127i	$0.581843\pi$
$824$	0	0
$825$	−41.7550	−1.45372
$826$	0	0
$827$	4.31402	0.150013	0.0750066	−	0.997183i	$-0.476102\pi$
$827$	4.31402	0.150013	0.0750066	+	0.997183i	$0.476102\pi$
$828$	0	0
$829$	−16.2207	−0.563367	−0.281683	−	0.959507i	$-0.590893\pi$
$829$	−16.2207	−0.563367	−0.281683	+	0.959507i	$0.590893\pi$
$830$	0	0
$831$	−42.2783	−1.46662
$832$	0	0
$833$	−7.95302	−0.275556
$834$	0	0
$835$	8.60396	0.297752
$836$	0	0
$837$	48.8733	1.68931
$838$	0	0
$839$	−25.0682	−0.865449	−0.432724	−	0.901526i	$-0.642448\pi$
$839$	−25.0682	−0.865449	−0.432724	+	0.901526i	$0.642448\pi$
$840$	0	0
$841$	−9.10305	−0.313898
$842$	0	0
$843$	55.0707	1.89674
$844$	0	0
$845$	−1.04496	−0.0359476
$846$	0	0
$847$	0.315668	0.0108465
$848$	0	0
$849$	−19.4915	−0.668945
$850$	0	0
$851$	−8.01006	−0.274581
$852$	0	0
$853$	19.7158	0.675055	0.337528	−	0.941316i	$-0.390409\pi$
$853$	19.7158	0.675055	0.337528	+	0.941316i	$0.390409\pi$
$854$	0	0
$855$	24.4504	0.836186
$856$	0	0
$857$	33.7262	1.15207	0.576033	−	0.817427i	$-0.304600\pi$
$857$	33.7262	1.15207	0.576033	+	0.817427i	$0.304600\pi$
$858$	0	0
$859$	48.4742	1.65392	0.826959	−	0.562262i	$-0.190069\pi$
$859$	48.4742	1.65392	0.826959	+	0.562262i	$0.190069\pi$
$860$	0	0
$861$	35.2177	1.20021
$862$	0	0
$863$	−22.9172	−0.780109	−0.390055	−	0.920792i	$-0.627544\pi$
$863$	−22.9172	−0.780109	−0.390055	+	0.920792i	$0.627544\pi$
$864$	0	0
$865$	−5.84703	−0.198805
$866$	0	0
$867$	−151.179	−5.13430
$868$	0	0
$869$	−48.2495	−1.63675
$870$	0	0
$871$	0.146952	0.00497929
$872$	0	0
$873$	−10.3390	−0.349923
$874$	0	0
$875$	−9.30855	−0.314686
$876$	0	0
$877$	−12.5212	−0.422810	−0.211405	−	0.977399i	$-0.567804\pi$
$877$	−12.5212	−0.422810	−0.211405	+	0.977399i	$0.567804\pi$
$878$	0	0
$879$	39.6946	1.33886
$880$	0	0
$881$	−2.45362	−0.0826645	−0.0413323	−	0.999145i	$-0.513160\pi$
$881$	−2.45362	−0.0826645	−0.0413323	+	0.999145i	$0.513160\pi$
$882$	0	0
$883$	17.2151	0.579334	0.289667	−	0.957127i	$-0.406455\pi$
$883$	17.2151	0.579334	0.289667	+	0.957127i	$0.406455\pi$
$884$	0	0
$885$	29.4677	0.990548
$886$	0	0
$887$	23.4323	0.786780	0.393390	−	0.919372i	$-0.371302\pi$
$887$	23.4323	0.786780	0.393390	+	0.919372i	$0.371302\pi$
$888$	0	0
$889$	1.78940	0.0600147
$890$	0	0
$891$	−88.2414	−2.95620
$892$	0	0
$893$	9.71923	0.325242
$894$	0	0
$895$	−20.4136	−0.682352
$896$	0	0
$897$	−20.6374	−0.689061
$898$	0	0
$899$	14.2378	0.474858
$900$	0	0
$901$	−67.2873	−2.24167
$902$	0	0
$903$	−22.5803	−0.751426
$904$	0	0
$905$	−11.1156	−0.369494
$906$	0	0
$907$	19.3818	0.643562	0.321781	−	0.946814i	$-0.395719\pi$
$907$	19.3818	0.643562	0.321781	+	0.946814i	$0.395719\pi$
$908$	0	0
$909$	−18.7298	−0.621229
$910$	0	0
$911$	13.2455	0.438843	0.219421	−	0.975630i	$-0.429583\pi$
$911$	13.2455	0.438843	0.219421	+	0.975630i	$0.429583\pi$
$912$	0	0
$913$	5.17217	0.171174
$914$	0	0
$915$	−31.8052	−1.05145
$916$	0	0
$917$	−19.7262	−0.651417
$918$	0	0
$919$	23.5530	0.776941	0.388471	−	0.921461i	$-0.373003\pi$
$919$	23.5530	0.776941	0.388471	+	0.921461i	$0.373003\pi$
$920$	0	0
$921$	67.0329	2.20881
$922$	0	0
$923$	16.4904	0.542788
$924$	0	0
$925$	4.95812	0.163022
$926$	0	0
$927$	−2.25846	−0.0741776
$928$	0	0
$929$	28.6041	0.938470	0.469235	−	0.883073i	$-0.344530\pi$
$929$	28.6041	0.938470	0.469235	+	0.883073i	$0.344530\pi$
$930$	0	0
$931$	−3.04496	−0.0997945
$932$	0	0
$933$	39.2787	1.28593
$934$	0	0
$935$	−27.1647	−0.888380
$936$	0	0
$937$	14.6501	0.478598	0.239299	−	0.970946i	$-0.423082\pi$
$937$	14.6501	0.478598	0.239299	+	0.970946i	$0.423082\pi$
$938$	0	0
$939$	63.4505	2.07063
$940$	0	0
$941$	−45.7056	−1.48996	−0.744980	−	0.667087i	$-0.767541\pi$
$941$	−45.7056	−1.48996	−0.744980	+	0.667087i	$0.767541\pi$
$942$	0	0
$943$	68.0248	2.21519
$944$	0	0
$945$	−16.0000	−0.520480
$946$	0	0
$947$	−21.5504	−0.700295	−0.350147	−	0.936695i	$-0.613868\pi$
$947$	−21.5504	−0.700295	−0.350147	+	0.936695i	$0.613868\pi$
$948$	0	0
$949$	−1.63937	−0.0532163
$950$	0	0
$951$	24.8733	0.806573
$952$	0	0
$953$	−1.26772	−0.0410656	−0.0205328	−	0.999789i	$-0.506536\pi$
$953$	−1.26772	−0.0410656	−0.0205328	+	0.999789i	$0.506536\pi$
$954$	0	0
$955$	17.5838	0.568997
$956$	0	0
$957$	−47.6585	−1.54058
$958$	0	0
$959$	19.4116	0.626833
$960$	0	0
$961$	−20.8117	−0.671345
$962$	0	0
$963$	77.5343	2.49851
$964$	0	0
$965$	−17.2809	−0.556291
$966$	0	0
$967$	33.0237	1.06197	0.530986	−	0.847381i	$-0.321822\pi$
$967$	33.0237	1.06197	0.530986	+	0.847381i	$0.321822\pi$
$968$	0	0
$969$	−79.1566	−2.54288
$970$	0	0
$971$	−7.25202	−0.232729	−0.116364	−	0.993207i	$-0.537124\pi$
$971$	−7.25202	−0.232729	−0.116364	+	0.993207i	$0.537124\pi$
$972$	0	0
$973$	−6.24693	−0.200267
$974$	0	0
$975$	12.7742	0.409103
$976$	0	0
$977$	30.3964	0.972468	0.486234	−	0.873829i	$-0.338370\pi$
$977$	30.3964	0.972468	0.486234	+	0.873829i	$0.338370\pi$
$978$	0	0
$979$	−8.69843	−0.278003
$980$	0	0
$981$	58.8552	1.87910
$982$	0	0
$983$	22.4096	0.714755	0.357377	−	0.933960i	$-0.383671\pi$
$983$	22.4096	0.714755	0.357377	+	0.933960i	$0.383671\pi$
$984$	0	0
$985$	−4.61526	−0.147054
$986$	0	0
$987$	−10.4334	−0.332098
$988$	0	0
$989$	−43.6151	−1.38688
$990$	0	0
$991$	−49.6147	−1.57606	−0.788031	−	0.615635i	$-0.788900\pi$
$991$	−49.6147	−1.57606	−0.788031	+	0.615635i	$0.788900\pi$
$992$	0	0
$993$	−36.4723	−1.15741
$994$	0	0
$995$	12.0893	0.383257
$996$	0	0
$997$	−11.5814	−0.366786	−0.183393	−	0.983040i	$-0.558708\pi$
$997$	−11.5814	−0.366786	−0.183393	+	0.983040i	$0.558708\pi$
$998$	0	0
$999$	19.4257	0.614602

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1456.2.a.u.1.1		4
4.3	odd	2		728.2.a.h.1.4	✓	4
8.3	odd	2		5824.2.a.cc.1.1		4
8.5	even	2		5824.2.a.cf.1.4		4
12.11	even	2		6552.2.a.bt.1.3		4
28.27	even	2		5096.2.a.t.1.1		4
52.51	odd	2		9464.2.a.ba.1.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
728.2.a.h.1.4	✓	4	4.3	odd	2
1456.2.a.u.1.1		4	1.1	even	1	trivial
5096.2.a.t.1.1		4	28.27	even	2
5824.2.a.cc.1.1		4	8.3	odd	2
5824.2.a.cf.1.4		4	8.5	even	2
6552.2.a.bt.1.3		4	12.11	even	2
9464.2.a.ba.1.4		4	52.51	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:	\( N \)	\(=\)	\( 1456 = 2^{4} \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1456.a (trivial)

\(f(q)\)	\(=\)	\(q-3.26869 q^{3} -1.04496 q^{5} -1.00000 q^{7} +7.68433 q^{9} +O(q^{10})\)	\(q-3.26869 q^{3} -1.04496 q^{5} -1.00000 q^{7} +7.68433 q^{9} -3.26869 q^{11} +1.00000 q^{13} +3.41564 q^{15} -7.95302 q^{17} -3.04496 q^{19} +3.26869 q^{21} +6.31365 q^{23} -3.90806 q^{25} -15.3116 q^{27} -4.46060 q^{29} -3.19191 q^{31} +10.6843 q^{33} +1.04496 q^{35} -1.26869 q^{37} -3.26869 q^{39} +10.7742 q^{41} -6.90806 q^{43} -8.02980 q^{45} -3.19191 q^{47} +1.00000 q^{49} +25.9960 q^{51} +8.46060 q^{53} +3.41564 q^{55} +9.95302 q^{57} +8.62729 q^{59} -9.31163 q^{61} -7.68433 q^{63} -1.04496 q^{65} +0.146952 q^{67} -20.6374 q^{69} +16.4904 q^{71} -1.63937 q^{73} +12.7742 q^{75} +3.26869 q^{77} +14.7611 q^{79} +26.9960 q^{81} -1.58234 q^{83} +8.31057 q^{85} +14.5803 q^{87} +2.66114 q^{89} -1.00000 q^{91} +10.4334 q^{93} +3.18185 q^{95} -1.34547 q^{97} -25.1177 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - q^{3} - 4 q^{7} + 9 q^{9}+O(q^{10}) \) 4 * q - q^3 - 4 * q^7 + 9 * q^9	\( 4 q - q^{3} - 4 q^{7} + 9 q^{9} - q^{11} + 4 q^{13} + 4 q^{15} + 2 q^{17} - 8 q^{19} + q^{21} + 9 q^{23} + 14 q^{25} - 7 q^{27} - 4 q^{29} - 11 q^{31} + 21 q^{33} + 7 q^{37} - q^{39} + 13 q^{41} + 2 q^{43} + 12 q^{45} - 11 q^{47} + 4 q^{49} + 28 q^{51} + 20 q^{53} + 4 q^{55} + 6 q^{57} + 2 q^{59} + 17 q^{61} - 9 q^{63} + 3 q^{67} - 27 q^{69} + 8 q^{71} + 11 q^{73} + 21 q^{75} + q^{77} + 27 q^{79} + 32 q^{81} + 22 q^{83} - 8 q^{85} - 8 q^{87} + 10 q^{89} - 4 q^{91} - 27 q^{93} + 34 q^{95} + 17 q^{97} - 10 q^{99}+O(q^{100}) \) 4 * q - q^3 - 4 * q^7 + 9 * q^9 - q^11 + 4 * q^13 + 4 * q^15 + 2 * q^17 - 8 * q^19 + q^21 + 9 * q^23 + 14 * q^25 - 7 * q^27 - 4 * q^29 - 11 * q^31 + 21 * q^33 + 7 * q^37 - q^39 + 13 * q^41 + 2 * q^43 + 12 * q^45 - 11 * q^47 + 4 * q^49 + 28 * q^51 + 20 * q^53 + 4 * q^55 + 6 * q^57 + 2 * q^59 + 17 * q^61 - 9 * q^63 + 3 * q^67 - 27 * q^69 + 8 * q^71 + 11 * q^73 + 21 * q^75 + q^77 + 27 * q^79 + 32 * q^81 + 22 * q^83 - 8 * q^85 - 8 * q^87 + 10 * q^89 - 4 * q^91 - 27 * q^93 + 34 * q^95 + 17 * q^97 - 10 * q^99

Introduction

L-functions

Modular forms

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